LMFDB - Embedded newform 3600.2.x.e.2143.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(2143,3600)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2143.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3600.2143
Dual form			3600.2.x.e.3007.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 + 1.73205i) q^{7} -3.46410i q^{11} +(-1.00000 - 1.00000i) q^{13} +(1.00000 - 1.00000i) q^{17} -6.92820 q^{19} +(1.73205 - 1.73205i) q^{23} -4.00000i q^{29} -3.46410i q^{31} +(-5.00000 + 5.00000i) q^{37} -2.00000 q^{41} +(-1.73205 + 1.73205i) q^{43} +(-1.73205 - 1.73205i) q^{47} -1.00000i q^{49} +(-7.00000 - 7.00000i) q^{53} -6.92820 q^{59} +6.00000 q^{61} +(-5.19615 - 5.19615i) q^{67} -10.3923i q^{71} +(7.00000 + 7.00000i) q^{73} +(6.00000 - 6.00000i) q^{77} +(-12.1244 + 12.1244i) q^{83} -8.00000i q^{89} -3.46410i q^{91} +(7.00000 - 7.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{13} + 4 q^{17} - 20 q^{37} - 8 q^{41} - 28 q^{53} + 24 q^{61} + 28 q^{73} + 24 q^{77} + 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^13 + 4 * q^17 - 20 * q^37 - 8 * q^41 - 28 * q^53 + 24 * q^61 + 28 * q^73 + 24 * q^77 + 28 * q^97

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
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	0.954110	−	0.299456i	$-0.0968053\pi$
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	−0.299456	+	0.954110i	$0.596805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.00000i	0.242536	−	0.242536i	−0.575363	−	0.817898i	$-0.695139\pi$
$17$	1.00000	−	1.00000i	0.242536	−	0.242536i	0.817898	+	0.575363i	$0.195139\pi$
$18$		0			0
$19$	−6.92820			−1.58944			−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820			−1.58944			−0.794719	+	0.606977i	$0.792382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	−	1.73205i	0.361158	−	0.361158i	−0.503081	−	0.864239i	$-0.667800\pi$
$23$	1.73205	−	1.73205i	0.361158	−	0.361158i	0.864239	+	0.503081i	$0.167800\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	−1.73205	+	1.73205i	−0.264135	+	0.264135i	−0.826732	−	0.562596i	$-0.809803\pi$
$43$	−1.73205	+	1.73205i	−0.264135	+	0.264135i	0.562596	+	0.826732i	$0.309803\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.73205	−	1.73205i	−0.252646	−	0.252646i	0.569409	−	0.822054i	$-0.307172\pi$
$47$	−1.73205	−	1.73205i	−0.252646	−	0.252646i	−0.822054	+	0.569409i	$0.807172\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	0.0377628	−	0.999287i	$-0.487977\pi$
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	−0.999287	+	0.0377628i	$0.987977\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.92820			−0.901975			−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820			−0.901975			−0.450988	+	0.892530i	$0.648928\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.19615	−	5.19615i	−0.634811	−	0.634811i	0.314460	−	0.949271i	$-0.398177\pi$
$67$	−5.19615	−	5.19615i	−0.634811	−	0.634811i	−0.949271	+	0.314460i	$0.898177\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	7.00000	+	7.00000i	0.819288	+	0.819288i	0.986005	−	0.166717i	$-0.0533166\pi$
$73$	7.00000	+	7.00000i	0.819288	+	0.819288i	−0.166717	+	0.986005i	$0.553317\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	−	6.00000i	0.683763	−	0.683763i
$78$		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
$83$	−12.1244	+	12.1244i	−1.33082	+	1.33082i	−0.426185	+	0.904636i	$0.640143\pi$
$83$	−12.1244	+	12.1244i	−1.33082	+	1.33082i	−0.904636	+	0.426185i	$0.859857\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.00000i		−	0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$89$		−	8.00000i		−	0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$90$		0			0
$91$		−	3.46410i		−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.00000	−	7.00000i	0.710742	−	0.710742i	−0.255948	−	0.966691i	$-0.582388\pi$
$97$	7.00000	−	7.00000i	0.710742	−	0.710742i	0.966691	+	0.255948i	$0.0823876\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$	−1.73205	+	1.73205i	−0.170664	+	0.170664i	−0.787271	−	0.616607i	$-0.788507\pi$
$103$	−1.73205	+	1.73205i	−0.170664	+	0.170664i	0.616607	+	0.787271i	$0.288507\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.66025	−	8.66025i	−0.837218	−	0.837218i	0.151274	−	0.988492i	$-0.451663\pi$
$107$	−8.66025	−	8.66025i	−0.837218	−	0.837218i	−0.988492	+	0.151274i	$0.951663\pi$
$108$		0			0
$109$			12.0000i			1.14939i	0.818367	+	0.574696i	$0.194880\pi$
$109$			12.0000i			1.14939i	−0.818367	+	0.574696i	$0.805120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.00000	+	9.00000i	0.846649	+	0.846649i	0.989713	−	0.143065i	$-0.0456957\pi$
