LMFDB - Embedded newform 1512.2.q.c.1369.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(793,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.793"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1512 = 2^{3} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1512.q (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [22,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1369.4
Character	$\chi$	$=$	1512.1369
Dual form			1512.2.q.c.793.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.918286 + 1.59052i) q^{5} +(-0.361656 - 2.62092i) q^{7} +(-1.54860 - 2.68225i) q^{11} +(2.40225 + 4.16081i) q^{13} +(-1.87185 + 3.24214i) q^{17} +(-2.71408 - 4.70093i) q^{19} +(-3.97914 + 6.89208i) q^{23} +(0.813503 + 1.40903i) q^{25} +(0.325267 - 0.563379i) q^{29} +1.03668 q^{31} +(4.50072 + 1.83153i) q^{35} +(0.873712 + 1.51331i) q^{37} +(-2.52260 - 4.36927i) q^{41} +(-6.09645 + 10.5594i) q^{43} +4.61383 q^{47} +(-6.73841 + 1.89574i) q^{49} +(-4.55082 + 7.88226i) q^{53} +5.68821 q^{55} +5.79727 q^{59} -4.81245 q^{61} -8.82379 q^{65} -14.4774 q^{67} +5.00714 q^{71} +(-1.81364 + 3.14131i) q^{73} +(-6.46989 + 5.02879i) q^{77} -14.3581 q^{79} +(-3.83139 + 6.63616i) q^{83} +(-3.43778 - 5.95441i) q^{85} +(5.76798 + 9.99043i) q^{89} +(10.0364 - 7.80087i) q^{91} +9.96922 q^{95} +(-1.04480 + 1.80964i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$22 q - 3 q^{5} - 5 q^{7} + 3 q^{11} - 3 q^{13} - 7 q^{17} - q^{19} - 2 q^{23} - 10 q^{25} - 9 q^{29} + 8 q^{31} - 14 q^{35} + 2 q^{37} - 16 q^{41} + 10 q^{47} + 15 q^{49} - 11 q^{53} + 22 q^{55} - 38 q^{59}+ \cdots - 29 q^{97}+O(q^{100})$ 22 * q - 3 * q^5 - 5 * q^7 + 3 * q^11 - 3 * q^13 - 7 * q^17 - q^19 - 2 * q^23 - 10 * q^25 - 9 * q^29 + 8 * q^31 - 14 * q^35 + 2 * q^37 - 16 * q^41 + 10 * q^47 + 15 * q^49 - 11 * q^53 + 22 * q^55 - 38 * q^59 + 26 * q^61 + 26 * q^65 - 52 * q^67 + 48 * q^71 - 35 * q^73 - 17 * q^77 - 20 * q^79 + 28 * q^83 - 20 * q^85 - 6 * q^89 - 37 * q^91 + 24 * q^95 - 29 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$ .

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−0.918286	+	1.59052i	−0.410670	+	0.711301i	−0.994963	−	0.100242i	$-0.968038\pi$
$5$	−0.918286	+	1.59052i	−0.410670	+	0.711301i	0.584293	+	0.811543i	$0.301372\pi$
$6$		0			0
$7$	−0.361656	−	2.62092i	−0.136693	−	0.990613i
$7$	−0.361656	−	2.62092i	−0.136693	−	0.990613i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.54860	−	2.68225i	−0.466919	−	0.808728i	0.532367	−	0.846514i	$-0.321303\pi$
$11$	−1.54860	−	2.68225i	−0.466919	−	0.808728i	−0.999286	+	0.0377862i	$0.987969\pi$
$12$		0			0
$13$	2.40225	+	4.16081i	0.666263	+	1.15400i	0.978941	+	0.204143i	$0.0654406\pi$
$13$	2.40225	+	4.16081i	0.666263	+	1.15400i	−0.312678	+	0.949859i	$0.601226\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.87185	+	3.24214i	−0.453990	+	0.786333i	−0.998629	−	0.0523367i	$-0.983333\pi$
$17$	−1.87185	+	3.24214i	−0.453990	+	0.786333i	0.544640	+	0.838670i	$0.316666\pi$
$18$		0			0
$19$	−2.71408	−	4.70093i	−0.622654	−	1.07847i	−0.988990	−	0.147985i	$-0.952721\pi$
$19$	−2.71408	−	4.70093i	−0.622654	−	1.07847i	0.366336	−	0.930483i	$-0.380612\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.97914	+	6.89208i	−0.829709	+	1.43710i	0.0685581	+	0.997647i	$0.478160\pi$
$23$	−3.97914	+	6.89208i	−0.829709	+	1.43710i	−0.898267	+	0.439450i	$0.855173\pi$
$24$		0			0
$25$	0.813503	+	1.40903i	0.162701	+	0.281806i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.325267	−	0.563379i	0.0604006	−	0.104617i	−0.834244	−	0.551396i	$-0.814096\pi$
$29$	0.325267	−	0.563379i	0.0604006	−	0.104617i	0.894645	+	0.446779i	$0.147429\pi$
$30$		0			0
$31$	1.03668			0.186194			0.0930970	−	0.995657i	$-0.470323\pi$
$31$	1.03668			0.186194			0.0930970	+	0.995657i	$0.470323\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.50072	+	1.83153i	0.760760	+	0.309585i
$36$		0			0
$37$	0.873712	+	1.51331i	0.143637	+	0.248787i	0.928864	−	0.370422i	$-0.120787\pi$
$37$	0.873712	+	1.51331i	0.143637	+	0.248787i	−0.785226	+	0.619209i	$0.787453\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.52260	−	4.36927i	−0.393964	−	0.682365i	0.599005	−	0.800745i	$-0.295563\pi$
$41$	−2.52260	−	4.36927i	−0.393964	−	0.682365i	−0.992968	+	0.118381i	$0.962230\pi$
$42$		0			0
$43$	−6.09645	+	10.5594i	−0.929699	+	1.61029i	−0.145876	+	0.989303i	$0.546600\pi$
$43$	−6.09645	+	10.5594i	−0.929699	+	1.61029i	−0.783824	+	0.620984i	$0.786733\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.61383			0.672996			0.336498	−	0.941684i	$-0.390757\pi$
$47$	4.61383			0.672996			0.336498	+	0.941684i	$0.390757\pi$
$48$		0			0
$49$	−6.73841	+	1.89574i	−0.962630	+	0.270820i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.55082	+	7.88226i	−0.625104	+	1.08271i	0.363417	+	0.931626i	$0.381610\pi$
$53$	−4.55082	+	7.88226i	−0.625104	+	1.08271i	−0.988521	+	0.151085i	$0.951723\pi$
$54$		0			0
$55$	5.68821			0.766998
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.79727			0.754740			0.377370	−	0.926063i	$-0.376828\pi$
$59$	5.79727			0.754740			0.377370	+	0.926063i	$0.376828\pi$
$60$		0			0
$61$	−4.81245			−0.616172			−0.308086	−	0.951359i	$-0.599688\pi$
$61$	−4.81245			−0.616172			−0.308086	+	0.951359i	$0.599688\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.82379			−1.09446
$66$		0			0
$67$	−14.4774			−1.76870			−0.884348	−	0.466828i	$-0.845397\pi$
$67$	−14.4774			−1.76870			−0.884348	+	0.466828i	$0.845397\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.00714			0.594238			0.297119	−	0.954840i	$-0.403974\pi$
$71$	5.00714			0.594238			0.297119	+	0.954840i	$0.403974\pi$
$72$		0			0
$73$	−1.81364	+	3.14131i	−0.212270	+	0.367662i	−0.952425	−	0.304774i	$-0.901419\pi$
