LMFDB - Embedded newform 2960.2.p.g.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(961,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("2960.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.p (of order $2$, degree $1$, 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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	2960.961
Dual form			2960.2.p.g.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24914 q^{3} -1.00000i q^{5} -1.52932 q^{7} +2.05863 q^{9} -2.71982 q^{11} -0.941367i q^{13} +2.24914i q^{15} +4.83709i q^{17} -0.249141i q^{19} +3.43965 q^{21} +0.941367i q^{23} -1.00000 q^{25} +2.11727 q^{27} -0.719824i q^{29} +4.02760i q^{31} +6.11727 q^{33} +1.52932i q^{35} +(-4.71982 + 3.83709i) q^{37} +2.11727i q^{39} -8.27674 q^{41} +2.71982i q^{43} -2.05863i q^{45} +3.30777 q^{47} -4.66119 q^{49} -10.8793i q^{51} +8.39400 q^{53} +2.71982i q^{55} +0.560352i q^{57} -7.30777i q^{59} -6.83709i q^{61} -3.14830 q^{63} -0.941367 q^{65} +7.68879 q^{67} -2.11727i q^{69} +3.05863 q^{71} +9.11383 q^{73} +2.24914 q^{75} +4.15947 q^{77} -1.75086i q^{79} -10.9379 q^{81} -0.131874 q^{83} +4.83709 q^{85} +1.61899i q^{87} +8.99656i q^{89} +1.43965i q^{91} -9.05863i q^{93} -0.249141 q^{95} -16.2767i q^{97} -5.59912 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 10 q^{7} + 14 q^{9} + 2 q^{11} - 16 q^{21} - 6 q^{25} + 16 q^{27} + 40 q^{33} - 10 q^{37} + 2 q^{41} + 4 q^{47} - 8 q^{49} + 2 q^{53} - 58 q^{63} - 4 q^{65} - 8 q^{67} + 20 q^{71} - 12 q^{73}+ \cdots + 58 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 10 * q^7 + 14 * q^9 + 2 * q^11 - 16 * q^21 - 6 * q^25 + 16 * q^27 + 40 * q^33 - 10 * q^37 + 2 * q^41 + 4 * q^47 - 8 * q^49 + 2 * q^53 - 58 * q^63 - 4 * q^65 - 8 * q^67 + 20 * q^71 - 12 * q^73 - 4 * q^75 - 30 * q^77 + 6 * q^81 + 20 * q^83 + 14 * q^85 + 16 * q^95 + 58 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times$.

$n$	$741$	$1777$	$2481$	$2591$
$\chi(n)$	$1$	$1$	$-1$	$1$

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			0
$2$		0			0
$3$	−2.24914			−1.29854			−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914			−1.29854			−0.649271	+	0.760557i	$0.724926\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−1.52932			−0.578027			−0.289014	−	0.957325i	$-0.593327\pi$
$7$	−1.52932			−0.578027			−0.289014	+	0.957325i	$0.593327\pi$
$8$		0			0
$9$	2.05863			0.686211
$10$		0			0
$11$	−2.71982			−0.820058			−0.410029	−	0.912073i	$-0.634481\pi$
$11$	−2.71982			−0.820058			−0.410029	+	0.912073i	$0.634481\pi$
$12$		0			0
$13$		−	0.941367i		−	0.261088i	−0.991443	−	0.130544i	$-0.958328\pi$
$13$		−	0.941367i		−	0.261088i	0.991443	−	0.130544i	$-0.0416724\pi$
$14$		0			0
$15$			2.24914i			0.580726i
$16$		0			0
$17$			4.83709i			1.17317i	0.809889	+	0.586583i	$0.199527\pi$
$17$			4.83709i			1.17317i	−0.809889	+	0.586583i	$0.800473\pi$
$18$		0			0
$19$		−	0.249141i		−	0.0571568i	−0.999592	−	0.0285784i	$-0.990902\pi$
$19$		−	0.249141i		−	0.0571568i	0.999592	−	0.0285784i	$-0.00909802\pi$
$20$		0			0
$21$	3.43965			0.750593
$22$		0			0
$23$			0.941367i			0.196289i	0.995172	+	0.0981443i	$0.0312907\pi$
$23$			0.941367i			0.196289i	−0.995172	+	0.0981443i	$0.968709\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	2.11727			0.407468
$28$		0			0
$29$		−	0.719824i		−	0.133668i	−0.997764	−	0.0668340i	$-0.978710\pi$
$29$		−	0.719824i		−	0.133668i	0.997764	−	0.0668340i	$-0.0212898\pi$
$30$		0			0
$31$			4.02760i			0.723378i	0.932299	+	0.361689i	$0.117800\pi$
$31$			4.02760i			0.723378i	−0.932299	+	0.361689i	$0.882200\pi$
$32$		0			0
$33$	6.11727			1.06488
$34$		0			0
$35$			1.52932i			0.258502i
$36$		0			0
$37$	−4.71982	+	3.83709i	−0.775934	+	0.630814i
$37$	−4.71982	+	3.83709i	−0.775934	+	0.630814i
$38$		0			0
$39$			2.11727i			0.339034i
$40$		0			0
$41$	−8.27674			−1.29261			−0.646305	−	0.763079i	$-0.723686\pi$
$41$	−8.27674			−1.29261			−0.646305	+	0.763079i	$0.723686\pi$
$42$		0			0
$43$			2.71982i			0.414769i	0.978259	+	0.207385i	$0.0664952\pi$
$43$			2.71982i			0.414769i	−0.978259	+	0.207385i	$0.933505\pi$
$44$		0			0
$45$		−	2.05863i		−	0.306883i
$46$		0			0
$47$	3.30777			0.482488			0.241244	−	0.970464i	$-0.422445\pi$
$47$	3.30777			0.482488			0.241244	+	0.970464i	$0.422445\pi$
$48$		0			0
$49$	−4.66119			−0.665884
$50$		0			0
$51$		−	10.8793i		−	1.52341i
$52$		0			0
$53$	8.39400			1.15301			0.576503	−	0.817095i	$-0.304417\pi$
$53$	8.39400			1.15301			0.576503	+	0.817095i	$0.304417\pi$
$54$		0			0
$55$			2.71982i			0.366741i
$56$		0			0
$57$			0.560352i			0.0742204i
$58$		0			0
$59$		−	7.30777i		−	0.951391i	−0.879610	−	0.475696i	$-0.842196\pi$
$59$		−	7.30777i		−	0.951391i	0.879610	−	0.475696i	$-0.157804\pi$
$60$		0			0
$61$		−	6.83709i		−	0.875400i	−0.899121	−	0.437700i	$-0.855793\pi$
$61$		−	6.83709i		−	0.875400i	0.899121	−	0.437700i	$-0.144207\pi$
$62$		0			0
$63$	−3.14830			−0.396649
$64$		0			0
$65$	−0.941367			−0.116762
$66$		0			0
$67$	7.68879			0.939335			0.469668	−	0.882843i	$-0.344374\pi$
$67$	7.68879			0.939335			0.469668	+	0.882843i	$0.344374\pi$
$68$		0			0
$69$		−	2.11727i		−	0.254889i
$70$		0			0
$71$	3.05863			0.362993			0.181496	−	0.983392i	$-0.441906\pi$
$71$	3.05863			0.362993			0.181496	+	0.983392i	$0.441906\pi$
$72$		0			0
$73$	9.11383			1.06669			0.533346	−	0.845897i	$-0.320934\pi$
