LMFDB - Embedded newform 9065.2.a.q.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:	$0$
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	$-0.157601\pi$
$3$	3.04811	1.75983	0.879913	+	0.475135i	$0.157601\pi$
$4$	−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	$-0.734463\pi$
$13$	−4.84416	−1.34353	−0.671764	+	0.740765i	$0.734463\pi$
$14$	0	0
$15$	−3.04811	−0.787018
$16$	−3.91773	−0.979433
$17$	5.93954	1.44055	0.720275	−	0.693689i	$-0.244016\pi$
$17$	5.93954	1.44055	0.720275	+	0.693689i	$0.244016\pi$
$18$	−8.80662	−2.07574
$19$	−2.80285	−0.643019	−0.321509	−	0.946906i	$-0.604190\pi$
$19$	−2.80285	−0.643019	−0.321509	+	0.946906i	$0.604190\pi$
$20$	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	$-0.756743\pi$
$31$	−8.03902	−1.44385	−0.721925	+	0.691971i	$0.756743\pi$
$32$	−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	$-0.364369\pi$
$41$	5.29308	0.826640	0.413320	+	0.910586i	$0.364369\pi$
$42$	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	$-0.389613\pi$
$47$	4.66023	0.679764	0.339882	+	0.940468i	$0.389613\pi$
$48$	−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	$-0.437861\pi$
$59$	2.97995	0.387956	0.193978	+	0.981006i	$0.437861\pi$
$60$	0.122900	0.0158664
$61$	13.4537	1.72257	0.861285	−	0.508122i	$-0.169660\pi$
$61$	13.4537	1.72257	0.861285	+	0.508122i	$0.169660\pi$
$62$	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	$-0.691437\pi$
$73$	−9.66860	−1.13162	−0.565812	+	0.824534i	$0.691437\pi$
$74$	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	$-0.681573\pi$
$83$	−9.83911	−1.07998	−0.539991	+	0.841670i	$0.681573\pi$
$84$	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	$-0.491168\pi$
$89$	0.523456	0.0554863	0.0277431	+	0.999615i	$0.491168\pi$
$90$	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	$-0.383439\pi$
$97$	7.05294	0.716117	0.358059	+	0.933699i	$0.383439\pi$
$98$	0	0
$99$	−22.9440	−2.30596
$100$	−0.0403202	−0.00403202
$101$	2.39273	0.238086	0.119043	−	0.992889i	$-0.462017\pi$
$101$	2.39273	0.238086	0.119043	+	0.992889i	$0.462017\pi$
$102$	−25.3440	−2.50943
$103$	19.5679	1.92808	0.964041	−	0.265755i	$-0.0856211\pi$
$103$	19.5679	1.92808	0.964041	+	0.265755i	$0.0856211\pi$
$104$	−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	$-0.191567\pi$
$131$	18.8692	1.64861	0.824303	+	0.566148i	$0.191567\pi$
$132$	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	$-0.336209\pi$
$139$	11.6048	0.984310	0.492155	+	0.870508i	$0.336209\pi$
$140$	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	$-0.585196\pi$
$157$	−6.62753	−0.528934	−0.264467	+	0.964395i	$0.585196\pi$
$158$	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	$-0.257493\pi$
$167$	17.8404	1.38053	0.690267	+	0.723554i	$0.257493\pi$
$168$	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	$-0.987425\pi$
$173$	−26.2854	−1.99844	−0.999220	+	0.0394945i	$0.987425\pi$
$174$	−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	$-0.307322\pi$
$181$	15.3108	1.13804	0.569021	+	0.822323i	$0.307322\pi$
$182$	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	$-0.389705\pi$
$199$	9.58159	0.679221	0.339611	+	0.940566i	$0.389705\pi$
$200$	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	$-0.439226\pi$
$223$	5.66767	0.379536	0.189768	+	0.981829i	$0.439226\pi$
$224$	0	0
$225$	6.29096	0.419397
$226$	−13.5132	−0.898887
$227$	−21.7868	−1.44604	−0.723021	−	0.690826i	$-0.757247\pi$
$227$	−21.7868	−1.44604	−0.723021	+	0.690826i	$0.757247\pi$
$228$	0.344472	0.0228132
$229$	11.8799	0.785045	0.392522	−	0.919742i	$-0.371603\pi$
$229$	11.8799	0.785045	0.392522	+	0.919742i	$0.371603\pi$
$230$	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	$-0.531442\pi$
$241$	−3.06191	−0.197235	−0.0986174	+	0.995125i	$0.531442\pi$
$242$	−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	$-0.333785\pi$
$251$	15.8041	0.997544	0.498772	+	0.866733i	$0.333785\pi$
$252$	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	$-0.563362\pi$
$257$	−6.34024	−0.395494	−0.197747	+	0.980253i	$0.563362\pi$
$258$	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	$-0.605474\pi$
