LMFDB - Embedded newform 9065.2.a.bc.1.9

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 = [34,-2,10,30,-34,8,0,-6,28,2,-30] 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:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	\(q-1.84557 q^{2} +1.65469 q^{3} +1.40612 q^{4} -1.00000 q^{5} -3.05385 q^{6} +1.09605 q^{8} -0.261986 q^{9} +1.84557 q^{10} +3.96820 q^{11} +2.32670 q^{12} +0.869614 q^{13} -1.65469 q^{15} -4.83507 q^{16} +6.02109 q^{17} +0.483512 q^{18} -2.90554 q^{19} -1.40612 q^{20} -7.32357 q^{22} +7.31882 q^{23} +1.81362 q^{24} +1.00000 q^{25} -1.60493 q^{26} -5.39759 q^{27} -1.33715 q^{29} +3.05385 q^{30} +4.18047 q^{31} +6.73135 q^{32} +6.56615 q^{33} -11.1123 q^{34} -0.368383 q^{36} +1.00000 q^{37} +5.36237 q^{38} +1.43895 q^{39} -1.09605 q^{40} +11.7386 q^{41} +0.185085 q^{43} +5.57976 q^{44} +0.261986 q^{45} -13.5074 q^{46} -6.01749 q^{47} -8.00056 q^{48} -1.84557 q^{50} +9.96306 q^{51} +1.22278 q^{52} -0.214255 q^{53} +9.96162 q^{54} -3.96820 q^{55} -4.80778 q^{57} +2.46781 q^{58} +6.27126 q^{59} -2.32670 q^{60} -5.58316 q^{61} -7.71533 q^{62} -2.75303 q^{64} -0.869614 q^{65} -12.1183 q^{66} -5.58798 q^{67} +8.46637 q^{68} +12.1104 q^{69} +7.81923 q^{71} -0.287148 q^{72} +1.24040 q^{73} -1.84557 q^{74} +1.65469 q^{75} -4.08554 q^{76} -2.65567 q^{78} +2.15195 q^{79} +4.83507 q^{80} -8.14541 q^{81} -21.6644 q^{82} -9.50582 q^{83} -6.02109 q^{85} -0.341587 q^{86} -2.21258 q^{87} +4.34932 q^{88} +13.7542 q^{89} -0.483512 q^{90} +10.2911 q^{92} +6.91740 q^{93} +11.1057 q^{94} +2.90554 q^{95} +11.1383 q^{96} +0.951656 q^{97} -1.03961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) $ 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * 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 $

$2$	−1.84557	−1.30501	−0.652507	−	0.757783i	$-0.726283\pi$
$2$	−1.84557	−1.30501	−0.652507	+	0.757783i	$0.726283\pi$
$3$	1.65469	0.955338	0.477669	−	0.878540i	$-0.341482\pi$
$3$	1.65469	0.955338	0.477669	+	0.878540i	$0.341482\pi$
$4$	1.40612	0.703060
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.05385	−1.24673
$7$	0	0
$7$	0	0
$8$	1.09605	0.387511
$9$	−0.261986	−0.0873285
$10$	1.84557	0.583620
$11$	3.96820	1.19646	0.598228	−	0.801326i	$-0.295872\pi$
$11$	3.96820	1.19646	0.598228	+	0.801326i	$0.295872\pi$
$12$	2.32670	0.671660
$13$	0.869614	0.241188	0.120594	−	0.992702i	$-0.461520\pi$
$13$	0.869614	0.241188	0.120594	+	0.992702i	$0.461520\pi$
$14$	0	0
$15$	−1.65469	−0.427240
$16$	−4.83507	−1.20877
$17$	6.02109	1.46033	0.730164	−	0.683272i	$-0.239444\pi$
$17$	6.02109	1.46033	0.730164	+	0.683272i	$0.239444\pi$
$18$	0.483512	0.113965
$19$	−2.90554	−0.666576	−0.333288	−	0.942825i	$-0.608158\pi$
$19$	−2.90554	−0.666576	−0.333288	+	0.942825i	$0.608158\pi$
$20$	−1.40612	−0.314418
$21$	0	0
$22$	−7.32357	−1.56139
$23$	7.31882	1.52608	0.763040	−	0.646352i	$-0.223706\pi$
$23$	7.31882	1.52608	0.763040	+	0.646352i	$0.223706\pi$
$24$	1.81362	0.370204
$25$	1.00000	0.200000
$26$	−1.60493	−0.314753
$27$	−5.39759	−1.03877
$28$	0	0
$29$	−1.33715	−0.248303	−0.124152	−	0.992263i	$-0.539621\pi$
$29$	−1.33715	−0.248303	−0.124152	+	0.992263i	$0.539621\pi$
$30$	3.05385	0.557554
$31$	4.18047	0.750834	0.375417	−	0.926856i	$-0.377499\pi$
$31$	4.18047	0.750834	0.375417	+	0.926856i	$0.377499\pi$
$32$	6.73135	1.18995
$33$	6.56615	1.14302
$34$	−11.1123	−1.90575
$35$	0	0
$36$	−0.368383	−0.0613972
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	5.36237	0.869891
$39$	1.43895	0.230416
$40$	−1.09605	−0.173300
$41$	11.7386	1.83327	0.916633	−	0.399731i	$-0.130896\pi$
$41$	11.7386	1.83327	0.916633	+	0.399731i	$0.130896\pi$
$42$	0	0
$43$	0.185085	0.0282252	0.0141126	−	0.999900i	$-0.495508\pi$
$43$	0.185085	0.0282252	0.0141126	+	0.999900i	$0.495508\pi$
$44$	5.57976	0.841180
$45$	0.261986	0.0390545
$46$	−13.5074	−1.99155
$47$	−6.01749	−0.877742	−0.438871	−	0.898550i	$-0.644621\pi$
$47$	−6.01749	−0.877742	−0.438871	+	0.898550i	$0.644621\pi$
$48$	−8.00056	−1.15478
$49$	0	0
$50$	−1.84557	−0.261003
$51$	9.96306	1.39511
$52$	1.22278	0.169569
$53$	−0.214255	−0.0294302	−0.0147151	−	0.999892i	$-0.504684\pi$
$53$	−0.214255	−0.0294302	−0.0147151	+	0.999892i	$0.504684\pi$
$54$	9.96162	1.35560
$55$	−3.96820	−0.535071
$56$	0	0
$57$	−4.80778	−0.636806
$58$	2.46781	0.324039
$59$	6.27126	0.816448	0.408224	−	0.912882i	$-0.366148\pi$
$59$	6.27126	0.816448	0.408224	+	0.912882i	$0.366148\pi$
$60$	−2.32670	−0.300376
$61$	−5.58316	−0.714850	−0.357425	−	0.933942i	$-0.616345\pi$
$61$	−5.58316	−0.714850	−0.357425	+	0.933942i	$0.616345\pi$
$62$	−7.71533	−0.979849
$63$	0	0
$64$	−2.75303	−0.344129
$65$	−0.869614	−0.107862
$66$	−12.1183	−1.49166
$67$	−5.58798	−0.682681	−0.341340	−	0.939940i	$-0.610881\pi$
$67$	−5.58798	−0.682681	−0.341340	+	0.939940i	$0.610881\pi$
$68$	8.46637	1.02670
$69$	12.1104	1.45792
$70$	0	0
$71$	7.81923	0.927972	0.463986	−	0.885843i	$-0.346419\pi$
$71$	7.81923	0.927972	0.463986	+	0.885843i	$0.346419\pi$
$72$	−0.287148	−0.0338407
$73$	1.24040	0.145177	0.0725886	−	0.997362i	$-0.476874\pi$
$73$	1.24040	0.145177	0.0725886	+	0.997362i	$0.476874\pi$
$74$	−1.84557	−0.214543
$75$	1.65469	0.191068
$76$	−4.08554	−0.468643
$77$	0	0
$78$	−2.65567	−0.300696
$79$	2.15195	0.242113	0.121057	−	0.992646i	$-0.461372\pi$
