LMFDB - Embedded newform 252.9.d.b.181.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(181,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.181");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.d (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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 2160x^{4} + 976392x^{2} + 85162752 $ x^6 + 2160x^4 + 976392x^2 + 85162752
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.5
Root			$-21.7042i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.9.d.b.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+685.916i q^{5} +(-1845.45 - 1535.94i) q^{7} +O(q^{10})$	$q+685.916i q^{5} +(-1845.45 - 1535.94i) q^{7} -24517.4 q^{11} -275.445i q^{13} +154429. i q^{17} -11190.3i q^{19} -285585. q^{23} -79856.0 q^{25} +266410. q^{29} -1.47857e6i q^{31} +(1.05352e6 - 1.26583e6i) q^{35} -2.49179e6 q^{37} +708210. i q^{41} -1.28062e6 q^{43} -1.67233e6i q^{47} +(1.04660e6 + 5.66900e6i) q^{49} +1.96046e6 q^{53} -1.68169e7i q^{55} +1.26883e7i q^{59} -4.91081e6i q^{61} +188932. q^{65} +2.67889e7 q^{67} +5.66931e6 q^{71} +4.14867e7i q^{73} +(4.52457e7 + 3.76572e7i) q^{77} +2.85253e7 q^{79} -6.35086e7i q^{83} -1.05925e8 q^{85} -1.16381e7i q^{89} +(-423066. + 508322. i) q^{91} +7.67560e6 q^{95} -8.14742e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2166 q^{7}+O(q^{10}) $ 6 * q + 2166 * q^7	$ 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95}+O(q^{100}) $ 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$5$			685.916i			1.09747i	0.835998	+	0.548733i	$0.184890\pi$
$5$			685.916i			1.09747i	−0.835998	+	0.548733i	$0.815110\pi$
$6$		0			0
$7$	−1845.45	−	1535.94i	−0.768619	−	0.639707i
$7$	−1845.45	−	1535.94i	−0.768619	−	0.639707i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−24517.4			−1.67457			−0.837286	−	0.546766i	$-0.815859\pi$
$11$	−24517.4			−1.67457			−0.837286	+	0.546766i	$0.815859\pi$
$12$		0			0
$13$		−	275.445i		−	0.00964410i	−0.999988	−	0.00482205i	$-0.998465\pi$
$13$		−	275.445i		−	0.00964410i	0.999988	−	0.00482205i	$-0.00153491\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			154429.i			1.84898i	0.381201	+	0.924492i	$0.375510\pi$
$17$			154429.i			1.84898i	−0.381201	+	0.924492i	$0.624490\pi$
$18$		0			0
$19$		−	11190.3i		−	0.0858671i	−0.999078	−	0.0429336i	$-0.986330\pi$
$19$		−	11190.3i		−	0.0858671i	0.999078	−	0.0429336i	$-0.0136704\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−285585.			−1.02053			−0.510263	−	0.860018i	$-0.670452\pi$
$23$	−285585.			−1.02053			−0.510263	+	0.860018i	$0.670452\pi$
$24$		0			0
$25$	−79856.0			−0.204431
$26$		0			0
$27$		0			0
$28$		0			0
$29$	266410.			0.376668			0.188334	−	0.982105i	$-0.439691\pi$
$29$	266410.			0.376668			0.188334	+	0.982105i	$0.439691\pi$
$30$		0			0
$31$		−	1.47857e6i		−	1.60102i	−0.599320	−	0.800510i	$-0.704562\pi$
$31$		−	1.47857e6i		−	1.60102i	0.599320	−	0.800510i	$-0.295438\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.05352e6	−	1.26583e6i	0.702056	−	0.843533i
$36$		0			0
$37$	−2.49179e6			−1.32955			−0.664774	−	0.747044i	$-0.731472\pi$
$37$	−2.49179e6			−1.32955			−0.664774	+	0.747044i	$0.731472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			708210.i			0.250626i	0.992117	+	0.125313i	$0.0399936\pi$
$41$			708210.i			0.250626i	−0.992117	+	0.125313i	$0.960006\pi$
$42$		0			0
$43$	−1.28062e6			−0.374582			−0.187291	−	0.982304i	$-0.559971\pi$
$43$	−1.28062e6			−0.374582			−0.187291	+	0.982304i	$0.559971\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.67233e6i		−	0.342713i	−0.985209	−	0.171356i	$-0.945185\pi$
$47$		−	1.67233e6i		−	0.342713i	0.985209	−	0.171356i	$-0.0548150\pi$
$48$		0			0
$49$	1.04660e6	+	5.66900e6i	0.181550	+	0.983382i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.96046e6			0.248459			0.124230	−	0.992253i	$-0.460354\pi$
$53$	1.96046e6			0.248459			0.124230	+	0.992253i	$0.460354\pi$
$54$		0			0
$55$		−	1.68169e7i		−	1.83778i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.26883e7i			1.04712i	0.851989	+	0.523559i	$0.175396\pi$
$59$			1.26883e7i			1.04712i	−0.851989	+	0.523559i	$0.824604\pi$
$60$		0			0
$61$		−	4.91081e6i		−	0.354678i	−0.984150	−	0.177339i	$-0.943251\pi$
$61$		−	4.91081e6i		−	0.354678i	0.984150	−	0.177339i	$-0.0567488\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	188932.			0.0105841
$66$		0			0
$67$	2.67889e7			1.32940			0.664701	−	0.747110i	$-0.268559\pi$
$67$	2.67889e7			1.32940			0.664701	+	0.747110i	$0.268559\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.66931e6			0.223098			0.111549	−	0.993759i	$-0.464419\pi$
$71$	5.66931e6			0.223098			0.111549	+	0.993759i	$0.464419\pi$
$72$		0			0
$73$			4.14867e7i			1.46089i	0.682972	+	0.730444i	$0.260687\pi$
$73$			4.14867e7i			1.46089i	−0.682972	+	0.730444i	$0.739313\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.52457e7	+	3.76572e7i	1.28711	+	1.07123i
$78$		0			0
$79$	2.85253e7			0.732356			0.366178	−	0.930545i	$-0.380666\pi$
$79$	2.85253e7			0.732356			0.366178	+	0.930545i	$0.380666\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.35086e7i		−	1.33820i	−0.743173	−	0.669099i	$-0.766680\pi$
$83$		−	6.35086e7i		−	1.33820i	0.743173	−	0.669099i	$-0.233320\pi$
