LMFDB - Embedded newform 1152.5.g.c.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(127,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, 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("1152.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.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:	$119.082197473$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Root			$0.500000 + 2.68460i$ of defining polynomial
Character	$\chi$	$=$	1152.127
Dual form			1152.5.g.c.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-38.1647 q^{5} +50.0160i q^{7} +O(q^{10})$	$q-38.1647 q^{5} +50.0160i q^{7} +14.9385i q^{11} -281.909 q^{13} +570.708 q^{17} -71.7400i q^{19} -746.211i q^{23} +831.546 q^{25} +558.040 q^{29} -1709.27i q^{31} -1908.85i q^{35} -210.427 q^{37} +35.0966 q^{41} +2340.97i q^{43} +193.166i q^{47} -100.598 q^{49} -1962.98 q^{53} -570.125i q^{55} -4860.39i q^{59} -478.284 q^{61} +10759.0 q^{65} +4762.37i q^{67} +4203.83i q^{71} -5327.22 q^{73} -747.166 q^{77} -6615.84i q^{79} +6872.82i q^{83} -21780.9 q^{85} +5237.05 q^{89} -14100.0i q^{91} +2737.93i q^{95} -5808.32 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{5}+O(q^{10}) $ 16 * q - 96 * q^5	$ 16 q - 96 q^{5} + 480 q^{13} + 480 q^{17} + 2672 q^{25} + 3360 q^{29} + 1120 q^{37} - 1440 q^{41} - 2480 q^{49} - 3552 q^{53} + 18272 q^{61} + 1344 q^{65} - 8480 q^{73} + 17280 q^{77} - 29888 q^{85} - 18720 q^{89} + 13088 q^{97}+O(q^{100}) $ 16 * q - 96 * q^5 + 480 * q^13 + 480 * q^17 + 2672 * q^25 + 3360 * q^29 + 1120 * q^37 - 1440 * q^41 - 2480 * q^49 - 3552 * q^53 + 18272 * q^61 + 1344 * q^65 - 8480 * q^73 + 17280 * q^77 - 29888 * q^85 - 18720 * q^89 + 13088 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	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
$9$	0	0
$10$	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	$-0.813984\pi$
$13$	−281.909	−1.66810	−0.834051	+	0.551687i	$0.813984\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	570.708	1.97477	0.987385	−	0.158340i	$-0.0506143\pi$
$17$	570.708	1.97477	0.987385	+	0.158340i	$0.0506143\pi$
$18$	0	0
$19$	− 71.7400i	− 0.198726i	−0.995051	−	0.0993628i	$-0.968320\pi$
$19$	− 71.7400i	− 0.198726i	0.995051	−	0.0993628i	$-0.0316804\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 746.211i	− 1.41061i	−0.708906	−	0.705303i	$-0.750811\pi$
$23$	− 746.211i	− 1.41061i	0.708906	−	0.705303i	$-0.249189\pi$
$24$	0	0
$25$	831.546	1.33047
$26$	0	0
$27$	0	0
$28$	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
$33$	0	0
$34$	0	0
$35$	− 1908.85i	− 1.55824i
$36$	0	0
$37$	−210.427	−0.153708	−0.0768541	−	0.997042i	$-0.524488\pi$
$37$	−210.427	−0.153708	−0.0768541	+	0.997042i	$0.524488\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	2340.97i	1.26607i	0.774122	+	0.633036i	$0.218191\pi$
$43$	2340.97i	1.26607i	−0.774122	+	0.633036i	$0.781809\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	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
$57$	0	0
$58$	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	$-0.520471\pi$
$61$	−478.284	−0.128536	−0.0642682	+	0.997933i	$0.520471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10759.0	2.54651
$66$	0	0
$67$	4762.37i	1.06090i	0.847717	+	0.530449i	$0.177977\pi$
$67$	4762.37i	1.06090i	−0.847717	+	0.530449i	$0.822023\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4203.83i	0.833928i	0.908923	+	0.416964i	$0.136906\pi$
$71$	4203.83i	0.833928i	−0.908923	+	0.416964i	$0.863094\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	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
$87$	0	0
$88$	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
$91$	− 14100.0i	− 1.70269i
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	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
$105$	0	0
