LMFDB - Embedded newform 252.9.bk.a.53.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(53,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("252.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bk (of order $6$, degree $2$, 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:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.7
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-349.907 + 202.019i) q^{5} +(-1821.32 + 1564.48i) q^{7} +O(q^{10})$	$q+(-349.907 + 202.019i) q^{5} +(-1821.32 + 1564.48i) q^{7} +(10579.1 + 6107.83i) q^{11} +7958.67 q^{13} +(-140355. - 81034.2i) q^{17} +(-35098.2 - 60791.9i) q^{19} +(-93110.9 + 53757.6i) q^{23} +(-113689. + 196916. i) q^{25} +813834. i q^{29} +(-876420. + 1.51800e6i) q^{31} +(321238. - 915362. i) q^{35} +(-1.67953e6 - 2.90903e6i) q^{37} +1.96551e6i q^{41} +3.97887e6 q^{43} +(3.18444e6 - 1.83854e6i) q^{47} +(869617. - 5.69883e6i) q^{49} +(-4.53376e6 - 2.61757e6i) q^{53} -4.93558e6 q^{55} +(1.47374e7 + 8.50867e6i) q^{59} +(7.69375e6 + 1.33260e7i) q^{61} +(-2.78479e6 + 1.60780e6i) q^{65} +(-1.19649e6 + 2.07238e6i) q^{67} -4.94900e7i q^{71} +(258100. - 447042. i) q^{73} +(-2.88234e7 + 5.42641e6i) q^{77} +(-2.00826e7 - 3.47840e7i) q^{79} +3.77285e7i q^{83} +6.54817e7 q^{85} +(8.00892e7 - 4.62395e7i) q^{89} +(-1.44953e7 + 1.24512e7i) q^{91} +(2.45622e7 + 1.41810e7i) q^{95} -1.54164e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 1230 q^{7}+O(q^{10}) $ 44 * q + 1230 * q^7	$ 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) $ 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−349.907	+	202.019i	−0.559850	+	0.323230i	−0.753085	−	0.657923i	$-0.771435\pi$
$5$	−349.907	+	202.019i	−0.559850	+	0.323230i	0.193235	+	0.981152i	$0.438102\pi$
$6$		0			0
$7$	−1821.32	+	1564.48i	−0.758568	+	0.651594i
$7$	−1821.32	+	1564.48i	−0.758568	+	0.651594i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10579.1	+	6107.83i	0.722565	+	0.417173i	0.815696	−	0.578481i	$-0.196354\pi$
$11$	10579.1	+	6107.83i	0.722565	+	0.417173i	−0.0931312	+	0.995654i	$0.529688\pi$
$12$		0			0
$13$	7958.67			0.278655			0.139327	−	0.990246i	$-0.455506\pi$
$13$	7958.67			0.278655			0.139327	+	0.990246i	$0.455506\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−140355.	−	81034.2i	−1.68048	−	0.970225i	−0.961342	−	0.275358i	$-0.911204\pi$
$17$	−140355.	−	81034.2i	−1.68048	−	0.970225i	−0.719138	−	0.694868i	$-0.755463\pi$
$18$		0			0
$19$	−35098.2	−	60791.9i	−0.269321	−	0.466478i	0.699365	−	0.714764i	$-0.253466\pi$
$19$	−35098.2	−	60791.9i	−0.269321	−	0.466478i	−0.968687	+	0.248286i	$0.920133\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−93110.9	+	53757.6i	−0.332728	+	0.192100i	−0.657051	−	0.753846i	$-0.728197\pi$
$23$	−93110.9	+	53757.6i	−0.332728	+	0.192100i	0.324324	+	0.945946i	$0.394863\pi$
$24$		0			0
$25$	−113689.	+	196916.i	−0.291045	+	0.504105i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			813834.i			1.15065i	0.817924	+	0.575326i	$0.195125\pi$
$29$			813834.i			1.15065i	−0.817924	+	0.575326i	$0.804875\pi$
$30$		0			0
$31$	−876420.	+	1.51800e6i	−0.948999	+	1.64371i	−0.201459	+	0.979497i	$0.564568\pi$
$31$	−876420.	+	1.51800e6i	−0.948999	+	1.64371i	−0.747539	+	0.664217i	$0.768765\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	321238.	−	915362.i	0.214070	−	0.609987i
$36$		0			0
$37$	−1.67953e6	−	2.90903e6i	−0.896149	−	1.55218i	−0.832376	−	0.554212i	$-0.813020\pi$
$37$	−1.67953e6	−	2.90903e6i	−0.896149	−	1.55218i	−0.0637733	−	0.997964i	$-0.520313\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.96551e6i			0.695569i	0.937575	+	0.347784i	$0.113066\pi$
$41$			1.96551e6i			0.695569i	−0.937575	+	0.347784i	$0.886934\pi$
$42$		0			0
$43$	3.97887e6			1.16382			0.581910	−	0.813253i	$-0.302305\pi$
$43$	3.97887e6			1.16382			0.581910	+	0.813253i	$0.302305\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.18444e6	−	1.83854e6i	0.652592	−	0.376774i	−0.136857	−	0.990591i	$-0.543700\pi$
$47$	3.18444e6	−	1.83854e6i	0.652592	−	0.376774i	0.789448	+	0.613817i	$0.210367\pi$
$48$		0			0
$49$	869617.	−	5.69883e6i	0.150850	−	0.988557i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.53376e6	−	2.61757e6i	−0.574586	−	0.331737i	0.184393	−	0.982853i	$-0.440968\pi$
$53$	−4.53376e6	−	2.61757e6i	−0.574586	−	0.331737i	−0.758979	+	0.651115i	$0.774301\pi$
$54$		0			0
$55$	−4.93558e6			−0.539371
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.47374e7	+	8.50867e6i	1.21623	+	0.702188i	0.964108	−	0.265509i	$-0.0855399\pi$
$59$	1.47374e7	+	8.50867e6i	1.21623	+	0.702188i	0.252117	+	0.967697i	$0.418873\pi$
$60$		0			0
$61$	7.69375e6	+	1.33260e7i	0.555673	+	0.962453i	0.997851	+	0.0655259i	$0.0208725\pi$
$61$	7.69375e6	+	1.33260e7i	0.555673	+	0.962453i	−0.442178	+	0.896927i	$0.645794\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.78479e6	+	1.60780e6i	−0.156005	+	0.0900696i
$66$		0			0
$67$	−1.19649e6	+	2.07238e6i	−0.0593759	+	0.102842i	−0.894185	−	0.447697i	$-0.852244\pi$
$67$	−1.19649e6	+	2.07238e6i	−0.0593759	+	0.102842i	0.834810	+	0.550539i	$0.185578\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.94900e7i		−	1.94753i	−0.227554	−	0.973765i	$-0.573073\pi$
$71$		−	4.94900e7i		−	1.94753i	0.227554	−	0.973765i	$-0.426927\pi$
$72$		0			0
$73$	258100.	−	447042.i	0.00908859	−	0.0157419i	−0.861445	−	0.507850i	$-0.830440\pi$
$73$	258100.	−	447042.i	0.00908859	−	0.0157419i	0.870534	+	0.492108i	$0.163774\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.88234e7	+	5.42641e6i	−0.819942	+	0.154365i
$78$		0			0
$79$	−2.00826e7	−	3.47840e7i	−0.515598	−	0.893042i	−0.999836	−	0.0181054i	$-0.994237\pi$
$79$	−2.00826e7	−	3.47840e7i	−0.515598	−	0.893042i	0.484238	−	0.874936i	$-0.339097\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.77285e7i			0.794981i	0.917606	+	0.397490i	$0.130119\pi$
$83$			3.77285e7i			0.794981i	−0.917606	+	0.397490i	$0.869881\pi$
$84$		0			0
$85$	6.54817e7			1.25442
