LMFDB - Embedded newform 378.2.f.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(127,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			127.2
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	378.127
Dual form			378.2.f.c.253.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.18614 + 3.78651i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.18614 + 3.78651i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -4.37228 q^{10} +(-0.686141 + 1.18843i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.37228 q^{17} +5.00000 q^{19} +(2.18614 - 3.78651i) q^{20} +(-0.686141 - 1.18843i) q^{22} +(-0.813859 - 1.40965i) q^{23} +(-7.05842 + 12.2255i) q^{25} +2.00000 q^{26} +1.00000 q^{28} +(-4.37228 + 7.57301i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.686141 - 1.18843i) q^{34} -4.37228 q^{35} +2.00000 q^{37} +(-2.50000 + 4.33013i) q^{38} +(2.18614 + 3.78651i) q^{40} +(2.31386 + 4.00772i) q^{41} +(4.05842 - 7.02939i) q^{43} +1.37228 q^{44} +1.62772 q^{46} +(-0.500000 - 0.866025i) q^{49} +(-7.05842 - 12.2255i) q^{50} +(-1.00000 + 1.73205i) q^{52} +8.74456 q^{53} -6.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-4.37228 - 7.57301i) q^{58} +(-5.05842 - 8.76144i) q^{59} +(-1.55842 + 2.69927i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(4.37228 - 7.57301i) q^{65} +(1.05842 + 1.83324i) q^{67} +(0.686141 + 1.18843i) q^{68} +(2.18614 - 3.78651i) q^{70} +7.11684 q^{71} +12.1168 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-2.50000 - 4.33013i) q^{76} +(-0.686141 - 1.18843i) q^{77} +(2.55842 - 4.43132i) q^{79} -4.37228 q^{80} -4.62772 q^{82} +(8.74456 - 15.1460i) q^{83} +(-3.00000 - 5.19615i) q^{85} +(4.05842 + 7.02939i) q^{86} +(-0.686141 + 1.18843i) q^{88} -14.7446 q^{89} +2.00000 q^{91} +(-0.813859 + 1.40965i) q^{92} +(10.9307 + 18.9325i) q^{95} +(4.05842 - 7.02939i) q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} + 3 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 - 2 * q^7 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} + 3 q^{5} - 2 q^{7} + 4 q^{8} - 6 q^{10} + 3 q^{11} - 4 q^{13} - 2 q^{14} - 2 q^{16} + 6 q^{17} + 20 q^{19} + 3 q^{20} + 3 q^{22} - 9 q^{23} - 11 q^{25} + 8 q^{26} + 4 q^{28} - 6 q^{29} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 6 q^{35} + 8 q^{37} - 10 q^{38} + 3 q^{40} + 15 q^{41} - q^{43} - 6 q^{44} + 18 q^{46} - 2 q^{49} - 11 q^{50} - 4 q^{52} + 12 q^{53} - 24 q^{55} - 2 q^{56} - 6 q^{58} - 3 q^{59} + 11 q^{61} + 8 q^{62} + 4 q^{64} + 6 q^{65} - 13 q^{67} - 3 q^{68} + 3 q^{70} - 6 q^{71} + 14 q^{73} - 4 q^{74} - 10 q^{76} + 3 q^{77} - 7 q^{79} - 6 q^{80} - 30 q^{82} + 12 q^{83} - 12 q^{85} - q^{86} + 3 q^{88} - 36 q^{89} + 8 q^{91} - 9 q^{92} + 15 q^{95} - q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 - 2 * q^7 + 4 * q^8 - 6 * q^10 + 3 * q^11 - 4 * q^13 - 2 * q^14 - 2 * q^16 + 6 * q^17 + 20 * q^19 + 3 * q^20 + 3 * q^22 - 9 * q^23 - 11 * q^25 + 8 * q^26 + 4 * q^28 - 6 * q^29 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 6 * q^35 + 8 * q^37 - 10 * q^38 + 3 * q^40 + 15 * q^41 - q^43 - 6 * q^44 + 18 * q^46 - 2 * q^49 - 11 * q^50 - 4 * q^52 + 12 * q^53 - 24 * q^55 - 2 * q^56 - 6 * q^58 - 3 * q^59 + 11 * q^61 + 8 * q^62 + 4 * q^64 + 6 * q^65 - 13 * q^67 - 3 * q^68 + 3 * q^70 - 6 * q^71 + 14 * q^73 - 4 * q^74 - 10 * q^76 + 3 * q^77 - 7 * q^79 - 6 * q^80 - 30 * q^82 + 12 * q^83 - 12 * q^85 - q^86 + 3 * q^88 - 36 * q^89 + 8 * q^91 - 9 * q^92 + 15 * q^95 - q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	2.18614	+	3.78651i	0.977672	+	1.69338i	0.670820	+	0.741620i	$0.265942\pi$
$5$	2.18614	+	3.78651i	0.977672	+	1.69338i	0.306851	+	0.951757i	$0.400725\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$	−4.37228			−1.38264
$11$	−0.686141	+	1.18843i	−0.206879	+	0.358325i	−0.950730	−	0.310021i	$-0.899664\pi$
$11$	−0.686141	+	1.18843i	−0.206879	+	0.358325i	0.743851	+	0.668346i	$0.232997\pi$
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−1.37228			−0.332827			−0.166414	−	0.986056i	$-0.553219\pi$
$17$	−1.37228			−0.332827			−0.166414	+	0.986056i	$0.553219\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$	2.18614	−	3.78651i	0.488836	−	0.846689i
$21$		0			0
$22$	−0.686141	−	1.18843i	−0.146286	−	0.253374i
$23$	−0.813859	−	1.40965i	−0.169701	−	0.293931i	0.768613	−	0.639713i	$-0.220947\pi$
$23$	−0.813859	−	1.40965i	−0.169701	−	0.293931i	−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	−7.05842	+	12.2255i	−1.41168	+	2.44511i
$26$	2.00000			0.392232
$27$		0			0
$28$	1.00000			0.188982
$29$	−4.37228	+	7.57301i	−0.811912	+	1.40627i	0.0996117	+	0.995026i	$0.468240\pi$
$29$	−4.37228	+	7.57301i	−0.811912	+	1.40627i	−0.911524	+	0.411247i	$0.865093\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	0.686141	−	1.18843i	0.117672	−	0.203814i
$35$	−4.37228			−0.739050
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.50000	+	4.33013i	−0.405554	+	0.702439i
$39$		0			0
$40$	2.18614	+	3.78651i	0.345659	+	0.598699i
$41$	2.31386	+	4.00772i	0.361364	+	0.625901i	0.988186	−	0.153262i	$-0.0489778\pi$
$41$	2.31386	+	4.00772i	0.361364	+	0.625901i	−0.626821	+	0.779163i	$0.715644\pi$
$42$		0			0
$43$	4.05842	−	7.02939i	0.618904	−	1.07197i	−0.370783	−	0.928720i	$-0.620910\pi$
$43$	4.05842	−	7.02939i	0.618904	−	1.07197i	0.989686	−	0.143253i	$-0.0457562\pi$
$44$	1.37228			0.206879
$45$		0			0
$46$	1.62772			0.239994
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$	−7.05842	−	12.2255i	−0.998212	−	1.72895i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	8.74456			1.20116			0.600579	−	0.799565i	$-0.294937\pi$
$53$	8.74456			1.20116			0.600579	+	0.799565i	$0.294937\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	−4.37228	−	7.57301i	−0.574109	−	0.994385i
$59$	−5.05842	−	8.76144i	−0.658550	−	1.14064i	−0.980991	−	0.194053i	$-0.937837\pi$
$59$	−5.05842	−	8.76144i	−0.658550	−	1.14064i	0.322441	−	0.946590i	$-0.395497\pi$
$60$		0			0
$61$	−1.55842	+	2.69927i	−0.199535	+	0.345606i	−0.948378	−	0.317142i	$-0.897277\pi$
$61$	−1.55842	+	2.69927i	−0.199535	+	0.345606i	0.748842	+	0.662748i	$0.230610\pi$
$62$	2.00000			0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	4.37228	−	7.57301i	0.542315	−	0.939317i
$66$		0			0
$67$	1.05842	+	1.83324i	0.129307	+	0.223966i	0.923408	−	0.383819i	$-0.125391\pi$
$67$	1.05842	+	1.83324i	0.129307	+	0.223966i	−0.794101	+	0.607785i	$0.792058\pi$
$68$	0.686141	+	1.18843i	0.0832068	+	0.144118i
$69$		0			0
$70$	2.18614	−	3.78651i	0.261294	−	0.452574i
$71$	7.11684			0.844614			0.422307	−	0.906453i	$-0.361220\pi$
$71$	7.11684			0.844614			0.422307	+	0.906453i	$0.361220\pi$
$72$		0			0
$73$	12.1168			1.41817			0.709085	−	0.705123i	$-0.249108\pi$
$73$	12.1168			1.41817			0.709085	+	0.705123i	$0.249108\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$		0			0
$76$	−2.50000	−	4.33013i	−0.286770	−	0.496700i
$77$	−0.686141	−	1.18843i	−0.0781930	−	0.135434i
$78$		0			0
$79$	2.55842	−	4.43132i	0.287845	−	0.498562i	−0.685450	−	0.728120i	$-0.740395\pi$
$79$	2.55842	−	4.43132i	0.287845	−	0.498562i	0.973295	+	0.229557i	$0.0737279\pi$
$80$	−4.37228			−0.488836
$81$		0			0
