LMFDB - Embedded newform 9065.2.a.bd.1.4

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 = [38,-2,-6,42,-38,-8,0,-6,48,2,34] 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:	$38$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	\(q-2.54479 q^{2} +2.35441 q^{3} +4.47595 q^{4} -1.00000 q^{5} -5.99147 q^{6} -6.30078 q^{8} +2.54323 q^{9} +2.54479 q^{10} -2.20460 q^{11} +10.5382 q^{12} +3.88869 q^{13} -2.35441 q^{15} +7.08224 q^{16} +6.64930 q^{17} -6.47198 q^{18} -7.22049 q^{19} -4.47595 q^{20} +5.61025 q^{22} -8.15756 q^{23} -14.8346 q^{24} +1.00000 q^{25} -9.89589 q^{26} -1.07542 q^{27} -3.93286 q^{29} +5.99147 q^{30} +0.890739 q^{31} -5.42127 q^{32} -5.19053 q^{33} -16.9211 q^{34} +11.3834 q^{36} -1.00000 q^{37} +18.3746 q^{38} +9.15555 q^{39} +6.30078 q^{40} +11.5171 q^{41} -5.23431 q^{43} -9.86769 q^{44} -2.54323 q^{45} +20.7593 q^{46} +7.64537 q^{47} +16.6745 q^{48} -2.54479 q^{50} +15.6552 q^{51} +17.4056 q^{52} +11.4750 q^{53} +2.73672 q^{54} +2.20460 q^{55} -17.0000 q^{57} +10.0083 q^{58} +7.15917 q^{59} -10.5382 q^{60} -4.43500 q^{61} -2.26674 q^{62} -0.368509 q^{64} -3.88869 q^{65} +13.2088 q^{66} -14.6358 q^{67} +29.7620 q^{68} -19.2062 q^{69} -9.40829 q^{71} -16.0243 q^{72} -4.68801 q^{73} +2.54479 q^{74} +2.35441 q^{75} -32.3186 q^{76} -23.2990 q^{78} -6.11966 q^{79} -7.08224 q^{80} -10.1617 q^{81} -29.3086 q^{82} +12.0500 q^{83} -6.64930 q^{85} +13.3202 q^{86} -9.25954 q^{87} +13.8907 q^{88} +0.282461 q^{89} +6.47198 q^{90} -36.5129 q^{92} +2.09716 q^{93} -19.4559 q^{94} +7.22049 q^{95} -12.7639 q^{96} +17.0402 q^{97} -5.60681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) $ 38 * q - 2 * q^2 - 6 * q^3 + 42 * q^4 - 38 * q^5 - 8 * q^6 - 6 * q^8 + 48 * q^9 + 2 * q^10 + 34 * q^11 - 20 * q^12 - 22 * q^13 + 6 * q^15 + 46 * q^16 - 22 * q^17 - 36 * q^18 - 40 * q^19 - 42 * q^20 - 4 * q^22 - 16 * q^23 - 24 * q^24 + 38 * q^25 - 20 * q^26 - 30 * q^27 + 26 * q^29 + 8 * q^30 - 52 * q^31 - 34 * q^32 - 38 * q^33 - 12 * q^34 + 74 * q^36 - 38 * q^37 - 16 * q^38 + 22 * q^39 + 6 * q^40 - 16 * q^41 + 72 * q^44 - 48 * q^45 + 12 * q^46 - 34 * q^47 - 34 * q^48 - 2 * q^50 + 46 * q^51 - 68 * q^52 + 4 * q^53 - 32 * q^54 - 34 * q^55 - 12 * q^57 - 40 * q^58 - 32 * q^59 + 20 * q^60 - 60 * q^61 + 38 * q^62 + 22 * q^64 + 22 * q^65 - 66 * q^66 - 20 * q^67 - 60 * q^68 - 24 * q^69 + 52 * q^71 - 78 * q^72 - 48 * q^73 + 2 * q^74 - 6 * q^75 - 86 * q^76 + 8 * q^78 + 22 * q^79 - 46 * q^80 + 58 * q^81 - 36 * q^82 - 44 * q^83 + 22 * q^85 + 20 * q^86 - 106 * q^87 - 66 * q^88 - 52 * q^89 + 36 * q^90 - 58 * q^92 - 48 * q^93 - 96 * q^94 + 40 * q^95 - 26 * q^96 - 38 * q^97 + 92 * 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 $

$2$	−2.54479	−1.79944	−0.899719	−	0.436470i	$-0.856229\pi$
$2$	−2.54479	−1.79944	−0.899719	+	0.436470i	$0.856229\pi$
$3$	2.35441	1.35932	0.679659	−	0.733529i	$-0.262128\pi$
$3$	2.35441	1.35932	0.679659	+	0.733529i	$0.262128\pi$
$4$	4.47595	2.23798
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−5.99147	−2.44601
$7$	0	0
$7$	0	0
$8$	−6.30078	−2.22766
$9$	2.54323	0.847743
$10$	2.54479	0.804733
$11$	−2.20460	−0.664713	−0.332356	−	0.943154i	$-0.607844\pi$
$11$	−2.20460	−0.664713	−0.332356	+	0.943154i	$0.607844\pi$
$12$	10.5382	3.04212
$13$	3.88869	1.07853	0.539264	−	0.842137i	$-0.318702\pi$
$13$	3.88869	1.07853	0.539264	+	0.842137i	$0.318702\pi$
$14$	0	0
$15$	−2.35441	−0.607905
$16$	7.08224	1.77056
$17$	6.64930	1.61269	0.806346	−	0.591444i	$-0.201442\pi$
$17$	6.64930	1.61269	0.806346	+	0.591444i	$0.201442\pi$
$18$	−6.47198	−1.52546
$19$	−7.22049	−1.65650	−0.828248	−	0.560362i	$-0.810662\pi$
$19$	−7.22049	−1.65650	−0.828248	+	0.560362i	$0.810662\pi$
$20$	−4.47595	−1.00085
$21$	0	0
$22$	5.61025	1.19611
$23$	−8.15756	−1.70097	−0.850485	−	0.526000i	$-0.823691\pi$
$23$	−8.15756	−1.70097	−0.850485	+	0.526000i	$0.823691\pi$
$24$	−14.8346	−3.02810
$25$	1.00000	0.200000
$26$	−9.89589	−1.94074
$27$	−1.07542	−0.206965
$28$	0	0
$29$	−3.93286	−0.730313	−0.365157	−	0.930946i	$-0.618985\pi$
$29$	−3.93286	−0.730313	−0.365157	+	0.930946i	$0.618985\pi$
$30$	5.99147	1.09389
$31$	0.890739	0.159981	0.0799907	−	0.996796i	$-0.474511\pi$
$31$	0.890739	0.159981	0.0799907	+	0.996796i	$0.474511\pi$
$32$	−5.42127	−0.958354
$33$	−5.19053	−0.903555
$34$	−16.9211	−2.90194
$35$	0	0
$36$	11.3834	1.89723
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	18.3746	2.98076
$39$	9.15555	1.46606
$40$	6.30078	0.996240
$41$	11.5171	1.79867	0.899334	−	0.437262i	$-0.144052\pi$
$41$	11.5171	1.79867	0.899334	+	0.437262i	$0.144052\pi$
$42$	0	0
$43$	−5.23431	−0.798224	−0.399112	−	0.916902i	$-0.630682\pi$
$43$	−5.23431	−0.798224	−0.399112	+	0.916902i	$0.630682\pi$
$44$	−9.86769	−1.48761
$45$	−2.54323	−0.379122
$46$	20.7593	3.06079
$47$	7.64537	1.11519	0.557596	−	0.830112i	$-0.311724\pi$
$47$	7.64537	1.11519	0.557596	+	0.830112i	$0.311724\pi$
$48$	16.6745	2.40675
$49$	0	0
$50$	−2.54479	−0.359888
$51$	15.6552	2.19216
$52$	17.4056	2.41372
$53$	11.4750	1.57621	0.788103	−	0.615543i	$-0.211063\pi$
$53$	11.4750	1.57621	0.788103	+	0.615543i	$0.211063\pi$
$54$	2.73672	0.372421
$55$	2.20460	0.297269
$56$	0	0
$57$	−17.0000	−2.25170
$58$	10.0083	1.31415
$59$	7.15917	0.932044	0.466022	−	0.884773i	$-0.345687\pi$
$59$	7.15917	0.932044	0.466022	+	0.884773i	$0.345687\pi$
$60$	−10.5382	−1.36048
$61$	−4.43500	−0.567844	−0.283922	−	0.958847i	$-0.591636\pi$
$61$	−4.43500	−0.567844	−0.283922	+	0.958847i	$0.591636\pi$
