LMFDB - Embedded newform 1440.2.bi.e.1423.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,2,Mod(847,1440)] 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("1440.847"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bi (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1423.3
Character	$\chi$	$=$	1440.1423
Dual form			1440.2.bi.e.847.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.51371 + 1.64581i) q^{5} +(3.43671 + 3.43671i) q^{7} +3.48120 q^{11} +(2.05033 - 2.05033i) q^{13} +(1.64963 - 1.64963i) q^{17} +0.642023i q^{19} +(-2.31024 + 2.31024i) q^{23} +(-0.417363 - 4.98255i) q^{25} +0.699613 q^{29} +1.56863i q^{31} +(-10.8583 + 0.453979i) q^{35} +(5.31751 + 5.31751i) q^{37} -4.92316 q^{41} +(-3.56519 - 3.56519i) q^{43} +(6.85586 + 6.85586i) q^{47} +16.6220i q^{49} +(1.94008 - 1.94008i) q^{53} +(-5.26953 + 5.72939i) q^{55} -2.74121i q^{59} -5.20943i q^{61} +(0.270842 + 6.47805i) q^{65} +(-6.92316 + 6.92316i) q^{67} +11.1548i q^{71} +(-6.56519 - 6.56519i) q^{73} +(11.9639 + 11.9639i) q^{77} +2.09702 q^{79} +(6.64648 + 6.64648i) q^{83} +(0.217911 + 5.21202i) q^{85} +0.733690i q^{89} +14.0928 q^{91} +(-1.05665 - 0.971837i) q^{95} +(8.79083 - 8.79083i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) $ 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.51371	+	1.64581i	−0.676952	+	0.736027i
$5$	−1.51371	+	1.64581i	−0.676952	+	0.736027i
$6$		0			0
$7$	3.43671	+	3.43671i	1.29895	+	1.29895i	0.929083	+	0.369871i	$0.120598\pi$
$7$	3.43671	+	3.43671i	1.29895	+	1.29895i	0.369871	+	0.929083i	$0.379402\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.48120			1.04962			0.524811	−	0.851219i	$-0.324136\pi$
$11$	3.48120			1.04962			0.524811	+	0.851219i	$0.324136\pi$
$12$		0			0
$13$	2.05033	−	2.05033i	0.568659	−	0.568659i	−0.363093	−	0.931753i	$-0.618279\pi$
$13$	2.05033	−	2.05033i	0.568659	−	0.568659i	0.931753	+	0.363093i	$0.118279\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.64963	−	1.64963i	0.400093	−	0.400093i	−0.478173	−	0.878266i	$-0.658701\pi$
$17$	1.64963	−	1.64963i	0.400093	−	0.400093i	0.878266	+	0.478173i	$0.158701\pi$
$18$		0			0
$19$			0.642023i			0.147290i	0.997285	+	0.0736451i	$0.0234632\pi$
$19$			0.642023i			0.147290i	−0.997285	+	0.0736451i	$0.976537\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.31024	+	2.31024i	−0.481719	+	0.481719i	−0.905680	−	0.423961i	$-0.860639\pi$
$23$	−2.31024	+	2.31024i	−0.481719	+	0.481719i	0.423961	+	0.905680i	$0.360639\pi$
$24$		0			0
$25$	−0.417363	−	4.98255i	−0.0834726	−	0.996510i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.699613			0.129915			0.0649574	−	0.997888i	$-0.479309\pi$
$29$	0.699613			0.129915			0.0649574	+	0.997888i	$0.479309\pi$
$30$		0			0
$31$			1.56863i			0.281734i	0.990029	+	0.140867i	$0.0449890\pi$
$31$			1.56863i			0.281734i	−0.990029	+	0.140867i	$0.955011\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.8583	+	0.453979i	−1.83540	+	0.0767365i
$36$		0			0
$37$	5.31751	+	5.31751i	0.874193	+	0.874193i	0.992926	−	0.118733i	$-0.0378834\pi$
$37$	5.31751	+	5.31751i	0.874193	+	0.874193i	−0.118733	+	0.992926i	$0.537883\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.92316			−0.768869			−0.384435	−	0.923152i	$-0.625604\pi$
$41$	−4.92316			−0.768869			−0.384435	+	0.923152i	$0.625604\pi$
$42$		0			0
$43$	−3.56519	−	3.56519i	−0.543686	−	0.543686i	0.380921	−	0.924607i	$-0.375607\pi$
$43$	−3.56519	−	3.56519i	−0.543686	−	0.543686i	−0.924607	+	0.380921i	$0.875607\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.85586	+	6.85586i	1.00003	+	1.00003i	1.00000		2.93703e-5i	$9.34887e-6\pi$
$47$	6.85586	+	6.85586i	1.00003	+	1.00003i	2.93703e−5		1.00000i	$0.499991\pi$
$48$		0			0
$49$			16.6220i			2.37457i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.94008	−	1.94008i	0.266490	−	0.266490i	−0.561194	−	0.827684i	$-0.689658\pi$
$53$	1.94008	−	1.94008i	0.266490	−	0.266490i	0.827684	+	0.561194i	$0.189658\pi$
$54$		0			0
$55$	−5.26953	+	5.72939i	−0.710544	+	0.772551i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.74121i		−	0.356876i	−0.983951	−	0.178438i	$-0.942896\pi$
$59$		−	2.74121i		−	0.356876i	0.983951	−	0.178438i	$-0.0571043\pi$
$60$		0			0
$61$		−	5.20943i		−	0.666999i	−0.942750	−	0.333500i	$-0.891770\pi$
$61$		−	5.20943i		−	0.666999i	0.942750	−	0.333500i	$-0.108230\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.270842	+	6.47805i	0.0335939	+	0.803504i
$66$		0			0
$67$	−6.92316	+	6.92316i	−0.845799	+	0.845799i	−0.989606	−	0.143807i	$-0.954066\pi$
$67$	−6.92316	+	6.92316i	−0.845799	+	0.845799i	0.143807	+	0.989606i	$0.454066\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.1548i			1.32384i	0.749576	+	0.661918i	$0.230257\pi$
$71$			11.1548i			1.32384i	−0.749576	+	0.661918i	$0.769743\pi$
$72$		0			0
$73$	−6.56519	−	6.56519i	−0.768397	−	0.768397i	0.209427	−	0.977824i	$-0.432840\pi$
$73$	−6.56519	−	6.56519i	−0.768397	−	0.768397i	−0.977824	+	0.209427i	$0.932840\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.9639	+	11.9639i	1.36341	+	1.36341i
$78$		0			0
$79$	2.09702			0.235933			0.117966	−	0.993018i	$-0.462363\pi$
$79$	2.09702			0.235933			0.117966	+	0.993018i	$0.462363\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.64648	+	6.64648i	0.729546	+	0.729546i	0.970529	−	0.240984i	$-0.0774700\pi$
$83$	6.64648	+	6.64648i	0.729546	+	0.729546i	−0.240984	+	0.970529i	$0.577470\pi$
$84$		0			0
$85$	0.217911	+	5.21202i	0.0236357	+	0.565323i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			0.733690i			0.0777710i	0.999244	+	0.0388855i	$0.0123808\pi$
$89$			0.733690i			0.0777710i	−0.999244	+	0.0388855i	$0.987619\pi$
$90$		0			0
$91$	14.0928			1.47732
