LMFDB - Embedded newform 2496.2.m.c.2209.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(2209,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.2209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.162447943996702457856.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.14
Root			$-0.140577 - 1.40721i$ of defining polynomial
Character	$\chi$	$=$	2496.2209
Dual form			2496.2.m.c.2209.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.646084 q^{5} +3.09557i q^{7} -1.00000 q^{9} +1.41421 q^{11} +(-2.44949 + 2.64575i) q^{13} +0.646084i q^{15} -5.58258 q^{17} -5.36169 q^{19} -3.09557 q^{21} +3.46410 q^{23} -4.58258 q^{25} -1.00000i q^{27} -4.37780i q^{29} -9.28672i q^{31} +1.41421i q^{33} +2.00000i q^{35} -4.89898 q^{37} +(-2.64575 - 2.44949i) q^{39} +12.4328i q^{41} -5.58258i q^{43} -0.646084 q^{45} +3.74166i q^{47} -2.58258 q^{49} -5.58258i q^{51} -10.5830i q^{53} +0.913701 q^{55} -5.36169i q^{57} +3.65231 q^{59} -4.37780i q^{61} -3.09557i q^{63} +(-1.58258 + 1.70938i) q^{65} +4.77136 q^{67} +3.46410i q^{69} +2.44949i q^{71} +5.65685i q^{73} -4.58258i q^{75} +4.37780i q^{77} +7.11890 q^{79} +1.00000 q^{81} -14.3757 q^{83} -3.60681 q^{85} +4.37780 q^{87} +15.2612i q^{89} +(-8.19012 - 7.58258i) q^{91} +9.28672 q^{93} -3.46410 q^{95} +5.06653i q^{97} -1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) $ 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	0.646084			0.288937			0.144469	−	0.989509i	$-0.453853\pi$
$5$	0.646084			0.288937			0.144469	+	0.989509i	$0.453853\pi$
$6$		0			0
$7$			3.09557i			1.17002i	0.811027	+	0.585008i	$0.198909\pi$
$7$			3.09557i			1.17002i	−0.811027	+	0.585008i	$0.801091\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		0			0
$13$	−2.44949	+	2.64575i	−0.679366	+	0.733799i
$13$	−2.44949	+	2.64575i	−0.679366	+	0.733799i
$14$		0			0
$15$			0.646084i			0.166818i
$16$		0			0
$17$	−5.58258			−1.35397			−0.676987	−	0.735995i	$-0.736715\pi$
$17$	−5.58258			−1.35397			−0.676987	+	0.735995i	$0.736715\pi$
$18$		0			0
$19$	−5.36169			−1.23006			−0.615028	−	0.788505i	$-0.710855\pi$
$19$	−5.36169			−1.23006			−0.615028	+	0.788505i	$0.710855\pi$
$20$		0			0
$21$	−3.09557			−0.675510
$22$		0			0
$23$	3.46410			0.722315			0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410			0.722315			0.361158	+	0.932505i	$0.382382\pi$
$24$		0			0
$25$	−4.58258			−0.916515
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	4.37780i		−	0.812937i	−0.913665	−	0.406469i	$-0.866760\pi$
$29$		−	4.37780i		−	0.812937i	0.913665	−	0.406469i	$-0.133240\pi$
$30$		0			0
$31$		−	9.28672i		−	1.66794i	−0.551807	−	0.833972i	$-0.686061\pi$
$31$		−	9.28672i		−	1.66794i	0.551807	−	0.833972i	$-0.313939\pi$
$32$		0			0
$33$			1.41421i			0.246183i
$34$		0			0
$35$			2.00000i			0.338062i
$36$		0			0
$37$	−4.89898			−0.805387			−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898			−0.805387			−0.402694	+	0.915335i	$0.631926\pi$
$38$		0			0
$39$	−2.64575	−	2.44949i	−0.423659	−	0.392232i
$40$		0			0
$41$			12.4328i			1.94167i	0.239747	+	0.970835i	$0.422935\pi$
$41$			12.4328i			1.94167i	−0.239747	+	0.970835i	$0.577065\pi$
$42$		0			0
$43$		−	5.58258i		−	0.851335i	−0.904880	−	0.425667i	$-0.860039\pi$
$43$		−	5.58258i		−	0.851335i	0.904880	−	0.425667i	$-0.139961\pi$
$44$		0			0
$45$	−0.646084			−0.0963125
$46$		0			0
$47$			3.74166i			0.545777i	0.962046	+	0.272888i	$0.0879790\pi$
$47$			3.74166i			0.545777i	−0.962046	+	0.272888i	$0.912021\pi$
$48$		0			0
$49$	−2.58258			−0.368939
$50$		0			0
$51$		−	5.58258i		−	0.781717i
$52$		0			0
$53$		−	10.5830i		−	1.45369i	−0.686803	−	0.726844i	$-0.740986\pi$
$53$		−	10.5830i		−	1.45369i	0.686803	−	0.726844i	$-0.259014\pi$
$54$		0			0
$55$	0.913701			0.123203
$56$		0			0
$57$		−	5.36169i		−	0.710173i
$58$		0			0
$59$	3.65231			0.475491			0.237745	−	0.971328i	$-0.423592\pi$
$59$	3.65231			0.475491			0.237745	+	0.971328i	$0.423592\pi$
$60$		0			0
$61$		−	4.37780i		−	0.560520i	−0.959924	−	0.280260i	$-0.909579\pi$
$61$		−	4.37780i		−	0.560520i	0.959924	−	0.280260i	$-0.0904207\pi$
$62$		0			0
$63$		−	3.09557i		−	0.390006i
$64$		0			0
$65$	−1.58258	+	1.70938i	−0.196294	+	0.212022i
$66$		0			0
$67$	4.77136			0.582915			0.291457	−	0.956584i	$-0.405860\pi$
$67$	4.77136			0.582915			0.291457	+	0.956584i	$0.405860\pi$
$68$		0			0
$69$			3.46410i			0.417029i
$70$		0			0
$71$			2.44949i			0.290701i	0.989380	+	0.145350i	$0.0464310\pi$
$71$			2.44949i			0.290701i	−0.989380	+	0.145350i	$0.953569\pi$
$72$		0			0
$73$			5.65685i			0.662085i	0.943616	+	0.331042i	$0.107400\pi$
$73$			5.65685i			0.662085i	−0.943616	+	0.331042i	$0.892600\pi$
$74$		0			0
$75$		−	4.58258i		−	0.529150i
$76$		0			0
$77$			4.37780i			0.498897i
$78$		0			0
$79$	7.11890			0.800939			0.400470	−	0.916310i	$-0.368847\pi$
$79$	7.11890			0.800939			0.400470	+	0.916310i	$0.368847\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−14.3757			−1.57794			−0.788969	−	0.614433i	$-0.789385\pi$
$83$	−14.3757			−1.57794			−0.788969	+	0.614433i	$0.789385\pi$
$84$		0			0
$85$	−3.60681			−0.391214
$86$		0			0
$87$	4.37780			0.469350
$88$		0			0
$89$			15.2612i			1.61768i	0.588027	+	0.808841i	$0.299905\pi$
