LMFDB - Embedded newform 2700.2.j.c.1457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(593,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.j (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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1457.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2700.1457
Dual form			2700.2.j.c.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{7} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{7} +(-4.89898 + 4.89898i) q^{13} -1.00000i q^{19} -11.0000 q^{31} +(-3.67423 - 3.67423i) q^{37} +(1.22474 - 1.22474i) q^{43} -4.00000i q^{49} -13.0000 q^{61} +(2.44949 + 2.44949i) q^{67} +(-11.0227 + 11.0227i) q^{73} -17.0000i q^{79} -12.0000 q^{91} +(13.4722 + 13.4722i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 44 q^{31} - 52 q^{61} - 48 q^{91}+O(q^{100}) $ 4 * q - 44 * q^31 - 52 * q^61 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	0.899608	−	0.436698i	$-0.143852\pi$
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	−0.436698	+	0.899608i	$0.643852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.89898	+	4.89898i	−1.35873	+	1.35873i	−0.483250	+	0.875482i	$0.660544\pi$
$13$	−4.89898	+	4.89898i	−1.35873	+	1.35873i	−0.875482	+	0.483250i	$0.839456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−11.0000			−1.97566			−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−11.0000			−1.97566			−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.67423	−	3.67423i	−0.604040	−	0.604040i	0.337342	−	0.941382i	$-0.390472\pi$
$37$	−3.67423	−	3.67423i	−0.604040	−	0.604040i	−0.941382	+	0.337342i	$0.890472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	1.22474	−	1.22474i	0.186772	−	0.186772i	−0.607527	−	0.794299i	$-0.707838\pi$
$43$	1.22474	−	1.22474i	0.186772	−	0.186772i	0.794299	+	0.607527i	$0.207838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$		−	4.00000i		−	0.571429i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.44949	+	2.44949i	0.299253	+	0.299253i	0.840721	−	0.541468i	$-0.182131\pi$
$67$	2.44949	+	2.44949i	0.299253	+	0.299253i	−0.541468	+	0.840721i	$0.682131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−11.0227	+	11.0227i	−1.29011	+	1.29011i	−0.355393	+	0.934717i	$0.615653\pi$
$73$	−11.0227	+	11.0227i	−1.29011	+	1.29011i	−0.934717	+	0.355393i	$0.884347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	17.0000i		−	1.91265i	−0.292306	−	0.956325i	$-0.594423\pi$
$79$		−	17.0000i		−	1.91265i	0.292306	−	0.956325i	$-0.405577\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.4722	+	13.4722i	1.36789	+	1.36789i	0.863437	+	0.504457i	$0.168307\pi$
$97$	13.4722	+	13.4722i	1.36789	+	1.36789i	0.504457	+	0.863437i	$0.331693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−11.0227	+	11.0227i	−1.08610	+	1.08610i	−0.0901732	+	0.995926i	$0.528742\pi$
$103$	−11.0227	+	11.0227i	−1.08610	+	1.08610i	−0.995926	+	0.0901732i	$0.971258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		−	17.0000i		−	1.62830i	−0.580651	−	0.814152i	$-0.697202\pi$
$109$		−	17.0000i		−	1.62830i	0.580651	−	0.814152i	$-0.302798\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	0.953491	−	0.301420i	$-0.0974607\pi$
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	−0.301420	+	0.953491i	$0.597461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	1.22474	−	1.22474i	0.106199	−	0.106199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$			23.0000i			1.95083i	0.220366	+	0.975417i	$0.429275\pi$
$139$			23.0000i			1.95083i	−0.220366	+	0.975417i	$0.570725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−23.0000			−1.87171			−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000			−1.87171			−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−15.9217	−	15.9217i	−1.27069	−	1.27069i	−0.945727	−	0.324961i	$-0.894649\pi$
$157$	−15.9217	−	15.9217i	−1.27069	−	1.27069i	−0.324961	−	0.945727i	$-0.605351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−17.1464	+	17.1464i	−1.34301	+	1.34301i	−0.449966	+	0.893045i	$0.648564\pi$
$163$	−17.1464	+	17.1464i	−1.34301	+	1.34301i	−0.893045	+	0.449966i	$0.851436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	35.0000i		−	2.69231i
$170$		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
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−8.57321	+	8.57321i	−0.617113	+	0.617113i	−0.944790	−	0.327677i	$-0.893734\pi$
$193$	−8.57321	+	8.57321i	−0.617113	+	0.617113i	0.327677	+	0.944790i	$0.393734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			28.0000i			1.98487i	0.122782	+	0.992434i	$0.460818\pi$
