LMFDB - Embedded newform 252.9.c.a.197.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(197,252)]

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, 0]))

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

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

chi := DirichletCharacter("252.197");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.c (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:	$16$
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} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 $ x^16 + 4002260x^14 + 6534459751956x^12 + 5613923146579405376x^10 + 2733728904154246859079616x^8 + 757873148017661341349205888000x^6 + 113644318422397913452531577312640000x^4 + 8098650340007618970326973663348480000000*x^2 + 199066230990417435753898292645889849600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{37}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.15
Root			$941.588i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+941.588i q^{5} +907.493 q^{7} +O(q^{10})$	$q+941.588i q^{5} +907.493 q^{7} +24986.1i q^{11} -53929.9 q^{13} +126144. i q^{17} +2191.95 q^{19} -499304. i q^{23} -495964. q^{25} +471537. i q^{29} -1.20831e6 q^{31} +854485. i q^{35} -90570.5 q^{37} +5.01160e6i q^{41} +1.25602e6 q^{43} -4.26407e6i q^{47} +823543. q^{49} +2.44768e6i q^{53} -2.35266e7 q^{55} -363348. i q^{59} +2.52850e7 q^{61} -5.07798e7i q^{65} +2.86054e7 q^{67} -6.10791e6i q^{71} +4.56049e7 q^{73} +2.26747e7i q^{77} -6.35845e6 q^{79} -4.96314e7i q^{83} -1.18775e8 q^{85} +6.47277e7i q^{89} -4.89410e7 q^{91} +2.06392e6i q^{95} -1.05856e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) $ 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

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
$5$	941.588i	1.50654i	0.657711	+	0.753271i	$0.271525\pi$
$5$	941.588i	1.50654i	−0.657711	+	0.753271i	$0.728475\pi$
$6$	0	0
$7$	907.493	0.377964
$7$	907.493	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24986.1i	1.70659i	0.521432	+	0.853293i	$0.325398\pi$
$11$	24986.1i	1.70659i	−0.521432	+	0.853293i	$0.674602\pi$
$12$	0	0
$13$	−53929.9	−1.88824	−0.944118	−	0.329607i	$-0.893084\pi$
$13$	−53929.9	−1.88824	−0.944118	+	0.329607i	$0.893084\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	126144.i	1.51032i	0.655538	+	0.755162i	$0.272442\pi$
$17$	126144.i	1.51032i	−0.655538	+	0.755162i	$0.727558\pi$
$18$	0	0
$19$	2191.95	0.0168196	0.00840982	−	0.999965i	$-0.497323\pi$
$19$	2191.95	0.0168196	0.00840982	+	0.999965i	$0.497323\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 499304.i	− 1.78424i	−0.451797	−	0.892121i	$-0.649217\pi$
$23$	− 499304.i	− 1.78424i	0.451797	−	0.892121i	$-0.350783\pi$
$24$	0	0
$25$	−495964.	−1.26967
$26$	0	0
$27$	0	0
$28$	0	0
$29$	471537.i	0.666690i	0.942805	+	0.333345i	$0.108177\pi$
$29$	471537.i	0.666690i	−0.942805	+	0.333345i	$0.891823\pi$
$30$	0	0
$31$	−1.20831e6	−1.30837	−0.654187	−	0.756333i	$-0.726989\pi$
$31$	−1.20831e6	−1.30837	−0.654187	+	0.756333i	$0.726989\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	854485.i	0.569419i
$36$	0	0
$37$	−90570.5	−0.0483259	−0.0241630	−	0.999708i	$-0.507692\pi$
$37$	−90570.5	−0.0483259	−0.0241630	+	0.999708i	$0.507692\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.01160e6i	1.77354i	0.462212	+	0.886770i	$0.347056\pi$
$41$	5.01160e6i	1.77354i	−0.462212	+	0.886770i	$0.652944\pi$
$42$	0	0
$43$	1.25602e6	0.367385	0.183692	−	0.982984i	$-0.441195\pi$
$43$	1.25602e6	0.367385	0.183692	+	0.982984i	$0.441195\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.26407e6i	− 0.873841i	−0.899500	−	0.436921i	$-0.856069\pi$
$47$	− 4.26407e6i	− 0.873841i	0.899500	−	0.436921i	$-0.143931\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.44768e6i	0.310207i	0.987898	+	0.155103i	$0.0495710\pi$
$53$	2.44768e6i	0.310207i	−0.987898	+	0.155103i	$0.950429\pi$
$54$	0	0
$55$	−2.35266e7	−2.57104
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 363348.i	− 0.0299857i	−0.999888	−	0.0149929i	$-0.995227\pi$
$59$	− 363348.i	− 0.0299857i	0.999888	−	0.0149929i	$-0.00477256\pi$
$60$	0	0
$61$	2.52850e7	1.82618	0.913089	−	0.407760i	$-0.133690\pi$
$61$	2.52850e7	1.82618	0.913089	+	0.407760i	$0.133690\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.07798e7i	− 2.84471i
$66$	0	0
$67$	2.86054e7	1.41954	0.709772	−	0.704431i	$-0.248798\pi$
$67$	2.86054e7	1.41954	0.709772	+	0.704431i	$0.248798\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 6.10791e6i	− 0.240358i	−0.992752	−	0.120179i	$-0.961653\pi$
$71$	− 6.10791e6i	− 0.240358i	0.992752	−	0.120179i	$-0.0383469\pi$
$72$	0	0
$73$	4.56049e7	1.60590	0.802952	−	0.596043i	$-0.203261\pi$
