LMFDB - Embedded newform 270.2.e.b.181.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,2,Mod(91,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("270.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	270.e (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:	$2.15596085457$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			181.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	270.181
Dual form			270.2.e.b.91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} -1.00000 q^{10} +(3.00000 - 5.19615i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -4.00000 q^{19} +(-0.500000 + 0.866025i) q^{20} +(-3.00000 - 5.19615i) q^{22} +(4.50000 + 7.79423i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.00000 q^{26} -1.00000 q^{28} +(1.50000 - 2.59808i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} -1.00000 q^{35} +8.00000 q^{37} +(-2.00000 + 3.46410i) q^{38} +(0.500000 + 0.866025i) q^{40} +(-1.50000 - 2.59808i) q^{41} +(-4.00000 + 6.92820i) q^{43} -6.00000 q^{44} +9.00000 q^{46} +(-1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} -6.00000 q^{53} -6.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-1.50000 - 2.59808i) q^{58} +(3.00000 + 5.19615i) q^{59} +(6.50000 - 11.2583i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-1.00000 + 1.73205i) q^{65} +(6.50000 + 11.2583i) q^{67} +(-0.500000 + 0.866025i) q^{70} +6.00000 q^{71} -4.00000 q^{73} +(4.00000 - 6.92820i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(5.00000 - 8.66025i) q^{79} +1.00000 q^{80} -3.00000 q^{82} +(-4.50000 + 7.79423i) q^{83} +(4.00000 + 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{88} -9.00000 q^{89} -2.00000 q^{91} +(4.50000 - 7.79423i) q^{92} +(1.50000 + 2.59808i) q^{94} +(2.00000 + 3.46410i) q^{95} +(-1.00000 + 1.73205i) q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - q^5 + q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} - q^{14} - q^{16} - 8 q^{19} - q^{20} - 6 q^{22} + 9 q^{23} - q^{25} - 4 q^{26} - 2 q^{28} + 3 q^{29} + 4 q^{31} + q^{32} - 2 q^{35} + 16 q^{37} - 4 q^{38} + q^{40} - 3 q^{41} - 8 q^{43} - 12 q^{44} + 18 q^{46} - 3 q^{47} + 6 q^{49} + q^{50} - 2 q^{52} - 12 q^{53} - 12 q^{55} - q^{56} - 3 q^{58} + 6 q^{59} + 13 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 13 q^{67} - q^{70} + 12 q^{71} - 8 q^{73} + 8 q^{74} + 4 q^{76} - 6 q^{77} + 10 q^{79} + 2 q^{80} - 6 q^{82} - 9 q^{83} + 8 q^{86} - 6 q^{88} - 18 q^{89} - 4 q^{91} + 9 q^{92} + 3 q^{94} + 4 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^5 + q^7 - 2 * q^8 - 2 * q^10 + 6 * q^11 - 2 * q^13 - q^14 - q^16 - 8 * q^19 - q^20 - 6 * q^22 + 9 * q^23 - q^25 - 4 * q^26 - 2 * q^28 + 3 * q^29 + 4 * q^31 + q^32 - 2 * q^35 + 16 * q^37 - 4 * q^38 + q^40 - 3 * q^41 - 8 * q^43 - 12 * q^44 + 18 * q^46 - 3 * q^47 + 6 * q^49 + q^50 - 2 * q^52 - 12 * q^53 - 12 * q^55 - q^56 - 3 * q^58 + 6 * q^59 + 13 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 + 13 * q^67 - q^70 + 12 * q^71 - 8 * q^73 + 8 * q^74 + 4 * q^76 - 6 * q^77 + 10 * q^79 + 2 * q^80 - 6 * q^82 - 9 * q^83 + 8 * q^86 - 6 * q^88 - 18 * q^89 - 4 * q^91 + 9 * q^92 + 3 * q^94 + 4 * q^95 - 2 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.821541	−	0.570149i	$-0.193114\pi$
$12$		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
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−0.500000	+	0.866025i	−0.111803	+	0.193649i
$21$		0			0
$22$	−3.00000	−	5.19615i	−0.639602	−	1.10782i
$23$	4.50000	+	7.79423i	0.938315	+	1.62521i	0.768613	+	0.639713i	$0.220947\pi$
$23$	4.50000	+	7.79423i	0.938315	+	1.62521i	0.169701	+	0.985496i	$0.445720\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	−2.00000			−0.392232
$27$		0			0
$28$	−1.00000			−0.188982
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$	−2.00000	+	3.46410i	−0.324443	+	0.561951i
$39$		0			0
$40$	0.500000	+	0.866025i	0.0790569	+	0.136931i
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	0.381246	+	0.924473i	$0.375495\pi$
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	−0.991241	+	0.132068i	$0.957838\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	9.00000			1.32698
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$	0.500000	+	0.866025i	0.0707107	+	0.122474i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	−1.50000	−	2.59808i	−0.196960	−	0.341144i
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	0.992524	−	0.122047i	$-0.0389457\pi$
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	−0.601958	+	0.798528i	$0.705612\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	−1.00000	+	1.73205i	−0.124035	+	0.214834i
$66$		0			0
$67$	6.50000	+	11.2583i	0.794101	+	1.37542i	0.923408	+	0.383819i	$0.125391\pi$
$67$	6.50000	+	11.2583i	0.794101	+	1.37542i	−0.129307	+	0.991605i	$0.541275\pi$
$68$		0			0
$69$		0			0
$70$	−0.500000	+	0.866025i	−0.0597614	+	0.103510i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$	4.00000	−	6.92820i	0.464991	−	0.805387i
$75$		0			0
$76$	2.00000	+	3.46410i	0.229416	+	0.397360i
$77$	−3.00000	−	5.19615i	−0.341882	−	0.592157i
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−3.00000			−0.331295
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		0			0
$85$		0			0
$86$	4.00000	+	6.92820i	0.431331	+	0.747087i
$87$		0			0
$88$	−3.00000	+	5.19615i	−0.319801	+	0.553912i
$89$	−9.00000			−0.953998			−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000			−0.953998			−0.476999	+	0.878904i	$0.658275\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$	4.50000	−	7.79423i	0.469157	−	0.812605i
$93$		0			0
$94$	1.50000	+	2.59808i	0.154713	+	0.267971i
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$	6.00000			0.606092
$99$		0			0
$100$	1.00000			0.100000
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$	1.00000	+	1.73205i	0.0980581	+	0.169842i