$73$	9.11383			1.06669			0.533346	+	0.845897i	$0.320934\pi$
$74$		0			0
$75$	2.24914			0.259708
$76$		0			0
$77$	4.15947			0.474016
$78$		0			0
$79$		−	1.75086i		−	0.196987i	−0.995138	−	0.0984935i	$-0.968598\pi$
$79$		−	1.75086i		−	0.196987i	0.995138	−	0.0984935i	$-0.0314024\pi$
$80$		0			0
$81$	−10.9379			−1.21533
$82$		0			0
$83$	−0.131874			−0.0144751			−0.00723754	−	0.999974i	$-0.502304\pi$
$83$	−0.131874			−0.0144751			−0.00723754	+	0.999974i	$0.502304\pi$
$84$		0			0
$85$	4.83709			0.524656
$86$		0			0
$87$			1.61899i			0.173573i
$88$		0			0
$89$			8.99656i			0.953634i	0.879003	+	0.476817i	$0.158210\pi$
$89$			8.99656i			0.953634i	−0.879003	+	0.476817i	$0.841790\pi$
$90$		0			0
$91$			1.43965i			0.150916i
$92$		0			0
$93$		−	9.05863i		−	0.939337i
$94$		0			0
$95$	−0.249141			−0.0255613
$96$		0			0
$97$		−	16.2767i		−	1.65265i	−0.563192	−	0.826326i	$-0.690427\pi$
$97$		−	16.2767i		−	1.65265i	0.563192	−	0.826326i	$-0.309573\pi$
$98$		0			0
$99$	−5.59912			−0.562733
$100$		0			0
$101$	−11.1138			−1.10587			−0.552934	−	0.833225i	$-0.686492\pi$
$101$	−11.1138			−1.10587			−0.552934	+	0.833225i	$0.686492\pi$
$102$		0			0
$103$			14.1725i			1.39645i	0.715876	+	0.698227i	$0.246027\pi$
$103$			14.1725i			1.39645i	−0.715876	+	0.698227i	$0.753973\pi$
$104$		0			0
$105$		−	3.43965i		−	0.335675i
$106$		0			0
$107$	−10.6922			−1.03366			−0.516828	−	0.856089i	$-0.672887\pi$
$107$	−10.6922			−1.03366			−0.516828	+	0.856089i	$0.672887\pi$
$108$		0			0
$109$		−	12.2767i		−	1.17590i	−0.808898	−	0.587949i	$-0.799936\pi$
$109$		−	12.2767i		−	1.17590i	0.808898	−	0.587949i	$-0.200064\pi$
$110$		0			0
$111$	10.6155	−	8.63016i	1.00758	−	0.819138i
$112$		0			0
$113$			2.95436i			0.277922i	0.990298	+	0.138961i	$0.0443763\pi$
$113$			2.95436i			0.277922i	−0.990298	+	0.138961i	$0.955624\pi$
$114$		0			0
$115$	0.941367			0.0877829
$116$		0			0
$117$		−	1.93793i		−	0.179162i
$118$		0			0
$119$		−	7.39744i		−	0.678122i
$120$		0			0
$121$	−3.60256			−0.327505
$122$		0			0
$123$	18.6155			1.67851
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−8.30434			−0.736891			−0.368445	−	0.929649i	$-0.620110\pi$
$127$	−8.30434			−0.736891			−0.368445	+	0.929649i	$0.620110\pi$
$128$		0			0
$129$		−	6.11727i		−	0.538595i
$130$		0			0
$131$		−	11.0732i		−	0.967474i	−0.875214	−	0.483737i	$-0.839279\pi$
$131$		−	11.0732i		−	0.967474i	0.875214	−	0.483737i	$-0.160721\pi$
$132$		0			0
$133$			0.381015i			0.0330382i
$134$		0			0
$135$		−	2.11727i		−	0.182225i
$136$		0			0
$137$	0.325819			0.0278366			0.0139183	−	0.999903i	$-0.495570\pi$
$137$	0.325819			0.0278366			0.0139183	+	0.999903i	$0.495570\pi$
$138$		0			0
$139$	13.3906			1.13577			0.567887	−	0.823107i	$-0.307761\pi$
$139$	13.3906			1.13577			0.567887	+	0.823107i	$0.307761\pi$
$140$		0			0
$141$	−7.43965			−0.626531
$142$		0			0
$143$			2.56035i			0.214107i
$144$		0			0
$145$	−0.719824			−0.0597781
$146$		0			0
$147$	10.4837			0.864679
$148$		0			0
$149$	8.44309			0.691685			0.345842	−	0.938293i	$-0.387593\pi$
$149$	8.44309			0.691685			0.345842	+	0.938293i	$0.387593\pi$
$150$		0			0
$151$	4.94137			0.402123			0.201061	−	0.979579i	$-0.435561\pi$
$151$	4.94137			0.402123			0.201061	+	0.979579i	$0.435561\pi$
$152$		0			0
$153$			9.95779i			0.805040i
$154$		0			0
$155$	4.02760			0.323504
$156$		0			0
$157$	19.2733			1.53818			0.769088	−	0.639142i	$-0.220711\pi$
$157$	19.2733			1.53818			0.769088	+	0.639142i	$0.220711\pi$
$158$		0			0
$159$	−18.8793			−1.49723
$160$		0			0
$161$		−	1.43965i		−	0.113460i
$162$		0			0
$163$		−	17.3354i		−	1.35781i	−0.734226	−	0.678906i	$-0.762455\pi$
$163$		−	17.3354i		−	1.35781i	0.734226	−	0.678906i	$-0.237545\pi$
$164$		0			0
$165$		−	6.11727i		−	0.476229i
$166$		0			0
$167$		−	17.0518i		−	1.31950i	−0.751483	−	0.659752i	$-0.770661\pi$
$167$		−	17.0518i		−	1.31950i	0.751483	−	0.659752i	$-0.229339\pi$
$168$		0			0
$169$	12.1138			0.931833
$170$		0			0
$171$		−	0.512889i		−	0.0392216i
$172$		0			0
$173$	10.7198			0.815013			0.407507	−	0.913202i	$-0.366398\pi$
$173$	10.7198			0.815013			0.407507	+	0.913202i	$0.366398\pi$
$174$		0			0
$175$	1.52932			0.115605
$176$		0			0
$177$			16.4362i			1.23542i
$178$		0			0
$179$		−	9.48024i		−	0.708586i	−0.935134	−	0.354293i	$-0.884722\pi$
$179$		−	9.48024i		−	0.708586i	0.935134	−	0.354293i	$-0.115278\pi$
$180$		0			0
$181$	20.9966			1.56066			0.780331	−	0.625367i	$-0.215051\pi$
$181$	20.9966			1.56066			0.780331	+	0.625367i	$0.215051\pi$
$182$		0			0
$183$			15.3776i			1.13674i
$184$		0			0
$185$	3.83709	+	4.71982i	0.282108	+	0.347008i
$186$		0			0
$187$		−	13.1560i		−	0.962064i
$188$		0			0
$189$	−3.23797			−0.235528
$190$		0			0
$191$		−	11.0862i		−	0.802172i	−0.916041	−	0.401086i	$-0.868633\pi$
$191$		−	11.0862i		−	0.802172i	0.916041	−	0.401086i	$-0.131367\pi$
$192$		0			0
$193$		−	2.23453i		−	0.160845i	−0.996761	−	0.0804226i	$-0.974373\pi$
$193$		−	2.23453i		−	0.160845i	0.996761	−	0.0804226i	$-0.0256270\pi$
$194$		0			0
$195$	2.11727			0.151621
$196$		0			0
$197$	13.4396			0.957535			0.478768	−	0.877942i	$-0.341084\pi$
$197$	13.4396			0.957535			0.478768	+	0.877942i	$0.341084\pi$
$198$		0			0
$199$		−	23.4802i		−	1.66447i	−0.554423	−	0.832235i	$-0.687061\pi$
$199$		−	23.4802i		−	1.66447i	0.554423	−	0.832235i	$-0.312939\pi$
$200$		0			0
$201$	−17.2932			−1.21977
$202$		0			0
$203$			1.10084i			0.0772637i