$73$	−1.81364	+	3.14131i	−0.212270	+	0.367662i	0.740155	+	0.672437i	$0.234752\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.46989	+	5.02879i	−0.737312	+	0.573084i
$78$		0			0
$79$	−14.3581			−1.61541			−0.807705	−	0.589587i	$-0.799290\pi$
$79$	−14.3581			−1.61541			−0.807705	+	0.589587i	$0.799290\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.83139	+	6.63616i	−0.420550	+	0.728414i	−0.995993	−	0.0894279i	$-0.971496\pi$
$83$	−3.83139	+	6.63616i	−0.420550	+	0.728414i	0.575443	+	0.817842i	$0.304829\pi$
$84$		0			0
$85$	−3.43778	−	5.95441i	−0.372880	−	0.645847i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.76798	+	9.99043i	0.611405	+	1.05898i	0.991004	+	0.133833i	$0.0427286\pi$
$89$	5.76798	+	9.99043i	0.611405	+	1.05898i	−0.379599	+	0.925151i	$0.623938\pi$
$90$		0			0
$91$	10.0364	−	7.80087i	1.05210	−	0.817753i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.96922			1.02282
$96$		0			0
$97$	−1.04480	+	1.80964i	−0.106083	+	0.183741i	−0.914180	−	0.405308i	$-0.867164\pi$
$97$	−1.04480	+	1.80964i	−0.106083	+	0.183741i	0.808097	+	0.589049i	$0.200498\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.22661	−	14.2489i	−0.818578	−	1.41782i	−0.906730	−	0.421712i	$-0.861429\pi$
$101$	−8.22661	−	14.2489i	−0.818578	−	1.41782i	0.0881520	−	0.996107i	$-0.471904\pi$
$102$		0			0
$103$	3.87346	−	6.70903i	0.381663	−	0.661060i	−0.609637	−	0.792681i	$-0.708685\pi$
$103$	3.87346	−	6.70903i	0.381663	−	0.661060i	0.991300	+	0.131621i	$0.0420181\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.74746	−	6.49080i	−0.362281	−	0.627489i	0.626055	−	0.779779i	$-0.284669\pi$
$107$	−3.74746	−	6.49080i	−0.362281	−	0.627489i	−0.988336	+	0.152290i	$0.951335\pi$
$108$		0			0
$109$	−4.30644	+	7.45897i	−0.412482	+	0.714440i	−0.995160	−	0.0982628i	$-0.968671\pi$
$109$	−4.30644	+	7.45897i	−0.412482	+	0.714440i	0.582678	+	0.812703i	$0.302005\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.55747	+	2.69762i	0.146514	+	0.253771i	0.929937	−	0.367719i	$-0.119861\pi$
$113$	1.55747	+	2.69762i	0.146514	+	0.253771i	−0.783422	+	0.621490i	$0.786528\pi$
$114$		0			0
$115$	−7.30798	−	12.6578i	−0.681473	−	1.18035i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.17433	+	3.73342i	0.841010	+	0.342242i
$120$		0			0
$121$	0.703704	−	1.21885i	0.0639731	−	0.110805i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.1710			−1.08860
$126$		0			0
$127$	10.8866			0.966033			0.483017	−	0.875611i	$-0.339541\pi$
$127$	10.8866			0.966033			0.483017	+	0.875611i	$0.339541\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.02790	+	13.9047i	−0.701401	+	1.21486i	0.266574	+	0.963815i	$0.414108\pi$
$131$	−8.02790	+	13.9047i	−0.701401	+	1.21486i	−0.967975	+	0.251048i	$0.919225\pi$
$132$		0			0
$133$	−11.3392	+	8.81351i	−0.983232	+	0.764228i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.72031	+	11.6399i	0.574155	+	0.994465i	0.996133	+	0.0878590i	$0.0280025\pi$
$137$	6.72031	+	11.6399i	0.574155	+	0.994465i	−0.421978	+	0.906606i	$0.638664\pi$
$138$		0			0
$139$	−4.06953	−	7.04863i	−0.345173	−	0.597857i	0.640212	−	0.768198i	$-0.278846\pi$
$139$	−4.06953	−	7.04863i	−0.345173	−	0.597857i	−0.985385	+	0.170341i	$0.945513\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.44022	−	12.8868i	0.622182	−	1.07765i
$144$		0			0
$145$	0.597376	+	1.03469i	0.0496094	+	0.0859260i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.76479	−	6.52081i	0.308424	−	0.534205i	−0.669594	−	0.742727i	$-0.733532\pi$
$149$	3.76479	−	6.52081i	0.308424	−	0.534205i	0.978018	+	0.208522i	$0.0668652\pi$
$150$		0			0
$151$	2.83616	+	4.91237i	0.230803	+	0.399763i	0.958045	−	0.286619i	$-0.0925313\pi$
$151$	2.83616	+	4.91237i	0.230803	+	0.399763i	−0.727241	+	0.686382i	$0.759198\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.951973	+	1.64886i	−0.0764643	+	0.132440i
$156$		0			0
$157$	0.436763			0.0348575			0.0174287	−	0.999848i	$-0.494452\pi$
$157$	0.436763			0.0348575			0.0174287	+	0.999848i	$0.494452\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	19.5026	+	7.93644i	1.53702	+	0.625479i
$162$		0			0
$163$	9.12649	+	15.8076i	0.714842	+	1.23814i	0.963020	+	0.269429i	$0.0868348\pi$
$163$	9.12649	+	15.8076i	0.714842	+	1.23814i	−0.248178	+	0.968714i	$0.579832\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.765108	−	1.32521i	−0.0592058	−	0.102548i	0.834903	−	0.550397i	$-0.185523\pi$
$167$	−0.765108	−	1.32521i	−0.0592058	−	0.102548i	−0.894109	+	0.447849i	$0.852190\pi$
$168$		0			0
$169$	−5.04157	+	8.73226i	−0.387813	+	0.671713i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.16949			−0.164943			−0.0824716	−	0.996593i	$-0.526281\pi$
$173$	−2.16949			−0.164943			−0.0824716	+	0.996593i	$0.526281\pi$
$174$		0			0
$175$	3.39874	−	2.64171i	0.256921	−	0.199694i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.08263	−	1.87517i	0.0809195	−	0.140157i	−0.822726	−	0.568439i	$-0.807548\pi$
$179$	1.08263	−	1.87517i	0.0809195	−	0.140157i	0.903645	+	0.428282i	$0.140881\pi$
$180$		0			0
$181$	0.557838			0.0414638			0.0207319	−	0.999785i	$-0.493400\pi$
$181$	0.557838			0.0414638			0.0207319	+	0.999785i	$0.493400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.20927			−0.235950
$186$		0			0
$187$	11.5949			0.847906
$188$		0			0
$189$		0			0
$190$		0			0
$191$	23.9997			1.73655			0.868277	−	0.496079i	$-0.165227\pi$
$191$	23.9997			1.73655			0.868277	+	0.496079i	$0.165227\pi$
$192$		0			0
$193$	−21.2794			−1.53172			−0.765862	−	0.643005i	$-0.777687\pi$
$193$	−21.2794			−1.53172			−0.765862	+	0.643005i	$0.777687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.8768			1.05993			0.529964	−	0.848020i	$-0.322205\pi$
$197$	14.8768			1.05993			0.529964	+	0.848020i	$0.322205\pi$
$198$		0			0
$199$	6.17884	−	10.7021i	0.438006	−	0.758649i	−0.559530	−	0.828810i	$-0.689018\pi$
