LMFDB - Embedded newform 1344.4.p.c.223.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.32
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -3.15485 q^{5} +(-6.11026 + 17.4833i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -3.15485 q^{5} +(-6.11026 + 17.4833i) q^{7} -9.00000 q^{9} -60.9479 q^{11} -59.6460 q^{13} -9.46455i q^{15} +21.4250i q^{17} -95.5205i q^{19} +(-52.4498 - 18.3308i) q^{21} -46.3236i q^{23} -115.047 q^{25} -27.0000i q^{27} -107.590i q^{29} +94.2549 q^{31} -182.844i q^{33} +(19.2770 - 55.1571i) q^{35} +131.650i q^{37} -178.938i q^{39} +283.782i q^{41} +373.255 q^{43} +28.3937 q^{45} +136.224 q^{47} +(-268.330 - 213.655i) q^{49} -64.2751 q^{51} -298.133i q^{53} +192.281 q^{55} +286.561 q^{57} +468.435i q^{59} +563.070 q^{61} +(54.9923 - 157.349i) q^{63} +188.174 q^{65} -160.990 q^{67} +138.971 q^{69} -409.730i q^{71} +930.545i q^{73} -345.141i q^{75} +(372.407 - 1065.57i) q^{77} +442.826i q^{79} +81.0000 q^{81} -190.709i q^{83} -67.5928i q^{85} +322.771 q^{87} +829.404i q^{89} +(364.453 - 1042.81i) q^{91} +282.765i q^{93} +301.353i q^{95} +1030.40i q^{97} +548.531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 288 q^{9}+O(q^{10}) $ 32 * q - 288 * q^9	$ 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) $ 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−3.15485			−0.282178			−0.141089	−	0.989997i	$-0.545060\pi$
$5$	−3.15485			−0.282178			−0.141089	+	0.989997i	$0.545060\pi$
$6$		0			0
$7$	−6.11026	+	17.4833i	−0.329923	+	0.944008i
$7$	−6.11026	+	17.4833i	−0.329923	+	0.944008i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−60.9479			−1.67059			−0.835294	−	0.549803i	$-0.814703\pi$
$11$	−60.9479			−1.67059			−0.835294	+	0.549803i	$0.814703\pi$
$12$		0			0
$13$	−59.6460			−1.27253			−0.636263	−	0.771472i	$-0.719521\pi$
$13$	−59.6460			−1.27253			−0.636263	+	0.771472i	$0.719521\pi$
$14$		0			0
$15$		−	9.46455i		−	0.162916i
$16$		0			0
$17$			21.4250i			0.305667i	0.988252	+	0.152833i	$0.0488398\pi$
$17$			21.4250i			0.305667i	−0.988252	+	0.152833i	$0.951160\pi$
$18$		0			0
$19$		−	95.5205i		−	1.15336i	−0.816969	−	0.576682i	$-0.804347\pi$
$19$		−	95.5205i		−	1.15336i	0.816969	−	0.576682i	$-0.195653\pi$
$20$		0			0
$21$	−52.4498	−	18.3308i	−0.545023	−	0.190481i
$22$		0			0
$23$		−	46.3236i		−	0.419963i	−0.977705	−	0.209981i	$-0.932660\pi$
$23$		−	46.3236i		−	0.419963i	0.977705	−	0.209981i	$-0.0673403\pi$
$24$		0			0
$25$	−115.047			−0.920375
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$		−	107.590i		−	0.688932i	−0.938799	−	0.344466i	$-0.888060\pi$
$29$		−	107.590i		−	0.688932i	0.938799	−	0.344466i	$-0.111940\pi$
$30$		0			0
$31$	94.2549			0.546086			0.273043	−	0.962002i	$-0.411970\pi$
$31$	94.2549			0.546086			0.273043	+	0.962002i	$0.411970\pi$
$32$		0			0
$33$		−	182.844i		−	0.964515i
$34$		0			0
$35$	19.2770	−	55.1571i	0.0930971	−	0.266379i
$36$		0			0
$37$			131.650i			0.584949i	0.956273	+	0.292475i	$0.0944787\pi$
$37$			131.650i			0.584949i	−0.956273	+	0.292475i	$0.905521\pi$
$38$		0			0
$39$		−	178.938i		−	0.734693i
$40$		0			0
$41$			283.782i			1.08096i	0.841358	+	0.540479i	$0.181757\pi$
$41$			283.782i			1.08096i	−0.841358	+	0.540479i	$0.818243\pi$
$42$		0			0
$43$	373.255			1.32374			0.661871	−	0.749618i	$-0.269763\pi$
$43$	373.255			1.32374			0.661871	+	0.749618i	$0.269763\pi$
$44$		0			0
$45$	28.3937			0.0940595
$46$		0			0
$47$	136.224			0.422773			0.211387	−	0.977403i	$-0.432202\pi$
$47$	136.224			0.422773			0.211387	+	0.977403i	$0.432202\pi$
$48$		0			0
$49$	−268.330	−	213.655i	−0.782302	−	0.622900i
$50$		0			0
$51$	−64.2751			−0.176477
$52$		0			0
$53$		−	298.133i		−	0.772674i	−0.922358	−	0.386337i	$-0.873740\pi$
$53$		−	298.133i		−	0.772674i	0.922358	−	0.386337i	$-0.126260\pi$
$54$		0			0
$55$	192.281			0.471404
$56$		0			0
$57$	286.561			0.665895
$58$		0			0
$59$			468.435i			1.03365i	0.856092	+	0.516823i	$0.172885\pi$
$59$			468.435i			1.03365i	−0.856092	+	0.516823i	$0.827115\pi$
$60$		0			0
$61$	563.070			1.18186			0.590932	−	0.806721i	$-0.298760\pi$
$61$	563.070			1.18186			0.590932	+	0.806721i	$0.298760\pi$
$62$		0			0
$63$	54.9923	−	157.349i	0.109974	−	0.314669i
$64$		0			0
$65$	188.174			0.359079
$66$		0			0
$67$	−160.990			−0.293553			−0.146777	−	0.989170i	$-0.546890\pi$
$67$	−160.990			−0.293553			−0.146777	+	0.989170i	$0.546890\pi$
$68$		0			0
$69$	138.971			0.242466
$70$		0			0
$71$		−	409.730i		−	0.684873i	−0.939541	−	0.342437i	$-0.888748\pi$
$71$		−	409.730i		−	0.684873i	0.939541	−	0.342437i	$-0.111252\pi$
$72$		0			0
$73$			930.545i			1.49195i	0.665976	+	0.745973i	$0.268015\pi$
$73$			930.545i			1.49195i	−0.665976	+	0.745973i	$0.731985\pi$
$74$		0			0
$75$		−	345.141i		−	0.531379i
$76$		0			0
$77$	372.407	−	1065.57i	0.551165	−	1.57705i
$78$		0			0
$79$			442.826i			0.630656i	0.948983	+	0.315328i	$0.102115\pi$
$79$			442.826i			0.630656i	−0.948983	+	0.315328i	$0.897885\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	190.709i		−	0.252205i	−0.992017	−	0.126102i	$-0.959753\pi$
$83$		−	190.709i		−	0.252205i	0.992017	−	0.126102i	$-0.0402468\pi$
$84$		0			0
$85$		−	67.5928i		−	0.0862526i
$86$		0			0
$87$	322.771			0.397755
$88$		0			0
$89$			829.404i			0.987828i	0.869511	+	0.493914i	$0.164434\pi$
$89$			829.404i			0.987828i	−0.869511	+	0.493914i	$0.835566\pi$
