LMFDB - Embedded newform 252.8.f.a.125.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(125,252)]

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, 1, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("252.125");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.f (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:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 71480 x^{18} + 1912507236 x^{16} + 23093807115120 x^{14} + \cdots + 18\!\cdots\!76 $ x^20 + 71480x^18 + 1912507236x^16 + 23093807115120x^14 + 114952238574376692x^12 + 126766863857546639424x^10 + 56585472804958733913504x^8 + 11203603854275179795151616x^6 + 830466508333016623708408896x^4 + 3850857622616571255220794368*x^2 + 1834425505624671455246181376
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{52}\cdot 3^{47}\cdot 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			125.10
Root			$-21.9392i$ of defining polynomial
Character	$\chi$	$=$	252.125
Dual form			252.8.f.a.125.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.8524 q^{5} +(-396.183 + 816.445i) q^{7} +643.885i q^{11} -6871.76i q^{13} -18271.5 q^{17} +40411.7i q^{19} -105170. i q^{23} -77556.1 q^{25} +212794. i q^{29} -103411. i q^{31} +(9449.90 - 19474.2i) q^{35} +498893. q^{37} +799807. q^{41} -376424. q^{43} -898022. q^{47} +(-509622. - 646922. i) q^{49} -594558. i q^{53} -15358.2i q^{55} +1.39034e6 q^{59} +1.59586e6i q^{61} +163908. i q^{65} +1.30668e6 q^{67} -2.67924e6i q^{71} -5.32408e6i q^{73} +(-525697. - 255096. i) q^{77} -7.54010e6 q^{79} +4.29740e6 q^{83} +435819. q^{85} +8.40954e6 q^{89} +(5.61042e6 + 2.72247e6i) q^{91} -963916. i q^{95} -8.49132e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2468 q^{7} + 548780 q^{25} - 257344 q^{37} - 1589224 q^{43} - 291556 q^{49} - 11860256 q^{67} - 1991600 q^{79} + 11742576 q^{85} - 11268816 q^{91}+O(q^{100}) $ 20 * q - 2468 * q^7 + 548780 * q^25 - 257344 * q^37 - 1589224 * q^43 - 291556 * q^49 - 11860256 * q^67 - 1991600 * q^79 + 11742576 * q^85 - 11268816 * q^91

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
$5$	−23.8524			−0.0853370			−0.0426685	−	0.999089i	$-0.513586\pi$
$5$	−23.8524			−0.0853370			−0.0426685	+	0.999089i	$0.513586\pi$
$6$		0			0
$7$	−396.183	+	816.445i	−0.436568	+	0.899671i
$7$	−396.183	+	816.445i	−0.436568	+	0.899671i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			643.885i			0.145859i	0.997337	+	0.0729297i	$0.0232349\pi$
$11$			643.885i			0.145859i	−0.997337	+	0.0729297i	$0.976765\pi$
$12$		0			0
$13$		−	6871.76i		−	0.867494i	−0.901035	−	0.433747i	$-0.857191\pi$
$13$		−	6871.76i		−	0.867494i	0.901035	−	0.433747i	$-0.142809\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−18271.5			−0.901993			−0.450997	−	0.892526i	$-0.648931\pi$
$17$	−18271.5			−0.901993			−0.450997	+	0.892526i	$0.648931\pi$
$18$		0			0
$19$			40411.7i			1.35167i	0.737055	+	0.675833i	$0.236216\pi$
$19$			40411.7i			1.35167i	−0.737055	+	0.675833i	$0.763784\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	105170.i		−	1.80238i	−0.433426	−	0.901189i	$-0.642696\pi$
$23$		−	105170.i		−	1.80238i	0.433426	−	0.901189i	$-0.357304\pi$
$24$		0			0
$25$	−77556.1			−0.992718
$26$		0			0
$27$		0			0
$28$		0			0
$29$			212794.i			1.62019i	0.586299	+	0.810094i	$0.300584\pi$
$29$			212794.i			1.62019i	−0.586299	+	0.810094i	$0.699416\pi$
$30$		0			0
$31$		−	103411.i		−	0.623448i	−0.950173	−	0.311724i	$-0.899094\pi$
$31$		−	103411.i		−	0.623448i	0.950173	−	0.311724i	$-0.100906\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9449.90	−	19474.2i	0.0372554	−	0.0767752i
$36$		0			0
$37$	498893.			1.61920			0.809602	−	0.586980i	$-0.199683\pi$
$37$	498893.			1.61920			0.809602	+	0.586980i	$0.199683\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	799807.			1.81235			0.906174	−	0.422904i	$-0.138989\pi$
$41$	799807.			1.81235			0.906174	+	0.422904i	$0.138989\pi$
$42$		0			0
$43$	−376424.			−0.722001			−0.361000	−	0.932566i	$-0.617565\pi$
$43$	−376424.			−0.722001			−0.361000	+	0.932566i	$0.617565\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−898022.			−1.26167			−0.630833	−	0.775919i	$-0.717287\pi$
$47$	−898022.			−1.26167			−0.630833	+	0.775919i	$0.717287\pi$
$48$		0			0
$49$	−509622.	−	646922.i	−0.618816	−	0.785536i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	594558.i		−	0.548566i	−0.961649	−	0.274283i	$-0.911560\pi$
$53$		−	594558.i		−	0.548566i	0.961649	−	0.274283i	$-0.0884405\pi$
$54$		0			0
$55$		−	15358.2i		−	0.0124472i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.39034e6			0.881330			0.440665	−	0.897672i	$-0.354743\pi$
$59$	1.39034e6			0.881330			0.440665	+	0.897672i	$0.354743\pi$
$60$		0			0
$61$			1.59586e6i			0.900202i	0.892978	+	0.450101i	$0.148612\pi$
$61$			1.59586e6i			0.900202i	−0.892978	+	0.450101i	$0.851388\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			163908.i			0.0740293i
$66$		0			0
$67$	1.30668e6			0.530773			0.265387	−	0.964142i	$-0.414500\pi$
$67$	1.30668e6			0.530773			0.265387	+	0.964142i	$0.414500\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.67924e6i		−	0.888398i	−0.895928	−	0.444199i	$-0.853488\pi$
$71$		−	2.67924e6i		−	0.888398i	0.895928	−	0.444199i	$-0.146512\pi$
