LMFDB - Embedded newform 1350.2.e.l.901.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(451,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, 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("1350.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			901.1
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	1350.901
Dual form			1350.2.e.l.451.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.18614 - 2.05446i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.18614 - 2.05446i) q^{7} -1.00000 q^{8} +(0.686141 + 1.18843i) q^{11} +(2.37228 - 4.10891i) q^{13} +(1.18614 - 2.05446i) q^{14} +(-0.500000 - 0.866025i) q^{16} -7.37228 q^{17} +3.37228 q^{19} +(-0.686141 + 1.18843i) q^{22} +(2.18614 - 3.78651i) q^{23} +4.74456 q^{26} +2.37228 q^{28} +(-2.18614 - 3.78651i) q^{29} +(3.37228 - 5.84096i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-3.68614 - 6.38458i) q^{34} +4.00000 q^{37} +(1.68614 + 2.92048i) q^{38} +(-1.50000 + 2.59808i) q^{41} +(-5.68614 - 9.84868i) q^{43} -1.37228 q^{44} +4.37228 q^{46} +(0.813859 + 1.40965i) q^{47} +(0.686141 - 1.18843i) q^{49} +(2.37228 + 4.10891i) q^{52} +11.4891 q^{53} +(1.18614 + 2.05446i) q^{56} +(2.18614 - 3.78651i) q^{58} +(-0.686141 + 1.18843i) q^{59} +(-4.55842 - 7.89542i) q^{61} +6.74456 q^{62} +1.00000 q^{64} +(-3.50000 + 6.06218i) q^{67} +(3.68614 - 6.38458i) q^{68} +6.00000 q^{71} +14.1168 q^{73} +(2.00000 + 3.46410i) q^{74} +(-1.68614 + 2.92048i) q^{76} +(1.62772 - 2.81929i) q^{77} +(-1.00000 - 1.73205i) q^{79} -3.00000 q^{82} +(-0.813859 - 1.40965i) q^{83} +(5.68614 - 9.84868i) q^{86} +(-0.686141 - 1.18843i) q^{88} +1.11684 q^{89} -11.2554 q^{91} +(2.18614 + 3.78651i) q^{92} +(-0.813859 + 1.40965i) q^{94} +(-1.31386 - 2.27567i) q^{97} +1.37228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8	$ 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} - 2 q^{16} - 18 q^{17} + 2 q^{19} + 3 q^{22} + 3 q^{23} - 4 q^{26} - 2 q^{28} - 3 q^{29} + 2 q^{31} + 2 q^{32} - 9 q^{34} + 16 q^{37} + q^{38} - 6 q^{41} - 17 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{49} - 2 q^{52} - q^{56} + 3 q^{58} + 3 q^{59} - q^{61} + 4 q^{62} + 4 q^{64} - 14 q^{67} + 9 q^{68} + 24 q^{71} + 22 q^{73} + 8 q^{74} - q^{76} + 18 q^{77} - 4 q^{79} - 12 q^{82} - 9 q^{83} + 17 q^{86} + 3 q^{88} - 30 q^{89} - 68 q^{91} + 3 q^{92} - 9 q^{94} - 11 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8 - 3 * q^11 - 2 * q^13 - q^14 - 2 * q^16 - 18 * q^17 + 2 * q^19 + 3 * q^22 + 3 * q^23 - 4 * q^26 - 2 * q^28 - 3 * q^29 + 2 * q^31 + 2 * q^32 - 9 * q^34 + 16 * q^37 + q^38 - 6 * q^41 - 17 * q^43 + 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^49 - 2 * q^52 - q^56 + 3 * q^58 + 3 * q^59 - q^61 + 4 * q^62 + 4 * q^64 - 14 * q^67 + 9 * q^68 + 24 * q^71 + 22 * q^73 + 8 * q^74 - q^76 + 18 * q^77 - 4 * q^79 - 12 * q^82 - 9 * q^83 + 17 * q^86 + 3 * q^88 - 30 * q^89 - 68 * q^91 + 3 * q^92 - 9 * q^94 - 11 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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			0
$5$		0			0
$6$		0			0
$7$	−1.18614	−	2.05446i	−0.448319	−	0.776511i	0.549958	−	0.835192i	$-0.314644\pi$
$7$	−1.18614	−	2.05446i	−0.448319	−	0.776511i	−0.998277	+	0.0586811i	$0.981310\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	0.686141	+	1.18843i	0.206879	+	0.358325i	0.950730	−	0.310021i	$-0.100336\pi$
$11$	0.686141	+	1.18843i	0.206879	+	0.358325i	−0.743851	+	0.668346i	$0.767003\pi$
$12$		0			0
$13$	2.37228	−	4.10891i	0.657952	−	1.13961i	−0.323192	−	0.946333i	$-0.604756\pi$
$13$	2.37228	−	4.10891i	0.657952	−	1.13961i	0.981145	−	0.193274i	$-0.0619106\pi$
$14$	1.18614	−	2.05446i	0.317009	−	0.549076i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−7.37228			−1.78804			−0.894020	−	0.448026i	$-0.852127\pi$
$17$	−7.37228			−1.78804			−0.894020	+	0.448026i	$0.852127\pi$
$18$		0			0
$19$	3.37228			0.773654			0.386827	−	0.922152i	$-0.373571\pi$
$19$	3.37228			0.773654			0.386827	+	0.922152i	$0.373571\pi$
$20$		0			0
$21$		0			0
$22$	−0.686141	+	1.18843i	−0.146286	+	0.253374i
$23$	2.18614	−	3.78651i	0.455842	−	0.789541i	−0.542894	−	0.839801i	$-0.682672\pi$
$23$	2.18614	−	3.78651i	0.455842	−	0.789541i	0.998736	+	0.0502598i	$0.0160049\pi$
$24$		0			0
$25$		0			0
$26$	4.74456			0.930485
$27$		0			0
$28$	2.37228			0.448319
$29$	−2.18614	−	3.78651i	−0.405956	−	0.703137i	0.588476	−	0.808515i	$-0.299728\pi$
$29$	−2.18614	−	3.78651i	−0.405956	−	0.703137i	−0.994432	+	0.105378i	$0.966395\pi$
$30$		0			0
$31$	3.37228	−	5.84096i	0.605680	−	1.04907i	−0.386264	−	0.922388i	$-0.626235\pi$
$31$	3.37228	−	5.84096i	0.605680	−	1.04907i	0.991944	−	0.126680i	$-0.0404320\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−3.68614	−	6.38458i	−0.632168	−	1.09495i
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$	1.68614	+	2.92048i	0.273528	+	0.473765i
$39$		0			0
$40$		0			0
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	$-0.908600\pi$
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	0.724797	+	0.688963i	$0.241934\pi$
$42$		0			0
$43$	−5.68614	−	9.84868i	−0.867128	−	1.50191i	−0.864918	−	0.501913i	$-0.832630\pi$
$43$	−5.68614	−	9.84868i	−0.867128	−	1.50191i	−0.00221007	−	0.999998i	$-0.500703\pi$
$44$	−1.37228			−0.206879
$45$		0			0
$46$	4.37228			0.644658
