LMFDB - Embedded newform 384.5.b.b.319.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(319,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.b (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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.5.b.b.319.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -12.0000i q^{5} +62.3538i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -12.0000i q^{5} +62.3538i q^{7} +27.0000 q^{9} +20.7846 q^{11} -264.000i q^{13} -62.3538i q^{15} +110.000 q^{17} +103.923 q^{19} +324.000i q^{21} -124.708i q^{23} +481.000 q^{25} +140.296 q^{27} +228.000i q^{29} +1434.14i q^{31} +108.000 q^{33} +748.246 q^{35} +1392.00i q^{37} -1371.78i q^{39} +1282.00 q^{41} +2514.94 q^{43} -324.000i q^{45} -2618.86i q^{47} -1487.00 q^{49} +571.577 q^{51} -4500.00i q^{53} -249.415i q^{55} +540.000 q^{57} +6339.31 q^{59} +960.000i q^{61} +1683.55i q^{63} -3168.00 q^{65} +3263.18 q^{67} -648.000i q^{69} +6110.68i q^{71} -3170.00 q^{73} +2499.35 q^{75} +1296.00i q^{77} -1558.85i q^{79} +729.000 q^{81} +4427.12 q^{83} -1320.00i q^{85} +1184.72i q^{87} +1550.00 q^{89} +16461.4 q^{91} +7452.00i q^{93} -1247.08i q^{95} -8018.00 q^{97} +561.184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9}+O(q^{10}) $ 4 * q + 108 * q^9	$ 4 q + 108 q^{9} + 440 q^{17} + 1924 q^{25} + 432 q^{33} + 5128 q^{41} - 5948 q^{49} + 2160 q^{57} - 12672 q^{65} - 12680 q^{73} + 2916 q^{81} + 6200 q^{89} - 32072 q^{97}+O(q^{100}) $ 4 * q + 108 * q^9 + 440 * q^17 + 1924 * q^25 + 432 * q^33 + 5128 * q^41 - 5948 * q^49 + 2160 * q^57 - 12672 * q^65 - 12680 * q^73 + 2916 * q^81 + 6200 * q^89 - 32072 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	− 12.0000i	− 0.480000i	−0.970773	−	0.240000i	$-0.922853\pi$
$5$	− 12.0000i	− 0.480000i	0.970773	−	0.240000i	$-0.0771474\pi$
$6$	0	0
$7$	62.3538i	1.27253i	0.771472	+	0.636264i	$0.219521\pi$
$7$	62.3538i	1.27253i	−0.771472	+	0.636264i	$0.780479\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	20.7846	0.171774	0.0858868	−	0.996305i	$-0.472628\pi$
$11$	20.7846	0.171774	0.0858868	+	0.996305i	$0.472628\pi$
$12$	0	0
$13$	− 264.000i	− 1.56213i	−0.624450	−	0.781065i	$-0.714677\pi$
$13$	− 264.000i	− 1.56213i	0.624450	−	0.781065i	$-0.285323\pi$
$14$	0	0
$15$	− 62.3538i	− 0.277128i
$16$	0	0
$17$	110.000	0.380623	0.190311	−	0.981724i	$-0.439050\pi$
$17$	110.000	0.380623	0.190311	+	0.981724i	$0.439050\pi$
$18$	0	0
$19$	103.923	0.287875	0.143938	−	0.989587i	$-0.454023\pi$
$19$	103.923	0.287875	0.143938	+	0.989587i	$0.454023\pi$
$20$	0	0
$21$	324.000i	0.734694i
$22$	0	0
$23$	− 124.708i	− 0.235742i	−0.993029	−	0.117871i	$-0.962393\pi$
$23$	− 124.708i	− 0.235742i	0.993029	−	0.117871i	$-0.0376070\pi$
$24$	0	0
$25$	481.000	0.769600
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	228.000i	0.271106i	0.990770	+	0.135553i	$0.0432811\pi$
$29$	228.000i	0.271106i	−0.990770	+	0.135553i	$0.956719\pi$
$30$	0	0
$31$	1434.14i	1.49234i	0.665756	+	0.746170i	$0.268109\pi$
$31$	1434.14i	1.49234i	−0.665756	+	0.746170i	$0.731891\pi$
$32$	0	0
$33$	108.000	0.0991736
$34$	0	0
$35$	748.246	0.610813
$36$	0	0
$37$	1392.00i	1.01680i	0.861121	+	0.508400i	$0.169763\pi$
$37$	1392.00i	1.01680i	−0.861121	+	0.508400i	$0.830237\pi$
$38$	0	0
$39$	− 1371.78i	− 0.901896i
$40$	0	0
$41$	1282.00	0.762641	0.381321	−	0.924443i	$-0.375469\pi$
$41$	1282.00	0.762641	0.381321	+	0.924443i	$0.375469\pi$
$42$	0	0
$43$	2514.94	1.36016	0.680081	−	0.733137i	$-0.261945\pi$
$43$	2514.94	1.36016	0.680081	+	0.733137i	$0.261945\pi$
$44$	0	0
$45$	− 324.000i	− 0.160000i
$46$	0	0
$47$	− 2618.86i	− 1.18554i	−0.805371	−	0.592771i	$-0.798034\pi$
$47$	− 2618.86i	− 1.18554i	0.805371	−	0.592771i	$-0.201966\pi$
$48$	0	0
$49$	−1487.00	−0.619325
$50$	0	0
$51$	571.577	0.219753
$52$	0	0
$53$	− 4500.00i	− 1.60199i	−0.598669	−	0.800997i	$-0.704303\pi$
$53$	− 4500.00i	− 1.60199i	0.598669	−	0.800997i	$-0.295697\pi$
$54$	0	0
$55$	− 249.415i	− 0.0824513i
$56$	0	0
$57$	540.000	0.166205
$58$	0	0
$59$	6339.31	1.82112	0.910558	−	0.413381i	$-0.135652\pi$
$59$	6339.31	1.82112	0.910558	+	0.413381i	$0.135652\pi$
$60$	0	0
$61$	960.000i	0.257995i	0.991645	+	0.128998i	$0.0411759\pi$
$61$	960.000i	0.257995i	−0.991645	+	0.128998i	$0.958824\pi$
$62$	0	0
$63$	1683.55i	0.424176i
$64$	0	0
$65$	−3168.00	−0.749822
$66$	0	0
$67$	3263.18	0.726929	0.363464	−	0.931608i	$-0.381594\pi$
$67$	3263.18	0.726929	0.363464	+	0.931608i	$0.381594\pi$
$68$	0	0
$69$	− 648.000i	− 0.136106i
$70$	0	0
$71$	6110.68i	1.21220i	0.795390	+	0.606098i	$0.207266\pi$
$71$	6110.68i	1.21220i	−0.795390	+	0.606098i	$0.792734\pi$
$72$	0	0
$73$	−3170.00	−0.594858	−0.297429	−	0.954744i	$-0.596129\pi$
$73$	−3170.00	−0.594858	−0.297429	+	0.954744i	$0.596129\pi$
$74$	0	0
$75$	2499.35	0.444329
$76$	0	0
$77$	1296.00i	0.218587i
$78$	0	0