$106$	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	$-0.503797\pi$
$109$	−283.406	−0.0238537	−0.0119269	+	0.999929i	$0.503797\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4755.34	−0.372413	−0.186206	−	0.982511i	$-0.559619\pi$
$113$	−4755.34	−0.372413	−0.186206	+	0.982511i	$0.559619\pi$
$114$	0	0
$115$	28478.9i	2.15341i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	28544.5i	2.01571i
$120$	0	0
$121$	14417.8	0.984758
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	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
$135$	0	0
$136$	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
$139$	37114.7i	1.92095i	0.278361	+	0.960477i	$0.410209\pi$
$139$	37114.7i	1.92095i	−0.278361	+	0.960477i	$0.589791\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4211.31i	− 0.205942i
$144$	0	0
$145$	−21297.4	−1.01296
$146$	0	0
$147$	0	0
$148$	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
$153$	0	0
$154$	0	0
$155$	65233.9i	2.71525i
$156$	0	0
$157$	7662.12	0.310849	0.155425	−	0.987848i	$-0.450325\pi$
$157$	7662.12	0.310849	0.155425	+	0.987848i	$0.450325\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	37322.5	1.43985
$162$	0	0
$163$	8483.09i	0.319285i	0.987175	+	0.159643i	$0.0510342\pi$
$163$	8483.09i	0.319285i	−0.987175	+	0.159643i	$0.948966\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24615.9i	0.882638i	0.897350	+	0.441319i	$0.145489\pi$
$167$	24615.9i	0.882638i	−0.897350	+	0.441319i	$0.854511\pi$
$168$	0	0
$169$	50911.9	1.78257
$170$	0	0
$171$	0	0
$172$	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
$177$	0	0
$178$	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	$-0.651904\pi$
$181$	−30095.0	−0.918623	−0.459311	+	0.888275i	$0.651904\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8030.87	0.234649
$186$	0	0
$187$	8525.55i	0.243803i
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	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
$201$	0	0
$202$	0	0
$203$	27910.9i	0.677301i
$204$	0	0
$205$	−1339.45	−0.0318727
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1071.69	0.0245345
$210$	0	0
$211$	− 51764.4i	− 1.16270i	−0.813655	−	0.581348i	$-0.802525\pi$
$211$	− 51764.4i	− 1.16270i	0.813655	−	0.581348i	$-0.197475\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 89342.3i	− 1.93277i
$216$	0	0
$217$	85491.0	1.81552
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	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	$-0.595771\pi$
$229$	−31082.1	−0.592706	−0.296353	+	0.955078i	$0.595771\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	− 7372.12i	− 0.133492i
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	3839.31	0.0639618
$246$	0	0
$247$	20224.2i	0.331495i
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	−10512.6	−0.159164	−0.0795819	−	0.996828i	$-0.525359\pi$
$257$	−10512.6	−0.159164	−0.0795819	+	0.996828i	$0.525359\pi$
$258$	0	0
$259$	− 10524.7i	− 0.156895i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	39630.6i	0.572954i	0.958087	+	0.286477i	$0.0924841\pi$
$263$	39630.6i	0.572954i	−0.958087	+	0.286477i	$0.907516\pi$
$264$	0	0
$265$	74916.6	1.06681
$266$	0	0
$267$	0	0
$268$	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
$273$	0	0
$274$	0	0
$275$	12422.1i	0.164259i
$276$	0	0
$277$	−13320.1	−0.173599	−0.0867996	−	0.996226i	$-0.527664\pi$
$277$	−13320.1	−0.173599	−0.0867996	+	0.996226i	$0.527664\pi$
$278$	0	0
$279$	0	0
$280$	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.i	1.75968i	0.475270	+	0.879840i	$0.342350\pi$
$283$	140931.i	1.75968i	−0.475270	+	0.879840i	$0.657650\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1755.39i	0.0213113i
$288$	0	0
$289$	242187.	2.89971
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	210364.i	2.35304i
$300$	0	0
$301$	−117086.	−1.29232