$82$	−4.62772			−0.511046
$83$	8.74456	−	15.1460i	0.959840	−	1.66249i	0.236960	−	0.971519i	$-0.423849\pi$
$83$	8.74456	−	15.1460i	0.959840	−	1.66249i	0.722881	−	0.690973i	$-0.242818\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$	4.05842	+	7.02939i	0.437631	+	0.757999i
$87$		0			0
$88$	−0.686141	+	1.18843i	−0.0731428	+	0.126687i
$89$	−14.7446			−1.56292			−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446			−1.56292			−0.781460	+	0.623955i	$0.785525\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$	−0.813859	+	1.40965i	−0.0848507	+	0.146966i
$93$		0			0
$94$		0			0
$95$	10.9307	+	18.9325i	1.12147	+	1.94244i
$96$		0			0
$97$	4.05842	−	7.02939i	0.412070	−	0.713727i	−0.583046	−	0.812439i	$-0.698139\pi$
$97$	4.05842	−	7.02939i	0.412070	−	0.713727i	0.995116	+	0.0987127i	$0.0314725\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	14.1168			1.41168
$101$	0.813859	−	1.40965i	0.0809820	−	0.140265i	−0.822690	−	0.568490i	$-0.807528\pi$
$101$	0.813859	−	1.40965i	0.0809820	−	0.140265i	0.903672	+	0.428225i	$0.140861\pi$
$102$		0			0
$103$	5.00000	+	8.66025i	0.492665	+	0.853320i	0.999964	−	0.00844953i	$-0.00268960\pi$
$103$	5.00000	+	8.66025i	0.492665	+	0.853320i	−0.507300	+	0.861770i	$0.669356\pi$
$104$	−1.00000	−	1.73205i	−0.0980581	−	0.169842i
$105$		0			0
$106$	−4.37228	+	7.57301i	−0.424674	+	0.735556i
$107$	7.37228			0.712705			0.356353	−	0.934352i	$-0.384020\pi$
$107$	7.37228			0.712705			0.356353	+	0.934352i	$0.384020\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	3.00000	−	5.19615i	0.286039	−	0.495434i
$111$		0			0
$112$	−0.500000	−	0.866025i	−0.0472456	−	0.0818317i
$113$	−2.18614	−	3.78651i	−0.205655	−	0.356205i	0.744686	−	0.667415i	$-0.232599\pi$
$113$	−2.18614	−	3.78651i	−0.205655	−	0.356205i	−0.950341	+	0.311210i	$0.899266\pi$
$114$		0			0
$115$	3.55842	−	6.16337i	0.331825	−	0.574737i
$116$	8.74456			0.811912
$117$		0			0
$118$	10.1168			0.931331
$119$	0.686141	−	1.18843i	0.0628984	−	0.108943i
$120$		0			0
$121$	4.55842	+	7.89542i	0.414402	+	0.717765i
$122$	−1.55842	−	2.69927i	−0.141093	−	0.244380i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$	−39.8614			−3.56531
$126$		0			0
$127$	3.11684			0.276575			0.138288	−	0.990392i	$-0.455840\pi$
$127$	3.11684			0.276575			0.138288	+	0.990392i	$0.455840\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	4.37228	+	7.57301i	0.383474	+	0.664197i
$131$	−0.813859	−	1.40965i	−0.0711072	−	0.123161i	0.828280	−	0.560315i	$-0.189320\pi$
$131$	−0.813859	−	1.40965i	−0.0711072	−	0.123161i	−0.899387	+	0.437154i	$0.855987\pi$
$132$		0			0
$133$	−2.50000	+	4.33013i	−0.216777	+	0.375470i
$134$	−2.11684			−0.182867
$135$		0			0
$136$	−1.37228			−0.117672
$137$	5.31386	−	9.20387i	0.453994	−	0.786340i	−0.544636	−	0.838672i	$-0.683332\pi$
$137$	5.31386	−	9.20387i	0.453994	−	0.786340i	0.998630	+	0.0523324i	$0.0166655\pi$
$138$		0			0
$139$	−6.61684	−	11.4607i	−0.561233	−	0.972085i	−0.997389	−	0.0722136i	$-0.976994\pi$
$139$	−6.61684	−	11.4607i	−0.561233	−	0.972085i	0.436156	−	0.899871i	$-0.356340\pi$
$140$	2.18614	+	3.78651i	0.184763	+	0.320018i
$141$		0			0
$142$	−3.55842	+	6.16337i	−0.298616	+	0.517218i
$143$	2.74456			0.229512
$144$		0			0
$145$	−38.2337			−3.17513
$146$	−6.05842	+	10.4935i	−0.501399	+	0.868448i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	1.62772	+	2.81929i	0.133348	+	0.230965i	0.924965	−	0.380052i	$-0.124094\pi$
$149$	1.62772	+	2.81929i	0.133348	+	0.230965i	−0.791617	+	0.611017i	$0.790761\pi$
$150$		0			0
$151$	−4.55842	+	7.89542i	−0.370959	+	0.642520i	−0.989713	−	0.143065i	$-0.954304\pi$
$151$	−4.55842	+	7.89542i	−0.370959	+	0.642520i	0.618754	+	0.785585i	$0.287638\pi$
$152$	5.00000			0.405554
$153$		0			0
$154$	1.37228			0.110582
$155$	4.37228	−	7.57301i	0.351190	−	0.608279i
$156$		0			0
$157$	−4.55842	−	7.89542i	−0.363802	−	0.630123i	0.624781	−	0.780800i	$-0.285188\pi$
$157$	−4.55842	−	7.89542i	−0.363802	−	0.630123i	−0.988583	+	0.150677i	$0.951855\pi$
$158$	2.55842	+	4.43132i	0.203537	+	0.352537i
$159$		0			0
$160$	2.18614	−	3.78651i	0.172830	−	0.299350i
$161$	1.62772			0.128282
$162$		0			0
$163$	−18.2337			−1.42817			−0.714086	−	0.700058i	$-0.753158\pi$
$163$	−18.2337			−1.42817			−0.714086	+	0.700058i	$0.753158\pi$
$164$	2.31386	−	4.00772i	0.180682	−	0.312951i
$165$		0			0
$166$	8.74456	+	15.1460i	0.678710	+	1.17556i
$167$	−2.74456	−	4.75372i	−0.212381	−	0.367854i	0.740078	−	0.672521i	$-0.234788\pi$
$167$	−2.74456	−	4.75372i	−0.212381	−	0.367854i	−0.952459	+	0.304666i	$0.901455\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$	6.00000			0.460179
$171$		0			0
$172$	−8.11684			−0.618904
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−7.05842	−	12.2255i	−0.533567	−	0.924164i
$176$	−0.686141	−	1.18843i	−0.0517198	−	0.0895813i
$177$		0			0
$178$	7.37228	−	12.7692i	0.552576	−	0.957089i
$179$	−3.25544			−0.243323			−0.121661	−	0.992572i	$-0.538822\pi$
$179$	−3.25544			−0.243323			−0.121661	+	0.992572i	$0.538822\pi$
$180$		0			0
$181$	0.883156			0.0656445			0.0328222	−	0.999461i	$-0.489550\pi$
$181$	0.883156			0.0656445			0.0328222	+	0.999461i	$0.489550\pi$
$182$	−1.00000	+	1.73205i	−0.0741249	+	0.128388i
$183$		0			0
$184$	−0.813859	−	1.40965i	−0.0599985	−	0.103920i
$185$	4.37228	+	7.57301i	0.321457	+	0.556779i
$186$		0			0
$187$	0.941578	−	1.63086i	0.0688550	−	0.119260i
$188$		0			0
$189$		0			0
$190$	−21.8614			−1.58599
$191$	−9.55842	+	16.5557i	−0.691623	+	1.19793i	0.279683	+	0.960092i	$0.409771\pi$
$191$	−9.55842	+	16.5557i	−0.691623	+	1.19793i	−0.971306	+	0.237834i	$0.923563\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$	4.05842	+	7.02939i	0.291378	+	0.504681i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.0357143	+	0.0618590i
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$	−7.05842	+	12.2255i	−0.499106	+	0.864477i
$201$		0			0
$202$	0.813859	+	1.40965i	0.0572629	+	0.0991823i
$203$	−4.37228	−	7.57301i	−0.306874	−	0.531521i
$204$		0			0
$205$	−10.1168	+	17.5229i	−0.706591	+	1.22385i
$206$	−10.0000			−0.696733
$207$		0			0
$208$	2.00000			0.138675
$209$	−3.43070	+	5.94215i	−0.237307	+	0.411027i
$210$		0			0
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	0.998221	+	0.0596196i	$0.0189888\pi$
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−4.37228	−	7.57301i	−0.300290	−	0.520117i
$213$		0			0
$214$	−3.68614	+	6.38458i	−0.251979	+	0.436441i
$215$	35.4891			2.42034
$216$		0			0
$217$	2.00000			0.135769
$218$	−7.00000	+	12.1244i	−0.474100	+	0.821165i
$219$		0			0
$220$	3.00000	+	5.19615i	0.202260	+	0.350325i
$221$	1.37228	+	2.37686i	0.0923096	+	0.159885i
$222$		0			0
$223$	2.00000	−	3.46410i	0.133930	−	0.231973i	−0.791258	−	0.611482i	$-0.790574\pi$
$223$	2.00000	−	3.46410i	0.133930	−	0.231973i	0.925188	+	0.379509i	$0.123907\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	4.37228			0.290840
$227$	−6.12772	+	10.6135i	−0.406711	+	0.704444i	−0.994519	−	0.104556i	$-0.966658\pi$
$227$	−6.12772	+	10.6135i	−0.406711	+	0.704444i	0.587808	+	0.809000i	$0.299991\pi$
$228$		0			0
$229$	1.44158	+	2.49689i	0.0952622	+	0.164999i	0.909718	−	0.415227i	$-0.136298\pi$