$79$	− 1558.85i	− 0.249775i	−0.992171	−	0.124887i	$-0.960143\pi$
$79$	− 1558.85i	− 0.249775i	0.992171	−	0.124887i	$-0.0398570\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	4427.12	0.642636	0.321318	−	0.946971i	$-0.395874\pi$
$83$	4427.12	0.642636	0.321318	+	0.946971i	$0.395874\pi$
$84$	0	0
$85$	− 1320.00i	− 0.182699i
$86$	0	0
$87$	1184.72i	0.156523i
$88$	0	0
$89$	1550.00	0.195682	0.0978412	−	0.995202i	$-0.468806\pi$
$89$	1550.00	0.195682	0.0978412	+	0.995202i	$0.468806\pi$
$90$	0	0
$91$	16461.4	1.98785
$92$	0	0
$93$	7452.00i	0.861602i
$94$	0	0
$95$	− 1247.08i	− 0.138180i
$96$	0	0
$97$	−8018.00	−0.852163	−0.426081	−	0.904685i	$-0.640106\pi$
$97$	−8018.00	−0.852163	−0.426081	+	0.904685i	$0.640106\pi$
$98$	0	0
$99$	561.184	0.0572579
$100$	0	0
$101$	15516.0i	1.52103i	0.649322	+	0.760514i	$0.275053\pi$
$101$	15516.0i	1.52103i	−0.649322	+	0.760514i	$0.724947\pi$
$102$	0	0
$103$	− 11535.5i	− 1.08733i	−0.839303	−	0.543664i	$-0.817037\pi$
$103$	− 11535.5i	− 1.08733i	0.839303	−	0.543664i	$-0.182963\pi$
$104$	0	0
$105$	3888.00	0.352653
$106$	0	0
$107$	19059.5	1.66473	0.832365	−	0.554228i	$-0.186987\pi$
$107$	19059.5	1.66473	0.832365	+	0.554228i	$0.186987\pi$
$108$	0	0
$109$	− 10968.0i	− 0.923155i	−0.887100	−	0.461577i	$-0.847284\pi$
$109$	− 10968.0i	− 0.923155i	0.887100	−	0.461577i	$-0.152716\pi$
$110$	0	0
$111$	7233.04i	0.587050i
$112$	0	0
$113$	10786.0	0.844702	0.422351	−	0.906432i	$-0.361205\pi$
$113$	10786.0	0.844702	0.422351	+	0.906432i	$0.361205\pi$
$114$	0	0
$115$	−1496.49	−0.113156
$116$	0	0
$117$	− 7128.00i	− 0.520710i
$118$	0	0
$119$	6858.92i	0.484353i
$120$	0	0
$121$	−14209.0	−0.970494
$122$	0	0
$123$	6661.47	0.440311
$124$	0	0
$125$	− 13272.0i	− 0.849408i
$126$	0	0
$127$	− 6422.44i	− 0.398192i	−0.979980	−	0.199096i	$-0.936199\pi$
$127$	− 6422.44i	− 0.398192i	0.979980	−	0.199096i	$-0.0638006\pi$
$128$	0	0
$129$	13068.0	0.785289
$130$	0	0
$131$	−16024.9	−0.933800	−0.466900	−	0.884310i	$-0.654629\pi$
$131$	−16024.9	−0.933800	−0.466900	+	0.884310i	$0.654629\pi$
$132$	0	0
$133$	6480.00i	0.366329i
$134$	0	0
$135$	− 1683.55i	− 0.0923760i
$136$	0	0
$137$	17794.0	0.948053	0.474026	−	0.880511i	$-0.342800\pi$
$137$	17794.0	0.948053	0.474026	+	0.880511i	$0.342800\pi$
$138$	0	0
$139$	17313.6	0.896102	0.448051	−	0.894008i	$-0.352118\pi$
$139$	17313.6	0.896102	0.448051	+	0.894008i	$0.352118\pi$
$140$	0	0
$141$	− 13608.0i	− 0.684473i
$142$	0	0
$143$	− 5487.14i	− 0.268333i
$144$	0	0
$145$	2736.00	0.130131
$146$	0	0
$147$	−7726.68	−0.357568
$148$	0	0
$149$	− 13596.0i	− 0.612405i	−0.951966	−	0.306202i	$-0.900942\pi$
$149$	− 13596.0i	− 0.612405i	0.951966	−	0.306202i	$-0.0990584\pi$
$150$	0	0
$151$	20763.8i	0.910654i	0.890324	+	0.455327i	$0.150478\pi$
$151$	20763.8i	0.910654i	−0.890324	+	0.455327i	$0.849522\pi$
$152$	0	0
$153$	2970.00	0.126874
$154$	0	0
$155$	17209.7	0.716323
$156$	0	0
$157$	12432.0i	0.504361i	0.967680	+	0.252181i	$0.0811477\pi$
$157$	12432.0i	0.504361i	−0.967680	+	0.252181i	$0.918852\pi$
$158$	0	0
$159$	− 23382.7i	− 0.924911i
$160$	0	0
$161$	7776.00	0.299988
$162$	0	0
$163$	−43710.0	−1.64515	−0.822576	−	0.568655i	$-0.807464\pi$
$163$	−43710.0	−1.64515	−0.822576	+	0.568655i	$0.807464\pi$
$164$	0	0
$165$	− 1296.00i	− 0.0476033i
$166$	0	0
$167$	− 43148.8i	− 1.54716i	−0.633696	−	0.773582i	$-0.718463\pi$
$167$	− 43148.8i	− 1.54716i	0.633696	−	0.773582i	$-0.281537\pi$
$168$	0	0
$169$	−41135.0	−1.44025
$170$	0	0
$171$	2805.92	0.0959585
$172$	0	0
$173$	53628.0i	1.79184i	0.444215	+	0.895920i	$0.353483\pi$
$173$	53628.0i	1.79184i	−0.444215	+	0.895920i	$0.646517\pi$
$174$	0	0
$175$	29992.2i	0.979337i
$176$	0	0
$177$	32940.0	1.05142
$178$	0	0
$179$	−40717.1	−1.27078	−0.635390	−	0.772192i	$-0.719161\pi$
$179$	−40717.1	−1.27078	−0.635390	+	0.772192i	$0.719161\pi$
$180$	0	0
$181$	8568.00i	0.261530i	0.991413	+	0.130765i	$0.0417434\pi$
$181$	8568.00i	0.261530i	−0.991413	+	0.130765i	$0.958257\pi$
$182$	0	0
$183$	4988.31i	0.148954i
$184$	0	0
$185$	16704.0	0.488064
$186$	0	0
$187$	2286.31	0.0653810
$188$	0	0
$189$	8748.00i	0.244898i
$190$	0	0
$191$	67092.7i	1.83911i	0.392957	+	0.919557i	$0.371452\pi$
$191$	67092.7i	1.83911i	−0.392957	+	0.919557i	$0.628548\pi$
$192$	0	0
$193$	−59234.0	−1.59022	−0.795109	−	0.606467i	$-0.792586\pi$
$193$	−59234.0	−1.59022	−0.795109	+	0.606467i	$0.792586\pi$
$194$	0	0
$195$	−16461.4	−0.432910
$196$	0	0
$197$	− 24996.0i	− 0.644077i	−0.946727	−	0.322039i	$-0.895632\pi$
$197$	− 24996.0i	− 0.644077i	0.946727	−	0.322039i	$-0.104368\pi$
$198$	0	0
$199$	28371.0i	0.716421i	0.933641	+	0.358211i	$0.116613\pi$
$199$	28371.0i	0.716421i	−0.933641	+	0.358211i	$0.883387\pi$
$200$	0	0
$201$	16956.0	0.419693
$202$	0	0
$203$	−14216.7	−0.344990
$204$	0	0
$205$	− 15384.0i	− 0.366068i
$206$	0	0
$207$	− 3367.11i	− 0.0785808i
$208$	0	0
