LMFDB - Embedded newform 1225.2.b.d.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not 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:	$9.78167424761$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.d.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} +4.00000 q^{16} -3.00000i q^{17} +2.00000 q^{19} +6.00000i q^{23} +5.00000i q^{27} -3.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -5.00000 q^{39} +12.0000 q^{41} +10.0000i q^{43} -6.00000 q^{44} -9.00000i q^{47} +4.00000i q^{48} +3.00000 q^{51} +10.0000i q^{52} -12.0000i q^{53} +2.00000i q^{57} -8.00000 q^{61} +8.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{69} +2.00000i q^{73} +4.00000 q^{76} +1.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{87} -12.0000 q^{89} +12.0000i q^{92} +4.00000i q^{93} +1.00000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 4 q^{9} - 6 q^{11} + 8 q^{16} + 4 q^{19} - 6 q^{29} + 8 q^{31} + 8 q^{36} - 10 q^{39} + 24 q^{41} - 12 q^{44} + 6 q^{51} - 16 q^{61} + 16 q^{64} - 12 q^{69} + 8 q^{76} + 2 q^{79} + 2 q^{81}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 + 4 * q^9 - 6 * q^11 + 8 * q^16 + 4 * q^19 - 6 * q^29 + 8 * q^31 + 8 * q^36 - 10 * q^39 + 24 * q^41 - 12 * q^44 + 6 * q^51 - 16 * q^61 + 16 * q^64 - 12 * q^69 + 8 * q^76 + 2 * q^79 + 2 * q^81 - 24 * q^89 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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			$\pi$
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000i	0.577350i
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	4.00000	0.666667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	4.00000i	0.577350i
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	10.0000i	1.38675i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−6.00000	−0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	12.0000i	1.25109i
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	5.00000i	0.492665i	0.969185	+	0.246332i	$0.0792255\pi$
$103$	5.00000i	0.492665i	−0.969185	+	0.246332i	$0.920775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	10.0000i	0.962250i
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	10.0000i	0.924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	12.0000i	1.08200i
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	− 6.00000i	− 0.522233i
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	− 15.0000i	− 1.25436i
$144$	8.00000	0.666667
$145$	0	0
$146$	0	0
$147$	0	0
$148$	4.00000i	0.328798i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	−10.0000	−0.800641
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	24.0000	1.87409
$165$	0	0
$166$	0	0
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	20.0000i	1.52499i
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	0	0
$175$	0	0
$176$	−12.0000	−0.904534
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	− 8.00000i	− 0.591377i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.00000i	0.658145i
$188$	− 18.0000i	− 1.31278i
$189$	0	0
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	8.00000i	0.577350i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	0	0
$207$	12.0000i	0.834058i
$208$	20.0000i	1.38675i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 24.0000i	− 1.64833i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	15.0000	1.00901
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	4.00000i	0.264906i
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.00000i	0.0649570i
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	−16.0000	−1.02430
$245$	0	0
$246$	0	0
$247$	10.0000i	0.636285i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	− 8.00000i	− 0.488678i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−12.0000	−0.722315
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	− 13.0000i	− 0.772770i	−0.922338	−	0.386385i	$-0.873724\pi$
$283$	− 13.0000i	− 0.772770i	0.922338	−	0.386385i	$-0.126276\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−1.00000	−0.0586210
$292$	4.00000i	0.234082i
$293$	− 21.0000i	− 1.22683i	−0.789760	−	0.613417i	$-0.789795\pi$
$293$	− 21.0000i	− 1.22683i	0.789760	−	0.613417i	$-0.210205\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	−30.0000	−1.73494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	8.00000	0.458831
$305$	0	0
$306$	0	0
$307$	− 11.0000i	− 0.627803i	−0.949456	−	0.313902i	$-0.898364\pi$