$113$	9.00000	+	9.00000i	0.846649	+	0.846649i	−0.143065	+	0.989713i	$0.545696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.46410			0.317554
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−12.1244	−	12.1244i	−1.07586	−	1.07586i	−0.996876	−	0.0789869i	$-0.974831\pi$
$127$	−12.1244	−	12.1244i	−1.07586	−	1.07586i	−0.0789869	−	0.996876i	$-0.525169\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.3923i			0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$131$			10.3923i			0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$132$		0			0
$133$	−12.0000	−	12.0000i	−1.04053	−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	9.00000i	0.768922	−	0.768922i	−0.208995	−	0.977917i	$-0.567019\pi$
$137$	9.00000	−	9.00000i	0.768922	−	0.768922i	0.977917	+	0.208995i	$0.0670192\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.46410	+	3.46410i	−0.289683	+	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	4.00000i		−	0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$		−	4.00000i		−	0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.0000	−	11.0000i	0.877896	−	0.877896i	−0.115421	−	0.993317i	$-0.536822\pi$
$157$	11.0000	−	11.0000i	0.877896	−	0.877896i	0.993317	+	0.115421i	$0.0368217\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−1.73205	+	1.73205i	−0.135665	+	0.135665i	−0.771678	−	0.636013i	$-0.780582\pi$
$163$	−1.73205	+	1.73205i	−0.135665	+	0.135665i	0.636013	+	0.771678i	$0.280582\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.5885	−	15.5885i	−1.20627	−	1.20627i	−0.972226	−	0.234045i	$-0.924804\pi$
$167$	−15.5885	−	15.5885i	−1.20627	−	1.20627i	−0.234045	−	0.972226i	$-0.575196\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.0000	−	15.0000i	−1.14043	−	1.14043i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−15.0000	−	15.0000i	−1.14043	−	1.14043i	−0.152057	−	0.988372i	$-0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.92820			0.517838			0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820			0.517838			0.258919	+	0.965899i	$0.416634\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.46410	−	3.46410i	−0.253320	−	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.2487i		−	1.75458i	−0.479965	−	0.877288i	$-0.659351\pi$
$191$		−	24.2487i		−	1.75458i	0.479965	−	0.877288i	$-0.340649\pi$
$192$		0			0
$193$	−1.00000	−	1.00000i	−0.0719816	−	0.0719816i	0.670199	−	0.742181i	$-0.266209\pi$
$193$	−1.00000	−	1.00000i	−0.0719816	−	0.0719816i	−0.742181	+	0.670199i	$0.766209\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.00000	+	3.00000i	−0.213741	+	0.213741i	−0.805855	−	0.592113i	$-0.798294\pi$
$197$	−3.00000	+	3.00000i	−0.213741	+	0.213741i	0.592113	+	0.805855i	$0.298294\pi$
$198$		0			0
$199$	13.8564			0.982255			0.491127	−	0.871088i	$-0.336585\pi$
$199$	13.8564			0.982255			0.491127	+	0.871088i	$0.336585\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.92820	−	6.92820i	0.486265	−	0.486265i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			24.0000i			1.66011i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.00000	−	6.00000i	0.407307	−	0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000			−0.134535
$222$		0			0
$223$	−15.5885	+	15.5885i	−1.04388	+	1.04388i	−0.0448883	+	0.998992i	$0.514293\pi$
$223$	−15.5885	+	15.5885i	−1.04388	+	1.04388i	−0.998992	+	0.0448883i	$0.985707\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.66025	−	8.66025i	−0.574801	−	0.574801i	0.358665	−	0.933466i	$-0.383232\pi$
$227$	−8.66025	−	8.66025i	−0.574801	−	0.574801i	−0.933466	+	0.358665i	$0.883232\pi$
$228$		0			0
$229$		−	20.0000i		−	1.32164i	−0.750546	−	0.660819i	$-0.770209\pi$
$229$		−	20.0000i		−	1.32164i	0.750546	−	0.660819i	$-0.229791\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	0.739104	−	0.673592i	$-0.235249\pi$
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	−0.673592	+	0.739104i	$0.735249\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−27.7128			−1.79259			−0.896296	−	0.443455i	$-0.853752\pi$
$239$	−27.7128			−1.79259			−0.896296	+	0.443455i	$0.853752\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.92820	+	6.92820i	0.440831	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	3.46410i		−	0.218652i	−0.994006	−	0.109326i	$-0.965131\pi$
$251$		−	3.46410i		−	0.218652i	0.994006	−	0.109326i	$-0.0348693\pi$
$252$		0			0
$253$	−6.00000	−	6.00000i	−0.377217	−	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	−	9.00000i	0.561405	−	0.561405i	−0.368302	−	0.929706i	$-0.620061\pi$
$257$	9.00000	−	9.00000i	0.561405	−	0.561405i	0.929706	+	0.368302i	$0.120061\pi$