$209$	2160.00	0.0494494
$210$	0	0
$211$	−54684.3	−1.22828	−0.614141	−	0.789197i	$-0.710497\pi$
$211$	−54684.3	−1.22828	−0.614141	+	0.789197i	$0.710497\pi$
$212$	0	0
$213$	31752.0i	0.699861i
$214$	0	0
$215$	− 30179.3i	− 0.652877i
$216$	0	0
$217$	−89424.0	−1.89904
$218$	0	0
$219$	−16471.8	−0.343442
$220$	0	0
$221$	− 29040.0i	− 0.594582i
$222$	0	0
$223$	− 38098.2i	− 0.766116i	−0.923724	−	0.383058i	$-0.874871\pi$
$223$	− 38098.2i	− 0.766116i	0.923724	−	0.383058i	$-0.125129\pi$
$224$	0	0
$225$	12987.0	0.256533
$226$	0	0
$227$	−92678.6	−1.79857	−0.899286	−	0.437362i	$-0.855913\pi$
$227$	−92678.6	−1.79857	−0.899286	+	0.437362i	$0.855913\pi$
$228$	0	0
$229$	− 66072.0i	− 1.25993i	−0.776623	−	0.629965i	$-0.783069\pi$
$229$	− 66072.0i	− 1.25993i	0.776623	−	0.629965i	$-0.216931\pi$
$230$	0	0
$231$	6734.21i	0.126201i
$232$	0	0
$233$	72686.0	1.33887	0.669436	−	0.742870i	$-0.266536\pi$
$233$	72686.0	1.33887	0.669436	+	0.742870i	$0.266536\pi$
$234$	0	0
$235$	−31426.3	−0.569060
$236$	0	0
$237$	− 8100.00i	− 0.144208i
$238$	0	0
$239$	− 83304.7i	− 1.45839i	−0.684306	−	0.729195i	$-0.739895\pi$
$239$	− 83304.7i	− 1.45839i	0.684306	−	0.729195i	$-0.260105\pi$
$240$	0	0
$241$	−44738.0	−0.770269	−0.385135	−	0.922860i	$-0.625845\pi$
$241$	−44738.0	−0.770269	−0.385135	+	0.922860i	$0.625845\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	17844.0i	0.297276i
$246$	0	0
$247$	− 27435.7i	− 0.449699i
$248$	0	0
$249$	23004.0	0.371026
$250$	0	0
$251$	−20181.9	−0.320342	−0.160171	−	0.987089i	$-0.551205\pi$
$251$	−20181.9	−0.320342	−0.160171	+	0.987089i	$0.551205\pi$
$252$	0	0
$253$	− 2592.00i	− 0.0404943i
$254$	0	0
$255$	− 6858.92i	− 0.105481i
$256$	0	0
$257$	−55166.0	−0.835228	−0.417614	−	0.908624i	$-0.637134\pi$
$257$	−55166.0	−0.835228	−0.417614	+	0.908624i	$0.637134\pi$
$258$	0	0
$259$	−86796.5	−1.29391
$260$	0	0
$261$	6156.00i	0.0903686i
$262$	0	0
$263$	113733.i	1.64428i	0.569283	+	0.822141i	$0.307221\pi$
$263$	113733.i	1.64428i	−0.569283	+	0.822141i	$0.692779\pi$
$264$	0	0
$265$	−54000.0	−0.768957
$266$	0	0
$267$	8054.04	0.112977
$268$	0	0
$269$	− 92460.0i	− 1.27776i	−0.769306	−	0.638880i	$-0.779398\pi$
$269$	− 92460.0i	− 1.27776i	0.769306	−	0.638880i	$-0.220602\pi$
$270$	0	0
$271$	− 50569.0i	− 0.688566i	−0.938866	−	0.344283i	$-0.888122\pi$
$271$	− 50569.0i	− 0.688566i	0.938866	−	0.344283i	$-0.111878\pi$
$272$	0	0
$273$	85536.0	1.14769
$274$	0	0
$275$	9997.40	0.132197
$276$	0	0
$277$	125208.i	1.63182i	0.578178	+	0.815911i	$0.303764\pi$
$277$	125208.i	1.63182i	−0.578178	+	0.815911i	$0.696236\pi$
$278$	0	0
$279$	38721.7i	0.497446i
$280$	0	0
$281$	130670.	1.65487	0.827434	−	0.561563i	$-0.189800\pi$
$281$	130670.	1.65487	0.827434	+	0.561563i	$0.189800\pi$
$282$	0	0
$283$	12408.4	0.154933	0.0774664	−	0.996995i	$-0.475317\pi$
$283$	12408.4	0.154933	0.0774664	+	0.996995i	$0.475317\pi$
$284$	0	0
$285$	− 6480.00i	− 0.0797784i
$286$	0	0
$287$	79937.6i	0.970482i
$288$	0	0
$289$	−71421.0	−0.855126
$290$	0	0
$291$	−41662.8	−0.491996
$292$	0	0
$293$	60204.0i	0.701278i	0.936511	+	0.350639i	$0.114036\pi$
$293$	60204.0i	0.701278i	−0.936511	+	0.350639i	$0.885964\pi$
$294$	0	0
$295$	− 76071.7i	− 0.874136i
$296$	0	0
$297$	2916.00	0.0330579
$298$	0	0
$299$	−32922.8	−0.368260
$300$	0	0
$301$	156816.i	1.73084i
$302$	0	0
$303$	80623.5i	0.878166i
$304$	0	0
$305$	11520.0	0.123838
$306$	0	0
$307$	88563.2	0.939673	0.469836	−	0.882754i	$-0.344313\pi$
$307$	88563.2	0.939673	0.469836	+	0.882754i	$0.344313\pi$
$308$	0	0
$309$	− 59940.0i	− 0.627769i
$310$	0	0
$311$	− 78067.0i	− 0.807136i	−0.914950	−	0.403568i	$-0.867770\pi$
$311$	− 78067.0i	− 0.807136i	0.914950	−	0.403568i	$-0.132230\pi$
$312$	0	0
$313$	−76366.0	−0.779491	−0.389746	−	0.920923i	$-0.627437\pi$
$313$	−76366.0	−0.779491	−0.389746	+	0.920923i	$0.627437\pi$
$314$	0	0
$315$	20202.6	0.203604
$316$	0	0
$317$	− 13836.0i	− 0.137687i	−0.997627	−	0.0688434i	$-0.978069\pi$
$317$	− 13836.0i	− 0.137687i	0.997627	−	0.0688434i	$-0.0219309\pi$
$318$	0	0
$319$	4738.89i	0.0465688i
$320$	0	0
$321$	99036.0	0.961132
$322$	0	0
$323$	11431.5	0.109572
$324$	0	0
$325$	− 126984.i	− 1.20222i
$326$	0	0
$327$	− 56991.4i	− 0.532984i
$328$	0	0
$329$	163296.	1.50863
$330$	0	0
$331$	−44458.3	−0.405786	−0.202893	−	0.979201i	$-0.565034\pi$
$331$	−44458.3	−0.405786	−0.202893	+	0.979201i	$0.565034\pi$
$332$	0	0
$333$	37584.0i	0.338934i
$334$	0	0
$335$	− 39158.2i	− 0.348926i
$336$	0	0
$337$	6386.00	0.0562301	0.0281151	−	0.999605i	$-0.491050\pi$
$337$	6386.00	0.0562301	0.0281151	+	0.999605i	$0.491050\pi$
$338$	0	0
$339$	56045.7	0.487689
$340$	0	0
$341$	29808.0i	0.256345i
$342$	0	0
$343$	56991.4i	0.484419i
$344$	0	0
$345$	−7776.00	−0.0653308
$346$	0	0
$347$	54642.7	0.453809	0.226905	−	0.973917i	$-0.427139\pi$
$347$	54642.7	0.453809	0.226905	+	0.973917i	$0.427139\pi$
