LMFDB - Embedded newform 252.9.bk.a.53.12

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			53.12
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.4272 - 13.5257i) q^{5} +(-2307.52 + 663.449i) q^{7} +O(q^{10})$	$q+(23.4272 - 13.5257i) q^{5} +(-2307.52 + 663.449i) q^{7} +(-6538.52 - 3775.02i) q^{11} +10734.9 q^{13} +(48780.7 + 28163.5i) q^{17} +(55457.2 + 96054.6i) q^{19} +(-409756. + 236573. i) q^{23} +(-194947. + 337657. i) q^{25} -386162. i q^{29} +(768340. - 1.33080e6i) q^{31} +(-45085.0 + 46753.5i) q^{35} +(792859. + 1.37327e6i) q^{37} +794246. i q^{41} -6.19734e6 q^{43} +(6.61379e6 - 3.81847e6i) q^{47} +(4.88447e6 - 3.06184e6i) q^{49} +(2.25899e6 + 1.30423e6i) q^{53} -204239. q^{55} +(-1.13342e7 - 6.54383e6i) q^{59} +(-2.09062e6 - 3.62105e6i) q^{61} +(251489. - 145197. i) q^{65} +(-3.05074e6 + 5.28404e6i) q^{67} +2.56254e7i q^{71} +(461382. - 799136. i) q^{73} +(1.75923e7 + 4.37294e6i) q^{77} +(-941806. - 1.63126e6i) q^{79} -8.96877e7i q^{83} +1.52372e6 q^{85} +(-2.07346e7 + 1.19711e7i) q^{89} +(-2.47710e7 + 7.12207e6i) q^{91} +(2.59841e6 + 1.50019e6i) q^{95} -1.05576e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	23.4272	−	13.5257i	0.0374835	−	0.0216411i	−0.481141	−	0.876643i	$-0.659778\pi$
$5$	23.4272	−	13.5257i	0.0374835	−	0.0216411i	0.518625	+	0.855002i	$0.326444\pi$
$6$		0			0
$7$	−2307.52	+	663.449i	−0.961065	+	0.276322i
$7$	−2307.52	+	663.449i	−0.961065	+	0.276322i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6538.52	−	3775.02i	−0.446590	−	0.257839i	0.259799	−	0.965663i	$-0.416344\pi$
$11$	−6538.52	−	3775.02i	−0.446590	−	0.257839i	−0.706389	+	0.707824i	$0.749677\pi$
$12$		0			0
$13$	10734.9			0.375860			0.187930	−	0.982182i	$-0.439822\pi$
$13$	10734.9			0.375860			0.187930	+	0.982182i	$0.439822\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	48780.7	+	28163.5i	0.584053	+	0.337203i	0.762742	−	0.646702i	$-0.223852\pi$
$17$	48780.7	+	28163.5i	0.584053	+	0.337203i	−0.178690	+	0.983906i	$0.557186\pi$
$18$		0			0
$19$	55457.2	+	96054.6i	0.425543	+	0.737062i	0.996471	−	0.0839383i	$-0.0267499\pi$
$19$	55457.2	+	96054.6i	0.425543	+	0.737062i	−0.570928	+	0.821000i	$0.693417\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−409756.	+	236573.i	−1.46425	+	0.845383i	−0.999203	−	0.0399055i	$-0.987294\pi$
$23$	−409756.	+	236573.i	−1.46425	+	0.845383i	−0.465043	+	0.885288i	$0.653961\pi$
$24$		0			0
$25$	−194947.	+	337657.i	−0.499063	+	0.864403i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	386162.i		−	0.545982i	−0.962017	−	0.272991i	$-0.911987\pi$
$29$		−	386162.i		−	0.545982i	0.962017	−	0.272991i	$-0.0880128\pi$
$30$		0			0
$31$	768340.	−	1.33080e6i	0.831968	−	1.44101i	−0.0645078	−	0.997917i	$-0.520548\pi$
$31$	768340.	−	1.33080e6i	0.831968	−	1.44101i	0.896476	−	0.443093i	$-0.146119\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−45085.0	+	46753.5i	−0.0300441	+	0.0311560i
$36$		0			0
$37$	792859.	+	1.37327e6i	0.423047	+	0.732739i	0.996236	−	0.0866841i	$-0.0276271\pi$
$37$	792859.	+	1.37327e6i	0.423047	+	0.732739i	−0.573189	+	0.819424i	$0.694294\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			794246.i			0.281073i	0.990075	+	0.140537i	$0.0448828\pi$
$41$			794246.i			0.281073i	−0.990075	+	0.140537i	$0.955117\pi$
$42$		0			0
$43$	−6.19734e6			−1.81272			−0.906362	−	0.422502i	$-0.861152\pi$
$43$	−6.19734e6			−1.81272			−0.906362	+	0.422502i	$0.861152\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.61379e6	−	3.81847e6i	1.35537	−	0.782525i	0.366377	−	0.930467i	$-0.380598\pi$
$47$	6.61379e6	−	3.81847e6i	1.35537	−	0.782525i	0.988996	+	0.147942i	$0.0472648\pi$
$48$		0			0
$49$	4.88447e6	−	3.06184e6i	0.847292	−	0.531127i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.25899e6	+	1.30423e6i	0.286293	+	0.165291i	0.636269	−	0.771467i	$-0.280477\pi$
$53$	2.25899e6	+	1.30423e6i	0.286293	+	0.165291i	−0.349976	+	0.936759i	$0.613810\pi$
$54$		0			0
$55$	−204239.			−0.0223197
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.13342e7	−	6.54383e6i	−0.935372	−	0.540037i	−0.0468654	−	0.998901i	$-0.514923\pi$
$59$	−1.13342e7	−	6.54383e6i	−0.935372	−	0.540037i	−0.888507	+	0.458864i	$0.848257\pi$
$60$		0			0
$61$	−2.09062e6	−	3.62105e6i	−0.150992	−	0.261527i	0.780600	−	0.625031i	$-0.214914\pi$
$61$	−2.09062e6	−	3.62105e6i	−0.150992	−	0.261527i	−0.931593	+	0.363504i	$0.881580\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	251489.	−	145197.i	0.0140885	−	0.00813401i
$66$		0			0
$67$	−3.05074e6	+	5.28404e6i	−0.151393	+	0.262221i	−0.931740	−	0.363126i	$-0.881709\pi$
$67$	−3.05074e6	+	5.28404e6i	−0.151393	+	0.262221i	0.780347	+	0.625347i	$0.215043\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.56254e7i			1.00841i	0.863584	+	0.504206i	$0.168215\pi$
$71$			2.56254e7i			1.00841i	−0.863584	+	0.504206i	$0.831785\pi$
$72$		0			0
$73$	461382.	−	799136.i	0.0162468	−	0.0281403i	−0.857788	−	0.514004i	$-0.828162\pi$
$73$	461382.	−	799136.i	0.0162468	−	0.0281403i	0.874035	+	0.485864i	$0.161495\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.75923e7	+	4.37294e6i	0.500449	+	0.124397i
$78$		0			0
$79$	−941806.	−	1.63126e6i	−0.0241798	−	0.0418807i	0.853682	−	0.520794i	$-0.174364\pi$
$79$	−941806.	−	1.63126e6i	−0.0241798	−	0.0418807i	−0.877862	+	0.478913i	$0.841031\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.96877e7i		−	1.88982i	−0.327330	−	0.944910i	$-0.606149\pi$
$83$		−	8.96877e7i		−	1.88982i	0.327330	−	0.944910i	$-0.393851\pi$
$84$		0			0
$85$	1.52372e6			0.0291898
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.07346e7	+	1.19711e7i	−0.330473	+	0.190798i	−0.656051	−	0.754717i	$-0.727774\pi$
$89$	−2.07346e7	+	1.19711e7i	−0.330473	+	0.190798i	0.325578	+	0.945515i	$0.394441\pi$
$90$		0			0
$91$	−2.47710e7	+	7.12207e6i	−0.361225	+	0.103858i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.59841e6	+	1.50019e6i	0.0319016	+	0.0184184i
$96$		0			0
