LMFDB - Embedded newform 768.5.b.h.127.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$79.3881316484$
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} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 $ x^16 - 8x^15 + 104x^14 - 588x^13 + 3846x^12 - 15796x^11 + 66992x^10 - 203224x^9 + 604877x^8 - 1351612x^7 + 2878620x^6 - 4608408x^5 + 6889612x^4 - 7344036x^3 + 7202376x^2 - 4122756*x + 2196513
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{70}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.10
Root			$0.500000 - 2.27039i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.5.b.h.127.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -38.1647i q^{5} +50.0160i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -38.1647i q^{5} +50.0160i q^{7} +27.0000 q^{9} +14.9385 q^{11} -281.909i q^{13} -198.310i q^{15} -570.708 q^{17} -71.7400 q^{19} +259.891i q^{21} +746.211i q^{23} -831.546 q^{25} +140.296 q^{27} -558.040i q^{29} +1709.27i q^{31} +77.6229 q^{33} +1908.85 q^{35} +210.427i q^{37} -1464.84i q^{39} +35.0966 q^{41} -2340.97 q^{43} -1030.45i q^{45} +193.166i q^{47} -100.598 q^{49} -2965.49 q^{51} -1962.98i q^{53} -570.125i q^{55} -372.772 q^{57} -4860.39 q^{59} -478.284i q^{61} +1350.43i q^{63} -10759.0 q^{65} +4762.37 q^{67} +3877.42i q^{69} -4203.83i q^{71} +5327.22 q^{73} -4320.84 q^{75} +747.166i q^{77} +6615.84i q^{79} +729.000 q^{81} -6872.82 q^{83} +21780.9i q^{85} -2899.66i q^{87} +5237.05 q^{89} +14100.0 q^{91} +8881.65i q^{93} +2737.93i q^{95} -5808.32 q^{97} +403.341 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 432 q^{9}+O(q^{10}) $ 16 * q + 432 * q^9	$ 16 q + 432 q^{9} - 576 q^{11} - 480 q^{17} - 192 q^{19} - 2672 q^{25} - 9024 q^{35} - 1440 q^{41} - 12224 q^{43} - 2480 q^{49} - 6336 q^{51} + 7488 q^{57} - 15360 q^{59} - 1344 q^{65} - 12288 q^{67} + 8480 q^{73} - 21888 q^{75} + 11664 q^{81} - 13248 q^{83} - 18720 q^{89} + 30272 q^{91} + 13088 q^{97} - 15552 q^{99}+O(q^{100}) $ 16 * q + 432 * q^9 - 576 * q^11 - 480 * q^17 - 192 * q^19 - 2672 * q^25 - 9024 * q^35 - 1440 * q^41 - 12224 * q^43 - 2480 * q^49 - 6336 * q^51 + 7488 * q^57 - 15360 * q^59 - 1344 * q^65 - 12288 * q^67 + 8480 * q^73 - 21888 * q^75 + 11664 * q^81 - 13248 * q^83 - 18720 * q^89 + 30272 * q^91 + 13088 * q^97 - 15552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 38.1647i	− 1.52659i	−0.646051	−	0.763294i	$-0.723581\pi$
$5$	− 38.1647i	− 1.52659i	0.646051	−	0.763294i	$-0.276419\pi$
$6$	0	0
$7$	50.0160i	1.02073i	0.859957	+	0.510367i	$0.170490\pi$
$7$	50.0160i	1.02073i	−0.859957	+	0.510367i	$0.829510\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	14.9385	0.123459	0.0617295	−	0.998093i	$-0.480338\pi$
$11$	14.9385	0.123459	0.0617295	+	0.998093i	$0.480338\pi$
$12$	0	0
$13$	− 281.909i	− 1.66810i	−0.551687	−	0.834051i	$-0.686016\pi$
$13$	− 281.909i	− 1.66810i	0.551687	−	0.834051i	$-0.313984\pi$
$14$	0	0
$15$	− 198.310i	− 0.881376i
$16$	0	0
$17$	−570.708	−1.97477	−0.987385	−	0.158340i	$-0.949386\pi$
$17$	−570.708	−1.97477	−0.987385	+	0.158340i	$0.949386\pi$
$18$	0	0
$19$	−71.7400	−0.198726	−0.0993628	−	0.995051i	$-0.531680\pi$
$19$	−71.7400	−0.198726	−0.0993628	+	0.995051i	$0.531680\pi$
$20$	0	0
$21$	259.891i	0.589321i
$22$	0	0
$23$	746.211i	1.41061i	0.708906	+	0.705303i	$0.249189\pi$
$23$	746.211i	1.41061i	−0.708906	+	0.705303i	$0.750811\pi$
$24$	0	0
$25$	−831.546	−1.33047
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	− 558.040i	− 0.663543i	−0.943360	−	0.331771i	$-0.892354\pi$
$29$	− 558.040i	− 0.663543i	0.943360	−	0.331771i	$-0.107646\pi$
$30$	0	0
$31$	1709.27i	1.77864i	0.457285	+	0.889320i	$0.348822\pi$
$31$	1709.27i	1.77864i	−0.457285	+	0.889320i	$0.651178\pi$
$32$	0	0
$33$	77.6229	0.0712791
$34$	0	0
$35$	1908.85	1.55824
$36$	0	0
$37$	210.427i	0.153708i	0.997042	+	0.0768541i	$0.0244876\pi$
$37$	210.427i	0.153708i	−0.997042	+	0.0768541i	$0.975512\pi$
$38$	0	0
$39$	− 1464.84i	− 0.963079i
$40$	0	0
$41$	35.0966	0.0208784	0.0104392	−	0.999946i	$-0.496677\pi$
$41$	35.0966	0.0208784	0.0104392	+	0.999946i	$0.496677\pi$
$42$	0	0
$43$	−2340.97	−1.26607	−0.633036	−	0.774122i	$-0.718191\pi$
$43$	−2340.97	−1.26607	−0.633036	+	0.774122i	$0.718191\pi$
$44$	0	0
$45$	− 1030.45i	− 0.508863i
$46$	0	0
$47$	193.166i	0.0874449i	0.999044	+	0.0437224i	$0.0139217\pi$
$47$	193.166i	0.0874449i	−0.999044	+	0.0437224i	$0.986078\pi$
$48$	0	0
$49$	−100.598	−0.0418985
$50$	0	0
$51$	−2965.49	−1.14013
$52$	0	0
$53$	− 1962.98i	− 0.698818i	−0.936970	−	0.349409i	$-0.886382\pi$
$53$	− 1962.98i	− 0.698818i	0.936970	−	0.349409i	$-0.113618\pi$
$54$	0	0
$55$	− 570.125i	− 0.188471i
$56$	0	0
$57$	−372.772	−0.114734
$58$	0	0
$59$	−4860.39	−1.39626	−0.698131	−	0.715970i	$-0.745985\pi$
$59$	−4860.39	−1.39626	−0.698131	+	0.715970i	$0.745985\pi$
$60$	0	0
$61$	− 478.284i	− 0.128536i	−0.997933	−	0.0642682i	$-0.979529\pi$
$61$	− 478.284i	− 0.128536i	0.997933	−	0.0642682i	$-0.0204713\pi$
