LMFDB - Embedded newform 921.2.x.a.131.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [921,2,Mod(5,921)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(921, base_ring=CyclotomicField(306))

chi = DirichletCharacter(H, H._module([153, 1]))

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

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

chi := DirichletCharacter("921.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 921 = 3 \cdot 307 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	921.x (of order $306$, degree $96$, 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:	$7.35422202616$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{306}]$

Embedding invariants

Embedding label			131.1
Character	$\chi$	$=$	921.131
Dual form			921.2.x.a.689.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72466 - 0.159813i) q^{3} +(0.306783 - 1.97633i) q^{4} +(-0.550381 + 4.85278i) q^{7} +(2.94892 + 0.551249i) q^{9} +O(q^{10})$	\(q+(-1.72466 - 0.159813i) q^{3} +(0.306783 - 1.97633i) q^{4} +(-0.550381 + 4.85278i) q^{7} +(2.94892 + 0.551249i) q^{9} +(-0.844942 + 3.35948i) q^{12} +(-2.36976 - 5.76291i) q^{13} +(-3.81177 - 1.21261i) q^{16} +(-0.0536532 - 0.244988i) q^{19} +(1.72476 - 8.28145i) q^{21} +(-0.665352 - 4.95553i) q^{25} +(-4.99779 - 1.42199i) q^{27} +(9.42185 + 2.57649i) q^{28} +(-0.229991 + 2.79393i) q^{31} +(1.99413 - 5.65893i) q^{36} +(-11.3421 - 3.35336i) q^{37} +(3.16605 + 10.3178i) q^{39} +(-9.76304 + 8.71835i) q^{43} +(6.38022 + 2.70052i) q^{48} +(-16.4243 - 3.77410i) q^{49} +(-12.1164 + 2.91547i) q^{52} +(0.0533813 + 0.431096i) q^{57} +(0.0174615 - 0.120648i) q^{61} +(-4.29812 + 14.0071i) q^{63} +(-3.56591 + 7.16131i) q^{64} +(-5.09098 - 10.4927i) q^{67} +(-3.95849 + 5.13148i) q^{73} +(0.355547 + 8.65295i) q^{75} +(-0.500638 + 0.0308783i) q^{76} +(0.287672 - 9.33708i) q^{79} +(8.39225 + 3.25117i) q^{81} +(-15.8377 - 5.94931i) q^{84} +(29.2704 - 8.32814i) q^{91} +(0.843165 - 4.78183i) q^{93} +(-3.92940 + 17.9422i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) $ 96 * q - 6 * q^4 + 18 * q^9	$ 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) $ 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/921\mathbb{Z}\right)^\times$.

$n$	$308$	$619$
$\chi(n)$	$-1$	$e\left(\frac{223}{306}\right)$

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			0		0.759405	−	0.650618i	$-0.225490\pi$
$2$		0			0		−0.759405	+	0.650618i	$0.774510\pi$
$3$	−1.72466	−	0.159813i	−0.995734	−	0.0922684i
$3$	−1.72466	−	0.159813i	−0.995734	−	0.0922684i
$4$	0.306783	−	1.97633i	0.153392	−	0.988165i
$5$		0			0		0.658380	−	0.752685i	$-0.271242\pi$
$5$		0			0		−0.658380	+	0.752685i	$0.728758\pi$
$6$		0			0
$7$	−0.550381	+	4.85278i	−0.208024	+	1.83418i	0.278412	+	0.960462i	$0.410192\pi$
$7$	−0.550381	+	4.85278i	−0.208024	+	1.83418i	−0.486436	+	0.873716i	$0.661703\pi$
$8$		0			0
$9$	2.94892	+	0.551249i	0.982973	+	0.183750i
$10$		0			0
$11$		0			0		0.964585	−	0.263774i	$-0.0849673\pi$
$11$		0			0		−0.964585	+	0.263774i	$0.915033\pi$
$12$	−0.844942	+	3.35948i	−0.243914	+	0.969797i
$13$	−2.36976	−	5.76291i	−0.657254	−	1.59834i	−0.794125	−	0.607754i	$-0.792071\pi$
$13$	−2.36976	−	5.76291i	−0.657254	−	1.59834i	0.136871	−	0.990589i	$-0.456295\pi$
$14$		0			0
$15$		0			0
$16$	−3.81177	−	1.21261i	−0.952942	−	0.303153i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−0.0536532	−	0.244988i	−0.0123089	−	0.0562042i	0.970290	−	0.241947i	$-0.0777859\pi$
$19$	−0.0536532	−	0.244988i	−0.0123089	−	0.0562042i	−0.982598	+	0.185742i	$0.940531\pi$
$20$		0			0
$21$	1.72476	−	8.28145i	0.376374	−	1.80716i
$22$		0			0
$23$		0			0		−0.163529	−	0.986539i	$-0.552288\pi$
$23$		0			0		0.163529	+	0.986539i	$0.447712\pi$
$24$		0			0
$25$	−0.665352	−	4.95553i	−0.133070	−	0.991107i
$26$		0			0
$27$	−4.99779	−	1.42199i	−0.961826	−	0.273663i
$28$	9.42185	+	2.57649i	1.78056	+	0.486910i
$29$		0			0		0.491083	−	0.871113i	$-0.336601\pi$
$29$		0			0		−0.491083	+	0.871113i	$0.663399\pi$
$30$		0			0
$31$	−0.229991	+	2.79393i	−0.0413077	+	0.501804i	0.943876	+	0.330301i	$0.107150\pi$
$31$	−0.229991	+	2.79393i	−0.0413077	+	0.501804i	−0.985183	+	0.171504i	$0.945137\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.99413	−	5.65893i	0.332355	−	0.943154i
$37$	−11.3421	−	3.35336i	−1.86464	−	0.551290i	−0.998691	−	0.0511517i	$-0.983711\pi$
$37$	−11.3421	−	3.35336i	−1.86464	−	0.551290i	−0.865946	−	0.500138i	$-0.833283\pi$
$38$		0			0
$39$	3.16605	+	10.3178i	0.506974	+	1.65217i
$40$		0			0
$41$		0			0		−0.725021	−	0.688727i	$-0.758170\pi$
$41$		0			0		0.725021	+	0.688727i	$0.241830\pi$
$42$		0			0
$43$	−9.76304	+	8.71835i	−1.48885	+	1.32954i	−0.702748	+	0.711439i	$0.748044\pi$
$43$	−9.76304	+	8.71835i	−1.48885	+	1.32954i	−0.786102	+	0.618097i	$0.787904\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.885823	−	0.464024i	$-0.846405\pi$
$47$		0			0		0.885823	+	0.464024i	$0.153595\pi$
$48$	6.38022	+	2.70052i	0.920906	+	0.389786i
$49$	−16.4243	−	3.77410i	−2.34633	−	0.539157i
$50$		0			0
$51$		0			0
$52$	−12.1164	+	2.91547i	−1.68024	+	0.404303i
$53$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$53$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.0533813	+	0.431096i	0.00707053	+	0.0571001i
$58$		0			0
$59$		0			0		−0.454905	−	0.890540i	$-0.650327\pi$
$59$		0			0		0.454905	+	0.890540i	$0.349673\pi$
