LMFDB - Embedded newform 2312.4.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$136.412415933$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.65088 q^{3} +11.4162 q^{5} +19.1986 q^{7} +17.2342 q^{9} +O(q^{10})$	$q-6.65088 q^{3} +11.4162 q^{5} +19.1986 q^{7} +17.2342 q^{9} -54.5575 q^{11} -18.9197 q^{13} -75.9278 q^{15} +133.541 q^{19} -127.688 q^{21} +136.398 q^{23} +5.32966 q^{25} +64.9512 q^{27} -150.773 q^{29} -207.164 q^{31} +362.855 q^{33} +219.176 q^{35} +218.351 q^{37} +125.833 q^{39} -71.1766 q^{41} -309.098 q^{43} +196.749 q^{45} -275.202 q^{47} +25.5877 q^{49} +41.9225 q^{53} -622.839 q^{55} -888.168 q^{57} -338.413 q^{59} -344.428 q^{61} +330.873 q^{63} -215.992 q^{65} +479.408 q^{67} -907.168 q^{69} +1.42783 q^{71} +688.236 q^{73} -35.4469 q^{75} -1047.43 q^{77} +93.3804 q^{79} -897.306 q^{81} +1440.50 q^{83} +1002.77 q^{87} +173.056 q^{89} -363.233 q^{91} +1377.82 q^{93} +1524.54 q^{95} -581.630 q^{97} -940.255 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Display 100 coefficients 10 coefficients

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
$3$	−6.65088	−1.27996	−0.639981	−	0.768391i	$-0.721058\pi$
$3$	−6.65088	−1.27996	−0.639981	+	0.768391i	$0.721058\pi$
$4$	0	0
$5$	11.4162	1.02110	0.510548	−	0.859849i	$-0.329443\pi$
$5$	11.4162	1.02110	0.510548	+	0.859849i	$0.329443\pi$
$6$	0	0
$7$	19.1986	1.03663	0.518314	−	0.855190i	$-0.326560\pi$
$7$	19.1986	1.03663	0.518314	+	0.855190i	$0.326560\pi$
$8$	0	0
$9$	17.2342	0.638304
$10$	0	0
$11$	−54.5575	−1.49543	−0.747714	−	0.664021i	$-0.768848\pi$
$11$	−54.5575	−1.49543	−0.747714	+	0.664021i	$0.768848\pi$
$12$	0	0
$13$	−18.9197	−0.403646	−0.201823	−	0.979422i	$-0.564687\pi$
$13$	−18.9197	−0.403646	−0.201823	+	0.979422i	$0.564687\pi$
$14$	0	0
$15$	−75.9278	−1.30696
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	133.541	1.61245	0.806224	−	0.591611i	$-0.201508\pi$
$19$	133.541	1.61245	0.806224	+	0.591611i	$0.201508\pi$
$20$	0	0
$21$	−127.688	−1.32685
$22$	0	0
$23$	136.398	1.23656	0.618282	−	0.785956i	$-0.287829\pi$
$23$	136.398	1.23656	0.618282	+	0.785956i	$0.287829\pi$
$24$	0	0
$25$	5.32966	0.0426373
$26$	0	0
$27$	64.9512	0.462958
$28$	0	0
$29$	−150.773	−0.965440	−0.482720	−	0.875775i	$-0.660351\pi$
$29$	−150.773	−0.965440	−0.482720	+	0.875775i	$0.660351\pi$
$30$	0	0
$31$	−207.164	−1.20025	−0.600125	−	0.799906i	$-0.704882\pi$
$31$	−207.164	−1.20025	−0.600125	+	0.799906i	$0.704882\pi$
$32$	0	0
$33$	362.855	1.91409
$34$	0	0
$35$	219.176	1.05850
$36$	0	0
$37$	218.351	0.970179	0.485089	−	0.874465i	$-0.338787\pi$
$37$	218.351	0.970179	0.485089	+	0.874465i	$0.338787\pi$
$38$	0	0
$39$	125.833	0.516651
$40$	0	0
$41$	−71.1766	−0.271120	−0.135560	−	0.990769i	$-0.543283\pi$
$41$	−71.1766	−0.271120	−0.135560	+	0.990769i	$0.543283\pi$
$42$	0	0
$43$	−309.098	−1.09621	−0.548104	−	0.836410i	$-0.684650\pi$
$43$	−309.098	−1.09621	−0.548104	+	0.836410i	$0.684650\pi$
$44$	0	0
$45$	196.749	0.651769
$46$	0	0
$47$	−275.202	−0.854093	−0.427046	−	0.904230i	$-0.640446\pi$
$47$	−275.202	−0.854093	−0.427046	+	0.904230i	$0.640446\pi$
$48$	0	0
$49$	25.5877	0.0745997
$50$	0	0
$51$	0	0
$52$	0	0
$53$	41.9225	0.108651	0.0543255	−	0.998523i	$-0.482699\pi$
$53$	41.9225	0.108651	0.0543255	+	0.998523i	$0.482699\pi$
$54$	0	0
$55$	−622.839	−1.52698
$56$	0	0
$57$	−888.168	−2.06387
$58$	0	0
$59$	−338.413	−0.746738	−0.373369	−	0.927683i	$-0.621798\pi$
$59$	−338.413	−0.746738	−0.373369	+	0.927683i	$0.621798\pi$
$60$	0	0
$61$	−344.428	−0.722942	−0.361471	−	0.932383i	$-0.617725\pi$
$61$	−344.428	−0.722942	−0.361471	+	0.932383i	$0.617725\pi$
$62$	0	0
$63$	330.873	0.661684
$64$	0	0
$65$	−215.992	−0.412161
$66$	0	0
$67$	479.408	0.874165	0.437082	−	0.899421i	$-0.356012\pi$
$67$	479.408	0.874165	0.437082	+	0.899421i	$0.356012\pi$
$68$	0	0
$69$	−907.168	−1.58276
$70$	0	0
$71$	1.42783	0.00238665	0.00119333	−	0.999999i	$-0.499620\pi$
$71$	1.42783	0.00238665	0.00119333	+	0.999999i	$0.499620\pi$
$72$	0	0
$73$	688.236	1.10345	0.551726	−	0.834026i	$-0.313969\pi$
$73$	688.236	1.10345	0.551726	+	0.834026i	$0.313969\pi$
$74$	0	0
$75$	−35.4469	−0.0545741
$76$	0	0
$77$	−1047.43	−1.55020
$78$	0	0
$79$	93.3804	0.132989	0.0664944	−	0.997787i	$-0.478819\pi$
$79$	93.3804	0.132989	0.0664944	+	0.997787i	$0.478819\pi$
$80$	0	0
$81$	−897.306	−1.23087
$82$	0	0
$83$	1440.50	1.90500	0.952502	−	0.304531i	$-0.0984998\pi$
$83$	1440.50	1.90500	0.952502	+	0.304531i	$0.0984998\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1002.77	1.23573