$62$	−2.26674	−0.287877
$63$	0	0
$64$	−0.368509	−0.0460636
$65$	−3.88869	−0.482333
$66$	13.2088	1.62589
$67$	−14.6358	−1.78805	−0.894024	−	0.448018i	$-0.852130\pi$
$67$	−14.6358	−1.78805	−0.894024	+	0.448018i	$0.852130\pi$
$68$	29.7620	3.60917
$69$	−19.2062	−2.31216
$70$	0	0
$71$	−9.40829	−1.11656	−0.558279	−	0.829653i	$-0.688538\pi$
$71$	−9.40829	−1.11656	−0.558279	+	0.829653i	$0.688538\pi$
$72$	−16.0243	−1.88848
$73$	−4.68801	−0.548690	−0.274345	−	0.961631i	$-0.588461\pi$
$73$	−4.68801	−0.548690	−0.274345	+	0.961631i	$0.588461\pi$
$74$	2.54479	0.295826
$75$	2.35441	0.271863
$76$	−32.3186	−3.70720
$77$	0	0
$78$	−23.2990	−2.63809
$79$	−6.11966	−0.688516	−0.344258	−	0.938875i	$-0.611869\pi$
$79$	−6.11966	−0.688516	−0.344258	+	0.938875i	$0.611869\pi$
$80$	−7.08224	−0.791819
$81$	−10.1617	−1.12907
$82$	−29.3086	−3.23659
$83$	12.0500	1.32266	0.661328	−	0.750097i	$-0.269993\pi$
$83$	12.0500	1.32266	0.661328	+	0.750097i	$0.269993\pi$
$84$	0	0
$85$	−6.64930	−0.721218
$86$	13.3202	1.43636
$87$	−9.25954	−0.992727
$88$	13.8907	1.48075
$89$	0.282461	0.0299408	0.0149704	−	0.999888i	$-0.495235\pi$
$89$	0.282461	0.0299408	0.0149704	+	0.999888i	$0.495235\pi$
$90$	6.47198	0.682207
$91$	0	0
$92$	−36.5129	−3.80673
$93$	2.09716	0.217466
$94$	−19.4559	−2.00672
$95$	7.22049	0.740807
$96$	−12.7639	−1.30271
$97$	17.0402	1.73017	0.865084	−	0.501628i	$-0.167265\pi$
$97$	17.0402	1.73017	0.865084	+	0.501628i	$0.167265\pi$
$98$	0	0
$99$	−5.60681	−0.563506
$100$	4.47595	0.447595
$101$	3.40767	0.339076	0.169538	−	0.985524i	$-0.445773\pi$
$101$	3.40767	0.339076	0.169538	+	0.985524i	$0.445773\pi$
$102$	−39.8391	−3.94466
$103$	−15.1214	−1.48996	−0.744980	−	0.667087i	$-0.767541\pi$
$103$	−15.1214	−1.48996	−0.744980	+	0.667087i	$0.767541\pi$
$104$	−24.5018	−2.40260
$105$	0	0
$106$	−29.2013	−2.83629
$107$	10.4790	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$107$	10.4790	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$108$	−4.81354	−0.463183
$109$	−5.74162	−0.549948	−0.274974	−	0.961452i	$-0.588669\pi$
$109$	−5.74162	−0.549948	−0.274974	+	0.961452i	$0.588669\pi$
$110$	−5.61025	−0.534916
$111$	−2.35441	−0.223470
$112$	0	0
$113$	−10.8932	−1.02474	−0.512371	−	0.858764i	$-0.671233\pi$
$113$	−10.8932	−1.02474	−0.512371	+	0.858764i	$0.671233\pi$
$114$	43.2614	4.05180
$115$	8.15756	0.760696
$116$	−17.6033	−1.63442
$117$	9.88983	0.914315
$118$	−18.2186	−1.67716
$119$	0	0
$120$	14.8346	1.35421
$121$	−6.13973	−0.558157
$122$	11.2862	1.02180
$123$	27.1159	2.44496
$124$	3.98691	0.358035
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.24161	0.110175	0.0550876	−	0.998482i	$-0.482456\pi$
$127$	1.24161	0.110175	0.0550876	+	0.998482i	$0.482456\pi$
$128$	11.7803	1.04124
$129$	−12.3237	−1.08504
$130$	9.89589	0.867927
$131$	−8.20271	−0.716674	−0.358337	−	0.933592i	$-0.616656\pi$
$131$	−8.20271	−0.716674	−0.358337	+	0.933592i	$0.616656\pi$
$132$	−23.2326	−2.02214
$133$	0	0
$134$	37.2451	3.21748
$135$	1.07542	0.0925576
$136$	−41.8958	−3.59253
$137$	−6.15675	−0.526007	−0.263003	−	0.964795i	$-0.584713\pi$
$137$	−6.15675	−0.526007	−0.263003	+	0.964795i	$0.584713\pi$
$138$	48.8758	4.16058
$139$	−2.28391	−0.193719	−0.0968596	−	0.995298i	$-0.530880\pi$
$139$	−2.28391	−0.193719	−0.0968596	+	0.995298i	$0.530880\pi$
$140$	0	0
$141$	18.0003	1.51590
$142$	23.9421	2.00918
$143$	−8.57301	−0.716911
$144$	18.0118	1.50098
$145$	3.93286	0.326606
$146$	11.9300	0.987334
$147$	0	0
$148$	−4.47595	−0.367921
$149$	11.8950	0.974475	0.487238	−	0.873269i	$-0.338005\pi$
$149$	11.8950	0.974475	0.487238	+	0.873269i	$0.338005\pi$
$150$	−5.99147	−0.489201
$151$	−19.5398	−1.59013	−0.795065	−	0.606525i	$-0.792563\pi$
$151$	−19.5398	−1.59013	−0.795065	+	0.606525i	$0.792563\pi$
$152$	45.4947	3.69011
$153$	16.9107	1.36715
$154$	0	0
$155$	−0.890739	−0.0715459
$156$	40.9798	3.28101
$157$	−14.9353	−1.19196	−0.595982	−	0.802998i	$-0.703237\pi$
$157$	−14.9353	−1.19196	−0.595982	+	0.802998i	$0.703237\pi$
$158$	15.5732	1.23894
$159$	27.0167	2.14256
$160$	5.42127	0.428589
$161$	0	0
$162$	25.8593	2.03170
$163$	2.41381	0.189064	0.0945320	−	0.995522i	$-0.469865\pi$
$163$	2.41381	0.189064	0.0945320	+	0.995522i	$0.469865\pi$
$164$	51.5500	4.02538
$165$	5.19053	0.404082
$166$	−30.6646	−2.38004
$167$	6.81805	0.527597	0.263798	−	0.964578i	$-0.415025\pi$
$167$	6.81805	0.527597	0.263798	+	0.964578i	$0.415025\pi$
$168$	0	0
$169$	2.12190	0.163223
$170$	16.9211	1.29779
$171$	−18.3634	−1.40428
$172$	−23.4285	−1.78641
$173$	−18.6792	−1.42015	−0.710077	−	0.704124i	$-0.751340\pi$
$173$	−18.6792	−1.42015	−0.710077	+	0.704124i	$0.751340\pi$
$174$	23.5636	1.78635
$175$	0	0
$176$	−15.6135	−1.17691
$177$	16.8556	1.26694
$178$	−0.718803	−0.0538766
$179$	−4.51287	−0.337308	−0.168654	−	0.985675i	$-0.553942\pi$
$179$	−4.51287	−0.337308	−0.168654	+	0.985675i	$0.553942\pi$
$180$	−11.3834	−0.848467
$181$	−10.1544	−0.754774	−0.377387	−	0.926056i	$-0.623177\pi$
$181$	−10.1544	−0.754774	−0.377387	+	0.926056i	$0.623177\pi$
$182$	0	0
$183$	−10.4418	−0.771880
$184$	51.3990	3.78918
$185$	1.00000	0.0735215
$186$	−5.33684	−0.391316
$187$	−14.6591	−1.07198
$188$	34.2203	2.49577
$189$	0	0
$190$	−18.3746	−1.33304
$191$	−8.33736	−0.603271	−0.301635	−	0.953423i	$-0.597532\pi$
$191$	−8.33736	−0.603271	−0.301635	+	0.953423i	$0.597532\pi$
$192$	−0.867620	−0.0626151
$193$	6.53651	0.470508	0.235254	−	0.971934i	$-0.424408\pi$
$193$	6.53651	0.470508	0.235254	+	0.971934i	$0.424408\pi$
