LMFDB - Embedded newform 725.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(1,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
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			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} +3.52543 q^{6} -1.52543 q^{7} -3.06668 q^{8} +5.42864 q^{9} +4.90321 q^{11} -1.52543 q^{12} +6.42864 q^{13} -1.85236 q^{14} -2.67307 q^{16} -2.14764 q^{17} +6.59210 q^{18} +2.28100 q^{19} -4.42864 q^{21} +5.95407 q^{22} -6.90321 q^{23} -8.90321 q^{24} +7.80642 q^{26} +7.05086 q^{27} +0.801502 q^{28} +1.00000 q^{29} +1.71900 q^{31} +2.88739 q^{32} +14.2351 q^{33} -2.60793 q^{34} -2.85236 q^{36} -7.95407 q^{37} +2.76986 q^{38} +18.6637 q^{39} -3.37778 q^{41} -5.37778 q^{42} +1.09679 q^{43} -2.57628 q^{44} -8.38271 q^{46} -12.7096 q^{47} -7.76049 q^{48} -4.67307 q^{49} -6.23506 q^{51} -3.37778 q^{52} -3.37778 q^{53} +8.56199 q^{54} +4.67799 q^{56} +6.62222 q^{57} +1.21432 q^{58} -3.18421 q^{59} -2.42864 q^{61} +2.08742 q^{62} -8.28100 q^{63} +8.85236 q^{64} +17.2859 q^{66} +1.09679 q^{67} +1.12843 q^{68} -20.0415 q^{69} +3.57136 q^{71} -16.6479 q^{72} -14.1891 q^{73} -9.65878 q^{74} -1.19850 q^{76} -7.47949 q^{77} +22.6637 q^{78} +0.341219 q^{79} +4.18421 q^{81} -4.10171 q^{82} +7.33185 q^{83} +2.32693 q^{84} +1.33185 q^{86} +2.90321 q^{87} -15.0366 q^{88} +2.94914 q^{89} -9.80642 q^{91} +3.62714 q^{92} +4.99063 q^{93} -15.4336 q^{94} +8.38271 q^{96} +18.5763 q^{97} -5.67460 q^{98} +26.6178 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 2 q^{7} - 9 q^{8} + 3 q^{9} + 8 q^{11} + 2 q^{12} + 6 q^{13} - 12 q^{14} + 5 q^{16} + 13 q^{18} - 2 q^{22} - 14 q^{23} - 20 q^{24} + 10 q^{26} + 8 q^{27} + 22 q^{28}+ \cdots + 20 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 2 * q^3 + 5 * q^4 + 4 * q^6 + 2 * q^7 - 9 * q^8 + 3 * q^9 + 8 * q^11 + 2 * q^12 + 6 * q^13 - 12 * q^14 + 5 * q^16 + 13 * q^18 - 2 * q^22 - 14 * q^23 - 20 * q^24 + 10 * q^26 + 8 * q^27 + 22 * q^28 + 3 * q^29 + 12 * q^31 - 11 * q^32 + 16 * q^33 - 14 * q^34 - 15 * q^36 - 4 * q^37 + 2 * q^38 + 16 * q^39 - 10 * q^41 - 16 * q^42 + 10 * q^43 + 12 * q^44 + 8 * q^46 - 18 * q^47 + 10 * q^48 - q^49 + 8 * q^51 - 10 * q^52 - 10 * q^53 + 12 * q^54 - 32 * q^56 + 20 * q^57 - 3 * q^58 + 4 * q^59 + 6 * q^61 - 14 * q^62 - 18 * q^63 + 33 * q^64 + 12 * q^66 + 10 * q^67 + 36 * q^68 - 20 * q^69 + 24 * q^71 + 3 * q^72 + 4 * q^73 - 22 * q^74 + 16 * q^76 + 4 * q^77 + 28 * q^78 + 8 * q^79 - q^81 + 14 * q^82 + 2 * q^83 + 20 * q^84 - 16 * q^86 + 2 * q^87 - 38 * q^88 + 22 * q^89 - 16 * q^91 - 22 * q^92 - 12 * q^93 - 8 * q^96 + 36 * q^97 - 23 * q^98 + 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	$-0.358751\pi$
$2$	1.21432	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$3$	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
$5$	0	0
$5$	0	0
$6$	3.52543	1.43925
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	−3.06668	−1.08423
$9$	5.42864	1.80955
$10$	0	0
$11$	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	$0.149659\pi$
$14$	−1.85236	−0.495063
$15$	0	0
$16$	−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	$-0.415734\pi$
$19$	2.28100	0.523296	0.261648	+	0.965163i	$0.415734\pi$
$20$	0	0
$21$	−4.42864	−0.966408
$22$	5.95407	1.26941
$23$	−6.90321	−1.43942	−0.719710	−	0.694275i	$-0.755725\pi$
$23$	−6.90321	−1.43942	−0.719710	+	0.694275i	$0.755725\pi$
$24$	−8.90321	−1.81736
$25$	0	0
$26$	7.80642	1.53097
$27$	7.05086	1.35694
$28$	0.801502	0.151470
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.71900	0.308742	0.154371	−	0.988013i	$-0.450665\pi$
$31$	1.71900	0.308742	0.154371	+	0.988013i	$0.450665\pi$
$32$	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	$-0.726835\pi$
$37$	−7.95407	−1.30764	−0.653820	+	0.756650i	$0.726835\pi$
$38$	2.76986	0.449330
$39$	18.6637	2.98858
$40$	0	0
$41$	−3.37778	−0.527521	−0.263761	−	0.964588i	$-0.584963\pi$
$41$	−3.37778	−0.527521	−0.263761	+	0.964588i	$0.584963\pi$
$42$	−5.37778	−0.829810
$43$	1.09679	0.167259	0.0836293	−	0.996497i	$-0.473349\pi$
$43$	1.09679	0.167259	0.0836293	+	0.996497i	$0.473349\pi$