$90$		0			0
$91$	364.453	−	1042.81i	0.419835	−	1.20127i
$92$		0			0
$93$			282.765i			0.315283i
$94$		0			0
$95$			301.353i			0.325454i
$96$		0			0
$97$			1030.40i			1.07857i	0.842123	+	0.539285i	$0.181306\pi$
$97$			1030.40i			1.07857i	−0.842123	+	0.539285i	$0.818694\pi$
$98$		0			0
$99$	548.531			0.556863
$100$		0			0
$101$	−1784.29			−1.75785			−0.878926	−	0.476958i	$-0.841739\pi$
$101$	−1784.29			−1.75785			−0.878926	+	0.476958i	$0.841739\pi$
$102$		0			0
$103$	288.271			0.275769			0.137885	−	0.990448i	$-0.455970\pi$
$103$	288.271			0.275769			0.137885	+	0.990448i	$0.455970\pi$
$104$		0			0
$105$	165.471	+	57.8309i	0.153794	+	0.0537497i
$106$		0			0
$107$	−292.479			−0.264252			−0.132126	−	0.991233i	$-0.542180\pi$
$107$	−292.479			−0.264252			−0.132126	+	0.991233i	$0.542180\pi$
$108$		0			0
$109$		−	518.765i		−	0.455860i	−0.973678	−	0.227930i	$-0.926804\pi$
$109$		−	518.765i		−	0.455860i	0.973678	−	0.227930i	$-0.0731957\pi$
$110$		0			0
$111$	−394.950			−0.337721
$112$		0			0
$113$	−1101.69			−0.917156			−0.458578	−	0.888654i	$-0.651641\pi$
$113$	−1101.69			−0.917156			−0.458578	+	0.888654i	$0.651641\pi$
$114$		0			0
$115$			146.144i			0.118504i
$116$		0			0
$117$	536.814			0.424175
$118$		0			0
$119$	−374.580	−	130.913i	−0.288552	−	0.100847i
$120$		0			0
$121$	2383.64			1.79087
$122$		0			0
$123$	−851.345			−0.624091
$124$		0			0
$125$	757.312			0.541889
$126$		0			0
$127$		−	1739.50i		−	1.21540i	−0.794168	−	0.607698i	$-0.792093\pi$
$127$		−	1739.50i		−	1.21540i	0.794168	−	0.607698i	$-0.207907\pi$
$128$		0			0
$129$			1119.77i			0.764263i
$130$		0			0
$131$		−	2546.59i		−	1.69845i	−0.528033	−	0.849224i	$-0.677070\pi$
$131$		−	2546.59i		−	1.69845i	0.528033	−	0.849224i	$-0.322930\pi$
$132$		0			0
$133$	1670.01	+	583.655i	1.08878	+	0.380521i
$134$		0			0
$135$			85.1810i			0.0543053i
$136$		0			0
$137$	−197.078			−0.122902			−0.0614509	−	0.998110i	$-0.519573\pi$
$137$	−197.078			−0.122902			−0.0614509	+	0.998110i	$0.519573\pi$
$138$		0			0
$139$			810.877i			0.494803i	0.968913	+	0.247402i	$0.0795767\pi$
$139$			810.877i			0.494803i	−0.968913	+	0.247402i	$0.920423\pi$
$140$		0			0
$141$			408.673i			0.244088i
$142$		0			0
$143$	3635.30			2.12587
$144$		0			0
$145$			339.432i			0.194402i
$146$		0			0
$147$	640.964	−	804.989i	0.359631	−	0.451662i
$148$		0			0
$149$			1838.58i			1.01089i	0.862860	+	0.505443i	$0.168671\pi$
$149$			1838.58i			1.01089i	−0.862860	+	0.505443i	$0.831329\pi$
$150$		0			0
$151$		−	43.6496i		−	0.0235242i	−0.999931	−	0.0117621i	$-0.996256\pi$
$151$		−	43.6496i		−	0.0235242i	0.999931	−	0.0117621i	$-0.00374408\pi$
$152$		0			0
$153$		−	192.825i		−	0.101889i
$154$		0			0
$155$	−297.360			−0.154094
$156$		0			0
$157$	193.518			0.0983721			0.0491860	−	0.998790i	$-0.484337\pi$
$157$	193.518			0.0983721			0.0491860	+	0.998790i	$0.484337\pi$
$158$		0			0
$159$	894.399			0.446103
$160$		0			0
$161$	809.888	+	283.049i	0.396448	+	0.138555i
$162$		0			0
$163$	−1656.29			−0.795893			−0.397946	−	0.917409i	$-0.630277\pi$
$163$	−1656.29			−0.795893			−0.397946	+	0.917409i	$0.630277\pi$
$164$		0			0
$165$			576.844i			0.272165i
$166$		0			0
$167$	4172.57			1.93343			0.966716	−	0.255851i	$-0.0823555\pi$
$167$	4172.57			1.93343			0.966716	+	0.255851i	$0.0823555\pi$
$168$		0			0
$169$	1360.65			0.619322
$170$		0			0
$171$			859.684i			0.384454i
$172$		0			0
$173$	3172.52			1.39423			0.697116	−	0.716958i	$-0.254466\pi$
$173$	3172.52			1.39423			0.697116	+	0.716958i	$0.254466\pi$
$174$		0			0
$175$	702.966	−	2011.40i	0.303653	−	0.868842i
$176$		0			0
$177$	−1405.31			−0.596776
$178$		0			0
$179$	137.757			0.0575221			0.0287611	−	0.999586i	$-0.490844\pi$
$179$	137.757			0.0575221			0.0287611	+	0.999586i	$0.490844\pi$
$180$		0			0
$181$	978.118			0.401674			0.200837	−	0.979625i	$-0.435634\pi$
$181$	978.118			0.401674			0.200837	+	0.979625i	$0.435634\pi$
$182$		0			0
$183$			1689.21i			0.682350i
$184$		0			0
$185$		−	415.336i		−	0.165060i
$186$		0			0
$187$		−	1305.81i		−	0.510644i
$188$		0			0
$189$	472.048	+	164.977i	0.181674	+	0.0634937i
$190$		0			0
$191$		−	4051.60i		−	1.53489i	−0.641116	−	0.767444i	$-0.721528\pi$
$191$		−	4051.60i		−	1.53489i	0.641116	−	0.767444i	$-0.278472\pi$
$192$		0			0
$193$	1175.58			0.438444			0.219222	−	0.975675i	$-0.429648\pi$
$193$	1175.58			0.438444			0.219222	+	0.975675i	$0.429648\pi$
$194$		0			0
$195$			564.523i			0.207315i
$196$		0			0
$197$		−	3303.51i		−	1.19475i	−0.801963	−	0.597374i	$-0.796211\pi$
$197$		−	3303.51i		−	1.19475i	0.801963	−	0.597374i	$-0.203789\pi$
$198$		0			0
$199$	4452.76			1.58617			0.793086	−	0.609110i	$-0.208473\pi$
$199$	4452.76			1.58617			0.793086	+	0.609110i	$0.208473\pi$
$200$		0			0
$201$		−	482.970i		−	0.169483i
$202$		0			0
$203$	1881.03	+	657.405i	0.650358	+	0.227295i
$204$		0			0
$205$		−	895.289i		−	0.305023i
$206$		0			0
$207$			416.912i			0.139988i
$208$		0			0
$209$			5821.77i			1.92680i
$210$		0			0
$211$	−5875.61			−1.91703			−0.958516	−	0.285038i	$-0.907994\pi$
$211$	−5875.61			−1.91703			−0.958516	+	0.285038i	$0.907994\pi$
$212$		0			0
$213$	1229.19			0.395412
$214$		0			0
$215$	−1177.57			−0.373532
$216$		0			0
$217$	−575.921	+	1647.88i	−0.180166	+	0.515510i
$218$		0			0
$219$	−2791.63			−0.861375
$220$		0			0
