LMFDB - Embedded newform 7225.2.a.bv.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
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} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
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			1.10
Root			$-0.970758$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.970758 q^{2} -1.98754 q^{3} -1.05763 q^{4} -1.92942 q^{6} -0.240703 q^{7} -2.96822 q^{8} +0.950332 q^{9} +O(q^{10})$	\(q+0.970758 q^{2} -1.98754 q^{3} -1.05763 q^{4} -1.92942 q^{6} -0.240703 q^{7} -2.96822 q^{8} +0.950332 q^{9} -0.146845 q^{11} +2.10209 q^{12} +2.69729 q^{13} -0.233664 q^{14} -0.766161 q^{16} +0.922542 q^{18} -7.91899 q^{19} +0.478408 q^{21} -0.142551 q^{22} -4.85909 q^{23} +5.89946 q^{24} +2.61841 q^{26} +4.07381 q^{27} +0.254574 q^{28} +3.57955 q^{29} +1.39787 q^{31} +5.19268 q^{32} +0.291860 q^{33} -1.00510 q^{36} -6.86057 q^{37} -7.68742 q^{38} -5.36098 q^{39} +5.41926 q^{41} +0.464418 q^{42} -2.90389 q^{43} +0.155307 q^{44} -4.71700 q^{46} -9.32800 q^{47} +1.52278 q^{48} -6.94206 q^{49} -2.85273 q^{52} -8.31198 q^{53} +3.95468 q^{54} +0.714458 q^{56} +15.7393 q^{57} +3.47488 q^{58} -4.44828 q^{59} -13.1377 q^{61} +1.35699 q^{62} -0.228748 q^{63} +6.57315 q^{64} +0.283326 q^{66} +8.35762 q^{67} +9.65766 q^{69} +15.0357 q^{71} -2.82079 q^{72} -0.848026 q^{73} -6.65995 q^{74} +8.37535 q^{76} +0.0353459 q^{77} -5.20421 q^{78} -8.67861 q^{79} -10.9479 q^{81} +5.26079 q^{82} +4.82738 q^{83} -0.505978 q^{84} -2.81897 q^{86} -7.11452 q^{87} +0.435867 q^{88} -16.8782 q^{89} -0.649245 q^{91} +5.13912 q^{92} -2.77832 q^{93} -9.05523 q^{94} -10.3207 q^{96} +13.8129 q^{97} -6.73906 q^{98} -0.139551 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.970758	0.686429	0.343215	−	0.939257i	$-0.388484\pi$
$2$	0.970758	0.686429	0.343215	+	0.939257i	$0.388484\pi$
$3$	−1.98754	−1.14751	−0.573755	−	0.819027i	$-0.694514\pi$
$3$	−1.98754	−1.14751	−0.573755	+	0.819027i	$0.694514\pi$
$4$	−1.05763	−0.528815
$5$	0	0
$5$	0	0
$6$	−1.92942	−0.787684
$7$	−0.240703	−0.0909771	−0.0454886	−	0.998965i	$-0.514484\pi$
$7$	−0.240703	−0.0909771	−0.0454886	+	0.998965i	$0.514484\pi$
$8$	−2.96822	−1.04942
$9$	0.950332	0.316777
$10$	0	0
$11$	−0.146845	−0.0442753	−0.0221377	−	0.999755i	$-0.507047\pi$
$11$	−0.146845	−0.0442753	−0.0221377	+	0.999755i	$0.507047\pi$
$12$	2.10209	0.606820
$13$	2.69729	0.748093	0.374047	−	0.927410i	$-0.377970\pi$
$13$	2.69729	0.748093	0.374047	+	0.927410i	$0.377970\pi$
$14$	−0.233664	−0.0624494
$15$	0	0
$16$	−0.766161	−0.191540
$17$	0	0
$17$	0	0
$18$	0.922542	0.217445
$19$	−7.91899	−1.81674	−0.908370	−	0.418167i	$-0.862673\pi$
$19$	−7.91899	−1.81674	−0.908370	+	0.418167i	$0.862673\pi$
$20$	0	0
$21$	0.478408	0.104397
$22$	−0.142551	−0.0303919
$23$	−4.85909	−1.01319	−0.506595	−	0.862184i	$-0.669096\pi$
$23$	−4.85909	−1.01319	−0.506595	+	0.862184i	$0.669096\pi$
$24$	5.89946	1.20422
$25$	0	0
$26$	2.61841	0.513513
$27$	4.07381	0.784004
$28$	0.254574	0.0481101
$29$	3.57955	0.664706	0.332353	−	0.943155i	$-0.392157\pi$
$29$	3.57955	0.664706	0.332353	+	0.943155i	$0.392157\pi$
$30$	0	0
$31$	1.39787	0.251064	0.125532	−	0.992090i	$-0.459936\pi$
$31$	1.39787	0.251064	0.125532	+	0.992090i	$0.459936\pi$
$32$	5.19268	0.917944
$33$	0.291860	0.0508064
$34$	0	0
$35$	0	0
$36$	−1.00510	−0.167517
$37$	−6.86057	−1.12787	−0.563936	−	0.825819i	$-0.690713\pi$
$37$	−6.86057	−1.12787	−0.563936	+	0.825819i	$0.690713\pi$
$38$	−7.68742	−1.24706
$39$	−5.36098	−0.858444
$40$	0	0
$41$	5.41926	0.846347	0.423173	−	0.906049i	$-0.360916\pi$
$41$	5.41926	0.846347	0.423173	+	0.906049i	$0.360916\pi$
$42$	0.464418	0.0716612
$43$	−2.90389	−0.442839	−0.221419	−	0.975179i	$-0.571069\pi$
$43$	−2.90389	−0.442839	−0.221419	+	0.975179i	$0.571069\pi$
$44$	0.155307	0.0234135
$45$	0	0
$46$	−4.71700	−0.695484
$47$	−9.32800	−1.36063	−0.680315	−	0.732920i	$-0.738157\pi$
$47$	−9.32800	−1.36063	−0.680315	+	0.732920i	$0.738157\pi$
$48$	1.52278	0.219794
$49$	−6.94206	−0.991723
$50$	0	0
$51$	0	0
$52$	−2.85273	−0.395603
$53$	−8.31198	−1.14174	−0.570869	−	0.821041i	$-0.693394\pi$
$53$	−8.31198	−1.14174	−0.570869	+	0.821041i	$0.693394\pi$
$54$	3.95468	0.538163
$55$	0	0
$56$	0.714458	0.0954735
$57$	15.7393	2.08473
$58$	3.47488	0.456274
$59$	−4.44828	−0.579117	−0.289559	−	0.957160i	$-0.593509\pi$
$59$	−4.44828	−0.579117	−0.289559	+	0.957160i	$0.593509\pi$
$60$	0	0
$61$	−13.1377	−1.68211	−0.841055	−	0.540949i	$-0.818065\pi$
$61$	−13.1377	−1.68211	−0.841055	+	0.540949i	$0.818065\pi$
$62$	1.35699	0.172338
$63$	−0.228748	−0.0288195
$64$	6.57315	0.821644
$65$	0	0
$66$	0.283326	0.0348750
$67$	8.35762	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$67$	8.35762	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$68$	0	0
$69$	9.65766	1.16265
$70$	0	0
$71$	15.0357	1.78441	0.892203	−	0.451634i	$-0.149159\pi$
$71$	15.0357	1.78441	0.892203	+	0.451634i	$0.149159\pi$
$72$	−2.82079	−0.332434
$73$	−0.848026	−0.0992540	−0.0496270	−	0.998768i	$-0.515803\pi$
$73$	−0.848026	−0.0992540	−0.0496270	+	0.998768i	$0.515803\pi$
$74$	−6.65995	−0.774204
$75$	0	0