$84$		0			0
$85$	−1.05925e8			−2.02920
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.16381e7i		−	0.185490i	−0.995690	−	0.0927451i	$-0.970436\pi$
$89$		−	1.16381e7i		−	0.185490i	0.995690	−	0.0927451i	$-0.0295642\pi$
$90$		0			0
$91$	−423066.	+	508322.i	−0.00616940	+	0.00741264i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.67560e6			0.0942362
$96$		0			0
$97$		−	8.14742e6i		−	0.0920308i	−0.998941	−	0.0460154i	$-0.985348\pi$
$97$		−	8.14742e6i		−	0.0920308i	0.998941	−	0.0460154i	$-0.0146523\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1.58323e8i		−	1.52145i	−0.649072	−	0.760727i	$-0.724843\pi$
$101$		−	1.58323e8i		−	1.52145i	0.649072	−	0.760727i	$-0.275157\pi$
$102$		0			0
$103$		−	1.24520e8i		−	1.10634i	−0.833067	−	0.553172i	$-0.813417\pi$
$103$		−	1.24520e8i		−	1.10634i	0.833067	−	0.553172i	$-0.186583\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.52196e8			−1.16109			−0.580547	−	0.814227i	$-0.697161\pi$
$107$	−1.52196e8			−1.16109			−0.580547	+	0.814227i	$0.697161\pi$
$108$		0			0
$109$	2.67210e8			1.89298			0.946492	−	0.322726i	$-0.104599\pi$
$109$	2.67210e8			1.89298			0.946492	+	0.322726i	$0.104599\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.04288e8			0.639619			0.319809	−	0.947482i	$-0.396381\pi$
$113$	1.04288e8			0.639619			0.319809	+	0.947482i	$0.396381\pi$
$114$		0			0
$115$		−	1.95887e8i		−	1.11999i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.37193e8	−	2.84992e8i	1.18281	−	1.42116i
$120$		0			0
$121$	3.86744e8			1.80419
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.13161e8i			0.873109i
$126$		0			0
$127$	−4.31517e8			−1.65876			−0.829380	−	0.558685i	$-0.811306\pi$
$127$	−4.31517e8			−1.65876			−0.829380	+	0.558685i	$0.811306\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	3.89610e8i		−	1.32296i	−0.749965	−	0.661478i	$-0.769930\pi$
$131$		−	3.89610e8i		−	1.32296i	0.749965	−	0.661478i	$-0.230070\pi$
$132$		0			0
$133$	−1.71876e7	+	2.06512e7i	−0.0549298	+	0.0659991i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.43787e8			1.25977			0.629886	−	0.776688i	$-0.283102\pi$
$137$	4.43787e8			1.25977			0.629886	+	0.776688i	$0.283102\pi$
$138$		0			0
$139$			5.79636e8i			1.55273i	0.630283	+	0.776366i	$0.282939\pi$
$139$			5.79636e8i			1.55273i	−0.630283	+	0.776366i	$0.717061\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			6.75320e6i			0.0161497i
$144$		0			0
$145$			1.82735e8i			0.413380i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.21249e8			−0.245999			−0.122999	−	0.992407i	$-0.539251\pi$
$149$	−1.21249e8			−0.245999			−0.122999	+	0.992407i	$0.539251\pi$
$150$		0			0
$151$	1.88599e8			0.362770			0.181385	−	0.983412i	$-0.441942\pi$
$151$	1.88599e8			0.362770			0.181385	+	0.983412i	$0.441942\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.01418e9			1.75706
$156$		0			0
$157$		−	7.62607e8i		−	1.25517i	−0.778548	−	0.627584i	$-0.784044\pi$
$157$		−	7.62607e8i		−	1.25517i	0.778548	−	0.627584i	$-0.215956\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.27034e8	+	4.38641e8i	0.784396	+	0.652838i
$162$		0			0
$163$	6.46851e8			0.916334			0.458167	−	0.888866i	$-0.348506\pi$
$163$	6.46851e8			0.916334			0.458167	+	0.888866i	$0.348506\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.45334e8i		−	0.701126i	−0.936539	−	0.350563i	$-0.885990\pi$
$167$		−	5.45334e8i		−	0.701126i	0.936539	−	0.350563i	$-0.114010\pi$
$168$		0			0
$169$	8.15655e8			0.999907
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	6.04799e8i		−	0.675191i	−0.941291	−	0.337596i	$-0.890386\pi$
$173$		−	6.04799e8i		−	0.675191i	0.941291	−	0.337596i	$-0.109614\pi$
$174$		0			0
$175$	1.47371e8	+	1.22654e8i	0.157130	+	0.130776i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.07880e8			0.299895			0.149947	−	0.988694i	$-0.452090\pi$
$179$	3.07880e8			0.299895			0.149947	+	0.988694i	$0.452090\pi$
$180$		0			0
$181$		−	1.25107e9i		−	1.16565i	−0.812598	−	0.582824i	$-0.801948\pi$
$181$		−	1.25107e9i		−	1.16565i	0.812598	−	0.582824i	$-0.198052\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.70916e9i		−	1.45913i
$186$		0			0
$187$		−	3.78620e9i		−	3.09626i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.69283e8			0.502894			0.251447	−	0.967871i	$-0.419094\pi$
$191$	6.69283e8			0.502894			0.251447	+	0.967871i	$0.419094\pi$
$192$		0			0
$193$	−1.47941e9			−1.06625			−0.533125	−	0.846036i	$-0.678982\pi$
$193$	−1.47941e9			−1.06625			−0.533125	+	0.846036i	$0.678982\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.47444e9			0.978954			0.489477	−	0.872016i	$-0.337188\pi$
$197$	1.47444e9			0.978954			0.489477	+	0.872016i	$0.337188\pi$
$198$		0			0
$199$			2.19395e9i			1.39899i	0.714637	+	0.699495i	$0.246592\pi$
$199$			2.19395e9i			1.39899i	−0.714637	+	0.699495i	$0.753408\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.91647e8	−	4.09189e8i	−0.289514	−	0.240957i
$204$		0			0
$205$	−4.85773e8			−0.275054
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.74357e8i			0.143791i
$210$		0			0
$211$	1.88972e9			0.953382			0.476691	−	0.879071i	$-0.341836\pi$
