LMFDB - Embedded newform 1525.2.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1525,2,Mod(1,1525)] 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("1525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1525 = 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-2,-6,8,0,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.1771863082$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 13x^{4} + 6x^{3} - 12x^{2} - 2x + 2 $ x^7 - 2x^6 - 7x^5 + 13x^4 + 6x^3 - 12x^2 - 2x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 305)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.833905$ of defining polynomial
Character	$\chi$	$=$	1525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.288620 q^{2} -3.12076 q^{3} -1.91670 q^{4} +0.900716 q^{6} -3.85131 q^{7} +1.13044 q^{8} +6.73917 q^{9} -6.15543 q^{11} +5.98156 q^{12} +6.12154 q^{13} +1.11157 q^{14} +3.50713 q^{16} -0.870521 q^{17} -1.94506 q^{18} +2.43225 q^{19} +12.0190 q^{21} +1.77658 q^{22} +2.39985 q^{23} -3.52783 q^{24} -1.76680 q^{26} -11.6691 q^{27} +7.38180 q^{28} +0.158802 q^{29} +5.36951 q^{31} -3.27311 q^{32} +19.2097 q^{33} +0.251250 q^{34} -12.9169 q^{36} +1.00949 q^{37} -0.701997 q^{38} -19.1039 q^{39} -6.84659 q^{41} -3.46894 q^{42} +11.0844 q^{43} +11.7981 q^{44} -0.692645 q^{46} +1.26117 q^{47} -10.9449 q^{48} +7.83258 q^{49} +2.71669 q^{51} -11.7331 q^{52} -3.61870 q^{53} +3.36793 q^{54} -4.35367 q^{56} -7.59047 q^{57} -0.0458335 q^{58} -2.50872 q^{59} -1.00000 q^{61} -1.54975 q^{62} -25.9546 q^{63} -6.06957 q^{64} -5.54430 q^{66} +3.34429 q^{67} +1.66853 q^{68} -7.48935 q^{69} -0.661011 q^{71} +7.61822 q^{72} -6.74081 q^{73} -0.291359 q^{74} -4.66189 q^{76} +23.7065 q^{77} +5.51377 q^{78} -4.75987 q^{79} +16.1989 q^{81} +1.97606 q^{82} +0.929595 q^{83} -23.0368 q^{84} -3.19919 q^{86} -0.495583 q^{87} -6.95834 q^{88} +7.87841 q^{89} -23.5759 q^{91} -4.59978 q^{92} -16.7570 q^{93} -0.364000 q^{94} +10.2146 q^{96} -7.49871 q^{97} -2.26064 q^{98} -41.4825 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} - 6 q^{3} + 8 q^{4} - 2 q^{6} - 8 q^{7} - 3 q^{8} + 5 q^{9} - 6 q^{11} - q^{12} - 5 q^{13} - 8 q^{14} + 6 q^{16} + 2 q^{17} + 11 q^{18} + 9 q^{19} + 4 q^{21} - 3 q^{22} - 5 q^{23} - 14 q^{24}+ \cdots - 35 q^{99}+O(q^{100}) $ 7 * q - 2 * q^2 - 6 * q^3 + 8 * q^4 - 2 * q^6 - 8 * q^7 - 3 * q^8 + 5 * q^9 - 6 * q^11 - q^12 - 5 * q^13 - 8 * q^14 + 6 * q^16 + 2 * q^17 + 11 * q^18 + 9 * q^19 + 4 * q^21 - 3 * q^22 - 5 * q^23 - 14 * q^24 - 13 * q^26 - 21 * q^27 + 7 * q^28 - 13 * q^29 + 17 * q^31 - 23 * q^32 + 15 * q^33 - 11 * q^34 - 18 * q^36 - q^37 + 8 * q^38 - 11 * q^39 - 8 * q^41 - 11 * q^42 - 5 * q^43 - 21 * q^44 + q^46 - 21 * q^47 + 25 * q^48 + q^49 - 5 * q^51 - 32 * q^52 + 9 * q^53 - 22 * q^54 - 22 * q^56 + 8 * q^57 + 25 * q^58 - 24 * q^59 - 7 * q^61 - 16 * q^62 - q^63 - 27 * q^64 + 9 * q^66 - 27 * q^67 + 39 * q^68 - 9 * q^69 - 13 * q^71 + 14 * q^72 - 4 * q^73 - 32 * q^74 + 23 * q^76 + 25 * q^77 + 54 * q^78 + q^79 + 15 * q^81 - 18 * q^82 - 4 * q^83 - 42 * q^84 - 14 * q^86 + 19 * q^87 + 23 * q^88 - 10 * q^89 - q^91 - 14 * q^92 - 36 * q^93 + q^94 + 11 * q^96 - 2 * q^97 + 39 * q^98 - 35 * 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 $

$2$	−0.288620	−0.204085	−0.102043	−	0.994780i	$-0.532538\pi$
$2$	−0.288620	−0.204085	−0.102043	+	0.994780i	$0.532538\pi$
$3$	−3.12076	−1.80177	−0.900887	−	0.434054i	$-0.857083\pi$
$3$	−3.12076	−1.80177	−0.900887	+	0.434054i	$0.857083\pi$
$4$	−1.91670	−0.958349
$5$	0	0
$5$	0	0
$6$	0.900716	0.367716
$7$	−3.85131	−1.45566	−0.727829	−	0.685759i	$-0.759471\pi$
$7$	−3.85131	−1.45566	−0.727829	+	0.685759i	$0.759471\pi$
$8$	1.13044	0.399671
$9$	6.73917	2.24639
$10$	0	0
$11$	−6.15543	−1.85593	−0.927966	−	0.372664i	$-0.878445\pi$
$11$	−6.15543	−1.85593	−0.927966	+	0.372664i	$0.878445\pi$
$12$	5.98156	1.72673
$13$	6.12154	1.69781	0.848905	−	0.528546i	$-0.177263\pi$
$13$	6.12154	1.69781	0.848905	+	0.528546i	$0.177263\pi$
$14$	1.11157	0.297079
$15$	0	0
$16$	3.50713	0.876782
$17$	−0.870521	−0.211132	−0.105566	−	0.994412i	$-0.533666\pi$
$17$	−0.870521	−0.211132	−0.105566	+	0.994412i	$0.533666\pi$
$18$	−1.94506	−0.458455
$19$	2.43225	0.557996	0.278998	−	0.960292i	$-0.409998\pi$
$19$	2.43225	0.557996	0.278998	+	0.960292i	$0.409998\pi$
$20$	0	0
$21$	12.0190	2.62277
$22$	1.77658	0.378769
$23$	2.39985	0.500403	0.250201	−	0.968194i	$-0.419503\pi$
$23$	2.39985	0.500403	0.250201	+	0.968194i	$0.419503\pi$
$24$	−3.52783	−0.720116
$25$	0	0
$26$	−1.76680	−0.346498
$27$	−11.6691	−2.24571
$28$	7.38180	1.39503
$29$	0.158802	0.0294888	0.0147444	−	0.999891i	$-0.495307\pi$
$29$	0.158802	0.0294888	0.0147444	+	0.999891i	$0.495307\pi$
$30$	0	0
$31$	5.36951	0.964393	0.482197	−	0.876063i	$-0.339839\pi$
$31$	5.36951	0.964393	0.482197	+	0.876063i	$0.339839\pi$
$32$	−3.27311	−0.578609
$33$	19.2097	3.34397
$34$	0.251250	0.0430891
$35$	0	0
$36$	−12.9169	−2.15282
$37$	1.00949	0.165959	0.0829796	−	0.996551i	$-0.473556\pi$
$37$	1.00949	0.165959	0.0829796	+	0.996551i	$0.473556\pi$
$38$	−0.701997	−0.113879
$39$	−19.1039	−3.05907
$40$	0	0
$41$	−6.84659	−1.06926	−0.534629	−	0.845087i	$-0.679548\pi$
$41$	−6.84659	−1.06926	−0.534629	+	0.845087i	$0.679548\pi$
$42$	−3.46894	−0.535268
$43$	11.0844	1.69036	0.845179	−	0.534484i	$-0.179494\pi$
$43$	11.0844	1.69036	0.845179	+	0.534484i	$0.179494\pi$
$44$	11.7981	1.77863
$45$	0	0
$46$	−0.692645	−0.102125
$47$	1.26117	0.183961	0.0919803	−	0.995761i	$-0.470680\pi$
$47$	1.26117	0.183961	0.0919803	+	0.995761i	$0.470680\pi$