$72$		0			0
$73$		−	5.32408e6i		−	1.60182i	−0.598782	−	0.800912i	$-0.704348\pi$
$73$		−	5.32408e6i		−	1.60182i	0.598782	−	0.800912i	$-0.295652\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−525697.	−	255096.i	−0.131225	−	0.0636776i
$78$		0			0
$79$	−7.54010e6			−1.72061			−0.860305	−	0.509780i	$-0.829727\pi$
$79$	−7.54010e6			−1.72061			−0.860305	+	0.509780i	$0.829727\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.29740e6			0.824958			0.412479	−	0.910967i	$-0.364663\pi$
$83$	4.29740e6			0.824958			0.412479	+	0.910967i	$0.364663\pi$
$84$		0			0
$85$	435819.			0.0769734
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.40954e6			1.26447			0.632233	−	0.774778i	$-0.282138\pi$
$89$	8.40954e6			1.26447			0.632233	+	0.774778i	$0.282138\pi$
$90$		0			0
$91$	5.61042e6	+	2.72247e6i	0.780459	+	0.378720i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	963916.i		−	0.115347i
$96$		0			0
$97$		−	8.49132e6i		−	0.944657i	−0.881422	−	0.472329i	$-0.843414\pi$
$97$		−	8.49132e6i		−	0.944657i	0.881422	−	0.472329i	$-0.156586\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.94682e7			1.88019			0.940094	−	0.340915i	$-0.110737\pi$
$101$	1.94682e7			1.88019			0.940094	+	0.340915i	$0.110737\pi$
$102$		0			0
$103$		−	1.97166e7i		−	1.77787i	−0.458029	−	0.888937i	$-0.651445\pi$
$103$		−	1.97166e7i		−	1.77787i	0.458029	−	0.888937i	$-0.348555\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.98543e6i		−	0.472337i	−0.971712	−	0.236169i	$-0.924108\pi$
$107$		−	5.98543e6i		−	0.472337i	0.971712	−	0.236169i	$-0.0758918\pi$
$108$		0			0
$109$	−1.69158e7			−1.25113			−0.625563	−	0.780174i	$-0.715131\pi$
$109$	−1.69158e7			−1.25113			−0.625563	+	0.780174i	$0.715131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.24813e6i		−	0.276964i	−0.990365	−	0.138482i	$-0.955778\pi$
$113$		−	4.24813e6i		−	0.276964i	0.990365	−	0.138482i	$-0.0442223\pi$
$114$		0			0
$115$			2.50857e6i			0.153809i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.23885e6	−	1.49177e7i	0.393782	−	0.811497i
$120$		0			0
$121$	1.90726e7			0.978725
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.71337e6			0.170052
$126$		0			0
$127$	−2.18423e7			−0.946206			−0.473103	−	0.881007i	$-0.656866\pi$
$127$	−2.18423e7			−0.946206			−0.473103	+	0.881007i	$0.656866\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.35785e7			1.69365			0.846824	−	0.531874i	$-0.178512\pi$
$131$	4.35785e7			1.69365			0.846824	+	0.531874i	$0.178512\pi$
$132$		0			0
$133$	−3.29939e7	−	1.60104e7i	−1.21605	−	0.590094i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.23150e7i		−	1.07370i	−0.843678	−	0.536849i	$-0.819615\pi$
$137$		−	3.23150e7i		−	1.07370i	0.843678	−	0.536849i	$-0.180385\pi$
$138$		0			0
$139$		−	3.34417e7i		−	1.05618i	−0.849189	−	0.528089i	$-0.822909\pi$
$139$		−	3.34417e7i		−	1.05618i	0.849189	−	0.528089i	$-0.177091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.42463e6			0.126532
$144$		0			0
$145$		−	5.07564e6i		−	0.138262i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	3.81672e7i		−	0.945233i	−0.881268	−	0.472616i	$-0.843310\pi$
$149$		−	3.81672e7i		−	0.945233i	0.881268	−	0.472616i	$-0.156690\pi$
$150$		0			0
$151$	2.99275e7			0.707378			0.353689	−	0.935363i	$-0.384927\pi$
$151$	2.99275e7			0.707378			0.353689	+	0.935363i	$0.384927\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2.46660e6i			0.0532032i
$156$		0			0
$157$		−	4.20140e7i		−	0.866454i	−0.901285	−	0.433227i	$-0.857375\pi$
$157$		−	4.20140e7i		−	0.866454i	0.901285	−	0.433227i	$-0.142625\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.58658e7	+	4.16667e7i	1.62155	+	0.786861i
$162$		0			0
$163$	−5.42394e7			−0.980975			−0.490488	−	0.871448i	$-0.663181\pi$
$163$	−5.42394e7			−0.980975			−0.490488	+	0.871448i	$0.663181\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.56191e7			1.58868			0.794341	−	0.607472i	$-0.207816\pi$
$167$	9.56191e7			1.58868			0.794341	+	0.607472i	$0.207816\pi$
$168$		0			0
$169$	1.55274e7			0.247454
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.21683e6			−0.0766030			−0.0383015	−	0.999266i	$-0.512195\pi$
$173$	−5.21683e6			−0.0766030			−0.0383015	+	0.999266i	$0.512195\pi$
$174$		0			0
$175$	3.07264e7	−	6.33203e7i	0.433389	−	0.893119i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.02021e6i			0.0132954i	0.999978	+	0.00664771i	$0.00211605\pi$
$179$			1.02021e6i			0.0132954i	−0.999978	+	0.00664771i	$0.997884\pi$
$180$		0			0
$181$			8.57303e6i			0.107463i	0.998555	+	0.0537316i	$0.0171115\pi$
$181$			8.57303e6i			0.107463i	−0.998555	+	0.0537316i	$0.982888\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.18998e7			−0.138178
$186$		0			0
$187$		−	1.17648e7i		−	0.131564i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			5.55929e7i			0.577301i	0.957435	+	0.288650i	$0.0932065\pi$
$191$			5.55929e7i			0.577301i	−0.957435	+	0.288650i	$0.906793\pi$
$192$		0			0
$193$	−9.46260e7			−0.947458			−0.473729	−	0.880671i	$-0.657092\pi$
$193$	−9.46260e7			−0.947458			−0.473729	+	0.880671i	$0.657092\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.49666e8i		−	1.39473i	−0.716714	−	0.697367i	$-0.754355\pi$
