LMFDB - Embedded newform 4900.2.e.b.2549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (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:	$39.1266969904$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.b.2549.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -6.00000 q^{9} -2.00000 q^{11} -6.00000i q^{13} -2.00000i q^{17} +9.00000i q^{23} +9.00000i q^{27} -3.00000 q^{29} -2.00000 q^{31} +6.00000i q^{33} +8.00000i q^{37} -18.0000 q^{39} -5.00000 q^{41} -1.00000i q^{43} -8.00000i q^{47} -6.00000 q^{51} -4.00000i q^{53} -8.00000 q^{59} -7.00000 q^{61} -3.00000i q^{67} +27.0000 q^{69} +8.00000 q^{71} +14.0000i q^{73} -4.00000 q^{79} +9.00000 q^{81} -1.00000i q^{83} +9.00000i q^{87} +13.0000 q^{89} +6.00000i q^{93} +10.0000i q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} - 4 q^{11} - 6 q^{29} - 4 q^{31} - 36 q^{39} - 10 q^{41} - 12 q^{51} - 16 q^{59} - 14 q^{61} + 54 q^{69} + 16 q^{71} - 8 q^{79} + 18 q^{81} + 26 q^{89} + 24 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 - 4 * q^11 - 6 * q^29 - 4 * q^31 - 36 * q^39 - 10 * q^41 - 12 * q^51 - 16 * q^59 - 14 * q^61 + 54 * q^69 + 16 * q^71 - 8 * q^79 + 18 * q^81 + 26 * q^89 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$2$	0	0
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	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
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	−18.0000	−2.88231
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	0	0
$69$	27.0000	3.25042
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 1.00000i	− 0.109764i	−0.998493	−	0.0548821i	$-0.982522\pi$
$83$	− 1.00000i	− 0.109764i	0.998493	−	0.0548821i	$-0.0174783\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000i	0.964901i
$88$	0	0
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000i	0.622171i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	36.0000i	3.32820i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	15.0000i	1.35250i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	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
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	0	0
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	12.0000i	0.970143i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.0000i	1.80395i
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	0	0
$183$	21.0000i	1.55236i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 54.0000i	− 3.75326i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	42.0000	2.83810
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.0000i	0.779484i
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	− 17.0000i	− 1.04826i	−0.851637	−	0.524132i	$-0.824390\pi$
$263$	− 17.0000i	− 1.04826i	0.851637	−	0.524132i	$-0.175610\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 39.0000i	− 2.38676i
$268$	0	0
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	30.0000	1.75863
$292$	0	0
$293$	4.00000i	0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$	4.00000i	0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 18.0000i	− 1.04447i
$298$	0	0
$299$	54.0000	3.12290
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 9.00000i	− 0.517036i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.00000i	− 0.0570730i	−0.999593	−	0.0285365i	$-0.990915\pi$
$307$	− 1.00000i	− 0.0570730i	0.999593	−	0.0285365i	$-0.00908469\pi$
$308$	0	0
$309$	39.0000	2.21863
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	− 4.00000i	− 0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$	− 4.00000i	− 0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−45.0000	−2.51166
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.0000i	1.49310i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	− 48.0000i	− 2.63038i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.0000i	1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$	28.0000i	1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.0000i	1.34207i	0.741426	+	0.671035i	$0.234150\pi$
$347$	25.0000i	1.34207i	−0.741426	+	0.671035i	$0.765850\pi$
$348$	0	0
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	54.0000	2.88231
$352$	0	0
$353$	− 36.0000i	− 1.91609i	−0.286623	−	0.958043i	$-0.592533\pi$
$353$	− 36.0000i	− 1.91609i	0.286623	−	0.958043i	$-0.407467\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	21.0000i	1.10221i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.00000i	− 0.0521996i	−0.999659	−	0.0260998i	$-0.991691\pi$