$48$	−10.9449	−1.57976
$49$	7.83258	1.11894
$50$	0	0
$51$	2.71669	0.380413
$52$	−11.7331	−1.62709
$53$	−3.61870	−0.497066	−0.248533	−	0.968623i	$-0.579948\pi$
$53$	−3.61870	−0.497066	−0.248533	+	0.968623i	$0.579948\pi$
$54$	3.36793	0.458317
$55$	0	0
$56$	−4.35367	−0.581784
$57$	−7.59047	−1.00538
$58$	−0.0458335	−0.00601823
$59$	−2.50872	−0.326607	−0.163303	−	0.986576i	$-0.552215\pi$
$59$	−2.50872	−0.326607	−0.163303	+	0.986576i	$0.552215\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−1.54975	−0.196819
$63$	−25.9546	−3.26997
$64$	−6.06957	−0.758696
$65$	0	0
$66$	−5.54430	−0.682456
$67$	3.34429	0.408570	0.204285	−	0.978911i	$-0.434513\pi$
$67$	3.34429	0.408570	0.204285	+	0.978911i	$0.434513\pi$
$68$	1.66853	0.202339
$69$	−7.48935	−0.901612
$70$	0	0
$71$	−0.661011	−0.0784476	−0.0392238	−	0.999230i	$-0.512489\pi$
$71$	−0.661011	−0.0784476	−0.0392238	+	0.999230i	$0.512489\pi$
$72$	7.61822	0.897815
$73$	−6.74081	−0.788952	−0.394476	−	0.918906i	$-0.629074\pi$
$73$	−6.74081	−0.788952	−0.394476	+	0.918906i	$0.629074\pi$
$74$	−0.291359	−0.0338698
$75$	0	0
$76$	−4.66189	−0.534755
$77$	23.7065	2.70160
$78$	5.51377	0.624311
$79$	−4.75987	−0.535527	−0.267764	−	0.963485i	$-0.586285\pi$
$79$	−4.75987	−0.535527	−0.267764	+	0.963485i	$0.586285\pi$
$80$	0	0
$81$	16.1989	1.79987
$82$	1.97606	0.218220
$83$	0.929595	0.102036	0.0510182	−	0.998698i	$-0.483753\pi$
$83$	0.929595	0.102036	0.0510182	+	0.998698i	$0.483753\pi$
$84$	−23.0368	−2.51353
$85$	0	0
$86$	−3.19919	−0.344977
$87$	−0.495583	−0.0531321
$88$	−6.95834	−0.741762
$89$	7.87841	0.835110	0.417555	−	0.908652i	$-0.362887\pi$
$89$	7.87841	0.835110	0.417555	+	0.908652i	$0.362887\pi$
$90$	0	0
$91$	−23.5759	−2.47143
$92$	−4.59978	−0.479560
$93$	−16.7570	−1.73762
$94$	−0.364000	−0.0375437
$95$	0	0
$96$	10.2146	1.04252
$97$	−7.49871	−0.761378	−0.380689	−	0.924703i	$-0.624313\pi$
$97$	−7.49871	−0.761378	−0.380689	+	0.924703i	$0.624313\pi$
$98$	−2.26064	−0.228359
$99$	−41.4825	−4.16915
$100$	0	0
$101$	−5.68884	−0.566061	−0.283031	−	0.959111i	$-0.591340\pi$
$101$	−5.68884	−0.566061	−0.283031	+	0.959111i	$0.591340\pi$
$102$	−0.784093	−0.0776368
$103$	−10.5897	−1.04343	−0.521717	−	0.853119i	$-0.674708\pi$
$103$	−10.5897	−1.04343	−0.521717	+	0.853119i	$0.674708\pi$
$104$	6.92003	0.678564
$105$	0	0
$106$	1.04443	0.101444
$107$	−13.1815	−1.27431	−0.637154	−	0.770737i	$-0.719888\pi$
$107$	−13.1815	−1.27431	−0.637154	+	0.770737i	$0.719888\pi$
$108$	22.3661	2.15217
$109$	15.5260	1.48712	0.743561	−	0.668668i	$-0.233135\pi$
$109$	15.5260	1.48712	0.743561	+	0.668668i	$0.233135\pi$
$110$	0	0
$111$	−3.15038	−0.299021
$112$	−13.5070	−1.27629
$113$	4.59129	0.431913	0.215956	−	0.976403i	$-0.430713\pi$
$113$	4.59129	0.431913	0.215956	+	0.976403i	$0.430713\pi$
$114$	2.19077	0.205184
$115$	0	0
$116$	−0.304375	−0.0282605
$117$	41.2541	3.81394
$118$	0.724067	0.0666557
$119$	3.35265	0.307337
$120$	0	0
$121$	26.8894	2.44449
$122$	0.288620	0.0261305
$123$	21.3666	1.92656
$124$	−10.2917	−0.924225
$125$	0	0
$126$	7.49103	0.667354
$127$	−9.51879	−0.844656	−0.422328	−	0.906443i	$-0.638787\pi$
$127$	−9.51879	−0.844656	−0.422328	+	0.906443i	$0.638787\pi$
$128$	8.29802	0.733448
$129$	−34.5918	−3.04564
$130$	0	0
$131$	−5.52114	−0.482384	−0.241192	−	0.970477i	$-0.577538\pi$
$131$	−5.52114	−0.482384	−0.241192	+	0.970477i	$0.577538\pi$
$132$	−36.8191	−3.20469
$133$	−9.36734	−0.812252
$134$	−0.965231	−0.0833832
$135$	0	0
$136$	−0.984072	−0.0843834
$137$	12.7676	1.09081	0.545404	−	0.838174i	$-0.316376\pi$
$137$	12.7676	1.09081	0.545404	+	0.838174i	$0.316376\pi$
$138$	2.16158	0.184006
$139$	−10.7078	−0.908222	−0.454111	−	0.890945i	$-0.650043\pi$
$139$	−10.7078	−0.908222	−0.454111	+	0.890945i	$0.650043\pi$
$140$	0	0
$141$	−3.93582	−0.331456
$142$	0.190781	0.0160100
$143$	−37.6807	−3.15102
$144$	23.6351	1.96959
$145$	0	0
$146$	1.94554	0.161014
$147$	−24.4436	−2.01608
$148$	−1.93489	−0.159047
$149$	−4.32349	−0.354194	−0.177097	−	0.984193i	$-0.556671\pi$
$149$	−4.32349	−0.354194	−0.177097	+	0.984193i	$0.556671\pi$
$150$	0	0
$151$	23.4446	1.90790	0.953948	−	0.299972i	$-0.0969773\pi$
$151$	23.4446	1.90790	0.953948	+	0.299972i	$0.0969773\pi$
$152$	2.74951	0.223015
$153$	−5.86659	−0.474286
$154$	−6.84217	−0.551358
$155$	0	0
$156$	36.6164	2.93166
$157$	−12.4175	−0.991024	−0.495512	−	0.868601i	$-0.665020\pi$
$157$	−12.4175	−0.991024	−0.495512	+	0.868601i	$0.665020\pi$
$158$	1.37380	0.109293
$159$	11.2931	0.895600
$160$	0	0
$161$	−9.24255	−0.728415
$162$	−4.67532	−0.367328
$163$	−14.2313	−1.11468	−0.557342	−	0.830283i	$-0.688179\pi$
$163$	−14.2313	−1.11468	−0.557342	+	0.830283i	$0.688179\pi$
$164$	13.1228	1.02472
$165$	0	0
$166$	−0.268300	−0.0208241
$167$	9.24289	0.715236	0.357618	−	0.933868i	$-0.383589\pi$
$167$	9.24289	0.715236	0.357618	+	0.933868i	$0.383589\pi$
$168$	13.5868	1.04824
$169$	24.4732	1.88256
$170$	0	0
$171$	16.3913	1.25348
$172$	−21.2455	−1.61995
$173$	20.7662	1.57883	0.789413	−	0.613863i	$-0.210385\pi$
$173$	20.7662	1.57883	0.789413	+	0.613863i	$0.210385\pi$
$174$	0.143035	0.0108435
$175$	0	0
$176$	−21.5879	−1.62725
$177$	7.82911	0.588472
$178$	−2.27387	−0.170434
$179$	−17.9528	−1.34185	−0.670927	−	0.741523i	$-0.734104\pi$
$179$	−17.9528	−1.34185	−0.670927	+	0.741523i	$0.734104\pi$
$180$	0	0
$181$	−11.1122	−0.825960	−0.412980	−	0.910740i	$-0.635512\pi$
$181$	−11.1122	−0.825960	−0.412980	+	0.910740i	$0.635512\pi$
$182$	6.80450	0.504383
$183$	3.12076	0.230693
$184$	2.71288	0.199996
$185$	0	0