$194$	−43.3636	−3.11333
$195$	−9.15555	−0.655643
$196$	0	0
$197$	−3.20355	−0.228243	−0.114122	−	0.993467i	$-0.536405\pi$
$197$	−3.20355	−0.228243	−0.114122	+	0.993467i	$0.536405\pi$
$198$	14.2682	1.01399
$199$	−10.3204	−0.731592	−0.365796	−	0.930695i	$-0.619203\pi$
$199$	−10.3204	−0.731592	−0.365796	+	0.930695i	$0.619203\pi$
$200$	−6.30078	−0.445532
$201$	−34.4587	−2.43053
$202$	−8.67180	−0.610145
$203$	0	0
$204$	70.0718	4.90600
$205$	−11.5171	−0.804389
$206$	38.4809	2.68109
$207$	−20.7466	−1.44199
$208$	27.5406	1.90960
$209$	15.9183	1.10109
$210$	0	0
$211$	4.17102	0.287145	0.143573	−	0.989640i	$-0.454141\pi$
$211$	4.17102	0.287145	0.143573	+	0.989640i	$0.454141\pi$
$212$	51.3614	3.52751
$213$	−22.1509	−1.51776
$214$	−26.6668	−1.82291
$215$	5.23431	0.356977
$216$	6.77600	0.461048
$217$	0	0
$218$	14.6112	0.989596
$219$	−11.0375	−0.745844
$220$	9.86769	0.665280
$221$	25.8571	1.73933
$222$	5.99147	0.402121
$223$	10.3965	0.696202	0.348101	−	0.937457i	$-0.386827\pi$
$223$	10.3965	0.696202	0.348101	+	0.937457i	$0.386827\pi$
$224$	0	0
$225$	2.54323	0.169549
$226$	27.7208	1.84396
$227$	−3.45351	−0.229217	−0.114609	−	0.993411i	$-0.536561\pi$
$227$	−3.45351	−0.229217	−0.114609	+	0.993411i	$0.536561\pi$
$228$	−76.0911	−5.03926
$229$	24.0355	1.58831	0.794155	−	0.607715i	$-0.207914\pi$
$229$	24.0355	1.58831	0.794155	+	0.607715i	$0.207914\pi$
$230$	−20.7593	−1.36883
$231$	0	0
$232$	24.7801	1.62689
$233$	−18.2982	−1.19876	−0.599379	−	0.800465i	$-0.704586\pi$
$233$	−18.2982	−1.19876	−0.599379	+	0.800465i	$0.704586\pi$
$234$	−25.1675	−1.64525
$235$	−7.64537	−0.498729
$236$	32.0441	2.08589
$237$	−14.4082	−0.935911
$238$	0	0
$239$	13.4593	0.870612	0.435306	−	0.900283i	$-0.356640\pi$
$239$	13.4593	0.870612	0.435306	+	0.900283i	$0.356640\pi$
$240$	−16.6745	−1.07633
$241$	−10.9640	−0.706252	−0.353126	−	0.935576i	$-0.614881\pi$
$241$	−10.9640	−0.706252	−0.353126	+	0.935576i	$0.614881\pi$
$242$	15.6243	1.00437
$243$	−20.6984	−1.32781
$244$	−19.8509	−1.27082
$245$	0	0
$246$	−69.0043	−4.39955
$247$	−28.0783	−1.78658
$248$	−5.61235	−0.356385
$249$	28.3705	1.79791
$250$	2.54479	0.160947
$251$	−31.0897	−1.96236	−0.981181	−	0.193089i	$-0.938149\pi$
$251$	−31.0897	−1.96236	−0.981181	+	0.193089i	$0.938149\pi$
$252$	0	0
$253$	17.9842	1.13066
$254$	−3.15964	−0.198253
$255$	−15.6552	−0.980364
$256$	−29.2414	−1.82759
$257$	12.8560	0.801936	0.400968	−	0.916092i	$-0.368674\pi$
$257$	12.8560	0.801936	0.400968	+	0.916092i	$0.368674\pi$
$258$	31.3612	1.95246
$259$	0	0
$260$	−17.4056	−1.07945
$261$	−10.0022	−0.619118
$262$	20.8742	1.28961
$263$	−16.9630	−1.04599	−0.522993	−	0.852337i	$-0.675185\pi$
$263$	−16.9630	−1.04599	−0.522993	+	0.852337i	$0.675185\pi$
$264$	32.7044	2.01281
$265$	−11.4750	−0.704901
$266$	0	0
$267$	0.665027	0.0406990
$268$	−65.5092	−4.00161
$269$	17.1329	1.04461	0.522307	−	0.852758i	$-0.325072\pi$
$269$	17.1329	1.04461	0.522307	+	0.852758i	$0.325072\pi$
$270$	−2.73672	−0.166552
$271$	21.5304	1.30788	0.653939	−	0.756547i	$-0.273115\pi$
$271$	21.5304	1.30788	0.653939	+	0.756547i	$0.273115\pi$
$272$	47.0920	2.85537
$273$	0	0
$274$	15.6676	0.946516
$275$	−2.20460	−0.132943
$276$	−85.9661	−5.17455
$277$	−26.8300	−1.61206	−0.806030	−	0.591875i	$-0.798388\pi$
$277$	−26.8300	−1.61206	−0.806030	+	0.591875i	$0.798388\pi$
$278$	5.81208	0.348586
$279$	2.26535	0.135623
$280$	0	0
$281$	−31.6042	−1.88535	−0.942675	−	0.333713i	$-0.891698\pi$
$281$	−31.6042	−1.88535	−0.942675	+	0.333713i	$0.891698\pi$
$282$	−45.8070	−2.72777
$283$	7.95534	0.472896	0.236448	−	0.971644i	$-0.424017\pi$
$283$	7.95534	0.472896	0.236448	+	0.971644i	$0.424017\pi$
$284$	−42.1111	−2.49883
$285$	17.0000	1.00699
$286$	21.8165	1.29004
$287$	0	0
$288$	−13.7875	−0.812438
$289$	27.2132	1.60078
$290$	−10.0083	−0.587707
$291$	40.1195	2.35185
$292$	−20.9833	−1.22796
$293$	1.88525	0.110138	0.0550688	−	0.998483i	$-0.482462\pi$
$293$	1.88525	0.110138	0.0550688	+	0.998483i	$0.482462\pi$
$294$	0	0
$295$	−7.15917	−0.416823
$296$	6.30078	0.366225
$297$	2.37088	0.137572
$298$	−30.2702	−1.75351
$299$	−31.7222	−1.83454
$300$	10.5382	0.608424
$301$	0	0
$302$	49.7248	2.86134
$303$	8.02304	0.460911
$304$	−51.1373	−2.93293
$305$	4.43500	0.253948
$306$	−43.0342	−2.46010
$307$	−27.6166	−1.57616	−0.788081	−	0.615571i	$-0.788925\pi$
$307$	−27.6166	−1.57616	−0.788081	+	0.615571i	$0.788925\pi$
$308$	0	0
$309$	−35.6020	−2.02533
$310$	2.26674	0.128742
$311$	11.8079	0.669563	0.334782	−	0.942296i	$-0.391337\pi$
$311$	11.8079	0.669563	0.334782	+	0.942296i	$0.391337\pi$
$312$	−57.6871	−3.26589
$313$	4.29805	0.242940	0.121470	−	0.992595i	$-0.461239\pi$
$313$	4.29805	0.242940	0.121470	+	0.992595i	$0.461239\pi$
$314$	38.0071	2.14487
$315$	0	0
$316$	−27.3913	−1.54088
$317$	−13.6201	−0.764980	−0.382490	−	0.923960i	$-0.624933\pi$
$317$	−13.6201	−0.764980	−0.382490	+	0.923960i	$0.624933\pi$
$318$	−68.7518	−3.85541
$319$	8.67039	0.485448
$320$	0.368509	0.0206003
$321$	24.6718	1.37704
$322$	0	0
$323$	−48.0113	−2.67142
$324$	−45.4832	−2.52684
$325$	3.88869	0.215706
$326$	−6.14263	−0.340209
$327$	−13.5181	−0.747553
$328$	−72.5667	−4.00682
$329$	0	0
$330$	−13.2088	−0.727121
$331$	10.2240	0.561964	0.280982	−	0.959713i	$-0.409340\pi$
$331$	10.2240	0.561964	0.280982	+	0.959713i	$0.409340\pi$
$332$	53.9350	2.96007
$333$	−2.54323	−0.139368
$334$	−17.3505	−0.949378
$335$	14.6358	0.799640
$336$	0	0
$337$	−10.0875	−0.549503	−0.274752	−	0.961515i	$-0.588596\pi$
