LMFDB - Embedded newform 9025.2.a.cf.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,6,9,6,0,12,0,6,6,0,0,18,9,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 $ x^9 - 3x^8 - 6x^7 + 16x^6 + 12x^5 - 27x^4 - 8x^3 + 15*x^2 - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.256961$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.25696 q^{2} +3.01225 q^{3} -0.420048 q^{4} +3.78628 q^{6} -3.72392 q^{7} -3.04191 q^{8} +6.07366 q^{9} -3.35588 q^{11} -1.26529 q^{12} +4.84131 q^{13} -4.68082 q^{14} -2.98346 q^{16} +2.67526 q^{17} +7.63436 q^{18} -11.2174 q^{21} -4.21821 q^{22} -1.87703 q^{23} -9.16299 q^{24} +6.08534 q^{26} +9.25865 q^{27} +1.56423 q^{28} +5.25131 q^{29} +3.11891 q^{31} +2.33372 q^{32} -10.1087 q^{33} +3.36269 q^{34} -2.55123 q^{36} +0.992927 q^{37} +14.5832 q^{39} -0.416383 q^{41} -14.0998 q^{42} +7.21877 q^{43} +1.40963 q^{44} -2.35935 q^{46} -2.24086 q^{47} -8.98694 q^{48} +6.86757 q^{49} +8.05854 q^{51} -2.03358 q^{52} +0.260252 q^{53} +11.6378 q^{54} +11.3278 q^{56} +6.60069 q^{58} +5.18883 q^{59} +0.768938 q^{61} +3.92035 q^{62} -22.6178 q^{63} +8.90032 q^{64} -12.7063 q^{66} +10.7775 q^{67} -1.12374 q^{68} -5.65409 q^{69} -2.00146 q^{71} -18.4755 q^{72} -4.56262 q^{73} +1.24807 q^{74} +12.4970 q^{77} +18.3306 q^{78} +12.0521 q^{79} +9.66841 q^{81} -0.523377 q^{82} +13.5711 q^{83} +4.71184 q^{84} +9.07371 q^{86} +15.8183 q^{87} +10.2083 q^{88} +7.66941 q^{89} -18.0286 q^{91} +0.788443 q^{92} +9.39494 q^{93} -2.81668 q^{94} +7.02975 q^{96} -7.22894 q^{97} +8.63226 q^{98} -20.3825 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 12 q^{6} + 6 q^{8} + 6 q^{9} + 18 q^{12} + 9 q^{13} + 12 q^{16} + 9 q^{17} + 24 q^{18} - 12 q^{21} + 24 q^{22} + 12 q^{23} + 3 q^{24} - 3 q^{26} + 24 q^{27} + 15 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) $ 9 * q + 6 * q^2 + 9 * q^3 + 6 * q^4 + 12 * q^6 + 6 * q^8 + 6 * q^9 + 18 * q^12 + 9 * q^13 + 12 * q^16 + 9 * q^17 + 24 * q^18 - 12 * q^21 + 24 * q^22 + 12 * q^23 + 3 * q^24 - 3 * q^26 + 24 * q^27 + 15 * q^28 - 9 * q^29 - 18 * q^31 + 3 * q^32 - 9 * q^33 + 24 * q^34 + 18 * q^36 + 18 * q^37 + 18 * q^39 - 6 * q^41 + 12 * q^43 + 48 * q^44 + 9 * q^46 - 15 * q^47 - 21 * q^48 - 9 * q^49 + 6 * q^51 + 33 * q^52 + 15 * q^53 + 63 * q^54 - 6 * q^58 - 21 * q^59 - 12 * q^61 + 36 * q^62 - 21 * q^63 - 36 * q^64 + 3 * q^66 + 60 * q^67 + 51 * q^68 + 15 * q^69 + 18 * q^71 - 27 * q^73 + 27 * q^74 + 30 * q^77 - 6 * q^78 - 15 * q^79 + 33 * q^81 - 24 * q^82 + 48 * q^84 + 39 * q^86 - 15 * q^87 + 27 * q^88 + 39 * q^89 - 21 * q^91 + 6 * q^92 - 15 * q^93 - 15 * q^94 - 33 * q^96 + 15 * q^97 - 15 * q^98 - 36 * q^99

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 $

$2$	1.25696	0.888806	0.444403	−	0.895827i	$-0.353416\pi$
$2$	1.25696	0.888806	0.444403	+	0.895827i	$0.353416\pi$
$3$	3.01225	1.73912	0.869562	−	0.493823i	$-0.164401\pi$
$3$	3.01225	1.73912	0.869562	+	0.493823i	$0.164401\pi$
$4$	−0.420048	−0.210024
$5$	0	0
$5$	0	0
$6$	3.78628	1.54574
$7$	−3.72392	−1.40751	−0.703754	−	0.710443i	$-0.748494\pi$
$7$	−3.72392	−1.40751	−0.703754	+	0.710443i	$0.748494\pi$
$8$	−3.04191	−1.07548
$9$	6.07366	2.02455
$10$	0	0
$11$	−3.35588	−1.01183	−0.505917	−	0.862582i	$-0.668846\pi$
$11$	−3.35588	−1.01183	−0.505917	+	0.862582i	$0.668846\pi$
$12$	−1.26529	−0.365258
$13$	4.84131	1.34274	0.671369	−	0.741124i	$-0.265707\pi$
$13$	4.84131	1.34274	0.671369	+	0.741124i	$0.265707\pi$
$14$	−4.68082	−1.25100
$15$	0	0
$16$	−2.98346	−0.745866
$17$	2.67526	0.648845	0.324422	−	0.945912i	$-0.394830\pi$
$17$	2.67526	0.648845	0.324422	+	0.945912i	$0.394830\pi$
$18$	7.63436	1.79944
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−11.2174	−2.44783
$22$	−4.21821	−0.899325
$23$	−1.87703	−0.391388	−0.195694	−	0.980665i	$-0.562696\pi$
$23$	−1.87703	−0.391388	−0.195694	+	0.980665i	$0.562696\pi$
$24$	−9.16299	−1.87039
$25$	0	0
$26$	6.08534	1.19343
$27$	9.25865	1.78183
$28$	1.56423	0.295611
$29$	5.25131	0.975143	0.487572	−	0.873083i	$-0.337883\pi$
$29$	5.25131	0.975143	0.487572	+	0.873083i	$0.337883\pi$
$30$	0	0
$31$	3.11891	0.560173	0.280086	−	0.959975i	$-0.409637\pi$
$31$	3.11891	0.560173	0.280086	+	0.959975i	$0.409637\pi$
$32$	2.33372	0.412547
$33$	−10.1087	−1.75971
$34$	3.36269	0.576697
$35$	0	0
$36$	−2.55123	−0.425206
$37$	0.992927	0.163236	0.0816181	−	0.996664i	$-0.473991\pi$
$37$	0.992927	0.163236	0.0816181	+	0.996664i	$0.473991\pi$
$38$	0	0
$39$	14.5832	2.33519
$40$	0	0
$41$	−0.416383	−0.0650281	−0.0325140	−	0.999471i	$-0.510351\pi$
$41$	−0.416383	−0.0650281	−0.0325140	+	0.999471i	$0.510351\pi$
$42$	−14.0998	−2.17565
$43$	7.21877	1.10085	0.550426	−	0.834884i	$-0.314465\pi$
$43$	7.21877	1.10085	0.550426	+	0.834884i	$0.314465\pi$
$44$	1.40963	0.212510
$45$	0	0
$46$	−2.35935	−0.347868
$47$	−2.24086	−0.326864	−0.163432	−	0.986555i	$-0.552256\pi$
$47$	−2.24086	−0.326864	−0.163432	+	0.986555i	$0.552256\pi$
$48$	−8.98694	−1.29715
$49$	6.86757	0.981081
$50$	0	0
$51$	8.05854	1.12842
$52$	−2.03358	−0.282007
$53$	0.260252	0.0357483	0.0178742	−	0.999840i	$-0.494310\pi$
$53$	0.260252	0.0357483	0.0178742	+	0.999840i	$0.494310\pi$
$54$	11.6378	1.58370
$55$	0	0
$56$	11.3278	1.51374
$57$	0	0
$58$	6.60069	0.866713
$59$	5.18883	0.675528	0.337764	−	0.941231i	$-0.390329\pi$
$59$	5.18883	0.675528	0.337764	+	0.941231i	$0.390329\pi$
$60$	0	0
$61$	0.768938	0.0984524	0.0492262	−	0.998788i	$-0.484324\pi$