$211$	1.88972e9			0.953382			0.476691	+	0.879071i	$0.341836\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.78399e8i		−	0.411091i
$216$		0			0
$217$	−2.27100e9	+	2.72864e9i	−1.02418	+	1.23057i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.25367e7			0.0178318
$222$		0			0
$223$			1.22281e9i			0.494470i	0.968956	+	0.247235i	$0.0795220\pi$
$223$			1.22281e9i			0.494470i	−0.968956	+	0.247235i	$0.920478\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3.45132e8i		−	0.129982i	−0.997886	−	0.0649908i	$-0.979298\pi$
$227$		−	3.45132e8i		−	0.129982i	0.997886	−	0.0649908i	$-0.0207018\pi$
$228$		0			0
$229$		−	7.64566e8i		−	0.278018i	−0.990291	−	0.139009i	$-0.955608\pi$
$229$		−	7.64566e8i		−	0.278018i	0.990291	−	0.139009i	$-0.0443917\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.10990e9			1.73376			0.866879	−	0.498519i	$-0.166123\pi$
$233$	5.10990e9			1.73376			0.866879	+	0.498519i	$0.166123\pi$
$234$		0			0
$235$	1.14708e9			0.376116
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.77466e9			−0.850391			−0.425196	−	0.905101i	$-0.639795\pi$
$239$	−2.77466e9			−0.850391			−0.425196	+	0.905101i	$0.639795\pi$
$240$		0			0
$241$		−	3.28161e9i		−	0.972790i	−0.873739	−	0.486395i	$-0.838312\pi$
$241$		−	3.28161e9i		−	0.972790i	0.873739	−	0.486395i	$-0.161688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.88846e9	+	7.17881e8i	−1.07923	+	0.199245i
$246$		0			0
$247$	−3.08231e6			−0.000828111
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2.86848e9i			0.722697i	0.932431	+	0.361349i	$0.117684\pi$
$251$			2.86848e9i			0.722697i	−0.932431	+	0.361349i	$0.882316\pi$
$252$		0			0
$253$	7.00180e9			1.70894
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	3.79617e9i		−	0.870189i	−0.900385	−	0.435095i	$-0.856715\pi$
$257$		−	3.79617e9i		−	0.870189i	0.900385	−	0.435095i	$-0.143285\pi$
$258$		0			0
$259$	4.59848e9	+	3.82723e9i	1.02192	+	0.850521i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.93571e9			−0.404592			−0.202296	−	0.979324i	$-0.564840\pi$
$263$	−1.93571e9			−0.404592			−0.202296	+	0.979324i	$0.564840\pi$
$264$		0			0
$265$			1.34471e9i			0.272676i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.02542e9i			0.577800i	0.957359	+	0.288900i	$0.0932894\pi$
$269$			3.02542e9i			0.577800i	−0.957359	+	0.288900i	$0.906711\pi$
$270$		0			0
$271$			3.52174e8i			0.0652950i	0.999467	+	0.0326475i	$0.0103939\pi$
$271$			3.52174e8i			0.0652950i	−0.999467	+	0.0326475i	$0.989606\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.95786e9			0.342335
$276$		0			0
$277$	−6.88783e8			−0.116994			−0.0584969	−	0.998288i	$-0.518631\pi$
$277$	−6.88783e8			−0.116994			−0.0584969	+	0.998288i	$0.518631\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.08996e9			−1.29754			−0.648771	−	0.760984i	$-0.724717\pi$
$281$	−8.08996e9			−1.29754			−0.648771	+	0.760984i	$0.724717\pi$
$282$		0			0
$283$			3.53180e9i			0.550617i	0.961356	+	0.275309i	$0.0887801\pi$
$283$			3.53180e9i			0.550617i	−0.961356	+	0.275309i	$0.911220\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.08777e9	−	1.30697e9i	0.160327	−	0.192636i
$288$		0			0
$289$	−1.68726e10			−2.41874
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.15872e9i			1.10701i	0.832846	+	0.553505i	$0.186710\pi$
$293$			8.15872e9i			1.10701i	−0.832846	+	0.553505i	$0.813290\pi$
$294$		0			0
$295$	−8.70312e9			−1.14918
$296$		0			0
$297$		0			0
$298$		0			0
$299$			7.86631e7i			0.00984206i
$300$		0			0
$301$	2.36333e9	+	1.96695e9i	0.287911	+	0.239623i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.36840e9			0.389247
$306$		0			0
$307$			3.48446e9i			0.392267i	0.980577	+	0.196134i	$0.0628387\pi$
$307$			3.48446e9i			0.392267i	−0.980577	+	0.196134i	$0.937161\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.27017e9i		−	0.884042i	−0.897005	−	0.442021i	$-0.854262\pi$
$311$		−	8.27017e9i		−	0.884042i	0.897005	−	0.442021i	$-0.145738\pi$
$312$		0			0
$313$		−	1.11984e10i		−	1.16675i	−0.812202	−	0.583376i	$-0.801732\pi$
$313$		−	1.11984e10i		−	1.16675i	0.812202	−	0.583376i	$-0.198268\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.71651e10			−1.69984			−0.849921	−	0.526911i	$-0.823350\pi$
$317$	−1.71651e10			−1.69984			−0.849921	+	0.526911i	$0.823350\pi$
$318$		0			0
$319$	−6.53168e9			−0.630757
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.72810e9			0.158767
$324$		0			0
$325$			2.19960e7i			0.00197156i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.56859e9	+	3.08621e9i	−0.219236	+	0.263416i
$330$		0			0
$331$	4.14281e9			0.345131			0.172565	−	0.984998i	$-0.444794\pi$
$331$	4.14281e9			0.345131			0.172565	+	0.984998i	$0.444794\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.83750e10i			1.45897i
$336$		0			0
$337$	−1.40450e10			−1.08894			−0.544469	−	0.838781i	$-0.683269\pi$
$337$	−1.40450e10			−1.08894			−0.544469	+	0.838781i	$0.683269\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			3.62508e10i			2.68102i
$342$		0			0
$343$	6.77577e9	−	1.20694e10i	0.489533	−	0.871985i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.93568e9			−0.616325			−0.308163	−	0.951334i	$-0.599714\pi$
$347$	−8.93568e9			−0.616325			−0.308163	+	0.951334i	$0.599714\pi$
$348$		0			0