$73$	4.56049e7	1.60590	0.802952	+	0.596043i	$0.203261\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.26747e7i	0.645029i
$78$	0	0
$79$	−6.35845e6	−0.163246	−0.0816231	−	0.996663i	$-0.526010\pi$
$79$	−6.35845e6	−0.163246	−0.0816231	+	0.996663i	$0.526010\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.96314e7i	− 1.04579i	−0.852398	−	0.522894i	$-0.824852\pi$
$83$	− 4.96314e7i	− 1.04579i	0.852398	−	0.522894i	$-0.175148\pi$
$84$	0	0
$85$	−1.18775e8	−2.27537
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.47277e7i	1.03164i	0.856696	+	0.515822i	$0.172514\pi$
$89$	6.47277e7i	1.03164i	−0.856696	+	0.515822i	$0.827486\pi$
$90$	0	0
$91$	−4.89410e7	−0.713686
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.06392e6i	0.0253395i
$96$	0	0
$97$	−1.05856e8	−1.19572	−0.597859	−	0.801601i	$-0.703982\pi$
$97$	−1.05856e8	−1.19572	−0.597859	+	0.801601i	$0.703982\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.22043e7i	− 0.213379i	−0.994292	−	0.106689i	$-0.965975\pi$
$101$	− 2.22043e7i	− 0.213379i	0.994292	−	0.106689i	$-0.0340250\pi$
$102$	0	0
$103$	−5.40767e7	−0.480464	−0.240232	−	0.970715i	$-0.577224\pi$
$103$	−5.40767e7	−0.480464	−0.240232	+	0.970715i	$0.577224\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.80582e8i	− 1.37766i	−0.724925	−	0.688828i	$-0.758126\pi$
$107$	− 1.80582e8i	− 1.37766i	0.724925	−	0.688828i	$-0.241874\pi$
$108$	0	0
$109$	−2.14679e8	−1.52084	−0.760421	−	0.649431i	$-0.775007\pi$
$109$	−2.14679e8	−1.52084	−0.760421	+	0.649431i	$0.775007\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.75305e7i	− 0.352845i	−0.984314	−	0.176423i	$-0.943547\pi$
$113$	− 5.75305e7i	− 0.352845i	0.984314	−	0.176423i	$-0.0564526\pi$
$114$	0	0
$115$	4.70139e8	2.68803
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.14475e8i	0.570849i
$120$	0	0
$121$	−4.09948e8	−1.91244
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.91857e7i	− 0.406265i
$126$	0	0
$127$	5.18469e7	0.199300	0.0996502	−	0.995023i	$-0.468228\pi$
$127$	5.18469e7	0.199300	0.0996502	+	0.995023i	$0.468228\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.49316e8i	− 0.846575i	−0.905995	−	0.423288i	$-0.860876\pi$
$131$	− 2.49316e8i	− 0.846575i	0.905995	−	0.423288i	$-0.139124\pi$
$132$	0	0
$133$	1.98918e6	0.00635723
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.39696e7i	0.209977i	0.994473	+	0.104988i	$0.0334805\pi$
$137$	7.39696e7i	0.209977i	−0.994473	+	0.104988i	$0.966519\pi$
$138$	0	0
$139$	−1.68301e8	−0.450845	−0.225423	−	0.974261i	$-0.572376\pi$
$139$	−1.68301e8	−0.450845	−0.225423	+	0.974261i	$0.572376\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.34750e9i	− 3.22244i
$144$	0	0
$145$	−4.43994e8	−1.00440
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.82928e7i	− 0.179135i	−0.995981	−	0.0895675i	$-0.971452\pi$
$149$	− 8.82928e7i	− 0.179135i	0.995981	−	0.0895675i	$-0.0285485\pi$
$150$	0	0
$151$	−6.06244e8	−1.16611	−0.583055	−	0.812433i	$-0.698143\pi$
$151$	−6.06244e8	−1.16611	−0.583055	+	0.812433i	$0.698143\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.13773e9i	− 1.97112i
$156$	0	0
$157$	−8.09783e8	−1.33282	−0.666408	−	0.745587i	$-0.732169\pi$
$157$	−8.09783e8	−1.33282	−0.666408	+	0.745587i	$0.732169\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.53115e8i	− 0.674380i
$162$	0	0
$163$	1.13241e9	1.60418	0.802091	−	0.597202i	$-0.203721\pi$
$163$	1.13241e9	1.60418	0.802091	+	0.597202i	$0.203721\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.62472e8i	0.723161i	0.932341	+	0.361580i	$0.117763\pi$
$167$	5.62472e8i	0.723161i	−0.932341	+	0.361580i	$0.882237\pi$
$168$	0	0
$169$	2.09271e9	2.56544
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.06825e8i	− 0.342536i	−0.985225	−	0.171268i	$-0.945214\pi$
$173$	− 3.06825e8i	− 0.342536i	0.985225	−	0.171268i	$-0.0547863\pi$
$174$	0	0
$175$	−4.50083e8	−0.479889
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.43870e9i	− 1.40139i	−0.713460	−	0.700696i	$-0.752873\pi$
$179$	− 1.43870e9i	− 1.40139i	0.713460	−	0.700696i	$-0.247127\pi$
$180$	0	0
$181$	−1.76557e9	−1.64502	−0.822511	−	0.568749i	$-0.807428\pi$
$181$	−1.76557e9	−1.64502	−0.822511	+	0.568749i	$0.807428\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 8.52802e7i	− 0.0728050i
$186$	0	0
$187$	−3.15184e9	−2.57750
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.13893e9i	1.60717i	0.595189	+	0.803586i	$0.297077\pi$
$191$	2.13893e9i	1.60717i	−0.595189	+	0.803586i	$0.702923\pi$
$192$	0	0
$193$	1.58669e9	1.14357	0.571786	−	0.820403i	$-0.306251\pi$
$193$	1.58669e9	1.14357	0.571786	+	0.820403i	$0.306251\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.13699e9i	− 1.41886i	−0.704778	−	0.709428i	$-0.748954\pi$
$197$	− 2.13699e9i	− 1.41886i	0.704778	−	0.709428i	$-0.251046\pi$
$198$	0	0