$61$	0.768938	0.0984524	0.0492262	+	0.998788i	$0.484324\pi$
$62$	3.92035	0.497885
$63$	−22.6178	−2.84958
$64$	8.90032	1.11254
$65$	0	0
$66$	−12.7063	−1.56404
$67$	10.7775	1.31668	0.658339	−	0.752722i	$-0.271260\pi$
$67$	10.7775	1.31668	0.658339	+	0.752722i	$0.271260\pi$
$68$	−1.12374	−0.136273
$69$	−5.65409	−0.680672
$70$	0	0
$71$	−2.00146	−0.237529	−0.118765	−	0.992922i	$-0.537893\pi$
$71$	−2.00146	−0.237529	−0.118765	+	0.992922i	$0.537893\pi$
$72$	−18.4755	−2.17736
$73$	−4.56262	−0.534015	−0.267007	−	0.963694i	$-0.586035\pi$
$73$	−4.56262	−0.534015	−0.267007	+	0.963694i	$0.586035\pi$
$74$	1.24807	0.145085
$75$	0	0
$76$	0	0
$77$	12.4970	1.42417
$78$	18.3306	2.07553
$79$	12.0521	1.35596	0.677981	−	0.735079i	$-0.262855\pi$
$79$	12.0521	1.35596	0.677981	+	0.735079i	$0.262855\pi$
$80$	0	0
$81$	9.66841	1.07427
$82$	−0.523377	−0.0577973
$83$	13.5711	1.48962	0.744809	−	0.667278i	$-0.232541\pi$
$83$	13.5711	1.48962	0.744809	+	0.667278i	$0.232541\pi$
$84$	4.71184	0.514104
$85$	0	0
$86$	9.07371	0.978443
$87$	15.8183	1.69590
$88$	10.2083	1.08820
$89$	7.66941	0.812956	0.406478	−	0.913661i	$-0.366757\pi$
$89$	7.66941	0.812956	0.406478	+	0.913661i	$0.366757\pi$
$90$	0	0
$91$	−18.0286	−1.88991
$92$	0.788443	0.0822009
$93$	9.39494	0.974210
$94$	−2.81668	−0.290518
$95$	0	0
$96$	7.02975	0.717471
$97$	−7.22894	−0.733988	−0.366994	−	0.930223i	$-0.619613\pi$
$97$	−7.22894	−0.733988	−0.366994	+	0.930223i	$0.619613\pi$
$98$	8.63226	0.871990
$99$	−20.3825	−2.04852
$100$	0	0
$101$	4.49184	0.446955	0.223477	−	0.974709i	$-0.428259\pi$
$101$	4.49184	0.446955	0.223477	+	0.974709i	$0.428259\pi$
$102$	10.1293	1.00295
$103$	10.7445	1.05869	0.529345	−	0.848407i	$-0.322438\pi$
$103$	10.7445	1.05869	0.529345	+	0.848407i	$0.322438\pi$
$104$	−14.7268	−1.44408
$105$	0	0
$106$	0.327127	0.0317733
$107$	1.37863	0.133277	0.0666386	−	0.997777i	$-0.478773\pi$
$107$	1.37863	0.133277	0.0666386	+	0.997777i	$0.478773\pi$
$108$	−3.88908	−0.374227
$109$	−9.49878	−0.909818	−0.454909	−	0.890538i	$-0.650328\pi$
$109$	−9.49878	−0.909818	−0.454909	+	0.890538i	$0.650328\pi$
$110$	0	0
$111$	2.99095	0.283888
$112$	11.1102	1.04981
$113$	−17.0436	−1.60333	−0.801666	−	0.597772i	$-0.796053\pi$
$113$	−17.0436	−1.60333	−0.801666	+	0.597772i	$0.796053\pi$
$114$	0	0
$115$	0	0
$116$	−2.20580	−0.204804
$117$	29.4045	2.71844
$118$	6.52216	0.600413
$119$	−9.96243	−0.913255
$120$	0	0
$121$	0.261910	0.0238100
$122$	0.966525	0.0875051
$123$	−1.25425	−0.113092
$124$	−1.31009	−0.117650
$125$	0	0
$126$	−28.4297	−2.53272
$127$	19.8598	1.76227	0.881137	−	0.472860i	$-0.156779\pi$
$127$	19.8598	1.76227	0.881137	+	0.472860i	$0.156779\pi$
$128$	6.51992	0.576285
$129$	21.7447	1.91452
$130$	0	0
$131$	5.42317	0.473824	0.236912	−	0.971531i	$-0.423865\pi$
$131$	5.42317	0.473824	0.236912	+	0.971531i	$0.423865\pi$
$132$	4.24616	0.369581
$133$	0	0
$134$	13.5469	1.17027
$135$	0	0
$136$	−8.13788	−0.697817
$137$	18.5615	1.58581	0.792906	−	0.609344i	$-0.208567\pi$
$137$	18.5615	1.58581	0.792906	+	0.609344i	$0.208567\pi$
$138$	−7.10697	−0.604985
$139$	−10.5127	−0.891675	−0.445838	−	0.895114i	$-0.647094\pi$
$139$	−10.5127	−0.891675	−0.445838	+	0.895114i	$0.647094\pi$
$140$	0	0
$141$	−6.75005	−0.568457
$142$	−2.51575	−0.211117
$143$	−16.2468	−1.35863
$144$	−18.1205	−1.51005
$145$	0	0
$146$	−5.73504	−0.474635
$147$	20.6868	1.70622
$148$	−0.417077	−0.0342835
$149$	15.1409	1.24039	0.620196	−	0.784447i	$-0.287053\pi$
$149$	15.1409	1.24039	0.620196	+	0.784447i	$0.287053\pi$
$150$	0	0
$151$	−11.3432	−0.923095	−0.461548	−	0.887115i	$-0.652706\pi$
$151$	−11.3432	−0.923095	−0.461548	+	0.887115i	$0.652706\pi$
$152$	0	0
$153$	16.2486	1.31362
$154$	15.7083	1.26581
$155$	0	0
$156$	−6.12567	−0.490446
$157$	4.21528	0.336416	0.168208	−	0.985752i	$-0.446202\pi$
$157$	4.21528	0.336416	0.168208	+	0.985752i	$0.446202\pi$
$158$	15.1490	1.20519
$159$	0.783944	0.0621708
$160$	0	0
$161$	6.98990	0.550882
$162$	12.1528	0.954815
$163$	13.9056	1.08917	0.544585	−	0.838706i	$-0.316687\pi$
$163$	13.9056	1.08917	0.544585	+	0.838706i	$0.316687\pi$
$164$	0.174901	0.0136575
$165$	0	0
$166$	17.0583	1.32398
$167$	17.6222	1.36365	0.681824	−	0.731516i	$-0.261187\pi$
$167$	17.6222	1.36365	0.681824	+	0.731516i	$0.261187\pi$
$168$	34.1222	2.63259
$169$	10.4383	0.802943
$170$	0	0
$171$	0	0
$172$	−3.03223	−0.231206
$173$	−13.3182	−1.01256	−0.506282	−	0.862368i	$-0.668981\pi$
$173$	−13.3182	−1.01256	−0.506282	+	0.862368i	$0.668981\pi$
$174$	19.8829	1.50732
$175$	0	0
$176$	10.0121	0.754693
$177$	15.6301	1.17483
$178$	9.64015	0.722560
$179$	−14.6264	−1.09323	−0.546616	−	0.837383i	$-0.684084\pi$
$179$	−14.6264	−1.09323	−0.546616	+	0.837383i	$0.684084\pi$
$180$	0	0
$181$	8.78476	0.652966	0.326483	−	0.945203i	$-0.394136\pi$
$181$	8.78476	0.652966	0.326483	+	0.945203i	$0.394136\pi$
$182$	−22.6613	−1.67977
$183$	2.31623	0.171221
$184$	5.70975	0.420928
$185$	0	0
$186$	11.8091	0.865883
$187$	−8.97783	−0.656524
$188$	0.941271	0.0686493
$189$	−34.4785	−2.50794
$190$	0	0
$191$	−9.70737	−0.702401	−0.351200	−	0.936300i	$-0.614226\pi$
$191$	−9.70737	−0.702401	−0.351200	+	0.936300i	$0.614226\pi$
$192$	26.8100	1.93485
$193$	−20.2546	−1.45796	−0.728981	−	0.684534i	$-0.760006\pi$
$193$	−20.2546	−1.45796	−0.728981	+	0.684534i	$0.760006\pi$
$194$	−9.08650	−0.652373
$195$	0	0
$196$	−2.88471	−0.206051
