LMFDB - Embedded newform 3344.2.o.b.1519.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1519,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3344.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.o (of order $2$, degree $1$, 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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.2
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22672 q^{3} -2.25376 q^{5} -1.49199i q^{7} +7.41170 q^{9} +O(q^{10})$	$q-3.22672 q^{3} -2.25376 q^{5} -1.49199i q^{7} +7.41170 q^{9} -1.00000i q^{11} -0.212830i q^{13} +7.27224 q^{15} +4.11960 q^{17} +(0.508616 - 4.32912i) q^{19} +4.81423i q^{21} -1.98895i q^{23} +0.0794300 q^{25} -14.2353 q^{27} -2.13403i q^{29} +0.884846 q^{31} +3.22672i q^{33} +3.36259i q^{35} +7.35472i q^{37} +0.686741i q^{39} +7.14665i q^{41} +5.93109i q^{43} -16.7042 q^{45} +0.443233i q^{47} +4.77396 q^{49} -13.2928 q^{51} +7.34494i q^{53} +2.25376i q^{55} +(-1.64116 + 13.9689i) q^{57} +8.31295 q^{59} +2.57867 q^{61} -11.0582i q^{63} +0.479667i q^{65} -11.0775 q^{67} +6.41778i q^{69} -13.0687 q^{71} +10.6781 q^{73} -0.256298 q^{75} -1.49199 q^{77} +11.0376 q^{79} +23.6982 q^{81} -0.844445i q^{83} -9.28458 q^{85} +6.88590i q^{87} -2.53135i q^{89} -0.317540 q^{91} -2.85515 q^{93} +(-1.14630 + 9.75680i) q^{95} -0.891640i q^{97} -7.41170i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 56 q^{9}+O(q^{10}) $ 64 * q + 56 * q^9	$ 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) $ 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times$.

$n$	$705$	$837$	$2433$	$2927$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−3.22672			−1.86295			−0.931473	−	0.363811i	$-0.881475\pi$
$3$	−3.22672			−1.86295			−0.931473	+	0.363811i	$0.881475\pi$
$4$		0			0
$5$	−2.25376			−1.00791			−0.503956	−	0.863729i	$-0.668123\pi$
$5$	−2.25376			−1.00791			−0.503956	+	0.863729i	$0.668123\pi$
$6$		0			0
$7$		−	1.49199i		−	0.563920i	−0.959426	−	0.281960i	$-0.909015\pi$
$7$		−	1.49199i		−	0.563920i	0.959426	−	0.281960i	$-0.0909845\pi$
$8$		0			0
$9$	7.41170			2.47057
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		−	0.212830i		−	0.0590283i	−0.999564	−	0.0295142i	$-0.990604\pi$
$13$		−	0.212830i		−	0.0590283i	0.999564	−	0.0295142i	$-0.00939601\pi$
$14$		0			0
$15$	7.27224			1.87768
$16$		0			0
$17$	4.11960			0.999149			0.499574	−	0.866271i	$-0.333490\pi$
$17$	4.11960			0.999149			0.499574	+	0.866271i	$0.333490\pi$
$18$		0			0
$19$	0.508616	−	4.32912i	0.116684	−	0.993169i
$19$	0.508616	−	4.32912i	0.116684	−	0.993169i
$20$		0			0
$21$			4.81423i			1.05055i
$22$		0			0
$23$		−	1.98895i		−	0.414725i	−0.978264	−	0.207363i	$-0.933512\pi$
$23$		−	1.98895i		−	0.414725i	0.978264	−	0.207363i	$-0.0664880\pi$
$24$		0			0
$25$	0.0794300			0.0158860
$26$		0			0
$27$	−14.2353			−2.73958
$28$		0			0
$29$		−	2.13403i		−	0.396279i	−0.980174	−	0.198140i	$-0.936510\pi$
$29$		−	2.13403i		−	0.396279i	0.980174	−	0.198140i	$-0.0634899\pi$
$30$		0			0
$31$	0.884846			0.158923			0.0794615	−	0.996838i	$-0.474680\pi$
$31$	0.884846			0.158923			0.0794615	+	0.996838i	$0.474680\pi$
$32$		0			0
$33$			3.22672i			0.561699i
$34$		0			0
$35$			3.36259i			0.568381i
$36$		0			0
$37$			7.35472i			1.20911i	0.796564	+	0.604555i	$0.206649\pi$
$37$			7.35472i			1.20911i	−0.796564	+	0.604555i	$0.793351\pi$
$38$		0			0
$39$			0.686741i			0.109967i
$40$		0			0
$41$			7.14665i			1.11612i	0.829801	+	0.558060i	$0.188454\pi$
$41$			7.14665i			1.11612i	−0.829801	+	0.558060i	$0.811546\pi$
$42$		0			0
$43$			5.93109i			0.904483i	0.891896	+	0.452241i	$0.149375\pi$
$43$			5.93109i			0.904483i	−0.891896	+	0.452241i	$0.850625\pi$
$44$		0			0
$45$	−16.7042			−2.49011
$46$		0			0
$47$			0.443233i			0.0646522i	0.999477	+	0.0323261i	$0.0102915\pi$
$47$			0.443233i			0.0646522i	−0.999477	+	0.0323261i	$0.989708\pi$
$48$		0			0
$49$	4.77396			0.681994
$50$		0			0
$51$	−13.2928			−1.86136
$52$		0			0
$53$			7.34494i			1.00891i	0.863439	+	0.504453i	$0.168306\pi$
$53$			7.34494i			1.00891i	−0.863439	+	0.504453i	$0.831694\pi$
$54$		0			0
$55$			2.25376i			0.303897i
$56$		0			0
$57$	−1.64116	+	13.9689i	−0.217377	+	1.85022i
$58$		0			0
$59$	8.31295			1.08225			0.541127	−	0.840941i	$-0.317998\pi$
$59$	8.31295			1.08225			0.541127	+	0.840941i	$0.317998\pi$
$60$		0			0
$61$	2.57867			0.330164			0.165082	−	0.986280i	$-0.447211\pi$
$61$	2.57867			0.330164			0.165082	+	0.986280i	$0.447211\pi$
$62$		0			0
$63$		−	11.0582i		−	1.39320i
$64$		0			0
$65$			0.479667i			0.0594953i
$66$		0			0
$67$	−11.0775			−1.35334			−0.676668	−	0.736288i	$-0.736577\pi$
$67$	−11.0775			−1.35334			−0.676668	+	0.736288i	$0.736577\pi$
$68$		0			0
$69$			6.41778i			0.772610i
$70$		0			0
$71$	−13.0687			−1.55097			−0.775483	−	0.631368i	$-0.782494\pi$
$71$	−13.0687			−1.55097			−0.775483	+	0.631368i	$0.782494\pi$
$72$		0			0
$73$	10.6781			1.24978			0.624891	−	0.780712i	$-0.285143\pi$
$73$	10.6781			1.24978			0.624891	+	0.780712i	$0.285143\pi$
$74$		0			0
$75$	−0.256298			−0.0295947
$76$		0			0
$77$	−1.49199			−0.170028
$78$		0			0
$79$	11.0376			1.24182			0.620912	−	0.783880i	$-0.286762\pi$
$79$	11.0376			1.24182			0.620912	+	0.783880i	$0.286762\pi$
$80$		0			0
$81$	23.6982			2.63313
