LMFDB - Embedded newform 384.5.b.d.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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + \cdots + 362673936 $ x^16 + 46x^14 + 1311x^12 + 24382x^10 + 338077x^8 + 3338772x^6 + 24662556x^4 + 120434256*x^2 + 362673936
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{84}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.3
Root			$1.45110 + 3.51338i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.5.b.d.319.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} -19.1142i q^{5} -6.40010i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} -19.1142i q^{5} -6.40010i q^{7} +27.0000 q^{9} +67.7877 q^{11} +154.346i q^{13} +99.3205i q^{15} +284.063 q^{17} -434.548 q^{19} +33.2559i q^{21} +462.698i q^{23} +259.646 q^{25} -140.296 q^{27} -350.108i q^{29} -88.4523i q^{31} -352.235 q^{33} -122.333 q^{35} -1636.60i q^{37} -802.006i q^{39} +395.726 q^{41} +350.192 q^{43} -516.084i q^{45} +2459.53i q^{47} +2360.04 q^{49} -1476.04 q^{51} -3827.15i q^{53} -1295.71i q^{55} +2257.98 q^{57} +1754.25 q^{59} -2335.10i q^{61} -172.803i q^{63} +2950.21 q^{65} +2198.93 q^{67} -2404.25i q^{69} -4463.62i q^{71} +2592.13 q^{73} -1349.16 q^{75} -433.848i q^{77} -2040.82i q^{79} +729.000 q^{81} +5245.36 q^{83} -5429.65i q^{85} +1819.22i q^{87} +14320.4 q^{89} +987.830 q^{91} +459.612i q^{93} +8306.04i q^{95} +16844.8 q^{97} +1830.27 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 432 q^{9}+O(q^{10}) $ 16 * q + 432 * q^9	$ 16 q + 432 q^{9} + 480 q^{17} + 144 q^{25} + 2880 q^{33} - 3552 q^{41} - 20080 q^{49} - 7488 q^{57} + 23040 q^{65} + 34400 q^{73} + 11664 q^{81} - 15648 q^{89} + 5088 q^{97}+O(q^{100}) $ 16 * q + 432 * q^9 + 480 * q^17 + 144 * q^25 + 2880 * q^33 - 3552 * q^41 - 20080 * q^49 - 7488 * q^57 + 23040 * q^65 + 34400 * q^73 + 11664 * q^81 - 15648 * q^89 + 5088 * 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
$5$	− 19.1142i	− 0.764569i	−0.924045	−	0.382285i	$-0.875137\pi$
$5$	− 19.1142i	− 0.764569i	0.924045	−	0.382285i	$-0.124863\pi$
$6$	0	0
$7$	− 6.40010i	− 0.130614i	−0.997865	−	0.0653071i	$-0.979197\pi$
$7$	− 6.40010i	− 0.130614i	0.997865	−	0.0653071i	$-0.0208027\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	67.7877	0.560229	0.280114	−	0.959967i	$-0.409628\pi$
$11$	67.7877	0.560229	0.280114	+	0.959967i	$0.409628\pi$
$12$	0	0
$13$	154.346i	0.913290i	0.889649	+	0.456645i	$0.150949\pi$
$13$	154.346i	0.913290i	−0.889649	+	0.456645i	$0.849051\pi$
$14$	0	0
$15$	99.3205i	0.441424i
$16$	0	0
$17$	284.063	0.982918	0.491459	−	0.870901i	$-0.336464\pi$
$17$	284.063	0.982918	0.491459	+	0.870901i	$0.336464\pi$
$18$	0	0
$19$	−434.548	−1.20373	−0.601866	−	0.798597i	$-0.705576\pi$
$19$	−434.548	−1.20373	−0.601866	+	0.798597i	$0.705576\pi$
$20$	0	0
$21$	33.2559i	0.0754102i
$22$	0	0
$23$	462.698i	0.874666i	0.899300	+	0.437333i	$0.144077\pi$
$23$	462.698i	0.874666i	−0.899300	+	0.437333i	$0.855923\pi$
$24$	0	0
$25$	259.646	0.415434
$26$	0	0
$27$	−140.296	−0.192450
$28$	0	0
$29$	− 350.108i	− 0.416300i	−0.978097	−	0.208150i	$-0.933256\pi$
$29$	− 350.108i	− 0.416300i	0.978097	−	0.208150i	$-0.0667442\pi$
$30$	0	0
$31$	− 88.4523i	− 0.0920420i	−0.998940	−	0.0460210i	$-0.985346\pi$
$31$	− 88.4523i	− 0.0920420i	0.998940	−	0.0460210i	$-0.0146541\pi$
$32$	0	0
$33$	−352.235	−0.323448
$34$	0	0
$35$	−122.333	−0.0998637
$36$	0	0
$37$	− 1636.60i	− 1.19547i	−0.801694	−	0.597734i	$-0.796068\pi$
$37$	− 1636.60i	− 1.19547i	0.801694	−	0.597734i	$-0.203932\pi$
$38$	0	0
$39$	− 802.006i	− 0.527288i
$40$	0	0
$41$	395.726	0.235411	0.117706	−	0.993049i	$-0.462446\pi$
$41$	395.726	0.235411	0.117706	+	0.993049i	$0.462446\pi$
$42$	0	0
$43$	350.192	0.189396	0.0946978	−	0.995506i	$-0.469812\pi$
$43$	350.192	0.189396	0.0946978	+	0.995506i	$0.469812\pi$
$44$	0	0
$45$	− 516.084i	− 0.254856i
$46$	0	0
$47$	2459.53i	1.11341i	0.830709	+	0.556707i	$0.187935\pi$
$47$	2459.53i	1.11341i	−0.830709	+	0.556707i	$0.812065\pi$
$48$	0	0
$49$	2360.04	0.982940
$50$	0	0
$51$	−1476.04	−0.567488
$52$	0	0
$53$	− 3827.15i	− 1.36246i	−0.732069	−	0.681230i	$-0.761445\pi$
$53$	− 3827.15i	− 1.36246i	0.732069	−	0.681230i	$-0.238555\pi$
$54$	0	0
$55$	− 1295.71i	− 0.428334i
$56$	0	0
$57$	2257.98	0.694975
$58$	0	0
$59$	1754.25	0.503951	0.251975	−	0.967734i	$-0.418920\pi$
$59$	1754.25	0.503951	0.251975	+	0.967734i	$0.418920\pi$
$60$	0	0
$61$	− 2335.10i	− 0.627546i	−0.949498	−	0.313773i	$-0.898407\pi$
$61$	− 2335.10i	− 0.627546i	0.949498	−	0.313773i	$-0.101593\pi$
$62$	0	0
$63$	− 172.803i	− 0.0435381i
$64$	0	0
$65$	2950.21	0.698274
$66$	0	0
$67$	2198.93	0.489848	0.244924	−	0.969542i	$-0.421237\pi$
$67$	2198.93	0.489848	0.244924	+	0.969542i	$0.421237\pi$
$68$	0	0
$69$	− 2404.25i	− 0.504989i
$70$	0	0
$71$	− 4463.62i	− 0.885463i	−0.896654	−	0.442732i	$-0.854009\pi$
$71$	− 4463.62i	− 0.885463i	0.896654	−	0.442732i	$-0.145991\pi$
