LMFDB - Embedded newform 2940.2.f.a.1469.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1469,2940)] 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("2940.1469"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1469.21
Character	$\chi$	$=$	2940.1469
Dual form			2940.2.f.a.1469.23

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.917271 - 1.46922i) q^{3} +(-0.536879 - 2.17066i) q^{5} +(-1.31723 - 2.69535i) q^{9} -0.915904i q^{11} -4.31153 q^{13} +(-3.68164 - 1.20229i) q^{15} -0.537062i q^{17} -1.30351i q^{19} -7.98961 q^{23} +(-4.42352 + 2.33076i) q^{25} +(-5.16832 - 0.537062i) q^{27} +3.46154i q^{29} +6.42144i q^{31} +(-1.34567 - 0.840132i) q^{33} +5.04345i q^{37} +(-3.95484 + 6.33460i) q^{39} +9.89809 q^{41} +8.22096i q^{43} +(-5.14349 + 4.30633i) q^{45} -4.34132i q^{47} +(-0.789063 - 0.492631i) q^{51} -1.96985 q^{53} +(-1.98812 + 0.491730i) q^{55} +(-1.91514 - 1.19567i) q^{57} +14.3073 q^{59} -9.68095i q^{61} +(2.31477 + 9.35886i) q^{65} -12.1070i q^{67} +(-7.32863 + 11.7385i) q^{69} +0.943900i q^{71} -10.2242 q^{73} +(-0.633157 + 8.63708i) q^{75} -11.4363 q^{79} +(-5.52981 + 7.10079i) q^{81} -9.35240i q^{83} +(-1.16578 + 0.288337i) q^{85} +(5.08577 + 3.17517i) q^{87} -1.74901 q^{89} +(9.43453 + 5.89020i) q^{93} +(-2.82947 + 0.699827i) q^{95} +10.7844 q^{97} +(-2.46868 + 1.20646i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) $ 32 * q + 2 * q^15 - 12 * q^25 - 48 * q^39 + 20 * q^51 + 56 * q^79 + 40 * q^81 - 4 * q^85 - 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	0.917271	−	1.46922i	0.529586	−	0.848256i
$3$	0.917271	−	1.46922i	0.529586	−	0.848256i
$4$		0			0
$5$	−0.536879	−	2.17066i	−0.240100	−	0.970748i
$5$	−0.536879	−	2.17066i	−0.240100	−	0.970748i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.31723	−	2.69535i	−0.439076	−	0.898450i
$10$		0			0
$11$		−	0.915904i		−	0.276155i	−0.990421	−	0.138078i	$-0.955908\pi$
$11$		−	0.915904i		−	0.276155i	0.990421	−	0.138078i	$-0.0440924\pi$
$12$		0			0
$13$	−4.31153			−1.19580			−0.597902	−	0.801569i	$-0.703999\pi$
$13$	−4.31153			−1.19580			−0.597902	+	0.801569i	$0.703999\pi$
$14$		0			0
$15$	−3.68164	−	1.20229i	−0.950597	−	0.310429i
$16$		0			0
$17$		−	0.537062i		−	0.130257i	−0.997877	−	0.0651283i	$-0.979254\pi$
$17$		−	0.537062i		−	0.130257i	0.997877	−	0.0651283i	$-0.0207457\pi$
$18$		0			0
$19$		−	1.30351i		−	0.299045i	−0.988758	−	0.149523i	$-0.952226\pi$
$19$		−	1.30351i		−	0.299045i	0.988758	−	0.149523i	$-0.0477737\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.98961			−1.66595			−0.832974	−	0.553312i	$-0.813364\pi$
$23$	−7.98961			−1.66595			−0.832974	+	0.553312i	$0.813364\pi$
$24$		0			0
$25$	−4.42352	+	2.33076i	−0.884704	+	0.466153i
$26$		0			0
$27$	−5.16832	−	0.537062i	−0.994644	−	0.103358i
$28$		0			0
$29$			3.46154i			0.642791i	0.946945	+	0.321396i	$0.104152\pi$
$29$			3.46154i			0.642791i	−0.946945	+	0.321396i	$0.895848\pi$
$30$		0			0
$31$			6.42144i			1.15333i	0.816982	+	0.576663i	$0.195645\pi$
$31$			6.42144i			1.15333i	−0.816982	+	0.576663i	$0.804355\pi$
$32$		0			0
$33$	−1.34567	−	0.840132i	−0.234251	−	0.146248i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.04345i			0.829137i	0.910018	+	0.414569i	$0.136068\pi$
$37$			5.04345i			0.829137i	−0.910018	+	0.414569i	$0.863932\pi$
$38$		0			0
$39$	−3.95484	+	6.33460i	−0.633281	+	1.01435i
$40$		0			0
$41$	9.89809			1.54582			0.772911	−	0.634515i	$-0.218800\pi$
$41$	9.89809			1.54582			0.772911	+	0.634515i	$0.218800\pi$
$42$		0			0
$43$			8.22096i			1.25369i	0.779146	+	0.626843i	$0.215653\pi$
$43$			8.22096i			1.25369i	−0.779146	+	0.626843i	$0.784347\pi$
$44$		0			0
$45$	−5.14349	+	4.30633i	−0.766746	+	0.641950i
$46$		0			0
$47$		−	4.34132i		−	0.633246i	−0.948551	−	0.316623i	$-0.897451\pi$
$47$		−	4.34132i		−	0.633246i	0.948551	−	0.316623i	$-0.102549\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−0.789063	−	0.492631i	−0.110491	−	0.0689821i
$52$		0			0
$53$	−1.96985			−0.270580			−0.135290	−	0.990806i	$-0.543197\pi$
$53$	−1.96985			−0.270580			−0.135290	+	0.990806i	$0.543197\pi$
$54$		0			0
$55$	−1.98812	+	0.491730i	−0.268077	+	0.0663048i
$56$		0			0
$57$	−1.91514	−	1.19567i	−0.253667	−	0.158370i
$58$		0			0
$59$	14.3073			1.86265			0.931323	−	0.364195i	$-0.118656\pi$
$59$	14.3073			1.86265			0.931323	+	0.364195i	$0.118656\pi$
$60$		0			0
$61$		−	9.68095i		−	1.23952i	−0.784792	−	0.619759i	$-0.787230\pi$
$61$		−	9.68095i		−	1.23952i	0.784792	−	0.619759i	$-0.212770\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.31477	+	9.35886i	0.287112	+	1.16082i
$66$		0			0
$67$		−	12.1070i		−	1.47910i	−0.673100	−	0.739551i	$-0.735038\pi$
$67$		−	12.1070i		−	1.47910i	0.673100	−	0.739551i	$-0.264962\pi$
$68$		0			0
$69$	−7.32863	+	11.7385i	−0.882264	+	1.41315i
$70$		0			0
$71$			0.943900i			0.112020i	0.998430	+	0.0560102i	$0.0178379\pi$
$71$			0.943900i			0.112020i	−0.998430	+	0.0560102i	$0.982162\pi$
$72$		0			0
$73$	−10.2242			−1.19665			−0.598326	−	0.801253i	$-0.704167\pi$
$73$	−10.2242			−1.19665			−0.598326	+	0.801253i	$0.704167\pi$
$74$		0			0
$75$	−0.633157	+	8.63708i	−0.0731106	+	0.997324i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−11.4363			−1.28669			−0.643345	−	0.765577i	$-0.722454\pi$
$79$	−11.4363			−1.28669			−0.643345	+	0.765577i	$0.722454\pi$
$80$		0			0
$81$	−5.52981	+	7.10079i	−0.614424	+	0.788976i
$82$		0			0
$83$		−	9.35240i		−	1.02656i	−0.858222	−	0.513279i	$-0.828430\pi$
$83$		−	9.35240i		−	1.02656i	0.858222	−	0.513279i	$-0.171570\pi$
$84$		0			0
$85$	−1.16578	+	0.288337i	−0.126446	+	0.0312746i
$86$		0			0
$87$	5.08577	+	3.17517i	0.545252	+	0.340414i
$88$		0			0
$89$	−1.74901			−0.185395			−0.0926976	−	0.995694i	$-0.529549\pi$