$79$	2.15195	0.242113	0.121057	+	0.992646i	$0.461372\pi$
$80$	4.83507	0.540577
$81$	−8.14541	−0.905045
$82$	−21.6644	−2.39244
$83$	−9.50582	−1.04340	−0.521700	−	0.853129i	$-0.674702\pi$
$83$	−9.50582	−1.04340	−0.521700	+	0.853129i	$0.674702\pi$
$84$	0	0
$85$	−6.02109	−0.653079
$86$	−0.341587	−0.0368342
$87$	−2.21258	−0.237213
$88$	4.34932	0.463640
$89$	13.7542	1.45795	0.728973	−	0.684542i	$-0.239998\pi$
$89$	13.7542	1.45795	0.728973	+	0.684542i	$0.239998\pi$
$90$	−0.483512	−0.0509667
$91$	0	0
$92$	10.2911	1.07293
$93$	6.91740	0.717301
$94$	11.1057	1.14546
$95$	2.90554	0.298102
$96$	11.1383	1.13680
$97$	0.951656	0.0966261	0.0483130	−	0.998832i	$-0.484615\pi$
$97$	0.951656	0.0966261	0.0483130	+	0.998832i	$0.484615\pi$
$98$	0	0
$99$	−1.03961	−0.104485
$100$	1.40612	0.140612
$101$	12.7081	1.26450	0.632251	−	0.774764i	$-0.282131\pi$
$101$	12.7081	1.26450	0.632251	+	0.774764i	$0.282131\pi$
$102$	−18.3875	−1.82063
$103$	−4.62275	−0.455493	−0.227747	−	0.973720i	$-0.573136\pi$
$103$	−4.62275	−0.455493	−0.227747	+	0.973720i	$0.573136\pi$
$104$	0.953137	0.0934628
$105$	0	0
$106$	0.395422	0.0384068
$107$	1.54914	0.149761	0.0748803	−	0.997193i	$-0.476143\pi$
$107$	1.54914	0.149761	0.0748803	+	0.997193i	$0.476143\pi$
$108$	−7.58966	−0.730315
$109$	−9.87238	−0.945602	−0.472801	−	0.881169i	$-0.656757\pi$
$109$	−9.87238	−0.945602	−0.472801	+	0.881169i	$0.656757\pi$
$110$	7.32357	0.698275
$111$	1.65469	0.157057
$112$	0	0
$113$	−5.56999	−0.523980	−0.261990	−	0.965071i	$-0.584379\pi$
$113$	−5.56999	−0.523980	−0.261990	+	0.965071i	$0.584379\pi$
$114$	8.87308	0.831040
$115$	−7.31882	−0.682483
$116$	−1.88020	−0.174572
$117$	−0.227826	−0.0210626
$118$	−11.5740	−1.06548
$119$	0	0
$120$	−1.81362	−0.165560
$121$	4.74658	0.431507
$122$	10.3041	0.932889
$123$	19.4238	1.75139
$124$	5.87824	0.527881
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.90985	0.879357	0.439679	−	0.898155i	$-0.355092\pi$
$127$	9.90985	0.879357	0.439679	+	0.898155i	$0.355092\pi$
$128$	−8.38180	−0.740853
$129$	0.306259	0.0269646
$130$	1.60493	0.140762
$131$	11.9205	1.04149	0.520747	−	0.853711i	$-0.325653\pi$
$131$	11.9205	1.04149	0.520747	+	0.853711i	$0.325653\pi$
$132$	9.23280	0.803612
$133$	0	0
$134$	10.3130	0.890908
$135$	5.39759	0.464551
$136$	6.59939	0.565893
$137$	−21.5628	−1.84223	−0.921117	−	0.389286i	$-0.872722\pi$
$137$	−21.5628	−1.84223	−0.921117	+	0.389286i	$0.872722\pi$
$138$	−22.3506	−1.90261
$139$	−4.87429	−0.413432	−0.206716	−	0.978401i	$-0.566278\pi$
$139$	−4.87429	−0.413432	−0.206716	+	0.978401i	$0.566278\pi$
$140$	0	0
$141$	−9.95711	−0.838540
$142$	−14.4309	−1.21102
$143$	3.45080	0.288570
$144$	1.26672	0.105560
$145$	1.33715	0.111045
$146$	−2.28923	−0.189458
$147$	0	0
$148$	1.40612	0.115582
$149$	−11.4059	−0.934410	−0.467205	−	0.884149i	$-0.654739\pi$
$149$	−11.4059	−0.934410	−0.467205	+	0.884149i	$0.654739\pi$
$150$	−3.05385	−0.249346
$151$	11.2279	0.913715	0.456857	−	0.889540i	$-0.348975\pi$
$151$	11.2279	0.913715	0.456857	+	0.889540i	$0.348975\pi$
$152$	−3.18460	−0.258305
$153$	−1.57744	−0.127528
$154$	0	0
$155$	−4.18047	−0.335783
$156$	2.02333	0.161996
$157$	21.6331	1.72651	0.863253	−	0.504771i	$-0.168423\pi$
$157$	21.6331	1.72651	0.863253	+	0.504771i	$0.168423\pi$
$158$	−3.97157	−0.315961
$159$	−0.354526	−0.0281158
$160$	−6.73135	−0.532160
$161$	0	0
$162$	15.0329	1.18110
$163$	15.6800	1.22816	0.614078	−	0.789246i	$-0.289528\pi$
$163$	15.6800	1.22816	0.614078	+	0.789246i	$0.289528\pi$
$164$	16.5059	1.28890
$165$	−6.56615	−0.511174
$166$	17.5436	1.36165
$167$	−9.05381	−0.700605	−0.350303	−	0.936637i	$-0.613921\pi$
$167$	−9.05381	−0.700605	−0.350303	+	0.936637i	$0.613921\pi$
$168$	0	0
$169$	−12.2438	−0.941829
$170$	11.1123	0.852277
$171$	0.761209	0.0582111
$172$	0.260251	0.0198440
$173$	−7.16901	−0.545049	−0.272525	−	0.962149i	$-0.587859\pi$
$173$	−7.16901	−0.545049	−0.272525	+	0.962149i	$0.587859\pi$
$174$	4.08347	0.309567
$175$	0	0
$176$	−19.1865	−1.44624
$177$	10.3770	0.779984
$178$	−25.3844	−1.90264
$179$	5.57734	0.416870	0.208435	−	0.978036i	$-0.433163\pi$
$179$	5.57734	0.416870	0.208435	+	0.978036i	$0.433163\pi$
$180$	0.368383	0.0274577
$181$	3.18262	0.236563	0.118281	−	0.992980i	$-0.462262\pi$
$181$	3.18262	0.236563	0.118281	+	0.992980i	$0.462262\pi$
$182$	0	0
$183$	−9.23842	−0.682924
$184$	8.02176	0.591372
$185$	−1.00000	−0.0735215
$186$	−12.7665	−0.936087
$187$	23.8929	1.74722
$188$	−8.46132	−0.617105
$189$	0	0
$190$	−5.36237	−0.389027
$191$	−11.2168	−0.811622	−0.405811	−	0.913957i	$-0.633011\pi$
$191$	−11.2168	−0.811622	−0.405811	+	0.913957i	$0.633011\pi$
$192$	−4.55542	−0.328759
$193$	−1.61325	−0.116124	−0.0580622	−	0.998313i	$-0.518492\pi$
$193$	−1.61325	−0.116124	−0.0580622	+	0.998313i	$0.518492\pi$
$194$	−1.75635	−0.126098
$195$	−1.43895	−0.103045
$196$	0	0
$197$	10.6372	0.757869	0.378935	−	0.925423i	$-0.376291\pi$
$197$	10.6372	0.757869	0.378935	+	0.925423i	$0.376291\pi$
$198$	1.91867	0.136354
$199$	−17.4516	−1.23711	−0.618555	−	0.785741i	$-0.712282\pi$
$199$	−17.4516	−1.23711	−0.618555	+	0.785741i	$0.712282\pi$
$200$	1.09605	0.0775021
$201$	−9.24641	−0.652191
$202$	−23.4536	−1.65019
$203$	0	0
$204$	14.0093	0.980844
$205$	−11.7386	−0.819861
