LMFDB - Embedded newform 252.8.k.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(37,252)]

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, 0, 2]))

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

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, 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:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 $ x^8 - 2x^7 + 659x^6 + 12718x^5 + 417701x^4 + 3735784x^3 + 32480596x^2 + 479136*x + 7056
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{5}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-0.00737575 + 0.0127752i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.8.k.b.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-80.0854 - 138.712i) q^{5} +(-254.682 - 871.022i) q^{7} +(1581.93 - 2739.98i) q^{11} -4771.53 q^{13} +(-6504.73 + 11266.5i) q^{17} +(-22058.1 - 38205.8i) q^{19} +(-32533.0 - 56348.8i) q^{23} +(26235.2 - 45440.6i) q^{25} +246115. q^{29} +(-150328. + 260376. i) q^{31} +(-100425. + 105084. i) q^{35} +(-258479. - 447698. i) q^{37} +377844. q^{41} -71420.3 q^{43} +(558378. + 967140. i) q^{47} +(-693817. + 443667. i) q^{49} +(-184365. + 319330. i) q^{53} -506758. q^{55} +(515977. - 893698. i) q^{59} +(161341. + 279450. i) q^{61} +(382130. + 661869. i) q^{65} +(11763.8 - 20375.5i) q^{67} -2.84199e6 q^{71} +(-336523. + 582875. i) q^{73} +(-2.78947e6 - 680072. i) q^{77} +(429448. + 743826. i) q^{79} -7.32276e6 q^{83} +2.08373e6 q^{85} +(2.31400e6 + 4.00797e6i) q^{89} +(1.21522e6 + 4.15611e6i) q^{91} +(-3.53307e6 + 6.11946e6i) q^{95} +951821. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 196 q^{5} - 434 q^{7} - 406 q^{11} + 3948 q^{13} + 7436 q^{17} + 15874 q^{19} + 6788 q^{23} + 69898 q^{25} + 189088 q^{29} - 55890 q^{31} + 750596 q^{35} + 93742 q^{37} - 13944 q^{41} - 487844 q^{43}+ \cdots + 23722580 q^{97}+O(q^{100}) $ 8 * q + 196 * q^5 - 434 * q^7 - 406 * q^11 + 3948 * q^13 + 7436 * q^17 + 15874 * q^19 + 6788 * q^23 + 69898 * q^25 + 189088 * q^29 - 55890 * q^31 + 750596 * q^35 + 93742 * q^37 - 13944 * q^41 - 487844 * q^43 + 650484 * q^47 - 1834756 * q^49 - 368880 * q^53 - 2391536 * q^55 - 96026 * q^59 - 3618156 * q^61 - 974160 * q^65 - 316006 * q^67 - 12436312 * q^71 - 1287286 * q^73 + 6614440 * q^77 + 8187282 * q^79 - 7387300 * q^83 + 21933928 * q^85 + 14489928 * q^89 - 10505670 * q^91 - 18406792 * q^95 + 23722580 * q^97

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
$5$	−80.0854	−	138.712i	−0.286522	−	0.496271i	0.686455	−	0.727172i	$-0.259166\pi$
$5$	−80.0854	−	138.712i	−0.286522	−	0.496271i	−0.972977	+	0.230901i	$0.925833\pi$
$6$		0			0
$7$	−254.682	−	871.022i	−0.280644	−	0.959812i
$7$	−254.682	−	871.022i	−0.280644	−	0.959812i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1581.93	−	2739.98i	0.358354	−	0.620688i	−0.629332	−	0.777137i	$-0.716671\pi$
$11$	1581.93	−	2739.98i	0.358354	−	0.620688i	0.987686	+	0.156449i	$0.0500046\pi$
$12$		0			0
$13$	−4771.53			−0.602360			−0.301180	−	0.953567i	$-0.597380\pi$
$13$	−4771.53			−0.602360			−0.301180	+	0.953567i	$0.597380\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6504.73	+	11266.5i	−0.321113	+	0.556184i	−0.980718	−	0.195429i	$-0.937390\pi$
$17$	−6504.73	+	11266.5i	−0.321113	+	0.556184i	0.659605	+	0.751612i	$0.270724\pi$
$18$		0			0
$19$	−22058.1	−	38205.8i	−0.737787	−	1.27789i	−0.953490	−	0.301426i	$-0.902537\pi$
$19$	−22058.1	−	38205.8i	−0.737787	−	1.27789i	0.215702	−	0.976459i	$-0.430796\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32533.0	−	56348.8i	−0.557541	−	0.965690i	−0.997701	−	0.0677701i	$-0.978412\pi$
$23$	−32533.0	−	56348.8i	−0.557541	−	0.965690i	0.440160	−	0.897919i	$-0.354922\pi$
$24$		0			0
$25$	26235.2	−	45440.6i	0.335810	−	0.581640i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	246115.			1.87390			0.936949	−	0.349466i	$-0.113637\pi$
$29$	246115.			1.87390			0.936949	+	0.349466i	$0.113637\pi$
$30$		0			0
$31$	−150328.	+	260376.i	−0.906306	+	1.56977i	−0.0871514	+	0.996195i	$0.527776\pi$
$31$	−150328.	+	260376.i	−0.906306	+	1.56977i	−0.819155	+	0.573573i	$0.805557\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−100425.	+	105084.i	−0.395916	+	0.414283i
$36$		0			0
$37$	−258479.	−	447698.i	−0.838916	−	1.45305i	−0.890801	−	0.454393i	$-0.849856\pi$
$37$	−258479.	−	447698.i	−0.838916	−	1.45305i	0.0518851	−	0.998653i	$-0.483477\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	377844.			0.856188			0.428094	−	0.903734i	$-0.359185\pi$
$41$	377844.			0.856188			0.428094	+	0.903734i	$0.359185\pi$
$42$		0			0
$43$	−71420.3			−0.136988			−0.0684940	−	0.997652i	$-0.521819\pi$
$43$	−71420.3			−0.136988			−0.0684940	+	0.997652i	$0.521819\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	558378.	+	967140.i	0.784488	+	1.35877i	0.929305	+	0.369314i	$0.120407\pi$
$47$	558378.	+	967140.i	0.784488	+	1.35877i	−0.144817	+	0.989458i	$0.546259\pi$
$48$		0			0
$49$	−693817.	+	443667.i	−0.842478	+	0.538730i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−184365.	+	319330.i	−0.170104	+	0.294628i	−0.938456	−	0.345399i	$-0.887744\pi$
$53$	−184365.	+	319330.i	−0.170104	+	0.294628i	0.768352	+	0.640027i	$0.221077\pi$
$54$		0			0
$55$	−506758.			−0.410706
$56$		0			0
$57$		0			0
$58$		0			0
$59$	515977.	−	893698.i	0.327076	−	0.566512i	−0.654855	−	0.755755i	$-0.727270\pi$
$59$	515977.	−	893698.i	0.327076	−	0.566512i	0.981930	+	0.189243i	$0.0606035\pi$
$60$		0			0
$61$	161341.	+	279450.i	0.0910101	+	0.157634i	0.907936	−	0.419108i	$-0.137657\pi$
$61$	161341.	+	279450.i	0.0910101	+	0.157634i	−0.816926	+	0.576742i	$0.804324\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	382130.	+	661869.i	0.172590	+	0.298934i
$66$		0			0
$67$	11763.8	−	20375.5i	0.00477844	−	0.00827650i	−0.863626	−	0.504133i	$-0.831812\pi$
$67$	11763.8	−	20375.5i	0.00477844	−	0.00827650i	0.868405	+	0.495856i	$0.165146\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.84199e6			−0.942363			−0.471181	−	0.882036i	$-0.656172\pi$
$71$	−2.84199e6			−0.942363			−0.471181	+	0.882036i	$0.656172\pi$
$72$		0			0