$89$	−1.74901			−0.185395			−0.0926976	+	0.995694i	$0.529549\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	9.43453	+	5.89020i	0.978315	+	0.610785i
$94$		0			0
$95$	−2.82947	+	0.699827i	−0.290298	+	0.0718007i
$96$		0			0
$97$	10.7844			1.09499			0.547494	−	0.836809i	$-0.315582\pi$
$97$	10.7844			1.09499			0.547494	+	0.836809i	$0.315582\pi$
$98$		0			0
$99$	−2.46868	+	1.20646i	−0.248112	+	0.121253i
$100$		0			0
$101$	−9.84960			−0.980072			−0.490036	−	0.871702i	$-0.663016\pi$
$101$	−9.84960			−0.980072			−0.490036	+	0.871702i	$0.663016\pi$
$102$		0			0
$103$	2.45793			0.242188			0.121094	−	0.992641i	$-0.461360\pi$
$103$	2.45793			0.242188			0.121094	+	0.992641i	$0.461360\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.49963			−0.144975			−0.0724873	−	0.997369i	$-0.523094\pi$
$107$	−1.49963			−0.144975			−0.0724873	+	0.997369i	$0.523094\pi$
$108$		0			0
$109$	−3.86452			−0.370154			−0.185077	−	0.982724i	$-0.559253\pi$
$109$	−3.86452			−0.370154			−0.185077	+	0.982724i	$0.559253\pi$
$110$		0			0
$111$	7.40994	+	4.62620i	0.703321	+	0.439100i
$112$		0			0
$113$	−10.1791			−0.957567			−0.478783	−	0.877933i	$-0.658922\pi$
$113$	−10.1791			−0.957567			−0.478783	+	0.877933i	$0.658922\pi$
$114$		0			0
$115$	4.28945	+	17.3427i	0.399994	+	1.61722i
$116$		0			0
$117$	5.67927	+	11.6211i	0.525049	+	1.07437i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.1611			0.923738
$122$		0			0
$123$	9.07922	−	14.5425i	0.818646	−	1.31125i
$124$		0			0
$125$	7.43419	+	8.35062i	0.664934	+	0.746902i
$126$		0			0
$127$		−	9.73798i		−	0.864106i	−0.901848	−	0.432053i	$-0.857789\pi$
$127$		−	9.73798i		−	0.864106i	0.901848	−	0.432053i	$-0.142211\pi$
$128$		0			0
$129$	12.0784	+	7.54085i	1.06345	+	0.663935i
$130$		0			0
$131$	−17.6002			−1.53773			−0.768867	−	0.639409i	$-0.779179\pi$
$131$	−17.6002			−1.53773			−0.768867	+	0.639409i	$0.779179\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.60899	+	11.5070i	0.138480	+	0.990365i
$136$		0			0
$137$	−10.4195			−0.890194			−0.445097	−	0.895482i	$-0.646831\pi$
$137$	−10.4195			−0.890194			−0.445097	+	0.895482i	$0.646831\pi$
$138$		0			0
$139$			7.09182i			0.601520i	0.953700	+	0.300760i	$0.0972403\pi$
$139$			7.09182i			0.601520i	−0.953700	+	0.300760i	$0.902760\pi$
$140$		0			0
$141$	−6.37836	−	3.98216i	−0.537155	−	0.335359i
$142$		0			0
$143$			3.94895i			0.330228i
$144$		0			0
$145$	7.51382	−	1.85843i	0.623988	−	0.154334i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.26524i		−	0.513269i	−0.966509	−	0.256634i	$-0.917386\pi$
$149$		−	6.26524i		−	0.513269i	0.966509	−	0.256634i	$-0.0826136\pi$
$150$		0			0
$151$	−17.7116			−1.44135			−0.720673	−	0.693275i	$-0.756167\pi$
$151$	−17.7116			−1.44135			−0.720673	+	0.693275i	$0.756167\pi$
$152$		0			0
$153$	−1.44757	+	0.707433i	−0.117029	+	0.0571926i
$154$		0			0
$155$	13.9388	−	3.44754i	1.11959	−	0.276913i
$156$		0			0
$157$	−0.773945			−0.0617676			−0.0308838	−	0.999523i	$-0.509832\pi$
$157$	−0.773945			−0.0617676			−0.0308838	+	0.999523i	$0.509832\pi$
$158$		0			0
$159$	−1.80689	+	2.89415i	−0.143296	+	0.229521i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	6.88703i		−	0.539434i	−0.962940	−	0.269717i	$-0.913070\pi$
$163$		−	6.88703i		−	0.539434i	0.962940	−	0.269717i	$-0.0869302\pi$
$164$		0			0
$165$	−1.10118	+	3.37203i	−0.0857267	+	0.262512i
$166$		0			0
$167$			9.49709i			0.734907i	0.930042	+	0.367453i	$0.119770\pi$
$167$			9.49709i			0.734907i	−0.930042	+	0.367453i	$0.880230\pi$
$168$		0			0
$169$	5.58930			0.429946
$170$		0			0
$171$	−3.51341	+	1.71702i	−0.268677	+	0.131304i
$172$		0			0
$173$		−	11.9498i		−	0.908529i	−0.890867	−	0.454264i	$-0.849902\pi$
$173$		−	11.9498i		−	0.908529i	0.890867	−	0.454264i	$-0.150098\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	13.1236	−	21.0205i	0.986432	−	1.58000i
$178$		0			0
$179$			18.7457i			1.40112i	0.713593	+	0.700561i	$0.247067\pi$
$179$			18.7457i			1.40112i	−0.713593	+	0.700561i	$0.752933\pi$
$180$		0			0
$181$			16.1024i			1.19688i	0.801167	+	0.598441i	$0.204213\pi$
$181$			16.1024i			1.19688i	−0.801167	+	0.598441i	$0.795787\pi$
$182$		0			0
$183$	−14.2235	−	8.88005i	−1.05143	−	0.656432i
$184$		0			0
$185$	10.9476	−	2.70772i	0.804884	−	0.199076i
$186$		0			0
$187$	−0.491897			−0.0359711
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	17.3181i		−	1.25309i	−0.779385	−	0.626546i	$-0.784468\pi$
$191$		−	17.3181i		−	1.25309i	0.779385	−	0.626546i	$-0.215532\pi$
$192$		0			0
$193$			15.1303i			1.08911i	0.838727	+	0.544553i	$0.183301\pi$
$193$			15.1303i			1.08911i	−0.838727	+	0.544553i	$0.816699\pi$
$194$		0			0
$195$	15.8735	+	5.18370i	1.13673	+	0.371212i
$196$		0			0
$197$	18.6287			1.32724			0.663619	−	0.748071i	$-0.269020\pi$
$197$	18.6287			1.32724			0.663619	+	0.748071i	$0.269020\pi$
$198$		0			0
$199$			10.0492i			0.712366i	0.934416	+	0.356183i	$0.115922\pi$
$199$			10.0492i			0.712366i	−0.934416	+	0.356183i	$0.884078\pi$
$200$		0			0
$201$	−17.7878	−	11.1054i	−1.25466	−	0.783313i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.31408	−	21.4854i	−0.371151	−	1.50060i
$206$		0			0
$207$	10.5241	+	21.5348i	0.731479	+	1.49677i
$208$		0			0
$209$	−1.19389			−0.0825830
$210$		0			0
$211$	−1.68592			−0.116064			−0.0580319	−	0.998315i	$-0.518483\pi$
$211$	−1.68592			−0.116064			−0.0580319	+	0.998315i	$0.518483\pi$
$212$		0			0
$213$	1.38680	+	0.865812i	0.0950219	+	0.0593244i
$214$		0			0
$215$	17.8449	−	4.41366i	1.21701	−	0.301009i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.37836	+	15.0216i	−0.633731	+	1.01507i
$220$		0			0
$221$			2.31556i			0.155761i
$222$		0			0
$223$	−14.2095			−0.951541			−0.475770	−	0.879570i	$-0.657831\pi$
$223$	−14.2095			−0.951541			−0.475770	+	0.879570i	$0.657831\pi$