$199$	6.17884	−	10.7021i	0.438006	−	0.758649i	0.997536	+	0.0701616i	$0.0223515\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.59420	−	0.648749i	−0.111891	−	0.0455332i
$204$		0			0
$205$	9.26586			0.647156
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.40604	+	14.5597i	−0.581458	+	1.00711i
$210$		0			0
$211$	8.65802	+	14.9961i	0.596043	+	1.03238i	0.993399	+	0.114712i	$0.0365944\pi$
$211$	8.65802	+	14.9961i	0.596043	+	1.03238i	−0.397356	+	0.917664i	$0.630072\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.1966	−	19.3930i	−0.763599	−	1.32259i
$216$		0			0
$217$	−0.374923	−	2.71706i	−0.0254514	−	0.184446i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.9866			−1.20991
$222$		0			0
$223$	1.14489	−	1.98301i	0.0766677	−	0.132792i	−0.825143	−	0.564925i	$-0.808905\pi$
$223$	1.14489	−	1.98301i	0.0766677	−	0.132792i	0.901810	+	0.432132i	$0.142239\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.78013	−	3.08328i	−0.118152	−	0.204644i	0.800884	−	0.598820i	$-0.204364\pi$
$227$	−1.78013	−	3.08328i	−0.118152	−	0.204644i	−0.919035	+	0.394176i	$0.871030\pi$
$228$		0			0
$229$	13.4799	−	23.3478i	0.890775	−	1.54287i	0.0518260	−	0.998656i	$-0.483496\pi$
$229$	13.4799	−	23.3478i	0.890775	−	1.54287i	0.838949	−	0.544211i	$-0.183171\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.7321	−	18.5885i	−0.703081	−	1.21777i	−0.967380	−	0.253332i	$-0.918474\pi$
$233$	−10.7321	−	18.5885i	−0.703081	−	1.21777i	0.264298	−	0.964441i	$-0.414860\pi$
$234$		0			0
$235$	−4.23681	+	7.33837i	−0.276379	+	0.478703i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.65970	−	8.07083i	−0.301411	−	0.522059i	0.675045	−	0.737777i	$-0.264124\pi$
$239$	−4.65970	−	8.07083i	−0.301411	−	0.522059i	−0.976456	+	0.215718i	$0.930791\pi$
$240$		0			0
$241$	10.1003	+	17.4943i	0.650620	+	1.12691i	0.982973	+	0.183752i	$0.0588242\pi$
$241$	10.1003	+	17.4943i	0.650620	+	1.12691i	−0.332353	+	0.943155i	$0.607842\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.17258	−	12.4584i	0.202689	−	0.795937i
$246$		0			0
$247$	13.0398	−	22.5856i	0.829702	−	1.43709i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.1837			−1.71582			−0.857910	−	0.513800i	$-0.828238\pi$
$251$	−27.1837			−1.71582			−0.857910	+	0.513800i	$0.828238\pi$
$252$		0			0
$253$	24.6483			1.54963
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.2411	−	24.6662i	0.888333	−	1.53864i	0.0464876	−	0.998919i	$-0.485197\pi$
$257$	14.2411	−	24.6662i	0.888333	−	1.53864i	0.841845	−	0.539719i	$-0.181469\pi$
$258$		0			0
$259$	3.65028	−	2.83722i	0.226818	−	0.176297i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.79907	+	3.11608i	0.110935	+	0.192146i	0.916148	−	0.400841i	$-0.131282\pi$
$263$	1.79907	+	3.11608i	0.110935	+	0.192146i	−0.805212	+	0.592987i	$0.797949\pi$
$264$		0			0
$265$	−8.35791	−	14.4763i	−0.513422	−	0.889274i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.2261	+	19.4443i	−0.684470	+	1.18554i	0.289133	+	0.957289i	$0.406633\pi$
$269$	−11.2261	+	19.4443i	−0.684470	+	1.18554i	−0.973603	+	0.228248i	$0.926700\pi$
$270$		0			0
$271$	−14.7935	−	25.6231i	−0.898642	−	1.55649i	−0.829231	−	0.558906i	$-0.811221\pi$
$271$	−14.7935	−	25.6231i	−0.898642	−	1.55649i	−0.0694115	−	0.997588i	$-0.522112\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.51958	−	4.36403i	0.151936	−	0.263161i
$276$		0			0
$277$	10.1933	+	17.6554i	0.612459	+	1.06081i	0.990825	+	0.135153i	$0.0431527\pi$
$277$	10.1933	+	17.6554i	0.612459	+	1.06081i	−0.378366	+	0.925656i	$0.623514\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.23968	−	3.87924i	0.133608	−	0.231416i	−0.791457	−	0.611225i	$-0.790677\pi$
$281$	2.23968	−	3.87924i	0.133608	−	0.231416i	0.925065	+	0.379809i	$0.124010\pi$
$282$		0			0
$283$	2.07680			0.123453			0.0617264	−	0.998093i	$-0.480339\pi$
$283$	2.07680			0.123453			0.0617264	+	0.998093i	$0.480339\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.5392	+	8.19169i	−0.622108	+	0.483540i
$288$		0			0
$289$	1.49237	+	2.58486i	0.0877865	+	0.152051i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.887340	+	1.53692i	0.0518389	+	0.0897877i	0.890780	−	0.454434i	$-0.150158\pi$
$293$	0.887340	+	1.53692i	0.0518389	+	0.0897877i	−0.838942	+	0.544222i	$0.816825\pi$
$294$		0			0
$295$	−5.32355	+	9.22066i	−0.309949	+	0.536847i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−38.2355			−2.21122
$300$		0			0
$301$	29.8800	+	12.1594i	1.72225	+	0.700858i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.41921	−	7.65429i	0.253043	−	0.438283i
$306$		0			0
$307$	19.6315			1.12043			0.560215	−	0.828347i	$-0.310718\pi$
$307$	19.6315			1.12043			0.560215	+	0.828347i	$0.310718\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.3159			0.755076			0.377538	−	0.925994i	$-0.376771\pi$
$311$	13.3159			0.755076			0.377538	+	0.925994i	$0.376771\pi$
$312$		0			0
$313$	4.65281			0.262992			0.131496	−	0.991317i	$-0.458022\pi$
$313$	4.65281			0.262992			0.131496	+	0.991317i	$0.458022\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.12552			0.231713			0.115856	−	0.993266i	$-0.463039\pi$
$317$	4.12552			0.231713			0.115856	+	0.993266i	$0.463039\pi$
$318$		0			0
$319$	−2.01483			−0.112809
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.3214			1.13071
$324$		0			0
$325$	−3.90847	+	6.76967i	−0.216803	+	0.375514i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.66862	−	12.0925i	−0.0919939	−	0.666679i
$330$		0			0
$331$	0.0440594			0.00242172			0.00121086	−	0.999999i	$-0.499615\pi$
$331$	0.0440594			0.00242172			0.00121086	+	0.999999i	$0.499615\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.2944	−	23.0266i	0.726350	−	1.25808i
$336$		0			0
$337$	−13.3351	−	23.0970i	−0.726407	−	1.25817i	−0.958392	−	0.285454i	$-0.907856\pi$
