LMFDB - Embedded newform 252.9.c.a.197.8

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			197.8
Root			$-221.696i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-221.696i q^{5} -907.493 q^{7} +O(q^{10})$	$q-221.696i q^{5} -907.493 q^{7} -19481.4i q^{11} -19141.3 q^{13} +134659. i q^{17} -194587. q^{19} -42602.7i q^{23} +341476. q^{25} -814906. i q^{29} -203649. q^{31} +201187. i q^{35} +1.43832e6 q^{37} +2.77633e6i q^{41} -5.04485e6 q^{43} +5.42366e6i q^{47} +823543. q^{49} -1.34818e7i q^{53} -4.31894e6 q^{55} +1.38302e7i q^{59} +2.26176e7 q^{61} +4.24355e6i q^{65} +3.91134e7 q^{67} -3.02970e7i q^{71} -5.42017e7 q^{73} +1.76792e7i q^{77} -2.10178e7 q^{79} +1.30858e7i q^{83} +2.98533e7 q^{85} +1.78236e7i q^{89} +1.73706e7 q^{91} +4.31392e7i q^{95} +6.89791e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 221.696i	− 0.354713i	−0.984147	−	0.177357i	$-0.943245\pi$
$5$	− 221.696i	− 0.354713i	0.984147	−	0.177357i	$-0.0567546\pi$
$6$	0	0
$7$	−907.493	−0.377964
$7$	−907.493	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 19481.4i	− 1.33060i	−0.746574	−	0.665302i	$-0.768303\pi$
$11$	− 19481.4i	− 1.33060i	0.746574	−	0.665302i	$-0.231697\pi$
$12$	0	0
$13$	−19141.3	−0.670191	−0.335095	−	0.942184i	$-0.608769\pi$
$13$	−19141.3	−0.670191	−0.335095	+	0.942184i	$0.608769\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	134659.i	1.61227i	0.591729	+	0.806137i	$0.298446\pi$
$17$	134659.i	1.61227i	−0.591729	+	0.806137i	$0.701554\pi$
$18$	0	0
$19$	−194587.	−1.49314	−0.746570	−	0.665307i	$-0.768301\pi$
$19$	−194587.	−1.49314	−0.746570	+	0.665307i	$0.768301\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 42602.7i	− 0.152239i	−0.997099	−	0.0761194i	$-0.975747\pi$
$23$	− 42602.7i	− 0.152239i	0.997099	−	0.0761194i	$-0.0242530\pi$
$24$	0	0
$25$	341476.	0.874178
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 814906.i	− 1.15217i	−0.817391	−	0.576084i	$-0.804580\pi$
$29$	− 814906.i	− 1.15217i	0.817391	−	0.576084i	$-0.195420\pi$
$30$	0	0
$31$	−203649.	−0.220514	−0.110257	−	0.993903i	$-0.535167\pi$
$31$	−203649.	−0.220514	−0.110257	+	0.993903i	$0.535167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	201187.i	0.134069i
$36$	0	0
$37$	1.43832e6	0.767449	0.383725	−	0.923448i	$-0.374641\pi$
$37$	1.43832e6	0.767449	0.383725	+	0.923448i	$0.374641\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.77633e6i	0.982507i	0.871017	+	0.491253i	$0.163461\pi$
$41$	2.77633e6i	0.982507i	−0.871017	+	0.491253i	$0.836539\pi$
$42$	0	0
$43$	−5.04485e6	−1.47562	−0.737810	−	0.675009i	$-0.764140\pi$
$43$	−5.04485e6	−1.47562	−0.737810	+	0.675009i	$0.764140\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.42366e6i	1.11148i	0.831357	+	0.555739i	$0.187565\pi$
$47$	5.42366e6i	1.11148i	−0.831357	+	0.555739i	$0.812435\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.34818e7i	− 1.70861i	−0.519772	−	0.854305i	$-0.673983\pi$
$53$	− 1.34818e7i	− 1.70861i	0.519772	−	0.854305i	$-0.326017\pi$
$54$	0	0
$55$	−4.31894e6	−0.471983
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.38302e7i	1.14135i	0.821175	+	0.570676i	$0.193319\pi$
$59$	1.38302e7i	1.14135i	−0.821175	+	0.570676i	$0.806681\pi$
$60$	0	0
$61$	2.26176e7	1.63353	0.816766	−	0.576969i	$-0.195765\pi$
$61$	2.26176e7	1.63353	0.816766	+	0.576969i	$0.195765\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.24355e6i	0.237726i
$66$	0	0
$67$	3.91134e7	1.94100	0.970502	−	0.241091i	$-0.0775054\pi$
$67$	3.91134e7	1.94100	0.970502	+	0.241091i	$0.0775054\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 3.02970e7i	− 1.19225i	−0.802893	−	0.596123i	$-0.796707\pi$
$71$	− 3.02970e7i	− 1.19225i	0.802893	−	0.596123i	$-0.203293\pi$
$72$	0	0
$73$	−5.42017e7	−1.90863	−0.954315	−	0.298802i	$-0.903413\pi$
$73$	−5.42017e7	−1.90863	−0.954315	+	0.298802i	$0.903413\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.76792e7i	0.502921i
$78$	0	0
$79$	−2.10178e7	−0.539609	−0.269804	−	0.962915i	$-0.586959\pi$
$79$	−2.10178e7	−0.539609	−0.269804	+	0.962915i	$0.586959\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.30858e7i	0.275732i	0.990451	+	0.137866i	$0.0440244\pi$
$83$	1.30858e7i	0.275732i	−0.990451	+	0.137866i	$0.955976\pi$
$84$	0	0
$85$	2.98533e7	0.571895
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.78236e7i	0.284077i	0.989861	+	0.142038i	$0.0453657\pi$
$89$	1.78236e7i	0.284077i	−0.989861	+	0.142038i	$0.954634\pi$
$90$	0	0
$91$	1.73706e7	0.253308
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.31392e7i	0.529636i
$96$	0	0
$97$	6.89791e7	0.779167	0.389583	−	0.920991i	$-0.372619\pi$