$224$		0			0
$225$	12.1090	+	8.85279i	0.807268	+	0.590186i
$226$		0			0
$227$			0.681753i			0.0452496i	0.999744	+	0.0226248i	$0.00720231\pi$
$227$			0.681753i			0.0452496i	−0.999744	+	0.0226248i	$0.992798\pi$
$228$		0			0
$229$		−	2.95734i		−	0.195427i	−0.995215	−	0.0977133i	$-0.968847\pi$
$229$		−	2.95734i		−	0.195427i	0.995215	−	0.0977133i	$-0.0311528\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.9371			−1.76471			−0.882353	−	0.470587i	$-0.844042\pi$
$233$	−26.9371			−1.76471			−0.882353	+	0.470587i	$0.844042\pi$
$234$		0			0
$235$	−9.42352	+	2.33076i	−0.614723	+	0.152042i
$236$		0			0
$237$	−10.4902	+	16.8025i	−0.681413	+	1.09144i
$238$		0			0
$239$			17.3461i			1.12202i	0.827808	+	0.561012i	$0.189588\pi$
$239$			17.3461i			1.12202i	−0.827808	+	0.561012i	$0.810412\pi$
$240$		0			0
$241$		−	1.52746i		−	0.0983921i	−0.998789	−	0.0491961i	$-0.984334\pi$
$241$		−	1.52746i		−	0.0983921i	0.998789	−	0.0491961i	$-0.0156659\pi$
$242$		0			0
$243$	5.36030	+	14.6379i	0.343863	+	0.939020i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			5.62012i			0.357600i
$248$		0			0
$249$	−13.7408	−	8.57868i	−0.870785	−	0.543652i
$250$		0			0
$251$	−6.56268			−0.414233			−0.207116	−	0.978316i	$-0.566408\pi$
$251$	−6.56268			−0.414233			−0.207116	+	0.978316i	$0.566408\pi$
$252$		0			0
$253$			7.31771i			0.460061i
$254$		0			0
$255$	−0.645702	+	1.97727i	−0.0404354	+	0.123821i
$256$		0			0
$257$		−	16.9729i		−	1.05874i	−0.848391	−	0.529370i	$-0.822428\pi$
$257$		−	16.9729i		−	1.05874i	0.848391	−	0.529370i	$-0.177572\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	9.33005	−	4.55964i	0.577516	−	0.282234i
$262$		0			0
$263$	−5.31969			−0.328026			−0.164013	−	0.986458i	$-0.552444\pi$
$263$	−5.31969			−0.328026			−0.164013	+	0.986458i	$0.552444\pi$
$264$		0			0
$265$	1.05757	+	4.27588i	0.0649662	+	0.262665i
$266$		0			0
$267$	−1.60432	+	2.56969i	−0.0981828	+	0.157263i
$268$		0			0
$269$	−6.55668			−0.399768			−0.199884	−	0.979820i	$-0.564057\pi$
$269$	−6.55668			−0.399768			−0.199884	+	0.979820i	$0.564057\pi$
$270$		0			0
$271$			5.24431i			0.318569i	0.987233	+	0.159285i	$0.0509187\pi$
$271$			5.24431i			0.318569i	−0.987233	+	0.159285i	$0.949081\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.13476	+	4.05152i	0.128731	+	0.244316i
$276$		0			0
$277$		−	2.77326i		−	0.166629i	−0.996523	−	0.0833145i	$-0.973449\pi$
$277$		−	2.77326i		−	0.166629i	0.996523	−	0.0833145i	$-0.0265506\pi$
$278$		0			0
$279$	17.3080	−	8.45851i	1.03620	−	0.506398i
$280$		0			0
$281$		−	22.4373i		−	1.33850i	−0.743038	−	0.669250i	$-0.766616\pi$
$281$		−	22.4373i		−	1.33850i	0.743038	−	0.669250i	$-0.233384\pi$
$282$		0			0
$283$	7.24680			0.430778			0.215389	−	0.976528i	$-0.430898\pi$
$283$	7.24680			0.430778			0.215389	+	0.976528i	$0.430898\pi$
$284$		0			0
$285$	−1.56719	+	4.79906i	−0.0928324	+	0.284272i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.7116			0.983033
$290$		0			0
$291$	9.89220	−	15.8447i	0.579891	−	0.928831i
$292$		0			0
$293$		−	26.1686i		−	1.52879i	−0.644750	−	0.764393i	$-0.723039\pi$
$293$		−	26.1686i		−	1.52879i	0.644750	−	0.764393i	$-0.276961\pi$
$294$		0			0
$295$	−7.68127	−	31.0562i	−0.447221	−	1.80816i
$296$		0			0
$297$	−0.491897	+	4.73369i	−0.0285428	+	0.274676i
$298$		0			0
$299$	34.4474			1.99215
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.03475	+	14.4712i	−0.519033	+	0.831352i
$304$		0			0
$305$	−21.0140	+	5.19750i	−1.20326	+	0.297608i
$306$		0			0
$307$	22.4144			1.27926			0.639630	−	0.768683i	$-0.279088\pi$
$307$	22.4144			1.27926			0.639630	+	0.768683i	$0.279088\pi$
$308$		0			0
$309$	2.25459	−	3.61125i	0.128259	−	0.205437i
$310$		0			0
$311$	19.6276			1.11298			0.556488	−	0.830855i	$-0.312148\pi$
$311$	19.6276			1.11298			0.556488	+	0.830855i	$0.312148\pi$
$312$		0			0
$313$	−12.8333			−0.725380			−0.362690	−	0.931910i	$-0.618142\pi$
$313$	−12.8333			−0.725380			−0.362690	+	0.931910i	$0.618142\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.9690			−0.896908			−0.448454	−	0.893806i	$-0.648025\pi$
$317$	−15.9690			−0.896908			−0.448454	+	0.893806i	$0.648025\pi$
$318$		0			0
$319$	3.17044			0.177510
$320$		0			0
$321$	−1.37557	+	2.20329i	−0.0767766	+	0.122976i
$322$		0			0
$323$	−0.700065			−0.0389526
$324$		0			0
$325$	19.0722	−	10.0492i	1.05793	−	0.557427i
$326$		0			0
$327$	−3.54481	+	5.67784i	−0.196029	+	0.313985i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−19.6378			−1.07939			−0.539694	−	0.841861i	$-0.681460\pi$
$331$	−19.6378			−1.07939			−0.539694	+	0.841861i	$0.681460\pi$
$332$		0			0
$333$	13.5938	−	6.64337i	0.744938	−	0.364055i
$334$		0			0
$335$	−26.2801	+	6.49998i	−1.43584	+	0.355132i
$336$		0			0
$337$			0.304237i			0.0165729i	0.999966	+	0.00828643i	$0.00263768\pi$
$337$			0.304237i			0.0165729i	−0.999966	+	0.00828643i	$0.997362\pi$
$338$		0			0
$339$	−9.33696	+	14.9553i	−0.507114	+	0.812262i
$340$		0			0
$341$	5.88143			0.318497
$342$		0			0
$343$		0			0
$344$		0			0
$345$	29.4149	+	9.60580i	1.58364	+	0.517159i
$346$		0			0
$347$	−28.4469			−1.52711			−0.763556	−	0.645742i	$-0.776548\pi$
$347$	−28.4469			−1.52711			−0.763556	+	0.645742i	$0.776548\pi$
$348$		0			0
$349$		−	13.9719i		−	0.747897i	−0.927450	−	0.373948i	$-0.878004\pi$
$349$		−	13.9719i		−	0.747897i	0.927450	−	0.373948i	$-0.121996\pi$
$350$		0			0
$351$	22.2834	+	2.31556i	1.18940	+	0.123595i
$352$		0			0
$353$		−	34.6036i		−	1.84176i	−0.389843	−	0.920881i	$-0.627471\pi$
$353$		−	34.6036i		−	1.84176i	0.389843	−	0.920881i	$-0.372529\pi$
$354$		0			0
$355$	2.04888	−	0.506760i	0.108744	−	0.0268960i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			22.8931i			1.20825i	0.796889	+	0.604126i	$0.206477\pi$