$62$	0	0
$63$	1350.43i	0.340245i
$64$	0	0
$65$	−10759.0	−2.54651
$66$	0	0
$67$	4762.37	1.06090	0.530449	−	0.847717i	$-0.322023\pi$
$67$	4762.37	1.06090	0.530449	+	0.847717i	$0.322023\pi$
$68$	0	0
$69$	3877.42i	0.814414i
$70$	0	0
$71$	− 4203.83i	− 0.833928i	−0.908923	−	0.416964i	$-0.863094\pi$
$71$	− 4203.83i	− 0.833928i	0.908923	−	0.416964i	$-0.136906\pi$
$72$	0	0
$73$	5327.22	0.999666	0.499833	−	0.866122i	$-0.333395\pi$
$73$	5327.22	0.999666	0.499833	+	0.866122i	$0.333395\pi$
$74$	0	0
$75$	−4320.84	−0.768149
$76$	0	0
$77$	747.166i	0.126019i
$78$	0	0
$79$	6615.84i	1.06006i	0.847979	+	0.530030i	$0.177819\pi$
$79$	6615.84i	1.06006i	−0.847979	+	0.530030i	$0.822181\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	−6872.82	−0.997651	−0.498826	−	0.866702i	$-0.666235\pi$
$83$	−6872.82	−0.997651	−0.498826	+	0.866702i	$0.666235\pi$
$84$	0	0
$85$	21780.9i	3.01466i
$86$	0	0
$87$	− 2899.66i	− 0.383097i
$88$	0	0
$89$	5237.05	0.661160	0.330580	−	0.943778i	$-0.392756\pi$
$89$	5237.05	0.661160	0.330580	+	0.943778i	$0.392756\pi$
$90$	0	0
$91$	14100.0	1.70269
$92$	0	0
$93$	8881.65i	1.02690i
$94$	0	0
$95$	2737.93i	0.303372i
$96$	0	0
$97$	−5808.32	−0.617315	−0.308658	−	0.951173i	$-0.599880\pi$
$97$	−5808.32	−0.617315	−0.308658	+	0.951173i	$0.599880\pi$
$98$	0	0
$99$	403.341	0.0411530
$100$	0	0
$101$	15431.3i	1.51272i	0.654154	+	0.756362i	$0.273025\pi$
$101$	15431.3i	1.51272i	−0.654154	+	0.756362i	$0.726975\pi$
$102$	0	0
$103$	11783.2i	1.11068i	0.831624	+	0.555339i	$0.187411\pi$
$103$	11783.2i	1.11068i	−0.831624	+	0.555339i	$0.812589\pi$
$104$	0	0
$105$	9918.65	0.899651
$106$	0	0
$107$	−14214.7	−1.24157	−0.620785	−	0.783981i	$-0.713186\pi$
$107$	−14214.7	−1.24157	−0.620785	+	0.783981i	$0.713186\pi$
$108$	0	0
$109$	− 283.406i	− 0.0238537i	−0.999929	−	0.0119269i	$-0.996203\pi$
$109$	− 283.406i	− 0.0238537i	0.999929	−	0.0119269i	$-0.00379652\pi$
$110$	0	0
$111$	1093.41i	0.0887435i
$112$	0	0
$113$	4755.34	0.372413	0.186206	−	0.982511i	$-0.440381\pi$
$113$	4755.34	0.372413	0.186206	+	0.982511i	$0.440381\pi$
$114$	0	0
$115$	28478.9	2.15341
$116$	0	0
$117$	− 7611.55i	− 0.556034i
$118$	0	0
$119$	− 28544.5i	− 2.01571i
$120$	0	0
$121$	−14417.8	−0.984758
$122$	0	0
$123$	182.367	0.0120541
$124$	0	0
$125$	7882.75i	0.504496i
$126$	0	0
$127$	8132.78i	0.504233i	0.967697	+	0.252117i	$0.0811267\pi$
$127$	8132.78i	0.504233i	−0.967697	+	0.252117i	$0.918873\pi$
$128$	0	0
$129$	−12164.0	−0.730967
$130$	0	0
$131$	9963.50	0.580590	0.290295	−	0.956937i	$-0.406247\pi$
$131$	9963.50	0.580590	0.290295	+	0.956937i	$0.406247\pi$
$132$	0	0
$133$	− 3588.14i	− 0.202846i
$134$	0	0
$135$	− 5354.36i	− 0.293792i
$136$	0	0
$137$	−34463.3	−1.83618	−0.918091	−	0.396369i	$-0.870270\pi$
$137$	−34463.3	−1.83618	−0.918091	+	0.396369i	$0.870270\pi$
$138$	0	0
$139$	−37114.7	−1.92095	−0.960477	−	0.278361i	$-0.910209\pi$
$139$	−37114.7	−1.92095	−0.960477	+	0.278361i	$0.910209\pi$
$140$	0	0
$141$	1003.72i	0.0504863i
$142$	0	0
$143$	− 4211.31i	− 0.205942i
$144$	0	0
$145$	−21297.4	−1.01296
$146$	0	0
$147$	−522.725	−0.0241901
$148$	0	0
$149$	15314.2i	0.689797i	0.938640	+	0.344899i	$0.112087\pi$
$149$	15314.2i	0.689797i	−0.938640	+	0.344899i	$0.887913\pi$
$150$	0	0
$151$	42542.8i	1.86583i	0.360093	+	0.932916i	$0.382745\pi$
$151$	42542.8i	1.86583i	−0.360093	+	0.932916i	$0.617255\pi$
$152$	0	0
$153$	−15409.1	−0.658256
$154$	0	0
$155$	65233.9	2.71525
$156$	0	0
$157$	7662.12i	0.310849i	0.987848	+	0.155425i	$0.0496745\pi$
$157$	7662.12i	0.310849i	−0.987848	+	0.155425i	$0.950325\pi$
$158$	0	0
$159$	− 10199.9i	− 0.403463i
$160$	0	0
$161$	−37322.5	−1.43985
$162$	0	0
$163$	8483.09	0.319285	0.159643	−	0.987175i	$-0.448966\pi$
$163$	8483.09	0.319285	0.159643	+	0.987175i	$0.448966\pi$
$164$	0	0
$165$	− 2962.46i	− 0.108814i
$166$	0	0
$167$	− 24615.9i	− 0.882638i	−0.897350	−	0.441319i	$-0.854511\pi$
$167$	− 24615.9i	− 0.882638i	0.897350	−	0.441319i	$-0.145489\pi$
$168$	0	0
$169$	−50911.9	−1.78257
$170$	0	0
$171$	−1936.98	−0.0662419
$172$	0	0
$173$	− 1104.54i	− 0.0369055i	−0.999830	−	0.0184527i	$-0.994126\pi$
$173$	− 1104.54i	− 0.0369055i	0.999830	−	0.0184527i	$-0.00587402\pi$
$174$	0	0
$175$	− 41590.6i	− 1.35806i
$176$	0	0
$177$	−25255.3	−0.806132
$178$	0	0
$179$	−35189.9	−1.09828	−0.549139	−	0.835731i	$-0.685044\pi$
$179$	−35189.9	−1.09828	−0.549139	+	0.835731i	$0.685044\pi$
$180$	0	0
$181$	30095.0i	0.918623i	0.888275	+	0.459311i	$0.151904\pi$
$181$	30095.0i	0.918623i	−0.888275	+	0.459311i	$0.848096\pi$
$182$	0	0
$183$	− 2485.24i	− 0.0742105i
$184$	0	0
$185$	8030.87	0.234649
$186$	0	0
$187$	−8525.55	−0.243803
$188$	0	0
$189$	7017.05i	0.196440i
$190$	0	0
$191$	− 15842.3i	− 0.434261i	−0.976143	−	0.217131i	$-0.930330\pi$
$191$	− 15842.3i	− 0.434261i	0.976143	−	0.217131i	$-0.0696697\pi$
$192$	0	0
$193$	49712.8	1.33461	0.667303	−	0.744786i	$-0.267448\pi$
$193$	49712.8	1.33461	0.667303	+	0.744786i	$0.267448\pi$
$194$	0	0
$195$	−55905.4	−1.47023