$349$			7.14294e9i			0.481477i	0.970590	+	0.240738i	$0.0773896\pi$
$349$			7.14294e9i			0.481477i	−0.970590	+	0.240738i	$0.922610\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1.26158e9i		−	0.0812485i	−0.999174	−	0.0406243i	$-0.987065\pi$
$353$		−	1.26158e9i		−	0.0812485i	0.999174	−	0.0406243i	$-0.0129347\pi$
$354$		0			0
$355$			3.88867e9i			0.244843i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.23232e9			−0.0741902			−0.0370951	−	0.999312i	$-0.511810\pi$
$359$	−1.23232e9			−0.0741902			−0.0370951	+	0.999312i	$0.511810\pi$
$360$		0			0
$361$	1.68583e10			0.992627
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.84564e10			−1.60328
$366$		0			0
$367$			1.17035e10i			0.645137i	0.946546	+	0.322569i	$0.104546\pi$
$367$			1.17035e10i			0.645137i	−0.946546	+	0.322569i	$0.895454\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.61795e9	−	3.01115e9i	−0.190971	−	0.158941i
$372$		0			0
$373$	1.27148e10			0.656863			0.328431	−	0.944528i	$-0.393480\pi$
$373$	1.27148e10			0.656863			0.328431	+	0.944528i	$0.393480\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.33814e7i		−	0.00363262i
$378$		0			0
$379$	−3.32897e10			−1.61344			−0.806720	−	0.590934i	$-0.798759\pi$
$379$	−3.32897e10			−1.61344			−0.806720	+	0.590934i	$0.798759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	2.38937e10i		−	1.11042i	−0.831710	−	0.555210i	$-0.812638\pi$
$383$		−	2.38937e10i		−	1.11042i	0.831710	−	0.555210i	$-0.187362\pi$
$384$		0			0
$385$	−2.58297e10	+	3.10348e10i	−1.17564	+	1.41256i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.14233e10			0.935597			0.467799	−	0.883835i	$-0.345047\pi$
$389$	2.14233e10			0.935597			0.467799	+	0.883835i	$0.345047\pi$
$390$		0			0
$391$		−	4.41026e10i		−	1.88694i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.95660e10i			0.803736i
$396$		0			0
$397$			3.68154e10i			1.48206i	0.671469	+	0.741032i	$0.265664\pi$
$397$			3.68154e10i			1.48206i	−0.671469	+	0.741032i	$0.734336\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.69191e10			−1.42782			−0.713910	−	0.700237i	$-0.753078\pi$
$401$	−3.69191e10			−1.42782			−0.713910	+	0.700237i	$0.753078\pi$
$402$		0			0
$403$	−4.07266e8			−0.0154404
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.10922e10			2.22642
$408$		0			0
$409$			3.78556e10i			1.35281i	0.736530	+	0.676405i	$0.236463\pi$
$409$			3.78556e10i			1.35281i	−0.736530	+	0.676405i	$0.763537\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.94884e10	−	2.34157e10i	0.669849	−	0.804835i
$414$		0			0
$415$	4.35616e10			1.46863
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.32942e9i		−	0.0431326i	−0.999767	−	0.0215663i	$-0.993135\pi$
$419$		−	1.32942e9i		−	0.0431326i	0.999767	−	0.0215663i	$-0.00686529\pi$
$420$		0			0
$421$	3.33376e10			1.06122			0.530611	−	0.847616i	$-0.321963\pi$
$421$	3.33376e10			1.06122			0.530611	+	0.847616i	$0.321963\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.23321e10i		−	0.377990i
$426$		0			0
$427$	−7.54269e9	+	9.06268e9i	−0.226890	+	0.272612i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.30517e10			−1.53741			−0.768707	−	0.639602i	$-0.779099\pi$
$431$	−5.30517e10			−1.53741			−0.768707	+	0.639602i	$0.779099\pi$
$432$		0			0
$433$		−	6.87668e10i		−	1.95626i	−0.207990	−	0.978131i	$-0.566692\pi$
$433$		−	6.87668e10i		−	1.95626i	0.207990	−	0.978131i	$-0.433308\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3.19578e9i			0.0876296i
$438$		0			0
$439$			6.06896e9i			0.163401i	0.996657	+	0.0817007i	$0.0260352\pi$
$439$			6.06896e9i			0.163401i	−0.996657	+	0.0817007i	$0.973965\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.05335e10			−0.533148			−0.266574	−	0.963814i	$-0.585892\pi$
$443$	−2.05335e10			−0.533148			−0.266574	+	0.963814i	$0.585892\pi$
$444$		0			0
$445$	7.98274e9			0.203569
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.27711e9			−0.203654			−0.101827	−	0.994802i	$-0.532469\pi$
$449$	−8.27711e9			−0.203654			−0.101827	+	0.994802i	$0.532469\pi$
$450$		0			0
$451$		−	1.73635e10i		−	0.419692i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.48666e8	−	2.90188e8i	−0.00813512	−	0.00677071i
$456$		0			0
$457$	−1.82904e10			−0.419332			−0.209666	−	0.977773i	$-0.567238\pi$
$457$	−1.82904e10			−0.419332			−0.209666	+	0.977773i	$0.567238\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.74969e10i		−	0.387398i	−0.981061	−	0.193699i	$-0.937952\pi$
$461$		−	1.74969e10i		−	0.387398i	0.981061	−	0.193699i	$-0.0620485\pi$
$462$		0			0
$463$	1.40067e10			0.304797			0.152399	−	0.988319i	$-0.451300\pi$
$463$	1.40067e10			0.304797			0.152399	+	0.988319i	$0.451300\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			6.77247e10i			1.42390i	0.702230	+	0.711950i	$0.252188\pi$
$467$			6.77247e10i			1.42390i	−0.702230	+	0.711950i	$0.747812\pi$
$468$		0			0
$469$	−4.94377e10	−	4.11461e10i	−1.02180	−	0.850427i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.13975e10			0.627264
$474$		0			0
$475$			8.93612e8i			0.0175539i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	7.00318e10i		−	1.33031i	−0.746705	−	0.665156i	$-0.768365\pi$
$479$		−	7.00318e10i		−	1.33031i	0.746705	−	0.665156i	$-0.231635\pi$
$480$		0			0
