LMFDB - Embedded newform 900.2.k.d.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,2,Mod(307,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("900.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.k (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:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			307.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	900.307
Dual form			900.2.k.d.343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-2.00000 + 2.00000i) q^{8} +(-5.00000 + 5.00000i) q^{13} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} -10.0000 q^{26} +10.0000i q^{29} +(-4.00000 - 4.00000i) q^{32} +6.00000i q^{34} +(-5.00000 - 5.00000i) q^{37} +10.0000 q^{41} +7.00000i q^{49} +(-10.0000 - 10.0000i) q^{52} +(9.00000 - 9.00000i) q^{53} +(-10.0000 + 10.0000i) q^{58} -12.0000 q^{61} -8.00000i q^{64} +(-6.00000 + 6.00000i) q^{68} +(5.00000 - 5.00000i) q^{73} -10.0000i q^{74} +(10.0000 + 10.0000i) q^{82} -10.0000i q^{89} +(5.00000 + 5.00000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{8} - 10 q^{13} - 8 q^{16} + 6 q^{17} - 20 q^{26} - 8 q^{32} - 10 q^{37} + 20 q^{41} - 20 q^{52} + 18 q^{53} - 20 q^{58} - 24 q^{61} - 12 q^{68} + 10 q^{73} + 20 q^{82} + 10 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^8 - 10 * q^13 - 8 * q^16 + 6 * q^17 - 20 * q^26 - 8 * q^32 - 10 * q^37 + 20 * q^41 - 20 * q^52 + 18 * q^53 - 20 * q^58 - 24 * q^61 - 12 * q^68 + 10 * q^73 + 20 * q^82 + 10 * q^97 - 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0
$3$		0			0
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−5.00000	+	5.00000i	−1.38675	+	1.38675i	−0.554700	+	0.832050i	$0.687167\pi$
$13$	−5.00000	+	5.00000i	−1.38675	+	1.38675i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		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
$26$	−10.0000			−1.96116
$27$		0			0
$28$		0			0
$29$			10.0000i			1.85695i	0.371391	+	0.928477i	$0.378881\pi$
$29$			10.0000i			1.85695i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$			6.00000i			1.02899i
$35$		0			0
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		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
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$	−10.0000	−	10.0000i	−1.38675	−	1.38675i
$53$	9.00000	−	9.00000i	1.23625	−	1.23625i	0.274721	−	0.961524i	$-0.411414\pi$
$53$	9.00000	−	9.00000i	1.23625	−	1.23625i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−10.0000	+	10.0000i	−1.31306	+	1.31306i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−6.00000	+	6.00000i	−0.727607	+	0.727607i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	0.936329	+	0.351123i	$0.114200\pi$
$74$		−	10.0000i		−	1.16248i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$	10.0000	+	10.0000i	1.10432	+	1.10432i
$83$		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
$89$		−	10.0000i		−	1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$		−	10.0000i		−	1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	0.913812	−	0.406138i	$-0.133125\pi$
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−7.00000	+	7.00000i	−0.707107	+	0.707107i
$99$		0			0
$100$		0			0
$101$	20.0000			1.99007			0.995037	−	0.0995037i	$-0.0317255\pi$
$101$	20.0000			1.99007			0.995037	+	0.0995037i	$0.0317255\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		−	20.0000i		−	1.96116i
$105$		0			0
$106$	18.0000			1.74831
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	−	1.00000i	0.0940721	−	0.0940721i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	1.00000	−	1.00000i	0.0940721	−	0.0940721i	0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$		0			0
$116$	−20.0000			−1.85695
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−12.0000	−	12.0000i	−1.08643	−	1.08643i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−12.0000			−1.02899
$137$	7.00000	+	7.00000i	0.598050	+	0.598050i	0.939793	−	0.341743i	$-0.111017\pi$
$137$	7.00000	+	7.00000i	0.598050	+	0.598050i	−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	10.0000			0.827606
$147$		0			0
$148$	10.0000	−	10.0000i	0.821995	−	0.821995i
$149$			20.0000i			1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$149$			20.0000i			1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	0.877896	−	0.478852i	$-0.158947\pi$
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$			20.0000i			1.56174i
$165$		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
$169$		−	37.0000i		−	2.84615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.0000	−	11.0000i	0.836315	−	0.836315i	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	11.0000	−	11.0000i	0.836315	−	0.836315i	0.988372	+	0.152057i	$0.0485898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	10.0000	−	10.0000i	0.749532	−	0.749532i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		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