$105$		0			0
$106$	−3.00000	+	5.19615i	−0.291386	+	0.504695i
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$	−3.00000	+	5.19615i	−0.286039	+	0.495434i
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$	4.50000	−	7.79423i	0.419627	−	0.726816i
$116$	−3.00000			−0.278543
$117$		0			0
$118$	6.00000			0.552345
$119$		0			0
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$	−6.50000	−	11.2583i	−0.588482	−	1.01928i
$123$		0			0
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	1.00000	+	1.73205i	0.0877058	+	0.151911i
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$	13.0000			1.12303
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0.500000	+	0.866025i	0.0422577	+	0.0731925i
$141$		0			0
$142$	3.00000	−	5.19615i	0.251754	−	0.436051i
$143$	−12.0000			−1.00349
$144$		0			0
$145$	−3.00000			−0.249136
$146$	−2.00000	+	3.46410i	−0.165521	+	0.286691i
$147$		0			0
$148$	−4.00000	−	6.92820i	−0.328798	−	0.569495i
$149$	−1.50000	−	2.59808i	−0.122885	−	0.212843i	0.798019	−	0.602632i	$-0.205881\pi$
$149$	−1.50000	−	2.59808i	−0.122885	−	0.212843i	−0.920904	+	0.389789i	$0.872548\pi$
$150$		0			0
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	0.426948	+	0.904276i	$0.359589\pi$
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	−0.996600	+	0.0823900i	$0.973745\pi$
$152$	4.00000			0.324443
$153$		0			0
$154$	−6.00000			−0.483494
$155$	2.00000	−	3.46410i	0.160644	−	0.278243i
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$	−5.00000	−	8.66025i	−0.397779	−	0.688973i
$159$		0			0
$160$	0.500000	−	0.866025i	0.0395285	−	0.0684653i
$161$	9.00000			0.709299
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	−1.50000	+	2.59808i	−0.117130	+	0.202876i
$165$		0			0
$166$	4.50000	+	7.79423i	0.349268	+	0.604949i
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	0.802135	−	0.597143i	$-0.203697\pi$
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	−0.918208	+	0.396098i	$0.870364\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$	12.0000	−	20.7846i	0.912343	−	1.58022i	0.101598	−	0.994826i	$-0.467605\pi$
$173$	12.0000	−	20.7846i	0.912343	−	1.58022i	0.810745	−	0.585399i	$-0.199062\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.0377964	+	0.0654654i
$176$	3.00000	+	5.19615i	0.226134	+	0.391675i
$177$		0			0
$178$	−4.50000	+	7.79423i	−0.337289	+	0.584202i
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$	−1.00000	+	1.73205i	−0.0741249	+	0.128388i
$183$		0			0
$184$	−4.50000	−	7.79423i	−0.331744	−	0.574598i
$185$	−4.00000	−	6.92820i	−0.294086	−	0.509372i
$186$		0			0
$187$		0			0
$188$	3.00000			0.218797
$189$		0			0
$190$	4.00000			0.290191
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$	1.00000	+	1.73205i	0.0717958	+	0.124354i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	0.500000	−	0.866025i	0.0353553	−	0.0612372i
$201$		0			0
$202$	−3.00000	−	5.19615i	−0.211079	−	0.365600i
$203$	−1.50000	−	2.59808i	−0.105279	−	0.182349i
$204$		0			0
$205$	−1.50000	+	2.59808i	−0.104765	+	0.181458i
$206$	−8.00000			−0.557386
$207$		0			0
$208$	2.00000			0.138675
$209$	−12.0000	+	20.7846i	−0.830057	+	1.43770i
$210$		0			0
$211$	−1.00000	−	1.73205i	−0.0688428	−	0.119239i	0.829549	−	0.558433i	$-0.188597\pi$
$211$	−1.00000	−	1.73205i	−0.0688428	−	0.119239i	−0.898392	+	0.439194i	$0.855264\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	1.50000	−	2.59808i	0.102538	−	0.177601i
$215$	8.00000			0.545595
$216$		0			0
$217$	4.00000			0.271538
$218$	−3.50000	+	6.06218i	−0.237050	+	0.410582i
$219$		0			0
$220$	3.00000	+	5.19615i	0.202260	+	0.350325i
$221$		0			0
$222$		0			0
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	0.882281	+	0.470723i	$0.156007\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$		0			0
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		0			0
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	0.996832	−	0.0795401i	$-0.0253452\pi$
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	−0.567300	+	0.823511i	$0.692012\pi$
$230$	−4.50000	−	7.79423i	−0.296721	−	0.513936i
$231$		0			0
$232$	−1.50000	+	2.59808i	−0.0984798	+	0.170572i
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$		0			0
$235$	3.00000			0.195698
$236$	3.00000	−	5.19615i	0.195283	−	0.338241i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.157667	+	0.987492i	$0.550397\pi$
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.776360	+	0.630290i	$0.782936\pi$
$242$	−25.0000			−1.60706
$243$		0			0
$244$	−13.0000			−0.832240
$245$	3.00000	−	5.19615i	0.191663	−	0.331970i
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$	−2.00000	−	3.46410i	−0.127000	−	0.219971i
$249$		0			0
$250$	0.500000	−	0.866025i	0.0316228	−	0.0547723i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	54.0000			3.39495
$254$	−3.50000	+	6.06218i	−0.219610	+	0.380375i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	4.00000	−	6.92820i	0.248548	−	0.430498i
$260$	2.00000			0.124035
$261$		0			0
$262$	−18.0000			−1.11204
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	3.00000	+	5.19615i	0.184289	+	0.319197i
$266$	2.00000	+	3.46410i	0.122628	+	0.212398i
$267$		0			0
$268$	6.50000	−	11.2583i	0.397051	−	0.687712i
$269$	−21.0000			−1.28039			−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		0			0
$271$	−4.00000			−0.242983			−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000			−0.242983			−0.121491	+	0.992592i	$0.538768\pi$
$272$		0			0
$273$		0			0
$274$	6.00000	+	10.3923i	0.362473	+	0.627822i
$275$	3.00000	+	5.19615i	0.180907	+	0.313340i
$276$		0			0
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	−0.960810	−	0.277207i	$-0.910591\pi$
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	0.720473	+	0.693482i	$0.243925\pi$
$278$	16.0000			0.959616
$279$		0			0
$280$	1.00000			0.0597614
