LMFDB - Embedded newform 252.8.t.a.89.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(17,252)]

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, 1]))

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

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

chi := DirichletCharacter("252.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.t (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:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			89.10
Character	$\chi$	$=$	252.89
Dual form			252.8.t.a.17.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(21.6450 - 37.4903i) q^{5} +(487.468 + 765.452i) q^{7} +(2464.10 - 1422.65i) q^{11} -1905.54i q^{13} +(-17180.3 - 29757.1i) q^{17} +(-27599.1 - 15934.3i) q^{19} +(35809.5 + 20674.6i) q^{23} +(38125.5 + 66035.3i) q^{25} -3997.05i q^{29} +(-110802. + 63971.8i) q^{31} +(39248.3 - 1707.10i) q^{35} +(243552. - 421845. i) q^{37} -11984.7 q^{41} +323185. q^{43} +(-471739. + 817076. i) q^{47} +(-348292. + 746268. i) q^{49} +(1.78030e6 - 1.02785e6i) q^{53} -123173. i q^{55} +(-558158. - 966758. i) q^{59} +(-2.64733e6 - 1.52844e6i) q^{61} +(-71439.2 - 41245.5i) q^{65} +(-502727. - 870748. i) q^{67} -2.34983e6i q^{71} +(1.94909e6 - 1.12531e6i) q^{73} +(2.29014e6 + 1.19266e6i) q^{77} +(-454834. + 787795. i) q^{79} +4.64507e6 q^{83} -1.48747e6 q^{85} +(4.97345e6 - 8.61427e6i) q^{89} +(1.45860e6 - 928890. i) q^{91} +(-1.19477e6 + 689798. i) q^{95} -7.88352e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2634 q^{7} - 47862 q^{19} - 360762 q^{25} + 486018 q^{31} - 972270 q^{37} + 298788 q^{43} + 1556886 q^{49} + 4324644 q^{61} - 2969562 q^{67} - 5157378 q^{73} + 7676514 q^{79} - 15214128 q^{85} - 6114678 q^{91}+O(q^{100}) $ 36 * q + 2634 * q^7 - 47862 * q^19 - 360762 * q^25 + 486018 * q^31 - 972270 * q^37 + 298788 * q^43 + 1556886 * q^49 + 4324644 * q^61 - 2969562 * q^67 - 5157378 * q^73 + 7676514 * q^79 - 15214128 * q^85 - 6114678 * q^91

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	21.6450	−	37.4903i	0.0774397	−	0.134129i	−0.824705	−	0.565563i	$-0.808659\pi$
$5$	21.6450	−	37.4903i	0.0774397	−	0.134129i	0.902145	+	0.431434i	$0.141992\pi$
$6$		0			0
$7$	487.468	+	765.452i	0.537160	+	0.843481i
$7$	487.468	+	765.452i	0.537160	+	0.843481i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2464.10	−	1422.65i	0.558193	−	0.322273i	−0.194227	−	0.980957i	$-0.562220\pi$
$11$	2464.10	−	1422.65i	0.558193	−	0.322273i	0.752420	+	0.658684i	$0.228886\pi$
$12$		0			0
$13$		−	1905.54i		−	0.240556i	−0.992740	−	0.120278i	$-0.961621\pi$
$13$		−	1905.54i		−	0.240556i	0.992740	−	0.120278i	$-0.0383786\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−17180.3	−	29757.1i	−0.848124	−	1.46899i	−0.882880	−	0.469598i	$-0.844399\pi$
$17$	−17180.3	−	29757.1i	−0.848124	−	1.46899i	0.0347563	−	0.999396i	$-0.488934\pi$
$18$		0			0
$19$	−27599.1	−	15934.3i	−0.923117	−	0.532962i	−0.0384887	−	0.999259i	$-0.512254\pi$
$19$	−27599.1	−	15934.3i	−0.923117	−	0.532962i	−0.884628	+	0.466297i	$0.845588\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	35809.5	+	20674.6i	0.613693	+	0.354316i	0.774409	−	0.632685i	$-0.218047\pi$
$23$	35809.5	+	20674.6i	0.613693	+	0.354316i	−0.160716	+	0.987001i	$0.551380\pi$
$24$		0			0
$25$	38125.5	+	66035.3i	0.488006	+	0.845252i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3997.05i		−	0.0304332i	−0.999884	−	0.0152166i	$-0.995156\pi$
$29$		−	3997.05i		−	0.0304332i	0.999884	−	0.0152166i	$-0.00484378\pi$
$30$		0			0
$31$	−110802.	+	63971.8i	−0.668011	+	0.385676i	−0.795322	−	0.606187i	$-0.792698\pi$
$31$	−110802.	+	63971.8i	−0.668011	+	0.385676i	0.127312	+	0.991863i	$0.459365\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	39248.3	−	1707.10i	0.154733	−	0.00673007i
$36$		0			0
$37$	243552.	−	421845.i	0.790471	−	1.36914i	−0.135205	−	0.990818i	$-0.543169\pi$
$37$	243552.	−	421845.i	0.790471	−	1.36914i	0.925676	−	0.378318i	$-0.123497\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11984.7			−0.0271571			−0.0135785	−	0.999908i	$-0.504322\pi$
$41$	−11984.7			−0.0271571			−0.0135785	+	0.999908i	$0.504322\pi$
$42$		0			0
$43$	323185.			0.619886			0.309943	−	0.950755i	$-0.399690\pi$
$43$	323185.			0.619886			0.309943	+	0.950755i	$0.399690\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−471739.	+	817076.i	−0.662765	+	1.14794i	0.317121	+	0.948385i	$0.397284\pi$
$47$	−471739.	+	817076.i	−0.662765	+	1.14794i	−0.979886	+	0.199557i	$0.936050\pi$
$48$		0			0
$49$	−348292.	+	746268.i	−0.422919	+	0.906168i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.78030e6	−	1.02785e6i	1.64258	−	0.948344i	0.662669	−	0.748913i	$-0.269424\pi$
$53$	1.78030e6	−	1.02785e6i	1.64258	−	0.948344i	0.979912	−	0.199432i	$-0.0639096\pi$
$54$		0			0
$55$		−	123173.i		−	0.0998268i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−558158.	−	966758.i	−0.353814	−	0.612824i	0.633100	−	0.774070i	$-0.281782\pi$
$59$	−558158.	−	966758.i	−0.353814	−	0.612824i	−0.986914	+	0.161246i	$0.948449\pi$
$60$		0			0
$61$	−2.64733e6	−	1.52844e6i	−1.49332	−	0.862170i	−0.493352	−	0.869830i	$-0.664229\pi$
$61$	−2.64733e6	−	1.52844e6i	−1.49332	−	0.862170i	−0.999971	+	0.00765904i	$0.997562\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−71439.2	−	41245.5i	−0.0322656	−	0.0186286i
$66$		0			0
$67$	−502727.	−	870748.i	−0.204207	−	0.353696i	0.745673	−	0.666312i	$-0.232128\pi$
$67$	−502727.	−	870748.i	−0.204207	−	0.353696i	−0.949880	+	0.312616i	$0.898795\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.34983e6i		−	0.779170i	−0.920990	−	0.389585i	$-0.872618\pi$
$71$		−	2.34983e6i		−	0.779170i	0.920990	−	0.389585i	$-0.127382\pi$
$72$		0			0
$73$	1.94909e6	−	1.12531e6i	0.586409	−	0.338564i	−0.177267	−	0.984163i	$-0.556726\pi$
$73$	1.94909e6	−	1.12531e6i	0.586409	−	0.338564i	0.763676	+	0.645599i	$0.223392\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.29014e6	+	1.19266e6i	0.571670	+	0.297713i
$78$		0			0
$79$	−454834.	+	787795.i	−0.103791	+	0.179771i	−0.913243	−	0.407414i	$-0.866431\pi$