$199$			28.0000i			1.98487i	−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.4722	−	13.4722i	−0.914552	−	0.914552i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	13.4722	−	13.4722i	0.902165	−	0.902165i	−0.0934584	−	0.995623i	$-0.529792\pi$
$223$	13.4722	−	13.4722i	0.902165	−	0.902165i	0.995623	+	0.0934584i	$0.0297922\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		−	7.00000i		−	0.462573i	−0.972886	−	0.231287i	$-0.925707\pi$
$229$		−	7.00000i		−	0.462573i	0.972886	−	0.231287i	$-0.0742935\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.89898	+	4.89898i	0.311715	+	0.311715i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		−	9.00000i		−	0.559233i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	29.0000			1.76162			0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000			1.76162			0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8.57321	+	8.57321i	0.515115	+	0.515115i	0.916089	−	0.400975i	$-0.131328\pi$
$277$	8.57321	+	8.57321i	0.515115	+	0.515115i	−0.400975	+	0.916089i	$0.631328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−23.2702	+	23.2702i	−1.38327	+	1.38327i	−0.544518	+	0.838749i	$0.683287\pi$
$283$	−23.2702	+	23.2702i	−1.38327	+	1.38327i	−0.838749	+	0.544518i	$0.816713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	3.00000			0.172917
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.8207	+	20.8207i	1.18830	+	1.18830i	0.977538	+	0.210760i	$0.0675939\pi$
$307$	20.8207	+	20.8207i	1.18830	+	1.18830i	0.210760	+	0.977538i	$0.432406\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	19.5959	−	19.5959i	1.10763	−	1.10763i	0.114165	−	0.993462i	$-0.463581\pi$
$313$	19.5959	−	19.5959i	1.10763	−	1.10763i	0.993462	−	0.114165i	$-0.0364192\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.00000			−0.0549650			−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000			−0.0549650			−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−9.79796	−	9.79796i	−0.533729	−	0.533729i	0.387951	−	0.921680i	$-0.373183\pi$
$337$	−9.79796	−	9.79796i	−0.533729	−	0.533729i	−0.921680	+	0.387951i	$0.873183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	13.4722	−	13.4722i	0.727430	−	0.727430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	37.0000i		−	1.98056i	−0.139072	−	0.990282i	$-0.544412\pi$
$349$		−	37.0000i		−	1.98056i	0.139072	−	0.990282i	$-0.455588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	26.9444	+	26.9444i	1.40649	+	1.40649i	0.777064	+	0.629421i	$0.216708\pi$
$367$	26.9444	+	26.9444i	1.40649	+	1.40649i	0.629421	+	0.777064i	$0.283292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−20.8207	+	20.8207i	−1.07805	+	1.07805i	−0.0813690	+	0.996684i	$0.525929\pi$
$373$	−20.8207	+	20.8207i	−1.07805	+	1.07805i	−0.996684	+	0.0813690i	$0.974071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−28.1691	−	28.1691i	−1.41377	−	1.41377i	−0.724628	−	0.689140i	$-0.757989\pi$
$397$	−28.1691	−	28.1691i	−1.41377	−	1.41377i	−0.689140	−	0.724628i	$-0.742011\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	53.8888	−	53.8888i	2.68439	−	2.68439i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			38.0000i			1.87898i	0.342578	+	0.939490i	$0.388700\pi$
$409$			38.0000i			1.87898i	−0.342578	+	0.939490i	$0.611300\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−15.9217	−	15.9217i	−0.770504	−	0.770504i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	13.4722	−	13.4722i	0.647432	−	0.647432i	−0.304939	−	0.952372i	$-0.598636\pi$
$433$	13.4722	−	13.4722i	0.647432	−	0.647432i	0.952372	+	0.304939i	$0.0986362\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			13.0000i			0.620456i	0.950662	+	0.310228i	$0.100405\pi$
$439$			13.0000i			0.620456i	−0.950662	+	0.310228i	$0.899595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.3939	−	29.3939i	−1.37499	−	1.37499i	−0.852879	−	0.522108i	$-0.825146\pi$
$457$	−29.3939	−	29.3939i	−1.37499	−	1.37499i	−0.522108	−	0.852879i	$-0.674854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	25.7196	−	25.7196i	1.19529	−	1.19529i	0.219733	−	0.975560i	$-0.429481\pi$
$463$	25.7196	−	25.7196i	1.19529	−	1.19529i	0.975560	−	0.219733i	$-0.0705187\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$			6.00000i			0.277054i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	36.0000			1.64146
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.44949	+	2.44949i	0.110997	+	0.110997i	0.760424	−	0.649427i	$-0.224991\pi$