$359$			22.8931i			1.20825i	−0.796889	+	0.604126i	$0.793523\pi$
$360$		0			0
$361$	17.3009			0.910572
$362$		0			0
$363$	9.32050	−	14.9289i	0.489199	−	0.783566i
$364$		0			0
$365$	5.48916	+	22.1933i	0.287316	+	1.16165i
$366$		0			0
$367$	−28.9997			−1.51377			−0.756887	−	0.653546i	$-0.773281\pi$
$367$	−28.9997			−1.51377			−0.756887	+	0.653546i	$0.773281\pi$
$368$		0			0
$369$	−13.0381	−	26.6788i	−0.678734	−	1.38884i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			24.0309i			1.24427i	0.782909	+	0.622136i	$0.213735\pi$
$373$			24.0309i			1.24427i	−0.782909	+	0.622136i	$0.786265\pi$
$374$		0			0
$375$	19.0881	−	3.26270i	0.985704	−	0.168485i
$376$		0			0
$377$		−	14.9245i		−	0.768652i
$378$		0			0
$379$	−14.0820			−0.723342			−0.361671	−	0.932306i	$-0.617794\pi$
$379$	−14.0820			−0.723342			−0.361671	+	0.932306i	$0.617794\pi$
$380$		0			0
$381$	−14.3073	−	8.93236i	−0.732983	−	0.457619i
$382$		0			0
$383$		−	29.4030i		−	1.50242i	−0.660062	−	0.751211i	$-0.729470\pi$
$383$		−	29.4030i		−	1.50242i	0.660062	−	0.751211i	$-0.270530\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.1584	−	10.8289i	1.12637	−	0.550464i
$388$		0			0
$389$		−	23.4092i		−	1.18690i	−0.804873	−	0.593448i	$-0.797766\pi$
$389$		−	23.4092i		−	1.18690i	0.804873	−	0.593448i	$-0.202234\pi$
$390$		0			0
$391$			4.29091i			0.217001i
$392$		0			0
$393$	−16.1441	+	25.8586i	−0.814363	+	1.30439i
$394$		0			0
$395$	6.13993	+	24.8244i	0.308934	+	1.24905i
$396$		0			0
$397$	−11.9378			−0.599140			−0.299570	−	0.954074i	$-0.596843\pi$
$397$	−11.9378			−0.599140			−0.299570	+	0.954074i	$0.596843\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.0390i		−	1.25039i	−0.780470	−	0.625193i	$-0.785020\pi$
$401$		−	25.0390i		−	1.25039i	0.780470	−	0.625193i	$-0.214980\pi$
$402$		0			0
$403$		−	27.6863i		−	1.37915i
$404$		0			0
$405$	18.3822	+	8.19108i	0.913420	+	0.407018i
$406$		0			0
$407$	4.61931			0.228971
$408$		0			0
$409$		−	30.1813i		−	1.49237i	−0.665740	−	0.746184i	$-0.731884\pi$
$409$		−	30.1813i		−	1.49237i	0.665740	−	0.746184i	$-0.268116\pi$
$410$		0			0
$411$	−9.55746	+	15.3085i	−0.471435	+	0.755113i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−20.3009	+	5.02111i	−0.996530	+	0.246476i
$416$		0			0
$417$	10.4195	+	6.50511i	0.510243	+	0.318557i
$418$		0			0
$419$	9.85560			0.481477			0.240739	−	0.970590i	$-0.422610\pi$
$419$	9.85560			0.481477			0.240739	+	0.970590i	$0.422610\pi$
$420$		0			0
$421$	−8.16112			−0.397749			−0.198874	−	0.980025i	$-0.563729\pi$
$421$	−8.16112			−0.397749			−0.198874	+	0.980025i	$0.563729\pi$
$422$		0			0
$423$	−11.7014	+	5.71851i	−0.568940	+	0.278044i
$424$		0			0
$425$	1.25176	+	2.37570i	0.0607194	+	0.115239i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	5.80188	+	3.62225i	0.280118	+	0.174884i
$430$		0			0
$431$		−	3.28745i		−	0.158351i	−0.996861	−	0.0791755i	$-0.974771\pi$
$431$		−	3.28745i		−	0.158351i	0.996861	−	0.0791755i	$-0.0252288\pi$
$432$		0			0
$433$	−31.9717			−1.53646			−0.768231	−	0.640173i	$-0.778863\pi$
$433$	−31.9717			−1.53646			−0.768231	+	0.640173i	$0.778863\pi$
$434$		0			0
$435$	4.16176	−	12.7441i	0.199541	−	0.611035i
$436$		0			0
$437$			10.4145i			0.498194i
$438$		0			0
$439$			41.7097i			1.99070i	0.0963398	+	0.995349i	$0.469286\pi$
$439$			41.7097i			1.99070i	−0.0963398	+	0.995349i	$0.530714\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.9577			−0.900708			−0.450354	−	0.892850i	$-0.648702\pi$
$443$	−18.9577			−0.900708			−0.450354	+	0.892850i	$0.648702\pi$
$444$		0			0
$445$	0.939010	+	3.79651i	0.0445133	+	0.179972i
$446$		0			0
$447$	−9.20503	−	5.74692i	−0.435383	−	0.271820i
$448$		0			0
$449$		−	40.0091i		−	1.88815i	−0.329737	−	0.944073i	$-0.606960\pi$
$449$		−	40.0091i		−	1.88815i	0.329737	−	0.944073i	$-0.393040\pi$
$450$		0			0
$451$		−	9.06570i		−	0.426887i
$452$		0			0
$453$	−16.2463	+	26.0222i	−0.763318	+	1.22263i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			11.3868i			0.532652i	0.963883	+	0.266326i	$0.0858098\pi$
$457$			11.3868i			0.532652i	−0.963883	+	0.266326i	$0.914190\pi$
$458$		0			0
$459$	−0.288435	+	2.77571i	−0.0134630	+	0.129559i
$460$		0			0
$461$	8.71020			0.405674			0.202837	−	0.979212i	$-0.434984\pi$
$461$	8.71020			0.405674			0.202837	+	0.979212i	$0.434984\pi$
$462$		0			0
$463$			16.1154i			0.748944i	0.927238	+	0.374472i	$0.122176\pi$
$463$			16.1154i			0.748944i	−0.927238	+	0.374472i	$0.877824\pi$
$464$		0			0
$465$	7.72042	−	23.6415i	0.358026	−	1.09635i
$466$		0			0
$467$			9.10472i			0.421316i	0.977560	+	0.210658i	$0.0675607\pi$
$467$			9.10472i			0.421316i	−0.977560	+	0.210658i	$0.932439\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.709917	+	1.13710i	−0.0327113	+	0.0523947i
$472$		0			0
$473$	7.52961			0.346212
$474$		0			0
$475$	3.03817	+	5.76610i	0.139401	+	0.264567i
$476$		0			0
$477$	2.59475	+	5.30944i	0.118805	+	0.243103i
$478$		0			0
$479$	−26.4185			−1.20709			−0.603546	−	0.797328i	$-0.706246\pi$
$479$	−26.4185			−1.20709			−0.603546	+	0.797328i	$0.706246\pi$
$480$		0			0
$481$		−	21.7450i		−	0.991485i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.78991	−	23.4092i	−0.262906	−	1.06296i
$486$		0			0
$487$			9.16130i			0.415138i	0.978220	+	0.207569i	$0.0665552\pi$
$487$			9.16130i			0.415138i	−0.978220	+	0.207569i	$0.933445\pi$
$488$		0			0
$489$	−10.1186	−	6.31727i	−0.457578	−	0.285677i
$490$		0			0
$491$			33.0800i			1.49288i	0.665453	+	0.746440i	$0.268239\pi$
$491$			33.0800i			1.49288i	−0.665453	+	0.746440i	$0.731761\pi$
$492$		0			0
$493$	1.85906			0.0837278
$494$		0			0
$495$	3.94419	+	4.71094i	0.177278	+	0.211741i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.0873			0.764933			0.382466	−	0.923969i	$-0.375075\pi$
