LMFDB - Embedded newform 39.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(1,39)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	39.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.74166 q^{2} -3.00000 q^{3} -0.483315 q^{4} +19.4833 q^{5} +8.22497 q^{6} +7.48331 q^{7} +23.2583 q^{8} +9.00000 q^{9} -53.4166 q^{10} +22.8999 q^{11} +1.44994 q^{12} -13.0000 q^{13} -20.5167 q^{14} -58.4499 q^{15} -59.8999 q^{16} +67.0334 q^{17} -24.6749 q^{18} +16.5167 q^{19} -9.41657 q^{20} -22.4499 q^{21} -62.7836 q^{22} -175.600 q^{23} -69.7750 q^{24} +254.600 q^{25} +35.6415 q^{26} -27.0000 q^{27} -3.61680 q^{28} +291.800 q^{29} +160.250 q^{30} +117.283 q^{31} -21.8418 q^{32} -68.6997 q^{33} -183.783 q^{34} +145.800 q^{35} -4.34983 q^{36} -154.766 q^{37} -45.2831 q^{38} +39.0000 q^{39} +453.150 q^{40} -251.716 q^{41} +61.5501 q^{42} -502.566 q^{43} -11.0679 q^{44} +175.350 q^{45} +481.434 q^{46} -281.733 q^{47} +179.700 q^{48} -287.000 q^{49} -698.025 q^{50} -201.100 q^{51} +6.28309 q^{52} +366.999 q^{53} +74.0247 q^{54} +446.166 q^{55} +174.049 q^{56} -49.5501 q^{57} -800.015 q^{58} -79.6663 q^{59} +28.2497 q^{60} -194.865 q^{61} -321.550 q^{62} +67.3498 q^{63} +539.082 q^{64} -253.283 q^{65} +188.351 q^{66} +400.082 q^{67} -32.3982 q^{68} +526.799 q^{69} -399.733 q^{70} +528.299 q^{71} +209.325 q^{72} -734.366 q^{73} +424.316 q^{74} -763.799 q^{75} -7.98276 q^{76} +171.367 q^{77} -106.925 q^{78} +113.266 q^{79} -1167.05 q^{80} +81.0000 q^{81} +690.118 q^{82} -933.466 q^{83} +10.8504 q^{84} +1306.03 q^{85} +1377.86 q^{86} -875.399 q^{87} +532.613 q^{88} +1190.91 q^{89} -480.749 q^{90} -97.2831 q^{91} +84.8699 q^{92} -351.849 q^{93} +772.415 q^{94} +321.800 q^{95} +65.5253 q^{96} +557.165 q^{97} +786.856 q^{98} +206.099 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20}+ \cdots - 396 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * 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 $

$2$	−2.74166	−0.969322	−0.484661	−	0.874702i	$-0.661057\pi$
$2$	−2.74166	−0.969322	−0.484661	+	0.874702i	$0.661057\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−0.483315	−0.0604143
$5$	19.4833	1.74264	0.871320	−	0.490715i	$-0.163264\pi$
$5$	19.4833	1.74264	0.871320	+	0.490715i	$0.163264\pi$
$6$	8.22497	0.559638
$7$	7.48331	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$7$	7.48331	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$8$	23.2583	1.02788
$9$	9.00000	0.333333
$10$	−53.4166	−1.68918
$11$	22.8999	0.627689	0.313844	−	0.949474i	$-0.398383\pi$
$11$	22.8999	0.627689	0.313844	+	0.949474i	$0.398383\pi$
$12$	1.44994	0.0348802
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−20.5167	−0.391665
$15$	−58.4499	−1.00611
$16$	−59.8999	−0.935936
$17$	67.0334	0.956352	0.478176	−	0.878264i	$-0.341298\pi$
$17$	67.0334	0.956352	0.478176	+	0.878264i	$0.341298\pi$
$18$	−24.6749	−0.323107
$19$	16.5167	0.199431	0.0997155	−	0.995016i	$-0.468207\pi$
$19$	16.5167	0.199431	0.0997155	+	0.995016i	$0.468207\pi$
$20$	−9.41657	−0.105280
$21$	−22.4499	−0.233285
$22$	−62.7836	−0.608433
$23$	−175.600	−1.59196	−0.795979	−	0.605324i	$-0.793044\pi$
$23$	−175.600	−1.59196	−0.795979	+	0.605324i	$0.793044\pi$
$24$	−69.7750	−0.593449
$25$	254.600	2.03680
$26$	35.6415	0.268842
$27$	−27.0000	−0.192450
$28$	−3.61680	−0.0244111
$29$	291.800	1.86848	0.934239	−	0.356648i	$-0.116080\pi$
$29$	291.800	1.86848	0.934239	+	0.356648i	$0.116080\pi$
$30$	160.250	0.975249
$31$	117.283	0.679505	0.339753	−	0.940515i	$-0.389657\pi$
$31$	117.283	0.679505	0.339753	+	0.940515i	$0.389657\pi$
$32$	−21.8418	−0.120660
$33$	−68.6997	−0.362396
$34$	−183.783	−0.927013
$35$	145.800	0.704133
$36$	−4.34983	−0.0201381
$37$	−154.766	−0.687661	−0.343830	−	0.939032i	$-0.611724\pi$
$37$	−154.766	−0.687661	−0.343830	+	0.939032i	$0.611724\pi$
$38$	−45.2831	−0.193313
$39$	39.0000	0.160128
$40$	453.150	1.79123
$41$	−251.716	−0.958815	−0.479407	−	0.877592i	$-0.659148\pi$
$41$	−251.716	−0.958815	−0.479407	+	0.877592i	$0.659148\pi$
$42$	61.5501	0.226128
$43$	−502.566	−1.78234	−0.891170	−	0.453669i	$-0.850115\pi$
$43$	−502.566	−1.78234	−0.891170	+	0.453669i	$0.850115\pi$
$44$	−11.0679	−0.0379214
$45$	175.350	0.580880
$46$	481.434	1.54312
$47$	−281.733	−0.874361	−0.437181	−	0.899374i	$-0.644023\pi$
$47$	−281.733	−0.874361	−0.437181	+	0.899374i	$0.644023\pi$
$48$	179.700	0.540363
$49$	−287.000	−0.836735
$50$	−698.025	−1.97431
$51$	−201.100	−0.552150
$52$	6.28309	0.0167559
$53$	366.999	0.951154	0.475577	−	0.879674i	$-0.342239\pi$
$53$	366.999	0.951154	0.475577	+	0.879674i	$0.342239\pi$
$54$	74.0247	0.186546
$55$	446.166	1.09384
$56$	174.049	0.415328
$57$	−49.5501	−0.115141
$58$	−800.015	−1.81116
$59$	−79.6663	−0.175791	−0.0878955	−	0.996130i	$-0.528014\pi$
$59$	−79.6663	−0.175791	−0.0878955	+	0.996130i	$0.528014\pi$
$60$	28.2497	0.0607837
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	−321.550	−0.658660
$63$	67.3498	0.134687
$64$	539.082	1.05289
$65$	−253.283	−0.483322
$66$	188.351	0.351279
$67$	400.082	0.729519	0.364759	−	0.931102i	$-0.381151\pi$
$67$	400.082	0.729519	0.364759	+	0.931102i	$0.381151\pi$
$68$	−32.3982	−0.0577774
$69$	526.799	0.919117
$70$	−399.733	−0.682532
$71$	528.299	0.883065	0.441532	−	0.897245i	$-0.354435\pi$
$71$	528.299	0.883065	0.441532	+	0.897245i	$0.354435\pi$
$72$	209.325	0.342628
$73$	−734.366	−1.17741	−0.588706	−	0.808347i	$-0.700362\pi$
$73$	−734.366	−1.17741	−0.588706	+	0.808347i	$0.700362\pi$
$74$	424.316	0.666565
$75$	−763.799	−1.17594
$76$	−7.98276	−0.0120485
$77$	171.367	0.253625
$78$	−106.925	−0.155216
$79$	113.266	0.161309	0.0806545	−	0.996742i	$-0.474299\pi$