$73$	−336523.	+	582875.i	−0.101248	+	0.175366i	−0.912199	−	0.409748i	$-0.865617\pi$
$73$	−336523.	+	582875.i	−0.101248	+	0.175366i	0.810951	+	0.585114i	$0.198950\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.78947e6	−	680072.i	−0.696314	−	0.169761i
$78$		0			0
$79$	429448.	+	743826.i	0.0979977	+	0.169737i	0.910856	−	0.412725i	$-0.135423\pi$
$79$	429448.	+	743826.i	0.0979977	+	0.169737i	−0.812858	+	0.582462i	$0.802090\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.32276e6			−1.40573			−0.702864	−	0.711324i	$-0.748096\pi$
$83$	−7.32276e6			−1.40573			−0.702864	+	0.711324i	$0.748096\pi$
$84$		0			0
$85$	2.08373e6			0.368024
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.31400e6	+	4.00797e6i	0.347935	+	0.602641i	0.985883	−	0.167439i	$-0.0535496\pi$
$89$	2.31400e6	+	4.00797e6i	0.347935	+	0.602641i	−0.637947	+	0.770080i	$0.720216\pi$
$90$		0			0
$91$	1.21522e6	+	4.15611e6i	0.169048	+	0.578152i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.53307e6	+	6.11946e6i	−0.422785	+	0.732285i
$96$		0			0
$97$	951821.			0.105890			0.0529449	−	0.998597i	$-0.483139\pi$
$97$	951821.			0.105890			0.0529449	+	0.998597i	$0.483139\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.54896e6	+	1.48072e7i	−0.825635	+	1.43004i	0.0757975	+	0.997123i	$0.475850\pi$
$101$	−8.54896e6	+	1.48072e7i	−0.825635	+	1.43004i	−0.901433	+	0.432919i	$0.857484\pi$
$102$		0			0
$103$	2.02111e6	+	3.50067e6i	0.182247	+	0.315661i	0.942645	−	0.333796i	$-0.108330\pi$
$103$	2.02111e6	+	3.50067e6i	0.182247	+	0.315661i	−0.760398	+	0.649457i	$0.774996\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.96974e6	−	6.87579e6i	−0.313270	−	0.542600i	0.665798	−	0.746132i	$-0.268091\pi$
$107$	−3.96974e6	−	6.87579e6i	−0.313270	−	0.542600i	−0.979068	+	0.203532i	$0.934758\pi$
$108$		0			0
$109$	3.49719e6	−	6.05732e6i	0.258659	−	0.448010i	−0.707224	−	0.706989i	$-0.750053\pi$
$109$	3.49719e6	−	6.05732e6i	0.258659	−	0.448010i	0.965883	+	0.258979i	$0.0833862\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.09031e7			−1.36281			−0.681407	−	0.731904i	$-0.738632\pi$
$113$	−2.09031e7			−1.36281			−0.681407	+	0.731904i	$0.738632\pi$
$114$		0			0
$115$	−5.21084e6	+	9.02544e6i	−0.319496	+	0.553383i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.14700e7	+	2.79638e6i	0.623950	+	0.152119i
$120$		0			0
$121$	4.73858e6	+	8.20747e6i	0.243164	+	0.421173i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.09176e7			−0.957913
$126$		0			0
$127$	−1.32156e7			−0.572496			−0.286248	−	0.958155i	$-0.592408\pi$
$127$	−1.32156e7			−0.572496			−0.286248	+	0.958155i	$0.592408\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.97459e6	+	1.38124e7i	0.309926	+	0.536808i	0.978346	−	0.206976i	$-0.0663622\pi$
$131$	7.97459e6	+	1.38124e7i	0.309926	+	0.536808i	−0.668420	+	0.743784i	$0.733029\pi$
$132$		0			0
$133$	−2.76603e7	+	2.89435e7i	−1.01947	+	1.06677i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.78389e7	+	3.08978e7i	−0.592714	+	1.02661i	0.401151	+	0.916012i	$0.368610\pi$
$137$	−1.78389e7	+	3.08978e7i	−0.592714	+	1.02661i	−0.993865	+	0.110599i	$0.964723\pi$
$138$		0			0
$139$	−1.12444e7			−0.355127			−0.177563	−	0.984109i	$-0.556821\pi$
$139$	−1.12444e7			−0.355127			−0.177563	+	0.984109i	$0.556821\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.54823e6	+	1.30739e7i	−0.215858	+	0.373878i
$144$		0			0
$145$	−1.97103e7	−	3.41392e7i	−0.536913	−	0.929961i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.72724e6	+	1.68481e7i	0.240900	+	0.417252i	0.960971	−	0.276649i	$-0.0892240\pi$
$149$	9.72724e6	+	1.68481e7i	0.240900	+	0.417252i	−0.720071	+	0.693901i	$0.755891\pi$
$150$		0			0
$151$	−4.01078e7	+	6.94687e7i	−0.948002	+	1.64199i	−0.198375	+	0.980126i	$0.563566\pi$
$151$	−4.01078e7	+	6.94687e7i	−0.948002	+	1.64199i	−0.749627	+	0.661861i	$0.769767\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.81564e7			1.03871
$156$		0			0
$157$	2.44031e7	−	4.22674e7i	0.503264	−	0.871679i	−0.496729	−	0.867906i	$-0.665466\pi$
$157$	2.44031e7	−	4.22674e7i	0.503264	−	0.871679i	0.999993	−	0.00377328i	$-0.00120108\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.07955e7	+	4.26880e7i	−0.770410	+	0.806149i
$162$		0			0
$163$	−6.59019e6	−	1.14145e7i	−0.119190	−	0.206444i	0.800257	−	0.599658i	$-0.204697\pi$
$163$	−6.59019e6	−	1.14145e7i	−0.119190	−	0.206444i	−0.919447	+	0.393214i	$0.871363\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.54304e7			1.08711			0.543553	−	0.839375i	$-0.317079\pi$
$167$	6.54304e7			1.08711			0.543553	+	0.839375i	$0.317079\pi$
$168$		0			0
$169$	−3.99810e7			−0.637163
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.07134e6	+	1.39800e7i	0.118518	+	0.205279i	0.919181	−	0.393836i	$-0.128852\pi$
$173$	8.07134e6	+	1.39800e7i	0.118518	+	0.205279i	−0.800663	+	0.599116i	$0.795519\pi$
$174$		0			0
$175$	−4.62614e7	−	1.12785e7i	−0.652508	−	0.159081i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.82833e7	+	8.36291e7i	−0.629233	+	1.08986i	0.358473	+	0.933540i	$0.383297\pi$
$179$	−4.82833e7	+	8.36291e7i	−0.629233	+	1.08986i	−0.987706	+	0.156323i	$0.950036\pi$
$180$		0			0
$181$	7.18706e7			0.900899			0.450450	−	0.892802i	$-0.351264\pi$
$181$	7.18706e7			0.900899			0.450450	+	0.892802i	$0.351264\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.14007e7	+	7.17082e7i	−0.480736	+	0.832660i
$186$		0			0
$187$	2.05800e7	+	3.56457e7i	0.230144	+	0.398622i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.37396e7	−	1.45041e8i	−0.869589	−	1.50617i	−0.862417	−	0.506198i	$-0.831050\pi$
$191$	−8.37396e7	−	1.45041e8i	−0.869589	−	1.50617i	−0.00717183	−	0.999974i	$-0.502283\pi$
$192$		0			0
$193$	−8.45041e6	+	1.46365e7i	−0.0846111	+	0.146551i	−0.905225	−	0.424932i	$-0.860298\pi$
$193$	−8.45041e6	+	1.46365e7i	−0.0846111	+	0.146551i	0.820614	+	0.571482i	$0.193631\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.41597e7			0.504713			0.252357	−	0.967634i	$-0.418794\pi$
$197$	5.41597e7			0.504713			0.252357	+	0.967634i	$0.418794\pi$
$198$		0			0
$199$	−7.69956e6	+	1.33360e7i	−0.0692596	+	0.119961i	−0.898576	−	0.438819i	$-0.855397\pi$