$337$	−10.0875	−0.549503	−0.274752	+	0.961515i	$0.588596\pi$
$338$	−5.39980	−0.293710
$339$	−25.6469	−1.39295
$340$	−29.7620	−1.61407
$341$	−1.96373	−0.106342
$342$	46.7309	2.52692
$343$	0	0
$344$	32.9802	1.77817
$345$	19.2062	1.03403
$346$	47.5346	2.55548
$347$	−12.6324	−0.678142	−0.339071	−	0.940761i	$-0.610113\pi$
$347$	−12.6324	−0.678142	−0.339071	+	0.940761i	$0.610113\pi$
$348$	−41.4453	−2.22170
$349$	36.0137	1.92777	0.963886	−	0.266316i	$-0.0858064\pi$
$349$	36.0137	1.92777	0.963886	+	0.266316i	$0.0858064\pi$
$350$	0	0
$351$	−4.18198	−0.223218
$352$	11.9517	0.637030
$353$	−35.0543	−1.86575	−0.932875	−	0.360200i	$-0.882709\pi$
$353$	−35.0543	−1.86575	−0.932875	+	0.360200i	$0.882709\pi$
$354$	−42.8939	−2.27979
$355$	9.40829	0.499340
$356$	1.26428	0.0670067
$357$	0	0
$358$	11.4843	0.606965
$359$	12.5814	0.664023	0.332011	−	0.943275i	$-0.392273\pi$
$359$	12.5814	0.664023	0.332011	+	0.943275i	$0.392273\pi$
$360$	16.0243	0.844556
$361$	33.1355	1.74398
$362$	25.8409	1.35817
$363$	−14.4554	−0.758713
$364$	0	0
$365$	4.68801	0.245382
$366$	26.5722	1.38895
$367$	2.61616	0.136562	0.0682812	−	0.997666i	$-0.478248\pi$
$367$	2.61616	0.136562	0.0682812	+	0.997666i	$0.478248\pi$
$368$	−57.7738	−3.01167
$369$	29.2906	1.52481
$370$	−2.54479	−0.132297
$371$	0	0
$372$	9.38680	0.486683
$373$	−33.8273	−1.75151	−0.875756	−	0.482754i	$-0.839637\pi$
$373$	−33.8273	−1.75151	−0.875756	+	0.482754i	$0.839637\pi$
$374$	37.3042	1.92896
$375$	−2.35441	−0.121581
$376$	−48.1718	−2.48427
$377$	−15.2937	−0.787663
$378$	0	0
$379$	−11.1547	−0.572978	−0.286489	−	0.958083i	$-0.592488\pi$
$379$	−11.1547	−0.572978	−0.286489	+	0.958083i	$0.592488\pi$
$380$	32.3186	1.65791
$381$	2.92326	0.149763
$382$	21.2168	1.08555
$383$	−17.5343	−0.895961	−0.447980	−	0.894043i	$-0.647857\pi$
$383$	−17.5343	−0.895961	−0.447980	+	0.894043i	$0.647857\pi$
$384$	27.7356	1.41538
$385$	0	0
$386$	−16.6340	−0.846651
$387$	−13.3120	−0.676689
$388$	76.2710	3.87207
$389$	1.79309	0.0909133	0.0454567	−	0.998966i	$-0.485526\pi$
$389$	1.79309	0.0909133	0.0454567	+	0.998966i	$0.485526\pi$
$390$	23.2990	1.17979
$391$	−54.2421	−2.74314
$392$	0	0
$393$	−19.3125	−0.974187
$394$	8.15236	0.410710
$395$	6.11966	0.307914
$396$	−25.0958	−1.26111
$397$	−6.96092	−0.349358	−0.174679	−	0.984625i	$-0.555889\pi$
$397$	−6.96092	−0.349358	−0.174679	+	0.984625i	$0.555889\pi$
$398$	26.2632	1.31646
$399$	0	0
$400$	7.08224	0.354112
$401$	9.29439	0.464139	0.232070	−	0.972699i	$-0.425450\pi$
$401$	9.29439	0.464139	0.232070	+	0.972699i	$0.425450\pi$
$402$	87.6900	4.37358
$403$	3.46381	0.172545
$404$	15.2526	0.758843
$405$	10.1617	0.504938
$406$	0	0
$407$	2.20460	0.109278
$408$	−98.6397	−4.88339
$409$	37.4916	1.85384	0.926921	−	0.375256i	$-0.122445\pi$
$409$	37.4916	1.85384	0.926921	+	0.375256i	$0.122445\pi$
$410$	29.3086	1.44745
$411$	−14.4955	−0.715010
$412$	−67.6829	−3.33450
$413$	0	0
$414$	52.7956	2.59476
$415$	−12.0500	−0.591509
$416$	−21.0816	−1.03361
$417$	−5.37726	−0.263326
$418$	−40.5088	−1.98135
$419$	13.2388	0.646758	0.323379	−	0.946270i	$-0.395181\pi$
$419$	13.2388	0.646758	0.323379	+	0.946270i	$0.395181\pi$
$420$	0	0
$421$	10.5732	0.515305	0.257652	−	0.966238i	$-0.417051\pi$
$421$	10.5732	0.515305	0.257652	+	0.966238i	$0.417051\pi$
$422$	−10.6144	−0.516700
$423$	19.4439	0.945397
$424$	−72.3011	−3.51125
$425$	6.64930	0.322539
$426$	56.3695	2.73111
$427$	0	0
$428$	46.9034	2.26716
$429$	−20.1844	−0.974510
$430$	−13.3202	−0.642358
$431$	−19.2295	−0.926252	−0.463126	−	0.886293i	$-0.653272\pi$
$431$	−19.2295	−0.926252	−0.463126	+	0.886293i	$0.653272\pi$
$432$	−7.61640	−0.366444
$433$	32.4074	1.55740	0.778699	−	0.627397i	$-0.215880\pi$
$433$	32.4074	1.55740	0.778699	+	0.627397i	$0.215880\pi$
$434$	0	0
$435$	9.25954	0.443961
$436$	−25.6992	−1.23077
$437$	58.9016	2.81765
$438$	28.0881	1.34210
$439$	−17.9700	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$439$	−17.9700	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$440$	−13.8907	−0.662213
$441$	0	0
$442$	−65.8008	−3.12982
$443$	−7.94973	−0.377703	−0.188851	−	0.982006i	$-0.560476\pi$
$443$	−7.94973	−0.377703	−0.188851	+	0.982006i	$0.560476\pi$
$444$	−10.5382	−0.500121
$445$	−0.282461	−0.0133899
$446$	−26.4569	−1.25277
$447$	28.0056	1.32462
$448$	0	0
$449$	1.29739	0.0612274	0.0306137	−	0.999531i	$-0.490254\pi$
$449$	1.29739	0.0612274	0.0306137	+	0.999531i	$0.490254\pi$
$450$	−6.47198	−0.305092
$451$	−25.3906	−1.19560
$452$	−48.7572	−2.29335
$453$	−46.0047	−2.16149
$454$	8.78845	0.412463
$455$	0	0
$456$	107.113	5.01603
$457$	−14.7423	−0.689614	−0.344807	−	0.938674i	$-0.612056\pi$
$457$	−14.7423	−0.689614	−0.344807	+	0.938674i	$0.612056\pi$
$458$	−61.1653	−2.85807
$459$	−7.15081	−0.333771
$460$	36.5129	1.70242
$461$	−4.62821	−0.215557	−0.107779	−	0.994175i	$-0.534374\pi$
$461$	−4.62821	−0.215557	−0.107779	+	0.994175i	$0.534374\pi$
$462$	0	0
$463$	−8.54874	−0.397293	−0.198647	−	0.980071i	$-0.563655\pi$
$463$	−8.54874	−0.397293	−0.198647	+	0.980071i	$0.563655\pi$
$464$	−27.8535	−1.29306
$465$	−2.09716	−0.0972536
$466$	46.5652	2.15709
$467$	−24.9465	−1.15438	−0.577192	−	0.816608i	$-0.695852\pi$
$467$	−24.9465	−1.15438	−0.577192	+	0.816608i	$0.695852\pi$
$468$	44.2664	2.04622
$469$	0	0
$470$	19.4559	0.897432
$471$	−35.1637	−1.62026
$472$	−45.1083	−2.07628
$473$	11.5396	0.530590
$474$	36.6657	1.68411
$475$	−7.22049	−0.331299
$476$	0	0
$477$	29.1835	1.33622
$478$	−34.2512	−1.56661
