LMFDB - Embedded newform 384.5.g.a.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(127,384)]

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, 0, 0]))

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

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.g (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} - 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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{86}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.500000 - 2.68460i$ of defining polynomial
Character	$\chi$	$=$	384.127
Dual form			384.5.g.a.127.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -38.1647 q^{5} +50.0160i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -38.1647 q^{5} +50.0160i q^{7} -27.0000 q^{9} +14.9385i q^{11} +281.909 q^{13} +198.310i q^{15} -570.708 q^{17} +71.7400i q^{19} +259.891 q^{21} +746.211i q^{23} +831.546 q^{25} +140.296i q^{27} +558.040 q^{29} -1709.27i q^{31} +77.6229 q^{33} -1908.85i q^{35} +210.427 q^{37} -1464.84i q^{39} -35.0966 q^{41} -2340.97i q^{43} +1030.45 q^{45} -193.166i q^{47} -100.598 q^{49} +2965.49i q^{51} -1962.98 q^{53} -570.125i q^{55} +372.772 q^{57} -4860.39i q^{59} +478.284 q^{61} -1350.43i q^{63} -10759.0 q^{65} -4762.37i q^{67} +3877.42 q^{69} -4203.83i q^{71} -5327.22 q^{73} -4320.84i q^{75} -747.166 q^{77} -6615.84i q^{79} +729.000 q^{81} +6872.82i q^{83} +21780.9 q^{85} -2899.66i q^{87} -5237.05 q^{89} +14100.0i q^{91} -8881.65 q^{93} -2737.93i q^{95} -5808.32 q^{97} -403.341i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{5} - 432 q^{9}+O(q^{10}) $ 16 * q - 96 * q^5 - 432 * q^9	$ 16 q - 96 q^{5} - 432 q^{9} - 480 q^{13} - 480 q^{17} + 2672 q^{25} + 3360 q^{29} - 1120 q^{37} + 1440 q^{41} + 2592 q^{45} - 2480 q^{49} - 3552 q^{53} - 7488 q^{57} - 18272 q^{61} - 1344 q^{65} - 8480 q^{73} + 17280 q^{77} + 11664 q^{81} + 29888 q^{85} + 18720 q^{89} - 28800 q^{93} + 13088 q^{97}+O(q^{100}) $ 16 * q - 96 * q^5 - 432 * q^9 - 480 * q^13 - 480 * q^17 + 2672 * q^25 + 3360 * q^29 - 1120 * q^37 + 1440 * q^41 + 2592 * q^45 - 2480 * q^49 - 3552 * q^53 - 7488 * q^57 - 18272 * q^61 - 1344 * q^65 - 8480 * q^73 + 17280 * q^77 + 11664 * q^81 + 29888 * q^85 + 18720 * q^89 - 28800 * q^93 + 13088 * 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
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−38.1647	−1.52659	−0.763294	−	0.646051i	$-0.776419\pi$
$5$	−38.1647	−1.52659	−0.763294	+	0.646051i	$0.776419\pi$
$6$	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
$11$	14.9385i	0.123459i	0.998093	+	0.0617295i	$0.0196616\pi$
$11$	14.9385i	0.123459i	−0.998093	+	0.0617295i	$0.980338\pi$
$12$	0	0
$13$	281.909	1.66810	0.834051	−	0.551687i	$-0.186016\pi$
$13$	281.909	1.66810	0.834051	+	0.551687i	$0.186016\pi$
$14$	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
$19$	71.7400i	0.198726i	0.995051	+	0.0993628i	$0.0316804\pi$
$19$	71.7400i	0.198726i	−0.995051	+	0.0993628i	$0.968320\pi$
$20$	0	0
$21$	259.891	0.589321
$22$	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
$27$	140.296i	0.192450i
$28$	0	0
$29$	558.040	0.663543	0.331771	−	0.943360i	$-0.392354\pi$
$29$	558.040	0.663543	0.331771	+	0.943360i	$0.392354\pi$
$30$	0	0
$31$	− 1709.27i	− 1.77864i	−0.457285	−	0.889320i	$-0.651178\pi$
$31$	− 1709.27i	− 1.77864i	0.457285	−	0.889320i	$-0.348822\pi$
$32$	0	0
$33$	77.6229	0.0712791
$34$	0	0
$35$	− 1908.85i	− 1.55824i
$36$	0	0
$37$	210.427	0.153708	0.0768541	−	0.997042i	$-0.475512\pi$
$37$	210.427	0.153708	0.0768541	+	0.997042i	$0.475512\pi$
$38$	0	0
$39$	− 1464.84i	− 0.963079i
$40$	0	0
$41$	−35.0966	−0.0208784	−0.0104392	−	0.999946i	$-0.503323\pi$
$41$	−35.0966	−0.0208784	−0.0104392	+	0.999946i	$0.503323\pi$
$42$	0	0
$43$	− 2340.97i	− 1.26607i	−0.774122	−	0.633036i	$-0.781809\pi$
$43$	− 2340.97i	− 1.26607i	0.774122	−	0.633036i	$-0.218191\pi$
$44$	0	0
$45$	1030.45	0.508863
$46$	0	0
$47$	− 193.166i	− 0.0874449i	−0.999044	−	0.0437224i	$-0.986078\pi$
$47$	− 193.166i	− 0.0874449i	0.999044	−	0.0437224i	$-0.0139217\pi$
$48$	0	0
$49$	−100.598	−0.0418985
$50$	0	0
$51$	2965.49i	1.14013i
$52$	0	0
$53$	−1962.98	−0.698818	−0.349409	−	0.936970i	$-0.613618\pi$