$199$	−7.69956e6	+	1.33360e7i	−0.0692596	+	0.119961i	0.829316	+	0.558780i	$0.188730\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.26812e7	−	2.14372e8i	−0.525897	−	1.79859i
$204$		0			0
$205$	−3.02598e7	−	5.24115e7i	−0.245317	−	0.424901i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.39578e8			−1.05756
$210$		0			0
$211$	−1.35532e8			−0.993238			−0.496619	−	0.867969i	$-0.665425\pi$
$211$	−1.35532e8			−0.993238			−0.496619	+	0.867969i	$0.665425\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.71973e6	+	9.90686e6i	0.0392501	+	0.0679832i
$216$		0			0
$217$	2.65079e8	+	6.46262e7i	1.76103	+	0.429338i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.10375e7	−	5.37585e7i	0.193426	−	0.335023i
$222$		0			0
$223$	2.14853e8			1.29740			0.648701	−	0.761044i	$-0.275313\pi$
$223$	2.14853e8			1.29740			0.648701	+	0.761044i	$0.275313\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.08630e7	+	8.80973e7i	−0.288610	+	0.499887i	−0.973478	−	0.228780i	$-0.926526\pi$
$227$	−5.08630e7	+	8.80973e7i	−0.288610	+	0.499887i	0.684868	+	0.728667i	$0.259860\pi$
$228$		0			0
$229$	6.57822e7	+	1.13938e8i	0.361980	+	0.626967i	0.988287	−	0.152610i	$-0.0487677\pi$
$229$	6.57822e7	+	1.13938e8i	0.361980	+	0.626967i	−0.626307	+	0.779576i	$0.715434\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.75854e7	+	4.77793e7i	0.142867	+	0.247454i	0.928575	−	0.371144i	$-0.121034\pi$
$233$	2.75854e7	+	4.77793e7i	0.142867	+	0.247454i	−0.785708	+	0.618598i	$0.787701\pi$
$234$		0			0
$235$	8.94359e7	−	1.54908e8i	0.449546	−	0.778637i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.89001e8			0.895513			0.447757	−	0.894155i	$-0.352223\pi$
$239$	1.89001e8			0.895513			0.447757	+	0.894155i	$0.352223\pi$
$240$		0			0
$241$	1.49348e8	−	2.58678e8i	0.687290	−	1.19042i	−0.285422	−	0.958402i	$-0.592134\pi$
$241$	1.49348e8	−	2.58678e8i	0.687290	−	1.19042i	0.972711	−	0.232019i	$-0.0745330\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.17107e8	+	6.07095e7i	0.508745	+	0.263739i
$246$		0			0
$247$	1.05251e8	+	1.82300e8i	0.444414	+	0.769747i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.63594e7			−0.105215			−0.0526074	−	0.998615i	$-0.516753\pi$
$251$	−2.63594e7			−0.105215			−0.0526074	+	0.998615i	$0.516753\pi$
$252$		0			0
$253$	−2.05860e8			−0.799189
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.05761e7	−	7.02799e7i	−0.149109	−	0.258265i	0.781789	−	0.623543i	$-0.214307\pi$
$257$	−4.05761e7	−	7.02799e7i	−0.149109	−	0.258265i	−0.930899	+	0.365278i	$0.880974\pi$
$258$		0			0
$259$	−3.24125e8	+	3.39161e8i	−1.15921	+	1.21299i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.71901e8	−	4.70947e8i	0.921651	−	1.59635i	0.124790	−	0.992183i	$-0.460174\pi$
$263$	2.71901e8	−	4.70947e8i	0.921651	−	1.59635i	0.796861	−	0.604163i	$-0.206492\pi$
$264$		0			0
$265$	5.90599e7			0.194954
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.06642e7	+	1.84709e7i	−0.0334037	+	0.0578569i	−0.882244	−	0.470793i	$-0.843968\pi$
$269$	−1.06642e7	+	1.84709e7i	−0.0334037	+	0.0578569i	0.848840	+	0.528649i	$0.177301\pi$
$270$		0			0
$271$	−2.35124e8	−	4.07246e8i	−0.717636	−	1.24298i	−0.961934	−	0.273281i	$-0.911891\pi$
$271$	−2.35124e8	−	4.07246e8i	−0.717636	−	1.24298i	0.244298	−	0.969700i	$-0.421442\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.30043e7	−	1.43768e8i	−0.240678	−	0.416866i
$276$		0			0
$277$	4.46999e7	−	7.74225e7i	0.126365	−	0.218871i	−0.795901	−	0.605427i	$-0.793002\pi$
$277$	4.46999e7	−	7.74225e7i	0.126365	−	0.218871i	0.922266	+	0.386557i	$0.126336\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.84556e8			1.84051			0.920253	−	0.391324i	$-0.127983\pi$
$281$	6.84556e8			1.84051			0.920253	+	0.391324i	$0.127983\pi$
$282$		0			0
$283$	3.43984e8	−	5.95798e8i	0.902165	−	1.56260i	0.0774878	−	0.996993i	$-0.475310\pi$
$283$	3.43984e8	−	5.95798e8i	0.902165	−	1.56260i	0.824678	−	0.565603i	$-0.191357\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.62300e7	−	3.29111e8i	−0.240284	−	0.821779i
$288$		0			0
$289$	1.20546e8	+	2.08793e8i	0.293773	+	0.508830i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.46969e8			1.27036			0.635179	−	0.772365i	$-0.280926\pi$
$293$	5.46969e8			1.27036			0.635179	+	0.772365i	$0.280926\pi$
$294$		0			0
$295$	−1.65289e8			−0.374858
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.55232e8	+	2.68870e8i	0.335840	+	0.581693i
$300$		0			0
$301$	1.81895e7	+	6.22087e7i	0.0384448	+	0.131483i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.58421e7	−	4.47598e7i	0.0521529	−	0.0903314i
$306$		0			0
$307$	−5.33032e8			−1.05140			−0.525702	−	0.850669i	$-0.676197\pi$
$307$	−5.33032e8			−1.05140			−0.525702	+	0.850669i	$0.676197\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.53305e7	−	1.30476e8i	0.142007	−	0.245963i	−0.786245	−	0.617914i	$-0.787978\pi$
$311$	7.53305e7	−	1.30476e8i	0.142007	−	0.245963i	0.928252	+	0.371951i	$0.121311\pi$
$312$		0			0
$313$	−2.13226e8	−	3.69318e8i	−0.393038	−	0.680762i	0.599810	−	0.800142i	$-0.295243\pi$
$313$	−2.13226e8	−	3.69318e8i	−0.393038	−	0.680762i	−0.992849	+	0.119380i	$0.961909\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.17845e8	−	8.96934e8i	−0.913045	−	1.58144i	−0.809738	−	0.586791i	$-0.800391\pi$
$317$	−5.17845e8	−	8.96934e8i	−0.913045	−	1.58144i	−0.103307	−	0.994650i	$-0.532942\pi$
$318$		0			0
$319$	3.89337e8	−	6.74352e8i	0.671519	−	1.16311i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.73929e8			0.947652
$324$		0			0
$325$	−1.25182e8	+	2.16821e8i	−0.202278	+	0.350357i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.00192e8	−	7.32673e8i	1.08400	−	1.13429i
$330$		0			0
$331$	−7.15548e7	−	1.23937e8i	−0.108453	−	0.187846i	0.806691	−	0.590974i	$-0.201256\pi$
$331$	−7.15548e7	−	1.23937e8i	−0.108453	−	0.187846i	−0.915144	+	0.403128i	$0.867923\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.76844e6			−0.00547652
$336$		0			0
$337$	−8.02807e8			−1.14263			−0.571316	−	0.820730i	$-0.693567\pi$
$337$	−8.02807e8			−1.14263			−0.571316	+	0.820730i	$0.693567\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.75618e8	+	8.23794e8i	0.649557	+	1.12507i