$258$		0			0
$259$	−17.3205			−1.07624
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1244	+	12.1244i	−0.747620	+	0.747620i	−0.974032	−	0.226412i	$-0.927300\pi$
$263$	−12.1244	+	12.1244i	−0.747620	+	0.747620i	0.226412	+	0.974032i	$0.427300\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.00000i			0.243884i	0.992537	+	0.121942i	$0.0389122\pi$
$269$			4.00000i			0.243884i	−0.992537	+	0.121942i	$0.961088\pi$
$270$		0			0
$271$		−	3.46410i		−	0.210429i	−0.994450	−	0.105215i	$-0.966447\pi$
$271$		−	3.46410i		−	0.210429i	0.994450	−	0.105215i	$-0.0335529\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0000	+	13.0000i	−0.781094	+	0.781094i	−0.980015	−	0.198921i	$-0.936256\pi$
$277$	−13.0000	+	13.0000i	−0.781094	+	0.781094i	0.198921	+	0.980015i	$0.436256\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.0000			0.835170			0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000			0.835170			0.417585	+	0.908638i	$0.362877\pi$
$282$		0			0
$283$	12.1244	−	12.1244i	0.720718	−	0.720718i	−0.248033	−	0.968751i	$-0.579784\pi$
$283$	12.1244	−	12.1244i	0.720718	−	0.720718i	0.968751	+	0.248033i	$0.0797843\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46410	−	3.46410i	−0.204479	−	0.204479i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.00000	+	9.00000i	0.525786	+	0.525786i	0.919313	−	0.393527i	$-0.128745\pi$
$293$	9.00000	+	9.00000i	0.525786	+	0.525786i	−0.393527	+	0.919313i	$0.628745\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.46410			−0.200334
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−5.19615	−	5.19615i	−0.296560	−	0.296560i	0.543105	−	0.839665i	$-0.317249\pi$
$307$	−5.19615	−	5.19615i	−0.296560	−	0.296560i	−0.839665	+	0.543105i	$0.817249\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			17.3205i			0.982156i	0.871116	+	0.491078i	$0.163397\pi$
$311$			17.3205i			0.982156i	−0.871116	+	0.491078i	$0.836603\pi$
$312$		0			0
$313$	7.00000	+	7.00000i	0.395663	+	0.395663i	0.876700	−	0.481037i	$-0.159740\pi$
$313$	7.00000	+	7.00000i	0.395663	+	0.395663i	−0.481037	+	0.876700i	$0.659740\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.0000	+	19.0000i	−1.06715	+	1.06715i	−0.0695692	+	0.997577i	$0.522162\pi$
$317$	−19.0000	+	19.0000i	−1.06715	+	1.06715i	−0.997577	+	0.0695692i	$0.977838\pi$
$318$		0			0
$319$	−13.8564			−0.775810
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.92820	+	6.92820i	−0.385496	+	0.385496i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	6.00000i		−	0.330791i
$330$		0			0
$331$		−	24.2487i		−	1.33283i	−0.745581	−	0.666415i	$-0.767828\pi$
$331$		−	24.2487i		−	1.33283i	0.745581	−	0.666415i	$-0.232172\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.00000	+	1.00000i	−0.0544735	+	0.0544735i	−0.733819	−	0.679345i	$-0.762264\pi$
$337$	−1.00000	+	1.00000i	−0.0544735	+	0.0544735i	0.679345	+	0.733819i	$0.262264\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	13.8564	−	13.8564i	0.748176	−	0.748176i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.19615	+	5.19615i	0.278944	+	0.278944i	0.832687	−	0.553743i	$-0.186801\pi$
$347$	5.19615	+	5.19615i	0.278944	+	0.278944i	−0.553743	+	0.832687i	$0.686801\pi$
$348$		0			0
$349$		−	12.0000i		−	0.642345i	−0.947021	−	0.321173i	$-0.895923\pi$
$349$		−	12.0000i		−	0.642345i	0.947021	−	0.321173i	$-0.104077\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	+	1.00000i	0.0532246	+	0.0532246i	0.733218	−	0.679994i	$-0.238017\pi$
$353$	1.00000	+	1.00000i	0.0532246	+	0.0532246i	−0.679994	+	0.733218i	$0.738017\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.8564			−0.731313			−0.365657	−	0.930750i	$-0.619156\pi$
$359$	−13.8564			−0.731313			−0.365657	+	0.930750i	$0.619156\pi$
$360$		0			0
$361$	29.0000			1.52632
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.5885	+	15.5885i	0.813711	+	0.813711i	0.985188	−	0.171477i	$-0.0548540\pi$
$367$	15.5885	+	15.5885i	0.813711	+	0.813711i	−0.171477	+	0.985188i	$0.554854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	24.2487i		−	1.25893i
$372$		0			0
$373$	−1.00000	−	1.00000i	−0.0517780	−	0.0517780i	0.680744	−	0.732522i	$-0.261657\pi$
$373$	−1.00000	−	1.00000i	−0.0517780	−	0.0517780i	−0.732522	+	0.680744i	$0.761657\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	+	4.00000i	−0.206010	+	0.206010i
$378$		0			0
$379$	6.92820			0.355878			0.177939	−	0.984042i	$-0.443057\pi$
$379$	6.92820			0.355878			0.177939	+	0.984042i	$0.443057\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.5885	−	15.5885i	0.796533	−	0.796533i	−0.186014	−	0.982547i	$-0.559557\pi$