$97$	6.89791e7	0.779167	0.389583	+	0.920991i	$0.372619\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.90594e8i	1.83157i	0.401670	+	0.915785i	$0.368430\pi$
$101$	1.90594e8i	1.83157i	−0.401670	+	0.915785i	$0.631570\pi$
$102$	0	0
$103$	−2.00729e7	−0.178345	−0.0891727	−	0.996016i	$-0.528422\pi$
$103$	−2.00729e7	−0.178345	−0.0891727	+	0.996016i	$0.528422\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.00201e7i	− 0.0764427i	−0.999269	−	0.0382214i	$-0.987831\pi$
$107$	− 1.00201e7i	− 0.0764427i	0.999269	−	0.0382214i	$-0.0121692\pi$
$108$	0	0
$109$	1.41310e8	1.00107	0.500537	−	0.865715i	$-0.333136\pi$
$109$	1.41310e8	1.00107	0.500537	+	0.865715i	$0.333136\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.29538e8i	1.40780i	0.710299	+	0.703901i	$0.248560\pi$
$113$	2.29538e8i	1.40780i	−0.710299	+	0.703901i	$0.751440\pi$
$114$	0	0
$115$	−9.44484e6	−0.0540012
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.22202e8i	− 0.609382i
$120$	0	0
$121$	−1.65165e8	−0.770508
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.62304e8i	− 0.664796i
$126$	0	0
$127$	4.60959e8	1.77193	0.885967	−	0.463748i	$-0.153496\pi$
$127$	4.60959e8	1.77193	0.885967	+	0.463748i	$0.153496\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.50919e8i	1.19158i	0.803141	+	0.595788i	$0.203160\pi$
$131$	3.50919e8i	1.19158i	−0.803141	+	0.595788i	$0.796840\pi$
$132$	0	0
$133$	1.76587e8	0.564354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.62021e8i	0.743796i	0.928274	+	0.371898i	$0.121293\pi$
$137$	2.62021e8i	0.743796i	−0.928274	+	0.371898i	$0.878707\pi$
$138$	0	0
$139$	3.38052e8	0.905576	0.452788	−	0.891618i	$-0.350430\pi$
$139$	3.38052e8	0.905576	0.452788	+	0.891618i	$0.350430\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.72899e8i	0.891759i
$144$	0	0
$145$	−1.80661e8	−0.408689
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.73781e8i	1.56990i	0.619557	+	0.784952i	$0.287312\pi$
$149$	7.73781e8i	1.56990i	−0.619557	+	0.784952i	$0.712688\pi$
$150$	0	0
$151$	−3.78380e8	−0.727814	−0.363907	−	0.931435i	$-0.618557\pi$
$151$	−3.78380e8	−0.727814	−0.363907	+	0.931435i	$0.618557\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.51482e7i	0.0782192i
$156$	0	0
$157$	7.38439e8	1.21539	0.607695	−	0.794170i	$-0.292094\pi$
$157$	7.38439e8	1.21539	0.607695	+	0.794170i	$0.292094\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.86616e7i	0.0575409i
$162$	0	0
$163$	1.26252e9	1.78850	0.894250	−	0.447567i	$-0.147709\pi$
$163$	1.26252e9	1.78850	0.894250	+	0.447567i	$0.147709\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.70075e8i	0.861505i	0.902470	+	0.430752i	$0.141752\pi$
$167$	6.70075e8i	0.861505i	−0.902470	+	0.430752i	$0.858248\pi$
$168$	0	0
$169$	−4.49341e8	−0.550845
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.97871e8i	− 0.220902i	−0.993882	−	0.110451i	$-0.964771\pi$
$173$	− 1.97871e8i	− 0.220902i	0.993882	−	0.110451i	$-0.0352294\pi$
$174$	0	0
$175$	−3.09887e8	−0.330408
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.26692e9i	1.23406i	0.786938	+	0.617031i	$0.211665\pi$
$179$	1.26692e9i	1.23406i	−0.786938	+	0.617031i	$0.788335\pi$
$180$	0	0
$181$	9.41676e8	0.877379	0.438689	−	0.898639i	$-0.355443\pi$
$181$	9.41676e8	0.877379	0.438689	+	0.898639i	$0.355443\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.18870e8i	− 0.272224i
$186$	0	0
$187$	2.62334e9	2.14530
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.87580e8i	0.291225i	0.989342	+	0.145612i	$0.0465152\pi$
$191$	3.87580e8i	0.291225i	−0.989342	+	0.145612i	$0.953485\pi$
$192$	0	0
$193$	5.95489e8	0.429185	0.214593	−	0.976704i	$-0.431158\pi$
$193$	5.95489e8	0.429185	0.214593	+	0.976704i	$0.431158\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.92777e9i	1.27994i	0.768400	+	0.639970i	$0.221053\pi$
$197$	1.92777e9i	1.27994i	−0.768400	+	0.639970i	$0.778947\pi$
$198$	0	0
$199$	−1.50314e9	−0.958491	−0.479246	−	0.877681i	$-0.659090\pi$
$199$	−1.50314e9	−0.958491	−0.479246	+	0.877681i	$0.659090\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.39521e8i	0.435478i
$204$	0	0
$205$	6.15501e8	0.348508
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.79083e9i	1.98678i
$210$	0	0
$211$	−2.59815e9	−1.31079	−0.655397	−	0.755285i	$-0.727499\pi$
$211$	−2.59815e9	−1.31079	−0.655397	+	0.755285i	$0.727499\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.11842e9i	0.523422i
$216$	0	0
$217$	1.84810e8	0.0833464
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.57755e9i	− 1.08053i
$222$	0	0
$223$	−1.32831e9	−0.537130	−0.268565	−	0.963262i	$-0.586549\pi$
