LMFDB - Embedded newform 252.8.t.a.17.2

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			17.2
Character	$\chi$	$=$	252.17
Dual form			252.8.t.a.89.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-219.318 - 379.869i) q^{5} +(-692.530 + 586.468i) q^{7} +(800.099 + 461.937i) q^{11} +953.385i q^{13} +(839.723 - 1454.44i) q^{17} +(-46643.8 + 26929.8i) q^{19} +(17245.2 - 9956.50i) q^{23} +(-57137.9 + 98965.8i) q^{25} +128175. i q^{29} +(-68359.9 - 39467.6i) q^{31} +(374665. + 134448. i) q^{35} +(-5749.72 - 9958.80i) q^{37} -173758. q^{41} +533517. q^{43} +(13529.2 + 23433.3i) q^{47} +(135653. - 812294. i) q^{49} +(-689588. - 398134. i) q^{53} -405244. i q^{55} +(683859. - 1.18448e6i) q^{59} +(1.73937e6 - 1.00423e6i) q^{61} +(362162. - 209094. i) q^{65} +(83680.5 - 144939. i) q^{67} +616804. i q^{71} +(-2.89674e6 - 1.67243e6i) q^{73} +(-825004. + 149327. i) q^{77} +(969407. + 1.67906e6i) q^{79} +8.54857e6 q^{83} -736664. q^{85} +(3.46905e6 + 6.00857e6i) q^{89} +(-559130. - 660248. i) q^{91} +(2.04596e7 + 1.18124e7i) q^{95} -6.45974e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

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

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
$5$	−219.318	−	379.869i	−0.784654	−	1.35906i	−0.929205	−	0.369564i	$-0.879507\pi$
$5$	−219.318	−	379.869i	−0.784654	−	1.35906i	0.144551	−	0.989497i	$-0.453826\pi$
$6$		0			0
$7$	−692.530	+	586.468i	−0.763125	+	0.646251i
$7$	−692.530	+	586.468i	−0.763125	+	0.646251i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	800.099	+	461.937i	0.181246	+	0.104643i	0.587878	−	0.808950i	$-0.299963\pi$
$11$	800.099	+	461.937i	0.181246	+	0.104643i	−0.406632	+	0.913592i	$0.633297\pi$
$12$		0			0
$13$			953.385i			0.120356i	0.998188	+	0.0601779i	$0.0191668\pi$
$13$			953.385i			0.120356i	−0.998188	+	0.0601779i	$0.980833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	839.723	−	1454.44i	0.0414538	−	0.0718001i	−0.844554	−	0.535470i	$-0.820134\pi$
$17$	839.723	−	1454.44i	0.0414538	−	0.0718001i	0.886008	+	0.463670i	$0.153468\pi$
$18$		0			0
$19$	−46643.8	+	26929.8i	−1.56011	+	0.900732i	−0.562870	+	0.826545i	$0.690303\pi$
$19$	−46643.8	+	26929.8i	−1.56011	+	0.900732i	−0.997244	+	0.0741870i	$0.976364\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17245.2	−	9956.50i	0.295542	−	0.170631i	−0.344896	−	0.938641i	$-0.612086\pi$
$23$	17245.2	−	9956.50i	0.295542	−	0.170631i	0.640439	+	0.768009i	$0.278753\pi$
$24$		0			0
$25$	−57137.9	+	98965.8i	−0.731365	+	1.26676i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			128175.i			0.975911i	0.872869	+	0.487956i	$0.162257\pi$
$29$			128175.i			0.975911i	−0.872869	+	0.487956i	$0.837743\pi$
$30$		0			0
$31$	−68359.9	−	39467.6i	−0.412131	−	0.237944i	0.279574	−	0.960124i	$-0.409807\pi$
$31$	−68359.9	−	39467.6i	−0.412131	−	0.237944i	−0.691705	+	0.722180i	$0.743140\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	374665.	+	134448.i	1.47708	+	0.530049i
$36$		0			0
$37$	−5749.72	−	9958.80i	−0.0186612	−	0.0323222i	0.856544	−	0.516074i	$-0.172607\pi$
$37$	−5749.72	−	9958.80i	−0.0186612	−	0.0323222i	−0.875205	+	0.483752i	$0.839274\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−173758.			−0.393732			−0.196866	−	0.980430i	$-0.563076\pi$
$41$	−173758.			−0.393732			−0.196866	+	0.980430i	$0.563076\pi$
$42$		0			0
$43$	533517.			1.02331			0.511657	−	0.859190i	$-0.329032\pi$
$43$	533517.			1.02331			0.511657	+	0.859190i	$0.329032\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13529.2	+	23433.3i	0.0190077	+	0.0329223i	0.875373	−	0.483448i	$-0.160616\pi$
$47$	13529.2	+	23433.3i	0.0190077	+	0.0329223i	−0.856365	+	0.516371i	$0.827283\pi$
$48$		0			0
$49$	135653.	−	812294.i	0.164719	−	0.986341i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−689588.	−	398134.i	−0.636245	−	0.367336i	0.146922	−	0.989148i	$-0.453063\pi$
$53$	−689588.	−	398134.i	−0.636245	−	0.367336i	−0.783166	+	0.621812i	$0.786397\pi$
$54$		0			0
$55$		−	405244.i		−	0.328433i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	683859.	−	1.18448e6i	0.433495	−	0.750836i	−0.563676	−	0.825996i	$-0.690613\pi$
$59$	683859.	−	1.18448e6i	0.433495	−	0.750836i	0.997171	+	0.0751598i	$0.0239467\pi$
$60$		0			0
$61$	1.73937e6	−	1.00423e6i	0.981156	−	0.566471i	0.0785369	−	0.996911i	$-0.474975\pi$
$61$	1.73937e6	−	1.00423e6i	0.981156	−	0.566471i	0.902619	+	0.430441i	$0.141642\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	362162.	−	209094.i	0.163571	−	0.0944377i
$66$		0			0
$67$	83680.5	−	144939.i	0.0339909	−	0.0588740i	−0.848530	−	0.529148i	$-0.822512\pi$
$67$	83680.5	−	144939.i	0.0339909	−	0.0588740i	0.882520	+	0.470274i	$0.155845\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			616804.i			0.204523i	0.994758	+	0.102262i	$0.0326079\pi$
$71$			616804.i			0.204523i	−0.994758	+	0.102262i	$0.967392\pi$
$72$		0			0
$73$	−2.89674e6	−	1.67243e6i	−0.871525	−	0.503175i	−0.00367025	−	0.999993i	$-0.501168\pi$
$73$	−2.89674e6	−	1.67243e6i	−0.871525	−	0.503175i	−0.867855	+	0.496818i	$0.834502\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−825004.	+	149327.i	−0.205939	+	0.0372753i
$78$		0			0
$79$	969407.	+	1.67906e6i	0.221213	+	0.383153i	0.955177	−	0.296036i	$-0.0956650\pi$
$79$	969407.	+	1.67906e6i	0.221213	+	0.383153i	−0.733963	+	0.679189i	$0.762332\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.54857e6			1.64104			0.820521	−	0.571616i	$-0.193683\pi$
$83$	8.54857e6			1.64104			0.820521	+	0.571616i	$0.193683\pi$
$84$		0			0
$85$	−736664.			−0.130108
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.46905e6	+	6.00857e6i	0.521610	+	0.903455i	0.999684	+	0.0251353i	$0.00800166\pi$
$89$	3.46905e6	+	6.00857e6i	0.521610	+	0.903455i	−0.478074	+	0.878319i	$0.658665\pi$
$90$		0			0
$91$	−559130.	−	660248.i	−0.0777800	−	0.0918464i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.04596e7	+	1.18124e7i	2.44830	+	1.41353i