$342$		0			0
$343$	5.63147e8	+	4.91336e8i	0.753516	+	0.657430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.45919e8	+	9.45559e8i	−0.701415	+	1.21489i	0.266555	+	0.963820i	$0.414115\pi$
$347$	−5.45919e8	+	9.45559e8i	−0.701415	+	1.21489i	−0.967970	+	0.251066i	$0.919219\pi$
$348$		0			0
$349$	1.03523e9			1.30361			0.651805	−	0.758386i	$-0.274012\pi$
$349$	1.03523e9			1.30361			0.651805	+	0.758386i	$0.274012\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.35935e8	+	1.27468e9i	−0.890487	+	1.54237i	−0.0511955	+	0.998689i	$0.516303\pi$
$353$	−7.35935e8	+	1.27468e9i	−0.890487	+	1.54237i	−0.839292	+	0.543681i	$0.817030\pi$
$354$		0			0
$355$	2.27602e8	+	3.94218e8i	0.270008	+	0.467667i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.08408e8	−	8.80589e8i	−0.579939	−	1.00448i	−0.995486	−	0.0949118i	$-0.969743\pi$
$359$	−5.08408e8	−	8.80589e8i	−0.579939	−	1.00448i	0.415547	−	0.909572i	$-0.363590\pi$
$360$		0			0
$361$	−5.26187e8	+	9.11383e8i	−0.588661	+	1.01959i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.07802e8			0.116039
$366$		0			0
$367$	3.81967e8	−	6.61586e8i	0.403362	−	0.698643i	−0.590768	−	0.806842i	$-0.701175\pi$
$367$	3.81967e8	−	6.61586e8i	0.403362	−	0.698643i	0.994129	+	0.108199i	$0.0345084\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.25099e8	+	7.92588e7i	0.330527	+	0.0805821i
$372$		0			0
$373$	5.15720e8	+	8.93254e8i	0.514557	+	0.891239i	0.999857	+	0.0168911i	$0.00537687\pi$
$373$	5.15720e8	+	8.93254e8i	0.514557	+	0.891239i	−0.485301	+	0.874347i	$0.661290\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.17435e9			−1.12876
$378$		0			0
$379$	5.88248e8			0.555038			0.277519	−	0.960720i	$-0.410488\pi$
$379$	5.88248e8			0.555038			0.277519	+	0.960720i	$0.410488\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.19545e8	−	1.07308e9i	−0.563479	−	0.975974i	−0.997189	−	0.0749216i	$-0.976129\pi$
$383$	−6.19545e8	−	1.07308e9i	−0.563479	−	0.975974i	0.433711	−	0.901052i	$-0.357204\pi$
$384$		0			0
$385$	1.29062e8	+	4.41398e8i	0.115262	+	0.394201i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.55841e8	−	4.43130e8i	0.220367	−	0.381687i	−0.734552	−	0.678552i	$-0.762608\pi$
$389$	2.55841e8	−	4.43130e8i	0.220367	−	0.381687i	0.954920	+	0.296865i	$0.0959411\pi$
$390$		0			0
$391$	8.46473e8			0.716135
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.87850e7	−	1.19139e8i	0.0561570	−	0.0972668i
$396$		0			0
$397$	−6.28008e8	−	1.08774e9i	−0.503730	−	0.872487i	−0.999991	−	0.00431285i	$-0.998627\pi$
$397$	−6.28008e8	−	1.08774e9i	−0.503730	−	0.872487i	0.496260	−	0.868174i	$-0.334706\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.87326e8	+	1.01728e9i	0.454856	+	0.787833i	0.998680	−	0.0513658i	$-0.0163575\pi$
$401$	5.87326e8	+	1.01728e9i	0.454856	+	0.787833i	−0.543824	+	0.839199i	$0.683024\pi$
$402$		0			0
$403$	7.17296e8	−	1.24239e9i	0.545922	−	0.945565i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.63558e9			−1.20252
$408$		0			0
$409$	−1.08611e9	+	1.88120e9i	−0.784953	+	1.35958i	0.144075	+	0.989567i	$0.453979\pi$
$409$	−1.08611e9	+	1.88120e9i	−0.784953	+	1.35958i	−0.929027	+	0.370011i	$0.879354\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.09841e8	−	2.21819e8i	−0.635536	−	0.154943i
$414$		0			0
$415$	5.86446e8	+	1.01575e9i	0.402772	+	0.697622i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.97088e9			−1.30892			−0.654458	−	0.756098i	$-0.727103\pi$
$419$	−1.97088e9			−1.30892			−0.654458	+	0.756098i	$0.727103\pi$
$420$		0			0
$421$	−1.34491e9			−0.878430			−0.439215	−	0.898382i	$-0.644743\pi$
$421$	−1.34491e9			−0.878430			−0.439215	+	0.898382i	$0.644743\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.41305e8	+	5.91157e8i	0.215666	+	0.373544i
$426$		0			0
$427$	2.02317e8	−	2.11702e8i	0.125758	−	0.131592i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.40654e8	+	5.90031e8i	−0.204948	+	0.354980i	−0.950116	−	0.311896i	$-0.899036\pi$
$431$	−3.40654e8	+	5.90031e8i	−0.204948	+	0.354980i	0.745168	+	0.666877i	$0.232369\pi$
$432$		0			0
$433$	−1.29953e9			−0.769272			−0.384636	−	0.923068i	$-0.625673\pi$
$433$	−1.29953e9			−0.769272			−0.384636	+	0.923068i	$0.625673\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.43524e9	+	2.48590e9i	−0.822694	+	1.42495i
$438$		0			0
$439$	−8.64679e8	−	1.49767e9i	−0.487786	−	0.844870i	0.512115	−	0.858917i	$-0.328862\pi$
$439$	−8.64679e8	−	1.49767e9i	−0.487786	−	0.844870i	−0.999901	+	0.0140465i	$0.995529\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.60591e8	+	7.97768e8i	0.251711	+	0.435977i	0.963997	−	0.265913i	$-0.0856733\pi$
$443$	4.60591e8	+	7.97768e8i	0.251711	+	0.435977i	−0.712286	+	0.701890i	$0.752340\pi$
$444$		0			0
$445$	3.70635e8	−	6.41959e8i	0.199382	−	0.345340i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.26736e9			0.660751			0.330376	−	0.943850i	$-0.392825\pi$
$449$	1.26736e9			0.660751			0.330376	+	0.943850i	$0.392825\pi$
$450$		0			0
$451$	5.97722e8	−	1.03529e9i	0.306819	−	0.531425i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.79181e8	−	5.01410e8i	0.238484	−	0.249547i
$456$		0			0
$457$	6.62958e8	+	1.14828e9i	0.324922	+	0.562782i	0.981497	−	0.191479i	$-0.0613285\pi$
$457$	6.62958e8	+	1.14828e9i	0.324922	+	0.562782i	−0.656574	+	0.754261i	$0.727995\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.44980e9			0.689215			0.344608	−	0.938747i	$-0.388012\pi$
$461$	1.44980e9			0.689215			0.344608	+	0.938747i	$0.388012\pi$
$462$		0			0
$463$	−3.49096e9			−1.63460			−0.817299	−	0.576214i	$-0.804530\pi$
$463$	−3.49096e9			−1.63460			−0.817299	+	0.576214i	$0.804530\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.23615e8	−	7.33723e8i	−0.192470	−	0.333368i	0.753598	−	0.657335i	$-0.228316\pi$
$467$	−4.23615e8	−	7.33723e8i	−0.192470	−	0.333368i	−0.946068	+	0.323968i	$0.894983\pi$
$468$		0			0
$469$	−2.07436e7	−	5.05726e6i	−0.00928492	−	0.00226366i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.12982e8	+	1.95690e8i	−0.0490902	+	0.0850268i
$474$		0			0
$475$	−2.31479e9			−0.991026
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.41330e9	+	2.44791e9i	−0.587572	+	1.01770i	0.406977	+	0.913438i	$0.366583\pi$
