LMFDB - Embedded newform 1440.5.e.b.991.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,5,Mod(991,1440)] 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("1440.991"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1440.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,560,0,0,0,0,0,0,0,1000,0,0, 0,2064,0,0,0,0,0,0,0,3616] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$148.852746841$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 $ x^8 + 35x^6 + 413x^4 + 2015*x^2 + 3481
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			991.6
Root			$-2.72964i$ of defining polynomial
Character	$\chi$	$=$	1440.991
Dual form			1440.5.e.b.991.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.1803 q^{5} -32.1829i q^{7} -1.80919i q^{11} +50.2406 q^{13} -374.600 q^{17} +364.448i q^{19} -499.886i q^{23} +125.000 q^{25} +118.609 q^{29} +128.914i q^{31} -359.816i q^{35} +1346.95 q^{37} -1779.09 q^{41} +3034.10i q^{43} +1843.73i q^{47} +1365.26 q^{49} -1604.66 q^{53} -20.2274i q^{55} +1463.79i q^{59} +3006.04 q^{61} +561.707 q^{65} +3120.66i q^{67} +5532.73i q^{71} -3377.26 q^{73} -58.2250 q^{77} -11766.1i q^{79} -6719.80i q^{83} -4188.15 q^{85} +12536.4 q^{89} -1616.89i q^{91} +4074.66i q^{95} -968.394 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{13} + 560 q^{17} + 1000 q^{25} + 2064 q^{29} + 3616 q^{37} + 5216 q^{41} + 4088 q^{49} - 5088 q^{53} - 5504 q^{61} + 7600 q^{65} + 17936 q^{73} + 15840 q^{77} - 9600 q^{85} + 50608 q^{89} + 3440 q^{97}+O(q^{100}) $ 8 * q + 32 * q^13 + 560 * q^17 + 1000 * q^25 + 2064 * q^29 + 3616 * q^37 + 5216 * q^41 + 4088 * q^49 - 5088 * q^53 - 5504 * q^61 + 7600 * q^65 + 17936 * q^73 + 15840 * q^77 - 9600 * q^85 + 50608 * q^89 + 3440 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	11.1803	0.447214
$5$	11.1803	0.447214
$6$	0	0
$7$	− 32.1829i	− 0.656794i	−0.944540	−	0.328397i	$-0.893492\pi$
$7$	− 32.1829i	− 0.656794i	0.944540	−	0.328397i	$-0.106508\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.80919i	− 0.0149520i	−0.999972	−	0.00747600i	$-0.997620\pi$
$11$	− 1.80919i	− 0.0149520i	0.999972	−	0.00747600i	$-0.00237971\pi$
$12$	0	0
$13$	50.2406	0.297282	0.148641	−	0.988891i	$-0.452510\pi$
$13$	50.2406	0.297282	0.148641	+	0.988891i	$0.452510\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−374.600	−1.29619	−0.648097	−	0.761558i	$-0.724435\pi$
$17$	−374.600	−1.29619	−0.648097	+	0.761558i	$0.724435\pi$
$18$	0	0
$19$	364.448i	1.00955i	0.863250	+	0.504776i	$0.168425\pi$
$19$	364.448i	1.00955i	−0.863250	+	0.504776i	$0.831575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 499.886i	− 0.944965i	−0.881340	−	0.472482i	$-0.843358\pi$
$23$	− 499.886i	− 0.944965i	0.881340	−	0.472482i	$-0.156642\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	118.609	0.141033	0.0705164	−	0.997511i	$-0.477535\pi$
$29$	118.609	0.141033	0.0705164	+	0.997511i	$0.477535\pi$
$30$	0	0
$31$	128.914i	0.134145i	0.997748	+	0.0670727i	$0.0213659\pi$
$31$	128.914i	0.134145i	−0.997748	+	0.0670727i	$0.978634\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 359.816i	− 0.293727i
$36$	0	0
$37$	1346.95	0.983894	0.491947	−	0.870625i	$-0.336285\pi$
$37$	1346.95	0.983894	0.491947	+	0.870625i	$0.336285\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1779.09	−1.05835	−0.529175	−	0.848512i	$-0.677499\pi$
$41$	−1779.09	−1.05835	−0.529175	+	0.848512i	$0.677499\pi$
$42$	0	0
$43$	3034.10i	1.64094i	0.571690	+	0.820470i	$0.306288\pi$
$43$	3034.10i	1.64094i	−0.571690	+	0.820470i	$0.693712\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1843.73i	0.834643i	0.908759	+	0.417322i	$0.137031\pi$
$47$	1843.73i	0.834643i	−0.908759	+	0.417322i	$0.862969\pi$
$48$	0	0
$49$	1365.26	0.568622
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1604.66	−0.571257	−0.285629	−	0.958340i	$-0.592202\pi$
$53$	−1604.66	−0.571257	−0.285629	+	0.958340i	$0.592202\pi$
$54$	0	0
$55$	− 20.2274i	− 0.00668674i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1463.79i	0.420508i	0.977647	+	0.210254i	$0.0674291\pi$
$59$	1463.79i	0.420508i	−0.977647	+	0.210254i	$0.932571\pi$
$60$	0	0
$61$	3006.04	0.807859	0.403929	−	0.914790i	$-0.367644\pi$
$61$	3006.04	0.807859	0.403929	+	0.914790i	$0.367644\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	561.707	0.132948
$66$	0	0
$67$	3120.66i	0.695179i	0.937647	+	0.347589i	$0.113000\pi$
$67$	3120.66i	0.695179i	−0.937647	+	0.347589i	$0.887000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5532.73i	1.09755i	0.835971	+	0.548773i	$0.184905\pi$
$71$	5532.73i	1.09755i	−0.835971	+	0.548773i	$0.815095\pi$
$72$	0	0
$73$	−3377.26	−0.633751	−0.316875	−	0.948467i	$-0.602634\pi$
