LMFDB - Embedded newform 9065.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] 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("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [17,1,-1,13,17,-4,0,-3,8,1,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 23 x^{15} + 23 x^{14} + 209 x^{13} - 205 x^{12} - 971 x^{11} + 907 x^{10} + 2497 x^{9} + \cdots + 14 $ x^17 - x^16 - 23x^15 + 23x^14 + 209x^13 - 205x^12 - 971x^11 + 907x^10 + 2497x^9 - 2138x^8 - 3578x^7 + 2652x^6 + 2728x^5 - 1558x^4 - 1010x^3 + 300x^2 + 154*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.39989$ of defining polynomial
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.39989 q^{2} -3.04811 q^{3} -0.0403202 q^{4} +1.00000 q^{5} +4.26700 q^{6} +2.85621 q^{8} +6.29096 q^{9} -1.39989 q^{10} -3.64714 q^{11} +0.122900 q^{12} +4.84416 q^{13} -3.04811 q^{15} -3.91773 q^{16} -5.93954 q^{17} -8.80662 q^{18} +2.80285 q^{19} -0.0403202 q^{20} +5.10558 q^{22} +6.78095 q^{23} -8.70605 q^{24} +1.00000 q^{25} -6.78127 q^{26} -10.0312 q^{27} +2.56349 q^{29} +4.26700 q^{30} +8.03902 q^{31} -0.228051 q^{32} +11.1169 q^{33} +8.31468 q^{34} -0.253653 q^{36} -1.00000 q^{37} -3.92368 q^{38} -14.7655 q^{39} +2.85621 q^{40} -5.29308 q^{41} -2.79133 q^{43} +0.147054 q^{44} +6.29096 q^{45} -9.49256 q^{46} -4.66023 q^{47} +11.9417 q^{48} -1.39989 q^{50} +18.1044 q^{51} -0.195317 q^{52} +5.56644 q^{53} +14.0425 q^{54} -3.64714 q^{55} -8.54340 q^{57} -3.58860 q^{58} -2.97995 q^{59} +0.122900 q^{60} -13.4537 q^{61} -11.2537 q^{62} +8.15471 q^{64} +4.84416 q^{65} -15.5624 q^{66} -9.91515 q^{67} +0.239484 q^{68} -20.6691 q^{69} +5.93981 q^{71} +17.9683 q^{72} +9.66860 q^{73} +1.39989 q^{74} -3.04811 q^{75} -0.113012 q^{76} +20.6700 q^{78} -14.6739 q^{79} -3.91773 q^{80} +11.7033 q^{81} +7.40970 q^{82} +9.83911 q^{83} -5.93954 q^{85} +3.90755 q^{86} -7.81380 q^{87} -10.4170 q^{88} -0.523456 q^{89} -8.80662 q^{90} -0.273409 q^{92} -24.5038 q^{93} +6.52379 q^{94} +2.80285 q^{95} +0.695123 q^{96} -7.05294 q^{97} -22.9440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + q^{2} - q^{3} + 13 q^{4} + 17 q^{5} - 4 q^{6} - 3 q^{8} + 8 q^{9} + q^{10} - 19 q^{11} - 4 q^{12} - 11 q^{13} - q^{15} + 9 q^{16} - 13 q^{17} - 2 q^{18} - 20 q^{19} + 13 q^{20} + 16 q^{22} + 4 q^{23}+ \cdots - 6 q^{99}+O(q^{100}) $ 17 * q + q^2 - q^3 + 13 * q^4 + 17 * q^5 - 4 * q^6 - 3 * q^8 + 8 * q^9 + q^10 - 19 * q^11 - 4 * q^12 - 11 * q^13 - q^15 + 9 * q^16 - 13 * q^17 - 2 * q^18 - 20 * q^19 + 13 * q^20 + 16 * q^22 + 4 * q^23 - 12 * q^24 + 17 * q^25 - 2 * q^26 - q^27 - 5 * q^29 - 4 * q^30 - 30 * q^31 - 15 * q^32 + 3 * q^33 - 6 * q^34 - 21 * q^36 - 17 * q^37 - 20 * q^38 - 33 * q^39 - 3 * q^40 - 4 * q^41 + 8 * q^43 - 42 * q^44 + 8 * q^45 - 8 * q^46 - 25 * q^47 - 15 * q^48 + q^50 - 19 * q^51 - 44 * q^52 + 16 * q^53 - 2 * q^54 - 19 * q^55 + 14 * q^57 - 18 * q^58 - 10 * q^59 - 4 * q^60 - 32 * q^61 - 61 * q^62 + 57 * q^64 - 11 * q^65 - 31 * q^66 + 4 * q^67 + 30 * q^68 - 32 * q^69 - 12 * q^71 - 21 * q^72 - q^74 - q^75 - 77 * q^76 + 14 * q^78 - 19 * q^79 + 9 * q^80 + 69 * q^81 + 18 * q^82 - 6 * q^83 - 13 * q^85 - 4 * q^86 - 53 * q^87 - 15 * q^88 + 36 * q^89 - 2 * q^90 + 33 * q^92 - 50 * q^93 - 20 * q^94 - 20 * q^95 + 83 * q^96 - 21 * q^97 - 6 * 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 $

$2$	−1.39989	−0.989869	−0.494934	−	0.868930i	$-0.664808\pi$
$2$	−1.39989	−0.989869	−0.494934	+	0.868930i	$0.664808\pi$
$3$	−3.04811	−1.75983	−0.879913	−	0.475135i	$-0.842399\pi$
$3$	−3.04811	−1.75983	−0.879913	+	0.475135i	$0.842399\pi$
$4$	−0.0403202	−0.0201601
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	4.26700	1.74200
$7$	0	0
$7$	0	0
$8$	2.85621	1.00982
$9$	6.29096	2.09699
$10$	−1.39989	−0.442683
$11$	−3.64714	−1.09966	−0.549828	−	0.835278i	$-0.685307\pi$
$11$	−3.64714	−1.09966	−0.549828	+	0.835278i	$0.685307\pi$
$12$	0.122900	0.0354783
$13$	4.84416	1.34353	0.671764	−	0.740765i	$-0.265537\pi$
$13$	4.84416	1.34353	0.671764	+	0.740765i	$0.265537\pi$
$14$	0	0
$15$	−3.04811	−0.787018
$16$	−3.91773	−0.979433
$17$	−5.93954	−1.44055	−0.720275	−	0.693689i	$-0.755984\pi$
$17$	−5.93954	−1.44055	−0.720275	+	0.693689i	$0.755984\pi$
$18$	−8.80662	−2.07574
$19$	2.80285	0.643019	0.321509	−	0.946906i	$-0.395810\pi$
$19$	2.80285	0.643019	0.321509	+	0.946906i	$0.395810\pi$
$20$	−0.0403202	−0.00901587
$21$	0	0
$22$	5.10558	1.08851
$23$	6.78095	1.41393	0.706963	−	0.707250i	$-0.250065\pi$
$23$	6.78095	1.41393	0.706963	+	0.707250i	$0.250065\pi$
$24$	−8.70605	−1.77711
$25$	1.00000	0.200000
$26$	−6.78127	−1.32992
$27$	−10.0312	−1.93050
$28$	0	0
$29$	2.56349	0.476029	0.238014	−	0.971262i	$-0.423503\pi$
$29$	2.56349	0.476029	0.238014	+	0.971262i	$0.423503\pi$
$30$	4.26700	0.779044
$31$	8.03902	1.44385	0.721925	−	0.691971i	$-0.243257\pi$
$31$	8.03902	1.44385	0.721925	+	0.691971i	$0.243257\pi$
$32$	−0.228051	−0.0403140
$33$	11.1169	1.93520
$34$	8.31468	1.42596
$35$	0	0
$36$	−0.253653	−0.0422755
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−3.92368	−0.636504
$39$	−14.7655	−2.36437
$40$	2.85621	0.451607
$41$	−5.29308	−0.826640	−0.413320	−	0.910586i	$-0.635631\pi$
$41$	−5.29308	−0.826640	−0.413320	+	0.910586i	$0.635631\pi$
$42$	0	0
$43$	−2.79133	−0.425675	−0.212837	−	0.977088i	$-0.568270\pi$
$43$	−2.79133	−0.425675	−0.212837	+	0.977088i	$0.568270\pi$
$44$	0.147054	0.0221692
$45$	6.29096	0.937801
$46$	−9.49256	−1.39960
$47$	−4.66023	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$47$	−4.66023	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$48$	11.9417	1.72363
$49$	0	0