$76$	8.37535	0.960719
$77$	0.0353459	0.00402804
$78$	−5.20421	−0.589261
$79$	−8.67861	−0.976420	−0.488210	−	0.872726i	$-0.662350\pi$
$79$	−8.67861	−0.976420	−0.488210	+	0.872726i	$0.662350\pi$
$80$	0	0
$81$	−10.9479	−1.21643
$82$	5.26079	0.580957
$83$	4.82738	0.529873	0.264937	−	0.964266i	$-0.414649\pi$
$83$	4.82738	0.529873	0.264937	+	0.964266i	$0.414649\pi$
$84$	−0.505978	−0.0552067
$85$	0	0
$86$	−2.81897	−0.303977
$87$	−7.11452	−0.762756
$88$	0.435867	0.0464636
$89$	−16.8782	−1.78908	−0.894541	−	0.446987i	$-0.852497\pi$
$89$	−16.8782	−1.78908	−0.894541	+	0.446987i	$0.852497\pi$
$90$	0	0
$91$	−0.649245	−0.0680594
$92$	5.13912	0.535790
$93$	−2.77832	−0.288098
$94$	−9.05523	−0.933976
$95$	0	0
$96$	−10.3207	−1.05335
$97$	13.8129	1.40249	0.701243	−	0.712922i	$-0.252629\pi$
$97$	13.8129	1.40249	0.701243	+	0.712922i	$0.252629\pi$
$98$	−6.73906	−0.680748
$99$	−0.139551	−0.0140254
$100$	0	0
$101$	−9.62404	−0.957627	−0.478814	−	0.877917i	$-0.658933\pi$
$101$	−9.62404	−0.957627	−0.478814	+	0.877917i	$0.658933\pi$
$102$	0	0
$103$	14.0838	1.38772	0.693859	−	0.720111i	$-0.255909\pi$
$103$	14.0838	1.38772	0.693859	+	0.720111i	$0.255909\pi$
$104$	−8.00614	−0.785067
$105$	0	0
$106$	−8.06892	−0.783723
$107$	3.76252	0.363737	0.181868	−	0.983323i	$-0.441786\pi$
$107$	3.76252	0.363737	0.181868	+	0.983323i	$0.441786\pi$
$108$	−4.30858	−0.414593
$109$	5.63129	0.539380	0.269690	−	0.962947i	$-0.413079\pi$
$109$	5.63129	0.539380	0.269690	+	0.962947i	$0.413079\pi$
$110$	0	0
$111$	13.6357	1.29424
$112$	0.184417	0.0174258
$113$	−9.85006	−0.926616	−0.463308	−	0.886197i	$-0.653338\pi$
$113$	−9.85006	−0.926616	−0.463308	+	0.886197i	$0.653338\pi$
$114$	15.2791	1.43102
$115$	0	0
$116$	−3.78584	−0.351506
$117$	2.56332	0.236979
$118$	−4.31821	−0.397523
$119$	0	0
$120$	0	0
$121$	−10.9784	−0.998040
$122$	−12.7535	−1.15465
$123$	−10.7710	−0.971191
$124$	−1.47842	−0.132766
$125$	0	0
$126$	−0.222059	−0.0197826
$127$	8.23666	0.730885	0.365443	−	0.930834i	$-0.380918\pi$
$127$	8.23666	0.730885	0.365443	+	0.930834i	$0.380918\pi$
$128$	−4.00442	−0.353944
$129$	5.77160	0.508161
$130$	0	0
$131$	1.70110	0.148626	0.0743129	−	0.997235i	$-0.476324\pi$
$131$	1.70110	0.148626	0.0743129	+	0.997235i	$0.476324\pi$
$132$	−0.308680	−0.0268672
$133$	1.90612	0.165282
$134$	8.11322	0.700876
$135$	0	0
$136$	0	0
$137$	10.1145	0.864136	0.432068	−	0.901841i	$-0.357784\pi$
$137$	10.1145	0.864136	0.432068	+	0.901841i	$0.357784\pi$
$138$	9.37525	0.798074
$139$	16.3291	1.38502	0.692509	−	0.721409i	$-0.256505\pi$
$139$	16.3291	1.38502	0.692509	+	0.721409i	$0.256505\pi$
$140$	0	0
$141$	18.5398	1.56133
$142$	14.5960	1.22487
$143$	−0.396083	−0.0331221
$144$	−0.728107	−0.0606756
$145$	0	0
$146$	−0.823228	−0.0681308
$147$	13.7977	1.13801
$148$	7.25594	0.596435
$149$	−8.96696	−0.734602	−0.367301	−	0.930102i	$-0.619718\pi$
$149$	−8.96696	−0.734602	−0.367301	+	0.930102i	$0.619718\pi$
$150$	0	0
$151$	−2.04218	−0.166190	−0.0830951	−	0.996542i	$-0.526481\pi$
$151$	−2.04218	−0.166190	−0.0830951	+	0.996542i	$0.526481\pi$
$152$	23.5053	1.90653
$153$	0	0
$154$	0.0343123	0.00276497
$155$	0	0
$156$	5.66993	0.453958
$157$	1.68353	0.134360	0.0671801	−	0.997741i	$-0.478600\pi$
$157$	1.68353	0.134360	0.0671801	+	0.997741i	$0.478600\pi$
$158$	−8.42483	−0.670244
$159$	16.5204	1.31016
$160$	0	0
$161$	1.16960	0.0921772
$162$	−10.6277	−0.834993
$163$	23.7785	1.86248	0.931240	−	0.364408i	$-0.118729\pi$
$163$	23.7785	1.86248	0.931240	+	0.364408i	$0.118729\pi$
$164$	−5.73157	−0.447561
$165$	0	0
$166$	4.68621	0.363721
$167$	22.1761	1.71604	0.858020	−	0.513616i	$-0.171694\pi$
$167$	22.1761	1.71604	0.858020	+	0.513616i	$0.171694\pi$
$168$	−1.42002	−0.109557
$169$	−5.72463	−0.440356
$170$	0	0
$171$	−7.52567	−0.575502
$172$	3.07124	0.234180
$173$	−11.1979	−0.851358	−0.425679	−	0.904874i	$-0.639965\pi$
$173$	−11.1979	−0.851358	−0.425679	+	0.904874i	$0.639965\pi$
$174$	−6.90647	−0.523578
$175$	0	0
$176$	0.112507	0.00848051
$177$	8.84116	0.664543
$178$	−16.3846	−1.22808
$179$	−15.5478	−1.16210	−0.581050	−	0.813868i	$-0.697358\pi$
$179$	−15.5478	−1.16210	−0.581050	+	0.813868i	$0.697358\pi$
$180$	0	0
$181$	−10.3425	−0.768753	−0.384376	−	0.923176i	$-0.625583\pi$
$181$	−10.3425	−0.768753	−0.384376	+	0.923176i	$0.625583\pi$
$182$	−0.630260	−0.0467180
$183$	26.1118	1.93024
$184$	14.4228	1.06327
$185$	0	0
$186$	−2.69708	−0.197759
$187$	0	0
$188$	9.86557	0.719521
$189$	−0.980577	−0.0713265
$190$	0	0
$191$	15.1208	1.09410	0.547052	−	0.837098i	$-0.315750\pi$
$191$	15.1208	1.09410	0.547052	+	0.837098i	$0.315750\pi$
$192$	−13.0644	−0.942844
$193$	−6.34671	−0.456846	−0.228423	−	0.973562i	$-0.573357\pi$
$193$	−6.34671	−0.456846	−0.228423	+	0.973562i	$0.573357\pi$
$194$	13.4090	0.962708
$195$	0	0
$196$	7.34213	0.524438
$197$	−18.2502	−1.30027	−0.650136	−	0.759818i	$-0.725288\pi$
$197$	−18.2502	−1.30027	−0.650136	+	0.759818i	$0.725288\pi$
$198$	−0.135470	−0.00962747
$199$	17.8949	1.26853	0.634266	−	0.773115i	$-0.281302\pi$
$199$	17.8949	1.26853	0.634266	+	0.773115i	$0.281302\pi$
$200$	0	0
$201$	−16.6111	−1.17166
$202$	−9.34261	−0.657344