$73$	−3377.26	−0.633751	−0.316875	+	0.948467i	$0.602634\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−58.2250	−0.00982038
$78$	0	0
$79$	− 11766.1i	− 1.88528i	−0.333806	−	0.942642i	$-0.608333\pi$
$79$	− 11766.1i	− 1.88528i	0.333806	−	0.942642i	$-0.391667\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6719.80i	− 0.975440i	−0.873000	−	0.487720i	$-0.837829\pi$
$83$	− 6719.80i	− 0.975440i	0.873000	−	0.487720i	$-0.162171\pi$
$84$	0	0
$85$	−4188.15	−0.579675
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12536.4	1.58268	0.791339	−	0.611378i	$-0.209384\pi$
$89$	12536.4	1.58268	0.791339	+	0.611378i	$0.209384\pi$
$90$	0	0
$91$	− 1616.89i	− 0.195253i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4074.66i	0.451485i
$96$	0	0
$97$	−968.394	−0.102922	−0.0514610	−	0.998675i	$-0.516388\pi$
$97$	−968.394	−0.102922	−0.0514610	+	0.998675i	$0.516388\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15104.8	1.48072	0.740360	−	0.672211i	$-0.234655\pi$
$101$	15104.8	1.48072	0.740360	+	0.672211i	$0.234655\pi$
$102$	0	0
$103$	35.7601i	0.00337074i	0.999999	+	0.00168537i	$0.000536470\pi$
$103$	35.7601i	0.00337074i	−0.999999	+	0.00168537i	$0.999464\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10691.1i	0.933806i	0.884308	+	0.466903i	$0.154630\pi$
$107$	10691.1i	0.933806i	−0.884308	+	0.466903i	$0.845370\pi$
$108$	0	0
$109$	−5435.32	−0.457480	−0.228740	−	0.973488i	$-0.573461\pi$
$109$	−5435.32	−0.457480	−0.228740	+	0.973488i	$0.573461\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15583.6	−1.22043	−0.610213	−	0.792237i	$-0.708916\pi$
$113$	−15583.6	−1.22043	−0.610213	+	0.792237i	$0.708916\pi$
$114$	0	0
$115$	− 5588.90i	− 0.422601i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12055.7i	0.851332i
$120$	0	0
$121$	14637.7	0.999776
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1397.54	0.0894427
$126$	0	0
$127$	1530.71i	0.0949042i	0.998874	+	0.0474521i	$0.0151102\pi$
$127$	1530.71i	0.0949042i	−0.998874	+	0.0474521i	$0.984890\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2511.55i	0.146352i	0.997319	+	0.0731761i	$0.0233135\pi$
$131$	2511.55i	0.146352i	−0.997319	+	0.0731761i	$0.976686\pi$
$132$	0	0
$133$	11729.0	0.663068
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8095.96	0.431347	0.215674	−	0.976465i	$-0.430805\pi$
$137$	8095.96	0.431347	0.215674	+	0.976465i	$0.430805\pi$
$138$	0	0
$139$	− 20207.8i	− 1.04590i	−0.852364	−	0.522949i	$-0.824832\pi$
$139$	− 20207.8i	− 1.04590i	0.852364	−	0.522949i	$-0.175168\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 90.8949i	− 0.00444495i
$144$	0	0
$145$	1326.08	0.0630718
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21223.9	0.955990	0.477995	−	0.878363i	$-0.341364\pi$
$149$	21223.9	0.955990	0.477995	+	0.878363i	$0.341364\pi$
$150$	0	0
$151$	39395.1i	1.72778i	0.503682	+	0.863889i	$0.331978\pi$
$151$	39395.1i	1.72778i	−0.503682	+	0.863889i	$0.668022\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1441.30i	0.0599916i
$156$	0	0
$157$	27492.0	1.11534	0.557670	−	0.830063i	$-0.311696\pi$
$157$	27492.0	1.11534	0.557670	+	0.830063i	$0.311696\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16087.8	−0.620647
$162$	0	0
$163$	36559.7i	1.37603i	0.725698	+	0.688014i	$0.241517\pi$
$163$	36559.7i	1.37603i	−0.725698	+	0.688014i	$0.758483\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 35981.9i	− 1.29018i	−0.764106	−	0.645091i	$-0.776819\pi$
$167$	− 35981.9i	− 1.29018i	0.764106	−	0.645091i	$-0.223181\pi$
$168$	0	0
$169$	−26036.9	−0.911624
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−41985.2	−1.40283	−0.701413	−	0.712755i	$-0.747447\pi$
$173$	−41985.2	−1.40283	−0.701413	+	0.712755i	$0.747447\pi$
$174$	0	0
$175$	− 4022.86i	− 0.131359i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	55301.6i	1.72596i	0.505235	+	0.862982i	$0.331406\pi$
$179$	55301.6i	1.72596i	−0.505235	+	0.862982i	$0.668594\pi$
$180$	0	0
$181$	6709.23	0.204793	0.102397	−	0.994744i	$-0.467349\pi$
$181$	6709.23	0.204793	0.102397	+	0.994744i	$0.467349\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15059.4	0.440011
$186$	0	0
$187$	677.723i	0.0193807i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 57528.3i	− 1.57694i	−0.615074	−	0.788469i	$-0.710874\pi$
$191$	− 57528.3i	− 1.57694i	0.615074	−	0.788469i	$-0.289126\pi$
$192$	0	0
$193$	54959.8	1.47547	0.737736	−	0.675090i	$-0.235895\pi$
$193$	54959.8	1.47547	0.737736	+	0.675090i	$0.235895\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	43745.7	1.12720	0.563602	−	0.826046i	$-0.309415\pi$
$197$	43745.7	1.12720	0.563602	+	0.826046i	$0.309415\pi$
$198$	0	0
$199$	15262.1i	0.385397i	0.981258	+	0.192699i	$0.0617240\pi$
$199$	15262.1i	0.385397i	−0.981258	+	0.192699i	$0.938276\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3817.17i	− 0.0926295i