$89$			15.2612i			1.61768i	−0.588027	+	0.808841i	$0.700095\pi$
$90$		0			0
$91$	−8.19012	−	7.58258i	−0.858558	−	0.794870i
$92$		0			0
$93$	9.28672			0.962988
$94$		0			0
$95$	−3.46410			−0.355409
$96$		0			0
$97$			5.06653i			0.514428i	0.966354	+	0.257214i	$0.0828045\pi$
$97$			5.06653i			0.514428i	−0.966354	+	0.257214i	$0.917195\pi$
$98$		0			0
$99$	−1.41421			−0.142134
$100$		0			0
$101$		−	10.5830i		−	1.05305i	−0.850160	−	0.526524i	$-0.823495\pi$
$101$		−	10.5830i		−	1.05305i	0.850160	−	0.526524i	$-0.176505\pi$
$102$		0			0
$103$	−11.3060			−1.11401			−0.557007	−	0.830508i	$-0.688050\pi$
$103$	−11.3060			−1.11401			−0.557007	+	0.830508i	$0.688050\pi$
$104$		0			0
$105$	−2.00000			−0.195180
$106$		0			0
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$	9.79796			0.938474			0.469237	−	0.883072i	$-0.344529\pi$
$109$	9.79796			0.938474			0.469237	+	0.883072i	$0.344529\pi$
$110$		0			0
$111$		−	4.89898i		−	0.464991i
$112$		0			0
$113$	−16.7477			−1.57549			−0.787747	−	0.615999i	$-0.788752\pi$
$113$	−16.7477			−1.57549			−0.787747	+	0.615999i	$0.788752\pi$
$114$		0			0
$115$	2.23810			0.208704
$116$		0			0
$117$	2.44949	−	2.64575i	0.226455	−	0.244600i
$118$		0			0
$119$		−	17.2813i		−	1.58417i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	−12.4328			−1.12102
$124$		0			0
$125$	−6.19115			−0.553753
$126$		0			0
$127$	20.9753			1.86126			0.930629	−	0.365964i	$-0.119261\pi$
$127$	20.9753			1.86126			0.930629	+	0.365964i	$0.119261\pi$
$128$		0			0
$129$	5.58258			0.491518
$130$		0			0
$131$		−	6.00000i		−	0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$		−	6.00000i		−	0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$		0			0
$133$		−	16.5975i		−	1.43919i
$134$		0			0
$135$		−	0.646084i		−	0.0556060i
$136$		0			0
$137$		−	14.0805i		−	1.20298i	−0.798880	−	0.601490i	$-0.794574\pi$
$137$		−	14.0805i		−	1.20298i	0.798880	−	0.601490i	$-0.205426\pi$
$138$		0			0
$139$		−	8.74773i		−	0.741973i	−0.928638	−	0.370986i	$-0.879020\pi$
$139$		−	8.74773i		−	0.741973i	0.928638	−	0.370986i	$-0.120980\pi$
$140$		0			0
$141$	−3.74166			−0.315104
$142$		0			0
$143$	−3.46410	+	3.74166i	−0.289683	+	0.312893i
$144$		0			0
$145$		−	2.82843i		−	0.234888i
$146$		0			0
$147$		−	2.58258i		−	0.213007i
$148$		0			0
$149$	−17.9274			−1.46867			−0.734333	−	0.678789i	$-0.762505\pi$
$149$	−17.9274			−1.46867			−0.734333	+	0.678789i	$0.762505\pi$
$150$		0			0
$151$			1.80341i			0.146759i	0.997304	+	0.0733795i	$0.0233784\pi$
$151$			1.80341i			0.146759i	−0.997304	+	0.0733795i	$0.976622\pi$
$152$		0			0
$153$	5.58258			0.451324
$154$		0			0
$155$		−	6.00000i		−	0.481932i
$156$		0			0
$157$			16.5975i			1.32463i	0.749228	+	0.662313i	$0.230425\pi$
$157$			16.5975i			1.32463i	−0.749228	+	0.662313i	$0.769575\pi$
$158$		0			0
$159$	10.5830			0.839287
$160$		0			0
$161$			10.7234i			0.845121i
$162$		0			0
$163$	−25.1607			−1.97074			−0.985368	−	0.170439i	$-0.945482\pi$
$163$	−25.1607			−1.97074			−0.985368	+	0.170439i	$0.945482\pi$
$164$		0			0
$165$			0.913701i			0.0711315i
$166$		0			0
$167$			19.7308i			1.52681i	0.645919	+	0.763406i	$0.276474\pi$
$167$			19.7308i			1.52681i	−0.645919	+	0.763406i	$0.723526\pi$
$168$		0			0
$169$	−1.00000	−	12.9615i	−0.0769231	−	0.997037i
$170$		0			0
$171$	5.36169			0.410019
$172$		0			0
$173$			13.8564i			1.05348i	0.850026	+	0.526742i	$0.176586\pi$
$173$			13.8564i			1.05348i	−0.850026	+	0.526742i	$0.823414\pi$
$174$		0			0
$175$		−	14.1857i		−	1.07234i
$176$		0			0
$177$			3.65231i			0.274525i
$178$		0			0
$179$			10.0000i			0.747435i	0.927543	+	0.373718i	$0.121917\pi$
$179$			10.0000i			0.747435i	−0.927543	+	0.373718i	$0.878083\pi$
$180$		0			0
$181$		−	0.913701i		−	0.0679148i	−0.999423	−	0.0339574i	$-0.989189\pi$
$181$		−	0.913701i		−	0.0679148i	0.999423	−	0.0339574i	$-0.0108111\pi$
$182$		0			0
$183$	4.37780			0.323616
$184$		0			0
$185$	−3.16515			−0.232707
$186$		0			0
$187$	−7.89495			−0.577336
$188$		0			0
$189$	3.09557			0.225170
$190$		0			0
$191$	10.3923			0.751961			0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923			0.751961			0.375980	+	0.926628i	$0.377306\pi$
$192$		0			0
$193$			7.89495i			0.568291i	0.958781	+	0.284146i	$0.0917099\pi$
$193$			7.89495i			0.568291i	−0.958781	+	0.284146i	$0.908290\pi$
$194$		0			0
$195$	−1.70938	−	1.58258i	−0.122411	−	0.113331i
$196$		0			0
$197$	−22.8263			−1.62631			−0.813155	−	0.582048i	$-0.802252\pi$
$197$	−22.8263			−1.62631			−0.813155	+	0.582048i	$0.802252\pi$
$198$		0			0
$199$	3.46410			0.245564			0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410			0.245564			0.122782	+	0.992434i	$0.460818\pi$
$200$		0			0
$201$			4.77136i			0.336546i
$202$		0			0
$203$	13.5518			0.951150
$204$		0			0
$205$			8.03260i			0.561021i
$206$		0			0
$207$	−3.46410			−0.240772
$208$		0			0
$209$	−7.58258			−0.524498
$210$		0			0
$211$		−	19.1652i		−	1.31938i	−0.751536	−	0.659692i	$-0.770687\pi$
$211$		−	19.1652i		−	1.31938i	0.751536	−	0.659692i	$-0.229313\pi$
$212$		0			0
$213$	−2.44949			−0.167836
$214$		0			0
$215$		−	3.60681i		−	0.245983i
$216$		0			0
$217$	28.7477			1.95152
$218$		0			0
$219$	−5.65685			−0.382255