$221$		−	1277.92i		−	0.388969i
$222$		0			0
$223$	−1909.89			−0.573524			−0.286762	−	0.958002i	$-0.592579\pi$
$223$	−1909.89			−0.573524			−0.286762	+	0.958002i	$0.592579\pi$
$224$		0			0
$225$	1035.42			0.306792
$226$		0			0
$227$		−	4211.95i		−	1.23153i	−0.787931	−	0.615764i	$-0.788848\pi$
$227$		−	4211.95i		−	1.23153i	0.787931	−	0.615764i	$-0.211152\pi$
$228$		0			0
$229$	−135.681			−0.0391532			−0.0195766	−	0.999808i	$-0.506232\pi$
$229$	−135.681			−0.0391532			−0.0195766	+	0.999808i	$0.506232\pi$
$230$		0			0
$231$	3196.70	+	1117.22i	0.910509	+	0.318215i
$232$		0			0
$233$	−613.936			−0.172619			−0.0863096	−	0.996268i	$-0.527507\pi$
$233$	−613.936			−0.172619			−0.0863096	+	0.996268i	$0.527507\pi$
$234$		0			0
$235$	−429.767			−0.119298
$236$		0			0
$237$	−1328.48			−0.364109
$238$		0			0
$239$			4611.45i			1.24807i	0.781395	+	0.624037i	$0.214508\pi$
$239$			4611.45i			1.24807i	−0.781395	+	0.624037i	$0.785492\pi$
$240$		0			0
$241$			4934.31i			1.31887i	0.751763	+	0.659433i	$0.229204\pi$
$241$			4934.31i			1.31887i	−0.751763	+	0.659433i	$0.770796\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	846.540	+	674.048i	0.220749	+	0.175769i
$246$		0			0
$247$			5697.42i			1.46768i
$248$		0			0
$249$	572.126			0.145611
$250$		0			0
$251$		−	3655.89i		−	0.919353i	−0.888086	−	0.459676i	$-0.847965\pi$
$251$		−	3655.89i		−	0.919353i	0.888086	−	0.459676i	$-0.152035\pi$
$252$		0			0
$253$			2823.32i			0.701585i
$254$		0			0
$255$	202.779			0.0497980
$256$		0			0
$257$			4419.08i			1.07259i	0.844032	+	0.536294i	$0.180176\pi$
$257$			4419.08i			1.07259i	−0.844032	+	0.536294i	$0.819824\pi$
$258$		0			0
$259$	−2301.67	−	804.415i	−0.552197	−	0.192988i
$260$		0			0
$261$			968.314i			0.229644i
$262$		0			0
$263$		−	5949.45i		−	1.39490i	−0.716633	−	0.697450i	$-0.754318\pi$
$263$		−	5949.45i		−	1.39490i	0.716633	−	0.697450i	$-0.245682\pi$
$264$		0			0
$265$			940.565i			0.218032i
$266$		0			0
$267$	−2488.21			−0.570323
$268$		0			0
$269$	735.471			0.166701			0.0833503	−	0.996520i	$-0.473438\pi$
$269$	735.471			0.166701			0.0833503	+	0.996520i	$0.473438\pi$
$270$		0			0
$271$	8337.62			1.86891			0.934454	−	0.356084i	$-0.115888\pi$
$271$	8337.62			1.86891			0.934454	+	0.356084i	$0.115888\pi$
$272$		0			0
$273$	3128.42	+	1093.36i	0.693556	+	0.242392i
$274$		0			0
$275$	7011.86			1.53757
$276$		0			0
$277$		−	3922.04i		−	0.850731i	−0.905022	−	0.425366i	$-0.860145\pi$
$277$		−	3922.04i		−	0.850731i	0.905022	−	0.425366i	$-0.139855\pi$
$278$		0			0
$279$	−848.294			−0.182029
$280$		0			0
$281$	6465.12			1.37251			0.686257	−	0.727359i	$-0.259253\pi$
$281$	6465.12			1.37251			0.686257	+	0.727359i	$0.259253\pi$
$282$		0			0
$283$			868.108i			0.182345i	0.995835	+	0.0911726i	$0.0290615\pi$
$283$			868.108i			0.182345i	−0.995835	+	0.0911726i	$0.970939\pi$
$284$		0			0
$285$	−904.059			−0.187901
$286$		0			0
$287$	−4961.43	−	1733.98i	−1.02043	−	0.356633i
$288$		0			0
$289$	4453.97			0.906568
$290$		0			0
$291$	−3091.20			−0.622713
$292$		0			0
$293$	4595.00			0.916186			0.458093	−	0.888904i	$-0.348533\pi$
$293$	4595.00			0.916186			0.458093	+	0.888904i	$0.348533\pi$
$294$		0			0
$295$		−	1477.84i		−	0.291673i
$296$		0			0
$297$			1645.59i			0.321505i
$298$		0			0
$299$			2763.02i			0.534413i
$300$		0			0
$301$	−2280.69	+	6525.73i	−0.436733	+	1.24962i
$302$		0			0
$303$		−	5352.86i		−	1.01490i
$304$		0			0
$305$	−1776.40			−0.333497
$306$		0			0
$307$		−	3217.92i		−	0.598230i	−0.954217	−	0.299115i	$-0.903309\pi$
$307$		−	3217.92i		−	0.598230i	0.954217	−	0.299115i	$-0.0966914\pi$
$308$		0			0
$309$			864.814i			0.159215i
$310$		0			0
$311$	−7883.65			−1.43743			−0.718715	−	0.695305i	$-0.755269\pi$
$311$	−7883.65			−1.43743			−0.718715	+	0.695305i	$0.755269\pi$
$312$		0			0
$313$		−	2670.55i		−	0.482263i	−0.970492	−	0.241131i	$-0.922482\pi$
$313$		−	2670.55i		−	0.482263i	0.970492	−	0.241131i	$-0.0775185\pi$
$314$		0			0
$315$	−173.493	+	496.414i	−0.0310324	+	0.0887929i
$316$		0			0
$317$		−	5005.09i		−	0.886794i	−0.896325	−	0.443397i	$-0.853773\pi$
$317$		−	5005.09i		−	0.886794i	0.896325	−	0.443397i	$-0.146227\pi$
$318$		0			0
$319$			6557.41i			1.15092i
$320$		0			0
$321$		−	877.436i		−	0.152566i
$322$		0			0
$323$	2046.53			0.352545
$324$		0			0
$325$	6862.09			1.17120
$326$		0			0
$327$	1556.30			0.263191
$328$		0			0
$329$	−832.365	+	2381.65i	−0.139483	+	0.399101i
$330$		0			0
$331$	10962.6			1.82042			0.910210	−	0.414148i	$-0.135920\pi$
$331$	10962.6			1.82042			0.910210	+	0.414148i	$0.135920\pi$
$332$		0			0
$333$		−	1184.85i		−	0.194983i
$334$		0			0
$335$	507.900			0.0828344
$336$		0			0
$337$	−10584.5			−1.71090			−0.855448	−	0.517888i	$-0.826718\pi$
$337$	−10584.5			−1.71090			−0.855448	+	0.517888i	$0.826718\pi$
$338$		0			0
$339$		−	3305.08i		−	0.529520i
$340$		0			0
$341$	−5744.63			−0.912285
$342$		0			0
$343$	5374.94	−	3385.79i	0.846121	−	0.532990i
$344$		0			0
$345$	−438.432			−0.0684185
$346$		0			0
$347$	3267.07			0.505434			0.252717	−	0.967540i	$-0.418676\pi$
$347$	3267.07			0.505434			0.252717	+	0.967540i	$0.418676\pi$
$348$		0			0
$349$	5781.37			0.886732			0.443366	−	0.896341i	$-0.353784\pi$
$349$	5781.37			0.886732			0.443366	+	0.896341i	$0.353784\pi$
$350$		0			0
$351$			1610.44i			0.244898i
$352$		0			0