$348$	0	0
$349$	− 96192.0i	− 0.789747i	−0.918735	−	0.394874i	$-0.870788\pi$
$349$	− 96192.0i	− 0.789747i	0.918735	−	0.394874i	$-0.129212\pi$
$350$	0	0
$351$	− 37038.2i	− 0.300632i
$352$	0	0
$353$	−73502.0	−0.589861	−0.294931	−	0.955519i	$-0.595297\pi$
$353$	−73502.0	−0.589861	−0.294931	+	0.955519i	$0.595297\pi$
$354$	0	0
$355$	73328.1	0.581854
$356$	0	0
$357$	35640.0i	0.279641i
$358$	0	0
$359$	127327.i	0.987939i	0.869479	+	0.493969i	$0.164454\pi$
$359$	127327.i	0.987939i	−0.869479	+	0.493969i	$0.835546\pi$
$360$	0	0
$361$	−119521.	−0.917128
$362$	0	0
$363$	−73832.1	−0.560315
$364$	0	0
$365$	38040.0i	0.285532i
$366$	0	0
$367$	227404.i	1.68837i	0.536055	+	0.844183i	$0.319914\pi$
$367$	227404.i	1.68837i	−0.536055	+	0.844183i	$0.680086\pi$
$368$	0	0
$369$	34614.0	0.254214
$370$	0	0
$371$	280592.	2.03858
$372$	0	0
$373$	− 196704.i	− 1.41382i	−0.707301	−	0.706912i	$-0.750088\pi$
$373$	− 196704.i	− 1.41382i	0.707301	−	0.706912i	$-0.249912\pi$
$374$	0	0
$375$	− 68963.3i	− 0.490406i
$376$	0	0
$377$	60192.0	0.423503
$378$	0	0
$379$	53395.7	0.371730	0.185865	−	0.982575i	$-0.440491\pi$
$379$	53395.7	0.371730	0.185865	+	0.982575i	$0.440491\pi$
$380$	0	0
$381$	− 33372.0i	− 0.229896i
$382$	0	0
$383$	− 193796.i	− 1.32113i	−0.750767	−	0.660567i	$-0.770316\pi$
$383$	− 193796.i	− 1.32113i	0.750767	−	0.660567i	$-0.229684\pi$
$384$	0	0
$385$	15552.0	0.104922
$386$	0	0
$387$	67903.3	0.453387
$388$	0	0
$389$	− 109404.i	− 0.722993i	−0.932373	−	0.361496i	$-0.882266\pi$
$389$	− 109404.i	− 0.722993i	0.932373	−	0.361496i	$-0.117734\pi$
$390$	0	0
$391$	− 13717.8i	− 0.0897289i
$392$	0	0
$393$	−83268.0	−0.539129
$394$	0	0
$395$	−18706.1	−0.119892
$396$	0	0
$397$	− 113952.i	− 0.723004i	−0.932371	−	0.361502i	$-0.882264\pi$
$397$	− 113952.i	− 0.723004i	0.932371	−	0.361502i	$-0.117736\pi$
$398$	0	0
$399$	33671.1i	0.211500i
$400$	0	0
$401$	−112594.	−0.700207	−0.350104	−	0.936711i	$-0.613854\pi$
$401$	−112594.	−0.700207	−0.350104	+	0.936711i	$0.613854\pi$
$402$	0	0
$403$	378612.	2.33123
$404$	0	0
$405$	− 8748.00i	− 0.0533333i
$406$	0	0
$407$	28932.2i	0.174660i
$408$	0	0
$409$	−187870.	−1.12308	−0.561540	−	0.827449i	$-0.689791\pi$
$409$	−187870.	−1.12308	−0.561540	+	0.827449i	$0.689791\pi$
$410$	0	0
$411$	92460.3	0.547358
$412$	0	0
$413$	395280.i	2.31742i
$414$	0	0
$415$	− 53125.5i	− 0.308465i
$416$	0	0
$417$	89964.0	0.517365
$418$	0	0
$419$	97957.9	0.557971	0.278985	−	0.960295i	$-0.410002\pi$
$419$	97957.9	0.557971	0.278985	+	0.960295i	$0.410002\pi$
$420$	0	0
$421$	307992.i	1.73770i	0.495074	+	0.868851i	$0.335141\pi$
$421$	307992.i	1.73770i	−0.495074	+	0.868851i	$0.664859\pi$
$422$	0	0
$423$	− 70709.2i	− 0.395180i
$424$	0	0
$425$	52910.0	0.292927
$426$	0	0
$427$	−59859.7	−0.328306
$428$	0	0
$429$	− 28512.0i	− 0.154922i
$430$	0	0
$431$	− 93655.5i	− 0.504172i	−0.967705	−	0.252086i	$-0.918883\pi$
$431$	− 93655.5i	− 0.504172i	0.967705	−	0.252086i	$-0.0811165\pi$
$432$	0	0
$433$	54574.0	0.291078	0.145539	−	0.989352i	$-0.453508\pi$
$433$	54574.0	0.291078	0.145539	+	0.989352i	$0.453508\pi$
$434$	0	0
$435$	14216.7	0.0751311
$436$	0	0
$437$	− 12960.0i	− 0.0678644i
$438$	0	0
$439$	− 41215.9i	− 0.213863i	−0.994266	−	0.106931i	$-0.965897\pi$
$439$	− 41215.9i	− 0.213863i	0.994266	−	0.106931i	$-0.0341025\pi$
$440$	0	0
$441$	−40149.0	−0.206442
$442$	0	0
$443$	61210.7	0.311903	0.155952	−	0.987765i	$-0.450156\pi$
$443$	61210.7	0.311903	0.155952	+	0.987765i	$0.450156\pi$
$444$	0	0
$445$	− 18600.0i	− 0.0939275i
$446$	0	0
$447$	− 70646.9i	− 0.353572i
$448$	0	0
$449$	−128914.	−0.639451	−0.319726	−	0.947510i	$-0.603591\pi$
$449$	−128914.	−0.639451	−0.319726	+	0.947510i	$0.603591\pi$
$450$	0	0
$451$	26645.9	0.131002
$452$	0	0
$453$	107892.i	0.525766i
$454$	0	0
$455$	− 197537.i	− 0.954169i
$456$	0	0
$457$	−160030.	−0.766247	−0.383124	−	0.923697i	$-0.625152\pi$
$457$	−160030.	−0.766247	−0.383124	+	0.923697i	$0.625152\pi$
$458$	0	0
$459$	15432.6	0.0732509
$460$	0	0
$461$	− 299988.i	− 1.41157i	−0.708427	−	0.705784i	$-0.750595\pi$
$461$	− 299988.i	− 1.41157i	0.708427	−	0.705784i	$-0.249405\pi$
$462$	0	0
$463$	60670.3i	0.283018i	0.989937	+	0.141509i	$0.0451954\pi$
$463$	60670.3i	0.283018i	−0.989937	+	0.141509i	$0.954805\pi$
$464$	0	0
$465$	89424.0	0.413569
$466$	0	0
$467$	351031.	1.60958	0.804789	−	0.593561i	$-0.202278\pi$
$467$	351031.	1.60958	0.804789	+	0.593561i	$0.202278\pi$
$468$	0	0
$469$	203472.i	0.925037i
$470$	0	0
$471$	64598.6i	0.291193i
$472$	0	0
$473$	52272.0	0.233640
$474$	0	0
$475$	49987.0	0.221549
$476$	0	0
$477$	− 121500.i	− 0.533998i
$478$	0	0
$479$	348059.i	1.51699i	0.651680	+	0.758494i	$0.274064\pi$
$479$	348059.i	1.51699i	−0.651680	+	0.758494i	$0.725936\pi$
$480$	0	0
$481$	367488.	1.58837
$482$	0	0
$483$	40405.3	0.173198
$484$	0	0
$485$	96216.0i	0.409038i
$486$	0	0
