LMFDB - Embedded newform 5225.2.a.x.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,5,4,17,0,-1,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 $ x^15 - 5x^14 - 11x^13 + 85x^12 + 6x^11 - 537x^10 + 327x^9 + 1556x^8 - 1451x^7 - 2033x^6 + 2316x^5 + 927x^4 - 1295x^3 + 17x^2 + 103x - 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.12341$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.12341 q^{2} -0.300215 q^{3} +2.50886 q^{4} -0.637479 q^{6} -1.66086 q^{7} +1.08053 q^{8} -2.90987 q^{9} +1.00000 q^{11} -0.753198 q^{12} +0.406405 q^{13} -3.52669 q^{14} -2.72333 q^{16} +4.20766 q^{17} -6.17885 q^{18} -1.00000 q^{19} +0.498615 q^{21} +2.12341 q^{22} +3.54516 q^{23} -0.324390 q^{24} +0.862965 q^{26} +1.77423 q^{27} -4.16688 q^{28} +7.47483 q^{29} +5.10468 q^{31} -7.94379 q^{32} -0.300215 q^{33} +8.93459 q^{34} -7.30047 q^{36} -1.45804 q^{37} -2.12341 q^{38} -0.122009 q^{39} +8.69765 q^{41} +1.05876 q^{42} -0.953442 q^{43} +2.50886 q^{44} +7.52782 q^{46} +5.58315 q^{47} +0.817583 q^{48} -4.24154 q^{49} -1.26320 q^{51} +1.01962 q^{52} -1.77008 q^{53} +3.76742 q^{54} -1.79461 q^{56} +0.300215 q^{57} +15.8721 q^{58} +6.26485 q^{59} -1.18689 q^{61} +10.8393 q^{62} +4.83290 q^{63} -11.4213 q^{64} -0.637479 q^{66} +12.1533 q^{67} +10.5565 q^{68} -1.06431 q^{69} +5.93683 q^{71} -3.14420 q^{72} +2.39287 q^{73} -3.09601 q^{74} -2.50886 q^{76} -1.66086 q^{77} -0.259075 q^{78} -14.2878 q^{79} +8.19696 q^{81} +18.4687 q^{82} +8.85637 q^{83} +1.25096 q^{84} -2.02455 q^{86} -2.24405 q^{87} +1.08053 q^{88} -2.79190 q^{89} -0.674984 q^{91} +8.89432 q^{92} -1.53250 q^{93} +11.8553 q^{94} +2.38484 q^{96} +6.33470 q^{97} -9.00651 q^{98} -2.90987 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * 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 $

$2$	2.12341	1.50148	0.750738	−	0.660600i	$-0.229698\pi$
$2$	2.12341	1.50148	0.750738	+	0.660600i	$0.229698\pi$
$3$	−0.300215	−0.173329	−0.0866645	−	0.996238i	$-0.527621\pi$
$3$	−0.300215	−0.173329	−0.0866645	+	0.996238i	$0.527621\pi$
$4$	2.50886	1.25443
$5$	0	0
$5$	0	0
$6$	−0.637479	−0.260250
$7$	−1.66086	−0.627747	−0.313874	−	0.949465i	$-0.601627\pi$
$7$	−1.66086	−0.627747	−0.313874	+	0.949465i	$0.601627\pi$
$8$	1.08053	0.382024
$9$	−2.90987	−0.969957
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.753198	−0.217430
$13$	0.406405	0.112717	0.0563583	−	0.998411i	$-0.482051\pi$
$13$	0.406405	0.112717	0.0563583	+	0.998411i	$0.482051\pi$
$14$	−3.52669	−0.942548
$15$	0	0
$16$	−2.72333	−0.680832
$17$	4.20766	1.02051	0.510254	−	0.860024i	$-0.329551\pi$
$17$	4.20766	1.02051	0.510254	+	0.860024i	$0.329551\pi$
$18$	−6.17885	−1.45637
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.498615	0.108807
$22$	2.12341	0.452712
$23$	3.54516	0.739217	0.369608	−	0.929188i	$-0.379492\pi$
$23$	3.54516	0.739217	0.369608	+	0.929188i	$0.379492\pi$
$24$	−0.324390	−0.0662159
$25$	0	0
$26$	0.862965	0.169241
$27$	1.77423	0.341451
$28$	−4.16688	−0.787466
$29$	7.47483	1.38804	0.694021	−	0.719955i	$-0.255838\pi$
$29$	7.47483	1.38804	0.694021	+	0.719955i	$0.255838\pi$
$30$	0	0
$31$	5.10468	0.916827	0.458414	−	0.888739i	$-0.348418\pi$
$31$	5.10468	0.916827	0.458414	+	0.888739i	$0.348418\pi$
$32$	−7.94379	−1.40428
$33$	−0.300215	−0.0522607
$34$	8.93459	1.53227
$35$	0	0
$36$	−7.30047	−1.21675
$37$	−1.45804	−0.239700	−0.119850	−	0.992792i	$-0.538241\pi$
$37$	−1.45804	−0.239700	−0.119850	+	0.992792i	$0.538241\pi$
$38$	−2.12341	−0.344462
$39$	−0.122009	−0.0195371
$40$	0	0
$41$	8.69765	1.35834	0.679172	−	0.733979i	$-0.262339\pi$
$41$	8.69765	1.35834	0.679172	+	0.733979i	$0.262339\pi$
$42$	1.05876	0.163371
$43$	−0.953442	−0.145399	−0.0726993	−	0.997354i	$-0.523161\pi$
$43$	−0.953442	−0.145399	−0.0726993	+	0.997354i	$0.523161\pi$
$44$	2.50886	0.378226
$45$	0	0
$46$	7.52782	1.10992
$47$	5.58315	0.814386	0.407193	−	0.913342i	$-0.366508\pi$
$47$	5.58315	0.814386	0.407193	+	0.913342i	$0.366508\pi$
$48$	0.817583	0.118008
$49$	−4.24154	−0.605934
$50$	0	0
$51$	−1.26320	−0.176884
$52$	1.01962	0.141395
$53$	−1.77008	−0.243139	−0.121570	−	0.992583i	$-0.538793\pi$
$53$	−1.77008	−0.243139	−0.121570	+	0.992583i	$0.538793\pi$
$54$	3.76742	0.512680
$55$	0	0
$56$	−1.79461	−0.239815
$57$	0.300215	0.0397644
$58$	15.8721	2.08411
$59$	6.26485	0.815614	0.407807	−	0.913068i	$-0.366294\pi$
$59$	6.26485	0.815614	0.407807	+	0.913068i	$0.366294\pi$
$60$	0	0
$61$	−1.18689	−0.151965	−0.0759827	−	0.997109i	$-0.524209\pi$
$61$	−1.18689	−0.151965	−0.0759827	+	0.997109i	$0.524209\pi$
$62$	10.8393	1.37660
$63$	4.83290	0.608888
$64$	−11.4213	−1.42766
$65$	0	0
$66$	−0.637479	−0.0784682
$67$	12.1533	1.48476	0.742381	−	0.669978i	$-0.233697\pi$
$67$	12.1533	1.48476	0.742381	+	0.669978i	$0.233697\pi$
$68$	10.5565	1.28016
$69$	−1.06431	−0.128128
$70$	0	0
$71$	5.93683	0.704572	0.352286	−	0.935892i	$-0.385405\pi$
$71$	5.93683	0.704572	0.352286	+	0.935892i	$0.385405\pi$
$72$	−3.14420	−0.370547
$73$	2.39287	0.280064	0.140032	−	0.990147i	$-0.455279\pi$
$73$	2.39287	0.280064	0.140032	+	0.990147i	$0.455279\pi$
$74$	−3.09601	−0.359904
$75$	0	0
$76$	−2.50886	−0.287787
$77$	−1.66086	−0.189273