$199$	−1.57138e9	−1.00200	−0.501001	−	0.865447i	$-0.667035\pi$
$199$	−1.57138e9	−1.00200	−0.501001	+	0.865447i	$0.667035\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.27917e8i	0.251985i
$204$	0	0
$205$	−4.71886e9	−2.67191
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.47684e7i	0.0287042i
$210$	0	0
$211$	2.35017e8	0.118569	0.0592843	−	0.998241i	$-0.481118\pi$
$211$	2.35017e8	0.118569	0.0592843	+	0.998241i	$0.481118\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.18265e9i	0.553481i
$216$	0	0
$217$	−1.09653e9	−0.494519
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.80292e9i	− 2.85185i
$222$	0	0
$223$	3.66686e8	0.148277	0.0741387	−	0.997248i	$-0.476379\pi$
$223$	3.66686e8	0.148277	0.0741387	+	0.997248i	$0.476379\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.56183e9i	0.588205i	0.955774	+	0.294103i	$0.0950208\pi$
$227$	1.56183e9i	0.588205i	−0.955774	+	0.294103i	$0.904979\pi$
$228$	0	0
$229$	−6.61334e8	−0.240480	−0.120240	−	0.992745i	$-0.538366\pi$
$229$	−6.61334e8	−0.240480	−0.120240	+	0.992745i	$0.538366\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.49089e9i	0.845146i	0.906329	+	0.422573i	$0.138873\pi$
$233$	2.49089e9i	0.845146i	−0.906329	+	0.422573i	$0.861127\pi$
$234$	0	0
$235$	4.01499e9	1.31648
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4.63087e9i	− 1.41929i	−0.704559	−	0.709645i	$-0.748855\pi$
$239$	− 4.63087e9i	− 1.41929i	0.704559	−	0.709645i	$-0.251145\pi$
$240$	0	0
$241$	1.26121e9	0.373870	0.186935	−	0.982372i	$-0.440145\pi$
$241$	1.26121e9	0.373870	0.186935	+	0.982372i	$0.440145\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.75439e8i	0.215220i
$246$	0	0
$247$	−1.18212e8	−0.0317595
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.07047e8i	0.152942i	0.997072	+	0.0764711i	$0.0243653\pi$
$251$	6.07047e8i	0.152942i	−0.997072	+	0.0764711i	$0.975635\pi$
$252$	0	0
$253$	1.24757e10	3.04496
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.23889e9i	0.283988i	0.989868	+	0.141994i	$0.0453513\pi$
$257$	1.23889e9i	0.283988i	−0.989868	+	0.141994i	$0.954649\pi$
$258$	0	0
$259$	−8.21921e7	−0.0182655
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.58471e9i	1.37630i	0.725568	+	0.688151i	$0.241577\pi$
$263$	6.58471e9i	1.37630i	−0.725568	+	0.688151i	$0.758423\pi$
$264$	0	0
$265$	−2.30471e9	−0.467339
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1.83280e9i	− 0.350031i	−0.984566	−	0.175016i	$-0.944002\pi$
$269$	− 1.83280e9i	− 0.350031i	0.984566	−	0.175016i	$-0.0559976\pi$
$270$	0	0
$271$	−2.24143e9	−0.415573	−0.207787	−	0.978174i	$-0.566626\pi$
$271$	−2.24143e9	−0.415573	−0.207787	+	0.978174i	$0.566626\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.23922e10i	− 2.16680i
$276$	0	0
$277$	2.12680e9	0.361250	0.180625	−	0.983552i	$-0.442188\pi$
$277$	2.12680e9	0.361250	0.180625	+	0.983552i	$0.442188\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.85899e9i	− 0.298161i	−0.988825	−	0.149081i	$-0.952369\pi$
$281$	− 1.85899e9i	− 0.298161i	0.988825	−	0.149081i	$-0.0476314\pi$
$282$	0	0
$283$	8.04461e9	1.25418	0.627089	−	0.778948i	$-0.284246\pi$
$283$	8.04461e9	1.25418	0.627089	+	0.778948i	$0.284246\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.54799e9i	0.670335i
$288$	0	0
$289$	−8.93649e9	−1.28108
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.40866e8i	0.0869554i	0.999054	+	0.0434777i	$0.0138437\pi$
$293$	6.40866e8i	0.0869554i	−0.999054	+	0.0434777i	$0.986156\pi$
$294$	0	0
$295$	3.42124e8	0.0451747
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.69274e10i	3.36907i
$300$	0	0
$301$	1.13983e9	0.138858
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.38080e10i	2.75121i
$306$	0	0
$307$	−1.33888e10	−1.50726	−0.753628	−	0.657302i	$-0.771698\pi$
$307$	−1.33888e10	−1.50726	−0.753628	+	0.657302i	$0.771698\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.34877e9i	− 0.357967i	−0.983852	−	0.178984i	$-0.942719\pi$
$311$	− 3.34877e9i	− 0.357967i	0.983852	−	0.178984i	$-0.0572809\pi$
$312$	0	0
$313$	5.14429e9	0.535979	0.267990	−	0.963422i	$-0.413641\pi$
$313$	5.14429e9	0.535979	0.267990	+	0.963422i	$0.413641\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.97576e9i	0.789833i	0.918717	+	0.394916i	$0.129226\pi$
$317$	7.97576e9i	0.789833i	−0.918717	+	0.394916i	$0.870774\pi$
$318$	0	0
$319$	−1.17819e10	−1.13776
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.76501e8i	0.0254031i
$324$	0	0
$325$	2.67473e10	2.39743
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.86961e9i	− 0.330281i
$330$	0	0
$331$	1.88409e10	1.56960	0.784802	−	0.619746i	$-0.212764\pi$
$331$	1.88409e10	1.56960	0.784802	+	0.619746i	$0.212764\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.69345e10i	2.13860i
$336$	0	0