$196$	0	0
$197$	− 17445.4i	− 0.449520i	−0.974414	−	0.224760i	$-0.927840\pi$
$197$	− 17445.4i	− 0.449520i	0.974414	−	0.224760i	$-0.0721598\pi$
$198$	0	0
$199$	− 33628.6i	− 0.849187i	−0.905384	−	0.424593i	$-0.860417\pi$
$199$	− 33628.6i	− 0.849187i	0.905384	−	0.424593i	$-0.139583\pi$
$200$	0	0
$201$	24746.0	0.612510
$202$	0	0
$203$	27910.9	0.677301
$204$	0	0
$205$	− 1339.45i	− 0.0318727i
$206$	0	0
$207$	20147.7i	0.470202i
$208$	0	0
$209$	−1071.69	−0.0245345
$210$	0	0
$211$	−51764.4	−1.16270	−0.581348	−	0.813655i	$-0.697475\pi$
$211$	−51764.4	−1.16270	−0.581348	+	0.813655i	$0.697475\pi$
$212$	0	0
$213$	− 21843.7i	− 0.481469i
$214$	0	0
$215$	89342.3i	1.93277i
$216$	0	0
$217$	−85491.0	−1.81552
$218$	0	0
$219$	27681.1	0.577158
$220$	0	0
$221$	160888.i	3.29412i
$222$	0	0
$223$	− 58417.9i	− 1.17472i	−0.809324	−	0.587362i	$-0.800166\pi$
$223$	− 58417.9i	− 1.17472i	0.809324	−	0.587362i	$-0.199834\pi$
$224$	0	0
$225$	−22451.7	−0.443491
$226$	0	0
$227$	60234.3	1.16894	0.584470	−	0.811415i	$-0.301302\pi$
$227$	60234.3	1.16894	0.584470	+	0.811415i	$0.301302\pi$
$228$	0	0
$229$	31082.1i	0.592706i	0.955078	+	0.296353i	$0.0957706\pi$
$229$	31082.1i	0.592706i	−0.955078	+	0.296353i	$0.904229\pi$
$230$	0	0
$231$	3882.39i	0.0727570i
$232$	0	0
$233$	25930.7	0.477643	0.238821	−	0.971064i	$-0.423239\pi$
$233$	25930.7	0.477643	0.238821	+	0.971064i	$0.423239\pi$
$234$	0	0
$235$	7372.12	0.133492
$236$	0	0
$237$	34376.9i	0.612026i
$238$	0	0
$239$	4425.21i	0.0774708i	0.999250	+	0.0387354i	$0.0123329\pi$
$239$	4425.21i	0.0774708i	−0.999250	+	0.0387354i	$0.987667\pi$
$240$	0	0
$241$	−34491.1	−0.593844	−0.296922	−	0.954902i	$-0.595960\pi$
$241$	−34491.1	−0.593844	−0.296922	+	0.954902i	$0.595960\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	3839.31i	0.0639618i
$246$	0	0
$247$	20224.2i	0.331495i
$248$	0	0
$249$	−35712.2	−0.575994
$250$	0	0
$251$	−21673.7	−0.344022	−0.172011	−	0.985095i	$-0.555026\pi$
$251$	−21673.7	−0.344022	−0.172011	+	0.985095i	$0.555026\pi$
$252$	0	0
$253$	11147.3i	0.174152i
$254$	0	0
$255$	113177.i	1.74051i
$256$	0	0
$257$	10512.6	0.159164	0.0795819	−	0.996828i	$-0.474641\pi$
$257$	10512.6	0.159164	0.0795819	+	0.996828i	$0.474641\pi$
$258$	0	0
$259$	−10524.7	−0.156895
$260$	0	0
$261$	− 15067.1i	− 0.221181i
$262$	0	0
$263$	− 39630.6i	− 0.572954i	−0.958087	−	0.286477i	$-0.907516\pi$
$263$	− 39630.6i	− 0.572954i	0.958087	−	0.286477i	$-0.0924841\pi$
$264$	0	0
$265$	−74916.6	−1.06681
$266$	0	0
$267$	27212.5	0.381721
$268$	0	0
$269$	80644.9i	1.11448i	0.830351	+	0.557240i	$0.188140\pi$
$269$	80644.9i	1.11448i	−0.830351	+	0.557240i	$0.811860\pi$
$270$	0	0
$271$	− 49892.7i	− 0.679357i	−0.940542	−	0.339678i	$-0.889682\pi$
$271$	− 49892.7i	− 0.679357i	0.940542	−	0.339678i	$-0.110318\pi$
$272$	0	0
$273$	73265.6	0.983048
$274$	0	0
$275$	−12422.1	−0.164259
$276$	0	0
$277$	13320.1i	0.173599i	0.996226	+	0.0867996i	$0.0276640\pi$
$277$	13320.1i	0.173599i	−0.996226	+	0.0867996i	$0.972336\pi$
$278$	0	0
$279$	46150.4i	0.592880i
$280$	0	0
$281$	18143.5	0.229778	0.114889	−	0.993378i	$-0.463349\pi$
$281$	18143.5	0.229778	0.114889	+	0.993378i	$0.463349\pi$
$282$	0	0
$283$	−140931.	−1.75968	−0.879840	−	0.475270i	$-0.842350\pi$
$283$	−140931.	−1.75968	−0.879840	+	0.475270i	$0.842350\pi$
$284$	0	0
$285$	14226.7i	0.175152i
$286$	0	0
$287$	1755.39i	0.0213113i
$288$	0	0
$289$	242187.	2.89971
$290$	0	0
$291$	−30180.9	−0.356407
$292$	0	0
$293$	− 108058.i	− 1.25870i	−0.777122	−	0.629350i	$-0.783321\pi$
$293$	− 108058.i	− 1.25870i	0.777122	−	0.629350i	$-0.216679\pi$
$294$	0	0
$295$	185495.i	2.13152i
$296$	0	0
$297$	2095.82	0.0237597
$298$	0	0
$299$	210364.	2.35304
$300$	0	0
$301$	− 117086.i	− 1.29232i
$302$	0	0
$303$	80183.3i	0.873371i
$304$	0	0
$305$	−18253.6	−0.196222
$306$	0	0
$307$	−27092.9	−0.287461	−0.143730	−	0.989617i	$-0.545910\pi$
$307$	−27092.9	−0.287461	−0.143730	+	0.989617i	$0.545910\pi$
$308$	0	0
$309$	61227.2i	0.641250i
$310$	0	0
$311$	− 114935.i	− 1.18832i	−0.804348	−	0.594158i	$-0.797485\pi$
$311$	− 114935.i	− 1.18832i	0.804348	−	0.594158i	$-0.202515\pi$
$312$	0	0
$313$	173983.	1.77590	0.887948	−	0.459945i	$-0.152131\pi$
$313$	173983.	1.77590	0.887948	+	0.459945i	$0.152131\pi$
$314$	0	0
$315$	51538.8	0.519414
$316$	0	0
$317$	− 154121.i	− 1.53371i	−0.641822	−	0.766854i	$-0.721821\pi$
$317$	− 154121.i	− 1.53371i	0.641822	−	0.766854i	$-0.278179\pi$
$318$	0	0
$319$	− 8336.29i	− 0.0819203i
$320$	0	0
$321$	−73861.9	−0.716821
$322$	0	0
$323$	40942.6	0.392437
$324$	0	0
$325$	234420.i	2.21937i
$326$	0	0
$327$	− 1472.62i	− 0.0137719i
$328$	0	0
$329$	−9661.38	−0.0892580
$330$	0	0
$331$	−146508.	−1.33723	−0.668613	−	0.743611i	$-0.733111\pi$
$331$	−146508.	−1.33723	−0.668613	+	0.743611i	$0.733111\pi$
$332$	0	0
$333$	5681.52i	0.0512361i
$334$	0	0
$335$	− 181755.i	− 1.61955i
$336$	0	0
$337$	−156868.	−1.38126	−0.690628	−	0.723210i	$-0.742666\pi$
$337$	−156868.	−1.38126	−0.690628	+	0.723210i	$0.742666\pi$