$44$	−2.57628	−0.388389
$45$	0	0
$46$	−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
$49$	−4.67307	−0.667582
$50$	0	0
$51$	−6.23506	−0.873084
$52$	−3.37778	−0.468414
$53$	−3.37778	−0.463974	−0.231987	−	0.972719i	$-0.574523\pi$
$53$	−3.37778	−0.463974	−0.231987	+	0.972719i	$0.574523\pi$
$54$	8.56199	1.16514
$55$	0	0
$56$	4.67799	0.625123
$57$	6.62222	0.877134
$58$	1.21432	0.159448
$59$	−3.18421	−0.414549	−0.207274	−	0.978283i	$-0.566459\pi$
$59$	−3.18421	−0.414549	−0.207274	+	0.978283i	$0.566459\pi$
$60$	0	0
$61$	−2.42864	−0.310955	−0.155478	−	0.987839i	$-0.549692\pi$
$61$	−2.42864	−0.310955	−0.155478	+	0.987839i	$0.549692\pi$
$62$	2.08742	0.265103
$63$	−8.28100	−1.04331
$64$	8.85236	1.10654
$65$	0	0
$66$	17.2859	2.12775
$67$	1.09679	0.133994	0.0669970	−	0.997753i	$-0.478658\pi$
$67$	1.09679	0.133994	0.0669970	+	0.997753i	$0.478658\pi$
$68$	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
$75$	0	0
$76$	−1.19850	−0.137477
$77$	−7.47949	−0.852368
$78$	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	$0.493890\pi$
$80$	0	0
$81$	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	$0.368180\pi$
$84$	2.32693	0.253889
$85$	0	0
$86$	1.33185	0.143617
$87$	2.90321	0.311257
$88$	−15.0366	−1.60290
$89$	2.94914	0.312609	0.156304	−	0.987709i	$-0.450042\pi$
$89$	2.94914	0.312609	0.156304	+	0.987709i	$0.450042\pi$
$90$	0	0
$91$	−9.80642	−1.02799
$92$	3.62714	0.378155
$93$	4.99063	0.517504
$94$	−15.4336	−1.59185
$95$	0	0
$96$	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	$0.107926\pi$
$98$	−5.67460	−0.573221
$99$	26.6178	2.67519
$100$	0	0
$101$	15.4193	1.53427	0.767137	−	0.641483i	$-0.221680\pi$
$101$	15.4193	1.53427	0.767137	+	0.641483i	$0.221680\pi$
$102$	−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	$-0.453110\pi$
$107$	3.03657	0.293556	0.146778	+	0.989169i	$0.453110\pi$
$108$	−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	$0.624161\pi$
$110$	0	0
$111$	−23.0923	−2.19183
$112$	4.07758	0.385295
$113$	−7.82071	−0.735711	−0.367855	−	0.929883i	$-0.619908\pi$
$113$	−7.82071	−0.735711	−0.367855	+	0.929883i	$0.619908\pi$
$114$	8.04149	0.753154
$115$	0	0
$116$	−0.525428	−0.0487847
$117$	34.8988	3.22639
$118$	−3.86665	−0.355954
$119$	3.27607	0.300317
$120$	0	0
$121$	13.0415	1.18559
$122$	−2.94914	−0.267003
$123$	−9.80642	−0.884215
$124$	−0.903212	−0.0811108
$125$	0	0
$126$	−10.0558	−0.895840
$127$	12.4429	1.10413	0.552066	−	0.833801i	$-0.313840\pi$
$127$	12.4429	1.10413	0.552066	+	0.833801i	$0.313840\pi$
$128$	4.97481	0.439715
$129$	3.18421	0.280354
$130$	0	0
$131$	4.08742	0.357120	0.178560	−	0.983929i	$-0.442856\pi$
$131$	4.08742	0.357120	0.178560	+	0.983929i	$0.442856\pi$
$132$	−7.47949	−0.651006
$133$	−3.47949	−0.301710
$134$	1.33185	0.115054
$135$	0	0
$136$	6.58613	0.564756
$137$	19.9541	1.70479	0.852395	−	0.522898i	$-0.175149\pi$
$137$	19.9541	1.70479	0.852395	+	0.522898i	$0.175149\pi$
$138$	−24.3368	−2.07168
$139$	7.90813	0.670759	0.335380	−	0.942083i	$-0.391135\pi$
$139$	7.90813	0.670759	0.335380	+	0.942083i	$0.391135\pi$
$140$	0	0
$141$	−36.8988	−3.10744
$142$	4.33677	0.363934
$143$	31.5210	2.63592
$144$	−14.5111	−1.20926
$145$	0	0
$146$	−17.2301	−1.42598
$147$	−13.5669	−1.11898
$148$	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	$0.729254\pi$
$150$	0	0
$151$	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
$154$	−9.08250	−0.731889
$155$	0	0
$156$	−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
$160$	0	0
$161$	10.5303	0.829908
$162$	5.08097	0.399198
$163$	5.95407	0.466359	0.233179	−	0.972434i	$-0.425087\pi$
$163$	5.95407	0.466359	0.233179	+	0.972434i	$0.425087\pi$
$164$	1.77478	0.138587
$165$	0	0
$166$	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	$0.552450\pi$
$168$	13.5812	1.04781
$169$	28.3274	2.17903
$170$	0	0
$171$	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	$0.543109\pi$
$180$	0	0
$181$	18.0415	1.34101	0.670507	−	0.741904i	$-0.266077\pi$
$181$	18.0415	1.34101	0.670507	+	0.741904i	$0.266077\pi$
$182$	−11.9081	−0.882690
$183$	−7.05086	−0.521214
$184$	21.1699	1.56067