$50$	−1.39989	−0.197974
$51$	18.1044	2.53512
$52$	−0.195317	−0.0270857
$53$	5.56644	0.764609	0.382305	−	0.924036i	$-0.375130\pi$
$53$	5.56644	0.764609	0.382305	+	0.924036i	$0.375130\pi$
$54$	14.0425	1.91095
$55$	−3.64714	−0.491781
$56$	0	0
$57$	−8.54340	−1.13160
$58$	−3.58860	−0.471206
$59$	−2.97995	−0.387956	−0.193978	−	0.981006i	$-0.562139\pi$
$59$	−2.97995	−0.387956	−0.193978	+	0.981006i	$0.562139\pi$
$60$	0.122900	0.0158664
$61$	−13.4537	−1.72257	−0.861285	−	0.508122i	$-0.830340\pi$
$61$	−13.4537	−1.72257	−0.861285	+	0.508122i	$0.830340\pi$
$62$	−11.2537	−1.42922
$63$	0	0
$64$	8.15471	1.01934
$65$	4.84416	0.600844
$66$	−15.5624	−1.91559
$67$	−9.91515	−1.21133	−0.605664	−	0.795720i	$-0.707092\pi$
$67$	−9.91515	−1.21133	−0.605664	+	0.795720i	$0.707092\pi$
$68$	0.239484	0.0290416
$69$	−20.6691	−2.48826
$70$	0	0
$71$	5.93981	0.704926	0.352463	−	0.935826i	$-0.385344\pi$
$71$	5.93981	0.704926	0.352463	+	0.935826i	$0.385344\pi$
$72$	17.9683	2.11759
$73$	9.66860	1.13162	0.565812	−	0.824534i	$-0.308563\pi$
$73$	9.66860	1.13162	0.565812	+	0.824534i	$0.308563\pi$
$74$	1.39989	0.162733
$75$	−3.04811	−0.351965
$76$	−0.113012	−0.0129633
$77$	0	0
$78$	20.6700	2.34042
$79$	−14.6739	−1.65094	−0.825470	−	0.564446i	$-0.809090\pi$
$79$	−14.6739	−1.65094	−0.825470	+	0.564446i	$0.809090\pi$
$80$	−3.91773	−0.438016
$81$	11.7033	1.30036
$82$	7.40970	0.818265
$83$	9.83911	1.07998	0.539991	−	0.841670i	$-0.318427\pi$
$83$	9.83911	1.07998	0.539991	+	0.841670i	$0.318427\pi$
$84$	0	0
$85$	−5.93954	−0.644234
$86$	3.90755	0.421362
$87$	−7.81380	−0.837728
$88$	−10.4170	−1.11046
$89$	−0.523456	−0.0554863	−0.0277431	−	0.999615i	$-0.508832\pi$
$89$	−0.523456	−0.0554863	−0.0277431	+	0.999615i	$0.508832\pi$
$90$	−8.80662	−0.928299
$91$	0	0
$92$	−0.273409	−0.0285049
$93$	−24.5038	−2.54093
$94$	6.52379	0.672877
$95$	2.80285	0.287567
$96$	0.695123	0.0709456
$97$	−7.05294	−0.716117	−0.358059	−	0.933699i	$-0.616561\pi$
$97$	−7.05294	−0.716117	−0.358059	+	0.933699i	$0.616561\pi$
$98$	0	0
$99$	−22.9440	−2.30596
$100$	−0.0403202	−0.00403202
$101$	−2.39273	−0.238086	−0.119043	−	0.992889i	$-0.537983\pi$
$101$	−2.39273	−0.238086	−0.119043	+	0.992889i	$0.537983\pi$
$102$	−25.3440	−2.50943
$103$	−19.5679	−1.92808	−0.964041	−	0.265755i	$-0.914379\pi$
$103$	−19.5679	−1.92808	−0.964041	+	0.265755i	$0.914379\pi$
$104$	13.8360	1.35673
$105$	0	0
$106$	−7.79238	−0.756863
$107$	13.2493	1.28086	0.640429	−	0.768018i	$-0.278757\pi$
$107$	13.2493	1.28086	0.640429	+	0.768018i	$0.278757\pi$
$108$	0.404460	0.0389192
$109$	−5.78796	−0.554386	−0.277193	−	0.960814i	$-0.589404\pi$
$109$	−5.78796	−0.554386	−0.277193	+	0.960814i	$0.589404\pi$
$110$	5.10558	0.486798
$111$	3.04811	0.289314
$112$	0	0
$113$	9.65310	0.908087	0.454044	−	0.890979i	$-0.349981\pi$
$113$	9.65310	0.908087	0.454044	+	0.890979i	$0.349981\pi$
$114$	11.9598	1.12014
$115$	6.78095	0.632327
$116$	−0.103361	−0.00959679
$117$	30.4744	2.81736
$118$	4.17158	0.384025
$119$	0	0
$120$	−8.70605	−0.794750
$121$	2.30165	0.209241
$122$	18.8336	1.70512
$123$	16.1339	1.45474
$124$	−0.324135	−0.0291082
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.0645	−0.893084	−0.446542	−	0.894763i	$-0.647345\pi$
$127$	−10.0645	−0.893084	−0.446542	+	0.894763i	$0.647345\pi$
$128$	−10.9596	−0.968698
$129$	8.50829	0.749113
$130$	−6.78127	−0.594756
$131$	−18.8692	−1.64861	−0.824303	−	0.566148i	$-0.808433\pi$
$131$	−18.8692	−1.64861	−0.824303	+	0.566148i	$0.808433\pi$
$132$	−0.448235	−0.0390139
$133$	0	0
$134$	13.8801	1.19906
$135$	−10.0312	−0.863348
$136$	−16.9646	−1.45470
$137$	−5.74501	−0.490830	−0.245415	−	0.969418i	$-0.578924\pi$
$137$	−5.74501	−0.490830	−0.245415	+	0.969418i	$0.578924\pi$
$138$	28.9343	2.46305
$139$	−11.6048	−0.984310	−0.492155	−	0.870508i	$-0.663791\pi$
$139$	−11.6048	−0.984310	−0.492155	+	0.870508i	$0.663791\pi$
$140$	0	0
$141$	14.2049	1.19627
$142$	−8.31505	−0.697784
$143$	−17.6673	−1.47742
$144$	−24.6463	−2.05386
$145$	2.56349	0.212887
$146$	−13.5349	−1.12016
$147$	0	0
$148$	0.0403202	0.00331430
$149$	12.3462	1.01144	0.505721	−	0.862697i	$-0.331226\pi$
$149$	12.3462	1.01144	0.505721	+	0.862697i	$0.331226\pi$
$150$	4.26700	0.348399
$151$	12.4619	1.01413	0.507067	−	0.861907i	$-0.330730\pi$
$151$	12.4619	1.01413	0.507067	+	0.861907i	$0.330730\pi$
$152$	8.00556	0.649336
$153$	−37.3654	−3.02081
$154$	0	0
$155$	8.03902	0.645710
$156$	0.595348	0.0476660
$157$	6.62753	0.528934	0.264467	−	0.964395i	$-0.414804\pi$
$157$	6.62753	0.528934	0.264467	+	0.964395i	$0.414804\pi$
$158$	20.5417	1.63421
$159$	−16.9671	−1.34558
$160$	−0.228051	−0.0180290
$161$	0	0
$162$	−16.3832	−1.28719
$163$	−13.4708	−1.05511	−0.527557	−	0.849520i	$-0.676892\pi$
$163$	−13.4708	−1.05511	−0.527557	+	0.849520i	$0.676892\pi$
$164$	0.213418	0.0166651
$165$	11.1169	0.865448
$166$	−13.7736	−1.06904
$167$	−17.8404	−1.38053	−0.690267	−	0.723554i	$-0.742507\pi$
$167$	−17.8404	−1.38053	−0.690267	+	0.723554i	$0.742507\pi$
$168$	0	0
$169$	10.4659	0.805066
$170$	8.31468	0.637707
$171$	17.6326	1.34840
$172$	0.112547	0.00858164
$173$	26.2854	1.99844	0.999220	−	0.0394945i	$-0.0125748\pi$
$173$	26.2854	1.99844	0.999220	+	0.0394945i	$0.0125748\pi$
$174$	10.9384	0.829240
$175$	0	0
$176$	14.2885	1.07704
$177$	9.08320	0.682735
$178$	0.732779	0.0549241
$179$	−17.7463	−1.32642	−0.663209	−	0.748435i	$-0.730806\pi$
$179$	−17.7463	−1.32642	−0.663209	+	0.748435i	$0.730806\pi$
$180$	−0.253653	−0.0189062
$181$	−15.3108	−1.13804	−0.569021	−	0.822323i	$-0.692678\pi$
$181$	−15.3108	−1.13804	−0.569021	+	0.822323i	$0.692678\pi$
$182$	0	0
$183$	41.0083	3.03142
$184$	19.3679	1.42782