$269$	−10.6715	−0.650654	−0.325327	+	0.945602i	$0.605474\pi$
$270$	14.0425	0.854601
$271$	3.12586	0.189882	0.0949411	−	0.995483i	$-0.469734\pi$
$271$	3.12586	0.189882	0.0949411	+	0.995483i	$0.469734\pi$
$272$	−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	$-0.404341\pi$
$283$	9.95956	0.592034	0.296017	+	0.955183i	$0.404341\pi$
$284$	−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	$-0.384647\pi$
$293$	12.1366	0.709026	0.354513	+	0.935051i	$0.384647\pi$
$294$	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	$-0.550635\pi$
$307$	−5.55093	−0.316808	−0.158404	+	0.987374i	$0.550635\pi$
$308$	0	0
$309$	59.6450	3.39309
$310$	−11.2537	−0.639168
$311$	−3.09306	−0.175391	−0.0876956	−	0.996147i	$-0.527950\pi$
$311$	−3.09306	−0.175391	−0.0876956	+	0.996147i	$0.527950\pi$
$312$	−42.1735	−2.38760
$313$	−24.9749	−1.41166	−0.705832	−	0.708380i	$-0.749426\pi$
$313$	−24.9749	−1.41166	−0.705832	+	0.708380i	$0.749426\pi$
$314$	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	$-0.675625\pi$
$349$	−19.5847	−1.04834	−0.524172	+	0.851613i	$0.675625\pi$
$350$	0	0
$351$	−48.5927	−2.59368
$352$	0.831733	0.0443315
$353$	−5.04907	−0.268735	−0.134367	−	0.990932i	$-0.542900\pi$
$353$	−5.04907	−0.268735	−0.134367	+	0.990932i	$0.542900\pi$
$354$	−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	$-0.370418\pi$
$367$	15.1703	0.791885	0.395943	+	0.918275i	$0.370418\pi$
$368$	−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	$-0.130415\pi$
$383$	35.9013	1.83447	0.917236	+	0.398345i	$0.130415\pi$
$384$	−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	$-0.407593\pi$
$397$	11.4068	0.572489	0.286244	+	0.958157i	$0.407593\pi$
$398$	−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	$-0.428958\pi$
$409$	8.95246	0.442671	0.221335	+	0.975198i	$0.428958\pi$
$410$	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	$-0.407834\pi$
$419$	11.6889	0.571038	0.285519	+	0.958373i	$0.407834\pi$
$420$	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	$-0.675935\pi$
$433$	−21.8491	−1.05000	−0.525001	+	0.851102i	$0.675935\pi$
$434$	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	$-0.168975\pi$
$439$	36.1376	1.72475	0.862376	+	0.506268i	$0.168975\pi$
$440$	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	$-0.384901\pi$
$461$	15.1913	0.707530	0.353765	+	0.935334i	$0.384901\pi$
$462$	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	$-0.627433\pi$
$467$	−16.8444	−0.779467	−0.389733	+	0.920928i	$0.627433\pi$
$468$	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	$-0.357154\pi$
$479$	18.9907	0.867706	0.433853	+	0.900984i	$0.357154\pi$
$480$	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	$-0.418693\pi$
$503$	11.3334	0.505331	0.252665	+	0.967554i	$0.418693\pi$
$504$	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	$-0.369244\pi$
$509$	18.0185	0.798656	0.399328	+	0.916808i	$0.369244\pi$
$510$	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	$-0.421310\pi$
$521$	11.1708	0.489403	0.244702	+	0.969598i	$0.421310\pi$
$522$	−22.5757	−0.988112
$523$	20.7450	0.907115	0.453557	−	0.891227i	$-0.350155\pi$
$523$	20.7450	0.907115	0.453557	+	0.891227i	$0.350155\pi$
$524$	−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	$-0.379518\pi$
$563$	17.5363	0.739066	0.369533	+	0.929218i	$0.379518\pi$
$564$	−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	$-0.619812\pi$
$577$	−17.6589	−0.735151	−0.367576	+	0.929994i	$0.619812\pi$
$578$	−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	$-0.508674\pi$
$587$	−1.32023	−0.0544918	−0.0272459	+	0.999629i	$0.508674\pi$
$588$	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	$-0.679904\pi$
$593$	−26.0840	−1.07114	−0.535572	+	0.844490i	$0.679904\pi$
$594$	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	$-0.662615\pi$
$601$	−23.9728	−0.977873	−0.488936	+	0.872319i	$0.662615\pi$
$602$	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	$-0.140859\pi$
$607$	44.5284	1.80735	0.903675	+	0.428219i	$0.140859\pi$
$608$	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	$-0.472097\pi$
$619$	4.35625	0.175093	0.0875463	+	0.996160i	$0.472097\pi$