$185$	0	0
$186$	6.06022	0.444357
$187$	−10.5303	−0.770055
$188$	6.67799	0.487043
$189$	−10.7556	−0.782353
$190$	0	0
$191$	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	$-0.525684\pi$
$193$	−2.23951	−0.161203	−0.0806017	+	0.996746i	$0.525684\pi$
$194$	22.5575	1.61954
$195$	0	0
$196$	2.45536	0.175383
$197$	10.5620	0.752511	0.376255	−	0.926516i	$-0.377212\pi$
$197$	10.5620	0.752511	0.376255	+	0.926516i	$0.377212\pi$
$198$	32.3225	2.29706
$199$	3.18421	0.225723	0.112861	−	0.993611i	$-0.463998\pi$
$199$	3.18421	0.225723	0.112861	+	0.993611i	$0.463998\pi$
$200$	0	0
$201$	3.18421	0.224597
$202$	18.7239	1.31741
$203$	−1.52543	−0.107064
$204$	3.27607	0.229371
$205$	0	0
$206$	−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
$215$	0	0
$216$	−21.6227	−1.47124
$217$	−2.62222	−0.178008
$218$	−9.64143	−0.653000
$219$	−41.1941	−2.78364
$220$	0	0
$221$	−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	$0.587951\pi$
$224$	−4.40451	−0.294288
$225$	0	0
$226$	−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	$-0.409506\pi$
$229$	8.48886	0.560960	0.280480	+	0.959860i	$0.409506\pi$
$230$	0	0
$231$	−21.7146	−1.42871
$232$	−3.06668	−0.201337
$233$	20.5718	1.34771	0.673853	−	0.738866i	$-0.264638\pi$
$233$	20.5718	1.34771	0.673853	+	0.738866i	$0.264638\pi$
$234$	42.3783	2.77035
$235$	0	0
$236$	1.67307	0.108908
$237$	0.990632	0.0643485
$238$	3.97820	0.257869
$239$	0.815792	0.0527692	0.0263846	−	0.999652i	$-0.491601\pi$
$239$	0.815792	0.0527692	0.0263846	+	0.999652i	$0.491601\pi$
$240$	0	0
$241$	−7.24443	−0.466655	−0.233327	−	0.972398i	$-0.574961\pi$
$241$	−7.24443	−0.466655	−0.233327	+	0.972398i	$0.574961\pi$
$242$	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
$250$	0	0
$251$	−20.4242	−1.28916	−0.644582	−	0.764535i	$-0.722969\pi$
$251$	−20.4242	−1.28916	−0.644582	+	0.764535i	$0.722969\pi$
$252$	4.35106	0.274091
$253$	−33.8479	−2.12800
$254$	15.1097	0.948067
$255$	0	0
$256$	−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
$259$	12.1334	0.753930
$260$	0	0
$261$	5.42864	0.336024
$262$	4.96343	0.306642
$263$	−20.0558	−1.23669	−0.618346	−	0.785906i	$-0.712197\pi$
$263$	−20.0558	−1.23669	−0.618346	+	0.785906i	$0.712197\pi$
$264$	−43.6543	−2.68674
$265$	0	0
$266$	−4.22522	−0.259065
$267$	8.56199	0.523985
$268$	−0.576283	−0.0352021
$269$	−23.4608	−1.43043	−0.715214	−	0.698906i	$-0.753671\pi$
$269$	−23.4608	−1.43043	−0.715214	+	0.698906i	$0.753671\pi$
$270$	0	0
$271$	−21.9353	−1.33248	−0.666238	−	0.745739i	$-0.732097\pi$
$271$	−21.9353	−1.33248	−0.666238	+	0.745739i	$0.732097\pi$
$272$	5.74080	0.348087
$273$	−28.4701	−1.72309
$274$	24.2306	1.46383
$275$	0	0
$276$	10.5303	0.633853
$277$	−18.3368	−1.10175	−0.550875	−	0.834588i	$-0.685706\pi$
$277$	−18.3368	−1.10175	−0.550875	+	0.834588i	$0.685706\pi$
$278$	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
$285$	0	0
$286$	38.2766	2.26334
$287$	5.15257	0.304146
$288$	15.6746	0.923635
$289$	−12.3876	−0.728684
$290$	0	0
$291$	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	$0.573091\pi$
$294$	−16.4746	−0.960817
$295$	0	0
$296$	24.3926	1.41779
$297$	34.5718	2.00606
$298$	−19.5526	−1.13265
$299$	−44.3783	−2.56646
$300$	0	0
$301$	−1.67307	−0.0964342
$302$	6.88892	0.396413
$303$	44.7654	2.57171
$304$	−6.09726	−0.349702
$305$	0	0
$306$	−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	$0.554084\pi$
$308$	3.92993	0.223929
$309$	−22.5303	−1.28171
$310$	0	0
$311$	8.94470	0.507207	0.253604	−	0.967308i	$-0.418384\pi$
$311$	8.94470	0.507207	0.253604	+	0.967308i	$0.418384\pi$
$312$	−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	$-0.285385\pi$
$317$	22.2306	1.24860	0.624298	+	0.781186i	$0.285385\pi$
$318$	−11.9081	−0.667775
$319$	4.90321	0.274527
$320$	0	0
$321$	8.81579	0.492050
$322$	12.7872	0.712603
$323$	−4.89877	−0.272575
$324$	−2.19850	−0.122139
$325$	0	0
$326$	7.23014	0.400440
$327$	−23.0509	−1.27472
$328$	10.3586	0.571956
$329$	19.3876	1.06887
$330$	0	0
$331$	−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
$335$	0	0
$336$	11.8381	0.645819