$204$	0	0
$205$	−19890.8	−0.473309
$206$	0	0
$207$	0	0
$208$	0	0
$209$	659.357	0.0150948
$210$	0	0
$211$	− 71215.5i	− 1.59959i	−0.600271	−	0.799796i	$-0.704941\pi$
$211$	− 71215.5i	− 1.59959i	0.600271	−	0.799796i	$-0.295059\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	33922.2i	0.733851i
$216$	0	0
$217$	4148.82	0.0881059
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−18820.1	−0.385334
$222$	0	0
$223$	37447.8i	0.753038i	0.926409	+	0.376519i	$0.122879\pi$
$223$	37447.8i	0.753038i	−0.926409	+	0.376519i	$0.877121\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29560.2i	0.573661i	0.957981	+	0.286831i	$0.0926017\pi$
$227$	29560.2i	0.573661i	−0.957981	+	0.286831i	$0.907398\pi$
$228$	0	0
$229$	−69948.9	−1.33386	−0.666929	−	0.745121i	$-0.732392\pi$
$229$	−69948.9	−1.33386	−0.666929	+	0.745121i	$0.732392\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	58289.6	1.07369	0.536846	−	0.843681i	$-0.319616\pi$
$233$	58289.6	1.07369	0.536846	+	0.843681i	$0.319616\pi$
$234$	0	0
$235$	20613.5i	0.373264i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	83095.5i	1.45473i	0.686252	+	0.727364i	$0.259255\pi$
$239$	83095.5i	1.45473i	−0.686252	+	0.727364i	$0.740745\pi$
$240$	0	0
$241$	46933.8	0.808076	0.404038	−	0.914742i	$-0.367606\pi$
$241$	46933.8	0.808076	0.404038	+	0.914742i	$0.367606\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15264.1	0.254295
$246$	0	0
$247$	18310.1i	0.300121i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	56227.9i	0.892493i	0.894910	+	0.446246i	$0.147240\pi$
$251$	56227.9i	0.892493i	−0.894910	+	0.446246i	$0.852760\pi$
$252$	0	0
$253$	−904.390	−0.0141291
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15534.8	−0.235201	−0.117601	−	0.993061i	$-0.537520\pi$
$257$	−15534.8	−0.235201	−0.117601	+	0.993061i	$0.537520\pi$
$258$	0	0
$259$	− 43348.8i	− 0.646215i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	126497.i	1.82881i	0.404802	+	0.914404i	$0.367340\pi$
$263$	126497.i	1.82881i	−0.404802	+	0.914404i	$0.632660\pi$
$264$	0	0
$265$	−17940.7	−0.255474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	107881.	1.49087	0.745433	−	0.666580i	$-0.232243\pi$
$269$	107881.	1.49087	0.745433	+	0.666580i	$0.232243\pi$
$270$	0	0
$271$	82356.7i	1.12140i	0.828019	+	0.560700i	$0.189468\pi$
$271$	82356.7i	1.12140i	−0.828019	+	0.560700i	$0.810532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 226.149i	− 0.00299040i
$276$	0	0
$277$	−72796.3	−0.948746	−0.474373	−	0.880324i	$-0.657325\pi$
$277$	−72796.3	−0.948746	−0.474373	+	0.880324i	$0.657325\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	116309.	1.47300	0.736500	−	0.676438i	$-0.236477\pi$
$281$	116309.	1.47300	0.736500	+	0.676438i	$0.236477\pi$
$282$	0	0
$283$	134321.i	1.67715i	0.544788	+	0.838574i	$0.316610\pi$
$283$	134321.i	1.67715i	−0.544788	+	0.838574i	$0.683390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	57256.2i	0.695118i
$288$	0	0
$289$	56804.1	0.680118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	124186.	1.44657	0.723284	−	0.690550i	$-0.242632\pi$
$293$	124186.	1.44657	0.723284	+	0.690550i	$0.242632\pi$
$294$	0	0
$295$	16365.6i	0.188057i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 25114.6i	− 0.280921i
$300$	0	0
$301$	97646.1	1.07776
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33608.6	0.361285
$306$	0	0
$307$	− 49865.3i	− 0.529080i	−0.964375	−	0.264540i	$-0.914780\pi$
$307$	− 49865.3i	− 0.529080i	0.964375	−	0.264540i	$-0.0852202\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	74202.6i	0.767182i	0.923503	+	0.383591i	$0.125313\pi$
$311$	74202.6i	0.767182i	−0.923503	+	0.383591i	$0.874687\pi$
$312$	0	0
$313$	−49425.8	−0.504504	−0.252252	−	0.967662i	$-0.581171\pi$
$313$	−49425.8	−0.504504	−0.252252	+	0.967662i	$0.581171\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	54528.3	0.542630	0.271315	−	0.962491i	$-0.412541\pi$
$317$	54528.3	0.542630	0.271315	+	0.962491i	$0.412541\pi$
$318$	0	0
$319$	− 214.586i	− 0.00210872i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 136522.i	− 1.30857i
$324$	0	0
$325$	6280.07	0.0594563
$326$	0	0
$327$	0	0
$328$	0	0
$329$	59336.5	0.548189
$330$	0	0
$331$	− 76903.2i	− 0.701921i	−0.936390	−	0.350961i	$-0.885855\pi$
$331$	− 76903.2i	− 0.701921i	0.936390	−	0.350961i	$-0.114145\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	34890.0i	0.310893i
$336$	0	0
$337$	59178.9	0.521083	0.260542	−	0.965463i	$-0.416099\pi$
$337$	59178.9	0.521083	0.260542	+	0.965463i	$0.416099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	233.230	0.00200574
$342$	0	0
$343$	− 121209.i	− 1.03026i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 44386.0i	− 0.368627i	−0.982867	−	0.184314i	$-0.940994\pi$
