LMFDB - Embedded newform 2400.3.p.b.1999.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(1999,2400)] 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("2400.1999"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,-96,0,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.14
Character	$\chi$	$=$	2400.1999
Dual form			2400.3.p.b.1999.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -7.58970 q^{7} -3.00000 q^{9} -16.7055 q^{11} +10.7512 q^{13} -8.61276i q^{17} +35.6616 q^{19} -13.1457i q^{21} -11.2913 q^{23} -5.19615i q^{27} -0.589229i q^{29} -45.0411i q^{31} -28.9347i q^{33} +36.6203 q^{37} +18.6216i q^{39} -2.44967 q^{41} +7.91524i q^{43} -44.3755 q^{47} +8.60348 q^{49} +14.9177 q^{51} +73.3176 q^{53} +61.7677i q^{57} -89.7651 q^{59} -98.7987i q^{61} +22.7691 q^{63} +89.5013i q^{67} -19.5571i q^{69} +90.0121i q^{71} +68.0795i q^{73} +126.790 q^{77} +36.2647i q^{79} +9.00000 q^{81} +95.5904i q^{83} +1.02058 q^{87} +7.89970 q^{89} -81.5982 q^{91} +78.0135 q^{93} +137.133i q^{97} +50.1164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 96 q^{9} - 128 q^{11} - 64 q^{19} + 160 q^{49} - 192 q^{51} + 256 q^{59} + 288 q^{81} + 384 q^{91} + 384 q^{99}+O(q^{100}) $ 32 * q - 96 * q^9 - 128 * q^11 - 64 * q^19 + 160 * q^49 - 192 * q^51 + 256 * q^59 + 288 * q^81 + 384 * q^91 + 384 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.58970	−1.08424	−0.542121	−	0.840300i	$-0.682379\pi$
$7$	−7.58970	−1.08424	−0.542121	+	0.840300i	$0.682379\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−16.7055	−1.51868	−0.759340	−	0.650694i	$-0.774478\pi$
$11$	−16.7055	−1.51868	−0.759340	+	0.650694i	$0.774478\pi$
$12$	0	0
$13$	10.7512	0.827014	0.413507	−	0.910501i	$-0.364304\pi$
$13$	10.7512	0.827014	0.413507	+	0.910501i	$0.364304\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 8.61276i	− 0.506633i	−0.967383	−	0.253316i	$-0.918479\pi$
$17$	− 8.61276i	− 0.506633i	0.967383	−	0.253316i	$-0.0815214\pi$
$18$	0	0
$19$	35.6616	1.87693	0.938464	−	0.345378i	$-0.112249\pi$
$19$	35.6616	1.87693	0.938464	+	0.345378i	$0.112249\pi$
$20$	0	0
$21$	− 13.1457i	− 0.625988i
$22$	0	0
$23$	−11.2913	−0.490925	−0.245462	−	0.969406i	$-0.578940\pi$
$23$	−11.2913	−0.490925	−0.245462	+	0.969406i	$0.578940\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	− 0.589229i	− 0.0203183i	−0.999948	−	0.0101591i	$-0.996766\pi$
$29$	− 0.589229i	− 0.0203183i	0.999948	−	0.0101591i	$-0.00323381\pi$
$30$	0	0
$31$	− 45.0411i	− 1.45294i	−0.687198	−	0.726470i	$-0.741160\pi$
$31$	− 45.0411i	− 1.45294i	0.687198	−	0.726470i	$-0.258840\pi$
$32$	0	0
$33$	− 28.9347i	− 0.876810i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	36.6203	0.989738	0.494869	−	0.868968i	$-0.335216\pi$
$37$	36.6203	0.989738	0.494869	+	0.868968i	$0.335216\pi$
$38$	0	0
$39$	18.6216i	0.477477i
$40$	0	0
$41$	−2.44967	−0.0597481	−0.0298740	−	0.999554i	$-0.509511\pi$
$41$	−2.44967	−0.0597481	−0.0298740	+	0.999554i	$0.509511\pi$
$42$	0	0
$43$	7.91524i	0.184075i	0.995756	+	0.0920377i	$0.0293380\pi$
$43$	7.91524i	0.184075i	−0.995756	+	0.0920377i	$0.970662\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−44.3755	−0.944160	−0.472080	−	0.881556i	$-0.656497\pi$
$47$	−44.3755	−0.944160	−0.472080	+	0.881556i	$0.656497\pi$
$48$	0	0
$49$	8.60348	0.175581
$50$	0	0
$51$	14.9177	0.292505
$52$	0	0
$53$	73.3176	1.38335	0.691675	−	0.722209i	$-0.256873\pi$
$53$	73.3176	1.38335	0.691675	+	0.722209i	$0.256873\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	61.7677i	1.08364i
$58$	0	0
$59$	−89.7651	−1.52144	−0.760721	−	0.649079i	$-0.775155\pi$
$59$	−89.7651	−1.52144	−0.760721	+	0.649079i	$0.775155\pi$
$60$	0	0
$61$	− 98.7987i	− 1.61965i	−0.586671	−	0.809826i	$-0.699562\pi$
$61$	− 98.7987i	− 1.61965i	0.586671	−	0.809826i	$-0.300438\pi$
$62$	0	0
$63$	22.7691	0.361414
$64$	0	0
$65$	0	0
$66$	0	0
$67$	89.5013i	1.33584i	0.744233	+	0.667920i	$0.232815\pi$
$67$	89.5013i	1.33584i	−0.744233	+	0.667920i	$0.767185\pi$
$68$	0	0
$69$	− 19.5571i	− 0.283436i
$70$	0	0
$71$	90.0121i	1.26778i	0.773425	+	0.633888i	$0.218542\pi$
$71$	90.0121i	1.26778i	−0.773425	+	0.633888i	$0.781458\pi$
$72$	0	0
$73$	68.0795i	0.932596i	0.884628	+	0.466298i	$0.154412\pi$
$73$	68.0795i	0.932596i	−0.884628	+	0.466298i	$0.845588\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	126.790	1.64662
$78$	0	0
$79$	36.2647i	0.459046i	0.973303	+	0.229523i	$0.0737167\pi$
$79$	36.2647i	0.459046i	−0.973303	+	0.229523i	$0.926283\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	95.5904i	1.15169i	0.817558	+	0.575846i	$0.195327\pi$
$83$	95.5904i	1.15169i	−0.817558	+	0.575846i	$0.804673\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.02058	0.0117307
$88$	0	0