$197$	9.26182	0.659877	0.329939	−	0.944002i	$-0.392972\pi$
$197$	9.26182	0.659877	0.329939	+	0.944002i	$0.392972\pi$
$198$	−25.6200	−1.82073
$199$	18.1521	1.28676	0.643382	−	0.765545i	$-0.277531\pi$
$199$	18.1521	1.28676	0.643382	+	0.765545i	$0.277531\pi$
$200$	0	0
$201$	32.4644	2.28987
$202$	5.64607	0.397256
$203$	−19.5554	−1.37252
$204$	−3.38498	−0.236996
$205$	0	0
$206$	13.5055	0.940970
$207$	−11.4004	−0.792386
$208$	−14.4439	−1.00150
$209$	0	0
$210$	0	0
$211$	−17.3820	−1.19662	−0.598311	−	0.801264i	$-0.704161\pi$
$211$	−17.3820	−1.19662	−0.598311	+	0.801264i	$0.704161\pi$
$212$	−0.109318	−0.00750802
$213$	−6.02889	−0.413093
$214$	1.73288	0.118458
$215$	0	0
$216$	−28.1640	−1.91631
$217$	−11.6146	−0.788448
$218$	−11.9396	−0.808651
$219$	−13.7438	−0.928718
$220$	0	0
$221$	12.9517	0.871228
$222$	3.75950	0.252321
$223$	−8.72670	−0.584383	−0.292191	−	0.956360i	$-0.594384\pi$
$223$	−8.72670	−0.584383	−0.292191	+	0.956360i	$0.594384\pi$
$224$	−8.69057	−0.580663
$225$	0	0
$226$	−21.4232	−1.42505
$227$	−20.1154	−1.33510	−0.667552	−	0.744563i	$-0.732658\pi$
$227$	−20.1154	−1.33510	−0.667552	+	0.744563i	$0.732658\pi$
$228$	0	0
$229$	−20.8410	−1.37721	−0.688607	−	0.725135i	$-0.741778\pi$
$229$	−20.8410	−1.37721	−0.688607	+	0.725135i	$0.741778\pi$
$230$	0	0
$231$	37.6442	2.47680
$232$	−15.9740	−1.04874
$233$	−0.544051	−0.0356420	−0.0178210	−	0.999841i	$-0.505673\pi$
$233$	−0.544051	−0.0356420	−0.0178210	+	0.999841i	$0.505673\pi$
$234$	36.9603	2.41617
$235$	0	0
$236$	−2.17956	−0.141877
$237$	36.3038	2.35819
$238$	−12.5224	−0.811706
$239$	−22.2235	−1.43752	−0.718758	−	0.695260i	$-0.755289\pi$
$239$	−22.2235	−1.43752	−0.718758	+	0.695260i	$0.755289\pi$
$240$	0	0
$241$	−21.7336	−1.39998	−0.699991	−	0.714152i	$-0.746813\pi$
$241$	−21.7336	−1.39998	−0.699991	+	0.714152i	$0.746813\pi$
$242$	0.329211	0.0211625
$243$	1.34772	0.0864565
$244$	−0.322991	−0.0206774
$245$	0	0
$246$	−1.57654	−0.100517
$247$	0	0
$248$	−9.48743	−0.602452
$249$	40.8795	2.59063
$250$	0	0
$251$	17.8239	1.12504	0.562518	−	0.826785i	$-0.309833\pi$
$251$	17.8239	1.12504	0.562518	+	0.826785i	$0.309833\pi$
$252$	9.50058	0.598481
$253$	6.29908	0.396020
$254$	24.9630	1.56632
$255$	0	0
$256$	−9.60535	−0.600334
$257$	15.7239	0.980829	0.490414	−	0.871489i	$-0.336845\pi$
$257$	15.7239	0.980829	0.490414	+	0.871489i	$0.336845\pi$
$258$	27.3323	1.70163
$259$	−3.69758	−0.229756
$260$	0	0
$261$	31.8947	1.97423
$262$	6.81671	0.421138
$263$	30.4504	1.87765	0.938826	−	0.344391i	$-0.111915\pi$
$263$	30.4504	1.87765	0.938826	+	0.344391i	$0.111915\pi$
$264$	30.7499	1.89252
$265$	0	0
$266$	0	0
$267$	23.1022	1.41383
$268$	−4.52706	−0.276534
$269$	−2.61192	−0.159251	−0.0796257	−	0.996825i	$-0.525373\pi$
$269$	−2.61192	−0.159251	−0.0796257	+	0.996825i	$0.525373\pi$
$270$	0	0
$271$	−9.60823	−0.583658	−0.291829	−	0.956470i	$-0.594264\pi$
$271$	−9.60823	−0.583658	−0.291829	+	0.956470i	$0.594264\pi$
$272$	−7.98152	−0.483951
$273$	−54.3068	−3.28680
$274$	23.3310	1.40948
$275$	0	0
$276$	2.37499	0.142958
$277$	−20.0672	−1.20572	−0.602860	−	0.797847i	$-0.705972\pi$
$277$	−20.0672	−1.20572	−0.602860	+	0.797847i	$0.705972\pi$
$278$	−13.2140	−0.792526
$279$	18.9432	1.13410
$280$	0	0
$281$	−26.5556	−1.58417	−0.792086	−	0.610409i	$-0.791005\pi$
$281$	−26.5556	−1.58417	−0.792086	+	0.610409i	$0.791005\pi$
$282$	−8.48455	−0.505247
$283$	16.7788	0.997397	0.498699	−	0.866776i	$-0.333811\pi$
$283$	16.7788	0.997397	0.498699	+	0.866776i	$0.333811\pi$
$284$	0.840709	0.0498869
$285$	0	0
$286$	−20.4216	−1.20756
$287$	1.55058	0.0915276
$288$	14.1742	0.835224
$289$	−9.84301	−0.579000
$290$	0	0
$291$	−21.7754	−1.27650
$292$	1.91652	0.112156
$293$	13.3788	0.781597	0.390799	−	0.920476i	$-0.372199\pi$
$293$	13.3788	0.781597	0.390799	+	0.920476i	$0.372199\pi$
$294$	26.0026	1.51650
$295$	0	0
$296$	−3.02039	−0.175557
$297$	−31.0709	−1.80292
$298$	19.0315	1.10247
$299$	−9.08728	−0.525531
$300$	0	0
$301$	−26.8821	−1.54946
$302$	−14.2579	−0.820452
$303$	13.5305	0.777310
$304$	0	0
$305$	0	0
$306$	20.4239	1.16755
$307$	27.9723	1.59646	0.798232	−	0.602350i	$-0.205769\pi$
$307$	27.9723	1.59646	0.798232	+	0.602350i	$0.205769\pi$
$308$	−5.24935	−0.299109
$309$	32.3653	1.84119
$310$	0	0
$311$	24.8350	1.40826	0.704132	−	0.710069i	$-0.251336\pi$
$311$	24.8350	1.40826	0.704132	+	0.710069i	$0.251336\pi$
$312$	−44.3609	−2.51144
$313$	22.2726	1.25892	0.629462	−	0.777031i	$-0.283275\pi$
$313$	22.2726	1.25892	0.629462	+	0.777031i	$0.283275\pi$
$314$	5.29845	0.299009
$315$	0	0
$316$	−5.06245	−0.284785
$317$	−14.0757	−0.790568	−0.395284	−	0.918559i	$-0.629354\pi$
$317$	−14.0757	−0.790568	−0.395284	+	0.918559i	$0.629354\pi$
$318$	0.985388	0.0552578
$319$	−17.6227	−0.986684
$320$	0	0
$321$	4.15278	0.231786
$322$	8.78604	0.489627
$323$	0	0
$324$	−4.06120	−0.225622
$325$	0	0
$326$	17.4788	0.968061
$327$	−28.6127	−1.58229
$328$	1.26660	0.0699362
$329$	8.34479	0.460063
$330$	0	0
$331$	5.28549	0.290517	0.145258	−	0.989394i	$-0.453599\pi$
$331$	5.28549	0.290517	0.145258	+	0.989394i	$0.453599\pi$
$332$	−5.70050	−0.312856
$333$	6.03070	0.330481
$334$	22.1505	1.21202
$335$	0	0
$336$	33.4666	1.82575
$337$	23.8881	1.30127	0.650633	−	0.759392i	$-0.274504\pi$
$337$	23.8881	1.30127	0.650633	+	0.759392i	$0.274504\pi$
$338$	13.1205	0.713660
$339$	−51.3398	−2.78839
$340$	0	0
$341$	−10.4667	−0.566802
$342$	0	0
$343$	0.493173	0.0266288