$337$	−1.75019e9	−0.135695	−0.0678477	−	0.997696i	$-0.521613\pi$
$337$	−1.75019e9	−0.135695	−0.0678477	+	0.997696i	$0.521613\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.01910e10i	− 2.23285i
$342$	0	0
$343$	7.47359e8	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.99335e10i	− 1.37488i	−0.726239	−	0.687442i	$-0.758734\pi$
$347$	− 1.99335e10i	− 1.37488i	0.726239	−	0.687442i	$-0.241266\pi$
$348$	0	0
$349$	8.56778e9	0.577519	0.288760	−	0.957402i	$-0.406757\pi$
$349$	8.56778e9	0.577519	0.288760	+	0.957402i	$0.406757\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.73330e9i	0.433640i	0.976212	+	0.216820i	$0.0695684\pi$
$353$	6.73330e9i	0.433640i	−0.976212	+	0.216820i	$0.930432\pi$
$354$	0	0
$355$	5.75114e9	0.362110
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.70314e10i	1.02535i	0.858583	+	0.512675i	$0.171345\pi$
$359$	1.70314e10i	1.02535i	−0.858583	+	0.512675i	$0.828655\pi$
$360$	0	0
$361$	−1.69788e10	−0.999717
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.29410e10i	2.41936i
$366$	0	0
$367$	3.12572e8	0.0172300	0.00861501	−	0.999963i	$-0.497258\pi$
$367$	3.12572e8	0.0172300	0.00861501	+	0.999963i	$0.497258\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.22125e9i	0.117247i
$372$	0	0
$373$	7.57080e9	0.391117	0.195558	−	0.980692i	$-0.437348\pi$
$373$	7.57080e9	0.391117	0.195558	+	0.980692i	$0.437348\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.54300e10i	− 1.25887i
$378$	0	0
$379$	−1.48377e10	−0.719135	−0.359568	−	0.933119i	$-0.617076\pi$
$379$	−1.48377e10	−0.719135	−0.359568	+	0.933119i	$0.617076\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.03636e10i	1.41110i	0.708659	+	0.705551i	$0.249300\pi$
$383$	3.03636e10i	1.41110i	−0.708659	+	0.705551i	$0.750700\pi$
$384$	0	0
$385$	−2.13503e10	−0.971763
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.98750e7i	0.00392501i	0.999998	+	0.00196250i	$0.000624684\pi$
$389$	8.98750e7i	0.00392501i	−0.999998	+	0.00196250i	$0.999375\pi$
$390$	0	0
$391$	6.29841e10	2.69478
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 5.98705e9i	− 0.245937i
$396$	0	0
$397$	2.44217e10	0.983135	0.491568	−	0.870839i	$-0.336424\pi$
$397$	2.44217e10	0.983135	0.491568	+	0.870839i	$0.336424\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.01909e9i	− 0.0394126i	−0.999806	−	0.0197063i	$-0.993727\pi$
$401$	− 1.01909e9i	− 0.0394126i	0.999806	−	0.0197063i	$-0.00627311\pi$
$402$	0	0
$403$	6.51641e10	2.47052
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.26301e9i	− 0.0824723i
$408$	0	0
$409$	−3.92392e10	−1.40226	−0.701128	−	0.713035i	$-0.747320\pi$
$409$	−3.92392e10	−1.40226	−0.701128	+	0.713035i	$0.747320\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 3.29736e8i	− 0.0113335i
$414$	0	0
$415$	4.67323e10	1.57552
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.93930e10i	0.629202i	0.949224	+	0.314601i	$0.101871\pi$
$419$	1.93930e10i	0.629202i	−0.949224	+	0.314601i	$0.898129\pi$
$420$	0	0
$421$	−2.13062e10	−0.678231	−0.339115	−	0.940745i	$-0.610128\pi$
$421$	−2.13062e10	−0.678231	−0.339115	+	0.940745i	$0.610128\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.25627e10i	− 1.91761i
$426$	0	0
$427$	2.29459e10	0.690231
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.16227e10i	− 0.916409i	−0.888847	−	0.458204i	$-0.848493\pi$
$431$	− 3.16227e10i	− 0.916409i	0.888847	−	0.458204i	$-0.151507\pi$
$432$	0	0
$433$	−3.53401e10	−1.00535	−0.502674	−	0.864476i	$-0.667650\pi$
$433$	−3.53401e10	−1.00535	−0.502674	+	0.864476i	$0.667650\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.09445e9i	− 0.0300103i
$438$	0	0
$439$	8.86334e9	0.238638	0.119319	−	0.992856i	$-0.461929\pi$
$439$	8.86334e9	0.238638	0.119319	+	0.992856i	$0.461929\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.94168e10i	0.763802i	0.924203	+	0.381901i	$0.124730\pi$
$443$	2.94168e10i	0.763802i	−0.924203	+	0.381901i	$0.875270\pi$
$444$	0	0
$445$	−6.09469e10	−1.55422
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.79813e10i	0.934511i	0.884122	+	0.467255i	$0.154757\pi$
$449$	3.79813e10i	0.934511i	−0.884122	+	0.467255i	$0.845243\pi$
$450$	0	0
$451$	−1.25220e11	−3.02670
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 4.60823e10i	− 1.07520i
$456$	0	0
$457$	−5.08022e10	−1.16471	−0.582355	−	0.812935i	$-0.697869\pi$
$457$	−5.08022e10	−1.16471	−0.582355	+	0.812935i	$0.697869\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.53198e10i	− 0.339195i	−0.985513	−	0.169598i	$-0.945753\pi$
$461$	− 1.53198e10i	− 0.339195i	0.985513	−	0.169598i	$-0.0542468\pi$
$462$	0	0
$463$	−4.94608e10	−1.07631	−0.538155	−	0.842846i	$-0.680878\pi$
$463$	−4.94608e10	−1.07631	−0.538155	+	0.842846i	$0.680878\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.38377e9i	− 0.0501184i	−0.999686	−	0.0250592i	$-0.992023\pi$