$337$	−2.66815	−0.145343	−0.0726717	−	0.997356i	$-0.523153\pi$
$337$	−2.66815	−0.145343	−0.0726717	+	0.997356i	$0.523153\pi$
$338$	34.3985	1.87103
$339$	−22.7052	−1.23318
$340$	0	0
$341$	8.42864	0.456436
$342$	15.0366	0.813084
$343$	17.8064	0.961457
$344$	−3.36349	−0.181347
$345$	0	0
$346$	−29.6414	−1.59353
$347$	−14.6780	−0.787956	−0.393978	−	0.919120i	$-0.628901\pi$
$347$	−14.6780	−0.787956	−0.393978	+	0.919120i	$0.628901\pi$
$348$	−1.52543	−0.0817715
$349$	−11.1240	−0.595453	−0.297727	−	0.954651i	$-0.596228\pi$
$349$	−11.1240	−0.595453	−0.297727	+	0.954651i	$0.596228\pi$
$350$	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
$355$	0	0
$356$	−1.54956	−0.0821266
$357$	9.51114	0.503383
$358$	−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	$0.257124\pi$
$360$	0	0
$361$	−13.7971	−0.726161
$362$	21.9081	1.15147
$363$	37.8622	1.98725
$364$	5.15257	0.270068
$365$	0	0
$366$	−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
$370$	0	0
$371$	5.15257	0.267508
$372$	−2.62222	−0.135956
$373$	24.2766	1.25699	0.628496	−	0.777813i	$-0.283671\pi$
$373$	24.2766	1.25699	0.628496	+	0.777813i	$0.283671\pi$
$374$	−12.7872	−0.661211
$375$	0	0
$376$	38.9763	2.01005
$377$	6.42864	0.331092
$378$	−13.0607	−0.671770
$379$	29.7605	1.52869	0.764347	−	0.644805i	$-0.223062\pi$
$379$	29.7605	1.52869	0.764347	+	0.644805i	$0.223062\pi$
$380$	0	0
$381$	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	$0.552860\pi$
$390$	0	0
$391$	14.8256	0.749765
$392$	14.3308	0.723815
$393$	11.8666	0.598593
$394$	12.8256	0.646146
$395$	0	0
$396$	−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
$399$	−10.1017	−0.505718
$400$	0	0
$401$	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
$405$	0	0
$406$	−1.85236	−0.0919309
$407$	−39.0005	−1.93318
$408$	19.1209	0.946627
$409$	−11.0825	−0.547994	−0.273997	−	0.961731i	$-0.588346\pi$
$409$	−11.0825	−0.547994	−0.273997	+	0.961731i	$0.588346\pi$
$410$	0	0
$411$	57.9309	2.85752
$412$	4.07758	0.200888
$413$	4.85728	0.239011
$414$	−45.5067	−2.23653
$415$	0	0
$416$	18.5620	0.910077
$417$	22.9590	1.12431
$418$	13.5812	0.664278
$419$	30.9719	1.51308	0.756538	−	0.653950i	$-0.226889\pi$
$419$	30.9719	1.51308	0.756538	+	0.653950i	$0.226889\pi$
$420$	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
$425$	0	0
$426$	12.5906	0.610015
$427$	3.70471	0.179284
$428$	−1.59549	−0.0771212
$429$	91.5121	4.41825
$430$	0	0
$431$	−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	$0.686430\pi$
$434$	−3.18421	−0.152847
$435$	0	0
$436$	4.17178	0.199792
$437$	−15.7462	−0.753243
$438$	−50.0228	−2.39018
$439$	11.7462	0.560616	0.280308	−	0.959910i	$-0.409563\pi$
$439$	11.7462	0.560616	0.280308	+	0.959910i	$0.409563\pi$
$440$	0	0
$441$	−25.3684	−1.20802
$442$	−16.7654	−0.797449
$443$	−17.6874	−0.840352	−0.420176	−	0.907443i	$-0.638032\pi$
$443$	−17.6874	−0.840352	−0.420176	+	0.907443i	$0.638032\pi$
$444$	12.1334	0.575823
$445$	0	0
$446$	−9.89384	−0.468487
$447$	−46.7467	−2.21104
$448$	−13.5036	−0.637987
$449$	1.57136	0.0741571	0.0370785	−	0.999312i	$-0.488195\pi$
$449$	1.57136	0.0741571	0.0370785	+	0.999312i	$0.488195\pi$
$450$	0	0
$451$	−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	$-0.511017\pi$
$457$	−1.47949	−0.0692078	−0.0346039	+	0.999401i	$0.511017\pi$
$458$	10.3082	0.481670
$459$	−15.1427	−0.706802
$460$	0	0
$461$	−41.2543	−1.92140	−0.960702	−	0.277583i	$-0.910467\pi$
$461$	−41.2543	−1.92140	−0.960702	+	0.277583i	$0.910467\pi$
$462$	−26.3684	−1.22677
$463$	34.4242	1.59983	0.799914	−	0.600115i	$-0.204878\pi$
$463$	34.4242	1.59983	0.799914	+	0.600115i	$0.204878\pi$
$464$	−2.67307	−0.124094
$465$	0	0
$466$	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
$469$	−1.67307	−0.0772552
$470$	0	0
$471$	−5.49823	−0.253345
$472$	9.76494	0.449468
$473$	5.37778	0.247271
$474$	1.20294	0.0552531
$475$	0	0
$476$	−1.72134	−0.0788975
$477$	−18.3368	−0.839583
$478$	0.990632	0.0453105
$479$	−18.9763	−0.867051	−0.433526	−	0.901141i	$-0.642731\pi$
$479$	−18.9763	−0.867051	−0.433526	+	0.901141i	$0.642731\pi$
$480$	0	0
$481$	−51.1338	−2.33150
$482$	−8.79706	−0.400695