$186$	4.83641	0.354623
$187$	5.35844	0.391848
$188$	−2.41728	−0.176299
$189$	44.9411	3.26899
$190$	0	0
$191$	7.20538	0.521363	0.260681	−	0.965425i	$-0.416053\pi$
$191$	7.20538	0.521363	0.260681	+	0.965425i	$0.416053\pi$
$192$	18.9417	1.36700
$193$	5.98754	0.430992	0.215496	−	0.976505i	$-0.430863\pi$
$193$	5.98754	0.430992	0.215496	+	0.976505i	$0.430863\pi$
$194$	2.16428	0.155386
$195$	0	0
$196$	−15.0127	−1.07233
$197$	−26.9516	−1.92022	−0.960110	−	0.279623i	$-0.909791\pi$
$197$	−26.9516	−1.92022	−0.960110	+	0.279623i	$0.909791\pi$
$198$	11.9727	0.850862
$199$	8.86829	0.628657	0.314328	−	0.949314i	$-0.398221\pi$
$199$	8.86829	0.628657	0.314328	+	0.949314i	$0.398221\pi$
$200$	0	0
$201$	−10.4367	−0.736151
$202$	1.64192	0.115525
$203$	−0.611595	−0.0429256
$204$	−5.20708	−0.364568
$205$	0	0
$206$	3.05640	0.212950
$207$	16.1730	1.12410
$208$	21.4690	1.48861
$209$	−14.9715	−1.03560
$210$	0	0
$211$	5.00712	0.344704	0.172352	−	0.985035i	$-0.444863\pi$
$211$	5.00712	0.344704	0.172352	+	0.985035i	$0.444863\pi$
$212$	6.93595	0.476363
$213$	2.06286	0.141345
$214$	3.80446	0.260068
$215$	0	0
$216$	−13.1912	−0.897544
$217$	−20.6797	−1.40383
$218$	−4.48113	−0.303500
$219$	21.0365	1.42151
$220$	0	0
$221$	−5.32893	−0.358463
$222$	0.909264	0.0610258
$223$	−17.4309	−1.16726	−0.583630	−	0.812019i	$-0.698368\pi$
$223$	−17.4309	−1.16726	−0.583630	+	0.812019i	$0.698368\pi$
$224$	12.6057	0.842257
$225$	0	0
$226$	−1.32514	−0.0881471
$227$	13.6653	0.906995	0.453498	−	0.891258i	$-0.350176\pi$
$227$	13.6653	0.906995	0.453498	+	0.891258i	$0.350176\pi$
$228$	14.5486	0.963508
$229$	13.4738	0.890372	0.445186	−	0.895438i	$-0.353138\pi$
$229$	13.4738	0.890372	0.445186	+	0.895438i	$0.353138\pi$
$230$	0	0
$231$	−73.9823	−4.86768
$232$	0.179516	0.0117858
$233$	0.441806	0.0289437	0.0144718	−	0.999895i	$-0.495393\pi$
$233$	0.441806	0.0289437	0.0144718	+	0.999895i	$0.495393\pi$
$234$	−11.9068	−0.778370
$235$	0	0
$236$	4.80845	0.313003
$237$	14.8544	0.964899
$238$	−0.967642	−0.0627229
$239$	13.7594	0.890023	0.445012	−	0.895525i	$-0.353200\pi$
$239$	13.7594	0.890023	0.445012	+	0.895525i	$0.353200\pi$
$240$	0	0
$241$	−18.0778	−1.16449	−0.582247	−	0.813012i	$-0.697826\pi$
$241$	−18.0778	−1.16449	−0.582247	+	0.813012i	$0.697826\pi$
$242$	−7.76082	−0.498884
$243$	−15.5457	−0.997254
$244$	1.91670	0.122704
$245$	0	0
$246$	−6.16683	−0.393183
$247$	14.8891	0.947371
$248$	6.06991	0.385440
$249$	−2.90105	−0.183846
$250$	0	0
$251$	29.2973	1.84923	0.924614	−	0.380906i	$-0.124388\pi$
$251$	29.2973	1.84923	0.924614	+	0.380906i	$0.124388\pi$
$252$	49.7472	3.13378
$253$	−14.7721	−0.928714
$254$	2.74732	0.172382
$255$	0	0
$256$	9.74417	0.609010
$257$	14.7718	0.921438	0.460719	−	0.887546i	$-0.347592\pi$
$257$	14.7718	0.921438	0.460719	+	0.887546i	$0.347592\pi$
$258$	9.98391	0.621571
$259$	−3.88786	−0.241580
$260$	0	0
$261$	1.07019	0.0662432
$262$	1.59351	0.0984475
$263$	−19.7499	−1.21783	−0.608917	−	0.793234i	$-0.708396\pi$
$263$	−19.7499	−1.21783	−0.608917	+	0.793234i	$0.708396\pi$
$264$	21.7153	1.33649
$265$	0	0
$266$	2.70361	0.165769
$267$	−24.5867	−1.50468
$268$	−6.41000	−0.391553
$269$	−28.5573	−1.74117	−0.870584	−	0.492020i	$-0.836259\pi$
$269$	−28.5573	−1.74117	−0.870584	+	0.492020i	$0.836259\pi$
$270$	0	0
$271$	1.78379	0.108358	0.0541788	−	0.998531i	$-0.482746\pi$
$271$	1.78379	0.108358	0.0541788	+	0.998531i	$0.482746\pi$
$272$	−3.05303	−0.185117
$273$	73.5749	4.45296
$274$	−3.68498	−0.222618
$275$	0	0
$276$	14.3548	0.864059
$277$	−2.51392	−0.151047	−0.0755235	−	0.997144i	$-0.524063\pi$
$277$	−2.51392	−0.151047	−0.0755235	+	0.997144i	$0.524063\pi$
$278$	3.09048	0.185355
$279$	36.1860	2.16640
$280$	0	0
$281$	−5.92662	−0.353552	−0.176776	−	0.984251i	$-0.556567\pi$
$281$	−5.92662	−0.353552	−0.176776	+	0.984251i	$0.556567\pi$
$282$	1.13596	0.0676453
$283$	9.92190	0.589795	0.294898	−	0.955529i	$-0.404714\pi$
$283$	9.92190	0.589795	0.294898	+	0.955529i	$0.404714\pi$
$284$	1.26696	0.0751802
$285$	0	0
$286$	10.8754	0.643077
$287$	26.3683	1.55647
$288$	−22.0580	−1.29978
$289$	−16.2422	−0.955423
$290$	0	0
$291$	23.4017	1.37183
$292$	12.9201	0.756092
$293$	−15.7449	−0.919827	−0.459913	−	0.887964i	$-0.652120\pi$
$293$	−15.7449	−0.919827	−0.459913	+	0.887964i	$0.652120\pi$
$294$	7.05493	0.411452
$295$	0	0
$296$	1.14117	0.0663290
$297$	71.8281	4.16789
$298$	1.24785	0.0722859
$299$	14.6908	0.849588
$300$	0	0
$301$	−42.6895	−2.46058
$302$	−6.76660	−0.389374
$303$	17.7535	1.01991
$304$	8.53021	0.489241
$305$	0	0
$306$	1.69322	0.0967948
$307$	15.0652	0.859819	0.429909	−	0.902872i	$-0.358546\pi$
$307$	15.0652	0.859819	0.429909	+	0.902872i	$0.358546\pi$
$308$	−45.4382	−2.58908
$309$	33.0480	1.88003
$310$	0	0
$311$	23.3413	1.32357	0.661783	−	0.749695i	$-0.269800\pi$
$311$	23.3413	1.32357	0.661783	+	0.749695i	$0.269800\pi$
$312$	−21.5958	−1.22262
$313$	34.2838	1.93784	0.968918	−	0.247381i	$-0.0795700\pi$
$313$	34.2838	1.93784	0.968918	+	0.247381i	$0.0795700\pi$
$314$	3.58394	0.202254
$315$	0	0
$316$	9.12324	0.513222
$317$	−10.8065	−0.606951	−0.303476	−	0.952839i	$-0.598147\pi$
$317$	−10.8065	−0.606951	−0.303476	+	0.952839i	$0.598147\pi$
$318$	−3.25942	−0.182779
$319$	−0.977494	−0.0547292
$320$	0	0
$321$	41.1365	2.29601
$322$	2.66759	0.148659
$323$	−2.11733	−0.117811
$324$	−31.0483	−1.72491
$325$	0	0
$326$	4.10745	0.227491
$327$	−48.4530	−2.67946
$328$	−7.73965	−0.427351
$329$	−4.85716	−0.267784
$330$	0	0
$331$	−4.58182	−0.251839	−0.125920	−	0.992040i	$-0.540188\pi$