$467$	− 2.38377e9i	− 0.0501184i	0.999686	−	0.0250592i	$-0.00797743\pi$
$468$	0	0
$469$	2.59592e10	0.536537
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.13830e10i	0.626974i
$474$	0	0
$475$	−1.08713e9	−0.0213554
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.23041e10i	0.613642i	0.951767	+	0.306821i	$0.0992653\pi$
$479$	3.23041e10i	0.613642i	−0.951767	+	0.306821i	$0.900735\pi$
$480$	0	0
$481$	4.88446e9	0.0912508
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.96729e10i	− 1.80140i
$486$	0	0
$487$	1.97404e10	0.350946	0.175473	−	0.984484i	$-0.443855\pi$
$487$	1.97404e10	0.350946	0.175473	+	0.984484i	$0.443855\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.44136e10i	0.936227i	0.883668	+	0.468113i	$0.155066\pi$
$491$	5.44136e10i	0.936227i	−0.883668	+	0.468113i	$0.844934\pi$
$492$	0	0
$493$	−5.94815e10	−1.00692
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.54288e9i	− 0.0908469i
$498$	0	0
$499$	−5.04107e10	−0.813057	−0.406528	−	0.913638i	$-0.633261\pi$
$499$	−5.04107e10	−0.813057	−0.406528	+	0.913638i	$0.633261\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 7.41420e10i	− 1.15822i	−0.815248	−	0.579112i	$-0.803399\pi$
$503$	− 7.41420e10i	− 1.15822i	0.815248	−	0.579112i	$-0.196601\pi$
$504$	0	0
$505$	2.09073e10	0.321464
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.62781e10i	0.838433i	0.907886	+	0.419216i	$0.137695\pi$
$509$	5.62781e10i	0.838433i	−0.907886	+	0.419216i	$0.862305\pi$
$510$	0	0
$511$	4.13861e10	0.606975
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5.09180e10i	− 0.723839i
$516$	0	0
$517$	1.06542e11	1.49128
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.91011e10i	1.07357i	0.843718	+	0.536786i	$0.180362\pi$
$521$	7.91011e10i	1.07357i	−0.843718	+	0.536786i	$0.819638\pi$
$522$	0	0
$523$	7.22997e10	0.966340	0.483170	−	0.875527i	$-0.339485\pi$
$523$	7.22997e10	0.966340	0.483170	+	0.875527i	$0.339485\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.52421e11i	− 1.97607i
$528$	0	0
$529$	−1.70994e11	−2.18352
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.70275e11i	− 3.34886i
$534$	0	0
$535$	1.70034e11	2.07549
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.05771e10i	0.243798i
$540$	0	0
$541$	−2.94191e10	−0.343432	−0.171716	−	0.985147i	$-0.554931\pi$
$541$	−2.94191e10	−0.343432	−0.171716	+	0.985147i	$0.554931\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.02139e11i	− 2.29121i
$546$	0	0
$547$	−1.03084e11	−1.15144	−0.575722	−	0.817645i	$-0.695279\pi$
$547$	−1.03084e11	−1.15144	−0.575722	+	0.817645i	$0.695279\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.03359e9i	0.0112135i
$552$	0	0
$553$	−5.77025e9	−0.0617013
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.96527e10i	− 0.619740i	−0.950779	−	0.309870i	$-0.899715\pi$
$557$	− 5.96527e10i	− 0.619740i	0.950779	−	0.309870i	$-0.100285\pi$
$558$	0	0
$559$	−6.77368e10	−0.693709
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.74296e10i	0.173482i	0.996231	+	0.0867411i	$0.0276453\pi$
$563$	1.74296e10i	0.173482i	−0.996231	+	0.0867411i	$0.972355\pi$
$564$	0	0
$565$	5.41701e10	0.531576
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.15223e11i	− 1.09923i	−0.835417	−	0.549617i	$-0.814774\pi$
$569$	− 1.15223e11i	− 1.09923i	0.835417	−	0.549617i	$-0.185226\pi$
$570$	0	0
$571$	−1.40213e10	−0.131900	−0.0659499	−	0.997823i	$-0.521008\pi$
$571$	−1.40213e10	−0.131900	−0.0659499	+	0.997823i	$0.521008\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.47637e11i	2.26539i
$576$	0	0
$577$	−9.05712e10	−0.817122	−0.408561	−	0.912731i	$-0.633969\pi$
$577$	−9.05712e10	−0.817122	−0.408561	+	0.912731i	$0.633969\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.50401e10i	− 0.395271i
$582$	0	0
$583$	−6.11580e10	−0.529394
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.21930e11i	1.02697i	0.858099	+	0.513484i	$0.171646\pi$
$587$	1.21930e11i	1.02697i	−0.858099	+	0.513484i	$0.828354\pi$
$588$	0	0
$589$	−2.64856e9	−0.0220064
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.79477e10i	0.387748i	0.981026	+	0.193874i	$0.0621052\pi$
$593$	4.79477e10i	0.387748i	−0.981026	+	0.193874i	$0.937895\pi$
$594$	0	0
$595$	−1.07788e11	−0.860007
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.32810e10i	0.646902i	0.946245	+	0.323451i	$0.104843\pi$
$599$	8.32810e10i	0.646902i	−0.946245	+	0.323451i	$0.895157\pi$
$600$	0	0
$601$	8.01565e10	0.614385	0.307193	−	0.951647i	$-0.400610\pi$
$601$	8.01565e10	0.614385	0.307193	+	0.951647i	$0.400610\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.86002e11i	− 2.88116i
$606$	0	0
$607$	−2.30847e10	−0.170047	−0.0850234	−	0.996379i	$-0.527097\pi$
