LMFDB - Embedded newform 1224.2.w.i.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(217,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1224.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1224.w (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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.269485056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{3} + 81x^{2} - 72x + 32 $ x^6 - 8x^3 + 81x^2 - 72*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			361.1
Root			$-2.31594 - 2.31594i$ of defining polynomial
Character	$\chi$	$=$	1224.361
Dual form			1224.2.w.i.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.31594 + 2.31594i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-4.31594 - 4.31594i) q^{11} +4.72716 q^{13} +(3.41122 - 2.31594i) q^{17} +6.53660i q^{19} +(-1.41122 - 1.41122i) q^{23} -5.72716i q^{25} +(5.72716 - 5.72716i) q^{31} +4.63188 q^{35} +(6.72716 - 6.72716i) q^{37} +(8.04310 + 8.04310i) q^{41} +2.53660i q^{43} +1.80944 q^{47} -5.00000i q^{49} -1.80944i q^{53} +19.9909 q^{55} +8.63188i q^{59} +(6.53660 + 6.53660i) q^{61} +(-10.9478 + 10.9478i) q^{65} +5.45432 q^{67} +(6.35904 - 6.35904i) q^{71} +(-7.53660 + 7.53660i) q^{73} +8.63188i q^{77} +(-12.2638 - 12.2638i) q^{79} -13.2638i q^{83} +(-2.53660 + 13.2638i) q^{85} +7.36812 q^{89} +(-4.72716 - 4.72716i) q^{91} +(-15.1384 - 15.1384i) q^{95} +(12.9909 - 12.9909i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 12 q^{11} + 6 q^{17} + 6 q^{23} + 6 q^{31} + 12 q^{37} + 6 q^{41} + 12 q^{47} + 36 q^{55} + 12 q^{61} - 24 q^{65} - 24 q^{67} - 18 q^{71} - 18 q^{73} - 18 q^{79} + 12 q^{85} + 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 12 * q^11 + 6 * q^17 + 6 * q^23 + 6 * q^31 + 12 * q^37 + 6 * q^41 + 12 * q^47 + 36 * q^55 + 12 * q^61 - 24 * q^65 - 24 * q^67 - 18 * q^71 - 18 * q^73 - 18 * q^79 + 12 * q^85 + 72 * q^89 - 48 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	−2.31594	+	2.31594i	−1.03572	+	1.03572i	−0.0363821	+	0.999338i	$0.511583\pi$
$5$	−2.31594	+	2.31594i	−1.03572	+	1.03572i	−0.999338	+	0.0363821i	$0.988417\pi$
$6$		0			0
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	0.492403	−	0.870367i	$-0.336119\pi$
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	−0.870367	+	0.492403i	$0.836119\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.31594	−	4.31594i	−1.30131	−	1.30131i	−0.927512	−	0.373793i	$-0.878057\pi$
$11$	−4.31594	−	4.31594i	−1.30131	−	1.30131i	−0.373793	−	0.927512i	$-0.621943\pi$
$12$		0			0
$13$	4.72716			1.31108			0.655539	−	0.755161i	$-0.272441\pi$
$13$	4.72716			1.31108			0.655539	+	0.755161i	$0.272441\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.41122	−	2.31594i	0.827342	−	0.561698i
$17$	3.41122	−	2.31594i	0.827342	−	0.561698i
$18$		0			0
$19$			6.53660i			1.49960i	0.661665	+	0.749800i	$0.269850\pi$
$19$			6.53660i			1.49960i	−0.661665	+	0.749800i	$0.730150\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.41122	−	1.41122i	−0.294260	−	0.294260i	0.544501	−	0.838760i	$-0.316719\pi$
$23$	−1.41122	−	1.41122i	−0.294260	−	0.294260i	−0.838760	+	0.544501i	$0.816719\pi$
$24$		0			0
$25$		−	5.72716i		−	1.14543i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	5.72716	−	5.72716i	1.02863	−	1.02863i	0.0290504	−	0.999578i	$-0.490752\pi$
$31$	5.72716	−	5.72716i	1.02863	−	1.02863i	0.999578	−	0.0290504i	$-0.00924832\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.63188			0.782931
$36$		0			0
$37$	6.72716	−	6.72716i	1.10594	−	1.10594i	0.112259	−	0.993679i	$-0.464191\pi$
$37$	6.72716	−	6.72716i	1.10594	−	1.10594i	0.993679	−	0.112259i	$-0.0358088\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.04310	+	8.04310i	1.25612	+	1.25612i	0.952930	+	0.303192i	$0.0980522\pi$
$41$	8.04310	+	8.04310i	1.25612	+	1.25612i	0.303192	+	0.952930i	$0.401948\pi$
$42$		0			0
$43$			2.53660i			0.386828i	0.981117	+	0.193414i	$0.0619561\pi$
$43$			2.53660i			0.386828i	−0.981117	+	0.193414i	$0.938044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.80944			0.263934			0.131967	−	0.991254i	$-0.457871\pi$
$47$	1.80944			0.263934			0.131967	+	0.991254i	$0.457871\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.80944i		−	0.248546i	−0.992248	−	0.124273i	$-0.960340\pi$
$53$		−	1.80944i		−	0.248546i	0.992248	−	0.124273i	$-0.0396598\pi$
$54$		0			0
$55$	19.9909			2.69558
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.63188i			1.12378i	0.827214	+	0.561888i	$0.189925\pi$
$59$			8.63188i			1.12378i	−0.827214	+	0.561888i	$0.810075\pi$
$60$		0			0
$61$	6.53660	+	6.53660i	0.836926	+	0.836926i	0.988453	−	0.151527i	$-0.0484191\pi$
$61$	6.53660	+	6.53660i	0.836926	+	0.836926i	−0.151527	+	0.988453i	$0.548419\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.9478	+	10.9478i	−1.35791	+	1.35791i
$66$		0			0
$67$	5.45432			0.666351			0.333176	−	0.942865i	$-0.391880\pi$
$67$	5.45432			0.666351			0.333176	+	0.942865i	$0.391880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.35904	−	6.35904i	0.754679	−	0.754679i	−0.220669	−	0.975349i	$-0.570824\pi$
$71$	6.35904	−	6.35904i	0.754679	−	0.754679i	0.975349	+	0.220669i	$0.0708242\pi$
$72$		0			0
$73$	−7.53660	+	7.53660i	−0.882092	+	0.882092i	−0.993747	−	0.111655i	$-0.964385\pi$
$73$	−7.53660	+	7.53660i	−0.882092	+	0.882092i	0.111655	+	0.993747i	$0.464385\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			8.63188i			0.983694i
$78$		0			0
$79$	−12.2638	−	12.2638i	−1.37978	−	1.37978i	−0.844975	−	0.534806i	$-0.820385\pi$
