LMFDB - Embedded newform 828.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [828,4,Mod(1,828)] 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("828.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 828 = 2^{2} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	828.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-10] 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:	yes
Analytic conductor:	$48.8535814848$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 194x^{4} + 484x^{3} + 7122x^{2} - 18036x + 6804 $ x^6 - 2x^5 - 194x^4 + 484x^3 + 7122x^2 - 18036*x + 6804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.00119$ of defining polynomial
Character	$\chi$	$=$	828.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0501 q^{5} +21.0723 q^{7} -10.5502 q^{11} +41.8166 q^{13} -115.917 q^{17} -50.8569 q^{19} -23.0000 q^{23} -23.9962 q^{25} +91.5077 q^{29} +301.066 q^{31} -211.778 q^{35} -214.916 q^{37} +308.983 q^{41} +182.267 q^{43} +158.506 q^{47} +101.042 q^{49} -571.124 q^{53} +106.030 q^{55} -567.292 q^{59} -268.341 q^{61} -420.259 q^{65} -770.390 q^{67} +494.651 q^{71} -825.946 q^{73} -222.317 q^{77} -429.811 q^{79} -104.411 q^{83} +1164.97 q^{85} -1416.58 q^{89} +881.171 q^{91} +511.115 q^{95} -269.580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{5} - 4 q^{7} + 2 q^{11} - 16 q^{13} + 12 q^{17} + 62 q^{19} - 138 q^{23} + 186 q^{25} - 392 q^{29} + 72 q^{31} - 508 q^{35} - 186 q^{37} - 276 q^{41} - 66 q^{43} - 748 q^{47} - 122 q^{49} - 998 q^{53}+ \cdots - 608 q^{97}+O(q^{100}) $ 6 * q - 10 * q^5 - 4 * q^7 + 2 * q^11 - 16 * q^13 + 12 * q^17 + 62 * q^19 - 138 * q^23 + 186 * q^25 - 392 * q^29 + 72 * q^31 - 508 * q^35 - 186 * q^37 - 276 * q^41 - 66 * q^43 - 748 * q^47 - 122 * q^49 - 998 * q^53 + 352 * q^55 - 1352 * q^59 - 354 * q^61 - 1764 * q^65 - 198 * q^67 - 1932 * q^71 + 340 * q^73 - 2364 * q^77 - 908 * q^79 - 3046 * q^83 - 824 * q^85 - 744 * q^89 + 1056 * q^91 - 5324 * q^95 - 608 * q^97

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
$5$	−10.0501	−0.898905	−0.449453	−	0.893304i	$-0.648381\pi$
$5$	−10.0501	−0.898905	−0.449453	+	0.893304i	$0.648381\pi$
$6$	0	0
$7$	21.0723	1.13780	0.568899	−	0.822408i	$-0.307370\pi$
$7$	21.0723	1.13780	0.568899	+	0.822408i	$0.307370\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−10.5502	−0.289183	−0.144591	−	0.989491i	$-0.546187\pi$
$11$	−10.5502	−0.289183	−0.144591	+	0.989491i	$0.546187\pi$
$12$	0	0
$13$	41.8166	0.892140	0.446070	−	0.894998i	$-0.352823\pi$
$13$	41.8166	0.892140	0.446070	+	0.894998i	$0.352823\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−115.917	−1.65377	−0.826883	−	0.562374i	$-0.809888\pi$
$17$	−115.917	−1.65377	−0.826883	+	0.562374i	$0.809888\pi$
$18$	0	0
$19$	−50.8569	−0.614073	−0.307036	−	0.951698i	$-0.599337\pi$
$19$	−50.8569	−0.614073	−0.307036	+	0.951698i	$0.599337\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−23.9962	−0.191969
$26$	0	0
$27$	0	0
$28$	0	0
$29$	91.5077	0.585950	0.292975	−	0.956120i	$-0.405355\pi$
$29$	91.5077	0.585950	0.292975	+	0.956120i	$0.405355\pi$
$30$	0	0
$31$	301.066	1.74429	0.872145	−	0.489247i	$-0.162728\pi$
$31$	301.066	1.74429	0.872145	+	0.489247i	$0.162728\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−211.778	−1.02277
$36$	0	0
$37$	−214.916	−0.954920	−0.477460	−	0.878653i	$-0.658443\pi$
$37$	−214.916	−0.954920	−0.477460	+	0.878653i	$0.658443\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	308.983	1.17695	0.588476	−	0.808515i	$-0.299728\pi$
$41$	308.983	1.17695	0.588476	+	0.808515i	$0.299728\pi$
$42$	0	0
$43$	182.267	0.646407	0.323203	−	0.946329i	$-0.395240\pi$
$43$	182.267	0.646407	0.323203	+	0.946329i	$0.395240\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	158.506	0.491924	0.245962	−	0.969280i	$-0.420896\pi$
$47$	158.506	0.491924	0.245962	+	0.969280i	$0.420896\pi$
$48$	0	0
$49$	101.042	0.294583
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−571.124	−1.48019	−0.740093	−	0.672504i	$-0.765219\pi$
$53$	−571.124	−1.48019	−0.740093	+	0.672504i	$0.765219\pi$
$54$	0	0
$55$	106.030	0.259948
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−567.292	−1.25178	−0.625891	−	0.779911i	$-0.715264\pi$
$59$	−567.292	−1.25178	−0.625891	+	0.779911i	$0.715264\pi$
$60$	0	0
$61$	−268.341	−0.563239	−0.281619	−	0.959526i	$-0.590872\pi$
$61$	−268.341	−0.563239	−0.281619	+	0.959526i	$0.590872\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−420.259	−0.801950
$66$	0	0
$67$	−770.390	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$67$	−770.390	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	494.651	0.826820	0.413410	−	0.910545i	$-0.364338\pi$