$82$		0			0
$83$		−	0.844445i		−	0.0926898i	−0.998925	−	0.0463449i	$-0.985243\pi$
$83$		−	0.844445i		−	0.0926898i	0.998925	−	0.0463449i	$-0.0147573\pi$
$84$		0			0
$85$	−9.28458			−1.00705
$86$		0			0
$87$			6.88590i			0.738246i
$88$		0			0
$89$		−	2.53135i		−	0.268322i	−0.990960	−	0.134161i	$-0.957166\pi$
$89$		−	2.53135i		−	0.268322i	0.990960	−	0.134161i	$-0.0428340\pi$
$90$		0			0
$91$	−0.317540			−0.0332872
$92$		0			0
$93$	−2.85515			−0.296065
$94$		0			0
$95$	−1.14630	+	9.75680i	−0.117608	+	1.00103i
$96$		0			0
$97$		−	0.891640i		−	0.0905323i	−0.998975	−	0.0452661i	$-0.985586\pi$
$97$		−	0.891640i		−	0.0905323i	0.998975	−	0.0452661i	$-0.0144136\pi$
$98$		0			0
$99$		−	7.41170i		−	0.744904i
$100$		0			0
$101$	−13.8174			−1.37488			−0.687439	−	0.726242i	$-0.741265\pi$
$101$	−13.8174			−1.37488			−0.687439	+	0.726242i	$0.741265\pi$
$102$		0			0
$103$	−0.195214			−0.0192350			−0.00961750	−	0.999954i	$-0.503061\pi$
$103$	−0.195214			−0.0192350			−0.00961750	+	0.999954i	$0.503061\pi$
$104$		0			0
$105$		−	10.8501i		−	1.05886i
$106$		0			0
$107$	9.09343			0.879095			0.439548	−	0.898219i	$-0.355139\pi$
$107$	9.09343			0.879095			0.439548	+	0.898219i	$0.355139\pi$
$108$		0			0
$109$		−	8.96005i		−	0.858217i	−0.903253	−	0.429109i	$-0.858828\pi$
$109$		−	8.96005i		−	0.858217i	0.903253	−	0.429109i	$-0.141172\pi$
$110$		0			0
$111$		−	23.7316i		−	2.25250i
$112$		0			0
$113$		−	8.05414i		−	0.757670i	−0.925464	−	0.378835i	$-0.876325\pi$
$113$		−	8.05414i		−	0.757670i	0.925464	−	0.378835i	$-0.123675\pi$
$114$		0			0
$115$			4.48262i			0.418006i
$116$		0			0
$117$		−	1.57743i		−	0.145833i
$118$		0			0
$119$		−	6.14640i		−	0.563440i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	23.0602i		−	2.07927i
$124$		0			0
$125$	11.0898			0.991900
$126$		0			0
$127$	−6.63249			−0.588538			−0.294269	−	0.955723i	$-0.595076\pi$
$127$	−6.63249			−0.588538			−0.294269	+	0.955723i	$0.595076\pi$
$128$		0			0
$129$		−	19.1379i		−	1.68500i
$130$		0			0
$131$		−	9.83770i		−	0.859524i	−0.902942	−	0.429762i	$-0.858597\pi$
$131$		−	9.83770i		−	0.859524i	0.902942	−	0.429762i	$-0.141403\pi$
$132$		0			0
$133$	−6.45902	−	0.758850i	−0.560068	−	0.0658006i
$134$		0			0
$135$	32.0829			2.76126
$136$		0			0
$137$	5.55120			0.474271			0.237135	−	0.971477i	$-0.423791\pi$
$137$	5.55120			0.474271			0.237135	+	0.971477i	$0.423791\pi$
$138$		0			0
$139$		−	7.10963i		−	0.603031i	−0.953461	−	0.301516i	$-0.902507\pi$
$139$		−	7.10963i		−	0.603031i	0.953461	−	0.301516i	$-0.0974925\pi$
$140$		0			0
$141$		−	1.43019i		−	0.120444i
$142$		0			0
$143$	−0.212830			−0.0177977
$144$		0			0
$145$			4.80959i			0.399414i
$146$		0			0
$147$	−15.4042			−1.27052
$148$		0			0
$149$	−3.84707			−0.315164			−0.157582	−	0.987506i	$-0.550370\pi$
$149$	−3.84707			−0.315164			−0.157582	+	0.987506i	$0.550370\pi$
$150$		0			0
$151$	−2.80968			−0.228648			−0.114324	−	0.993444i	$-0.536470\pi$
$151$	−2.80968			−0.228648			−0.114324	+	0.993444i	$0.536470\pi$
$152$		0			0
$153$	30.5332			2.46846
$154$		0			0
$155$	−1.99423			−0.160180
$156$		0			0
$157$	17.7816			1.41913			0.709565	−	0.704640i	$-0.248892\pi$
$157$	17.7816			1.41913			0.709565	+	0.704640i	$0.248892\pi$
$158$		0			0
$159$		−	23.7000i		−	1.87954i
$160$		0			0
$161$	−2.96750			−0.233872
$162$		0			0
$163$		−	5.22993i		−	0.409640i	−0.978800	−	0.204820i	$-0.934339\pi$
$163$		−	5.22993i		−	0.409640i	0.978800	−	0.204820i	$-0.0656608\pi$
$164$		0			0
$165$		−	7.27224i		−	0.566143i
$166$		0			0
$167$	7.67783			0.594128			0.297064	−	0.954857i	$-0.403992\pi$
$167$	7.67783			0.594128			0.297064	+	0.954857i	$0.403992\pi$
$168$		0			0
$169$	12.9547			0.996516
$170$		0			0
$171$	3.76970	−	32.0861i	0.288276	−	2.45369i
$172$		0			0
$173$			17.1167i			1.30136i	0.759353	+	0.650679i	$0.225516\pi$
$173$			17.1167i			1.30136i	−0.759353	+	0.650679i	$0.774484\pi$
$174$		0			0
$175$		−	0.118509i		−	0.00895843i
$176$		0			0
$177$	−26.8235			−2.01618
$178$		0			0
$179$	−3.29233			−0.246080			−0.123040	−	0.992402i	$-0.539264\pi$
$179$	−3.29233			−0.246080			−0.123040	+	0.992402i	$0.539264\pi$
$180$		0			0
$181$		−	19.8163i		−	1.47294i	−0.676472	−	0.736468i	$-0.736492\pi$
$181$		−	19.8163i		−	1.47294i	0.676472	−	0.736468i	$-0.263508\pi$
$182$		0			0
$183$	−8.32062			−0.615078
$184$		0			0
$185$		−	16.5758i		−	1.21868i
$186$		0			0
$187$		−	4.11960i		−	0.301255i
$188$		0			0
$189$			21.2389i			1.54491i
$190$		0			0
$191$			25.6765i			1.85789i	0.370224	+	0.928943i	$0.379281\pi$
$191$			25.6765i			1.85789i	−0.370224	+	0.928943i	$0.620719\pi$
$192$		0			0
$193$			23.2303i			1.67215i	0.548611	+	0.836077i	$0.315157\pi$
$193$			23.2303i			1.67215i	−0.548611	+	0.836077i	$0.684843\pi$
$194$		0			0
$195$		−	1.54775i		−	0.110837i
$196$		0			0
$197$	−9.16834			−0.653217			−0.326609	−	0.945160i	$-0.605906\pi$
$197$	−9.16834			−0.653217			−0.326609	+	0.945160i	$0.605906\pi$
$198$		0			0
$199$		−	20.7718i		−	1.47247i	−0.676726	−	0.736235i	$-0.736602\pi$
$199$		−	20.7718i		−	1.47247i	0.676726	−	0.736235i	$-0.263398\pi$
$200$		0			0
$201$	35.7440			2.52119
$202$		0			0
$203$	−3.18395			−0.223470
$204$		0			0
$205$		−	16.1068i		−	1.12495i
$206$		0			0
$207$		−	14.7415i		−	1.02461i
