LMFDB - Embedded newform 525.2.b.c.251.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,2,Mod(251,525)] 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("525.251"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,4,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	525.251
Dual form			525.2.b.c.251.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +2.00000 q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} -3.46410i q^{12} -5.19615i q^{13} +4.00000 q^{16} -8.66025i q^{19} +(-1.50000 + 4.33013i) q^{21} +5.19615i q^{27} +(-5.00000 - 1.73205i) q^{28} +8.66025i q^{31} -6.00000 q^{36} +10.0000 q^{37} -9.00000 q^{39} -5.00000 q^{43} -6.92820i q^{48} +(5.50000 + 4.33013i) q^{49} -10.3923i q^{52} -15.0000 q^{57} -8.66025i q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +5.00000 q^{67} +13.8564i q^{73} -17.3205i q^{76} -4.00000 q^{79} +9.00000 q^{81} +(-3.00000 + 8.66025i) q^{84} +(-4.50000 + 12.9904i) q^{91} +15.0000 q^{93} +19.0526i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 5 q^{7} - 6 q^{9} + 8 q^{16} - 3 q^{21} - 10 q^{28} - 12 q^{36} + 20 q^{37} - 18 q^{39} - 10 q^{43} + 11 q^{49} - 30 q^{57} + 15 q^{63} + 16 q^{64} + 10 q^{67} - 8 q^{79} + 18 q^{81} - 6 q^{84}+ \cdots + 30 q^{93}+O(q^{100}) $ 2 * q + 4 * q^4 - 5 * q^7 - 6 * q^9 + 8 * q^16 - 3 * q^21 - 10 * q^28 - 12 * q^36 + 20 * q^37 - 18 * q^39 - 10 * q^43 + 11 * q^49 - 30 * q^57 + 15 * q^63 + 16 * q^64 + 10 * q^67 - 8 * q^79 + 18 * q^81 - 6 * q^84 - 9 * q^91 + 30 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$1$	$-1$	$-1$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	2.00000			1.00000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	3.46410i		−	1.00000i
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	8.66025i		−	1.98680i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$		−	8.66025i		−	1.98680i	0.114708	−	0.993399i	$-0.463407\pi$
$20$		0			0
$21$	−1.50000	+	4.33013i	−0.327327	+	0.944911i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.19615i			1.00000i
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			8.66025i			1.55543i	0.628619	+	0.777714i	$0.283621\pi$
$31$			8.66025i			1.55543i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$	−9.00000			−1.44115
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		−	6.92820i		−	1.00000i
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		−	10.3923i		−	1.44115i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−15.0000			−1.98680
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	8.66025i		−	1.10883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$		−	8.66025i		−	1.10883i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	5.00000			0.610847			0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000			0.610847			0.305424	+	0.952217i	$0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			13.8564i			1.62177i	0.585206	+	0.810885i	$0.301014\pi$
$73$			13.8564i			1.62177i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		−	17.3205i		−	1.98680i
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.00000	+	8.66025i	−0.327327	+	0.944911i
$85$		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
$91$	−4.50000	+	12.9904i	−0.471728	+	1.36176i
$92$		0			0
$93$	15.0000			1.55543
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.0526i			1.93449i	0.253837	+	0.967247i	$0.418307\pi$
$97$			19.0526i			1.93449i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		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
$103$			3.46410i			0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$			3.46410i			0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			10.3923i			1.00000i
$109$	19.0000			1.81987			0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000			1.81987			0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$		−	17.3205i		−	1.64399i
$112$	−10.0000	−	3.46410i	−0.944911	−	0.327327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			15.5885i			1.44115i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$			17.3205i			1.55543i
$125$		0			0
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$			8.66025i			0.762493i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−7.50000	+	21.6506i	−0.650332	+	1.87735i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	17.3205i		−	1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$		−	17.3205i		−	1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	7.50000	−	9.52628i	0.618590	−	0.785714i
$148$	20.0000			1.64399
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		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
$156$	−18.0000			−1.44115
