LMFDB - Embedded newform 3600.2.x.m.2143.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3600,2,Mod(2143,3600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3600.2143"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-32,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(101)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2143.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3600.2143
Dual form			3600.2.x.m.3007.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 - 2.82843i) q^{7} -5.19615i q^{11} +(2.44949 + 2.44949i) q^{13} +(3.67423 - 3.67423i) q^{17} -1.73205 q^{19} +(4.24264 - 4.24264i) q^{23} +6.00000i q^{29} +3.46410i q^{31} +3.00000 q^{41} +(2.82843 - 2.82843i) q^{43} +(-4.24264 - 4.24264i) q^{47} +9.00000i q^{49} +(-7.34847 - 7.34847i) q^{53} -10.3923 q^{59} -4.00000 q^{61} +(4.94975 + 4.94975i) q^{67} +10.3923i q^{71} +(-11.0227 - 11.0227i) q^{73} +(-14.6969 + 14.6969i) q^{77} +(2.12132 - 2.12132i) q^{83} -3.00000i q^{89} -13.8564i q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{41} - 32 q^{61}+O(q^{100}) $ 8 * q + 24 * q^41 - 32 * q^61

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.82843	−	2.82843i	−1.06904	−	1.06904i	−0.997433	−	0.0716124i	$-0.977186\pi$
$7$	−2.82843	−	2.82843i	−1.06904	−	1.06904i	−0.0716124	−	0.997433i	$-0.522814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.19615i		−	1.56670i	−0.621582	−	0.783349i	$-0.713510\pi$
$11$		−	5.19615i		−	1.56670i	0.621582	−	0.783349i	$-0.286490\pi$
$12$		0			0
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	0.959857	−	0.280491i	$-0.0904971\pi$
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	−0.280491	+	0.959857i	$0.590497\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.67423	−	3.67423i	0.891133	−	0.891133i	−0.103497	−	0.994630i	$-0.533003\pi$
$17$	3.67423	−	3.67423i	0.891133	−	0.891133i	0.994630	+	0.103497i	$0.0330032\pi$
$18$		0			0
$19$	−1.73205			−0.397360			−0.198680	−	0.980064i	$-0.563665\pi$
$19$	−1.73205			−0.397360			−0.198680	+	0.980064i	$0.563665\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.24264	−	4.24264i	0.884652	−	0.884652i	−0.109351	−	0.994003i	$-0.534877\pi$
$23$	4.24264	−	4.24264i	0.884652	−	0.884652i	0.994003	+	0.109351i	$0.0348774\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000			0.468521			0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000			0.468521			0.234261	+	0.972174i	$0.424733\pi$
$42$		0			0
$43$	2.82843	−	2.82843i	0.431331	−	0.431331i	−0.457750	−	0.889081i	$-0.651344\pi$
$43$	2.82843	−	2.82843i	0.431331	−	0.431331i	0.889081	+	0.457750i	$0.151344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.24264	−	4.24264i	−0.618853	−	0.618853i	0.326384	−	0.945237i	$-0.394170\pi$
$47$	−4.24264	−	4.24264i	−0.618853	−	0.618853i	−0.945237	+	0.326384i	$0.894170\pi$
$48$		0			0
$49$			9.00000i			1.28571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.34847	−	7.34847i	−1.00939	−	1.00939i	−0.999955	−	0.00943438i	$-0.996997\pi$
$53$	−7.34847	−	7.34847i	−1.00939	−	1.00939i	−0.00943438	−	0.999955i	$-0.503003\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.3923			−1.35296			−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923			−1.35296			−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.94975	+	4.94975i	0.604708	+	0.604708i	0.941558	−	0.336850i	$-0.109362\pi$
$67$	4.94975	+	4.94975i	0.604708	+	0.604708i	−0.336850	+	0.941558i	$0.609362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.3923i			1.23334i	0.787222	+	0.616670i	$0.211519\pi$
$71$			10.3923i			1.23334i	−0.787222	+	0.616670i	$0.788481\pi$
$72$		0			0
$73$	−11.0227	−	11.0227i	−1.29011	−	1.29011i	−0.934717	−	0.355393i	$-0.884347\pi$
$73$	−11.0227	−	11.0227i	−1.29011	−	1.29011i	−0.355393	−	0.934717i	$-0.615653\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.6969	+	14.6969i	−1.67487	+	1.67487i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.12132	−	2.12132i	0.232845	−	0.232845i	−0.581034	−	0.813879i	$-0.697352\pi$
$83$	2.12132	−	2.12132i	0.232845	−	0.232845i	0.813879	+	0.581034i	$0.197352\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	3.00000i		−	0.317999i	−0.987279	−	0.159000i	$-0.949173\pi$
$89$		−	3.00000i		−	0.317999i	0.987279	−	0.159000i	$-0.0508269\pi$
$90$		0			0
$91$		−	13.8564i		−	1.45255i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	−0.910633	−	0.413217i	$-0.864405\pi$