$79$	113.266	0.161309	0.0806545	+	0.996742i	$0.474299\pi$
$80$	−1167.05	−1.63100
$81$	81.0000	0.111111
$82$	690.118	0.929400
$83$	−933.466	−1.23447	−0.617236	−	0.786778i	$-0.711748\pi$
$83$	−933.466	−1.23447	−0.617236	+	0.786778i	$0.711748\pi$
$84$	10.8504	0.0140937
$85$	1306.03	1.66658
$86$	1377.86	1.72766
$87$	−875.399	−1.07877
$88$	532.613	0.645191
$89$	1190.91	1.41839	0.709195	−	0.705012i	$-0.249059\pi$
$89$	1190.91	1.41839	0.709195	+	0.705012i	$0.249059\pi$
$90$	−480.749	−0.563060
$91$	−97.2831	−0.112066
$92$	84.8699	0.0961771
$93$	−351.849	−0.392313
$94$	772.415	0.847538
$95$	321.800	0.347536
$96$	65.5253	0.0696630
$97$	557.165	0.583211	0.291606	−	0.956539i	$-0.405811\pi$
$97$	557.165	0.583211	0.291606	+	0.956539i	$0.405811\pi$
$98$	786.856	0.811066
$99$	206.099	0.209230
$100$	−123.052	−0.123052
$101$	−286.766	−0.282518	−0.141259	−	0.989973i	$-0.545115\pi$
$101$	−286.766	−0.282518	−0.141259	+	0.989973i	$0.545115\pi$
$102$	551.348	0.535211
$103$	−1911.36	−1.82847	−0.914234	−	0.405187i	$-0.867206\pi$
$103$	−1911.36	−1.82847	−0.914234	+	0.405187i	$0.867206\pi$
$104$	−302.358	−0.285084
$105$	−437.399	−0.406531
$106$	−1006.19	−0.921975
$107$	834.334	0.753814	0.376907	−	0.926251i	$-0.376988\pi$
$107$	834.334	0.753814	0.376907	+	0.926251i	$0.376988\pi$
$108$	13.0495	0.0116267
$109$	−1077.66	−0.946986	−0.473493	−	0.880798i	$-0.657007\pi$
$109$	−1077.66	−0.946986	−0.473493	+	0.880798i	$0.657007\pi$
$110$	−1223.23	−1.06028
$111$	464.299	0.397021
$112$	−448.250	−0.378175
$113$	−166.065	−0.138248	−0.0691241	−	0.997608i	$-0.522020\pi$
$113$	−166.065	−0.138248	−0.0691241	+	0.997608i	$0.522020\pi$
$114$	135.849	0.111609
$115$	−3421.26	−2.77421
$116$	−141.031	−0.112883
$117$	−117.000	−0.0924500
$118$	218.418	0.170398
$119$	501.632	0.386424
$120$	−1359.45	−1.03417
$121$	−806.595	−0.606007
$122$	534.254	0.396468
$123$	755.147	0.553572
$124$	−56.6847	−0.0410519
$125$	2525.03	1.80676
$126$	−184.650	−0.130555
$127$	1296.16	0.905637	0.452819	−	0.891603i	$-0.350419\pi$
$127$	1296.16	0.905637	0.452819	+	0.891603i	$0.350419\pi$
$128$	−1303.24	−0.899934
$129$	1507.70	1.02903
$130$	694.415	0.468494
$131$	−197.201	−0.131523	−0.0657617	−	0.997835i	$-0.520948\pi$
$131$	−197.201	−0.131523	−0.0657617	+	0.997835i	$0.520948\pi$
$132$	33.2036	0.0218939
$133$	123.600	0.0805823
$134$	−1096.89	−0.707139
$135$	−526.049	−0.335371
$136$	1559.09	0.983018
$137$	−546.915	−0.341066	−0.170533	−	0.985352i	$-0.554549\pi$
$137$	−546.915	−0.341066	−0.170533	+	0.985352i	$0.554549\pi$
$138$	−1444.30	−0.890921
$139$	609.666	0.372023	0.186012	−	0.982548i	$-0.440444\pi$
$139$	609.666	0.372023	0.186012	+	0.982548i	$0.440444\pi$
$140$	−70.4672	−0.0425397
$141$	845.199	0.504813
$142$	−1448.42	−0.855974
$143$	−297.699	−0.174090
$144$	−539.099	−0.311979
$145$	5685.23	3.25609
$146$	2013.38	1.14129
$147$	861.000	0.483089
$148$	74.8009	0.0415446
$149$	−2165.08	−1.19040	−0.595202	−	0.803576i	$-0.702928\pi$
$149$	−2165.08	−1.19040	−0.595202	+	0.803576i	$0.702928\pi$
$150$	2094.07	1.13987
$151$	−846.549	−0.456233	−0.228116	−	0.973634i	$-0.573257\pi$
$151$	−846.549	−0.456233	−0.228116	+	0.973634i	$0.573257\pi$
$152$	384.151	0.204992
$153$	603.300	0.318784
$154$	−469.830	−0.245844
$155$	2285.06	1.18413
$156$	−18.8493	−0.00967404
$157$	1653.60	0.840581	0.420291	−	0.907390i	$-0.361928\pi$
$157$	1653.60	0.840581	0.420291	+	0.907390i	$0.361928\pi$
$158$	−310.536	−0.156360
$159$	−1101.00	−0.549149
$160$	−425.550	−0.210267
$161$	−1314.07	−0.643248
$162$	−222.074	−0.107702
$163$	−2866.51	−1.37744	−0.688720	−	0.725027i	$-0.741827\pi$
$163$	−2866.51	−1.37744	−0.688720	+	0.725027i	$0.741827\pi$
$164$	121.658	0.0579262
$165$	−1338.50	−0.631526
$166$	2559.24	1.19660
$167$	729.066	0.337825	0.168913	−	0.985631i	$-0.445974\pi$
$167$	729.066	0.337825	0.168913	+	0.985631i	$0.445974\pi$
$168$	−522.148	−0.239789
$169$	169.000	0.0769231
$170$	−3580.69	−1.61545
$171$	148.650	0.0664770
$172$	242.898	0.107679
$173$	−3834.83	−1.68530	−0.842650	−	0.538462i	$-0.819005\pi$
$173$	−3834.83	−1.68530	−0.842650	+	0.538462i	$0.819005\pi$
$174$	2400.05	1.04567
$175$	1905.25	0.822990
$176$	−1371.70	−0.587476
$177$	238.999	0.101493
$178$	−3265.08	−1.37488
$179$	−283.862	−0.118530	−0.0592649	−	0.998242i	$-0.518876\pi$
$179$	−283.862	−0.118530	−0.0592649	+	0.998242i	$0.518876\pi$
$180$	−84.7492	−0.0350935
$181$	2363.60	0.970634	0.485317	−	0.874338i	$-0.338704\pi$
$181$	2363.60	0.970634	0.485317	+	0.874338i	$0.338704\pi$
$182$	266.717	0.108628
$183$	584.596	0.236145
$184$	−4084.15	−1.63635
$185$	−3015.36	−1.19835
$186$	964.650	0.380277
$187$	1535.06	0.600291
$188$	136.166	0.0528240
$189$	−202.049	−0.0777616
$190$	−882.265	−0.336875
$191$	2514.26	0.952491	0.476246	−	0.879312i	$-0.341997\pi$
$191$	2514.26	0.952491	0.476246	+	0.879312i	$0.341997\pi$
$192$	−1617.25	−0.607889
$193$	2420.73	0.902839	0.451420	−	0.892312i	$-0.350918\pi$
$193$	2420.73	0.902839	0.451420	+	0.892312i	$0.350918\pi$
$194$	−1527.55	−0.565320
$195$	759.849	0.279046
$196$	138.711	0.0505508
$197$	−4633.65	−1.67581	−0.837903	−	0.545819i	$-0.816219\pi$
$197$	−4633.65	−1.67581	−0.837903	+	0.545819i	$0.816219\pi$
$198$	−565.053	−0.202811
$199$	3054.17	1.08796	0.543980	−	0.839098i	$-0.316917\pi$
$199$	3054.17	1.08796	0.543980	+	0.839098i	$0.316917\pi$
$200$	5921.56	2.09359
$201$	−1200.25	−0.421188
$202$	786.215	0.273851
$203$	2183.63	0.754979
$204$	97.1947	0.0333578
$205$	−4904.26	−1.67087
$206$	5240.30	1.77237
$207$	−1580.40	−0.530653
$208$	778.699	0.259582
$209$	378.230	0.125181
$210$	1199.20	0.394060
$211$	−4031.60	−1.31539	−0.657694	−	0.753285i	$-0.728468\pi$