$60$		0			0
$61$	0.0174615	−	0.120648i	0.00223572	−	0.0154474i	−0.988540	−	0.150958i	$-0.951764\pi$
$61$	0.0174615	−	0.120648i	0.00223572	−	0.0154474i	0.990776	+	0.135511i	$0.0432675\pi$
$62$		0			0
$63$	−4.29812	+	14.0071i	−0.541512	+	1.76472i
$64$	−3.56591	+	7.16131i	−0.445738	+	0.895163i
$65$		0			0
$66$		0			0
$67$	−5.09098	−	10.4927i	−0.621962	−	1.28189i	−0.942359	−	0.334602i	$-0.891398\pi$
$67$	−5.09098	−	10.4927i	−0.621962	−	1.28189i	0.320397	−	0.947283i	$-0.396184\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.312920	−	0.949779i	$-0.601307\pi$
$71$		0			0		0.312920	+	0.949779i	$0.398693\pi$
$72$		0			0
$73$	−3.95849	+	5.13148i	−0.463306	+	0.600594i	−0.964285	−	0.264865i	$-0.914673\pi$
$73$	−3.95849	+	5.13148i	−0.463306	+	0.600594i	0.500980	+	0.865459i	$0.332973\pi$
$74$		0			0
$75$	0.355547	+	8.65295i	0.0410550	+	0.999157i
$76$	−0.500638	+	0.0308783i	−0.0574271	+	0.00354198i
$77$		0			0
$78$		0			0
$79$	0.287672	−	9.33708i	0.0323657	−	1.05050i	−0.834207	−	0.551452i	$-0.814074\pi$
$79$	0.287672	−	9.33708i	0.0323657	−	1.05050i	0.866572	−	0.499051i	$-0.166318\pi$
$80$		0			0
$81$	8.39225	+	3.25117i	0.932472	+	0.361242i
$82$		0			0
$83$		0			0		0.958965	−	0.283523i	$-0.0915033\pi$
$83$		0			0		−0.958965	+	0.283523i	$0.908497\pi$
$84$	−15.8377	−	5.94931i	−1.72804	−	0.649123i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.0718047	−	0.997419i	$-0.477124\pi$
$89$		0			0		−0.0718047	+	0.997419i	$0.522876\pi$
$90$		0			0
$91$	29.2704	−	8.32814i	3.06837	−	0.873027i
$92$		0			0
$93$	0.843165	−	4.78183i	0.0874321	−	0.495852i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.92940	+	17.9422i	−0.398970	+	1.82175i	0.153413	+	0.988162i	$0.450974\pi$
$97$	−3.92940	+	17.9422i	−0.398970	+	1.82175i	−0.552383	+	0.833591i	$0.686281\pi$
$98$		0			0
$99$		0			0
$100$	−9.99789	−	0.205318i	−0.999789	−	0.0205318i
$101$		0			0		−0.998103	−	0.0615609i	$-0.980392\pi$
$101$		0			0		0.998103	+	0.0615609i	$0.0196078\pi$
$102$		0			0
$103$	13.2124	−	13.6257i	1.30186	−	1.34258i	0.396440	−	0.918060i	$-0.370245\pi$
$103$	13.2124	−	13.6257i	1.30186	−	1.34258i	0.905417	−	0.424524i	$-0.139559\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.634889	−	0.772603i	$-0.718954\pi$
$107$		0			0		0.634889	+	0.772603i	$0.281046\pi$
$108$	−4.34357	+	9.44105i	−0.417960	+	0.908465i
$109$	−14.7946	−	14.6435i	−1.41707	−	1.40259i	−0.763771	−	0.645488i	$-0.776654\pi$
$109$	−14.7946	−	14.6435i	−1.41707	−	1.40259i	−0.653295	−	0.757104i	$-0.726614\pi$
$110$		0			0
$111$	19.0255	+	7.59605i	1.80582	+	0.720985i
$112$	7.98246	−	17.8303i	0.754271	−	1.68480i
$113$		0			0		0.881012	−	0.473094i	$-0.156863\pi$
$113$		0			0		−0.881012	+	0.473094i	$0.843137\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.81145	−	18.3007i	−0.352368	−	1.69190i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.46931	−	5.59751i	0.860847	−	0.508865i
$122$		0			0
$123$		0			0
$124$	5.45117	+	1.31167i	0.489529	+	0.117791i
$125$		0			0
$126$		0			0
$127$	−9.93489	+	5.60071i	−0.881579	+	0.496983i	−0.864363	−	0.502869i	$-0.832278\pi$
$127$	−9.93489	+	5.60071i	−0.881579	+	0.496983i	−0.0172166	+	0.999852i	$0.505480\pi$
$128$		0			0
$129$	18.2313	−	13.4759i	1.60517	−	1.18649i
$130$		0			0
$131$		0			0		0.0410550	−	0.999157i	$-0.486928\pi$
$131$		0			0		−0.0410550	+	0.999157i	$0.513072\pi$
$132$		0			0
$133$	1.21840	−	0.125530i	0.105649	−	0.0108849i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.508865	−	0.860847i	$-0.330065\pi$
$137$		0			0		−0.508865	+	0.860847i	$0.669935\pi$
$138$		0			0
$139$	−8.04172	−	22.0944i	−0.682089	−	1.87403i	−0.411964	−	0.911200i	$-0.635157\pi$
$139$	−8.04172	−	22.0944i	−0.682089	−	1.87403i	−0.270125	−	0.962825i	$-0.587065\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10.5721	−	5.67712i	−0.881012	−	0.473094i
$145$		0			0
$146$		0			0
$147$	27.7233	+	9.13388i	2.28658	+	0.753350i
$148$	−10.1069	+	21.3871i	−0.830785	+	1.75801i
$149$		0			0		0.779081	−	0.626924i	$-0.215686\pi$
$149$		0			0		−0.779081	+	0.626924i	$0.784314\pi$
$150$		0			0
$151$	1.17258	+	8.10178i	0.0954232	+	0.659314i	0.980883	+	0.194598i	$0.0623402\pi$
$151$	1.17258	+	8.10178i	0.0954232	+	0.659314i	−0.885460	+	0.464716i	$0.846157\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	21.3627	−	3.09184i	1.71038	−	0.247545i
$157$	−9.53913	+	8.87709i	−0.761306	+	0.708469i	−0.962560	−	0.271070i	$-0.912623\pi$
$157$	−9.53913	+	8.87709i	−0.761306	+	0.708469i	0.201254	+	0.979539i	$0.435498\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.35928	+	8.11801i	−0.341445	+	0.635852i	−0.992371	−	0.123290i	$-0.960656\pi$
$163$	−4.35928	+	8.11801i	−0.341445	+	0.635852i	0.650925	+	0.759142i	$0.274381\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.552365	−	0.833602i	$-0.686275\pi$
$167$		0			0		0.552365	+	0.833602i	$0.313725\pi$
$168$		0			0
$169$	−18.3559	+	18.1684i	−1.41199	+	1.39757i
$170$		0			0
$171$	−0.0231697	−	0.752027i	−0.00177183	−	0.0575089i
$172$	14.2352	+	21.9696i	1.08542	+	1.67517i
$173$		0			0		0.399220	−	0.916855i	$-0.369281\pi$
$173$		0			0		−0.399220	+	0.916855i	$0.630719\pi$
$174$		0			0
$175$	24.4143	−	0.501376i	1.84555	−	0.0379005i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.969797	−	0.243914i	$-0.0784314\pi$
$179$		0			0		−0.969797	+	0.243914i	$0.921569\pi$
$180$		0			0