$479$	−1.41330e9	+	2.44791e9i	−0.587572	+	1.01770i	−0.994549	+	0.104266i	$0.966751\pi$
$480$		0			0
$481$	1.23334e9	+	2.13621e9i	0.505330	+	0.875257i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.62269e7	−	1.32029e8i	−0.0303398	−	0.0525500i
$486$		0			0
$487$	−1.92842e9	+	3.34012e9i	−0.756572	+	1.31042i	0.188016	+	0.982166i	$0.439794\pi$
$487$	−1.92842e9	+	3.34012e9i	−0.756572	+	1.31042i	−0.944589	+	0.328256i	$0.893539\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.98226e9			−0.755746			−0.377873	−	0.925857i	$-0.623344\pi$
$491$	−1.98226e9			−0.755746			−0.377873	+	0.925857i	$0.623344\pi$
$492$		0			0
$493$	−1.60091e9	+	2.77286e9i	−0.601733	+	1.04223i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.23803e8	+	2.47544e9i	0.264468	+	0.904491i
$498$		0			0
$499$	−1.75830e9	−	3.04547e9i	−0.633492	−	1.09724i	−0.986832	−	0.161746i	$-0.948287\pi$
$499$	−1.75830e9	−	3.04547e9i	−0.633492	−	1.09724i	0.353340	−	0.935495i	$-0.385046\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.01692e9			1.75772			0.878859	−	0.477081i	$-0.158305\pi$
$503$	5.01692e9			1.75772			0.878859	+	0.477081i	$0.158305\pi$
$504$		0			0
$505$	2.73859e9			0.946252
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.77610e9	+	4.80834e9i	0.933088	+	1.61616i	0.778008	+	0.628254i	$0.216230\pi$
$509$	2.77610e9	+	4.80834e9i	0.933088	+	1.61616i	0.155080	+	0.987902i	$0.450436\pi$
$510$		0			0
$511$	5.93404e8	+	1.44671e8i	0.196733	+	0.0479634i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.23723e8	−	5.60705e8i	0.104436	−	0.180888i
$516$		0			0
$517$	3.53326e9			1.12450
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.99849e8	+	1.55858e9i	−0.278765	+	0.482835i	−0.971078	−	0.238762i	$-0.923258\pi$
$521$	−8.99849e8	+	1.55858e9i	−0.278765	+	0.482835i	0.692313	+	0.721597i	$0.256592\pi$
$522$		0			0
$523$	6.38363e8	+	1.10568e9i	0.195124	+	0.337965i	0.946941	−	0.321407i	$-0.104156\pi$
$523$	6.38363e8	+	1.10568e9i	0.195124	+	0.337965i	−0.751817	+	0.659372i	$0.770822\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.95569e9	−	3.38735e9i	−0.582053	−	1.00815i
$528$		0			0
$529$	−4.14382e8	+	7.17730e8i	−0.121704	+	0.210798i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.80289e9			−0.515733
$534$		0			0
$535$	−6.35837e8	+	1.10130e9i	−0.179518	+	0.310934i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.18071e8	+	2.60290e9i	0.0324776	+	0.715973i
$540$		0			0
$541$	−3.19521e9	−	5.53426e9i	−0.867579	−	1.50269i	−0.864464	−	0.502695i	$-0.832342\pi$
$541$	−3.19521e9	−	5.53426e9i	−0.867579	−	1.50269i	−0.00311482	−	0.999995i	$-0.500991\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.12030e9			−0.296446
$546$		0			0
$547$	−4.83104e9			−1.26207			−0.631037	−	0.775752i	$-0.717371\pi$
$547$	−4.83104e9			−1.26207			−0.631037	+	0.775752i	$0.717371\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.42885e9	−	9.40304e9i	−1.38254	−	2.39463i
$552$		0			0
$553$	5.38516e8	−	5.63498e8i	0.135413	−	0.141695i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.26607e6	+	9.12110e6i	−0.00129120	+	0.00223642i	−0.866670	−	0.498881i	$-0.833744\pi$
$557$	−5.26607e6	+	9.12110e6i	−0.00129120	+	0.00223642i	0.865379	+	0.501118i	$0.167078\pi$
$558$		0			0
$559$	3.40784e8			0.0825160
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.21000e9	−	2.09579e9i	0.285764	−	0.494957i	−0.687030	−	0.726629i	$-0.741086\pi$
$563$	1.21000e9	−	2.09579e9i	0.285764	−	0.494957i	0.972794	+	0.231671i	$0.0744194\pi$
$564$		0			0
$565$	1.67404e9	+	2.89952e9i	0.390477	+	0.676326i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.21669e8	−	2.10737e8i	−0.0276877	−	0.0479565i	0.851850	−	0.523787i	$-0.175481\pi$
$569$	−1.21669e8	−	2.10737e8i	−0.0276877	−	0.0479565i	−0.879537	+	0.475830i	$0.842148\pi$
$570$		0			0
$571$	2.81425e9	−	4.87442e9i	0.632610	−	1.09571i	−0.354406	−	0.935092i	$-0.615317\pi$
$571$	2.81425e9	−	4.87442e9i	0.632610	−	1.09571i	0.987016	−	0.160621i	$-0.0513497\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.41403e9			−0.748911
$576$		0			0
$577$	1.89860e9	−	3.28847e9i	0.411451	−	0.712653i	−0.583598	−	0.812043i	$-0.698356\pi$
$577$	1.89860e9	−	3.28847e9i	0.411451	−	0.712653i	0.995049	+	0.0993895i	$0.0316890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.86497e9	+	6.37829e9i	0.394509	+	1.34923i
$582$		0			0
$583$	5.83306e8	+	1.01032e9i	0.121915	+	0.211163i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.00190e9			0.408516			0.204258	−	0.978917i	$-0.434522\pi$
$587$	2.00190e9			0.408516			0.204258	+	0.978917i	$0.434522\pi$
$588$		0			0
$589$	1.32639e10			2.67464
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.06563e8	−	1.39701e9i	−0.158835	−	0.275111i	0.775614	−	0.631208i	$-0.217441\pi$
$593$	−8.06563e8	−	1.39701e9i	−0.158835	−	0.275111i	−0.934449	+	0.356097i	$0.884107\pi$
$594$		0			0
$595$	−5.30690e8	−	1.81498e9i	−0.103284	−	0.353234i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.69338e9	−	8.12917e9i	0.892261	−	1.54544i	0.0551024	−	0.998481i	$-0.482451\pi$
$599$	4.69338e9	−	8.12917e9i	0.892261	−	1.54544i	0.837158	−	0.546960i	$-0.184215\pi$
$600$		0			0
$601$	−4.79642e9			−0.901274			−0.450637	−	0.892707i	$-0.648803\pi$
$601$	−4.79642e9			−0.901274			−0.450637	+	0.892707i	$0.648803\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.58983e8	−	1.31460e9i	0.139344	−	0.241351i
$606$		0			0
$607$	2.00844e9	+	3.47872e9i	0.364500	+	0.631333i	0.988696	−	0.149935i	$-0.0479064\pi$
$607$	2.00844e9	+	3.47872e9i	0.364500	+	0.631333i	−0.624195	+	0.781268i	$0.714573\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.66432e9	−	4.61474e9i	−0.472544	−	0.818470i
$612$		0			0
$613$	−1.16046e9	+	2.00997e9i	−0.203478	+	0.352434i	−0.949647	−	0.313323i	$-0.898558\pi$
$613$	−1.16046e9	+	2.00997e9i	−0.203478	+	0.352434i	0.746169	+	0.665757i	$0.231891\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.65061e9			−1.31129			−0.655644	−	0.755070i	$-0.727603\pi$
$617$	−7.65061e9			−1.31129			−0.655644	+	0.755070i	$0.727603\pi$
$618$		0			0