$353$		−	10650.3i		−	1.60583i	−0.596093	−	0.802915i	$-0.703281\pi$
$353$		−	10650.3i		−	1.60583i	0.596093	−	0.802915i	$-0.296719\pi$
$354$		0			0
$355$			1292.64i			0.193256i
$356$		0			0
$357$	392.738	−	1123.74i	0.0582238	−	0.166596i
$358$		0			0
$359$		−	9179.33i		−	1.34949i	−0.738051	−	0.674745i	$-0.764254\pi$
$359$		−	9179.33i		−	1.34949i	0.738051	−	0.674745i	$-0.235746\pi$
$360$		0			0
$361$	−2265.16			−0.330247
$362$		0			0
$363$			7150.93i			1.03396i
$364$		0			0
$365$		−	2935.73i		−	0.420995i
$366$		0			0
$367$	−2199.66			−0.312865			−0.156433	−	0.987689i	$-0.549999\pi$
$367$	−2199.66			−0.312865			−0.156433	+	0.987689i	$0.549999\pi$
$368$		0			0
$369$		−	2554.03i		−	0.360319i
$370$		0			0
$371$	5212.34	+	1821.67i	0.729410	+	0.254923i
$372$		0			0
$373$		−	1961.66i		−	0.272309i	−0.990688	−	0.136154i	$-0.956526\pi$
$373$		−	1961.66i		−	0.272309i	0.990688	−	0.136154i	$-0.0434743\pi$
$374$		0			0
$375$			2271.94i			0.312859i
$376$		0			0
$377$			6417.34i			0.876684i
$378$		0			0
$379$	−4839.03			−0.655842			−0.327921	−	0.944705i	$-0.606348\pi$
$379$	−4839.03			−0.655842			−0.327921	+	0.944705i	$0.606348\pi$
$380$		0			0
$381$	5218.49			0.701709
$382$		0			0
$383$	14701.3			1.96136			0.980678	−	0.195628i	$-0.0626744\pi$
$383$	14701.3			1.96136			0.980678	+	0.195628i	$0.0626744\pi$
$384$		0			0
$385$	−1174.89	+	3361.71i	−0.155527	+	0.445009i
$386$		0			0
$387$	−3359.30			−0.441248
$388$		0			0
$389$		−	15110.1i		−	1.96945i	−0.174126	−	0.984723i	$-0.555710\pi$
$389$		−	15110.1i		−	1.96945i	0.174126	−	0.984723i	$-0.444290\pi$
$390$		0			0
$391$	992.486			0.128369
$392$		0			0
$393$	7639.77			0.980599
$394$		0			0
$395$		−	1397.05i		−	0.177958i
$396$		0			0
$397$	−11309.2			−1.42971			−0.714854	−	0.699274i	$-0.753507\pi$
$397$	−11309.2			−1.42971			−0.714854	+	0.699274i	$0.753507\pi$
$398$		0			0
$399$	−1750.96	+	5010.03i	−0.219694	+	0.628610i
$400$		0			0
$401$	−12823.1			−1.59690			−0.798450	−	0.602061i	$-0.794346\pi$
$401$	−12823.1			−1.59690			−0.798450	+	0.602061i	$0.794346\pi$
$402$		0			0
$403$	−5621.93			−0.694909
$404$		0			0
$405$	−255.543			−0.0313532
$406$		0			0
$407$		−	8023.79i		−	0.977210i
$408$		0			0
$409$		−	3805.30i		−	0.460049i	−0.973185	−	0.230025i	$-0.926119\pi$
$409$		−	3805.30i		−	0.460049i	0.973185	−	0.230025i	$-0.0738807\pi$
$410$		0			0
$411$		−	591.235i		−	0.0709574i
$412$		0			0
$413$	−8189.78	−	2862.26i	−0.975770	−	0.341023i
$414$		0			0
$415$			601.658i			0.0711668i
$416$		0			0
$417$	−2432.63			−0.285675
$418$		0			0
$419$		−	8027.30i		−	0.935941i	−0.883744	−	0.467970i	$-0.844985\pi$
$419$		−	8027.30i		−	0.935941i	0.883744	−	0.467970i	$-0.155015\pi$
$420$		0			0
$421$			16646.2i			1.92705i	0.267622	+	0.963524i	$0.413762\pi$
$421$			16646.2i			1.92705i	−0.267622	+	0.963524i	$0.586238\pi$
$422$		0			0
$423$	−1226.02			−0.140924
$424$		0			0
$425$		−	2464.89i		−	0.281328i
$426$		0			0
$427$	−3440.50	+	9844.30i	−0.389924	+	1.11569i
$428$		0			0
$429$			10905.9i			1.22737i
$430$		0			0
$431$			602.051i			0.0672849i	0.999434	+	0.0336425i	$0.0107107\pi$
$431$			602.051i			0.0672849i	−0.999434	+	0.0336425i	$0.989289\pi$
$432$		0			0
$433$		−	9231.81i		−	1.02460i	−0.858806	−	0.512301i	$-0.828793\pi$
$433$		−	9231.81i		−	1.02460i	0.858806	−	0.512301i	$-0.171207\pi$
$434$		0			0
$435$	−1018.30			−0.112238
$436$		0			0
$437$	−4424.85			−0.484369
$438$		0			0
$439$	−11079.3			−1.20452			−0.602262	−	0.798298i	$-0.705734\pi$
$439$	−11079.3			−1.20452			−0.602262	+	0.798298i	$0.705734\pi$
$440$		0			0
$441$	2414.97	+	1922.89i	0.260767	+	0.207633i
$442$		0			0
$443$	3757.07			0.402943			0.201471	−	0.979494i	$-0.435428\pi$
$443$	3757.07			0.402943			0.201471	+	0.979494i	$0.435428\pi$
$444$		0			0
$445$		−	2616.65i		−	0.278744i
$446$		0			0
$447$	−5515.73			−0.583635
$448$		0			0
$449$	−5309.74			−0.558090			−0.279045	−	0.960278i	$-0.590018\pi$
$449$	−5309.74			−0.558090			−0.279045	+	0.960278i	$0.590018\pi$
$450$		0			0
$451$		−	17295.9i		−	1.80583i
$452$		0			0
$453$	130.949			0.0135817
$454$		0			0
$455$	−1149.79	+	3289.90i	−0.118469	+	0.338974i
$456$		0			0
$457$	17826.5			1.82470			0.912350	−	0.409412i	$-0.134266\pi$
$457$	17826.5			1.82470			0.912350	+	0.409412i	$0.134266\pi$
$458$		0			0
$459$	578.476			0.0588256
$460$		0			0
$461$	−7613.64			−0.769203			−0.384601	−	0.923083i	$-0.625661\pi$
$461$	−7613.64			−0.769203			−0.384601	+	0.923083i	$0.625661\pi$
$462$		0			0
$463$		−	831.086i		−	0.0834208i	−0.999130	−	0.0417104i	$-0.986719\pi$
$463$		−	831.086i		−	0.0834208i	0.999130	−	0.0417104i	$-0.0132807\pi$
$464$		0			0
$465$		−	892.080i		−	0.0889661i
$466$		0			0
$467$		−	11189.4i		−	1.10875i	−0.832267	−	0.554374i	$-0.812958\pi$
$467$		−	11189.4i		−	1.10875i	0.832267	−	0.554374i	$-0.187042\pi$
$468$		0			0
$469$	983.691	−	2814.63i	0.0968500	−	0.277117i
$470$		0			0
$471$			580.554i			0.0567951i
$472$		0			0
$473$	−22749.1			−2.21143
$474$		0			0
$475$			10989.3i			1.06153i
$476$		0			0
$477$			2683.20i			0.257558i
$478$		0			0
$479$	−8503.67			−0.811154			−0.405577	−	0.914061i	$-0.632929\pi$
$479$	−8503.67			−0.811154			−0.405577	+	0.914061i	$0.632929\pi$
$480$		0			0
$481$		−	7852.40i		−	0.744363i
$482$		0			0
$483$	−849.147	+	2429.66i	−0.0799949	+	0.228889i
$484$		0			0
$485$		−	3250.76i		−	0.304349i
$486$		0			0
