LMFDB - Embedded newform 9025.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.17557$ 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)

$f(q)$	$=$	$q-1.17557 q^{2} -1.17557 q^{3} -0.618034 q^{4} +1.38197 q^{6} -0.236068 q^{7} +3.07768 q^{8} -1.61803 q^{9} +O(q^{10})$	\(q-1.17557 q^{2} -1.17557 q^{3} -0.618034 q^{4} +1.38197 q^{6} -0.236068 q^{7} +3.07768 q^{8} -1.61803 q^{9} +0.854102 q^{11} +0.726543 q^{12} -0.726543 q^{13} +0.277515 q^{14} -2.38197 q^{16} -0.763932 q^{17} +1.90211 q^{18} +0.277515 q^{21} -1.00406 q^{22} +7.09017 q^{23} -3.61803 q^{24} +0.854102 q^{26} +5.42882 q^{27} +0.145898 q^{28} -8.78402 q^{29} -1.17557 q^{31} -3.35520 q^{32} -1.00406 q^{33} +0.898056 q^{34} +1.00000 q^{36} +8.78402 q^{37} +0.854102 q^{39} +1.62460 q^{41} -0.326238 q^{42} -2.61803 q^{43} -0.527864 q^{44} -8.33499 q^{46} -7.47214 q^{47} +2.80017 q^{48} -6.94427 q^{49} +0.898056 q^{51} +0.449028 q^{52} -1.00406 q^{53} -6.38197 q^{54} -0.726543 q^{56} +10.3262 q^{58} -11.3067 q^{59} -13.9443 q^{61} +1.38197 q^{62} +0.381966 q^{63} +8.70820 q^{64} +1.18034 q^{66} +11.5842 q^{67} +0.472136 q^{68} -8.33499 q^{69} +13.0373 q^{71} -4.97980 q^{72} +1.00000 q^{73} -10.3262 q^{74} -0.201626 q^{77} -1.00406 q^{78} +8.50651 q^{79} -1.52786 q^{81} -1.90983 q^{82} +13.2361 q^{83} -0.171513 q^{84} +3.07768 q^{86} +10.3262 q^{87} +2.62866 q^{88} +8.33499 q^{89} +0.171513 q^{91} -4.38197 q^{92} +1.38197 q^{93} +8.78402 q^{94} +3.94427 q^{96} -4.25325 q^{97} +8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9	$ 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{58} - 20 q^{61} + 10 q^{62} + 6 q^{63} + 8 q^{64} - 40 q^{66} - 16 q^{68} + 4 q^{73} - 10 q^{74} - 50 q^{77} - 24 q^{81} - 30 q^{82} + 44 q^{83} + 10 q^{87} - 22 q^{92} + 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * 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 $

$2$	−1.17557	−0.831254	−0.415627	−	0.909535i	$-0.636438\pi$
$2$	−1.17557	−0.831254	−0.415627	+	0.909535i	$0.636438\pi$
$3$	−1.17557	−0.678716	−0.339358	−	0.940657i	$-0.610210\pi$
$3$	−1.17557	−0.678716	−0.339358	+	0.940657i	$0.610210\pi$
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	1.38197	0.564185
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	3.07768	1.08813
$9$	−1.61803	−0.539345
$10$	0	0
$11$	0.854102	0.257521	0.128761	−	0.991676i	$-0.458900\pi$
$11$	0.854102	0.257521	0.128761	+	0.991676i	$0.458900\pi$
$12$	0.726543	0.209735
$13$	−0.726543	−0.201507	−0.100753	−	0.994911i	$-0.532125\pi$
$13$	−0.726543	−0.201507	−0.100753	+	0.994911i	$0.532125\pi$
$14$	0.277515	0.0741689
$15$	0	0
$16$	−2.38197	−0.595492
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	1.90211	0.448332
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.277515	0.0605586
$22$	−1.00406	−0.214066
$23$	7.09017	1.47840	0.739201	−	0.673485i	$-0.235203\pi$
$23$	7.09017	1.47840	0.739201	+	0.673485i	$0.235203\pi$
$24$	−3.61803	−0.738528
$25$	0	0
$26$	0.854102	0.167503
$27$	5.42882	1.04478
$28$	0.145898	0.0275721
$29$	−8.78402	−1.63115	−0.815576	−	0.578650i	$-0.803580\pi$
$29$	−8.78402	−1.63115	−0.815576	+	0.578650i	$0.803580\pi$
$30$	0	0
$31$	−1.17557	−0.211139	−0.105569	−	0.994412i	$-0.533667\pi$
$31$	−1.17557	−0.211139	−0.105569	+	0.994412i	$0.533667\pi$
$32$	−3.35520	−0.593121
$33$	−1.00406	−0.174784
$34$	0.898056	0.154015
$35$	0	0
$36$	1.00000	0.166667
$37$	8.78402	1.44408	0.722042	−	0.691849i	$-0.243204\pi$
$37$	8.78402	1.44408	0.722042	+	0.691849i	$0.243204\pi$
$38$	0	0
$39$	0.854102	0.136766
$40$	0	0
$41$	1.62460	0.253720	0.126860	−	0.991921i	$-0.459510\pi$
$41$	1.62460	0.253720	0.126860	+	0.991921i	$0.459510\pi$
$42$	−0.326238	−0.0503396
$43$	−2.61803	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$43$	−2.61803	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$44$	−0.527864	−0.0795785
$45$	0	0
$46$	−8.33499	−1.22893
$47$	−7.47214	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$47$	−7.47214	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$48$	2.80017	0.404170
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0.898056	0.125753
$52$	0.449028	0.0622690
$53$	−1.00406	−0.137918	−0.0689589	−	0.997620i	$-0.521968\pi$
$53$	−1.00406	−0.137918	−0.0689589	+	0.997620i	$0.521968\pi$
$54$	−6.38197	−0.868476
$55$	0	0
$56$	−0.726543	−0.0970883
$57$	0	0
$58$	10.3262	1.35590
$59$	−11.3067	−1.47200	−0.736002	−	0.676979i	$-0.763289\pi$
$59$	−11.3067	−1.47200	−0.736002	+	0.676979i	$0.763289\pi$
$60$	0	0
$61$	−13.9443	−1.78538	−0.892691	−	0.450670i	$-0.851185\pi$
$61$	−13.9443	−1.78538	−0.892691	+	0.450670i	$0.851185\pi$
$62$	1.38197	0.175510
$63$	0.381966	0.0481232
$64$	8.70820	1.08853
$65$	0	0
$66$	1.18034	0.145290
$67$	11.5842	1.41523	0.707617	−	0.706596i	$-0.249770\pi$
$67$	11.5842	1.41523	0.707617	+	0.706596i	$0.249770\pi$
$68$	0.472136	0.0572549
$69$	−8.33499	−1.00342
$70$	0	0
$71$	13.0373	1.54724	0.773620	−	0.633650i	$-0.218444\pi$
$71$	13.0373	1.54724	0.773620	+	0.633650i	$0.218444\pi$
$72$	−4.97980	−0.586875
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	−10.3262	−1.20040
$75$	0	0