$383$	15.5885	−	15.5885i	0.796533	−	0.796533i	0.982547	+	0.186014i	$0.0595570\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$		−	3.46410i		−	0.175187i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.0000	−	11.0000i	0.552074	−	0.552074i	−0.374965	−	0.927039i	$-0.622345\pi$
$397$	11.0000	−	11.0000i	0.552074	−	0.552074i	0.927039	+	0.374965i	$0.122345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$	−3.46410	+	3.46410i	−0.172559	+	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.3205	+	17.3205i	0.858546	+	0.858546i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000	−	12.0000i	−0.590481	−	0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.92820			0.338465			0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820			0.338465			0.169232	+	0.985576i	$0.445871\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	10.3923	+	10.3923i	0.502919	+	0.502919i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			3.46410i			0.166860i	0.996514	+	0.0834300i	$0.0265875\pi$
$431$			3.46410i			0.166860i	−0.996514	+	0.0834300i	$0.973413\pi$
$432$		0			0
$433$	7.00000	+	7.00000i	0.336399	+	0.336399i	0.855010	−	0.518611i	$-0.173551\pi$
$433$	7.00000	+	7.00000i	0.336399	+	0.336399i	−0.518611	+	0.855010i	$0.673551\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	12.0000i	−0.574038	+	0.574038i
$438$		0			0
$439$	−13.8564			−0.661330			−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564			−0.661330			−0.330665	+	0.943748i	$0.607273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.1244	+	12.1244i	−0.576046	+	0.576046i	−0.933811	−	0.357766i	$-0.883539\pi$
$443$	−12.1244	+	12.1244i	−0.576046	+	0.576046i	0.357766	+	0.933811i	$0.383539\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$		0			0
$451$			6.92820i			0.326236i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$	−1.73205	+	1.73205i	−0.0804952	+	0.0804952i	−0.746208	−	0.665713i	$-0.768128\pi$
$463$	−1.73205	+	1.73205i	−0.0804952	+	0.0804952i	0.665713	+	0.746208i	$0.268128\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.0526	+	19.0526i	0.881647	+	0.881647i	0.993702	−	0.112055i	$-0.0357432\pi$
$467$	19.0526	+	19.0526i	0.881647	+	0.881647i	−0.112055	+	0.993702i	$0.535743\pi$
$468$		0			0
$469$		−	18.0000i		−	0.831163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	+	6.00000i	0.275880	+	0.275880i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.7128			1.26623			0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128			1.26623			0.633115	+	0.774057i	$0.281776\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	1.73205	+	1.73205i	0.0784867	+	0.0784867i	0.745260	−	0.666774i	$-0.232325\pi$
$487$	1.73205	+	1.73205i	0.0784867	+	0.0784867i	−0.666774	+	0.745260i	$0.732325\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.2487i			1.09433i	0.837025	+	0.547165i	$0.184293\pi$
$491$			24.2487i			1.09433i	−0.837025	+	0.547165i	$0.815707\pi$
$492$		0			0
$493$	−4.00000	−	4.00000i	−0.180151	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	−	18.0000i	0.807410	−	0.807410i
$498$		0			0
$499$	20.7846			0.930447			0.465223	−	0.885193i	$-0.345974\pi$
$499$	20.7846			0.930447			0.465223	+	0.885193i	$0.345974\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.73205	−	1.73205i	0.0772283	−	0.0772283i	−0.667437	−	0.744666i	$-0.732609\pi$
$503$	1.73205	−	1.73205i	0.0772283	−	0.0772283i	0.744666	+	0.667437i	$0.232609\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	20.0000i		−	0.886484i	−0.896402	−	0.443242i	$-0.853828\pi$
$509$		−	20.0000i		−	0.886484i	0.896402	−	0.443242i	$-0.146172\pi$
$510$		0			0
$511$			24.2487i			1.07270i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.00000	+	6.00000i	−0.263880	+	0.263880i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.0000			1.66481			0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000			1.66481			0.832405	+	0.554168i	$0.186963\pi$
$522$		0			0
$523$	12.1244	−	12.1244i	0.530161	−	0.530161i	−0.390459	−	0.920620i	$-0.627684\pi$
$523$	12.1244	−	12.1244i	0.530161	−	0.530161i	0.920620	+	0.390459i	$0.127684\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.46410	−	3.46410i	−0.150899	−	0.150899i
$528$		0			0
$529$			17.0000i			0.739130i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.00000	+	2.00000i	0.0866296	+	0.0866296i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.46410			−0.149209
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.66025	+	8.66025i	0.370286	+	0.370286i	0.867581	−	0.497296i	$-0.165674\pi$