$619$	−2.52375e9	+	4.37126e9i	−0.427689	+	0.740780i	−0.996667	−	0.0815729i	$-0.974006\pi$
$619$	−2.52375e9	+	4.37126e9i	−0.427689	+	0.740780i	0.568978	+	0.822353i	$0.307339\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.90169e9	−	3.03630e9i	0.480777	−	0.503080i
$624$		0			0
$625$	−3.74430e8	−	6.48532e8i	−0.0613466	−	0.106255i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.72533e9			1.07755
$630$		0			0
$631$	9.61793e9			1.52398			0.761989	−	0.647590i	$-0.224223\pi$
$631$	9.61793e9			1.52398			0.761989	+	0.647590i	$0.224223\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.05837e9	+	1.83316e9i	0.164033	+	0.284113i
$636$		0			0
$637$	3.31057e9	−	2.11697e9i	0.507475	−	0.324509i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.57668e9	+	4.46294e9i	−0.386418	+	0.669296i	−0.991965	−	0.126514i	$-0.959621\pi$
$641$	−2.57668e9	+	4.46294e9i	−0.386418	+	0.669296i	0.605547	+	0.795810i	$0.292955\pi$
$642$		0			0
$643$	6.36581e9			0.944311			0.472155	−	0.881515i	$-0.343476\pi$
$643$	6.36581e9			0.944311			0.472155	+	0.881515i	$0.343476\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.22790e9	−	3.85884e9i	0.323394	−	0.560134i	−0.657792	−	0.753199i	$-0.728510\pi$
$647$	2.22790e9	−	3.85884e9i	0.323394	−	0.560134i	0.981186	+	0.193065i	$0.0618428\pi$
$648$		0			0
$649$	−1.63248e9	−	2.82753e9i	−0.234418	−	0.406024i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.47326e8	+	4.28382e8i	0.0347596	+	0.0602053i	0.882882	−	0.469595i	$-0.155600\pi$
$653$	2.47326e8	+	4.28382e8i	0.0347596	+	0.0602053i	−0.848122	+	0.529801i	$0.822267\pi$
$654$		0			0
$655$	1.27730e9	−	2.21234e9i	0.177602	−	0.307615i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.33078e9			0.997818			0.498909	−	0.866654i	$-0.333734\pi$
$659$	7.33078e9			0.997818			0.498909	+	0.866654i	$0.333734\pi$
$660$		0			0
$661$	−3.49714e9	+	6.05723e9i	−0.470986	+	0.815772i	−0.999449	−	0.0331843i	$-0.989435\pi$
$661$	−3.49714e9	+	6.05723e9i	−0.470986	+	0.815772i	0.528463	+	0.848956i	$0.322769\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.22999e9	+	1.51887e9i	0.821508	+	0.200283i
$666$		0			0
$667$	−8.00688e9	−	1.38683e10i	−1.04478	−	1.80960i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.02092e9			0.130455
$672$		0			0
$673$	−2.05913e9			−0.260394			−0.130197	−	0.991488i	$-0.541561\pi$
$673$	−2.05913e9			−0.260394			−0.130197	+	0.991488i	$0.541561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.01355e9	−	1.75552e9i	−0.125541	−	0.217443i	0.796403	−	0.604766i	$-0.206733\pi$
$677$	−1.01355e9	−	1.75552e9i	−0.125541	−	0.217443i	−0.921944	+	0.387323i	$0.873400\pi$
$678$		0			0
$679$	−2.42412e8	−	8.29057e8i	−0.0297173	−	0.101634i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.72964e9	−	8.19198e9i	0.568009	−	0.983821i	−0.428753	−	0.903422i	$-0.641047\pi$
$683$	4.72964e9	−	8.19198e9i	0.568009	−	0.983821i	0.996763	−	0.0803996i	$-0.0256196\pi$
$684$		0			0
$685$	5.71454e9			0.679304
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.79706e8	−	1.52370e9i	0.102464	−	0.177472i
$690$		0			0
$691$	−5.57572e9	−	9.65742e9i	−0.642876	−	1.11349i	−0.984788	−	0.173762i	$-0.944408\pi$
$691$	−5.57572e9	−	9.65742e9i	−0.642876	−	1.11349i	0.341911	−	0.939732i	$-0.388926\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.00510e8	+	1.55973e9i	0.101752	+	0.176239i
$696$		0			0
$697$	−2.45777e9	+	4.25698e9i	−0.274933	+	0.476198i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.01555e10			−1.11349			−0.556746	−	0.830683i	$-0.687950\pi$
$701$	−1.01555e10			−1.11349			−0.556746	+	0.830683i	$0.687950\pi$
$702$		0			0
$703$	−1.14031e10	+	1.97508e10i	−1.23788	+	2.14408i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.50747e10	+	3.67520e9i	1.60428	+	0.391122i
$708$		0			0
$709$	−5.65609e9	−	9.79664e9i	−0.596012	−	1.03232i	−0.993403	−	0.114674i	$-0.963418\pi$
$709$	−5.65609e9	−	9.79664e9i	−0.596012	−	1.03232i	0.397391	−	0.917649i	$-0.369916\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.95625e10			2.02121
$714$		0			0
$715$	2.41801e9			0.247393
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.36466e9	−	7.55981e9i	−0.437924	−	0.758507i	0.559605	−	0.828760i	$-0.310953\pi$
$719$	−4.36466e9	−	7.55981e9i	−0.437924	−	0.758507i	−0.997529	+	0.0702522i	$0.977620\pi$
$720$		0			0
$721$	2.53442e9	−	2.65199e9i	0.251829	−	0.263511i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.45688e9	−	1.11836e10i	0.629274	−	1.08993i
$726$		0			0
$727$	5.43142e9			0.524255			0.262128	−	0.965033i	$-0.415576\pi$
$727$	5.43142e9			0.524255			0.262128	+	0.965033i	$0.415576\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.64570e8	−	8.04658e8i	0.0439886	−	0.0761905i
$732$		0			0
$733$	−1.84738e9	−	3.19976e9i	−0.173258	−	0.300092i	0.766299	−	0.642484i	$-0.222096\pi$
$733$	−1.84738e9	−	3.19976e9i	−0.173258	−	0.300092i	−0.939557	+	0.342392i	$0.888763\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.72190e7	−	6.44652e7i	−0.00342475	−	0.00593184i
$738$		0			0
$739$	5.50425e9	−	9.53363e9i	0.501698	−	0.868966i	−0.498300	−	0.867005i	$-0.666042\pi$
$739$	5.50425e9	−	9.53363e9i	0.501698	−	0.868966i	0.999998	−	0.00196163i	$-0.000624407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.40281e10			−1.25469			−0.627346	−	0.778741i	$-0.715859\pi$
$743$	−1.40281e10			−1.25469			−0.627346	+	0.778741i	$0.715859\pi$
$744$		0			0
$745$	1.55802e9	−	2.69857e9i	0.138047	−	0.239104i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.97795e9	+	5.20887e9i	−0.432876	+	0.452957i
$750$		0			0
$751$	2.72345e9	+	4.71716e9i	0.234628	+	0.406388i	0.959165	−	0.282849i	$-0.0912794\pi$
$751$	2.72345e9	+	4.71716e9i	0.234628	+	0.406388i	−0.724536	+	0.689237i	$0.757946\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.28482e10			1.08649
$756$		0			0
$757$	−1.04051e10			−0.871785			−0.435892	−	0.899999i	$-0.643567\pi$
$757$	−1.04051e10			−0.871785			−0.435892	+	0.899999i	$0.643567\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.35065e8	+	7.53554e8i	0.0357855	+	0.0619823i	0.883363	−	0.468689i	$-0.155273\pi$
$761$	4.35065e8	+	7.53554e8i	0.0357855	+	0.0619823i	−0.847578	+	0.530671i	$0.821940\pi$