$76$	0	0
$77$	−0.201626	−0.0229774
$78$	−1.00406	−0.113687
$79$	8.50651	0.957057	0.478528	−	0.878072i	$-0.341170\pi$
$79$	8.50651	0.957057	0.478528	+	0.878072i	$0.341170\pi$
$80$	0	0
$81$	−1.52786	−0.169763
$82$	−1.90983	−0.210905
$83$	13.2361	1.45285	0.726424	−	0.687247i	$-0.241181\pi$
$83$	13.2361	1.45285	0.726424	+	0.687247i	$0.241181\pi$
$84$	−0.171513	−0.0187136
$85$	0	0
$86$	3.07768	0.331875
$87$	10.3262	1.10709
$88$	2.62866	0.280216
$89$	8.33499	0.883508	0.441754	−	0.897136i	$-0.354356\pi$
$89$	8.33499	0.883508	0.441754	+	0.897136i	$0.354356\pi$
$90$	0	0
$91$	0.171513	0.0179795
$92$	−4.38197	−0.456852
$93$	1.38197	0.143303
$94$	8.78402	0.906003
$95$	0	0
$96$	3.94427	0.402561
$97$	−4.25325	−0.431853	−0.215926	−	0.976410i	$-0.569277\pi$
$97$	−4.25325	−0.431853	−0.215926	+	0.976410i	$0.569277\pi$
$98$	8.16348	0.824636
$99$	−1.38197	−0.138893
$100$	0	0
$101$	16.7082	1.66253	0.831264	−	0.555878i	$-0.187618\pi$
$101$	16.7082	1.66253	0.831264	+	0.555878i	$0.187618\pi$
$102$	−1.05573	−0.104533
$103$	−1.34708	−0.132732	−0.0663661	−	0.997795i	$-0.521141\pi$
$103$	−1.34708	−0.132732	−0.0663661	+	0.997795i	$0.521141\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	1.18034	0.114645
$107$	−7.77997	−0.752118	−0.376059	−	0.926596i	$-0.622721\pi$
$107$	−7.77997	−0.752118	−0.376059	+	0.926596i	$0.622721\pi$
$108$	−3.35520	−0.322854
$109$	9.23305	0.884366	0.442183	−	0.896925i	$-0.354204\pi$
$109$	9.23305	0.884366	0.442183	+	0.896925i	$0.354204\pi$
$110$	0	0
$111$	−10.3262	−0.980123
$112$	0.562306	0.0531329
$113$	−4.35926	−0.410084	−0.205042	−	0.978753i	$-0.565733\pi$
$113$	−4.35926	−0.410084	−0.205042	+	0.978753i	$0.565733\pi$
$114$	0	0
$115$	0	0
$116$	5.42882	0.504054
$117$	1.17557	0.108682
$118$	13.2918	1.22361
$119$	0.180340	0.0165317
$120$	0	0
$121$	−10.2705	−0.933683
$122$	16.3925	1.48410
$123$	−1.90983	−0.172204
$124$	0.726543	0.0652454
$125$	0	0
$126$	−0.449028	−0.0400026
$127$	−2.17963	−0.193411	−0.0967053	−	0.995313i	$-0.530830\pi$
$127$	−2.17963	−0.193411	−0.0967053	+	0.995313i	$0.530830\pi$
$128$	−3.52671	−0.311720
$129$	3.07768	0.270975
$130$	0	0
$131$	−13.3820	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$131$	−13.3820	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$132$	0.620541	0.0540112
$133$	0	0
$134$	−13.6180	−1.17642
$135$	0	0
$136$	−2.35114	−0.201609
$137$	−14.2361	−1.21627	−0.608135	−	0.793834i	$-0.708082\pi$
$137$	−14.2361	−1.21627	−0.608135	+	0.793834i	$0.708082\pi$
$138$	9.79837	0.834093
$139$	−10.9098	−0.925360	−0.462680	−	0.886525i	$-0.653112\pi$
$139$	−10.9098	−0.925360	−0.462680	+	0.886525i	$0.653112\pi$
$140$	0	0
$141$	8.78402	0.739748
$142$	−15.3262	−1.28615
$143$	−0.620541	−0.0518923
$144$	3.85410	0.321175
$145$	0	0
$146$	−1.17557	−0.0972909
$147$	8.16348	0.673313
$148$	−5.42882	−0.446247
$149$	−4.09017	−0.335080	−0.167540	−	0.985865i	$-0.553582\pi$
$149$	−4.09017	−0.335080	−0.167540	+	0.985865i	$0.553582\pi$
$150$	0	0
$151$	18.0171	1.46621	0.733104	−	0.680116i	$-0.238071\pi$
$151$	18.0171	1.46621	0.733104	+	0.680116i	$0.238071\pi$
$152$	0	0
$153$	1.23607	0.0999302
$154$	0.237026	0.0191001
$155$	0	0
$156$	−0.527864	−0.0422629
$157$	−17.2705	−1.37834	−0.689168	−	0.724601i	$-0.742024\pi$
$157$	−17.2705	−1.37834	−0.689168	+	0.724601i	$0.742024\pi$
$158$	−10.0000	−0.795557
$159$	1.18034	0.0936070
$160$	0	0
$161$	−1.67376	−0.131911
$162$	1.79611	0.141116
$163$	2.81966	0.220853	0.110426	−	0.993884i	$-0.464778\pi$
$163$	2.81966	0.220853	0.110426	+	0.993884i	$0.464778\pi$
$164$	−1.00406	−0.0784037
$165$	0	0
$166$	−15.5599	−1.20768
$167$	−13.0373	−1.00885	−0.504427	−	0.863454i	$-0.668296\pi$
$167$	−13.0373	−1.00885	−0.504427	+	0.863454i	$0.668296\pi$
$168$	0.854102	0.0658954
$169$	−12.4721	−0.959395
$170$	0	0
$171$	0	0
$172$	1.61803	0.123374
$173$	12.8658	0.978166	0.489083	−	0.872237i	$-0.337332\pi$
$173$	12.8658	0.978166	0.489083	+	0.872237i	$0.337332\pi$
$174$	−12.1392	−0.920272
$175$	0	0
$176$	−2.03444	−0.153352
$177$	13.2918	0.999073
$178$	−9.79837	−0.734419
$179$	12.1392	0.907328	0.453664	−	0.891173i	$-0.350117\pi$
$179$	12.1392	0.907328	0.453664	+	0.891173i	$0.350117\pi$
$180$	0	0
$181$	14.6619	1.08981	0.544904	−	0.838498i	$-0.316566\pi$
$181$	14.6619	1.08981	0.544904	+	0.838498i	$0.316566\pi$
$182$	−0.201626	−0.0149455
$183$	16.3925	1.21177
$184$	21.8213	1.60869
$185$	0	0
$186$	−1.62460	−0.119121
$187$	−0.652476	−0.0477138
$188$	4.61803	0.336805
$189$	−1.28157	−0.0932206
$190$	0	0
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	−10.2371	−0.738800
$193$	6.15537	0.443073	0.221537	−	0.975152i	$-0.428893\pi$
$193$	6.15537	0.443073	0.221537	+	0.975152i	$0.428893\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	4.29180	0.306557
$197$	20.8885	1.48825	0.744124	−	0.668042i	$-0.232867\pi$
$197$	20.8885	1.48825	0.744124	+	0.668042i	$0.232867\pi$
$198$	1.62460	0.115455
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	−13.6180	−0.960542
$202$	−19.6417	−1.38198
$203$	2.07363	0.145540
$204$	−0.555029	−0.0388598
$205$	0	0
$206$	1.58359	0.110334
$207$	−11.4721	−0.797369