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.00892e7	−	4.62395e7i	1.27648	−	0.736976i	0.300280	−	0.953851i	$-0.402920\pi$
$89$	8.00892e7	−	4.62395e7i	1.27648	−	0.736976i	0.976199	+	0.216875i	$0.0695864\pi$
$90$		0			0
$91$	−1.44953e7	+	1.24512e7i	−0.211379	+	0.181570i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.45622e7	+	1.41810e7i	0.301559	+	0.174105i
$96$		0			0
$97$	−1.54164e8			−1.74139			−0.870697	−	0.491821i	$-0.836332\pi$
$97$	−1.54164e8			−1.74139			−0.870697	+	0.491821i	$0.836332\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.65420e7	+	1.53240e7i	0.255063	+	0.147261i	0.622080	−	0.782953i	$-0.286288\pi$
$101$	2.65420e7	+	1.53240e7i	0.255063	+	0.147261i	−0.367017	+	0.930214i	$0.619621\pi$
$102$		0			0
$103$	−5.10742e7	−	8.84631e7i	−0.453787	−	0.785983i	0.544830	−	0.838546i	$-0.316594\pi$
$103$	−5.10742e7	−	8.84631e7i	−0.453787	−	0.785983i	−0.998618	+	0.0525635i	$0.983261\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.11907e8	−	6.46093e7i	0.853730	−	0.492901i	−0.00817787	−	0.999967i	$-0.502603\pi$
$107$	1.11907e8	−	6.46093e7i	0.853730	−	0.492901i	0.861908	+	0.507066i	$0.169270\pi$
$108$		0			0
$109$	1.08266e8	−	1.87523e8i	0.766987	−	1.32846i	−0.172203	−	0.985061i	$-0.555089\pi$
$109$	1.08266e8	−	1.87523e8i	0.766987	−	1.32846i	0.939190	−	0.343398i	$-0.111578\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.69205e8i			1.03777i	0.854845	+	0.518883i	$0.173652\pi$
$113$			1.69205e8i			1.03777i	−0.854845	+	0.518883i	$0.826348\pi$
$114$		0			0
$115$	2.17201e7	−	3.76203e7i	0.124185	−	0.215095i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.82408e8	−	7.19936e7i	1.90695	−	0.359010i
$120$		0			0
$121$	−3.25683e7	−	5.64099e7i	−0.151933	−	0.263157i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	2.49697e8i		−	1.02276i
$126$		0			0
$127$	−3.83510e8			−1.47422			−0.737109	−	0.675774i	$-0.763809\pi$
$127$	−3.83510e8			−1.47422			−0.737109	+	0.675774i	$0.763809\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.17169e8	−	6.76477e7i	0.397858	−	0.229704i	−0.287701	−	0.957720i	$-0.592891\pi$
$131$	1.17169e8	−	6.76477e7i	0.397858	−	0.229704i	0.685559	+	0.728017i	$0.259558\pi$
$132$		0			0
$133$	1.59033e8	+	5.58111e7i	0.508253	+	0.178367i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.07925e8	−	6.23103e7i	−0.306364	−	0.176880i	0.338934	−	0.940810i	$-0.389934\pi$
$137$	−1.07925e8	−	6.23103e7i	−0.306364	−	0.176880i	−0.645298	+	0.763931i	$0.723267\pi$
$138$		0			0
$139$	9.77388e6			0.0261823			0.0130912	−	0.999914i	$-0.495833\pi$
$139$	9.77388e6			0.0261823			0.0130912	+	0.999914i	$0.495833\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.41953e7	+	4.86102e7i	0.201346	+	0.116247i
$144$		0			0
$145$	−1.64410e8	−	2.84766e8i	−0.371925	−	0.644193i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.34818e8	−	4.24247e8i	1.49085	−	0.860744i	0.490907	−	0.871212i	$-0.336666\pi$
$149$	7.34818e8	−	4.24247e8i	1.49085	−	0.860744i	0.999945	+	0.0104686i	$0.00333231\pi$
$150$		0			0
$151$	1.79475e8	−	3.10859e8i	0.345220	−	0.597938i	−0.640174	−	0.768230i	$-0.721138\pi$
$151$	1.79475e8	−	3.10859e8i	0.345220	−	0.597938i	0.985394	+	0.170292i	$0.0544711\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	7.08213e8i		−	1.22698i
$156$		0			0
$157$	4.59164e8	−	7.95296e8i	0.755735	−	1.30897i	−0.189273	−	0.981925i	$-0.560613\pi$
$157$	4.59164e8	−	7.95296e8i	0.755735	−	1.30897i	0.945008	−	0.327047i	$-0.106054\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.54822e7	−	2.43580e8i	0.127225	−	0.362525i
$162$		0			0
$163$	9.56877e7	+	1.65736e8i	0.135552	+	0.234783i	0.925808	−	0.377994i	$-0.123386\pi$
$163$	9.56877e7	+	1.65736e8i	0.135552	+	0.234783i	−0.790256	+	0.612777i	$0.790053\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.09896e8i			0.655565i	0.944753	+	0.327782i	$0.106301\pi$
$167$			5.09896e8i			0.655565i	−0.944753	+	0.327782i	$0.893699\pi$
$168$		0			0
$169$	−7.52390e8			−0.922351
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.29299e9	−	7.46508e8i	1.44348	−	0.833393i	0.445400	−	0.895332i	$-0.353062\pi$
$173$	1.29299e9	−	7.46508e8i	1.44348	−	0.833393i	0.998080	+	0.0619383i	$0.0197282\pi$
$174$		0			0
$175$	−1.01006e8	−	5.36512e8i	−0.107695	−	0.572041i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.50876e9	+	8.71085e8i	1.46963	+	0.848493i	0.999420	−	0.0340600i	$-0.0108437\pi$
$179$	1.50876e9	+	8.71085e8i	1.46963	+	0.848493i	0.470213	+	0.882553i	$0.344177\pi$
$180$		0			0
$181$	−7.60559e8			−0.708628			−0.354314	−	0.935126i	$-0.615286\pi$
$181$	−7.60559e8			−0.708628			−0.354314	+	0.935126i	$0.615286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.17536e9	+	6.78592e8i	1.00342	+	0.579324i
$186$		0			0
$187$	−9.89886e8	−	1.71453e9i	−0.809503	−	1.40210i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.55485e9	−	8.97692e8i	1.16830	−	0.674518i	0.215021	−	0.976609i	$-0.431018\pi$
$191$	1.55485e9	−	8.97692e8i	1.16830	−	0.674518i	0.953279	+	0.302091i	$0.0976847\pi$
$192$		0			0
$193$	3.97709e8	−	6.88852e8i	0.286640	−	0.496474i	−0.686366	−	0.727256i	$-0.740795\pi$
$193$	3.97709e8	−	6.88852e8i	0.286640	−	0.496474i	0.973005	+	0.230782i	$0.0741284\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.47329e8i			0.297004i	0.988912	+	0.148502i	$0.0474451\pi$
$197$			4.47329e8i			0.297004i	−0.988912	+	0.148502i	$0.952555\pi$
$198$		0			0
$199$	−7.84277e8	+	1.35841e9i	−0.500100	+	0.866199i	0.499900	+	0.866083i	$0.333370\pi$
$199$	−7.84277e8	+	1.35841e9i	−0.500100	+	0.866199i	−1.00000		0.000115831i	$0.999963\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.27323e9	−	1.48225e9i	−0.749758	−	0.872847i
$204$		0			0
$205$	−3.97070e8	−	6.87745e8i	−0.224829	−	0.389415i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.57496e8i		−	0.449414i
$210$		0			0
$211$	2.18001e9			1.09984			0.549919	−	0.835218i	$-0.314659\pi$
$211$	2.18001e9			1.09984			0.549919	+	0.835218i	$0.314659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.39223e9	+	8.03806e8i	−0.651565	+	0.376181i
$216$		0			0