$71$	494.651	0.826820	0.413410	+	0.910545i	$0.364338\pi$
$72$	0	0
$73$	−825.946	−1.32424	−0.662121	−	0.749397i	$-0.730343\pi$
$73$	−825.946	−1.32424	−0.662121	+	0.749397i	$0.730343\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−222.317	−0.329031
$78$	0	0
$79$	−429.811	−0.612120	−0.306060	−	0.952012i	$-0.599011\pi$
$79$	−429.811	−0.612120	−0.306060	+	0.952012i	$0.599011\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−104.411	−0.138080	−0.0690400	−	0.997614i	$-0.521994\pi$
$83$	−104.411	−0.138080	−0.0690400	+	0.997614i	$0.521994\pi$
$84$	0	0
$85$	1164.97	1.48658
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1416.58	−1.68716	−0.843580	−	0.537004i	$-0.819556\pi$
$89$	−1416.58	−1.68716	−0.843580	+	0.537004i	$0.819556\pi$
$90$	0	0
$91$	881.171	1.01507
$92$	0	0
$93$	0	0
$94$	0	0
$95$	511.115	0.551993
$96$	0	0
$97$	−269.580	−0.282183	−0.141091	−	0.989997i	$-0.545061\pi$
$97$	−269.580	−0.282183	−0.141091	+	0.989997i	$0.545061\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1552.28	−1.52929	−0.764643	−	0.644454i	$-0.777085\pi$
$101$	−1552.28	−1.52929	−0.764643	+	0.644454i	$0.777085\pi$
$102$	0	0
$103$	−973.086	−0.930884	−0.465442	−	0.885078i	$-0.654105\pi$
$103$	−973.086	−0.930884	−0.465442	+	0.885078i	$0.654105\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1417.10	−1.28034	−0.640171	−	0.768232i	$-0.721137\pi$
$107$	−1417.10	−1.28034	−0.640171	+	0.768232i	$0.721137\pi$
$108$	0	0
$109$	680.458	0.597946	0.298973	−	0.954262i	$-0.403356\pi$
$109$	680.458	0.597946	0.298973	+	0.954262i	$0.403356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1601.39	−1.33315	−0.666575	−	0.745438i	$-0.732240\pi$
$113$	−1601.39	−1.33315	−0.666575	+	0.745438i	$0.732240\pi$
$114$	0	0
$115$	231.152	0.187435
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2442.64	−1.88165
$120$	0	0
$121$	−1219.69	−0.916373
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1497.42	1.07147
$126$	0	0
$127$	1685.21	1.17746	0.588732	−	0.808328i	$-0.299627\pi$
$127$	1685.21	1.17746	0.588732	+	0.808328i	$0.299627\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−280.924	−0.187362	−0.0936812	−	0.995602i	$-0.529863\pi$
$131$	−280.924	−0.187362	−0.0936812	+	0.995602i	$0.529863\pi$
$132$	0	0
$133$	−1071.67	−0.698690
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−951.068	−0.593104	−0.296552	−	0.955017i	$-0.595837\pi$
$137$	−951.068	−0.593104	−0.296552	+	0.955017i	$0.595837\pi$
$138$	0	0
$139$	2018.16	1.23150	0.615749	−	0.787942i	$-0.288853\pi$
$139$	2018.16	1.23150	0.615749	+	0.787942i	$0.288853\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−441.174	−0.257992
$144$	0	0
$145$	−919.659	−0.526714
$146$	0	0
$147$	0	0
$148$	0	0
$149$	528.089	0.290354	0.145177	−	0.989406i	$-0.453625\pi$
$149$	528.089	0.290354	0.145177	+	0.989406i	$0.453625\pi$
$150$	0	0
$151$	1914.52	1.03180	0.515900	−	0.856649i	$-0.327458\pi$
$151$	1914.52	1.03180	0.515900	+	0.856649i	$0.327458\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3025.73	−1.56795
$156$	0	0
$157$	888.146	0.451476	0.225738	−	0.974188i	$-0.427521\pi$
$157$	888.146	0.451476	0.225738	+	0.974188i	$0.427521\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−484.663	−0.237247
$162$	0	0
$163$	−1574.09	−0.756392	−0.378196	−	0.925725i	$-0.623456\pi$
$163$	−1574.09	−0.756392	−0.378196	+	0.925725i	$0.623456\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	637.390	0.295346	0.147673	−	0.989036i	$-0.452822\pi$
$167$	637.390	0.295346	0.147673	+	0.989036i	$0.452822\pi$
$168$	0	0
$169$	−448.376	−0.204086
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3567.77	1.56793	0.783966	−	0.620804i	$-0.213194\pi$
$173$	3567.77	1.56793	0.783966	+	0.620804i	$0.213194\pi$
$174$	0	0
$175$	−505.655	−0.218422
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3091.81	−1.29102	−0.645510	−	0.763752i	$-0.723355\pi$
$179$	−3091.81	−1.29102	−0.645510	+	0.763752i	$0.723355\pi$
$180$	0	0
$181$	1633.71	0.670898	0.335449	−	0.942058i	$-0.391112\pi$
$181$	1633.71	0.670898	0.335449	+	0.942058i	$0.391112\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2159.93	0.858383
$186$	0	0
$187$	1222.95	0.478241
$188$	0	0
$189$	0	0
$190$	0	0
$191$	581.977	0.220473	0.110237	−	0.993905i	$-0.464839\pi$
$191$	581.977	0.220473	0.110237	+	0.993905i	$0.464839\pi$
$192$	0	0
$193$	131.712	0.0491235	0.0245618	−	0.999698i	$-0.492181\pi$
$193$	131.712	0.0491235	0.0245618	+	0.999698i	$0.492181\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1328.16	0.480344	0.240172	−	0.970730i	$-0.422796\pi$