$499$	17.0873			0.764933			0.382466	+	0.923969i	$0.375075\pi$
$500$		0			0
$501$	13.9533	+	8.71140i	0.623389	+	0.389197i
$502$		0			0
$503$		−	21.1753i		−	0.944158i	−0.881556	−	0.472079i	$-0.843504\pi$
$503$		−	21.1753i		−	0.944158i	0.881556	−	0.472079i	$-0.156496\pi$
$504$		0			0
$505$	5.28804	+	21.3801i	0.235315	+	0.951403i
$506$		0			0
$507$	5.12690	−	8.21192i	0.227694	−	0.364704i
$508$		0			0
$509$	−31.1837			−1.38219			−0.691096	−	0.722763i	$-0.742872\pi$
$509$	−31.1837			−1.38219			−0.691096	+	0.722763i	$0.742872\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.700065	+	6.73696i	−0.0309086	+	0.297444i
$514$		0			0
$515$	−1.31961	−	5.33534i	−0.0581491	−	0.235103i
$516$		0			0
$517$	−3.97623			−0.174874
$518$		0			0
$519$	−17.5570	−	10.9612i	−0.770665	−	0.481145i
$520$		0			0
$521$	−38.9111			−1.70473			−0.852363	−	0.522951i	$-0.824831\pi$
$521$	−38.9111			−1.70473			−0.852363	+	0.522951i	$0.824831\pi$
$522$		0			0
$523$	3.76247			0.164521			0.0822606	−	0.996611i	$-0.473786\pi$
$523$	3.76247			0.164521			0.0822606	+	0.996611i	$0.473786\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.44871			0.150228
$528$		0			0
$529$	40.8338			1.77538
$530$		0			0
$531$	−18.8459	−	38.5630i	−0.817844	−	1.67349i
$532$		0			0
$533$	−42.6759			−1.84850
$534$		0			0
$535$	0.805120	+	3.25518i	0.0348084	+	0.140734i
$536$		0			0
$537$	27.5416	+	17.1949i	1.18851	+	0.742015i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	27.2445			1.17133			0.585667	−	0.810552i	$-0.300833\pi$
$541$	27.2445			1.17133			0.585667	+	0.810552i	$0.300833\pi$
$542$		0			0
$543$	23.6580	+	14.7703i	1.01526	+	0.633852i
$544$		0			0
$545$	2.07478	+	8.38856i	0.0888739	+	0.359326i
$546$		0			0
$547$			13.4516i			0.575148i	0.957758	+	0.287574i	$0.0928487\pi$
$547$			13.4516i			0.575148i	−0.957758	+	0.287574i	$0.907151\pi$
$548$		0			0
$549$	−26.0935	+	12.7520i	−1.11365	+	0.544243i
$550$		0			0
$551$	4.51214			0.192224
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6.06367	−	18.5682i	0.257388	−	0.788175i
$556$		0			0
$557$	37.6174			1.59390			0.796950	−	0.604046i	$-0.206446\pi$
$557$	37.6174			1.59390			0.796950	+	0.604046i	$0.206446\pi$
$558$		0			0
$559$		−	35.4449i		−	1.49916i
$560$		0			0
$561$	−0.451203	+	0.722706i	−0.0190498	+	0.0305127i
$562$		0			0
$563$		−	10.1788i		−	0.428987i	−0.976725	−	0.214494i	$-0.931190\pi$
$563$		−	10.1788i		−	0.428987i	0.976725	−	0.214494i	$-0.0688101\pi$
$564$		0			0
$565$	5.46493	+	22.0953i	0.229911	+	0.929556i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			0.859912i			0.0360494i	0.999838	+	0.0180247i	$0.00573775\pi$
$569$			0.859912i			0.0360494i	−0.999838	+	0.0180247i	$0.994262\pi$
$570$		0			0
$571$	3.43634			0.143806			0.0719032	−	0.997412i	$-0.477093\pi$
$571$	3.43634			0.143806			0.0719032	+	0.997412i	$0.477093\pi$
$572$		0			0
$573$	−25.4441	−	15.8854i	−1.06294	−	0.663620i
$574$		0			0
$575$	35.3422	−	18.6219i	1.47387	−	0.776586i
$576$		0			0
$577$	25.6996			1.06989			0.534944	−	0.844888i	$-0.320333\pi$
$577$	25.6996			1.06989			0.534944	+	0.844888i	$0.320333\pi$
$578$		0			0
$579$	22.2298	+	13.8786i	0.923840	+	0.576775i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			1.80420i			0.0747222i
$584$		0			0
$585$	22.1763	−	18.5669i	0.916878	−	0.767646i
$586$		0			0
$587$			22.5090i			0.929048i	0.885561	+	0.464524i	$0.153775\pi$
$587$			22.5090i			0.929048i	−0.885561	+	0.464524i	$0.846225\pi$
$588$		0			0
$589$	8.37041			0.344897
$590$		0			0
$591$	17.0875	−	27.3697i	0.702887	−	1.12584i
$592$		0			0
$593$		−	23.7727i		−	0.976227i	−0.872780	−	0.488114i	$-0.837685\pi$
$593$		−	23.7727i		−	0.976227i	0.872780	−	0.488114i	$-0.162315\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	14.7644	+	9.21780i	0.604269	+	0.377259i
$598$		0			0
$599$		−	43.5310i		−	1.77863i	−0.457296	−	0.889314i	$-0.651182\pi$
$599$		−	43.5310i		−	1.77863i	0.457296	−	0.889314i	$-0.348818\pi$
$600$		0			0
$601$		−	19.9347i		−	0.813153i	−0.913617	−	0.406577i	$-0.866722\pi$
$601$		−	19.9347i		−	0.813153i	0.913617	−	0.406577i	$-0.133278\pi$
$602$		0			0
$603$	−32.6325	+	15.9477i	−1.32890	+	0.649439i
$604$		0			0
$605$	−5.45529	−	22.0563i	−0.221789	−	0.896717i
$606$		0			0
$607$	−10.6333			−0.431590			−0.215795	−	0.976439i	$-0.569234\pi$
$607$	−10.6333			−0.431590			−0.215795	+	0.976439i	$0.569234\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			18.7177i			0.757238i
$612$		0			0
$613$		−	15.1344i		−	0.611273i	−0.952148	−	0.305637i	$-0.901131\pi$
$613$		−	15.1344i		−	0.611273i	0.952148	−	0.305637i	$-0.0988693\pi$
$614$		0			0
$615$	−36.4412	−	11.9003i	−1.46945	−	0.479868i
$616$		0			0
$617$	7.39865			0.297858			0.148929	−	0.988848i	$-0.452417\pi$
$617$	7.39865			0.297858			0.148929	+	0.988848i	$0.452417\pi$
$618$		0			0
$619$		−	18.5279i		−	0.744700i	−0.928092	−	0.372350i	$-0.878552\pi$
$619$		−	18.5279i		−	0.744700i	0.928092	−	0.372350i	$-0.121448\pi$
$620$		0			0
$621$	41.2929	+	4.29091i	1.65703	+	0.172188i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.1351	−	20.6204i	0.565403	−	0.824814i
$626$		0			0
$627$	−1.09512	+	1.75409i	−0.0437349	+	0.0700516i
$628$		0			0
$629$	2.70864			0.108001
$630$		0			0
$631$	13.1786			0.524632			0.262316	−	0.964982i	$-0.415514\pi$
$631$	13.1786			0.524632			0.262316	+	0.964982i	$0.415514\pi$
$632$		0			0
$633$	−1.54645	+	2.47700i	−0.0614658	+	0.0984518i
$634$		0			0
$635$	−21.1378	+	5.22812i	−0.838829	+	0.207471i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2.54414	−	1.24333i	0.100645	−	0.0491855i
$640$		0			0
$641$		−	14.1322i		−	0.558189i	−0.960264	−	0.279094i	$-0.909966\pi$
$641$		−	14.1322i		−	0.558189i	0.960264	−	0.279094i	$-0.0900343\pi$