$78$	−0.259075	−0.0293344
$79$	−14.2878	−1.60751	−0.803754	−	0.594962i	$-0.797167\pi$
$79$	−14.2878	−1.60751	−0.803754	+	0.594962i	$0.797167\pi$
$80$	0	0
$81$	8.19696	0.910774
$82$	18.4687	2.03952
$83$	8.85637	0.972113	0.486057	−	0.873927i	$-0.338435\pi$
$83$	8.85637	0.972113	0.486057	+	0.873927i	$0.338435\pi$
$84$	1.25096	0.136491
$85$	0	0
$86$	−2.02455	−0.218313
$87$	−2.24405	−0.240588
$88$	1.08053	0.115185
$89$	−2.79190	−0.295940	−0.147970	−	0.988992i	$-0.547274\pi$
$89$	−2.79190	−0.295940	−0.147970	+	0.988992i	$0.547274\pi$
$90$	0	0
$91$	−0.674984	−0.0707575
$92$	8.89432	0.927297
$93$	−1.53250	−0.158913
$94$	11.8553	1.22278
$95$	0	0
$96$	2.38484	0.243402
$97$	6.33470	0.643192	0.321596	−	0.946877i	$-0.395781\pi$
$97$	6.33470	0.643192	0.321596	+	0.946877i	$0.395781\pi$
$98$	−9.00651	−0.909795
$99$	−2.90987	−0.292453
$100$	0	0
$101$	12.8093	1.27458	0.637289	−	0.770625i	$-0.280056\pi$
$101$	12.8093	1.27458	0.637289	+	0.770625i	$0.280056\pi$
$102$	−2.68229	−0.265587
$103$	−13.6053	−1.34057	−0.670284	−	0.742105i	$-0.733828\pi$
$103$	−13.6053	−1.34057	−0.670284	+	0.742105i	$0.733828\pi$
$104$	0.439132	0.0430605
$105$	0	0
$106$	−3.75860	−0.365068
$107$	10.3908	1.00452	0.502258	−	0.864718i	$-0.332503\pi$
$107$	10.3908	1.00452	0.502258	+	0.864718i	$0.332503\pi$
$108$	4.45130	0.428327
$109$	6.56231	0.628555	0.314277	−	0.949331i	$-0.398238\pi$
$109$	6.56231	0.628555	0.314277	+	0.949331i	$0.398238\pi$
$110$	0	0
$111$	0.437725	0.0415470
$112$	4.52307	0.427390
$113$	1.73328	0.163053	0.0815267	−	0.996671i	$-0.474020\pi$
$113$	1.73328	0.163053	0.0815267	+	0.996671i	$0.474020\pi$
$114$	0.637479	0.0597053
$115$	0	0
$116$	18.7533	1.74120
$117$	−1.18259	−0.109330
$118$	13.3028	1.22463
$119$	−6.98835	−0.640621
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.52025	−0.228172
$123$	−2.61116	−0.235441
$124$	12.8070	1.15010
$125$	0	0
$126$	10.2622	0.914231
$127$	6.49967	0.576753	0.288376	−	0.957517i	$-0.406885\pi$
$127$	6.49967	0.576753	0.288376	+	0.957517i	$0.406885\pi$
$128$	−8.36443	−0.739318
$129$	0.286237	0.0252018
$130$	0	0
$131$	1.28315	0.112109	0.0560546	−	0.998428i	$-0.482148\pi$
$131$	1.28315	0.112109	0.0560546	+	0.998428i	$0.482148\pi$
$132$	−0.753198	−0.0655575
$133$	1.66086	0.144015
$134$	25.8064	2.22934
$135$	0	0
$136$	4.54650	0.389859
$137$	6.33302	0.541067	0.270533	−	0.962711i	$-0.412800\pi$
$137$	6.33302	0.541067	0.270533	+	0.962711i	$0.412800\pi$
$138$	−2.25996	−0.192381
$139$	−4.92149	−0.417435	−0.208717	−	0.977976i	$-0.566929\pi$
$139$	−4.92149	−0.417435	−0.208717	+	0.977976i	$0.566929\pi$
$140$	0	0
$141$	−1.67614	−0.141157
$142$	12.6063	1.05790
$143$	0.406405	0.0339853
$144$	7.92453	0.660378
$145$	0	0
$146$	5.08104	0.420510
$147$	1.27337	0.105026
$148$	−3.65802	−0.300688
$149$	−11.1592	−0.914194	−0.457097	−	0.889417i	$-0.651111\pi$
$149$	−11.1592	−0.914194	−0.457097	+	0.889417i	$0.651111\pi$
$150$	0	0
$151$	−9.38884	−0.764053	−0.382027	−	0.924151i	$-0.624774\pi$
$151$	−9.38884	−0.764053	−0.382027	+	0.924151i	$0.624774\pi$
$152$	−1.08053	−0.0876424
$153$	−12.2438	−0.989849
$154$	−3.52669	−0.284189
$155$	0	0
$156$	−0.306104	−0.0245079
$157$	19.9255	1.59022	0.795112	−	0.606462i	$-0.207412\pi$
$157$	19.9255	1.59022	0.795112	+	0.606462i	$0.207412\pi$
$158$	−30.3389	−2.41364
$159$	0.531404	0.0421431
$160$	0	0
$161$	−5.88802	−0.464041
$162$	17.4055	1.36751
$163$	−18.6951	−1.46431	−0.732156	−	0.681137i	$-0.761486\pi$
$163$	−18.6951	−1.46431	−0.732156	+	0.681137i	$0.761486\pi$
$164$	21.8212	1.70395
$165$	0	0
$166$	18.8057	1.45961
$167$	5.92467	0.458465	0.229233	−	0.973372i	$-0.426378\pi$
$167$	5.92467	0.458465	0.229233	+	0.973372i	$0.426378\pi$
$168$	0.538768	0.0415668
$169$	−12.8348	−0.987295
$170$	0	0
$171$	2.90987	0.222523
$172$	−2.39206	−0.182393
$173$	4.45145	0.338438	0.169219	−	0.985579i	$-0.445876\pi$
$173$	4.45145	0.338438	0.169219	+	0.985579i	$0.445876\pi$
$174$	−4.76504	−0.361237
$175$	0	0
$176$	−2.72333	−0.205279
$177$	−1.88080	−0.141370
$178$	−5.92834	−0.444348
$179$	−5.29409	−0.395699	−0.197850	−	0.980232i	$-0.563396\pi$
$179$	−5.29409	−0.395699	−0.197850	+	0.980232i	$0.563396\pi$
$180$	0	0
$181$	−24.6556	−1.83264	−0.916319	−	0.400450i	$-0.868854\pi$
$181$	−24.6556	−1.83264	−0.916319	+	0.400450i	$0.868854\pi$
$182$	−1.43327	−0.106241
$183$	0.356321	0.0263400
$184$	3.83064	0.282399
$185$	0	0
$186$	−3.25412	−0.238604
$187$	4.20766	0.307695
$188$	14.0074	1.02159
$189$	−2.94675	−0.214345
$190$	0	0
$191$	25.2282	1.82545	0.912724	−	0.408577i	$-0.133975\pi$
$191$	25.2282	1.82545	0.912724	+	0.408577i	$0.133975\pi$
$192$	3.42883	0.247455
$193$	0.939382	0.0676182	0.0338091	−	0.999428i	$-0.489236\pi$
$193$	0.939382	0.0676182	0.0338091	+	0.999428i	$0.489236\pi$
$194$	13.4512	0.965737
$195$	0	0
$196$	−10.6414	−0.760103
$197$	−6.91170	−0.492438	−0.246219	−	0.969214i	$-0.579188\pi$
$197$	−6.91170	−0.492438	−0.246219	+	0.969214i	$0.579188\pi$
$198$	−6.17885	−0.439111
$199$	22.5252	1.59677	0.798385	−	0.602147i	$-0.205688\pi$
$199$	22.5252	1.59677	0.798385	+	0.602147i	$0.205688\pi$
$200$	0	0
$201$	−3.64860	−0.257352
$202$	27.1995	1.91375
$203$	−12.4147	−0.871339
$204$	−3.16920	−0.221889