$181$	24.5561	−	10.6923i	1.82524	−	0.794752i	0.874294	−	0.485397i	$-0.161325\pi$
$181$	24.5561	−	10.6923i	1.82524	−	0.794752i	0.950947	−	0.309354i	$-0.100113\pi$
$182$		0			0
$183$	−0.0493964	+	0.205286i	−0.00365149	+	0.0151752i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.65132	−	23.4705i	0.702030	−	1.70723i
$190$		0			0
$191$		0			0		0.618892	−	0.785476i	$-0.287582\pi$
$191$		0			0		−0.618892	+	0.785476i	$0.712418\pi$
$192$	7.29446	−	11.7810i	0.526432	−	0.850217i
$193$	18.5525	−	20.3512i	1.33544	−	1.46491i	0.569510	−	0.821985i	$-0.307133\pi$
$193$	18.5525	−	20.3512i	1.33544	−	1.46491i	0.765930	−	0.642923i	$-0.222279\pi$
$194$		0			0
$195$		0			0
$196$	−12.4976	+	31.3021i	−0.892685	+	2.23586i
$197$		0			0		0.967242	−	0.253857i	$-0.0816993\pi$
$197$		0			0		−0.967242	+	0.253857i	$0.918301\pi$
$198$		0			0
$199$	25.3671	+	11.6707i	1.79822	+	0.827314i	0.958935	+	0.283626i	$0.0915374\pi$
$199$	25.3671	+	11.6707i	1.79822	+	0.827314i	0.839288	+	0.543688i	$0.182972\pi$
$200$		0			0
$201$	7.10335	+	18.9099i	0.501032	+	1.33380i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2.04483	+	24.8405i	0.141783	+	1.72238i
$209$		0			0
$210$		0			0
$211$	−3.69657	−	11.6199i	−0.254482	−	0.799950i	−0.992180	−	0.124818i	$-0.960165\pi$
$211$	−3.69657	−	11.6199i	−0.254482	−	0.799950i	0.737697	−	0.675132i	$-0.235913\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.4317	−	2.65382i	−0.911805	−	0.180153i
$218$		0			0
$219$	7.64713	−	8.21744i	0.516745	−	0.555283i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.0282	+	26.6963i	−0.671539	+	1.78771i	−0.0576298	+	0.998338i	$0.518354\pi$
$223$	−10.0282	+	26.6963i	−0.671539	+	1.78771i	−0.613909	+	0.789377i	$0.710404\pi$
$224$		0			0
$225$	0.769660	−	14.9802i	0.0513107	−	0.998683i
$226$		0			0
$227$		0			0		−0.517676	−	0.855577i	$-0.673203\pi$
$227$		0			0		0.517676	+	0.855577i	$0.326797\pi$
$228$	0.868366	+	0.0267541i	0.0575089	+	0.00177183i
$229$	−7.17956	+	11.3337i	−0.474439	+	0.748955i	−0.994451	−	0.105204i	$-0.966450\pi$
$229$	−7.17956	+	11.3337i	−0.474439	+	0.748955i	0.520012	+	0.854159i	$0.325927\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.408612	−	0.912708i	$-0.633987\pi$
$233$		0			0		0.408612	+	0.912708i	$0.366013\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.98833	+	16.0573i	−0.129156	+	1.04304i
$238$		0			0
$239$		0			0		0.626924	−	0.779081i	$-0.284314\pi$
$239$		0			0		−0.626924	+	0.779081i	$0.715686\pi$
$240$		0			0
$241$	13.3396	+	10.9619i	0.859282	+	0.706117i	0.957309	−	0.289066i	$-0.0933448\pi$
$241$	13.3396	+	10.9619i	0.859282	+	0.706117i	−0.0980274	+	0.995184i	$0.531253\pi$
$242$		0			0
$243$	−13.9542	−	6.94837i	−0.895163	−	0.445738i
$244$	−0.233083	−	0.0715225i	−0.0149216	−	0.00457876i
$245$		0			0
$246$		0			0
$247$	−1.28470	+	0.889763i	−0.0817434	+	0.0566143i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.427265	−	0.904126i	$-0.640523\pi$
$251$		0			0		0.427265	+	0.904126i	$0.359477\pi$
$252$	26.3640	+	12.7916i	1.66078	+	0.805797i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	13.0592	+	9.24438i	0.816197	+	0.577774i
$257$		0			0		0.666073	−	0.745886i	$-0.267974\pi$
$257$		0			0		−0.666073	+	0.745886i	$0.732026\pi$
$258$		0			0
$259$	22.5156	−	53.1953i	1.39905	−	3.30539i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.993630	−	0.112693i	$-0.0359477\pi$
$263$		0			0		−0.993630	+	0.112693i	$0.964052\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−22.2988	+	6.84248i	−1.36212	+	0.417971i
$269$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$269$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$270$		0			0
$271$	−16.8967	−	5.95417i	−1.02640	−	0.361690i	−0.234355	−	0.972151i	$-0.575298\pi$
$271$	−16.8967	−	5.95417i	−1.02640	−	0.361690i	−0.792046	+	0.610461i	$0.790984\pi$
$272$		0			0
$273$	−51.8125	+	9.68543i	−3.13583	+	0.586189i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.16381	−	0.944135i	−0.550600	−	0.0567276i	−0.176817	−	0.984244i	$-0.556580\pi$
$277$	−9.16381	−	0.944135i	−0.550600	−	0.0567276i	−0.373783	+	0.927516i	$0.621940\pi$
$278$		0			0
$279$	−2.21837	+	8.11228i	−0.132811	+	0.485670i
$280$		0			0
$281$		0			0		0.454905	−	0.890540i	$-0.349673\pi$
$281$		0			0		−0.454905	+	0.890540i	$0.650327\pi$
$282$		0			0
$283$	28.8455	−	14.3634i	1.71469	−	0.853813i	0.729151	−	0.684353i	$-0.239915\pi$
$283$	28.8455	−	14.3634i	1.71469	−	0.853813i	0.985537	−	0.169460i	$-0.0542022\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	−	14.7224i	−0.500000	−	0.866025i
$290$		0			0
$291$	9.64429	−	30.3162i	0.565358	−	1.77717i
$292$	8.92710	+	9.39753i	0.522419	+	0.549949i
$293$		0			0		0.827888	−	0.560894i	$-0.189542\pi$
$293$		0			0		−0.827888	+	0.560894i	$0.810458\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	17.2102	+	1.95190i	0.993630	+	0.112693i
$301$	−36.9348	−	52.1763i	−2.12889	−	3.00739i
$302$		0			0
$303$		0			0
$304$	−0.0925616	+	0.998899i	−0.00530877	+	0.0572908i
$305$		0			0
$306$		0			0
$307$	−2.47703	+	17.3454i	−0.141371	+	0.989957i
$307$	−2.47703	+	17.3454i	−0.141371	+	0.989957i
$308$		0			0
$309$	−24.9645	+	21.3883i	−1.42018	+	1.21674i
$310$		0			0
$311$		0			0		0.153392	−	0.988165i	$-0.450980\pi$
$311$		0			0		−0.153392	+	0.988165i	$0.549020\pi$
$312$		0			0
$313$	15.3654	−	10.8770i	0.868506	−	0.614803i	−0.0549575	−	0.998489i	$-0.517502\pi$
$313$	15.3654	−	10.8770i	0.868506	−	0.614803i	0.923464	+	0.383686i	$0.125345\pi$