$479$	−23.7322	−1.08435	−0.542176	−	0.840265i	$-0.682399\pi$
$479$	−23.7322	−1.08435	−0.542176	+	0.840265i	$0.682399\pi$
$480$	12.7639	0.582588
$481$	−3.88869	−0.177309
$482$	27.9010	1.27086
$483$	0	0
$484$	−27.4811	−1.24914
$485$	−17.0402	−0.773754
$486$	52.6732	2.38930
$487$	−1.49225	−0.0676205	−0.0338102	−	0.999428i	$-0.510764\pi$
$487$	−1.49225	−0.0676205	−0.0338102	+	0.999428i	$0.510764\pi$
$488$	27.9440	1.26496
$489$	5.68308	0.256998
$490$	0	0
$491$	11.3886	0.513958	0.256979	−	0.966417i	$-0.417273\pi$
$491$	11.3886	0.513958	0.256979	+	0.966417i	$0.417273\pi$
$492$	121.370	5.47176
$493$	−26.1508	−1.17777
$494$	71.4533	3.21483
$495$	5.60681	0.252007
$496$	6.30843	0.283257
$497$	0	0
$498$	−72.1970	−3.23522
$499$	32.5950	1.45915	0.729576	−	0.683900i	$-0.239718\pi$
$499$	32.5950	1.45915	0.729576	+	0.683900i	$0.239718\pi$
$500$	−4.47595	−0.200171
$501$	16.0525	0.717172
$502$	79.1166	3.53115
$503$	−5.12412	−0.228473	−0.114237	−	0.993454i	$-0.536442\pi$
$503$	−5.12412	−0.228473	−0.114237	+	0.993454i	$0.536442\pi$
$504$	0	0
$505$	−3.40767	−0.151639
$506$	−45.7659	−2.03454
$507$	4.99582	0.221872
$508$	5.55739	0.246569
$509$	17.9004	0.793421	0.396710	−	0.917944i	$-0.370152\pi$
$509$	17.9004	0.793421	0.396710	+	0.917944i	$0.370152\pi$
$510$	39.8391	1.76410
$511$	0	0
$512$	50.8526	2.24739
$513$	7.76508	0.342837
$514$	−32.7158	−1.44303
$515$	15.1214	0.666330
$516$	−55.1602	−2.42829
$517$	−16.8550	−0.741282
$518$	0	0
$519$	−43.9784	−1.93044
$520$	24.5018	1.07447
$521$	−1.88188	−0.0824466	−0.0412233	−	0.999150i	$-0.513125\pi$
$521$	−1.88188	−0.0824466	−0.0412233	+	0.999150i	$0.513125\pi$
$522$	25.4534	1.11406
$523$	−11.9217	−0.521299	−0.260649	−	0.965434i	$-0.583937\pi$
$523$	−11.9217	−0.521299	−0.260649	+	0.965434i	$0.583937\pi$
$524$	−36.7149	−1.60390
$525$	0	0
$526$	43.1674	1.88219
$527$	5.92279	0.258001
$528$	−36.7606	−1.59980
$529$	43.5458	1.89330
$530$	29.2013	1.26843
$531$	18.2074	0.790134
$532$	0	0
$533$	44.7864	1.93991
$534$	−1.69235	−0.0732353
$535$	−10.4790	−0.453046
$536$	92.2170	3.98317
$537$	−10.6251	−0.458509
$538$	−43.5997	−1.87972
$539$	0	0
$540$	4.81354	0.207142
$541$	−18.1448	−0.780108	−0.390054	−	0.920792i	$-0.627544\pi$
$541$	−18.1448	−0.780108	−0.390054	+	0.920792i	$0.627544\pi$
$542$	−54.7903	−2.35345
$543$	−23.9077	−1.02598
$544$	−36.0476	−1.54553
$545$	5.74162	0.245944
$546$	0	0
$547$	1.37089	0.0586149	0.0293074	−	0.999570i	$-0.490670\pi$
$547$	1.37089	0.0586149	0.0293074	+	0.999570i	$0.490670\pi$
$548$	−27.5573	−1.17719
$549$	−11.2792	−0.481386
$550$	5.61025	0.239222
$551$	28.3972	1.20976
$552$	121.014	5.15070
$553$	0	0
$554$	68.2767	2.90080
$555$	2.35441	0.0999390
$556$	−10.2227	−0.433539
$557$	−35.1843	−1.49081	−0.745404	−	0.666613i	$-0.767744\pi$
$557$	−35.1843	−1.49081	−0.745404	+	0.666613i	$0.767744\pi$
$558$	−5.76485	−0.244046
$559$	−20.3546	−0.860908
$560$	0	0
$561$	−34.5134	−1.45716
$562$	80.4261	3.39257
$563$	4.82881	0.203510	0.101755	−	0.994809i	$-0.467554\pi$
$563$	4.82881	0.203510	0.101755	+	0.994809i	$0.467554\pi$
$564$	80.5685	3.39255
$565$	10.8932	0.458278
$566$	−20.2447	−0.850947
$567$	0	0
$568$	59.2795	2.48731
$569$	22.3966	0.938915	0.469458	−	0.882955i	$-0.344449\pi$
$569$	22.3966	0.938915	0.469458	+	0.882955i	$0.344449\pi$
$570$	−43.2614	−1.81202
$571$	−32.9511	−1.37896	−0.689481	−	0.724304i	$-0.742161\pi$
$571$	−32.9511	−1.37896	−0.689481	+	0.724304i	$0.742161\pi$
$572$	−38.3724	−1.60443
$573$	−19.6295	−0.820036
$574$	0	0
$575$	−8.15756	−0.340194
$576$	−0.937203	−0.0390501
$577$	2.82197	0.117480	0.0587400	−	0.998273i	$-0.481292\pi$
$577$	2.82197	0.117480	0.0587400	+	0.998273i	$0.481292\pi$
$578$	−69.2519	−2.88050
$579$	15.3896	0.639570
$580$	17.6033	0.730936
$581$	0	0
$582$	−102.096	−4.23200
$583$	−25.2977	−1.04772
$584$	29.5381	1.22230
$585$	−9.88983	−0.408894
$586$	−4.79757	−0.198186
$587$	3.36188	0.138760	0.0693798	−	0.997590i	$-0.477898\pi$
$587$	3.36188	0.138760	0.0693798	+	0.997590i	$0.477898\pi$
$588$	0	0
$589$	−6.43158	−0.265009
$590$	18.2186	0.750047
$591$	−7.54246	−0.310255
$592$	−7.08224	−0.291078
$593$	8.42812	0.346102	0.173051	−	0.984913i	$-0.444638\pi$
$593$	8.42812	0.346102	0.173051	+	0.984913i	$0.444638\pi$
$594$	−6.03339	−0.247553
$595$	0	0
$596$	53.2414	2.18085
$597$	−24.2984	−0.994466
$598$	80.7264	3.30115
$599$	−33.3191	−1.36138	−0.680692	−	0.732570i	$-0.738321\pi$
$599$	−33.3191	−1.36138	−0.680692	+	0.732570i	$0.738321\pi$
$600$	−14.8346	−0.605620
$601$	11.6412	0.474853	0.237426	−	0.971406i	$-0.423696\pi$
$601$	11.6412	0.474853	0.237426	+	0.971406i	$0.423696\pi$
$602$	0	0
$603$	−37.2222	−1.51581
$604$	−87.4594	−3.55867
$605$	6.13973	0.249615
$606$	−20.4169	−0.829381
$607$	−36.3921	−1.47711	−0.738556	−	0.674192i	$-0.764492\pi$
$607$	−36.3921	−1.47711	−0.738556	+	0.674192i	$0.764492\pi$
$608$	39.1442	1.58751
$609$	0	0
$610$	−11.2862	−0.456963
$611$	29.7305	1.20277
$612$	75.6915	3.05965
$613$	10.3344	0.417401	0.208701	−	0.977980i	$-0.433077\pi$
$613$	10.3344	0.417401	0.208701	+	0.977980i	$0.433077\pi$
$614$	70.2784	2.83621
$615$	−27.1159	−1.09342
$616$	0	0
$617$	2.00704	0.0808004	0.0404002	−	0.999184i	$-0.487137\pi$
$617$	2.00704	0.0808004	0.0404002	+	0.999184i	$0.487137\pi$
$618$	90.5997	3.64445
$619$	4.83034	0.194148	0.0970739	−	0.995277i	$-0.469052\pi$
$619$	4.83034	0.194148	0.0970739	+	0.995277i	$0.469052\pi$
$620$	−3.98691	−0.160118
$621$	8.77282	0.352041
$622$	−30.0486	−1.20484