$185$	−1.00000	−0.0735215
$186$	34.3025	2.51518
$187$	21.6624	1.58411
$188$	0.187901	0.0137041
$189$	0	0
$190$	−3.92368	−0.284653
$191$	−9.37242	−0.678164	−0.339082	−	0.940757i	$-0.610116\pi$
$191$	−9.37242	−0.678164	−0.339082	+	0.940757i	$0.610116\pi$
$192$	−24.8564	−1.79386
$193$	20.8859	1.50340	0.751700	−	0.659506i	$-0.229234\pi$
$193$	20.8859	1.50340	0.751700	+	0.659506i	$0.229234\pi$
$194$	9.87330	0.708862
$195$	−14.7655	−1.05738
$196$	0	0
$197$	25.4155	1.81078	0.905388	−	0.424585i	$-0.139580\pi$
$197$	25.4155	1.81078	0.905388	+	0.424585i	$0.139580\pi$
$198$	32.1190	2.28260
$199$	−9.58159	−0.679221	−0.339611	−	0.940566i	$-0.610295\pi$
$199$	−9.58159	−0.679221	−0.339611	+	0.940566i	$0.610295\pi$
$200$	2.85621	0.201965
$201$	30.2224	2.13173
$202$	3.34955	0.235674
$203$	0	0
$204$	−0.729971	−0.0511082
$205$	−5.29308	−0.369685
$206$	27.3928	1.90855
$207$	42.6587	2.96498
$208$	−18.9781	−1.31590
$209$	−10.2224	−0.707099
$210$	0	0
$211$	26.4828	1.82315	0.911576	−	0.411132i	$-0.134867\pi$
$211$	26.4828	1.82315	0.911576	+	0.411132i	$0.134867\pi$
$212$	−0.224440	−0.0154146
$213$	−18.1052	−1.24055
$214$	−18.5475	−1.26788
$215$	−2.79133	−0.190367
$216$	−28.6512	−1.94947
$217$	0	0
$218$	8.10248	0.548769
$219$	−29.4709	−1.99146
$220$	0.147054	0.00991435
$221$	−28.7721	−1.93542
$222$	−4.26700	−0.286382
$223$	−5.66767	−0.379536	−0.189768	−	0.981829i	$-0.560774\pi$
$223$	−5.66767	−0.379536	−0.189768	+	0.981829i	$0.560774\pi$
$224$	0	0
$225$	6.29096	0.419397
$226$	−13.5132	−0.898887
$227$	21.7868	1.44604	0.723021	−	0.690826i	$-0.242753\pi$
$227$	21.7868	1.44604	0.723021	+	0.690826i	$0.242753\pi$
$228$	0.344472	0.0228132
$229$	−11.8799	−0.785045	−0.392522	−	0.919742i	$-0.628397\pi$
$229$	−11.8799	−0.785045	−0.392522	+	0.919742i	$0.628397\pi$
$230$	−9.49256	−0.625921
$231$	0	0
$232$	7.32189	0.480706
$233$	13.9670	0.915007	0.457503	−	0.889208i	$-0.348744\pi$
$233$	13.9670	0.915007	0.457503	+	0.889208i	$0.348744\pi$
$234$	−42.6607	−2.78881
$235$	−4.66023	−0.304000
$236$	0.120152	0.00782123
$237$	44.7275	2.90537
$238$	0	0
$239$	3.79988	0.245794	0.122897	−	0.992419i	$-0.460782\pi$
$239$	3.79988	0.245794	0.122897	+	0.992419i	$0.460782\pi$
$240$	11.9417	0.770832
$241$	3.06191	0.197235	0.0986174	−	0.995125i	$-0.468558\pi$
$241$	3.06191	0.197235	0.0986174	+	0.995125i	$0.468558\pi$
$242$	−3.22205	−0.207121
$243$	−5.57926	−0.357910
$244$	0.542456	0.0347272
$245$	0	0
$246$	−22.5856	−1.44000
$247$	13.5775	0.863914
$248$	22.9612	1.45804
$249$	−29.9907	−1.90058
$250$	−1.39989	−0.0885365
$251$	−15.8041	−0.997544	−0.498772	−	0.866733i	$-0.666215\pi$
$251$	−15.8041	−0.997544	−0.498772	+	0.866733i	$0.666215\pi$
$252$	0	0
$253$	−24.7311	−1.55483
$254$	14.0892	0.884036
$255$	18.1044	1.13374
$256$	−0.967289	−0.0604556
$257$	6.34024	0.395494	0.197747	−	0.980253i	$-0.436638\pi$
$257$	6.34024	0.395494	0.197747	+	0.980253i	$0.436638\pi$
$258$	−11.9106	−0.741523
$259$	0	0
$260$	−0.195317	−0.0121131
$261$	16.1268	0.998226
$262$	26.4147	1.63190
$263$	24.2607	1.49598	0.747988	−	0.663712i	$-0.231020\pi$
$263$	24.2607	1.49598	0.747988	+	0.663712i	$0.231020\pi$
$264$	31.7522	1.95421
$265$	5.56644	0.341944
$266$	0	0
$267$	1.59555	0.0976462
$268$	0.399781	0.0244205
$269$	10.6715	0.650654	0.325327	−	0.945602i	$-0.394526\pi$
$269$	10.6715	0.650654	0.325327	+	0.945602i	$0.394526\pi$
$270$	14.0425	0.854601
$271$	−3.12586	−0.189882	−0.0949411	−	0.995483i	$-0.530266\pi$
$271$	−3.12586	−0.189882	−0.0949411	+	0.995483i	$0.530266\pi$
$272$	23.2695	1.41092
$273$	0	0
$274$	8.04236	0.485857
$275$	−3.64714	−0.219931
$276$	0.833381	0.0501636
$277$	15.4510	0.928362	0.464181	−	0.885740i	$-0.346349\pi$
$277$	15.4510	0.928362	0.464181	+	0.885740i	$0.346349\pi$
$278$	16.2455	0.974337
$279$	50.5731	3.02773
$280$	0	0
$281$	9.86108	0.588263	0.294131	−	0.955765i	$-0.404970\pi$
$281$	9.86108	0.588263	0.294131	+	0.955765i	$0.404970\pi$
$282$	−19.8852	−1.18415
$283$	−9.95956	−0.592034	−0.296017	−	0.955183i	$-0.595659\pi$
$283$	−9.95956	−0.592034	−0.296017	+	0.955183i	$0.595659\pi$
$284$	−0.239494	−0.0142114
$285$	−8.54340	−0.506067
$286$	24.7322	1.46245
$287$	0	0
$288$	−1.43466	−0.0845379
$289$	18.2781	1.07518
$290$	−3.58860	−0.210730
$291$	21.4981	1.26024
$292$	−0.389840	−0.0228137
$293$	−12.1366	−0.709026	−0.354513	−	0.935051i	$-0.615353\pi$
$293$	−12.1366	−0.709026	−0.354513	+	0.935051i	$0.615353\pi$
$294$	0	0
$295$	−2.97995	−0.173499
$296$	−2.85621	−0.166014
$297$	36.5852	2.12289
$298$	−17.2833	−1.00120
$299$	32.8480	1.89965
$300$	0.122900	0.00709565
$301$	0	0
$302$	−17.4452	−1.00386
$303$	7.29331	0.418990
$304$	−10.9808	−0.629794
$305$	−13.4537	−0.770357
$306$	52.3073	2.99021
$307$	5.55093	0.316808	0.158404	−	0.987374i	$-0.449365\pi$
$307$	5.55093	0.316808	0.158404	+	0.987374i	$0.449365\pi$
$308$	0	0
$309$	59.6450	3.39309
$310$	−11.2537	−0.639168
$311$	3.09306	0.175391	0.0876956	−	0.996147i	$-0.472050\pi$
$311$	3.09306	0.175391	0.0876956	+	0.996147i	$0.472050\pi$
$312$	−42.1735	−2.38760
$313$	24.9749	1.41166	0.705832	−	0.708380i	$-0.250574\pi$
$313$	24.9749	1.41166	0.705832	+	0.708380i	$0.250574\pi$
$314$	−9.27778	−0.523575
$315$	0	0
$316$	0.591654	0.0332831
$317$	−12.5972	−0.707529	−0.353765	−	0.935335i	$-0.615099\pi$
$317$	−12.5972	−0.707529	−0.353765	+	0.935335i	$0.615099\pi$
$318$	23.7520	1.33195
$319$	−9.34943	−0.523468
$320$	8.15471	0.455862
$321$	−40.3853	−2.25409
$322$	0	0
$323$	−16.6477	−0.926301
$324$	−0.471878	−0.0262155
$325$	4.84416	0.268705
$326$	18.8576	1.04442
$327$	17.6423	0.975623
$328$	−15.1182	−0.834761
$329$	0	0
$330$	−15.5624	−0.856680