$220$		0			0
$221$	13.6745	−	14.7701i	0.919844	−	0.993545i
$222$		0			0
$223$			14.4554i			0.968005i	0.875067	+	0.484002i	$0.160817\pi$
$223$			14.4554i			0.968005i	−0.875067	+	0.484002i	$0.839183\pi$
$224$		0			0
$225$	4.58258			0.305505
$226$		0			0
$227$	23.4513			1.55652			0.778259	−	0.627944i	$-0.216103\pi$
$227$	23.4513			1.55652			0.778259	+	0.627944i	$0.216103\pi$
$228$		0			0
$229$	−1.29217			−0.0853888			−0.0426944	−	0.999088i	$-0.513594\pi$
$229$	−1.29217			−0.0853888			−0.0426944	+	0.999088i	$0.513594\pi$
$230$		0			0
$231$	−4.37780			−0.288038
$232$		0			0
$233$	−24.3303			−1.59393			−0.796966	−	0.604025i	$-0.793563\pi$
$233$	−24.3303			−1.59393			−0.796966	+	0.604025i	$0.793563\pi$
$234$		0			0
$235$			2.41742i			0.157695i
$236$		0			0
$237$			7.11890i			0.462422i
$238$		0			0
$239$		−	7.61816i		−	0.492778i	−0.969171	−	0.246389i	$-0.920756\pi$
$239$		−	7.61816i		−	0.492778i	0.969171	−	0.246389i	$-0.0792441\pi$
$240$		0			0
$241$		−	7.30463i		−	0.470532i	−0.971931	−	0.235266i	$-0.924404\pi$
$241$		−	7.30463i		−	0.470532i	0.971931	−	0.235266i	$-0.0755962\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.66856			−0.106600
$246$		0			0
$247$	13.1334	−	14.1857i	0.835659	−	0.902614i
$248$		0			0
$249$		−	14.3757i		−	0.911023i
$250$		0			0
$251$			7.16515i			0.452260i	0.974097	+	0.226130i	$0.0726075\pi$
$251$			7.16515i			0.452260i	−0.974097	+	0.226130i	$0.927393\pi$
$252$		0			0
$253$	4.89898			0.307996
$254$		0			0
$255$		−	3.60681i		−	0.225867i
$256$		0			0
$257$	−9.16515			−0.571706			−0.285853	−	0.958273i	$-0.592277\pi$
$257$	−9.16515			−0.571706			−0.285853	+	0.958273i	$0.592277\pi$
$258$		0			0
$259$		−	15.1652i		−	0.942317i
$260$		0			0
$261$			4.37780i			0.270979i
$262$		0			0
$263$	−8.75560			−0.539894			−0.269947	−	0.962875i	$-0.587006\pi$
$263$	−8.75560			−0.539894			−0.269947	+	0.962875i	$0.587006\pi$
$264$		0			0
$265$		−	6.83751i		−	0.420025i
$266$		0			0
$267$	−15.2612			−0.933969
$268$		0			0
$269$		−	0.723000i		−	0.0440821i	−0.999757	−	0.0220410i	$-0.992984\pi$
$269$		−	0.723000i		−	0.0440821i	0.999757	−	0.0220410i	$-0.00701645\pi$
$270$		0			0
$271$		−	5.41022i		−	0.328647i	−0.986406	−	0.164324i	$-0.947456\pi$
$271$		−	5.41022i		−	0.328647i	0.986406	−	0.164324i	$-0.0525441\pi$
$272$		0			0
$273$	7.58258	−	8.19012i	0.458918	−	0.495688i
$274$		0			0
$275$	−6.48074			−0.390803
$276$		0			0
$277$			10.2016i			0.612955i	0.951878	+	0.306478i	$0.0991504\pi$
$277$			10.2016i			0.612955i	−0.951878	+	0.306478i	$0.900850\pi$
$278$		0			0
$279$			9.28672i			0.555981i
$280$		0			0
$281$			30.4608i			1.81714i	0.417736	+	0.908569i	$0.362824\pi$
$281$			30.4608i			1.81714i	−0.417736	+	0.908569i	$0.637176\pi$
$282$		0			0
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$		−	3.46410i		−	0.205196i
$286$		0			0
$287$	−38.4865			−2.27179
$288$		0			0
$289$	14.1652			0.833244
$290$		0			0
$291$	−5.06653			−0.297005
$292$		0			0
$293$	7.10692			0.415191			0.207595	−	0.978215i	$-0.433436\pi$
$293$	7.10692			0.415191			0.207595	+	0.978215i	$0.433436\pi$
$294$		0			0
$295$	2.35970			0.137387
$296$		0			0
$297$		−	1.41421i		−	0.0820610i
$298$		0			0
$299$	−8.48528	+	9.16515i	−0.490716	+	0.530034i
$300$		0			0
$301$	17.2813			0.996076
$302$		0			0
$303$	10.5830			0.607978
$304$		0			0
$305$		−	2.82843i		−	0.161955i
$306$		0			0
$307$	0.885491			0.0505376			0.0252688	−	0.999681i	$-0.491956\pi$
$307$	0.885491			0.0505376			0.0252688	+	0.999681i	$0.491956\pi$
$308$		0			0
$309$		−	11.3060i		−	0.643176i
$310$		0			0
$311$	−8.56490			−0.485671			−0.242836	−	0.970067i	$-0.578078\pi$
$311$	−8.56490			−0.485671			−0.242836	+	0.970067i	$0.578078\pi$
$312$		0			0
$313$	−15.1652			−0.857185			−0.428593	−	0.903498i	$-0.640990\pi$
$313$	−15.1652			−0.857185			−0.428593	+	0.903498i	$0.640990\pi$
$314$		0			0
$315$		−	2.00000i		−	0.112687i
$316$		0			0
$317$	−8.12940			−0.456593			−0.228296	−	0.973592i	$-0.573315\pi$
$317$	−8.12940			−0.456593			−0.228296	+	0.973592i	$0.573315\pi$
$318$		0			0
$319$		−	6.19115i		−	0.346638i
$320$		0			0
$321$	−2.00000			−0.111629
$322$		0			0
$323$	29.9320			1.66546
$324$		0			0
$325$	11.2250	−	12.1244i	0.622649	−	0.672538i
$326$		0			0
$327$			9.79796i			0.541828i
$328$		0			0
$329$	−11.5826			−0.638568
$330$		0			0
$331$	−12.0760			−0.663756			−0.331878	−	0.943322i	$-0.607682\pi$
$331$	−12.0760			−0.663756			−0.331878	+	0.943322i	$0.607682\pi$
$332$		0			0
$333$	4.89898			0.268462
$334$		0			0
$335$	3.08270			0.168426
$336$		0			0
$337$	14.3303			0.780621			0.390311	−	0.920683i	$-0.372368\pi$
$337$	14.3303			0.780621			0.390311	+	0.920683i	$0.372368\pi$
$338$		0			0
$339$		−	16.7477i		−	0.909612i
$340$		0			0
$341$		−	13.1334i		−	0.711214i
$342$		0			0
$343$			13.6745i			0.738352i
$344$		0			0
$345$			2.23810i			0.120495i
$346$		0			0
$347$			29.4955i			1.58340i	0.610911	+	0.791699i	$0.290803\pi$
$347$			29.4955i			1.58340i	−0.610911	+	0.791699i	$0.709197\pi$
$348$		0			0
$349$	−17.5510			−0.939482			−0.469741	−	0.882804i	$-0.655653\pi$
$349$	−17.5510			−0.939482			−0.469741	+	0.882804i	$0.655653\pi$
$350$		0			0
$351$	2.64575	+	2.44949i	0.141220	+	0.130744i