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	0.863779	+	0.503871i	$0.168091\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	−14.0000			−1.00000
$197$	−13.0000	−	13.0000i	−0.926212	−	0.926212i	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	−13.0000	−	13.0000i	−0.926212	−	0.926212i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$	20.0000	+	20.0000i	1.40720	+	1.40720i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	20.0000	−	20.0000i	1.38675	−	1.38675i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	18.0000	+	18.0000i	1.23625	+	1.23625i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−6.00000	+	6.00000i	−0.406371	+	0.406371i
$219$		0			0
$220$		0			0
$221$	−30.0000			−2.01802
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$	2.00000			0.133038
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$	−20.0000	−	20.0000i	−1.31306	−	1.31306i
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.524097	+	0.851658i	$0.675597\pi$
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		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
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$		−	24.0000i		−	1.53644i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		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
$256$	16.0000			1.00000
$257$	17.0000	+	17.0000i	1.06043	+	1.06043i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	17.0000	+	17.0000i	1.06043	+	1.06043i	0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$		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
$269$		−	20.0000i		−	1.21942i	−0.792624	−	0.609711i	$-0.791286\pi$
$269$		−	20.0000i		−	1.21942i	0.792624	−	0.609711i	$-0.208714\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−12.0000	−	12.0000i	−0.727607	−	0.727607i
$273$		0			0
$274$			14.0000i			0.845771i
$275$		0			0
$276$		0			0
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$		0			0
$292$	10.0000	+	10.0000i	0.585206	+	0.585206i
$293$	−19.0000	+	19.0000i	−1.10999	+	1.10999i	−0.116841	+	0.993151i	$0.537277\pi$
$293$	−19.0000	+	19.0000i	−1.10999	+	1.10999i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$	20.0000			1.16248
$297$		0			0
$298$	−20.0000	+	20.0000i	−1.15857	+	1.15857i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	25.0000	−	25.0000i	1.41308	−	1.41308i	0.678280	−	0.734803i	$-0.262726\pi$
$313$	25.0000	−	25.0000i	1.41308	−	1.41308i	0.734803	−	0.678280i	$-0.237274\pi$
$314$			10.0000i			0.564333i
$315$		0			0
$316$		0			0
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	0.786318	−	0.617822i	$-0.211985\pi$
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		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
$328$	−20.0000	+	20.0000i	−1.10432	+	1.10432i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	25.0000	+	25.0000i	1.36184	+	1.36184i	0.871576	+	0.490261i	$0.163099\pi$
$337$	25.0000	+	25.0000i	1.36184	+	1.36184i	0.490261	+	0.871576i	$0.336901\pi$
$338$	37.0000	−	37.0000i	2.01253	−	2.01253i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	22.0000			1.18273
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			36.0000i			1.92704i	0.267644	+	0.963518i	$0.413755\pi$
$349$			36.0000i			1.92704i	−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	0.904819	+	0.425797i	$0.140006\pi$
$354$		0			0
$355$		0			0
$356$	20.0000			1.06000
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	18.0000	+	18.0000i	0.946059	+	0.946059i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.362446	+	0.932005i	$0.618058\pi$
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−50.0000	−	50.0000i	−2.57513	−	2.57513i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		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
$386$	10.0000			0.508987
$387$		0			0
$388$	−10.0000	+	10.0000i	−0.507673	+	0.507673i
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$		0			0
$392$	−14.0000	−	14.0000i	−0.707107	−	0.707107i
$393$		0			0
$394$		−	26.0000i		−	1.30986i
$395$		0			0
$396$		0			0
$397$	−25.0000	−	25.0000i	−1.25471	−	1.25471i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−25.0000	−	25.0000i	−1.25471	−	1.25471i	−0.301131	−	0.953583i	$-0.597364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−40.0000			−1.99750			−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−40.0000			−1.99750			−0.998752	+	0.0499376i	$0.984098\pi$
$402$		0			0
$403$		0			0
$404$			40.0000i			1.99007i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	40.0000			1.96116
