LMFDB - Embedded newform 225.3.g.b.118.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.g (of order $4$, 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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			118.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	225.118
Dual form			225.3.g.b.82.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.44949 - 2.44949i) q^{2} -8.00000i q^{4} +(6.12372 - 6.12372i) q^{7} +(-9.79796 - 9.79796i) q^{8} -6.00000 q^{11} +(3.67423 + 3.67423i) q^{13} -30.0000i q^{14} -16.0000 q^{16} +(-17.1464 + 17.1464i) q^{17} -23.0000i q^{19} +(-14.6969 + 14.6969i) q^{22} +(12.2474 + 12.2474i) q^{23} +18.0000 q^{26} +(-48.9898 - 48.9898i) q^{28} +6.00000i q^{29} +25.0000 q^{31} +84.0000i q^{34} +(-24.4949 + 24.4949i) q^{37} +(-56.3383 - 56.3383i) q^{38} +60.0000 q^{41} +(60.0125 + 60.0125i) q^{43} +48.0000i q^{44} +60.0000 q^{46} +(-7.34847 + 7.34847i) q^{47} -26.0000i q^{49} +(29.3939 - 29.3939i) q^{52} +(24.4949 + 24.4949i) q^{53} -120.000 q^{56} +(14.6969 + 14.6969i) q^{58} -18.0000i q^{59} -37.0000 q^{61} +(61.2372 - 61.2372i) q^{62} -64.0000i q^{64} +(-25.7196 + 25.7196i) q^{67} +(137.171 + 137.171i) q^{68} -132.000 q^{71} +(-24.4949 - 24.4949i) q^{73} +120.000i q^{74} -184.000 q^{76} +(-36.7423 + 36.7423i) q^{77} -10.0000i q^{79} +(146.969 - 146.969i) q^{82} +(-2.44949 - 2.44949i) q^{83} +294.000 q^{86} +(58.7878 + 58.7878i) q^{88} -132.000i q^{89} +45.0000 q^{91} +(97.9796 - 97.9796i) q^{92} +36.0000i q^{94} +(23.2702 - 23.2702i) q^{97} +(-63.6867 - 63.6867i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{11} - 64 q^{16} + 72 q^{26} + 100 q^{31} + 240 q^{41} + 240 q^{46} - 480 q^{56} - 148 q^{61} - 528 q^{71} - 736 q^{76} + 1176 q^{86} + 180 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 - 64 * q^16 + 72 * q^26 + 100 * q^31 + 240 * q^41 + 240 * q^46 - 480 * q^56 - 148 * q^61 - 528 * q^71 - 736 * q^76 + 1176 * q^86 + 180 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.44949	−	2.44949i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$2$	2.44949	−	2.44949i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0
$3$		0			0
$4$		−	8.00000i		−	2.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.12372	−	6.12372i	0.874818	−	0.874818i	−0.118175	−	0.992993i	$-0.537704\pi$
$7$	6.12372	−	6.12372i	0.874818	−	0.874818i	0.992993	+	0.118175i	$0.0377044\pi$
$8$	−9.79796	−	9.79796i	−1.22474	−	1.22474i
$9$		0			0
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	0.834158	−	0.551525i	$-0.185954\pi$
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	−0.551525	+	0.834158i	$0.685954\pi$
$14$		−	30.0000i		−	2.14286i
$15$		0			0
$16$	−16.0000			−1.00000
$17$	−17.1464	+	17.1464i	−1.00861	+	1.00861i	−0.00865084	+	0.999963i	$0.502754\pi$
$17$	−17.1464	+	17.1464i	−1.00861	+	1.00861i	−0.999963	+	0.00865084i	$0.997246\pi$
$18$		0			0
$19$		−	23.0000i		−	1.21053i	−0.796025	−	0.605263i	$-0.793068\pi$
$19$		−	23.0000i		−	1.21053i	0.796025	−	0.605263i	$-0.206932\pi$
$20$		0			0
$21$		0			0
$22$	−14.6969	+	14.6969i	−0.668043	+	0.668043i
$23$	12.2474	+	12.2474i	0.532498	+	0.532498i	0.921315	−	0.388817i	$-0.127116\pi$
$23$	12.2474	+	12.2474i	0.532498	+	0.532498i	−0.388817	+	0.921315i	$0.627116\pi$
$24$		0			0
$25$		0			0
$26$	18.0000			0.692308
$27$		0			0
$28$	−48.9898	−	48.9898i	−1.74964	−	1.74964i
$29$			6.00000i			0.206897i	0.994635	+	0.103448i	$0.0329876\pi$
$29$			6.00000i			0.206897i	−0.994635	+	0.103448i	$0.967012\pi$
$30$		0			0
$31$	25.0000			0.806452			0.403226	−	0.915101i	$-0.367889\pi$
$31$	25.0000			0.806452			0.403226	+	0.915101i	$0.367889\pi$
$32$		0			0
$33$		0			0
$34$			84.0000i			2.47059i
$35$		0			0
$36$		0			0
$37$	−24.4949	+	24.4949i	−0.662024	+	0.662024i	−0.955857	−	0.293833i	$-0.905069\pi$
$37$	−24.4949	+	24.4949i	−0.662024	+	0.662024i	0.293833	+	0.955857i	$0.405069\pi$
$38$	−56.3383	−	56.3383i	−1.48259	−	1.48259i
$39$		0			0
$40$		0			0
$41$	60.0000			1.46341			0.731707	−	0.681619i	$-0.238724\pi$
$41$	60.0000			1.46341			0.731707	+	0.681619i	$0.238724\pi$
$42$		0			0
$43$	60.0125	+	60.0125i	1.39564	+	1.39564i	0.812046	+	0.583594i	$0.198354\pi$
$43$	60.0125	+	60.0125i	1.39564	+	1.39564i	0.583594	+	0.812046i	$0.301646\pi$
$44$			48.0000i			1.09091i
$45$		0			0
$46$	60.0000			1.30435
$47$	−7.34847	+	7.34847i	−0.156350	+	0.156350i	−0.780947	−	0.624597i	$-0.785263\pi$
$47$	−7.34847	+	7.34847i	−0.156350	+	0.156350i	0.624597	+	0.780947i	$0.285263\pi$
$48$		0			0
$49$		−	26.0000i		−	0.530612i
$50$		0			0
$51$		0			0
$52$	29.3939	−	29.3939i	0.565267	−	0.565267i
$53$	24.4949	+	24.4949i	0.462168	+	0.462168i	0.899365	−	0.437198i	$-0.144029\pi$
$53$	24.4949	+	24.4949i	0.462168	+	0.462168i	−0.437198	+	0.899365i	$0.644029\pi$
$54$		0			0
$55$		0			0
$56$	−120.000			−2.14286
$57$		0			0
$58$	14.6969	+	14.6969i	0.253395	+	0.253395i
$59$		−	18.0000i		−	0.305085i	−0.988297	−	0.152542i	$-0.951254\pi$
$59$		−	18.0000i		−	0.305085i	0.988297	−	0.152542i	$-0.0487461\pi$
$60$		0			0
$61$	−37.0000			−0.606557			−0.303279	−	0.952902i	$-0.598081\pi$
$61$	−37.0000			−0.606557			−0.303279	+	0.952902i	$0.598081\pi$
$62$	61.2372	−	61.2372i	0.987697	−	0.987697i
$63$		0			0
$64$		−	64.0000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−25.7196	+	25.7196i	−0.383875	+	0.383875i	−0.872496	−	0.488621i	$-0.837500\pi$
$67$	−25.7196	+	25.7196i	−0.383875	+	0.383875i	0.488621	+	0.872496i	$0.337500\pi$
$68$	137.171	+	137.171i	2.01723	+	2.01723i
$69$		0			0
$70$		0			0
$71$	−132.000			−1.85915			−0.929577	−	0.368627i	$-0.879828\pi$
$71$	−132.000			−1.85915			−0.929577	+	0.368627i	$0.879828\pi$
$72$		0			0
$73$	−24.4949	−	24.4949i	−0.335547	−	0.335547i	0.519142	−	0.854688i	$-0.326252\pi$
$73$	−24.4949	−	24.4949i	−0.335547	−	0.335547i	−0.854688	+	0.519142i	$0.826252\pi$
$74$			120.000i			1.62162i
$75$		0			0
$76$	−184.000			−2.42105
$77$	−36.7423	+	36.7423i	−0.477173	+	0.477173i
$78$		0			0
$79$		−	10.0000i		−	0.126582i	−0.997995	−	0.0632911i	$-0.979840\pi$
$79$		−	10.0000i		−	0.126582i	0.997995	−	0.0632911i	$-0.0201597\pi$
$80$		0			0
$81$		0			0
$82$	146.969	−	146.969i	1.79231	−	1.79231i