$208$	1.73060	0.119995
$209$	0	0
$210$	0	0
$211$	14.2128	0.978453	0.489226	−	0.872157i	$-0.337279\pi$
$211$	14.2128	0.978453	0.489226	+	0.872157i	$0.337279\pi$
$212$	0.620541	0.0426190
$213$	−15.3262	−1.05014
$214$	9.14590	0.625201
$215$	0	0
$216$	16.7082	1.13685
$217$	0.277515	0.0188389
$218$	−10.8541	−0.735133
$219$	−1.17557	−0.0794377
$220$	0	0
$221$	0.555029	0.0373353
$222$	12.1392	0.814731
$223$	19.7477	1.32240	0.661201	−	0.750209i	$-0.270047\pi$
$223$	19.7477	1.32240	0.661201	+	0.750209i	$0.270047\pi$
$224$	0.792055	0.0529214
$225$	0	0
$226$	5.12461	0.340884
$227$	18.6376	1.23702	0.618511	−	0.785776i	$-0.287736\pi$
$227$	18.6376	1.23702	0.618511	+	0.785776i	$0.287736\pi$
$228$	0	0
$229$	−24.8541	−1.64241	−0.821203	−	0.570637i	$-0.806696\pi$
$229$	−24.8541	−1.64241	−0.821203	+	0.570637i	$0.806696\pi$
$230$	0	0
$231$	0.237026	0.0155951
$232$	−27.0344	−1.77490
$233$	−6.23607	−0.408538	−0.204269	−	0.978915i	$-0.565482\pi$
$233$	−6.23607	−0.408538	−0.204269	+	0.978915i	$0.565482\pi$
$234$	−1.38197	−0.0903419
$235$	0	0
$236$	6.98791	0.454874
$237$	−10.0000	−0.649570
$238$	−0.212002	−0.0137421
$239$	28.2148	1.82506	0.912531	−	0.409007i	$-0.134125\pi$
$239$	28.2148	1.82506	0.912531	+	0.409007i	$0.134125\pi$
$240$	0	0
$241$	−14.4904	−0.933406	−0.466703	−	0.884414i	$-0.654558\pi$
$241$	−14.4904	−0.933406	−0.466703	+	0.884414i	$0.654558\pi$
$242$	12.0737	0.776127
$243$	−14.4904	−0.929557
$244$	8.61803	0.551713
$245$	0	0
$246$	2.24514	0.143145
$247$	0	0
$248$	−3.61803	−0.229745
$249$	−15.5599	−0.986071
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	−0.236068	−0.0148709
$253$	6.05573	0.380720
$254$	2.56231	0.160773
$255$	0	0
$256$	−13.2705	−0.829407
$257$	13.7638	0.858563	0.429282	−	0.903171i	$-0.358767\pi$
$257$	13.7638	0.858563	0.429282	+	0.903171i	$0.358767\pi$
$258$	−3.61803	−0.225249
$259$	−2.07363	−0.128849
$260$	0	0
$261$	14.2128	0.879753
$262$	15.7314	0.971892
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	−3.09017	−0.190187
$265$	0	0
$266$	0	0
$267$	−9.79837	−0.599651
$268$	−7.15942	−0.437331
$269$	6.26137	0.381762	0.190881	−	0.981613i	$-0.438865\pi$
$269$	6.26137	0.381762	0.190881	+	0.981613i	$0.438865\pi$
$270$	0	0
$271$	6.38197	0.387677	0.193838	−	0.981033i	$-0.437906\pi$
$271$	6.38197	0.387677	0.193838	+	0.981033i	$0.437906\pi$
$272$	1.81966	0.110333
$273$	−0.201626	−0.0122030
$274$	16.7355	1.01103
$275$	0	0
$276$	5.15131	0.310072
$277$	28.8328	1.73240	0.866198	−	0.499701i	$-0.166557\pi$
$277$	28.8328	1.73240	0.866198	+	0.499701i	$0.166557\pi$
$278$	12.8253	0.769209
$279$	1.90211	0.113877
$280$	0	0
$281$	−10.7921	−0.643805	−0.321902	−	0.946773i	$-0.604322\pi$
$281$	−10.7921	−0.643805	−0.321902	+	0.946773i	$0.604322\pi$
$282$	−10.3262	−0.614919
$283$	−5.27051	−0.313299	−0.156650	−	0.987654i	$-0.550069\pi$
$283$	−5.27051	−0.313299	−0.156650	+	0.987654i	$0.550069\pi$
$284$	−8.05748	−0.478123
$285$	0	0
$286$	0.729490	0.0431357
$287$	−0.383516	−0.0226382
$288$	5.42882	0.319897
$289$	−16.4164	−0.965671
$290$	0	0
$291$	5.00000	0.293105
$292$	−0.618034	−0.0361677
$293$	−30.8828	−1.80419	−0.902097	−	0.431533i	$-0.857973\pi$
$293$	−30.8828	−1.80419	−0.902097	+	0.431533i	$0.857973\pi$
$294$	−9.59675	−0.559694
$295$	0	0
$296$	27.0344	1.57135
$297$	4.63677	0.269053
$298$	4.80828	0.278536
$299$	−5.15131	−0.297908
$300$	0	0
$301$	0.618034	0.0356229
$302$	−21.1803	−1.21879
$303$	−19.6417	−1.12838
$304$	0	0
$305$	0	0
$306$	−1.45309	−0.0830673
$307$	−25.4540	−1.45274	−0.726369	−	0.687305i	$-0.758793\pi$
$307$	−25.4540	−1.45274	−0.726369	+	0.687305i	$0.758793\pi$
$308$	0.124612	0.00710042
$309$	1.58359	0.0900874
$310$	0	0
$311$	−7.23607	−0.410320	−0.205160	−	0.978728i	$-0.565771\pi$
$311$	−7.23607	−0.410320	−0.205160	+	0.978728i	$0.565771\pi$
$312$	2.62866	0.148818
$313$	30.2705	1.71099	0.855495	−	0.517811i	$-0.173253\pi$
$313$	30.2705	1.71099	0.855495	+	0.517811i	$0.173253\pi$
$314$	20.3027	1.14575
$315$	0	0
$316$	−5.25731	−0.295747
$317$	26.6296	1.49567	0.747833	−	0.663887i	$-0.231094\pi$
$317$	26.6296	1.49567	0.747833	+	0.663887i	$0.231094\pi$
$318$	−1.38757	−0.0778112
$319$	−7.50245	−0.420057
$320$	0	0
$321$	9.14590	0.510474
$322$	1.96763	0.109651
$323$	0	0
$324$	0.944272	0.0524596
$325$	0	0
$326$	−3.31471	−0.183585
$327$	−10.8541	−0.600233
$328$	5.00000	0.276079
$329$	1.76393	0.0972487
$330$	0	0
$331$	1.79611	0.0987232	0.0493616	−	0.998781i	$-0.484281\pi$
$331$	1.79611	0.0987232	0.0493616	+	0.998781i	$0.484281\pi$
$332$	−8.18034	−0.448954
$333$	−14.2128	−0.778859
$334$	15.3262	0.838614
$335$	0	0
$336$	−0.661030	−0.0360622
$337$	−3.46120	−0.188544	−0.0942718	−	0.995547i	$-0.530052\pi$
$337$	−3.46120	−0.188544	−0.0942718	+	0.995547i	$0.530052\pi$
$338$	14.6619	0.797501
$339$	5.12461	0.278331
$340$	0	0
$341$	−1.00406	−0.0543727
$342$	0	0
$343$	3.29180	0.177740
$344$	−8.05748	−0.434430
$345$	0	0
$346$	−15.1246	−0.813104
$347$	−7.52786	−0.404117	−0.202058	−	0.979373i	$-0.564763\pi$
$347$	−7.52786	−0.404117	−0.202058	+	0.979373i	$0.564763\pi$
$348$	−6.38197	−0.342109
$349$	17.5623	0.940089	0.470044	−	0.882643i	$-0.344238\pi$