$623$	0	0
$624$	64.8419	2.59575
$625$	1.00000	0.0400000
$626$	−10.9376	−0.437155
$627$	37.4782	1.49673
$628$	−66.8496	−2.66759
$629$	−6.64930	−0.265125
$630$	0	0
$631$	−31.0999	−1.23807	−0.619034	−	0.785364i	$-0.712476\pi$
$631$	−31.0999	−1.23807	−0.619034	+	0.785364i	$0.712476\pi$
$632$	38.5586	1.53378
$633$	9.82028	0.390321
$634$	34.6602	1.37653
$635$	−1.24161	−0.0492718
$636$	120.926	4.79501
$637$	0	0
$638$	−22.0643	−0.873534
$639$	−23.9274	−0.946555
$640$	−11.7803	−0.465658
$641$	17.5240	0.692155	0.346077	−	0.938206i	$-0.387513\pi$
$641$	17.5240	0.692155	0.346077	+	0.938206i	$0.387513\pi$
$642$	−62.7845	−2.47791
$643$	18.2968	0.721554	0.360777	−	0.932652i	$-0.382511\pi$
$643$	18.2968	0.721554	0.360777	+	0.932652i	$0.382511\pi$
$644$	0	0
$645$	12.3237	0.485245
$646$	122.179	4.80705
$647$	−34.2314	−1.34577	−0.672887	−	0.739745i	$-0.734946\pi$
$647$	−34.2314	−1.34577	−0.672887	+	0.739745i	$0.734946\pi$
$648$	64.0264	2.51520
$649$	−15.7831	−0.619542
$650$	−9.89589	−0.388149
$651$	0	0
$652$	10.8041	0.423121
$653$	−50.3002	−1.96840	−0.984200	−	0.177063i	$-0.943340\pi$
$653$	−50.3002	−1.96840	−0.984200	+	0.177063i	$0.943340\pi$
$654$	34.4007	1.34518
$655$	8.20271	0.320506
$656$	81.5669	3.18465
$657$	−11.9227	−0.465148
$658$	0	0
$659$	11.3334	0.441485	0.220743	−	0.975332i	$-0.429152\pi$
$659$	11.3334	0.441485	0.220743	+	0.975332i	$0.429152\pi$
$660$	23.2326	0.904326
$661$	−14.4522	−0.562127	−0.281063	−	0.959689i	$-0.590687\pi$
$661$	−14.4522	−0.562127	−0.281063	+	0.959689i	$0.590687\pi$
$662$	−26.0180	−1.01122
$663$	60.8781	2.36431
$664$	−75.9241	−2.94643
$665$	0	0
$666$	6.47198	0.250784
$667$	32.0825	1.24224
$668$	30.5173	1.18075
$669$	24.4776	0.946359
$670$	−37.2451	−1.43890
$671$	9.77742	0.377453
$672$	0	0
$673$	33.8004	1.30291	0.651454	−	0.758688i	$-0.274159\pi$
$673$	33.8004	1.30291	0.651454	+	0.758688i	$0.274159\pi$
$674$	25.6707	0.988797
$675$	−1.07542	−0.0413930
$676$	9.49754	0.365290
$677$	39.9941	1.53710	0.768548	−	0.639792i	$-0.220979\pi$
$677$	39.9941	1.53710	0.768548	+	0.639792i	$0.220979\pi$
$678$	65.2660	2.50652
$679$	0	0
$680$	41.8958	1.60663
$681$	−8.13096	−0.311579
$682$	4.99727	0.191355
$683$	−29.3953	−1.12478	−0.562390	−	0.826872i	$-0.690118\pi$
$683$	−29.3953	−1.12478	−0.562390	+	0.826872i	$0.690118\pi$
$684$	−82.1936	−3.14275
$685$	6.15675	0.235237
$686$	0	0
$687$	56.5893	2.15902
$688$	−37.0706	−1.41331
$689$	44.6225	1.69998
$690$	−48.8758	−1.86067
$691$	49.3473	1.87726	0.938630	−	0.344927i	$-0.112096\pi$
$691$	49.3473	1.87726	0.938630	+	0.344927i	$0.112096\pi$
$692$	−83.6072	−3.17827
$693$	0	0
$694$	32.1467	1.22027
$695$	2.28391	0.0866338
$696$	58.3423	2.21146
$697$	76.5807	2.90070
$698$	−91.6474	−3.46891
$699$	−43.0815	−1.62949
$700$	0	0
$701$	−0.114605	−0.00432857	−0.00216429	−	0.999998i	$-0.500689\pi$
$701$	−0.114605	−0.00432857	−0.00216429	+	0.999998i	$0.500689\pi$
$702$	10.6423	0.401666
$703$	7.22049	0.272326
$704$	0.812416	0.0306191
$705$	−18.0003	−0.677931
$706$	89.2057	3.35730
$707$	0	0
$708$	75.4448	2.83539
$709$	26.7017	1.00280	0.501402	−	0.865214i	$-0.332818\pi$
$709$	26.7017	1.00280	0.501402	+	0.865214i	$0.332818\pi$
$710$	−23.9421	−0.898531
$711$	−15.5637	−0.583684
$712$	−1.77972	−0.0666979
$713$	−7.26626	−0.272124
$714$	0	0
$715$	8.57301	0.320613
$716$	−20.1994	−0.754887
$717$	31.6887	1.18344
$718$	−32.0171	−1.19487
$719$	17.2223	0.642283	0.321142	−	0.947031i	$-0.395933\pi$
$719$	17.2223	0.642283	0.321142	+	0.947031i	$0.395933\pi$
$720$	−18.0118	−0.671259
$721$	0	0
$722$	−84.3230	−3.13818
$723$	−25.8137	−0.960020
$724$	−45.4508	−1.68917
$725$	−3.93286	−0.146063
$726$	36.7860	1.36526
$727$	−25.8423	−0.958438	−0.479219	−	0.877695i	$-0.659080\pi$
$727$	−25.8423	−0.958438	−0.479219	+	0.877695i	$0.659080\pi$
$728$	0	0
$729$	−18.2475	−0.675834
$730$	−11.9300	−0.441549
$731$	−34.8045	−1.28729
$732$	−46.7370	−1.72745
$733$	−30.6178	−1.13090	−0.565448	−	0.824784i	$-0.691297\pi$
$733$	−30.6178	−1.13090	−0.565448	+	0.824784i	$0.691297\pi$
$734$	−6.65758	−0.245736
$735$	0	0
$736$	44.2243	1.63013
$737$	32.2661	1.18854
$738$	−74.5385	−2.74380
$739$	−7.73298	−0.284462	−0.142231	−	0.989833i	$-0.545428\pi$
$739$	−7.73298	−0.284462	−0.142231	+	0.989833i	$0.545428\pi$
$740$	4.47595	0.164539
$741$	−66.1076	−2.42852
$742$	0	0
$743$	−13.4302	−0.492708	−0.246354	−	0.969180i	$-0.579233\pi$
$743$	−13.4302	−0.492708	−0.246354	+	0.969180i	$0.579233\pi$
$744$	−13.2138	−0.484440
$745$	−11.8950	−0.435798
$746$	86.0834	3.15174
$747$	30.6458	1.12127
$748$	−65.6133	−2.39906
$749$	0	0
$750$	5.99147	0.218777
$751$	21.5083	0.784848	0.392424	−	0.919784i	$-0.371637\pi$
$751$	21.5083	0.784848	0.392424	+	0.919784i	$0.371637\pi$
$752$	54.1464	1.97452
$753$	−73.1977	−2.66747
$754$	38.9191	1.41735
$755$	19.5398	0.711127
$756$	0	0
$757$	49.3938	1.79525	0.897624	−	0.440761i	$-0.145291\pi$
$757$	49.3938	1.79525	0.897624	+	0.440761i	$0.145291\pi$
$758$	28.3864	1.03104
$759$	42.3421	1.53692
$760$	−45.4947	−1.65027
$761$	−49.1115	−1.78029	−0.890144	−	0.455679i	$-0.849397\pi$
$761$	−49.1115	−1.78029	−0.890144	+	0.455679i	$0.849397\pi$
$762$	−7.43907	−0.269489
$763$	0	0
$764$	−37.3176	−1.35011
$765$	−16.9107	−0.611408
$766$	44.6211	1.61223
$767$	27.8398	1.00524
$768$	−68.8461	−2.48427
$769$	−37.6430	−1.35744	−0.678721	−	0.734396i	$-0.737466\pi$
$769$	−37.6430	−1.35744	−0.678721	+	0.734396i	$0.737466\pi$
$770$	0	0
$771$	30.2683	1.09009
$772$	29.2571	1.05299