$352$		0			0
$353$			10.7850i			0.574027i	0.957927	+	0.287013i	$0.0926624\pi$
$353$			10.7850i			0.574027i	−0.957927	+	0.287013i	$0.907338\pi$
$354$		0			0
$355$			1.58258i			0.0839944i
$356$		0			0
$357$	17.2813			0.914622
$358$		0			0
$359$		−	5.03383i		−	0.265675i	−0.991138	−	0.132838i	$-0.957591\pi$
$359$		−	5.03383i		−	0.265675i	0.991138	−	0.132838i	$-0.0424089\pi$
$360$		0			0
$361$	9.74773			0.513038
$362$		0			0
$363$		−	9.00000i		−	0.472377i
$364$		0			0
$365$			3.65480i			0.191301i
$366$		0			0
$367$	−21.8890			−1.14260			−0.571299	−	0.820742i	$-0.693560\pi$
$367$	−21.8890			−1.14260			−0.571299	+	0.820742i	$0.693560\pi$
$368$		0			0
$369$		−	12.4328i		−	0.647224i
$370$		0			0
$371$	32.7605			1.70084
$372$		0			0
$373$		−	22.8027i		−	1.18068i	−0.807155	−	0.590340i	$-0.798994\pi$
$373$		−	22.8027i		−	1.18068i	0.807155	−	0.590340i	$-0.201006\pi$
$374$		0			0
$375$		−	6.19115i		−	0.319709i
$376$		0			0
$377$	11.5826	+	10.7234i	0.596533	+	0.552282i
$378$		0			0
$379$	7.00946			0.360052			0.180026	−	0.983662i	$-0.442382\pi$
$379$	7.00946			0.360052			0.180026	+	0.983662i	$0.442382\pi$
$380$		0			0
$381$			20.9753i			1.07460i
$382$		0			0
$383$		−	4.01135i		−	0.204970i	−0.994735	−	0.102485i	$-0.967321\pi$
$383$		−	4.01135i		−	0.204970i	0.994735	−	0.102485i	$-0.0326794\pi$
$384$		0			0
$385$			2.82843i			0.144150i
$386$		0			0
$387$			5.58258i			0.283778i
$388$		0			0
$389$			5.82380i			0.295278i	0.989041	+	0.147639i	$0.0471675\pi$
$389$			5.82380i			0.295278i	−0.989041	+	0.147639i	$0.952833\pi$
$390$		0			0
$391$	−19.3386			−0.977996
$392$		0			0
$393$	6.00000			0.302660
$394$		0			0
$395$	4.59941			0.231421
$396$		0			0
$397$	23.2027			1.16451			0.582256	−	0.813006i	$-0.302170\pi$
$397$	23.2027			1.16451			0.582256	+	0.813006i	$0.302170\pi$
$398$		0			0
$399$	16.5975			0.830915
$400$		0			0
$401$		−	14.6709i		−	0.732628i	−0.930491	−	0.366314i	$-0.880620\pi$
$401$		−	14.6709i		−	0.732628i	0.930491	−	0.366314i	$-0.119380\pi$
$402$		0			0
$403$	24.5704	+	22.7477i	1.22394	+	1.13314i
$404$		0			0
$405$	0.646084			0.0321042
$406$		0			0
$407$	−6.92820			−0.343418
$408$		0			0
$409$		−	10.7234i		−	0.530237i	−0.964216	−	0.265119i	$-0.914589\pi$
$409$		−	10.7234i		−	0.530237i	0.964216	−	0.265119i	$-0.0854111\pi$
$410$		0			0
$411$	14.0805			0.694541
$412$		0			0
$413$			11.3060i			0.556332i
$414$		0			0
$415$	−9.28790			−0.455925
$416$		0			0
$417$	8.74773			0.428378
$418$		0			0
$419$			8.33030i			0.406962i	0.979079	+	0.203481i	$0.0652255\pi$
$419$			8.33030i			0.406962i	−0.979079	+	0.203481i	$0.934774\pi$
$420$		0			0
$421$	−22.4499			−1.09414			−0.547072	−	0.837086i	$-0.684257\pi$
$421$	−22.4499			−1.09414			−0.547072	+	0.837086i	$0.684257\pi$
$422$		0			0
$423$		−	3.74166i		−	0.181926i
$424$		0			0
$425$	25.5826			1.24094
$426$		0			0
$427$	13.5518			0.655818
$428$		0			0
$429$	−3.74166	−	3.46410i	−0.180649	−	0.167248i
$430$		0			0
$431$			26.9444i			1.29787i	0.760846	+	0.648933i	$0.224784\pi$
$431$			26.9444i			1.29787i	−0.760846	+	0.648933i	$0.775216\pi$
$432$		0			0
$433$	−3.25227			−0.156294			−0.0781471	−	0.996942i	$-0.524900\pi$
$433$	−3.25227			−0.156294			−0.0781471	+	0.996942i	$0.524900\pi$
$434$		0			0
$435$	2.82843			0.135613
$436$		0			0
$437$	−18.5734			−0.888488
$438$		0			0
$439$	11.3060			0.539606			0.269803	−	0.962916i	$-0.413041\pi$
$439$	11.3060			0.539606			0.269803	+	0.962916i	$0.413041\pi$
$440$		0			0
$441$	2.58258			0.122980
$442$		0			0
$443$		−	2.83485i		−	0.134688i	−0.997730	−	0.0673439i	$-0.978548\pi$
$443$		−	2.83485i		−	0.134688i	0.997730	−	0.0673439i	$-0.0214525\pi$
$444$		0			0
$445$			9.86001i			0.467409i
$446$		0			0
$447$		−	17.9274i		−	0.847935i
$448$		0			0
$449$		−	9.01400i		−	0.425397i	−0.977118	−	0.212699i	$-0.931775\pi$
$449$		−	9.01400i		−	0.425397i	0.977118	−	0.212699i	$-0.0682252\pi$
$450$		0			0
$451$			17.5826i			0.827931i
$452$		0			0
$453$	−1.80341			−0.0847314
$454$		0			0
$455$	−5.29150	−	4.89898i	−0.248069	−	0.229668i
$456$		0			0
$457$		−	6.24718i		−	0.292231i	−0.989268	−	0.146115i	$-0.953323\pi$
$457$		−	6.24718i		−	0.292231i	0.989268	−	0.146115i	$-0.0466771\pi$
$458$		0			0
$459$			5.58258i			0.260572i
$460$		0			0
$461$	36.2311			1.68745			0.843725	−	0.536775i	$-0.180358\pi$
$461$	36.2311			1.68745			0.843725	+	0.536775i	$0.180358\pi$
$462$		0			0
$463$		−	2.07310i		−	0.0963450i	−0.998839	−	0.0481725i	$-0.984660\pi$
$463$		−	2.07310i		−	0.0963450i	0.998839	−	0.0481725i	$-0.0153397\pi$
$464$		0			0
$465$	6.00000			0.278243
$466$		0			0
$467$			36.6606i			1.69645i	0.529636	+	0.848225i	$0.322329\pi$
$467$			36.6606i			1.69645i	−0.529636	+	0.848225i	$0.677671\pi$
$468$		0			0
$469$			14.7701i			0.682020i
$470$		0			0
$471$	−16.5975			−0.764773
$472$		0			0
$473$		−	7.89495i		−	0.363010i
$474$		0			0
$475$	24.5704			1.12737
$476$		0			0
$477$			10.5830i			0.484563i
$478$		0			0
$479$			33.4052i			1.52632i	0.646207	+	0.763162i	$0.276354\pi$
$479$			33.4052i			1.52632i	−0.646207	+	0.763162i	$0.723646\pi$
$480$		0			0
$481$	12.0000	−	12.9615i	0.547153	−	0.590993i
$482$		0			0
$483$	−10.7234			−0.487931
$484$		0			0
$485$			3.27340i			0.148637i