$206$	8.53160	0.594425
$207$	−1.91742	−0.133270
$208$	−4.20464	−0.291540
$209$	−11.5297	−0.797529
$210$	0	0
$211$	−27.2673	−1.87716	−0.938578	−	0.345067i	$-0.887856\pi$
$211$	−27.2673	−1.87716	−0.938578	+	0.345067i	$0.887856\pi$
$212$	−0.301268	−0.0206912
$213$	12.9384	0.886527
$214$	−2.85904	−0.195440
$215$	−0.185085	−0.0126227
$216$	−5.91601	−0.402533
$217$	0	0
$218$	18.2201	1.23402
$219$	2.05248	0.138693
$220$	−5.57976	−0.376187
$221$	5.23603	0.352213
$222$	−3.05385	−0.204961
$223$	17.7442	1.18824	0.594120	−	0.804376i	$-0.297500\pi$
$223$	17.7442	1.18824	0.594120	+	0.804376i	$0.297500\pi$
$224$	0	0
$225$	−0.261986	−0.0174657
$226$	10.2798	0.683801
$227$	6.08629	0.403962	0.201981	−	0.979389i	$-0.435262\pi$
$227$	6.08629	0.403962	0.201981	+	0.979389i	$0.435262\pi$
$228$	−6.76031	−0.447713
$229$	−17.6354	−1.16538	−0.582689	−	0.812695i	$-0.697999\pi$
$229$	−17.6354	−1.16538	−0.582689	+	0.812695i	$0.697999\pi$
$230$	13.5074	0.890650
$231$	0	0
$232$	−1.46558	−0.0962201
$233$	20.9227	1.37069	0.685346	−	0.728218i	$-0.259651\pi$
$233$	20.9227	1.37069	0.685346	+	0.728218i	$0.259651\pi$
$234$	0.420469	0.0274869
$235$	6.01749	0.392538
$236$	8.81814	0.574012
$237$	3.56082	0.231300
$238$	0	0
$239$	5.74203	0.371421	0.185711	−	0.982604i	$-0.440541\pi$
$239$	5.74203	0.371421	0.185711	+	0.982604i	$0.440541\pi$
$240$	8.00056	0.516434
$241$	−15.9695	−1.02868	−0.514342	−	0.857585i	$-0.671964\pi$
$241$	−15.9695	−1.02868	−0.514342	+	0.857585i	$0.671964\pi$
$242$	−8.76013	−0.563123
$243$	2.71461	0.174142
$244$	−7.85059	−0.502582
$245$	0	0
$246$	−35.8480	−2.28559
$247$	−2.52670	−0.160770
$248$	4.58198	0.290956
$249$	−15.7292	−0.996799
$250$	1.84557	0.116724
$251$	4.11001	0.259421	0.129711	−	0.991552i	$-0.458595\pi$
$251$	4.11001	0.259421	0.129711	+	0.991552i	$0.458595\pi$
$252$	0	0
$253$	29.0425	1.82589
$254$	−18.2893	−1.14757
$255$	−9.96306	−0.623911
$256$	20.9752	1.31095
$257$	−17.7339	−1.10621	−0.553106	−	0.833111i	$-0.686557\pi$
$257$	−17.7339	−1.10621	−0.553106	+	0.833111i	$0.686557\pi$
$258$	−0.565222	−0.0351892
$259$	0	0
$260$	−1.22278	−0.0758337
$261$	0.350315	0.0216839
$262$	−22.0000	−1.35916
$263$	4.16495	0.256822	0.128411	−	0.991721i	$-0.459012\pi$
$263$	4.16495	0.256822	0.128411	+	0.991721i	$0.459012\pi$
$264$	7.19680	0.442933
$265$	0.214255	0.0131616
$266$	0	0
$267$	22.7591	1.39283
$268$	−7.85737	−0.479966
$269$	−26.5332	−1.61776	−0.808878	−	0.587977i	$-0.799925\pi$
$269$	−26.5332	−1.61776	−0.808878	+	0.587977i	$0.799925\pi$
$270$	−9.96162	−0.606245
$271$	11.7732	0.715171	0.357586	−	0.933880i	$-0.383600\pi$
$271$	11.7732	0.715171	0.357586	+	0.933880i	$0.383600\pi$
$272$	−29.1124	−1.76520
$273$	0	0
$274$	39.7956	2.40414
$275$	3.96820	0.239291
$276$	17.0287	1.02501
$277$	−9.07298	−0.545142	−0.272571	−	0.962136i	$-0.587874\pi$
$277$	−9.07298	−0.545142	−0.272571	+	0.962136i	$0.587874\pi$
$278$	8.99584	0.539534
$279$	−1.09522	−0.0655692
$280$	0	0
$281$	6.48980	0.387149	0.193574	−	0.981086i	$-0.437992\pi$
$281$	6.48980	0.387149	0.193574	+	0.981086i	$0.437992\pi$
$282$	18.3765	1.09431
$283$	−1.48142	−0.0880612	−0.0440306	−	0.999030i	$-0.514020\pi$
$283$	−1.48142	−0.0880612	−0.0440306	+	0.999030i	$0.514020\pi$
$284$	10.9948	0.652420
$285$	4.80778	0.284788
$286$	−6.36869	−0.376588
$287$	0	0
$288$	−1.76352	−0.103916
$289$	19.2535	1.13256
$290$	−2.46781	−0.144915
$291$	1.57470	0.0923106
$292$	1.74414	0.102068
$293$	2.81077	0.164207	0.0821034	−	0.996624i	$-0.473836\pi$
$293$	2.81077	0.164207	0.0821034	+	0.996624i	$0.473836\pi$
$294$	0	0
$295$	−6.27126	−0.365127
$296$	1.09605	0.0637064
$297$	−21.4187	−1.24284
$298$	21.0504	1.21942
$299$	6.36455	0.368071
$300$	2.32670	0.134332
$301$	0	0
$302$	−20.7219	−1.19241
$303$	21.0280	1.20803
$304$	14.0485	0.805735
$305$	5.58316	0.319691
$306$	2.91127	0.166426
$307$	−17.5728	−1.00294	−0.501468	−	0.865176i	$-0.667206\pi$
$307$	−17.5728	−1.00294	−0.501468	+	0.865176i	$0.667206\pi$
$308$	0	0
$309$	−7.64924	−0.435150
$310$	7.71533	0.438202
$311$	18.8487	1.06881	0.534405	−	0.845228i	$-0.320536\pi$
$311$	18.8487	1.06881	0.534405	+	0.845228i	$0.320536\pi$
$312$	1.57715	0.0892886
$313$	23.3544	1.32007	0.660034	−	0.751236i	$-0.270542\pi$
$313$	23.3544	1.32007	0.660034	+	0.751236i	$0.270542\pi$
$314$	−39.9253	−2.25311
$315$	0	0
$316$	3.02590	0.170220
$317$	30.6296	1.72033	0.860165	−	0.510016i	$-0.170360\pi$
$317$	30.6296	1.72033	0.860165	+	0.510016i	$0.170360\pi$
$318$	0.654302	0.0366914
$319$	−5.30608	−0.297084
$320$	2.75303	0.153899
$321$	2.56335	0.143072
$322$	0	0
$323$	−17.4945	−0.973420
$324$	−11.4534	−0.636301
$325$	0.869614	0.0482375
$326$	−28.9386	−1.60276
$327$	−16.3358	−0.903370
$328$	12.8661	0.710410
$329$	0	0
$330$	12.1183	0.667089
$331$	17.6814	0.971856	0.485928	−	0.873999i	$-0.338482\pi$
$331$	17.6814	0.971856	0.485928	+	0.873999i	$0.338482\pi$
$332$	−13.3663	−0.733572
$333$	−0.261986	−0.0143567
$334$	16.7094	0.914299
$335$	5.58798	0.305304
$336$	0	0
$337$	14.7730	0.804738	0.402369	−	0.915478i	$-0.368187\pi$
$337$	14.7730	0.804738	0.402369	+	0.915478i	$0.368187\pi$
$338$	22.5967	1.22910
$339$	−9.21663	−0.500578
$340$	−8.46637	−0.459154
$341$	16.5889	0.898340
$342$	−1.40486	−0.0759663
$343$	0	0
$344$	0.202861	0.0109376
$345$	−12.1104	−0.652003