$96$		0			0
$97$		−	6.45974e6i		−	0.718644i	−0.933214	−	0.359322i	$-0.883008\pi$
$97$		−	6.45974e6i		−	0.718644i	0.933214	−	0.359322i	$-0.116992\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.19425e6	+	1.41928e7i	−0.791378	+	1.37071i	0.133735	+	0.991017i	$0.457303\pi$
$101$	−8.19425e6	+	1.41928e7i	−0.791378	+	1.37071i	−0.925114	+	0.379690i	$0.876031\pi$
$102$		0			0
$103$	3.23869e6	−	1.86986e6i	0.292038	−	0.168608i	−0.346823	−	0.937931i	$-0.612739\pi$
$103$	3.23869e6	−	1.86986e6i	0.292038	−	0.168608i	0.638860	+	0.769323i	$0.279406\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.72520e7	−	9.96042e6i	1.36143	−	0.786022i	0.371615	−	0.928387i	$-0.378804\pi$
$107$	1.72520e7	−	9.96042e6i	1.36143	−	0.786022i	0.989814	+	0.142365i	$0.0454708\pi$
$108$		0			0
$109$	3.69069e6	−	6.39247e6i	0.272970	−	0.472798i	−0.696651	−	0.717410i	$-0.745327\pi$
$109$	3.69069e6	−	6.39247e6i	0.272970	−	0.472798i	0.969621	+	0.244612i	$0.0786606\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.55547e7i		−	1.01412i	−0.861912	−	0.507058i	$-0.830733\pi$
$113$		−	1.55547e7i		−	1.01412i	0.861912	−	0.507058i	$-0.169267\pi$
$114$		0			0
$115$	−7.56433e6	−	4.36727e6i	−0.463797	−	0.267773i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	271451.	+	1.49972e6i	0.0147665	+	0.0815820i
$120$		0			0
$121$	−9.31681e6	−	1.61372e7i	−0.478100	−	0.828093i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.58570e7			0.726167
$126$		0			0
$127$	3.94656e7			1.70965			0.854823	−	0.518920i	$-0.173666\pi$
$127$	3.94656e7			1.70965			0.854823	+	0.518920i	$0.173666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.56489e6	−	1.13707e7i	−0.255140	−	0.441915i	0.709794	−	0.704410i	$-0.248788\pi$
$131$	−6.56489e6	−	1.13707e7i	−0.255140	−	0.441915i	−0.964933	+	0.262495i	$0.915455\pi$
$132$		0			0
$133$	1.65088e7	−	4.60048e7i	0.608462	−	1.69560i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.31739e7	+	1.91530e7i	1.10224	+	0.636377i	0.936808	−	0.349845i	$-0.113766\pi$
$137$	3.31739e7	+	1.91530e7i	1.10224	+	0.636377i	0.165429	+	0.986222i	$0.447099\pi$
$138$		0			0
$139$		−	2.49090e7i		−	0.786691i	−0.919391	−	0.393345i	$-0.871318\pi$
$139$		−	2.49090e7i		−	0.786691i	0.919391	−	0.393345i	$-0.128682\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−440404.	+	762803.i	−0.0125943	+	0.0218140i
$144$		0			0
$145$	4.86897e7	−	2.81110e7i	1.32632	−	0.765753i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.26648e7	−	2.46326e7i	1.05662	−	0.610039i	0.132123	−	0.991233i	$-0.457820\pi$
$149$	4.26648e7	−	2.46326e7i	1.05662	−	0.610039i	0.924495	+	0.381194i	$0.124487\pi$
$150$		0			0
$151$	−2.32405e7	+	4.02538e7i	−0.549322	+	0.951453i	0.448999	+	0.893532i	$0.351781\pi$
$151$	−2.32405e7	+	4.02538e7i	−0.549322	+	0.951453i	−0.998321	+	0.0579211i	$0.981553\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.46237e7i			0.746815i
$156$		0			0
$157$	−8.03468e7	−	4.63883e7i	−1.65699	−	0.956664i	−0.974092	−	0.226150i	$-0.927386\pi$
$157$	−8.03468e7	−	4.63883e7i	−1.65699	−	0.956664i	−0.682898	−	0.730514i	$-0.739281\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.10362e6	+	1.70089e7i	−0.115265	+	0.321208i
$162$		0			0
$163$	2.14996e7	+	3.72383e7i	0.388842	+	0.673493i	0.992294	−	0.123906i	$-0.0395420\pi$
$163$	2.14996e7	+	3.72383e7i	0.388842	+	0.673493i	−0.603452	+	0.797399i	$0.706209\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.56737e7			1.42344			0.711721	−	0.702462i	$-0.247916\pi$
$167$	8.56737e7			1.42344			0.711721	+	0.702462i	$0.247916\pi$
$168$		0			0
$169$	6.18396e7			0.985514
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.16950e7	+	1.06859e8i	0.905917	+	1.56909i	0.819681	+	0.572820i	$0.194151\pi$
$173$	6.16950e7	+	1.06859e8i	0.905917	+	1.56909i	0.0862361	+	0.996275i	$0.472516\pi$
$174$		0			0
$175$	−1.84706e7	−	1.02046e8i	−0.260523	−	1.43934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.10108e7	+	5.25451e7i	1.18606	+	0.684773i	0.957409	−	0.288735i	$-0.0932347\pi$
$179$	9.10108e7	+	5.25451e7i	1.18606	+	0.684773i	0.228653	+	0.973508i	$0.426568\pi$
$180$		0			0
$181$			6.85865e7i			0.859733i	0.902893	+	0.429866i	$0.141439\pi$
$181$			6.85865e7i			0.859733i	−0.902893	+	0.429866i	$0.858561\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.52203e6	+	4.36828e6i	−0.0292852	+	0.0507235i
$186$		0			0
$187$	1.34372e6	−	775798.i	0.0150267	−	0.00867567i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.83159e7	+	2.78952e7i	−0.501734	+	0.289676i	−0.729429	−	0.684056i	$-0.760214\pi$
$191$	−4.83159e7	+	2.78952e7i	−0.501734	+	0.289676i	0.227695	+	0.973732i	$0.426881\pi$
$192$		0			0
$193$	−7.61119e7	+	1.31830e8i	−0.762082	+	1.31997i	0.179693	+	0.983723i	$0.442490\pi$
$193$	−7.61119e7	+	1.31830e8i	−0.762082	+	1.31997i	−0.941775	+	0.336243i	$0.890844\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.13704e8i		−	1.05960i	−0.848122	−	0.529802i	$-0.822266\pi$
$197$		−	1.13704e8i		−	1.05960i	0.848122	−	0.529802i	$-0.177734\pi$
$198$		0			0
$199$	−1.67973e8	−	9.69794e7i	−1.51097	−	0.872356i	−0.999918	−	0.0128038i	$-0.995924\pi$
$199$	−1.67973e8	−	9.69794e7i	−1.51097	−	0.872356i	−0.511047	−	0.859553i	$-0.670742\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.51705e7	−	8.87650e7i	−0.630684	−	0.744742i
$204$		0			0
$205$	3.81081e7	+	6.60052e7i	0.308944	+	0.535106i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.97595e7			−0.377020
$210$		0			0
$211$	1.45609e8			1.06709			0.533543	−	0.845773i	$-0.320860\pi$
$211$	1.45609e8			1.06709			0.533543	+	0.845773i	$0.320860\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.17010e8	−	2.02667e8i	−0.802948	−	1.39075i
$216$		0			0
$217$	7.04878e7	−	1.27584e7i	0.468279	−	0.0847593i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.38664e6	+	800579.i	0.00864156	+	0.00498921i
$222$		0			0
$223$		−	1.07368e8i		−	0.648350i	−0.945997	−	0.324175i	$-0.894913\pi$