$157$			1.73205i			0.138233i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$			1.73205i			0.138233i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−25.0000			−1.95815			−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000			−1.95815			−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		0			0
$171$			25.9808i			1.98680i
$172$	−10.0000			−0.762493
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	25.9808i		−	1.93113i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$		−	25.9808i		−	1.93113i	0.260153	−	0.965567i	$-0.416227\pi$
$182$		0			0
$183$	−15.0000			−1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.50000	−	12.9904i	0.327327	−	0.944911i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	13.8564i		−	1.00000i
$193$	25.0000			1.79954			0.899770	−	0.436365i	$-0.143734\pi$
$193$	25.0000			1.79954			0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	+	8.66025i	0.785714	+	0.618590i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			25.9808i			1.84173i	0.389885	+	0.920864i	$0.372515\pi$
$199$			25.9808i			1.84173i	−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$		−	8.66025i		−	0.610847i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	20.7846i		−	1.44115i
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.50000	−	21.6506i	0.509133	−	1.46974i
$218$		0			0
$219$	24.0000			1.62177
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	29.4449i		−	1.97177i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$		−	29.4449i		−	1.97177i	0.167412	−	0.985887i	$-0.446459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−30.0000			−1.98680
$229$		−	8.66025i		−	0.572286i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$		−	8.66025i		−	0.572286i	0.958187	−	0.286143i	$-0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			6.92820i			0.450035i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			25.9808i			1.67357i	0.547533	+	0.836784i	$0.315567\pi$
$241$			25.9808i			1.67357i	−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$		−	15.5885i		−	1.00000i
$244$		−	17.3205i		−	1.10883i
$245$		0			0
$246$		0			0
$247$	−45.0000			−2.86328
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	15.0000	+	5.19615i	0.944911	+	0.327327i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−25.0000	−	8.66025i	−1.55342	−	0.538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	10.0000			0.610847
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			17.3205i			1.05215i	0.850439	+	0.526073i	$0.176336\pi$
$271$			17.3205i			1.05215i	−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$	22.5000	+	7.79423i	1.36176	+	0.471728i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.00000			0.300421			0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000			0.300421			0.150210	+	0.988654i	$0.452005\pi$
$278$		0			0
$279$		−	25.9808i		−	1.55543i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	22.5167i		−	1.33848i	−0.743048	−	0.669238i	$-0.766621\pi$
$283$		−	22.5167i		−	1.33848i	0.743048	−	0.669238i	$-0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	33.0000			1.93449
$292$			27.7128i			1.62177i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	12.5000	+	4.33013i	0.720488	+	0.249584i
$302$		0			0
$303$		0			0
$304$		−	34.6410i		−	1.98680i
$305$		0			0
$306$		0			0
$307$		−	1.73205i		−	0.0988534i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$		−	1.73205i		−	0.0988534i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$		0			0
$309$	6.00000			0.341328
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			5.19615i			0.293704i	0.989158	+	0.146852i	$0.0469141\pi$
$313$			5.19615i			0.293704i	−0.989158	+	0.146852i	$0.953086\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$		0			0
$326$		0			0
$327$		−	32.9090i		−	1.81987i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.0000			−1.75888			−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000			−1.75888			−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−30.0000			−1.64399
$334$		0			0
$335$		0			0
$336$	−6.00000	+	17.3205i	−0.327327	+	0.944911i
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			34.6410i			1.85429i	0.374701	+	0.927146i	$0.377745\pi$
$349$			34.6410i			1.85429i	−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	27.0000			1.44115
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−56.0000			−2.94737
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$	−9.00000	+	25.9808i	−0.471728	+	1.36176i
$365$		0			0
$366$		0			0
$367$		−	15.5885i		−	0.813711i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$		−	15.5885i		−	0.813711i	0.913493	−	0.406855i	$-0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.0000			1.55543
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$		−	34.6410i		−	1.77471i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	15.0000			0.762493