$83$	−2.44949	−	2.44949i	−0.0295119	−	0.0295119i	0.692197	−	0.721709i	$-0.256643\pi$
$83$	−2.44949	−	2.44949i	−0.0295119	−	0.0295119i	−0.721709	+	0.692197i	$0.756643\pi$
$84$		0			0
$85$		0			0
$86$	294.000			3.41860
$87$		0			0
$88$	58.7878	+	58.7878i	0.668043	+	0.668043i
$89$		−	132.000i		−	1.48315i	−0.670872	−	0.741573i	$-0.734080\pi$
$89$		−	132.000i		−	1.48315i	0.670872	−	0.741573i	$-0.265920\pi$
$90$		0			0
$91$	45.0000			0.494505
$92$	97.9796	−	97.9796i	1.06500	−	1.06500i
$93$		0			0
$94$			36.0000i			0.382979i
$95$		0			0
$96$		0			0
$97$	23.2702	−	23.2702i	0.239898	−	0.239898i	−0.576910	−	0.816808i	$-0.695741\pi$
$97$	23.2702	−	23.2702i	0.239898	−	0.239898i	0.816808	+	0.576910i	$0.195741\pi$
$98$	−63.6867	−	63.6867i	−0.649865	−	0.649865i
$99$		0			0
$100$		0			0
$101$	−96.0000			−0.950495			−0.475248	−	0.879852i	$-0.657641\pi$
$101$	−96.0000			−0.950495			−0.475248	+	0.879852i	$0.657641\pi$
$102$		0			0
$103$	−19.5959	−	19.5959i	−0.190252	−	0.190252i	0.605553	−	0.795805i	$-0.292952\pi$
$103$	−19.5959	−	19.5959i	−0.190252	−	0.190252i	−0.795805	+	0.605553i	$0.792952\pi$
$104$		−	72.0000i		−	0.692308i
$105$		0			0
$106$	120.000			1.13208
$107$	29.3939	−	29.3939i	0.274709	−	0.274709i	−0.556283	−	0.830993i	$-0.687773\pi$
$107$	29.3939	−	29.3939i	0.274709	−	0.274709i	0.830993	+	0.556283i	$0.187773\pi$
$108$		0			0
$109$			167.000i			1.53211i	0.642775	+	0.766055i	$0.277783\pi$
$109$			167.000i			1.53211i	−0.642775	+	0.766055i	$0.722217\pi$
$110$		0			0
$111$		0			0
$112$	−97.9796	+	97.9796i	−0.874818	+	0.874818i
$113$	58.7878	+	58.7878i	0.520246	+	0.520246i	0.917646	−	0.397400i	$-0.130087\pi$
$113$	58.7878	+	58.7878i	0.520246	+	0.520246i	−0.397400	+	0.917646i	$0.630087\pi$
$114$		0			0
$115$		0			0
$116$	48.0000			0.413793
$117$		0			0
$118$	−44.0908	−	44.0908i	−0.373651	−	0.373651i
$119$			210.000i			1.76471i
$120$		0			0
$121$	−85.0000			−0.702479
$122$	−90.6311	+	90.6311i	−0.742878	+	0.742878i
$123$		0			0
$124$		−	200.000i		−	1.61290i
$125$		0			0
$126$		0			0
$127$	−44.0908	+	44.0908i	−0.347172	+	0.347172i	−0.859055	−	0.511883i	$-0.828948\pi$
$127$	−44.0908	+	44.0908i	−0.347172	+	0.347172i	0.511883	+	0.859055i	$0.328948\pi$
$128$	−156.767	−	156.767i	−1.22474	−	1.22474i
$129$		0			0
$130$		0			0
$131$	−108.000			−0.824427			−0.412214	−	0.911087i	$-0.635244\pi$
$131$	−108.000			−0.824427			−0.412214	+	0.911087i	$0.635244\pi$
$132$		0			0
$133$	−140.846	−	140.846i	−1.05899	−	1.05899i
$134$			126.000i			0.940299i
$135$		0			0
$136$	336.000			2.47059
$137$	−120.025	+	120.025i	−0.876095	+	0.876095i	−0.993128	−	0.117033i	$-0.962662\pi$
$137$	−120.025	+	120.025i	−0.876095	+	0.876095i	0.117033	+	0.993128i	$0.462662\pi$
$138$		0			0
$139$			58.0000i			0.417266i	0.977994	+	0.208633i	$0.0669014\pi$
$139$			58.0000i			0.417266i	−0.977994	+	0.208633i	$0.933099\pi$
$140$		0			0
$141$		0			0
$142$	−323.333	+	323.333i	−2.27699	+	2.27699i
$143$	−22.0454	−	22.0454i	−0.154164	−	0.154164i
$144$		0			0
$145$		0			0
$146$	−120.000			−0.821918
$147$		0			0
$148$	195.959	+	195.959i	1.32405	+	1.32405i
$149$		−	186.000i		−	1.24832i	−0.781296	−	0.624161i	$-0.785441\pi$
$149$		−	186.000i		−	1.24832i	0.781296	−	0.624161i	$-0.214559\pi$
$150$		0			0
$151$	−83.0000			−0.549669			−0.274834	−	0.961492i	$-0.588623\pi$
$151$	−83.0000			−0.549669			−0.274834	+	0.961492i	$0.588623\pi$
$152$	−225.353	+	225.353i	−1.48259	+	1.48259i
$153$		0			0
$154$			180.000i			1.16883i
$155$		0			0
$156$		0			0
$157$	−45.3156	+	45.3156i	−0.288634	+	0.288634i	−0.836540	−	0.547906i	$-0.815425\pi$
$157$	−45.3156	+	45.3156i	−0.288634	+	0.288634i	0.547906	+	0.836540i	$0.315425\pi$
$158$	−24.4949	−	24.4949i	−0.155031	−	0.155031i
$159$		0			0
$160$		0			0
$161$	150.000			0.931677
$162$		0			0
$163$	99.2043	+	99.2043i	0.608616	+	0.608616i	0.942584	−	0.333969i	$-0.108388\pi$
$163$	99.2043	+	99.2043i	0.608616	+	0.608616i	−0.333969	+	0.942584i	$0.608388\pi$
$164$		−	480.000i		−	2.92683i
$165$		0			0
$166$	−12.0000			−0.0722892
$167$	97.9796	−	97.9796i	0.586704	−	0.586704i	−0.350033	−	0.936737i	$-0.613830\pi$
$167$	97.9796	−	97.9796i	0.586704	−	0.586704i	0.936737	+	0.350033i	$0.113830\pi$
$168$		0			0
$169$		−	142.000i		−	0.840237i
$170$		0			0
$171$		0			0
$172$	480.100	−	480.100i	2.79128	−	2.79128i
$173$	−173.914	−	173.914i	−1.00528	−	1.00528i	−0.999986	−	0.00529594i	$-0.998314\pi$
$173$	−173.914	−	173.914i	−1.00528	−	1.00528i	−0.00529594	−	0.999986i	$-0.501686\pi$
$174$		0			0
$175$		0			0
$176$	96.0000			0.545455
$177$		0			0
$178$	−323.333	−	323.333i	−1.81648	−	1.81648i
$179$			150.000i			0.837989i	0.907989	+	0.418994i	$0.137617\pi$
$179$			150.000i			0.837989i	−0.907989	+	0.418994i	$0.862383\pi$
$180$		0			0
$181$	−215.000			−1.18785			−0.593923	−	0.804522i	$-0.702421\pi$
$181$	−215.000			−1.18785			−0.593923	+	0.804522i	$0.702421\pi$
$182$	110.227	−	110.227i	0.605643	−	0.605643i
$183$		0			0
$184$		−	240.000i		−	1.30435i
$185$		0			0
$186$		0			0
$187$	102.879	−	102.879i	0.550153	−	0.550153i
$188$	58.7878	+	58.7878i	0.312701	+	0.312701i
$189$		0			0
$190$		0			0
$191$	234.000			1.22513			0.612565	−	0.790420i	$-0.290138\pi$
$191$	234.000			1.22513			0.612565	+	0.790420i	$0.290138\pi$
$192$		0			0
$193$	−64.9115	−	64.9115i	−0.336329	−	0.336329i	0.518655	−	0.854984i	$-0.326433\pi$
$193$	−64.9115	−	64.9115i	−0.336329	−	0.336329i	−0.854984	+	0.518655i	$0.826433\pi$
$194$		−	114.000i		−	0.587629i
$195$		0			0
$196$	−208.000			−1.06122
$197$	193.510	−	193.510i	0.982283	−	0.982283i	−0.0175631	−	0.999846i	$-0.505591\pi$
$197$	193.510	−	193.510i	0.982283	−	0.982283i	0.999846	+	0.0175631i	$0.00559079\pi$
$198$		0			0
$199$			11.0000i			0.0552764i	0.999618	+	0.0276382i	$0.00879863\pi$
$199$			11.0000i			0.0552764i	−0.999618	+	0.0276382i	$0.991201\pi$
$200$		0			0
$201$		0			0
$202$	−235.151	+	235.151i	−1.16411	+	1.16411i
$203$	36.7423	+	36.7423i	0.180997	+	0.180997i
$204$		0			0
$205$		0			0
$206$	−96.0000			−0.466019
$207$		0			0
$208$	−58.7878	−	58.7878i	−0.282633	−	0.282633i
$209$			138.000i			0.660287i
$210$		0			0
$211$	−85.0000			−0.402844			−0.201422	−	0.979505i	$-0.564556\pi$
$211$	−85.0000			−0.402844			−0.201422	+	0.979505i	$0.564556\pi$