$79$	−12.2638	−	12.2638i	−1.37978	−	1.37978i	−0.534806	−	0.844975i	$-0.679615\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	13.2638i		−	1.45589i	−0.685637	−	0.727943i	$-0.740476\pi$
$83$		−	13.2638i		−	1.45589i	0.685637	−	0.727943i	$-0.259524\pi$
$84$		0			0
$85$	−2.53660	+	13.2638i	−0.275133	+	1.43866i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.36812			0.781019			0.390510	−	0.920599i	$-0.372299\pi$
$89$	7.36812			0.781019			0.390510	+	0.920599i	$0.372299\pi$
$90$		0			0
$91$	−4.72716	−	4.72716i	−0.495541	−	0.495541i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.1384	−	15.1384i	−1.55317	−	1.55317i
$96$		0			0
$97$	12.9909	−	12.9909i	1.31903	−	1.31903i	0.404483	−	0.914546i	$-0.367452\pi$
$97$	12.9909	−	12.9909i	1.31903	−	1.31903i	0.914546	−	0.404483i	$-0.132548\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.4543			1.13975			0.569874	−	0.821732i	$-0.306992\pi$
$101$	11.4543			1.13975			0.569874	+	0.821732i	$0.306992\pi$
$102$		0			0
$103$	3.46340			0.341259			0.170629	−	0.985335i	$-0.445420\pi$
$103$	3.46340			0.341259			0.170629	+	0.985335i	$0.445420\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.68406	−	3.68406i	0.356151	−	0.356151i	−0.506241	−	0.862392i	$-0.668965\pi$
$107$	3.68406	−	3.68406i	0.356151	−	0.356151i	0.862392	+	0.506241i	$0.168965\pi$
$108$		0			0
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	0.358741	−	0.933437i	$-0.383206\pi$
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	−0.933437	+	0.358741i	$0.883206\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.58878	−	4.58878i	−0.431676	−	0.431676i	0.457522	−	0.889198i	$-0.348737\pi$
$113$	−4.58878	−	4.58878i	−0.431676	−	0.431676i	−0.889198	+	0.457522i	$0.848737\pi$
$114$		0			0
$115$	6.53660			0.609541
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.72716	−	1.09528i	−0.525008	−	0.100404i
$120$		0			0
$121$			26.2547i			2.38679i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.68406	+	1.68406i	0.150627	+	0.150627i
$126$		0			0
$127$		−	6.53660i		−	0.580030i	−0.957022	−	0.290015i	$-0.906340\pi$
$127$		−	6.53660i		−	0.580030i	0.957022	−	0.290015i	$-0.0936602\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.31594	+	4.31594i	−0.377085	+	0.377085i	−0.870050	−	0.492964i	$-0.835913\pi$
$131$	−4.31594	+	4.31594i	−0.377085	+	0.377085i	0.492964	+	0.870050i	$0.335913\pi$
$132$		0			0
$133$	6.53660	−	6.53660i	0.566795	−	0.566795i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.44132			−0.550319			−0.275160	−	0.961399i	$-0.588731\pi$
$137$	−6.44132			−0.550319			−0.275160	+	0.961399i	$0.588731\pi$
$138$		0			0
$139$	−4.72716	+	4.72716i	−0.400952	+	0.400952i	−0.878569	−	0.477616i	$-0.841501\pi$
$139$	−4.72716	+	4.72716i	−0.400952	+	0.400952i	0.477616	+	0.878569i	$0.341501\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−20.4021	−	20.4021i	−1.70611	−	1.70611i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.44132			−0.527694			−0.263847	−	0.964565i	$-0.584991\pi$
$149$	−6.44132			−0.527694			−0.263847	+	0.964565i	$0.584991\pi$
$150$		0			0
$151$		−	5.45432i		−	0.443866i	−0.975062	−	0.221933i	$-0.928763\pi$
$151$		−	5.45432i		−	0.443866i	0.975062	−	0.221933i	$-0.0712367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			26.5275i			2.13074i
$156$		0			0
$157$	−4.72716			−0.377268			−0.188634	−	0.982047i	$-0.560406\pi$
$157$	−4.72716			−0.377268			−0.188634	+	0.982047i	$0.560406\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.82244i			0.222439i
$162$		0			0
$163$	5.45432	+	5.45432i	0.427215	+	0.427215i	0.887679	−	0.460463i	$-0.152317\pi$
$163$	5.45432	+	5.45432i	0.427215	+	0.427215i	−0.460463	+	0.887679i	$0.652317\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.4844	−	12.4844i	0.966074	−	0.966074i	−0.0333694	−	0.999443i	$-0.510624\pi$
$167$	12.4844	−	12.4844i	0.966074	−	0.966074i	0.999443	+	0.0333694i	$0.0106238\pi$
$168$		0			0
$169$	9.34604			0.718926
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.9478	+	10.9478i	−0.832347	+	0.832347i	−0.987837	−	0.155490i	$-0.950304\pi$
$173$	−10.9478	+	10.9478i	−0.832347	+	0.832347i	0.155490	+	0.987837i	$0.450304\pi$
$174$		0			0
$175$	−5.72716	+	5.72716i	−0.432933	+	0.432933i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.26376i		−	0.0944580i	−0.998884	−	0.0472290i	$-0.984961\pi$
$179$		−	1.26376i		−	0.0944580i	0.998884	−	0.0472290i	$-0.0150390\pi$
$180$		0			0
$181$	−11.9909	−	11.9909i	−0.891278	−	0.891278i	0.103365	−	0.994643i	$-0.467039\pi$
$181$	−11.9909	−	11.9909i	−0.891278	−	0.891278i	−0.994643	+	0.103365i	$0.967039\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			31.1594i			2.29088i
$186$		0			0
$187$	−24.7181	−	4.72716i	−1.80757	−	0.345684i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.4413			0.755508			0.377754	−	0.925906i	$-0.376697\pi$
$191$	10.4413			0.755508			0.377754	+	0.925906i	$0.376697\pi$
$192$		0			0
$193$	−3.34604	−	3.34604i	−0.240853	−	0.240853i	0.576350	−	0.817203i	$-0.304477\pi$
$193$	−3.34604	−	3.34604i	−0.240853	−	0.240853i	−0.817203	+	0.576350i	$0.804477\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.13838	+	1.13838i	0.0811062	+	0.0811062i	0.746496	−	0.665390i	$-0.231735\pi$
$197$	1.13838	+	1.13838i	0.0811062	+	0.0811062i	−0.665390	+	0.746496i	$0.731735\pi$
$198$		0			0
$199$	1.53660	−	1.53660i	0.108927	−	0.108927i	−0.650543	−	0.759470i	$-0.725459\pi$
$199$	1.53660	−	1.53660i	0.108927	−	0.108927i	0.759470	+	0.650543i	$0.225459\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−37.2547			−2.60198
$206$		0			0