$547$	8.66025	+	8.66025i	0.370286	+	0.370286i	−0.497296	+	0.867581i	$0.665674\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			27.7128i			1.18061i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.0000	−	29.0000i	1.22877	−	1.22877i	0.264340	−	0.964430i	$-0.414846\pi$
$557$	29.0000	−	29.0000i	1.22877	−	1.22877i	0.964430	−	0.264340i	$-0.0851541\pi$
$558$		0			0
$559$	3.46410			0.146516
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.4449	−	29.4449i	1.24095	−	1.24095i	0.281347	−	0.959606i	$-0.409219\pi$
$563$	29.4449	−	29.4449i	1.24095	−	1.24095i	0.959606	−	0.281347i	$-0.0907812\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			16.0000i			0.670755i	0.942084	+	0.335377i	$0.108864\pi$
$569$			16.0000i			0.670755i	−0.942084	+	0.335377i	$0.891136\pi$
$570$		0			0
$571$			31.1769i			1.30471i	0.757912	+	0.652357i	$0.226220\pi$
$571$			31.1769i			1.30471i	−0.757912	+	0.652357i	$0.773780\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	7.00000	−	7.00000i	0.291414	−	0.291414i	−0.546225	−	0.837639i	$-0.683936\pi$
$577$	7.00000	−	7.00000i	0.291414	−	0.291414i	0.837639	+	0.546225i	$0.183936\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.0000			−1.74245
$582$		0			0
$583$	−24.2487	+	24.2487i	−1.00428	+	1.00428i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.19615	+	5.19615i	0.214468	+	0.214468i	0.806162	−	0.591694i	$-0.201541\pi$
$587$	5.19615	+	5.19615i	0.214468	+	0.214468i	−0.591694	+	0.806162i	$0.701541\pi$
$588$		0			0
$589$			24.0000i			0.988903i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	33.0000	+	33.0000i	1.35515	+	1.35515i	0.879796	+	0.475352i	$0.157679\pi$
$593$	33.0000	+	33.0000i	1.35515	+	1.35515i	0.475352	+	0.879796i	$0.342321\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.8564			0.566157			0.283079	−	0.959097i	$-0.408644\pi$
$599$	13.8564			0.566157			0.283079	+	0.959097i	$0.408644\pi$
$600$		0			0
$601$	18.0000			0.734235			0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000			0.734235			0.367118	+	0.930175i	$0.380345\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.9808	−	25.9808i	−1.05453	−	1.05453i	−0.998425	−	0.0561015i	$-0.982133\pi$
$607$	−25.9808	−	25.9808i	−1.05453	−	1.05453i	−0.0561015	−	0.998425i	$-0.517867\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.46410i			0.140143i
$612$		0			0
$613$	31.0000	+	31.0000i	1.25208	+	1.25208i	0.954787	+	0.297291i	$0.0960833\pi$
$613$	31.0000	+	31.0000i	1.25208	+	1.25208i	0.297291	+	0.954787i	$0.403917\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.00000	+	7.00000i	−0.281809	+	0.281809i	−0.833830	−	0.552021i	$-0.813857\pi$
$617$	−7.00000	+	7.00000i	−0.281809	+	0.281809i	0.552021	+	0.833830i	$0.313857\pi$
$618$		0			0
$619$	−20.7846			−0.835404			−0.417702	−	0.908584i	$-0.637164\pi$
$619$	−20.7846			−0.835404			−0.417702	+	0.908584i	$0.637164\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.8564	−	13.8564i	0.555145	−	0.555145i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.0000i			0.398726i
$630$		0			0
$631$		−	17.3205i		−	0.689519i	−0.938691	−	0.344759i	$-0.887961\pi$
$631$		−	17.3205i		−	0.689519i	0.938691	−	0.344759i	$-0.112039\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.00000	+	1.00000i	−0.0396214	+	0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−26.0000			−1.02694			−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000			−1.02694			−0.513469	+	0.858108i	$0.671640\pi$
$642$		0			0
$643$	−15.5885	+	15.5885i	−0.614749	+	0.614749i	−0.944180	−	0.329431i	$-0.893143\pi$
$643$	−15.5885	+	15.5885i	−0.614749	+	0.614749i	0.329431	+	0.944180i	$0.393143\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.73205	−	1.73205i	−0.0680939	−	0.0680939i	0.672240	−	0.740334i	$-0.265332\pi$
$647$	−1.73205	−	1.73205i	−0.0680939	−	0.0680939i	−0.740334	+	0.672240i	$0.765332\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.0000	−	23.0000i	−0.900060	−	0.900060i	0.0953813	−	0.995441i	$-0.469593\pi$
$653$	−23.0000	−	23.0000i	−0.900060	−	0.900060i	−0.995441	+	0.0953813i	$0.969593\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	34.6410			1.34942			0.674711	−	0.738082i	$-0.264268\pi$
$659$	34.6410			1.34942			0.674711	+	0.738082i	$0.264268\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.92820	−	6.92820i	−0.268261	−	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	20.7846i		−	0.802381i
$672$		0			0
$673$	−25.0000	−	25.0000i	−0.963679	−	0.963679i	0.0356839	−	0.999363i	$-0.488639\pi$
$673$	−25.0000	−	25.0000i	−0.963679	−	0.963679i	−0.999363	+	0.0356839i	$0.988639\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−35.0000	+	35.0000i	−1.34516	+	1.34516i	−0.454321	+	0.890838i	$0.650118\pi$