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.05665	−	0.971837i	−0.108410	−	0.0997084i
$96$		0			0
$97$	8.79083	−	8.79083i	0.892574	−	0.892574i	−0.102191	−	0.994765i	$-0.532585\pi$
$97$	8.79083	−	8.79083i	0.892574	−	0.892574i	0.994765	+	0.102191i	$0.0325853\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.40933i			0.140233i	0.997539	+	0.0701167i	$0.0223372\pi$
$101$			1.40933i			0.140233i	−0.997539	+	0.0701167i	$0.977663\pi$
$102$		0			0
$103$	2.41334	−	2.41334i	0.237793	−	0.237793i	−0.578143	−	0.815936i	$-0.696222\pi$
$103$	2.41334	−	2.41334i	0.237793	−	0.237793i	0.815936	+	0.578143i	$0.196222\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.56073	−	1.56073i	0.150882	−	0.150882i	−0.627630	−	0.778512i	$-0.715975\pi$
$107$	1.56073	−	1.56073i	0.150882	−	0.150882i	0.778512	+	0.627630i	$0.215975\pi$
$108$		0			0
$109$	−14.6177			−1.40012			−0.700060	−	0.714084i	$-0.746843\pi$
$109$	−14.6177			−1.40012			−0.700060	+	0.714084i	$0.746843\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.55758	+	4.55758i	0.428742	+	0.428742i	0.888199	−	0.459458i	$-0.151956\pi$
$113$	4.55758	+	4.55758i	0.428742	+	0.428742i	−0.459458	+	0.888199i	$0.651956\pi$
$114$		0			0
$115$	−0.305176	−	7.29925i	−0.0284578	−	0.680659i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.3386			1.03941
$120$		0			0
$121$	1.11877			0.101707
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.83208	+	6.85524i	0.789966	+	0.613151i
$126$		0			0
$127$	−9.62582	−	9.62582i	−0.854154	−	0.854154i	0.136488	−	0.990642i	$-0.456418\pi$
$127$	−9.62582	−	9.62582i	−0.854154	−	0.854154i	−0.990642	+	0.136488i	$0.956418\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.26769			0.110759			0.0553794	−	0.998465i	$-0.482363\pi$
$131$	1.26769			0.110759			0.0553794	+	0.998465i	$0.482363\pi$
$132$		0			0
$133$	−2.20645	+	2.20645i	−0.191323	+	0.191323i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.2296	−	12.2296i	1.04485	−	1.04485i	0.0459032	−	0.998946i	$-0.485383\pi$
$137$	12.2296	−	12.2296i	1.04485	−	1.04485i	0.998946	−	0.0459032i	$-0.0146166\pi$
$138$		0			0
$139$			8.13630i			0.690112i	0.938582	+	0.345056i	$0.112140\pi$
$139$			8.13630i			0.690112i	−0.938582	+	0.345056i	$0.887860\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.13761	−	7.13761i	0.596877	−	0.596877i
$144$		0			0
$145$	−1.05901	+	1.15143i	−0.0879461	+	0.0956209i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.75071			0.225347			0.112674	−	0.993632i	$-0.464059\pi$
$149$	2.75071			0.225347			0.112674	+	0.993632i	$0.464059\pi$
$150$		0			0
$151$			10.2020i			0.830226i	0.909770	+	0.415113i	$0.136258\pi$
$151$			10.2020i			0.830226i	−0.909770	+	0.415113i	$0.863742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.58166	−	2.37445i	−0.207364	−	0.190720i
$156$		0			0
$157$	−1.29115	−	1.29115i	−0.103045	−	0.103045i	0.653705	−	0.756750i	$-0.273214\pi$
$157$	−1.29115	−	1.29115i	−0.103045	−	0.103045i	−0.756750	+	0.653705i	$0.773214\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.8793			−1.25146
$162$		0			0
$163$	−9.30787	−	9.30787i	−0.729049	−	0.729049i	0.241382	−	0.970430i	$-0.422399\pi$
$163$	−9.30787	−	9.30787i	−0.729049	−	0.729049i	−0.970430	+	0.241382i	$0.922399\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.8845	−	11.8845i	−0.919652	−	0.919652i	0.0773515	−	0.997004i	$-0.475354\pi$
$167$	−11.8845	−	11.8845i	−0.919652	−	0.919652i	−0.997004	+	0.0773515i	$0.975354\pi$
$168$		0			0
$169$			4.59229i			0.353253i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.2536	+	13.2536i	−1.00765	+	1.00765i	−0.00768426	+	0.999970i	$0.502446\pi$
$173$	−13.2536	+	13.2536i	−1.00765	+	1.00765i	−0.999970	+	0.00768426i	$0.997554\pi$
$174$		0			0
$175$	15.6892	−	18.5579i	1.18599	−	1.40285i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	3.61084i		−	0.269887i	−0.990853	−	0.134943i	$-0.956915\pi$
$179$		−	3.61084i		−	0.269887i	0.990853	−	0.134943i	$-0.0430853\pi$
$180$		0			0
$181$			21.8993i			1.62776i	0.581032	+	0.813881i	$0.302649\pi$
$181$			21.8993i			1.62776i	−0.581032	+	0.813881i	$0.697351\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.8008	+	0.702427i	−1.23522	+	0.0516434i
$186$		0			0
$187$	5.74268	−	5.74268i	0.419947	−	0.419947i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.1309i		−	0.733045i	−0.930409	−	0.366522i	$-0.880548\pi$
$191$		−	10.1309i		−	0.733045i	0.930409	−	0.366522i	$-0.119452\pi$
$192$		0			0
$193$	−14.3560	−	14.3560i	−1.03337	−	1.03337i	−0.999424	−	0.0339453i	$-0.989193\pi$
$193$	−14.3560	−	14.3560i	−1.03337	−	1.03337i	−0.0339453	−	0.999424i	$-0.510807\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.6884	−	14.6884i	−1.04650	−	1.04650i	−0.998865	−	0.0476396i	$-0.984830\pi$
$197$	−14.6884	−	14.6884i	−1.04650	−	1.04650i	−0.0476396	−	0.998865i	$-0.515170\pi$
$198$		0			0
$199$	−5.08593			−0.360532			−0.180266	−	0.983618i	$-0.557696\pi$
$199$	−5.08593			−0.360532			−0.180266	+	0.983618i	$0.557696\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.40437	+	2.40437i	0.168753	+	0.168753i
$204$		0			0
$205$	7.45224	−	8.10258i	0.520487	−	0.565909i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.23501i			0.154599i
$210$		0			0
$211$	21.7932			1.50030			0.750151	−	0.661266i	$-0.229981\pi$
$211$	21.7932			1.50030			0.750151	+	0.661266i	$0.229981\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.2643	−	0.470951i	0.768217	−	0.0321186i
$216$		0			0
$217$	−5.39092	+	5.39092i	−0.365959	+	0.365959i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.76456i		−	0.455033i
$222$		0			0
$223$	7.63273	−	7.63273i	0.511126	−	0.511126i	−0.403746	−	0.914871i	$-0.632292\pi$
