LMFDB - Embedded newform 7105.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,2,5,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} +1.00000 q^{5} -3.52543 q^{6} +3.06668 q^{8} +5.42864 q^{9} -1.21432 q^{10} +4.90321 q^{11} -1.52543 q^{12} +6.42864 q^{13} +2.90321 q^{15} -2.67307 q^{16} -2.14764 q^{17} -6.59210 q^{18} -2.28100 q^{19} -0.525428 q^{20} -5.95407 q^{22} +6.90321 q^{23} +8.90321 q^{24} +1.00000 q^{25} -7.80642 q^{26} +7.05086 q^{27} +1.00000 q^{29} -3.52543 q^{30} -1.71900 q^{31} -2.88739 q^{32} +14.2351 q^{33} +2.60793 q^{34} -2.85236 q^{36} +7.95407 q^{37} +2.76986 q^{38} +18.6637 q^{39} +3.06668 q^{40} +3.37778 q^{41} -1.09679 q^{43} -2.57628 q^{44} +5.42864 q^{45} -8.38271 q^{46} -12.7096 q^{47} -7.76049 q^{48} -1.21432 q^{50} -6.23506 q^{51} -3.37778 q^{52} +3.37778 q^{53} -8.56199 q^{54} +4.90321 q^{55} -6.62222 q^{57} -1.21432 q^{58} +3.18421 q^{59} -1.52543 q^{60} +2.42864 q^{61} +2.08742 q^{62} +8.85236 q^{64} +6.42864 q^{65} -17.2859 q^{66} -1.09679 q^{67} +1.12843 q^{68} +20.0415 q^{69} +3.57136 q^{71} +16.6479 q^{72} -14.1891 q^{73} -9.65878 q^{74} +2.90321 q^{75} +1.19850 q^{76} -22.6637 q^{78} +0.341219 q^{79} -2.67307 q^{80} +4.18421 q^{81} -4.10171 q^{82} +7.33185 q^{83} -2.14764 q^{85} +1.33185 q^{86} +2.90321 q^{87} +15.0366 q^{88} -2.94914 q^{89} -6.59210 q^{90} -3.62714 q^{92} -4.99063 q^{93} +15.4336 q^{94} -2.28100 q^{95} -8.38271 q^{96} +18.5763 q^{97} +26.6178 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{15} + 5 q^{16} - 13 q^{18} + 5 q^{20} + 2 q^{22} + 14 q^{23} + 20 q^{24} + 3 q^{25}+ \cdots + 20 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^15 + 5 * q^16 - 13 * q^18 + 5 * q^20 + 2 * q^22 + 14 * q^23 + 20 * q^24 + 3 * q^25 - 10 * q^26 + 8 * q^27 + 3 * q^29 - 4 * q^30 - 12 * q^31 + 11 * q^32 + 16 * q^33 + 14 * q^34 - 15 * q^36 + 4 * q^37 + 2 * q^38 + 16 * q^39 + 9 * q^40 + 10 * q^41 - 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 - 18 * q^47 + 10 * q^48 + 3 * q^50 + 8 * q^51 - 10 * q^52 + 10 * q^53 - 12 * q^54 + 8 * q^55 - 20 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 - 6 * q^61 - 14 * q^62 + 33 * q^64 + 6 * q^65 - 12 * q^66 - 10 * q^67 + 36 * q^68 + 20 * q^69 + 24 * q^71 - 3 * q^72 + 4 * q^73 - 22 * q^74 + 2 * q^75 - 16 * q^76 - 28 * q^78 + 8 * q^79 + 5 * q^80 - q^81 + 14 * q^82 + 2 * q^83 - 16 * q^86 + 2 * q^87 + 38 * q^88 - 22 * q^89 - 13 * q^90 + 22 * q^92 + 12 * q^93 + 8 * q^96 + 36 * q^97 + 20 * 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 $

$2$	−1.21432	−0.858654	−0.429327	−	0.903149i	$-0.641249\pi$
$2$	−1.21432	−0.858654	−0.429327	+	0.903149i	$0.641249\pi$
$3$	2.90321	1.67617	0.838085	−	0.545540i	$-0.183675\pi$
$3$	2.90321	1.67617	0.838085	+	0.545540i	$0.183675\pi$
$4$	−0.525428	−0.262714
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.52543	−1.43925
$7$	0	0
$7$	0	0
$8$	3.06668	1.08423
$9$	5.42864	1.80955
$10$	−1.21432	−0.384002
$11$	4.90321	1.47837	0.739187	−	0.673500i	$-0.235210\pi$
$11$	4.90321	1.47837	0.739187	+	0.673500i	$0.235210\pi$
$12$	−1.52543	−0.440353
$13$	6.42864	1.78298	0.891492	−	0.453037i	$-0.149659\pi$
$13$	6.42864	1.78298	0.891492	+	0.453037i	$0.149659\pi$
$14$	0	0
$15$	2.90321	0.749606
$16$	−2.67307	−0.668268
$17$	−2.14764	−0.520880	−0.260440	−	0.965490i	$-0.583868\pi$
$17$	−2.14764	−0.520880	−0.260440	+	0.965490i	$0.583868\pi$
$18$	−6.59210	−1.55377
$19$	−2.28100	−0.523296	−0.261648	−	0.965163i	$-0.584266\pi$
$19$	−2.28100	−0.523296	−0.261648	+	0.965163i	$0.584266\pi$
$20$	−0.525428	−0.117489
$21$	0	0
$22$	−5.95407	−1.26941
$23$	6.90321	1.43942	0.719710	−	0.694275i	$-0.244275\pi$
$23$	6.90321	1.43942	0.719710	+	0.694275i	$0.244275\pi$
$24$	8.90321	1.81736
$25$	1.00000	0.200000
$26$	−7.80642	−1.53097
$27$	7.05086	1.35694
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.52543	−0.643652
$31$	−1.71900	−0.308742	−0.154371	−	0.988013i	$-0.549335\pi$
$31$	−1.71900	−0.308742	−0.154371	+	0.988013i	$0.549335\pi$
$32$	−2.88739	−0.510423
$33$	14.2351	2.47801
$34$	2.60793	0.447256
$35$	0	0
$36$	−2.85236	−0.475393
$37$	7.95407	1.30764	0.653820	−	0.756650i	$-0.273165\pi$
$37$	7.95407	1.30764	0.653820	+	0.756650i	$0.273165\pi$
$38$	2.76986	0.449330
$39$	18.6637	2.98858
$40$	3.06668	0.484884
$41$	3.37778	0.527521	0.263761	−	0.964588i	$-0.415037\pi$
$41$	3.37778	0.527521	0.263761	+	0.964588i	$0.415037\pi$
$42$	0	0
$43$	−1.09679	−0.167259	−0.0836293	−	0.996497i	$-0.526651\pi$
$43$	−1.09679	−0.167259	−0.0836293	+	0.996497i	$0.526651\pi$
$44$	−2.57628	−0.388389
$45$	5.42864	0.809254
$46$	−8.38271	−1.23596
$47$	−12.7096	−1.85389	−0.926945	−	0.375196i	$-0.877575\pi$
$47$	−12.7096	−1.85389	−0.926945	+	0.375196i	$0.877575\pi$
$48$	−7.76049	−1.12013
$49$	0	0
$50$	−1.21432	−0.171731
$51$	−6.23506	−0.873084
$52$	−3.37778	−0.468414
$53$	3.37778	0.463974	0.231987	−	0.972719i	$-0.425477\pi$
$53$	3.37778	0.463974	0.231987	+	0.972719i	$0.425477\pi$
$54$	−8.56199	−1.16514
$55$	4.90321	0.661149
$56$	0	0
$57$	−6.62222	−0.877134
$58$	−1.21432	−0.159448
$59$	3.18421	0.414549	0.207274	−	0.978283i	$-0.433541\pi$
$59$	3.18421	0.414549	0.207274	+	0.978283i	$0.433541\pi$
$60$	−1.52543	−0.196932
$61$	2.42864	0.310955	0.155478	−	0.987839i	$-0.450308\pi$
$61$	2.42864	0.310955	0.155478	+	0.987839i	$0.450308\pi$
$62$	2.08742	0.265103
$63$	0	0
$64$	8.85236	1.10654
$65$	6.42864	0.797375
$66$	−17.2859	−2.12775
$67$	−1.09679	−0.133994	−0.0669970	−	0.997753i	$-0.521342\pi$
$67$	−1.09679	−0.133994	−0.0669970	+	0.997753i	$0.521342\pi$
