LMFDB - Embedded newform 4050.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(1,4050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4050.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,0,0,0,4,0,0,0,0,0,0,0,4,10,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 810)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31342$ of defining polynomial
Character	$\chi$	$=$	4050.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.31342 q^{7} +1.00000 q^{8} +2.62685 q^{11} -1.73205 q^{13} +1.31342 q^{14} +1.00000 q^{16} -1.27492 q^{17} -2.27492 q^{19} +2.62685 q^{22} +6.27492 q^{23} -1.73205 q^{26} +1.31342 q^{28} -9.13642 q^{29} +8.54983 q^{31} +1.00000 q^{32} -1.27492 q^{34} +9.97368 q^{37} -2.27492 q^{38} +5.61478 q^{41} -2.62685 q^{43} +2.62685 q^{44} +6.27492 q^{46} +10.2749 q^{47} -5.27492 q^{49} -1.73205 q^{52} +8.27492 q^{53} +1.31342 q^{56} -9.13642 q^{58} +8.24163 q^{59} +7.27492 q^{61} +8.54983 q^{62} +1.00000 q^{64} -12.1819 q^{67} -1.27492 q^{68} -6.92820 q^{71} -4.83507 q^{73} +9.97368 q^{74} -2.27492 q^{76} +3.45017 q^{77} +5.61478 q^{82} +4.00000 q^{83} -2.62685 q^{86} +2.62685 q^{88} -3.04547 q^{89} -2.27492 q^{91} +6.27492 q^{92} +10.2749 q^{94} -19.1101 q^{97} -5.27492 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 10 q^{17} + 6 q^{19} + 10 q^{23} + 4 q^{31} + 4 q^{32} + 10 q^{34} + 6 q^{38} + 10 q^{46} + 26 q^{47} - 6 q^{49} + 18 q^{53} + 14 q^{61} + 4 q^{62} + 4 q^{64}+ \cdots - 6 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 + 10 * q^17 + 6 * q^19 + 10 * q^23 + 4 * q^31 + 4 * q^32 + 10 * q^34 + 6 * q^38 + 10 * q^46 + 26 * q^47 - 6 * q^49 + 18 * q^53 + 14 * q^61 + 4 * q^62 + 4 * q^64 + 10 * q^68 + 6 * q^76 + 44 * q^77 + 16 * q^83 + 6 * q^91 + 10 * q^92 + 26 * q^94 - 6 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.31342	0.496428	0.248214	−	0.968705i	$-0.420156\pi$
$7$	1.31342	0.496428	0.248214	+	0.968705i	$0.420156\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.62685	0.792025	0.396012	−	0.918245i	$-0.370394\pi$
$11$	2.62685	0.792025	0.396012	+	0.918245i	$0.370394\pi$
$12$	0	0
$13$	−1.73205	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$14$	1.31342	0.351027
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.27492	−0.309213	−0.154606	−	0.987976i	$-0.549411\pi$
$17$	−1.27492	−0.309213	−0.154606	+	0.987976i	$0.549411\pi$
$18$	0	0
$19$	−2.27492	−0.521902	−0.260951	−	0.965352i	$-0.584036\pi$
$19$	−2.27492	−0.521902	−0.260951	+	0.965352i	$0.584036\pi$
$20$	0	0
$21$	0	0
$22$	2.62685	0.560046
$23$	6.27492	1.30841	0.654205	−	0.756317i	$-0.273003\pi$
$23$	6.27492	1.30841	0.654205	+	0.756317i	$0.273003\pi$
$24$	0	0
$25$	0	0
$26$	−1.73205	−0.339683
$27$	0	0
$28$	1.31342	0.248214
$29$	−9.13642	−1.69659	−0.848296	−	0.529523i	$-0.822371\pi$
$29$	−9.13642	−1.69659	−0.848296	+	0.529523i	$0.822371\pi$
$30$	0	0
$31$	8.54983	1.53560	0.767798	−	0.640692i	$-0.221353\pi$
$31$	8.54983	1.53560	0.767798	+	0.640692i	$0.221353\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.27492	−0.218646
$35$	0	0
$36$	0	0
$37$	9.97368	1.63966	0.819831	−	0.572605i	$-0.194067\pi$
$37$	9.97368	1.63966	0.819831	+	0.572605i	$0.194067\pi$
$38$	−2.27492	−0.369040
$39$	0	0
$40$	0	0
$41$	5.61478	0.876881	0.438441	−	0.898760i	$-0.355531\pi$
$41$	5.61478	0.876881	0.438441	+	0.898760i	$0.355531\pi$
$42$	0	0
$43$	−2.62685	−0.400591	−0.200295	−	0.979736i	$-0.564190\pi$
$43$	−2.62685	−0.400591	−0.200295	+	0.979736i	$0.564190\pi$
$44$	2.62685	0.396012
$45$	0	0
$46$	6.27492	0.925186
$47$	10.2749	1.49875	0.749375	−	0.662145i	$-0.230354\pi$
$47$	10.2749	1.49875	0.749375	+	0.662145i	$0.230354\pi$
$48$	0	0
$49$	−5.27492	−0.753560
$50$	0	0
$51$	0	0
$52$	−1.73205	−0.240192
$53$	8.27492	1.13665	0.568324	−	0.822805i	$-0.307592\pi$
$53$	8.27492	1.13665	0.568324	+	0.822805i	$0.307592\pi$
$54$	0	0
$55$	0	0
$56$	1.31342	0.175514
$57$	0	0
$58$	−9.13642	−1.19967
$59$	8.24163	1.07297	0.536484	−	0.843910i	$-0.319752\pi$
$59$	8.24163	1.07297	0.536484	+	0.843910i	$0.319752\pi$
$60$	0	0
$61$	7.27492	0.931458	0.465729	−	0.884927i	$-0.345792\pi$
$61$	7.27492	0.931458	0.465729	+	0.884927i	$0.345792\pi$
$62$	8.54983	1.08583
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−12.1819	−1.48826	−0.744128	−	0.668037i	$-0.767135\pi$
$67$	−12.1819	−1.48826	−0.744128	+	0.668037i	$0.767135\pi$
$68$	−1.27492	−0.154606
$69$	0	0
$70$	0	0
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	−4.83507	−0.565902	−0.282951	−	0.959134i	$-0.591313\pi$
$73$	−4.83507	−0.565902	−0.282951	+	0.959134i	$0.591313\pi$
$74$	9.97368	1.15942
$75$	0	0
$76$	−2.27492	−0.260951
