LMFDB - Embedded newform 525.2.g.a.524.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,2,Mod(524,525)] 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("525.524"); 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, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			524.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	525.524
Dual form			525.2.g.a.524.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -2.00000 q^{4} +(-0.866025 - 2.50000i) q^{7} +3.00000 q^{9} -3.46410 q^{12} +5.19615 q^{13} +4.00000 q^{16} -8.66025i q^{19} +(-1.50000 - 4.33013i) q^{21} +5.19615 q^{27} +(1.73205 + 5.00000i) q^{28} -8.66025i q^{31} -6.00000 q^{36} +10.0000i q^{37} +9.00000 q^{39} +5.00000i q^{43} +6.92820 q^{48} +(-5.50000 + 4.33013i) q^{49} -10.3923 q^{52} -15.0000i q^{57} +8.66025i q^{61} +(-2.59808 - 7.50000i) q^{63} -8.00000 q^{64} +5.00000i q^{67} -13.8564 q^{73} +17.3205i q^{76} +4.00000 q^{79} +9.00000 q^{81} +(3.00000 + 8.66025i) q^{84} +(-4.50000 - 12.9904i) q^{91} -15.0000i q^{93} +19.0526 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 12 q^{9} + 16 q^{16} - 6 q^{21} - 24 q^{36} + 36 q^{39} - 22 q^{49} - 32 q^{64} + 16 q^{79} + 36 q^{81} + 12 q^{84} - 18 q^{91}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^9 + 16 * q^16 - 6 * q^21 - 24 * q^36 + 36 * q^39 - 22 * q^49 - 32 * q^64 + 16 * q^79 + 36 * q^81 + 12 * q^84 - 18 * q^91

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.866025	−	2.50000i	−0.327327	−	0.944911i
$7$	−0.866025	−	2.50000i	−0.327327	−	0.944911i
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−3.46410			−1.00000
$13$	5.19615			1.44115			0.720577	−	0.693375i	$-0.243877\pi$
$13$	5.19615			1.44115			0.720577	+	0.693375i	$0.243877\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		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			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.19615			1.00000
$28$	1.73205	+	5.00000i	0.327327	+	0.944911i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	8.66025i		−	1.55543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$		−	8.66025i		−	1.55543i	0.628619	−	0.777714i	$-0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$		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
$43$			5.00000i			0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$			5.00000i			0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	6.92820			1.00000
$49$	−5.50000	+	4.33013i	−0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$	−10.3923			−1.44115
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	15.0000i		−	1.98680i
$58$		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	$0.187058\pi$
$61$			8.66025i			1.10883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	−2.59808	−	7.50000i	−0.327327	−	0.944911i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$			5.00000i			0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$			5.00000i			0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−13.8564			−1.62177			−0.810885	−	0.585206i	$-0.801014\pi$
$73$	−13.8564			−1.62177			−0.810885	+	0.585206i	$0.801014\pi$
$74$		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	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	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
$93$		−	15.0000i		−	1.55543i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	19.0526			1.93449			0.967247	−	0.253837i	$-0.0816925\pi$
$97$	19.0526			1.93449			0.967247	+	0.253837i	$0.0816925\pi$
$98$		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
$103$	−3.46410			−0.341328			−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410			−0.341328			−0.170664	+	0.985329i	$0.554591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−10.3923			−1.00000
$109$	−19.0000			−1.81987			−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000			−1.81987			−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$			17.3205i			1.64399i
$112$	−3.46410	−	10.0000i	−0.327327	−	0.944911i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	15.5885			1.44115
$118$		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
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		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
$133$	−21.6506	+	7.50000i	−1.87735	+	0.650332i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		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
$147$	−9.52628	+	7.50000i	−0.785714	+	0.618590i
$148$		−	20.0000i		−	1.64399i
$149$		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
$157$	1.73205			0.138233			0.0691164	−	0.997609i	$-0.477982\pi$
$157$	1.73205			0.138233			0.0691164	+	0.997609i	$0.477982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			25.0000i			1.95815i	0.203497	+	0.979076i	$0.434769\pi$
$163$			25.0000i			1.95815i	−0.203497	+	0.979076i	$0.565231\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	14.0000			1.07692
$170$		0			0
$171$		−	25.9808i		−	1.98680i
$172$		−	10.0000i		−	0.762493i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		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	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$			15.0000i			1.10883i
$184$		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			$\pi$
$192$	−13.8564			−1.00000