$68$	1.12843	0.136842
$69$	20.0415	2.41271
$70$	0	0
$71$	3.57136	0.423843	0.211921	−	0.977287i	$-0.432028\pi$
$71$	3.57136	0.423843	0.211921	+	0.977287i	$0.432028\pi$
$72$	16.6479	1.96197
$73$	−14.1891	−1.66071	−0.830356	−	0.557233i	$-0.811863\pi$
$73$	−14.1891	−1.66071	−0.830356	+	0.557233i	$0.811863\pi$
$74$	−9.65878	−1.12281
$75$	2.90321	0.335234
$76$	1.19850	0.137477
$77$	0	0
$78$	−22.6637	−2.56616
$79$	0.341219	0.0383902	0.0191951	−	0.999816i	$-0.493890\pi$
$79$	0.341219	0.0383902	0.0191951	+	0.999816i	$0.493890\pi$
$80$	−2.67307	−0.298858
$81$	4.18421	0.464912
$82$	−4.10171	−0.452958
$83$	7.33185	0.804775	0.402388	−	0.915469i	$-0.368180\pi$
$83$	7.33185	0.804775	0.402388	+	0.915469i	$0.368180\pi$
$84$	0	0
$85$	−2.14764	−0.232945
$86$	1.33185	0.143617
$87$	2.90321	0.311257
$88$	15.0366	1.60290
$89$	−2.94914	−0.312609	−0.156304	−	0.987709i	$-0.549958\pi$
$89$	−2.94914	−0.312609	−0.156304	+	0.987709i	$0.549958\pi$
$90$	−6.59210	−0.694869
$91$	0	0
$92$	−3.62714	−0.378155
$93$	−4.99063	−0.517504
$94$	15.4336	1.59185
$95$	−2.28100	−0.234025
$96$	−8.38271	−0.855556
$97$	18.5763	1.88614	0.943068	−	0.332600i	$-0.107926\pi$
$97$	18.5763	1.88614	0.943068	+	0.332600i	$0.107926\pi$
$98$	0	0
$99$	26.6178	2.67519
$100$	−0.525428	−0.0525428
$101$	−15.4193	−1.53427	−0.767137	−	0.641483i	$-0.778320\pi$
$101$	−15.4193	−1.53427	−0.767137	+	0.641483i	$0.778320\pi$
$102$	7.57136	0.749676
$103$	−7.76049	−0.764664	−0.382332	−	0.924025i	$-0.624879\pi$
$103$	−7.76049	−0.764664	−0.382332	+	0.924025i	$0.624879\pi$
$104$	19.7146	1.93317
$105$	0	0
$106$	−4.10171	−0.398393
$107$	−3.03657	−0.293556	−0.146778	−	0.989169i	$-0.546890\pi$
$107$	−3.03657	−0.293556	−0.146778	+	0.989169i	$0.546890\pi$
$108$	−3.70471	−0.356486
$109$	−7.93978	−0.760493	−0.380246	−	0.924885i	$-0.624161\pi$
$109$	−7.93978	−0.760493	−0.380246	+	0.924885i	$0.624161\pi$
$110$	−5.95407	−0.567698
$111$	23.0923	2.19183
$112$	0	0
$113$	7.82071	0.735711	0.367855	−	0.929883i	$-0.380092\pi$
$113$	7.82071	0.735711	0.367855	+	0.929883i	$0.380092\pi$
$114$	8.04149	0.753154
$115$	6.90321	0.643728
$116$	−0.525428	−0.0487847
$117$	34.8988	3.22639
$118$	−3.86665	−0.355954
$119$	0	0
$120$	8.90321	0.812748
$121$	13.0415	1.18559
$122$	−2.94914	−0.267003
$123$	9.80642	0.884215
$124$	0.903212	0.0811108
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.4429	−1.10413	−0.552066	−	0.833801i	$-0.686160\pi$
$127$	−12.4429	−1.10413	−0.552066	+	0.833801i	$0.686160\pi$
$128$	−4.97481	−0.439715
$129$	−3.18421	−0.280354
$130$	−7.80642	−0.684669
$131$	−4.08742	−0.357120	−0.178560	−	0.983929i	$-0.557144\pi$
$131$	−4.08742	−0.357120	−0.178560	+	0.983929i	$0.557144\pi$
$132$	−7.47949	−0.651006
$133$	0	0
$134$	1.33185	0.115054
$135$	7.05086	0.606841
$136$	−6.58613	−0.564756
$137$	−19.9541	−1.70479	−0.852395	−	0.522898i	$-0.824851\pi$
$137$	−19.9541	−1.70479	−0.852395	+	0.522898i	$0.824851\pi$
$138$	−24.3368	−2.07168
$139$	−7.90813	−0.670759	−0.335380	−	0.942083i	$-0.608865\pi$
$139$	−7.90813	−0.670759	−0.335380	+	0.942083i	$0.608865\pi$
$140$	0	0
$141$	−36.8988	−3.10744
$142$	−4.33677	−0.363934
$143$	31.5210	2.63592
$144$	−14.5111	−1.20926
$145$	1.00000	0.0830455
$146$	17.2301	1.42598
$147$	0	0
$148$	−4.17929	−0.343535
$149$	−16.1017	−1.31910	−0.659552	−	0.751659i	$-0.729254\pi$
$149$	−16.1017	−1.31910	−0.659552	+	0.751659i	$0.729254\pi$
$150$	−3.52543	−0.287850
$151$	5.67307	0.461668	0.230834	−	0.972993i	$-0.425855\pi$
$151$	5.67307	0.461668	0.230834	+	0.972993i	$0.425855\pi$
$152$	−6.99508	−0.567376
$153$	−11.6588	−0.942557
$154$	0	0
$155$	−1.71900	−0.138074
$156$	−9.80642	−0.785142
$157$	−1.89384	−0.151145	−0.0755726	−	0.997140i	$-0.524078\pi$
$157$	−1.89384	−0.151145	−0.0755726	+	0.997140i	$0.524078\pi$
$158$	−0.414349	−0.0329639
$159$	9.80642	0.777700
$160$	−2.88739	−0.228268
$161$	0	0
$162$	−5.08097	−0.399198
$163$	−5.95407	−0.466359	−0.233179	−	0.972434i	$-0.574913\pi$
$163$	−5.95407	−0.466359	−0.233179	+	0.972434i	$0.574913\pi$
$164$	−1.77478	−0.138587
$165$	14.2351	1.10820
$166$	−8.90321	−0.691023
$167$	−4.23951	−0.328063	−0.164032	−	0.986455i	$-0.552450\pi$
$167$	−4.23951	−0.328063	−0.164032	+	0.986455i	$0.552450\pi$
$168$	0	0
$169$	28.3274	2.17903
$170$	2.60793	0.200019
$171$	−12.3827	−0.946929
$172$	0.576283	0.0439411
$173$	−24.4099	−1.85585	−0.927925	−	0.372766i	$-0.878409\pi$
$173$	−24.4099	−1.85585	−0.927925	+	0.372766i	$0.878409\pi$
$174$	−3.52543	−0.267262
$175$	0	0
$176$	−13.1066	−0.987950
$177$	9.24443	0.694854
$178$	3.58120	0.268423
$179$	−3.61285	−0.270037	−0.135018	−	0.990843i	$-0.543109\pi$
$179$	−3.61285	−0.270037	−0.135018	+	0.990843i	$0.543109\pi$
$180$	−2.85236	−0.212602
$181$	−18.0415	−1.34101	−0.670507	−	0.741904i	$-0.733923\pi$
$181$	−18.0415	−1.34101	−0.670507	+	0.741904i	$0.733923\pi$
$182$	0	0
$183$	7.05086	0.521214
$184$	21.1699	1.56067
$185$	7.95407	0.584795
$186$	6.06022	0.444357
$187$	−10.5303	−0.770055
$188$	6.67799	0.487043
$189$	0	0
$190$	2.76986	0.200947
$191$	9.85236	0.712892	0.356446	−	0.934316i	$-0.383988\pi$
$191$	9.85236	0.712892	0.356446	+	0.934316i	$0.383988\pi$
$192$	25.7003	1.85476
$193$	2.23951	0.161203	0.0806017	−	0.996746i	$-0.474316\pi$
$193$	2.23951	0.161203	0.0806017	+	0.996746i	$0.474316\pi$
$194$	−22.5575	−1.61954
$195$	18.6637	1.33654
$196$	0	0
$197$	−10.5620	−0.752511	−0.376255	−	0.926516i	$-0.622788\pi$
$197$	−10.5620	−0.752511	−0.376255	+	0.926516i	$0.622788\pi$
$198$	−32.3225	−2.29706
$199$	−3.18421	−0.225723	−0.112861	−	0.993611i	$-0.536002\pi$