$77$	3.45017	0.393183
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	5.61478	0.620049
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	−2.62685	−0.283260
$87$	0	0
$88$	2.62685	0.280023
$89$	−3.04547	−0.322820	−0.161410	−	0.986887i	$-0.551604\pi$
$89$	−3.04547	−0.322820	−0.161410	+	0.986887i	$0.551604\pi$
$90$	0	0
$91$	−2.27492	−0.238476
$92$	6.27492	0.654205
$93$	0	0
$94$	10.2749	1.05978
$95$	0	0
$96$	0	0
$97$	−19.1101	−1.94034	−0.970168	−	0.242432i	$-0.922055\pi$
$97$	−19.1101	−1.94034	−0.970168	+	0.242432i	$0.922055\pi$
$98$	−5.27492	−0.532847
$99$	0	0
$100$	0	0
$101$	−9.55505	−0.950763	−0.475382	−	0.879780i	$-0.657690\pi$
$101$	−9.55505	−0.950763	−0.475382	+	0.879780i	$0.657690\pi$
$102$	0	0
$103$	6.56712	0.647078	0.323539	−	0.946215i	$-0.395127\pi$
$103$	6.56712	0.647078	0.323539	+	0.946215i	$0.395127\pi$
$104$	−1.73205	−0.169842
$105$	0	0
$106$	8.27492	0.803731
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	7.82475	0.749475	0.374738	−	0.927131i	$-0.377733\pi$
$109$	7.82475	0.749475	0.374738	+	0.927131i	$0.377733\pi$
$110$	0	0
$111$	0	0
$112$	1.31342	0.124107
$113$	0.725083	0.0682101	0.0341050	−	0.999418i	$-0.489142\pi$
$113$	0.725083	0.0682101	0.0341050	+	0.999418i	$0.489142\pi$
$114$	0	0
$115$	0	0
$116$	−9.13642	−0.848296
$117$	0	0
$118$	8.24163	0.758703
$119$	−1.67451	−0.153502
$120$	0	0
$121$	−4.09967	−0.372697
$122$	7.27492	0.658640
$123$	0	0
$124$	8.54983	0.767798
$125$	0	0
$126$	0	0
$127$	22.0980	1.96088	0.980442	−	0.196810i	$-0.0630581\pi$
$127$	22.0980	1.96088	0.980442	+	0.196810i	$0.0630581\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	17.7967	1.55490	0.777452	−	0.628943i	$-0.216512\pi$
$131$	17.7967	1.55490	0.777452	+	0.628943i	$0.216512\pi$
$132$	0	0
$133$	−2.98793	−0.259086
$134$	−12.1819	−1.05236
$135$	0	0
$136$	−1.27492	−0.109323
$137$	−3.82475	−0.326771	−0.163385	−	0.986562i	$-0.552241\pi$
$137$	−3.82475	−0.326771	−0.163385	+	0.986562i	$0.552241\pi$
$138$	0	0
$139$	−10.2749	−0.871507	−0.435754	−	0.900066i	$-0.643518\pi$
$139$	−10.2749	−0.871507	−0.435754	+	0.900066i	$0.643518\pi$
$140$	0	0
$141$	0	0
$142$	−6.92820	−0.581402
$143$	−4.54983	−0.380476
$144$	0	0
$145$	0	0
$146$	−4.83507	−0.400153
$147$	0	0
$148$	9.97368	0.819831
$149$	−1.37097	−0.112314	−0.0561570	−	0.998422i	$-0.517885\pi$
$149$	−1.37097	−0.112314	−0.0561570	+	0.998422i	$0.517885\pi$
$150$	0	0
$151$	−17.0997	−1.39155	−0.695776	−	0.718259i	$-0.744939\pi$
$151$	−17.0997	−1.39155	−0.695776	+	0.718259i	$0.744939\pi$
$152$	−2.27492	−0.184520
$153$	0	0
$154$	3.45017	0.278022
$155$	0	0
$156$	0	0
$157$	−6.03341	−0.481518	−0.240759	−	0.970585i	$-0.577396\pi$
$157$	−6.03341	−0.481518	−0.240759	+	0.970585i	$0.577396\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.24163	0.649531
$162$	0	0
$163$	24.3638	1.90832	0.954160	−	0.299297i	$-0.0967521\pi$
$163$	24.3638	1.90832	0.954160	+	0.299297i	$0.0967521\pi$
$164$	5.61478	0.438441
$165$	0	0
$166$	4.00000	0.310460
$167$	21.0997	1.63274	0.816371	−	0.577528i	$-0.195983\pi$
$167$	21.0997	1.63274	0.816371	+	0.577528i	$0.195983\pi$
$168$	0	0
$169$	−10.0000	−0.769231
$170$	0	0
$171$	0	0
$172$	−2.62685	−0.200295
$173$	20.0997	1.52815	0.764075	−	0.645128i	$-0.223196\pi$
$173$	20.0997	1.52815	0.764075	+	0.645128i	$0.223196\pi$
$174$	0	0
$175$	0	0
$176$	2.62685	0.198006
$177$	0	0
$178$	−3.04547	−0.228268
$179$	1.31342	0.0981699	0.0490850	−	0.998795i	$-0.484369\pi$
$179$	1.31342	0.0981699	0.0490850	+	0.998795i	$0.484369\pi$
$180$	0	0
$181$	6.54983	0.486845	0.243423	−	0.969920i	$-0.421730\pi$
$181$	6.54983	0.486845	0.243423	+	0.969920i	$0.421730\pi$
$182$	−2.27492	−0.168628
$183$	0	0
$184$	6.27492	0.462593
$185$	0	0
$186$	0	0
$187$	−3.34901	−0.244904
$188$	10.2749	0.749375
$189$	0	0
$190$	0	0
$191$	−14.8087	−1.07152	−0.535762	−	0.844369i	$-0.679975\pi$
$191$	−14.8087	−1.07152	−0.535762	+	0.844369i	$0.679975\pi$
$192$	0	0
$193$	−8.29917	−0.597387	−0.298694	−	0.954349i	$-0.596551\pi$
$193$	−8.29917	−0.597387	−0.298694	+	0.954349i	$0.596551\pi$
$194$	−19.1101	−1.37203
$195$	0	0
$196$	−5.27492	−0.376780
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	−7.45017	−0.528128	−0.264064	−	0.964505i	$-0.585063\pi$
$199$	−7.45017	−0.528128	−0.264064	+	0.964505i	$0.585063\pi$
$200$	0	0
$201$	0	0
$202$	−9.55505	−0.672291
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	6.56712	0.457553
$207$	0	0
$208$	−1.73205	−0.120096
$209$	−5.97586	−0.413359
$210$	0	0
$211$	−9.72508	−0.669502	−0.334751	−	0.942307i	$-0.608652\pi$
$211$	−9.72508	−0.669502	−0.334751	+	0.942307i	$0.608652\pi$
$212$	8.27492	0.568324