$223$	7.63273	−	7.63273i	0.511126	−	0.511126i	0.914871	+	0.403746i	$0.132292\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.84363	+	8.84363i	−0.586973	+	0.586973i	−0.936810	−	0.349838i	$-0.886237\pi$
$227$	−8.84363	+	8.84363i	−0.586973	+	0.586973i	0.349838	+	0.936810i	$0.386237\pi$
$228$		0			0
$229$	23.9520			1.58279			0.791397	−	0.611302i	$-0.209354\pi$
$229$	23.9520			1.58279			0.791397	+	0.611302i	$0.209354\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.38332	−	4.38332i	−0.287161	−	0.287161i	0.548796	−	0.835956i	$-0.315086\pi$
$233$	−4.38332	−	4.38332i	−0.287161	−	0.287161i	−0.835956	+	0.548796i	$0.815086\pi$
$234$		0			0
$235$	−21.6612	+	0.905638i	−1.41302	+	0.0590773i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.7993			−1.08666			−0.543328	−	0.839521i	$-0.682836\pi$
$239$	−16.7993			−1.08666			−0.543328	+	0.839521i	$0.682836\pi$
$240$		0			0
$241$	3.47277			0.223701			0.111850	−	0.993725i	$-0.464322\pi$
$241$	3.47277			0.223701			0.111850	+	0.993725i	$0.464322\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−27.3565	−	25.1608i	−1.74774	−	1.60747i
$246$		0			0
$247$	1.31636	+	1.31636i	0.0837580	+	0.0837580i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.6116			0.669797			0.334898	−	0.942254i	$-0.391298\pi$
$251$	10.6116			0.669797			0.334898	+	0.942254i	$0.391298\pi$
$252$		0			0
$253$	−8.04242	+	8.04242i	−0.505623	+	0.505623i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.4495	−	16.4495i	1.02609	−	1.02609i	0.0264382	−	0.999650i	$-0.491583\pi$
$257$	16.4495	−	16.4495i	1.02609	−	1.02609i	0.999650	−	0.0264382i	$-0.00841653\pi$
$258$		0			0
$259$			36.5495i			2.27107i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−17.3101	+	17.3101i	−1.06739	+	1.06739i	−0.0698281	+	0.997559i	$0.522245\pi$
$263$	−17.3101	+	17.3101i	−1.06739	+	1.06739i	−0.997559	+	0.0698281i	$0.977755\pi$
$264$		0			0
$265$	0.256279	+	6.12971i	0.0157431	+	0.376545i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.2178			−1.04979			−0.524895	−	0.851167i	$-0.675895\pi$
$269$	−17.2178			−1.04979			−0.524895	+	0.851167i	$0.675895\pi$
$270$		0			0
$271$		−	8.37293i		−	0.508619i	−0.967123	−	0.254310i	$-0.918152\pi$
$271$		−	8.37293i		−	0.508619i	0.967123	−	0.254310i	$-0.0818482\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.45292	−	17.3453i	−0.0876146	−	1.04596i
$276$		0			0
$277$	−11.2504	−	11.2504i	−0.675971	−	0.675971i	0.283115	−	0.959086i	$-0.408632\pi$
$277$	−11.2504	−	11.2504i	−0.675971	−	0.675971i	−0.959086	+	0.283115i	$0.908632\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	25.9447			1.54773			0.773864	−	0.633352i	$-0.218321\pi$
$281$	25.9447			1.54773			0.773864	+	0.633352i	$0.218321\pi$
$282$		0			0
$283$	−0.0392414	−	0.0392414i	−0.00233266	−	0.00233266i	0.705939	−	0.708272i	$-0.250525\pi$
$283$	−0.0392414	−	0.0392414i	−0.00233266	−	0.00233266i	−0.708272	+	0.705939i	$0.750525\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.9195	−	16.9195i	−0.998726	−	0.998726i
$288$		0			0
$289$			11.5575i			0.679851i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.6367	−	21.6367i	1.26403	−	1.26403i	0.314905	−	0.949123i	$-0.398027\pi$
$293$	21.6367	−	21.6367i	1.26403	−	1.26403i	0.949123	−	0.314905i	$-0.101973\pi$
$294$		0			0
$295$	4.51151	+	4.14940i	0.262670	+	0.241588i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			9.47352i			0.547868i
$300$		0			0
$301$		−	24.5050i		−	1.41245i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.57372	+	7.88557i	0.490930	+	0.451526i
$306$		0			0
$307$	18.1166	−	18.1166i	1.03397	−	1.03397i	0.0345669	−	0.999402i	$-0.488995\pi$
$307$	18.1166	−	18.1166i	1.03397	−	1.03397i	0.999402	−	0.0345669i	$-0.0110052\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9.63648i		−	0.546435i	−0.961952	−	0.273217i	$-0.911912\pi$
$311$		−	9.63648i		−	0.546435i	0.961952	−	0.273217i	$-0.0880878\pi$
$312$		0			0
$313$	18.2372	+	18.2372i	1.03083	+	1.03083i	0.999509	+	0.0313208i	$0.00997135\pi$
$313$	18.2372	+	18.2372i	1.03083	+	1.03083i	0.0313208	+	0.999509i	$0.490029\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.866839	+	0.866839i	0.0486865	+	0.0486865i	0.731031	−	0.682344i	$-0.239040\pi$
$317$	0.866839	+	0.866839i	0.0486865	+	0.0486865i	−0.682344	+	0.731031i	$0.739040\pi$
$318$		0			0
$319$	2.43549			0.136362
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.05910	+	1.05910i	0.0589298	+	0.0589298i
$324$		0			0
$325$	−11.0716	−	9.36014i	−0.614142	−	0.519207i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			47.1232i			2.59799i
$330$		0			0
$331$	−14.4884			−0.796352			−0.398176	−	0.917309i	$-0.630357\pi$
$331$	−14.4884			−0.796352			−0.398176	+	0.917309i	$0.630357\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.914529	−	21.8739i	−0.0499661	−	1.19510i
$336$		0			0
$337$	21.9977	−	21.9977i	1.19829	−	1.19829i	0.223611	−	0.974679i	$-0.428216\pi$
$337$	21.9977	−	21.9977i	1.19829	−	1.19829i	0.974679	−	0.223611i	$-0.0717845\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.46071i			0.295714i
$342$		0			0
$343$	−33.0679	+	33.0679i	−1.78550	+	1.78550i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.52917	−	3.52917i	0.189456	−	0.189456i	−0.606005	−	0.795461i	$-0.707229\pi$
$347$	3.52917	−	3.52917i	0.189456	−	0.189456i	0.795461	+	0.606005i	$0.207229\pi$
$348$		0			0
$349$	3.26043			0.174527			0.0872635	−	0.996185i	$-0.472188\pi$
$349$	3.26043			0.174527			0.0872635	+	0.996185i	$0.472188\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.6567	−	22.6567i	−1.20589	−	1.20589i	−0.972346	−	0.233547i	$-0.924967\pi$
$353$	−22.6567	−	22.6567i	−1.20589	−	1.20589i	−0.233547	−	0.972346i	$-0.575033\pi$
$354$		0			0
$355$	−18.3587	−	16.8852i	−0.974379	−	0.896173i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.9181			−0.892903			−0.446451	−	0.894808i	$-0.647312\pi$