$199$	−3.18421	−0.225723	−0.112861	+	0.993611i	$0.536002\pi$
$200$	3.06668	0.216847
$201$	−3.18421	−0.224597
$202$	18.7239	1.31741
$203$	0	0
$204$	3.27607	0.229371
$205$	3.37778	0.235915
$206$	9.42372	0.656581
$207$	37.4750	2.60470
$208$	−17.1842	−1.19151
$209$	−11.1842	−0.773628
$210$	0	0
$211$	−5.76049	−0.396569	−0.198284	−	0.980145i	$-0.563537\pi$
$211$	−5.76049	−0.396569	−0.198284	+	0.980145i	$0.563537\pi$
$212$	−1.77478	−0.121892
$213$	10.3684	0.710432
$214$	3.68736	0.252063
$215$	−1.09679	−0.0748003
$216$	21.6227	1.47124
$217$	0	0
$218$	9.64143	0.653000
$219$	−41.1941	−2.78364
$220$	−2.57628	−0.173693
$221$	−13.8064	−0.928721
$222$	−28.0415	−1.88202
$223$	−8.14764	−0.545607	−0.272803	−	0.962070i	$-0.587951\pi$
$223$	−8.14764	−0.545607	−0.272803	+	0.962070i	$0.587951\pi$
$224$	0	0
$225$	5.42864	0.361909
$226$	−9.49685	−0.631721
$227$	−10.5161	−0.697975	−0.348988	−	0.937127i	$-0.613475\pi$
$227$	−10.5161	−0.697975	−0.348988	+	0.937127i	$0.613475\pi$
$228$	3.47949	0.230435
$229$	−8.48886	−0.560960	−0.280480	−	0.959860i	$-0.590494\pi$
$229$	−8.48886	−0.560960	−0.280480	+	0.959860i	$0.590494\pi$
$230$	−8.38271	−0.552739
$231$	0	0
$232$	3.06668	0.201337
$233$	−20.5718	−1.34771	−0.673853	−	0.738866i	$-0.735362\pi$
$233$	−20.5718	−1.34771	−0.673853	+	0.738866i	$0.735362\pi$
$234$	−42.3783	−2.77035
$235$	−12.7096	−0.829085
$236$	−1.67307	−0.108908
$237$	0.990632	0.0643485
$238$	0	0
$239$	0.815792	0.0527692	0.0263846	−	0.999652i	$-0.491601\pi$
$239$	0.815792	0.0527692	0.0263846	+	0.999652i	$0.491601\pi$
$240$	−7.76049	−0.500938
$241$	7.24443	0.466655	0.233327	−	0.972398i	$-0.425039\pi$
$241$	7.24443	0.466655	0.233327	+	0.972398i	$0.425039\pi$
$242$	−15.8365	−1.01801
$243$	−9.00492	−0.577666
$244$	−1.27607	−0.0816923
$245$	0	0
$246$	−11.9081	−0.759235
$247$	−14.6637	−0.933029
$248$	−5.27163	−0.334749
$249$	21.2859	1.34894
$250$	−1.21432	−0.0768003
$251$	20.4242	1.28916	0.644582	−	0.764535i	$-0.277031\pi$
$251$	20.4242	1.28916	0.644582	+	0.764535i	$0.277031\pi$
$252$	0	0
$253$	33.8479	2.12800
$254$	15.1097	0.948067
$255$	−6.23506	−0.390455
$256$	−11.6637	−0.728981
$257$	−3.08250	−0.192281	−0.0961405	−	0.995368i	$-0.530650\pi$
$257$	−3.08250	−0.192281	−0.0961405	+	0.995368i	$0.530650\pi$
$258$	3.86665	0.240727
$259$	0	0
$260$	−3.37778	−0.209481
$261$	5.42864	0.336024
$262$	4.96343	0.306642
$263$	20.0558	1.23669	0.618346	−	0.785906i	$-0.287803\pi$
$263$	20.0558	1.23669	0.618346	+	0.785906i	$0.287803\pi$
$264$	43.6543	2.68674
$265$	3.37778	0.207496
$266$	0	0
$267$	−8.56199	−0.523985
$268$	0.576283	0.0352021
$269$	23.4608	1.43043	0.715214	−	0.698906i	$-0.246329\pi$
$269$	23.4608	1.43043	0.715214	+	0.698906i	$0.246329\pi$
$270$	−8.56199	−0.521066
$271$	21.9353	1.33248	0.666238	−	0.745739i	$-0.267903\pi$
$271$	21.9353	1.33248	0.666238	+	0.745739i	$0.267903\pi$
$272$	5.74080	0.348087
$273$	0	0
$274$	24.2306	1.46383
$275$	4.90321	0.295675
$276$	−10.5303	−0.633853
$277$	18.3368	1.10175	0.550875	−	0.834588i	$-0.314294\pi$
$277$	18.3368	1.10175	0.550875	+	0.834588i	$0.314294\pi$
$278$	9.60300	0.575950
$279$	−9.33185	−0.558683
$280$	0	0
$281$	6.89877	0.411546	0.205773	−	0.978600i	$-0.434029\pi$
$281$	6.89877	0.411546	0.205773	+	0.978600i	$0.434029\pi$
$282$	44.8069	2.66821
$283$	29.0049	1.72416	0.862082	−	0.506769i	$-0.169160\pi$
$283$	29.0049	1.72416	0.862082	+	0.506769i	$0.169160\pi$
$284$	−1.87649	−0.111349
$285$	−6.62222	−0.392266
$286$	−38.2766	−2.26334
$287$	0	0
$288$	−15.6746	−0.923635
$289$	−12.3876	−0.728684
$290$	−1.21432	−0.0713073
$291$	53.9309	3.16148
$292$	7.45536	0.436292
$293$	−7.79213	−0.455221	−0.227611	−	0.973752i	$-0.573091\pi$
$293$	−7.79213	−0.455221	−0.227611	+	0.973752i	$0.573091\pi$
$294$	0	0
$295$	3.18421	0.185392
$296$	24.3926	1.41779
$297$	34.5718	2.00606
$298$	19.5526	1.13265
$299$	44.3783	2.56646
$300$	−1.52543	−0.0880706
$301$	0	0
$302$	−6.88892	−0.396413
$303$	−44.7654	−2.57171
$304$	6.09726	0.349702
$305$	2.42864	0.139063
$306$	14.1575	0.809330
$307$	−5.92549	−0.338185	−0.169093	−	0.985600i	$-0.554084\pi$
$307$	−5.92549	−0.338185	−0.169093	+	0.985600i	$0.554084\pi$
$308$	0	0
$309$	−22.5303	−1.28171
$310$	2.08742	0.118557
$311$	−8.94470	−0.507207	−0.253604	−	0.967308i	$-0.581616\pi$
$311$	−8.94470	−0.507207	−0.253604	+	0.967308i	$0.581616\pi$
$312$	57.2355	3.24032
$313$	30.5116	1.72462	0.862309	−	0.506382i	$-0.169017\pi$
$313$	30.5116	1.72462	0.862309	+	0.506382i	$0.169017\pi$
$314$	2.29973	0.129781
$315$	0	0
$316$	−0.179286	−0.0100856
$317$	−22.2306	−1.24860	−0.624298	−	0.781186i	$-0.714615\pi$
$317$	−22.2306	−1.24860	−0.624298	+	0.781186i	$0.714615\pi$
$318$	−11.9081	−0.667775
$319$	4.90321	0.274527
$320$	8.85236	0.494862
$321$	−8.81579	−0.492050
$322$	0	0
$323$	4.89877	0.272575
$324$	−2.19850	−0.122139
$325$	6.42864	0.356597
$326$	7.23014	0.400440
$327$	−23.0509	−1.27472
$328$	10.3586	0.571956
$329$	0	0
$330$	−17.2859	−0.951558
$331$	−6.54770	−0.359894	−0.179947	−	0.983676i	$-0.557593\pi$
$331$	−6.54770	−0.359894	−0.179947	+	0.983676i	$0.557593\pi$
$332$	−3.85236	−0.211426
$333$	43.1798	2.36624
$334$	5.14812	0.281693
$335$	−1.09679	−0.0599239
$336$	0	0
$337$	2.66815	0.145343	0.0726717	−	0.997356i	$-0.476847\pi$
$337$	2.66815	0.145343	0.0726717	+	0.997356i	$0.476847\pi$
$338$	−34.3985	−1.87103
$339$	22.7052	1.23318
$340$	1.12843	0.0611978
$341$	−8.42864	−0.456436
$342$	15.0366	0.813084
$343$	0	0
$344$	−3.36349	−0.181347
$345$	20.0415	1.07900
$346$	29.6414	1.59353
$347$	14.6780	0.787956	0.393978	−	0.919120i	$-0.371099\pi$