$212$	195.959	−	195.959i	0.924336	−	0.924336i
$213$		0			0
$214$		−	144.000i		−	0.672897i
$215$		0			0
$216$		0			0
$217$	153.093	−	153.093i	0.705498	−	0.705498i
$218$	409.065	+	409.065i	1.87644	+	1.87644i
$219$		0			0
$220$		0			0
$221$	−126.000			−0.570136
$222$		0			0
$223$	101.654	+	101.654i	0.455847	+	0.455847i	0.897289	−	0.441443i	$-0.145533\pi$
$223$	101.654	+	101.654i	0.455847	+	0.455847i	−0.441443	+	0.897289i	$0.645533\pi$
$224$		0			0
$225$		0			0
$226$	288.000			1.27434
$227$	−195.959	+	195.959i	−0.863256	+	0.863256i	−0.991715	−	0.128459i	$-0.958997\pi$
$227$	−195.959	+	195.959i	−0.863256	+	0.863256i	0.128459	+	0.991715i	$0.458997\pi$
$228$		0			0
$229$		−	227.000i		−	0.991266i	−0.868532	−	0.495633i	$-0.834936\pi$
$229$		−	227.000i		−	0.991266i	0.868532	−	0.495633i	$-0.165064\pi$
$230$		0			0
$231$		0			0
$232$	58.7878	−	58.7878i	0.253395	−	0.253395i
$233$	−102.879	−	102.879i	−0.441539	−	0.441539i	0.450990	−	0.892529i	$-0.351071\pi$
$233$	−102.879	−	102.879i	−0.441539	−	0.441539i	−0.892529	+	0.450990i	$0.851071\pi$
$234$		0			0
$235$		0			0
$236$	−144.000			−0.610169
$237$		0			0
$238$	514.393	+	514.393i	2.16131	+	2.16131i
$239$			228.000i			0.953975i	0.878910	+	0.476987i	$0.158271\pi$
$239$			228.000i			0.953975i	−0.878910	+	0.476987i	$0.841729\pi$
$240$		0			0
$241$	191.000			0.792531			0.396266	−	0.918136i	$-0.370306\pi$
$241$	191.000			0.792531			0.396266	+	0.918136i	$0.370306\pi$
$242$	−208.207	+	208.207i	−0.860358	+	0.860358i
$243$		0			0
$244$			296.000i			1.21311i
$245$		0			0
$246$		0			0
$247$	84.5074	−	84.5074i	0.342135	−	0.342135i
$248$	−244.949	−	244.949i	−0.987697	−	0.987697i
$249$		0			0
$250$		0			0
$251$	−192.000			−0.764940			−0.382470	−	0.923968i	$-0.624927\pi$
$251$	−192.000			−0.764940			−0.382470	+	0.923968i	$0.624927\pi$
$252$		0			0
$253$	−73.4847	−	73.4847i	−0.290453	−	0.290453i
$254$			216.000i			0.850394i
$255$		0			0
$256$	−512.000			−2.00000
$257$	−142.070	+	142.070i	−0.552803	+	0.552803i	−0.927249	−	0.374446i	$-0.877833\pi$
$257$	−142.070	+	142.070i	−0.552803	+	0.552803i	0.374446	+	0.927249i	$0.377833\pi$
$258$		0			0
$259$			300.000i			1.15830i
$260$		0			0
$261$		0			0
$262$	−264.545	+	264.545i	−1.00971	+	1.00971i
$263$	249.848	+	249.848i	0.949992	+	0.949992i	0.998808	−	0.0488156i	$-0.0155447\pi$
$263$	249.848	+	249.848i	0.949992	+	0.949992i	−0.0488156	+	0.998808i	$0.515545\pi$
$264$		0			0
$265$		0			0
$266$	−690.000			−2.59398
$267$		0			0
$268$	205.757	+	205.757i	0.767751	+	0.767751i
$269$		−	462.000i		−	1.71747i	−0.512418	−	0.858736i	$-0.671250\pi$
$269$		−	462.000i		−	1.71747i	0.512418	−	0.858736i	$-0.328750\pi$
$270$		0			0
$271$	446.000			1.64576			0.822878	−	0.568218i	$-0.192367\pi$
$271$	446.000			1.64576			0.822878	+	0.568218i	$0.192367\pi$
$272$	274.343	−	274.343i	1.00861	−	1.00861i
$273$		0			0
$274$			588.000i			2.14599i
$275$		0			0
$276$		0			0
$277$	−175.139	+	175.139i	−0.632269	+	0.632269i	−0.948637	−	0.316368i	$-0.897537\pi$
$277$	−175.139	+	175.139i	−0.632269	+	0.632269i	0.316368	+	0.948637i	$0.397537\pi$
$278$	142.070	+	142.070i	0.511045	+	0.511045i
$279$		0			0
$280$		0			0
$281$	546.000			1.94306			0.971530	−	0.236916i	$-0.0761365\pi$
$281$	546.000			1.94306			0.971530	+	0.236916i	$0.0761365\pi$
$282$		0			0
$283$	−192.285	−	192.285i	−0.679452	−	0.679452i	0.280424	−	0.959876i	$-0.409525\pi$
$283$	−192.285	−	192.285i	−0.679452	−	0.679452i	−0.959876	+	0.280424i	$0.909525\pi$
$284$			1056.00i			3.71831i
$285$		0			0
$286$	−108.000			−0.377622
$287$	367.423	−	367.423i	1.28022	−	1.28022i
$288$		0			0
$289$		−	299.000i		−	1.03460i
$290$		0			0
$291$		0			0
$292$	−195.959	+	195.959i	−0.671093	+	0.671093i
$293$	−276.792	−	276.792i	−0.944684	−	0.944684i	0.0538645	−	0.998548i	$-0.482846\pi$
$293$	−276.792	−	276.792i	−0.944684	−	0.944684i	−0.998548	+	0.0538645i	$0.982846\pi$
$294$		0			0
$295$		0			0
$296$	480.000			1.62162
$297$		0			0
$298$	−455.605	−	455.605i	−1.52888	−	1.52888i
$299$			90.0000i			0.301003i
$300$		0			0
$301$	735.000			2.44186
$302$	−203.308	+	203.308i	−0.673204	+	0.673204i
$303$		0			0
$304$			368.000i			1.21053i
$305$		0			0
$306$		0			0
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	−0.551155	−	0.834403i	$-0.685813\pi$
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	0.834403	+	0.551155i	$0.185813\pi$
$308$	293.939	+	293.939i	0.954347	+	0.954347i
$309$		0			0
$310$		0			0
$311$	294.000			0.945338			0.472669	−	0.881240i	$-0.343291\pi$
$311$	294.000			0.945338			0.472669	+	0.881240i	$0.343291\pi$
$312$		0			0
$313$	393.143	+	393.143i	1.25605	+	1.25605i	0.952964	+	0.303085i	$0.0980165\pi$
$313$	393.143	+	393.143i	1.25605	+	1.25605i	0.303085	+	0.952964i	$0.401984\pi$
$314$			222.000i			0.707006i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	−93.0806	+	93.0806i	−0.293630	+	0.293630i	−0.838512	−	0.544883i	$-0.816574\pi$
$317$	−93.0806	+	93.0806i	−0.293630	+	0.293630i	0.544883	+	0.838512i	$0.316574\pi$
$318$		0			0
$319$		−	36.0000i		−	0.112853i
$320$		0			0
$321$		0			0
$322$	367.423	−	367.423i	1.14107	−	1.14107i
$323$	394.368	+	394.368i	1.22095	+	1.22095i
$324$		0			0
$325$		0			0
$326$	486.000			1.49080
$327$		0			0
$328$	−587.878	−	587.878i	−1.79231	−	1.79231i
$329$			90.0000i			0.273556i
$330$		0			0
$331$	−178.000			−0.537764			−0.268882	−	0.963173i	$-0.586654\pi$
$331$	−178.000			−0.537764			−0.268882	+	0.963173i	$0.586654\pi$
$332$	−19.5959	+	19.5959i	−0.0590238	+	0.0590238i
$333$		0			0
$334$		−	480.000i		−	1.43713i
$335$		0			0
$336$		0			0
$337$	−150.644	+	150.644i	−0.447014	+	0.447014i	−0.894361	−	0.447347i	$-0.852369\pi$
$337$	−150.644	+	150.644i	−0.447014	+	0.447014i	0.447347	+	0.894361i	$0.352369\pi$
$338$	−347.828	−	347.828i	−1.02908	−	1.02908i
$339$		0			0
$340$		0			0
$341$	−150.000			−0.439883
$342$		0			0
$343$	140.846	+	140.846i	0.410629	+	0.410629i
$344$		−	1176.00i		−	3.41860i
$345$		0			0
$346$	−852.000			−2.46243
$347$	374.772	−	374.772i	1.08003	−	1.08003i	0.0835290	−	0.996505i	$-0.473381\pi$
$347$	374.772	−	374.772i	1.08003	−	1.08003i	0.996505	−	0.0835290i	$-0.0266191\pi$
$348$		0			0
