LMFDB - Embedded newform 1805.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, 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("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.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:	$14.4129975648$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	1805.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.17557 q^{2} +1.17557 q^{3} -0.618034 q^{4} -1.00000 q^{5} +1.38197 q^{6} +0.236068 q^{7} -3.07768 q^{8} -1.61803 q^{9} -1.17557 q^{10} +0.854102 q^{11} -0.726543 q^{12} +0.726543 q^{13} +0.277515 q^{14} -1.17557 q^{15} -2.38197 q^{16} +0.763932 q^{17} -1.90211 q^{18} +0.618034 q^{20} +0.277515 q^{21} +1.00406 q^{22} -7.09017 q^{23} -3.61803 q^{24} +1.00000 q^{25} +0.854102 q^{26} -5.42882 q^{27} -0.145898 q^{28} -8.78402 q^{29} -1.38197 q^{30} -1.17557 q^{31} +3.35520 q^{32} +1.00406 q^{33} +0.898056 q^{34} -0.236068 q^{35} +1.00000 q^{36} -8.78402 q^{37} +0.854102 q^{39} +3.07768 q^{40} +1.62460 q^{41} +0.326238 q^{42} +2.61803 q^{43} -0.527864 q^{44} +1.61803 q^{45} -8.33499 q^{46} +7.47214 q^{47} -2.80017 q^{48} -6.94427 q^{49} +1.17557 q^{50} +0.898056 q^{51} -0.449028 q^{52} +1.00406 q^{53} -6.38197 q^{54} -0.854102 q^{55} -0.726543 q^{56} -10.3262 q^{58} -11.3067 q^{59} +0.726543 q^{60} -13.9443 q^{61} -1.38197 q^{62} -0.381966 q^{63} +8.70820 q^{64} -0.726543 q^{65} +1.18034 q^{66} -11.5842 q^{67} -0.472136 q^{68} -8.33499 q^{69} -0.277515 q^{70} +13.0373 q^{71} +4.97980 q^{72} -1.00000 q^{73} -10.3262 q^{74} +1.17557 q^{75} +0.201626 q^{77} +1.00406 q^{78} +8.50651 q^{79} +2.38197 q^{80} -1.52786 q^{81} +1.90983 q^{82} -13.2361 q^{83} -0.171513 q^{84} -0.763932 q^{85} +3.07768 q^{86} -10.3262 q^{87} -2.62866 q^{88} +8.33499 q^{89} +1.90211 q^{90} +0.171513 q^{91} +4.38197 q^{92} -1.38197 q^{93} +8.78402 q^{94} +3.94427 q^{96} +4.25325 q^{97} -8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{23} - 10 q^{24} + 4 q^{25} - 10 q^{26} - 14 q^{28} - 10 q^{30} + 8 q^{35} + 4 q^{36} - 10 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 4 * q^5 + 10 * q^6 - 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 + 12 * q^17 - 2 * q^20 - 6 * q^23 - 10 * q^24 + 4 * q^25 - 10 * q^26 - 14 * q^28 - 10 * q^30 + 8 * q^35 + 4 * q^36 - 10 * q^39 - 30 * q^42 + 6 * q^43 - 20 * q^44 + 2 * q^45 + 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^55 - 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 + 14 * q^80 - 24 * q^81 + 30 * q^82 - 44 * q^83 - 12 * q^85 - 10 * q^87 + 22 * q^92 - 10 * q^93 - 20 * q^96 - 10 * q^99

Coefficient data

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

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

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

$2$	1.17557	0.831254	0.415627	−	0.909535i	$-0.363562\pi$
$2$	1.17557	0.831254	0.415627	+	0.909535i	$0.363562\pi$
$3$	1.17557	0.678716	0.339358	−	0.940657i	$-0.389790\pi$
$3$	1.17557	0.678716	0.339358	+	0.940657i	$0.389790\pi$
$4$	−0.618034	−0.309017
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.38197	0.564185
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−3.07768	−1.08813
$9$	−1.61803	−0.539345
$10$	−1.17557	−0.371748
$11$	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	$-0.467875\pi$
$13$	0.726543	0.201507	0.100753	+	0.994911i	$0.467875\pi$
$14$	0.277515	0.0741689
$15$	−1.17557	−0.303531
$16$	−2.38197	−0.595492
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	−1.90211	−0.448332
$19$	0	0
$19$	0	0
$20$	0.618034	0.138197
$21$	0.277515	0.0605586
$22$	1.00406	0.214066
$23$	−7.09017	−1.47840	−0.739201	−	0.673485i	$-0.764797\pi$
$23$	−7.09017	−1.47840	−0.739201	+	0.673485i	$0.764797\pi$
$24$	−3.61803	−0.738528
$25$	1.00000	0.200000
$26$	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	$0.803580\pi$
$30$	−1.38197	−0.252311
$31$	−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
$35$	−0.236068	−0.0399028
$36$	1.00000	0.166667
$37$	−8.78402	−1.44408	−0.722042	−	0.691849i	$-0.756796\pi$
$37$	−8.78402	−1.44408	−0.722042	+	0.691849i	$0.756796\pi$
$38$	0	0
$39$	0.854102	0.136766
$40$	3.07768	0.486624
$41$	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	$-0.436028\pi$
$43$	2.61803	0.399246	0.199623	+	0.979873i	$0.436028\pi$
$44$	−0.527864	−0.0795785
$45$	1.61803	0.241202
$46$	−8.33499	−1.22893
$47$	7.47214	1.08992	0.544962	−	0.838461i	$-0.316544\pi$