$347$	− 44386.0i	− 0.368627i	0.982867	−	0.184314i	$-0.0590062\pi$
$348$	0	0
$349$	222882.	1.82988	0.914941	−	0.403587i	$-0.132237\pi$
$349$	222882.	1.82988	0.914941	+	0.403587i	$0.132237\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28966.3	−0.232458	−0.116229	−	0.993222i	$-0.537081\pi$
$353$	−28966.3	−0.232458	−0.116229	+	0.993222i	$0.537081\pi$
$354$	0	0
$355$	61857.8i	0.490837i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1777.59i	− 0.0137925i	−0.999976	−	0.00689624i	$-0.997805\pi$
$359$	− 1777.59i	− 0.0137925i	0.999976	−	0.00689624i	$-0.00219516\pi$
$360$	0	0
$361$	−2501.53	−0.0191951
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−37758.9	−0.283422
$366$	0	0
$367$	190911.i	1.41742i	0.705498	+	0.708712i	$0.250724\pi$
$367$	190911.i	1.41742i	−0.705498	+	0.708712i	$0.749276\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	51642.7i	0.375198i
$372$	0	0
$373$	−10573.7	−0.0759994	−0.0379997	−	0.999278i	$-0.512099\pi$
$373$	−10573.7	−0.0759994	−0.0379997	+	0.999278i	$0.512099\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5958.97	0.0419265
$378$	0	0
$379$	96649.6i	0.672855i	0.941709	+	0.336428i	$0.109219\pi$
$379$	96649.6i	0.672855i	−0.941709	+	0.336428i	$0.890781\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	223806.i	1.52572i	0.646564	+	0.762860i	$0.276205\pi$
$383$	223806.i	1.52572i	−0.646564	+	0.762860i	$0.723795\pi$
$384$	0	0
$385$	−650.976	−0.00439181
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−128478.	−0.849045	−0.424523	−	0.905417i	$-0.639558\pi$
$389$	−128478.	−0.849045	−0.424523	+	0.905417i	$0.639558\pi$
$390$	0	0
$391$	187257.i	1.22486i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 131548.i	− 0.843124i
$396$	0	0
$397$	38124.0	0.241890	0.120945	−	0.992659i	$-0.461408\pi$
$397$	38124.0	0.241890	0.120945	+	0.992659i	$0.461408\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	153511.	0.954667	0.477334	−	0.878722i	$-0.341603\pi$
$401$	153511.	0.954667	0.477334	+	0.878722i	$0.341603\pi$
$402$	0	0
$403$	6476.70i	0.0398789i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2436.89i	− 0.0147112i
$408$	0	0
$409$	112134.	0.670335	0.335167	−	0.942159i	$-0.391207\pi$
$409$	112134.	0.670335	0.335167	+	0.942159i	$0.391207\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	47108.9	0.276187
$414$	0	0
$415$	− 75129.7i	− 0.436230i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 95891.1i	− 0.546198i	−0.961986	−	0.273099i	$-0.911951\pi$
$419$	− 95891.1i	− 0.546198i	0.961986	−	0.273099i	$-0.0880487\pi$
$420$	0	0
$421$	−95023.9	−0.536128	−0.268064	−	0.963401i	$-0.586384\pi$
$421$	−95023.9	−0.536128	−0.268064	+	0.963401i	$0.586384\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−46825.0	−0.259239
$426$	0	0
$427$	− 96743.2i	− 0.530597i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10032.0i	0.0540051i	0.999635	+	0.0270026i	$0.00859623\pi$
$431$	10032.0i	0.0540051i	−0.999635	+	0.0270026i	$0.991404\pi$
$432$	0	0
$433$	−9765.08	−0.0520835	−0.0260418	−	0.999661i	$-0.508290\pi$
$433$	−9765.08	−0.0520835	−0.0260418	+	0.999661i	$0.508290\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	182183.	0.953991
$438$	0	0
$439$	334316.i	1.73471i	0.497686	+	0.867357i	$0.334183\pi$
$439$	334316.i	1.73471i	−0.497686	+	0.867357i	$0.665817\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	92435.9i	0.471013i	0.971873	+	0.235507i	$0.0756749\pi$
$443$	92435.9i	0.471013i	−0.971873	+	0.235507i	$0.924325\pi$
$444$	0	0
$445$	140161.	0.707795
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−254591.	−1.26285	−0.631423	−	0.775438i	$-0.717529\pi$
$449$	−254591.	−1.26285	−0.631423	+	0.775438i	$0.717529\pi$
$450$	0	0
$451$	3218.71i	0.0158245i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 18077.4i	− 0.0873197i
$456$	0	0
$457$	−195259.	−0.934930	−0.467465	−	0.884012i	$-0.654833\pi$
$457$	−195259.	−0.934930	−0.467465	+	0.884012i	$0.654833\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	58243.8	0.274062	0.137031	−	0.990567i	$-0.456244\pi$
$461$	58243.8	0.274062	0.137031	+	0.990567i	$0.456244\pi$
$462$	0	0
$463$	− 140061.i	− 0.653363i	−0.945135	−	0.326681i	$-0.894070\pi$
$463$	− 140061.i	− 0.653363i	0.945135	−	0.326681i	$-0.105930\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	151762.i	0.695872i	0.937518	+	0.347936i	$0.113117\pi$
$467$	151762.i	0.695872i	−0.937518	+	0.347936i	$0.886883\pi$
$468$	0	0
$469$	100432.	0.456589
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5489.26	0.0245353
$474$	0	0
$475$	45556.0i	0.201910i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 67796.2i	− 0.295484i	−0.989026	−	0.147742i	$-0.952799\pi$