$88$	0	0
$89$	173.056	0.206111	0.103056	−	0.994676i	$-0.467138\pi$
$89$	173.056	0.206111	0.103056	+	0.994676i	$0.467138\pi$
$90$	0	0
$91$	−363.233	−0.418431
$92$	0	0
$93$	1377.82	1.53627
$94$	0	0
$95$	1524.54	1.64646
$96$	0	0
$97$	−581.630	−0.608820	−0.304410	−	0.952541i	$-0.598459\pi$
$97$	−581.630	−0.608820	−0.304410	+	0.952541i	$0.598459\pi$
$98$	0	0
$99$	−940.255	−0.954537
$100$	0	0
$101$	−1671.62	−1.64686	−0.823428	−	0.567420i	$-0.807942\pi$
$101$	−1671.62	−1.64686	−0.823428	+	0.567420i	$0.807942\pi$
$102$	0	0
$103$	−11.1266	−0.0106441	−0.00532205	−	0.999986i	$-0.501694\pi$
$103$	−11.1266	−0.0106441	−0.00532205	+	0.999986i	$0.501694\pi$
$104$	0	0
$105$	−1457.71	−1.35484
$106$	0	0
$107$	1704.87	1.54034	0.770168	−	0.637841i	$-0.220172\pi$
$107$	1704.87	1.54034	0.770168	+	0.637841i	$0.220172\pi$
$108$	0	0
$109$	−1681.93	−1.47797	−0.738987	−	0.673719i	$-0.764696\pi$
$109$	−1681.93	−1.47797	−0.738987	+	0.673719i	$0.764696\pi$
$110$	0	0
$111$	−1452.22	−1.24179
$112$	0	0
$113$	−578.123	−0.481285	−0.240643	−	0.970614i	$-0.577358\pi$
$113$	−578.123	−0.481285	−0.240643	+	0.970614i	$0.577358\pi$
$114$	0	0
$115$	1557.15	1.26265
$116$	0	0
$117$	−326.067	−0.257649
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1645.52	1.23630
$122$	0	0
$123$	473.387	0.347023
$124$	0	0
$125$	−1366.18	−0.977559
$126$	0	0
$127$	−1889.53	−1.32023	−0.660114	−	0.751166i	$-0.729492\pi$
$127$	−1889.53	−1.32023	−0.660114	+	0.751166i	$0.729492\pi$
$128$	0	0
$129$	2055.77	1.40311
$130$	0	0
$131$	1705.99	1.13781	0.568904	−	0.822404i	$-0.307368\pi$
$131$	1705.99	1.13781	0.568904	+	0.822404i	$0.307368\pi$
$132$	0	0
$133$	2563.81	1.67151
$134$	0	0
$135$	741.496	0.472724
$136$	0	0
$137$	1755.68	1.09488	0.547438	−	0.836846i	$-0.315603\pi$
$137$	1755.68	1.09488	0.547438	+	0.836846i	$0.315603\pi$
$138$	0	0
$139$	2765.63	1.68761	0.843804	−	0.536651i	$-0.180311\pi$
$139$	2765.63	1.68761	0.843804	+	0.536651i	$0.180311\pi$
$140$	0	0
$141$	1830.34	1.09321
$142$	0	0
$143$	1032.21	0.603623
$144$	0	0
$145$	−1721.25	−0.985807
$146$	0	0
$147$	−170.181	−0.0954848
$148$	0	0
$149$	−2597.29	−1.42804	−0.714020	−	0.700125i	$-0.753127\pi$
$149$	−2597.29	−1.42804	−0.714020	+	0.700125i	$0.753127\pi$
$150$	0	0
$151$	−2386.79	−1.28632	−0.643159	−	0.765732i	$-0.722377\pi$
$151$	−2386.79	−1.28632	−0.643159	+	0.765732i	$0.722377\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2365.03	−1.22557
$156$	0	0
$157$	−1973.55	−1.00323	−0.501614	−	0.865092i	$-0.667260\pi$
$157$	−1973.55	−1.00323	−0.501614	+	0.865092i	$0.667260\pi$
$158$	0	0
$159$	−278.822	−0.139069
$160$	0	0
$161$	2618.66	1.28186
$162$	0	0
$163$	2254.55	1.08338	0.541688	−	0.840580i	$-0.317785\pi$
$163$	2254.55	1.08338	0.541688	+	0.840580i	$0.317785\pi$
$164$	0	0
$165$	4142.43	1.95447
$166$	0	0
$167$	312.157	0.144643	0.0723217	−	0.997381i	$-0.476959\pi$
$167$	312.157	0.144643	0.0723217	+	0.997381i	$0.476959\pi$
$168$	0	0
$169$	−1839.04	−0.837070
$170$	0	0
$171$	2301.48	1.02923
$172$	0	0
$173$	4497.19	1.97639	0.988194	−	0.153205i	$-0.0489594\pi$
$173$	4497.19	1.97639	0.988194	+	0.153205i	$0.0489594\pi$
$174$	0	0
$175$	102.322	0.0441991
$176$	0	0
$177$	2250.74	0.955797
$178$	0	0
$179$	−309.838	−0.129377	−0.0646883	−	0.997906i	$-0.520605\pi$
$179$	−309.838	−0.129377	−0.0646883	+	0.997906i	$0.520605\pi$
$180$	0	0
$181$	2429.95	0.997881	0.498941	−	0.866636i	$-0.333723\pi$
$181$	2429.95	0.997881	0.498941	+	0.866636i	$0.333723\pi$
$182$	0	0
$183$	2290.75	0.925339
$184$	0	0
$185$	2492.73	0.990646
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1246.97	0.479915
$190$	0	0
$191$	−292.230	−0.110707	−0.0553535	−	0.998467i	$-0.517629\pi$
$191$	−292.230	−0.110707	−0.0553535	+	0.998467i	$0.517629\pi$
$192$	0	0
$193$	−2362.89	−0.881268	−0.440634	−	0.897687i	$-0.645246\pi$
$193$	−2362.89	−0.881268	−0.440634	+	0.897687i	$0.645246\pi$
$194$	0	0
$195$	1436.53	0.527551
$196$	0	0
$197$	−1421.57	−0.514125	−0.257062	−	0.966395i	$-0.582755\pi$
$197$	−1421.57	−0.514125	−0.257062	+	0.966395i	$0.582755\pi$
$198$	0	0
$199$	2610.85	0.930041	0.465021	−	0.885300i	$-0.346047\pi$
$199$	2610.85	0.930041	0.465021	+	0.885300i	$0.346047\pi$
$200$	0	0
$201$	−3188.49	−1.11890
$202$	0	0
$203$	−2894.63	−1.00080
$204$	0	0
$205$	−812.566	−0.276839
$206$	0	0
$207$	2350.71	0.789303
$208$	0	0
$209$	−7285.68	−2.41130
$210$	0	0
$211$	−2668.76	−0.870734	−0.435367	−	0.900253i	$-0.643381\pi$