$347$	14.6780	0.787956	0.393978	+	0.919120i	$0.371099\pi$
$348$	−1.52543	−0.0817715
$349$	11.1240	0.595453	0.297727	−	0.954651i	$-0.403772\pi$
$349$	11.1240	0.595453	0.297727	+	0.954651i	$0.403772\pi$
$350$	0	0
$351$	45.3274	2.41940
$352$	−14.1575	−0.754597
$353$	13.4795	0.717441	0.358721	−	0.933445i	$-0.383213\pi$
$353$	13.4795	0.717441	0.358721	+	0.933445i	$0.383213\pi$
$354$	−11.2257	−0.596639
$355$	3.57136	0.189548
$356$	1.54956	0.0821266
$357$	0	0
$358$	4.38715	0.231868
$359$	26.1891	1.38221	0.691105	−	0.722755i	$-0.257124\pi$
$359$	26.1891	1.38221	0.691105	+	0.722755i	$0.257124\pi$
$360$	16.6479	0.877420
$361$	−13.7971	−0.726161
$362$	21.9081	1.15147
$363$	37.8622	1.98725
$364$	0	0
$365$	−14.1891	−0.742693
$366$	−8.56199	−0.447543
$367$	22.9862	1.19987	0.599935	−	0.800049i	$-0.295193\pi$
$367$	22.9862	1.19987	0.599935	+	0.800049i	$0.295193\pi$
$368$	−18.4528	−0.961917
$369$	18.3368	0.954574
$370$	−9.65878	−0.502136
$371$	0	0
$372$	2.62222	0.135956
$373$	−24.2766	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$373$	−24.2766	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$374$	12.7872	0.661211
$375$	2.90321	0.149921
$376$	−38.9763	−2.01005
$377$	6.42864	0.331092
$378$	0	0
$379$	29.7605	1.52869	0.764347	−	0.644805i	$-0.223062\pi$
$379$	29.7605	1.52869	0.764347	+	0.644805i	$0.223062\pi$
$380$	1.19850	0.0614817
$381$	−36.1245	−1.85071
$382$	−11.9639	−0.612127
$383$	−23.8020	−1.21622	−0.608112	−	0.793851i	$-0.708073\pi$
$383$	−23.8020	−1.21622	−0.608112	+	0.793851i	$0.708073\pi$
$384$	−14.4429	−0.737038
$385$	0	0
$386$	−2.71948	−0.138418
$387$	−5.95407	−0.302662
$388$	−9.76049	−0.495514
$389$	−6.52051	−0.330603	−0.165301	−	0.986243i	$-0.552860\pi$
$389$	−6.52051	−0.330603	−0.165301	+	0.986243i	$0.552860\pi$
$390$	−22.6637	−1.14762
$391$	−14.8256	−0.749765
$392$	0	0
$393$	−11.8666	−0.598593
$394$	12.8256	0.646146
$395$	0.341219	0.0171686
$396$	−13.9857	−0.702808
$397$	14.7654	0.741055	0.370527	−	0.928822i	$-0.379177\pi$
$397$	14.7654	0.741055	0.370527	+	0.928822i	$0.379177\pi$
$398$	3.86665	0.193817
$399$	0	0
$400$	−2.67307	−0.133654
$401$	6.81579	0.340364	0.170182	−	0.985413i	$-0.445564\pi$
$401$	6.81579	0.340364	0.170182	+	0.985413i	$0.445564\pi$
$402$	3.86665	0.192851
$403$	−11.0509	−0.550482
$404$	8.10171	0.403075
$405$	4.18421	0.207915
$406$	0	0
$407$	39.0005	1.93318
$408$	−19.1209	−0.946627
$409$	11.0825	0.547994	0.273997	−	0.961731i	$-0.411654\pi$
$409$	11.0825	0.547994	0.273997	+	0.961731i	$0.411654\pi$
$410$	−4.10171	−0.202569
$411$	−57.9309	−2.85752
$412$	4.07758	0.200888
$413$	0	0
$414$	−45.5067	−2.23653
$415$	7.33185	0.359906
$416$	−18.5620	−0.910077
$417$	−22.9590	−1.12431
$418$	13.5812	0.664278
$419$	−30.9719	−1.51308	−0.756538	−	0.653950i	$-0.773111\pi$
$419$	−30.9719	−1.51308	−0.756538	+	0.653950i	$0.773111\pi$
$420$	0	0
$421$	22.8988	1.11602	0.558009	−	0.829835i	$-0.311566\pi$
$421$	22.8988	1.11602	0.558009	+	0.829835i	$0.311566\pi$
$422$	6.99508	0.340515
$423$	−68.9960	−3.35470
$424$	10.3586	0.503057
$425$	−2.14764	−0.104176
$426$	−12.5906	−0.610015
$427$	0	0
$428$	1.59549	0.0771212
$429$	91.5121	4.41825
$430$	1.33185	0.0642276
$431$	−28.0830	−1.35271	−0.676355	−	0.736576i	$-0.736441\pi$
$431$	−28.0830	−1.35271	−0.676355	+	0.736576i	$0.736441\pi$
$432$	−18.8474	−0.906798
$433$	−23.0049	−1.10555	−0.552773	−	0.833332i	$-0.686430\pi$
$433$	−23.0049	−1.10555	−0.552773	+	0.833332i	$0.686430\pi$
$434$	0	0
$435$	2.90321	0.139198
$436$	4.17178	0.199792
$437$	−15.7462	−0.753243
$438$	50.0228	2.39018
$439$	−11.7462	−0.560616	−0.280308	−	0.959910i	$-0.590437\pi$
$439$	−11.7462	−0.560616	−0.280308	+	0.959910i	$0.590437\pi$
$440$	15.0366	0.716840
$441$	0	0
$442$	16.7654	0.797449
$443$	17.6874	0.840352	0.420176	−	0.907443i	$-0.361968\pi$
$443$	17.6874	0.840352	0.420176	+	0.907443i	$0.361968\pi$
$444$	−12.1334	−0.575823
$445$	−2.94914	−0.139803
$446$	9.89384	0.468487
$447$	−46.7467	−2.21104
$448$	0	0
$449$	1.57136	0.0741571	0.0370785	−	0.999312i	$-0.488195\pi$
$449$	1.57136	0.0741571	0.0370785	+	0.999312i	$0.488195\pi$
$450$	−6.59210	−0.310755
$451$	16.5620	0.779874
$452$	−4.10922	−0.193281
$453$	16.4701	0.773834
$454$	12.7699	0.599319
$455$	0	0
$456$	−20.3082	−0.951018
$457$	1.47949	0.0692078	0.0346039	−	0.999401i	$-0.488983\pi$
$457$	1.47949	0.0692078	0.0346039	+	0.999401i	$0.488983\pi$
$458$	10.3082	0.481670
$459$	−15.1427	−0.706802
$460$	−3.62714	−0.169116
$461$	41.2543	1.92140	0.960702	−	0.277583i	$-0.0895334\pi$
$461$	41.2543	1.92140	0.960702	+	0.277583i	$0.0895334\pi$
$462$	0	0
$463$	−34.4242	−1.59983	−0.799914	−	0.600115i	$-0.795122\pi$
$463$	−34.4242	−1.59983	−0.799914	+	0.600115i	$0.795122\pi$
$464$	−2.67307	−0.124094
$465$	−4.99063	−0.231435
$466$	24.9808	1.15721
$467$	−15.1699	−0.701980	−0.350990	−	0.936379i	$-0.614155\pi$
$467$	−15.1699	−0.701980	−0.350990	+	0.936379i	$0.614155\pi$
$468$	−18.3368	−0.847618
$469$	0	0
$470$	15.4336	0.711897
$471$	−5.49823	−0.253345
$472$	9.76494	0.449468
$473$	−5.37778	−0.247271
$474$	−1.20294	−0.0552531
$475$	−2.28100	−0.104659
$476$	0	0
$477$	18.3368	0.839583
$478$	−0.990632	−0.0453105
$479$	18.9763	0.867051	0.433526	−	0.901141i	$-0.357269\pi$
$479$	18.9763	0.867051	0.433526	+	0.901141i	$0.357269\pi$
$480$	−8.38271	−0.382616
$481$	51.1338	2.33150
$482$	−8.79706	−0.400695
$483$	0	0
$484$	−6.85236	−0.311471
$485$	18.5763	0.843506
$486$	10.9349	0.496015
$487$	−32.3926	−1.46785	−0.733923	−	0.679232i	$-0.762313\pi$
$487$	−32.3926	−1.46785	−0.733923	+	0.679232i	$0.762313\pi$