$229$	1.44158	+	2.49689i	0.0952622	+	0.164999i	−0.814456	+	0.580226i	$0.802964\pi$
$230$	3.55842	+	6.16337i	0.234635	+	0.406400i
$231$		0			0
$232$	−4.37228	+	7.57301i	−0.287054	+	0.497193i
$233$	0.255437			0.0167343			0.00836713	−	0.999965i	$-0.497337\pi$
$233$	0.255437			0.0167343			0.00836713	+	0.999965i	$0.497337\pi$
$234$		0			0
$235$		0			0
$236$	−5.05842	+	8.76144i	−0.329275	+	0.570321i
$237$		0			0
$238$	0.686141	+	1.18843i	0.0444759	+	0.0770345i
$239$	4.93070	+	8.54023i	0.318941	+	0.552421i	0.980267	−	0.197677i	$-0.0633396\pi$
$239$	4.93070	+	8.54023i	0.318941	+	0.552421i	−0.661327	+	0.750098i	$0.730006\pi$
$240$		0			0
$241$	−9.05842	+	15.6896i	−0.583504	+	1.01066i	0.411556	+	0.911385i	$0.364986\pi$
$241$	−9.05842	+	15.6896i	−0.583504	+	1.01066i	−0.995060	+	0.0992745i	$0.968348\pi$
$242$	−9.11684			−0.586053
$243$		0			0
$244$	3.11684			0.199535
$245$	2.18614	−	3.78651i	0.139667	−	0.241911i
$246$		0			0
$247$	−5.00000	−	8.66025i	−0.318142	−	0.551039i
$248$	−1.00000	−	1.73205i	−0.0635001	−	0.109985i
$249$		0			0
$250$	19.9307	−	34.5210i	1.26053	−	2.18330i
$251$	9.00000			0.568075			0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000			0.568075			0.284037	+	0.958813i	$0.408326\pi$
$252$		0			0
$253$	2.23369			0.140431
$254$	−1.55842	+	2.69927i	−0.0977841	+	0.169367i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−3.43070	−	5.94215i	−0.214001	−	0.370661i	0.738962	−	0.673747i	$-0.235316\pi$
$257$	−3.43070	−	5.94215i	−0.214001	−	0.370661i	−0.952963	+	0.303086i	$0.901983\pi$
$258$		0			0
$259$	−1.00000	+	1.73205i	−0.0621370	+	0.107624i
$260$	−8.74456			−0.542315
$261$		0			0
$262$	1.62772			0.100561
$263$	3.81386	−	6.60580i	0.235173	−	0.407331i	−0.724150	−	0.689642i	$-0.757768\pi$
$263$	3.81386	−	6.60580i	0.235173	−	0.407331i	0.959323	+	0.282311i	$0.0911011\pi$
$264$		0			0
$265$	19.1168	+	33.1113i	1.17434	+	2.03401i
$266$	−2.50000	−	4.33013i	−0.153285	−	0.265497i
$267$		0			0
$268$	1.05842	−	1.83324i	0.0646534	−	0.111983i
$269$	1.62772			0.0992438			0.0496219	−	0.998768i	$-0.484198\pi$
$269$	1.62772			0.0992438			0.0496219	+	0.998768i	$0.484198\pi$
$270$		0			0
$271$	16.2337			0.986126			0.493063	−	0.869994i	$-0.335877\pi$
$271$	16.2337			0.986126			0.493063	+	0.869994i	$0.335877\pi$
$272$	0.686141	−	1.18843i	0.0416034	−	0.0720592i
$273$		0			0
$274$	5.31386	+	9.20387i	0.321022	+	0.556026i
$275$	−9.68614	−	16.7769i	−0.584096	−	1.01168i
$276$		0			0
$277$	6.11684	−	10.5947i	0.367526	−	0.636573i	−0.621652	−	0.783293i	$-0.713538\pi$
$277$	6.11684	−	10.5947i	0.367526	−	0.636573i	0.989178	+	0.146720i	$0.0468717\pi$
$278$	13.2337			0.793704
$279$		0			0
$280$	−4.37228			−0.261294
$281$	8.18614	−	14.1788i	0.488344	−	0.845837i	−0.511566	−	0.859244i	$-0.670934\pi$
$281$	8.18614	−	14.1788i	0.488344	−	0.845837i	0.999910	+	0.0134071i	$0.00426773\pi$
$282$		0			0
$283$	−13.5584	−	23.4839i	−0.805965	−	1.39597i	−0.915638	−	0.402004i	$-0.868314\pi$
$283$	−13.5584	−	23.4839i	−0.805965	−	1.39597i	0.109673	−	0.993968i	$-0.465019\pi$
$284$	−3.55842	−	6.16337i	−0.211153	−	0.365729i
$285$		0			0
$286$	−1.37228	+	2.37686i	−0.0811447	+	0.140547i
$287$	−4.62772			−0.273166
$288$		0			0
$289$	−15.1168			−0.889226
$290$	19.1168	−	33.1113i	1.12258	−	1.94437i
$291$		0			0
$292$	−6.05842	−	10.4935i	−0.354542	−	0.614085i
$293$	−5.18614	−	8.98266i	−0.302978	−	0.524773i	0.673831	−	0.738885i	$-0.264647\pi$
$293$	−5.18614	−	8.98266i	−0.302978	−	0.524773i	−0.976809	+	0.214113i	$0.931314\pi$
$294$		0			0
$295$	22.1168	−	38.3075i	1.28769	−	2.23035i
$296$	2.00000			0.116248
$297$		0			0
$298$	−3.25544			−0.188582
$299$	−1.62772	+	2.81929i	−0.0941334	+	0.163044i
$300$		0			0
$301$	4.05842	+	7.02939i	0.233924	+	0.405167i
$302$	−4.55842	−	7.89542i	−0.262308	−	0.454330i
$303$		0			0
$304$	−2.50000	+	4.33013i	−0.143385	+	0.248350i
$305$	−13.6277			−0.780321
$306$		0			0
$307$	−13.0000			−0.741949			−0.370975	−	0.928643i	$-0.620976\pi$
$307$	−13.0000			−0.741949			−0.370975	+	0.928643i	$0.620976\pi$
$308$	−0.686141	+	1.18843i	−0.0390965	+	0.0677171i
$309$		0			0
$310$	4.37228	+	7.57301i	0.248329	+	0.430118i
$311$	4.11684	+	7.13058i	0.233445	+	0.404338i	0.958820	−	0.284016i	$-0.0916668\pi$
$311$	4.11684	+	7.13058i	0.233445	+	0.404338i	−0.725375	+	0.688354i	$0.758334\pi$
$312$		0			0
$313$	10.0584	−	17.4217i	0.568536	−	0.984733i	−0.428175	−	0.903696i	$-0.640843\pi$
$313$	10.0584	−	17.4217i	0.568536	−	0.984733i	0.996711	−	0.0810370i	$-0.0258232\pi$
$314$	9.11684			0.514493
$315$		0			0
$316$	−5.11684			−0.287845
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	−0.937892	−	0.346929i	$-0.887225\pi$
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	0.769395	+	0.638774i	$0.220558\pi$
$318$		0			0
$319$	−6.00000	−	10.3923i	−0.335936	−	0.581857i
$320$	2.18614	+	3.78651i	0.122209	+	0.211672i
$321$		0			0
$322$	−0.813859	+	1.40965i	−0.0453546	+	0.0785565i
$323$	−6.86141			−0.381779
$324$		0			0
$325$	28.2337			1.56612
$326$	9.11684	−	15.7908i	0.504935	−	0.874574i
$327$		0			0
$328$	2.31386	+	4.00772i	0.127762	+	0.221289i
$329$		0			0
$330$		0			0
$331$	−11.1168	+	19.2549i	−0.611037	+	1.05835i	0.380029	+	0.924975i	$0.375914\pi$
$331$	−11.1168	+	19.2549i	−0.611037	+	1.05835i	−0.991066	+	0.133373i	$0.957419\pi$
$332$	−17.4891			−0.959840
$333$		0			0
$334$	5.48913			0.300352
$335$	−4.62772	+	8.01544i	−0.252839	+	0.437930i
$336$		0			0
$337$	4.05842	+	7.02939i	0.221076	+	0.382915i	0.955135	−	0.296171i	$-0.0957097\pi$
$337$	4.05842	+	7.02939i	0.221076	+	0.382915i	−0.734059	+	0.679086i	$0.762376\pi$
$338$	4.50000	+	7.79423i	0.244768	+	0.423950i
$339$		0			0
$340$	−3.00000	+	5.19615i	−0.162698	+	0.281801i
$341$	2.74456			0.148626
$342$		0			0
$343$	1.00000			0.0539949
$344$	4.05842	−	7.02939i	0.218815	−	0.378999i
$345$		0			0
$346$	−3.00000	−	5.19615i	−0.161281	−	0.279347i
$347$	−5.05842	−	8.76144i	−0.271550	−	0.470339i	0.697709	−	0.716382i	$-0.254203\pi$
$347$	−5.05842	−	8.76144i	−0.271550	−	0.470339i	−0.969259	+	0.246043i	$0.920870\pi$
$348$		0			0
$349$	11.0000	−	19.0526i	0.588817	−	1.01986i	−0.405571	−	0.914063i	$-0.632927\pi$
$349$	11.0000	−	19.0526i	0.588817	−	1.01986i	0.994388	−	0.105797i	$-0.0337393\pi$
$350$	14.1168			0.754577
$351$		0			0
$352$	1.37228			0.0731428
$353$	−6.68614	+	11.5807i	−0.355867	+	0.616380i	−0.987266	−	0.159078i	$-0.949148\pi$
$353$	−6.68614	+	11.5807i	−0.355867	+	0.616380i	0.631399	+	0.775458i	$0.282481\pi$
$354$		0			0
$355$	15.5584	+	26.9480i	0.825755	+	1.43025i
$356$	7.37228	+	12.7692i	0.390730	+	0.676764i
$357$		0			0
$358$	1.62772	−	2.81929i	0.0860276	−	0.149004i
$359$	−21.8614			−1.15380			−0.576900	−	0.816814i	$-0.695738\pi$
$359$	−21.8614			−1.15380			−0.576900	+	0.816814i	$0.695738\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$	−0.441578	+	0.764836i	−0.0232088	+	0.0401989i
$363$		0			0
$364$	−1.00000	−	1.73205i	−0.0524142	−	0.0907841i
$365$	26.4891	+	45.8805i	1.38650	+	2.40150i
$366$		0			0
$367$	6.11684	−	10.5947i	0.319297	−	0.553038i	−0.661045	−	0.750346i	$-0.729887\pi$
$367$	6.11684	−	10.5947i	0.319297	−	0.553038i	0.980341	+	0.197308i	$0.0632200\pi$
$368$	1.62772			0.0848507