$762$		0			0
$763$	−6.16673e9	−	1.50344e9i	−0.502596	−	0.122533i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.46200e9	+	4.26431e9i	−0.197017	+	0.341244i
$768$		0			0
$769$	1.17740e9			0.0933643			0.0466822	−	0.998910i	$-0.485135\pi$
$769$	1.17740e9			0.0933643			0.0466822	+	0.998910i	$0.485135\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.40843e9	+	1.45638e10i	−0.654767	+	1.13409i	0.327185	+	0.944960i	$0.393900\pi$
$773$	−8.40843e9	+	1.45638e10i	−0.654767	+	1.13409i	−0.981952	+	0.189129i	$0.939433\pi$
$774$		0			0
$775$	7.88777e9	+	1.36620e10i	0.608693	+	1.05429i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.33453e9	−	1.44358e10i	−0.631684	−	1.09411i
$780$		0			0
$781$	−4.49583e9	+	7.78700e9i	−0.337700	+	0.584913i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.81733e9			−0.576786
$786$		0			0
$787$	7.43513e9	−	1.28780e10i	0.543722	−	0.941755i	−0.454964	−	0.890510i	$-0.650348\pi$
$787$	7.43513e9	−	1.28780e10i	0.543722	−	0.941755i	0.998686	−	0.0512448i	$-0.0163189\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.32365e9	+	1.82071e10i	0.382465	+	1.30805i
$792$		0			0
$793$	−7.69843e8	−	1.33341e9i	−0.0548208	−	0.0949525i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.62043e10			1.13377			0.566886	−	0.823796i	$-0.308148\pi$
$797$	1.62043e10			1.13377			0.566886	+	0.823796i	$0.308148\pi$
$798$		0			0
$799$	−1.45284e10			−1.00764
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.06471e9	+	1.84414e9i	0.0725651	+	0.125686i
$804$		0			0
$805$	9.18847e9	+	2.24014e9i	0.620808	+	0.151353i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.10784e9	−	1.91883e9i	0.0735624	−	0.127414i	−0.826898	−	0.562352i	$-0.809897\pi$
$809$	1.10784e9	−	1.91883e9i	0.0735624	−	0.127414i	0.900460	+	0.434938i	$0.143230\pi$
$810$		0			0
$811$	4.87572e9			0.320971			0.160485	−	0.987038i	$-0.448694\pi$
$811$	4.87572e9			0.320971			0.160485	+	0.987038i	$0.448694\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.05556e9	+	1.82828e9i	−0.0683014	+	0.118301i
$816$		0			0
$817$	1.57540e9	+	2.72867e9i	0.101068	+	0.175055i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.48439e9	−	9.49925e9i	−0.345882	−	0.599084i	0.639632	−	0.768681i	$-0.279087\pi$
$821$	−5.48439e9	−	9.49925e9i	−0.345882	−	0.599084i	−0.985514	+	0.169597i	$0.945753\pi$
$822$		0			0
$823$	−1.08079e8	+	1.87198e8i	−0.00675834	+	0.0117058i	−0.869385	−	0.494136i	$-0.835485\pi$
$823$	−1.08079e8	+	1.87198e8i	−0.00675834	+	0.0117058i	0.862626	+	0.505841i	$0.168818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.31243e10			−1.42167			−0.710836	−	0.703357i	$-0.751683\pi$
$827$	−2.31243e10			−1.42167			−0.710836	+	0.703357i	$0.751683\pi$
$828$		0			0
$829$	−1.17820e10	+	2.04071e10i	−0.718256	+	1.24406i	0.243434	+	0.969917i	$0.421726\pi$
$829$	−1.17820e10	+	2.04071e10i	−0.718256	+	1.24406i	−0.961690	+	0.274139i	$0.911607\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.85496e8	−	1.07028e10i	−0.0291024	−	0.641566i
$834$		0			0
$835$	−5.24002e9	−	9.07598e9i	−0.311480	−	0.539499i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.50625e10			−1.46507			−0.732534	−	0.680730i	$-0.761663\pi$
$839$	−2.50625e10			−1.46507			−0.732534	+	0.680730i	$0.761663\pi$
$840$		0			0
$841$	4.33229e10			2.51149
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.20190e9	+	5.54585e9i	0.182561	+	0.316205i
$846$		0			0
$847$	5.94206e9	−	6.21771e9i	0.336004	−	0.351591i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.68182e10	+	2.91299e10i	−0.935461	+	1.62027i
$852$		0			0
$853$	−1.90748e10			−1.05230			−0.526149	−	0.850392i	$-0.676365\pi$
$853$	−1.90748e10			−1.05230			−0.526149	+	0.850392i	$0.676365\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.67098e10	+	2.89422e10i	−0.906856	+	1.57072i	−0.0884496	+	0.996081i	$0.528191\pi$
$857$	−1.67098e10	+	2.89422e10i	−0.906856	+	1.57072i	−0.818406	+	0.574640i	$0.805142\pi$
$858$		0			0
$859$	3.98428e9	+	6.90097e9i	0.214473	+	0.371479i	0.953110	−	0.302625i	$-0.0978631\pi$
$859$	3.98428e9	+	6.90097e9i	0.214473	+	0.371479i	−0.738636	+	0.674104i	$0.764530\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.08562e10	−	1.88034e10i	−0.574961	−	0.995861i	−0.996046	−	0.0888399i	$-0.971684\pi$
$863$	−1.08562e10	−	1.88034e10i	−0.574961	−	0.995861i	0.421085	−	0.907021i	$-0.361649\pi$
$864$		0			0
$865$	1.29279e9	−	2.23918e9i	0.0679161	−	0.117634i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.71743e9			0.140472
$870$		0			0
$871$	−5.61314e7	+	9.72224e7i	−0.00287834	+	0.00498543i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.32733e9	+	1.82197e10i	0.268832	+	0.919416i
$876$		0			0
$877$	9.33106e9	+	1.61619e10i	0.467124	+	0.809083i	0.999295	−	0.0375541i	$-0.0119567\pi$
$877$	9.33106e9	+	1.61619e10i	0.467124	+	0.809083i	−0.532170	+	0.846637i	$0.678623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.50045e10			−0.739273			−0.369636	−	0.929176i	$-0.620518\pi$
$881$	−1.50045e10			−0.739273			−0.369636	+	0.929176i	$0.620518\pi$
$882$		0			0
$883$	−2.65989e10			−1.30017			−0.650086	−	0.759861i	$-0.725267\pi$
$883$	−2.65989e10			−1.30017			−0.650086	+	0.759861i	$0.725267\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.16407e9	−	1.06765e10i	−0.296575	−	0.513683i	0.678775	−	0.734346i	$-0.262511\pi$
$887$	−6.16407e9	−	1.06765e10i	−0.296575	−	0.513683i	−0.975350	+	0.220663i	$0.929178\pi$
$888$		0			0
$889$	3.36577e9	+	1.15111e10i	0.160667	+	0.549489i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.46336e10	−	4.26666e10i	1.15757	−	2.00497i
$894$		0			0
$895$	1.54672e10			0.721157
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.69981e10	+	6.40826e10i	−1.69832	+	2.94158i
$900$		0			0
$901$	−2.39849e9	−	4.15431e9i	−0.109245	−	0.189218i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.75579e9	−	9.96932e9i	−0.258128	−	0.447090i
$906$		0			0
$907$	2.27456e9	−	3.93965e9i	0.101221	−	0.175320i	−0.810967	−	0.585092i	$-0.801058\pi$
$907$	2.27456e9	−	3.93965e9i	0.101221	−	0.175320i	0.912188	+	0.409772i	$0.134392\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.78309e9			0.384887			0.192443	−	0.981308i	$-0.438359\pi$