$338$	0	0
$339$	24709.4	0.215012
$340$	0	0
$341$	25534.0i	0.219589i
$342$	0	0
$343$	115057.i	0.977967i
$344$	0	0
$345$	147981.	1.24327
$346$	0	0
$347$	−125592.	−1.04304	−0.521521	−	0.853239i	$-0.674635\pi$
$347$	−125592.	−1.04304	−0.521521	+	0.853239i	$0.674635\pi$
$348$	0	0
$349$	44311.9i	0.363806i	0.983316	+	0.181903i	$0.0582257\pi$
$349$	44311.9i	0.363806i	−0.983316	+	0.181903i	$0.941774\pi$
$350$	0	0
$351$	− 39550.8i	− 0.321026i
$352$	0	0
$353$	−155580.	−1.24855	−0.624273	−	0.781206i	$-0.714605\pi$
$353$	−155580.	−1.24855	−0.624273	+	0.781206i	$0.714605\pi$
$354$	0	0
$355$	−160438.	−1.27307
$356$	0	0
$357$	− 148322.i	− 1.16377i
$358$	0	0
$359$	− 208069.i	− 1.61443i	−0.590259	−	0.807214i	$-0.700974\pi$
$359$	− 208069.i	− 1.61443i	0.590259	−	0.807214i	$-0.299026\pi$
$360$	0	0
$361$	−125174.	−0.960508
$362$	0	0
$363$	−74917.3	−0.568550
$364$	0	0
$365$	− 203312.i	− 1.52608i
$366$	0	0
$367$	22943.3i	0.170343i	0.996366	+	0.0851714i	$0.0271438\pi$
$367$	22943.3i	0.170343i	−0.996366	+	0.0851714i	$0.972856\pi$
$368$	0	0
$369$	947.607	0.00695946
$370$	0	0
$371$	98180.3	0.713307
$372$	0	0
$373$	− 214254.i	− 1.53997i	−0.638062	−	0.769985i	$-0.720264\pi$
$373$	− 214254.i	− 1.53997i	0.638062	−	0.769985i	$-0.279736\pi$
$374$	0	0
$375$	40960.0i	0.291271i
$376$	0	0
$377$	−157317.	−1.10686
$378$	0	0
$379$	−135220.	−0.941376	−0.470688	−	0.882300i	$-0.655994\pi$
$379$	−135220.	−0.941376	−0.470688	+	0.882300i	$0.655994\pi$
$380$	0	0
$381$	42259.2i	0.291119i
$382$	0	0
$383$	99667.3i	0.679446i	0.940525	+	0.339723i	$0.110333\pi$
$383$	99667.3i	0.679446i	−0.940525	+	0.339723i	$0.889667\pi$
$384$	0	0
$385$	28515.4	0.192379
$386$	0	0
$387$	−63206.1	−0.422024
$388$	0	0
$389$	− 83760.5i	− 0.553529i	−0.960938	−	0.276764i	$-0.910738\pi$
$389$	− 83760.5i	− 0.553529i	0.960938	−	0.276764i	$-0.0892621\pi$
$390$	0	0
$391$	− 425869.i	− 2.78562i
$392$	0	0
$393$	51771.9	0.335204
$394$	0	0
$395$	252491.	1.61828
$396$	0	0
$397$	− 222509.i	− 1.41178i	−0.708322	−	0.705890i	$-0.750547\pi$
$397$	− 222509.i	− 1.41178i	0.708322	−	0.705890i	$-0.249453\pi$
$398$	0	0
$399$	− 18644.5i	− 0.117113i
$400$	0	0
$401$	119674.	0.744237	0.372119	−	0.928185i	$-0.378631\pi$
$401$	119674.	0.744237	0.372119	+	0.928185i	$0.378631\pi$
$402$	0	0
$403$	481860.	2.96695
$404$	0	0
$405$	− 27822.1i	− 0.169621i
$406$	0	0
$407$	3143.46i	0.0189767i
$408$	0	0
$409$	34233.7	0.204648	0.102324	−	0.994751i	$-0.467372\pi$
$409$	34233.7	0.204648	0.102324	+	0.994751i	$0.467372\pi$
$410$	0	0
$411$	−179077.	−1.06012
$412$	0	0
$413$	− 243097.i	− 1.42521i
$414$	0	0
$415$	262299.i	1.52300i
$416$	0	0
$417$	−192854.	−1.10906
$418$	0	0
$419$	90685.6	0.516547	0.258274	−	0.966072i	$-0.416846\pi$
$419$	90685.6	0.516547	0.258274	+	0.966072i	$0.416846\pi$
$420$	0	0
$421$	− 278121.i	− 1.56917i	−0.620022	−	0.784584i	$-0.712876\pi$
$421$	− 278121.i	− 1.56917i	0.620022	−	0.784584i	$-0.287124\pi$
$422$	0	0
$423$	5215.48i	0.0291483i
$424$	0	0
$425$	474570.	2.62738
$426$	0	0
$427$	23921.8	0.131201
$428$	0	0
$429$	− 21882.6i	− 0.118901i
$430$	0	0
$431$	183909.i	0.990032i	0.868884	+	0.495016i	$0.164838\pi$
$431$	183909.i	0.990032i	−0.868884	+	0.495016i	$0.835162\pi$
$432$	0	0
$433$	119776.	0.638843	0.319421	−	0.947613i	$-0.396511\pi$
$433$	119776.	0.638843	0.319421	+	0.947613i	$0.396511\pi$
$434$	0	0
$435$	−110665.	−0.584831
$436$	0	0
$437$	− 53533.1i	− 0.280324i
$438$	0	0
$439$	288630.i	1.49766i	0.662763	+	0.748829i	$0.269384\pi$
$439$	288630.i	1.49766i	−0.662763	+	0.748829i	$0.730616\pi$
$440$	0	0
$441$	−2716.16	−0.0139662
$442$	0	0
$443$	230448.	1.17427	0.587133	−	0.809491i	$-0.300257\pi$
$443$	230448.	1.17427	0.587133	+	0.809491i	$0.300257\pi$
$444$	0	0
$445$	− 199870.i	− 1.00932i
$446$	0	0
$447$	79574.8i	0.398255i
$448$	0	0
$449$	73745.4	0.365799	0.182899	−	0.983132i	$-0.441452\pi$
$449$	73745.4	0.365799	0.182899	+	0.983132i	$0.441452\pi$
$450$	0	0
$451$	524.291	0.00257762
$452$	0	0
$453$	221059.i	1.07724i
$454$	0	0
$455$	− 538121.i	− 2.59931i
$456$	0	0
$457$	303453.	1.45298	0.726489	−	0.687179i	$-0.241151\pi$
$457$	303453.	1.45298	0.726489	+	0.687179i	$0.241151\pi$
$458$	0	0
$459$	−80068.2	−0.380045
$460$	0	0
$461$	147882.i	0.695849i	0.937523	+	0.347924i	$0.113113\pi$
$461$	147882.i	0.695849i	−0.937523	+	0.347924i	$0.886887\pi$
$462$	0	0
$463$	− 106150.i	− 0.495174i	−0.968866	−	0.247587i	$-0.920362\pi$
$463$	− 106150.i	− 0.495174i	0.968866	−	0.247587i	$-0.0796376\pi$
$464$	0	0
$465$	338965.	1.56765
$466$	0	0
$467$	158470.	0.726630	0.363315	−	0.931666i	$-0.381645\pi$
$467$	158470.	0.726630	0.363315	+	0.931666i	$0.381645\pi$
$468$	0	0
$469$	238195.i	1.08289i
$470$	0	0
$471$	39813.5i	0.179469i
$472$	0	0
$473$	−34970.6	−0.156308
$474$	0	0
$475$	59655.0	0.264399
$476$	0	0
$477$	− 53000.4i	− 0.232939i
$478$	0	0
$479$	− 87871.4i	− 0.382981i	−0.981495	−	0.191490i	$-0.938668\pi$
$479$	− 87871.4i	− 0.382981i	0.981495	−	0.191490i	$-0.0613320\pi$
$480$	0	0