$677$	−35.0000	+	35.0000i	−1.34516	+	1.34516i	−0.890838	+	0.454321i	$0.849882\pi$
$678$		0			0
$679$	24.2487			0.930580
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.73205	−	1.73205i	0.0662751	−	0.0662751i	−0.673192	−	0.739467i	$-0.735077\pi$
$683$	1.73205	−	1.73205i	0.0662751	−	0.0662751i	0.739467	+	0.673192i	$0.235077\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			14.0000i			0.533358i
$690$		0			0
$691$			45.0333i			1.71315i	0.516024	+	0.856574i	$0.327412\pi$
$691$			45.0333i			1.71315i	−0.516024	+	0.856574i	$0.672588\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.00000	+	2.00000i	−0.0757554	+	0.0757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$	34.6410	−	34.6410i	1.30651	−	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.3205	+	17.3205i	0.651405	+	0.651405i
$708$		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
$713$	−6.00000	−	6.00000i	−0.224702	−	0.224702i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.7128			1.03351			0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128			1.03351			0.516757	+	0.856132i	$0.327139\pi$
$720$		0			0
$721$	−6.00000			−0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.73205	+	1.73205i	0.0642382	+	0.0642382i	0.738496	−	0.674258i	$-0.235536\pi$
$727$	1.73205	+	1.73205i	0.0642382	+	0.0642382i	−0.674258	+	0.738496i	$0.735536\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			3.46410i			0.128124i
$732$		0			0
$733$	−1.00000	−	1.00000i	−0.0369358	−	0.0369358i	0.688398	−	0.725333i	$-0.258314\pi$
$733$	−1.00000	−	1.00000i	−0.0369358	−	0.0369358i	−0.725333	+	0.688398i	$0.758314\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18.0000	+	18.0000i	−0.663039	+	0.663039i
$738$		0			0
$739$	20.7846			0.764574			0.382287	−	0.924044i	$-0.375137\pi$
$739$	20.7846			0.764574			0.382287	+	0.924044i	$0.375137\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.1244	+	12.1244i	−0.444799	+	0.444799i	−0.893621	−	0.448822i	$-0.851844\pi$
$743$	−12.1244	+	12.1244i	−0.444799	+	0.444799i	0.448822	+	0.893621i	$0.351844\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	30.0000i		−	1.09618i
$750$		0			0
$751$		−	3.46410i		−	0.126407i	−0.998001	−	0.0632034i	$-0.979868\pi$
$751$		−	3.46410i		−	0.126407i	0.998001	−	0.0632034i	$-0.0201317\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−29.0000	+	29.0000i	−1.05402	+	1.05402i	−0.0555680	+	0.998455i	$0.517697\pi$
$757$	−29.0000	+	29.0000i	−1.05402	+	1.05402i	−0.998455	+	0.0555680i	$0.982303\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.0000			−0.362500			−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000			−0.362500			−0.181250	+	0.983437i	$0.558014\pi$
$762$		0			0
$763$	−20.7846	+	20.7846i	−0.752453	+	0.752453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.92820	+	6.92820i	0.250163	+	0.250163i
$768$		0			0
$769$			48.0000i			1.73092i	0.500974	+	0.865462i	$0.332975\pi$
$769$			48.0000i			1.73092i	−0.500974	+	0.865462i	$0.667025\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.00000	+	1.00000i	0.0359675	+	0.0359675i	0.724862	−	0.688894i	$-0.241904\pi$
$773$	1.00000	+	1.00000i	0.0359675	+	0.0359675i	−0.688894	+	0.724862i	$0.741904\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.8564			0.496457
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−5.19615	−	5.19615i	−0.185223	−	0.185223i	0.608404	−	0.793627i	$-0.291810\pi$
$787$	−5.19615	−	5.19615i	−0.185223	−	0.185223i	−0.793627	+	0.608404i	$0.791810\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			31.1769i			1.10852i
$792$		0			0
$793$	−6.00000	−	6.00000i	−0.213066	−	0.213066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	+	3.00000i	−0.106265	+	0.106265i	−0.758240	−	0.651975i	$-0.773941\pi$
$797$	−3.00000	+	3.00000i	−0.106265	+	0.106265i	0.651975	+	0.758240i	$0.273941\pi$
$798$		0			0
$799$	−3.46410			−0.122551
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.2487	−	24.2487i	0.855718	−	0.855718i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			8.00000i			0.281265i	0.990062	+	0.140633i	$0.0449136\pi$
$809$			8.00000i			0.281265i	−0.990062	+	0.140633i	$0.955086\pi$
$810$		0			0
$811$		−	24.2487i		−	0.851487i	−0.904844	−	0.425744i	$-0.860013\pi$
$811$		−	24.2487i		−	0.851487i	0.904844	−	0.425744i	$-0.139987\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	12.0000	−	12.0000i	0.419827	−	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.00000			0.0698005			0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000			0.0698005			0.0349002	+	0.999391i	$0.488889\pi$
$822$		0			0
$823$	25.9808	−	25.9808i	0.905632	−	0.905632i	−0.0902837	−	0.995916i	$-0.528777\pi$