$223$		−	1.07368e8i		−	0.648350i	0.945997	−	0.324175i	$-0.105087\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.55616e8	+	2.69535e8i	−0.883006	+	1.52941i	−0.0350247	+	0.999386i	$0.511151\pi$
$227$	−1.55616e8	+	2.69535e8i	−0.883006	+	1.52941i	−0.847982	+	0.530025i	$0.822182\pi$
$228$		0			0
$229$	−121331.	+	70050.7i	−0.000667650	+	0.000385468i	−0.500334	−	0.865833i	$-0.666789\pi$
$229$	−121331.	+	70050.7i	−0.000667650	+	0.000385468i	0.499666	+	0.866218i	$0.333456\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.37071e8	−	7.91380e7i	0.709904	−	0.409863i	−0.101121	−	0.994874i	$-0.532243\pi$
$233$	1.37071e8	−	7.91380e7i	0.709904	−	0.409863i	0.811026	+	0.585011i	$0.198910\pi$
$234$		0			0
$235$	5.93439e6	−	1.02787e7i	0.0298290	−	0.0516653i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.49554e8i		−	1.18242i	−0.806517	−	0.591211i	$-0.798650\pi$
$239$		−	2.49554e8i		−	1.18242i	0.806517	−	0.591211i	$-0.201350\pi$
$240$		0			0
$241$	1.58394e8	+	9.14488e7i	0.728918	+	0.420841i	0.818026	−	0.575181i	$-0.195068\pi$
$241$	1.58394e8	+	9.14488e7i	0.728918	+	0.420841i	−0.0891080	+	0.996022i	$0.528402\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.38317e8	+	1.26620e8i	−1.46974	+	0.550074i
$246$		0			0
$247$	−2.56745e7	−	4.44695e7i	−0.108408	−	0.187769i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.42709e8			−0.569631			−0.284815	−	0.958582i	$-0.591932\pi$
$251$	−1.42709e8			−0.569631			−0.284815	+	0.958582i	$0.591932\pi$
$252$		0			0
$253$	1.83971e7			0.0714213
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.56486e8	−	2.71042e8i	−0.575055	−	0.996025i	−0.996036	−	0.0889553i	$-0.971647\pi$
$257$	−1.56486e8	−	2.71042e8i	−0.575055	−	0.996025i	0.420980	−	0.907070i	$-0.361686\pi$
$258$		0			0
$259$	9.82237e6	+	3.52474e6i	0.0351291	+	0.0126060i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.60111e6	+	3.23380e6i	0.0189858	+	0.0109615i	0.509463	−	0.860493i	$-0.329844\pi$
$263$	5.60111e6	+	3.23380e6i	0.0189858	+	0.0109615i	−0.490477	+	0.871454i	$0.663177\pi$
$264$		0			0
$265$			3.49271e8i			1.15293i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.11896e8	+	1.93810e8i	−0.350496	+	0.607077i	−0.986336	−	0.164744i	$-0.947320\pi$
$269$	−1.11896e8	+	1.93810e8i	−0.350496	+	0.607077i	0.635840	+	0.771821i	$0.280654\pi$
$270$		0			0
$271$	−1.78985e8	+	1.03337e8i	−0.546291	+	0.315401i	−0.747625	−	0.664121i	$-0.768806\pi$
$271$	−1.78985e8	+	1.03337e8i	−0.546291	+	0.315401i	0.201333	+	0.979523i	$0.435473\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.14319e7	+	5.27883e7i	−0.265115	+	0.153064i
$276$		0			0
$277$	−2.66507e8	+	4.61603e8i	−0.753406	+	1.30494i	0.192757	+	0.981247i	$0.438257\pi$
$277$	−2.66507e8	+	4.61603e8i	−0.753406	+	1.30494i	−0.946163	+	0.323691i	$0.895076\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.69161e8i			0.723668i	0.932242	+	0.361834i	$0.117849\pi$
$281$			2.69161e8i			0.723668i	−0.932242	+	0.361834i	$0.882151\pi$
$282$		0			0
$283$	−7.75191e7	−	4.47556e7i	−0.203309	−	0.117380i	0.394889	−	0.918729i	$-0.370783\pi$
$283$	−7.75191e7	−	4.47556e7i	−0.203309	−	0.117380i	−0.598198	+	0.801348i	$0.704116\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.20333e8	−	1.01903e8i	0.300467	−	0.254450i
$288$		0			0
$289$	2.03759e8	+	3.52921e8i	0.496563	+	0.860073i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.51978e8			−0.585230			−0.292615	−	0.956230i	$-0.594525\pi$
$293$	−2.51978e8			−0.585230			−0.292615	+	0.956230i	$0.594525\pi$
$294$		0			0
$295$	−5.99929e8			−1.36058
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.49238e6	+	1.64413e7i	0.0205365	+	0.0355702i
$300$		0			0
$301$	−3.69477e8	+	3.12891e8i	−0.780916	+	0.661318i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.62950e8	−	4.40489e8i	−1.53974	−	0.888967i
$306$		0			0
$307$			3.89412e8i			0.768112i	0.923310	+	0.384056i	$0.125473\pi$
$307$			3.89412e8i			0.768112i	−0.923310	+	0.384056i	$0.874527\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.50469e8	−	7.80236e8i	0.849188	−	1.47084i	−0.0327457	−	0.999464i	$-0.510425\pi$
$311$	4.50469e8	−	7.80236e8i	0.849188	−	1.47084i	0.881934	−	0.471373i	$-0.156242\pi$
$312$		0			0
$313$	−2.22284e8	+	1.28336e8i	−0.409735	+	0.236560i	−0.690676	−	0.723165i	$-0.742687\pi$
$313$	−2.22284e8	+	1.28336e8i	−0.409735	+	0.236560i	0.280941	+	0.959725i	$0.409353\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.85342e8	+	5.11153e8i	−1.56100	+	0.901246i	−0.563847	+	0.825879i	$0.690679\pi$
$317$	−8.85342e8	+	5.11153e8i	−1.56100	+	0.901246i	−0.997156	+	0.0753667i	$0.975987\pi$
$318$		0			0
$319$	−5.92088e7	+	1.02553e8i	−0.102122	+	0.176880i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			9.04543e7i			0.149355i
$324$		0			0
$325$	−9.43525e7	−	5.44744e7i	−0.152462	−	0.0880240i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.31123e7	−	8.29380e6i	−0.0357814	−	0.0128401i
$330$		0			0
$331$	−3.66905e6	−	6.35498e6i	−0.00556103	−	0.00963199i	0.863231	−	0.504808i	$-0.168437\pi$
$331$	−3.66905e6	−	6.35498e6i	−0.00556103	−	0.00963199i	−0.868793	+	0.495176i	$0.835103\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.34104e7			−0.106684
$336$		0			0
$337$	7.95420e8			1.13212			0.566059	−	0.824365i	$-0.308467\pi$
$337$	7.95420e8			1.13212			0.566059	+	0.824365i	$0.308467\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.64631e7	−	6.31560e7i	−0.0497982	−	0.0862530i
$342$		0			0
$343$	3.82441e8	+	6.42094e8i	0.511723	+	0.859151i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.53338e8	−	4.34940e8i	−0.967914	−	0.558825i	−0.0693141	−	0.997595i	$-0.522081\pi$
$347$	−7.53338e8	−	4.34940e8i	−0.967914	−	0.558825i	−0.898600	+	0.438770i	$0.855414\pi$
$348$		0			0
$349$		−	1.33914e9i		−	1.68631i	−0.537674	−	0.843153i	$-0.680697\pi$
$349$		−	1.33914e9i		−	1.68631i	0.537674	−	0.843153i	$-0.319303\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.65300e8	−	4.59513e8i	0.321016	−	0.556015i	−0.659682	−	0.751545i	$-0.729309\pi$