$481$	59321.2	0.256401
$482$	0	0
$483$	−193933.	−0.831300
$484$	0	0
$485$	221673.i	0.942386i
$486$	0	0
$487$	11919.2i	0.0502561i	0.999684	+	0.0251281i	$0.00799935\pi$
$487$	11919.2i	0.0502561i	−0.999684	+	0.0251281i	$0.992001\pi$
$488$	0	0
$489$	44079.4	0.184339
$490$	0	0
$491$	−116945.	−0.485088	−0.242544	−	0.970140i	$-0.577982\pi$
$491$	−116945.	−0.485088	−0.242544	+	0.970140i	$0.577982\pi$
$492$	0	0
$493$	318478.i	1.31034i
$494$	0	0
$495$	− 15393.4i	− 0.0628237i
$496$	0	0
$497$	210259.	0.851219
$498$	0	0
$499$	−99656.8	−0.400227	−0.200113	−	0.979773i	$-0.564131\pi$
$499$	−99656.8	−0.400227	−0.200113	+	0.979773i	$0.564131\pi$
$500$	0	0
$501$	− 127908.i	− 0.509591i
$502$	0	0
$503$	166780.i	0.659186i	0.944123	+	0.329593i	$0.106912\pi$
$503$	166780.i	0.659186i	−0.944123	+	0.329593i	$0.893088\pi$
$504$	0	0
$505$	588931.	2.30931
$506$	0	0
$507$	−264546.	−1.02917
$508$	0	0
$509$	− 94288.3i	− 0.363934i	−0.983305	−	0.181967i	$-0.941754\pi$
$509$	− 94288.3i	− 0.363934i	0.983305	−	0.181967i	$-0.0582463\pi$
$510$	0	0
$511$	266446.i	1.02039i
$512$	0	0
$513$	−10064.8	−0.0382448
$514$	0	0
$515$	449702.	1.69555
$516$	0	0
$517$	2885.61i	0.0107959i
$518$	0	0
$519$	− 5739.38i	− 0.0213074i
$520$	0	0
$521$	252399.	0.929848	0.464924	−	0.885351i	$-0.346082\pi$
$521$	252399.	0.929848	0.464924	+	0.885351i	$0.346082\pi$
$522$	0	0
$523$	360986.	1.31973	0.659867	−	0.751382i	$-0.270612\pi$
$523$	360986.	1.31973	0.659867	+	0.751382i	$0.270612\pi$
$524$	0	0
$525$	− 216111.i	− 0.784076i
$526$	0	0
$527$	− 975497.i	− 3.51240i
$528$	0	0
$529$	−276989.	−0.989809
$530$	0	0
$531$	−131230.	−0.465421
$532$	0	0
$533$	− 9894.05i	− 0.0348273i
$534$	0	0
$535$	542501.i	1.89537i
$536$	0	0
$537$	−182852.	−0.634091
$538$	0	0
$539$	−1502.79	−0.00517275
$540$	0	0
$541$	− 212815.i	− 0.727123i	−0.931570	−	0.363561i	$-0.881561\pi$
$541$	− 212815.i	− 0.727123i	0.931570	−	0.363561i	$-0.118439\pi$
$542$	0	0
$543$	156378.i	0.530367i
$544$	0	0
$545$	−10816.1	−0.0364148
$546$	0	0
$547$	−84823.3	−0.283492	−0.141746	−	0.989903i	$-0.545272\pi$
$547$	−84823.3	−0.283492	−0.141746	+	0.989903i	$0.545272\pi$
$548$	0	0
$549$	− 12913.7i	− 0.0428455i
$550$	0	0
$551$	40033.7i	0.131863i
$552$	0	0
$553$	−330898.	−1.08204
$554$	0	0
$555$	41729.6	0.135475
$556$	0	0
$557$	8677.73i	0.0279702i	0.999902	+	0.0139851i	$0.00445174\pi$
$557$	8677.73i	0.0279702i	−0.999902	+	0.0139851i	$0.995548\pi$
$558$	0	0
$559$	659940.i	2.11194i
$560$	0	0
$561$	−44300.0	−0.140760
$562$	0	0
$563$	−114230.	−0.360381	−0.180190	−	0.983632i	$-0.557671\pi$
$563$	−114230.	−0.360381	−0.180190	+	0.983632i	$0.557671\pi$
$564$	0	0
$565$	− 181486.i	− 0.568521i
$566$	0	0
$567$	36461.7i	0.113415i
$568$	0	0
$569$	29567.0	0.0913236	0.0456618	−	0.998957i	$-0.485460\pi$
$569$	29567.0	0.0913236	0.0456618	+	0.998957i	$0.485460\pi$
$570$	0	0
$571$	229583.	0.704155	0.352078	−	0.935971i	$-0.385475\pi$
$571$	229583.	0.704155	0.352078	+	0.935971i	$0.385475\pi$
$572$	0	0
$573$	− 82318.9i	− 0.250721i
$574$	0	0
$575$	− 620508.i	− 1.87677i
$576$	0	0
$577$	522988.	1.57087	0.785434	−	0.618945i	$-0.212440\pi$
$577$	522988.	1.57087	0.785434	+	0.618945i	$0.212440\pi$
$578$	0	0
$579$	258315.	0.770536
$580$	0	0
$581$	− 343751.i	− 1.01834i
$582$	0	0
$583$	− 29324.0i	− 0.0862754i
$584$	0	0
$585$	−290493.	−0.848835
$586$	0	0
$587$	−548252.	−1.59112	−0.795561	−	0.605873i	$-0.792824\pi$
$587$	−548252.	−1.59112	−0.795561	+	0.605873i	$0.792824\pi$
$588$	0	0
$589$	− 122623.i	− 0.353461i
$590$	0	0
$591$	− 90649.0i	− 0.259530i
$592$	0	0
$593$	509890.	1.45000	0.724998	−	0.688751i	$-0.241841\pi$
$593$	509890.	1.45000	0.724998	+	0.688751i	$0.241841\pi$
$594$	0	0
$595$	−1.08939e6	−3.07717
$596$	0	0
$597$	− 174740.i	− 0.490278i
$598$	0	0
$599$	− 163569.i	− 0.455877i	−0.973676	−	0.227939i	$-0.926801\pi$
$599$	− 163569.i	− 0.455877i	0.973676	−	0.227939i	$-0.0731985\pi$
$600$	0	0
$601$	−359449.	−0.995151	−0.497575	−	0.867421i	$-0.665776\pi$
$601$	−359449.	−0.995151	−0.497575	+	0.867421i	$0.665776\pi$
$602$	0	0
$603$	128584.	0.353633
$604$	0	0
$605$	550253.i	1.50332i
$606$	0	0
$607$	333341.i	0.904714i	0.891837	+	0.452357i	$0.149417\pi$
$607$	333341.i	0.904714i	−0.891837	+	0.452357i	$0.850583\pi$
$608$	0	0
$609$	145029.	0.391040
$610$	0	0
$611$	54455.2	0.145867
$612$	0	0
$613$	− 357269.i	− 0.950768i	−0.879778	−	0.475384i	$-0.842309\pi$
$613$	− 357269.i	− 0.950768i	0.879778	−	0.475384i	$-0.157691\pi$
$614$	0	0
$615$	− 6959.99i	− 0.0184017i
$616$	0	0
$617$	−271334.	−0.712744	−0.356372	−	0.934344i	$-0.615986\pi$
$617$	−271334.	−0.712744	−0.356372	+	0.934344i	$0.615986\pi$
$618$	0	0
$619$	−383666.	−1.00132	−0.500659	−	0.865645i	$-0.666909\pi$
$619$	−383666.	−1.00132	−0.500659	+	0.865645i	$0.666909\pi$
$620$	0	0
$621$	104690.i	0.271471i
$622$	0	0
$623$	261936.i	0.674869i
$624$	0	0
$625$	−218873.	−0.560315
$626$	0	0
$627$	−5568.66	−0.0141650
$628$	0	0
$629$	− 120092.i	− 0.303538i
$630$	0	0