$607$	−2.30847e10	−0.170047	−0.0850234	+	0.996379i	$0.527097\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.29961e11i	1.65002i
$612$	0	0
$613$	5.52096e10	0.390996	0.195498	−	0.980704i	$-0.437368\pi$
$613$	5.52096e10	0.390996	0.195498	+	0.980704i	$0.437368\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.43940e11i	0.993208i	0.867977	+	0.496604i	$0.165420\pi$
$617$	1.43940e11i	0.993208i	−0.867977	+	0.496604i	$0.834580\pi$
$618$	0	0
$619$	9.10695e10	0.620312	0.310156	−	0.950686i	$-0.399619\pi$
$619$	9.10695e10	0.620312	0.310156	+	0.950686i	$0.399619\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.87399e10i	0.389925i
$624$	0	0
$625$	−1.00344e11	−0.657612
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.14249e10i	− 0.0729878i
$630$	0	0
$631$	−1.61095e11	−1.01616	−0.508081	−	0.861309i	$-0.669645\pi$
$631$	−1.61095e11	−1.01616	−0.508081	+	0.861309i	$0.669645\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.88184e10i	0.300254i
$636$	0	0
$637$	−4.44136e10	−0.269748
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.19074e11i	− 0.705317i	−0.935752	−	0.352659i	$-0.885278\pi$
$641$	− 1.19074e11i	− 0.705317i	0.935752	−	0.352659i	$-0.114722\pi$
$642$	0	0
$643$	−2.89565e11	−1.69396	−0.846979	−	0.531627i	$-0.821581\pi$
$643$	−2.89565e11	−1.69396	−0.846979	+	0.531627i	$0.821581\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.67714e10i	− 0.438109i	−0.975713	−	0.219055i	$-0.929703\pi$
$647$	− 7.67714e10i	− 0.438109i	0.975713	−	0.219055i	$-0.0702973\pi$
$648$	0	0
$649$	9.07866e9	0.0511732
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.10422e11i	− 0.607302i	−0.952783	−	0.303651i	$-0.901794\pi$
$653$	− 1.10422e11i	− 0.607302i	0.952783	−	0.303651i	$-0.0982057\pi$
$654$	0	0
$655$	2.34753e11	1.27540
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.91552e10i	0.154588i	0.997008	+	0.0772938i	$0.0246280\pi$
$659$	2.91552e10i	0.154588i	−0.997008	+	0.0772938i	$0.975372\pi$
$660$	0	0
$661$	5.18282e10	0.271494	0.135747	−	0.990744i	$-0.456657\pi$
$661$	5.18282e10	0.271494	0.135747	+	0.990744i	$0.456657\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.87299e9i	0.00957743i
$666$	0	0
$667$	2.35441e11	1.18954
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.31774e11i	3.11653i
$672$	0	0
$673$	2.54468e11	1.24043	0.620217	−	0.784431i	$-0.287045\pi$
$673$	2.54468e11	1.24043	0.620217	+	0.784431i	$0.287045\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.05798e11i	1.45573i	0.685723	+	0.727863i	$0.259486\pi$
$677$	3.05798e11i	1.45573i	−0.685723	+	0.727863i	$0.740514\pi$
$678$	0	0
$679$	−9.60637e10	−0.451939
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.74827e11i	− 0.803387i	−0.915774	−	0.401693i	$-0.868422\pi$
$683$	− 1.74827e11i	− 0.803387i	0.915774	−	0.401693i	$-0.131578\pi$
$684$	0	0
$685$	−6.96490e10	−0.316339
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.32003e11i	− 0.585744i
$690$	0	0
$691$	2.89579e11	1.27015	0.635074	−	0.772451i	$-0.280970\pi$
$691$	2.89579e11	1.27015	0.635074	+	0.772451i	$0.280970\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.58470e11i	− 0.679217i
$696$	0	0
$697$	−6.32182e11	−2.67862
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.11338e11i	− 0.461073i	−0.973064	−	0.230537i	$-0.925952\pi$
$701$	− 1.11338e11i	− 0.461073i	0.973064	−	0.230537i	$-0.0740481\pi$
$702$	0	0
$703$	−1.98526e8	−0.000812825 0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.01502e10i	− 0.0806496i
$708$	0	0
$709$	4.37749e11	1.73237	0.866183	−	0.499727i	$-0.166566\pi$
$709$	4.37749e11	1.73237	0.866183	+	0.499727i	$0.166566\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.03315e11i	2.33446i
$714$	0	0
$715$	1.26879e12	4.85474
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.41051e10i	0.352126i	0.984379	+	0.176063i	$0.0563362\pi$
$719$	9.41051e10i	0.352126i	−0.984379	+	0.176063i	$0.943664\pi$
$720$	0	0
$721$	−4.90742e10	−0.181598
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.33865e11i	− 0.846475i
$726$	0	0
$727$	−3.88630e9	−0.0139123	−0.00695614	−	0.999976i	$-0.502214\pi$
$727$	−3.88630e9	−0.0139123	−0.00695614	+	0.999976i	$0.502214\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.58439e11i	0.554870i
$732$	0	0
$733$	−5.00744e11	−1.73460	−0.867302	−	0.497783i	$-0.834148\pi$
$733$	−5.00744e11	−1.73460	−0.867302	+	0.497783i	$0.834148\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.14738e11i	2.42257i
$738$	0	0
$739$	3.79955e11	1.27396	0.636978	−	0.770882i	$-0.280184\pi$
$739$	3.79955e11	1.27396	0.636978	+	0.770882i	$0.280184\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.86403e11i	0.611643i	0.952089	+	0.305822i	$0.0989312\pi$
$743$	1.86403e11i	0.611643i	−0.952089	+	0.305822i	$0.901069\pi$