$47$	0.813859	+	1.40965i	0.118714	+	0.205618i	0.919258	−	0.393655i	$-0.128790\pi$
$47$	0.813859	+	1.40965i	0.118714	+	0.205618i	−0.800545	+	0.599273i	$0.795456\pi$
$48$		0			0
$49$	0.686141	−	1.18843i	0.0980201	−	0.169776i
$50$		0			0
$51$		0			0
$52$	2.37228	+	4.10891i	0.328976	+	0.569804i
$53$	11.4891			1.57815			0.789076	−	0.614295i	$-0.210560\pi$
$53$	11.4891			1.57815			0.789076	+	0.614295i	$0.210560\pi$
$54$		0			0
$55$		0			0
$56$	1.18614	+	2.05446i	0.158505	+	0.274538i
$57$		0			0
$58$	2.18614	−	3.78651i	0.287054	−	0.497193i
$59$	−0.686141	+	1.18843i	−0.0893279	+	0.154720i	−0.907227	−	0.420641i	$-0.861805\pi$
$59$	−0.686141	+	1.18843i	−0.0893279	+	0.154720i	0.817899	+	0.575361i	$0.195139\pi$
$60$		0			0
$61$	−4.55842	−	7.89542i	−0.583646	−	1.01090i	−0.995043	−	0.0994483i	$-0.968292\pi$
$61$	−4.55842	−	7.89542i	−0.583646	−	1.01090i	0.411397	−	0.911456i	$-0.365041\pi$
$62$	6.74456			0.856560
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	−0.996659	−	0.0816792i	$-0.973972\pi$
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	0.569066	+	0.822292i	$0.307305\pi$
$68$	3.68614	−	6.38458i	0.447010	−	0.774244i
$69$		0			0
$70$		0			0
$71$	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
$73$	14.1168			1.65225			0.826126	−	0.563486i	$-0.190540\pi$
$73$	14.1168			1.65225			0.826126	+	0.563486i	$0.190540\pi$
$74$	2.00000	+	3.46410i	0.232495	+	0.402694i
$75$		0			0
$76$	−1.68614	+	2.92048i	−0.193414	+	0.335002i
$77$	1.62772	−	2.81929i	0.185496	−	0.321288i
$78$		0			0
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	0.804272	−	0.594261i	$-0.202555\pi$
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	−0.916781	+	0.399390i	$0.869222\pi$
$80$		0			0
$81$		0			0
$82$	−3.00000			−0.331295
$83$	−0.813859	−	1.40965i	−0.0893327	−	0.154729i	0.817897	−	0.575365i	$-0.195140\pi$
$83$	−0.813859	−	1.40965i	−0.0893327	−	0.154729i	−0.907229	+	0.420637i	$0.861807\pi$
$84$		0			0
$85$		0			0
$86$	5.68614	−	9.84868i	0.613152	−	1.06201i
$87$		0			0
$88$	−0.686141	−	1.18843i	−0.0731428	−	0.126687i
$89$	1.11684			0.118385			0.0591926	−	0.998247i	$-0.481147\pi$
$89$	1.11684			0.118385			0.0591926	+	0.998247i	$0.481147\pi$
$90$		0			0
$91$	−11.2554			−1.17989
$92$	2.18614	+	3.78651i	0.227921	+	0.394771i
$93$		0			0
$94$	−0.813859	+	1.40965i	−0.0839432	+	0.145394i
$95$		0			0
$96$		0			0
$97$	−1.31386	−	2.27567i	−0.133402	−	0.231059i	0.791584	−	0.611061i	$-0.209257\pi$
$97$	−1.31386	−	2.27567i	−0.133402	−	0.231059i	−0.924986	+	0.380001i	$0.875924\pi$
$98$	1.37228			0.138621
$99$		0			0
$100$		0			0
$101$	−4.37228	−	7.57301i	−0.435058	−	0.753543i	0.562242	−	0.826973i	$-0.309939\pi$
$101$	−4.37228	−	7.57301i	−0.435058	−	0.753543i	−0.997300	+	0.0734297i	$0.976606\pi$
$102$		0			0
$103$	−8.00000	+	13.8564i	−0.788263	+	1.36531i	0.138767	+	0.990325i	$0.455686\pi$
$103$	−8.00000	+	13.8564i	−0.788263	+	1.36531i	−0.927030	+	0.374987i	$0.877647\pi$
$104$	−2.37228	+	4.10891i	−0.232621	+	0.402912i
$105$		0			0
$106$	5.74456	+	9.94987i	0.557961	+	0.966417i
$107$	−14.4891			−1.40072			−0.700358	−	0.713791i	$-0.746976\pi$
$107$	−14.4891			−1.40072			−0.700358	+	0.713791i	$0.746976\pi$
$108$		0			0
$109$	9.62772			0.922168			0.461084	−	0.887356i	$-0.347461\pi$
$109$	9.62772			0.922168			0.461084	+	0.887356i	$0.347461\pi$
$110$		0			0
$111$		0			0
$112$	−1.18614	+	2.05446i	−0.112080	+	0.194128i
$113$	−7.37228	+	12.7692i	−0.693526	+	1.20122i	0.277149	+	0.960827i	$0.410610\pi$
$113$	−7.37228	+	12.7692i	−0.693526	+	1.20122i	−0.970675	+	0.240395i	$0.922723\pi$
$114$		0			0
$115$		0			0
$116$	4.37228			0.405956
$117$		0			0
$118$	−1.37228			−0.126329
$119$	8.74456	+	15.1460i	0.801613	+	1.38843i
$120$		0			0
$121$	4.55842	−	7.89542i	0.414402	−	0.717765i
$122$	4.55842	−	7.89542i	0.412700	−	0.714818i
$123$		0			0
$124$	3.37228	+	5.84096i	0.302840	+	0.524534i
$125$		0			0
$126$		0			0
$127$	−9.11684			−0.808989			−0.404495	−	0.914540i	$-0.632553\pi$
$127$	−9.11684			−0.808989			−0.404495	+	0.914540i	$0.632553\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	4.37228	−	7.57301i	0.382008	−	0.661657i	−0.609341	−	0.792908i	$-0.708566\pi$
$131$	4.37228	−	7.57301i	0.382008	−	0.661657i	0.991349	+	0.131251i	$0.0418993\pi$
$132$		0			0
$133$	−4.00000	−	6.92820i	−0.346844	−	0.600751i
$134$	−7.00000			−0.604708
$135$		0			0
$136$	7.37228			0.632168
$137$	0.941578	+	1.63086i	0.0804444	+	0.139334i	0.903441	−	0.428713i	$-0.141033\pi$
$137$	0.941578	+	1.63086i	0.0804444	+	0.139334i	−0.822996	+	0.568046i	$0.807699\pi$
$138$		0			0
$139$	−9.05842	+	15.6896i	−0.768325	+	1.33078i	0.170145	+	0.985419i	$0.445576\pi$
$139$	−9.05842	+	15.6896i	−0.768325	+	1.33078i	−0.938470	+	0.345359i	$0.887757\pi$
$140$		0			0
$141$		0			0
$142$	3.00000	+	5.19615i	0.251754	+	0.436051i
$143$	6.51087			0.544467
$144$		0			0
$145$		0			0
$146$	7.05842	+	12.2255i	0.584159	+	1.01179i
$147$		0			0
$148$	−2.00000	+	3.46410i	−0.164399	+	0.284747i
$149$	9.55842	−	16.5557i	0.783056	−	1.35629i	−0.147097	−	0.989122i	$-0.546993\pi$
$149$	9.55842	−	16.5557i	0.783056	−	1.35629i	0.930153	−	0.367171i	$-0.119674\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$	−3.37228			−0.273528
$153$		0			0
$154$	3.25544			0.262331
$155$		0			0
$156$		0			0
$157$	2.37228	−	4.10891i	0.189329	−	0.327927i	−0.755698	−	0.654920i	$-0.772702\pi$