$211$	−2668.76	−0.870734	−0.435367	+	0.900253i	$0.643381\pi$
$212$	0	0
$213$	−9.49633	−0.00305483
$214$	0	0
$215$	−3528.72	−1.11933
$216$	0	0
$217$	−3977.27	−1.24421
$218$	0	0
$219$	−4577.38	−1.41238
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5855.95	−1.75849	−0.879246	−	0.476368i	$-0.841953\pi$
$223$	−5855.95	−1.75849	−0.879246	+	0.476368i	$0.841953\pi$
$224$	0	0
$225$	91.8525	0.0272155
$226$	0	0
$227$	5585.74	1.63321	0.816605	−	0.577197i	$-0.195853\pi$
$227$	5585.74	1.63321	0.816605	+	0.577197i	$0.195853\pi$
$228$	0	0
$229$	987.588	0.284986	0.142493	−	0.989796i	$-0.454488\pi$
$229$	987.588	0.284986	0.142493	+	0.989796i	$0.454488\pi$
$230$	0	0
$231$	6966.33	1.98420
$232$	0	0
$233$	3968.99	1.11595	0.557977	−	0.829856i	$-0.311578\pi$
$233$	3968.99	1.11595	0.557977	+	0.829856i	$0.311578\pi$
$234$	0	0
$235$	−3141.76	−0.872111
$236$	0	0
$237$	−621.062	−0.170221
$238$	0	0
$239$	−4250.47	−1.15038	−0.575188	−	0.818021i	$-0.695071\pi$
$239$	−4250.47	−1.15038	−0.575188	+	0.818021i	$0.695071\pi$
$240$	0	0
$241$	−5210.19	−1.39260	−0.696302	−	0.717749i	$-0.745173\pi$
$241$	−5210.19	−1.39260	−0.696302	+	0.717749i	$0.745173\pi$
$242$	0	0
$243$	4214.19	1.11251
$244$	0	0
$245$	292.114	0.0761734
$246$	0	0
$247$	−2526.57	−0.650857
$248$	0	0
$249$	−9580.59	−2.43833
$250$	0	0
$251$	−3972.67	−0.999016	−0.499508	−	0.866309i	$-0.666486\pi$
$251$	−3972.67	−0.999016	−0.499508	+	0.866309i	$0.666486\pi$
$252$	0	0
$253$	−7441.54	−1.84919
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5518.71	−1.33949	−0.669743	−	0.742593i	$-0.733596\pi$
$257$	−5518.71	−1.33949	−0.669743	+	0.742593i	$0.733596\pi$
$258$	0	0
$259$	4192.03	1.00572
$260$	0	0
$261$	−2598.44	−0.616244
$262$	0	0
$263$	2306.88	0.540867	0.270434	−	0.962739i	$-0.412833\pi$
$263$	2306.88	0.540867	0.270434	+	0.962739i	$0.412833\pi$
$264$	0	0
$265$	478.596	0.110943
$266$	0	0
$267$	−1150.97	−0.263815
$268$	0	0
$269$	822.818	0.186499	0.0932493	−	0.995643i	$-0.470275\pi$
$269$	822.818	0.186499	0.0932493	+	0.995643i	$0.470275\pi$
$270$	0	0
$271$	−2859.40	−0.640946	−0.320473	−	0.947258i	$-0.603842\pi$
$271$	−2859.40	−0.640946	−0.320473	+	0.947258i	$0.603842\pi$
$272$	0	0
$273$	2415.82	0.535576
$274$	0	0
$275$	−290.773	−0.0637610
$276$	0	0
$277$	1314.46	0.285120	0.142560	−	0.989786i	$-0.454467\pi$
$277$	1314.46	0.285120	0.142560	+	0.989786i	$0.454467\pi$
$278$	0	0
$279$	−3570.30	−0.766124
$280$	0	0
$281$	3829.62	0.813012	0.406506	−	0.913648i	$-0.366747\pi$
$281$	3829.62	0.813012	0.406506	+	0.913648i	$0.366747\pi$
$282$	0	0
$283$	−6051.99	−1.27121	−0.635607	−	0.772013i	$-0.719250\pi$
$283$	−6051.99	−1.27121	−0.635607	+	0.772013i	$0.719250\pi$
$284$	0	0
$285$	−10139.5	−2.10741
$286$	0	0
$287$	−1366.49	−0.281051
$288$	0	0
$289$	0	0
$290$	0	0
$291$	3868.35	0.779267
$292$	0	0
$293$	−4769.11	−0.950902	−0.475451	−	0.879742i	$-0.657715\pi$
$293$	−4769.11	−0.950902	−0.475451	+	0.879742i	$0.657715\pi$
$294$	0	0
$295$	−3863.39	−0.762492
$296$	0	0
$297$	−3543.57	−0.692320
$298$	0	0
$299$	−2580.62	−0.499134
$300$	0	0
$301$	−5934.26	−1.13636
$302$	0	0
$303$	11117.8	2.10791
$304$	0	0
$305$	−3932.06	−0.738193
$306$	0	0
$307$	−2514.42	−0.467444	−0.233722	−	0.972303i	$-0.575091\pi$
$307$	−2514.42	−0.467444	−0.233722	+	0.972303i	$0.575091\pi$
$308$	0	0
$309$	74.0020	0.0136240
$310$	0	0
$311$	2706.90	0.493550	0.246775	−	0.969073i	$-0.420629\pi$
$311$	2706.90	0.493550	0.246775	+	0.969073i	$0.420629\pi$
$312$	0	0
$313$	−10486.5	−1.89372	−0.946859	−	0.321649i	$-0.895763\pi$
$313$	−10486.5	−1.89372	−0.946859	+	0.321649i	$0.895763\pi$
$314$	0	0
$315$	3777.31	0.675643
$316$	0	0
$317$	−5670.90	−1.00476	−0.502381	−	0.864646i	$-0.667543\pi$
$317$	−5670.90	−1.00476	−0.502381	+	0.864646i	$0.667543\pi$
$318$	0	0
$319$	8225.77	1.44375
$320$	0	0
$321$	−11338.9	−1.97157
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−100.836	−0.0172104
$326$	0	0
$327$	11186.3	1.89175
$328$	0	0
$329$	−5283.51	−0.885377
$330$	0	0
$331$	−1567.51	−0.260297	−0.130149	−	0.991494i	$-0.541545\pi$
$331$	−1567.51	−0.260297	−0.130149	+	0.991494i	$0.541545\pi$
$332$	0	0
$333$	3763.10	0.619269
$334$	0	0
$335$	5473.02	0.892606
$336$	0	0
$337$	−4904.30	−0.792742	−0.396371	−	0.918090i	$-0.629731\pi$
$337$	−4904.30	−0.792742	−0.396371	+	0.918090i	$0.629731\pi$
$338$	0	0
$339$	3845.03	0.616027
$340$	0	0
$341$	11302.3	1.79489
$342$	0	0
$343$	−6093.88	−0.959297
$344$	0	0
$345$	−10356.4	−1.61615
$346$	0	0
$347$	−7061.76	−1.09249	−0.546247	−	0.837624i	$-0.683944\pi$