$388$			38.1051i			1.93449i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	19.0526i		−	0.956221i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$		−	19.0526i		−	0.956221i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$	37.5000	+	12.9904i	1.87735	+	0.650332i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	45.0000			2.24161
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			25.9808i			1.28467i	0.766426	+	0.642333i	$0.222033\pi$
$409$			25.9808i			1.28467i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$			6.92820i			0.341328i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−30.0000			−1.46911
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.50000	+	21.6506i	−0.362950	+	1.04775i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			20.7846i			1.00000i
$433$			22.5167i			1.08208i	0.840996	+	0.541041i	$0.181970\pi$
$433$			22.5167i			1.08208i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$		0			0
$436$	38.0000			1.81987
$437$		0			0
$438$		0			0
$439$		−	8.66025i		−	0.413331i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$		−	8.66025i		−	0.413331i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$	−16.5000	−	12.9904i	−0.785714	−	0.618590i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		−	34.6410i		−	1.64399i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.0000	−	6.92820i	−0.944911	−	0.327327i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			39.8372i			1.87171i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			31.1769i			1.44115i
$469$	−12.5000	−	4.33013i	−0.577196	−	0.199947i
$470$		0			0
$471$	3.00000			0.138233
$472$		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
$481$		−	51.9615i		−	2.36924i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	25.0000			1.13286			0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000			1.13286			0.566429	+	0.824110i	$0.308325\pi$
$488$		0			0
$489$			43.3013i			1.95815i
$490$		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
$496$			34.6410i			1.55543i
$497$		0			0
$498$		0			0
$499$	−11.0000			−0.492428			−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000			−0.492428			−0.246214	+	0.969216i	$0.579187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			24.2487i			1.07692i
$508$	40.0000			1.77471
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	12.0000	−	34.6410i	0.530849	−	1.53243i
$512$		0			0
$513$	45.0000			1.98680
$514$		0			0
$515$		0			0
$516$			17.3205i			0.762493i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			29.4449i			1.28753i	0.765222	+	0.643767i	$0.222629\pi$
$523$			29.4449i			1.28753i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	−15.0000	+	43.3013i	−0.650332	+	1.87735i
$533$		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
$543$	−45.0000			−1.93113
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$			25.9808i			1.10883i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	10.0000	+	3.46410i	0.425243	+	0.147309i
$554$		0			0
$555$		0			0
$556$		−	34.6410i		−	1.46911i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			25.9808i			1.09887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.5000	−	7.79423i	−0.944911	−	0.327327i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		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
$576$	−24.0000			−1.00000
$577$			32.9090i			1.37002i	0.728535	+	0.685009i	$0.240202\pi$
$577$			32.9090i			1.37002i	−0.728535	+	0.685009i	$0.759798\pi$
$578$		0			0
$579$		−	43.3013i		−	1.79954i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	15.0000	−	19.0526i	0.618590	−	0.785714i
$589$	75.0000			3.09032
$590$		0			0
$591$		0			0
$592$	40.0000			1.64399
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	45.0000			1.84173
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			43.3013i			1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$			43.3013i			1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$	−15.0000			−0.610847
$604$	−46.0000			−1.87171
$605$		0			0
$606$		0			0
$607$		−	45.0333i		−	1.82785i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$		−	45.0333i		−	1.82785i	0.405887	−	0.913923i	$-0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			8.66025i			0.348085i	0.984738	+	0.174042i	$0.0556830\pi$
$619$			8.66025i			0.348085i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−36.0000			−1.44115
$625$		0			0
$626$		0			0
$627$		0			0
$628$			3.46410i			0.138233i
$629$		0			0
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$		−	22.5167i		−	0.894957i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.5000	−	28.5788i	0.891482	−	1.13233i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			31.1769i			1.22950i	0.788723	+	0.614749i	$0.210743\pi$