$211$	−4031.60	−1.31539	−0.657694	+	0.753285i	$0.728468\pi$
$212$	−177.376	−0.0574634
$213$	−1584.90	−0.509838
$214$	−2287.46	−0.730689
$215$	−9791.66	−3.10598
$216$	−627.975	−0.197816
$217$	877.666	0.274562
$218$	2954.59	0.917935
$219$	2203.10	0.679779
$220$	−215.638	−0.0660834
$221$	−871.434	−0.265244
$222$	−1272.95	−0.384841
$223$	3784.95	1.13659	0.568294	−	0.822826i	$-0.307604\pi$
$223$	3784.95	1.13659	0.568294	+	0.822826i	$0.307604\pi$
$224$	−163.449	−0.0487539
$225$	2291.40	0.678932
$226$	455.292	0.134007
$227$	2013.83	0.588821	0.294411	−	0.955679i	$-0.404877\pi$
$227$	2013.83	0.588821	0.294411	+	0.955679i	$0.404877\pi$
$228$	23.9483	0.00695620
$229$	−3050.73	−0.880340	−0.440170	−	0.897915i	$-0.645082\pi$
$229$	−3050.73	−0.880340	−0.440170	+	0.897915i	$0.645082\pi$
$230$	9379.93	2.68910
$231$	−514.101	−0.146430
$232$	6786.78	1.92058
$233$	5587.49	1.57103	0.785513	−	0.618846i	$-0.212399\pi$
$233$	5587.49	1.57103	0.785513	+	0.618846i	$0.212399\pi$
$234$	320.774	0.0896139
$235$	−5489.09	−1.52370
$236$	38.5039	0.0106203
$237$	−339.798	−0.0931317
$238$	−1375.30	−0.374570
$239$	−1335.69	−0.361501	−0.180750	−	0.983529i	$-0.557853\pi$
$239$	−1335.69	−0.361501	−0.180750	+	0.983529i	$0.557853\pi$
$240$	3501.15	0.941658
$241$	−571.558	−0.152769	−0.0763845	−	0.997078i	$-0.524338\pi$
$241$	−571.558	−0.152769	−0.0763845	+	0.997078i	$0.524338\pi$
$242$	2211.41	0.587416
$243$	−243.000	−0.0641500
$244$	94.1813	0.0247104
$245$	−5591.71	−1.45813
$246$	−2070.36	−0.536590
$247$	−214.717	−0.0553122
$248$	2727.81	0.698452
$249$	2800.40	0.712723
$250$	−6922.76	−1.75134
$251$	4088.60	1.02817	0.514084	−	0.857740i	$-0.328132\pi$
$251$	4088.60	1.02817	0.514084	+	0.857740i	$0.328132\pi$
$252$	−32.5512	−0.00813703
$253$	−4021.21	−0.999254
$254$	−3553.64	−0.877854
$255$	−3918.10	−0.962199
$256$	−739.607	−0.180568
$257$	3050.23	0.740342	0.370171	−	0.928964i	$-0.379299\pi$
$257$	3050.23	0.740342	0.370171	+	0.928964i	$0.379299\pi$
$258$	−4133.59	−0.997466
$259$	−1158.17	−0.277857
$260$	122.415	0.0291996
$261$	2626.20	0.622826
$262$	540.659	0.127489
$263$	5770.99	1.35306	0.676530	−	0.736415i	$-0.263483\pi$
$263$	5770.99	1.35306	0.676530	+	0.736415i	$0.263483\pi$
$264$	−1597.84	−0.372501
$265$	7150.35	1.65752
$266$	−338.868	−0.0781102
$267$	−3572.74	−0.818908
$268$	−193.365	−0.0440734
$269$	−2079.40	−0.471314	−0.235657	−	0.971836i	$-0.575724\pi$
$269$	−2079.40	−0.471314	−0.235657	+	0.971836i	$0.575724\pi$
$270$	1442.25	0.325083
$271$	6012.00	1.34761	0.673807	−	0.738908i	$-0.264658\pi$
$271$	6012.00	1.34761	0.673807	+	0.738908i	$0.264658\pi$
$272$	−4015.29	−0.895084
$273$	291.849	0.0647015
$274$	1499.45	0.330603
$275$	5830.30	1.27847
$276$	−254.610	−0.0555279
$277$	−735.201	−0.159473	−0.0797364	−	0.996816i	$-0.525408\pi$
$277$	−735.201	−0.159473	−0.0797364	+	0.996816i	$0.525408\pi$
$278$	−1671.50	−0.360610
$279$	1055.55	0.226502
$280$	3391.06	0.723767
$281$	−1902.92	−0.403981	−0.201990	−	0.979387i	$-0.564741\pi$
$281$	−1902.92	−0.403981	−0.201990	+	0.979387i	$0.564741\pi$
$282$	−2317.25	−0.489326
$283$	2125.71	0.446502	0.223251	−	0.974761i	$-0.428333\pi$
$283$	2125.71	0.446502	0.223251	+	0.974761i	$0.428333\pi$
$284$	−255.335	−0.0533498
$285$	−965.399	−0.200650
$286$	816.187	0.168749
$287$	−1883.67	−0.387420
$288$	−196.576	−0.0402200
$289$	−419.527	−0.0853913
$290$	−15586.9	−3.15620
$291$	−1671.49	−0.336717
$292$	354.930	0.0711325
$293$	−1641.03	−0.327200	−0.163600	−	0.986527i	$-0.552311\pi$
$293$	−1641.03	−0.327200	−0.163600	+	0.986527i	$0.552311\pi$
$294$	−2360.57	−0.468269
$295$	−1552.16	−0.306341
$296$	−3599.61	−0.706835
$297$	−618.297	−0.120799
$298$	5935.91	1.15389
$299$	2282.79	0.441530
$300$	369.155	0.0710439
$301$	−3760.86	−0.720174
$302$	2320.95	0.442237
$303$	860.299	0.163112
$304$	−989.348	−0.186655
$305$	−3796.62	−0.712767
$306$	−1654.04	−0.309004
$307$	−3373.27	−0.627111	−0.313555	−	0.949570i	$-0.601520\pi$
$307$	−3373.27	−0.627111	−0.313555	+	0.949570i	$0.601520\pi$
$308$	−82.8242	−0.0153226
$309$	5734.09	1.05567
$310$	−6264.86	−1.14781
$311$	−868.525	−0.158359	−0.0791793	−	0.996860i	$-0.525230\pi$
$311$	−868.525	−0.158359	−0.0791793	+	0.996860i	$0.525230\pi$
$312$	907.075	0.164593
$313$	−4343.19	−0.784319	−0.392159	−	0.919897i	$-0.628272\pi$
$313$	−4343.19	−0.784319	−0.392159	+	0.919897i	$0.628272\pi$
$314$	−4533.59	−0.814794
$315$	1312.20	0.234711
$316$	−54.7431	−0.00974537
$317$	−3277.65	−0.580730	−0.290365	−	0.956916i	$-0.593777\pi$
$317$	−3277.65	−0.580730	−0.290365	+	0.956916i	$0.593777\pi$
$318$	3018.56	0.532302
$319$	6682.18	1.17282
$320$	10503.1	1.83482
$321$	−2503.00	−0.435215
$322$	3602.72	0.623515
$323$	1107.17	0.190726
$324$	−39.1485	−0.00671271
$325$	−3309.79	−0.564906
$326$	7859.00	1.33518
$327$	3232.99	0.546743
$328$	−5854.49	−0.985550
$329$	−2108.30	−0.353295
$330$	3669.70	0.612153
$331$	5589.62	0.928197	0.464099	−	0.885784i	$-0.346378\pi$
$331$	5589.62	0.928197	0.464099	+	0.885784i	$0.346378\pi$
$332$	451.158	0.0745798
$333$	−1392.90	−0.229220
$334$	−1998.85	−0.327461
$335$	7794.92	1.27129
$336$	1344.75	0.218340
$337$	901.544	0.145728	0.0728638	−	0.997342i	$-0.476786\pi$
$337$	901.544	0.145728	0.0728638	+	0.997342i	$0.476786\pi$
$338$	−463.340	−0.0745633
$339$	498.194	0.0798176
$340$	−631.225	−0.100685
$341$	2685.77	0.426518
$342$	−407.548	−0.0644376
$343$	−4714.49	−0.742153
$344$	−11688.9	−1.83204
$345$	10263.8	1.60169
$346$	10513.8	1.63360
$347$	−812.318	−0.125670	−0.0628350	−	0.998024i	$-0.520014\pi$
$347$	−812.318	−0.125670	−0.0628350	+	0.998024i	$0.520014\pi$
$348$	423.093	0.0651730
$349$	4437.96	0.680683	0.340342	−	0.940302i	$-0.389457\pi$