$631$	− 199171.i	− 0.500227i	−0.968216	−	0.250114i	$-0.919532\pi$
$631$	− 199171.i	− 0.500227i	0.968216	−	0.250114i	$-0.0804680\pi$
$632$	0	0
$633$	−268976.	−0.671283
$634$	0	0
$635$	310385.	0.769757
$636$	0	0
$637$	28359.6i	0.0698911i
$638$	0	0
$639$	− 113503.i	− 0.277976i
$640$	0	0
$641$	46510.1	0.113196	0.0565980	−	0.998397i	$-0.481975\pi$
$641$	46510.1	0.113196	0.0565980	+	0.998397i	$0.481975\pi$
$642$	0	0
$643$	103482.	0.250290	0.125145	−	0.992138i	$-0.460060\pi$
$643$	103482.	0.250290	0.125145	+	0.992138i	$0.460060\pi$
$644$	0	0
$645$	464236.i	1.11589i
$646$	0	0
$647$	− 512475.i	− 1.22423i	−0.790767	−	0.612117i	$-0.790318\pi$
$647$	− 512475.i	− 1.22423i	0.790767	−	0.612117i	$-0.209682\pi$
$648$	0	0
$649$	−72607.1	−0.172381
$650$	0	0
$651$	−444224.	−1.04819
$652$	0	0
$653$	227855.i	0.534358i	0.963647	+	0.267179i	$0.0860915\pi$
$653$	227855.i	0.534358i	−0.963647	+	0.267179i	$0.913909\pi$
$654$	0	0
$655$	− 380254.i	− 0.886322i
$656$	0	0
$657$	143835.	0.333222
$658$	0	0
$659$	182148.	0.419423	0.209712	−	0.977763i	$-0.432747\pi$
$659$	182148.	0.419423	0.209712	+	0.977763i	$0.432747\pi$
$660$	0	0
$661$	− 606405.i	− 1.38791i	−0.720020	−	0.693953i	$-0.755868\pi$
$661$	− 606405.i	− 1.38791i	0.720020	−	0.693953i	$-0.244132\pi$
$662$	0	0
$663$	835999.i	1.90186i
$664$	0	0
$665$	−136941.	−0.309663
$666$	0	0
$667$	416415.	0.935997
$668$	0	0
$669$	− 303548.i	− 0.678228i
$670$	0	0
$671$	− 7144.86i	− 0.0158690i
$672$	0	0
$673$	−635313.	−1.40268	−0.701339	−	0.712828i	$-0.747414\pi$
$673$	−635313.	−1.40268	−0.701339	+	0.712828i	$0.747414\pi$
$674$	0	0
$675$	−116663.	−0.256050
$676$	0	0
$677$	134998.i	0.294543i	0.989096	+	0.147272i	$0.0470491\pi$
$677$	134998.i	0.294543i	−0.989096	+	0.147272i	$0.952951\pi$
$678$	0	0
$679$	− 290509.i	− 0.630115i
$680$	0	0
$681$	312987.	0.674888
$682$	0	0
$683$	566455.	1.21429	0.607147	−	0.794589i	$-0.292314\pi$
$683$	566455.	1.21429	0.607147	+	0.794589i	$0.292314\pi$
$684$	0	0
$685$	1.31528e6i	2.80310i
$686$	0	0
$687$	161507.i	0.342199i
$688$	0	0
$689$	−553382.	−1.16570
$690$	0	0
$691$	−543725.	−1.13874	−0.569368	−	0.822083i	$-0.692812\pi$
$691$	−543725.	−1.13874	−0.569368	+	0.822083i	$0.692812\pi$
$692$	0	0
$693$	20173.5i	0.0420063i
$694$	0	0
$695$	1.41647e6i	2.93251i
$696$	0	0
$697$	−20029.9	−0.0412300
$698$	0	0
$699$	134740.	0.275767
$700$	0	0
$701$	− 767758.i	− 1.56239i	−0.624289	−	0.781193i	$-0.714611\pi$
$701$	− 767758.i	− 1.56239i	0.624289	−	0.781193i	$-0.285389\pi$
$702$	0	0
$703$	− 15096.0i	− 0.0305458i
$704$	0	0
$705$	38306.6	0.0770719
$706$	0	0
$707$	−771811.	−1.54409
$708$	0	0
$709$	− 560272.i	− 1.11457i	−0.830322	−	0.557284i	$-0.811843\pi$
$709$	− 560272.i	− 1.11457i	0.830322	−	0.557284i	$-0.188157\pi$
$710$	0	0
$711$	178628.i	0.353353i
$712$	0	0
$713$	−1.27548e6	−2.50896
$714$	0	0
$715$	−160724.	−0.314389
$716$	0	0
$717$	22994.1i	0.0447278i
$718$	0	0
$719$	− 759774.i	− 1.46969i	−0.678234	−	0.734846i	$-0.737254\pi$
$719$	− 759774.i	− 1.46969i	0.678234	−	0.734846i	$-0.262746\pi$
$720$	0	0
$721$	−589347.	−1.13371
$722$	0	0
$723$	−179221.	−0.342856
$724$	0	0
$725$	464035.i	0.882826i
$726$	0	0
$727$	472137.i	0.893305i	0.894708	+	0.446652i	$0.147384\pi$
$727$	472137.i	0.893305i	−0.894708	+	0.446652i	$0.852616\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	1.33601e6	2.50020
$732$	0	0
$733$	600287.i	1.11725i	0.829420	+	0.558626i	$0.188671\pi$
$733$	600287.i	1.11725i	−0.829420	+	0.558626i	$0.811329\pi$
$734$	0	0
$735$	19949.6i	0.0369284i
$736$	0	0
$737$	71142.9	0.130977
$738$	0	0
$739$	116503.	0.213327	0.106664	−	0.994295i	$-0.465983\pi$
$739$	116503.	0.213327	0.106664	+	0.994295i	$0.465983\pi$
$740$	0	0
$741$	105088.i	0.191389i
$742$	0	0
$743$	− 362499.i	− 0.656642i	−0.944566	−	0.328321i	$-0.893517\pi$
$743$	− 362499.i	− 0.656642i	0.944566	−	0.328321i	$-0.106483\pi$
$744$	0	0
$745$	584462.	1.05304
$746$	0	0
$747$	−185566.	−0.332550
$748$	0	0
$749$	− 710964.i	− 1.26731i
$750$	0	0
$751$	− 255543.i	− 0.453089i	−0.974001	−	0.226545i	$-0.927257\pi$
$751$	− 255543.i	− 0.453089i	0.974001	−	0.226545i	$-0.0727429\pi$
$752$	0	0
$753$	−112620.	−0.198621
$754$	0	0
$755$	1.62364e6	2.84836
$756$	0	0
$757$	157849.i	0.275454i	0.990470	+	0.137727i	$0.0439797\pi$
$757$	157849.i	0.275454i	−0.990470	+	0.137727i	$0.956020\pi$
$758$	0	0
$759$	57923.0i	0.100547i
$760$	0	0
$761$	436931.	0.754474	0.377237	−	0.926117i	$-0.376874\pi$
$761$	436931.	0.754474	0.377237	+	0.926117i	$0.376874\pi$
$762$	0	0
$763$	14174.8	0.0243483
$764$	0	0
$765$	588085.i	1.00489i
$766$	0	0
$767$	1.37019e6i	2.32911i
$768$	0	0
$769$	−42172.0	−0.0713135	−0.0356568	−	0.999364i	$-0.511352\pi$
$769$	−42172.0	−0.0713135	−0.0356568	+	0.999364i	$0.511352\pi$
$770$	0	0
$771$	54625.1	0.0918932
$772$	0	0
$773$	143407.i	0.240000i	0.992774	+	0.120000i	$0.0382895\pi$
$773$	143407.i	0.240000i	−0.992774	+	0.120000i	$0.961711\pi$
$774$	0	0
$775$	− 1.42134e6i	− 2.36643i
$776$	0	0
$777$	−54687.9	−0.0905835
$778$	0	0