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$			36.0000i			1.74831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−5.00000	+	5.00000i	−0.240285	+	0.240285i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−5.00000	+	5.00000i	−0.240285	+	0.240285i	0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$	−12.0000			−0.574696
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$	−30.0000	−	30.0000i	−1.42695	−	1.42695i
$443$		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
$449$		−	40.0000i		−	1.88772i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$		−	40.0000i		−	1.88772i	0.330350	−	0.943858i	$-0.392833\pi$
$450$		0			0
$451$		0			0
$452$	2.00000	+	2.00000i	0.0940721	+	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.187112	−	0.982339i	$-0.559913\pi$
$458$	−4.00000	+	4.00000i	−0.186908	+	0.186908i
$459$		0			0
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		−	40.0000i		−	1.85695i
$465$		0			0
$466$	−42.0000			−1.94561
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		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
$481$	50.0000			2.27980
$482$	−8.00000	−	8.00000i	−0.364390	−	0.364390i
$483$		0			0
$484$			22.0000i			1.00000i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	24.0000	−	24.0000i	1.08643	−	1.08643i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−30.0000	+	30.0000i	−1.35113	+	1.35113i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		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
$509$		−	10.0000i		−	0.443242i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$		−	10.0000i		−	0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$			34.0000i			1.49968i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	40.0000			1.75243			0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000			1.75243			0.876216	+	0.481919i	$0.160060\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		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
$533$	−50.0000	+	50.0000i	−2.16574	+	2.16574i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	20.0000	−	20.0000i	0.862261	−	0.862261i
$539$		0			0
$540$		0			0
$541$	−42.0000			−1.80572			−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000			−1.80572			−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$		−	24.0000i		−	1.02899i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−14.0000	+	14.0000i	−0.598050	+	0.598050i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	10.0000i		−	0.424859i
$555$		0			0
$556$		0			0
$557$	33.0000	+	33.0000i	1.39825	+	1.39825i	0.805056	+	0.593199i	$0.202135\pi$
$557$	33.0000	+	33.0000i	1.39825	+	1.39825i	0.593199	+	0.805056i	$0.297865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−10.0000	−	10.0000i	−0.421825	−	0.421825i
$563$		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
$569$			40.0000i			1.67689i	0.544988	+	0.838444i	$0.316534\pi$
$569$			40.0000i			1.67689i	−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	25.0000	+	25.0000i	1.04076	+	1.04076i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	25.0000	+	25.0000i	1.04076	+	1.04076i	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−1.00000	+	1.00000i	−0.0415945	+	0.0415945i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			20.0000i			0.827606i
$585$		0			0
$586$	−38.0000			−1.56977
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	20.0000	+	20.0000i	0.821995	+	0.821995i
$593$	31.0000	−	31.0000i	1.27302	−	1.27302i	0.328521	−	0.944497i	$-0.393450\pi$
$593$	31.0000	−	31.0000i	1.27302	−	1.27302i	0.944497	−	0.328521i	$-0.106550\pi$
$594$		0			0
$595$		0			0
$596$	−40.0000			−1.63846
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	48.0000			1.95796			0.978980	−	0.203954i	$-0.0653794\pi$
$601$	48.0000			1.95796			0.978980	+	0.203954i	$0.0653794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−35.0000	+	35.0000i	−1.41364	+	1.41364i	−0.686624	+	0.727013i	$0.740908\pi$
$613$	−35.0000	+	35.0000i	−1.41364	+	1.41364i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	0.764911	−	0.644136i	$-0.222783\pi$
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	50.0000			1.99840
$627$		0			0
$628$	−10.0000	+	10.0000i	−0.399043	+	0.399043i
$629$		−	30.0000i		−	1.19618i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			6.00000i			0.238290i
$635$		0			0
$636$		0			0
$637$	−35.0000	−	35.0000i	−1.38675	−	1.38675i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−50.0000			−1.97488			−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000			−1.97488			−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		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