$314$		0			0
$315$		0			0
$316$	−18.3649	−	3.43300i	−1.03311	−	0.193121i
$317$		0			0		0.0102665	−	0.999947i	$-0.496732\pi$
$317$		0			0		−0.0102665	+	0.999947i	$0.503268\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	−	15.5885i	0.500000	−	0.866025i
$325$	−26.9816	+	15.5778i	−1.49667	+	0.864101i
$326$		0			0
$327$	23.1755	+	27.6194i	1.28161	+	1.52736i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.1723	−	24.4454i	−0.669053	−	1.34364i	−0.926114	−	0.377244i	$-0.876872\pi$
$331$	−12.1723	−	24.4454i	−0.669053	−	1.34364i	0.257061	−	0.966395i	$-0.417246\pi$
$332$		0			0
$333$	−31.5985	−	16.1411i	−1.73159	−	0.884529i
$334$		0			0
$335$		0			0
$336$	−16.6166	+	29.4755i	−0.906508	+	1.60802i
$337$	3.52661	−	34.2294i	0.192107	−	1.86459i	−0.250817	−	0.968034i	$-0.580699\pi$
$337$	3.52661	−	34.2294i	0.192107	−	1.86459i	0.442924	−	0.896559i	$-0.353941\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.9922	−	45.3827i	0.863499	−	2.45043i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.986539	−	0.163529i	$-0.947712\pi$
$347$		0			0		0.986539	+	0.163529i	$0.0522876\pi$
$348$		0			0
$349$	−0.109119	−	0.198293i	−0.00584098	−	0.0106144i	0.873172	−	0.487412i	$-0.162059\pi$
$349$	−0.109119	−	0.198293i	−0.00584098	−	0.0106144i	−0.879013	+	0.476797i	$0.841798\pi$
$350$		0			0
$351$	3.64876	+	32.1716i	0.194757	+	1.71719i
$352$		0			0
$353$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$353$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.972250	−	0.233944i	$-0.0751634\pi$
$359$		0			0		−0.972250	+	0.233944i	$0.924837\pi$
$360$		0			0
$361$	17.2037	−	7.91496i	0.905458	−	0.416577i
$362$		0			0
$363$	−17.2259	+	8.14049i	−0.904126	+	0.427265i
$364$	−7.47950	−	60.4029i	−0.392032	−	3.16597i
$365$		0			0
$366$		0			0
$367$	18.1617	+	26.2231i	0.948035	+	1.36884i	0.929426	+	0.369009i	$0.120303\pi$
$367$	18.1617	+	26.2231i	0.948035	+	1.36884i	0.0186089	+	0.999827i	$0.494076\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−9.19180	−	3.13336i	−0.476573	−	0.162457i
$373$	−8.72596	+	10.6187i	−0.451813	+	0.549816i	−0.947946	−	0.318430i	$-0.896844\pi$
$373$	−8.72596	+	10.6187i	−0.451813	+	0.549816i	0.496133	+	0.868246i	$0.334753\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	13.5731	+	35.0361i	0.697202	+	1.79968i	0.593117	+	0.805116i	$0.297897\pi$
$379$	13.5731	+	35.0361i	0.697202	+	1.79968i	0.104085	+	0.994568i	$0.466809\pi$
$380$		0			0
$381$	18.0294	−	8.07161i	0.923674	−	0.413521i
$382$		0			0
$383$		0			0		0.998103	−	0.0615609i	$-0.0196078\pi$
$383$		0			0		−0.998103	+	0.0615609i	$0.980392\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−33.5964	+	20.3278i	−1.70780	+	1.03332i
$388$	34.2542	+	13.2702i	1.73899	+	0.673690i
$389$		0			0		−0.998683	−	0.0513107i	$-0.983660\pi$
$389$		0			0		0.998683	+	0.0513107i	$0.0163399\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.17761	+	26.2053i	−0.259857	+	1.31521i	0.598074	+	0.801441i	$0.295933\pi$
$397$	−5.17761	+	26.2053i	−0.259857	+	1.31521i	−0.857930	+	0.513766i	$0.828250\pi$
$398$		0			0
$399$	−2.12140	+	0.0217804i	−0.106203	+	0.00109038i
$400$	−3.47296	+	19.6962i	−0.173648	+	0.984808i
$401$		0			0		0.974601	−	0.223951i	$-0.0718954\pi$
$401$		0			0		−0.974601	+	0.223951i	$0.928105\pi$
$402$		0			0
$403$	16.6462	−	5.29553i	0.829205	−	0.263789i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.61985	+	39.0643i	0.178990	+	1.93161i	0.331795	+	0.943351i	$0.392346\pi$
$409$	3.61985	+	39.0643i	0.178990	+	1.93161i	−0.152805	+	0.988256i	$0.548831\pi$
$410$		0			0
$411$		0			0
$412$	−22.8756	−	30.2922i	−1.12700	−	1.49239i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	10.3383	+	39.3906i	0.506266	+	1.92897i
$418$		0			0
$419$		0			0		0.408612	−	0.912708i	$-0.366013\pi$
$419$		0			0		−0.408612	+	0.912708i	$0.633987\pi$
$420$		0			0
$421$	25.0872	+	22.8700i	1.22268	+	1.11462i	0.989853	+	0.142097i	$0.0453845\pi$
$421$	25.0872	+	22.8700i	1.22268	+	1.11462i	0.232823	+	0.972519i	$0.425204\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.575867	+	0.151139i	0.0278682	+	0.00731414i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.972250	−	0.233944i	$-0.924837\pi$
$431$		0			0		0.972250	+	0.233944i	$0.0751634\pi$
$432$	17.3261	+	11.4807i	0.833602	+	0.552365i
$433$	−4.43465	−	10.1847i	−0.213115	−	0.489445i	0.777034	−	0.629459i	$-0.216723\pi$
$433$	−4.43465	−	10.1847i	−0.213115	−	0.489445i	−0.990150	+	0.140014i	$0.955285\pi$
$434$		0			0
$435$		0			0
$436$	−33.4791	+	24.7466i	−1.60336	+	1.18515i
$437$		0			0
$438$		0			0
$439$	−0.763272	−	37.1672i	−0.0364290	−	1.77389i	−0.479630	−	0.877471i	$-0.659229\pi$
$439$	−0.763272	−	37.1672i	−0.0364290	−	1.77389i	0.443201	−	0.896422i	$-0.353843\pi$
$440$		0			0
$441$	−46.3536	−	20.1834i	−2.20731	−	0.961115i
$442$		0			0
$443$		0			0		0.999526	−	0.0307951i	$-0.00980392\pi$
$443$		0			0		−0.999526	+	0.0307951i	$0.990196\pi$
$444$	20.8490	−	35.2702i	0.989449	−	1.67385i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−32.7896	−	21.2460i	−1.54916	−	1.00378i
$449$		0			0		−0.855577	−	0.517676i	$-0.826797\pi$
$449$		0			0		0.855577	+	0.517676i	$0.173203\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.727529	−	14.1602i	−0.0341823	−	0.665306i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.96936	+	8.56370i	0.372791	+	0.400593i	0.890515	−	0.454955i	$-0.150345\pi$