$337$	−13.3351	−	23.0970i	−0.726407	−	1.25817i	0.231986	−	0.972719i	$-0.425478\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.60541	−	2.78064i	−0.0869376	−	0.150580i
$342$		0			0
$343$	7.40556	+	16.9752i	0.399863	+	0.916575i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.8252			−0.581126			−0.290563	−	0.956856i	$-0.593843\pi$
$347$	−10.8252			−0.581126			−0.290563	+	0.956856i	$0.593843\pi$
$348$		0			0
$349$	−2.69555	+	4.66884i	−0.144290	+	0.249917i	−0.929108	−	0.369809i	$-0.879423\pi$
$349$	−2.69555	+	4.66884i	−0.144290	+	0.249917i	0.784818	+	0.619726i	$0.212756\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.47307	−	7.74759i	−0.238078	−	0.412362i	0.722085	−	0.691804i	$-0.243184\pi$
$353$	−4.47307	−	7.74759i	−0.238078	−	0.412362i	−0.960163	+	0.279442i	$0.909851\pi$
$354$		0			0
$355$	−4.59798	+	7.96394i	−0.244036	+	0.422682i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.84157	−	3.18969i	−0.0971942	−	0.168345i	0.813328	−	0.581805i	$-0.197653\pi$
$359$	−1.84157	−	3.18969i	−0.0971942	−	0.168345i	−0.910522	+	0.413460i	$0.864320\pi$
$360$		0			0
$361$	−5.23251	+	9.06297i	−0.275395	+	0.476998i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.33087	−	5.76924i	−0.174346	−	0.301976i
$366$		0			0
$367$	3.74988	+	6.49498i	0.195742	+	0.339035i	0.947144	−	0.320810i	$-0.103955\pi$
$367$	3.74988	+	6.49498i	0.195742	+	0.339035i	−0.751401	+	0.659845i	$0.770622\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	22.3046	+	9.07666i	1.15800	+	0.471237i
$372$		0			0
$373$	−4.11917	+	7.13461i	−0.213282	+	0.369416i	−0.952740	−	0.303787i	$-0.901749\pi$
$373$	−4.11917	+	7.13461i	−0.213282	+	0.369416i	0.739458	+	0.673203i	$0.235082\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.12549			0.160971
$378$		0			0
$379$	−3.92853			−0.201795			−0.100897	−	0.994897i	$-0.532171\pi$
$379$	−3.92853			−0.201795			−0.100897	+	0.994897i	$0.532171\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.9632	−	20.7210i	0.611293	−	1.05879i	−0.379729	−	0.925098i	$-0.623983\pi$
$383$	11.9632	−	20.7210i	0.611293	−	1.05879i	0.991023	−	0.133694i	$-0.0426838\pi$
$384$		0			0
$385$	−2.05718	−	14.9083i	−0.104843	−	0.759799i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.32875	+	10.9617i	0.320881	+	0.555781i	0.980670	−	0.195669i	$-0.0626879\pi$
$389$	6.32875	+	10.9617i	0.320881	+	0.555781i	−0.659789	+	0.751451i	$0.729355\pi$
$390$		0			0
$391$	−14.8967	−	25.8018i	−0.753359	−	1.30486i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.1848	−	22.8368i	0.663400	−	1.14904i
$396$		0			0
$397$	−17.7703	−	30.7791i	−0.891866	−	1.54476i	−0.837636	−	0.546229i	$-0.816063\pi$
$397$	−17.7703	−	30.7791i	−0.891866	−	1.54476i	−0.0542297	−	0.998528i	$-0.517270\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.66166	+	2.87808i	−0.0829794	+	0.143724i	−0.904528	−	0.426413i	$-0.859777\pi$
$401$	−1.66166	+	2.87808i	−0.0829794	+	0.143724i	0.821549	+	0.570138i	$0.193110\pi$
$402$		0			0
$403$	2.49037	+	4.31345i	0.124054	+	0.214868i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.70605	−	4.68702i	0.134134	−	0.232327i
$408$		0			0
$409$	22.5129			1.11319			0.556595	−	0.830784i	$-0.312107\pi$
$409$	22.5129			1.11319			0.556595	+	0.830784i	$0.312107\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.09662	−	15.1942i	−0.103168	−	0.747656i
$414$		0			0
$415$	−7.03662	−	12.1878i	−0.345414	−	0.598275i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.59772	+	6.23144i	0.175760	+	0.304426i	0.940424	−	0.340004i	$-0.110428\pi$
$419$	3.59772	+	6.23144i	0.175760	+	0.304426i	−0.764664	+	0.644429i	$0.777095\pi$
$420$		0			0
$421$	16.8121	−	29.1193i	0.819370	−	1.41919i	−0.0867773	−	0.996228i	$-0.527657\pi$
$421$	16.8121	−	29.1193i	0.819370	−	1.41919i	0.906147	−	0.422962i	$-0.139010\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.09102			−0.295458
$426$		0			0
$427$	1.74045	+	12.6130i	0.0842264	+	0.610388i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.4871	+	28.5565i	−0.794156	+	1.37552i	0.129217	+	0.991616i	$0.458754\pi$
$431$	−16.4871	+	28.5565i	−0.794156	+	1.37552i	−0.923373	+	0.383903i	$0.874580\pi$
$432$		0			0
$433$	19.8977			0.956221			0.478110	−	0.878300i	$-0.341322\pi$
$433$	19.8977			0.956221			0.478110	+	0.878300i	$0.341322\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	43.1989			2.06648
$438$		0			0
$439$	29.1268			1.39015			0.695074	−	0.718938i	$-0.255372\pi$
$439$	29.1268			1.39015			0.695074	+	0.718938i	$0.255372\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.7663			−0.654058			−0.327029	−	0.945014i	$-0.606048\pi$
$443$	−13.7663			−0.654058			−0.327029	+	0.945014i	$0.606048\pi$
$444$		0			0
$445$	−21.1866			−1.00434
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.0958			0.570838			0.285419	−	0.958403i	$-0.407867\pi$
$449$	12.0958			0.570838			0.285419	+	0.958403i	$0.407867\pi$
$450$		0			0
$451$	−7.81297	+	13.5325i	−0.367898	+	0.637218i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.19118	+	23.1264i	0.149605	+	1.08418i
$456$		0			0
$457$	−8.35476			−0.390819			−0.195410	−	0.980722i	$-0.562604\pi$
$457$	−8.35476			−0.390819			−0.195410	+	0.980722i	$0.562604\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.1673	+	19.3423i	−0.520112	+	0.900860i	0.479615	+	0.877479i	$0.340776\pi$
$461$	−11.1673	+	19.3423i	−0.520112	+	0.900860i	−0.999727	+	0.0233807i	$0.992557\pi$
$462$		0			0
$463$	−0.0370790	−	0.0642228i	−0.00172321	−	0.00298469i	0.865163	−	0.501492i	$-0.167215\pi$
$463$	−0.0370790	−	0.0642228i	−0.00172321	−	0.00298469i	−0.866886	+	0.498507i	$0.833882\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.5828	−	25.2581i	−0.674810	−	1.16880i	−0.976524	−	0.215407i	$-0.930892\pi$
$467$	−14.5828	−	25.2581i	−0.674810	−	1.16880i	0.301715	−	0.953398i	$-0.402441\pi$