$349$	17.5623	0.940089	0.470044	+	0.882643i	$0.344238\pi$
$350$	0	0
$351$	−3.94427	−0.210530
$352$	−2.86568	−0.152741
$353$	−19.3262	−1.02863	−0.514316	−	0.857601i	$-0.671954\pi$
$353$	−19.3262	−1.02863	−0.514316	+	0.857601i	$0.671954\pi$
$354$	−15.6254	−0.830483
$355$	0	0
$356$	−5.15131	−0.273019
$357$	−0.212002	−0.0112203
$358$	−14.2705	−0.754220
$359$	9.61803	0.507620	0.253810	−	0.967254i	$-0.418316\pi$
$359$	9.61803	0.507620	0.253810	+	0.967254i	$0.418316\pi$
$360$	0	0
$361$	0	0
$362$	−17.2361	−0.905908
$363$	12.0737	0.633705
$364$	−0.106001	−0.00555597
$365$	0	0
$366$	−19.2705	−1.00729
$367$	27.0689	1.41298	0.706492	−	0.707721i	$-0.250277\pi$
$367$	27.0689	1.41298	0.706492	+	0.707721i	$0.250277\pi$
$368$	−16.8885	−0.880376
$369$	−2.62866	−0.136842
$370$	0	0
$371$	0.237026	0.0123058
$372$	−0.854102	−0.0442831
$373$	−4.53077	−0.234594	−0.117297	−	0.993097i	$-0.537423\pi$
$373$	−4.53077	−0.234594	−0.117297	+	0.993097i	$0.537423\pi$
$374$	0.767031	0.0396622
$375$	0	0
$376$	−22.9969	−1.18597
$377$	6.38197	0.328688
$378$	1.50658	0.0774900
$379$	13.2088	0.678490	0.339245	−	0.940698i	$-0.389828\pi$
$379$	13.2088	0.678490	0.339245	+	0.940698i	$0.389828\pi$
$380$	0	0
$381$	2.56231	0.131271
$382$	−15.2824	−0.781916
$383$	2.62866	0.134318	0.0671590	−	0.997742i	$-0.478607\pi$
$383$	2.62866	0.134318	0.0671590	+	0.997742i	$0.478607\pi$
$384$	4.14590	0.211569
$385$	0	0
$386$	−7.23607	−0.368306
$387$	4.23607	0.215331
$388$	2.62866	0.133450
$389$	−14.1459	−0.717226	−0.358613	−	0.933486i	$-0.616750\pi$
$389$	−14.1459	−0.717226	−0.358613	+	0.933486i	$0.616750\pi$
$390$	0	0
$391$	−5.41641	−0.273920
$392$	−21.3723	−1.07946
$393$	15.7314	0.793546
$394$	−24.5560	−1.23711
$395$	0	0
$396$	0.854102	0.0429202
$397$	−29.8328	−1.49727	−0.748633	−	0.662985i	$-0.769289\pi$
$397$	−29.8328	−1.49727	−0.748633	+	0.662985i	$0.769289\pi$
$398$	15.7719	0.790576
$399$	0	0
$400$	0	0
$401$	−10.8576	−0.542205	−0.271103	−	0.962550i	$-0.587388\pi$
$401$	−10.8576	−0.542205	−0.271103	+	0.962550i	$0.587388\pi$
$402$	16.0090	0.798454
$403$	0.854102	0.0425458
$404$	−10.3262	−0.513750
$405$	0	0
$406$	−2.43769	−0.120981
$407$	7.50245	0.371883
$408$	2.76393	0.136835
$409$	18.1231	0.896128	0.448064	−	0.894001i	$-0.352114\pi$
$409$	18.1231	0.896128	0.448064	+	0.894001i	$0.352114\pi$
$410$	0	0
$411$	16.7355	0.825501
$412$	0.832544	0.0410165
$413$	2.66914	0.131340
$414$	13.4863	0.662816
$415$	0	0
$416$	2.43769	0.119518
$417$	12.8253	0.628056
$418$	0	0
$419$	3.81966	0.186603	0.0933013	−	0.995638i	$-0.470258\pi$
$419$	3.81966	0.186603	0.0933013	+	0.995638i	$0.470258\pi$
$420$	0	0
$421$	−5.98385	−0.291635	−0.145818	−	0.989311i	$-0.546581\pi$
$421$	−5.98385	−0.291635	−0.145818	+	0.989311i	$0.546581\pi$
$422$	−16.7082	−0.813343
$423$	12.0902	0.587844
$424$	−3.09017	−0.150072
$425$	0	0
$426$	18.0171	0.872930
$427$	3.29180	0.159301
$428$	4.80828	0.232417
$429$	0.729490	0.0352201
$430$	0	0
$431$	−1.62460	−0.0782542	−0.0391271	−	0.999234i	$-0.512458\pi$
$431$	−1.62460	−0.0782542	−0.0391271	+	0.999234i	$0.512458\pi$
$432$	−12.9313	−0.622156
$433$	22.5478	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$433$	22.5478	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$434$	−0.326238	−0.0156599
$435$	0	0
$436$	−5.70634	−0.273284
$437$	0	0
$438$	1.38197	0.0660329
$439$	−37.6587	−1.79735	−0.898677	−	0.438611i	$-0.855471\pi$
$439$	−37.6587	−1.79735	−0.898677	+	0.438611i	$0.855471\pi$
$440$	0	0
$441$	11.2361	0.535051
$442$	−0.652476	−0.0310351
$443$	36.7771	1.74733	0.873666	−	0.486526i	$-0.161736\pi$
$443$	36.7771	1.74733	0.873666	+	0.486526i	$0.161736\pi$
$444$	6.38197	0.302875
$445$	0	0
$446$	−23.2148	−1.09925
$447$	4.80828	0.227424
$448$	−2.05573	−0.0971240
$449$	−37.6587	−1.77723	−0.888613	−	0.458658i	$-0.848330\pi$
$449$	−37.6587	−1.77723	−0.888613	+	0.458658i	$0.848330\pi$
$450$	0	0
$451$	1.38757	0.0653382
$452$	2.69417	0.126723
$453$	−21.1803	−0.995139
$454$	−21.9098	−1.02828
$455$	0	0
$456$	0	0
$457$	−0.763932	−0.0357352	−0.0178676	−	0.999840i	$-0.505688\pi$
$457$	−0.763932	−0.0357352	−0.0178676	+	0.999840i	$0.505688\pi$
$458$	29.2177	1.36526
$459$	−4.14725	−0.193577
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	−0.278640	−0.0129635
$463$	−6.85410	−0.318537	−0.159269	−	0.987235i	$-0.550914\pi$
$463$	−6.85410	−0.318537	−0.159269	+	0.987235i	$0.550914\pi$
$464$	20.9232	0.971337
$465$	0	0
$466$	7.33094	0.339599
$467$	−13.1803	−0.609913	−0.304957	−	0.952366i	$-0.598642\pi$
$467$	−13.1803	−0.609913	−0.304957	+	0.952366i	$0.598642\pi$
$468$	−0.726543	−0.0335844
$469$	−2.73466	−0.126275
$470$	0	0
$471$	20.3027	0.935499
$472$	−34.7984	−1.60172
$473$	−2.23607	−0.102815
$474$	11.7557	0.539957
$475$	0	0
$476$	−0.111456	−0.00510859
$477$	1.62460	0.0743853
$478$	−33.1685	−1.51709
$479$	−31.5623	−1.44212	−0.721059	−	0.692873i	$-0.756344\pi$
$479$	−31.5623	−1.44212	−0.721059	+	0.692873i	$0.756344\pi$
$480$	0	0
$481$	−6.38197	−0.290993
$482$	17.0344	0.775898
$483$	1.96763	0.0895301
$484$	6.34752	0.288524
$485$	0	0
$486$	17.0344	0.772698
$487$	−34.0260	−1.54187	−0.770933	−	0.636916i	$-0.780210\pi$