$72$	0	0
$73$	2592.13	0.486420	0.243210	−	0.969974i	$-0.421800\pi$
$73$	2592.13	0.486420	0.243210	+	0.969974i	$0.421800\pi$
$74$	0	0
$75$	−1349.16	−0.239851
$76$	0	0
$77$	− 433.848i	− 0.0731738i
$78$	0	0
$79$	− 2040.82i	− 0.327002i	−0.986543	−	0.163501i	$-0.947721\pi$
$79$	− 2040.82i	− 0.327002i	0.986543	−	0.163501i	$-0.0522788\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	5245.36	0.761411	0.380706	−	0.924696i	$-0.375681\pi$
$83$	5245.36	0.761411	0.380706	+	0.924696i	$0.375681\pi$
$84$	0	0
$85$	− 5429.65i	− 0.751509i
$86$	0	0
$87$	1819.22i	0.240351i
$88$	0	0
$89$	14320.4	1.80790	0.903952	−	0.427635i	$-0.140653\pi$
$89$	14320.4	1.80790	0.903952	+	0.427635i	$0.140653\pi$
$90$	0	0
$91$	987.830	0.119289
$92$	0	0
$93$	459.612i	0.0531405i
$94$	0	0
$95$	8306.04i	0.920337i
$96$	0	0
$97$	16844.8	1.79029	0.895145	−	0.445776i	$-0.147072\pi$
$97$	16844.8	1.79029	0.895145	+	0.445776i	$0.147072\pi$
$98$	0	0
$99$	1830.27	0.186743
$100$	0	0
$101$	− 8909.06i	− 0.873352i	−0.899619	−	0.436676i	$-0.856156\pi$
$101$	− 8909.06i	− 0.873352i	0.899619	−	0.436676i	$-0.143844\pi$
$102$	0	0
$103$	− 19586.8i	− 1.84624i	−0.384507	−	0.923122i	$-0.625629\pi$
$103$	− 19586.8i	− 1.84624i	0.384507	−	0.923122i	$-0.374371\pi$
$104$	0	0
$105$	635.661	0.0576563
$106$	0	0
$107$	−13789.3	−1.20441	−0.602207	−	0.798340i	$-0.705712\pi$
$107$	−13789.3	−1.20441	−0.602207	+	0.798340i	$0.705712\pi$
$108$	0	0
$109$	− 13179.9i	− 1.10933i	−0.832074	−	0.554665i	$-0.812847\pi$
$109$	− 13179.9i	− 1.10933i	0.832074	−	0.554665i	$-0.187153\pi$
$110$	0	0
$111$	8504.00i	0.690204i
$112$	0	0
$113$	−2732.48	−0.213993	−0.106997	−	0.994259i	$-0.534123\pi$
$113$	−2732.48	−0.213993	−0.106997	+	0.994259i	$0.534123\pi$
$114$	0	0
$115$	8844.13	0.668743
$116$	0	0
$117$	4167.34i	0.304430i
$118$	0	0
$119$	− 1818.03i	− 0.128383i
$120$	0	0
$121$	−10045.8	−0.686144
$122$	0	0
$123$	−2056.25	−0.135915
$124$	0	0
$125$	− 16909.3i	− 1.08220i
$126$	0	0
$127$	23919.4i	1.48301i	0.670949	+	0.741503i	$0.265887\pi$
$127$	23919.4i	1.48301i	−0.670949	+	0.741503i	$0.734113\pi$
$128$	0	0
$129$	−1819.65	−0.109348
$130$	0	0
$131$	−25059.5	−1.46026	−0.730129	−	0.683309i	$-0.760540\pi$
$131$	−25059.5	−1.46026	−0.730129	+	0.683309i	$0.760540\pi$
$132$	0	0
$133$	2781.15i	0.157225i
$134$	0	0
$135$	2681.65i	0.147141i
$136$	0	0
$137$	528.551	0.0281609	0.0140804	−	0.999901i	$-0.495518\pi$
$137$	528.551	0.0281609	0.0140804	+	0.999901i	$0.495518\pi$
$138$	0	0
$139$	24051.5	1.24484	0.622419	−	0.782684i	$-0.286150\pi$
$139$	24051.5	1.24484	0.622419	+	0.782684i	$0.286150\pi$
$140$	0	0
$141$	− 12780.1i	− 0.642830i
$142$	0	0
$143$	10462.8i	0.511651i
$144$	0	0
$145$	−6692.06	−0.318290
$146$	0	0
$147$	−12263.1	−0.567501
$148$	0	0
$149$	5229.61i	0.235557i	0.993040	+	0.117779i	$0.0375773\pi$
$149$	5229.61i	0.235557i	−0.993040	+	0.117779i	$0.962423\pi$
$150$	0	0
$151$	− 26586.7i	− 1.16603i	−0.812460	−	0.583017i	$-0.801872\pi$
$151$	− 26586.7i	− 1.16603i	0.812460	−	0.583017i	$-0.198128\pi$
$152$	0	0
$153$	7669.71	0.327639
$154$	0	0
$155$	−1690.70	−0.0703725
$156$	0	0
$157$	10616.7i	0.430717i	0.976535	+	0.215359i	$0.0690920\pi$
$157$	10616.7i	0.430717i	−0.976535	+	0.215359i	$0.930908\pi$
$158$	0	0
$159$	19886.5i	0.786617i
$160$	0	0
$161$	2961.32	0.114244
$162$	0	0
$163$	13539.3	0.509589	0.254794	−	0.966995i	$-0.417992\pi$
$163$	13539.3	0.509589	0.254794	+	0.966995i	$0.417992\pi$
$164$	0	0
$165$	6732.70i	0.247299i
$166$	0	0
$167$	− 31228.7i	− 1.11975i	−0.828577	−	0.559876i	$-0.810849\pi$
$167$	− 31228.7i	− 1.11975i	0.828577	−	0.559876i	$-0.189151\pi$
$168$	0	0
$169$	4738.29	0.165901
$170$	0	0
$171$	−11732.8	−0.401244
$172$	0	0
$173$	42767.2i	1.42895i	0.699659	+	0.714477i	$0.253335\pi$
$173$	42767.2i	1.42895i	−0.699659	+	0.714477i	$0.746665\pi$
$174$	0	0
$175$	− 1661.76i	− 0.0542615i
$176$	0	0
$177$	−9115.36	−0.290956
$178$	0	0
$179$	−16667.9	−0.520206	−0.260103	−	0.965581i	$-0.583756\pi$
$179$	−16667.9	−0.520206	−0.260103	+	0.965581i	$0.583756\pi$
$180$	0	0
$181$	23689.5i	0.723101i	0.932353	+	0.361550i	$0.117752\pi$
$181$	23689.5i	0.723101i	−0.932353	+	0.361550i	$0.882248\pi$
$182$	0	0
$183$	12133.5i	0.362314i
$184$	0	0
$185$	−31282.3	−0.914019
$186$	0	0
$187$	19256.0	0.550659
$188$	0	0
$189$	897.909i	0.0251367i
$190$	0	0
$191$	67099.3i	1.83929i	0.392747	+	0.919647i	$0.371525\pi$
$191$	67099.3i	1.83929i	−0.392747	+	0.919647i	$0.628475\pi$
$192$	0	0
$193$	−11239.9	−0.301750	−0.150875	−	0.988553i	$-0.548209\pi$
$193$	−11239.9	−0.301750	−0.150875	+	0.988553i	$0.548209\pi$
$194$	0	0
$195$	−15329.7	−0.403149
$196$	0	0
$197$	14984.6i	0.386113i	0.981188	+	0.193056i	$0.0618400\pi$
$197$	14984.6i	0.386113i	−0.981188	+	0.193056i	$0.938160\pi$
$198$	0	0
$199$	− 71104.7i	− 1.79553i	−0.440477	−	0.897764i	$-0.645191\pi$
$199$	− 71104.7i	− 1.79553i	0.440477	−	0.897764i	$-0.354809\pi$