$349$			514.000i			1.47278i	0.676558	+	0.736390i	$0.263471\pi$
$349$			514.000i			1.47278i	−0.676558	+	0.736390i	$0.736529\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−242.499	−	242.499i	−0.686967	−	0.686967i	0.274593	−	0.961561i	$-0.411457\pi$
$353$	−242.499	−	242.499i	−0.686967	−	0.686967i	−0.961561	+	0.274593i	$0.911457\pi$
$354$		0			0
$355$		0			0
$356$	−1056.00			−2.96629
$357$		0			0
$358$	367.423	+	367.423i	1.02632	+	1.02632i
$359$		−	648.000i		−	1.80501i	−0.430675	−	0.902507i	$-0.641725\pi$
$359$		−	648.000i		−	1.80501i	0.430675	−	0.902507i	$-0.358275\pi$
$360$		0			0
$361$	−168.000			−0.465374
$362$	−526.640	+	526.640i	−1.45481	+	1.45481i
$363$		0			0
$364$		−	360.000i		−	0.989011i
$365$		0			0
$366$		0			0
$367$	143.295	−	143.295i	0.390450	−	0.390450i	−0.484398	−	0.874848i	$-0.660961\pi$
$367$	143.295	−	143.295i	0.390450	−	0.390450i	0.874848	+	0.484398i	$0.160961\pi$
$368$	−195.959	−	195.959i	−0.532498	−	0.532498i
$369$		0			0
$370$		0			0
$371$	300.000			0.808625
$372$		0			0
$373$	35.5176	+	35.5176i	0.0952215	+	0.0952215i	0.753113	−	0.657891i	$-0.228551\pi$
$373$	35.5176	+	35.5176i	0.0952215	+	0.0952215i	−0.657891	+	0.753113i	$0.728551\pi$
$374$		−	504.000i		−	1.34759i
$375$		0			0
$376$	144.000			0.382979
$377$	−22.0454	+	22.0454i	−0.0584759	+	0.0584759i
$378$		0			0
$379$		−	215.000i		−	0.567282i	−0.958930	−	0.283641i	$-0.908458\pi$
$379$		−	215.000i		−	0.567282i	0.958930	−	0.283641i	$-0.0915425\pi$
$380$		0			0
$381$		0			0
$382$	573.181	−	573.181i	1.50047	−	1.50047i
$383$	−19.5959	−	19.5959i	−0.0511643	−	0.0511643i	0.681062	−	0.732226i	$-0.261519\pi$
$383$	−19.5959	−	19.5959i	−0.0511643	−	0.0511643i	−0.732226	+	0.681062i	$0.761519\pi$
$384$		0			0
$385$		0			0
$386$	−318.000			−0.823834
$387$		0			0
$388$	−186.161	−	186.161i	−0.479797	−	0.479797i
$389$			192.000i			0.493573i	0.969070	+	0.246787i	$0.0793747\pi$
$389$			192.000i			0.493573i	−0.969070	+	0.246787i	$0.920625\pi$
$390$		0			0
$391$	−420.000			−1.07417
$392$	−254.747	+	254.747i	−0.649865	+	0.649865i
$393$		0			0
$394$		−	948.000i		−	2.40609i
$395$		0			0
$396$		0			0
$397$	−366.199	+	366.199i	−0.922415	+	0.922415i	−0.997200	−	0.0747848i	$-0.976173\pi$
$397$	−366.199	+	366.199i	−0.922415	+	0.922415i	0.0747848	+	0.997200i	$0.476173\pi$
$398$	26.9444	+	26.9444i	0.0676995	+	0.0676995i
$399$		0			0
$400$		0			0
$401$	−228.000			−0.568579			−0.284289	−	0.958739i	$-0.591758\pi$
$401$	−228.000			−0.568579			−0.284289	+	0.958739i	$0.591758\pi$
$402$		0			0
$403$	91.8559	+	91.8559i	0.227930	+	0.227930i
$404$			768.000i			1.90099i
$405$		0			0
$406$	180.000			0.443350
$407$	146.969	−	146.969i	0.361104	−	0.361104i
$408$		0			0
$409$		−	707.000i		−	1.72861i	−0.502971	−	0.864303i	$-0.667760\pi$
$409$		−	707.000i		−	1.72861i	0.502971	−	0.864303i	$-0.332240\pi$
$410$		0			0
$411$		0			0
$412$	−156.767	+	156.767i	−0.380503	+	0.380503i
$413$	−110.227	−	110.227i	−0.266894	−	0.266894i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	338.030	+	338.030i	0.808683	+	0.808683i
$419$			48.0000i			0.114558i	0.998358	+	0.0572792i	$0.0182425\pi$
$419$			48.0000i			0.114558i	−0.998358	+	0.0572792i	$0.981757\pi$
$420$		0			0
$421$	514.000			1.22090			0.610451	−	0.792054i	$-0.290988\pi$
$421$	514.000			1.22090			0.610451	+	0.792054i	$0.290988\pi$
$422$	−208.207	+	208.207i	−0.493381	+	0.493381i
$423$		0			0
$424$		−	480.000i		−	1.13208i
$425$		0			0
$426$		0			0
$427$	−226.578	+	226.578i	−0.530627	+	0.530627i
$428$	−235.151	−	235.151i	−0.549418	−	0.549418i
$429$		0			0
$430$		0			0
$431$	−30.0000			−0.0696056			−0.0348028	−	0.999394i	$-0.511080\pi$
$431$	−30.0000			−0.0696056			−0.0348028	+	0.999394i	$0.511080\pi$
$432$		0			0
$433$	−353.951	−	353.951i	−0.817439	−	0.817439i	0.168297	−	0.985736i	$-0.446173\pi$
$433$	−353.951	−	353.951i	−0.817439	−	0.817439i	−0.985736	+	0.168297i	$0.946173\pi$
$434$		−	750.000i		−	1.72811i
$435$		0			0
$436$	1336.00			3.06422
$437$	281.691	−	281.691i	0.644603	−	0.644603i
$438$		0			0
$439$			575.000i			1.30979i	0.755718	+	0.654897i	$0.227288\pi$
$439$			575.000i			1.30979i	−0.755718	+	0.654897i	$0.772712\pi$
$440$		0			0
$441$		0			0
$442$	−308.636	+	308.636i	−0.698271	+	0.698271i
$443$	112.677	+	112.677i	0.254349	+	0.254349i	0.822751	−	0.568402i	$-0.192438\pi$
$443$	112.677	+	112.677i	0.254349	+	0.254349i	−0.568402	+	0.822751i	$0.692438\pi$
$444$		0			0
$445$		0			0
$446$	498.000			1.11659
$447$		0			0
$448$	−391.918	−	391.918i	−0.874818	−	0.874818i
$449$			204.000i			0.454343i	0.973855	+	0.227171i	$0.0729478\pi$
$449$			204.000i			0.454343i	−0.973855	+	0.227171i	$0.927052\pi$
$450$		0			0
$451$	−360.000			−0.798226
$452$	470.302	−	470.302i	1.04049	−	1.04049i
$453$		0			0
$454$			960.000i			2.11454i
$455$		0			0
$456$		0			0
$457$	440.908	−	440.908i	0.964788	−	0.964788i	−0.0346127	−	0.999401i	$-0.511020\pi$
$457$	440.908	−	440.908i	0.964788	−	0.964788i	0.999401	+	0.0346127i	$0.0110198\pi$
$458$	−556.034	−	556.034i	−1.21405	−	1.21405i
$459$		0			0
$460$		0			0
$461$	−132.000			−0.286334			−0.143167	−	0.989699i	$-0.545729\pi$
$461$	−132.000			−0.286334			−0.143167	+	0.989699i	$0.545729\pi$
$462$		0			0
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	0.0566875	−	0.998392i	$-0.481946\pi$
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	−0.998392	+	0.0566875i	$0.981946\pi$
$464$		−	96.0000i		−	0.206897i
$465$		0			0
$466$	−504.000			−1.08155
$467$	−276.792	+	276.792i	−0.592703	+	0.592703i	−0.938361	−	0.345658i	$-0.887656\pi$
$467$	−276.792	+	276.792i	−0.592703	+	0.592703i	0.345658	+	0.938361i	$0.387656\pi$
$468$		0			0
$469$			315.000i			0.671642i
$470$		0			0
$471$		0			0
$472$	−176.363	+	176.363i	−0.373651	+	0.373651i
$473$	−360.075	−	360.075i	−0.761258	−	0.761258i
$474$		0			0
$475$		0			0
$476$	1680.00			3.52941
$477$		0			0
$478$	558.484	+	558.484i	1.16838	+	1.16838i
$479$			810.000i			1.69102i	0.533957	+	0.845511i	$0.320704\pi$
$479$			810.000i			1.69102i	−0.533957	+	0.845511i	$0.679296\pi$
$480$		0			0
$481$	−180.000			−0.374220
$482$	467.853	−	467.853i	0.970648	−	0.970648i
$483$		0			0
$484$			680.000i			1.40496i
$485$		0			0
$486$		0			0