$367$	− 1.00000i	− 0.0521996i	0.999659	−	0.0260998i	$-0.00830876\pi$
$368$	0	0
$369$	30.0000	1.56174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0000i	0.927047i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	48.0000	2.45911
$382$	0	0
$383$	− 9.00000i	− 0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$	− 9.00000i	− 0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.0000	−1.44819	−0.724095	−	0.689700i	$-0.757743\pi$
$401$	−29.0000	−1.44819	−0.724095	+	0.689700i	$0.757743\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 16.0000i	− 0.793091i
$408$	0	0
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 30.0000i	− 1.46911i
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	48.0000i	2.33384i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	36.0000	1.73810
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 37.0000i	− 1.75792i	−0.476893	−	0.878962i	$-0.658237\pi$
$443$	− 37.0000i	− 1.75792i	0.476893	−	0.878962i	$-0.341763\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	27.0000i	1.27706i
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	30.0000i	1.40952i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	0	0
$459$	18.0000	0.840168
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	− 17.0000i	− 0.790057i	−0.918669	−	0.395029i	$-0.870735\pi$
$463$	− 17.0000i	− 0.790057i	0.918669	−	0.395029i	$-0.129265\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.00000i	− 0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$	− 5.00000i	− 0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	2.00000i	0.0919601i
$474$	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
$481$	48.0000	2.18861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	27.0000	1.20627
$502$	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
$507$	69.0000i	3.06440i
$508$	0	0
$509$	−41.0000	−1.81729	−0.908647	−	0.417566i	$-0.862883\pi$
$509$	−41.0000	−1.81729	−0.908647	+	0.417566i	$0.862883\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−48.0000	−2.10697
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000i	0.174243i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	48.0000	2.08302
$532$	0	0
$533$	30.0000i	1.29944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 18.0000i	− 0.776757i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	0	0
$543$	3.00000i	0.128742i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.0000i	− 0.641354i	−0.947189	−	0.320677i	$-0.896090\pi$
$547$	− 15.0000i	− 0.641354i	0.947189	−	0.320677i	$-0.103910\pi$
$548$	0	0
$549$	42.0000	1.79252
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	− 13.0000i	− 0.547885i	−0.961746	−	0.273942i	$-0.911672\pi$
$563$	− 13.0000i	− 0.547885i	0.961746	−	0.273942i	$-0.0883277\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.00000i	0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	42.0000	1.72765
$592$	0	0
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 60.0000i	− 2.45564i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	18.0000i	0.733017i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.00000i	− 0.0405887i	−0.999794	−	0.0202944i	$-0.993540\pi$
$607$	− 1.00000i	− 0.0405887i	0.999794	−	0.0202944i	$-0.00646034\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.0000i	0.805170i	0.915383	+	0.402585i	$0.131888\pi$
$617$	20.0000i	0.805170i	−0.915383	+	0.402585i	$0.868112\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−81.0000	−3.25042
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	− 12.0000i	− 0.476957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−48.0000	−1.89885
$640$	0	0
$641$	−31.0000	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$641$	−31.0000	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.00000i	− 0.0393141i	−0.999807	−	0.0196570i	$-0.993743\pi$
$647$	− 1.00000i	− 0.0393141i	0.999807	−	0.0196570i	$-0.00625743\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 84.0000i	− 3.27715i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	0	0
$663$	36.0000i	1.39812i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 27.0000i	− 1.04544i
$668$	0	0
$669$	48.0000	1.85579
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 29.0000i	− 1.10965i	−0.831966	−	0.554827i	$-0.812784\pi$