$487$			1829.08i			0.170192i	0.996373	+	0.0850962i	$0.0271198\pi$
$487$			1829.08i			0.170192i	−0.996373	+	0.0850962i	$0.972880\pi$
$488$		0			0
$489$		−	4968.87i		−	0.459509i
$490$		0			0
$491$	15970.1			1.46786			0.733931	−	0.679224i	$-0.237684\pi$
$491$	15970.1			1.46786			0.733931	+	0.679224i	$0.237684\pi$
$492$		0			0
$493$	2305.13			0.210584
$494$		0			0
$495$	−1730.53			−0.157135
$496$		0			0
$497$	7163.42	+	2503.55i	0.646526	+	0.225955i
$498$		0			0
$499$	1903.84			0.170796			0.0853982	−	0.996347i	$-0.472784\pi$
$499$	1903.84			0.170796			0.0853982	+	0.996347i	$0.472784\pi$
$500$		0			0
$501$			12517.7i			1.11627i
$502$		0			0
$503$	−5674.20			−0.502982			−0.251491	−	0.967860i	$-0.580921\pi$
$503$	−5674.20			−0.502982			−0.251491	+	0.967860i	$0.580921\pi$
$504$		0			0
$505$	5629.16			0.496028
$506$		0			0
$507$			4081.95i			0.357566i
$508$		0			0
$509$	−10624.8			−0.925219			−0.462609	−	0.886562i	$-0.653087\pi$
$509$	−10624.8			−0.925219			−0.462609	+	0.886562i	$0.653087\pi$
$510$		0			0
$511$	−16269.0	−	5685.87i	−1.40841	−	0.492227i
$512$		0			0
$513$	−2579.05			−0.221965
$514$		0			0
$515$	−909.453			−0.0778161
$516$		0			0
$517$	−8302.58			−0.706280
$518$		0			0
$519$			9517.56i			0.804961i
$520$		0			0
$521$		−	695.563i		−	0.0584898i	−0.999572	−	0.0292449i	$-0.990690\pi$
$521$		−	695.563i		−	0.0584898i	0.999572	−	0.0292449i	$-0.00931026\pi$
$522$		0			0
$523$			11831.5i			0.989211i	0.869118	+	0.494605i	$0.164687\pi$
$523$			11831.5i			0.989211i	−0.869118	+	0.494605i	$0.835313\pi$
$524$		0			0
$525$	6034.19	+	2108.90i	0.501626	+	0.175314i
$526$		0			0
$527$			2019.42i			0.166921i
$528$		0			0
$529$	10021.1			0.823631
$530$		0			0
$531$		−	4215.92i		−	0.344549i
$532$		0			0
$533$		−	16926.5i		−	1.37555i
$534$		0			0
$535$	922.726			0.0745662
$536$		0			0
$537$			413.272i			0.0332104i
$538$		0			0
$539$	16354.1	+	13021.8i	1.30690	+	1.04061i
$540$		0			0
$541$			10553.5i			0.838687i	0.907828	+	0.419343i	$0.137740\pi$
$541$			10553.5i			0.838687i	−0.907828	+	0.419343i	$0.862260\pi$
$542$		0			0
$543$			2934.36i			0.231907i
$544$		0			0
$545$			1636.63i			0.128634i
$546$		0			0
$547$	−1001.89			−0.0783136			−0.0391568	−	0.999233i	$-0.512467\pi$
$547$	−1001.89			−0.0783136			−0.0391568	+	0.999233i	$0.512467\pi$
$548$		0			0
$549$	−5067.63			−0.393955
$550$		0			0
$551$	−10277.1			−0.794589
$552$		0			0
$553$	−7742.05	−	2705.78i	−0.595344	−	0.208068i
$554$		0			0
$555$	1246.01			0.0952975
$556$		0			0
$557$		−	12853.2i		−	0.977752i	−0.872353	−	0.488876i	$-0.837407\pi$
$557$		−	12853.2i		−	0.977752i	0.872353	−	0.488876i	$-0.162593\pi$
$558$		0			0
$559$	−22263.2			−1.68450
$560$		0			0
$561$	3917.43			0.294820
$562$		0			0
$563$			10159.5i			0.760518i	0.924880	+	0.380259i	$0.124165\pi$
$563$			10159.5i			0.760518i	−0.924880	+	0.380259i	$0.875835\pi$
$564$		0			0
$565$	3475.68			0.258802
$566$		0			0
$567$	−494.931	+	1416.14i	−0.0366581	+	0.104890i
$568$		0			0
$569$	−9104.92			−0.670823			−0.335411	−	0.942072i	$-0.608875\pi$
$569$	−9104.92			−0.670823			−0.335411	+	0.942072i	$0.608875\pi$
$570$		0			0
$571$	−21714.0			−1.59143			−0.795713	−	0.605674i	$-0.792903\pi$
$571$	−21714.0			−1.59143			−0.795713	+	0.605674i	$0.792903\pi$
$572$		0			0
$573$	12154.8			0.886168
$574$		0			0
$575$			5329.39i			0.386523i
$576$		0			0
$577$		−	12878.6i		−	0.929191i	−0.885523	−	0.464595i	$-0.846200\pi$
$577$		−	12878.6i		−	0.929191i	0.885523	−	0.464595i	$-0.153800\pi$
$578$		0			0
$579$			3526.73i			0.253136i
$580$		0			0
$581$	3334.21	+	1165.28i	0.238083	+	0.0832081i
$582$		0			0
$583$			18170.6i			1.29082i
$584$		0			0
$585$	−1693.57			−0.119693
$586$		0			0
$587$			19024.7i			1.33771i	0.743394	+	0.668854i	$0.233215\pi$
$587$			19024.7i			1.33771i	−0.743394	+	0.668854i	$0.766785\pi$
$588$		0			0
$589$		−	9003.27i		−	0.629836i
$590$		0			0
$591$	9910.53			0.689788
$592$		0			0
$593$		−	25477.9i		−	1.76434i	−0.470934	−	0.882169i	$-0.656083\pi$
$593$		−	25477.9i		−	1.76434i	0.470934	−	0.882169i	$-0.343917\pi$
$594$		0			0
$595$	1181.74	+	413.010i	0.0814232	+	0.0284567i
$596$		0			0
$597$			13358.3i			0.915776i
$598$		0			0
$599$			12475.5i			0.850976i	0.904964	+	0.425488i	$0.139898\pi$
$599$			12475.5i			0.850976i	−0.904964	+	0.425488i	$0.860102\pi$
$600$		0			0
$601$			947.642i			0.0643180i	0.999483	+	0.0321590i	$0.0102383\pi$
$601$			947.642i			0.0643180i	−0.999483	+	0.0321590i	$0.989762\pi$
$602$		0			0
$603$	1448.91			0.0978511
$604$		0			0
$605$	−7520.04			−0.505344
$606$		0			0
$607$	17586.1			1.17594			0.587971	−	0.808882i	$-0.299927\pi$
$607$	17586.1			1.17594			0.587971	+	0.808882i	$0.299927\pi$
$608$		0			0
$609$	−1972.22	+	5643.10i	−0.131229	+	0.375484i
$610$		0			0
$611$	−8125.24			−0.537990
$612$		0			0
$613$			9936.44i			0.654697i	0.944904	+	0.327348i	$0.106155\pi$
$613$			9936.44i			0.654697i	−0.944904	+	0.327348i	$0.893845\pi$
$614$		0			0
$615$	2685.87			0.176105
$616$		0			0
$617$	−23950.9			−1.56277			−0.781383	−	0.624051i	$-0.785486\pi$
$617$	−23950.9			−1.56277			−0.781383	+	0.624051i	$0.785486\pi$
$618$		0			0
$619$			26145.1i			1.69767i	0.528654	+	0.848837i	$0.322697\pi$
$619$			26145.1i			1.69767i	−0.528654	+	0.848837i	$0.677303\pi$
$620$		0			0
$621$	−1250.74			−0.0808218
$622$		0			0