$207$		0			0
$208$		0			0
$209$	28.2116	−	28.2116i	1.95144	−	1.95144i
$210$		0			0
$211$	6.00000	+	6.00000i	0.413057	+	0.413057i	0.882802	−	0.469745i	$-0.155654\pi$
$211$	6.00000	+	6.00000i	0.413057	+	0.413057i	−0.469745	+	0.882802i	$0.655654\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.87462	−	5.87462i	−0.400646	−	0.400646i
$216$		0			0
$217$	−11.4543			−0.777570
$218$		0			0
$219$		0			0
$220$		0			0
$221$	16.1254	−	10.9478i	1.08471	−	0.736430i
$222$		0			0
$223$			17.9909i			1.20476i	0.798209	+	0.602381i	$0.205781\pi$
$223$			17.9909i			1.20476i	−0.798209	+	0.602381i	$0.794219\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.1384	+	15.1384i	1.00477	+	1.00477i	0.999989	+	0.00478099i	$0.00152184\pi$
$227$	15.1384	+	15.1384i	1.00477	+	1.00477i	0.00478099	+	0.999989i	$0.498478\pi$
$228$		0			0
$229$		−	3.07320i		−	0.203083i	−0.994831	−	0.101541i	$-0.967623\pi$
$229$		−	3.07320i		−	0.203083i	0.994831	−	0.101541i	$-0.0323775\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.77934	+	2.77934i	−0.182081	+	0.182081i	−0.792262	−	0.610181i	$-0.791097\pi$
$233$	−2.77934	+	2.77934i	−0.182081	+	0.182081i	0.610181	+	0.792262i	$0.291097\pi$
$234$		0			0
$235$	−4.19056	+	4.19056i	−0.273362	+	0.273362i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.07320			0.198789			0.0993945	−	0.995048i	$-0.468309\pi$
$239$	3.07320			0.198789			0.0993945	+	0.995048i	$0.468309\pi$
$240$		0			0
$241$	0.809441	−	0.809441i	0.0521407	−	0.0521407i	−0.680556	−	0.732696i	$-0.738262\pi$
$241$	0.809441	−	0.809441i	0.0521407	−	0.0521407i	0.732696	+	0.680556i	$0.238262\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.5797	+	11.5797i	0.739800	+	0.739800i
$246$		0			0
$247$			30.8996i			1.96609i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.0130			0.821373			0.410687	−	0.911777i	$-0.365289\pi$
$251$	13.0130			0.821373			0.410687	+	0.911777i	$0.365289\pi$
$252$		0			0
$253$			12.1815i			0.765843i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	15.5405i		−	0.969391i	−0.874683	−	0.484696i	$-0.838930\pi$
$257$		−	15.5405i		−	0.969391i	0.874683	−	0.484696i	$-0.161070\pi$
$258$		0			0
$259$	−13.4543			−0.836011
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	21.3500i		−	1.31650i	−0.752802	−	0.658248i	$-0.771298\pi$
$263$		−	21.3500i		−	1.31650i	0.752802	−	0.658248i	$-0.228702\pi$
$264$		0			0
$265$	4.19056	+	4.19056i	0.257424	+	0.257424i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.3159	+	10.3159i	−0.628974	+	0.628974i	−0.947810	−	0.318836i	$-0.896708\pi$
$269$	−10.3159	+	10.3159i	−0.628974	+	0.628974i	0.318836	+	0.947810i	$0.396708\pi$
$270$		0			0
$271$	−20.3720			−1.23751			−0.618757	−	0.785583i	$-0.712363\pi$
$271$	−20.3720			−1.23751			−0.618757	+	0.785583i	$0.712363\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−24.7181	+	24.7181i	−1.49056	+	1.49056i
$276$		0			0
$277$	1.80944	−	1.80944i	0.108719	−	0.108719i	−0.650655	−	0.759374i	$-0.725506\pi$
$277$	1.80944	−	1.80944i	0.108719	−	0.108719i	0.759374	+	0.650655i	$0.225506\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.36812i			0.439545i	0.975551	+	0.219773i	$0.0705315\pi$
$281$			7.36812i			0.439545i	−0.975551	+	0.219773i	$0.929468\pi$
$282$		0			0
$283$	8.91772	+	8.91772i	0.530103	+	0.530103i	0.920603	−	0.390500i	$-0.127698\pi$
$283$	8.91772	+	8.91772i	0.530103	+	0.530103i	−0.390500	+	0.920603i	$0.627698\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.0862i		−	0.949538i
$288$		0			0
$289$	6.27284	−	15.8004i	0.368991	−	0.929433i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.5587			0.792106			0.396053	−	0.918228i	$-0.370380\pi$
$293$	13.5587			0.792106			0.396053	+	0.918228i	$0.370380\pi$
$294$		0			0
$295$	−19.9909	−	19.9909i	−1.16392	−	1.16392i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.67106	−	6.67106i	−0.385797	−	0.385797i
$300$		0			0
$301$	2.53660	−	2.53660i	0.146207	−	0.146207i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−30.2768			−1.73364
$306$		0			0
$307$	−19.9818			−1.14042			−0.570212	−	0.821498i	$-0.693139\pi$
$307$	−19.9818			−1.14042			−0.570212	+	0.821498i	$0.693139\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.64096	−	5.64096i	0.319869	−	0.319869i	−0.528847	−	0.848717i	$-0.677376\pi$
$311$	5.64096	−	5.64096i	0.319869	−	0.319869i	0.848717	+	0.528847i	$0.177376\pi$
$312$		0			0
$313$	7.53660	+	7.53660i	0.425994	+	0.425994i	0.887261	−	0.461267i	$-0.152605\pi$
$313$	7.53660	+	7.53660i	0.425994	+	0.425994i	−0.461267	+	0.887261i	$0.652605\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.07320	+	3.07320i	0.172608	+	0.172608i	0.788124	−	0.615516i	$-0.211052\pi$
$317$	3.07320	+	3.07320i	0.172608	+	0.172608i	−0.615516	+	0.788124i	$0.711052\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.1384	+	22.2978i	0.842322	+	1.24068i
$324$		0			0
$325$		−	27.0732i		−	1.50175i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.80944	−	1.80944i	−0.0997577	−	0.0997577i
$330$		0			0
$331$		−	19.9909i		−	1.09880i	−0.835559	−	0.549400i	$-0.814856\pi$
$331$		−	19.9909i		−	1.09880i	0.835559	−	0.549400i	$-0.185144\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.6319	+	12.6319i	−0.690153	+	0.690153i
$336$		0			0
$337$	−12.2638	+	12.2638i	−0.668050	+	0.668050i	−0.957264	−	0.289215i	$-0.906606\pi$
$337$	−12.2638	+	12.2638i	−0.668050	+	0.668050i	0.289215	+	0.957264i	$0.406606\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−49.4362			−2.67712