$197$	1328.16	0.480344	0.240172	+	0.970730i	$0.422796\pi$
$198$	0	0
$199$	2450.09	0.872775	0.436387	−	0.899759i	$-0.356258\pi$
$199$	2450.09	0.872775	0.436387	+	0.899759i	$0.356258\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1928.28	0.666693
$204$	0	0
$205$	−3105.30	−1.05797
$206$	0	0
$207$	0	0
$208$	0	0
$209$	536.552	0.177579
$210$	0	0
$211$	−1675.11	−0.546537	−0.273269	−	0.961938i	$-0.588105\pi$
$211$	−1675.11	−0.546537	−0.273269	+	0.961938i	$0.588105\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1831.80	−0.581059
$216$	0	0
$217$	6344.15	1.98465
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4847.25	−1.47539
$222$	0	0
$223$	−6208.77	−1.86444	−0.932220	−	0.361892i	$-0.882131\pi$
$223$	−6208.77	−1.86444	−0.932220	+	0.361892i	$0.882131\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6542.25	−1.91288	−0.956442	−	0.291923i	$-0.905705\pi$
$227$	−6542.25	−1.91288	−0.956442	+	0.291923i	$0.905705\pi$
$228$	0	0
$229$	−6250.87	−1.80380	−0.901898	−	0.431949i	$-0.857826\pi$
$229$	−6250.87	−1.80380	−0.901898	+	0.431949i	$0.857826\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1924.65	0.541150	0.270575	−	0.962699i	$-0.412786\pi$
$233$	1924.65	0.541150	0.270575	+	0.962699i	$0.412786\pi$
$234$	0	0
$235$	−1592.99	−0.442193
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4814.56	−1.30305	−0.651523	−	0.758629i	$-0.725870\pi$
$239$	−4814.56	−1.30305	−0.651523	+	0.758629i	$0.725870\pi$
$240$	0	0
$241$	−1862.61	−0.497847	−0.248924	−	0.968523i	$-0.580077\pi$
$241$	−1862.61	−0.497847	−0.248924	+	0.968523i	$0.580077\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1015.48	−0.264802
$246$	0	0
$247$	−2126.66	−0.547839
$248$	0	0
$249$	0	0
$250$	0	0
$251$	887.868	0.223274	0.111637	−	0.993749i	$-0.464391\pi$
$251$	887.868	0.223274	0.111637	+	0.993749i	$0.464391\pi$
$252$	0	0
$253$	242.655	0.0602988
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5339.18	1.29591	0.647956	−	0.761678i	$-0.275624\pi$
$257$	5339.18	1.29591	0.647956	+	0.761678i	$0.275624\pi$
$258$	0	0
$259$	−4528.79	−1.08651
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1578.75	−0.370151	−0.185076	−	0.982724i	$-0.559253\pi$
$263$	−1578.75	−0.370151	−0.185076	+	0.982724i	$0.559253\pi$
$264$	0	0
$265$	5739.83	1.33055
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5706.26	1.29337	0.646686	−	0.762757i	$-0.276155\pi$
$269$	5706.26	1.29337	0.646686	+	0.762757i	$0.276155\pi$
$270$	0	0
$271$	608.499	0.136397	0.0681986	−	0.997672i	$-0.478275\pi$
$271$	608.499	0.136397	0.0681986	+	0.997672i	$0.478275\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	253.165	0.0555142
$276$	0	0
$277$	−3164.79	−0.686476	−0.343238	−	0.939248i	$-0.611524\pi$
$277$	−3164.79	−0.686476	−0.343238	+	0.939248i	$0.611524\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1560.78	0.331346	0.165673	−	0.986181i	$-0.447020\pi$
$281$	1560.78	0.331346	0.165673	+	0.986181i	$0.447020\pi$
$282$	0	0
$283$	3715.59	0.780455	0.390228	−	0.920718i	$-0.372396\pi$
$283$	3715.59	0.780455	0.390228	+	0.920718i	$0.372396\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6510.98	1.33913
$288$	0	0
$289$	8523.77	1.73494
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3710.76	0.739881	0.369940	−	0.929055i	$-0.379378\pi$
$293$	3710.76	0.739881	0.369940	+	0.929055i	$0.379378\pi$
$294$	0	0
$295$	5701.32	1.12523
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−961.781	−0.186024
$300$	0	0
$301$	3840.79	0.735480
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2696.85	0.506298
$306$	0	0
$307$	−8198.86	−1.52421	−0.762107	−	0.647452i	$-0.775835\pi$
$307$	−8198.86	−1.52421	−0.762107	+	0.647452i	$0.775835\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6918.59	−1.26147	−0.630735	−	0.775998i	$-0.717247\pi$
$311$	−6918.59	−1.26147	−0.630735	+	0.775998i	$0.717247\pi$
$312$	0	0
$313$	3115.03	0.562531	0.281265	−	0.959630i	$-0.409246\pi$
$313$	3115.03	0.562531	0.281265	+	0.959630i	$0.409246\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5834.81	1.03380	0.516901	−	0.856045i	$-0.327085\pi$
$317$	5834.81	1.03380	0.516901	+	0.856045i	$0.327085\pi$
$318$	0	0
$319$	−965.427	−0.169447
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5895.19	1.01553
$324$	0	0
$325$	−1003.44	−0.171264
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3340.08	0.559709
$330$	0	0
$331$	715.158	0.118757	0.0593786	−	0.998236i	$-0.481088\pi$
$331$	715.158	0.118757	0.0593786	+	0.998236i	$0.481088\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7742.47	1.26274