$217$	−7.78642e8	−	4.13591e9i	−0.351155	−	1.86523i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.11704e9	−	6.44924e8i	−0.468274	−	0.270358i
$222$		0			0
$223$	−1.17447e9			−0.474921			−0.237461	−	0.971397i	$-0.576315\pi$
$223$	−1.17447e9			−0.474921			−0.237461	+	0.971397i	$0.576315\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.97460e9	+	1.14003e9i	0.743661	+	0.429353i	0.823399	−	0.567463i	$-0.192075\pi$
$227$	1.97460e9	+	1.14003e9i	0.743661	+	0.429353i	−0.0797378	+	0.996816i	$0.525408\pi$
$228$		0			0
$229$	1.23665e9	+	2.14195e9i	0.449682	+	0.778873i	0.998365	−	0.0571578i	$-0.0182038\pi$
$229$	1.23665e9	+	2.14195e9i	0.449682	+	0.778873i	−0.548683	+	0.836031i	$0.684870\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.39582e9	−	8.05875e8i	0.473592	−	0.273429i	−0.244150	−	0.969737i	$-0.578509\pi$
$233$	1.39582e9	−	8.05875e8i	0.473592	−	0.273429i	0.717742	+	0.696309i	$0.245176\pi$
$234$		0			0
$235$	−7.42837e8	+	1.28663e9i	−0.243569	+	0.421874i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.26329e9i		−	0.693662i	−0.937928	−	0.346831i	$-0.887258\pi$
$239$		−	2.26329e9i		−	0.693662i	0.937928	−	0.346831i	$-0.112742\pi$
$240$		0			0
$241$	−2.25951e9	+	3.91359e9i	−0.669802	+	1.16013i	0.308157	+	0.951336i	$0.400288\pi$
$241$	−2.25951e9	+	3.91359e9i	−0.669802	+	1.16013i	−0.977959	+	0.208796i	$0.933046\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.46986e8	+	2.16974e9i	0.235078	+	0.602203i
$246$		0			0
$247$	−2.79335e8	−	4.83822e8i	−0.0750477	−	0.129986i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	6.83085e8i		−	0.172100i	−0.996291	−	0.0860499i	$-0.972576\pi$
$251$		−	6.83085e8i		−	0.172100i	0.996291	−	0.0860499i	$-0.0274244\pi$
$252$		0			0
$253$	−1.31337e9			−0.320556
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.48843e7	+	8.59346e6i	−0.00341190	+	0.00196986i	−0.501705	−	0.865039i	$-0.667294\pi$
$257$	−1.48843e7	+	8.59346e6i	−0.00341190	+	0.00196986i	0.498293	+	0.867009i	$0.333960\pi$
$258$		0			0
$259$	7.61007e9	+	2.67069e9i	1.69118	+	0.593505i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.49429e8	−	4.32683e8i	−0.156642	−	0.0904371i	0.419630	−	0.907695i	$-0.362160\pi$
$263$	−7.49429e8	−	4.32683e8i	−0.156642	−	0.0904371i	−0.576272	+	0.817258i	$0.695493\pi$
$264$		0			0
$265$	2.11519e9			0.428909
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.41936e9	−	1.39682e9i	−0.462053	−	0.266766i	0.250854	−	0.968025i	$-0.419289\pi$
$269$	−2.41936e9	−	1.39682e9i	−0.462053	−	0.266766i	−0.712907	+	0.701258i	$0.752622\pi$
$270$		0			0
$271$	−3.46401e9	−	5.99983e9i	−0.642246	−	1.11240i	−0.984930	−	0.172952i	$-0.944669\pi$
$271$	−3.46401e9	−	5.99983e9i	−0.642246	−	1.11240i	0.342684	−	0.939451i	$-0.388664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.40546e9	+	1.38879e9i	−0.420598	+	0.242832i
$276$		0			0
$277$	3.02498e9	−	5.23942e9i	0.513811	−	0.889947i	−0.486061	−	0.873925i	$-0.661567\pi$
$277$	3.02498e9	−	5.23942e9i	0.513811	−	0.889947i	0.999872	−	0.0160216i	$-0.00510006\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.17380e9i			0.669432i	0.942319	+	0.334716i	$0.108640\pi$
$281$			4.17380e9i			0.669432i	−0.942319	+	0.334716i	$0.891360\pi$
$282$		0			0
$283$	3.92835e9	−	6.80410e9i	0.612441	−	1.06078i	−0.378387	−	0.925648i	$-0.623521\pi$
$283$	3.92835e9	−	6.80410e9i	0.612441	−	1.06078i	0.990828	−	0.135131i	$-0.0431456\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.07500e9	−	3.57983e9i	−0.453229	−	0.527636i
$288$		0			0
$289$	9.64520e9	+	1.67060e10i	1.38267	+	2.39486i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.07386e10i			1.45706i	0.685016	+	0.728528i	$0.259795\pi$
$293$			1.07386e10i			1.45706i	−0.685016	+	0.728528i	$0.740205\pi$
$294$		0			0
$295$	−6.87564e9			−0.907872
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.41038e8	+	4.27839e8i	−0.0927162	+	0.0535297i
$300$		0			0
$301$	−7.24680e9	+	6.22486e9i	−0.882836	+	0.758339i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.38419e9	−	3.10856e9i	−0.622187	−	0.359220i
$306$		0			0
$307$	−6.62105e9			−0.745372			−0.372686	−	0.927957i	$-0.621563\pi$
$307$	−6.62105e9			−0.745372			−0.372686	+	0.927957i	$0.621563\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.98990e9	−	5.19032e9i	−0.960977	−	0.554820i	−0.0645035	−	0.997917i	$-0.520546\pi$
$311$	−8.98990e9	−	5.19032e9i	−0.960977	−	0.554820i	−0.896474	+	0.443097i	$0.853880\pi$
$312$		0			0
$313$	4.95239e9	+	8.57780e9i	0.515986	+	0.893714i	0.999828	+	0.0185583i	$0.00590761\pi$
$313$	4.95239e9	+	8.57780e9i	0.515986	+	0.893714i	−0.483842	+	0.875155i	$0.660759\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.20277e9	+	4.15852e9i	−0.713284	+	0.411814i	−0.812276	−	0.583274i	$-0.801771\pi$
$317$	−7.20277e9	+	4.15852e9i	−0.713284	+	0.411814i	0.0989920	+	0.995088i	$0.468438\pi$
$318$		0			0
$319$	−4.97076e9	+	8.60960e9i	−0.480021	+	0.831420i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.13766e10i			1.04521i
$324$		0			0
$325$	−9.04816e8	+	1.56719e9i	−0.0811011	+	0.140471i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.92353e9	+	8.33055e9i	−0.249531	+	0.711034i
$330$		0			0
$331$	−1.72675e9	−	2.99081e9i	−0.143852	−	0.249159i	0.785092	−	0.619379i	$-0.212616\pi$
$331$	−1.72675e9	−	2.99081e9i	−0.143852	−	0.249159i	−0.928944	+	0.370220i	$0.879282\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.66854e8i		−	0.0767682i
$336$		0			0
$337$	6.00768e9			0.465787			0.232893	−	0.972502i	$-0.425181\pi$
$337$	6.00768e9			0.465787			0.232893	+	0.972502i	$0.425181\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.85434e10	+	1.07061e10i	−1.37143	+	0.791793i
$342$		0			0
$343$	7.33185e9	+	1.17399e10i	0.529708	+	0.848180i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.66076e10	−	9.58841e9i	−1.14548	−	0.661346i	−0.197701	−	0.980262i	$-0.563348\pi$
$347$	−1.66076e10	−	9.58841e9i	−1.14548	−	0.661346i	−0.947783	+	0.318917i	$0.896681\pi$
$348$		0			0
$349$	1.40166e9			0.0944803			0.0472401	−	0.998884i	$-0.484957\pi$