$331$	32.8035	1.80304	0.901521	−	0.432736i	$-0.142452\pi$
$331$	32.8035	1.80304	0.901521	+	0.432736i	$0.142452\pi$
$332$	−0.396715	−0.0217726
$333$	−6.29096	−0.344742
$334$	24.9746	1.36655
$335$	−9.91515	−0.541722
$336$	0	0
$337$	−7.88075	−0.429292	−0.214646	−	0.976692i	$-0.568860\pi$
$337$	−7.88075	−0.429292	−0.214646	+	0.976692i	$0.568860\pi$
$338$	−14.6510	−0.796910
$339$	−29.4237	−1.59807
$340$	0.239484	0.0129878
$341$	−29.3195	−1.58774
$342$	−24.6837	−1.33474
$343$	0	0
$344$	−7.97265	−0.429857
$345$	−20.6691	−1.11279
$346$	−36.7965	−1.97819
$347$	17.2018	0.923442	0.461721	−	0.887025i	$-0.347232\pi$
$347$	17.2018	0.923442	0.461721	+	0.887025i	$0.347232\pi$
$348$	0.315054	0.0168887
$349$	19.5847	1.04834	0.524172	−	0.851613i	$-0.324375\pi$
$349$	19.5847	1.04834	0.524172	+	0.851613i	$0.324375\pi$
$350$	0	0
$351$	−48.5927	−2.59368
$352$	0.831733	0.0443315
$353$	5.04907	0.268735	0.134367	−	0.990932i	$-0.457100\pi$
$353$	5.04907	0.268735	0.134367	+	0.990932i	$0.457100\pi$
$354$	−12.7154	−0.675818
$355$	5.93981	0.315252
$356$	0.0211059	0.00111861
$357$	0	0
$358$	24.8427	1.31298
$359$	18.4078	0.971528	0.485764	−	0.874090i	$-0.338541\pi$
$359$	18.4078	0.971528	0.485764	+	0.874090i	$0.338541\pi$
$360$	17.9683	0.947014
$361$	−11.1440	−0.586527
$362$	21.4334	1.12651
$363$	−7.01569	−0.368228
$364$	0	0
$365$	9.66860	0.506078
$366$	−57.4070	−3.00071
$367$	−15.1703	−0.791885	−0.395943	−	0.918275i	$-0.629582\pi$
$367$	−15.1703	−0.791885	−0.395943	+	0.918275i	$0.629582\pi$
$368$	−26.5660	−1.38485
$369$	−33.2985	−1.73345
$370$	1.39989	0.0727766
$371$	0	0
$372$	0.987998	0.0512253
$373$	−11.4128	−0.590935	−0.295467	−	0.955353i	$-0.595475\pi$
$373$	−11.4128	−0.590935	−0.295467	+	0.955353i	$0.595475\pi$
$374$	−30.3248	−1.56806
$375$	−3.04811	−0.157404
$376$	−13.3106	−0.686442
$377$	12.4180	0.639558
$378$	0	0
$379$	−25.7791	−1.32419	−0.662093	−	0.749422i	$-0.730331\pi$
$379$	−25.7791	−1.32419	−0.662093	+	0.749422i	$0.730331\pi$
$380$	−0.113012	−0.00579738
$381$	30.6778	1.57167
$382$	13.1203	0.671294
$383$	−35.9013	−1.83447	−0.917236	−	0.398345i	$-0.869585\pi$
$383$	−35.9013	−1.83447	−0.917236	+	0.398345i	$0.869585\pi$
$384$	33.4059	1.70474
$385$	0	0
$386$	−29.2379	−1.48817
$387$	−17.5602	−0.892633
$388$	0.284376	0.0144370
$389$	13.7326	0.696270	0.348135	−	0.937444i	$-0.386815\pi$
$389$	13.7326	0.696270	0.348135	+	0.937444i	$0.386815\pi$
$390$	20.6700	1.04667
$391$	−40.2757	−2.03683
$392$	0	0
$393$	57.5152	2.90126
$394$	−35.5787	−1.79243
$395$	−14.6739	−0.738323
$396$	0.925108	0.0464884
$397$	−11.4068	−0.572489	−0.286244	−	0.958157i	$-0.592407\pi$
$397$	−11.4068	−0.572489	−0.286244	+	0.958157i	$0.592407\pi$
$398$	13.4131	0.672340
$399$	0	0
$400$	−3.91773	−0.195887
$401$	−2.75338	−0.137497	−0.0687485	−	0.997634i	$-0.521901\pi$
$401$	−2.75338	−0.137497	−0.0687485	+	0.997634i	$0.521901\pi$
$402$	−42.3080	−2.11013
$403$	38.9423	1.93985
$404$	0.0964755	0.00479984
$405$	11.7033	0.581540
$406$	0	0
$407$	3.64714	0.180782
$408$	51.7099	2.56002
$409$	−8.95246	−0.442671	−0.221335	−	0.975198i	$-0.571042\pi$
$409$	−8.95246	−0.442671	−0.221335	+	0.975198i	$0.571042\pi$
$410$	7.40970	0.365939
$411$	17.5114	0.863775
$412$	0.788981	0.0388703
$413$	0	0
$414$	−59.7173	−2.93494
$415$	9.83911	0.482983
$416$	−1.10471	−0.0541630
$417$	35.3728	1.73221
$418$	14.3102	0.699935
$419$	−11.6889	−0.571038	−0.285519	−	0.958373i	$-0.592166\pi$
$419$	−11.6889	−0.571038	−0.285519	+	0.958373i	$0.592166\pi$
$420$	0	0
$421$	−10.1867	−0.496468	−0.248234	−	0.968700i	$-0.579850\pi$
$421$	−10.1867	−0.496468	−0.248234	+	0.968700i	$0.579850\pi$
$422$	−37.0729	−1.80468
$423$	−29.3173	−1.42546
$424$	15.8989	0.772121
$425$	−5.93954	−0.288110
$426$	25.3452	1.22798
$427$	0	0
$428$	−0.534214	−0.0258222
$429$	53.8519	2.60000
$430$	3.90755	0.188439
$431$	−32.1101	−1.54669	−0.773344	−	0.633986i	$-0.781418\pi$
$431$	−32.1101	−1.54669	−0.773344	+	0.633986i	$0.781418\pi$
$432$	39.2995	1.89080
$433$	21.8491	1.05000	0.525001	−	0.851102i	$-0.324065\pi$
$433$	21.8491	1.05000	0.525001	+	0.851102i	$0.324065\pi$
$434$	0	0
$435$	−7.81380	−0.374643
$436$	0.233372	0.0111765
$437$	19.0060	0.909181
$438$	41.2559	1.97128
$439$	−36.1376	−1.72475	−0.862376	−	0.506268i	$-0.831025\pi$
$439$	−36.1376	−1.72475	−0.862376	+	0.506268i	$0.831025\pi$
$440$	−10.4170	−0.496612
$441$	0	0
$442$	40.2776	1.91581
$443$	−5.66883	−0.269334	−0.134667	−	0.990891i	$-0.542997\pi$
$443$	−5.66883	−0.269334	−0.134667	+	0.990891i	$0.542997\pi$
$444$	−0.122900	−0.00583259
$445$	−0.523456	−0.0248142
$446$	7.93410	0.375690
$447$	−37.6326	−1.77996
$448$	0	0
$449$	−2.14411	−0.101187	−0.0505935	−	0.998719i	$-0.516111\pi$
$449$	−2.14411	−0.101187	−0.0505935	+	0.998719i	$0.516111\pi$
$450$	−8.80662	−0.415148
$451$	19.3046	0.909019
$452$	−0.389215	−0.0183071
$453$	−37.9851	−1.78470
$454$	−30.4991	−1.43139
$455$	0	0
$456$	−24.4018	−1.14272
$457$	3.63740	0.170150	0.0850752	−	0.996375i	$-0.472887\pi$
$457$	3.63740	0.170150	0.0850752	+	0.996375i	$0.472887\pi$
$458$	16.6305	0.777091
$459$	59.5807	2.78099
$460$	−0.273409	−0.0127478
$461$	−15.1913	−0.707530	−0.353765	−	0.935334i	$-0.615099\pi$
$461$	−15.1913	−0.707530	−0.353765	+	0.935334i	$0.615099\pi$
$462$	0	0
$463$	13.1805	0.612548	0.306274	−	0.951943i	$-0.400918\pi$
$463$	13.1805	0.612548	0.306274	+	0.951943i	$0.400918\pi$
$464$	−10.0431	−0.466239
$465$	−24.5038	−1.13634
$466$	−19.5522	−0.905737
$467$	16.8444	0.779467	0.389733	−	0.920928i	$-0.372567\pi$
$467$	16.8444	0.779467	0.389733	+	0.920928i	$0.372567\pi$
$468$	−1.22873	−0.0567982
$469$	0	0
$470$	6.52379	0.300920