$353$	2.65300e8	−	4.59513e8i	0.321016	−	0.556015i	0.980698	+	0.195529i	$0.0626425\pi$
$354$		0			0
$355$	2.34305e8	−	1.35276e8i	0.277960	−	0.160480i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.64282e8	−	2.10318e8i	0.415535	−	0.239909i	−0.277630	−	0.960688i	$-0.589549\pi$
$359$	3.64282e8	−	2.10318e8i	0.415535	−	0.239909i	0.693165	+	0.720779i	$0.256216\pi$
$360$		0			0
$361$	1.00349e9	−	1.73810e9i	1.12264	−	1.94447i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.46718e9i			1.57927i
$366$		0			0
$367$	7.81478e8	+	4.51186e8i	0.825250	+	0.476458i	0.852223	−	0.523178i	$-0.175254\pi$
$367$	7.81478e8	+	4.51186e8i	0.825250	+	0.476458i	−0.0269737	+	0.999636i	$0.508587\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.11053e8	−	1.28702e8i	0.722925	−	0.130851i
$372$		0			0
$373$	−1.90564e8	−	3.30066e8i	−0.190134	−	0.329322i	0.755160	−	0.655540i	$-0.227559\pi$
$373$	−1.90564e8	−	3.30066e8i	−0.190134	−	0.329322i	−0.945295	+	0.326218i	$0.894226\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.22200e8			−0.117456
$378$		0			0
$379$	−5.83269e8			−0.550341			−0.275170	−	0.961395i	$-0.588734\pi$
$379$	−5.83269e8			−0.550341			−0.275170	+	0.961395i	$0.588734\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.48081e8	+	2.56484e8i	0.134680	+	0.233273i	0.925475	−	0.378808i	$-0.123666\pi$
$383$	1.48081e8	+	2.56484e8i	0.134680	+	0.233273i	−0.790795	+	0.612081i	$0.790333\pi$
$384$		0			0
$385$	2.37663e8	+	2.80644e8i	0.212250	+	0.250636i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.44771e8	−	8.35836e7i	−0.124698	−	0.0719941i	0.436354	−	0.899775i	$-0.356270\pi$
$389$	−1.44771e8	−	8.35836e7i	−0.124698	−	0.0719941i	−0.561051	+	0.827781i	$0.689603\pi$
$390$		0			0
$391$		−	3.34428e7i		−	0.0282933i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.25216e8	−	7.36495e8i	0.347152	−	0.601285i
$396$		0			0
$397$	−1.34379e9	+	7.75840e8i	−1.07787	+	0.622308i	−0.930321	−	0.366747i	$-0.880471\pi$
$397$	−1.34379e9	+	7.75840e8i	−1.07787	+	0.622308i	−0.147548	+	0.989055i	$0.547138\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.08899e8	−	1.78343e8i	0.239228	−	0.138118i	−0.375594	−	0.926784i	$-0.622561\pi$
$401$	3.08899e8	−	1.78343e8i	0.239228	−	0.138118i	0.614822	+	0.788666i	$0.289228\pi$
$402$		0			0
$403$	3.76278e7	−	6.51733e7i	0.0286379	−	0.0496023i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.06240e7i		−	0.00781104i
$408$		0			0
$409$	2.13271e9	+	1.23132e9i	1.54134	+	0.889895i	0.998754	+	0.0498965i	$0.0158891\pi$
$409$	2.13271e9	+	1.23132e9i	1.54134	+	0.889895i	0.542589	+	0.839998i	$0.317444\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.21066e8	+	1.22135e9i	0.154418	+	0.853129i
$414$		0			0
$415$	−1.87485e9	−	3.24734e9i	−1.28765	−	2.23028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.47671e9			1.64485			0.822425	−	0.568874i	$-0.192621\pi$
$419$	2.47671e9			1.64485			0.822425	+	0.568874i	$0.192621\pi$
$420$		0			0
$421$	−2.27835e9			−1.48810			−0.744052	−	0.668122i	$-0.767098\pi$
$421$	−2.27835e9			−1.48810			−0.744052	+	0.668122i	$0.767098\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.59600e7	+	1.66208e8i	0.0606358	+	0.105024i
$426$		0			0
$427$	−6.15620e8	+	1.71554e9i	−0.382662	+	1.06636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.87647e9	+	1.08338e9i	1.12894	+	0.651795i	0.943669	−	0.330892i	$-0.107350\pi$
$431$	1.87647e9	+	1.08338e9i	1.12894	+	0.651795i	0.185273	+	0.982687i	$0.440683\pi$
$432$		0			0
$433$			3.62409e8i			0.214531i	0.994230	+	0.107266i	$0.0342095\pi$
$433$			3.62409e8i			0.214531i	−0.994230	+	0.107266i	$0.965790\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.36253e8	+	9.28818e8i	−0.307387	+	0.532409i
$438$		0			0
$439$	9.72933e7	−	5.61723e7i	0.0548854	−	0.0316881i	−0.472306	−	0.881435i	$-0.656578\pi$
$439$	9.72933e7	−	5.61723e7i	0.0548854	−	0.0316881i	0.527192	+	0.849746i	$0.323245\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.36348e9	−	1.36455e9i	1.29163	−	0.745724i	0.312688	−	0.949856i	$-0.398771\pi$
$443$	2.36348e9	−	1.36455e9i	1.29163	−	0.745724i	0.978943	+	0.204132i	$0.0654373\pi$
$444$		0			0
$445$	1.52165e9	−	2.63557e9i	0.818567	−	1.41780i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.13737e9i			1.11434i	0.830398	+	0.557170i	$0.188113\pi$
$449$			2.13737e9i			1.11434i	−0.830398	+	0.557170i	$0.811887\pi$
$450$		0			0
$451$	−1.39023e8	−	8.02652e7i	−0.0713625	−	0.0412012i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.28181e8	+	3.57200e8i	−0.0637945	+	0.177776i
$456$		0			0
$457$	1.35919e9	+	2.35419e9i	0.666153	+	1.15381i	0.978971	+	0.203997i	$0.0653934\pi$
$457$	1.35919e9	+	2.35419e9i	0.666153	+	1.15381i	−0.312819	+	0.949813i	$0.601273\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.13481e9			1.01486			0.507430	−	0.861693i	$-0.330596\pi$
$461$	2.13481e9			1.01486			0.507430	+	0.861693i	$0.330596\pi$
$462$		0			0
$463$	2.18177e8			0.102159			0.0510794	−	0.998695i	$-0.483734\pi$
$463$	2.18177e8			0.102159			0.0510794	+	0.998695i	$0.483734\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.26844e7	−	5.66110e7i	−0.0148502	−	0.0257212i	0.858505	−	0.512805i	$-0.171394\pi$
$467$	−3.26844e7	−	5.66110e7i	−0.0148502	−	0.0257212i	−0.873355	+	0.487084i	$0.838060\pi$
$468$		0			0
$469$	2.70508e7	+	1.49451e8i	0.0121081	+	0.0668948i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.26867e8	+	2.46452e8i	0.185472	+	0.107082i
$474$		0			0
$475$		−	6.15485e9i		−	2.63506i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.16962e9	−	3.75790e9i	0.902008	−	1.56232i	0.0771231	−	0.997022i	$-0.475427\pi$
$479$	2.16962e9	−	3.75790e9i	0.902008	−	1.56232i	0.824884	−	0.565301i	$-0.191240\pi$
$480$		0			0
$481$	9.49458e6	−	5.48170e6i	0.00389016	−	0.00224599i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.45385e9	+	1.41673e9i	−0.976681	+	0.563887i