$479$	− 67796.2i	− 0.295484i	0.989026	−	0.147742i	$-0.0472006\pi$
$480$	0	0
$481$	67671.6	0.292493
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10827.0	−0.0460281
$486$	0	0
$487$	82482.5i	0.347779i	0.984765	+	0.173890i	$0.0556336\pi$
$487$	82482.5i	0.347779i	−0.984765	+	0.173890i	$0.944366\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	432389.i	1.79354i	0.442496	+	0.896771i	$0.354093\pi$
$491$	432389.i	1.79354i	−0.442496	+	0.896771i	$0.645907\pi$
$492$	0	0
$493$	−44430.8	−0.182806
$494$	0	0
$495$	0	0
$496$	0	0
$497$	178059.	0.720861
$498$	0	0
$499$	− 457217.i	− 1.83621i	−0.396343	−	0.918103i	$-0.629721\pi$
$499$	− 457217.i	− 1.83621i	0.396343	−	0.918103i	$-0.370279\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	409308.i	1.61776i	0.587973	+	0.808881i	$0.299926\pi$
$503$	409308.i	1.61776i	−0.587973	+	0.808881i	$0.700074\pi$
$504$	0	0
$505$	168877.	0.662198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	69419.2	0.267944	0.133972	−	0.990985i	$-0.457227\pi$
$509$	69419.2	0.267944	0.133972	+	0.990985i	$0.457227\pi$
$510$	0	0
$511$	108690.i	0.416244i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	399.811i	0.00150744i
$516$	0	0
$517$	3335.65	0.0124796
$518$	0	0
$519$	0	0
$520$	0	0
$521$	326815.	1.20400	0.602001	−	0.798496i	$-0.294370\pi$
$521$	326815.	1.20400	0.602001	+	0.798496i	$0.294370\pi$
$522$	0	0
$523$	− 541900.i	− 1.98114i	−0.137002	−	0.990571i	$-0.543747\pi$
$523$	− 541900.i	− 1.98114i	0.137002	−	0.990571i	$-0.456253\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 48291.1i	− 0.173878i
$528$	0	0
$529$	29954.5	0.107041
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−89382.4	−0.314628
$534$	0	0
$535$	119531.i	0.417611i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2470.02i	− 0.00850203i
$540$	0	0
$541$	−253798.	−0.867150	−0.433575	−	0.901118i	$-0.642748\pi$
$541$	−253798.	−0.867150	−0.433575	+	0.901118i	$0.642748\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−60768.8	−0.204591
$546$	0	0
$547$	− 421251.i	− 1.40788i	−0.710259	−	0.703940i	$-0.751422\pi$
$547$	− 421251.i	− 1.40788i	0.710259	−	0.703940i	$-0.248578\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	43226.7i	0.142380i
$552$	0	0
$553$	−378666.	−1.23824
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−117888.	−0.379979	−0.189990	−	0.981786i	$-0.560845\pi$
$557$	−117888.	−0.379979	−0.189990	+	0.981786i	$0.560845\pi$
$558$	0	0
$559$	152435.i	0.487821i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 160395.i	− 0.506026i	−0.967463	−	0.253013i	$-0.918578\pi$
$563$	− 160395.i	− 0.506026i	0.967463	−	0.253013i	$-0.0814216\pi$
$564$	0	0
$565$	−174230.	−0.545791
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−288064.	−0.889744	−0.444872	−	0.895594i	$-0.646751\pi$
$569$	−288064.	−0.889744	−0.444872	+	0.895594i	$0.646751\pi$
$570$	0	0
$571$	− 225085.i	− 0.690358i	−0.938537	−	0.345179i	$-0.887818\pi$
$571$	− 225085.i	− 0.690358i	0.938537	−	0.345179i	$-0.112182\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 62485.8i	− 0.188993i
$576$	0	0
$577$	−583308.	−1.75205	−0.876024	−	0.482267i	$-0.839814\pi$
$577$	−583308.	−1.75205	−0.876024	+	0.482267i	$0.839814\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−216263.	−0.640663
$582$	0	0
$583$	2903.14i	0.00854143i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 211122.i	− 0.612713i	−0.951917	−	0.306357i	$-0.900890\pi$
$587$	− 211122.i	− 0.612713i	0.951917	−	0.306357i	$-0.0991100\pi$
$588$	0	0
$589$	−46982.4	−0.135427
$590$	0	0
$591$	0	0
$592$	0	0
$593$	283082.	0.805014	0.402507	−	0.915417i	$-0.368139\pi$
$593$	283082.	0.805014	0.402507	+	0.915417i	$0.368139\pi$
$594$	0	0
$595$	134787.i	0.380727i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	123497.i	0.344194i	0.985080	+	0.172097i	$0.0550542\pi$
$599$	123497.i	0.344194i	−0.985080	+	0.172097i	$0.944946\pi$
$600$	0	0
$601$	266514.	0.737856	0.368928	−	0.929458i	$-0.379725\pi$
$601$	266514.	0.737856	0.368928	+	0.929458i	$0.379725\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	163655.	0.447114
$606$	0	0
$607$	436937.i	1.18588i	0.805246	+	0.592940i	$0.202033\pi$
$607$	436937.i	1.18588i	−0.805246	+	0.592940i	$0.797967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	92629.9i	0.248124i
$612$	0	0
$613$	371598.	0.988900	0.494450	−	0.869206i	$-0.335370\pi$
$613$	371598.	0.988900	0.494450	+	0.869206i	$0.335370\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−185390.	−0.486986	−0.243493	−	0.969903i	$-0.578293\pi$
$617$	−185390.	−0.486986	−0.243493	+	0.969903i	$0.578293\pi$
$618$	0	0
$619$	175656.i	0.458439i	0.973375	+	0.229219i	$0.0736173\pi$