$47$	7.47214	1.08992	0.544962	+	0.838461i	$0.316544\pi$
$48$	−2.80017	−0.404170
$49$	−6.94427	−0.992039
$50$	1.17557	0.166251
$51$	0.898056	0.125753
$52$	−0.449028	−0.0622690
$53$	1.00406	0.137918	0.0689589	−	0.997620i	$-0.478032\pi$
$53$	1.00406	0.137918	0.0689589	+	0.997620i	$0.478032\pi$
$54$	−6.38197	−0.868476
$55$	−0.854102	−0.115167
$56$	−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	$0.763289\pi$
$60$	0.726543	0.0937962
$61$	−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
$65$	−0.726543	−0.0901165
$66$	1.18034	0.145290
$67$	−11.5842	−1.41523	−0.707617	−	0.706596i	$-0.750230\pi$
$67$	−11.5842	−1.41523	−0.707617	+	0.706596i	$0.750230\pi$
$68$	−0.472136	−0.0572549
$69$	−8.33499	−1.00342
$70$	−0.277515	−0.0331693
$71$	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	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−10.3262	−1.20040
$75$	1.17557	0.135743
$76$	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	$0.341170\pi$
$80$	2.38197	0.266312
$81$	−1.52786	−0.169763
$82$	1.90983	0.210905
$83$	−13.2361	−1.45285	−0.726424	−	0.687247i	$-0.758819\pi$
$83$	−13.2361	−1.45285	−0.726424	+	0.687247i	$0.758819\pi$
$84$	−0.171513	−0.0187136
$85$	−0.763932	−0.0828601
$86$	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	$0.354356\pi$
$90$	1.90211	0.200500
$91$	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	$-0.430723\pi$
$97$	4.25325	0.431853	0.215926	+	0.976410i	$0.430723\pi$
$98$	−8.16348	−0.824636
$99$	−1.38197	−0.138893
$100$	−0.618034	−0.0618034
$101$	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	$-0.478859\pi$
$103$	1.34708	0.132732	0.0663661	+	0.997795i	$0.478859\pi$
$104$	−2.23607	−0.219265
$105$	−0.277515	−0.0270826
$106$	1.18034	0.114645
$107$	7.77997	0.752118	0.376059	−	0.926596i	$-0.377279\pi$
$107$	7.77997	0.752118	0.376059	+	0.926596i	$0.377279\pi$
$108$	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	$0.354204\pi$
$110$	−1.00406	−0.0957331
$111$	−10.3262	−0.980123
$112$	−0.562306	−0.0531329
$113$	4.35926	0.410084	0.205042	−	0.978753i	$-0.434267\pi$
$113$	4.35926	0.410084	0.205042	+	0.978753i	$0.434267\pi$
$114$	0	0
$115$	7.09017	0.661162
$116$	5.42882	0.504054
$117$	−1.17557	−0.108682
$118$	−13.2918	−1.22361
$119$	0.180340	0.0165317
$120$	3.61803	0.330280
$121$	−10.2705	−0.933683
$122$	−16.3925	−1.48410
$123$	1.90983	0.172204
$124$	0.726543	0.0652454
$125$	−1.00000	−0.0894427
$126$	−0.449028	−0.0400026
$127$	2.17963	0.193411	0.0967053	−	0.995313i	$-0.469170\pi$
$127$	2.17963	0.193411	0.0967053	+	0.995313i	$0.469170\pi$
$128$	3.52671	0.311720
$129$	3.07768	0.270975
$130$	−0.854102	−0.0749097
$131$	−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
$135$	5.42882	0.467239
$136$	−2.35114	−0.201609
$137$	14.2361	1.21627	0.608135	−	0.793834i	$-0.291918\pi$
$137$	14.2361	1.21627	0.608135	+	0.793834i	$0.291918\pi$
$138$	−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	$0.653112\pi$
$140$	0.145898	0.0123306
$141$	8.78402	0.739748
$142$	15.3262	1.28615
$143$	0.620541	0.0518923
$144$	3.85410	0.321175
$145$	8.78402	0.729473
$146$	−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	$0.553582\pi$
$150$	1.38197	0.112837
$151$	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
$155$	1.17557	0.0944241
$156$	−0.527864	−0.0422629
$157$	17.2705	1.37834	0.689168	−	0.724601i	$-0.257976\pi$
$157$	17.2705	1.37834	0.689168	+	0.724601i	$0.257976\pi$
$158$	10.0000	0.795557
$159$	1.18034	0.0936070
$160$	−3.35520	−0.265252
$161$	−1.67376	−0.131911
$162$	−1.79611	−0.141116
$163$	−2.81966	−0.220853	−0.110426	−	0.993884i	$-0.535222\pi$
$163$	−2.81966	−0.220853	−0.110426	+	0.993884i	$0.535222\pi$
$164$	−1.00406	−0.0784037
$165$	−1.00406	−0.0781657
$166$	−15.5599	−1.20768
$167$	13.0373	1.00885	0.504427	−	0.863454i	$-0.331704\pi$
$167$	13.0373	1.00885	0.504427	+	0.863454i	$0.331704\pi$
$168$	−0.854102	−0.0658954
$169$	−12.4721	−0.959395
$170$	−0.898056	−0.0688777
$171$	0	0
$172$	−1.61803	−0.123374
$173$	−12.8658	−0.978166	−0.489083	−	0.872237i	$-0.662668\pi$
$173$	−12.8658	−0.978166	−0.489083	+	0.872237i	$0.662668\pi$
$174$	−12.1392	−0.920272
$175$	0.236068	0.0178451
$176$	−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	$0.350117\pi$
$180$	−1.00000	−0.0745356
$181$	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
$185$	8.78402	0.645814
$186$	−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	$-0.571107\pi$