$208$		0			0
$209$	−4.32912	−	0.508616i	−0.299452	−	0.0351817i
$210$		0			0
$211$	27.6995			1.90691			0.953456	−	0.301533i	$-0.0974983\pi$
$211$	27.6995			1.90691			0.953456	+	0.301533i	$0.0974983\pi$
$212$		0			0
$213$	42.1689			2.88937
$214$		0			0
$215$		−	13.3672i		−	0.911639i
$216$		0			0
$217$		−	1.32018i		−	0.0896198i
$218$		0			0
$219$	−34.4553			−2.32828
$220$		0			0
$221$		−	0.876772i		−	0.0589781i
$222$		0			0
$223$	10.8286			0.725135			0.362567	−	0.931958i	$-0.381900\pi$
$223$	10.8286			0.725135			0.362567	+	0.931958i	$0.381900\pi$
$224$		0			0
$225$	0.588711			0.0392474
$226$		0			0
$227$	12.4026			0.823188			0.411594	−	0.911367i	$-0.364972\pi$
$227$	12.4026			0.823188			0.411594	+	0.911367i	$0.364972\pi$
$228$		0			0
$229$	16.1355			1.06627			0.533134	−	0.846031i	$-0.321014\pi$
$229$	16.1355			1.06627			0.533134	+	0.846031i	$0.321014\pi$
$230$		0			0
$231$	4.81423			0.316753
$232$		0			0
$233$	1.28992			0.0845052			0.0422526	−	0.999107i	$-0.486547\pi$
$233$	1.28992			0.0845052			0.0422526	+	0.999107i	$0.486547\pi$
$234$		0			0
$235$		−	0.998941i		−	0.0651637i
$236$		0			0
$237$	−35.6151			−2.31345
$238$		0			0
$239$		−	1.40739i		−	0.0910363i	−0.998964	−	0.0455182i	$-0.985506\pi$
$239$		−	1.40739i		−	0.0910363i	0.998964	−	0.0455182i	$-0.0144939\pi$
$240$		0			0
$241$		−	24.2913i		−	1.56474i	−0.622815	−	0.782369i	$-0.714011\pi$
$241$		−	24.2913i		−	1.56474i	0.622815	−	0.782369i	$-0.285989\pi$
$242$		0			0
$243$	−33.7613			−2.16579
$244$		0			0
$245$	−10.7594			−0.687390
$246$		0			0
$247$	−0.921366	−	0.108248i	−0.0586251	−	0.00688768i
$248$		0			0
$249$			2.72478i			0.172676i
$250$		0			0
$251$			26.6289i			1.68080i	0.541964	+	0.840401i	$0.317681\pi$
$251$			26.6289i			1.68080i	−0.541964	+	0.840401i	$0.682319\pi$
$252$		0			0
$253$	−1.98895			−0.125044
$254$		0			0
$255$	29.9587			1.87609
$256$		0			0
$257$		−	20.3958i		−	1.27226i	−0.771583	−	0.636128i	$-0.780535\pi$
$257$		−	20.3958i		−	1.27226i	0.771583	−	0.636128i	$-0.219465\pi$
$258$		0			0
$259$	10.9732			0.681841
$260$		0			0
$261$		−	15.8168i		−	0.979034i
$262$		0			0
$263$			22.7166i			1.40076i	0.713768	+	0.700382i	$0.246987\pi$
$263$			22.7166i			1.40076i	−0.713768	+	0.700382i	$0.753013\pi$
$264$		0			0
$265$		−	16.5537i		−	1.01689i
$266$		0			0
$267$			8.16794i			0.499870i
$268$		0			0
$269$		−	23.6670i		−	1.44300i	−0.692414	−	0.721501i	$-0.743453\pi$
$269$		−	23.6670i		−	1.44300i	0.692414	−	0.721501i	$-0.256547\pi$
$270$		0			0
$271$		−	14.5541i		−	0.884099i	−0.896991	−	0.442050i	$-0.854252\pi$
$271$		−	14.5541i		−	0.884099i	0.896991	−	0.442050i	$-0.145748\pi$
$272$		0			0
$273$	1.02461			0.0620123
$274$		0			0
$275$		−	0.0794300i		−	0.00478981i
$276$		0			0
$277$	16.3511			0.982443			0.491222	−	0.871035i	$-0.336550\pi$
$277$	16.3511			0.982443			0.491222	+	0.871035i	$0.336550\pi$
$278$		0			0
$279$	6.55821			0.392630
$280$		0			0
$281$		−	8.07529i		−	0.481731i	−0.970558	−	0.240866i	$-0.922569\pi$
$281$		−	8.07529i		−	0.481731i	0.970558	−	0.240866i	$-0.0774313\pi$
$282$		0			0
$283$		−	28.0607i		−	1.66804i	−0.551738	−	0.834018i	$-0.686035\pi$
$283$		−	28.0607i		−	1.66804i	0.551738	−	0.834018i	$-0.313965\pi$
$284$		0			0
$285$	3.69877	−	31.4824i	0.219096	−	1.86486i
$286$		0			0
$287$	10.6627			0.629402
$288$		0			0
$289$	−0.0289268			−0.00170158
$290$		0			0
$291$			2.87707i			0.168657i
$292$		0			0
$293$		−	22.8260i		−	1.33351i	−0.745277	−	0.666755i	$-0.767683\pi$
$293$		−	22.8260i		−	1.33351i	0.745277	−	0.666755i	$-0.232317\pi$
$294$		0			0
$295$	−18.7354			−1.09082
$296$		0			0
$297$			14.2353i			0.826015i
$298$		0			0
$299$	−0.423308			−0.0244805
$300$		0			0
$301$	8.84913			0.510056
$302$		0			0
$303$	44.5847			2.56132
$304$		0			0
$305$	−5.81169			−0.332777
$306$		0			0
$307$	−16.1097			−0.919428			−0.459714	−	0.888067i	$-0.652048\pi$
$307$	−16.1097			−0.919428			−0.459714	+	0.888067i	$0.652048\pi$
$308$		0			0
$309$	0.629900			0.0358338
$310$		0			0
$311$		−	8.34358i		−	0.473121i	−0.971617	−	0.236561i	$-0.923980\pi$
$311$		−	8.34358i		−	0.473121i	0.971617	−	0.236561i	$-0.0760202\pi$
$312$		0			0
$313$	−27.6670			−1.56383			−0.781915	−	0.623385i	$-0.785757\pi$
$313$	−27.6670			−1.56383			−0.781915	+	0.623385i	$0.785757\pi$
$314$		0			0
$315$			24.9225i			1.40422i
$316$		0			0
$317$		−	11.5481i		−	0.648604i	−0.945954	−	0.324302i	$-0.894871\pi$
$317$		−	11.5481i		−	0.648604i	0.945954	−	0.324302i	$-0.105129\pi$
$318$		0			0
$319$	−2.13403			−0.119483
$320$		0			0
$321$	−29.3419			−1.63771
$322$		0			0
$323$	2.09529	−	17.8342i	0.116585	−	0.992324i
$324$		0			0
$325$		−	0.0169051i		−	0.000937724i
$326$		0			0
$327$			28.9115i			1.59881i
$328$		0			0
$329$	0.661300			0.0364587
$330$		0			0
$331$	9.88119			0.543119			0.271560	−	0.962422i	$-0.412461\pi$
$331$	9.88119			0.543119			0.271560	+	0.962422i	$0.412461\pi$
$332$		0			0
$333$			54.5110i			2.98718i
$334$		0			0
$335$	24.9661			1.36404
$336$		0			0
$337$			1.90543i			0.103795i	0.998652	+	0.0518977i	$0.0165270\pi$
$337$			1.90543i			0.103795i	−0.998652	+	0.0518977i	$0.983473\pi$
$338$		0			0
$339$			25.9884i			1.41150i
$340$		0			0
$341$		−	0.884846i		−	0.0479171i
$342$		0			0
$343$		−	17.5667i		−	0.948510i