$487$	247732.i	1.04454i	0.852781	+	0.522268i	$0.174914\pi$
$487$	247732.i	1.04454i	−0.852781	+	0.522268i	$0.825086\pi$
$488$	0	0
$489$	−227124.	−0.949829
$490$	0	0
$491$	−243409.	−1.00965	−0.504827	−	0.863220i	$-0.668444\pi$
$491$	−243409.	−1.00965	−0.504827	+	0.863220i	$0.668444\pi$
$492$	0	0
$493$	25080.0i	0.103189i
$494$	0	0
$495$	− 6734.21i	− 0.0274838i
$496$	0	0
$497$	−381024.	−1.54255
$498$	0	0
$499$	341055.	1.36969	0.684846	−	0.728688i	$-0.259869\pi$
$499$	341055.	1.36969	0.684846	+	0.728688i	$0.259869\pi$
$500$	0	0
$501$	− 224208.i	− 0.893255i
$502$	0	0
$503$	44021.8i	0.173993i	0.996209	+	0.0869965i	$0.0277269\pi$
$503$	44021.8i	0.173993i	−0.996209	+	0.0869965i	$0.972273\pi$
$504$	0	0
$505$	186192.	0.730093
$506$	0	0
$507$	−213744.	−0.831529
$508$	0	0
$509$	− 166428.i	− 0.642378i	−0.947015	−	0.321189i	$-0.895917\pi$
$509$	− 166428.i	− 0.642378i	0.947015	−	0.321189i	$-0.104083\pi$
$510$	0	0
$511$	− 197662.i	− 0.756973i
$512$	0	0
$513$	14580.0	0.0554017
$514$	0	0
$515$	−138426.	−0.521917
$516$	0	0
$517$	− 54432.0i	− 0.203645i
$518$	0	0
$519$	278659.i	1.03452i
$520$	0	0
$521$	444130.	1.63619	0.818097	−	0.575081i	$-0.195029\pi$
$521$	444130.	1.63619	0.818097	+	0.575081i	$0.195029\pi$
$522$	0	0
$523$	−175152.	−0.640341	−0.320171	−	0.947360i	$-0.603740\pi$
$523$	−175152.	−0.640341	−0.320171	+	0.947360i	$0.603740\pi$
$524$	0	0
$525$	155844.i	0.565420i
$526$	0	0
$527$	157755.i	0.568018i
$528$	0	0
$529$	264289.	0.944426
$530$	0	0
$531$	171161.	0.607039
$532$	0	0
$533$	− 338448.i	− 1.19134i
$534$	0	0
$535$	− 228714.i	− 0.799070i
$536$	0	0
$537$	−211572.	−0.733685
$538$	0	0
$539$	−30906.7	−0.106384
$540$	0	0
$541$	− 138360.i	− 0.472733i	−0.971664	−	0.236367i	$-0.924043\pi$
$541$	− 138360.i	− 0.472733i	0.971664	−	0.236367i	$-0.0759566\pi$
$542$	0	0
$543$	44520.6i	0.150995i
$544$	0	0
$545$	−131616.	−0.443114
$546$	0	0
$547$	−341096.	−1.13999	−0.569997	−	0.821647i	$-0.693055\pi$
$547$	−341096.	−1.13999	−0.569997	+	0.821647i	$0.693055\pi$
$548$	0	0
$549$	25920.0i	0.0859984i
$550$	0	0
$551$	23694.5i	0.0780447i
$552$	0	0
$553$	97200.0	0.317845
$554$	0	0
$555$	86796.5	0.281784
$556$	0	0
$557$	61260.0i	0.197454i	0.995115	+	0.0987272i	$0.0314771\pi$
$557$	61260.0i	0.197454i	−0.995115	+	0.0987272i	$0.968523\pi$
$558$	0	0
$559$	− 663944.i	− 2.12475i
$560$	0	0
$561$	11880.0	0.0377477
$562$	0	0
$563$	331577.	1.04609	0.523043	−	0.852306i	$-0.324797\pi$
$563$	331577.	1.04609	0.523043	+	0.852306i	$0.324797\pi$
$564$	0	0
$565$	− 129432.i	− 0.405457i
$566$	0	0
$567$	45455.9i	0.141392i
$568$	0	0
$569$	137026.	0.423232	0.211616	−	0.977353i	$-0.432127\pi$
$569$	137026.	0.423232	0.211616	+	0.977353i	$0.432127\pi$
$570$	0	0
$571$	−259122.	−0.794752	−0.397376	−	0.917656i	$-0.630079\pi$
$571$	−259122.	−0.794752	−0.397376	+	0.917656i	$0.630079\pi$
$572$	0	0
$573$	348624.i	1.06181i
$574$	0	0
$575$	− 59984.4i	− 0.181427i
$576$	0	0
$577$	276050.	0.829156	0.414578	−	0.910014i	$-0.363929\pi$
$577$	276050.	0.829156	0.414578	+	0.910014i	$0.363929\pi$
$578$	0	0
$579$	−307789.	−0.918112
$580$	0	0
$581$	276048.i	0.817772i
$582$	0	0
$583$	− 93530.7i	− 0.275180i
$584$	0	0
$585$	−85536.0	−0.249941
$586$	0	0
$587$	−405944.	−1.17812	−0.589061	−	0.808089i	$-0.700502\pi$
$587$	−405944.	−1.17812	−0.589061	+	0.808089i	$0.700502\pi$
$588$	0	0
$589$	149040.i	0.429608i
$590$	0	0
$591$	− 129883.i	− 0.371858i
$592$	0	0
$593$	−430238.	−1.22349	−0.611744	−	0.791056i	$-0.709532\pi$
$593$	−430238.	−1.22349	−0.611744	+	0.791056i	$0.709532\pi$
$594$	0	0
$595$	82307.1	0.232489
$596$	0	0
$597$	147420.i	0.413626i
$598$	0	0
$599$	50007.8i	0.139375i	0.997569	+	0.0696873i	$0.0222002\pi$
$599$	50007.8i	0.139375i	−0.997569	+	0.0696873i	$0.977800\pi$
$600$	0	0
$601$	530110.	1.46763	0.733816	−	0.679348i	$-0.237738\pi$
$601$	530110.	1.46763	0.733816	+	0.679348i	$0.237738\pi$
$602$	0	0
$603$	88106.0	0.242310
$604$	0	0
$605$	170508.i	0.465837i
$606$	0	0
$607$	516851.i	1.40277i	0.712780	+	0.701387i	$0.247436\pi$
$607$	516851.i	1.40277i	−0.712780	+	0.701387i	$0.752564\pi$
$608$	0	0
$609$	−73872.0	−0.199180
$610$	0	0
$611$	−691379.	−1.85197
$612$	0	0
$613$	− 411840.i	− 1.09599i	−0.836481	−	0.547996i	$-0.815391\pi$
$613$	− 411840.i	− 1.09599i	0.836481	−	0.547996i	$-0.184609\pi$
$614$	0	0
$615$	− 79937.6i	− 0.211349i
$616$	0	0
$617$	−150610.	−0.395625	−0.197812	−	0.980240i	$-0.563384\pi$
$617$	−150610.	−0.395625	−0.197812	+	0.980240i	$0.563384\pi$
$618$	0	0
$619$	−314492.	−0.820783	−0.410391	−	0.911909i	$-0.634608\pi$
$619$	−314492.	−0.820783	−0.410391	+	0.911909i	$0.634608\pi$
$620$	0	0
$621$	− 17496.0i	− 0.0453686i
$622$	0	0
$623$	96648.4i	0.249011i
$624$	0	0
$625$	141361.	0.361884
$626$	0	0
$627$	11223.7	0.0285496
$628$	0	0
$629$	153120.i	0.387018i
$630$	0	0
$631$	381418.i	0.957950i	0.877829	+	0.478975i	$0.158992\pi$