$331$	−4.58182	−0.251839	−0.125920	+	0.992040i	$0.540188\pi$
$332$	−1.78175	−0.0977864
$333$	6.80312	0.372809
$334$	−2.66769	−0.145969
$335$	0	0
$336$	42.1523	2.29959
$337$	5.74367	0.312878	0.156439	−	0.987688i	$-0.449999\pi$
$337$	5.74367	0.312878	0.156439	+	0.987688i	$0.449999\pi$
$338$	−7.06348	−0.384202
$339$	−14.3283	−0.778209
$340$	0	0
$341$	−33.0517	−1.78985
$342$	−4.73087	−0.255816
$343$	−3.20652	−0.173136
$344$	12.5303	0.675586
$345$	0	0
$346$	−5.99355	−0.322215
$347$	−33.1472	−1.77944	−0.889718	−	0.456511i	$-0.849099\pi$
$347$	−33.1472	−1.77944	−0.889718	+	0.456511i	$0.849099\pi$
$348$	0.949883	0.0509191
$349$	−16.6773	−0.892717	−0.446358	−	0.894854i	$-0.647279\pi$
$349$	−16.6773	−0.892717	−0.446358	+	0.894854i	$0.647279\pi$
$350$	0	0
$351$	−71.4326	−3.81279
$352$	20.1474	1.07386
$353$	13.5520	0.721299	0.360650	−	0.932701i	$-0.382555\pi$
$353$	13.5520	0.721299	0.360650	+	0.932701i	$0.382555\pi$
$354$	−2.25964	−0.120099
$355$	0	0
$356$	−15.1005	−0.800327
$357$	−10.4628	−0.553751
$358$	5.18154	0.273853
$359$	−32.2510	−1.70214	−0.851072	−	0.525049i	$-0.824047\pi$
$359$	−32.2510	−1.70214	−0.851072	+	0.525049i	$0.824047\pi$
$360$	0	0
$361$	−13.0842	−0.688640
$362$	3.20720	0.168566
$363$	−83.9153	−4.40441
$364$	45.1880	2.36849
$365$	0	0
$366$	−0.900716	−0.0470812
$367$	−7.73041	−0.403524	−0.201762	−	0.979435i	$-0.564667\pi$
$367$	−7.73041	−0.403524	−0.201762	+	0.979435i	$0.564667\pi$
$368$	8.41657	0.438744
$369$	−46.1403	−2.40197
$370$	0	0
$371$	13.9367	0.723558
$372$	32.1181	1.66525
$373$	13.2899	0.688126	0.344063	−	0.938947i	$-0.388197\pi$
$373$	13.2899	0.688126	0.344063	+	0.938947i	$0.388197\pi$
$374$	−1.54655	−0.0799704
$375$	0	0
$376$	1.42568	0.0735237
$377$	0.972112	0.0500663
$378$	−12.9709	−0.667153
$379$	10.2805	0.528076	0.264038	−	0.964512i	$-0.414946\pi$
$379$	10.2805	0.528076	0.264038	+	0.964512i	$0.414946\pi$
$380$	0	0
$381$	29.7059	1.52188
$382$	−2.07962	−0.106403
$383$	−6.45427	−0.329798	−0.164899	−	0.986310i	$-0.552730\pi$
$383$	−6.45427	−0.329798	−0.164899	+	0.986310i	$0.552730\pi$
$384$	−25.8961	−1.32151
$385$	0	0
$386$	−1.72813	−0.0879593
$387$	74.6997	3.79720
$388$	14.3728	0.729666
$389$	−20.0995	−1.01909	−0.509543	−	0.860445i	$-0.670186\pi$
$389$	−20.0995	−1.01909	−0.509543	+	0.860445i	$0.670186\pi$
$390$	0	0
$391$	−2.08912	−0.105651
$392$	8.85425	0.447207
$393$	17.2302	0.869147
$394$	7.77877	0.391889
$395$	0	0
$396$	79.5094	3.99550
$397$	−33.4682	−1.67972	−0.839860	−	0.542803i	$-0.817363\pi$
$397$	−33.4682	−1.67972	−0.839860	+	0.542803i	$0.817363\pi$
$398$	−2.55957	−0.128300
$399$	29.2333	1.46349
$400$	0	0
$401$	−21.3012	−1.06373	−0.531866	−	0.846829i	$-0.678509\pi$
$401$	−21.3012	−1.06373	−0.531866	+	0.846829i	$0.678509\pi$
$402$	3.01226	0.150238
$403$	32.8697	1.63736
$404$	10.9038	0.542484
$405$	0	0
$406$	0.176519	0.00876048
$407$	−6.21385	−0.308009
$408$	3.07105	0.152040
$409$	−10.0153	−0.495224	−0.247612	−	0.968859i	$-0.579646\pi$
$409$	−10.0153	−0.495224	−0.247612	+	0.968859i	$0.579646\pi$
$410$	0	0
$411$	−39.8446	−1.96539
$412$	20.2973	0.999974
$413$	9.66184	0.475428
$414$	−4.66785	−0.229412
$415$	0	0
$416$	−20.0365	−0.982368
$417$	33.4165	1.63641
$418$	4.32109	0.211352
$419$	−7.69379	−0.375866	−0.187933	−	0.982182i	$-0.560179\pi$
$419$	−7.69379	−0.375866	−0.187933	+	0.982182i	$0.560179\pi$
$420$	0	0
$421$	−7.62095	−0.371422	−0.185711	−	0.982604i	$-0.559459\pi$
$421$	−7.62095	−0.371422	−0.185711	+	0.982604i	$0.559459\pi$
$422$	−1.44516	−0.0703492
$423$	8.49924	0.413247
$424$	−4.09071	−0.198663
$425$	0	0
$426$	−0.595384	−0.0288464
$427$	3.85131	0.186378
$428$	25.2650	1.22123
$429$	117.593	5.67743
$430$	0	0
$431$	7.60427	0.366285	0.183142	−	0.983086i	$-0.441373\pi$
$431$	7.60427	0.366285	0.183142	+	0.983086i	$0.441373\pi$
$432$	−40.9249	−1.96900
$433$	−21.6297	−1.03946	−0.519729	−	0.854331i	$-0.673967\pi$
$433$	−21.6297	−1.03946	−0.519729	+	0.854331i	$0.673967\pi$
$434$	5.96857	0.286501
$435$	0	0
$436$	−29.7587	−1.42518
$437$	5.83703	0.279223
$438$	−6.07156	−0.290110
$439$	9.52323	0.454519	0.227260	−	0.973834i	$-0.427023\pi$
$439$	9.52323	0.454519	0.227260	+	0.973834i	$0.427023\pi$
$440$	0	0
$441$	52.7850	2.51357
$442$	1.53804	0.0731570
$443$	−33.8993	−1.61061	−0.805303	−	0.592864i	$-0.797997\pi$
$443$	−33.8993	−1.61061	−0.805303	+	0.592864i	$0.797997\pi$
$444$	6.03833	0.286566
$445$	0	0
$446$	5.03092	0.238221
$447$	13.4926	0.638178
$448$	23.3758	1.10440
$449$	28.7537	1.35697	0.678487	−	0.734613i	$-0.262636\pi$
$449$	28.7537	1.35697	0.678487	+	0.734613i	$0.262636\pi$
$450$	0	0
$451$	42.1437	1.98447
$452$	−8.80012	−0.413923
$453$	−73.1651	−3.43760
$454$	−3.94407	−0.185105
$455$	0	0
$456$	−8.58057	−0.401822
$457$	11.9021	0.556755	0.278377	−	0.960472i	$-0.410203\pi$
$457$	11.9021	0.556755	0.278377	+	0.960472i	$0.410203\pi$
$458$	−3.88881	−0.181712
$459$	10.1582	0.474142
$460$	0	0
$461$	−21.9751	−1.02348	−0.511742	−	0.859139i	$-0.671000\pi$
$461$	−21.9751	−1.02348	−0.511742	+	0.859139i	$0.671000\pi$
$462$	21.3528	0.993422
$463$	−20.9303	−0.972713	−0.486357	−	0.873760i	$-0.661674\pi$
$463$	−20.9303	−0.972713	−0.486357	+	0.873760i	$0.661674\pi$
$464$	0.556939	0.0258552
$465$	0	0
$466$	−0.127514	−0.00590698
$467$	−17.6217	−0.815433	−0.407717	−	0.913108i	$-0.633675\pi$
$467$	−17.6217	−0.815433	−0.407717	+	0.913108i	$0.633675\pi$
$468$	−79.0716	−3.65509
$469$	−12.8799	−0.594739
$470$	0	0
$471$	38.7521	1.78560
$472$	−2.83595	−0.130535
$473$	−68.2294	−3.13719
$474$	−4.28729	−0.196922
$475$	0	0