$307$	− 11.0000i	− 0.627803i	0.949456	−	0.313902i	$-0.101636\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	0	0
$315$	0	0
$316$	2.00000	0.112509
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	7.00000i	0.387101i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	24.0000i	1.31717i
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	− 6.00000i	− 0.321634i
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	15.0000i	0.798369i	0.916871	+	0.399185i	$0.130707\pi$
$353$	15.0000i	0.798369i	−0.916871	+	0.399185i	$0.869293\pi$
$354$	0	0
$355$	0	0
$356$	−24.0000	−1.27200
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 2.00000i	− 0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	24.0000i	1.25109i
$369$	24.0000	1.24939
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.0000i	− 0.772539i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000i	1.01666i
$388$	2.00000i	0.101535i
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	6.00000i	0.302660i
$394$	0	0
$395$	0	0
$396$	−12.0000	−0.603023
$397$	25.0000i	1.25471i	0.778732	+	0.627357i	$0.215863\pi$
$397$	25.0000i	1.25471i	−0.778732	+	0.627357i	$0.784137\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	20.0000i	0.996271i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	10.0000i	0.492665i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.0000i	0.685583i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	− 18.0000i	− 0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000i	0.580042i
$429$	15.0000	0.724207
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	20.0000i	0.962250i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	12.0000i	0.574038i
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.0000i	0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$	18.0000i	0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	− 12.0000i	− 0.564433i
$453$	− 1.00000i	− 0.0469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	−12.0000	−0.557086
$465$	0	0
$466$	0	0
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	20.0000i	0.924500i
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	− 30.0000i	− 1.37940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 24.0000i	− 1.09888i
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	0	0
$484$	−4.00000	−0.181818
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	24.0000i	1.08200i
$493$	9.00000i	0.405340i
$494$	0	0
$495$	0	0
$496$	16.0000	0.718421
$497$	0	0
$498$	0	0
$499$	31.0000	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	27.0000i	1.20387i	0.798545	+	0.601935i	$0.205603\pi$
$503$	27.0000i	1.20387i	−0.798545	+	0.601935i	$0.794397\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	− 32.0000i	− 1.41977i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.0000i	0.441511i
$514$	0	0
$515$	0	0
$516$	−20.0000	−0.880451
$517$	27.0000i	1.18746i
$518$	0	0
$519$	9.00000	0.395056
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	− 12.0000i	− 0.522233i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	60.0000i	2.59889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	− 20.0000i	− 0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 24.0000i	− 1.02523i
$549$	−16.0000	−0.682863
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	28.0000	1.18746
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	−50.0000	−2.11477
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	18.0000	0.757937
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 30.0000i	− 1.25436i
$573$	9.00000i	0.375980i
$574$	0	0
$575$	0	0
$576$	16.0000	0.666667
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	0	0
$582$	0	0
$583$	36.0000i	1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	− 39.0000i	− 1.60154i	−0.598973	−	0.800769i	$-0.704424\pi$
$593$	− 39.0000i	− 1.60154i	0.598973	−	0.800769i	$-0.295576\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	13.0000i	0.527654i	0.964570	+	0.263827i	$0.0849848\pi$
$607$	13.0000i	0.527654i	−0.964570	+	0.263827i	$0.915015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.0000	1.82051
$612$	− 12.0000i	− 0.485071i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	0	0
$624$	−20.0000	−0.800641
$625$	0	0
$626$	0	0
$627$	− 6.00000i	− 0.239617i
$628$	− 28.0000i	− 1.11732i
$629$	6.00000	0.239236
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	0	0
$633$	− 13.0000i	− 0.516704i