$205$	0	0
$206$	−28.8896	−2.01283
$207$	−10.3160	−0.717008
$208$	−1.10678	−0.0767410
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−13.5534	−0.933054	−0.466527	−	0.884507i	$-0.654495\pi$
$211$	−13.5534	−0.933054	−0.466527	+	0.884507i	$0.654495\pi$
$212$	−4.44089	−0.305002
$213$	−1.78232	−0.122123
$214$	22.0639	1.50826
$215$	0	0
$216$	1.91711	0.130442
$217$	−8.47817	−0.575536
$218$	13.9345	0.943761
$219$	−0.718375	−0.0485433
$220$	0	0
$221$	1.71002	0.115028
$222$	0.929469	0.0623819
$223$	3.13674	0.210052	0.105026	−	0.994469i	$-0.466507\pi$
$223$	3.13674	0.210052	0.105026	+	0.994469i	$0.466507\pi$
$224$	13.1935	0.881531
$225$	0	0
$226$	3.68046	0.244821
$227$	−17.7305	−1.17681	−0.588407	−	0.808565i	$-0.700245\pi$
$227$	−17.7305	−1.17681	−0.588407	+	0.808565i	$0.700245\pi$
$228$	0.753198	0.0498818
$229$	−1.58042	−0.104437	−0.0522186	−	0.998636i	$-0.516629\pi$
$229$	−1.58042	−0.104437	−0.0522186	+	0.998636i	$0.516629\pi$
$230$	0	0
$231$	0.498615	0.0328065
$232$	8.07676	0.530265
$233$	10.1587	0.665519	0.332760	−	0.943012i	$-0.392020\pi$
$233$	10.1587	0.665519	0.332760	+	0.943012i	$0.392020\pi$
$234$	−2.51112	−0.164157
$235$	0	0
$236$	15.7177	1.02313
$237$	4.28942	0.278628
$238$	−14.8391	−0.961878
$239$	−6.60437	−0.427201	−0.213601	−	0.976921i	$-0.568519\pi$
$239$	−6.60437	−0.427201	−0.213601	+	0.976921i	$0.568519\pi$
$240$	0	0
$241$	−27.4790	−1.77008	−0.885041	−	0.465514i	$-0.845869\pi$
$241$	−27.4790	−1.77008	−0.885041	+	0.465514i	$0.845869\pi$
$242$	2.12341	0.136498
$243$	−7.78354	−0.499314
$244$	−2.97774	−0.190630
$245$	0	0
$246$	−5.54457	−0.353509
$247$	−0.406405	−0.0258590
$248$	5.51575	0.350250
$249$	−2.65881	−0.168495
$250$	0	0
$251$	−6.10901	−0.385597	−0.192799	−	0.981238i	$-0.561756\pi$
$251$	−6.10901	−0.385597	−0.192799	+	0.981238i	$0.561756\pi$
$252$	12.1251	0.763808
$253$	3.54516	0.222882
$254$	13.8015	0.865981
$255$	0	0
$256$	5.08143	0.317589
$257$	0.740664	0.0462014	0.0231007	−	0.999733i	$-0.492646\pi$
$257$	0.740664	0.0462014	0.0231007	+	0.999733i	$0.492646\pi$
$258$	0.607799	0.0378399
$259$	2.42160	0.150471
$260$	0	0
$261$	−21.7508	−1.34634
$262$	2.72465	0.168329
$263$	−0.968172	−0.0597001	−0.0298500	−	0.999554i	$-0.509503\pi$
$263$	−0.968172	−0.0597001	−0.0298500	+	0.999554i	$0.509503\pi$
$264$	−0.324390	−0.0199648
$265$	0	0
$266$	3.52669	0.216235
$267$	0.838168	0.0512951
$268$	30.4910	1.86253
$269$	−4.52415	−0.275842	−0.137921	−	0.990443i	$-0.544042\pi$
$269$	−4.52415	−0.275842	−0.137921	+	0.990443i	$0.544042\pi$
$270$	0	0
$271$	3.44651	0.209361	0.104680	−	0.994506i	$-0.466618\pi$
$271$	3.44651	0.209361	0.104680	+	0.994506i	$0.466618\pi$
$272$	−11.4588	−0.694794
$273$	0.202640	0.0122643
$274$	13.4476	0.812399
$275$	0	0
$276$	−2.67021	−0.160728
$277$	−5.47688	−0.329074	−0.164537	−	0.986371i	$-0.552613\pi$
$277$	−5.47688	−0.329074	−0.164537	+	0.986371i	$0.552613\pi$
$278$	−10.4503	−0.626769
$279$	−14.8540	−0.889283
$280$	0	0
$281$	20.1704	1.20327	0.601633	−	0.798773i	$-0.294517\pi$
$281$	20.1704	1.20327	0.601633	+	0.798773i	$0.294517\pi$
$282$	−3.55914	−0.211944
$283$	−7.06912	−0.420215	−0.210108	−	0.977678i	$-0.567381\pi$
$283$	−7.06912	−0.420215	−0.210108	+	0.977678i	$0.567381\pi$
$284$	14.8947	0.883837
$285$	0	0
$286$	0.862965	0.0510282
$287$	−14.4456	−0.852697
$288$	23.1154	1.36209
$289$	0.704429	0.0414370
$290$	0	0
$291$	−1.90177	−0.111484
$292$	6.00339	0.351322
$293$	25.8891	1.51246	0.756228	−	0.654308i	$-0.227040\pi$
$293$	25.8891	1.51246	0.756228	+	0.654308i	$0.227040\pi$
$294$	2.70389	0.157694
$295$	0	0
$296$	−1.57545	−0.0915713
$297$	1.77423	0.102951
$298$	−23.6954	−1.37264
$299$	1.44077	0.0833220
$300$	0	0
$301$	1.58354	0.0912735
$302$	−19.9364	−1.14721
$303$	−3.84555	−0.220921
$304$	2.72333	0.156194
$305$	0	0
$306$	−25.9985	−1.48624
$307$	34.3162	1.95853	0.979264	−	0.202590i	$-0.0649358\pi$
$307$	34.3162	1.95853	0.979264	+	0.202590i	$0.0649358\pi$
$308$	−4.16688	−0.237430
$309$	4.08450	0.232359
$310$	0	0
$311$	12.9422	0.733887	0.366944	−	0.930243i	$-0.380404\pi$
$311$	12.9422	0.733887	0.366944	+	0.930243i	$0.380404\pi$
$312$	−0.131834	−0.00746363
$313$	6.03044	0.340860	0.170430	−	0.985370i	$-0.445484\pi$
$313$	6.03044	0.340860	0.170430	+	0.985370i	$0.445484\pi$
$314$	42.3099	2.38769
$315$	0	0
$316$	−35.8463	−2.01651
$317$	7.55456	0.424306	0.212153	−	0.977236i	$-0.431952\pi$
$317$	7.55456	0.424306	0.212153	+	0.977236i	$0.431952\pi$
$318$	1.12839	0.0632768
$319$	7.47483	0.418510
$320$	0	0
$321$	−3.11947	−0.174112
$322$	−12.5027	−0.696747
$323$	−4.20766	−0.234121
$324$	20.5651	1.14250
$325$	0	0
$326$	−39.6973	−2.19863
$327$	−1.97010	−0.108947
$328$	9.39805	0.518921
$329$	−9.27284	−0.511228
$330$	0	0
$331$	−21.3753	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$331$	−21.3753	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$332$	22.2194	1.21945
$333$	4.24271	0.232499
$334$	12.5805	0.688375
$335$	0	0
$336$	−1.35789	−0.0740791
$337$	−20.6132	−1.12287	−0.561436	−	0.827520i	$-0.689751\pi$
$337$	−20.6132	−1.12287	−0.561436	+	0.827520i	$0.689751\pi$
$338$	−27.2536	−1.48240
$339$	−0.520357	−0.0282619
$340$	0	0
$341$	5.10468	0.276434