$281$	−7.50000	+	12.9904i	−0.447412	+	0.774941i	−0.998217	−	0.0596933i	$-0.980988\pi$
$281$	−7.50000	+	12.9904i	−0.447412	+	0.774941i	0.550804	+	0.834634i	$0.314321\pi$
$282$		0			0
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	0.991960	−	0.126550i	$-0.0403903\pi$
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	−0.605575	+	0.795788i	$0.707057\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$		0			0
$286$	−6.00000	+	10.3923i	−0.354787	+	0.614510i
$287$	−3.00000			−0.177084
$288$		0			0
$289$	−17.0000			−1.00000
$290$	−1.50000	+	2.59808i	−0.0880830	+	0.152564i
$291$		0			0
$292$	2.00000	+	3.46410i	0.117041	+	0.202721i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	3.00000	−	5.19615i	0.174667	−	0.302532i
$296$	−8.00000			−0.464991
$297$		0			0
$298$	−3.00000			−0.173785
$299$	9.00000	−	15.5885i	0.520483	−	0.901504i
$300$		0			0
$301$	4.00000	+	6.92820i	0.230556	+	0.399335i
$302$	7.00000	+	12.1244i	0.402805	+	0.697678i
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	−13.0000			−0.744378
$306$		0			0
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$	−3.00000	+	5.19615i	−0.170941	+	0.296078i
$309$		0			0
$310$	−2.00000	−	3.46410i	−0.113592	−	0.196748i
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	0.984475	−	0.175525i	$-0.0561621\pi$
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	−0.644246	+	0.764818i	$0.722829\pi$
$312$		0			0
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	−0.892903	−	0.450250i	$-0.851335\pi$
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	0.836379	+	0.548151i	$0.184668\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	−10.0000			−0.562544
$317$	−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
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$	−0.500000	−	0.866025i	−0.0279508	−	0.0484123i
$321$		0			0
$322$	4.50000	−	7.79423i	0.250775	−	0.434355i
$323$		0			0
$324$		0			0
$325$	2.00000			0.110940
$326$	−2.00000	+	3.46410i	−0.110770	+	0.191859i
$327$		0			0
$328$	1.50000	+	2.59808i	0.0828236	+	0.143455i
$329$	1.50000	+	2.59808i	0.0826977	+	0.143237i
$330$		0			0
$331$	5.00000	−	8.66025i	0.274825	−	0.476011i	−0.695266	−	0.718752i	$-0.744713\pi$
$331$	5.00000	−	8.66025i	0.274825	−	0.476011i	0.970091	+	0.242742i	$0.0780468\pi$
$332$	9.00000			0.493939
$333$		0			0
$334$	−3.00000			−0.164153
$335$	6.50000	−	11.2583i	0.355133	−	0.615108i
$336$		0			0
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	0.736270	−	0.676688i	$-0.236585\pi$
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	−0.954164	+	0.299285i	$0.903252\pi$
$338$	−4.50000	−	7.79423i	−0.244768	−	0.423950i
$339$		0			0
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$	13.0000			0.701934
$344$	4.00000	−	6.92820i	0.215666	−	0.373544i
$345$		0			0
$346$	−12.0000	−	20.7846i	−0.645124	−	1.11739i
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	−11.5000	+	19.9186i	−0.615581	+	1.06622i	0.374701	+	0.927146i	$0.377745\pi$
$349$	−11.5000	+	19.9186i	−0.615581	+	1.06622i	−0.990282	+	0.139072i	$0.955588\pi$
$350$	1.00000			0.0534522
$351$		0			0
$352$	6.00000			0.319801
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	−0.347024	−	0.937856i	$-0.612808\pi$
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	0.985719	−	0.168397i	$-0.0538590\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$	4.50000	+	7.79423i	0.238500	+	0.413093i
$357$		0			0
$358$	9.00000	−	15.5885i	0.475665	−	0.823876i
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	2.50000	−	4.33013i	0.131397	−	0.227586i
$363$		0			0
$364$	1.00000	+	1.73205i	0.0524142	+	0.0907841i
$365$	2.00000	+	3.46410i	0.104685	+	0.181319i
$366$		0			0
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	−0.951336	−	0.308155i	$-0.900289\pi$
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	0.742538	+	0.669804i	$0.233622\pi$
$368$	−9.00000			−0.469157
$369$		0			0
$370$	−8.00000			−0.415900
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$		0			0
$375$		0			0
$376$	1.50000	−	2.59808i	0.0773566	−	0.133986i
$377$	−6.00000			−0.309016
$378$		0			0
$379$	−22.0000			−1.13006			−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000			−1.13006			−0.565032	+	0.825069i	$0.691136\pi$
$380$	2.00000	−	3.46410i	0.102598	−	0.177705i
$381$		0			0
$382$	6.00000	+	10.3923i	0.306987	+	0.531717i
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		0			0
$385$	−3.00000	+	5.19615i	−0.152894	+	0.264820i
$386$	−2.00000			−0.101797
$387$		0			0
$388$	2.00000			0.101535
$389$	10.5000	−	18.1865i	0.532371	−	0.922094i	−0.466915	−	0.884302i	$-0.654634\pi$
$389$	10.5000	−	18.1865i	0.532371	−	0.922094i	0.999286	−	0.0377914i	$-0.0120322\pi$
$390$		0			0
$391$		0			0
$392$	−3.00000	−	5.19615i	−0.151523	−	0.262445i
$393$		0			0
$394$	6.00000	−	10.3923i	0.302276	−	0.523557i
$395$	−10.0000			−0.503155
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$	4.00000	−	6.92820i	0.200502	−	0.347279i
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.0250000	−	0.0433013i
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$	−6.00000			−0.298511
$405$		0			0
$406$	−3.00000			−0.148888
$407$	24.0000	−	41.5692i	1.18964	−	2.06051i
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$	1.50000	+	2.59808i	0.0740797	+	0.128310i
$411$		0			0
$412$	−4.00000	+	6.92820i	−0.197066	+	0.341328i
$413$	6.00000			0.295241
$414$		0			0
$415$	9.00000			0.441793
$416$	1.00000	−	1.73205i	0.0490290	−	0.0849208i
$417$		0			0
$418$	12.0000	+	20.7846i	0.586939	+	1.01661i
$419$	−15.0000	−	25.9808i	−0.732798	−	1.26924i	−0.955683	−	0.294398i	$-0.904881\pi$
$419$	−15.0000	−	25.9808i	−0.732798	−	1.26924i	0.222885	−	0.974845i	$-0.428453\pi$
$420$		0			0
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	0.999109	−	0.0422075i	$-0.0134391\pi$
$422$	−2.00000			−0.0973585