$483$	30.5718	1.39107
$484$	−6.85236	−0.311471
$485$	0	0
$486$	−10.9349	−0.496015
$487$	32.3926	1.46785	0.733923	−	0.679232i	$-0.237687\pi$
$487$	32.3926	1.46785	0.733923	+	0.679232i	$0.237687\pi$
$488$	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
$495$	0	0
$496$	−4.59502	−0.206322
$497$	−5.44785	−0.244370
$498$	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	0
$501$	−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	$0.334792\pi$
$504$	25.3951	1.13119
$505$	0	0
$506$	−41.1022	−1.82722
$507$	82.2405	3.65243
$508$	−6.53786	−0.290071
$509$	−9.18421	−0.407083	−0.203541	−	0.979066i	$-0.565245\pi$
$509$	−9.18421	−0.407083	−0.203541	+	0.979066i	$0.565245\pi$
$510$	0	0
$511$	21.6445	0.957496
$512$	−24.1131	−1.06566
$513$	16.0830	0.710081
$514$	−3.74314	−0.165103
$515$	0	0
$516$	−1.67307	−0.0736528
$517$	−62.3180	−2.74074
$518$	14.7338	0.647365
$519$	−70.8671	−3.11072
$520$	0	0
$521$	7.01921	0.307517	0.153759	−	0.988108i	$-0.450862\pi$
$521$	7.01921	0.307517	0.153759	+	0.988108i	$0.450862\pi$
$522$	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
$530$	0	0
$531$	−17.2859	−0.750145
$532$	1.82822	0.0792635
$533$	−21.7146	−0.940562
$534$	10.3970	0.449922
$535$	0	0
$536$	−3.36349	−0.145281
$537$	−10.4889	−0.452628
$538$	−28.4889	−1.22824
$539$	−22.9131	−0.986935
$540$	0	0
$541$	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
$545$	0	0
$546$	−34.5718	−1.47954
$547$	19.4237	0.830498	0.415249	−	0.909708i	$-0.363694\pi$
$547$	19.4237	0.830498	0.415249	+	0.909708i	$0.363694\pi$
$548$	−10.4844	−0.447872
$549$	−13.1842	−0.562688
$550$	0	0
$551$	2.28100	0.0971737
$552$	61.4608	2.61594
$553$	−0.520505	−0.0221341
$554$	−22.2667	−0.946022
$555$	0	0
$556$	−4.15515	−0.176218
$557$	−30.3497	−1.28596	−0.642979	−	0.765884i	$-0.722302\pi$
$557$	−30.3497	−1.28596	−0.642979	+	0.765884i	$0.722302\pi$
$558$	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
$565$	0	0
$566$	35.2212	1.48046
$567$	−6.38271	−0.268048
$568$	−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	$0.527123\pi$
$570$	0	0
$571$	−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
$574$	6.25686	0.261156
$575$	0	0
$576$	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	0
$581$	−11.1842	−0.463999
$582$	65.4893	2.71462
$583$	−16.5620	−0.685928
$584$	43.5135	1.80060
$585$	0	0
$586$	−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	$0.323081\pi$
$588$	7.12843	0.293972
$589$	3.92104	0.161564
$590$	0	0
$591$	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	$0.224395\pi$
$600$	0	0
$601$	−29.9496	−1.22167	−0.610835	−	0.791758i	$-0.709166\pi$
$601$	−29.9496	−1.22167	−0.610835	+	0.791758i	$0.709166\pi$
$602$	−2.03164	−0.0828036
$603$	5.95407	0.242468
$604$	−2.98079	−0.121287
$605$	0	0
$606$	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
$609$	−4.42864	−0.179458
$610$	0	0
$611$	−81.7057	−3.30546
$612$	6.12584	0.247623
$613$	2.65386	0.107188	0.0535942	−	0.998563i	$-0.482932\pi$
$613$	2.65386	0.107188	0.0535942	+	0.998563i	$0.482932\pi$
$614$	−7.19544	−0.290384
$615$	0	0
$616$	22.9372	0.924166
$617$	18.3096	0.737116	0.368558	−	0.929605i	$-0.379852\pi$
$617$	18.3096	0.737116	0.368558	+	0.929605i	$0.379852\pi$
$618$	−27.3590	−1.10054
$619$	12.8113	0.514931	0.257466	−	0.966287i	$-0.417113\pi$
$619$	12.8113	0.514931	0.257466	+	0.966287i	$0.417113\pi$
$620$	0	0
$621$	−48.6735	−1.95320
$622$	10.8617	0.435515
$623$	−4.49871	−0.180237
$624$	−49.8894	−1.99717
$625$	0	0
$626$	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
$635$	0	0
$636$	5.15257	0.204313
$637$	−30.0415	−1.19029
$638$	5.95407	0.235724
$639$	19.3876	0.766963
$640$	0	0
$641$	−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	$0.210465\pi$
$644$	−5.53294	−0.218028
$645$	0	0
$646$	−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
$650$	0	0
$651$	−7.61285	−0.298371
$652$	−3.12843	−0.122519
$653$	−22.8430	−0.893915	−0.446958	−	0.894555i	$-0.647493\pi$
$653$	−22.8430	−0.893915	−0.446958	+	0.894555i	$0.647493\pi$
$654$	−27.9911	−1.09454
$655$	0	0
$656$	9.02906	0.352525
$657$	−77.0277	−3.00514