$642$		0			0
$643$	−17.9278			−0.707005			−0.353503	−	0.935434i	$-0.615009\pi$
$643$	−17.9278			−0.707005			−0.353503	+	0.935434i	$0.615009\pi$
$644$		0			0
$645$	9.88396	−	30.2667i	0.389180	−	1.19175i
$646$		0			0
$647$		−	24.0053i		−	0.943744i	−0.881667	−	0.471872i	$-0.843578\pi$
$647$		−	24.0053i		−	0.943744i	0.881667	−	0.471872i	$-0.156422\pi$
$648$		0			0
$649$		−	13.1041i		−	0.514380i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.9489			−1.13286			−0.566430	−	0.824110i	$-0.691676\pi$
$653$	−28.9489			−1.13286			−0.566430	+	0.824110i	$0.691676\pi$
$654$		0			0
$655$	9.44916	+	38.2040i	0.369209	+	1.49275i
$656$		0			0
$657$	13.4676	+	27.5578i	0.525422	+	1.07513i
$658$		0			0
$659$			5.09127i			0.198328i	0.995071	+	0.0991638i	$0.0316168\pi$
$659$			5.09127i			0.198328i	−0.995071	+	0.0991638i	$0.968383\pi$
$660$		0			0
$661$		−	29.0717i		−	1.13076i	−0.824832	−	0.565378i	$-0.808730\pi$
$661$		−	29.0717i		−	1.13076i	0.824832	−	0.565378i	$-0.191270\pi$
$662$		0			0
$663$	3.40207	+	2.12399i	0.132125	+	0.0824891i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	27.6563i		−	1.07086i
$668$		0			0
$669$	−13.0340	+	20.8770i	−0.503923	+	0.807150i
$670$		0			0
$671$	−8.86682			−0.342300
$672$		0			0
$673$			8.51862i			0.328369i	0.986430	+	0.164184i	$0.0524992\pi$
$673$			8.51862i			0.328369i	−0.986430	+	0.164184i	$0.947501\pi$
$674$		0			0
$675$	24.1140	−	9.67043i	0.928146	−	0.372215i
$676$		0			0
$677$		−	18.9765i		−	0.729325i	−0.931140	−	0.364662i	$-0.881184\pi$
$677$		−	18.9765i		−	0.729325i	0.931140	−	0.364662i	$-0.118816\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.00165	+	0.625352i	0.0383832	+	0.0239636i
$682$		0			0
$683$	20.9173			0.800379			0.400190	−	0.916432i	$-0.368944\pi$
$683$	20.9173			0.800379			0.400190	+	0.916432i	$0.368944\pi$
$684$		0			0
$685$	5.59399	+	22.6171i	0.213735	+	0.864154i
$686$		0			0
$687$	−4.34499	−	2.71268i	−0.165772	−	0.103495i
$688$		0			0
$689$	8.49309			0.323561
$690$		0			0
$691$			34.0523i			1.29541i	0.761891	+	0.647705i	$0.224271\pi$
$691$			34.0523i			1.29541i	−0.761891	+	0.647705i	$0.775729\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.3939	−	3.80745i	0.583924	−	0.144425i
$696$		0			0
$697$		−	5.31588i		−	0.201353i
$698$		0			0
$699$	−24.7086	+	39.5766i	−0.934565	+	1.49692i
$700$		0			0
$701$			18.9758i			0.716706i	0.933586	+	0.358353i	$0.116662\pi$
$701$			18.9758i			0.716706i	−0.933586	+	0.358353i	$0.883338\pi$
$702$		0			0
$703$	6.57418			0.247950
$704$		0			0
$705$	−5.21951	+	15.9832i	−0.196578	+	0.601962i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.07381			0.115439			0.0577197	−	0.998333i	$-0.481617\pi$
$709$	3.07381			0.115439			0.0577197	+	0.998333i	$0.481617\pi$
$710$		0			0
$711$	15.0643	+	30.8249i	0.564955	+	1.15603i
$712$		0			0
$713$		−	51.3048i		−	1.92138i
$714$		0			0
$715$	8.57182	−	2.12011i	0.320568	−	0.0792875i
$716$		0			0
$717$	25.4852	+	15.9110i	0.951764	+	0.594209i
$718$		0			0
$719$	13.4021			0.499815			0.249907	−	0.968270i	$-0.419600\pi$
$719$	13.4021			0.499815			0.249907	+	0.968270i	$0.419600\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−2.24417	−	1.40109i	−0.0834617	−	0.0521071i
$724$		0			0
$725$	−8.06802	−	15.3122i	−0.299639	−	0.568680i
$726$		0			0
$727$	4.88627			0.181222			0.0906109	−	0.995886i	$-0.471118\pi$
$727$	4.88627			0.181222			0.0906109	+	0.995886i	$0.471118\pi$
$728$		0			0
$729$	26.4231	+	5.55142i	0.978634	+	0.205608i
$730$		0			0
$731$	4.41516			0.163301
$732$		0			0
$733$	36.9909			1.36629			0.683145	−	0.730283i	$-0.260612\pi$
$733$	36.9909			1.36629			0.683145	+	0.730283i	$0.260612\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.0888			−0.408462
$738$		0			0
$739$	37.9518			1.39608			0.698040	−	0.716058i	$-0.254056\pi$
$739$	37.9518			1.39608			0.698040	+	0.716058i	$0.254056\pi$
$740$		0			0
$741$	8.25721	+	5.15517i	0.303336	+	0.189380i
$742$		0			0
$743$	−14.0878			−0.516830			−0.258415	−	0.966034i	$-0.583200\pi$
$743$	−14.0878			−0.516830			−0.258415	+	0.966034i	$0.583200\pi$
$744$		0			0
$745$	−13.5997	+	3.36368i	−0.498255	+	0.123236i
$746$		0			0
$747$	−25.2080	+	12.3193i	−0.922312	+	0.450738i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−25.7504			−0.939646			−0.469823	−	0.882761i	$-0.655682\pi$
$751$	−25.7504			−0.939646			−0.469823	+	0.882761i	$0.655682\pi$
$752$		0			0
$753$	−6.01975	+	9.64204i	−0.219372	+	0.351375i
$754$		0			0
$755$	9.50897	+	38.4458i	0.346067	+	1.39918i
$756$		0			0
$757$		−	40.3111i		−	1.46513i	−0.680697	−	0.732565i	$-0.738323\pi$
$757$		−	40.3111i		−	1.46513i	0.680697	−	0.732565i	$-0.261677\pi$
$758$		0			0
$759$	10.7513	+	6.71232i	0.390249	+	0.243642i
$760$		0			0
$761$	19.5306			0.707983			0.353991	−	0.935249i	$-0.384824\pi$
$761$	19.5306			0.707983			0.353991	+	0.935249i	$0.384824\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.31277	+	2.76237i	0.0836182	+	0.0998738i
$766$		0			0
$767$	−61.6862			−2.22736
$768$		0			0
$769$		−	47.6475i		−	1.71821i	−0.511797	−	0.859107i	$-0.671020\pi$
$769$		−	47.6475i		−	1.71821i	0.511797	−	0.859107i	$-0.328980\pi$
$770$		0			0
$771$	−24.9370	−	15.5687i	−0.898083	−	0.560695i
$772$		0			0
$773$			21.1247i			0.759803i	0.925027	+	0.379901i	$0.124042\pi$
$773$			21.1247i			0.759803i	−0.925027	+	0.379901i	$0.875958\pi$
$774$		0			0
$775$	−14.9669	−	28.4054i	−0.537626	−	1.02035i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	12.9022i		−	0.462271i
$780$		0			0
$781$	0.864522			0.0309350
$782$		0			0
$783$	1.85906	−	17.8903i	0.0664373	−	0.639349i
$784$		0			0
$785$	0.415515	+	1.67997i	0.0148304	+	0.0599608i
$786$		0			0
$787$	6.54661			0.233361			0.116681	−	0.993169i	$-0.462775\pi$
$787$	6.54661			0.233361			0.116681	+	0.993169i	$0.462775\pi$