$486$		0			0
$487$	6.47502e8	−	1.12151e9i	0.254033	−	0.439997i	−0.710600	−	0.703597i	$-0.751576\pi$
$487$	6.47502e8	−	1.12151e9i	0.254033	−	0.439997i	0.964632	+	0.263599i	$0.0849096\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.50530e9i			0.955158i	0.878589	+	0.477579i	$0.158486\pi$
$491$			2.50530e9i			0.955158i	−0.878589	+	0.477579i	$0.841514\pi$
$492$		0			0
$493$	1.86423e8	+	1.07631e8i	0.0700705	+	0.0404552i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.61736e8	−	4.27155e8i	−0.132174	−	0.156077i
$498$		0			0
$499$	−2.17412e9	−	3.76569e9i	−0.783308	−	1.35673i	−0.930005	−	0.367548i	$-0.880198\pi$
$499$	−2.17412e9	−	3.76569e9i	−0.783308	−	1.35673i	0.146697	−	0.989181i	$-0.453136\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.66394e9			1.28369			0.641845	−	0.766834i	$-0.278169\pi$
$503$	3.66394e9			1.28369			0.641845	+	0.766834i	$0.278169\pi$
$504$		0			0
$505$	7.18857e9			2.48383
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.39277e9	+	2.41235e9i	0.468131	+	0.810827i	0.999337	−	0.0364160i	$-0.0115941\pi$
$509$	1.39277e9	+	2.41235e9i	0.468131	+	0.810827i	−0.531206	+	0.847243i	$0.678261\pi$
$510$		0			0
$511$	2.98691e9	−	5.40636e8i	0.990260	−	0.179239i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.42060e9	−	8.20186e8i	−0.458297	−	0.264598i
$516$		0			0
$517$			2.49986e7i			0.00795607i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.80207e8	−	1.00495e9i	0.179743	−	0.311323i	−0.762050	−	0.647518i	$-0.775807\pi$
$521$	5.80207e8	−	1.00495e9i	0.179743	−	0.311323i	0.941792	+	0.336195i	$0.109140\pi$
$522$		0			0
$523$	3.66163e9	−	2.11404e9i	1.11923	−	0.646186i	0.178023	−	0.984026i	$-0.443030\pi$
$523$	3.66163e9	−	2.11404e9i	1.11923	−	0.646186i	0.941203	+	0.337841i	$0.109696\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.14807e8	+	6.62837e7i	−0.0341688	+	0.0197274i
$528$		0			0
$529$	−1.50415e9	+	2.60526e9i	−0.441770	+	0.765168i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.65658e8i		−	0.0473879i
$534$		0			0
$535$	−7.56731e9	−	4.36899e9i	−2.13650	−	1.23351i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.83765e8	−	5.87252e8i	0.133068	−	0.161534i
$540$		0			0
$541$	2.95692e9	+	5.12154e9i	0.802877	+	1.39062i	0.917715	+	0.397240i	$0.130032\pi$
$541$	2.95692e9	+	5.12154e9i	0.802877	+	1.39062i	−0.114837	+	0.993384i	$0.536635\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.23774e9			−0.856749
$546$		0			0
$547$	−1.18986e9			−0.310842			−0.155421	−	0.987848i	$-0.549673\pi$
$547$	−1.18986e9			−0.310842			−0.155421	+	0.987848i	$0.549673\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.45173e9	−	5.97857e9i	−0.879035	−	1.52253i
$552$		0			0
$553$	−1.65606e9	−	5.94275e8i	−0.416426	−	0.149434i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.07356e9	−	1.77452e9i	−0.753614	−	0.435099i	0.0733842	−	0.997304i	$-0.476620\pi$
$557$	−3.07356e9	−	1.77452e9i	−0.753614	−	0.435099i	−0.826998	+	0.562204i	$0.809953\pi$
$558$		0			0
$559$			5.08648e8i			0.123162i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.42466e8	+	5.93169e8i	−0.0808795	+	0.140087i	−0.903628	−	0.428318i	$-0.859106\pi$
$563$	−3.42466e8	+	5.93169e8i	−0.0808795	+	0.140087i	0.822748	+	0.568406i	$0.192440\pi$
$564$		0			0
$565$	−5.90876e9	+	3.41142e9i	−1.37825	+	0.795731i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.58143e9	−	2.64509e9i	1.04258	−	0.601932i	0.122015	−	0.992528i	$-0.461064\pi$
$569$	4.58143e9	−	2.64509e9i	1.04258	−	0.601932i	0.920562	+	0.390596i	$0.127731\pi$
$570$		0			0
$571$	−3.61867e9	+	6.26772e9i	−0.813435	+	1.40891i	0.0970118	+	0.995283i	$0.469072\pi$
$571$	−3.61867e9	+	6.26772e9i	−0.813435	+	1.40891i	−0.910446	+	0.413627i	$0.864262\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.27557e9i			0.499176i
$576$		0			0
$577$	9.44590e8	+	5.45359e8i	0.204705	+	0.118186i	0.598848	−	0.800863i	$-0.295625\pi$
$577$	9.44590e8	+	5.45359e8i	0.204705	+	0.118186i	−0.394143	+	0.919049i	$0.628959\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.92014e9	+	5.01346e9i	−1.25232	+	1.06053i
$582$		0			0
$583$	−3.67826e8	−	6.37093e8i	−0.0768780	−	0.133157i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.59205e9			1.34520			0.672601	−	0.740006i	$-0.265177\pi$
$587$	6.59205e9			1.34520			0.672601	+	0.740006i	$0.265177\pi$
$588$		0			0
$589$	4.25142e9			0.857295
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.20101e8	−	7.27636e8i	−0.0827298	−	0.143292i	0.821692	−	0.569932i	$-0.193031\pi$
$593$	−4.20101e8	−	7.27636e8i	−0.0827298	−	0.143292i	−0.904422	+	0.426640i	$0.859697\pi$
$594$		0			0
$595$	5.10162e8	−	4.32030e8i	0.0992884	−	0.0840823i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.11648e9	+	2.37665e9i	0.782587	+	0.451827i	0.837346	−	0.546673i	$-0.184106\pi$
$599$	4.11648e9	+	2.37665e9i	0.782587	+	0.451827i	−0.0547595	+	0.998500i	$0.517439\pi$
$600$		0			0
$601$			3.92620e9i			0.737754i	0.929478	+	0.368877i	$0.120258\pi$
$601$			3.92620e9i			0.737754i	−0.929478	+	0.368877i	$0.879742\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.08668e9	+	7.07834e9i	−0.750286	+	1.29953i
$606$		0			0
$607$	−3.70044e9	+	2.13645e9i	−0.671572	+	0.387732i	−0.796672	−	0.604412i	$-0.793408\pi$
$607$	−3.70044e9	+	2.13645e9i	−0.671572	+	0.387732i	0.125100	+	0.992144i	$0.460075\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.23410e7	+	1.28986e7i	−0.00396239	+	0.00228769i
$612$		0			0
$613$	−3.47209e9	+	6.01384e9i	−0.608806	+	1.05448i	0.382631	+	0.923901i	$0.375018\pi$
$613$	−3.47209e9	+	6.01384e9i	−0.608806	+	1.05448i	−0.991437	+	0.130582i	$0.958315\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.70403e9i			0.806255i	0.915144	+	0.403127i	$0.132077\pi$
$617$			4.70403e9i			0.806255i	−0.915144	+	0.403127i	$0.867923\pi$
$618$		0			0
$619$	−5.24157e9	−	3.02622e9i	−0.888269	−	0.512842i	−0.0148931	−	0.999889i	$-0.504741\pi$