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	11.2296	0.762312
$218$	7.82475	0.529959
$219$	0	0
$220$	0	0
$221$	2.20822	0.148541
$222$	0	0
$223$	−16.4833	−1.10380	−0.551900	−	0.833910i	$-0.686097\pi$
$223$	−16.4833	−1.10380	−0.551900	+	0.833910i	$0.686097\pi$
$224$	1.31342	0.0877568
$225$	0	0
$226$	0.725083	0.0482318
$227$	−0.549834	−0.0364938	−0.0182469	−	0.999834i	$-0.505808\pi$
$227$	−0.549834	−0.0364938	−0.0182469	+	0.999834i	$0.505808\pi$
$228$	0	0
$229$	22.3746	1.47855	0.739277	−	0.673401i	$-0.235167\pi$
$229$	22.3746	1.47855	0.739277	+	0.673401i	$0.235167\pi$
$230$	0	0
$231$	0	0
$232$	−9.13642	−0.599836
$233$	−0.175248	−0.0114809	−0.00574045	−	0.999984i	$-0.501827\pi$
$233$	−0.175248	−0.0114809	−0.00574045	+	0.999984i	$0.501827\pi$
$234$	0	0
$235$	0	0
$236$	8.24163	0.536484
$237$	0	0
$238$	−1.67451	−0.108542
$239$	9.55505	0.618065	0.309032	−	0.951051i	$-0.399995\pi$
$239$	9.55505	0.618065	0.309032	+	0.951051i	$0.399995\pi$
$240$	0	0
$241$	13.8248	0.890531	0.445265	−	0.895399i	$-0.353109\pi$
$241$	13.8248	0.890531	0.445265	+	0.895399i	$0.353109\pi$
$242$	−4.09967	−0.263537
$243$	0	0
$244$	7.27492	0.465729
$245$	0	0
$246$	0	0
$247$	3.94027	0.250714
$248$	8.54983	0.542915
$249$	0	0
$250$	0	0
$251$	−2.98793	−0.188597	−0.0942983	−	0.995544i	$-0.530061\pi$
$251$	−2.98793	−0.188597	−0.0942983	+	0.995544i	$0.530061\pi$
$252$	0	0
$253$	16.4833	1.03629
$254$	22.0980	1.38655
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.725083	0.0452294	0.0226147	−	0.999744i	$-0.492801\pi$
$257$	0.725083	0.0452294	0.0226147	+	0.999744i	$0.492801\pi$
$258$	0	0
$259$	13.0997	0.813974
$260$	0	0
$261$	0	0
$262$	17.7967	1.09948
$263$	−19.3746	−1.19469	−0.597344	−	0.801985i	$-0.703777\pi$
$263$	−19.3746	−1.19469	−0.597344	+	0.801985i	$0.703777\pi$
$264$	0	0
$265$	0	0
$266$	−2.98793	−0.183202
$267$	0	0
$268$	−12.1819	−0.744128
$269$	11.6482	0.710202	0.355101	−	0.934828i	$-0.384446\pi$
$269$	11.6482	0.710202	0.355101	+	0.934828i	$0.384446\pi$
$270$	0	0
$271$	−12.5498	−0.762348	−0.381174	−	0.924503i	$-0.624480\pi$
$271$	−12.5498	−0.762348	−0.381174	+	0.924503i	$0.624480\pi$
$272$	−1.27492	−0.0773032
$273$	0	0
$274$	−3.82475	−0.231062
$275$	0	0
$276$	0	0
$277$	−20.4235	−1.22713	−0.613565	−	0.789644i	$-0.710265\pi$
$277$	−20.4235	−1.22713	−0.613565	+	0.789644i	$0.710265\pi$
$278$	−10.2749	−0.616249
$279$	0	0
$280$	0	0
$281$	2.68439	0.160137	0.0800687	−	0.996789i	$-0.474486\pi$
$281$	2.68439	0.160137	0.0800687	+	0.996789i	$0.474486\pi$
$282$	0	0
$283$	9.55505	0.567989	0.283994	−	0.958826i	$-0.408340\pi$
$283$	9.55505	0.567989	0.283994	+	0.958826i	$0.408340\pi$
$284$	−6.92820	−0.411113
$285$	0	0
$286$	−4.54983	−0.269037
$287$	7.37459	0.435308
$288$	0	0
$289$	−15.3746	−0.904387
$290$	0	0
$291$	0	0
$292$	−4.83507	−0.282951
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	0	0
$295$	0	0
$296$	9.97368	0.579708
$297$	0	0
$298$	−1.37097	−0.0794180
$299$	−10.8685	−0.628540
$300$	0	0
$301$	−3.45017	−0.198864
$302$	−17.0997	−0.983975
$303$	0	0
$304$	−2.27492	−0.130475
$305$	0	0
$306$	0	0
$307$	−0.952341	−0.0543530	−0.0271765	−	0.999631i	$-0.508652\pi$
$307$	−0.952341	−0.0543530	−0.0271765	+	0.999631i	$0.508652\pi$
$308$	3.45017	0.196591
$309$	0	0
$310$	0	0
$311$	−21.7370	−1.23259	−0.616295	−	0.787516i	$-0.711367\pi$
$311$	−21.7370	−1.23259	−0.616295	+	0.787516i	$0.711367\pi$
$312$	0	0
$313$	17.8542	1.00918	0.504590	−	0.863359i	$-0.331644\pi$
$313$	17.8542	1.00918	0.504590	+	0.863359i	$0.331644\pi$
$314$	−6.03341	−0.340485
$315$	0	0
$316$	0	0
$317$	22.0997	1.24124	0.620621	−	0.784111i	$-0.286881\pi$
$317$	22.0997	1.24124	0.620621	+	0.784111i	$0.286881\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	8.24163	0.459288
$323$	2.90033	0.161379
$324$	0	0
$325$	0	0
$326$	24.3638	1.34939
$327$	0	0
$328$	5.61478	0.310024
$329$	13.4953	0.744021
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	21.0997	1.15452
$335$	0	0
$336$	0	0
$337$	5.25370	0.286187	0.143094	−	0.989709i	$-0.454295\pi$
$337$	5.25370	0.286187	0.143094	+	0.989709i	$0.454295\pi$
$338$	−10.0000	−0.543928
$339$	0	0
$340$	0	0
$341$	22.4591	1.21623
$342$	0	0
$343$	−16.1222	−0.870515
$344$	−2.62685	−0.141630
$345$	0	0
$346$	20.0997	1.08056
$347$	7.45017	0.399946	0.199973	−	0.979801i	$-0.435915\pi$
$347$	7.45017	0.399946	0.199973	+	0.979801i	$0.435915\pi$
$348$	0	0
$349$	−32.1993	−1.72359	−0.861796	−	0.507256i	$-0.830660\pi$
$349$	−32.1993	−1.72359	−0.861796	+	0.507256i	$0.830660\pi$
$350$	0	0
$351$	0	0
$352$	2.62685	0.140011
$353$	−14.5498	−0.774410	−0.387205	−	0.921994i	$-0.626559\pi$
$353$	−14.5498	−0.774410	−0.387205	+	0.921994i	$0.626559\pi$