$336$	0	0
$337$	8758.96	1.41582	0.707910	−	0.706303i	$-0.249638\pi$
$337$	8758.96	1.41582	0.707910	+	0.706303i	$0.249638\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3176.31	−0.504419
$342$	0	0
$343$	−5098.61	−0.802622
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2425.48	0.375235	0.187617	−	0.982242i	$-0.439924\pi$
$347$	2425.48	0.375235	0.187617	+	0.982242i	$0.439924\pi$
$348$	0	0
$349$	−5751.59	−0.882164	−0.441082	−	0.897467i	$-0.645405\pi$
$349$	−5751.59	−0.882164	−0.441082	+	0.897467i	$0.645405\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11519.5	1.73689	0.868445	−	0.495785i	$-0.165120\pi$
$353$	11519.5	1.73689	0.868445	+	0.495785i	$0.165120\pi$
$354$	0	0
$355$	−4971.27	−0.743233
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11263.7	1.65592	0.827960	−	0.560788i	$-0.189502\pi$
$359$	11263.7	1.65592	0.827960	+	0.560788i	$0.189502\pi$
$360$	0	0
$361$	−4272.57	−0.622915
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8300.81	1.19037
$366$	0	0
$367$	445.585	0.0633770	0.0316885	−	0.999498i	$-0.489912\pi$
$367$	445.585	0.0633770	0.0316885	+	0.999498i	$0.489912\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12034.9	−1.68415
$372$	0	0
$373$	4177.64	0.579920	0.289960	−	0.957039i	$-0.406358\pi$
$373$	4177.64	0.579920	0.289960	+	0.957039i	$0.406358\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3826.54	0.522750
$378$	0	0
$379$	11964.5	1.62157	0.810784	−	0.585346i	$-0.199041\pi$
$379$	11964.5	1.62157	0.810784	+	0.585346i	$0.199041\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6849.49	0.913818	0.456909	−	0.889513i	$-0.348956\pi$
$383$	6849.49	0.913818	0.456909	+	0.889513i	$0.348956\pi$
$384$	0	0
$385$	2234.30	0.295768
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−235.323	−0.0306718	−0.0153359	−	0.999882i	$-0.504882\pi$
$389$	−235.323	−0.0306718	−0.0153359	+	0.999882i	$0.504882\pi$
$390$	0	0
$391$	2666.09	0.344834
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4319.63	0.550238
$396$	0	0
$397$	−6847.36	−0.865640	−0.432820	−	0.901480i	$-0.642482\pi$
$397$	−6847.36	−0.865640	−0.432820	+	0.901480i	$0.642482\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1593.43	−0.198434	−0.0992169	−	0.995066i	$-0.531634\pi$
$401$	−1593.43	−0.198434	−0.0992169	+	0.995066i	$0.531634\pi$
$402$	0	0
$403$	12589.5	1.55615
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2267.42	0.276147
$408$	0	0
$409$	−11753.6	−1.42097	−0.710486	−	0.703712i	$-0.751525\pi$
$409$	−11753.6	−1.42097	−0.710486	+	0.703712i	$0.751525\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11954.1	−1.42427
$414$	0	0
$415$	1049.34	0.124121
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9922.84	−1.15695	−0.578476	−	0.815700i	$-0.696352\pi$
$419$	−9922.84	−1.15695	−0.578476	+	0.815700i	$0.696352\pi$
$420$	0	0
$421$	−7515.70	−0.870054	−0.435027	−	0.900418i	$-0.643261\pi$
$421$	−7515.70	−0.870054	−0.435027	+	0.900418i	$0.643261\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2781.57	0.317472
$426$	0	0
$427$	−5654.57	−0.640852
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10245.3	1.14501	0.572506	−	0.819901i	$-0.305971\pi$
$431$	10245.3	1.14501	0.572506	+	0.819901i	$0.305971\pi$
$432$	0	0
$433$	−393.145	−0.0436336	−0.0218168	−	0.999762i	$-0.506945\pi$
$433$	−393.145	−0.0436336	−0.0218168	+	0.999762i	$0.506945\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1169.71	0.128043
$438$	0	0
$439$	5868.32	0.637994	0.318997	−	0.947756i	$-0.396654\pi$
$439$	5868.32	0.637994	0.318997	+	0.947756i	$0.396654\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	609.869	0.0654080	0.0327040	−	0.999465i	$-0.489588\pi$
$443$	609.869	0.0654080	0.0327040	+	0.999465i	$0.489588\pi$
$444$	0	0
$445$	14236.7	1.51660
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8305.70	0.872985	0.436492	−	0.899708i	$-0.356221\pi$
$449$	8305.70	0.872985	0.436492	+	0.899708i	$0.356221\pi$
$450$	0	0
$451$	−3259.84	−0.340354
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8855.83	−0.912456
$456$	0	0
$457$	8497.82	0.869828	0.434914	−	0.900472i	$-0.356779\pi$
$457$	8497.82	0.869828	0.434914	+	0.900472i	$0.356779\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15657.8	1.58190	0.790951	−	0.611879i	$-0.209586\pi$
$461$	15657.8	1.58190	0.790951	+	0.611879i	$0.209586\pi$
$462$	0	0
$463$	4132.58	0.414810	0.207405	−	0.978255i	$-0.433498\pi$
$463$	4132.58	0.414810	0.207405	+	0.978255i	$0.433498\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11747.8	−1.16408	−0.582039	−	0.813161i	$-0.697745\pi$
$467$	−11747.8	−1.16408	−0.582039	+	0.813161i	$0.697745\pi$