$342$	6.17885	0.334114
$343$	18.6706	1.00812
$344$	−1.03022	−0.0555458
$345$	0	0
$346$	9.45225	0.508156
$347$	32.0956	1.72298	0.861490	−	0.507774i	$-0.169532\pi$
$347$	32.0956	1.72298	0.861490	+	0.507774i	$0.169532\pi$
$348$	−5.63003	−0.301801
$349$	−1.86111	−0.0996232	−0.0498116	−	0.998759i	$-0.515862\pi$
$349$	−1.86111	−0.0996232	−0.0498116	+	0.998759i	$0.515862\pi$
$350$	0	0
$351$	0.721057	0.0384872
$352$	−7.94379	−0.423406
$353$	−11.1724	−0.594648	−0.297324	−	0.954777i	$-0.596094\pi$
$353$	−11.1724	−0.594648	−0.297324	+	0.954777i	$0.596094\pi$
$354$	−3.99371	−0.212263
$355$	0	0
$356$	−7.00449	−0.371237
$357$	2.09801	0.111038
$358$	−11.2415	−0.594133
$359$	8.55089	0.451299	0.225649	−	0.974209i	$-0.427550\pi$
$359$	8.55089	0.451299	0.225649	+	0.974209i	$0.427550\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−52.3540	−2.75166
$363$	−0.300215	−0.0157572
$364$	−1.69344	−0.0887605
$365$	0	0
$366$	0.756615	0.0395489
$367$	−6.53421	−0.341083	−0.170542	−	0.985350i	$-0.554552\pi$
$367$	−6.53421	−0.341083	−0.170542	+	0.985350i	$0.554552\pi$
$368$	−9.65463	−0.503282
$369$	−25.3090	−1.31754
$370$	0	0
$371$	2.93986	0.152630
$372$	−3.84483	−0.199345
$373$	−29.9850	−1.55257	−0.776283	−	0.630385i	$-0.782897\pi$
$373$	−29.9850	−1.55257	−0.776283	+	0.630385i	$0.782897\pi$
$374$	8.93459	0.461997
$375$	0	0
$376$	6.03275	0.311115
$377$	3.03781	0.156455
$378$	−6.25716	−0.321834
$379$	5.14929	0.264501	0.132251	−	0.991216i	$-0.457780\pi$
$379$	5.14929	0.264501	0.132251	+	0.991216i	$0.457780\pi$
$380$	0	0
$381$	−1.95130	−0.0999680
$382$	53.5698	2.74087
$383$	5.87340	0.300117	0.150058	−	0.988677i	$-0.452054\pi$
$383$	5.87340	0.300117	0.150058	+	0.988677i	$0.452054\pi$
$384$	2.51112	0.128145
$385$	0	0
$386$	1.99469	0.101527
$387$	2.77439	0.141030
$388$	15.8929	0.806841
$389$	17.3191	0.878113	0.439057	−	0.898459i	$-0.355313\pi$
$389$	17.3191	0.878113	0.439057	+	0.898459i	$0.355313\pi$
$390$	0	0
$391$	14.9168	0.754377
$392$	−4.58310	−0.231481
$393$	−0.385220	−0.0194318
$394$	−14.6764	−0.739385
$395$	0	0
$396$	−7.30047	−0.366863
$397$	−2.93936	−0.147522	−0.0737610	−	0.997276i	$-0.523500\pi$
$397$	−2.93936	−0.147522	−0.0737610	+	0.997276i	$0.523500\pi$
$398$	47.8302	2.39751
$399$	−0.498615	−0.0249620
$400$	0	0
$401$	−39.8344	−1.98924	−0.994618	−	0.103609i	$-0.966961\pi$
$401$	−39.8344	−1.98924	−0.994618	+	0.103609i	$0.966961\pi$
$402$	−7.74747	−0.386409
$403$	2.07457	0.103342
$404$	32.1369	1.59887
$405$	0	0
$406$	−26.3614	−1.30829
$407$	−1.45804	−0.0722724
$408$	−1.36493	−0.0675739
$409$	−34.7653	−1.71904	−0.859518	−	0.511106i	$-0.829236\pi$
$409$	−34.7653	−1.71904	−0.859518	+	0.511106i	$0.829236\pi$
$410$	0	0
$411$	−1.90127	−0.0937826
$412$	−34.1338	−1.68165
$413$	−10.4051	−0.511999
$414$	−21.9050	−1.07657
$415$	0	0
$416$	−3.22840	−0.158285
$417$	1.47750	0.0723536
$418$	−2.12341	−0.103859
$419$	4.20222	0.205292	0.102646	−	0.994718i	$-0.467269\pi$
$419$	4.20222	0.205292	0.102646	+	0.994718i	$0.467269\pi$
$420$	0	0
$421$	20.9129	1.01923	0.509616	−	0.860402i	$-0.329788\pi$
$421$	20.9129	1.01923	0.509616	+	0.860402i	$0.329788\pi$
$422$	−28.7794	−1.40096
$423$	−16.2462	−0.789919
$424$	−1.91262	−0.0928850
$425$	0	0
$426$	−3.78460	−0.183364
$427$	1.97126	0.0953958
$428$	26.0691	1.26010
$429$	−0.122009	−0.00589065
$430$	0	0
$431$	10.5831	0.509771	0.254886	−	0.966971i	$-0.417962\pi$
$431$	10.5831	0.509771	0.254886	+	0.966971i	$0.417962\pi$
$432$	−4.83181	−0.232471
$433$	11.8075	0.567430	0.283715	−	0.958909i	$-0.408433\pi$
$433$	11.8075	0.567430	0.283715	+	0.958909i	$0.408433\pi$
$434$	−18.0026	−0.864154
$435$	0	0
$436$	16.4639	0.788480
$437$	−3.54516	−0.169588
$438$	−1.52540	−0.0728866
$439$	−9.66940	−0.461495	−0.230748	−	0.973014i	$-0.574117\pi$
$439$	−9.66940	−0.461495	−0.230748	+	0.973014i	$0.574117\pi$
$440$	0	0
$441$	12.3423	0.587730
$442$	3.63107	0.172712
$443$	26.2089	1.24522	0.622610	−	0.782532i	$-0.286072\pi$
$443$	26.2089	1.24522	0.622610	+	0.782532i	$0.286072\pi$
$444$	1.09819	0.0521179
$445$	0	0
$446$	6.66059	0.315388
$447$	3.35014	0.158456
$448$	18.9691	0.896208
$449$	1.35633	0.0640090	0.0320045	−	0.999488i	$-0.489811\pi$
$449$	1.35633	0.0640090	0.0320045	+	0.999488i	$0.489811\pi$
$450$	0	0
$451$	8.69765	0.409556
$452$	4.34857	0.204539
$453$	2.81867	0.132433
$454$	−37.6491	−1.76696
$455$	0	0
$456$	0.324390	0.0151910
$457$	0.925283	0.0432829	0.0216415	−	0.999766i	$-0.493111\pi$
$457$	0.925283	0.0432829	0.0216415	+	0.999766i	$0.493111\pi$
$458$	−3.35588	−0.156810
$459$	7.46536	0.348453
$460$	0	0
$461$	−17.9298	−0.835073	−0.417536	−	0.908660i	$-0.637106\pi$
$461$	−17.9298	−0.835073	−0.417536	+	0.908660i	$0.637106\pi$
$462$	1.05876	0.0492582
$463$	−18.3406	−0.852360	−0.426180	−	0.904638i	$-0.640141\pi$
$463$	−18.3406	−0.852360	−0.426180	+	0.904638i	$0.640141\pi$
$464$	−20.3564	−0.945023
$465$	0	0
$466$	21.5711	0.999262
$467$	−9.78809	−0.452939	−0.226469	−	0.974018i	$-0.572718\pi$
$467$	−9.78809	−0.452939	−0.226469	+	0.974018i	$0.572718\pi$
$468$	−2.96695	−0.137147
$469$	−20.1850	−0.932055
$470$	0	0
$471$	−5.98192	−0.275632
$472$	6.76934	0.311584
$473$	−0.953442	−0.0438393
$474$	9.10820	0.418353