$369$		0			0
$370$	−8.74456			−0.454608
$371$	−4.37228	+	7.57301i	−0.226998	+	0.393171i
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$	0.941578	+	1.63086i	0.0486878	+	0.0843298i
$375$		0			0
$376$		0			0
$377$	17.4891			0.900736
$378$		0			0
$379$	−8.11684			−0.416934			−0.208467	−	0.978029i	$-0.566847\pi$
$379$	−8.11684			−0.416934			−0.208467	+	0.978029i	$0.566847\pi$
$380$	10.9307	−	18.9325i	0.560733	−	0.971218i
$381$		0			0
$382$	−9.55842	−	16.5557i	−0.489051	−	0.847062i
$383$	−16.3723	−	28.3576i	−0.836584	−	1.44901i	−0.892734	−	0.450584i	$-0.851216\pi$
$383$	−16.3723	−	28.3576i	−0.836584	−	1.44901i	0.0561493	−	0.998422i	$-0.482118\pi$
$384$		0			0
$385$	3.00000	−	5.19615i	0.152894	−	0.264820i
$386$	−7.00000			−0.356291
$387$		0			0
$388$	−8.11684			−0.412070
$389$	5.48913	−	9.50744i	0.278310	−	0.482047i	−0.692655	−	0.721269i	$-0.743559\pi$
$389$	5.48913	−	9.50744i	0.278310	−	0.482047i	0.970965	+	0.239222i	$0.0768925\pi$
$390$		0			0
$391$	1.11684	+	1.93443i	0.0564812	+	0.0978284i
$392$	−0.500000	−	0.866025i	−0.0252538	−	0.0437409i
$393$		0			0
$394$	−3.00000	+	5.19615i	−0.151138	+	0.261778i
$395$	22.3723			1.12567
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$	5.00000	−	8.66025i	0.250627	−	0.434099i
$399$		0			0
$400$	−7.05842	−	12.2255i	−0.352921	−	0.611277i
$401$	−5.87228	−	10.1711i	−0.293248	−	0.507920i	0.681328	−	0.731978i	$-0.261403\pi$
$401$	−5.87228	−	10.1711i	−0.293248	−	0.507920i	−0.974576	+	0.224058i	$0.928069\pi$
$402$		0			0
$403$	−2.00000	+	3.46410i	−0.0996271	+	0.172559i
$404$	−1.62772			−0.0809820
$405$		0			0
$406$	8.74456			0.433985
$407$	−1.37228	+	2.37686i	−0.0680215	+	0.117817i
$408$		0			0
$409$	11.1753	+	19.3561i	0.552581	+	0.957099i	0.998087	+	0.0618200i	$0.0196905\pi$
$409$	11.1753	+	19.3561i	0.552581	+	0.957099i	−0.445506	+	0.895279i	$0.646976\pi$
$410$	−10.1168	−	17.5229i	−0.499635	−	0.865394i
$411$		0			0
$412$	5.00000	−	8.66025i	0.246332	−	0.426660i
$413$	10.1168			0.497817
$414$		0			0
$415$	76.4674			3.75364
$416$	−1.00000	+	1.73205i	−0.0490290	+	0.0849208i
$417$		0			0
$418$	−3.43070	−	5.94215i	−0.167801	−	0.290640i
$419$	−6.30298	−	10.9171i	−0.307921	−	0.533335i	0.669986	−	0.742373i	$-0.266300\pi$
$419$	−6.30298	−	10.9171i	−0.307921	−	0.533335i	−0.977907	+	0.209039i	$0.932967\pi$
$420$		0			0
$421$	−17.1168	+	29.6472i	−0.834224	+	1.44492i	0.0604368	+	0.998172i	$0.480751\pi$
$421$	−17.1168	+	29.6472i	−0.834224	+	1.44492i	−0.894661	+	0.446746i	$0.852583\pi$
$422$	−16.0000			−0.778868
$423$		0			0
$424$	8.74456			0.424674
$425$	9.68614	−	16.7769i	0.469847	−	0.813799i
$426$		0			0
$427$	−1.55842	−	2.69927i	−0.0754173	−	0.130627i
$428$	−3.68614	−	6.38458i	−0.178176	−	0.308610i
$429$		0			0
$430$	−17.7446	+	30.7345i	−0.855719	+	1.48215i
$431$	6.51087			0.313618			0.156809	−	0.987629i	$-0.449879\pi$
$431$	6.51087			0.313618			0.156809	+	0.987629i	$0.449879\pi$
$432$		0			0
$433$	−20.1168			−0.966754			−0.483377	−	0.875412i	$-0.660590\pi$
$433$	−20.1168			−0.966754			−0.483377	+	0.875412i	$0.660590\pi$
$434$	−1.00000	+	1.73205i	−0.0480015	+	0.0831411i
$435$		0			0
$436$	−7.00000	−	12.1244i	−0.335239	−	0.580651i
$437$	−4.06930	−	7.04823i	−0.194661	−	0.337162i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$	−6.00000			−0.286039
$441$		0			0
$442$	−2.74456			−0.130546
$443$	−20.0584	+	34.7422i	−0.953004	+	1.65065i	−0.214134	+	0.976804i	$0.568693\pi$
$443$	−20.0584	+	34.7422i	−0.953004	+	1.65065i	−0.738870	+	0.673848i	$0.764640\pi$
$444$		0			0
$445$	−32.2337	−	55.8304i	−1.52802	−	2.64661i
$446$	2.00000	+	3.46410i	0.0947027	+	0.164030i
$447$		0			0
$448$	−0.500000	+	0.866025i	−0.0236228	+	0.0409159i
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$	−6.35053			−0.299035
$452$	−2.18614	+	3.78651i	−0.102827	+	0.178102i
$453$		0			0
$454$	−6.12772	−	10.6135i	−0.287588	−	0.498117i
$455$	4.37228	+	7.57301i	0.204976	+	0.355028i
$456$		0			0
$457$	17.7337	−	30.7156i	0.829547	−	1.43682i	−0.0688472	−	0.997627i	$-0.521932\pi$
$457$	17.7337	−	30.7156i	0.829547	−	1.43682i	0.898394	−	0.439190i	$-0.144735\pi$
$458$	−2.88316			−0.134721
$459$		0			0
$460$	−7.11684			−0.331825
$461$	1.06930	−	1.85208i	0.0498021	−	0.0862598i	−0.840050	−	0.542509i	$-0.817474\pi$
$461$	1.06930	−	1.85208i	0.0498021	−	0.0862598i	0.889852	+	0.456250i	$0.150808\pi$
$462$		0			0
$463$	11.5584	+	20.0198i	0.537165	+	0.930398i	0.999055	+	0.0434604i	$0.0138382\pi$
$463$	11.5584	+	20.0198i	0.537165	+	0.930398i	−0.461890	+	0.886937i	$0.652828\pi$
$464$	−4.37228	−	7.57301i	−0.202978	−	0.351568i
$465$		0			0
$466$	−0.127719	+	0.221215i	−0.00591645	+	0.0102476i
$467$	−33.0951			−1.53146			−0.765729	−	0.643163i	$-0.777622\pi$
$467$	−33.0951			−1.53146			−0.765729	+	0.643163i	$0.777622\pi$
$468$		0			0
$469$	−2.11684			−0.0977468
$470$		0			0
$471$		0			0
$472$	−5.05842	−	8.76144i	−0.232833	−	0.403278i
$473$	5.56930	+	9.64630i	0.256077	+	0.443538i
$474$		0			0
$475$	−35.2921	+	61.1277i	−1.61931	+	2.80473i
$476$	−1.37228			−0.0628984
$477$		0			0
$478$	−9.86141			−0.451050
$479$	16.3723	−	28.3576i	0.748069	−	1.29569i	−0.200679	−	0.979657i	$-0.564315\pi$
$479$	16.3723	−	28.3576i	0.748069	−	1.29569i	0.948747	−	0.316036i	$-0.102352\pi$
$480$		0			0
$481$	−2.00000	−	3.46410i	−0.0911922	−	0.157949i
$482$	−9.05842	−	15.6896i	−0.412600	−	0.714644i
$483$		0			0
$484$	4.55842	−	7.89542i	0.207201	−	0.358883i
$485$	35.4891			1.61148
$486$		0			0
$487$	35.3505			1.60189			0.800943	−	0.598741i	$-0.204332\pi$
$487$	35.3505			1.60189			0.800943	+	0.598741i	$0.204332\pi$
$488$	−1.55842	+	2.69927i	−0.0705464	+	0.122190i
$489$		0			0
$490$	2.18614	+	3.78651i	0.0987598	+	0.171057i
$491$	−12.6861	−	21.9730i	−0.572518	−	0.991629i	−0.996306	−	0.0858685i	$-0.972634\pi$
$491$	−12.6861	−	21.9730i	−0.572518	−	0.991629i	0.423789	−	0.905761i	$-0.360700\pi$
$492$		0			0
$493$	6.00000	−	10.3923i	0.270226	−	0.468046i
$494$	10.0000			0.449921
$495$		0			0
$496$	2.00000			0.0898027
$497$	−3.55842	+	6.16337i	−0.159617	+	0.276465i
$498$		0			0
$499$	−9.05842	−	15.6896i	−0.405511	−	0.702365i	0.588870	−	0.808228i	$-0.299573\pi$
$499$	−9.05842	−	15.6896i	−0.405511	−	0.702365i	−0.994381	+	0.105863i	$0.966240\pi$
$500$	19.9307	+	34.5210i	0.891328	+	1.54383i
$501$		0			0
$502$	−4.50000	+	7.79423i	−0.200845	+	0.347873i
$503$	32.2337			1.43723			0.718615	−	0.695409i	$-0.244777\pi$
$503$	32.2337			1.43723			0.718615	+	0.695409i	$0.244777\pi$
$504$		0			0
$505$	7.11684			0.316695
$506$	−1.11684	+	1.93443i	−0.0496498	+	0.0859959i
$507$		0			0
$508$	−1.55842	−	2.69927i	−0.0691438	−	0.119761i
$509$	−14.4891	−	25.0959i	−0.642219	−	1.11236i	−0.984936	−	0.172918i	$-0.944681\pi$
$509$	−14.4891	−	25.0959i	−0.642219	−	1.11236i	0.342717	−	0.939439i	$-0.388653\pi$
$510$		0			0
$511$	−6.05842	+	10.4935i	−0.268009	+	0.464205i
$512$	1.00000			0.0441942
$513$		0			0
$514$	6.86141			0.302644
$515$	−21.8614	+	37.8651i	−0.963329	+	1.66853i
$516$		0			0
$517$		0			0
$518$	−1.00000	−	1.73205i	−0.0439375	−	0.0761019i
$519$		0			0
$520$	4.37228	−	7.57301i	0.191737	−	0.332099i
$521$	24.8614			1.08920			0.544599	−	0.838697i	$-0.316682\pi$