$471$	−20.2014	−0.930832
$472$	−8.51137	−0.391767
$473$	10.1804	0.468095
$474$	−62.6134	−2.87593
$475$	2.80285	0.128604
$476$	0	0
$477$	35.0182	1.60337
$478$	−5.31939	−0.243303
$479$	−18.9907	−0.867706	−0.433853	−	0.900984i	$-0.642846\pi$
$479$	−18.9907	−0.867706	−0.433853	+	0.900984i	$0.642846\pi$
$480$	0.695123	0.0317279
$481$	−4.84416	−0.220875
$482$	−4.28632	−0.195236
$483$	0	0
$484$	−0.0928031	−0.00421832
$485$	−7.05294	−0.320257
$486$	7.81033	0.354284
$487$	−36.5957	−1.65831	−0.829156	−	0.559018i	$-0.811178\pi$
$487$	−36.5957	−1.65831	−0.829156	+	0.559018i	$0.811178\pi$
$488$	−38.4267	−1.73949
$489$	41.0604	1.85682
$490$	0	0
$491$	−36.8339	−1.66229	−0.831146	−	0.556054i	$-0.812315\pi$
$491$	−36.8339	−1.66229	−0.831146	+	0.556054i	$0.812315\pi$
$492$	−0.650521	−0.0293278
$493$	−15.2260	−0.685743
$494$	−19.0069	−0.855161
$495$	−22.9440	−1.03126
$496$	−31.4947	−1.41416
$497$	0	0
$498$	41.9835	1.88133
$499$	3.91663	0.175332	0.0876661	−	0.996150i	$-0.472059\pi$
$499$	3.91663	0.175332	0.0876661	+	0.996150i	$0.472059\pi$
$500$	−0.0403202	−0.00180317
$501$	54.3796	2.42950
$502$	22.1239	0.987437
$503$	−11.3334	−0.505331	−0.252665	−	0.967554i	$-0.581307\pi$
$503$	−11.3334	−0.505331	−0.252665	+	0.967554i	$0.581307\pi$
$504$	0	0
$505$	−2.39273	−0.106475
$506$	34.6207	1.53908
$507$	−31.9011	−1.41678
$508$	0.405805	0.0180047
$509$	−18.0185	−0.798656	−0.399328	−	0.916808i	$-0.630756\pi$
$509$	−18.0185	−0.798656	−0.399328	+	0.916808i	$0.630756\pi$
$510$	−25.3440	−1.12225
$511$	0	0
$512$	23.2732	1.02854
$513$	−28.1160	−1.24135
$514$	−8.87562	−0.391487
$515$	−19.5679	−0.862264
$516$	−0.343056	−0.0151022
$517$	16.9965	0.747506
$518$	0	0
$519$	−80.1206	−3.51690
$520$	13.8360	0.606747
$521$	−11.1708	−0.489403	−0.244702	−	0.969598i	$-0.578690\pi$
$521$	−11.1708	−0.489403	−0.244702	+	0.969598i	$0.578690\pi$
$522$	−22.5757	−0.988112
$523$	−20.7450	−0.907115	−0.453557	−	0.891227i	$-0.649845\pi$
$523$	−20.7450	−0.907115	−0.453557	+	0.891227i	$0.649845\pi$
$524$	0.760809	0.0332361
$525$	0	0
$526$	−33.9621	−1.48082
$527$	−47.7481	−2.07994
$528$	−43.5530	−1.89540
$529$	22.9813	0.999187
$530$	−7.79238	−0.338479
$531$	−18.7467	−0.813538
$532$	0	0
$533$	−25.6405	−1.11061
$534$	−2.23359	−0.0966569
$535$	13.2493	0.572817
$536$	−28.3198	−1.22323
$537$	54.0925	2.33426
$538$	−14.9389	−0.644062
$539$	0	0
$540$	0.404460	0.0174052
$541$	−15.4967	−0.666257	−0.333128	−	0.942881i	$-0.608104\pi$
$541$	−15.4967	−0.666257	−0.333128	+	0.942881i	$0.608104\pi$
$542$	4.37584	0.187958
$543$	46.6689	2.00276
$544$	1.35452	0.0580744
$545$	−5.78796	−0.247929
$546$	0	0
$547$	15.7560	0.673677	0.336838	−	0.941562i	$-0.390642\pi$
$547$	15.7560	0.673677	0.336838	+	0.941562i	$0.390642\pi$
$548$	0.231640	0.00989518
$549$	−84.6366	−3.61220
$550$	5.10558	0.217703
$551$	7.18510	0.306096
$552$	−59.0353	−2.51271
$553$	0	0
$554$	−21.6297	−0.918956
$555$	3.04811	0.129385
$556$	0.467910	0.0198438
$557$	−20.8340	−0.882765	−0.441382	−	0.897319i	$-0.645512\pi$
$557$	−20.8340	−0.882765	−0.441382	+	0.897319i	$0.645512\pi$
$558$	−70.7966	−2.99706
$559$	−13.5217	−0.571905
$560$	0	0
$561$	−66.0292	−2.78775
$562$	−13.8044	−0.582303
$563$	−17.5363	−0.739066	−0.369533	−	0.929218i	$-0.620482\pi$
$563$	−17.5363	−0.739066	−0.369533	+	0.929218i	$0.620482\pi$
$564$	−0.572744	−0.0241169
$565$	9.65310	0.406109
$566$	13.9422	0.586036
$567$	0	0
$568$	16.9654	0.711851
$569$	27.6209	1.15793	0.578964	−	0.815353i	$-0.303457\pi$
$569$	27.6209	1.15793	0.578964	+	0.815353i	$0.303457\pi$
$570$	11.9598	0.500940
$571$	6.93846	0.290366	0.145183	−	0.989405i	$-0.453623\pi$
$571$	6.93846	0.290366	0.145183	+	0.989405i	$0.453623\pi$
$572$	0.712351	0.0297849
$573$	28.5681	1.19345
$574$	0	0
$575$	6.78095	0.282785
$576$	51.3009	2.13754
$577$	17.6589	0.735151	0.367576	−	0.929994i	$-0.380188\pi$
$577$	17.6589	0.735151	0.367576	+	0.929994i	$0.380188\pi$
$578$	−25.5873	−1.06429
$579$	−63.6624	−2.64572
$580$	−0.103361	−0.00429182
$581$	0	0
$582$	−30.0949	−1.24747
$583$	−20.3016	−0.840806
$584$	27.6156	1.14274
$585$	30.4744	1.25996
$586$	16.9898	0.701843
$587$	1.32023	0.0544918	0.0272459	−	0.999629i	$-0.491326\pi$
$587$	1.32023	0.0544918	0.0272459	+	0.999629i	$0.491326\pi$
$588$	0	0
$589$	22.5322	0.928423
$590$	4.17158	0.171741
$591$	−77.4690	−3.18665
$592$	3.91773	0.161018
$593$	26.0840	1.07114	0.535572	−	0.844490i	$-0.320096\pi$
$593$	26.0840	1.07114	0.535572	+	0.844490i	$0.320096\pi$
$594$	−51.2151	−2.10138
$595$	0	0
$596$	−0.497803	−0.0203908
$597$	29.2057	1.19531
$598$	−45.9834	−1.88040
$599$	28.0723	1.14700	0.573502	−	0.819204i	$-0.305585\pi$
$599$	28.0723	1.14700	0.573502	+	0.819204i	$0.305585\pi$
$600$	−8.70605	−0.355423
$601$	23.9728	0.977873	0.488936	−	0.872319i	$-0.337385\pi$
$601$	23.9728	0.977873	0.488936	+	0.872319i	$0.337385\pi$
$602$	0	0
$603$	−62.3758	−2.54014
$604$	−0.502466	−0.0204450
$605$	2.30165	0.0935755
$606$	−10.2098	−0.414745
$607$	−44.5284	−1.80735	−0.903675	−	0.428219i	$-0.859141\pi$
$607$	−44.5284	−1.80735	−0.903675	+	0.428219i	$0.859141\pi$
$608$	−0.639193	−0.0259227
$609$	0	0
$610$	18.8336	0.762552
$611$	−22.5749	−0.913282
$612$	1.50658	0.0608999
$613$	−1.95068	−0.0787871	−0.0393936	−	0.999224i	$-0.512543\pi$
$613$	−1.95068	−0.0787871	−0.0393936	+	0.999224i	$0.512543\pi$
$614$	−7.77067	−0.313599
$615$	16.1339	0.650580
$616$	0	0
$617$	−35.3685	−1.42388	−0.711940	−	0.702240i	$-0.752183\pi$
$617$	−35.3685	−1.42388	−0.711940	+	0.702240i	$0.752183\pi$
$618$	−83.4962	−3.35871
$619$	−4.35625	−0.175093	−0.0875463	−	0.996160i	$-0.527903\pi$
$619$	−4.35625	−0.175093	−0.0875463	+	0.996160i	$0.527903\pi$