$468$	0	0
$469$	−16233.9	−1.59832
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1922.96	−0.186930
$474$	0	0
$475$	1220.37	0.117883
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2914.71	0.278031	0.139015	−	0.990290i	$-0.455606\pi$
$479$	2914.71	0.278031	0.139015	+	0.990290i	$0.455606\pi$
$480$	0	0
$481$	−8987.07	−0.851923
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2709.30	0.253655
$486$	0	0
$487$	−13172.6	−1.22568	−0.612839	−	0.790208i	$-0.709973\pi$
$487$	−13172.6	−1.22568	−0.612839	+	0.790208i	$0.709973\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3958.95	−0.363879	−0.181940	−	0.983310i	$-0.558238\pi$
$491$	−3958.95	−0.363879	−0.181940	+	0.983310i	$0.558238\pi$
$492$	0	0
$493$	−10607.3	−0.969025
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10423.4	0.940754
$498$	0	0
$499$	−4578.47	−0.410742	−0.205371	−	0.978684i	$-0.565840\pi$
$499$	−4578.47	−0.410742	−0.205371	+	0.978684i	$0.565840\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8176.22	0.724770	0.362385	−	0.932028i	$-0.381962\pi$
$503$	8176.22	0.724770	0.362385	+	0.932028i	$0.381962\pi$
$504$	0	0
$505$	15600.5	1.37468
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18826.6	−1.63944	−0.819719	−	0.572766i	$-0.805870\pi$
$509$	−18826.6	−1.63944	−0.819719	+	0.572766i	$0.805870\pi$
$510$	0	0
$511$	−17404.6	−1.50672
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9779.58	0.836776
$516$	0	0
$517$	−1672.27	−0.142256
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17310.7	1.45566	0.727829	−	0.685759i	$-0.240530\pi$
$521$	17310.7	1.45566	0.727829	+	0.685759i	$0.240530\pi$
$522$	0	0
$523$	−8705.84	−0.727878	−0.363939	−	0.931423i	$-0.618568\pi$
$523$	−8705.84	−0.727878	−0.363939	+	0.931423i	$0.618568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34898.7	−2.88465
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12920.6	1.05001
$534$	0	0
$535$	14242.0	1.15091
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1066.01	−0.0851883
$540$	0	0
$541$	−5590.18	−0.444252	−0.222126	−	0.975018i	$-0.571300\pi$
$541$	−5590.18	−0.444252	−0.222126	+	0.975018i	$0.571300\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6838.65	−0.537496
$546$	0	0
$547$	−3567.66	−0.278870	−0.139435	−	0.990231i	$-0.544529\pi$
$547$	−3567.66	−0.278870	−0.139435	+	0.990231i	$0.544529\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4653.80	−0.359816
$552$	0	0
$553$	−9057.11	−0.696469
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9701.80	−0.738022	−0.369011	−	0.929425i	$-0.620304\pi$
$557$	−9701.80	−0.738022	−0.369011	+	0.929425i	$0.620304\pi$
$558$	0	0
$559$	7621.79	0.576686
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14576.2	1.09114	0.545572	−	0.838064i	$-0.316312\pi$
$563$	14576.2	1.09114	0.545572	+	0.838064i	$0.316312\pi$
$564$	0	0
$565$	16094.1	1.19838
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21542.1	1.58716	0.793578	−	0.608469i	$-0.208216\pi$
$569$	21542.1	1.58716	0.793578	+	0.608469i	$0.208216\pi$
$570$	0	0
$571$	15256.7	1.11817	0.559083	−	0.829112i	$-0.311153\pi$
$571$	15256.7	1.11817	0.559083	+	0.829112i	$0.311153\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	551.912	0.0400284
$576$	0	0
$577$	8388.02	0.605196	0.302598	−	0.953118i	$-0.402146\pi$
$577$	8388.02	0.605196	0.302598	+	0.953118i	$0.402146\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2200.19	−0.157107
$582$	0	0
$583$	6025.48	0.428044
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24869.1	−1.74865	−0.874325	−	0.485342i	$-0.838695\pi$
$587$	−24869.1	−1.74865	−0.874325	+	0.485342i	$0.838695\pi$
$588$	0	0
$589$	−15311.3	−1.07112
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5110.64	−0.353911	−0.176955	−	0.984219i	$-0.556625\pi$
$593$	−5110.64	−0.353911	−0.176955	+	0.984219i	$0.556625\pi$
$594$	0	0
$595$	24548.7	1.69143
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19254.8	−1.31340	−0.656702	−	0.754150i	$-0.728049\pi$
$599$	−19254.8	−1.31340	−0.656702	+	0.754150i	$0.728049\pi$
$600$	0	0
$601$	4918.63	0.333836	0.166918	−	0.985971i	$-0.446619\pi$
$601$	4918.63	0.333836	0.166918	+	0.985971i	$0.446619\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12258.0	0.823733
$606$	0	0
$607$	−10528.1	−0.703994	−0.351997	−	0.936001i	$-0.614497\pi$
$607$	−10528.1	−0.703994	−0.351997	+	0.936001i	$0.614497\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6628.16	0.438865
$612$	0	0
$613$	11500.4	0.757742	0.378871	−	0.925449i	$-0.376312\pi$