$481$			6.86351e8i			0.0128223i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.58845e9			0.101001
$486$		0			0
$487$	−3.05578e10			−0.543259			−0.271629	−	0.962402i	$-0.587562\pi$
$487$	−3.05578e10			−0.543259			−0.271629	+	0.962402i	$0.587562\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.30623e10			1.25709			0.628546	−	0.777772i	$-0.283650\pi$
$491$	7.30623e10			1.25709			0.628546	+	0.777772i	$0.283650\pi$
$492$		0			0
$493$			4.11414e10i			0.696453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.04624e10	−	8.70769e9i	−0.171478	−	0.142718i
$498$		0			0
$499$	−2.46330e10			−0.397297			−0.198649	−	0.980071i	$-0.563655\pi$
$499$	−2.46330e10			−0.397297			−0.198649	+	0.980071i	$0.563655\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.46256e10i		−	1.16578i	−0.812552	−	0.582889i	$-0.801922\pi$
$503$		−	7.46256e10i		−	1.16578i	0.812552	−	0.582889i	$-0.198078\pi$
$504$		0			0
$505$	1.08596e11			1.66974
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	2.44558e10i		−	0.364343i	−0.983267	−	0.182171i	$-0.941687\pi$
$509$		−	2.44558e10i		−	0.364343i	0.983267	−	0.182171i	$-0.0583126\pi$
$510$		0			0
$511$	6.37209e10	−	7.65617e10i	0.934540	−	1.12287i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.54103e10			1.21417
$516$		0			0
$517$			4.10012e10i			0.573897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.97227e10i		−	0.810566i	−0.914191	−	0.405283i	$-0.867173\pi$
$521$		−	5.97227e10i		−	0.810566i	0.914191	−	0.405283i	$-0.132827\pi$
$522$		0			0
$523$		−	5.39772e9i		−	0.0721446i	−0.999349	−	0.0360723i	$-0.988515\pi$
$523$		−	5.39772e9i		−	0.0721446i	0.999349	−	0.0360723i	$-0.0114847\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.28335e11			2.96026
$528$		0			0
$529$	3.24789e9			0.0414743
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.95073e8			0.00241707
$534$		0			0
$535$		−	1.04394e11i		−	1.27426i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.56599e10	−	1.38989e11i	−0.304019	−	1.64674i
$540$		0			0
$541$	−9.77217e9			−0.114078			−0.0570390	−	0.998372i	$-0.518166\pi$
$541$	−9.77217e9			−0.114078			−0.0570390	+	0.998372i	$0.518166\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.83284e11i			2.07749i
$546$		0			0
$547$	1.72001e10			0.192124			0.0960620	−	0.995375i	$-0.469375\pi$
$547$	1.72001e10			0.192124			0.0960620	+	0.995375i	$0.469375\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.98120e9i		−	0.0323434i
$552$		0			0
$553$	−5.26422e10	−	4.38131e10i	−0.562903	−	0.468493i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.51876e10			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$557$	7.51876e10			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$558$		0			0
$559$			3.52741e8i			0.00361251i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	5.53132e10i		−	0.550548i	−0.961366	−	0.275274i	$-0.911231\pi$
$563$		−	5.53132e10i		−	0.550548i	0.961366	−	0.275274i	$-0.0887685\pi$
$564$		0			0
$565$			7.15329e10i			0.701960i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.29842e11			−1.23870			−0.619352	−	0.785113i	$-0.712605\pi$
$569$	−1.29842e11			−1.23870			−0.619352	+	0.785113i	$0.712605\pi$
$570$		0			0
$571$	1.36941e11			1.28821			0.644107	−	0.764935i	$-0.277229\pi$
$571$	1.36941e11			1.28821			0.644107	+	0.764935i	$0.277229\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.28057e10			0.208628
$576$		0			0
$577$			5.86892e10i			0.529486i	0.964319	+	0.264743i	$0.0852871\pi$
$577$			5.86892e10i			0.529486i	−0.964319	+	0.264743i	$0.914713\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.75452e10	+	1.17202e11i	−0.856054	+	1.02856i
$582$		0			0
$583$	−4.80655e10			−0.416063
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	6.05036e10i		−	0.509600i	−0.966994	−	0.254800i	$-0.917990\pi$
$587$		−	6.05036e10i		−	0.509600i	0.966994	−	0.254800i	$-0.0820096\pi$
$588$		0			0
$589$	−1.65457e10			−0.137475
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.80905e9i			0.0631509i	0.999501	+	0.0315754i	$0.0100524\pi$
$593$			7.80905e9i			0.0631509i	−0.999501	+	0.0315754i	$0.989948\pi$
$594$		0			0
$595$	1.95480e11	+	1.62695e11i	1.55968	+	1.29809i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.07074e11			1.60849			0.804247	−	0.594296i	$-0.202569\pi$
$599$	2.07074e11			1.60849			0.804247	+	0.594296i	$0.202569\pi$
$600$		0			0
$601$		−	2.04173e11i		−	1.56495i	−0.622680	−	0.782476i	$-0.713956\pi$
$601$		−	2.04173e11i		−	1.56495i	0.622680	−	0.782476i	$-0.286044\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			2.65274e11i			1.98004i
$606$		0			0
$607$			7.16270e10i			0.527621i	0.964575	+	0.263810i	$0.0849793\pi$
$607$			7.16270e10i			0.527621i	−0.964575	+	0.263810i	$0.915021\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.60635e8			−0.00330516
$612$		0			0
$613$	−9.87464e10			−0.699325			−0.349663	−	0.936876i	$-0.613704\pi$
$613$	−9.87464e10			−0.699325			−0.349663	+	0.936876i	$0.613704\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.68662e11			−1.85381			−0.926906	−	0.375294i	$-0.877542\pi$
$617$	−2.68662e11			−1.85381			−0.926906	+	0.375294i	$0.877542\pi$
$618$		0			0
$619$		−	1.55484e11i		−	1.05907i	−0.848289	−	0.529533i	$-0.822367\pi$