$223$	−1.32831e9	−0.537130	−0.268565	+	0.963262i	$0.586549\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.87981e9i	− 0.707964i	−0.935252	−	0.353982i	$-0.884827\pi$
$227$	− 1.87981e9i	− 0.707964i	0.935252	−	0.353982i	$-0.115173\pi$
$228$	0	0
$229$	−5.36880e7	−0.0195225	−0.00976125	−	0.999952i	$-0.503107\pi$
$229$	−5.36880e7	−0.0195225	−0.00976125	+	0.999952i	$0.503107\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.03976e9i	− 0.352784i	−0.984320	−	0.176392i	$-0.943557\pi$
$233$	− 1.03976e9i	− 0.352784i	0.984320	−	0.176392i	$-0.0564426\pi$
$234$	0	0
$235$	1.20240e9	0.394256
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.06002e9i	0.631364i	0.948865	+	0.315682i	$0.102233\pi$
$239$	2.06002e9i	0.631364i	−0.948865	+	0.315682i	$0.897767\pi$
$240$	0	0
$241$	5.91153e9	1.75239	0.876197	−	0.481954i	$-0.160073\pi$
$241$	5.91153e9	1.75239	0.876197	+	0.481954i	$0.160073\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.82576e8i	− 0.0506733i
$246$	0	0
$247$	3.72466e9	1.00069
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.27398e9i	− 1.58070i	−0.612658	−	0.790348i	$-0.709900\pi$
$251$	− 6.27398e9i	− 1.58070i	0.612658	−	0.790348i	$-0.290100\pi$
$252$	0	0
$253$	−8.29959e8	−0.202570
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.16469e9i	− 0.954663i	−0.878723	−	0.477331i	$-0.841604\pi$
$257$	− 4.16469e9i	− 0.954663i	0.878723	−	0.477331i	$-0.158396\pi$
$258$	0	0
$259$	−1.30527e9	−0.290069
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.30124e9i	1.10804i	0.832504	+	0.554019i	$0.186906\pi$
$263$	5.30124e9i	1.10804i	−0.832504	+	0.554019i	$0.813094\pi$
$264$	0	0
$265$	−2.98885e9	−0.606067
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.25848e9i	1.19525i	0.801775	+	0.597626i	$0.203889\pi$
$269$	6.25848e9i	1.19525i	−0.801775	+	0.597626i	$0.796111\pi$
$270$	0	0
$271$	−1.62468e9	−0.301226	−0.150613	−	0.988593i	$-0.548125\pi$
$271$	−1.62468e9	−0.301226	−0.150613	+	0.988593i	$0.548125\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.65242e9i	− 1.16319i
$276$	0	0
$277$	−2.99114e9	−0.508063	−0.254032	−	0.967196i	$-0.581757\pi$
$277$	−2.99114e9	−0.508063	−0.254032	+	0.967196i	$0.581757\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.36189e9i	0.378822i	0.981898	+	0.189411i	$0.0606579\pi$
$281$	2.36189e9i	0.378822i	−0.981898	+	0.189411i	$0.939342\pi$
$282$	0	0
$283$	−6.63475e9	−1.03438	−0.517188	−	0.855872i	$-0.673021\pi$
$283$	−6.63475e9	−1.03438	−0.517188	+	0.855872i	$0.673021\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.51950e9i	− 0.371353i
$288$	0	0
$289$	−1.11572e10	−1.59943
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.55688e9i	− 0.889665i	−0.895614	−	0.444833i	$-0.853263\pi$
$293$	− 6.55688e9i	− 0.889665i	0.895614	−	0.444833i	$-0.146737\pi$
$294$	0	0
$295$	3.06609e9	0.404853
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.15471e8i	0.102029i
$300$	0	0
$301$	4.57816e9	0.557732
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.01423e9i	− 0.579436i
$306$	0	0
$307$	1.12109e10	1.26208	0.631042	−	0.775749i	$-0.282628\pi$
$307$	1.12109e10	1.26208	0.631042	+	0.775749i	$0.282628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 7.89689e8i	− 0.0844140i	−0.999109	−	0.0422070i	$-0.986561\pi$
$311$	− 7.89689e8i	− 0.0844140i	0.999109	−	0.0422070i	$-0.0134389\pi$
$312$	0	0
$313$	−1.75039e9	−0.182371	−0.0911857	−	0.995834i	$-0.529066\pi$
$313$	−1.75039e9	−0.182371	−0.0911857	+	0.995834i	$0.529066\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.03386e8i	0.0102383i	0.999987	+	0.00511913i	$0.00162948\pi$
$317$	1.03386e8i	0.0102383i	−0.999987	+	0.00511913i	$0.998371\pi$
$318$	0	0
$319$	−1.58755e10	−1.53308
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.62029e10i	− 2.40735i
$324$	0	0
$325$	−6.53630e9	−0.585866
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.92193e9i	− 0.420099i
$330$	0	0
$331$	−1.71555e10	−1.42919	−0.714596	−	0.699538i	$-0.753389\pi$
$331$	−1.71555e10	−1.42919	−0.714596	+	0.699538i	$0.753389\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 8.67128e9i	− 0.688500i
$336$	0	0
$337$	3.46235e9	0.268443	0.134221	−	0.990951i	$-0.457147\pi$
$337$	3.46235e9	0.268443	0.134221	+	0.990951i	$0.457147\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.96737e9i	0.293417i
$342$	0	0
$343$	−7.47359e8	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.02435e9i	− 0.622441i	−0.950338	−	0.311220i	$-0.899262\pi$
$347$	− 9.02435e9i	− 0.622441i	0.950338	−	0.311220i	$-0.100738\pi$
$348$	0	0
$349$	−1.16999e10	−0.788646	−0.394323	−	0.918972i	$-0.629021\pi$
$349$	−1.16999e10	−0.788646	−0.394323	+	0.918972i	$0.629021\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.18232e10i	− 1.40546i	−0.711456	−	0.702731i	$-0.751964\pi$