$623$	−14500.7	−	5067.87i	−0.932517	−	0.325907i
$624$		0			0
$625$	11991.7			0.767466
$626$		0			0
$627$	−17465.3			−1.11244
$628$		0			0
$629$	−2820.61			−0.178800
$630$		0			0
$631$			6978.35i			0.440260i	0.975471	+	0.220130i	$0.0706481\pi$
$631$			6978.35i			0.440260i	−0.975471	+	0.220130i	$0.929352\pi$
$632$		0			0
$633$		−	17626.8i		−	1.10680i
$634$		0			0
$635$			5487.85i			0.342958i
$636$		0			0
$637$	16004.8	+	12743.6i	0.995499	+	0.792656i
$638$		0			0
$639$			3687.57i			0.228291i
$640$		0			0
$641$	9353.09			0.576326			0.288163	−	0.957581i	$-0.406956\pi$
$641$	9353.09			0.576326			0.288163	+	0.957581i	$0.406956\pi$
$642$		0			0
$643$			11216.2i			0.687906i	0.938987	+	0.343953i	$0.111766\pi$
$643$			11216.2i			0.687906i	−0.938987	+	0.343953i	$0.888234\pi$
$644$		0			0
$645$		−	3532.70i		−	0.215659i
$646$		0			0
$647$	−8875.46			−0.539305			−0.269653	−	0.962958i	$-0.586909\pi$
$647$	−8875.46			−0.539305			−0.269653	+	0.962958i	$0.586909\pi$
$648$		0			0
$649$		−	28550.1i		−	1.72680i
$650$		0			0
$651$	−4943.65	−	1727.76i	−0.297630	−	0.104019i
$652$		0			0
$653$			12.5387i			0.000751419i	1.00000		0.000375710i	$0.000119592\pi$
$653$			12.5387i			0.000751419i	−1.00000		0.000375710i	$0.999880\pi$
$654$		0			0
$655$			8034.11i			0.479265i
$656$		0			0
$657$		−	8374.90i		−	0.497315i
$658$		0			0
$659$	127.456			0.00753414			0.00376707	−	0.999993i	$-0.498801\pi$
$659$	127.456			0.00753414			0.00376707	+	0.999993i	$0.498801\pi$
$660$		0			0
$661$	−18113.0			−1.06583			−0.532916	−	0.846168i	$-0.678904\pi$
$661$	−18113.0			−1.06583			−0.532916	+	0.846168i	$0.678904\pi$
$662$		0			0
$663$	3833.76			0.224571
$664$		0			0
$665$	−5268.63	−	1841.34i	−0.307231	−	0.107375i
$666$		0			0
$667$	−4983.98			−0.289326
$668$		0			0
$669$		−	5729.67i		−	0.331124i
$670$		0			0
$671$	−34317.9			−1.97441
$672$		0			0
$673$	−11689.6			−0.669540			−0.334770	−	0.942300i	$-0.608659\pi$
$673$	−11689.6			−0.669540			−0.334770	+	0.942300i	$0.608659\pi$
$674$		0			0
$675$			3106.27i			0.177126i
$676$		0			0
$677$	−16101.4			−0.914070			−0.457035	−	0.889449i	$-0.651089\pi$
$677$	−16101.4			−0.914070			−0.457035	+	0.889449i	$0.651089\pi$
$678$		0			0
$679$	−18014.8	−	6296.02i	−1.01818	−	0.355845i
$680$		0			0
$681$	12635.8			0.711023
$682$		0			0
$683$	23914.0			1.33974			0.669870	−	0.742478i	$-0.266350\pi$
$683$	23914.0			1.33974			0.669870	+	0.742478i	$0.266350\pi$
$684$		0			0
$685$	621.753			0.0346802
$686$		0			0
$687$		−	407.044i		−	0.0226051i
$688$		0			0
$689$			17782.4i			0.983247i
$690$		0			0
$691$		−	27330.4i		−	1.50463i	−0.658805	−	0.752314i	$-0.728938\pi$
$691$		−	27330.4i		−	1.50463i	0.658805	−	0.752314i	$-0.271062\pi$
$692$		0			0
$693$	−3351.66	+	9590.11i	−0.183722	+	0.525683i
$694$		0			0
$695$		−	2558.20i		−	0.139623i
$696$		0			0
$697$	−6080.04			−0.330413
$698$		0			0
$699$		−	1841.81i		−	0.0996617i
$700$		0			0
$701$		−	13641.6i		−	0.735002i	−0.930023	−	0.367501i	$-0.880213\pi$
$701$		−	13641.6i		−	0.735002i	0.930023	−	0.367501i	$-0.119787\pi$
$702$		0			0
$703$	12575.3			0.674659
$704$		0			0
$705$		−	1289.30i		−	0.0688765i
$706$		0			0
$707$	10902.4	−	31195.1i	0.579956	−	1.65943i
$708$		0			0
$709$			25345.6i			1.34256i	0.741205	+	0.671278i	$0.234254\pi$
$709$			25345.6i			1.34256i	−0.741205	+	0.671278i	$0.765746\pi$
$710$		0			0
$711$		−	3985.43i		−	0.210219i
$712$		0			0
$713$		−	4366.23i		−	0.229336i
$714$		0			0
$715$	−11468.8			−0.599874
$716$		0			0
$717$	−13834.3			−0.720576
$718$		0			0
$719$	13143.0			0.681715			0.340857	−	0.940115i	$-0.389283\pi$
$719$	13143.0			0.681715			0.340857	+	0.940115i	$0.389283\pi$
$720$		0			0
$721$	−1761.41	+	5039.93i	−0.0909826	+	0.260328i
$722$		0			0
$723$	−14802.9			−0.761447
$724$		0			0
$725$			12377.9i			0.634076i
$726$		0			0
$727$	7313.19			0.373083			0.186541	−	0.982447i	$-0.440272\pi$
$727$	7313.19			0.373083			0.186541	+	0.982447i	$0.440272\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			7997.02i			0.404624i
$732$		0			0
$733$	−2577.28			−0.129869			−0.0649344	−	0.997890i	$-0.520684\pi$
$733$	−2577.28			−0.129869			−0.0649344	+	0.997890i	$0.520684\pi$
$734$		0			0
$735$	−2022.15	+	2539.62i	−0.101480	+	0.127449i
$736$		0			0
$737$	9812.00			0.490407
$738$		0			0
$739$	5495.97			0.273576			0.136788	−	0.990600i	$-0.456322\pi$
$739$	5495.97			0.273576			0.136788	+	0.990600i	$0.456322\pi$
$740$		0			0
$741$	−17092.3			−0.847368
$742$		0			0
$743$		−	16818.0i		−	0.830409i	−0.909728	−	0.415204i	$-0.863710\pi$
$743$		−	16818.0i		−	0.830409i	0.909728	−	0.415204i	$-0.136290\pi$
$744$		0			0
$745$		−	5800.43i		−	0.285250i
$746$		0			0
$747$			1716.38i			0.0840683i
$748$		0			0
$749$	1787.12	−	5113.48i	0.0871828	−	0.249456i
$750$		0			0
$751$			10217.3i			0.496450i	0.968702	+	0.248225i	$0.0798472\pi$
$751$			10217.3i			0.496450i	−0.968702	+	0.248225i	$0.920153\pi$
$752$		0			0
$753$	10967.7			0.530789
$754$		0			0
$755$			137.708i			0.00663802i
$756$		0			0
$757$			2257.96i			0.108411i	0.998530	+	0.0542055i	$0.0172626\pi$
$757$			2257.96i			0.108411i	−0.998530	+	0.0542055i	$0.982737\pi$
$758$		0			0
$759$	−8469.97			−0.405060
$760$		0			0
$761$		−	27933.8i		−	1.33061i	−0.746570	−	0.665307i	$-0.768301\pi$
$761$		−	27933.8i		−	1.33061i	0.746570	−	0.665307i	$-0.231699\pi$
$762$		0			0