$619$	−5.24157e9	−	3.02622e9i	−0.888269	−	0.512842i	−0.873376	+	0.487047i	$0.838074\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.92626e9	−	2.12663e9i	−0.981912	−	0.352358i
$624$		0			0
$625$	9.86177e8	+	1.70811e9i	0.161575	+	0.279856i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.93127e7			−0.00309432
$630$		0			0
$631$	6.17668e9			0.978707			0.489353	−	0.872086i	$-0.337233\pi$
$631$	6.17668e9			0.978707			0.489353	+	0.872086i	$0.337233\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.65551e9	−	1.49918e10i	−1.34148	−	2.32351i
$636$		0			0
$637$	7.74429e8	+	1.29330e8i	0.118712	+	0.0198248i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.64927e9	−	1.52956e9i	−0.397304	−	0.229383i	0.288016	−	0.957626i	$-0.407004\pi$
$641$	−2.64927e9	−	1.52956e9i	−0.397304	−	0.229383i	−0.685320	+	0.728242i	$0.740338\pi$
$642$		0			0
$643$			1.31512e9i			0.195086i	0.995231	+	0.0975430i	$0.0310983\pi$
$643$			1.31512e9i			0.195086i	−0.995231	+	0.0975430i	$0.968902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.03171e9	+	1.78697e9i	−0.149758	+	0.259389i	−0.931138	−	0.364667i	$-0.881183\pi$
$647$	−1.03171e9	+	1.78697e9i	−0.149758	+	0.259389i	0.781380	+	0.624056i	$0.214516\pi$
$648$		0			0
$649$	1.09431e9	−	6.31800e8i	0.157139	−	0.0907242i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.71504e9	−	1.56753e9i	0.381575	−	0.220302i	−0.296929	−	0.954900i	$-0.595962\pi$
$653$	2.71504e9	−	1.56753e9i	0.381575	−	0.220302i	0.678503	+	0.734598i	$0.262629\pi$
$654$		0			0
$655$	−2.87959e9	+	4.98760e9i	−0.400393	+	0.693501i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3.30749e9i		−	0.450194i	−0.974336	−	0.225097i	$-0.927730\pi$
$659$		−	3.30749e9i		−	0.450194i	0.974336	−	0.225097i	$-0.0722698\pi$
$660$		0			0
$661$	−1.75333e9	−	1.01228e9i	−0.236134	−	0.136332i	0.377265	−	0.926105i	$-0.376865\pi$
$661$	−1.75333e9	−	1.01228e9i	−0.236134	−	0.136332i	−0.613398	+	0.789774i	$0.710198\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.10965e10	+	3.81850e9i	−2.78185	+	0.503520i
$666$		0			0
$667$	1.27617e9	+	2.21040e9i	0.166521	+	0.288423i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.85556e9			0.237108
$672$		0			0
$673$	9.94913e9			1.25815			0.629075	−	0.777344i	$-0.283434\pi$
$673$	9.94913e9			1.25815			0.629075	+	0.777344i	$0.283434\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.99749e9	−	5.19180e9i	−0.371276	−	0.643069i	0.618486	−	0.785796i	$-0.287746\pi$
$677$	−2.99749e9	−	5.19180e9i	−0.371276	−	0.643069i	−0.989762	+	0.142727i	$0.954413\pi$
$678$		0			0
$679$	3.78843e9	+	4.47356e9i	0.464424	+	0.548415i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.74010e9	+	2.15935e9i	0.449170	+	0.259328i	0.707480	−	0.706734i	$-0.249832\pi$
$683$	3.74010e9	+	2.15935e9i	0.449170	+	0.259328i	−0.258310	+	0.966062i	$0.583165\pi$
$684$		0			0
$685$		−	1.68023e10i		−	1.99734i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.79575e8	−	6.57443e8i	0.0442110	−	0.0765757i
$690$		0			0
$691$	−2.23630e9	+	1.29113e9i	−0.257844	+	0.148866i	−0.623351	−	0.781942i	$-0.714229\pi$
$691$	−2.23630e9	+	1.29113e9i	−0.257844	+	0.148866i	0.365507	+	0.930809i	$0.380896\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.46215e9	+	5.46298e9i	−1.06916	+	0.617280i
$696$		0			0
$697$	−1.45908e8	+	2.52721e8i	−0.0163217	+	0.0282700i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	9.48518e9i		−	1.04000i	−0.854167	−	0.519999i	$-0.825932\pi$
$701$		−	9.48518e9i		−	1.04000i	0.854167	−	0.519999i	$-0.174068\pi$
$702$		0			0
$703$	5.36377e8	+	3.09678e8i	0.0582273	+	0.0336176i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.64889e9	−	1.46346e10i	−0.281901	−	1.55745i
$708$		0			0
$709$	2.78014e9	+	4.81535e9i	0.292958	+	0.507418i	0.974508	−	0.224354i	$-0.0720271\pi$
$709$	2.78014e9	+	4.81535e9i	0.292958	+	0.507418i	−0.681550	+	0.731772i	$0.738694\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.57184e9			−0.162403
$714$		0			0
$715$	3.86354e8			0.0395288
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.44576e9	−	7.70028e9i	−0.446061	−	0.772601i	0.552064	−	0.833802i	$-0.313840\pi$
$719$	−4.44576e9	−	7.70028e9i	−0.446061	−	0.772601i	−0.998125	+	0.0612008i	$0.980507\pi$
$720$		0			0
$721$	−1.14628e9	+	3.19432e9i	−0.113898	+	0.317399i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.26849e10	−	7.32365e9i	−1.23625	−	0.713747i
$726$		0			0
$727$			8.49812e9i			0.820262i	0.912027	+	0.410131i	$0.134517\pi$
$727$			8.49812e9i			0.820262i	−0.912027	+	0.410131i	$0.865483\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.48006e8	−	7.75970e8i	0.0424203	−	0.0734741i
$732$		0			0
$733$	−1.31884e10	+	7.61435e9i	−1.23688	+	0.714116i	−0.968456	−	0.249184i	$-0.919838\pi$
$733$	−1.31884e10	+	7.61435e9i	−1.23688	+	0.714116i	−0.268429	+	0.963300i	$0.586504\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.33905e8	−	7.73103e7i	0.0123215	−	0.00711379i
$738$		0			0
$739$	−7.65585e9	+	1.32603e10i	−0.697811	+	1.20864i	0.271413	+	0.962463i	$0.412509\pi$
$739$	−7.65585e9	+	1.32603e10i	−0.697811	+	1.20864i	−0.969224	+	0.246181i	$0.920824\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.33226e10i			1.19160i	0.803135	+	0.595798i	$0.203164\pi$
$743$			1.33226e10i			1.19160i	−0.803135	+	0.595798i	$0.796836\pi$
$744$		0			0
$745$	−1.87143e10	−	1.08047e10i	−1.65816	−	0.957339i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.10603e9	+	1.70156e10i	−0.530973	+	1.47966i
$750$		0			0
$751$	−1.67547e9	−	2.90200e9i	−0.144343	−	0.250010i	0.784784	−	0.619769i	$-0.212774\pi$
$751$	−1.67547e9	−	2.90200e9i	−0.144343	−	0.250010i	−0.929128	+	0.369759i	$0.879440\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.03882e10			1.72411
$756$		0			0
$757$	1.76437e10			1.47827			0.739137	−	0.673555i	$-0.235233\pi$
$757$	1.76437e10			1.47827			0.739137	+	0.673555i	$0.235233\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.13252e9	+	7.15773e9i	0.339914	+	0.588747i	0.984416	−	0.175855i	$-0.0562689\pi$