$53$	−1962.98	−0.698818	−0.349409	+	0.936970i	$0.613618\pi$
$54$	0	0
$55$	− 570.125i	− 0.188471i
$56$	0	0
$57$	372.772	0.114734
$58$	0	0
$59$	− 4860.39i	− 1.39626i	−0.715970	−	0.698131i	$-0.754015\pi$
$59$	− 4860.39i	− 1.39626i	0.715970	−	0.698131i	$-0.245985\pi$
$60$	0	0
$61$	478.284	0.128536	0.0642682	−	0.997933i	$-0.479529\pi$
$61$	478.284	0.128536	0.0642682	+	0.997933i	$0.479529\pi$
$62$	0	0
$63$	− 1350.43i	− 0.340245i
$64$	0	0
$65$	−10759.0	−2.54651
$66$	0	0
$67$	− 4762.37i	− 1.06090i	−0.847717	−	0.530449i	$-0.822023\pi$
$67$	− 4762.37i	− 1.06090i	0.847717	−	0.530449i	$-0.177977\pi$
$68$	0	0
$69$	3877.42	0.814414
$70$	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	$-0.666605\pi$
$73$	−5327.22	−0.999666	−0.499833	+	0.866122i	$0.666605\pi$
$74$	0	0
$75$	− 4320.84i	− 0.768149i
$76$	0	0
$77$	−747.166	−0.126019
$78$	0	0
$79$	− 6615.84i	− 1.06006i	−0.847979	−	0.530030i	$-0.822181\pi$
$79$	− 6615.84i	− 1.06006i	0.847979	−	0.530030i	$-0.177819\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	6872.82i	0.997651i	0.866702	+	0.498826i	$0.166235\pi$
$83$	6872.82i	0.997651i	−0.866702	+	0.498826i	$0.833765\pi$
$84$	0	0
$85$	21780.9	3.01466
$86$	0	0
$87$	− 2899.66i	− 0.383097i
$88$	0	0
$89$	−5237.05	−0.661160	−0.330580	−	0.943778i	$-0.607244\pi$
$89$	−5237.05	−0.661160	−0.330580	+	0.943778i	$0.607244\pi$
$90$	0	0
$91$	14100.0i	1.70269i
$92$	0	0
$93$	−8881.65	−1.02690
$94$	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
$99$	− 403.341i	− 0.0411530i
$100$	0	0
$101$	15431.3	1.51272	0.756362	−	0.654154i	$-0.226975\pi$
$101$	15431.3	1.51272	0.756362	+	0.654154i	$0.226975\pi$
$102$	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
$107$	− 14214.7i	− 1.24157i	−0.783981	−	0.620785i	$-0.786814\pi$
$107$	− 14214.7i	− 1.24157i	0.783981	−	0.620785i	$-0.213186\pi$
$108$	0	0
$109$	283.406	0.0238537	0.0119269	−	0.999929i	$-0.496203\pi$
$109$	283.406	0.0238537	0.0119269	+	0.999929i	$0.496203\pi$
$110$	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
$115$	− 28478.9i	− 2.15341i
$116$	0	0
$117$	−7611.55	−0.556034
$118$	0	0
$119$	− 28544.5i	− 2.01571i
$120$	0	0
$121$	14417.8	0.984758
$122$	0	0
$123$	182.367i	0.0120541i
$124$	0	0
$125$	−7882.75	−0.504496
$126$	0	0
$127$	− 8132.78i	− 0.504233i	−0.967697	−	0.252117i	$-0.918873\pi$
$127$	− 8132.78i	− 0.504233i	0.967697	−	0.252117i	$-0.0811267\pi$
$128$	0	0
$129$	−12164.0	−0.730967
$130$	0	0
$131$	− 9963.50i	− 0.580590i	−0.956937	−	0.290295i	$-0.906247\pi$
$131$	− 9963.50i	− 0.580590i	0.956937	−	0.290295i	$-0.0937534\pi$
$132$	0	0
$133$	−3588.14	−0.202846
$134$	0	0
$135$	− 5354.36i	− 0.293792i
$136$	0	0
$137$	34463.3	1.83618	0.918091	−	0.396369i	$-0.129730\pi$
$137$	34463.3	1.83618	0.918091	+	0.396369i	$0.129730\pi$
$138$	0	0
$139$	− 37114.7i	− 1.92095i	−0.278361	−	0.960477i	$-0.589791\pi$
$139$	− 37114.7i	− 1.92095i	0.278361	−	0.960477i	$-0.410209\pi$
$140$	0	0
$141$	−1003.72	−0.0504863
$142$	0	0
$143$	4211.31i	0.205942i
$144$	0	0
$145$	−21297.4	−1.01296
$146$	0	0
$147$	522.725i	0.0241901i
$148$	0	0
$149$	15314.2	0.689797	0.344899	−	0.938640i	$-0.387913\pi$
$149$	15314.2	0.689797	0.344899	+	0.938640i	$0.387913\pi$
$150$	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
$155$	65233.9i	2.71525i
$156$	0	0
$157$	−7662.12	−0.310849	−0.155425	−	0.987848i	$-0.549675\pi$
$157$	−7662.12	−0.310849	−0.155425	+	0.987848i	$0.549675\pi$
$158$	0	0
$159$	10199.9i	0.403463i
$160$	0	0
$161$	−37322.5	−1.43985
$162$	0	0
$163$	− 8483.09i	− 0.319285i	−0.987175	−	0.159643i	$-0.948966\pi$
$163$	− 8483.09i	− 0.319285i	0.987175	−	0.159643i	$-0.0510342\pi$
$164$	0	0
$165$	−2962.46	−0.108814
$166$	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
$171$	− 1936.98i	− 0.0662419i
$172$	0	0
$173$	1104.54	0.0369055	0.0184527	−	0.999830i	$-0.494126\pi$
$173$	1104.54	0.0369055	0.0184527	+	0.999830i	$0.494126\pi$
$174$	0	0
$175$	41590.6i	1.35806i
$176$	0	0
$177$	−25255.3	−0.806132
$178$	0	0
$179$	35189.9i	1.09828i	0.835731	+	0.549139i	$0.185044\pi$
$179$	35189.9i	1.09828i	−0.835731	+	0.549139i	$0.814956\pi$
$180$	0	0
$181$	30095.0	0.918623	0.459311	−	0.888275i	$-0.348096\pi$
$181$	30095.0	0.918623	0.459311	+	0.888275i	$0.348096\pi$
$182$	0	0
$183$	− 2485.24i	− 0.0742105i
$184$	0	0