$347$	−7061.76	−1.09249	−0.546247	+	0.837624i	$0.683944\pi$
$348$	0	0
$349$	−9026.29	−1.38443	−0.692215	−	0.721691i	$-0.743365\pi$
$349$	−9026.29	−1.38443	−0.692215	+	0.721691i	$0.743365\pi$
$350$	0	0
$351$	−1228.86	−0.186871
$352$	0	0
$353$	−3120.19	−0.470456	−0.235228	−	0.971940i	$-0.575584\pi$
$353$	−3120.19	−0.470456	−0.235228	+	0.971940i	$0.575584\pi$
$354$	0	0
$355$	16.3004	0.00243700
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1675.07	−0.246258	−0.123129	−	0.992391i	$-0.539293\pi$
$359$	−1675.07	−0.246258	−0.123129	+	0.992391i	$0.539293\pi$
$360$	0	0
$361$	10974.3	1.59999
$362$	0	0
$363$	−10944.1	−1.58242
$364$	0	0
$365$	7857.04	1.12673
$366$	0	0
$367$	−3827.87	−0.544450	−0.272225	−	0.962234i	$-0.587760\pi$
$367$	−3827.87	−0.544450	−0.272225	+	0.962234i	$0.587760\pi$
$368$	0	0
$369$	−1226.67	−0.173057
$370$	0	0
$371$	804.855	0.112631
$372$	0	0
$373$	5331.33	0.740069	0.370035	−	0.929018i	$-0.379346\pi$
$373$	5331.33	0.740069	0.370035	+	0.929018i	$0.379346\pi$
$374$	0	0
$375$	9086.30	1.25124
$376$	0	0
$377$	2852.58	0.389696
$378$	0	0
$379$	−4887.06	−0.662352	−0.331176	−	0.943569i	$-0.607445\pi$
$379$	−4887.06	−0.662352	−0.331176	+	0.943569i	$0.607445\pi$
$380$	0	0
$381$	12567.1	1.68984
$382$	0	0
$383$	9101.53	1.21427	0.607137	−	0.794598i	$-0.292318\pi$
$383$	9101.53	1.21427	0.607137	+	0.794598i	$0.292318\pi$
$384$	0	0
$385$	−11957.7	−1.58291
$386$	0	0
$387$	−5327.05	−0.699714
$388$	0	0
$389$	−14524.0	−1.89304	−0.946522	−	0.322640i	$-0.895430\pi$
$389$	−14524.0	−1.89304	−0.946522	+	0.322640i	$0.895430\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−11346.3	−1.45635
$394$	0	0
$395$	1066.05	0.135794
$396$	0	0
$397$	937.947	0.118575	0.0592874	−	0.998241i	$-0.481117\pi$
$397$	937.947	0.118575	0.0592874	+	0.998241i	$0.481117\pi$
$398$	0	0
$399$	−17051.6	−2.13947
$400$	0	0
$401$	−7185.60	−0.894842	−0.447421	−	0.894323i	$-0.647657\pi$
$401$	−7185.60	−0.894842	−0.447421	+	0.894323i	$0.647657\pi$
$402$	0	0
$403$	3919.49	0.484476
$404$	0	0
$405$	−10243.8	−1.25684
$406$	0	0
$407$	−11912.7	−1.45083
$408$	0	0
$409$	−2791.91	−0.337533	−0.168767	−	0.985656i	$-0.553978\pi$
$409$	−2791.91	−0.337533	−0.168767	+	0.985656i	$0.553978\pi$
$410$	0	0
$411$	−11676.8	−1.40140
$412$	0	0
$413$	−6497.06	−0.774091
$414$	0	0
$415$	16445.0	1.94519
$416$	0	0
$417$	−18393.9	−2.16008
$418$	0	0
$419$	−8398.60	−0.979233	−0.489616	−	0.871938i	$-0.662863\pi$
$419$	−8398.60	−0.979233	−0.489616	+	0.871938i	$0.662863\pi$
$420$	0	0
$421$	6187.51	0.716296	0.358148	−	0.933665i	$-0.383408\pi$
$421$	6187.51	0.716296	0.358148	+	0.933665i	$0.383408\pi$
$422$	0	0
$423$	−4742.89	−0.545170
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6612.55	−0.749423
$428$	0	0
$429$	−6865.13	−0.772614
$430$	0	0
$431$	−8447.29	−0.944064	−0.472032	−	0.881582i	$-0.656479\pi$
$431$	−8447.29	−0.944064	−0.472032	+	0.881582i	$0.656479\pi$
$432$	0	0
$433$	980.412	0.108812	0.0544060	−	0.998519i	$-0.482673\pi$
$433$	980.412	0.108812	0.0544060	+	0.998519i	$0.482673\pi$
$434$	0	0
$435$	11447.8	1.26180
$436$	0	0
$437$	18214.8	1.99389
$438$	0	0
$439$	−11660.0	−1.26766	−0.633829	−	0.773473i	$-0.718518\pi$
$439$	−11660.0	−1.26766	−0.633829	+	0.773473i	$0.718518\pi$
$440$	0	0
$441$	440.983	0.0476173
$442$	0	0
$443$	0.580982	6.23099e−5 0	3.11550e−5	−	1.00000i	$-0.499990\pi$
$443$	0.580982	6.23099e−5 0	3.11550e−5		1.00000i	$0.499990\pi$
$444$	0	0
$445$	1975.64	0.210459
$446$	0	0
$447$	17274.2	1.82784
$448$	0	0
$449$	6555.73	0.689051	0.344526	−	0.938777i	$-0.388040\pi$
$449$	6555.73	0.689051	0.344526	+	0.938777i	$0.388040\pi$
$450$	0	0
$451$	3883.21	0.405440
$452$	0	0
$453$	15874.2	1.64644
$454$	0	0
$455$	−4146.75	−0.427258
$456$	0	0
$457$	−10406.7	−1.06522	−0.532610	−	0.846361i	$-0.678789\pi$
$457$	−10406.7	−1.06522	−0.532610	+	0.846361i	$0.678789\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15855.1	1.60183	0.800916	−	0.598777i	$-0.204346\pi$
$461$	15855.1	1.60183	0.800916	+	0.598777i	$0.204346\pi$
$462$	0	0
$463$	10253.0	1.02915	0.514574	−	0.857446i	$-0.327950\pi$
$463$	10253.0	1.02915	0.514574	+	0.857446i	$0.327950\pi$
$464$	0	0
$465$	15729.5	1.56868
$466$	0	0
$467$	1871.34	0.185429	0.0927147	−	0.995693i	$-0.470446\pi$
$467$	1871.34	0.185429	0.0927147	+	0.995693i	$0.470446\pi$
$468$	0	0
$469$	9203.98	0.906185
$470$	0	0
$471$	13125.9	1.28409
$472$	0	0
$473$	16863.6	1.63930
$474$	0	0
$475$	711.731	0.0687504
$476$	0	0
$477$	722.501	0.0693523