$89$	7.89970	0.0887607	0.0443803	−	0.999015i	$-0.485869\pi$
$89$	7.89970	0.0887607	0.0443803	+	0.999015i	$0.485869\pi$
$90$	0	0
$91$	−81.5982	−0.896683
$92$	0	0
$93$	78.0135	0.838855
$94$	0	0
$95$	0	0
$96$	0	0
$97$	137.133i	1.41375i	0.707341	+	0.706873i	$0.249895\pi$
$97$	137.133i	1.41375i	−0.707341	+	0.706873i	$0.750105\pi$
$98$	0	0
$99$	50.1164	0.506227
$100$	0	0
$101$	92.4295i	0.915144i	0.889173	+	0.457572i	$0.151281\pi$
$101$	92.4295i	0.915144i	−0.889173	+	0.457572i	$0.848719\pi$
$102$	0	0
$103$	123.314	1.19722	0.598610	−	0.801040i	$-0.295720\pi$
$103$	123.314	1.19722	0.598610	+	0.801040i	$0.295720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.240104i	− 0.00224396i	−0.999999	−	0.00112198i	$-0.999643\pi$
$107$	− 0.240104i	− 0.00224396i	0.999999	−	0.00112198i	$-0.000357137\pi$
$108$	0	0
$109$	117.597i	1.07887i	0.842028	+	0.539435i	$0.181362\pi$
$109$	117.597i	1.07887i	−0.842028	+	0.539435i	$0.818638\pi$
$110$	0	0
$111$	63.4282i	0.571425i
$112$	0	0
$113$	41.1728i	0.364361i	0.983265	+	0.182180i	$0.0583155\pi$
$113$	41.1728i	0.364361i	−0.983265	+	0.182180i	$0.941685\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−32.2535	−0.275671
$118$	0	0
$119$	65.3682i	0.549313i
$120$	0	0
$121$	158.073	1.30639
$122$	0	0
$123$	− 4.24295i	− 0.0344956i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.15264	−0.00907591	−0.00453795	−	0.999990i	$-0.501444\pi$
$127$	−1.15264	−0.00907591	−0.00453795	+	0.999990i	$0.501444\pi$
$128$	0	0
$129$	−13.7096	−0.106276
$130$	0	0
$131$	191.878	1.46472	0.732359	−	0.680919i	$-0.238419\pi$
$131$	191.878	1.46472	0.732359	+	0.680919i	$0.238419\pi$
$132$	0	0
$133$	−270.661	−2.03504
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.0127i	− 0.124180i	−0.998071	−	0.0620901i	$-0.980223\pi$
$137$	− 17.0127i	− 0.124180i	0.998071	−	0.0620901i	$-0.0197766\pi$
$138$	0	0
$139$	−176.378	−1.26890	−0.634452	−	0.772963i	$-0.718774\pi$
$139$	−176.378	−1.26890	−0.634452	+	0.772963i	$0.718774\pi$
$140$	0	0
$141$	− 76.8607i	− 0.545111i
$142$	0	0
$143$	−179.604	−1.25597
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.9017i	0.101372i
$148$	0	0
$149$	− 212.046i	− 1.42313i	−0.702620	−	0.711565i	$-0.747987\pi$
$149$	− 212.046i	− 1.42313i	0.702620	−	0.711565i	$-0.252013\pi$
$150$	0	0
$151$	143.361i	0.949408i	0.880146	+	0.474704i	$0.157445\pi$
$151$	143.361i	0.949408i	−0.880146	+	0.474704i	$0.842555\pi$
$152$	0	0
$153$	25.8383i	0.168878i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	82.2710	0.524019	0.262009	−	0.965065i	$-0.415615\pi$
$157$	82.2710	0.524019	0.262009	+	0.965065i	$0.415615\pi$
$158$	0	0
$159$	126.990i	0.798678i
$160$	0	0
$161$	85.6973	0.532282
$162$	0	0
$163$	− 27.8115i	− 0.170623i	−0.996354	−	0.0853113i	$-0.972812\pi$
$163$	− 27.8115i	− 0.170623i	0.996354	−	0.0853113i	$-0.0271885\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	207.819	1.24443	0.622214	−	0.782848i	$-0.286234\pi$
$167$	207.819	1.24443	0.622214	+	0.782848i	$0.286234\pi$
$168$	0	0
$169$	−53.4122	−0.316048
$170$	0	0
$171$	−106.985	−0.625642
$172$	0	0
$173$	45.7625	0.264523	0.132262	−	0.991215i	$-0.457776\pi$
$173$	45.7625	0.264523	0.132262	+	0.991215i	$0.457776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 155.478i	− 0.878405i
$178$	0	0
$179$	−33.0983	−0.184907	−0.0924533	−	0.995717i	$-0.529471\pi$
$179$	−33.0983	−0.184907	−0.0924533	+	0.995717i	$0.529471\pi$
$180$	0	0
$181$	− 177.147i	− 0.978711i	−0.872084	−	0.489356i	$-0.837232\pi$
$181$	− 177.147i	− 0.978711i	0.872084	−	0.489356i	$-0.162768\pi$
$182$	0	0
$183$	171.124	0.935106
$184$	0	0
$185$	0	0
$186$	0	0
$187$	143.880i	0.769413i
$188$	0	0
$189$	39.4372i	0.208663i
$190$	0	0
$191$	28.8000i	0.150785i	0.997154	+	0.0753927i	$0.0240210\pi$
$191$	28.8000i	0.150785i	−0.997154	+	0.0753927i	$0.975979\pi$
$192$	0	0
$193$	87.7961i	0.454902i	0.973789	+	0.227451i	$0.0730392\pi$
$193$	87.7961i	0.454902i	−0.973789	+	0.227451i	$0.926961\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	347.625	1.76459	0.882296	−	0.470696i	$-0.155997\pi$
$197$	347.625	1.76459	0.882296	+	0.470696i	$0.155997\pi$
$198$	0	0
$199$	− 129.347i	− 0.649985i	−0.945717	−	0.324992i	$-0.894638\pi$
$199$	− 129.347i	− 0.649985i	0.945717	−	0.324992i	$-0.105362\pi$
$200$	0	0
$201$	−155.021	−0.771247
$202$	0	0
$203$	4.47207i	0.0220299i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	33.8738	0.163642
$208$	0	0
$209$	−595.744	−2.85045
$210$	0	0
$211$	190.507	0.902877	0.451439	−	0.892302i	$-0.350911\pi$
$211$	190.507	0.902877	0.451439	+	0.892302i	$0.350911\pi$
$212$	0	0
$213$	−155.905	−0.731951
$214$	0	0
$215$	0	0
$216$	0	0
$217$	341.848i	1.57534i
$218$	0	0
$219$	−117.917	−0.538434
$220$	0	0
$221$	− 92.5973i	− 0.418992i
$222$	0	0