$476$	−6.42601	−0.294536
$477$	−24.3870	−1.11660
$478$	−3.97125	−0.181641
$479$	−39.0391	−1.78374	−0.891871	−	0.452290i	$-0.850607\pi$
$479$	−39.0391	−1.78374	−0.891871	+	0.452290i	$0.850607\pi$
$480$	0	0
$481$	6.17963	0.281767
$482$	5.21763	0.237656
$483$	28.8438	1.31244
$484$	−51.5388	−2.34267
$485$	0	0
$486$	4.48679	0.203525
$487$	−5.26567	−0.238610	−0.119305	−	0.992858i	$-0.538067\pi$
$487$	−5.26567	−0.238610	−0.119305	+	0.992858i	$0.538067\pi$
$488$	−1.13044	−0.0511726
$489$	44.4126	2.00841
$490$	0	0
$491$	−42.2901	−1.90853	−0.954263	−	0.298970i	$-0.903357\pi$
$491$	−42.2901	−1.90853	−0.954263	+	0.298970i	$0.903357\pi$
$492$	−40.9533	−1.84632
$493$	−0.138240	−0.00622604
$494$	−4.29730	−0.193345
$495$	0	0
$496$	18.8316	0.845563
$497$	2.54576	0.114193
$498$	0.837301	0.0375204
$499$	−19.5493	−0.875145	−0.437573	−	0.899183i	$-0.644162\pi$
$499$	−19.5493	−0.875145	−0.437573	+	0.899183i	$0.644162\pi$
$500$	0	0
$501$	−28.8449	−1.28869
$502$	−8.45579	−0.377401
$503$	23.6259	1.05343	0.526713	−	0.850043i	$-0.323424\pi$
$503$	23.6259	1.05343	0.526713	+	0.850043i	$0.323424\pi$
$504$	−29.3401	−1.30691
$505$	0	0
$506$	4.26353	0.189537
$507$	−76.3752	−3.39194
$508$	18.2446	0.809475
$509$	−14.1735	−0.628230	−0.314115	−	0.949385i	$-0.601708\pi$
$509$	−14.1735	−0.628230	−0.314115	+	0.949385i	$0.601708\pi$
$510$	0	0
$511$	25.9609	1.14844
$512$	−19.4084	−0.857738
$513$	−28.3820	−1.25310
$514$	−4.26343	−0.188052
$515$	0	0
$516$	66.3021	2.91879
$517$	−7.76305	−0.341419
$518$	1.12212	0.0493029
$519$	−64.8064	−2.84469
$520$	0	0
$521$	−15.8457	−0.694215	−0.347107	−	0.937825i	$-0.612836\pi$
$521$	−15.8457	−0.694215	−0.347107	+	0.937825i	$0.612836\pi$
$522$	−0.308879	−0.0135193
$523$	7.16268	0.313202	0.156601	−	0.987662i	$-0.449946\pi$
$523$	7.16268	0.313202	0.156601	+	0.987662i	$0.449946\pi$
$524$	10.5824	0.462292
$525$	0	0
$526$	5.70024	0.248542
$527$	−4.67428	−0.203615
$528$	67.3707	2.93193
$529$	−17.2407	−0.749597
$530$	0	0
$531$	−16.9067	−0.733686
$532$	17.9544	0.778421
$533$	−41.9116	−1.81539
$534$	7.09621	0.307083
$535$	0	0
$536$	3.78052	0.163294
$537$	56.0264	2.41772
$538$	8.24221	0.355347
$539$	−48.2129	−2.07668
$540$	0	0
$541$	−39.5868	−1.70197	−0.850984	−	0.525191i	$-0.823994\pi$
$541$	−39.5868	−1.70197	−0.850984	+	0.525191i	$0.823994\pi$
$542$	−0.514839	−0.0221142
$543$	34.6784	1.48819
$544$	2.84931	0.122163
$545$	0	0
$546$	−21.2352	−0.908784
$547$	−10.9821	−0.469562	−0.234781	−	0.972048i	$-0.575437\pi$
$547$	−10.9821	−0.469562	−0.234781	+	0.972048i	$0.575437\pi$
$548$	−24.4716	−1.04537
$549$	−6.73917	−0.287621
$550$	0	0
$551$	0.386246	0.0164546
$552$	−8.46626	−0.360348
$553$	18.3317	0.779545
$554$	0.725570	0.0308265
$555$	0	0
$556$	20.5236	0.870394
$557$	−9.05385	−0.383624	−0.191812	−	0.981432i	$-0.561436\pi$
$557$	−9.05385	−0.383624	−0.191812	+	0.981432i	$0.561436\pi$
$558$	−10.4440	−0.442131
$559$	67.8537	2.86990
$560$	0	0
$561$	−16.7224	−0.706021
$562$	1.71054	0.0721549
$563$	8.71600	0.367336	0.183668	−	0.982988i	$-0.441203\pi$
$563$	8.71600	0.367336	0.183668	+	0.982988i	$0.441203\pi$
$564$	7.54377	0.317650
$565$	0	0
$566$	−2.86366	−0.120369
$567$	−62.3868	−2.62000
$568$	−0.747233	−0.0313532
$569$	−3.59318	−0.150634	−0.0753169	−	0.997160i	$-0.523997\pi$
$569$	−3.59318	−0.150634	−0.0753169	+	0.997160i	$0.523997\pi$
$570$	0	0
$571$	−16.0433	−0.671391	−0.335696	−	0.941970i	$-0.608971\pi$
$571$	−16.0433	−0.671391	−0.335696	+	0.941970i	$0.608971\pi$
$572$	72.2226	3.01978
$573$	−22.4863	−0.939378
$574$	−7.61043	−0.317653
$575$	0	0
$576$	−40.9039	−1.70433
$577$	8.07205	0.336044	0.168022	−	0.985783i	$-0.446262\pi$
$577$	8.07205	0.336044	0.168022	+	0.985783i	$0.446262\pi$
$578$	4.68783	0.194988
$579$	−18.6857	−0.776551
$580$	0	0
$581$	−3.58016	−0.148530
$582$	−6.75421	−0.279971
$583$	22.2746	0.922521
$584$	−7.62008	−0.315321
$585$	0	0
$586$	4.54430	0.187723
$587$	−12.3447	−0.509521	−0.254760	−	0.967004i	$-0.581997\pi$
$587$	−12.3447	−0.509521	−0.254760	+	0.967004i	$0.581997\pi$
$588$	46.8511	1.93210
$589$	13.0600	0.538128
$590$	0	0
$591$	84.1095	3.45980
$592$	3.54041	0.145510
$593$	−31.1544	−1.27936	−0.639680	−	0.768641i	$-0.720933\pi$
$593$	−31.1544	−1.27936	−0.639680	+	0.768641i	$0.720933\pi$
$594$	−20.7311	−0.850605
$595$	0	0
$596$	8.28682	0.339442
$597$	−27.6758	−1.13270
$598$	−4.24005	−0.173389
$599$	18.3649	0.750371	0.375186	−	0.926950i	$-0.377579\pi$
$599$	18.3649	0.750371	0.375186	+	0.926950i	$0.377579\pi$
$600$	0	0
$601$	32.3863	1.32106	0.660531	−	0.750798i	$-0.270331\pi$
$601$	32.3863	1.32106	0.660531	+	0.750798i	$0.270331\pi$
$602$	12.3211	0.502169
$603$	22.5377	0.917808
$604$	−44.9363	−1.82843
$605$	0	0
$606$	−5.12403	−0.208150
$607$	13.8753	0.563181	0.281591	−	0.959535i	$-0.409138\pi$
$607$	13.8753	0.563181	0.281591	+	0.959535i	$0.409138\pi$
$608$	−7.96101	−0.322862
$609$	1.90864	0.0773421
$610$	0	0
$611$	7.72031	0.312330
$612$	11.2445	0.454531
$613$	−2.07841	−0.0839464	−0.0419732	−	0.999119i	$-0.513364\pi$
$613$	−2.07841	−0.0839464	−0.0419732	+	0.999119i	$0.513364\pi$
$614$	−4.34814	−0.175476
$615$	0	0
$616$	26.7987	1.07975
$617$	26.7159	1.07554	0.537772	−	0.843090i	$-0.319266\pi$
$617$	26.7159	1.07554	0.537772	+	0.843090i	$0.319266\pi$
$618$	−9.53831	−0.383687
$619$	12.0352	0.483734	0.241867	−	0.970309i	$-0.422240\pi$
$619$	12.0352	0.483734	0.241867	+	0.970309i	$0.422240\pi$
$620$	0	0
$621$	−28.0039	−1.12376
$622$	−6.73679	−0.270121
$623$	−30.3422	−1.21563
$624$	−66.9998	−2.68214
$625$	0	0
$626$	−9.89501	−0.395484