$204$		0			0
$205$			8.27674i			0.578072i
$206$		0			0
$207$			1.93793i			0.134695i
$208$		0			0
$209$			0.677618i			0.0468718i
$210$		0			0
$211$	19.7716			1.36113			0.680566	−	0.732687i	$-0.261734\pi$
$211$	19.7716			1.36113			0.680566	+	0.732687i	$0.261734\pi$
$212$		0			0
$213$	−6.87930			−0.471362
$214$		0			0
$215$	2.71982			0.185490
$216$		0			0
$217$		−	6.15947i		−	0.418132i
$218$		0			0
$219$	−20.4983			−1.38515
$220$		0			0
$221$	4.55348			0.306300
$222$		0			0
$223$	9.14143			0.612155			0.306078	−	0.952007i	$-0.400983\pi$
$223$	9.14143			0.612155			0.306078	+	0.952007i	$0.400983\pi$
$224$		0			0
$225$	−2.05863			−0.137242
$226$		0			0
$227$		−	18.7198i		−	1.24248i	−0.783621	−	0.621239i	$-0.786630\pi$
$227$		−	18.7198i		−	1.24248i	0.783621	−	0.621239i	$-0.213370\pi$
$228$		0			0
$229$	−25.1138			−1.65957			−0.829784	−	0.558084i	$-0.811537\pi$
$229$	−25.1138			−1.65957			−0.829784	+	0.558084i	$0.811537\pi$
$230$		0			0
$231$	−9.35524			−0.615529
$232$		0			0
$233$	20.4362			1.33882			0.669410	−	0.742893i	$-0.266547\pi$
$233$	20.4362			1.33882			0.669410	+	0.742893i	$0.266547\pi$
$234$		0			0
$235$		−	3.30777i		−	0.215775i
$236$		0			0
$237$			3.93793i			0.255796i
$238$		0			0
$239$			8.91377i			0.576584i	0.957543	+	0.288292i	$0.0930873\pi$
$239$			8.91377i			0.576584i	−0.957543	+	0.288292i	$0.906913\pi$
$240$		0			0
$241$		−	13.8827i		−	0.894265i	−0.894468	−	0.447133i	$-0.852445\pi$
$241$		−	13.8827i		−	0.894265i	0.894468	−	0.447133i	$-0.147555\pi$
$242$		0			0
$243$	18.2491			1.17068
$244$		0			0
$245$			4.66119i			0.297793i
$246$		0			0
$247$	−0.234533			−0.0149230
$248$		0			0
$249$	0.296604			0.0187965
$250$		0			0
$251$			3.80605i			0.240236i	0.992760	+	0.120118i	$0.0383273\pi$
$251$			3.80605i			0.240236i	−0.992760	+	0.120118i	$0.961673\pi$
$252$		0			0
$253$		−	2.56035i		−	0.160968i
$254$		0			0
$255$	−10.8793			−0.681288
$256$		0			0
$257$			8.78801i			0.548181i	0.961704	+	0.274090i	$0.0883768\pi$
$257$			8.78801i			0.548181i	−0.961704	+	0.274090i	$0.911623\pi$
$258$		0			0
$259$	7.21811	−	5.86813i	0.448511	−	0.364628i
$260$		0			0
$261$		−	1.48185i		−	0.0917244i
$262$		0			0
$263$	10.4707			0.645650			0.322825	−	0.946459i	$-0.395367\pi$
$263$	10.4707			0.645650			0.322825	+	0.946459i	$0.395367\pi$
$264$		0			0
$265$		−	8.39400i		−	0.515640i
$266$		0			0
$267$		−	20.2345i		−	1.23833i
$268$		0			0
$269$	−5.88273			−0.358677			−0.179338	−	0.983787i	$-0.557396\pi$
$269$	−5.88273			−0.358677			−0.179338	+	0.983787i	$0.557396\pi$
$270$		0			0
$271$	−7.61211			−0.462403			−0.231201	−	0.972906i	$-0.574266\pi$
$271$	−7.61211			−0.462403			−0.231201	+	0.972906i	$0.574266\pi$
$272$		0			0
$273$		−	3.23797i		−	0.195971i
$274$		0			0
$275$	2.71982			0.164012
$276$		0			0
$277$		−	26.4914i		−	1.59171i	−0.605484	−	0.795857i	$-0.707021\pi$
$277$		−	26.4914i		−	1.59171i	0.605484	−	0.795857i	$-0.292979\pi$
$278$		0			0
$279$			8.29135i			0.496390i
$280$		0			0
$281$		−	26.6707i		−	1.59104i	−0.605925	−	0.795522i	$-0.707197\pi$
$281$		−	26.6707i		−	1.59104i	0.605925	−	0.795522i	$-0.292803\pi$
$282$		0			0
$283$			8.73281i			0.519112i	0.965728	+	0.259556i	$0.0835762\pi$
$283$			8.73281i			0.519112i	−0.965728	+	0.259556i	$0.916424\pi$
$284$		0			0
$285$	0.560352			0.0331924
$286$		0			0
$287$	12.6578			0.747164
$288$		0			0
$289$	−6.39744			−0.376320
$290$		0			0
$291$			36.6087i			2.14604i
$292$		0			0
$293$	23.4819			1.37182			0.685912	−	0.727684i	$-0.259403\pi$
$293$	23.4819			1.37182			0.685912	+	0.727684i	$0.259403\pi$
$294$		0			0
$295$	−7.30777			−0.425475
$296$		0			0
$297$	−5.75859			−0.334147
$298$		0			0
$299$	0.886172			0.0512486
$300$		0			0
$301$		−	4.15947i		−	0.239748i
$302$		0			0
$303$	24.9966			1.43601
$304$		0			0
$305$	−6.83709			−0.391491
$306$		0			0
$307$	−27.2457			−1.55499			−0.777497	−	0.628886i	$-0.783511\pi$
$307$	−27.2457			−1.55499			−0.777497	+	0.628886i	$0.783511\pi$
$308$		0			0
$309$		−	31.8759i		−	1.81335i
$310$		0			0
$311$			17.7018i			1.00378i	0.864933	+	0.501888i	$0.167361\pi$
$311$			17.7018i			1.00378i	−0.864933	+	0.501888i	$0.832639\pi$
$312$		0			0
$313$			23.9931i			1.35617i	0.734983	+	0.678086i	$0.237190\pi$
$313$			23.9931i			1.35617i	−0.734983	+	0.678086i	$0.762810\pi$
$314$		0			0
$315$			3.14830i			0.177387i
$316$		0			0
$317$	−3.16291			−0.177647			−0.0888234	−	0.996047i	$-0.528311\pi$
$317$	−3.16291			−0.177647			−0.0888234	+	0.996047i	$0.528311\pi$
$318$		0			0
$319$			1.95779i			0.109615i
$320$		0			0
$321$	24.0483			1.34225
$322$		0			0
$323$	1.20512			0.0670544
$324$		0			0
$325$			0.941367i			0.0522176i
$326$		0			0
$327$			27.6121i			1.52695i
$328$		0			0
$329$	−5.05863			−0.278891
$330$		0			0
$331$			4.48367i			0.246445i	0.992379	+	0.123222i	$0.0393229\pi$
$331$			4.48367i			0.246445i	−0.992379	+	0.123222i	$0.960677\pi$
$332$		0			0
$333$	−9.71639	+	7.89916i	−0.532455	+	0.432871i
$334$		0			0
$335$		−	7.68879i		−	0.420083i
$336$		0			0
$337$	−11.2051			−0.610382			−0.305191	−	0.952291i	$-0.598720\pi$
$337$	−11.2051			−0.610382			−0.305191	+	0.952291i	$0.598720\pi$
$338$		0			0
$339$		−	6.64476i		−	0.360894i
$340$		0			0
$341$		−	10.9544i		−	0.593212i
$342$		0			0
$343$	17.8337			0.962927
$344$		0			0
$345$	−2.11727			−0.113990
$346$		0			0
$347$			13.9931i			0.751190i	0.926784	+	0.375595i	$0.122562\pi$
