LMFDB - Embedded newform 5239.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5239,2,Mod(1,5239)] 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("5239.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5239, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.a (trivial)

Newform invariants

comment:select newform

sage:traces = [36,2,-5,28,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.8336256189$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	5239.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.51712 q^{2} -2.55317 q^{3} +4.33588 q^{4} +0.350569 q^{5} +6.42663 q^{6} +1.25323 q^{7} -5.87968 q^{8} +3.51869 q^{9} -0.882423 q^{10} +0.660450 q^{11} -11.0702 q^{12} -3.15453 q^{14} -0.895063 q^{15} +6.12809 q^{16} +3.74858 q^{17} -8.85696 q^{18} +2.46144 q^{19} +1.52003 q^{20} -3.19971 q^{21} -1.66243 q^{22} -2.92240 q^{23} +15.0118 q^{24} -4.87710 q^{25} -1.32430 q^{27} +5.43386 q^{28} +2.76396 q^{29} +2.25298 q^{30} -1.00000 q^{31} -3.66576 q^{32} -1.68624 q^{33} -9.43562 q^{34} +0.439344 q^{35} +15.2566 q^{36} +6.15389 q^{37} -6.19573 q^{38} -2.06123 q^{40} -5.55631 q^{41} +8.05405 q^{42} +3.46032 q^{43} +2.86363 q^{44} +1.23354 q^{45} +7.35603 q^{46} +9.57856 q^{47} -15.6461 q^{48} -5.42941 q^{49} +12.2762 q^{50} -9.57077 q^{51} -5.19908 q^{53} +3.33343 q^{54} +0.231533 q^{55} -7.36860 q^{56} -6.28448 q^{57} -6.95722 q^{58} -9.92816 q^{59} -3.88089 q^{60} -11.0598 q^{61} +2.51712 q^{62} +4.40973 q^{63} -3.02903 q^{64} +4.24447 q^{66} -0.0241667 q^{67} +16.2534 q^{68} +7.46139 q^{69} -1.10588 q^{70} -5.14755 q^{71} -20.6888 q^{72} +9.68885 q^{73} -15.4901 q^{74} +12.4521 q^{75} +10.6725 q^{76} +0.827696 q^{77} -7.79772 q^{79} +2.14832 q^{80} -7.17489 q^{81} +13.9859 q^{82} -16.5885 q^{83} -13.8736 q^{84} +1.31414 q^{85} -8.71004 q^{86} -7.05687 q^{87} -3.88323 q^{88} +4.33965 q^{89} -3.10497 q^{90} -12.6712 q^{92} +2.55317 q^{93} -24.1104 q^{94} +0.862904 q^{95} +9.35932 q^{96} +7.85953 q^{97} +13.6665 q^{98} +2.32392 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2 q^{2} - 5 q^{3} + 28 q^{4} + 5 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 15 q^{10} - q^{11} - 13 q^{12} - 19 q^{14} - 10 q^{15} + 4 q^{16} - 46 q^{17} - 9 q^{18} + 8 q^{19} - 5 q^{20} - 16 q^{21}+ \cdots - 47 q^{99}+O(q^{100}) $ 36 * q + 2 * q^2 - 5 * q^3 + 28 * q^4 + 5 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 - 15 * q^10 - q^11 - 13 * q^12 - 19 * q^14 - 10 * q^15 + 4 * q^16 - 46 * q^17 - 9 * q^18 + 8 * q^19 - 5 * q^20 - 16 * q^21 - 21 * q^22 - 24 * q^23 + 57 * q^24 + 5 * q^25 - 11 * q^27 - 32 * q^28 - 73 * q^29 - 31 * q^30 - 36 * q^31 + 3 * q^32 - 8 * q^33 - 37 * q^34 - 31 * q^35 - 31 * q^36 + 6 * q^37 - 25 * q^38 - 22 * q^40 + 6 * q^41 - 37 * q^42 - 37 * q^43 - 16 * q^45 - 45 * q^46 + 13 * q^47 - 46 * q^48 - 5 * q^49 + 24 * q^50 - 46 * q^51 - 42 * q^53 - 26 * q^54 - 47 * q^55 - 95 * q^56 + 33 * q^57 + 9 * q^58 + 20 * q^59 - 40 * q^60 - 48 * q^61 - 2 * q^62 + 52 * q^63 + 21 * q^64 - 4 * q^66 - 50 * q^67 - 76 * q^68 - 33 * q^69 + 93 * q^70 + 16 * q^71 - 52 * q^72 - 43 * q^73 - 76 * q^74 - 40 * q^75 + 62 * q^76 - 47 * q^77 - 43 * q^79 - 46 * q^80 - 96 * q^81 + 7 * q^82 + 9 * q^83 - 77 * q^84 - 5 * q^85 + 64 * q^86 - 8 * q^87 - 54 * q^88 + 6 * q^89 + 27 * q^90 - 35 * q^92 + 5 * q^93 - 76 * q^94 - 63 * q^95 + 72 * q^96 - 4 * q^97 - 41 * q^98 - 47 * 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 $

$2$	−2.51712	−1.77987	−0.889935	−	0.456087i	$-0.849251\pi$
$2$	−2.51712	−1.77987	−0.889935	+	0.456087i	$0.849251\pi$
$3$	−2.55317	−1.47407	−0.737037	−	0.675852i	$-0.763776\pi$
$3$	−2.55317	−1.47407	−0.737037	+	0.675852i	$0.763776\pi$
$4$	4.33588	2.16794
$5$	0.350569	0.156779	0.0783896	−	0.996923i	$-0.475022\pi$
$5$	0.350569	0.156779	0.0783896	+	0.996923i	$0.475022\pi$
$6$	6.42663	2.62366
$7$	1.25323	0.473677	0.236838	−	0.971549i	$-0.423889\pi$
$7$	1.25323	0.473677	0.236838	+	0.971549i	$0.423889\pi$
$8$	−5.87968	−2.07878
$9$	3.51869	1.17290
$10$	−0.882423	−0.279047
$11$	0.660450	0.199133	0.0995665	−	0.995031i	$-0.468254\pi$
$11$	0.660450	0.199133	0.0995665	+	0.995031i	$0.468254\pi$
$12$	−11.0702	−3.19571
$13$	0	0
$13$	0	0
$14$	−3.15453	−0.843083
$15$	−0.895063	−0.231104
$16$	6.12809	1.53202
$17$	3.74858	0.909164	0.454582	−	0.890705i	$-0.349789\pi$
$17$	3.74858	0.909164	0.454582	+	0.890705i	$0.349789\pi$
$18$	−8.85696	−2.08760
$19$	2.46144	0.564693	0.282346	−	0.959312i	$-0.408887\pi$
$19$	2.46144	0.564693	0.282346	+	0.959312i	$0.408887\pi$
$20$	1.52003	0.339888
$21$	−3.19971	−0.698235
$22$	−1.66243	−0.354431
$23$	−2.92240	−0.609363	−0.304681	−	0.952454i	$-0.598550\pi$
$23$	−2.92240	−0.609363	−0.304681	+	0.952454i	$0.598550\pi$
$24$	15.0118	3.06428
$25$	−4.87710	−0.975420
$26$	0	0
$27$	−1.32430	−0.254863
$28$	5.43386	1.02690
$29$	2.76396	0.513255	0.256627	−	0.966510i	$-0.417389\pi$
$29$	2.76396	0.513255	0.256627	+	0.966510i	$0.417389\pi$
$30$	2.25298	0.411336
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−3.66576	−0.648021
$33$	−1.68624	−0.293537
$34$	−9.43562	−1.61819
$35$	0.439344	0.0742627
$36$	15.2566	2.54277
$37$	6.15389	1.01169	0.505847	−	0.862623i	$-0.331180\pi$
$37$	6.15389	1.01169	0.505847	+	0.862623i	$0.331180\pi$
$38$	−6.19573	−1.00508
$39$	0	0
$40$	−2.06123	−0.325910
$41$	−5.55631	−0.867751	−0.433875	−	0.900973i	$-0.642854\pi$
$41$	−5.55631	−0.867751	−0.433875	+	0.900973i	$0.642854\pi$
$42$	8.05405	1.24277
$43$	3.46032	0.527694	0.263847	−	0.964564i	$-0.415009\pi$
$43$	3.46032	0.527694	0.263847	+	0.964564i	$0.415009\pi$
$44$	2.86363	0.431709
$45$	1.23354	0.183886
$46$	7.35603	1.08459
$47$	9.57856	1.39718	0.698588	−	0.715524i	$-0.253812\pi$
$47$	9.57856	1.39718	0.698588	+	0.715524i	$0.253812\pi$
$48$	−15.6461	−2.25832