$346$	13.2309	0.711297
$347$	−13.7413	−0.737673	−0.368837	−	0.929494i	$-0.620244\pi$
$347$	−13.7413	−0.737673	−0.368837	+	0.929494i	$0.620244\pi$
$348$	−3.11115	−0.166775
$349$	0.0246967	0.00132198	0.000660992	−	1.00000i	$-0.499790\pi$
$349$	0.0246967	0.00132198	0.000660992		1.00000i	$0.499790\pi$
$350$	0	0
$351$	−4.69382	−0.250538
$352$	26.7113	1.42372
$353$	−16.2966	−0.867381	−0.433691	−	0.901062i	$-0.642789\pi$
$353$	−16.2966	−0.867381	−0.433691	+	0.901062i	$0.642789\pi$
$354$	−19.1515	−1.01789
$355$	−7.81923	−0.415002
$356$	19.3401	1.02502
$357$	0	0
$358$	−10.2934	−0.544021
$359$	17.1951	0.907524	0.453762	−	0.891123i	$-0.350082\pi$
$359$	17.1951	0.907524	0.453762	+	0.891123i	$0.350082\pi$
$360$	0.287148	0.0151340
$361$	−10.5578	−0.555676
$362$	−5.87375	−0.308717
$363$	7.85414	0.412235
$364$	0	0
$365$	−1.24040	−0.0649252
$366$	17.0501	0.891224
$367$	26.7076	1.39412	0.697062	−	0.717011i	$-0.254490\pi$
$367$	26.7076	1.39412	0.697062	+	0.717011i	$0.254490\pi$
$368$	−35.3870	−1.84467
$369$	−3.07535	−0.160096
$370$	1.84557	0.0959465
$371$	0	0
$372$	9.72669	0.504305
$373$	32.7572	1.69610	0.848051	−	0.529914i	$-0.177776\pi$
$373$	32.7572	1.69610	0.848051	+	0.529914i	$0.177776\pi$
$374$	−44.0959	−2.28014
$375$	−1.65469	−0.0854481
$376$	−6.59545	−0.340134
$377$	−1.16281	−0.0598876
$378$	0	0
$379$	−11.6241	−0.597089	−0.298545	−	0.954396i	$-0.596501\pi$
$379$	−11.6241	−0.597089	−0.298545	+	0.954396i	$0.596501\pi$
$380$	4.08554	0.209584
$381$	16.3978	0.840084
$382$	20.7014	1.05918
$383$	−27.6815	−1.41446	−0.707229	−	0.706984i	$-0.750055\pi$
$383$	−27.6815	−1.41446	−0.707229	+	0.706984i	$0.750055\pi$
$384$	−13.8693	−0.707766
$385$	0	0
$386$	2.97736	0.151544
$387$	−0.0484896	−0.00246486
$388$	1.33814	0.0679339
$389$	23.4655	1.18975	0.594874	−	0.803819i	$-0.297202\pi$
$389$	23.4655	1.18975	0.594874	+	0.803819i	$0.297202\pi$
$390$	2.65567	0.134475
$391$	44.0673	2.22858
$392$	0	0
$393$	19.7247	0.994980
$394$	−19.6317	−0.989029
$395$	−2.15195	−0.108276
$396$	−1.46182	−0.0734590
$397$	5.61581	0.281849	0.140925	−	0.990020i	$-0.454992\pi$
$397$	5.61581	0.281849	0.140925	+	0.990020i	$0.454992\pi$
$398$	32.2081	1.61445
$399$	0	0
$400$	−4.83507	−0.241753
$401$	−15.9477	−0.796391	−0.398195	−	0.917301i	$-0.630363\pi$
$401$	−15.9477	−0.796391	−0.398195	+	0.917301i	$0.630363\pi$
$402$	17.0649	0.851118
$403$	3.63539	0.181092
$404$	17.8691	0.889021
$405$	8.14541	0.404749
$406$	0	0
$407$	3.96820	0.196696
$408$	10.9200	0.540619
$409$	27.7487	1.37209	0.686043	−	0.727561i	$-0.259346\pi$
$409$	27.7487	1.37209	0.686043	+	0.727561i	$0.259346\pi$
$410$	21.6644	1.06993
$411$	−35.6798	−1.75996
$412$	−6.50014	−0.320239
$413$	0	0
$414$	3.53874	0.173919
$415$	9.50582	0.466622
$416$	5.85368	0.287000
$417$	−8.06547	−0.394968
$418$	21.2789	1.04079
$419$	12.9229	0.631326	0.315663	−	0.948871i	$-0.397773\pi$
$419$	12.9229	0.631326	0.315663	+	0.948871i	$0.397773\pi$
$420$	0	0
$421$	18.3270	0.893204	0.446602	−	0.894733i	$-0.352634\pi$
$421$	18.3270	0.893204	0.446602	+	0.894733i	$0.352634\pi$
$422$	50.3236	2.44971
$423$	1.57650	0.0766519
$424$	−0.234833	−0.0114045
$425$	6.02109	0.292066
$426$	−23.8788	−1.15693
$427$	0	0
$428$	2.17827	0.105291
$429$	5.71002	0.275682
$430$	0.341587	0.0164728
$431$	2.99719	0.144370	0.0721849	−	0.997391i	$-0.477003\pi$
$431$	2.99719	0.144370	0.0721849	+	0.997391i	$0.477003\pi$
$432$	26.0977	1.25563
$433$	15.7953	0.759073	0.379536	−	0.925177i	$-0.376083\pi$
$433$	15.7953	0.759073	0.379536	+	0.925177i	$0.376083\pi$
$434$	0	0
$435$	2.21258	0.106085
$436$	−13.8817	−0.664815
$437$	−21.2651	−1.01725
$438$	−3.78798	−0.180997
$439$	−11.5418	−0.550861	−0.275430	−	0.961321i	$-0.588820\pi$
$439$	−11.5418	−0.550861	−0.275430	+	0.961321i	$0.588820\pi$
$440$	−4.34932	−0.207346
$441$	0	0
$442$	−9.66344	−0.459643
$443$	−19.3930	−0.921389	−0.460694	−	0.887559i	$-0.652400\pi$
$443$	−19.3930	−0.921389	−0.460694	+	0.887559i	$0.652400\pi$
$444$	2.32670	0.110420
$445$	−13.7542	−0.652013
$446$	−32.7482	−1.55067
$447$	−18.8733	−0.892677
$448$	0	0
$449$	4.34795	0.205192	0.102596	−	0.994723i	$-0.467285\pi$
$449$	4.34795	0.205192	0.102596	+	0.994723i	$0.467285\pi$
$450$	0.483512	0.0227930
$451$	46.5812	2.19342
$452$	−7.83207	−0.368389
$453$	18.5788	0.872907
$454$	−11.2327	−0.527175
$455$	0	0
$456$	−5.26955	−0.246769
$457$	−20.5120	−0.959513	−0.479756	−	0.877402i	$-0.659275\pi$
$457$	−20.5120	−0.959513	−0.479756	+	0.877402i	$0.659275\pi$
$458$	32.5472	1.52083
$459$	−32.4994	−1.51694
$460$	−10.2911	−0.479827
$461$	15.5418	0.723855	0.361928	−	0.932206i	$-0.382119\pi$
$461$	15.5418	0.723855	0.361928	+	0.932206i	$0.382119\pi$
$462$	0	0
$463$	36.4628	1.69457	0.847284	−	0.531139i	$-0.178236\pi$
$463$	36.4628	1.69457	0.847284	+	0.531139i	$0.178236\pi$
$464$	6.46522	0.300140
$465$	−6.91740	−0.320787
$466$	−38.6143	−1.78877
$467$	0.843160	0.0390168	0.0195084	−	0.999810i	$-0.493790\pi$
$467$	0.843160	0.0390168	0.0195084	+	0.999810i	$0.493790\pi$
$468$	−0.320351	−0.0148082
$469$	0	0
$470$	−11.1057	−0.512267
$471$	35.7961	1.64940
$472$	6.87359	0.316383
$473$	0.734453	0.0337702
$474$	−6.57173	−0.301850
$475$	−2.90554	−0.133315
$476$	0	0
$477$	0.0561317	0.00257009
$478$	−10.5973	−0.484710
$479$	11.3567	0.518902	0.259451	−	0.965756i	$-0.416458\pi$