$197$		−	1.49666e8i		−	1.39473i	0.716714	−	0.697367i	$-0.245645\pi$
$198$		0			0
$199$			6.75438e7i			0.607575i	0.952740	+	0.303787i	$0.0982512\pi$
$199$			6.75438e7i			0.607575i	−0.952740	+	0.303787i	$0.901749\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.73734e8	−	8.43051e7i	−1.45764	−	0.707323i
$204$		0			0
$205$	−1.90773e7			−0.154660
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.60205e7			−0.197153
$210$		0			0
$211$	1.08106e7			0.0792251			0.0396125	−	0.999215i	$-0.487388\pi$
$211$	1.08106e7			0.0792251			0.0396125	+	0.999215i	$0.487388\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.97861e6			0.0616133
$216$		0			0
$217$	8.44293e7	+	4.09696e7i	0.560899	+	0.272178i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.25558e8i			0.782474i
$222$		0			0
$223$			7.39119e7i			0.446321i	0.974782	+	0.223160i	$0.0716374\pi$
$223$			7.39119e7i			0.446321i	−0.974782	+	0.223160i	$0.928363\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.26000e7			0.128238			0.0641192	−	0.997942i	$-0.479576\pi$
$227$	2.26000e7			0.128238			0.0641192	+	0.997942i	$0.479576\pi$
$228$		0			0
$229$			1.28839e8i			0.708963i	0.935063	+	0.354481i	$0.115343\pi$
$229$			1.28839e8i			0.708963i	−0.935063	+	0.354481i	$0.884657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.44998e7i			0.385842i	0.981214	+	0.192921i	$0.0617961\pi$
$233$			7.44998e7i			0.385842i	−0.981214	+	0.192921i	$0.938204\pi$
$234$		0			0
$235$	2.14200e7			0.107667
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.92646e8i			1.38659i	0.720652	+	0.693297i	$0.243843\pi$
$239$			2.92646e8i			1.38659i	−0.720652	+	0.693297i	$0.756157\pi$
$240$		0			0
$241$		−	1.90802e8i		−	0.878058i	−0.898473	−	0.439029i	$-0.855323\pi$
$241$		−	1.90802e8i		−	0.878058i	0.898473	−	0.439029i	$-0.144677\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.21557e7	+	1.54307e7i	0.0528079	+	0.0670352i
$246$		0			0
$247$	2.77700e8			1.17256
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.64024e8			−1.85218			−0.926089	−	0.377305i	$-0.876851\pi$
$251$	−4.64024e8			−1.85218			−0.926089	+	0.377305i	$0.876851\pi$
$252$		0			0
$253$	6.77177e7			0.262894
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.55825e8			−0.572627			−0.286313	−	0.958136i	$-0.592430\pi$
$257$	−1.55825e8			−0.572627			−0.286313	+	0.958136i	$0.592430\pi$
$258$		0			0
$259$	−1.97653e8	+	4.07319e8i	−0.706893	+	1.45675i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2.42817e8i		−	0.823066i	−0.911395	−	0.411533i	$-0.864994\pi$
$263$		−	2.42817e8i		−	0.823066i	0.911395	−	0.411533i	$-0.135006\pi$
$264$		0			0
$265$			1.41816e7i			0.0468129i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.99083e6			0.0312946			0.0156473	−	0.999878i	$-0.495019\pi$
$269$	9.99083e6			0.0312946			0.0156473	+	0.999878i	$0.495019\pi$
$270$		0			0
$271$		−	9.85005e6i		−	0.0300640i	−0.999887	−	0.0150320i	$-0.995215\pi$
$271$		−	9.85005e6i		−	0.0300640i	0.999887	−	0.0150320i	$-0.00478501\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.99372e7i		−	0.144797i
$276$		0			0
$277$	3.25387e8			0.919859			0.459929	−	0.887955i	$-0.347875\pi$
$277$	3.25387e8			0.919859			0.459929	+	0.887955i	$0.347875\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.28153e8i			0.882276i	0.897439	+	0.441138i	$0.145425\pi$
$281$			3.28153e8i			0.882276i	−0.897439	+	0.441138i	$0.854575\pi$
$282$		0			0
$283$			5.56460e8i			1.45942i	0.683755	+	0.729711i	$0.260346\pi$
$283$			5.56460e8i			1.45942i	−0.683755	+	0.729711i	$0.739654\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.16870e8	+	6.52999e8i	−0.791214	+	1.63052i
$288$		0			0
$289$	−7.64905e7			−0.186408
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.56994e7			−0.222266			−0.111133	−	0.993806i	$-0.535448\pi$
$293$	−9.56994e7			−0.222266			−0.111133	+	0.993806i	$0.535448\pi$
$294$		0			0
$295$	−3.31629e7			−0.0752100
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.22706e8			−1.56355
$300$		0			0
$301$	1.49133e8	−	3.07329e8i	0.315203	−	0.649563i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.80651e7i		−	0.0768205i
$306$		0			0
$307$			3.40920e8i			0.672462i	0.941780	+	0.336231i	$0.109152\pi$
$307$			3.40920e8i			0.672462i	−0.941780	+	0.336231i	$0.890848\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.46606e8			−1.21893			−0.609464	−	0.792814i	$-0.708615\pi$
$311$	−6.46606e8			−1.21893			−0.609464	+	0.792814i	$0.708615\pi$
$312$		0			0
$313$			5.36531e8i			0.988985i	0.869182	+	0.494492i	$0.164646\pi$
$313$			5.36531e8i			0.988985i	−0.869182	+	0.494492i	$0.835354\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.13662e8i			0.729354i	0.931134	+	0.364677i	$0.118821\pi$
$317$			4.13662e8i			0.729354i	−0.931134	+	0.364677i	$0.881179\pi$
$318$		0			0
$319$	−1.37015e8			−0.236320
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	7.38382e8i		−	1.21919i
$324$		0			0
$325$			5.32947e8i			0.861177i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.55780e8	−	7.33185e8i	0.550803	−	1.13508i
$330$		0			0
$331$	3.27670e8			0.496637			0.248318	−	0.968678i	$-0.420122\pi$
$331$	3.27670e8			0.496637			0.248318	+	0.968678i	$0.420122\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.11676e7			−0.0452946