$475$	0	0
$476$	−17.5328	−0.803616
$477$	5.15070	0.235834
$478$	−14.0238	−0.641433
$479$	−23.2528	−1.06245	−0.531223	−	0.847232i	$-0.678267\pi$
$479$	−23.2528	−1.06245	−0.531223	+	0.847232i	$0.678267\pi$
$480$	0	0
$481$	−0.592555	−0.0270182
$482$	−58.3492	−2.65774
$483$	1.76767	0.0804318
$484$	2.50886	0.114039
$485$	0	0
$486$	−16.5276	−0.749709
$487$	20.1880	0.914807	0.457403	−	0.889259i	$-0.348780\pi$
$487$	20.1880	0.914807	0.457403	+	0.889259i	$0.348780\pi$
$488$	−1.28247	−0.0580545
$489$	5.61254	0.253808
$490$	0	0
$491$	−39.7054	−1.79188	−0.895940	−	0.444176i	$-0.853497\pi$
$491$	−39.7054	−1.79188	−0.895940	+	0.444176i	$0.853497\pi$
$492$	−6.55105	−0.295344
$493$	31.4516	1.41651
$494$	−0.862965	−0.0388266
$495$	0	0
$496$	−13.9017	−0.624205
$497$	−9.86025	−0.442293
$498$	−5.64575	−0.252992
$499$	−3.54572	−0.158728	−0.0793640	−	0.996846i	$-0.525289\pi$
$499$	−3.54572	−0.158728	−0.0793640	+	0.996846i	$0.525289\pi$
$500$	0	0
$501$	−1.77867	−0.0794653
$502$	−12.9719	−0.578966
$503$	14.6798	0.654542	0.327271	−	0.944931i	$-0.393871\pi$
$503$	14.6798	0.654542	0.327271	+	0.944931i	$0.393871\pi$
$504$	5.22208	0.232610
$505$	0	0
$506$	7.52782	0.334652
$507$	3.85321	0.171127
$508$	16.3068	0.723497
$509$	−16.2586	−0.720651	−0.360325	−	0.932827i	$-0.617334\pi$
$509$	−16.2586	−0.720651	−0.360325	+	0.932827i	$0.617334\pi$
$510$	0	0
$511$	−3.97423	−0.175810
$512$	27.5188	1.21617
$513$	−1.77423	−0.0783342
$514$	1.57273	0.0693703
$515$	0	0
$516$	0.718131	0.0316139
$517$	5.58315	0.245547
$518$	5.14205	0.225929
$519$	−1.33639	−0.0586611
$520$	0	0
$521$	30.2820	1.32668	0.663338	−	0.748320i	$-0.269139\pi$
$521$	30.2820	1.32668	0.663338	+	0.748320i	$0.269139\pi$
$522$	−46.1858	−2.02150
$523$	22.9391	1.00305	0.501527	−	0.865142i	$-0.332772\pi$
$523$	22.9391	1.00305	0.501527	+	0.865142i	$0.332772\pi$
$524$	3.21925	0.140634
$525$	0	0
$526$	−2.05583	−0.0896383
$527$	21.4788	0.935630
$528$	0.817583	0.0355807
$529$	−10.4319	−0.453559
$530$	0	0
$531$	−18.2299	−0.791111
$532$	4.16688	0.180657
$533$	3.53477	0.153108
$534$	1.77977	0.0770184
$535$	0	0
$536$	13.1320	0.567215
$537$	1.58936	0.0685861
$538$	−9.60661	−0.414170
$539$	−4.24154	−0.182696
$540$	0	0
$541$	−41.8092	−1.79752	−0.898758	−	0.438445i	$-0.855529\pi$
$541$	−41.8092	−1.79752	−0.898758	+	0.438445i	$0.855529\pi$
$542$	7.31835	0.314350
$543$	7.40198	0.317649
$544$	−33.4248	−1.43308
$545$	0	0
$546$	0.430288	0.0184146
$547$	18.4227	0.787698	0.393849	−	0.919175i	$-0.371143\pi$
$547$	18.4227	0.787698	0.393849	+	0.919175i	$0.371143\pi$
$548$	15.8887	0.678731
$549$	3.45369	0.147400
$550$	0	0
$551$	−7.47483	−0.318438
$552$	−1.15002	−0.0489479
$553$	23.7301	1.00911
$554$	−11.6296	−0.494096
$555$	0	0
$556$	−12.3473	−0.523644
$557$	−25.4381	−1.07785	−0.538923	−	0.842355i	$-0.681168\pi$
$557$	−25.4381	−1.07785	−0.538923	+	0.842355i	$0.681168\pi$
$558$	−31.5410	−1.33524
$559$	−0.387484	−0.0163888
$560$	0	0
$561$	−1.26320	−0.0533324
$562$	42.8300	1.80668
$563$	17.9758	0.757589	0.378795	−	0.925481i	$-0.376339\pi$
$563$	17.9758	0.757589	0.378795	+	0.925481i	$0.376339\pi$
$564$	−4.20522	−0.177072
$565$	0	0
$566$	−15.0106	−0.630943
$567$	−13.6140	−0.571735
$568$	6.41491	0.269163
$569$	−6.12536	−0.256789	−0.128394	−	0.991723i	$-0.540982\pi$
$569$	−6.12536	−0.256789	−0.128394	+	0.991723i	$0.540982\pi$
$570$	0	0
$571$	16.5581	0.692936	0.346468	−	0.938062i	$-0.387381\pi$
$571$	16.5581	0.692936	0.346468	+	0.938062i	$0.387381\pi$
$572$	1.01962	0.0426323
$573$	−7.57387	−0.316403
$574$	−30.6739	−1.28030
$575$	0	0
$576$	33.2344	1.38477
$577$	43.0297	1.79135	0.895676	−	0.444708i	$-0.146693\pi$
$577$	43.0297	1.79135	0.895676	+	0.444708i	$0.146693\pi$
$578$	1.49579	0.0622167
$579$	−0.282016	−0.0117202
$580$	0	0
$581$	−14.7092	−0.610241
$582$	−4.03824	−0.167390
$583$	−1.77008	−0.0733092
$584$	2.58556	0.106991
$585$	0	0
$586$	54.9731	2.27092
$587$	5.74599	0.237162	0.118581	−	0.992944i	$-0.462165\pi$
$587$	5.74599	0.237162	0.118581	+	0.992944i	$0.462165\pi$
$588$	3.19472	0.131748
$589$	−5.10468	−0.210335
$590$	0	0
$591$	2.07499	0.0853539
$592$	3.97072	0.163196
$593$	−11.8917	−0.488334	−0.244167	−	0.969733i	$-0.578514\pi$
$593$	−11.8917	−0.488334	−0.244167	+	0.969733i	$0.578514\pi$
$594$	3.76742	0.154579
$595$	0	0
$596$	−27.9968	−1.14679
$597$	−6.76240	−0.276767
$598$	3.05935	0.125106
$599$	34.5106	1.41007	0.705033	−	0.709174i	$-0.250932\pi$
$599$	34.5106	1.41007	0.705033	+	0.709174i	$0.250932\pi$
$600$	0	0
$601$	−28.8714	−1.17769	−0.588845	−	0.808246i	$-0.700417\pi$
$601$	−28.8714	−1.17769	−0.588845	+	0.808246i	$0.700417\pi$
$602$	3.36249	0.137045
$603$	−35.3645	−1.44016
$604$	−23.5553	−0.958453
$605$	0	0
$606$	−8.16568	−0.331708
$607$	−19.5305	−0.792718	−0.396359	−	0.918096i	$-0.629726\pi$
$607$	−19.5305	−0.792718	−0.396359	+	0.918096i	$0.629726\pi$
$608$	7.94379	0.322163
$609$	3.72706	0.151028
$610$	0	0
$611$	2.26902	0.0917948
$612$	−30.7179	−1.24170
$613$	34.5585	1.39581	0.697903	−	0.716193i	$-0.254117\pi$
$613$	34.5585	1.39581	0.697903	+	0.716193i	$0.254117\pi$
$614$	72.8673	2.94068
$615$	0	0
$616$	−1.79461	−0.0723068
$617$	22.1066	0.889980	0.444990	−	0.895536i	$-0.353207\pi$