$223$	211.437	0.948149	0.474075	−	0.880485i	$-0.342783\pi$
$223$	211.437	0.948149	0.474075	+	0.880485i	$0.342783\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	88.8880i	0.391577i	0.980646	+	0.195788i	$0.0627266\pi$
$227$	88.8880i	0.391577i	−0.980646	+	0.195788i	$0.937273\pi$
$228$	0	0
$229$	152.872i	0.667565i	0.942650	+	0.333782i	$0.108325\pi$
$229$	152.872i	0.667565i	−0.942650	+	0.333782i	$0.891675\pi$
$230$	0	0
$231$	219.606i	0.950675i
$232$	0	0
$233$	188.417i	0.808656i	0.914614	+	0.404328i	$0.132495\pi$
$233$	188.417i	0.808656i	−0.914614	+	0.404328i	$0.867505\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−62.8122	−0.265030
$238$	0	0
$239$	227.979i	0.953888i	0.878934	+	0.476944i	$0.158255\pi$
$239$	227.979i	0.953888i	−0.878934	+	0.476944i	$0.841745\pi$
$240$	0	0
$241$	−265.794	−1.10288	−0.551439	−	0.834215i	$-0.685921\pi$
$241$	−265.794	−1.10288	−0.551439	+	0.834215i	$0.685921\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	383.404	1.55224
$248$	0	0
$249$	−165.567	−0.664929
$250$	0	0
$251$	−154.894	−0.617106	−0.308553	−	0.951207i	$-0.599845\pi$
$251$	−154.894	−0.617106	−0.308553	+	0.951207i	$0.599845\pi$
$252$	0	0
$253$	188.626	0.745558
$254$	0	0
$255$	0	0
$256$	0	0
$257$	422.025i	1.64212i	0.570842	+	0.821060i	$0.306617\pi$
$257$	422.025i	1.64212i	−0.570842	+	0.821060i	$0.693383\pi$
$258$	0	0
$259$	−277.937	−1.07312
$260$	0	0
$261$	1.76769i	0.00677275i
$262$	0	0
$263$	−118.083	−0.448983	−0.224492	−	0.974476i	$-0.572072\pi$
$263$	−118.083	−0.448983	−0.224492	+	0.974476i	$0.572072\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.6827i	0.0512460i
$268$	0	0
$269$	493.643i	1.83510i	0.397617	+	0.917551i	$0.369837\pi$
$269$	493.643i	1.83510i	−0.397617	+	0.917551i	$0.630163\pi$
$270$	0	0
$271$	− 362.955i	− 1.33932i	−0.742669	−	0.669659i	$-0.766440\pi$
$271$	− 362.955i	− 1.33932i	0.742669	−	0.669659i	$-0.233560\pi$
$272$	0	0
$273$	− 141.332i	− 0.517700i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	321.011	1.15888	0.579442	−	0.815013i	$-0.303270\pi$
$277$	321.011	1.15888	0.579442	+	0.815013i	$0.303270\pi$
$278$	0	0
$279$	135.123i	0.484313i
$280$	0	0
$281$	−185.799	−0.661206	−0.330603	−	0.943770i	$-0.607252\pi$
$281$	−185.799	−0.661206	−0.330603	+	0.943770i	$0.607252\pi$
$282$	0	0
$283$	208.628i	0.737202i	0.929588	+	0.368601i	$0.120163\pi$
$283$	208.628i	0.737202i	−0.929588	+	0.368601i	$0.879837\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.5923	0.0647814
$288$	0	0
$289$	214.820	0.743323
$290$	0	0
$291$	−237.522	−0.816226
$292$	0	0
$293$	306.925	1.04752	0.523762	−	0.851864i	$-0.324528\pi$
$293$	306.925	1.04752	0.523762	+	0.851864i	$0.324528\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	86.8042i	0.292270i
$298$	0	0
$299$	−121.394	−0.406002
$300$	0	0
$301$	− 60.0743i	− 0.199582i
$302$	0	0
$303$	−160.093	−0.528358
$304$	0	0
$305$	0	0
$306$	0	0
$307$	283.517i	0.923510i	0.887008	+	0.461755i	$0.152780\pi$
$307$	283.517i	0.923510i	−0.887008	+	0.461755i	$0.847220\pi$
$308$	0	0
$309$	213.586i	0.691216i
$310$	0	0
$311$	− 614.016i	− 1.97433i	−0.159709	−	0.987164i	$-0.551056\pi$
$311$	− 614.016i	− 1.97433i	0.159709	−	0.987164i	$-0.448944\pi$
$312$	0	0
$313$	− 511.920i	− 1.63553i	−0.575555	−	0.817763i	$-0.695214\pi$
$313$	− 511.920i	− 1.63553i	0.575555	−	0.817763i	$-0.304786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−429.605	−1.35522	−0.677611	−	0.735421i	$-0.736985\pi$
$317$	−429.605	−1.35522	−0.677611	+	0.735421i	$0.736985\pi$
$318$	0	0
$319$	9.84336i	0.0308569i
$320$	0	0
$321$	0.415872	0.00129555
$322$	0	0
$323$	− 307.145i	− 0.950913i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−203.684	−0.622885
$328$	0	0
$329$	336.797	1.02370
$330$	0	0
$331$	537.298	1.62326	0.811628	−	0.584175i	$-0.198582\pi$
$331$	537.298	1.62326	0.811628	+	0.584175i	$0.198582\pi$
$332$	0	0
$333$	−109.861	−0.329913
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 158.578i	− 0.470559i	−0.971928	−	0.235279i	$-0.924400\pi$
$337$	− 158.578i	− 0.470559i	0.971928	−	0.235279i	$-0.0756005\pi$
$338$	0	0
$339$	−71.3133	−0.210364
$340$	0	0
$341$	752.434i	2.20655i
$342$	0	0
$343$	306.597	0.893870
$344$	0	0
$345$	0	0
$346$	0	0
$347$	169.697i	0.489041i	0.969644	+	0.244520i	$0.0786305\pi$
$347$	169.697i	0.489041i	−0.969644	+	0.244520i	$0.921370\pi$
$348$	0	0
$349$	296.370i	0.849196i	0.905382	+	0.424598i	$0.139585\pi$
$349$	296.370i	0.849196i	−0.905382	+	0.424598i	$0.860415\pi$
$350$	0	0
$351$	− 55.8648i	− 0.159159i
$352$	0	0
$353$	133.075i	0.376982i	0.982075	+	0.188491i	$0.0603596\pi$
$353$	133.075i	0.376982i	−0.982075	+	0.188491i	$0.939640\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−113.221	−0.317146
$358$	0	0