$185$	−8030.87	−0.234649
$186$	0	0
$187$	− 8525.55i	− 0.243803i
$188$	0	0
$189$	−7017.05	−0.196440
$190$	0	0
$191$	15842.3i	0.434261i	0.976143	+	0.217131i	$0.0696697\pi$
$191$	15842.3i	0.434261i	−0.976143	+	0.217131i	$0.930330\pi$
$192$	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
$195$	55905.4i	1.47023i
$196$	0	0
$197$	−17445.4	−0.449520	−0.224760	−	0.974414i	$-0.572160\pi$
$197$	−17445.4	−0.449520	−0.224760	+	0.974414i	$0.572160\pi$
$198$	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
$203$	27910.9i	0.677301i
$204$	0	0
$205$	1339.45	0.0318727
$206$	0	0
$207$	− 20147.7i	− 0.470202i
$208$	0	0
$209$	−1071.69	−0.0245345
$210$	0	0
$211$	51764.4i	1.16270i	0.813655	+	0.581348i	$0.197475\pi$
$211$	51764.4i	1.16270i	−0.813655	+	0.581348i	$0.802525\pi$
$212$	0	0
$213$	−21843.7	−0.481469
$214$	0	0
$215$	89342.3i	1.93277i
$216$	0	0
$217$	85491.0	1.81552
$218$	0	0
$219$	27681.1i	0.577158i
$220$	0	0
$221$	−160888.	−3.29412
$222$	0	0
$223$	58417.9i	1.17472i	0.809324	+	0.587362i	$0.199834\pi$
$223$	58417.9i	1.17472i	−0.809324	+	0.587362i	$0.800166\pi$
$224$	0	0
$225$	−22451.7	−0.443491
$226$	0	0
$227$	− 60234.3i	− 1.16894i	−0.811415	−	0.584470i	$-0.801302\pi$
$227$	− 60234.3i	− 1.16894i	0.811415	−	0.584470i	$-0.198698\pi$
$228$	0	0
$229$	31082.1	0.592706	0.296353	−	0.955078i	$-0.404229\pi$
$229$	31082.1	0.592706	0.296353	+	0.955078i	$0.404229\pi$
$230$	0	0
$231$	3882.39i	0.0727570i
$232$	0	0
$233$	−25930.7	−0.477643	−0.238821	−	0.971064i	$-0.576761\pi$
$233$	−25930.7	−0.477643	−0.238821	+	0.971064i	$0.576761\pi$
$234$	0	0
$235$	7372.12i	0.133492i
$236$	0	0
$237$	−34376.9	−0.612026
$238$	0	0
$239$	− 4425.21i	− 0.0774708i	−0.999250	−	0.0387354i	$-0.987667\pi$
$239$	− 4425.21i	− 0.0774708i	0.999250	−	0.0387354i	$-0.0123329\pi$
$240$	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
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	3839.31	0.0639618
$246$	0	0
$247$	20224.2i	0.331495i
$248$	0	0
$249$	35712.2	0.575994
$250$	0	0
$251$	− 21673.7i	− 0.344022i	−0.985095	−	0.172011i	$-0.944974\pi$
$251$	− 21673.7i	− 0.344022i	0.985095	−	0.172011i	$-0.0550265\pi$
$252$	0	0
$253$	−11147.3	−0.174152
$254$	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
$259$	10524.7i	0.156895i
$260$	0	0
$261$	−15067.1	−0.221181
$262$	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
$267$	27212.5i	0.381721i
$268$	0	0
$269$	−80644.9	−1.11448	−0.557240	−	0.830351i	$-0.688140\pi$
$269$	−80644.9	−1.11448	−0.557240	+	0.830351i	$0.688140\pi$
$270$	0	0
$271$	49892.7i	0.679357i	0.940542	+	0.339678i	$0.110318\pi$
$271$	49892.7i	0.679357i	−0.940542	+	0.339678i	$0.889682\pi$
$272$	0	0
$273$	73265.6	0.983048
$274$	0	0
$275$	12422.1i	0.164259i
$276$	0	0
$277$	13320.1	0.173599	0.0867996	−	0.996226i	$-0.472336\pi$
$277$	13320.1	0.173599	0.0867996	+	0.996226i	$0.472336\pi$
$278$	0	0
$279$	46150.4i	0.592880i
$280$	0	0
$281$	−18143.5	−0.229778	−0.114889	−	0.993378i	$-0.536651\pi$
$281$	−18143.5	−0.229778	−0.114889	+	0.993378i	$0.536651\pi$
$282$	0	0
$283$	− 140931.i	− 1.75968i	−0.475270	−	0.879840i	$-0.657650\pi$
$283$	− 140931.i	− 1.75968i	0.475270	−	0.879840i	$-0.342350\pi$
$284$	0	0
$285$	−14226.7	−0.175152
$286$	0	0
$287$	− 1755.39i	− 0.0213113i
$288$	0	0
$289$	242187.	2.89971
$290$	0	0
$291$	30180.9i	0.356407i
$292$	0	0
$293$	−108058.	−1.25870	−0.629350	−	0.777122i	$-0.716679\pi$
$293$	−108058.	−1.25870	−0.629350	+	0.777122i	$0.716679\pi$
$294$	0	0
$295$	185495.i	2.13152i
$296$	0	0
$297$	−2095.82	−0.0237597
$298$	0	0
$299$	210364.i	2.35304i
$300$	0	0
$301$	117086.	1.29232
$302$	0	0
$303$	− 80183.3i	− 0.873371i
$304$	0	0
$305$	−18253.6	−0.196222
$306$	0	0
$307$	27092.9i	0.287461i	0.989617	+	0.143730i	$0.0459098\pi$
$307$	27092.9i	0.287461i	−0.989617	+	0.143730i	$0.954090\pi$
$308$	0	0
$309$	61227.2	0.641250
$310$	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	$-0.847869\pi$
$313$	−173983.	−1.77590	−0.887948	+	0.459945i	$0.847869\pi$
$314$	0	0
$315$	51538.8i	0.519414i
$316$	0	0
$317$	154121.	1.53371	0.766854	−	0.641822i	$-0.221821\pi$
$317$	154121.	1.53371	0.766854	+	0.641822i	$0.221821\pi$
$318$	0	0
$319$	8336.29i	0.0819203i
$320$	0	0
$321$	−73861.9	−0.716821
$322$	0	0