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.25076	−	4.25076i	−0.228193	−	0.228193i	0.583744	−	0.811937i	$-0.301587\pi$
$347$	−4.25076	−	4.25076i	−0.228193	−	0.228193i	−0.811937	+	0.583744i	$0.801587\pi$
$348$		0			0
$349$			33.8004i			1.80929i	0.426164	+	0.904646i	$0.359865\pi$
$349$			33.8004i			1.80929i	−0.426164	+	0.904646i	$0.640135\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.17756			0.488472			0.244236	−	0.969716i	$-0.421463\pi$
$353$	9.17756			0.488472			0.244236	+	0.969716i	$0.421463\pi$
$354$		0			0
$355$			29.4543i			1.56327i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.2508i			0.646571i	0.946302	+	0.323285i	$0.104787\pi$
$359$			12.2508i			0.646571i	−0.946302	+	0.323285i	$0.895213\pi$
$360$		0			0
$361$	−23.7272			−1.24880
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	34.9086i		−	1.82720i
$366$		0			0
$367$	−14.6449	−	14.6449i	−0.764456	−	0.764456i	0.212668	−	0.977125i	$-0.431785\pi$
$367$	−14.6449	−	14.6449i	−0.764456	−	0.764456i	−0.977125	+	0.212668i	$0.931785\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.80944	+	1.80944i	−0.0939415	+	0.0939415i
$372$		0			0
$373$	19.8354			1.02704			0.513520	−	0.858078i	$-0.328341\pi$
$373$	19.8354			1.02704			0.513520	+	0.858078i	$0.328341\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6.72716	−	6.72716i	0.345551	−	0.345551i	−0.512898	−	0.858449i	$-0.671428\pi$
$379$	6.72716	−	6.72716i	0.345551	−	0.345551i	0.858449	+	0.512898i	$0.171428\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			25.8094i			1.31880i	0.751792	+	0.659400i	$0.229190\pi$
$383$			25.8094i			1.31880i	−0.751792	+	0.659400i	$0.770810\pi$
$384$		0			0
$385$	−19.9909	−	19.9909i	−1.01883	−	1.01883i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	18.4413i		−	0.935012i	−0.883990	−	0.467506i	$-0.845153\pi$
$389$		−	18.4413i		−	0.935012i	0.883990	−	0.467506i	$-0.154847\pi$
$390$		0			0
$391$	−8.08228	−	1.54568i	−0.408739	−	0.0781684i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	56.8043			2.85813
$396$		0			0
$397$	16.8826	+	16.8826i	0.847316	+	0.847316i	0.989797	−	0.142482i	$-0.0455082\pi$
$397$	16.8826	+	16.8826i	0.847316	+	0.847316i	−0.142482	+	0.989797i	$0.545508\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.6018	−	13.6018i	−0.679240	−	0.679240i	0.280588	−	0.959828i	$-0.409471\pi$
$401$	−13.6018	−	13.6018i	−0.679240	−	0.679240i	−0.959828	+	0.280588i	$0.909471\pi$
$402$		0			0
$403$	27.0732	−	27.0732i	1.34861	−	1.34861i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−58.0680			−2.87833
$408$		0			0
$409$	−13.2728			−0.656300			−0.328150	−	0.944626i	$-0.606425\pi$
$409$	−13.2728			−0.656300			−0.328150	+	0.944626i	$0.606425\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.63188	−	8.63188i	0.424747	−	0.424747i
$414$		0			0
$415$	30.7181	+	30.7181i	1.50789	+	1.50789i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.5145	−	17.5145i	−0.855641	−	0.855641i	0.135180	−	0.990821i	$-0.456839\pi$
$419$	−17.5145	−	17.5145i	−0.855641	−	0.855641i	−0.990821	+	0.135180i	$0.956839\pi$
$420$		0			0
$421$	9.80036			0.477640			0.238820	−	0.971064i	$-0.423239\pi$
$421$	9.80036			0.477640			0.238820	+	0.971064i	$0.423239\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.2638	−	19.5366i	−0.643387	−	0.947664i
$426$		0			0
$427$		−	13.0732i		−	0.632657i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.90472	+	6.90472i	0.332589	+	0.332589i	0.853569	−	0.520980i	$-0.174433\pi$
$431$	6.90472	+	6.90472i	0.332589	+	0.332589i	−0.520980	+	0.853569i	$0.674433\pi$
$432$		0			0
$433$			37.2547i			1.79035i	0.445719	+	0.895173i	$0.352948\pi$
$433$			37.2547i			1.79035i	−0.445719	+	0.895173i	$0.647052\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.22458	−	9.22458i	0.441272	−	0.441272i
$438$		0			0
$439$	5.72716	−	5.72716i	0.273342	−	0.273342i	−0.557102	−	0.830444i	$-0.688087\pi$
$439$	5.72716	−	5.72716i	0.273342	−	0.273342i	0.830444	+	0.557102i	$0.188087\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.64488			0.458242			0.229121	−	0.973398i	$-0.426415\pi$
$443$	9.64488			0.458242			0.229121	+	0.973398i	$0.426415\pi$
$444$		0			0
$445$	−17.0641	+	17.0641i	−0.808917	+	0.808917i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.52360	+	2.52360i	0.119096	+	0.119096i	0.764143	−	0.645047i	$-0.223162\pi$
$449$	2.52360	+	2.52360i	0.119096	+	0.119096i	−0.645047	+	0.764143i	$0.723162\pi$
$450$		0			0
$451$		−	69.4271i		−	3.26919i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21.8956			1.02648
$456$		0			0
$457$		−	18.7272i		−	0.876020i	−0.898970	−	0.438010i	$-0.855684\pi$
$457$		−	18.7272i		−	0.876020i	0.898970	−	0.438010i	$-0.144316\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	35.8775i		−	1.67098i	−0.549505	−	0.835491i	$-0.685184\pi$
$461$		−	35.8775i		−	1.67098i	0.549505	−	0.835491i	$-0.314816\pi$
$462$		0			0
$463$	−12.1464			−0.564491			−0.282246	−	0.959342i	$-0.591079\pi$
$463$	−12.1464			−0.564491			−0.282246	+	0.959342i	$0.591079\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			30.0680i			1.39138i	0.718341	+	0.695692i	$0.244902\pi$
$467$			30.0680i			1.39138i	−0.718341	+	0.695692i	$0.755098\pi$
$468$		0			0
$469$	−5.45432	−	5.45432i	−0.251857	−	0.251857i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.9478	−	10.9478i	0.503381	−	0.503381i
$474$		0			0
$475$	37.4362			1.71769
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.5576	−	27.5576i	1.25914	−	1.25914i	0.307636	−	0.951504i	$-0.400462\pi$
$479$	27.5576	−	27.5576i	1.25914	−	1.25914i	0.951504	−	0.307636i	$-0.0995379\pi$