$354$	0	0
$355$	0	0
$356$	−3.04547	−0.161410
$357$	0	0
$358$	1.31342	0.0694166
$359$	21.7370	1.14723	0.573616	−	0.819124i	$-0.305540\pi$
$359$	21.7370	1.14723	0.573616	+	0.819124i	$0.305540\pi$
$360$	0	0
$361$	−13.8248	−0.727619
$362$	6.54983	0.344252
$363$	0	0
$364$	−2.27492	−0.119238
$365$	0	0
$366$	0	0
$367$	−2.62685	−0.137120	−0.0685602	−	0.997647i	$-0.521841\pi$
$367$	−2.62685	−0.137120	−0.0685602	+	0.997647i	$0.521841\pi$
$368$	6.27492	0.327103
$369$	0	0
$370$	0	0
$371$	10.8685	0.564263
$372$	0	0
$373$	−22.6893	−1.17481	−0.587404	−	0.809294i	$-0.699850\pi$
$373$	−22.6893	−1.17481	−0.587404	+	0.809294i	$0.699850\pi$
$374$	−3.34901	−0.173173
$375$	0	0
$376$	10.2749	0.529888
$377$	15.8248	0.815016
$378$	0	0
$379$	−27.3746	−1.40614	−0.703069	−	0.711122i	$-0.748188\pi$
$379$	−27.3746	−1.40614	−0.703069	+	0.711122i	$0.748188\pi$
$380$	0	0
$381$	0	0
$382$	−14.8087	−0.757681
$383$	−3.37459	−0.172433	−0.0862166	−	0.996276i	$-0.527478\pi$
$383$	−3.37459	−0.172433	−0.0862166	+	0.996276i	$0.527478\pi$
$384$	0	0
$385$	0	0
$386$	−8.29917	−0.422417
$387$	0	0
$388$	−19.1101	−0.970168
$389$	−33.9189	−1.71975	−0.859877	−	0.510501i	$-0.829460\pi$
$389$	−33.9189	−1.71975	−0.859877	+	0.510501i	$0.829460\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−5.27492	−0.266424
$393$	0	0
$394$	11.0000	0.554172
$395$	0	0
$396$	0	0
$397$	14.3901	0.722219	0.361110	−	0.932523i	$-0.382398\pi$
$397$	14.3901	0.722219	0.361110	+	0.932523i	$0.382398\pi$
$398$	−7.45017	−0.373443
$399$	0	0
$400$	0	0
$401$	−21.6794	−1.08262	−0.541309	−	0.840824i	$-0.682071\pi$
$401$	−21.6794	−1.08262	−0.541309	+	0.840824i	$0.682071\pi$
$402$	0	0
$403$	−14.8087	−0.737676
$404$	−9.55505	−0.475382
$405$	0	0
$406$	−12.0000	−0.595550
$407$	26.1993	1.29865
$408$	0	0
$409$	−16.0997	−0.796077	−0.398039	−	0.917369i	$-0.630309\pi$
$409$	−16.0997	−0.796077	−0.398039	+	0.917369i	$0.630309\pi$
$410$	0	0
$411$	0	0
$412$	6.56712	0.323539
$413$	10.8248	0.532651
$414$	0	0
$415$	0	0
$416$	−1.73205	−0.0849208
$417$	0	0
$418$	−5.97586	−0.292289
$419$	16.4833	0.805260	0.402630	−	0.915363i	$-0.368096\pi$
$419$	16.4833	0.805260	0.402630	+	0.915363i	$0.368096\pi$
$420$	0	0
$421$	9.27492	0.452032	0.226016	−	0.974124i	$-0.427430\pi$
$421$	9.27492	0.452032	0.226016	+	0.974124i	$0.427430\pi$
$422$	−9.72508	−0.473410
$423$	0	0
$424$	8.27492	0.401866
$425$	0	0
$426$	0	0
$427$	9.55505	0.462401
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	−33.9189	−1.63381	−0.816907	−	0.576770i	$-0.804313\pi$
$431$	−33.9189	−1.63381	−0.816907	+	0.576770i	$0.804313\pi$
$432$	0	0
$433$	−10.8109	−0.519540	−0.259770	−	0.965670i	$-0.583647\pi$
$433$	−10.8109	−0.519540	−0.259770	+	0.965670i	$0.583647\pi$
$434$	11.2296	0.539036
$435$	0	0
$436$	7.82475	0.374738
$437$	−14.2749	−0.682862
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	0	0
$441$	0	0
$442$	2.20822	0.105034
$443$	7.45017	0.353968	0.176984	−	0.984214i	$-0.443366\pi$
$443$	7.45017	0.353968	0.176984	+	0.984214i	$0.443366\pi$
$444$	0	0
$445$	0	0
$446$	−16.4833	−0.780505
$447$	0	0
$448$	1.31342	0.0620535
$449$	10.8685	0.512915	0.256458	−	0.966555i	$-0.417445\pi$
$449$	10.8685	0.512915	0.256458	+	0.966555i	$0.417445\pi$
$450$	0	0
$451$	14.7492	0.694511
$452$	0.725083	0.0341050
$453$	0	0
$454$	−0.549834	−0.0258050
$455$	0	0
$456$	0	0
$457$	15.9495	0.746088	0.373044	−	0.927814i	$-0.378314\pi$
$457$	15.9495	0.746088	0.373044	+	0.927814i	$0.378314\pi$
$458$	22.3746	1.04550
$459$	0	0
$460$	0	0
$461$	−6.92820	−0.322679	−0.161339	−	0.986899i	$-0.551581\pi$
$461$	−6.92820	−0.322679	−0.161339	+	0.986899i	$0.551581\pi$
$462$	0	0
$463$	−16.8443	−0.782823	−0.391411	−	0.920216i	$-0.628013\pi$
$463$	−16.8443	−0.782823	−0.391411	+	0.920216i	$0.628013\pi$
$464$	−9.13642	−0.424148
$465$	0	0
$466$	−0.175248	−0.00811822
$467$	−13.6495	−0.631624	−0.315812	−	0.948822i	$-0.602277\pi$
$467$	−13.6495	−0.631624	−0.315812	+	0.948822i	$0.602277\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	8.24163	0.379352
$473$	−6.90033	−0.317278
$474$	0	0
$475$	0	0
$476$	−1.67451	−0.0767509
$477$	0	0
$478$	9.55505	0.437038
$479$	1.67451	0.0765102	0.0382551	−	0.999268i	$-0.487820\pi$
$479$	1.67451	0.0765102	0.0382551	+	0.999268i	$0.487820\pi$
$480$	0	0
$481$	−17.2749	−0.787668
$482$	13.8248	0.629700
$483$	0	0
$484$	−4.09967	−0.186349
$485$	0	0
$486$	0	0
$487$	−9.91613	−0.449343	−0.224671	−	0.974435i	$-0.572131\pi$
$487$	−9.91613	−0.449343	−0.224671	+	0.974435i	$0.572131\pi$
$488$	7.27492	0.329320
$489$	0	0
$490$	0	0
$491$	−18.7490	−0.846131	−0.423066	−	0.906099i	$-0.639046\pi$
$491$	−18.7490	−0.846131	−0.423066	+	0.906099i	$0.639046\pi$