$359$	238.095i	0.663218i	0.943417	+	0.331609i	$0.107591\pi$
$359$	238.095i	0.663218i	−0.943417	+	0.331609i	$0.892409\pi$
$360$	0	0
$361$	910.751	2.52286
$362$	0	0
$363$	273.791i	0.754244i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	381.665	1.03996	0.519979	−	0.854179i	$-0.325940\pi$
$367$	381.665	1.03996	0.519979	+	0.854179i	$0.325940\pi$
$368$	0	0
$369$	7.34901	0.0199160
$370$	0	0
$371$	−556.458	−1.49989
$372$	0	0
$373$	−42.2729	−0.113332	−0.0566661	−	0.998393i	$-0.518047\pi$
$373$	−42.2729	−0.113332	−0.0566661	+	0.998393i	$0.518047\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.33491i	− 0.0168035i
$378$	0	0
$379$	412.302	1.08787	0.543934	−	0.839128i	$-0.316934\pi$
$379$	412.302	1.08787	0.543934	+	0.839128i	$0.316934\pi$
$380$	0	0
$381$	− 1.99643i	− 0.00523998i
$382$	0	0
$383$	−403.711	−1.05408	−0.527038	−	0.849842i	$-0.676697\pi$
$383$	−403.711	−1.05408	−0.527038	+	0.849842i	$0.676697\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 23.7457i	− 0.0613585i
$388$	0	0
$389$	− 78.5053i	− 0.201813i	−0.994896	−	0.100907i	$-0.967826\pi$
$389$	− 78.5053i	− 0.201813i	0.994896	−	0.100907i	$-0.0321743\pi$
$390$	0	0
$391$	97.2490i	0.248719i
$392$	0	0
$393$	332.342i	0.845655i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−323.080	−0.813803	−0.406902	−	0.913472i	$-0.633391\pi$
$397$	−323.080	−0.813803	−0.406902	+	0.913472i	$0.633391\pi$
$398$	0	0
$399$	− 468.798i	− 1.17493i
$400$	0	0
$401$	−621.389	−1.54960	−0.774799	−	0.632207i	$-0.782149\pi$
$401$	−621.389	−1.54960	−0.774799	+	0.632207i	$0.782149\pi$
$402$	0	0
$403$	− 484.245i	− 1.20160i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−611.760	−1.50310
$408$	0	0
$409$	−322.888	−0.789458	−0.394729	−	0.918798i	$-0.629161\pi$
$409$	−322.888	−0.789458	−0.394729	+	0.918798i	$0.629161\pi$
$410$	0	0
$411$	29.4668	0.0716954
$412$	0	0
$413$	681.290	1.64961
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 305.495i	− 0.732602i
$418$	0	0
$419$	19.2316	0.0458988	0.0229494	−	0.999737i	$-0.492694\pi$
$419$	19.2316	0.0458988	0.0229494	+	0.999737i	$0.492694\pi$
$420$	0	0
$421$	312.791i	0.742970i	0.928439	+	0.371485i	$0.121151\pi$
$421$	312.791i	0.742970i	−0.928439	+	0.371485i	$0.878849\pi$
$422$	0	0
$423$	133.127	0.314720
$424$	0	0
$425$	0	0
$426$	0	0
$427$	749.852i	1.75609i
$428$	0	0
$429$	− 311.083i	− 0.725134i
$430$	0	0
$431$	− 417.415i	− 0.968479i	−0.874935	−	0.484240i	$-0.839096\pi$
$431$	− 417.415i	− 0.968479i	0.874935	−	0.484240i	$-0.160904\pi$
$432$	0	0
$433$	279.474i	0.645436i	0.946495	+	0.322718i	$0.104596\pi$
$433$	279.474i	0.645436i	−0.946495	+	0.322718i	$0.895404\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−402.665	−0.921430
$438$	0	0
$439$	− 30.1336i	− 0.0686415i	−0.999411	−	0.0343207i	$-0.989073\pi$
$439$	− 30.1336i	− 0.0686415i	0.999411	−	0.0343207i	$-0.0109268\pi$
$440$	0	0
$441$	−25.8104	−0.0585271
$442$	0	0
$443$	56.2858i	0.127056i	0.997980	+	0.0635279i	$0.0202352\pi$
$443$	56.2858i	0.127056i	−0.997980	+	0.0635279i	$0.979765\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	367.275	0.821644
$448$	0	0
$449$	283.497	0.631396	0.315698	−	0.948860i	$-0.397761\pi$
$449$	283.497	0.631396	0.315698	+	0.948860i	$0.397761\pi$
$450$	0	0
$451$	40.9229	0.0907382
$452$	0	0
$453$	−248.308	−0.548141
$454$	0	0
$455$	0	0
$456$	0	0
$457$	265.204i	0.580315i	0.956979	+	0.290158i	$0.0937078\pi$
$457$	265.204i	0.580315i	−0.956979	+	0.290158i	$0.906292\pi$
$458$	0	0
$459$	−44.7532	−0.0975015
$460$	0	0
$461$	709.643i	1.53936i	0.638432	+	0.769678i	$0.279583\pi$
$461$	709.643i	1.53936i	−0.638432	+	0.769678i	$0.720417\pi$
$462$	0	0
$463$	−417.392	−0.901495	−0.450747	−	0.892652i	$-0.648843\pi$
$463$	−417.392	−0.901495	−0.450747	+	0.892652i	$0.648843\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 679.287i	− 1.45458i	−0.686332	−	0.727288i	$-0.740780\pi$
$467$	− 679.287i	− 1.45458i	0.686332	−	0.727288i	$-0.259220\pi$
$468$	0	0
$469$	− 679.287i	− 1.44837i
$470$	0	0
$471$	142.498i	0.302543i
$472$	0	0
$473$	− 132.228i	− 0.279552i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−219.953	−0.461117
$478$	0	0
$479$	− 157.578i	− 0.328974i	−0.986379	−	0.164487i	$-0.947403\pi$
$479$	− 157.578i	− 0.328974i	0.986379	−	0.164487i	$-0.0525968\pi$
$480$	0	0
$481$	393.711	0.818527
$482$	0	0
$483$	148.432i	0.307313i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	736.646	1.51262	0.756311	−	0.654213i	$-0.227000\pi$
$487$	736.646	1.51262	0.756311	+	0.654213i	$0.227000\pi$
$488$	0	0
$489$	48.1709	0.0985090
$490$	0	0
$491$	362.408	0.738101	0.369050	−	0.929409i	$-0.379683\pi$
$491$	362.408	0.738101	0.369050	+	0.929409i	$0.379683\pi$
$492$	0	0
$493$	−5.07489	−0.0102939
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 683.164i	− 1.37458i