$486$		0			0
$487$			20.1072i			0.911142i	0.890199	+	0.455571i	$0.150565\pi$
$487$			20.1072i			0.911142i	−0.890199	+	0.455571i	$0.849435\pi$
$488$		0			0
$489$		−	25.1607i		−	1.13781i
$490$		0			0
$491$			34.3303i			1.54930i	0.632387	+	0.774652i	$0.282075\pi$
$491$			34.3303i			1.54930i	−0.632387	+	0.774652i	$0.717925\pi$
$492$		0			0
$493$			24.4394i			1.10070i
$494$		0			0
$495$	−0.913701			−0.0410678
$496$		0			0
$497$	−7.58258			−0.340125
$498$		0			0
$499$	−14.4373			−0.646302			−0.323151	−	0.946347i	$-0.604742\pi$
$499$	−14.4373			−0.646302			−0.323151	+	0.946347i	$0.604742\pi$
$500$		0			0
$501$	−19.7308			−0.881506
$502$		0			0
$503$	40.1232			1.78901			0.894503	−	0.447062i	$-0.147530\pi$
$503$	40.1232			1.78901			0.894503	+	0.447062i	$0.147530\pi$
$504$		0			0
$505$		−	6.83751i		−	0.304265i
$506$		0			0
$507$	12.9615	−	1.00000i	0.575640	−	0.0444116i
$508$		0			0
$509$	−11.4665			−0.508245			−0.254122	−	0.967172i	$-0.581787\pi$
$509$	−11.4665			−0.508245			−0.254122	+	0.967172i	$0.581787\pi$
$510$		0			0
$511$	−17.5112			−0.774650
$512$		0			0
$513$			5.36169i			0.236724i
$514$		0			0
$515$	−7.30463			−0.321880
$516$		0			0
$517$			5.29150i			0.232720i
$518$		0			0
$519$	−13.8564			−0.608229
$520$		0			0
$521$	2.41742			0.105909			0.0529546	−	0.998597i	$-0.483136\pi$
$521$	2.41742			0.105909			0.0529546	+	0.998597i	$0.483136\pi$
$522$		0			0
$523$		−	1.58258i		−	0.0692012i	−0.999401	−	0.0346006i	$-0.988984\pi$
$523$		−	1.58258i		−	0.0692012i	0.999401	−	0.0346006i	$-0.0110159\pi$
$524$		0			0
$525$	14.1857			0.619115
$526$		0			0
$527$			51.8438i			2.25835i
$528$		0			0
$529$	−11.0000			−0.478261
$530$		0			0
$531$	−3.65231			−0.158497
$532$		0			0
$533$	−32.8940	−	30.4539i	−1.42480	−	1.31911i
$534$		0			0
$535$			1.29217i			0.0558653i
$536$		0			0
$537$	−10.0000			−0.431532
$538$		0			0
$539$	−3.65231			−0.157316
$540$		0			0
$541$	38.4391			1.65262			0.826312	−	0.563213i	$-0.190435\pi$
$541$	38.4391			1.65262			0.826312	+	0.563213i	$0.190435\pi$
$542$		0			0
$543$	0.913701			0.0392106
$544$		0			0
$545$	6.33030			0.271160
$546$		0			0
$547$			11.9129i			0.509358i	0.967026	+	0.254679i	$0.0819698\pi$
$547$			11.9129i			0.509358i	−0.967026	+	0.254679i	$0.918030\pi$
$548$		0			0
$549$			4.37780i			0.186840i
$550$		0			0
$551$			23.4724i			0.999959i
$552$		0			0
$553$			22.0371i			0.937112i
$554$		0			0
$555$		−	3.16515i		−	0.134353i
$556$		0			0
$557$	34.9389			1.48041			0.740205	−	0.672381i	$-0.234728\pi$
$557$	34.9389			1.48041			0.740205	+	0.672381i	$0.234728\pi$
$558$		0			0
$559$	14.7701	+	13.6745i	0.624709	+	0.578368i
$560$		0			0
$561$		−	7.89495i		−	0.333325i
$562$		0			0
$563$			1.16515i			0.0491053i	0.999699	+	0.0245526i	$0.00781613\pi$
$563$			1.16515i			0.0491053i	−0.999699	+	0.0245526i	$0.992184\pi$
$564$		0			0
$565$	−10.8204			−0.455219
$566$		0			0
$567$			3.09557i			0.130002i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$		−	9.66970i		−	0.404664i	−0.979317	−	0.202332i	$-0.935148\pi$
$571$		−	9.66970i		−	0.404664i	0.979317	−	0.202332i	$-0.0648520\pi$
$572$		0			0
$573$			10.3923i			0.434145i
$574$		0			0
$575$	−15.8745			−0.662013
$576$		0			0
$577$		−	23.2177i		−	0.966567i	−0.875464	−	0.483284i	$-0.839444\pi$
$577$		−	23.2177i		−	0.966567i	0.875464	−	0.483284i	$-0.160556\pi$
$578$		0			0
$579$	−7.89495			−0.328103
$580$		0			0
$581$		−	44.5010i		−	1.84621i
$582$		0			0
$583$		−	14.9666i		−	0.619854i
$584$		0			0
$585$	1.58258	−	1.70938i	0.0654315	−	0.0706740i
$586$		0			0
$587$	8.25172			0.340585			0.170293	−	0.985394i	$-0.445529\pi$
$587$	8.25172			0.340585			0.170293	+	0.985394i	$0.445529\pi$
$588$		0			0
$589$			49.7925i			2.05167i
$590$		0			0
$591$		−	22.8263i		−	0.938950i
$592$		0			0
$593$			14.6709i			0.602460i	0.953552	+	0.301230i	$0.0973972\pi$
$593$			14.6709i			0.602460i	−0.953552	+	0.301230i	$0.902603\pi$
$594$		0			0
$595$		−	11.1652i		−	0.457727i
$596$		0			0
$597$			3.46410i			0.141776i
$598$		0			0
$599$	0.381401			0.0155836			0.00779181	−	0.999970i	$-0.497520\pi$
$599$	0.381401			0.0155836			0.00779181	+	0.999970i	$0.497520\pi$
$600$		0			0
$601$	23.1652			0.944926			0.472463	−	0.881350i	$-0.343365\pi$
$601$	23.1652			0.944926			0.472463	+	0.881350i	$0.343365\pi$
$602$		0			0
$603$	−4.77136			−0.194305
$604$		0			0
$605$	−5.81475			−0.236403
$606$		0			0
$607$	25.1624			1.02131			0.510655	−	0.859785i	$-0.329403\pi$
$607$	25.1624			1.02131			0.510655	+	0.859785i	$0.329403\pi$
$608$		0			0
$609$			13.5518i			0.549147i
$610$		0			0
$611$	−9.89949	−	9.16515i	−0.400491	−	0.370782i
$612$		0			0
$613$	−25.0343			−1.01113			−0.505563	−	0.862790i	$-0.668715\pi$
$613$	−25.0343			−1.01113			−0.505563	+	0.862790i	$0.668715\pi$
$614$		0			0
$615$	−8.03260			−0.323906
$616$		0			0
$617$			0.528723i			0.0212856i	0.999943	+	0.0106428i	$0.00338777\pi$
$617$			0.528723i			0.0212856i	−0.999943	+	0.0106428i	$0.996612\pi$
$618$		0			0
$619$	−15.4947			−0.622786			−0.311393	−	0.950281i	$-0.600796\pi$
$619$	−15.4947			−0.622786			−0.311393	+	0.950281i	$0.600796\pi$
$620$		0			0
$621$		−	3.46410i		−	0.139010i
$622$		0			0
$623$	−47.2421			−1.89272
$624$		0			0