$157$	2.37228	−	4.10891i	0.189329	−	0.327927i	0.945027	+	0.326993i	$0.106036\pi$
$158$	1.00000	−	1.73205i	0.0795557	−	0.137795i
$159$		0			0
$160$		0			0
$161$	−10.3723			−0.817450
$162$		0			0
$163$	−1.48913			−0.116637			−0.0583186	−	0.998298i	$-0.518574\pi$
$163$	−1.48913			−0.116637			−0.0583186	+	0.998298i	$0.518574\pi$
$164$	−1.50000	−	2.59808i	−0.117130	−	0.202876i
$165$		0			0
$166$	0.813859	−	1.40965i	0.0631677	−	0.109410i
$167$	−3.81386	+	6.60580i	−0.295125	+	0.511172i	−0.975014	−	0.222143i	$-0.928695\pi$
$167$	−3.81386	+	6.60580i	−0.295125	+	0.511172i	0.679889	+	0.733315i	$0.262028\pi$
$168$		0			0
$169$	−4.75544	−	8.23666i	−0.365803	−	0.633589i
$170$		0			0
$171$		0			0
$172$	11.3723			0.867128
$173$	−4.62772	−	8.01544i	−0.351839	−	0.609403i	0.634733	−	0.772732i	$-0.281110\pi$
$173$	−4.62772	−	8.01544i	−0.351839	−	0.609403i	−0.986572	+	0.163329i	$0.947777\pi$
$174$		0			0
$175$		0			0
$176$	0.686141	−	1.18843i	0.0517198	−	0.0895813i
$177$		0			0
$178$	0.558422	+	0.967215i	0.0418555	+	0.0724958i
$179$	3.25544			0.243323			0.121661	−	0.992572i	$-0.461178\pi$
$179$	3.25544			0.243323			0.121661	+	0.992572i	$0.461178\pi$
$180$		0			0
$181$	−7.86141			−0.584334			−0.292167	−	0.956367i	$-0.594376\pi$
$181$	−7.86141			−0.584334			−0.292167	+	0.956367i	$0.594376\pi$
$182$	−5.62772	−	9.74749i	−0.417154	−	0.722532i
$183$		0			0
$184$	−2.18614	+	3.78651i	−0.161164	+	0.279145i
$185$		0			0
$186$		0			0
$187$	−5.05842	−	8.76144i	−0.369908	−	0.640700i
$188$	−1.62772			−0.118714
$189$		0			0
$190$		0			0
$191$	−2.74456	−	4.75372i	−0.198590	−	0.343967i	0.749482	−	0.662025i	$-0.230303\pi$
$191$	−2.74456	−	4.75372i	−0.198590	−	0.343967i	−0.948071	+	0.318058i	$0.896969\pi$
$192$		0			0
$193$	1.94158	−	3.36291i	0.139758	−	0.242068i	−0.787647	−	0.616127i	$-0.788701\pi$
$193$	1.94158	−	3.36291i	0.139758	−	0.242068i	0.927405	+	0.374059i	$0.122034\pi$
$194$	1.31386	−	2.27567i	0.0943296	−	0.163384i
$195$		0			0
$196$	0.686141	+	1.18843i	0.0490100	+	0.0848879i
$197$	17.4891			1.24605			0.623024	−	0.782202i	$-0.285904\pi$
$197$	17.4891			1.24605			0.623024	+	0.782202i	$0.285904\pi$
$198$		0			0
$199$	−9.48913			−0.672666			−0.336333	−	0.941743i	$-0.609187\pi$
$199$	−9.48913			−0.672666			−0.336333	+	0.941743i	$0.609187\pi$
$200$		0			0
$201$		0			0
$202$	4.37228	−	7.57301i	0.307633	−	0.532835i
$203$	−5.18614	+	8.98266i	−0.363996	+	0.630459i
$204$		0			0
$205$		0			0
$206$	−16.0000			−1.11477
$207$		0			0
$208$	−4.74456			−0.328976
$209$	2.31386	+	4.00772i	0.160053	+	0.277220i
$210$		0			0
$211$	3.62772	−	6.28339i	0.249742	−	0.432567i	−0.713712	−	0.700439i	$-0.752987\pi$
$211$	3.62772	−	6.28339i	0.249742	−	0.432567i	0.963454	+	0.267873i	$0.0863207\pi$
$212$	−5.74456	+	9.94987i	−0.394538	+	0.683360i
$213$		0			0
$214$	−7.24456	−	12.5480i	−0.495228	−	0.857760i
$215$		0			0
$216$		0			0
$217$	−16.0000			−1.08615
$218$	4.81386	+	8.33785i	0.326036	+	0.564710i
$219$		0			0
$220$		0			0
$221$	−17.4891	+	30.2921i	−1.17645	+	2.03766i
$222$		0			0
$223$	6.18614	+	10.7147i	0.414255	+	0.717510i	0.995350	−	0.0963255i	$-0.0307090\pi$
$223$	6.18614	+	10.7147i	0.414255	+	0.717510i	−0.581095	+	0.813836i	$0.697376\pi$
$224$	−2.37228			−0.158505
$225$		0			0
$226$	−14.7446			−0.980794
$227$	−0.941578	−	1.63086i	−0.0624947	−	0.108244i	0.833085	−	0.553145i	$-0.186572\pi$
$227$	−0.941578	−	1.63086i	−0.0624947	−	0.108244i	−0.895580	+	0.444901i	$0.853239\pi$
$228$		0			0
$229$	−9.18614	+	15.9109i	−0.607037	+	1.05142i	0.384689	+	0.923046i	$0.374309\pi$
$229$	−9.18614	+	15.9109i	−0.607037	+	1.05142i	−0.991726	+	0.128373i	$0.959025\pi$
$230$		0			0
$231$		0			0
$232$	2.18614	+	3.78651i	0.143527	+	0.248596i
$233$	−10.1168			−0.662776			−0.331388	−	0.943494i	$-0.607517\pi$
$233$	−10.1168			−0.662776			−0.331388	+	0.943494i	$0.607517\pi$
$234$		0			0
$235$		0			0
$236$	−0.686141	−	1.18843i	−0.0446640	−	0.0773602i
$237$		0			0
$238$	−8.74456	+	15.1460i	−0.566826	+	0.981771i
$239$	−7.37228	+	12.7692i	−0.476873	+	0.825969i	−0.999649	−	0.0265017i	$-0.991563\pi$
$239$	−7.37228	+	12.7692i	−0.476873	+	0.825969i	0.522776	+	0.852470i	$0.324897\pi$
$240$		0			0
$241$	−5.24456	−	9.08385i	−0.337832	−	0.585142i	0.646193	−	0.763174i	$-0.276360\pi$
$241$	−5.24456	−	9.08385i	−0.337832	−	0.585142i	−0.984025	+	0.178032i	$0.943027\pi$
$242$	9.11684			0.586053
$243$		0			0
$244$	9.11684			0.583646
$245$		0			0
$246$		0			0
$247$	8.00000	−	13.8564i	0.509028	−	0.881662i
$248$	−3.37228	+	5.84096i	−0.214140	+	0.370901i
$249$		0			0
$250$		0			0
$251$	−15.6060			−0.985040			−0.492520	−	0.870301i	$-0.663924\pi$
$251$	−15.6060			−0.985040			−0.492520	+	0.870301i	$0.663924\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$	−4.55842	−	7.89542i	−0.286021	−	0.495403i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	0.686141	−	1.18843i	0.0428003	−	0.0741323i	−0.843832	−	0.536608i	$-0.819705\pi$
$257$	0.686141	−	1.18843i	0.0428003	−	0.0741323i	0.886632	+	0.462476i	$0.153039\pi$
$258$		0			0
$259$	−4.74456	−	8.21782i	−0.294813	−	0.510631i
$260$		0			0
$261$		0			0
$262$	8.74456			0.540241
$263$	−2.74456	−	4.75372i	−0.169237	−	0.293127i	0.768915	−	0.639351i	$-0.220797\pi$
$263$	−2.74456	−	4.75372i	−0.169237	−	0.293127i	−0.938152	+	0.346224i	$0.887464\pi$
$264$		0			0
$265$		0			0
$266$	4.00000	−	6.92820i	0.245256	−	0.424795i