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	0.413217	+	0.910633i	$0.364405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	9.89949	−	9.89949i	0.975426	−	0.975426i	−0.0242790	−	0.999705i	$-0.507729\pi$
$103$	9.89949	−	9.89949i	0.975426	−	0.975426i	0.999705	+	0.0242790i	$0.00772901\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.6066	−	10.6066i	−1.02538	−	1.02538i	−0.999669	−	0.0257094i	$-0.991816\pi$
$107$	−10.6066	−	10.6066i	−1.02538	−	1.02538i	−0.0257094	−	0.999669i	$-0.508184\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.67423	−	3.67423i	−0.345643	−	0.345643i	0.512841	−	0.858484i	$-0.328593\pi$
$113$	−3.67423	−	3.67423i	−0.345643	−	0.345643i	−0.858484	+	0.512841i	$0.828593\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.7846			−1.90532
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.41421	−	1.41421i	−0.125491	−	0.125491i	0.641572	−	0.767063i	$-0.278283\pi$
$127$	−1.41421	−	1.41421i	−0.125491	−	0.125491i	−0.767063	+	0.641572i	$0.778283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.3923i		−	0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$		−	10.3923i		−	0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$		0			0
$133$	4.89898	+	4.89898i	0.424795	+	0.424795i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.67423	+	3.67423i	−0.313911	+	0.313911i	−0.846423	−	0.532512i	$-0.821248\pi$
$137$	−3.67423	+	3.67423i	−0.313911	+	0.313911i	0.532512	+	0.846423i	$0.321248\pi$
$138$		0			0
$139$	−15.5885			−1.32220			−0.661098	−	0.750300i	$-0.729909\pi$
$139$	−15.5885			−1.32220			−0.661098	+	0.750300i	$0.729909\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.7279	−	12.7279i	1.06436	−	1.06436i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$			6.92820i			0.563809i	0.959442	+	0.281905i	$0.0909662\pi$
$151$			6.92820i			0.563809i	−0.959442	+	0.281905i	$0.909034\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.44949	+	2.44949i	−0.195491	+	0.195491i	−0.798064	−	0.602573i	$-0.794142\pi$
$157$	−2.44949	+	2.44949i	−0.195491	+	0.195491i	0.602573	+	0.798064i	$0.294142\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	−0.707107	+	0.707107i	−0.0553849	+	0.0553849i	−0.734257	−	0.678872i	$-0.762469\pi$
$163$	−0.707107	+	0.707107i	−0.0553849	+	0.0553849i	0.678872	+	0.734257i	$0.262469\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.24264	+	4.24264i	0.328305	+	0.328305i	0.851942	−	0.523636i	$-0.175425\pi$
$167$	4.24264	+	4.24264i	0.328305	+	0.328305i	−0.523636	+	0.851942i	$0.675425\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.5885			−1.16514			−0.582568	−	0.812782i	$-0.697952\pi$
$179$	−15.5885			−1.16514			−0.582568	+	0.812782i	$0.697952\pi$
$180$		0			0
$181$	26.0000			1.93256			0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000			1.93256			0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−19.0919	−	19.0919i	−1.39614	−	1.39614i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.3923i		−	0.751961i	−0.926628	−	0.375980i	$-0.877306\pi$
$191$		−	10.3923i		−	0.751961i	0.926628	−	0.375980i	$-0.122694\pi$
$192$		0			0
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	0.944790	−	0.327677i	$-0.106266\pi$
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	−0.327677	+	0.944790i	$0.606266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.34847	−	7.34847i	0.523557	−	0.523557i	−0.395087	−	0.918644i	$-0.629286\pi$
$197$	7.34847	−	7.34847i	0.523557	−	0.523557i	0.918644	+	0.395087i	$0.129286\pi$
$198$		0			0
$199$	−13.8564			−0.982255			−0.491127	−	0.871088i	$-0.663415\pi$
$199$	−13.8564			−0.982255			−0.491127	+	0.871088i	$0.663415\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.9706	−	16.9706i	1.19110	−	1.19110i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			9.00000i			0.622543i
$210$		0			0
$211$			19.0526i			1.31163i	0.754921	+	0.655816i	$0.227675\pi$
$211$			19.0526i			1.31163i	−0.754921	+	0.655816i	$0.772325\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.79796	−	9.79796i	0.665129	−	0.665129i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.0000			1.21081
$222$		0			0
$223$	−2.82843	+	2.82843i	−0.189405	+	0.189405i	−0.795439	−	0.606034i	$-0.792760\pi$
$223$	−2.82843	+	2.82843i	−0.189405	+	0.189405i	0.606034	+	0.795439i	$0.292760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.6969	−	14.6969i	−0.962828	−	0.962828i	0.0365050	−	0.999333i	$-0.488378\pi$