$634$	0	0
$635$	0	0
$636$	24.0000	0.951662
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	41.0000i	1.61688i	0.588577	+	0.808441i	$0.299688\pi$
$643$	41.0000i	1.61688i	−0.588577	+	0.808441i	$0.700312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	48.0000	1.87409
$657$	4.00000i	0.156055i
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	15.0000i	0.582552i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.0000i	− 0.696963i
$668$	6.00000i	0.232147i
$669$	19.0000	0.734582
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	28.0000i	1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$	28.0000i	1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$	0	0
$675$	0	0
$676$	−24.0000	−0.923077
$677$	− 45.0000i	− 1.72949i	−0.502211	−	0.864745i	$-0.667480\pi$
$677$	− 45.0000i	− 1.72949i	0.502211	−	0.864745i	$-0.332520\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	0	0
$687$	− 4.00000i	− 0.152610i
$688$	40.0000i	1.52499i
$689$	60.0000	2.28582
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	4.00000i	0.150863i
$704$	−24.0000	−0.904534
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	21.0000i	0.784259i
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.0000i	0.371904i
$724$	−40.0000	−1.48659
$725$	0	0
$726$	0	0
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	30.0000	1.10959
$732$	− 16.0000i	− 0.591377i
$733$	− 31.0000i	− 1.14501i	−0.819901	−	0.572506i	$-0.805971\pi$
$733$	− 31.0000i	− 1.14501i	0.819901	−	0.572506i	$-0.194029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0000i	0.442026i
$738$	0	0
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	−10.0000	−0.367359
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.0000i	0.878114i
$748$	18.0000i	0.658145i
$749$	0	0
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	− 36.0000i	− 1.31278i
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	18.0000	0.651217
$765$	0	0
$766$	0	0
$767$	0	0
$768$	16.0000i	0.577350i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	8.00000i	0.287926i
$773$	21.0000i	0.755318i	0.925945	+	0.377659i	$0.123271\pi$
$773$	21.0000i	0.755318i	−0.925945	+	0.377659i	$0.876729\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5.00000i	− 0.178231i	−0.996021	−	0.0891154i	$-0.971596\pi$
$787$	− 5.00000i	− 0.178231i	0.996021	−	0.0891154i	$-0.0284040\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	−32.0000	−1.13421
$797$	− 15.0000i	− 0.531327i	−0.964066	−	0.265664i	$-0.914409\pi$
$797$	− 15.0000i	− 0.531327i	0.964066	−	0.265664i	$-0.0855911\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	−24.0000	−0.847998
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	− 6.00000i	− 0.211210i
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	12.0000	0.420084
$817$	20.0000i	0.699711i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.0000	−0.942306	−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000	−0.942306	−0.471153	+	0.882051i	$0.656162\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0000i	1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$	54.0000i	1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$	24.0000i	0.834058i
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	40.0000i	1.38675i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	20.0000i	0.691301i
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	3.00000i	0.103325i
$844$	−26.0000	−0.894957
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 48.0000i	− 1.64833i
$849$	13.0000	0.446159
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0000i	1.02478i	0.858753	+	0.512390i	$0.171240\pi$
$857$	30.0000i	1.02478i	−0.858753	+	0.512390i	$0.828760\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−3.00000	−0.101768
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	0	0
$879$	21.0000	0.708312
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	0	0
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 38.0000i	− 1.27233i
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 30.0000i	− 1.00167i
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.0000i	0.863316i	0.902037	+	0.431658i	$0.142071\pi$
$907$	26.0000i	0.863316i	−0.902037	+	0.431658i	$0.857929\pi$
$908$	6.00000i	0.199117i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	8.00000i	0.264906i
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	−8.00000	−0.264327