$468$		0			0
$469$	5.23584	+	37.9441i	0.241769	+	1.75209i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	37.7637			1.73638
$474$		0			0
$475$	4.41583	−	7.64845i	0.202612	−	0.350935i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.9551	+	24.1710i	0.637626	+	1.10440i	0.985952	+	0.167027i	$0.0534168\pi$
$479$	13.9551	+	24.1710i	0.637626	+	1.10440i	−0.348326	+	0.937373i	$0.613250\pi$
$480$		0			0
$481$	−4.19774	+	7.27070i	−0.191401	+	0.331515i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.91884	−	3.32353i	−0.0871301	−	0.150914i
$486$		0			0
$487$	−2.14409	+	3.71367i	−0.0971580	+	0.168283i	−0.910507	−	0.413493i	$-0.864309\pi$
$487$	−2.14409	+	3.71367i	−0.0971580	+	0.168283i	0.813349	+	0.581776i	$0.197642\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.22215	+	9.04503i	0.235672	+	0.408196i	0.959468	−	0.281818i	$-0.0909375\pi$
$491$	5.22215	+	9.04503i	0.235672	+	0.408196i	−0.723796	+	0.690015i	$0.757604\pi$
$492$		0			0
$493$	1.21770	+	2.10912i	0.0548425	+	0.0949900i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.81086	−	13.1233i	−0.0812282	−	0.588660i
$498$		0			0
$499$	3.06312	−	5.30548i	0.137124	−	0.237506i	−0.789283	−	0.614030i	$-0.789547\pi$
$499$	3.06312	−	5.30548i	0.137124	−	0.237506i	0.926407	+	0.376524i	$0.122881\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.4469			0.554982			0.277491	−	0.960728i	$-0.410497\pi$
$503$	12.4469			0.554982			0.277491	+	0.960728i	$0.410497\pi$
$504$		0			0
$505$	30.2175			1.34466
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.90450	−	10.2269i	0.261712	−	0.453299i	−0.704985	−	0.709222i	$-0.749046\pi$
$509$	5.90450	−	10.2269i	0.261712	−	0.453299i	0.966697	+	0.255923i	$0.0823795\pi$
$510$		0			0
$511$	8.88902	+	3.61731i	0.393227	+	0.160021i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.11388	+	12.3216i	0.313475	+	0.542955i
$516$		0			0
$517$	−7.14495	−	12.3754i	−0.314235	−	0.544271i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.54828	+	9.60991i	−0.243075	+	0.421018i	−0.961589	−	0.274495i	$-0.911489\pi$
$521$	−5.54828	+	9.60991i	−0.243075	+	0.421018i	0.718514	+	0.695513i	$0.244823\pi$
$522$		0			0
$523$	10.6209	+	18.3960i	0.464421	+	0.804401i	0.999175	−	0.0406065i	$-0.0129290\pi$
$523$	10.6209	+	18.3960i	0.464421	+	0.804401i	−0.534754	+	0.845008i	$0.679596\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.94052	+	3.36107i	−0.0845302	+	0.146411i
$528$		0			0
$529$	−20.1672	−	34.9305i	−0.876833	−	1.51872i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.1198	−	20.9921i	0.524967	−	0.909269i
$534$		0			0
$535$	13.7650			0.595111
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.5199	+	15.1383i	0.668490	+	0.652054i
$540$		0			0
$541$	−6.33567	−	10.9737i	−0.272392	−	0.471796i	0.697082	−	0.716991i	$-0.254481\pi$
$541$	−6.33567	−	10.9737i	−0.272392	−	0.471796i	−0.969474	+	0.245195i	$0.921148\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.90908	−	13.6989i	−0.338788	−	0.586798i
$546$		0			0
$547$	−21.4805	+	37.2053i	−0.918438	+	1.59078i	−0.116651	+	0.993173i	$0.537216\pi$
$547$	−21.4805	+	37.2053i	−0.918438	+	1.59078i	−0.801788	+	0.597609i	$0.796117\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.53121			−0.150435
$552$		0			0
$553$	5.19268	+	37.6313i	0.220815	+	1.60025i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.5129	+	28.6012i	−0.699673	+	1.21187i	0.268906	+	0.963166i	$0.413338\pi$
$557$	−16.5129	+	28.6012i	−0.699673	+	1.21187i	−0.968580	+	0.248703i	$0.919996\pi$
$558$		0			0
$559$	−58.5807			−2.47770
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.8132			−1.55149			−0.775746	−	0.631046i	$-0.782626\pi$
$563$	−36.8132			−1.55149			−0.775746	+	0.631046i	$0.782626\pi$
$564$		0			0
$565$	−5.72081			−0.240676
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−44.3571			−1.85955			−0.929774	−	0.368132i	$-0.879998\pi$
$569$	−44.3571			−1.85955			−0.929774	+	0.368132i	$0.879998\pi$
$570$		0			0
$571$	−42.5872			−1.78222			−0.891110	−	0.453787i	$-0.850073\pi$
$571$	−42.5872			−1.78222			−0.891110	+	0.453787i	$0.850073\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.9482			−0.539977
$576$		0			0
$577$	16.3209	−	28.2687i	0.679450	−	1.17684i	−0.295697	−	0.955282i	$-0.595552\pi$
$577$	16.3209	−	28.2687i	0.679450	−	1.17684i	0.975147	−	0.221559i	$-0.0711147\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.7785	+	7.64175i	0.779063	+	0.317033i
$582$		0			0
$583$	28.1895			1.16749
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.1270	−	22.7366i	0.541809	−	0.938441i	−0.456991	−	0.889471i	$-0.651073\pi$
$587$	13.1270	−	22.7366i	0.541809	−	0.938441i	0.998800	−	0.0489701i	$-0.0155939\pi$
$588$		0			0
$589$	−2.81365	−	4.87338i	−0.115934	−	0.200804i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.59998	+	4.50330i	0.106768	+	0.184928i	0.914459	−	0.404678i	$-0.132616\pi$
$593$	2.59998	+	4.50330i	0.106768	+	0.184928i	−0.807691	+	0.589606i	$0.799283\pi$
$594$		0			0
$595$	−14.3627	+	11.1636i	−0.588814	+	0.457663i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−26.3675			−1.07735			−0.538673	−	0.842515i	$-0.681074\pi$
$599$	−26.3675			−1.07735			−0.538673	+	0.842515i	$0.681074\pi$
$600$		0			0
$601$	−15.4505	+	26.7611i	−0.630239	+	1.09161i	0.357263	+	0.934004i	$0.383710\pi$
$601$	−15.4505	+	26.7611i	−0.630239	+	1.09161i	−0.987503	+	0.157603i	$0.949623\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.29240	+	2.23851i	0.0525436	+	0.0910082i
$606$		0			0
$607$	−3.83661	+	6.64519i	−0.155723	+	0.269720i	−0.933322	−	0.359040i	$-0.883104\pi$
$607$	−3.83661	+	6.64519i	−0.155723	+	0.269720i	0.777599	+	0.628760i	$0.216437\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.0836	+	19.1973i	0.448393	+	0.776639i