$349$	4437.96	0.680683	0.340342	+	0.940302i	$0.389457\pi$
$350$	−5223.54	−0.797743
$351$	351.000	0.0533761
$352$	−500.174	−0.0757368
$353$	7115.35	1.07284	0.536419	−	0.843952i	$-0.319777\pi$
$353$	7115.35	1.07284	0.536419	+	0.843952i	$0.319777\pi$
$354$	−655.253	−0.0983794
$355$	10293.0	1.53886
$356$	−575.587	−0.0856911
$357$	−1504.90	−0.223102
$358$	778.253	0.114894
$359$	4693.98	0.690081	0.345040	−	0.938588i	$-0.387865\pi$
$359$	4693.98	0.690081	0.345040	+	0.938588i	$0.387865\pi$
$360$	4078.35	0.597077
$361$	−6586.20	−0.960227
$362$	−6480.17	−0.940857
$363$	2419.79	0.349878
$364$	47.0184	0.00677042
$365$	−14307.9	−2.05181
$366$	−1602.76	−0.228901
$367$	9243.98	1.31480	0.657400	−	0.753542i	$-0.271656\pi$
$367$	9243.98	1.31480	0.657400	+	0.753542i	$0.271656\pi$
$368$	10518.4	1.48997
$369$	−2265.44	−0.319605
$370$	8267.09	1.16158
$371$	2746.37	0.384324
$372$	170.054	0.0237013
$373$	−4311.99	−0.598569	−0.299285	−	0.954164i	$-0.596748\pi$
$373$	−4311.99	−0.598569	−0.299285	+	0.954164i	$0.596748\pi$
$374$	−4208.60	−0.581876
$375$	−7575.09	−1.04314
$376$	−6552.64	−0.898741
$377$	−3793.40	−0.518223
$378$	553.951	0.0753760
$379$	−2382.73	−0.322936	−0.161468	−	0.986878i	$-0.551623\pi$
$379$	−2382.73	−0.322936	−0.161468	+	0.986878i	$0.551623\pi$
$380$	−155.531	−0.0209962
$381$	−3888.49	−0.522870
$382$	−6893.25	−0.923271
$383$	4845.81	0.646499	0.323250	−	0.946314i	$-0.395225\pi$
$383$	4845.81	0.646499	0.323250	+	0.946314i	$0.395225\pi$
$384$	3909.73	0.519577
$385$	3338.80	0.441976
$386$	−6636.81	−0.875142
$387$	−4523.10	−0.594113
$388$	−269.286	−0.0352343
$389$	9561.50	1.24624	0.623120	−	0.782127i	$-0.285865\pi$
$389$	9561.50	1.24624	0.623120	+	0.782127i	$0.285865\pi$
$390$	−2083.25	−0.270485
$391$	−11771.0	−1.52247
$392$	−6675.14	−0.860066
$393$	591.604	0.0759350
$394$	12703.9	1.62440
$395$	2206.79	0.281103
$396$	−99.6107	−0.0126405
$397$	−7440.11	−0.940575	−0.470287	−	0.882513i	$-0.655850\pi$
$397$	−7440.11	−0.940575	−0.470287	+	0.882513i	$0.655850\pi$
$398$	−8373.48	−1.05458
$399$	−370.799	−0.0465242
$400$	−15250.5	−1.90631
$401$	−8687.80	−1.08192	−0.540958	−	0.841050i	$-0.681938\pi$
$401$	−8687.80	−1.08192	−0.540958	+	0.841050i	$0.681938\pi$
$402$	3290.66	0.408267
$403$	−1524.68	−0.188461
$404$	138.598	0.0170681
$405$	1578.15	0.193627
$406$	−5986.76	−0.731818
$407$	−3544.13	−0.431637
$408$	−4677.26	−0.567546
$409$	2556.10	0.309024	0.154512	−	0.987991i	$-0.450619\pi$
$409$	2556.10	0.309024	0.154512	+	0.987991i	$0.450619\pi$
$410$	13445.8	1.61961
$411$	1640.74	0.196915
$412$	923.790	0.110466
$413$	−596.168	−0.0710303
$414$	4332.90	0.514374
$415$	−18187.0	−2.15124
$416$	283.943	0.0334650
$417$	−1829.00	−0.214788
$418$	−1036.98	−0.121340
$419$	−3347.46	−0.390296	−0.195148	−	0.980774i	$-0.562519\pi$
$419$	−3347.46	−0.390296	−0.195148	+	0.980774i	$0.562519\pi$
$420$	211.402	0.0245603
$421$	−1854.48	−0.214684	−0.107342	−	0.994222i	$-0.534234\pi$
$421$	−1854.48	−0.214684	−0.107342	+	0.994222i	$0.534234\pi$
$422$	11053.3	1.27503
$423$	−2535.60	−0.291454
$424$	8535.79	0.977675
$425$	17066.7	1.94789
$426$	4345.25	0.494197
$427$	−1458.24	−0.165267
$428$	−403.246	−0.0455412
$429$	893.096	0.100511
$430$	26845.4	3.01069
$431$	−14043.1	−1.56945	−0.784725	−	0.619844i	$-0.787196\pi$
$431$	−14043.1	−1.56945	−0.784725	+	0.619844i	$0.787196\pi$
$432$	1617.30	0.180121
$433$	3086.47	0.342555	0.171278	−	0.985223i	$-0.445210\pi$
$433$	3086.47	0.342555	0.171278	+	0.985223i	$0.445210\pi$
$434$	−2406.26	−0.266139
$435$	−17055.7	−1.87990
$436$	520.851	0.0572116
$437$	−2900.32	−0.317486
$438$	−6040.14	−0.658925
$439$	2837.68	0.308508	0.154254	−	0.988031i	$-0.450703\pi$
$439$	2837.68	0.308508	0.154254	+	0.988031i	$0.450703\pi$
$440$	10377.1	1.12434
$441$	−2583.00	−0.278912
$442$	2389.17	0.257107
$443$	18309.4	1.96367	0.981834	−	0.189744i	$-0.0607658\pi$
$443$	18309.4	1.96367	0.981834	+	0.189744i	$0.0607658\pi$
$444$	−224.403	−0.0239858
$445$	23203.0	2.47174
$446$	−10377.0	−1.10172
$447$	6495.24	0.687281
$448$	4034.12	0.425433
$449$	13861.2	1.45690	0.728451	−	0.685098i	$-0.240241\pi$
$449$	13861.2	1.45690	0.728451	+	0.685098i	$0.240241\pi$
$450$	−6282.22	−0.658104
$451$	−5764.26	−0.601837
$452$	80.2614	0.00835217
$453$	2539.65	0.263406
$454$	−5521.23	−0.570758
$455$	−1895.40	−0.195291
$456$	−1152.45	−0.118352
$457$	−8990.36	−0.920243	−0.460122	−	0.887856i	$-0.652194\pi$
$457$	−8990.36	−0.920243	−0.460122	+	0.887856i	$0.652194\pi$
$458$	8364.05	0.853333
$459$	−1809.90	−0.184050
$460$	1653.55	0.167602
$461$	−3406.90	−0.344198	−0.172099	−	0.985080i	$-0.555055\pi$
$461$	−3406.90	−0.344198	−0.172099	+	0.985080i	$0.555055\pi$
$462$	1409.49	0.141938
$463$	−7498.45	−0.752662	−0.376331	−	0.926485i	$-0.622814\pi$
$463$	−7498.45	−0.752662	−0.376331	+	0.926485i	$0.622814\pi$
$464$	−17478.8	−1.74878
$465$	−6855.19	−0.683660
$466$	−15319.0	−1.52283
$467$	7711.38	0.764112	0.382056	−	0.924139i	$-0.375216\pi$
$467$	7711.38	0.764112	0.382056	+	0.924139i	$0.375216\pi$
$468$	56.5478	0.00558531
$469$	2993.94	0.294770
$470$	15049.2	1.47695
$471$	−4960.79	−0.485310
$472$	−1852.91	−0.180693
$473$	−11508.7	−1.11875
$474$	931.608	0.0902747
$475$	4205.14	0.406200
$476$	−242.446	−0.0233456
$477$	3302.99	0.317051
$478$	3662.01	0.350411
$479$	−9439.82	−0.900451	−0.450226	−	0.892915i	$-0.648656\pi$
$479$	−9439.82	−0.900451	−0.450226	+	0.892915i	$0.648656\pi$
$480$	1276.65	0.121398
$481$	2011.96	0.190723
$482$	1567.02	0.148082
$483$	3942.20	0.371380
$484$	389.839	0.0366115
$485$	10855.4	1.01633
$486$	666.223	0.0621821
$487$	−6156.20	−0.572821	−0.286411	−	0.958107i	$-0.592462\pi$
$487$	−6156.20	−0.572821	−0.286411	+	0.958107i	$0.592462\pi$