$344$	−21.9588	−1.18394
$345$	0	0
$346$	−16.7405	−0.899973
$347$	8.76704	0.470639	0.235320	−	0.971918i	$-0.424386\pi$
$347$	8.76704	0.470639	0.235320	+	0.971918i	$0.424386\pi$
$348$	−6.64444	−0.356179
$349$	2.04011	0.109205	0.0546023	−	0.998508i	$-0.482611\pi$
$349$	2.04011	0.109205	0.0546023	+	0.998508i	$0.482611\pi$
$350$	0	0
$351$	44.8240	2.39253
$352$	−7.83167	−0.417429
$353$	−9.06450	−0.482455	−0.241227	−	0.970469i	$-0.577550\pi$
$353$	−9.06450	−0.482455	−0.241227	+	0.970469i	$0.577550\pi$
$354$	19.6464	1.04419
$355$	0	0
$356$	−3.22152	−0.170740
$357$	−30.0094	−1.58826
$358$	−18.3849	−0.971671
$359$	−23.6209	−1.24667	−0.623333	−	0.781956i	$-0.714222\pi$
$359$	−23.6209	−1.24667	−0.623333	+	0.781956i	$0.714222\pi$
$360$	0	0
$361$	0	0
$362$	11.0421	0.580360
$363$	0.788939	0.0414086
$364$	7.57290	0.396928
$365$	0	0
$366$	2.91142	0.152182
$367$	−10.4211	−0.543977	−0.271988	−	0.962301i	$-0.587681\pi$
$367$	−10.4211	−0.543977	−0.271988	+	0.962301i	$0.587681\pi$
$368$	5.60005	0.291923
$369$	−2.52897	−0.131653
$370$	0	0
$371$	−0.969157	−0.0503161
$372$	−3.94633	−0.204608
$373$	9.20454	0.476593	0.238297	−	0.971192i	$-0.423411\pi$
$373$	9.20454	0.476593	0.238297	+	0.971192i	$0.423411\pi$
$374$	−11.2848	−0.583522
$375$	0	0
$376$	6.81650	0.351534
$377$	25.4232	1.30936
$378$	−43.3381	−2.22907
$379$	−12.6143	−0.647954	−0.323977	−	0.946065i	$-0.605020\pi$
$379$	−12.6143	−0.647954	−0.323977	+	0.946065i	$0.605020\pi$
$380$	0	0
$381$	59.8228	3.06482
$382$	−12.2018	−0.624298
$383$	0.529736	0.0270682	0.0135341	−	0.999908i	$-0.495692\pi$
$383$	0.529736	0.0270682	0.0135341	+	0.999908i	$0.495692\pi$
$384$	19.6396	1.00223
$385$	0	0
$386$	−25.4593	−1.29584
$387$	43.8444	2.22873
$388$	3.03651	0.154155
$389$	30.3866	1.54066	0.770331	−	0.637644i	$-0.220091\pi$
$389$	30.3866	1.54066	0.770331	+	0.637644i	$0.220091\pi$
$390$	0	0
$391$	−5.02153	−0.253950
$392$	−20.8905	−1.05513
$393$	16.3359	0.824040
$394$	11.6417	0.586503
$395$	0	0
$396$	8.56163	0.430238
$397$	12.1258	0.608577	0.304288	−	0.952580i	$-0.401581\pi$
$397$	12.1258	0.608577	0.304288	+	0.952580i	$0.401581\pi$
$398$	22.8164	1.14368
$399$	0	0
$400$	0	0
$401$	11.5906	0.578806	0.289403	−	0.957207i	$-0.406543\pi$
$401$	11.5906	0.578806	0.289403	+	0.957207i	$0.406543\pi$
$402$	40.8066	2.03525
$403$	15.0996	0.752164
$404$	−1.88679	−0.0938713
$405$	0	0
$406$	−24.5804	−1.21991
$407$	−3.33214	−0.165168
$408$	−24.5133	−1.21359
$409$	−4.83657	−0.239153	−0.119577	−	0.992825i	$-0.538154\pi$
$409$	−4.83657	−0.239153	−0.119577	+	0.992825i	$0.538154\pi$
$410$	0	0
$411$	55.9118	2.75793
$412$	−4.51323	−0.222351
$413$	−19.3228	−0.950812
$414$	−14.3299	−0.704277
$415$	0	0
$416$	11.2982	0.553942
$417$	−31.6669	−1.55073
$418$	0	0
$419$	12.4329	0.607385	0.303693	−	0.952770i	$-0.401780\pi$
$419$	12.4329	0.607385	0.303693	+	0.952770i	$0.401780\pi$
$420$	0	0
$421$	−27.6160	−1.34592	−0.672961	−	0.739678i	$-0.734978\pi$
$421$	−27.6160	−1.34592	−0.672961	+	0.739678i	$0.734978\pi$
$422$	−21.8484	−1.06357
$423$	−13.6103	−0.661753
$424$	−0.791662	−0.0384465
$425$	0	0
$426$	−7.57809	−0.367159
$427$	−2.86346	−0.138573
$428$	−0.579091	−0.0279914
$429$	−48.9396	−2.36282
$430$	0	0
$431$	−7.09165	−0.341593	−0.170796	−	0.985306i	$-0.554634\pi$
$431$	−7.09165	−0.341593	−0.170796	+	0.985306i	$0.554634\pi$
$432$	−27.6228	−1.32900
$433$	−6.58386	−0.316400	−0.158200	−	0.987407i	$-0.550569\pi$
$433$	−6.58386	−0.316400	−0.158200	+	0.987407i	$0.550569\pi$
$434$	−14.5991	−0.700777
$435$	0	0
$436$	3.98995	0.191084
$437$	0	0
$438$	−17.2754	−0.825450
$439$	−4.42825	−0.211349	−0.105674	−	0.994401i	$-0.533700\pi$
$439$	−4.42825	−0.211349	−0.105674	+	0.994401i	$0.533700\pi$
$440$	0	0
$441$	41.7113	1.98625
$442$	16.2798	0.774352
$443$	2.77214	0.131708	0.0658541	−	0.997829i	$-0.479023\pi$
$443$	2.77214	0.131708	0.0658541	+	0.997829i	$0.479023\pi$
$444$	−1.25634	−0.0596234
$445$	0	0
$446$	−10.9691	−0.519403
$447$	45.6083	2.15720
$448$	−33.1441	−1.56591
$449$	−26.3569	−1.24386	−0.621930	−	0.783073i	$-0.713651\pi$
$449$	−26.3569	−1.24386	−0.621930	+	0.783073i	$0.713651\pi$
$450$	0	0
$451$	1.39733	0.0657977
$452$	7.15916	0.336739
$453$	−34.1685	−1.60538
$454$	−25.2842	−1.18665
$455$	0	0
$456$	0	0
$457$	−33.5784	−1.57073	−0.785366	−	0.619032i	$-0.787525\pi$
$457$	−33.5784	−1.57073	−0.785366	+	0.619032i	$0.787525\pi$
$458$	−26.1964	−1.22408
$459$	24.7693	1.15613
$460$	0	0
$461$	30.4525	1.41831	0.709156	−	0.705051i	$-0.249076\pi$
$461$	30.4525	1.41831	0.709156	+	0.705051i	$0.249076\pi$
$462$	47.3172	2.20140
$463$	28.5762	1.32805	0.664025	−	0.747710i	$-0.268847\pi$
$463$	28.5762	1.32805	0.664025	+	0.747710i	$0.268847\pi$
$464$	−15.6671	−0.727326
$465$	0	0
$466$	−0.683851	−0.0316788
$467$	11.2173	0.519073	0.259536	−	0.965733i	$-0.416430\pi$
$467$	11.2173	0.519073	0.259536	+	0.965733i	$0.416430\pi$
$468$	−12.3513	−0.570939
$469$	−40.1344	−1.85323
$470$	0	0
$471$	12.6975	0.585070
$472$	−15.7839	−0.726515
$473$	−24.2253	−1.11388
$474$	45.6325	2.09597
$475$	0	0
$476$	4.18470	0.191806
$477$	1.58068	0.0723745
$478$	−27.9340	−1.27767
$479$	−17.3905	−0.794594	−0.397297	−	0.917690i	$-0.630052\pi$
$479$	−17.3905	−0.794594	−0.397297	+	0.917690i	$0.630052\pi$
$480$	0	0
$481$	4.80706	0.219183
$482$	−27.3182	−1.24431
$483$	21.0554	0.958052
$484$	−0.110015	−0.00500068
$485$	0	0
$486$	1.69404	0.0768430
$487$	15.3473	0.695451	0.347726	−	0.937596i	$-0.386954\pi$