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18253.6	0.196222
$306$	0	0
$307$	− 27092.9i	− 0.287461i	−0.989617	−	0.143730i	$-0.954090\pi$
$307$	− 27092.9i	− 0.287461i	0.989617	−	0.143730i	$-0.0459098\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	114935.i	1.18832i	0.804348	+	0.594158i	$0.202515\pi$
$311$	114935.i	1.18832i	−0.804348	+	0.594158i	$0.797485\pi$
$312$	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
$315$	0	0
$316$	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
$321$	0	0
$322$	0	0
$323$	− 40942.6i	− 0.392437i
$324$	0	0
$325$	−234420.	−2.21937
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9661.38	−0.0892580
$330$	0	0
$331$	146508.i	1.33723i	0.743611	+	0.668613i	$0.233111\pi$
$331$	146508.i	1.33723i	−0.743611	+	0.668613i	$0.766889\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	25534.0	0.219589
$342$	0	0
$343$	115057.i	0.977967i
$344$	0	0
$345$	0	0
$346$	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	$-0.441774\pi$
$349$	44311.9	0.363806	0.181903	+	0.983316i	$0.441774\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	155580.	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$353$	155580.	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$354$	0	0
$355$	− 160438.i	− 1.27307i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	208069.i	1.61443i	0.590259	+	0.807214i	$0.299026\pi$
$359$	208069.i	1.61443i	−0.590259	+	0.807214i	$0.700974\pi$
$360$	0	0
$361$	125174.	0.960508
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	0	0
$371$	− 98180.3i	− 0.713307i
$372$	0	0
$373$	214254.	1.53997	0.769985	−	0.638062i	$-0.220264\pi$
$373$	214254.	1.53997	0.769985	+	0.638062i	$0.220264\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−157317.	−1.10686
$378$	0	0
$379$	135220.i	0.941376i	0.882300	+	0.470688i	$0.155994\pi$
$379$	135220.i	0.941376i	−0.882300	+	0.470688i	$0.844006\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	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
$393$	0	0
$394$	0	0
$395$	252491.i	1.61828i
$396$	0	0
$397$	−222509.	−1.41178	−0.705890	−	0.708322i	$-0.749453\pi$
$397$	−222509.	−1.41178	−0.705890	+	0.708322i	$0.749453\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−119674.	−0.744237	−0.372119	−	0.928185i	$-0.621369\pi$
$401$	−119674.	−0.744237	−0.372119	+	0.928185i	$0.621369\pi$
$402$	0	0
$403$	481860.i	2.96695i
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	243097.	1.42521
$414$	0	0
$415$	− 262299.i	− 1.52300i
$416$	0	0
$417$	0	0
$418$	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	$-0.212876\pi$
$421$	278121.	1.56917	0.784584	+	0.620022i	$0.212876\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	474570.	2.62738
$426$	0	0
$427$	− 23921.8i	− 0.131201i
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	−73745.4	−0.365799	−0.182899	−	0.983132i	$-0.558548\pi$
$449$	−73745.4	−0.365799	−0.182899	+	0.983132i	$0.558548\pi$
$450$	0	0
$451$	524.291i	0.00257762i
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	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
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	−34970.6	−0.156308
$474$	0	0
$475$	− 59655.0i	− 0.264399i
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	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
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−210259.	−0.851219
$498$	0	0
$499$	− 99656.8i	− 0.400227i	−0.979773	−	0.200113i	$-0.935869\pi$
$499$	− 99656.8i	− 0.400227i	0.979773	−	0.200113i	$-0.0641310\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 166780.i	− 0.659186i	−0.944123	−	0.329593i	$-0.893088\pi$
$503$	− 166780.i	− 0.659186i	0.944123	−	0.329593i	$-0.106912\pi$
$504$	0	0
$505$	−588931.	−2.30931
$506$	0	0
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	− 449702.i	− 1.69555i