$49$	−5.42941	−0.775630
$50$	12.2762	1.73612
$51$	−9.57077	−1.34018
$52$	0	0
$53$	−5.19908	−0.714148	−0.357074	−	0.934076i	$-0.616226\pi$
$53$	−5.19908	−0.714148	−0.357074	+	0.934076i	$0.616226\pi$
$54$	3.33343	0.453622
$55$	0.231533	0.0312199
$56$	−7.36860	−0.984670
$57$	−6.28448	−0.832400
$58$	−6.95722	−0.913527
$59$	−9.92816	−1.29254	−0.646268	−	0.763110i	$-0.723671\pi$
$59$	−9.92816	−1.29254	−0.646268	+	0.763110i	$0.723671\pi$
$60$	−3.88089	−0.501020
$61$	−11.0598	−1.41606	−0.708031	−	0.706181i	$-0.750416\pi$
$61$	−11.0598	−1.41606	−0.708031	+	0.706181i	$0.750416\pi$
$62$	2.51712	0.319674
$63$	4.40973	0.555574
$64$	−3.02903	−0.378629
$65$	0	0
$66$	4.24447	0.522458
$67$	−0.0241667	−0.00295243	−0.00147621	−	0.999999i	$-0.500470\pi$
$67$	−0.0241667	−0.00295243	−0.00147621	+	0.999999i	$0.500470\pi$
$68$	16.2534	1.97101
$69$	7.46139	0.898246
$70$	−1.10588	−0.132178
$71$	−5.14755	−0.610902	−0.305451	−	0.952208i	$-0.598807\pi$
$71$	−5.14755	−0.610902	−0.305451	+	0.952208i	$0.598807\pi$
$72$	−20.6888	−2.43820
$73$	9.68885	1.13399	0.566997	−	0.823720i	$-0.308105\pi$
$73$	9.68885	1.13399	0.566997	+	0.823720i	$0.308105\pi$
$74$	−15.4901	−1.80068
$75$	12.4521	1.43784
$76$	10.6725	1.22422
$77$	0.827696	0.0943247
$78$	0	0
$79$	−7.79772	−0.877312	−0.438656	−	0.898655i	$-0.644545\pi$
$79$	−7.79772	−0.877312	−0.438656	+	0.898655i	$0.644545\pi$
$80$	2.14832	0.240189
$81$	−7.17489	−0.797210
$82$	13.9859	1.54448
$83$	−16.5885	−1.82082	−0.910412	−	0.413704i	$-0.864235\pi$
$83$	−16.5885	−1.82082	−0.910412	+	0.413704i	$0.864235\pi$
$84$	−13.8736	−1.51373
$85$	1.31414	0.142538
$86$	−8.71004	−0.939228
$87$	−7.05687	−0.756576
$88$	−3.88323	−0.413954
$89$	4.33965	0.460002	0.230001	−	0.973190i	$-0.426127\pi$
$89$	4.33965	0.460002	0.230001	+	0.973190i	$0.426127\pi$
$90$	−3.10497	−0.327293
$91$	0	0
$92$	−12.6712	−1.32106
$93$	2.55317	0.264752
$94$	−24.1104	−2.48679
$95$	0.862904	0.0885321
$96$	9.35932	0.955232
$97$	7.85953	0.798015	0.399007	−	0.916948i	$-0.369355\pi$
$97$	7.85953	0.798015	0.399007	+	0.916948i	$0.369355\pi$
$98$	13.6665	1.38052
$99$	2.32392	0.233563
$100$	−21.1465	−2.11465
$101$	−14.4907	−1.44188	−0.720939	−	0.692998i	$-0.756289\pi$
$101$	−14.4907	−1.44188	−0.720939	+	0.692998i	$0.756289\pi$
$102$	24.0908	2.38534
$103$	10.2006	1.00510	0.502549	−	0.864549i	$-0.332396\pi$
$103$	10.2006	1.00510	0.502549	+	0.864549i	$0.332396\pi$
$104$	0	0
$105$	−1.12172	−0.109469
$106$	13.0867	1.27109
$107$	−4.71529	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$107$	−4.71529	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$108$	−5.74203	−0.552527
$109$	15.0493	1.44146	0.720729	−	0.693217i	$-0.243807\pi$
$109$	15.0493	1.44146	0.720729	+	0.693217i	$0.243807\pi$
$110$	−0.582796	−0.0555674
$111$	−15.7119	−1.49131
$112$	7.67991	0.725683
$113$	−5.24457	−0.493367	−0.246684	−	0.969096i	$-0.579341\pi$
$113$	−5.24457	−0.493367	−0.246684	+	0.969096i	$0.579341\pi$
$114$	15.8188	1.48156
$115$	−1.02450	−0.0955354
$116$	11.9842	1.11271
$117$	0	0
$118$	24.9903	2.30055
$119$	4.69784	0.430650
$120$	5.26269	0.480415
$121$	−10.5638	−0.960346
$122$	27.8388	2.52041
$123$	14.1862	1.27913
$124$	−4.33588	−0.389373
$125$	−3.46261	−0.309705
$126$	−11.0998	−0.988849
$127$	1.49078	0.132285	0.0661427	−	0.997810i	$-0.478931\pi$
$127$	1.49078	0.132285	0.0661427	+	0.997810i	$0.478931\pi$
$128$	14.9560	1.32193
$129$	−8.83480	−0.777861
$130$	0	0
$131$	−19.6433	−1.71624	−0.858122	−	0.513446i	$-0.828369\pi$
$131$	−19.6433	−1.71624	−0.858122	+	0.513446i	$0.828369\pi$
$132$	−7.31134	−0.636371
$133$	3.08475	0.267482
$134$	0.0608303	0.00525494
$135$	−0.464260	−0.0399572
$136$	−22.0405	−1.88995
$137$	−15.8536	−1.35446	−0.677231	−	0.735770i	$-0.736820\pi$
$137$	−15.8536	−1.35446	−0.677231	+	0.735770i	$0.736820\pi$
$138$	−18.7812	−1.59876
$139$	16.1612	1.37077	0.685387	−	0.728179i	$-0.259633\pi$
$139$	16.1612	1.37077	0.685387	+	0.728179i	$0.259633\pi$
$140$	1.90494	0.160997
$141$	−24.4557	−2.05954
$142$	12.9570	1.08733
$143$	0	0
$144$	21.5629	1.79690
$145$	0.968959	0.0804677
$146$	−24.3880	−2.01836
$147$	13.8622	1.14334
$148$	26.6825	2.19329
$149$	−0.986393	−0.0808084	−0.0404042	−	0.999183i	$-0.512865\pi$
$149$	−0.986393	−0.0808084	−0.0404042	+	0.999183i	$0.512865\pi$
$150$	−31.3433	−2.55917
$151$	8.16626	0.664560	0.332280	−	0.943181i	$-0.392182\pi$
$151$	8.16626	0.664560	0.332280	+	0.943181i	$0.392182\pi$
$152$	−14.4725	−1.17387
$153$	13.1901	1.06636
$154$	−2.08341	−0.167886
$155$	−0.350569	−0.0281584
$156$	0	0
$157$	−14.3662	−1.14655	−0.573276	−	0.819362i	$-0.694328\pi$
$157$	−14.3662	−1.14655	−0.573276	+	0.819362i	$0.694328\pi$
$158$	19.6278	1.56150
$159$	13.2741	1.05271
$160$	−1.28510	−0.101596
$161$	−3.66244	−0.288641
$162$	18.0600	1.41893
$163$	23.2207	1.81878	0.909391	−	0.415942i	$-0.136548\pi$
$163$	23.2207	1.81878	0.909391	+	0.415942i	$0.136548\pi$
$164$	−24.0915	−1.88123
$165$	−0.591144	−0.0460205
$166$	41.7552	3.24083
$167$	−6.10167	−0.472161	−0.236081	−	0.971733i	$-0.575863\pi$
$167$	−6.10167	−0.472161	−0.236081	+	0.971733i	$0.575863\pi$
$168$	18.8133	1.45148
$169$	0	0
$170$	−3.30784	−0.253699
$171$	8.66104	0.662327
$172$	15.0035	1.14401
$173$	−0.0478257	−0.00363612	−0.00181806	−	0.999998i	$-0.500579\pi$
$173$	−0.0478257	−0.00363612	−0.00181806	+	0.999998i	$0.500579\pi$
$174$	17.7630	1.34661
$175$	−6.11213	−0.462034
$176$	4.04730	0.305076
$177$	25.3483	1.90530
$178$	−10.9234	−0.818744
$179$	−10.3782	−0.775700	−0.387850	−	0.921722i	$-0.626782\pi$
$179$	−10.3782	−0.775700	−0.387850	+	0.921722i	$0.626782\pi$
$180$	5.34850	0.398653
$181$	−10.0926	−0.750176	−0.375088	−	0.926989i	$-0.622387\pi$
$181$	−10.0926	−0.750176	−0.375088	+	0.926989i	$0.622387\pi$
$182$	0	0
$183$	28.2376	2.08738