$193$	−6.15537	−0.443073	−0.221537	+	0.975152i	$0.571107\pi$
$194$	5.00000	0.358979
$195$	−0.854102	−0.0611635
$196$	4.29180	0.306557
$197$	−20.8885	−1.48825	−0.744124	−	0.668042i	$-0.767133\pi$
$197$	−20.8885	−1.48825	−0.744124	+	0.668042i	$0.767133\pi$
$198$	−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	$0.657744\pi$
$200$	−3.07768	−0.217625
$201$	−13.6180	−0.960542
$202$	19.6417	1.38198
$203$	−2.07363	−0.145540
$204$	−0.555029	−0.0388598
$205$	−1.62460	−0.113467
$206$	1.58359	0.110334
$207$	11.4721	0.797369
$208$	−1.73060	−0.119995
$209$	0	0
$210$	−0.326238	−0.0225126
$211$	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
$215$	−2.61803	−0.178548
$216$	16.7082	1.13685
$217$	−0.277515	−0.0188389
$218$	10.8541	0.735133
$219$	−1.17557	−0.0794377
$220$	0.527864	0.0355886
$221$	0.555029	0.0373353
$222$	−12.1392	−0.814731
$223$	−19.7477	−1.32240	−0.661201	−	0.750209i	$-0.729953\pi$
$223$	−19.7477	−1.32240	−0.661201	+	0.750209i	$0.729953\pi$
$224$	0.792055	0.0529214
$225$	−1.61803	−0.107869
$226$	5.12461	0.340884
$227$	−18.6376	−1.23702	−0.618511	−	0.785776i	$-0.712264\pi$
$227$	−18.6376	−1.23702	−0.618511	+	0.785776i	$0.712264\pi$
$228$	0	0
$229$	−24.8541	−1.64241	−0.821203	−	0.570637i	$-0.806696\pi$
$229$	−24.8541	−1.64241	−0.821203	+	0.570637i	$0.806696\pi$
$230$	8.33499	0.549593
$231$	0.237026	0.0155951
$232$	27.0344	1.77490
$233$	6.23607	0.408538	0.204269	−	0.978915i	$-0.434518\pi$
$233$	6.23607	0.408538	0.204269	+	0.978915i	$0.434518\pi$
$234$	−1.38197	−0.0903419
$235$	−7.47214	−0.487428
$236$	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	$0.134125\pi$
$240$	2.80017	0.180750
$241$	−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
$245$	6.94427	0.443653
$246$	2.24514	0.143145
$247$	0	0
$248$	3.61803	0.229745
$249$	−15.5599	−0.986071
$250$	−1.17557	−0.0743496
$251$	−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
$255$	−0.898056	−0.0562384
$256$	−13.2705	−0.829407
$257$	−13.7638	−0.858563	−0.429282	−	0.903171i	$-0.641233\pi$
$257$	−13.7638	−0.858563	−0.429282	+	0.903171i	$0.641233\pi$
$258$	3.61803	0.225249
$259$	−2.07363	−0.128849
$260$	0.449028	0.0278475
$261$	14.2128	0.879753
$262$	−15.7314	−0.971892
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−3.09017	−0.190187
$265$	−1.00406	−0.0616787
$266$	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	$0.438865\pi$
$270$	6.38197	0.388394
$271$	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
$275$	0.854102	0.0515043
$276$	5.15131	0.310072
$277$	−28.8328	−1.73240	−0.866198	−	0.499701i	$-0.833443\pi$
$277$	−28.8328	−1.73240	−0.866198	+	0.499701i	$0.833443\pi$
$278$	−12.8253	−0.769209
$279$	1.90211	0.113877
$280$	0.726543	0.0434192
$281$	−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	$-0.449931\pi$
$283$	5.27051	0.313299	0.156650	+	0.987654i	$0.449931\pi$
$284$	−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
$290$	10.3262	0.606378
$291$	5.00000	0.293105
$292$	0.618034	0.0361677
$293$	30.8828	1.80419	0.902097	−	0.431533i	$-0.142027\pi$
$293$	30.8828	1.80419	0.902097	+	0.431533i	$0.142027\pi$
$294$	−9.59675	−0.559694
$295$	11.3067	0.658300
$296$	27.0344	1.57135
$297$	−4.63677	−0.269053
$298$	−4.80828	−0.278536
$299$	−5.15131	−0.297908
$300$	−0.726543	−0.0419470
$301$	0.618034	0.0356229
$302$	21.1803	1.21879
$303$	19.6417	1.12838
$304$	0	0
$305$	13.9443	0.798447
$306$	−1.45309	−0.0830673
$307$	25.4540	1.45274	0.726369	−	0.687305i	$-0.241207\pi$
$307$	25.4540	1.45274	0.726369	+	0.687305i	$0.241207\pi$
$308$	−0.124612	−0.00710042
$309$	1.58359	0.0900874
$310$	1.38197	0.0784904
$311$	−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	$-0.826747\pi$
$313$	−30.2705	−1.71099	−0.855495	+	0.517811i	$0.826747\pi$
$314$	20.3027	1.14575
$315$	0.381966	0.0215213
$316$	−5.25731	−0.295747
$317$	−26.6296	−1.49567	−0.747833	−	0.663887i	$-0.768906\pi$
$317$	−26.6296	−1.49567	−0.747833	+	0.663887i	$0.768906\pi$
$318$	1.38757	0.0778112
$319$	−7.50245	−0.420057
$320$	−8.70820	−0.486803
$321$	9.14590	0.510474
$322$	−1.96763	−0.109651
$323$	0	0
$324$	0.944272	0.0524596
$325$	0.726543	0.0403013
$326$	−3.31471	−0.183585
$327$	10.8541	0.600233
$328$	−5.00000	−0.276079
$329$	1.76393	0.0972487
$330$	−1.18034	−0.0649756
$331$	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
$335$	11.5842	0.632912
$336$	−0.661030	−0.0360622