$79$	−454834.	+	787795.i	−0.103791	+	0.179771i	0.809453	+	0.587185i	$0.199764\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.64507e6			0.891701			0.445851	−	0.895107i	$-0.352901\pi$
$83$	4.64507e6			0.891701			0.445851	+	0.895107i	$0.352901\pi$
$84$		0			0
$85$	−1.48747e6			−0.262714
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.97345e6	−	8.61427e6i	0.747813	−	1.29525i	−0.201056	−	0.979580i	$-0.564437\pi$
$89$	4.97345e6	−	8.61427e6i	0.747813	−	1.29525i	0.948869	−	0.315670i	$-0.102229\pi$
$90$		0			0
$91$	1.45860e6	−	928890.i	0.202904	−	0.129217i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.19477e6	+	689798.i	−0.142972	+	0.0825448i
$96$		0			0
$97$		−	7.88352e6i		−	0.877040i	−0.898722	−	0.438520i	$-0.855503\pi$
$97$		−	7.88352e6i		−	0.877040i	0.898722	−	0.438520i	$-0.144497\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.92951e6	+	3.34201e6i	0.186347	+	0.322763i	0.944030	−	0.329861i	$-0.107002\pi$
$101$	1.92951e6	+	3.34201e6i	0.186347	+	0.322763i	−0.757682	+	0.652623i	$0.773668\pi$
$102$		0			0
$103$	−4.38680e6	−	2.53272e6i	−0.395564	−	0.228379i	0.289004	−	0.957328i	$-0.406676\pi$
$103$	−4.38680e6	−	2.53272e6i	−0.395564	−	0.228379i	−0.684568	+	0.728949i	$0.740009\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.61374e7	+	9.31692e6i	1.27347	+	0.735240i	0.975640	−	0.219378i	$-0.0704029\pi$
$107$	1.61374e7	+	9.31692e6i	1.27347	+	0.735240i	0.297833	+	0.954618i	$0.403736\pi$
$108$		0			0
$109$	−9.71053e6	−	1.68191e7i	−0.718208	−	1.24397i	−0.961709	−	0.274072i	$-0.911629\pi$
$109$	−9.71053e6	−	1.68191e7i	−0.718208	−	1.24397i	0.243501	−	0.969901i	$-0.421704\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.51561e7i		−	0.988131i	−0.869425	−	0.494065i	$-0.835510\pi$
$113$		−	1.51561e7i		−	0.988131i	0.869425	−	0.494065i	$-0.164490\pi$
$114$		0			0
$115$	1.55020e6	−	895007.i	0.0950484	−	0.0548762i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.44028e7	−	2.76564e7i	0.783490	−	1.50446i
$120$		0			0
$121$	−5.69572e6	+	9.86528e6i	−0.292280	+	0.506245i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.68295e6			0.306043
$126$		0			0
$127$	−2.69027e7			−1.16542			−0.582711	−	0.812679i	$-0.698008\pi$
$127$	−2.69027e7			−1.16542			−0.582711	+	0.812679i	$0.698008\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.43289e7	−	2.48184e7i	0.556883	−	0.964549i	−0.440872	−	0.897570i	$-0.645331\pi$
$131$	1.43289e7	−	2.48184e7i	0.556883	−	0.964549i	0.997754	−	0.0669791i	$-0.0213361\pi$
$132$		0			0
$133$	−1.25670e6	−	2.88932e7i	−0.0463182	−	1.06492i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.15280e7	−	2.39762e7i	1.37981	−	0.796634i	0.387674	−	0.921796i	$-0.373279\pi$
$137$	4.15280e7	−	2.39762e7i	1.37981	−	0.796634i	0.992136	+	0.125163i	$0.0399453\pi$
$138$		0			0
$139$		−	2.81496e7i		−	0.889038i	−0.895769	−	0.444519i	$-0.853375\pi$
$139$		−	2.81496e7i		−	0.889038i	0.895769	−	0.444519i	$-0.146625\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.71091e6	−	4.69544e6i	−0.0775246	−	0.134277i
$144$		0			0
$145$	−149851.	−	86516.4i	−0.00408198	−	0.00235673i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.96356e6	−	1.71101e6i	−0.0733942	−	0.0423741i	0.462854	−	0.886435i	$-0.346825\pi$
$149$	−2.96356e6	−	1.71101e6i	−0.0733942	−	0.0423741i	−0.536248	+	0.844060i	$0.680159\pi$
$150$		0			0
$151$	−1.47357e7	−	2.55229e7i	−0.348297	−	0.603269i	0.637650	−	0.770326i	$-0.279907\pi$
$151$	−1.47357e7	−	2.55229e7i	−0.348297	−	0.603269i	−0.985947	+	0.167058i	$0.946573\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			5.53869e6i			0.119467i
$156$		0			0
$157$	−5.76835e7	+	3.33036e7i	−1.18961	+	0.686819i	−0.958217	−	0.286042i	$-0.907660\pi$
$157$	−5.76835e7	+	3.33036e7i	−1.18961	+	0.686819i	−0.231388	+	0.972861i	$0.574327\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.63056e6	+	3.74887e7i	0.0307926	+	0.707962i
$162$		0			0
$163$	4.48038e7	−	7.76025e7i	0.810324	−	1.40352i	−0.102314	−	0.994752i	$-0.532625\pi$
$163$	4.48038e7	−	7.76025e7i	0.810324	−	1.40352i	0.912638	−	0.408770i	$-0.134042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.83745e7			1.63446			0.817231	−	0.576310i	$-0.195508\pi$
$167$	9.83745e7			1.63446			0.817231	+	0.576310i	$0.195508\pi$
$168$		0			0
$169$	5.91174e7			0.942133
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.35455e7	−	2.34615e7i	0.198899	−	0.344503i	−0.749273	−	0.662262i	$-0.769597\pi$
$173$	1.35455e7	−	2.34615e7i	0.198899	−	0.344503i	0.948172	+	0.317758i	$0.102930\pi$
$174$		0			0
$175$	−3.19619e7	+	6.13734e7i	−0.450816	+	0.865659i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.09526e7	−	2.36440e7i	0.533698	−	0.308131i	−0.208823	−	0.977953i	$-0.566963\pi$
$179$	4.09526e7	−	2.36440e7i	0.533698	−	0.308131i	0.742521	+	0.669823i	$0.233630\pi$
$180$		0			0
$181$		−	3.98951e6i		−	0.0500086i	−0.999687	−	0.0250043i	$-0.992040\pi$
$181$		−	3.98951e6i		−	0.0500086i	0.999687	−	0.0250043i	$-0.00795994\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.05434e7	−	1.82617e7i	−0.122428	−	0.212051i
$186$		0			0
$187$	−8.46680e7	−	4.88831e7i	−0.946834	−	0.546655i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.42463e7	−	3.13191e7i	−0.563317	−	0.325231i	0.191159	−	0.981559i	$-0.438776\pi$
$191$	−5.42463e7	−	3.13191e7i	−0.563317	−	0.325231i	−0.754476	+	0.656328i	$0.772109\pi$
$192$		0			0
$193$	1.09529e7	+	1.89710e7i	0.109668	+	0.189950i	0.915636	−	0.402009i	$-0.131688\pi$
$193$	1.09529e7	+	1.89710e7i	0.109668	+	0.189950i	−0.805968	+	0.591959i	$0.798355\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.54338e7i		−	0.516586i	−0.966067	−	0.258293i	$-0.916840\pi$
$197$		−	5.54338e7i		−	0.516586i	0.966067	−	0.258293i	$-0.0831600\pi$
$198$		0			0
$199$	8.58186e7	−	4.95474e7i	0.771962	−	0.445692i	−0.0616124	−	0.998100i	$-0.519624\pi$
$199$	8.58186e7	−	4.95474e7i	0.771962	−	0.445692i	0.833574	+	0.552408i	$0.186291\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.05956e6	−	1.94844e6i	0.0256698	−	0.0163475i
$204$		0			0