$184$	17.1828	1.26673
$185$	2.15736	0.158612
$186$	−6.42663	−0.471224
$187$	2.47575	0.181045
$188$	41.5315	3.02900
$189$	−1.65966	−0.120722
$190$	−2.17203	−0.157576
$191$	1.73739	0.125713	0.0628565	−	0.998023i	$-0.479979\pi$
$191$	1.73739	0.125713	0.0628565	+	0.998023i	$0.479979\pi$
$192$	7.73364	0.558128
$193$	−3.38782	−0.243861	−0.121930	−	0.992539i	$-0.538908\pi$
$193$	−3.38782	−0.243861	−0.121930	+	0.992539i	$0.538908\pi$
$194$	−19.7834	−1.42036
$195$	0	0
$196$	−23.5413	−1.68152
$197$	−0.518299	−0.0369273	−0.0184637	−	0.999830i	$-0.505877\pi$
$197$	−0.518299	−0.0369273	−0.0184637	+	0.999830i	$0.505877\pi$
$198$	−5.84957	−0.415711
$199$	12.1683	0.862586	0.431293	−	0.902212i	$-0.358058\pi$
$199$	12.1683	0.862586	0.431293	+	0.902212i	$0.358058\pi$
$200$	28.6758	2.02769
$201$	0.0617017	0.00435210
$202$	36.4748	2.56636
$203$	3.46388	0.243117
$204$	−41.4977	−2.90542
$205$	−1.94787	−0.136045
$206$	−25.6762	−1.78894
$207$	−10.2830	−0.714720
$208$	0	0
$209$	1.62566	0.112449
$210$	2.82350	0.194840
$211$	13.6044	0.936565	0.468282	−	0.883579i	$-0.344873\pi$
$211$	13.6044	0.936565	0.468282	+	0.883579i	$0.344873\pi$
$212$	−22.5426	−1.54823
$213$	13.1426	0.900515
$214$	11.8689	0.811344
$215$	1.21308	0.0827315
$216$	7.78649	0.529804
$217$	−1.25323	−0.0850748
$218$	−37.8808	−2.56561
$219$	−24.7373	−1.67159
$220$	1.00390	0.0676829
$221$	0	0
$222$	39.5488	2.65434
$223$	3.63268	0.243262	0.121631	−	0.992575i	$-0.461188\pi$
$223$	3.63268	0.243262	0.121631	+	0.992575i	$0.461188\pi$
$224$	−4.59404	−0.306952
$225$	−17.1610	−1.14407
$226$	13.2012	0.878130
$227$	14.2559	0.946196	0.473098	−	0.881010i	$-0.343136\pi$
$227$	14.2559	0.946196	0.473098	+	0.881010i	$0.343136\pi$
$228$	−27.2487	−1.80459
$229$	17.4447	1.15278	0.576388	−	0.817176i	$-0.304462\pi$
$229$	17.4447	1.15278	0.576388	+	0.817176i	$0.304462\pi$
$230$	2.57880	0.170041
$231$	−2.11325	−0.139042
$232$	−16.2512	−1.06694
$233$	10.6659	0.698747	0.349373	−	0.936984i	$-0.386395\pi$
$233$	10.6659	0.698747	0.349373	+	0.936984i	$0.386395\pi$
$234$	0	0
$235$	3.35795	0.219048
$236$	−43.0473	−2.80214
$237$	19.9089	1.29322
$238$	−11.8250	−0.766501
$239$	−6.36483	−0.411707	−0.205853	−	0.978583i	$-0.565997\pi$
$239$	−6.36483	−0.411707	−0.205853	+	0.978583i	$0.565997\pi$
$240$	−5.48503	−0.354057
$241$	16.5659	1.06710	0.533551	−	0.845768i	$-0.320857\pi$
$241$	16.5659	1.06710	0.533551	+	0.845768i	$0.320857\pi$
$242$	26.5903	1.70929
$243$	22.2916	1.43001
$244$	−47.9540	−3.06994
$245$	−1.90338	−0.121603
$246$	−35.7084	−2.27668
$247$	0	0
$248$	5.87968	0.373360
$249$	42.3533	2.68403
$250$	8.71578	0.551235
$251$	13.7076	0.865216	0.432608	−	0.901582i	$-0.357593\pi$
$251$	13.7076	0.865216	0.432608	+	0.901582i	$0.357593\pi$
$252$	19.1201	1.20445
$253$	−1.93010	−0.121344
$254$	−3.75247	−0.235451
$255$	−3.35522	−0.210112
$256$	−31.5878	−1.97424
$257$	−0.110362	−0.00688416	−0.00344208	−	0.999994i	$-0.501096\pi$
$257$	−0.110362	−0.00688416	−0.00344208	+	0.999994i	$0.501096\pi$
$258$	22.2382	1.38449
$259$	7.71224	0.479215
$260$	0	0
$261$	9.72552	0.601995
$262$	49.4445	3.05469
$263$	−0.332118	−0.0204793	−0.0102396	−	0.999948i	$-0.503259\pi$
$263$	−0.332118	−0.0204793	−0.0102396	+	0.999948i	$0.503259\pi$
$264$	9.91457	0.610200
$265$	−1.82264	−0.111964
$266$	−7.76468	−0.476083
$267$	−11.0799	−0.678077
$268$	−0.104784	−0.00640069
$269$	21.0062	1.28077	0.640384	−	0.768055i	$-0.278775\pi$
$269$	21.0062	1.28077	0.640384	+	0.768055i	$0.278775\pi$
$270$	1.16860	0.0711186
$271$	−23.9353	−1.45396	−0.726982	−	0.686657i	$-0.759078\pi$
$271$	−23.9353	−1.45396	−0.726982	+	0.686657i	$0.759078\pi$
$272$	22.9716	1.39286
$273$	0	0
$274$	39.9053	2.41077
$275$	−3.22108	−0.194238
$276$	32.3517	1.94734
$277$	14.6717	0.881537	0.440769	−	0.897621i	$-0.354706\pi$
$277$	14.6717	0.881537	0.440769	+	0.897621i	$0.354706\pi$
$278$	−40.6796	−2.43980
$279$	−3.51869	−0.210658
$280$	−2.58320	−0.154376
$281$	21.9817	1.31132	0.655658	−	0.755058i	$-0.272391\pi$
$281$	21.9817	1.31132	0.655658	+	0.755058i	$0.272391\pi$
$282$	61.5579	3.66572
$283$	16.8398	1.00102	0.500511	−	0.865730i	$-0.333146\pi$
$283$	16.8398	1.00102	0.500511	+	0.865730i	$0.333146\pi$
$284$	−22.3192	−1.32440
$285$	−2.20314	−0.130503
$286$	0	0
$287$	−6.96334	−0.411033
$288$	−12.8987	−0.760062
$289$	−2.94814	−0.173420
$290$	−2.43898	−0.143222
$291$	−20.0667	−1.17633
$292$	42.0097	2.45843
$293$	−16.5859	−0.968957	−0.484478	−	0.874803i	$-0.660991\pi$
$293$	−16.5859	−0.968957	−0.484478	+	0.874803i	$0.660991\pi$
$294$	−34.8929	−2.03499
$295$	−3.48051	−0.202643
$296$	−36.1829	−2.10309
$297$	−0.874637	−0.0507516
$298$	2.48287	0.143829
$299$	0	0
$300$	53.9907	3.11716
$301$	4.33658	0.249956
$302$	−20.5554	−1.18283
$303$	36.9973	2.12544
$304$	15.0839	0.865123
$305$	−3.87722	−0.222009
$306$	−33.2010	−1.89798
$307$	−9.59813	−0.547794	−0.273897	−	0.961759i	$-0.588313\pi$
$307$	−9.59813	−0.547794	−0.273897	+	0.961759i	$0.588313\pi$
$308$	3.58879	0.204490
$309$	−26.0440	−1.48159
$310$	0.882423	0.0501183
$311$	8.51620	0.482909	0.241455	−	0.970412i	$-0.422376\pi$
$311$	8.51620	0.482909	0.241455	+	0.970412i	$0.422376\pi$
$312$	0	0
$313$	−19.5627	−1.10575	−0.552875	−	0.833264i	$-0.686469\pi$
$313$	−19.5627	−1.10575	−0.552875	+	0.833264i	$0.686469\pi$
$314$	36.1615	2.04071
$315$	1.54591	0.0871024
$316$	−33.8100	−1.90196
$317$	32.0733	1.80142	0.900708	−	0.434425i	$-0.143048\pi$
$317$	32.0733	1.80142	0.900708	+	0.434425i	$0.143048\pi$
$318$	−33.4126	−1.87368
$319$	1.82546	0.102206
$320$	−1.06189	−0.0593612
$321$	12.0389	0.671949
$322$	9.21880	0.513743
$323$	9.22690	0.513399
$324$	−31.1095	−1.72830
$325$	0	0
$326$	−58.4491	−3.23720
$327$	−38.4234	−2.12482
$328$	32.6694	1.80386
$329$	12.0041	0.661810
$330$	1.48798	0.0819106