$97$	−1.05576e7			−0.119255			−0.0596275	−	0.998221i	$-0.518991\pi$
$97$	−1.05576e7			−0.119255			−0.0596275	+	0.998221i	$0.518991\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.14873e8	+	6.63218e7i	1.10390	+	0.637339i	0.937244	−	0.348675i	$-0.113368\pi$
$101$	1.14873e8	+	6.63218e7i	1.10390	+	0.637339i	0.166660	+	0.986014i	$0.446702\pi$
$102$		0			0
$103$	−9.51652e7	−	1.64831e8i	−0.845530	−	1.46450i	−0.885160	−	0.465287i	$-0.845951\pi$
$103$	−9.51652e7	−	1.64831e8i	−0.845530	−	1.46450i	0.0396300	−	0.999214i	$-0.487382\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.13902e8	−	1.23496e8i	1.63185	−	0.942147i	0.648324	−	0.761365i	$-0.275470\pi$
$107$	2.13902e8	−	1.23496e8i	1.63185	−	0.942147i	0.983523	−	0.180782i	$-0.0578629\pi$
$108$		0			0
$109$	6.32864e6	−	1.09615e7i	0.0448337	−	0.0776542i	−0.842738	−	0.538324i	$-0.819058\pi$
$109$	6.32864e6	−	1.09615e7i	0.0448337	−	0.0776542i	0.887571	+	0.460670i	$0.152391\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.79320e8i		−	1.71312i	−0.516045	−	0.856561i	$-0.672596\pi$
$113$		−	2.79320e8i		−	1.71312i	0.516045	−	0.856561i	$-0.327404\pi$
$114$		0			0
$115$	−6.39962e6	+	1.10845e7i	−0.0365900	+	0.0633758i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.31247e8	−	3.26244e7i	−0.654489	−	0.162688i
$120$		0			0
$121$	−7.86779e7	−	1.36274e8i	−0.367038	−	0.635729i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.11141e7i			0.0864833i
$126$		0			0
$127$	−1.23251e8			−0.473778			−0.236889	−	0.971537i	$-0.576128\pi$
$127$	−1.23251e8			−0.473778			−0.236889	+	0.971537i	$0.576128\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.20361e8	−	2.42695e8i	1.42737	−	0.824093i	0.430459	−	0.902610i	$-0.358352\pi$
$131$	4.20361e8	−	2.42695e8i	1.42737	−	0.824093i	0.996913	+	0.0785169i	$0.0250185\pi$
$132$		0			0
$133$	−1.91696e8	−	1.84855e8i	−0.612641	−	0.590778i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.51097e7	+	4.33646e7i	0.213213	+	0.123099i	0.602804	−	0.797889i	$-0.294050\pi$
$137$	7.51097e7	+	4.33646e7i	0.213213	+	0.123099i	−0.389591	+	0.920988i	$0.627384\pi$
$138$		0			0
$139$	5.81965e6			0.0155897			0.00779485	−	0.999970i	$-0.497519\pi$
$139$	5.81965e6			0.0155897			0.00779485	+	0.999970i	$0.497519\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.01905e7	−	4.05245e7i	−0.167855	−	0.0969112i
$144$		0			0
$145$	−5.22311e6	−	9.04669e6i	−0.0118156	−	0.0204653i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.20908e8	−	6.98065e7i	0.245308	−	0.141629i	−0.372306	−	0.928110i	$-0.621433\pi$
$149$	1.20908e8	−	6.98065e7i	0.245308	−	0.141629i	0.617614	+	0.786481i	$0.288099\pi$
$150$		0			0
$151$	−1.96107e8	+	3.39667e8i	−0.377212	+	0.653350i	−0.990655	−	0.136389i	$-0.956450\pi$
$151$	−1.96107e8	+	3.39667e8i	−0.377212	+	0.653350i	0.613444	+	0.789739i	$0.289784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.15693e7i		−	0.0720188i
$156$		0			0
$157$	1.27122e8	−	2.20182e8i	0.209229	−	0.362396i	−0.742243	−	0.670131i	$-0.766238\pi$
$157$	1.27122e8	−	2.20182e8i	0.209229	−	0.362396i	0.951472	+	0.307735i	$0.0995712\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.88565e8	−	8.17748e8i	1.17364	−	1.21707i
$162$		0			0
$163$	−4.75798e8	−	8.24106e8i	−0.674019	−	1.16743i	−0.976755	−	0.214360i	$-0.931233\pi$
$163$	−4.75798e8	−	8.24106e8i	−0.674019	−	1.16743i	0.302736	−	0.953074i	$-0.402100\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.84071e8i		−	1.26520i	−0.774477	−	0.632602i	$-0.781987\pi$
$167$		−	9.84071e8i		−	1.26520i	0.774477	−	0.632602i	$-0.218013\pi$
$168$		0			0
$169$	−7.00492e8			−0.858730
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.96642e8	−	5.17677e8i	1.00100	−	0.577929i	0.0924573	−	0.995717i	$-0.470528\pi$
$173$	8.96642e8	−	5.17677e8i	1.00100	−	0.577929i	0.908545	+	0.417788i	$0.137195\pi$
$174$		0			0
$175$	2.25824e8	−	9.08488e8i	0.240779	−	0.968650i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.97156e8	+	2.87033e8i	0.484262	+	0.279589i	0.722191	−	0.691694i	$-0.243135\pi$
$179$	4.97156e8	+	2.87033e8i	0.484262	+	0.279589i	−0.237929	+	0.971283i	$0.576469\pi$
$180$		0			0
$181$	−1.63207e9			−1.52063			−0.760315	−	0.649554i	$-0.774956\pi$
$181$	−1.63207e9			−1.52063			−0.760315	+	0.649554i	$0.774956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.71489e7	+	2.14479e7i	0.0317146	+	0.0183104i
$186$		0			0
$187$	−2.12636e8	−	3.68296e8i	−0.173888	−	0.301183i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.57175e9	−	9.07449e8i	1.18100	−	0.681850i	0.224753	−	0.974416i	$-0.427842\pi$
$191$	1.57175e9	−	9.07449e8i	1.18100	−	0.681850i	0.956245	+	0.292566i	$0.0945090\pi$
$192$		0			0
$193$	1.04863e9	−	1.81629e9i	0.755778	−	1.30905i	−0.189209	−	0.981937i	$-0.560592\pi$
$193$	1.04863e9	−	1.81629e9i	0.755778	−	1.30905i	0.944987	−	0.327109i	$-0.106074\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.93160e9i		−	1.28248i	−0.767340	−	0.641241i	$-0.778420\pi$
$197$		−	1.93160e9i		−	1.28248i	0.767340	−	0.641241i	$-0.221580\pi$
$198$		0			0
$199$	−2.10563e8	+	3.64705e8i	−0.134267	+	0.232557i	−0.925317	−	0.379194i	$-0.876201\pi$
$199$	−2.10563e8	+	3.64705e8i	−0.134267	+	0.232557i	0.791050	+	0.611751i	$0.209535\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.56199e8	+	8.91076e8i	0.150867	+	0.524724i
$204$		0			0
$205$	1.07427e7	+	1.86069e7i	0.00608274	+	0.0105356i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.37407e8i		−	0.438886i
$210$		0			0
$211$	−9.78430e8			−0.493628			−0.246814	−	0.969063i	$-0.579384\pi$
$211$	−9.78430e8			−0.493628			−0.246814	+	0.969063i	$0.579384\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.45186e8	+	8.38233e7i	−0.0679472	+	0.0392293i
$216$		0			0
$217$	−8.90037e8	+	3.58061e9i	−0.401392	+	1.61480i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.23657e8	+	3.02333e8i	0.219522	+	0.126741i
$222$		0			0
$223$	7.61547e8			0.307948			0.153974	−	0.988075i	$-0.450793\pi$