$267$		0			0
$268$	−3.50000	−	6.06218i	−0.213797	−	0.370306i
$269$	−4.37228			−0.266583			−0.133291	−	0.991077i	$-0.542555\pi$
$269$	−4.37228			−0.266583			−0.133291	+	0.991077i	$0.542555\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$	3.68614	+	6.38458i	0.223505	+	0.387122i
$273$		0			0
$274$	−0.941578	+	1.63086i	−0.0568828	+	0.0985239i
$275$		0			0
$276$		0			0
$277$	2.62772	+	4.55134i	0.157884	+	0.273464i	0.934106	−	0.356997i	$-0.116199\pi$
$277$	2.62772	+	4.55134i	0.157884	+	0.273464i	−0.776221	+	0.630461i	$0.782866\pi$
$278$	−18.1168			−1.08658
$279$		0			0
$280$		0			0
$281$	2.18614	+	3.78651i	0.130414	+	0.225884i	0.923836	−	0.382788i	$-0.125036\pi$
$281$	2.18614	+	3.78651i	0.130414	+	0.225884i	−0.793422	+	0.608672i	$0.791703\pi$
$282$		0			0
$283$	−15.9307	+	27.5928i	−0.946982	+	1.64022i	−0.195249	+	0.980754i	$0.562551\pi$
$283$	−15.9307	+	27.5928i	−0.946982	+	1.64022i	−0.751733	+	0.659467i	$0.770782\pi$
$284$	−3.00000	+	5.19615i	−0.178017	+	0.308335i
$285$		0			0
$286$	3.25544	+	5.63858i	0.192498	+	0.333416i
$287$	7.11684			0.420094
$288$		0			0
$289$	37.3505			2.19709
$290$		0			0
$291$		0			0
$292$	−7.05842	+	12.2255i	−0.413063	+	0.715446i
$293$	4.11684	−	7.13058i	0.240509	−	0.416573i	−0.720351	−	0.693610i	$-0.756019\pi$
$293$	4.11684	−	7.13058i	0.240509	−	0.416573i	0.960859	+	0.277037i	$0.0893524\pi$
$294$		0			0
$295$		0			0
$296$	−4.00000			−0.232495
$297$		0			0
$298$	19.1168			1.10741
$299$	−10.3723	−	17.9653i	−0.599845	−	1.03896i
$300$		0			0
$301$	−13.4891	+	23.3639i	−0.777500	+	1.34667i
$302$	−5.00000	+	8.66025i	−0.287718	+	0.498342i
$303$		0			0
$304$	−1.68614	−	2.92048i	−0.0967068	−	0.167501i
$305$		0			0
$306$		0			0
$307$	33.2337			1.89675			0.948373	−	0.317156i	$-0.102728\pi$
$307$	33.2337			1.89675			0.948373	+	0.317156i	$0.102728\pi$
$308$	1.62772	+	2.81929i	0.0927479	+	0.160644i
$309$		0			0
$310$		0			0
$311$	−4.62772	+	8.01544i	−0.262414	+	0.454514i	−0.966883	−	0.255221i	$-0.917852\pi$
$311$	−4.62772	+	8.01544i	−0.262414	+	0.454514i	0.704469	+	0.709735i	$0.251185\pi$
$312$		0			0
$313$	4.68614	+	8.11663i	0.264876	+	0.458779i	0.967531	−	0.252752i	$-0.0813355\pi$
$313$	4.68614	+	8.11663i	0.264876	+	0.458779i	−0.702655	+	0.711531i	$0.748002\pi$
$314$	4.74456			0.267751
$315$		0			0
$316$	2.00000			0.112509
$317$	4.37228	+	7.57301i	0.245572	+	0.425343i	0.962292	−	0.272018i	$-0.0876910\pi$
$317$	4.37228	+	7.57301i	0.245572	+	0.425343i	−0.716720	+	0.697361i	$0.754358\pi$
$318$		0			0
$319$	3.00000	−	5.19615i	0.167968	−	0.290929i
$320$		0			0
$321$		0			0
$322$	−5.18614	−	8.98266i	−0.289012	−	0.500584i
$323$	−24.8614			−1.38333
$324$		0			0
$325$		0			0
$326$	−0.744563	−	1.28962i	−0.0412375	−	0.0714255i
$327$		0			0
$328$	1.50000	−	2.59808i	0.0828236	−	0.143455i
$329$	1.93070	−	3.34408i	0.106443	−	0.184365i
$330$		0			0
$331$	9.11684	+	15.7908i	0.501107	+	0.867943i	0.999999	+	0.00127880i	$0.000407055\pi$
$331$	9.11684	+	15.7908i	0.501107	+	0.867943i	−0.498892	+	0.866664i	$0.666260\pi$
$332$	1.62772			0.0893327
$333$		0			0
$334$	−7.62772			−0.417370
$335$		0			0
$336$		0			0
$337$	1.68614	−	2.92048i	0.0918499	−	0.159089i	−0.816440	−	0.577431i	$-0.804055\pi$
$337$	1.68614	−	2.92048i	0.0918499	−	0.159089i	0.908290	+	0.418342i	$0.137389\pi$
$338$	4.75544	−	8.23666i	0.258662	−	0.448015i
$339$		0			0
$340$		0			0
$341$	9.25544			0.501210
$342$		0			0
$343$	−19.8614			−1.07242
$344$	5.68614	+	9.84868i	0.306576	+	0.531005i
$345$		0			0
$346$	4.62772	−	8.01544i	0.248788	−	0.430913i
$347$	11.0584	−	19.1537i	0.593647	−	1.02823i	−0.400089	−	0.916476i	$-0.631021\pi$
$347$	11.0584	−	19.1537i	0.593647	−	1.02823i	0.993736	−	0.111751i	$-0.0356459\pi$
$348$		0			0
$349$	−0.441578	−	0.764836i	−0.0236371	−	0.0409407i	0.853965	−	0.520331i	$-0.174191\pi$
$349$	−0.441578	−	0.764836i	−0.0236371	−	0.0409407i	−0.877602	+	0.479390i	$0.840858\pi$
$350$		0			0
$351$		0			0
$352$	1.37228			0.0731428
$353$	−15.1753	−	26.2843i	−0.807698	−	1.39897i	−0.914455	−	0.404689i	$-0.867380\pi$
$353$	−15.1753	−	26.2843i	−0.807698	−	1.39897i	0.106757	−	0.994285i	$-0.465953\pi$
$354$		0			0
$355$		0			0
$356$	−0.558422	+	0.967215i	−0.0295963	+	0.0512623i
$357$		0			0
$358$	1.62772	+	2.81929i	0.0860276	+	0.149004i
$359$	−5.48913			−0.289705			−0.144852	−	0.989453i	$-0.546271\pi$
$359$	−5.48913			−0.289705			−0.144852	+	0.989453i	$0.546271\pi$
$360$		0			0
$361$	−7.62772			−0.401459
$362$	−3.93070	−	6.80818i	−0.206593	−	0.357830i
$363$		0			0
$364$	5.62772	−	9.74749i	0.294973	−	0.510908i
$365$		0			0
$366$		0			0
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	0.578101	−	0.815966i	$-0.303794\pi$
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	−0.995697	+	0.0926670i	$0.970461\pi$
$368$	−4.37228			−0.227921
$369$		0			0
$370$		0			0
$371$	−13.6277	−	23.6039i	−0.707516	−	1.22545i
$372$		0			0
$373$	3.74456	−	6.48577i	0.193886	−	0.335821i	−0.752649	−	0.658422i	$-0.771224\pi$
$373$	3.74456	−	6.48577i	0.193886	−	0.335821i	0.946535	+	0.322602i	$0.104557\pi$
$374$	5.05842	−	8.76144i	0.261565	−	0.453043i
$375$		0			0
$376$	−0.813859	−	1.40965i	−0.0419716	−	0.0726969i
$377$	−20.7446			−1.06840
$378$		0			0
$379$	−10.8614			−0.557913			−0.278956	−	0.960304i	$-0.589989\pi$
$379$	−10.8614			−0.557913			−0.278956	+	0.960304i	$0.589989\pi$