$619$		−	1.55484e11i		−	1.05907i	0.848289	−	0.529533i	$-0.177633\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.78753e10	+	2.14775e10i	−0.118659	+	0.142571i
$624$		0			0
$625$	−1.77405e11			−1.16264
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.84804e11i		−	2.45831i
$630$		0			0
$631$	−1.48914e11			−0.939327			−0.469663	−	0.882846i	$-0.655625\pi$
$631$	−1.48914e11			−0.939327			−0.469663	+	0.882846i	$0.655625\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	2.95985e11i		−	1.82043i
$636$		0			0
$637$	1.56150e9	−	2.88281e8i	0.00948383	−	0.00175089i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.64534e10			0.156693			0.0783463	−	0.996926i	$-0.475036\pi$
$641$	2.64534e10			0.156693			0.0783463	+	0.996926i	$0.475036\pi$
$642$		0			0
$643$			1.31229e11i			0.767689i	0.923398	+	0.383845i	$0.125400\pi$
$643$			1.31229e11i			0.767689i	−0.923398	+	0.383845i	$0.874600\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5.77378e10i		−	0.329491i	−0.986336	−	0.164745i	$-0.947320\pi$
$647$		−	5.77378e10i		−	0.329491i	0.986336	−	0.164745i	$-0.0526802\pi$
$648$		0			0
$649$		−	3.11084e11i		−	1.75347i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.72475e10			0.314850			0.157425	−	0.987531i	$-0.449681\pi$
$653$	5.72475e10			0.314850			0.157425	+	0.987531i	$0.449681\pi$
$654$		0			0
$655$	2.67240e11			1.45190
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.42215e10			0.393539			0.196770	−	0.980450i	$-0.436955\pi$
$659$	7.42215e10			0.393539			0.196770	+	0.980450i	$0.436955\pi$
$660$		0			0
$661$			2.36706e11i			1.23995i	0.784623	+	0.619973i	$0.212856\pi$
$661$			2.36706e11i			1.23995i	−0.784623	+	0.619973i	$0.787144\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.41650e10	−	1.17892e10i	−0.0724317	−	0.0602836i
$666$		0			0
$667$	−7.60827e10			−0.384399
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.20400e11i			0.593933i
$672$		0			0
$673$	−3.95895e10			−0.192983			−0.0964917	−	0.995334i	$-0.530762\pi$
$673$	−3.95895e10			−0.192983			−0.0964917	+	0.995334i	$0.530762\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			9.91459e10i			0.471976i	0.971756	+	0.235988i	$0.0758326\pi$
$677$			9.91459e10i			0.471976i	−0.971756	+	0.235988i	$0.924167\pi$
$678$		0			0
$679$	−1.25139e10	+	1.50357e10i	−0.0588727	+	0.0707366i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.00192e10			0.0460417			0.0230208	−	0.999735i	$-0.492672\pi$
$683$	1.00192e10			0.0460417			0.0230208	+	0.999735i	$0.492672\pi$
$684$		0			0
$685$			3.04400e11i			1.38256i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	5.40001e8i		−	0.00239617i
$690$		0			0
$691$		−	9.19247e10i		−	0.403199i	−0.979468	−	0.201600i	$-0.935386\pi$
$691$		−	9.19247e10i		−	0.403199i	0.979468	−	0.201600i	$-0.0646140\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.97582e11			−1.70407
$696$		0			0
$697$	−1.09368e11			−0.463404
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.64528e10			−0.316608			−0.158304	−	0.987390i	$-0.550603\pi$
$701$	−7.64528e10			−0.316608			−0.158304	+	0.987390i	$0.550603\pi$
$702$		0			0
$703$			2.78838e10i			0.114164i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.43174e11	+	2.92178e11i	−0.973284	+	1.16942i
$708$		0			0
$709$	1.66811e11			0.660145			0.330072	−	0.943956i	$-0.392927\pi$
$709$	1.66811e11			0.660145			0.330072	+	0.943956i	$0.392927\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.22259e11i			1.63388i
$714$		0			0
$715$	−4.63213e9			−0.0177238
$716$		0			0
$717$		0			0
$718$		0			0
$719$			4.43477e11i			1.65942i	0.558198	+	0.829708i	$0.311493\pi$
$719$			4.43477e11i			1.65942i	−0.558198	+	0.829708i	$0.688507\pi$
$720$		0			0
$721$	−1.91255e11	+	2.29796e11i	−0.707736	+	0.850357i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.12744e10			−0.0770027
$726$		0			0
$727$			1.75400e11i			0.627902i	0.949439	+	0.313951i	$0.101653\pi$
$727$			1.75400e11i			0.627902i	−0.949439	+	0.313951i	$0.898347\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.97765e11i		−	0.692596i
$732$		0			0
$733$		−	4.65826e10i		−	0.161365i	−0.996740	−	0.0806823i	$-0.974290\pi$
$733$		−	4.65826e10i		−	0.161365i	0.996740	−	0.0806823i	$-0.0257099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.56795e11			−2.22618
$738$		0			0
$739$	3.54893e11			1.18992			0.594962	−	0.803754i	$-0.297167\pi$
$739$	3.54893e11			1.18992			0.594962	+	0.803754i	$0.297167\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.71203e11			0.561767			0.280884	−	0.959742i	$-0.409372\pi$
$743$	1.71203e11			0.561767			0.280884	+	0.959742i	$0.409372\pi$
$744$		0			0
$745$		−	8.31665e10i		−	0.269975i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.80870e11	+	2.33763e11i	0.892439	+	0.742760i
$750$		0			0
$751$	−1.53650e11			−0.483029			−0.241514	−	0.970397i	$-0.577644\pi$
$751$	−1.53650e11			−0.483029			−0.241514	+	0.970397i	$0.577644\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.29363e11i			0.398128i
$756$		0			0
$757$	−7.15192e10			−0.217791			−0.108895	−	0.994053i	$-0.534731\pi$
$757$	−7.15192e10			−0.217791			−0.108895	+	0.994053i	$0.534731\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3.28287e10i		−	0.0978848i	−0.998802	−	0.0489424i	$-0.984415\pi$
