LMFDB - Embedded newform 2496.4.a.bt.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,4,Mod(1,2496)] 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("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-12,0,2,0,10,0,36,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$147.268767374$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.236113.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 15x^{2} + 16x + 47 $ x^4 - 2x^3 - 15x^2 + 16*x + 47
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.01754$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +6.51212 q^{5} +22.5823 q^{7} +9.00000 q^{9} -13.1609 q^{11} +13.0000 q^{13} -19.5364 q^{15} -125.275 q^{17} -32.9509 q^{19} -67.7469 q^{21} -116.368 q^{23} -82.5922 q^{25} -27.0000 q^{27} +260.556 q^{29} +77.2094 q^{31} +39.4827 q^{33} +147.059 q^{35} +61.4140 q^{37} -39.0000 q^{39} -17.5300 q^{41} -103.599 q^{43} +58.6091 q^{45} +415.804 q^{47} +166.960 q^{49} +375.824 q^{51} +86.9305 q^{53} -85.7053 q^{55} +98.8526 q^{57} +461.382 q^{59} +169.230 q^{61} +203.241 q^{63} +84.6576 q^{65} +20.0114 q^{67} +349.103 q^{69} +552.787 q^{71} +127.948 q^{73} +247.777 q^{75} -297.203 q^{77} +447.402 q^{79} +81.0000 q^{81} +568.017 q^{83} -815.804 q^{85} -781.669 q^{87} +128.368 q^{89} +293.570 q^{91} -231.628 q^{93} -214.580 q^{95} -791.936 q^{97} -118.448 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 2 q^{5} + 10 q^{7} + 36 q^{9} + 56 q^{11} + 52 q^{13} - 6 q^{15} + 36 q^{17} + 186 q^{19} - 30 q^{21} + 52 q^{23} - 32 q^{25} - 108 q^{27} + 316 q^{29} + 130 q^{31} - 168 q^{33} + 348 q^{35}+ \cdots + 504 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 2 * q^5 + 10 * q^7 + 36 * q^9 + 56 * q^11 + 52 * q^13 - 6 * q^15 + 36 * q^17 + 186 * q^19 - 30 * q^21 + 52 * q^23 - 32 * q^25 - 108 * q^27 + 316 * q^29 + 130 * q^31 - 168 * q^33 + 348 * q^35 + 404 * q^37 - 156 * q^39 - 382 * q^41 + 268 * q^43 + 18 * q^45 - 64 * q^47 - 600 * q^49 - 108 * q^51 - 156 * q^53 - 436 * q^55 - 558 * q^57 + 1132 * q^59 + 100 * q^61 + 90 * q^63 + 26 * q^65 + 534 * q^67 - 156 * q^69 - 60 * q^71 - 388 * q^73 + 96 * q^75 + 204 * q^77 + 888 * q^79 + 324 * q^81 + 1716 * q^83 - 964 * q^85 - 948 * q^87 - 2554 * q^89 + 130 * q^91 - 390 * q^93 - 2284 * q^95 - 1084 * q^97 + 504 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	6.51212	0.582462	0.291231	−	0.956653i	$-0.405935\pi$
$5$	6.51212	0.582462	0.291231	+	0.956653i	$0.405935\pi$
$6$	0	0
$7$	22.5823	1.21933	0.609665	−	0.792659i	$-0.291304\pi$
$7$	22.5823	1.21933	0.609665	+	0.792659i	$0.291304\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−13.1609	−0.360741	−0.180371	−	0.983599i	$-0.557730\pi$
$11$	−13.1609	−0.360741	−0.180371	+	0.983599i	$0.557730\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−19.5364	−0.336285
$16$	0	0
$17$	−125.275	−1.78727	−0.893634	−	0.448796i	$-0.851853\pi$
$17$	−125.275	−1.78727	−0.893634	+	0.448796i	$0.851853\pi$
$18$	0	0
$19$	−32.9509	−0.397866	−0.198933	−	0.980013i	$-0.563748\pi$
$19$	−32.9509	−0.397866	−0.198933	+	0.980013i	$0.563748\pi$
$20$	0	0
$21$	−67.7469	−0.703980
$22$	0	0
$23$	−116.368	−1.05497	−0.527485	−	0.849564i	$-0.676865\pi$
$23$	−116.368	−1.05497	−0.527485	+	0.849564i	$0.676865\pi$
$24$	0	0
$25$	−82.5922	−0.660738
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	260.556	1.66842	0.834209	−	0.551449i	$-0.185925\pi$
$29$	260.556	1.66842	0.834209	+	0.551449i	$0.185925\pi$
$30$	0	0
$31$	77.2094	0.447330	0.223665	−	0.974666i	$-0.428198\pi$
$31$	77.2094	0.447330	0.223665	+	0.974666i	$0.428198\pi$
$32$	0	0
$33$	39.4827	0.208274
$34$	0	0
$35$	147.059	0.710213
$36$	0	0
$37$	61.4140	0.272876	0.136438	−	0.990649i	$-0.456435\pi$