$337$	3.46120	0.188544	0.0942718	−	0.995547i	$-0.469948\pi$
$337$	3.46120	0.188544	0.0942718	+	0.995547i	$0.469948\pi$
$338$	−14.6619	−0.797501
$339$	5.12461	0.278331
$340$	0.472136	0.0256052
$341$	−1.00406	−0.0543727
$342$	0	0
$343$	−3.29180	−0.177740
$344$	−8.05748	−0.434430
$345$	8.33499	0.448741
$346$	−15.1246	−0.813104
$347$	7.52786	0.404117	0.202058	−	0.979373i	$-0.435237\pi$
$347$	7.52786	0.404117	0.202058	+	0.979373i	$0.435237\pi$
$348$	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	$0.344238\pi$
$350$	0.277515	0.0148338
$351$	−3.94427	−0.210530
$352$	2.86568	0.152741
$353$	19.3262	1.02863	0.514316	−	0.857601i	$-0.328046\pi$
$353$	19.3262	1.02863	0.514316	+	0.857601i	$0.328046\pi$
$354$	−15.6254	−0.830483
$355$	−13.0373	−0.691947
$356$	−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	$0.418316\pi$
$360$	−4.97980	−0.262458
$361$	0	0
$362$	17.2361	0.905908
$363$	−12.0737	−0.633705
$364$	−0.106001	−0.00555597
$365$	1.00000	0.0523424
$366$	−19.2705	−1.00729
$367$	−27.0689	−1.41298	−0.706492	−	0.707721i	$-0.749723\pi$
$367$	−27.0689	−1.41298	−0.706492	+	0.707721i	$0.749723\pi$
$368$	16.8885	0.880376
$369$	−2.62866	−0.136842
$370$	10.3262	0.536836
$371$	0.237026	0.0123058
$372$	0.854102	0.0442831
$373$	4.53077	0.234594	0.117297	−	0.993097i	$-0.462577\pi$
$373$	4.53077	0.234594	0.117297	+	0.993097i	$0.462577\pi$
$374$	0.767031	0.0396622
$375$	−1.17557	−0.0607062
$376$	−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	$-0.521393\pi$
$383$	−2.62866	−0.134318	−0.0671590	+	0.997742i	$0.521393\pi$
$384$	4.14590	0.211569
$385$	−0.201626	−0.0102758
$386$	−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	$0.616750\pi$
$390$	−1.00406	−0.0508424
$391$	−5.41641	−0.273920
$392$	21.3723	1.07946
$393$	−15.7314	−0.793546
$394$	−24.5560	−1.23711
$395$	−8.50651	−0.428009
$396$	0.854102	0.0429202
$397$	29.8328	1.49727	0.748633	−	0.662985i	$-0.230711\pi$
$397$	29.8328	1.49727	0.748633	+	0.662985i	$0.230711\pi$
$398$	−15.7719	−0.790576
$399$	0	0
$400$	−2.38197	−0.119098
$401$	−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
$405$	1.52786	0.0759202
$406$	−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	$0.352114\pi$
$410$	−1.90983	−0.0943198
$411$	16.7355	0.825501
$412$	−0.832544	−0.0410165
$413$	−2.66914	−0.131340
$414$	13.4863	0.662816
$415$	13.2361	0.649733
$416$	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	$0.470258\pi$
$420$	0.171513	0.00836900
$421$	−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
$425$	0.763932	0.0370561
$426$	18.0171	0.872930
$427$	−3.29180	−0.159301
$428$	−4.80828	−0.232417
$429$	0.729490	0.0352201
$430$	−3.07768	−0.148419
$431$	−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	$-0.682253\pi$
$433$	−22.5478	−1.08358	−0.541790	+	0.840514i	$0.682253\pi$
$434$	−0.326238	−0.0156599
$435$	10.3262	0.495105
$436$	−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	$0.855471\pi$
$440$	2.62866	0.125316
$441$	11.2361	0.535051
$442$	0.652476	0.0310351
$443$	−36.7771	−1.74733	−0.873666	−	0.486526i	$-0.838264\pi$
$443$	−36.7771	−1.74733	−0.873666	+	0.486526i	$0.838264\pi$
$444$	6.38197	0.302875
$445$	−8.33499	−0.395117
$446$	−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	$0.848330\pi$
$450$	−1.90211	−0.0896665
$451$	1.38757	0.0653382
$452$	−2.69417	−0.126723
$453$	21.1803	0.995139
$454$	−21.9098	−1.02828
$455$	−0.171513	−0.00804067
$456$	0	0
$457$	0.763932	0.0357352	0.0178676	−	0.999840i	$-0.494312\pi$
$457$	0.763932	0.0357352	0.0178676	+	0.999840i	$0.494312\pi$
$458$	−29.2177	−1.36526
$459$	−4.14725	−0.193577
$460$	−4.38197	−0.204310
$461$	−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	$-0.449086\pi$
$463$	6.85410	0.318537	0.159269	+	0.987235i	$0.449086\pi$
$464$	20.9232	0.971337
$465$	1.38197	0.0640871
$466$	7.33094	0.339599
$467$	13.1803	0.609913	0.304957	−	0.952366i	$-0.401358\pi$
$467$	13.1803	0.609913	0.304957	+	0.952366i	$0.401358\pi$
$468$	0.726543	0.0335844
$469$	−2.73466	−0.126275
$470$	−8.78402	−0.405177
$471$	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	$0.756344\pi$
$480$	−3.94427	−0.180031
$481$	−6.38197	−0.290993
$482$	−17.0344	−0.775898
$483$	−1.96763	−0.0895301
$484$	6.34752	0.288524
$485$	−4.25325	−0.193130
$486$	17.0344	0.772698
$487$	34.0260	1.54187	0.770933	−	0.636916i	$-0.219790\pi$
$487$	34.0260	1.54187	0.770933	+	0.636916i	$0.219790\pi$
$488$	42.9161	1.94272