$620$	−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	$-0.365039\pi$
$643$	20.8643	0.822807	0.411403	+	0.911453i	$0.365039\pi$
$644$	0	0
$645$	8.50829	0.335013
$646$	23.3048	0.916916
$647$	−25.4420	−1.00023	−0.500113	−	0.865960i	$-0.666708\pi$
$647$	−25.4420	−1.00023	−0.500113	+	0.865960i	$0.666708\pi$
$648$	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	$-0.174106\pi$
$661$	43.9179	1.70821	0.854104	+	0.520102i	$0.174106\pi$
$662$	−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	$-0.788145\pi$
$677$	−40.9319	−1.57314	−0.786571	+	0.617500i	$0.788145\pi$
$678$	−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	$-0.531559\pi$
$691$	−5.20391	−0.197966	−0.0989830	+	0.995089i	$0.531559\pi$
$692$	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	$-0.546685\pi$
$719$	−7.83731	−0.292282	−0.146141	+	0.989264i	$0.546685\pi$
$720$	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	$-0.335770\pi$
$727$	26.6047	0.986712	0.493356	+	0.869827i	$0.335770\pi$
$728$	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	$-0.946037\pi$
$733$	−53.3717	−1.97133	−0.985664	+	0.168718i	$0.946037\pi$
$734$	−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	$-0.229169\pi$
$761$	41.4805	1.50367	0.751834	+	0.659352i	$0.229169\pi$
$762$	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	$-0.471359\pi$
$769$	4.98364	0.179715	0.0898574	+	0.995955i	$0.471359\pi$
$770$	0	0
$771$	−19.3257	−0.696000
$772$	−0.842123	−0.0303087
$773$	−7.58656	−0.272869	−0.136435	−	0.990649i	$-0.543564\pi$
$773$	−7.58656	−0.272869	−0.136435	+	0.990649i	$0.543564\pi$
$774$	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	$-0.562429\pi$
$787$	−10.9337	−0.389744	−0.194872	+	0.980829i	$0.562429\pi$
$788$	−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	$-0.534089\pi$
$797$	−6.03515	−0.213776	−0.106888	+	0.994271i	$0.534089\pi$
$798$	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	$-0.180370\pi$
$811$	48.0542	1.68741	0.843705	+	0.536807i	$0.180370\pi$
$812$	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	$-0.547533\pi$
$829$	−8.56716	−0.297550	−0.148775	+	0.988871i	$0.547533\pi$
$830$	−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	$-0.516625\pi$
$839$	−3.02431	−0.104411	−0.0522053	+	0.998636i	$0.516625\pi$
$840$	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	$-0.296259\pi$
$853$	34.8869	1.19451	0.597253	+	0.802053i	$0.296259\pi$
$854$	0	0
$855$	17.6326	0.603024
$856$	37.8428	1.29344
$857$	−1.35026	−0.0461240	−0.0230620	−	0.999734i	$-0.507342\pi$
$857$	−1.35026	−0.0461240	−0.0230620	+	0.999734i	$0.507342\pi$
$858$	−75.3865	−2.57365
$859$	−31.2917	−1.06766	−0.533829	−	0.845592i	$-0.679247\pi$
$859$	−31.2917	−1.06766	−0.533829	+	0.845592i	$0.679247\pi$
$860$	−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	$-0.667553\pi$
$881$	−29.8247	−1.00482	−0.502409	+	0.864630i	$0.667553\pi$
$882$	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	$-0.420857\pi$
$887$	14.6579	0.492164	0.246082	+	0.969249i	$0.420857\pi$
$888$	−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	$-0.405079\pi$
$929$	17.9099	0.587606	0.293803	+	0.955866i	$0.405079\pi$
$930$	−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	$-0.626240\pi$
$937$	−23.6483	−0.772557	−0.386279	+	0.922382i	$0.626240\pi$
$938$	0	0
$939$	−76.1261	−2.48428
$940$	0.187901	0.00612867
$941$	−58.6328	−1.91138	−0.955688	−	0.294381i	$-0.904886\pi$
$941$	−58.6328	−1.91138	−0.955688	+	0.294381i	$0.904886\pi$
$942$	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	$-0.355698\pi$
$971$	27.2949	0.875935	0.437968	+	0.898991i	$0.355698\pi$
$972$	−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	$-0.375775\pi$
$983$	23.8553	0.760865	0.380432	+	0.924809i	$0.375775\pi$
$984$	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	$-0.739908\pi$
$997$	−43.2163	−1.36867	−0.684336	+	0.729166i	$0.739908\pi$
$998$	−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.q.1.5	yes	17
7.6	odd	2		9065.2.a.p.1.5	✓	17

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

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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