$331$	−7.32907	−0.402842	−0.201421	−	0.979505i	$-0.564556\pi$
$331$	−7.32907	−0.402842	−0.201421	+	0.979505i	$0.564556\pi$
$332$	−71.9257	−3.94744
$333$	21.6536	1.18661
$334$	15.3586	0.840386
$335$	−0.00847209	−0.000462879 0
$336$	−19.6081	−1.06971
$337$	−30.9133	−1.68396	−0.841978	−	0.539512i	$-0.818609\pi$
$337$	−30.9133	−1.68396	−0.841978	+	0.539512i	$0.818609\pi$
$338$	0	0
$339$	13.3903	0.727260
$340$	5.69794	0.309014
$341$	−0.660450	−0.0357654
$342$	−21.8009	−1.17886
$343$	−15.5769	−0.841075
$344$	−20.3456	−1.09696
$345$	2.61573	0.140826
$346$	0.120383	0.00647183
$347$	3.29960	0.177132	0.0885658	−	0.996070i	$-0.471772\pi$
$347$	3.29960	0.177132	0.0885658	+	0.996070i	$0.471772\pi$
$348$	−30.5977	−1.64021
$349$	−27.0059	−1.44559	−0.722796	−	0.691061i	$-0.757144\pi$
$349$	−27.0059	−1.44559	−0.722796	+	0.691061i	$0.757144\pi$
$350$	15.3850	0.822360
$351$	0	0
$352$	−2.42105	−0.129042
$353$	−28.3874	−1.51091	−0.755454	−	0.655201i	$-0.772584\pi$
$353$	−28.3874	−1.51091	−0.755454	+	0.655201i	$0.772584\pi$
$354$	−63.8047	−3.39118
$355$	−1.80457	−0.0957767
$356$	18.8162	0.997256
$357$	−11.9944	−0.634810
$358$	26.1231	1.38065
$359$	−5.15728	−0.272191	−0.136095	−	0.990696i	$-0.543455\pi$
$359$	−5.15728	−0.272191	−0.136095	+	0.990696i	$0.543455\pi$
$360$	−7.25285	−0.382259
$361$	−12.9413	−0.681122
$362$	25.4042	1.33522
$363$	26.9712	1.41562
$364$	0	0
$365$	3.39661	0.177787
$366$	−71.0773	−3.71527
$367$	−8.82770	−0.460802	−0.230401	−	0.973096i	$-0.574004\pi$
$367$	−8.82770	−0.460802	−0.230401	+	0.973096i	$0.574004\pi$
$368$	−17.9087	−0.933558
$369$	−19.5509	−1.01778
$370$	−5.43034	−0.282310
$371$	−6.51564	−0.338275
$372$	11.0702	0.573966
$373$	26.7130	1.38315	0.691573	−	0.722307i	$-0.256918\pi$
$373$	26.7130	1.38315	0.691573	+	0.722307i	$0.256918\pi$
$374$	−6.23175	−0.322236
$375$	8.84063	0.456528
$376$	−56.3189	−2.90443
$377$	0	0
$378$	4.17756	0.214870
$379$	26.9190	1.38274	0.691369	−	0.722502i	$-0.257008\pi$
$379$	26.9190	1.38274	0.691369	+	0.722502i	$0.257008\pi$
$380$	3.74145	0.191932
$381$	−3.80622	−0.194998
$382$	−4.37321	−0.223753
$383$	−21.3494	−1.09090	−0.545451	−	0.838143i	$-0.683642\pi$
$383$	−21.3494	−1.09090	−0.545451	+	0.838143i	$0.683642\pi$
$384$	−38.1851	−1.94863
$385$	0.290164	0.0147882
$386$	8.52755	0.434041
$387$	12.1758	0.618931
$388$	34.0780	1.73005
$389$	−32.2747	−1.63639	−0.818197	−	0.574938i	$-0.805026\pi$
$389$	−32.2747	−1.63639	−0.818197	+	0.574938i	$0.805026\pi$
$390$	0	0
$391$	−10.9549	−0.554011
$392$	31.9232	1.61237
$393$	50.1527	2.52987
$394$	1.30462	0.0657258
$395$	−2.73364	−0.137544
$396$	10.0762	0.506349
$397$	−34.3213	−1.72254	−0.861268	−	0.508151i	$-0.830329\pi$
$397$	−34.3213	−1.72254	−0.861268	+	0.508151i	$0.830329\pi$
$398$	−30.6290	−1.53529
$399$	−7.87590	−0.394288
$400$	−29.8873	−1.49437
$401$	−12.5193	−0.625186	−0.312593	−	0.949887i	$-0.601198\pi$
$401$	−12.5193	−0.625186	−0.312593	+	0.949887i	$0.601198\pi$
$402$	−0.155310	−0.00774618
$403$	0	0
$404$	−62.8299	−3.12591
$405$	−2.51529	−0.124986
$406$	−8.71899	−0.432716
$407$	4.06433	0.201462
$408$	56.2731	2.78593
$409$	14.3723	0.710665	0.355332	−	0.934740i	$-0.384368\pi$
$409$	14.3723	0.710665	0.355332	+	0.934740i	$0.384368\pi$
$410$	4.90302	0.242143
$411$	40.4769	1.99658
$412$	44.2287	2.17899
$413$	−12.4423	−0.612244
$414$	25.8836	1.27211
$415$	−5.81541	−0.285467
$416$	0	0
$417$	−41.2623	−2.02062
$418$	−4.09197	−0.200145
$419$	−36.0418	−1.76075	−0.880377	−	0.474274i	$-0.842711\pi$
$419$	−36.0418	−1.76075	−0.880377	+	0.474274i	$0.842711\pi$
$420$	−4.86364	−0.237322
$421$	−37.0184	−1.80417	−0.902083	−	0.431562i	$-0.857963\pi$
$421$	−37.0184	−1.80417	−0.902083	+	0.431562i	$0.857963\pi$
$422$	−34.2438	−1.66696
$423$	33.7040	1.63874
$424$	30.5689	1.48456
$425$	−18.2822	−0.886817
$426$	−33.0814	−1.60280
$427$	−13.8605	−0.670756
$428$	−20.4449	−0.988243
$429$	0	0
$430$	−3.05347	−0.147251
$431$	19.1893	0.924315	0.462157	−	0.886798i	$-0.347075\pi$
$431$	19.1893	0.924315	0.462157	+	0.886798i	$0.347075\pi$
$432$	−8.11546	−0.390455
$433$	−14.5012	−0.696885	−0.348442	−	0.937330i	$-0.613289\pi$
$433$	−14.5012	−0.696885	−0.348442	+	0.937330i	$0.613289\pi$
$434$	3.15453	0.151422
$435$	−2.47392	−0.118615
$436$	65.2518	3.12499
$437$	−7.19331	−0.344103
$438$	62.2667	2.97522
$439$	7.67460	0.366289	0.183144	−	0.983086i	$-0.441372\pi$
$439$	7.67460	0.366289	0.183144	+	0.983086i	$0.441372\pi$
$440$	−1.36134	−0.0648994
$441$	−19.1044	−0.909734
$442$	0	0
$443$	−35.0863	−1.66700	−0.833500	−	0.552519i	$-0.813667\pi$
$443$	−35.0863	−1.66700	−0.833500	+	0.552519i	$0.813667\pi$
$444$	−68.1251	−3.23307
$445$	1.52135	0.0721187
$446$	−9.14387	−0.432975
$447$	2.51843	0.119118
$448$	−3.79608	−0.179348
$449$	−6.40721	−0.302375	−0.151187	−	0.988505i	$-0.548310\pi$
$449$	−6.40721	−0.302375	−0.151187	+	0.988505i	$0.548310\pi$
$450$	43.1963	2.03629
$451$	−3.66967	−0.172798
$452$	−22.7398	−1.06959
$453$	−20.8499	−0.979612
$454$	−35.8837	−1.68411
$455$	0	0
$456$	36.9507	1.73038
$457$	8.21687	0.384369	0.192185	−	0.981359i	$-0.438443\pi$
$457$	8.21687	0.384369	0.192185	+	0.981359i	$0.438443\pi$
$458$	−43.9103	−2.05179
$459$	−4.96426	−0.231712
$460$	−4.44212	−0.207115
$461$	−12.9030	−0.600953	−0.300476	−	0.953789i	$-0.597146\pi$
$461$	−12.9030	−0.600953	−0.300476	+	0.953789i	$0.597146\pi$
$462$	5.31930	0.247476
$463$	−10.2654	−0.477072	−0.238536	−	0.971134i	$-0.576668\pi$
$463$	−10.2654	−0.477072	−0.238536	+	0.971134i	$0.576668\pi$
$464$	16.9378	0.786318
$465$	0.895063	0.0415076
$466$	−26.8473	−1.24368
$467$	18.7975	0.869844	0.434922	−	0.900468i	$-0.356776\pi$
$467$	18.7975	0.869844	0.434922	+	0.900468i	$0.356776\pi$
$468$	0	0
$469$	−0.0302864	−0.00139850
$470$	−8.45235	−0.389878
$471$	36.6795	1.69010
$472$	58.3744	2.68690