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	0.860927	+	0.508729i	$0.169885\pi$
$654$		0			0
$655$		0			0
$656$	−40.0000			−1.56174
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−12.0000			−0.466746			−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000			−0.466746			−0.233373	+	0.972387i	$0.574976\pi$
$662$		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
$673$	35.0000	−	35.0000i	1.34915	−	1.34915i	0.462566	−	0.886585i	$-0.346929\pi$
$673$	35.0000	−	35.0000i	1.34915	−	1.34915i	0.886585	−	0.462566i	$-0.153071\pi$
$674$			50.0000i			1.92593i
$675$		0			0
$676$	74.0000			2.84615
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.0384331	−	0.999261i	$-0.512237\pi$
$678$		0			0
$679$		0			0
$680$		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
$689$			90.0000i			3.42873i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	22.0000	+	22.0000i	0.836315	+	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	30.0000	+	30.0000i	1.13633	+	1.13633i
$698$	−36.0000	+	36.0000i	−1.36262	+	1.36262i
$699$		0			0
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	18.0000			0.677439
$707$		0			0
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	20.0000	+	20.0000i	0.749532	+	0.749532i
$713$		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
$721$		0			0
$722$	−19.0000	−	19.0000i	−0.707107	−	0.707107i
$723$		0			0
$724$			36.0000i			1.33793i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	0.997268	+	0.0738717i	$0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		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
$746$	−50.0000			−1.83063
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		−	100.000i		−	3.64179i
$755$		0			0
$756$		0			0
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.944986	+	0.327111i	$0.106075\pi$
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.0000			1.45000			0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000			1.45000			0.724999	+	0.688749i	$0.241840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$		0			0
$772$	10.0000	+	10.0000i	0.359908	+	0.359908i
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$		0			0
$776$	−20.0000			−0.717958
$777$		0			0
$778$	20.0000	−	20.0000i	0.717035	−	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	26.0000	−	26.0000i	0.926212	−	0.926212i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	60.0000	−	60.0000i	2.13066	−	2.13066i
$794$		−	50.0000i		−	1.77443i
$795$		0			0
$796$		0			0
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.920967	+	0.389640i	$0.127401\pi$
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−40.0000	−	40.0000i	−1.41245	−	1.41245i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−40.0000	+	40.0000i	−1.40720	+	1.40720i
$809$			10.0000i			0.351581i	0.984428	+	0.175791i	$0.0562482\pi$
$809$			10.0000i			0.351581i	−0.984428	+	0.175791i	$0.943752\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−6.00000	+	6.00000i	−0.209785	+	0.209785i
$819$		0			0
$820$		0			0
$821$	50.0000			1.74501			0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000			1.74501			0.872506	+	0.488603i	$0.162493\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		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
$829$		−	54.0000i		−	1.87550i	−0.347314	−	0.937749i	$-0.612906\pi$
$829$		−	54.0000i		−	1.87550i	0.347314	−	0.937749i	$-0.387094\pi$
$830$		0			0
$831$		0			0
$832$	40.0000	+	40.0000i	1.38675	+	1.38675i
$833$	−21.0000	+	21.0000i	−0.727607	+	0.727607i
$834$		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
$841$	−71.0000			−2.44828
$842$	−28.0000	−	28.0000i	−0.964944	−	0.964944i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−36.0000	+	36.0000i	−1.23625	+	1.23625i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.00000	−	5.00000i	0.171197	−	0.171197i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	5.00000	−	5.00000i	0.171197	−	0.171197i	0.787505	+	0.616308i	$0.211372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.0000	+	33.0000i	1.12726	+	1.12726i	0.990621	+	0.136637i	$0.0436295\pi$
$857$	33.0000	+	33.0000i	1.12726	+	1.12726i	0.136637	+	0.990621i	$0.456370\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		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
$866$	−10.0000			−0.339814
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−12.0000	−	12.0000i	−0.406371	−	0.406371i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−35.0000	−	35.0000i	−1.18187	−	1.18187i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−35.0000	−	35.0000i	−1.18187	−	1.18187i	−0.202606	−	0.979260i	$-0.564941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	50.0000			1.68454			0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000			1.68454			0.842271	+	0.539054i	$0.181218\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		−	60.0000i		−	2.01802i