$489$	−3.31471	−0.149896
$490$	8.16348	0.368788
$491$	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
$495$	1.38197	0.0621148
$496$	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	$0.830239\pi$
$500$	0.618034	0.0276393
$501$	15.3262	0.684726
$502$	−3.52671	−0.157405
$503$	0.347524	0.0154953	0.00774767	−	0.999970i	$-0.497534\pi$
$503$	0.347524	0.0154953	0.00774767	+	0.999970i	$0.497534\pi$
$504$	1.17557	0.0523641
$505$	−16.7082	−0.743505
$506$	−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	$0.576583\pi$
$510$	−1.05573	−0.0467484
$511$	−0.236068	−0.0104430
$512$	−22.6538	−1.00117
$513$	0	0
$514$	−16.1803	−0.713684
$515$	−1.34708	−0.0593596
$516$	−1.90211	−0.0837359
$517$	6.38197	0.280679
$518$	−2.43769	−0.107106
$519$	−15.1246	−0.663897
$520$	2.23607	0.0980581
$521$	−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	$-0.292933\pi$
$523$	27.6992	1.21120	0.605600	+	0.795769i	$0.292933\pi$
$524$	8.27051	0.361299
$525$	0.277515	0.0121117
$526$	−7.05342	−0.307544
$527$	−0.898056	−0.0391199
$528$	−2.39163	−0.104082
$529$	27.2705	1.18567
$530$	−1.18034	−0.0512707
$531$	18.2946	0.793917
$532$	0	0
$533$	1.18034	0.0511262
$534$	11.5187	0.498462
$535$	−7.77997	−0.336357
$536$	35.6525	1.53995
$537$	14.2705	0.615818
$538$	7.36068	0.317341
$539$	−5.93112	−0.255471
$540$	−3.35520	−0.144385
$541$	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
$545$	−9.23305	−0.395500
$546$	0.237026	0.0101438
$547$	−13.1433	−0.561966	−0.280983	−	0.959713i	$-0.590660\pi$
$547$	−13.1433	−0.561966	−0.280983	+	0.959713i	$0.590660\pi$
$548$	−8.79837	−0.375848
$549$	22.5623	0.962936
$550$	1.00406	0.0428131
$551$	0	0
$552$	25.6525	1.09184
$553$	2.00811	0.0853937
$554$	−33.8950	−1.44006
$555$	10.3262	0.438324
$556$	6.74265	0.285952
$557$	27.0689	1.14695	0.573473	−	0.819225i	$-0.305596\pi$
$557$	27.0689	1.14695	0.573473	+	0.819225i	$0.305596\pi$
$558$	2.23607	0.0946603
$559$	1.90211	0.0804508
$560$	0.562306	0.0237618
$561$	0.767031	0.0323841
$562$	−12.6869	−0.535165
$563$	18.4661	0.778253	0.389127	−	0.921184i	$-0.372777\pi$
$563$	18.4661	0.778253	0.389127	+	0.921184i	$0.372777\pi$
$564$	−5.42882	−0.228595
$565$	−4.35926	−0.183395
$566$	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
$575$	−7.09017	−0.295681
$576$	−14.0902	−0.587090
$577$	17.4721	0.727375	0.363687	−	0.931521i	$-0.381518\pi$
$577$	17.4721	0.727375	0.363687	+	0.931521i	$0.381518\pi$
$578$	−19.2986	−0.802718
$579$	−7.23607	−0.300721
$580$	−5.42882	−0.225420
$581$	−3.12461	−0.129631
$582$	5.87785	0.243645
$583$	0.857567	0.0355168
$584$	3.07768	0.127355
$585$	1.17557	0.0486039
$586$	36.3050	1.49974
$587$	9.58359	0.395557	0.197779	−	0.980247i	$-0.436627\pi$
$587$	9.58359	0.395557	0.197779	+	0.980247i	$0.436627\pi$
$588$	5.04531	0.208065
$589$	0	0
$590$	13.2918	0.547215
$591$	−24.5560	−1.01010
$592$	20.9232	0.859940
$593$	7.29180	0.299438	0.149719	−	0.988729i	$-0.452163\pi$
$593$	7.29180	0.299438	0.149719	+	0.988729i	$0.452163\pi$
$594$	−5.45085	−0.223651
$595$	−0.180340	−0.00739321
$596$	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	$0.567771\pi$
$600$	−3.61803	−0.147706
$601$	−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
$605$	10.2705	0.417556
$606$	23.0902	0.937974
$607$	20.5397	0.833682	0.416841	−	0.908979i	$-0.363137\pi$
$607$	20.5397	0.833682	0.416841	+	0.908979i	$0.363137\pi$
$608$	0	0
$609$	−2.43769	−0.0987803
$610$	16.3925	0.663712
$611$	5.42882	0.219627
$612$	0.763932	0.0308801
$613$	35.4721	1.43271	0.716353	−	0.697738i	$-0.245810\pi$
$613$	35.4721	1.43271	0.716353	+	0.697738i	$0.245810\pi$
$614$	29.9230	1.20759
$615$	−1.90983	−0.0770118
$616$	−0.620541	−0.0250023
$617$	19.3820	0.780289	0.390144	−	0.920754i	$-0.372425\pi$
$617$	19.3820	0.780289	0.390144	+	0.920754i	$0.372425\pi$
$618$	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	$0.686702\pi$
$620$	−0.726543	−0.0291787
$621$	38.4913	1.54460
$622$	−8.50651	−0.341080
$623$	1.96763	0.0788312
$624$	−2.03444	−0.0814429
$625$	1.00000	0.0400000
$626$	−35.5851	−1.42227
$627$	0	0
$628$	−10.6738	−0.425929
$629$	−6.71040	−0.267561
$630$	0.449028	0.0178897
$631$	−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
$635$	−2.17963	−0.0864959
$636$	−0.729490	−0.0289262
$637$	−5.04531	−0.199902
$638$	−8.81966	−0.349174
$639$	−21.0948	−0.834496
$640$	−3.52671	−0.139406
$641$	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	$-0.697821\pi$