$761$	4.13252e9	+	7.15773e9i	0.339914	+	0.588747i	−0.644503	+	0.764602i	$0.722936\pi$
$762$		0			0
$763$	1.19306e9	+	6.59145e9i	0.0972362	+	0.537212i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.12926e9	+	6.51981e8i	0.0903674	+	0.0521737i
$768$		0			0
$769$		−	3.98977e9i		−	0.316378i	−0.987409	−	0.158189i	$-0.949434\pi$
$769$		−	3.98977e9i		−	0.316378i	0.987409	−	0.158189i	$-0.0505655\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.10559e9	−	1.40393e10i	0.631184	−	1.09324i	−0.356126	−	0.934438i	$-0.615902\pi$
$773$	8.10559e9	−	1.40393e10i	0.631184	−	1.09324i	0.987310	−	0.158805i	$-0.0507642\pi$
$774$		0			0
$775$	7.81188e9	−	4.51019e9i	0.602837	−	0.348048i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.10473e9	−	4.67927e9i	0.614267	−	0.354647i
$780$		0			0
$781$	−2.84925e8	+	4.93504e8i	−0.0214019	+	0.0370691i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			4.06951e10i			3.00260i
$786$		0			0
$787$	−2.99721e9	−	1.73044e9i	−0.219183	−	0.126545i	0.386389	−	0.922336i	$-0.373722\pi$
$787$	−2.99721e9	−	1.73044e9i	−0.219183	−	0.126545i	−0.605572	+	0.795791i	$0.707056\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.12235e9	+	1.07721e10i	0.655374	+	0.773897i
$792$		0			0
$793$	9.57415e8	+	1.65829e9i	0.0681780	+	0.118088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.85903e10			1.30071			0.650357	−	0.759629i	$-0.274619\pi$
$797$	1.85903e10			1.30071			0.650357	+	0.759629i	$0.274619\pi$
$798$		0			0
$799$	4.54432e7			0.00315177
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.54512e9	−	2.67623e9i	−0.105307	−	0.182397i
$804$		0			0
$805$	7.79979e9	−	1.41177e9i	0.526984	−	0.0953850i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.14014e10	−	6.58259e9i	−0.757073	−	0.437096i	0.0711712	−	0.997464i	$-0.477326\pi$
$809$	−1.14014e10	−	6.58259e9i	−0.757073	−	0.437096i	−0.828244	+	0.560368i	$0.810660\pi$
$810$		0			0
$811$		−	1.72854e10i		−	1.13790i	−0.822371	−	0.568952i	$-0.807349\pi$
$811$		−	1.72854e10i		−	1.13790i	0.822371	−	0.568952i	$-0.192651\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.43046e9	−	1.63340e10i	0.610213	−	1.05692i
$816$		0			0
$817$	−2.48853e10	+	1.43675e10i	−1.59649	+	0.921732i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.69551e9	−	3.86565e9i	0.422262	−	0.243793i	−0.273782	−	0.961792i	$-0.588275\pi$
$821$	6.69551e9	−	3.86565e9i	0.422262	−	0.243793i	0.696045	+	0.717998i	$0.254941\pi$
$822$		0			0
$823$	−2.54038e9	+	4.40007e9i	−0.158855	+	0.275144i	−0.934456	−	0.356079i	$-0.884113\pi$
$823$	−2.54038e9	+	4.40007e9i	−0.158855	+	0.275144i	0.775601	+	0.631223i	$0.217447\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.75664e10i			1.69477i	0.530980	+	0.847384i	$0.321824\pi$
$827$			2.75664e10i			1.69477i	−0.530980	+	0.847384i	$0.678176\pi$
$828$		0			0
$829$	2.18567e10	+	1.26189e10i	1.33242	+	0.769276i	0.985671	−	0.168680i	$-0.0539505\pi$
$829$	2.18567e10	+	1.26189e10i	1.33242	+	0.769276i	0.346754	+	0.937956i	$0.387284\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.06752e9	−	8.79401e8i	−0.0639912	−	0.0527144i
$834$		0			0
$835$	−1.87898e10	−	3.25448e10i	−1.11691	−	1.93455i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.82396e10			1.06623			0.533113	−	0.846044i	$-0.321022\pi$
$839$	1.82396e10			1.06623			0.533113	+	0.846044i	$0.321022\pi$
$840$		0			0
$841$	8.21054e8			0.0475977
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.35625e10	−	2.34909e10i	−0.773288	−	1.33937i
$846$		0			0
$847$	1.59161e10	+	5.71148e9i	0.900006	+	0.322966i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.98310e8	−	1.14494e8i	−0.0110304	−	0.00636839i
$852$		0			0
$853$		−	1.45235e10i		−	0.801215i	−0.916250	−	0.400608i	$-0.868799\pi$
$853$		−	1.45235e10i		−	0.801215i	0.916250	−	0.400608i	$-0.131201\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.50173e9	−	6.06518e9i	0.190042	−	0.329163i	−0.755222	−	0.655469i	$-0.772471\pi$
$857$	3.50173e9	−	6.06518e9i	0.190042	−	0.329163i	0.945264	+	0.326306i	$0.105804\pi$
$858$		0			0
$859$	7.92680e9	−	4.57654e9i	0.426699	−	0.246355i	−0.271240	−	0.962512i	$-0.587434\pi$
$859$	7.92680e9	−	4.57654e9i	0.426699	−	0.246355i	0.697939	+	0.716157i	$0.254100\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.28341e9	−	7.40976e8i	0.0679715	−	0.0392433i	−0.465629	−	0.884980i	$-0.654172\pi$
$863$	1.28341e9	−	7.40976e8i	0.0679715	−	0.0392433i	0.533601	+	0.845737i	$0.320839\pi$
$864$		0			0
$865$	2.70616e10	−	4.68721e10i	1.42166	−	2.46239i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.79122e9i			0.0925934i
$870$		0			0
$871$	1.38183e8	+	7.97798e7i	0.00708582	+	0.00409100i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.09815e10	+	9.29964e9i	−0.554156	+	0.469286i
$876$		0			0
$877$	8.02873e9	+	1.39062e10i	0.401928	+	0.696160i	0.993959	−	0.109756i	$-0.0350068\pi$
$877$	8.02873e9	+	1.39062e10i	0.401928	+	0.696160i	−0.592031	+	0.805916i	$0.701674\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.50565e10			1.23454			0.617270	−	0.786751i	$-0.288239\pi$
$881$	2.50565e10			1.23454			0.617270	+	0.786751i	$0.288239\pi$
$882$		0			0
$883$	1.37876e10			0.673946			0.336973	−	0.941514i	$-0.390597\pi$
$883$	1.37876e10			0.673946			0.336973	+	0.941514i	$0.390597\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.73012e9	−	1.16569e10i	−0.323810	−	0.560856i	0.657461	−	0.753489i	$-0.271631\pi$
$887$	−6.73012e9	−	1.16569e10i	−0.323810	−	0.560856i	−0.981271	+	0.192633i	$0.938297\pi$
$888$		0			0
$889$	−2.73311e10	+	2.31453e10i	−1.30467	+	1.10486i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.26211e9	−	7.28679e8i	−0.0593085	−	0.0342418i
$894$		0			0
$895$		−	4.60963e10i		−	2.14924i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.05876e9	−	8.76203e9i	0.232212	−	0.402203i
$900$		0			0
