LMFDB - Embedded newform 4900.2.e.m.2549.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,0,0,0,0,0,0,4,0,12,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,-6, 0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.1266969904$
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 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.m.2549.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000 q^{9} +6.00000 q^{11} -2.00000i q^{13} -6.00000i q^{17} -8.00000 q^{19} -3.00000i q^{23} -5.00000i q^{27} -3.00000 q^{29} +2.00000 q^{31} -6.00000i q^{33} +8.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} -5.00000i q^{43} -6.00000 q^{51} -12.0000i q^{53} +8.00000i q^{57} -1.00000 q^{61} -7.00000i q^{67} -3.00000 q^{69} +10.0000i q^{73} +4.00000 q^{79} +1.00000 q^{81} -3.00000i q^{83} +3.00000i q^{87} +3.00000 q^{89} -2.00000i q^{93} -10.0000i q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} + 12 q^{11} - 16 q^{19} - 6 q^{29} + 4 q^{31} - 4 q^{39} - 6 q^{41} - 12 q^{51} - 2 q^{61} - 6 q^{69} + 8 q^{79} + 2 q^{81} + 6 q^{89} + 24 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 12 * q^11 - 16 * q^19 - 6 * q^29 + 4 * q^31 - 4 * q^39 - 6 * q^41 - 12 * q^51 - 2 * q^61 - 6 * q^69 + 8 * q^79 + 2 * q^81 + 6 * q^89 + 24 * q^99

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	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
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	− 5.00000i	− 0.762493i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	− 5.00000i	− 0.762493i	0.924473	−	0.381246i	$-0.124505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 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
$57$	8.00000i	1.05963i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000i	0.321634i
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	7.00000i	0.689730i	0.938652	+	0.344865i	$0.112075\pi$
$103$	7.00000i	0.689730i	−0.938652	+	0.344865i	$0.887925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	0	0
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	3.00000i	0.270501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	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
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	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
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 21.0000i	− 1.62503i	−0.582941	−	0.812514i	$-0.698098\pi$
$167$	− 21.0000i	− 1.62503i	0.582941	−	0.812514i	$-0.301902\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−16.0000	−1.22355
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	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
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.0000i	− 2.63258i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	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
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.00000i	− 0.417029i
$208$	0	0
$209$	−48.0000	−3.32023
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	0	0
$249$	−3.00000	−0.190117
$250$	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
$256$	0	0
$257$	− 24.0000i	− 1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$	− 24.0000i	− 1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	3.00000i	0.184988i	0.995713	+	0.0924940i	$0.0294839\pi$
$263$	3.00000i	0.184988i	−0.995713	+	0.0924940i	$0.970516\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3.00000i	− 0.183597i
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 30.0000i	− 1.74078i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.00000i	0.172345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	48.0000i	2.67079i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.0000i	0.940102i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.0000i	1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$	20.0000i	1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.0000i	− 1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$	− 27.0000i	− 1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.00000i	0.260998i	0.991448	+	0.130499i	$0.0416579\pi$
$367$	5.00000i	0.260998i	−0.991448	+	0.130499i	$0.958342\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 3.00000i	− 0.153293i	−0.997058	−	0.0766464i	$-0.975579\pi$
$383$	− 3.00000i	− 0.153293i	0.997058	−	0.0766464i	$-0.0244213\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.0000i	− 0.508329i
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.0000i	2.37927i
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.00000i	0.0979404i
$418$	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
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12.0000	−0.579365
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	28.0000i	1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$	28.0000i	1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.0000i	1.14808i