$479$	11.3567	0.518902	0.259451	+	0.965756i	$0.416458\pi$
$480$	−11.1383	−0.508393
$481$	0.869614	0.0396510
$482$	29.4728	1.34245
$483$	0	0
$484$	6.67426	0.303375
$485$	−0.951656	−0.0432125
$486$	−5.01000	−0.227258
$487$	−1.48578	−0.0673270	−0.0336635	−	0.999433i	$-0.510717\pi$
$487$	−1.48578	−0.0673270	−0.0336635	+	0.999433i	$0.510717\pi$
$488$	−6.11940	−0.277012
$489$	25.9457	1.17330
$490$	0	0
$491$	−13.8261	−0.623965	−0.311982	−	0.950088i	$-0.600993\pi$
$491$	−13.8261	−0.623965	−0.311982	+	0.950088i	$0.600993\pi$
$492$	27.3123	1.23133
$493$	−8.05112	−0.362604
$494$	4.66319	0.209807
$495$	1.03961	0.0467270
$496$	−20.2128	−0.907583
$497$	0	0
$498$	29.0294	1.30084
$499$	−35.5263	−1.59038	−0.795188	−	0.606363i	$-0.792628\pi$
$499$	−35.5263	−1.59038	−0.795188	+	0.606363i	$0.792628\pi$
$500$	−1.40612	−0.0628836
$501$	−14.9813	−0.669315
$502$	−7.58529	−0.338548
$503$	7.80987	0.348225	0.174112	−	0.984726i	$-0.444294\pi$
$503$	7.80987	0.348225	0.174112	+	0.984726i	$0.444294\pi$
$504$	0	0
$505$	−12.7081	−0.565503
$506$	−53.5999	−2.38281
$507$	−20.2597	−0.899765
$508$	13.9344	0.618241
$509$	−2.54686	−0.112888	−0.0564439	−	0.998406i	$-0.517976\pi$
$509$	−2.54686	−0.112888	−0.0564439	+	0.998406i	$0.517976\pi$
$510$	18.3875	0.814212
$511$	0	0
$512$	−21.9476	−0.969957
$513$	15.6829	0.692417
$514$	32.7291	1.44362
$515$	4.62275	0.203703
$516$	0.430637	0.0189577
$517$	−23.8786	−1.05018
$518$	0	0
$519$	−11.8625	−0.520707
$520$	−0.953137	−0.0417978
$521$	41.5847	1.82186	0.910929	−	0.412562i	$-0.135366\pi$
$521$	41.5847	1.82186	0.910929	+	0.412562i	$0.135366\pi$
$522$	−0.646530	−0.0282978
$523$	7.19471	0.314603	0.157301	−	0.987551i	$-0.449721\pi$
$523$	7.19471	0.314603	0.157301	+	0.987551i	$0.449721\pi$
$524$	16.7616	0.732233
$525$	0	0
$526$	−7.68669	−0.335156
$527$	25.1710	1.09646
$528$	−31.7478	−1.38165
$529$	30.5651	1.32892
$530$	−0.395422	−0.0171760
$531$	−1.64298	−0.0712992
$532$	0	0
$533$	10.2081	0.442161
$534$	−42.0034	−1.81766
$535$	−1.54914	−0.0669750
$536$	−6.12469	−0.264546
$537$	9.22880	0.398252
$538$	48.9688	2.11119
$539$	0	0
$540$	7.58966	0.326607
$541$	−44.6388	−1.91917	−0.959585	−	0.281418i	$-0.909195\pi$
$541$	−44.6388	−1.91917	−0.959585	+	0.281418i	$0.909195\pi$
$542$	−21.7282	−0.933308
$543$	5.26627	0.225997
$544$	40.5301	1.73771
$545$	9.87238	0.422886
$546$	0	0
$547$	−3.79670	−0.162335	−0.0811677	−	0.996700i	$-0.525865\pi$
$547$	−3.79670	−0.162335	−0.0811677	+	0.996700i	$0.525865\pi$
$548$	−30.3199	−1.29520
$549$	1.46271	0.0624268
$550$	−7.32357	−0.312278
$551$	3.88515	0.165513
$552$	13.2736	0.564960
$553$	0	0
$554$	16.7448	0.711418
$555$	−1.65469	−0.0702379
$556$	−6.85384	−0.290668
$557$	6.61553	0.280309	0.140154	−	0.990130i	$-0.455240\pi$
$557$	6.61553	0.280309	0.140154	+	0.990130i	$0.455240\pi$
$558$	2.02131	0.0855687
$559$	0.160952	0.00680756
$560$	0	0
$561$	39.5354	1.66919
$562$	−11.9774	−0.505234
$563$	−32.0822	−1.35210	−0.676051	−	0.736855i	$-0.736310\pi$
$563$	−32.0822	−1.35210	−0.676051	+	0.736855i	$0.736310\pi$
$564$	−14.0009	−0.589544
$565$	5.56999	0.234331
$566$	2.73406	0.114921
$567$	0	0
$568$	8.57024	0.359599
$569$	−45.1921	−1.89455	−0.947275	−	0.320422i	$-0.896175\pi$
$569$	−45.1921	−1.89455	−0.947275	+	0.320422i	$0.896175\pi$
$570$	−8.87308	−0.371652
$571$	−10.5553	−0.441724	−0.220862	−	0.975305i	$-0.570887\pi$
$571$	−10.5553	−0.441724	−0.220862	+	0.975305i	$0.570887\pi$
$572$	4.85224	0.202882
$573$	−18.5604	−0.775374
$574$	0	0
$575$	7.31882	0.305216
$576$	0.721254	0.0300523
$577$	36.6297	1.52491	0.762457	−	0.647039i	$-0.223993\pi$
$577$	36.6297	1.52491	0.762457	+	0.647039i	$0.223993\pi$
$578$	−35.5336	−1.47800
$579$	−2.66944	−0.110938
$580$	1.88020	0.0780709
$581$	0	0
$582$	−2.90622	−0.120467
$583$	−0.850205	−0.0352119
$584$	1.35953	0.0562577
$585$	0.227826	0.00941946
$586$	−5.18746	−0.214292
$587$	26.1481	1.07925	0.539624	−	0.841906i	$-0.318566\pi$
$587$	26.1481	1.07925	0.539624	+	0.841906i	$0.318566\pi$
$588$	0	0
$589$	−12.1465	−0.500488
$590$	11.5740	0.476495
$591$	17.6013	0.724021
$592$	−4.83507	−0.198720
$593$	29.7296	1.22085	0.610424	−	0.792075i	$-0.290999\pi$
$593$	29.7296	1.22085	0.610424	+	0.792075i	$0.290999\pi$
$594$	39.5297	1.62192
$595$	0	0
$596$	−16.0381	−0.656946
$597$	−28.8770	−1.18186
$598$	−11.7462	−0.480338
$599$	42.1008	1.72019	0.860096	−	0.510132i	$-0.170404\pi$
$599$	42.1008	1.72019	0.860096	+	0.510132i	$0.170404\pi$
$600$	1.81362	0.0740408
$601$	−15.1347	−0.617356	−0.308678	−	0.951167i	$-0.599887\pi$
$601$	−15.1347	−0.617356	−0.308678	+	0.951167i	$0.599887\pi$
$602$	0	0
$603$	1.46397	0.0596175
$604$	15.7878	0.642396
$605$	−4.74658	−0.192976
$606$	−38.8086	−1.57649
$607$	32.5551	1.32137	0.660686	−	0.750662i	$-0.270265\pi$
$607$	32.5551	1.32137	0.660686	+	0.750662i	$0.270265\pi$
$608$	−19.5582	−0.793190
$609$	0	0
$610$	−10.3041	−0.417201
$611$	−5.23290	−0.211700
$612$	−2.21807	−0.0896601
$613$	−18.4294	−0.744355	−0.372177	−	0.928162i	$-0.621389\pi$
$613$	−18.4294	−0.744355	−0.372177	+	0.928162i	$0.621389\pi$
$614$	32.4319	1.30884
$615$	−19.4238	−0.783245
$616$	0	0
$617$	0.172985	0.00696412	0.00348206	−	0.999994i	$-0.498892\pi$
$617$	0.172985	0.00696412	0.00348206	+	0.999994i	$0.498892\pi$