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$		0			0
$427$	−6.50000	−	11.2583i	−0.314557	−	0.544829i
$428$	−1.50000	−	2.59808i	−0.0725052	−	0.125583i
$429$		0			0
$430$	4.00000	−	6.92820i	0.192897	−	0.334108i
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$	2.00000	−	3.46410i	0.0960031	−	0.166282i
$435$		0			0
$436$	3.50000	+	6.06218i	0.167620	+	0.290326i
$437$	−18.0000	−	31.1769i	−0.861057	−	1.49139i
$438$		0			0
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$	6.00000			0.286039
$441$		0			0
$442$		0			0
$443$	4.50000	−	7.79423i	0.213801	−	0.370315i	−0.739100	−	0.673596i	$-0.764749\pi$
$443$	4.50000	−	7.79423i	0.213801	−	0.370315i	0.952901	+	0.303281i	$0.0980821\pi$
$444$		0			0
$445$	4.50000	+	7.79423i	0.213320	+	0.369482i
$446$	−0.500000	−	0.866025i	−0.0236757	−	0.0410075i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$		0			0
$454$	6.00000	+	10.3923i	0.281594	+	0.487735i
$455$	1.00000	+	1.73205i	0.0468807	+	0.0811998i
$456$		0			0
$457$	−4.00000	+	6.92820i	−0.187112	+	0.324088i	−0.944286	−	0.329125i	$-0.893246\pi$
$457$	−4.00000	+	6.92820i	−0.187112	+	0.324088i	0.757174	+	0.653213i	$0.226579\pi$
$458$	13.0000			0.607450
$459$		0			0
$460$	−9.00000			−0.419627
$461$	−13.5000	+	23.3827i	−0.628758	+	1.08904i	0.359044	+	0.933321i	$0.383103\pi$
$461$	−13.5000	+	23.3827i	−0.628758	+	1.08904i	−0.987801	+	0.155719i	$0.950230\pi$
$462$		0			0
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	0.908750	−	0.417340i	$-0.137038\pi$
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	−0.815802	+	0.578331i	$0.803704\pi$
$464$	1.50000	+	2.59808i	0.0696358	+	0.120613i
$465$		0			0
$466$	6.00000	−	10.3923i	0.277945	−	0.481414i
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	13.0000			0.600284
$470$	1.50000	−	2.59808i	0.0691898	−	0.119840i
$471$		0			0
$472$	−3.00000	−	5.19615i	−0.138086	−	0.239172i
$473$	24.0000	+	41.5692i	1.10352	+	1.91135i
$474$		0			0
$475$	2.00000	−	3.46410i	0.0917663	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.0000	+	25.9808i	−0.685367	+	1.18709i	0.287954	+	0.957644i	$0.407025\pi$
$479$	−15.0000	+	25.9808i	−0.685367	+	1.18709i	−0.973321	+	0.229447i	$0.926308\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$	14.5000	+	25.1147i	0.660457	+	1.14394i
$483$		0			0
$484$	−12.5000	+	21.6506i	−0.568182	+	0.984120i
$485$	2.00000			0.0908153
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	−6.50000	+	11.2583i	−0.294241	+	0.509641i
$489$		0			0
$490$	−3.00000	−	5.19615i	−0.135526	−	0.234738i
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	0.969061	−	0.246822i	$-0.0793863\pi$
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	−0.698285	+	0.715820i	$0.746053\pi$
$492$		0			0
$493$		0			0
$494$	8.00000			0.359937
$495$		0			0
$496$	−4.00000			−0.179605
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	0.246214	−	0.969216i	$-0.420813\pi$
$500$	−0.500000	−	0.866025i	−0.0223607	−	0.0387298i
$501$		0			0
$502$	−6.00000	+	10.3923i	−0.267793	+	0.463831i
$503$	−27.0000			−1.20387			−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000			−1.20387			−0.601935	+	0.798545i	$0.705603\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$	27.0000	−	46.7654i	1.20030	−	2.07897i
$507$		0			0
$508$	3.50000	+	6.06218i	0.155287	+	0.268966i
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$	−2.00000	+	3.46410i	−0.0884748	+	0.153243i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	18.0000			0.793946
$515$	−4.00000	+	6.92820i	−0.176261	+	0.305293i
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$	−4.00000	−	6.92820i	−0.175750	−	0.304408i
$519$		0			0
$520$	1.00000	−	1.73205i	0.0438529	−	0.0759555i
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	−19.0000			−0.830812			−0.415406	−	0.909636i	$-0.636360\pi$
$523$	−19.0000			−0.830812			−0.415406	+	0.909636i	$0.636360\pi$
$524$	−9.00000	+	15.5885i	−0.393167	+	0.680985i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−29.0000	+	50.2295i	−1.26087	+	2.18389i
$530$	6.00000			0.260623
$531$		0			0
$532$	4.00000			0.173422
$533$	−3.00000	+	5.19615i	−0.129944	+	0.225070i
$534$		0			0
$535$	−1.50000	−	2.59808i	−0.0648507	−	0.112325i
$536$	−6.50000	−	11.2583i	−0.280757	−	0.486286i
$537$		0			0
$538$	−10.5000	+	18.1865i	−0.452687	+	0.784077i
$539$	36.0000			1.55063
$540$		0			0
$541$	29.0000			1.24681			0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000			1.24681			0.623404	+	0.781900i	$0.285749\pi$
$542$	−2.00000	+	3.46410i	−0.0859074	+	0.148796i
$543$		0			0
$544$		0			0
$545$	3.50000	+	6.06218i	0.149924	+	0.259675i
$546$		0			0
$547$	21.5000	−	37.2391i	0.919274	−	1.59223i	0.118753	−	0.992924i	$-0.462110\pi$
$547$	21.5000	−	37.2391i	0.919274	−	1.59223i	0.800521	−	0.599305i	$-0.204556\pi$
$548$	12.0000			0.512615
$549$		0			0
$550$	6.00000			0.255841
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	−5.00000	−	8.66025i	−0.212622	−	0.368271i
$554$	4.00000	+	6.92820i	0.169944	+	0.294351i
$555$		0			0
$556$	8.00000	−	13.8564i	0.339276	−	0.587643i
$557$	6.00000			0.254228			0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000			0.254228			0.127114	+	0.991888i	$0.459429\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	0.500000	−	0.866025i	0.0211289	−	0.0365963i
$561$		0			0
$562$	7.50000	+	12.9904i	0.316368	+	0.547966i
$563$	−1.50000	−	2.59808i	−0.0632175	−	0.109496i	0.832684	−	0.553748i	$-0.186803\pi$
$563$	−1.50000	−	2.59808i	−0.0632175	−	0.109496i	−0.895902	+	0.444252i	$0.853470\pi$
$564$		0			0
$565$		0			0
$566$	13.0000			0.546431
$567$		0			0
$568$	−6.00000			−0.251754
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	0.892413	+	0.451219i	$0.149011\pi$
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	−0.0554391	+	0.998462i	$0.517656\pi$
$572$	6.00000	+	10.3923i	0.250873	+	0.434524i
$573$		0			0