$344$		0			0
$345$		−	14.4641i		−	0.778723i
$346$		0			0
$347$			28.2111i			1.51445i	0.653153	+	0.757226i	$0.273446\pi$
$347$			28.2111i			1.51445i	−0.653153	+	0.757226i	$0.726554\pi$
$348$		0			0
$349$	−3.97377			−0.212711			−0.106355	−	0.994328i	$-0.533918\pi$
$349$	−3.97377			−0.212711			−0.106355	+	0.994328i	$0.533918\pi$
$350$		0			0
$351$			3.02969i			0.161713i
$352$		0			0
$353$	−14.2758			−0.759825			−0.379913	−	0.925022i	$-0.624046\pi$
$353$	−14.2758			−0.759825			−0.379913	+	0.925022i	$0.624046\pi$
$354$		0			0
$355$	29.4537			1.56324
$356$		0			0
$357$			19.8327i			1.04966i
$358$		0			0
$359$			29.5820i			1.56128i	0.624981	+	0.780640i	$0.285107\pi$
$359$			29.5820i			1.56128i	−0.624981	+	0.780640i	$0.714893\pi$
$360$		0			0
$361$	−18.4826	−	4.40372i	−0.972770	−	0.231775i
$362$		0			0
$363$	3.22672			0.169359
$364$		0			0
$365$	−24.0660			−1.25967
$366$		0			0
$367$			16.3602i			0.853998i	0.904252	+	0.426999i	$0.140429\pi$
$367$			16.3602i			0.853998i	−0.904252	+	0.426999i	$0.859571\pi$
$368$		0			0
$369$			52.9688i			2.75745i
$370$		0			0
$371$	10.9586			0.568942
$372$		0			0
$373$		−	2.55903i		−	0.132502i	−0.997803	−	0.0662509i	$-0.978896\pi$
$373$		−	2.55903i		−	0.132502i	0.997803	−	0.0662509i	$-0.0211038\pi$
$374$		0			0
$375$	−35.7836			−1.84786
$376$		0			0
$377$	−0.454185			−0.0233917
$378$		0			0
$379$	−1.73501			−0.0891216			−0.0445608	−	0.999007i	$-0.514189\pi$
$379$	−1.73501			−0.0891216			−0.0445608	+	0.999007i	$0.514189\pi$
$380$		0			0
$381$	21.4012			1.09641
$382$		0			0
$383$	−10.6587			−0.544632			−0.272316	−	0.962208i	$-0.587790\pi$
$383$	−10.6587			−0.544632			−0.272316	+	0.962208i	$0.587790\pi$
$384$		0			0
$385$	3.36259			0.171373
$386$		0			0
$387$			43.9594i			2.23458i
$388$		0			0
$389$	−4.82783			−0.244781			−0.122390	−	0.992482i	$-0.539056\pi$
$389$	−4.82783			−0.244781			−0.122390	+	0.992482i	$0.539056\pi$
$390$		0			0
$391$		−	8.19368i		−	0.414372i
$392$		0			0
$393$			31.7435i			1.60125i
$394$		0			0
$395$	−24.8760			−1.25165
$396$		0			0
$397$	−15.9354			−0.799776			−0.399888	−	0.916564i	$-0.630951\pi$
$397$	−15.9354			−0.799776			−0.399888	+	0.916564i	$0.630951\pi$
$398$		0			0
$399$	20.8414	+	2.44859i	1.04338	+	0.122583i
$400$		0			0
$401$		−	7.84623i		−	0.391822i	−0.980622	−	0.195911i	$-0.937234\pi$
$401$		−	7.84623i		−	0.391822i	0.980622	−	0.195911i	$-0.0627664\pi$
$402$		0			0
$403$		−	0.188321i		−	0.00938096i
$404$		0			0
$405$	−53.4099			−2.65396
$406$		0			0
$407$	7.35472			0.364560
$408$		0			0
$409$		−	17.3874i		−	0.859752i	−0.902888	−	0.429876i	$-0.858557\pi$
$409$		−	17.3874i		−	0.859752i	0.902888	−	0.429876i	$-0.141443\pi$
$410$		0			0
$411$	−17.9121			−0.883541
$412$		0			0
$413$		−	12.4029i		−	0.610305i
$414$		0			0
$415$			1.90317i			0.0934232i
$416$		0			0
$417$			22.9408i			1.12341i
$418$		0			0
$419$			25.0428i			1.22342i	0.791082	+	0.611711i	$0.209518\pi$
$419$			25.0428i			1.22342i	−0.791082	+	0.611711i	$0.790482\pi$
$420$		0			0
$421$		−	16.3580i		−	0.797242i	−0.917116	−	0.398621i	$-0.869489\pi$
$421$		−	16.3580i		−	0.797242i	0.917116	−	0.398621i	$-0.130511\pi$
$422$		0			0
$423$			3.28511i			0.159727i
$424$		0			0
$425$	0.327220			0.0158725
$426$		0			0
$427$		−	3.84735i		−	0.186186i
$428$		0			0
$429$	0.686741			0.0331562
$430$		0			0
$431$	18.9288			0.911769			0.455885	−	0.890039i	$-0.349323\pi$
$431$	18.9288			0.911769			0.455885	+	0.890039i	$0.349323\pi$
$432$		0			0
$433$		−	39.3963i		−	1.89327i	−0.322312	−	0.946634i	$-0.604460\pi$
$433$		−	39.3963i		−	1.89327i	0.322312	−	0.946634i	$-0.395540\pi$
$434$		0			0
$435$		−	15.5192i		−	0.744087i
$436$		0			0
$437$	−8.61042	−	1.01161i	−0.411892	−	0.0483919i
$438$		0			0
$439$	−16.4509			−0.785156			−0.392578	−	0.919719i	$-0.628417\pi$
$439$	−16.4509			−0.785156			−0.392578	+	0.919719i	$0.628417\pi$
$440$		0			0
$441$	35.3832			1.68491
$442$		0			0
$443$			11.0114i			0.523168i	0.965181	+	0.261584i	$0.0842448\pi$
$443$			11.0114i			0.523168i	−0.965181	+	0.261584i	$0.915755\pi$
$444$		0			0
$445$			5.70505i			0.270445i
$446$		0			0
$447$	12.4134			0.587133
$448$		0			0
$449$		−	11.9523i		−	0.564065i	−0.959405	−	0.282032i	$-0.908991\pi$
$449$		−	11.9523i		−	0.564065i	0.959405	−	0.282032i	$-0.0910085\pi$
$450$		0			0
$451$	7.14665			0.336523
$452$		0			0
$453$	9.06603			0.425959
$454$		0			0
$455$	0.715659			0.0335506
$456$		0			0
$457$	13.1309			0.614236			0.307118	−	0.951671i	$-0.400635\pi$
$457$	13.1309			0.614236			0.307118	+	0.951671i	$0.400635\pi$
$458$		0			0
$459$	−58.6437			−2.73725
$460$		0			0
$461$	−16.5961			−0.772956			−0.386478	−	0.922299i	$-0.626309\pi$
$461$	−16.5961			−0.772956			−0.386478	+	0.922299i	$0.626309\pi$
$462$		0			0
$463$		−	13.1824i		−	0.612637i	−0.951929	−	0.306318i	$-0.900903\pi$
$463$		−	13.1824i		−	0.612637i	0.951929	−	0.306318i	$-0.0990972\pi$
$464$		0			0
$465$	6.43481			0.298407
$466$		0			0
$467$		−	2.69293i		−	0.124614i	−0.998057	−	0.0623070i	$-0.980154\pi$
$467$		−	2.69293i		−	0.124614i	0.998057	−	0.0623070i	$-0.0198458\pi$
$468$		0			0
$469$			16.5276i			0.763173i
$470$		0			0
$471$	−57.3763			−2.64376
$472$		0			0
$473$	5.93109			0.272712
$474$		0			0