$498$	0	0
$499$	648.812	1.30022	0.650112	−	0.759838i	$-0.274722\pi$
$499$	648.812	1.30022	0.650112	+	0.759838i	$0.274722\pi$
$500$	0	0
$501$	359.954i	0.718470i
$502$	0	0
$503$	402.612	0.800421	0.400210	−	0.916423i	$-0.368937\pi$
$503$	402.612	0.800421	0.400210	+	0.916423i	$0.368937\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 92.5126i	− 0.182471i
$508$	0	0
$509$	470.175i	0.923724i	0.886952	+	0.461862i	$0.152818\pi$
$509$	470.175i	0.923724i	−0.886952	+	0.461862i	$0.847182\pi$
$510$	0	0
$511$	− 516.702i	− 1.01116i
$512$	0	0
$513$	− 185.303i	− 0.361215i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	741.315	1.43388
$518$	0	0
$519$	79.2630i	0.152723i
$520$	0	0
$521$	−155.978	−0.299382	−0.149691	−	0.988733i	$-0.547828\pi$
$521$	−155.978	−0.299382	−0.149691	+	0.988733i	$0.547828\pi$
$522$	0	0
$523$	313.578i	0.599576i	0.954006	+	0.299788i	$0.0969159\pi$
$523$	313.578i	0.599576i	−0.954006	+	0.299788i	$0.903084\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−387.928	−0.736107
$528$	0	0
$529$	−401.507	−0.758993
$530$	0	0
$531$	269.295	0.507147
$532$	0	0
$533$	−26.3369	−0.0494125
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 57.3279i	− 0.106756i
$538$	0	0
$539$	−143.725	−0.266652
$540$	0	0
$541$	− 892.335i	− 1.64942i	−0.565557	−	0.824709i	$-0.691339\pi$
$541$	− 892.335i	− 1.64942i	0.565557	−	0.824709i	$-0.308661\pi$
$542$	0	0
$543$	306.827	0.565059
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 671.435i	− 1.22749i	−0.789506	−	0.613743i	$-0.789663\pi$
$547$	− 671.435i	− 1.22749i	0.789506	−	0.613743i	$-0.210337\pi$
$548$	0	0
$549$	296.396i	0.539884i
$550$	0	0
$551$	− 21.0129i	− 0.0381359i
$552$	0	0
$553$	− 275.238i	− 0.497717i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−296.095	−0.531590	−0.265795	−	0.964030i	$-0.585634\pi$
$557$	−296.095	−0.531590	−0.265795	+	0.964030i	$0.585634\pi$
$558$	0	0
$559$	85.0982i	0.152233i
$560$	0	0
$561$	−249.208	−0.444221
$562$	0	0
$563$	707.222i	1.25617i	0.778146	+	0.628084i	$0.216161\pi$
$563$	707.222i	1.25617i	−0.778146	+	0.628084i	$0.783839\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−68.3073	−0.120471
$568$	0	0
$569$	−230.263	−0.404680	−0.202340	−	0.979315i	$-0.564855\pi$
$569$	−230.263	−0.404680	−0.202340	+	0.979315i	$0.564855\pi$
$570$	0	0
$571$	439.428	0.769575	0.384788	−	0.923005i	$-0.374275\pi$
$571$	439.428	0.769575	0.384788	+	0.923005i	$0.374275\pi$
$572$	0	0
$573$	−49.8831	−0.0870560
$574$	0	0
$575$	0	0
$576$	0	0
$577$	184.670i	0.320053i	0.987113	+	0.160026i	$0.0511579\pi$
$577$	184.670i	0.320053i	−0.987113	+	0.160026i	$0.948842\pi$
$578$	0	0
$579$	−152.067	−0.262638
$580$	0	0
$581$	− 725.502i	− 1.24871i
$582$	0	0
$583$	−1224.81	−2.10087
$584$	0	0
$585$	0	0
$586$	0	0
$587$	297.341i	0.506543i	0.967395	+	0.253271i	$0.0815066\pi$
$587$	297.341i	0.506543i	−0.967395	+	0.253271i	$0.918493\pi$
$588$	0	0
$589$	− 1606.24i	− 2.72706i
$590$	0	0
$591$	602.103i	1.01879i
$592$	0	0
$593$	− 607.065i	− 1.02372i	−0.859070	−	0.511859i	$-0.828957\pi$
$593$	− 607.065i	− 1.02372i	0.859070	−	0.511859i	$-0.171043\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	224.036	0.375269
$598$	0	0
$599$	− 448.633i	− 0.748970i	−0.927233	−	0.374485i	$-0.877820\pi$
$599$	− 448.633i	− 0.748970i	0.927233	−	0.374485i	$-0.122180\pi$
$600$	0	0
$601$	580.584	0.966030	0.483015	−	0.875612i	$-0.339542\pi$
$601$	580.584	0.966030	0.483015	+	0.875612i	$0.339542\pi$
$602$	0	0
$603$	− 268.504i	− 0.445280i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−132.576	−0.218413	−0.109206	−	0.994019i	$-0.534831\pi$
$607$	−132.576	−0.218413	−0.109206	+	0.994019i	$0.534831\pi$
$608$	0	0
$609$	−7.74585	−0.0127190
$610$	0	0
$611$	−477.089	−0.780834
$612$	0	0
$613$	−175.738	−0.286685	−0.143343	−	0.989673i	$-0.545785\pi$
$613$	−175.738	−0.286685	−0.143343	+	0.989673i	$0.545785\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 283.254i	− 0.459082i	−0.973299	−	0.229541i	$-0.926277\pi$
$617$	− 283.254i	− 0.459082i	0.973299	−	0.229541i	$-0.0737226\pi$
$618$	0	0
$619$	−80.2105	−0.129581	−0.0647904	−	0.997899i	$-0.520638\pi$
$619$	−80.2105	−0.129581	−0.0647904	+	0.997899i	$0.520638\pi$
$620$	0	0
$621$	58.6712i	0.0944785i
$622$	0	0
$623$	−59.9563	−0.0962381
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1031.86i	− 1.64571i
$628$	0	0
$629$	− 315.402i	− 0.501434i
$630$	0	0
$631$	− 178.580i	− 0.283011i	−0.989937	−	0.141505i	$-0.954806\pi$
$631$	− 178.580i	− 0.283011i	0.989937	−	0.141505i	$-0.0451942\pi$
$632$	0	0
$633$	329.968i	0.521277i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	92.4976	0.145208
$638$	0	0
$639$	− 270.036i	− 0.422592i
$640$	0	0
$641$	−379.689	−0.592339	−0.296169	−	0.955135i	$-0.595709\pi$