$618$	14.1172	0.567877
$619$	−32.7554	−1.31655	−0.658274	−	0.752778i	$-0.728713\pi$
$619$	−32.7554	−1.31655	−0.658274	+	0.752778i	$0.728713\pi$
$620$	−5.87824	−0.236076
$621$	−39.5040	−1.58524
$622$	−34.7865	−1.39481
$623$	0	0
$624$	−6.95740	−0.278519
$625$	1.00000	0.0400000
$626$	−43.1021	−1.72271
$627$	−19.0782	−0.761910
$628$	30.4187	1.21384
$629$	6.02109	0.240077
$630$	0	0
$631$	36.8148	1.46557	0.732786	−	0.680459i	$-0.238219\pi$
$631$	36.8148	1.46557	0.732786	+	0.680459i	$0.238219\pi$
$632$	2.35863	0.0938215
$633$	−45.1190	−1.79332
$634$	−56.5290	−2.24505
$635$	−9.90985	−0.393261
$636$	−0.498506	−0.0197671
$637$	0	0
$638$	9.79274	0.387698
$639$	−2.04853	−0.0810384
$640$	8.38180	0.331320
$641$	9.85676	0.389319	0.194659	−	0.980871i	$-0.437640\pi$
$641$	9.85676	0.389319	0.194659	+	0.980871i	$0.437640\pi$
$642$	−4.73083	−0.186711
$643$	22.1002	0.871548	0.435774	−	0.900056i	$-0.356475\pi$
$643$	22.1002	0.871548	0.435774	+	0.900056i	$0.356475\pi$
$644$	0	0
$645$	−0.306259	−0.0120589
$646$	32.2873	1.27033
$647$	−15.0811	−0.592899	−0.296449	−	0.955049i	$-0.595803\pi$
$647$	−15.0811	−0.592899	−0.296449	+	0.955049i	$0.595803\pi$
$648$	−8.92774	−0.350715
$649$	24.8856	0.976845
$650$	−1.60493	−0.0629506
$651$	0	0
$652$	22.0480	0.863467
$653$	−34.3588	−1.34457	−0.672283	−	0.740294i	$-0.734686\pi$
$653$	−34.3588	−1.34457	−0.672283	+	0.740294i	$0.734686\pi$
$654$	30.1488	1.17891
$655$	−11.9205	−0.465771
$656$	−56.7570	−2.21599
$657$	−0.324966	−0.0126781
$658$	0	0
$659$	−41.2526	−1.60697	−0.803487	−	0.595322i	$-0.797025\pi$
$659$	−41.2526	−1.60697	−0.803487	+	0.595322i	$0.797025\pi$
$660$	−9.23280	−0.359386
$661$	−21.4484	−0.834245	−0.417122	−	0.908850i	$-0.636961\pi$
$661$	−21.4484	−0.834245	−0.417122	+	0.908850i	$0.636961\pi$
$662$	−32.6322	−1.26828
$663$	8.66402	0.336483
$664$	−10.4188	−0.404328
$665$	0	0
$666$	0.483512	0.0187357
$667$	−9.78638	−0.378930
$668$	−12.7307	−0.492567
$669$	29.3613	1.13517
$670$	−10.3130	−0.398426
$671$	−22.1551	−0.855287
$672$	0	0
$673$	22.9067	0.882989	0.441495	−	0.897264i	$-0.354448\pi$
$673$	22.9067	0.882989	0.441495	+	0.897264i	$0.354448\pi$
$674$	−27.2646	−1.05019
$675$	−5.39759	−0.207753
$676$	−17.2162	−0.662162
$677$	21.3096	0.818993	0.409497	−	0.912312i	$-0.365704\pi$
$677$	21.3096	0.818993	0.409497	+	0.912312i	$0.365704\pi$
$678$	17.0099	0.653261
$679$	0	0
$680$	−6.59939	−0.253075
$681$	10.0710	0.385920
$682$	−30.6160	−1.17235
$683$	34.9123	1.33588	0.667942	−	0.744214i	$-0.267176\pi$
$683$	34.9123	1.33588	0.667942	+	0.744214i	$0.267176\pi$
$684$	1.07035	0.0409259
$685$	21.5628	0.823872
$686$	0	0
$687$	−29.1811	−1.11333
$688$	−0.894898	−0.0341177
$689$	−0.186319	−0.00709819
$690$	22.3506	0.850872
$691$	−0.752275	−0.0286179	−0.0143090	−	0.999898i	$-0.504555\pi$
$691$	−0.752275	−0.0286179	−0.0143090	+	0.999898i	$0.504555\pi$
$692$	−10.0805	−0.383202
$693$	0	0
$694$	25.3606	0.962674
$695$	4.87429	0.184892
$696$	−2.42509	−0.0919228
$697$	70.6793	2.67717
$698$	−0.0455794	−0.00172521
$699$	34.6207	1.30947
$700$	0	0
$701$	−25.0969	−0.947896	−0.473948	−	0.880553i	$-0.657172\pi$
$701$	−25.0969	−0.947896	−0.473948	+	0.880553i	$0.657172\pi$
$702$	8.66277	0.326955
$703$	−2.90554	−0.109584
$704$	−10.9246	−0.411735
$705$	9.95711	0.375007
$706$	30.0765	1.13194
$707$	0	0
$708$	14.5913	0.548376
$709$	23.4450	0.880497	0.440248	−	0.897876i	$-0.354890\pi$
$709$	23.4450	0.880497	0.440248	+	0.897876i	$0.354890\pi$
$710$	14.4309	0.541583
$711$	−0.563780	−0.0211434
$712$	15.0753	0.564970
$713$	30.5961	1.14583
$714$	0	0
$715$	−3.45080	−0.129053
$716$	7.84241	0.293085
$717$	9.50131	0.354833
$718$	−31.7348	−1.18433
$719$	9.20788	0.343396	0.171698	−	0.985150i	$-0.445075\pi$
$719$	9.20788	0.343396	0.171698	+	0.985150i	$0.445075\pi$
$720$	−1.26672	−0.0472078
$721$	0	0
$722$	19.4852	0.725165
$723$	−26.4246	−0.982742
$724$	4.47515	0.166318
$725$	−1.33715	−0.0496606
$726$	−14.4953	−0.537973
$727$	−21.3870	−0.793199	−0.396599	−	0.917992i	$-0.629810\pi$
$727$	−21.3870	−0.793199	−0.396599	+	0.917992i	$0.629810\pi$
$728$	0	0
$729$	28.9281	1.07141
$730$	2.28923	0.0847283
$731$	1.11441	0.0412180
$732$	−12.9903	−0.480136
$733$	27.7328	1.02433	0.512166	−	0.858886i	$-0.328843\pi$
$733$	27.7328	1.02433	0.512166	+	0.858886i	$0.328843\pi$
$734$	−49.2906	−1.81935
$735$	0	0
$736$	49.2655	1.81595
$737$	−22.1742	−0.816798
$738$	5.67577	0.208928
$739$	−14.7858	−0.543904	−0.271952	−	0.962311i	$-0.587669\pi$
$739$	−14.7858	−0.543904	−0.271952	+	0.962311i	$0.587669\pi$
$740$	−1.40612	−0.0516900
$741$	−4.18091	−0.153590
$742$	0	0
$743$	4.73605	0.173749	0.0868745	−	0.996219i	$-0.472312\pi$
$743$	4.73605	0.173749	0.0868745	+	0.996219i	$0.472312\pi$
$744$	7.58178	0.277962
$745$	11.4059	0.417881
$746$	−60.4556	−2.21344
$747$	2.49039	0.0911185
$748$	33.5962	1.22840
$749$	0	0
$750$	3.05385	0.111511
$751$	39.7179	1.44933	0.724663	−	0.689103i	$-0.241995\pi$
$751$	39.7179	1.44933	0.724663	+	0.689103i	$0.241995\pi$
$752$	29.0950	1.06098
$753$	6.80080	0.247835
$754$	2.14604	0.0781542
$755$	−11.2279	−0.408626
$756$	0	0
$757$	−31.0771	−1.12952	−0.564758	−	0.825256i	$-0.691031\pi$
$757$	−31.0771	−1.12952	−0.564758	+	0.825256i	$0.691031\pi$
$758$	21.4530	0.779209
$759$	48.0565	1.74434
$760$	3.18460	0.115518