$763$	9069.71	+	3169.79i	0.430335	+	0.150399i
$764$		0			0
$765$			608.336i			0.0287509i
$766$		0			0
$767$		−	27940.3i		−	1.31534i
$768$		0			0
$769$			33790.5i			1.58455i	0.610166	+	0.792274i	$0.291103\pi$
$769$			33790.5i			1.58455i	−0.610166	+	0.792274i	$0.708897\pi$
$770$		0			0
$771$	−13257.3			−0.619259
$772$		0			0
$773$	−18955.5			−0.881996			−0.440998	−	0.897508i	$-0.645375\pi$
$773$	−18955.5			−0.881996			−0.440998	+	0.897508i	$0.645375\pi$
$774$		0			0
$775$	−10843.7			−0.502604
$776$		0			0
$777$	2413.25	−	6905.02i	0.111422	−	0.318811i
$778$		0			0
$779$	27107.0			1.24674
$780$		0			0
$781$			24972.2i			1.14414i
$782$		0			0
$783$	−2904.94			−0.132585
$784$		0			0
$785$	−610.520			−0.0277585
$786$		0			0
$787$		−	3001.54i		−	0.135951i	−0.997687	−	0.0679754i	$-0.978346\pi$
$787$		−	3001.54i		−	0.135951i	0.997687	−	0.0679754i	$-0.0216539\pi$
$788$		0			0
$789$	17848.3			0.805346
$790$		0			0
$791$	6731.63	−	19261.2i	0.302591	−	0.865803i
$792$		0			0
$793$	−33584.9			−1.50395
$794$		0			0
$795$	−2821.69			−0.125881
$796$		0			0
$797$	14068.6			0.625265			0.312633	−	0.949874i	$-0.398789\pi$
$797$	14068.6			0.625265			0.312633	+	0.949874i	$0.398789\pi$
$798$		0			0
$799$			2918.61i			0.129228i
$800$		0			0
$801$		−	7464.64i		−	0.329276i
$802$		0			0
$803$		−	56714.7i		−	2.49243i
$804$		0			0
$805$	−2555.08	−	892.978i	−0.111869	−	0.0390973i
$806$		0			0
$807$			2206.41i			0.0962446i
$808$		0			0
$809$	−4579.59			−0.199023			−0.0995117	−	0.995036i	$-0.531728\pi$
$809$	−4579.59			−0.199023			−0.0995117	+	0.995036i	$0.531728\pi$
$810$		0			0
$811$			28588.1i			1.23781i	0.785465	+	0.618906i	$0.212424\pi$
$811$			28588.1i			1.23781i	−0.785465	+	0.618906i	$0.787576\pi$
$812$		0			0
$813$			25012.8i			1.07901i
$814$		0			0
$815$	5225.34			0.224584
$816$		0			0
$817$		−	35653.5i		−	1.52676i
$818$		0			0
$819$	−3280.07	+	9385.27i	−0.139945	+	0.400425i
$820$		0			0
$821$		−	13834.8i		−	0.588107i	−0.955789	−	0.294054i	$-0.904996\pi$
$821$		−	13834.8i		−	0.588107i	0.955789	−	0.294054i	$-0.0950045\pi$
$822$		0			0
$823$			33259.1i			1.40867i	0.709866	+	0.704337i	$0.248756\pi$
$823$			33259.1i			1.40867i	−0.709866	+	0.704337i	$0.751244\pi$
$824$		0			0
$825$			21035.6i			0.887716i
$826$		0			0
$827$	17593.0			0.739744			0.369872	−	0.929083i	$-0.379402\pi$
$827$	17593.0			0.739744			0.369872	+	0.929083i	$0.379402\pi$
$828$		0			0
$829$	−10912.0			−0.457164			−0.228582	−	0.973525i	$-0.573409\pi$
$829$	−10912.0			−0.457164			−0.228582	+	0.973525i	$0.573409\pi$
$830$		0			0
$831$	11766.1			0.491170
$832$		0			0
$833$	4577.56	−	5748.97i	0.190400	−	0.239124i
$834$		0			0
$835$	−13163.8			−0.545573
$836$		0			0
$837$		−	2544.88i		−	0.105094i
$838$		0			0
$839$	−24129.6			−0.992905			−0.496452	−	0.868064i	$-0.665364\pi$
$839$	−24129.6			−0.992905			−0.496452	+	0.868064i	$0.665364\pi$
$840$		0			0
$841$	12813.3			0.525372
$842$		0			0
$843$			19395.3i			0.792421i
$844$		0			0
$845$	−4292.65			−0.174759
$846$		0			0
$847$	−14564.7	+	41673.9i	−0.590848	+	1.69059i
$848$		0			0
$849$	−2604.32			−0.105277
$850$		0			0
$851$	6098.50			0.245657
$852$		0			0
$853$	15204.6			0.610310			0.305155	−	0.952303i	$-0.401292\pi$
$853$	15204.6			0.610310			0.305155	+	0.952303i	$0.401292\pi$
$854$		0			0
$855$		−	2712.18i		−	0.108485i
$856$		0			0
$857$			13467.4i			0.536799i	0.963308	+	0.268400i	$0.0864948\pi$
$857$			13467.4i			0.536799i	−0.963308	+	0.268400i	$0.913505\pi$
$858$		0			0
$859$			796.377i			0.0316322i	0.999875	+	0.0158161i	$0.00503463\pi$
$859$			796.377i			0.0316322i	−0.999875	+	0.0158161i	$0.994965\pi$
$860$		0			0
$861$	5201.94	−	14884.3i	0.205902	−	0.589147i
$862$		0			0
$863$		−	33680.2i		−	1.32849i	−0.747514	−	0.664246i	$-0.768753\pi$
$863$		−	33680.2i		−	1.32849i	0.747514	−	0.664246i	$-0.231247\pi$
$864$		0			0
$865$	−10008.8			−0.393422
$866$		0			0
$867$			13361.9i			0.523407i
$868$		0			0
$869$		−	26989.3i		−	1.05357i
$870$		0			0
$871$	9602.42			0.373554
$872$		0			0
$873$		−	9273.61i		−	0.359524i
$874$		0			0
$875$	−4627.37	+	13240.3i	−0.178781	+	0.511547i
$876$		0			0
$877$		−	11140.6i		−	0.428954i	−0.976729	−	0.214477i	$-0.931195\pi$
$877$		−	11140.6i		−	0.428954i	0.976729	−	0.214477i	$-0.0688047\pi$
$878$		0			0
$879$			13785.0i			0.528960i
$880$		0			0
$881$			2372.73i			0.0907370i	0.998970	+	0.0453685i	$0.0144462\pi$
$881$			2372.73i			0.0907370i	−0.998970	+	0.0453685i	$0.985554\pi$
$882$		0			0
$883$	17403.2			0.663267			0.331634	−	0.943408i	$-0.392400\pi$
$883$	17403.2			0.663267			0.331634	+	0.943408i	$0.392400\pi$
$884$		0			0
$885$	4433.53			0.168397
$886$		0			0
$887$	31804.3			1.20393			0.601965	−	0.798523i	$-0.294385\pi$
$887$	31804.3			1.20393			0.601965	+	0.798523i	$0.294385\pi$
$888$		0			0
$889$	30412.1	+	10628.8i	1.14734	+	0.400987i
$890$		0			0
$891$	−4936.78			−0.185621
$892$		0			0
$893$		−	13012.2i		−	0.487611i
$894$		0			0
$895$	−434.603			−0.0162315
$896$		0			0
$897$	−8289.06			−0.308544
$898$		0			0
$899$		−	10140.9i		−	0.376217i
$900$		0			0
$901$	6387.51			0.236181
$902$		0			0
$903$	−19577.2	−	6842.06i	−0.721470	−	0.252148i
$904$		0			0
$905$	−3085.82			−0.113344
$906$		0			0
$907$	1321.37			0.0483741			0.0241871	−	0.999707i	$-0.492300\pi$
$907$	1321.37			0.0483741			0.0241871	+	0.999707i	$0.492300\pi$