$480$		0			0
$481$	31.8004	−	31.8004i	1.44997	−	1.44997i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			60.1724i			2.73229i
$486$		0			0
$487$	21.8826	+	21.8826i	0.991597	+	0.991597i	0.999965	−	0.00836771i	$-0.00266356\pi$
$487$	21.8826	+	21.8826i	0.991597	+	0.991597i	−0.00836771	+	0.999965i	$0.502664\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			34.9086i			1.57540i	0.616056	+	0.787702i	$0.288730\pi$
$491$			34.9086i			1.57540i	−0.616056	+	0.787702i	$0.711270\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.7181			−0.570484
$498$		0			0
$499$	15.6449	+	15.6449i	0.700361	+	0.700361i	0.964488	−	0.264127i	$-0.0850838\pi$
$499$	15.6449	+	15.6449i	0.700361	+	0.700361i	−0.264127	+	0.964488i	$0.585084\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.9387	+	11.9387i	0.532322	+	0.532322i	0.921263	−	0.388941i	$-0.127159\pi$
$503$	11.9387	+	11.9387i	0.532322	+	0.532322i	−0.388941	+	0.921263i	$0.627159\pi$
$504$		0			0
$505$	−26.5275	+	26.5275i	−1.18046	+	1.18046i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.6007			0.780138			0.390069	−	0.920786i	$-0.372451\pi$
$509$	17.6007			0.780138			0.390069	+	0.920786i	$0.372451\pi$
$510$		0			0
$511$	15.0732			0.666799
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.02102	+	8.02102i	−0.353449	+	0.353449i
$516$		0			0
$517$	−7.80944	−	7.80944i	−0.343459	−	0.343459i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.8525	−	21.8525i	−0.957377	−	0.957377i	0.0417508	−	0.999128i	$-0.486706\pi$
$521$	−21.8525	−	21.8525i	−0.957377	−	0.957377i	−0.999128	+	0.0417508i	$0.986706\pi$
$522$		0			0
$523$	−12.6921			−0.554986			−0.277493	−	0.960728i	$-0.589504\pi$
$523$	−12.6921			−0.554986			−0.277493	+	0.960728i	$0.589504\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.27284	−	32.8004i	0.273249	−	1.42881i
$528$		0			0
$529$		−	19.0169i		−	0.826823i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	38.0210	+	38.0210i	1.64687	+	1.64687i
$534$		0			0
$535$			17.0641i			0.737746i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.5797	+	21.5797i	−0.929504	+	0.929504i
$540$		0			0
$541$	12.1815	−	12.1815i	0.523723	−	0.523723i	−0.394971	−	0.918694i	$-0.629245\pi$
$541$	12.1815	−	12.1815i	0.523723	−	0.523723i	0.918694	+	0.394971i	$0.129245\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	27.7913			1.19045
$546$		0			0
$547$	6.18148	−	6.18148i	0.264301	−	0.264301i	−0.562498	−	0.826799i	$-0.690160\pi$
$547$	6.18148	−	6.18148i	0.264301	−	0.264301i	0.826799	+	0.562498i	$0.190160\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			24.5275i			1.04302i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.0602			0.934721			0.467360	−	0.884067i	$-0.345205\pi$
$557$	22.0602			0.934721			0.467360	+	0.884067i	$0.345205\pi$
$558$		0			0
$559$			11.9909i			0.507162i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.0260i			0.759705i	0.925047	+	0.379853i	$0.124025\pi$
$563$			18.0260i			0.759705i	−0.925047	+	0.379853i	$0.875975\pi$
$564$		0			0
$565$	21.2547			0.894191
$566$		0			0
$567$		0			0
$568$		0			0
$569$			28.3370i			1.18795i	0.804484	+	0.593974i	$0.202442\pi$
$569$			28.3370i			1.18795i	−0.804484	+	0.593974i	$0.797558\pi$
$570$		0			0
$571$	1.26376	+	1.26376i	0.0528868	+	0.0528868i	0.733056	−	0.680169i	$-0.238094\pi$
$571$	1.26376	+	1.26376i	0.0528868	+	0.0528868i	−0.680169	+	0.733056i	$0.738094\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.08228	+	8.08228i	−0.337054	+	0.337054i
$576$		0			0
$577$	10.7272			0.446578			0.223289	−	0.974752i	$-0.428321\pi$
$577$	10.7272			0.446578			0.223289	+	0.974752i	$0.428321\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.2638	+	13.2638i	−0.550274	+	0.550274i
$582$		0			0
$583$	−7.80944	+	7.80944i	−0.323434	+	0.323434i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			45.6449i			1.88397i	0.335661	+	0.941983i	$0.391040\pi$
$587$			45.6449i			1.88397i	−0.335661	+	0.941983i	$0.608960\pi$
$588$		0			0
$589$	37.4362	+	37.4362i	1.54253	+	1.54253i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			0.545680i			0.0224084i	0.999937	+	0.0112042i	$0.00356648\pi$
$593$			0.545680i			0.0224084i	−0.999937	+	0.0112042i	$0.996434\pi$
$594$		0			0
$595$	15.8004	−	10.7272i	0.647752	−	0.439771i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.10436			0.0859817			0.0429909	−	0.999075i	$-0.486311\pi$
$599$	2.10436			0.0859817			0.0429909	+	0.999075i	$0.486311\pi$
$600$		0			0
$601$	−29.1815	−	29.1815i	−1.19034	−	1.19034i	−0.976973	−	0.213365i	$-0.931558\pi$
$601$	−29.1815	−	29.1815i	−1.19034	−	1.19034i	−0.213365	−	0.976973i	$-0.568442\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−60.8043	−	60.8043i	−2.47205	−	2.47205i
$606$		0			0
$607$	12.8354	−	12.8354i	0.520974	−	0.520974i	−0.396891	−	0.917866i	$-0.629911\pi$
$607$	12.8354	−	12.8354i	0.520974	−	0.520974i	0.917866	+	0.396891i	$0.129911\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.55352			0.346038
$612$		0			0
$613$	15.6358			0.631524			0.315762	−	0.948838i	$-0.397740\pi$
$613$	15.6358			0.631524			0.315762	+	0.948838i	$0.397740\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.7272	−	29.7272i	1.19677	−	1.19677i	0.221642	−	0.975128i	$-0.428858\pi$
$617$	29.7272	−	29.7272i	1.19677	−	1.19677i	0.975128	−	0.221642i	$-0.0711418\pi$
$618$		0			0
$619$	−8.37204	−	8.37204i	−0.336501	−	0.336501i	0.518548	−	0.855049i	$-0.326473\pi$
$619$	−8.37204	−	8.37204i	−0.336501	−	0.336501i	−0.855049	+	0.518548i	$0.826473\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.36812	−	7.36812i	−0.295197	−	0.295197i