$205$	−259409.	+	449309.i	−0.00210303	+	0.00364256i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.06759e7			−0.687036
$210$		0			0
$211$	−4.96415e7			−0.363795			−0.181897	−	0.983318i	$-0.558224\pi$
$211$	−4.96415e7			−0.363795			−0.181897	+	0.983318i	$0.558224\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.99535e6	−	1.21163e7i	0.0480037	−	0.0831449i
$216$		0			0
$217$	−1.02980e8	−	5.36298e7i	−0.684139	−	0.356284i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.67034e7	+	3.27377e7i	−0.353375	+	0.204021i
$222$		0			0
$223$			1.75096e8i			1.05733i	0.848832	+	0.528663i	$0.177306\pi$
$223$			1.75096e8i			1.05733i	−0.848832	+	0.528663i	$0.822694\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.87276e7	+	1.19040e8i	0.389978	+	0.675463i	0.992446	−	0.122681i	$-0.0391491\pi$
$227$	6.87276e7	+	1.19040e8i	0.389978	+	0.675463i	−0.602468	+	0.798143i	$0.705816\pi$
$228$		0			0
$229$	1.46965e8	+	8.48504e7i	0.808705	+	0.466906i	0.846506	−	0.532379i	$-0.178702\pi$
$229$	1.46965e8	+	8.48504e7i	0.808705	+	0.466906i	−0.0378008	+	0.999285i	$0.512035\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.06488e7	−	1.19216e7i	−0.106942	−	0.0617431i	0.445575	−	0.895245i	$-0.352999\pi$
$233$	−2.06488e7	−	1.19216e7i	−0.106942	−	0.0617431i	−0.552517	+	0.833502i	$0.686333\pi$
$234$		0			0
$235$	2.04216e7	+	3.53713e7i	0.102649	+	0.177793i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.59527e8i			0.755861i	0.925834	+	0.377930i	$0.123364\pi$
$239$			1.59527e8i			0.755861i	−0.925834	+	0.377930i	$0.876636\pi$
$240$		0			0
$241$	−1.01027e8	+	5.83280e7i	−0.464920	+	0.268422i	−0.714111	−	0.700033i	$-0.753169\pi$
$241$	−1.01027e8	+	5.83280e7i	−0.464920	+	0.268422i	0.249191	+	0.968454i	$0.419835\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.04390e7	+	2.92106e7i	0.0887930	+	0.126899i
$246$		0			0
$247$	−3.03635e7	+	5.25911e7i	−0.128207	+	0.222061i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.09877e8			−1.23689			−0.618446	−	0.785827i	$-0.712237\pi$
$251$	−3.09877e8			−1.23689			−0.618446	+	0.785827i	$0.712237\pi$
$252$		0			0
$253$	1.17651e8			0.456745
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.73890e8	+	3.01186e8i	−0.639012	+	1.10680i	0.346638	+	0.937999i	$0.387323\pi$
$257$	−1.73890e8	+	3.01186e8i	−0.639012	+	1.10680i	−0.985650	+	0.168802i	$0.946010\pi$
$258$		0			0
$259$	4.41626e8	−	1.92084e7i	1.57945	−	0.0686977i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.88562e8	+	2.82072e8i	−1.65605	+	0.956124i	−0.681546	+	0.731775i	$0.738692\pi$
$263$	−4.88562e8	+	2.82072e8i	−1.65605	+	0.956124i	−0.974509	+	0.224348i	$0.927975\pi$
$264$		0			0
$265$		−	8.89918e7i		−	0.293758i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.46286e7	−	7.72990e7i	−0.139791	−	0.242126i	0.787626	−	0.616153i	$-0.211310\pi$
$269$	−4.46286e7	−	7.72990e7i	−0.139791	−	0.242126i	−0.927418	+	0.374028i	$0.877976\pi$
$270$		0			0
$271$	2.75949e8	+	1.59319e8i	0.842241	+	0.486268i	0.858025	−	0.513607i	$-0.171691\pi$
$271$	2.75949e8	+	1.59319e8i	0.842241	+	0.486268i	−0.0157845	+	0.999875i	$0.505025\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.87890e8	+	1.08478e8i	0.544803	+	0.314542i
$276$		0			0
$277$	9.35365e7	+	1.62010e8i	0.264425	+	0.457997i	0.967413	−	0.253205i	$-0.0814846\pi$
$277$	9.35365e7	+	1.62010e8i	0.264425	+	0.457997i	−0.702988	+	0.711202i	$0.748151\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.05899e7i		−	0.189789i	−0.995487	−	0.0948944i	$-0.969749\pi$
$281$		−	7.05899e7i		−	0.189789i	0.995487	−	0.0948944i	$-0.0302513\pi$
$282$		0			0
$283$	4.70136e7	−	2.71433e7i	0.123302	−	0.0711885i	−0.437080	−	0.899422i	$-0.643987\pi$
$283$	4.70136e7	−	2.71433e7i	0.123302	−	0.0711885i	0.560383	+	0.828234i	$0.310654\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.84215e6	−	9.17370e6i	−0.0145877	−	0.0229064i
$288$		0			0
$289$	−3.85156e8	+	6.67109e8i	−0.938629	+	1.62575i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.51658e8			−0.816739			−0.408370	−	0.912817i	$-0.633903\pi$
$293$	−3.51658e8			−0.816739			−0.408370	+	0.912817i	$0.633903\pi$
$294$		0			0
$295$	−4.83254e7			−0.109597
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.93963e7	−	6.82364e7i	0.0852327	−	0.147627i
$300$		0			0
$301$	1.57543e8	+	2.47383e8i	0.332978	+	0.522862i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.14603e8	+	6.61662e7i	−0.231285	+	0.133532i
$306$		0			0
$307$			1.07950e8i			0.212932i	0.994316	+	0.106466i	$0.0339535\pi$
$307$			1.07950e8i			0.212932i	−0.994316	+	0.106466i	$0.966047\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.09487e7	−	1.40207e8i	−0.152598	−	0.264307i	0.779584	−	0.626298i	$-0.215431\pi$
$311$	−8.09487e7	−	1.40207e8i	−0.152598	−	0.264307i	−0.932182	+	0.361991i	$0.882097\pi$
$312$		0			0
$313$	−2.62241e8	−	1.51405e8i	−0.483388	−	0.279084i	0.238439	−	0.971157i	$-0.423364\pi$
$313$	−2.62241e8	−	1.51405e8i	−0.483388	−	0.279084i	−0.721827	+	0.692073i	$0.756697\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.55921e8	−	2.05491e8i	−0.627548	−	0.362315i	0.152254	−	0.988341i	$-0.451347\pi$
$317$	−3.55921e8	−	2.05491e8i	−0.627548	−	0.362315i	−0.779802	+	0.626027i	$0.784680\pi$
$318$		0			0
$319$	−5.68641e6	−	9.84915e6i	−0.00980778	−	0.0169876i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.09503e9i			1.80807i
$324$		0			0
$325$	1.25833e8	−	7.26496e7i	0.203330	−	0.117393i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.55391e8	+	3.72050e7i	−1.32428	+	0.0575991i
$330$		0			0
$331$	2.90842e8	−	5.03754e8i	0.440818	−	0.763520i	−0.556932	−	0.830558i	$-0.688022\pi$
$331$	2.90842e8	−	5.03754e8i	0.440818	−	0.763520i	0.997750	+	0.0670382i	$0.0213549\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.35262e7			−0.0632548
$336$		0			0
$337$	9.16894e8			1.30501			0.652506	−	0.757784i	$-0.273718\pi$
$337$	9.16894e8			1.30501			0.652506	+	0.757784i	$0.273718\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.82019e8	+	3.15266e8i	−0.248586	+	0.430563i
$342$		0			0