$336$		0			0
$337$	−3.51326e8			−0.500041			−0.250020	−	0.968241i	$-0.580437\pi$
$337$	−3.51326e8			−0.500041			−0.250020	+	0.968241i	$0.580437\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.65848e7			0.0909358
$342$		0			0
$343$	7.30080e8	−	1.59779e8i	0.976879	−	0.213791i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.08160e9i			1.38967i	0.719169	+	0.694836i	$0.244523\pi$
$347$			1.08160e9i			1.38967i	−0.719169	+	0.694836i	$0.755477\pi$
$348$		0			0
$349$		−	5.84964e8i		−	0.736615i	−0.929704	−	0.368307i	$-0.879937\pi$
$349$		−	5.84964e8i		−	0.736615i	0.929704	−	0.368307i	$-0.120063\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.22041e8			−0.873676			−0.436838	−	0.899540i	$-0.643902\pi$
$353$	−7.22041e8			−0.873676			−0.436838	+	0.899540i	$0.643902\pi$
$354$		0			0
$355$			6.39063e7i			0.0758132i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8.98675e8i			1.02511i	0.858653	+	0.512557i	$0.171302\pi$
$359$			8.98675e8i			1.02511i	−0.858653	+	0.512557i	$0.828698\pi$
$360$		0			0
$361$	−7.39232e8			−0.827000
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.26992e8i			0.136695i
$366$		0			0
$367$		−	1.09069e9i		−	1.15178i	−0.817527	−	0.575890i	$-0.804656\pi$
$367$		−	1.09069e9i		−	1.15178i	0.817527	−	0.575890i	$-0.195344\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.85424e8	+	2.35554e8i	0.493529	+	0.239486i
$372$		0			0
$373$	−3.97934e8			−0.397037			−0.198518	−	0.980097i	$-0.563613\pi$
$373$	−3.97934e8			−0.397037			−0.198518	+	0.980097i	$0.563613\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.46227e9			1.40550
$378$		0			0
$379$	1.40632e9			1.32692			0.663462	−	0.748210i	$-0.269086\pi$
$379$	1.40632e9			1.32692			0.663462	+	0.748210i	$0.269086\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.05857e9			0.962776			0.481388	−	0.876508i	$-0.340133\pi$
$383$	1.05857e9			0.962776			0.481388	+	0.876508i	$0.340133\pi$
$384$		0			0
$385$	1.25391e7	+	6.08466e6i	0.0111984	+	0.00543405i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.13515e9i		−	1.83909i	−0.392980	−	0.919547i	$-0.628556\pi$
$389$		−	2.13515e9i		−	1.83909i	0.392980	−	0.919547i	$-0.371444\pi$
$390$		0			0
$391$			1.92162e9i			1.62573i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.79849e8			0.146832
$396$		0			0
$397$		−	1.86451e9i		−	1.49554i	−0.663956	−	0.747772i	$-0.731124\pi$
$397$		−	1.86451e9i		−	1.49554i	0.663956	−	0.747772i	$-0.268876\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.44179e9i		−	1.11659i	−0.829641	−	0.558297i	$-0.811455\pi$
$401$		−	1.44179e9i		−	1.11659i	0.829641	−	0.558297i	$-0.188545\pi$
$402$		0			0
$403$	−7.10615e8			−0.540838
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.21230e8i			0.236176i
$408$		0			0
$409$		−	3.60787e8i		−	0.260747i	−0.991465	−	0.130374i	$-0.958382\pi$
$409$		−	3.60787e8i		−	0.260747i	0.991465	−	0.130374i	$-0.0416177\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.50828e8	+	1.13513e9i	−0.384761	+	0.792907i
$414$		0			0
$415$	−1.02503e8			−0.0703994
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.28039e9			−1.51447			−0.757235	−	0.653142i	$-0.773450\pi$
$419$	−2.28039e9			−1.51447			−0.757235	+	0.653142i	$0.773450\pi$
$420$		0			0
$421$	−1.73520e9			−1.13335			−0.566674	−	0.823942i	$-0.691770\pi$
$421$	−1.73520e9			−1.13335			−0.566674	+	0.823942i	$0.691770\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.41707e9			0.895425
$426$		0			0
$427$	−1.30293e9	−	6.32251e8i	−0.809886	−	0.393000i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.65594e9i		−	1.59790i	−0.601400	−	0.798948i	$-0.705390\pi$
$431$		−	2.65594e9i		−	1.59790i	0.601400	−	0.798948i	$-0.294610\pi$
$432$		0			0
$433$		−	2.00338e9i		−	1.18592i	−0.805231	−	0.592961i	$-0.797959\pi$
$433$		−	2.00338e9i		−	1.18592i	0.805231	−	0.592961i	$-0.202041\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.25011e9			2.43621
$438$		0			0
$439$			4.07192e8i			0.229706i	0.993382	+	0.114853i	$0.0366398\pi$
$439$			4.07192e8i			0.229706i	−0.993382	+	0.114853i	$0.963360\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			8.65872e8i			0.473196i	0.971608	+	0.236598i	$0.0760324\pi$
$443$			8.65872e8i			0.473196i	−0.971608	+	0.236598i	$0.923968\pi$
$444$		0			0
$445$	−2.00588e8			−0.107906
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.29775e9i		−	0.676596i	−0.941039	−	0.338298i	$-0.890149\pi$
$449$		−	1.29775e9i		−	0.676596i	0.941039	−	0.338298i	$-0.109851\pi$
$450$		0			0
$451$			5.14984e8i			0.264348i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.33822e8	−	6.49375e7i	−0.0666020	−	0.0323188i
$456$		0			0
$457$	2.28635e9			1.12056			0.560282	−	0.828302i	$-0.310693\pi$
$457$	2.28635e9			1.12056			0.560282	+	0.828302i	$0.310693\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.04333e9			1.92214			0.961071	−	0.276301i	$-0.0891088\pi$
$461$	4.04333e9			1.92214			0.961071	+	0.276301i	$0.0891088\pi$
$462$		0			0
$463$	−1.49148e9			−0.698367			−0.349184	−	0.937054i	$-0.613541\pi$
$463$	−1.49148e9			−0.698367			−0.349184	+	0.937054i	$0.613541\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.14324e7			0.0188248			0.00941241	−	0.999956i	$-0.497004\pi$