$487$	−34.0260	−1.54187	−0.770933	+	0.636916i	$0.780210\pi$
$488$	−42.9161	−1.94272
$489$	−3.31471	−0.149896
$490$	0	0
$491$	3.72949	0.168310	0.0841548	−	0.996453i	$-0.473181\pi$
$491$	3.72949	0.168310	0.0841548	+	0.996453i	$0.473181\pi$
$492$	1.18034	0.0532138
$493$	6.71040	0.302221
$494$	0	0
$495$	0	0
$496$	2.80017	0.125731
$497$	−3.07768	−0.138053
$498$	18.2918	0.819675
$499$	−38.4721	−1.72225	−0.861125	−	0.508394i	$-0.830239\pi$
$499$	−38.4721	−1.72225	−0.861125	+	0.508394i	$0.830239\pi$
$500$	0	0
$501$	15.3262	0.684726
$502$	3.52671	0.157405
$503$	−0.347524	−0.0154953	−0.00774767	−	0.999970i	$-0.502466\pi$
$503$	−0.347524	−0.0154953	−0.00774767	+	0.999970i	$0.502466\pi$
$504$	1.17557	0.0523641
$505$	0	0
$506$	−7.11894	−0.316475
$507$	14.6619	0.651157
$508$	1.34708	0.0597672
$509$	−10.7516	−0.476558	−0.238279	−	0.971197i	$-0.576583\pi$
$509$	−10.7516	−0.476558	−0.238279	+	0.971197i	$0.576583\pi$
$510$	0	0
$511$	−0.236068	−0.0104430
$512$	22.6538	1.00117
$513$	0	0
$514$	−16.1803	−0.713684
$515$	0	0
$516$	−1.90211	−0.0837359
$517$	−6.38197	−0.280679
$518$	2.43769	0.107106
$519$	−15.1246	−0.663897
$520$	0	0
$521$	−6.08985	−0.266801	−0.133401	−	0.991062i	$-0.542590\pi$
$521$	−6.08985	−0.266801	−0.133401	+	0.991062i	$0.542590\pi$
$522$	−16.7082	−0.731298
$523$	−27.6992	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$523$	−27.6992	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$524$	8.27051	0.361299
$525$	0	0
$526$	−7.05342	−0.307544
$527$	0.898056	0.0391199
$528$	2.39163	0.104082
$529$	27.2705	1.18567
$530$	0	0
$531$	18.2946	0.793917
$532$	0	0
$533$	−1.18034	−0.0511262
$534$	11.5187	0.498462
$535$	0	0
$536$	35.6525	1.53995
$537$	−14.2705	−0.615818
$538$	−7.36068	−0.317341
$539$	−5.93112	−0.255471
$540$	0	0
$541$	24.5967	1.05750	0.528748	−	0.848779i	$-0.322662\pi$
$541$	24.5967	1.05750	0.528748	+	0.848779i	$0.322662\pi$
$542$	−7.50245	−0.322258
$543$	−17.2361	−0.739670
$544$	2.56314	0.109894
$545$	0	0
$546$	0.237026	0.0101438
$547$	13.1433	0.561966	0.280983	−	0.959713i	$-0.409340\pi$
$547$	13.1433	0.561966	0.280983	+	0.959713i	$0.409340\pi$
$548$	8.79837	0.375848
$549$	22.5623	0.962936
$550$	0	0
$551$	0	0
$552$	−25.6525	−1.09184
$553$	−2.00811	−0.0853937
$554$	−33.8950	−1.44006
$555$	0	0
$556$	6.74265	0.285952
$557$	−27.0689	−1.14695	−0.573473	−	0.819225i	$-0.694404\pi$
$557$	−27.0689	−1.14695	−0.573473	+	0.819225i	$0.694404\pi$
$558$	−2.23607	−0.0946603
$559$	1.90211	0.0804508
$560$	0	0
$561$	0.767031	0.0323841
$562$	12.6869	0.535165
$563$	−18.4661	−0.778253	−0.389127	−	0.921184i	$-0.627223\pi$
$563$	−18.4661	−0.778253	−0.389127	+	0.921184i	$0.627223\pi$
$564$	−5.42882	−0.228595
$565$	0	0
$566$	6.19586	0.260431
$567$	0.360680	0.0151471
$568$	40.1246	1.68359
$569$	−7.71445	−0.323407	−0.161703	−	0.986839i	$-0.551699\pi$
$569$	−7.71445	−0.323407	−0.161703	+	0.986839i	$0.551699\pi$
$570$	0	0
$571$	−19.1459	−0.801231	−0.400615	−	0.916246i	$-0.631204\pi$
$571$	−19.1459	−0.801231	−0.400615	+	0.916246i	$0.631204\pi$
$572$	0.383516	0.0160356
$573$	−15.2824	−0.638432
$574$	0.450850	0.0188181
$575$	0	0
$576$	−14.0902	−0.587090
$577$	−17.4721	−0.727375	−0.363687	−	0.931521i	$-0.618482\pi$
$577$	−17.4721	−0.727375	−0.363687	+	0.931521i	$0.618482\pi$
$578$	19.2986	0.802718
$579$	−7.23607	−0.300721
$580$	0	0
$581$	−3.12461	−0.129631
$582$	−5.87785	−0.243645
$583$	−0.857567	−0.0355168
$584$	3.07768	0.127355
$585$	0	0
$586$	36.3050	1.49974
$587$	−9.58359	−0.395557	−0.197779	−	0.980247i	$-0.563373\pi$
$587$	−9.58359	−0.395557	−0.197779	+	0.980247i	$0.563373\pi$
$588$	−5.04531	−0.208065
$589$	0	0
$590$	0	0
$591$	−24.5560	−1.01010
$592$	−20.9232	−0.859940
$593$	−7.29180	−0.299438	−0.149719	−	0.988729i	$-0.547837\pi$
$593$	−7.29180	−0.299438	−0.149719	+	0.988729i	$0.547837\pi$
$594$	−5.45085	−0.223651
$595$	0	0
$596$	2.52786	0.103545
$597$	15.7719	0.645502
$598$	6.05573	0.247637
$599$	−10.3431	−0.422608	−0.211304	−	0.977420i	$-0.567771\pi$
$599$	−10.3431	−0.422608	−0.211304	+	0.977420i	$0.567771\pi$
$600$	0	0
$601$	−17.0130	−0.693975	−0.346988	−	0.937870i	$-0.612795\pi$
$601$	−17.0130	−0.693975	−0.346988	+	0.937870i	$0.612795\pi$
$602$	−0.726543	−0.0296117
$603$	−18.7436	−0.763299
$604$	−11.1352	−0.453083
$605$	0	0
$606$	23.0902	0.937974
$607$	−20.5397	−0.833682	−0.416841	−	0.908979i	$-0.636863\pi$
$607$	−20.5397	−0.833682	−0.416841	+	0.908979i	$0.636863\pi$
$608$	0	0
$609$	−2.43769	−0.0987803
$610$	0	0
$611$	5.42882	0.219627
$612$	−0.763932	−0.0308801
$613$	−35.4721	−1.43271	−0.716353	−	0.697738i	$-0.754190\pi$
$613$	−35.4721	−1.43271	−0.716353	+	0.697738i	$0.754190\pi$
$614$	29.9230	1.20759
$615$	0	0
$616$	−0.620541	−0.0250023
$617$	−19.3820	−0.780289	−0.390144	−	0.920754i	$-0.627575\pi$
$617$	−19.3820	−0.780289	−0.390144	+	0.920754i	$0.627575\pi$
$618$	−1.86162	−0.0748855
$619$	−27.5410	−1.10697	−0.553484	−	0.832860i	$-0.686702\pi$
$619$	−27.5410	−1.10697	−0.553484	+	0.832860i	$0.686702\pi$
$620$	0	0
$621$	38.4913	1.54460
$622$	8.50651	0.341080
$623$	−1.96763	−0.0788312
$624$	−2.03444	−0.0814429