$343$	−7.41014e8	+	9.71811e7i	−0.991510	+	0.130033i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.92103e8	−	5.15056e8i	1.14620	−	0.661761i	0.198245	−	0.980153i	$-0.436476\pi$
$347$	8.92103e8	−	5.15056e8i	1.14620	−	0.661761i	0.947959	+	0.318391i	$0.103143\pi$
$348$		0			0
$349$			1.08302e9i			1.36379i	0.731452	+	0.681893i	$0.238843\pi$
$349$			1.08302e9i			1.36379i	−0.731452	+	0.681893i	$0.761157\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.38235e8	−	2.39430e8i	−0.167266	−	0.289713i	0.770192	−	0.637812i	$-0.220160\pi$
$353$	−1.38235e8	−	2.39430e8i	−0.167266	−	0.289713i	−0.937458	+	0.348100i	$0.886827\pi$
$354$		0			0
$355$	−8.80959e7	−	5.08622e7i	−0.104510	−	0.0603387i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.58248e8	−	9.13644e7i	−0.180512	−	0.104219i	0.407021	−	0.913419i	$-0.366568\pi$
$359$	−1.58248e8	−	9.13644e7i	−0.180512	−	0.104219i	−0.587533	+	0.809200i	$0.699901\pi$
$360$		0			0
$361$	6.08694e7	+	1.05429e8i	0.0680964	+	0.117946i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	9.74291e7i		−	0.104873i
$366$		0			0
$367$	2.08340e8	−	1.20285e8i	0.220010	−	0.127023i	−0.385945	−	0.922522i	$-0.626125\pi$
$367$	2.08340e8	−	1.20285e8i	0.220010	−	0.127023i	0.605955	+	0.795499i	$0.292791\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.65461e9	+	8.61685e8i	1.68224	+	0.876072i
$372$		0			0
$373$	2.45693e8	−	4.25552e8i	0.245138	−	0.424592i	−0.717032	−	0.697040i	$-0.754500\pi$
$373$	2.45693e8	−	4.25552e8i	0.245138	−	0.424592i	0.962170	+	0.272448i	$0.0878333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.61654e6			−0.00732088
$378$		0			0
$379$	2.08396e8			0.196631			0.0983157	−	0.995155i	$-0.468655\pi$
$379$	2.08396e8			0.196631			0.0983157	+	0.995155i	$0.468655\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.53213e7	+	2.65373e7i	−0.0139348	+	0.0241358i	−0.872909	−	0.487884i	$-0.837769\pi$
$383$	−1.53213e7	+	2.65373e7i	−0.0139348	+	0.0241358i	0.858974	+	0.512019i	$0.171102\pi$
$384$		0			0
$385$	9.42833e7	−	6.00431e7i	0.0842020	−	0.0536229i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.45940e8	+	4.88404e8i	−0.728645	+	0.420683i	−0.817926	−	0.575323i	$-0.804876\pi$
$389$	−8.45940e8	+	4.88404e8i	−0.728645	+	0.420683i	0.0892814	+	0.996006i	$0.471543\pi$
$390$		0			0
$391$		−	1.42079e9i		−	1.20201i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.96898e7	+	3.41037e7i	0.0160750	+	0.0278428i
$396$		0			0
$397$	−1.12127e9	−	6.47367e8i	−0.899383	−	0.519259i	−0.0223827	−	0.999749i	$-0.507125\pi$
$397$	−1.12127e9	−	6.47367e8i	−0.899383	−	0.519259i	−0.877000	+	0.480491i	$0.840459\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.33786e8	−	1.92711e8i	−0.258501	−	0.149246i	0.365150	−	0.930949i	$-0.381018\pi$
$401$	−3.33786e8	−	1.92711e8i	−0.258501	−	0.149246i	−0.623651	+	0.781703i	$0.714351\pi$
$402$		0			0
$403$	1.21901e8	+	2.11138e8i	0.0927766	+	0.160694i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.38596e9i		−	1.01899i
$408$		0			0
$409$	−9.82181e8	+	5.67063e8i	−0.709839	+	0.409826i	−0.811001	−	0.585044i	$-0.801077\pi$
$409$	−9.82181e8	+	5.67063e8i	−0.709839	+	0.409826i	0.101162	+	0.994870i	$0.467744\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.67923e8	−	8.98507e8i	0.326850	−	0.627619i
$414$		0			0
$415$	1.00543e8	−	1.74145e8i	0.0690530	−	0.119603i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.39165e9			0.924233			0.462116	−	0.886819i	$-0.347090\pi$
$419$	1.39165e9			0.924233			0.462116	+	0.886819i	$0.347090\pi$
$420$		0			0
$421$	−2.39488e9			−1.56421			−0.782107	−	0.623145i	$-0.785855\pi$
$421$	−2.39488e9			−1.56421			−0.782107	+	0.623145i	$0.785855\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.31001e9	−	2.26901e9i	0.827780	−	1.43376i
$426$		0			0
$427$	−1.20544e8	−	2.77147e9i	−0.0749289	−	1.72271i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.66289e9	−	1.53742e9i	1.60208	−	0.924959i	0.611004	−	0.791627i	$-0.290766\pi$
$431$	2.66289e9	−	1.53742e9i	1.60208	−	0.924959i	0.991071	−	0.133332i	$-0.0425676\pi$
$432$		0			0
$433$			1.66956e9i			0.988315i	0.869372	+	0.494158i	$0.164523\pi$
$433$			1.66956e9i			0.988315i	−0.869372	+	0.494158i	$0.835477\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.58873e8	−	1.14120e9i	−0.377673	−	0.654150i
$438$		0			0
$439$	5.41459e8	+	3.12611e8i	0.305450	+	0.176351i	0.644888	−	0.764277i	$-0.276904\pi$
$439$	5.41459e8	+	3.12611e8i	0.305450	+	0.176351i	−0.339439	+	0.940628i	$0.610237\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.19878e9	+	6.92117e8i	0.655130	+	0.378239i	0.790419	−	0.612567i	$-0.209863\pi$
$443$	1.19878e9	+	6.92117e8i	0.655130	+	0.378239i	−0.135289	+	0.990806i	$0.543196\pi$
$444$		0			0
$445$	−2.15301e8	−	3.72912e8i	−0.115821	−	0.200607i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.09094e9i			0.568772i	0.958710	+	0.284386i	$0.0917897\pi$
$449$			1.09094e9i			0.568772i	−0.958710	+	0.284386i	$0.908210\pi$
$450$		0			0
$451$	−2.95315e7	+	1.70500e7i	−0.0151589	+	0.00875198i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.25293e6	−	7.47892e7i	−0.00161896	−	0.0372219i
$456$		0			0
$457$	3.61198e8	−	6.25613e8i	0.177026	−	0.306619i	−0.763834	−	0.645412i	$-0.776686\pi$
$457$	3.61198e8	−	6.25613e8i	0.177026	−	0.306619i	0.940861	+	0.338794i	$0.110019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.99228e9			−0.947103			−0.473552	−	0.880766i	$-0.657028\pi$
$461$	−1.99228e9			−0.947103			−0.473552	+	0.880766i	$0.657028\pi$
$462$		0			0
$463$	1.28043e9			0.599547			0.299774	−	0.954010i	$-0.403089\pi$
$463$	1.28043e9			0.599547			0.299774	+	0.954010i	$0.403089\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.69659e9	−	2.93858e9i	0.770847	−	1.33515i	−0.166252	−	0.986083i	$-0.553167\pi$
$467$	1.69659e9	−	2.93858e9i	0.770847	−	1.33515i	0.937099	−	0.349063i	$-0.113500\pi$
$468$		0			0
$469$	4.21453e8	−	8.09275e8i	0.188644	−	0.362236i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.96361e8	−	4.59779e8i	0.346016	−	0.199772i
$474$		0			0
$475$		−	2.43002e9i		−	1.04035i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.14510e9	+	3.71542e9i	0.891812	+	1.54466i	0.837702	+	0.546128i	$0.183899\pi$