$353$	− 2.18232e10i	− 1.40546i	0.711456	−	0.702731i	$-0.248036\pi$
$354$	0	0
$355$	−6.71672e9	−0.422906
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.38983e9i	0.264283i	0.991231	+	0.132142i	$0.0421854\pi$
$359$	4.38983e9i	0.264283i	−0.991231	+	0.132142i	$0.957815\pi$
$360$	0	0
$361$	2.08807e10	1.22947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.20163e10i	0.677017i
$366$	0	0
$367$	−2.80399e10	−1.54565	−0.772826	−	0.634618i	$-0.781158\pi$
$367$	−2.80399e10	−1.54565	−0.772826	+	0.634618i	$0.781158\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.22346e10i	0.645794i
$372$	0	0
$373$	2.38135e10	1.23023	0.615117	−	0.788436i	$-0.289109\pi$
$373$	2.38135e10	1.23023	0.615117	+	0.788436i	$0.289109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.55984e10i	0.772172i
$378$	0	0
$379$	−1.46006e10	−0.707644	−0.353822	−	0.935313i	$-0.615118\pi$
$379$	−1.46006e10	−0.707644	−0.353822	+	0.935313i	$0.615118\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.86050e10i	− 0.864639i	−0.901721	−	0.432319i	$-0.857695\pi$
$383$	− 1.86050e10i	− 0.864639i	0.901721	−	0.432319i	$-0.142305\pi$
$384$	0	0
$385$	3.91941e9	0.178393
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.88545e10i	− 1.26013i	−0.776542	−	0.630065i	$-0.783028\pi$
$389$	− 2.88545e10i	− 1.26013i	0.776542	−	0.630065i	$-0.216972\pi$
$390$	0	0
$391$	5.73682e9	0.245451
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.65956e9i	0.191406i
$396$	0	0
$397$	−1.40951e10	−0.567421	−0.283711	−	0.958910i	$-0.591566\pi$
$397$	−1.40951e10	−0.567421	−0.283711	+	0.958910i	$0.591566\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.40495e10i	− 0.543354i	−0.962388	−	0.271677i	$-0.912422\pi$
$401$	− 1.40495e10i	− 0.543354i	0.962388	−	0.271677i	$-0.0875783\pi$
$402$	0	0
$403$	3.89811e9	0.147786
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.80205e10i	− 1.02117i
$408$	0	0
$409$	−1.92375e9	−0.0687474	−0.0343737	−	0.999409i	$-0.510944\pi$
$409$	−1.92375e9	−0.0687474	−0.0343737	+	0.999409i	$0.510944\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.25508e10i	− 0.431391i
$414$	0	0
$415$	2.90107e9	0.0978060
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.86258e9i	0.287544i	0.989611	+	0.143772i	$0.0459232\pi$
$419$	8.86258e9i	0.287544i	−0.989611	+	0.143772i	$0.954077\pi$
$420$	0	0
$421$	−7.06857e9	−0.225011	−0.112505	−	0.993651i	$-0.535888\pi$
$421$	−7.06857e9	−0.225011	−0.112505	+	0.993651i	$0.535888\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.59827e10i	1.40942i
$426$	0	0
$427$	−2.05253e10	−0.617417
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 2.15168e10i	− 0.623546i	−0.950157	−	0.311773i	$-0.899077\pi$
$431$	− 2.15168e10i	− 0.623546i	0.950157	−	0.311773i	$-0.100923\pi$
$432$	0	0
$433$	−1.27339e10	−0.362250	−0.181125	−	0.983460i	$-0.557974\pi$
$433$	−1.27339e10	−0.362250	−0.181125	+	0.983460i	$0.557974\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.28995e9i	0.227314i
$438$	0	0
$439$	6.14824e10	1.65536	0.827681	−	0.561199i	$-0.189660\pi$
$439$	6.14824e10	1.65536	0.827681	+	0.561199i	$0.189660\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.95753e10i	0.508269i	0.967169	+	0.254135i	$0.0817906\pi$
$443$	1.95753e10i	0.508269i	−0.967169	+	0.254135i	$0.918209\pi$
$444$	0	0
$445$	3.95142e9	0.100766
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.91515e10i	1.45539i	0.685899	+	0.727697i	$0.259409\pi$
$449$	5.91515e10i	1.45539i	−0.685899	+	0.727697i	$0.740591\pi$
$450$	0	0
$451$	5.40867e10	1.30733
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 3.85099e9i	− 0.0898518i
$456$	0	0
$457$	−1.05670e10	−0.242262	−0.121131	−	0.992637i	$-0.538652\pi$
$457$	−1.05670e10	−0.242262	−0.121131	+	0.992637i	$0.538652\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.94330e10i	0.430265i	0.976585	+	0.215132i	$0.0690183\pi$
$461$	1.94330e10i	0.430265i	−0.976585	+	0.215132i	$0.930982\pi$
$462$	0	0
$463$	5.45181e10	1.18636	0.593180	−	0.805070i	$-0.297872\pi$
$463$	5.45181e10	1.18636	0.593180	+	0.805070i	$0.297872\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.48698e10i	− 1.36388i	−0.731410	−	0.681938i	$-0.761137\pi$
$467$	− 6.48698e10i	− 1.36388i	0.731410	−	0.681938i	$-0.238863\pi$
$468$	0	0
$469$	−3.54951e10	−0.733631
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.82807e10i	1.96347i
$474$	0	0
$475$	−6.64469e10	−1.30527
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 3.58840e10i	− 0.681647i	−0.940127	−	0.340823i	$-0.889294\pi$
$479$	− 3.58840e10i	− 0.681647i	0.940127	−	0.340823i	$-0.110706\pi$