$773$	35.7119	1.28447	0.642233	−	0.766509i	$-0.278008\pi$
$773$	35.7119	1.28447	0.642233	+	0.766509i	$0.278008\pi$
$774$	33.8764	1.21766
$775$	0.890739	0.0319963
$776$	−107.366	−3.85423
$777$	0	0
$778$	−4.56304	−0.163593
$779$	−83.1591	−2.97949
$780$	−40.9798	−1.46731
$781$	20.7415	0.742190
$782$	138.035	4.93611
$783$	4.22948	0.151149
$784$	0	0
$785$	14.9353	0.533063
$786$	49.1463	1.75299
$787$	−23.4157	−0.834679	−0.417340	−	0.908751i	$-0.637037\pi$
$787$	−23.4157	−0.834679	−0.417340	+	0.908751i	$0.637037\pi$
$788$	−14.3389	−0.510804
$789$	−39.9379	−1.42183
$790$	−15.5732	−0.554071
$791$	0	0
$792$	35.3273	1.25530
$793$	−17.2464	−0.612436
$794$	17.7141	0.628649
$795$	−27.0167	−0.958184
$796$	−46.1935	−1.63729
$797$	3.59111	0.127204	0.0636018	−	0.997975i	$-0.479741\pi$
$797$	3.59111	0.127204	0.0636018	+	0.997975i	$0.479741\pi$
$798$	0	0
$799$	50.8364	1.79846
$800$	−5.42127	−0.191671
$801$	0.718362	0.0253821
$802$	−23.6523	−0.835190
$803$	10.3352	0.364721
$804$	−154.235	−5.43946
$805$	0	0
$806$	−8.81466	−0.310483
$807$	40.3379	1.41996
$808$	−21.4710	−0.755346
$809$	23.1592	0.814234	0.407117	−	0.913376i	$-0.366534\pi$
$809$	23.1592	0.814234	0.407117	+	0.913376i	$0.366534\pi$
$810$	−25.8593	−0.908604
$811$	−24.4422	−0.858282	−0.429141	−	0.903238i	$-0.641184\pi$
$811$	−24.4422	−0.858282	−0.429141	+	0.903238i	$0.641184\pi$
$812$	0	0
$813$	50.6913	1.77782
$814$	−5.61025	−0.196639
$815$	−2.41381	−0.0845520
$816$	110.874	3.88136
$817$	37.7943	1.32225
$818$	−95.4083	−3.33587
$819$	0	0
$820$	−51.5500	−1.80020
$821$	−19.1107	−0.666968	−0.333484	−	0.942756i	$-0.608224\pi$
$821$	−19.1107	−0.666968	−0.333484	+	0.942756i	$0.608224\pi$
$822$	36.8880	1.28662
$823$	21.1105	0.735866	0.367933	−	0.929852i	$-0.380066\pi$
$823$	21.1105	0.735866	0.367933	+	0.929852i	$0.380066\pi$
$824$	95.2768	3.31913
$825$	−5.19053	−0.180711
$826$	0	0
$827$	−4.63103	−0.161037	−0.0805184	−	0.996753i	$-0.525658\pi$
$827$	−4.63103	−0.161037	−0.0805184	+	0.996753i	$0.525658\pi$
$828$	−92.8606	−3.22713
$829$	33.8483	1.17560	0.587799	−	0.809007i	$-0.299994\pi$
$829$	33.8483	1.17560	0.587799	+	0.809007i	$0.299994\pi$
$830$	30.6646	1.06438
$831$	−63.1687	−2.19130
$832$	−1.43302	−0.0496809
$833$	0	0
$834$	13.6840	0.473838
$835$	−6.81805	−0.235948
$836$	71.2496	2.46422
$837$	−0.957921	−0.0331106
$838$	−33.6899	−1.16380
$839$	42.8414	1.47905	0.739524	−	0.673130i	$-0.235051\pi$
$839$	42.8414	1.47905	0.739524	+	0.673130i	$0.235051\pi$
$840$	0	0
$841$	−13.5326	−0.466643
$842$	−26.9065	−0.927259
$843$	−74.4092	−2.56279
$844$	18.6693	0.642624
$845$	−2.12190	−0.0729957
$846$	−49.4807	−1.70118
$847$	0	0
$848$	81.2685	2.79077
$849$	18.7301	0.642816
$850$	−16.9211	−0.580388
$851$	8.15756	0.279638
$852$	−99.1465	−3.39670
$853$	17.0137	0.582539	0.291270	−	0.956641i	$-0.405922\pi$
$853$	17.0137	0.582539	0.291270	+	0.956641i	$0.405922\pi$
$854$	0	0
$855$	18.3634	0.628014
$856$	−66.0257	−2.25671
$857$	−11.6747	−0.398800	−0.199400	−	0.979918i	$-0.563899\pi$
$857$	−11.6747	−0.398800	−0.199400	+	0.979918i	$0.563899\pi$
$858$	51.3649	1.75357
$859$	56.7067	1.93481	0.967404	−	0.253238i	$-0.0814957\pi$
$859$	56.7067	1.93481	0.967404	+	0.253238i	$0.0814957\pi$
$860$	23.4285	0.798906
$861$	0	0
$862$	48.9350	1.66673
$863$	−9.26836	−0.315499	−0.157749	−	0.987479i	$-0.550424\pi$
$863$	−9.26836	−0.315499	−0.157749	+	0.987479i	$0.550424\pi$
$864$	5.83015	0.198346
$865$	18.6792	0.635112
$866$	−82.4699	−2.80244
$867$	64.0710	2.17597
$868$	0	0
$869$	13.4914	0.457665
$870$	−23.5636	−0.798880
$871$	−56.9141	−1.92846
$872$	36.1767	1.22510
$873$	43.3371	1.46674
$874$	−149.892	−5.07018
$875$	0	0
$876$	−49.4032	−1.66918
$877$	1.85155	0.0625225	0.0312612	−	0.999511i	$-0.490048\pi$
$877$	1.85155	0.0625225	0.0312612	+	0.999511i	$0.490048\pi$
$878$	45.7298	1.54331
$879$	4.43865	0.149712
$880$	15.6135	0.526332
$881$	−30.3014	−1.02088	−0.510439	−	0.859914i	$-0.670517\pi$
$881$	−30.3014	−1.02088	−0.510439	+	0.859914i	$0.670517\pi$
$882$	0	0
$883$	22.7042	0.764056	0.382028	−	0.924151i	$-0.375226\pi$
$883$	22.7042	0.764056	0.382028	+	0.924151i	$0.375226\pi$
$884$	115.735	3.89259
$885$	−16.8556	−0.566595
$886$	20.2304	0.679653
$887$	−6.56187	−0.220326	−0.110163	−	0.993914i	$-0.535137\pi$
$887$	−6.56187	−0.220326	−0.110163	+	0.993914i	$0.535137\pi$
$888$	14.8346	0.497816
$889$	0	0
$890$	0.718803	0.0240943
$891$	22.4024	0.750510
$892$	46.5343	1.55808
$893$	−55.2034	−1.84731
$894$	−71.2684	−2.38357
$895$	4.51287	0.150849
$896$	0	0
$897$	−74.6870	−2.49373
$898$	−3.30157	−0.110175
$899$	−3.50315	−0.116837
$900$	11.3834	0.379446
$901$	76.3005	2.54194
$902$	64.6138	2.15140
$903$	0	0
$904$	68.6353	2.28278
$905$	10.1544	0.337545
$906$	117.072	3.88947
$907$	13.0689	0.433947	0.216973	−	0.976178i	$-0.430382\pi$
$907$	13.0689	0.433947	0.216973	+	0.976178i	$0.430382\pi$
$908$	−15.4577	−0.512983
$909$	8.66648	0.287449
$910$	0	0
$911$	−33.1053	−1.09683	−0.548414	−	0.836207i	$-0.684768\pi$
$911$	−33.1053	−1.09683	−0.548414	+	0.836207i	$0.684768\pi$
$912$	−120.398	−3.98678
$913$	−26.5654	−0.879185
$914$	37.5160	1.24092
$915$	10.4418	0.345195
$916$	107.582	3.55460
$917$	0	0
$918$	18.1973	0.600600
$919$	−26.3425	−0.868958	−0.434479	−	0.900682i	$-0.643067\pi$
$919$	−26.3425	−0.868958	−0.434479	+	0.900682i	$0.643067\pi$
$920$	−51.3990	−1.69457
$921$	−65.0207	−2.14250
$922$	11.7778	0.387882
$923$	−36.5859	−1.20424
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	21.7547	0.714905