$627$	46.7227	1.86592
$628$	23.8006	0.949747
$629$	−0.878783	−0.0350394
$630$	0	0
$631$	−14.6217	−0.582081	−0.291040	−	0.956711i	$-0.594001\pi$
$631$	−14.6217	−0.582081	−0.291040	+	0.956711i	$0.594001\pi$
$632$	−5.38074	−0.214035
$633$	−15.6260	−0.621079
$634$	3.11896	0.123870
$635$	0	0
$636$	−21.6454	−0.858298
$637$	47.9474	1.89975
$638$	0.282125	0.0111694
$639$	−4.45467	−0.176224
$640$	0	0
$641$	−31.1534	−1.23049	−0.615243	−	0.788337i	$-0.710942\pi$
$641$	−31.1534	−1.23049	−0.615243	+	0.788337i	$0.710942\pi$
$642$	−11.8728	−0.468583
$643$	21.8872	0.863145	0.431572	−	0.902078i	$-0.357959\pi$
$643$	21.8872	0.863145	0.431572	+	0.902078i	$0.357959\pi$
$644$	17.7152	0.698076
$645$	0	0
$646$	0.611103	0.0240435
$647$	−9.41543	−0.370159	−0.185079	−	0.982724i	$-0.559254\pi$
$647$	−9.41543	−0.370159	−0.185079	+	0.982724i	$0.559254\pi$
$648$	18.3118	0.719356
$649$	15.4422	0.606161
$650$	0	0
$651$	64.5363	2.52938
$652$	27.2772	1.06826
$653$	28.7782	1.12618	0.563088	−	0.826397i	$-0.309613\pi$
$653$	28.7782	1.12618	0.563088	+	0.826397i	$0.309613\pi$
$654$	13.9845	0.546838
$655$	0	0
$656$	−24.0119	−0.937506
$657$	−45.4274	−1.77229
$658$	1.40188	0.0546508
$659$	15.3672	0.598622	0.299311	−	0.954156i	$-0.403243\pi$
$659$	15.3672	0.598622	0.299311	+	0.954156i	$0.403243\pi$
$660$	0	0
$661$	21.4817	0.835540	0.417770	−	0.908553i	$-0.362812\pi$
$661$	21.4817	0.835540	0.417770	+	0.908553i	$0.362812\pi$
$662$	1.32241	0.0513968
$663$	16.6303	0.645869
$664$	1.05085	0.0407809
$665$	0	0
$666$	−1.96352	−0.0760848
$667$	0.381100	0.0147563
$668$	−17.7158	−0.685446
$669$	54.3978	2.10314
$670$	0	0
$671$	6.15543	0.237628
$672$	−39.3396	−1.51756
$673$	29.5559	1.13930	0.569649	−	0.821888i	$-0.307079\pi$
$673$	29.5559	1.13930	0.569649	+	0.821888i	$0.307079\pi$
$674$	−1.65774	−0.0638538
$675$	0	0
$676$	−46.9078	−1.80415
$677$	1.70021	0.0653443	0.0326721	−	0.999466i	$-0.489598\pi$
$677$	1.70021	0.0653443	0.0326721	+	0.999466i	$0.489598\pi$
$678$	4.13545	0.158821
$679$	28.8798	1.10831
$680$	0	0
$681$	−42.6460	−1.63420
$682$	9.53939	0.365282
$683$	36.2148	1.38572	0.692860	−	0.721073i	$-0.256351\pi$
$683$	36.2148	1.38572	0.692860	+	0.721073i	$0.256351\pi$
$684$	−31.4172	−1.20127
$685$	0	0
$686$	0.925466	0.0353345
$687$	−42.0485	−1.60425
$688$	38.8745	1.48208
$689$	−22.1520	−0.843923
$690$	0	0
$691$	48.0345	1.82732	0.913659	−	0.406482i	$-0.133244\pi$
$691$	48.0345	1.82732	0.913659	+	0.406482i	$0.133244\pi$
$692$	−39.8026	−1.51307
$693$	159.762	6.06885
$694$	9.56696	0.363157
$695$	0	0
$696$	−0.560226	−0.0212353
$697$	5.96010	0.225755
$698$	4.81342	0.182190
$699$	−1.37877	−0.0521499
$700$	0	0
$701$	−4.88732	−0.184591	−0.0922957	−	0.995732i	$-0.529420\pi$
$701$	−4.88732	−0.184591	−0.0922957	+	0.995732i	$0.529420\pi$
$702$	20.6169	0.778135
$703$	2.45533	0.0926046
$704$	37.3608	1.40809
$705$	0	0
$706$	−3.91138	−0.147207
$707$	21.9095	0.823991
$708$	−15.0060	−0.563961
$709$	26.1967	0.983836	0.491918	−	0.870641i	$-0.336296\pi$
$709$	26.1967	0.983836	0.491918	+	0.870641i	$0.336296\pi$
$710$	0	0
$711$	−32.0776	−1.20300
$712$	8.90607	0.333769
$713$	12.8860	0.482585
$714$	3.01978	0.113013
$715$	0	0
$716$	34.4101	1.28596
$717$	−42.9399	−1.60362
$718$	9.30830	0.347383
$719$	−32.9406	−1.22848	−0.614239	−	0.789120i	$-0.710537\pi$
$719$	−32.9406	−1.22848	−0.614239	+	0.789120i	$0.710537\pi$
$720$	0	0
$721$	40.7842	1.51888
$722$	3.77636	0.140541
$723$	56.4166	2.09816
$724$	21.2987	0.791558
$725$	0	0
$726$	24.2197	0.898877
$727$	−24.3742	−0.903988	−0.451994	−	0.892021i	$-0.649287\pi$
$727$	−24.3742	−0.903988	−0.451994	+	0.892021i	$0.649287\pi$
$728$	−26.6512	−0.987758
$729$	−0.0822600	−0.00304667
$730$	0	0
$731$	−9.64922	−0.356889
$732$	−5.98156	−0.221085
$733$	−15.7737	−0.582616	−0.291308	−	0.956629i	$-0.594091\pi$
$733$	−15.7737	−0.582616	−0.291308	+	0.956629i	$0.594091\pi$
$734$	2.23115	0.0823534
$735$	0	0
$736$	−7.85496	−0.289538
$737$	−20.5856	−0.758279
$738$	13.3170	0.490207
$739$	−12.7545	−0.469181	−0.234591	−	0.972094i	$-0.575375\pi$
$739$	−12.7545	−0.469181	−0.234591	+	0.972094i	$0.575375\pi$
$740$	0	0
$741$	−46.4654	−1.70695
$742$	−4.02242	−0.147668
$743$	−16.8624	−0.618620	−0.309310	−	0.950961i	$-0.600098\pi$
$743$	−16.8624	−0.618620	−0.309310	+	0.950961i	$0.600098\pi$
$744$	−18.9428	−0.694475
$745$	0	0
$746$	−3.83574	−0.140437
$747$	6.26470	0.229213
$748$	−10.2705	−0.375527
$749$	50.7662	1.85496
$750$	0	0
$751$	11.8725	0.433235	0.216617	−	0.976257i	$-0.430498\pi$
$751$	11.8725	0.433235	0.216617	+	0.976257i	$0.430498\pi$
$752$	4.42309	0.161293
$753$	−91.4299	−3.33189
$754$	−0.280571	−0.0102178
$755$	0	0
$756$	−86.1386	−3.13283
$757$	−31.6244	−1.14941	−0.574703	−	0.818362i	$-0.694883\pi$
$757$	−31.6244	−1.14941	−0.574703	+	0.818362i	$0.694883\pi$
$758$	−2.96718	−0.107773
$759$	46.1002	1.67333
$760$	0	0
$761$	37.6068	1.36325	0.681623	−	0.731703i	$-0.261274\pi$
$761$	37.6068	1.36325	0.681623	+	0.731703i	$0.261274\pi$
$762$	−8.57373	−0.310593
$763$	−59.7955	−2.16474
$764$	−13.8105	−0.499648
$765$	0	0
$766$	1.86283	0.0673069
$767$	−15.3572	−0.554516
$768$	−30.4092	−1.09730
$769$	−42.9677	−1.54946	−0.774728	−	0.632295i	$-0.782113\pi$
$769$	−42.9677	−1.54946	−0.774728	+	0.632295i	$0.782113\pi$
$770$	0	0
$771$	−46.0992	−1.66022
$772$	−11.4763	−0.413041
$773$	−25.7568	−0.926407	−0.463203	−	0.886252i	$-0.653300\pi$
$773$	−25.7568	−0.926407	−0.463203	+	0.886252i	$0.653300\pi$
$774$	−21.5599	−0.774953
$775$	0	0
$776$	−8.47683	−0.304301
$777$	12.1331	0.435272
$778$	5.80113	0.207981