$200$	0	0
$201$	−11426.0	−0.282814
$202$	0	0
$203$	−2240.73	−0.0543747
$204$	0	0
$205$	− 7564.01i	− 0.179988i
$206$	0	0
$207$	12492.9i	0.291555i
$208$	0	0
$209$	−29457.0	−0.674365
$210$	0	0
$211$	60965.8	1.36937	0.684686	−	0.728838i	$-0.259939\pi$
$211$	60965.8	1.36937	0.684686	+	0.728838i	$0.259939\pi$
$212$	0	0
$213$	23193.7i	0.511222i
$214$	0	0
$215$	− 6693.66i	− 0.144806i
$216$	0	0
$217$	−566.104	−0.0120220
$218$	0	0
$219$	−13469.1	−0.280835
$220$	0	0
$221$	43844.0i	0.897689i
$222$	0	0
$223$	− 87167.1i	− 1.75284i	−0.481545	−	0.876422i	$-0.659924\pi$
$223$	− 87167.1i	− 1.75284i	0.481545	−	0.876422i	$-0.340076\pi$
$224$	0	0
$225$	7010.44	0.138478
$226$	0	0
$227$	14968.7	0.290492	0.145246	−	0.989396i	$-0.453603\pi$
$227$	14968.7	0.290492	0.145246	+	0.989396i	$0.453603\pi$
$228$	0	0
$229$	78306.4i	1.49323i	0.665257	+	0.746615i	$0.268322\pi$
$229$	78306.4i	1.49323i	−0.665257	+	0.746615i	$0.731678\pi$
$230$	0	0
$231$	2254.34i	0.0422469i
$232$	0	0
$233$	−54896.5	−1.01119	−0.505595	−	0.862771i	$-0.668727\pi$
$233$	−54896.5	−1.01119	−0.505595	+	0.862771i	$0.668727\pi$
$234$	0	0
$235$	47012.1	0.851283
$236$	0	0
$237$	10604.4i	0.188795i
$238$	0	0
$239$	66861.2i	1.17052i	0.810846	+	0.585259i	$0.199007\pi$
$239$	66861.2i	1.17052i	−0.810846	+	0.585259i	$0.800993\pi$
$240$	0	0
$241$	−99876.4	−1.71961	−0.859803	−	0.510626i	$-0.829414\pi$
$241$	−99876.4	−1.71961	−0.859803	+	0.510626i	$0.829414\pi$
$242$	0	0
$243$	−3788.00	−0.0641500
$244$	0	0
$245$	− 45110.3i	− 0.751526i
$246$	0	0
$247$	− 67070.7i	− 1.09936i
$248$	0	0
$249$	−27255.7	−0.439601
$250$	0	0
$251$	−18469.6	−0.293164	−0.146582	−	0.989199i	$-0.546827\pi$
$251$	−18469.6	−0.293164	−0.146582	+	0.989199i	$0.546827\pi$
$252$	0	0
$253$	31365.2i	0.490013i
$254$	0	0
$255$	28213.3i	0.433884i
$256$	0	0
$257$	12430.9	0.188208	0.0941040	−	0.995562i	$-0.470001\pi$
$257$	12430.9	0.188208	0.0941040	+	0.995562i	$0.470001\pi$
$258$	0	0
$259$	−10474.4	−0.156145
$260$	0	0
$261$	− 9452.93i	− 0.138767i
$262$	0	0
$263$	50546.1i	0.730762i	0.930858	+	0.365381i	$0.119061\pi$
$263$	50546.1i	0.730762i	−0.930858	+	0.365381i	$0.880939\pi$
$264$	0	0
$265$	−73153.1	−1.04170
$266$	0	0
$267$	−74411.0	−1.04379
$268$	0	0
$269$	88678.2i	1.22550i	0.790278	+	0.612748i	$0.209936\pi$
$269$	88678.2i	1.22550i	−0.790278	+	0.612748i	$0.790064\pi$
$270$	0	0
$271$	18805.0i	0.256056i	0.991771	+	0.128028i	$0.0408647\pi$
$271$	18805.0i	0.256056i	−0.991771	+	0.128028i	$0.959135\pi$
$272$	0	0
$273$	−5132.92	−0.0688714
$274$	0	0
$275$	17600.8	0.232738
$276$	0	0
$277$	− 37803.3i	− 0.492686i	−0.969183	−	0.246343i	$-0.920771\pi$
$277$	− 37803.3i	− 0.492686i	0.969183	−	0.246343i	$-0.0792289\pi$
$278$	0	0
$279$	− 2388.21i	− 0.0306807i
$280$	0	0
$281$	25583.9	0.324006	0.162003	−	0.986790i	$-0.448205\pi$
$281$	25583.9	0.324006	0.162003	+	0.986790i	$0.448205\pi$
$282$	0	0
$283$	42546.3	0.531237	0.265619	−	0.964078i	$-0.414424\pi$
$283$	42546.3	0.531237	0.265619	+	0.964078i	$0.414424\pi$
$284$	0	0
$285$	− 43159.5i	− 0.531357i
$286$	0	0
$287$	− 2532.69i	− 0.0307481i
$288$	0	0
$289$	−2829.08	−0.0338726
$290$	0	0
$291$	−87528.3	−1.03362
$292$	0	0
$293$	− 101982.i	− 1.18793i	−0.804492	−	0.593963i	$-0.797562\pi$
$293$	− 101982.i	− 1.18793i	0.804492	−	0.593963i	$-0.202438\pi$
$294$	0	0
$295$	− 33531.2i	− 0.385305i
$296$	0	0
$297$	−9510.34	−0.107816
$298$	0	0
$299$	−71415.7	−0.798824
$300$	0	0
$301$	− 2241.27i	− 0.0247378i
$302$	0	0
$303$	46292.9i	0.504230i
$304$	0	0
$305$	−44633.7	−0.479803
$306$	0	0
$307$	170124.	1.80505	0.902523	−	0.430642i	$-0.141713\pi$
$307$	170124.	1.80505	0.902523	+	0.430642i	$0.141713\pi$
$308$	0	0
$309$	101776.i	1.06593i
$310$	0	0
$311$	41592.7i	0.430028i	0.976611	+	0.215014i	$0.0689797\pi$
$311$	41592.7i	0.430028i	−0.976611	+	0.215014i	$0.931020\pi$
$312$	0	0
$313$	121687.	1.24210	0.621049	−	0.783772i	$-0.286707\pi$
$313$	121687.	1.24210	0.621049	+	0.783772i	$0.286707\pi$
$314$	0	0
$315$	−3302.99	−0.0332879
$316$	0	0
$317$	126498.i	1.25882i	0.777073	+	0.629411i	$0.216704\pi$
$317$	126498.i	1.25882i	−0.777073	+	0.629411i	$0.783296\pi$
$318$	0	0
$319$	− 23733.0i	− 0.233223i
$320$	0	0
$321$	71651.5	0.695368
$322$	0	0
$323$	−123439.	−1.18317
$324$	0	0
$325$	40075.3i	0.379411i
$326$	0	0
$327$	68485.0i	0.640471i
$328$	0	0
$329$	15741.3	0.145428
$330$	0	0
$331$	−33699.5	−0.307586	−0.153793	−	0.988103i	$-0.549149\pi$
$331$	−33699.5	−0.307586	−0.153793	+	0.988103i	$0.549149\pi$
$332$	0	0
$333$	− 44188.1i	− 0.398489i
$334$	0	0
$335$	− 42030.8i	− 0.374523i
$336$	0	0
$337$	92980.8	0.818717	0.409358	−	0.912374i	$-0.365753\pi$
$337$	92980.8	0.818717	0.409358	+	0.912374i	$0.365753\pi$
$338$	0	0
$339$	14198.4	0.123549
$340$	0	0
$341$	− 5995.98i	− 0.0515645i
$342$	0	0
$343$	− 30471.1i	− 0.259000i
$344$	0	0
$345$	−45955.4	−0.386099
$346$	0	0