$620$	−0.324135	−0.0130176
$621$	−68.0210	−2.72959
$622$	−4.32993	−0.173614
$623$	0	0
$624$	57.8473	2.31575
$625$	1.00000	0.0400000
$626$	−34.9620	−1.39736
$627$	31.1590	1.24437
$628$	−0.267223	−0.0106634
$629$	5.93954	0.236825
$630$	0	0
$631$	26.0056	1.03526	0.517632	−	0.855603i	$-0.326813\pi$
$631$	26.0056	1.03526	0.517632	+	0.855603i	$0.326813\pi$
$632$	−41.9117	−1.66716
$633$	−80.7224	−3.20843
$634$	17.6346	0.700361
$635$	−10.0645	−0.399399
$636$	0.684117	0.0271270
$637$	0	0
$638$	13.0881	0.518164
$639$	37.3671	1.47822
$640$	−10.9596	−0.433215
$641$	2.10114	0.0829900	0.0414950	−	0.999139i	$-0.486788\pi$
$641$	2.10114	0.0829900	0.0414950	+	0.999139i	$0.486788\pi$
$642$	56.5348	2.23125
$643$	−20.8643	−0.822807	−0.411403	−	0.911453i	$-0.634961\pi$
$643$	−20.8643	−0.822807	−0.411403	+	0.911453i	$0.634961\pi$
$644$	0	0
$645$	8.50829	0.335013
$646$	23.3048	0.916916
$647$	25.4420	1.00023	0.500113	−	0.865960i	$-0.333292\pi$
$647$	25.4420	1.00023	0.500113	+	0.865960i	$0.333292\pi$
$648$	33.4271	1.31314
$649$	10.8683	0.426618
$650$	−6.78127	−0.265983
$651$	0	0
$652$	0.543145	0.0212712
$653$	31.2907	1.22450	0.612249	−	0.790665i	$-0.290265\pi$
$653$	31.2907	1.22450	0.612249	+	0.790665i	$0.290265\pi$
$654$	−24.6972	−0.965738
$655$	−18.8692	−0.737279
$656$	20.7369	0.809639
$657$	60.8247	2.37300
$658$	0	0
$659$	35.9609	1.40084	0.700419	−	0.713732i	$-0.252997\pi$
$659$	35.9609	1.40084	0.700419	+	0.713732i	$0.252997\pi$
$660$	−0.448235	−0.0174475
$661$	−43.9179	−1.70821	−0.854104	−	0.520102i	$-0.825894\pi$
$661$	−43.9179	−1.70821	−0.854104	+	0.520102i	$0.825894\pi$
$662$	−45.9211	−1.78477
$663$	87.7003	3.40600
$664$	28.1026	1.09059
$665$	0	0
$666$	8.80662	0.341250
$667$	17.3829	0.673070
$668$	0.719330	0.0278317
$669$	17.2757	0.667916
$670$	13.8801	0.536234
$671$	49.0676	1.89423
$672$	0	0
$673$	−26.5814	−1.02464	−0.512319	−	0.858796i	$-0.671213\pi$
$673$	−26.5814	−1.02464	−0.512319	+	0.858796i	$0.671213\pi$
$674$	11.0321	0.424942
$675$	−10.0312	−0.386101
$676$	−0.421986	−0.0162302
$677$	40.9319	1.57314	0.786571	−	0.617500i	$-0.211855\pi$
$677$	40.9319	1.57314	0.786571	+	0.617500i	$0.211855\pi$
$678$	41.1898	1.58188
$679$	0	0
$680$	−16.9646	−0.650563
$681$	−66.4086	−2.54478
$682$	41.0439	1.57165
$683$	−21.6347	−0.827830	−0.413915	−	0.910315i	$-0.635839\pi$
$683$	−21.6347	−0.827830	−0.413915	+	0.910315i	$0.635839\pi$
$684$	−0.710952	−0.0271839
$685$	−5.74501	−0.219506
$686$	0	0
$687$	36.2112	1.38154
$688$	10.9357	0.416920
$689$	26.9647	1.02727
$690$	28.9343	1.10151
$691$	5.20391	0.197966	0.0989830	−	0.995089i	$-0.468441\pi$
$691$	5.20391	0.197966	0.0989830	+	0.995089i	$0.468441\pi$
$692$	−1.05983	−0.0402888
$693$	0	0
$694$	−24.0806	−0.914086
$695$	−11.6048	−0.440197
$696$	−22.3179	−0.845958
$697$	31.4385	1.19082
$698$	−27.4163	−1.03772
$699$	−42.5728	−1.61025
$700$	0	0
$701$	−9.61311	−0.363082	−0.181541	−	0.983383i	$-0.558109\pi$
$701$	−9.61311	−0.363082	−0.181541	+	0.983383i	$0.558109\pi$
$702$	68.0242	2.56741
$703$	−2.80285	−0.105712
$704$	−29.7414	−1.12092
$705$	14.2049	0.534987
$706$	−7.06812	−0.266012
$707$	0	0
$708$	−0.366236	−0.0137640
$709$	28.4078	1.06688	0.533439	−	0.845839i	$-0.320899\pi$
$709$	28.4078	1.06688	0.533439	+	0.845839i	$0.320899\pi$
$710$	−8.31505	−0.312058
$711$	−92.3127	−3.46200
$712$	−1.49510	−0.0560314
$713$	54.5122	2.04150
$714$	0	0
$715$	−17.6673	−0.660721
$716$	0.715533	0.0267407
$717$	−11.5824	−0.432554
$718$	−25.7689	−0.961685
$719$	7.83731	0.292282	0.146141	−	0.989264i	$-0.453315\pi$
$719$	7.83731	0.292282	0.146141	+	0.989264i	$0.453315\pi$
$720$	−24.6463	−0.918513
$721$	0	0
$722$	15.6003	0.580584
$723$	−9.33302	−0.347099
$724$	0.617335	0.0229431
$725$	2.56349	0.0952058
$726$	9.82116	0.364497
$727$	−26.6047	−0.986712	−0.493356	−	0.869827i	$-0.664230\pi$
$727$	−26.6047	−0.986712	−0.493356	+	0.869827i	$0.664230\pi$
$728$	0	0
$729$	−18.1036	−0.670505
$730$	−13.5349	−0.500950
$731$	16.5792	0.613205
$732$	−1.65346	−0.0611138
$733$	53.3717	1.97133	0.985664	−	0.168718i	$-0.0539627\pi$
$733$	53.3717	1.97133	0.985664	+	0.168718i	$0.0539627\pi$
$734$	21.2367	0.783862
$735$	0	0
$736$	−1.54640	−0.0570011
$737$	36.1620	1.33204
$738$	46.6141	1.71589
$739$	−19.9661	−0.734466	−0.367233	−	0.930129i	$-0.619695\pi$
$739$	−19.9661	−0.734466	−0.367233	+	0.930129i	$0.619695\pi$
$740$	0.0403202	0.00148220
$741$	−41.3856	−1.52034
$742$	0	0
$743$	4.54143	0.166609	0.0833044	−	0.996524i	$-0.473453\pi$
$743$	4.54143	0.166609	0.0833044	+	0.996524i	$0.473453\pi$
$744$	−69.9881	−2.56589
$745$	12.3462	0.452331
$746$	15.9767	0.584948
$747$	61.8974	2.26471
$748$	−0.873431	−0.0319358
$749$	0	0
$750$	4.26700	0.155809
$751$	−15.2284	−0.555692	−0.277846	−	0.960626i	$-0.589620\pi$
$751$	−15.2284	−0.555692	−0.277846	+	0.960626i	$0.589620\pi$
$752$	18.2575	0.665784
$753$	48.1725	1.75550
$754$	−17.3837	−0.633078
$755$	12.4619	0.453534
$756$	0	0
$757$	−32.8000	−1.19214	−0.596069	−	0.802934i	$-0.703271\pi$
$757$	−32.8000	−1.19214	−0.596069	+	0.802934i	$0.703271\pi$
$758$	36.0878	1.31077
$759$	75.3830	2.73623
$760$	8.00556	0.290392
$761$	−41.4805	−1.50367	−0.751834	−	0.659352i	$-0.770831\pi$
$761$	−41.4805	−1.50367	−0.751834	+	0.659352i	$0.770831\pi$
$762$	−42.9454	−1.55575
$763$	0	0
$764$	0.377898	0.0136719
$765$	−37.3654	−1.35095
$766$	50.2577	1.81589
$767$	−14.4353	−0.521229
$768$	2.94840	0.106391
$769$	−4.98364	−0.179715	−0.0898574	−	0.995955i	$-0.528641\pi$
$769$	−4.98364	−0.179715	−0.0898574	+	0.995955i	$0.528641\pi$
$770$	0	0
$771$	−19.3257	−0.696000
$772$	−0.842123	−0.0303087