$479$	2.14510e9	+	3.71542e9i	0.891812	+	1.54466i	0.0541102	+	0.998535i	$0.482768\pi$
$480$		0			0
$481$	−8.03841e8	−	4.64098e8i	−0.329354	−	0.190152i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.95556e8	−	1.70639e8i	−0.117637	−	0.0679177i
$486$		0			0
$487$	−2.28287e9	−	3.95405e9i	−0.895633	−	1.55128i	−0.833019	−	0.553244i	$-0.813390\pi$
$487$	−2.28287e9	−	3.95405e9i	−0.895633	−	1.55128i	−0.0626142	−	0.998038i	$-0.519944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.37398e9i			0.905089i	0.891742	+	0.452545i	$0.149484\pi$
$491$			2.37398e9i			0.905089i	−0.891742	+	0.452545i	$0.850516\pi$
$492$		0			0
$493$	−1.18941e8	+	6.86706e7i	−0.0447061	+	0.0258111i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.79868e9	−	1.14547e9i	0.657215	−	0.418539i
$498$		0			0
$499$	−1.51572e9	+	2.62530e9i	−0.546092	+	0.945859i	0.452445	+	0.891792i	$0.350552\pi$
$499$	−1.51572e9	+	2.62530e9i	−0.546092	+	0.945859i	−0.998537	+	0.0540667i	$0.982782\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.14489e8			−0.145219			−0.0726097	−	0.997360i	$-0.523133\pi$
$503$	−4.14489e8			−0.145219			−0.0726097	+	0.997360i	$0.523133\pi$
$504$		0			0
$505$	1.67058e8			0.0577227
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.25908e9	−	2.18079e9i	0.423196	−	0.732996i	−0.573054	−	0.819517i	$-0.694242\pi$
$509$	1.25908e9	−	2.18079e9i	0.423196	−	0.732996i	0.996250	+	0.0865209i	$0.0275749\pi$
$510$		0			0
$511$	1.81149e9	+	9.43382e8i	0.600567	+	0.312762i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.89905e8	+	1.09642e8i	−0.0612648	+	0.0353712i
$516$		0			0
$517$			2.68448e9i			0.854364i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.49988e9	−	4.32992e9i	−0.774439	−	1.34137i	−0.935109	−	0.354360i	$-0.884699\pi$
$521$	−2.49988e9	−	4.32992e9i	−0.774439	−	1.34137i	0.160670	−	0.987008i	$-0.448635\pi$
$522$		0			0
$523$	−2.84190e9	−	1.64077e9i	−0.868667	−	0.501525i	−0.00176184	−	0.999998i	$-0.500561\pi$
$523$	−2.84190e9	−	1.64077e9i	−0.868667	−	0.501525i	−0.866905	+	0.498473i	$0.833894\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.80724e9	+	2.19811e9i	1.13311	+	0.654202i
$528$		0			0
$529$	−8.47531e8	−	1.46797e9i	−0.248921	−	0.431143i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.28373e7i			0.00653279i
$534$		0			0
$535$	6.98588e8	−	4.03330e8i	0.197235	−	0.113873i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.03451e8	+	2.33438e9i	0.0559628	+	0.642112i
$540$		0			0
$541$	−4.90462e8	+	8.49506e8i	−0.133173	+	0.230662i	−0.924898	−	0.380215i	$-0.875850\pi$
$541$	−4.90462e8	+	8.49506e8i	−0.133173	+	0.230662i	0.791725	+	0.610877i	$0.209183\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.40739e8			−0.222471
$546$		0			0
$547$	3.59461e9			0.939065			0.469533	−	0.882915i	$-0.344422\pi$
$547$	3.59461e9			0.939065			0.469533	+	0.882915i	$0.344422\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.36904e7	+	1.10315e8i	−0.0162197	+	0.0280934i
$552$		0			0
$553$	−8.24737e8	+	3.58717e7i	−0.207385	+	0.00902016i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.04811e9	−	1.18248e9i	0.502181	−	0.289934i	−0.227433	−	0.973794i	$-0.573033\pi$
$557$	2.04811e9	−	1.18248e9i	0.502181	−	0.289934i	0.729614	+	0.683859i	$0.239700\pi$
$558$		0			0
$559$		−	6.15841e8i		−	0.149117i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.46203e9	−	5.99641e9i	−0.817619	−	1.41616i	−0.907432	−	0.420199i	$-0.861960\pi$
$563$	−3.46203e9	−	5.99641e9i	−0.817619	−	1.41616i	0.0898129	−	0.995959i	$-0.471373\pi$
$564$		0			0
$565$	−5.68209e8	−	3.28056e8i	−0.132537	−	0.0765205i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.69528e9	−	3.86552e9i	−1.52362	−	0.879661i	−0.999609	−	0.0279480i	$-0.991103\pi$
$569$	−6.69528e9	−	3.86552e9i	−1.52362	−	0.879661i	−0.524008	−	0.851713i	$-0.675564\pi$
$570$		0			0
$571$	1.26315e9	+	2.18784e9i	0.283941	+	0.491800i	0.972352	−	0.233521i	$-0.0750248\pi$
$571$	1.26315e9	+	2.18784e9i	0.283941	+	0.491800i	−0.688411	+	0.725321i	$0.741691\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.15292e9i			0.691633i
$576$		0			0
$577$	−4.84925e9	+	2.79972e9i	−1.05089	+	0.606734i	−0.922900	−	0.385040i	$-0.874187\pi$
$577$	−4.84925e9	+	2.79972e9i	−1.05089	+	0.606734i	−0.127995	+	0.991775i	$0.540854\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.26433e9	+	3.55558e9i	0.478986	+	0.752133i
$582$		0			0
$583$	2.92455e9	−	5.06548e9i	0.611251	−	1.05872i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.61031e9			−0.328607			−0.164304	−	0.986410i	$-0.552538\pi$
$587$	−1.61031e9			−0.328607			−0.164304	+	0.986410i	$0.552538\pi$
$588$		0			0
$589$	4.07739e9			0.822202
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.22039e9	+	5.57787e9i	−0.634186	+	1.09844i	0.352501	+	0.935811i	$0.385331\pi$
$593$	−3.22039e9	+	5.57787e9i	−0.634186	+	1.09844i	−0.986687	+	0.162631i	$0.948002\pi$
$594$		0			0
$595$	−7.25096e8	−	1.13859e9i	−0.141119	−	0.221594i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.37179e8	−	4.25611e8i	0.140146	−	0.0809131i	−0.428288	−	0.903642i	$-0.640883\pi$
$599$	7.37179e8	−	4.25611e8i	0.140146	−	0.0809131i	0.568433	+	0.822729i	$0.307550\pi$
$600$		0			0
$601$			7.56509e9i			1.42152i	0.703433	+	0.710761i	$0.251649\pi$
$601$			7.56509e9i			1.42152i	−0.703433	+	0.710761i	$0.748351\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.46568e8	+	4.27069e8i	0.0452682	+	0.0784068i
$606$		0			0
$607$	−1.78625e9	−	1.03129e9i	−0.324178	−	0.187164i	0.329076	−	0.944304i	$-0.393263\pi$
$607$	−1.78625e9	−	1.03129e9i	−0.324178	−	0.187164i	−0.653253	+	0.757140i	$0.726596\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.55697e9	+	8.98917e8i	0.276144	+	0.159432i
$612$		0			0
$613$	3.72749e9	+	6.45620e9i	0.653589	+	1.13205i	0.982246	+	0.187600i	$0.0600708\pi$
$613$	3.72749e9	+	6.45620e9i	0.653589	+	1.13205i	−0.328657	+	0.944450i	$0.606596\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.52998e9i			1.11922i	0.828757	+	0.559608i	$0.189048\pi$