$347$	6718.56	0.0557978	0.0278989	−	0.999611i	$-0.491118\pi$
$347$	6718.56	0.0557978	0.0278989	+	0.999611i	$0.491118\pi$
$348$	0	0
$349$	− 79498.6i	− 0.652693i	−0.945250	−	0.326346i	$-0.894182\pi$
$349$	− 79498.6i	− 0.652693i	0.945250	−	0.326346i	$-0.105818\pi$
$350$	0	0
$351$	− 21654.2i	− 0.175763i
$352$	0	0
$353$	45108.7	0.362002	0.181001	−	0.983483i	$-0.442066\pi$
$353$	45108.7	0.362002	0.181001	+	0.983483i	$0.442066\pi$
$354$	0	0
$355$	−85318.7	−0.676998
$356$	0	0
$357$	9446.78i	0.0741220i
$358$	0	0
$359$	171192.i	1.32829i	0.747602	+	0.664147i	$0.231205\pi$
$359$	171192.i	1.32829i	−0.747602	+	0.664147i	$0.768795\pi$
$360$	0	0
$361$	58510.5	0.448972
$362$	0	0
$363$	52199.7	0.396145
$364$	0	0
$365$	− 49546.6i	− 0.371902i
$366$	0	0
$367$	− 197479.i	− 1.46618i	−0.680130	−	0.733091i	$-0.738077\pi$
$367$	− 197479.i	− 1.46618i	0.680130	−	0.733091i	$-0.261923\pi$
$368$	0	0
$369$	10684.6	0.0784704
$370$	0	0
$371$	−24494.1	−0.177957
$372$	0	0
$373$	− 78469.2i	− 0.564003i	−0.959414	−	0.282002i	$-0.909002\pi$
$373$	− 78469.2i	− 0.564003i	0.959414	−	0.282002i	$-0.0909984\pi$
$374$	0	0
$375$	87863.5i	0.624807i
$376$	0	0
$377$	54037.9	0.380203
$378$	0	0
$379$	248236.	1.72817	0.864086	−	0.503344i	$-0.167897\pi$
$379$	248236.	1.72817	0.864086	+	0.503344i	$0.167897\pi$
$380$	0	0
$381$	− 124289.i	− 0.856214i
$382$	0	0
$383$	− 75091.5i	− 0.511910i	−0.966689	−	0.255955i	$-0.917610\pi$
$383$	− 75091.5i	− 0.511910i	0.966689	−	0.255955i	$-0.0823898\pi$
$384$	0	0
$385$	−8292.67	−0.0559465
$386$	0	0
$387$	9455.19	0.0631319
$388$	0	0
$389$	− 280557.i	− 1.85405i	−0.375000	−	0.927025i	$-0.622357\pi$
$389$	− 280557.i	− 1.85405i	0.375000	−	0.927025i	$-0.377643\pi$
$390$	0	0
$391$	131436.i	0.859725i
$392$	0	0
$393$	130213.	0.843081
$394$	0	0
$395$	−39008.8	−0.250016
$396$	0	0
$397$	− 225848.i	− 1.43296i	−0.697605	−	0.716482i	$-0.745751\pi$
$397$	− 225848.i	− 1.43296i	0.697605	−	0.716482i	$-0.254249\pi$
$398$	0	0
$399$	− 14451.3i	− 0.0907737i
$400$	0	0
$401$	165029.	1.02629	0.513147	−	0.858301i	$-0.328480\pi$
$401$	165029.	1.02629	0.513147	+	0.858301i	$0.328480\pi$
$402$	0	0
$403$	13652.3	0.0840611
$404$	0	0
$405$	− 13934.3i	− 0.0849522i
$406$	0	0
$407$	− 110941.i	− 0.669735i
$408$	0	0
$409$	−29698.0	−0.177534	−0.0887669	−	0.996052i	$-0.528293\pi$
$409$	−29698.0	−0.177534	−0.0887669	+	0.996052i	$0.528293\pi$
$410$	0	0
$411$	−2746.43	−0.0162587
$412$	0	0
$413$	− 11227.4i	− 0.0658231i
$414$	0	0
$415$	− 100261.i	− 0.582152i
$416$	0	0
$417$	−124975.	−0.718708
$418$	0	0
$419$	−100933.	−0.574918	−0.287459	−	0.957793i	$-0.592810\pi$
$419$	−100933.	−0.574918	−0.287459	+	0.957793i	$0.592810\pi$
$420$	0	0
$421$	211728.i	1.19457i	0.802028	+	0.597287i	$0.203755\pi$
$421$	211728.i	1.19457i	−0.802028	+	0.597287i	$0.796245\pi$
$422$	0	0
$423$	66407.4i	0.371138i
$424$	0	0
$425$	73755.9	0.408337
$426$	0	0
$427$	−14944.9	−0.0819665
$428$	0	0
$429$	− 54366.1i	− 0.295402i
$430$	0	0
$431$	− 147606.i	− 0.794603i	−0.917688	−	0.397302i	$-0.869947\pi$
$431$	− 147606.i	− 0.794603i	0.917688	−	0.397302i	$-0.130053\pi$
$432$	0	0
$433$	98534.8	0.525550	0.262775	−	0.964857i	$-0.415362\pi$
$433$	98534.8	0.525550	0.262775	+	0.964857i	$0.415362\pi$
$434$	0	0
$435$	34772.9	0.183765
$436$	0	0
$437$	− 201064.i	− 1.05286i
$438$	0	0
$439$	27484.6i	0.142613i	0.997454	+	0.0713067i	$0.0227169\pi$
$439$	27484.6i	0.142613i	−0.997454	+	0.0713067i	$0.977283\pi$
$440$	0	0
$441$	63721.0	0.327647
$442$	0	0
$443$	53561.9	0.272928	0.136464	−	0.990645i	$-0.456426\pi$
$443$	53561.9	0.272928	0.136464	+	0.990645i	$0.456426\pi$
$444$	0	0
$445$	− 273724.i	− 1.38227i
$446$	0	0
$447$	− 27173.8i	− 0.135999i
$448$	0	0
$449$	69100.4	0.342758	0.171379	−	0.985205i	$-0.445178\pi$
$449$	69100.4	0.342758	0.171379	+	0.985205i	$0.445178\pi$
$450$	0	0
$451$	26825.4	0.131884
$452$	0	0
$453$	138149.i	0.673210i
$454$	0	0
$455$	− 18881.6i	− 0.0912045i
$456$	0	0
$457$	115106.	0.551146	0.275573	−	0.961280i	$-0.411132\pi$
$457$	115106.	0.551146	0.275573	+	0.961280i	$0.411132\pi$
$458$	0	0
$459$	−39853.0	−0.189163
$460$	0	0
$461$	108198.i	0.509119i	0.967057	+	0.254559i	$0.0819304\pi$
$461$	108198.i	0.509119i	−0.967057	+	0.254559i	$0.918070\pi$
$462$	0	0
$463$	− 63779.4i	− 0.297522i	−0.988873	−	0.148761i	$-0.952472\pi$
$463$	− 63779.4i	− 0.297522i	0.988873	−	0.148761i	$-0.0475285\pi$
$464$	0	0
$465$	8785.13	0.0406296
$466$	0	0
$467$	392929.	1.80169	0.900844	−	0.434142i	$-0.142948\pi$
$467$	392929.	1.80169	0.900844	+	0.434142i	$0.142948\pi$
$468$	0	0
$469$	− 14073.3i	− 0.0639811i
$470$	0	0
$471$	− 55166.2i	− 0.248675i
$472$	0	0
$473$	23738.7	0.106105
$474$	0	0
$475$	−112829.	−0.500071
$476$	0	0
$477$	− 103333.i	− 0.454153i
$478$	0	0
$479$	− 249509.i	− 1.08746i	−0.839259	−	0.543732i	$-0.817011\pi$
$479$	− 249509.i	− 1.08746i	0.839259	−	0.543732i	$-0.182989\pi$
$480$	0	0
$481$	252602.	1.09181
$482$	0	0