$380$		0			0
$381$		0			0
$382$	2.74456	−	4.75372i	0.140424	−	0.243222i
$383$	−11.4891	+	19.8997i	−0.587067	+	1.01683i	0.407547	+	0.913184i	$0.366384\pi$
$383$	−11.4891	+	19.8997i	−0.587067	+	1.01683i	−0.994614	+	0.103646i	$0.966949\pi$
$384$		0			0
$385$		0			0
$386$	3.88316			0.197647
$387$		0			0
$388$	2.62772			0.133402
$389$	−5.18614	−	8.98266i	−0.262948	−	0.455439i	0.704076	−	0.710124i	$-0.251361\pi$
$389$	−5.18614	−	8.98266i	−0.262948	−	0.455439i	−0.967024	+	0.254686i	$0.918028\pi$
$390$		0			0
$391$	−16.1168	+	27.9152i	−0.815064	+	1.41173i
$392$	−0.686141	+	1.18843i	−0.0346553	+	0.0600248i
$393$		0			0
$394$	8.74456	+	15.1460i	0.440545	+	0.763046i
$395$		0			0
$396$		0			0
$397$	−11.2554			−0.564894			−0.282447	−	0.959283i	$-0.591146\pi$
$397$	−11.2554			−0.564894			−0.282447	+	0.959283i	$0.591146\pi$
$398$	−4.74456	−	8.21782i	−0.237823	−	0.411922i
$399$		0			0
$400$		0			0
$401$	8.05842	−	13.9576i	0.402418	−	0.697009i	−0.591599	−	0.806232i	$-0.701503\pi$
$401$	8.05842	−	13.9576i	0.402418	−	0.697009i	0.994017	+	0.109223i	$0.0348364\pi$
$402$		0			0
$403$	−16.0000	−	27.7128i	−0.797017	−	1.38047i
$404$	8.74456			0.435058
$405$		0			0
$406$	−10.3723			−0.514768
$407$	2.74456	+	4.75372i	0.136043	+	0.235633i
$408$		0			0
$409$	5.43070	−	9.40625i	0.268531	−	0.465109i	−0.699952	−	0.714190i	$-0.746795\pi$
$409$	5.43070	−	9.40625i	0.268531	−	0.465109i	0.968483	+	0.249081i	$0.0801285\pi$
$410$		0			0
$411$		0			0
$412$	−8.00000	−	13.8564i	−0.394132	−	0.682656i
$413$	3.25544			0.160190
$414$		0			0
$415$		0			0
$416$	−2.37228	−	4.10891i	−0.116311	−	0.201456i
$417$		0			0
$418$	−2.31386	+	4.00772i	−0.113175	+	0.196024i
$419$	−15.8614	+	27.4728i	−0.774880	+	1.34213i	0.159981	+	0.987120i	$0.448857\pi$
$419$	−15.8614	+	27.4728i	−0.774880	+	1.34213i	−0.934862	+	0.355012i	$0.884477\pi$
$420$		0			0
$421$	19.2337	+	33.3137i	0.937393	+	1.62361i	0.770311	+	0.637669i	$0.220101\pi$
$421$	19.2337	+	33.3137i	0.937393	+	1.62361i	0.167082	+	0.985943i	$0.446566\pi$
$422$	7.25544			0.353189
$423$		0			0
$424$	−11.4891			−0.557961
$425$		0			0
$426$		0			0
$427$	−10.8139	+	18.7302i	−0.523319	+	0.906416i
$428$	7.24456	−	12.5480i	0.350179	−	0.606528i
$429$		0			0
$430$		0			0
$431$	26.2337			1.26363			0.631816	−	0.775118i	$-0.282310\pi$
$431$	26.2337			1.26363			0.631816	+	0.775118i	$0.282310\pi$
$432$		0			0
$433$	−0.627719			−0.0301662			−0.0150831	−	0.999886i	$-0.504801\pi$
$433$	−0.627719			−0.0301662			−0.0150831	+	0.999886i	$0.504801\pi$
$434$	−8.00000	−	13.8564i	−0.384012	−	0.665129i
$435$		0			0
$436$	−4.81386	+	8.33785i	−0.230542	+	0.399311i
$437$	7.37228	−	12.7692i	0.352664	−	0.610832i
$438$		0			0
$439$	−8.11684	−	14.0588i	−0.387396	−	0.670989i	0.604703	−	0.796451i	$-0.293292\pi$
$439$	−8.11684	−	14.0588i	−0.387396	−	0.670989i	−0.992098	+	0.125462i	$0.959959\pi$
$440$		0			0
$441$		0			0
$442$	−34.9783			−1.66375
$443$	13.2446	+	22.9403i	0.629268	+	1.08992i	0.987699	+	0.156368i	$0.0499786\pi$
$443$	13.2446	+	22.9403i	0.629268	+	1.08992i	−0.358431	+	0.933556i	$0.616688\pi$
$444$		0			0
$445$		0			0
$446$	−6.18614	+	10.7147i	−0.292922	+	0.507356i
$447$		0			0
$448$	−1.18614	−	2.05446i	−0.0560399	−	0.0970639i
$449$	18.8614			0.890125			0.445062	−	0.895500i	$-0.353181\pi$
$449$	18.8614			0.890125			0.445062	+	0.895500i	$0.353181\pi$
$450$		0			0
$451$	−4.11684			−0.193855
$452$	−7.37228	−	12.7692i	−0.346763	−	0.600611i
$453$		0			0
$454$	0.941578	−	1.63086i	0.0441904	−	0.0765401i
$455$		0			0
$456$		0			0
$457$	15.0584	+	26.0820i	0.704403	+	1.22006i	0.966906	+	0.255131i	$0.0821186\pi$
$457$	15.0584	+	26.0820i	0.704403	+	1.22006i	−0.262503	+	0.964931i	$0.584548\pi$
$458$	−18.3723			−0.858480
$459$		0			0
$460$		0			0
$461$	9.55842	+	16.5557i	0.445180	+	0.771075i	0.998065	−	0.0621833i	$-0.0198063\pi$
$461$	9.55842	+	16.5557i	0.445180	+	0.771075i	−0.552885	+	0.833258i	$0.686473\pi$
$462$		0			0
$463$	10.0000	−	17.3205i	0.464739	−	0.804952i	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	10.0000	−	17.3205i	0.464739	−	0.804952i	0.999190	+	0.0402476i	$0.0128147\pi$
$464$	−2.18614	+	3.78651i	−0.101489	+	0.175784i
$465$		0			0
$466$	−5.05842	−	8.76144i	−0.234327	−	0.405866i
$467$	25.8832			1.19773			0.598865	−	0.800850i	$-0.295619\pi$
$467$	25.8832			1.19773			0.598865	+	0.800850i	$0.295619\pi$
$468$		0			0
$469$	16.6060			0.766792
$470$		0			0
$471$		0			0
$472$	0.686141	−	1.18843i	0.0315822	−	0.0547019i
$473$	7.80298	−	13.5152i	0.358782	−	0.621428i
$474$		0			0
$475$		0			0
$476$	−17.4891			−0.801613
$477$		0			0
$478$	−14.7446			−0.674401
$479$	−11.7446	−	20.3422i	−0.536623	−	0.929458i	−0.999083	−	0.0428178i	$-0.986366\pi$
$479$	−11.7446	−	20.3422i	−0.536623	−	0.929458i	0.462460	−	0.886640i	$-0.346967\pi$
$480$		0			0
$481$	9.48913	−	16.4356i	0.432667	−	0.749401i
$482$	5.24456	−	9.08385i	0.238883	−	0.413758i
$483$		0			0
$484$	4.55842	+	7.89542i	0.207201	+	0.358883i
$485$		0			0
$486$		0			0
$487$	1.25544			0.0568893			0.0284446	−	0.999595i	$-0.490945\pi$
$487$	1.25544			0.0568893			0.0284446	+	0.999595i	$0.490945\pi$
$488$	4.55842	+	7.89542i	0.206350	+	0.357409i
$489$		0			0
$490$		0			0
$491$	1.80298	−	3.12286i	0.0813676	−	0.140933i	−0.822470	−	0.568808i	$-0.807405\pi$
$491$	1.80298	−	3.12286i	0.0813676	−	0.140933i	0.903838	+	0.427876i	$0.140738\pi$