$624$		0			0
$625$	20.8354			0.833417
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.36812	−	38.5275i	0.293786	−	1.53619i
$630$		0			0
$631$		−	40.8996i		−	1.62819i	−0.580735	−	0.814093i	$-0.697235\pi$
$631$		−	40.8996i		−	1.62819i	0.580735	−	0.814093i	$-0.302765\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.1384	+	15.1384i	0.600748	+	0.600748i
$636$		0			0
$637$		−	23.6358i		−	0.936485i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.6018	+	13.6018i	−0.537238	+	0.537238i	−0.922717	−	0.385479i	$-0.874036\pi$
$641$	−13.6018	+	13.6018i	−0.537238	+	0.537238i	0.385479	+	0.922717i	$0.374036\pi$
$642$		0			0
$643$	−27.7913	+	27.7913i	−1.09598	+	1.09598i	−0.101106	+	0.994876i	$0.532238\pi$
$643$	−27.7913	+	27.7913i	−1.09598	+	1.09598i	−0.994876	+	0.101106i	$0.967762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.55868			−0.218534			−0.109267	−	0.994012i	$-0.534850\pi$
$647$	−5.55868			−0.218534			−0.109267	+	0.994012i	$0.534850\pi$
$648$		0			0
$649$	37.2547	−	37.2547i	1.46237	−	1.46237i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.9478	−	26.9478i	−1.05455	−	1.05455i	−0.998424	−	0.0561260i	$-0.982125\pi$
$653$	−26.9478	−	26.9478i	−1.05455	−	1.05455i	−0.0561260	−	0.998424i	$-0.517875\pi$
$654$		0			0
$655$		−	19.9909i		−	0.781110i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.1044			0.705246			0.352623	−	0.935766i	$-0.385290\pi$
$659$	18.1044			0.705246			0.352623	+	0.935766i	$0.385290\pi$
$660$		0			0
$661$		−	14.5626i		−	0.566419i	−0.959058	−	0.283210i	$-0.908601\pi$
$661$		−	14.5626i		−	0.566419i	0.959058	−	0.283210i	$-0.0913992\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			30.2768i			1.17408i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	56.4232i		−	2.17819i
$672$		0			0
$673$	21.9177	+	21.9177i	0.844866	+	0.844866i	0.989487	−	0.144621i	$-0.0461963\pi$
$673$	21.9177	+	21.9177i	0.844866	+	0.844866i	−0.144621	+	0.989487i	$0.546196\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.125382	+	0.125382i	−0.00481882	+	0.00481882i	−0.709512	−	0.704693i	$-0.751085\pi$
$677$	−0.125382	+	0.125382i	−0.00481882	+	0.00481882i	0.704693	+	0.709512i	$0.251085\pi$
$678$		0			0
$679$	−25.9818			−0.997092
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.5195	−	15.5195i	0.593837	−	0.593837i	−0.344829	−	0.938666i	$-0.612063\pi$
$683$	15.5195	−	15.5195i	0.593837	−	0.593837i	0.938666	+	0.344829i	$0.112063\pi$
$684$		0			0
$685$	14.9177	−	14.9177i	0.569977	−	0.569977i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	8.55352i		−	0.325863i
$690$		0			0
$691$	−14.3811	−	14.3811i	−0.547083	−	0.547083i	0.378513	−	0.925596i	$-0.376436\pi$
$691$	−14.3811	−	14.3811i	−0.547083	−	0.547083i	−0.925596	+	0.378513i	$0.876436\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	21.8956i		−	0.830549i
$696$		0			0
$697$	46.0641	+	8.80944i	1.74480	+	0.333681i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−35.6267			−1.34560			−0.672801	−	0.739823i	$-0.734909\pi$
$701$	−35.6267			−1.34560			−0.672801	+	0.739823i	$0.734909\pi$
$702$		0			0
$703$	43.9728	+	43.9728i	1.65846	+	1.65846i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.4543	−	11.4543i	−0.430784	−	0.430784i
$708$		0			0
$709$	−7.26376	+	7.26376i	−0.272796	+	0.272796i	−0.830225	−	0.557429i	$-0.811788\pi$
$709$	−7.26376	+	7.26376i	−0.272796	+	0.272796i	0.557429	+	0.830225i	$0.311788\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.1646			−0.605368
$714$		0			0
$715$	94.5003			3.53411
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.96990	−	6.96990i	0.259933	−	0.259933i	−0.565094	−	0.825027i	$-0.691160\pi$
$719$	6.96990	−	6.96990i	0.259933	−	0.259933i	0.825027	+	0.565094i	$0.191160\pi$
$720$		0			0
$721$	−3.46340	−	3.46340i	−0.128984	−	0.128984i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.381117			−0.0141349			−0.00706743	−	0.999975i	$-0.502250\pi$
$727$	−0.381117			−0.0141349			−0.00706743	+	0.999975i	$0.502250\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.87462	+	8.65291i	0.217281	+	0.320039i
$732$		0			0
$733$		−	15.0732i		−	0.556741i	−0.960474	−	0.278371i	$-0.910206\pi$
$733$		−	15.0732i		−	0.556741i	0.960474	−	0.278371i	$-0.0897944\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−23.5405	−	23.5405i	−0.867126	−	0.867126i
$738$		0			0
$739$		−	12.3902i		−	0.455781i	−0.973687	−	0.227890i	$-0.926817\pi$
$739$		−	12.3902i		−	0.455781i	0.973687	−	0.227890i	$-0.0731828\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−26.9909	+	26.9909i	−0.990201	+	0.990201i	−0.999952	−	0.00975157i	$-0.996896\pi$
$743$	−26.9909	+	26.9909i	−0.990201	+	0.990201i	0.00975157	+	0.999952i	$0.496896\pi$
$744$		0			0
$745$	14.9177	−	14.9177i	0.546543	−	0.546543i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.36812			−0.269225
$750$		0			0
$751$	−6.27284	+	6.27284i	−0.228899	+	0.228899i	−0.812233	−	0.583334i	$-0.801748\pi$
$751$	−6.27284	+	6.27284i	−0.228899	+	0.228899i	0.583334	+	0.812233i	$0.301748\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.6319	+	12.6319i	0.459721	+	0.459721i
$756$		0			0
$757$		−	24.3279i		−	0.884212i	−0.896963	−	0.442106i	$-0.854231\pi$
$757$		−	24.3279i		−	0.884212i	0.896963	−	0.442106i	$-0.145769\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	36.7181			1.33103			0.665515	−	0.746385i	$-0.268212\pi$
$761$	36.7181			1.33103			0.665515	+	0.746385i	$0.268212\pi$
$762$		0			0
$763$			12.0000i			0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			40.8043i			1.47336i