$619$	175656.i	0.458439i	−0.973375	+	0.229219i	$0.926383\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 403457.i	− 1.03949i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−504567.	−1.27532
$630$	0	0
$631$	234115.i	0.587991i	0.955807	+	0.293996i	$0.0949851\pi$
$631$	234115.i	0.587991i	−0.955807	+	0.293996i	$0.905015\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17113.9i	0.0424425i
$636$	0	0
$637$	68591.5	0.169041
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−243706.	−0.593130	−0.296565	−	0.955013i	$-0.595841\pi$
$641$	−243706.	−0.593130	−0.296565	+	0.955013i	$0.595841\pi$
$642$	0	0
$643$	− 340234.i	− 0.822917i	−0.911428	−	0.411459i	$-0.865019\pi$
$643$	− 340234.i	− 0.822917i	0.911428	−	0.411459i	$-0.134981\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 143252.i	− 0.342210i	−0.985253	−	0.171105i	$-0.945266\pi$
$647$	− 143252.i	− 0.342210i	0.985253	−	0.171105i	$-0.0547337\pi$
$648$	0	0
$649$	2648.27	0.00628743
$650$	0	0
$651$	0	0
$652$	0	0
$653$	246942.	0.579121	0.289561	−	0.957160i	$-0.406491\pi$
$653$	246942.	0.579121	0.289561	+	0.957160i	$0.406491\pi$
$654$	0	0
$655$	28080.0i	0.0654507i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 87444.3i	− 0.201354i	−0.994919	−	0.100677i	$-0.967899\pi$
$659$	− 87444.3i	− 0.201354i	0.994919	−	0.100677i	$-0.0321009\pi$
$660$	0	0
$661$	566207.	1.29590	0.647952	−	0.761681i	$-0.275626\pi$
$661$	566207.	1.29590	0.647952	+	0.761681i	$0.275626\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	131134.	0.296533
$666$	0	0
$667$	− 59290.8i	− 0.133271i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 5438.51i	− 0.0120791i
$672$	0	0
$673$	401892.	0.887318	0.443659	−	0.896196i	$-0.353680\pi$
$673$	401892.	0.887318	0.443659	+	0.896196i	$0.353680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−339705.	−0.741182	−0.370591	−	0.928796i	$-0.620845\pi$
$677$	−339705.	−0.741182	−0.370591	+	0.928796i	$0.620845\pi$
$678$	0	0
$679$	31165.7i	0.0675986i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 360329.i	− 0.772428i	−0.922409	−	0.386214i	$-0.873783\pi$
$683$	− 360329.i	− 0.772428i	0.922409	−	0.386214i	$-0.126217\pi$
$684$	0	0
$685$	90515.6	0.192904
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−80619.1	−0.169824
$690$	0	0
$691$	419757.i	0.879108i	0.898216	+	0.439554i	$0.144863\pi$
$691$	419757.i	0.879108i	−0.898216	+	0.439554i	$0.855137\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 225930.i	− 0.467740i
$696$	0	0
$697$	666446.	1.37183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−123094.	−0.250495	−0.125248	−	0.992126i	$-0.539973\pi$
$701$	−123094.	−0.250495	−0.125248	+	0.992126i	$0.539973\pi$
$702$	0	0
$703$	490894.i	0.993292i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 486117.i	− 0.972528i
$708$	0	0
$709$	−330597.	−0.657668	−0.328834	−	0.944388i	$-0.606656\pi$
$709$	−330597.	−0.657668	−0.328834	+	0.944388i	$0.606656\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	64442.2	0.126763
$714$	0	0
$715$	− 1016.24i	− 0.00198784i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 78007.6i	− 0.150896i	−0.997150	−	0.0754482i	$-0.975961\pi$
$719$	− 78007.6i	− 0.150896i	0.997150	−	0.0754482i	$-0.0240388\pi$
$720$	0	0
$721$	1150.87	0.00221388
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14826.1	0.0282066
$726$	0	0
$727$	− 635393.i	− 1.20219i	−0.799177	−	0.601096i	$-0.794731\pi$
$727$	− 635393.i	− 1.20219i	0.799177	−	0.601096i	$-0.205269\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.13657e6i	− 2.12698i
$732$	0	0
$733$	−73723.1	−0.137213	−0.0686066	−	0.997644i	$-0.521855\pi$
$733$	−73723.1	−0.137213	−0.0686066	+	0.997644i	$0.521855\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5645.87	0.0103943
$738$	0	0
$739$	− 523323.i	− 0.958255i	−0.877745	−	0.479128i	$-0.840953\pi$
$739$	− 523323.i	− 0.958255i	0.877745	−	0.479128i	$-0.159047\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 281437.i	− 0.509804i	−0.966967	−	0.254902i	$-0.917957\pi$
$743$	− 281437.i	− 0.509804i	0.966967	−	0.254902i	$-0.0820432\pi$
$744$	0	0
$745$	237291.	0.427532
$746$	0	0
$747$	0	0
$748$	0	0
$749$	344072.	0.613318
$750$	0	0
$751$	165291.i	0.293069i	0.989206	+	0.146535i	$0.0468119\pi$
$751$	165291.i	0.293069i	−0.989206	+	0.146535i	$0.953188\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	440450.i	0.772686i
$756$	0	0
$757$	−790794.	−1.37998	−0.689988	−	0.723821i	$-0.742384\pi$
$757$	−790794.	−1.37998	−0.689988	+	0.723821i	$0.742384\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−899919.	−1.55394	−0.776970	−	0.629538i	$-0.783244\pi$
$761$	−899919.	−1.55394	−0.776970	+	0.629538i	$0.783244\pi$
$762$	0	0
$763$	174925.i	0.300470i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	73541.6i	0.125009i
$768$	0	0
$769$	−415583.	−0.702757	−0.351378	−	0.936234i	$-0.614287\pi$