$488$	−4532.25	−0.420420
$489$	8599.54	0.795265
$490$	15330.6	1.41340
$491$	3842.74	0.353198	0.176599	−	0.984283i	$-0.443490\pi$
$491$	3842.74	0.353198	0.176599	+	0.984283i	$0.443490\pi$
$492$	−364.974	−0.0334437
$493$	19560.3	1.78692
$494$	588.680	0.0536153
$495$	4015.49	0.364612
$496$	−7025.24	−0.635973
$497$	3953.43	0.356812
$498$	−7677.73	−0.690858
$499$	−12842.4	−1.15211	−0.576056	−	0.817410i	$-0.695409\pi$
$499$	−12842.4	−1.15211	−0.576056	+	0.817410i	$0.695409\pi$
$500$	−1220.38	−0.109154
$501$	−2187.20	−0.195043
$502$	−11209.5	−0.996627
$503$	8580.11	0.760573	0.380287	−	0.924869i	$-0.375825\pi$
$503$	8580.11	0.760573	0.380287	+	0.924869i	$0.375825\pi$
$504$	1566.45	0.138443
$505$	−5587.16	−0.492327
$506$	11024.8	0.968599
$507$	−507.000	−0.0444116
$508$	−626.455	−0.0547135
$509$	−43.5957	−0.00379635	−0.00189818	−	0.999998i	$-0.500604\pi$
$509$	−43.5957	−0.00379635	−0.00189818	+	0.999998i	$0.500604\pi$
$510$	10742.1	0.932681
$511$	−5495.49	−0.475746
$512$	12453.7	1.07496
$513$	−445.951	−0.0383805
$514$	−8362.68	−0.717630
$515$	−37239.7	−3.18636
$516$	−728.693	−0.0621684
$517$	−6451.66	−0.548827
$518$	3175.29	0.269333
$519$	11504.5	0.973008
$520$	−5890.94	−0.496798
$521$	11368.1	0.955939	0.477969	−	0.878377i	$-0.341373\pi$
$521$	11368.1	0.955939	0.477969	+	0.878377i	$0.341373\pi$
$522$	−7200.14	−0.603719
$523$	−5229.53	−0.437230	−0.218615	−	0.975811i	$-0.570154\pi$
$523$	−5229.53	−0.437230	−0.218615	+	0.975811i	$0.570154\pi$
$524$	95.3103	0.00794590
$525$	−5715.75	−0.475154
$526$	−15822.1	−1.31155
$527$	7861.88	0.649846
$528$	4115.10	0.339180
$529$	18668.2	1.53433
$530$	−19603.8	−1.60667
$531$	−716.997	−0.0585970
$532$	−59.7375	−0.00486832
$533$	3272.31	0.265927
$534$	9795.24	0.793786
$535$	16255.6	1.31363
$536$	9305.24	0.749860
$537$	851.586	0.0684333
$538$	5701.01	0.456855
$539$	−6572.27	−0.525209
$540$	254.247	0.0202612
$541$	−6567.99	−0.521959	−0.260980	−	0.965344i	$-0.584046\pi$
$541$	−6567.99	−0.521959	−0.260980	+	0.965344i	$0.584046\pi$
$542$	−16482.9	−1.30627
$543$	−7090.79	−0.560396
$544$	−1464.13	−0.115393
$545$	−20996.5	−1.65026
$546$	−800.151	−0.0627166
$547$	−13675.7	−1.06897	−0.534487	−	0.845177i	$-0.679495\pi$
$547$	−13675.7	−1.06897	−0.534487	+	0.845177i	$0.679495\pi$
$548$	264.332	0.0206053
$549$	−1753.79	−0.136339
$550$	−15984.7	−1.23925
$551$	4819.57	0.372632
$552$	12252.5	0.944745
$553$	847.604	0.0651786
$554$	2015.67	0.154581
$555$	9046.09	0.691865
$556$	−294.661	−0.0224755
$557$	4527.96	0.344445	0.172222	−	0.985058i	$-0.444905\pi$
$557$	4527.96	0.344445	0.172222	+	0.985058i	$0.444905\pi$
$558$	−2893.95	−0.219553
$559$	6533.36	0.494332
$560$	−8733.39	−0.659023
$561$	−4605.17	−0.346578
$562$	5217.15	0.391588
$563$	18441.8	1.38051	0.690256	−	0.723566i	$-0.257498\pi$
$563$	18441.8	1.38051	0.690256	+	0.723566i	$0.257498\pi$
$564$	−408.497	−0.0304979
$565$	−3235.49	−0.240917
$566$	−5827.96	−0.432804
$567$	606.148	0.0448957
$568$	12287.4	0.907687
$569$	−13553.5	−0.998578	−0.499289	−	0.866436i	$-0.666405\pi$
$569$	−13553.5	−0.998578	−0.499289	+	0.866436i	$0.666405\pi$
$570$	2646.79	0.194495
$571$	14815.5	1.08583	0.542915	−	0.839788i	$-0.317321\pi$
$571$	14815.5	1.08583	0.542915	+	0.839788i	$0.317321\pi$
$572$	143.882	0.0105175
$573$	−7542.79	−0.549921
$574$	5164.37	0.375534
$575$	−44707.6	−3.24249
$576$	4851.74	0.350965
$577$	21596.2	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$577$	21596.2	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$578$	1150.20	0.0827716
$579$	−7262.19	−0.521254
$580$	−2747.75	−0.196714
$581$	−6985.42	−0.498802
$582$	4582.66	0.326387
$583$	8404.23	0.597029
$584$	−17080.1	−1.21024
$585$	−2279.55	−0.161107
$586$	4499.13	0.317163
$587$	−918.801	−0.0646047	−0.0323024	−	0.999478i	$-0.510284\pi$
$587$	−918.801	−0.0646047	−0.0323024	+	0.999478i	$0.510284\pi$
$588$	−416.134	−0.0291855
$589$	1937.13	0.135514
$590$	4255.50	0.296943
$591$	13900.9	0.967527
$592$	9270.49	0.643606
$593$	19816.0	1.37226	0.686128	−	0.727481i	$-0.259309\pi$
$593$	19816.0	1.37226	0.686128	+	0.727481i	$0.259309\pi$
$594$	1695.16	0.117093
$595$	9773.45	0.673399
$596$	1046.42	0.0719175
$597$	−9162.50	−0.628134
$598$	−6258.64	−0.427985
$599$	−5141.86	−0.350736	−0.175368	−	0.984503i	$-0.556111\pi$
$599$	−5141.86	−0.350736	−0.175368	+	0.984503i	$0.556111\pi$
$600$	−17764.7	−1.20873
$601$	12380.9	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$601$	12380.9	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$602$	10311.0	0.698081
$603$	3600.74	0.243173
$604$	409.150	0.0275630
$605$	−15715.1	−1.05605
$606$	−2358.65	−0.158108
$607$	−23717.0	−1.58590	−0.792951	−	0.609286i	$-0.791456\pi$
$607$	−23717.0	−1.58590	−0.792951	+	0.609286i	$0.791456\pi$
$608$	−360.754	−0.0240633
$609$	−6550.89	−0.435887
$610$	10409.0	0.690901
$611$	3662.53	0.242504
$612$	−291.584	−0.0192591
$613$	−26157.1	−1.72345	−0.861726	−	0.507373i	$-0.830617\pi$
$613$	−26157.1	−1.72345	−0.861726	+	0.507373i	$0.830617\pi$
$614$	9248.36	0.607872
$615$	14712.8	0.964677
$616$	3985.71	0.260696
$617$	23613.9	1.54077	0.770387	−	0.637576i	$-0.220063\pi$
$617$	23613.9	1.54077	0.770387	+	0.637576i	$0.220063\pi$
$618$	−15720.9	−1.02328
$619$	23345.4	1.51588	0.757940	−	0.652324i	$-0.226206\pi$
$619$	23345.4	1.51588	0.757940	+	0.652324i	$0.226206\pi$
$620$	−1104.40	−0.0715387
$621$	4741.19	0.306372
$622$	2381.20	0.153501
$623$	8911.99	0.573116
$624$	−2336.10	−0.149870
$625$	17371.0	1.11174
$626$	11907.5	0.760258
$627$	−1134.69	−0.0722730
$628$	−799.207	−0.0507832
$629$	−10374.5	−0.657645
$630$	−3597.60	−0.227511
$631$	15245.7	0.961841	0.480921	−	0.876764i	$-0.340302\pi$