$483$	−15387.4	−0.0659587
$484$	0	0
$485$	− 321976.i	− 1.36880i
$486$	0	0
$487$	401706.i	1.69375i	0.531789	+	0.846877i	$0.321520\pi$
$487$	401706.i	1.69375i	−0.531789	+	0.846877i	$0.678480\pi$
$488$	0	0
$489$	−70352.1	−0.294211
$490$	0	0
$491$	−110571.	−0.458645	−0.229322	−	0.973351i	$-0.573651\pi$
$491$	−110571.	−0.458645	−0.229322	+	0.973351i	$0.573651\pi$
$492$	0	0
$493$	− 99452.9i	− 0.409189i
$494$	0	0
$495$	− 34984.1i	− 0.142778i
$496$	0	0
$497$	−28567.6	−0.115654
$498$	0	0
$499$	186945.	0.750779	0.375390	−	0.926867i	$-0.377509\pi$
$499$	186945.	0.750779	0.375390	+	0.926867i	$0.377509\pi$
$500$	0	0
$501$	162269.i	0.646489i
$502$	0	0
$503$	439468.i	1.73697i	0.495719	+	0.868483i	$0.334905\pi$
$503$	439468.i	1.73697i	−0.495719	+	0.868483i	$0.665095\pi$
$504$	0	0
$505$	−170290.	−0.667738
$506$	0	0
$507$	−24620.9	−0.0957828
$508$	0	0
$509$	339847.i	1.31174i	0.754874	+	0.655870i	$0.227698\pi$
$509$	339847.i	1.31174i	−0.754874	+	0.655870i	$0.772302\pi$
$510$	0	0
$511$	− 16589.9i	− 0.0635334i
$512$	0	0
$513$	60965.3	0.231658
$514$	0	0
$515$	−374387.	−1.41158
$516$	0	0
$517$	166726.i	0.623767i
$518$	0	0
$519$	− 222225.i	− 0.825007i
$520$	0	0
$521$	107168.	0.394812	0.197406	−	0.980322i	$-0.436748\pi$
$521$	107168.	0.394812	0.197406	+	0.980322i	$0.436748\pi$
$522$	0	0
$523$	−372679.	−1.36248	−0.681242	−	0.732058i	$-0.738560\pi$
$523$	−372679.	−1.36248	−0.681242	+	0.732058i	$0.738560\pi$
$524$	0	0
$525$	8634.76i	0.0313279i
$526$	0	0
$527$	− 25126.1i	− 0.0904697i
$528$	0	0
$529$	65751.1	0.234959
$530$	0	0
$531$	47364.8	0.167984
$532$	0	0
$533$	61078.8i	0.214999i
$534$	0	0
$535$	263573.i	0.920858i
$536$	0	0
$537$	86609.0	0.300341
$538$	0	0
$539$	159981.	0.550671
$540$	0	0
$541$	105010.i	0.358785i	0.983778	+	0.179393i	$0.0574132\pi$
$541$	105010.i	0.358785i	−0.983778	+	0.179393i	$0.942587\pi$
$542$	0	0
$543$	− 123094.i	− 0.417482i
$544$	0	0
$545$	−251924.	−0.848159
$546$	0	0
$547$	−556117.	−1.85862	−0.929311	−	0.369298i	$-0.879598\pi$
$547$	−556117.	−1.85862	−0.929311	+	0.369298i	$0.879598\pi$
$548$	0	0
$549$	− 63047.7i	− 0.209182i
$550$	0	0
$551$	152139.i	0.501114i
$552$	0	0
$553$	−13061.5	−0.0427112
$554$	0	0
$555$	162548.	0.527709
$556$	0	0
$557$	336211.i	1.08368i	0.840481	+	0.541841i	$0.182273\pi$
$557$	336211.i	1.08368i	−0.840481	+	0.541841i	$0.817727\pi$
$558$	0	0
$559$	54050.8i	0.172973i
$560$	0	0
$561$	−100057.	−0.317923
$562$	0	0
$563$	558606.	1.76234	0.881168	−	0.472803i	$-0.156758\pi$
$563$	558606.	1.76234	0.881168	+	0.472803i	$0.156758\pi$
$564$	0	0
$565$	52229.2i	0.163613i
$566$	0	0
$567$	− 4665.67i	− 0.0145127i
$568$	0	0
$569$	190223.	0.587542	0.293771	−	0.955876i	$-0.405090\pi$
$569$	190223.	0.587542	0.293771	+	0.955876i	$0.405090\pi$
$570$	0	0
$571$	−72652.1	−0.222831	−0.111416	−	0.993774i	$-0.535538\pi$
$571$	−72652.1	−0.222831	−0.111416	+	0.993774i	$0.535538\pi$
$572$	0	0
$573$	− 348658.i	− 1.06192i
$574$	0	0
$575$	120138.i	0.363366i
$576$	0	0
$577$	−294340.	−0.884091	−0.442046	−	0.896993i	$-0.645747\pi$
$577$	−294340.	−0.884091	−0.442046	+	0.896993i	$0.645747\pi$
$578$	0	0
$579$	58404.2	0.174216
$580$	0	0
$581$	− 33570.8i	− 0.0994512i
$582$	0	0
$583$	− 259434.i	− 0.763289i
$584$	0	0
$585$	79655.6	0.232758
$586$	0	0
$587$	189152.	0.548953	0.274477	−	0.961594i	$-0.411495\pi$
$587$	189152.	0.548953	0.274477	+	0.961594i	$0.411495\pi$
$588$	0	0
$589$	38436.7i	0.110794i
$590$	0	0
$591$	− 77862.5i	− 0.222922i
$592$	0	0
$593$	−50118.2	−0.142523	−0.0712617	−	0.997458i	$-0.522703\pi$
$593$	−50118.2	−0.142523	−0.0712617	+	0.997458i	$0.522703\pi$
$594$	0	0
$595$	−34750.3	−0.0981578
$596$	0	0
$597$	369471.i	1.03665i
$598$	0	0
$599$	− 142995.i	− 0.398536i	−0.979945	−	0.199268i	$-0.936144\pi$
$599$	− 142995.i	− 0.398536i	0.979945	−	0.199268i	$-0.0638564\pi$
$600$	0	0
$601$	−582870.	−1.61370	−0.806849	−	0.590757i	$-0.798829\pi$
$601$	−582870.	−1.61370	−0.806849	+	0.590757i	$0.798829\pi$
$602$	0	0
$603$	59371.0	0.163283
$604$	0	0
$605$	192018.i	0.524605i
$606$	0	0
$607$	44407.7i	0.120526i	0.998183	+	0.0602630i	$0.0191939\pi$
$607$	44407.7i	0.120526i	−0.998183	+	0.0602630i	$0.980806\pi$
$608$	0	0
$609$	11643.2	0.0313933
$610$	0	0
$611$	−379619.	−1.01687
$612$	0	0
$613$	108875.i	0.289738i	0.989451	+	0.144869i	$0.0462761\pi$
$613$	108875.i	0.289738i	−0.989451	+	0.144869i	$0.953724\pi$
$614$	0	0
$615$	39303.7i	0.103916i
$616$	0	0
$617$	−698860.	−1.83578	−0.917888	−	0.396839i	$-0.870107\pi$
$617$	−698860.	−1.83578	−0.917888	+	0.396839i	$0.870107\pi$
$618$	0	0
$619$	−441728.	−1.15285	−0.576427	−	0.817149i	$-0.695553\pi$
$619$	−441728.	−1.15285	−0.576427	+	0.817149i	$0.695553\pi$
$620$	0	0
$621$	− 64914.8i	− 0.168330i
$622$	0	0
$623$	− 91652.0i	− 0.236138i
$624$	0	0
$625$	−160930.	−0.411981
$626$	0	0
$627$	153063.	0.389345
$628$	0	0
$629$	− 464897.i	− 1.17505i
$630$	0	0