$473$	2.28537	0.105081
$474$	−50.1131	−2.30177
$475$	−12.0047	−0.550813
$476$	20.3692	0.933623
$477$	−18.2939	−0.837622
$478$	16.0210	0.732785
$479$	24.6891	1.12807	0.564037	−	0.825750i	$-0.309248\pi$
$479$	24.6891	1.12807	0.564037	+	0.825750i	$0.309248\pi$
$480$	3.28109	0.149761
$481$	0	0
$482$	−41.6983	−1.89930
$483$	9.35085	0.425478
$484$	−45.8034	−2.08197
$485$	2.75531	0.125112
$486$	−56.1107	−2.54523
$487$	−21.2392	−0.962438	−0.481219	−	0.876600i	$-0.659806\pi$
$487$	−21.2392	−0.962438	−0.481219	+	0.876600i	$0.659806\pi$
$488$	65.0281	2.94368
$489$	−59.2863	−2.68102
$490$	4.79104	0.216437
$491$	13.3221	0.601220	0.300610	−	0.953747i	$-0.402810\pi$
$491$	13.3221	0.601220	0.300610	+	0.953747i	$0.402810\pi$
$492$	61.5098	2.77308
$493$	10.3609	0.466633
$494$	0	0
$495$	0.814694	0.0366178
$496$	−6.12809	−0.275159
$497$	−6.45107	−0.289370
$498$	−106.608	−4.77723
$499$	43.7894	1.96028	0.980140	−	0.198305i	$-0.0635437\pi$
$499$	43.7894	1.96028	0.980140	+	0.198305i	$0.0635437\pi$
$500$	−15.0134	−0.671422
$501$	15.5786	0.696001
$502$	−34.5036	−1.53997
$503$	27.6290	1.23192	0.615958	−	0.787779i	$-0.288769\pi$
$503$	27.6290	1.23192	0.615958	+	0.787779i	$0.288769\pi$
$504$	−25.9278	−1.15492
$505$	−5.07999	−0.226057
$506$	4.85829	0.215977
$507$	0	0
$508$	6.46384	0.286787
$509$	−18.5810	−0.823586	−0.411793	−	0.911277i	$-0.635097\pi$
$509$	−18.5810	−0.823586	−0.411793	+	0.911277i	$0.635097\pi$
$510$	8.44547	0.373972
$511$	12.1424	0.537147
$512$	49.5984	2.19196
$513$	−3.25970	−0.143919
$514$	0.277793	0.0122529
$515$	3.57602	0.157578
$516$	−38.3066	−1.68636
$517$	6.32616	0.278224
$518$	−19.4126	−0.852941
$519$	0.122107	0.00535992
$520$	0	0
$521$	19.3235	0.846579	0.423290	−	0.905994i	$-0.360875\pi$
$521$	19.3235	0.846579	0.423290	+	0.905994i	$0.360875\pi$
$522$	−24.4803	−1.07147
$523$	−16.1930	−0.708070	−0.354035	−	0.935232i	$-0.615191\pi$
$523$	−16.1930	−0.708070	−0.354035	+	0.935232i	$0.615191\pi$
$524$	−85.1710	−3.72071
$525$	15.6053	0.681072
$526$	0.835979	0.0364504
$527$	−3.74858	−0.163291
$528$	−10.3334	−0.449706
$529$	−14.4596	−0.628677
$530$	4.58779	0.199281
$531$	−34.9341	−1.51601
$532$	13.3751	0.579885
$533$	0	0
$534$	27.8893	1.20689
$535$	−1.65303	−0.0714669
$536$	0.142092	0.00613745
$537$	26.4972	1.14344
$538$	−52.8750	−2.27960
$539$	−3.58585	−0.154454
$540$	−2.01298	−0.0866247
$541$	−26.3360	−1.13227	−0.566136	−	0.824312i	$-0.691562\pi$
$541$	−26.3360	−1.13227	−0.566136	+	0.824312i	$0.691562\pi$
$542$	60.2479	2.58787
$543$	25.7681	1.10582
$544$	−13.7414	−0.589158
$545$	5.27581	0.225991
$546$	0	0
$547$	24.0327	1.02756	0.513782	−	0.857921i	$-0.328244\pi$
$547$	24.0327	1.02756	0.513782	+	0.857921i	$0.328244\pi$
$548$	−68.7392	−2.93639
$549$	−38.9160	−1.66089
$550$	8.10784	0.345719
$551$	6.80332	0.289831
$552$	−43.8706	−1.86726
$553$	−9.77234	−0.415562
$554$	−36.9304	−1.56902
$555$	−5.50812	−0.233807
$556$	70.0730	2.97176
$557$	−36.9008	−1.56354	−0.781769	−	0.623569i	$-0.785682\pi$
$557$	−36.9008	−1.56354	−0.781769	+	0.623569i	$0.785682\pi$
$558$	8.85696	0.374945
$559$	0	0
$560$	2.69234	0.113772
$561$	−6.32101	−0.266873
$562$	−55.3304	−2.33397
$563$	17.4323	0.734685	0.367343	−	0.930086i	$-0.380268\pi$
$563$	17.4323	0.734685	0.367343	+	0.930086i	$0.380268\pi$
$564$	−106.037	−4.46497
$565$	−1.83858	−0.0773497
$566$	−42.3877	−1.78169
$567$	−8.99179	−0.377620
$568$	30.2660	1.26993
$569$	20.6086	0.863957	0.431978	−	0.901884i	$-0.357816\pi$
$569$	20.6086	0.863957	0.431978	+	0.901884i	$0.357816\pi$
$570$	5.54557	0.232278
$571$	35.3415	1.47900	0.739498	−	0.673159i	$-0.235063\pi$
$571$	35.3415	1.47900	0.739498	+	0.673159i	$0.235063\pi$
$572$	0	0
$573$	−4.43585	−0.185310
$574$	17.5276	0.731586
$575$	14.2528	0.594385
$576$	−10.6582	−0.444093
$577$	14.4494	0.601535	0.300768	−	0.953697i	$-0.402757\pi$
$577$	14.4494	0.601535	0.300768	+	0.953697i	$0.402757\pi$
$578$	7.42082	0.308666
$579$	8.64970	0.359469
$580$	4.20129	0.174449
$581$	−20.7892	−0.862481
$582$	50.5104	2.09372
$583$	−3.43373	−0.142211
$584$	−56.9674	−2.35733
$585$	0	0
$586$	41.7486	1.72462
$587$	−36.2202	−1.49497	−0.747483	−	0.664281i	$-0.768738\pi$
$587$	−36.2202	−1.49497	−0.747483	+	0.664281i	$0.768738\pi$
$588$	60.1050	2.47869
$589$	−2.46144	−0.101422
$590$	8.76084	0.360678
$591$	1.32331	0.0544336
$592$	37.7116	1.54994
$593$	−14.5311	−0.596720	−0.298360	−	0.954453i	$-0.596440\pi$
$593$	−14.5311	−0.596720	−0.298360	+	0.954453i	$0.596440\pi$
$594$	2.20156	0.0903312
$595$	1.64692	0.0675170
$596$	−4.27688	−0.175188
$597$	−31.0677	−1.27152
$598$	0	0
$599$	−20.5843	−0.841053	−0.420526	−	0.907280i	$-0.638155\pi$
$599$	−20.5843	−0.841053	−0.420526	+	0.907280i	$0.638155\pi$
$600$	−73.2143	−2.98896
$601$	−17.9939	−0.733988	−0.366994	−	0.930223i	$-0.619613\pi$
$601$	−17.9939	−0.733988	−0.366994	+	0.930223i	$0.619613\pi$
$602$	−10.9157	−0.444890
$603$	−0.0850350	−0.00346289
$604$	35.4079	1.44073
$605$	−3.70334	−0.150562
$606$	−93.1264	−3.78300
$607$	−29.6147	−1.20202	−0.601012	−	0.799240i	$-0.705236\pi$
$607$	−29.6147	−1.20202	−0.601012	+	0.799240i	$0.705236\pi$
$608$	−9.02305	−0.365933
$609$	−8.84388	−0.358372
$610$	9.75943	0.395148
$611$	0	0
$612$	57.1907	2.31180
$613$	−0.980934	−0.0396196	−0.0198098	−	0.999804i	$-0.506306\pi$
$613$	−0.980934	−0.0396196	−0.0198098	+	0.999804i	$0.506306\pi$
$614$	24.1596	0.975003
$615$	4.97325	0.200541
$616$	−4.86659	−0.196080
$617$	−16.3254	−0.657236	−0.328618	−	0.944463i	$-0.606583\pi$
$617$	−16.3254	−0.657236	−0.328618	+	0.944463i	$0.606583\pi$
$618$	65.5557	2.63704
$619$	−32.9615	−1.32483	−0.662417	−	0.749135i	$-0.730469\pi$
$619$	−32.9615	−1.32483	−0.662417	+	0.749135i	$0.730469\pi$
$620$	−1.52003	−0.0610457
$621$	3.87015	0.155304
$622$	−21.4363	−0.859516