$574$	−1.50000	+	2.59808i	−0.0626088	+	0.108442i
$575$	−9.00000			−0.375326
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$	−8.50000	+	14.7224i	−0.353553	+	0.612372i
$579$		0			0
$580$	1.50000	+	2.59808i	0.0622841	+	0.107879i
$581$	4.50000	+	7.79423i	0.186691	+	0.323359i
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$	4.00000			0.165521
$585$		0			0
$586$		0			0
$587$	−7.50000	+	12.9904i	−0.309558	+	0.536170i	−0.978266	−	0.207355i	$-0.933514\pi$
$587$	−7.50000	+	12.9904i	−0.309558	+	0.536170i	0.668708	+	0.743525i	$0.266848\pi$
$588$		0			0
$589$	−8.00000	−	13.8564i	−0.329634	−	0.570943i
$590$	−3.00000	−	5.19615i	−0.123508	−	0.213922i
$591$		0			0
$592$	−4.00000	+	6.92820i	−0.164399	+	0.284747i
$593$	−24.0000			−0.985562			−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000			−0.985562			−0.492781	+	0.870153i	$0.664020\pi$
$594$		0			0
$595$		0			0
$596$	−1.50000	+	2.59808i	−0.0614424	+	0.106421i
$597$		0			0
$598$	−9.00000	−	15.5885i	−0.368037	−	0.637459i
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	0.798206	−	0.602384i	$-0.205782\pi$
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	−0.920783	+	0.390075i	$0.872449\pi$
$600$		0			0
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	0.469095	+	0.883148i	$0.344580\pi$
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$	8.00000			0.326056
$603$		0			0
$604$	14.0000			0.569652
$605$	−12.5000	+	21.6506i	−0.508197	+	0.880223i
$606$		0			0
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	0.405887	−	0.913923i	$-0.366962\pi$
$608$	−2.00000	−	3.46410i	−0.0811107	−	0.140488i
$609$		0			0
$610$	−6.50000	+	11.2583i	−0.263177	+	0.455836i
$611$	6.00000			0.242734
$612$		0			0
$613$	−40.0000			−1.61558			−0.807792	−	0.589467i	$-0.799338\pi$
$613$	−40.0000			−1.61558			−0.807792	+	0.589467i	$0.799338\pi$
$614$	−3.50000	+	6.06218i	−0.141249	+	0.244650i
$615$		0			0
$616$	3.00000	+	5.19615i	0.120873	+	0.209359i
$617$	6.00000	+	10.3923i	0.241551	+	0.418378i	0.961156	−	0.276005i	$-0.0890106\pi$
$617$	6.00000	+	10.3923i	0.241551	+	0.418378i	−0.719605	+	0.694383i	$0.755677\pi$
$618$		0			0
$619$	20.0000	−	34.6410i	0.803868	−	1.39234i	−0.113185	−	0.993574i	$-0.536105\pi$
$619$	20.0000	−	34.6410i	0.803868	−	1.39234i	0.917053	−	0.398766i	$-0.130561\pi$
$620$	−4.00000			−0.160644
$621$		0			0
$622$	12.0000			0.481156
$623$	−4.50000	+	7.79423i	−0.180289	+	0.312269i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	1.00000	+	1.73205i	0.0399680	+	0.0692267i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	−5.00000	+	8.66025i	−0.198889	+	0.344486i
$633$		0			0
$634$	3.00000	+	5.19615i	0.119145	+	0.206366i
$635$	3.50000	+	6.06218i	0.138893	+	0.240570i
$636$		0			0
$637$	6.00000	−	10.3923i	0.237729	−	0.411758i
$638$	−18.0000			−0.712627
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	−0.330997	−	0.943632i	$-0.607385\pi$
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	0.982708	−	0.185164i	$-0.0592817\pi$
$642$		0			0
$643$	15.5000	+	26.8468i	0.611260	+	1.05873i	0.991028	+	0.133652i	$0.0426705\pi$
$643$	15.5000	+	26.8468i	0.611260	+	1.05873i	−0.379768	+	0.925082i	$0.623996\pi$
$644$	−4.50000	−	7.79423i	−0.177325	−	0.307136i
$645$		0			0
$646$		0			0
$647$	3.00000			0.117942			0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000			0.117942			0.0589711	+	0.998260i	$0.481218\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$	1.00000	−	1.73205i	0.0392232	−	0.0679366i
$651$		0			0
$652$	2.00000	+	3.46410i	0.0783260	+	0.135665i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−9.00000	+	15.5885i	−0.351659	+	0.609091i
$656$	3.00000			0.117130
$657$		0			0
$658$	3.00000			0.116952
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	23.0000	+	39.8372i	0.894596	+	1.54949i	0.834303	+	0.551306i	$0.185870\pi$
$661$	23.0000	+	39.8372i	0.894596	+	1.54949i	0.0602929	+	0.998181i	$0.480797\pi$
$662$	−5.00000	−	8.66025i	−0.194331	−	0.336590i
$663$		0			0
$664$	4.50000	−	7.79423i	0.174634	−	0.302475i
$665$	4.00000			0.155113
$666$		0			0
$667$	27.0000			1.04544
$668$	−1.50000	+	2.59808i	−0.0580367	+	0.100523i
$669$		0			0
$670$	−6.50000	−	11.2583i	−0.251117	−	0.434947i
$671$	−39.0000	−	67.5500i	−1.50558	−	2.60774i
$672$		0			0
$673$	23.0000	−	39.8372i	0.886585	−	1.53561i	0.0426985	−	0.999088i	$-0.486405\pi$
$673$	23.0000	−	39.8372i	0.886585	−	1.53561i	0.843886	−	0.536522i	$-0.180262\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	−9.00000			−0.346154
$677$	9.00000	−	15.5885i	0.345898	−	0.599113i	−0.639618	−	0.768693i	$-0.720908\pi$
$677$	9.00000	−	15.5885i	0.345898	−	0.599113i	0.985517	+	0.169580i	$0.0542410\pi$
$678$		0			0
$679$	1.00000	+	1.73205i	0.0383765	+	0.0664700i
$680$		0			0
$681$		0			0
$682$	12.0000	−	20.7846i	0.459504	−	0.795884i
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	−0.755110	−	0.655598i	$-0.772417\pi$
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	0.945319	+	0.326146i	$0.105750\pi$
$692$	−24.0000			−0.912343
$693$		0			0
$694$	12.0000			0.455514
$695$	8.00000	−	13.8564i	0.303457	−	0.525603i
$696$		0			0
$697$		0			0
$698$	11.5000	+	19.9186i	0.435281	+	0.753930i
$699$		0			0
$700$	0.500000	−	0.866025i	0.0188982	−	0.0327327i
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$	−32.0000			−1.20690
$704$	3.00000	−	5.19615i	0.113067	−	0.195837i
$705$		0			0
$706$	−12.0000	−	20.7846i	−0.451626	−	0.782239i
$707$	−3.00000	−	5.19615i	−0.112827	−	0.195421i
$708$		0			0
$709$	−5.50000	+	9.52628i	−0.206557	+	0.357767i	−0.950628	−	0.310334i	$-0.899559\pi$
$709$	−5.50000	+	9.52628i	−0.206557	+	0.357767i	0.744071	+	0.668101i	$0.232892\pi$
$710$	−6.00000			−0.225176
$711$		0			0
$712$	9.00000			0.337289
$713$	−18.0000	+	31.1769i	−0.674105	+	1.16758i
$714$		0			0
$715$	6.00000	+	10.3923i	0.224387	+	0.388650i
$716$	−9.00000	−	15.5885i	−0.336346	−	0.582568i