$193$		−	25.0000i		−	1.79954i	−0.436365	−	0.899770i	$-0.643734\pi$
$193$		−	25.0000i		−	1.79954i	0.436365	−	0.899770i	$-0.356266\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		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
$208$	20.7846			1.44115
$209$		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
$217$	−21.6506	+	7.50000i	−1.46974	+	0.509133i
$218$		0			0
$219$	−24.0000			−1.62177
$220$		0			0
$221$		0			0
$222$		0			0
$223$	29.4449			1.97177			0.985887	−	0.167412i	$-0.0535411\pi$
$223$	29.4449			1.97177			0.985887	+	0.167412i	$0.0535411\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$			30.0000i			1.98680i
$229$		−	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			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.92820			0.450035
$238$		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	$-0.684433\pi$
$241$		−	25.9808i		−	1.67357i	0.547533	−	0.836784i	$-0.315567\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$		−	17.3205i		−	1.10883i
$245$		0			0
$246$		0			0
$247$		−	45.0000i		−	2.86328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	5.19615	+	15.0000i	0.327327	+	0.944911i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	25.0000	−	8.66025i	1.55342	−	0.538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		−	10.0000i		−	0.610847i
$269$		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	$-0.823664\pi$
$271$		−	17.3205i		−	1.05215i	0.850439	−	0.526073i	$-0.176336\pi$
$272$		0			0
$273$	−7.79423	−	22.5000i	−0.471728	−	1.36176i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			5.00000i			0.300421i	0.988654	+	0.150210i	$0.0479951\pi$
$277$			5.00000i			0.300421i	−0.988654	+	0.150210i	$0.952005\pi$
$278$		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
$283$	22.5167			1.33848			0.669238	−	0.743048i	$-0.266621\pi$
$283$	22.5167			1.33848			0.669238	+	0.743048i	$0.266621\pi$
$284$		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
$292$	27.7128			1.62177
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		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
$307$	−1.73205			−0.0988534			−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205			−0.0988534			−0.0494267	+	0.998778i	$0.515739\pi$
$308$		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
$313$	−5.19615			−0.293704			−0.146852	−	0.989158i	$-0.546914\pi$
$313$	−5.19615			−0.293704			−0.146852	+	0.989158i	$0.546914\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		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
$327$	−32.9090			−1.81987
$328$		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
$333$			30.0000i			1.64399i
$334$		0			0
$335$		0			0
$336$	−6.00000	−	17.3205i	−0.327327	−	0.944911i
$337$		−	5.00000i		−	0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$		−	5.00000i		−	0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		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		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		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
$363$	19.0526			1.00000
$364$	9.00000	+	25.9808i	0.471728	+	1.36176i
$365$		0			0
$366$		0			0
$367$	−15.5885			−0.813711			−0.406855	−	0.913493i	$-0.633375\pi$
$367$	−15.5885			−0.813711			−0.406855	+	0.913493i	$0.633375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$			30.0000i			1.55543i
$373$			25.0000i			1.29445i	0.762299	+	0.647225i	$0.224071\pi$
$373$			25.0000i			1.29445i	−0.762299	+	0.647225i	$0.775929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$			34.6410i			1.77471i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			15.0000i			0.762493i
$388$	−38.1051			−1.93449
$389$		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
$397$	−19.0526			−0.956221			−0.478110	−	0.878300i	$-0.658678\pi$
$397$	−19.0526			−0.956221			−0.478110	+	0.878300i	$0.658678\pi$
$398$		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
$403$		−	45.0000i		−	2.24161i
$404$		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
$412$	6.92820			0.341328
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	30.0000i		−	1.46911i
$418$		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
$427$	21.6506	−	7.50000i	1.04775	−	0.362950i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	20.7846			1.00000
$433$	−22.5167			−1.08208			−0.541041	−	0.840996i	$-0.681970\pi$
$433$	−22.5167			−1.08208			−0.541041	+	0.840996i	$0.681970\pi$
$434$		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			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		−	34.6410i		−	1.64399i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	6.92820	+	20.0000i	0.327327	+	0.944911i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−39.8372			−1.87171
$454$		0			0
$455$		0			0
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		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
$463$		−	20.0000i		−	0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$		−	20.0000i		−	0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−31.1769			−1.44115
$469$	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
$487$			25.0000i			1.13286i	0.824110	+	0.566429i	$0.191675\pi$
$487$			25.0000i			1.13286i	−0.824110	+	0.566429i	$0.808325\pi$