$478$	0	0
$479$	−19466.9	−1.85692	−0.928462	−	0.371427i	$-0.878869\pi$
$479$	−19466.9	−1.85692	−0.928462	+	0.371427i	$0.878869\pi$
$480$	0	0
$481$	−4131.14	−0.391609
$482$	0	0
$483$	−17416.4	−1.64073
$484$	0	0
$485$	−6640.00	−0.621664
$486$	0	0
$487$	4307.41	0.400795	0.200398	−	0.979715i	$-0.435777\pi$
$487$	4307.41	0.400795	0.200398	+	0.979715i	$0.435777\pi$
$488$	0	0
$489$	−14994.8	−1.38668
$490$	0	0
$491$	−14313.7	−1.31562	−0.657809	−	0.753185i	$-0.728517\pi$
$491$	−14313.7	−1.31562	−0.657809	+	0.753185i	$0.728517\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−10734.1	−0.974674
$496$	0	0
$497$	27.4124	0.00247407
$498$	0	0
$499$	2831.95	0.254059	0.127030	−	0.991899i	$-0.459456\pi$
$499$	2831.95	0.254059	0.127030	+	0.991899i	$0.459456\pi$
$500$	0	0
$501$	−2076.12	−0.185138
$502$	0	0
$503$	−3568.16	−0.316295	−0.158147	−	0.987416i	$-0.550552\pi$
$503$	−3568.16	−0.316295	−0.158147	+	0.987416i	$0.550552\pi$
$504$	0	0
$505$	−19083.6	−1.68160
$506$	0	0
$507$	12231.3	1.07142
$508$	0	0
$509$	3351.94	0.291891	0.145945	−	0.989293i	$-0.453378\pi$
$509$	3351.94	0.291891	0.145945	+	0.989293i	$0.453378\pi$
$510$	0	0
$511$	13213.2	1.14387
$512$	0	0
$513$	8673.67	0.746495
$514$	0	0
$515$	−127.024	−0.0108686
$516$	0	0
$517$	15014.3	1.27723
$518$	0	0
$519$	−29910.3	−2.52970
$520$	0	0
$521$	7426.07	0.624457	0.312228	−	0.950007i	$-0.398925\pi$
$521$	7426.07	0.624457	0.312228	+	0.950007i	$0.398925\pi$
$522$	0	0
$523$	10251.0	0.857067	0.428533	−	0.903526i	$-0.359030\pi$
$523$	10251.0	0.857067	0.428533	+	0.903526i	$0.359030\pi$
$524$	0	0
$525$	−680.533	−0.0565731
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6437.45	0.529091
$530$	0	0
$531$	−5832.27	−0.476646
$532$	0	0
$533$	1346.64	0.109436
$534$	0	0
$535$	19463.1	1.57283
$536$	0	0
$537$	2060.70	0.165597
$538$	0	0
$539$	−1396.00	−0.111558
$540$	0	0
$541$	−11062.7	−0.879153	−0.439577	−	0.898205i	$-0.644872\pi$
$541$	−11062.7	−0.879153	−0.439577	+	0.898205i	$0.644872\pi$
$542$	0	0
$543$	−16161.3	−1.27725
$544$	0	0
$545$	−19201.2	−1.50915
$546$	0	0
$547$	18345.5	1.43400	0.717000	−	0.697074i	$-0.245515\pi$
$547$	18345.5	1.43400	0.717000	+	0.697074i	$0.245515\pi$
$548$	0	0
$549$	−5935.94	−0.461457
$550$	0	0
$551$	−20134.4	−1.55672
$552$	0	0
$553$	1792.78	0.137860
$554$	0	0
$555$	−16578.9	−1.26799
$556$	0	0
$557$	−4676.86	−0.355772	−0.177886	−	0.984051i	$-0.556926\pi$
$557$	−4676.86	−0.355772	−0.177886	+	0.984051i	$0.556926\pi$
$558$	0	0
$559$	5848.05	0.442480
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19373.2	1.45024	0.725119	−	0.688624i	$-0.241785\pi$
$563$	19373.2	1.45024	0.725119	+	0.688624i	$0.241785\pi$
$564$	0	0
$565$	−6599.97	−0.491438
$566$	0	0
$567$	−17227.0	−1.27596
$568$	0	0
$569$	−10753.9	−0.792314	−0.396157	−	0.918183i	$-0.629656\pi$
$569$	−10753.9	−0.792314	−0.396157	+	0.918183i	$0.629656\pi$
$570$	0	0
$571$	8304.84	0.608663	0.304332	−	0.952566i	$-0.401567\pi$
$571$	8304.84	0.608663	0.304332	+	0.952566i	$0.401567\pi$
$572$	0	0
$573$	1943.59	0.141701
$574$	0	0
$575$	726.956	0.0527238
$576$	0	0
$577$	−9480.83	−0.684042	−0.342021	−	0.939692i	$-0.611111\pi$
$577$	−9480.83	−0.684042	−0.342021	+	0.939692i	$0.611111\pi$
$578$	0	0
$579$	15715.3	1.12799
$580$	0	0
$581$	27655.6	1.97478
$582$	0	0
$583$	−2287.19	−0.162480
$584$	0	0
$585$	−3722.44	−0.263084
$586$	0	0
$587$	2092.70	0.147146	0.0735731	−	0.997290i	$-0.476560\pi$
$587$	2092.70	0.147146	0.0735731	+	0.997290i	$0.476560\pi$
$588$	0	0
$589$	−27665.0	−1.93534
$590$	0	0
$591$	9454.68	0.658060
$592$	0	0
$593$	7719.04	0.534541	0.267271	−	0.963622i	$-0.413878\pi$
$593$	7719.04	0.534541	0.267271	+	0.963622i	$0.413878\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17364.4	−1.19042
$598$	0	0
$599$	−9585.15	−0.653821	−0.326910	−	0.945055i	$-0.606008\pi$
$599$	−9585.15	−0.653821	−0.326910	+	0.945055i	$0.606008\pi$
$600$	0	0
$601$	−24693.8	−1.67601	−0.838004	−	0.545664i	$-0.816277\pi$
$601$	−24693.8	−1.67601	−0.838004	+	0.545664i	$0.816277\pi$
$602$	0	0
$603$	8262.22	0.557983
$604$	0	0
$605$	18785.6	1.26238
$606$	0	0
$607$	10410.2	0.696105	0.348053	−	0.937475i	$-0.386843\pi$
$607$	10410.2	0.696105	0.348053	+	0.937475i	$0.386843\pi$
$608$	0	0
$609$	19251.8	1.28099
$610$	0	0
$611$	5206.75	0.344751
$612$	0	0
$613$	−27045.1	−1.78196	−0.890980	−	0.454043i	$-0.849981\pi$
$613$	−27045.1	−1.78196	−0.890980	+	0.454043i	$0.849981\pi$
$614$	0	0
$615$	5404.28	0.354344
$616$	0	0