$683$	− 29.0000i	− 1.10965i	0.831966	−	0.554827i	$-0.187216\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.0000i	1.60240i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0000i	0.378777i
$698$	0	0
$699$	54.0000	2.04247
$700$	0	0
$701$	−29.0000	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$702$	0	0
$703$	0	0
$704$	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
$711$	24.0000	0.900070
$712$	0	0
$713$	− 18.0000i	− 0.674105i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 78.0000i	− 2.91296i
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	30.0000i	1.11571i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.00000i	− 0.111264i	−0.998451	−	0.0556319i	$-0.982283\pi$
$727$	− 3.00000i	− 0.111264i	0.998451	−	0.0556319i	$-0.0177173\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.00000i	0.221013i
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.00000i	0.110059i	0.998485	+	0.0550297i	$0.0175253\pi$
$743$	3.00000i	0.110059i	−0.998485	+	0.0550297i	$0.982475\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	90.0000i	3.27978i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	−54.0000	−1.96008
$760$	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
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.0000i	1.73318i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	− 27.0000i	− 0.964901i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.0000i	0.392108i	0.980593	+	0.196054i	$0.0628127\pi$
$787$	11.0000i	0.392108i	−0.980593	+	0.196054i	$0.937187\pi$
$788$	0	0
$789$	−51.0000	−1.81565
$790$	0	0
$791$	0	0
$792$	0	0
$793$	42.0000i	1.49146i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36.0000i	− 1.27519i	−0.770374	−	0.637593i	$-0.779930\pi$
$797$	− 36.0000i	− 1.27519i	0.770374	−	0.637593i	$-0.220070\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−78.0000	−2.75599
$802$	0	0
$803$	− 28.0000i	− 0.988099i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 27.0000i	− 0.950445i
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	−50.0000	−1.75574	−0.877869	−	0.478901i	$-0.841035\pi$
$811$	−50.0000	−1.75574	−0.877869	+	0.478901i	$0.841035\pi$
$812$	0	0
$813$	− 72.0000i	− 2.52515i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 19.0000i	− 0.662298i	−0.943578	−	0.331149i	$-0.892564\pi$
$823$	− 19.0000i	− 0.662298i	0.943578	−	0.331149i	$-0.107436\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.0000i	0.799788i	0.916561	+	0.399894i	$0.130953\pi$
$827$	23.0000i	0.799788i	−0.916561	+	0.399894i	$0.869047\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	54.0000	1.87324
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 18.0000i	− 0.622171i
$838$	0	0
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	66.0000i	2.27316i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−72.0000	−2.46813
$852$	0	0
$853$	− 40.0000i	− 1.36957i	−0.728743	−	0.684787i	$-0.759895\pi$
$853$	− 40.0000i	− 1.36957i	0.728743	−	0.684787i	$-0.240105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 37.0000i	− 1.25949i	−0.776800	−	0.629747i	$-0.783158\pi$
$863$	− 37.0000i	− 1.25949i	0.776800	−	0.629747i	$-0.216842\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 39.0000i	− 1.32451i
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−18.0000	−0.609907
$872$	0	0
$873$	− 60.0000i	− 2.03069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000i	1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$	32.0000i	1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	31.0000	1.04442	0.522208	−	0.852818i	$-0.325108\pi$
$881$	31.0000	1.04442	0.522208	+	0.852818i	$0.325108\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 53.0000i	− 1.77957i	−0.456384	−	0.889783i	$-0.650856\pi$
$887$	− 53.0000i	− 1.77957i	0.456384	−	0.889783i	$-0.349144\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.0000	−0.603023
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 162.000i	− 5.40902i
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37.0000i	− 1.22856i	−0.789086	−	0.614282i	$-0.789446\pi$
$907$	− 37.0000i	− 1.22856i	0.789086	−	0.614282i	$-0.210554\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	0	0
$913$	2.00000i	0.0661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−3.00000	−0.0988534
$922$	0	0
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 78.0000i	− 2.56186i
$928$	0	0
$929$	−5.00000	−0.164045	−0.0820223	−	0.996630i	$-0.526138\pi$