$641$	−379.689	−0.592339	−0.296169	+	0.955135i	$0.595709\pi$
$642$	0	0
$643$	847.921i	1.31869i	0.751839	+	0.659347i	$0.229167\pi$
$643$	847.921i	1.31869i	−0.751839	+	0.659347i	$0.770833\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	372.369	0.575532	0.287766	−	0.957701i	$-0.407087\pi$
$647$	372.369	0.575532	0.287766	+	0.957701i	$0.407087\pi$
$648$	0	0
$649$	1499.57	2.31058
$650$	0	0
$651$	−592.099	−0.909522
$652$	0	0
$653$	−650.726	−0.996517	−0.498259	−	0.867028i	$-0.666027\pi$
$653$	−650.726	−0.996517	−0.498259	+	0.867028i	$0.666027\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 204.238i	− 0.310865i
$658$	0	0
$659$	589.211	0.894099	0.447049	−	0.894509i	$-0.352475\pi$
$659$	589.211	0.894099	0.447049	+	0.894509i	$0.352475\pi$
$660$	0	0
$661$	− 402.340i	− 0.608684i	−0.952563	−	0.304342i	$-0.901563\pi$
$661$	− 402.340i	− 0.608684i	0.952563	−	0.304342i	$-0.0984366\pi$
$662$	0	0
$663$	160.383	0.241905
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.65315i	0.00997473i
$668$	0	0
$669$	366.220i	0.547414i
$670$	0	0
$671$	1650.48i	2.45973i
$672$	0	0
$673$	1237.67i	1.83903i	0.393056	+	0.919515i	$0.371418\pi$
$673$	1237.67i	1.83903i	−0.393056	+	0.919515i	$0.628582\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	163.462	0.241451	0.120725	−	0.992686i	$-0.461478\pi$
$677$	163.462	0.241451	0.120725	+	0.992686i	$0.461478\pi$
$678$	0	0
$679$	− 1040.80i	− 1.53284i
$680$	0	0
$681$	−153.958	−0.226077
$682$	0	0
$683$	− 992.033i	− 1.45246i	−0.687449	−	0.726232i	$-0.741270\pi$
$683$	− 992.033i	− 1.45246i	0.687449	−	0.726232i	$-0.258730\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−264.783	−0.385419
$688$	0	0
$689$	788.250	1.14405
$690$	0	0
$691$	465.365	0.673466	0.336733	−	0.941600i	$-0.390678\pi$
$691$	465.365	0.673466	0.336733	+	0.941600i	$0.390678\pi$
$692$	0	0
$693$	−380.369	−0.548872
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.0984i	0.0302703i
$698$	0	0
$699$	−326.347	−0.466878
$700$	0	0
$701$	261.352i	0.372827i	0.982471	+	0.186414i	$0.0596864\pi$
$701$	261.352i	0.372827i	−0.982471	+	0.186414i	$0.940314\pi$
$702$	0	0
$703$	1305.94	1.85767
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 701.512i	− 0.992238i
$708$	0	0
$709$	186.604i	0.263194i	0.991303	+	0.131597i	$0.0420104\pi$
$709$	186.604i	0.263194i	−0.991303	+	0.131597i	$0.957990\pi$
$710$	0	0
$711$	− 108.794i	− 0.153015i
$712$	0	0
$713$	508.572i	0.713284i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−394.871	−0.550727
$718$	0	0
$719$	− 448.275i	− 0.623470i	−0.950169	−	0.311735i	$-0.899090\pi$
$719$	− 448.275i	− 0.623470i	0.950169	−	0.311735i	$-0.100910\pi$
$720$	0	0
$721$	−935.914	−1.29808
$722$	0	0
$723$	− 460.368i	− 0.636747i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−956.205	−1.31527	−0.657637	−	0.753335i	$-0.728444\pi$
$727$	−956.205	−1.31527	−0.657637	+	0.753335i	$0.728444\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	68.1721	0.0932586
$732$	0	0
$733$	−398.986	−0.544320	−0.272160	−	0.962252i	$-0.587738\pi$
$733$	−398.986	−0.544320	−0.272160	+	0.962252i	$0.587738\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1495.16i	− 2.02871i
$738$	0	0
$739$	1011.87	1.36924	0.684622	−	0.728899i	$-0.259967\pi$
$739$	1011.87	1.36924	0.684622	+	0.728899i	$0.259967\pi$
$740$	0	0
$741$	664.076i	0.896189i
$742$	0	0
$743$	346.145	0.465875	0.232938	−	0.972492i	$-0.425166\pi$
$743$	346.145	0.465875	0.232938	+	0.972492i	$0.425166\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 286.771i	− 0.383897i
$748$	0	0
$749$	1.82231i	0.00243300i
$750$	0	0
$751$	71.2058i	0.0948146i	0.998876	+	0.0474073i	$0.0150959\pi$
$751$	71.2058i	0.0948146i	−0.998876	+	0.0474073i	$0.984904\pi$
$752$	0	0
$753$	− 268.283i	− 0.356286i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1452.44	−1.91867	−0.959337	−	0.282264i	$-0.908915\pi$
$757$	−1452.44	−1.91867	−0.959337	+	0.282264i	$0.908915\pi$
$758$	0	0
$759$	326.710i	0.430448i
$760$	0	0
$761$	414.416	0.544567	0.272284	−	0.962217i	$-0.412221\pi$
$761$	414.416	0.544567	0.272284	+	0.962217i	$0.412221\pi$
$762$	0	0
$763$	− 892.523i	− 1.16976i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−965.081	−1.25825
$768$	0	0
$769$	132.780	0.172665	0.0863326	−	0.996266i	$-0.472485\pi$
$769$	132.780	0.172665	0.0863326	+	0.996266i	$0.472485\pi$
$770$	0	0
$771$	−730.968	−0.948078
$772$	0	0
$773$	135.449	0.175225	0.0876126	−	0.996155i	$-0.472076\pi$
$773$	135.449	0.175225	0.0876126	+	0.996155i	$0.472076\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 481.401i	− 0.619564i
$778$	0	0
$779$	−87.3592	−0.112143
$780$	0	0
$781$	− 1503.70i	− 1.92535i
$782$	0	0
$783$	−3.06173	−0.00391025