$761$	−11.9745	−0.434075	−0.217037	−	0.976163i	$-0.569639\pi$
$761$	−11.9745	−0.434075	−0.217037	+	0.976163i	$0.569639\pi$
$762$	−30.2632	−1.09632
$763$	0	0
$764$	−15.7722	−0.570619
$765$	1.57744	0.0570324
$766$	51.0881	1.84589
$767$	5.45358	0.196917
$768$	34.7076	1.25240
$769$	10.3532	0.373346	0.186673	−	0.982422i	$-0.440230\pi$
$769$	10.3532	0.373346	0.186673	+	0.982422i	$0.440230\pi$
$770$	0	0
$771$	−29.3442	−1.05681
$772$	−2.26842	−0.0816424
$773$	27.2250	0.979215	0.489607	−	0.871943i	$-0.337140\pi$
$773$	27.2250	0.979215	0.489607	+	0.871943i	$0.337140\pi$
$774$	0.0894908	0.00321668
$775$	4.18047	0.150167
$776$	1.04306	0.0374436
$777$	0	0
$778$	−43.3072	−1.55264
$779$	−34.1070	−1.22201
$780$	−2.02333	−0.0724469
$781$	31.0282	1.11028
$782$	−81.3291	−2.90832
$783$	7.21740	0.257929
$784$	0	0
$785$	−21.6331	−0.772117
$786$	−36.4033	−1.29846
$787$	46.7007	1.66470	0.832350	−	0.554250i	$-0.186995\pi$
$787$	46.7007	1.66470	0.832350	+	0.554250i	$0.186995\pi$
$788$	14.9572	0.532827
$789$	6.89171	0.245351
$790$	3.97157	0.141302
$791$	0	0
$792$	−1.13946	−0.0404890
$793$	−4.85519	−0.172413
$794$	−10.3643	−0.367817
$795$	0.354526	0.0125738
$796$	−24.5390	−0.869763
$797$	−41.8395	−1.48203	−0.741016	−	0.671488i	$-0.765656\pi$
$797$	−41.8395	−1.48203	−0.741016	+	0.671488i	$0.765656\pi$
$798$	0	0
$799$	−36.2319	−1.28179
$800$	6.73135	0.237989
$801$	−3.60341	−0.127320
$802$	29.4326	1.03930
$803$	4.92213	0.173698
$804$	−13.0016	−0.458530
$805$	0	0
$806$	−6.70937	−0.236327
$807$	−43.9043	−1.54550
$808$	13.9286	0.490008
$809$	20.3158	0.714264	0.357132	−	0.934054i	$-0.383755\pi$
$809$	20.3158	0.714264	0.357132	+	0.934054i	$0.383755\pi$
$810$	−15.0329	−0.528202
$811$	19.5256	0.685638	0.342819	−	0.939402i	$-0.388618\pi$
$811$	19.5256	0.685638	0.342819	+	0.939402i	$0.388618\pi$
$812$	0	0
$813$	19.4811	0.683231
$814$	−7.32357	−0.256691
$815$	−15.6800	−0.549248
$816$	−48.1721	−1.68636
$817$	−0.537771	−0.0188142
$818$	−51.2121	−1.79059
$819$	0	0
$820$	−16.5059	−0.576412
$821$	−52.7916	−1.84244	−0.921220	−	0.389041i	$-0.872806\pi$
$821$	−52.7916	−1.84244	−0.921220	+	0.389041i	$0.872806\pi$
$822$	65.8496	2.29677
$823$	−3.11210	−0.108481	−0.0542406	−	0.998528i	$-0.517274\pi$
$823$	−3.11210	−0.108481	−0.0542406	+	0.998528i	$0.517274\pi$
$824$	−5.06675	−0.176509
$825$	6.56615	0.228604
$826$	0	0
$827$	44.8394	1.55922	0.779609	−	0.626267i	$-0.215418\pi$
$827$	44.8394	1.55922	0.779609	+	0.626267i	$0.215418\pi$
$828$	−2.69613	−0.0936970
$829$	12.4898	0.433788	0.216894	−	0.976195i	$-0.430407\pi$
$829$	12.4898	0.433788	0.216894	+	0.976195i	$0.430407\pi$
$830$	−17.5436	−0.608948
$831$	−15.0130	−0.520796
$832$	−2.39407	−0.0829996
$833$	0	0
$834$	14.8854	0.515438
$835$	9.05381	0.313320
$836$	−16.2122	−0.560711
$837$	−22.5644	−0.779941
$838$	−23.8501	−0.823889
$839$	−25.7810	−0.890058	−0.445029	−	0.895516i	$-0.646807\pi$
$839$	−25.7810	−0.890058	−0.445029	+	0.895516i	$0.646807\pi$
$840$	0	0
$841$	−27.2120	−0.938346
$842$	−33.8237	−1.16564
$843$	10.7386	0.369858
$844$	−38.3411	−1.31975
$845$	12.2438	0.421199
$846$	−2.90953	−0.100032
$847$	0	0
$848$	1.03594	0.0355742
$849$	−2.45130	−0.0841283
$850$	−11.1123	−0.381150
$851$	7.31882	0.250886
$852$	18.1930	0.623282
$853$	36.7263	1.25748	0.628742	−	0.777614i	$-0.283570\pi$
$853$	36.7263	1.25748	0.628742	+	0.777614i	$0.283570\pi$
$854$	0	0
$855$	−0.761209	−0.0260328
$856$	1.69792	0.0580339
$857$	−45.1191	−1.54124	−0.770619	−	0.637296i	$-0.780053\pi$
$857$	−45.1191	−1.54124	−0.770619	+	0.637296i	$0.780053\pi$
$858$	−10.5382	−0.359769
$859$	39.4779	1.34697	0.673485	−	0.739201i	$-0.264797\pi$
$859$	39.4779	1.34697	0.673485	+	0.739201i	$0.264797\pi$
$860$	−0.260251	−0.00887450
$861$	0	0
$862$	−5.53152	−0.188404
$863$	10.9988	0.374405	0.187202	−	0.982321i	$-0.440058\pi$
$863$	10.9988	0.374405	0.187202	+	0.982321i	$0.440058\pi$
$864$	−36.3331	−1.23608
$865$	7.16901	0.243754
$866$	−29.1513	−0.990600
$867$	31.8587	1.08198
$868$	0	0
$869$	8.53936	0.289678
$870$	−4.08347	−0.138442
$871$	−4.85939	−0.164654
$872$	−10.8206	−0.366431
$873$	−0.249320	−0.00843821
$874$	39.2462	1.32752
$875$	0	0
$876$	2.88603	0.0975098
$877$	15.5935	0.526555	0.263277	−	0.964720i	$-0.415197\pi$
$877$	15.5935	0.526555	0.263277	+	0.964720i	$0.415197\pi$
$878$	21.3012	0.718881
$879$	4.65096	0.156873
$880$	19.1865	0.646777
$881$	34.0718	1.14791	0.573955	−	0.818887i	$-0.305409\pi$
$881$	34.0718	1.14791	0.573955	+	0.818887i	$0.305409\pi$
$882$	0	0
$883$	−35.1622	−1.18330	−0.591651	−	0.806194i	$-0.701524\pi$
$883$	−35.1622	−1.18330	−0.591651	+	0.806194i	$0.701524\pi$
$884$	7.36248	0.247627
$885$	−10.3770	−0.348820
$886$	35.7911	1.20242
$887$	−55.4484	−1.86177	−0.930887	−	0.365308i	$-0.880964\pi$
$887$	−55.4484	−1.86177	−0.930887	+	0.365308i	$0.880964\pi$
$888$	1.81362	0.0608611
$889$	0	0
$890$	25.3844	0.850886
$891$	−32.3226	−1.08285
$892$	24.9505	0.835404
$893$	17.4841	0.585082
$894$	34.8320	1.16496
$895$	−5.57734	−0.186430
$896$	0	0
$897$	10.5314	0.351633
$898$	−8.02443	−0.267779
$899$	−5.58992	−0.186434
$900$	−0.368383	−0.0122794
$901$	−1.29005	−0.0429777
$902$	−85.9687	−2.86244
$903$	0	0
$904$	−6.10496	−0.203048
$905$	−3.18262	−0.105794
$906$	−34.2884	−1.13916