$516$	0	0
$517$	−2885.61	−0.0107959
$518$	0	0
$519$	0	0
$520$	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.i	− 1.31973i	−0.751382	−	0.659867i	$-0.770612\pi$
$523$	− 360986.i	− 1.31973i	0.751382	−	0.659867i	$-0.229388\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 975497.i	− 3.51240i
$528$	0	0
$529$	−276989.	−0.989809
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9894.05	−0.0348273
$534$	0	0
$535$	542501.i	1.89537i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1502.79i	− 0.00517275i
$540$	0	0
$541$	−212815.	−0.727123	−0.363561	−	0.931570i	$-0.618439\pi$
$541$	−212815.	−0.727123	−0.363561	+	0.931570i	$0.618439\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10816.1	0.0364148
$546$	0	0
$547$	− 84823.3i	− 0.283492i	−0.989903	−	0.141746i	$-0.954728\pi$
$547$	− 84823.3i	− 0.283492i	0.989903	−	0.141746i	$-0.0452716\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 40033.7i	− 0.131863i
$552$	0	0
$553$	330898.	1.08204
$554$	0	0
$555$	0	0
$556$	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
$561$	0	0
$562$	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
$567$	0	0
$568$	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.i	− 0.704155i	−0.935971	−	0.352078i	$-0.885475\pi$
$571$	− 229583.i	− 0.704155i	0.935971	−	0.352078i	$-0.114525\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	−343751.	−1.01834
$582$	0	0
$583$	− 29324.0i	− 0.0862754i
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	−509890.	−1.45000	−0.724998	−	0.688751i	$-0.758159\pi$
$593$	−509890.	−1.45000	−0.724998	+	0.688751i	$0.758159\pi$
$594$	0	0
$595$	− 1.08939e6i	− 3.07717i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	163569.i	0.455877i	0.973676	+	0.227939i	$0.0731985\pi$
$599$	163569.i	0.455877i	−0.973676	+	0.227939i	$0.926801\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	0	0
$611$	− 54455.2i	− 0.145867i
$612$	0	0
$613$	357269.	0.950768	0.475384	−	0.879778i	$-0.342309\pi$
$613$	357269.	0.950768	0.475384	+	0.879778i	$0.342309\pi$
$614$	0	0
$615$	0	0
$616$	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.i	1.00132i	0.865645	+	0.500659i	$0.166909\pi$
$619$	383666.i	1.00132i	−0.865645	+	0.500659i	$0.833091\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	261936.i	0.674869i
$624$	0	0
$625$	−218873.	−0.560315
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	310385.i	0.769757i
$636$	0	0
$637$	28359.6	0.0698911
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−46510.1	−0.113196	−0.0565980	−	0.998397i	$-0.518025\pi$
$641$	−46510.1	−0.113196	−0.0565980	+	0.998397i	$0.518025\pi$
$642$	0	0
$643$	103482.i	0.250290i	0.992138	+	0.125145i	$0.0399396\pi$
$643$	103482.i	0.250290i	−0.992138	+	0.125145i	$0.960060\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	512475.i	1.22423i	0.790767	+	0.612117i	$0.209682\pi$
$647$	512475.i	1.22423i	−0.790767	+	0.612117i	$0.790318\pi$
$648$	0	0
$649$	72607.1	0.172381
$650$	0	0
$651$	0	0
$652$	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
$657$	0	0
$658$	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	$-0.255868\pi$
$661$	606405.	1.38791	0.693953	+	0.720020i	$0.255868\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−136941.	−0.309663
$666$	0	0
$667$	− 416415.i	− 0.935997i
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	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
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	553382.	1.16570
$690$	0	0
$691$	− 543725.i	− 1.13874i	−0.822083	−	0.569368i	$-0.807188\pi$
$691$	− 543725.i	− 1.13874i	0.822083	−	0.569368i	$-0.192812\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.41647e6i	− 2.93251i
$696$	0	0
$697$	20029.9	0.0412300
$698$	0	0
$699$	0	0
$700$	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
$705$	0	0