$643$			31.1769i			1.22950i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−37.5000	−	12.9904i	−1.46974	−	0.509133i
$652$	−50.0000			−1.95815
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	41.5692i		−	1.62177i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	34.6410i		−	1.34738i	−0.739014	−	0.673690i	$-0.764708\pi$
$661$		−	34.6410i		−	1.34738i	0.739014	−	0.673690i	$-0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−51.0000			−1.97177
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$		0			0
$676$	−28.0000			−1.07692
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	16.5000	−	47.6314i	0.633212	−	1.82793i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$			51.9615i			1.98680i
$685$		0			0
$686$		0			0
$687$	−15.0000			−0.572286
$688$	−20.0000			−0.762493
$689$		0			0
$690$		0			0
$691$			51.9615i			1.97671i	0.152167	+	0.988355i	$0.451375\pi$
$691$			51.9615i			1.97671i	−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
$703$		−	86.6025i		−	3.26628i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	31.0000			1.16423			0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000			1.16423			0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		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
$721$	3.00000	−	8.66025i	0.111726	−	0.322525i
$722$		0			0
$723$	45.0000			1.67357
$724$		−	51.9615i		−	1.93113i
$725$		0			0
$726$		0			0
$727$		−	53.6936i		−	1.99138i	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$		−	53.6936i		−	1.99138i	0.0927199	−	0.995692i	$-0.470444\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−30.0000			−1.10883
$733$		−	20.7846i		−	0.767697i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$		−	20.7846i		−	0.767697i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$			77.9423i			2.86328i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	52.0000			1.89751			0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000			1.89751			0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	9.00000	−	25.9808i	0.327327	−	0.944911i
$757$	55.0000			1.99901			0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	55.0000			1.99901			0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−47.5000	−	16.4545i	−1.71962	−	0.595692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	27.7128i		−	1.00000i
$769$		−	25.9808i		−	0.936890i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$		−	25.9808i		−	0.936890i	0.883493	−	0.468445i	$-0.155186\pi$
$770$		0			0
$771$		0			0
$772$	50.0000			1.79954
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−15.0000	+	43.3013i	−0.538122	+	1.55342i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$		0			0
$786$		0			0
$787$			50.2295i			1.79049i	0.445577	+	0.895244i	$0.352999\pi$
$787$			50.2295i			1.79049i	−0.445577	+	0.895244i	$0.647001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−45.0000			−1.59800
$794$		0			0
$795$		0			0
$796$			51.9615i			1.84173i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	17.3205i		−	0.610847i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			43.3013i			1.52051i	0.649623	+	0.760257i	$0.274927\pi$
$811$			43.3013i			1.52051i	−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$	30.0000			1.05215
$814$		0			0
$815$		0			0
$816$		0			0
$817$			43.3013i			1.51492i
$818$		0			0
$819$	13.5000	−	38.9711i	0.471728	−	1.36176i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	5.00000			0.174289			0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000			0.174289			0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$		−	8.66025i		−	0.300421i
$832$		−	41.5692i		−	1.44115i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−45.0000			−1.55543
$838$		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
$844$	26.0000			0.894957
$845$		0			0
$846$		0			0
$847$	−27.5000	−	9.52628i	−0.944911	−	0.327327i
$848$		0			0
$849$	−39.0000			−1.33848
$850$		0			0
$851$		0			0
$852$		0			0
$853$			46.7654i			1.60122i	0.599189	+	0.800608i	$0.295490\pi$
$853$			46.7654i			1.60122i	−0.599189	+	0.800608i	$0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			17.3205i			0.590968i	0.955348	+	0.295484i	$0.0954809\pi$
$859$			17.3205i			0.590968i	−0.955348	+	0.295484i	$0.904519\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			29.4449i			1.00000i
$868$	15.0000	−	43.3013i	0.509133	−	1.46974i
$869$		0			0
$870$		0			0
$871$		−	25.9808i		−	0.880325i
$872$		0			0
$873$		−	57.1577i		−	1.93449i
$874$		0			0
$875$		0			0
$876$	48.0000			1.62177