$613$	11500.4	0.757742	0.378871	+	0.925449i	$0.376312\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20010.3	1.30565	0.652823	−	0.757511i	$-0.273585\pi$
$617$	20010.3	1.30565	0.652823	+	0.757511i	$0.273585\pi$
$618$	0	0
$619$	9307.13	0.604338	0.302169	−	0.953254i	$-0.402289\pi$
$619$	9307.13	0.604338	0.302169	+	0.953254i	$0.402289\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−29850.6	−1.91965
$624$	0	0
$625$	−12049.7	−0.771178
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24912.5	1.57921
$630$	0	0
$631$	−18599.5	−1.17343	−0.586715	−	0.809793i	$-0.699579\pi$
$631$	−18599.5	−1.17343	−0.586715	+	0.809793i	$0.699579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16936.4	−1.05843
$636$	0	0
$637$	4225.22	0.262809
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22439.0	−1.38266	−0.691332	−	0.722537i	$-0.742976\pi$
$641$	−22439.0	−1.38266	−0.691332	+	0.722537i	$0.742976\pi$
$642$	0	0
$643$	−1985.24	−0.121758	−0.0608789	−	0.998145i	$-0.519390\pi$
$643$	−1985.24	−0.121758	−0.0608789	+	0.998145i	$0.519390\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13290.3	−0.807569	−0.403784	−	0.914854i	$-0.632305\pi$
$647$	−13290.3	−0.807569	−0.403784	+	0.914854i	$0.632305\pi$
$648$	0	0
$649$	5985.05	0.361994
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12750.4	−0.764104	−0.382052	−	0.924141i	$-0.624783\pi$
$653$	−12750.4	−0.764104	−0.382052	+	0.924141i	$0.624783\pi$
$654$	0	0
$655$	2823.31	0.168421
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1488.75	0.0880024	0.0440012	−	0.999031i	$-0.485989\pi$
$659$	1488.75	0.0880024	0.0440012	+	0.999031i	$0.485989\pi$
$660$	0	0
$661$	9420.14	0.554313	0.277157	−	0.960825i	$-0.410608\pi$
$661$	9420.14	0.554313	0.277157	+	0.960825i	$0.410608\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10770.4	0.628056
$666$	0	0
$667$	−2104.68	−0.122179
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2831.06	0.162879
$672$	0	0
$673$	18487.0	1.05887	0.529437	−	0.848349i	$-0.322403\pi$
$673$	18487.0	1.05887	0.529437	+	0.848349i	$0.322403\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18621.1	−1.05712	−0.528558	−	0.848897i	$-0.677267\pi$
$677$	−18621.1	−1.05712	−0.528558	+	0.848897i	$0.677267\pi$
$678$	0	0
$679$	−5680.67	−0.321067
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28246.0	1.58244	0.791218	−	0.611534i	$-0.209447\pi$
$683$	28246.0	1.58244	0.791218	+	0.611534i	$0.209447\pi$
$684$	0	0
$685$	9558.30	0.533144
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23882.4	−1.32053
$690$	0	0
$691$	−7452.93	−0.410308	−0.205154	−	0.978730i	$-0.565770\pi$
$691$	−7452.93	−0.410308	−0.205154	+	0.978730i	$0.565770\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20282.7	−1.10700
$696$	0	0
$697$	−35816.4	−1.94640
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10226.1	−0.550976	−0.275488	−	0.961305i	$-0.588839\pi$
$701$	−10226.1	−0.550976	−0.275488	+	0.961305i	$0.588839\pi$
$702$	0	0
$703$	10930.0	0.586390
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−32710.2	−1.74002
$708$	0	0
$709$	4511.15	0.238956	0.119478	−	0.992837i	$-0.461878\pi$
$709$	4511.15	0.238956	0.119478	+	0.992837i	$0.461878\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6924.51	−0.363710
$714$	0	0
$715$	4433.83	0.231910
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4439.57	0.230275	0.115138	−	0.993350i	$-0.463269\pi$
$719$	4439.57	0.230275	0.115138	+	0.993350i	$0.463269\pi$
$720$	0	0
$721$	−20505.2	−1.05916
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2195.84	−0.112485
$726$	0	0
$727$	25447.5	1.29820	0.649102	−	0.760702i	$-0.275145\pi$
$727$	25447.5	1.29820	0.649102	+	0.760702i	$0.275145\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21127.9	−1.06901
$732$	0	0
$733$	−33876.9	−1.70706	−0.853528	−	0.521046i	$-0.825542\pi$
$733$	−33876.9	−1.70706	−0.853528	+	0.521046i	$0.825542\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8127.78	0.406229
$738$	0	0
$739$	−16480.5	−0.820360	−0.410180	−	0.912005i	$-0.634534\pi$
$739$	−16480.5	−0.820360	−0.410180	+	0.912005i	$0.634534\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9618.42	−0.474920	−0.237460	−	0.971397i	$-0.576315\pi$
$743$	−9618.42	−0.474920	−0.237460	+	0.971397i	$0.576315\pi$
$744$	0	0
$745$	−5307.33	−0.261001
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−29861.7	−1.45677
$750$	0	0
$751$	26032.3	1.26489	0.632446	−	0.774605i	$-0.282051\pi$
$751$	26032.3	1.26489	0.632446	+	0.774605i	$0.282051\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−19241.1	−0.927490
$756$	0	0
$757$	15205.4	0.730052	0.365026	−	0.930997i	$-0.381060\pi$