$706$	0	0
$707$	771811.i	1.54409i
$708$	0	0
$709$	560272.	1.11457	0.557284	−	0.830322i	$-0.311843\pi$
$709$	560272.	1.11457	0.557284	+	0.830322i	$0.311843\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.27548e6	−2.50896
$714$	0	0
$715$	160724.i	0.314389i
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	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
$729$	0	0
$730$	0	0
$731$	1.33601e6i	2.50020i
$732$	0	0
$733$	600287.	1.11725	0.558626	−	0.829420i	$-0.311329\pi$
$733$	600287.	1.11725	0.558626	+	0.829420i	$0.311329\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−71142.9	−0.130977
$738$	0	0
$739$	116503.i	0.213327i	0.994295	+	0.106664i	$0.0340168\pi$
$739$	116503.i	0.213327i	−0.994295	+	0.106664i	$0.965983\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	362499.i	0.656642i	0.944566	+	0.328321i	$0.106483\pi$
$743$	362499.i	0.656642i	−0.944566	+	0.328321i	$0.893517\pi$
$744$	0	0
$745$	−584462.	−1.05304
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	0	0
$755$	− 1.62364e6i	− 2.84836i
$756$	0	0
$757$	−157849.	−0.275454	−0.137727	−	0.990470i	$-0.543980\pi$
$757$	−157849.	−0.275454	−0.137727	+	0.990470i	$0.543980\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	− 14174.8i	− 0.0243483i
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	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
$777$	0	0
$778$	0	0
$779$	− 2517.83i	− 0.00414907i
$780$	0	0
$781$	−62799.1	−0.102956
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−292423.	−0.474539
$786$	0	0
$787$	− 295695.i	− 0.477414i	−0.971092	−	0.238707i	$-0.923276\pi$
$787$	− 295695.i	− 0.477414i	0.971092	−	0.238707i	$-0.0767235\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 237843.i	− 0.380134i
$792$	0	0
$793$	134833.	0.214412
$794$	0	0
$795$	0	0
$796$	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
$801$	0	0
$802$	0	0
$803$	− 79580.9i	− 0.123418i
$804$	0	0
$805$	−1.42440e6	−2.19806
$806$	0	0
$807$	0	0
$808$	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.i	0.322744i	0.986894	+	0.161372i	$0.0515920\pi$
$811$	212276.i	0.322744i	−0.986894	+	0.161372i	$0.948408\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 323755.i	− 0.487417i
$816$	0	0
$817$	167941.	0.251601
$818$	0	0
$819$	0	0
$820$	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
$825$	0	0
$826$	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	$-0.225010\pi$
$829$	1.04514e6	1.52077	0.760386	+	0.649472i	$0.225010\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−57412.3	−0.0827399
$834$	0	0
$835$	− 939459.i	− 1.34743i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	765515.i	1.08750i	0.839247	+	0.543751i	$0.182996\pi$
$839$	765515.i	1.08750i	−0.839247	+	0.543751i	$0.817004\pi$
$840$	0	0
$841$	−395873.	−0.559711
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.94304e6	−2.72125
$846$	0	0
$847$	721122.i	1.00518i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	157022.i	0.216822i
$852$	0	0
$853$	−445462.	−0.612227	−0.306113	−	0.951995i	$-0.599029\pi$
$853$	−445462.	−0.612227	−0.306113	+	0.951995i	$0.599029\pi$
$854$	0	0
$855$	0	0
$856$	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.i	0.744386i	0.928155	+	0.372193i	$0.121394\pi$
$859$	549268.i	0.744386i	−0.928155	+	0.372193i	$0.878606\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	98830.9	0.130874
$870$	0	0
$871$	− 1.34256e6i	− 1.76969i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 394264.i	− 0.514957i
$876$	0	0
$877$	−467000.	−0.607180	−0.303590	−	0.952803i	$-0.598185\pi$
$877$	−467000.	−0.607180	−0.303590	+	0.952803i	$0.598185\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	916522.	1.18084	0.590420	−	0.807096i	$-0.298962\pi$
$881$	916522.	1.18084	0.590420	+	0.807096i	$0.298962\pi$
$882$	0	0