$717$		0			0
$718$	6.00000	−	10.3923i	0.223918	−	0.387837i
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$	−1.50000	+	2.59808i	−0.0558242	+	0.0966904i
$723$		0			0
$724$	−2.50000	−	4.33013i	−0.0929118	−	0.160928i
$725$	1.50000	+	2.59808i	0.0557086	+	0.0964901i
$726$		0			0
$727$	−26.5000	+	45.8993i	−0.982831	+	1.70231i	−0.331625	+	0.943411i	$0.607597\pi$
$727$	−26.5000	+	45.8993i	−0.982831	+	1.70231i	−0.651206	+	0.758901i	$0.725737\pi$
$728$	2.00000			0.0741249
$729$		0			0
$730$	4.00000			0.148047
$731$		0			0
$732$		0			0
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	0.707303	−	0.706910i	$-0.249912\pi$
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	−0.965854	+	0.259087i	$0.916578\pi$
$734$	4.00000	+	6.92820i	0.147643	+	0.255725i
$735$		0			0
$736$	−4.50000	+	7.79423i	−0.165872	+	0.287299i
$737$	78.0000			2.87317
$738$		0			0
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$	−4.00000	+	6.92820i	−0.147043	+	0.254686i
$741$		0			0
$742$	3.00000	+	5.19615i	0.110133	+	0.190757i
$743$	−7.50000	−	12.9904i	−0.275148	−	0.476571i	0.695024	−	0.718986i	$-0.255394\pi$
$743$	−7.50000	−	12.9904i	−0.275148	−	0.476571i	−0.970173	+	0.242415i	$0.922060\pi$
$744$		0			0
$745$	−1.50000	+	2.59808i	−0.0549557	+	0.0951861i
$746$	−26.0000			−0.951928
$747$		0			0
$748$		0			0
$749$	1.50000	−	2.59808i	0.0548088	−	0.0949316i
$750$		0			0
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	0.847203	−	0.531269i	$-0.178285\pi$
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	−0.883694	+	0.468065i	$0.844951\pi$
$752$	−1.50000	−	2.59808i	−0.0546994	−	0.0947421i
$753$		0			0
$754$	−3.00000	+	5.19615i	−0.109254	+	0.189233i
$755$	14.0000			0.509512
$756$		0			0
$757$	−46.0000			−1.67190			−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000			−1.67190			−0.835949	+	0.548807i	$0.815082\pi$
$758$	−11.0000	+	19.0526i	−0.399538	+	0.692020i
$759$		0			0
$760$	−2.00000	−	3.46410i	−0.0725476	−	0.125656i
$761$	−16.5000	−	28.5788i	−0.598125	−	1.03598i	−0.993098	−	0.117289i	$-0.962579\pi$
$761$	−16.5000	−	28.5788i	−0.598125	−	1.03598i	0.394973	−	0.918693i	$-0.370754\pi$
$762$		0			0
$763$	−3.50000	+	6.06218i	−0.126709	+	0.219466i
$764$	12.0000			0.434145
$765$		0			0
$766$	−12.0000			−0.433578
$767$	6.00000	−	10.3923i	0.216647	−	0.375244i
$768$		0			0
$769$	−14.5000	−	25.1147i	−0.522883	−	0.905661i	−0.999645	−	0.0266282i	$-0.991523\pi$
$769$	−14.5000	−	25.1147i	−0.522883	−	0.905661i	0.476762	−	0.879032i	$-0.341810\pi$
$770$	3.00000	+	5.19615i	0.108112	+	0.187256i
$771$		0			0
$772$	−1.00000	+	1.73205i	−0.0359908	+	0.0623379i
$773$	48.0000			1.72644			0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000			1.72644			0.863220	+	0.504828i	$0.168444\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$	1.00000	−	1.73205i	0.0358979	−	0.0621770i
$777$		0			0
$778$	−10.5000	−	18.1865i	−0.376443	−	0.652019i
$779$	6.00000	+	10.3923i	0.214972	+	0.372343i
$780$		0			0
$781$	18.0000	−	31.1769i	0.644091	−	1.11560i
$782$		0			0
$783$		0			0
$784$	−6.00000			−0.214286
$785$	−7.00000	+	12.1244i	−0.249841	+	0.432737i
$786$		0			0
$787$	−10.0000	−	17.3205i	−0.356462	−	0.617409i	0.630905	−	0.775860i	$-0.282684\pi$
$787$	−10.0000	−	17.3205i	−0.356462	−	0.617409i	−0.987367	+	0.158450i	$0.949350\pi$
$788$	−6.00000	−	10.3923i	−0.213741	−	0.370211i
$789$		0			0
$790$	−5.00000	+	8.66025i	−0.177892	+	0.308118i
$791$		0			0
$792$		0			0
$793$	−26.0000			−0.923287
$794$	1.00000	−	1.73205i	0.0354887	−	0.0614682i
$795$		0			0
$796$	−4.00000	−	6.92820i	−0.141776	−	0.245564i
$797$	−21.0000	−	36.3731i	−0.743858	−	1.28840i	−0.950726	−	0.310031i	$-0.899660\pi$
$797$	−21.0000	−	36.3731i	−0.743858	−	1.28840i	0.206868	−	0.978369i	$-0.433673\pi$
$798$		0			0
$799$		0			0
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	6.00000			0.211867
$803$	−12.0000	+	20.7846i	−0.423471	+	0.733473i
$804$		0			0
$805$	−4.50000	−	7.79423i	−0.158604	−	0.274710i
$806$	−4.00000	−	6.92820i	−0.140894	−	0.244036i
$807$		0			0
$808$	−3.00000	+	5.19615i	−0.105540	+	0.182800i
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$	−1.50000	+	2.59808i	−0.0526397	+	0.0911746i
$813$		0			0
$814$	−24.0000	−	41.5692i	−0.841200	−	1.45700i
$815$	2.00000	+	3.46410i	0.0700569	+	0.121342i
$816$		0			0
$817$	16.0000	−	27.7128i	0.559769	−	0.969549i
$818$	10.0000			0.349642
$819$		0			0
$820$	3.00000			0.104765
$821$	13.5000	−	23.3827i	0.471153	−	0.816061i	−0.528302	−	0.849056i	$-0.677171\pi$
$821$	13.5000	−	23.3827i	0.471153	−	0.816061i	0.999456	+	0.0329950i	$0.0105045\pi$
$822$		0			0
$823$	12.5000	+	21.6506i	0.435723	+	0.754694i	0.997354	−	0.0726937i	$-0.0231595\pi$
$823$	12.5000	+	21.6506i	0.435723	+	0.754694i	−0.561632	+	0.827387i	$0.689826\pi$
$824$	4.00000	+	6.92820i	0.139347	+	0.241355i
$825$		0			0
$826$	3.00000	−	5.19615i	0.104383	−	0.180797i
$827$	−3.00000			−0.104320			−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000			−0.104320			−0.0521601	+	0.998639i	$0.516611\pi$
$828$		0			0
$829$	−7.00000			−0.243120			−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000			−0.243120			−0.121560	+	0.992584i	$0.538790\pi$
$830$	4.50000	−	7.79423i	0.156197	−	0.270542i
$831$		0			0
$832$	−1.00000	−	1.73205i	−0.0346688	−	0.0600481i
$833$		0			0
$834$		0			0
$835$	−1.50000	+	2.59808i	−0.0519096	+	0.0899101i
$836$	24.0000			0.830057
$837$		0			0
$838$	−30.0000			−1.03633
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	−0.367791	−	0.929909i	$-0.619886\pi$
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	0.989220	−	0.146438i	$-0.0467809\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$	−11.0000	−	19.0526i	−0.379085	−	0.656595i
$843$		0			0
$844$	−1.00000	+	1.73205i	−0.0344214	+	0.0596196i
$845$	−9.00000			−0.309609
$846$		0			0
$847$	−25.0000			−0.859010