$631$	381418.i	0.957950i	−0.877829	+	0.478975i	$0.841008\pi$
$632$	0	0
$633$	−284148.	−0.709148
$634$	0	0
$635$	−77069.3	−0.191132
$636$	0	0
$637$	392568.i	0.967467i
$638$	0	0
$639$	164988.i	0.404065i
$640$	0	0
$641$	−264754.	−0.644357	−0.322178	−	0.946679i	$-0.604415\pi$
$641$	−264754.	−0.644357	−0.322178	+	0.946679i	$0.604415\pi$
$642$	0	0
$643$	−655360.	−1.58510	−0.792552	−	0.609805i	$-0.791248\pi$
$643$	−655360.	−1.58510	−0.792552	+	0.609805i	$0.791248\pi$
$644$	0	0
$645$	− 156816.i	− 0.376939i
$646$	0	0
$647$	− 593733.i	− 1.41835i	−0.705034	−	0.709174i	$-0.749068\pi$
$647$	− 593733.i	− 1.41835i	0.705034	−	0.709174i	$-0.250932\pi$
$648$	0	0
$649$	131760.	0.312820
$650$	0	0
$651$	−464661.	−1.09641
$652$	0	0
$653$	− 72612.0i	− 0.170287i	−0.996369	−	0.0851436i	$-0.972865\pi$
$653$	− 72612.0i	− 0.170287i	0.996369	−	0.0851436i	$-0.0271349\pi$
$654$	0	0
$655$	192299.i	0.448224i
$656$	0	0
$657$	−85590.0	−0.198286
$658$	0	0
$659$	−163429.	−0.376322	−0.188161	−	0.982138i	$-0.560253\pi$
$659$	−163429.	−0.376322	−0.188161	+	0.982138i	$0.560253\pi$
$660$	0	0
$661$	360816.i	0.825815i	0.910773	+	0.412908i	$0.135487\pi$
$661$	360816.i	0.825815i	−0.910773	+	0.412908i	$0.864513\pi$
$662$	0	0
$663$	− 150896.i	− 0.343282i
$664$	0	0
$665$	77760.0	0.175838
$666$	0	0
$667$	28433.3	0.0639111
$668$	0	0
$669$	− 197964.i	− 0.442317i
$670$	0	0
$671$	19953.2i	0.0443168i
$672$	0	0
$673$	406702.	0.897938	0.448969	−	0.893547i	$-0.351791\pi$
$673$	406702.	0.897938	0.448969	+	0.893547i	$0.351791\pi$
$674$	0	0
$675$	67482.4	0.148110
$676$	0	0
$677$	− 181548.i	− 0.396108i	−0.980191	−	0.198054i	$-0.936538\pi$
$677$	− 181548.i	− 0.396108i	0.980191	−	0.198054i	$-0.0634622\pi$
$678$	0	0
$679$	− 499953.i	− 1.08440i
$680$	0	0
$681$	−481572.	−1.03841
$682$	0	0
$683$	15567.7	0.0333720	0.0166860	−	0.999861i	$-0.494688\pi$
$683$	15567.7	0.0333720	0.0166860	+	0.999861i	$0.494688\pi$
$684$	0	0
$685$	− 213528.i	− 0.455065i
$686$	0	0
$687$	− 343320.i	− 0.727421i
$688$	0	0
$689$	−1.18800e6	−2.50252
$690$	0	0
$691$	5674.20	0.0118836	0.00594181	−	0.999982i	$-0.498109\pi$
$691$	5674.20	0.0118836	0.00594181	+	0.999982i	$0.498109\pi$
$692$	0	0
$693$	34992.0i	0.0728622i
$694$	0	0
$695$	− 207763.i	− 0.430129i
$696$	0	0
$697$	141020.	0.290279
$698$	0	0
$699$	377688.	0.772998
$700$	0	0
$701$	− 868188.i	− 1.76676i	−0.468657	−	0.883380i	$-0.655262\pi$
$701$	− 868188.i	− 1.76676i	0.468657	−	0.883380i	$-0.344738\pi$
$702$	0	0
$703$	144661.i	0.292712i
$704$	0	0
$705$	−163296.	−0.328547
$706$	0	0
$707$	−967482.	−1.93555
$708$	0	0
$709$	47112.0i	0.0937215i	0.998901	+	0.0468607i	$0.0149217\pi$
$709$	47112.0i	0.0937215i	−0.998901	+	0.0468607i	$0.985078\pi$
$710$	0	0
$711$	− 42088.8i	− 0.0832583i
$712$	0	0
$713$	178848.	0.351807
$714$	0	0
$715$	−65845.6	−0.128800
$716$	0	0
$717$	− 432864.i	− 0.842002i
$718$	0	0
$719$	324365.i	0.627445i	0.949515	+	0.313723i	$0.101576\pi$
$719$	324365.i	0.627445i	−0.949515	+	0.313723i	$0.898424\pi$
$720$	0	0
$721$	719280.	1.38365
$722$	0	0
$723$	−232465.	−0.444715
$724$	0	0
$725$	109668.i	0.208643i
$726$	0	0
$727$	544287.i	1.02981i	0.857246	+	0.514907i	$0.172174\pi$
$727$	544287.i	1.02981i	−0.857246	+	0.514907i	$0.827826\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	276643.	0.517708
$732$	0	0
$733$	− 455976.i	− 0.848661i	−0.905507	−	0.424330i	$-0.860510\pi$
$733$	− 455976.i	− 0.848661i	0.905507	−	0.424330i	$-0.139490\pi$
$734$	0	0
$735$	92720.1i	0.171632i
$736$	0	0
$737$	67824.0	0.124867
$738$	0	0
$739$	−684042.	−1.25255	−0.626274	−	0.779603i	$-0.715421\pi$
$739$	−684042.	−1.25255	−0.626274	+	0.779603i	$0.715421\pi$
$740$	0	0
$741$	− 142560.i	− 0.259634i
$742$	0	0
$743$	863476.i	1.56413i	0.623198	+	0.782064i	$0.285833\pi$
$743$	863476.i	1.56413i	−0.623198	+	0.782064i	$0.714167\pi$
$744$	0	0
$745$	−163152.	−0.293954
$746$	0	0
$747$	119532.	0.214212
$748$	0	0
$749$	1.18843e6i	2.11841i
$750$	0	0
$751$	546905.i	0.969689i	0.874600	+	0.484844i	$0.161124\pi$
$751$	546905.i	0.969689i	−0.874600	+	0.484844i	$0.838876\pi$
$752$	0	0
$753$	−104868.	−0.184949
$754$	0	0
$755$	249166.	0.437114
$756$	0	0
$757$	359496.i	0.627339i	0.949532	+	0.313670i	$0.101558\pi$
$757$	359496.i	0.627339i	−0.949532	+	0.313670i	$0.898442\pi$
$758$	0	0
$759$	− 13468.4i	− 0.0233794i
$760$	0	0
$761$	−315806.	−0.545320	−0.272660	−	0.962111i	$-0.587903\pi$
$761$	−315806.	−0.545320	−0.272660	+	0.962111i	$0.587903\pi$
$762$	0	0
$763$	683897.	1.17474
$764$	0	0
$765$	− 35640.0i	− 0.0608997i
$766$	0	0
$767$	− 1.67358e6i	− 2.84482i
$768$	0	0
$769$	303070.	0.512496	0.256248	−	0.966611i	$-0.417514\pi$
$769$	303070.	0.512496	0.256248	+	0.966611i	$0.417514\pi$
$770$	0	0
$771$	−286651.	−0.482219
$772$	0	0
$773$	− 379668.i	− 0.635397i	−0.948192	−	0.317698i	$-0.897090\pi$
$773$	− 379668.i	− 0.635397i	0.948192	−	0.317698i	$-0.102910\pi$