$488$	7.44785	0.337148
$489$	−17.2859	−0.781696
$490$	0	0
$491$	2.69673	0.121702	0.0608508	−	0.998147i	$-0.480619\pi$
$491$	2.69673	0.121702	0.0608508	+	0.998147i	$0.480619\pi$
$492$	−5.15257	−0.232296
$493$	−2.14764	−0.0967250
$494$	17.8064	0.801149
$495$	26.6178	1.19638
$496$	4.59502	0.206322
$497$	0	0
$498$	−25.8479	−1.15827
$499$	14.5718	0.652325	0.326163	−	0.945314i	$-0.394244\pi$
$499$	14.5718	0.652325	0.326163	+	0.945314i	$0.394244\pi$
$500$	−0.525428	−0.0234978
$501$	−12.3082	−0.549890
$502$	−24.8015	−1.10695
$503$	22.2494	0.992050	0.496025	−	0.868308i	$-0.334792\pi$
$503$	22.2494	0.992050	0.496025	+	0.868308i	$0.334792\pi$
$504$	0	0
$505$	−15.4193	−0.686149
$506$	−41.1022	−1.82722
$507$	82.2405	3.65243
$508$	6.53786	0.290071
$509$	9.18421	0.407083	0.203541	−	0.979066i	$-0.434755\pi$
$509$	9.18421	0.407083	0.203541	+	0.979066i	$0.434755\pi$
$510$	7.57136	0.335265
$511$	0	0
$512$	24.1131	1.06566
$513$	−16.0830	−0.710081
$514$	3.74314	0.165103
$515$	−7.76049	−0.341968
$516$	1.67307	0.0736528
$517$	−62.3180	−2.74074
$518$	0	0
$519$	−70.8671	−3.11072
$520$	19.7146	0.864541
$521$	−7.01921	−0.307517	−0.153759	−	0.988108i	$-0.549138\pi$
$521$	−7.01921	−0.307517	−0.153759	+	0.988108i	$0.549138\pi$
$522$	−6.59210	−0.288529
$523$	7.29036	0.318785	0.159393	−	0.987215i	$-0.449046\pi$
$523$	7.29036	0.318785	0.159393	+	0.987215i	$0.449046\pi$
$524$	2.14764	0.0938202
$525$	0	0
$526$	−24.3541	−1.06189
$527$	3.69181	0.160818
$528$	−38.0513	−1.65597
$529$	24.6543	1.07193
$530$	−4.10171	−0.178167
$531$	17.2859	0.750145
$532$	0	0
$533$	21.7146	0.940562
$534$	10.3970	0.449922
$535$	−3.03657	−0.131282
$536$	−3.36349	−0.145281
$537$	−10.4889	−0.452628
$538$	−28.4889	−1.22824
$539$	0	0
$540$	−3.70471	−0.159425
$541$	30.9491	1.33061	0.665304	−	0.746573i	$-0.268302\pi$
$541$	30.9491	1.33061	0.665304	+	0.746573i	$0.268302\pi$
$542$	−26.6365	−1.14414
$543$	−52.3783	−2.24777
$544$	6.20108	0.265869
$545$	−7.93978	−0.340103
$546$	0	0
$547$	−19.4237	−0.830498	−0.415249	−	0.909708i	$-0.636306\pi$
$547$	−19.4237	−0.830498	−0.415249	+	0.909708i	$0.636306\pi$
$548$	10.4844	0.447872
$549$	13.1842	0.562688
$550$	−5.95407	−0.253882
$551$	−2.28100	−0.0971737
$552$	61.4608	2.61594
$553$	0	0
$554$	−22.2667	−0.946022
$555$	23.0923	0.980215
$556$	4.15515	0.176218
$557$	30.3497	1.28596	0.642979	−	0.765884i	$-0.277698\pi$
$557$	30.3497	1.28596	0.642979	+	0.765884i	$0.277698\pi$
$558$	11.3319	0.479716
$559$	−7.05086	−0.298219
$560$	0	0
$561$	−30.5718	−1.29074
$562$	−8.37731	−0.353375
$563$	33.1798	1.39836	0.699180	−	0.714946i	$-0.253549\pi$
$563$	33.1798	1.39836	0.699180	+	0.714946i	$0.253549\pi$
$564$	19.3876	0.816366
$565$	7.82071	0.329020
$566$	−35.2212	−1.48046
$567$	0	0
$568$	10.9522	0.459544
$569$	−4.06022	−0.170213	−0.0851067	−	0.996372i	$-0.527123\pi$
$569$	−4.06022	−0.170213	−0.0851067	+	0.996372i	$0.527123\pi$
$570$	8.04149	0.336821
$571$	−31.5496	−1.32031	−0.660154	−	0.751130i	$-0.729509\pi$
$571$	−31.5496	−1.32031	−0.660154	+	0.751130i	$0.729509\pi$
$572$	−16.5620	−0.692492
$573$	28.6035	1.19493
$574$	0	0
$575$	6.90321	0.287884
$576$	48.0563	2.00234
$577$	−33.7891	−1.40666	−0.703329	−	0.710865i	$-0.748304\pi$
$577$	−33.7891	−1.40666	−0.703329	+	0.710865i	$0.748304\pi$
$578$	15.0425	0.625687
$579$	6.50177	0.270204
$580$	−0.525428	−0.0218172
$581$	0	0
$582$	−65.4893	−2.71462
$583$	16.5620	0.685928
$584$	−43.5135	−1.80060
$585$	34.8988	1.44289
$586$	9.46214	0.390877
$587$	25.5669	1.05526	0.527630	−	0.849474i	$-0.323081\pi$
$587$	25.5669	1.05526	0.527630	+	0.849474i	$0.323081\pi$
$588$	0	0
$589$	3.92104	0.161564
$590$	−3.86665	−0.159187
$591$	−30.6637	−1.26134
$592$	−21.2618	−0.873854
$593$	−7.96836	−0.327221	−0.163611	−	0.986525i	$-0.552314\pi$
$593$	−7.96836	−0.327221	−0.163611	+	0.986525i	$0.552314\pi$
$594$	−41.9813	−1.72251
$595$	0	0
$596$	8.46028	0.346547
$597$	−9.24443	−0.378349
$598$	−53.8894	−2.20370
$599$	37.2815	1.52328	0.761640	−	0.648001i	$-0.224395\pi$
$599$	37.2815	1.52328	0.761640	+	0.648001i	$0.224395\pi$
$600$	8.90321	0.363472
$601$	29.9496	1.22167	0.610835	−	0.791758i	$-0.290834\pi$
$601$	29.9496	1.22167	0.610835	+	0.791758i	$0.290834\pi$
$602$	0	0
$603$	−5.95407	−0.242468
$604$	−2.98079	−0.121287
$605$	13.0415	0.530212
$606$	54.3595	2.20820
$607$	31.8435	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$607$	31.8435	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$608$	6.58613	0.267103
$609$	0	0
$610$	−2.94914	−0.119407
$611$	−81.7057	−3.30546
$612$	6.12584	0.247623
$613$	−2.65386	−0.107188	−0.0535942	−	0.998563i	$-0.517068\pi$
$613$	−2.65386	−0.107188	−0.0535942	+	0.998563i	$0.517068\pi$
$614$	7.19544	0.290384
$615$	9.80642	0.395433
$616$	0	0
$617$	−18.3096	−0.737116	−0.368558	−	0.929605i	$-0.620148\pi$
$617$	−18.3096	−0.737116	−0.368558	+	0.929605i	$0.620148\pi$
$618$	27.3590	1.10054
$619$	−12.8113	−0.514931	−0.257466	−	0.966287i	$-0.582887\pi$
$619$	−12.8113	−0.514931	−0.257466	+	0.966287i	$0.582887\pi$
$620$	0.903212	0.0362739
$621$	48.6735	1.95320
$622$	10.8617	0.435515
$623$	0	0
$624$	−49.8894	−1.99717
$625$	1.00000	0.0400000
$626$	−37.0509	−1.48085
$627$	−32.4701	−1.29673
$628$	0.995078	0.0397079
$629$	−17.0825	−0.681124
$630$	0	0
$631$	30.2766	1.20529	0.602645	−	0.798009i	$-0.294113\pi$
$631$	30.2766	1.20529	0.602645	+	0.798009i	$0.294113\pi$
$632$	1.04641	0.0416239
$633$	−16.7239	−0.664716
$634$	26.9951	1.07211
$635$	−12.4429	−0.493783
$636$	−5.15257	−0.204313
$637$	0	0
$638$	−5.95407	−0.235724