$487$	2.44949	+	2.44949i	0.110997	+	0.110997i	−0.649427	+	0.760424i	$0.724991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			43.0000i			1.92494i	0.271380	+	0.962472i	$0.412520\pi$
$499$			43.0000i			1.92494i	−0.271380	+	0.962472i	$0.587480\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−27.0000			−1.19441
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−20.8207	+	20.8207i	−0.910424	+	0.910424i	−0.996305	−	0.0858814i	$-0.972629\pi$
$523$	−20.8207	+	20.8207i	−0.910424	+	0.910424i	0.0858814	+	0.996305i	$0.472629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	33.0681	+	33.0681i	1.41389	+	1.41389i	0.722062	+	0.691828i	$0.243195\pi$
$547$	33.0681	+	33.0681i	1.41389	+	1.41389i	0.691828	+	0.722062i	$0.256805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	20.8207	−	20.8207i	0.885385	−	0.885385i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	47.0000			1.96689			0.983444	−	0.181210i	$-0.0580014\pi$
$571$	47.0000			1.96689			0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−23.2702	−	23.2702i	−0.968749	−	0.968749i	0.0307771	−	0.999526i	$-0.490202\pi$
$577$	−23.2702	−	23.2702i	−0.968749	−	0.968749i	−0.999526	+	0.0307771i	$0.990202\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$			11.0000i			0.453247i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	49.0000			1.99875			0.999376	−	0.0353259i	$-0.0112469\pi$
$601$	49.0000			1.99875			0.999376	+	0.0353259i	$0.0112469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.67423	−	3.67423i	−0.149133	−	0.149133i	0.628598	−	0.777730i	$-0.283629\pi$
$607$	−3.67423	−	3.67423i	−0.149133	−	0.149133i	−0.777730	+	0.628598i	$0.783629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−11.0227	+	11.0227i	−0.445203	+	0.445203i	−0.893756	−	0.448553i	$-0.851939\pi$
$613$	−11.0227	+	11.0227i	−0.445203	+	0.445203i	0.448553	+	0.893756i	$0.351939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$			49.0000i			1.96948i	0.174042	+	0.984738i	$0.444317\pi$
$619$			49.0000i			1.96948i	−0.174042	+	0.984738i	$0.555683\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19.5959	+	19.5959i	0.776419	+	0.776419i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	−0.123018	−	0.992404i	$-0.539257\pi$
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	0.992404	+	0.123018i	$0.0392574\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	49.0000			1.90588			0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000			1.90588			0.952940	+	0.303160i	$0.0980418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.7196	−	25.7196i	0.991419	−	0.991419i	−0.00854415	−	0.999963i	$-0.502720\pi$
$673$	25.7196	−	25.7196i	0.991419	−	0.991419i	0.999963	+	0.00854415i	$0.00271972\pi$
$674$		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
$679$			33.0000i			1.26642i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−3.67423	+	3.67423i	−0.138576	+	0.138576i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	31.0000i		−	1.16423i	−0.813107	−	0.582115i	$-0.802225\pi$
$709$		−	31.0000i		−	1.16423i	0.813107	−	0.582115i	$-0.197775\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−27.0000			−1.00553
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.9217	−	15.9217i	−0.590503	−	0.590503i	0.347265	−	0.937767i	$-0.387111\pi$
$727$	−15.9217	−	15.9217i	−0.590503	−	0.590503i	−0.937767	+	0.347265i	$0.887111\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−14.6969	+	14.6969i	−0.542844	+	0.542844i	−0.924362	−	0.381518i	$-0.875402\pi$
$733$	−14.6969	+	14.6969i	−0.542844	+	0.542844i	0.381518	+	0.924362i	$0.375402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	16.0000i		−	0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$		−	16.0000i		−	0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−41.0000			−1.49611			−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000			−1.49611			−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.22474	+	1.22474i	0.0445141	+	0.0445141i	0.729013	−	0.684499i	$-0.239979\pi$
$757$	1.22474	+	1.22474i	0.0445141	+	0.0445141i	−0.684499	+	0.729013i	$0.739979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	20.8207	−	20.8207i	0.753759	−	0.753759i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	2.00000i		−	0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$		−	2.00000i		−	0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−35.5176	−	35.5176i	−1.26607	−	1.26607i	−0.948103	−	0.317962i	$-0.897001\pi$