$347$			13.9931i			0.751190i	−0.926784	+	0.375595i	$0.877438\pi$
$348$		0			0
$349$	−6.56035			−0.351168			−0.175584	−	0.984464i	$-0.556181\pi$
$349$	−6.56035			−0.351168			−0.175584	+	0.984464i	$0.556181\pi$
$350$		0			0
$351$		−	1.99312i		−	0.106385i
$352$		0			0
$353$			36.5957i			1.94779i	0.226995	+	0.973896i	$0.427110\pi$
$353$			36.5957i			1.94779i	−0.226995	+	0.973896i	$0.572890\pi$
$354$		0			0
$355$		−	3.05863i		−	0.162335i
$356$		0			0
$357$			16.6379i			0.880570i
$358$		0			0
$359$	−21.2311			−1.12053			−0.560267	−	0.828312i	$-0.689302\pi$
$359$	−21.2311			−1.12053			−0.560267	+	0.828312i	$0.689302\pi$
$360$		0			0
$361$	18.9379			0.996733
$362$		0			0
$363$	8.10266			0.425279
$364$		0			0
$365$		−	9.11383i		−	0.477040i
$366$		0			0
$367$	−23.4932			−1.22634			−0.613168	−	0.789952i	$-0.710105\pi$
$367$	−23.4932			−1.22634			−0.613168	+	0.789952i	$0.710105\pi$
$368$		0			0
$369$	−17.0388			−0.887003
$370$		0			0
$371$	−12.8371			−0.666469
$372$		0			0
$373$	−7.55691			−0.391282			−0.195641	−	0.980676i	$-0.562679\pi$
$373$	−7.55691			−0.391282			−0.195641	+	0.980676i	$0.562679\pi$
$374$		0			0
$375$		−	2.24914i		−	0.116145i
$376$		0			0
$377$	−0.677618			−0.0348991
$378$		0			0
$379$	−12.2637			−0.629946			−0.314973	−	0.949101i	$-0.601996\pi$
$379$	−12.2637			−0.629946			−0.314973	+	0.949101i	$0.601996\pi$
$380$		0			0
$381$	18.6776			0.956883
$382$		0			0
$383$			33.5208i			1.71283i	0.516285	+	0.856417i	$0.327315\pi$
$383$			33.5208i			1.71283i	−0.516285	+	0.856417i	$0.672685\pi$
$384$		0			0
$385$		−	4.15947i		−	0.211986i
$386$		0			0
$387$			5.59912i			0.284619i
$388$		0			0
$389$		−	9.95092i		−	0.504532i	−0.967658	−	0.252266i	$-0.918824\pi$
$389$		−	9.95092i		−	0.504532i	0.967658	−	0.252266i	$-0.0811757\pi$
$390$		0			0
$391$	−4.55348			−0.230279
$392$		0			0
$393$			24.9053i			1.25630i
$394$		0			0
$395$	−1.75086			−0.0880953
$396$		0			0
$397$	8.67762			0.435517			0.217759	−	0.976003i	$-0.430125\pi$
$397$	8.67762			0.435517			0.217759	+	0.976003i	$0.430125\pi$
$398$		0			0
$399$		−	0.856956i		−	0.0429014i
$400$		0			0
$401$			11.5569i			0.577125i	0.957461	+	0.288562i	$0.0931773\pi$
$401$			11.5569i			0.577125i	−0.957461	+	0.288562i	$0.906823\pi$
$402$		0			0
$403$	3.79145			0.188865
$404$		0			0
$405$			10.9379i			0.543510i
$406$		0			0
$407$	12.8371	−	10.4362i	0.636311	−	0.517304i
$408$		0			0
$409$		−	3.34836i		−	0.165566i	−0.996568	−	0.0827829i	$-0.973619\pi$
$409$		−	3.34836i		−	0.165566i	0.996568	−	0.0827829i	$-0.0263808\pi$
$410$		0			0
$411$	−0.732814			−0.0361470
$412$		0			0
$413$			11.1759i			0.549930i
$414$		0			0
$415$			0.131874i			0.00647345i
$416$		0			0
$417$	−30.1173			−1.47485
$418$		0			0
$419$	11.3776			0.555831			0.277916	−	0.960606i	$-0.410356\pi$
$419$	11.3776			0.555831			0.277916	+	0.960606i	$0.410356\pi$
$420$		0			0
$421$			3.23109i			0.157474i	0.996895	+	0.0787370i	$0.0250887\pi$
$421$			3.23109i			0.157474i	−0.996895	+	0.0787370i	$0.974911\pi$
$422$		0			0
$423$	6.80949			0.331089
$424$		0			0
$425$		−	4.83709i		−	0.234633i
$426$		0			0
$427$			10.4561i			0.506005i
$428$		0			0
$429$		−	5.75859i		−	0.278027i
$430$		0			0
$431$			29.2327i			1.40809i	0.710155	+	0.704045i	$0.248625\pi$
$431$			29.2327i			1.40809i	−0.710155	+	0.704045i	$0.751375\pi$
$432$		0			0
$433$	17.7655			0.853754			0.426877	−	0.904310i	$-0.359614\pi$
$433$	17.7655			0.853754			0.426877	+	0.904310i	$0.359614\pi$
$434$		0			0
$435$	1.61899			0.0776244
$436$		0			0
$437$	0.234533			0.0112192
$438$		0			0
$439$		−	32.7604i		−	1.56357i	−0.623549	−	0.781785i	$-0.714310\pi$
$439$		−	32.7604i		−	1.56357i	0.623549	−	0.781785i	$-0.285690\pi$
$440$		0			0
$441$	−9.59568			−0.456937
$442$		0			0
$443$	0.600939			0.0285515			0.0142757	−	0.999898i	$-0.495456\pi$
$443$	0.600939			0.0285515			0.0142757	+	0.999898i	$0.495456\pi$
$444$		0			0
$445$	8.99656			0.426478
$446$		0			0
$447$	−18.9897			−0.898181
$448$		0			0
$449$		−	34.0191i		−	1.60546i	−0.596342	−	0.802730i	$-0.703380\pi$
$449$		−	34.0191i		−	1.60546i	0.596342	−	0.802730i	$-0.296620\pi$
$450$		0			0
$451$	22.5113			1.06001
$452$		0			0
$453$	−11.1138			−0.522173
$454$		0			0
$455$	1.43965			0.0674917
$456$		0			0
$457$		−	30.9215i		−	1.44645i	−0.690614	−	0.723223i	$-0.742660\pi$
$457$		−	30.9215i		−	1.44645i	0.690614	−	0.723223i	$-0.257340\pi$
$458$		0			0
$459$			10.2414i			0.478028i
$460$		0			0
$461$			1.27330i			0.0593035i	0.999560	+	0.0296518i	$0.00943983\pi$
$461$			1.27330i			0.0593035i	−0.999560	+	0.0296518i	$0.990560\pi$
$462$		0			0
$463$		−	2.61555i		−	0.121555i	−0.998151	−	0.0607774i	$-0.980642\pi$
$463$		−	2.61555i		−	0.121555i	0.998151	−	0.0607774i	$-0.0193580\pi$
$464$		0			0
$465$	−9.05863			−0.420084
$466$		0			0
$467$		−	19.0096i		−	0.879657i	−0.898082	−	0.439829i	$-0.855039\pi$
$467$		−	19.0096i		−	0.879657i	0.898082	−	0.439829i	$-0.144961\pi$
$468$		0			0
$469$	−11.7586			−0.542961
$470$		0			0
$471$	−43.3484			−1.99739
$472$		0			0
$473$		−	7.39744i		−	0.340135i
$474$		0			0
$475$			0.249141i			0.0114314i
$476$		0			0
$477$	17.2802			0.791205
$478$		0			0
$479$			38.8578i			1.77546i	0.460366	+	0.887729i	$0.347718\pi$
$479$			38.8578i			1.77546i	−0.460366	+	0.887729i	$0.652282\pi$
$480$		0			0
$481$	3.61211	+	4.44309i	0.164698	+	0.202587i
$482$		0			0
$483$			3.23797i			0.147333i