$492$		0			0
$493$	16.1168	+	27.9152i	0.725866	+	1.25724i
$494$	16.0000			0.719874
$495$		0			0
$496$	−6.74456			−0.302840
$497$	−7.11684	−	12.3267i	−0.319234	−	0.552930i
$498$		0			0
$499$	1.05842	−	1.83324i	0.0473815	−	0.0820671i	−0.841362	−	0.540472i	$-0.818246\pi$
$499$	1.05842	−	1.83324i	0.0473815	−	0.0820671i	0.888743	+	0.458405i	$0.151579\pi$
$500$		0			0
$501$		0			0
$502$	−7.80298	−	13.5152i	−0.348264	−	0.603211i
$503$	21.8614			0.974752			0.487376	−	0.873192i	$-0.337954\pi$
$503$	21.8614			0.974752			0.487376	+	0.873192i	$0.337954\pi$
$504$		0			0
$505$		0			0
$506$	3.00000	+	5.19615i	0.133366	+	0.230997i
$507$		0			0
$508$	4.55842	−	7.89542i	0.202247	−	0.350303i
$509$	4.67527	−	8.09780i	0.207228	−	0.358929i	−0.743613	−	0.668611i	$-0.766889\pi$
$509$	4.67527	−	8.09780i	0.207228	−	0.358929i	0.950840	+	0.309682i	$0.100223\pi$
$510$		0			0
$511$	−16.7446	−	29.0024i	−0.740736	−	1.28299i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	1.37228			0.0605287
$515$		0			0
$516$		0			0
$517$	−1.11684	+	1.93443i	−0.0491187	+	0.0850762i
$518$	4.74456	−	8.21782i	0.208464	−	0.361070i
$519$		0			0
$520$		0			0
$521$	41.2337			1.80648			0.903240	−	0.429135i	$-0.141182\pi$
$521$	41.2337			1.80648			0.903240	+	0.429135i	$0.141182\pi$
$522$		0			0
$523$	11.1168			0.486106			0.243053	−	0.970013i	$-0.421851\pi$
$523$	11.1168			0.486106			0.243053	+	0.970013i	$0.421851\pi$
$524$	4.37228	+	7.57301i	0.191004	+	0.330829i
$525$		0			0
$526$	2.74456	−	4.75372i	0.119669	−	0.207272i
$527$	−24.8614	+	43.0612i	−1.08298	+	1.87578i
$528$		0			0
$529$	1.94158	+	3.36291i	0.0844164	+	0.146214i
$530$		0			0
$531$		0			0
$532$	8.00000			0.346844
$533$	7.11684	+	12.3267i	0.308265	+	0.533930i
$534$		0			0
$535$		0			0
$536$	3.50000	−	6.06218i	0.151177	−	0.261846i
$537$		0			0
$538$	−2.18614	−	3.78651i	−0.0942512	−	0.163248i
$539$	1.88316			0.0811133
$540$		0			0
$541$	21.6277			0.929848			0.464924	−	0.885351i	$-0.346082\pi$
$541$	21.6277			0.929848			0.464924	+	0.885351i	$0.346082\pi$
$542$	4.00000	+	6.92820i	0.171815	+	0.297592i
$543$		0			0
$544$	−3.68614	+	6.38458i	−0.158042	+	0.273737i
$545$		0			0
$546$		0			0
$547$	19.7337	+	34.1798i	0.843752	+	1.46142i	0.886701	+	0.462343i	$0.152991\pi$
$547$	19.7337	+	34.1798i	0.843752	+	1.46142i	−0.0429494	+	0.999077i	$0.513675\pi$
$548$	−1.88316			−0.0804444
$549$		0			0
$550$		0			0
$551$	−7.37228	−	12.7692i	−0.314070	−	0.543985i
$552$		0			0
$553$	−2.37228	+	4.10891i	−0.100880	+	0.174729i
$554$	−2.62772	+	4.55134i	−0.111641	+	0.193368i
$555$		0			0
$556$	−9.05842	−	15.6896i	−0.384163	−	0.665389i
$557$	−9.76631			−0.413812			−0.206906	−	0.978361i	$-0.566339\pi$
$557$	−9.76631			−0.413812			−0.206906	+	0.978361i	$0.566339\pi$
$558$		0			0
$559$	−53.9565			−2.28212
$560$		0			0
$561$		0			0
$562$	−2.18614	+	3.78651i	−0.0922168	+	0.159724i
$563$	−8.36141	+	14.4824i	−0.352391	+	0.610360i	−0.986668	−	0.162747i	$-0.947965\pi$
$563$	−8.36141	+	14.4824i	−0.352391	+	0.610360i	0.634277	+	0.773106i	$0.281298\pi$
$564$		0			0
$565$		0			0
$566$	−31.8614			−1.33923
$567$		0			0
$568$	−6.00000			−0.251754
$569$	4.80298	+	8.31901i	0.201352	+	0.348751i	0.948964	−	0.315384i	$-0.102133\pi$
$569$	4.80298	+	8.31901i	0.201352	+	0.348751i	−0.747613	+	0.664135i	$0.768800\pi$
$570$		0			0
$571$	15.8030	−	27.3716i	0.661334	−	1.14546i	−0.318931	−	0.947778i	$-0.603324\pi$
$571$	15.8030	−	27.3716i	0.661334	−	1.14546i	0.980265	−	0.197687i	$-0.0633429\pi$
$572$	−3.25544	+	5.63858i	−0.136117	+	0.235761i
$573$		0			0
$574$	3.55842	+	6.16337i	0.148526	+	0.257254i
$575$		0			0
$576$		0			0
$577$	23.8832			0.994269			0.497134	−	0.867674i	$-0.334386\pi$
$577$	23.8832			0.994269			0.497134	+	0.867674i	$0.334386\pi$
$578$	18.6753	+	32.3465i	0.776789	+	1.34544i
$579$		0			0
$580$		0			0
$581$	−1.93070	+	3.34408i	−0.0800991	+	0.138736i
$582$		0			0
$583$	7.88316	+	13.6540i	0.326487	+	0.565492i
$584$	−14.1168			−0.584159
$585$		0			0
$586$	8.23369			0.340131
$587$	−13.5000	−	23.3827i	−0.557205	−	0.965107i	−0.997728	−	0.0673658i	$-0.978541\pi$
$587$	−13.5000	−	23.3827i	−0.557205	−	0.965107i	0.440524	−	0.897741i	$-0.354793\pi$
$588$		0			0
$589$	11.3723	−	19.6974i	0.468587	−	0.811616i
$590$		0			0
$591$		0			0
$592$	−2.00000	−	3.46410i	−0.0821995	−	0.142374i
$593$	37.7228			1.54909			0.774545	−	0.632519i	$-0.217979\pi$
$593$	37.7228			1.54909			0.774545	+	0.632519i	$0.217979\pi$
$594$		0			0
$595$		0			0
$596$	9.55842	+	16.5557i	0.391528	+	0.678147i
$597$		0			0
$598$	10.3723	−	17.9653i	0.424154	−	0.734656i
$599$	−19.1168	+	33.1113i	−0.781093	+	1.35289i	0.150212	+	0.988654i	$0.452004\pi$
$599$	−19.1168	+	33.1113i	−0.781093	+	1.35289i	−0.931305	+	0.364239i	$0.881329\pi$
$600$		0			0
$601$	−13.4307	−	23.2627i	−0.547850	−	0.948904i	−0.998422	−	0.0561635i	$-0.982113\pi$
$601$	−13.4307	−	23.2627i	−0.547850	−	0.948904i	0.450572	−	0.892740i	$-0.351220\pi$
$602$	−26.9783			−1.09955
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$	0.441578	−	0.764836i	0.0179231	−	0.0310437i	−0.856925	−	0.515442i	$-0.827628\pi$
$607$	0.441578	−	0.764836i	0.0179231	−	0.0310437i	0.874848	+	0.484398i	$0.160961\pi$
$608$	1.68614	−	2.92048i	0.0683820	−	0.118441i
$609$		0			0
$610$		0			0
$611$	7.72281			0.312432
$612$		0			0
$613$	0.233688			0.00943857			0.00471928	−	0.999989i	$-0.498498\pi$