$643$	−29.5279	−1.16447	−0.582233	+	0.813022i	$0.697821\pi$
$644$	1.03444	0.0407627
$645$	−3.07768	−0.121184
$646$	0	0
$647$	17.4721	0.686901	0.343450	−	0.939171i	$-0.388404\pi$
$647$	17.4721	0.686901	0.343450	+	0.939171i	$0.388404\pi$
$648$	4.70228	0.184723
$649$	−9.65706	−0.379073
$650$	0.854102	0.0335006
$651$	−0.326238	−0.0127863
$652$	1.74265	0.0682473
$653$	10.7984	0.422573	0.211287	−	0.977424i	$-0.432235\pi$
$653$	10.7984	0.422573	0.211287	+	0.977424i	$0.432235\pi$
$654$	12.7598	0.498946
$655$	13.3820	0.522877
$656$	−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	$0.704613\pi$
$660$	0.620541	0.0241545
$661$	−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
$670$	13.6180	0.526111
$671$	−11.9098	−0.459774
$672$	0.931116	0.0359186
$673$	34.9646	1.34779	0.673893	−	0.738829i	$-0.264621\pi$
$673$	34.9646	1.34779	0.673893	+	0.738829i	$0.264621\pi$
$674$	4.06888	0.156728
$675$	−5.42882	−0.208956
$676$	7.70820	0.296469
$677$	16.0494	0.616830	0.308415	−	0.951252i	$-0.400201\pi$
$677$	16.0494	0.616830	0.308415	+	0.951252i	$0.400201\pi$
$678$	6.02434	0.231363
$679$	1.00406	0.0385322
$680$	2.35114	0.0901621
$681$	−21.9098	−0.839587
$682$	−1.18034	−0.0451976
$683$	32.6134	1.24792	0.623959	−	0.781457i	$-0.285523\pi$
$683$	32.6134	1.24792	0.623959	+	0.781457i	$0.285523\pi$
$684$	0	0
$685$	−14.2361	−0.543932
$686$	−3.86974	−0.147747
$687$	−29.2177	−1.11473
$688$	−6.23607	−0.237748
$689$	0.729490	0.0277914
$690$	9.79837	0.373018
$691$	−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
$695$	10.9098	0.413833
$696$	31.7809	1.20465
$697$	1.24108	0.0470094
$698$	20.6457	0.781452
$699$	7.33094	0.277282
$700$	−0.145898	−0.00551443
$701$	−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
$705$	−8.78402	−0.330825
$706$	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	$0.416441\pi$
$710$	−15.3262	−0.575183
$711$	−13.7638	−0.516184
$712$	−25.6525	−0.961367
$713$	8.33499	0.312148
$714$	0.249224	0.00932696
$715$	−0.620541	−0.0232069
$716$	−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	$0.787260\pi$
$720$	−3.85410	−0.143634
$721$	0.318003	0.0118431
$722$	0	0
$723$	−17.0344	−0.633518
$724$	−9.06154	−0.336769
$725$	−8.78402	−0.326230
$726$	−14.1935	−0.526770
$727$	−10.3607	−0.384256	−0.192128	−	0.981370i	$-0.561539\pi$
$727$	−10.3607	−0.384256	−0.192128	+	0.981370i	$0.561539\pi$
$728$	−0.527864	−0.0195639
$729$	21.6180	0.800668
$730$	1.17557	0.0435098
$731$	2.00000	0.0739727
$732$	10.1311	0.374456
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	−31.8214	−1.17455
$735$	8.16348	0.301115
$736$	−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	$0.702022\pi$
$740$	−5.42882	−0.199568
$741$	0	0
$742$	0.278640	0.0102292
$743$	11.5842	0.424983	0.212491	−	0.977163i	$-0.431842\pi$
$743$	11.5842	0.424983	0.212491	+	0.977163i	$0.431842\pi$
$744$	4.25325	0.155932
$745$	4.09017	0.149852
$746$	5.32624	0.195007
$747$	21.4164	0.783585
$748$	−0.403252	−0.0147444
$749$	1.83660	0.0671079
$750$	−1.38197	−0.0504623
$751$	−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
$755$	−18.0171	−0.655708
$756$	0.792055	0.0288068
$757$	−43.9787	−1.59843	−0.799217	−	0.601043i	$-0.794752\pi$
$757$	−43.9787	−1.59843	−0.799217	+	0.601043i	$0.794752\pi$
$758$	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
$765$	1.23607	0.0446901
$766$	−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	$0.380903\pi$
$770$	−0.237026	−0.00854181
$771$	−16.1803	−0.582721
$772$	3.80423	0.136917
$773$	6.94742	0.249881	0.124941	−	0.992164i	$-0.460126\pi$
$773$	6.94742	0.249881	0.124941	+	0.992164i	$0.460126\pi$
$774$	−4.97980	−0.178995
$775$	−1.17557	−0.0422277
$776$	−13.0902	−0.469910
$777$	−2.43769	−0.0874518
$778$	−16.6295	−0.596196
$779$	0	0
$780$	0.527864	0.0189006
$781$	11.1352	0.398447
$782$	−6.36737	−0.227697
$783$	47.6869	1.70419
$784$	16.5410	0.590751
$785$	−17.2705	−0.616411
$786$	−18.4934	−0.659639
$787$	−43.9856	−1.56792	−0.783959	−	0.620812i	$-0.786803\pi$
$787$	−43.9856	−1.56792	−0.783959	+	0.620812i	$0.786803\pi$
$788$	12.9098	0.459894
$789$	−7.05342	−0.251109
$790$	−10.0000	−0.355784
$791$	1.02908	0.0365899
$792$	4.25325	0.151133
$793$	−10.1311	−0.359766
$794$	35.0706	1.24461
$795$	−1.18034	−0.0418623
$796$	8.29180	0.293895
$797$	4.14725	0.146903	0.0734516	−	0.997299i	$-0.476599\pi$
$797$	4.14725	0.146903	0.0734516	+	0.997299i	$0.476599\pi$