$487$	15.3473	0.695451	0.347726	+	0.937596i	$0.386954\pi$
$488$	−2.33904	−0.105883
$489$	41.8872	1.89420
$490$	0	0
$491$	8.35401	0.377011	0.188506	−	0.982072i	$-0.439636\pi$
$491$	8.35401	0.377011	0.188506	+	0.982072i	$0.439636\pi$
$492$	0.526846	0.0237520
$493$	14.0486	0.632717
$494$	0	0
$495$	0	0
$496$	−9.30515	−0.417813
$497$	7.45326	0.334324
$498$	51.3839	2.30257
$499$	0.270833	0.0121241	0.00606207	−	0.999982i	$-0.498070\pi$
$499$	0.270833	0.0121241	0.00606207	+	0.999982i	$0.498070\pi$
$500$	0	0
$501$	53.0826	2.37156
$502$	22.4040	0.999939
$503$	−41.5995	−1.85483	−0.927415	−	0.374034i	$-0.877974\pi$
$503$	−41.5995	−1.85483	−0.927415	+	0.374034i	$0.877974\pi$
$504$	68.8013	3.06466
$505$	0	0
$506$	7.91770	0.351985
$507$	31.4427	1.39642
$508$	−8.34209	−0.370120
$509$	−1.62776	−0.0721493	−0.0360747	−	0.999349i	$-0.511485\pi$
$509$	−1.62776	−0.0721493	−0.0360747	+	0.999349i	$0.511485\pi$
$510$	0	0
$511$	16.9908	0.751630
$512$	−25.1134	−1.10987
$513$	0	0
$514$	19.7643	0.871766
$515$	0	0
$516$	−9.13385	−0.402095
$517$	7.52006	0.330732
$518$	−4.64771	−0.204209
$519$	−40.1178	−1.76098
$520$	0	0
$521$	9.14124	0.400485	0.200243	−	0.979746i	$-0.435827\pi$
$521$	9.14124	0.400485	0.200243	+	0.979746i	$0.435827\pi$
$522$	40.0904	1.75471
$523$	37.2995	1.63099	0.815497	−	0.578762i	$-0.196464\pi$
$523$	37.2995	1.63099	0.815497	+	0.578762i	$0.196464\pi$
$524$	−2.27799	−0.0995146
$525$	0	0
$526$	38.2750	1.66887
$527$	8.34388	0.363465
$528$	30.1591	1.31251
$529$	−19.4768	−0.846816
$530$	0	0
$531$	31.5152	1.36764
$532$	0	0
$533$	−2.01584	−0.0873156
$534$	29.0386	1.25662
$535$	0	0
$536$	−32.7841	−1.41606
$537$	−44.0585	−1.90127
$538$	−3.28308	−0.141544
$539$	−23.0467	−0.992692
$540$	0	0
$541$	−36.2441	−1.55826	−0.779128	−	0.626864i	$-0.784338\pi$
$541$	−36.2441	−1.55826	−0.779128	+	0.626864i	$0.784338\pi$
$542$	−12.0772	−0.518759
$543$	26.4619	1.13559
$544$	6.24329	0.267679
$545$	0	0
$546$	−68.2615	−2.92132
$547$	−15.1033	−0.645772	−0.322886	−	0.946438i	$-0.604653\pi$
$547$	−15.1033	−0.645772	−0.322886	+	0.946438i	$0.604653\pi$
$548$	−7.79671	−0.333059
$549$	4.67027	0.199322
$550$	0	0
$551$	0	0
$552$	17.1992	0.732047
$553$	−44.8809	−1.90853
$554$	−25.2236	−1.07165
$555$	0	0
$556$	4.41584	0.187273
$557$	9.36275	0.396712	0.198356	−	0.980130i	$-0.436440\pi$
$557$	9.36275	0.396712	0.198356	+	0.980130i	$0.436440\pi$
$558$	23.8109	1.00799
$559$	34.9483	1.47815
$560$	0	0
$561$	−27.0435	−1.14178
$562$	−33.3793	−1.40802
$563$	−14.4454	−0.608802	−0.304401	−	0.952544i	$-0.598456\pi$
$563$	−14.4454	−0.608802	−0.304401	+	0.952544i	$0.598456\pi$
$564$	2.83535	0.119390
$565$	0	0
$566$	21.0903	0.886492
$567$	−36.0044	−1.51204
$568$	6.08825	0.255457
$569$	37.8642	1.58735	0.793675	−	0.608342i	$-0.208165\pi$
$569$	37.8642	1.58735	0.793675	+	0.608342i	$0.208165\pi$
$570$	0	0
$571$	−18.5203	−0.775048	−0.387524	−	0.921860i	$-0.626670\pi$
$571$	−18.5203	−0.775048	−0.387524	+	0.921860i	$0.626670\pi$
$572$	6.82446	0.285345
$573$	−29.2411	−1.22156
$574$	1.94901	0.0813502
$575$	0	0
$576$	54.0575	2.25240
$577$	9.08095	0.378045	0.189022	−	0.981973i	$-0.439468\pi$
$577$	9.08095	0.378045	0.189022	+	0.981973i	$0.439468\pi$
$578$	−12.3723	−0.514619
$579$	−61.0121	−2.53558
$580$	0	0
$581$	−50.5375	−2.09665
$582$	−27.3708	−1.13456
$583$	−0.873373	−0.0361714
$584$	13.8791	0.574320
$585$	0	0
$586$	16.8166	0.694688
$587$	23.0373	0.950852	0.475426	−	0.879756i	$-0.342294\pi$
$587$	23.0373	0.950852	0.475426	+	0.879756i	$0.342294\pi$
$588$	−8.68948	−0.358348
$589$	0	0
$590$	0	0
$591$	27.8989	1.14761
$592$	−2.96236	−0.121752
$593$	−40.2431	−1.65259	−0.826294	−	0.563239i	$-0.809555\pi$
$593$	−40.2431	−1.65259	−0.826294	+	0.563239i	$0.809555\pi$
$594$	−39.0549	−1.60244
$595$	0	0
$596$	−6.35992	−0.260512
$597$	54.6786	2.23784
$598$	−11.4224	−0.467095
$599$	37.7691	1.54320	0.771602	−	0.636106i	$-0.219456\pi$
$599$	37.7691	1.54320	0.771602	+	0.636106i	$0.219456\pi$
$600$	0	0
$601$	−43.8389	−1.78823	−0.894113	−	0.447841i	$-0.852193\pi$
$601$	−43.8389	−1.78823	−0.894113	+	0.447841i	$0.852193\pi$
$602$	−33.7898	−1.37717
$603$	65.4587	2.66568
$604$	4.76469	0.193872
$605$	0	0
$606$	17.0074	0.690877
$607$	−7.80810	−0.316921	−0.158461	−	0.987365i	$-0.550653\pi$
$607$	−7.80810	−0.316921	−0.158461	+	0.987365i	$0.550653\pi$
$608$	0	0
$609$	−58.9059	−2.38699
$610$	0	0
$611$	−10.8487	−0.438892
$612$	−6.82520	−0.275892
$613$	38.8957	1.57098	0.785491	−	0.618873i	$-0.212410\pi$
$613$	38.8957	1.57098	0.785491	+	0.618873i	$0.212410\pi$
$614$	35.1601	1.41895
$615$	0	0
$616$	−38.0147	−1.53166
$617$	−12.4615	−0.501681	−0.250841	−	0.968028i	$-0.580707\pi$
$617$	−12.4615	−0.501681	−0.250841	+	0.968028i	$0.580707\pi$
$618$	40.6819	1.63646
$619$	−2.75741	−0.110830	−0.0554148	−	0.998463i	$-0.517648\pi$
$619$	−2.75741	−0.110830	−0.0554148	+	0.998463i	$0.517648\pi$
$620$	0	0
$621$	−17.3788	−0.697386
$622$	31.2167	1.25167
$623$	−28.5602	−1.14424
$624$	−43.5085	−1.74174
$625$	0	0
$626$	27.9958	1.11894
$627$	0	0
$628$	−1.77062	−0.0706555
$629$	2.65633	0.105915
$630$	0	0
$631$	−39.9220	−1.58927	−0.794636	−	0.607087i	$-0.792338\pi$
$631$	−39.9220	−1.58927	−0.794636	+	0.607087i	$0.792338\pi$
$632$	−36.6612	−1.45831
$633$	−52.3588	−2.08108
$634$	−17.6926	−0.702661
$635$	0	0
$636$	−0.329295	−0.0130574
$637$	33.2480	1.31733
$638$	−22.1511	−0.876970