$233$	−14.6969	−	14.6969i	−0.962828	−	0.962828i	−0.999333	+	0.0365050i	$0.988378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.3923			0.672222			0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923			0.672222			0.336111	+	0.941822i	$0.390888\pi$
$240$		0			0
$241$	13.0000			0.837404			0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000			0.837404			0.418702	+	0.908124i	$0.362485\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.24264	−	4.24264i	−0.269953	−	0.269953i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	5.19615i		−	0.327978i	−0.986462	−	0.163989i	$-0.947564\pi$
$251$		−	5.19615i		−	0.327978i	0.986462	−	0.163989i	$-0.0524362\pi$
$252$		0			0
$253$	−22.0454	−	22.0454i	−1.38598	−	1.38598i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.6969	−	14.6969i	0.916770	−	0.916770i	−0.0800232	−	0.996793i	$-0.525499\pi$
$257$	14.6969	−	14.6969i	0.916770	−	0.916770i	0.996793	+	0.0800232i	$0.0254994\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7279	−	12.7279i	0.784837	−	0.784837i	−0.195805	−	0.980643i	$-0.562732\pi$
$263$	12.7279	−	12.7279i	0.784837	−	0.784837i	0.980643	+	0.195805i	$0.0627321\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			24.0000i			1.46331i	0.681677	+	0.731653i	$0.261251\pi$
$269$			24.0000i			1.46331i	−0.681677	+	0.731653i	$0.738749\pi$
$270$		0			0
$271$		−	31.1769i		−	1.89386i	−0.321436	−	0.946931i	$-0.604165\pi$
$271$		−	31.1769i		−	1.89386i	0.321436	−	0.946931i	$-0.395835\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−17.1464	+	17.1464i	−1.03023	+	1.03023i	−0.0307004	+	0.999529i	$0.509774\pi$
$277$	−17.1464	+	17.1464i	−1.03023	+	1.03023i	−0.999529	+	0.0307004i	$0.990226\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	4.94975	−	4.94975i	0.294232	−	0.294232i	−0.544518	−	0.838749i	$-0.683287\pi$
$283$	4.94975	−	4.94975i	0.294232	−	0.294232i	0.838749	+	0.544518i	$0.183287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.48528	−	8.48528i	−0.500870	−	0.500870i
$288$		0			0
$289$		−	10.0000i		−	0.588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.6969	+	14.6969i	0.858604	+	0.858604i	0.991174	−	0.132569i	$-0.0423227\pi$
$293$	14.6969	+	14.6969i	0.858604	+	0.858604i	−0.132569	+	0.991174i	$0.542323\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	20.7846			1.20201
$300$		0			0
$301$	−16.0000			−0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.0208	+	12.0208i	0.686064	+	0.686064i	0.961360	−	0.275295i	$-0.0887757\pi$
$307$	12.0208	+	12.0208i	0.686064	+	0.686064i	−0.275295	+	0.961360i	$0.588776\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−4.89898	−	4.89898i	−0.276907	−	0.276907i	0.554966	−	0.831873i	$-0.312731\pi$
$313$	−4.89898	−	4.89898i	−0.276907	−	0.276907i	−0.831873	+	0.554966i	$0.812731\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.6969	+	14.6969i	−0.825462	+	0.825462i	−0.986885	−	0.161423i	$-0.948392\pi$
$317$	−14.6969	+	14.6969i	−0.825462	+	0.825462i	0.161423	+	0.986885i	$0.448392\pi$
$318$		0			0
$319$	31.1769			1.74557
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.36396	+	6.36396i	−0.354100	+	0.354100i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			24.0000i			1.32316i
$330$		0			0
$331$			15.5885i			0.856819i	0.903585	+	0.428410i	$0.140926\pi$
$331$			15.5885i			0.856819i	−0.903585	+	0.428410i	$0.859074\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.57321	−	8.57321i	0.467013	−	0.467013i	−0.433933	−	0.900945i	$-0.642874\pi$
$337$	8.57321	−	8.57321i	0.467013	−	0.467013i	0.900945	+	0.433933i	$0.142874\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.0000			0.974755
$342$		0			0
$343$	5.65685	−	5.65685i	0.305441	−	0.305441i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.0919	−	19.0919i	−1.02491	−	1.02491i	−0.999682	−	0.0252242i	$-0.991970\pi$
$347$	−19.0919	−	19.0919i	−1.02491	−	1.02491i	−0.0252242	−	0.999682i	$-0.508030\pi$
$348$		0			0
$349$			28.0000i			1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$			28.0000i			1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.6969	−	14.6969i	−0.782239	−	0.782239i	0.197969	−	0.980208i	$-0.436565\pi$
$353$	−14.6969	−	14.6969i	−0.782239	−	0.782239i	−0.980208	+	0.197969i	$0.936565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.7846			−1.09697			−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846			−1.09697			−0.548485	+	0.836160i	$0.684795\pi$
$360$		0			0
$361$	−16.0000			−0.842105
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.82843	+	2.82843i	0.147643	+	0.147643i	0.777064	−	0.629421i	$-0.216708\pi$