$323$	− 40942.6i	− 0.392437i
$324$	0	0
$325$	234420.	2.21937
$326$	0	0
$327$	− 1472.62i	− 0.0137719i
$328$	0	0
$329$	9661.38	0.0892580
$330$	0	0
$331$	− 146508.i	− 1.33723i	−0.743611	−	0.668613i	$-0.766889\pi$
$331$	− 146508.i	− 1.33723i	0.743611	−	0.668613i	$-0.233111\pi$
$332$	0	0
$333$	−5681.52	−0.0512361
$334$	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
$339$	− 24709.4i	− 0.215012i
$340$	0	0
$341$	25534.0	0.219589
$342$	0	0
$343$	115057.i	0.977967i
$344$	0	0
$345$	−147981.	−1.24327
$346$	0	0
$347$	− 125592.i	− 1.04304i	−0.853239	−	0.521521i	$-0.825365\pi$
$347$	− 125592.i	− 1.04304i	0.853239	−	0.521521i	$-0.174635\pi$
$348$	0	0
$349$	−44311.9	−0.363806	−0.181903	−	0.983316i	$-0.558226\pi$
$349$	−44311.9	−0.363806	−0.181903	+	0.983316i	$0.558226\pi$
$350$	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.i	1.27307i
$356$	0	0
$357$	−148322.	−1.16377
$358$	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
$363$	− 74917.3i	− 0.568550i
$364$	0	0
$365$	203312.	1.52608
$366$	0	0
$367$	− 22943.3i	− 0.170343i	−0.996366	−	0.0851714i	$-0.972856\pi$
$367$	− 22943.3i	− 0.170343i	0.996366	−	0.0851714i	$-0.0271438\pi$
$368$	0	0
$369$	947.607	0.00695946
$370$	0	0
$371$	− 98180.3i	− 0.713307i
$372$	0	0
$373$	−214254.	−1.53997	−0.769985	−	0.638062i	$-0.779736\pi$
$373$	−214254.	−1.53997	−0.769985	+	0.638062i	$0.779736\pi$
$374$	0	0
$375$	40960.0i	0.291271i
$376$	0	0
$377$	157317.	1.10686
$378$	0	0
$379$	− 135220.i	− 0.941376i	−0.882300	−	0.470688i	$-0.844006\pi$
$379$	− 135220.i	− 0.941376i	0.882300	−	0.470688i	$-0.155994\pi$
$380$	0	0
$381$	−42259.2	−0.291119
$382$	0	0
$383$	− 99667.3i	− 0.679446i	−0.940525	−	0.339723i	$-0.889667\pi$
$383$	− 99667.3i	− 0.679446i	0.940525	−	0.339723i	$-0.110333\pi$
$384$	0	0
$385$	28515.4	0.192379
$386$	0	0
$387$	63206.1i	0.422024i
$388$	0	0
$389$	−83760.5	−0.553529	−0.276764	−	0.960938i	$-0.589262\pi$
$389$	−83760.5	−0.553529	−0.276764	+	0.960938i	$0.589262\pi$
$390$	0	0
$391$	− 425869.i	− 2.78562i
$392$	0	0
$393$	−51771.9	−0.335204
$394$	0	0
$395$	252491.i	1.61828i
$396$	0	0
$397$	222509.	1.41178	0.705890	−	0.708322i	$-0.250547\pi$
$397$	222509.	1.41178	0.705890	+	0.708322i	$0.250547\pi$
$398$	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.i	− 2.96695i
$404$	0	0
$405$	−27822.1	−0.169621
$406$	0	0
$407$	3143.46i	0.0189767i
$408$	0	0
$409$	−34233.7	−0.204648	−0.102324	−	0.994751i	$-0.532628\pi$
$409$	−34233.7	−0.204648	−0.102324	+	0.994751i	$0.532628\pi$
$410$	0	0
$411$	− 179077.i	− 1.06012i
$412$	0	0
$413$	243097.	1.42521
$414$	0	0
$415$	− 262299.i	− 1.52300i
$416$	0	0
$417$	−192854.	−1.10906
$418$	0	0
$419$	− 90685.6i	− 0.516547i	−0.966072	−	0.258274i	$-0.916846\pi$
$419$	− 90685.6i	− 0.516547i	0.966072	−	0.258274i	$-0.0831536\pi$
$420$	0	0
$421$	−278121.	−1.56917	−0.784584	−	0.620022i	$-0.787124\pi$
$421$	−278121.	−1.56917	−0.784584	+	0.620022i	$0.787124\pi$
$422$	0	0
$423$	5215.48i	0.0291483i
$424$	0	0
$425$	−474570.	−2.62738
$426$	0	0
$427$	23921.8i	0.131201i
$428$	0	0
$429$	21882.6	0.118901
$430$	0	0
$431$	− 183909.i	− 0.990032i	−0.868884	−	0.495016i	$-0.835162\pi$
$431$	− 183909.i	− 0.990032i	0.868884	−	0.495016i	$-0.164838\pi$
$432$	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.i	0.584831i
$436$	0	0
$437$	−53533.1	−0.280324
$438$	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.i	1.17427i	0.809491	+	0.587133i	$0.199743\pi$
$443$	230448.i	1.17427i	−0.809491	+	0.587133i	$0.800257\pi$
$444$	0	0
$445$	199870.	1.00932
$446$	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
$451$	− 524.291i	− 0.00257762i
$452$	0	0
$453$	221059.	1.07724
$454$	0	0
$455$	− 538121.i	− 2.59931i
$456$	0	0
$457$	−303453.	−1.45298	−0.726489	−	0.687179i	$-0.758849\pi$
$457$	−303453.	−1.45298	−0.726489	+	0.687179i	$0.758849\pi$
$458$	0	0
$459$	− 80068.2i	− 0.380045i
$460$	0	0
$461$	−147882.	−0.695849	−0.347924	−	0.937523i	$-0.613113\pi$
$461$	−147882.	−0.695849	−0.347924	+	0.937523i	$0.613113\pi$
$462$	0	0
$463$	106150.i	0.495174i	0.968866	+	0.247587i	$0.0796376\pi$
$463$	106150.i	0.495174i	−0.968866	+	0.247587i	$0.920362\pi$
$464$	0	0
$465$	338965.	1.56765
$466$	0	0
$467$	− 158470.i	− 0.726630i	−0.931666	−	0.363315i	$-0.881645\pi$