$487$	488.673	−	488.673i	1.00344	−	1.00344i	0.00344166	−	0.999994i	$-0.498904\pi$
$487$	488.673	−	488.673i	1.00344	−	1.00344i	0.999994	−	0.00344166i	$-0.00109552\pi$
$488$	362.524	+	362.524i	0.742878	+	0.742878i
$489$		0			0
$490$		0			0
$491$	−348.000			−0.708758			−0.354379	−	0.935102i	$-0.615308\pi$
$491$	−348.000			−0.708758			−0.354379	+	0.935102i	$0.615308\pi$
$492$		0			0
$493$	−102.879	−	102.879i	−0.208679	−	0.208679i
$494$		−	414.000i		−	0.838057i
$495$		0			0
$496$	−400.000			−0.806452
$497$	−808.332	+	808.332i	−1.62642	+	1.62642i
$498$		0			0
$499$			227.000i			0.454910i	0.973789	+	0.227455i	$0.0730404\pi$
$499$			227.000i			0.454910i	−0.973789	+	0.227455i	$0.926960\pi$
$500$		0			0
$501$		0			0
$502$	−470.302	+	470.302i	−0.936857	+	0.936857i
$503$	137.171	+	137.171i	0.272707	+	0.272707i	0.830189	−	0.557482i	$-0.188233\pi$
$503$	137.171	+	137.171i	0.272707	+	0.272707i	−0.557482	+	0.830189i	$0.688233\pi$
$504$		0			0
$505$		0			0
$506$	−360.000			−0.711462
$507$		0			0
$508$	352.727	+	352.727i	0.694344	+	0.694344i
$509$		−	618.000i		−	1.21415i	−0.794646	−	0.607073i	$-0.792344\pi$
$509$		−	618.000i		−	1.21415i	0.794646	−	0.607073i	$-0.207656\pi$
$510$		0			0
$511$	−300.000			−0.587084
$512$	−627.069	+	627.069i	−1.22474	+	1.22474i
$513$		0			0
$514$			696.000i			1.35409i
$515$		0			0
$516$		0			0
$517$	44.0908	−	44.0908i	0.0852820	−	0.0852820i
$518$	734.847	+	734.847i	1.41862	+	1.41862i
$519$		0			0
$520$		0			0
$521$	−690.000			−1.32438			−0.662188	−	0.749338i	$-0.730372\pi$
$521$	−690.000			−1.32438			−0.662188	+	0.749338i	$0.730372\pi$
$522$		0			0
$523$	255.972	+	255.972i	0.489430	+	0.489430i	0.908126	−	0.418697i	$-0.137513\pi$
$523$	255.972	+	255.972i	0.489430	+	0.489430i	−0.418697	+	0.908126i	$0.637513\pi$
$524$			864.000i			1.64885i
$525$		0			0
$526$	1224.00			2.32700
$527$	−428.661	+	428.661i	−0.813398	+	0.813398i
$528$		0			0
$529$		−	229.000i		−	0.432892i
$530$		0			0
$531$		0			0
$532$	−1126.77	+	1126.77i	−2.11798	+	2.11798i
$533$	220.454	+	220.454i	0.413610	+	0.413610i
$534$		0			0
$535$		0			0
$536$	504.000			0.940299
$537$		0			0
$538$	−1131.66	−	1131.66i	−2.10347	−	2.10347i
$539$			156.000i			0.289425i
$540$		0			0
$541$	−325.000			−0.600739			−0.300370	−	0.953823i	$-0.597110\pi$
$541$	−325.000			−0.600739			−0.300370	+	0.953823i	$0.597110\pi$
$542$	1092.47	−	1092.47i	2.01563	−	2.01563i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−489.898	+	489.898i	−0.895609	+	0.895609i	−0.995044	−	0.0994353i	$-0.968296\pi$
$547$	−489.898	+	489.898i	−0.895609	+	0.895609i	0.0994353	+	0.995044i	$0.468296\pi$
$548$	960.200	+	960.200i	1.75219	+	1.75219i
$549$		0			0
$550$		0			0
$551$	138.000			0.250454
$552$		0			0
$553$	−61.2372	−	61.2372i	−0.110736	−	0.110736i
$554$			858.000i			1.54874i
$555$		0			0
$556$	464.000			0.834532
$557$	154.318	−	154.318i	0.277052	−	0.277052i	−0.554879	−	0.831931i	$-0.687236\pi$
$557$	154.318	−	154.318i	0.277052	−	0.277052i	0.831931	+	0.554879i	$0.187236\pi$
$558$		0			0
$559$			441.000i			0.788909i
$560$		0			0
$561$		0			0
$562$	1337.42	−	1337.42i	2.37975	−	2.37975i
$563$	609.923	+	609.923i	1.08334	+	1.08334i	0.996195	+	0.0871492i	$0.0277757\pi$
$563$	609.923	+	609.923i	1.08334	+	1.08334i	0.0871492	+	0.996195i	$0.472224\pi$
$564$		0			0
$565$		0			0
$566$	−942.000			−1.66431
$567$		0			0
$568$	1293.33	+	1293.33i	2.27699	+	2.27699i
$569$			198.000i			0.347979i	0.984748	+	0.173989i	$0.0556659\pi$
$569$			198.000i			0.347979i	−0.984748	+	0.173989i	$0.944334\pi$
$570$		0			0
$571$	−169.000			−0.295972			−0.147986	−	0.988989i	$-0.547279\pi$
$571$	−169.000			−0.295972			−0.147986	+	0.988989i	$0.547279\pi$
$572$	−176.363	+	176.363i	−0.308327	+	0.308327i
$573$		0			0
$574$		−	1800.00i		−	3.13589i
$575$		0			0
$576$		0			0
$577$	243.724	−	243.724i	0.422399	−	0.422399i	−0.463630	−	0.886029i	$-0.653453\pi$
$577$	243.724	−	243.724i	0.422399	−	0.422399i	0.886029	+	0.463630i	$0.153453\pi$
$578$	−732.397	−	732.397i	−1.26712	−	1.26712i
$579$		0			0
$580$		0			0
$581$	−30.0000			−0.0516351
$582$		0			0
$583$	−146.969	−	146.969i	−0.252092	−	0.252092i
$584$			480.000i			0.821918i
$585$		0			0
$586$	−1356.00			−2.31399
$587$	−289.040	+	289.040i	−0.492402	+	0.492402i	−0.909062	−	0.416660i	$-0.863200\pi$
$587$	−289.040	+	289.040i	−0.492402	+	0.492402i	0.416660	+	0.909062i	$0.363200\pi$
$588$		0			0
$589$		−	575.000i		−	0.976231i
$590$		0			0
$591$		0			0
$592$	391.918	−	391.918i	0.662024	−	0.662024i
$593$	−421.312	−	421.312i	−0.710476	−	0.710476i	0.256159	−	0.966635i	$-0.417543\pi$
$593$	−421.312	−	421.312i	−0.710476	−	0.710476i	−0.966635	+	0.256159i	$0.917543\pi$
$594$		0			0
$595$		0			0
$596$	−1488.00			−2.49664
$597$		0			0
$598$	220.454	+	220.454i	0.368652	+	0.368652i
$599$			144.000i			0.240401i	0.992750	+	0.120200i	$0.0383537\pi$
$599$			144.000i			0.240401i	−0.992750	+	0.120200i	$0.961646\pi$
$600$		0			0
$601$	−899.000			−1.49584			−0.747920	−	0.663789i	$-0.768947\pi$
$601$	−899.000			−1.49584			−0.747920	+	0.663789i	$0.768947\pi$
$602$	1800.37	−	1800.37i	2.99066	−	2.99066i
$603$		0			0
$604$			664.000i			1.09934i
$605$		0			0
$606$		0			0
$607$	4.89898	−	4.89898i	0.00807081	−	0.00807081i	−0.703060	−	0.711131i	$-0.748183\pi$
$607$	4.89898	−	4.89898i	0.00807081	−	0.00807081i	0.711131	+	0.703060i	$0.248183\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−54.0000			−0.0883797
$612$		0			0
$613$	259.646	+	259.646i	0.423566	+	0.423566i	0.886430	−	0.462864i	$-0.153178\pi$
$613$	259.646	+	259.646i	0.423566	+	0.423566i	−0.462864	+	0.886430i	$0.653178\pi$
$614$		−	426.000i		−	0.693811i
$615$		0			0
$616$	720.000			1.16883
$617$	−9.79796	+	9.79796i	−0.0158800	+	0.0158800i	−0.715002	−	0.699122i	$-0.753574\pi$
$617$	−9.79796	+	9.79796i	−0.0158800	+	0.0158800i	0.699122	+	0.715002i	$0.253574\pi$
$618$		0			0
$619$		−	637.000i		−	1.02908i	−0.857467	−	0.514540i	$-0.827963\pi$
$619$		−	637.000i		−	1.02908i	0.857467	−	0.514540i	$-0.172037\pi$
$620$		0			0
$621$		0			0
$622$	720.150	−	720.150i	1.15780	−	1.15780i
$623$	−808.332	−	808.332i	−1.29748	−	1.29748i
$624$		0			0
$625$		0			0
$626$	1926.00			3.07668
$627$		0			0