$848$	3.00000	−	5.19615i	0.103020	−	0.178437i
$849$		0			0
$850$		0			0
$851$	36.0000	+	62.3538i	1.23406	+	2.13746i
$852$		0			0
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	−0.767642	−	0.640879i	$-0.778570\pi$
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	0.938839	+	0.344358i	$0.111903\pi$
$854$	−13.0000			−0.444851
$855$		0			0
$856$	−3.00000			−0.102538
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	0.244701	+	0.969599i	$0.421310\pi$
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	−0.962048	+	0.272882i	$0.912023\pi$
$858$		0			0
$859$	17.0000	+	29.4449i	0.580033	+	1.00465i	0.995475	+	0.0950262i	$0.0302935\pi$
$859$	17.0000	+	29.4449i	0.580033	+	1.00465i	−0.415442	+	0.909620i	$0.636373\pi$
$860$	−4.00000	−	6.92820i	−0.136399	−	0.236250i
$861$		0			0
$862$	−3.00000	+	5.19615i	−0.102180	+	0.176982i
$863$	−3.00000			−0.102121			−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000			−0.102121			−0.0510606	+	0.998696i	$0.516260\pi$
$864$		0			0
$865$	−24.0000			−0.816024
$866$	−8.00000	+	13.8564i	−0.271851	+	0.470860i
$867$		0			0
$868$	−2.00000	−	3.46410i	−0.0678844	−	0.117579i
$869$	−30.0000	−	51.9615i	−1.01768	−	1.76267i
$870$		0			0
$871$	13.0000	−	22.5167i	0.440488	−	0.762948i
$872$	7.00000			0.237050
$873$		0			0
$874$	−36.0000			−1.21772
$875$	0.500000	−	0.866025i	0.0169031	−	0.0292770i
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$	−14.0000	−	24.2487i	−0.472477	−	0.818354i
$879$		0			0
$880$	3.00000	−	5.19615i	0.101130	−	0.175162i
$881$	−21.0000			−0.707508			−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000			−0.707508			−0.353754	+	0.935339i	$0.615095\pi$
$882$		0			0
$883$	−31.0000			−1.04323			−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000			−1.04323			−0.521617	+	0.853180i	$0.674671\pi$
$884$		0			0
$885$		0			0
$886$	−4.50000	−	7.79423i	−0.151180	−	0.261852i
$887$	18.0000	+	31.1769i	0.604381	+	1.04682i	0.992149	+	0.125061i	$0.0399128\pi$
$887$	18.0000	+	31.1769i	0.604381	+	1.04682i	−0.387768	+	0.921757i	$0.626754\pi$
$888$		0			0
$889$	−3.50000	+	6.06218i	−0.117386	+	0.203319i
$890$	9.00000			0.301681
$891$		0			0
$892$	−1.00000			−0.0334825
$893$	6.00000	−	10.3923i	0.200782	−	0.347765i
$894$		0			0
$895$	−9.00000	−	15.5885i	−0.300837	−	0.521065i
$896$	−0.500000	−	0.866025i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	−3.00000	+	5.19615i	−0.100111	+	0.173398i
$899$	12.0000			0.400222
$900$		0			0
$901$		0			0
$902$	−9.00000	+	15.5885i	−0.299667	+	0.519039i
$903$		0			0
$904$		0			0
$905$	−2.50000	−	4.33013i	−0.0831028	−	0.143938i
$906$		0			0
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	−0.376228	−	0.926527i	$-0.622779\pi$
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	0.990510	−	0.137441i	$-0.0438878\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$	2.00000			0.0662994
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	−0.999994	−	0.00349271i	$-0.998888\pi$
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	0.503022	+	0.864274i	$0.332222\pi$
$912$		0			0
$913$	27.0000	+	46.7654i	0.893570	+	1.54771i
$914$	4.00000	+	6.92820i	0.132308	+	0.229165i
$915$		0			0
$916$	6.50000	−	11.2583i	0.214766	−	0.371986i
$917$	−18.0000			−0.594412
$918$		0			0
$919$	38.0000			1.25350			0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000			1.25350			0.626752	+	0.779219i	$0.284384\pi$
$920$	−4.50000	+	7.79423i	−0.148361	+	0.256968i
$921$		0			0
$922$	13.5000	+	23.3827i	0.444599	+	0.770068i
$923$	−6.00000	−	10.3923i	−0.197492	−	0.342067i
$924$		0			0
$925$	−4.00000	+	6.92820i	−0.131519	+	0.227798i
$926$	4.00000			0.131448
$927$		0			0
$928$	3.00000			0.0984798
$929$	3.00000	−	5.19615i	0.0984268	−	0.170480i	−0.812607	−	0.582812i	$-0.801952\pi$
$929$	3.00000	−	5.19615i	0.0984268	−	0.170480i	0.911034	+	0.412332i	$0.135286\pi$
$930$		0			0
$931$	−12.0000	−	20.7846i	−0.393284	−	0.681188i
$932$	−6.00000	−	10.3923i	−0.196537	−	0.340411i
$933$		0			0
$934$	−18.0000	+	31.1769i	−0.588978	+	1.02014i
$935$		0			0
$936$		0			0
$937$	56.0000			1.82944			0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000			1.82944			0.914720	+	0.404088i	$0.132411\pi$
$938$	6.50000	−	11.2583i	0.212233	−	0.367598i
$939$		0			0
$940$	−1.50000	−	2.59808i	−0.0489246	−	0.0847399i
$941$	10.5000	+	18.1865i	0.342290	+	0.592864i	0.984858	−	0.173365i	$-0.0554641\pi$
$941$	10.5000	+	18.1865i	0.342290	+	0.592864i	−0.642567	+	0.766229i	$0.722131\pi$
$942$		0			0
$943$	13.5000	−	23.3827i	0.439620	−	0.761445i
$944$	−6.00000			−0.195283
$945$		0			0
$946$	48.0000			1.56061
$947$	19.5000	−	33.7750i	0.633665	−	1.09754i	−0.353131	−	0.935574i	$-0.614883\pi$
$947$	19.5000	−	33.7750i	0.633665	−	1.09754i	0.986796	−	0.161966i	$-0.0517835\pi$
$948$		0			0
$949$	4.00000	+	6.92820i	0.129845	+	0.224899i
$950$	−2.00000	−	3.46410i	−0.0648886	−	0.112390i
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	12.0000			0.388311
$956$		0			0
$957$		0			0
$958$	15.0000	+	25.9808i	0.484628	+	0.839400i
$959$	6.00000	+	10.3923i	0.193750	+	0.335585i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$	−16.0000			−0.515861
$963$		0			0
$964$	29.0000			0.934027
$965$	−1.00000	+	1.73205i	−0.0321911	+	0.0557567i
$966$		0			0
$967$	18.5000	+	32.0429i	0.594920	+	1.03043i	0.993558	+	0.113323i	$0.0361496\pi$
$967$	18.5000	+	32.0429i	0.594920	+	1.03043i	−0.398638	+	0.917108i	$0.630517\pi$
$968$	12.5000	+	21.6506i	0.401765	+	0.695878i
$969$		0			0
$970$	1.00000	−	1.73205i	0.0321081	−	0.0556128i
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$	4.00000	−	6.92820i	0.128168	−	0.221994i
$975$		0			0
$976$	6.50000	+	11.2583i	0.208060	+	0.360370i
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	−0.977379	−	0.211495i	$-0.932167\pi$
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	0.305530	−	0.952183i	$-0.401167\pi$