$467$	− 158470.i	− 0.726630i	0.931666	−	0.363315i	$-0.118355\pi$
$468$	0	0
$469$	238195.	1.08289
$470$	0	0
$471$	39813.5i	0.179469i
$472$	0	0
$473$	34970.6	0.156308
$474$	0	0
$475$	59655.0i	0.264399i
$476$	0	0
$477$	53000.4	0.232939
$478$	0	0
$479$	87871.4i	0.382981i	0.981495	+	0.191490i	$0.0613320\pi$
$479$	87871.4i	0.382981i	−0.981495	+	0.191490i	$0.938668\pi$
$480$	0	0
$481$	59321.2	0.256401
$482$	0	0
$483$	193933.i	0.831300i
$484$	0	0
$485$	221673.	0.942386
$486$	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.i	− 0.485088i	−0.970140	−	0.242544i	$-0.922018\pi$
$491$	− 116945.i	− 0.485088i	0.970140	−	0.242544i	$-0.0779819\pi$
$492$	0	0
$493$	−318478.	−1.31034
$494$	0	0
$495$	15393.4i	0.0628237i
$496$	0	0
$497$	210259.	0.851219
$498$	0	0
$499$	99656.8i	0.400227i	0.979773	+	0.200113i	$0.0641310\pi$
$499$	99656.8i	0.400227i	−0.979773	+	0.200113i	$0.935869\pi$
$500$	0	0
$501$	−127908.	−0.509591
$502$	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.i	− 1.02917i
$508$	0	0
$509$	94288.3	0.363934	0.181967	−	0.983305i	$-0.441754\pi$
$509$	94288.3	0.363934	0.181967	+	0.983305i	$0.441754\pi$
$510$	0	0
$511$	− 266446.i	− 1.02039i
$512$	0	0
$513$	−10064.8	−0.0382448
$514$	0	0
$515$	− 449702.i	− 1.69555i
$516$	0	0
$517$	2885.61	0.0107959
$518$	0	0
$519$	− 5739.38i	− 0.0213074i
$520$	0	0
$521$	−252399.	−0.929848	−0.464924	−	0.885351i	$-0.653918\pi$
$521$	−252399.	−0.929848	−0.464924	+	0.885351i	$0.653918\pi$
$522$	0	0
$523$	360986.i	1.31973i	0.751382	+	0.659867i	$0.229388\pi$
$523$	360986.i	1.31973i	−0.751382	+	0.659867i	$0.770612\pi$
$524$	0	0
$525$	216111.	0.784076
$526$	0	0
$527$	975497.i	3.51240i
$528$	0	0
$529$	−276989.	−0.989809
$530$	0	0
$531$	131230.i	0.465421i
$532$	0	0
$533$	−9894.05	−0.0348273
$534$	0	0
$535$	542501.i	1.89537i
$536$	0	0
$537$	182852.	0.634091
$538$	0	0
$539$	− 1502.79i	− 0.00517275i
$540$	0	0
$541$	212815.	0.727123	0.363561	−	0.931570i	$-0.381561\pi$
$541$	212815.	0.727123	0.363561	+	0.931570i	$0.381561\pi$
$542$	0	0
$543$	− 156378.i	− 0.530367i
$544$	0	0
$545$	−10816.1	−0.0364148
$546$	0	0
$547$	84823.3i	0.283492i	0.989903	+	0.141746i	$0.0452716\pi$
$547$	84823.3i	0.283492i	−0.989903	+	0.141746i	$0.954728\pi$
$548$	0	0
$549$	−12913.7	−0.0428455
$550$	0	0
$551$	40033.7i	0.131863i
$552$	0	0
$553$	330898.	1.08204
$554$	0	0
$555$	41729.6i	0.135475i
$556$	0	0
$557$	−8677.73	−0.0279702	−0.0139851	−	0.999902i	$-0.504452\pi$
$557$	−8677.73	−0.0279702	−0.0139851	+	0.999902i	$0.504452\pi$
$558$	0	0
$559$	− 659940.i	− 2.11194i
$560$	0	0
$561$	−44300.0	−0.140760
$562$	0	0
$563$	114230.i	0.360381i	0.983632	+	0.180190i	$0.0576714\pi$
$563$	114230.i	0.360381i	−0.983632	+	0.180190i	$0.942329\pi$
$564$	0	0
$565$	−181486.	−0.568521
$566$	0	0
$567$	36461.7i	0.113415i
$568$	0	0
$569$	−29567.0	−0.0913236	−0.0456618	−	0.998957i	$-0.514540\pi$
$569$	−29567.0	−0.0913236	−0.0456618	+	0.998957i	$0.514540\pi$
$570$	0	0
$571$	229583.i	0.704155i	0.935971	+	0.352078i	$0.114525\pi$
$571$	229583.i	0.704155i	−0.935971	+	0.352078i	$0.885475\pi$
$572$	0	0
$573$	82318.9	0.250721
$574$	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.i	− 0.770536i
$580$	0	0
$581$	−343751.	−1.01834
$582$	0	0
$583$	− 29324.0i	− 0.0862754i
$584$	0	0
$585$	290493.	0.848835
$586$	0	0
$587$	− 548252.i	− 1.59112i	−0.605873	−	0.795561i	$-0.707176\pi$
$587$	− 548252.i	− 1.59112i	0.605873	−	0.795561i	$-0.292824\pi$
$588$	0	0
$589$	122623.	0.353461
$590$	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
$595$	1.08939e6i	3.07717i
$596$	0	0
$597$	−174740.	−0.490278
$598$	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	$-0.334224\pi$
$601$	359449.	0.995151	0.497575	+	0.867421i	$0.334224\pi$
$602$	0	0
$603$	128584.i	0.353633i
$604$	0	0
$605$	−550253.	−1.50332
$606$	0	0
$607$	− 333341.i	− 0.904714i	−0.891837	−	0.452357i	$-0.850583\pi$
$607$	− 333341.i	− 0.904714i	0.891837	−	0.452357i	$-0.149417\pi$
$608$	0	0
$609$	145029.	0.391040
$610$	0	0
$611$	− 54455.2i	− 0.145867i
$612$	0	0
$613$	−357269.	−0.950768	−0.475384	−	0.879778i	$-0.657691\pi$
$613$	−357269.	−0.950768	−0.475384	+	0.879778i	$0.657691\pi$
$614$	0	0
$615$	− 6959.99i	− 0.0184017i
$616$	0	0
$617$	271334.	0.712744	0.356372	−	0.934344i	$-0.384014\pi$