$349$	1.40166e9			0.0944803			0.0472401	+	0.998884i	$0.484957\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.16222e10	−	6.71006e9i	−0.748493	−	0.432143i	0.0766559	−	0.997058i	$-0.475576\pi$
$353$	−1.16222e10	−	6.71006e9i	−0.748493	−	0.432143i	−0.825149	+	0.564915i	$0.808909\pi$
$354$		0			0
$355$	9.99791e9	+	1.73169e10i	0.629500	+	1.09033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.26658e9	+	7.31259e8i	−0.0762525	+	0.0440244i	−0.537641	−	0.843174i	$-0.680685\pi$
$359$	−1.26658e9	+	7.31259e8i	−0.0762525	+	0.0440244i	0.461389	+	0.887198i	$0.347351\pi$
$360$		0			0
$361$	6.02801e9	−	1.04408e10i	0.354932	−	0.614760i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			2.08564e8i			0.0117508i
$366$		0			0
$367$	−1.01014e10	+	1.74961e10i	−0.556823	+	0.964445i	0.440936	+	0.897538i	$0.354646\pi$
$367$	−1.01014e10	+	1.74961e10i	−0.556823	+	0.964445i	−0.997759	+	0.0669071i	$0.978687\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.23525e10	−	2.32554e9i	0.652020	−	0.122752i
$372$		0			0
$373$	−8.22226e9	−	1.42414e10i	−0.424772	−	0.735727i	0.571627	−	0.820514i	$-0.306312\pi$
$373$	−8.22226e9	−	1.42414e10i	−0.424772	−	0.735727i	−0.996399	+	0.0847868i	$0.972979\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.47703e9i			0.320635i
$378$		0			0
$379$	1.50115e10			0.727559			0.363780	−	0.931485i	$-0.381486\pi$
$379$	1.50115e10			0.727559			0.363780	+	0.931485i	$0.381486\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.85696e9	+	5.69092e9i	−0.458087	+	0.264477i	−0.711240	−	0.702950i	$-0.751866\pi$
$383$	−9.85696e9	+	5.69092e9i	−0.458087	+	0.264477i	0.253153	+	0.967426i	$0.418533\pi$
$384$		0			0
$385$	8.98928e9	−	7.72161e9i	0.409149	−	0.351451i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.57517e10	+	9.09424e9i	0.687905	+	0.397162i	0.802827	−	0.596212i	$-0.203328\pi$
$389$	1.57517e10	+	9.09424e9i	0.687905	+	0.397162i	−0.114922	+	0.993375i	$0.536662\pi$
$390$		0			0
$391$	1.74248e10			0.745523
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.40540e10	+	8.11411e9i	0.577315	+	0.333313i
$396$		0			0
$397$	4.04840e9	+	7.01203e9i	0.162975	+	0.282281i	0.935934	−	0.352175i	$-0.114558\pi$
$397$	4.04840e9	+	7.01203e9i	0.162975	+	0.282281i	−0.772959	+	0.634456i	$0.781224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.54624e9	+	3.77947e9i	−0.253171	+	0.146169i	−0.621215	−	0.783640i	$-0.713361\pi$
$401$	−6.54624e9	+	3.77947e9i	−0.253171	+	0.146169i	0.368044	+	0.929808i	$0.380028\pi$
$402$		0			0
$403$	−6.97514e9	+	1.20813e10i	−0.264443	+	0.458029i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.10331e10i		−	1.49540i
$408$		0			0
$409$	−2.30352e9	+	3.98981e9i	−0.0823186	+	0.142580i	−0.904245	−	0.427013i	$-0.859566\pi$
$409$	−2.30352e9	+	3.98981e9i	−0.0823186	+	0.142580i	0.821927	+	0.569593i	$0.192899\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.01532e10	+	7.55940e9i	−1.38013	+	0.259829i
$414$		0			0
$415$	−7.62185e9	−	1.32014e10i	−0.256961	−	0.445070i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.03520e10i			0.335868i	0.985798	+	0.167934i	$0.0537096\pi$
$419$			1.03520e10i			0.335868i	−0.985798	+	0.167934i	$0.946290\pi$
$420$		0			0
$421$	3.21723e10			1.02413			0.512063	−	0.858948i	$-0.328882\pi$
$421$	3.21723e10			1.02413			0.512063	+	0.858948i	$0.328882\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.19138e10	−	1.84255e10i	0.978190	−	0.564758i
$426$		0			0
$427$	−3.48610e10	−	1.22342e10i	−1.04864	−	0.368013i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.46168e10	+	8.43903e9i	0.423588	+	0.244559i	0.696611	−	0.717449i	$-0.254690\pi$
$431$	1.46168e10	+	8.43903e9i	0.423588	+	0.244559i	−0.273023	+	0.962007i	$0.588024\pi$
$432$		0			0
$433$	1.06664e10			0.303435			0.151717	−	0.988424i	$-0.451520\pi$
$433$	1.06664e10			0.303435			0.151717	+	0.988424i	$0.451520\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.53605e9	+	3.77359e9i	0.179221	+	0.103473i
$438$		0			0
$439$	−1.32618e10	−	2.29702e10i	−0.357063	−	0.618452i	0.630405	−	0.776266i	$-0.282889\pi$
$439$	−1.32618e10	−	2.29702e10i	−0.357063	−	0.618452i	−0.987469	+	0.157814i	$0.949555\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.27728e10	+	2.46949e10i	−1.11059	+	0.641198i	−0.938981	−	0.343968i	$-0.888229\pi$
$443$	−4.27728e10	+	2.46949e10i	−1.11059	+	0.641198i	−0.171606	+	0.985166i	$0.554895\pi$
$444$		0			0
$445$	−1.86825e10	+	3.23590e10i	−0.476425	+	0.825193i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.24668e8i			0.00798829i	0.999992	+	0.00399415i	$0.00127138\pi$
$449$			3.24668e8i			0.00798829i	−0.999992	+	0.00399415i	$0.998729\pi$
$450$		0			0
$451$	−1.20050e10	+	2.07933e10i	−0.290173	+	0.502594i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.55663e9	−	7.28506e9i	0.0596516	−	0.169976i
$456$		0			0
$457$	3.80358e9	+	6.58799e9i	0.0872022	+	0.151039i	0.906327	−	0.422576i	$-0.138874\pi$
$457$	3.80358e9	+	6.58799e9i	0.0872022	+	0.151039i	−0.819125	+	0.573615i	$0.805541\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.90203e10i		−	0.421127i	−0.977580	−	0.210563i	$-0.932470\pi$
$461$		−	1.90203e10i		−	0.421127i	0.977580	−	0.210563i	$-0.0675298\pi$
$462$		0			0
$463$	−7.80548e10			−1.69854			−0.849270	−	0.527959i	$-0.822957\pi$
$463$	−7.80548e10			−1.69854			−0.849270	+	0.527959i	$0.822957\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.41733e10	−	2.55034e10i	0.928735	−	0.536206i	0.0423239	−	0.999104i	$-0.486524\pi$
$467$	4.41733e10	−	2.55034e10i	0.928735	−	0.536206i	0.886411	+	0.462898i	$0.153191\pi$
$468$		0			0
$469$	−1.06300e9	−	5.64636e9i	−0.0219707	−	0.116702i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.20928e10	+	2.43023e10i	0.840936	+	0.485514i
$474$		0			0
$475$	1.59612e10			0.313538
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.45961e10	−	2.57475e10i	−0.847139	−	0.489096i	0.0125458	−	0.999921i	$-0.496006\pi$
$479$	−4.45961e10	−	2.57475e10i	−0.847139	−	0.489096i	−0.859684	+	0.510826i	$0.829340\pi$
$480$		0			0