$978$		0			0
$979$	−27.0000	+	46.7654i	−0.862924	+	1.49463i
$980$	−6.00000			−0.191663
$981$		0			0
$982$	12.0000			0.382935
$983$	4.50000	−	7.79423i	0.143528	−	0.248597i	−0.785295	−	0.619122i	$-0.787489\pi$
$983$	4.50000	−	7.79423i	0.143528	−	0.248597i	0.928823	+	0.370525i	$0.120822\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$		0			0
$987$		0			0
$988$	4.00000	−	6.92820i	0.127257	−	0.220416i
$989$	−72.0000			−2.28947
$990$		0			0
$991$	−10.0000			−0.317660			−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000			−0.317660			−0.158830	+	0.987306i	$0.550772\pi$
$992$	−2.00000	+	3.46410i	−0.0635001	+	0.109985i
$993$		0			0
$994$	−3.00000	−	5.19615i	−0.0951542	−	0.164812i
$995$	−4.00000	−	6.92820i	−0.126809	−	0.219639i
$996$		0			0
$997$	5.00000	−	8.66025i	0.158352	−	0.274273i	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	5.00000	−	8.66025i	0.158352	−	0.274273i	0.934274	+	0.356555i	$0.116049\pi$
$998$	−32.0000			−1.01294
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.e.b.181.1		2
3.2	odd	2		90.2.e.a.61.1	yes	2
4.3	odd	2		2160.2.q.b.721.1		2
5.2	odd	4		1350.2.j.e.1099.2		4
5.3	odd	4		1350.2.j.e.1099.1		4
5.4	even	2		1350.2.e.b.451.1		2
9.2	odd	6		810.2.a.g.1.1		1
9.4	even	3	inner	270.2.e.b.91.1		2
9.5	odd	6		90.2.e.a.31.1	✓	2
9.7	even	3		810.2.a.b.1.1		1
12.11	even	2		720.2.q.b.241.1		2
15.2	even	4		450.2.j.c.349.1		4
15.8	even	4		450.2.j.c.349.2		4
15.14	odd	2		450.2.e.e.151.1		2
36.7	odd	6		6480.2.a.v.1.1		1
36.11	even	6		6480.2.a.g.1.1		1
36.23	even	6		720.2.q.b.481.1		2
36.31	odd	6		2160.2.q.b.1441.1		2
45.2	even	12		4050.2.c.t.649.2		2
45.4	even	6		1350.2.e.b.901.1		2
45.7	odd	12		4050.2.c.a.649.1		2
45.13	odd	12		1350.2.j.e.199.2		4
45.14	odd	6		450.2.e.e.301.1		2
45.22	odd	12		1350.2.j.e.199.1		4
45.23	even	12		450.2.j.c.49.1		4
45.29	odd	6		4050.2.a.n.1.1		1
45.32	even	12		450.2.j.c.49.2		4
45.34	even	6		4050.2.a.ba.1.1		1
45.38	even	12		4050.2.c.t.649.1		2
45.43	odd	12		4050.2.c.a.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.a.31.1	✓	2	9.5	odd	6
90.2.e.a.61.1	yes	2	3.2	odd	2
270.2.e.b.91.1		2	9.4	even	3	inner
270.2.e.b.181.1		2	1.1	even	1	trivial
450.2.e.e.151.1		2	15.14	odd	2
450.2.e.e.301.1		2	45.14	odd	6
450.2.j.c.49.1		4	45.23	even	12
450.2.j.c.49.2		4	45.32	even	12
450.2.j.c.349.1		4	15.2	even	4
450.2.j.c.349.2		4	15.8	even	4
720.2.q.b.241.1		2	12.11	even	2
720.2.q.b.481.1		2	36.23	even	6
810.2.a.b.1.1		1	9.7	even	3
810.2.a.g.1.1		1	9.2	odd	6
1350.2.e.b.451.1		2	5.4	even	2
1350.2.e.b.901.1		2	45.4	even	6
1350.2.j.e.199.1		4	45.22	odd	12
1350.2.j.e.199.2		4	45.13	odd	12
1350.2.j.e.1099.1		4	5.3	odd	4
1350.2.j.e.1099.2		4	5.2	odd	4
2160.2.q.b.721.1		2	4.3	odd	2
2160.2.q.b.1441.1		2	36.31	odd	6
4050.2.a.n.1.1		1	45.29	odd	6
4050.2.a.ba.1.1		1	45.34	even	6
4050.2.c.a.649.1		2	45.7	odd	12
4050.2.c.a.649.2		2	45.43	odd	12
4050.2.c.t.649.1		2	45.38	even	12
4050.2.c.t.649.2		2	45.2	even	12
6480.2.a.g.1.1		1	36.11	even	6
6480.2.a.v.1.1		1	36.7	odd	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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} -1.00000 q^{10} +(3.00000 - 5.19615i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -4.00000 q^{19} +(-0.500000 + 0.866025i) q^{20} +(-3.00000 - 5.19615i) q^{22} +(4.50000 + 7.79423i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.00000 q^{26} -1.00000 q^{28} +(1.50000 - 2.59808i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} -1.00000 q^{35} +8.00000 q^{37} +(-2.00000 + 3.46410i) q^{38} +(0.500000 + 0.866025i) q^{40} +(-1.50000 - 2.59808i) q^{41} +(-4.00000 + 6.92820i) q^{43} -6.00000 q^{44} +9.00000 q^{46} +(-1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} -6.00000 q^{53} -6.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-1.50000 - 2.59808i) q^{58} +(3.00000 + 5.19615i) q^{59} +(6.50000 - 11.2583i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-1.00000 + 1.73205i) q^{65} +(6.50000 + 11.2583i) q^{67} +(-0.500000 + 0.866025i) q^{70} +6.00000 q^{71} -4.00000 q^{73} +(4.00000 - 6.92820i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(5.00000 - 8.66025i) q^{79} +1.00000 q^{80} -3.00000 q^{82} +(-4.50000 + 7.79423i) q^{83} +(4.00000 + 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{88} -9.00000 q^{89} -2.00000 q^{91} +(4.50000 - 7.79423i) q^{92} +(1.50000 + 2.59808i) q^{94} +(2.00000 + 3.46410i) q^{95} +(-1.00000 + 1.73205i) q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - q^5 + q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} - q^{14} - q^{16} - 8 q^{19} - q^{20} - 6 q^{22} + 9 q^{23} - q^{25} - 4 q^{26} - 2 q^{28} + 3 q^{29} + 4 q^{31} + q^{32} - 2 q^{35} + 16 q^{37} - 4 q^{38} + q^{40} - 3 q^{41} - 8 q^{43} - 12 q^{44} + 18 q^{46} - 3 q^{47} + 6 q^{49} + q^{50} - 2 q^{52} - 12 q^{53} - 12 q^{55} - q^{56} - 3 q^{58} + 6 q^{59} + 13 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 13 q^{67} - q^{70} + 12 q^{71} - 8 q^{73} + 8 q^{74} + 4 q^{76} - 6 q^{77} + 10 q^{79} + 2 q^{80} - 6 q^{82} - 9 q^{83} + 8 q^{86} - 6 q^{88} - 18 q^{89} - 4 q^{91} + 9 q^{92} + 3 q^{94} + 4 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^5 + q^7 - 2 * q^8 - 2 * q^10 + 6 * q^11 - 2 * q^13 - q^14 - q^16 - 8 * q^19 - q^20 - 6 * q^22 + 9 * q^23 - q^25 - 4 * q^26 - 2 * q^28 + 3 * q^29 + 4 * q^31 + q^32 - 2 * q^35 + 16 * q^37 - 4 * q^38 + q^40 - 3 * q^41 - 8 * q^43 - 12 * q^44 + 18 * q^46 - 3 * q^47 + 6 * q^49 + q^50 - 2 * q^52 - 12 * q^53 - 12 * q^55 - q^56 - 3 * q^58 + 6 * q^59 + 13 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 + 13 * q^67 - q^70 + 12 * q^71 - 8 * q^73 + 8 * q^74 + 4 * q^76 - 6 * q^77 + 10 * q^79 + 2 * q^80 - 6 * q^82 - 9 * q^83 + 8 * q^86 - 6 * q^88 - 18 * q^89 - 4 * q^91 + 9 * q^92 + 3 * q^94 + 4 * q^95 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more