$761$		−	3.28287e10i		−	0.0978848i	0.998802	−	0.0489424i	$-0.0155851\pi$
$762$		0			0
$763$	−4.93124e11	−	4.10418e11i	−1.45498	−	1.21096i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.49494e9			0.0100985
$768$		0			0
$769$			3.53063e11i			1.00960i	0.863238	+	0.504798i	$0.168433\pi$
$769$			3.53063e11i			1.00960i	−0.863238	+	0.504798i	$0.831567\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			5.74666e11i			1.60952i	0.593597	+	0.804762i	$0.297707\pi$
$773$			5.74666e11i			1.60952i	−0.593597	+	0.804762i	$0.702293\pi$
$774$		0			0
$775$			1.18073e11i			0.327299i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.92507e9			0.0215206
$780$		0			0
$781$	−1.38997e11			−0.373594
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.23085e11			1.37751
$786$		0			0
$787$		−	4.04284e11i		−	1.05387i	−0.849905	−	0.526936i	$-0.823341\pi$
$787$		−	4.04284e11i		−	1.05387i	0.849905	−	0.526936i	$-0.176659\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.92459e11	−	1.60180e11i	−0.491623	−	0.409169i
$792$		0			0
$793$	−1.35266e9			−0.00342055
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.43229e11i		−	0.354975i	−0.984123	−	0.177488i	$-0.943203\pi$
$797$		−	1.43229e11i		−	0.354975i	0.984123	−	0.177488i	$-0.0567969\pi$
$798$		0			0
$799$	2.58256e11			0.633671
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.01714e12i		−	2.44636i
$804$		0			0
$805$	−3.00871e11	+	3.61501e11i	−0.716467	+	0.860848i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.28105e10			0.0765983			0.0382991	−	0.999266i	$-0.487806\pi$
$809$	3.28105e10			0.0765983			0.0382991	+	0.999266i	$0.487806\pi$
$810$		0			0
$811$			6.27249e11i			1.44996i	0.688769	+	0.724981i	$0.258151\pi$
$811$			6.27249e11i			1.44996i	−0.688769	+	0.724981i	$0.741849\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			4.43686e11i			1.00565i
$816$		0			0
$817$			1.43305e10i			0.0321643i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.85180e11			−1.50811			−0.754054	−	0.656813i	$-0.771904\pi$
$821$	−6.85180e11			−1.50811			−0.754054	+	0.656813i	$0.771904\pi$
$822$		0			0
$823$	−4.40814e11			−0.960851			−0.480425	−	0.877036i	$-0.659518\pi$
$823$	−4.40814e11			−0.960851			−0.480425	+	0.877036i	$0.659518\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.10595e11			0.664007			0.332004	−	0.943278i	$-0.392275\pi$
$827$	3.10595e11			0.664007			0.332004	+	0.943278i	$0.392275\pi$
$828$		0			0
$829$		−	8.03539e11i		−	1.70133i	−0.525707	−	0.850666i	$-0.676199\pi$
$829$		−	8.03539e11i		−	1.70133i	0.525707	−	0.850666i	$-0.323801\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.75458e11	+	1.61626e11i	−1.81826	+	0.335684i
$834$		0			0
$835$	3.74053e11			0.769462
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	2.60770e11i		−	0.526271i	−0.964759	−	0.263136i	$-0.915243\pi$
$839$		−	2.60770e11i		−	0.526271i	0.964759	−	0.263136i	$-0.0847567\pi$
$840$		0			0
$841$	−4.29272e11			−0.858121
$842$		0			0
$843$		0			0
$844$		0			0
$845$			5.59471e11i			1.09736i
$846$		0			0
$847$	−7.13718e11	−	5.94014e11i	−1.38673	−	1.15415i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.11618e11			1.35684
$852$		0			0
$853$		−	2.90865e11i		−	0.549409i	−0.961529	−	0.274704i	$-0.911420\pi$
$853$		−	2.90865e11i		−	0.549409i	0.961529	−	0.274704i	$-0.0885800\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.89300e10i		−	0.127787i	−0.997957	−	0.0638933i	$-0.979648\pi$
$857$		−	6.89300e10i		−	0.127787i	0.997957	−	0.0638933i	$-0.0203517\pi$
$858$		0			0
$859$		−	2.65953e11i		−	0.488464i	−0.969717	−	0.244232i	$-0.921464\pi$
$859$		−	2.65953e11i		−	0.488464i	0.969717	−	0.244232i	$-0.0785358\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.83047e11			−0.870855			−0.435427	−	0.900224i	$-0.643403\pi$
$863$	−4.83047e11			−0.870855			−0.435427	+	0.900224i	$0.643403\pi$
$864$		0			0
$865$	4.14842e11			0.740999
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.99367e11			−1.22638
$870$		0			0
$871$		−	7.37888e9i		−	0.0128209i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.27402e11	−	3.93380e11i	0.558534	−	0.671088i
$876$		0			0
$877$	2.77389e11			0.468911			0.234456	−	0.972127i	$-0.424669\pi$
$877$	2.77389e11			0.468911			0.234456	+	0.972127i	$0.424669\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	2.40890e11i		−	0.399866i	−0.979810	−	0.199933i	$-0.935928\pi$
$881$		−	2.40890e11i		−	0.399866i	0.979810	−	0.199933i	$-0.0640725\pi$
$882$		0			0
$883$	7.97420e11			1.31173			0.655865	−	0.754879i	$-0.272304\pi$
$883$	7.97420e11			1.31173			0.655865	+	0.754879i	$0.272304\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.06442e10i			0.0333507i	0.999861	+	0.0166753i	$0.00530817\pi$
$887$			2.06442e10i			0.0333507i	−0.999861	+	0.0166753i	$0.994692\pi$
$888$		0			0
$889$	7.96346e11	+	6.62783e11i	1.27495	+	1.06112i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.87139e10			−0.0294278
$894$		0			0
$895$			2.11180e11i			0.329124i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	3.93907e11i		−	0.603052i
$900$		0			0
$901$			3.02753e11i			0.459398i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.58130e11			1.27926