$617$	22.1066	0.889980	0.444990	+	0.895536i	$0.353207\pi$
$618$	8.67307	0.348882
$619$	45.0191	1.80947	0.904736	−	0.425973i	$-0.140068\pi$
$619$	45.0191	1.80947	0.904736	+	0.425973i	$0.140068\pi$
$620$	0	0
$621$	6.28993	0.252406
$622$	27.4817	1.10191
$623$	4.63696	0.185776
$624$	0.332270	0.0133015
$625$	0	0
$626$	12.8051	0.511794
$627$	0.300215	0.0119894
$628$	49.9903	1.99483
$629$	−6.13494	−0.244616
$630$	0	0
$631$	1.39480	0.0555261	0.0277631	−	0.999615i	$-0.491162\pi$
$631$	1.39480	0.0555261	0.0277631	+	0.999615i	$0.491162\pi$
$632$	−15.4384	−0.614107
$633$	4.06893	0.161725
$634$	16.0414	0.637086
$635$	0	0
$636$	1.33322	0.0528656
$637$	−1.72378	−0.0682988
$638$	15.8721	0.628383
$639$	−17.2754	−0.683404
$640$	0	0
$641$	46.7278	1.84564	0.922819	−	0.385234i	$-0.125879\pi$
$641$	46.7278	1.84564	0.922819	+	0.385234i	$0.125879\pi$
$642$	−6.62390	−0.261425
$643$	−26.6503	−1.05098	−0.525492	−	0.850798i	$-0.676119\pi$
$643$	−26.6503	−1.05098	−0.525492	+	0.850798i	$0.676119\pi$
$644$	−14.7722	−0.582108
$645$	0	0
$646$	−8.93459	−0.351527
$647$	−48.3556	−1.90105	−0.950526	−	0.310644i	$-0.899455\pi$
$647$	−48.3556	−1.90105	−0.950526	+	0.310644i	$0.899455\pi$
$648$	8.85705	0.347938
$649$	6.26485	0.245917
$650$	0	0
$651$	2.54527	0.0997571
$652$	−46.9034	−1.83688
$653$	11.5581	0.452304	0.226152	−	0.974092i	$-0.427385\pi$
$653$	11.5581	0.452304	0.226152	+	0.974092i	$0.427385\pi$
$654$	−4.18333	−0.163581
$655$	0	0
$656$	−23.6865	−0.924804
$657$	−6.96294	−0.271650
$658$	−19.6900	−0.767597
$659$	−12.2772	−0.478251	−0.239125	−	0.970989i	$-0.576861\pi$
$659$	−12.2772	−0.478251	−0.239125	+	0.970989i	$0.576861\pi$
$660$	0	0
$661$	2.16082	0.0840461	0.0420231	−	0.999117i	$-0.486620\pi$
$661$	2.16082	0.0840461	0.0420231	+	0.999117i	$0.486620\pi$
$662$	−45.3884	−1.76407
$663$	−0.513372	−0.0199377
$664$	9.56956	0.371371
$665$	0	0
$666$	9.00900	0.349092
$667$	26.4995	1.02606
$668$	14.8642	0.575113
$669$	−0.941696	−0.0364081
$670$	0	0
$671$	−1.18689	−0.0458193
$672$	−3.96090	−0.152795
$673$	24.2038	0.932988	0.466494	−	0.884524i	$-0.345517\pi$
$673$	24.2038	0.932988	0.466494	+	0.884524i	$0.345517\pi$
$674$	−43.7702	−1.68597
$675$	0	0
$676$	−32.2009	−1.23849
$677$	−17.0810	−0.656476	−0.328238	−	0.944595i	$-0.606455\pi$
$677$	−17.0810	−0.656476	−0.328238	+	0.944595i	$0.606455\pi$
$678$	−1.10493	−0.0424346
$679$	−10.5211	−0.403762
$680$	0	0
$681$	5.32295	0.203976
$682$	10.8393	0.415059
$683$	−15.6456	−0.598663	−0.299332	−	0.954149i	$-0.596764\pi$
$683$	−15.6456	−0.598663	−0.299332	+	0.954149i	$0.596764\pi$
$684$	7.30047	0.279141
$685$	0	0
$686$	39.6454	1.51367
$687$	0.474465	0.0181020
$688$	2.59653	0.0989919
$689$	−0.719370	−0.0274058
$690$	0	0
$691$	−32.7991	−1.24774	−0.623869	−	0.781529i	$-0.714440\pi$
$691$	−32.7991	−1.24774	−0.623869	+	0.781529i	$0.714440\pi$
$692$	11.1681	0.424547
$693$	4.83290	0.183587
$694$	68.1520	2.58702
$695$	0	0
$696$	−2.42476	−0.0919104
$697$	36.5968	1.38620
$698$	−3.95191	−0.149582
$699$	−3.04979	−0.115354
$700$	0	0
$701$	45.3838	1.71412	0.857061	−	0.515216i	$-0.172288\pi$
$701$	45.3838	1.71412	0.857061	+	0.515216i	$0.172288\pi$
$702$	1.53110	0.0577876
$703$	1.45804	0.0549910
$704$	−11.4213	−0.430455
$705$	0	0
$706$	−23.7236	−0.892851
$707$	−21.2746	−0.800112
$708$	−4.71867	−0.177339
$709$	−22.2346	−0.835037	−0.417519	−	0.908668i	$-0.637100\pi$
$709$	−22.2346	−0.835037	−0.417519	+	0.908668i	$0.637100\pi$
$710$	0	0
$711$	41.5758	1.55921
$712$	−3.01672	−0.113056
$713$	18.0969	0.677734
$714$	4.45492	0.166721
$715$	0	0
$716$	−13.2822	−0.496378
$717$	1.98273	0.0740464
$718$	18.1570	0.677615
$719$	−39.4219	−1.47019	−0.735094	−	0.677965i	$-0.762862\pi$
$719$	−39.4219	−1.47019	−0.735094	+	0.677965i	$0.762862\pi$
$720$	0	0
$721$	22.5965	0.841537
$722$	2.12341	0.0790251
$723$	8.24961	0.306806
$724$	−61.8576	−2.29892
$725$	0	0
$726$	−0.637479	−0.0236590
$727$	14.3222	0.531182	0.265591	−	0.964086i	$-0.414433\pi$
$727$	14.3222	0.531182	0.265591	+	0.964086i	$0.414433\pi$
$728$	−0.729339	−0.0270311
$729$	−22.2542	−0.824228
$730$	0	0
$731$	−4.01176	−0.148380
$732$	0.893961	0.0330418
$733$	45.2907	1.67285	0.836425	−	0.548082i	$-0.184642\pi$
$733$	45.2907	1.67285	0.836425	+	0.548082i	$0.184642\pi$
$734$	−13.8748	−0.512128
$735$	0	0
$736$	−28.1620	−1.03807
$737$	12.1533	0.447672
$738$	−53.7414	−1.97825
$739$	−38.1634	−1.40386	−0.701931	−	0.712245i	$-0.747679\pi$
$739$	−38.1634	−1.40386	−0.701931	+	0.712245i	$0.747679\pi$
$740$	0	0
$741$	0.122009	0.00448211
$742$	6.24252	0.229170
$743$	−10.5543	−0.387201	−0.193600	−	0.981080i	$-0.562017\pi$
$743$	−10.5543	−0.387201	−0.193600	+	0.981080i	$0.562017\pi$
$744$	−1.65591	−0.0607086
$745$	0	0
$746$	−63.6705	−2.33114
$747$	−25.7709	−0.942908
$748$	10.5565	0.385982
$749$	−17.2577	−0.630581
$750$	0	0
$751$	−53.4076	−1.94887	−0.974436	−	0.224665i	$-0.927871\pi$
$751$	−53.4076	−1.94887	−0.974436	+	0.224665i	$0.927871\pi$
$752$	−15.2047	−0.554460
$753$	1.83402	0.0668352
$754$	6.45052	0.234914
$755$	0	0
$756$	−7.39300	−0.268881
$757$	−20.4970	−0.744977	−0.372488	−	0.928037i	$-0.621495\pi$
$757$	−20.4970	−0.744977	−0.372488	+	0.928037i	$0.621495\pi$