$359$	−16.9181			−0.892903			−0.446451	+	0.894808i	$0.647312\pi$
$360$		0			0
$361$	18.5878			0.978306
$362$		0			0
$363$		0			0
$364$		0			0
$365$	20.7428	−	0.867242i	1.08573	−	0.0453935i
$366$		0			0
$367$	2.82093	+	2.82093i	0.147252	+	0.147252i	0.776889	−	0.629637i	$-0.216797\pi$
$367$	2.82093	+	2.82093i	0.147252	+	0.147252i	−0.629637	+	0.776889i	$0.716797\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.3350			0.692318
$372$		0			0
$373$	22.7300	−	22.7300i	1.17691	−	1.17691i	0.196388	−	0.980526i	$-0.437079\pi$
$373$	22.7300	−	22.7300i	1.17691	−	1.17691i	0.980526	−	0.196388i	$-0.0629213\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.43444	−	1.43444i	0.0738773	−	0.0738773i
$378$		0			0
$379$			2.55793i			0.131392i	0.997840	+	0.0656959i	$0.0209267\pi$
$379$			2.55793i			0.131392i	−0.997840	+	0.0656959i	$0.979073\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.48289	−	5.48289i	0.280163	−	0.280163i	−0.553011	−	0.833174i	$-0.686521\pi$
$383$	5.48289	−	5.48289i	0.280163	−	0.280163i	0.833174	+	0.553011i	$0.186521\pi$
$384$		0			0
$385$	−37.8001	+	1.58039i	−1.92647	+	0.0805443i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.02529			0.254792			0.127396	−	0.991852i	$-0.459338\pi$
$389$	5.02529			0.254792			0.127396	+	0.991852i	$0.459338\pi$
$390$		0			0
$391$			7.62208i			0.385465i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.17427	+	3.45128i	−0.159715	+	0.173653i
$396$		0			0
$397$	−8.56361	−	8.56361i	−0.429795	−	0.429795i	0.458763	−	0.888558i	$-0.348293\pi$
$397$	−8.56361	−	8.56361i	−0.429795	−	0.429795i	−0.888558	+	0.458763i	$0.848293\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.5710			0.627763			0.313882	−	0.949462i	$-0.398370\pi$
$401$	12.5710			0.627763			0.313882	+	0.949462i	$0.398370\pi$
$402$		0			0
$403$	3.21620	+	3.21620i	0.160211	+	0.160211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.5113	+	18.5113i	0.917572	+	0.917572i
$408$		0			0
$409$		−	9.94711i		−	0.491853i	−0.969288	−	0.245927i	$-0.920908\pi$
$409$		−	9.94711i		−	0.491853i	0.969288	−	0.245927i	$-0.0790922\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.42076	−	9.42076i	0.463565	−	0.463565i
$414$		0			0
$415$	−20.9997	+	0.877980i	−1.03083	+	0.0430983i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.92920i		−	0.0942475i	−0.998889	−	0.0471238i	$-0.984994\pi$
$419$		−	1.92920i		−	0.0942475i	0.998889	−	0.0471238i	$-0.0150055\pi$
$420$		0			0
$421$			0.454084i			0.0221307i	0.999939	+	0.0110654i	$0.00352229\pi$
$421$			0.454084i			0.0221307i	−0.999939	+	0.0110654i	$0.996478\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.90784	−	7.53085i	−0.432094	−	0.365300i
$426$		0			0
$427$	17.9033	−	17.9033i	0.866402	−	0.866402i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			7.07961i			0.341013i	0.985357	+	0.170506i	$0.0545403\pi$
$431$			7.07961i			0.341013i	−0.985357	+	0.170506i	$0.945460\pi$
$432$		0			0
$433$	−15.1484	−	15.1484i	−0.727987	−	0.727987i	0.242231	−	0.970219i	$-0.422121\pi$
$433$	−15.1484	−	15.1484i	−0.727987	−	0.727987i	−0.970219	+	0.242231i	$0.922121\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.48323	−	1.48323i	−0.0709525	−	0.0709525i
$438$		0			0
$439$	−25.2936			−1.20720			−0.603599	−	0.797288i	$-0.706267\pi$
$439$	−25.2936			−1.20720			−0.603599	+	0.797288i	$0.706267\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.9275	−	26.9275i	−1.27936	−	1.27936i	−0.941024	−	0.338341i	$-0.890134\pi$
$443$	−26.9275	−	26.9275i	−1.27936	−	1.27936i	−0.338341	−	0.941024i	$-0.609866\pi$
$444$		0			0
$445$	−1.20751	−	1.11059i	−0.0572416	−	0.0526472i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	22.4863i		−	1.06120i	−0.847624	−	0.530598i	$-0.821967\pi$
$449$		−	22.4863i		−	1.06120i	0.847624	−	0.530598i	$-0.178033\pi$
$450$		0			0
$451$	−17.1385			−0.807022
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−21.3324	+	23.1940i	−1.00008	+	1.08735i
$456$		0			0
$457$	−0.0735546	+	0.0735546i	−0.00344074	+	0.00344074i	−0.708825	−	0.705384i	$-0.750774\pi$
$457$	−0.0735546	+	0.0735546i	−0.00344074	+	0.00344074i	0.705384	+	0.708825i	$0.250774\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	33.7901i		−	1.57376i	−0.617105	−	0.786881i	$-0.711695\pi$
$461$		−	33.7901i		−	1.57376i	0.617105	−	0.786881i	$-0.288305\pi$
$462$		0			0
$463$	27.4189	−	27.4189i	1.27427	−	1.27427i	0.330439	−	0.943827i	$-0.392803\pi$
$463$	27.4189	−	27.4189i	1.27427	−	1.27427i	0.943827	−	0.330439i	$-0.107197\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.8191	−	11.8191i	0.546923	−	0.546923i	−0.378626	−	0.925550i	$-0.623603\pi$
$467$	11.8191	−	11.8191i	0.546923	−	0.546923i	0.925550	+	0.378626i	$0.123603\pi$
$468$		0			0
$469$	−47.5858			−2.19731
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.4111	−	12.4111i	−0.570665	−	0.570665i
$474$		0			0
$475$	3.19891	−	0.267957i	0.146776	−	0.0122947i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	39.9242			1.82418			0.912091	−	0.409988i	$-0.134467\pi$
$479$	39.9242			1.82418			0.912091	+	0.409988i	$0.134467\pi$
$480$		0			0
$481$	21.8053			0.994236
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.16124	+	27.7748i	0.0527293	+	1.26119i
$486$		0			0
$487$	10.5650	+	10.5650i	0.478745	+	0.478745i	0.904730	−	0.425985i	$-0.140072\pi$
$487$	10.5650	+	10.5650i	0.478745	+	0.478745i	−0.425985	+	0.904730i	$0.640072\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.7875			−1.25403			−0.627016	−	0.779006i	$-0.715724\pi$
$491$	−27.7875			−1.25403			−0.627016	+	0.779006i	$0.715724\pi$
$492$		0			0
$493$	1.15410	−	1.15410i	0.0519780	−	0.0519780i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−38.3360	+	38.3360i	−1.71960	+	1.71960i