$769$	−415583.	−0.702757	−0.351378	+	0.936234i	$0.614287\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−739588.	−1.23774	−0.618872	−	0.785492i	$-0.712410\pi$
$773$	−739588.	−1.23774	−0.618872	+	0.785492i	$0.712410\pi$
$774$	0	0
$775$	16114.2i	0.0268291i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 648385.i	− 1.06846i
$780$	0	0
$781$	10009.8	0.0164105
$782$	0	0
$783$	0	0
$784$	0	0
$785$	307370.	0.498795
$786$	0	0
$787$	258121.i	0.416749i	0.978049	+	0.208375i	$0.0668173\pi$
$787$	258121.i	0.416749i	−0.978049	+	0.208375i	$0.933183\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	501526.i	0.801568i
$792$	0	0
$793$	151025.	0.240161
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−61353.4	−0.0965878	−0.0482939	−	0.998833i	$-0.515378\pi$
$797$	−61353.4	−0.0965878	−0.0482939	+	0.998833i	$0.515378\pi$
$798$	0	0
$799$	− 690660.i	− 1.08186i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6110.10i	0.00947584i
$804$	0	0
$805$	−179867.	−0.277562
$806$	0	0
$807$	0	0
$808$	0	0
$809$	847065.	1.29425	0.647127	−	0.762382i	$-0.275970\pi$
$809$	847065.	1.29425	0.647127	+	0.762382i	$0.275970\pi$
$810$	0	0
$811$	147315.i	0.223978i	0.993709	+	0.111989i	$0.0357221\pi$
$811$	147315.i	0.223978i	−0.993709	+	0.111989i	$0.964278\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	408750.i	0.615378i
$816$	0	0
$817$	−1.10577e6	−1.65661
$818$	0	0
$819$	0	0
$820$	0	0
$821$	168969.	0.250681	0.125340	−	0.992114i	$-0.459998\pi$
$821$	168969.	0.250681	0.125340	+	0.992114i	$0.459998\pi$
$822$	0	0
$823$	542532.i	0.800988i	0.916299	+	0.400494i	$0.131161\pi$
$823$	542532.i	0.800988i	−0.916299	+	0.400494i	$0.868839\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 432833.i	− 0.632862i	−0.948616	−	0.316431i	$-0.897515\pi$
$827$	− 432833.i	− 0.632862i	0.948616	−	0.316431i	$-0.102485\pi$
$828$	0	0
$829$	−1.32620e6	−1.92975	−0.964876	−	0.262708i	$-0.915385\pi$
$829$	−1.32620e6	−1.92975	−0.964876	+	0.262708i	$0.915385\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−511427.	−0.737044
$834$	0	0
$835$	− 402290.i	− 0.576987i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	879452.i	1.24936i	0.780880	+	0.624681i	$0.214771\pi$
$839$	879452.i	1.24936i	−0.780880	+	0.624681i	$0.785229\pi$
$840$	0	0
$841$	−693213.	−0.980110
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−291101.	−0.407690
$846$	0	0
$847$	− 471085.i	− 0.656647i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 673322.i	− 0.929745i
$852$	0	0
$853$	543116.	0.746440	0.373220	−	0.927743i	$-0.378254\pi$
$853$	543116.	0.746440	0.373220	+	0.927743i	$0.378254\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.45508e6	−1.98119	−0.990593	−	0.136842i	$-0.956305\pi$
$857$	−1.45508e6	−1.98119	−0.990593	+	0.136842i	$0.956305\pi$
$858$	0	0
$859$	− 1.03526e6i	− 1.40302i	−0.712658	−	0.701512i	$-0.752509\pi$
$859$	− 1.03526e6i	− 1.40302i	0.712658	−	0.701512i	$-0.247491\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 133280.i	− 0.178955i	−0.995989	−	0.0894776i	$-0.971480\pi$
$863$	− 133280.i	− 0.178955i	0.995989	−	0.0894776i	$-0.0285198\pi$
$864$	0	0
$865$	−469409.	−0.627363
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21287.1	−0.0281888
$870$	0	0
$871$	156784.i	0.206664i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 44977.0i	− 0.0587454i
$876$	0	0
$877$	−379461.	−0.493365	−0.246682	−	0.969096i	$-0.579340\pi$
$877$	−379461.	−0.493365	−0.246682	+	0.969096i	$0.579340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.04171e6	1.34213	0.671063	−	0.741400i	$-0.265838\pi$
$881$	1.04171e6	1.34213	0.671063	+	0.741400i	$0.265838\pi$
$882$	0	0
$883$	− 828406.i	− 1.06248i	−0.847220	−	0.531241i	$-0.821726\pi$
$883$	− 828406.i	− 1.06248i	0.847220	−	0.531241i	$-0.178274\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 758303.i	− 0.963819i	−0.876221	−	0.481910i	$-0.839943\pi$
$887$	− 758303.i	− 0.963819i	0.876221	−	0.481910i	$-0.160057\pi$
$888$	0	0
$889$	49262.7	0.0623325
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−671943.	−0.842616
$894$	0	0
$895$	618291.i	0.771875i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15290.3i	0.0189189i
$900$	0	0
$901$	601106.	0.740460
$902$	0	0
$903$	0	0
$904$	0	0
$905$	75011.5	0.0915864
$906$	0	0
$907$	− 449282.i	− 0.546141i	−0.961994	−	0.273070i	$-0.911961\pi$
$907$	− 449282.i	− 0.546141i	0.961994	−	0.273070i	$-0.0880392\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	828118.i	0.997827i	0.866652	+	0.498914i	$0.166267\pi$
$911$	828118.i	0.997827i	−0.866652	+	0.498914i	$0.833733\pi$
$912$	0	0
$913$	−12157.4	−0.0145848
$914$	0	0
$915$	0	0
$916$	0	0
$917$	80829.0	0.0961232
$918$	0	0
$919$	− 1.27981e6i	− 1.51535i	−0.652630	−	0.757677i	$-0.726334\pi$