$223$	7.61547e8			0.307948			0.153974	+	0.988075i	$0.450793\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.50466e7	−	5.48752e7i	−0.0357959	−	0.0206668i	0.481995	−	0.876174i	$-0.339912\pi$
$227$	−9.50466e7	−	5.48752e7i	−0.0357959	−	0.0206668i	−0.517791	+	0.855507i	$0.673246\pi$
$228$		0			0
$229$	−1.30437e9	−	2.25923e9i	−0.474305	−	0.821520i	0.525262	−	0.850941i	$-0.323967\pi$
$229$	−1.30437e9	−	2.25923e9i	−0.474305	−	0.821520i	−0.999567	+	0.0294201i	$0.990634\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.63617e9	+	9.44645e8i	−0.555144	+	0.320513i	−0.751194	−	0.660081i	$-0.770522\pi$
$233$	−1.63617e9	+	9.44645e8i	−0.555144	+	0.320513i	0.196050	+	0.980594i	$0.437188\pi$
$234$		0			0
$235$	1.03295e8	−	1.78912e8i	0.0338694	−	0.0586635i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.24465e8i		−	0.130092i	−0.997882	−	0.0650460i	$-0.979281\pi$
$239$		−	4.24465e8i		−	0.130092i	0.997882	−	0.0650460i	$-0.0207194\pi$
$240$		0			0
$241$	1.12453e9	−	1.94775e9i	0.333353	−	0.577385i	−0.649814	−	0.760093i	$-0.725153\pi$
$241$	1.12453e9	−	1.94775e9i	0.333353	−	0.577385i	0.983167	+	0.182709i	$0.0584865\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.30159e7	−	1.37796e8i	0.0202653	−	0.0382448i
$246$		0			0
$247$	5.95328e8	+	1.03114e9i	0.159944	+	0.277032i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.27429e9i			0.321050i	0.987032	+	0.160525i	$0.0513188\pi$
$251$			1.27429e9i			0.321050i	−0.987032	+	0.160525i	$0.948681\pi$
$252$		0			0
$253$	3.57227e9			0.871890
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.93069e8	−	3.42409e8i	0.135948	−	0.0784896i	−0.430483	−	0.902598i	$-0.641657\pi$
$257$	5.93069e8	−	3.42409e8i	0.135948	−	0.0784896i	0.566431	+	0.824109i	$0.308324\pi$
$258$		0			0
$259$	−2.74063e9	−	2.64283e9i	−0.609048	−	0.587313i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.99809e9	−	2.30830e9i	−0.835659	−	0.482468i	0.0201272	−	0.999797i	$-0.493593\pi$
$263$	−3.99809e9	−	2.30830e9i	−0.835659	−	0.482468i	−0.855786	+	0.517329i	$0.826926\pi$
$264$		0			0
$265$	7.05622e7			0.0143083
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.96006e9	+	1.13164e9i	0.374335	+	0.216123i	0.675351	−	0.737497i	$-0.263992\pi$
$269$	1.96006e9	+	1.13164e9i	0.374335	+	0.216123i	−0.301015	+	0.953619i	$0.597326\pi$
$270$		0			0
$271$	2.08280e9	+	3.60752e9i	0.386164	+	0.668855i	0.991930	−	0.126788i	$-0.0404667\pi$
$271$	2.08280e9	+	3.60752e9i	0.386164	+	0.668855i	−0.605766	+	0.795643i	$0.707133\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.54933e9	−	1.47185e9i	0.445753	−	0.257356i
$276$		0			0
$277$	−3.02982e9	+	5.24781e9i	−0.514634	+	0.891372i	0.485222	+	0.874391i	$0.338739\pi$
$277$	−3.02982e9	+	5.24781e9i	−0.514634	+	0.891372i	−0.999856	+	0.0169809i	$0.994595\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.10028e9i			1.13881i	0.822058	+	0.569404i	$0.192826\pi$
$281$			7.10028e9i			1.13881i	−0.822058	+	0.569404i	$0.807174\pi$
$282$		0			0
$283$	−1.76462e9	+	3.05642e9i	−0.275110	+	0.476504i	−0.970163	−	0.242454i	$-0.922048\pi$
$283$	−1.76462e9	+	3.05642e9i	−0.275110	+	0.476504i	0.695053	+	0.718958i	$0.255381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.26942e8	−	1.83274e9i	−0.0776667	−	0.270130i
$288$		0			0
$289$	−1.90151e9	−	3.29351e9i	−0.272588	−	0.472137i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.26537e9i			1.12148i	0.827992	+	0.560740i	$0.189483\pi$
$293$			8.26537e9i			1.12148i	−0.827992	+	0.560740i	$0.810517\pi$
$294$		0			0
$295$	−3.54039e8			−0.0467480
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.39870e9	+	2.53959e9i	−0.550351	+	0.317745i
$300$		0			0
$301$	1.43005e10	−	4.11162e9i	1.74215	−	0.500895i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.79545e7	−	5.65540e7i	−0.0113194	−	0.00653528i
$306$		0			0
$307$	6.33317e9			0.712964			0.356482	−	0.934302i	$-0.383976\pi$
$307$	6.33317e9			0.712964			0.356482	+	0.934302i	$0.383976\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.27635e9	−	5.35571e9i	−0.991598	−	0.572500i	−0.0858466	−	0.996308i	$-0.527360\pi$
$311$	−9.27635e9	−	5.35571e9i	−0.991598	−	0.572500i	−0.905752	+	0.423809i	$0.860693\pi$
$312$		0			0
$313$	5.49028e9	+	9.50945e9i	0.572028	+	0.990782i	0.996358	+	0.0852740i	$0.0271766\pi$
$313$	5.49028e9	+	9.50945e9i	0.572028	+	0.990782i	−0.424329	+	0.905508i	$0.639490\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.09639e10	+	6.33003e9i	−1.08575	+	0.626858i	−0.932442	−	0.361321i	$-0.882326\pi$
$317$	−1.09639e10	+	6.33003e9i	−1.08575	+	0.626858i	−0.153308	+	0.988178i	$0.548993\pi$
$318$		0			0
$319$	−1.45777e9	+	2.52493e9i	−0.140775	+	0.243830i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.24748e9i			0.573977i
$324$		0			0
$325$	−2.09274e9	+	3.62473e9i	−0.187578	+	0.324894i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.27281e10	+	1.31991e10i	−1.08637	+	1.12658i
$330$		0			0
$331$	3.83211e9	+	6.63742e9i	0.319247	+	0.552952i	0.980331	−	0.197360i	$-0.0632368\pi$
$331$	3.83211e9	+	6.63742e9i	0.319247	+	0.552952i	−0.661084	+	0.750312i	$0.729903\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.65054e8i			0.0131053i
$336$		0			0
$337$	2.58097e9			0.200107			0.100054	−	0.994982i	$-0.468099\pi$
$337$	2.58097e9			0.200107			0.100054	+	0.994982i	$0.468099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.00476e10	+	5.80099e9i	−0.743097	+	0.429027i
$342$		0			0
$343$	−9.23963e9	+	1.03058e10i	−0.667541	+	0.744573i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.85388e9	+	3.95709e9i	0.472736	+	0.272934i	0.717384	−	0.696678i	$-0.245339\pi$
$347$	6.85388e9	+	3.95709e9i	0.472736	+	0.272934i	−0.244649	+	0.969612i	$0.578673\pi$
$348$		0			0
$349$	1.58298e10			1.06702			0.533511	−	0.845793i	$-0.320872\pi$
$349$	1.58298e10			1.06702			0.533511	+	0.845793i	$0.320872\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.30037e9	+	4.21487e9i	0.470160	+	0.271447i	0.716307	−	0.697785i	$-0.245831\pi$