$475$	0.0403993	−	0.343862i	0.00185365	−	0.0157775i
$476$		0			0
$477$			54.4385i			2.49257i
$478$		0			0
$479$		−	38.3493i		−	1.75222i	−0.482108	−	0.876112i	$-0.660129\pi$
$479$		−	38.3493i		−	1.75222i	0.482108	−	0.876112i	$-0.339871\pi$
$480$		0			0
$481$	1.56530			0.0713717
$482$		0			0
$483$	9.57528			0.435690
$484$		0			0
$485$			2.00954i			0.0912486i
$486$		0			0
$487$	−16.3485			−0.740822			−0.370411	−	0.928868i	$-0.620783\pi$
$487$	−16.3485			−0.740822			−0.370411	+	0.928868i	$0.620783\pi$
$488$		0			0
$489$			16.8755i			0.763136i
$490$		0			0
$491$			6.40059i			0.288854i	0.989515	+	0.144427i	$0.0461340\pi$
$491$			6.40059i			0.288854i	−0.989515	+	0.144427i	$0.953866\pi$
$492$		0			0
$493$		−	8.79134i		−	0.395942i
$494$		0			0
$495$			16.7042i			0.750797i
$496$		0			0
$497$			19.4984i			0.874621i
$498$		0			0
$499$		−	5.84955i		−	0.261862i	−0.991391	−	0.130931i	$-0.958203\pi$
$499$		−	5.84955i		−	0.261862i	0.991391	−	0.130931i	$-0.0417966\pi$
$500$		0			0
$501$	−24.7742			−1.10683
$502$		0			0
$503$			30.4292i			1.35677i	0.734706	+	0.678386i	$0.237320\pi$
$503$			30.4292i			1.35677i	−0.734706	+	0.678386i	$0.762680\pi$
$504$		0			0
$505$	31.1410			1.38576
$506$		0			0
$507$	−41.8011			−1.85645
$508$		0			0
$509$			40.3885i			1.79019i	0.445875	+	0.895095i	$0.352893\pi$
$509$			40.3885i			1.79019i	−0.445875	+	0.895095i	$0.647107\pi$
$510$		0			0
$511$		−	15.9317i		−	0.704777i
$512$		0			0
$513$	−7.24029	+	61.6263i	−0.319667	+	2.72087i
$514$		0			0
$515$	0.439965			0.0193872
$516$		0			0
$517$	0.443233			0.0194934
$518$		0			0
$519$		−	55.2307i		−	2.42436i
$520$		0			0
$521$		−	29.5678i		−	1.29539i	−0.761901	−	0.647694i	$-0.775734\pi$
$521$		−	29.5678i		−	1.29539i	0.761901	−	0.647694i	$-0.224266\pi$
$522$		0			0
$523$	−10.6681			−0.466482			−0.233241	−	0.972419i	$-0.574933\pi$
$523$	−10.6681			−0.466482			−0.233241	+	0.972419i	$0.574933\pi$
$524$		0			0
$525$			0.382394i			0.0166891i
$526$		0			0
$527$	3.64521			0.158788
$528$		0			0
$529$	19.0441			0.828003
$530$		0			0
$531$	61.6131			2.67378
$532$		0			0
$533$	1.52102			0.0658826
$534$		0			0
$535$	−20.4944			−0.886050
$536$		0			0
$537$	10.6234			0.458434
$538$		0			0
$539$		−	4.77396i		−	0.205629i
$540$		0			0
$541$	30.4981			1.31122			0.655608	−	0.755102i	$-0.272413\pi$
$541$	30.4981			1.31122			0.655608	+	0.755102i	$0.272413\pi$
$542$		0			0
$543$			63.9417i			2.74400i
$544$		0			0
$545$			20.1938i			0.865007i
$546$		0			0
$547$	21.9064			0.936651			0.468325	−	0.883556i	$-0.344858\pi$
$547$	21.9064			0.936651			0.468325	+	0.883556i	$0.344858\pi$
$548$		0			0
$549$	19.1123			0.815693
$550$		0			0
$551$	−9.23847	−	1.08540i	−0.393572	−	0.0462396i
$552$		0			0
$553$		−	16.4680i		−	0.700290i
$554$		0			0
$555$			53.4853i			2.27033i
$556$		0			0
$557$	−19.2670			−0.816368			−0.408184	−	0.912900i	$-0.633838\pi$
$557$	−19.2670			−0.816368			−0.408184	+	0.912900i	$0.633838\pi$
$558$		0			0
$559$	1.26231			0.0533901
$560$		0			0
$561$			13.2928i			0.561221i
$562$		0			0
$563$	34.2292			1.44259			0.721295	−	0.692628i	$-0.243547\pi$
$563$	34.2292			1.44259			0.721295	+	0.692628i	$0.243547\pi$
$564$		0			0
$565$			18.1521i			0.763664i
$566$		0			0
$567$		−	35.3574i		−	1.48487i
$568$		0			0
$569$			26.8387i			1.12514i	0.826751	+	0.562569i	$0.190187\pi$
$569$			26.8387i			1.12514i	−0.826751	+	0.562569i	$0.809813\pi$
$570$		0			0
$571$		−	40.3305i		−	1.68778i	−0.536516	−	0.843890i	$-0.680260\pi$
$571$		−	40.3305i		−	1.68778i	0.536516	−	0.843890i	$-0.319740\pi$
$572$		0			0
$573$		−	82.8507i		−	3.46114i
$574$		0			0
$575$		−	0.157982i		−	0.00658832i
$576$		0			0
$577$	−12.9901			−0.540785			−0.270392	−	0.962750i	$-0.587153\pi$
$577$	−12.9901			−0.540785			−0.270392	+	0.962750i	$0.587153\pi$
$578$		0			0
$579$		−	74.9576i		−	3.11513i
$580$		0			0
$581$	−1.25990			−0.0522696
$582$		0			0
$583$	7.34494			0.304196
$584$		0			0
$585$			3.55514i			0.146987i
$586$		0			0
$587$		−	20.6950i		−	0.854172i	−0.904211	−	0.427086i	$-0.859540\pi$
$587$		−	20.6950i		−	0.854172i	0.904211	−	0.427086i	$-0.140460\pi$
$588$		0			0
$589$	0.450046	−	3.83061i	0.0185438	−	0.157837i
$590$		0			0
$591$	29.5836			1.21691
$592$		0			0
$593$	11.2452			0.461786			0.230893	−	0.972979i	$-0.425835\pi$
$593$	11.2452			0.461786			0.230893	+	0.972979i	$0.425835\pi$
$594$		0			0
$595$			13.8525i			0.567898i
$596$		0			0
$597$			67.0245i			2.74313i
$598$		0			0
$599$	−35.4118			−1.44689			−0.723443	−	0.690384i	$-0.757442\pi$
$599$	−35.4118			−1.44689			−0.723443	+	0.690384i	$0.757442\pi$
$600$		0			0
$601$		−	5.96012i		−	0.243118i	−0.992584	−	0.121559i	$-0.961211\pi$
$601$		−	5.96012i		−	0.243118i	0.992584	−	0.121559i	$-0.0387894\pi$
$602$		0			0
$603$	−82.1033			−3.34350
$604$		0			0
$605$	2.25376			0.0916283
$606$		0			0
$607$	17.1832			0.697444			0.348722	−	0.937226i	$-0.386616\pi$
$607$	17.1832			0.697444			0.348722	+	0.937226i	$0.386616\pi$
$608$		0			0
$609$	10.2737			0.416312
$610$		0			0
$611$	0.0943331			0.00381631
$612$		0			0
$613$	46.8679			1.89298			0.946489	−	0.322737i	$-0.104603\pi$
$613$	46.8679			1.89298			0.946489	+	0.322737i	$0.104603\pi$
$614$		0			0
$615$			51.9722i			2.09572i
$616$		0			0