$625$	0	0
$626$	−35.5851	−1.42227
$627$	0	0
$628$	10.6738	0.425929
$629$	−6.71040	−0.267561
$630$	0	0
$631$	−18.8197	−0.749199	−0.374599	−	0.927187i	$-0.622220\pi$
$631$	−18.8197	−0.749199	−0.374599	+	0.927187i	$0.622220\pi$
$632$	26.1803	1.04140
$633$	−16.7082	−0.664091
$634$	−31.3050	−1.24328
$635$	0	0
$636$	−0.729490	−0.0289262
$637$	5.04531	0.199902
$638$	8.81966	0.349174
$639$	−21.0948	−0.834496
$640$	0	0
$641$	38.4508	1.51872	0.759358	−	0.650673i	$-0.225513\pi$
$641$	38.4508	1.51872	0.759358	+	0.650673i	$0.225513\pi$
$642$	−10.7516	−0.424334
$643$	29.5279	1.16447	0.582233	−	0.813022i	$-0.302179\pi$
$643$	29.5279	1.16447	0.582233	+	0.813022i	$0.302179\pi$
$644$	1.03444	0.0407627
$645$	0	0
$646$	0	0
$647$	−17.4721	−0.686901	−0.343450	−	0.939171i	$-0.611596\pi$
$647$	−17.4721	−0.686901	−0.343450	+	0.939171i	$0.611596\pi$
$648$	−4.70228	−0.184723
$649$	−9.65706	−0.379073
$650$	0	0
$651$	−0.326238	−0.0127863
$652$	−1.74265	−0.0682473
$653$	−10.7984	−0.422573	−0.211287	−	0.977424i	$-0.567765\pi$
$653$	−10.7984	−0.422573	−0.211287	+	0.977424i	$0.567765\pi$
$654$	12.7598	0.498946
$655$	0	0
$656$	−3.86974	−0.151088
$657$	−1.61803	−0.0631255
$658$	−2.07363	−0.0808384
$659$	−30.7768	−1.19890	−0.599448	−	0.800414i	$-0.704613\pi$
$659$	−30.7768	−1.19890	−0.599448	+	0.800414i	$0.704613\pi$
$660$	0	0
$661$	−13.4208	−0.522008	−0.261004	−	0.965338i	$-0.584054\pi$
$661$	−13.4208	−0.522008	−0.261004	+	0.965338i	$0.584054\pi$
$662$	−2.11146	−0.0820641
$663$	−0.652476	−0.0253401
$664$	40.7364	1.58088
$665$	0	0
$666$	16.7082	0.647430
$667$	−62.2802	−2.41150
$668$	8.05748	0.311753
$669$	−23.2148	−0.897535
$670$	0	0
$671$	−11.9098	−0.459774
$672$	−0.931116	−0.0359186
$673$	−34.9646	−1.34779	−0.673893	−	0.738829i	$-0.735379\pi$
$673$	−34.9646	−1.34779	−0.673893	+	0.738829i	$0.735379\pi$
$674$	4.06888	0.156728
$675$	0	0
$676$	7.70820	0.296469
$677$	−16.0494	−0.616830	−0.308415	−	0.951252i	$-0.599799\pi$
$677$	−16.0494	−0.616830	−0.308415	+	0.951252i	$0.599799\pi$
$678$	−6.02434	−0.231363
$679$	1.00406	0.0385322
$680$	0	0
$681$	−21.9098	−0.839587
$682$	1.18034	0.0451976
$683$	−32.6134	−1.24792	−0.623959	−	0.781457i	$-0.714477\pi$
$683$	−32.6134	−1.24792	−0.623959	+	0.781457i	$0.714477\pi$
$684$	0	0
$685$	0	0
$686$	−3.86974	−0.147747
$687$	29.2177	1.11473
$688$	6.23607	0.237748
$689$	0.729490	0.0277914
$690$	0	0
$691$	−7.76393	−0.295354	−0.147677	−	0.989036i	$-0.547180\pi$
$691$	−7.76393	−0.295354	−0.147677	+	0.989036i	$0.547180\pi$
$692$	−7.95148	−0.302270
$693$	0.326238	0.0123928
$694$	8.84953	0.335924
$695$	0	0
$696$	31.7809	1.20465
$697$	−1.24108	−0.0470094
$698$	−20.6457	−0.781452
$699$	7.33094	0.277282
$700$	0	0
$701$	−43.0902	−1.62749	−0.813747	−	0.581220i	$-0.802576\pi$
$701$	−43.0902	−1.62749	−0.813747	+	0.581220i	$0.802576\pi$
$702$	4.63677	0.175004
$703$	0	0
$704$	7.43769	0.280319
$705$	0	0
$706$	22.7194	0.855054
$707$	−3.94427	−0.148340
$708$	−8.21478	−0.308730
$709$	13.8197	0.519008	0.259504	−	0.965742i	$-0.416441\pi$
$709$	13.8197	0.519008	0.259504	+	0.965742i	$0.416441\pi$
$710$	0	0
$711$	−13.7638	−0.516184
$712$	25.6525	0.961367
$713$	−8.33499	−0.312148
$714$	0.249224	0.00932696
$715$	0	0
$716$	−7.50245	−0.280380
$717$	−33.1685	−1.23870
$718$	−11.3067	−0.421961
$719$	−42.0902	−1.56970	−0.784849	−	0.619687i	$-0.787260\pi$
$719$	−42.0902	−1.56970	−0.784849	+	0.619687i	$0.787260\pi$
$720$	0	0
$721$	0.318003	0.0118431
$722$	0	0
$723$	17.0344	0.633518
$724$	−9.06154	−0.336769
$725$	0	0
$726$	−14.1935	−0.526770
$727$	10.3607	0.384256	0.192128	−	0.981370i	$-0.438461\pi$
$727$	10.3607	0.384256	0.192128	+	0.981370i	$0.438461\pi$
$728$	0.527864	0.0195639
$729$	21.6180	0.800668
$730$	0	0
$731$	2.00000	0.0739727
$732$	−10.1311	−0.374456
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	−31.8214	−1.17455
$735$	0	0
$736$	−23.7889	−0.876871
$737$	9.89408	0.364453
$738$	3.09017	0.113751
$739$	−32.2361	−1.18582	−0.592911	−	0.805268i	$-0.702022\pi$
$739$	−32.2361	−1.18582	−0.592911	+	0.805268i	$0.702022\pi$
$740$	0	0
$741$	0	0
$742$	−0.278640	−0.0102292
$743$	−11.5842	−0.424983	−0.212491	−	0.977163i	$-0.568158\pi$
$743$	−11.5842	−0.424983	−0.212491	+	0.977163i	$0.568158\pi$
$744$	4.25325	0.155932
$745$	0	0
$746$	5.32624	0.195007
$747$	−21.4164	−0.783585
$748$	0.403252	0.0147444
$749$	1.83660	0.0671079
$750$	0	0
$751$	−41.7000	−1.52165	−0.760827	−	0.648954i	$-0.775207\pi$
$751$	−41.7000	−1.52165	−0.760827	+	0.648954i	$0.775207\pi$
$752$	17.7984	0.649040
$753$	3.52671	0.128521
$754$	−7.50245	−0.273223
$755$	0	0
$756$	0.792055	0.0288068
$757$	43.9787	1.59843	0.799217	−	0.601043i	$-0.205248\pi$
$757$	43.9787	1.59843	0.799217	+	0.601043i	$0.205248\pi$
$758$	−15.5279	−0.563997
$759$	−7.11894	−0.258401
$760$	0	0
$761$	−41.1803	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$761$	−41.1803	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$762$	−3.01217	−0.109119
$763$	−2.17963	−0.0789078
$764$	−8.03444	−0.290676
$765$	0	0
$766$	−3.09017	−0.111652
$767$	8.21478	0.296619
$768$	15.6004	0.562932
$769$	20.2705	0.730973	0.365487	−	0.930817i	$-0.380903\pi$