$623$	5.43858	0.217892
$624$	0	0
$625$	23.1716	0.926865
$626$	49.2417	1.96809
$627$	−4.15058	−0.165758
$628$	−62.2903	−2.48565
$629$	23.0684	0.919795
$630$	−3.89125	−0.155031
$631$	−17.1980	−0.684643	−0.342322	−	0.939583i	$-0.611213\pi$
$631$	−17.1980	−0.684643	−0.342322	+	0.939583i	$0.611213\pi$
$632$	45.8481	1.82374
$633$	−34.7343	−1.38057
$634$	−80.7322	−3.20629
$635$	0.522621	0.0207396
$636$	57.5551	2.28221
$637$	0	0
$638$	−4.59489	−0.181913
$639$	−18.1126	−0.716525
$640$	5.24309	0.207251
$641$	5.66344	0.223692	0.111846	−	0.993726i	$-0.464324\pi$
$641$	5.66344	0.223692	0.111846	+	0.993726i	$0.464324\pi$
$642$	−30.3034	−1.19598
$643$	20.7917	0.819945	0.409973	−	0.912098i	$-0.365538\pi$
$643$	20.7917	0.819945	0.409973	+	0.912098i	$0.365538\pi$
$644$	−15.8799	−0.625756
$645$	−3.09721	−0.121952
$646$	−23.2252	−0.913783
$647$	10.1424	0.398739	0.199369	−	0.979924i	$-0.436111\pi$
$647$	10.1424	0.398739	0.199369	+	0.979924i	$0.436111\pi$
$648$	42.1861	1.65723
$649$	−6.55705	−0.257387
$650$	0	0
$651$	3.19971	0.125407
$652$	100.682	3.94301
$653$	−18.1719	−0.711121	−0.355560	−	0.934653i	$-0.615710\pi$
$653$	−18.1719	−0.711121	−0.355560	+	0.934653i	$0.615710\pi$
$654$	96.7161	3.78190
$655$	−6.88633	−0.269071
$656$	−34.0496	−1.32941
$657$	34.0921	1.33006
$658$	−30.2158	−1.17794
$659$	−46.8319	−1.82431	−0.912155	−	0.409845i	$-0.865583\pi$
$659$	−46.8319	−1.82431	−0.912155	+	0.409845i	$0.865583\pi$
$660$	−2.56313	−0.0997697
$661$	−40.2748	−1.56651	−0.783253	−	0.621703i	$-0.786441\pi$
$661$	−40.2748	−1.56651	−0.783253	+	0.621703i	$0.786441\pi$
$662$	18.4481	0.717007
$663$	0	0
$664$	97.5351	3.78509
$665$	1.08142	0.0419356
$666$	−54.5047	−2.11202
$667$	−8.07740	−0.312758
$668$	−26.4561	−1.02362
$669$	−9.27485	−0.358586
$670$	0.0213252	0.000823866 0
$671$	−7.30444	−0.281985
$672$	11.7294	0.452471
$673$	7.43778	0.286706	0.143353	−	0.989672i	$-0.454212\pi$
$673$	7.43778	0.286706	0.143353	+	0.989672i	$0.454212\pi$
$674$	77.8125	2.99722
$675$	6.45877	0.248598
$676$	0	0
$677$	24.2747	0.932952	0.466476	−	0.884534i	$-0.345523\pi$
$677$	24.2747	0.932952	0.466476	+	0.884534i	$0.345523\pi$
$678$	−33.7049	−1.29443
$679$	9.84981	0.378001
$680$	−7.72670	−0.296306
$681$	−36.3977	−1.39476
$682$	1.66243	0.0636577
$683$	16.4054	0.627733	0.313867	−	0.949467i	$-0.398376\pi$
$683$	16.4054	0.627733	0.313867	+	0.949467i	$0.398376\pi$
$684$	37.5532	1.43588
$685$	−5.55777	−0.212352
$686$	39.2089	1.49700
$687$	−44.5393	−1.69928
$688$	21.2052	0.808440
$689$	0	0
$690$	−6.58411	−0.250653
$691$	2.24074	0.0852419	0.0426210	−	0.999091i	$-0.486429\pi$
$691$	2.24074	0.0852419	0.0426210	+	0.999091i	$0.486429\pi$
$692$	−0.207367	−0.00788289
$693$	2.91240	0.110633
$694$	−8.30547	−0.315271
$695$	5.66561	0.214909
$696$	41.4922	1.57276
$697$	−20.8283	−0.788928
$698$	67.9770	2.57297
$699$	−27.2319	−1.03000
$700$	−26.5015	−1.00166
$701$	13.5475	0.511683	0.255841	−	0.966719i	$-0.417647\pi$
$701$	13.5475	0.511683	0.255841	+	0.966719i	$0.417647\pi$
$702$	0	0
$703$	15.1474	0.571296
$704$	−2.00052	−0.0753976
$705$	−8.57342	−0.322894
$706$	71.4544	2.68922
$707$	−18.1602	−0.682984
$708$	109.907	4.13057
$709$	7.58705	0.284938	0.142469	−	0.989799i	$-0.454496\pi$
$709$	7.58705	0.284938	0.142469	+	0.989799i	$0.454496\pi$
$710$	4.54232	0.170470
$711$	−27.4378	−1.02900
$712$	−25.5158	−0.956244
$713$	2.92240	0.109445
$714$	30.1913	1.12988
$715$	0	0
$716$	−44.9985	−1.68167
$717$	16.2505	0.606887
$718$	12.9815	0.484464
$719$	5.25694	0.196051	0.0980254	−	0.995184i	$-0.468747\pi$
$719$	5.25694	0.196051	0.0980254	+	0.995184i	$0.468747\pi$
$720$	7.55927	0.281717
$721$	12.7837	0.476091
$722$	32.5748	1.21231
$723$	−42.2956	−1.57299
$724$	−43.7602	−1.62634
$725$	−13.4801	−0.500639
$726$	−67.8897	−2.51962
$727$	36.7845	1.36426	0.682131	−	0.731230i	$-0.261053\pi$
$727$	36.7845	1.36426	0.682131	+	0.731230i	$0.261053\pi$
$728$	0	0
$729$	−35.3898	−1.31073
$730$	−8.54967	−0.316437
$731$	12.9713	0.479761
$732$	122.435	4.52532
$733$	7.99448	0.295283	0.147642	−	0.989041i	$-0.452832\pi$
$733$	7.99448	0.295283	0.147642	+	0.989041i	$0.452832\pi$
$734$	22.2203	0.820168
$735$	4.85967	0.179252
$736$	10.7128	0.394880
$737$	−0.0159609	−0.000587926 0
$738$	49.2120	1.81152
$739$	12.0376	0.442811	0.221406	−	0.975182i	$-0.428936\pi$
$739$	12.0376	0.442811	0.221406	+	0.975182i	$0.428936\pi$
$740$	9.35407	0.343862
$741$	0	0
$742$	16.4006	0.602086
$743$	−9.28620	−0.340678	−0.170339	−	0.985386i	$-0.554486\pi$
$743$	−9.28620	−0.340678	−0.170339	+	0.985386i	$0.554486\pi$
$744$	−15.0118	−0.550361
$745$	−0.345799	−0.0126691
$746$	−67.2397	−2.46182
$747$	−58.3698	−2.13564
$748$	10.7345	0.392494
$749$	−5.90934	−0.215923
$750$	−22.2529	−0.812561
$751$	−14.3900	−0.525099	−0.262550	−	0.964919i	$-0.584563\pi$
$751$	−14.3900	−0.525099	−0.262550	+	0.964919i	$0.584563\pi$
$752$	58.6983	2.14051
$753$	−34.9979	−1.27539
$754$	0	0
$755$	2.86284	0.104189
$756$	−7.19608	−0.261719
$757$	−22.6602	−0.823598	−0.411799	−	0.911275i	$-0.635099\pi$
$757$	−22.6602	−0.823598	−0.411799	+	0.911275i	$0.635099\pi$
$758$	−67.7583	−2.46109
$759$	4.92788	0.178871
$760$	−5.07360	−0.184039
$761$	−14.4013	−0.522046	−0.261023	−	0.965333i	$-0.584060\pi$
$761$	−14.4013	−0.522046	−0.261023	+	0.965333i	$0.584060\pi$
$762$	9.58070	0.347072
$763$	18.8602	0.682785
$764$	7.53311	0.272538
$765$	4.62404	0.167182
$766$	53.7389	1.94167
$767$	0	0
$768$	80.6492	2.91018
$769$	4.03329	0.145444	0.0727220	−	0.997352i	$-0.476831\pi$
$769$	4.03329	0.145444	0.0727220	+	0.997352i	$0.476831\pi$
$770$	−0.730378	−0.0263210
$771$	0.281772	0.0101478
$772$	−14.6892	−0.528676
$773$	−41.1403	−1.47971	−0.739857	−	0.672764i	$-0.765107\pi$
$773$	−41.1403	−1.47971	−0.739857	+	0.672764i	$0.765107\pi$
$774$	−30.6479	−1.10162