$877$	−25.0000			−0.844190			−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−25.0000			−0.844190			−0.422095	+	0.906552i	$0.638705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	55.0000			1.85090			0.925449	−	0.378873i	$-0.123688\pi$
$883$	55.0000			1.85090			0.925449	+	0.378873i	$0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−50.0000	−	17.3205i	−1.67695	−	0.580911i
$890$		0			0
$891$		0			0
$892$		−	58.8897i		−	1.97177i
$893$		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
$903$	7.50000	−	21.6506i	0.249584	−	0.720488i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	40.0000			1.32818			0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000			1.32818			0.664089	+	0.747653i	$0.268820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−60.0000			−1.98680
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	17.3205i		−	0.572286i
$917$		0			0
$918$		0			0
$919$	1.00000			0.0329870			0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000			0.0329870			0.0164935	+	0.999864i	$0.494750\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	10.3923i		−	0.341328i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	37.5000	−	47.6314i	1.22901	−	1.56106i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	50.2295i		−	1.64093i	−0.571700	−	0.820463i	$-0.693716\pi$
$937$		−	50.2295i		−	1.64093i	0.571700	−	0.820463i	$-0.306284\pi$
$938$		0			0
$939$	9.00000			0.293704
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$			13.8564i			0.450035i
$949$	72.0000			2.33722
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−44.0000			−1.41935
$962$		0			0
$963$		0			0
$964$			51.9615i			1.67357i
$965$		0			0
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	31.1769i		−	1.00000i
$973$	−15.0000	+	43.3013i	−0.480878	+	1.38817i
$974$		0			0
$975$		0			0
$976$		−	34.6410i		−	1.10883i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−57.0000			−1.81987
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−90.0000			−2.86328
$989$		0			0
$990$		0			0
$991$	−17.0000			−0.540023			−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000			−0.540023			−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$			55.4256i			1.75888i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			62.3538i			1.97477i	0.158352	+	0.987383i	$0.449382\pi$
$997$			62.3538i			1.97477i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$			51.9615i			1.64399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.b.c.251.1	✓	2
3.2	odd	2	CM	525.2.b.c.251.1	✓	2
5.2	odd	4		525.2.g.a.524.1		4
5.3	odd	4		525.2.g.a.524.4		4
5.4	even	2		525.2.b.d.251.2	yes	2
7.6	odd	2	inner	525.2.b.c.251.2	yes	2
15.2	even	4		525.2.g.a.524.1		4
15.8	even	4		525.2.g.a.524.4		4
15.14	odd	2		525.2.b.d.251.2	yes	2
21.20	even	2	inner	525.2.b.c.251.2	yes	2
35.13	even	4		525.2.g.a.524.2		4
35.27	even	4		525.2.g.a.524.3		4
35.34	odd	2		525.2.b.d.251.1	yes	2
105.62	odd	4		525.2.g.a.524.3		4
105.83	odd	4		525.2.g.a.524.2		4
105.104	even	2		525.2.b.d.251.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.b.c.251.1	✓	2	1.1	even	1	trivial
525.2.b.c.251.1	✓	2	3.2	odd	2	CM
525.2.b.c.251.2	yes	2	7.6	odd	2	inner
525.2.b.c.251.2	yes	2	21.20	even	2	inner
525.2.b.d.251.1	yes	2	35.34	odd	2
525.2.b.d.251.1	yes	2	105.104	even	2
525.2.b.d.251.2	yes	2	5.4	even	2
525.2.b.d.251.2	yes	2	15.14	odd	2
525.2.g.a.524.1		4	5.2	odd	4
525.2.g.a.524.1		4	15.2	even	4
525.2.g.a.524.2		4	35.13	even	4
525.2.g.a.524.2		4	105.83	odd	4
525.2.g.a.524.3		4	35.27	even	4
525.2.g.a.524.3		4	105.62	odd	4
525.2.g.a.524.4		4	5.3	odd	4
525.2.g.a.524.4		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +2.00000 q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} -3.46410i q^{12} -5.19615i q^{13} +4.00000 q^{16} -8.66025i q^{19} +(-1.50000 + 4.33013i) q^{21} +5.19615i q^{27} +(-5.00000 - 1.73205i) q^{28} +8.66025i q^{31} -6.00000 q^{36} +10.0000 q^{37} -9.00000 q^{39} -5.00000 q^{43} -6.92820i q^{48} +(5.50000 + 4.33013i) q^{49} -10.3923i q^{52} -15.0000 q^{57} -8.66025i q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +5.00000 q^{67} +13.8564i q^{73} -17.3205i q^{76} -4.00000 q^{79} +9.00000 q^{81} +(-3.00000 + 8.66025i) q^{84} +(-4.50000 + 12.9904i) q^{91} +15.0000 q^{93} +19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 5 q^{7} - 6 q^{9} + 8 q^{16} - 3 q^{21} - 10 q^{28} - 12 q^{36} + 20 q^{37} - 18 q^{39} - 10 q^{43} + 11 q^{49} - 30 q^{57} + 15 q^{63} + 16 q^{64} + 10 q^{67} - 8 q^{79} + 18 q^{81} - 6 q^{84}+ \cdots + 30 q^{93}+O(q^{100}) \) 2 * q + 4 * q^4 - 5 * q^7 - 6 * q^9 + 8 * q^16 - 3 * q^21 - 10 * q^28 - 12 * q^36 + 20 * q^37 - 18 * q^39 - 10 * q^43 + 11 * q^49 - 30 * q^57 + 15 * q^63 + 16 * q^64 + 10 * q^67 - 8 * q^79 + 18 * q^81 - 6 * q^84 - 9 * q^91 + 30 * q^93

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