$492$	0	0
$493$	11.6482	0.524608
$494$	3.94027	0.177281
$495$	0	0
$496$	8.54983	0.383899
$497$	−9.09967	−0.408176
$498$	0	0
$499$	14.2749	0.639033	0.319517	−	0.947581i	$-0.396480\pi$
$499$	14.2749	0.639033	0.319517	+	0.947581i	$0.396480\pi$
$500$	0	0
$501$	0	0
$502$	−2.98793	−0.133358
$503$	−33.0997	−1.47584	−0.737921	−	0.674887i	$-0.764192\pi$
$503$	−33.0997	−1.47584	−0.737921	+	0.674887i	$0.764192\pi$
$504$	0	0
$505$	0	0
$506$	16.4833	0.732770
$507$	0	0
$508$	22.0980	0.980442
$509$	−6.92820	−0.307087	−0.153544	−	0.988142i	$-0.549069\pi$
$509$	−6.92820	−0.307087	−0.153544	+	0.988142i	$0.549069\pi$
$510$	0	0
$511$	−6.35050	−0.280929
$512$	1.00000	0.0441942
$513$	0	0
$514$	0.725083	0.0319820
$515$	0	0
$516$	0	0
$517$	26.9906	1.18705
$518$	13.0997	0.575566
$519$	0	0
$520$	0	0
$521$	43.8350	1.92045	0.960223	−	0.279235i	$-0.0900809\pi$
$521$	43.8350	1.92045	0.960223	+	0.279235i	$0.0900809\pi$
$522$	0	0
$523$	10.5074	0.459456	0.229728	−	0.973255i	$-0.426216\pi$
$523$	10.5074	0.459456	0.229728	+	0.973255i	$0.426216\pi$
$524$	17.7967	0.777452
$525$	0	0
$526$	−19.3746	−0.844772
$527$	−10.9003	−0.474826
$528$	0	0
$529$	16.3746	0.711939
$530$	0	0
$531$	0	0
$532$	−2.98793	−0.129543
$533$	−9.72508	−0.421240
$534$	0	0
$535$	0	0
$536$	−12.1819	−0.526178
$537$	0	0
$538$	11.6482	0.502189
$539$	−13.8564	−0.596838
$540$	0	0
$541$	10.7251	0.461107	0.230554	−	0.973060i	$-0.425946\pi$
$541$	10.7251	0.461107	0.230554	+	0.973060i	$0.425946\pi$
$542$	−12.5498	−0.539062
$543$	0	0
$544$	−1.27492	−0.0546616
$545$	0	0
$546$	0	0
$547$	33.9189	1.45027	0.725133	−	0.688609i	$-0.241778\pi$
$547$	33.9189	1.45027	0.725133	+	0.688609i	$0.241778\pi$
$548$	−3.82475	−0.163385
$549$	0	0
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	0	0
$554$	−20.4235	−0.867713
$555$	0	0
$556$	−10.2749	−0.435754
$557$	−45.1993	−1.91516	−0.957579	−	0.288172i	$-0.906953\pi$
$557$	−45.1993	−1.91516	−0.957579	+	0.288172i	$0.906953\pi$
$558$	0	0
$559$	4.54983	0.192437
$560$	0	0
$561$	0	0
$562$	2.68439	0.113234
$563$	29.0997	1.22640	0.613202	−	0.789926i	$-0.289881\pi$
$563$	29.0997	1.22640	0.613202	+	0.789926i	$0.289881\pi$
$564$	0	0
$565$	0	0
$566$	9.55505	0.401629
$567$	0	0
$568$	−6.92820	−0.290701
$569$	−12.9616	−0.543379	−0.271689	−	0.962385i	$-0.587582\pi$
$569$	−12.9616	−0.543379	−0.271689	+	0.962385i	$0.587582\pi$
$570$	0	0
$571$	−17.0997	−0.715599	−0.357799	−	0.933798i	$-0.616473\pi$
$571$	−17.0997	−0.715599	−0.357799	+	0.933798i	$0.616473\pi$
$572$	−4.54983	−0.190238
$573$	0	0
$574$	7.37459	0.307809
$575$	0	0
$576$	0	0
$577$	−29.0838	−1.21077	−0.605387	−	0.795931i	$-0.706982\pi$
$577$	−29.0838	−1.21077	−0.605387	+	0.795931i	$0.706982\pi$
$578$	−15.3746	−0.639498
$579$	0	0
$580$	0	0
$581$	5.25370	0.217960
$582$	0	0
$583$	21.7370	0.900253
$584$	−4.83507	−0.200077
$585$	0	0
$586$	15.0000	0.619644
$587$	−33.0997	−1.36617	−0.683085	−	0.730339i	$-0.739362\pi$
$587$	−33.0997	−1.36617	−0.683085	+	0.730339i	$0.739362\pi$
$588$	0	0
$589$	−19.4502	−0.801430
$590$	0	0
$591$	0	0
$592$	9.97368	0.409916
$593$	20.1752	0.828498	0.414249	−	0.910164i	$-0.364044\pi$
$593$	20.1752	0.828498	0.414249	+	0.910164i	$0.364044\pi$
$594$	0	0
$595$	0	0
$596$	−1.37097	−0.0561570
$597$	0	0
$598$	−10.8685	−0.444445
$599$	6.92820	0.283079	0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820	0.283079	0.141539	+	0.989933i	$0.454795\pi$
$600$	0	0
$601$	0.450166	0.0183626	0.00918132	−	0.999958i	$-0.497077\pi$
$601$	0.450166	0.0183626	0.00918132	+	0.999958i	$0.497077\pi$
$602$	−3.45017	−0.140618
$603$	0	0
$604$	−17.0997	−0.695776
$605$	0	0
$606$	0	0
$607$	−33.9189	−1.37672	−0.688362	−	0.725367i	$-0.741670\pi$
$607$	−33.9189	−1.37672	−0.688362	+	0.725367i	$0.741670\pi$
$608$	−2.27492	−0.0922601
$609$	0	0
$610$	0	0
$611$	−17.7967	−0.719977
$612$	0	0
$613$	−12.5430	−0.506606	−0.253303	−	0.967387i	$-0.581517\pi$
$613$	−12.5430	−0.506606	−0.253303	+	0.967387i	$0.581517\pi$
$614$	−0.952341	−0.0384334
$615$	0	0
$616$	3.45017	0.139011
$617$	7.82475	0.315013	0.157506	−	0.987518i	$-0.449655\pi$
$617$	7.82475	0.315013	0.157506	+	0.987518i	$0.449655\pi$
$618$	0	0
$619$	22.2749	0.895305	0.447652	−	0.894208i	$-0.352260\pi$
$619$	22.2749	0.895305	0.447652	+	0.894208i	$0.352260\pi$
$620$	0	0
$621$	0	0
$622$	−21.7370	−0.871572
$623$	−4.00000	−0.160257
$624$	0	0
$625$	0	0
$626$	17.8542	0.713598
$627$	0	0
$628$	−6.03341	−0.240759
$629$	−12.7156	−0.507005
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	22.0997	0.877690
$635$	0	0
$636$	0	0
$637$	9.13642	0.361998
$638$	−24.0000	−0.950169
$639$	0	0
$640$	0	0