$203$	−0.861609	−0.0604731
$204$	0	0
$205$	0	0
$206$	13.6720	0.952570
$207$	−4.61775	−0.320956
$208$	−2.06656	−0.143290
$209$	1.16286	0.0804368
$210$	0	0
$211$	−7.63456	−0.525585	−0.262792	−	0.964852i	$-0.584643\pi$
$211$	−7.63456	−0.525585	−0.262792	+	0.964852i	$0.584643\pi$
$212$	8.79100	0.603768
$213$	−29.8841	−2.04762
$214$	3.65250	0.249680
$215$	0	0
$216$	−12.0919	−0.822752
$217$	−0.336470	−0.0228411
$218$	5.46662	0.370246
$219$	1.68549	0.113895
$220$	0	0
$221$	0	0
$222$	13.2370	0.888406
$223$	−2.01600	−0.135002	−0.0675008	−	0.997719i	$-0.521503\pi$
$223$	−2.01600	−0.135002	−0.0675008	+	0.997719i	$0.521503\pi$
$224$	−1.24989	−0.0835120
$225$	0	0
$226$	−9.56203	−0.636056
$227$	6.34906	0.421402	0.210701	−	0.977551i	$-0.432425\pi$
$227$	6.34906	0.421402	0.210701	+	0.977551i	$0.432425\pi$
$228$	−16.6464	−1.10243
$229$	11.3405	0.749403	0.374702	−	0.927145i	$-0.377745\pi$
$229$	11.3405	0.749403	0.374702	+	0.927145i	$0.377745\pi$
$230$	0	0
$231$	−0.0702516	−0.00462222
$232$	−10.6249	−0.697558
$233$	−7.33456	−0.480503	−0.240251	−	0.970711i	$-0.577230\pi$
$233$	−7.33456	−0.480503	−0.240251	+	0.970711i	$0.577230\pi$
$234$	2.48836	0.162669
$235$	0	0
$236$	4.70464	0.306246
$237$	17.2491	1.12045
$238$	0	0
$239$	23.3261	1.50884	0.754420	−	0.656392i	$-0.227918\pi$
$239$	23.3261	1.50884	0.754420	+	0.656392i	$0.227918\pi$
$240$	0	0
$241$	6.02533	0.388125	0.194063	−	0.980989i	$-0.437833\pi$
$241$	6.02533	0.388125	0.194063	+	0.980989i	$0.437833\pi$
$242$	−10.6574	−0.685084
$243$	9.53795	0.611860
$244$	13.8948	0.889525
$245$	0	0
$246$	−10.4561	−0.666654
$247$	−21.3598	−1.35909
$248$	−4.14917	−0.263473
$249$	−9.59462	−0.608035
$250$	0	0
$251$	−16.2255	−1.02414	−0.512072	−	0.858943i	$-0.671122\pi$
$251$	−16.2255	−1.02414	−0.512072	+	0.858943i	$0.671122\pi$
$252$	0.241930	0.0152402
$253$	0.713532	0.0448594
$254$	7.99580	0.501701
$255$	0	0
$256$	−17.0336	−1.06460
$257$	−2.71922	−0.169620	−0.0848101	−	0.996397i	$-0.527028\pi$
$257$	−2.71922	−0.169620	−0.0848101	+	0.996397i	$0.527028\pi$
$258$	5.60283	0.348817
$259$	1.65136	0.102610
$260$	0	0
$261$	3.40176	0.210564
$262$	1.65136	0.102021
$263$	16.1534	0.996061	0.498031	−	0.867159i	$-0.334057\pi$
$263$	16.1534	0.996061	0.498031	+	0.867159i	$0.334057\pi$
$264$	−0.866305	−0.0533174
$265$	0	0
$266$	1.85038	0.113454
$267$	33.5461	2.05299
$268$	−8.83926	−0.539944
$269$	24.9598	1.52183	0.760914	−	0.648852i	$-0.224751\pi$
$269$	24.9598	1.52183	0.760914	+	0.648852i	$0.224751\pi$
$270$	0	0
$271$	−28.6964	−1.74318	−0.871591	−	0.490234i	$-0.836911\pi$
$271$	−28.6964	−1.74318	−0.871591	+	0.490234i	$0.836911\pi$
$272$	0	0
$273$	1.29040	0.0780988
$274$	9.81868	0.593168
$275$	0	0
$276$	−10.2142	−0.614824
$277$	−12.7839	−0.768111	−0.384056	−	0.923310i	$-0.625473\pi$
$277$	−12.7839	−0.768111	−0.384056	+	0.923310i	$0.625473\pi$
$278$	15.8516	0.950717
$279$	1.32844	0.0795314
$280$	0	0
$281$	18.8504	1.12452	0.562260	−	0.826960i	$-0.309932\pi$
$281$	18.8504	1.12452	0.562260	+	0.826960i	$0.309932\pi$
$282$	17.9977	1.07175
$283$	17.1927	1.02200	0.510999	−	0.859581i	$-0.329276\pi$
$283$	17.1927	1.02200	0.510999	+	0.859581i	$0.329276\pi$
$284$	−15.9022	−0.943620
$285$	0	0
$286$	−0.384500	−0.0227360
$287$	−1.30443	−0.0769982
$288$	4.93477	0.290784
$289$	0	0
$290$	0	0
$291$	−27.4537	−1.60937
$292$	0.896898	0.0524870
$293$	14.0384	0.820130	0.410065	−	0.912056i	$-0.365506\pi$
$293$	14.0384	0.820130	0.410065	+	0.912056i	$0.365506\pi$
$294$	13.3942	0.781164
$295$	0	0
$296$	20.3637	1.18361
$297$	−0.598217	−0.0347121
$298$	−8.70474	−0.504252
$299$	−13.1064	−0.757961
$300$	0	0
$301$	0.698974	0.0402882
$302$	−1.98246	−0.114078
$303$	19.1282	1.09889
$304$	6.06722	0.347979
$305$	0	0
$306$	0	0
$307$	22.1973	1.26687	0.633434	−	0.773797i	$-0.281645\pi$
$307$	22.1973	1.26687	0.633434	+	0.773797i	$0.281645\pi$
$308$	−0.0373829	−0.00213009
$309$	−27.9922	−1.59242
$310$	0	0
$311$	11.2832	0.639813	0.319906	−	0.947449i	$-0.396349\pi$
$311$	11.2832	0.639813	0.319906	+	0.947449i	$0.396349\pi$
$312$	15.9126	0.900871
$313$	16.9144	0.956060	0.478030	−	0.878343i	$-0.341351\pi$
$313$	16.9144	0.956060	0.478030	+	0.878343i	$0.341351\pi$
$314$	1.63430	0.0922288
$315$	0	0
$316$	9.17876	0.516345
$317$	12.8953	0.724270	0.362135	−	0.932126i	$-0.382048\pi$
$317$	12.8953	0.724270	0.362135	+	0.932126i	$0.382048\pi$
$318$	16.0373	0.899329
$319$	−0.525638	−0.0294301
$320$	0	0
$321$	−7.47818	−0.417391
$322$	1.13540	0.0632731
$323$	0	0
$324$	11.5788	0.643266
$325$	0	0
$326$	23.0832	1.27846
$327$	−11.1924	−0.618943
$328$	−16.0855	−0.888176
$329$	2.24528	0.123786
$330$	0	0
$331$	18.6809	1.02679	0.513396	−	0.858152i	$-0.328387\pi$
$331$	18.6809	1.02679	0.513396	+	0.858152i	$0.328387\pi$
$332$	−5.10558	−0.280205
$333$	−6.51982	−0.357284
$334$	21.5277	1.17794
$335$	0	0
$336$	−0.366537	−0.0199962
$337$	34.5147	1.88013	0.940067	−	0.340989i	$-0.110762\pi$
$337$	34.5147	1.88013	0.940067	+	0.340989i	$0.110762\pi$
$338$	−5.55723	−0.302273
$339$	19.5774	1.06330
$340$	0	0
$341$	−0.205269	−0.0111159
$342$	−7.30560	−0.395042
$343$	3.35589	0.181201
$344$	8.61937	0.464725
$345$	0	0
