LMFDB - Embedded newform 4900.2.e.k.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,-2,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,-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 980)
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.k.2549.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000 q^{9} -1.00000 q^{11} +5.00000i q^{13} +1.00000i q^{17} +6.00000 q^{19} +4.00000i q^{23} -5.00000i q^{27} -3.00000 q^{29} +2.00000 q^{31} +1.00000i q^{33} +8.00000i q^{37} +5.00000 q^{39} -10.0000 q^{41} +2.00000i q^{43} -7.00000i q^{47} +1.00000 q^{51} +2.00000i q^{53} -6.00000i q^{57} -14.0000 q^{59} -8.00000 q^{61} +14.0000i q^{67} +4.00000 q^{69} +10.0000i q^{73} +11.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +3.00000i q^{87} -4.00000 q^{89} -2.00000i q^{93} -3.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 2 q^{11} + 12 q^{19} - 6 q^{29} + 4 q^{31} + 10 q^{39} - 20 q^{41} + 2 q^{51} - 28 q^{59} - 16 q^{61} + 8 q^{69} + 22 q^{79} + 2 q^{81} - 8 q^{89} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 2 * q^11 + 12 * q^19 - 6 * q^29 + 4 * q^31 + 10 * q^39 - 20 * q^41 + 2 * q^51 - 28 * q^59 - 16 * q^61 + 8 * q^69 + 22 * q^79 + 2 * q^81 - 8 * q^89 - 4 * 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
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	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
$16$	0	0
$17$	1.00000i	0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$	1.00000i	0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	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
$33$	1.00000i	0.174078i
$34$	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
$39$	5.00000	0.800641
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	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
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	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
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000i	0.321634i
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.00000i	− 0.304604i	−0.988334	−	0.152302i	$-0.951331\pi$
$97$	− 3.00000i	− 0.304604i	0.988334	−	0.152302i	$-0.0486686\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	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
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.0000i	0.924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	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
$137$	16.0000i	1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$137$	16.0000i	1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	0	0
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 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
$159$	2.00000	0.158610
$160$	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	$0.301902\pi$
$167$	21.0000i	1.62503i	−0.582941	+	0.812514i	$0.698098\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	12.0000	0.917663
$172$	0	0
$173$	− 21.0000i	− 1.59660i	−0.602260	−	0.798300i	$-0.705733\pi$
$173$	− 21.0000i	− 1.59660i	0.602260	−	0.798300i	$-0.294267\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.0000i	1.05230i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	8.00000i	0.591377i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.00000i	− 0.0731272i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	12.0000i	0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$	12.0000i	0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	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
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000i	0.556038i
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	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
$221$	−5.00000	−0.336336
$222$	0	0
$223$	− 29.0000i	− 1.94198i	−0.239113	−	0.970992i	$-0.576857\pi$
$223$	− 29.0000i	− 1.94198i	0.239113	−	0.970992i	$-0.423143\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.00000i	0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$	9.00000i	0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$	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
$233$	8.00000i	0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$	8.00000i	0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 11.0000i	− 0.714527i
$238$	0	0
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.0000i	1.90885i
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.00000i	0.244796i
$268$	0	0
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	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
$277$	20.0000i	1.20168i	0.799368	+	0.600842i	$0.205168\pi$
$277$	20.0000i	1.20168i	−0.799368	+	0.600842i	$0.794832\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	0	0
$283$	1.00000i	0.0594438i	0.999558	+	0.0297219i	$0.00946217\pi$
$283$	1.00000i	0.0594438i	−0.999558	+	0.0297219i	$0.990538\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	−3.00000	−0.175863
$292$	0	0
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000i	0.290129i
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.0000i	0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.0000i	1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$	23.0000i	1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	25.0000i	1.41308i	0.707671	+	0.706542i	$0.249746\pi$
$313$	25.0000i	1.41308i	−0.707671	+	0.706542i	$0.750254\pi$
$314$	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
$319$	3.00000	0.167968
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.0000i	− 0.608301i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	25.0000	1.33440