$823$	25.9808	−	25.9808i	0.905632	−	0.905632i	0.995916	+	0.0902837i	$0.0287774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.19615	+	5.19615i	0.180688	+	0.180688i	0.791656	−	0.610968i	$-0.209219\pi$
$827$	5.19615	+	5.19615i	0.180688	+	0.180688i	−0.610968	+	0.791656i	$0.709219\pi$
$828$		0			0
$829$			12.0000i			0.416777i	0.978046	+	0.208389i	$0.0668219\pi$
$829$			12.0000i			0.416777i	−0.978046	+	0.208389i	$0.933178\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.00000	−	1.00000i	−0.0346479	−	0.0346479i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−41.5692			−1.43513			−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692			−1.43513			−0.717564	+	0.696492i	$0.754743\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.73205	−	1.73205i	−0.0595140	−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			17.3205i			0.593739i
$852$		0			0
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	0.689780	−	0.724019i	$-0.257707\pi$
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	−0.724019	+	0.689780i	$0.757707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	0.990621	+	0.136637i	$0.0436295\pi$
$858$		0			0
$859$	6.92820			0.236387			0.118194	−	0.992991i	$-0.462290\pi$
$859$	6.92820			0.236387			0.118194	+	0.992991i	$0.462290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.73205	−	1.73205i	0.0589597	−	0.0589597i	−0.677012	−	0.735972i	$-0.736726\pi$
$863$	1.73205	−	1.73205i	0.0589597	−	0.0589597i	0.735972	+	0.677012i	$0.236726\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$			10.3923i			0.352130i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.00000	+	5.00000i	−0.168838	+	0.168838i	−0.786468	−	0.617630i	$-0.788093\pi$
$877$	−5.00000	+	5.00000i	−0.168838	+	0.168838i	0.617630	+	0.786468i	$0.288093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.0000			−0.336909			−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000			−0.336909			−0.168454	+	0.985709i	$0.553878\pi$
$882$		0			0
$883$	25.9808	−	25.9808i	0.874322	−	0.874322i	−0.118618	−	0.992940i	$-0.537846\pi$
$883$	25.9808	−	25.9808i	0.874322	−	0.874322i	0.992940	+	0.118618i	$0.0378463\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.9808	+	25.9808i	0.872349	+	0.872349i	0.992728	−	0.120379i	$-0.0384111\pi$
$887$	25.9808	+	25.9808i	0.872349	+	0.872349i	−0.120379	+	0.992728i	$0.538411\pi$
$888$		0			0
$889$		−	42.0000i		−	1.40863i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.0000	+	12.0000i	0.401565	+	0.401565i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.8564			−0.462137
$900$		0			0
$901$	−14.0000			−0.466408
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	36.3731	+	36.3731i	1.20775	+	1.20775i	0.971755	+	0.235993i	$0.0758343\pi$
$907$	36.3731	+	36.3731i	1.20775	+	1.20775i	0.235993	+	0.971755i	$0.424166\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			31.1769i			1.03294i	0.856306	+	0.516469i	$0.172754\pi$
$911$			31.1769i			1.03294i	−0.856306	+	0.516469i	$0.827246\pi$
$912$		0			0
$913$	42.0000	+	42.0000i	1.39000	+	1.39000i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.0000	+	18.0000i	−0.594412	+	0.594412i
$918$		0			0
$919$	41.5692			1.37124			0.685621	−	0.727959i	$-0.259531\pi$
$919$	41.5692			1.37124			0.685621	+	0.727959i	$0.259531\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3923	+	10.3923i	−0.342067	+	0.342067i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			56.0000i			1.83730i	0.395072	+	0.918650i	$0.370720\pi$
$929$			56.0000i			1.83730i	−0.395072	+	0.918650i	$0.629280\pi$
$930$		0			0
$931$			6.92820i			0.227063i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	23.0000	−	23.0000i	0.751377	−	0.751377i	−0.223359	−	0.974736i	$-0.571702\pi$
$937$	23.0000	−	23.0000i	0.751377	−	0.751377i	0.974736	+	0.223359i	$0.0717022\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	34.0000			1.10837			0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000			1.10837			0.554184	+	0.832394i	$0.313030\pi$
$942$		0			0
$943$	−3.46410	+	3.46410i	−0.112807	+	0.112807i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.19615	+	5.19615i	0.168852	+	0.168852i	0.786475	−	0.617622i	$-0.211904\pi$
$947$	5.19615	+	5.19615i	0.168852	+	0.168852i	−0.617622	+	0.786475i	$0.711904\pi$
$948$		0			0
$949$		−	14.0000i		−	0.454459i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.00000	+	9.00000i	0.291539	+	0.291539i	0.837688	−	0.546149i	$-0.183907\pi$
$953$	9.00000	+	9.00000i	0.291539	+	0.291539i	−0.546149	+	0.837688i	$0.683907\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	31.1769			1.00676