$612$		0			0
$613$	7.97498	−	13.8131i	0.322106	−	0.557905i	−0.658816	−	0.752304i	$-0.728942\pi$
$613$	7.97498	−	13.8131i	0.322106	−	0.557905i	0.980922	+	0.194399i	$0.0622758\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.67011	+	6.35682i	0.147753	+	0.255916i	0.930397	−	0.366554i	$-0.119463\pi$
$617$	3.67011	+	6.35682i	0.147753	+	0.255916i	−0.782644	+	0.622470i	$0.786129\pi$
$618$		0			0
$619$	−10.2842	−	17.8127i	−0.413357	−	0.715955i	0.581898	−	0.813262i	$-0.302310\pi$
$619$	−10.2842	−	17.8127i	−0.413357	−	0.715955i	−0.995254	+	0.0973072i	$0.968977\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0981	−	18.7305i	0.965469	−	0.750421i
$624$		0			0
$625$	7.10891	−	12.3130i	0.284356	−	0.492520i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.54182			−0.260840
$630$		0			0
$631$	−5.09394			−0.202787			−0.101393	−	0.994846i	$-0.532330\pi$
$631$	−5.09394			−0.202787			−0.101393	+	0.994846i	$0.532330\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.99705	+	17.3154i	−0.396721	+	0.687141i
$636$		0			0
$637$	−24.0751	−	23.4832i	−0.953892	−	0.930439i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.31861	+	10.9442i	0.249570	+	0.432268i	0.963407	−	0.268044i	$-0.0863773\pi$
$641$	6.31861	+	10.9442i	0.249570	+	0.432268i	−0.713836	+	0.700312i	$0.753044\pi$
$642$		0			0
$643$	12.4329	+	21.5344i	0.490306	+	0.849235i	0.999938	−	0.0111579i	$-0.00355174\pi$
$643$	12.4329	+	21.5344i	0.490306	+	0.849235i	−0.509632	+	0.860393i	$0.670218\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.12339	+	1.94577i	−0.0441650	+	0.0764960i	−0.887263	−	0.461264i	$-0.847396\pi$
$647$	−1.12339	+	1.94577i	−0.0441650	+	0.0764960i	0.843098	+	0.537760i	$0.180729\pi$
$648$		0			0
$649$	−8.97762	−	15.5497i	−0.352403	−	0.610379i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.02881	−	1.78195i	0.0402604	−	0.0697330i	−0.845193	−	0.534461i	$-0.820515\pi$
$653$	1.02881	−	1.78195i	0.0402604	−	0.0697330i	0.885453	+	0.464728i	$0.153848\pi$
$654$		0			0
$655$	−14.7438	−	25.5370i	−0.576088	−	0.997815i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.16599	−	7.21571i	0.162284	−	0.281084i	−0.773403	−	0.633914i	$-0.781447\pi$
$659$	4.16599	−	7.21571i	0.162284	−	0.281084i	0.935687	+	0.352830i	$0.114781\pi$
$660$		0			0
$661$	34.0926			1.32605			0.663024	−	0.748598i	$-0.269273\pi$
$661$	34.0926			1.32605			0.663024	+	0.748598i	$0.269273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.60543	−	26.1285i	−0.139812	−	1.01322i
$666$		0			0
$667$	2.58857	+	4.48353i	0.100230	+	0.173603i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.45255	+	12.9082i	0.287702	+	0.498315i
$672$		0			0
$673$	0.571008	−	0.989016i	0.0220108	−	0.0381237i	−0.854810	−	0.518941i	$-0.826327\pi$
$673$	0.571008	−	0.989016i	0.0220108	−	0.0381237i	0.876821	+	0.480817i	$0.159660\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−36.3812			−1.39824			−0.699121	−	0.715004i	$-0.746425\pi$
$677$	−36.3812			−1.39824			−0.699121	+	0.715004i	$0.746425\pi$
$678$		0			0
$679$	5.12077	+	2.08386i	0.196517	+	0.0799711i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.11274	+	5.39142i	−0.119106	+	0.206297i	−0.919414	−	0.393292i	$-0.871336\pi$
$683$	−3.11274	+	5.39142i	−0.119106	+	0.206297i	0.800308	+	0.599589i	$0.204669\pi$
$684$		0			0
$685$	−24.6846			−0.943152
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−43.7288			−1.66593
$690$		0			0
$691$	−39.8259			−1.51505			−0.757525	−	0.652806i	$-0.773592\pi$
$691$	−39.8259			−1.51505			−0.757525	+	0.652806i	$0.773592\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.9480			0.567008
$696$		0			0
$697$	18.8877			0.715422
$698$		0			0
$699$		0			0
$700$		0			0
$701$	48.3337			1.82554			0.912769	−	0.408477i	$-0.133940\pi$
$701$	48.3337			1.82554			0.912769	+	0.408477i	$0.133940\pi$
$702$		0			0
$703$	4.74265	−	8.21452i	0.178873	−	0.309816i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.3700	+	26.7145i	−1.29262	+	1.00470i
$708$		0			0
$709$	−16.0840			−0.604046			−0.302023	−	0.953301i	$-0.597662\pi$
$709$	−16.0840			−0.604046			−0.302023	+	0.953301i	$0.597662\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.12512	+	7.14491i	−0.154487	+	0.267579i
$714$		0			0
$715$	13.6645	+	23.6676i	0.511023	+	0.885117i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.0734	−	36.5002i	−0.785906	−	1.36123i	−0.928456	−	0.371442i	$-0.878864\pi$
$719$	−21.0734	−	36.5002i	−0.785906	−	1.36123i	0.142550	−	0.989788i	$-0.454470\pi$
$720$		0			0
$721$	−18.9847	−	7.72566i	−0.707026	−	0.287718i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.05842			0.0393089
$726$		0			0
$727$	−12.9548	+	22.4384i	−0.480467	+	0.832192i	−0.999749	−	0.0224103i	$-0.992866\pi$
$727$	−12.9548	+	22.4384i	−0.480467	+	0.832192i	0.519282	+	0.854603i	$0.326199\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−22.8232	−	39.5310i	−0.844148	−	1.46211i
$732$		0			0
$733$	−10.2027	+	17.6716i	−0.376846	+	0.652716i	−0.990601	−	0.136780i	$-0.956325\pi$
$733$	−10.2027	+	17.6716i	−0.376846	+	0.652716i	0.613756	+	0.789496i	$0.289658\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	22.4196	+	38.8320i	0.825838	+	1.43039i
$738$		0			0
$739$	−11.8953	+	20.6033i	−0.437576	+	0.757903i	−0.997502	−	0.0706392i	$-0.977496\pi$
$739$	−11.8953	+	20.6033i	−0.437576	+	0.757903i	0.559926	+	0.828542i	$0.310829\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.6320	−	37.4678i	−0.793603	−	1.37456i	−0.923723	−	0.383062i	$-0.874870\pi$
$743$	−21.6320	−	37.4678i	−0.793603	−	1.37456i	0.130120	−	0.991498i	$-0.458464\pi$
$744$		0			0
$745$	6.91430	+	11.9759i	0.253321	+	0.438764i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.6565	+	12.1692i	−0.572078	+	0.444654i
$750$		0			0