$885$		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
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	40.0000	−	40.0000i	1.33482	−	1.33482i
$899$		0			0
$900$		0			0
$901$	54.0000			1.79900
$902$		0			0
$903$		0			0
$904$			4.00000i			0.133038i
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		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
$914$		−	50.0000i		−	1.65385i
$915$		0			0
$916$	−8.00000			−0.264327
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	−20.0000	−	20.0000i	−0.658665	−	0.658665i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	40.0000	−	40.0000i	1.31306	−	1.31306i
$929$		−	40.0000i		−	1.31236i	−0.754606	−	0.656179i	$-0.772172\pi$
$929$		−	40.0000i		−	1.31236i	0.754606	−	0.656179i	$-0.227828\pi$
$930$		0			0
$931$		0			0
$932$	−42.0000	−	42.0000i	−1.37576	−	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	0.784046	−	0.620703i	$-0.213153\pi$
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.0000			0.651981			0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000			0.651981			0.325991	+	0.945373i	$0.394302\pi$
$942$		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
$949$			50.0000i			1.62307i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	41.0000	−	41.0000i	1.32812	−	1.32812i	0.421111	−	0.907009i	$-0.361640\pi$
$953$	41.0000	−	41.0000i	1.32812	−	1.32812i	0.907009	−	0.421111i	$-0.138360\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	50.0000	+	50.0000i	1.61206	+	1.61206i
$963$		0			0
$964$		−	16.0000i		−	0.515325i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−22.0000	+	22.0000i	−0.707107	+	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	48.0000			1.53644
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	0.127971	−	0.991778i	$-0.459153\pi$
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		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
$986$	−60.0000			−1.91079
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	25.0000	+	25.0000i	0.791758	+	0.791758i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	25.0000	+	25.0000i	0.791758	+	0.791758i	−0.190022	+	0.981780i	$0.560856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.k.d.307.1		2
3.2	odd	2		900.2.k.b.307.1		2
4.3	odd	2	CM	900.2.k.d.307.1		2
5.2	odd	4		180.2.k.a.163.1	yes	2
5.3	odd	4	inner	900.2.k.d.343.1		2
5.4	even	2		180.2.k.a.127.1	✓	2
12.11	even	2		900.2.k.b.307.1		2
15.2	even	4		180.2.k.b.163.1	yes	2
15.8	even	4		900.2.k.b.343.1		2
15.14	odd	2		180.2.k.b.127.1	yes	2
20.3	even	4	inner	900.2.k.d.343.1		2
20.7	even	4		180.2.k.a.163.1	yes	2
20.19	odd	2		180.2.k.a.127.1	✓	2
60.23	odd	4		900.2.k.b.343.1		2
60.47	odd	4		180.2.k.b.163.1	yes	2
60.59	even	2		180.2.k.b.127.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.k.a.127.1	✓	2	5.4	even	2
180.2.k.a.127.1	✓	2	20.19	odd	2
180.2.k.a.163.1	yes	2	5.2	odd	4
180.2.k.a.163.1	yes	2	20.7	even	4
180.2.k.b.127.1	yes	2	15.14	odd	2
180.2.k.b.127.1	yes	2	60.59	even	2
180.2.k.b.163.1	yes	2	15.2	even	4
180.2.k.b.163.1	yes	2	60.47	odd	4
900.2.k.b.307.1		2	3.2	odd	2
900.2.k.b.307.1		2	12.11	even	2
900.2.k.b.343.1		2	15.8	even	4
900.2.k.b.343.1		2	60.23	odd	4
900.2.k.d.307.1		2	1.1	even	1	trivial
900.2.k.d.307.1		2	4.3	odd	2	CM
900.2.k.d.343.1		2	5.3	odd	4	inner
900.2.k.d.343.1		2	20.3	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-2.00000 + 2.00000i) q^{8} +(-5.00000 + 5.00000i) q^{13} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} -10.0000 q^{26} +10.0000i q^{29} +(-4.00000 - 4.00000i) q^{32} +6.00000i q^{34} +(-5.00000 - 5.00000i) q^{37} +10.0000 q^{41} +7.00000i q^{49} +(-10.0000 - 10.0000i) q^{52} +(9.00000 - 9.00000i) q^{53} +(-10.0000 + 10.0000i) q^{58} -12.0000 q^{61} -8.00000i q^{64} +(-6.00000 + 6.00000i) q^{68} +(5.00000 - 5.00000i) q^{73} -10.0000i q^{74} +(10.0000 + 10.0000i) q^{82} -10.0000i q^{89} +(5.00000 + 5.00000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{8} - 10 q^{13} - 8 q^{16} + 6 q^{17} - 20 q^{26} - 8 q^{32} - 10 q^{37} + 20 q^{41} - 20 q^{52} + 18 q^{53} - 20 q^{58} - 24 q^{61} - 12 q^{68} + 10 q^{73} + 20 q^{82} + 10 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^8 - 10 * q^13 - 8 * q^16 + 6 * q^17 - 20 * q^26 - 8 * q^32 - 10 * q^37 + 20 * q^41 - 20 * q^52 + 18 * q^53 - 20 * q^58 - 24 * q^61 - 12 * q^68 + 10 * q^73 + 20 * q^82 + 10 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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