$758$	10.9340	0.397142
$759$	−1.06431	−0.0386320
$760$	0	0
$761$	23.3188	0.845305	0.422653	−	0.906292i	$-0.361099\pi$
$761$	23.3188	0.845305	0.422653	+	0.906292i	$0.361099\pi$
$762$	−4.14340	−0.150100
$763$	−10.8991	−0.394573
$764$	63.2941	2.28990
$765$	0	0
$766$	12.4716	0.450619
$767$	2.54607	0.0919332
$768$	−1.52552	−0.0550475
$769$	−33.4479	−1.20616	−0.603081	−	0.797680i	$-0.706060\pi$
$769$	−33.4479	−1.20616	−0.603081	+	0.797680i	$0.706060\pi$
$770$	0	0
$771$	−0.222358	−0.00800804
$772$	2.35678	0.0848225
$773$	−0.0528628	−0.00190134	−0.000950671	−	1.00000i	$-0.500303\pi$
$773$	−0.0528628	−0.00190134	−0.000950671		1.00000i	$0.500303\pi$
$774$	5.89117	0.211754
$775$	0	0
$776$	6.84482	0.245715
$777$	−0.727001	−0.0260810
$778$	36.7755	1.31847
$779$	−8.69765	−0.311626
$780$	0	0
$781$	5.93683	0.212436
$782$	31.6745	1.13268
$783$	13.2621	0.473948
$784$	11.5511	0.412539
$785$	0	0
$786$	−0.817980	−0.0291764
$787$	26.7570	0.953785	0.476893	−	0.878962i	$-0.341763\pi$
$787$	26.7570	0.953785	0.476893	+	0.878962i	$0.341763\pi$
$788$	−17.3405	−0.617730
$789$	0.290660	0.0103478
$790$	0	0
$791$	−2.87874	−0.102356
$792$	−3.14420	−0.111724
$793$	−0.482358	−0.0171290
$794$	−6.24145	−0.221501
$795$	0	0
$796$	56.5127	2.00304
$797$	−15.8450	−0.561258	−0.280629	−	0.959816i	$-0.590543\pi$
$797$	−15.8450	−0.561258	−0.280629	+	0.959816i	$0.590543\pi$
$798$	−1.05876	−0.0374798
$799$	23.4920	0.831087
$800$	0	0
$801$	8.12406	0.287049
$802$	−84.5848	−2.98679
$803$	2.39287	0.0844426
$804$	−9.15384	−0.322831
$805$	0	0
$806$	4.40516	0.155165
$807$	1.35822	0.0478114
$808$	13.8409	0.486920
$809$	31.2953	1.10028	0.550142	−	0.835071i	$-0.314573\pi$
$809$	31.2953	1.10028	0.550142	+	0.835071i	$0.314573\pi$
$810$	0	0
$811$	2.29851	0.0807116	0.0403558	−	0.999185i	$-0.487151\pi$
$811$	2.29851	0.0807116	0.0403558	+	0.999185i	$0.487151\pi$
$812$	−31.1467	−1.09304
$813$	−1.03469	−0.0362883
$814$	−3.09601	−0.108515
$815$	0	0
$816$	3.44011	0.120428
$817$	0.953442	0.0333567
$818$	−73.8210	−2.58109
$819$	1.96412	0.0686317
$820$	0	0
$821$	3.75192	0.130943	0.0654714	−	0.997854i	$-0.479145\pi$
$821$	3.75192	0.130943	0.0654714	+	0.997854i	$0.479145\pi$
$822$	−4.03717	−0.140812
$823$	40.2049	1.40145	0.700727	−	0.713429i	$-0.252859\pi$
$823$	40.2049	1.40145	0.700727	+	0.713429i	$0.252859\pi$
$824$	−14.7009	−0.512129
$825$	0	0
$826$	−22.0942	−0.768755
$827$	36.4082	1.26604	0.633019	−	0.774136i	$-0.281815\pi$
$827$	36.4082	1.26604	0.633019	+	0.774136i	$0.281815\pi$
$828$	−25.8813	−0.899438
$829$	−37.9222	−1.31709	−0.658546	−	0.752540i	$-0.728828\pi$
$829$	−37.9222	−1.31709	−0.658546	+	0.752540i	$0.728828\pi$
$830$	0	0
$831$	1.64424	0.0570380
$832$	−4.64166	−0.160921
$833$	−17.8470	−0.618360
$834$	3.13734	0.108637
$835$	0	0
$836$	−2.50886	−0.0867709
$837$	9.05688	0.313051
$838$	8.92304	0.308241
$839$	21.3401	0.736743	0.368372	−	0.929679i	$-0.379915\pi$
$839$	21.3401	0.736743	0.368372	+	0.929679i	$0.379915\pi$
$840$	0	0
$841$	26.8731	0.926658
$842$	44.4066	1.53035
$843$	−6.05545	−0.208561
$844$	−34.0036	−1.17045
$845$	0	0
$846$	−34.4974	−1.18605
$847$	−1.66086	−0.0570679
$848$	4.82050	0.165537
$849$	2.12225	0.0728355
$850$	0	0
$851$	−5.16898	−0.177190
$852$	−4.47161	−0.153195
$853$	−35.0632	−1.20054	−0.600270	−	0.799798i	$-0.704940\pi$
$853$	−35.0632	−1.20054	−0.600270	+	0.799798i	$0.704940\pi$
$854$	4.18578	0.143235
$855$	0	0
$856$	11.2275	0.383749
$857$	44.9677	1.53607	0.768034	−	0.640409i	$-0.221235\pi$
$857$	44.9677	1.53607	0.768034	+	0.640409i	$0.221235\pi$
$858$	−0.259075	−0.00884467
$859$	29.2008	0.996319	0.498160	−	0.867085i	$-0.334009\pi$
$859$	29.2008	0.996319	0.498160	+	0.867085i	$0.334009\pi$
$860$	0	0
$861$	4.33678	0.147797
$862$	22.4723	0.765409
$863$	40.8821	1.39164	0.695822	−	0.718214i	$-0.255040\pi$
$863$	40.8821	1.39164	0.695822	+	0.718214i	$0.255040\pi$
$864$	−14.0941	−0.479492
$865$	0	0
$866$	25.0720	0.851983
$867$	−0.211480	−0.00718224
$868$	−21.2706	−0.721971
$869$	−14.2878	−0.484682
$870$	0	0
$871$	4.93917	0.167357
$872$	7.09075	0.240123
$873$	−18.4332	−0.623868
$874$	−7.52782	−0.254632
$875$	0	0
$876$	−1.80231	−0.0608942
$877$	−8.58398	−0.289860	−0.144930	−	0.989442i	$-0.546296\pi$
$877$	−8.58398	−0.289860	−0.144930	+	0.989442i	$0.546296\pi$
$878$	−20.5321	−0.692924
$879$	−7.77228	−0.262153
$880$	0	0
$881$	−6.82174	−0.229830	−0.114915	−	0.993375i	$-0.536660\pi$
$881$	−6.82174	−0.229830	−0.114915	+	0.993375i	$0.536660\pi$
$882$	26.2078	0.882462
$883$	27.8523	0.937305	0.468653	−	0.883383i	$-0.344740\pi$
$883$	27.8523	0.937305	0.468653	+	0.883383i	$0.344740\pi$
$884$	4.29020	0.144295
$885$	0	0
$886$	55.6521	1.86967
$887$	−13.9072	−0.466958	−0.233479	−	0.972362i	$-0.575011\pi$
$887$	−13.9072	−0.466958	−0.233479	+	0.972362i	$0.575011\pi$
$888$	0.472974	0.0158720
$889$	−10.7951	−0.362055
$890$	0	0
$891$	8.19696	0.274609
$892$	7.86966	0.263496
$893$	−5.58315	−0.186833
$894$	7.11372	0.237918
$895$	0	0
$896$	13.8922	0.464105
$897$	−0.432541	−0.0144421
$898$	2.88003	0.0961080
$899$	38.1566	1.27259
$900$	0	0
$901$	−7.44790	−0.248125
$902$	18.4687	0.614939
$903$	−0.475401	−0.0158203