$481$	−1.33668e10	−	2.31520e10i	−0.249716	−	0.432522i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.39431e10	−	3.11441e10i	0.974920	−	0.562870i
$486$		0			0
$487$	2.39213e10	−	4.14328e10i	0.425273	−	0.736595i	−0.571172	−	0.820830i	$-0.693511\pi$
$487$	2.39213e10	−	4.14328e10i	0.425273	−	0.736595i	0.996446	+	0.0842348i	$0.0268446\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9.81309e9i			0.168842i	0.996430	+	0.0844209i	$0.0269040\pi$
$491$			9.81309e9i			0.168842i	−0.996430	+	0.0844209i	$0.973096\pi$
$492$		0			0
$493$	6.59484e10	−	1.14226e11i	1.11639	−	1.93365i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.74261e10	+	9.01372e10i	1.26900	+	1.47733i
$498$		0			0
$499$	−3.88196e10	−	6.72376e10i	−0.626108	−	1.08445i	−0.988325	−	0.152357i	$-0.951313\pi$
$499$	−3.88196e10	−	6.72376e10i	−0.626108	−	1.08445i	0.362217	−	0.932094i	$-0.382020\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1.09735e11i		−	1.71424i	−0.515114	−	0.857122i	$-0.672250\pi$
$503$		−	1.09735e11i		−	1.71424i	0.515114	−	0.857122i	$-0.327750\pi$
$504$		0			0
$505$	−1.23829e10			−0.190396
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.55015e10	+	1.47233e10i	−0.379922	+	0.219348i	−0.677784	−	0.735261i	$-0.737060\pi$
$509$	−2.55015e10	+	1.47233e10i	−0.379922	+	0.219348i	0.297862	+	0.954609i	$0.403726\pi$
$510$		0			0
$511$	2.29305e8	+	1.21800e9i	0.00336302	+	0.0178634i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.57424e10	+	2.06359e10i	0.508106	+	0.293355i
$516$		0			0
$517$	4.49179e10			0.628720
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.05467e9	−	4.07302e9i	−0.0957472	−	0.0552797i	0.451362	−	0.892341i	$-0.350938\pi$
$521$	−7.05467e9	−	4.07302e9i	−0.0957472	−	0.0552797i	−0.547109	+	0.837061i	$0.684272\pi$
$522$		0			0
$523$	−1.87140e10	−	3.24137e10i	−0.250127	−	0.433233i	0.713434	−	0.700723i	$-0.247139\pi$
$523$	−1.87140e10	−	3.24137e10i	−0.250127	−	0.433233i	−0.963561	+	0.267490i	$0.913806\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.46021e11	−	1.42040e11i	3.18955	−	1.84149i
$528$		0			0
$529$	−3.33757e10	+	5.78085e10i	−0.426195	+	0.738191i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1.56428e10i			0.193824i
$534$		0			0
$535$	−2.61046e10	+	4.52144e10i	−0.318641	+	0.551902i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.40072e10	−	5.49769e10i	0.521398	−	0.651366i
$540$		0			0
$541$	−3.06590e10	−	5.31030e10i	−0.357906	−	0.619912i	0.629704	−	0.776835i	$-0.283176\pi$
$541$	−3.06590e10	−	5.31030e10i	−0.357906	−	0.619912i	−0.987611	+	0.156923i	$0.949843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			8.74873e10i			0.991652i
$546$		0			0
$547$	−8.06663e10			−0.901038			−0.450519	−	0.892767i	$-0.648761\pi$
$547$	−8.06663e10			−0.901038			−0.450519	+	0.892767i	$0.648761\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.94745e10	−	2.85641e10i	0.536754	−	0.309895i
$552$		0			0
$553$	9.09957e10	+	3.19341e10i	0.973017	+	0.341472i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.34971e10	+	2.51131e10i	0.451897	+	0.260903i	0.708631	−	0.705579i	$-0.249313\pi$
$557$	4.34971e10	+	2.51131e10i	0.451897	+	0.260903i	−0.256734	+	0.966482i	$0.582646\pi$
$558$		0			0
$559$	3.16665e10			0.324304
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.37176e9	+	5.41079e9i	0.0932798	+	0.0538551i	0.545914	−	0.837841i	$-0.316182\pi$
$563$	9.37176e9	+	5.41079e9i	0.0932798	+	0.0538551i	−0.452634	+	0.891696i	$0.649516\pi$
$564$		0			0
$565$	−3.41826e10	−	5.92059e10i	−0.335437	−	0.580994i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.46519e11	+	8.45927e10i	−1.39780	+	0.807019i	−0.994162	−	0.107901i	$-0.965587\pi$
$569$	−1.46519e11	+	8.45927e10i	−1.39780	+	0.807019i	−0.403636	+	0.914920i	$0.632254\pi$
$570$		0			0
$571$	8.81136e10	−	1.52617e11i	0.828893	−	1.43569i	−0.0700140	−	0.997546i	$-0.522304\pi$
$571$	8.81136e10	−	1.52617e11i	0.828893	−	1.43569i	0.898907	−	0.438139i	$-0.144362\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.44467e10i		−	0.223639i
$576$		0			0
$577$	−1.60526e10	+	2.78040e10i	−0.144825	+	0.250844i	−0.929308	−	0.369307i	$-0.879595\pi$
$577$	−1.60526e10	+	2.78040e10i	−0.144825	+	0.250844i	0.784483	+	0.620151i	$0.212929\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.90253e10	−	6.87156e10i	−0.518005	−	0.603047i
$582$		0			0
$583$	−3.19753e10	−	5.53828e10i	−0.276784	−	0.479403i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.86668e11i		−	1.57223i	−0.618079	−	0.786116i	$-0.712089\pi$
$587$		−	1.86668e11i		−	1.57223i	0.618079	−	0.786116i	$-0.287911\pi$
$588$		0			0
$589$	1.23043e11			1.02234
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.63636e10	−	5.56356e10i	0.779282	−	0.449918i	−0.0568940	−	0.998380i	$-0.518120\pi$
$593$	9.63636e10	−	5.56356e10i	0.779282	−	0.449918i	0.836176	+	0.548462i	$0.184786\pi$
$594$		0			0
$595$	−1.19263e11	+	1.02445e11i	−0.951565	+	0.817375i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.32077e11	−	7.62549e10i	−1.02594	−	0.592326i	−0.110120	−	0.993918i	$-0.535123\pi$
$599$	−1.32077e11	−	7.62549e10i	−1.02594	−	0.592326i	−0.915818	+	0.401593i	$0.868457\pi$
$600$		0			0
$601$	−3.43616e10			−0.263375			−0.131688	−	0.991291i	$-0.542040\pi$
$601$	−3.43616e10			−0.263375			−0.131688	+	0.991291i	$0.542040\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.27917e10	+	1.31588e10i	0.170120	+	0.0982189i
$606$		0			0
$607$	−1.62186e10	−	2.80914e10i	−0.119470	−	0.206927i	0.800088	−	0.599883i	$-0.204786\pi$
$607$	−1.62186e10	−	2.80914e10i	−0.119470	−	0.206927i	−0.919558	+	0.392955i	$0.871453\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.53439e10	−	1.46323e10i	0.181848	−	0.104990i
$612$		0			0
$613$	−8.55918e9	+	1.48249e10i	−0.0606164	+	0.104991i	−0.894741	−	0.446585i	$-0.852640\pi$
$613$	−8.55918e9	+	1.48249e10i	−0.0606164	+	0.104991i	0.834125	+	0.551576i	$0.185973\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.04980e11i		−	0.724376i	−0.932105	−	0.362188i	$-0.882030\pi$