$744$	0	0
$745$	8.31355e10	0.269874
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.63877e11i	− 0.520705i
$750$	0	0
$751$	−2.41035e11	−0.757740	−0.378870	−	0.925450i	$-0.623687\pi$
$751$	−2.41035e11	−0.757740	−0.378870	+	0.925450i	$0.623687\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.70832e11i	− 1.75679i
$756$	0	0
$757$	−1.99372e11	−0.607127	−0.303564	−	0.952811i	$-0.598177\pi$
$757$	−1.99372e11	−0.607127	−0.303564	+	0.952811i	$0.598177\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.99662e11i	1.19166i	0.803109	+	0.595832i	$0.203178\pi$
$761$	3.99662e11i	1.19166i	−0.803109	+	0.595832i	$0.796822\pi$
$762$	0	0
$763$	−1.94820e11	−0.574824
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.95953e10i	0.0566201i
$768$	0	0
$769$	5.96711e11	1.70631	0.853156	−	0.521656i	$-0.174685\pi$
$769$	5.96711e11	1.70631	0.853156	+	0.521656i	$0.174685\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.35295e11i	− 0.939095i	−0.882907	−	0.469548i	$-0.844417\pi$
$773$	− 3.35295e11i	− 0.939095i	0.882907	−	0.469548i	$-0.155583\pi$
$774$	0	0
$775$	5.99278e11	1.66120
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.09852e10i	0.0298303i
$780$	0	0
$781$	1.52613e11	0.410192
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7.62483e11i	− 2.00794i
$786$	0	0
$787$	2.10892e11	0.549745	0.274873	−	0.961481i	$-0.411364\pi$
$787$	2.10892e11	0.549745	0.274873	+	0.961481i	$0.411364\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 5.22085e10i	− 0.133363i
$792$	0	0
$793$	−1.36362e12	−3.44826
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.55490e11i	− 0.385363i	−0.981261	−	0.192681i	$-0.938282\pi$
$797$	− 1.55490e11i	− 0.385363i	0.981261	−	0.192681i	$-0.0617184\pi$
$798$	0	0
$799$	5.37885e11	1.31978
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.13949e12i	2.74061i
$804$	0	0
$805$	4.26648e11	1.01598
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.07779e10i	− 0.188581i	−0.995545	−	0.0942906i	$-0.969942\pi$
$809$	− 8.07779e10i	− 0.188581i	0.995545	−	0.0942906i	$-0.0300583\pi$
$810$	0	0
$811$	1.70633e10	0.0394439	0.0197220	−	0.999806i	$-0.493722\pi$
$811$	1.70633e10	0.0394439	0.0197220	+	0.999806i	$0.493722\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.06626e12i	2.41677i
$816$	0	0
$817$	2.75313e9	0.00617928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.91523e11i	− 0.641654i	−0.947138	−	0.320827i	$-0.896039\pi$
$821$	− 2.91523e11i	− 0.641654i	0.947138	−	0.320827i	$-0.103961\pi$
$822$	0	0
$823$	−3.90563e11	−0.851319	−0.425659	−	0.904883i	$-0.639958\pi$
$823$	−3.90563e11	−0.851319	−0.425659	+	0.904883i	$0.639958\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.59920e11i	1.41081i	0.708804	+	0.705406i	$0.249235\pi$
$827$	6.59920e11i	1.41081i	−0.708804	+	0.705406i	$0.750765\pi$
$828$	0	0
$829$	1.61290e10	0.0341500	0.0170750	−	0.999854i	$-0.494565\pi$
$829$	1.61290e10	0.0341500	0.0170750	+	0.999854i	$0.494565\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.03885e11i	0.215761i
$834$	0	0
$835$	−5.29617e11	−1.08947
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.03605e11i	1.21816i	0.793108	+	0.609081i	$0.208462\pi$
$839$	6.03605e11i	1.21816i	−0.793108	+	0.609081i	$0.791538\pi$
$840$	0	0
$841$	2.77899e11	0.555524
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.97047e12i	3.86494i
$846$	0	0
$847$	−3.72024e11	−0.722833
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.52222e10i	0.0862251i
$852$	0	0
$853$	−6.39297e11	−1.20755	−0.603777	−	0.797153i	$-0.706338\pi$
$853$	−6.39297e11	−1.20755	−0.603777	+	0.797153i	$0.706338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.07084e11i	− 1.12545i	−0.826645	−	0.562724i	$-0.809753\pi$
$857$	− 6.07084e11i	− 1.12545i	0.826645	−	0.562724i	$-0.190247\pi$
$858$	0	0
$859$	2.42260e11	0.444947	0.222474	−	0.974939i	$-0.428587\pi$
$859$	2.42260e11	0.444947	0.222474	+	0.974939i	$0.428587\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 7.54726e11i	− 1.36065i	−0.732911	−	0.680324i	$-0.761839\pi$
$863$	− 7.54726e11i	− 1.36065i	0.732911	−	0.680324i	$-0.238161\pi$
$864$	0	0
$865$	2.88903e11	0.516044
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.58873e11i	− 0.278594i
$870$	0	0
$871$	−1.54269e12	−2.68043
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 9.00103e10i	− 0.153554i
$876$	0	0
$877$	−2.98896e11	−0.505268	−0.252634	−	0.967562i	$-0.581297\pi$
$877$	−2.98896e11	−0.505268	−0.252634	+	0.967562i	$0.581297\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.04520e12i	− 1.73499i	−0.497443	−	0.867497i	$-0.665728\pi$
$881$	− 1.04520e12i	− 1.73499i	0.497443	−	0.867497i	$-0.334272\pi$
$882$	0	0
$883$	1.05259e11	0.173148	0.0865740	−	0.996245i	$-0.472408\pi$