$617$			6.52998e9i			1.11922i	−0.828757	+	0.559608i	$0.810952\pi$
$618$		0			0
$619$	4.56575e9	−	2.63603e9i	0.773739	−	0.446718i	−0.0604678	−	0.998170i	$-0.519259\pi$
$619$	4.56575e9	−	2.63603e9i	0.773739	−	0.446718i	0.834207	+	0.551452i	$0.185926\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.01821e9	−	3.92245e8i	1.49421	−	0.0649904i
$624$		0			0
$625$	−2.83390e9	+	4.90846e9i	−0.464306	+	0.804202i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.67372e10			−2.68167
$630$		0			0
$631$	5.93765e9			0.940831			0.470416	−	0.882445i	$-0.344104\pi$
$631$	5.93765e9			0.940831			0.470416	+	0.882445i	$0.344104\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.82311e8	+	1.00859e9i	−0.0902499	+	0.156317i
$636$		0			0
$637$	1.42204e9	+	6.63684e8i	0.217984	+	0.101736i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.36252e9	+	5.40545e9i	−1.40407	+	0.810642i	−0.994808	−	0.101774i	$-0.967548\pi$
$641$	−9.36252e9	+	5.40545e9i	−1.40407	+	0.810642i	−0.409265	+	0.912416i	$0.634215\pi$
$642$		0			0
$643$			8.11643e9i			1.20400i	0.798496	+	0.602000i	$0.205629\pi$
$643$			8.11643e9i			1.20400i	−0.798496	+	0.602000i	$0.794371\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.21229e9	+	9.02796e9i	0.756596	+	1.31046i	0.944577	+	0.328290i	$0.106472\pi$
$647$	5.21229e9	+	9.02796e9i	0.756596	+	1.31046i	−0.187981	+	0.982173i	$0.560194\pi$
$648$		0			0
$649$	−2.75072e9	−	1.58813e9i	−0.394993	−	0.228049i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.58422e9	−	4.37875e9i	−1.06590	−	0.615395i	−0.138839	−	0.990315i	$-0.544337\pi$
$653$	−7.58422e9	−	4.37875e9i	−1.06590	−	0.615395i	−0.927057	+	0.374920i	$0.877670\pi$
$654$		0			0
$655$	−6.20300e8	−	1.07439e9i	−0.0862496	−	0.149389i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			8.30490e9i			1.13041i	0.824951	+	0.565204i	$0.191203\pi$
$659$			8.30490e9i			1.13041i	−0.824951	+	0.565204i	$0.808797\pi$
$660$		0			0
$661$	4.95141e8	−	2.85870e8i	0.0666843	−	0.0385002i	−0.466287	−	0.884633i	$-0.654409\pi$
$661$	4.95141e8	−	2.85870e8i	0.0666843	−	0.0385002i	0.532971	+	0.846133i	$0.321075\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.11042e9	−	5.78281e8i	−0.146424	−	0.0762541i
$666$		0			0
$667$	8.26377e7	−	1.43133e8i	0.0107830	−	0.0186766i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.69772e9			−1.11142
$672$		0			0
$673$	9.17478e9			1.16023			0.580114	−	0.814536i	$-0.303008\pi$
$673$	9.17478e9			1.16023			0.580114	+	0.814536i	$0.303008\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.13865e9	−	7.16835e9i	0.512623	−	0.887890i	−0.487269	−	0.873252i	$-0.662007\pi$
$677$	4.13865e9	−	7.16835e9i	0.512623	−	0.887890i	0.999893	−	0.0146381i	$-0.00465962\pi$
$678$		0			0
$679$	6.03446e9	−	3.84297e9i	0.739766	−	0.471110i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.72121e8	+	4.45784e8i	−0.0927285	+	0.0535368i	−0.545647	−	0.838015i	$-0.683716\pi$
$683$	−7.72121e8	+	4.45784e8i	−0.0927285	+	0.0535368i	0.452919	+	0.891552i	$0.350383\pi$
$684$		0			0
$685$		−	2.07587e9i		−	0.246764i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.95862e9	−	3.39242e9i	−0.228130	−	0.395132i
$690$		0			0
$691$	4.95007e8	+	2.85792e8i	0.0570739	+	0.0329516i	0.528265	−	0.849079i	$-0.322843\pi$
$691$	4.95007e8	+	2.85792e8i	0.0570739	+	0.0329516i	−0.471192	+	0.882031i	$0.656176\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.05534e9	−	6.09299e8i	−0.119246	−	0.0688468i
$696$		0			0
$697$	2.05900e8	+	3.56630e8i	0.0230326	+	0.0398935i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5.60772e9i			0.614855i	0.951571	+	0.307428i	$0.0994682\pi$
$701$			5.60772e9i			0.614855i	−0.951571	+	0.307428i	$0.900532\pi$
$702$		0			0
$703$	−1.34436e10	+	7.76168e9i	−1.45939	+	0.842581i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.61758e9	+	3.10608e9i	−0.172146	+	0.330555i
$708$		0			0
$709$	−4.99219e9	+	8.64672e9i	−0.526052	+	0.911149i	0.473487	+	0.880801i	$0.342995\pi$
$709$	−4.99219e9	+	8.64672e9i	−0.526052	+	0.911149i	−0.999539	+	0.0303486i	$0.990338\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.29038e9			−0.546605
$714$		0			0
$715$	−2.34711e8			−0.0240139
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.47450e9	−	1.12142e10i	0.649613	−	1.12516i	−0.333602	−	0.942714i	$-0.608264\pi$
$719$	6.47450e9	−	1.12142e10i	0.649613	−	1.12516i	0.983215	−	0.182449i	$-0.0584025\pi$
$720$		0			0
$721$	−1.99750e8	−	4.59250e9i	−0.0198478	−	0.456327i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.63947e8	−	1.52390e8i	0.0257237	−	0.0148516i
$726$		0			0
$727$			1.58437e10i			1.52927i	0.644461	+	0.764637i	$0.277082\pi$
$727$			1.58437e10i			1.52927i	−0.644461	+	0.764637i	$0.722918\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.55241e9	−	9.61706e9i	−0.525740	−	0.910608i
$732$		0			0
$733$	1.19719e10	+	6.91198e9i	1.12279	+	0.648244i	0.942112	−	0.335298i	$-0.108837\pi$
$733$	1.19719e10	+	6.91198e9i	1.12279	+	0.648244i	0.180680	+	0.983542i	$0.442170\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.47754e9	−	1.43041e9i	−0.227973	−	0.131621i
$738$		0			0
$739$	−2.93942e9	−	5.09122e9i	−0.267921	−	0.464052i	0.700404	−	0.713747i	$-0.253003\pi$
$739$	−2.93942e9	−	5.09122e9i	−0.267921	−	0.464052i	−0.968325	+	0.249694i	$0.919670\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.16143e10i			1.93321i	0.256268	+	0.966606i	$0.417507\pi$
$743$			2.16143e10i			1.93321i	−0.256268	+	0.966606i	$0.582493\pi$
$744$		0			0
$745$	−1.28293e8	+	7.40698e7i	−0.0113672	+	0.00656288i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.34804e8	+	1.68941e10i	0.0638977	+	1.46909i
$750$		0			0
$751$	2.47556e9	−	4.28779e9i	0.213272	−	0.369397i	−0.739465	−	0.673195i	$-0.764921\pi$
$751$	2.47556e9	−	4.28779e9i	0.213272	−	0.369397i	0.952737	+	0.303798i	$0.0982547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.27582e9			−0.107888
$756$		0			0
$757$	−8.29744e9			−0.695198			−0.347599	−	0.937643i	$-0.613003\pi$