$641$	−30.7583	−1.21488	−0.607440	−	0.794366i	$-0.707803\pi$
$641$	−30.7583	−1.21488	−0.607440	+	0.794366i	$0.707803\pi$
$642$	0	0
$643$	−0.952341	−0.0375567	−0.0187783	−	0.999824i	$-0.505978\pi$
$643$	−0.952341	−0.0375567	−0.0187783	+	0.999824i	$0.505978\pi$
$644$	8.24163	0.324766
$645$	0	0
$646$	2.90033	0.114112
$647$	21.0997	0.829514	0.414757	−	0.909932i	$-0.363867\pi$
$647$	21.0997	0.829514	0.414757	+	0.909932i	$0.363867\pi$
$648$	0	0
$649$	21.6495	0.849817
$650$	0	0
$651$	0	0
$652$	24.3638	0.954160
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	0	0
$656$	5.61478	0.219220
$657$	0	0
$658$	13.4953	0.526102
$659$	−4.66244	−0.181623	−0.0908114	−	0.995868i	$-0.528946\pi$
$659$	−4.66244	−0.181623	−0.0908114	+	0.995868i	$0.528946\pi$
$660$	0	0
$661$	31.2749	1.21645	0.608227	−	0.793763i	$-0.291881\pi$
$661$	31.2749	1.21645	0.608227	+	0.793763i	$0.291881\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	−57.3303	−2.21984
$668$	21.0997	0.816371
$669$	0	0
$670$	0	0
$671$	19.1101	0.737737
$672$	0	0
$673$	−5.67232	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$673$	−5.67232	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$674$	5.25370	0.202365
$675$	0	0
$676$	−10.0000	−0.384615
$677$	−23.7251	−0.911829	−0.455915	−	0.890024i	$-0.650688\pi$
$677$	−23.7251	−0.911829	−0.455915	+	0.890024i	$0.650688\pi$
$678$	0	0
$679$	−25.0997	−0.963237
$680$	0	0
$681$	0	0
$682$	22.4591	0.860004
$683$	26.7492	1.02353	0.511764	−	0.859126i	$-0.328992\pi$
$683$	26.7492	1.02353	0.511764	+	0.859126i	$0.328992\pi$
$684$	0	0
$685$	0	0
$686$	−16.1222	−0.615547
$687$	0	0
$688$	−2.62685	−0.100148
$689$	−14.3326	−0.546028
$690$	0	0
$691$	−23.9244	−0.910128	−0.455064	−	0.890459i	$-0.650384\pi$
$691$	−23.9244	−0.910128	−0.455064	+	0.890459i	$0.650384\pi$
$692$	20.0997	0.764075
$693$	0	0
$694$	7.45017	0.282804
$695$	0	0
$696$	0	0
$697$	−7.15838	−0.271143
$698$	−32.1993	−1.21876
$699$	0	0
$700$	0	0
$701$	18.6915	0.705967	0.352984	−	0.935629i	$-0.385167\pi$
$701$	18.6915	0.705967	0.352984	+	0.935629i	$0.385167\pi$
$702$	0	0
$703$	−22.6893	−0.855743
$704$	2.62685	0.0990031
$705$	0	0
$706$	−14.5498	−0.547590
$707$	−12.5498	−0.471985
$708$	0	0
$709$	−3.82475	−0.143642	−0.0718208	−	0.997418i	$-0.522881\pi$
$709$	−3.82475	−0.143642	−0.0718208	+	0.997418i	$0.522881\pi$
$710$	0	0
$711$	0	0
$712$	−3.04547	−0.114134
$713$	53.6495	2.00919
$714$	0	0
$715$	0	0
$716$	1.31342	0.0490850
$717$	0	0
$718$	21.7370	0.811216
$719$	8.60271	0.320827	0.160413	−	0.987050i	$-0.448717\pi$
$719$	8.60271	0.320827	0.160413	+	0.987050i	$0.448717\pi$
$720$	0	0
$721$	8.62541	0.321227
$722$	−13.8248	−0.514504
$723$	0	0
$724$	6.54983	0.243423
$725$	0	0
$726$	0	0
$727$	−11.8208	−0.438410	−0.219205	−	0.975679i	$-0.570346\pi$
$727$	−11.8208	−0.438410	−0.219205	+	0.975679i	$0.570346\pi$
$728$	−2.27492	−0.0843140
$729$	0	0
$730$	0	0
$731$	3.34901	0.123868
$732$	0	0
$733$	47.0531	1.73795	0.868973	−	0.494860i	$-0.164781\pi$
$733$	47.0531	1.73795	0.868973	+	0.494860i	$0.164781\pi$
$734$	−2.62685	−0.0969587
$735$	0	0
$736$	6.27492	0.231297
$737$	−32.0000	−1.17874
$738$	0	0
$739$	−10.9003	−0.400975	−0.200488	−	0.979696i	$-0.564253\pi$
$739$	−10.9003	−0.400975	−0.200488	+	0.979696i	$0.564253\pi$
$740$	0	0
$741$	0	0
$742$	10.8685	0.398994
$743$	−45.0997	−1.65455	−0.827273	−	0.561800i	$-0.810109\pi$
$743$	−45.0997	−1.65455	−0.827273	+	0.561800i	$0.810109\pi$
$744$	0	0
$745$	0	0
$746$	−22.6893	−0.830714
$747$	0	0
$748$	−3.34901	−0.122452
$749$	15.7611	0.575898
$750$	0	0
$751$	−32.5498	−1.18776	−0.593880	−	0.804554i	$-0.702405\pi$
$751$	−32.5498	−1.18776	−0.593880	+	0.804554i	$0.702405\pi$
$752$	10.2749	0.374688
$753$	0	0
$754$	15.8248	0.576303
$755$	0	0
$756$	0	0
$757$	−26.3994	−0.959502	−0.479751	−	0.877405i	$-0.659273\pi$
$757$	−26.3994	−0.959502	−0.479751	+	0.877405i	$0.659273\pi$
$758$	−27.3746	−0.994290
$759$	0	0
$760$	0	0
$761$	17.4931	0.634126	0.317063	−	0.948405i	$-0.397303\pi$
$761$	17.4931	0.634126	0.317063	+	0.948405i	$0.397303\pi$
$762$	0	0
$763$	10.2772	0.372060
$764$	−14.8087	−0.535762
$765$	0	0
$766$	−3.37459	−0.121929
$767$	−14.2749	−0.515437
$768$	0	0
$769$	35.2749	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$769$	35.2749	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$770$	0	0
$771$	0	0
$772$	−8.29917	−0.298694
$773$	−45.8248	−1.64820	−0.824101	−	0.566443i	$-0.808319\pi$
$773$	−45.8248	−1.64820	−0.824101	+	0.566443i	$0.808319\pi$
$774$	0	0
$775$	0	0
$776$	−19.1101	−0.686013
$777$	0	0
$778$	−33.9189	−1.21605
$779$	−12.7732	−0.457646
$780$	0	0
$781$	−18.1993	−0.651224
$782$	−8.00000	−0.286079
$783$	0	0