$457$	7.96936	+	8.56370i	0.372791	+	0.400593i	−0.517724	+	0.855548i	$0.673220\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.759405	−	0.650618i	$-0.774510\pi$
$461$		0			0		0.759405	+	0.650618i	$0.225490\pi$
$462$		0			0
$463$	−39.0361	+	5.64974i	−1.81416	+	0.262566i	−0.964371	−	0.264553i	$-0.914775\pi$
$463$	−39.0361	+	5.64974i	−1.81416	+	0.262566i	−0.849791	+	0.527119i	$0.823272\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.312920	−	0.949779i	$-0.398693\pi$
$467$		0			0		−0.312920	+	0.949779i	$0.601307\pi$
$468$	−37.3375	+	1.91834i	−1.72593	+	0.0886752i
$469$	53.7207	−	18.9304i	2.48059	−	0.874126i
$470$		0			0
$471$	17.8705	−	13.7855i	0.823427	−	0.635203i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.17835	+	0.428884i	−0.0540664	+	0.0196785i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.543778	−	0.839229i	$-0.683007\pi$
$479$		0			0		0.543778	+	0.839229i	$0.316993\pi$
$480$		0			0
$481$	7.55307	+	73.3104i	0.344390	+	3.34267i
$482$		0			0
$483$		0			0
$484$	−8.15751	−	20.4317i	−0.370796	−	0.928714i
$485$		0			0
$486$		0			0
$487$	−7.34045	−	13.0209i	−0.332627	−	0.590035i	0.653322	−	0.757080i	$-0.273375\pi$
$487$	−7.34045	−	13.0209i	−0.332627	−	0.590035i	−0.985949	+	0.167045i	$0.946577\pi$
$488$		0			0
$489$	8.81566	−	13.3042i	0.398658	−	0.601635i
$490$		0			0
$491$		0			0		0.752685	−	0.658380i	$-0.228758\pi$
$491$		0			0		−0.752685	+	0.658380i	$0.771242\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	4.26462	−	10.3709i	0.191487	−	0.465668i
$497$		0			0
$498$		0			0
$499$	−15.2596	+	24.6451i	−0.683114	+	1.10327i	0.305413	+	0.952220i	$0.401206\pi$
$499$	−15.2596	+	24.6451i	−0.683114	+	1.10327i	−0.988527	+	0.151047i	$0.951736\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.370796	−	0.928714i	$-0.379085\pi$
$503$		0			0		−0.370796	+	0.928714i	$0.620915\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	34.5612	−	28.4008i	1.53492	−	1.26132i
$508$	8.02100	+	21.3528i	0.355874	+	0.947379i
$509$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$509$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$510$		0			0
$511$	−22.7232	−	22.0339i	−1.00522	−	0.974724i
$512$		0			0
$513$	−0.0802241	+	1.30069i	−0.00354198	+	0.0574271i
$514$		0			0
$515$		0			0
$516$	−21.0399	−	40.1652i	−0.926229	−	1.76817i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$521$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$522$		0			0
$523$	1.64976	+	5.79830i	0.0721390	+	0.253542i	0.989504	−	0.144504i	$-0.0461588\pi$
$523$	1.64976	+	5.79830i	0.0721390	+	0.253542i	−0.917365	+	0.398047i	$0.869688\pi$
$524$		0			0
$525$	−42.1866	−	3.03703i	−1.84117	−	0.132547i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.7699	+	7.42105i	−0.946517	+	0.322654i
$530$		0			0
$531$		0			0
$532$	0.125696	−	2.44648i	0.00544962	−	0.106068i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−36.3713	−	28.0573i	−1.56372	−	1.20628i	−0.878274	−	0.478157i	$-0.841305\pi$
$541$	−36.3713	−	28.0573i	−1.56372	−	1.20628i	−0.685450	−	0.728120i	$-0.740394\pi$
$542$		0			0
$543$	−44.0597	+	14.5162i	−1.89078	+	0.622949i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	42.0822	−	20.4180i	1.79930	−	0.873011i	0.895540	−	0.444980i	$-0.146789\pi$
$547$	42.0822	−	20.4180i	1.79930	−	0.873011i	0.903764	−	0.428031i	$-0.140792\pi$
$548$		0			0
$549$	0.118000	−	0.346156i	0.00503610	−	0.0147736i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	45.1524	+	6.53496i	1.92008	+	0.277895i
$554$		0			0
$555$		0			0
$556$	−46.1330	+	9.11489i	−1.95647	+	0.386557i
$557$		0			0		0.992421	−	0.122888i	$-0.0392157\pi$
$557$		0			0		−0.992421	+	0.122888i	$0.960784\pi$
$558$		0			0
$559$	73.3792	+	35.6031i	3.10361	+	1.50585i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.816197	−	0.577774i	$-0.803922\pi$
$563$		0			0		0.816197	+	0.577774i	$0.196078\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−20.3962	+	38.9363i	−0.856559	+	1.63517i
$568$		0			0
$569$		0			0		−0.618892	−	0.785476i	$-0.712418\pi$
$569$		0			0		0.618892	+	0.785476i	$0.287582\pi$
$570$		0			0
$571$	31.6731	+	35.4684i	1.32548	+	1.48431i	0.737266	+	0.675602i	$0.236116\pi$
$571$	31.6731	+	35.4684i	1.32548	+	1.48431i	0.588213	+	0.808706i	$0.299831\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−14.4632	+	19.1524i	−0.602635	+	0.798017i
$577$	11.3928	−	38.5340i	0.474287	−	1.60419i	−0.288459	−	0.957492i	$-0.593143\pi$
$577$	11.3928	−	38.5340i	0.474287	−	1.60419i	0.762746	−	0.646698i	$-0.223851\pi$
$578$		0			0
$579$	−35.2492	+	32.1339i	−1.46491	+	1.33544i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$587$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$588$	26.5566	−	51.9883i	1.09518	−	2.14396i
$589$	0.696819	−	0.0935581i	0.0287119	−	0.00385499i
$590$		0			0
$591$		0			0
$592$	39.1673	+	26.5358i	1.60977	+	1.09062i
$593$		0			0		−0.978993	−	0.203893i	$-0.934641\pi$
$593$		0			0		0.978993	+	0.203893i	$0.0653595\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−41.8845	−	24.1820i	−1.71422	−	0.989704i
$598$		0			0
$599$		0			0		−0.688727	−	0.725021i	$-0.741830\pi$
$599$		0			0		0.688727	+	0.725021i	$0.258170\pi$
$600$		0			0
$601$	−37.9140	+	15.5906i	−1.54654	+	0.635954i	−0.981903	−	0.189382i	$-0.939352\pi$
$601$	−37.9140	+	15.5906i	−1.54654	+	0.635954i	−0.564641	+	0.825336i	$0.690985\pi$