$467$	4.14324e7			0.0188248			0.00941241	+	0.999956i	$0.497004\pi$
$468$		0			0
$469$	−5.17686e8	+	1.06684e9i	−0.231719	+	0.477521i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.42374e8i		−	0.105311i
$474$		0			0
$475$		−	3.13417e9i		−	1.34182i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.81530e8			0.0754700			0.0377350	−	0.999288i	$-0.487986\pi$
$479$	1.81530e8			0.0754700			0.0377350	+	0.999288i	$0.487986\pi$
$480$		0			0
$481$		−	3.42828e9i		−	1.40465i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			2.02538e8i			0.0806142i
$486$		0			0
$487$	−1.63416e8			−0.0641127			−0.0320563	−	0.999486i	$-0.510206\pi$
$487$	−1.63416e8			−0.0641127			−0.0320563	+	0.999486i	$0.510206\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.82974e8i		−	0.374763i	−0.982287	−	0.187381i	$-0.940000\pi$
$491$		−	9.82974e8i		−	0.374763i	0.982287	−	0.187381i	$-0.0600001\pi$
$492$		0			0
$493$		−	3.88806e9i		−	1.46140i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.18745e9	+	1.06147e9i	0.799266	+	0.387846i
$498$		0			0
$499$	1.28099e8			0.0461524			0.0230762	−	0.999734i	$-0.492654\pi$
$499$	1.28099e8			0.0461524			0.0230762	+	0.999734i	$0.492654\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.87368e8			0.135717			0.0678587	−	0.997695i	$-0.478383\pi$
$503$	3.87368e8			0.135717			0.0678587	+	0.997695i	$0.478383\pi$
$504$		0			0
$505$	−4.64364e8			−0.160450
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.74362e9			0.586055			0.293028	−	0.956104i	$-0.405337\pi$
$509$	1.74362e9			0.586055			0.293028	+	0.956104i	$0.405337\pi$
$510$		0			0
$511$	4.34682e9	+	2.10931e9i	1.44112	+	0.699306i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4.70287e8i			0.151718i
$516$		0			0
$517$		−	5.78223e8i		−	0.184026i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.83093e9			0.567206			0.283603	−	0.958942i	$-0.408470\pi$
$521$	1.83093e9			0.567206			0.283603	+	0.958942i	$0.408470\pi$
$522$		0			0
$523$			5.32924e9i			1.62896i	0.580194	+	0.814478i	$0.302977\pi$
$523$			5.32924e9i			1.62896i	−0.580194	+	0.814478i	$0.697023\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.88947e9i			0.562346i
$528$		0			0
$529$	−7.65598e9			−2.24857
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.49609e9i		−	1.57220i
$534$		0			0
$535$			1.42767e8i			0.0403078i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.16544e8	−	3.28138e8i	0.114578	−	0.0902601i
$540$		0			0
$541$	−1.47651e9			−0.400910			−0.200455	−	0.979703i	$-0.564242\pi$
$541$	−1.47651e9			−0.400910			−0.200455	+	0.979703i	$0.564242\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.03483e8			0.106767
$546$		0			0
$547$	5.74676e8			0.150130			0.0750650	−	0.997179i	$-0.476084\pi$
$547$	5.74676e8			0.150130			0.0750650	+	0.997179i	$0.476084\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.59935e9			−2.18995
$552$		0			0
$553$	2.98726e9	−	6.15608e9i	0.751163	−	1.54798i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6.87265e9i		−	1.68512i	−0.538602	−	0.842561i	$-0.681047\pi$
$557$		−	6.87265e9i		−	1.68512i	0.538602	−	0.842561i	$-0.318953\pi$
$558$		0			0
$559$			2.58670e9i			0.626331i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.82726e9			−1.14004			−0.570022	−	0.821629i	$-0.693065\pi$
$563$	−4.82726e9			−1.14004			−0.570022	+	0.821629i	$0.693065\pi$
$564$		0			0
$565$			1.01328e8i			0.0236353i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.11400e9i			0.936205i	0.883674	+	0.468102i	$0.155062\pi$
$569$			4.11400e9i			0.936205i	−0.883674	+	0.468102i	$0.844938\pi$
$570$		0			0
$571$	1.90302e8			0.0427777			0.0213889	−	0.999771i	$-0.493191\pi$
$571$	1.90302e8			0.0427777			0.0213889	+	0.999771i	$0.493191\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			8.15660e9i			1.78925i
$576$		0			0
$577$			6.16463e9i			1.33595i	0.744182	+	0.667977i	$0.232840\pi$
$577$			6.16463e9i			1.33595i	−0.744182	+	0.667977i	$0.767160\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.70255e9	+	3.50859e9i	−0.360151	+	0.742191i
$582$		0			0
$583$	3.82827e8			0.0800135
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.39623e8			−0.110118			−0.0550588	−	0.998483i	$-0.517535\pi$
$587$	−5.39623e8			−0.110118			−0.0550588	+	0.998483i	$0.517535\pi$
$588$		0			0
$589$	4.17901e9			0.842694
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.36499e9			1.25345			0.626724	−	0.779241i	$-0.284395\pi$
$593$	6.36499e9			1.25345			0.626724	+	0.779241i	$0.284395\pi$
$594$		0			0
$595$	−1.72664e8	+	3.55823e8i	−0.0336041	+	0.0692507i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	7.11729e9i		−	1.35307i	−0.736410	−	0.676536i	$-0.763480\pi$
$599$		−	7.11729e9i		−	1.35307i	0.736410	−	0.676536i	$-0.236520\pi$
$600$		0			0
$601$			2.08445e9i			0.391680i	0.980636	+	0.195840i	$0.0627433\pi$
$601$			2.08445e9i			0.391680i	−0.980636	+	0.195840i	$0.937257\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.54927e8			−0.0835214
$606$		0			0
$607$		−	1.02932e9i		−	0.186806i	−0.995628	−	0.0934031i	$-0.970225\pi$
$607$		−	1.02932e9i		−	0.186806i	0.995628	−	0.0934031i	$-0.0297745\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.17099e9i			1.09449i
$612$		0			0