$757$	15205.4	0.730052	0.365026	+	0.930997i	$0.381060\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26797.8	1.27650	0.638251	−	0.769828i	$-0.279658\pi$
$761$	26797.8	1.27650	0.638251	+	0.769828i	$0.279658\pi$
$762$	0	0
$763$	14338.8	0.680341
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23722.2	−1.11676
$768$	0	0
$769$	23815.5	1.11679	0.558394	−	0.829576i	$-0.311418\pi$
$769$	23815.5	1.11679	0.558394	+	0.829576i	$0.311418\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−268.163	−0.0124776	−0.00623879	−	0.999981i	$-0.501986\pi$
$773$	−268.163	−0.0124776	−0.00623879	+	0.999981i	$0.501986\pi$
$774$	0	0
$775$	−7224.42	−0.334850
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15713.9	−0.722734
$780$	0	0
$781$	−5218.67	−0.239102
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8925.92	−0.405834
$786$	0	0
$787$	−7005.51	−0.317305	−0.158653	−	0.987334i	$-0.550715\pi$
$787$	−7005.51	−0.317305	−0.158653	+	0.987334i	$0.550715\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−33744.9	−1.51685
$792$	0	0
$793$	−11221.1	−0.502488
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30691.0	−1.36403	−0.682013	−	0.731340i	$-0.738895\pi$
$797$	−30691.0	−1.36403	−0.682013	+	0.731340i	$0.738895\pi$
$798$	0	0
$799$	−18373.5	−0.813527
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8713.91	0.382948
$804$	0	0
$805$	4870.89	0.213263
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20261.3	0.880529	0.440264	−	0.897868i	$-0.354885\pi$
$809$	20261.3	0.880529	0.440264	+	0.897868i	$0.354885\pi$
$810$	0	0
$811$	−26133.5	−1.13153	−0.565765	−	0.824566i	$-0.691419\pi$
$811$	−26133.5	−1.13153	−0.565765	+	0.824566i	$0.691419\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15819.7	0.679925
$816$	0	0
$817$	−9269.55	−0.396941
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15605.6	0.663385	0.331693	−	0.943388i	$-0.392380\pi$
$821$	15605.6	0.663385	0.331693	+	0.943388i	$0.392380\pi$
$822$	0	0
$823$	22973.6	0.973035	0.486518	−	0.873671i	$-0.338267\pi$
$823$	22973.6	0.973035	0.486518	+	0.873671i	$0.338267\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3890.28	0.163577	0.0817886	−	0.996650i	$-0.473937\pi$
$827$	3890.28	0.163577	0.0817886	+	0.996650i	$0.473937\pi$
$828$	0	0
$829$	−5116.57	−0.214362	−0.107181	−	0.994240i	$-0.534182\pi$
$829$	−5116.57	−0.214362	−0.107181	+	0.994240i	$0.534182\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11712.5	−0.487171
$834$	0	0
$835$	−6405.82	−0.265488
$836$	0	0
$837$	0	0
$838$	0	0
$839$	886.569	0.0364812	0.0182406	−	0.999834i	$-0.494194\pi$
$839$	886.569	0.0364812	0.0182406	+	0.999834i	$0.494194\pi$
$840$	0	0
$841$	−16015.3	−0.656662
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4506.21	0.183454
$846$	0	0
$847$	−25701.7	−1.04265
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4943.08	0.199115
$852$	0	0
$853$	−8601.64	−0.345269	−0.172635	−	0.984986i	$-0.555228\pi$
$853$	−8601.64	−0.345269	−0.172635	+	0.984986i	$0.555228\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1027.43	−0.0409527	−0.0204763	−	0.999790i	$-0.506518\pi$
$857$	−1027.43	−0.0409527	−0.0204763	+	0.999790i	$0.506518\pi$
$858$	0	0
$859$	13875.7	0.551143	0.275571	−	0.961281i	$-0.411133\pi$
$859$	13875.7	0.551143	0.275571	+	0.961281i	$0.411133\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16938.0	0.668109	0.334054	−	0.942554i	$-0.391583\pi$
$863$	16938.0	0.668109	0.334054	+	0.942554i	$0.391583\pi$
$864$	0	0
$865$	−35856.3	−1.40942
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4534.60	0.177015
$870$	0	0
$871$	−32215.1	−1.25323
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31554.1	1.21911
$876$	0	0
$877$	24252.0	0.933788	0.466894	−	0.884313i	$-0.345373\pi$
$877$	24252.0	0.933788	0.466894	+	0.884313i	$0.345373\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18566.9	0.710026	0.355013	−	0.934861i	$-0.384476\pi$
$881$	18566.9	0.710026	0.355013	+	0.934861i	$0.384476\pi$
$882$	0	0
$883$	−43432.3	−1.65528	−0.827641	−	0.561258i	$-0.810317\pi$
$883$	−43432.3	−1.65528	−0.827641	+	0.561258i	$0.810317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45736.3	1.73131	0.865657	−	0.500637i	$-0.166901\pi$
$887$	45736.3	1.73131	0.865657	+	0.500637i	$0.166901\pi$
$888$	0	0
$889$	35511.2	1.33971
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8061.11	−0.302077
$894$	0	0
$895$	31072.9	1.16050
$896$	0	0
$897$	0	0
$898$	0	0
$899$	27549.8	1.02207
$900$	0	0
$901$	66203.0	2.44788
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16418.9	−0.603074
$906$	0	0
$907$	−2379.40	−0.0871075	−0.0435538	−	0.999051i	$-0.513868\pi$