$346$	−10.8704	−0.584397
$347$	31.2266	1.67633	0.838166	−	0.545416i	$-0.183628\pi$
$347$	31.2266	1.67633	0.838166	+	0.545416i	$0.183628\pi$
$348$	7.52452	0.403357
$349$	−13.6005	−0.728021	−0.364010	−	0.931395i	$-0.618593\pi$
$349$	−13.6005	−0.728021	−0.364010	+	0.931395i	$0.618593\pi$
$350$	0	0
$351$	10.9882	0.586508
$352$	−0.762517	−0.0406423
$353$	−1.69075	−0.0899896	−0.0449948	−	0.998987i	$-0.514327\pi$
$353$	−1.69075	−0.0899896	−0.0449948	+	0.998987i	$0.514327\pi$
$354$	8.58263	0.456161
$355$	0	0
$356$	17.8508	0.946092
$357$	0	0
$358$	−15.0932	−0.797700
$359$	6.93766	0.366156	0.183078	−	0.983098i	$-0.441394\pi$
$359$	6.93766	0.366156	0.183078	+	0.983098i	$0.441394\pi$
$360$	0	0
$361$	43.7103	2.30054
$362$	−10.0401	−0.527694
$363$	21.8201	1.14526
$364$	0.686661	0.0359908
$365$	0	0
$366$	25.3482	1.32497
$367$	−0.892281	−0.0465767	−0.0232883	−	0.999729i	$-0.507414\pi$
$367$	−0.892281	−0.0465767	−0.0232883	+	0.999729i	$0.507414\pi$
$368$	3.72285	0.194067
$369$	5.15010	0.268104
$370$	0	0
$371$	2.00072	0.103872
$372$	2.93843	0.152351
$373$	25.1118	1.30024	0.650120	−	0.759831i	$-0.274718\pi$
$373$	25.1118	1.30024	0.650120	+	0.759831i	$0.274718\pi$
$374$	0	0
$375$	0	0
$376$	27.6875	1.42788
$377$	9.65509	0.497262
$378$	−0.951902	−0.0489606
$379$	−24.6505	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$379$	−24.6505	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$380$	0	0
$381$	−16.3707	−0.838698
$382$	14.6787	0.751026
$383$	26.0826	1.33276	0.666380	−	0.745612i	$-0.267843\pi$
$383$	26.0826	1.33276	0.666380	+	0.745612i	$0.267843\pi$
$384$	7.95895	0.406154
$385$	0	0
$386$	−6.16112	−0.313593
$387$	−2.75966	−0.140281
$388$	−14.6089	−0.741655
$389$	−10.7671	−0.545911	−0.272956	−	0.962027i	$-0.588001\pi$
$389$	−10.7671	−0.545911	−0.272956	+	0.962027i	$0.588001\pi$
$390$	0	0
$391$	0	0
$392$	20.6055	1.04074
$393$	−3.38101	−0.170549
$394$	−17.7165	−0.892545
$395$	0	0
$396$	0.147594	0.00741685
$397$	0.939360	0.0471451	0.0235726	−	0.999722i	$-0.492496\pi$
$397$	0.939360	0.0471451	0.0235726	+	0.999722i	$0.492496\pi$
$398$	17.3716	0.870758
$399$	−3.78850	−0.189662
$400$	0	0
$401$	−16.7876	−0.838331	−0.419166	−	0.907910i	$-0.637677\pi$
$401$	−16.7876	−0.838331	−0.419166	+	0.907910i	$0.637677\pi$
$402$	−16.1254	−0.804261
$403$	3.77045	0.187819
$404$	10.1787	0.506407
$405$	0	0
$406$	−0.836413	−0.0415105
$407$	1.00744	0.0499369
$408$	0	0
$409$	−11.6683	−0.576962	−0.288481	−	0.957486i	$-0.593150\pi$
$409$	−11.6683	−0.576962	−0.288481	+	0.957486i	$0.593150\pi$
$410$	0	0
$411$	−20.1029	−0.991604
$412$	−14.8954	−0.733846
$413$	1.07071	0.0526864
$414$	−4.48272	−0.220314
$415$	0	0
$416$	14.0062	0.686708
$417$	−32.4548	−1.58932
$418$	1.12886	0.0552142
$419$	34.2911	1.67523	0.837615	−	0.546261i	$-0.183949\pi$
$419$	34.2911	1.67523	0.837615	+	0.546261i	$0.183949\pi$
$420$	0	0
$421$	10.3116	0.502558	0.251279	−	0.967915i	$-0.419149\pi$
$421$	10.3116	0.502558	0.251279	+	0.967915i	$0.419149\pi$
$422$	−7.41131	−0.360777
$423$	−8.86470	−0.431017
$424$	24.6718	1.19817
$425$	0	0
$426$	−29.0102	−1.40555
$427$	3.16228	0.153034
$428$	−3.97935	−0.192349
$429$	0.787232	0.0380079
$430$	0	0
$431$	−1.62573	−0.0783086	−0.0391543	−	0.999233i	$-0.512466\pi$
$431$	−1.62573	−0.0783086	−0.0391543	+	0.999233i	$0.512466\pi$
$432$	−3.12119	−0.150168
$433$	−5.42024	−0.260480	−0.130240	−	0.991482i	$-0.541575\pi$
$433$	−5.42024	−0.260480	−0.130240	+	0.991482i	$0.541575\pi$
$434$	−0.326631	−0.0156788
$435$	0	0
$436$	−5.95582	−0.285232
$437$	38.4791	1.84070
$438$	1.63620	0.0781808
$439$	−21.8895	−1.04473	−0.522364	−	0.852723i	$-0.674950\pi$
$439$	−21.8895	−1.04473	−0.522364	+	0.852723i	$0.674950\pi$
$440$	0	0
$441$	−6.59727	−0.314156
$442$	0	0
$443$	−5.84785	−0.277840	−0.138920	−	0.990304i	$-0.544363\pi$
$443$	−5.84785	−0.277840	−0.138920	+	0.990304i	$0.544363\pi$
$444$	−14.4215	−0.684415
$445$	0	0
$446$	−1.95705	−0.0926691
$447$	17.8222	0.842962
$448$	−1.58218	−0.0747508
$449$	−34.5670	−1.63132	−0.815659	−	0.578533i	$-0.803626\pi$
$449$	−34.5670	−1.63132	−0.815659	+	0.578533i	$0.803626\pi$
$450$	0	0
$451$	−0.795790	−0.0374723
$452$	10.4177	0.490008
$453$	4.05892	0.190705
$454$	6.16340	0.289263
$455$	0	0
$456$	−46.7178	−2.18776
$457$	23.8789	1.11701	0.558504	−	0.829502i	$-0.311376\pi$
$457$	23.8789	1.11701	0.558504	+	0.829502i	$0.311376\pi$
$458$	11.0089	0.514412
$459$	0	0
$460$	0	0
$461$	6.54778	0.304961	0.152480	−	0.988307i	$-0.451274\pi$
$461$	6.54778	0.304961	0.152480	+	0.988307i	$0.451274\pi$
$462$	−0.0681973	−0.00317283
$463$	16.9598	0.788190	0.394095	−	0.919070i	$-0.371058\pi$
$463$	16.9598	0.788190	0.394095	+	0.919070i	$0.371058\pi$
$464$	−2.74251	−0.127318
$465$	0	0
$466$	−7.12008	−0.329831
$467$	−29.0276	−1.34324	−0.671619	−	0.740897i	$-0.734401\pi$
$467$	−29.0276	−1.34324	−0.671619	+	0.740897i	$0.734401\pi$
$468$	−2.71104	−0.125318
$469$	−2.01170	−0.0928918
$470$	0	0
$471$	−3.34609	−0.154180
$472$	13.2035	0.607739
$473$	0.426420	0.0196068
$474$	16.7447	0.769111
$475$	0	0
$476$	0	0
$477$	−7.89915	−0.361677
$478$	22.6440	1.03571