$631$	649021.i	1.63005i	0.579428	+	0.815023i	$0.303276\pi$
$631$	649021.i	1.63005i	−0.579428	+	0.815023i	$0.696724\pi$
$632$	0	0
$633$	−316788.	−0.790608
$634$	0	0
$635$	457201.	1.13386
$636$	0	0
$637$	364263.i	0.897710i
$638$	0	0
$639$	− 120518.i	− 0.295154i
$640$	0	0
$641$	−774491.	−1.88495	−0.942477	−	0.334272i	$-0.891510\pi$
$641$	−774491.	−1.88495	−0.942477	+	0.334272i	$0.891510\pi$
$642$	0	0
$643$	−611676.	−1.47945	−0.739723	−	0.672911i	$-0.765044\pi$
$643$	−611676.	−1.47945	−0.739723	+	0.672911i	$0.765044\pi$
$644$	0	0
$645$	34781.3i	0.0836038i
$646$	0	0
$647$	− 518940.i	− 1.23968i	−0.784729	−	0.619838i	$-0.787198\pi$
$647$	− 518940.i	− 1.23968i	0.784729	−	0.619838i	$-0.212802\pi$
$648$	0	0
$649$	118917.	0.282327
$650$	0	0
$651$	2941.56	0.00694090
$652$	0	0
$653$	648450.i	1.52072i	0.649500	+	0.760362i	$0.274978\pi$
$653$	648450.i	1.52072i	−0.649500	+	0.760362i	$0.725022\pi$
$654$	0	0
$655$	478993.i	1.11647i
$656$	0	0
$657$	69987.6	0.162140
$658$	0	0
$659$	611892.	1.40898	0.704489	−	0.709715i	$-0.251176\pi$
$659$	611892.	1.40898	0.704489	+	0.709715i	$0.251176\pi$
$660$	0	0
$661$	157591.i	0.360684i	0.983604	+	0.180342i	$0.0577205\pi$
$661$	157591.i	0.360684i	−0.983604	+	0.180342i	$0.942280\pi$
$662$	0	0
$663$	− 227820.i	− 0.518281i
$664$	0	0
$665$	53159.5	0.120209
$666$	0	0
$667$	161995.	0.364124
$668$	0	0
$669$	452934.i	1.01200i
$670$	0	0
$671$	− 158291.i	− 0.351569i
$672$	0	0
$673$	−96816.9	−0.213757	−0.106879	−	0.994272i	$-0.534086\pi$
$673$	−96816.9	−0.213757	−0.106879	+	0.994272i	$0.534086\pi$
$674$	0	0
$675$	−36427.3	−0.0799502
$676$	0	0
$677$	50973.9i	0.111217i	0.998453	+	0.0556084i	$0.0177098\pi$
$677$	50973.9i	0.111217i	−0.998453	+	0.0556084i	$0.982290\pi$
$678$	0	0
$679$	− 107809.i	− 0.233837i
$680$	0	0
$681$	−77779.9	−0.167715
$682$	0	0
$683$	−179552.	−0.384901	−0.192451	−	0.981307i	$-0.561644\pi$
$683$	−179552.	−0.384901	−0.192451	+	0.981307i	$0.561644\pi$
$684$	0	0
$685$	− 10102.9i	− 0.0215309i
$686$	0	0
$687$	− 406892.i	− 0.862116i
$688$	0	0
$689$	590706.	1.24432
$690$	0	0
$691$	189721.	0.397338	0.198669	−	0.980067i	$-0.436338\pi$
$691$	189721.	0.397338	0.198669	+	0.980067i	$0.436338\pi$
$692$	0	0
$693$	− 11713.9i	− 0.0243913i
$694$	0	0
$695$	− 459726.i	− 0.951765i
$696$	0	0
$697$	112411.	0.231390
$698$	0	0
$699$	285251.	0.583811
$700$	0	0
$701$	499395.i	1.01627i	0.861278	+	0.508133i	$0.169664\pi$
$701$	499395.i	1.01627i	−0.861278	+	0.508133i	$0.830336\pi$
$702$	0	0
$703$	711179.i	1.43902i
$704$	0	0
$705$	−244282.	−0.491488
$706$	0	0
$707$	−57018.9	−0.114072
$708$	0	0
$709$	328533.i	0.653561i	0.945100	+	0.326781i	$0.105964\pi$
$709$	328533.i	0.653561i	−0.945100	+	0.326781i	$0.894036\pi$
$710$	0	0
$711$	− 55102.2i	− 0.109001i
$712$	0	0
$713$	40926.8	0.0805060
$714$	0	0
$715$	199988.	0.391193
$716$	0	0
$717$	− 347421.i	− 0.675799i
$718$	0	0
$719$	− 70795.6i	− 0.136946i	−0.997653	−	0.0684728i	$-0.978187\pi$
$719$	− 70795.6i	− 0.136946i	0.997653	−	0.0684728i	$-0.0218126\pi$
$720$	0	0
$721$	−125357.	−0.241146
$722$	0	0
$723$	518973.	0.992815
$724$	0	0
$725$	− 90904.2i	− 0.172945i
$726$	0	0
$727$	− 264078.i	− 0.499646i	−0.968291	−	0.249823i	$-0.919627\pi$
$727$	− 264078.i	− 0.499646i	0.968291	−	0.249823i	$-0.0803725\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	99476.8	0.186160
$732$	0	0
$733$	− 216452.i	− 0.402859i	−0.979503	−	0.201430i	$-0.935441\pi$
$733$	− 216452.i	− 0.402859i	0.979503	−	0.201430i	$-0.0645587\pi$
$734$	0	0
$735$	234400.i	0.433894i
$736$	0	0
$737$	149060.	0.274427
$738$	0	0
$739$	−515590.	−0.944094	−0.472047	−	0.881573i	$-0.656485\pi$
$739$	−515590.	−0.944094	−0.472047	+	0.881573i	$0.656485\pi$
$740$	0	0
$741$	348510.i	0.634714i
$742$	0	0
$743$	− 622349.i	− 1.12734i	−0.825999	−	0.563672i	$-0.809388\pi$
$743$	− 622349.i	− 1.12734i	0.825999	−	0.563672i	$-0.190612\pi$
$744$	0	0
$745$	99960.0	0.180100
$746$	0	0
$747$	141625.	0.253804
$748$	0	0
$749$	88253.1i	0.157314i
$750$	0	0
$751$	583086.i	1.03384i	0.856034	+	0.516919i	$0.172921\pi$
$751$	583086.i	1.03384i	−0.856034	+	0.516919i	$0.827079\pi$
$752$	0	0
$753$	95970.8	0.169258
$754$	0	0
$755$	−508185.	−0.891513
$756$	0	0
$757$	884570.i	1.54362i	0.635853	+	0.771810i	$0.280649\pi$
$757$	884570.i	1.54362i	−0.635853	+	0.771810i	$0.719351\pi$
$758$	0	0
$759$	− 162979.i	− 0.282909i
$760$	0	0
$761$	−343529.	−0.593190	−0.296595	−	0.955003i	$-0.595851\pi$
$761$	−343529.	−0.593190	−0.296595	+	0.955003i	$0.595851\pi$
$762$	0	0
$763$	−84352.9	−0.144894
$764$	0	0
$765$	− 146601.i	− 0.250503i
$766$	0	0
$767$	270762.i	0.460253i
$768$	0	0
$769$	−694796.	−1.17491	−0.587455	−	0.809257i	$-0.699870\pi$
$769$	−694796.	−1.17491	−0.587455	+	0.809257i	$0.699870\pi$
$770$	0	0
$771$	−64593.1	−0.108662
$772$	0	0
$773$	− 49585.9i	− 0.0829850i	−0.999139	−	0.0414925i	$-0.986789\pi$
$773$	− 49585.9i	− 0.0829850i	0.999139	−	0.0414925i	$-0.0132113\pi$