$883$	1.05259e11	0.173148	0.0865740	+	0.996245i	$0.472408\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.58413e11i	− 1.54831i	−0.632995	−	0.774156i	$-0.718175\pi$
$887$	− 9.58413e11i	− 1.54831i	0.632995	−	0.774156i	$-0.281825\pi$
$888$	0	0
$889$	4.70507e10	0.0753284
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.34663e9i	− 0.0146977i
$894$	0	0
$895$	1.35467e12	2.11125
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.69764e11i	− 0.872280i
$900$	0	0
$901$	−3.08759e11	−0.468512
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.66244e12i	− 2.47829i
$906$	0	0
$907$	7.68631e11	1.13577	0.567883	−	0.823109i	$-0.307763\pi$
$907$	7.68631e11	1.13577	0.567883	+	0.823109i	$0.307763\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.13758e11i	0.891095i	0.895258	+	0.445547i	$0.146991\pi$
$911$	6.13758e11i	0.891095i	−0.895258	+	0.445547i	$0.853009\pi$
$912$	0	0
$913$	1.24010e12	1.78473
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.26253e11i	− 0.319975i
$918$	0	0
$919$	4.29816e11	0.602589	0.301294	−	0.953531i	$-0.402581\pi$
$919$	4.29816e11	0.602589	0.301294	+	0.953531i	$0.402581\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.29399e11i	0.453853i
$924$	0	0
$925$	4.49197e10	0.0613578
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 4.85111e11i	− 0.651296i	−0.945491	−	0.325648i	$-0.894418\pi$
$929$	− 4.85111e11i	− 0.651296i	0.945491	−	0.325648i	$-0.105582\pi$
$930$	0	0
$931$	1.80517e9	0.00240281
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2.96774e12i	− 3.88311i
$936$	0	0
$937$	−5.41736e11	−0.702796	−0.351398	−	0.936226i	$-0.614294\pi$
$937$	−5.41736e11	−0.702796	−0.351398	+	0.936226i	$0.614294\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 4.75823e11i	− 0.606857i	−0.952854	−	0.303429i	$-0.901869\pi$
$941$	− 4.75823e11i	− 0.606857i	0.952854	−	0.303429i	$-0.0981314\pi$
$942$	0	0
$943$	2.50231e12	3.16442
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.66078e11i	− 0.828181i	−0.910236	−	0.414091i	$-0.864100\pi$
$947$	− 6.66078e11i	− 0.828181i	0.910236	−	0.414091i	$-0.135900\pi$
$948$	0	0
$949$	−2.45947e12	−3.03233
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.63438e12i	1.98145i	0.135895	+	0.990723i	$0.456609\pi$
$953$	1.63438e12i	1.98145i	−0.135895	+	0.990723i	$0.543391\pi$
$954$	0	0
$955$	−2.01399e12	−2.42127
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.71269e10i	0.0793638i
$960$	0	0
$961$	6.07125e11	0.711843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.49401e12i	1.72284i
$966$	0	0
$967$	−1.67087e12	−1.91090	−0.955449	−	0.295155i	$-0.904629\pi$
$967$	−1.67087e12	−1.91090	−0.955449	+	0.295155i	$0.904629\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.93837e11i	0.893006i	0.894782	+	0.446503i	$0.147331\pi$
$971$	7.93837e11i	0.893006i	−0.894782	+	0.446503i	$0.852669\pi$
$972$	0	0
$973$	−1.52732e11	−0.170404
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.20378e11i	0.351629i	0.984423	+	0.175815i	$0.0562559\pi$
$977$	3.20378e11i	0.351629i	−0.984423	+	0.175815i	$0.943744\pi$
$978$	0	0
$979$	−1.61729e12	−1.76059
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.87282e11i	0.736073i	0.929811	+	0.368037i	$0.119970\pi$
$983$	6.87282e11i	0.736073i	−0.929811	+	0.368037i	$0.880030\pi$
$984$	0	0
$985$	2.01217e12	2.13756
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 6.27134e11i	− 0.655503i
$990$	0	0
$991$	−1.00613e12	−1.04318	−0.521590	−	0.853196i	$-0.674661\pi$
$991$	−1.00613e12	−1.04318	−0.521590	+	0.853196i	$0.674661\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.47959e12i	− 1.50956i
$996$	0	0
$997$	1.08814e12	1.10130	0.550648	−	0.834738i	$-0.314381\pi$
$997$	1.08814e12	1.10130	0.550648	+	0.834738i	$0.314381\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.c.a.197.15	yes	16
3.2	odd	2	inner	252.9.c.a.197.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.2	✓	16	3.2	odd	2	inner
252.9.c.a.197.15	yes	16	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+941.588i q^{5} +907.493 q^{7} +O(q^{10})\)	\(q+941.588i q^{5} +907.493 q^{7} +24986.1i q^{11} -53929.9 q^{13} +126144. i q^{17} +2191.95 q^{19} -499304. i q^{23} -495964. q^{25} +471537. i q^{29} -1.20831e6 q^{31} +854485. i q^{35} -90570.5 q^{37} +5.01160e6i q^{41} +1.25602e6 q^{43} -4.26407e6i q^{47} +823543. q^{49} +2.44768e6i q^{53} -2.35266e7 q^{55} -363348. i q^{59} +2.52850e7 q^{61} -5.07798e7i q^{65} +2.86054e7 q^{67} -6.10791e6i q^{71} +4.56049e7 q^{73} +2.26747e7i q^{77} -6.35845e6 q^{79} -4.96314e7i q^{83} -1.18775e8 q^{85} +6.47277e7i q^{89} -4.89410e7 q^{91} +2.06392e6i q^{95} -1.05856e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) \) 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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