$367$	2.82843	+	2.82843i	0.147643	+	0.147643i	−0.629421	+	0.777064i	$0.716708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			41.5692i			2.15817i
$372$		0			0
$373$	2.44949	+	2.44949i	0.126830	+	0.126830i	0.767672	−	0.640843i	$-0.221415\pi$
$373$	2.44949	+	2.44949i	0.126830	+	0.126830i	−0.640843	+	0.767672i	$0.721415\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.6969	+	14.6969i	−0.756931	+	0.756931i
$378$		0			0
$379$	1.73205			0.0889695			0.0444847	−	0.999010i	$-0.485835\pi$
$379$	1.73205			0.0889695			0.0444847	+	0.999010i	$0.485835\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−25.4558	+	25.4558i	−1.30073	+	1.30073i	−0.372835	+	0.927898i	$0.621614\pi$
$383$	−25.4558	+	25.4558i	−1.30073	+	1.30073i	−0.927898	+	0.372835i	$0.878386\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	30.0000i		−	1.52106i	−0.649303	−	0.760530i	$-0.724939\pi$
$389$		−	30.0000i		−	1.52106i	0.649303	−	0.760530i	$-0.275061\pi$
$390$		0			0
$391$		−	31.1769i		−	1.57668i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.79796	−	9.79796i	0.491745	−	0.491745i	−0.417110	−	0.908856i	$-0.636957\pi$
$397$	9.79796	−	9.79796i	0.491745	−	0.491745i	0.908856	+	0.417110i	$0.136957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.0000			−1.34832			−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000			−1.34832			−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$	−8.48528	+	8.48528i	−0.422682	+	0.422682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	25.0000i		−	1.23617i	−0.786111	−	0.618085i	$-0.787909\pi$
$409$		−	25.0000i		−	1.23617i	0.786111	−	0.618085i	$-0.212091\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	29.3939	+	29.3939i	1.44638	+	1.44638i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.5885			−0.761546			−0.380773	−	0.924669i	$-0.624342\pi$
$419$	−15.5885			−0.761546			−0.380773	+	0.924669i	$0.624342\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	11.3137	+	11.3137i	0.547509	+	0.547509i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.7846i		−	1.00116i	−0.865690	−	0.500580i	$-0.833120\pi$
$431$		−	20.7846i		−	1.00116i	0.865690	−	0.500580i	$-0.166880\pi$
$432$		0			0
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	0.735923	−	0.677065i	$-0.236749\pi$
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	−0.677065	+	0.735923i	$0.736749\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.34847	+	7.34847i	−0.351525	+	0.351525i
$438$		0			0
$439$	−38.1051			−1.81866			−0.909329	−	0.416078i	$-0.863404\pi$
$439$	−38.1051			−1.81866			−0.909329	+	0.416078i	$0.863404\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.12132	−	2.12132i	0.100787	−	0.100787i	−0.654915	−	0.755702i	$-0.727296\pi$
$443$	2.12132	−	2.12132i	0.100787	−	0.100787i	0.755702	+	0.654915i	$0.227296\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.0000i		−	1.27421i	−0.770778	−	0.637104i	$-0.780132\pi$
$449$		−	27.0000i		−	1.27421i	0.770778	−	0.637104i	$-0.219868\pi$
$450$		0			0
$451$		−	15.5885i		−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0227	−	11.0227i	0.515620	−	0.515620i	−0.400623	−	0.916243i	$-0.631206\pi$
$457$	11.0227	−	11.0227i	0.515620	−	0.515620i	0.916243	+	0.400623i	$0.131206\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$	2.82843	−	2.82843i	0.131448	−	0.131448i	−0.638322	−	0.769770i	$-0.720371\pi$
$463$	2.82843	−	2.82843i	0.131448	−	0.131448i	0.769770	+	0.638322i	$0.220371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.4558	+	25.4558i	1.17796	+	1.17796i	0.980264	+	0.197692i	$0.0633445\pi$
$467$	25.4558	+	25.4558i	1.17796	+	1.17796i	0.197692	+	0.980264i	$0.436655\pi$
$468$		0			0
$469$		−	28.0000i		−	1.29292i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.6969	−	14.6969i	−0.675766	−	0.675766i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.3923			−0.474837			−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923			−0.474837			−0.237418	+	0.971408i	$0.576301\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.3848	+	18.3848i	0.833094	+	0.833094i	0.987939	−	0.154845i	$-0.0494878\pi$
$487$	18.3848	+	18.3848i	0.833094	+	0.833094i	−0.154845	+	0.987939i	$0.549488\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			10.3923i			0.468998i	0.972116	+	0.234499i	$0.0753450\pi$
$491$			10.3923i			0.468998i	−0.972116	+	0.234499i	$0.924655\pi$
$492$		0			0
$493$	22.0454	+	22.0454i	0.992875	+	0.992875i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	29.3939	−	29.3939i	1.31850	−	1.31850i