$775$	4.87710	0.175191
$776$	−46.2116	−1.65890
$777$	−19.6907	−0.706399
$778$	81.2393	2.91257
$779$	−13.6765	−0.490013
$780$	0	0
$781$	−3.39970	−0.121651
$782$	27.5747	0.986068
$783$	−3.66033	−0.130809
$784$	−33.2719	−1.18828
$785$	−5.03636	−0.179755
$786$	−126.240	−4.50284
$787$	−8.73381	−0.311327	−0.155663	−	0.987810i	$-0.549751\pi$
$787$	−8.73381	−0.311327	−0.155663	+	0.987810i	$0.549751\pi$
$788$	−2.24728	−0.0800562
$789$	0.847954	0.0301880
$790$	6.88089	0.244811
$791$	−6.57265	−0.233696
$792$	−13.6639	−0.485526
$793$	0	0
$794$	86.3907	3.06589
$795$	4.65350	0.165043
$796$	52.7602	1.87003
$797$	14.9825	0.530708	0.265354	−	0.964151i	$-0.414511\pi$
$797$	14.9825	0.530708	0.265354	+	0.964151i	$0.414511\pi$
$798$	19.8246	0.701782
$799$	35.9060	1.27026
$800$	17.8783	0.632093
$801$	15.2699	0.539535
$802$	31.5126	1.11275
$803$	6.39900	0.225816
$804$	0.267531	0.00943509
$805$	−1.28394	−0.0452529
$806$	0	0
$807$	−53.6324	−1.88795
$808$	85.2007	2.99735
$809$	−20.7682	−0.730171	−0.365085	−	0.930974i	$-0.618960\pi$
$809$	−20.7682	−0.730171	−0.365085	+	0.930974i	$0.618960\pi$
$810$	6.33129	0.222459
$811$	26.2346	0.921222	0.460611	−	0.887602i	$-0.347630\pi$
$811$	26.2346	0.921222	0.460611	+	0.887602i	$0.347630\pi$
$812$	15.0190	0.527062
$813$	61.1109	2.14325
$814$	−10.2304	−0.358576
$815$	8.14044	0.285147
$816$	−58.6506	−2.05318
$817$	8.51738	0.297985
$818$	−36.1768	−1.26489
$819$	0	0
$820$	−8.44574	−0.294938
$821$	33.4654	1.16795	0.583975	−	0.811772i	$-0.301497\pi$
$821$	33.4654	1.16795	0.583975	+	0.811772i	$0.301497\pi$
$822$	−101.885	−3.55365
$823$	−46.1063	−1.60716	−0.803582	−	0.595195i	$-0.797075\pi$
$823$	−46.1063	−1.60716	−0.803582	+	0.595195i	$0.797075\pi$
$824$	−59.9764	−2.08938
$825$	8.22397	0.286322
$826$	31.3187	1.08972
$827$	28.8236	1.00230	0.501148	−	0.865362i	$-0.332911\pi$
$827$	28.8236	1.00230	0.501148	+	0.865362i	$0.332911\pi$
$828$	−44.5860	−1.54947
$829$	50.2001	1.74352	0.871761	−	0.489931i	$-0.162978\pi$
$829$	50.2001	1.74352	0.871761	+	0.489931i	$0.162978\pi$
$830$	14.6381	0.508095
$831$	−37.4594	−1.29945
$832$	0	0
$833$	−20.3526	−0.705176
$834$	103.862	3.59645
$835$	−2.13906	−0.0740251
$836$	7.04865	0.243783
$837$	1.32430	0.0457747
$838$	90.7213	3.13392
$839$	−1.12676	−0.0389001	−0.0194500	−	0.999811i	$-0.506192\pi$
$839$	−1.12676	−0.0389001	−0.0194500	+	0.999811i	$0.506192\pi$
$840$	6.59536	0.227562
$841$	−21.3605	−0.736570
$842$	93.1797	3.21118
$843$	−56.1230	−1.93298
$844$	58.9870	2.03042
$845$	0	0
$846$	−84.8369	−2.91675
$847$	−13.2389	−0.454893
$848$	−31.8604	−1.09409
$849$	−42.9949	−1.47558
$850$	46.0185	1.57842
$851$	−17.9841	−0.616488
$852$	56.9846	1.95226
$853$	27.2824	0.934131	0.467065	−	0.884223i	$-0.345311\pi$
$853$	27.2824	0.934131	0.467065	+	0.884223i	$0.345311\pi$
$854$	34.8885	1.19386
$855$	3.03629	0.103839
$856$	27.7244	0.947601
$857$	26.3374	0.899667	0.449834	−	0.893112i	$-0.351483\pi$
$857$	26.3374	0.899667	0.449834	+	0.893112i	$0.351483\pi$
$858$	0	0
$859$	−33.7799	−1.15256	−0.576278	−	0.817254i	$-0.695495\pi$
$859$	−33.7799	−1.15256	−0.576278	+	0.817254i	$0.695495\pi$
$860$	5.25978	0.179357
$861$	17.7786	0.605894
$862$	−48.3017	−1.64516
$863$	2.60079	0.0885318	0.0442659	−	0.999020i	$-0.485905\pi$
$863$	2.60079	0.0885318	0.0442659	+	0.999020i	$0.485905\pi$
$864$	4.85459	0.165156
$865$	−0.0167662	−0.000570068 0
$866$	36.5013	1.24036
$867$	7.52712	0.255634
$868$	−5.43386	−0.184437
$869$	−5.15000	−0.174702
$870$	6.22715	0.211120
$871$	0	0
$872$	−88.4849	−2.99648
$873$	27.6553	0.935989
$874$	18.1064	0.612459
$875$	−4.33944	−0.146700
$876$	−107.258	−3.62391
$877$	−13.2524	−0.447500	−0.223750	−	0.974647i	$-0.571830\pi$
$877$	−13.2524	−0.447500	−0.223750	+	0.974647i	$0.571830\pi$
$878$	−19.3179	−0.651947
$879$	42.3466	1.42831
$880$	1.41886	0.0478297
$881$	−38.6677	−1.30275	−0.651374	−	0.758757i	$-0.725807\pi$
$881$	−38.6677	−1.30275	−0.651374	+	0.758757i	$0.725807\pi$
$882$	48.0881	1.61921
$883$	21.7830	0.733055	0.366527	−	0.930407i	$-0.380547\pi$
$883$	21.7830	0.733055	0.366527	+	0.930407i	$0.380547\pi$
$884$	0	0
$885$	8.88633	0.298711
$886$	88.3163	2.96704
$887$	−34.5354	−1.15958	−0.579792	−	0.814764i	$-0.696866\pi$
$887$	−34.5354	−1.15958	−0.579792	+	0.814764i	$0.696866\pi$
$888$	92.3812	3.10011
$889$	1.86829	0.0626605
$890$	−3.82941	−0.128362
$891$	−4.73865	−0.158751
$892$	15.7509	0.527377
$893$	23.5771	0.788976
$894$	−6.33918	−0.212014
$895$	−3.63826	−0.121614
$896$	18.7433	0.626168
$897$	0	0
$898$	16.1277	0.538188
$899$	−2.76396	−0.0921833
$900$	−74.4081	−2.48027
$901$	−19.4892	−0.649278
$902$	9.23698	0.307558
$903$	−11.0720	−0.368455
$904$	30.8364	1.02560
$905$	−3.53815	−0.117612
$906$	52.4815	1.74358
$907$	8.76013	0.290875	0.145438	−	0.989367i	$-0.453541\pi$
$907$	8.76013	0.290875	0.145438	+	0.989367i	$0.453541\pi$
$908$	61.8118	2.05130
$909$	−50.9883	−1.69117
$910$	0	0
$911$	−15.5179	−0.514131	−0.257065	−	0.966394i	$-0.582756\pi$
$911$	−15.5179	−0.514131	−0.257065	+	0.966394i	$0.582756\pi$
$912$	−38.5119	−1.27526
$913$	−10.9559	−0.362586
$914$	−20.6828	−0.684127
$915$	9.89922	0.327258
$916$	75.6380	2.49915
$917$	−24.6176	−0.812944
$918$	12.4956	0.412417
$919$	4.49211	0.148181	0.0740905	−	0.997252i	$-0.476395\pi$
$919$	4.49211	0.148181	0.0740905	+	0.997252i	$0.476395\pi$
$920$	6.02375	0.198597
$921$	24.5057	0.807490
$922$	32.4784	1.06962
$923$	0	0
$924$	−9.16280	−0.301434
$925$	−30.0131	−0.986826
$926$	25.8391	0.849127
$927$	35.8928	1.17888
$928$	−10.1320	−0.332600
$929$	−20.9694	−0.687984	−0.343992	−	0.938973i	$-0.611779\pi$
$929$	−20.9694	−0.687984	−0.343992	+	0.938973i	$0.611779\pi$
$930$	−2.25298	−0.0738781
$931$	−13.3642	−0.437993
$932$	46.2461	1.51484
$933$	−21.7433	−0.711844
$934$	−47.3155	−1.54821