$602$		0			0
$603$	−9.22882	−	33.7485i	−0.375826	−	1.37434i
$604$	16.3715	+	0.168087i	0.666148	+	0.00683934i
$605$		0			0
$606$		0			0
$607$	−46.5475	−	5.27921i	−1.88930	−	0.214277i	−0.909708	−	0.415249i	$-0.863695\pi$
$607$	−46.5475	−	5.27921i	−1.88930	−	0.214277i	−0.979597	+	0.200972i	$0.935590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	23.7869i		−	0.960745i	−0.877065	−	0.480372i	$-0.840502\pi$
$613$		−	23.7869i		−	0.960745i	0.877065	−	0.480372i	$-0.159498\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$617$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$618$		0			0
$619$	9.62298	−	11.0014i	0.386780	−	0.442182i	−0.526092	−	0.850428i	$-0.676343\pi$
$619$	9.62298	−	11.0014i	0.386780	−	0.442182i	0.912872	+	0.408246i	$0.133859\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.443208	−	43.1682i	0.0177425	−	1.72811i
$625$	−24.1146	+	6.59435i	−0.964585	+	0.263774i
$626$		0			0
$627$		0			0
$628$	14.6176	+	21.5758i	0.583307	+	0.860969i
$629$		0			0
$630$		0			0
$631$	7.63012	−	13.2158i	0.303750	−	0.526111i	−0.673232	−	0.739431i	$-0.735094\pi$
$631$	7.63012	−	13.2158i	0.303750	−	0.526111i	0.976982	+	0.213320i	$0.0684278\pi$
$632$		0			0
$633$	4.51831	+	20.6312i	0.179587	+	0.820018i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	17.1720	+	103.596i	0.680380	+	4.10461i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$641$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$642$		0			0
$643$	−15.7105	+	27.8683i	−0.619563	+	1.09902i	0.366146	+	0.930558i	$0.380677\pi$
$643$	−15.7105	+	27.8683i	−0.619563	+	1.09902i	−0.985708	+	0.168461i	$0.946120\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$647$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	22.7411	+	6.72352i	0.891293	+	0.263515i
$652$	14.7065	+	11.1059i	0.575952	+	0.434939i
$653$		0			0		−0.293353	−	0.956004i	$-0.594771\pi$
$653$		0			0		0.293353	+	0.956004i	$0.405229\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−14.5020	+	12.9502i	−0.565776	+	0.505235i
$658$		0			0
$659$		0			0		0.785476	−	0.618892i	$-0.212418\pi$
$659$		0			0		−0.785476	+	0.618892i	$0.787582\pi$
$660$		0			0
$661$	−24.7681	−	12.9744i	−0.963369	−	0.504645i	−0.0922603	−	0.995735i	$-0.529409\pi$
$661$	−24.7681	−	12.9744i	−0.963369	−	0.504645i	−0.871109	+	0.491090i	$0.836599\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	21.5617	−	44.4394i	0.833624	−	1.71813i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−22.8088	−	44.6513i	−0.879213	−	1.72118i	−0.668613	−	0.743610i	$-0.733112\pi$
$673$	−22.8088	−	44.6513i	−0.879213	−	1.72118i	−0.210600	−	0.977572i	$-0.567542\pi$
$674$		0			0
$675$	−3.72145	+	25.7129i	−0.143239	+	0.989688i
$676$	30.2755	+	41.8510i	1.16444	+	1.60966i
$677$		0			0		0.293353	−	0.956004i	$-0.405229\pi$
$677$		0			0		−0.293353	+	0.956004i	$0.594771\pi$
$678$		0			0
$679$	−84.9068	−	28.9435i	−3.25842	−	1.11075i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.0718047	−	0.997419i	$-0.522876\pi$
$683$		0			0		0.0718047	+	0.997419i	$0.477124\pi$
$684$	−1.49336	−	0.184918i	−0.0571001	−	0.00707053i
$685$		0			0
$686$		0			0
$687$	14.1936	−	18.3995i	0.541520	−	0.701984i
$688$	47.7864	−	21.3936i	1.82184	−	0.815622i
$689$		0			0
$690$		0			0
$691$	6.39138	+	41.1739i	0.243139	+	1.56633i	0.724167	+	0.689625i	$0.242224\pi$
$691$	6.39138	+	41.1739i	0.243139	+	1.56633i	−0.481028	+	0.876705i	$0.659736\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	6.49902	−	48.4046i	0.245640	−	1.82952i
$701$		0			0		0.569364	−	0.822086i	$-0.307190\pi$
$701$		0			0		−0.569364	+	0.822086i	$0.692810\pi$
$702$		0			0
$703$	−0.212992	+	2.95861i	−0.00803314	+	0.111586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−43.9271	−	24.1726i	−1.64972	−	0.907821i	−0.987334	−	0.158655i	$-0.949284\pi$
$709$	−43.9271	−	24.1726i	−1.64972	−	0.907821i	−0.662382	−	0.749166i	$-0.730454\pi$
$710$		0			0
$711$	5.99537	−	27.3757i	0.224844	−	1.02667i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$719$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$720$		0			0
$721$	58.8509	+	71.6162i	2.19172	+	2.66713i
$722$		0			0
$723$	−21.2545	−	21.0374i	−0.790464	−	0.782390i
$724$	−13.5981	−	51.8112i	−0.505369	−	1.92555i
$725$		0			0
$726$		0			0
$727$	−12.0398	+	6.46521i	−0.446530	+	0.239781i	−0.680915	−	0.732362i	$-0.738418\pi$
$727$	−12.0398	+	6.46521i	−0.446530	+	0.239781i	0.234385	+	0.972144i	$0.424692\pi$
$728$		0			0
$729$	22.9559	+	14.2137i	0.850217	+	0.526432i
$730$		0			0
$731$		0			0
$732$	0.390560	+	0.160602i	0.0144355	+	0.00593602i
$733$	24.1908	+	17.8810i	0.893508	+	0.660451i	0.940995	−	0.338420i	$-0.109893\pi$
$733$	24.1908	+	17.8810i	0.893508	+	0.660451i	−0.0474873	+	0.998872i	$0.515121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.53560	−	4.33064i	−0.240416	−	0.159305i	0.426384	−	0.904542i	$-0.359787\pi$
$739$	−6.53560	−	4.33064i	−0.240416	−	0.159305i	−0.666800	+	0.745237i	$0.732336\pi$
$740$		0			0
$741$	2.35787	−	1.32923i	0.0866185	−	0.0488304i
$742$		0			0
$743$		0			0		0.804162	−	0.594410i	$-0.202614\pi$
$743$		0			0		−0.804162	+	0.594410i	$0.797386\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−22.3741	+	37.8502i	−0.816441	+	1.38117i	0.105731	+	0.994395i	$0.466282\pi$
$751$	−22.3741	+	37.8502i	−0.816441	+	1.38117i	−0.922172	+	0.386779i	$0.873588\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−43.4247	−	26.2746i	−1.57934	−	0.955597i