$353$	7.30037e9	+	4.21487e9i	0.470160	+	0.271447i	−0.246146	+	0.969233i	$0.579164\pi$
$354$		0			0
$355$	3.46601e8	+	6.00331e8i	0.0218231	+	0.0377988i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.75753e10	+	1.01471e10i	−1.05810	+	0.610892i	−0.924905	−	0.380199i	$-0.875856\pi$
$359$	−1.75753e10	+	1.01471e10i	−1.05810	+	0.610892i	−0.133191	+	0.991090i	$0.542522\pi$
$360$		0			0
$361$	2.34079e9	−	4.05436e9i	0.137827	−	0.238723i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.49620e7i		−	0.00140640i
$366$		0			0
$367$	2.76794e9	−	4.79421e9i	0.152578	−	0.264273i	−0.779596	−	0.626282i	$-0.784576\pi$
$367$	2.76794e9	−	4.79421e9i	0.152578	−	0.264273i	0.932175	+	0.362009i	$0.117909\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.07794e9	−	1.51080e9i	−0.320819	−	0.0797466i
$372$		0			0
$373$	1.37790e10	+	2.38660e10i	0.711841	+	1.23294i	0.964165	+	0.265302i	$0.0854717\pi$
$373$	1.37790e10	+	2.38660e10i	0.711841	+	1.23294i	−0.252324	+	0.967643i	$0.581195\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.14542e9i		−	0.205212i
$378$		0			0
$379$	1.52111e10			0.737233			0.368616	−	0.929582i	$-0.379832\pi$
$379$	1.52111e10			0.737233			0.368616	+	0.929582i	$0.379832\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.72406e9	+	5.61419e9i	−0.451911	+	0.260911i	−0.708637	−	0.705573i	$-0.750689\pi$
$383$	−9.72406e9	+	5.61419e9i	−0.451911	+	0.260911i	0.256726	+	0.966484i	$0.417356\pi$
$384$		0			0
$385$	4.71285e8	−	1.35502e8i	0.0214506	−	0.00616741i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.22888e10	+	7.09493e9i	0.536674	+	0.309849i	0.743730	−	0.668480i	$-0.233055\pi$
$389$	1.22888e10	+	7.09493e9i	0.536674	+	0.309849i	−0.207056	+	0.978329i	$0.566388\pi$
$390$		0			0
$391$	−2.66509e10			−1.14026
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.41277e7	−	2.54771e7i	−0.00181269	−	0.00104656i
$396$		0			0
$397$	−1.44616e10	−	2.50481e10i	−0.582174	−	1.00835i	−0.995221	−	0.0976454i	$-0.968869\pi$
$397$	−1.44616e10	−	2.50481e10i	−0.582174	−	1.00835i	0.413047	−	0.910710i	$-0.364464\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.69181e10	−	9.76765e9i	0.654294	−	0.377757i	−0.135805	−	0.990736i	$-0.543362\pi$
$401$	1.69181e10	−	9.76765e9i	0.654294	−	0.377757i	0.790100	+	0.612979i	$0.210029\pi$
$402$		0			0
$403$	8.24807e9	−	1.42861e10i	0.312703	−	0.541617i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.19722e10i		−	0.436312i
$408$		0			0
$409$	−1.07739e10	+	1.86609e10i	−0.385016	+	0.666867i	−0.991771	−	0.128021i	$-0.959137\pi$
$409$	−1.07739e10	+	1.86609e10i	−0.385016	+	0.666867i	0.606755	+	0.794889i	$0.292471\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.04954e10	+	7.58030e9i	1.04818	+	0.260547i
$414$		0			0
$415$	−1.21309e9	−	2.10113e9i	−0.0408978	−	0.0708370i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			2.08301e10i			0.675826i	0.941177	+	0.337913i	$0.109721\pi$
$419$			2.08301e10i			0.675826i	−0.941177	+	0.337913i	$0.890279\pi$
$420$		0			0
$421$	−1.80612e10			−0.574934			−0.287467	−	0.957791i	$-0.592813\pi$
$421$	−1.80612e10			−0.574934			−0.287467	+	0.957791i	$0.592813\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.90193e10	+	1.09808e10i	−0.582959	+	0.336571i
$426$		0			0
$427$	7.22652e9	+	6.96863e9i	0.217379	+	0.209622i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.37026e10	−	2.52317e10i	−1.26648	−	0.731202i	−0.292159	−	0.956370i	$-0.594374\pi$
$431$	−4.37026e10	−	2.52317e10i	−1.26648	−	0.731202i	−0.974320	+	0.225167i	$0.927707\pi$
$432$		0			0
$433$	4.97105e10			1.41415			0.707076	−	0.707137i	$-0.250014\pi$
$433$	4.97105e10			1.41415			0.707076	+	0.707137i	$0.250014\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.54478e10	−	2.62393e10i	−1.24620	−	0.719493i
$438$		0			0
$439$	1.76507e10	+	3.05719e10i	0.475230	+	0.823123i	0.999598	−	0.0283692i	$-0.00903140\pi$
$439$	1.76507e10	+	3.05719e10i	0.475230	+	0.823123i	−0.524367	+	0.851492i	$0.675698\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.32356e10	−	1.91886e10i	0.862955	−	0.498227i	−0.00204575	−	0.999998i	$-0.500651\pi$
$443$	3.32356e10	−	1.91886e10i	0.862955	−	0.498227i	0.865001	+	0.501771i	$0.167318\pi$
$444$		0			0
$445$	−3.23835e8	+	5.60899e8i	−0.00825817	+	0.0143036i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.18319e10i			0.783209i	0.920134	+	0.391604i	$0.128080\pi$
$449$			3.18319e10i			0.783209i	−0.920134	+	0.391604i	$0.871920\pi$
$450$		0			0
$451$	2.99829e9	−	5.19320e9i	0.0724716	−	0.125525i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.83984e8	+	5.01895e8i	−0.0112924	+	0.0117103i
$456$		0			0
$457$	−3.24032e10	−	5.61240e10i	−0.742887	−	1.28672i	−0.951175	−	0.308651i	$-0.900123\pi$
$457$	−3.24032e10	−	5.61240e10i	−0.742887	−	1.28672i	0.208288	−	0.978068i	$-0.433211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.65098e10i			0.586953i	0.955966	+	0.293476i	$0.0948122\pi$
$461$			2.65098e10i			0.586953i	−0.955966	+	0.293476i	$0.905188\pi$
$462$		0			0
$463$	−2.98576e10			−0.649727			−0.324864	−	0.945761i	$-0.605318\pi$
$463$	−2.98576e10			−0.649727			−0.324864	+	0.945761i	$0.605318\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.64542e10	−	3.83673e10i	1.39719	−	0.806667i	0.403091	−	0.915160i	$-0.367936\pi$
$467$	6.64542e10	−	3.83673e10i	1.39719	−	0.806667i	0.994097	+	0.108493i	$0.0346026\pi$
$468$		0			0
$469$	3.53395e9	−	1.42170e10i	0.0730414	−	0.293844i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.05215e10	+	2.33951e10i	0.809544	+	0.467390i
$474$		0			0
$475$	−4.32447e10			−0.849491
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.17717e9	+	4.14374e9i	0.136336	+	0.0787137i	0.566617	−	0.823981i	$-0.308252\pi$
$479$	7.17717e9	+	4.14374e9i	0.136336	+	0.0787137i	−0.430281	+	0.902695i	$0.641585\pi$
$480$		0			0
$481$	8.51128e9	+	1.47420e10i	0.159006	+	0.275407i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.47334e8	+	1.42798e8i	−0.00447009	+	0.00258081i
$486$		0			0