$352$	0	0
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 23.0000i	− 1.20059i	−0.799779	−	0.600295i	$-0.795050\pi$
$367$	− 23.0000i	− 1.20059i	0.799779	−	0.600295i	$-0.204950\pi$
$368$	0	0
$369$	−20.0000	−1.04116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.0000i	− 0.772539i
$378$	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
$381$	−6.00000	−0.307389
$382$	0	0
$383$	32.0000i	1.63512i	0.575841	+	0.817562i	$0.304675\pi$
$383$	32.0000i	1.63512i	−0.575841	+	0.817562i	$0.695325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 23.0000i	− 1.15434i	−0.816625	−	0.577168i	$-0.804158\pi$
$397$	− 23.0000i	− 1.15434i	0.816625	−	0.577168i	$-0.195842\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	0	0
$403$	10.0000i	0.498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	16.0000	0.789222
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.0000i	0.783523i
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	− 14.0000i	− 0.680703i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	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
$443$	26.0000i	1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$	26.0000i	1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 22.0000i	− 1.04056i
$448$	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
$451$	10.0000	0.470882
$452$	0	0
$453$	3.00000i	0.140952i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.0000i	0.879215i	0.898190	+	0.439608i	$0.144882\pi$
$467$	19.0000i	0.879215i	−0.898190	+	0.439608i	$0.855118\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.00000i	0.183147i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	− 3.00000i	− 0.135113i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	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
$507$	12.0000i	0.532939i
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 30.0000i	− 1.32453i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.00000i	0.307860i
$518$	0	0
$519$	−21.0000	−0.921798
$520$	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
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.00000i	0.0871214i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−28.0000	−1.21510
$532$	0	0
$533$	− 50.0000i	− 2.16574i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20.0000i	− 0.863064i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	− 10.0000i	− 0.429141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	0	0
$549$	−16.0000	−0.682863
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	− 3.00000i	− 0.125327i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	− 23.0000i	− 0.944497i	−0.881466	−	0.472248i	$-0.843443\pi$
$593$	− 23.0000i	− 0.944497i	0.881466	−	0.472248i	$-0.156557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 22.0000i	− 0.900400i
$598$	0	0
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	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
$603$	28.0000i	1.14025i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 29.0000i	− 1.17707i	−0.808470	−	0.588537i	$-0.799704\pi$
$607$	− 29.0000i	− 1.17707i	0.808470	−	0.588537i	$-0.200296\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.0000	1.41595
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	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
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.00000i	0.239617i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	0	0
$633$	− 17.0000i	− 0.675689i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	15.0000i	0.591542i	0.955259	+	0.295771i	$0.0955766\pi$
$643$	15.0000i	0.591542i	−0.955259	+	0.295771i	$0.904423\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20.0000i	0.780274i
$658$	0	0
$659$	37.0000	1.44132	0.720658	−	0.693291i	$-0.243840\pi$
$659$	37.0000	1.44132	0.720658	+	0.693291i	$0.243840\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	0	0
$663$	5.00000i	0.194184i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	−29.0000	−1.12120
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.0000i	1.19143i	0.803197	+	0.595713i	$0.203131\pi$
$677$	31.0000i	1.19143i	−0.803197	+	0.595713i	$0.796869\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	− 48.0000i	− 1.83667i	−0.395805	−	0.918334i	$-0.629534\pi$
$683$	− 48.0000i	− 1.83667i	0.395805	−	0.918334i	$-0.370466\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.0000i	− 0.378777i
$698$	0	0
$699$	8.00000	0.302588
$700$	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
$703$	48.0000i	1.81035i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	0	0
$711$	22.0000	0.825064
$712$	0	0
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.0000i	0.485494i
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.0000i	0.595046i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	33.0000i	1.21888i	0.792831	+	0.609441i	$0.208606\pi$
$733$	33.0000i	1.21888i	−0.792831	+	0.609441i	$0.791394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.0000i	− 0.515697i
$738$	0	0
$739$	−3.00000	−0.110357	−0.0551784	−	0.998477i	$-0.517573\pi$
$739$	−3.00000	−0.110357	−0.0551784	+	0.998477i	$0.517573\pi$
$740$	0	0
$741$	30.0000	1.10208
$742$	0	0
$743$	18.0000i	0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$743$	18.0000i	0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000i	0.292705i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	0	0
$753$	4.00000i	0.145768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 70.0000i	− 2.52755i