$917$	0	0
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	11.0000	0.362462
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.0000i	0.328443i
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	− 48.0000i	− 1.57229i
$933$	− 18.0000i	− 0.589294i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 47.0000i	− 1.53542i	−0.640796	−	0.767712i	$-0.721395\pi$
$937$	− 47.0000i	− 1.53542i	0.640796	−	0.767712i	$-0.278605\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	72.0000i	2.34464i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	2.00000i	0.0649570i
$949$	−10.0000	−0.324614
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	42.0000	1.35838
$957$	9.00000i	0.290929i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000i	0.386695i
$964$	20.0000	0.644157
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	32.0000i	1.02640i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−32.0000	−1.02430
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	− 21.0000i	− 0.669796i	−0.942254	−	0.334898i	$-0.891298\pi$
$983$	− 21.0000i	− 0.669796i	0.942254	−	0.334898i	$-0.108702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	20.0000i	0.636285i
$989$	−60.0000	−1.90789
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	−24.0000	−0.760469
$997$	37.0000i	1.17180i	0.810383	+	0.585901i	$0.199259\pi$
$997$	37.0000i	1.17180i	−0.810383	+	0.585901i	$0.800741\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.d.99.2		2
5.2	odd	4		1225.2.a.e.1.1		1
5.3	odd	4		245.2.a.c.1.1		1
5.4	even	2	inner	1225.2.b.d.99.1		2
7.6	odd	2		175.2.b.a.99.1		2
15.8	even	4		2205.2.a.e.1.1		1
20.3	even	4		3920.2.a.ba.1.1		1
21.20	even	2		1575.2.d.c.1324.2		2
28.27	even	2		2800.2.g.l.449.2		2
35.3	even	12		245.2.e.a.226.1		2
35.13	even	4		35.2.a.a.1.1	✓	1
35.18	odd	12		245.2.e.b.226.1		2
35.23	odd	12		245.2.e.b.116.1		2
35.27	even	4		175.2.a.b.1.1		1
35.33	even	12		245.2.e.a.116.1		2
35.34	odd	2		175.2.b.a.99.2		2
105.62	odd	4		1575.2.a.f.1.1		1
105.83	odd	4		315.2.a.b.1.1		1
105.104	even	2		1575.2.d.c.1324.1		2
140.27	odd	4		2800.2.a.z.1.1		1
140.83	odd	4		560.2.a.b.1.1		1
140.139	even	2		2800.2.g.l.449.1		2
280.13	even	4		2240.2.a.k.1.1		1
280.83	odd	4		2240.2.a.u.1.1		1
385.153	odd	4		4235.2.a.c.1.1		1
420.83	even	4		5040.2.a.v.1.1		1
455.363	even	4		5915.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.a.1.1	✓	1	35.13	even	4
175.2.a.b.1.1		1	35.27	even	4
175.2.b.a.99.1		2	7.6	odd	2
175.2.b.a.99.2		2	35.34	odd	2
245.2.a.c.1.1		1	5.3	odd	4
245.2.e.a.116.1		2	35.33	even	12
245.2.e.a.226.1		2	35.3	even	12
245.2.e.b.116.1		2	35.23	odd	12
245.2.e.b.226.1		2	35.18	odd	12
315.2.a.b.1.1		1	105.83	odd	4
560.2.a.b.1.1		1	140.83	odd	4
1225.2.a.e.1.1		1	5.2	odd	4
1225.2.b.d.99.1		2	5.4	even	2	inner
1225.2.b.d.99.2		2	1.1	even	1	trivial
1575.2.a.f.1.1		1	105.62	odd	4
1575.2.d.c.1324.1		2	105.104	even	2
1575.2.d.c.1324.2		2	21.20	even	2
2205.2.a.e.1.1		1	15.8	even	4
2240.2.a.k.1.1		1	280.13	even	4
2240.2.a.u.1.1		1	280.83	odd	4
2800.2.a.z.1.1		1	140.27	odd	4
2800.2.g.l.449.1		2	140.139	even	2
2800.2.g.l.449.2		2	28.27	even	2
3920.2.a.ba.1.1		1	20.3	even	4
4235.2.a.c.1.1		1	385.153	odd	4
5040.2.a.v.1.1		1	420.83	even	4
5915.2.a.f.1.1		1	455.363	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} +4.00000 q^{16} -3.00000i q^{17} +2.00000 q^{19} +6.00000i q^{23} +5.00000i q^{27} -3.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -5.00000 q^{39} +12.0000 q^{41} +10.0000i q^{43} -6.00000 q^{44} -9.00000i q^{47} +4.00000i q^{48} +3.00000 q^{51} +10.0000i q^{52} -12.0000i q^{53} +2.00000i q^{57} -8.00000 q^{61} +8.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{69} +2.00000i q^{73} +4.00000 q^{76} +1.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{87} -12.0000 q^{89} +12.0000i q^{92} +4.00000i q^{93} +1.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 4 q^{9} - 6 q^{11} + 8 q^{16} + 4 q^{19} - 6 q^{29} + 8 q^{31} + 8 q^{36} - 10 q^{39} + 24 q^{41} - 12 q^{44} + 6 q^{51} - 16 q^{61} + 16 q^{64} - 12 q^{69} + 8 q^{76} + 2 q^{79} + 2 q^{81}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 + 4 * q^9 - 6 * q^11 + 8 * q^16 + 4 * q^19 - 6 * q^29 + 8 * q^31 + 8 * q^36 - 10 * q^39 + 24 * q^41 - 12 * q^44 + 6 * q^51 - 16 * q^61 + 16 * q^64 - 12 * q^69 + 8 * q^76 + 2 * q^79 + 2 * q^81 - 24 * q^89 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more