$639$	19.3876	0.766963
$640$	−4.97481	−0.196647
$641$	−4.50177	−0.177809	−0.0889046	−	0.996040i	$-0.528337\pi$
$641$	−4.50177	−0.177809	−0.0889046	+	0.996040i	$0.528337\pi$
$642$	10.7052	0.422500
$643$	40.0272	1.57852	0.789259	−	0.614060i	$-0.210465\pi$
$643$	40.0272	1.57852	0.789259	+	0.614060i	$0.210465\pi$
$644$	0	0
$645$	−3.18421	−0.125378
$646$	−5.94867	−0.234047
$647$	−27.3604	−1.07565	−0.537825	−	0.843057i	$-0.680754\pi$
$647$	−27.3604	−1.07565	−0.537825	+	0.843057i	$0.680754\pi$
$648$	12.8316	0.504073
$649$	15.6128	0.612858
$650$	−7.80642	−0.306193
$651$	0	0
$652$	3.12843	0.122519
$653$	22.8430	0.893915	0.446958	−	0.894555i	$-0.352507\pi$
$653$	22.8430	0.893915	0.446958	+	0.894555i	$0.352507\pi$
$654$	27.9911	1.09454
$655$	−4.08742	−0.159709
$656$	−9.02906	−0.352525
$657$	−77.0277	−3.00514
$658$	0	0
$659$	15.0178	0.585012	0.292506	−	0.956264i	$-0.405511\pi$
$659$	15.0178	0.585012	0.292506	+	0.956264i	$0.405511\pi$
$660$	−7.47949	−0.291139
$661$	4.65080	0.180895	0.0904475	−	0.995901i	$-0.471170\pi$
$661$	4.65080	0.180895	0.0904475	+	0.995901i	$0.471170\pi$
$662$	7.95100	0.309025
$663$	−40.0830	−1.55669
$664$	22.4844	0.872565
$665$	0	0
$666$	−52.4340	−2.03178
$667$	6.90321	0.267293
$668$	2.22755	0.0861867
$669$	−23.6543	−0.914529
$670$	1.33185	0.0514539
$671$	11.9081	0.459708
$672$	0	0
$673$	−44.8671	−1.72950	−0.864750	−	0.502202i	$-0.832523\pi$
$673$	−44.8671	−1.72950	−0.864750	+	0.502202i	$0.832523\pi$
$674$	−3.23999	−0.124800
$675$	7.05086	0.271388
$676$	−14.8840	−0.572462
$677$	−27.2212	−1.04620	−0.523099	−	0.852272i	$-0.675224\pi$
$677$	−27.2212	−1.04620	−0.523099	+	0.852272i	$0.675224\pi$
$678$	−27.5714	−1.05887
$679$	0	0
$680$	−6.58613	−0.252566
$681$	−30.5303	−1.16993
$682$	10.2351	0.391921
$683$	12.0558	0.461301	0.230651	−	0.973037i	$-0.425915\pi$
$683$	12.0558	0.461301	0.230651	+	0.973037i	$0.425915\pi$
$684$	6.50622	0.248771
$685$	−19.9541	−0.762406
$686$	0	0
$687$	−24.6450	−0.940264
$688$	2.93179	0.111774
$689$	21.7146	0.827259
$690$	−24.3368	−0.926485
$691$	−37.5812	−1.42966	−0.714828	−	0.699300i	$-0.753495\pi$
$691$	−37.5812	−1.42966	−0.714828	+	0.699300i	$0.753495\pi$
$692$	12.8256	0.487558
$693$	0	0
$694$	−17.8238	−0.676581
$695$	−7.90813	−0.299973
$696$	8.90321	0.337475
$697$	−7.25428	−0.274775
$698$	−13.5081	−0.511288
$699$	−59.7244	−2.25898
$700$	0	0
$701$	2.04149	0.0771059	0.0385530	−	0.999257i	$-0.487725\pi$
$701$	2.04149	0.0771059	0.0385530	+	0.999257i	$0.487725\pi$
$702$	−55.0420	−2.07743
$703$	−18.1432	−0.684284
$704$	43.4050	1.63589
$705$	−36.8988	−1.38969
$706$	−16.3684	−0.616033
$707$	0	0
$708$	−4.85728	−0.182548
$709$	−32.1432	−1.20716	−0.603582	−	0.797301i	$-0.706260\pi$
$709$	−32.1432	−1.20716	−0.603582	+	0.797301i	$0.706260\pi$
$710$	−4.33677	−0.162756
$711$	1.85236	0.0694688
$712$	−9.04407	−0.338941
$713$	−11.8666	−0.444409
$714$	0	0
$715$	31.5210	1.17882
$716$	1.89829	0.0709424
$717$	2.36842	0.0884501
$718$	−31.8020	−1.18684
$719$	−1.01921	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$719$	−1.01921	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$720$	−14.5111	−0.540798
$721$	0	0
$722$	16.7540	0.623521
$723$	21.0321	0.782193
$724$	9.47949	0.352303
$725$	1.00000	0.0371391
$726$	−45.9768	−1.70636
$727$	24.1476	0.895587	0.447793	−	0.894137i	$-0.352210\pi$
$727$	24.1476	0.895587	0.447793	+	0.894137i	$0.352210\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	17.2301	0.637716
$731$	2.35551	0.0871217
$732$	−3.70471	−0.136930
$733$	−18.8845	−0.697514	−0.348757	−	0.937213i	$-0.613396\pi$
$733$	−18.8845	−0.697514	−0.348757	+	0.937213i	$0.613396\pi$
$734$	−27.9126	−1.03027
$735$	0	0
$736$	−19.9323	−0.734713
$737$	−5.37778	−0.198093
$738$	−22.2667	−0.819649
$739$	−3.31312	−0.121875	−0.0609375	−	0.998142i	$-0.519409\pi$
$739$	−3.31312	−0.121875	−0.0609375	+	0.998142i	$0.519409\pi$
$740$	−4.17929	−0.153634
$741$	−42.5718	−1.56392
$742$	0	0
$743$	2.99508	0.109879	0.0549394	−	0.998490i	$-0.482503\pi$
$743$	2.99508	0.109879	0.0549394	+	0.998490i	$0.482503\pi$
$744$	−15.3047	−0.561096
$745$	−16.1017	−0.589921
$746$	29.4795	1.07932
$747$	39.8020	1.45628
$748$	5.53294	0.202304
$749$	0	0
$750$	−3.52543	−0.128730
$751$	11.9956	0.437724	0.218862	−	0.975756i	$-0.429766\pi$
$751$	11.9956	0.437724	0.218862	+	0.975756i	$0.429766\pi$
$752$	33.9738	1.23890
$753$	59.2958	2.16086
$754$	−7.80642	−0.284293
$755$	5.67307	0.206464
$756$	0	0
$757$	18.5763	0.675166	0.337583	−	0.941296i	$-0.390391\pi$
$757$	18.5763	0.675166	0.337583	+	0.941296i	$0.390391\pi$
$758$	−36.1388	−1.31262
$759$	98.2677	3.56689
$760$	−6.99508	−0.253738
$761$	−2.59057	−0.0939082	−0.0469541	−	0.998897i	$-0.514951\pi$
$761$	−2.59057	−0.0939082	−0.0469541	+	0.998897i	$0.514951\pi$
$762$	43.8666	1.58912
$763$	0	0
$764$	−5.17670	−0.187286
$765$	−11.6588	−0.421524
$766$	28.9032	1.04432
$767$	20.4701	0.739133
$768$	−33.8622	−1.22190
$769$	−32.7467	−1.18088	−0.590438	−	0.807083i	$-0.701045\pi$
$769$	−32.7467	−1.18088	−0.590438	+	0.807083i	$0.701045\pi$
$770$	0	0
$771$	−8.94914	−0.322296
$772$	−1.17670	−0.0423504
$773$	−28.2208	−1.01503	−0.507515	−	0.861643i	$-0.669436\pi$
$773$	−28.2208	−1.01503	−0.507515	+	0.861643i	$0.669436\pi$
$774$	7.23014	0.259882
$775$	−1.71900	−0.0617484
$776$	56.9675	2.04501
$777$	0	0
$778$	7.91798	0.283873
$779$	−7.70471	−0.276050
$780$	−9.80642	−0.351126
$781$	17.5111	0.626598
$782$	18.0031	0.643788
$783$	7.05086	0.251977
$784$	0	0
$785$	−1.89384	−0.0675942
$786$	14.4099	0.513984