$487$	−2.74730e10	+	4.75845e10i	−0.488416	+	0.845961i	−0.999911	−	0.0133253i	$-0.995758\pi$
$487$	−2.74730e10	+	4.75845e10i	−0.488416	+	0.845961i	0.511496	+	0.859286i	$0.329092\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.77959e10i		−	1.68265i	−0.540527	−	0.841327i	$-0.681775\pi$
$491$		−	9.77959e10i		−	1.68265i	0.540527	−	0.841327i	$-0.318225\pi$
$492$		0			0
$493$	1.08757e10	−	1.88373e10i	0.184107	−	0.318882i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.70012e10	−	5.91311e10i	−0.278646	−	0.969149i
$498$		0			0
$499$	−3.63645e10	−	6.29852e10i	−0.586511	−	1.01587i	−0.994685	−	0.102962i	$-0.967168\pi$
$499$	−3.63645e10	−	6.29852e10i	−0.586511	−	1.01587i	0.408175	−	0.912904i	$-0.366165\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	6.16837e10i		−	0.963603i	−0.876280	−	0.481802i	$-0.839983\pi$
$503$		−	6.16837e10i		−	0.963603i	0.876280	−	0.481802i	$-0.160017\pi$
$504$		0			0
$505$	3.58819e9			0.0551709
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.53141e10	−	4.92561e10i	1.27101	−	0.733820i	0.295834	−	0.955239i	$-0.404402\pi$
$509$	8.53141e10	−	4.92561e10i	1.27101	−	0.733820i	0.975179	+	0.221420i	$0.0710691\pi$
$510$		0			0
$511$	−5.34460e8	+	2.15012e9i	−0.00783847	+	0.0315341i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.45890e9	−	2.57435e9i	−0.0633868	−	0.0365964i
$516$		0			0
$517$	−5.76592e10			−0.807061
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.02652e11	+	5.92661e10i	1.39321	+	0.804369i	0.993669	−	0.112347i	$-0.0358367\pi$
$521$	1.02652e11	+	5.92661e10i	1.39321	+	0.804369i	0.399540	+	0.916716i	$0.369170\pi$
$522$		0			0
$523$	−5.48450e10	−	9.49944e10i	−0.733045	−	1.26967i	−0.955576	−	0.294745i	$-0.904765\pi$
$523$	−5.48450e10	−	9.49944e10i	−0.733045	−	1.26967i	0.222531	−	0.974926i	$-0.428568\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.49603e10	−	4.32783e10i	0.971826	−	0.561084i
$528$		0			0
$529$	7.27779e10	−	1.26055e11i	0.929344	−	1.60967i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8.52617e9i			0.105644i
$534$		0			0
$535$	3.34074e9	−	5.78634e9i	0.0407782	−	0.0706299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.34957e10	+	1.58094e9i	−0.515337	+	0.0187310i
$540$		0			0
$541$	1.37169e10	+	2.37584e10i	0.160128	+	0.277350i	0.934915	−	0.354873i	$-0.115476\pi$
$541$	1.37169e10	+	2.37584e10i	0.160128	+	0.277350i	−0.774786	+	0.632223i	$0.782143\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	3.42397e8i		−	0.00388100i
$546$		0			0
$547$	−9.19133e10			−1.02667			−0.513333	−	0.858190i	$-0.671589\pi$
$547$	−9.19133e10			−1.02667			−0.513333	+	0.858190i	$0.671589\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.70927e10	−	2.14155e10i	0.402422	−	0.232339i
$552$		0			0
$553$	3.25549e9	+	3.13931e9i	0.0348109	+	0.0335687i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.13798e10	+	2.38907e10i	0.429901	+	0.248203i	0.699304	−	0.714824i	$-0.253493\pi$
$557$	4.13798e10	+	2.38907e10i	0.429901	+	0.248203i	−0.269404	+	0.963027i	$0.586827\pi$
$558$		0			0
$559$	−6.65280e10			−0.681329
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.57197e11	−	9.07576e10i	−1.56462	−	0.903336i	−0.996779	−	0.0802008i	$-0.974444\pi$
$563$	−1.57197e11	−	9.07576e10i	−1.56462	−	0.903336i	−0.567845	−	0.823135i	$-0.692223\pi$
$564$		0			0
$565$	−3.77800e9	−	6.54368e9i	−0.0370738	−	0.0642138i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.35498e10	−	4.82375e10i	0.797069	−	0.460188i	−0.0453760	−	0.998970i	$-0.514449\pi$
$569$	8.35498e10	−	4.82375e10i	0.797069	−	0.460188i	0.842445	+	0.538782i	$0.181115\pi$
$570$		0			0
$571$	−3.71303e10	+	6.43115e10i	−0.349288	+	0.604985i	−0.986123	−	0.166015i	$-0.946910\pi$
$571$	−3.71303e10	+	6.43115e10i	−0.349288	+	0.604985i	0.636835	+	0.771000i	$0.280243\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.84476e11i		−	1.68760i
$576$		0			0
$577$	−4.72547e10	+	8.18476e10i	−0.426326	+	0.738418i	−0.996543	−	0.0830756i	$-0.973526\pi$
$577$	−4.72547e10	+	8.18476e10i	−0.426326	+	0.738418i	0.570217	+	0.821494i	$0.306859\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.95032e10	+	2.06956e11i	0.522199	+	1.81624i
$582$		0			0
$583$	−9.84696e9	−	1.70554e10i	−0.0852369	−	0.147635i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	7.97139e10i		−	0.671401i	−0.941969	−	0.335700i	$-0.891027\pi$
$587$		−	7.97139e10i		−	0.671401i	0.941969	−	0.335700i	$-0.108973\pi$
$588$		0			0
$589$	1.70440e11			1.41615
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.01085e11	+	5.83614e10i	−0.817463	+	0.471962i	−0.849541	−	0.527523i	$-0.823121\pi$
$593$	−1.01085e11	+	5.83614e10i	−0.817463	+	0.471962i	0.0320781	+	0.999485i	$0.489787\pi$
$594$		0			0
$595$	−3.51602e9	+	1.01091e9i	−0.0280533	+	0.00806577i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.28602e10	+	3.05189e10i	0.410603	+	0.237062i	0.691049	−	0.722808i	$-0.257149\pi$
$599$	5.28602e10	+	3.05189e10i	0.410603	+	0.237062i	−0.280446	+	0.959870i	$0.590482\pi$
$600$		0			0
$601$	1.36720e11			1.04793			0.523967	−	0.851739i	$-0.324452\pi$
$601$	1.36720e11			1.04793			0.523967	+	0.851739i	$0.324452\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.68640e9	−	2.12834e9i	−0.0275157	−	0.0158862i
$606$		0			0
$607$	−7.02143e10	−	1.21615e11i	−0.517215	−	0.895842i	−0.999800	−	0.0199934i	$-0.993635\pi$
$607$	−7.02143e10	−	1.21615e11i	−0.517215	−	0.895842i	0.482585	−	0.875849i	$-0.339698\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.09985e10	−	4.09910e10i	0.509430	−	0.294119i
$612$		0			0
$613$	1.09232e10	−	1.89195e10i	0.0773585	−	0.133989i	−0.824751	−	0.565496i	$-0.808685\pi$
$613$	1.09232e10	−	1.89195e10i	0.0773585	−	0.133989i	0.902109	+	0.431507i	$0.142018\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.01700e11i			0.701749i	0.936423	+	0.350874i	$0.114116\pi$
$617$			1.01700e11i			0.701749i	−0.936423	+	0.350874i	$0.885884\pi$
$618$		0			0
$619$	−1.15225e11	+	1.99575e11i	−0.784844	+	1.35939i	0.144248	+	0.989542i	$0.453924\pi$