$774$	0	0
$775$	− 22966.3i	− 0.0382373i
$776$	0	0
$777$	54426.5	0.0901505
$778$	0	0
$779$	−171962.	−0.283372
$780$	0	0
$781$	− 302578.i	− 0.496062i
$782$	0	0
$783$	49118.9i	0.0801170i
$784$	0	0
$785$	202931.	0.329313
$786$	0	0
$787$	557614.	0.900293	0.450147	−	0.892955i	$-0.351372\pi$
$787$	557614.	0.900293	0.450147	+	0.892955i	$0.351372\pi$
$788$	0	0
$789$	− 262645.i	− 0.421906i
$790$	0	0
$791$	17488.1i	0.0279506i
$792$	0	0
$793$	360414.	0.573132
$794$	0	0
$795$	380114.	0.601423
$796$	0	0
$797$	− 459192.i	− 0.722899i	−0.932392	−	0.361449i	$-0.882282\pi$
$797$	− 459192.i	− 0.722899i	0.932392	−	0.361449i	$-0.117718\pi$
$798$	0	0
$799$	698663.i	1.09439i
$800$	0	0
$801$	386651.	0.602634
$802$	0	0
$803$	175715.	0.272506
$804$	0	0
$805$	− 56603.3i	− 0.0873474i
$806$	0	0
$807$	− 460785.i	− 0.707541i
$808$	0	0
$809$	−594379.	−0.908168	−0.454084	−	0.890959i	$-0.650033\pi$
$809$	−594379.	−0.908168	−0.454084	+	0.890959i	$0.650033\pi$
$810$	0	0
$811$	−1.17014e6	−1.77908	−0.889541	−	0.456855i	$-0.848976\pi$
$811$	−1.17014e6	−1.77908	−0.889541	+	0.456855i	$0.848976\pi$
$812$	0	0
$813$	− 97713.7i	− 0.147834i
$814$	0	0
$815$	− 258793.i	− 0.389616i
$816$	0	0
$817$	−152175.	−0.227982
$818$	0	0
$819$	26671.4	0.0397629
$820$	0	0
$821$	− 63648.5i	− 0.0944282i	−0.998885	−	0.0472141i	$-0.984966\pi$
$821$	− 63648.5i	− 0.0944282i	0.998885	−	0.0472141i	$-0.0150343\pi$
$822$	0	0
$823$	− 79834.3i	− 0.117866i	−0.998262	−	0.0589331i	$-0.981230\pi$
$823$	− 79834.3i	− 0.117866i	0.998262	−	0.0589331i	$-0.0187699\pi$
$824$	0	0
$825$	−91456.4	−0.134371
$826$	0	0
$827$	−242691.	−0.354848	−0.177424	−	0.984134i	$-0.556776\pi$
$827$	−242691.	−0.354848	−0.177424	+	0.984134i	$0.556776\pi$
$828$	0	0
$829$	− 417062.i	− 0.606864i	−0.952853	−	0.303432i	$-0.901867\pi$
$829$	− 417062.i	− 0.606864i	0.952853	−	0.303432i	$-0.0981326\pi$
$830$	0	0
$831$	196432.i	0.284452i
$832$	0	0
$833$	670400.	0.966149
$834$	0	0
$835$	−596914.	−0.856128
$836$	0	0
$837$	12409.5i	0.0177135i
$838$	0	0
$839$	− 31642.8i	− 0.0449522i	−0.999747	−	0.0224761i	$-0.992845\pi$
$839$	− 31642.8i	− 0.0449522i	0.999747	−	0.0224761i	$-0.00715497\pi$
$840$	0	0
$841$	584705.	0.826694
$842$	0	0
$843$	−132938.	−0.187065
$844$	0	0
$845$	− 90568.8i	− 0.126843i
$846$	0	0
$847$	64294.3i	0.0896202i
$848$	0	0
$849$	−221077.	−0.306710
$850$	0	0
$851$	757251.	1.04564
$852$	0	0
$853$	− 1.10273e6i	− 1.51555i	−0.652513	−	0.757777i	$-0.726285\pi$
$853$	− 1.10273e6i	− 1.51555i	0.652513	−	0.757777i	$-0.273715\pi$
$854$	0	0
$855$	224263.i	0.306779i
$856$	0	0
$857$	505940.	0.688870	0.344435	−	0.938810i	$-0.388071\pi$
$857$	505940.	0.688870	0.344435	+	0.938810i	$0.388071\pi$
$858$	0	0
$859$	−741942.	−1.00550	−0.502752	−	0.864431i	$-0.667679\pi$
$859$	−741942.	−1.00550	−0.502752	+	0.864431i	$0.667679\pi$
$860$	0	0
$861$	13160.2i	0.0177524i
$862$	0	0
$863$	536150.i	0.719888i	0.932974	+	0.359944i	$0.117204\pi$
$863$	536150.i	0.719888i	−0.932974	+	0.359944i	$0.882796\pi$
$864$	0	0
$865$	817462.	1.09253
$866$	0	0
$867$	14700.3	0.0195564
$868$	0	0
$869$	− 138343.i	− 0.183196i
$870$	0	0
$871$	339396.i	0.447373i
$872$	0	0
$873$	454811.	0.596763
$874$	0	0
$875$	−108221.	−0.141350
$876$	0	0
$877$	− 59941.4i	− 0.0779341i	−0.999240	−	0.0389671i	$-0.987593\pi$
$877$	− 59941.4i	− 0.0779341i	0.999240	−	0.0389671i	$-0.0124067\pi$
$878$	0	0
$879$	529916.i	0.685850i
$880$	0	0
$881$	−828350.	−1.06724	−0.533620	−	0.845724i	$-0.679169\pi$
$881$	−828350.	−1.06724	−0.533620	+	0.845724i	$0.679169\pi$
$882$	0	0
$883$	245690.	0.315113	0.157556	−	0.987510i	$-0.449638\pi$
$883$	245690.	0.315113	0.157556	+	0.987510i	$0.449638\pi$
$884$	0	0
$885$	174233.i	0.222456i
$886$	0	0
$887$	− 758015.i	− 0.963453i	−0.876322	−	0.481726i	$-0.840010\pi$
$887$	− 758015.i	− 0.963453i	0.876322	−	0.481726i	$-0.159990\pi$
$888$	0	0
$889$	153087.	0.193702
$890$	0	0
$891$	49417.2	0.0622476
$892$	0	0
$893$	− 1.06878e6i	− 1.34025i
$894$	0	0
$895$	318594.i	0.397733i
$896$	0	0
$897$	371087.	0.461201
$898$	0	0
$899$	−30967.9	−0.0383171
$900$	0	0
$901$	− 1.08715e6i	− 1.33919i
$902$	0	0
$903$	11646.0i	0.0142824i
$904$	0	0
$905$	452807.	0.552861
$906$	0	0
$907$	520331.	0.632506	0.316253	−	0.948675i	$-0.397575\pi$
$907$	520331.	0.632506	0.316253	+	0.948675i	$0.397575\pi$
$908$	0	0
$909$	− 240545.i	− 0.291117i
$910$	0	0
$911$	1.10094e6i	1.32656i	0.748370	+	0.663282i	$0.230837\pi$
$911$	1.10094e6i	1.32656i	−0.748370	+	0.663282i	$0.769163\pi$
$912$	0	0
$913$	355571.	0.426564
$914$	0	0
$915$	231923.	0.277014
$916$	0	0
$917$	160383.i	0.190731i
$918$	0	0
$919$	− 990050.i	− 1.17227i	−0.810215	−	0.586133i	$-0.800650\pi$
$919$	− 990050.i	− 1.17227i	0.810215	−	0.586133i	$-0.199350\pi$
$920$	0	0
$921$	−883989.	−1.04214
$922$	0	0
$923$	688942.	0.808685
$924$	0	0
$925$	− 424936.i	− 0.496638i
$926$	0	0
$927$	− 528844.i	− 0.615415i