$617$	271334.	0.712744	0.356372	+	0.934344i	$0.384014\pi$
$618$	0	0
$619$	− 383666.i	− 1.00132i	−0.865645	−	0.500659i	$-0.833091\pi$
$619$	− 383666.i	− 1.00132i	0.865645	−	0.500659i	$-0.166909\pi$
$620$	0	0
$621$	−104690.	−0.271471
$622$	0	0
$623$	− 261936.i	− 0.674869i
$624$	0	0
$625$	−218873.	−0.560315
$626$	0	0
$627$	5568.66i	0.0141650i
$628$	0	0
$629$	−120092.	−0.303538
$630$	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.i	0.769757i
$636$	0	0
$637$	−28359.6	−0.0698911
$638$	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.i	− 0.250290i	−0.992138	−	0.125145i	$-0.960060\pi$
$643$	− 103482.i	− 0.250290i	0.992138	−	0.125145i	$-0.0399396\pi$
$644$	0	0
$645$	464236.	1.11589
$646$	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.i	− 1.04819i
$652$	0	0
$653$	−227855.	−0.534358	−0.267179	−	0.963647i	$-0.586091\pi$
$653$	−227855.	−0.534358	−0.267179	+	0.963647i	$0.586091\pi$
$654$	0	0
$655$	380254.i	0.886322i
$656$	0	0
$657$	143835.	0.333222
$658$	0	0
$659$	− 182148.i	− 0.419423i	−0.977763	−	0.209712i	$-0.932747\pi$
$659$	− 182148.i	− 0.419423i	0.977763	−	0.209712i	$-0.0672525\pi$
$660$	0	0
$661$	−606405.	−1.38791	−0.693953	−	0.720020i	$-0.744132\pi$
$661$	−606405.	−1.38791	−0.693953	+	0.720020i	$0.744132\pi$
$662$	0	0
$663$	835999.i	1.90186i
$664$	0	0
$665$	136941.	0.309663
$666$	0	0
$667$	416415.i	0.935997i
$668$	0	0
$669$	303548.	0.678228
$670$	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.i	0.256050i
$676$	0	0
$677$	134998.	0.294543	0.147272	−	0.989096i	$-0.452951\pi$
$677$	134998.	0.294543	0.147272	+	0.989096i	$0.452951\pi$
$678$	0	0
$679$	− 290509.i	− 0.630115i
$680$	0	0
$681$	−312987.	−0.674888
$682$	0	0
$683$	566455.i	1.21429i	0.794589	+	0.607147i	$0.207686\pi$
$683$	566455.i	1.21429i	−0.794589	+	0.607147i	$0.792314\pi$
$684$	0	0
$685$	−1.31528e6	−2.80310
$686$	0	0
$687$	− 161507.i	− 0.342199i
$688$	0	0
$689$	−553382.	−1.16570
$690$	0	0
$691$	543725.i	1.13874i	0.822083	+	0.569368i	$0.192812\pi$
$691$	543725.i	1.13874i	−0.822083	+	0.569368i	$0.807188\pi$
$692$	0	0
$693$	20173.5	0.0420063
$694$	0	0
$695$	1.41647e6i	2.93251i
$696$	0	0
$697$	20029.9	0.0412300
$698$	0	0
$699$	134740.i	0.275767i
$700$	0	0
$701$	767758.	1.56239	0.781193	−	0.624289i	$-0.214611\pi$
$701$	767758.	1.56239	0.781193	+	0.624289i	$0.214611\pi$
$702$	0	0
$703$	15096.0i	0.0305458i
$704$	0	0
$705$	38306.6	0.0770719
$706$	0	0
$707$	771811.i	1.54409i
$708$	0	0
$709$	−560272.	−1.11457	−0.557284	−	0.830322i	$-0.688157\pi$
$709$	−560272.	−1.11457	−0.557284	+	0.830322i	$0.688157\pi$
$710$	0	0
$711$	178628.i	0.353353i
$712$	0	0
$713$	1.27548e6	2.50896
$714$	0	0
$715$	− 160724.i	− 0.314389i
$716$	0	0
$717$	−22994.1	−0.0447278
$718$	0	0
$719$	759774.i	1.46969i	0.678234	+	0.734846i	$0.262746\pi$
$719$	759774.i	1.46969i	−0.678234	+	0.734846i	$0.737254\pi$
$720$	0	0
$721$	−589347.	−1.13371
$722$	0	0
$723$	179221.i	0.342856i
$724$	0	0
$725$	464035.	0.882826
$726$	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
$731$	1.33601e6i	2.50020i
$732$	0	0
$733$	−600287.	−1.11725	−0.558626	−	0.829420i	$-0.688671\pi$
$733$	−600287.	−1.11725	−0.558626	+	0.829420i	$0.688671\pi$
$734$	0	0
$735$	− 19949.6i	− 0.0369284i
$736$	0	0
$737$	71142.9	0.130977
$738$	0	0
$739$	− 116503.i	− 0.213327i	−0.994295	−	0.106664i	$-0.965983\pi$
$739$	− 116503.i	− 0.213327i	0.994295	−	0.106664i	$-0.0340168\pi$
$740$	0	0
$741$	105088.	0.191389
$742$	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.i	− 0.332550i
$748$	0	0
$749$	710964.	1.26731
$750$	0	0
$751$	255543.i	0.453089i	0.974001	+	0.226545i	$0.0727429\pi$
$751$	255543.i	0.453089i	−0.974001	+	0.226545i	$0.927257\pi$
$752$	0	0
$753$	−112620.	−0.198621
$754$	0	0
$755$	− 1.62364e6i	− 2.84836i
$756$	0	0
$757$	157849.	0.275454	0.137727	−	0.990470i	$-0.456020\pi$
$757$	157849.	0.275454	0.137727	+	0.990470i	$0.456020\pi$
$758$	0	0
$759$	57923.0i	0.100547i
$760$	0	0
$761$	−436931.	−0.754474	−0.377237	−	0.926117i	$-0.623126\pi$
$761$	−436931.	−0.754474	−0.377237	+	0.926117i	$0.623126\pi$
$762$	0	0
$763$	14174.8i	0.0243483i
$764$	0	0
$765$	−588085.	−1.00489
$766$	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
$771$	− 54625.1i	− 0.0918932i
$772$	0	0