$907$	−31.1167	−1.03321	−0.516606	−	0.856223i	$-0.672805\pi$
$907$	−31.1167	−1.03321	−0.516606	+	0.856223i	$0.672805\pi$
$908$	8.55806	0.284009
$909$	−3.32934	−0.110427
$910$	0	0
$911$	−20.3199	−0.673230	−0.336615	−	0.941642i	$-0.609282\pi$
$911$	−20.3199	−0.673230	−0.336615	+	0.941642i	$0.609282\pi$
$912$	23.2459	0.769750
$913$	−37.7209	−1.24838
$914$	37.8564	1.25218
$915$	9.23842	0.305413
$916$	−24.7974	−0.819330
$917$	0	0
$918$	59.9798	1.97963
$919$	20.2527	0.668076	0.334038	−	0.942560i	$-0.391589\pi$
$919$	20.2527	0.668076	0.334038	+	0.942560i	$0.391589\pi$
$920$	−8.02176	−0.264470
$921$	−29.0777	−0.958143
$922$	−28.6835	−0.944641
$923$	6.79972	0.223815
$924$	0	0
$925$	1.00000	0.0328798
$926$	−67.2945	−2.21144
$927$	1.21109	0.0397776
$928$	−9.00085	−0.295467
$929$	−12.3676	−0.405768	−0.202884	−	0.979203i	$-0.565031\pi$
$929$	−12.3676	−0.405768	−0.202884	+	0.979203i	$0.565031\pi$
$930$	12.7665	0.418631
$931$	0	0
$932$	29.4198	0.963678
$933$	31.1888	1.02108
$934$	−1.55611	−0.0509174
$935$	−23.8929	−0.781380
$936$	−0.249708	−0.00816197
$937$	32.6616	1.06701	0.533505	−	0.845797i	$-0.320875\pi$
$937$	32.6616	1.06701	0.533505	+	0.845797i	$0.320875\pi$
$938$	0	0
$939$	38.6444	1.26111
$940$	8.46132	0.275978
$941$	50.1886	1.63610	0.818051	−	0.575146i	$-0.195055\pi$
$941$	50.1886	1.63610	0.818051	+	0.575146i	$0.195055\pi$
$942$	−66.0641	−2.15249
$943$	85.9129	2.79771
$944$	−30.3220	−0.986896
$945$	0	0
$946$	−1.35548	−0.0440705
$947$	−43.9638	−1.42863	−0.714315	−	0.699824i	$-0.753262\pi$
$947$	−43.9638	−1.42863	−0.714315	+	0.699824i	$0.753262\pi$
$948$	5.00694	0.162618
$949$	1.07867	0.0350150
$950$	5.36237	0.173978
$951$	50.6826	1.64350
$952$	0	0
$953$	−17.4827	−0.566319	−0.283159	−	0.959073i	$-0.591383\pi$
$953$	−17.4827	−0.566319	−0.283159	+	0.959073i	$0.591383\pi$
$954$	−0.103595	−0.00335401
$955$	11.2168	0.362968
$956$	8.07399	0.261131
$957$	−8.77995	−0.283815
$958$	−20.9596	−0.677174
$959$	0	0
$960$	4.55542	0.147026
$961$	−13.5237	−0.436248
$962$	−1.60493	−0.0517451
$963$	−0.405851	−0.0130784
$964$	−22.4550	−0.723227
$965$	1.61325	0.0519324
$966$	0	0
$967$	−59.9966	−1.92936	−0.964681	−	0.263422i	$-0.915149\pi$
$967$	−59.9966	−1.92936	−0.964681	+	0.263422i	$0.915149\pi$
$968$	5.20247	0.167214
$969$	−28.9481	−0.929946
$970$	1.75635	0.0563929
$971$	55.1365	1.76941	0.884707	−	0.466147i	$-0.154358\pi$
$971$	55.1365	1.76941	0.884707	+	0.466147i	$0.154358\pi$
$972$	3.81707	0.122432
$973$	0	0
$974$	2.74210	0.0878627
$975$	1.43895	0.0460832
$976$	26.9949	0.864087
$977$	28.7274	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$977$	28.7274	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$978$	−47.8845	−1.53118
$979$	54.5795	1.74437
$980$	0	0
$981$	2.58642	0.0825781
$982$	25.5171	0.814282
$983$	−24.3437	−0.776444	−0.388222	−	0.921566i	$-0.626911\pi$
$983$	−24.3437	−0.776444	−0.388222	+	0.921566i	$0.626911\pi$
$984$	21.2894	0.678682
$985$	−10.6372	−0.338929
$986$	14.8589	0.473203
$987$	0	0
$988$	−3.55284	−0.113031
$989$	1.35460	0.0430738
$990$	−1.91867	−0.0609794
$991$	−12.6966	−0.403322	−0.201661	−	0.979455i	$-0.564634\pi$
$991$	−12.6966	−0.403322	−0.201661	+	0.979455i	$0.564634\pi$
$992$	28.1402	0.893452
$993$	29.2573	0.928451
$994$	0	0
$995$	17.4516	0.553252
$996$	−22.1172	−0.700810
$997$	−20.0671	−0.635533	−0.317766	−	0.948169i	$-0.602933\pi$
$997$	−20.0671	−0.635533	−0.317766	+	0.948169i	$0.602933\pi$
$998$	65.5662	2.07546
$999$	−5.39759	−0.170772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bc.1.9	yes	34
7.6	odd	2		9065.2.a.bb.1.9	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bb.1.9	✓	34	7.6	odd	2
9065.2.a.bc.1.9	yes	34	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)

\(f(q)\)	\(=\)	\(q-1.84557 q^{2} +1.65469 q^{3} +1.40612 q^{4} -1.00000 q^{5} -3.05385 q^{6} +1.09605 q^{8} -0.261986 q^{9} +1.84557 q^{10} +3.96820 q^{11} +2.32670 q^{12} +0.869614 q^{13} -1.65469 q^{15} -4.83507 q^{16} +6.02109 q^{17} +0.483512 q^{18} -2.90554 q^{19} -1.40612 q^{20} -7.32357 q^{22} +7.31882 q^{23} +1.81362 q^{24} +1.00000 q^{25} -1.60493 q^{26} -5.39759 q^{27} -1.33715 q^{29} +3.05385 q^{30} +4.18047 q^{31} +6.73135 q^{32} +6.56615 q^{33} -11.1123 q^{34} -0.368383 q^{36} +1.00000 q^{37} +5.36237 q^{38} +1.43895 q^{39} -1.09605 q^{40} +11.7386 q^{41} +0.185085 q^{43} +5.57976 q^{44} +0.261986 q^{45} -13.5074 q^{46} -6.01749 q^{47} -8.00056 q^{48} -1.84557 q^{50} +9.96306 q^{51} +1.22278 q^{52} -0.214255 q^{53} +9.96162 q^{54} -3.96820 q^{55} -4.80778 q^{57} +2.46781 q^{58} +6.27126 q^{59} -2.32670 q^{60} -5.58316 q^{61} -7.71533 q^{62} -2.75303 q^{64} -0.869614 q^{65} -12.1183 q^{66} -5.58798 q^{67} +8.46637 q^{68} +12.1104 q^{69} +7.81923 q^{71} -0.287148 q^{72} +1.24040 q^{73} -1.84557 q^{74} +1.65469 q^{75} -4.08554 q^{76} -2.65567 q^{78} +2.15195 q^{79} +4.83507 q^{80} -8.14541 q^{81} -21.6644 q^{82} -9.50582 q^{83} -6.02109 q^{85} -0.341587 q^{86} -2.21258 q^{87} +4.34932 q^{88} +13.7542 q^{89} -0.483512 q^{90} +10.2911 q^{92} +6.91740 q^{93} +11.1057 q^{94} +2.90554 q^{95} +11.1383 q^{96} +0.951656 q^{97} -1.03961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) \) 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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