$928$	0	0
$929$	112392.	0.130228	0.0651140	−	0.997878i	$-0.479259\pi$
$929$	112392.	0.130228	0.0651140	+	0.997878i	$0.479259\pi$
$930$	0	0
$931$	−1.02555e6	−1.18320
$932$	0	0
$933$	− 216122.i	− 0.248277i
$934$	0	0
$935$	− 368063.i	− 0.421017i
$936$	0	0
$937$	−293444.	−0.334231	−0.167115	−	0.985937i	$-0.553445\pi$
$937$	−293444.	−0.334231	−0.167115	+	0.985937i	$0.553445\pi$
$938$	0	0
$939$	−632304.	−0.717125
$940$	0	0
$941$	− 531934.i	− 0.600729i	−0.953825	−	0.300364i	$-0.902892\pi$
$941$	− 531934.i	− 0.600729i	0.953825	−	0.300364i	$-0.0971083\pi$
$942$	0	0
$943$	183102.i	0.205906i
$944$	0	0
$945$	17162.8	0.0192188
$946$	0	0
$947$	378760.	0.422342	0.211171	−	0.977449i	$-0.432272\pi$
$947$	378760.	0.422342	0.211171	+	0.977449i	$0.432272\pi$
$948$	0	0
$949$	400085.i	0.444243i
$950$	0	0
$951$	− 657301.i	− 0.726781i
$952$	0	0
$953$	1.75746e6	1.93508	0.967539	−	0.252721i	$-0.0813256\pi$
$953$	1.75746e6	1.93508	0.967539	+	0.252721i	$0.0813256\pi$
$954$	0	0
$955$	1.28255e6	1.40627
$956$	0	0
$957$	123320.i	0.134651i
$958$	0	0
$959$	− 3382.78i	− 0.00367821i
$960$	0	0
$961$	915697.	0.991528
$962$	0	0
$963$	−372312.	−0.401471
$964$	0	0
$965$	214842.i	0.230709i
$966$	0	0
$967$	− 59833.4i	− 0.0639868i	−0.999488	−	0.0319934i	$-0.989814\pi$
$967$	− 59833.4i	− 0.0639868i	0.999488	−	0.0319934i	$-0.0101856\pi$
$968$	0	0
$969$	641408.	0.683104
$970$	0	0
$971$	−687930.	−0.729635	−0.364817	−	0.931079i	$-0.618869\pi$
$971$	−687930.	−0.729635	−0.364817	+	0.931079i	$0.618869\pi$
$972$	0	0
$973$	− 153932.i	− 0.162594i
$974$	0	0
$975$	− 208238.i	− 0.219053i
$976$	0	0
$977$	1.20414e6	1.26150	0.630752	−	0.775985i	$-0.282747\pi$
$977$	1.20414e6	1.26150	0.630752	+	0.775985i	$0.282747\pi$
$978$	0	0
$979$	970746.	1.01284
$980$	0	0
$981$	− 355858.i	− 0.369776i
$982$	0	0
$983$	− 202530.i	− 0.209596i	−0.994494	−	0.104798i	$-0.966580\pi$
$983$	− 202530.i	− 0.209596i	0.994494	−	0.104798i	$-0.0334196\pi$
$984$	0	0
$985$	286420.	0.295210
$986$	0	0
$987$	−81793.9	−0.0839628
$988$	0	0
$989$	162033.i	0.165658i
$990$	0	0
$991$	− 762037.i	− 0.775941i	−0.921672	−	0.387970i	$-0.873176\pi$
$991$	− 762037.i	− 0.775941i	0.921672	−	0.387970i	$-0.126824\pi$
$992$	0	0
$993$	175108.	0.177585
$994$	0	0
$995$	−1.35911e6	−1.37281
$996$	0	0
$997$	− 75700.5i	− 0.0761567i	−0.999275	−	0.0380784i	$-0.987876\pi$
$997$	− 75700.5i	− 0.0761567i	0.999275	−	0.0380784i	$-0.0121237\pi$
$998$	0	0
$999$	229608.i	0.230068i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.b.d.319.3	✓	16
3.2	odd	2		1152.5.b.m.703.11		16
4.3	odd	2	inner	384.5.b.d.319.11	yes	16
8.3	odd	2	inner	384.5.b.d.319.6	yes	16
8.5	even	2	inner	384.5.b.d.319.14	yes	16
12.11	even	2		1152.5.b.m.703.12		16
16.3	odd	4		768.5.g.j.511.5		8
16.5	even	4		768.5.g.h.511.8		8
16.11	odd	4		768.5.g.h.511.4		8
16.13	even	4		768.5.g.j.511.1		8
24.5	odd	2		1152.5.b.m.703.5		16
24.11	even	2		1152.5.b.m.703.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.d.319.3	✓	16	1.1	even	1	trivial
384.5.b.d.319.6	yes	16	8.3	odd	2	inner
384.5.b.d.319.11	yes	16	4.3	odd	2	inner
384.5.b.d.319.14	yes	16	8.5	even	2	inner
768.5.g.h.511.4		8	16.11	odd	4
768.5.g.h.511.8		8	16.5	even	4
768.5.g.j.511.1		8	16.13	even	4
768.5.g.j.511.5		8	16.3	odd	4
1152.5.b.m.703.5		16	24.5	odd	2
1152.5.b.m.703.6		16	24.11	even	2
1152.5.b.m.703.11		16	3.2	odd	2
1152.5.b.m.703.12		16	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} -19.1142i q^{5} -6.40010i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} -19.1142i q^{5} -6.40010i q^{7} +27.0000 q^{9} +67.7877 q^{11} +154.346i q^{13} +99.3205i q^{15} +284.063 q^{17} -434.548 q^{19} +33.2559i q^{21} +462.698i q^{23} +259.646 q^{25} -140.296 q^{27} -350.108i q^{29} -88.4523i q^{31} -352.235 q^{33} -122.333 q^{35} -1636.60i q^{37} -802.006i q^{39} +395.726 q^{41} +350.192 q^{43} -516.084i q^{45} +2459.53i q^{47} +2360.04 q^{49} -1476.04 q^{51} -3827.15i q^{53} -1295.71i q^{55} +2257.98 q^{57} +1754.25 q^{59} -2335.10i q^{61} -172.803i q^{63} +2950.21 q^{65} +2198.93 q^{67} -2404.25i q^{69} -4463.62i q^{71} +2592.13 q^{73} -1349.16 q^{75} -433.848i q^{77} -2040.82i q^{79} +729.000 q^{81} +5245.36 q^{83} -5429.65i q^{85} +1819.22i q^{87} +14320.4 q^{89} +987.830 q^{91} +459.612i q^{93} +8306.04i q^{95} +16844.8 q^{97} +1830.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 432 q^{9}+O(q^{10}) \) 16 * q + 432 * q^9	\( 16 q + 432 q^{9} + 480 q^{17} + 144 q^{25} + 2880 q^{33} - 3552 q^{41} - 20080 q^{49} - 7488 q^{57} + 23040 q^{65} + 34400 q^{73} + 11664 q^{81} - 15648 q^{89} + 5088 q^{97}+O(q^{100}) \) 16 * q + 432 * q^9 + 480 * q^17 + 144 * q^25 + 2880 * q^33 - 3552 * q^41 - 20080 * q^49 - 7488 * q^57 + 23040 * q^65 + 34400 * q^73 + 11664 * q^81 - 15648 * q^89 + 5088 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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