$498$		0			0
$499$	−3.46410			−0.155074			−0.0775372	−	0.996989i	$-0.524706\pi$
$499$	−3.46410			−0.155074			−0.0775372	+	0.996989i	$0.524706\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.9706	+	16.9706i	−0.756680	+	0.756680i	−0.975717	−	0.219037i	$-0.929709\pi$
$503$	−16.9706	+	16.9706i	−0.756680	+	0.756680i	0.219037	+	0.975717i	$0.429709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	30.0000i		−	1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$		−	30.0000i		−	1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$		0			0
$511$			62.3538i			2.75837i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−22.0454	+	22.0454i	−0.969556	+	0.969556i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	12.0208	−	12.0208i	0.525634	−	0.525634i	−0.393634	−	0.919267i	$-0.628782\pi$
$523$	12.0208	−	12.0208i	0.525634	−	0.525634i	0.919267	+	0.393634i	$0.128782\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.7279	+	12.7279i	0.554437	+	0.554437i
$528$		0			0
$529$		−	13.0000i		−	0.565217i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.34847	+	7.34847i	0.318298	+	0.318298i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	46.7654			2.01433
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−24.7487	−	24.7487i	−1.05818	−	1.05818i	−0.998200	−	0.0599800i	$-0.980896\pi$
$547$	−24.7487	−	24.7487i	−1.05818	−	1.05818i	−0.0599800	−	0.998200i	$-0.519104\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.3923i		−	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.0454	+	22.0454i	−0.934094	+	0.934094i	−0.997959	−	0.0638647i	$-0.979657\pi$
$557$	−22.0454	+	22.0454i	−0.934094	+	0.934094i	0.0638647	+	0.997959i	$0.479657\pi$
$558$		0			0
$559$	13.8564			0.586064
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.48528	−	8.48528i	0.357612	−	0.357612i	−0.505320	−	0.862932i	$-0.668626\pi$
$563$	8.48528	−	8.48528i	0.357612	−	0.357612i	0.862932	+	0.505320i	$0.168626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	9.00000i		−	0.377300i	−0.982044	−	0.188650i	$-0.939589\pi$
$569$		−	9.00000i		−	0.377300i	0.982044	−	0.188650i	$-0.0604111\pi$
$570$		0			0
$571$			3.46410i			0.144968i	0.997370	+	0.0724841i	$0.0230926\pi$
$571$			3.46410i			0.144968i	−0.997370	+	0.0724841i	$0.976907\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.22474	−	1.22474i	0.0509868	−	0.0509868i	−0.681154	−	0.732140i	$-0.738521\pi$
$577$	1.22474	−	1.22474i	0.0509868	−	0.0509868i	0.732140	+	0.681154i	$0.238521\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$	−38.1838	+	38.1838i	−1.58141	+	1.58141i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.3345	+	23.3345i	0.963119	+	0.963119i	0.999344	−	0.0362248i	$-0.0115332\pi$
$587$	23.3345	+	23.3345i	0.963119	+	0.963119i	−0.0362248	+	0.999344i	$0.511533\pi$
$588$		0			0
$589$		−	6.00000i		−	0.247226i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.0227	+	11.0227i	0.452648	+	0.452648i	0.896233	−	0.443584i	$-0.146293\pi$
$593$	11.0227	+	11.0227i	0.452648	+	0.452648i	−0.443584	+	0.896233i	$0.646293\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.7846			0.849236			0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846			0.849236			0.424618	+	0.905373i	$0.360408\pi$
$600$		0			0
$601$	13.0000			0.530281			0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000			0.530281			0.265141	+	0.964210i	$0.414582\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−28.2843	−	28.2843i	−1.14802	−	1.14802i	−0.986941	−	0.161082i	$-0.948502\pi$
$607$	−28.2843	−	28.2843i	−1.14802	−	1.14802i	−0.161082	−	0.986941i	$-0.551498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	20.7846i		−	0.840855i
$612$		0			0
$613$	22.0454	+	22.0454i	0.890406	+	0.890406i	0.994561	−	0.104155i	$-0.0332139\pi$
$613$	22.0454	+	22.0454i	0.890406	+	0.890406i	−0.104155	+	0.994561i	$0.533214\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.3939	−	29.3939i	1.18335	−	1.18335i	0.204483	−	0.978870i	$-0.434449\pi$
$617$	29.3939	−	29.3939i	1.18335	−	1.18335i	0.978870	−	0.204483i	$-0.0655512\pi$
$618$		0			0
$619$	38.1051			1.53157			0.765787	−	0.643094i	$-0.222350\pi$
$619$	38.1051			1.53157			0.765787	+	0.643094i	$0.222350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.48528	+	8.48528i	−0.339956	+	0.339956i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			17.3205i			0.689519i	0.938691	+	0.344759i	$0.112039\pi$