$617$	13.9474			0.561502			0.280751	−	0.959781i	$-0.409417\pi$
$617$	13.9474			0.561502			0.280751	+	0.959781i	$0.409417\pi$
$618$		0			0
$619$		−	4.63870i		−	0.186445i	−0.995645	−	0.0932226i	$-0.970283\pi$
$619$		−	4.63870i		−	0.186445i	0.995645	−	0.0932226i	$-0.0297168\pi$
$620$		0			0
$621$			28.3133i			1.13617i
$622$		0			0
$623$	−3.77675			−0.151312
$624$		0			0
$625$	−25.3908			−1.01563
$626$		0			0
$627$	13.9689	+	1.64116i	0.557862	+	0.0655415i
$628$		0			0
$629$			30.2985i			1.20808i
$630$		0			0
$631$		−	33.0356i		−	1.31512i	−0.753400	−	0.657562i	$-0.771588\pi$
$631$		−	33.0356i		−	1.31512i	0.753400	−	0.657562i	$-0.228412\pi$
$632$		0			0
$633$	−89.3784			−3.55247
$634$		0			0
$635$	14.9480			0.593195
$636$		0			0
$637$		−	1.01604i		−	0.0402570i
$638$		0			0
$639$	−96.8611			−3.83177
$640$		0			0
$641$		−	42.4536i		−	1.67682i	−0.545042	−	0.838409i	$-0.683486\pi$
$641$		−	42.4536i		−	1.67682i	0.545042	−	0.838409i	$-0.316514\pi$
$642$		0			0
$643$		−	15.1481i		−	0.597381i	−0.954350	−	0.298691i	$-0.903450\pi$
$643$		−	15.1481i		−	0.597381i	0.954350	−	0.298691i	$-0.0965499\pi$
$644$		0			0
$645$			43.1323i			1.69833i
$646$		0			0
$647$			4.39296i			0.172705i	0.996265	+	0.0863526i	$0.0275212\pi$
$647$			4.39296i			0.172705i	−0.996265	+	0.0863526i	$0.972479\pi$
$648$		0			0
$649$		−	8.31295i		−	0.326312i
$650$		0			0
$651$			4.25985i			0.166957i
$652$		0			0
$653$	0.858331			0.0335891			0.0167946	−	0.999859i	$-0.494654\pi$
$653$	0.858331			0.0335891			0.0167946	+	0.999859i	$0.494654\pi$
$654$		0			0
$655$			22.1718i			0.866324i
$656$		0			0
$657$	79.1432			3.08767
$658$		0			0
$659$	34.8927			1.35923			0.679613	−	0.733571i	$-0.262148\pi$
$659$	34.8927			1.35923			0.679613	+	0.733571i	$0.262148\pi$
$660$		0			0
$661$		−	18.4582i		−	0.717942i	−0.933349	−	0.358971i	$-0.883128\pi$
$661$		−	18.4582i		−	0.717942i	0.933349	−	0.358971i	$-0.116872\pi$
$662$		0			0
$663$			2.82909i			0.109873i
$664$		0			0
$665$	14.5571	+	1.71027i	0.564499	+	0.0663212i
$666$		0			0
$667$	−4.24448			−0.164347
$668$		0			0
$669$	−34.9407			−1.35089
$670$		0			0
$671$		−	2.57867i		−	0.0995483i
$672$		0			0
$673$			14.8309i			0.571691i	0.958276	+	0.285845i	$0.0922744\pi$
$673$			14.8309i			0.571691i	−0.958276	+	0.285845i	$0.907726\pi$
$674$		0			0
$675$	−1.13071			−0.0435210
$676$		0			0
$677$		−	3.33808i		−	0.128293i	−0.997940	−	0.0641465i	$-0.979568\pi$
$677$		−	3.33808i		−	0.128293i	0.997940	−	0.0641465i	$-0.0204325\pi$
$678$		0			0
$679$	−1.33032			−0.0510529
$680$		0			0
$681$	−40.0196			−1.53355
$682$		0			0
$683$	1.97487			0.0755664			0.0377832	−	0.999286i	$-0.487970\pi$
$683$	1.97487			0.0755664			0.0377832	+	0.999286i	$0.487970\pi$
$684$		0			0
$685$	−12.5111			−0.478023
$686$		0			0
$687$	−52.0648			−1.98640
$688$		0			0
$689$	1.56322			0.0595540
$690$		0			0
$691$		−	18.3951i		−	0.699781i	−0.936791	−	0.349891i	$-0.886219\pi$
$691$		−	18.3951i		−	0.699781i	0.936791	−	0.349891i	$-0.113781\pi$
$692$		0			0
$693$	−11.0582			−0.420066
$694$		0			0
$695$			16.0234i			0.607802i
$696$		0			0
$697$			29.4413i			1.11517i
$698$		0			0
$699$	−4.16219			−0.157429
$700$		0			0
$701$	17.0942			0.645639			0.322820	−	0.946461i	$-0.395369\pi$
$701$	17.0942			0.645639			0.322820	+	0.946461i	$0.395369\pi$
$702$		0			0
$703$	31.8395	+	3.74073i	1.20085	+	0.141084i
$704$		0			0
$705$			3.22330i			0.121396i
$706$		0			0
$707$			20.6154i			0.775321i
$708$		0			0
$709$	−13.6991			−0.514482			−0.257241	−	0.966347i	$-0.582813\pi$
$709$	−13.6991			−0.514482			−0.257241	+	0.966347i	$0.582813\pi$
$710$		0			0
$711$	81.8072			3.06801
$712$		0			0
$713$		−	1.75992i		−	0.0659093i
$714$		0			0
$715$	0.479667			0.0179385
$716$		0			0
$717$			4.54124i			0.169596i
$718$		0			0
$719$		−	23.9826i		−	0.894401i	−0.894434	−	0.447200i	$-0.852421\pi$
$719$		−	23.9826i		−	0.894401i	0.894434	−	0.447200i	$-0.147579\pi$
$720$		0			0
$721$			0.291258i			0.0108470i
$722$		0			0
$723$			78.3810i			2.91502i
$724$		0			0
$725$		−	0.169506i		−	0.00629529i
$726$		0			0
$727$			40.4348i			1.49964i	0.661640	+	0.749822i	$0.269861\pi$
$727$			40.4348i			1.49964i	−0.661640	+	0.749822i	$0.730139\pi$
$728$		0			0
$729$	37.8438			1.40162
$730$		0			0
$731$			24.4337i			0.903713i
$732$		0			0
$733$	−15.5909			−0.575861			−0.287931	−	0.957651i	$-0.592967\pi$
$733$	−15.5909			−0.575861			−0.287931	+	0.957651i	$0.592967\pi$
$734$		0			0
$735$	34.7174			1.28057
$736$		0			0
$737$			11.0775i			0.408046i
$738$		0			0
$739$			19.3079i			0.710254i	0.934818	+	0.355127i	$0.115562\pi$
$739$			19.3079i			0.710254i	−0.934818	+	0.355127i	$0.884438\pi$
$740$		0			0
$741$	2.97299	+	0.349287i	0.109215	+	0.0128314i
$742$		0			0
$743$	−0.594749			−0.0218192			−0.0109096	−	0.999940i	$-0.503473\pi$
$743$	−0.594749			−0.0218192			−0.0109096	+	0.999940i	$0.503473\pi$
$744$		0			0
$745$	8.67036			0.317657
$746$		0			0
$747$		−	6.25877i		−	0.228996i
$748$		0			0
$749$		−	13.5673i		−	0.495739i
$750$		0			0
$751$	−31.7606			−1.15896			−0.579481	−	0.814986i	$-0.696745\pi$
$751$	−31.7606			−1.15896			−0.579481	+	0.814986i	$0.696745\pi$
$752$		0			0
$753$		−	85.9240i		−	3.13124i
$754$		0			0
$755$	6.33233			0.230457
$756$		0			0
$757$	−50.2832			−1.82757			−0.913787	−	0.406193i	$-0.866856\pi$