$613$	1.87594e9			0.328933			0.164466	−	0.986383i	$-0.447410\pi$
$613$	1.87594e9			0.328933			0.164466	+	0.986383i	$0.447410\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.48916e9i		−	0.255236i	−0.991823	−	0.127618i	$-0.959267\pi$
$617$		−	1.48916e9i		−	0.255236i	0.991823	−	0.127618i	$-0.0407332\pi$
$618$		0			0
$619$			2.10140e9i			0.356116i	0.984020	+	0.178058i	$0.0569814\pi$
$619$			2.10140e9i			0.356116i	−0.984020	+	0.178058i	$0.943019\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.33171e9	+	6.86593e9i	−0.552026	+	1.13760i
$624$		0			0
$625$	5.97049e9			0.978206
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.11553e9			−1.46051
$630$		0			0
$631$	−5.80511e9			−0.919830			−0.459915	−	0.887963i	$-0.652120\pi$
$631$	−5.80511e9			−0.919830			−0.459915	+	0.887963i	$0.652120\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.20992e8			0.0807463
$636$		0			0
$637$	−4.44550e9	+	3.50200e9i	−0.681448	+	0.536819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	2.22507e9i		−	0.333689i	−0.985983	−	0.166844i	$-0.946642\pi$
$641$		−	2.22507e9i		−	0.333689i	0.985983	−	0.166844i	$-0.0533577\pi$
$642$		0			0
$643$			9.40778e9i			1.39556i	0.716312	+	0.697780i	$0.245829\pi$
$643$			9.40778e9i			1.39556i	−0.716312	+	0.697780i	$0.754171\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.21664e10			−1.76603			−0.883015	−	0.469345i	$-0.844490\pi$
$647$	−1.21664e10			−1.76603			−0.883015	+	0.469345i	$0.844490\pi$
$648$		0			0
$649$			8.95218e8i			0.128550i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8.44756e9i			1.18723i	0.804749	+	0.593615i	$0.202300\pi$
$653$			8.44756e9i			1.18723i	−0.804749	+	0.593615i	$0.797700\pi$
$654$		0			0
$655$	−1.03945e9			−0.144531
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.44574e9i		−	0.332898i	−0.986050	−	0.166449i	$-0.946770\pi$
$659$		−	2.44574e9i		−	0.332898i	0.986050	−	0.166449i	$-0.0532300\pi$
$660$		0			0
$661$		−	8.94751e9i		−	1.20503i	−0.798108	−	0.602514i	$-0.794166\pi$
$661$		−	8.94751e9i		−	1.20503i	0.798108	−	0.602514i	$-0.205834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.86984e8	+	3.81887e8i	0.103774	+	0.0503568i
$666$		0			0
$667$	2.23796e10			2.92019
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.02755e9			−0.131303
$672$		0			0
$673$	−3.50385e9			−0.443091			−0.221545	−	0.975150i	$-0.571110\pi$
$673$	−3.50385e9			−0.443091			−0.221545	+	0.975150i	$0.571110\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.03313e9			0.127966			0.0639828	−	0.997951i	$-0.479620\pi$
$677$	1.03313e9			0.127966			0.0639828	+	0.997951i	$0.479620\pi$
$678$		0			0
$679$	6.93270e9	+	3.36411e9i	0.849881	+	0.412407i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	9.17949e8i		−	0.110242i	−0.998480	−	0.0551209i	$-0.982446\pi$
$683$		−	9.17949e8i		−	0.110242i	0.998480	−	0.0551209i	$-0.0175544\pi$
$684$		0			0
$685$			7.70790e8i			0.0916261i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.08566e9			−0.475878
$690$		0			0
$691$			9.56290e8i			0.110260i	0.998479	+	0.0551298i	$0.0175573\pi$
$691$			9.56290e8i			0.110260i	−0.998479	+	0.0551298i	$0.982443\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.97666e8i			0.0901310i
$696$		0			0
$697$	−1.46137e10			−1.63473
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.86676e9i			0.314325i	0.987573	+	0.157162i	$0.0502346\pi$
$701$			2.86676e9i			0.314325i	−0.987573	+	0.157162i	$0.949765\pi$
$702$		0			0
$703$			2.01611e10i			2.18862i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.71297e9	+	1.58947e10i	−0.820830	+	1.69155i
$708$		0			0
$709$	−6.32278e9			−0.666263			−0.333132	−	0.942880i	$-0.608105\pi$
$709$	−6.32278e9			−0.666263			−0.333132	+	0.942880i	$0.608105\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.08758e10			−1.12369
$714$		0			0
$715$	−1.05538e8			−0.0107979
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.14699e10			−1.15083			−0.575414	−	0.817862i	$-0.695159\pi$
$719$	−1.14699e10			−1.15083			−0.575414	+	0.817862i	$0.695159\pi$
$720$		0			0
$721$	1.60975e10	+	7.81136e9i	1.59950	+	0.776163i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.65034e10i		−	1.60839i
$726$		0			0
$727$		−	1.28262e10i		−	1.23802i	−0.785385	−	0.619008i	$-0.787535\pi$
$727$		−	1.28262e10i		−	1.23802i	0.785385	−	0.619008i	$-0.212465\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.87783e9			0.651240
$732$		0			0
$733$			5.83532e9i			0.547269i	0.961834	+	0.273634i	$0.0882259\pi$
$733$			5.83532e9i			0.547269i	−0.961834	+	0.273634i	$0.911774\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8.41355e8i			0.0774182i
$738$		0			0
$739$	−1.57289e10			−1.43365			−0.716823	−	0.697255i	$-0.754404\pi$
$739$	−1.57289e10			−1.43365			−0.716823	+	0.697255i	$0.754404\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.27677e10i		−	1.14196i	−0.820963	−	0.570981i	$-0.806563\pi$
$743$		−	1.27677e10i		−	1.14196i	0.820963	−	0.570981i	$-0.193437\pi$
$744$		0			0
$745$			9.10380e8i			0.0806633i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.88678e9	+	2.37132e9i	0.424948	+	0.206207i
$750$		0			0
$751$	−4.66554e9			−0.401941			−0.200971	−	0.979597i	$-0.564410\pi$