$484$		0			0
$485$	−16.2767			−0.739089
$486$		0			0
$487$			38.4914i			1.74421i	0.489317	+	0.872106i	$0.337246\pi$
$487$			38.4914i			1.74421i	−0.489317	+	0.872106i	$0.662754\pi$
$488$		0			0
$489$			38.9897i			1.76317i
$490$		0			0
$491$	11.2672			0.508481			0.254240	−	0.967141i	$-0.418175\pi$
$491$	11.2672			0.508481			0.254240	+	0.967141i	$0.418175\pi$
$492$		0			0
$493$	3.48185			0.156815
$494$		0			0
$495$			5.59912i			0.251662i
$496$		0			0
$497$	−4.67762			−0.209820
$498$		0			0
$499$		−	20.8578i		−	0.933724i	−0.884330	−	0.466862i	$-0.845384\pi$
$499$		−	20.8578i		−	0.933724i	0.884330	−	0.466862i	$-0.154616\pi$
$500$		0			0
$501$			38.3518i			1.71343i
$502$		0			0
$503$			6.64476i			0.296275i	0.988967	+	0.148138i	$0.0473279\pi$
$503$			6.64476i			0.296275i	−0.988967	+	0.148138i	$0.952672\pi$
$504$		0			0
$505$			11.1138i			0.494559i
$506$		0			0
$507$	−27.2457			−1.21002
$508$		0			0
$509$	−18.5535			−0.822368			−0.411184	−	0.911552i	$-0.634885\pi$
$509$	−18.5535			−0.822368			−0.411184	+	0.911552i	$0.634885\pi$
$510$		0			0
$511$	−13.9379			−0.616578
$512$		0			0
$513$		−	0.527497i		−	0.0232896i
$514$		0			0
$515$	14.1725			0.624513
$516$		0			0
$517$	−8.99656			−0.395668
$518$		0			0
$519$	−24.1104			−1.05833
$520$		0			0
$521$	−42.6509			−1.86857			−0.934284	−	0.356529i	$-0.883960\pi$
$521$	−42.6509			−1.86857			−0.934284	+	0.356529i	$0.883960\pi$
$522$		0			0
$523$		−	12.3879i		−	0.541685i	−0.962624	−	0.270842i	$-0.912698\pi$
$523$		−	12.3879i		−	0.541685i	0.962624	−	0.270842i	$-0.0873022\pi$
$524$		0			0
$525$	−3.43965			−0.150119
$526$		0			0
$527$	−19.4819			−0.848643
$528$		0			0
$529$	22.1138			0.961471
$530$		0			0
$531$		−	15.0440i		−	0.652855i
$532$		0			0
$533$			7.79145i			0.337485i
$534$		0			0
$535$			10.6922i			0.462265i
$536$		0			0
$537$			21.3224i			0.920129i
$538$		0			0
$539$	12.6776			0.546064
$540$		0			0
$541$		−	7.64820i		−	0.328822i	−0.986392	−	0.164411i	$-0.947428\pi$
$541$		−	7.64820i		−	0.328822i	0.986392	−	0.164411i	$-0.0525723\pi$
$542$		0			0
$543$	−47.2242			−2.02659
$544$		0			0
$545$	−12.2767			−0.525878
$546$		0			0
$547$		−	12.6837i		−	0.542317i	−0.962535	−	0.271159i	$-0.912593\pi$
$547$		−	12.6837i		−	0.542317i	0.962535	−	0.271159i	$-0.0874068\pi$
$548$		0			0
$549$		−	14.0751i		−	0.600709i
$550$		0			0
$551$	−0.179337			−0.00764003
$552$		0			0
$553$			2.67762i			0.113864i
$554$		0			0
$555$	−8.63016	−	10.6155i	−0.366330	−	0.450605i
$556$		0			0
$557$			2.61555i			0.110824i	0.998464	+	0.0554121i	$0.0176473\pi$
$557$			2.61555i			0.110824i	−0.998464	+	0.0554121i	$0.982353\pi$
$558$		0			0
$559$	2.56035			0.108291
$560$		0			0
$561$			29.5898i			1.24928i
$562$		0			0
$563$		−	35.3285i		−	1.48892i	−0.667668	−	0.744459i	$-0.732707\pi$
$563$		−	35.3285i		−	1.48892i	0.667668	−	0.744459i	$-0.267293\pi$
$564$		0			0
$565$	2.95436			0.124291
$566$		0			0
$567$	16.7276			0.702491
$568$		0			0
$569$			28.8724i			1.21039i	0.796075	+	0.605197i	$0.206906\pi$
$569$			28.8724i			1.21039i	−0.796075	+	0.605197i	$0.793094\pi$
$570$		0			0
$571$	8.15947			0.341463			0.170732	−	0.985318i	$-0.445387\pi$
$571$	8.15947			0.341463			0.170732	+	0.985318i	$0.445387\pi$
$572$		0			0
$573$			24.9345i			1.04165i
$574$		0			0
$575$		−	0.941367i		−	0.0392577i
$576$		0			0
$577$		−	37.5500i		−	1.56323i	−0.623762	−	0.781614i	$-0.714397\pi$
$577$		−	37.5500i		−	1.56323i	0.623762	−	0.781614i	$-0.285603\pi$
$578$		0			0
$579$			5.02578i			0.208864i
$580$		0			0
$581$	0.201677			0.00836699
$582$		0			0
$583$	−22.8302			−0.945531
$584$		0			0
$585$	−1.93793			−0.0801235
$586$		0			0
$587$			19.5078i			0.805174i	0.915382	+	0.402587i	$0.131889\pi$
$587$			19.5078i			0.805174i	−0.915382	+	0.402587i	$0.868111\pi$
$588$		0			0
$589$	1.00344			0.0413459
$590$		0			0
$591$	−30.2277			−1.24340
$592$		0			0
$593$	41.5569			1.70654			0.853269	−	0.521471i	$-0.174617\pi$
$593$	41.5569			1.70654			0.853269	+	0.521471i	$0.174617\pi$
$594$		0			0
$595$	−7.39744			−0.303266
$596$		0			0
$597$			52.8103i			2.16138i
$598$		0			0
$599$	−27.6381			−1.12926			−0.564631	−	0.825344i	$-0.690981\pi$
$599$	−27.6381			−1.12926			−0.564631	+	0.825344i	$0.690981\pi$
$600$		0			0
$601$	14.4301			0.588616			0.294308	−	0.955711i	$-0.404911\pi$
$601$	14.4301			0.588616			0.294308	+	0.955711i	$0.404911\pi$
$602$		0			0
$603$	15.8284			0.644582
$604$		0			0
$605$			3.60256i			0.146465i
$606$		0			0
$607$			47.4880i			1.92748i	0.266848	+	0.963739i	$0.414018\pi$
$607$			47.4880i			1.92748i	−0.266848	+	0.963739i	$0.585982\pi$
$608$		0			0
$609$		−	2.47594i		−	0.100330i
$610$		0			0
$611$		−	3.11383i		−	0.125972i
$612$		0			0
$613$	−18.7198			−0.756087			−0.378043	−	0.925788i	$-0.623403\pi$
$613$	−18.7198			−0.756087			−0.378043	+	0.925788i	$0.623403\pi$
$614$		0			0
$615$		−	18.6155i		−	0.750651i
$616$		0			0
$617$	−15.0225			−0.604785			−0.302392	−	0.953184i	$-0.597785\pi$
$617$	−15.0225			−0.604785			−0.302392	+	0.953184i	$0.597785\pi$
$618$		0			0
$619$	−16.6318			−0.668487			−0.334244	−	0.942487i	$-0.608481\pi$
$619$	−16.6318			−0.668487			−0.334244	+	0.942487i	$0.608481\pi$
$620$		0			0
$621$			1.99312i			0.0799813i
$622$		0			0
$623$		−	13.7586i		−	0.551226i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	1.52406i		−	0.0608651i
$628$		0			0
$629$	−18.5604	−	22.8302i	−0.740050	−	0.910300i