$779$	−2517.83	−0.00414907
$780$	0	0
$781$	− 62799.1i	− 0.102956i
$782$	0	0
$783$	− 78290.8i	− 0.127699i
$784$	0	0
$785$	292423.	0.474539
$786$	0	0
$787$	−295695.	−0.477414	−0.238707	−	0.971092i	$-0.576724\pi$
$787$	−295695.	−0.477414	−0.238707	+	0.971092i	$0.576724\pi$
$788$	0	0
$789$	− 205927.i	− 0.330795i
$790$	0	0
$791$	237843.i	0.380134i
$792$	0	0
$793$	−134833.	−0.214412
$794$	0	0
$795$	−389278.	−0.615922
$796$	0	0
$797$	− 41548.0i	− 0.0654084i	−0.999465	−	0.0327042i	$-0.989588\pi$
$797$	− 41548.0i	− 0.0654084i	0.999465	−	0.0327042i	$-0.0104119\pi$
$798$	0	0
$799$	− 110241.i	− 0.172683i
$800$	0	0
$801$	141400.	0.220387
$802$	0	0
$803$	79580.9	0.123418
$804$	0	0
$805$	1.42440e6i	2.19806i
$806$	0	0
$807$	419043.i	0.643445i
$808$	0	0
$809$	−1.20959e6	−1.84817	−0.924083	−	0.382191i	$-0.875170\pi$
$809$	−1.20959e6	−1.84817	−0.924083	+	0.382191i	$0.875170\pi$
$810$	0	0
$811$	−212276.	−0.322744	−0.161372	−	0.986894i	$-0.551592\pi$
$811$	−212276.	−0.322744	−0.161372	+	0.986894i	$0.551592\pi$
$812$	0	0
$813$	− 259250.i	− 0.392227i
$814$	0	0
$815$	− 323755.i	− 0.487417i
$816$	0	0
$817$	167941.	0.251601
$818$	0	0
$819$	380699.	0.567563
$820$	0	0
$821$	1.12848e6i	1.67420i	0.547048	+	0.837101i	$0.315752\pi$
$821$	1.12848e6i	1.67420i	−0.547048	+	0.837101i	$0.684248\pi$
$822$	0	0
$823$	− 51078.0i	− 0.0754109i	−0.999289	−	0.0377055i	$-0.987995\pi$
$823$	− 51078.0i	− 0.0754109i	0.999289	−	0.0377055i	$-0.0120049\pi$
$824$	0	0
$825$	−64547.0	−0.0948349
$826$	0	0
$827$	−461932.	−0.675409	−0.337705	−	0.941252i	$-0.609651\pi$
$827$	−461932.	−0.675409	−0.337705	+	0.941252i	$0.609651\pi$
$828$	0	0
$829$	1.04514e6i	1.52077i	0.649472	+	0.760386i	$0.274990\pi$
$829$	1.04514e6i	1.52077i	−0.649472	+	0.760386i	$0.725010\pi$
$830$	0	0
$831$	69213.3i	0.100228i
$832$	0	0
$833$	57412.3	0.0827399
$834$	0	0
$835$	−939459.	−1.34743
$836$	0	0
$837$	239804.i	0.342300i
$838$	0	0
$839$	− 765515.i	− 1.08750i	−0.839247	−	0.543751i	$-0.817004\pi$
$839$	− 765515.i	− 1.08750i	0.839247	−	0.543751i	$-0.182996\pi$
$840$	0	0
$841$	395873.	0.559711
$842$	0	0
$843$	94276.5	0.132662
$844$	0	0
$845$	1.94304e6i	2.72125i
$846$	0	0
$847$	− 721122.i	− 1.00518i
$848$	0	0
$849$	−732299.	−1.01595
$850$	0	0
$851$	−157022.	−0.216822
$852$	0	0
$853$	445462.i	0.612227i	0.951995	+	0.306113i	$0.0990286\pi$
$853$	445462.i	0.612227i	−0.951995	+	0.306113i	$0.900971\pi$
$854$	0	0
$855$	73924.2i	0.101124i
$856$	0	0
$857$	−255454.	−0.347817	−0.173909	−	0.984762i	$-0.555640\pi$
$857$	−255454.	−0.347817	−0.173909	+	0.984762i	$0.555640\pi$
$858$	0	0
$859$	−549268.	−0.744386	−0.372193	−	0.928155i	$-0.621394\pi$
$859$	−549268.	−0.744386	−0.372193	+	0.928155i	$0.621394\pi$
$860$	0	0
$861$	9121.27i	0.0123041i
$862$	0	0
$863$	− 217330.i	− 0.291808i	−0.989299	−	0.145904i	$-0.953391\pi$
$863$	− 217330.i	− 0.291808i	0.989299	−	0.145904i	$-0.0466091\pi$
$864$	0	0
$865$	−42154.6	−0.0563395
$866$	0	0
$867$	1.25844e6	1.67415
$868$	0	0
$869$	98830.9i	0.130874i
$870$	0	0
$871$	− 1.34256e6i	− 1.76969i
$872$	0	0
$873$	−156825.	−0.205772
$874$	0	0
$875$	−394264.	−0.514957
$876$	0	0
$877$	− 467000.i	− 0.607180i	−0.952803	−	0.303590i	$-0.901815\pi$
$877$	− 467000.i	− 0.607180i	0.952803	−	0.303590i	$-0.0981853\pi$
$878$	0	0
$879$	− 561487.i	− 0.726711i
$880$	0	0
$881$	−916522.	−1.18084	−0.590420	−	0.807096i	$-0.701038\pi$
$881$	−916522.	−1.18084	−0.590420	+	0.807096i	$0.701038\pi$
$882$	0	0
$883$	633021.	0.811890	0.405945	−	0.913898i	$-0.366942\pi$
$883$	633021.	0.811890	0.405945	+	0.913898i	$0.366942\pi$
$884$	0	0
$885$	963862.i	1.23063i
$886$	0	0
$887$	− 666258.i	− 0.846828i	−0.905936	−	0.423414i	$-0.860832\pi$
$887$	− 666258.i	− 0.846828i	0.905936	−	0.423414i	$-0.139168\pi$
$888$	0	0
$889$	−406769.	−0.514688
$890$	0	0
$891$	10890.2	0.0137177
$892$	0	0
$893$	− 13857.7i	− 0.0173775i
$894$	0	0
$895$	1.34301e6i	1.67662i
$896$	0	0
$897$	1.09308e6	1.35853
$898$	0	0
$899$	953842.	1.18020
$900$	0	0
$901$	1.12029e6i	1.38000i
$902$	0	0
$903$	− 608395.i	− 0.746123i
$904$	0	0
$905$	1.14857e6	1.40236
$906$	0	0
$907$	−374365.	−0.455072	−0.227536	−	0.973770i	$-0.573067\pi$
$907$	−374365.	−0.455072	−0.227536	+	0.973770i	$0.573067\pi$
$908$	0	0
$909$	416645.i	0.504241i
$910$	0	0
$911$	790211.i	0.952152i	0.879404	+	0.476076i	$0.157941\pi$
$911$	790211.i	0.952152i	−0.879404	+	0.476076i	$0.842059\pi$
$912$	0	0
$913$	−102670.	−0.123169
$914$	0	0
$915$	−94848.3	−0.113289
$916$	0	0
$917$	498334.i	0.592628i
$918$	0	0
$919$	1.00581e6i	1.19093i	0.803380	+	0.595466i	$0.203033\pi$
$919$	1.00581e6i	1.19093i	−0.803380	+	0.595466i	$0.796967\pi$
$920$	0	0
$921$	−140779.	−0.165966
$922$	0	0
$923$	−1.18510e6	−1.39108
$924$	0	0
$925$	− 174979.i	− 0.204505i
$926$	0	0
$927$	318146.i	0.370226i
$928$	0	0
$929$	−645243.	−0.747639	−0.373820	−	0.927501i	$-0.621952\pi$
$929$	−645243.	−0.747639	−0.373820	+	0.927501i	$0.621952\pi$
$930$	0	0
$931$	7216.92	0.00832631
$932$	0	0
$933$	− 597221.i	− 0.686075i