$751$	−4.66554e9			−0.401941			−0.200971	+	0.979597i	$0.564410\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.13843e8			−0.0603654
$756$		0			0
$757$	1.65338e9			0.138528			0.0692639	−	0.997598i	$-0.477935\pi$
$757$	1.65338e9			0.138528			0.0692639	+	0.997598i	$0.477935\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.63897e9			−0.792838			−0.396419	−	0.918070i	$-0.629747\pi$
$761$	−9.63897e9			−0.792838			−0.396419	+	0.918070i	$0.629747\pi$
$762$		0			0
$763$	6.70176e9	−	1.38109e10i	0.546202	−	1.12560i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	9.55407e9i		−	0.764548i
$768$		0			0
$769$		−	2.55518e8i		−	0.0202619i	−0.999949	−	0.0101309i	$-0.996775\pi$
$769$		−	2.55518e8i		−	0.0202619i	0.999949	−	0.0101309i	$-0.00322483\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.04766e10			1.59452			0.797258	−	0.603639i	$-0.206283\pi$
$773$	2.04766e10			1.59452			0.797258	+	0.603639i	$0.206283\pi$
$774$		0			0
$775$			8.02014e9i			0.618908i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			3.23215e10i			2.44969i
$780$		0			0
$781$	1.72512e9			0.129581
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.00214e9i			0.0739406i
$786$		0			0
$787$		−	1.44231e10i		−	1.05475i	−0.849634	−	0.527373i	$-0.823177\pi$
$787$		−	1.44231e10i		−	1.05475i	0.849634	−	0.527373i	$-0.176823\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.46837e9	+	1.68304e9i	0.249177	+	0.120914i
$792$		0			0
$793$	1.09664e10			0.780920
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.22429e10			1.55628			0.778138	−	0.628093i	$-0.216164\pi$
$797$	2.22429e10			1.55628			0.778138	+	0.628093i	$0.216164\pi$
$798$		0			0
$799$	1.64082e10			1.13801
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.42810e9			0.233641
$804$		0			0
$805$	−2.04811e9	−	9.93850e8i	−0.138378	−	0.0671483i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.47353e10i		−	0.978452i	−0.872157	−	0.489226i	$-0.837279\pi$
$809$		−	1.47353e10i		−	0.978452i	0.872157	−	0.489226i	$-0.162721\pi$
$810$		0			0
$811$			3.21939e9i			0.211934i	0.994370	+	0.105967i	$0.0337938\pi$
$811$			3.21939e9i			0.211934i	−0.994370	+	0.105967i	$0.966206\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.29374e9			0.0837134
$816$		0			0
$817$		−	1.52119e10i		−	0.975904i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1.77772e10i			1.12114i	0.828106	+	0.560572i	$0.189419\pi$
$821$			1.77772e10i			1.12114i	−0.828106	+	0.560572i	$0.810581\pi$
$822$		0			0
$823$	−2.41352e10			−1.50922			−0.754609	−	0.656174i	$-0.772174\pi$
$823$	−2.41352e10			−1.50922			−0.754609	+	0.656174i	$0.772174\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.45903e10i		−	0.897006i	−0.893781	−	0.448503i	$-0.851957\pi$
$827$		−	1.45903e10i		−	0.897006i	0.893781	−	0.448503i	$-0.148043\pi$
$828$		0			0
$829$		−	1.50353e10i		−	0.916583i	−0.888802	−	0.458291i	$-0.848462\pi$
$829$		−	1.50353e10i		−	0.916583i	0.888802	−	0.458291i	$-0.151538\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.31156e9	+	1.18203e10i	0.558168	+	0.708548i
$834$		0			0
$835$	−2.28075e9			−0.135573
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.23475e10			0.721789			0.360895	−	0.932607i	$-0.382471\pi$
$839$	1.23475e10			0.721789			0.360895	+	0.932607i	$0.382471\pi$
$840$		0			0
$841$	−2.80313e10			−1.62501
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.70365e8			−0.0211170
$846$		0			0
$847$	−7.55622e9	+	1.55717e10i	−0.427280	+	0.880531i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	5.24688e10i		−	2.91842i
$852$		0			0
$853$		−	1.46116e10i		−	0.806079i	−0.915183	−	0.403039i	$-0.867954\pi$
$853$		−	1.46116e10i		−	0.806079i	0.915183	−	0.403039i	$-0.132046\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.87295e10			−1.01647			−0.508233	−	0.861220i	$-0.669701\pi$
$857$	−1.87295e10			−1.01647			−0.508233	+	0.861220i	$0.669701\pi$
$858$		0			0
$859$			2.83604e9i			0.152664i	0.997082	+	0.0763319i	$0.0243209\pi$
$859$			2.83604e9i			0.152664i	−0.997082	+	0.0763319i	$0.975679\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	5.18114e9i		−	0.274402i	−0.990543	−	0.137201i	$-0.956189\pi$
$863$		−	5.18114e9i		−	0.274402i	0.990543	−	0.137201i	$-0.0438107\pi$
$864$		0			0
$865$	1.24434e8			0.00653706
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	4.85496e9i		−	0.250967i
$870$		0			0
$871$		−	8.97923e9i		−	0.460443i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.47117e9	+	3.03176e9i	−0.0742395	+	0.152991i
$876$		0			0
$877$	2.88914e10			1.44634			0.723169	−	0.690671i	$-0.242685\pi$
$877$	2.88914e10			1.44634			0.723169	+	0.690671i	$0.242685\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.73613e10			0.855393			0.427697	−	0.903922i	$-0.359325\pi$
$881$	1.73613e10			0.855393			0.427697	+	0.903922i	$0.359325\pi$
$882$		0			0
$883$	−5.09174e9			−0.248888			−0.124444	−	0.992227i	$-0.539715\pi$
$883$	−5.09174e9			−0.248888			−0.124444	+	0.992227i	$0.539715\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.20796e10			1.54346			0.771732	−	0.635948i	$-0.219391\pi$
$887$	3.20796e10			1.54346			0.771732	+	0.635948i	$0.219391\pi$
$888$		0			0
$889$	8.65354e9	−	1.78331e10i	0.413084	−	0.851274i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	3.62906e10i		−	1.70535i