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 15.0000i	− 0.709476i
$448$	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
$451$	−18.0000	−0.847587
$452$	0	0
$453$	10.0000i	0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.00000i	− 0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$	− 4.00000i	− 0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$	0	0
$459$	−30.0000	−1.40028
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 29.0000i	− 1.34774i	−0.738848	−	0.673872i	$-0.764630\pi$
$463$	− 29.0000i	− 1.34774i	0.738848	−	0.673872i	$-0.235370\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	0	0
$469$	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	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	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
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	9.00000i	0.401290i	0.979664	+	0.200645i	$0.0643038\pi$
$503$	9.00000i	0.401290i	−0.979664	+	0.200645i	$0.935696\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	−39.0000	−1.72864	−0.864322	−	0.502938i	$-0.832252\pi$
$509$	−39.0000	−1.72864	−0.864322	+	0.502938i	$0.832252\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	40.0000i	1.76604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.00000i	0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	− 17.0000i	− 0.729540i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19.0000i	− 0.812381i	−0.913788	−	0.406191i	$-0.866857\pi$
$547$	− 19.0000i	− 0.812381i	0.913788	−	0.406191i	$-0.133143\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−36.0000	−1.51992
$562$	0	0
$563$	9.00000i	0.379305i	0.981851	+	0.189652i	$0.0607361\pi$
$563$	9.00000i	0.379305i	−0.981851	+	0.189652i	$0.939264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 72.0000i	− 2.98194i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 14.0000i	− 0.570124i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 43.0000i	− 1.74532i	−0.488332	−	0.872658i	$-0.662394\pi$
$607$	− 43.0000i	− 1.74532i	0.488332	−	0.872658i	$-0.337606\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 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
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	48.0000i	1.91694i
$628$	0	0
$629$	48.0000	1.91389
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.0000	−1.54041	−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000	−1.54041	−0.770204	+	0.637798i	$0.779845\pi$
$642$	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
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20.0000i	0.780274i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	0	0
$663$	12.0000i	0.466041i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	− 32.0000i	− 1.23351i	−0.787155	−	0.616755i	$-0.788447\pi$
$673$	− 32.0000i	− 1.23351i	0.787155	−	0.616755i	$-0.211553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	15.0000i	0.573959i	0.957937	+	0.286980i	$0.0926512\pi$
$683$	15.0000i	0.573959i	−0.957937	+	0.286980i	$0.907349\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	0	0
$703$	− 64.0000i	− 2.41381i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	− 6.00000i	− 0.224702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000i	0.224074i
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 26.0000i	− 0.966950i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−30.0000	−1.10959
$732$	0	0
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 42.0000i	− 1.54709i
$738$	0	0
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	39.0000i	1.43077i	0.698730	+	0.715386i	$0.253749\pi$
$743$	39.0000i	1.43077i	−0.698730	+	0.715386i	$0.746251\pi$
$744$	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
$753$	18.0000i	0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	0	0
$759$	−18.0000	−0.653359
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	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
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	0	0
$789$	3.00000	0.106803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	60.0000i	2.11735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15.0000i	− 0.528025i
$808$	0	0
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40.0000i	1.39942i
$818$	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
$823$	49.0000i	1.70803i	0.520246	+	0.854016i	$0.325840\pi$
$823$	49.0000i	1.70803i	−0.520246	+	0.854016i	$0.674160\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.0000i	− 1.56480i	−0.622774	−	0.782402i	$-0.713994\pi$
$827$	− 45.0000i	− 1.56480i	0.622774	−	0.782402i	$-0.286006\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10.0000i	− 0.345651i
$838$	0	0
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	− 32.0000i	− 1.09566i	−0.836590	−	0.547830i	$-0.815454\pi$