$480$	0	0
$481$	−2.75314e10	−0.514337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.52924e10i	− 0.276381i
$486$	0	0
$487$	−8.43804e10	−1.50012	−0.750060	−	0.661370i	$-0.769975\pi$
$487$	−8.43804e10	−1.50012	−0.750060	+	0.661370i	$0.769975\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.79053e10i	1.34042i	0.742172	+	0.670210i	$0.233796\pi$
$491$	7.79053e10i	1.34042i	−0.742172	+	0.670210i	$0.766204\pi$
$492$	0	0
$493$	1.09734e11	1.85761
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.74943e10i	0.450627i
$498$	0	0
$499$	−2.86613e10	−0.462268	−0.231134	−	0.972922i	$-0.574244\pi$
$499$	−2.86613e10	−0.462268	−0.231134	+	0.972922i	$0.574244\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.24664e10i	0.350963i	0.984483	+	0.175481i	$0.0561482\pi$
$503$	2.24664e10i	0.350963i	−0.984483	+	0.175481i	$0.943852\pi$
$504$	0	0
$505$	4.22538e10	0.649682
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.20669e11i	1.79773i	0.438227	+	0.898864i	$0.355607\pi$
$509$	1.20669e11i	1.79773i	−0.438227	+	0.898864i	$0.644393\pi$
$510$	0	0
$511$	4.91877e10	0.721394
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.45008e9i	0.0632615i
$516$	0	0
$517$	1.05660e11	1.47894
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.64705e10i	0.630705i	0.948975	+	0.315353i	$0.102123\pi$
$521$	4.64705e10i	0.630705i	−0.948975	+	0.315353i	$0.897877\pi$
$522$	0	0
$523$	2.55571e10	0.341590	0.170795	−	0.985307i	$-0.445366\pi$
$523$	2.55571e10	0.341590	0.170795	+	0.985307i	$0.445366\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.74231e10i	− 0.355529i
$528$	0	0
$529$	7.64960e10	0.976823
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.31426e10i	− 0.658467i
$534$	0	0
$535$	−2.22141e9	−0.0271152
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.60438e10i	− 0.190086i
$540$	0	0
$541$	1.38276e11	1.61421	0.807104	−	0.590410i	$-0.201034\pi$
$541$	1.38276e11	1.61421	0.807104	+	0.590410i	$0.201034\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.13278e10i	− 0.355094i
$546$	0	0
$547$	6.33492e9	0.0707606	0.0353803	−	0.999374i	$-0.488736\pi$
$547$	6.33492e9	0.0707606	0.0353803	+	0.999374i	$0.488736\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.58571e11i	1.72035i
$552$	0	0
$553$	1.90735e10	0.203953
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.12825e10i	0.844455i	0.906490	+	0.422227i	$0.138752\pi$
$557$	8.12825e10i	0.844455i	−0.906490	+	0.422227i	$0.861248\pi$
$558$	0	0
$559$	9.65651e10	0.988947
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 6.53719e10i	− 0.650665i	−0.945600	−	0.325332i	$-0.894524\pi$
$563$	− 6.53719e10i	− 0.650665i	0.945600	−	0.325332i	$-0.105476\pi$
$564$	0	0
$565$	5.08877e10	0.499366
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.59552e10i	0.533815i	0.963722	+	0.266908i	$0.0860019\pi$
$569$	5.59552e10i	0.533815i	−0.963722	+	0.266908i	$0.913998\pi$
$570$	0	0
$571$	−1.07046e11	−1.00699	−0.503496	−	0.863997i	$-0.667953\pi$
$571$	−1.07046e11	−1.00699	−0.503496	+	0.863997i	$0.667953\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.45478e10i	− 0.133084i
$576$	0	0
$577$	−9.68411e10	−0.873688	−0.436844	−	0.899537i	$-0.643904\pi$
$577$	−9.68411e10	−0.873688	−0.436844	+	0.899537i	$0.643904\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.18753e10i	− 0.104217i
$582$	0	0
$583$	−2.62643e11	−2.27348
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.19009e11i	1.00236i	0.865342	+	0.501182i	$0.167101\pi$
$587$	1.19009e11i	1.00236i	−0.865342	+	0.501182i	$0.832899\pi$
$588$	0	0
$589$	3.96276e10	0.329258
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.43498e11i	1.16045i	0.814456	+	0.580226i	$0.197036\pi$
$593$	1.43498e11i	1.16045i	−0.814456	+	0.580226i	$0.802964\pi$
$594$	0	0
$595$	−2.70916e10	−0.216156
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.87645e11i	− 1.45757i	−0.684741	−	0.728787i	$-0.740085\pi$
$599$	− 1.87645e11i	− 1.45757i	0.684741	−	0.728787i	$-0.259915\pi$
$600$	0	0
$601$	−3.22645e10	−0.247302	−0.123651	−	0.992326i	$-0.539460\pi$
$601$	−3.22645e10	−0.247302	−0.123651	+	0.992326i	$0.539460\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.66165e10i	0.273310i
$606$	0	0
$607$	4.80704e10	0.354097	0.177049	−	0.984202i	$-0.443345\pi$
$607$	4.80704e10	0.354097	0.177049	+	0.984202i	$0.443345\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.03816e11i	− 0.744902i
$612$	0	0
$613$	1.76119e11	1.24728	0.623640	−	0.781712i	$-0.285653\pi$
$613$	1.76119e11	1.24728	0.623640	+	0.781712i	$0.285653\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.55999e10i	0.590653i	0.955396	+	0.295327i	$0.0954285\pi$