$934$	0	0
$935$	325375.i	0.372187i
$936$	0	0
$937$	801226.	0.912591	0.456295	−	0.889828i	$-0.349176\pi$
$937$	801226.	0.912591	0.456295	+	0.889828i	$0.349176\pi$
$938$	0	0
$939$	904040.	1.02531
$940$	0	0
$941$	− 504479.i	− 0.569724i	−0.958569	−	0.284862i	$-0.908052\pi$
$941$	− 504479.i	− 0.569724i	0.958569	−	0.284862i	$-0.0919477\pi$
$942$	0	0
$943$	26189.4i	0.0294512i
$944$	0	0
$945$	267804.	0.299884
$946$	0	0
$947$	170424.	0.190034	0.0950169	−	0.995476i	$-0.469709\pi$
$947$	170424.	0.190034	0.0950169	+	0.995476i	$0.469709\pi$
$948$	0	0
$949$	− 1.50179e6i	− 1.66755i
$950$	0	0
$951$	− 800835.i	− 0.885486i
$952$	0	0
$953$	352354.	0.387965	0.193983	−	0.981005i	$-0.437859\pi$
$953$	352354.	0.387965	0.193983	+	0.981005i	$0.437859\pi$
$954$	0	0
$955$	−604616.	−0.662938
$956$	0	0
$957$	− 43316.7i	− 0.0472967i
$958$	0	0
$959$	− 1.72372e6i	− 1.87425i
$960$	0	0
$961$	−1.99809e6	−2.16356
$962$	0	0
$963$	−383798.	−0.413857
$964$	0	0
$965$	− 1.89727e6i	− 2.03740i
$966$	0	0
$967$	1.37663e6i	1.47219i	0.676879	+	0.736094i	$0.263332\pi$
$967$	1.37663e6i	1.47219i	−0.676879	+	0.736094i	$0.736668\pi$
$968$	0	0
$969$	212744.	0.226574
$970$	0	0
$971$	−534975.	−0.567408	−0.283704	−	0.958912i	$-0.591563\pi$
$971$	−534975.	−0.567408	−0.283704	+	0.958912i	$0.591563\pi$
$972$	0	0
$973$	− 1.85633e6i	− 1.96078i
$974$	0	0
$975$	1.21808e6i	1.28135i
$976$	0	0
$977$	−1.19772e6	−1.25477	−0.627386	−	0.778708i	$-0.715875\pi$
$977$	−1.19772e6	−1.25477	−0.627386	+	0.778708i	$0.715875\pi$
$978$	0	0
$979$	78233.9	0.0816262
$980$	0	0
$981$	− 7651.96i	− 0.00795123i
$982$	0	0
$983$	149071.i	0.154272i	0.997021	+	0.0771360i	$0.0245776\pi$
$983$	149071.i	0.154272i	−0.997021	+	0.0771360i	$0.975422\pi$
$984$	0	0
$985$	−665799.	−0.686232
$986$	0	0
$987$	−50202.0	−0.0515331
$988$	0	0
$989$	− 1.74685e6i	− 1.78593i
$990$	0	0
$991$	− 564204.i	− 0.574498i	−0.957856	−	0.287249i	$-0.907259\pi$
$991$	− 564204.i	− 0.574498i	0.957856	−	0.287249i	$-0.0927407\pi$
$992$	0	0
$993$	−761276.	−0.772047
$994$	0	0
$995$	−1.28343e6	−1.29636
$996$	0	0
$997$	1.31568e6i	1.32361i	0.749678	+	0.661803i	$0.230208\pi$
$997$	1.31568e6i	1.32361i	−0.749678	+	0.661803i	$0.769792\pi$
$998$	0	0
$999$	29522.0i	0.0295812i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.b.h.127.10		16
4.3	odd	2		768.5.b.i.127.2		16
8.3	odd	2	inner	768.5.b.h.127.15		16
8.5	even	2		768.5.b.i.127.7		16
16.3	odd	4		384.5.g.a.127.2	✓	16
16.5	even	4		384.5.g.b.127.7	yes	16
16.11	odd	4		384.5.g.b.127.15	yes	16
16.13	even	4		384.5.g.a.127.10	yes	16
48.5	odd	4		1152.5.g.c.127.3		16
48.11	even	4		1152.5.g.c.127.4		16
48.29	odd	4		1152.5.g.f.127.13		16
48.35	even	4		1152.5.g.f.127.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.g.a.127.2	✓	16	16.3	odd	4
384.5.g.a.127.10	yes	16	16.13	even	4
384.5.g.b.127.7	yes	16	16.5	even	4
384.5.g.b.127.15	yes	16	16.11	odd	4
768.5.b.h.127.10		16	1.1	even	1	trivial
768.5.b.h.127.15		16	8.3	odd	2	inner
768.5.b.i.127.2		16	4.3	odd	2
768.5.b.i.127.7		16	8.5	even	2
1152.5.g.c.127.3		16	48.5	odd	4
1152.5.g.c.127.4		16	48.11	even	4
1152.5.g.f.127.13		16	48.29	odd	4
1152.5.g.f.127.14		16	48.35	even	4

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -38.1647i q^{5} +50.0160i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -38.1647i q^{5} +50.0160i q^{7} +27.0000 q^{9} +14.9385 q^{11} -281.909i q^{13} -198.310i q^{15} -570.708 q^{17} -71.7400 q^{19} +259.891i q^{21} +746.211i q^{23} -831.546 q^{25} +140.296 q^{27} -558.040i q^{29} +1709.27i q^{31} +77.6229 q^{33} +1908.85 q^{35} +210.427i q^{37} -1464.84i q^{39} +35.0966 q^{41} -2340.97 q^{43} -1030.45i q^{45} +193.166i q^{47} -100.598 q^{49} -2965.49 q^{51} -1962.98i q^{53} -570.125i q^{55} -372.772 q^{57} -4860.39 q^{59} -478.284i q^{61} +1350.43i q^{63} -10759.0 q^{65} +4762.37 q^{67} +3877.42i q^{69} -4203.83i q^{71} +5327.22 q^{73} -4320.84 q^{75} +747.166i q^{77} +6615.84i q^{79} +729.000 q^{81} -6872.82 q^{83} +21780.9i q^{85} -2899.66i q^{87} +5237.05 q^{89} +14100.0 q^{91} +8881.65i q^{93} +2737.93i q^{95} -5808.32 q^{97} +403.341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 432 q^{9}+O(q^{10}) \) 16 * q + 432 * q^9	\( 16 q + 432 q^{9} - 576 q^{11} - 480 q^{17} - 192 q^{19} - 2672 q^{25} - 9024 q^{35} - 1440 q^{41} - 12224 q^{43} - 2480 q^{49} - 6336 q^{51} + 7488 q^{57} - 15360 q^{59} - 1344 q^{65} - 12288 q^{67} + 8480 q^{73} - 21888 q^{75} + 11664 q^{81} - 13248 q^{83} - 18720 q^{89} + 30272 q^{91} + 13088 q^{97} - 15552 q^{99}+O(q^{100}) \) 16 * q + 432 * q^9 - 576 * q^11 - 480 * q^17 - 192 * q^19 - 2672 * q^25 - 9024 * q^35 - 1440 * q^41 - 12224 * q^43 - 2480 * q^49 - 6336 * q^51 + 7488 * q^57 - 15360 * q^59 - 1344 * q^65 - 12288 * q^67 + 8480 * q^73 - 21888 * q^75 + 11664 * q^81 - 13248 * q^83 - 18720 * q^89 + 30272 * q^91 + 13088 * q^97 - 15552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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