$927$	−38.4573	−1.26310
$928$	21.3211	0.699898
$929$	14.9371	0.490071	0.245035	−	0.969514i	$-0.421200\pi$
$929$	14.9371	0.490071	0.245035	+	0.969514i	$0.421200\pi$
$930$	5.33684	0.175002
$931$	0	0
$932$	−81.9021	−2.68279
$933$	27.8005	0.910149
$934$	63.4835	2.07724
$935$	14.6591	0.479403
$936$	−62.3136	−2.03678
$937$	3.29147	0.107528	0.0537638	−	0.998554i	$-0.482878\pi$
$937$	3.29147	0.107528	0.0537638	+	0.998554i	$0.482878\pi$
$938$	0	0
$939$	10.1193	0.330232
$940$	−34.2203	−1.11614
$941$	16.6749	0.543585	0.271792	−	0.962356i	$-0.412384\pi$
$941$	16.6749	0.543585	0.271792	+	0.962356i	$0.412384\pi$
$942$	89.4842	2.91555
$943$	−93.9514	−3.05948
$944$	50.7030	1.65024
$945$	0	0
$946$	−29.3658	−0.954763
$947$	28.0659	0.912020	0.456010	−	0.889975i	$-0.349278\pi$
$947$	28.0659	0.912020	0.456010	+	0.889975i	$0.349278\pi$
$948$	−64.4903	−2.09455
$949$	−18.2302	−0.591778
$950$	18.3746	0.596152
$951$	−32.0672	−1.03985
$952$	0	0
$953$	18.9875	0.615066	0.307533	−	0.951537i	$-0.400497\pi$
$953$	18.9875	0.615066	0.307533	+	0.951537i	$0.400497\pi$
$954$	−74.2657	−2.40444
$955$	8.33736	0.269791
$956$	60.2433	1.94841
$957$	20.4136	0.659878
$958$	60.3934	1.95122
$959$	0	0
$960$	0.867620	0.0280023
$961$	−30.2066	−0.974406
$962$	9.89589	0.319056
$963$	26.6505	0.858799
$964$	−49.0742	−1.58057
$965$	−6.53651	−0.210418
$966$	0	0
$967$	35.8455	1.15271	0.576356	−	0.817199i	$-0.304474\pi$
$967$	35.8455	1.15271	0.576356	+	0.817199i	$0.304474\pi$
$968$	38.6851	1.24338
$969$	−113.038	−3.63130
$970$	43.3636	1.39232
$971$	−30.7807	−0.987799	−0.493899	−	0.869519i	$-0.664429\pi$
$971$	−30.7807	−0.987799	−0.493899	+	0.869519i	$0.664429\pi$
$972$	−92.6452	−2.97160
$973$	0	0
$974$	3.79747	0.121679
$975$	9.15555	0.293212
$976$	−31.4098	−1.00540
$977$	−44.2971	−1.41719	−0.708596	−	0.705615i	$-0.750671\pi$
$977$	−44.2971	−1.41719	−0.708596	+	0.705615i	$0.750671\pi$
$978$	−14.4622	−0.462452
$979$	−0.622714	−0.0199020
$980$	0	0
$981$	−14.6023	−0.466214
$982$	−28.9815	−0.924836
$983$	−11.9697	−0.381775	−0.190888	−	0.981612i	$-0.561137\pi$
$983$	−11.9697	−0.381775	−0.190888	+	0.981612i	$0.561137\pi$
$984$	−170.851	−5.44654
$985$	3.20355	0.102074
$986$	66.5482	2.11933
$987$	0	0
$988$	−125.677	−3.99832
$989$	42.6992	1.35776
$990$	−14.2682	−0.453472
$991$	−46.9285	−1.49073	−0.745367	−	0.666654i	$-0.767726\pi$
$991$	−46.9285	−1.49073	−0.745367	+	0.666654i	$0.767726\pi$
$992$	−4.82893	−0.153319
$993$	24.0715	0.763887
$994$	0	0
$995$	10.3204	0.327178
$996$	126.985	4.02367
$997$	−61.7824	−1.95667	−0.978335	−	0.207029i	$-0.933621\pi$
$997$	−61.7824	−1.95667	−0.978335	+	0.207029i	$0.933621\pi$
$998$	−82.9473	−2.62565
$999$	1.07542	0.0340249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bd.1.4	✓	38
7.6	odd	2		9065.2.a.be.1.4	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bd.1.4	✓	38	1.1	even	1	trivial
9065.2.a.be.1.4	yes	38	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)

\(f(q)\)	\(=\)	\(q-2.54479 q^{2} +2.35441 q^{3} +4.47595 q^{4} -1.00000 q^{5} -5.99147 q^{6} -6.30078 q^{8} +2.54323 q^{9} +2.54479 q^{10} -2.20460 q^{11} +10.5382 q^{12} +3.88869 q^{13} -2.35441 q^{15} +7.08224 q^{16} +6.64930 q^{17} -6.47198 q^{18} -7.22049 q^{19} -4.47595 q^{20} +5.61025 q^{22} -8.15756 q^{23} -14.8346 q^{24} +1.00000 q^{25} -9.89589 q^{26} -1.07542 q^{27} -3.93286 q^{29} +5.99147 q^{30} +0.890739 q^{31} -5.42127 q^{32} -5.19053 q^{33} -16.9211 q^{34} +11.3834 q^{36} -1.00000 q^{37} +18.3746 q^{38} +9.15555 q^{39} +6.30078 q^{40} +11.5171 q^{41} -5.23431 q^{43} -9.86769 q^{44} -2.54323 q^{45} +20.7593 q^{46} +7.64537 q^{47} +16.6745 q^{48} -2.54479 q^{50} +15.6552 q^{51} +17.4056 q^{52} +11.4750 q^{53} +2.73672 q^{54} +2.20460 q^{55} -17.0000 q^{57} +10.0083 q^{58} +7.15917 q^{59} -10.5382 q^{60} -4.43500 q^{61} -2.26674 q^{62} -0.368509 q^{64} -3.88869 q^{65} +13.2088 q^{66} -14.6358 q^{67} +29.7620 q^{68} -19.2062 q^{69} -9.40829 q^{71} -16.0243 q^{72} -4.68801 q^{73} +2.54479 q^{74} +2.35441 q^{75} -32.3186 q^{76} -23.2990 q^{78} -6.11966 q^{79} -7.08224 q^{80} -10.1617 q^{81} -29.3086 q^{82} +12.0500 q^{83} -6.64930 q^{85} +13.3202 q^{86} -9.25954 q^{87} +13.8907 q^{88} +0.282461 q^{89} +6.47198 q^{90} -36.5129 q^{92} +2.09716 q^{93} -19.4559 q^{94} +7.22049 q^{95} -12.7639 q^{96} +17.0402 q^{97} -5.60681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 - 6 * q^3 + 42 * q^4 - 38 * q^5 - 8 * q^6 - 6 * q^8 + 48 * q^9 + 2 * q^10 + 34 * q^11 - 20 * q^12 - 22 * q^13 + 6 * q^15 + 46 * q^16 - 22 * q^17 - 36 * q^18 - 40 * q^19 - 42 * q^20 - 4 * q^22 - 16 * q^23 - 24 * q^24 + 38 * q^25 - 20 * q^26 - 30 * q^27 + 26 * q^29 + 8 * q^30 - 52 * q^31 - 34 * q^32 - 38 * q^33 - 12 * q^34 + 74 * q^36 - 38 * q^37 - 16 * q^38 + 22 * q^39 + 6 * q^40 - 16 * q^41 + 72 * q^44 - 48 * q^45 + 12 * q^46 - 34 * q^47 - 34 * q^48 - 2 * q^50 + 46 * q^51 - 68 * q^52 + 4 * q^53 - 32 * q^54 - 34 * q^55 - 12 * q^57 - 40 * q^58 - 32 * q^59 + 20 * q^60 - 60 * q^61 + 38 * q^62 + 22 * q^64 + 22 * q^65 - 66 * q^66 - 20 * q^67 - 60 * q^68 - 24 * q^69 + 52 * q^71 - 78 * q^72 - 48 * q^73 + 2 * q^74 - 6 * q^75 - 86 * q^76 + 8 * q^78 + 22 * q^79 - 46 * q^80 + 58 * q^81 - 36 * q^82 - 44 * q^83 + 22 * q^85 + 20 * q^86 - 106 * q^87 - 66 * q^88 - 52 * q^89 + 36 * q^90 - 58 * q^92 - 48 * q^93 - 96 * q^94 + 40 * q^95 - 26 * q^96 - 38 * q^97 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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