$919$	− 1.27981e6i	− 1.51535i	0.652630	−	0.757677i	$-0.273666\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	277968.i	0.326280i
$924$	0	0
$925$	168369.	0.196779
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.12164e6	−1.29964	−0.649819	−	0.760089i	$-0.725155\pi$
$929$	−1.12164e6	−1.29964	−0.649819	+	0.760089i	$0.725155\pi$
$930$	0	0
$931$	497567.i	0.574053i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7577.17i	0.00866731i
$936$	0	0
$937$	−746398.	−0.850142	−0.425071	−	0.905160i	$-0.639751\pi$
$937$	−746398.	−0.850142	−0.425071	+	0.905160i	$0.639751\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1071.73	0.00121034	0.000605169	−	1.00000i	$-0.499807\pi$
$941$	1071.73	0.00121034	0.000605169		1.00000i	$0.499807\pi$
$942$	0	0
$943$	889342.i	1.00010i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 383292.i	− 0.427395i	−0.976900	−	0.213697i	$-0.931449\pi$
$947$	− 383292.i	− 0.427395i	0.976900	−	0.213697i	$-0.0685507\pi$
$948$	0	0
$949$	−169675.	−0.188402
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.09716e6	−1.20805	−0.604025	−	0.796965i	$-0.706437\pi$
$953$	−1.09716e6	−1.20805	−0.604025	+	0.796965i	$0.706437\pi$
$954$	0	0
$955$	− 643186.i	− 0.705228i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 260552.i	− 0.283306i
$960$	0	0
$961$	906902.	0.982005
$962$	0	0
$963$	0	0
$964$	0	0
$965$	614470.	0.659851
$966$	0	0
$967$	− 135722.i	− 0.145143i	−0.997363	−	0.0725716i	$-0.976879\pi$
$967$	− 135722.i	− 0.145143i	0.997363	−	0.0725716i	$-0.0231206\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 914675.i	− 0.970126i	−0.874479	−	0.485063i	$-0.838797\pi$
$971$	− 914675.i	− 0.970126i	0.874479	−	0.485063i	$-0.161203\pi$
$972$	0	0
$973$	−650346.	−0.686940
$974$	0	0
$975$	0	0
$976$	0	0
$977$	714583.	0.748624	0.374312	−	0.927303i	$-0.377879\pi$
$977$	714583.	0.748624	0.374312	+	0.927303i	$0.377879\pi$
$978$	0	0
$979$	− 22680.7i	− 0.0236642i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.74144e6i	− 1.80219i	−0.433618	−	0.901097i	$-0.642763\pi$
$983$	− 1.74144e6i	− 1.80219i	0.433618	−	0.901097i	$-0.357237\pi$
$984$	0	0
$985$	489092.	0.504101
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.51670e6	1.55063
$990$	0	0
$991$	1.40429e6i	1.42991i	0.699170	+	0.714955i	$0.253553\pi$
$991$	1.40429e6i	1.42991i	−0.699170	+	0.714955i	$0.746447\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	170636.i	0.172355i
$996$	0	0
$997$	−927646.	−0.933237	−0.466619	−	0.884459i	$-0.654528\pi$
$997$	−927646.	−0.933237	−0.466619	+	0.884459i	$0.654528\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.5.e.b.991.6		8
3.2	odd	2		160.5.b.a.31.5	yes	8
4.3	odd	2	inner	1440.5.e.b.991.7		8
12.11	even	2		160.5.b.a.31.4	✓	8
15.2	even	4		800.5.h.l.799.6		8
15.8	even	4		800.5.h.k.799.4		8
15.14	odd	2		800.5.b.g.351.4		8
24.5	odd	2		320.5.b.c.191.4		8
24.11	even	2		320.5.b.c.191.5		8
60.23	odd	4		800.5.h.l.799.5		8
60.47	odd	4		800.5.h.k.799.3		8
60.59	even	2		800.5.b.g.351.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.5.b.a.31.4	✓	8	12.11	even	2
160.5.b.a.31.5	yes	8	3.2	odd	2
320.5.b.c.191.4		8	24.5	odd	2
320.5.b.c.191.5		8	24.11	even	2
800.5.b.g.351.4		8	15.14	odd	2
800.5.b.g.351.5		8	60.59	even	2
800.5.h.k.799.3		8	60.47	odd	4
800.5.h.k.799.4		8	15.8	even	4
800.5.h.l.799.5		8	60.23	odd	4
800.5.h.l.799.6		8	15.2	even	4
1440.5.e.b.991.6		8	1.1	even	1	trivial
1440.5.e.b.991.7		8	4.3	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1440.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.1803 q^{5} -32.1829i q^{7} -1.80919i q^{11} +50.2406 q^{13} -374.600 q^{17} +364.448i q^{19} -499.886i q^{23} +125.000 q^{25} +118.609 q^{29} +128.914i q^{31} -359.816i q^{35} +1346.95 q^{37} -1779.09 q^{41} +3034.10i q^{43} +1843.73i q^{47} +1365.26 q^{49} -1604.66 q^{53} -20.2274i q^{55} +1463.79i q^{59} +3006.04 q^{61} +561.707 q^{65} +3120.66i q^{67} +5532.73i q^{71} -3377.26 q^{73} -58.2250 q^{77} -11766.1i q^{79} -6719.80i q^{83} -4188.15 q^{85} +12536.4 q^{89} -1616.89i q^{91} +4074.66i q^{95} -968.394 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{13} + 560 q^{17} + 1000 q^{25} + 2064 q^{29} + 3616 q^{37} + 5216 q^{41} + 4088 q^{49} - 5088 q^{53} - 5504 q^{61} + 7600 q^{65} + 17936 q^{73} + 15840 q^{77} - 9600 q^{85} + 50608 q^{89} + 3440 q^{97}+O(q^{100}) \) 8 * q + 32 * q^13 + 560 * q^17 + 1000 * q^25 + 2064 * q^29 + 3616 * q^37 + 5216 * q^41 + 4088 * q^49 - 5088 * q^53 - 5504 * q^61 + 7600 * q^65 + 17936 * q^73 + 15840 * q^77 - 9600 * q^85 + 50608 * q^89 + 3440 * q^97

Introduction

L-functions

Modular forms

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Database

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