$625$	18.9129			0.756515
$626$		0			0
$627$		−	7.58258i		−	0.302819i
$628$		0			0
$629$	27.3489			1.09047
$630$		0			0
$631$		−	28.8826i		−	1.14980i	−0.818224	−	0.574900i	$-0.805041\pi$
$631$		−	28.8826i		−	1.14980i	0.818224	−	0.574900i	$-0.194959\pi$
$632$		0			0
$633$	19.1652			0.761746
$634$		0			0
$635$	13.5518			0.537787
$636$		0			0
$637$	6.32599	−	6.83285i	0.250645	−	0.270727i
$638$		0			0
$639$		−	2.44949i		−	0.0969003i
$640$		0			0
$641$	−6.41742			−0.253473			−0.126737	−	0.991936i	$-0.540450\pi$
$641$	−6.41742			−0.253473			−0.126737	+	0.991936i	$0.540450\pi$
$642$		0			0
$643$	6.54234			0.258005			0.129002	−	0.991644i	$-0.458822\pi$
$643$	6.54234			0.258005			0.129002	+	0.991644i	$0.458822\pi$
$644$		0			0
$645$	3.60681			0.142018
$646$		0			0
$647$	43.3966			1.70610			0.853049	−	0.521831i	$-0.174751\pi$
$647$	43.3966			1.70610			0.853049	+	0.521831i	$0.174751\pi$
$648$		0			0
$649$	5.16515			0.202750
$650$		0			0
$651$			28.7477i			1.12671i
$652$		0			0
$653$			18.6156i			0.728485i	0.931304	+	0.364243i	$0.118672\pi$
$653$			18.6156i			0.728485i	−0.931304	+	0.364243i	$0.881328\pi$
$654$		0			0
$655$		−	3.87650i		−	0.151468i
$656$		0			0
$657$		−	5.65685i		−	0.220695i
$658$		0			0
$659$			24.8348i			0.967428i	0.875226	+	0.483714i	$0.160713\pi$
$659$			24.8348i			0.967428i	−0.875226	+	0.483714i	$0.839287\pi$
$660$		0			0
$661$	10.0677			0.391586			0.195793	−	0.980645i	$-0.437272\pi$
$661$	10.0677			0.391586			0.195793	+	0.980645i	$0.437272\pi$
$662$		0			0
$663$	14.7701	+	13.6745i	0.573623	+	0.531072i
$664$		0			0
$665$		−	10.7234i		−	0.415835i
$666$		0			0
$667$		−	15.1652i		−	0.587197i
$668$		0			0
$669$	−14.4554			−0.558878
$670$		0			0
$671$		−	6.19115i		−	0.239007i
$672$		0			0
$673$	16.4174			0.632845			0.316423	−	0.948618i	$-0.397518\pi$
$673$	16.4174			0.632845			0.316423	+	0.948618i	$0.397518\pi$
$674$		0			0
$675$			4.58258i			0.176383i
$676$		0			0
$677$		−	6.92820i		−	0.266272i	−0.991098	−	0.133136i	$-0.957495\pi$
$677$		−	6.92820i		−	0.266272i	0.991098	−	0.133136i	$-0.0425048\pi$
$678$		0			0
$679$	−15.6838			−0.601889
$680$		0			0
$681$			23.4513i			0.898656i
$682$		0			0
$683$	33.5844			1.28507			0.642535	−	0.766256i	$-0.277883\pi$
$683$	33.5844			1.28507			0.642535	+	0.766256i	$0.277883\pi$
$684$		0			0
$685$		−	9.09720i		−	0.347586i
$686$		0			0
$687$		−	1.29217i		−	0.0492993i
$688$		0			0
$689$	28.0000	+	25.9230i	1.06672	+	0.987586i
$690$		0			0
$691$	26.2181			0.997385			0.498692	−	0.866779i	$-0.333814\pi$
$691$	26.2181			0.997385			0.498692	+	0.866779i	$0.333814\pi$
$692$		0			0
$693$		−	4.37780i		−	0.166299i
$694$		0			0
$695$		−	5.65176i		−	0.214384i
$696$		0			0
$697$		−	69.4068i		−	2.62897i
$698$		0			0
$699$		−	24.3303i		−	0.920257i
$700$		0			0
$701$			39.0188i			1.47372i	0.676045	+	0.736860i	$0.263692\pi$
$701$			39.0188i			1.47372i	−0.676045	+	0.736860i	$0.736308\pi$
$702$		0			0
$703$	26.2668			0.990672
$704$		0			0
$705$	−2.41742			−0.0910455
$706$		0			0
$707$	32.7605			1.23208
$708$		0			0
$709$	12.6520			0.475155			0.237578	−	0.971369i	$-0.423647\pi$
$709$	12.6520			0.475155			0.237578	+	0.971369i	$0.423647\pi$
$710$		0			0
$711$	−7.11890			−0.266980
$712$		0			0
$713$		−	32.1701i		−	1.20478i
$714$		0			0
$715$	−2.23810	+	2.41742i	−0.0837002	+	0.0904065i
$716$		0			0
$717$	7.61816			0.284505
$718$		0			0
$719$	−38.4865			−1.43530			−0.717652	−	0.696401i	$-0.754783\pi$
$719$	−38.4865			−1.43530			−0.717652	+	0.696401i	$0.754783\pi$
$720$		0			0
$721$		−	34.9986i		−	1.30341i
$722$		0			0
$723$	7.30463			0.271662
$724$		0			0
$725$			20.0616i			0.745069i
$726$		0			0
$727$	−46.8607			−1.73797			−0.868984	−	0.494840i	$-0.835227\pi$
$727$	−46.8607			−1.73797			−0.868984	+	0.494840i	$0.835227\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			31.1652i			1.15268i
$732$		0			0
$733$	9.79796			0.361896			0.180948	−	0.983493i	$-0.442083\pi$
$733$	9.79796			0.361896			0.180948	+	0.983493i	$0.442083\pi$
$734$		0			0
$735$		−	1.66856i		−	0.0615458i
$736$		0			0
$737$	6.74773			0.248556
$738$		0			0
$739$	−43.7790			−1.61044			−0.805219	−	0.592978i	$-0.797952\pi$
$739$	−43.7790			−1.61044			−0.805219	+	0.592978i	$0.797952\pi$
$740$		0			0
$741$	14.1857	+	13.1334i	0.521125	+	0.482468i
$742$		0			0
$743$		−	21.0229i		−	0.771257i	−0.922654	−	0.385628i	$-0.873985\pi$
$743$		−	21.0229i		−	0.771257i	0.922654	−	0.385628i	$-0.126015\pi$
$744$		0			0
$745$	−11.5826			−0.424353
$746$		0			0
$747$	14.3757			0.525979
$748$		0			0
$749$	−6.19115			−0.226220
$750$		0			0
$751$	2.55040			0.0930655			0.0465327	−	0.998917i	$-0.485183\pi$
$751$	2.55040			0.0930655			0.0465327	+	0.998917i	$0.485183\pi$
$752$		0			0
$753$	−7.16515			−0.261113
$754$		0			0
$755$			1.16515i			0.0424042i
$756$		0			0
$757$		−	25.1624i		−	0.914543i	−0.889327	−	0.457272i	$-0.848827\pi$
$757$		−	25.1624i		−	0.914543i	0.889327	−	0.457272i	$-0.151173\pi$
$758$		0			0
$759$			4.89898i			0.177822i
$760$		0			0
$761$		−	20.3277i		−	0.736879i	−0.929652	−	0.368440i	$-0.879892\pi$
$761$		−	20.3277i		−	0.736879i	0.929652	−	0.368440i	$-0.120108\pi$
$762$		0			0
$763$			30.3303i			1.09803i
$764$		0			0