$617$	15027.1	0.980502	0.490251	−	0.871581i	$-0.336905\pi$
$617$	15027.1	0.980502	0.490251	+	0.871581i	$0.336905\pi$
$618$	0	0
$619$	−20991.5	−1.36303	−0.681517	−	0.731802i	$-0.738680\pi$
$619$	−20991.5	−1.36303	−0.681517	+	0.731802i	$0.738680\pi$
$620$	0	0
$621$	8859.22	0.572477
$622$	0	0
$623$	3322.44	0.213661
$624$	0	0
$625$	−16262.8	−1.04082
$626$	0	0
$627$	48456.2	3.08637
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1816.02	−0.114572	−0.0572858	−	0.998358i	$-0.518245\pi$
$631$	−1816.02	−0.114572	−0.0572858	+	0.998358i	$0.518245\pi$
$632$	0	0
$633$	17749.6	1.11451
$634$	0	0
$635$	−21571.3	−1.34808
$636$	0	0
$637$	−484.113	−0.0301118
$638$	0	0
$639$	24.6075	0.00152341
$640$	0	0
$641$	−611.831	−0.0377002	−0.0188501	−	0.999822i	$-0.506001\pi$
$641$	−611.831	−0.0377002	−0.0188501	+	0.999822i	$0.506001\pi$
$642$	0	0
$643$	−10047.0	−0.616198	−0.308099	−	0.951354i	$-0.599693\pi$
$643$	−10047.0	−0.616198	−0.308099	+	0.951354i	$0.599693\pi$
$644$	0	0
$645$	23469.1	1.43271
$646$	0	0
$647$	10673.1	0.648539	0.324270	−	0.945965i	$-0.394882\pi$
$647$	10673.1	0.648539	0.324270	+	0.945965i	$0.394882\pi$
$648$	0	0
$649$	18462.9	1.11669
$650$	0	0
$651$	26452.3	1.59255
$652$	0	0
$653$	22021.4	1.31970	0.659849	−	0.751398i	$-0.270620\pi$
$653$	22021.4	1.31970	0.659849	+	0.751398i	$0.270620\pi$
$654$	0	0
$655$	19475.9	1.16181
$656$	0	0
$657$	11861.2	0.704337
$658$	0	0
$659$	−2694.29	−0.159264	−0.0796318	−	0.996824i	$-0.525374\pi$
$659$	−2694.29	−0.159264	−0.0796318	+	0.996824i	$0.525374\pi$
$660$	0	0
$661$	14617.8	0.860162	0.430081	−	0.902790i	$-0.358485\pi$
$661$	14617.8	0.860162	0.430081	+	0.902790i	$0.358485\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29269.0	1.70677
$666$	0	0
$667$	−20565.1	−1.19383
$668$	0	0
$669$	38947.2	2.25080
$670$	0	0
$671$	18791.1	1.08111
$672$	0	0
$673$	21286.5	1.21922	0.609609	−	0.792702i	$-0.291326\pi$
$673$	21286.5	1.21922	0.609609	+	0.792702i	$0.291326\pi$
$674$	0	0
$675$	346.168	0.0197393
$676$	0	0
$677$	−21179.4	−1.20235	−0.601176	−	0.799117i	$-0.705301\pi$
$677$	−21179.4	−1.20235	−0.601176	+	0.799117i	$0.705301\pi$
$678$	0	0
$679$	−11166.5	−0.631120
$680$	0	0
$681$	−37150.1	−2.09045
$682$	0	0
$683$	−25878.7	−1.44981	−0.724904	−	0.688850i	$-0.758116\pi$
$683$	−25878.7	−1.44981	−0.724904	+	0.688850i	$0.758116\pi$
$684$	0	0
$685$	20043.2	1.11797
$686$	0	0
$687$	−6568.33	−0.364771
$688$	0	0
$689$	−793.163	−0.0438565
$690$	0	0
$691$	23444.9	1.29072	0.645360	−	0.763878i	$-0.276707\pi$
$691$	23444.9	1.29072	0.645360	+	0.763878i	$0.276707\pi$
$692$	0	0
$693$	−18051.6	−0.989500
$694$	0	0
$695$	31573.0	1.72321
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−26397.3	−1.42838
$700$	0	0
$701$	−23744.0	−1.27931	−0.639657	−	0.768660i	$-0.720924\pi$
$701$	−23744.0	−1.27931	−0.639657	+	0.768660i	$0.720924\pi$
$702$	0	0
$703$	29158.8	1.56436
$704$	0	0
$705$	20895.5	1.11627
$706$	0	0
$707$	−32092.9	−1.70718
$708$	0	0
$709$	−1358.15	−0.0719413	−0.0359706	−	0.999353i	$-0.511452\pi$
$709$	−1358.15	−0.0719413	−0.0359706	+	0.999353i	$0.511452\pi$
$710$	0	0
$711$	1609.34	0.0848872
$712$	0	0
$713$	−28256.8	−1.48419
$714$	0	0
$715$	11784.0	0.616357
$716$	0	0
$717$	28269.4	1.47244
$718$	0	0
$719$	−27589.1	−1.43102	−0.715508	−	0.698605i	$-0.753805\pi$
$719$	−27589.1	−1.43102	−0.715508	+	0.698605i	$0.753805\pi$
$720$	0	0
$721$	−213.617	−0.0110340
$722$	0	0
$723$	34652.3	1.78248
$724$	0	0
$725$	−803.567	−0.0411638
$726$	0	0
$727$	4578.49	0.233572	0.116786	−	0.993157i	$-0.462741\pi$
$727$	4578.49	0.233572	0.116786	+	0.993157i	$0.462741\pi$
$728$	0	0
$729$	−3800.82	−0.193102
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1608.02	0.0810282	0.0405141	−	0.999179i	$-0.487100\pi$
$733$	1608.02	0.0810282	0.0405141	+	0.999179i	$0.487100\pi$
$734$	0	0
$735$	−1942.82	−0.0974991
$736$	0	0
$737$	−26155.3	−1.30725
$738$	0	0
$739$	−10735.2	−0.534370	−0.267185	−	0.963645i	$-0.586093\pi$
$739$	−10735.2	−0.534370	−0.267185	+	0.963645i	$0.586093\pi$
$740$	0	0
$741$	16803.9	0.833073
$742$	0	0
$743$	14947.9	0.738067	0.369033	−	0.929416i	$-0.379689\pi$
$743$	14947.9	0.738067	0.369033	+	0.929416i	$0.379689\pi$
$744$	0	0
$745$	−29651.1	−1.45817
$746$	0	0
$747$	24825.9	1.21597
$748$	0	0
$749$	32731.2	1.59676
$750$	0	0
$751$	−2290.98	−0.111317	−0.0556584	−	0.998450i	$-0.517726\pi$
$751$	−2290.98	−0.111317	−0.0556584	+	0.998450i	$0.517726\pi$
$752$	0	0
$753$	26421.8	1.27870
$754$	0	0
$755$	−27248.1	−1.31346
$756$	0	0