$628$	362.524	+	362.524i	0.577268	+	0.577268i
$629$		−	840.000i		−	1.33545i
$630$		0			0
$631$	287.000			0.454834			0.227417	−	0.973798i	$-0.426972\pi$
$631$	287.000			0.454834			0.227417	+	0.973798i	$0.426972\pi$
$632$	−97.9796	+	97.9796i	−0.155031	+	0.155031i
$633$		0			0
$634$			456.000i			0.719243i
$635$		0			0
$636$		0			0
$637$	95.5301	−	95.5301i	0.149969	−	0.149969i
$638$	−88.1816	−	88.1816i	−0.138216	−	0.138216i
$639$		0			0
$640$		0			0
$641$	60.0000			0.0936037			0.0468019	−	0.998904i	$-0.485097\pi$
$641$	60.0000			0.0936037			0.0468019	+	0.998904i	$0.485097\pi$
$642$		0			0
$643$	−191.060	−	191.060i	−0.297139	−	0.297139i	0.542753	−	0.839892i	$-0.317382\pi$
$643$	−191.060	−	191.060i	−0.297139	−	0.297139i	−0.839892	+	0.542753i	$0.817382\pi$
$644$		−	1200.00i		−	1.86335i
$645$		0			0
$646$	1932.00			2.99071
$647$	656.463	−	656.463i	1.01463	−	1.01463i	0.0147349	−	0.999891i	$-0.495310\pi$
$647$	656.463	−	656.463i	1.01463	−	1.01463i	0.999891	−	0.0147349i	$-0.00469044\pi$
$648$		0			0
$649$			108.000i			0.166410i
$650$		0			0
$651$		0			0
$652$	793.635	−	793.635i	1.21723	−	1.21723i
$653$	751.993	+	751.993i	1.15160	+	1.15160i	0.986232	+	0.165365i	$0.0528803\pi$
$653$	751.993	+	751.993i	1.15160	+	1.15160i	0.165365	+	0.986232i	$0.447120\pi$
$654$		0			0
$655$		0			0
$656$	−960.000			−1.46341
$657$		0			0
$658$	220.454	+	220.454i	0.335037	+	0.335037i
$659$			420.000i			0.637329i	0.947868	+	0.318665i	$0.103234\pi$
$659$			420.000i			0.637329i	−0.947868	+	0.318665i	$0.896766\pi$
$660$		0			0
$661$	−158.000			−0.239032			−0.119516	−	0.992832i	$-0.538134\pi$
$661$	−158.000			−0.239032			−0.119516	+	0.992832i	$0.538134\pi$
$662$	−436.009	+	436.009i	−0.658624	+	0.658624i
$663$		0			0
$664$			48.0000i			0.0722892i
$665$		0			0
$666$		0			0
$667$	−73.4847	+	73.4847i	−0.110172	+	0.110172i
$668$	−783.837	−	783.837i	−1.17341	−	1.17341i
$669$		0			0
$670$		0			0
$671$	222.000			0.330849
$672$		0			0
$673$	122.474	+	122.474i	0.181983	+	0.181983i	0.792219	−	0.610236i	$-0.208926\pi$
$673$	122.474	+	122.474i	0.181983	+	0.181983i	−0.610236	+	0.792219i	$0.708926\pi$
$674$			738.000i			1.09496i
$675$		0			0
$676$	−1136.00			−1.68047
$677$	242.499	−	242.499i	0.358197	−	0.358197i	−0.504951	−	0.863148i	$-0.668489\pi$
$677$	242.499	−	242.499i	0.358197	−	0.358197i	0.863148	+	0.504951i	$0.168489\pi$
$678$		0			0
$679$		−	285.000i		−	0.419735i
$680$		0			0
$681$		0			0
$682$	−367.423	+	367.423i	−0.538744	+	0.538744i
$683$	−19.5959	−	19.5959i	−0.0286909	−	0.0286909i	0.692616	−	0.721307i	$-0.256458\pi$
$683$	−19.5959	−	19.5959i	−0.0286909	−	0.0286909i	−0.721307	+	0.692616i	$0.756458\pi$
$684$		0			0
$685$		0			0
$686$	690.000			1.00583
$687$		0			0
$688$	−960.200	−	960.200i	−1.39564	−	1.39564i
$689$			180.000i			0.261248i
$690$		0			0
$691$	−74.0000			−0.107091			−0.0535456	−	0.998565i	$-0.517052\pi$
$691$	−74.0000			−0.107091			−0.0535456	+	0.998565i	$0.517052\pi$
$692$	−1391.31	+	1391.31i	−2.01056	+	2.01056i
$693$		0			0
$694$		−	1836.00i		−	2.64553i
$695$		0			0
$696$		0			0
$697$	−1028.79	+	1028.79i	−1.47602	+	1.47602i
$698$	1259.04	+	1259.04i	1.80378	+	1.80378i
$699$		0			0
$700$		0			0
$701$	102.000			0.145506			0.0727532	−	0.997350i	$-0.476821\pi$
$701$	102.000			0.145506			0.0727532	+	0.997350i	$0.476821\pi$
$702$		0			0
$703$	563.383	+	563.383i	0.801398	+	0.801398i
$704$			384.000i			0.545455i
$705$		0			0
$706$	−1188.00			−1.68272
$707$	−587.878	+	587.878i	−0.831510	+	0.831510i
$708$		0			0
$709$			817.000i			1.15233i	0.817334	+	0.576164i	$0.195451\pi$
$709$			817.000i			1.15233i	−0.817334	+	0.576164i	$0.804549\pi$
$710$		0			0
$711$		0			0
$712$	−1293.33	+	1293.33i	−1.81648	+	1.81648i
$713$	306.186	+	306.186i	0.429434	+	0.429434i
$714$		0			0
$715$		0			0
$716$	1200.00			1.67598
$717$		0			0
$718$	−1587.27	−	1587.27i	−2.21068	−	2.21068i
$719$		−	6.00000i		−	0.00834492i	−0.999991	−	0.00417246i	$-0.998672\pi$
$719$		−	6.00000i		−	0.00834492i	0.999991	−	0.00417246i	$-0.00132814\pi$
$720$		0			0
$721$	−240.000			−0.332871
$722$	−411.514	+	411.514i	−0.569964	+	0.569964i
$723$		0			0
$724$			1720.00i			2.37569i
$725$		0			0
$726$		0			0
$727$	−45.3156	+	45.3156i	−0.0623323	+	0.0623323i	−0.737586	−	0.675253i	$-0.764034\pi$
$727$	−45.3156	+	45.3156i	−0.0623323	+	0.0623323i	0.675253	+	0.737586i	$0.264034\pi$
$728$	−440.908	−	440.908i	−0.605643	−	0.605643i
$729$		0			0
$730$		0			0
$731$	−2058.00			−2.81532
$732$		0			0
$733$	−484.999	−	484.999i	−0.661663	−	0.661663i	0.294109	−	0.955772i	$-0.404977\pi$
$733$	−484.999	−	484.999i	−0.661663	−	0.661663i	−0.955772	+	0.294109i	$0.904977\pi$
$734$		−	702.000i		−	0.956403i
$735$		0			0
$736$		0			0
$737$	154.318	−	154.318i	0.209387	−	0.209387i
$738$		0			0
$739$			298.000i			0.403248i	0.979463	+	0.201624i	$0.0646218\pi$
$739$			298.000i			0.403248i	−0.979463	+	0.201624i	$0.935378\pi$
$740$		0			0
$741$		0			0
$742$	734.847	−	734.847i	0.990360	−	0.990360i
$743$	−249.848	−	249.848i	−0.336269	−	0.336269i	0.518692	−	0.854961i	$-0.326419\pi$
$743$	−249.848	−	249.848i	−0.336269	−	0.336269i	−0.854961	+	0.518692i	$0.826419\pi$
$744$		0			0
$745$		0			0
$746$	174.000			0.233244
$747$		0			0
$748$	−823.029	−	823.029i	−1.10031	−	1.10031i
$749$		−	360.000i		−	0.480641i
$750$		0			0
$751$	1186.00			1.57923			0.789614	−	0.613604i	$-0.210281\pi$
$751$	1186.00			1.57923			0.789614	+	0.613604i	$0.210281\pi$
$752$	117.576	−	117.576i	0.156350	−	0.156350i
$753$		0			0
$754$			108.000i			0.143236i
$755$		0			0
$756$		0			0
$757$	−743.420	+	743.420i	−0.982061	+	0.982061i	−0.999842	−	0.0177810i	$-0.994340\pi$
$757$	−743.420	+	743.420i	−0.982061	+	0.982061i	0.0177810	+	0.999842i	$0.494340\pi$
$758$	−526.640	−	526.640i	−0.694776	−	0.694776i
$759$		0			0
$760$		0			0
$761$	576.000			0.756899			0.378449	−	0.925622i	$-0.376457\pi$
$761$	576.000			0.756899			0.378449	+	0.925622i	$0.376457\pi$
$762$		0			0
$763$	1022.66	+	1022.66i	1.34032	+	1.34032i
$764$		−	1872.00i		−	2.45026i
$765$		0			0
$766$	−96.0000			−0.125326
$767$	66.1362	−	66.1362i	0.0862271	−	0.0862271i
$768$		0			0
$769$			491.000i			0.638492i	0.947672	+	0.319246i	$0.103430\pi$
$769$			491.000i			0.638492i	−0.947672	+	0.319246i	$0.896570\pi$