$779$	−16.6526	−0.596641
$780$	0	0
$781$	4.06881	0.145594
$782$	0.602962	0.0215619
$783$	−1.85307	−0.0662232
$784$	27.4699	0.981066
$785$	0	0
$786$	−4.97298	−0.177380
$787$	48.1802	1.71744	0.858720	−	0.512445i	$-0.171260\pi$
$787$	48.1802	1.71744	0.858720	+	0.512445i	$0.171260\pi$
$788$	51.6580	1.84024
$789$	61.6349	2.19426
$790$	0	0
$791$	−17.6825	−0.628717
$792$	−46.8934	−1.66629
$793$	−6.12154	−0.217382
$794$	9.65960	0.342806
$795$	0	0
$796$	−16.9978	−0.602472
$797$	28.8142	1.02065	0.510325	−	0.859981i	$-0.329525\pi$
$797$	28.8142	1.02065	0.510325	+	0.859981i	$0.329525\pi$
$798$	−8.43732	−0.298678
$799$	−1.09788	−0.0388401
$800$	0	0
$801$	53.0939	1.87598
$802$	6.14797	0.217092
$803$	41.4926	1.46424
$804$	20.0041	0.705490
$805$	0	0
$806$	−9.48686	−0.334161
$807$	89.1205	3.13719
$808$	−6.43089	−0.226238
$809$	−4.74308	−0.166758	−0.0833790	−	0.996518i	$-0.526571\pi$
$809$	−4.74308	−0.166758	−0.0833790	+	0.996518i	$0.526571\pi$
$810$	0	0
$811$	1.30841	0.0459445	0.0229722	−	0.999736i	$-0.492687\pi$
$811$	1.30841	0.0459445	0.0229722	+	0.999736i	$0.492687\pi$
$812$	1.17224	0.0411377
$813$	−5.56679	−0.195236
$814$	1.79344	0.0628602
$815$	0	0
$816$	9.52779	0.333539
$817$	26.9601	0.943213
$818$	2.89062	0.101068
$819$	−158.882	−5.55179
$820$	0	0
$821$	−28.4629	−0.993362	−0.496681	−	0.867933i	$-0.665448\pi$
$821$	−28.4629	−0.993362	−0.496681	+	0.867933i	$0.665448\pi$
$822$	11.5000	0.401107
$823$	−19.2130	−0.669724	−0.334862	−	0.942267i	$-0.608690\pi$
$823$	−19.2130	−0.669724	−0.334862	+	0.942267i	$0.608690\pi$
$824$	−11.9710	−0.417030
$825$	0	0
$826$	−2.78860	−0.0970279
$827$	−44.4278	−1.54490	−0.772452	−	0.635073i	$-0.780970\pi$
$827$	−44.4278	−1.54490	−0.772452	+	0.635073i	$0.780970\pi$
$828$	−30.9987	−1.07728
$829$	−52.3280	−1.81743	−0.908714	−	0.417420i	$-0.862934\pi$
$829$	−52.3280	−1.81743	−0.908714	+	0.417420i	$0.862934\pi$
$830$	0	0
$831$	7.84536	0.272153
$832$	−37.1551	−1.28812
$833$	−6.81843	−0.236245
$834$	−9.64467	−0.333968
$835$	0	0
$836$	28.6959	0.992470
$837$	−62.6571	−2.16575
$838$	2.22059	0.0767089
$839$	41.8636	1.44529	0.722646	−	0.691218i	$-0.242926\pi$
$839$	41.8636	1.44529	0.722646	+	0.691218i	$0.242926\pi$
$840$	0	0
$841$	−28.9748	−0.999130
$842$	2.19956	0.0758019
$843$	18.4956	0.637021
$844$	−9.59714	−0.330347
$845$	0	0
$846$	−2.45305	−0.0843377
$847$	−103.559	−3.55834
$848$	−12.6912	−0.435819
$849$	−30.9639	−1.06268
$850$	0	0
$851$	2.42262	0.0830464
$852$	−3.95388	−0.135458
$853$	−17.2610	−0.591006	−0.295503	−	0.955342i	$-0.595487\pi$
$853$	−17.2610	−0.591006	−0.295503	+	0.955342i	$0.595487\pi$
$854$	−1.11157	−0.0380370
$855$	0	0
$856$	−14.9009	−0.509303
$857$	−45.2701	−1.54640	−0.773199	−	0.634164i	$-0.781345\pi$
$857$	−45.2701	−1.54640	−0.773199	+	0.634164i	$0.781345\pi$
$858$	−33.9396	−1.15868
$859$	−20.2274	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$859$	−20.2274	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$860$	0	0
$861$	−82.2893	−2.80441
$862$	−2.19475	−0.0747534
$863$	0.227214	0.00773445	0.00386722	−	0.999993i	$-0.498769\pi$
$863$	0.227214	0.00773445	0.00386722	+	0.999993i	$0.498769\pi$
$864$	38.1941	1.29939
$865$	0	0
$866$	6.24278	0.212138
$867$	50.6880	1.72146
$868$	39.6367	1.34536
$869$	29.2991	0.993903
$870$	0	0
$871$	20.4722	0.693674
$872$	17.5512	0.594359
$873$	−50.5350	−1.71035
$874$	−1.68468	−0.0569853
$875$	0	0
$876$	−40.3206	−1.36231
$877$	32.2041	1.08745	0.543727	−	0.839262i	$-0.317013\pi$
$877$	32.2041	1.08745	0.543727	+	0.839262i	$0.317013\pi$
$878$	−2.74860	−0.0927607
$879$	49.1361	1.65732
$880$	0	0
$881$	22.7100	0.765119	0.382560	−	0.923931i	$-0.375043\pi$
$881$	22.7100	0.765119	0.382560	+	0.923931i	$0.375043\pi$
$882$	−15.2348	−0.512984
$883$	15.9426	0.536511	0.268256	−	0.963348i	$-0.413553\pi$
$883$	15.9426	0.536511	0.268256	+	0.963348i	$0.413553\pi$
$884$	10.2140	0.343532
$885$	0	0
$886$	9.78404	0.328701
$887$	51.1098	1.71610	0.858049	−	0.513568i	$-0.171677\pi$
$887$	51.1098	1.71610	0.858049	+	0.513568i	$0.171677\pi$
$888$	−3.56131	−0.119510
$889$	36.6598	1.22953
$890$	0	0
$891$	−99.7110	−3.34044
$892$	33.4098	1.11864
$893$	3.06748	0.102649
$894$	−3.89424	−0.130243
$895$	0	0
$896$	−31.9582	−1.06765
$897$	−45.8464	−1.53077
$898$	−8.29892	−0.276939
$899$	0.852689	0.0284388
$900$	0	0
$901$	3.15015	0.104947
$902$	−12.1635	−0.405001
$903$	133.224	4.43341
$904$	5.19018	0.172623
$905$	0	0
$906$	21.1169	0.701564
$907$	−23.9276	−0.794504	−0.397252	−	0.917709i	$-0.630036\pi$
$907$	−23.9276	−0.794504	−0.397252	+	0.917709i	$0.630036\pi$
$908$	−26.1922	−0.869218
$909$	−38.3381	−1.27159
$910$	0	0
$911$	31.6630	1.04904	0.524521	−	0.851397i	$-0.324244\pi$
$911$	31.6630	1.04904	0.524521	+	0.851397i	$0.324244\pi$
$912$	−26.6208	−0.881502
$913$	−5.72206	−0.189373
$914$	−3.43518	−0.113626
$915$	0	0
$916$	−25.8252	−0.853287
$917$	21.2636	0.702186
$918$	−2.93185	−0.0967656
$919$	−22.0411	−0.727068	−0.363534	−	0.931581i	$-0.618430\pi$
$919$	−22.0411	−0.727068	−0.363534	+	0.931581i	$0.618430\pi$
$920$	0	0
$921$	−47.0150	−1.54920
$922$	6.34247	0.208878
$923$	−4.04641	−0.133189
$924$	141.802	4.66493
$925$	0	0
$926$	6.04091	0.198517
$927$	−71.3657	−2.34396
$928$	−0.519776	−0.0170625
$929$	16.1290	0.529174	0.264587	−	0.964362i	$-0.414764\pi$
$929$	16.1290	0.529174	0.264587	+	0.964362i	$0.414764\pi$
$930$	0	0
$931$	19.0508	0.624364
$932$	−0.846809	−0.0277381
$933$	−72.8428	−2.38477
$934$	5.08597	0.166418
$935$	0	0
$936$	46.6352	1.52432
$937$	−17.3914	−0.568151	−0.284076	−	0.958802i	$-0.591687\pi$