$788$		0			0
$789$	−4.87959	+	7.81581i	−0.173718	+	0.278250i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			41.7397i			1.48222i
$794$		0			0
$795$	7.25230	+	2.36833i	0.257213	+	0.0839960i
$796$		0			0
$797$			6.50728i			0.230500i	0.993337	+	0.115250i	$0.0367669\pi$
$797$			6.50728i			0.230500i	−0.993337	+	0.115250i	$0.963233\pi$
$798$		0			0
$799$	−2.33156			−0.0824845
$800$		0			0
$801$	2.30385	+	4.71421i	0.0814027	+	0.166568i
$802$		0			0
$803$			9.36439i			0.330462i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6.01425	+	9.63322i	−0.211712	+	0.339106i
$808$		0			0
$809$			1.63413i			0.0574528i	0.999587	+	0.0287264i	$0.00914515\pi$
$809$			1.63413i			0.0574528i	−0.999587	+	0.0287264i	$0.990855\pi$
$810$		0			0
$811$		−	29.4411i		−	1.03382i	−0.856041	−	0.516909i	$-0.827083\pi$
$811$		−	29.4411i		−	1.03382i	0.856041	−	0.516909i	$-0.172917\pi$
$812$		0			0
$813$	7.70506	+	4.81045i	0.270228	+	0.168710i
$814$		0			0
$815$	−14.9494	+	3.69750i	−0.523655	+	0.129518i
$816$		0			0
$817$	10.7161			0.374909
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22.6734i			0.791307i	0.918400	+	0.395654i	$0.129482\pi$
$821$			22.6734i			0.791307i	−0.918400	+	0.395654i	$0.870518\pi$
$822$		0			0
$823$			19.1888i			0.668878i	0.942417	+	0.334439i	$0.108547\pi$
$823$			19.1888i			0.668878i	−0.942417	+	0.334439i	$0.891453\pi$
$824$		0			0
$825$	7.91073	+	0.579911i	0.275416	+	0.0201899i
$826$		0			0
$827$	32.5074			1.13039			0.565196	−	0.824957i	$-0.308800\pi$
$827$	32.5074			1.13039			0.565196	+	0.824957i	$0.308800\pi$
$828$		0			0
$829$		−	51.6859i		−	1.79513i	−0.440887	−	0.897563i	$-0.645336\pi$
$829$		−	51.6859i		−	1.79513i	0.440887	−	0.897563i	$-0.354664\pi$
$830$		0			0
$831$	−4.07454	−	2.54383i	−0.141344	−	0.0882445i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	20.6149	−	5.09879i	0.713409	−	0.176451i
$836$		0			0
$837$	3.44871	−	33.1881i	0.119205	−	1.14715i
$838$		0			0
$839$	2.88841			0.0997192			0.0498596	−	0.998756i	$-0.484123\pi$
$839$	2.88841			0.0997192			0.0498596	+	0.998756i	$0.484123\pi$
$840$		0			0
$841$	17.0178			0.586819
$842$		0			0
$843$	−32.9654	−	20.5811i	−1.13539	−	0.708851i
$844$		0			0
$845$	−3.00078	−	12.1325i	−0.103230	−	0.417369i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	6.64728	−	10.6472i	0.228134	−	0.365410i
$850$		0			0
$851$		−	40.2952i		−	1.38130i
$852$		0			0
$853$	49.3673			1.69031			0.845153	−	0.534525i	$-0.179509\pi$
$853$	49.3673			1.69031			0.845153	+	0.534525i	$0.179509\pi$
$854$		0			0
$855$	5.61334	+	6.70459i	0.191972	+	0.229292i
$856$		0			0
$857$			17.5037i			0.597915i	0.954266	+	0.298957i	$0.0966389\pi$
$857$			17.5037i			0.597915i	−0.954266	+	0.298957i	$0.903361\pi$
$858$		0			0
$859$		−	6.04085i		−	0.206111i	−0.994676	−	0.103056i	$-0.967138\pi$
$859$		−	6.04085i		−	0.206111i	0.994676	−	0.103056i	$-0.0328619\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−34.0169			−1.15795			−0.578975	−	0.815345i	$-0.696547\pi$
$863$	−34.0169			−1.15795			−0.578975	+	0.815345i	$0.696547\pi$
$864$		0			0
$865$	−25.9390	+	6.41562i	−0.881953	+	0.218137i
$866$		0			0
$867$	15.3290	−	24.5530i	0.520601	−	0.833864i
$868$		0			0
$869$			10.4746i			0.355326i
$870$		0			0
$871$			52.1996i			1.76872i
$872$		0			0
$873$	−14.2055	−	29.0677i	−0.480784	−	0.983792i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			7.24003i			0.244478i	0.992501	+	0.122239i	$0.0390075\pi$
$877$			7.24003i			0.244478i	−0.992501	+	0.122239i	$0.960992\pi$
$878$		0			0
$879$	−38.4475	−	24.0037i	−1.29680	−	0.809625i
$880$		0			0
$881$	−25.9119			−0.872993			−0.436496	−	0.899706i	$-0.643781\pi$
$881$	−25.9119			−0.872993			−0.436496	+	0.899706i	$0.643781\pi$
$882$		0			0
$883$			17.5664i			0.591155i	0.955319	+	0.295577i	$0.0955120\pi$
$883$			17.5664i			0.591155i	−0.955319	+	0.295577i	$0.904488\pi$
$884$		0			0
$885$	−52.6742	−	17.2014i	−1.77062	−	0.578219i
$886$		0			0
$887$		−	14.7678i		−	0.495855i	−0.968779	−	0.247928i	$-0.920250\pi$
$887$		−	14.7678i		−	0.495855i	0.968779	−	0.247928i	$-0.0797496\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	6.50364	+	5.06478i	0.217880	+	0.169676i
$892$		0			0
$893$	−5.65895			−0.189369
$894$		0			0
$895$	40.6906	−	10.0642i	1.36014	−	0.336409i
$896$		0			0
$897$	31.5976	−	50.6110i	1.05501	−	1.68985i
$898$		0			0
$899$	−22.2281			−0.741347
$900$		0			0
$901$			1.05793i			0.0352449i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.9528	−	8.64504i	1.16187	−	0.287371i
$906$		0			0
$907$			29.3244i			0.973701i	0.873485	+	0.486851i	$0.161854\pi$
$907$			29.3244i			0.973701i	−0.873485	+	0.486851i	$0.838146\pi$
$908$		0			0
$909$	12.9742	+	26.5481i	0.430326	+	0.880545i
$910$		0			0
$911$		−	30.3043i		−	1.00403i	−0.864860	−	0.502013i	$-0.832593\pi$
$911$		−	30.3043i		−	1.00403i	0.864860	−	0.502013i	$-0.167407\pi$
$912$		0			0
$913$	−8.56590			−0.283490
$914$		0			0
$915$	−11.6393	+	35.6418i	−0.384783	+	1.17828i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−7.63775			−0.251946			−0.125973	−	0.992034i	$-0.540205\pi$
$919$	−7.63775			−0.251946			−0.125973	+	0.992034i	$0.540205\pi$
$920$		0			0
$921$	20.5601	−	32.9318i	0.677479	−	1.08514i
$922$		0			0
$923$		−	4.06965i		−	0.133954i
$924$		0			0
$925$	−11.7551	−	22.3098i	−0.386505	−	0.733541i
$926$		0			0
$927$	−3.23766	−	6.62499i	−0.106339	−	0.217593i
$928$		0			0
$929$	−17.2502			−0.565959			−0.282980	−	0.959126i	$-0.591323\pi$
$929$	−17.2502			−0.565959			−0.282980	+	0.959126i	$0.591323\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	18.0038	−	28.8372i	0.589417	−	0.944089i
$934$		0			0
$935$	0.264089	+	1.06774i	0.00863664	+	0.0349188i
$936$		0			0