$935$	0.867921	0.0283840
$936$	0	0
$937$	18.7410	0.612242	0.306121	−	0.951993i	$-0.400969\pi$
$937$	18.7410	0.612242	0.306121	+	0.951993i	$0.400969\pi$
$938$	0.0762344	0.00248914
$939$	49.9470	1.62996
$940$	14.5597	0.474884
$941$	−3.52462	−0.114899	−0.0574497	−	0.998348i	$-0.518297\pi$
$941$	−3.52462	−0.114899	−0.0574497	+	0.998348i	$0.518297\pi$
$942$	−92.3266	−3.00816
$943$	16.2378	0.528775
$944$	−60.8407	−1.98020
$945$	−0.581825	−0.0189268
$946$	−5.75254	−0.187031
$947$	−3.38821	−0.110102	−0.0550510	−	0.998484i	$-0.517532\pi$
$947$	−3.38821	−0.110102	−0.0550510	+	0.998484i	$0.517532\pi$
$948$	86.3227	2.80363
$949$	0	0
$950$	30.2172	0.980376
$951$	−81.8887	−2.65542
$952$	−27.6218	−0.895227
$953$	−56.0860	−1.81680	−0.908401	−	0.418099i	$-0.862696\pi$
$953$	−56.0860	−1.81680	−0.908401	+	0.418099i	$0.862696\pi$
$954$	46.0480	1.49086
$955$	0.609075	0.0197092
$956$	−27.5972	−0.892556
$957$	−4.66071	−0.150659
$958$	−62.1453	−2.00783
$959$	−19.8682	−0.641577
$960$	2.71118	0.0875028
$961$	1.00000	0.0322581
$962$	0	0
$963$	−16.5916	−0.534658
$964$	71.8277	2.31341
$965$	−1.18767	−0.0382323
$966$	−23.5372	−0.757296
$967$	−32.2364	−1.03665	−0.518326	−	0.855183i	$-0.673445\pi$
$967$	−32.2364	−1.03665	−0.518326	+	0.855183i	$0.673445\pi$
$968$	62.1118	1.99635
$969$	−23.5579	−0.756788
$970$	−6.93544	−0.222683
$971$	29.3898	0.943163	0.471581	−	0.881823i	$-0.343683\pi$
$971$	29.3898	0.943163	0.471581	+	0.881823i	$0.343683\pi$
$972$	96.6539	3.10018
$973$	20.2537	0.649304
$974$	53.4615	1.71302
$975$	0	0
$976$	−67.7755	−2.16944
$977$	41.8636	1.33933	0.669667	−	0.742661i	$-0.266437\pi$
$977$	41.8636	1.33933	0.669667	+	0.742661i	$0.266437\pi$
$978$	149.231	4.77187
$979$	2.86612	0.0916016
$980$	−8.25284	−0.263627
$981$	52.9537	1.69068
$982$	−33.5334	−1.07009
$983$	59.9742	1.91288	0.956440	−	0.291930i	$-0.0942974\pi$
$983$	59.9742	1.91288	0.956440	+	0.291930i	$0.0942974\pi$
$984$	−83.4105	−2.65903
$985$	−0.181700	−0.00578943
$986$	−26.0797	−0.830546
$987$	−30.6487	−0.975557
$988$	0	0
$989$	−10.1125	−0.321557
$990$	−2.05068	−0.0651749
$991$	−54.5197	−1.73188	−0.865938	−	0.500151i	$-0.833278\pi$
$991$	−54.5197	−1.73188	−0.865938	+	0.500151i	$0.833278\pi$
$992$	3.66576	0.116388
$993$	18.7124	0.593820
$994$	16.2381	0.515041
$995$	4.26582	0.135236
$996$	183.639	5.81881
$997$	−38.3292	−1.21390	−0.606948	−	0.794741i	$-0.707606\pi$
$997$	−38.3292	−1.21390	−0.606948	+	0.794741i	$0.707606\pi$
$998$	−110.223	−3.48905
$999$	−8.14963	−0.257843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.t.1.3	yes	36
13.12	even	2		5239.2.a.s.1.34	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5239.2.a.s.1.34	✓	36	13.12	even	2
5239.2.a.t.1.3	yes	36	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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51712 q^{2} -2.55317 q^{3} +4.33588 q^{4} +0.350569 q^{5} +6.42663 q^{6} +1.25323 q^{7} -5.87968 q^{8} +3.51869 q^{9} -0.882423 q^{10} +0.660450 q^{11} -11.0702 q^{12} -3.15453 q^{14} -0.895063 q^{15} +6.12809 q^{16} +3.74858 q^{17} -8.85696 q^{18} +2.46144 q^{19} +1.52003 q^{20} -3.19971 q^{21} -1.66243 q^{22} -2.92240 q^{23} +15.0118 q^{24} -4.87710 q^{25} -1.32430 q^{27} +5.43386 q^{28} +2.76396 q^{29} +2.25298 q^{30} -1.00000 q^{31} -3.66576 q^{32} -1.68624 q^{33} -9.43562 q^{34} +0.439344 q^{35} +15.2566 q^{36} +6.15389 q^{37} -6.19573 q^{38} -2.06123 q^{40} -5.55631 q^{41} +8.05405 q^{42} +3.46032 q^{43} +2.86363 q^{44} +1.23354 q^{45} +7.35603 q^{46} +9.57856 q^{47} -15.6461 q^{48} -5.42941 q^{49} +12.2762 q^{50} -9.57077 q^{51} -5.19908 q^{53} +3.33343 q^{54} +0.231533 q^{55} -7.36860 q^{56} -6.28448 q^{57} -6.95722 q^{58} -9.92816 q^{59} -3.88089 q^{60} -11.0598 q^{61} +2.51712 q^{62} +4.40973 q^{63} -3.02903 q^{64} +4.24447 q^{66} -0.0241667 q^{67} +16.2534 q^{68} +7.46139 q^{69} -1.10588 q^{70} -5.14755 q^{71} -20.6888 q^{72} +9.68885 q^{73} -15.4901 q^{74} +12.4521 q^{75} +10.6725 q^{76} +0.827696 q^{77} -7.79772 q^{79} +2.14832 q^{80} -7.17489 q^{81} +13.9859 q^{82} -16.5885 q^{83} -13.8736 q^{84} +1.31414 q^{85} -8.71004 q^{86} -7.05687 q^{87} -3.88323 q^{88} +4.33965 q^{89} -3.10497 q^{90} -12.6712 q^{92} +2.55317 q^{93} -24.1104 q^{94} +0.862904 q^{95} +9.35932 q^{96} +7.85953 q^{97} +13.6665 q^{98} +2.32392 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2 q^{2} - 5 q^{3} + 28 q^{4} + 5 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 15 q^{10} - q^{11} - 13 q^{12} - 19 q^{14} - 10 q^{15} + 4 q^{16} - 46 q^{17} - 9 q^{18} + 8 q^{19} - 5 q^{20} - 16 q^{21}+ \cdots - 47 q^{99}+O(q^{100}) \) 36 * q + 2 * q^2 - 5 * q^3 + 28 * q^4 + 5 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 - 15 * q^10 - q^11 - 13 * q^12 - 19 * q^14 - 10 * q^15 + 4 * q^16 - 46 * q^17 - 9 * q^18 + 8 * q^19 - 5 * q^20 - 16 * q^21 - 21 * q^22 - 24 * q^23 + 57 * q^24 + 5 * q^25 - 11 * q^27 - 32 * q^28 - 73 * q^29 - 31 * q^30 - 36 * q^31 + 3 * q^32 - 8 * q^33 - 37 * q^34 - 31 * q^35 - 31 * q^36 + 6 * q^37 - 25 * q^38 - 22 * q^40 + 6 * q^41 - 37 * q^42 - 37 * q^43 - 16 * q^45 - 45 * q^46 + 13 * q^47 - 46 * q^48 - 5 * q^49 + 24 * q^50 - 46 * q^51 - 42 * q^53 - 26 * q^54 - 47 * q^55 - 95 * q^56 + 33 * q^57 + 9 * q^58 + 20 * q^59 - 40 * q^60 - 48 * q^61 - 2 * q^62 + 52 * q^63 + 21 * q^64 - 4 * q^66 - 50 * q^67 - 76 * q^68 - 33 * q^69 + 93 * q^70 + 16 * q^71 - 52 * q^72 - 43 * q^73 - 76 * q^74 - 40 * q^75 + 62 * q^76 - 47 * q^77 - 43 * q^79 - 46 * q^80 - 96 * q^81 + 7 * q^82 + 9 * q^83 - 77 * q^84 - 5 * q^85 + 64 * q^86 - 8 * q^87 - 54 * q^88 + 6 * q^89 + 27 * q^90 - 35 * q^92 + 5 * q^93 - 76 * q^94 - 63 * q^95 + 72 * q^96 - 4 * q^97 - 41 * q^98 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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