$631$			17.3205i			0.689519i	−0.938691	+	0.344759i	$0.887961\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−22.0454	+	22.0454i	−0.873471	+	0.873471i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$	−2.82843	+	2.82843i	−0.111542	+	0.111542i	−0.760675	−	0.649133i	$-0.775132\pi$
$643$	−2.82843	+	2.82843i	−0.111542	+	0.111542i	0.649133	+	0.760675i	$0.275132\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.9706	+	16.9706i	0.667182	+	0.667182i	0.957063	−	0.289881i	$-0.0936157\pi$
$647$	16.9706	+	16.9706i	0.667182	+	0.667182i	−0.289881	+	0.957063i	$0.593616\pi$
$648$		0			0
$649$			54.0000i			2.11969i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−29.3939	−	29.3939i	−1.15027	−	1.15027i	−0.986498	−	0.163773i	$-0.947633\pi$
$653$	−29.3939	−	29.3939i	−1.15027	−	1.15027i	−0.163773	−	0.986498i	$-0.552367\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	25.9808			1.01207			0.506033	−	0.862514i	$-0.331111\pi$
$659$	25.9808			1.01207			0.506033	+	0.862514i	$0.331111\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	25.4558	+	25.4558i	0.985654	+	0.985654i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			20.7846i			0.802381i
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$	27.7128			1.06352
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.36396	+	6.36396i	−0.243510	+	0.243510i	−0.818301	−	0.574790i	$-0.805084\pi$
$683$	−6.36396	+	6.36396i	−0.243510	+	0.243510i	0.574790	+	0.818301i	$0.305084\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	36.0000i		−	1.37149i
$690$		0			0
$691$			15.5885i			0.593013i	0.955031	+	0.296506i	$0.0958216\pi$
$691$			15.5885i			0.593013i	−0.955031	+	0.296506i	$0.904178\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	11.0227	−	11.0227i	0.417515	−	0.417515i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	4.00000i		−	0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$		−	4.00000i		−	0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.6969	+	14.6969i	0.550405	+	0.550405i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	41.5692			1.55027			0.775135	−	0.631795i	$-0.217682\pi$
$719$	41.5692			1.55027			0.775135	+	0.631795i	$0.217682\pi$
$720$		0			0
$721$	−56.0000			−2.08555
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−9.89949	−	9.89949i	−0.367152	−	0.367152i	0.499286	−	0.866437i	$-0.333596\pi$
$727$	−9.89949	−	9.89949i	−0.367152	−	0.367152i	−0.866437	+	0.499286i	$0.833596\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	20.7846i		−	0.768747i
$732$		0			0
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	0.171005	−	0.985270i	$-0.445299\pi$
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	−0.985270	+	0.171005i	$0.945299\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.7196	−	25.7196i	0.947395	−	0.947395i
$738$		0			0
$739$	31.1769			1.14686			0.573431	−	0.819254i	$-0.305612\pi$
$739$	31.1769			1.14686			0.573431	+	0.819254i	$0.305612\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−29.6985	+	29.6985i	−1.08953	+	1.08953i	−0.0939553	+	0.995576i	$0.529951\pi$
$743$	−29.6985	+	29.6985i	−1.08953	+	1.08953i	−0.995576	+	0.0939553i	$0.970049\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			60.0000i			2.19235i
$750$		0			0
$751$		−	31.1769i		−	1.13766i	−0.822455	−	0.568831i	$-0.807396\pi$
$751$		−	31.1769i		−	1.13766i	0.822455	−	0.568831i	$-0.192604\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34.2929	−	34.2929i	1.24640	−	1.24640i	0.289095	−	0.957301i	$-0.406646\pi$
$757$	34.2929	−	34.2929i	1.24640	−	1.24640i	0.957301	−	0.289095i	$-0.0933542\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.0000			−0.543750			−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000			−0.543750			−0.271875	+	0.962333i	$0.587644\pi$
$762$		0			0
$763$	−22.6274	+	22.6274i	−0.819167	+	0.819167i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.4558	−	25.4558i	−0.919157	−	0.919157i
$768$		0			0
$769$		−	7.00000i		−	0.252426i	−0.992003	−	0.126213i	$-0.959718\pi$
$769$		−	7.00000i		−	0.252426i	0.992003	−	0.126213i	$-0.0402824\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.0454	+	22.0454i	0.792918	+	0.792918i	0.981968	−	0.189049i	$-0.0605406\pi$
$773$	22.0454	+	22.0454i	0.792918	+	0.792918i	−0.189049	+	0.981968i	$0.560541\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.19615			−0.186171
$780$		0			0
$781$	54.0000			1.93227
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	36.7696	+	36.7696i	1.31069	+	1.31069i	0.920904	+	0.389789i	$0.127452\pi$