$787$	−11.9857	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$787$	−11.9857	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$788$	5.54956	0.197695
$789$	58.2262	2.07291
$790$	−0.414349	−0.0147419
$791$	0	0
$792$	81.6281	2.90053
$793$	15.6128	0.554428
$794$	−17.9299	−0.636309
$795$	9.80642	0.347798
$796$	1.67307	0.0593004
$797$	33.0366	1.17022	0.585108	−	0.810956i	$-0.301052\pi$
$797$	33.0366	1.17022	0.585108	+	0.810956i	$0.301052\pi$
$798$	0	0
$799$	27.2958	0.965655
$800$	−2.88739	−0.102085
$801$	−16.0098	−0.565680
$802$	−8.27655	−0.292255
$803$	−69.5723	−2.45515
$804$	1.67307	0.0590047
$805$	0	0
$806$	13.4193	0.472674
$807$	68.1116	2.39764
$808$	−47.2859	−1.66351
$809$	−32.1303	−1.12964	−0.564820	−	0.825214i	$-0.691055\pi$
$809$	−32.1303	−1.12964	−0.564820	+	0.825214i	$0.691055\pi$
$810$	−5.08097	−0.178527
$811$	−15.3176	−0.537872	−0.268936	−	0.963158i	$-0.586672\pi$
$811$	−15.3176	−0.537872	−0.268936	+	0.963158i	$0.586672\pi$
$812$	0	0
$813$	63.6829	2.23346
$814$	−47.3590	−1.65993
$815$	−5.95407	−0.208562
$816$	16.6668	0.583453
$817$	2.50177	0.0875258
$818$	−13.4577	−0.470537
$819$	0	0
$820$	−1.77478	−0.0619780
$821$	−17.1427	−0.598285	−0.299143	−	0.954208i	$-0.596701\pi$
$821$	−17.1427	−0.598285	−0.299143	+	0.954208i	$0.596701\pi$
$822$	70.3466	2.45362
$823$	−35.6400	−1.24233	−0.621167	−	0.783678i	$-0.713341\pi$
$823$	−35.6400	−1.24233	−0.621167	+	0.783678i	$0.713341\pi$
$824$	−23.7989	−0.829075
$825$	14.2351	0.495601
$826$	0	0
$827$	4.70964	0.163770	0.0818850	−	0.996642i	$-0.473906\pi$
$827$	4.70964	0.163770	0.0818850	+	0.996642i	$0.473906\pi$
$828$	−19.6904	−0.684290
$829$	−2.25380	−0.0782777	−0.0391388	−	0.999234i	$-0.512461\pi$
$829$	−2.25380	−0.0782777	−0.0391388	+	0.999234i	$0.512461\pi$
$830$	−8.90321	−0.309035
$831$	53.2355	1.84672
$832$	56.9086	1.97295
$833$	0	0
$834$	27.8796	0.965390
$835$	−4.23951	−0.146714
$836$	5.87649	0.203243
$837$	−12.1204	−0.418944
$838$	37.6098	1.29921
$839$	8.42419	0.290835	0.145418	−	0.989370i	$-0.453547\pi$
$839$	8.42419	0.290835	0.145418	+	0.989370i	$0.453547\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−27.8064	−0.958273
$843$	20.0286	0.689821
$844$	3.02672	0.104184
$845$	28.3274	0.974492
$846$	83.7832	2.88053
$847$	0	0
$848$	−9.02906	−0.310059
$849$	84.2074	2.88999
$850$	2.60793	0.0894511
$851$	54.9086	1.88224
$852$	−5.44785	−0.186640
$853$	−51.8247	−1.77444	−0.887222	−	0.461342i	$-0.847368\pi$
$853$	−51.8247	−1.77444	−0.887222	+	0.461342i	$0.847368\pi$
$854$	0	0
$855$	−12.3827	−0.423480
$856$	−9.31216	−0.318283
$857$	30.0415	1.02620	0.513099	−	0.858330i	$-0.328497\pi$
$857$	30.0415	1.02620	0.513099	+	0.858330i	$0.328497\pi$
$858$	−111.125	−3.79374
$859$	−19.6874	−0.671724	−0.335862	−	0.941911i	$-0.609028\pi$
$859$	−19.6874	−0.671724	−0.335862	+	0.941911i	$0.609028\pi$
$860$	0.576283	0.0196511
$861$	0	0
$862$	34.1017	1.16151
$863$	19.5986	0.667143	0.333571	−	0.942725i	$-0.391746\pi$
$863$	19.5986	0.667143	0.333571	+	0.942725i	$0.391746\pi$
$864$	−20.3586	−0.692613
$865$	−24.4099	−0.829962
$866$	27.9353	0.949281
$867$	−35.9639	−1.22140
$868$	0	0
$869$	1.67307	0.0567550
$870$	−3.52543	−0.119523
$871$	−7.05086	−0.238909
$872$	−24.3487	−0.824552
$873$	100.844	3.41305
$874$	19.1209	0.646775
$875$	0	0
$876$	21.6445	0.731300
$877$	−16.2351	−0.548219	−0.274110	−	0.961698i	$-0.588383\pi$
$877$	−16.2351	−0.548219	−0.274110	+	0.961698i	$0.588383\pi$
$878$	14.2636	0.481375
$879$	−22.6222	−0.763028
$880$	−13.1066	−0.441824
$881$	20.7052	0.697576	0.348788	−	0.937202i	$-0.386593\pi$
$881$	20.7052	0.697576	0.348788	+	0.937202i	$0.386593\pi$
$882$	0	0
$883$	8.75112	0.294499	0.147249	−	0.989099i	$-0.452958\pi$
$883$	8.75112	0.294499	0.147249	+	0.989099i	$0.452958\pi$
$884$	7.25428	0.243988
$885$	9.24443	0.310748
$886$	−21.4781	−0.721571
$887$	−0.414349	−0.0139125	−0.00695625	−	0.999976i	$-0.502214\pi$
$887$	−0.414349	−0.0139125	−0.00695625	+	0.999976i	$0.502214\pi$
$888$	70.8167	2.37645
$889$	0	0
$890$	3.58120	0.120042
$891$	20.5161	0.687314
$892$	4.28100	0.143338
$893$	28.9906	0.970135
$894$	56.7654	1.89852
$895$	−3.61285	−0.120764
$896$	0	0
$897$	128.839	4.30183
$898$	−1.90813	−0.0636753
$899$	−1.71900	−0.0573320
$900$	−2.85236	−0.0950786
$901$	−7.25428	−0.241675
$902$	−20.1116	−0.669642
$903$	0	0
$904$	23.9836	0.797683
$905$	−18.0415	−0.599719
$906$	−20.0000	−0.664455
$907$	−46.9862	−1.56015	−0.780075	−	0.625686i	$-0.784819\pi$
$907$	−46.9862	−1.56015	−0.780075	+	0.625686i	$0.784819\pi$
$908$	5.52543	0.183368
$909$	−83.7057	−2.77634
$910$	0	0
$911$	−12.1704	−0.403223	−0.201612	−	0.979466i	$-0.564618\pi$
$911$	−12.1704	−0.403223	−0.201612	+	0.979466i	$0.564618\pi$
$912$	17.7017	0.586160
$913$	35.9496	1.18976
$914$	−1.79658	−0.0594256
$915$	7.05086	0.233094
$916$	4.46028	0.147372
$917$	0	0
$918$	18.3881	0.606898
$919$	−23.2672	−0.767514	−0.383757	−	0.923434i	$-0.625370\pi$
$919$	−23.2672	−0.767514	−0.383757	+	0.923434i	$0.625370\pi$
$920$	21.1699	0.697952
$921$	−17.2029	−0.566856
$922$	−50.0959	−1.64982
$923$	22.9590	0.755704
$924$	0	0
$925$	7.95407	0.261528
$926$	41.8020	1.37370
$927$	−42.1289	−1.38369
$928$	−2.88739	−0.0947832
$929$	−44.7556	−1.46838	−0.734191	−	0.678943i	$-0.762438\pi$
$929$	−44.7556	−1.46838	−0.734191	+	0.678943i	$0.762438\pi$
$930$	6.06022	0.198723
$931$	0	0
$932$	10.8090	0.354061
$933$	−25.9684	−0.850166
$934$	18.4211	0.602758
$935$	−10.5303	−0.344379
$936$	107.023	3.49816
$937$	22.7239	0.742358	0.371179	−	0.928561i	$-0.378954\pi$
$937$	22.7239	0.742358	0.371179	+	0.928561i	$0.378954\pi$