$630$		0			0
$631$		−	35.9294i		−	1.43033i	−0.698957	−	0.715164i	$-0.746352\pi$
$631$		−	35.9294i		−	1.43033i	0.698957	−	0.715164i	$-0.253648\pi$
$632$		0			0
$633$	−44.4691			−1.76749
$634$		0			0
$635$			8.30434i			0.329548i
$636$		0			0
$637$			4.38789i			0.173855i
$638$		0			0
$639$	6.29660			0.249090
$640$		0			0
$641$	29.3354			1.15868			0.579339	−	0.815087i	$-0.303311\pi$
$641$	29.3354			1.15868			0.579339	+	0.815087i	$0.303311\pi$
$642$		0			0
$643$		−	0.368025i		−	0.0145135i	−0.999974	−	0.00725674i	$-0.997690\pi$
$643$		−	0.368025i		−	0.0145135i	0.999974	−	0.00725674i	$-0.00230991\pi$
$644$		0			0
$645$	−6.11727			−0.240867
$646$		0			0
$647$			26.7811i			1.05287i	0.850214	+	0.526437i	$0.176473\pi$
$647$			26.7811i			1.05287i	−0.850214	+	0.526437i	$0.823527\pi$
$648$		0			0
$649$			19.8759i			0.780196i
$650$		0			0
$651$			13.8535i			0.542962i
$652$		0			0
$653$		−	2.00000i		−	0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$		−	2.00000i		−	0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$		0			0
$655$	−11.0732			−0.432667
$656$		0			0
$657$	18.7620			0.731976
$658$		0			0
$659$	12.9966			0.506274			0.253137	−	0.967430i	$-0.418538\pi$
$659$	12.9966			0.506274			0.253137	+	0.967430i	$0.418538\pi$
$660$		0			0
$661$			18.6026i			0.723556i	0.932264	+	0.361778i	$0.117830\pi$
$661$			18.6026i			0.723556i	−0.932264	+	0.361778i	$0.882170\pi$
$662$		0			0
$663$	−10.2414			−0.397743
$664$		0			0
$665$	0.381015			0.0147751
$666$		0			0
$667$	0.677618			0.0262375
$668$		0			0
$669$	−20.5604			−0.794909
$670$		0			0
$671$			18.5957i			0.717878i
$672$		0			0
$673$	20.4622			0.788759			0.394380	−	0.918948i	$-0.370959\pi$
$673$	20.4622			0.788759			0.394380	+	0.918948i	$0.370959\pi$
$674$		0			0
$675$	−2.11727			−0.0814936
$676$		0			0
$677$	−39.9018			−1.53355			−0.766776	−	0.641915i	$-0.778140\pi$
$677$	−39.9018			−1.53355			−0.766776	+	0.641915i	$0.778140\pi$
$678$		0			0
$679$			24.8923i			0.955278i
$680$		0			0
$681$			42.1035i			1.61341i
$682$		0			0
$683$			12.3940i			0.474243i	0.971480	+	0.237122i	$0.0762040\pi$
$683$			12.3940i			0.474243i	−0.971480	+	0.237122i	$0.923796\pi$
$684$		0			0
$685$		−	0.325819i		−	0.0124489i
$686$		0			0
$687$	56.4845			2.15502
$688$		0			0
$689$		−	7.90184i		−	0.301036i
$690$		0			0
$691$	51.5630			1.96155			0.980775	−	0.195142i	$-0.0625167\pi$
$691$	51.5630			1.96155			0.980775	+	0.195142i	$0.0625167\pi$
$692$		0			0
$693$	8.56283			0.325275
$694$		0			0
$695$		−	13.3906i		−	0.507933i
$696$		0			0
$697$		−	40.0353i		−	1.51645i
$698$		0			0
$699$	−45.9639			−1.73851
$700$		0			0
$701$			9.76547i			0.368837i	0.982848	+	0.184418i	$0.0590401\pi$
$701$			9.76547i			0.368837i	−0.982848	+	0.184418i	$0.940960\pi$
$702$		0			0
$703$	0.955975	+	1.17590i	0.0360553	+	0.0443499i
$704$		0			0
$705$			7.43965i			0.280193i
$706$		0			0
$707$	16.9966			0.639222
$708$		0			0
$709$			49.8268i			1.87128i	0.352951	+	0.935642i	$0.385178\pi$
$709$			49.8268i			1.87128i	−0.352951	+	0.935642i	$0.614822\pi$
$710$		0			0
$711$		−	3.60438i		−	0.135175i
$712$		0			0
$713$	−3.79145			−0.141991
$714$		0			0
$715$	2.56035			0.0957517
$716$		0			0
$717$		−	20.0483i		−	0.748718i
$718$		0			0
$719$	4.26375			0.159011			0.0795055	−	0.996834i	$-0.474666\pi$
$719$	4.26375			0.159011			0.0795055	+	0.996834i	$0.474666\pi$
$720$		0			0
$721$		−	21.6742i		−	0.807189i
$722$		0			0
$723$			31.2242i			1.16124i
$724$		0			0
$725$			0.719824i			0.0267336i
$726$		0			0
$727$		−	13.1759i		−	0.488667i	−0.969691	−	0.244334i	$-0.921431\pi$
$727$		−	13.1759i		−	0.488667i	0.969691	−	0.244334i	$-0.0785692\pi$
$728$		0			0
$729$	−8.23109			−0.304855
$730$		0			0
$731$	−13.1560			−0.486593
$732$		0			0
$733$	9.39057			0.346848			0.173424	−	0.984847i	$-0.444517\pi$
$733$	9.39057			0.346848			0.173424	+	0.984847i	$0.444517\pi$
$734$		0			0
$735$		−	10.4837i		−	0.386696i
$736$		0			0
$737$	−20.9122			−0.770309
$738$		0			0
$739$	9.62510			0.354065			0.177033	−	0.984205i	$-0.443350\pi$
$739$	9.62510			0.354065			0.177033	+	0.984205i	$0.443350\pi$
$740$		0			0
$741$	0.527497			0.0193781
$742$		0			0
$743$	25.6137			0.939677			0.469838	−	0.882753i	$-0.344312\pi$
$743$	25.6137			0.939677			0.469838	+	0.882753i	$0.344312\pi$
$744$		0			0
$745$		−	8.44309i		−	0.309331i
$746$		0			0
$747$	−0.271481			−0.00993296
$748$		0			0
$749$	16.3518			0.597482
$750$		0			0
$751$	−11.9740			−0.436938			−0.218469	−	0.975844i	$-0.570106\pi$
$751$	−11.9740			−0.436938			−0.218469	+	0.975844i	$0.570106\pi$
$752$		0			0
$753$		−	8.56035i		−	0.311957i
$754$		0			0
$755$		−	4.94137i		−	0.179835i
$756$		0			0
$757$			19.2051i			0.698022i	0.937119	+	0.349011i	$0.113482\pi$
$757$			19.2051i			0.698022i	−0.937119	+	0.349011i	$0.886518\pi$
$758$		0			0
$759$			5.75859i			0.209024i
$760$		0			0
$761$	−31.8596			−1.15491			−0.577455	−	0.816422i	$-0.695954\pi$
$761$	−31.8596			−1.15491			−0.577455	+	0.816422i	$0.695954\pi$
$762$		0			0
$763$			18.7750i			0.679701i
$764$		0			0
$765$	9.95779			0.360025
$766$		0			0
$767$	−6.87930			−0.248397
$768$		0			0
$769$		−	35.2242i		−	1.27022i	−0.772423	−	0.635109i	$-0.780955\pi$
$769$		−	35.2242i		−	1.27022i	0.772423	−	0.635109i	$-0.219045\pi$
$770$		0			0
$771$		−	19.7655i		−	0.711836i
$772$		0			0
$773$	−50.2630			−1.80783			−0.903917	−	0.427708i	$-0.859321\pi$