$787$	36.7696	+	36.7696i	1.31069	+	1.31069i	0.389789	+	0.920904i	$0.372548\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			20.7846i			0.739016i
$792$		0			0
$793$	−9.79796	−	9.79796i	−0.347936	−	0.347936i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.34847	−	7.34847i	0.260296	−	0.260296i	−0.564878	−	0.825174i	$-0.691077\pi$
$797$	7.34847	−	7.34847i	0.260296	−	0.260296i	0.825174	+	0.564878i	$0.191077\pi$
$798$		0			0
$799$	−31.1769			−1.10296
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−57.2756	+	57.2756i	−2.02121	+	2.02121i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.0000i			0.632846i	0.948618	+	0.316423i	$0.102482\pi$
$809$			18.0000i			0.632846i	−0.948618	+	0.316423i	$0.897518\pi$
$810$		0			0
$811$		−	45.0333i		−	1.58133i	−0.612247	−	0.790667i	$-0.709734\pi$
$811$		−	45.0333i		−	1.58133i	0.612247	−	0.790667i	$-0.290266\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.89898	+	4.89898i	−0.171394	+	0.171394i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.0000			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$822$		0			0
$823$	−14.1421	+	14.1421i	−0.492964	+	0.492964i	−0.909239	−	0.416275i	$-0.863335\pi$
$823$	−14.1421	+	14.1421i	−0.492964	+	0.492964i	0.416275	+	0.909239i	$0.363335\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.12132	+	2.12132i	0.0737655	+	0.0737655i	0.743027	−	0.669261i	$-0.233389\pi$
$827$	2.12132	+	2.12132i	0.0737655	+	0.0737655i	−0.669261	+	0.743027i	$0.733389\pi$
$828$		0			0
$829$			2.00000i			0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$			2.00000i			0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.0681	+	33.0681i	1.14574	+	1.14574i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	41.5692			1.43513			0.717564	−	0.696492i	$-0.245257\pi$
$839$	41.5692			1.43513			0.717564	+	0.696492i	$0.245257\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	45.2548	+	45.2548i	1.55497	+	1.55497i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	26.9444	+	26.9444i	0.922558	+	0.922558i	0.997210	−	0.0746514i	$-0.0237844\pi$
$853$	26.9444	+	26.9444i	0.922558	+	0.922558i	−0.0746514	+	0.997210i	$0.523784\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.3712	−	18.3712i	0.627547	−	0.627547i	−0.319903	−	0.947450i	$-0.603650\pi$
$857$	18.3712	−	18.3712i	0.627547	−	0.627547i	0.947450	+	0.319903i	$0.103650\pi$
$858$		0			0
$859$	−15.5885			−0.531871			−0.265936	−	0.963991i	$-0.585681\pi$
$859$	−15.5885			−0.531871			−0.265936	+	0.963991i	$0.585681\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.24264	−	4.24264i	0.144421	−	0.144421i	−0.631199	−	0.775621i	$-0.717437\pi$
$863$	4.24264	−	4.24264i	0.144421	−	0.144421i	0.775621	+	0.631199i	$0.217437\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$			24.2487i			0.821636i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	24.4949	−	24.4949i	0.827134	−	0.827134i	−0.159985	−	0.987119i	$-0.551145\pi$
$877$	24.4949	−	24.4949i	0.827134	−	0.827134i	0.987119	+	0.159985i	$0.0511447\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	38.8909	−	38.8909i	1.30878	−	1.30878i	0.386487	−	0.922295i	$-0.373688\pi$
$883$	38.8909	−	38.8909i	1.30878	−	1.30878i	0.922295	−	0.386487i	$-0.126312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.34847	+	7.34847i	0.245907	+	0.245907i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.7846			−0.693206
$900$		0			0
$901$	−54.0000			−1.79900
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−31.1127	−	31.1127i	−1.03308	−	1.03308i	−0.999434	−	0.0336464i	$-0.989288\pi$
$907$	−31.1127	−	31.1127i	−1.03308	−	1.03308i	−0.0336464	−	0.999434i	$-0.510712\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.7846i			0.688625i	0.938855	+	0.344312i	$0.111888\pi$
$911$			20.7846i			0.688625i	−0.938855	+	0.344312i	$0.888112\pi$
$912$		0			0
$913$	−11.0227	−	11.0227i	−0.364798	−	0.364798i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.3939	+	29.3939i	−0.970671	+	0.970671i
$918$		0			0
$919$	45.0333			1.48551			0.742756	−	0.669562i	$-0.233518\pi$
$919$	45.0333			1.48551			0.742756	+	0.669562i	$0.233518\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−25.4558	+	25.4558i	−0.837889	+	0.837889i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$		−	15.5885i		−	0.510891i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.22474	+	1.22474i	−0.0400107	+	0.0400107i	−0.726829	−	0.686818i	$-0.759007\pi$