$751$	7.18465	−	12.4442i	0.262172	−	0.454095i	−0.704647	−	0.709558i	$-0.748895\pi$
$751$	7.18465	−	12.4442i	0.262172	−	0.454095i	0.966819	+	0.255463i	$0.0822280\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.4176			−0.379136
$756$		0			0
$757$	39.7854			1.44603			0.723013	−	0.690835i	$-0.242757\pi$
$757$	39.7854			1.44603			0.723013	+	0.690835i	$0.242757\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.1966	+	19.3932i	−0.405878	+	0.703002i	−0.994423	−	0.105463i	$-0.966368\pi$
$761$	−11.1966	+	19.3932i	−0.405878	+	0.703002i	0.588545	+	0.808464i	$0.299701\pi$
$762$		0			0
$763$	21.1068	+	8.58924i	0.764117	+	0.310951i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.9265	+	24.1213i	0.502856	+	0.870971i
$768$		0			0
$769$	−1.45546	−	2.52093i	−0.0524853	−	0.0909071i	0.838589	−	0.544764i	$-0.183381\pi$
$769$	−1.45546	−	2.52093i	−0.0524853	−	0.0909071i	−0.891074	+	0.453857i	$0.850048\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.68612	−	11.5807i	0.240483	−	0.416529i	−0.720369	−	0.693591i	$-0.756028\pi$
$773$	6.68612	−	11.5807i	0.240483	−	0.416529i	0.960852	+	0.277062i	$0.0893609\pi$
$774$		0			0
$775$	0.843347	+	1.46072i	0.0302939	+	0.0524706i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.6931	+	23.7171i	−0.490606	+	0.849754i
$780$		0			0
$781$	−7.75403	−	13.4304i	−0.277461	−	0.480577i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.401073	+	0.694679i	−0.0143149	+	0.0247941i
$786$		0			0
$787$	−23.8528			−0.850260			−0.425130	−	0.905132i	$-0.639772\pi$
$787$	−23.8528			−0.850260			−0.425130	+	0.905132i	$0.639772\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.50696	−	5.05761i	0.231361	−	0.179828i
$792$		0			0
$793$	−11.5607	−	20.0237i	−0.410533	−	0.711063i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.10559	+	10.5752i	0.216271	+	0.374593i	0.953665	−	0.300870i	$-0.0972772\pi$
$797$	6.10559	+	10.5752i	0.216271	+	0.374593i	−0.737394	+	0.675463i	$0.763944\pi$
$798$		0			0
$799$	−8.63639	+	14.9587i	−0.305533	+	0.529199i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.2344			0.396452
$804$		0			0
$805$	−30.5320	+	23.7314i	−1.07611	+	0.836421i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.7838	+	46.3910i	−0.941669	+	1.63102i	−0.179383	+	0.983779i	$0.557410\pi$
$809$	−26.7838	+	46.3910i	−0.941669	+	1.63102i	−0.762287	+	0.647240i	$0.775923\pi$
$810$		0			0
$811$	1.81310			0.0636667			0.0318334	−	0.999493i	$-0.489865\pi$
$811$	1.81310			0.0636667			0.0318334	+	0.999493i	$0.489865\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.5229			−1.17426
$816$		0			0
$817$	66.1851			2.31552
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.6743			1.38464			0.692321	−	0.721590i	$-0.256588\pi$
$821$	39.6743			1.38464			0.692321	+	0.721590i	$0.256588\pi$
$822$		0			0
$823$	16.8131			0.586069			0.293034	−	0.956102i	$-0.405335\pi$
$823$	16.8131			0.586069			0.293034	+	0.956102i	$0.405335\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.2198			−1.29426			−0.647130	−	0.762379i	$-0.724031\pi$
$827$	−37.2198			−1.29426			−0.647130	+	0.762379i	$0.724031\pi$
$828$		0			0
$829$	−11.4365	+	19.8086i	−0.397206	+	0.687981i	−0.993380	−	0.114874i	$-0.963354\pi$
$829$	−11.4365	+	19.8086i	−0.397206	+	0.687981i	0.596174	+	0.802855i	$0.296687\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.46703	−	25.3954i	0.224069	−	0.879898i
$834$		0			0
$835$	2.81035			0.0972562
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−18.4071	+	31.8820i	−0.635484	+	1.10069i	0.350929	+	0.936402i	$0.385866\pi$
$839$	−18.4071	+	31.8820i	−0.635484	+	1.10069i	−0.986412	+	0.164288i	$0.947467\pi$
$840$		0			0
$841$	14.2884	+	24.7482i	0.492704	+	0.853388i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.25921	−	16.0374i	−0.318527	−	0.551704i
$846$		0			0
$847$	−3.44901	−	1.40354i	−0.118509	−	0.0482264i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.9065			−0.476709
$852$		0			0
$853$	6.98355	−	12.0959i	0.239112	−	0.414155i	−0.721347	−	0.692573i	$-0.756477\pi$
$853$	6.98355	−	12.0959i	0.239112	−	0.414155i	0.960460	+	0.278419i	$0.0898102\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.78386	+	15.2141i	0.300051	+	0.519703i	0.976147	−	0.217110i	$-0.0696630\pi$
$857$	8.78386	+	15.2141i	0.300051	+	0.519703i	−0.676096	+	0.736813i	$0.736330\pi$
$858$		0			0
$859$	−1.42288	+	2.46451i	−0.0485482	+	0.0840879i	−0.889278	−	0.457366i	$-0.848793\pi$
$859$	−1.42288	+	2.46451i	−0.0485482	+	0.0840879i	0.840730	+	0.541454i	$0.182126\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.7115	+	47.9977i	0.943310	+	1.63386i	0.759100	+	0.650974i	$0.225639\pi$
$863$	27.7115	+	47.9977i	0.943310	+	1.63386i	0.184210	+	0.982887i	$0.441027\pi$
$864$		0			0
$865$	1.99221	−	3.45061i	0.0677372	−	0.117324i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	22.2348	+	38.5119i	0.754265	+	1.30643i
$870$		0			0
$871$	−34.7783	−	60.2378i	−1.17842	−	2.04108i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.40170	+	31.8991i	0.148805	+	1.07839i
$876$		0			0
$877$	−27.4345	+	47.5179i	−0.926396	+	1.60456i	−0.137095	+	0.990558i	$0.543776\pi$
$877$	−27.4345	+	47.5179i	−0.926396	+	1.60456i	−0.789301	+	0.614006i	$0.789557\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	51.1572			1.72353			0.861765	−	0.507307i	$-0.169359\pi$
$881$	51.1572			1.72353			0.861765	+	0.507307i	$0.169359\pi$
$882$		0			0
$883$	38.6438			1.30047			0.650234	−	0.759734i	$-0.274671\pi$
$883$	38.6438			1.30047			0.650234	+	0.759734i	$0.274671\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.92626	+	15.4607i	−0.299714	+	0.519121i	−0.976071	−	0.217454i	$-0.930225\pi$
$887$	−8.92626	+	15.4607i	−0.299714	+	0.519121i	0.676356	+	0.736575i	$0.263558\pi$
$888$		0			0
$889$	−3.93722	−	28.5330i	−0.132050	−	0.956966i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.5223	−	21.6893i	−0.419043	−	0.725805i