$773$	−50.2630			−1.80783			−0.903917	+	0.427708i	$0.859321\pi$
$774$		0			0
$775$		−	4.02760i		−	0.144676i
$776$		0			0
$777$	−16.2345	+	13.1982i	−0.582411	+	0.473484i
$778$		0			0
$779$			2.06207i			0.0738814i
$780$		0			0
$781$	−8.31894			−0.297675
$782$		0			0
$783$		−	1.52406i		−	0.0544654i
$784$		0			0
$785$		−	19.2733i		−	0.687894i
$786$		0			0
$787$	51.9525			1.85191			0.925954	−	0.377636i	$-0.123263\pi$
$787$	51.9525			1.85191			0.925954	+	0.377636i	$0.123263\pi$
$788$		0			0
$789$	−23.5500			−0.838404
$790$		0			0
$791$		−	4.51815i		−	0.160647i
$792$		0			0
$793$	−6.43621			−0.228557
$794$		0			0
$795$			18.8793i			0.669580i
$796$		0			0
$797$			35.4880i			1.25705i	0.777790	+	0.628524i	$0.216341\pi$
$797$			35.4880i			1.25705i	−0.777790	+	0.628524i	$0.783659\pi$
$798$		0			0
$799$			16.0000i			0.566039i
$800$		0			0
$801$			18.5206i			0.654394i
$802$		0			0
$803$	−24.7880			−0.874750
$804$		0			0
$805$	−1.43965			−0.0507409
$806$		0			0
$807$	13.2311			0.465757
$808$		0			0
$809$			13.2051i			0.464267i	0.972684	+	0.232134i	$0.0745706\pi$
$809$			13.2051i			0.464267i	−0.972684	+	0.232134i	$0.925429\pi$
$810$		0			0
$811$	10.1173			0.355265			0.177633	−	0.984097i	$-0.443156\pi$
$811$	10.1173			0.355265			0.177633	+	0.984097i	$0.443156\pi$
$812$		0			0
$813$	17.1207			0.600449
$814$		0			0
$815$	−17.3354			−0.607232
$816$		0			0
$817$	0.677618			0.0237069
$818$		0			0
$819$			2.96371i			0.103560i
$820$		0			0
$821$	5.68106			0.198270			0.0991351	−	0.995074i	$-0.468392\pi$
$821$	5.68106			0.198270			0.0991351	+	0.995074i	$0.468392\pi$
$822$		0			0
$823$	22.3404			0.778738			0.389369	−	0.921082i	$-0.372693\pi$
$823$	22.3404			0.778738			0.389369	+	0.921082i	$0.372693\pi$
$824$		0			0
$825$	−6.11727			−0.212976
$826$		0			0
$827$		−	3.71639i		−	0.129231i	−0.997910	−	0.0646157i	$-0.979418\pi$
$827$		−	3.71639i		−	0.129231i	0.997910	−	0.0646157i	$-0.0205822\pi$
$828$		0			0
$829$			15.8596i			0.550828i	0.961326	+	0.275414i	$0.0888149\pi$
$829$			15.8596i			0.550828i	−0.961326	+	0.275414i	$0.911185\pi$
$830$		0			0
$831$			59.5829i			2.06691i
$832$		0			0
$833$		−	22.5466i		−	0.781193i
$834$		0			0
$835$	−17.0518			−0.590100
$836$		0			0
$837$			8.52750i			0.294753i
$838$		0			0
$839$	33.8950			1.17018			0.585092	−	0.810967i	$-0.301058\pi$
$839$	33.8950			1.17018			0.585092	+	0.810967i	$0.301058\pi$
$840$		0			0
$841$	28.4819			0.982133
$842$		0			0
$843$			59.9862i			2.06604i
$844$		0			0
$845$		−	12.1138i		−	0.416728i
$846$		0			0
$847$	5.50945			0.189307
$848$		0			0
$849$		−	19.6413i		−	0.674089i
$850$		0			0
$851$	−3.61211	−	4.44309i	−0.123822	−	0.152307i
$852$		0			0
$853$		−	35.7586i		−	1.22435i	−0.790722	−	0.612175i	$-0.790295\pi$
$853$		−	35.7586i		−	1.22435i	0.790722	−	0.612175i	$-0.209705\pi$
$854$		0			0
$855$	−0.512889			−0.0175404
$856$		0			0
$857$		−	17.3837i		−	0.593816i	−0.954906	−	0.296908i	$-0.904045\pi$
$857$		−	17.3837i		−	0.593816i	0.954906	−	0.296908i	$-0.0959554\pi$
$858$		0			0
$859$		−	43.3922i		−	1.48052i	−0.672319	−	0.740261i	$-0.734702\pi$
$859$		−	43.3922i		−	1.48052i	0.672319	−	0.740261i	$-0.265298\pi$
$860$		0			0
$861$	−28.4691			−0.970223
$862$		0			0
$863$	−34.8708			−1.18702			−0.593508	−	0.804828i	$-0.702257\pi$
$863$	−34.8708			−1.18702			−0.593508	+	0.804828i	$0.702257\pi$
$864$		0			0
$865$		−	10.7198i		−	0.364485i
$866$		0			0
$867$	14.3887			0.488667
$868$		0			0
$869$			4.76203i			0.161541i
$870$		0			0
$871$		−	7.23797i		−	0.245249i
$872$		0			0
$873$		−	33.5078i		−	1.13407i
$874$		0			0
$875$		−	1.52932i		−	0.0517003i
$876$		0			0
$877$	47.0907			1.59014			0.795070	−	0.606517i	$-0.207434\pi$
$877$	47.0907			1.59014			0.795070	+	0.606517i	$0.207434\pi$
$878$		0			0
$879$	−52.8140			−1.78137
$880$		0			0
$881$	29.0027			0.977125			0.488562	−	0.872529i	$-0.337521\pi$
$881$	29.0027			0.977125			0.488562	+	0.872529i	$0.337521\pi$
$882$		0			0
$883$		−	11.0388i		−	0.371484i	−0.982599	−	0.185742i	$-0.940531\pi$
$883$		−	11.0388i		−	0.371484i	0.982599	−	0.185742i	$-0.0594689\pi$
$884$		0			0
$885$	16.4362			0.552497
$886$		0			0
$887$	5.76709			0.193640			0.0968199	−	0.995302i	$-0.469133\pi$
$887$	5.76709			0.193640			0.0968199	+	0.995302i	$0.469133\pi$
$888$		0			0
$889$	12.7000			0.425943
$890$		0			0
$891$	29.7492			0.996637
$892$		0			0
$893$		−	0.824101i		−	0.0275775i
$894$		0			0
$895$	−9.48024			−0.316889
$896$		0			0
$897$	−1.99312			−0.0665485
$898$		0			0
$899$	2.89916			0.0966924
$900$		0			0
$901$			40.6026i			1.35267i
$902$		0			0
$903$			9.35524i			0.311323i
$904$		0			0
$905$		−	20.9966i		−	0.697949i
$906$		0			0
$907$		−	54.2139i		−	1.80014i	−0.435742	−	0.900072i	$-0.643514\pi$
$907$		−	54.2139i		−	1.80014i	0.435742	−	0.900072i	$-0.356486\pi$
$908$		0			0
$909$	−22.8793			−0.758858
$910$		0			0
$911$		−	37.1284i		−	1.23012i	−0.788480	−	0.615060i	$-0.789132\pi$
$911$		−	37.1284i		−	1.23012i	0.788480	−	0.615060i	$-0.210868\pi$
$912$		0			0
$913$	0.358675			0.0118704
$914$		0			0
$915$	15.3776			0.508367
$916$		0			0
$917$			16.9345i			0.559226i
$918$		0			0
$919$		−	20.0889i		−	0.662672i	−0.943513	−	0.331336i	$-0.892501\pi$
$919$		−	20.0889i		−	0.662672i	0.943513	−	0.331336i	$-0.107499\pi$
$920$		0			0
$921$	61.2794			2.01923
$922$		0			0
$923$		−	2.87930i		−	0.0947732i
$924$		0			0