$631$	15245.7	0.961841	0.480921	+	0.876764i	$0.340302\pi$
$632$	2634.38	0.165807
$633$	12094.8	0.759439
$634$	8986.20	0.562914
$635$	25253.6	1.57820
$636$	532.128	0.0331765
$637$	3731.00	0.232068
$638$	−18320.3	−1.13684
$639$	4754.69	0.294355
$640$	−25391.5	−1.56826
$641$	10192.7	0.628063	0.314032	−	0.949413i	$-0.398320\pi$
$641$	10192.7	0.628063	0.314032	+	0.949413i	$0.398320\pi$
$642$	6862.37	0.421863
$643$	−5506.31	−0.337710	−0.168855	−	0.985641i	$-0.554007\pi$
$643$	−5506.31	−0.337710	−0.168855	+	0.985641i	$0.554007\pi$
$644$	635.108	0.0388614
$645$	29375.0	1.79324
$646$	−3035.48	−0.184875
$647$	−13297.5	−0.808005	−0.404003	−	0.914758i	$-0.632381\pi$
$647$	−13297.5	−0.808005	−0.404003	+	0.914758i	$0.632381\pi$
$648$	1883.93	0.114209
$649$	−1824.35	−0.110342
$650$	9074.32	0.547576
$651$	−2633.00	−0.158518
$652$	1385.43	0.0832171
$653$	−12440.2	−0.745519	−0.372760	−	0.927928i	$-0.621588\pi$
$653$	−12440.2	−0.745519	−0.372760	+	0.927928i	$0.621588\pi$
$654$	−8863.76	−0.529970
$655$	−3842.14	−0.229198
$656$	15077.7	0.897389
$657$	−6609.29	−0.392470
$658$	5780.23	0.342457
$659$	−9562.87	−0.565276	−0.282638	−	0.959227i	$-0.591209\pi$
$659$	−9562.87	−0.565276	−0.282638	+	0.959227i	$0.591209\pi$
$660$	646.915	0.0381533
$661$	2409.69	0.141795	0.0708973	−	0.997484i	$-0.477414\pi$
$661$	2409.69	0.141795	0.0708973	+	0.997484i	$0.477414\pi$
$662$	−15324.8	−0.899722
$663$	2614.30	0.153139
$664$	−21710.9	−1.26889
$665$	2408.13	0.140426
$666$	3818.85	0.222188
$667$	−51239.9	−2.97454
$668$	−352.368	−0.0204095
$669$	−11354.9	−0.656209
$670$	−21371.0	−1.23229
$671$	−4462.40	−0.256735
$672$	490.346	0.0281481
$673$	7929.02	0.454147	0.227074	−	0.973878i	$-0.427084\pi$
$673$	7929.02	0.454147	0.227074	+	0.973878i	$0.427084\pi$
$674$	−2471.72	−0.141257
$675$	−6874.19	−0.391982
$676$	−81.6802	−0.00464726
$677$	−2628.26	−0.149206	−0.0746030	−	0.997213i	$-0.523769\pi$
$677$	−2628.26	−0.149206	−0.0746030	+	0.997213i	$0.523769\pi$
$678$	−1365.88	−0.0773690
$679$	4169.44	0.235653
$680$	30376.1	1.71305
$681$	−6041.48	−0.339956
$682$	−7363.46	−0.413433
$683$	10021.5	0.561437	0.280719	−	0.959790i	$-0.409427\pi$
$683$	10021.5	0.561437	0.280719	+	0.959790i	$0.409427\pi$
$684$	−71.8448	−0.00401616
$685$	−10655.7	−0.594356
$686$	12925.5	0.719385
$687$	9152.18	0.508264
$688$	30103.7	1.66816
$689$	−4770.99	−0.263803
$690$	−28139.8	−1.55256
$691$	23987.2	1.32057	0.660286	−	0.751014i	$-0.270435\pi$
$691$	23987.2	1.32057	0.660286	+	0.751014i	$0.270435\pi$
$692$	1853.43	0.101816
$693$	1542.30	0.0845415
$694$	2227.10	0.121815
$695$	11878.3	0.648303
$696$	−20360.3	−1.10885
$697$	−16873.4	−0.916964
$698$	−12167.4	−0.659801
$699$	−16762.5	−0.907032
$700$	−920.835	−0.0497204
$701$	−3763.71	−0.202787	−0.101393	−	0.994846i	$-0.532330\pi$
$701$	−3763.71	−0.202787	−0.101393	+	0.994846i	$0.532330\pi$
$702$	−962.322	−0.0517386
$703$	−2556.23	−0.137141
$704$	12344.9	0.660890
$705$	16467.3	0.879707
$706$	−19507.8	−1.03993
$707$	−2145.96	−0.114155
$708$	−115.512	−0.00613163
$709$	−36047.8	−1.90946	−0.954728	−	0.297479i	$-0.903854\pi$
$709$	−36047.8	−1.90946	−0.954728	+	0.297479i	$0.903854\pi$
$710$	−28219.9	−1.49166
$711$	1019.39	0.0537696
$712$	27698.7	1.45794
$713$	−20594.9	−1.08174
$714$	4125.91	0.216258
$715$	−5800.15	−0.303376
$716$	137.195	0.00716090
$717$	4007.08	0.208713
$718$	−12869.3	−0.668910
$719$	−3944.18	−0.204580	−0.102290	−	0.994755i	$-0.532617\pi$
$719$	−3944.18	−0.204580	−0.102290	+	0.994755i	$0.532617\pi$
$720$	−10503.4	−0.543667
$721$	−14303.3	−0.738812
$722$	18057.1	0.930770
$723$	1714.68	0.0882012
$724$	−1142.36	−0.0586402
$725$	74292.1	3.80571
$726$	−6634.22	−0.339145
$727$	−20447.8	−1.04315	−0.521573	−	0.853206i	$-0.674655\pi$
$727$	−20447.8	−1.04315	−0.521573	+	0.853206i	$0.674655\pi$
$728$	−2262.64	−0.115191
$729$	729.000	0.0370370
$730$	39227.3	1.98886
$731$	−33688.7	−1.70454
$732$	−282.544	−0.0142666
$733$	−13536.2	−0.682089	−0.341045	−	0.940047i	$-0.610781\pi$
$733$	−13536.2	−0.682089	−0.341045	+	0.940047i	$0.610781\pi$
$734$	−25343.8	−1.27447
$735$	16775.1	0.841851
$736$	3835.40	0.192085
$737$	9161.83	0.457911
$738$	6211.07	0.309800
$739$	15839.1	0.788433	0.394217	−	0.919018i	$-0.371016\pi$
$739$	15839.1	0.788433	0.394217	+	0.919018i	$0.371016\pi$
$740$	1457.37	0.0723972
$741$	644.151	0.0319345
$742$	−7529.60	−0.372534
$743$	−1664.92	−0.0822075	−0.0411037	−	0.999155i	$-0.513087\pi$
$743$	−1664.92	−0.0822075	−0.0411037	+	0.999155i	$0.513087\pi$
$744$	−8183.43	−0.403252
$745$	−42182.9	−2.07445
$746$	11822.0	0.580206
$747$	−8401.19	−0.411491
$748$	−741.916	−0.0362662
$749$	6243.58	0.304587
$750$	20768.3	1.01113
$751$	22399.1	1.08835	0.544177	−	0.838970i	$-0.316842\pi$
$751$	22399.1	1.08835	0.544177	+	0.838970i	$0.316842\pi$
$752$	16875.8	0.818346
$753$	−12265.8	−0.593613
$754$	10400.2	0.502325
$755$	−16493.6	−0.795050
$756$	97.6535	0.00469792
$757$	23798.9	1.14265	0.571326	−	0.820723i	$-0.306429\pi$
$757$	23798.9	1.14265	0.571326	+	0.820723i	$0.306429\pi$
$758$	6532.64	0.313029
$759$	12063.6	0.576920
$760$	7484.53	0.357227
$761$	13693.5	0.652285	0.326142	−	0.945321i	$-0.394251\pi$
$761$	13693.5	0.652285	0.326142	+	0.945321i	$0.394251\pi$
$762$	10660.9	0.506829
$763$	−8064.50	−0.382640
$764$	−1215.18	−0.0575441
$765$	11754.3	0.555526
$766$	−13285.5	−0.626666
$767$	1035.66	0.0487556
$768$	2218.82	0.104251
$769$	16299.9	0.764358	0.382179	−	0.924088i	$-0.375174\pi$
$769$	16299.9	0.764358	0.382179	+	0.924088i	$0.375174\pi$
$770$	−9153.84	−0.428418
$771$	−9150.68	−0.427437
$772$	−1169.97	−0.0545444
$773$	33532.2	1.56024	0.780122	−	0.625628i	$-0.215157\pi$
$773$	33532.2	1.56024	0.780122	+	0.625628i	$0.215157\pi$