$929$	−5.00000	−0.164045	−0.0820223	+	0.996630i	$0.526138\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	72.0000i	2.35717i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.0000i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$937$	52.0000i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	− 45.0000i	− 1.46540i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.0000i	0.942373i	0.882034	+	0.471187i	$0.156174\pi$
$947$	29.0000i	0.942373i	−0.882034	+	0.471187i	$0.843826\pi$
$948$	0	0
$949$	84.0000	2.72676
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	60.0000i	1.94359i	0.235826	+	0.971795i	$0.424220\pi$
$953$	60.0000i	1.94359i	−0.235826	+	0.971795i	$0.575780\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 18.0000i	− 0.581857i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	90.0000i	2.90021i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.0000i	1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$	37.0000i	1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	−26.0000	−0.830964
$980$	0	0
$981$	54.0000	1.72409
$982$	0	0
$983$	− 19.0000i	− 0.606006i	−0.952990	−	0.303003i	$-0.902011\pi$
$983$	− 19.0000i	− 0.606006i	0.952990	−	0.303003i	$-0.0979892\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.00000	0.286183
$990$	0	0
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	0	0
$993$	− 30.0000i	− 0.952021i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.0000i	1.26681i	0.773819	+	0.633406i	$0.218344\pi$
$997$	40.0000i	1.26681i	−0.773819	+	0.633406i	$0.781656\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.b.2549.1		2
5.2	odd	4		4900.2.a.a.1.1		1
5.3	odd	4		980.2.a.i.1.1		1
5.4	even	2	inner	4900.2.e.b.2549.2		2
7.3	odd	6		700.2.r.c.149.2		4
7.5	odd	6		700.2.r.c.249.1		4
7.6	odd	2		4900.2.e.c.2549.2		2
15.8	even	4		8820.2.a.k.1.1		1
20.3	even	4		3920.2.a.d.1.1		1
35.3	even	12		140.2.i.b.121.1	yes	2
35.12	even	12		700.2.i.a.501.1		2
35.13	even	4		980.2.a.a.1.1		1
35.17	even	12		700.2.i.a.401.1		2
35.18	odd	12		980.2.i.a.961.1		2
35.19	odd	6		700.2.r.c.249.2		4
35.23	odd	12		980.2.i.a.361.1		2
35.24	odd	6		700.2.r.c.149.1		4
35.27	even	4		4900.2.a.v.1.1		1
35.33	even	12		140.2.i.b.81.1	✓	2
35.34	odd	2		4900.2.e.c.2549.1		2
105.38	odd	12		1260.2.s.b.541.1		2
105.68	odd	12		1260.2.s.b.361.1		2
105.83	odd	4		8820.2.a.w.1.1		1
140.3	odd	12		560.2.q.a.401.1		2
140.83	odd	4		3920.2.a.bi.1.1		1
140.103	odd	12		560.2.q.a.81.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.i.b.81.1	✓	2	35.33	even	12
140.2.i.b.121.1	yes	2	35.3	even	12
560.2.q.a.81.1		2	140.103	odd	12
560.2.q.a.401.1		2	140.3	odd	12
700.2.i.a.401.1		2	35.17	even	12
700.2.i.a.501.1		2	35.12	even	12
700.2.r.c.149.1		4	35.24	odd	6
700.2.r.c.149.2		4	7.3	odd	6
700.2.r.c.249.1		4	7.5	odd	6
700.2.r.c.249.2		4	35.19	odd	6
980.2.a.a.1.1		1	35.13	even	4
980.2.a.i.1.1		1	5.3	odd	4
980.2.i.a.361.1		2	35.23	odd	12
980.2.i.a.961.1		2	35.18	odd	12
1260.2.s.b.361.1		2	105.68	odd	12
1260.2.s.b.541.1		2	105.38	odd	12
3920.2.a.d.1.1		1	20.3	even	4
3920.2.a.bi.1.1		1	140.83	odd	4
4900.2.a.a.1.1		1	5.2	odd	4
4900.2.a.v.1.1		1	35.27	even	4
4900.2.e.b.2549.1		2	1.1	even	1	trivial
4900.2.e.b.2549.2		2	5.4	even	2	inner
4900.2.e.c.2549.1		2	35.34	odd	2
4900.2.e.c.2549.2		2	7.6	odd	2
8820.2.a.k.1.1		1	15.8	even	4
8820.2.a.w.1.1		1	105.83	odd	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -6.00000 q^{9} -2.00000 q^{11} -6.00000i q^{13} -2.00000i q^{17} +9.00000i q^{23} +9.00000i q^{27} -3.00000 q^{29} -2.00000 q^{31} +6.00000i q^{33} +8.00000i q^{37} -18.0000 q^{39} -5.00000 q^{41} -1.00000i q^{43} -8.00000i q^{47} -6.00000 q^{51} -4.00000i q^{53} -8.00000 q^{59} -7.00000 q^{61} -3.00000i q^{67} +27.0000 q^{69} +8.00000 q^{71} +14.0000i q^{73} -4.00000 q^{79} +9.00000 q^{81} -1.00000i q^{83} +9.00000i q^{87} +13.0000 q^{89} +6.00000i q^{93} +10.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} - 4 q^{11} - 6 q^{29} - 4 q^{31} - 36 q^{39} - 10 q^{41} - 12 q^{51} - 16 q^{59} - 14 q^{61} + 54 q^{69} + 16 q^{71} - 8 q^{79} + 18 q^{81} + 26 q^{89} + 24 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 - 4 * q^11 - 6 * q^29 - 4 * q^31 - 36 * q^39 - 10 * q^41 - 12 * q^51 - 16 * q^59 - 14 * q^61 + 54 * q^69 + 16 * q^71 - 8 * q^79 + 18 * q^81 + 26 * q^89 + 24 * q^99

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