$937$	−17.3914	−0.568151	−0.284076	+	0.958802i	$0.591687\pi$
$938$	3.71740	0.121377
$939$	−106.992	−3.49154
$940$	0	0
$941$	20.7937	0.677857	0.338928	−	0.940812i	$-0.389936\pi$
$941$	20.7937	0.677857	0.338928	+	0.940812i	$0.389936\pi$
$942$	−11.1846	−0.364415
$943$	−16.4308	−0.535059
$944$	−8.79839	−0.286363
$945$	0	0
$946$	19.6924	0.640255
$947$	−33.4988	−1.08857	−0.544283	−	0.838902i	$-0.683198\pi$
$947$	−33.4988	−1.08857	−0.544283	+	0.838902i	$0.683198\pi$
$948$	−28.4715	−0.924710
$949$	−41.2641	−1.33949
$950$	0	0
$951$	33.7244	1.09359
$952$	3.78996	0.122833
$953$	29.5151	0.956089	0.478044	−	0.878336i	$-0.341346\pi$
$953$	29.5151	0.956089	0.478044	+	0.878336i	$0.341346\pi$
$954$	7.03858	0.227883
$955$	0	0
$956$	−26.3727	−0.852953
$957$	3.05053	0.0986096
$958$	11.2675	0.364036
$959$	−49.1718	−1.58784
$960$	0	0
$961$	−2.16832	−0.0699458
$962$	−1.78357	−0.0575045
$963$	−88.8326	−2.86259
$964$	34.6497	1.11599
$965$	0	0
$966$	−8.32491	−0.267850
$967$	23.2818	0.748694	0.374347	−	0.927289i	$-0.377867\pi$
$967$	23.2818	0.748694	0.374347	+	0.927289i	$0.377867\pi$
$968$	30.3968	0.976990
$969$	6.60767	0.212269
$970$	0	0
$971$	−39.5281	−1.26852	−0.634259	−	0.773121i	$-0.718695\pi$
$971$	−39.5281	−1.26852	−0.634259	+	0.773121i	$0.718695\pi$
$972$	29.7963	0.955718
$973$	41.2390	1.32206
$974$	1.51978	0.0486969
$975$	0	0
$976$	−3.50713	−0.112260
$977$	−33.5738	−1.07412	−0.537060	−	0.843544i	$-0.680465\pi$
$977$	−33.5738	−1.07412	−0.537060	+	0.843544i	$0.680465\pi$
$978$	−12.8184	−0.409887
$979$	−48.4950	−1.54991
$980$	0	0
$981$	104.632	3.34066
$982$	12.2058	0.389502
$983$	−36.8536	−1.17545	−0.587724	−	0.809062i	$-0.699976\pi$
$983$	−36.8536	−1.17545	−0.587724	+	0.809062i	$0.699976\pi$
$984$	24.1536	0.769989
$985$	0	0
$986$	0.0398990	0.00127064
$987$	15.1580	0.482486
$988$	−28.5379	−0.907912
$989$	26.6009	0.845859
$990$	0	0
$991$	−24.8754	−0.790193	−0.395096	−	0.918640i	$-0.629289\pi$
$991$	−24.8754	−0.790193	−0.395096	+	0.918640i	$0.629289\pi$
$992$	−17.5750	−0.558007
$993$	14.2988	0.453758
$994$	−0.734758	−0.0233051
$995$	0	0
$996$	5.56043	0.176189
$997$	19.1635	0.606914	0.303457	−	0.952845i	$-0.401859\pi$
$997$	19.1635	0.606914	0.303457	+	0.952845i	$0.401859\pi$
$998$	5.64232	0.178604
$999$	−11.7798	−0.372696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1525.2.a.h.1.4		7
5.2	odd	4		1525.2.b.f.1099.6		14
5.3	odd	4		1525.2.b.f.1099.9		14
5.4	even	2		305.2.a.d.1.4	✓	7
15.14	odd	2		2745.2.a.m.1.4		7
20.19	odd	2		4880.2.a.z.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
305.2.a.d.1.4	✓	7	5.4	even	2
1525.2.a.h.1.4		7	1.1	even	1	trivial
1525.2.b.f.1099.6		14	5.2	odd	4
1525.2.b.f.1099.9		14	5.3	odd	4
2745.2.a.m.1.4		7	15.14	odd	2
4880.2.a.z.1.1		7	20.19	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:	\( N \)	\(=\)	\( 1525 = 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1525.a (trivial)

\(f(q)\)	\(=\)	\(q-0.288620 q^{2} -3.12076 q^{3} -1.91670 q^{4} +0.900716 q^{6} -3.85131 q^{7} +1.13044 q^{8} +6.73917 q^{9} -6.15543 q^{11} +5.98156 q^{12} +6.12154 q^{13} +1.11157 q^{14} +3.50713 q^{16} -0.870521 q^{17} -1.94506 q^{18} +2.43225 q^{19} +12.0190 q^{21} +1.77658 q^{22} +2.39985 q^{23} -3.52783 q^{24} -1.76680 q^{26} -11.6691 q^{27} +7.38180 q^{28} +0.158802 q^{29} +5.36951 q^{31} -3.27311 q^{32} +19.2097 q^{33} +0.251250 q^{34} -12.9169 q^{36} +1.00949 q^{37} -0.701997 q^{38} -19.1039 q^{39} -6.84659 q^{41} -3.46894 q^{42} +11.0844 q^{43} +11.7981 q^{44} -0.692645 q^{46} +1.26117 q^{47} -10.9449 q^{48} +7.83258 q^{49} +2.71669 q^{51} -11.7331 q^{52} -3.61870 q^{53} +3.36793 q^{54} -4.35367 q^{56} -7.59047 q^{57} -0.0458335 q^{58} -2.50872 q^{59} -1.00000 q^{61} -1.54975 q^{62} -25.9546 q^{63} -6.06957 q^{64} -5.54430 q^{66} +3.34429 q^{67} +1.66853 q^{68} -7.48935 q^{69} -0.661011 q^{71} +7.61822 q^{72} -6.74081 q^{73} -0.291359 q^{74} -4.66189 q^{76} +23.7065 q^{77} +5.51377 q^{78} -4.75987 q^{79} +16.1989 q^{81} +1.97606 q^{82} +0.929595 q^{83} -23.0368 q^{84} -3.19919 q^{86} -0.495583 q^{87} -6.95834 q^{88} +7.87841 q^{89} -23.5759 q^{91} -4.59978 q^{92} -16.7570 q^{93} -0.364000 q^{94} +10.2146 q^{96} -7.49871 q^{97} -2.26064 q^{98} -41.4825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} - 6 q^{3} + 8 q^{4} - 2 q^{6} - 8 q^{7} - 3 q^{8} + 5 q^{9} - 6 q^{11} - q^{12} - 5 q^{13} - 8 q^{14} + 6 q^{16} + 2 q^{17} + 11 q^{18} + 9 q^{19} + 4 q^{21} - 3 q^{22} - 5 q^{23} - 14 q^{24}+ \cdots - 35 q^{99}+O(q^{100}) \) 7 * q - 2 * q^2 - 6 * q^3 + 8 * q^4 - 2 * q^6 - 8 * q^7 - 3 * q^8 + 5 * q^9 - 6 * q^11 - q^12 - 5 * q^13 - 8 * q^14 + 6 * q^16 + 2 * q^17 + 11 * q^18 + 9 * q^19 + 4 * q^21 - 3 * q^22 - 5 * q^23 - 14 * q^24 - 13 * q^26 - 21 * q^27 + 7 * q^28 - 13 * q^29 + 17 * q^31 - 23 * q^32 + 15 * q^33 - 11 * q^34 - 18 * q^36 - q^37 + 8 * q^38 - 11 * q^39 - 8 * q^41 - 11 * q^42 - 5 * q^43 - 21 * q^44 + q^46 - 21 * q^47 + 25 * q^48 + q^49 - 5 * q^51 - 32 * q^52 + 9 * q^53 - 22 * q^54 - 22 * q^56 + 8 * q^57 + 25 * q^58 - 24 * q^59 - 7 * q^61 - 16 * q^62 - q^63 - 27 * q^64 + 9 * q^66 - 27 * q^67 + 39 * q^68 - 9 * q^69 - 13 * q^71 + 14 * q^72 - 4 * q^73 - 32 * q^74 + 23 * q^76 + 25 * q^77 + 54 * q^78 + q^79 + 15 * q^81 - 18 * q^82 - 4 * q^83 - 42 * q^84 - 14 * q^86 + 19 * q^87 + 23 * q^88 - 10 * q^89 - q^91 - 14 * q^92 - 36 * q^93 + q^94 + 11 * q^96 - 2 * q^97 + 39 * q^98 - 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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