$757$	−26514.2	−1.27302	−0.636509	−	0.771270i	$-0.719622\pi$
$757$	−26514.2	−1.27302	−0.636509	+	0.771270i	$0.719622\pi$
$758$	0	0
$759$	49492.8	2.36690
$760$	0	0
$761$	21096.2	1.00491	0.502454	−	0.864604i	$-0.332430\pi$
$761$	21096.2	1.00491	0.502454	+	0.864604i	$0.332430\pi$
$762$	0	0
$763$	−32290.7	−1.53211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6402.68	0.301418
$768$	0	0
$769$	−13109.6	−0.614750	−0.307375	−	0.951588i	$-0.599451\pi$
$769$	−13109.6	−0.614750	−0.307375	+	0.951588i	$0.599451\pi$
$770$	0	0
$771$	36704.3	1.71449
$772$	0	0
$773$	−7809.66	−0.363382	−0.181691	−	0.983356i	$-0.558157\pi$
$773$	−7809.66	−0.363382	−0.181691	+	0.983356i	$0.558157\pi$
$774$	0	0
$775$	−1104.11	−0.0511754
$776$	0	0
$777$	−27880.7	−1.28728
$778$	0	0
$779$	−9505.02	−0.437166
$780$	0	0
$781$	−77.8988	−0.00356907
$782$	0	0
$783$	−9792.85	−0.446958
$784$	0	0
$785$	−22530.5	−1.02439
$786$	0	0
$787$	32225.9	1.45963	0.729815	−	0.683644i	$-0.239606\pi$
$787$	32225.9	1.45963	0.729815	+	0.683644i	$0.239606\pi$
$788$	0	0
$789$	−15342.7	−0.692289
$790$	0	0
$791$	−11099.2	−0.498914
$792$	0	0
$793$	6516.49	0.291813
$794$	0	0
$795$	−3183.08	−0.142003
$796$	0	0
$797$	4011.32	0.178279	0.0891394	−	0.996019i	$-0.471588\pi$
$797$	4011.32	0.178279	0.0891394	+	0.996019i	$0.471588\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2982.48	0.131562
$802$	0	0
$803$	−37548.4	−1.65013
$804$	0	0
$805$	29895.1	1.30890
$806$	0	0
$807$	−5472.46	−0.238711
$808$	0	0
$809$	3725.51	0.161906	0.0809530	−	0.996718i	$-0.474204\pi$
$809$	3725.51	0.161906	0.0809530	+	0.996718i	$0.474204\pi$
$810$	0	0
$811$	29099.4	1.25995	0.629973	−	0.776617i	$-0.283066\pi$
$811$	29099.4	1.25995	0.629973	+	0.776617i	$0.283066\pi$
$812$	0	0
$813$	19017.5	0.820386
$814$	0	0
$815$	25738.4	1.10623
$816$	0	0
$817$	−41277.3	−1.76758
$818$	0	0
$819$	−6260.04	−0.267086
$820$	0	0
$821$	−44432.5	−1.88880	−0.944399	−	0.328801i	$-0.893356\pi$
$821$	−44432.5	−1.88880	−0.944399	+	0.328801i	$0.893356\pi$
$822$	0	0
$823$	13477.8	0.570845	0.285422	−	0.958402i	$-0.407866\pi$
$823$	13477.8	0.570845	0.285422	+	0.958402i	$0.407866\pi$
$824$	0	0
$825$	1933.90	0.0816117
$826$	0	0
$827$	−17446.8	−0.733597	−0.366798	−	0.930300i	$-0.619546\pi$
$827$	−17446.8	−0.733597	−0.366798	+	0.930300i	$0.619546\pi$
$828$	0	0
$829$	44421.0	1.86104	0.930521	−	0.366238i	$-0.119354\pi$
$829$	44421.0	1.86104	0.930521	+	0.366238i	$0.119354\pi$
$830$	0	0
$831$	−8742.32	−0.364943
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3563.65	0.147695
$836$	0	0
$837$	−13455.5	−0.555665
$838$	0	0
$839$	−14146.6	−0.582117	−0.291059	−	0.956705i	$-0.594007\pi$
$839$	−14146.6	−0.582117	−0.291059	+	0.956705i	$0.594007\pi$
$840$	0	0
$841$	−1656.64	−0.0679258
$842$	0	0
$843$	−25470.4	−1.04062
$844$	0	0
$845$	−20994.9	−0.854729
$846$	0	0
$847$	31591.7	1.28159
$848$	0	0
$849$	40251.1	1.62711
$850$	0	0
$851$	29782.6	1.19969
$852$	0	0
$853$	15839.1	0.635782	0.317891	−	0.948127i	$-0.397025\pi$
$853$	15839.1	0.635782	0.317891	+	0.948127i	$0.397025\pi$
$854$	0	0
$855$	26274.1	1.05094
$856$	0	0
$857$	23833.1	0.949969	0.474985	−	0.879994i	$-0.342454\pi$
$857$	23833.1	0.949969	0.474985	+	0.879994i	$0.342454\pi$
$858$	0	0
$859$	27854.2	1.10637	0.553186	−	0.833058i	$-0.313412\pi$
$859$	27854.2	1.10637	0.553186	+	0.833058i	$0.313412\pi$
$860$	0	0
$861$	9088.38	0.359734
$862$	0	0
$863$	26506.9	1.04555	0.522773	−	0.852472i	$-0.324898\pi$
$863$	26506.9	1.04555	0.522773	+	0.852472i	$0.324898\pi$
$864$	0	0
$865$	51340.9	2.01808
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5094.60	−0.198875
$870$	0	0
$871$	−9070.28	−0.352853
$872$	0	0
$873$	−10023.9	−0.388612
$874$	0	0
$875$	−26228.8	−1.01337
$876$	0	0
$877$	−33173.0	−1.27728	−0.638639	−	0.769507i	$-0.720502\pi$
$877$	−33173.0	−1.27728	−0.638639	+	0.769507i	$0.720502\pi$
$878$	0	0
$879$	31718.8	1.21712
$880$	0	0
$881$	7824.81	0.299233	0.149617	−	0.988744i	$-0.452196\pi$
$881$	7824.81	0.299233	0.149617	+	0.988744i	$0.452196\pi$
$882$	0	0
$883$	−45820.6	−1.74630	−0.873152	−	0.487448i	$-0.837928\pi$
$883$	−45820.6	−1.74630	−0.873152	+	0.487448i	$0.837928\pi$
$884$	0	0
$885$	25694.9	0.975961
$886$	0	0
$887$	−12739.5	−0.482246	−0.241123	−	0.970495i	$-0.577516\pi$
$887$	−12739.5	−0.482246	−0.241123	+	0.970495i	$0.577516\pi$
$888$	0	0
$889$	−36276.4	−1.36859
$890$	0	0
$891$	48954.7	1.84068
$892$	0	0
$893$	−36750.9	−1.37718
$894$	0	0
$895$	−3537.18	−0.132106
$896$	0	0
$897$	17163.4	0.638873
$898$	0	0