$773$	143407.	0.240000	0.120000	−	0.992774i	$-0.461711\pi$
$773$	143407.	0.240000	0.120000	+	0.992774i	$0.461711\pi$
$774$	0	0
$775$	− 1.42134e6i	− 2.36643i
$776$	0	0
$777$	54687.9	0.0905835
$778$	0	0
$779$	− 2517.83i	− 0.00414907i
$780$	0	0
$781$	62799.1	0.102956
$782$	0	0
$783$	78290.8i	0.127699i
$784$	0	0
$785$	292423.	0.474539
$786$	0	0
$787$	295695.i	0.477414i	0.971092	+	0.238707i	$0.0767235\pi$
$787$	295695.i	0.477414i	−0.971092	+	0.238707i	$0.923276\pi$
$788$	0	0
$789$	−205927.	−0.330795
$790$	0	0
$791$	237843.i	0.380134i
$792$	0	0
$793$	134833.	0.214412
$794$	0	0
$795$	− 389278.i	− 0.615922i
$796$	0	0
$797$	41548.0	0.0654084	0.0327042	−	0.999465i	$-0.489588\pi$
$797$	41548.0	0.0654084	0.0327042	+	0.999465i	$0.489588\pi$
$798$	0	0
$799$	110241.i	0.172683i
$800$	0	0
$801$	141400.	0.220387
$802$	0	0
$803$	− 79580.9i	− 0.123418i
$804$	0	0
$805$	1.42440e6	2.19806
$806$	0	0
$807$	419043.i	0.643445i
$808$	0	0
$809$	1.20959e6	1.84817	0.924083	−	0.382191i	$-0.124830\pi$
$809$	1.20959e6	1.84817	0.924083	+	0.382191i	$0.124830\pi$
$810$	0	0
$811$	− 212276.i	− 0.322744i	−0.986894	−	0.161372i	$-0.948408\pi$
$811$	− 212276.i	− 0.322744i	0.986894	−	0.161372i	$-0.0515920\pi$
$812$	0	0
$813$	259250.	0.392227
$814$	0	0
$815$	323755.i	0.487417i
$816$	0	0
$817$	167941.	0.251601
$818$	0	0
$819$	− 380699.i	− 0.567563i
$820$	0	0
$821$	1.12848e6	1.67420	0.837101	−	0.547048i	$-0.184248\pi$
$821$	1.12848e6	1.67420	0.837101	+	0.547048i	$0.184248\pi$
$822$	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.i	− 0.675409i	−0.941252	−	0.337705i	$-0.890349\pi$
$827$	− 461932.i	− 0.675409i	0.941252	−	0.337705i	$-0.109651\pi$
$828$	0	0
$829$	−1.04514e6	−1.52077	−0.760386	−	0.649472i	$-0.774990\pi$
$829$	−1.04514e6	−1.52077	−0.760386	+	0.649472i	$0.774990\pi$
$830$	0	0
$831$	− 69213.3i	− 0.100228i
$832$	0	0
$833$	57412.3	0.0827399
$834$	0	0
$835$	939459.i	1.34743i
$836$	0	0
$837$	239804.	0.342300
$838$	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
$843$	94276.5i	0.132662i
$844$	0	0
$845$	−1.94304e6	−2.72125
$846$	0	0
$847$	721122.i	1.00518i
$848$	0	0
$849$	−732299.	−1.01595
$850$	0	0
$851$	157022.i	0.216822i
$852$	0	0
$853$	445462.	0.612227	0.306113	−	0.951995i	$-0.400971\pi$
$853$	445462.	0.612227	0.306113	+	0.951995i	$0.400971\pi$
$854$	0	0
$855$	73924.2i	0.101124i
$856$	0	0
$857$	255454.	0.347817	0.173909	−	0.984762i	$-0.444360\pi$
$857$	255454.	0.347817	0.173909	+	0.984762i	$0.444360\pi$
$858$	0	0
$859$	− 549268.i	− 0.744386i	−0.928155	−	0.372193i	$-0.878606\pi$
$859$	− 549268.i	− 0.744386i	0.928155	−	0.372193i	$-0.121394\pi$
$860$	0	0
$861$	−9121.27	−0.0123041
$862$	0	0
$863$	217330.i	0.291808i	0.989299	+	0.145904i	$0.0466091\pi$
$863$	217330.i	0.291808i	−0.989299	+	0.145904i	$0.953391\pi$
$864$	0	0
$865$	−42154.6	−0.0563395
$866$	0	0
$867$	− 1.25844e6i	− 1.67415i
$868$	0	0
$869$	98830.9	0.130874
$870$	0	0
$871$	− 1.34256e6i	− 1.76969i
$872$	0	0
$873$	156825.	0.205772
$874$	0	0
$875$	− 394264.i	− 0.514957i
$876$	0	0
$877$	467000.	0.607180	0.303590	−	0.952803i	$-0.401815\pi$
$877$	467000.	0.607180	0.303590	+	0.952803i	$0.401815\pi$
$878$	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.i	− 0.811890i	−0.913898	−	0.405945i	$-0.866942\pi$
$883$	− 633021.i	− 0.811890i	0.913898	−	0.405945i	$-0.133058\pi$
$884$	0	0
$885$	963862.	1.23063
$886$	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
$891$	10890.2i	0.0137177i
$892$	0	0
$893$	13857.7	0.0173775
$894$	0	0
$895$	− 1.34301e6i	− 1.67662i
$896$	0	0
$897$	1.09308e6	1.35853
$898$	0	0
$899$	− 953842.i	− 1.18020i
$900$	0	0
$901$	1.12029e6	1.38000
$902$	0	0
$903$	− 608395.i	− 0.746123i
$904$	0	0
$905$	−1.14857e6	−1.40236
$906$	0	0
$907$	− 374365.i	− 0.455072i	−0.973770	−	0.227536i	$-0.926933\pi$
$907$	− 374365.i	− 0.455072i	0.973770	−	0.227536i	$-0.0730670\pi$
$908$	0	0
$909$	−416645.	−0.504241
$910$	0	0
$911$	− 790211.i	− 0.952152i	−0.879404	−	0.476076i	$-0.842059\pi$
$911$	− 790211.i	− 0.952152i	0.879404	−	0.476076i	$-0.157941\pi$
$912$	0	0
$913$	−102670.	−0.123169
$914$	0	0
$915$	94848.3i	0.113289i
$916$	0	0
$917$	498334.	0.592628
$918$	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
$923$	− 1.18510e6i	− 1.39108i