$757$	−20.2365	−	26.2331i	−0.735509	−	0.953457i	0.264431	−	0.964405i	$-0.414816\pi$
$757$	−20.2365	−	26.2331i	−0.735509	−	0.953457i	−0.999940	+	0.0109479i	$0.996515\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.949779	−	0.312920i	$-0.898693\pi$
$761$		0			0		0.949779	+	0.312920i	$0.101307\pi$
$762$		0			0
$763$	79.2042	−	63.7354i	2.86739	−	2.30738i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−21.0452	−	18.0305i	−0.759405	−	0.650618i
$769$	−5.33923	−	5.50626i	−0.192537	−	0.198561i	0.614745	−	0.788726i	$-0.289259\pi$
$769$	−5.33923	−	5.50626i	−0.192537	−	0.198561i	−0.807283	+	0.590165i	$0.799063\pi$
$770$		0			0
$771$		0			0
$772$	−34.5290	−	42.9093i	−1.24273	−	1.54434i
$773$		0			0		−0.904126	−	0.427265i	$-0.859477\pi$
$773$		0			0		0.904126	+	0.427265i	$0.140523\pi$
$774$		0			0
$775$	13.9984	−	0.719216i	0.502838	−	0.0258350i
$776$		0			0
$777$	−47.3332	+	88.1456i	−1.69807	+	3.16221i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	58.0293	+	34.3023i	2.07247	+	1.22508i
$785$		0			0
$786$		0			0
$787$	5.46735	−	12.5564i	0.194890	−	0.447588i	−0.791678	−	0.610939i	$-0.790792\pi$
$787$	5.46735	−	12.5564i	0.194890	−	0.447588i	0.986568	+	0.163351i	$0.0522302\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.736663	+	0.185278i	−0.0261597	+	0.00657942i
$794$		0			0
$795$		0			0
$796$	30.8473	−	46.5533i	1.09336	−	1.65004i
$797$		0			0		0.233944	−	0.972250i	$-0.424837\pi$
$797$		0			0		−0.233944	+	0.972250i	$0.575163\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	39.5515	−	8.23732i	1.39487	−	0.290508i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.912708	−	0.408612i	$-0.866013\pi$
$809$		0			0		0.912708	+	0.408612i	$0.133987\pi$
$810$		0			0
$811$	53.5784	−	14.0619i	1.88139	−	0.493781i	0.881631	−	0.471939i	$-0.156446\pi$
$811$	53.5784	−	14.0619i	1.88139	−	0.493781i	0.999761	−	0.0218416i	$-0.00695296\pi$
$812$		0			0
$813$	28.1895	+	12.9692i	0.988650	+	0.454851i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.65971	+	1.92406i	0.0930515	+	0.0673144i
$818$		0			0
$819$	90.9069	−	8.42376i	3.17654	−	0.294350i
$820$		0			0
$821$		0			0		0.0205318	−	0.999789i	$-0.493464\pi$
$821$		0			0		−0.0205318	+	0.999789i	$0.506536\pi$
$822$		0			0
$823$	26.6150	+	50.8082i	0.927742	+	1.77106i	0.486614	+	0.873617i	$0.338232\pi$
$823$	26.6150	+	50.8082i	0.927742	+	1.77106i	0.441127	+	0.897445i	$0.354579\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.223951	−	0.974601i	$-0.571895\pi$
$827$		0			0		0.223951	+	0.974601i	$0.428105\pi$
$828$		0			0
$829$	−0.529565	−	51.5793i	−0.0183926	−	1.79142i	−0.453713	−	0.891148i	$-0.649901\pi$
$829$	−0.529565	−	51.5793i	−0.0183926	−	1.79142i	0.435320	−	0.900276i	$-0.356635\pi$
$830$		0			0
$831$	15.6536	+	3.09281i	0.543017	+	0.107289i
$832$	49.7203	+	3.57939i	1.72374	+	0.124093i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5.12240	−	13.6364i	0.177056	−	0.471344i
$838$		0			0
$839$		0			0		0.0513107	−	0.998683i	$-0.483660\pi$
$839$		0			0		−0.0513107	+	0.998683i	$0.516340\pi$
$840$		0			0
$841$	−15.0126	−	24.8117i	−0.517676	−	0.855577i
$842$		0			0
$843$		0			0
$844$	−24.0989	+	3.74084i	−0.829518	+	0.128765i
$845$		0			0
$846$		0			0
$847$	21.9518	+	49.0332i	0.754271	+	1.68480i
$848$		0			0
$849$	−52.0442	+	20.1620i	−1.78615	+	0.691959i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	24.2286	−	30.1089i	0.829570	−	1.03091i	−0.169401	−	0.985547i	$-0.554183\pi$
$853$	24.2286	−	30.1089i	0.829570	−	1.03091i	0.998971	−	0.0453623i	$-0.0144442\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$857$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$858$		0			0
$859$	−27.1279	+	19.6246i	−0.925590	+	0.669581i	−0.943893	−	0.330251i	$-0.892867\pi$
$859$	−27.1279	+	19.6246i	−0.925590	+	0.669581i	0.0183027	+	0.999832i	$0.494174\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.981035	−	0.193831i	$-0.0620915\pi$
$863$		0			0		−0.981035	+	0.193831i	$0.937908\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.3068	+	26.7496i	0.417960	+	0.908465i
$868$	−9.36546	+	25.7314i	−0.317884	+	0.873380i
$869$		0			0
$870$		0			0
$871$	−48.4040	+	54.2040i	−1.64011	+	1.83663i
$872$		0			0
$873$	−21.4781	+	50.7440i	−0.726923	+	1.71742i
$874$		0			0
$875$		0			0
$876$	−13.8944	−	17.6342i	−0.469447	−	0.595806i
$877$	56.1624	−	6.36969i	1.89647	−	0.215089i	0.914561	−	0.404449i	$-0.132537\pi$
$877$	56.1624	−	6.36969i	1.89647	−	0.215089i	0.981908	+	0.189360i	$0.0606413\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.163529	−	0.986539i	$-0.447712\pi$
$881$		0			0		−0.163529	+	0.986539i	$0.552288\pi$
$882$		0			0
$883$	−23.8969	+	31.6446i	−0.804194	+	1.06492i	0.192306	+	0.981335i	$0.438403\pi$
$883$	−23.8969	+	31.6446i	−0.804194	+	1.06492i	−0.996500	+	0.0835897i	$0.973361\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$887$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$888$		0			0
$889$	−21.7110	−	51.2944i	−0.728165	−	1.72036i
$890$		0			0
$891$		0			0
$892$	49.6842	+	28.0090i	1.66355	+	0.937812i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−29.3698	−	6.11679i	−0.978993	−	0.203893i
$901$		0			0
$902$		0			0
$903$	55.3616	+	95.8892i	1.84232	+	3.19099i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	36.7621	−	24.9063i	1.22066	−	0.827001i	0.231361	−	0.972868i	$-0.425682\pi$