$906$		0			0
$907$	8.06876e11			1.19228			0.596139	−	0.802881i	$-0.296701\pi$
$907$	8.06876e11			1.19228			0.596139	+	0.802881i	$0.296701\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.47775e11			0.650109			0.325054	−	0.945695i	$-0.394617\pi$
$911$	4.47775e11			0.650109			0.325054	+	0.945695i	$0.394617\pi$
$912$		0			0
$913$			1.55707e12i			2.24091i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.98416e11	+	7.19008e11i	−0.846304	+	1.01685i
$918$		0			0
$919$	−5.61848e11			−0.787693			−0.393846	−	0.919176i	$-0.628856\pi$
$919$	−5.61848e11			−0.787693			−0.393846	+	0.919176i	$0.628856\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.56158e9i		−	0.00215158i
$924$		0			0
$925$	1.98984e11			0.271802
$926$		0			0
$927$		0			0
$928$		0			0
$929$			5.30983e11i			0.712882i	0.934318	+	0.356441i	$0.116010\pi$
$929$			5.30983e11i			0.712882i	−0.934318	+	0.356441i	$0.883990\pi$
$930$		0			0
$931$	6.34377e10	−	1.17118e10i	0.0844401	−	0.0155892i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.59701e12			3.39803
$936$		0			0
$937$		−	1.19762e12i		−	1.55368i	−0.629699	−	0.776839i	$-0.716822\pi$
$937$		−	1.19762e12i		−	1.55368i	0.629699	−	0.776839i	$-0.283178\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.36372e12i		−	1.73927i	−0.493695	−	0.869635i	$-0.664354\pi$
$941$		−	1.36372e12i		−	1.73927i	0.493695	−	0.869635i	$-0.335646\pi$
$942$		0			0
$943$		−	2.02254e11i		−	0.255771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.95853e11			−1.11388			−0.556938	−	0.830554i	$-0.688024\pi$
$947$	−8.95853e11			−1.11388			−0.556938	+	0.830554i	$0.688024\pi$
$948$		0			0
$949$	1.14273e10			0.0140890
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.10721e12			1.34232			0.671161	−	0.741311i	$-0.265796\pi$
$953$	1.10721e12			1.34232			0.671161	+	0.741311i	$0.265796\pi$
$954$		0			0
$955$			4.59072e11i			0.551909i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.18988e11	−	6.81628e11i	−0.968285	−	0.805885i
$960$		0			0
$961$	−1.33329e12			−1.56326
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.01475e12i		−	1.17017i
$966$		0			0
$967$	4.92082e11			0.562771			0.281385	−	0.959595i	$-0.409206\pi$
$967$	4.92082e11			0.562771			0.281385	+	0.959595i	$0.409206\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.93428e11i		−	0.555069i	−0.960716	−	0.277535i	$-0.910483\pi$
$971$		−	4.93428e11i		−	0.555069i	0.960716	−	0.277535i	$-0.0895173\pi$
$972$		0			0
$973$	8.90284e11	−	1.06969e12i	0.993293	−	1.19346i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.13514e12			1.24587			0.622934	−	0.782274i	$-0.285941\pi$
$977$	1.13514e12			1.24587			0.622934	+	0.782274i	$0.285941\pi$
$978$		0			0
$979$			2.85335e11i			0.310616i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.18371e12i			1.26774i	0.773438	+	0.633872i	$0.218535\pi$
$983$			1.18371e12i			1.26774i	−0.773438	+	0.633872i	$0.781465\pi$
$984$		0			0
$985$			1.01134e12i			1.07437i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.65727e11			0.382271
$990$		0			0
$991$	9.78462e11			1.01449			0.507247	−	0.861801i	$-0.330663\pi$
$991$	9.78462e11			1.01449			0.507247	+	0.861801i	$0.330663\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.50487e12			−1.53534
$996$		0			0
$997$		−	3.05757e11i		−	0.309454i	−0.987957	−	0.154727i	$-0.950550\pi$
$997$		−	3.05757e11i		−	0.309454i	0.987957	−	0.154727i	$-0.0494498\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.d.b.181.5		6
3.2	odd	2		28.9.b.a.13.5	yes	6
7.6	odd	2	inner	252.9.d.b.181.2		6
12.11	even	2		112.9.c.d.97.2		6
21.2	odd	6		196.9.h.b.129.2		12
21.5	even	6		196.9.h.b.129.5		12
21.11	odd	6		196.9.h.b.117.5		12
21.17	even	6		196.9.h.b.117.2		12
21.20	even	2		28.9.b.a.13.2	✓	6
84.83	odd	2		112.9.c.d.97.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.9.b.a.13.2	✓	6	21.20	even	2
28.9.b.a.13.5	yes	6	3.2	odd	2
112.9.c.d.97.2		6	12.11	even	2
112.9.c.d.97.5		6	84.83	odd	2
196.9.h.b.117.2		12	21.17	even	6
196.9.h.b.117.5		12	21.11	odd	6
196.9.h.b.129.2		12	21.2	odd	6
196.9.h.b.129.5		12	21.5	even	6
252.9.d.b.181.2		6	7.6	odd	2	inner
252.9.d.b.181.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+685.916i q^{5} +(-1845.45 - 1535.94i) q^{7} +O(q^{10})\)	\(q+685.916i q^{5} +(-1845.45 - 1535.94i) q^{7} -24517.4 q^{11} -275.445i q^{13} +154429. i q^{17} -11190.3i q^{19} -285585. q^{23} -79856.0 q^{25} +266410. q^{29} -1.47857e6i q^{31} +(1.05352e6 - 1.26583e6i) q^{35} -2.49179e6 q^{37} +708210. i q^{41} -1.28062e6 q^{43} -1.67233e6i q^{47} +(1.04660e6 + 5.66900e6i) q^{49} +1.96046e6 q^{53} -1.68169e7i q^{55} +1.26883e7i q^{59} -4.91081e6i q^{61} +188932. q^{65} +2.67889e7 q^{67} +5.66931e6 q^{71} +4.14867e7i q^{73} +(4.52457e7 + 3.76572e7i) q^{77} +2.85253e7 q^{79} -6.35086e7i q^{83} -1.05925e8 q^{85} -1.16381e7i q^{89} +(-423066. + 508322. i) q^{91} +7.67560e6 q^{95} -8.14742e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2166 q^{7}+O(q^{10}) \) 6 * q + 2166 * q^7	\( 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95}+O(q^{100}) \) 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Introduction

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