$639$	−12.1562	−0.480891
$640$	0	0
$641$	5.01268	0.197989	0.0989945	−	0.995088i	$-0.468437\pi$
$641$	5.01268	0.197989	0.0989945	+	0.995088i	$0.468437\pi$
$642$	5.21989	0.206012
$643$	−49.9753	−1.97083	−0.985417	−	0.170157i	$-0.945572\pi$
$643$	−49.9753	−1.97083	−0.985417	+	0.170157i	$0.945572\pi$
$644$	−2.93610	−0.115698
$645$	0	0
$646$	0	0
$647$	−32.7681	−1.28825	−0.644123	−	0.764922i	$-0.722777\pi$
$647$	−32.7681	−1.28825	−0.644123	+	0.764922i	$0.722777\pi$
$648$	−29.4104	−1.15535
$649$	−17.4131	−0.683523
$650$	0	0
$651$	−34.9860	−1.37121
$652$	−5.84102	−0.228752
$653$	−15.7996	−0.618284	−0.309142	−	0.951016i	$-0.600042\pi$
$653$	−15.7996	−0.618284	−0.309142	+	0.951016i	$0.600042\pi$
$654$	−35.9651	−1.40635
$655$	0	0
$656$	1.24226	0.0485022
$657$	−27.7118	−1.08114
$658$	10.4891	0.408907
$659$	13.5485	0.527773	0.263887	−	0.964554i	$-0.414996\pi$
$659$	13.5485	0.527773	0.263887	+	0.964554i	$0.414996\pi$
$660$	0	0
$661$	−3.65778	−0.142271	−0.0711355	−	0.997467i	$-0.522662\pi$
$661$	−3.65778	−0.142271	−0.0711355	+	0.997467i	$0.522662\pi$
$662$	6.64365	0.258213
$663$	39.0139	1.51517
$664$	−41.2819	−1.60205
$665$	0	0
$666$	7.58036	0.293733
$667$	−9.85686	−0.381659
$668$	−7.40219	−0.286399
$669$	−26.2870	−1.01631
$670$	0	0
$671$	−2.58046	−0.0996176
$672$	−26.1782	−1.00985
$673$	−26.9303	−1.03809	−0.519043	−	0.854748i	$-0.673712\pi$
$673$	−26.9303	−1.03809	−0.519043	+	0.854748i	$0.673712\pi$
$674$	30.0264	1.15657
$675$	0	0
$676$	−4.38457	−0.168637
$677$	20.6933	0.795308	0.397654	−	0.917535i	$-0.369824\pi$
$677$	20.6933	0.795308	0.397654	+	0.917535i	$0.369824\pi$
$678$	−64.5321	−2.47834
$679$	26.9200	1.03309
$680$	0	0
$681$	−60.5926	−2.32191
$682$	−13.1562	−0.503777
$683$	26.9573	1.03149	0.515745	−	0.856742i	$-0.327515\pi$
$683$	26.9573	1.03149	0.515745	+	0.856742i	$0.327515\pi$
$684$	0	0
$685$	0	0
$686$	0.619899	0.0236679
$687$	−62.7785	−2.39515
$688$	−21.5369	−0.821087
$689$	1.25996	0.0480006
$690$	0	0
$691$	8.70043	0.330980	0.165490	−	0.986211i	$-0.447079\pi$
$691$	8.70043	0.330980	0.165490	+	0.986211i	$0.447079\pi$
$692$	5.59429	0.212663
$693$	75.9027	2.88330
$694$	11.0198	0.418307
$695$	0	0
$696$	−48.1177	−1.82390
$697$	−1.11393	−0.0421931
$698$	2.56434	0.0970616
$699$	−1.63882	−0.0619858
$700$	0	0
$701$	−0.403070	−0.0152237	−0.00761187	−	0.999971i	$-0.502423\pi$
$701$	−0.403070	−0.0152237	−0.00761187	+	0.999971i	$0.502423\pi$
$702$	56.3420	2.12649
$703$	0	0
$704$	−29.8684	−1.12571
$705$	0	0
$706$	−11.3937	−0.428808
$707$	−16.7272	−0.629092
$708$	−6.56538	−0.246742
$709$	40.0543	1.50427	0.752135	−	0.659010i	$-0.229024\pi$
$709$	40.0543	1.50427	0.752135	+	0.659010i	$0.229024\pi$
$710$	0	0
$711$	73.2002	2.74522
$712$	−23.3296	−0.874315
$713$	−5.85428	−0.219245
$714$	−37.7206	−1.41166
$715$	0	0
$716$	6.14382	0.229605
$717$	−66.9427	−2.50002
$718$	−29.6906	−1.10804
$719$	18.0756	0.674108	0.337054	−	0.941485i	$-0.390570\pi$
$719$	18.0756	0.674108	0.337054	+	0.941485i	$0.390570\pi$
$720$	0	0
$721$	−40.0118	−1.49012
$722$	0	0
$723$	−65.4670	−2.43474
$724$	−3.69002	−0.137139
$725$	0	0
$726$	0.991666	0.0368042
$727$	−26.3614	−0.977689	−0.488844	−	0.872371i	$-0.662581\pi$
$727$	−26.3614	−0.977689	−0.488844	+	0.872371i	$0.662581\pi$
$728$	54.8414	2.03256
$729$	−24.9455	−0.923909
$730$	0	0
$731$	19.3120	0.714282
$732$	−0.972931	−0.0359606
$733$	3.30919	0.122228	0.0611139	−	0.998131i	$-0.480535\pi$
$733$	3.30919	0.122228	0.0611139	+	0.998131i	$0.480535\pi$
$734$	−13.0989	−0.483490
$735$	0	0
$736$	−4.38046	−0.161466
$737$	−36.1678	−1.33226
$738$	−3.17882	−0.117014
$739$	17.6943	0.650894	0.325447	−	0.945560i	$-0.394485\pi$
$739$	17.6943	0.650894	0.325447	+	0.945560i	$0.394485\pi$
$740$	0	0
$741$	0	0
$742$	−1.21819	−0.0447212
$743$	37.2532	1.36669	0.683343	−	0.730097i	$-0.260525\pi$
$743$	37.2532	1.36669	0.683343	+	0.730097i	$0.260525\pi$
$744$	−28.5785	−1.04774
$745$	0	0
$746$	11.5698	0.423599
$747$	82.4261	3.01581
$748$	3.77112	0.137886
$749$	−5.13391	−0.187589
$750$	0	0
$751$	13.5647	0.494982	0.247491	−	0.968890i	$-0.420394\pi$
$751$	13.5647	0.494982	0.247491	+	0.968890i	$0.420394\pi$
$752$	6.68553	0.243796
$753$	53.6902	1.95658
$754$	31.9560	1.16377
$755$	0	0
$756$	14.4826	0.526728
$757$	−12.5370	−0.455665	−0.227833	−	0.973700i	$-0.573164\pi$
$757$	−12.5370	−0.455665	−0.227833	+	0.973700i	$0.573164\pi$
$758$	−15.8557	−0.575905
$759$	18.9744	0.688728
$760$	0	0
$761$	−27.2282	−0.987022	−0.493511	−	0.869740i	$-0.664287\pi$
$761$	−27.2282	−0.987022	−0.493511	+	0.869740i	$0.664287\pi$
$762$	75.1950	2.72403
$763$	35.3727	1.28058
$764$	4.07757	0.147521
$765$	0	0
$766$	0.665858	0.0240584
$767$	25.1207	0.907056
$768$	−28.9337	−1.04406
$769$	−0.920184	−0.0331827	−0.0165913	−	0.999862i	$-0.505281\pi$
$769$	−0.920184	−0.0331827	−0.0165913	+	0.999862i	$0.505281\pi$
$770$	0	0
$771$	47.3643	1.70578
$772$	8.50793	0.306207
$773$	−1.82987	−0.0658157	−0.0329079	−	0.999458i	$-0.510477\pi$
$773$	−1.82987	−0.0658157	−0.0329079	+	0.999458i	$0.510477\pi$
$774$	55.1107	1.98091
$775$	0	0
$776$	21.9898	0.789387
$777$	−11.1380	−0.399575
$778$	38.1948	1.36935
$779$	0	0
$780$	0	0
$781$	6.71664	0.240340
$782$	−6.31187	−0.225712
$783$	48.6200	1.73754
$784$	−20.4891	−0.731754
$785$	0	0
$786$	20.5337	0.732411
$787$	−2.90489	−0.103548	−0.0517740	−	0.998659i	$-0.516488\pi$
$787$	−2.90489	−0.103548	−0.0517740	+	0.998659i	$0.516488\pi$