$769$	20.2705	0.730973	0.365487	+	0.930817i	$0.380903\pi$
$770$	0	0
$771$	−16.1803	−0.582721
$772$	−3.80423	−0.136917
$773$	−6.94742	−0.249881	−0.124941	−	0.992164i	$-0.539874\pi$
$773$	−6.94742	−0.249881	−0.124941	+	0.992164i	$0.539874\pi$
$774$	−4.97980	−0.178995
$775$	0	0
$776$	−13.0902	−0.469910
$777$	2.43769	0.0874518
$778$	16.6295	0.596196
$779$	0	0
$780$	0	0
$781$	11.1352	0.398447
$782$	6.36737	0.227697
$783$	−47.6869	−1.70419
$784$	16.5410	0.590751
$785$	0	0
$786$	−18.4934	−0.659639
$787$	43.9856	1.56792	0.783959	−	0.620812i	$-0.213197\pi$
$787$	43.9856	1.56792	0.783959	+	0.620812i	$0.213197\pi$
$788$	−12.9098	−0.459894
$789$	−7.05342	−0.251109
$790$	0	0
$791$	1.02908	0.0365899
$792$	−4.25325	−0.151133
$793$	10.1311	0.359766
$794$	35.0706	1.24461
$795$	0	0
$796$	8.29180	0.293895
$797$	−4.14725	−0.146903	−0.0734516	−	0.997299i	$-0.523401\pi$
$797$	−4.14725	−0.146903	−0.0734516	+	0.997299i	$0.523401\pi$
$798$	0	0
$799$	5.70820	0.201942
$800$	0	0
$801$	−13.4863	−0.476515
$802$	12.7639	0.450710
$803$	0.854102	0.0301406
$804$	8.41641	0.296824
$805$	0	0
$806$	−1.00406	−0.0353664
$807$	−7.36068	−0.259108
$808$	51.4226	1.80904
$809$	−1.78522	−0.0627649	−0.0313825	−	0.999507i	$-0.509991\pi$
$809$	−1.78522	−0.0627649	−0.0313825	+	0.999507i	$0.509991\pi$
$810$	0	0
$811$	21.2663	0.746760	0.373380	−	0.927679i	$-0.378199\pi$
$811$	21.2663	0.746760	0.373380	+	0.927679i	$0.378199\pi$
$812$	−1.28157	−0.0449743
$813$	−7.50245	−0.263122
$814$	−8.81966	−0.309129
$815$	0	0
$816$	−2.13914	−0.0748848
$817$	0	0
$818$	−21.3050	−0.744910
$819$	−0.277515	−0.00969714
$820$	0	0
$821$	28.4508	0.992942	0.496471	−	0.868053i	$-0.334629\pi$
$821$	28.4508	0.992942	0.496471	+	0.868053i	$0.334629\pi$
$822$	−19.6738	−0.686201
$823$	21.2361	0.740243	0.370121	−	0.928983i	$-0.379316\pi$
$823$	21.2361	0.740243	0.370121	+	0.928983i	$0.379316\pi$
$824$	−4.14590	−0.144429
$825$	0	0
$826$	−3.13777	−0.109177
$827$	2.17963	0.0757931	0.0378965	−	0.999282i	$-0.487934\pi$
$827$	2.17963	0.0757931	0.0378965	+	0.999282i	$0.487934\pi$
$828$	7.09017	0.246400
$829$	20.7517	0.720737	0.360369	−	0.932810i	$-0.382651\pi$
$829$	20.7517	0.720737	0.360369	+	0.932810i	$0.382651\pi$
$830$	0	0
$831$	−33.8950	−1.17580
$832$	−6.32688	−0.219345
$833$	5.30495	0.183806
$834$	−15.0770	−0.522074
$835$	0	0
$836$	0	0
$837$	−6.38197	−0.220593
$838$	−4.49028	−0.155114
$839$	10.4086	0.359346	0.179673	−	0.983726i	$-0.442496\pi$
$839$	10.4086	0.359346	0.179673	+	0.983726i	$0.442496\pi$
$840$	0	0
$841$	48.1591	1.66066
$842$	7.03444	0.242423
$843$	12.6869	0.436961
$844$	−8.78402	−0.302359
$845$	0	0
$846$	−14.2128	−0.488648
$847$	2.42454	0.0833081
$848$	2.39163	0.0821289
$849$	6.19586	0.212641
$850$	0	0
$851$	62.2802	2.13494
$852$	9.47214	0.324510
$853$	6.52786	0.223510	0.111755	−	0.993736i	$-0.464353\pi$
$853$	6.52786	0.223510	0.111755	+	0.993736i	$0.464353\pi$
$854$	−3.86974	−0.132420
$855$	0	0
$856$	−23.9443	−0.818398
$857$	−35.5851	−1.21556	−0.607782	−	0.794104i	$-0.707941\pi$
$857$	−35.5851	−1.21556	−0.607782	+	0.794104i	$0.707941\pi$
$858$	−0.857567	−0.0292769
$859$	−27.2361	−0.929283	−0.464641	−	0.885499i	$-0.653817\pi$
$859$	−27.2361	−0.929283	−0.464641	+	0.885499i	$0.653817\pi$
$860$	0	0
$861$	0.450850	0.0153649
$862$	1.90983	0.0650491
$863$	−24.8335	−0.845341	−0.422671	−	0.906283i	$-0.638907\pi$
$863$	−24.8335	−0.845341	−0.422671	+	0.906283i	$0.638907\pi$
$864$	−18.2148	−0.619679
$865$	0	0
$866$	−26.5066	−0.900730
$867$	19.2986	0.655416
$868$	−0.171513	−0.00582154
$869$	7.26543	0.246463
$870$	0	0
$871$	−8.41641	−0.285179
$872$	28.4164	0.962301
$873$	6.88191	0.232917
$874$	0	0
$875$	0	0
$876$	0.726543	0.0245476
$877$	−23.1684	−0.782341	−0.391170	−	0.920318i	$-0.627930\pi$
$877$	−23.1684	−0.782341	−0.391170	+	0.920318i	$0.627930\pi$
$878$	44.2705	1.49406
$879$	36.3050	1.22454
$880$	0	0
$881$	−5.32624	−0.179446	−0.0897228	−	0.995967i	$-0.528598\pi$
$881$	−5.32624	−0.179446	−0.0897228	+	0.995967i	$0.528598\pi$
$882$	−13.2088	−0.444763
$883$	57.2148	1.92543	0.962715	−	0.270516i	$-0.0871944\pi$
$883$	57.2148	1.92543	0.962715	+	0.270516i	$0.0871944\pi$
$884$	−0.343027	−0.0115372
$885$	0	0
$886$	−43.2341	−1.45248
$887$	−19.3642	−0.650185	−0.325092	−	0.945682i	$-0.605395\pi$
$887$	−19.3642	−0.650185	−0.325092	+	0.945682i	$0.605395\pi$
$888$	−31.7809	−1.06650
$889$	0.514540	0.0172571
$890$	0	0
$891$	−1.30495	−0.0437175
$892$	−12.2047	−0.408645
$893$	0	0
$894$	−5.65248	−0.189047
$895$	0	0
$896$	0.832544	0.0278133
$897$	6.05573	0.202195
$898$	44.2705	1.47733
$899$	10.3262	0.344399
$900$	0	0
$901$	0.767031	0.0255535
$902$	−1.63119	−0.0543127
$903$	−0.726543	−0.0241778
$904$	−13.4164	−0.446223
$905$	0	0
$906$	24.8990	0.827213
$907$	41.5035	1.37810	0.689050	−	0.724714i	$-0.258028\pi$
$907$	41.5035	1.37810	0.689050	+	0.724714i	$0.258028\pi$
$908$	−11.5187	−0.382261
$909$	−27.0344	−0.896676
$910$	0	0
$911$	12.9718	0.429774	0.214887	−	0.976639i	$-0.431062\pi$
$911$	12.9718	0.429774	0.214887	+	0.976639i	$0.431062\pi$
$912$	0	0
$913$	11.3050	0.374139
$914$	0.898056	0.0297051