$521$	24.8614			1.08920			0.544599	+	0.838697i	$0.316682\pi$
$522$		0			0
$523$	−35.1168			−1.53555			−0.767776	−	0.640718i	$-0.778637\pi$
$523$	−35.1168			−1.53555			−0.767776	+	0.640718i	$0.778637\pi$
$524$	−0.813859	+	1.40965i	−0.0355536	+	0.0615807i
$525$		0			0
$526$	3.81386	+	6.60580i	0.166292	+	0.288026i
$527$	1.37228	+	2.37686i	0.0597775	+	0.103538i
$528$		0			0
$529$	10.1753	−	17.6241i	0.442403	−	0.766264i
$530$	−38.2337			−1.66077
$531$		0			0
$532$	5.00000			0.216777
$533$	4.62772	−	8.01544i	0.200449	−	0.347187i
$534$		0			0
$535$	16.1168	+	27.9152i	0.696792	+	1.20688i
$536$	1.05842	+	1.83324i	0.0457169	+	0.0791839i
$537$		0			0
$538$	−0.813859	+	1.40965i	−0.0350880	+	0.0607741i
$539$	1.37228			0.0591083
$540$		0			0
$541$	−6.23369			−0.268007			−0.134004	−	0.990981i	$-0.542783\pi$
$541$	−6.23369			−0.268007			−0.134004	+	0.990981i	$0.542783\pi$
$542$	−8.11684	+	14.0588i	−0.348648	+	0.603877i
$543$		0			0
$544$	0.686141	+	1.18843i	0.0294180	+	0.0509535i
$545$	30.6060	+	53.0111i	1.31102	+	2.27075i
$546$		0			0
$547$	−9.05842	+	15.6896i	−0.387310	+	0.670841i	−0.992087	−	0.125554i	$-0.959929\pi$
$547$	−9.05842	+	15.6896i	−0.387310	+	0.670841i	0.604777	+	0.796395i	$0.293262\pi$
$548$	−10.6277			−0.453994
$549$		0			0
$550$	19.3723			0.826037
$551$	−21.8614	+	37.8651i	−0.931327	+	1.61311i
$552$		0			0
$553$	2.55842	+	4.43132i	0.108795	+	0.188439i
$554$	6.11684	+	10.5947i	0.259880	+	0.450125i
$555$		0			0
$556$	−6.61684	+	11.4607i	−0.280617	+	0.486042i
$557$	−29.4891			−1.24949			−0.624747	−	0.780827i	$-0.714798\pi$
$557$	−29.4891			−1.24949			−0.624747	+	0.780827i	$0.714798\pi$
$558$		0			0
$559$	−16.2337			−0.686612
$560$	2.18614	−	3.78651i	0.0923813	−	0.160009i
$561$		0			0
$562$	8.18614	+	14.1788i	0.345312	+	0.598097i
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	0.895902	−	0.444252i	$-0.146530\pi$
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	−0.832684	+	0.553748i	$0.813197\pi$
$564$		0			0
$565$	9.55842	−	16.5557i	0.402126	−	0.696502i
$566$	27.1168			1.13981
$567$		0			0
$568$	7.11684			0.298616
$569$	8.05842	−	13.9576i	0.337827	−	0.585133i	−0.646197	−	0.763171i	$-0.723642\pi$
$569$	8.05842	−	13.9576i	0.337827	−	0.585133i	0.984024	+	0.178038i	$0.0569749\pi$
$570$		0			0
$571$	11.1753	+	19.3561i	0.467670	+	0.810029i	0.999318	−	0.0369371i	$-0.0117601\pi$
$571$	11.1753	+	19.3561i	0.467670	+	0.810029i	−0.531647	+	0.846966i	$0.678427\pi$
$572$	−1.37228	−	2.37686i	−0.0573780	−	0.0993815i
$573$		0			0
$574$	2.31386	−	4.00772i	0.0965786	−	0.167279i
$575$	22.9783			0.958259
$576$		0			0
$577$	9.88316			0.411441			0.205721	−	0.978611i	$-0.434046\pi$
$577$	9.88316			0.411441			0.205721	+	0.978611i	$0.434046\pi$
$578$	7.55842	−	13.0916i	0.314389	−	0.544538i
$579$		0			0
$580$	19.1168	+	33.1113i	0.793784	+	1.37487i
$581$	8.74456	+	15.1460i	0.362786	+	0.628363i
$582$		0			0
$583$	−6.00000	+	10.3923i	−0.248495	+	0.430405i
$584$	12.1168			0.501399
$585$		0			0
$586$	10.3723			0.428475
$587$	7.24456	−	12.5480i	0.299015	−	0.517909i	−0.676896	−	0.736079i	$-0.736675\pi$
$587$	7.24456	−	12.5480i	0.299015	−	0.517909i	0.975911	+	0.218170i	$0.0700086\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$	22.1168	+	38.3075i	0.910536	+	1.57709i
$591$		0			0
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	−14.7446			−0.605487			−0.302743	−	0.953072i	$-0.597902\pi$
$593$	−14.7446			−0.605487			−0.302743	+	0.953072i	$0.597902\pi$
$594$		0			0
$595$	6.00000			0.245976
$596$	1.62772	−	2.81929i	0.0666740	−	0.115483i
$597$		0			0
$598$	−1.62772	−	2.81929i	−0.0665624	−	0.115289i
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	0.999938	−	0.0111569i	$-0.00355143\pi$
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	−0.509631	+	0.860393i	$0.670218\pi$
$600$		0			0
$601$	−12.0584	+	20.8858i	−0.491873	+	0.851950i	−0.999956	−	0.00935863i	$-0.997021\pi$
$601$	−12.0584	+	20.8858i	−0.491873	+	0.851950i	0.508083	+	0.861308i	$0.330354\pi$
$602$	−8.11684			−0.330818
$603$		0			0
$604$	9.11684			0.370959
$605$	−19.9307	+	34.5210i	−0.810298	+	1.40348i
$606$		0			0
$607$	−11.1168	−	19.2549i	−0.451219	−	0.781534i	0.547243	−	0.836974i	$-0.315677\pi$
$607$	−11.1168	−	19.2549i	−0.451219	−	0.781534i	−0.998462	+	0.0554398i	$0.982344\pi$
$608$	−2.50000	−	4.33013i	−0.101388	−	0.175610i
$609$		0			0
$610$	6.81386	−	11.8020i	0.275885	−	0.477847i
$611$		0			0
$612$		0			0
$613$	−36.2337			−1.46346			−0.731732	−	0.681592i	$-0.761288\pi$
$613$	−36.2337			−1.46346			−0.731732	+	0.681592i	$0.761288\pi$
$614$	6.50000	−	11.2583i	0.262319	−	0.454349i
$615$		0			0
$616$	−0.686141	−	1.18843i	−0.0276454	−	0.0478832i
$617$	9.43070	+	16.3345i	0.379666	+	0.657600i	0.991014	−	0.133762i	$-0.0427056\pi$
$617$	9.43070	+	16.3345i	0.379666	+	0.657600i	−0.611348	+	0.791362i	$0.709372\pi$
$618$		0			0
$619$	−22.7337	+	39.3759i	−0.913744	+	1.58265i	−0.105014	+	0.994471i	$0.533489\pi$
$619$	−22.7337	+	39.3759i	−0.913744	+	1.58265i	−0.808730	+	0.588180i	$0.799844\pi$
$620$	−8.74456			−0.351190
$621$		0			0
$622$	−8.23369			−0.330141
$623$	7.37228	−	12.7692i	0.295364	−	0.511586i
$624$		0			0
$625$	−51.8505	−	89.8078i	−2.07402	−	3.59231i
$626$	10.0584	+	17.4217i	0.402015	+	0.696311i
$627$		0			0
$628$	−4.55842	+	7.89542i	−0.181901	+	0.315061i
$629$	−2.74456			−0.109433
$630$		0			0
$631$	−37.3505			−1.48690			−0.743451	−	0.668791i	$-0.766812\pi$
$631$	−37.3505			−1.48690			−0.743451	+	0.668791i	$0.766812\pi$
$632$	2.55842	−	4.43132i	0.101769	−	0.176268i
$633$		0			0
$634$	−3.00000	−	5.19615i	−0.119145	−	0.206366i
$635$	6.81386	+	11.8020i	0.270400	+	0.468346i
$636$		0			0
$637$	−1.00000	+	1.73205i	−0.0396214	+	0.0686264i
$638$	12.0000			0.475085
$639$		0			0
$640$	−4.37228			−0.172830
$641$	17.1060	−	29.6284i	0.675645	−	1.17025i	−0.300635	−	0.953739i	$-0.597198\pi$
$641$	17.1060	−	29.6284i	0.675645	−	1.17025i	0.976280	−	0.216512i	$-0.0694682\pi$
$642$		0			0
$643$	−13.1753	−	22.8202i	−0.519582	−	0.899942i	−0.999741	−	0.0227606i	$-0.992754\pi$
$643$	−13.1753	−	22.8202i	−0.519582	−	0.899942i	0.480159	−	0.877181i	$-0.340579\pi$
$644$	−0.813859	−	1.40965i	−0.0320706	−	0.0555478i
$645$		0			0
$646$	3.43070	−	5.94215i	0.134979	−	0.233791i
$647$	−5.48913			−0.215800			−0.107900	−	0.994162i	$-0.534413\pi$
$647$	−5.48913			−0.215800			−0.107900	+	0.994162i	$0.534413\pi$
$648$		0			0
$649$	13.8832			0.544962
$650$	−14.1168	+	24.4511i	−0.553708	+	0.959051i
$651$		0			0
$652$	9.11684	+	15.7908i	0.357043	+	0.618417i
$653$	−13.3723	−	23.1615i	−0.523298	−	0.906378i	−0.999632	−	0.0271143i	$-0.991368\pi$
$653$	−13.3723	−	23.1615i	−0.523298	−	0.906378i	0.476335	−	0.879264i	$-0.341965\pi$
$654$		0			0
$655$	3.55842	−	6.16337i	0.139039	−	0.240823i
$656$	−4.62772			−0.180682
$657$		0			0
$658$		0			0
$659$	−10.3723	+	17.9653i	−0.404047	+	0.699829i	−0.994210	−	0.107454i	$-0.965730\pi$
$659$	−10.3723	+	17.9653i	−0.404047	+	0.699829i	0.590163	+	0.807284i	$0.299063\pi$
$660$		0			0
$661$	−13.5584	−	23.4839i	−0.527361	−	0.913417i	−0.999491	−	0.0318879i	$-0.989848\pi$
$661$	−13.5584	−	23.4839i	−0.527361	−	0.913417i	0.472130	−	0.881529i	$-0.343485\pi$