$757$	−8.29744e9			−0.695198			−0.347599	+	0.937643i	$0.613003\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.06968e9	−	1.05130e10i	0.499252	−	0.864729i	−0.500748	−	0.865593i	$-0.666942\pi$
$761$	6.06968e9	−	1.05130e10i	0.499252	−	0.864729i	1.00000		0.000863849i	$0.000274972\pi$
$762$		0			0
$763$	8.14067e9	−	1.56317e10i	0.663474	−	1.27401i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.84219e9	+	1.06359e9i	−0.147418	+	0.0851120i
$768$		0			0
$769$		−	2.20938e9i		−	0.175198i	−0.996156	−	0.0875989i	$-0.972081\pi$
$769$		−	2.20938e9i		−	0.175198i	0.996156	−	0.0875989i	$-0.0279194\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.64144e9	−	2.84306e9i	−0.127819	−	0.221390i	0.795012	−	0.606594i	$-0.207464\pi$
$773$	−1.64144e9	−	2.84306e9i	−0.127819	−	0.221390i	−0.922831	+	0.385204i	$0.874131\pi$
$774$		0			0
$775$	−8.44879e9	−	4.87791e9i	−0.651987	−	0.376425i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.30766e8	+	1.90968e8i	0.0250691	+	0.0144737i
$780$		0			0
$781$	−3.34299e9	−	5.79022e9i	−0.251105	−	0.434927i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.88343e9i			0.212748i
$786$		0			0
$787$	3.49909e9	−	2.02020e9i	0.255884	−	0.147735i	−0.366572	−	0.930390i	$-0.619468\pi$
$787$	3.49909e9	−	2.02020e9i	0.255884	−	0.147735i	0.622456	+	0.782655i	$0.286135\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.16013e10	−	7.38814e9i	0.833469	−	0.530784i
$792$		0			0
$793$	−2.91250e9	+	5.04459e9i	−0.207400	+	0.359228i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.09339e10			−1.46469			−0.732346	−	0.680933i	$-0.761575\pi$
$797$	−2.09339e10			−1.46469			−0.732346	+	0.680933i	$0.761575\pi$
$798$		0			0
$799$	3.24185e10			2.24843
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.20183e9	−	5.54573e9i	0.218220	−	0.377968i
$804$		0			0
$805$	1.44076e9	+	7.50315e8i	0.0973431	+	0.0506942i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.25479e10	−	1.30181e10i	1.49722	−	0.864423i	0.497230	−	0.867619i	$-0.334351\pi$
$809$	2.25479e10	−	1.30181e10i	1.49722	−	0.864423i	0.999995	+	0.00319580i	$0.00101726\pi$
$810$		0			0
$811$			1.96611e10i			1.29430i	0.762364	+	0.647149i	$0.224039\pi$
$811$			1.96611e10i			1.29430i	−0.762364	+	0.647149i	$0.775961\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.93956e9	−	3.35942e9i	−0.125502	−	0.217377i
$816$		0			0
$817$	−8.91960e9	−	5.14973e9i	−0.572227	−	0.330375i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.79598e10	+	1.03691e10i	1.13266	+	0.653942i	0.944603	−	0.328216i	$-0.106447\pi$
$821$	1.79598e10	+	1.03691e10i	1.13266	+	0.653942i	0.188058	+	0.982158i	$0.439781\pi$
$822$		0			0
$823$	−1.14202e10	−	1.97803e10i	−0.714123	−	1.23690i	−0.963297	−	0.268438i	$-0.913492\pi$
$823$	−1.14202e10	−	1.97803e10i	−0.714123	−	1.23690i	0.249174	−	0.968459i	$-0.419841\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.34616e10i			0.827612i	0.910365	+	0.413806i	$0.135801\pi$
$827$			1.34616e10i			0.827612i	−0.910365	+	0.413806i	$0.864199\pi$
$828$		0			0
$829$	−1.51871e10	+	8.76825e9i	−0.925833	+	0.534530i	−0.885491	−	0.464656i	$-0.846178\pi$
$829$	−1.51871e10	+	8.76825e9i	−0.925833	+	0.534530i	−0.0403417	+	0.999186i	$0.512845\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.81906e10	−	2.45693e9i	1.68984	−	0.147277i
$834$		0			0
$835$	2.12932e9	−	3.68809e9i	0.126572	−	0.219229i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.28377e9			−0.484240			−0.242120	−	0.970246i	$-0.577843\pi$
$839$	−8.28377e9			−0.484240			−0.242120	+	0.970246i	$0.577843\pi$
$840$		0			0
$841$	1.72339e10			0.999074
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.27960e9	−	2.21633e9i	0.0729585	−	0.126368i
$846$		0			0
$847$	−1.03279e10	+	4.49208e8i	−0.584009	+	0.0254013i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.74430e10	−	1.00707e10i	0.970213	−	0.560153i
$852$		0			0
$853$		−	1.38127e8i		−	0.00762006i	−0.999993	−	0.00381003i	$-0.998787\pi$
$853$		−	1.38127e8i		−	0.00762006i	0.999993	−	0.00381003i	$-0.00121277\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.27091e10	−	2.20128e10i	−0.689734	−	1.19465i	−0.971924	−	0.235296i	$-0.924394\pi$
$857$	−1.27091e10	−	2.20128e10i	−0.689734	−	1.19465i	0.282190	−	0.959359i	$-0.408939\pi$
$858$		0			0
$859$	−1.93666e10	−	1.11813e10i	−1.04250	−	0.601889i	−0.121961	−	0.992535i	$-0.538918\pi$
$859$	−1.93666e10	−	1.11813e10i	−1.04250	−	0.601889i	−0.920541	+	0.390646i	$0.872252\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.93553e10	−	1.11748e10i	−1.02509	−	0.591835i	−0.109515	−	0.993985i	$-0.534930\pi$
$863$	−1.93553e10	−	1.11748e10i	−1.02509	−	0.591835i	−0.915574	+	0.402150i	$0.868263\pi$
$864$		0			0
$865$	−5.86385e8	−	1.01565e9i	−0.0308054	−	0.0533565i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2.58828e9i			0.133796i
$870$		0			0
$871$	−1.65924e9	+	9.57965e8i	−0.0850837	+	0.0491231i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.25773e9	+	5.11548e9i	0.164394	+	0.258142i
$876$		0			0
$877$	1.79827e10	−	3.11469e10i	0.900235	−	1.55925i	0.0730458	−	0.997329i	$-0.476728\pi$
$877$	1.79827e10	−	3.11469e10i	0.900235	−	1.55925i	0.827189	−	0.561924i	$-0.189939\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.26706e9			−0.0624283			−0.0312142	−	0.999513i	$-0.509937\pi$
$881$	−1.26706e9			−0.0624283			−0.0312142	+	0.999513i	$0.509937\pi$
$882$		0			0
$883$	−3.76489e10			−1.84030			−0.920152	−	0.391561i	$-0.871935\pi$
$883$	−3.76489e10			−1.84030			−0.920152	+	0.391561i	$0.871935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.64364e10	−	2.84687e10i	0.790813	−	1.36973i	−0.134651	−	0.990893i	$-0.542991\pi$
$887$	1.64364e10	−	2.84687e10i	0.790813	−	1.36973i	0.925464	−	0.378835i	$-0.123675\pi$
$888$		0			0
$889$	−1.31142e10	−	2.05928e10i	−0.626018	−	0.983011i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.60391e10	−	1.50337e10i	1.22362	−	0.706456i
$894$		0			0
$895$		−	2.04710e9i		−	0.0954462i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.55699e8	+	4.42883e8i	0.0117373	+	0.0203297i
$900$		0			0