$617$		−	1.04980e11i		−	0.724376i	0.932105	−	0.362188i	$-0.117970\pi$
$618$		0			0
$619$	1.24195e10	−	2.15112e10i	0.0845945	−	0.146522i	−0.820624	−	0.571469i	$-0.806374\pi$
$619$	1.24195e10	−	2.15112e10i	0.0845945	−	0.146522i	0.905218	+	0.424947i	$0.139707\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.35274e10	+	2.09515e11i	−0.488087	+	1.39079i
$624$		0			0
$625$	6.03342e9	+	1.04502e10i	0.0395406	+	0.0684864i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.44397e11i			3.47787i
$630$		0			0
$631$	−1.64527e11			−1.03782			−0.518908	−	0.854830i	$-0.673661\pi$
$631$	−1.64527e11			−1.03782			−0.518908	+	0.854830i	$0.673661\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.34193e11	−	7.74761e10i	0.825341	−	0.476511i
$636$		0			0
$637$	6.92099e9	−	4.53551e10i	0.0420350	−	0.275466i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.85646e11	+	1.07183e11i	1.09965	+	0.634882i	0.936128	−	0.351659i	$-0.114382\pi$
$641$	1.85646e11	+	1.07183e11i	1.09965	+	0.634882i	0.163519	+	0.986540i	$0.447716\pi$
$642$		0			0
$643$	−1.59773e11			−0.934672			−0.467336	−	0.884080i	$-0.654786\pi$
$643$	−1.59773e11			−0.934672			−0.467336	+	0.884080i	$0.654786\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.38290e11	−	7.98419e10i	−0.789177	−	0.455632i	0.0504956	−	0.998724i	$-0.483920\pi$
$647$	−1.38290e11	−	7.98419e10i	−0.789177	−	0.455632i	−0.839673	+	0.543093i	$0.817253\pi$
$648$		0			0
$649$	1.03939e11	+	1.80028e11i	0.585868	+	1.01475i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.93727e10	−	5.73729e10i	0.546530	−	0.315540i	−0.201191	−	0.979552i	$-0.564481\pi$
$653$	9.93727e10	−	5.73729e10i	0.546530	−	0.315540i	0.747721	+	0.664013i	$0.231148\pi$
$654$		0			0
$655$	−2.73322e10	+	4.73407e10i	−0.148494	+	0.257199i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3.04429e11i		−	1.61415i	−0.590449	−	0.807075i	$-0.701049\pi$
$659$		−	3.04429e11i		−	1.61415i	0.590449	−	0.807075i	$-0.298951\pi$
$660$		0			0
$661$	−1.54169e11	+	2.67028e11i	−0.807590	+	1.39879i	0.106939	+	0.994266i	$0.465895\pi$
$661$	−1.54169e11	+	2.67028e11i	−0.807590	+	1.39879i	−0.914529	+	0.404521i	$0.867438\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.69215e10	+	1.25989e10i	−0.342199	+	0.0644237i
$666$		0			0
$667$	−4.37497e10	−	7.57768e10i	−0.221041	−	0.382854i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.87969e11i			0.927246i
$672$		0			0
$673$	3.19224e11			1.55609			0.778047	−	0.628206i	$-0.216211\pi$
$673$	3.19224e11			1.55609			0.778047	+	0.628206i	$0.216211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.27203e11	+	1.88910e11i	−1.55762	+	0.899293i	−0.560138	+	0.828399i	$0.689252\pi$
$677$	−3.27203e11	+	1.88910e11i	−1.55762	+	0.899293i	−0.997484	+	0.0708939i	$0.977415\pi$
$678$		0			0
$679$	2.80783e11	−	2.41187e11i	1.32096	−	1.13468i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.23257e11	−	7.11627e10i	−0.566410	−	0.327017i	0.189305	−	0.981918i	$-0.439377\pi$
$683$	−1.23257e11	−	7.11627e10i	−0.566410	−	0.327017i	−0.755714	+	0.654902i	$0.772710\pi$
$684$		0			0
$685$	5.03514e10			0.228691
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.60827e10	−	2.08323e10i	−0.160111	−	0.0924402i
$690$		0			0
$691$	4.68290e10	+	8.11101e10i	0.205401	+	0.355765i	0.950260	−	0.311457i	$-0.100817\pi$
$691$	4.68290e10	+	8.11101e10i	0.205401	+	0.355765i	−0.744860	+	0.667221i	$0.767484\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.41994e9	+	1.97451e9i	−0.0146582	+	0.00846290i
$696$		0			0
$697$	1.59274e11	−	2.75870e11i	0.674859	−	1.16889i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.68356e11i			0.697199i	0.937272	+	0.348599i	$0.113343\pi$
$701$			1.68356e11i			0.697199i	−0.937272	+	0.348599i	$0.886657\pi$
$702$		0			0
$703$	−1.17897e11	+	2.04203e11i	−0.482704	+	0.836068i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.23155e10	+	1.36144e10i	−0.289437	+	0.0544905i
$708$		0			0
$709$	2.66170e9	+	4.61020e9i	0.0105335	+	0.0182446i	0.871244	−	0.490850i	$-0.163314\pi$
$709$	2.66170e9	+	4.61020e9i	0.0105335	+	0.0182446i	−0.860711	+	0.509095i	$0.829980\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.88457e11i		−	0.729212i
$714$		0			0
$715$	−3.92806e10			−0.150298
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.42127e10	+	1.97527e10i	−0.128018	+	0.0739114i	−0.562642	−	0.826701i	$-0.690215\pi$
$719$	−3.42127e10	+	1.97527e10i	−0.128018	+	0.0739114i	0.434623	+	0.900612i	$0.356882\pi$
$720$		0			0
$721$	2.31421e11	+	8.12152e10i	0.856370	+	0.300536i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.60257e11	−	9.25243e10i	−0.580049	−	0.334891i
$726$		0			0
$727$	−6.32747e9			−0.0226513			−0.0113256	−	0.999936i	$-0.503605\pi$
$727$	−6.32747e9			−0.0226513			−0.0113256	+	0.999936i	$0.503605\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.58456e11	−	3.22425e11i	−1.95578	−	1.12917i
$732$		0			0
$733$	9.68943e10	+	1.67826e11i	0.335647	+	0.581357i	0.983609	−	0.180315i	$-0.0577118\pi$
$733$	9.68943e10	+	1.67826e11i	0.335647	+	0.581357i	−0.647962	+	0.761673i	$0.724378\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.53155e10	+	1.46159e10i	−0.0858058	+	0.0495400i
$738$		0			0
$739$	−2.33208e11	+	4.03928e11i	−0.781927	+	1.35434i	0.148891	+	0.988854i	$0.452430\pi$
$739$	−2.33208e11	+	4.03928e11i	−0.781927	+	1.35434i	−0.930818	+	0.365483i	$0.880904\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.78453e11i			0.585556i	0.956181	+	0.292778i	$0.0945796\pi$
$743$			1.78453e11i			0.585556i	−0.956181	+	0.292778i	$0.905420\pi$
$744$		0			0
$745$	−1.71412e11	+	2.96894e11i	−0.556436	+	0.963775i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.02738e11	+	2.92750e11i	−0.326440	+	0.930184i
$750$		0			0
$751$	2.81487e11	+	4.87549e11i	0.884908	+	1.53271i	0.845819	+	0.533470i	$0.179112\pi$
$751$	2.81487e11	+	4.87549e11i	0.884908	+	1.53271i	0.0390887	+	0.999236i	$0.487555\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.45029e11i			0.446341i
$756$		0			0