$662$	−11.1168	−	19.2549i	−0.432068	−	0.748364i
$663$		0			0
$664$	8.74456	−	15.1460i	0.339355	−	0.587780i
$665$	−21.8614			−0.847749
$666$		0			0
$667$	14.2337			0.551131
$668$	−2.74456	+	4.75372i	−0.106190	+	0.183927i
$669$		0			0
$670$	−4.62772	−	8.01544i	−0.178784	−	0.309664i
$671$	−2.13859	−	3.70415i	−0.0825595	−	0.142997i
$672$		0			0
$673$	1.44158	−	2.49689i	0.0555687	−	0.0962479i	−0.836903	−	0.547351i	$-0.815636\pi$
$673$	1.44158	−	2.49689i	0.0555687	−	0.0962479i	0.892472	+	0.451103i	$0.148969\pi$
$674$	−8.11684			−0.312649
$675$		0			0
$676$	−9.00000			−0.346154
$677$	17.2337	−	29.8496i	0.662344	−	1.14721i	−0.317654	−	0.948207i	$-0.602895\pi$
$677$	17.2337	−	29.8496i	0.662344	−	1.14721i	0.979998	−	0.199007i	$-0.0637718\pi$
$678$		0			0
$679$	4.05842	+	7.02939i	0.155748	+	0.269763i
$680$	−3.00000	−	5.19615i	−0.115045	−	0.199263i
$681$		0			0
$682$	−1.37228	+	2.37686i	−0.0525474	+	0.0910147i
$683$	−29.8397			−1.14178			−0.570891	−	0.821026i	$-0.693402\pi$
$683$	−29.8397			−1.14178			−0.570891	+	0.821026i	$0.693402\pi$
$684$		0			0
$685$	46.4674			1.77543
$686$	−0.500000	+	0.866025i	−0.0190901	+	0.0330650i
$687$		0			0
$688$	4.05842	+	7.02939i	0.154726	+	0.267993i
$689$	−8.74456	−	15.1460i	−0.333141	−	0.577018i
$690$		0			0
$691$	11.5584	−	20.0198i	0.439703	−	0.761588i	−0.557963	−	0.829866i	$-0.688417\pi$
$691$	11.5584	−	20.0198i	0.439703	−	0.761588i	0.997666	+	0.0682775i	$0.0217503\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$	10.1168			0.384030
$695$	28.9307	−	50.1094i	1.09740	−	1.90076i
$696$		0			0
$697$	−3.17527	−	5.49972i	−0.120272	−	0.208317i
$698$	11.0000	+	19.0526i	0.416356	+	0.721150i
$699$		0			0
$700$	−7.05842	+	12.2255i	−0.266783	+	0.462082i
$701$	38.2337			1.44407			0.722033	−	0.691858i	$-0.243208\pi$
$701$	38.2337			1.44407			0.722033	+	0.691858i	$0.243208\pi$
$702$		0			0
$703$	10.0000			0.377157
$704$	−0.686141	+	1.18843i	−0.0258599	+	0.0447907i
$705$		0			0
$706$	−6.68614	−	11.5807i	−0.251636	−	0.435847i
$707$	0.813859	+	1.40965i	0.0306083	+	0.0530152i
$708$		0			0
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	0.0747503	+	0.997202i	$0.476184\pi$
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	−0.900978	+	0.433865i	$0.857149\pi$
$710$	−31.1168			−1.16779
$711$		0			0
$712$	−14.7446			−0.552576
$713$	−1.62772	+	2.81929i	−0.0609585	+	0.105583i
$714$		0			0
$715$	6.00000	+	10.3923i	0.224387	+	0.388650i
$716$	1.62772	+	2.81929i	0.0608307	+	0.105362i
$717$		0			0
$718$	10.9307	−	18.9325i	0.407930	−	0.706556i
$719$	−2.74456			−0.102355			−0.0511775	−	0.998690i	$-0.516297\pi$
$719$	−2.74456			−0.102355			−0.0511775	+	0.998690i	$0.516297\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$	−3.00000	+	5.19615i	−0.111648	+	0.193381i
$723$		0			0
$724$	−0.441578	−	0.764836i	−0.0164111	−	0.0284249i
$725$	−61.7228	−	106.907i	−2.29233	−	3.97043i
$726$		0			0
$727$	18.1168	−	31.3793i	0.671917	−	1.16379i	−0.305443	−	0.952210i	$-0.598805\pi$
$727$	18.1168	−	31.3793i	0.671917	−	1.16379i	0.977360	−	0.211583i	$-0.0678620\pi$
$728$	2.00000			0.0741249
$729$		0			0
$730$	−52.9783			−1.96081
$731$	−5.56930	+	9.64630i	−0.205988	+	0.356781i
$732$		0			0
$733$	20.5584	+	35.6082i	0.759343	+	1.31522i	0.943186	+	0.332265i	$0.107813\pi$
$733$	20.5584	+	35.6082i	0.759343	+	1.31522i	−0.183844	+	0.982956i	$0.558854\pi$
$734$	6.11684	+	10.5947i	0.225777	+	0.391057i
$735$		0			0
$736$	−0.813859	+	1.40965i	−0.0299993	+	0.0519602i
$737$	−2.90491			−0.107004
$738$		0			0
$739$	−8.11684			−0.298583			−0.149291	−	0.988793i	$-0.547699\pi$
$739$	−8.11684			−0.298583			−0.149291	+	0.988793i	$0.547699\pi$
$740$	4.37228	−	7.57301i	0.160728	−	0.278390i
$741$		0			0
$742$	−4.37228	−	7.57301i	−0.160511	−	0.278014i
$743$	−6.86141	−	11.8843i	−0.251721	−	0.435993i	0.712279	−	0.701896i	$-0.247663\pi$
$743$	−6.86141	−	11.8843i	−0.251721	−	0.435993i	−0.964000	+	0.265904i	$0.914330\pi$
$744$		0			0
$745$	−7.11684	+	12.3267i	−0.260741	+	0.451617i
$746$	−10.0000			−0.366126
$747$		0			0
$748$	−1.88316			−0.0688550
$749$	−3.68614	+	6.38458i	−0.134689	+	0.233288i
$750$		0			0
$751$	8.55842	+	14.8236i	0.312301	+	0.540922i	0.978860	−	0.204531i	$-0.0655668\pi$
$751$	8.55842	+	14.8236i	0.312301	+	0.540922i	−0.666559	+	0.745452i	$0.732234\pi$
$752$		0			0
$753$		0			0
$754$	−8.74456	+	15.1460i	−0.318458	+	0.551586i
$755$	−39.8614			−1.45071
$756$		0			0
$757$	46.2337			1.68039			0.840196	−	0.542283i	$-0.182440\pi$
$757$	46.2337			1.68039			0.840196	+	0.542283i	$0.182440\pi$
$758$	4.05842	−	7.02939i	0.147409	−	0.255319i
$759$		0			0
$760$	10.9307	+	18.9325i	0.396498	+	0.686755i
$761$	17.7446	+	30.7345i	0.643240	+	1.11412i	0.984705	+	0.174230i	$0.0557435\pi$
$761$	17.7446	+	30.7345i	0.643240	+	1.11412i	−0.341465	+	0.939894i	$0.610923\pi$
$762$		0			0
$763$	−7.00000	+	12.1244i	−0.253417	+	0.438931i
$764$	19.1168			0.691623
$765$		0			0
$766$	32.7446			1.18311
$767$	−10.1168	+	17.5229i	−0.365298	+	0.632715i
$768$		0			0
$769$	5.00000	+	8.66025i	0.180305	+	0.312297i	0.941984	−	0.335657i	$-0.108958\pi$
$769$	5.00000	+	8.66025i	0.180305	+	0.312297i	−0.761680	+	0.647954i	$0.775625\pi$
$770$	3.00000	+	5.19615i	0.108112	+	0.187256i
$771$		0			0
$772$	3.50000	−	6.06218i	0.125968	−	0.218183i
$773$	39.8614			1.43372			0.716858	−	0.697220i	$-0.245580\pi$
$773$	39.8614			1.43372			0.716858	+	0.697220i	$0.245580\pi$
$774$		0			0
$775$	28.2337			1.01418
$776$	4.05842	−	7.02939i	0.145689	−	0.252341i
$777$		0			0
$778$	5.48913	+	9.50744i	0.196795	+	0.340858i
$779$	11.5693	+	20.0386i	0.414513	+	0.717958i
$780$		0			0
$781$	−4.88316	+	8.45787i	−0.174733	+	0.302647i
$782$	−2.23369			−0.0798765
$783$		0			0
$784$	1.00000			0.0357143
$785$	19.9307	−	34.5210i	0.711357	−	1.23211i
$786$		0			0
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	0.899468	−	0.436987i	$-0.143954\pi$
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	−0.828176	+	0.560469i	$0.810621\pi$
$788$	−3.00000	−	5.19615i	−0.106871	−	0.185105i
$789$		0			0
$790$	−11.1861	+	19.3750i	−0.397985	+	0.689330i
$791$	4.37228			0.155460
$792$		0			0
$793$	6.23369			0.221365
$794$	11.0000	−	19.0526i	0.390375	−	0.676150i
$795$		0			0
$796$	5.00000	+	8.66025i	0.177220	+	0.306955i
$797$	−4.06930	−	7.04823i	−0.144142	−	0.249661i	0.784911	−	0.619609i	$-0.212709\pi$
$797$	−4.06930	−	7.04823i	−0.144142	−	0.249661i	−0.929052	+	0.369948i	$0.879376\pi$
$798$		0			0
$799$		0			0
$800$	14.1168			0.499106
$801$		0			0
$802$	11.7446			0.414715
$803$	−8.31386	+	14.4000i	−0.293390	+	0.508166i
$804$		0			0
$805$	3.55842	+	6.16337i	0.125418	+	0.217230i
$806$	−2.00000	−	3.46410i	−0.0704470	−	0.122018i
$807$		0			0
$808$	0.813859	−	1.40965i	0.0286315	−	0.0495912i
$809$	6.86141			0.241234			0.120617	−	0.992699i	$-0.461513\pi$
$809$	6.86141			0.241234			0.120617	+	0.992699i	$0.461513\pi$
$810$		0			0
$811$	42.1168			1.47892			0.739461	−	0.673199i	$-0.235080\pi$
$811$	42.1168			1.47892			0.739461	+	0.673199i	$0.235080\pi$
$812$	−4.37228	+	7.57301i	−0.153437	+	0.265761i
$813$		0			0
$814$	−1.37228	−	2.37686i	−0.0480984	−	0.0833089i
$815$	−39.8614	−	69.0420i	−1.39628	−	2.41844i