$894$		0			0
$895$		−	2.43343e7i		−	0.00113459i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.20052e10			1.01010
$900$		0			0
$901$			1.08635e10i			0.494803i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.04487e8i		−	0.00917057i
$906$		0			0
$907$	−3.98435e10			−1.77309			−0.886547	−	0.462639i	$-0.846903\pi$
$907$	−3.98435e10			−1.77309			−0.886547	+	0.462639i	$0.846903\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.81415e9i			0.210962i	0.994421	+	0.105481i	$0.0336383\pi$
$911$			4.81415e9i			0.210962i	−0.994421	+	0.105481i	$0.966362\pi$
$912$		0			0
$913$			2.76703e9i			0.120328i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.72650e10	+	3.55795e10i	−0.739393	+	1.52373i
$918$		0			0
$919$	2.30402e10			0.979223			0.489612	−	0.871941i	$-0.337139\pi$
$919$	2.30402e10			0.979223			0.489612	+	0.871941i	$0.337139\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.84111e10			−0.770680
$924$		0			0
$925$	−3.86922e10			−1.60741
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.46870e10			1.41942			0.709712	−	0.704492i	$-0.248825\pi$
$929$	3.46870e10			1.41942			0.709712	+	0.704492i	$0.248825\pi$
$930$		0			0
$931$	2.61432e10	−	2.05947e10i	1.06178	−	0.836433i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.80618e8i			0.0112273i
$936$		0			0
$937$			2.19036e10i			0.869815i	0.900475	+	0.434908i	$0.143219\pi$
$937$			2.19036e10i			0.869815i	−0.900475	+	0.434908i	$0.856781\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.40379e9			0.367908			0.183954	−	0.982935i	$-0.441110\pi$
$941$	9.40379e9			0.367908			0.183954	+	0.982935i	$0.441110\pi$
$942$		0			0
$943$		−	8.41160e10i		−	3.26654i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	9.05372e9i		−	0.346419i	−0.984885	−	0.173210i	$-0.944586\pi$
$947$		−	9.05372e9i		−	0.346419i	0.984885	−	0.173210i	$-0.0554138\pi$
$948$		0			0
$949$	−3.65858e10			−1.38957
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	2.49734e10i		−	0.934659i	−0.884083	−	0.467329i	$-0.845216\pi$
$953$		−	2.49734e10i		−	0.934659i	0.884083	−	0.467329i	$-0.154784\pi$
$954$		0			0
$955$		−	1.32602e9i		−	0.0492651i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.63834e10	+	1.28026e10i	0.965975	+	0.468742i
$960$		0			0
$961$	1.68188e10			0.611312
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.25706e9			0.0808531
$966$		0			0
$967$	−1.04896e10			−0.373049			−0.186524	−	0.982450i	$-0.559722\pi$
$967$	−1.04896e10			−0.373049			−0.186524	+	0.982450i	$0.559722\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.13205e10			0.747361			0.373681	−	0.927557i	$-0.378096\pi$
$971$	2.13205e10			0.747361			0.373681	+	0.927557i	$0.378096\pi$
$972$		0			0
$973$	2.73033e10	+	1.32490e10i	0.950212	+	0.461094i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.48845e10i		−	0.510627i	−0.966858	−	0.255314i	$-0.917821\pi$
$977$		−	1.48845e10i		−	0.510627i	0.966858	−	0.255314i	$-0.0821787\pi$
$978$		0			0
$979$			5.41478e9i			0.184434i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.24046e10			0.752314			0.376157	−	0.926556i	$-0.377245\pi$
$983$	2.24046e10			0.752314			0.376157	+	0.926556i	$0.377245\pi$
$984$		0			0
$985$			3.56989e9i			0.119022i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3.95886e10i			1.30132i
$990$		0			0
$991$	4.80897e9			0.156962			0.0784809	−	0.996916i	$-0.474993\pi$
$991$	4.80897e9			0.156962			0.0784809	+	0.996916i	$0.474993\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1.61108e9i		−	0.0518486i
$996$		0			0
$997$			3.11939e10i			0.996867i	0.866928	+	0.498434i	$0.166091\pi$
$997$			3.11939e10i			0.996867i	−0.866928	+	0.498434i	$0.833909\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.f.a.125.10	yes	20
3.2	odd	2	inner	252.8.f.a.125.12	yes	20
7.6	odd	2	inner	252.8.f.a.125.11	yes	20
21.20	even	2	inner	252.8.f.a.125.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.f.a.125.9	✓	20	21.20	even	2	inner
252.8.f.a.125.10	yes	20	1.1	even	1	trivial
252.8.f.a.125.11	yes	20	7.6	odd	2	inner
252.8.f.a.125.12	yes	20	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-23.8524 q^{5} +(-396.183 + 816.445i) q^{7} +643.885i q^{11} -6871.76i q^{13} -18271.5 q^{17} +40411.7i q^{19} -105170. i q^{23} -77556.1 q^{25} +212794. i q^{29} -103411. i q^{31} +(9449.90 - 19474.2i) q^{35} +498893. q^{37} +799807. q^{41} -376424. q^{43} -898022. q^{47} +(-509622. - 646922. i) q^{49} -594558. i q^{53} -15358.2i q^{55} +1.39034e6 q^{59} +1.59586e6i q^{61} +163908. i q^{65} +1.30668e6 q^{67} -2.67924e6i q^{71} -5.32408e6i q^{73} +(-525697. - 255096. i) q^{77} -7.54010e6 q^{79} +4.29740e6 q^{83} +435819. q^{85} +8.40954e6 q^{89} +(5.61042e6 + 2.72247e6i) q^{91} -963916. i q^{95} -8.49132e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2468 q^{7} + 548780 q^{25} - 257344 q^{37} - 1589224 q^{43} - 291556 q^{49} - 11860256 q^{67} - 1991600 q^{79} + 11742576 q^{85} - 11268816 q^{91}+O(q^{100}) \) 20 * q - 2468 * q^7 + 548780 * q^25 - 257344 * q^37 - 1589224 * q^43 - 291556 * q^49 - 11860256 * q^67 - 1991600 * q^79 + 11742576 * q^85 - 11268816 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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