$658$	23.5428	0.917793
$659$	15.0178	0.585012	0.292506	−	0.956264i	$-0.405511\pi$
$659$	15.0178	0.585012	0.292506	+	0.956264i	$0.405511\pi$
$660$	0	0
$661$	−4.65080	−0.180895	−0.0904475	−	0.995901i	$-0.528830\pi$
$661$	−4.65080	−0.180895	−0.0904475	+	0.995901i	$0.528830\pi$
$662$	−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
$670$	0	0
$671$	−11.9081	−0.459708
$672$	−12.7872	−0.493277
$673$	44.8671	1.72950	0.864750	−	0.502202i	$-0.167477\pi$
$673$	44.8671	1.72950	0.864750	+	0.502202i	$0.167477\pi$
$674$	−3.23999	−0.124800
$675$	0	0
$676$	−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
$679$	−28.3368	−1.08747
$680$	0	0
$681$	−30.5303	−1.16993
$682$	10.2351	0.391921
$683$	−12.0558	−0.461301	−0.230651	−	0.973037i	$-0.574085\pi$
$683$	−12.0558	−0.461301	−0.230651	+	0.973037i	$0.574085\pi$
$684$	−6.50622	−0.248771
$685$	0	0
$686$	21.6227	0.825558
$687$	24.6450	0.940264
$688$	−2.93179	−0.111774
$689$	−21.7146	−0.827259
$690$	0	0
$691$	37.5812	1.42966	0.714828	−	0.699300i	$-0.246505\pi$
$691$	37.5812	1.42966	0.714828	+	0.699300i	$0.246505\pi$
$692$	12.8256	0.487558
$693$	−40.6035	−1.54240
$694$	−17.8238	−0.676581
$695$	0	0
$696$	−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
$705$	0	0
$706$	16.3684	0.616033
$707$	−23.5210	−0.884598
$708$	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	$0.706260\pi$
$710$	0	0
$711$	1.85236	0.0694688
$712$	−9.04407	−0.338941
$713$	−11.8666	−0.444409
$714$	11.5496	0.432231
$715$	0	0
$716$	1.89829	0.0709424
$717$	2.36842	0.0884501
$718$	31.8020	1.18684
$719$	1.01921	0.0380102	0.0190051	−	0.999819i	$-0.493950\pi$
$719$	1.01921	0.0380102	0.0190051	+	0.999819i	$0.493950\pi$
$720$	0	0
$721$	11.8381	0.440873
$722$	−16.7540	−0.623521
$723$	−21.0321	−0.782193
$724$	−9.47949	−0.352303
$725$	0	0
$726$	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	$0.352210\pi$
$728$	30.0731	1.11458
$729$	−38.6958	−1.43318
$730$	0	0
$731$	−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	$0.519409\pi$
$740$	0	0
$741$	42.5718	1.56392
$742$	6.25686	0.229697
$743$	−2.99508	−0.109879	−0.0549394	−	0.998490i	$-0.517497\pi$
$743$	−2.99508	−0.109879	−0.0549394	+	0.998490i	$0.517497\pi$
$744$	−15.3047	−0.561096
$745$	0	0
$746$	29.4795	1.07932
$747$	39.8020	1.45628
$748$	5.53294	0.202304
$749$	−4.63206	−0.169252
$750$	0	0
$751$	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
$755$	0	0
$756$	5.65127	0.205535
$757$	−18.5763	−0.675166	−0.337583	−	0.941296i	$-0.609609\pi$
$757$	−18.5763	−0.675166	−0.337583	+	0.941296i	$0.609609\pi$
$758$	36.1388	1.31262
$759$	−98.2677	−3.56689
$760$	0	0
$761$	2.59057	0.0939082	0.0469541	−	0.998897i	$-0.485049\pi$
$761$	2.59057	0.0939082	0.0469541	+	0.998897i	$0.485049\pi$
$762$	43.8666	1.58912
$763$	12.1116	0.438468
$764$	−5.17670	−0.187286
$765$	0	0
$766$	−28.9032	−1.04432
$767$	−20.4701	−0.739133
$768$	−33.8622	−1.22190
$769$	32.7467	1.18088	0.590438	−	0.807083i	$-0.298955\pi$
$769$	32.7467	1.18088	0.590438	+	0.807083i	$0.298955\pi$
$770$	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
$775$	0	0
$776$	−56.9675	−2.04501
$777$	35.2257	1.26371
$778$	−7.91798	−0.283873
$779$	−7.70471	−0.276050
$780$	0	0
$781$	17.5111	0.626598
$782$	18.0031	0.643788
$783$	7.05086	0.251977
$784$	12.4914	0.446123
$785$	0	0
$786$	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	0
$791$	11.9299	0.424180
$792$	−81.6281	−2.90053
$793$	−15.6128	−0.554428
$794$	17.9299	0.636309
$795$	0	0
$796$	−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	$0.301052\pi$
$798$	−12.2667	−0.434237
$799$	27.2958	0.965655
$800$	0	0
$801$	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	$0.691055\pi$
$810$	0	0
$811$	15.3176	0.537872	0.268936	−	0.963158i	$-0.413328\pi$
$811$	15.3176	0.537872	0.268936	+	0.963158i	$0.413328\pi$
$812$	0.801502	0.0281272
$813$	−63.6829	−2.23346
$814$	−47.3590	−1.65993
$815$	0	0
$816$	16.6668	0.583453
$817$	2.50177	0.0875258
$818$	−13.4577	−0.470537
$819$	−53.2355	−1.86020
$820$	0	0
$821$	−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	$-0.286659\pi$