$816$		0			0
$817$	20.2921	−	35.1470i	0.709931	−	1.22964i
$818$	−22.3505			−0.781468
$819$		0			0
$820$	20.2337			0.706591
$821$	−1.88316	+	3.26172i	−0.0657226	+	0.113835i	−0.897014	−	0.442002i	$-0.854269\pi$
$821$	−1.88316	+	3.26172i	−0.0657226	+	0.113835i	0.831292	+	0.555836i	$0.187602\pi$
$822$		0			0
$823$	6.11684	+	10.5947i	0.213220	+	0.369307i	0.952720	−	0.303848i	$-0.0982716\pi$
$823$	6.11684	+	10.5947i	0.213220	+	0.369307i	−0.739501	+	0.673156i	$0.764938\pi$
$824$	5.00000	+	8.66025i	0.174183	+	0.301694i
$825$		0			0
$826$	−5.05842	+	8.76144i	−0.176005	+	0.304850i
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	−13.7663			−0.478124			−0.239062	−	0.971004i	$-0.576840\pi$
$829$	−13.7663			−0.478124			−0.239062	+	0.971004i	$0.576840\pi$
$830$	−38.2337	+	66.2227i	−1.32711	+	2.29862i
$831$		0			0
$832$	−1.00000	−	1.73205i	−0.0346688	−	0.0600481i
$833$	0.686141	+	1.18843i	0.0237734	+	0.0411767i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$	6.86141			0.237307
$837$		0			0
$838$	12.6060			0.435466
$839$	2.74456	−	4.75372i	0.0947528	−	0.164117i	−0.814753	−	0.579809i	$-0.803127\pi$
$839$	2.74456	−	4.75372i	0.0947528	−	0.164117i	0.909505	+	0.415692i	$0.136461\pi$
$840$		0			0
$841$	−23.7337	−	41.1080i	−0.818403	−	1.41752i
$842$	−17.1168	−	29.6472i	−0.589885	−	1.02171i
$843$		0			0
$844$	8.00000	−	13.8564i	0.275371	−	0.476957i
$845$	39.3505			1.35370
$846$		0			0
$847$	−9.11684			−0.313258
$848$	−4.37228	+	7.57301i	−0.150145	+	0.260058i
$849$		0			0
$850$	9.68614	+	16.7769i	0.332232	+	0.575443i
$851$	−1.62772	−	2.81929i	−0.0557975	−	0.0966441i
$852$		0			0
$853$	17.5584	−	30.4121i	0.601189	−	1.04129i	−0.391452	−	0.920198i	$-0.628027\pi$
$853$	17.5584	−	30.4121i	0.601189	−	1.04129i	0.992641	−	0.121091i	$-0.0386394\pi$
$854$	3.11684			0.106656
$855$		0			0
$856$	7.37228			0.251979
$857$	−19.9783	+	34.6033i	−0.682444	+	1.18203i	0.291789	+	0.956483i	$0.405750\pi$
$857$	−19.9783	+	34.6033i	−0.682444	+	1.18203i	−0.974233	+	0.225545i	$0.927584\pi$
$858$		0			0
$859$	−16.9416	−	29.3437i	−0.578039	−	1.00119i	−0.995704	−	0.0925921i	$-0.970485\pi$
$859$	−16.9416	−	29.3437i	−0.578039	−	1.00119i	0.417665	−	0.908601i	$-0.362849\pi$
$860$	−17.7446	−	30.7345i	−0.605085	−	1.04804i
$861$		0			0
$862$	−3.25544	+	5.63858i	−0.110881	+	0.192051i
$863$	−9.86141			−0.335686			−0.167843	−	0.985814i	$-0.553680\pi$
$863$	−9.86141			−0.335686			−0.167843	+	0.985814i	$0.553680\pi$
$864$		0			0
$865$	−26.2337			−0.891972
$866$	10.0584	−	17.4217i	0.341799	−	0.592013i
$867$		0			0
$868$	−1.00000	−	1.73205i	−0.0339422	−	0.0587896i
$869$	3.51087	+	6.08101i	0.119098	+	0.206284i
$870$		0			0
$871$	2.11684	−	3.66648i	0.0717265	−	0.124234i
$872$	14.0000			0.474100
$873$		0			0
$874$	8.13859			0.275292
$875$	19.9307	−	34.5210i	0.673781	−	1.16702i
$876$		0			0
$877$	29.3505	+	50.8366i	0.991097	+	1.71663i	0.610852	+	0.791744i	$0.290827\pi$
$877$	29.3505	+	50.8366i	0.991097	+	1.71663i	0.380245	+	0.924886i	$0.375840\pi$
$878$	−4.00000	−	6.92820i	−0.134993	−	0.233816i
$879$		0			0
$880$	3.00000	−	5.19615i	0.101130	−	0.175162i
$881$	20.2337			0.681690			0.340845	−	0.940119i	$-0.389287\pi$
$881$	20.2337			0.681690			0.340845	+	0.940119i	$0.389287\pi$
$882$		0			0
$883$	−40.3505			−1.35790			−0.678952	−	0.734183i	$-0.737565\pi$
$883$	−40.3505			−1.35790			−0.678952	+	0.734183i	$0.737565\pi$
$884$	1.37228	−	2.37686i	0.0461548	−	0.0799425i
$885$		0			0
$886$	−20.0584	−	34.7422i	−0.673876	−	1.16719i
$887$	12.8614	+	22.2766i	0.431844	+	0.747975i	0.997032	−	0.0769865i	$-0.0245298\pi$
$887$	12.8614	+	22.2766i	0.431844	+	0.747975i	−0.565188	+	0.824962i	$0.691197\pi$
$888$		0			0
$889$	−1.55842	+	2.69927i	−0.0522678	+	0.0905305i
$890$	64.4674			2.16095
$891$		0			0
$892$	−4.00000			−0.133930
$893$		0			0
$894$		0			0
$895$	−7.11684	−	12.3267i	−0.237890	−	0.412037i
$896$	−0.500000	−	0.866025i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	16.5000	−	28.5788i	0.550612	−	0.953688i
$899$	17.4891			0.583295
$900$		0			0
$901$	−12.0000			−0.399778
$902$	3.17527	−	5.49972i	0.105725	−	0.183121i
$903$		0			0
$904$	−2.18614	−	3.78651i	−0.0727100	−	0.125937i
$905$	1.93070	+	3.34408i	0.0641787	+	0.111161i
$906$		0			0
$907$	13.0584	−	22.6179i	0.433598	−	0.751013i	−0.563582	−	0.826060i	$-0.690577\pi$
$907$	13.0584	−	22.6179i	0.433598	−	0.751013i	0.997180	+	0.0750466i	$0.0239105\pi$
$908$	12.2554			0.406711
$909$		0			0
$910$	−8.74456			−0.289879
$911$	−18.8139	+	32.5866i	−0.623331	+	1.07964i	0.365530	+	0.930800i	$0.380888\pi$
$911$	−18.8139	+	32.5866i	−0.623331	+	1.07964i	−0.988861	+	0.148841i	$0.952446\pi$
$912$		0			0
$913$	12.0000	+	20.7846i	0.397142	+	0.687870i
$914$	17.7337	+	30.7156i	0.586578	+	1.01598i
$915$		0			0
$916$	1.44158	−	2.49689i	0.0476311	−	0.0824994i
$917$	1.62772			0.0537520
$918$		0			0
$919$	−47.1168			−1.55424			−0.777121	−	0.629352i	$-0.783321\pi$
$919$	−47.1168			−1.55424			−0.777121	+	0.629352i	$0.783321\pi$
$920$	3.55842	−	6.16337i	0.117318	−	0.203200i
$921$		0			0
$922$	1.06930	+	1.85208i	0.0352154	+	0.0609949i
$923$	−7.11684	−	12.3267i	−0.234254	−	0.405739i
$924$		0			0
$925$	−14.1168	+	24.4511i	−0.464159	+	0.803947i
$926$	−23.1168			−0.759667
$927$		0			0
$928$	8.74456			0.287054
$929$	22.1168	−	38.3075i	0.725630	−	1.25683i	−0.233084	−	0.972457i	$-0.574882\pi$
$929$	22.1168	−	38.3075i	0.725630	−	1.25683i	0.958714	−	0.284372i	$-0.0917850\pi$
$930$		0			0
$931$	−2.50000	−	4.33013i	−0.0819342	−	0.141914i
$932$	−0.127719	−	0.221215i	−0.00418356	−	0.00724615i
$933$		0			0
$934$	16.5475	−	28.6612i	0.541452	−	0.937823i
$935$	8.23369			0.269270
$936$		0			0
$937$	30.4674			0.995326			0.497663	−	0.867371i	$-0.334192\pi$
$937$	30.4674			0.995326			0.497663	+	0.867371i	$0.334192\pi$
$938$	1.05842	−	1.83324i	0.0345587	−	0.0598574i
$939$		0			0
$940$		0			0
$941$	9.55842	+	16.5557i	0.311596	+	0.539699i	0.978708	−	0.205258i	$-0.0658032\pi$
$941$	9.55842	+	16.5557i	0.311596	+	0.539699i	−0.667112	+	0.744957i	$0.732470\pi$
$942$		0			0
$943$	3.76631	−	6.52344i	0.122648	−	0.212433i
$944$	10.1168			0.329275
$945$		0			0
$946$	−11.1386			−0.362147
$947$	17.0584	−	29.5461i	0.554324	−	0.960118i	−0.443632	−	0.896209i	$-0.646310\pi$
$947$	17.0584	−	29.5461i	0.554324	−	0.960118i	0.997956	−	0.0639085i	$-0.0203566\pi$
$948$		0			0
$949$	−12.1168	−	20.9870i	−0.393329	−	0.681267i
$950$	−35.2921	−	61.1277i	−1.14503	−	1.98325i
$951$		0			0
$952$	0.686141	−	1.18843i	0.0222379	−	0.0385173i
$953$	−28.1168			−0.910794			−0.455397	−	0.890288i	$-0.650503\pi$
$953$	−28.1168			−0.910794			−0.455397	+	0.890288i	$0.650503\pi$
$954$		0			0
$955$	−83.5842			−2.70472
$956$	4.93070	−	8.54023i	0.159470	−	0.276211i
$957$		0			0
$958$	16.3723	+	28.3576i	0.528964	+	0.916193i
$959$	5.31386	+	9.20387i	0.171593	+	0.297209i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	4.00000			0.128965
$963$		0			0
$964$	18.1168			0.583504
$965$	−15.3030	+	26.5055i	−0.492621	+	0.853244i
$966$		0			0
$967$	−15.4416	−	26.7456i	−0.496568	−	0.860080i	0.503424	−	0.864039i	$-0.332073\pi$
$967$	−15.4416	−	26.7456i	−0.496568	−	0.860080i	−0.999992	+	0.00395879i	$0.998740\pi$