$925$	4.71982	−	3.83709i	0.155187	−	0.126163i
$926$		0			0
$927$			29.1759i			0.958262i
$928$		0			0
$929$	12.9803			0.425871			0.212936	−	0.977066i	$-0.431698\pi$
$929$	12.9803			0.425871			0.212936	+	0.977066i	$0.431698\pi$
$930$		0			0
$931$			1.16129i			0.0380598i
$932$		0			0
$933$		−	39.8138i		−	1.30344i
$934$		0			0
$935$	−13.1560			−0.430248
$936$		0			0
$937$	−9.32238			−0.304549			−0.152274	−	0.988338i	$-0.548660\pi$
$937$	−9.32238			−0.304549			−0.152274	+	0.988338i	$0.548660\pi$
$938$		0			0
$939$		−	53.9639i		−	1.76105i
$940$		0			0
$941$	−17.7586			−0.578914			−0.289457	−	0.957191i	$-0.593475\pi$
$941$	−17.7586			−0.578914			−0.289457	+	0.957191i	$0.593475\pi$
$942$		0			0
$943$		−	7.79145i		−	0.253724i
$944$		0			0
$945$			3.23797i			0.105331i
$946$		0			0
$947$		−	17.3354i		−	0.563324i	−0.959514	−	0.281662i	$-0.909114\pi$
$947$		−	17.3354i		−	0.563324i	0.959514	−	0.281662i	$-0.0908857\pi$
$948$		0			0
$949$		−	8.57946i		−	0.278501i
$950$		0			0
$951$	7.11383			0.230682
$952$		0			0
$953$	−5.88961			−0.190783			−0.0953916	−	0.995440i	$-0.530410\pi$
$953$	−5.88961			−0.190783			−0.0953916	+	0.995440i	$0.530410\pi$
$954$		0			0
$955$	−11.0862			−0.358742
$956$		0			0
$957$		−	4.40335i		−	0.142340i
$958$		0			0
$959$	−0.498281			−0.0160903
$960$		0			0
$961$	14.7785			0.476724
$962$		0			0
$963$	−22.0114			−0.709307
$964$		0			0
$965$	−2.23453			−0.0719322
$966$		0			0
$967$		−	32.3189i		−	1.03931i	−0.854377	−	0.519654i	$-0.826061\pi$
$967$		−	32.3189i		−	1.03931i	0.854377	−	0.519654i	$-0.173939\pi$
$968$		0			0
$969$	−2.71047			−0.0870730
$970$		0			0
$971$	−6.11115			−0.196116			−0.0980581	−	0.995181i	$-0.531263\pi$
$971$	−6.11115			−0.196116			−0.0980581	+	0.995181i	$0.531263\pi$
$972$		0			0
$973$	−20.4784			−0.656508
$974$		0			0
$975$		−	2.11727i		−	0.0678068i
$976$		0			0
$977$		−	47.3906i		−	1.51616i	−0.652162	−	0.758079i	$-0.726138\pi$
$977$		−	47.3906i		−	1.51616i	0.652162	−	0.758079i	$-0.273862\pi$
$978$		0			0
$979$		−	24.4691i		−	0.782035i
$980$		0			0
$981$		−	25.2733i		−	0.806914i
$982$		0			0
$983$	55.7862			1.77930			0.889652	−	0.456640i	$-0.150947\pi$
$983$	55.7862			1.77930			0.889652	+	0.456640i	$0.150947\pi$
$984$		0			0
$985$		−	13.4396i		−	0.428223i
$986$		0			0
$987$	11.3776			0.362152
$988$		0			0
$989$	−2.56035			−0.0814145
$990$		0			0
$991$			29.9655i			0.951886i	0.879476	+	0.475943i	$0.157893\pi$
$991$			29.9655i			0.951886i	−0.879476	+	0.475943i	$0.842107\pi$
$992$		0			0
$993$		−	10.0844i		−	0.320019i
$994$		0			0
$995$	−23.4802			−0.744374
$996$		0			0
$997$			20.7811i			0.658145i	0.944305	+	0.329073i	$0.106736\pi$
$997$			20.7811i			0.658145i	−0.944305	+	0.329073i	$0.893264\pi$
$998$		0			0
$999$	−9.99312	+	8.12414i	−0.316168	+	0.257036i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.p.g.961.1		6
4.3	odd	2		370.2.d.c.221.6	yes	6
12.11	even	2		3330.2.h.n.2071.2		6
20.3	even	4		1850.2.c.j.1849.5		6
20.7	even	4		1850.2.c.i.1849.2		6
20.19	odd	2		1850.2.d.f.1701.1		6
37.36	even	2	inner	2960.2.p.g.961.2		6
148.147	odd	2		370.2.d.c.221.3	✓	6
444.443	even	2		3330.2.h.n.2071.5		6
740.147	even	4		1850.2.c.j.1849.2		6
740.443	even	4		1850.2.c.i.1849.5		6
740.739	odd	2		1850.2.d.f.1701.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.d.c.221.3	✓	6	148.147	odd	2
370.2.d.c.221.6	yes	6	4.3	odd	2
1850.2.c.i.1849.2		6	20.7	even	4
1850.2.c.i.1849.5		6	740.443	even	4
1850.2.c.j.1849.2		6	740.147	even	4
1850.2.c.j.1849.5		6	20.3	even	4
1850.2.d.f.1701.1		6	20.19	odd	2
1850.2.d.f.1701.4		6	740.739	odd	2
2960.2.p.g.961.1		6	1.1	even	1	trivial
2960.2.p.g.961.2		6	37.36	even	2	inner
3330.2.h.n.2071.2		6	12.11	even	2
3330.2.h.n.2071.5		6	444.443	even	2

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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.24914 q^{3} -1.00000i q^{5} -1.52932 q^{7} +2.05863 q^{9} -2.71982 q^{11} -0.941367i q^{13} +2.24914i q^{15} +4.83709i q^{17} -0.249141i q^{19} +3.43965 q^{21} +0.941367i q^{23} -1.00000 q^{25} +2.11727 q^{27} -0.719824i q^{29} +4.02760i q^{31} +6.11727 q^{33} +1.52932i q^{35} +(-4.71982 + 3.83709i) q^{37} +2.11727i q^{39} -8.27674 q^{41} +2.71982i q^{43} -2.05863i q^{45} +3.30777 q^{47} -4.66119 q^{49} -10.8793i q^{51} +8.39400 q^{53} +2.71982i q^{55} +0.560352i q^{57} -7.30777i q^{59} -6.83709i q^{61} -3.14830 q^{63} -0.941367 q^{65} +7.68879 q^{67} -2.11727i q^{69} +3.05863 q^{71} +9.11383 q^{73} +2.24914 q^{75} +4.15947 q^{77} -1.75086i q^{79} -10.9379 q^{81} -0.131874 q^{83} +4.83709 q^{85} +1.61899i q^{87} +8.99656i q^{89} +1.43965i q^{91} -9.05863i q^{93} -0.249141 q^{95} -16.2767i q^{97} -5.59912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 10 q^{7} + 14 q^{9} + 2 q^{11} - 16 q^{21} - 6 q^{25} + 16 q^{27} + 40 q^{33} - 10 q^{37} + 2 q^{41} + 4 q^{47} - 8 q^{49} + 2 q^{53} - 58 q^{63} - 4 q^{65} - 8 q^{67} + 20 q^{71} - 12 q^{73}+ \cdots + 58 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 10 * q^7 + 14 * q^9 + 2 * q^11 - 16 * q^21 - 6 * q^25 + 16 * q^27 + 40 * q^33 - 10 * q^37 + 2 * q^41 + 4 * q^47 - 8 * q^49 + 2 * q^53 - 58 * q^63 - 4 * q^65 - 8 * q^67 + 20 * q^71 - 12 * q^73 - 4 * q^75 - 30 * q^77 + 6 * q^81 + 20 * q^83 + 14 * q^85 + 16 * q^95 + 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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