$498$		0			0
$499$		−	19.6133i		−	0.878014i	−0.898484	−	0.439007i	$-0.855330\pi$
$499$		−	19.6133i		−	0.878014i	0.898484	−	0.439007i	$-0.144670\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.2004	−	24.2004i	1.07904	−	1.07904i	0.0824469	−	0.996595i	$-0.473727\pi$
$503$	24.2004	−	24.2004i	1.07904	−	1.07904i	0.996595	−	0.0824469i	$-0.0262735\pi$
$504$		0			0
$505$	−2.31948	−	2.13332i	−0.103216	−	0.0949313i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.8167			−0.479444			−0.239722	−	0.970842i	$-0.577056\pi$
$509$	−10.8167			−0.479444			−0.239722	+	0.970842i	$0.577056\pi$
$510$		0			0
$511$		−	45.1253i		−	1.99623i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.318795	+	7.62498i	0.0140478	+	0.335997i
$516$		0			0
$517$	23.8666	+	23.8666i	1.04965	+	1.04965i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	29.5691			1.29544			0.647722	−	0.761877i	$-0.275722\pi$
$521$	29.5691			1.29544			0.647722	+	0.761877i	$0.275722\pi$
$522$		0			0
$523$	26.2357	+	26.2357i	1.14721	+	1.14721i	0.987100	+	0.160106i	$0.0511835\pi$
$523$	26.2357	+	26.2357i	1.14721	+	1.14721i	0.160106	+	0.987100i	$0.448816\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.58765	+	2.58765i	0.112720	+	0.112720i
$528$		0			0
$529$			12.3256i			0.535894i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.0941	+	10.0941i	−0.437224	+	0.437224i
$534$		0			0
$535$	0.206168	+	4.93117i	0.00891343	+	0.213193i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			57.8644i			2.49240i
$540$		0			0
$541$			9.66967i			0.415731i	0.978157	+	0.207866i	$0.0666517\pi$
$541$			9.66967i			0.415731i	−0.978157	+	0.207866i	$0.933348\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	22.1269	−	24.0579i	0.947814	−	1.03053i
$546$		0			0
$547$	−4.45632	+	4.45632i	−0.190538	+	0.190538i	−0.795929	−	0.605390i	$-0.793017\pi$
$547$	−4.45632	+	4.45632i	−0.190538	+	0.190538i	0.605390	+	0.795929i	$0.293017\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			0.449168i			0.0191352i
$552$		0			0
$553$	7.20684	+	7.20684i	0.306466	+	0.306466i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.10582	−	7.10582i	−0.301083	−	0.301083i	0.540354	−	0.841438i	$-0.318290\pi$
$557$	−7.10582	−	7.10582i	−0.301083	−	0.301083i	−0.841438	+	0.540354i	$0.818290\pi$
$558$		0			0
$559$	−14.6196			−0.618344
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.1395	+	12.1395i	0.511620	+	0.511620i	0.915023	−	0.403403i	$-0.132173\pi$
$563$	12.1395	+	12.1395i	0.511620	+	0.511620i	−0.403403	+	0.915023i	$0.632173\pi$
$564$		0			0
$565$	−14.3998	+	0.602043i	−0.605803	+	0.0253282i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	9.17667i		−	0.384706i	−0.981326	−	0.192353i	$-0.938388\pi$
$569$		−	9.17667i		−	0.384706i	0.981326	−	0.192353i	$-0.0616118\pi$
$570$		0			0
$571$	−14.1460			−0.591990			−0.295995	−	0.955190i	$-0.595651\pi$
$571$	−14.1460			−0.591990			−0.295995	+	0.955190i	$0.595651\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.4751	+	10.5467i	0.520248	+	0.439827i
$576$		0			0
$577$	6.88789	−	6.88789i	0.286747	−	0.286747i	−0.549046	−	0.835792i	$-0.685009\pi$
$577$	6.88789	−	6.88789i	0.286747	−	0.286747i	0.835792	+	0.549046i	$0.185009\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			45.6840i			1.89529i
$582$		0			0
$583$	6.75381	−	6.75381i	0.279714	−	0.279714i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.90740	−	5.90740i	0.243824	−	0.243824i	−0.574606	−	0.818430i	$-0.694845\pi$
$587$	5.90740	−	5.90740i	0.243824	−	0.243824i	0.818430	+	0.574606i	$0.194845\pi$
$588$		0			0
$589$	−1.00710			−0.0414967
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.6480	+	22.6480i	0.930042	+	0.930042i	0.997708	−	0.0676664i	$-0.0215554\pi$
$593$	22.6480	+	22.6480i	0.930042	+	0.930042i	−0.0676664	+	0.997708i	$0.521555\pi$
$594$		0			0
$595$	−17.1633	+	18.6611i	−0.703627	+	0.765031i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.4493			−0.712958			−0.356479	−	0.934303i	$-0.616023\pi$
$599$	−17.4493			−0.712958			−0.356479	+	0.934303i	$0.616023\pi$
$600$		0			0
$601$	−26.2079			−1.06904			−0.534521	−	0.845155i	$-0.679508\pi$
$601$	−26.2079			−1.06904			−0.534521	+	0.845155i	$0.679508\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.69350	+	1.84128i	−0.0688505	+	0.0748588i
$606$		0			0
$607$	8.79177	+	8.79177i	0.356847	+	0.356847i	0.862649	−	0.505802i	$-0.168804\pi$
$607$	8.79177	+	8.79177i	0.356847	+	0.356847i	−0.505802	+	0.862649i	$0.668804\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	28.1135			1.13735
$612$		0			0
$613$	−10.9270	+	10.9270i	−0.441339	+	0.441339i	−0.892462	−	0.451123i	$-0.851024\pi$
$613$	−10.9270	+	10.9270i	−0.441339	+	0.441339i	0.451123	+	0.892462i	$0.351024\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.6319	+	13.6319i	−0.548799	+	0.548799i	−0.926093	−	0.377294i	$-0.876855\pi$
$617$	−13.6319	+	13.6319i	−0.548799	+	0.548799i	0.377294	+	0.926093i	$0.376855\pi$
$618$		0			0
$619$			1.50796i			0.0606100i	0.999541	+	0.0303050i	$0.00964785\pi$
$619$			1.50796i			0.0606100i	−0.999541	+	0.0303050i	$0.990352\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.52148	+	2.52148i	−0.101021	+	0.101021i
$624$		0			0
$625$	−24.6516	+	4.15906i	−0.986065	+	0.166362i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.5438			0.699517
$630$		0			0
$631$			42.8319i			1.70511i	0.522638	+	0.852555i	$0.324948\pi$
$631$			42.8319i			1.70511i	−0.522638	+	0.852555i	$0.675052\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	30.4130	−	1.27154i	1.20690	−	0.0504596i
$636$		0			0
$637$	34.0805	+	34.0805i	1.35032	+	1.35032i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.93597			−0.392447			−0.196224	−	0.980559i	$-0.562868\pi$