$757$	−50.2832			−1.82757			−0.913787	+	0.406193i	$0.866856\pi$
$758$		0			0
$759$	6.41778			0.232951
$760$		0			0
$761$	16.3026			0.590969			0.295485	−	0.955347i	$-0.404519\pi$
$761$	16.3026			0.590969			0.295485	+	0.955347i	$0.404519\pi$
$762$		0			0
$763$	−13.3683			−0.483966
$764$		0			0
$765$	−68.8145			−2.48799
$766$		0			0
$767$		−	1.76924i		−	0.0638836i
$768$		0			0
$769$	26.8303			0.967524			0.483762	−	0.875200i	$-0.339270\pi$
$769$	26.8303			0.967524			0.483762	+	0.875200i	$0.339270\pi$
$770$		0			0
$771$			65.8115i			2.37014i
$772$		0			0
$773$		−	46.2761i		−	1.66444i	−0.554449	−	0.832218i	$-0.687071\pi$
$773$		−	46.2761i		−	1.66444i	0.554449	−	0.832218i	$-0.312929\pi$
$774$		0			0
$775$	0.0702833			0.00252465
$776$		0			0
$777$	−35.4074			−1.27023
$778$		0			0
$779$	30.9387	+	3.63490i	1.10849	+	0.130234i
$780$		0			0
$781$			13.0687i			0.467634i
$782$		0			0
$783$			30.3785i			1.08564i
$784$		0			0
$785$	−40.0755			−1.43036
$786$		0			0
$787$	16.9354			0.603681			0.301840	−	0.953358i	$-0.402399\pi$
$787$	16.9354			0.603681			0.301840	+	0.953358i	$0.402399\pi$
$788$		0			0
$789$		−	73.2999i		−	2.60955i
$790$		0			0
$791$	−12.0167			−0.427265
$792$		0			0
$793$		−	0.548817i		−	0.0194890i
$794$		0			0
$795$			53.4142i			1.89441i
$796$		0			0
$797$		−	1.70367i		−	0.0603470i	−0.999545	−	0.0301735i	$-0.990394\pi$
$797$		−	1.70367i		−	0.0603470i	0.999545	−	0.0301735i	$-0.00960597\pi$
$798$		0			0
$799$			1.82594i			0.0645972i
$800$		0			0
$801$		−	18.7616i		−	0.662908i
$802$		0			0
$803$		−	10.6781i		−	0.376824i
$804$		0			0
$805$	6.68803			0.235722
$806$		0			0
$807$			76.3666i			2.68823i
$808$		0			0
$809$	−36.9563			−1.29931			−0.649656	−	0.760228i	$-0.725087\pi$
$809$	−36.9563			−1.29931			−0.649656	+	0.760228i	$0.725087\pi$
$810$		0			0
$811$	54.8593			1.92637			0.963185	−	0.268841i	$-0.0866405\pi$
$811$	54.8593			1.92637			0.963185	+	0.268841i	$0.0866405\pi$
$812$		0			0
$813$			46.9620i			1.64703i
$814$		0			0
$815$			11.7870i			0.412881i
$816$		0			0
$817$	25.6764	+	3.01664i	0.898304	+	0.105539i
$818$		0			0
$819$	−2.35351			−0.0822383
$820$		0			0
$821$	−38.7052			−1.35082			−0.675410	−	0.737442i	$-0.736033\pi$
$821$	−38.7052			−1.35082			−0.675410	+	0.737442i	$0.736033\pi$
$822$		0			0
$823$		−	53.5270i		−	1.86583i	−0.360091	−	0.932917i	$-0.617254\pi$
$823$		−	53.5270i		−	1.86583i	0.360091	−	0.932917i	$-0.382746\pi$
$824$		0			0
$825$			0.256298i			0.00892315i
$826$		0			0
$827$	−0.279527			−0.00972011			−0.00486005	−	0.999988i	$-0.501547\pi$
$827$	−0.279527			−0.00972011			−0.00486005	+	0.999988i	$0.501547\pi$
$828$		0			0
$829$			23.9033i			0.830195i	0.909777	+	0.415097i	$0.136253\pi$
$829$			23.9033i			0.830195i	−0.909777	+	0.415097i	$0.863747\pi$
$830$		0			0
$831$	−52.7604			−1.83024
$832$		0			0
$833$	19.6668			0.681414
$834$		0			0
$835$	−17.3040			−0.598829
$836$		0			0
$837$	−12.5960			−0.435383
$838$		0			0
$839$	11.4707			0.396012			0.198006	−	0.980201i	$-0.436554\pi$
$839$	11.4707			0.396012			0.198006	+	0.980201i	$0.436554\pi$
$840$		0			0
$841$	24.4459			0.842963
$842$		0			0
$843$			26.0567i			0.897439i
$844$		0			0
$845$	−29.1968			−1.00440
$846$		0			0
$847$			1.49199i			0.0512654i
$848$		0			0
$849$			90.5439i			3.10746i
$850$		0			0
$851$	14.6282			0.501448
$852$		0			0
$853$	−6.59626			−0.225852			−0.112926	−	0.993603i	$-0.536022\pi$
$853$	−6.59626			−0.225852			−0.112926	+	0.993603i	$0.536022\pi$
$854$		0			0
$855$	−8.49600	+	72.3144i	−0.290557	+	2.47310i
$856$		0			0
$857$			3.47477i			0.118696i	0.998237	+	0.0593479i	$0.0189021\pi$
$857$			3.47477i			0.118696i	−0.998237	+	0.0593479i	$0.981098\pi$
$858$		0			0
$859$		−	4.69804i		−	0.160295i	−0.996783	−	0.0801475i	$-0.974461\pi$
$859$		−	4.69804i		−	0.160295i	0.996783	−	0.0801475i	$-0.0255391\pi$
$860$		0			0
$861$	−34.4056			−1.17254
$862$		0			0
$863$	3.35196			0.114102			0.0570509	−	0.998371i	$-0.481830\pi$
$863$	3.35196			0.114102			0.0570509	+	0.998371i	$0.481830\pi$
$864$		0			0
$865$		−	38.5769i		−	1.31165i
$866$		0			0
$867$	0.0933387			0.00316995
$868$		0			0
$869$		−	11.0376i		−	0.374424i
$870$		0			0
$871$			2.35763i			0.0798851i
$872$		0			0
$873$		−	6.60856i		−	0.223666i
$874$		0			0
$875$		−	16.5459i		−	0.559352i
$876$		0			0
$877$			21.6563i			0.731280i	0.930756	+	0.365640i	$0.119150\pi$
$877$			21.6563i			0.731280i	−0.930756	+	0.365640i	$0.880850\pi$
$878$		0			0
$879$			73.6531i			2.48426i
$880$		0			0
$881$	32.0889			1.08110			0.540551	−	0.841311i	$-0.318216\pi$
$881$	32.0889			1.08110			0.540551	+	0.841311i	$0.318216\pi$
$882$		0			0
$883$		−	19.8289i		−	0.667297i	−0.942698	−	0.333648i	$-0.891720\pi$
$883$		−	19.8289i		−	0.667297i	0.942698	−	0.333648i	$-0.108280\pi$
$884$		0			0
$885$	60.4538			2.03213
$886$		0			0
$887$	−52.3234			−1.75685			−0.878424	−	0.477881i	$-0.841405\pi$
$887$	−52.3234			−1.75685			−0.878424	+	0.477881i	$0.841405\pi$
$888$		0			0
$889$			9.89562i			0.331888i
$890$		0			0
$891$		−	23.6982i		−	0.793918i
$892$		0			0
$893$	1.91881	+	0.225435i	0.0642106	+	0.00754390i
$894$		0			0
$895$	7.42012			0.248027
$896$		0			0
$897$	1.36589			0.0456059
$898$		0			0
$899$		−	1.88829i		−	0.0629779i