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 370.900i	− 0.471283i	−0.971840	−	0.235642i	$-0.924281\pi$
$787$	− 370.900i	− 0.471283i	0.971840	−	0.235642i	$-0.0757192\pi$
$788$	0	0
$789$	− 204.525i	− 0.259220i
$790$	0	0
$791$	− 312.489i	− 0.395055i
$792$	0	0
$793$	− 1062.20i	− 1.33947i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	980.881	1.23072	0.615358	−	0.788247i	$-0.289011\pi$
$797$	980.881	1.23072	0.615358	+	0.788247i	$0.289011\pi$
$798$	0	0
$799$	382.196i	0.478343i
$800$	0	0
$801$	−23.6991	−0.0295869
$802$	0	0
$803$	− 1137.30i	− 1.41631i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−855.014	−1.05950
$808$	0	0
$809$	464.769	0.574498	0.287249	−	0.957856i	$-0.407259\pi$
$809$	464.769	0.574498	0.287249	+	0.957856i	$0.407259\pi$
$810$	0	0
$811$	49.1816	0.0606431	0.0303216	−	0.999540i	$-0.490347\pi$
$811$	49.1816	0.0606431	0.0303216	+	0.999540i	$0.490347\pi$
$812$	0	0
$813$	628.657	0.773255
$814$	0	0
$815$	0	0
$816$	0	0
$817$	282.270i	0.345496i
$818$	0	0
$819$	244.795	0.298894
$820$	0	0
$821$	− 911.027i	− 1.10965i	−0.831965	−	0.554827i	$-0.812784\pi$
$821$	− 911.027i	− 1.10965i	0.831965	−	0.554827i	$-0.187216\pi$
$822$	0	0
$823$	−1278.02	−1.55287	−0.776437	−	0.630195i	$-0.782975\pi$
$823$	−1278.02	−1.55287	−0.776437	+	0.630195i	$0.782975\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 671.016i	− 0.811386i	−0.914009	−	0.405693i	$-0.867030\pi$
$827$	− 671.016i	− 0.811386i	0.914009	−	0.405693i	$-0.132970\pi$
$828$	0	0
$829$	− 31.8486i	− 0.0384181i	−0.999815	−	0.0192091i	$-0.993885\pi$
$829$	− 31.8486i	− 0.0384181i	0.999815	−	0.0192091i	$-0.00611481\pi$
$830$	0	0
$831$	556.007i	0.669082i
$832$	0	0
$833$	− 74.0997i	− 0.0889552i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−234.041	−0.279618
$838$	0	0
$839$	112.707i	0.134335i	0.997742	+	0.0671674i	$0.0213961\pi$
$839$	112.707i	0.134335i	−0.997742	+	0.0671674i	$0.978604\pi$
$840$	0	0
$841$	840.653	0.999587
$842$	0	0
$843$	− 321.813i	− 0.381748i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1199.73	−1.41644
$848$	0	0
$849$	−361.354	−0.425623
$850$	0	0
$851$	−413.490	−0.485887
$852$	0	0
$853$	−512.714	−0.601072	−0.300536	−	0.953770i	$-0.597166\pi$
$853$	−512.714	−0.601072	−0.300536	+	0.953770i	$0.597166\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1620.81i	1.89126i	0.325243	+	0.945631i	$0.394554\pi$
$857$	1620.81i	1.89126i	−0.325243	+	0.945631i	$0.605446\pi$
$858$	0	0
$859$	225.172	0.262132	0.131066	−	0.991374i	$-0.458160\pi$
$859$	225.172	0.262132	0.131066	+	0.991374i	$0.458160\pi$
$860$	0	0
$861$	32.2027i	0.0374016i
$862$	0	0
$863$	1029.21	1.19260	0.596299	−	0.802763i	$-0.296637\pi$
$863$	1029.21	1.19260	0.596299	+	0.802763i	$0.296637\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	372.080i	0.429158i
$868$	0	0
$869$	− 605.818i	− 0.697144i
$870$	0	0
$871$	962.244i	1.10476i
$872$	0	0
$873$	− 411.400i	− 0.471248i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	570.056	0.650007	0.325003	−	0.945713i	$-0.394635\pi$
$877$	570.056	0.650007	0.325003	+	0.945713i	$0.394635\pi$
$878$	0	0
$879$	531.609i	0.604789i
$880$	0	0
$881$	596.788	0.677398	0.338699	−	0.940895i	$-0.390013\pi$
$881$	596.788	0.677398	0.338699	+	0.940895i	$0.390013\pi$
$882$	0	0
$883$	− 856.738i	− 0.970258i	−0.874442	−	0.485129i	$-0.838773\pi$
$883$	− 856.738i	− 0.970258i	0.874442	−	0.485129i	$-0.161227\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1648.76	−1.85880	−0.929402	−	0.369068i	$-0.879677\pi$
$887$	−1648.76	−1.85880	−0.929402	+	0.369068i	$0.879677\pi$
$888$	0	0
$889$	8.74819	0.00984048
$890$	0	0
$891$	−150.349	−0.168742
$892$	0	0
$893$	−1582.50	−1.77212
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 210.261i	− 0.234405i
$898$	0	0
$899$	−26.5395	−0.0295212
$900$	0	0
$901$	− 631.467i	− 0.700851i
$902$	0	0
$903$	104.052	0.115229
$904$	0	0
$905$	0	0
$906$	0	0
$907$	844.509i	0.931101i	0.885021	+	0.465551i	$0.154144\pi$
$907$	844.509i	0.931101i	−0.885021	+	0.465551i	$0.845856\pi$
$908$	0	0
$909$	− 277.289i	− 0.305048i
$910$	0	0
$911$	1074.78i	1.17978i	0.807483	+	0.589891i	$0.200829\pi$
$911$	1074.78i	1.17978i	−0.807483	+	0.589891i	$0.799171\pi$
$912$	0	0
$913$	− 1596.88i	− 1.74905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1456.30	−1.58811
$918$	0	0
$919$	1142.19i	1.24286i	0.783471	+	0.621428i	$0.213447\pi$
$919$	1142.19i	1.24286i	−0.783471	+	0.621428i	$0.786553\pi$
$920$	0	0
$921$	−491.067	−0.533188
$922$	0	0
$923$	967.736i	1.04847i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−369.941	−0.399074
$928$	0	0
$929$	−415.953	−0.447742	−0.223871	−	0.974619i	$-0.571869\pi$
$929$	−415.953	−0.447742	−0.223871	+	0.974619i	$0.571869\pi$
$930$	0	0
$931$	306.814	0.329553
$932$	0	0
$933$	1063.51	1.13988