$641$	−9.93597			−0.392447			−0.196224	+	0.980559i	$0.562868\pi$
$642$		0			0
$643$	20.3751	+	20.3751i	0.803517	+	0.803517i	0.983643	−	0.180127i	$-0.0576508\pi$
$643$	20.3751	+	20.3751i	0.803517	+	0.803517i	−0.180127	+	0.983643i	$0.557651\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.4648	+	25.4648i	1.00112	+	1.00112i	0.999999	+	0.00112374i	$0.000357696\pi$
$647$	25.4648	+	25.4648i	1.00112	+	1.00112i	0.00112374	+	0.999999i	$0.499642\pi$
$648$		0			0
$649$		−	9.54272i		−	0.374585i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.9677	+	16.9677i	−0.663999	+	0.663999i	−0.956320	−	0.292321i	$-0.905572\pi$
$653$	−16.9677	+	16.9677i	−0.663999	+	0.663999i	0.292321	+	0.956320i	$0.405572\pi$
$654$		0			0
$655$	−1.91892	+	2.08638i	−0.0749784	+	0.0815216i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.958005i		−	0.0373186i	−0.999826	−	0.0186593i	$-0.994060\pi$
$659$		−	0.958005i		−	0.0373186i	0.999826	−	0.0186593i	$-0.00593978\pi$
$660$		0			0
$661$		−	5.22656i		−	0.203289i	−0.994821	−	0.101645i	$-0.967590\pi$
$661$		−	5.22656i		−	0.203289i	0.994821	−	0.101645i	$-0.0324105\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.291465	−	6.97131i	−0.0113025	−	0.270336i
$666$		0			0
$667$	−1.61628	+	1.61628i	−0.0625824	+	0.0625824i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	18.1351i		−	0.700097i
$672$		0			0
$673$	16.3145	+	16.3145i	0.628876	+	0.628876i	0.947785	−	0.318909i	$-0.103316\pi$
$673$	16.3145	+	16.3145i	0.628876	+	0.628876i	−0.318909	+	0.947785i	$0.603316\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.9712	+	21.9712i	0.844423	+	0.844423i	0.989431	−	0.145008i	$-0.0463207\pi$
$677$	21.9712	+	21.9712i	0.844423	+	0.844423i	−0.145008	+	0.989431i	$0.546321\pi$
$678$		0			0
$679$	60.4231			2.31883
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−27.9871	−	27.9871i	−1.07090	−	1.07090i	−0.997287	−	0.0736087i	$-0.976548\pi$
$683$	−27.9871	−	27.9871i	−1.07090	−	1.07090i	−0.0736087	−	0.997287i	$-0.523452\pi$
$684$		0			0
$685$	1.61550	+	38.6398i	0.0617251	+	1.47635i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	7.95560i		−	0.303084i
$690$		0			0
$691$	39.3415			1.49662			0.748310	−	0.663349i	$-0.230866\pi$
$691$	39.3415			1.49662			0.748310	+	0.663349i	$0.230866\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.3908	−	12.3160i	−0.507941	−	0.467172i
$696$		0			0
$697$	−8.12138	+	8.12138i	−0.307619	+	0.307619i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			10.3150i			0.389594i	0.980844	+	0.194797i	$0.0624048\pi$
$701$			10.3150i			0.389594i	−0.980844	+	0.194797i	$0.937595\pi$
$702$		0			0
$703$	−3.41396	+	3.41396i	−0.128760	+	0.128760i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.84346	+	4.84346i	−0.182157	+	0.182157i
$708$		0			0
$709$	45.3404			1.70279			0.851397	−	0.524522i	$-0.175756\pi$
$709$	45.3404			1.70279			0.851397	+	0.524522i	$0.175756\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.62391	−	3.62391i	−0.135717	−	0.135717i
$714$		0			0
$715$	0.942858	+	22.5514i	0.0352609	+	0.843375i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−43.8494			−1.63531			−0.817653	−	0.575711i	$-0.804725\pi$
$719$	−43.8494			−1.63531			−0.817653	+	0.575711i	$0.804725\pi$
$720$		0			0
$721$	16.5879			0.617765
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.291992	−	3.48586i	−0.0108443	−	0.129461i
$726$		0			0
$727$	2.07567	+	2.07567i	0.0769824	+	0.0769824i	0.744550	−	0.667567i	$-0.232664\pi$
$727$	2.07567	+	2.07567i	0.0769824	+	0.0769824i	−0.667567	+	0.744550i	$0.732664\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11.7625			−0.435050
$732$		0			0
$733$	2.13221	−	2.13221i	0.0787549	−	0.0787549i	−0.666632	−	0.745387i	$-0.732265\pi$
$733$	2.13221	−	2.13221i	0.0787549	−	0.0787549i	0.745387	+	0.666632i	$0.232265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.1009	+	24.1009i	−0.887769	+	0.887769i
$738$		0			0
$739$		−	32.9512i		−	1.21213i	−0.795416	−	0.606064i	$-0.792747\pi$
$739$		−	32.9512i		−	1.21213i	0.795416	−	0.606064i	$-0.207253\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.0344	+	17.0344i	−0.624931	+	0.624931i	−0.946788	−	0.321857i	$-0.895693\pi$
$743$	−17.0344	+	17.0344i	−0.624931	+	0.624931i	0.321857	+	0.946788i	$0.395693\pi$
$744$		0			0
$745$	−4.16378	+	4.52714i	−0.152549	+	0.165862i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.7276			0.391977
$750$		0			0
$751$		−	38.0634i		−	1.38895i	−0.719515	−	0.694477i	$-0.755636\pi$
$751$		−	38.0634i		−	1.38895i	0.719515	−	0.694477i	$-0.244364\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.7905	−	15.4429i	−0.611069	−	0.562023i
$756$		0			0
$757$	−31.5125	−	31.5125i	−1.14534	−	1.14534i	−0.987458	−	0.157885i	$-0.949533\pi$
$757$	−31.5125	−	31.5125i	−1.14534	−	1.14534i	−0.157885	−	0.987458i	$-0.550467\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.1731			−1.31128			−0.655638	−	0.755076i	$-0.727600\pi$
$761$	−36.1731			−1.31128			−0.655638	+	0.755076i	$0.727600\pi$
$762$		0			0
$763$	−50.2367	−	50.2367i	−1.81869	−	1.81869i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.62039	−	5.62039i	−0.202941	−	0.202941i
$768$		0			0
$769$			29.7981i			1.07455i	0.843408	+	0.537273i	$0.180546\pi$
$769$			29.7981i			1.07455i	−0.843408	+	0.537273i	$0.819454\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.11081	−	4.11081i	0.147856	−	0.147856i	−0.629304	−	0.777159i	$-0.716660\pi$
$773$	4.11081	−	4.11081i	0.147856	−	0.147856i	0.777159	+	0.629304i	$0.216660\pi$
$774$		0			0
$775$	7.81577	−	0.654687i	0.280751	−	0.0235170i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	3.16079i		−	0.113247i
$780$		0			0