$798$	0	0
$799$	5.70820	0.201942
$800$	3.35520	0.118624
$801$	−13.4863	−0.476515
$802$	−12.7639	−0.450710
$803$	−0.854102	−0.0301406
$804$	8.41641	0.296824
$805$	1.67376	0.0589924
$806$	−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	$0.509991\pi$
$810$	1.79611	0.0631089
$811$	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
$815$	2.81966	0.0987684
$816$	−2.13914	−0.0748848
$817$	0	0
$818$	21.3050	0.744910
$819$	−0.277515	−0.00969714
$820$	1.00406	0.0350632
$821$	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	$-0.620684\pi$
$823$	−21.2361	−0.740243	−0.370121	+	0.928983i	$0.620684\pi$
$824$	−4.14590	−0.144429
$825$	1.00406	0.0349568
$826$	−3.13777	−0.109177
$827$	−2.17963	−0.0757931	−0.0378965	−	0.999282i	$-0.512066\pi$
$827$	−2.17963	−0.0757931	−0.0378965	+	0.999282i	$0.512066\pi$
$828$	−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	$0.382651\pi$
$830$	15.5599	0.540093
$831$	−33.8950	−1.17580
$832$	6.32688	0.219345
$833$	−5.30495	−0.183806
$834$	−15.0770	−0.522074
$835$	−13.0373	−0.451174
$836$	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	$0.442496\pi$
$840$	0.854102	0.0294693
$841$	48.1591	1.66066
$842$	−7.03444	−0.242423
$843$	−12.6869	−0.436961
$844$	−8.78402	−0.302359
$845$	12.4721	0.429055
$846$	−14.2128	−0.488648
$847$	−2.42454	−0.0833081
$848$	−2.39163	−0.0821289
$849$	6.19586	0.212641
$850$	0.898056	0.0308031
$851$	62.2802	2.13494
$852$	−9.47214	−0.324510
$853$	−6.52786	−0.223510	−0.111755	−	0.993736i	$-0.535647\pi$
$853$	−6.52786	−0.223510	−0.111755	+	0.993736i	$0.535647\pi$
$854$	−3.86974	−0.132420
$855$	0	0
$856$	−23.9443	−0.818398
$857$	35.5851	1.21556	0.607782	−	0.794104i	$-0.292059\pi$
$857$	35.5851	1.21556	0.607782	+	0.794104i	$0.292059\pi$
$858$	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	$0.653817\pi$
$860$	1.61803	0.0551745
$861$	0.450850	0.0153649
$862$	−1.90983	−0.0650491
$863$	24.8335	0.845341	0.422671	−	0.906283i	$-0.361093\pi$
$863$	24.8335	0.845341	0.422671	+	0.906283i	$0.361093\pi$
$864$	−18.2148	−0.619679
$865$	12.8658	0.437449
$866$	−26.5066	−0.900730
$867$	−19.2986	−0.655416
$868$	0.171513	0.00582154
$869$	7.26543	0.246463
$870$	12.1392	0.411558
$871$	−8.41641	−0.285179
$872$	−28.4164	−0.962301
$873$	−6.88191	−0.232917
$874$	0	0
$875$	−0.236068	−0.00798055
$876$	0.726543	0.0245476
$877$	23.1684	0.782341	0.391170	−	0.920318i	$-0.372070\pi$
$877$	23.1684	0.782341	0.391170	+	0.920318i	$0.372070\pi$
$878$	−44.2705	−1.49406
$879$	36.3050	1.22454
$880$	2.03444	0.0685810
$881$	−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	$-0.912806\pi$
$883$	−57.2148	−1.92543	−0.962715	+	0.270516i	$0.912806\pi$
$884$	−0.343027	−0.0115372
$885$	13.2918	0.446799
$886$	−43.2341	−1.45248
$887$	19.3642	0.650185	0.325092	−	0.945682i	$-0.394605\pi$
$887$	19.3642	0.650185	0.325092	+	0.945682i	$0.394605\pi$
$888$	31.7809	1.06650
$889$	0.514540	0.0172571
$890$	−9.79837	−0.328442
$891$	−1.30495	−0.0437175
$892$	12.2047	0.408645
$893$	0	0
$894$	−5.65248	−0.189047
$895$	−12.1392	−0.405769
$896$	0.832544	0.0278133
$897$	−6.05573	−0.202195
$898$	−44.2705	−1.47733
$899$	10.3262	0.344399
$900$	1.00000	0.0333333
$901$	0.767031	0.0255535
$902$	1.63119	0.0543127
$903$	0.726543	0.0241778
$904$	−13.4164	−0.446223
$905$	−14.6619	−0.487377
$906$	24.8990	0.827213
$907$	−41.5035	−1.37810	−0.689050	−	0.724714i	$-0.741972\pi$
$907$	−41.5035	−1.37810	−0.689050	+	0.724714i	$0.741972\pi$
$908$	11.5187	0.382261
$909$	−27.0344	−0.896676
$910$	−0.201626	−0.00668384
$911$	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
$915$	16.3925	0.541918
$916$	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	$0.711332\pi$
$920$	−21.8213	−0.719427
$921$	29.9230	0.985996
$922$	−9.40456	−0.309723
$923$	9.47214	0.311779
$924$	−0.146490	−0.00481917
$925$	−8.78402	−0.288817
$926$	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	$0.407656\pi$
$930$	1.62460	0.0532727
$931$	0	0
$932$	−3.85410	−0.126245
$933$	−8.50651	−0.278491
$934$	15.4944	0.506993
$935$	−0.652476	−0.0213382
$936$	3.61803	0.118259
$937$	47.7984	1.56150	0.780752	−	0.624841i	$-0.214836\pi$
$937$	47.7984	1.56150	0.780752	+	0.624841i	$0.214836\pi$
$938$	−3.21478	−0.104966
$939$	−35.5851	−1.16128
$940$	4.61803	0.150624
$941$	−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
$945$	1.28157	0.0416895
$946$	2.62866	0.0854650