$904$	1.87286	0.0622904
$905$	0	0
$906$	5.98519	0.198844
$907$	10.8762	0.361139	0.180569	−	0.983562i	$-0.442206\pi$
$907$	10.8762	0.361139	0.180569	+	0.983562i	$0.442206\pi$
$908$	−44.4834	−1.47623
$909$	−37.2735	−1.23629
$910$	0	0
$911$	13.6506	0.452265	0.226133	−	0.974097i	$-0.427392\pi$
$911$	13.6506	0.452265	0.226133	+	0.974097i	$0.427392\pi$
$912$	−0.817583	−0.0270729
$913$	8.85637	0.293103
$914$	1.96475	0.0649883
$915$	0	0
$916$	−3.96506	−0.131009
$917$	−2.13113	−0.0703763
$918$	15.8520	0.523195
$919$	16.0698	0.530093	0.265047	−	0.964236i	$-0.414613\pi$
$919$	16.0698	0.530093	0.265047	+	0.964236i	$0.414613\pi$
$920$	0	0
$921$	−10.3022	−0.339470
$922$	−38.0722	−1.25384
$923$	2.41276	0.0794169
$924$	1.25096	0.0411535
$925$	0	0
$926$	−38.9446	−1.27980
$927$	39.5896	1.30029
$928$	−59.3785	−1.94919
$929$	−4.90033	−0.160775	−0.0803873	−	0.996764i	$-0.525616\pi$
$929$	−4.90033	−0.160775	−0.0803873	+	0.996764i	$0.525616\pi$
$930$	0	0
$931$	4.24154	0.139011
$932$	25.4868	0.834849
$933$	−3.88545	−0.127204
$934$	−20.7841	−0.680077
$935$	0	0
$936$	−1.27782	−0.0417668
$937$	−18.5173	−0.604934	−0.302467	−	0.953160i	$-0.597810\pi$
$937$	−18.5173	−0.604934	−0.302467	+	0.953160i	$0.597810\pi$
$938$	−42.8609	−1.39946
$939$	−1.81043	−0.0590810
$940$	0	0
$941$	−4.56469	−0.148805	−0.0744024	−	0.997228i	$-0.523705\pi$
$941$	−4.56469	−0.148805	−0.0744024	+	0.997228i	$0.523705\pi$
$942$	−12.7021	−0.413855
$943$	30.8345	1.00411
$944$	−17.0612	−0.555296
$945$	0	0
$946$	−2.02455	−0.0658237
$947$	−32.7129	−1.06302	−0.531512	−	0.847050i	$-0.678376\pi$
$947$	−32.7129	−1.06302	−0.531512	+	0.847050i	$0.678376\pi$
$948$	10.7616	0.349520
$949$	0.972476	0.0315679
$950$	0	0
$951$	−2.26799	−0.0735446
$952$	−7.55111	−0.244733
$953$	17.7014	0.573403	0.286702	−	0.958020i	$-0.407441\pi$
$953$	17.7014	0.573403	0.286702	+	0.958020i	$0.407441\pi$
$954$	10.9370	0.354100
$955$	0	0
$956$	−16.5695	−0.535895
$957$	−2.24405	−0.0725400
$958$	−49.3751	−1.59524
$959$	−10.5183	−0.339653
$960$	0	0
$961$	−4.94225	−0.159427
$962$	−1.25824	−0.0405672
$963$	−30.2358	−0.974337
$964$	−68.9412	−2.22045
$965$	0	0
$966$	3.75349	0.120766
$967$	1.49038	0.0479273	0.0239637	−	0.999713i	$-0.492371\pi$
$967$	1.49038	0.0479273	0.0239637	+	0.999713i	$0.492371\pi$
$968$	1.08053	0.0347295
$969$	1.26320	0.0405799
$970$	0	0
$971$	−14.0384	−0.450515	−0.225258	−	0.974299i	$-0.572322\pi$
$971$	−14.0384	−0.450515	−0.225258	+	0.974299i	$0.572322\pi$
$972$	−19.5278	−0.626356
$973$	8.17391	0.262044
$974$	42.8674	1.37356
$975$	0	0
$976$	3.23228	0.103463
$977$	−52.4306	−1.67740	−0.838701	−	0.544592i	$-0.816685\pi$
$977$	−52.4306	−1.67740	−0.838701	+	0.544592i	$0.816685\pi$
$978$	11.9177	0.381087
$979$	−2.79190	−0.0892294
$980$	0	0
$981$	−19.0955	−0.609671
$982$	−84.3108	−2.69046
$983$	−42.2403	−1.34726	−0.673628	−	0.739071i	$-0.735265\pi$
$983$	−42.2403	−1.34726	−0.673628	+	0.739071i	$0.735265\pi$
$984$	−2.82143	−0.0899440
$985$	0	0
$986$	66.7845	2.12685
$987$	2.78384	0.0886107
$988$	−1.01962	−0.0324383
$989$	−3.38010	−0.107481
$990$	0	0
$991$	−11.3461	−0.360420	−0.180210	−	0.983628i	$-0.557678\pi$
$991$	−11.3461	−0.360420	−0.180210	+	0.983628i	$0.557678\pi$
$992$	−40.5505	−1.28748
$993$	6.41716	0.203643
$994$	−20.9373	−0.664092
$995$	0	0
$996$	−6.67060	−0.211366
$997$	27.2874	0.864200	0.432100	−	0.901826i	$-0.357773\pi$
$997$	27.2874	0.864200	0.432100	+	0.901826i	$0.357773\pi$
$998$	−7.52901	−0.238327
$999$	−2.58690	−0.0818458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.x.1.12	yes	15
5.4	even	2		5225.2.a.s.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.4	✓	15	5.4	even	2
5225.2.a.x.1.12	yes	15	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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.12341 q^{2} -0.300215 q^{3} +2.50886 q^{4} -0.637479 q^{6} -1.66086 q^{7} +1.08053 q^{8} -2.90987 q^{9} +1.00000 q^{11} -0.753198 q^{12} +0.406405 q^{13} -3.52669 q^{14} -2.72333 q^{16} +4.20766 q^{17} -6.17885 q^{18} -1.00000 q^{19} +0.498615 q^{21} +2.12341 q^{22} +3.54516 q^{23} -0.324390 q^{24} +0.862965 q^{26} +1.77423 q^{27} -4.16688 q^{28} +7.47483 q^{29} +5.10468 q^{31} -7.94379 q^{32} -0.300215 q^{33} +8.93459 q^{34} -7.30047 q^{36} -1.45804 q^{37} -2.12341 q^{38} -0.122009 q^{39} +8.69765 q^{41} +1.05876 q^{42} -0.953442 q^{43} +2.50886 q^{44} +7.52782 q^{46} +5.58315 q^{47} +0.817583 q^{48} -4.24154 q^{49} -1.26320 q^{51} +1.01962 q^{52} -1.77008 q^{53} +3.76742 q^{54} -1.79461 q^{56} +0.300215 q^{57} +15.8721 q^{58} +6.26485 q^{59} -1.18689 q^{61} +10.8393 q^{62} +4.83290 q^{63} -11.4213 q^{64} -0.637479 q^{66} +12.1533 q^{67} +10.5565 q^{68} -1.06431 q^{69} +5.93683 q^{71} -3.14420 q^{72} +2.39287 q^{73} -3.09601 q^{74} -2.50886 q^{76} -1.66086 q^{77} -0.259075 q^{78} -14.2878 q^{79} +8.19696 q^{81} +18.4687 q^{82} +8.85637 q^{83} +1.25096 q^{84} -2.02455 q^{86} -2.24405 q^{87} +1.08053 q^{88} -2.79190 q^{89} -0.674984 q^{91} +8.89432 q^{92} -1.53250 q^{93} +11.8553 q^{94} +2.38484 q^{96} +6.33470 q^{97} -9.00651 q^{98} -2.90987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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