$899$	31234.6	1.15877
$900$	0	0
$901$	0	0
$902$	0	0
$903$	39468.0	1.45450
$904$	0	0
$905$	27740.8	1.01893
$906$	0	0
$907$	−8724.96	−0.319413	−0.159707	−	0.987165i	$-0.551055\pi$
$907$	−8724.96	−0.319413	−0.159707	+	0.987165i	$0.551055\pi$
$908$	0	0
$909$	−28809.1	−1.05119
$910$	0	0
$911$	−50321.9	−1.83012	−0.915060	−	0.403319i	$-0.867857\pi$
$911$	−50321.9	−1.83012	−0.915060	+	0.403319i	$0.867857\pi$
$912$	0	0
$913$	−78590.0	−2.84880
$914$	0	0
$915$	26151.6	0.944860
$916$	0	0
$917$	32752.6	1.17948
$918$	0	0
$919$	−10055.8	−0.360948	−0.180474	−	0.983580i	$-0.557763\pi$
$919$	−10055.8	−0.360948	−0.180474	+	0.983580i	$0.557763\pi$
$920$	0	0
$921$	16723.1	0.598311
$922$	0	0
$923$	−27.0142	−0.000963362 0
$924$	0	0
$925$	1163.74	0.0413658
$926$	0	0
$927$	−191.759	−0.00679416
$928$	0	0
$929$	−438.911	−0.0155007	−0.00775037	−	0.999970i	$-0.502467\pi$
$929$	−438.911	−0.0155007	−0.00775037	+	0.999970i	$0.502467\pi$
$930$	0	0
$931$	3417.02	0.120288
$932$	0	0
$933$	−18003.2	−0.631725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47589.9	1.65923	0.829613	−	0.558339i	$-0.188561\pi$
$937$	47589.9	1.65923	0.829613	+	0.558339i	$0.188561\pi$
$938$	0	0
$939$	69744.7	2.42389
$940$	0	0
$941$	36715.4	1.27193	0.635966	−	0.771717i	$-0.280602\pi$
$941$	36715.4	1.27193	0.635966	+	0.771717i	$0.280602\pi$
$942$	0	0
$943$	−9708.35	−0.335257
$944$	0	0
$945$	14235.7	0.490040
$946$	0	0
$947$	−42106.3	−1.44485	−0.722424	−	0.691451i	$-0.756972\pi$
$947$	−42106.3	−1.44485	−0.722424	+	0.691451i	$0.756972\pi$
$948$	0	0
$949$	−13021.3	−0.445404
$950$	0	0
$951$	37716.5	1.28606
$952$	0	0
$953$	−44400.0	−1.50919	−0.754595	−	0.656191i	$-0.772166\pi$
$953$	−44400.0	−1.50919	−0.754595	+	0.656191i	$0.772166\pi$
$954$	0	0
$955$	−3336.16	−0.113042
$956$	0	0
$957$	−54708.6	−1.84794
$958$	0	0
$959$	33706.7	1.13498
$960$	0	0
$961$	13125.9	0.440600
$962$	0	0
$963$	29382.1	0.983203
$964$	0	0
$965$	−26975.2	−0.899859
$966$	0	0
$967$	−17278.2	−0.574590	−0.287295	−	0.957842i	$-0.592756\pi$
$967$	−17278.2	−0.574590	−0.287295	+	0.957842i	$0.592756\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5241.74	0.173239	0.0866197	−	0.996241i	$-0.472394\pi$
$971$	5241.74	0.173239	0.0866197	+	0.996241i	$0.472394\pi$
$972$	0	0
$973$	53096.3	1.74942
$974$	0	0
$975$	670.647	0.0220286
$976$	0	0
$977$	39147.0	1.28191	0.640954	−	0.767579i	$-0.278539\pi$
$977$	39147.0	1.28191	0.640954	+	0.767579i	$0.278539\pi$
$978$	0	0
$979$	−9441.50	−0.308224
$980$	0	0
$981$	−28986.6	−0.943397
$982$	0	0
$983$	3737.58	0.121272	0.0606360	−	0.998160i	$-0.480687\pi$
$983$	3737.58	0.121272	0.0606360	+	0.998160i	$0.480687\pi$
$984$	0	0
$985$	−16228.9	−0.524971
$986$	0	0
$987$	35140.0	1.13325
$988$	0	0
$989$	−42160.4	−1.35553
$990$	0	0
$991$	−29632.5	−0.949856	−0.474928	−	0.880025i	$-0.657526\pi$
$991$	−29632.5	−0.949856	−0.474928	+	0.880025i	$0.657526\pi$
$992$	0	0
$993$	10425.4	0.333171
$994$	0	0
$995$	29806.0	0.949662
$996$	0	0
$997$	54812.4	1.74115	0.870575	−	0.492036i	$-0.163747\pi$
$997$	54812.4	1.74115	0.870575	+	0.492036i	$0.163747\pi$
$998$	0	0
$999$	14182.1	0.449152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.r.1.4		24
17.10	odd	16		136.4.n.a.49.1	yes	24
17.12	odd	16		136.4.n.a.25.1	✓	24
17.16	even	2	inner	2312.4.a.r.1.21		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.a.25.1	✓	24	17.12	odd	16
136.4.n.a.49.1	yes	24	17.10	odd	16
2312.4.a.r.1.4		24	1.1	even	1	trivial
2312.4.a.r.1.21		24	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-6.65088 q^{3} +11.4162 q^{5} +19.1986 q^{7} +17.2342 q^{9} +O(q^{10})\)	\(q-6.65088 q^{3} +11.4162 q^{5} +19.1986 q^{7} +17.2342 q^{9} -54.5575 q^{11} -18.9197 q^{13} -75.9278 q^{15} +133.541 q^{19} -127.688 q^{21} +136.398 q^{23} +5.32966 q^{25} +64.9512 q^{27} -150.773 q^{29} -207.164 q^{31} +362.855 q^{33} +219.176 q^{35} +218.351 q^{37} +125.833 q^{39} -71.1766 q^{41} -309.098 q^{43} +196.749 q^{45} -275.202 q^{47} +25.5877 q^{49} +41.9225 q^{53} -622.839 q^{55} -888.168 q^{57} -338.413 q^{59} -344.428 q^{61} +330.873 q^{63} -215.992 q^{65} +479.408 q^{67} -907.168 q^{69} +1.42783 q^{71} +688.236 q^{73} -35.4469 q^{75} -1047.43 q^{77} +93.3804 q^{79} -897.306 q^{81} +1440.50 q^{83} +1002.77 q^{87} +173.056 q^{89} -363.233 q^{91} +1377.82 q^{93} +1524.54 q^{95} -581.630 q^{97} -940.255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Introduction

L-functions

Modular forms

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