$901$	−1.15813e9	+	6.68644e8i	−0.0527496	+	0.0304550i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.60539e10	−	1.50422e10i	1.16843	−	0.674593i
$906$		0			0
$907$	1.21813e10	−	2.10986e10i	0.542085	−	0.938919i	−0.456699	−	0.889621i	$-0.650968\pi$
$907$	1.21813e10	−	2.10986e10i	0.542085	−	0.938919i	0.998784	−	0.0492980i	$-0.0156984\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.88875e10i			0.827677i	0.910350	+	0.413838i	$0.135812\pi$
$911$			1.88875e10i			0.827677i	−0.910350	+	0.413838i	$0.864188\pi$
$912$		0			0
$913$	6.83970e9	+	3.94890e9i	0.297433	+	0.171723i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.12150e10	+	4.02447e9i	0.480292	+	0.172352i
$918$		0			0
$919$	−1.31764e10	−	2.28221e10i	−0.560004	−	0.969955i	−0.997495	−	0.0707328i	$-0.977466\pi$
$919$	−1.31764e10	−	2.28221e10i	−0.560004	−	0.969955i	0.437491	−	0.899223i	$-0.355867\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.88052e8			−0.0246156
$924$		0			0
$925$	1.31411e9			0.0545927
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.34465e10	+	2.32900e10i	0.550241	+	0.953046i	0.998257	+	0.0590200i	$0.0187976\pi$
$929$	1.34465e10	+	2.32900e10i	0.550241	+	0.953046i	−0.448016	+	0.894026i	$0.647869\pi$
$930$		0			0
$931$	1.55476e10	+	4.15416e10i	0.631449	+	1.68717i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.89404e8	−	3.40292e8i	−0.0235815	−	0.0136148i
$936$		0			0
$937$		−	3.38586e10i		−	1.34456i	−0.740297	−	0.672280i	$-0.765315\pi$
$937$		−	3.38586e10i		−	1.34456i	0.740297	−	0.672280i	$-0.234685\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.74142e10	−	3.01624e10i	0.681304	−	1.18005i	−0.293279	−	0.956027i	$-0.594746\pi$
$941$	1.74142e10	−	3.01624e10i	0.681304	−	1.18005i	0.974583	−	0.224026i	$-0.0719202\pi$
$942$		0			0
$943$	−2.99648e9	+	1.73002e9i	−0.116365	+	0.0671831i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.22819e10	+	1.28645e10i	−0.852565	+	0.492229i	−0.861516	−	0.507731i	$-0.830484\pi$
$947$	−2.22819e10	+	1.28645e10i	−0.852565	+	0.492229i	0.00895017	+	0.999960i	$0.497151\pi$
$948$		0			0
$949$	1.59447e9	−	2.76171e9i	0.0605600	−	0.104893i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.87255e10i			0.700822i	0.936596	+	0.350411i	$0.113958\pi$
$953$			1.87255e10i			0.700822i	−0.936596	+	0.350411i	$0.886042\pi$
$954$		0			0
$955$	2.11931e10	+	1.22358e10i	0.787376	+	0.454592i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.42066e10	+	6.19144e9i	−1.25240	+	0.226687i
$960$		0			0
$961$	−1.06409e10	−	1.84306e10i	−0.386765	−	0.669897i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.67707e10			2.39189
$966$		0			0
$967$	−1.68719e10			−0.600029			−0.300014	−	0.953935i	$-0.596992\pi$
$967$	−1.68719e10			−0.600029			−0.300014	+	0.953935i	$0.596992\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.03180e9	+	1.78713e9i	0.0361683	+	0.0626453i	0.883543	−	0.468350i	$-0.155151\pi$
$971$	1.03180e9	+	1.78713e9i	0.0361683	+	0.0626453i	−0.847375	+	0.530996i	$0.821818\pi$
$972$		0			0
$973$	1.46083e10	+	1.72502e10i	0.508400	+	0.600343i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.64419e10	+	2.10398e10i	1.25017	+	0.721788i	0.971144	−	0.238495i	$-0.0766539\pi$
$977$	3.64419e10	+	2.10398e10i	1.25017	+	0.721788i	0.279030	+	0.960283i	$0.409987\pi$
$978$		0			0
$979$			6.40994e9i			0.218331i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.93963e8	−	6.82364e8i	0.0132287	−	0.0229128i	−0.859335	−	0.511413i	$-0.829122\pi$
$983$	3.93963e8	−	6.82364e8i	0.0132287	−	0.0229128i	0.872564	+	0.488500i	$0.162456\pi$
$984$		0			0
$985$	−4.31926e10	+	2.49372e10i	−1.44007	+	0.831422i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.20059e9	−	5.31196e9i	0.302433	−	0.174610i
$990$		0			0
$991$	4.05407e9	−	7.02185e9i	0.132322	−	0.229189i	−0.792249	−	0.610198i	$-0.791090\pi$
$991$	4.05407e9	−	7.02185e9i	0.132322	−	0.229189i	0.924571	+	0.381009i	$0.124423\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			8.50772e10i			2.73799i
$996$		0			0
$997$	−1.83876e10	−	1.06161e10i	−0.587615	−	0.339259i	0.176539	−	0.984294i	$-0.443510\pi$
$997$	−1.83876e10	−	1.06161e10i	−0.587615	−	0.339259i	−0.764154	+	0.645034i	$0.776843\pi$
$998$		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.17.2	✓	36
3.2	odd	2	inner	252.8.t.a.17.17	yes	36
7.5	odd	6	inner	252.8.t.a.89.17	yes	36
21.5	even	6	inner	252.8.t.a.89.2	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.t.a.17.2	✓	36	1.1	even	1	trivial
252.8.t.a.17.17	yes	36	3.2	odd	2	inner
252.8.t.a.89.2	yes	36	21.5	even	6	inner
252.8.t.a.89.17	yes	36	7.5	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-219.318 - 379.869i) q^{5} +(-692.530 + 586.468i) q^{7} +(800.099 + 461.937i) q^{11} +953.385i q^{13} +(839.723 - 1454.44i) q^{17} +(-46643.8 + 26929.8i) q^{19} +(17245.2 - 9956.50i) q^{23} +(-57137.9 + 98965.8i) q^{25} +128175. i q^{29} +(-68359.9 - 39467.6i) q^{31} +(374665. + 134448. i) q^{35} +(-5749.72 - 9958.80i) q^{37} -173758. q^{41} +533517. q^{43} +(13529.2 + 23433.3i) q^{47} +(135653. - 812294. i) q^{49} +(-689588. - 398134. i) q^{53} -405244. i q^{55} +(683859. - 1.18448e6i) q^{59} +(1.73937e6 - 1.00423e6i) q^{61} +(362162. - 209094. i) q^{65} +(83680.5 - 144939. i) q^{67} +616804. i q^{71} +(-2.89674e6 - 1.67243e6i) q^{73} +(-825004. + 149327. i) q^{77} +(969407. + 1.67906e6i) q^{79} +8.54857e6 q^{83} -736664. q^{85} +(3.46905e6 + 6.00857e6i) q^{89} +(-559130. - 660248. i) q^{91} +(2.04596e7 + 1.18124e7i) q^{95} -6.45974e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2634 q^{7} - 47862 q^{19} - 360762 q^{25} + 486018 q^{31} - 972270 q^{37} + 298788 q^{43} + 1556886 q^{49} + 4324644 q^{61} - 2969562 q^{67} - 5157378 q^{73} + 7676514 q^{79} - 15214128 q^{85} - 6114678 q^{91}+O(q^{100}) \) 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

L-functions

Modular forms

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Database

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