$908$		0			0
$909$	16058.6			0.585951
$910$		0			0
$911$			15070.8i			0.548098i	0.961716	+	0.274049i	$0.0883631\pi$
$911$			15070.8i			0.548098i	−0.961716	+	0.274049i	$0.911637\pi$
$912$		0			0
$913$			11623.3i			0.421330i
$914$		0			0
$915$		−	5329.20i		−	0.192544i
$916$		0			0
$917$	44522.7	+	15560.3i	1.60335	+	0.560357i
$918$		0			0
$919$		−	42833.7i		−	1.53749i	−0.639556	−	0.768744i	$-0.720882\pi$
$919$		−	42833.7i		−	1.53749i	0.639556	−	0.768744i	$-0.279118\pi$
$920$		0			0
$921$	9653.77			0.345388
$922$		0			0
$923$			24438.8i			0.871519i
$924$		0			0
$925$		−	15145.9i		−	0.538373i
$926$		0			0
$927$	−2594.44			−0.0919231
$928$		0			0
$929$		−	19434.7i		−	0.686362i	−0.939269	−	0.343181i	$-0.888496\pi$
$929$		−	19434.7i		−	0.686362i	0.939269	−	0.343181i	$-0.111504\pi$
$930$		0			0
$931$	−20408.4	+	25631.0i	−0.718429	+	0.902278i
$932$		0			0
$933$		−	23651.0i		−	0.829901i
$934$		0			0
$935$			4119.64i			0.144093i
$936$		0			0
$937$			42125.9i			1.46872i	0.678759	+	0.734361i	$0.262518\pi$
$937$			42125.9i			1.46872i	−0.678759	+	0.734361i	$0.737482\pi$
$938$		0			0
$939$	8011.64			0.278435
$940$		0			0
$941$	−1325.03			−0.0459030			−0.0229515	−	0.999737i	$-0.507306\pi$
$941$	−1325.03			−0.0459030			−0.0229515	+	0.999737i	$0.507306\pi$
$942$		0			0
$943$	13145.8			0.453962
$944$		0			0
$945$	−1489.24	−	520.478i	−0.0512646	−	0.0179166i
$946$		0			0
$947$	42522.6			1.45913			0.729566	−	0.683910i	$-0.239722\pi$
$947$	42522.6			1.45913			0.729566	+	0.683910i	$0.239722\pi$
$948$		0			0
$949$		−	55503.3i		−	1.89854i
$950$		0			0
$951$	15015.3			0.511991
$952$		0			0
$953$	−30507.0			−1.03696			−0.518478	−	0.855091i	$-0.673501\pi$
$953$	−30507.0			−1.03696			−0.518478	+	0.855091i	$0.673501\pi$
$954$		0			0
$955$			12782.2i			0.433112i
$956$		0			0
$957$	−19672.2			−0.664486
$958$		0			0
$959$	1204.20	−	3445.57i	0.0405481	−	0.116020i
$960$		0			0
$961$	−20907.0			−0.701790
$962$		0			0
$963$	2632.31			0.0880840
$964$		0			0
$965$	−3708.77			−0.123720
$966$		0			0
$967$		−	15544.9i		−	0.516950i	−0.966018	−	0.258475i	$-0.916780\pi$
$967$		−	15544.9i		−	0.516950i	0.966018	−	0.258475i	$-0.0832199\pi$
$968$		0			0
$969$			6139.59i			0.203542i
$970$		0			0
$971$			939.518i			0.0310510i	0.999879	+	0.0155255i	$0.00494213\pi$
$971$			939.518i			0.0310510i	−0.999879	+	0.0155255i	$0.995058\pi$
$972$		0			0
$973$	−14176.8	−	4954.67i	−0.467098	−	0.163247i
$974$		0			0
$975$			20586.3i			0.676193i
$976$		0			0
$977$	7969.39			0.260966			0.130483	−	0.991451i	$-0.458347\pi$
$977$	7969.39			0.260966			0.130483	+	0.991451i	$0.458347\pi$
$978$		0			0
$979$		−	50550.4i		−	1.65025i
$980$		0			0
$981$			4668.89i			0.151953i
$982$		0			0
$983$	35911.0			1.16519			0.582595	−	0.812762i	$-0.302037\pi$
$983$	35911.0			1.16519			0.582595	+	0.812762i	$0.302037\pi$
$984$		0			0
$985$			10422.1i			0.337132i
$986$		0			0
$987$	−7144.94	−	2497.10i	−0.230421	−	0.0805303i
$988$		0			0
$989$		−	17290.5i		−	0.555922i
$990$		0			0
$991$		−	29077.2i		−	0.932055i	−0.884770	−	0.466027i	$-0.845685\pi$
$991$		−	29077.2i		−	0.932055i	0.884770	−	0.466027i	$-0.154315\pi$
$992$		0			0
$993$			32887.8i			1.05102i
$994$		0			0
$995$	−14047.8			−0.447583
$996$		0			0
$997$	18798.3			0.597141			0.298570	−	0.954388i	$-0.403490\pi$
$997$	18798.3			0.597141			0.298570	+	0.954388i	$0.403490\pi$
$998$		0			0
$999$	3554.55			0.112574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.c.223.32	yes	32
4.3	odd	2	inner	1344.4.p.c.223.19	✓	32
7.6	odd	2		1344.4.p.d.223.3	yes	32
8.3	odd	2		1344.4.p.d.223.4	yes	32
8.5	even	2		1344.4.p.d.223.21	yes	32
28.27	even	2		1344.4.p.d.223.22	yes	32
56.13	odd	2	inner	1344.4.p.c.223.20	yes	32
56.27	even	2	inner	1344.4.p.c.223.31	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.19	✓	32	4.3	odd	2	inner
1344.4.p.c.223.20	yes	32	56.13	odd	2	inner
1344.4.p.c.223.31	yes	32	56.27	even	2	inner
1344.4.p.c.223.32	yes	32	1.1	even	1	trivial
1344.4.p.d.223.3	yes	32	7.6	odd	2
1344.4.p.d.223.4	yes	32	8.3	odd	2
1344.4.p.d.223.21	yes	32	8.5	even	2
1344.4.p.d.223.22	yes	32	28.27	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -3.15485 q^{5} +(-6.11026 + 17.4833i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -3.15485 q^{5} +(-6.11026 + 17.4833i) q^{7} -9.00000 q^{9} -60.9479 q^{11} -59.6460 q^{13} -9.46455i q^{15} +21.4250i q^{17} -95.5205i q^{19} +(-52.4498 - 18.3308i) q^{21} -46.3236i q^{23} -115.047 q^{25} -27.0000i q^{27} -107.590i q^{29} +94.2549 q^{31} -182.844i q^{33} +(19.2770 - 55.1571i) q^{35} +131.650i q^{37} -178.938i q^{39} +283.782i q^{41} +373.255 q^{43} +28.3937 q^{45} +136.224 q^{47} +(-268.330 - 213.655i) q^{49} -64.2751 q^{51} -298.133i q^{53} +192.281 q^{55} +286.561 q^{57} +468.435i q^{59} +563.070 q^{61} +(54.9923 - 157.349i) q^{63} +188.174 q^{65} -160.990 q^{67} +138.971 q^{69} -409.730i q^{71} +930.545i q^{73} -345.141i q^{75} +(372.407 - 1065.57i) q^{77} +442.826i q^{79} +81.0000 q^{81} -190.709i q^{83} -67.5928i q^{85} +322.771 q^{87} +829.404i q^{89} +(364.453 - 1042.81i) q^{91} +282.765i q^{93} +301.353i q^{95} +1030.40i q^{97} +548.531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 32 * q - 288 * q^9	\( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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