$924$	0	0
$925$	174979.	0.204505
$926$	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
$931$	− 7216.92i	− 0.00832631i
$932$	0	0
$933$	−597221.	−0.686075
$934$	0	0
$935$	325375.i	0.372187i
$936$	0	0
$937$	−801226.	−0.912591	−0.456295	−	0.889828i	$-0.650824\pi$
$937$	−801226.	−0.912591	−0.456295	+	0.889828i	$0.650824\pi$
$938$	0	0
$939$	904040.i	1.02531i
$940$	0	0
$941$	504479.	0.569724	0.284862	−	0.958569i	$-0.408052\pi$
$941$	504479.	0.569724	0.284862	+	0.958569i	$0.408052\pi$
$942$	0	0
$943$	− 26189.4i	− 0.0294512i
$944$	0	0
$945$	267804.	0.299884
$946$	0	0
$947$	− 170424.i	− 0.190034i	−0.995476	−	0.0950169i	$-0.969709\pi$
$947$	− 170424.i	− 0.190034i	0.995476	−	0.0950169i	$-0.0302905\pi$
$948$	0	0
$949$	−1.50179e6	−1.66755
$950$	0	0
$951$	− 800835.i	− 0.885486i
$952$	0	0
$953$	−352354.	−0.387965	−0.193983	−	0.981005i	$-0.562141\pi$
$953$	−352354.	−0.387965	−0.193983	+	0.981005i	$0.562141\pi$
$954$	0	0
$955$	− 604616.i	− 0.662938i
$956$	0	0
$957$	43316.7	0.0472967
$958$	0	0
$959$	1.72372e6i	1.87425i
$960$	0	0
$961$	−1.99809e6	−2.16356
$962$	0	0
$963$	383798.i	0.413857i
$964$	0	0
$965$	−1.89727e6	−2.03740
$966$	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.i	− 0.567408i	−0.958912	−	0.283704i	$-0.908437\pi$
$971$	− 534975.i	− 0.567408i	0.958912	−	0.283704i	$-0.0915633\pi$
$972$	0	0
$973$	1.85633e6	1.96078
$974$	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
$979$	− 78233.9i	− 0.0816262i
$980$	0	0
$981$	−7651.96	−0.00795123
$982$	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
$987$	− 50202.0i	− 0.0515331i
$988$	0	0
$989$	1.74685e6	1.78593
$990$	0	0
$991$	564204.i	0.574498i	0.957856	+	0.287249i	$0.0927407\pi$
$991$	564204.i	0.574498i	−0.957856	+	0.287249i	$0.907259\pi$
$992$	0	0
$993$	−761276.	−0.772047
$994$	0	0
$995$	1.28343e6i	1.29636i
$996$	0	0
$997$	1.31568e6	1.32361	0.661803	−	0.749678i	$-0.269792\pi$
$997$	1.31568e6	1.32361	0.661803	+	0.749678i	$0.269792\pi$
$998$	0	0
$999$	29522.0i	0.0295812i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.g.a.127.2	✓	16	1.1	even	1	trivial
384.5.g.a.127.10	yes	16	4.3	odd	2	inner
384.5.g.b.127.7	yes	16	8.3	odd	2
384.5.g.b.127.15	yes	16	8.5	even	2
768.5.b.h.127.10		16	16.11	odd	4
768.5.b.h.127.15		16	16.13	even	4
768.5.b.i.127.2		16	16.5	even	4
768.5.b.i.127.7		16	16.3	odd	4
1152.5.g.c.127.3		16	24.11	even	2
1152.5.g.c.127.4		16	24.5	odd	2
1152.5.g.f.127.13		16	12.11	even	2
1152.5.g.f.127.14		16	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -38.1647 q^{5} +50.0160i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -38.1647 q^{5} +50.0160i q^{7} -27.0000 q^{9} +14.9385i q^{11} +281.909 q^{13} +198.310i q^{15} -570.708 q^{17} +71.7400i q^{19} +259.891 q^{21} +746.211i q^{23} +831.546 q^{25} +140.296i q^{27} +558.040 q^{29} -1709.27i q^{31} +77.6229 q^{33} -1908.85i q^{35} +210.427 q^{37} -1464.84i q^{39} -35.0966 q^{41} -2340.97i q^{43} +1030.45 q^{45} -193.166i q^{47} -100.598 q^{49} +2965.49i q^{51} -1962.98 q^{53} -570.125i q^{55} +372.772 q^{57} -4860.39i q^{59} +478.284 q^{61} -1350.43i q^{63} -10759.0 q^{65} -4762.37i q^{67} +3877.42 q^{69} -4203.83i q^{71} -5327.22 q^{73} -4320.84i q^{75} -747.166 q^{77} -6615.84i q^{79} +729.000 q^{81} +6872.82i q^{83} +21780.9 q^{85} -2899.66i q^{87} -5237.05 q^{89} +14100.0i q^{91} -8881.65 q^{93} -2737.93i q^{95} -5808.32 q^{97} -403.341i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{5} - 432 q^{9}+O(q^{10}) \) 16 * q - 96 * q^5 - 432 * q^9	\( 16 q - 96 q^{5} - 432 q^{9} - 480 q^{13} - 480 q^{17} + 2672 q^{25} + 3360 q^{29} - 1120 q^{37} + 1440 q^{41} + 2592 q^{45} - 2480 q^{49} - 3552 q^{53} - 7488 q^{57} - 18272 q^{61} - 1344 q^{65} - 8480 q^{73} + 17280 q^{77} + 11664 q^{81} + 29888 q^{85} + 18720 q^{89} - 28800 q^{93} + 13088 q^{97}+O(q^{100}) \) 16 * q - 96 * q^5 - 432 * q^9 - 480 * q^13 - 480 * q^17 + 2672 * q^25 + 3360 * q^29 - 1120 * q^37 + 1440 * q^41 + 2592 * q^45 - 2480 * q^49 - 3552 * q^53 - 7488 * q^57 - 18272 * q^61 - 1344 * q^65 - 8480 * q^73 + 17280 * q^77 + 11664 * q^81 + 29888 * q^85 + 18720 * q^89 - 28800 * q^93 + 13088 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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