$619$	−1.15225e11	+	1.99575e11i	−0.784844	+	1.35939i	−0.929092	+	0.369849i	$0.879409\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.99032e10	−	4.13799e10i	0.264884	−	0.274687i
$624$		0			0
$625$	−7.58654e10	−	1.31403e11i	−0.497192	−	0.861161i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			8.93188e10i			0.570611i
$630$		0			0
$631$	−1.70816e11			−1.07748			−0.538741	−	0.842471i	$-0.681100\pi$
$631$	−1.70816e11			−1.07748			−0.538741	+	0.842471i	$0.681100\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.88742e9	+	1.66705e9i	−0.0177589	+	0.0102531i
$636$		0			0
$637$	5.24344e10	−	3.28686e10i	0.318463	−	0.199629i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.20728e10	−	4.73848e10i	−0.486146	−	0.280677i	0.236828	−	0.971552i	$-0.423892\pi$
$641$	−8.20728e10	−	4.73848e10i	−0.486146	−	0.280677i	−0.722974	+	0.690875i	$0.757226\pi$
$642$		0			0
$643$	2.13833e11			1.25093			0.625463	−	0.780254i	$-0.284910\pi$
$643$	2.13833e11			1.25093			0.625463	+	0.780254i	$0.284910\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.74854e11	−	1.00952e11i	−0.997834	−	0.576099i	−0.0902270	−	0.995921i	$-0.528759\pi$
$647$	−1.74854e11	−	1.00952e11i	−0.997834	−	0.576099i	−0.907607	+	0.419822i	$0.862093\pi$
$648$		0			0
$649$	4.94061e10	+	8.55739e10i	0.278485	+	0.482350i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.24715e10	+	3.60680e10i	−0.343581	+	0.198367i	−0.661855	−	0.749632i	$-0.730230\pi$
$653$	−6.24715e10	+	3.60680e10i	−0.343581	+	0.198367i	0.318273	+	0.947999i	$0.396897\pi$
$654$		0			0
$655$	6.56524e9	−	1.13713e10i	0.0356686	−	0.0617798i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6.33845e10i			0.336079i	0.985780	+	0.168039i	$0.0537436\pi$
$659$			6.33845e10i			0.336079i	−0.985780	+	0.168039i	$0.946256\pi$
$660$		0			0
$661$	1.25682e11	−	2.17687e11i	0.658365	−	1.14032i	−0.322674	−	0.946510i	$-0.604582\pi$
$661$	1.25682e11	−	2.17687e11i	0.658365	−	1.14032i	0.981039	−	0.193811i	$-0.0620850\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.99117e9	−	1.73781e9i	−0.0357490	−	0.00888618i
$666$		0			0
$667$	9.13555e10	+	1.58232e11i	0.461563	+	0.799451i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.15685e10i			0.155727i
$672$		0			0
$673$	−1.56945e11			−0.765043			−0.382522	−	0.923947i	$-0.624944\pi$
$673$	−1.56945e11			−0.765043			−0.382522	+	0.923947i	$0.624944\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.18203e11	−	6.82443e10i	0.562694	−	0.324872i	−0.191532	−	0.981486i	$-0.561346\pi$
$677$	1.18203e11	−	6.82443e10i	0.562694	−	0.324872i	0.754226	+	0.656615i	$0.228012\pi$
$678$		0			0
$679$	2.43617e10	−	7.00440e9i	0.114612	−	0.0329528i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.12774e11	+	6.51104e10i	0.518236	+	0.299204i	0.736213	−	0.676750i	$-0.236612\pi$
$683$	1.12774e11	+	6.51104e10i	0.518236	+	0.299204i	−0.217976	+	0.975954i	$0.569946\pi$
$684$		0			0
$685$	2.34614e9			0.0106560
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.42500e10	+	1.40008e10i	0.107606	+	0.0621262i
$690$		0			0
$691$	2.22049e10	+	3.84599e10i	0.0973948	+	0.168693i	0.910606	−	0.413277i	$-0.135616\pi$
$691$	2.22049e10	+	3.84599e10i	0.0973948	+	0.168693i	−0.813211	+	0.581969i	$0.802282\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.36338e8	−	7.87148e7i	0.000584356	−	0.000337378i
$696$		0			0
$697$	−2.23688e10	+	3.87439e10i	−0.0947788	+	0.164162i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			8.01068e10i			0.331740i	0.986148	+	0.165870i	$0.0530431\pi$
$701$			8.01068e10i			0.331740i	−0.986148	+	0.165870i	$0.946957\pi$
$702$		0			0
$703$	−8.79394e10	+	1.52316e11i	−0.360049	+	0.623624i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.09072e11	−	7.68265e10i	−1.23703	−	0.307492i
$708$		0			0
$709$	6.38242e10	+	1.10547e11i	0.252581	+	0.437483i	0.964236	−	0.265047i	$-0.0853873\pi$
$709$	6.38242e10	+	1.10547e11i	0.252581	+	0.437483i	−0.711655	+	0.702529i	$0.752054\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			7.27073e11i			2.81333i
$714$		0			0
$715$	−2.19249e9			−0.00838905
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.78418e11	+	1.60745e11i	−1.04179	+	0.601479i	−0.920340	−	0.391118i	$-0.872088\pi$
$719$	−2.78418e11	+	1.60745e11i	−1.04179	+	0.601479i	−0.121452	+	0.992597i	$0.538755\pi$
$720$		0			0
$721$	3.28952e11	+	3.17213e11i	1.21728	+	1.17384i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.30391e11	+	7.52810e10i	0.471948	+	0.272479i
$726$		0			0
$727$	1.63606e11			0.585682			0.292841	−	0.956161i	$-0.405399\pi$
$727$	1.63606e11			0.585682			0.292841	+	0.956161i	$0.405399\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.02311e11	−	1.74539e11i	−1.05873	−	0.611256i
$732$		0			0
$733$	1.28002e11	+	2.21705e11i	0.443404	+	0.767999i	0.997940	−	0.0641615i	$-0.0204373\pi$
$733$	1.28002e11	+	2.21705e11i	0.443404	+	0.767999i	−0.554535	+	0.832160i	$0.687104\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.98947e10	−	2.30332e10i	0.135221	−	0.0780701i
$738$		0			0
$739$	−1.79996e10	+	3.11762e10i	−0.0603510	+	0.104531i	−0.894622	−	0.446823i	$-0.852555\pi$
$739$	−1.79996e10	+	3.11762e10i	−0.0603510	+	0.104531i	0.834271	+	0.551354i	$0.185889\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.01755e11i		−	0.333888i	−0.985966	−	0.166944i	$-0.946610\pi$
$743$		−	1.01755e11i		−	0.333888i	0.985966	−	0.166944i	$-0.0533899\pi$
$744$		0			0
$745$	1.88836e9	−	3.27074e9i	0.00612999	−	0.0106175i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.11649e11	+	4.26883e11i	−1.30798	+	1.35638i
$750$		0			0
$751$	−2.41349e11	−	4.18029e11i	−0.758727	−	1.31415i	−0.943500	−	0.331373i	$-0.892488\pi$
$751$	−2.41349e11	−	4.18029e11i	−0.758727	−	1.31415i	0.184772	−	0.982781i	$-0.440845\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.06099e10i			0.0326531i
$756$		0			0
$757$	1.09685e11			0.334013			0.167007	−	0.985956i	$-0.446590\pi$