$488$		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	$-0.420813\pi$
$499$	11.0000			0.492428			0.246214	+	0.969216i	$0.420813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	24.2487			1.07692
$508$		−	40.0000i		−	1.77471i
$509$		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
$513$		−	45.0000i		−	1.98680i
$514$		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
$523$	−29.4449			−1.28753			−0.643767	−	0.765222i	$-0.722629\pi$
$523$	−29.4449			−1.28753			−0.643767	+	0.765222i	$0.722629\pi$
$524$		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
$532$	43.3013	−	15.0000i	1.87735	−	0.650332i
$533$		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
$543$			45.0000i			1.93113i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$			25.9808i			1.10883i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.46410	−	10.0000i	−0.147309	−	0.425243i
$554$		0			0
$555$		0			0
$556$			34.6410i			1.46911i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$			25.9808i			1.09887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−7.79423	−	22.5000i	−0.327327	−	0.944911i
$568$		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
$577$	32.9090			1.37002			0.685009	−	0.728535i	$-0.259798\pi$
$577$	32.9090			1.37002			0.685009	+	0.728535i	$0.259798\pi$
$578$		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		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	19.0526	−	15.0000i	0.785714	−	0.618590i
$589$	−75.0000			−3.09032
$590$		0			0
$591$		0			0
$592$			40.0000i			1.64399i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			45.0000i			1.84173i
$598$		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	$-0.655420\pi$
$601$		−	43.3013i		−	1.76630i	0.469095	−	0.883148i	$-0.344580\pi$
$602$		0			0
$603$			15.0000i			0.610847i
$604$	46.0000			1.87171
$605$		0			0
$606$		0			0
$607$	−45.0333			−1.82785			−0.913923	−	0.405887i	$-0.866962\pi$
$607$	−45.0333			−1.82785			−0.913923	+	0.405887i	$0.866962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			10.0000i			0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$			10.0000i			0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		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
$628$	−3.46410			−0.138233
$629$		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
$633$	22.5167			0.894957
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−28.5788	+	22.5000i	−1.13233	+	0.891482i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	−31.1769			−1.22950			−0.614749	−	0.788723i	$-0.710743\pi$
$643$	−31.1769			−1.22950			−0.614749	+	0.788723i	$0.710743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−37.5000	+	12.9904i	−1.46974	+	0.509133i
$652$		−	50.0000i		−	1.95815i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−41.5692			−1.62177
$658$		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	$0.235292\pi$
$661$			34.6410i			1.34738i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		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
$673$			50.0000i			1.92736i	0.267063	+	0.963679i	$0.413947\pi$
$673$			50.0000i			1.92736i	−0.267063	+	0.963679i	$0.586053\pi$
$674$		0			0
$675$		0			0
$676$	−28.0000			−1.07692
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−16.5000	−	47.6314i	−0.633212	−	1.82793i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$			51.9615i			1.98680i
$685$		0			0
$686$		0			0
$687$		−	15.0000i		−	0.572286i
$688$			20.0000i			0.762493i
$689$		0			0
$690$		0			0
$691$		−	51.9615i		−	1.97671i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$		−	51.9615i		−	1.97671i	0.152167	−	0.988355i	$-0.451375\pi$
$692$		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
$703$	86.6025			3.26628
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−31.0000			−1.16423			−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000			−1.16423			−0.582115	+	0.813107i	$0.697775\pi$
$710$		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
$723$		−	45.0000i		−	1.67357i
$724$		−	51.9615i		−	1.93113i
$725$		0			0
$726$		0			0
$727$	−53.6936			−1.99138			−0.995692	−	0.0927199i	$-0.970444\pi$
$727$	−53.6936			−1.99138			−0.995692	+	0.0927199i	$0.970444\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		−	30.0000i		−	1.10883i
$733$	20.7846			0.767697			0.383849	−	0.923396i	$-0.374598\pi$
$733$	20.7846			0.767697			0.383849	+	0.923396i	$0.374598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		−	77.9423i		−	2.86328i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		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
$757$			55.0000i			1.99901i	0.0314762	+	0.999505i	$0.489979\pi$
$757$			55.0000i			1.99901i	−0.0314762	+	0.999505i	$0.510021\pi$
$758$		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
$763$	16.4545	+	47.5000i	0.595692	+	1.71962i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.7128			1.00000
$769$		−	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
$772$			50.0000i			1.79954i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	43.3013	−	15.0000i	1.55342	−	0.538122i
$778$		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
$787$	50.2295			1.79049			0.895244	−	0.445577i	$-0.147001\pi$
$787$	50.2295			1.79049			0.895244	+	0.445577i	$0.147001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			45.0000i			1.59800i