$774$	12400.8	0.575887
$775$	29860.2	1.38401
$776$	12958.7	0.599473
$777$	3474.50	0.160421
$778$	−26214.3	−1.20801
$779$	−4157.51	−0.191217
$780$	−367.246	−0.0168584
$781$	12098.0	0.554290
$782$	32272.1	1.47577
$783$	−7878.59	−0.359589
$784$	17191.3	0.783130
$785$	32217.5	1.46483
$786$	−1621.98	−0.0736055
$787$	16163.3	0.732097	0.366049	−	0.930596i	$-0.380710\pi$
$787$	16163.3	0.732097	0.366049	+	0.930596i	$0.380710\pi$
$788$	2239.51	0.101243
$789$	−17313.0	−0.781189
$790$	−6050.27	−0.272480
$791$	−1242.71	−0.0558607
$792$	4793.52	0.215064
$793$	2533.25	0.113441
$794$	20398.2	0.911720
$795$	−21451.1	−0.956970
$796$	−1476.12	−0.0657284
$797$	−39636.4	−1.76160	−0.880798	−	0.473492i	$-0.842993\pi$
$797$	−39636.4	−1.76160	−0.880798	+	0.473492i	$0.842993\pi$
$798$	1016.60	0.0450969
$799$	−18885.5	−0.836197
$800$	−5560.90	−0.245760
$801$	10718.2	0.472797
$802$	23819.0	1.04872
$803$	−16816.9	−0.739048
$804$	580.096	0.0254458
$805$	−25602.4	−1.12095
$806$	4180.15	0.182679
$807$	6238.21	0.272113
$808$	−6669.71	−0.290396
$809$	−23811.2	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$809$	−23811.2	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$810$	−4326.74	−0.187687
$811$	27218.6	1.17851	0.589256	−	0.807946i	$-0.299421\pi$
$811$	27218.6	1.17851	0.589256	+	0.807946i	$0.299421\pi$
$812$	−1055.38	−0.0456116
$813$	−18036.0	−0.778045
$814$	9716.80	0.418395
$815$	−55849.2	−2.40038
$816$	12045.9	0.516777
$817$	−8300.73	−0.355454
$818$	−7007.94	−0.299544
$819$	−875.548	−0.0373555
$820$	2370.30	0.100944
$821$	−43094.8	−1.83193	−0.915967	−	0.401253i	$-0.868575\pi$
$821$	−43094.8	−1.83193	−0.915967	+	0.401253i	$0.868575\pi$
$822$	−4498.36	−0.190874
$823$	26541.1	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$823$	26541.1	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$824$	−44455.1	−1.87945
$825$	−17490.9	−0.738127
$826$	1634.49	0.0688512
$827$	−44898.7	−1.88788	−0.943942	−	0.330112i	$-0.892913\pi$
$827$	−44898.7	−1.88788	−0.943942	+	0.330112i	$0.892913\pi$
$828$	763.829	0.0320590
$829$	−7137.48	−0.299029	−0.149514	−	0.988760i	$-0.547771\pi$
$829$	−7137.48	−0.299029	−0.149514	+	0.988760i	$0.547771\pi$
$830$	49862.6	2.08525
$831$	2205.60	0.0920717
$832$	−7008.06	−0.292020
$833$	−19238.6	−0.800213
$834$	5014.49	0.208198
$835$	14204.6	0.588708
$836$	−182.804	−0.00756270
$837$	−3166.64	−0.130771
$838$	9177.59	0.378323
$839$	−4387.17	−0.180527	−0.0902634	−	0.995918i	$-0.528771\pi$
$839$	−4387.17	−0.180527	−0.0902634	+	0.995918i	$0.528771\pi$
$840$	−10173.2	−0.417867
$841$	60758.1	2.49121
$842$	5084.36	0.208098
$843$	5708.76	0.233239
$844$	1948.53	0.0794683
$845$	3292.68	0.134049
$846$	6951.74	0.282513
$847$	−6036.01	−0.244864
$848$	−21983.2	−0.890219
$849$	−6377.12	−0.257788
$850$	−46791.0	−1.88814
$851$	27176.9	1.09473
$852$	766.004	0.0308015
$853$	−9328.85	−0.374459	−0.187230	−	0.982316i	$-0.559951\pi$
$853$	−9328.85	−0.374459	−0.187230	+	0.982316i	$0.559951\pi$
$854$	3997.99	0.160197
$855$	2896.20	0.115845
$856$	19405.2	0.774833
$857$	−5010.39	−0.199710	−0.0998552	−	0.995002i	$-0.531838\pi$
$857$	−5010.39	−0.199710	−0.0998552	+	0.995002i	$0.531838\pi$
$858$	−2448.56	−0.0974272
$859$	30233.4	1.20088	0.600438	−	0.799672i	$-0.294993\pi$
$859$	30233.4	1.20088	0.600438	+	0.799672i	$0.294993\pi$
$860$	4732.45	0.187646
$861$	5651.01	0.223677
$862$	38501.4	1.52130
$863$	4334.93	0.170988	0.0854940	−	0.996339i	$-0.472753\pi$
$863$	4334.93	0.170988	0.0854940	+	0.996339i	$0.472753\pi$
$864$	589.728	0.0232210
$865$	−74715.2	−2.93687
$866$	−8462.05	−0.332047
$867$	1258.58	0.0493007
$868$	−424.189	−0.0165875
$869$	2593.78	0.101252
$870$	46760.8	1.82223
$871$	−5201.06	−0.202332
$872$	−25064.7	−0.973391
$873$	5014.48	0.194404
$874$	7951.69	0.307746
$875$	18895.6	0.730043
$876$	−1064.79	−0.0410684
$877$	34683.3	1.33543	0.667716	−	0.744416i	$-0.267272\pi$
$877$	34683.3	1.33543	0.667716	+	0.744416i	$0.267272\pi$
$878$	−7779.95	−0.299044
$879$	4923.08	0.188909
$880$	−26725.3	−1.02376
$881$	−18269.2	−0.698642	−0.349321	−	0.937003i	$-0.613588\pi$
$881$	−18269.2	−0.698642	−0.349321	+	0.937003i	$0.613588\pi$
$882$	7081.70	0.270355
$883$	−14592.0	−0.556128	−0.278064	−	0.960563i	$-0.589693\pi$
$883$	−14592.0	−0.556128	−0.278064	+	0.960563i	$0.589693\pi$
$884$	421.177	0.0160246
$885$	4656.49	0.176866
$886$	−50198.0	−1.90343
$887$	30459.3	1.15301	0.576507	−	0.817092i	$-0.304415\pi$
$887$	30459.3	1.15301	0.576507	+	0.817092i	$0.304415\pi$
$888$	10798.8	0.408091
$889$	9699.60	0.365933
$890$	−63614.6	−2.39592
$891$	1854.89	0.0697432
$892$	−1829.32	−0.0686662
$893$	−4653.30	−0.174375
$894$	−17807.7	−0.666196
$895$	−5530.57	−0.206555
$896$	−9752.58	−0.363628
$897$	−6848.38	−0.254917
$898$	−38002.6	−1.41221
$899$	34223.2	1.26964
$900$	−1107.47	−0.0410172
$901$	24601.2	0.909638
$902$	15803.6	0.583374
$903$	11282.6	0.415793
$904$	−3862.39	−0.142103
$905$	46050.7	1.69147
$906$	−6962.84	−0.255326
$907$	−9364.89	−0.342840	−0.171420	−	0.985198i	$-0.554836\pi$
$907$	−9364.89	−0.342840	−0.171420	+	0.985198i	$0.554836\pi$
$908$	−973.313	−0.0355733
$909$	−2580.90	−0.0941727
$910$	5196.53	0.189300
$911$	32479.8	1.18123	0.590616	−	0.806952i	$-0.298885\pi$
$911$	32479.8	1.18123	0.590616	+	0.806952i	$0.298885\pi$
$912$	2968.04	0.107765
$913$	−21376.3	−0.774864
$914$	24648.5	0.892012
$915$	11389.9	0.411516
$916$	1474.46	0.0531851
$917$	−1475.72	−0.0531435
$918$	4962.13	0.178404
$919$	295.958	0.0106232	0.00531161	−	0.999986i	$-0.498309\pi$
$919$	295.958	0.0106232	0.00531161	+	0.999986i	$0.498309\pi$
$920$	−79572.9	−2.85157
$921$	10119.8	0.362062
$922$	9340.56	0.333639