$770$		0			0
$771$		0			0
$772$	−519.292	+	519.292i	−0.672658	+	0.672658i
$773$	519.292	+	519.292i	0.671788	+	0.671788i	0.958128	−	0.286340i	$-0.0924388\pi$
$773$	519.292	+	519.292i	0.671788	+	0.671788i	−0.286340	+	0.958128i	$0.592439\pi$
$774$		0			0
$775$		0			0
$776$	−456.000			−0.587629
$777$		0			0
$778$	470.302	+	470.302i	0.604501	+	0.604501i
$779$		−	1380.00i		−	1.77150i
$780$		0			0
$781$	792.000			1.01408
$782$	−1028.79	+	1028.79i	−1.31558	+	1.31558i
$783$		0			0
$784$			416.000i			0.530612i
$785$		0			0
$786$		0			0
$787$	123.699	−	123.699i	0.157178	−	0.157178i	−0.624137	−	0.781315i	$-0.714549\pi$
$787$	123.699	−	123.699i	0.157178	−	0.157178i	0.781315	+	0.624137i	$0.214549\pi$
$788$	−1548.08	−	1548.08i	−1.96457	−	1.96457i
$789$		0			0
$790$		0			0
$791$	720.000			0.910240
$792$		0			0
$793$	−135.947	−	135.947i	−0.171433	−	0.171433i
$794$			1794.00i			2.25945i
$795$		0			0
$796$	88.0000			0.110553
$797$	−460.504	+	460.504i	−0.577797	+	0.577797i	−0.934296	−	0.356499i	$-0.883970\pi$
$797$	−460.504	+	460.504i	−0.577797	+	0.577797i	0.356499	+	0.934296i	$0.383970\pi$
$798$		0			0
$799$		−	252.000i		−	0.315394i
$800$		0			0
$801$		0			0
$802$	−558.484	+	558.484i	−0.696364	+	0.696364i
$803$	146.969	+	146.969i	0.183025	+	0.183025i
$804$		0			0
$805$		0			0
$806$	450.000			0.558313
$807$		0			0
$808$	940.604	+	940.604i	1.16411	+	1.16411i
$809$			900.000i			1.11248i	0.831020	+	0.556242i	$0.187757\pi$
$809$			900.000i			1.11248i	−0.831020	+	0.556242i	$0.812243\pi$
$810$		0			0
$811$	1477.00			1.82121			0.910604	−	0.413280i	$-0.135617\pi$
$811$	1477.00			1.82121			0.910604	+	0.413280i	$0.135617\pi$
$812$	293.939	−	293.939i	0.361994	−	0.361994i
$813$		0			0
$814$		−	720.000i		−	0.884521i
$815$		0			0
$816$		0			0
$817$	1380.29	−	1380.29i	1.68946	−	1.68946i
$818$	−1731.79	−	1731.79i	−2.11710	−	2.11710i
$819$		0			0
$820$		0			0
$821$	786.000			0.957369			0.478685	−	0.877987i	$-0.341114\pi$
$821$	786.000			0.957369			0.478685	+	0.877987i	$0.341114\pi$
$822$		0			0
$823$	−371.098	−	371.098i	−0.450909	−	0.450909i	0.444747	−	0.895656i	$-0.353293\pi$
$823$	−371.098	−	371.098i	−0.450909	−	0.450909i	−0.895656	+	0.444747i	$0.853293\pi$
$824$			384.000i			0.466019i
$825$		0			0
$826$	−540.000			−0.653753
$827$	−788.736	+	788.736i	−0.953731	+	0.953731i	−0.998976	−	0.0452447i	$-0.985593\pi$
$827$	−788.736	+	788.736i	−0.953731	+	0.953731i	0.0452447	+	0.998976i	$0.485593\pi$
$828$		0			0
$829$		−	170.000i		−	0.205066i	−0.994730	−	0.102533i	$-0.967305\pi$
$829$		−	170.000i		−	0.205066i	0.994730	−	0.102533i	$-0.0326948\pi$
$830$		0			0
$831$		0			0
$832$	235.151	−	235.151i	0.282633	−	0.282633i
$833$	445.807	+	445.807i	0.535183	+	0.535183i
$834$		0			0
$835$		0			0
$836$	1104.00			1.32057
$837$		0			0
$838$	117.576	+	117.576i	0.140305	+	0.140305i
$839$		−	990.000i		−	1.17998i	−0.807412	−	0.589988i	$-0.799132\pi$
$839$		−	990.000i		−	1.17998i	0.807412	−	0.589988i	$-0.200868\pi$
$840$		0			0
$841$	805.000			0.957194
$842$	1259.04	−	1259.04i	1.49529	−	1.49529i
$843$		0			0
$844$			680.000i			0.805687i
$845$		0			0
$846$		0			0
$847$	−520.517	+	520.517i	−0.614541	+	0.614541i
$848$	−391.918	−	391.918i	−0.462168	−	0.462168i
$849$		0			0
$850$		0			0
$851$	−600.000			−0.705053
$852$		0			0
$853$	723.824	+	723.824i	0.848563	+	0.848563i	0.989954	−	0.141391i	$-0.0451574\pi$
$853$	723.824	+	723.824i	0.848563	+	0.848563i	−0.141391	+	0.989954i	$0.545157\pi$
$854$			1110.00i			1.29977i
$855$		0			0
$856$	−576.000			−0.672897
$857$	739.746	−	739.746i	0.863181	−	0.863181i	−0.128525	−	0.991706i	$-0.541024\pi$
$857$	739.746	−	739.746i	0.863181	−	0.863181i	0.991706	+	0.128525i	$0.0410244\pi$
$858$		0			0
$859$		−	1510.00i		−	1.75786i	−0.476953	−	0.878929i	$-0.658259\pi$
$859$		−	1510.00i		−	1.75786i	0.476953	−	0.878929i	$-0.341741\pi$
$860$		0			0
$861$		0			0
$862$	−73.4847	+	73.4847i	−0.0852491	+	0.0852491i
$863$	−372.322	−	372.322i	−0.431428	−	0.431428i	0.457686	−	0.889114i	$-0.348678\pi$
$863$	−372.322	−	372.322i	−0.431428	−	0.431428i	−0.889114	+	0.457686i	$0.848678\pi$
$864$		0			0
$865$		0			0
$866$	−1734.00			−2.00231
$867$		0			0
$868$	−1224.74	−	1224.74i	−1.41100	−	1.41100i
$869$			60.0000i			0.0690449i
$870$		0			0
$871$	−189.000			−0.216992
$872$	1636.26	−	1636.26i	1.87644	−	1.87644i
$873$		0			0
$874$		−	1380.00i		−	1.57895i
$875$		0			0
$876$		0			0
$877$	300.062	−	300.062i	0.342147	−	0.342147i	−0.515027	−	0.857174i	$-0.672218\pi$
$877$	300.062	−	300.062i	0.342147	−	0.342147i	0.857174	+	0.515027i	$0.172218\pi$
$878$	1408.46	+	1408.46i	1.60416	+	1.60416i
$879$		0			0
$880$		0			0
$881$	216.000			0.245176			0.122588	−	0.992458i	$-0.460881\pi$
$881$	216.000			0.245176			0.122588	+	0.992458i	$0.460881\pi$
$882$		0			0
$883$	−846.299	−	846.299i	−0.958436	−	0.958436i	0.0407343	−	0.999170i	$-0.487030\pi$
$883$	−846.299	−	846.299i	−0.958436	−	0.958436i	−0.999170	+	0.0407343i	$0.987030\pi$
$884$			1008.00i			1.14027i
$885$		0			0
$886$	552.000			0.623025
$887$	996.942	−	996.942i	1.12395	−	1.12395i	0.132807	−	0.991142i	$-0.457601\pi$
$887$	996.942	−	996.942i	1.12395	−	1.12395i	0.991142	−	0.132807i	$-0.0423989\pi$
$888$		0			0
$889$			540.000i			0.607424i
$890$		0			0
$891$		0			0
$892$	813.231	−	813.231i	0.911693	−	0.911693i
$893$	169.015	+	169.015i	0.189266	+	0.189266i
$894$		0			0
$895$		0			0
$896$	−1920.00			−2.14286
$897$		0			0
$898$	499.696	+	499.696i	0.556454	+	0.556454i
$899$			150.000i			0.166852i
$900$		0			0
$901$	−840.000			−0.932297
$902$	−881.816	+	881.816i	−0.977623	+	0.977623i
$903$		0			0
$904$		−	1152.00i		−	1.27434i
$905$		0			0
$906$		0			0
$907$	−53.8888	+	53.8888i	−0.0594143	+	0.0594143i	−0.736190	−	0.676775i	$-0.763377\pi$
$907$	−53.8888	+	53.8888i	−0.0594143	+	0.0594143i	0.676775	+	0.736190i	$0.263377\pi$
$908$	1567.67	+	1567.67i	1.72651	+	1.72651i
$909$		0			0
$910$		0			0
$911$	558.000			0.612514			0.306257	−	0.951949i	$-0.400923\pi$
$911$	558.000			0.612514			0.306257	+	0.951949i	$0.400923\pi$
$912$		0			0
$913$	14.6969	+	14.6969i	0.0160974	+	0.0160974i
$914$		−	2160.00i		−	2.36324i
$915$		0			0
$916$	−1816.00			−1.98253