$911$	8.78309e9			0.384887			0.192443	+	0.981308i	$0.438359\pi$
$912$		0			0
$913$	−1.15841e10	+	2.00642e10i	−0.503749	+	0.872519i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.99992e9	−	1.04638e10i	0.428256	−	0.448123i
$918$		0			0
$919$	1.15521e10	+	2.00088e10i	0.490971	+	0.850388i	0.999946	−	0.0103940i	$-0.00330859\pi$
$919$	1.15521e10	+	2.00088e10i	0.490971	+	0.850388i	−0.508974	+	0.860782i	$0.669975\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.35606e10			0.567642
$924$		0			0
$925$	−2.71249e10			−1.12687
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.42081e10	−	2.46091e10i	−0.581407	−	1.00703i	−0.995313	−	0.0967070i	$-0.969169\pi$
$929$	−1.42081e10	−	2.46091e10i	−0.581407	−	1.00703i	0.413906	−	0.910320i	$-0.364164\pi$
$930$		0			0
$931$	3.22550e10	+	1.67214e10i	1.31001	+	0.679122i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.29632e9	−	5.70940e9i	0.131883	−	0.228428i
$936$		0			0
$937$	−2.66810e10			−1.05953			−0.529766	−	0.848144i	$-0.677720\pi$
$937$	−2.66810e10			−1.05953			−0.529766	+	0.848144i	$0.677720\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.47193e10	+	4.28151e10i	−0.967103	+	1.67507i	−0.263249	+	0.964728i	$0.584794\pi$
$941$	−2.47193e10	+	4.28151e10i	−0.967103	+	1.67507i	−0.703854	+	0.710344i	$0.748539\pi$
$942$		0			0
$943$	−1.22924e10	−	2.12911e10i	−0.477360	−	0.826811i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.04792e10	+	1.81505e10i	0.400961	+	0.694486i	0.993842	−	0.110805i	$-0.0353428\pi$
$947$	1.04792e10	+	1.81505e10i	0.400961	+	0.694486i	−0.592881	+	0.805290i	$0.702009\pi$
$948$		0			0
$949$	1.60573e9	−	2.78121e9i	0.0609875	−	0.105634i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.44763e10			0.541791			0.270895	−	0.962609i	$-0.412680\pi$
$953$	1.44763e10			0.541791			0.270895	+	0.962609i	$0.412680\pi$
$954$		0			0
$955$	−1.34126e10	+	2.32314e10i	−0.498313	+	0.863104i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.14560e10	+	7.66894e9i	1.15170	+	0.280782i
$960$		0			0
$961$	−3.14409e10	−	5.44572e10i	−1.14278	−	1.97935i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.70702e9			0.0969718
$966$		0			0
$967$	−3.47169e10			−1.23466			−0.617332	−	0.786703i	$-0.711786\pi$
$967$	−3.47169e10			−1.23466			−0.617332	+	0.786703i	$0.711786\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.62040e9	+	1.14669e10i	0.232069	+	0.401955i	0.958417	−	0.285372i	$-0.0921172\pi$
$971$	6.62040e9	+	1.14669e10i	0.232069	+	0.401955i	−0.726348	+	0.687327i	$0.758784\pi$
$972$		0			0
$973$	2.86374e9	+	9.79409e9i	0.0996640	+	0.340855i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.03510e10	+	1.79284e10i	−0.355099	+	0.615050i	−0.987135	−	0.159890i	$-0.948886\pi$
$977$	−1.03510e10	+	1.79284e10i	−0.355099	+	0.615050i	0.632036	+	0.774939i	$0.282220\pi$
$978$		0			0
$979$	1.46423e10			0.498736
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.02622e9	−	3.50951e9i	0.0680376	−	0.117845i	−0.830000	−	0.557764i	$-0.811660\pi$
$983$	2.02622e9	−	3.50951e9i	0.0680376	−	0.117845i	0.898037	+	0.439919i	$0.144993\pi$
$984$		0			0
$985$	−4.33741e9	−	7.51261e9i	−0.144612	−	0.250475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.32352e9	+	4.02445e9i	0.0763764	+	0.132288i
$990$		0			0
$991$	2.45811e10	−	4.25757e10i	0.802313	−	1.38965i	−0.115778	−	0.993275i	$-0.536936\pi$
$991$	2.45811e10	−	4.25757e10i	0.802313	−	1.38965i	0.918090	−	0.396371i	$-0.129731\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.46649e9			0.0793777
$996$		0			0
$997$	5.36989e9	−	9.30093e9i	0.171606	−	0.297230i	−0.767375	−	0.641198i	$-0.778438\pi$
$997$	5.36989e9	−	9.30093e9i	0.171606	−	0.297230i	0.938981	+	0.343968i	$0.111771\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.b.109.1		8
3.2	odd	2		84.8.i.a.25.4	✓	8
7.2	even	3	inner	252.8.k.b.37.1		8
21.2	odd	6		84.8.i.a.37.4	yes	8
21.5	even	6		588.8.i.o.373.1		8
21.11	odd	6		588.8.a.i.1.1		4
21.17	even	6		588.8.a.j.1.4		4
21.20	even	2		588.8.i.o.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.i.a.25.4	✓	8	3.2	odd	2
84.8.i.a.37.4	yes	8	21.2	odd	6
252.8.k.b.37.1		8	7.2	even	3	inner
252.8.k.b.109.1		8	1.1	even	1	trivial
588.8.a.i.1.1		4	21.11	odd	6
588.8.a.j.1.4		4	21.17	even	6
588.8.i.o.361.1		8	21.20	even	2
588.8.i.o.373.1		8	21.5	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-80.0854 - 138.712i) q^{5} +(-254.682 - 871.022i) q^{7} +(1581.93 - 2739.98i) q^{11} -4771.53 q^{13} +(-6504.73 + 11266.5i) q^{17} +(-22058.1 - 38205.8i) q^{19} +(-32533.0 - 56348.8i) q^{23} +(26235.2 - 45440.6i) q^{25} +246115. q^{29} +(-150328. + 260376. i) q^{31} +(-100425. + 105084. i) q^{35} +(-258479. - 447698. i) q^{37} +377844. q^{41} -71420.3 q^{43} +(558378. + 967140. i) q^{47} +(-693817. + 443667. i) q^{49} +(-184365. + 319330. i) q^{53} -506758. q^{55} +(515977. - 893698. i) q^{59} +(161341. + 279450. i) q^{61} +(382130. + 661869. i) q^{65} +(11763.8 - 20375.5i) q^{67} -2.84199e6 q^{71} +(-336523. + 582875. i) q^{73} +(-2.78947e6 - 680072. i) q^{77} +(429448. + 743826. i) q^{79} -7.32276e6 q^{83} +2.08373e6 q^{85} +(2.31400e6 + 4.00797e6i) q^{89} +(1.21522e6 + 4.15611e6i) q^{91} +(-3.53307e6 + 6.11946e6i) q^{95} +951821. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 196 q^{5} - 434 q^{7} - 406 q^{11} + 3948 q^{13} + 7436 q^{17} + 15874 q^{19} + 6788 q^{23} + 69898 q^{25} + 189088 q^{29} - 55890 q^{31} + 750596 q^{35} + 93742 q^{37} - 13944 q^{41} - 487844 q^{43}+ \cdots + 23722580 q^{97}+O(q^{100}) \) 8 * q + 196 * q^5 - 434 * q^7 - 406 * q^11 + 3948 * q^13 + 7436 * q^17 + 15874 * q^19 + 6788 * q^23 + 69898 * q^25 + 189088 * q^29 - 55890 * q^31 + 750596 * q^35 + 93742 * q^37 - 13944 * q^41 - 487844 * q^43 + 650484 * q^47 - 1834756 * q^49 - 368880 * q^53 - 2391536 * q^55 - 96026 * q^59 - 3618156 * q^61 - 974160 * q^65 - 316006 * q^67 - 12436312 * q^71 - 1287286 * q^73 + 6614440 * q^77 + 8187282 * q^79 - 7387300 * q^83 + 21933928 * q^85 + 14489928 * q^89 - 10505670 * q^91 - 18406792 * q^95 + 23722580 * q^97

Introduction

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