$934$	0	0
$935$	0	0
$936$	0	0
$937$	846.979i	0.903927i	0.892037	+	0.451963i	$0.149276\pi$
$937$	846.979i	0.903927i	−0.892037	+	0.451963i	$0.850724\pi$
$938$	0	0
$939$	886.671	0.944271
$940$	0	0
$941$	149.279i	0.158638i	0.996849	+	0.0793192i	$0.0252746\pi$
$941$	149.279i	0.158638i	−0.996849	+	0.0793192i	$0.974725\pi$
$942$	0	0
$943$	27.6599	0.0293318
$944$	0	0
$945$	0	0
$946$	0	0
$947$	543.005i	0.573395i	0.958021	+	0.286697i	$0.0925574\pi$
$947$	543.005i	0.573395i	−0.958021	+	0.286697i	$0.907443\pi$
$948$	0	0
$949$	731.935i	0.771269i
$950$	0	0
$951$	− 744.098i	− 0.782438i
$952$	0	0
$953$	− 344.292i	− 0.361272i	−0.983550	−	0.180636i	$-0.942184\pi$
$953$	− 344.292i	− 0.361272i	0.983550	−	0.180636i	$-0.0578156\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.0492	−0.0178153
$958$	0	0
$959$	129.121i	0.134641i
$960$	0	0
$961$	−1067.70	−1.11103
$962$	0	0
$963$	0.720311i	0 0.000747987i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1044.78	1.08044	0.540219	−	0.841524i	$-0.318341\pi$
$967$	1044.78	1.08044	0.540219	+	0.841524i	$0.318341\pi$
$968$	0	0
$969$	531.991	0.549010
$970$	0	0
$971$	−306.656	−0.315814	−0.157907	−	0.987454i	$-0.550475\pi$
$971$	−306.656	−0.315814	−0.157907	+	0.987454i	$0.550475\pi$
$972$	0	0
$973$	1338.65	1.37580
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1753.81i	− 1.79509i	−0.440919	−	0.897547i	$-0.645347\pi$
$977$	− 1753.81i	− 1.79509i	0.440919	−	0.897547i	$-0.354653\pi$
$978$	0	0
$979$	−131.968	−0.134799
$980$	0	0
$981$	− 352.790i	− 0.359623i
$982$	0	0
$983$	−951.541	−0.967997	−0.483999	−	0.875069i	$-0.660816\pi$
$983$	−951.541	−0.967997	−0.483999	+	0.875069i	$0.660816\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	583.349i	0.591033i
$988$	0	0
$989$	− 89.3731i	− 0.0903672i
$990$	0	0
$991$	− 1786.06i	− 1.80229i	−0.433523	−	0.901143i	$-0.642730\pi$
$991$	− 1786.06i	− 1.80229i	0.433523	−	0.901143i	$-0.357270\pi$
$992$	0	0
$993$	930.627i	0.937187i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1375.80	1.37994	0.689969	−	0.723839i	$-0.257624\pi$
$997$	1375.80	1.37994	0.689969	+	0.723839i	$0.257624\pi$
$998$	0	0
$999$	− 190.285i	− 0.190475i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.p.b.1999.14		32
4.3	odd	2		600.3.p.b.499.8		32
5.2	odd	4		480.3.g.a.271.9		16
5.3	odd	4		2400.3.g.b.751.7		16
5.4	even	2	inner	2400.3.p.b.1999.31		32
8.3	odd	2	inner	2400.3.p.b.1999.32		32
8.5	even	2		600.3.p.b.499.26		32
15.2	even	4		1440.3.g.c.271.10		16
20.3	even	4		600.3.g.d.451.12		16
20.7	even	4		120.3.g.a.91.5	✓	16
20.19	odd	2		600.3.p.b.499.25		32
40.3	even	4		2400.3.g.b.751.2		16
40.13	odd	4		600.3.g.d.451.11		16
40.19	odd	2	inner	2400.3.p.b.1999.13		32
40.27	even	4		480.3.g.a.271.16		16
40.29	even	2		600.3.p.b.499.7		32
40.37	odd	4		120.3.g.a.91.6	yes	16
60.47	odd	4		360.3.g.c.91.12		16
120.77	even	4		360.3.g.c.91.11		16
120.107	odd	4		1440.3.g.c.271.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.g.a.91.5	✓	16	20.7	even	4
120.3.g.a.91.6	yes	16	40.37	odd	4
360.3.g.c.91.11		16	120.77	even	4
360.3.g.c.91.12		16	60.47	odd	4
480.3.g.a.271.9		16	5.2	odd	4
480.3.g.a.271.16		16	40.27	even	4
600.3.g.d.451.11		16	40.13	odd	4
600.3.g.d.451.12		16	20.3	even	4
600.3.p.b.499.7		32	40.29	even	2
600.3.p.b.499.8		32	4.3	odd	2
600.3.p.b.499.25		32	20.19	odd	2
600.3.p.b.499.26		32	8.5	even	2
1440.3.g.c.271.7		16	120.107	odd	4
1440.3.g.c.271.10		16	15.2	even	4
2400.3.g.b.751.2		16	40.3	even	4
2400.3.g.b.751.7		16	5.3	odd	4
2400.3.p.b.1999.13		32	40.19	odd	2	inner
2400.3.p.b.1999.14		32	1.1	even	1	trivial
2400.3.p.b.1999.31		32	5.4	even	2	inner
2400.3.p.b.1999.32		32	8.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -7.58970 q^{7} -3.00000 q^{9} -16.7055 q^{11} +10.7512 q^{13} -8.61276i q^{17} +35.6616 q^{19} -13.1457i q^{21} -11.2913 q^{23} -5.19615i q^{27} -0.589229i q^{29} -45.0411i q^{31} -28.9347i q^{33} +36.6203 q^{37} +18.6216i q^{39} -2.44967 q^{41} +7.91524i q^{43} -44.3755 q^{47} +8.60348 q^{49} +14.9177 q^{51} +73.3176 q^{53} +61.7677i q^{57} -89.7651 q^{59} -98.7987i q^{61} +22.7691 q^{63} +89.5013i q^{67} -19.5571i q^{69} +90.0121i q^{71} +68.0795i q^{73} +126.790 q^{77} +36.2647i q^{79} +9.00000 q^{81} +95.5904i q^{83} +1.02058 q^{87} +7.89970 q^{89} -81.5982 q^{91} +78.0135 q^{93} +137.133i q^{97} +50.1164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 96 q^{9} - 128 q^{11} - 64 q^{19} + 160 q^{49} - 192 q^{51} + 256 q^{59} + 288 q^{81} + 384 q^{91} + 384 q^{99}+O(q^{100}) \) 32 * q - 96 * q^9 - 128 * q^11 - 64 * q^19 + 160 * q^49 - 192 * q^51 + 256 * q^59 + 288 * q^81 + 384 * q^91 + 384 * q^99

Introduction

L-functions

Modular forms

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