$823$	35.6400	1.24233	0.621167	+	0.783678i	$0.286659\pi$
$824$	23.7989	0.829075
$825$	0	0
$826$	5.89829	0.205228
$827$	−4.70964	−0.163770	−0.0818850	−	0.996642i	$-0.526094\pi$
$827$	−4.70964	−0.163770	−0.0818850	+	0.996642i	$0.526094\pi$
$828$	19.6904	0.684290
$829$	2.25380	0.0782777	0.0391388	−	0.999234i	$-0.487539\pi$
$829$	2.25380	0.0782777	0.0391388	+	0.999234i	$0.487539\pi$
$830$	0	0
$831$	−53.2355	−1.84672
$832$	56.9086	1.97295
$833$	10.0361	0.347730
$834$	27.8796	0.965390
$835$	0	0
$836$	−5.87649	−0.203243
$837$	12.1204	0.418944
$838$	37.6098	1.29921
$839$	−8.42419	−0.290835	−0.145418	−	0.989370i	$-0.546453\pi$
$839$	−8.42419	−0.290835	−0.145418	+	0.989370i	$0.546453\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	27.8064	0.958273
$843$	20.0286	0.689821
$844$	3.02672	0.104184
$845$	0	0
$846$	−83.7832	−2.88053
$847$	−19.8938	−0.683561
$848$	9.02906	0.310059
$849$	84.2074	2.88999
$850$	0	0
$851$	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	$0.847368\pi$
$854$	4.49871	0.153943
$855$	0	0
$856$	−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	$-0.390972\pi$
$859$	19.6874	0.671724	0.335862	+	0.941911i	$0.390972\pi$
$860$	0	0
$861$	14.9590	0.509801
$862$	−34.1017	−1.16151
$863$	−19.5986	−0.667143	−0.333571	−	0.942725i	$-0.608254\pi$
$863$	−19.5986	−0.667143	−0.333571	+	0.942725i	$0.608254\pi$
$864$	20.3586	0.692613
$865$	0	0
$866$	−27.9353	−0.949281
$867$	−35.9639	−1.22140
$868$	1.37778	0.0467650
$869$	1.67307	0.0567550
$870$	0	0
$871$	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	$-0.411617\pi$
$877$	16.2351	0.548219	0.274110	+	0.961698i	$0.411617\pi$
$878$	14.2636	0.481375
$879$	−22.6222	−0.763028
$880$	0	0
$881$	−20.7052	−0.697576	−0.348788	−	0.937202i	$-0.613407\pi$
$881$	−20.7052	−0.697576	−0.348788	+	0.937202i	$0.613407\pi$
$882$	−30.8054	−1.03727
$883$	−8.75112	−0.294499	−0.147249	−	0.989099i	$-0.547042\pi$
$883$	−8.75112	−0.294499	−0.147249	+	0.989099i	$0.547042\pi$
$884$	7.25428	0.243988
$885$	0	0
$886$	−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
$889$	−18.9808	−0.636595
$890$	0	0
$891$	20.5161	0.687314
$892$	4.28100	0.143338
$893$	−28.9906	−0.970135
$894$	−56.7654	−1.89852
$895$	0	0
$896$	−7.58871	−0.253521
$897$	−128.839	−4.30183
$898$	1.90813	0.0636753
$899$	1.71900	0.0573320
$900$	0	0
$901$	7.25428	0.241675
$902$	−20.1116	−0.669642
$903$	−4.85728	−0.161640
$904$	23.9836	0.797683
$905$	0	0
$906$	20.0000	0.664455
$907$	46.9862	1.56015	0.780075	−	0.625686i	$-0.215181\pi$
$907$	46.9862	1.56015	0.780075	+	0.625686i	$0.215181\pi$
$908$	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
$915$	0	0
$916$	−4.46028	−0.147372
$917$	−6.23506	−0.205900
$918$	−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	$0.625370\pi$
$920$	0	0
$921$	−17.2029	−0.566856
$922$	−50.0959	−1.64982
$923$	22.9590	0.755704
$924$	11.4094	0.375343
$925$	0	0
$926$	41.8020	1.37370
$927$	−42.1289	−1.38369
$928$	2.88739	0.0947832
$929$	44.7556	1.46838	0.734191	−	0.678943i	$-0.237562\pi$
$929$	44.7556	1.46838	0.734191	+	0.678943i	$0.237562\pi$
$930$	0	0
$931$	−10.6593	−0.349343
$932$	−10.8090	−0.354061
$933$	25.9684	0.850166
$934$	−18.4211	−0.602758
$935$	0	0
$936$	−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	$0.378954\pi$
$938$	−2.03164	−0.0663355
$939$	88.5817	2.89075
$940$	0	0
$941$	4.10171	0.133712	0.0668560	−	0.997763i	$-0.478703\pi$
$941$	4.10171	0.133712	0.0668560	+	0.997763i	$0.478703\pi$
$942$	−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	$-0.587025\pi$
$947$	−16.6178	−0.540005	−0.270002	+	0.962860i	$0.587025\pi$
$948$	−0.520505	−0.0169052
$949$	−91.2168	−2.96102
$950$	0	0
$951$	64.5402	2.09286
$952$	−10.0467	−0.325614
$953$	−2.85728	−0.0925563	−0.0462782	−	0.998929i	$-0.514736\pi$
$953$	−2.85728	−0.0925563	−0.0462782	+	0.998929i	$0.514736\pi$
$954$	−22.2667	−0.720911
$955$	0	0
$956$	−0.428639	−0.0138632
$957$	14.2351	0.460154
$958$	−23.0433	−0.744497
$959$	−30.4385	−0.982910
$960$	0	0
$961$	−28.0450	−0.904678
$962$	−62.0928	−2.00195
$963$	16.4844	0.531203
$964$	3.80642	0.122597
$965$	0	0
$966$	37.1240	1.19444