$774$	0	0
$775$	689820.i	1.14850i
$776$	0	0
$777$	−451008.	−0.747037
$778$	0	0
$779$	133229.	0.219546
$780$	0	0
$781$	127008.i	0.208223i
$782$	0	0
$783$	31987.5i	0.0521743i
$784$	0	0
$785$	149184.	0.242093
$786$	0	0
$787$	−52689.0	−0.0850688	−0.0425344	−	0.999095i	$-0.513543\pi$
$787$	−52689.0	−0.0850688	−0.0425344	+	0.999095i	$0.513543\pi$
$788$	0	0
$789$	590976.i	0.949327i
$790$	0	0
$791$	672548.i	1.07491i
$792$	0	0
$793$	253440.	0.403022
$794$	0	0
$795$	−280592.	−0.443957
$796$	0	0
$797$	51900.0i	0.0817054i	0.999165	+	0.0408527i	$0.0130074\pi$
$797$	51900.0i	0.0817054i	−0.999165	+	0.0408527i	$0.986993\pi$
$798$	0	0
$799$	− 288075.i	− 0.451244i
$800$	0	0
$801$	41850.0	0.0652275
$802$	0	0
$803$	−65887.2	−0.102181
$804$	0	0
$805$	− 93312.0i	− 0.143994i
$806$	0	0
$807$	− 480436.i	− 0.737715i
$808$	0	0
$809$	498562.	0.761767	0.380883	−	0.924623i	$-0.375620\pi$
$809$	498562.	0.761767	0.380883	+	0.924623i	$0.375620\pi$
$810$	0	0
$811$	69441.4	0.105579	0.0527894	−	0.998606i	$-0.483189\pi$
$811$	69441.4	0.105579	0.0527894	+	0.998606i	$0.483189\pi$
$812$	0	0
$813$	− 262764.i	− 0.397544i
$814$	0	0
$815$	524520.i	0.789673i
$816$	0	0
$817$	261360.	0.391557
$818$	0	0
$819$	444458.	0.662618
$820$	0	0
$821$	763068.i	1.13208i	0.824378	+	0.566040i	$0.191525\pi$
$821$	763068.i	1.13208i	−0.824378	+	0.566040i	$0.808475\pi$
$822$	0	0
$823$	− 1.23404e6i	− 1.82193i	−0.412486	−	0.910964i	$-0.635339\pi$
$823$	− 1.23404e6i	− 1.82193i	0.412486	−	0.910964i	$-0.364661\pi$
$824$	0	0
$825$	51948.0	0.0763240
$826$	0	0
$827$	587560.	0.859095	0.429548	−	0.903044i	$-0.358673\pi$
$827$	587560.	0.859095	0.429548	+	0.903044i	$0.358673\pi$
$828$	0	0
$829$	518184.i	0.754006i	0.926212	+	0.377003i	$0.123045\pi$
$829$	518184.i	0.754006i	−0.926212	+	0.377003i	$0.876955\pi$
$830$	0	0
$831$	650600.i	0.942132i
$832$	0	0
$833$	−163570.	−0.235729
$834$	0	0
$835$	−517786.	−0.742639
$836$	0	0
$837$	201204.i	0.287201i
$838$	0	0
$839$	− 608698.i	− 0.864725i	−0.901700	−	0.432362i	$-0.857680\pi$
$839$	− 608698.i	− 0.864725i	0.901700	−	0.432362i	$-0.142320\pi$
$840$	0	0
$841$	655297.	0.926502
$842$	0	0
$843$	678981.	0.955438
$844$	0	0
$845$	493620.i	0.691320i
$846$	0	0
$847$	− 885986.i	− 1.23498i
$848$	0	0
$849$	64476.0	0.0894505
$850$	0	0
$851$	173593.	0.239703
$852$	0	0
$853$	− 1.28203e6i	− 1.76198i	−0.473136	−	0.880990i	$-0.656878\pi$
$853$	− 1.28203e6i	− 1.76198i	0.473136	−	0.880990i	$-0.343122\pi$
$854$	0	0
$855$	− 33671.1i	− 0.0460601i
$856$	0	0
$857$	−1.15264e6	−1.56939	−0.784696	−	0.619881i	$-0.787181\pi$
$857$	−1.15264e6	−1.56939	−0.784696	+	0.619881i	$0.787181\pi$
$858$	0	0
$859$	1.24011e6	1.68064	0.840321	−	0.542089i	$-0.182367\pi$
$859$	1.24011e6	1.68064	0.840321	+	0.542089i	$0.182367\pi$
$860$	0	0
$861$	415368.i	0.560308i
$862$	0	0
$863$	− 251161.i	− 0.337234i	−0.985682	−	0.168617i	$-0.946070\pi$
$863$	− 251161.i	− 0.337234i	0.985682	−	0.168617i	$-0.0539301\pi$
$864$	0	0
$865$	643536.	0.860084
$866$	0	0
$867$	−371114.	−0.493707
$868$	0	0
$869$	− 32400.0i	− 0.0429048i
$870$	0	0
$871$	− 861481.i	− 1.13556i
$872$	0	0
$873$	−216486.	−0.284054
$874$	0	0
$875$	827560.	1.08089
$876$	0	0
$877$	− 661728.i	− 0.860360i	−0.902743	−	0.430180i	$-0.858450\pi$
$877$	− 661728.i	− 0.860360i	0.902743	−	0.430180i	$-0.141550\pi$
$878$	0	0
$879$	312829.i	0.404883i
$880$	0	0
$881$	858562.	1.10616	0.553082	−	0.833127i	$-0.313451\pi$
$881$	858562.	1.10616	0.553082	+	0.833127i	$0.313451\pi$
$882$	0	0
$883$	1.09620e6	1.40595	0.702973	−	0.711216i	$-0.251855\pi$
$883$	1.09620e6	1.40595	0.702973	+	0.711216i	$0.251855\pi$
$884$	0	0
$885$	− 395280.i	− 0.504683i
$886$	0	0
$887$	937802.i	1.19197i	0.802997	+	0.595983i	$0.203237\pi$
$887$	937802.i	1.19197i	−0.802997	+	0.595983i	$0.796763\pi$
$888$	0	0
$889$	400464.	0.506711
$890$	0	0
$891$	15152.0	0.0190860
$892$	0	0
$893$	− 272160.i	− 0.341288i
$894$	0	0
$895$	488605.i	0.609974i
$896$	0	0
$897$	−171072.	−0.212615
$898$	0	0
$899$	−326983.	−0.404582
$900$	0	0
$901$	− 495000.i	− 0.609755i
$902$	0	0
$903$	814840.i	0.999302i
$904$	0	0
$905$	102816.	0.125535
$906$	0	0
$907$	533104.	0.648034	0.324017	−	0.946051i	$-0.394967\pi$
$907$	533104.	0.648034	0.324017	+	0.946051i	$0.394967\pi$
$908$	0	0
$909$	418932.i	0.507009i
$910$	0	0
$911$	148153.i	0.178514i	0.996009	+	0.0892571i	$0.0284493\pi$
$911$	148153.i	0.178514i	−0.996009	+	0.0892571i	$0.971551\pi$
$912$	0	0
$913$	92016.0	0.110388
$914$	0	0
$915$	59859.7	0.0714977
$916$	0	0
$917$	− 999216.i	− 1.18829i
$918$	0	0
$919$	− 183008.i	− 0.216691i	−0.994113	−	0.108345i	$-0.965445\pi$
$919$	− 183008.i	− 0.216691i	0.994113	−	0.108345i	$-0.0345552\pi$
$920$	0	0
$921$	460188.	0.542520
$922$	0	0
$923$	1.61322e6	1.89361
$924$	0	0
$925$	669552.i	0.782530i
$926$	0	0
$927$	− 311457.i	− 0.362443i
$928$	0	0