$768$		0			0
$769$	7.10828			0.256331			0.128166	−	0.991753i	$-0.459091\pi$
$769$	7.10828			0.256331			0.128166	+	0.991753i	$0.459091\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	4.16456i		−	0.149789i	−0.997191	−	0.0748945i	$-0.976138\pi$
$773$		−	4.16456i		−	0.149789i	0.997191	−	0.0748945i	$-0.0238620\pi$
$774$		0			0
$775$	−32.8004	−	32.8004i	−1.17822	−	1.17822i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−52.5745	+	52.5745i	−1.88368	+	1.88368i
$780$		0			0
$781$	−54.8905			−1.96414
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.9478	−	10.9478i	0.390744	−	0.390744i
$786$		0			0
$787$	−25.8094	+	25.8094i	−0.920007	+	0.920007i	−0.997029	−	0.0770222i	$-0.975459\pi$
$787$	−25.8094	+	25.8094i	−0.920007	+	0.920007i	0.0770222	+	0.997029i	$0.475459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			9.17756i			0.326316i
$792$		0			0
$793$	30.8996	+	30.8996i	1.09728	+	1.09728i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33.0992i		−	1.17243i	−0.810154	−	0.586217i	$-0.800617\pi$
$797$		−	33.0992i		−	1.17243i	0.810154	−	0.586217i	$-0.199383\pi$
$798$		0			0
$799$	6.17240	−	4.19056i	0.218364	−	0.148251i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	65.0550			2.29574
$804$		0			0
$805$	−6.53660	−	6.53660i	−0.230385	−	0.230385i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.2077	−	12.2077i	−0.429199	−	0.429199i	0.459157	−	0.888355i	$-0.348152\pi$
$809$	−12.2077	−	12.2077i	−0.429199	−	0.429199i	−0.888355	+	0.459157i	$0.848152\pi$
$810$		0			0
$811$	−36.8645	+	36.8645i	−1.29449	+	1.29449i	−0.362506	+	0.931982i	$0.618079\pi$
$811$	−36.8645	+	36.8645i	−1.29449	+	1.29449i	−0.931982	+	0.362506i	$0.881921\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−25.2638			−0.884951
$816$		0			0
$817$	−16.5808			−0.580087
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.4935	−	11.4935i	0.401126	−	0.401126i	−0.477504	−	0.878630i	$-0.658458\pi$
$821$	11.4935	−	11.4935i	0.401126	−	0.401126i	0.878630	+	0.477504i	$0.158458\pi$
$822$		0			0
$823$	−3.88264	−	3.88264i	−0.135340	−	0.135340i	0.636191	−	0.771532i	$-0.280509\pi$
$823$	−3.88264	−	3.88264i	−0.135340	−	0.135340i	−0.771532	+	0.636191i	$0.780509\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.9478	−	16.9478i	−0.589333	−	0.589333i	0.348117	−	0.937451i	$-0.386821\pi$
$827$	−16.9478	−	16.9478i	−0.589333	−	0.589333i	−0.937451	+	0.348117i	$0.886821\pi$
$828$		0			0
$829$	−34.3630			−1.19347			−0.596737	−	0.802437i	$-0.703537\pi$
$829$	−34.3630			−1.19347			−0.596737	+	0.802437i	$0.703537\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.5797	−	17.0561i	−0.401213	−	0.590959i
$834$		0			0
$835$			57.8264i			2.00116i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.2025	−	25.2025i	−0.870087	−	0.870087i	0.122394	−	0.992482i	$-0.460943\pi$
$839$	−25.2025	−	25.2025i	−0.870087	−	0.870087i	−0.992482	+	0.122394i	$0.960943\pi$
$840$		0			0
$841$			29.0000i			1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.6449	+	21.6449i	−0.744606	+	0.744606i
$846$		0			0
$847$	26.2547	−	26.2547i	0.902122	−	0.902122i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.9870			−0.650866
$852$		0			0
$853$	23.2728	−	23.2728i	0.796847	−	0.796847i	−0.185750	−	0.982597i	$-0.559472\pi$
$853$	23.2728	−	23.2728i	0.796847	−	0.796847i	0.982597	+	0.185750i	$0.0594715\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.6358	+	28.6358i	0.978180	+	0.978180i	0.999767	−	0.0215867i	$-0.00687178\pi$
$857$	28.6358	+	28.6358i	0.978180	+	0.978180i	−0.0215867	+	0.999767i	$0.506872\pi$
$858$		0			0
$859$		−	2.92680i		−	0.0998610i	−0.998753	−	0.0499305i	$-0.984100\pi$
$859$		−	2.92680i		−	0.0998610i	0.998753	−	0.0499305i	$-0.0159000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.63188			0.157671			0.0788355	−	0.996888i	$-0.474880\pi$
$863$	4.63188			0.157671			0.0788355	+	0.996888i	$0.474880\pi$
$864$		0			0
$865$		−	50.7090i		−	1.72416i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			105.859i			3.59103i
$870$		0			0
$871$	25.7834			0.873639
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	3.36812i		−	0.113863i
$876$		0			0
$877$	16.9177	+	16.9177i	0.571271	+	0.571271i	0.932483	−	0.361213i	$-0.117637\pi$
$877$	16.9177	+	16.9177i	0.571271	+	0.571271i	−0.361213	+	0.932483i	$0.617637\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−5.47640	+	5.47640i	−0.184504	+	0.184504i	−0.793315	−	0.608811i	$-0.791647\pi$
$881$	−5.47640	+	5.47640i	−0.184504	+	0.184504i	0.608811	+	0.793315i	$0.291647\pi$
$882$		0			0
$883$	−55.5916			−1.87081			−0.935404	−	0.353581i	$-0.884964\pi$
$883$	−55.5916			−1.87081			−0.935404	+	0.353581i	$0.884964\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13.9569	+	13.9569i	−0.468627	+	0.468627i	−0.901469	−	0.432843i	$-0.857511\pi$
$887$	−13.9569	+	13.9569i	−0.468627	+	0.468627i	0.432843	+	0.901469i	$0.357511\pi$
$888$		0			0
$889$	−6.53660	+	6.53660i	−0.219231	+	0.219231i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			11.8276i			0.395795i
$894$		0			0
$895$	2.92680	+	2.92680i	0.0978320	+	0.0978320i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−4.19056	−	6.17240i	−0.139608	−	0.205633i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	55.5405			1.84623
$906$		0			0
$907$	20.3460	+	20.3460i	0.675579	+	0.675579i	0.958997	−	0.283417i	$-0.0914682\pi$
$907$	20.3460	+	20.3460i	0.675579	+	0.675579i	−0.283417	+	0.958997i	$0.591468\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−10.3380	−	10.3380i	−0.342514	−	0.342514i	0.514798	−	0.857312i	$-0.327867\pi$