$781$			38.8323i			1.38953i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.07941	−	0.170557i	0.145600	−	0.00608744i
$786$		0			0
$787$	−14.4907	+	14.4907i	−0.516537	+	0.516537i	−0.916522	−	0.399985i	$-0.869015\pi$
$787$	−14.4907	+	14.4907i	−0.516537	+	0.516537i	0.399985	+	0.916522i	$0.369015\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			31.3262i			1.11383i
$792$		0			0
$793$	−10.6811	−	10.6811i	−0.379295	−	0.379295i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.99359	−	3.99359i	−0.141460	−	0.141460i	0.632830	−	0.774291i	$-0.281893\pi$
$797$	−3.99359	−	3.99359i	−0.141460	−	0.141460i	−0.774291	+	0.632830i	$0.781893\pi$
$798$		0			0
$799$	22.6192			0.800210
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.8547	−	22.8547i	−0.806527	−	0.806527i
$804$		0			0
$805$	24.0366	−	26.1342i	0.847179	−	0.921110i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.83726i		−	0.0645947i	−0.999478	−	0.0322974i	$-0.989718\pi$
$809$		−	1.83726i		−	0.0645947i	0.999478	−	0.0322974i	$-0.0102824\pi$
$810$		0			0
$811$	−19.2559			−0.676166			−0.338083	−	0.941116i	$-0.609778\pi$
$811$	−19.2559			−0.676166			−0.338083	+	0.941116i	$0.609778\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	29.4084	−	1.22954i	1.03013	−	0.0430690i
$816$		0			0
$817$	2.28893	−	2.28893i	0.0800797	−	0.0800797i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			29.3331i			1.02373i	0.859066	+	0.511865i	$0.171045\pi$
$821$			29.3331i			1.02373i	−0.859066	+	0.511865i	$0.828955\pi$
$822$		0			0
$823$	−11.8788	+	11.8788i	−0.414067	+	0.414067i	−0.883153	−	0.469085i	$-0.844584\pi$
$823$	−11.8788	+	11.8788i	−0.414067	+	0.414067i	0.469085	+	0.883153i	$0.344584\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.1429	−	24.1429i	0.839530	−	0.839530i	−0.149267	−	0.988797i	$-0.547691\pi$
$827$	24.1429	−	24.1429i	0.839530	−	0.839530i	0.988797	+	0.149267i	$0.0476913\pi$
$828$		0			0
$829$	30.6528			1.06462			0.532309	−	0.846550i	$-0.321325\pi$
$829$	30.6528			1.06462			0.532309	+	0.846550i	$0.321325\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.4200	+	27.4200i	0.950047	+	0.950047i
$834$		0			0
$835$	37.5494	−	1.56991i	1.29945	−	0.0543290i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.59796			−0.0896914			−0.0448457	−	0.998994i	$-0.514280\pi$
$839$	−2.59796			−0.0896914			−0.0448457	+	0.998994i	$0.514280\pi$
$840$		0			0
$841$	−28.5105			−0.983122
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.55803	−	6.95140i	−0.260004	−	0.239135i
$846$		0			0
$847$	3.84490	+	3.84490i	0.132112	+	0.132112i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.5695			−0.842230
$852$		0			0
$853$	−20.7909	+	20.7909i	−0.711866	+	0.711866i	−0.966925	−	0.255059i	$-0.917905\pi$
$853$	−20.7909	+	20.7909i	−0.711866	+	0.711866i	0.255059	+	0.966925i	$0.417905\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.7632	−	33.7632i	1.15333	−	1.15333i	0.167446	−	0.985881i	$-0.446448\pi$
$857$	33.7632	−	33.7632i	1.15333	−	1.15333i	0.985881	−	0.167446i	$-0.0535521\pi$
$858$		0			0
$859$			31.9834i			1.09126i	0.838027	+	0.545629i	$0.183709\pi$
$859$			31.9834i			1.09126i	−0.838027	+	0.545629i	$0.816291\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.78203	+	2.78203i	−0.0947013	+	0.0947013i	−0.752870	−	0.658169i	$-0.771331\pi$
$863$	−2.78203	+	2.78203i	−0.0947013	+	0.0947013i	0.658169	+	0.752870i	$0.271331\pi$
$864$		0			0
$865$	−1.75076	−	41.8751i	−0.0595278	−	1.42380i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.30014			0.247640
$870$		0			0
$871$			28.3895i			0.961943i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.79384	+	53.9128i	0.229674	+	1.82258i
$876$		0			0
$877$	3.46500	+	3.46500i	0.117005	+	0.117005i	0.763185	−	0.646180i	$-0.223635\pi$
$877$	3.46500	+	3.46500i	0.117005	+	0.117005i	−0.646180	+	0.763185i	$0.723635\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11.9117			−0.401316			−0.200658	−	0.979661i	$-0.564308\pi$
$881$	−11.9117			−0.401316			−0.200658	+	0.979661i	$0.564308\pi$
$882$		0			0
$883$	−14.6270	−	14.6270i	−0.492237	−	0.492237i	0.416774	−	0.909010i	$-0.363161\pi$
$883$	−14.6270	−	14.6270i	−0.492237	−	0.492237i	−0.909010	+	0.416774i	$0.863161\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.84575	+	5.84575i	0.196281	+	0.196281i	0.798404	−	0.602123i	$-0.205678\pi$
$887$	5.84575	+	5.84575i	0.196281	+	0.196281i	−0.602123	+	0.798404i	$0.705678\pi$
$888$		0			0
$889$		−	66.1623i		−	2.21901i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.40162	+	4.40162i	−0.147295	+	0.147295i
$894$		0			0
$895$	5.94274	+	5.46576i	0.198644	+	0.182700i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1.09743i			0.0366014i
$900$		0			0
$901$		−	6.40081i		−	0.213242i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.0420	−	33.1492i	−1.19808	−	1.10192i
$906$		0			0
$907$	19.5550	−	19.5550i	0.649314	−	0.649314i	−0.303513	−	0.952827i	$-0.598160\pi$
$907$	19.5550	−	19.5550i	0.649314	−	0.649314i	0.952827	+	0.303513i	$0.0981597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.2174i			0.669832i	0.942248	+	0.334916i	$0.108708\pi$
$911$			20.2174i			0.669832i	−0.942248	+	0.334916i	$0.891292\pi$
$912$		0			0
$913$	23.1377	+	23.1377i	0.765747	+	0.765747i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.35669	+	4.35669i	0.143871	+	0.143871i
$918$		0			0
$919$	−58.3013			−1.92318			−0.961591	−	0.274485i	$-0.911493\pi$
$919$	−58.3013			−1.92318			−0.961591	+	0.274485i	$0.911493\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	22.8711	+	22.8711i	0.752812	+	0.752812i
$924$		0			0
$925$	24.2754	−	28.7141i	0.798171	−	0.944113i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			5.05985i			0.166008i	0.996549	+	0.0830041i	$0.0264515\pi$