$917$	−661.362	+	661.362i	−0.721224	+	0.721224i
$918$		0			0
$919$		−	755.000i		−	0.821545i	−0.911738	−	0.410773i	$-0.865259\pi$
$919$		−	755.000i		−	0.821545i	0.911738	−	0.410773i	$-0.134741\pi$
$920$		0			0
$921$		0			0
$922$	−323.333	+	323.333i	−0.350686	+	0.350686i
$923$	−484.999	−	484.999i	−0.525459	−	0.525459i
$924$		0			0
$925$		0			0
$926$	−2136.00			−2.30670
$927$		0			0
$928$		0			0
$929$		−	762.000i		−	0.820237i	−0.912032	−	0.410118i	$-0.865487\pi$
$929$		−	762.000i		−	0.820237i	0.912032	−	0.410118i	$-0.134513\pi$
$930$		0			0
$931$	−598.000			−0.642320
$932$	−823.029	+	823.029i	−0.883078	+	0.883078i
$933$		0			0
$934$			1356.00i			1.45182i
$935$		0			0
$936$		0			0
$937$	388.244	−	388.244i	0.414348	−	0.414348i	−0.468902	−	0.883250i	$-0.655350\pi$
$937$	388.244	−	388.244i	0.414348	−	0.414348i	0.883250	+	0.468902i	$0.155350\pi$
$938$	771.589	+	771.589i	0.822590	+	0.822590i
$939$		0			0
$940$		0			0
$941$	690.000			0.733262			0.366631	−	0.930366i	$-0.380511\pi$
$941$	690.000			0.733262			0.366631	+	0.930366i	$0.380511\pi$
$942$		0			0
$943$	734.847	+	734.847i	0.779265	+	0.779265i
$944$			288.000i			0.305085i
$945$		0			0
$946$	−1764.00			−1.86469
$947$	100.429	−	100.429i	0.106050	−	0.106050i	−0.652091	−	0.758141i	$-0.726108\pi$
$947$	100.429	−	100.429i	0.106050	−	0.106050i	0.758141	+	0.652091i	$0.226108\pi$
$948$		0			0
$949$		−	180.000i		−	0.189673i
$950$		0			0
$951$		0			0
$952$	2057.57	−	2057.57i	2.16131	−	2.16131i
$953$	749.544	+	749.544i	0.786510	+	0.786510i	0.980920	−	0.194410i	$-0.0622794\pi$
$953$	749.544	+	749.544i	0.786510	+	0.786510i	−0.194410	+	0.980920i	$0.562279\pi$
$954$		0			0
$955$		0			0
$956$	1824.00			1.90795
$957$		0			0
$958$	1984.09	+	1984.09i	2.07107	+	2.07107i
$959$			1470.00i			1.53285i
$960$		0			0
$961$	−336.000			−0.349636
$962$	−440.908	+	440.908i	−0.458324	+	0.458324i
$963$		0			0
$964$		−	1528.00i		−	1.58506i
$965$		0			0
$966$		0			0
$967$	4.89898	−	4.89898i	0.00506616	−	0.00506616i	−0.704569	−	0.709635i	$-0.748860\pi$
$967$	4.89898	−	4.89898i	0.00506616	−	0.00506616i	0.709635	+	0.704569i	$0.248860\pi$
$968$	832.827	+	832.827i	0.860358	+	0.860358i
$969$		0			0
$970$		0			0
$971$	−90.0000			−0.0926880			−0.0463440	−	0.998926i	$-0.514757\pi$
$971$	−90.0000			−0.0926880			−0.0463440	+	0.998926i	$0.514757\pi$
$972$		0			0
$973$	355.176	+	355.176i	0.365032	+	0.365032i
$974$		−	2394.00i		−	2.45791i
$975$		0			0
$976$	592.000			0.606557
$977$	301.287	−	301.287i	0.308380	−	0.308380i	−0.535901	−	0.844281i	$-0.680028\pi$
$977$	301.287	−	301.287i	0.308380	−	0.308380i	0.844281	+	0.535901i	$0.180028\pi$
$978$		0			0
$979$			792.000i			0.808989i
$980$		0			0
$981$		0			0
$982$	−852.422	+	852.422i	−0.868047	+	0.868047i
$983$	−465.403	−	465.403i	−0.473452	−	0.473452i	0.429578	−	0.903030i	$-0.358662\pi$
$983$	−465.403	−	465.403i	−0.473452	−	0.473452i	−0.903030	+	0.429578i	$0.858662\pi$
$984$		0			0
$985$		0			0
$986$	−504.000			−0.511156
$987$		0			0
$988$	−676.059	−	676.059i	−0.684270	−	0.684270i
$989$			1470.00i			1.48635i
$990$		0			0
$991$	1067.00			1.07669			0.538345	−	0.842724i	$-0.319050\pi$
$991$	1067.00			1.07669			0.538345	+	0.842724i	$0.319050\pi$
$992$		0			0
$993$		0			0
$994$			3960.00i			3.98390i
$995$		0			0
$996$		0			0
$997$	279.242	−	279.242i	0.280082	−	0.280082i	−0.553060	−	0.833142i	$-0.686540\pi$
$997$	279.242	−	279.242i	0.280082	−	0.280082i	0.833142	+	0.553060i	$0.186540\pi$
$998$	556.034	+	556.034i	0.557148	+	0.557148i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.g.b.118.2		4
3.2	odd	2		75.3.f.b.43.1	yes	4
5.2	odd	4	inner	225.3.g.b.82.2		4
5.3	odd	4	inner	225.3.g.b.82.1		4
5.4	even	2	inner	225.3.g.b.118.1		4
12.11	even	2		1200.3.bg.g.193.2		4
15.2	even	4		75.3.f.b.7.1	✓	4
15.8	even	4		75.3.f.b.7.2	yes	4
15.14	odd	2		75.3.f.b.43.2	yes	4
60.23	odd	4		1200.3.bg.g.1057.1		4
60.47	odd	4		1200.3.bg.g.1057.2		4
60.59	even	2		1200.3.bg.g.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.b.7.1	✓	4	15.2	even	4
75.3.f.b.7.2	yes	4	15.8	even	4
75.3.f.b.43.1	yes	4	3.2	odd	2
75.3.f.b.43.2	yes	4	15.14	odd	2
225.3.g.b.82.1		4	5.3	odd	4	inner
225.3.g.b.82.2		4	5.2	odd	4	inner
225.3.g.b.118.1		4	5.4	even	2	inner
225.3.g.b.118.2		4	1.1	even	1	trivial
1200.3.bg.g.193.1		4	60.59	even	2
1200.3.bg.g.193.2		4	12.11	even	2
1200.3.bg.g.1057.1		4	60.23	odd	4
1200.3.bg.g.1057.2		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 - 2.44949i) q^{2} -8.00000i q^{4} +(6.12372 - 6.12372i) q^{7} +(-9.79796 - 9.79796i) q^{8} -6.00000 q^{11} +(3.67423 + 3.67423i) q^{13} -30.0000i q^{14} -16.0000 q^{16} +(-17.1464 + 17.1464i) q^{17} -23.0000i q^{19} +(-14.6969 + 14.6969i) q^{22} +(12.2474 + 12.2474i) q^{23} +18.0000 q^{26} +(-48.9898 - 48.9898i) q^{28} +6.00000i q^{29} +25.0000 q^{31} +84.0000i q^{34} +(-24.4949 + 24.4949i) q^{37} +(-56.3383 - 56.3383i) q^{38} +60.0000 q^{41} +(60.0125 + 60.0125i) q^{43} +48.0000i q^{44} +60.0000 q^{46} +(-7.34847 + 7.34847i) q^{47} -26.0000i q^{49} +(29.3939 - 29.3939i) q^{52} +(24.4949 + 24.4949i) q^{53} -120.000 q^{56} +(14.6969 + 14.6969i) q^{58} -18.0000i q^{59} -37.0000 q^{61} +(61.2372 - 61.2372i) q^{62} -64.0000i q^{64} +(-25.7196 + 25.7196i) q^{67} +(137.171 + 137.171i) q^{68} -132.000 q^{71} +(-24.4949 - 24.4949i) q^{73} +120.000i q^{74} -184.000 q^{76} +(-36.7423 + 36.7423i) q^{77} -10.0000i q^{79} +(146.969 - 146.969i) q^{82} +(-2.44949 - 2.44949i) q^{83} +294.000 q^{86} +(58.7878 + 58.7878i) q^{88} -132.000i q^{89} +45.0000 q^{91} +(97.9796 - 97.9796i) q^{92} +36.0000i q^{94} +(23.2702 - 23.2702i) q^{97} +(-63.6867 - 63.6867i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{11} - 64 q^{16} + 72 q^{26} + 100 q^{31} + 240 q^{41} + 240 q^{46} - 480 q^{56} - 148 q^{61} - 528 q^{71} - 736 q^{76} + 1176 q^{86} + 180 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 - 64 * q^16 + 72 * q^26 + 100 * q^31 + 240 * q^41 + 240 * q^46 - 480 * q^56 - 148 * q^61 - 528 * q^71 - 736 * q^76 + 1176 * q^86 + 180 * q^91

Introduction

L-functions

Modular forms

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Database

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