$765$	3.60681			0.130405
$766$		0			0
$767$	−8.94630	+	9.66311i	−0.323032	+	0.348915i
$768$		0			0
$769$		−	10.2563i		−	0.369850i	−0.982753	−	0.184925i	$-0.940796\pi$
$769$		−	10.2563i		−	0.369850i	0.982753	−	0.184925i	$-0.0592043\pi$
$770$		0			0
$771$		−	9.16515i		−	0.330075i
$772$		0			0
$773$	−26.4331			−0.950734			−0.475367	−	0.879788i	$-0.657685\pi$
$773$	−26.4331			−0.950734			−0.475367	+	0.879788i	$0.657685\pi$
$774$		0			0
$775$			42.5571i			1.52870i
$776$		0			0
$777$	15.1652			0.544047
$778$		0			0
$779$		−	66.6606i		−	2.38836i
$780$		0			0
$781$			3.46410i			0.123955i
$782$		0			0
$783$	−4.37780			−0.156450
$784$		0			0
$785$			10.7234i			0.382734i
$786$		0			0
$787$	−12.6663			−0.451505			−0.225753	−	0.974185i	$-0.572484\pi$
$787$	−12.6663			−0.451505			−0.225753	+	0.974185i	$0.572484\pi$
$788$		0			0
$789$		−	8.75560i		−	0.311708i
$790$		0			0
$791$		−	51.8438i		−	1.84335i
$792$		0			0
$793$	11.5826	+	10.7234i	0.411309	+	0.380798i
$794$		0			0
$795$	6.83751			0.242501
$796$		0			0
$797$			13.1334i			0.465209i	0.972571	+	0.232605i	$0.0747248\pi$
$797$			13.1334i			0.465209i	−0.972571	+	0.232605i	$0.925275\pi$
$798$		0			0
$799$		−	20.8881i		−	0.738967i
$800$		0			0
$801$		−	15.2612i		−	0.539227i
$802$		0			0
$803$			8.00000i			0.282314i
$804$		0			0
$805$			6.92820i			0.244187i
$806$		0			0
$807$	0.723000			0.0254508
$808$		0			0
$809$	16.7477			0.588819			0.294409	−	0.955679i	$-0.404877\pi$
$809$	16.7477			0.588819			0.294409	+	0.955679i	$0.404877\pi$
$810$		0			0
$811$	41.4177			1.45437			0.727186	−	0.686440i	$-0.240828\pi$
$811$	41.4177			1.45437			0.727186	+	0.686440i	$0.240828\pi$
$812$		0			0
$813$	5.41022			0.189745
$814$		0			0
$815$	−16.2559			−0.569420
$816$		0			0
$817$			29.9320i			1.04719i
$818$		0			0
$819$	8.19012	+	7.58258i	0.286186	+	0.264957i
$820$		0			0
$821$	48.6134			1.69662			0.848310	−	0.529500i	$-0.177621\pi$
$821$	48.6134			1.69662			0.848310	+	0.529500i	$0.177621\pi$
$822$		0			0
$823$	2.93180			0.102196			0.0510981	−	0.998694i	$-0.483728\pi$
$823$	2.93180			0.102196			0.0510981	+	0.998694i	$0.483728\pi$
$824$		0			0
$825$		−	6.48074i		−	0.225630i
$826$		0			0
$827$	−31.3463			−1.09002			−0.545008	−	0.838431i	$-0.683473\pi$
$827$	−31.3463			−1.09002			−0.545008	+	0.838431i	$0.683473\pi$
$828$		0			0
$829$		−	42.6736i		−	1.48212i	−0.671441	−	0.741058i	$-0.734324\pi$
$829$		−	42.6736i		−	1.48212i	0.671441	−	0.741058i	$-0.265676\pi$
$830$		0			0
$831$	−10.2016			−0.353890
$832$		0			0
$833$	14.4174			0.499534
$834$		0			0
$835$			12.7477i			0.441153i
$836$		0			0
$837$	−9.28672			−0.320996
$838$		0			0
$839$		−	39.5964i		−	1.36702i	−0.729942	−	0.683509i	$-0.760453\pi$
$839$		−	39.5964i		−	1.36702i	0.729942	−	0.683509i	$-0.239547\pi$
$840$		0			0
$841$	9.83485			0.339133
$842$		0			0
$843$	−30.4608			−1.04912
$844$		0			0
$845$	−0.646084	−	8.37420i	−0.0222260	−	0.288081i
$846$		0			0
$847$		−	27.8602i		−	0.957286i
$848$		0			0
$849$	−12.0000			−0.411839
$850$		0			0
$851$	−16.9706			−0.581743
$852$		0			0
$853$	−14.6969			−0.503214			−0.251607	−	0.967830i	$-0.580959\pi$
$853$	−14.6969			−0.503214			−0.251607	+	0.967830i	$0.580959\pi$
$854$		0			0
$855$	3.46410			0.118470
$856$		0			0
$857$	−26.8348			−0.916661			−0.458330	−	0.888782i	$-0.651552\pi$
$857$	−26.8348			−0.916661			−0.458330	+	0.888782i	$0.651552\pi$
$858$		0			0
$859$		−	44.6606i		−	1.52380i	−0.647695	−	0.761900i	$-0.724267\pi$
$859$		−	44.6606i		−	1.52380i	0.647695	−	0.761900i	$-0.275733\pi$
$860$		0			0
$861$		−	38.4865i		−	1.31162i
$862$		0			0
$863$		−	13.5396i		−	0.460894i	−0.973085	−	0.230447i	$-0.925981\pi$
$863$		−	13.5396i		−	0.460894i	0.973085	−	0.230447i	$-0.0740188\pi$
$864$		0			0
$865$			8.95240i			0.304391i
$866$		0			0
$867$			14.1652i			0.481074i
$868$		0			0
$869$	10.0677			0.341522
$870$		0			0
$871$	−11.6874	+	12.6238i	−0.396013	+	0.427743i
$872$		0			0
$873$		−	5.06653i		−	0.171476i
$874$		0			0
$875$		−	19.1652i		−	0.647900i
$876$		0			0
$877$	−26.0568			−0.879874			−0.439937	−	0.898029i	$-0.644999\pi$
$877$	−26.0568			−0.879874			−0.439937	+	0.898029i	$0.644999\pi$
$878$		0			0
$879$			7.10692i			0.239711i
$880$		0			0
$881$	40.3303			1.35876			0.679381	−	0.733786i	$-0.262248\pi$
$881$	40.3303			1.35876			0.679381	+	0.733786i	$0.262248\pi$
$882$		0			0
$883$		−	49.4955i		−	1.66565i	−0.553533	−	0.832827i	$-0.686721\pi$
$883$		−	49.4955i		−	1.66565i	0.553533	−	0.832827i	$-0.313279\pi$
$884$		0			0
$885$			2.35970i			0.0793205i
$886$		0			0
$887$	54.5517			1.83167			0.915834	−	0.401557i	$-0.131531\pi$
$887$	54.5517			1.83167			0.915834	+	0.401557i	$0.131531\pi$
$888$		0			0
$889$			64.9306i			2.17770i
$890$		0			0
$891$	1.41421			0.0473779
$892$		0			0
$893$		−	20.0616i		−	0.671336i
$894$		0			0
$895$			6.46084i			0.215962i
$896$		0			0
$897$	−9.16515	−	8.48528i	−0.306015	−	0.283315i
$898$		0			0
$899$	−40.6554			−1.35593
$900$		0			0
$901$			59.0804i			1.96825i
$902$		0			0
$903$			17.2813i			0.575085i
$904$		0			0
$905$		−	0.590327i		−	0.0196231i
$906$		0			0
$907$			40.6606i			1.35011i	0.737766	+	0.675057i	$0.235881\pi$
$907$			40.6606i			1.35011i	−0.737766	+	0.675057i	$0.764119\pi$