$967$	−16.8015	−0.540300	−0.270150	−	0.962818i	$-0.587073\pi$
$967$	−16.8015	−0.540300	−0.270150	+	0.962818i	$0.587073\pi$
$968$	−39.9940	−1.28546
$969$	−14.2222	−0.456881
$970$	0	0
$971$	38.3640	1.23116	0.615579	−	0.788075i	$-0.288922\pi$
$971$	38.3640	1.23116	0.615579	+	0.788075i	$0.288922\pi$
$972$	4.73143	0.151761
$973$	−12.0633	−0.386731
$974$	39.3349	1.26037
$975$	0	0
$976$	6.49193	0.207801
$977$	−31.6356	−1.01211	−0.506056	−	0.862500i	$-0.668897\pi$
$977$	−31.6356	−1.01211	−0.506056	+	0.862500i	$0.668897\pi$
$978$	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
$985$	0	0
$986$	−2.60793	−0.0830533
$987$	56.2864	1.79162
$988$	−7.70471	−0.245120
$989$	−7.57136	−0.240755
$990$	0	0
$991$	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
$994$	−6.61543	−0.209829
$995$	0	0
$996$	−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	725.2.a.d.1.3		3
3.2	odd	2		6525.2.a.bh.1.1		3
5.2	odd	4		725.2.b.d.349.4		6
5.3	odd	4		725.2.b.d.349.3		6
5.4	even	2		145.2.a.d.1.1	✓	3
15.14	odd	2		1305.2.a.o.1.3		3
20.19	odd	2		2320.2.a.s.1.3		3
35.34	odd	2		7105.2.a.p.1.1		3
40.19	odd	2		9280.2.a.bm.1.1		3
40.29	even	2		9280.2.a.bu.1.3		3
145.144	even	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	5.4	even	2
725.2.a.d.1.3		3	1.1	even	1	trivial
725.2.b.d.349.3		6	5.3	odd	4
725.2.b.d.349.4		6	5.2	odd	4
1305.2.a.o.1.3		3	15.14	odd	2
2320.2.a.s.1.3		3	20.19	odd	2
4205.2.a.e.1.3		3	145.144	even	2
6525.2.a.bh.1.1		3	3.2	odd	2
7105.2.a.p.1.1		3	35.34	odd	2
9280.2.a.bm.1.1		3	40.19	odd	2
9280.2.a.bu.1.3		3	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} +3.52543 q^{6} -1.52543 q^{7} -3.06668 q^{8} +5.42864 q^{9} +4.90321 q^{11} -1.52543 q^{12} +6.42864 q^{13} -1.85236 q^{14} -2.67307 q^{16} -2.14764 q^{17} +6.59210 q^{18} +2.28100 q^{19} -4.42864 q^{21} +5.95407 q^{22} -6.90321 q^{23} -8.90321 q^{24} +7.80642 q^{26} +7.05086 q^{27} +0.801502 q^{28} +1.00000 q^{29} +1.71900 q^{31} +2.88739 q^{32} +14.2351 q^{33} -2.60793 q^{34} -2.85236 q^{36} -7.95407 q^{37} +2.76986 q^{38} +18.6637 q^{39} -3.37778 q^{41} -5.37778 q^{42} +1.09679 q^{43} -2.57628 q^{44} -8.38271 q^{46} -12.7096 q^{47} -7.76049 q^{48} -4.67307 q^{49} -6.23506 q^{51} -3.37778 q^{52} -3.37778 q^{53} +8.56199 q^{54} +4.67799 q^{56} +6.62222 q^{57} +1.21432 q^{58} -3.18421 q^{59} -2.42864 q^{61} +2.08742 q^{62} -8.28100 q^{63} +8.85236 q^{64} +17.2859 q^{66} +1.09679 q^{67} +1.12843 q^{68} -20.0415 q^{69} +3.57136 q^{71} -16.6479 q^{72} -14.1891 q^{73} -9.65878 q^{74} -1.19850 q^{76} -7.47949 q^{77} +22.6637 q^{78} +0.341219 q^{79} +4.18421 q^{81} -4.10171 q^{82} +7.33185 q^{83} +2.32693 q^{84} +1.33185 q^{86} +2.90321 q^{87} -15.0366 q^{88} +2.94914 q^{89} -9.80642 q^{91} +3.62714 q^{92} +4.99063 q^{93} -15.4336 q^{94} +8.38271 q^{96} +18.5763 q^{97} -5.67460 q^{98} +26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 2 q^{7} - 9 q^{8} + 3 q^{9} + 8 q^{11} + 2 q^{12} + 6 q^{13} - 12 q^{14} + 5 q^{16} + 13 q^{18} - 2 q^{22} - 14 q^{23} - 20 q^{24} + 10 q^{26} + 8 q^{27} + 22 q^{28}+ \cdots + 20 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 2 * q^3 + 5 * q^4 + 4 * q^6 + 2 * q^7 - 9 * q^8 + 3 * q^9 + 8 * q^11 + 2 * q^12 + 6 * q^13 - 12 * q^14 + 5 * q^16 + 13 * q^18 - 2 * q^22 - 14 * q^23 - 20 * q^24 + 10 * q^26 + 8 * q^27 + 22 * q^28 + 3 * q^29 + 12 * q^31 - 11 * q^32 + 16 * q^33 - 14 * q^34 - 15 * q^36 - 4 * q^37 + 2 * q^38 + 16 * q^39 - 10 * q^41 - 16 * q^42 + 10 * q^43 + 12 * q^44 + 8 * q^46 - 18 * q^47 + 10 * q^48 - q^49 + 8 * q^51 - 10 * q^52 - 10 * q^53 + 12 * q^54 - 32 * q^56 + 20 * q^57 - 3 * q^58 + 4 * q^59 + 6 * q^61 - 14 * q^62 - 18 * q^63 + 33 * q^64 + 12 * q^66 + 10 * q^67 + 36 * q^68 - 20 * q^69 + 24 * q^71 + 3 * q^72 + 4 * q^73 - 22 * q^74 + 16 * q^76 + 4 * q^77 + 28 * q^78 + 8 * q^79 - q^81 + 14 * q^82 + 2 * q^83 + 20 * q^84 - 16 * q^86 + 2 * q^87 - 38 * q^88 + 22 * q^89 - 16 * q^91 - 22 * q^92 - 12 * q^93 - 8 * q^96 + 36 * q^97 - 23 * q^98 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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