$479$	−38.0757	−1.73972	−0.869861	−	0.493297i	$-0.835791\pi$
$479$	−38.0757	−1.73972	−0.869861	+	0.493297i	$0.835791\pi$
$480$	0	0
$481$	−18.5049	−0.843753
$482$	5.84913	0.266421
$483$	−2.32463	−0.105774
$484$	11.6111	0.527778
$485$	0	0
$486$	9.25904	0.419999
$487$	−7.17552	−0.325154	−0.162577	−	0.986696i	$-0.551981\pi$
$487$	−7.17552	−0.325154	−0.162577	+	0.986696i	$0.551981\pi$
$488$	38.9956	1.76525
$489$	−47.2609	−2.13721
$490$	0	0
$491$	−9.82560	−0.443423	−0.221711	−	0.975112i	$-0.571164\pi$
$491$	−9.82560	−0.443423	−0.221711	+	0.975112i	$0.571164\pi$
$492$	11.3918	0.513580
$493$	0	0
$494$	−20.7352	−0.932920
$495$	0	0
$496$	−1.07099	−0.0480889
$497$	−3.61913	−0.162340
$498$	−9.31405	−0.417373
$499$	34.7167	1.55413	0.777066	−	0.629419i	$-0.216707\pi$
$499$	34.7167	1.55413	0.777066	+	0.629419i	$0.216707\pi$
$500$	0	0
$501$	−44.0760	−1.96917
$502$	−15.7510	−0.703002
$503$	12.1638	0.542359	0.271179	−	0.962529i	$-0.412586\pi$
$503$	12.1638	0.542359	0.271179	+	0.962529i	$0.412586\pi$
$504$	0.678973	0.0302439
$505$	0	0
$506$	0.692666	0.0307928
$507$	11.3780	0.505313
$508$	−8.71133	−0.386503
$509$	44.6026	1.97697	0.988487	−	0.151304i	$-0.0483473\pi$
$509$	44.6026	1.97697	0.988487	+	0.151304i	$0.0483473\pi$
$510$	0	0
$511$	0.204122	0.00902984
$512$	−8.52669	−0.376830
$513$	−32.2604	−1.42433
$514$	−2.63970	−0.116432
$515$	0	0
$516$	−6.10422	−0.268723
$517$	1.36977	0.0602423
$518$	1.60307	0.0704348
$519$	22.2563	0.976941
$520$	0	0
$521$	16.8083	0.736387	0.368193	−	0.929749i	$-0.379976\pi$
$521$	16.8083	0.736387	0.368193	+	0.929749i	$0.379976\pi$
$522$	3.30229	0.144537
$523$	13.1468	0.574870	0.287435	−	0.957800i	$-0.407197\pi$
$523$	13.1468	0.574870	0.287435	+	0.957800i	$0.407197\pi$
$524$	−1.79913	−0.0785955
$525$	0	0
$526$	15.6810	0.683726
$527$	0	0
$528$	−0.223612	−0.00973146
$529$	0.610767	0.0265551
$530$	0	0
$531$	−4.22735	−0.183451
$532$	−2.01597	−0.0874035
$533$	14.6173	0.633146
$534$	32.5651	1.40923
$535$	0	0
$536$	−24.8072	−1.07151
$537$	30.9020	1.33352
$538$	24.2300	1.04463
$539$	1.01941	0.0439089
$540$	0	0
$541$	22.2593	0.957002	0.478501	−	0.878087i	$-0.341180\pi$
$541$	22.2593	0.957002	0.478501	+	0.878087i	$0.341180\pi$
$542$	−27.8572	−1.19657
$543$	20.5562	0.882151
$544$	0	0
$545$	0	0
$546$	1.25267	0.0536093
$547$	−26.1639	−1.11869	−0.559343	−	0.828936i	$-0.688947\pi$
$547$	−26.1639	−1.11869	−0.559343	+	0.828936i	$0.688947\pi$
$548$	−10.6973	−0.456968
$549$	−12.4852	−0.532855
$550$	0	0
$551$	−28.3464	−1.20760
$552$	−28.6660	−1.22011
$553$	2.08897	0.0888319
$554$	−12.4101	−0.527254
$555$	0	0
$556$	−17.2702	−0.732418
$557$	35.7714	1.51568	0.757841	−	0.652440i	$-0.226254\pi$
$557$	35.7714	1.51568	0.757841	+	0.652440i	$0.226254\pi$
$558$	1.28959	0.0545927
$559$	−7.83262	−0.331285
$560$	0	0
$561$	0	0
$562$	18.2992	0.771904
$563$	−24.4033	−1.02848	−0.514239	−	0.857647i	$-0.671926\pi$
$563$	−24.4033	−1.02848	−0.514239	+	0.857647i	$0.671926\pi$
$564$	−19.6083	−0.825657
$565$	0	0
$566$	16.6899	0.701530
$567$	2.63518	0.110667
$568$	−44.6291	−1.87260
$569$	1.48958	0.0624464	0.0312232	−	0.999512i	$-0.490060\pi$
$569$	1.48958	0.0624464	0.0312232	+	0.999512i	$0.490060\pi$
$570$	0	0
$571$	−15.4142	−0.645066	−0.322533	−	0.946558i	$-0.604534\pi$
$571$	−15.4142	−0.645066	−0.322533	+	0.946558i	$0.604534\pi$
$572$	0.418909	0.0175155
$573$	−30.0533	−1.25550
$574$	−1.26629	−0.0528538
$575$	0	0
$576$	6.24668	0.260278
$577$	−45.7514	−1.90465	−0.952327	−	0.305079i	$-0.901317\pi$
$577$	−45.7514	−1.90465	−0.952327	+	0.305079i	$0.901317\pi$
$578$	0	0
$579$	12.6144	0.524235
$580$	0	0
$581$	−1.16196	−0.0482064
$582$	−26.6509	−1.10472
$583$	1.22057	0.0505509
$584$	2.51713	0.104159
$585$	0	0
$586$	13.6279	0.562961
$587$	−11.7659	−0.485633	−0.242816	−	0.970072i	$-0.578071\pi$
$587$	−11.7659	−0.485633	−0.242816	+	0.970072i	$0.578071\pi$
$588$	−14.5928	−0.601797
$589$	−11.0697	−0.456118
$590$	0	0
$591$	36.2730	1.49207
$592$	5.25630	0.216033
$593$	12.7825	0.524912	0.262456	−	0.964944i	$-0.415467\pi$
$593$	12.7825	0.524912	0.262456	+	0.964944i	$0.415467\pi$
$594$	−0.580723	−0.0238274
$595$	0	0
$596$	9.48372	0.388468
$597$	−35.5668	−1.45565
$598$	−12.7231	−0.520287
$599$	2.28863	0.0935109	0.0467554	−	0.998906i	$-0.485112\pi$
$599$	2.28863	0.0935109	0.0467554	+	0.998906i	$0.485112\pi$
$600$	0	0
$601$	−30.7826	−1.25565	−0.627824	−	0.778355i	$-0.716054\pi$
$601$	−30.7826	−1.25565	−0.627824	+	0.778355i	$0.716054\pi$
$602$	0.678534	0.0276550
$603$	7.94252	0.323444
$604$	2.15987	0.0878839
$605$	0	0
$606$	18.5688	0.754308
$607$	−15.8045	−0.641485	−0.320743	−	0.947166i	$-0.603932\pi$
$607$	−15.8045	−0.641485	−0.320743	+	0.947166i	$0.603932\pi$
$608$	−41.1207	−1.66767
$609$	1.71249	0.0693934
$610$	0	0
$611$	−25.1603	−1.01788
$612$	0	0
$613$	2.57732	0.104097	0.0520485	−	0.998645i	$-0.483425\pi$
$613$	2.57732	0.104097	0.0520485	+	0.998645i	$0.483425\pi$
$614$	21.5482	0.869615
$615$	0	0
$616$	−0.104914	−0.00422712
$617$	−31.9384	−1.28579	−0.642895	−	0.765954i	$-0.722267\pi$
$617$	−31.9384	−1.28579	−0.642895	+	0.765954i	$0.722267\pi$
$618$	−27.1736	−1.09308
$619$	−29.7559	−1.19599	−0.597995	−	0.801500i	$-0.704036\pi$