$915$	0	0
$916$	15.3607	0.507531
$917$	3.15905	0.104321
$918$	4.87539	0.160912
$919$	−37.3607	−1.23242	−0.616208	−	0.787584i	$-0.711332\pi$
$919$	−37.3607	−1.23242	−0.616208	+	0.787584i	$0.711332\pi$
$920$	0	0
$921$	29.9230	0.985996
$922$	9.40456	0.309723
$923$	−9.47214	−0.311779
$924$	−0.146490	−0.00481917
$925$	0	0
$926$	8.05748	0.264785
$927$	2.17963	0.0715884
$928$	29.4721	0.967470
$929$	17.4377	0.572112	0.286056	−	0.958213i	$-0.407656\pi$
$929$	17.4377	0.572112	0.286056	+	0.958213i	$0.407656\pi$
$930$	0	0
$931$	0	0
$932$	3.85410	0.126245
$933$	8.50651	0.278491
$934$	15.4944	0.506993
$935$	0	0
$936$	3.61803	0.118259
$937$	−47.7984	−1.56150	−0.780752	−	0.624841i	$-0.785164\pi$
$937$	−47.7984	−1.56150	−0.780752	+	0.624841i	$0.785164\pi$
$938$	3.21478	0.104966
$939$	−35.5851	−1.16128
$940$	0	0
$941$	−39.6013	−1.29097	−0.645484	−	0.763774i	$-0.723344\pi$
$941$	−39.6013	−1.29097	−0.645484	+	0.763774i	$0.723344\pi$
$942$	−23.8673	−0.777637
$943$	11.5187	0.375100
$944$	26.9321	0.876566
$945$	0	0
$946$	2.62866	0.0854650
$947$	25.4164	0.825922	0.412961	−	0.910749i	$-0.364495\pi$
$947$	25.4164	0.825922	0.412961	+	0.910749i	$0.364495\pi$
$948$	6.18034	0.200728
$949$	−0.726543	−0.0235846
$950$	0	0
$951$	−31.3050	−1.01513
$952$	0.555029	0.0179886
$953$	55.0148	1.78210	0.891052	−	0.453901i	$-0.149968\pi$
$953$	55.0148	1.78210	0.891052	+	0.453901i	$0.149968\pi$
$954$	−1.90983	−0.0618330
$955$	0	0
$956$	−17.4377	−0.563975
$957$	8.81966	0.285099
$958$	37.1037	1.19877
$959$	3.36068	0.108522
$960$	0	0
$961$	−29.6180	−0.955420
$962$	7.50245	0.241889
$963$	12.5882	0.405651
$964$	8.95554	0.288438
$965$	0	0
$966$	−2.31308	−0.0744222
$967$	46.9443	1.50963	0.754813	−	0.655940i	$-0.227728\pi$
$967$	46.9443	1.50963	0.754813	+	0.655940i	$0.227728\pi$
$968$	−31.6094	−1.01596
$969$	0	0
$970$	0	0
$971$	−14.5964	−0.468420	−0.234210	−	0.972186i	$-0.575250\pi$
$971$	−14.5964	−0.468420	−0.234210	+	0.972186i	$0.575250\pi$
$972$	8.95554	0.287249
$973$	2.57546	0.0825655
$974$	40.0000	1.28168
$975$	0	0
$976$	33.2148	1.06318
$977$	−4.31877	−0.138170	−0.0690848	−	0.997611i	$-0.522008\pi$
$977$	−4.31877	−0.138170	−0.0690848	+	0.997611i	$0.522008\pi$
$978$	3.89667	0.124602
$979$	7.11894	0.227522
$980$	0	0
$981$	−14.9394	−0.476978
$982$	−4.38428	−0.139908
$983$	6.26137	0.199707	0.0998533	−	0.995002i	$-0.468163\pi$
$983$	6.26137	0.199707	0.0998533	+	0.995002i	$0.468163\pi$
$984$	−5.87785	−0.187379
$985$	0	0
$986$	−7.88854	−0.251222
$987$	−2.07363	−0.0660043
$988$	0	0
$989$	−18.5623	−0.590247
$990$	0	0
$991$	−17.2500	−0.547966	−0.273983	−	0.961735i	$-0.588341\pi$
$991$	−17.2500	−0.547966	−0.273983	+	0.961735i	$0.588341\pi$
$992$	3.94427	0.125231
$993$	−2.11146	−0.0670050
$994$	3.61803	0.114757
$995$	0	0
$996$	9.61657	0.304713
$997$	19.0557	0.603501	0.301750	−	0.953387i	$-0.402429\pi$
$997$	19.0557	0.603501	0.301750	+	0.953387i	$0.402429\pi$
$998$	45.2267	1.43163
$999$	47.6869	1.50875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bl.1.2		4
5.4	even	2		1805.2.a.l.1.3	yes	4
19.18	odd	2	inner	9025.2.a.bl.1.3		4
95.94	odd	2		1805.2.a.l.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.l.1.2	✓	4	95.94	odd	2
1805.2.a.l.1.3	yes	4	5.4	even	2
9025.2.a.bl.1.2		4	1.1	even	1	trivial
9025.2.a.bl.1.3		4	19.18	odd	2	inner

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

\(f(q)\)	\(=\)	\(q-1.17557 q^{2} -1.17557 q^{3} -0.618034 q^{4} +1.38197 q^{6} -0.236068 q^{7} +3.07768 q^{8} -1.61803 q^{9} +O(q^{10})\)	\(q-1.17557 q^{2} -1.17557 q^{3} -0.618034 q^{4} +1.38197 q^{6} -0.236068 q^{7} +3.07768 q^{8} -1.61803 q^{9} +0.854102 q^{11} +0.726543 q^{12} -0.726543 q^{13} +0.277515 q^{14} -2.38197 q^{16} -0.763932 q^{17} +1.90211 q^{18} +0.277515 q^{21} -1.00406 q^{22} +7.09017 q^{23} -3.61803 q^{24} +0.854102 q^{26} +5.42882 q^{27} +0.145898 q^{28} -8.78402 q^{29} -1.17557 q^{31} -3.35520 q^{32} -1.00406 q^{33} +0.898056 q^{34} +1.00000 q^{36} +8.78402 q^{37} +0.854102 q^{39} +1.62460 q^{41} -0.326238 q^{42} -2.61803 q^{43} -0.527864 q^{44} -8.33499 q^{46} -7.47214 q^{47} +2.80017 q^{48} -6.94427 q^{49} +0.898056 q^{51} +0.449028 q^{52} -1.00406 q^{53} -6.38197 q^{54} -0.726543 q^{56} +10.3262 q^{58} -11.3067 q^{59} -13.9443 q^{61} +1.38197 q^{62} +0.381966 q^{63} +8.70820 q^{64} +1.18034 q^{66} +11.5842 q^{67} +0.472136 q^{68} -8.33499 q^{69} +13.0373 q^{71} -4.97980 q^{72} +1.00000 q^{73} -10.3262 q^{74} -0.201626 q^{77} -1.00406 q^{78} +8.50651 q^{79} -1.52786 q^{81} -1.90983 q^{82} +13.2361 q^{83} -0.171513 q^{84} +3.07768 q^{86} +10.3262 q^{87} +2.62866 q^{88} +8.33499 q^{89} +0.171513 q^{91} -4.38197 q^{92} +1.38197 q^{93} +8.78402 q^{94} +3.94427 q^{96} -4.25325 q^{97} +8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9	\( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{58} - 20 q^{61} + 10 q^{62} + 6 q^{63} + 8 q^{64} - 40 q^{66} - 16 q^{68} + 4 q^{73} - 10 q^{74} - 50 q^{77} - 24 q^{81} - 30 q^{82} + 44 q^{83} + 10 q^{87} - 22 q^{92} + 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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