$900$		0			0
$901$			30.2582i			1.00805i
$902$		0			0
$903$	−28.5536			−0.950206
$904$		0			0
$905$			44.6612i			1.48459i
$906$		0			0
$907$	−29.2126			−0.969989			−0.484995	−	0.874517i	$-0.661178\pi$
$907$	−29.2126			−0.969989			−0.484995	+	0.874517i	$0.661178\pi$
$908$		0			0
$909$	−102.410			−3.39673
$910$		0			0
$911$	−44.9660			−1.48979			−0.744895	−	0.667182i	$-0.767500\pi$
$911$	−44.9660			−1.48979			−0.744895	+	0.667182i	$0.767500\pi$
$912$		0			0
$913$	−0.844445			−0.0279470
$914$		0			0
$915$	18.7527			0.619945
$916$		0			0
$917$	−14.6778			−0.484703
$918$		0			0
$919$			7.71753i			0.254578i	0.991866	+	0.127289i	$0.0406275\pi$
$919$			7.71753i			0.254578i	−0.991866	+	0.127289i	$0.959372\pi$
$920$		0			0
$921$	51.9814			1.71284
$922$		0			0
$923$			2.78140i			0.0915510i
$924$		0			0
$925$			0.584186i			0.0192079i
$926$		0			0
$927$	−1.44687			−0.0475213
$928$		0			0
$929$	48.6799			1.59713			0.798567	−	0.601906i	$-0.205592\pi$
$929$	48.6799			1.59713			0.798567	+	0.601906i	$0.205592\pi$
$930$		0			0
$931$	2.42811	−	20.6671i	0.0795781	−	0.677336i
$932$		0			0
$933$			26.9224i			0.881399i
$934$		0			0
$935$			9.28458i			0.303638i
$936$		0			0
$937$	28.6611			0.936317			0.468159	−	0.883644i	$-0.344918\pi$
$937$	28.6611			0.936317			0.468159	+	0.883644i	$0.344918\pi$
$938$		0			0
$939$	89.2735			2.91333
$940$		0			0
$941$		−	21.5537i		−	0.702631i	−0.936257	−	0.351315i	$-0.885735\pi$
$941$		−	21.5537i		−	0.702631i	0.936257	−	0.351315i	$-0.114265\pi$
$942$		0			0
$943$	14.2143			0.462883
$944$		0			0
$945$		−	47.8674i		−	1.55713i
$946$		0			0
$947$		−	19.6707i		−	0.639213i	−0.947550	−	0.319607i	$-0.896449\pi$
$947$		−	19.6707i		−	0.639213i	0.947550	−	0.319607i	$-0.103551\pi$
$948$		0			0
$949$		−	2.27263i		−	0.0737726i
$950$		0			0
$951$			37.2623i			1.20831i
$952$		0			0
$953$		−	50.5327i		−	1.63692i	−0.574566	−	0.818458i	$-0.694829\pi$
$953$		−	50.5327i		−	1.63692i	0.574566	−	0.818458i	$-0.305171\pi$
$954$		0			0
$955$		−	57.8686i		−	1.87258i
$956$		0			0
$957$	6.88590			0.222590
$958$		0			0
$959$		−	8.28234i		−	0.267451i
$960$		0			0
$961$	−30.2170			−0.974743
$962$		0			0
$963$	67.3977			2.17186
$964$		0			0
$965$		−	52.3555i		−	1.68538i
$966$		0			0
$967$			19.3091i			0.620937i	0.950584	+	0.310469i	$0.100486\pi$
$967$			19.3091i			0.620937i	−0.950584	+	0.310469i	$0.899514\pi$
$968$		0			0
$969$	−6.76091	+	57.5460i	−0.217192	+	1.84864i
$970$		0			0
$971$	44.4504			1.42648			0.713240	−	0.700920i	$-0.247227\pi$
$971$	44.4504			1.42648			0.713240	+	0.700920i	$0.247227\pi$
$972$		0			0
$973$	−10.6075			−0.340061
$974$		0			0
$975$			0.0545478i			0.00174693i
$976$		0			0
$977$			37.1402i			1.18822i	0.804384	+	0.594110i	$0.202496\pi$
$977$			37.1402i			1.18822i	−0.804384	+	0.594110i	$0.797504\pi$
$978$		0			0
$979$	−2.53135			−0.0809022
$980$		0			0
$981$		−	66.4092i		−	2.12028i
$982$		0			0
$983$	−28.8053			−0.918745			−0.459373	−	0.888244i	$-0.651926\pi$
$983$	−28.8053			−0.918745			−0.459373	+	0.888244i	$0.651926\pi$
$984$		0			0
$985$	20.6632			0.658385
$986$		0			0
$987$	−2.13383			−0.0679205
$988$		0			0
$989$	11.7966			0.375112
$990$		0			0
$991$	50.2559			1.59643			0.798216	−	0.602371i	$-0.205777\pi$
$991$	50.2559			1.59643			0.798216	+	0.602371i	$0.205777\pi$
$992$		0			0
$993$	−31.8838			−1.01180
$994$		0			0
$995$			46.8145i			1.48412i
$996$		0			0
$997$	21.3135			0.675007			0.337503	−	0.941324i	$-0.390418\pi$
$997$	21.3135			0.675007			0.337503	+	0.941324i	$0.390418\pi$
$998$		0			0
$999$		−	104.697i		−	3.31246i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.2	yes	64
4.3	odd	2	inner	3344.2.o.b.1519.64	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.63	yes	64
76.75	even	2	inner	3344.2.o.b.1519.1	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.1	✓	64	76.75	even	2	inner
3344.2.o.b.1519.2	yes	64	1.1	even	1	trivial
3344.2.o.b.1519.63	yes	64	19.18	odd	2	inner
3344.2.o.b.1519.64	yes	64	4.3	odd	2	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.22672 q^{3} -2.25376 q^{5} -1.49199i q^{7} +7.41170 q^{9} +O(q^{10})\)	\(q-3.22672 q^{3} -2.25376 q^{5} -1.49199i q^{7} +7.41170 q^{9} -1.00000i q^{11} -0.212830i q^{13} +7.27224 q^{15} +4.11960 q^{17} +(0.508616 - 4.32912i) q^{19} +4.81423i q^{21} -1.98895i q^{23} +0.0794300 q^{25} -14.2353 q^{27} -2.13403i q^{29} +0.884846 q^{31} +3.22672i q^{33} +3.36259i q^{35} +7.35472i q^{37} +0.686741i q^{39} +7.14665i q^{41} +5.93109i q^{43} -16.7042 q^{45} +0.443233i q^{47} +4.77396 q^{49} -13.2928 q^{51} +7.34494i q^{53} +2.25376i q^{55} +(-1.64116 + 13.9689i) q^{57} +8.31295 q^{59} +2.57867 q^{61} -11.0582i q^{63} +0.479667i q^{65} -11.0775 q^{67} +6.41778i q^{69} -13.0687 q^{71} +10.6781 q^{73} -0.256298 q^{75} -1.49199 q^{77} +11.0376 q^{79} +23.6982 q^{81} -0.844445i q^{83} -9.28458 q^{85} +6.88590i q^{87} -2.53135i q^{89} -0.317540 q^{91} -2.85515 q^{93} +(-1.14630 + 9.75680i) q^{95} -0.891640i q^{97} -7.41170i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 64 * q + 56 * q^9	\( 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) \) 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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