$937$	−1.22474	+	1.22474i	−0.0400107	+	0.0400107i	0.686818	+	0.726829i	$0.259007\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.00000			−0.195594			−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000			−0.195594			−0.0977972	+	0.995206i	$0.531180\pi$
$942$		0			0
$943$	12.7279	−	12.7279i	0.414478	−	0.414478i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.48528	−	8.48528i	−0.275735	−	0.275735i	0.555669	−	0.831404i	$-0.312462\pi$
$947$	−8.48528	−	8.48528i	−0.275735	−	0.275735i	−0.831404	+	0.555669i	$0.812462\pi$
$948$		0			0
$949$		−	54.0000i		−	1.75291i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.67423	−	3.67423i	−0.119020	−	0.119020i	0.645088	−	0.764108i	$-0.276821\pi$
$953$	−3.67423	−	3.67423i	−0.119020	−	0.119020i	−0.764108	+	0.645088i	$0.776821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	20.7846			0.671170
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.82843	+	2.82843i	0.0909561	+	0.0909561i	0.751121	−	0.660165i	$-0.229514\pi$
$967$	2.82843	+	2.82843i	0.0909561	+	0.0909561i	−0.660165	+	0.751121i	$0.729514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			46.7654i			1.50077i	0.661000	+	0.750386i	$0.270132\pi$
$971$			46.7654i			1.50077i	−0.661000	+	0.750386i	$0.729868\pi$
$972$		0			0
$973$	44.0908	+	44.0908i	1.41349	+	1.41349i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.0227	+	11.0227i	−0.352648	+	0.352648i	−0.861094	−	0.508446i	$-0.830220\pi$
$977$	−11.0227	+	11.0227i	−0.352648	+	0.352648i	0.508446	+	0.861094i	$0.330220\pi$
$978$		0			0
$979$	−15.5885			−0.498209
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.9706	−	16.9706i	0.541277	−	0.541277i	−0.382626	−	0.923903i	$-0.624980\pi$
$983$	16.9706	−	16.9706i	0.541277	−	0.541277i	0.923903	+	0.382626i	$0.124980\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$		−	13.8564i		−	0.440163i	−0.975481	−	0.220082i	$-0.929368\pi$
$991$		−	13.8564i		−	0.440163i	0.975481	−	0.220082i	$-0.0706324\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.x.m.2143.1		8
3.2	odd	2		400.2.n.d.143.2	yes	8
4.3	odd	2	inner	3600.2.x.m.2143.4		8
5.2	odd	4	inner	3600.2.x.m.3007.3		8
5.3	odd	4	inner	3600.2.x.m.3007.1		8
5.4	even	2	inner	3600.2.x.m.2143.3		8
12.11	even	2		400.2.n.d.143.3	yes	8
15.2	even	4		400.2.n.d.207.4	yes	8
15.8	even	4		400.2.n.d.207.2	yes	8
15.14	odd	2		400.2.n.d.143.4	yes	8
20.3	even	4	inner	3600.2.x.m.3007.4		8
20.7	even	4	inner	3600.2.x.m.3007.2		8
20.19	odd	2	inner	3600.2.x.m.2143.2		8
24.5	odd	2		1600.2.n.u.1343.3		8
24.11	even	2		1600.2.n.u.1343.2		8
60.23	odd	4		400.2.n.d.207.3	yes	8
60.47	odd	4		400.2.n.d.207.1	yes	8
60.59	even	2		400.2.n.d.143.1	✓	8
120.29	odd	2		1600.2.n.u.1343.1		8
120.53	even	4		1600.2.n.u.1407.3		8
120.59	even	2		1600.2.n.u.1343.4		8
120.77	even	4		1600.2.n.u.1407.1		8
120.83	odd	4		1600.2.n.u.1407.2		8
120.107	odd	4		1600.2.n.u.1407.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.n.d.143.1	✓	8	60.59	even	2
400.2.n.d.143.2	yes	8	3.2	odd	2
400.2.n.d.143.3	yes	8	12.11	even	2
400.2.n.d.143.4	yes	8	15.14	odd	2
400.2.n.d.207.1	yes	8	60.47	odd	4
400.2.n.d.207.2	yes	8	15.8	even	4
400.2.n.d.207.3	yes	8	60.23	odd	4
400.2.n.d.207.4	yes	8	15.2	even	4
1600.2.n.u.1343.1		8	120.29	odd	2
1600.2.n.u.1343.2		8	24.11	even	2
1600.2.n.u.1343.3		8	24.5	odd	2
1600.2.n.u.1343.4		8	120.59	even	2
1600.2.n.u.1407.1		8	120.77	even	4
1600.2.n.u.1407.2		8	120.83	odd	4
1600.2.n.u.1407.3		8	120.53	even	4
1600.2.n.u.1407.4		8	120.107	odd	4
3600.2.x.m.2143.1		8	1.1	even	1	trivial
3600.2.x.m.2143.2		8	20.19	odd	2	inner
3600.2.x.m.2143.3		8	5.4	even	2	inner
3600.2.x.m.2143.4		8	4.3	odd	2	inner
3600.2.x.m.3007.1		8	5.3	odd	4	inner
3600.2.x.m.3007.2		8	20.7	even	4	inner
3600.2.x.m.3007.3		8	5.2	odd	4	inner
3600.2.x.m.3007.4		8	20.3	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 - 2.82843i) q^{7} -5.19615i q^{11} +(2.44949 + 2.44949i) q^{13} +(3.67423 - 3.67423i) q^{17} -1.73205 q^{19} +(4.24264 - 4.24264i) q^{23} +6.00000i q^{29} +3.46410i q^{31} +3.00000 q^{41} +(2.82843 - 2.82843i) q^{43} +(-4.24264 - 4.24264i) q^{47} +9.00000i q^{49} +(-7.34847 - 7.34847i) q^{53} -10.3923 q^{59} -4.00000 q^{61} +(4.94975 + 4.94975i) q^{67} +10.3923i q^{71} +(-11.0227 - 11.0227i) q^{73} +(-14.6969 + 14.6969i) q^{77} +(2.12132 - 2.12132i) q^{83} -3.00000i q^{89} -13.8564i q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{41} - 32 q^{61}+O(q^{100}) \) 8 * q + 24 * q^41 - 32 * q^61

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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