$788$	−3.89041	−0.138590
$789$	91.7243	3.26547
$790$	0	0
$791$	63.4692	2.25670
$792$	62.0016	2.20313
$793$	3.72266	0.132196
$794$	15.2417	0.540907
$795$	0	0
$796$	−7.62474	−0.270252
$797$	19.9042	0.705044	0.352522	−	0.935804i	$-0.385324\pi$
$797$	19.9042	0.705044	0.352522	+	0.935804i	$0.385324\pi$
$798$	0	0
$799$	−5.99488	−0.212084
$800$	0	0
$801$	46.5814	1.64587
$802$	14.5689	0.514447
$803$	15.3116	0.540335
$804$	−13.6366	−0.480927
$805$	0	0
$806$	18.9796	0.668528
$807$	−7.86775	−0.276958
$808$	−13.6638	−0.480689
$809$	44.4186	1.56168	0.780838	−	0.624734i	$-0.214792\pi$
$809$	44.4186	1.56168	0.780838	+	0.624734i	$0.214792\pi$
$810$	0	0
$811$	−34.0494	−1.19564	−0.597819	−	0.801631i	$-0.703966\pi$
$811$	−34.0494	−1.19564	−0.597819	+	0.801631i	$0.703966\pi$
$812$	8.21423	0.288263
$813$	−28.9424	−1.01505
$814$	−4.18837	−0.146802
$815$	0	0
$816$	−24.0424	−0.841651
$817$	0	0
$818$	−6.07938	−0.212561
$819$	−109.500	−3.82623
$820$	0	0
$821$	−26.8065	−0.935553	−0.467776	−	0.883847i	$-0.654945\pi$
$821$	−26.8065	−0.935553	−0.467776	+	0.883847i	$0.654945\pi$
$822$	70.2790	2.45126
$823$	−11.0407	−0.384853	−0.192426	−	0.981311i	$-0.561636\pi$
$823$	−11.0407	−0.384853	−0.192426	+	0.981311i	$0.561636\pi$
$824$	−32.6839	−1.13860
$825$	0	0
$826$	−24.2880	−0.845087
$827$	11.8987	0.413757	0.206878	−	0.978367i	$-0.433670\pi$
$827$	11.8987	0.413757	0.206878	+	0.978367i	$0.433670\pi$
$828$	4.78874	0.166420
$829$	−11.2194	−0.389665	−0.194832	−	0.980837i	$-0.562416\pi$
$829$	−11.2194	−0.389665	−0.194832	+	0.980837i	$0.562416\pi$
$830$	0	0
$831$	−60.4474	−2.09690
$832$	43.0892	1.49385
$833$	18.3725	0.636569
$834$	−39.8040	−1.37830
$835$	0	0
$836$	0	0
$837$	28.8769	0.998132
$838$	15.6276	0.539848
$839$	35.8294	1.23697	0.618484	−	0.785797i	$-0.287747\pi$
$839$	35.8294	1.23697	0.618484	+	0.785797i	$0.287747\pi$
$840$	0	0
$841$	−1.42378	−0.0490959
$842$	−34.7123	−1.19626
$843$	−79.9921	−2.75507
$844$	7.30126	0.251320
$845$	0	0
$846$	−17.1076	−0.588170
$847$	−0.975332	−0.0335128
$848$	−0.776452	−0.0266635
$849$	50.5420	1.73460
$850$	0	0
$851$	−1.86375	−0.0638886
$852$	2.53243	0.0867595
$853$	−13.0632	−0.447276	−0.223638	−	0.974672i	$-0.571793\pi$
$853$	−13.0632	−0.447276	−0.223638	+	0.974672i	$0.571793\pi$
$854$	−3.59926	−0.123164
$855$	0	0
$856$	−4.19366	−0.143337
$857$	−24.1902	−0.826323	−0.413161	−	0.910658i	$-0.635575\pi$
$857$	−24.1902	−0.826323	−0.413161	+	0.910658i	$0.635575\pi$
$858$	−61.5151	−2.10009
$859$	54.1525	1.84766	0.923830	−	0.382804i	$-0.125041\pi$
$859$	54.1525	1.84766	0.923830	+	0.382804i	$0.125041\pi$
$860$	0	0
$861$	4.67072	0.159178
$862$	−8.91393	−0.303610
$863$	49.6852	1.69131	0.845653	−	0.533733i	$-0.179211\pi$
$863$	49.6852	1.69131	0.845653	+	0.533733i	$0.179211\pi$
$864$	21.6071	0.735088
$865$	0	0
$866$	−8.27566	−0.281218
$867$	−29.6496	−1.00695
$868$	4.87868	0.165593
$869$	−40.4452	−1.37201
$870$	0	0
$871$	52.1770	1.76795
$872$	28.8944	0.978488
$873$	−43.9062	−1.48600
$874$	0	0
$875$	0	0
$876$	5.77305	0.195053
$877$	−8.27138	−0.279305	−0.139652	−	0.990201i	$-0.544598\pi$
$877$	−8.27138	−0.279305	−0.139652	+	0.990201i	$0.544598\pi$
$878$	−5.56614	−0.187848
$879$	40.3003	1.35929
$880$	0	0
$881$	−24.5193	−0.826075	−0.413037	−	0.910714i	$-0.635532\pi$
$881$	−24.5193	−0.826075	−0.413037	+	0.910714i	$0.635532\pi$
$882$	52.4295	1.76539
$883$	−49.1833	−1.65515	−0.827575	−	0.561355i	$-0.810280\pi$
$883$	−49.1833	−1.65515	−0.827575	+	0.561355i	$0.810280\pi$
$884$	−5.44036	−0.182979
$885$	0	0
$886$	3.48447	0.117063
$887$	21.6466	0.726821	0.363411	−	0.931629i	$-0.381612\pi$
$887$	21.6466	0.726821	0.363411	+	0.931629i	$0.381612\pi$
$888$	−9.09818	−0.305315
$889$	−73.9564	−2.48042
$890$	0	0
$891$	−32.4460	−1.08698
$892$	3.66564	0.122735
$893$	0	0
$894$	57.3278	1.91733
$895$	0	0
$896$	−24.2796	−0.811126
$897$	−27.3732	−0.913964
$898$	−33.1296	−1.10555
$899$	16.3783	0.546248
$900$	0	0
$901$	0.696240	0.0231951
$902$	1.75639	0.0584813
$903$	−80.9757	−2.69470
$904$	51.8452	1.72435
$905$	0	0
$906$	−42.9485	−1.42687
$907$	13.4933	0.448039	0.224020	−	0.974585i	$-0.428082\pi$
$907$	13.4933	0.448039	0.224020	+	0.974585i	$0.428082\pi$
$908$	8.44943	0.280404
$909$	27.2819	0.904884
$910$	0	0
$911$	−16.2069	−0.536958	−0.268479	−	0.963286i	$-0.586521\pi$
$911$	−16.2069	−0.536958	−0.268479	+	0.963286i	$0.586521\pi$
$912$	0	0
$913$	−45.5428	−1.50725
$914$	−42.2068	−1.39608
$915$	0	0
$916$	8.75424	0.289248
$917$	−20.1954	−0.666912
$918$	31.1340	1.02758
$919$	19.7373	0.651075	0.325537	−	0.945529i	$-0.394455\pi$
$919$	19.7373	0.651075	0.325537	+	0.945529i	$0.394455\pi$
$920$	0	0
$921$	84.2596	2.77645
$922$	38.2776	1.26060
$923$	−9.68967	−0.318939
$924$	−15.8124	−0.520189
$925$	0	0
$926$	35.9192	1.18038
$927$	65.2587	2.14338
$928$	12.2551	0.402292
$929$	6.48784	0.212859	0.106430	−	0.994320i	$-0.466058\pi$
$929$	6.48784	0.212859	0.106430	+	0.994320i	$0.466058\pi$
$930$	0	0
$931$	0	0
$932$	0.228528	0.00748568
$933$	74.8094	2.44915
$934$	14.0997	0.461355
$935$	0	0
$936$	−89.4457	−2.92362
$937$	7.40129	0.241790	0.120895	−	0.992665i	$-0.461424\pi$
$937$	7.40129	0.241790	0.120895	+	0.992665i	$0.461424\pi$
$938$	−50.4474	−1.64717
$939$	67.0908	2.18943
$940$	0	0
$941$	−53.1005	−1.73103	−0.865513	−	0.500887i	$-0.833007\pi$
$941$	−53.1005	−1.73103	−0.865513	+	0.500887i	$0.833007\pi$
$942$	15.9603	0.520013
$943$	0.781563	0.0254512