$757$	1.09685e11			0.334013			0.167007	+	0.985956i	$0.446590\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.65424e11	+	9.55079e10i	−0.493243	+	0.284774i	−0.725919	−	0.687780i	$-0.758585\pi$
$761$	−1.65424e11	+	9.55079e10i	−0.493243	+	0.284774i	0.232676	+	0.972554i	$0.425252\pi$
$762$		0			0
$763$	−7.33104e9	+	2.94926e10i	−0.0216305	+	0.0870193i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.21672e11	−	7.02475e10i	−0.351568	−	0.202978i
$768$		0			0
$769$	4.65887e11			1.33222			0.666109	−	0.745855i	$-0.267959\pi$
$769$	4.65887e11			1.33222			0.666109	+	0.745855i	$0.267959\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.86382e11	−	2.23078e11i	−1.08218	−	0.624796i	−0.150696	−	0.988580i	$-0.548151\pi$
$773$	−3.86382e11	−	2.23078e11i	−1.08218	−	0.624796i	−0.931483	+	0.363784i	$0.881485\pi$
$774$		0			0
$775$	2.99570e11	+	5.18871e11i	0.830409	+	1.43831i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.62910e10	+	4.40466e10i	−0.207168	+	0.119609i
$780$		0			0
$781$	9.67364e10	−	1.67552e11i	0.260008	−	0.450346i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	6.87765e9i		−	0.0181118i
$786$		0			0
$787$	−3.20655e11	+	5.55390e11i	−0.835870	+	1.44777i	0.0574510	+	0.998348i	$0.481703\pi$
$787$	−3.20655e11	+	5.55390e11i	−0.835870	+	1.44777i	−0.893321	+	0.449420i	$0.851631\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.85315e11	+	6.44536e11i	0.473373	+	1.64642i
$792$		0			0
$793$	−2.24426e10	−	3.88717e10i	−0.0567519	−	0.0982972i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			2.23443e11i			0.553776i	0.960902	+	0.276888i	$0.0893031\pi$
$797$			2.23443e11i			0.553776i	−0.960902	+	0.276888i	$0.910697\pi$
$798$		0			0
$799$	4.30167e11			1.05548
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.03351e9	+	3.48345e9i	−0.0145113	+	0.00837813i
$804$		0			0
$805$	7.41325e9	−	2.98234e10i	0.0176533	−	0.0710189i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.18621e11	+	1.83956e11i	0.743842	+	0.429457i	0.823465	−	0.567368i	$-0.192038\pi$
$809$	3.18621e11	+	1.83956e11i	0.743842	+	0.429457i	−0.0796224	+	0.996825i	$0.525371\pi$
$810$		0			0
$811$	−5.14769e11			−1.18995			−0.594975	−	0.803744i	$-0.702838\pi$
$811$	−5.14769e11			−1.18995			−0.594975	+	0.803744i	$0.702838\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.22932e10	−	1.28710e10i	−0.0505291	−	0.0291730i
$816$		0			0
$817$	−3.43687e11	−	5.95283e11i	−0.771391	−	1.33609i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.27822e11	−	1.89268e11i	0.721549	−	0.416587i	−0.0937735	−	0.995594i	$-0.529893\pi$
$821$	3.27822e11	−	1.89268e11i	0.721549	−	0.416587i	0.815323	+	0.579007i	$0.196560\pi$
$822$		0			0
$823$	1.44961e11	−	2.51080e11i	0.315975	−	0.547284i	−0.663670	−	0.748026i	$-0.731002\pi$
$823$	1.44961e11	−	2.51080e11i	0.315975	−	0.547284i	0.979644	+	0.200742i	$0.0643352\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	2.65534e11i		−	0.567673i	−0.958873	−	0.283836i	$-0.908393\pi$
$827$		−	2.65534e11i		−	0.567673i	0.958873	−	0.283836i	$-0.0916072\pi$
$828$		0			0
$829$	−2.06382e11	+	3.57463e11i	−0.436971	+	0.756856i	−0.997454	−	0.0713096i	$-0.977282\pi$
$829$	−2.06382e11	+	3.57463e11i	−0.436971	+	0.756856i	0.560483	+	0.828166i	$0.310615\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.24500e11	−	1.17946e10i	0.673961	−	0.0244965i
$834$		0			0
$835$	−1.33102e10	−	2.30540e10i	−0.0273804	−	0.0474242i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	7.82771e10i		−	0.157974i	−0.996876	−	0.0789872i	$-0.974831\pi$
$839$		−	7.82771e10i		−	0.157974i	0.996876	−	0.0789872i	$-0.0251686\pi$
$840$		0			0
$841$	3.51125e11			0.701904
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.64105e10	+	9.47463e9i	−0.0321882	+	0.0185838i
$846$		0			0
$847$	2.71962e11	+	2.62256e11i	0.528414	+	0.509556i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.49757e11	−	3.75138e11i	−1.23889	−	0.715274i
$852$		0			0
$853$	2.71336e11			0.512521			0.256261	−	0.966608i	$-0.417509\pi$
$853$	2.71336e11			0.512521			0.256261	+	0.966608i	$0.417509\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.68531e11	−	2.12772e11i	−0.683205	−	0.394449i	0.117857	−	0.993031i	$-0.462398\pi$
$857$	−3.68531e11	−	2.12772e11i	−0.683205	−	0.394449i	−0.801061	+	0.598582i	$0.795731\pi$
$858$		0			0
$859$	−2.66317e11	−	4.61275e11i	−0.489133	−	0.847203i	0.510789	−	0.859706i	$-0.329353\pi$
$859$	−2.66317e11	−	4.61275e11i	−0.489133	−	0.847203i	−0.999922	+	0.0125034i	$0.996020\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.35013e11	−	1.93420e11i	0.603975	−	0.348705i	−0.166629	−	0.986020i	$-0.553288\pi$
$863$	3.35013e11	−	1.93420e11i	0.603975	−	0.348705i	0.770604	+	0.637315i	$0.219955\pi$
$864$		0			0
$865$	1.40039e10	−	2.42554e10i	0.0250140	−	0.0433255i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.42213e10i			0.0249380i
$870$		0			0
$871$	−3.27495e10	+	5.67238e10i	−0.0569026	+	0.0985582i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.40081e10	−	4.87211e10i	−0.0238972	−	0.0831161i
$876$		0			0
$877$	4.55166e10	+	7.88371e10i	0.0769435	+	0.133270i	0.901930	−	0.431883i	$-0.142151\pi$
$877$	4.55166e10	+	7.88371e10i	0.0769435	+	0.133270i	−0.824986	+	0.565153i	$0.808817\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			6.62395e11i			1.09955i	0.835314	+	0.549773i	$0.185286\pi$
$881$			6.62395e11i			1.09955i	−0.835314	+	0.549773i	$0.814714\pi$
$882$		0			0
$883$	−9.27751e11			−1.52612			−0.763060	−	0.646327i	$-0.776304\pi$
$883$	−9.27751e11			−1.52612			−0.763060	+	0.646327i	$0.776304\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.74649e11	−	4.47244e11i	1.25144	−	0.722520i	0.280046	−	0.959987i	$-0.409650\pi$
$887$	7.74649e11	−	4.47244e11i	1.25144	−	0.722520i	0.971396	+	0.237467i	$0.0763171\pi$
$888$		0			0
$889$	2.84404e11	−	8.17707e10i	0.455332	−	0.130915i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.33564e11	+	4.23523e11i	1.15354	+	0.665996i
$894$		0			0