$911$	−10.3380	−	10.3380i	−0.342514	−	0.342514i	−0.857312	+	0.514798i	$0.827867\pi$
$912$		0			0
$913$	−57.2456	+	57.2456i	−1.89455	+	1.89455i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.63188			0.285050
$918$		0			0
$919$	44.3720			1.46370			0.731849	−	0.681467i	$-0.238658\pi$
$919$	44.3720			1.46370			0.731849	+	0.681467i	$0.238658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.0602	−	30.0602i	0.989444	−	0.989444i
$924$		0			0
$925$	−38.5275	−	38.5275i	−1.26678	−	1.26678i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.293864	+	0.293864i	0.00964137	+	0.00964137i	0.711911	−	0.702270i	$-0.247830\pi$
$929$	0.293864	+	0.293864i	0.00964137	+	0.00964137i	−0.702270	+	0.711911i	$0.747830\pi$
$930$		0			0
$931$	32.6830			1.07114
$932$		0			0
$933$		0			0
$934$		0			0
$935$	68.1934	−	46.2978i	2.23016	−	1.51410i
$936$		0			0
$937$		−	5.23777i		−	0.171110i	−0.996333	−	0.0855552i	$-0.972734\pi$
$937$		−	5.23777i		−	0.171110i	0.996333	−	0.0855552i	$-0.0272664\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.8094	+	13.8094i	0.450175	+	0.450175i	0.895412	−	0.445238i	$-0.146881\pi$
$941$	13.8094	+	13.8094i	0.450175	+	0.450175i	−0.445238	+	0.895412i	$0.646881\pi$
$942$		0			0
$943$		−	22.7012i		−	0.739252i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.6137	−	14.6137i	0.474882	−	0.474882i	−0.428608	−	0.903490i	$-0.640996\pi$
$947$	14.6137	−	14.6137i	0.474882	−	0.474882i	0.903490	+	0.428608i	$0.140996\pi$
$948$		0			0
$949$	−35.6267	+	35.6267i	−1.15649	+	1.15649i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.7913			−0.900248			−0.450124	−	0.892966i	$-0.648620\pi$
$953$	−27.7913			−0.900248			−0.450124	+	0.892966i	$0.648620\pi$
$954$		0			0
$955$	−24.1815	+	24.1815i	−0.782494	+	0.782494i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.44132	+	6.44132i	0.208001	+	0.208001i
$960$		0			0
$961$		−	34.6007i		−	1.11615i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.4985			0.498913
$966$		0			0
$967$		−	29.6098i		−	0.952187i	−0.879395	−	0.476094i	$-0.842052\pi$
$967$		−	29.6098i		−	0.952187i	0.879395	−	0.476094i	$-0.157948\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	47.0810i		−	1.51090i	−0.655205	−	0.755451i	$-0.727418\pi$
$971$		−	47.0810i		−	1.51090i	0.655205	−	0.755451i	$-0.272582\pi$
$972$		0			0
$973$	9.45432			0.303092
$974$		0			0
$975$		0			0
$976$		0			0
$977$			11.7051i			0.374479i	0.982314	+	0.187239i	$0.0599540\pi$
$977$			11.7051i			0.374479i	−0.982314	+	0.187239i	$0.940046\pi$
$978$		0			0
$979$	−31.8004	−	31.8004i	−1.01634	−	1.01634i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.10331	−	8.10331i	0.258455	−	0.258455i	−0.565970	−	0.824426i	$-0.691498\pi$
$983$	8.10331	−	8.10331i	0.258455	−	0.258455i	0.824426	+	0.565970i	$0.191498\pi$
$984$		0			0
$985$	−5.27284			−0.168007
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.57970	−	3.57970i	0.113828	−	0.113828i
$990$		0			0
$991$	−6.84452	+	6.84452i	−0.217423	+	0.217423i	−0.807412	−	0.589988i	$-0.799132\pi$
$991$	−6.84452	+	6.84452i	−0.217423	+	0.217423i	0.589988	+	0.807412i	$0.299132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			7.11736i			0.225635i
$996$		0			0
$997$	−15.1083	−	15.1083i	−0.478484	−	0.478484i	0.426163	−	0.904647i	$-0.359865\pi$
$997$	−15.1083	−	15.1083i	−0.478484	−	0.478484i	−0.904647	+	0.426163i	$0.859865\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.w.i.361.1	yes	6
3.2	odd	2		1224.2.w.j.361.3	yes	6
4.3	odd	2		2448.2.be.w.1585.1		6
12.11	even	2		2448.2.be.v.1585.3		6
17.13	even	4	inner	1224.2.w.i.217.1	✓	6
51.47	odd	4		1224.2.w.j.217.3	yes	6
68.47	odd	4		2448.2.be.w.1441.1		6
204.47	even	4		2448.2.be.v.1441.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.2.w.i.217.1	✓	6	17.13	even	4	inner
1224.2.w.i.361.1	yes	6	1.1	even	1	trivial
1224.2.w.j.217.3	yes	6	51.47	odd	4
1224.2.w.j.361.3	yes	6	3.2	odd	2
2448.2.be.v.1441.3		6	204.47	even	4
2448.2.be.v.1585.3		6	12.11	even	2
2448.2.be.w.1441.1		6	68.47	odd	4
2448.2.be.w.1585.1		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1224.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.31594 + 2.31594i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-4.31594 - 4.31594i) q^{11} +4.72716 q^{13} +(3.41122 - 2.31594i) q^{17} +6.53660i q^{19} +(-1.41122 - 1.41122i) q^{23} -5.72716i q^{25} +(5.72716 - 5.72716i) q^{31} +4.63188 q^{35} +(6.72716 - 6.72716i) q^{37} +(8.04310 + 8.04310i) q^{41} +2.53660i q^{43} +1.80944 q^{47} -5.00000i q^{49} -1.80944i q^{53} +19.9909 q^{55} +8.63188i q^{59} +(6.53660 + 6.53660i) q^{61} +(-10.9478 + 10.9478i) q^{65} +5.45432 q^{67} +(6.35904 - 6.35904i) q^{71} +(-7.53660 + 7.53660i) q^{73} +8.63188i q^{77} +(-12.2638 - 12.2638i) q^{79} -13.2638i q^{83} +(-2.53660 + 13.2638i) q^{85} +7.36812 q^{89} +(-4.72716 - 4.72716i) q^{91} +(-15.1384 - 15.1384i) q^{95} +(12.9909 - 12.9909i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 12 q^{11} + 6 q^{17} + 6 q^{23} + 6 q^{31} + 12 q^{37} + 6 q^{41} + 12 q^{47} + 36 q^{55} + 12 q^{61} - 24 q^{65} - 24 q^{67} - 18 q^{71} - 18 q^{73} - 18 q^{79} + 12 q^{85} + 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 12 * q^11 + 6 * q^17 + 6 * q^23 + 6 * q^31 + 12 * q^37 + 6 * q^41 + 12 * q^47 + 36 * q^55 + 12 * q^61 - 24 * q^65 - 24 * q^67 - 18 * q^71 - 18 * q^73 - 18 * q^79 + 12 * q^85 + 72 * q^89 - 48 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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