$787$	−35.5176	−	35.5176i	−1.26607	−	1.26607i	−0.317962	−	0.948103i	$-0.602999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	63.6867	−	63.6867i	2.26158	−	2.26158i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	19.0000			0.667180			0.333590	−	0.942718i	$-0.391740\pi$
$811$	19.0000			0.667180			0.333590	+	0.942718i	$0.391740\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.22474	−	1.22474i	−0.0428484	−	0.0428484i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−17.1464	+	17.1464i	−0.597687	+	0.597687i	−0.939696	−	0.342010i	$-0.888892\pi$
$823$	−17.1464	+	17.1464i	−0.597687	+	0.597687i	0.342010	+	0.939696i	$0.388892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		−	7.00000i		−	0.243120i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$		−	7.00000i		−	0.243120i	0.992584	−	0.121560i	$-0.0387897\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.4722	+	13.4722i	0.462910	+	0.462910i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−4.89898	+	4.89898i	−0.167738	+	0.167738i	−0.785984	−	0.618246i	$-0.787843\pi$
$853$	−4.89898	+	4.89898i	−0.167738	+	0.167738i	0.618246	+	0.785984i	$0.287843\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$			13.0000i			0.443554i	0.975097	+	0.221777i	$0.0711857\pi$
$859$			13.0000i			0.443554i	−0.975097	+	0.221777i	$0.928814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	37.9671	+	37.9671i	1.28206	+	1.28206i	0.939495	+	0.342563i	$0.111295\pi$
$877$	37.9671	+	37.9671i	1.28206	+	1.28206i	0.342563	+	0.939495i	$0.388705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	15.9217	−	15.9217i	0.535807	−	0.535807i	−0.386487	−	0.922295i	$-0.626312\pi$
$883$	15.9217	−	15.9217i	0.535807	−	0.535807i	0.922295	+	0.386487i	$0.126312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$			18.0000i			0.603701i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−40.4166	−	40.4166i	−1.34201	−	1.34201i	−0.894057	−	0.447954i	$-0.852153\pi$
$907$	−40.4166	−	40.4166i	−1.34201	−	1.34201i	−0.447954	−	0.894057i	$-0.647847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			53.0000i			1.74831i	0.485648	+	0.874154i	$0.338584\pi$
$919$			53.0000i			1.74831i	−0.485648	+	0.874154i	$0.661416\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−4.00000			−0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−35.5176	−	35.5176i	−1.16031	−	1.16031i	−0.984408	−	0.175902i	$-0.943716\pi$
$937$	−35.5176	−	35.5176i	−1.16031	−	1.16031i	−0.175902	−	0.984408i	$-0.556284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		−	108.000i		−	3.50583i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	90.0000			2.90323
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	33.0681	+	33.0681i	1.06340	+	1.06340i	0.997849	+	0.0655495i	$0.0208800\pi$
$967$	33.0681	+	33.0681i	1.06340	+	1.06340i	0.0655495	+	0.997849i	$0.479120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−28.1691	+	28.1691i	−0.903061	+	0.903061i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	17.0000			0.540023			0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000			0.540023			0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	44.0908	+	44.0908i	1.39637	+	1.39637i	0.810157	+	0.586214i	$0.199382\pi$
$997$	44.0908	+	44.0908i	1.39637	+	1.39637i	0.586214	+	0.810157i	$0.300618\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.c.1457.2	yes	4
3.2	odd	2	CM	2700.2.j.c.1457.2	yes	4
5.2	odd	4	inner	2700.2.j.c.593.1	✓	4
5.3	odd	4	inner	2700.2.j.c.593.2	yes	4
5.4	even	2	inner	2700.2.j.c.1457.1	yes	4
15.2	even	4	inner	2700.2.j.c.593.1	✓	4
15.8	even	4	inner	2700.2.j.c.593.2	yes	4
15.14	odd	2	inner	2700.2.j.c.1457.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2700.2.j.c.593.1	✓	4	5.2	odd	4	inner
2700.2.j.c.593.1	✓	4	15.2	even	4	inner
2700.2.j.c.593.2	yes	4	5.3	odd	4	inner
2700.2.j.c.593.2	yes	4	15.8	even	4	inner
2700.2.j.c.1457.1	yes	4	5.4	even	2	inner
2700.2.j.c.1457.1	yes	4	15.14	odd	2	inner
2700.2.j.c.1457.2	yes	4	1.1	even	1	trivial
2700.2.j.c.1457.2	yes	4	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{7} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{7} +(-4.89898 + 4.89898i) q^{13} -1.00000i q^{19} -11.0000 q^{31} +(-3.67423 - 3.67423i) q^{37} +(1.22474 - 1.22474i) q^{43} -4.00000i q^{49} -13.0000 q^{61} +(2.44949 + 2.44949i) q^{67} +(-11.0227 + 11.0227i) q^{73} -17.0000i q^{79} -12.0000 q^{91} +(13.4722 + 13.4722i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 44 q^{31} - 52 q^{61} - 48 q^{91}+O(q^{100}) \) 4 * q - 44 * q^31 - 52 * q^61 - 48 * q^91

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