$908$		0			0
$909$			10.5830i			0.351016i
$910$		0			0
$911$	−17.3205			−0.573854			−0.286927	−	0.957952i	$-0.592634\pi$
$911$	−17.3205			−0.573854			−0.286927	+	0.957952i	$0.592634\pi$
$912$		0			0
$913$	−20.3303			−0.672835
$914$		0			0
$915$	2.82843			0.0935049
$916$		0			0
$917$	18.5734			0.613349
$918$		0			0
$919$	54.1703			1.78691			0.893457	−	0.449149i	$-0.148273\pi$
$919$	54.1703			1.78691			0.893457	+	0.449149i	$0.148273\pi$
$920$		0			0
$921$			0.885491i			0.0291779i
$922$		0			0
$923$	−6.48074	−	6.00000i	−0.213316	−	0.197492i
$924$		0			0
$925$	22.4499			0.738150
$926$		0			0
$927$	11.3060			0.371338
$928$		0			0
$929$			31.0511i			1.01875i	0.860544	+	0.509377i	$0.170124\pi$
$929$			31.0511i			1.01875i	−0.860544	+	0.509377i	$0.829876\pi$
$930$		0			0
$931$	13.8470			0.453816
$932$		0			0
$933$		−	8.56490i		−	0.280402i
$934$		0			0
$935$	−5.10080			−0.166814
$936$		0			0
$937$	−0.0871215			−0.00284614			−0.00142307	−	0.999999i	$-0.500453\pi$
$937$	−0.0871215			−0.00284614			−0.00142307	+	0.999999i	$0.500453\pi$
$938$		0			0
$939$		−	15.1652i		−	0.494896i
$940$		0			0
$941$	−31.3321			−1.02140			−0.510699	−	0.859760i	$-0.670613\pi$
$941$	−31.3321			−1.02140			−0.510699	+	0.859760i	$0.670613\pi$
$942$		0			0
$943$			43.0683i			1.40250i
$944$		0			0
$945$	2.00000			0.0650600
$946$		0			0
$947$	−14.3757			−0.467147			−0.233574	−	0.972339i	$-0.575042\pi$
$947$	−14.3757			−0.467147			−0.233574	+	0.972339i	$0.575042\pi$
$948$		0			0
$949$	−14.9666	−	13.8564i	−0.485837	−	0.449798i
$950$		0			0
$951$		−	8.12940i		−	0.263614i
$952$		0			0
$953$	−25.1652			−0.815179			−0.407590	−	0.913165i	$-0.633631\pi$
$953$	−25.1652			−0.815179			−0.407590	+	0.913165i	$0.633631\pi$
$954$		0			0
$955$	6.71430			0.217270
$956$		0			0
$957$	6.19115			0.200131
$958$		0			0
$959$	43.5873			1.40751
$960$		0			0
$961$	−55.2432			−1.78204
$962$		0			0
$963$		−	2.00000i		−	0.0644491i
$964$		0			0
$965$			5.10080i			0.164201i
$966$		0			0
$967$			40.5121i			1.30278i	0.758742	+	0.651391i	$0.225814\pi$
$967$			40.5121i			1.30278i	−0.758742	+	0.651391i	$0.774186\pi$
$968$		0			0
$969$			29.9320i			0.961556i
$970$		0			0
$971$		−	33.4955i		−	1.07492i	−0.843289	−	0.537460i	$-0.819384\pi$
$971$		−	33.4955i		−	1.07492i	0.843289	−	0.537460i	$-0.180616\pi$
$972$		0			0
$973$	27.0792			0.868120
$974$		0			0
$975$	12.1244	+	11.2250i	0.388290	+	0.359487i
$976$		0			0
$977$			0.528723i			0.0169153i	0.999964	+	0.00845767i	$0.00269219\pi$
$977$			0.528723i			0.0169153i	−0.999964	+	0.00845767i	$0.997308\pi$
$978$		0			0
$979$			21.5826i			0.689782i
$980$		0			0
$981$	−9.79796			−0.312825
$982$		0			0
$983$		−	13.2699i		−	0.423245i	−0.977351	−	0.211622i	$-0.932125\pi$
$983$		−	13.2699i		−	0.423245i	0.977351	−	0.211622i	$-0.0678747\pi$
$984$		0			0
$985$	−14.7477			−0.469902
$986$		0			0
$987$		−	11.5826i		−	0.368677i
$988$		0			0
$989$		−	19.3386i		−	0.614932i
$990$		0			0
$991$	29.1986			0.927525			0.463762	−	0.885960i	$-0.346499\pi$
$991$	29.1986			0.927525			0.463762	+	0.885960i	$0.346499\pi$
$992$		0			0
$993$		−	12.0760i		−	0.383220i
$994$		0			0
$995$	2.23810			0.0709525
$996$		0			0
$997$			0.381401i			0.0120791i	0.999982	+	0.00603954i	$0.00192246\pi$
$997$			0.381401i			0.0120791i	−0.999982	+	0.00603954i	$0.998078\pi$
$998$		0			0
$999$			4.89898i			0.154997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.m.c.2209.14	yes	16
4.3	odd	2	inner	2496.2.m.c.2209.5	yes	16
8.3	odd	2	inner	2496.2.m.c.2209.12	yes	16
8.5	even	2	inner	2496.2.m.c.2209.3	✓	16
13.12	even	2	inner	2496.2.m.c.2209.11	yes	16
52.51	odd	2	inner	2496.2.m.c.2209.4	yes	16
104.51	odd	2	inner	2496.2.m.c.2209.13	yes	16
104.77	even	2	inner	2496.2.m.c.2209.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.m.c.2209.3	✓	16	8.5	even	2	inner
2496.2.m.c.2209.4	yes	16	52.51	odd	2	inner
2496.2.m.c.2209.5	yes	16	4.3	odd	2	inner
2496.2.m.c.2209.6	yes	16	104.77	even	2	inner
2496.2.m.c.2209.11	yes	16	13.12	even	2	inner
2496.2.m.c.2209.12	yes	16	8.3	odd	2	inner
2496.2.m.c.2209.13	yes	16	104.51	odd	2	inner
2496.2.m.c.2209.14	yes	16	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.646084 q^{5} +3.09557i q^{7} -1.00000 q^{9} +1.41421 q^{11} +(-2.44949 + 2.64575i) q^{13} +0.646084i q^{15} -5.58258 q^{17} -5.36169 q^{19} -3.09557 q^{21} +3.46410 q^{23} -4.58258 q^{25} -1.00000i q^{27} -4.37780i q^{29} -9.28672i q^{31} +1.41421i q^{33} +2.00000i q^{35} -4.89898 q^{37} +(-2.64575 - 2.44949i) q^{39} +12.4328i q^{41} -5.58258i q^{43} -0.646084 q^{45} +3.74166i q^{47} -2.58258 q^{49} -5.58258i q^{51} -10.5830i q^{53} +0.913701 q^{55} -5.36169i q^{57} +3.65231 q^{59} -4.37780i q^{61} -3.09557i q^{63} +(-1.58258 + 1.70938i) q^{65} +4.77136 q^{67} +3.46410i q^{69} +2.44949i q^{71} +5.65685i q^{73} -4.58258i q^{75} +4.37780i q^{77} +7.11890 q^{79} +1.00000 q^{81} -14.3757 q^{83} -3.60681 q^{85} +4.37780 q^{87} +15.2612i q^{89} +(-8.19012 - 7.58258i) q^{91} +9.28672 q^{93} -3.46410 q^{95} +5.06653i q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) \) 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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