$853$	− 32.0000i	− 1.09566i	0.836590	−	0.547830i	$-0.184546\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 33.0000i	− 1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$	− 33.0000i	− 1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 40.0000i	− 1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$	− 40.0000i	− 1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	57.0000i	1.91387i	0.290298	+	0.956936i	$0.406246\pi$
$887$	57.0000i	1.91387i	−0.290298	+	0.956936i	$0.593754\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.00000i	0.200334i
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−72.0000	−2.39867
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.0000i	0.763702i	0.924224	+	0.381851i	$0.124713\pi$
$907$	23.0000i	0.763702i	−0.924224	+	0.381851i	$0.875287\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	− 18.0000i	− 0.595713i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.0000i	0.459820i
$928$	0	0
$929$	45.0000	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$929$	45.0000	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.0000i	1.43742i	0.695311	+	0.718709i	$0.255266\pi$
$937$	44.0000i	1.43742i	−0.695311	+	0.718709i	$0.744734\pi$
$938$	0	0
$939$	4.00000	0.130535
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	9.00000i	0.293080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.00000i	0.292461i	0.989251	+	0.146230i	$0.0467141\pi$
$947$	9.00000i	0.292461i	−0.989251	+	0.146230i	$0.953286\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	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
$963$	− 6.00000i	− 0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.00000i	− 0.225105i	−0.993646	−	0.112552i	$-0.964097\pi$
$967$	− 7.00000i	− 0.225105i	0.993646	−	0.112552i	$-0.0359026\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	−34.0000	−1.08554
$982$	0	0
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15.0000	−0.476972
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	22.0000i	0.698149i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.0000i	1.01345i	0.862108	+	0.506725i	$0.169144\pi$
$997$	32.0000i	1.01345i	−0.862108	+	0.506725i	$0.830856\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.m.2549.1		2
5.2	odd	4		4900.2.a.i.1.1		1
5.3	odd	4		980.2.a.g.1.1		1
5.4	even	2	inner	4900.2.e.m.2549.2		2
7.2	even	3		700.2.r.a.249.2		4
7.4	even	3		700.2.r.a.149.1		4
7.6	odd	2		4900.2.e.n.2549.2		2
15.8	even	4		8820.2.a.p.1.1		1
20.3	even	4		3920.2.a.k.1.1		1
35.2	odd	12		700.2.i.b.501.1		2
35.3	even	12		980.2.i.f.961.1		2
35.4	even	6		700.2.r.a.149.2		4
35.9	even	6		700.2.r.a.249.1		4
35.13	even	4		980.2.a.e.1.1		1
35.18	odd	12		140.2.i.a.121.1	yes	2
35.23	odd	12		140.2.i.a.81.1	✓	2
35.27	even	4		4900.2.a.q.1.1		1
35.32	odd	12		700.2.i.b.401.1		2
35.33	even	12		980.2.i.f.361.1		2
35.34	odd	2		4900.2.e.n.2549.1		2
105.23	even	12		1260.2.s.c.361.1		2
105.53	even	12		1260.2.s.c.541.1		2
105.83	odd	4		8820.2.a.a.1.1		1
140.23	even	12		560.2.q.f.81.1		2
140.83	odd	4		3920.2.a.w.1.1		1
140.123	even	12		560.2.q.f.401.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.i.a.81.1	✓	2	35.23	odd	12
140.2.i.a.121.1	yes	2	35.18	odd	12
560.2.q.f.81.1		2	140.23	even	12
560.2.q.f.401.1		2	140.123	even	12
700.2.i.b.401.1		2	35.32	odd	12
700.2.i.b.501.1		2	35.2	odd	12
700.2.r.a.149.1		4	7.4	even	3
700.2.r.a.149.2		4	35.4	even	6
700.2.r.a.249.1		4	35.9	even	6
700.2.r.a.249.2		4	7.2	even	3
980.2.a.e.1.1		1	35.13	even	4
980.2.a.g.1.1		1	5.3	odd	4
980.2.i.f.361.1		2	35.33	even	12
980.2.i.f.961.1		2	35.3	even	12
1260.2.s.c.361.1		2	105.23	even	12
1260.2.s.c.541.1		2	105.53	even	12
3920.2.a.k.1.1		1	20.3	even	4
3920.2.a.w.1.1		1	140.83	odd	4
4900.2.a.i.1.1		1	5.2	odd	4
4900.2.a.q.1.1		1	35.27	even	4
4900.2.e.m.2549.1		2	1.1	even	1	trivial
4900.2.e.m.2549.2		2	5.4	even	2	inner
4900.2.e.n.2549.1		2	35.34	odd	2
4900.2.e.n.2549.2		2	7.6	odd	2
8820.2.a.a.1.1		1	105.83	odd	4
8820.2.a.p.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000 q^{9} +6.00000 q^{11} -2.00000i q^{13} -6.00000i q^{17} -8.00000 q^{19} -3.00000i q^{23} -5.00000i q^{27} -3.00000 q^{29} +2.00000 q^{31} -6.00000i q^{33} +8.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} -5.00000i q^{43} -6.00000 q^{51} -12.0000i q^{53} +8.00000i q^{57} -1.00000 q^{61} -7.00000i q^{67} -3.00000 q^{69} +10.0000i q^{73} +4.00000 q^{79} +1.00000 q^{81} -3.00000i q^{83} +3.00000i q^{87} +3.00000 q^{89} -2.00000i q^{93} -10.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} + 12 q^{11} - 16 q^{19} - 6 q^{29} + 4 q^{31} - 4 q^{39} - 6 q^{41} - 12 q^{51} - 2 q^{61} - 6 q^{69} + 8 q^{79} + 2 q^{81} + 6 q^{89} + 24 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 12 * q^11 - 16 * q^19 - 6 * q^29 + 4 * q^31 - 4 * q^39 - 6 * q^41 - 12 * q^51 - 2 * q^61 - 6 * q^69 + 8 * q^79 + 2 * q^81 + 6 * q^89 + 24 * q^99

Introduction

L-functions

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