$938$	0	0
$939$	88.5817	2.89075
$940$	6.67799	0.217812
$941$	−4.10171	−0.133712	−0.0668560	−	0.997763i	$-0.521297\pi$
$941$	−4.10171	−0.133712	−0.0668560	+	0.997763i	$0.521297\pi$
$942$	6.67661	0.217536
$943$	23.3176	0.759324
$944$	−8.51161	−0.277029
$945$	0	0
$946$	6.53035	0.212320
$947$	16.6178	0.540005	0.270002	−	0.962860i	$-0.412975\pi$
$947$	16.6178	0.540005	0.270002	+	0.962860i	$0.412975\pi$
$948$	−0.520505	−0.0169052
$949$	−91.2168	−2.96102
$950$	2.76986	0.0898661
$951$	−64.5402	−2.09286
$952$	0	0
$953$	2.85728	0.0925563	0.0462782	−	0.998929i	$-0.485264\pi$
$953$	2.85728	0.0925563	0.0462782	+	0.998929i	$0.485264\pi$
$954$	−22.2667	−0.720911
$955$	9.85236	0.318815
$956$	−0.428639	−0.0138632
$957$	14.2351	0.460154
$958$	−23.0433	−0.744497
$959$	0	0
$960$	25.7003	0.829473
$961$	−28.0450	−0.904678
$962$	−62.0928	−2.00195
$963$	−16.4844	−0.531203
$964$	−3.80642	−0.122597
$965$	2.23951	0.0720923
$966$	0	0
$967$	16.8015	0.540300	0.270150	−	0.962818i	$-0.412927\pi$
$967$	16.8015	0.540300	0.270150	+	0.962818i	$0.412927\pi$
$968$	39.9940	1.28546
$969$	14.2222	0.456881
$970$	−22.5575	−0.724279
$971$	−38.3640	−1.23116	−0.615579	−	0.788075i	$-0.711078\pi$
$971$	−38.3640	−1.23116	−0.615579	+	0.788075i	$0.711078\pi$
$972$	4.73143	0.151761
$973$	0	0
$974$	39.3349	1.26037
$975$	18.6637	0.597717
$976$	−6.49193	−0.207801
$977$	31.6356	1.01211	0.506056	−	0.862500i	$-0.331103\pi$
$977$	31.6356	1.01211	0.506056	+	0.862500i	$0.331103\pi$
$978$	20.9906	0.671206
$979$	−14.4603	−0.462153
$980$	0	0
$981$	−43.1022	−1.37615
$982$	−3.27469	−0.104500
$983$	35.3733	1.12823	0.564117	−	0.825695i	$-0.309217\pi$
$983$	35.3733	1.12823	0.564117	+	0.825695i	$0.309217\pi$
$984$	30.0731	0.958696
$985$	−10.5620	−0.336533
$986$	2.60793	0.0830533
$987$	0	0
$988$	7.70471	0.245120
$989$	−7.57136	−0.240755
$990$	−32.3225	−1.02728
$991$	24.6953	0.784474	0.392237	−	0.919864i	$-0.371701\pi$
$991$	24.6953	0.784474	0.392237	+	0.919864i	$0.371701\pi$
$992$	4.96343	0.157589
$993$	−19.0094	−0.603244
$994$	0	0
$995$	−3.18421	−0.100946
$996$	−11.1842	−0.354385
$997$	11.4050	0.361199	0.180600	−	0.983557i	$-0.442196\pi$
$997$	11.4050	0.361199	0.180600	+	0.983557i	$0.442196\pi$
$998$	−17.6949	−0.560121
$999$	56.0830	1.77439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.p.1.1		3
7.6	odd	2		145.2.a.d.1.1	✓	3
21.20	even	2		1305.2.a.o.1.3		3
28.27	even	2		2320.2.a.s.1.3		3
35.13	even	4		725.2.b.d.349.4		6
35.27	even	4		725.2.b.d.349.3		6
35.34	odd	2		725.2.a.d.1.3		3
56.13	odd	2		9280.2.a.bu.1.3		3
56.27	even	2		9280.2.a.bm.1.1		3
105.104	even	2		6525.2.a.bh.1.1		3
203.202	odd	2		4205.2.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.d.1.1	✓	3	7.6	odd	2
725.2.a.d.1.3		3	35.34	odd	2
725.2.b.d.349.3		6	35.27	even	4
725.2.b.d.349.4		6	35.13	even	4
1305.2.a.o.1.3		3	21.20	even	2
2320.2.a.s.1.3		3	28.27	even	2
4205.2.a.e.1.3		3	203.202	odd	2
6525.2.a.bh.1.1		3	105.104	even	2
7105.2.a.p.1.1		3	1.1	even	1	trivial
9280.2.a.bm.1.1		3	56.27	even	2
9280.2.a.bu.1.3		3	56.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} +1.00000 q^{5} -3.52543 q^{6} +3.06668 q^{8} +5.42864 q^{9} -1.21432 q^{10} +4.90321 q^{11} -1.52543 q^{12} +6.42864 q^{13} +2.90321 q^{15} -2.67307 q^{16} -2.14764 q^{17} -6.59210 q^{18} -2.28100 q^{19} -0.525428 q^{20} -5.95407 q^{22} +6.90321 q^{23} +8.90321 q^{24} +1.00000 q^{25} -7.80642 q^{26} +7.05086 q^{27} +1.00000 q^{29} -3.52543 q^{30} -1.71900 q^{31} -2.88739 q^{32} +14.2351 q^{33} +2.60793 q^{34} -2.85236 q^{36} +7.95407 q^{37} +2.76986 q^{38} +18.6637 q^{39} +3.06668 q^{40} +3.37778 q^{41} -1.09679 q^{43} -2.57628 q^{44} +5.42864 q^{45} -8.38271 q^{46} -12.7096 q^{47} -7.76049 q^{48} -1.21432 q^{50} -6.23506 q^{51} -3.37778 q^{52} +3.37778 q^{53} -8.56199 q^{54} +4.90321 q^{55} -6.62222 q^{57} -1.21432 q^{58} +3.18421 q^{59} -1.52543 q^{60} +2.42864 q^{61} +2.08742 q^{62} +8.85236 q^{64} +6.42864 q^{65} -17.2859 q^{66} -1.09679 q^{67} +1.12843 q^{68} +20.0415 q^{69} +3.57136 q^{71} +16.6479 q^{72} -14.1891 q^{73} -9.65878 q^{74} +2.90321 q^{75} +1.19850 q^{76} -22.6637 q^{78} +0.341219 q^{79} -2.67307 q^{80} +4.18421 q^{81} -4.10171 q^{82} +7.33185 q^{83} -2.14764 q^{85} +1.33185 q^{86} +2.90321 q^{87} +15.0366 q^{88} -2.94914 q^{89} -6.59210 q^{90} -3.62714 q^{92} -4.99063 q^{93} +15.4336 q^{94} -2.28100 q^{95} -8.38271 q^{96} +18.5763 q^{97} +26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{15} + 5 q^{16} - 13 q^{18} + 5 q^{20} + 2 q^{22} + 14 q^{23} + 20 q^{24} + 3 q^{25}+ \cdots + 20 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^15 + 5 * q^16 - 13 * q^18 + 5 * q^20 + 2 * q^22 + 14 * q^23 + 20 * q^24 + 3 * q^25 - 10 * q^26 + 8 * q^27 + 3 * q^29 - 4 * q^30 - 12 * q^31 + 11 * q^32 + 16 * q^33 + 14 * q^34 - 15 * q^36 + 4 * q^37 + 2 * q^38 + 16 * q^39 + 9 * q^40 + 10 * q^41 - 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 - 18 * q^47 + 10 * q^48 + 3 * q^50 + 8 * q^51 - 10 * q^52 + 10 * q^53 - 12 * q^54 + 8 * q^55 - 20 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 - 6 * q^61 - 14 * q^62 + 33 * q^64 + 6 * q^65 - 12 * q^66 - 10 * q^67 + 36 * q^68 + 20 * q^69 + 24 * q^71 - 3 * q^72 + 4 * q^73 - 22 * q^74 + 2 * q^75 - 16 * q^76 - 28 * q^78 + 8 * q^79 + 5 * q^80 - q^81 + 14 * q^82 + 2 * q^83 - 16 * q^86 + 2 * q^87 + 38 * q^88 - 22 * q^89 - 13 * q^90 + 22 * q^92 + 12 * q^93 + 8 * q^96 + 36 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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