$619$	−29.7559	−1.19599	−0.597995	+	0.801500i	$0.704036\pi$
$620$	0	0
$621$	−19.7950	−0.794346
$622$	10.9533	0.439186
$623$	4.06262	0.162765
$624$	4.10737	0.164427
$625$	0	0
$626$	16.4198	0.656268
$627$	−2.31124	−0.0923020
$628$	−1.78055	−0.0710516
$629$	0	0
$630$	0	0
$631$	31.5372	1.25548	0.627738	−	0.778425i	$-0.283981\pi$
$631$	31.5372	1.25548	0.627738	+	0.778425i	$0.283981\pi$
$632$	25.7600	1.02468
$633$	15.1740	0.603113
$634$	12.5182	0.497160
$635$	0	0
$636$	−17.4725	−0.692830
$637$	−18.7247	−0.741902
$638$	−0.510267	−0.0202017
$639$	14.2889	0.565260
$640$	0	0
$641$	−42.6851	−1.68596	−0.842980	−	0.537944i	$-0.819201\pi$
$641$	−42.6851	−1.68596	−0.842980	+	0.537944i	$0.819201\pi$
$642$	−7.25950	−0.286510
$643$	−12.3279	−0.486166	−0.243083	−	0.970005i	$-0.578159\pi$
$643$	−12.3279	−0.486166	−0.243083	+	0.970005i	$0.578159\pi$
$644$	−1.23700	−0.0487446
$645$	0	0
$646$	0	0
$647$	−0.415176	−0.0163223	−0.00816114	−	0.999967i	$-0.502598\pi$
$647$	−0.415176	−0.0163223	−0.00816114	+	0.999967i	$0.502598\pi$
$648$	32.4956	1.27655
$649$	0.653207	0.0256406
$650$	0	0
$651$	0.668750	0.0262104
$652$	−25.1489	−0.984906
$653$	32.6828	1.27898	0.639488	−	0.768801i	$-0.279146\pi$
$653$	32.6828	1.27898	0.639488	+	0.768801i	$0.279146\pi$
$654$	−10.8651	−0.424861
$655$	0	0
$656$	−4.15203	−0.162109
$657$	−0.805907	−0.0314414
$658$	2.17962	0.0849704
$659$	26.0263	1.01384	0.506920	−	0.861993i	$-0.330784\pi$
$659$	26.0263	1.01384	0.506920	+	0.861993i	$0.330784\pi$
$660$	0	0
$661$	−25.1438	−0.977981	−0.488991	−	0.872289i	$-0.662635\pi$
$661$	−25.1438	−0.977981	−0.488991	+	0.872289i	$0.662635\pi$
$662$	18.1346	0.704821
$663$	0	0
$664$	−14.3287	−0.556061
$665$	0	0
$666$	−6.32917	−0.245250
$667$	−17.3934	−0.673474
$668$	−23.4541	−0.907468
$669$	4.00690	0.154916
$670$	0	0
$671$	1.92920	0.0744760
$672$	2.48422	0.0958307
$673$	30.2113	1.16456	0.582280	−	0.812988i	$-0.302161\pi$
$673$	30.2113	1.16456	0.582280	+	0.812988i	$0.302161\pi$
$674$	33.5054	1.29058
$675$	0	0
$676$	6.05454	0.232867
$677$	−27.6783	−1.06376	−0.531881	−	0.846819i	$-0.678515\pi$
$677$	−27.6783	−1.06376	−0.531881	+	0.846819i	$0.678515\pi$
$678$	19.0049	0.729881
$679$	−3.32480	−0.127594
$680$	0	0
$681$	−12.6190	−0.483563
$682$	−0.199267	−0.00763031
$683$	5.41853	0.207334	0.103667	−	0.994612i	$-0.466942\pi$
$683$	5.41853	0.207334	0.103667	+	0.994612i	$0.466942\pi$
$684$	7.95937	0.304334
$685$	0	0
$686$	3.25776	0.124382
$687$	−22.5398	−0.859947
$688$	2.22484	0.0848214
$689$	−22.4198	−0.854127
$690$	0	0
$691$	21.6328	0.822950	0.411475	−	0.911421i	$-0.365014\pi$
$691$	21.6328	0.822950	0.411475	+	0.911421i	$0.365014\pi$
$692$	11.8432	0.450211
$693$	0.0335904	0.00127599
$694$	30.3135	1.15068
$695$	0	0
$696$	21.1174	0.800454
$697$	0	0
$698$	−13.2028	−0.499735
$699$	14.5778	0.551381
$700$	0	0
$701$	12.5437	0.473767	0.236884	−	0.971538i	$-0.423874\pi$
$701$	12.5437	0.473767	0.236884	+	0.971538i	$0.423874\pi$
$702$	10.6669	0.402597
$703$	54.3288	2.04905
$704$	−0.965233	−0.0363786
$705$	0	0
$706$	−1.64131	−0.0617715
$707$	2.31653	0.0871222
$708$	−9.35067	−0.351420
$709$	7.23503	0.271717	0.135859	−	0.990728i	$-0.456621\pi$
$709$	7.23503	0.271717	0.135859	+	0.990728i	$0.456621\pi$
$710$	0	0
$711$	−8.24757	−0.309308
$712$	50.0980	1.87750
$713$	−6.79236	−0.254376
$714$	0	0
$715$	0	0
$716$	16.4439	0.614536
$717$	−46.3617	−1.73141
$718$	6.73479	0.251340
$719$	2.17282	0.0810326	0.0405163	−	0.999179i	$-0.487100\pi$
$719$	2.17282	0.0810326	0.0405163	+	0.999179i	$0.487100\pi$
$720$	0	0
$721$	−3.39001	−0.126251
$722$	42.4322	1.57916
$723$	−11.9756	−0.445378
$724$	10.9385	0.406528
$725$	0	0
$726$	21.1821	0.786140
$727$	−20.3983	−0.756530	−0.378265	−	0.925697i	$-0.623479\pi$
$727$	−20.3983	−0.756530	−0.378265	+	0.925697i	$0.623479\pi$
$728$	1.92710	0.0714231
$729$	13.8865	0.514315
$730$	0	0
$731$	0	0
$732$	−27.6166	−1.02074
$733$	−35.3414	−1.30536	−0.652682	−	0.757632i	$-0.726356\pi$
$733$	−35.3414	−1.30536	−0.652682	+	0.757632i	$0.726356\pi$
$734$	−0.866189	−0.0319716
$735$	0	0
$736$	−25.2317	−0.930053
$737$	−1.22727	−0.0452071
$738$	4.99950	0.184034
$739$	18.7727	0.690564	0.345282	−	0.938499i	$-0.387783\pi$
$739$	18.7727	0.690564	0.345282	+	0.938499i	$0.387783\pi$
$740$	0	0
$741$	42.4535	1.55957
$742$	1.94221	0.0713009
$743$	−10.6842	−0.391966	−0.195983	−	0.980607i	$-0.562790\pi$
$743$	−10.6842	−0.391966	−0.195983	+	0.980607i	$0.562790\pi$
$744$	8.24666	0.302337
$745$	0	0
$746$	24.3775	0.892523
$747$	4.58761	0.167852
$748$	0	0
$749$	−0.905650	−0.0330917
$750$	0	0
$751$	−15.0841	−0.550427	−0.275214	−	0.961383i	$-0.588749\pi$
$751$	−15.0841	−0.550427	−0.275214	+	0.961383i	$0.588749\pi$
$752$	7.14675	0.260615
$753$	32.2489	1.17521
$754$	9.37275	0.341335
$755$	0	0
$756$	1.03709	0.0377185
$757$	16.6888	0.606565	0.303283	−	0.952901i	$-0.401917\pi$
$757$	16.6888	0.606565	0.303283	+	0.952901i	$0.401917\pi$
$758$	−23.9296	−0.869163
$759$	−1.41818	−0.0514765
$760$	0	0
$761$	−23.6474	−0.857217	−0.428609	−	0.903490i	$-0.640996\pi$
$761$	−23.6474	−0.857217	−0.428609	+	0.903490i	$0.640996\pi$