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.5885	+	15.5885i	0.501291	+	0.501291i	0.911839	−	0.410548i	$-0.134663\pi$
$967$	15.5885	+	15.5885i	0.501291	+	0.501291i	−0.410548	+	0.911839i	$0.634663\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	3.46410i		−	0.111168i	−0.998454	−	0.0555842i	$-0.982298\pi$
$971$		−	3.46410i		−	0.111168i	0.998454	−	0.0555842i	$-0.0177021\pi$
$972$		0			0
$973$	12.0000	+	12.0000i	0.384702	+	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.0000	+	23.0000i	−0.735835	+	0.735835i	−0.971769	−	0.235934i	$-0.924185\pi$
$977$	−23.0000	+	23.0000i	−0.735835	+	0.735835i	0.235934	+	0.971769i	$0.424185\pi$
$978$		0			0
$979$	−27.7128			−0.885705
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.5885	−	15.5885i	0.497195	−	0.497195i	−0.413369	−	0.910564i	$-0.635648\pi$
$983$	15.5885	−	15.5885i	0.497195	−	0.497195i	0.910564	+	0.413369i	$0.135648\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.00000i			0.190789i
$990$		0			0
$991$		−	3.46410i		−	0.110041i	−0.998485	−	0.0550204i	$-0.982478\pi$
$991$		−	3.46410i		−	0.110041i	0.998485	−	0.0550204i	$-0.0175224\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.00000	+	5.00000i	−0.158352	+	0.158352i	−0.781836	−	0.623484i	$-0.785717\pi$
$997$	−5.00000	+	5.00000i	−0.158352	+	0.158352i	0.623484	+	0.781836i	$0.285717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.x.e.2143.2		4
3.2	odd	2		400.2.n.b.143.1		4
4.3	odd	2	inner	3600.2.x.e.2143.1		4
5.2	odd	4	inner	3600.2.x.e.3007.1		4
5.3	odd	4		720.2.x.d.127.2		4
5.4	even	2		720.2.x.d.703.1		4
12.11	even	2		400.2.n.b.143.2		4
15.2	even	4		400.2.n.b.207.2		4
15.8	even	4		80.2.n.b.47.1	✓	4
15.14	odd	2		80.2.n.b.63.2	yes	4
20.3	even	4		720.2.x.d.127.1		4
20.7	even	4	inner	3600.2.x.e.3007.2		4
20.19	odd	2		720.2.x.d.703.2		4
24.5	odd	2		1600.2.n.r.1343.2		4
24.11	even	2		1600.2.n.r.1343.1		4
60.23	odd	4		80.2.n.b.47.2	yes	4
60.47	odd	4		400.2.n.b.207.1		4
60.59	even	2		80.2.n.b.63.1	yes	4
120.29	odd	2		320.2.n.i.63.1		4
120.53	even	4		320.2.n.i.127.2		4
120.59	even	2		320.2.n.i.63.2		4
120.77	even	4		1600.2.n.r.1407.1		4
120.83	odd	4		320.2.n.i.127.1		4
120.107	odd	4		1600.2.n.r.1407.2		4
240.29	odd	4		1280.2.o.r.383.2		4
240.53	even	4		1280.2.o.r.127.1		4
240.59	even	4		1280.2.o.q.383.2		4
240.83	odd	4		1280.2.o.q.127.1		4
240.149	odd	4		1280.2.o.q.383.1		4
240.173	even	4		1280.2.o.q.127.2		4
240.179	even	4		1280.2.o.r.383.1		4
240.203	odd	4		1280.2.o.r.127.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.n.b.47.1	✓	4	15.8	even	4
80.2.n.b.47.2	yes	4	60.23	odd	4
80.2.n.b.63.1	yes	4	60.59	even	2
80.2.n.b.63.2	yes	4	15.14	odd	2
320.2.n.i.63.1		4	120.29	odd	2
320.2.n.i.63.2		4	120.59	even	2
320.2.n.i.127.1		4	120.83	odd	4
320.2.n.i.127.2		4	120.53	even	4
400.2.n.b.143.1		4	3.2	odd	2
400.2.n.b.143.2		4	12.11	even	2
400.2.n.b.207.1		4	60.47	odd	4
400.2.n.b.207.2		4	15.2	even	4
720.2.x.d.127.1		4	20.3	even	4
720.2.x.d.127.2		4	5.3	odd	4
720.2.x.d.703.1		4	5.4	even	2
720.2.x.d.703.2		4	20.19	odd	2
1280.2.o.q.127.1		4	240.83	odd	4
1280.2.o.q.127.2		4	240.173	even	4
1280.2.o.q.383.1		4	240.149	odd	4
1280.2.o.q.383.2		4	240.59	even	4
1280.2.o.r.127.1		4	240.53	even	4
1280.2.o.r.127.2		4	240.203	odd	4
1280.2.o.r.383.1		4	240.179	even	4
1280.2.o.r.383.2		4	240.29	odd	4
1600.2.n.r.1343.1		4	24.11	even	2
1600.2.n.r.1343.2		4	24.5	odd	2
1600.2.n.r.1407.1		4	120.77	even	4
1600.2.n.r.1407.2		4	120.107	odd	4
3600.2.x.e.2143.1		4	4.3	odd	2	inner
3600.2.x.e.2143.2		4	1.1	even	1	trivial
3600.2.x.e.3007.1		4	5.2	odd	4	inner
3600.2.x.e.3007.2		4	20.7	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.73205i) q^{7} -3.46410i q^{11} +(-1.00000 - 1.00000i) q^{13} +(1.00000 - 1.00000i) q^{17} -6.92820 q^{19} +(1.73205 - 1.73205i) q^{23} -4.00000i q^{29} -3.46410i q^{31} +(-5.00000 + 5.00000i) q^{37} -2.00000 q^{41} +(-1.73205 + 1.73205i) q^{43} +(-1.73205 - 1.73205i) q^{47} -1.00000i q^{49} +(-7.00000 - 7.00000i) q^{53} -6.92820 q^{59} +6.00000 q^{61} +(-5.19615 - 5.19615i) q^{67} -10.3923i q^{71} +(7.00000 + 7.00000i) q^{73} +(6.00000 - 6.00000i) q^{77} +(-12.1244 + 12.1244i) q^{83} -8.00000i q^{89} -3.46410i q^{91} +(7.00000 - 7.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{13} + 4 q^{17} - 20 q^{37} - 8 q^{41} - 28 q^{53} + 24 q^{61} + 28 q^{73} + 24 q^{77} + 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^13 + 4 * q^17 - 20 * q^37 - 8 * q^41 - 28 * q^53 + 24 * q^61 + 28 * q^73 + 24 * q^77 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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