$773$	7.58656	0.272869	0.136435	−	0.990649i	$-0.456436\pi$
$773$	7.58656	0.272869	0.136435	+	0.990649i	$0.456436\pi$
$774$	24.5822	0.883590
$775$	8.03902	0.288770
$776$	−20.1447	−0.723153
$777$	0	0
$778$	−19.2241	−0.689216
$779$	−14.8357	−0.531545
$780$	0.595348	0.0213169
$781$	−21.6633	−0.775175
$782$	56.3814	2.01620
$783$	−25.7149	−0.918975
$784$	0	0
$785$	6.62753	0.236547
$786$	−80.5148	−2.87187
$787$	10.9337	0.389744	0.194872	−	0.980829i	$-0.437571\pi$
$787$	10.9337	0.389744	0.194872	+	0.980829i	$0.437571\pi$
$788$	−1.02476	−0.0365054
$789$	−73.9491	−2.63266
$790$	20.5417	0.730842
$791$	0	0
$792$	−65.5331	−2.32862
$793$	−65.1718	−2.31432
$794$	15.9682	0.566689
$795$	−16.9671	−0.601761
$796$	0.386332	0.0136932
$797$	6.03515	0.213776	0.106888	−	0.994271i	$-0.465911\pi$
$797$	6.03515	0.213776	0.106888	+	0.994271i	$0.465911\pi$
$798$	0	0
$799$	27.6796	0.979234
$800$	−0.228051	−0.00806280
$801$	−3.29304	−0.116354
$802$	3.85441	0.136104
$803$	−35.2628	−1.24440
$804$	−1.21857	−0.0429758
$805$	0	0
$806$	−54.5147	−1.92020
$807$	−32.5279	−1.14504
$808$	−6.83416	−0.240425
$809$	−38.0682	−1.33841	−0.669203	−	0.743080i	$-0.733364\pi$
$809$	−38.0682	−1.33841	−0.669203	+	0.743080i	$0.733364\pi$
$810$	−16.3832	−0.575649
$811$	−48.0542	−1.68741	−0.843705	−	0.536807i	$-0.819630\pi$
$811$	−48.0542	−1.68741	−0.843705	+	0.536807i	$0.819630\pi$
$812$	0	0
$813$	9.52794	0.334160
$814$	−5.10558	−0.178951
$815$	−13.4708	−0.471861
$816$	−70.9280	−2.48298
$817$	−7.82371	−0.273717
$818$	12.5324	0.438186
$819$	0	0
$820$	0.213418	0.00745288
$821$	−36.5629	−1.27605	−0.638027	−	0.770014i	$-0.720249\pi$
$821$	−36.5629	−1.27605	−0.638027	+	0.770014i	$0.720249\pi$
$822$	−24.5140	−0.855023
$823$	−39.2913	−1.36961	−0.684804	−	0.728727i	$-0.740112\pi$
$823$	−39.2913	−1.36961	−0.684804	+	0.728727i	$0.740112\pi$
$824$	−55.8901	−1.94702
$825$	11.1169	0.387040
$826$	0	0
$827$	−2.05042	−0.0713002	−0.0356501	−	0.999364i	$-0.511350\pi$
$827$	−2.05042	−0.0713002	−0.0356501	+	0.999364i	$0.511350\pi$
$828$	−1.72001	−0.0597744
$829$	8.56716	0.297550	0.148775	−	0.988871i	$-0.452467\pi$
$829$	8.56716	0.297550	0.148775	+	0.988871i	$0.452467\pi$
$830$	−13.7736	−0.478090
$831$	−47.0964	−1.63375
$832$	39.5027	1.36951
$833$	0	0
$834$	−49.5179	−1.71466
$835$	−17.8404	−0.617394
$836$	0.412170	0.0142552
$837$	−80.6410	−2.78736
$838$	16.3631	0.565253
$839$	3.02431	0.104411	0.0522053	−	0.998636i	$-0.483375\pi$
$839$	3.02431	0.104411	0.0522053	+	0.998636i	$0.483375\pi$
$840$	0	0
$841$	−22.4285	−0.773397
$842$	14.2602	0.491438
$843$	−30.0576	−1.03524
$844$	−1.06779	−0.0367549
$845$	10.4659	0.360036
$846$	41.0409	1.41101
$847$	0	0
$848$	−21.8078	−0.748884
$849$	30.3578	1.04188
$850$	8.31468	0.285191
$851$	−6.78095	−0.232448
$852$	0.730005	0.0250095
$853$	−34.8869	−1.19451	−0.597253	−	0.802053i	$-0.703741\pi$
$853$	−34.8869	−1.19451	−0.597253	+	0.802053i	$0.703741\pi$
$854$	0	0
$855$	17.6326	0.603024
$856$	37.8428	1.29344
$857$	1.35026	0.0461240	0.0230620	−	0.999734i	$-0.492658\pi$
$857$	1.35026	0.0461240	0.0230620	+	0.999734i	$0.492658\pi$
$858$	−75.3865	−2.57365
$859$	31.2917	1.06766	0.533829	−	0.845592i	$-0.320753\pi$
$859$	31.2917	1.06766	0.533829	+	0.845592i	$0.320753\pi$
$860$	0.112547	0.00383783
$861$	0	0
$862$	44.9505	1.53102
$863$	14.1149	0.480476	0.240238	−	0.970714i	$-0.422774\pi$
$863$	14.1149	0.480476	0.240238	+	0.970714i	$0.422774\pi$
$864$	2.28762	0.0778264
$865$	26.2854	0.893729
$866$	−30.5863	−1.03936
$867$	−55.7137	−1.89214
$868$	0	0
$869$	53.5177	1.81546
$870$	10.9384	0.370848
$871$	−48.0305	−1.62745
$872$	−16.5317	−0.559833
$873$	−44.3697	−1.50169
$874$	−26.6063	−0.899970
$875$	0	0
$876$	1.18827	0.0401480
$877$	40.0006	1.35072	0.675361	−	0.737487i	$-0.263988\pi$
$877$	40.0006	1.35072	0.675361	+	0.737487i	$0.263988\pi$
$878$	50.5885	1.70728
$879$	36.9936	1.24776
$880$	14.2885	0.481666
$881$	29.8247	1.00482	0.502409	−	0.864630i	$-0.332447\pi$
$881$	29.8247	1.00482	0.502409	+	0.864630i	$0.332447\pi$
$882$	0	0
$883$	−32.7577	−1.10239	−0.551193	−	0.834378i	$-0.685827\pi$
$883$	−32.7577	−1.10239	−0.551193	+	0.834378i	$0.685827\pi$
$884$	1.16010	0.0390182
$885$	9.08320	0.305328
$886$	7.93572	0.266606
$887$	−14.6579	−0.492164	−0.246082	−	0.969249i	$-0.579143\pi$
$887$	−14.6579	−0.492164	−0.246082	+	0.969249i	$0.579143\pi$
$888$	8.70605	0.292156
$889$	0	0
$890$	0.732779	0.0245628
$891$	−42.6835	−1.42995
$892$	0.228522	0.00765148
$893$	−13.0619	−0.437101
$894$	52.6814	1.76193
$895$	−17.7463	−0.593192
$896$	0	0
$897$	−100.124	−3.34305
$898$	3.00151	0.100162
$899$	20.6080	0.687315
$900$	−0.253653	−0.00845509
$901$	−33.0621	−1.10146
$902$	−27.0243	−0.899809
$903$	0	0
$904$	27.5713	0.917009
$905$	−15.3108	−0.508948
$906$	53.1748	1.76662
$907$	9.66145	0.320803	0.160402	−	0.987052i	$-0.448721\pi$
$907$	9.66145	0.320803	0.160402	+	0.987052i	$0.448721\pi$
$908$	−0.878449	−0.0291524
$909$	−15.0526	−0.499263
$910$	0	0
$911$	15.1053	0.500460	0.250230	−	0.968186i	$-0.419494\pi$
$911$	15.1053	0.500460	0.250230	+	0.968186i	$0.419494\pi$
$912$	33.4708	1.10833
$913$	−35.8847	−1.18761
$914$	−5.09195	−0.168427
$915$	41.0083	1.35569
$916$	0.478999	0.0158266
$917$	0	0
$918$	−83.4061	−2.75281
$919$	−44.7051	−1.47469	−0.737343	−	0.675519i	$-0.763920\pi$
$919$	−44.7051	−1.47469	−0.737343	+	0.675519i	$0.763920\pi$
$920$	19.3679	0.638539
$921$	−16.9198	−0.557527
$922$	21.2661	0.700361
$923$	28.7734	0.947087
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−18.4511	−0.606342
$927$	−123.101	−4.04316
$928$	−0.584606	−0.0191906
$929$	−17.9099	−0.587606	−0.293803	−	0.955866i	$-0.594921\pi$