$929$			5.05985i			0.166008i	−0.996549	+	0.0830041i	$0.973549\pi$
$930$		0			0
$931$	−10.6717			−0.349750
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.758591	+	18.1441i	0.0248086	+	0.593376i
$936$		0			0
$937$	−10.0055	+	10.0055i	−0.326864	+	0.326864i	−0.851393	−	0.524529i	$-0.824242\pi$
$937$	−10.0055	+	10.0055i	−0.326864	+	0.326864i	0.524529	+	0.851393i	$0.324242\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	23.9169i		−	0.779667i	−0.920885	−	0.389834i	$-0.872532\pi$
$941$		−	23.9169i		−	0.779667i	0.920885	−	0.389834i	$-0.127468\pi$
$942$		0			0
$943$	11.3737	−	11.3737i	0.370379	−	0.370379i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.5395	+	24.5395i	−0.797426	+	0.797426i	−0.982689	−	0.185263i	$-0.940686\pi$
$947$	−24.5395	+	24.5395i	−0.797426	+	0.797426i	0.185263	+	0.982689i	$0.440686\pi$
$948$		0			0
$949$	−26.9216			−0.873912
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.855191	+	0.855191i	0.0277023	+	0.0277023i	0.720822	−	0.693120i	$-0.243764\pi$
$953$	0.855191	+	0.855191i	0.0277023	+	0.0277023i	−0.693120	+	0.720822i	$0.743764\pi$
$954$		0			0
$955$	16.6735	+	15.3352i	0.539541	+	0.496236i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	84.0595			2.71442
$960$		0			0
$961$	28.5394			0.920626
$962$		0			0
$963$		0			0
$964$		0			0
$965$	45.3581	−	1.89639i	1.46013	−	0.0610469i
$966$		0			0
$967$	−15.5174	−	15.5174i	−0.499007	−	0.499007i	0.412122	−	0.911129i	$-0.364788\pi$
$967$	−15.5174	−	15.5174i	−0.499007	−	0.499007i	−0.911129	+	0.412122i	$0.864788\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.2086			−0.359701			−0.179851	−	0.983694i	$-0.557561\pi$
$971$	−11.2086			−0.359701			−0.179851	+	0.983694i	$0.557561\pi$
$972$		0			0
$973$	−27.9621	+	27.9621i	−0.896424	+	0.896424i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.3780	−	19.3780i	0.619957	−	0.619957i	−0.325563	−	0.945520i	$-0.605554\pi$
$977$	19.3780	−	19.3780i	0.619957	−	0.619957i	0.945520	+	0.325563i	$0.105554\pi$
$978$		0			0
$979$			2.55412i			0.0816302i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	14.1500	−	14.1500i	0.451314	−	0.451314i	−0.444476	−	0.895791i	$-0.646610\pi$
$983$	14.1500	−	14.1500i	0.451314	−	0.451314i	0.895791	+	0.444476i	$0.146610\pi$
$984$		0			0
$985$	46.4082	−	1.94029i	1.47869	−	0.0618228i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.4729			0.523808
$990$		0			0
$991$			45.3242i			1.43977i	0.694093	+	0.719885i	$0.255805\pi$
$991$			45.3242i			1.43977i	−0.694093	+	0.719885i	$0.744195\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.69862	−	8.37046i	0.244063	−	0.265361i
$996$		0			0
$997$	−14.5950	−	14.5950i	−0.462228	−	0.462228i	0.437157	−	0.899385i	$-0.355985\pi$
$997$	−14.5950	−	14.5950i	−0.462228	−	0.462228i	−0.899385	+	0.437157i	$0.855985\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bi.e.1423.3		24
3.2	odd	2		480.2.bh.a.463.11		24
4.3	odd	2		360.2.w.e.163.8		24
5.2	odd	4	inner	1440.2.bi.e.847.10		24
8.3	odd	2	inner	1440.2.bi.e.1423.10		24
8.5	even	2		360.2.w.e.163.10		24
12.11	even	2		120.2.v.a.43.5	yes	24
15.2	even	4		480.2.bh.a.367.8		24
15.8	even	4		2400.2.bh.b.1807.6		24
15.14	odd	2		2400.2.bh.b.943.5		24
20.7	even	4		360.2.w.e.307.10		24
24.5	odd	2		120.2.v.a.43.3	✓	24
24.11	even	2		480.2.bh.a.463.8		24
40.27	even	4	inner	1440.2.bi.e.847.3		24
40.37	odd	4		360.2.w.e.307.8		24
60.23	odd	4		600.2.v.b.307.10		24
60.47	odd	4		120.2.v.a.67.3	yes	24
60.59	even	2		600.2.v.b.43.8		24
120.29	odd	2		600.2.v.b.43.10		24
120.53	even	4		600.2.v.b.307.8		24
120.59	even	2		2400.2.bh.b.943.6		24
120.77	even	4		120.2.v.a.67.5	yes	24
120.83	odd	4		2400.2.bh.b.1807.5		24
120.107	odd	4		480.2.bh.a.367.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.v.a.43.3	✓	24	24.5	odd	2
120.2.v.a.43.5	yes	24	12.11	even	2
120.2.v.a.67.3	yes	24	60.47	odd	4
120.2.v.a.67.5	yes	24	120.77	even	4
360.2.w.e.163.8		24	4.3	odd	2
360.2.w.e.163.10		24	8.5	even	2
360.2.w.e.307.8		24	40.37	odd	4
360.2.w.e.307.10		24	20.7	even	4
480.2.bh.a.367.8		24	15.2	even	4
480.2.bh.a.367.11		24	120.107	odd	4
480.2.bh.a.463.8		24	24.11	even	2
480.2.bh.a.463.11		24	3.2	odd	2
600.2.v.b.43.8		24	60.59	even	2
600.2.v.b.43.10		24	120.29	odd	2
600.2.v.b.307.8		24	120.53	even	4
600.2.v.b.307.10		24	60.23	odd	4
1440.2.bi.e.847.3		24	40.27	even	4	inner
1440.2.bi.e.847.10		24	5.2	odd	4	inner
1440.2.bi.e.1423.3		24	1.1	even	1	trivial
1440.2.bi.e.1423.10		24	8.3	odd	2	inner
2400.2.bh.b.943.5		24	15.14	odd	2
2400.2.bh.b.943.6		24	120.59	even	2
2400.2.bh.b.1807.5		24	120.83	odd	4
2400.2.bh.b.1807.6		24	15.8	even	4

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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.51371 + 1.64581i) q^{5} +(3.43671 + 3.43671i) q^{7} +3.48120 q^{11} +(2.05033 - 2.05033i) q^{13} +(1.64963 - 1.64963i) q^{17} +0.642023i q^{19} +(-2.31024 + 2.31024i) q^{23} +(-0.417363 - 4.98255i) q^{25} +0.699613 q^{29} +1.56863i q^{31} +(-10.8583 + 0.453979i) q^{35} +(5.31751 + 5.31751i) q^{37} -4.92316 q^{41} +(-3.56519 - 3.56519i) q^{43} +(6.85586 + 6.85586i) q^{47} +16.6220i q^{49} +(1.94008 - 1.94008i) q^{53} +(-5.26953 + 5.72939i) q^{55} -2.74121i q^{59} -5.20943i q^{61} +(0.270842 + 6.47805i) q^{65} +(-6.92316 + 6.92316i) q^{67} +11.1548i q^{71} +(-6.56519 - 6.56519i) q^{73} +(11.9639 + 11.9639i) q^{77} +2.09702 q^{79} +(6.64648 + 6.64648i) q^{83} +(0.217911 + 5.21202i) q^{85} +0.733690i q^{89} +14.0928 q^{91} +(-1.05665 - 0.971837i) q^{95} +(8.79083 - 8.79083i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) \) 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Introduction

L-functions

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