$907$	36.7621	−	24.9063i	1.22066	−	0.827001i	0.989304	+	0.145868i	$0.0465973\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.999947	−	0.0102665i	$-0.996732\pi$
$911$		0			0		0.999947	+	0.0102665i	$0.00326797\pi$
$912$	0.319275	−	1.70797i	0.0105723	−	0.0565566i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	20.1967	+	17.6662i	0.667316	+	0.583707i
$917$		0			0
$918$		0			0
$919$	−30.3755	−	35.4544i	−1.00199	−	1.16953i	−0.985501	−	0.169669i	$-0.945730\pi$
$919$	−30.3755	−	35.4544i	−1.00199	−	1.16953i	−0.0164935	−	0.999864i	$-0.505250\pi$
$920$		0			0
$921$	7.04407	−	29.5192i	0.232110	−	0.972690i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−9.07118	+	58.4375i	−0.298259	+	1.92141i
$926$		0			0
$927$	46.4735	−	32.8979i	1.52639	−	1.08051i
$928$		0			0
$929$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$929$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$930$		0			0
$931$	−0.0433911	+	4.22626i	−0.00142208	+	0.138510i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40.7314	+	12.9576i	1.33064	+	0.423306i	0.882354	−	0.470586i	$-0.155957\pi$
$937$	40.7314	+	12.9576i	1.33064	+	0.423306i	0.448283	+	0.893892i	$0.352036\pi$
$938$		0			0
$939$	−28.2385	+	16.3035i	−0.921528	+	0.532045i
$940$		0			0
$941$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$941$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.890540	−	0.454905i	$-0.849673\pi$
$947$		0			0		0.890540	+	0.454905i	$0.150327\pi$
$948$	31.1246	+	8.85572i	1.01088	+	0.287620i
$949$	38.9529	+	10.6520i	1.26446	+	0.345779i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.920906	−	0.389786i	$-0.127451\pi$
$953$		0			0		−0.920906	+	0.389786i	$0.872549\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.8296	+	3.78423i	0.736438	+	0.122072i
$962$		0			0
$963$		0			0
$964$	25.7567	−	23.0006i	0.829568	−	0.740800i
$965$		0			0
$966$		0			0
$967$	−29.0571	−	24.3818i	−0.934414	−	0.784066i	0.0421907	−	0.999110i	$-0.486566\pi$
$967$	−29.0571	−	24.3818i	−0.934414	−	0.784066i	−0.976605	+	0.215043i	$0.931011\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.745886	−	0.666073i	$-0.767974\pi$
$971$		0			0		0.745886	+	0.666073i	$0.232026\pi$
$972$	−18.0132	+	25.4465i	−0.577774	+	0.816197i
$973$	111.645	−	26.8643i	3.57919	−	0.861230i
$974$		0			0
$975$	49.0236	−	22.5544i	1.57001	−	0.722320i
$976$	−0.212858	+	0.438708i	−0.00681343	+	0.0140427i
$977$		0			0		0.904126	−	0.427265i	$-0.140523\pi$
$977$		0			0		−0.904126	+	0.427265i	$0.859477\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−35.5559	−	51.3379i	−1.13521	−	1.63909i
$982$		0			0
$983$		0			0		−0.586123	−	0.810222i	$-0.699346\pi$
$983$		0			0		0.586123	+	0.810222i	$0.300654\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.36434	+	2.81196i	0.0434055	+	0.0894602i
$989$		0			0
$990$		0			0
$991$	−10.6165	−	1.31461i	−0.337245	−	0.0417599i	−0.0475208	−	0.998870i	$-0.515132\pi$
$991$	−10.6165	−	1.31461i	−0.337245	−	0.0417599i	−0.289724	+	0.957110i	$0.593563\pi$
$992$		0			0
$993$	17.0865	+	44.1053i	0.542223	+	1.39964i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.4228	−	1.19795i	0.615125	−	0.0379396i	0.248997	−	0.968504i	$-0.419899\pi$
$997$	19.4228	−	1.19795i	0.615125	−	0.0379396i	0.366128	+	0.930565i	$0.380683\pi$
$998$		0			0
$999$	51.9172	+	32.8879i	1.64259	+	1.04053i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	921.2.x.a.131.1	✓	96
3.2	odd	2	CM	921.2.x.a.131.1	✓	96
307.75	odd	306	inner	921.2.x.a.689.1	yes	96
921.689	even	306	inner	921.2.x.a.689.1	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
921.2.x.a.131.1	✓	96	1.1	even	1	trivial
921.2.x.a.131.1	✓	96	3.2	odd	2	CM
921.2.x.a.689.1	yes	96	307.75	odd	306	inner
921.2.x.a.689.1	yes	96	921.689	even	306	inner

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:	\( N \)	\(=\)	\( 921 = 3 \cdot 307 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	921.x (of order \(306\), degree \(96\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72466 - 0.159813i) q^{3} +(0.306783 - 1.97633i) q^{4} +(-0.550381 + 4.85278i) q^{7} +(2.94892 + 0.551249i) q^{9} +O(q^{10})\)	\(q+(-1.72466 - 0.159813i) q^{3} +(0.306783 - 1.97633i) q^{4} +(-0.550381 + 4.85278i) q^{7} +(2.94892 + 0.551249i) q^{9} +(-0.844942 + 3.35948i) q^{12} +(-2.36976 - 5.76291i) q^{13} +(-3.81177 - 1.21261i) q^{16} +(-0.0536532 - 0.244988i) q^{19} +(1.72476 - 8.28145i) q^{21} +(-0.665352 - 4.95553i) q^{25} +(-4.99779 - 1.42199i) q^{27} +(9.42185 + 2.57649i) q^{28} +(-0.229991 + 2.79393i) q^{31} +(1.99413 - 5.65893i) q^{36} +(-11.3421 - 3.35336i) q^{37} +(3.16605 + 10.3178i) q^{39} +(-9.76304 + 8.71835i) q^{43} +(6.38022 + 2.70052i) q^{48} +(-16.4243 - 3.77410i) q^{49} +(-12.1164 + 2.91547i) q^{52} +(0.0533813 + 0.431096i) q^{57} +(0.0174615 - 0.120648i) q^{61} +(-4.29812 + 14.0071i) q^{63} +(-3.56591 + 7.16131i) q^{64} +(-5.09098 - 10.4927i) q^{67} +(-3.95849 + 5.13148i) q^{73} +(0.355547 + 8.65295i) q^{75} +(-0.500638 + 0.0308783i) q^{76} +(0.287672 - 9.33708i) q^{79} +(8.39225 + 3.25117i) q^{81} +(-15.8377 - 5.94931i) q^{84} +(29.2704 - 8.32814i) q^{91} +(0.843165 - 4.78183i) q^{93} +(-3.92940 + 17.9422i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) \) 96 * q - 6 * q^4 + 18 * q^9	\( 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) \) 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more