$617$	8.55999e10i	0.590653i	−0.955396	+	0.295327i	$0.904572\pi$
$618$	0	0
$619$	−7.64495e10	−0.520729	−0.260365	−	0.965510i	$-0.583843\pi$
$619$	−7.64495e10	−0.520729	−0.260365	+	0.965510i	$0.583843\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.61748e10i	− 0.107371i
$624$	0	0
$625$	9.74070e10	0.638367
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.93683e11i	1.23734i
$630$	0	0
$631$	−1.98446e10	−0.125177	−0.0625885	−	0.998039i	$-0.519936\pi$
$631$	−1.98446e10	−0.125177	−0.0625885	+	0.998039i	$0.519936\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.02193e11i	− 0.628529i
$636$	0	0
$637$	−1.57637e10	−0.0957415
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2.72891e11i	− 1.61643i	−0.588888	−	0.808214i	$-0.700434\pi$
$641$	− 2.72891e11i	− 1.61643i	0.588888	−	0.808214i	$-0.299566\pi$
$642$	0	0
$643$	−4.55006e10	−0.266179	−0.133089	−	0.991104i	$-0.542490\pi$
$643$	−4.55006e10	−0.266179	−0.133089	+	0.991104i	$0.542490\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.73260e10i	0.555408i	0.960667	+	0.277704i	$0.0895734\pi$
$647$	9.73260e10i	0.555408i	−0.960667	+	0.277704i	$0.910427\pi$
$648$	0	0
$649$	2.69431e11	1.51869
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.60986e11i	0.885392i	0.896672	+	0.442696i	$0.145978\pi$
$653$	1.60986e11i	0.885392i	−0.896672	+	0.442696i	$0.854022\pi$
$654$	0	0
$655$	7.77973e10	0.422668
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.02994e11i	− 1.07632i	−0.842843	−	0.538160i	$-0.819120\pi$
$659$	− 2.02994e11i	− 1.07632i	0.842843	−	0.538160i	$-0.180880\pi$
$660$	0	0
$661$	−2.97091e11	−1.55627	−0.778133	−	0.628099i	$-0.783833\pi$
$661$	−2.97091e11	−1.55627	−0.778133	+	0.628099i	$0.783833\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.91485e10i	− 0.200184i
$666$	0	0
$667$	−3.47172e10	−0.175405
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 4.40623e11i	− 2.17359i
$672$	0	0
$673$	2.16935e10	0.105747	0.0528737	−	0.998601i	$-0.483162\pi$
$673$	2.16935e10	0.105747	0.0528737	+	0.998601i	$0.483162\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.42130e10i	0.0676596i	0.999428	+	0.0338298i	$0.0107704\pi$
$677$	1.42130e10i	0.0676596i	−0.999428	+	0.0338298i	$0.989230\pi$
$678$	0	0
$679$	−6.25980e10	−0.294497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 7.61548e9i	− 0.0349957i	−0.999847	−	0.0174978i	$-0.994430\pi$
$683$	− 7.61548e9i	− 0.0349957i	0.999847	−	0.0174978i	$-0.00557002\pi$
$684$	0	0
$685$	5.80890e10	0.263834
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.58059e11i	1.14509i
$690$	0	0
$691$	−1.09041e11	−0.478273	−0.239136	−	0.970986i	$-0.576864\pi$
$691$	−1.09041e11	−0.478273	−0.239136	+	0.970986i	$0.576864\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 7.49448e10i	− 0.321220i
$696$	0	0
$697$	−3.73857e11	−1.58407
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.22792e11i	1.33675i	0.743823	+	0.668376i	$0.233010\pi$
$701$	3.22792e11i	1.33675i	−0.743823	+	0.668376i	$0.766990\pi$
$702$	0	0
$703$	−2.79880e11	−1.14591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.72962e11i	− 0.692268i
$708$	0	0
$709$	−1.28670e11	−0.509205	−0.254602	−	0.967046i	$-0.581945\pi$
$709$	−1.28670e11	−0.509205	−0.254602	+	0.967046i	$0.581945\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.67600e9i	0.0335708i
$714$	0	0
$715$	8.26702e10	0.316319
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1.19347e11i	− 0.446577i	−0.974752	−	0.223288i	$-0.928321\pi$
$719$	− 1.19347e11i	− 0.446577i	0.974752	−	0.223288i	$-0.0716792\pi$
$720$	0	0
$721$	1.82160e10	0.0674082
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.78271e11i	− 1.00720i
$726$	0	0
$727$	−1.26145e11	−0.451578	−0.225789	−	0.974176i	$-0.572496\pi$
$727$	−1.26145e11	−0.451578	−0.225789	+	0.974176i	$0.572496\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.79333e11i	− 2.37910i
$732$	0	0
$733$	−2.14006e10	−0.0741327	−0.0370664	−	0.999313i	$-0.511801\pi$
$733$	−2.14006e10	−0.0741327	−0.0370664	+	0.999313i	$0.511801\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.61984e11i	− 2.58271i
$738$	0	0
$739$	9.32712e9	0.0312730	0.0156365	−	0.999878i	$-0.495023\pi$
$739$	9.32712e9	0.0312730	0.0156365	+	0.999878i	$0.495023\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.47858e11i	1.79768i	0.438275	+	0.898841i	$0.355590\pi$
$743$	5.47858e11i	1.79768i	−0.438275	+	0.898841i	$0.644410\pi$
$744$	0	0
$745$	1.71544e11	0.556866
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.09315e9i	0.0288926i
$750$	0	0
$751$	−3.25787e10	−0.102418	−0.0512088	−	0.998688i	$-0.516307\pi$