$784$	−5.27492	−0.188390
$785$	0	0
$786$	0	0
$787$	3.34901	0.119379	0.0596897	−	0.998217i	$-0.480989\pi$
$787$	3.34901	0.119379	0.0596897	+	0.998217i	$0.480989\pi$
$788$	11.0000	0.391859
$789$	0	0
$790$	0	0
$791$	0.952341	0.0338614
$792$	0	0
$793$	−12.6005	−0.447458
$794$	14.3901	0.510686
$795$	0	0
$796$	−7.45017	−0.264064
$797$	−28.3746	−1.00508	−0.502540	−	0.864554i	$-0.667601\pi$
$797$	−28.3746	−1.00508	−0.502540	+	0.864554i	$0.667601\pi$
$798$	0	0
$799$	−13.0997	−0.463433
$800$	0	0
$801$	0	0
$802$	−21.6794	−0.765526
$803$	−12.7010	−0.448208
$804$	0	0
$805$	0	0
$806$	−14.8087	−0.521616
$807$	0	0
$808$	−9.55505	−0.336146
$809$	49.6224	1.74463	0.872315	−	0.488944i	$-0.162618\pi$
$809$	49.6224	1.74463	0.872315	+	0.488944i	$0.162618\pi$
$810$	0	0
$811$	6.90033	0.242303	0.121152	−	0.992634i	$-0.461341\pi$
$811$	6.90033	0.242303	0.121152	+	0.992634i	$0.461341\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	26.1993	0.918286
$815$	0	0
$816$	0	0
$817$	5.97586	0.209069
$818$	−16.0997	−0.562912
$819$	0	0
$820$	0	0
$821$	−21.2032	−0.739998	−0.369999	−	0.929032i	$-0.620642\pi$
$821$	−21.2032	−0.739998	−0.369999	+	0.929032i	$0.620642\pi$
$822$	0	0
$823$	42.5216	1.48221	0.741104	−	0.671390i	$-0.234302\pi$
$823$	42.5216	1.48221	0.741104	+	0.671390i	$0.234302\pi$
$824$	6.56712	0.228776
$825$	0	0
$826$	10.8248	0.376641
$827$	−27.4502	−0.954536	−0.477268	−	0.878758i	$-0.658373\pi$
$827$	−27.4502	−0.954536	−0.477268	+	0.878758i	$0.658373\pi$
$828$	0	0
$829$	46.5498	1.61674	0.808371	−	0.588673i	$-0.200349\pi$
$829$	46.5498	1.61674	0.808371	+	0.588673i	$0.200349\pi$
$830$	0	0
$831$	0	0
$832$	−1.73205	−0.0600481
$833$	6.72508	0.233010
$834$	0	0
$835$	0	0
$836$	−5.97586	−0.206680
$837$	0	0
$838$	16.4833	0.569405
$839$	48.7276	1.68226	0.841132	−	0.540830i	$-0.181890\pi$
$839$	48.7276	1.68226	0.841132	+	0.540830i	$0.181890\pi$
$840$	0	0
$841$	54.4743	1.87842
$842$	9.27492	0.319635
$843$	0	0
$844$	−9.72508	−0.334751
$845$	0	0
$846$	0	0
$847$	−5.38460	−0.185017
$848$	8.27492	0.284162
$849$	0	0
$850$	0	0
$851$	62.5840	2.14535
$852$	0	0
$853$	−10.2772	−0.351885	−0.175943	−	0.984400i	$-0.556297\pi$
$853$	−10.2772	−0.351885	−0.175943	+	0.984400i	$0.556297\pi$
$854$	9.55505	0.326967
$855$	0	0
$856$	12.0000	0.410152
$857$	−33.8248	−1.15543	−0.577716	−	0.816238i	$-0.696056\pi$
$857$	−33.8248	−1.15543	−0.577716	+	0.816238i	$0.696056\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	−33.9189	−1.15528
$863$	9.72508	0.331046	0.165523	−	0.986206i	$-0.447069\pi$
$863$	9.72508	0.331046	0.165523	+	0.986206i	$0.447069\pi$
$864$	0	0
$865$	0	0
$866$	−10.8109	−0.367370
$867$	0	0
$868$	11.2296	0.381156
$869$	0	0
$870$	0	0
$871$	21.0997	0.714935
$872$	7.82475	0.264980
$873$	0	0
$874$	−14.2749	−0.482856
$875$	0	0
$876$	0	0
$877$	10.3348	0.348980	0.174490	−	0.984659i	$-0.444172\pi$
$877$	10.3348	0.348980	0.174490	+	0.984659i	$0.444172\pi$
$878$	−36.0000	−1.21494
$879$	0	0
$880$	0	0
$881$	15.7611	0.531005	0.265502	−	0.964110i	$-0.414462\pi$
$881$	15.7611	0.531005	0.265502	+	0.964110i	$0.414462\pi$
$882$	0	0
$883$	0.952341	0.0320488	0.0160244	−	0.999872i	$-0.494899\pi$
$883$	0.952341	0.0320488	0.0160244	+	0.999872i	$0.494899\pi$
$884$	2.20822	0.0742705
$885$	0	0
$886$	7.45017	0.250293
$887$	−18.8248	−0.632073	−0.316037	−	0.948747i	$-0.602352\pi$
$887$	−18.8248	−0.632073	−0.316037	+	0.948747i	$0.602352\pi$
$888$	0	0
$889$	29.0241	0.973437
$890$	0	0
$891$	0	0
$892$	−16.4833	−0.551900
$893$	−23.3746	−0.782201
$894$	0	0
$895$	0	0
$896$	1.31342	0.0438784
$897$	0	0
$898$	10.8685	0.362686
$899$	−78.1149	−2.60528
$900$	0	0
$901$	−10.5498	−0.351466
$902$	14.7492	0.491094
$903$	0	0
$904$	0.725083	0.0241159
$905$	0	0
$906$	0	0
$907$	41.5692	1.38028	0.690142	−	0.723674i	$-0.257548\pi$
$907$	41.5692	1.38028	0.690142	+	0.723674i	$0.257548\pi$
$908$	−0.549834	−0.0182469
$909$	0	0
$910$	0	0
$911$	5.97586	0.197989	0.0989946	−	0.995088i	$-0.468437\pi$
$911$	5.97586	0.197989	0.0989946	+	0.995088i	$0.468437\pi$
$912$	0	0
$913$	10.5074	0.347744
$914$	15.9495	0.527564
$915$	0	0
$916$	22.3746	0.739277
$917$	23.3746	0.771897
$918$	0	0
$919$	33.0997	1.09186	0.545929	−	0.837832i	$-0.316177\pi$
$919$	33.0997	1.09186	0.545929	+	0.837832i	$0.316177\pi$
$920$	0	0
$921$	0	0
$922$	−6.92820	−0.228168
$923$	12.0000	0.394985
$924$	0	0
$925$	0	0
$926$	−16.8443	−0.553539
$927$	0	0
$928$	−9.13642	−0.299918
$929$	−13.5529	−0.444655	−0.222328	−	0.974972i	$-0.571365\pi$
$929$	−13.5529	−0.444655	−0.222328	+	0.974972i	$0.571365\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−0.175248	−0.00574045
$933$	0	0
$934$	−13.6495	−0.446625