$757$	−2.75340e11			−0.838466			−0.419233	−	0.907879i	$-0.637701\pi$
$757$	−2.75340e11			−0.838466			−0.419233	+	0.907879i	$0.637701\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.54738e11	+	2.04808e11i	−1.05772	+	0.610673i	−0.924800	−	0.380454i	$-0.875768\pi$
$761$	−3.54738e11	+	2.04808e11i	−1.05772	+	0.610673i	−0.132917	+	0.991127i	$0.542434\pi$
$762$		0			0
$763$	9.61877e10	+	5.10920e11i	0.283806	+	1.50749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.17290e11	+	6.77176e10i	0.338907	+	0.195668i
$768$		0			0
$769$	4.47102e11			1.27850			0.639250	−	0.768999i	$-0.279245\pi$
$769$	4.47102e11			1.27850			0.639250	+	0.768999i	$0.279245\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.97227e11	−	1.71604e11i	−0.832472	−	0.480628i	0.0222260	−	0.999753i	$-0.492925\pi$
$773$	−2.97227e11	−	1.71604e11i	−0.832472	−	0.480628i	−0.854698	+	0.519125i	$0.826258\pi$
$774$		0			0
$775$	−1.99280e11	−	3.45162e11i	−0.552403	−	0.956790i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.19487e11	−	6.89859e10i	0.324468	−	0.187331i
$780$		0			0
$781$	3.02277e11	−	5.23559e11i	0.812457	−	1.40722i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			3.71039e11i			0.977105i
$786$		0			0
$787$	5.26115e10	−	9.11257e10i	0.137145	−	0.237543i	−0.789270	−	0.614047i	$-0.789541\pi$
$787$	5.26115e10	−	9.11257e10i	0.137145	−	0.237543i	0.926415	+	0.376504i	$0.122874\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.64717e11	−	3.08177e11i	−0.676202	−	0.787215i
$792$		0			0
$793$	6.12320e10	+	1.06057e11i	0.154841	+	0.268192i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.01194e11i			0.746471i	0.927737	+	0.373236i	$0.121752\pi$
$797$			3.01194e11i			0.746471i	−0.927737	+	0.373236i	$0.878248\pi$
$798$		0			0
$799$	−5.95937e11			−1.46222
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.46091e9	−	3.15286e9i	0.0131342	−	0.00758302i
$804$		0			0
$805$	1.92969e10	+	1.02499e11i	0.0459519	+	0.244082i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.30511e10	−	1.33086e10i	−0.0538143	−	0.0310697i	0.472851	−	0.881142i	$-0.343225\pi$
$809$	−2.30511e10	−	1.33086e10i	−0.0538143	−	0.0310697i	−0.526666	+	0.850072i	$0.676558\pi$
$810$		0			0
$811$	2.25219e11			0.520622			0.260311	−	0.965525i	$-0.416175\pi$
$811$	2.25219e11			0.520622			0.260311	+	0.965525i	$0.416175\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.69635e10	−	3.86614e10i	−0.151778	−	0.0876289i
$816$		0			0
$817$	−1.39651e11	−	2.41883e11i	−0.313442	−	0.542897i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.98848e11	−	1.72540e11i	0.657776	−	0.379767i	−0.133653	−	0.991028i	$-0.542671\pi$
$821$	2.98848e11	−	1.72540e11i	0.657776	−	0.379767i	0.791429	+	0.611261i	$0.209337\pi$
$822$		0			0
$823$	−2.74635e10	+	4.75681e10i	−0.0598626	+	0.103685i	−0.894404	−	0.447261i	$-0.852400\pi$
$823$	−2.74635e10	+	4.75681e10i	−0.0598626	+	0.103685i	0.834541	+	0.550946i	$0.185733\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.15387e11i		−	0.674251i	−0.941460	−	0.337126i	$-0.890545\pi$
$827$		−	3.15387e11i		−	0.674251i	0.941460	−	0.337126i	$-0.109455\pi$
$828$		0			0
$829$	−1.83826e11	+	3.18396e11i	−0.389214	+	0.674139i	−0.992344	−	0.123504i	$-0.960587\pi$
$829$	−1.83826e11	+	3.18396e11i	−0.389214	+	0.674139i	0.603130	+	0.797643i	$0.293920\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.83856e11	+	7.29393e11i	−1.21262	+	1.51489i
$834$		0			0
$835$	−1.03008e11	−	1.78416e11i	−0.211898	−	0.367018i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			4.42480e11i			0.892988i	0.894787	+	0.446494i	$0.147328\pi$
$839$			4.42480e11i			0.892988i	−0.894787	+	0.446494i	$0.852672\pi$
$840$		0			0
$841$	−1.62079e11			−0.323998
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.63266e11	−	1.51997e11i	0.516379	−	0.298131i
$846$		0			0
$847$	1.47569e11	+	5.17882e10i	0.286723	+	0.100623i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.12765e11	+	1.80575e11i	0.596347	+	0.344301i
$852$		0			0
$853$	8.10112e11			1.53020			0.765102	−	0.643910i	$-0.222689\pi$
$853$	8.10112e11			1.53020			0.765102	+	0.643910i	$0.222689\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.23929e10	+	1.87020e10i	0.0600518	+	0.0346709i	0.529725	−	0.848169i	$-0.322295\pi$
$857$	3.23929e10	+	1.87020e10i	0.0600518	+	0.0346709i	−0.469673	+	0.882840i	$0.655628\pi$
$858$		0			0
$859$	−2.53311e11	−	4.38747e11i	−0.465244	−	0.805827i	0.533968	−	0.845504i	$-0.320700\pi$
$859$	−2.53311e11	−	4.38747e11i	−0.465244	−	0.805827i	−0.999213	+	0.0396779i	$0.987367\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.18246e11	−	5.30150e11i	1.65545	−	0.955774i	0.680675	−	0.732586i	$-0.261687\pi$
$863$	9.18246e11	−	5.30150e11i	1.65545	−	0.955774i	0.974775	−	0.223189i	$-0.0716466\pi$
$864$		0			0
$865$	−3.01617e11	+	5.22416e11i	−0.538755	+	0.933151i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	4.90644e11i		−	0.860374i
$870$		0			0
$871$	−9.52247e9	+	1.64934e10i	−0.0165454	+	0.0286574i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.90645e11	+	4.54778e11i	0.666423	+	0.775830i
$876$		0			0
$877$	−3.16539e11	−	5.48262e11i	−0.535093	−	0.926809i	−0.999159	−	0.0410078i	$-0.986943\pi$
$877$	−3.16539e11	−	5.48262e11i	−0.535093	−	0.926809i	0.464066	−	0.885801i	$-0.346390\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.39180e11i			0.895015i	0.894280	+	0.447507i	$0.147688\pi$
$881$			5.39180e11i			0.895015i	−0.894280	+	0.447507i	$0.852312\pi$
$882$		0			0
$883$	−1.39105e11			−0.228823			−0.114412	−	0.993433i	$-0.536498\pi$
$883$	−1.39105e11			−0.228823			−0.114412	+	0.993433i	$0.536498\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.45611e11	+	2.57274e11i	−0.719882	+	0.415624i	−0.814709	−	0.579870i	$-0.803103\pi$
$887$	−4.45611e11	+	2.57274e11i	−0.719882	+	0.415624i	0.0948272	+	0.995494i	$0.469770\pi$
$888$		0			0
$889$	6.98494e11	−	5.99992e11i	1.11829	−	0.960592i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.23536e11	−	1.29059e11i	−0.351514	−	0.202946i
$894$		0			0