$894$		0			0
$895$	1.98833	+	3.44388i	0.0664624	+	0.115116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.337199	−	0.584047i	0.0112462	−	0.0194790i
$900$		0			0
$901$	−17.0369	−	29.5088i	−0.567581	−	0.983080i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.512255	+	0.887252i	−0.0170279	+	0.0294932i
$906$		0			0
$907$	18.2332	+	31.5807i	0.605422	+	1.04862i	0.991985	+	0.126358i	$0.0403289\pi$
$907$	18.2332	+	31.5807i	0.605422	+	1.04862i	−0.386563	+	0.922263i	$0.626338\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.9847	+	32.8825i	−0.628993	+	1.08945i	0.358762	+	0.933429i	$0.383199\pi$
$911$	−18.9847	+	32.8825i	−0.628993	+	1.08945i	−0.987754	+	0.156018i	$0.950134\pi$
$912$		0			0
$913$	23.7331			0.785451
$914$		0			0
$915$		0			0
$916$		0			0
$917$	39.3465	+	16.0117i	1.29934	+	0.528754i
$918$		0			0
$919$	−1.21770	−	2.10911i	−0.0401681	−	0.0695732i	0.845242	−	0.534383i	$-0.179456\pi$
$919$	−1.21770	−	2.10911i	−0.0401681	−	0.0695732i	−0.885411	+	0.464810i	$0.846123\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0284	+	20.8338i	0.395919	+	0.685752i
$924$		0			0
$925$	−1.42153	+	2.46217i	−0.0467398	+	0.0809557i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.4808			1.16409			0.582044	−	0.813157i	$-0.302253\pi$
$929$	35.4808			1.16409			0.582044	+	0.813157i	$0.302253\pi$
$930$		0			0
$931$	27.2004	+	26.5316i	0.891456	+	0.869538i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.6475	+	18.4420i	−0.348209	+	0.603116i
$936$		0			0
$937$	−30.0427			−0.981452			−0.490726	−	0.871314i	$-0.663268\pi$
$937$	−30.0427			−0.981452			−0.490726	+	0.871314i	$0.663268\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.2499			0.986119			0.493060	−	0.869996i	$-0.335878\pi$
$941$	30.2499			0.986119			0.493060	+	0.869996i	$0.335878\pi$
$942$		0			0
$943$	40.1511			1.30750
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.9889			−1.10449			−0.552245	−	0.833682i	$-0.686229\pi$
$947$	−33.9889			−1.10449			−0.552245	+	0.833682i	$0.686229\pi$
$948$		0			0
$949$	−17.4272			−0.565711
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.4324			0.694265			0.347132	−	0.937816i	$-0.387155\pi$
$953$	21.4324			0.694265			0.347132	+	0.937816i	$0.387155\pi$
$954$		0			0
$955$	−22.0385	+	38.1719i	−0.713151	+	1.23521i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	28.0768	−	21.8230i	0.906647	−	0.704702i
$960$		0			0
$961$	−29.9253			−0.965332
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.5406	−	33.8452i	0.629033	−	1.08952i
$966$		0			0
$967$	12.4095	+	21.4938i	0.399061	+	0.691194i	0.993610	−	0.112865i	$-0.0360028\pi$
$967$	12.4095	+	21.4938i	0.399061	+	0.691194i	−0.594549	+	0.804059i	$0.702669\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.9437	+	24.1512i	0.447475	+	0.775050i	0.998221	−	0.0596234i	$-0.0189900\pi$
$971$	13.9437	+	24.1512i	0.447475	+	0.775050i	−0.550746	+	0.834673i	$0.685657\pi$
$972$		0			0
$973$	−17.0021	+	13.2151i	−0.545062	+	0.423656i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27.2473			−0.871719			−0.435859	−	0.900015i	$-0.643555\pi$
$977$	−27.2473			−0.871719			−0.435859	+	0.900015i	$0.643555\pi$
$978$		0			0
$979$	17.8645	−	30.9423i	0.570953	−	0.988920i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.94539	−	12.0298i	−0.221523	−	0.383690i	0.733747	−	0.679422i	$-0.237770\pi$
$983$	−6.94539	−	12.0298i	−0.221523	−	0.383690i	−0.955271	+	0.295733i	$0.904436\pi$
$984$		0			0
$985$	−13.6611	+	23.6618i	−0.435280	+	0.753928i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−48.5173	−	84.0344i	−1.54276	−	2.67214i
$990$		0			0
$991$	21.3271	−	36.9397i	0.677479	−	1.17343i	−0.298259	−	0.954485i	$-0.596406\pi$
$991$	21.3271	−	36.9397i	0.677479	−	1.17343i	0.975738	−	0.218942i	$-0.0702607\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.3479	+	19.6551i	0.359752	+	0.623108i
$996$		0			0
$997$	−21.5905	−	37.3959i	−0.683779	−	1.18434i	−0.973819	−	0.227325i	$-0.927002\pi$
$997$	−21.5905	−	37.3959i	−0.683779	−	1.18434i	0.290040	−	0.957014i	$-0.406331\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.q.c.1369.4		22
3.2	odd	2		504.2.q.d.25.9	✓	22
4.3	odd	2		3024.2.q.k.2881.4		22
7.2	even	3		1512.2.t.d.289.8		22
9.4	even	3		1512.2.t.d.361.8		22
9.5	odd	6		504.2.t.d.193.1	yes	22
12.11	even	2		1008.2.q.k.529.3		22
21.2	odd	6		504.2.t.d.457.1	yes	22
28.23	odd	6		3024.2.t.l.289.8		22
36.23	even	6		1008.2.t.k.193.11		22
36.31	odd	6		3024.2.t.l.1873.8		22
63.23	odd	6		504.2.q.d.121.9	yes	22
63.58	even	3	inner	1512.2.q.c.793.4		22
84.23	even	6		1008.2.t.k.961.11		22
252.23	even	6		1008.2.q.k.625.3		22
252.247	odd	6		3024.2.q.k.2305.4		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.q.d.25.9	✓	22	3.2	odd	2
504.2.q.d.121.9	yes	22	63.23	odd	6
504.2.t.d.193.1	yes	22	9.5	odd	6
504.2.t.d.457.1	yes	22	21.2	odd	6
1008.2.q.k.529.3		22	12.11	even	2
1008.2.q.k.625.3		22	252.23	even	6
1008.2.t.k.193.11		22	36.23	even	6
1008.2.t.k.961.11		22	84.23	even	6
1512.2.q.c.793.4		22	63.58	even	3	inner
1512.2.q.c.1369.4		22	1.1	even	1	trivial
1512.2.t.d.289.8		22	7.2	even	3
1512.2.t.d.361.8		22	9.4	even	3
3024.2.q.k.2305.4		22	252.247	odd	6
3024.2.q.k.2881.4		22	4.3	odd	2
3024.2.t.l.289.8		22	28.23	odd	6
3024.2.t.l.1873.8		22	36.31	odd	6

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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

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Newspace parameters

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$q$ -expansion

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1512.2.q.c.1369.4

Level	$1512$
Weight	$2$
Character	1512.1369
Analytic conductor	$12.073$
Analytic rank	$0$
Dimension	$22$
Inner twists	$2$