$762$	−15.8920	−0.575707
$763$	−1.35547	−0.0490712
$764$	−15.9922	−0.578579
$765$	0	0
$766$	25.3199	0.914845
$767$	−11.9983	−0.433234
$768$	33.8551	1.22164
$769$	33.2995	1.20081	0.600405	−	0.799696i	$-0.295006\pi$
$769$	33.2995	1.20081	0.600405	+	0.799696i	$0.295006\pi$
$770$	0	0
$771$	5.40457	0.194641
$772$	6.71247	0.241587
$773$	34.8608	1.25386	0.626928	−	0.779077i	$-0.284312\pi$
$773$	34.8608	1.25386	0.626928	+	0.779077i	$0.284312\pi$
$774$	−2.67896	−0.0962932
$775$	0	0
$776$	−40.9997	−1.47180
$777$	−3.28215	−0.117746
$778$	−10.4522	−0.374730
$779$	−42.9151	−1.53759
$780$	0	0
$781$	−2.20791	−0.0790052
$782$	0	0
$783$	14.5824	0.521132
$784$	5.31874	0.189955
$785$	0	0
$786$	−3.28214	−0.117070
$787$	4.06098	0.144758	0.0723792	−	0.997377i	$-0.476941\pi$
$787$	4.06098	0.144758	0.0723792	+	0.997377i	$0.476941\pi$
$788$	19.3019	0.687603
$789$	−32.1056	−1.14299
$790$	0	0
$791$	2.37094	0.0843009
$792$	0.414218	0.0147186
$793$	−35.4362	−1.25838
$794$	0.911891	0.0323618
$795$	0	0
$796$	−18.9261	−0.670819
$797$	−36.0341	−1.27639	−0.638196	−	0.769874i	$-0.720319\pi$
$797$	−36.0341	−1.27639	−0.638196	+	0.769874i	$0.720319\pi$
$798$	−3.67772	−0.130190
$799$	0	0
$800$	0	0
$801$	−16.0399	−0.566741
$802$	−16.2967	−0.575455
$803$	0.124528	0.00439450
$804$	17.5684	0.619591
$805$	0	0
$806$	3.66019	0.128925
$807$	−49.6088	−1.74631
$808$	28.5662	1.00496
$809$	33.7903	1.18800	0.594002	−	0.804464i	$-0.297547\pi$
$809$	33.7903	1.18800	0.594002	+	0.804464i	$0.297547\pi$
$810$	0	0
$811$	0.0332766	0.00116850	0.000584250	−	1.00000i	$-0.499814\pi$
$811$	0.0332766	0.00116850	0.000584250		1.00000i	$0.499814\pi$
$812$	0.911263	0.0319790
$813$	57.0354	2.00032
$814$	0.977979	0.0342781
$815$	0	0
$816$	0	0
$817$	22.9958	0.804523
$818$	−11.3271	−0.396044
$819$	−0.616999	−0.0215597
$820$	0	0
$821$	28.9203	1.00932	0.504662	−	0.863317i	$-0.331617\pi$
$821$	28.9203	1.00932	0.504662	+	0.863317i	$0.331617\pi$
$822$	−19.5151	−0.680666
$823$	23.1687	0.807608	0.403804	−	0.914845i	$-0.367688\pi$
$823$	23.1687	0.807608	0.403804	+	0.914845i	$0.367688\pi$
$824$	−41.8038	−1.45630
$825$	0	0
$826$	1.03940	0.0361655
$827$	7.74018	0.269152	0.134576	−	0.990903i	$-0.457033\pi$
$827$	7.74018	0.269152	0.134576	+	0.990903i	$0.457033\pi$
$828$	4.88387	0.169726
$829$	11.3412	0.393895	0.196947	−	0.980414i	$-0.436897\pi$
$829$	11.3412	0.393895	0.196947	+	0.980414i	$0.436897\pi$
$830$	0	0
$831$	25.4086	0.881415
$832$	17.7297	0.614667
$833$	0	0
$834$	−31.5058	−1.09096
$835$	0	0
$836$	−1.22988	−0.0425362
$837$	5.69463	0.196835
$838$	33.2883	1.14993
$839$	−21.5577	−0.744256	−0.372128	−	0.928181i	$-0.621372\pi$
$839$	−21.5577	−0.744256	−0.372128	+	0.928181i	$0.621372\pi$
$840$	0	0
$841$	−16.1868	−0.558166
$842$	10.0101	0.344970
$843$	−37.4660	−1.29040
$844$	8.07453	0.277937
$845$	0	0
$846$	−8.60548	−0.295862
$847$	2.64254	0.0907988
$848$	6.36832	0.218689
$849$	−34.1712	−1.17275
$850$	0	0
$851$	33.3361	1.14275
$852$	31.6063	1.08281
$853$	−2.70374	−0.0925741	−0.0462871	−	0.998928i	$-0.514739\pi$
$853$	−2.70374	−0.0925741	−0.0462871	+	0.998928i	$0.514739\pi$
$854$	3.06981	0.105047
$855$	0	0
$856$	−11.1680	−0.381714
$857$	2.92448	0.0998982	0.0499491	−	0.998752i	$-0.484094\pi$
$857$	2.92448	0.0998982	0.0499491	+	0.998752i	$0.484094\pi$
$858$	0.764211	0.0260897
$859$	−14.1956	−0.484346	−0.242173	−	0.970233i	$-0.577860\pi$
$859$	−14.1956	−0.484346	−0.242173	+	0.970233i	$0.577860\pi$
$860$	0	0
$861$	2.59262	0.0883561
$862$	−1.57819	−0.0537533
$863$	−30.9097	−1.05218	−0.526090	−	0.850429i	$-0.676342\pi$
$863$	−30.9097	−1.05218	−0.526090	+	0.850429i	$0.676342\pi$
$864$	21.1540	0.719672
$865$	0	0
$866$	−5.26174	−0.178801
$867$	0	0
$868$	0.355861	0.0120787
$869$	1.27441	0.0432313
$870$	0	0
$871$	22.5429	0.763838
$872$	−16.7149	−0.566038
$873$	13.1268	0.444276
$874$	37.3539	1.26351
$875$	0	0
$876$	−1.78262	−0.0602293
$877$	−42.4009	−1.43178	−0.715889	−	0.698214i	$-0.753978\pi$
$877$	−42.4009	−1.43178	−0.715889	+	0.698214i	$0.753978\pi$
$878$	−21.2494	−0.717132
$879$	−27.9019	−0.941107
$880$	0	0
$881$	39.0565	1.31585	0.657924	−	0.753084i	$-0.271435\pi$
$881$	39.0565	1.31585	0.657924	+	0.753084i	$0.271435\pi$
$882$	−6.40435	−0.215646
$883$	−45.5807	−1.53391	−0.766957	−	0.641699i	$-0.778230\pi$
$883$	−45.5807	−1.53391	−0.766957	+	0.641699i	$0.778230\pi$
$884$	0	0
$885$	0	0
$886$	−5.67685	−0.190717
$887$	42.6982	1.43367	0.716833	−	0.697245i	$-0.245591\pi$
$887$	42.6982	1.43367	0.716833	+	0.697245i	$0.245591\pi$
$888$	−40.4737	−1.35821
$889$	−1.98259	−0.0664939
$890$	0	0
$891$	1.60764	0.0538578
$892$	2.13219	0.0713909
$893$	73.8683	2.47191
$894$	17.3011	0.578634
$895$	0	0
$896$	0.963875	0.0322008
$897$	26.0495	0.869767
$898$	−33.5562	−1.11978
$899$	5.00373	0.166884
$900$	0	0
$901$	0	0
$902$	−0.772519	−0.0257221
$903$	−1.38924	−0.0462311
$904$	29.2371	0.972413
$905$	0	0
$906$	3.94023	0.130905
$907$	−34.8391	−1.15681	−0.578406	−	0.815749i	$-0.696325\pi$
$907$	−34.8391	−1.15681	−0.578406	+	0.815749i	$0.696325\pi$
$908$	−6.71495	−0.222844
$909$	−9.14603	−0.303355
$910$	0	0