$901$	−6.11720e10	−	3.53177e10i	−2.78622	−	1.60863i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.49568e8	−	8.63531e7i	−0.00670762	−	0.00387265i
$906$		0			0
$907$	1.72901e10	+	2.99473e10i	0.769434	+	1.33270i	0.937870	+	0.346986i	$0.112795\pi$
$907$	1.72901e10	+	2.99473e10i	0.769434	+	1.33270i	−0.168436	+	0.985713i	$0.553872\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.19186e9i			0.0960502i	0.998846	+	0.0480251i	$0.0152927\pi$
$911$			2.19186e9i			0.0960502i	−0.998846	+	0.0480251i	$0.984707\pi$
$912$		0			0
$913$	1.14459e10	−	6.60831e9i	0.497741	−	0.287371i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.59822e10	−	1.13009e9i	1.11271	−	0.0483972i
$918$		0			0
$919$	−1.51503e10	+	2.62411e10i	−0.643897	+	1.11526i	0.340658	+	0.940187i	$0.389350\pi$
$919$	−1.51503e10	+	2.62411e10i	−0.643897	+	1.11526i	−0.984555	+	0.175075i	$0.943983\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.47769e9			−0.187434
$924$		0			0
$925$	3.71422e10			1.54302
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.64848e10	−	2.85525e10i	0.674573	−	1.16839i	−0.302021	−	0.953301i	$-0.597661\pi$
$929$	1.64848e10	−	2.85525e10i	0.674573	−	1.16839i	0.976594	−	0.215093i	$-0.0690055\pi$
$930$		0			0
$931$	2.15038e10	−	1.50465e10i	0.873356	−	0.611099i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.66529e9	+	2.11615e9i	−0.146645	+	0.0846655i
$936$		0			0
$937$			1.83470e10i			0.728580i	0.931286	+	0.364290i	$0.118688\pi$
$937$			1.83470e10i			0.728580i	−0.931286	+	0.364290i	$0.881312\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.38257e9	+	1.10549e10i	0.249708	+	0.432507i	0.963445	−	0.267907i	$-0.0863321\pi$
$941$	6.38257e9	+	1.10549e10i	0.249708	+	0.432507i	−0.713737	+	0.700414i	$0.752999\pi$
$942$		0			0
$943$	−4.29165e8	−	2.47779e8i	−0.0166661	−	0.00962217i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.95416e9	−	2.86029e9i	−0.189559	−	0.109442i	0.402217	−	0.915544i	$-0.368240\pi$
$947$	−4.95416e9	−	2.86029e9i	−0.189559	−	0.109442i	−0.591776	+	0.806102i	$0.701573\pi$
$948$		0			0
$949$	−2.14431e9	−	3.71406e9i	−0.0814434	−	0.141064i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.08823e10i			1.15580i	0.816106	+	0.577902i	$0.196128\pi$
$953$			3.08823e10i			1.15580i	−0.816106	+	0.577902i	$0.803872\pi$
$954$		0			0
$955$	−2.34833e9	+	1.35581e9i	−0.0872462	+	0.0503716i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.85963e10	+	2.01001e10i	1.41312	+	0.735924i
$960$		0			0
$961$	−5.57152e9	+	9.65015e9i	−0.202508	+	0.350754i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.48304e8			0.0339705
$966$		0			0
$967$	2.58173e10			0.918161			0.459080	−	0.888395i	$-0.348179\pi$
$967$	2.58173e10			0.918161			0.459080	+	0.888395i	$0.348179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.25087e9	−	7.36273e9i	0.149008	−	0.258090i	−0.781853	−	0.623463i	$-0.785725\pi$
$971$	4.25087e9	−	7.36273e9i	0.149008	−	0.258090i	0.930861	+	0.365373i	$0.119059\pi$
$972$		0			0
$973$	2.15472e10	−	1.37220e10i	0.749886	−	0.477555i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.42421e10	+	1.39962e10i	−0.831646	+	0.480151i	−0.854416	−	0.519590i	$-0.826085\pi$
$977$	−2.42421e10	+	1.39962e10i	−0.831646	+	0.480151i	0.0227699	+	0.999741i	$0.492751\pi$
$978$		0			0
$979$		−	2.83019e10i		−	0.963999i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.86631e10	−	3.23255e10i	−0.626682	−	1.08545i	−0.988213	−	0.153085i	$-0.951079\pi$
$983$	−1.86631e10	−	3.23255e10i	−0.626682	−	1.08545i	0.361531	−	0.932360i	$-0.382254\pi$
$984$		0			0
$985$	−2.07823e9	−	1.19987e9i	−0.0692894	−	0.0400043i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.15731e10	+	6.68173e9i	0.380419	+	0.219635i
$990$		0			0
$991$	−2.79170e10	−	4.83536e10i	−0.911194	−	1.57823i	−0.812381	−	0.583128i	$-0.801829\pi$
$991$	−2.79170e10	−	4.83536e10i	−0.911194	−	1.57823i	−0.0988131	−	0.995106i	$-0.531505\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.28982e9i		−	0.138057i
$996$		0			0
$997$	1.10510e10	−	6.38028e9i	0.353157	−	0.203895i	−0.312918	−	0.949780i	$-0.601307\pi$
$997$	1.10510e10	−	6.38028e9i	0.353157	−	0.203895i	0.666075	+	0.745885i	$0.267973\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.t.a.89.10	yes	36
3.2	odd	2	inner	252.8.t.a.89.9	yes	36
7.3	odd	6	inner	252.8.t.a.17.9	✓	36
21.17	even	6	inner	252.8.t.a.17.10	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.t.a.17.9	✓	36	7.3	odd	6	inner
252.8.t.a.17.10	yes	36	21.17	even	6	inner
252.8.t.a.89.9	yes	36	3.2	odd	2	inner
252.8.t.a.89.10	yes	36	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(21.6450 - 37.4903i) q^{5} +(487.468 + 765.452i) q^{7} +(2464.10 - 1422.65i) q^{11} -1905.54i q^{13} +(-17180.3 - 29757.1i) q^{17} +(-27599.1 - 15934.3i) q^{19} +(35809.5 + 20674.6i) q^{23} +(38125.5 + 66035.3i) q^{25} -3997.05i q^{29} +(-110802. + 63971.8i) q^{31} +(39248.3 - 1707.10i) q^{35} +(243552. - 421845. i) q^{37} -11984.7 q^{41} +323185. q^{43} +(-471739. + 817076. i) q^{47} +(-348292. + 746268. i) q^{49} +(1.78030e6 - 1.02785e6i) q^{53} -123173. i q^{55} +(-558158. - 966758. i) q^{59} +(-2.64733e6 - 1.52844e6i) q^{61} +(-71439.2 - 41245.5i) q^{65} +(-502727. - 870748. i) q^{67} -2.34983e6i q^{71} +(1.94909e6 - 1.12531e6i) q^{73} +(2.29014e6 + 1.19266e6i) q^{77} +(-454834. + 787795. i) q^{79} +4.64507e6 q^{83} -1.48747e6 q^{85} +(4.97345e6 - 8.61427e6i) q^{89} +(1.45860e6 - 928890. i) q^{91} +(-1.19477e6 + 689798. i) q^{95} -7.88352e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2634 q^{7} - 47862 q^{19} - 360762 q^{25} + 486018 q^{31} - 972270 q^{37} + 298788 q^{43} + 1556886 q^{49} + 4324644 q^{61} - 2969562 q^{67} - 5157378 q^{73} + 7676514 q^{79} - 15214128 q^{85} - 6114678 q^{91}+O(q^{100}) \) 36 * q + 2634 * q^7 - 47862 * q^19 - 360762 * q^25 + 486018 * q^31 - 972270 * q^37 + 298788 * q^43 + 1556886 * q^49 + 4324644 * q^61 - 2969562 * q^67 - 5157378 * q^73 + 7676514 * q^79 - 15214128 * q^85 - 6114678 * q^91

Introduction

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Database

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