$935$	0	0
$936$	0	0
$937$	−5.55724	−0.181547	−0.0907735	−	0.995872i	$-0.528934\pi$
$937$	−5.55724	−0.181547	−0.0907735	+	0.995872i	$0.528934\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	0	0
$941$	−10.9260	−0.356178	−0.178089	−	0.984014i	$-0.556992\pi$
$941$	−10.9260	−0.356178	−0.178089	+	0.984014i	$0.556992\pi$
$942$	0	0
$943$	35.2323	1.14732
$944$	8.24163	0.268242
$945$	0	0
$946$	−6.90033	−0.224349
$947$	−16.5498	−0.537797	−0.268899	−	0.963168i	$-0.586660\pi$
$947$	−16.5498	−0.537797	−0.268899	+	0.963168i	$0.586660\pi$
$948$	0	0
$949$	8.37459	0.271851
$950$	0	0
$951$	0	0
$952$	−1.67451	−0.0542711
$953$	−39.2749	−1.27224	−0.636120	−	0.771590i	$-0.719462\pi$
$953$	−39.2749	−1.27224	−0.636120	+	0.771590i	$0.719462\pi$
$954$	0	0
$955$	0	0
$956$	9.55505	0.309032
$957$	0	0
$958$	1.67451	0.0541009
$959$	−5.02352	−0.162218
$960$	0	0
$961$	42.0997	1.35805
$962$	−17.2749	−0.556966
$963$	0	0
$964$	13.8248	0.445265
$965$	0	0
$966$	0	0
$967$	9.55505	0.307270	0.153635	−	0.988128i	$-0.450902\pi$
$967$	9.55505	0.307270	0.153635	+	0.988128i	$0.450902\pi$
$968$	−4.09967	−0.131768
$969$	0	0
$970$	0	0
$971$	−37.6289	−1.20757	−0.603785	−	0.797147i	$-0.706342\pi$
$971$	−37.6289	−1.20757	−0.603785	+	0.797147i	$0.706342\pi$
$972$	0	0
$973$	−13.4953	−0.432640
$974$	−9.91613	−0.317733
$975$	0	0
$976$	7.27492	0.232864
$977$	0.199338	0.00637738	0.00318869	−	0.999995i	$-0.498985\pi$
$977$	0.199338	0.00637738	0.00318869	+	0.999995i	$0.498985\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	−18.7490	−0.598305
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	0	0
$985$	0	0
$986$	11.6482	0.370954
$987$	0	0
$988$	3.94027	0.125357
$989$	−16.4833	−0.524137
$990$	0	0
$991$	−47.2990	−1.50250	−0.751251	−	0.660016i	$-0.770549\pi$
$991$	−47.2990	−1.50250	−0.751251	+	0.660016i	$0.770549\pi$
$992$	8.54983	0.271458
$993$	0	0
$994$	−9.09967	−0.288624
$995$	0	0
$996$	0	0
$997$	46.1583	1.46185	0.730924	−	0.682459i	$-0.239089\pi$
$997$	46.1583	1.46185	0.730924	+	0.682459i	$0.239089\pi$
$998$	14.2749	0.451865
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.cb.1.3		4
3.2	odd	2		4050.2.a.ca.1.3		4
5.2	odd	4		810.2.c.g.649.5	yes	8
5.3	odd	4		810.2.c.g.649.1	✓	8
5.4	even	2		4050.2.a.ca.1.2		4
15.2	even	4		810.2.c.g.649.4	yes	8
15.8	even	4		810.2.c.g.649.8	yes	8
15.14	odd	2	inner	4050.2.a.cb.1.2		4
45.2	even	12		810.2.i.g.109.1		8
45.7	odd	12		810.2.i.g.109.4		8
45.13	odd	12		810.2.i.g.379.4		8
45.22	odd	12		810.2.i.i.379.1		8
45.23	even	12		810.2.i.g.379.1		8
45.32	even	12		810.2.i.i.379.4		8
45.38	even	12		810.2.i.i.109.4		8
45.43	odd	12		810.2.i.i.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.c.g.649.1	✓	8	5.3	odd	4
810.2.c.g.649.4	yes	8	15.2	even	4
810.2.c.g.649.5	yes	8	5.2	odd	4
810.2.c.g.649.8	yes	8	15.8	even	4
810.2.i.g.109.1		8	45.2	even	12
810.2.i.g.109.4		8	45.7	odd	12
810.2.i.g.379.1		8	45.23	even	12
810.2.i.g.379.4		8	45.13	odd	12
810.2.i.i.109.1		8	45.43	odd	12
810.2.i.i.109.4		8	45.38	even	12
810.2.i.i.379.1		8	45.22	odd	12
810.2.i.i.379.4		8	45.32	even	12
4050.2.a.ca.1.2		4	5.4	even	2
4050.2.a.ca.1.3		4	3.2	odd	2
4050.2.a.cb.1.2		4	15.14	odd	2	inner
4050.2.a.cb.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.31342 q^{7} +1.00000 q^{8} +2.62685 q^{11} -1.73205 q^{13} +1.31342 q^{14} +1.00000 q^{16} -1.27492 q^{17} -2.27492 q^{19} +2.62685 q^{22} +6.27492 q^{23} -1.73205 q^{26} +1.31342 q^{28} -9.13642 q^{29} +8.54983 q^{31} +1.00000 q^{32} -1.27492 q^{34} +9.97368 q^{37} -2.27492 q^{38} +5.61478 q^{41} -2.62685 q^{43} +2.62685 q^{44} +6.27492 q^{46} +10.2749 q^{47} -5.27492 q^{49} -1.73205 q^{52} +8.27492 q^{53} +1.31342 q^{56} -9.13642 q^{58} +8.24163 q^{59} +7.27492 q^{61} +8.54983 q^{62} +1.00000 q^{64} -12.1819 q^{67} -1.27492 q^{68} -6.92820 q^{71} -4.83507 q^{73} +9.97368 q^{74} -2.27492 q^{76} +3.45017 q^{77} +5.61478 q^{82} +4.00000 q^{83} -2.62685 q^{86} +2.62685 q^{88} -3.04547 q^{89} -2.27492 q^{91} +6.27492 q^{92} +10.2749 q^{94} -19.1101 q^{97} -5.27492 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 10 q^{17} + 6 q^{19} + 10 q^{23} + 4 q^{31} + 4 q^{32} + 10 q^{34} + 6 q^{38} + 10 q^{46} + 26 q^{47} - 6 q^{49} + 18 q^{53} + 14 q^{61} + 4 q^{62} + 4 q^{64}+ \cdots - 6 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 + 10 * q^17 + 6 * q^19 + 10 * q^23 + 4 * q^31 + 4 * q^32 + 10 * q^34 + 6 * q^38 + 10 * q^46 + 26 * q^47 - 6 * q^49 + 18 * q^53 + 14 * q^61 + 4 * q^62 + 4 * q^64 + 10 * q^68 + 6 * q^76 + 44 * q^77 + 16 * q^83 + 6 * q^91 + 10 * q^92 + 26 * q^94 - 6 * q^98

Introduction

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