$895$	1.55293e10			0.0242024
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.13906e11	−	2.96704e11i	−0.786765	−	0.454239i
$900$		0			0
$901$	7.34633e10	+	1.27242e11i	0.111473	+	0.193078i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.82347e10	+	2.20748e10i	−0.0569985	+	0.0329081i
$906$		0			0
$907$	1.21717e11	−	2.10820e11i	0.179855	−	0.311518i	−0.761976	−	0.647606i	$-0.775771\pi$
$907$	1.21717e11	−	2.10820e11i	0.179855	−	0.311518i	0.941831	+	0.336088i	$0.109104\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	5.03684e11i		−	0.731281i	−0.930756	−	0.365641i	$-0.880850\pi$
$911$		−	5.03684e11i		−	0.731281i	0.930756	−	0.365641i	$-0.119150\pi$
$912$		0			0
$913$	−3.38573e11	+	5.86425e11i	−0.487269	+	0.843975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.08974e11	+	8.38912e11i	−1.14408	+	1.18642i
$918$		0			0
$919$	−3.56741e11	−	6.17893e11i	−0.500139	−	0.866266i	−1.00000		0.000160119i	$-0.999949\pi$
$919$	−3.56741e11	−	6.17893e11i	−0.500139	−	0.866266i	0.499861	−	0.866105i	$-0.333384\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.75087e11i			0.379021i
$924$		0			0
$925$	−6.18261e11			−0.844510
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.03821e11	−	1.75411e11i	0.407901	−	0.235502i	−0.281987	−	0.959418i	$-0.590993\pi$
$929$	3.03821e11	−	1.75411e11i	0.407901	−	0.235502i	0.689887	+	0.723917i	$0.257660\pi$
$930$		0			0
$931$	5.64983e11	+	2.99375e11i	0.752032	+	0.398490i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.96291e9	−	5.75209e9i	−0.0130359	−	0.00752626i
$936$		0			0
$937$	1.13829e12			1.47671			0.738354	−	0.674413i	$-0.235603\pi$
$937$	1.13829e12			1.47671			0.738354	+	0.674413i	$0.235603\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.37867e11	−	3.10538e11i	−0.685988	−	0.396055i	0.116119	−	0.993235i	$-0.462954\pi$
$941$	−5.37867e11	−	3.10538e11i	−0.685988	−	0.396055i	−0.802107	+	0.597180i	$0.796288\pi$
$942$		0			0
$943$	−1.87897e11	−	3.25447e11i	−0.237615	−	0.411561i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.54959e11	−	5.51346e11i	1.18737	−	0.685526i	0.229659	−	0.973271i	$-0.426239\pi$
$947$	9.54959e11	−	5.51346e11i	1.18737	−	0.685526i	0.957707	+	0.287745i	$0.0929056\pi$
$948$		0			0
$949$	4.95290e9	−	8.57867e9i	0.00610653	−	0.0105768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.30554e12i		−	1.58277i	−0.611320	−	0.791384i	$-0.709361\pi$
$953$		−	1.30554e12i		−	1.58277i	0.611320	−	0.791384i	$-0.290639\pi$
$954$		0			0
$955$	2.45477e10	−	4.25179e10i	0.0295120	−	0.0511162i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.02087e11	−	5.02331e10i	−0.238927	−	0.0593904i
$960$		0			0
$961$	−7.54246e11	−	1.30639e12i	−0.884341	−	1.53172i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	5.67339e10i		−	0.0654234i
$966$		0			0
$967$	−1.41138e12			−1.61413			−0.807066	−	0.590461i	$-0.798946\pi$
$967$	−1.41138e12			−1.61413			−0.807066	+	0.590461i	$0.798946\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.73135e11	−	5.61840e11i	1.09470	−	0.632027i	0.159878	−	0.987137i	$-0.448890\pi$
$971$	9.73135e11	−	5.61840e11i	1.09470	−	0.632027i	0.934825	+	0.355110i	$0.115556\pi$
$972$		0			0
$973$	−1.34290e10	+	3.86104e9i	−0.0149827	+	0.00430778i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.06817e12	+	6.16708e11i	1.17236	+	0.676864i	0.954235	−	0.299056i	$-0.0966719\pi$
$977$	1.06817e12	+	6.16708e11i	1.17236	+	0.676864i	0.218127	+	0.975920i	$0.430005\pi$
$978$		0			0
$979$	1.80765e11			0.196781
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.24581e12	−	7.19268e11i	−1.33425	−	0.770330i	−0.348302	−	0.937382i	$-0.613242\pi$
$983$	−1.24581e12	−	7.19268e11i	−1.33425	−	0.770330i	−0.985948	+	0.167052i	$0.946575\pi$
$984$		0			0
$985$	−2.61261e10	−	4.52518e10i	−0.0277543	−	0.0480719i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.53940e12	−	1.46612e12i	2.65427	−	1.53245i
$990$		0			0
$991$	−3.42784e11	+	5.93719e11i	−0.355407	+	0.615582i	−0.987187	−	0.159565i	$-0.948991\pi$
$991$	−3.42784e11	+	5.93719e11i	−0.355407	+	0.615582i	0.631781	+	0.775147i	$0.282324\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1.13920e10i			0.0116227i
$996$		0			0
$997$	5.31832e11	−	9.21160e11i	0.538262	−	0.932298i	−0.460735	−	0.887538i	$-0.652414\pi$
$997$	5.31832e11	−	9.21160e11i	0.538262	−	0.932298i	0.998998	−	0.0447602i	$-0.0142524\pi$
$998$		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.53.12	yes	44
3.2	odd	2	inner	252.9.bk.a.53.11	✓	44
7.2	even	3	inner	252.9.bk.a.233.11	yes	44
21.2	odd	6	inner	252.9.bk.a.233.12	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.11	✓	44	3.2	odd	2	inner
252.9.bk.a.53.12	yes	44	1.1	even	1	trivial
252.9.bk.a.233.11	yes	44	7.2	even	3	inner
252.9.bk.a.233.12	yes	44	21.2	odd	6	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)

\(f(q)\)	\(=\)	\(q+(23.4272 - 13.5257i) q^{5} +(-2307.52 + 663.449i) q^{7} +O(q^{10})\)	\(q+(23.4272 - 13.5257i) q^{5} +(-2307.52 + 663.449i) q^{7} +(-6538.52 - 3775.02i) q^{11} +10734.9 q^{13} +(48780.7 + 28163.5i) q^{17} +(55457.2 + 96054.6i) q^{19} +(-409756. + 236573. i) q^{23} +(-194947. + 337657. i) q^{25} -386162. i q^{29} +(768340. - 1.33080e6i) q^{31} +(-45085.0 + 46753.5i) q^{35} +(792859. + 1.37327e6i) q^{37} +794246. i q^{41} -6.19734e6 q^{43} +(6.61379e6 - 3.81847e6i) q^{47} +(4.88447e6 - 3.06184e6i) q^{49} +(2.25899e6 + 1.30423e6i) q^{53} -204239. q^{55} +(-1.13342e7 - 6.54383e6i) q^{59} +(-2.09062e6 - 3.62105e6i) q^{61} +(251489. - 145197. i) q^{65} +(-3.05074e6 + 5.28404e6i) q^{67} +2.56254e7i q^{71} +(461382. - 799136. i) q^{73} +(1.75923e7 + 4.37294e6i) q^{77} +(-941806. - 1.63126e6i) q^{79} -8.96877e7i q^{83} +1.52372e6 q^{85} +(-2.07346e7 + 1.19711e7i) q^{89} +(-2.47710e7 + 7.12207e6i) q^{91} +(2.59841e6 + 1.50019e6i) q^{95} -1.05576e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 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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