$895$	−7.03901e11			−1.09703
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.23540e12	−	7.13261e11i	−1.89134	−	1.09197i
$900$		0			0
$901$	4.24225e11	+	7.34779e11i	0.643720	+	1.11496i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.66124e11	−	1.53647e11i	0.396726	−	0.229050i
$906$		0			0
$907$	1.91938e11	−	3.32447e11i	0.283617	−	0.491239i	−0.688656	−	0.725088i	$-0.741799\pi$
$907$	1.91938e11	−	3.32447e11i	0.283617	−	0.491239i	0.972273	+	0.233849i	$0.0751322\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.05117e12i		−	1.52616i	−0.646306	−	0.763078i	$-0.723687\pi$
$911$		−	1.05117e12i		−	1.52616i	0.646306	−	0.763078i	$-0.276313\pi$
$912$		0			0
$913$	−2.30439e11	+	3.99132e11i	−0.331644	+	0.574425i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.07569e11	+	3.06517e11i	−0.152129	+	0.433488i
$918$		0			0
$919$	−8.68948e10	−	1.50506e11i	−0.121824	−	0.211005i	0.798663	−	0.601778i	$-0.205541\pi$
$919$	−8.68948e10	−	1.50506e11i	−0.121824	−	0.211005i	−0.920487	+	0.390773i	$0.872208\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	3.93875e11i		−	0.542689i
$924$		0			0
$925$	7.63778e11			1.04328
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.06698e12	−	6.16020e11i	1.43249	−	0.827050i	0.435182	−	0.900343i	$-0.356684\pi$
$929$	1.06698e12	−	6.16020e11i	1.43249	−	0.827050i	0.997310	+	0.0732927i	$0.0233507\pi$
$930$		0			0
$931$	−3.76965e11	+	1.47153e11i	−0.501767	+	0.195871i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.92735e11	+	3.99951e11i	0.906402	+	0.523311i
$936$		0			0
$937$	−2.45315e11			−0.318248			−0.159124	−	0.987259i	$-0.550867\pi$
$937$	−2.45315e11			−0.318248			−0.159124	+	0.987259i	$0.550867\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.40790e11	−	8.12850e10i	−0.179561	−	0.103670i	0.407525	−	0.913194i	$-0.366392\pi$
$941$	−1.40790e11	−	8.12850e10i	−0.179561	−	0.103670i	−0.587086	+	0.809524i	$0.699725\pi$
$942$		0			0
$943$	−1.05661e11	−	1.83010e11i	−0.133619	−	0.231435i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.47524e11	−	4.31583e11i	0.929448	−	0.536617i	0.0428111	−	0.999083i	$-0.486369\pi$
$947$	7.47524e11	−	4.31583e11i	0.929448	−	0.536617i	0.886637	+	0.462466i	$0.153035\pi$
$948$		0			0
$949$	2.05413e9	−	3.55786e9i	0.00253258	−	0.00438656i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			4.03315e11i			0.488960i	0.969654	+	0.244480i	$0.0786172\pi$
$953$			4.03315e11i			0.488960i	−0.969654	+	0.244480i	$0.921383\pi$
$954$		0			0
$955$	−3.62701e11	+	6.28216e11i	−0.436049	+	0.755259i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.94049e11	−	5.53587e10i	0.347652	−	0.0654502i
$960$		0			0
$961$	−1.10978e12	−	1.92220e12i	−1.30120	−	2.25374i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			3.21379e11i			0.370602i
$966$		0			0
$967$	8.52582e11			0.975058			0.487529	−	0.873107i	$-0.337898\pi$
$967$	8.52582e11			0.975058			0.487529	+	0.873107i	$0.337898\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.21106e12	−	6.99203e11i	1.36234	−	0.786550i	0.372409	−	0.928069i	$-0.378532\pi$
$971$	1.21106e12	−	6.99203e11i	1.36234	−	0.786550i	0.989936	+	0.141518i	$0.0451984\pi$
$972$		0			0
$973$	−1.78014e10	+	1.52910e10i	−0.0198610	+	0.0170602i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.30450e11	+	3.06255e11i	0.582192	+	0.336129i	0.762004	−	0.647572i	$-0.224216\pi$
$977$	5.30450e11	+	3.06255e11i	0.582192	+	0.336129i	−0.179812	+	0.983701i	$0.557549\pi$
$978$		0			0
$979$	1.12969e12			1.22979
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.19275e10	+	5.30744e10i	0.0984536	+	0.0568422i	0.548418	−	0.836204i	$-0.315230\pi$
$983$	9.19275e10	+	5.30744e10i	0.0984536	+	0.0568422i	−0.449965	+	0.893046i	$0.648563\pi$
$984$		0			0
$985$	−9.03688e10	−	1.56523e11i	−0.0960005	−	0.166278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.70476e11	+	2.13894e11i	−0.387235	+	0.223570i
$990$		0			0
$991$	6.81982e11	−	1.18123e12i	0.707096	−	1.22473i	−0.258833	−	0.965922i	$-0.583338\pi$
$991$	6.81982e11	−	1.18123e12i	0.707096	−	1.22473i	0.965930	−	0.258805i	$-0.0833286\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	6.33754e11i		−	0.646589i
$996$		0			0
$997$	−6.95115e11	+	1.20397e12i	−0.703519	+	1.21853i	0.263704	+	0.964604i	$0.415056\pi$
$997$	−6.95115e11	+	1.20397e12i	−0.703519	+	1.21853i	−0.967223	+	0.253928i	$0.918278\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bk.a.53.7	✓	44
3.2	odd	2	inner	252.9.bk.a.53.16	yes	44
7.2	even	3	inner	252.9.bk.a.233.16	yes	44
21.2	odd	6	inner	252.9.bk.a.233.7	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.7	✓	44	1.1	even	1	trivial
252.9.bk.a.53.16	yes	44	3.2	odd	2	inner
252.9.bk.a.233.7	yes	44	21.2	odd	6	inner
252.9.bk.a.233.16	yes	44	7.2	even	3	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.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-349.907 + 202.019i) q^{5} +(-1821.32 + 1564.48i) q^{7} +O(q^{10})\)	\(q+(-349.907 + 202.019i) q^{5} +(-1821.32 + 1564.48i) q^{7} +(10579.1 + 6107.83i) q^{11} +7958.67 q^{13} +(-140355. - 81034.2i) q^{17} +(-35098.2 - 60791.9i) q^{19} +(-93110.9 + 53757.6i) q^{23} +(-113689. + 196916. i) q^{25} +813834. i q^{29} +(-876420. + 1.51800e6i) q^{31} +(321238. - 915362. i) q^{35} +(-1.67953e6 - 2.90903e6i) q^{37} +1.96551e6i q^{41} +3.97887e6 q^{43} +(3.18444e6 - 1.83854e6i) q^{47} +(869617. - 5.69883e6i) q^{49} +(-4.53376e6 - 2.61757e6i) q^{53} -4.93558e6 q^{55} +(1.47374e7 + 8.50867e6i) q^{59} +(7.69375e6 + 1.33260e7i) q^{61} +(-2.78479e6 + 1.60780e6i) q^{65} +(-1.19649e6 + 2.07238e6i) q^{67} -4.94900e7i q^{71} +(258100. - 447042. i) q^{73} +(-2.88234e7 + 5.42641e6i) q^{77} +(-2.00826e7 - 3.47840e7i) q^{79} +3.77285e7i q^{83} +6.54817e7 q^{85} +(8.00892e7 - 4.62395e7i) q^{89} +(-1.44953e7 + 1.24512e7i) q^{91} +(2.45622e7 + 1.41810e7i) q^{95} -1.54164e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 44 * q + 1230 * q^7	\( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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