$929$	−17.9099	−0.587606	−0.293803	+	0.955866i	$0.594921\pi$
$930$	34.3025	1.12482
$931$	0	0
$932$	−0.563151	−0.0184466
$933$	−9.42797	−0.308658
$934$	−23.5803	−0.771569
$935$	21.6624	0.708435
$936$	87.0414	2.84504
$937$	23.6483	0.772557	0.386279	−	0.922382i	$-0.373760\pi$
$937$	23.6483	0.772557	0.386279	+	0.922382i	$0.373760\pi$
$938$	0	0
$939$	−76.1261	−2.48428
$940$	0.187901	0.00612867
$941$	58.6328	1.91138	0.955688	−	0.294381i	$-0.0951135\pi$
$941$	58.6328	1.91138	0.955688	+	0.294381i	$0.0951135\pi$
$942$	28.2797	0.921401
$943$	−35.8921	−1.16881
$944$	11.6746	0.379977
$945$	0	0
$946$	−14.2514	−0.463353
$947$	38.8086	1.26111	0.630556	−	0.776144i	$-0.282827\pi$
$947$	38.8086	1.26111	0.630556	+	0.776144i	$0.282827\pi$
$948$	−1.80342	−0.0585725
$949$	46.8362	1.52037
$950$	−3.92368	−0.127301
$951$	38.3976	1.24513
$952$	0	0
$953$	28.7794	0.932255	0.466127	−	0.884718i	$-0.345649\pi$
$953$	28.7794	0.932255	0.466127	+	0.884718i	$0.345649\pi$
$954$	−49.0215	−1.58713
$955$	−9.37242	−0.303284
$956$	−0.153212	−0.00495522
$957$	28.4981	0.921211
$958$	26.5848	0.858915
$959$	0	0
$960$	−24.8564	−0.802238
$961$	33.6259	1.08471
$962$	6.78127	0.218637
$963$	83.3507	2.68594
$964$	−0.123457	−0.00397627
$965$	20.8859	0.672341
$966$	0	0
$967$	49.8680	1.60365	0.801823	−	0.597562i	$-0.203864\pi$
$967$	49.8680	1.60365	0.801823	+	0.597562i	$0.203864\pi$
$968$	6.57402	0.211297
$969$	50.7439	1.63013
$970$	9.87330	0.317013
$971$	−27.2949	−0.875935	−0.437968	−	0.898991i	$-0.644302\pi$
$971$	−27.2949	−0.875935	−0.437968	+	0.898991i	$0.644302\pi$
$972$	0.224957	0.00721550
$973$	0	0
$974$	51.2298	1.64151
$975$	−14.7655	−0.472875
$976$	52.7080	1.68714
$977$	3.98645	0.127538	0.0637689	−	0.997965i	$-0.479688\pi$
$977$	3.98645	0.127538	0.0637689	+	0.997965i	$0.479688\pi$
$978$	−57.4799	−1.83800
$979$	1.90912	0.0610158
$980$	0	0
$981$	−36.4118	−1.16254
$982$	51.5633	1.64545
$983$	−23.8553	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$983$	−23.8553	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$984$	46.0818	1.46903
$985$	25.4155	0.809804
$986$	21.3146	0.678796
$987$	0	0
$988$	−0.547446	−0.0174166
$989$	−18.9279	−0.601872
$990$	32.1190	1.02081
$991$	27.1694	0.863064	0.431532	−	0.902098i	$-0.357973\pi$
$991$	27.1694	0.863064	0.431532	+	0.902098i	$0.357973\pi$
$992$	−1.83330	−0.0582074
$993$	−99.9885	−3.17304
$994$	0	0
$995$	−9.58159	−0.303757
$996$	1.20923	0.0383159
$997$	43.2163	1.36867	0.684336	−	0.729166i	$-0.260092\pi$
$997$	43.2163	1.36867	0.684336	+	0.729166i	$0.260092\pi$
$998$	−5.48283	−0.173556
$999$	10.0312	0.317373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.p.1.5	✓	17
7.6	odd	2		9065.2.a.q.1.5	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.p.1.5	✓	17	1.1	even	1	trivial
9065.2.a.q.1.5	yes	17	7.6	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:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39989 q^{2} -3.04811 q^{3} -0.0403202 q^{4} +1.00000 q^{5} +4.26700 q^{6} +2.85621 q^{8} +6.29096 q^{9} -1.39989 q^{10} -3.64714 q^{11} +0.122900 q^{12} +4.84416 q^{13} -3.04811 q^{15} -3.91773 q^{16} -5.93954 q^{17} -8.80662 q^{18} +2.80285 q^{19} -0.0403202 q^{20} +5.10558 q^{22} +6.78095 q^{23} -8.70605 q^{24} +1.00000 q^{25} -6.78127 q^{26} -10.0312 q^{27} +2.56349 q^{29} +4.26700 q^{30} +8.03902 q^{31} -0.228051 q^{32} +11.1169 q^{33} +8.31468 q^{34} -0.253653 q^{36} -1.00000 q^{37} -3.92368 q^{38} -14.7655 q^{39} +2.85621 q^{40} -5.29308 q^{41} -2.79133 q^{43} +0.147054 q^{44} +6.29096 q^{45} -9.49256 q^{46} -4.66023 q^{47} +11.9417 q^{48} -1.39989 q^{50} +18.1044 q^{51} -0.195317 q^{52} +5.56644 q^{53} +14.0425 q^{54} -3.64714 q^{55} -8.54340 q^{57} -3.58860 q^{58} -2.97995 q^{59} +0.122900 q^{60} -13.4537 q^{61} -11.2537 q^{62} +8.15471 q^{64} +4.84416 q^{65} -15.5624 q^{66} -9.91515 q^{67} +0.239484 q^{68} -20.6691 q^{69} +5.93981 q^{71} +17.9683 q^{72} +9.66860 q^{73} +1.39989 q^{74} -3.04811 q^{75} -0.113012 q^{76} +20.6700 q^{78} -14.6739 q^{79} -3.91773 q^{80} +11.7033 q^{81} +7.40970 q^{82} +9.83911 q^{83} -5.93954 q^{85} +3.90755 q^{86} -7.81380 q^{87} -10.4170 q^{88} -0.523456 q^{89} -8.80662 q^{90} -0.273409 q^{92} -24.5038 q^{93} +6.52379 q^{94} +2.80285 q^{95} +0.695123 q^{96} -7.05294 q^{97} -22.9440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + q^{2} - q^{3} + 13 q^{4} + 17 q^{5} - 4 q^{6} - 3 q^{8} + 8 q^{9} + q^{10} - 19 q^{11} - 4 q^{12} - 11 q^{13} - q^{15} + 9 q^{16} - 13 q^{17} - 2 q^{18} - 20 q^{19} + 13 q^{20} + 16 q^{22} + 4 q^{23}+ \cdots - 6 q^{99}+O(q^{100}) \) 17 * q + q^2 - q^3 + 13 * q^4 + 17 * q^5 - 4 * q^6 - 3 * q^8 + 8 * q^9 + q^10 - 19 * q^11 - 4 * q^12 - 11 * q^13 - q^15 + 9 * q^16 - 13 * q^17 - 2 * q^18 - 20 * q^19 + 13 * q^20 + 16 * q^22 + 4 * q^23 - 12 * q^24 + 17 * q^25 - 2 * q^26 - q^27 - 5 * q^29 - 4 * q^30 - 30 * q^31 - 15 * q^32 + 3 * q^33 - 6 * q^34 - 21 * q^36 - 17 * q^37 - 20 * q^38 - 33 * q^39 - 3 * q^40 - 4 * q^41 + 8 * q^43 - 42 * q^44 + 8 * q^45 - 8 * q^46 - 25 * q^47 - 15 * q^48 + q^50 - 19 * q^51 - 44 * q^52 + 16 * q^53 - 2 * q^54 - 19 * q^55 + 14 * q^57 - 18 * q^58 - 10 * q^59 - 4 * q^60 - 32 * q^61 - 61 * q^62 + 57 * q^64 - 11 * q^65 - 31 * q^66 + 4 * q^67 + 30 * q^68 - 32 * q^69 - 12 * q^71 - 21 * q^72 - q^74 - q^75 - 77 * q^76 + 14 * q^78 - 19 * q^79 + 9 * q^80 + 69 * q^81 + 18 * q^82 - 6 * q^83 - 13 * q^85 - 4 * q^86 - 53 * q^87 - 15 * q^88 + 36 * q^89 - 2 * q^90 + 33 * q^92 - 50 * q^93 - 20 * q^94 - 20 * q^95 + 83 * q^96 - 21 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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