$794$		0			0
$795$		0			0
$796$		−	51.9615i		−	1.84173i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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	$-0.725073\pi$
$811$		−	43.3013i		−	1.52051i	0.649623	−	0.760257i	$-0.274927\pi$
$812$		0			0
$813$		−	30.0000i		−	1.05215i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	43.3013			1.51492
$818$		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
$823$		−	5.00000i		−	0.174289i	−0.996196	−	0.0871445i	$-0.972226\pi$
$823$		−	5.00000i		−	0.174289i	0.996196	−	0.0871445i	$-0.0277742\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		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
$832$	−41.5692			−1.44115
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	45.0000i		−	1.55543i
$838$		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
$847$	−9.52628	−	27.5000i	−0.327327	−	0.944911i
$848$		0			0
$849$	39.0000			1.33848
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−46.7654			−1.60122			−0.800608	−	0.599189i	$-0.795490\pi$
$853$	−46.7654			−1.60122			−0.800608	+	0.599189i	$0.795490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		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			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.4449			1.00000
$868$	43.3013	−	15.0000i	1.46974	−	0.509133i
$869$		0			0
$870$		0			0
$871$			25.9808i			0.880325i
$872$		0			0
$873$	57.1577			1.93449
$874$		0			0
$875$		0			0
$876$	48.0000			1.62177
$877$		−	25.0000i		−	0.844190i	−0.906552	−	0.422095i	$-0.861295\pi$
$877$		−	25.0000i		−	0.844190i	0.906552	−	0.422095i	$-0.138705\pi$
$878$		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
$883$		−	55.0000i		−	1.85090i	−0.378873	−	0.925449i	$-0.623688\pi$
$883$		−	55.0000i		−	1.85090i	0.378873	−	0.925449i	$-0.376312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	50.0000	−	17.3205i	1.67695	−	0.580911i
$890$		0			0
$891$		0			0
$892$	−58.8897			−1.97177
$893$		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
$903$	21.6506	−	7.50000i	0.720488	−	0.249584i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			40.0000i			1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$			40.0000i			1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	60.0000i		−	1.98680i
$913$		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	$-0.505250\pi$
$919$	−1.00000			−0.0329870			−0.0164935	+	0.999864i	$0.505250\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.3923			−0.341328
$928$		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
$937$	−50.2295			−1.64093			−0.820463	−	0.571700i	$-0.806284\pi$
$937$	−50.2295			−1.64093			−0.820463	+	0.571700i	$0.806284\pi$
$938$		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			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$	−13.8564			−0.450035
$949$	−72.0000			−2.33722
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		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
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−31.1769			−1.00000
$973$	−43.3013	+	15.0000i	−1.38817	+	0.480878i
$974$		0			0
$975$		0			0
$976$			34.6410i			1.10883i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−57.0000			−1.81987
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$			90.0000i			2.86328i
$989$		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
$993$	−55.4256			−1.75888
$994$		0			0
$995$		0			0
$996$		0			0
$997$	62.3538			1.97477			0.987383	−	0.158352i	$-0.0506179\pi$
$997$	62.3538			1.97477			0.987383	+	0.158352i	$0.0506179\pi$
$998$		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.g.a.524.3		4
3.2	odd	2	CM	525.2.g.a.524.3		4
5.2	odd	4		525.2.b.d.251.1	yes	2
5.3	odd	4		525.2.b.c.251.2	yes	2
5.4	even	2	inner	525.2.g.a.524.2		4
7.6	odd	2	inner	525.2.g.a.524.1		4
15.2	even	4		525.2.b.d.251.1	yes	2
15.8	even	4		525.2.b.c.251.2	yes	2
15.14	odd	2	inner	525.2.g.a.524.2		4
21.20	even	2	inner	525.2.g.a.524.1		4
35.13	even	4		525.2.b.c.251.1	✓	2
35.27	even	4		525.2.b.d.251.2	yes	2
35.34	odd	2	inner	525.2.g.a.524.4		4
105.62	odd	4		525.2.b.d.251.2	yes	2
105.83	odd	4		525.2.b.c.251.1	✓	2
105.104	even	2	inner	525.2.g.a.524.4		4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -2.00000 q^{4} +(-0.866025 - 2.50000i) q^{7} +3.00000 q^{9} -3.46410 q^{12} +5.19615 q^{13} +4.00000 q^{16} -8.66025i q^{19} +(-1.50000 - 4.33013i) q^{21} +5.19615 q^{27} +(1.73205 + 5.00000i) q^{28} -8.66025i q^{31} -6.00000 q^{36} +10.0000i q^{37} +9.00000 q^{39} +5.00000i q^{43} +6.92820 q^{48} +(-5.50000 + 4.33013i) q^{49} -10.3923 q^{52} -15.0000i q^{57} +8.66025i q^{61} +(-2.59808 - 7.50000i) q^{63} -8.00000 q^{64} +5.00000i q^{67} -13.8564 q^{73} +17.3205i q^{76} +4.00000 q^{79} +9.00000 q^{81} +(3.00000 + 8.66025i) q^{84} +(-4.50000 - 12.9904i) q^{91} -15.0000i q^{93} +19.0526 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 12 q^{9} + 16 q^{16} - 6 q^{21} - 24 q^{36} + 36 q^{39} - 22 q^{49} - 32 q^{64} + 16 q^{79} + 36 q^{81} + 12 q^{84} - 18 q^{91}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^9 + 16 * q^16 - 6 * q^21 - 24 * q^36 + 36 * q^39 - 22 * q^49 - 32 * q^64 + 16 * q^79 + 36 * q^81 + 12 * q^84 - 18 * q^91

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