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	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
$779$	−60.0000	−2.14972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	15.0000i	0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.00000i	− 0.106938i	−0.998569	−	0.0534692i	$-0.982972\pi$
$787$	− 3.00000i	− 0.106938i	0.998569	−	0.0534692i	$-0.0170279\pi$
$788$	0	0
$789$	−18.0000	−0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.0000i	1.31061i	0.755366	+	0.655304i	$0.227459\pi$
$797$	37.0000i	1.31061i	−0.755366	+	0.655304i	$0.772541\pi$
$798$	0	0
$799$	7.00000	0.247642
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8.00000i	− 0.281613i
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0000i	0.419827i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10.0000i	− 0.345651i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	13.0000i	0.447744i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.00000	0.0343199
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.0000i	1.02121i	0.859815	+	0.510606i	$0.170579\pi$
$863$	30.0000i	1.02121i	−0.859815	+	0.510606i	$0.829421\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.0000i	− 0.543388i
$868$	0	0
$869$	−11.0000	−0.373149
$870$	0	0
$871$	−70.0000	−2.37186
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 54.0000i	− 1.82345i	−0.410801	−	0.911725i	$-0.634751\pi$
$877$	− 54.0000i	− 1.82345i	0.410801	−	0.911725i	$-0.365249\pi$
$878$	0	0
$879$	−9.00000	−0.303562
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	0	0
$883$	56.0000i	1.88455i	0.334840	+	0.942275i	$0.391318\pi$
$883$	56.0000i	1.88455i	−0.334840	+	0.942275i	$0.608682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	− 42.0000i	− 1.40548i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.0000i	0.667781i
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.0000i	0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$	16.0000i	0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$	0	0
$909$	−20.0000	−0.663358
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	− 4.00000i	− 0.132381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.0000i	0.459820i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 14.0000i	− 0.458339i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.00000i	0.294017i	0.989135	+	0.147009i	$0.0469645\pi$
$937$	9.00000i	0.294017i	−0.989135	+	0.147009i	$0.953036\pi$
$938$	0	0
$939$	25.0000	0.815844
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 54.0000i	− 1.75476i	−0.479792	−	0.877382i	$-0.659288\pi$
$947$	− 54.0000i	− 1.75476i	0.479792	−	0.877382i	$-0.340712\pi$
$948$	0	0
$949$	−50.0000	−1.62307
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	− 50.0000i	− 1.61966i	−0.586665	−	0.809829i	$-0.699560\pi$
$953$	− 50.0000i	− 1.61966i	0.586665	−	0.809829i	$-0.300440\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.00000i	− 0.0969762i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	36.0000i	1.16008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	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
$979$	4.00000	0.127841
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	53.0000i	1.69044i	0.534421	+	0.845219i	$0.320530\pi$
$983$	53.0000i	1.69044i	−0.534421	+	0.845219i	$0.679470\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.00000i	− 0.0950110i	−0.998871	−	0.0475055i	$-0.984873\pi$
$997$	− 3.00000i	− 0.0950110i	0.998871	−	0.0475055i	$-0.0151272\pi$
$998$	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.k.2549.1		2
5.2	odd	4		4900.2.a.h.1.1		1
5.3	odd	4		980.2.a.f.1.1	yes	1
5.4	even	2	inner	4900.2.e.k.2549.2		2
7.6	odd	2		4900.2.e.j.2549.2		2
15.8	even	4		8820.2.a.v.1.1		1
20.3	even	4		3920.2.a.n.1.1		1
35.3	even	12		980.2.i.g.961.1		2
35.13	even	4		980.2.a.d.1.1	✓	1
35.18	odd	12		980.2.i.e.961.1		2
35.23	odd	12		980.2.i.e.361.1		2
35.27	even	4		4900.2.a.o.1.1		1
35.33	even	12		980.2.i.g.361.1		2
35.34	odd	2		4900.2.e.j.2549.1		2
105.83	odd	4		8820.2.a.i.1.1		1
140.83	odd	4		3920.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.d.1.1	✓	1	35.13	even	4
980.2.a.f.1.1	yes	1	5.3	odd	4
980.2.i.e.361.1		2	35.23	odd	12
980.2.i.e.961.1		2	35.18	odd	12
980.2.i.g.361.1		2	35.33	even	12
980.2.i.g.961.1		2	35.3	even	12
3920.2.a.n.1.1		1	20.3	even	4
3920.2.a.z.1.1		1	140.83	odd	4
4900.2.a.h.1.1		1	5.2	odd	4
4900.2.a.o.1.1		1	35.27	even	4
4900.2.e.j.2549.1		2	35.34	odd	2
4900.2.e.j.2549.2		2	7.6	odd	2
4900.2.e.k.2549.1		2	1.1	even	1	trivial
4900.2.e.k.2549.2		2	5.4	even	2	inner
8820.2.a.i.1.1		1	105.83	odd	4
8820.2.a.v.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000 q^{9} -1.00000 q^{11} +5.00000i q^{13} +1.00000i q^{17} +6.00000 q^{19} +4.00000i q^{23} -5.00000i q^{27} -3.00000 q^{29} +2.00000 q^{31} +1.00000i q^{33} +8.00000i q^{37} +5.00000 q^{39} -10.0000 q^{41} +2.00000i q^{43} -7.00000i q^{47} +1.00000 q^{51} +2.00000i q^{53} -6.00000i q^{57} -14.0000 q^{59} -8.00000 q^{61} +14.0000i q^{67} +4.00000 q^{69} +10.0000i q^{73} +11.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +3.00000i q^{87} -4.00000 q^{89} -2.00000i q^{93} -3.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 2 q^{11} + 12 q^{19} - 6 q^{29} + 4 q^{31} + 10 q^{39} - 20 q^{41} + 2 q^{51} - 28 q^{59} - 16 q^{61} + 8 q^{69} + 22 q^{79} + 2 q^{81} - 8 q^{89} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 2 * q^11 + 12 * q^19 - 6 * q^29 + 4 * q^31 + 10 * q^39 - 20 * q^41 + 2 * q^51 - 28 * q^59 - 16 * q^61 + 8 * q^69 + 22 * q^79 + 2 * q^81 - 8 * q^89 - 4 * q^99

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