$923$	−6867.89	−0.244918
$924$	248.473	0.00884649
$925$	−39403.5	−1.40062
$926$	20558.2	0.729572
$927$	−17202.3	−0.609489
$928$	−6373.42	−0.225450
$929$	−5620.38	−0.198492	−0.0992458	−	0.995063i	$-0.531643\pi$
$929$	−5620.38	−0.198492	−0.0992458	+	0.995063i	$0.531643\pi$
$930$	18794.6	0.662687
$931$	−4740.29	−0.166871
$932$	−2700.52	−0.0949125
$933$	2605.58	0.0914284
$934$	−21142.0	−0.740670
$935$	29908.0	1.04609
$936$	−2721.23	−0.0950278
$937$	−32583.1	−1.13601	−0.568006	−	0.823024i	$-0.692285\pi$
$937$	−32583.1	−1.13601	−0.568006	+	0.823024i	$0.692285\pi$
$938$	−8208.35	−0.285727
$939$	13029.6	0.452827
$940$	2652.96	0.0920532
$941$	8812.99	0.305308	0.152654	−	0.988280i	$-0.451218\pi$
$941$	8812.99	0.305308	0.152654	+	0.988280i	$0.451218\pi$
$942$	13600.8	0.470422
$943$	44201.2	1.52639
$944$	4772.00	0.164529
$945$	−3936.59	−0.135510
$946$	31552.9	1.08443
$947$	13426.8	0.460732	0.230366	−	0.973104i	$-0.426008\pi$
$947$	13426.8	0.460732	0.230366	+	0.973104i	$0.426008\pi$
$948$	164.229	0.00562649
$949$	9546.76	0.326555
$950$	−11529.1	−0.393739
$951$	9832.96	0.335285
$952$	11667.1	0.397199
$953$	−13394.6	−0.455293	−0.227647	−	0.973744i	$-0.573103\pi$
$953$	−13394.6	−0.455293	−0.227647	+	0.973744i	$0.573103\pi$
$954$	−9055.67	−0.307325
$955$	48986.2	1.65985
$956$	645.560	0.0218398
$957$	−20046.5	−0.677129
$958$	25880.7	0.872828
$959$	−4092.74	−0.137812
$960$	−31509.3	−1.05933
$961$	−16035.7	−0.538273
$962$	−5516.11	−0.184872
$963$	7509.00	0.251271
$964$	276.243	0.00922944
$965$	47163.8	1.57332
$966$	−10808.2	−0.359986
$967$	45590.8	1.51613	0.758066	−	0.652178i	$-0.226144\pi$
$967$	45590.8	1.51613	0.758066	+	0.652178i	$0.226144\pi$
$968$	−18760.1	−0.622904
$969$	−3321.51	−0.110116
$970$	−29761.8	−0.985149
$971$	264.763	0.00875041	0.00437521	−	0.999990i	$-0.498607\pi$
$971$	264.763	0.00875041	0.00437521	+	0.999990i	$0.498607\pi$
$972$	117.445	0.00387558
$973$	4562.32	0.150320
$974$	16878.2	0.555248
$975$	9929.38	0.326148
$976$	11672.4	0.382812
$977$	610.521	0.0199921	0.00999606	−	0.999950i	$-0.496818\pi$
$977$	610.521	0.0199921	0.00999606	+	0.999950i	$0.496818\pi$
$978$	−23577.0	−0.770868
$979$	27271.8	0.890308
$980$	2702.56	0.0880918
$981$	−9698.98	−0.315662
$982$	−10535.5	−0.342363
$983$	−57829.7	−1.87638	−0.938190	−	0.346121i	$-0.887499\pi$
$983$	−57829.7	−1.87638	−0.938190	+	0.346121i	$0.887499\pi$
$984$	17563.5	0.569007
$985$	−90278.8	−2.92033
$986$	−53627.7	−1.73210
$987$	6324.89	0.203975
$988$	103.776	0.00334165
$989$	88250.4	2.83741
$990$	−11009.1	−0.353427
$991$	−56780.7	−1.82008	−0.910039	−	0.414522i	$-0.863949\pi$
$991$	−56780.7	−1.82008	−0.910039	+	0.414522i	$0.863949\pi$
$992$	−2561.67	−0.0819890
$993$	−16768.9	−0.535895
$994$	−10838.9	−0.345866
$995$	59505.3	1.89592
$996$	−1353.47	−0.0430587
$997$	18616.6	0.591369	0.295684	−	0.955286i	$-0.404452\pi$
$997$	18616.6	0.591369	0.295684	+	0.955286i	$0.404452\pi$
$998$	35209.4	1.11677
$999$	4178.69	0.132340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.a.b.1.1	✓	2
3.2	odd	2		117.4.a.c.1.2		2
4.3	odd	2		624.4.a.r.1.2		2
5.4	even	2		975.4.a.j.1.2		2
7.6	odd	2		1911.4.a.h.1.1		2
8.3	odd	2		2496.4.a.s.1.1		2
8.5	even	2		2496.4.a.bc.1.1		2
12.11	even	2		1872.4.a.t.1.1		2
13.5	odd	4		507.4.b.f.337.3		4
13.8	odd	4		507.4.b.f.337.2		4
13.12	even	2		507.4.a.f.1.2		2
39.38	odd	2		1521.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	1.1	even	1	trivial
117.4.a.c.1.2		2	3.2	odd	2
507.4.a.f.1.2		2	13.12	even	2
507.4.b.f.337.2		4	13.8	odd	4
507.4.b.f.337.3		4	13.5	odd	4
624.4.a.r.1.2		2	4.3	odd	2
975.4.a.j.1.2		2	5.4	even	2
1521.4.a.s.1.1		2	39.38	odd	2
1872.4.a.t.1.1		2	12.11	even	2
1911.4.a.h.1.1		2	7.6	odd	2
2496.4.a.s.1.1		2	8.3	odd	2
2496.4.a.bc.1.1		2	8.5	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74166 q^{2} -3.00000 q^{3} -0.483315 q^{4} +19.4833 q^{5} +8.22497 q^{6} +7.48331 q^{7} +23.2583 q^{8} +9.00000 q^{9} -53.4166 q^{10} +22.8999 q^{11} +1.44994 q^{12} -13.0000 q^{13} -20.5167 q^{14} -58.4499 q^{15} -59.8999 q^{16} +67.0334 q^{17} -24.6749 q^{18} +16.5167 q^{19} -9.41657 q^{20} -22.4499 q^{21} -62.7836 q^{22} -175.600 q^{23} -69.7750 q^{24} +254.600 q^{25} +35.6415 q^{26} -27.0000 q^{27} -3.61680 q^{28} +291.800 q^{29} +160.250 q^{30} +117.283 q^{31} -21.8418 q^{32} -68.6997 q^{33} -183.783 q^{34} +145.800 q^{35} -4.34983 q^{36} -154.766 q^{37} -45.2831 q^{38} +39.0000 q^{39} +453.150 q^{40} -251.716 q^{41} +61.5501 q^{42} -502.566 q^{43} -11.0679 q^{44} +175.350 q^{45} +481.434 q^{46} -281.733 q^{47} +179.700 q^{48} -287.000 q^{49} -698.025 q^{50} -201.100 q^{51} +6.28309 q^{52} +366.999 q^{53} +74.0247 q^{54} +446.166 q^{55} +174.049 q^{56} -49.5501 q^{57} -800.015 q^{58} -79.6663 q^{59} +28.2497 q^{60} -194.865 q^{61} -321.550 q^{62} +67.3498 q^{63} +539.082 q^{64} -253.283 q^{65} +188.351 q^{66} +400.082 q^{67} -32.3982 q^{68} +526.799 q^{69} -399.733 q^{70} +528.299 q^{71} +209.325 q^{72} -734.366 q^{73} +424.316 q^{74} -763.799 q^{75} -7.98276 q^{76} +171.367 q^{77} -106.925 q^{78} +113.266 q^{79} -1167.05 q^{80} +81.0000 q^{81} +690.118 q^{82} -933.466 q^{83} +10.8504 q^{84} +1306.03 q^{85} +1377.86 q^{86} -875.399 q^{87} +532.613 q^{88} +1190.91 q^{89} -480.749 q^{90} -97.2831 q^{91} +84.8699 q^{92} -351.849 q^{93} +772.415 q^{94} +321.800 q^{95} +65.5253 q^{96} +557.165 q^{97} +786.856 q^{98} +206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20}+ \cdots - 396 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * q^99

Introduction

L-functions

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Database

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