$929$	−23122.0	−0.0267913	−0.0133957	−	0.999910i	$-0.504264\pi$
$929$	−23122.0	−0.0267913	−0.0133957	+	0.999910i	$0.504264\pi$
$930$	0	0
$931$	−154534.	−0.178289
$932$	0	0
$933$	− 405648.i	− 0.466000i
$934$	0	0
$935$	− 27435.7i	− 0.0313829i
$936$	0	0
$937$	−1.03090e6	−1.17418	−0.587092	−	0.809520i	$-0.699728\pi$
$937$	−1.03090e6	−1.17418	−0.587092	+	0.809520i	$0.699728\pi$
$938$	0	0
$939$	−396809.	−0.450040
$940$	0	0
$941$	− 1.68690e6i	− 1.90507i	−0.304435	−	0.952533i	$-0.598468\pi$
$941$	− 1.68690e6i	− 1.90507i	0.304435	−	0.952533i	$-0.401532\pi$
$942$	0	0
$943$	− 159875.i	− 0.179787i
$944$	0	0
$945$	104976.	0.117551
$946$	0	0
$947$	381876.	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$947$	381876.	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$948$	0	0
$949$	836880.i	0.929246i
$950$	0	0
$951$	− 71894.0i	− 0.0794935i
$952$	0	0
$953$	88738.0	0.0977066	0.0488533	−	0.998806i	$-0.484443\pi$
$953$	88738.0	0.0977066	0.0488533	+	0.998806i	$0.484443\pi$
$954$	0	0
$955$	805113.	0.882775
$956$	0	0
$957$	24624.0i	0.0268865i
$958$	0	0
$959$	1.10952e6i	1.20642i
$960$	0	0
$961$	−1.13323e6	−1.22708
$962$	0	0
$963$	514606.	0.554910
$964$	0	0
$965$	710808.i	0.763304i
$966$	0	0
$967$	1.31062e6i	1.40159i	0.713361	+	0.700797i	$0.247172\pi$
$967$	1.31062e6i	1.40159i	−0.713361	+	0.700797i	$0.752828\pi$
$968$	0	0
$969$	59400.0	0.0632614
$970$	0	0
$971$	−190616.	−0.202172	−0.101086	−	0.994878i	$-0.532232\pi$
$971$	−190616.	−0.202172	−0.101086	+	0.994878i	$0.532232\pi$
$972$	0	0
$973$	1.07957e6i	1.14031i
$974$	0	0
$975$	− 659828.i	− 0.694099i
$976$	0	0
$977$	−7186.00	−0.00752832	−0.00376416	−	0.999993i	$-0.501198\pi$
$977$	−7186.00	−0.00752832	−0.00376416	+	0.999993i	$0.501198\pi$
$978$	0	0
$979$	32216.1	0.0336131
$980$	0	0
$981$	− 296136.i	− 0.307718i
$982$	0	0
$983$	− 810101.i	− 0.838363i	−0.907902	−	0.419182i	$-0.862317\pi$
$983$	− 810101.i	− 0.838363i	0.907902	−	0.419182i	$-0.137683\pi$
$984$	0	0
$985$	−299952.	−0.309157
$986$	0	0
$987$	848511.	0.871010
$988$	0	0
$989$	− 313632.i	− 0.320647i
$990$	0	0
$991$	− 313079.i	− 0.318791i	−0.987215	−	0.159395i	$-0.949045\pi$
$991$	− 313079.i	− 0.318791i	0.987215	−	0.159395i	$-0.0509545\pi$
$992$	0	0
$993$	−231012.	−0.234280
$994$	0	0
$995$	340452.	0.343882
$996$	0	0
$997$	1.13174e6i	1.13857i	0.822142	+	0.569283i	$0.192779\pi$
$997$	1.13174e6i	1.13857i	−0.822142	+	0.569283i	$0.807221\pi$
$998$	0	0
$999$	195292.i	0.195683i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.b.b.319.3	yes	4
3.2	odd	2		1152.5.b.e.703.4		4
4.3	odd	2	inner	384.5.b.b.319.1	✓	4
8.3	odd	2	inner	384.5.b.b.319.4	yes	4
8.5	even	2	inner	384.5.b.b.319.2	yes	4
12.11	even	2		1152.5.b.e.703.3		4
16.3	odd	4		768.5.g.a.511.1		2
16.5	even	4		768.5.g.b.511.1		2
16.11	odd	4		768.5.g.b.511.2		2
16.13	even	4		768.5.g.a.511.2		2
24.5	odd	2		1152.5.b.e.703.2		4
24.11	even	2		1152.5.b.e.703.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.b.319.1	✓	4	4.3	odd	2	inner
384.5.b.b.319.2	yes	4	8.5	even	2	inner
384.5.b.b.319.3	yes	4	1.1	even	1	trivial
384.5.b.b.319.4	yes	4	8.3	odd	2	inner
768.5.g.a.511.1		2	16.3	odd	4
768.5.g.a.511.2		2	16.13	even	4
768.5.g.b.511.1		2	16.5	even	4
768.5.g.b.511.2		2	16.11	odd	4
1152.5.b.e.703.1		4	24.11	even	2
1152.5.b.e.703.2		4	24.5	odd	2
1152.5.b.e.703.3		4	12.11	even	2
1152.5.b.e.703.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -12.0000i q^{5} +62.3538i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -12.0000i q^{5} +62.3538i q^{7} +27.0000 q^{9} +20.7846 q^{11} -264.000i q^{13} -62.3538i q^{15} +110.000 q^{17} +103.923 q^{19} +324.000i q^{21} -124.708i q^{23} +481.000 q^{25} +140.296 q^{27} +228.000i q^{29} +1434.14i q^{31} +108.000 q^{33} +748.246 q^{35} +1392.00i q^{37} -1371.78i q^{39} +1282.00 q^{41} +2514.94 q^{43} -324.000i q^{45} -2618.86i q^{47} -1487.00 q^{49} +571.577 q^{51} -4500.00i q^{53} -249.415i q^{55} +540.000 q^{57} +6339.31 q^{59} +960.000i q^{61} +1683.55i q^{63} -3168.00 q^{65} +3263.18 q^{67} -648.000i q^{69} +6110.68i q^{71} -3170.00 q^{73} +2499.35 q^{75} +1296.00i q^{77} -1558.85i q^{79} +729.000 q^{81} +4427.12 q^{83} -1320.00i q^{85} +1184.72i q^{87} +1550.00 q^{89} +16461.4 q^{91} +7452.00i q^{93} -1247.08i q^{95} -8018.00 q^{97} +561.184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9}+O(q^{10}) \) 4 * q + 108 * q^9	\( 4 q + 108 q^{9} + 440 q^{17} + 1924 q^{25} + 432 q^{33} + 5128 q^{41} - 5948 q^{49} + 2160 q^{57} - 12672 q^{65} - 12680 q^{73} + 2916 q^{81} + 6200 q^{89} - 32072 q^{97}+O(q^{100}) \) 4 * q + 108 * q^9 + 440 * q^17 + 1924 * q^25 + 432 * q^33 + 5128 * q^41 - 5948 * q^49 + 2160 * q^57 - 12672 * q^65 - 12680 * q^73 + 2916 * q^81 + 6200 * q^89 - 32072 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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