$883$	633021.i	0.811890i	0.913898	+	0.405945i	$0.133058\pi$
$883$	633021.i	0.811890i	−0.913898	+	0.405945i	$0.866942\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	666258.i	0.846828i	0.905936	+	0.423414i	$0.139168\pi$
$887$	666258.i	0.846828i	−0.905936	+	0.423414i	$0.860832\pi$
$888$	0	0
$889$	406769.	0.514688
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13857.7	0.0173775
$894$	0	0
$895$	− 1.34301e6i	− 1.67662i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 953842.i	− 1.18020i
$900$	0	0
$901$	−1.12029e6	−1.38000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.14857e6	1.40236
$906$	0	0
$907$	374365.i	0.455072i	0.973770	+	0.227536i	$0.0730670\pi$
$907$	374365.i	0.455072i	−0.973770	+	0.227536i	$0.926933\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	− 1.18510e6i	− 1.39108i
$924$	0	0
$925$	−174979.	−0.204505
$926$	0	0
$927$	0	0
$928$	0	0
$929$	645243.	0.747639	0.373820	−	0.927501i	$-0.378048\pi$
$929$	645243.	0.747639	0.373820	+	0.927501i	$0.378048\pi$
$930$	0	0
$931$	7216.92i	0.00832631i
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	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
$945$	0	0
$946$	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
$951$	0	0
$952$	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.i	0.662938i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.72372e6i	− 1.87425i
$960$	0	0
$961$	−1.99809e6	−2.16356
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	1.19772e6	1.25477	0.627386	−	0.778708i	$-0.284125\pi$
$977$	1.19772e6	1.25477	0.627386	+	0.778708i	$0.284125\pi$
$978$	0	0
$979$	78233.9i	0.0816262i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 149071.i	− 0.154272i	−0.997021	−	0.0771360i	$-0.975422\pi$
$983$	− 149071.i	− 0.154272i	0.997021	−	0.0771360i	$-0.0245776\pi$
$984$	0	0
$985$	665799.	0.686232
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	0	0
$995$	1.28343e6i	1.29636i
$996$	0	0
$997$	−1.31568e6	−1.32361	−0.661803	−	0.749678i	$-0.730208\pi$
$997$	−1.31568e6	−1.32361	−0.661803	+	0.749678i	$0.730208\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.g.a.127.2	✓	16	24.5	odd	2
384.5.g.a.127.10	yes	16	24.11	even	2
384.5.g.b.127.7	yes	16	12.11	even	2
384.5.g.b.127.15	yes	16	3.2	odd	2
768.5.b.h.127.10		16	48.35	even	4
768.5.b.h.127.15		16	48.5	odd	4
768.5.b.i.127.2		16	48.29	odd	4
768.5.b.i.127.7		16	48.11	even	4
1152.5.g.c.127.3		16	4.3	odd	2	inner
1152.5.g.c.127.4		16	1.1	even	1	trivial
1152.5.g.f.127.13		16	8.3	odd	2
1152.5.g.f.127.14		16	8.5	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-38.1647 q^{5} +50.0160i q^{7} +O(q^{10})\)	\(q-38.1647 q^{5} +50.0160i q^{7} +14.9385i q^{11} -281.909 q^{13} +570.708 q^{17} -71.7400i q^{19} -746.211i q^{23} +831.546 q^{25} +558.040 q^{29} -1709.27i q^{31} -1908.85i q^{35} -210.427 q^{37} +35.0966 q^{41} +2340.97i q^{43} +193.166i q^{47} -100.598 q^{49} -1962.98 q^{53} -570.125i q^{55} -4860.39i q^{59} -478.284 q^{61} +10759.0 q^{65} +4762.37i q^{67} +4203.83i q^{71} -5327.22 q^{73} -747.166 q^{77} -6615.84i q^{79} +6872.82i q^{83} -21780.9 q^{85} +5237.05 q^{89} -14100.0i q^{91} +2737.93i q^{95} -5808.32 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{5}+O(q^{10}) \) 16 * q - 96 * q^5	\( 16 q - 96 q^{5} + 480 q^{13} + 480 q^{17} + 2672 q^{25} + 3360 q^{29} + 1120 q^{37} - 1440 q^{41} - 2480 q^{49} - 3552 q^{53} + 18272 q^{61} + 1344 q^{65} - 8480 q^{73} + 17280 q^{77} - 29888 q^{85} - 18720 q^{89} + 13088 q^{97}+O(q^{100}) \) 16 * q - 96 * q^5 + 480 * q^13 + 480 * q^17 + 2672 * q^25 + 3360 * q^29 + 1120 * q^37 - 1440 * q^41 - 2480 * q^49 - 3552 * q^53 + 18272 * q^61 + 1344 * q^65 - 8480 * q^73 + 17280 * q^77 - 29888 * q^85 - 18720 * q^89 + 13088 * q^97

Introduction

L-functions

Modular forms

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