$911$	−14.5521	−0.482133	−0.241067	−	0.970509i	$-0.577497\pi$
$911$	−14.5521	−0.482133	−0.241067	+	0.970509i	$0.577497\pi$
$912$	−12.0589	−0.399309
$913$	−0.708875	−0.0234603
$914$	23.1806	0.766746
$915$	0	0
$916$	−11.9941	−0.396295
$917$	−0.409460	−0.0135215
$918$	0	0
$919$	31.8817	1.05168	0.525840	−	0.850583i	$-0.323751\pi$
$919$	31.8817	1.05168	0.525840	+	0.850583i	$0.323751\pi$
$920$	0	0
$921$	−44.1181	−1.45374
$922$	6.35631	0.209334
$923$	40.5556	1.33490
$924$	0.0743002	0.00244430
$925$	0	0
$926$	16.4639	0.541037
$927$	13.3843	0.439598
$928$	18.5875	0.610163
$929$	−60.2926	−1.97813	−0.989067	−	0.147465i	$-0.952889\pi$
$929$	−60.2926	−1.97813	−0.989067	+	0.147465i	$0.952889\pi$
$930$	0	0
$931$	54.9741	1.80170
$932$	7.75724	0.254097
$933$	−22.4259	−0.734191
$934$	−28.1788	−0.922038
$935$	0	0
$936$	−7.60849	−0.248691
$937$	35.8629	1.17159	0.585794	−	0.810460i	$-0.300783\pi$
$937$	35.8629	1.17159	0.585794	+	0.810460i	$0.300783\pi$
$938$	−1.95288	−0.0637637
$939$	−33.6182	−1.09709
$940$	0	0
$941$	10.8635	0.354138	0.177069	−	0.984198i	$-0.443338\pi$
$941$	10.8635	0.354138	0.177069	+	0.984198i	$0.443338\pi$
$942$	−3.24824	−0.105833
$943$	−26.3327	−0.857510
$944$	3.40810	0.110924
$945$	0	0
$946$	0.413951	0.0134587
$947$	−44.0130	−1.43023	−0.715116	−	0.699006i	$-0.753626\pi$
$947$	−44.0130	−1.43023	−0.715116	+	0.699006i	$0.753626\pi$
$948$	−18.2432	−0.592511
$949$	−2.28737	−0.0742512
$950$	0	0
$951$	−25.6299	−0.831106
$952$	0	0
$953$	−24.8072	−0.803584	−0.401792	−	0.915731i	$-0.631612\pi$
$953$	−24.8072	−0.803584	−0.401792	+	0.915731i	$0.631612\pi$
$954$	−7.66816	−0.248266
$955$	0	0
$956$	−24.6704	−0.797897
$957$	1.04473	0.0337713
$958$	−36.9622	−1.19420
$959$	−2.43458	−0.0786166
$960$	0	0
$961$	−29.0460	−0.936967
$962$	−17.9638	−0.579177
$963$	3.57565	0.115224
$964$	−6.37256	−0.205246
$965$	0	0
$966$	−2.25665	−0.0726065
$967$	−12.4941	−0.401782	−0.200891	−	0.979614i	$-0.564384\pi$
$967$	−12.4941	−0.401782	−0.200891	+	0.979614i	$0.564384\pi$
$968$	32.5864	1.04737
$969$	0	0
$970$	0	0
$971$	1.92487	0.0617719	0.0308859	−	0.999523i	$-0.490167\pi$
$971$	1.92487	0.0617719	0.0308859	+	0.999523i	$0.490167\pi$
$972$	−10.0876	−0.323561
$973$	−3.93047	−0.126005
$974$	−6.96569	−0.223195
$975$	0	0
$976$	10.0656	0.322192
$977$	43.8303	1.40225	0.701127	−	0.713036i	$-0.252681\pi$
$977$	43.8303	1.40225	0.701127	+	0.713036i	$0.252681\pi$
$978$	−45.8789	−1.46704
$979$	2.47847	0.0792122
$980$	0	0
$981$	5.35160	0.170863
$982$	−9.53827	−0.304379
$983$	2.35222	0.0750241	0.0375120	−	0.999296i	$-0.488057\pi$
$983$	2.35222	0.0750241	0.0375120	+	0.999296i	$0.488057\pi$
$984$	31.9707	1.01919
$985$	0	0
$986$	0	0
$987$	−4.46259	−0.142046
$988$	22.5908	0.718708
$989$	14.1102	0.448680
$990$	0	0
$991$	−14.1464	−0.449376	−0.224688	−	0.974431i	$-0.572136\pi$
$991$	−14.1464	−0.449376	−0.224688	+	0.974431i	$0.572136\pi$
$992$	7.25867	0.230463
$993$	−37.1290	−1.17825
$994$	−3.51330	−0.111435
$995$	0	0
$996$	10.1476	0.321538
$997$	19.2113	0.608428	0.304214	−	0.952604i	$-0.401606\pi$
$997$	19.2113	0.608428	0.304214	+	0.952604i	$0.401606\pi$
$998$	33.7015	1.06680
$999$	−27.9486	−0.884256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bv.1.10	yes	15
5.4	even	2		7225.2.a.bt.1.6	✓	15
17.16	even	2		7225.2.a.bu.1.10	yes	15
85.84	even	2		7225.2.a.bw.1.6	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.6	✓	15	5.4	even	2
7225.2.a.bu.1.10	yes	15	17.16	even	2
7225.2.a.bv.1.10	yes	15	1.1	even	1	trivial
7225.2.a.bw.1.6	yes	15	85.84	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.970758 q^{2} -1.98754 q^{3} -1.05763 q^{4} -1.92942 q^{6} -0.240703 q^{7} -2.96822 q^{8} +0.950332 q^{9} +O(q^{10})\)	\(q+0.970758 q^{2} -1.98754 q^{3} -1.05763 q^{4} -1.92942 q^{6} -0.240703 q^{7} -2.96822 q^{8} +0.950332 q^{9} -0.146845 q^{11} +2.10209 q^{12} +2.69729 q^{13} -0.233664 q^{14} -0.766161 q^{16} +0.922542 q^{18} -7.91899 q^{19} +0.478408 q^{21} -0.142551 q^{22} -4.85909 q^{23} +5.89946 q^{24} +2.61841 q^{26} +4.07381 q^{27} +0.254574 q^{28} +3.57955 q^{29} +1.39787 q^{31} +5.19268 q^{32} +0.291860 q^{33} -1.00510 q^{36} -6.86057 q^{37} -7.68742 q^{38} -5.36098 q^{39} +5.41926 q^{41} +0.464418 q^{42} -2.90389 q^{43} +0.155307 q^{44} -4.71700 q^{46} -9.32800 q^{47} +1.52278 q^{48} -6.94206 q^{49} -2.85273 q^{52} -8.31198 q^{53} +3.95468 q^{54} +0.714458 q^{56} +15.7393 q^{57} +3.47488 q^{58} -4.44828 q^{59} -13.1377 q^{61} +1.35699 q^{62} -0.228748 q^{63} +6.57315 q^{64} +0.283326 q^{66} +8.35762 q^{67} +9.65766 q^{69} +15.0357 q^{71} -2.82079 q^{72} -0.848026 q^{73} -6.65995 q^{74} +8.37535 q^{76} +0.0353459 q^{77} -5.20421 q^{78} -8.67861 q^{79} -10.9479 q^{81} +5.26079 q^{82} +4.82738 q^{83} -0.505978 q^{84} -2.81897 q^{86} -7.11452 q^{87} +0.435867 q^{88} -16.8782 q^{89} -0.649245 q^{91} +5.13912 q^{92} -2.77832 q^{93} -9.05523 q^{94} -10.3207 q^{96} +13.8129 q^{97} -6.73906 q^{98} -0.139551 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * q^99

Introduction

L-functions

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Database

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