$751$	−3.25787e10	−0.102418	−0.0512088	+	0.998688i	$0.516307\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.38853e10i	0.258165i
$756$	0	0
$757$	1.83680e11	0.559342	0.279671	−	0.960096i	$-0.409775\pi$
$757$	1.83680e11	0.559342	0.279671	+	0.960096i	$0.409775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.15436e11i	0.344194i	0.985080	+	0.172097i	$0.0550543\pi$
$761$	1.15436e11i	0.344194i	−0.985080	+	0.172097i	$0.944946\pi$
$762$	0	0
$763$	−1.28238e11	−0.378371
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.64728e11i	− 0.764924i
$768$	0	0
$769$	1.80169e11	0.515198	0.257599	−	0.966252i	$-0.417069\pi$
$769$	1.80169e11	0.515198	0.257599	+	0.966252i	$0.417069\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6.65686e11i	− 1.86445i	−0.361874	−	0.932227i	$-0.617863\pi$
$773$	− 6.65686e11i	− 1.86445i	0.361874	−	0.932227i	$-0.382137\pi$
$774$	0	0
$775$	−6.95413e10	−0.192768
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.40239e11i	− 1.46702i
$780$	0	0
$781$	−5.90227e11	−1.58641
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.63709e11i	− 0.431115i
$786$	0	0
$787$	4.36920e11	1.13895	0.569473	−	0.822010i	$-0.307147\pi$
$787$	4.36920e11	1.13895	0.569473	+	0.822010i	$0.307147\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2.08304e11i	− 0.532099i
$792$	0	0
$793$	−4.32931e11	−1.09478
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.59644e11i	0.891333i	0.895199	+	0.445667i	$0.147033\pi$
$797$	3.59644e11i	0.891333i	−0.895199	+	0.445667i	$0.852967\pi$
$798$	0	0
$799$	−7.30343e11	−1.79201
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.05592e12i	2.53963i
$804$	0	0
$805$	8.57112e9	0.0204105
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.25571e11i	0.993522i	0.867887	+	0.496761i	$0.165477\pi$
$809$	4.25571e11i	0.993522i	−0.867887	+	0.496761i	$0.834523\pi$
$810$	0	0
$811$	−1.09416e11	−0.252929	−0.126465	−	0.991971i	$-0.540363\pi$
$811$	−1.09416e11	−0.252929	−0.126465	+	0.991971i	$0.540363\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 2.79896e11i	− 0.634405i
$816$	0	0
$817$	9.81665e11	2.20331
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.79455e11i	1.49551i	0.663977	+	0.747753i	$0.268867\pi$
$821$	6.79455e11i	1.49551i	−0.663977	+	0.747753i	$0.731133\pi$
$822$	0	0
$823$	−7.80059e11	−1.70031	−0.850155	−	0.526533i	$-0.823492\pi$
$823$	−7.80059e11	−1.70031	−0.850155	+	0.526533i	$0.823492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.01868e11i	− 1.28671i	−0.765569	−	0.643353i	$-0.777543\pi$
$827$	− 6.01868e11i	− 1.28671i	0.765569	−	0.643353i	$-0.222457\pi$
$828$	0	0
$829$	1.26918e11	0.268722	0.134361	−	0.990932i	$-0.457102\pi$
$829$	1.26918e11	0.268722	0.134361	+	0.990932i	$0.457102\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.10897e11i	0.230325i
$834$	0	0
$835$	1.48553e11	0.305587
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 8.99511e11i	− 1.81534i	−0.419682	−	0.907671i	$-0.637858\pi$
$839$	− 8.99511e11i	− 1.81534i	0.419682	−	0.907671i	$-0.362142\pi$
$840$	0	0
$841$	−1.63826e11	−0.327490
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.96170e10i	0.195392i
$846$	0	0
$847$	1.49886e11	0.291225
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 6.12764e10i	− 0.116836i
$852$	0	0
$853$	3.68583e11	0.696209	0.348105	−	0.937456i	$-0.386825\pi$
$853$	3.68583e11	0.696209	0.348105	+	0.937456i	$0.386825\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.55843e11i	0.474297i	0.971473	+	0.237148i	$0.0762127\pi$
$857$	2.55843e11i	0.474297i	−0.971473	+	0.237148i	$0.923787\pi$
$858$	0	0
$859$	−1.52059e11	−0.279279	−0.139640	−	0.990202i	$-0.544594\pi$
$859$	−1.52059e11	−0.279279	−0.139640	+	0.990202i	$0.544594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 8.82719e11i	− 1.59140i	−0.605691	−	0.795700i	$-0.707103\pi$
$863$	− 8.82719e11i	− 1.59140i	0.605691	−	0.795700i	$-0.292897\pi$
$864$	0	0
$865$	−4.38673e10	−0.0783567
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.09456e11i	0.718006i
$870$	0	0
$871$	−7.48682e11	−1.30084
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.47289e11i	0.251269i
$876$	0	0
$877$	−6.67630e11	−1.12859	−0.564297	−	0.825572i	$-0.690853\pi$
$877$	−6.67630e11	−1.12859	−0.564297	+	0.825572i	$0.690853\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.80883e11i	1.62822i	0.580709	+	0.814111i	$0.302775\pi$
$881$	9.80883e11i	1.62822i	−0.580709	+	0.814111i	$0.697225\pi$
$882$	0	0
$883$	5.60537e11	0.922065	0.461033	−	0.887383i	$-0.347479\pi$
$883$	5.60537e11	0.922065	0.461033	+	0.887383i	$0.347479\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.43060e11i	0.877311i	0.898655	+	0.438655i	$0.144545\pi$