$613$	0.233688			0.00943857			0.00471928	+	0.999989i	$0.498498\pi$
$614$	16.6168	+	28.7812i	0.670601	+	1.16152i
$615$		0			0
$616$	−1.62772	+	2.81929i	−0.0655827	+	0.113592i
$617$	11.0584	−	19.1537i	0.445195	−	0.771101i	−0.552870	−	0.833267i	$-0.686468\pi$
$617$	11.0584	−	19.1537i	0.445195	−	0.771101i	0.998066	+	0.0621663i	$0.0198009\pi$
$618$		0			0
$619$	19.0584	+	33.0102i	0.766023	+	1.32679i	0.939704	+	0.341988i	$0.111100\pi$
$619$	19.0584	+	33.0102i	0.766023	+	1.32679i	−0.173682	+	0.984802i	$0.555566\pi$
$620$		0			0
$621$		0			0
$622$	−9.25544			−0.371109
$623$	−1.32473	−	2.29451i	−0.0530743	−	0.0919275i
$624$		0			0
$625$		0			0
$626$	−4.68614	+	8.11663i	−0.187296	+	0.324406i
$627$		0			0
$628$	2.37228	+	4.10891i	0.0946643	+	0.163963i
$629$	−29.4891			−1.17581
$630$		0			0
$631$	33.7228			1.34248			0.671242	−	0.741238i	$-0.265761\pi$
$631$	33.7228			1.34248			0.671242	+	0.741238i	$0.265761\pi$
$632$	1.00000	+	1.73205i	0.0397779	+	0.0688973i
$633$		0			0
$634$	−4.37228	+	7.57301i	−0.173645	+	0.300763i
$635$		0			0
$636$		0			0
$637$	−3.25544	−	5.63858i	−0.128985	−	0.223409i
$638$	6.00000			0.237542
$639$		0			0
$640$		0			0
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	−0.937451	−	0.348117i	$-0.886821\pi$
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	0.167247	−	0.985915i	$-0.446512\pi$
$642$		0			0
$643$	5.50000	−	9.52628i	0.216899	−	0.375680i	−0.736959	−	0.675937i	$-0.763739\pi$
$643$	5.50000	−	9.52628i	0.216899	−	0.375680i	0.953858	+	0.300257i	$0.0970725\pi$
$644$	5.18614	−	8.98266i	0.204363	−	0.353966i
$645$		0			0
$646$	−12.4307	−	21.5306i	−0.489079	−	0.847111i
$647$	24.0951			0.947276			0.473638	−	0.880720i	$-0.342941\pi$
$647$	24.0951			0.947276			0.473638	+	0.880720i	$0.342941\pi$
$648$		0			0
$649$	−1.88316			−0.0739203
$650$		0			0
$651$		0			0
$652$	0.744563	−	1.28962i	0.0291593	−	0.0505054i
$653$	18.8614	−	32.6689i	0.738104	−	1.27843i	−0.215244	−	0.976560i	$-0.569055\pi$
$653$	18.8614	−	32.6689i	0.738104	−	1.27843i	0.953348	−	0.301873i	$-0.0976119\pi$
$654$		0			0
$655$		0			0
$656$	3.00000			0.117130
$657$		0			0
$658$	3.86141			0.150533
$659$	−2.74456	−	4.75372i	−0.106913	−	0.185179i	0.807605	−	0.589724i	$-0.200763\pi$
$659$	−2.74456	−	4.75372i	−0.106913	−	0.185179i	−0.914518	+	0.404545i	$0.867430\pi$
$660$		0			0
$661$	−11.1168	+	19.2549i	−0.432395	+	0.748930i	−0.997079	−	0.0763770i	$-0.975665\pi$
$661$	−11.1168	+	19.2549i	−0.432395	+	0.748930i	0.564684	+	0.825307i	$0.308998\pi$
$662$	−9.11684	+	15.7908i	−0.354336	+	0.613728i
$663$		0			0
$664$	0.813859	+	1.40965i	0.0315839	+	0.0547049i
$665$		0			0
$666$		0			0
$667$	−19.1168			−0.740207
$668$	−3.81386	−	6.60580i	−0.147563	−	0.255586i
$669$		0			0
$670$		0			0
$671$	6.25544	−	10.8347i	0.241488	−	0.418270i
$672$		0			0
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	0.753420	−	0.657539i	$-0.228403\pi$
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	−0.946156	+	0.323711i	$0.895069\pi$
$674$	3.37228			0.129895
$675$		0			0
$676$	9.51087			0.365803
$677$	21.8614	+	37.8651i	0.840202	+	1.45527i	0.889724	+	0.456500i	$0.150897\pi$
$677$	21.8614	+	37.8651i	0.840202	+	1.45527i	−0.0495215	+	0.998773i	$0.515770\pi$
$678$		0			0
$679$	−3.11684	+	5.39853i	−0.119613	+	0.207177i
$680$		0			0
$681$		0			0
$682$	4.62772	+	8.01544i	0.177205	+	0.306927i
$683$	33.0951			1.26635			0.633174	−	0.774009i	$-0.281752\pi$
$683$	33.0951			1.26635			0.633174	+	0.774009i	$0.281752\pi$
$684$		0			0
$685$		0			0
$686$	−9.93070	−	17.2005i	−0.379156	−	0.656717i
$687$		0			0
$688$	−5.68614	+	9.84868i	−0.216782	+	0.375478i
$689$	27.2554	−	47.2078i	1.03835	−	1.79847i
$690$		0			0
$691$	0.883156	+	1.52967i	0.0335968	+	0.0581914i	0.882335	−	0.470622i	$-0.155970\pi$
$691$	0.883156	+	1.52967i	0.0335968	+	0.0581914i	−0.848738	+	0.528813i	$0.822637\pi$
$692$	9.25544			0.351839
$693$		0			0
$694$	22.1168			0.839544
$695$		0			0
$696$		0			0
$697$	11.0584	−	19.1537i	0.418868	−	0.725500i
$698$	0.441578	−	0.764836i	0.0167140	−	0.0289495i
$699$		0			0
$700$		0			0
$701$	−14.1386			−0.534007			−0.267004	−	0.963696i	$-0.586034\pi$
$701$	−14.1386			−0.534007			−0.267004	+	0.963696i	$0.586034\pi$
$702$		0			0
$703$	13.4891			0.508752
$704$	0.686141	+	1.18843i	0.0258599	+	0.0447907i
$705$		0			0
$706$	15.1753	−	26.2843i	0.571129	−	0.989224i
$707$	−10.3723	+	17.9653i	−0.390090	+	0.675655i
$708$		0			0
$709$	12.9307	+	22.3966i	0.485623	+	0.841123i	0.999863	−	0.0165226i	$-0.00525955\pi$
$709$	12.9307	+	22.3966i	0.485623	+	0.841123i	−0.514241	+	0.857646i	$0.671926\pi$
$710$		0			0
$711$		0			0
$712$	−1.11684			−0.0418555
$713$	−14.7446	−	25.5383i	−0.552188	−	0.956418i
$714$		0			0
$715$		0			0
$716$	−1.62772	+	2.81929i	−0.0608307	+	0.105362i
$717$		0			0
$718$	−2.74456	−	4.75372i	−0.102426	−	0.177407i
$719$	38.2337			1.42588			0.712938	−	0.701227i	$-0.247364\pi$
$719$	38.2337			1.42588			0.712938	+	0.701227i	$0.247364\pi$
$720$		0			0
$721$	37.9565			1.41357
$722$	−3.81386	−	6.60580i	−0.141937	−	0.245842i
$723$		0			0
$724$	3.93070	−	6.80818i	0.146083	−	0.253024i
$725$		0			0
$726$		0			0
$727$	0.441578	+	0.764836i	0.0163772	+	0.0283662i	0.874098	−	0.485750i	$-0.161453\pi$
$727$	0.441578	+	0.764836i	0.0163772	+	0.0283662i	−0.857721	+	0.514116i	$0.828120\pi$