$944$	−15.4807	−0.503853
$945$	0	0
$946$	−30.4503	−0.990023
$947$	−15.2771	−0.496438	−0.248219	−	0.968704i	$-0.579845\pi$
$947$	−15.2771	−0.496438	−0.248219	+	0.968704i	$0.579845\pi$
$948$	−15.2494	−0.495277
$949$	−22.0891	−0.717041
$950$	0	0
$951$	−42.3994	−1.37490
$952$	30.3048	0.982184
$953$	−16.9121	−0.547836	−0.273918	−	0.961753i	$-0.588320\pi$
$953$	−16.9121	−0.547836	−0.273918	+	0.961753i	$0.588320\pi$
$954$	1.98686	0.0643269
$955$	0	0
$956$	9.33494	0.301913
$957$	−53.0841	−1.71597
$958$	−21.8592	−0.706240
$959$	−69.1214	−2.23205
$960$	0	0
$961$	−21.2724	−0.686207
$962$	6.04229	0.194811
$963$	8.37334	0.269827
$964$	9.12915	0.294030
$965$	0	0
$966$	26.4658	0.851522
$967$	−14.5256	−0.467110	−0.233555	−	0.972344i	$-0.575036\pi$
$967$	−14.5256	−0.467110	−0.233555	+	0.972344i	$0.575036\pi$
$968$	−0.796706	−0.0256071
$969$	0	0
$970$	0	0
$971$	−30.6047	−0.982151	−0.491076	−	0.871117i	$-0.663396\pi$
$971$	−30.6047	−0.982151	−0.491076	+	0.871117i	$0.663396\pi$
$972$	−0.566109	−0.0181580
$973$	39.1484	1.25504
$974$	19.2909	0.618121
$975$	0	0
$976$	−2.29410	−0.0734323
$977$	−44.0014	−1.40773	−0.703865	−	0.710333i	$-0.748544\pi$
$977$	−44.0014	−1.40773	−0.703865	+	0.710333i	$0.748544\pi$
$978$	52.6505	1.68358
$979$	−25.7376	−0.822577
$980$	0	0
$981$	−57.6924	−1.84198
$982$	10.5007	0.335090
$983$	35.3620	1.12787	0.563936	−	0.825819i	$-0.309286\pi$
$983$	35.3620	1.12787	0.563936	+	0.825819i	$0.309286\pi$
$984$	3.81531	0.121628
$985$	0	0
$986$	17.6585	0.562362
$987$	25.1366	0.800108
$988$	0	0
$989$	−13.5498	−0.430860
$990$	0	0
$991$	−32.1513	−1.02132	−0.510660	−	0.859783i	$-0.670599\pi$
$991$	−32.1513	−1.02132	−0.510660	+	0.859783i	$0.670599\pi$
$992$	7.27865	0.231097
$993$	15.9212	0.505245
$994$	9.36846	0.297150
$995$	0	0
$996$	−17.1714	−0.544095
$997$	8.31350	0.263291	0.131646	−	0.991297i	$-0.457974\pi$
$997$	8.31350	0.263291	0.131646	+	0.991297i	$0.457974\pi$
$998$	0.340427	0.0107760
$999$	9.19316	0.290859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cf.1.6		9
5.4	even	2		1805.2.a.s.1.4		9
19.2	odd	18		475.2.l.c.251.2		18
19.10	odd	18		475.2.l.c.176.2		18
19.18	odd	2		9025.2.a.cc.1.4		9
95.2	even	36		475.2.u.b.99.2		36
95.29	odd	18		95.2.k.a.81.2	yes	18
95.48	even	36		475.2.u.b.24.2		36
95.59	odd	18		95.2.k.a.61.2	✓	18
95.67	even	36		475.2.u.b.24.5		36
95.78	even	36		475.2.u.b.99.5		36
95.94	odd	2		1805.2.a.v.1.6		9
285.29	even	18		855.2.bs.c.271.2		18
285.59	even	18		855.2.bs.c.631.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.k.a.61.2	✓	18	95.59	odd	18
95.2.k.a.81.2	yes	18	95.29	odd	18
475.2.l.c.176.2		18	19.10	odd	18
475.2.l.c.251.2		18	19.2	odd	18
475.2.u.b.24.2		36	95.48	even	36
475.2.u.b.24.5		36	95.67	even	36
475.2.u.b.99.2		36	95.2	even	36
475.2.u.b.99.5		36	95.78	even	36
855.2.bs.c.271.2		18	285.29	even	18
855.2.bs.c.631.2		18	285.59	even	18
1805.2.a.s.1.4		9	5.4	even	2
1805.2.a.v.1.6		9	95.94	odd	2
9025.2.a.cc.1.4		9	19.18	odd	2
9025.2.a.cf.1.6		9	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25696 q^{2} +3.01225 q^{3} -0.420048 q^{4} +3.78628 q^{6} -3.72392 q^{7} -3.04191 q^{8} +6.07366 q^{9} -3.35588 q^{11} -1.26529 q^{12} +4.84131 q^{13} -4.68082 q^{14} -2.98346 q^{16} +2.67526 q^{17} +7.63436 q^{18} -11.2174 q^{21} -4.21821 q^{22} -1.87703 q^{23} -9.16299 q^{24} +6.08534 q^{26} +9.25865 q^{27} +1.56423 q^{28} +5.25131 q^{29} +3.11891 q^{31} +2.33372 q^{32} -10.1087 q^{33} +3.36269 q^{34} -2.55123 q^{36} +0.992927 q^{37} +14.5832 q^{39} -0.416383 q^{41} -14.0998 q^{42} +7.21877 q^{43} +1.40963 q^{44} -2.35935 q^{46} -2.24086 q^{47} -8.98694 q^{48} +6.86757 q^{49} +8.05854 q^{51} -2.03358 q^{52} +0.260252 q^{53} +11.6378 q^{54} +11.3278 q^{56} +6.60069 q^{58} +5.18883 q^{59} +0.768938 q^{61} +3.92035 q^{62} -22.6178 q^{63} +8.90032 q^{64} -12.7063 q^{66} +10.7775 q^{67} -1.12374 q^{68} -5.65409 q^{69} -2.00146 q^{71} -18.4755 q^{72} -4.56262 q^{73} +1.24807 q^{74} +12.4970 q^{77} +18.3306 q^{78} +12.0521 q^{79} +9.66841 q^{81} -0.523377 q^{82} +13.5711 q^{83} +4.71184 q^{84} +9.07371 q^{86} +15.8183 q^{87} +10.2083 q^{88} +7.66941 q^{89} -18.0286 q^{91} +0.788443 q^{92} +9.39494 q^{93} -2.81668 q^{94} +7.02975 q^{96} -7.22894 q^{97} +8.63226 q^{98} -20.3825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 12 q^{6} + 6 q^{8} + 6 q^{9} + 18 q^{12} + 9 q^{13} + 12 q^{16} + 9 q^{17} + 24 q^{18} - 12 q^{21} + 24 q^{22} + 12 q^{23} + 3 q^{24} - 3 q^{26} + 24 q^{27} + 15 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 9 * q^3 + 6 * q^4 + 12 * q^6 + 6 * q^8 + 6 * q^9 + 18 * q^12 + 9 * q^13 + 12 * q^16 + 9 * q^17 + 24 * q^18 - 12 * q^21 + 24 * q^22 + 12 * q^23 + 3 * q^24 - 3 * q^26 + 24 * q^27 + 15 * q^28 - 9 * q^29 - 18 * q^31 + 3 * q^32 - 9 * q^33 + 24 * q^34 + 18 * q^36 + 18 * q^37 + 18 * q^39 - 6 * q^41 + 12 * q^43 + 48 * q^44 + 9 * q^46 - 15 * q^47 - 21 * q^48 - 9 * q^49 + 6 * q^51 + 33 * q^52 + 15 * q^53 + 63 * q^54 - 6 * q^58 - 21 * q^59 - 12 * q^61 + 36 * q^62 - 21 * q^63 - 36 * q^64 + 3 * q^66 + 60 * q^67 + 51 * q^68 + 15 * q^69 + 18 * q^71 - 27 * q^73 + 27 * q^74 + 30 * q^77 - 6 * q^78 - 15 * q^79 + 33 * q^81 - 24 * q^82 + 48 * q^84 + 39 * q^86 - 15 * q^87 + 27 * q^88 + 39 * q^89 - 21 * q^91 + 6 * q^92 - 15 * q^93 - 15 * q^94 - 33 * q^96 + 15 * q^97 - 15 * q^98 - 36 * q^99

Introduction

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