$937$	36.0492			1.17768			0.588839	−	0.808251i	$-0.299585\pi$
$937$	36.0492			1.17768			0.588839	+	0.808251i	$0.299585\pi$
$938$		0			0
$939$	−11.7716	+	18.8549i	−0.384151	+	0.615308i
$940$		0			0
$941$	31.6307			1.03113			0.515565	−	0.856850i	$-0.327582\pi$
$941$	31.6307			1.03113			0.515565	+	0.856850i	$0.327582\pi$
$942$		0			0
$943$	−79.0818			−2.57526
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.6174			1.31989			0.659944	−	0.751315i	$-0.270580\pi$
$947$	40.6174			1.31989			0.659944	+	0.751315i	$0.270580\pi$
$948$		0			0
$949$	44.0820			1.43096
$950$		0			0
$951$	−14.6479	+	23.4620i	−0.474990	+	0.760807i
$952$		0			0
$953$	−5.31938			−0.172312			−0.0861558	−	0.996282i	$-0.527458\pi$
$953$	−5.31938			−0.172312			−0.0861558	+	0.996282i	$0.527458\pi$
$954$		0			0
$955$	−37.5916	+	9.29771i	−1.21644	+	0.300867i
$956$		0			0
$957$	2.90815	−	4.65807i	0.0940070	−	0.150574i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−10.2349			−0.330159
$962$		0			0
$963$	1.97536	+	4.04203i	0.0636550	+	0.130252i
$964$		0			0
$965$	32.8428	−	8.12316i	1.05725	−	0.261494i
$966$		0			0
$967$			6.75930i			0.217364i	0.994077	+	0.108682i	$0.0346631\pi$
$967$			6.75930i			0.217364i	−0.994077	+	0.108682i	$0.965337\pi$
$968$		0			0
$969$	−0.642149	+	1.02855i	−0.0206288	+	0.0330418i
$970$		0			0
$971$	43.9702			1.41107			0.705535	−	0.708675i	$-0.250707\pi$
$971$	43.9702			1.41107			0.705535	+	0.708675i	$0.250707\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	2.72987	−	37.2390i	0.0874260	−	1.19260i
$976$		0			0
$977$	9.24885			0.295897			0.147949	−	0.988995i	$-0.452733\pi$
$977$	9.24885			0.295897			0.147949	+	0.988995i	$0.452733\pi$
$978$		0			0
$979$			1.60193i			0.0511979i
$980$		0			0
$981$	5.09046	+	10.4162i	0.162526	+	0.332565i
$982$		0			0
$983$		−	13.1389i		−	0.419067i	−0.977801	−	0.209533i	$-0.932805\pi$
$983$		−	13.1389i		−	0.419067i	0.977801	−	0.209533i	$-0.0671945\pi$
$984$		0			0
$985$	−10.0013	−	40.4365i	−0.318669	−	1.28841i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	65.6823i		−	2.08857i
$990$		0			0
$991$	−6.37041			−0.202363			−0.101181	−	0.994868i	$-0.532262\pi$
$991$	−6.37041			−0.202363			−0.101181	+	0.994868i	$0.532262\pi$
$992$		0			0
$993$	−18.0131	+	28.8522i	−0.571630	+	0.915598i
$994$		0			0
$995$	21.8133	−	5.39518i	0.691528	−	0.171039i
$996$		0			0
$997$	23.5091			0.744542			0.372271	−	0.928124i	$-0.378579\pi$
$997$	23.5091			0.744542			0.372271	+	0.928124i	$0.378579\pi$
$998$		0			0
$999$	2.70864	−	26.0662i	0.0856976	−	0.824697i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.f.a.1469.21		32
3.2	odd	2	inner	2940.2.f.a.1469.24		32
5.4	even	2	inner	2940.2.f.a.1469.11		32
7.2	even	3		420.2.bn.a.269.11	yes	32
7.3	odd	6		420.2.bn.a.89.16	yes	32
7.6	odd	2	inner	2940.2.f.a.1469.12		32
15.14	odd	2	inner	2940.2.f.a.1469.10		32
21.2	odd	6		420.2.bn.a.269.1	yes	32
21.17	even	6		420.2.bn.a.89.6	yes	32
21.20	even	2	inner	2940.2.f.a.1469.9		32
35.2	odd	12		2100.2.bi.n.101.14		32
35.3	even	12		2100.2.bi.n.1601.8		32
35.9	even	6		420.2.bn.a.269.6	yes	32
35.17	even	12		2100.2.bi.n.1601.9		32
35.23	odd	12		2100.2.bi.n.101.3		32
35.24	odd	6		420.2.bn.a.89.1	✓	32
35.34	odd	2	inner	2940.2.f.a.1469.22		32
105.2	even	12		2100.2.bi.n.101.9		32
105.17	odd	12		2100.2.bi.n.1601.14		32
105.23	even	12		2100.2.bi.n.101.8		32
105.38	odd	12		2100.2.bi.n.1601.3		32
105.44	odd	6		420.2.bn.a.269.16	yes	32
105.59	even	6		420.2.bn.a.89.11	yes	32
105.104	even	2	inner	2940.2.f.a.1469.23		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bn.a.89.1	✓	32	35.24	odd	6
420.2.bn.a.89.6	yes	32	21.17	even	6
420.2.bn.a.89.11	yes	32	105.59	even	6
420.2.bn.a.89.16	yes	32	7.3	odd	6
420.2.bn.a.269.1	yes	32	21.2	odd	6
420.2.bn.a.269.6	yes	32	35.9	even	6
420.2.bn.a.269.11	yes	32	7.2	even	3
420.2.bn.a.269.16	yes	32	105.44	odd	6
2100.2.bi.n.101.3		32	35.23	odd	12
2100.2.bi.n.101.8		32	105.23	even	12
2100.2.bi.n.101.9		32	105.2	even	12
2100.2.bi.n.101.14		32	35.2	odd	12
2100.2.bi.n.1601.3		32	105.38	odd	12
2100.2.bi.n.1601.8		32	35.3	even	12
2100.2.bi.n.1601.9		32	35.17	even	12
2100.2.bi.n.1601.14		32	105.17	odd	12
2940.2.f.a.1469.9		32	21.20	even	2	inner
2940.2.f.a.1469.10		32	15.14	odd	2	inner
2940.2.f.a.1469.11		32	5.4	even	2	inner
2940.2.f.a.1469.12		32	7.6	odd	2	inner
2940.2.f.a.1469.21		32	1.1	even	1	trivial
2940.2.f.a.1469.22		32	35.34	odd	2	inner
2940.2.f.a.1469.23		32	105.104	even	2	inner
2940.2.f.a.1469.24		32	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.917271 - 1.46922i) q^{3} +(-0.536879 - 2.17066i) q^{5} +(-1.31723 - 2.69535i) q^{9} -0.915904i q^{11} -4.31153 q^{13} +(-3.68164 - 1.20229i) q^{15} -0.537062i q^{17} -1.30351i q^{19} -7.98961 q^{23} +(-4.42352 + 2.33076i) q^{25} +(-5.16832 - 0.537062i) q^{27} +3.46154i q^{29} +6.42144i q^{31} +(-1.34567 - 0.840132i) q^{33} +5.04345i q^{37} +(-3.95484 + 6.33460i) q^{39} +9.89809 q^{41} +8.22096i q^{43} +(-5.14349 + 4.30633i) q^{45} -4.34132i q^{47} +(-0.789063 - 0.492631i) q^{51} -1.96985 q^{53} +(-1.98812 + 0.491730i) q^{55} +(-1.91514 - 1.19567i) q^{57} +14.3073 q^{59} -9.68095i q^{61} +(2.31477 + 9.35886i) q^{65} -12.1070i q^{67} +(-7.32863 + 11.7385i) q^{69} +0.943900i q^{71} -10.2242 q^{73} +(-0.633157 + 8.63708i) q^{75} -11.4363 q^{79} +(-5.52981 + 7.10079i) q^{81} -9.35240i q^{83} +(-1.16578 + 0.288337i) q^{85} +(5.08577 + 3.17517i) q^{87} -1.74901 q^{89} +(9.43453 + 5.89020i) q^{93} +(-2.82947 + 0.699827i) q^{95} +10.7844 q^{97} +(-2.46868 + 1.20646i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) \) 32 * q + 2 * q^15 - 12 * q^25 - 48 * q^39 + 20 * q^51 + 56 * q^79 + 40 * q^81 - 4 * q^85 - 28 * q^99

Introduction

L-functions

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