$728$	11.2554			0.417154
$729$		0			0
$730$		0			0
$731$	41.9198	+	72.6073i	1.55046	+	2.68548i
$732$		0			0
$733$	17.1168	−	29.6472i	0.632225	−	1.09505i	−0.354871	−	0.934915i	$-0.615475\pi$
$733$	17.1168	−	29.6472i	0.632225	−	1.09505i	0.987096	−	0.160131i	$-0.0511915\pi$
$734$	8.00000	−	13.8564i	0.295285	−	0.511449i
$735$		0			0
$736$	−2.18614	−	3.78651i	−0.0805822	−	0.139572i
$737$	−9.60597			−0.353840
$738$		0			0
$739$	−23.8832			−0.878556			−0.439278	−	0.898351i	$-0.644766\pi$
$739$	−23.8832			−0.878556			−0.439278	+	0.898351i	$0.644766\pi$
$740$		0			0
$741$		0			0
$742$	13.6277	−	23.6039i	0.500289	−	0.866526i
$743$	−12.5584	+	21.7518i	−0.460724	+	0.797997i	−0.998997	−	0.0447732i	$-0.985743\pi$
$743$	−12.5584	+	21.7518i	−0.460724	+	0.797997i	0.538273	+	0.842770i	$0.319077\pi$
$744$		0			0
$745$		0			0
$746$	7.48913			0.274196
$747$		0			0
$748$	10.1168			0.369908
$749$	17.1861	+	29.7673i	0.627968	+	1.08767i
$750$		0			0
$751$	9.11684	−	15.7908i	0.332678	−	0.576216i	−0.650358	−	0.759628i	$-0.725381\pi$
$751$	9.11684	−	15.7908i	0.332678	−	0.576216i	0.983036	+	0.183412i	$0.0587143\pi$
$752$	0.813859	−	1.40965i	0.0296784	−	0.0514045i
$753$		0			0
$754$	−10.3723	−	17.9653i	−0.377736	−	0.654258i
$755$		0			0
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$	−5.43070	−	9.40625i	−0.197252	−	0.341651i
$759$		0			0
$760$		0			0
$761$	6.04755	−	10.4747i	0.219223	−	0.379706i	−0.735347	−	0.677690i	$-0.762981\pi$
$761$	6.04755	−	10.4747i	0.219223	−	0.379706i	0.954571	+	0.297984i	$0.0963143\pi$
$762$		0			0
$763$	−11.4198	−	19.7797i	−0.413426	−	0.716074i
$764$	5.48913			0.198590
$765$		0			0
$766$	−22.9783			−0.830238
$767$	3.25544	+	5.63858i	0.117547	+	0.203597i
$768$		0			0
$769$	9.06930	−	15.7085i	0.327047	−	0.566462i	−0.654877	−	0.755735i	$-0.727280\pi$
$769$	9.06930	−	15.7085i	0.327047	−	0.566462i	0.981925	+	0.189273i	$0.0606131\pi$
$770$		0			0
$771$		0			0
$772$	1.94158	+	3.36291i	0.0698789	+	0.121034i
$773$	14.7446			0.530325			0.265163	−	0.964204i	$-0.414574\pi$
$773$	14.7446			0.530325			0.265163	+	0.964204i	$0.414574\pi$
$774$		0			0
$775$		0			0
$776$	1.31386	+	2.27567i	0.0471648	+	0.0816918i
$777$		0			0
$778$	5.18614	−	8.98266i	0.185932	−	0.322044i
$779$	−5.05842	+	8.76144i	−0.181237	+	0.313911i
$780$		0			0
$781$	4.11684	+	7.13058i	0.147312	+	0.255152i
$782$	−32.2337			−1.15267
$783$		0			0
$784$	−1.37228			−0.0490100
$785$		0			0
$786$		0			0
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	−0.999999	−	0.00110111i	$-0.999650\pi$
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	0.500953	+	0.865474i	$0.332983\pi$
$788$	−8.74456	+	15.1460i	−0.311512	+	0.539555i
$789$		0

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.18614 - 2.05446i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.18614 - 2.05446i) q^{7} -1.00000 q^{8} +(0.686141 + 1.18843i) q^{11} +(2.37228 - 4.10891i) q^{13} +(1.18614 - 2.05446i) q^{14} +(-0.500000 - 0.866025i) q^{16} -7.37228 q^{17} +3.37228 q^{19} +(-0.686141 + 1.18843i) q^{22} +(2.18614 - 3.78651i) q^{23} +4.74456 q^{26} +2.37228 q^{28} +(-2.18614 - 3.78651i) q^{29} +(3.37228 - 5.84096i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-3.68614 - 6.38458i) q^{34} +4.00000 q^{37} +(1.68614 + 2.92048i) q^{38} +(-1.50000 + 2.59808i) q^{41} +(-5.68614 - 9.84868i) q^{43} -1.37228 q^{44} +4.37228 q^{46} +(0.813859 + 1.40965i) q^{47} +(0.686141 - 1.18843i) q^{49} +(2.37228 + 4.10891i) q^{52} +11.4891 q^{53} +(1.18614 + 2.05446i) q^{56} +(2.18614 - 3.78651i) q^{58} +(-0.686141 + 1.18843i) q^{59} +(-4.55842 - 7.89542i) q^{61} +6.74456 q^{62} +1.00000 q^{64} +(-3.50000 + 6.06218i) q^{67} +(3.68614 - 6.38458i) q^{68} +6.00000 q^{71} +14.1168 q^{73} +(2.00000 + 3.46410i) q^{74} +(-1.68614 + 2.92048i) q^{76} +(1.62772 - 2.81929i) q^{77} +(-1.00000 - 1.73205i) q^{79} -3.00000 q^{82} +(-0.813859 - 1.40965i) q^{83} +(5.68614 - 9.84868i) q^{86} +(-0.686141 - 1.18843i) q^{88} +1.11684 q^{89} -11.2554 q^{91} +(2.18614 + 3.78651i) q^{92} +(-0.813859 + 1.40965i) q^{94} +(-1.31386 - 2.27567i) q^{97} +1.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8	\( 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} - 2 q^{16} - 18 q^{17} + 2 q^{19} + 3 q^{22} + 3 q^{23} - 4 q^{26} - 2 q^{28} - 3 q^{29} + 2 q^{31} + 2 q^{32} - 9 q^{34} + 16 q^{37} + q^{38} - 6 q^{41} - 17 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{49} - 2 q^{52} - q^{56} + 3 q^{58} + 3 q^{59} - q^{61} + 4 q^{62} + 4 q^{64} - 14 q^{67} + 9 q^{68} + 24 q^{71} + 22 q^{73} + 8 q^{74} - q^{76} + 18 q^{77} - 4 q^{79} - 12 q^{82} - 9 q^{83} + 17 q^{86} + 3 q^{88} - 30 q^{89} - 68 q^{91} + 3 q^{92} - 9 q^{94} - 11 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8 - 3 * q^11 - 2 * q^13 - q^14 - 2 * q^16 - 18 * q^17 + 2 * q^19 + 3 * q^22 + 3 * q^23 - 4 * q^26 - 2 * q^28 - 3 * q^29 + 2 * q^31 + 2 * q^32 - 9 * q^34 + 16 * q^37 + q^38 - 6 * q^41 - 17 * q^43 + 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^49 - 2 * q^52 - q^56 + 3 * q^58 + 3 * q^59 - q^61 + 4 * q^62 + 4 * q^64 - 14 * q^67 + 9 * q^68 + 24 * q^71 + 22 * q^73 + 8 * q^74 - q^76 + 18 * q^77 - 4 * q^79 - 12 * q^82 - 9 * q^83 + 17 * q^86 + 3 * q^88 - 30 * q^89 - 68 * q^91 + 3 * q^92 - 9 * q^94 - 11 * q^97 - 6 * q^98

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