$968$	4.55842	+	7.89542i	0.146513	+	0.253768i
$969$		0			0
$970$	−17.7446	+	30.7345i	−0.569744	+	0.986825i
$971$	−1.62772			−0.0522360			−0.0261180	−	0.999659i	$-0.508315\pi$
$971$	−1.62772			−0.0522360			−0.0261180	+	0.999659i	$0.508315\pi$
$972$		0			0
$973$	13.2337			0.424253
$974$	−17.6753	+	30.6145i	−0.566352	+	0.980951i
$975$		0			0
$976$	−1.55842	−	2.69927i	−0.0498839	−	0.0864014i
$977$	20.0584	+	34.7422i	0.641726	+	1.11150i	0.985047	+	0.172284i	$0.0551146\pi$
$977$	20.0584	+	34.7422i	0.641726	+	1.11150i	−0.343322	+	0.939218i	$0.611552\pi$
$978$		0			0
$979$	10.1168	−	17.5229i	0.323336	−	0.560034i
$980$	−4.37228			−0.139667
$981$		0			0
$982$	25.3723			0.809662
$983$	−19.6277	+	33.9962i	−0.626027	+	1.08431i	0.362314	+	0.932056i	$0.381987\pi$
$983$	−19.6277	+	33.9962i	−0.626027	+	1.08431i	−0.988341	+	0.152255i	$0.951347\pi$
$984$		0			0
$985$	13.1168	+	22.7190i	0.417937	+	0.723889i
$986$	6.00000	+	10.3923i	0.191079	+	0.330958i
$987$		0			0
$988$	−5.00000	+	8.66025i	−0.159071	+	0.275519i
$989$	−13.2119			−0.420115
$990$		0			0
$991$	48.4674			1.53962			0.769808	−	0.638275i	$-0.220352\pi$
$991$	48.4674			1.53962			0.769808	+	0.638275i	$0.220352\pi$
$992$	−1.00000	+	1.73205i	−0.0317500	+	0.0549927i
$993$		0			0
$994$	−3.55842	−	6.16337i	−0.112866	−	0.195490i
$995$	−21.8614	−	37.8651i	−0.693053	−	1.20040i
$996$		0			0
$997$	2.55842	−	4.43132i	0.0810260	−	0.140341i	−0.822665	−	0.568527i	$-0.807514\pi$
$997$	2.55842	−	4.43132i	0.0810260	−	0.140341i	0.903691	+	0.428185i	$0.140847\pi$
$998$	18.1168			0.573479
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.f.c.127.2		4
3.2	odd	2		126.2.f.d.43.1	✓	4
4.3	odd	2		3024.2.r.f.2017.2		4
7.2	even	3		2646.2.e.n.2125.2		4
7.3	odd	6		2646.2.h.l.667.2		4
7.4	even	3		2646.2.h.k.667.1		4
7.5	odd	6		2646.2.e.m.2125.1		4
7.6	odd	2		2646.2.f.j.883.1		4
9.2	odd	6		1134.2.a.k.1.2		2
9.4	even	3	inner	378.2.f.c.253.2		4
9.5	odd	6		126.2.f.d.85.1	yes	4
9.7	even	3		1134.2.a.n.1.1		2
12.11	even	2		1008.2.r.f.673.2		4
21.2	odd	6		882.2.e.l.655.1		4
21.5	even	6		882.2.e.k.655.2		4
21.11	odd	6		882.2.h.m.79.2		4
21.17	even	6		882.2.h.n.79.1		4
21.20	even	2		882.2.f.k.295.2		4
36.7	odd	6		9072.2.a.bb.1.1		2
36.11	even	6		9072.2.a.bm.1.2		2
36.23	even	6		1008.2.r.f.337.2		4
36.31	odd	6		3024.2.r.f.1009.2		4
63.4	even	3		2646.2.e.n.1549.2		4
63.5	even	6		882.2.h.n.67.1		4
63.13	odd	6		2646.2.f.j.1765.1		4
63.20	even	6		7938.2.a.bh.1.1		2
63.23	odd	6		882.2.h.m.67.2		4
63.31	odd	6		2646.2.e.m.1549.1		4
63.32	odd	6		882.2.e.l.373.2		4
63.34	odd	6		7938.2.a.bs.1.2		2
63.40	odd	6		2646.2.h.l.361.2		4
63.41	even	6		882.2.f.k.589.2		4
63.58	even	3		2646.2.h.k.361.1		4
63.59	even	6		882.2.e.k.373.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.d.43.1	✓	4	3.2	odd	2
126.2.f.d.85.1	yes	4	9.5	odd	6
378.2.f.c.127.2		4	1.1	even	1	trivial
378.2.f.c.253.2		4	9.4	even	3	inner
882.2.e.k.373.1		4	63.59	even	6
882.2.e.k.655.2		4	21.5	even	6
882.2.e.l.373.2		4	63.32	odd	6
882.2.e.l.655.1		4	21.2	odd	6
882.2.f.k.295.2		4	21.20	even	2
882.2.f.k.589.2		4	63.41	even	6
882.2.h.m.67.2		4	63.23	odd	6
882.2.h.m.79.2		4	21.11	odd	6
882.2.h.n.67.1		4	63.5	even	6
882.2.h.n.79.1		4	21.17	even	6
1008.2.r.f.337.2		4	36.23	even	6
1008.2.r.f.673.2		4	12.11	even	2
1134.2.a.k.1.2		2	9.2	odd	6
1134.2.a.n.1.1		2	9.7	even	3
2646.2.e.m.1549.1		4	63.31	odd	6
2646.2.e.m.2125.1		4	7.5	odd	6
2646.2.e.n.1549.2		4	63.4	even	3
2646.2.e.n.2125.2		4	7.2	even	3
2646.2.f.j.883.1		4	7.6	odd	2
2646.2.f.j.1765.1		4	63.13	odd	6
2646.2.h.k.361.1		4	63.58	even	3
2646.2.h.k.667.1		4	7.4	even	3
2646.2.h.l.361.2		4	63.40	odd	6
2646.2.h.l.667.2		4	7.3	odd	6
3024.2.r.f.1009.2		4	36.31	odd	6
3024.2.r.f.2017.2		4	4.3	odd	2
7938.2.a.bh.1.1		2	63.20	even	6
7938.2.a.bs.1.2		2	63.34	odd	6
9072.2.a.bb.1.1		2	36.7	odd	6
9072.2.a.bm.1.2		2	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.18614 + 3.78651i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.18614 + 3.78651i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -4.37228 q^{10} +(-0.686141 + 1.18843i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.37228 q^{17} +5.00000 q^{19} +(2.18614 - 3.78651i) q^{20} +(-0.686141 - 1.18843i) q^{22} +(-0.813859 - 1.40965i) q^{23} +(-7.05842 + 12.2255i) q^{25} +2.00000 q^{26} +1.00000 q^{28} +(-4.37228 + 7.57301i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.686141 - 1.18843i) q^{34} -4.37228 q^{35} +2.00000 q^{37} +(-2.50000 + 4.33013i) q^{38} +(2.18614 + 3.78651i) q^{40} +(2.31386 + 4.00772i) q^{41} +(4.05842 - 7.02939i) q^{43} +1.37228 q^{44} +1.62772 q^{46} +(-0.500000 - 0.866025i) q^{49} +(-7.05842 - 12.2255i) q^{50} +(-1.00000 + 1.73205i) q^{52} +8.74456 q^{53} -6.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-4.37228 - 7.57301i) q^{58} +(-5.05842 - 8.76144i) q^{59} +(-1.55842 + 2.69927i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(4.37228 - 7.57301i) q^{65} +(1.05842 + 1.83324i) q^{67} +(0.686141 + 1.18843i) q^{68} +(2.18614 - 3.78651i) q^{70} +7.11684 q^{71} +12.1168 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-2.50000 - 4.33013i) q^{76} +(-0.686141 - 1.18843i) q^{77} +(2.55842 - 4.43132i) q^{79} -4.37228 q^{80} -4.62772 q^{82} +(8.74456 - 15.1460i) q^{83} +(-3.00000 - 5.19615i) q^{85} +(4.05842 + 7.02939i) q^{86} +(-0.686141 + 1.18843i) q^{88} -14.7446 q^{89} +2.00000 q^{91} +(-0.813859 + 1.40965i) q^{92} +(10.9307 + 18.9325i) q^{95} +(4.05842 - 7.02939i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 3 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 - 2 * q^7 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} + 3 q^{5} - 2 q^{7} + 4 q^{8} - 6 q^{10} + 3 q^{11} - 4 q^{13} - 2 q^{14} - 2 q^{16} + 6 q^{17} + 20 q^{19} + 3 q^{20} + 3 q^{22} - 9 q^{23} - 11 q^{25} + 8 q^{26} + 4 q^{28} - 6 q^{29} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 6 q^{35} + 8 q^{37} - 10 q^{38} + 3 q^{40} + 15 q^{41} - q^{43} - 6 q^{44} + 18 q^{46} - 2 q^{49} - 11 q^{50} - 4 q^{52} + 12 q^{53} - 24 q^{55} - 2 q^{56} - 6 q^{58} - 3 q^{59} + 11 q^{61} + 8 q^{62} + 4 q^{64} + 6 q^{65} - 13 q^{67} - 3 q^{68} + 3 q^{70} - 6 q^{71} + 14 q^{73} - 4 q^{74} - 10 q^{76} + 3 q^{77} - 7 q^{79} - 6 q^{80} - 30 q^{82} + 12 q^{83} - 12 q^{85} - q^{86} + 3 q^{88} - 36 q^{89} + 8 q^{91} - 9 q^{92} + 15 q^{95} - q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 + 3 * q^5 - 2 * q^7 + 4 * q^8 - 6 * q^10 + 3 * q^11 - 4 * q^13 - 2 * q^14 - 2 * q^16 + 6 * q^17 + 20 * q^19 + 3 * q^20 + 3 * q^22 - 9 * q^23 - 11 * q^25 + 8 * q^26 + 4 * q^28 - 6 * q^29 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 6 * q^35 + 8 * q^37 - 10 * q^38 + 3 * q^40 + 15 * q^41 - q^43 - 6 * q^44 + 18 * q^46 - 2 * q^49 - 11 * q^50 - 4 * q^52 + 12 * q^53 - 24 * q^55 - 2 * q^56 - 6 * q^58 - 3 * q^59 + 11 * q^61 + 8 * q^62 + 4 * q^64 + 6 * q^65 - 13 * q^67 - 3 * q^68 + 3 * q^70 - 6 * q^71 + 14 * q^73 - 4 * q^74 - 10 * q^76 + 3 * q^77 - 7 * q^79 - 6 * q^80 - 30 * q^82 + 12 * q^83 - 12 * q^85 - q^86 + 3 * q^88 - 36 * q^89 + 8 * q^91 - 9 * q^92 + 15 * q^95 - q^97 + 4 * q^98

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