$887$	5.43060e11i	0.877311i	−0.898655	+	0.438655i	$0.855455\pi$
$888$	0	0
$889$	−4.18317e11	−0.669728
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.05538e12i	− 1.65959i
$894$	0	0
$895$	2.80871e11	0.437739
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.65955e11i	0.254069i
$900$	0	0
$901$	1.81544e12	2.75475
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.08766e11i	− 0.311218i
$906$	0	0
$907$	−1.56480e11	−0.231222	−0.115611	−	0.993295i	$-0.536883\pi$
$907$	−1.56480e11	−0.231222	−0.115611	+	0.993295i	$0.536883\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.05213e12i	1.52755i	0.645483	+	0.763775i	$0.276656\pi$
$911$	1.05213e12i	1.52755i	−0.645483	+	0.763775i	$0.723344\pi$
$912$	0	0
$913$	2.54929e11	0.366891
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.18457e11i	− 0.450374i
$918$	0	0
$919$	1.04578e12	1.46615	0.733077	−	0.680146i	$-0.238084\pi$
$919$	1.04578e12	1.46615	0.733077	+	0.680146i	$0.238084\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.79924e11i	0.799033i
$924$	0	0
$925$	4.91153e11	0.670888
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.30474e12i	1.75171i	0.482578	+	0.875853i	$0.339700\pi$
$929$	1.30474e12i	1.75171i	−0.482578	+	0.875853i	$0.660300\pi$
$930$	0	0
$931$	−1.60251e11	−0.213306
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 5.81583e11i	− 0.760966i
$936$	0	0
$937$	−7.72155e11	−1.00172	−0.500860	−	0.865528i	$-0.666983\pi$
$937$	−7.72155e11	−1.00172	−0.500860	+	0.865528i	$0.666983\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.85580e11i	− 0.236686i	−0.992973	−	0.118343i	$-0.962242\pi$
$941$	− 1.85580e11i	− 0.236686i	0.992973	−	0.118343i	$-0.0377582\pi$
$942$	0	0
$943$	1.18279e11	0.149576
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.65591e11i	− 0.578902i	−0.957193	−	0.289451i	$-0.906527\pi$
$947$	− 4.65591e11i	− 0.578902i	0.957193	−	0.289451i	$-0.0934727\pi$
$948$	0	0
$949$	1.03749e12	1.27915
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.01476e11i	0.365495i	0.983160	+	0.182748i	$0.0584991\pi$
$953$	3.01476e11i	0.365495i	−0.983160	+	0.182748i	$0.941501\pi$
$954$	0	0
$955$	8.59249e10	0.103301
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.37782e11i	− 0.281128i
$960$	0	0
$961$	−8.11418e11	−0.951374
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.32017e11i	− 0.152238i
$966$	0	0
$967$	8.13827e11	0.930735	0.465368	−	0.885118i	$-0.345922\pi$
$967$	8.13827e11	0.930735	0.465368	+	0.885118i	$0.345922\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.34616e12i	1.51433i	0.653225	+	0.757164i	$0.273416\pi$
$971$	1.34616e12i	1.51433i	−0.653225	+	0.757164i	$0.726584\pi$
$972$	0	0
$973$	−3.06780e11	−0.342275
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8.92037e11i	− 0.979050i	−0.871989	−	0.489525i	$-0.837170\pi$
$977$	− 8.92037e11i	− 0.979050i	0.871989	−	0.489525i	$-0.162830\pi$
$978$	0	0
$979$	3.47229e11	0.377994
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.53170e11i	0.378242i	0.981954	+	0.189121i	$0.0605638\pi$
$983$	3.53170e11i	0.378242i	−0.981954	+	0.189121i	$0.939436\pi$
$984$	0	0
$985$	4.27377e11	0.454011
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.14924e11i	0.224647i
$990$	0	0
$991$	7.61424e11	0.789463	0.394732	−	0.918796i	$-0.370838\pi$
$991$	7.61424e11	0.789463	0.394732	+	0.918796i	$0.370838\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.33241e11i	0.339990i
$996$	0	0
$997$	−1.57737e12	−1.59644	−0.798221	−	0.602365i	$-0.794225\pi$
$997$	−1.57737e12	−1.59644	−0.798221	+	0.602365i	$0.794225\pi$
$998$	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.197.8	✓	16
3.2	odd	2	inner	252.9.c.a.197.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.8	✓	16	1.1	even	1	trivial
252.9.c.a.197.9	yes	16	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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-221.696i q^{5} -907.493 q^{7} +O(q^{10})\)	\(q-221.696i q^{5} -907.493 q^{7} -19481.4i q^{11} -19141.3 q^{13} +134659. i q^{17} -194587. q^{19} -42602.7i q^{23} +341476. q^{25} -814906. i q^{29} -203649. q^{31} +201187. i q^{35} +1.43832e6 q^{37} +2.77633e6i q^{41} -5.04485e6 q^{43} +5.42366e6i q^{47} +823543. q^{49} -1.34818e7i q^{53} -4.31894e6 q^{55} +1.38302e7i q^{59} +2.26176e7 q^{61} +4.24355e6i q^{65} +3.91134e7 q^{67} -3.02970e7i q^{71} -5.42017e7 q^{73} +1.76792e7i q^{77} -2.10178e7 q^{79} +1.30858e7i q^{83} +2.98533e7 q^{85} +1.78236e7i q^{89} +1.73706e7 q^{91} +4.31392e7i q^{95} +6.89791e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

Properties

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