$947$	−25.4164	−0.825922	−0.412961	−	0.910749i	$-0.635505\pi$
$947$	−25.4164	−0.825922	−0.412961	+	0.910749i	$0.635505\pi$
$948$	−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	$-0.850032\pi$
$953$	−55.0148	−1.78210	−0.891052	+	0.453901i	$0.850032\pi$
$954$	−1.90983	−0.0618330
$955$	−13.0000	−0.420670
$956$	−17.4377	−0.563975
$957$	−8.81966	−0.285099
$958$	−37.1037	−1.19877
$959$	3.36068	0.108522
$960$	−10.2371	−0.330401
$961$	−29.6180	−0.955420
$962$	−7.50245	−0.241889
$963$	−12.5882	−0.405651
$964$	8.95554	0.288438
$965$	6.15537	0.198148
$966$	−2.31308	−0.0744222
$967$	−46.9443	−1.50963	−0.754813	−	0.655940i	$-0.772272\pi$
$967$	−46.9443	−1.50963	−0.754813	+	0.655940i	$0.772272\pi$
$968$	31.6094	1.01596
$969$	0	0
$970$	−5.00000	−0.160540
$971$	−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
$975$	0.854102	0.0273532
$976$	33.2148	1.06318
$977$	4.31877	0.138170	0.0690848	−	0.997611i	$-0.477992\pi$
$977$	4.31877	0.138170	0.0690848	+	0.997611i	$0.477992\pi$
$978$	−3.89667	−0.124602
$979$	7.11894	0.227522
$980$	−4.29180	−0.137096
$981$	−14.9394	−0.476978
$982$	4.38428	0.139908
$983$	−6.26137	−0.199707	−0.0998533	−	0.995002i	$-0.531837\pi$
$983$	−6.26137	−0.199707	−0.0998533	+	0.995002i	$0.531837\pi$
$984$	−5.87785	−0.187379
$985$	20.8885	0.665564
$986$	−7.88854	−0.251222
$987$	2.07363	0.0660043
$988$	0	0
$989$	−18.5623	−0.590247
$990$	1.62460	0.0516331
$991$	−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
$995$	13.4164	0.425329
$996$	9.61657	0.304713
$997$	−19.0557	−0.603501	−0.301750	−	0.953387i	$-0.597571\pi$
$997$	−19.0557	−0.603501	−0.301750	+	0.953387i	$0.597571\pi$
$998$	−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	1805.2.a.l.1.3	yes	4
5.4	even	2		9025.2.a.bl.1.2		4
19.18	odd	2	inner	1805.2.a.l.1.2	✓	4
95.94	odd	2		9025.2.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.l.1.2	✓	4	19.18	odd	2	inner
1805.2.a.l.1.3	yes	4	1.1	even	1	trivial
9025.2.a.bl.1.2		4	5.4	even	2
9025.2.a.bl.1.3		4	95.94	odd	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:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1805.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17557 q^{2} +1.17557 q^{3} -0.618034 q^{4} -1.00000 q^{5} +1.38197 q^{6} +0.236068 q^{7} -3.07768 q^{8} -1.61803 q^{9} -1.17557 q^{10} +0.854102 q^{11} -0.726543 q^{12} +0.726543 q^{13} +0.277515 q^{14} -1.17557 q^{15} -2.38197 q^{16} +0.763932 q^{17} -1.90211 q^{18} +0.618034 q^{20} +0.277515 q^{21} +1.00406 q^{22} -7.09017 q^{23} -3.61803 q^{24} +1.00000 q^{25} +0.854102 q^{26} -5.42882 q^{27} -0.145898 q^{28} -8.78402 q^{29} -1.38197 q^{30} -1.17557 q^{31} +3.35520 q^{32} +1.00406 q^{33} +0.898056 q^{34} -0.236068 q^{35} +1.00000 q^{36} -8.78402 q^{37} +0.854102 q^{39} +3.07768 q^{40} +1.62460 q^{41} +0.326238 q^{42} +2.61803 q^{43} -0.527864 q^{44} +1.61803 q^{45} -8.33499 q^{46} +7.47214 q^{47} -2.80017 q^{48} -6.94427 q^{49} +1.17557 q^{50} +0.898056 q^{51} -0.449028 q^{52} +1.00406 q^{53} -6.38197 q^{54} -0.854102 q^{55} -0.726543 q^{56} -10.3262 q^{58} -11.3067 q^{59} +0.726543 q^{60} -13.9443 q^{61} -1.38197 q^{62} -0.381966 q^{63} +8.70820 q^{64} -0.726543 q^{65} +1.18034 q^{66} -11.5842 q^{67} -0.472136 q^{68} -8.33499 q^{69} -0.277515 q^{70} +13.0373 q^{71} +4.97980 q^{72} -1.00000 q^{73} -10.3262 q^{74} +1.17557 q^{75} +0.201626 q^{77} +1.00406 q^{78} +8.50651 q^{79} +2.38197 q^{80} -1.52786 q^{81} +1.90983 q^{82} -13.2361 q^{83} -0.171513 q^{84} -0.763932 q^{85} +3.07768 q^{86} -10.3262 q^{87} -2.62866 q^{88} +8.33499 q^{89} +1.90211 q^{90} +0.171513 q^{91} +4.38197 q^{92} -1.38197 q^{93} +8.78402 q^{94} +3.94427 q^{96} +4.25325 q^{97} -8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{23} - 10 q^{24} + 4 q^{25} - 10 q^{26} - 14 q^{28} - 10 q^{30} + 8 q^{35} + 4 q^{36} - 10 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 4 * q^5 + 10 * q^6 - 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 + 12 * q^17 - 2 * q^20 - 6 * q^23 - 10 * q^24 + 4 * q^25 - 10 * q^26 - 14 * q^28 - 10 * q^30 + 8 * q^35 + 4 * q^36 - 10 * q^39 - 30 * q^42 + 6 * q^43 - 20 * q^44 + 2 * q^45 + 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^55 - 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 + 14 * q^80 - 24 * q^81 + 30 * q^82 - 44 * q^83 - 12 * q^85 - 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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