$907$	−2379.40	−0.0871075	−0.0435538	+	0.999051i	$0.513868\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1883.03	−0.0684826	−0.0342413	−	0.999414i	$-0.510901\pi$
$911$	−1883.03	−0.0684826	−0.0342413	+	0.999414i	$0.510901\pi$
$912$	0	0
$913$	1101.56	0.0399304
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5919.72	−0.213180
$918$	0	0
$919$	1490.34	0.0534948	0.0267474	−	0.999642i	$-0.491485\pi$
$919$	1490.34	0.0534948	0.0267474	+	0.999642i	$0.491485\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20684.6	0.737640
$924$	0	0
$925$	5157.17	0.183315
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24046.2	−0.849226	−0.424613	−	0.905375i	$-0.639590\pi$
$929$	−24046.2	−0.849226	−0.424613	+	0.905375i	$0.639590\pi$
$930$	0	0
$931$	−5138.68	−0.180895
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12290.7	−0.429893
$936$	0	0
$937$	−5900.85	−0.205734	−0.102867	−	0.994695i	$-0.532802\pi$
$937$	−5900.85	−0.205734	−0.102867	+	0.994695i	$0.532802\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37280.9	−1.29152	−0.645761	−	0.763540i	$-0.723460\pi$
$941$	−37280.9	−1.29152	−0.645761	+	0.763540i	$0.723460\pi$
$942$	0	0
$943$	−7106.61	−0.245411
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28912.1	−0.992100	−0.496050	−	0.868294i	$-0.665217\pi$
$947$	−28912.1	−0.992100	−0.496050	+	0.868294i	$0.665217\pi$
$948$	0	0
$949$	−34538.2	−1.18141
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32283.8	1.09735	0.548675	−	0.836036i	$-0.315132\pi$
$953$	32283.8	1.09735	0.548675	+	0.836036i	$0.315132\pi$
$954$	0	0
$955$	−5848.91	−0.198185
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20041.2	−0.674832
$960$	0	0
$961$	60849.6	2.04255
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1323.72	−0.0441574
$966$	0	0
$967$	−10887.1	−0.362053	−0.181026	−	0.983478i	$-0.557942\pi$
$967$	−10887.1	−0.362053	−0.181026	+	0.983478i	$0.557942\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23712.6	0.783700	0.391850	−	0.920029i	$-0.371835\pi$
$971$	23712.6	0.783700	0.391850	+	0.920029i	$0.371835\pi$
$972$	0	0
$973$	42527.3	1.40120
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19159.9	0.627410	0.313705	−	0.949521i	$-0.398430\pi$
$977$	19159.9	0.627410	0.313705	+	0.949521i	$0.398430\pi$
$978$	0	0
$979$	14945.2	0.487898
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43304.9	−1.40510	−0.702549	−	0.711636i	$-0.747955\pi$
$983$	−43304.9	−1.40510	−0.702549	+	0.711636i	$0.747955\pi$
$984$	0	0
$985$	−13348.1	−0.431784
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4192.15	−0.134785
$990$	0	0
$991$	21119.8	0.676985	0.338492	−	0.940969i	$-0.390083\pi$
$991$	21119.8	0.676985	0.338492	+	0.940969i	$0.390083\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24623.6	−0.784542
$996$	0	0
$997$	222.283	0.00706095	0.00353047	−	0.999994i	$-0.498876\pi$
$997$	222.283	0.00706095	0.00353047	+	0.999994i	$0.498876\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	828.4.a.h.1.2	✓	6
3.2	odd	2		828.4.a.i.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
828.4.a.h.1.2	✓	6	1.1	even	1	trivial
828.4.a.i.1.5	yes	6	3.2	odd	2

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:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	828.a (trivial)

\(f(q)\)	\(=\)	\(q-10.0501 q^{5} +21.0723 q^{7} -10.5502 q^{11} +41.8166 q^{13} -115.917 q^{17} -50.8569 q^{19} -23.0000 q^{23} -23.9962 q^{25} +91.5077 q^{29} +301.066 q^{31} -211.778 q^{35} -214.916 q^{37} +308.983 q^{41} +182.267 q^{43} +158.506 q^{47} +101.042 q^{49} -571.124 q^{53} +106.030 q^{55} -567.292 q^{59} -268.341 q^{61} -420.259 q^{65} -770.390 q^{67} +494.651 q^{71} -825.946 q^{73} -222.317 q^{77} -429.811 q^{79} -104.411 q^{83} +1164.97 q^{85} -1416.58 q^{89} +881.171 q^{91} +511.115 q^{95} -269.580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{5} - 4 q^{7} + 2 q^{11} - 16 q^{13} + 12 q^{17} + 62 q^{19} - 138 q^{23} + 186 q^{25} - 392 q^{29} + 72 q^{31} - 508 q^{35} - 186 q^{37} - 276 q^{41} - 66 q^{43} - 748 q^{47} - 122 q^{49} - 998 q^{53}+ \cdots - 608 q^{97}+O(q^{100}) \) 6 * q - 10 * q^5 - 4 * q^7 + 2 * q^11 - 16 * q^13 + 12 * q^17 + 62 * q^19 - 138 * q^23 + 186 * q^25 - 392 * q^29 + 72 * q^31 - 508 * q^35 - 186 * q^37 - 276 * q^41 - 66 * q^43 - 748 * q^47 - 122 * q^49 - 998 * q^53 + 352 * q^55 - 1352 * q^59 - 354 * q^61 - 1764 * q^65 - 198 * q^67 - 1932 * q^71 + 340 * q^73 - 2364 * q^77 - 908 * q^79 - 3046 * q^83 - 824 * q^85 - 744 * q^89 + 1056 * q^91 - 5324 * q^95 - 608 * q^97

Introduction

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