$37$	61.4140	0.272876	0.136438	+	0.990649i	$0.456435\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−17.5300	−0.0667737	−0.0333868	−	0.999443i	$-0.510629\pi$
$41$	−17.5300	−0.0667737	−0.0333868	+	0.999443i	$0.510629\pi$
$42$	0	0
$43$	−103.599	−0.367410	−0.183705	−	0.982981i	$-0.558809\pi$
$43$	−103.599	−0.367410	−0.183705	+	0.982981i	$0.558809\pi$
$44$	0	0
$45$	58.6091	0.194154
$46$	0	0
$47$	415.804	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$47$	415.804	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$48$	0	0
$49$	166.960	0.486764
$50$	0	0
$51$	375.824	1.03188
$52$	0	0
$53$	86.9305	0.225298	0.112649	−	0.993635i	$-0.464066\pi$
$53$	86.9305	0.225298	0.112649	+	0.993635i	$0.464066\pi$
$54$	0	0
$55$	−85.7053	−0.210118
$56$	0	0
$57$	98.8526	0.229708
$58$	0	0
$59$	461.382	1.01808	0.509040	−	0.860743i	$-0.330000\pi$
$59$	461.382	1.01808	0.509040	+	0.860743i	$0.330000\pi$
$60$	0	0
$61$	169.230	0.355207	0.177604	−	0.984102i	$-0.443166\pi$
$61$	169.230	0.355207	0.177604	+	0.984102i	$0.443166\pi$
$62$	0	0
$63$	203.241	0.406443
$64$	0	0
$65$	84.6576	0.161546
$66$	0	0
$67$	20.0114	0.0364892	0.0182446	−	0.999834i	$-0.494192\pi$
$67$	20.0114	0.0364892	0.0182446	+	0.999834i	$0.494192\pi$
$68$	0	0
$69$	349.103	0.609087
$70$	0	0
$71$	552.787	0.923997	0.461998	−	0.886881i	$-0.347133\pi$
$71$	552.787	0.923997	0.461998	+	0.886881i	$0.347133\pi$
$72$	0	0
$73$	127.948	0.205139	0.102569	−	0.994726i	$-0.467294\pi$
$73$	127.948	0.205139	0.102569	+	0.994726i	$0.467294\pi$
$74$	0	0
$75$	247.777	0.381477
$76$	0	0
$77$	−297.203	−0.439863
$78$	0	0
$79$	447.402	0.637173	0.318587	−	0.947894i	$-0.396792\pi$
$79$	447.402	0.637173	0.318587	+	0.947894i	$0.396792\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	568.017	0.751181	0.375590	−	0.926786i	$-0.377440\pi$
$83$	568.017	0.751181	0.375590	+	0.926786i	$0.377440\pi$
$84$	0	0
$85$	−815.804	−1.04102
$86$	0	0
$87$	−781.669	−0.963261
$88$	0	0
$89$	128.368	0.152887	0.0764436	−	0.997074i	$-0.475643\pi$
$89$	128.368	0.152887	0.0764436	+	0.997074i	$0.475643\pi$
$90$	0	0
$91$	293.570	0.338181
$92$	0	0
$93$	−231.628	−0.258266
$94$	0	0
$95$	−214.580	−0.231742
$96$	0	0
$97$	−791.936	−0.828958	−0.414479	−	0.910059i	$-0.636036\pi$
$97$	−791.936	−0.828958	−0.414479	+	0.910059i	$0.636036\pi$
$98$	0	0
$99$	−118.448	−0.120247
$100$	0	0
$101$	1133.39	1.11660	0.558298	−	0.829641i	$-0.311455\pi$
$101$	1133.39	1.11660	0.558298	+	0.829641i	$0.311455\pi$
$102$	0	0
$103$	−1089.54	−1.04229	−0.521144	−	0.853469i	$-0.674494\pi$
$103$	−1089.54	−1.04229	−0.521144	+	0.853469i	$0.674494\pi$
$104$	0	0
$105$	−441.176	−0.410042
$106$	0	0
$107$	1650.59	1.49129	0.745647	−	0.666341i	$-0.232140\pi$
$107$	1650.59	1.49129	0.745647	+	0.666341i	$0.232140\pi$
$108$	0	0
$109$	−1013.17	−0.890311	−0.445155	−	0.895453i	$-0.646852\pi$
$109$	−1013.17	−0.890311	−0.445155	+	0.895453i	$0.646852\pi$
$110$	0	0
$111$	−184.242	−0.157545
$112$	0	0
$113$	308.918	0.257173	0.128586	−	0.991698i	$-0.458956\pi$
$113$	308.918	0.257173	0.128586	+	0.991698i	$0.458956\pi$
$114$	0	0
$115$	−757.800	−0.614480
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−2828.99	−2.17927
$120$	0	0
$121$	−1157.79	−0.869866
$122$	0	0
$123$	52.5899	0.0385518
$124$	0	0
$125$	−1351.87	−0.967317
$126$	0	0
$127$	−1580.96	−1.10462	−0.552312	−	0.833637i	$-0.686254\pi$
$127$	−1580.96	−1.10462	−0.552312	+	0.833637i	$0.686254\pi$
$128$	0	0
$129$	310.796	0.212124
$130$	0	0
$131$	1193.91	0.796280	0.398140	−	0.917325i	$-0.369656\pi$
$131$	1193.91	0.796280	0.398140	+	0.917325i	$0.369656\pi$
$132$	0	0
$133$	−744.106	−0.485129
$134$	0	0
$135$	−175.827	−0.112095
$136$	0	0
$137$	−1717.73	−1.07121	−0.535606	−	0.844468i	$-0.679917\pi$
$137$	−1717.73	−1.07121	−0.535606	+	0.844468i	$0.679917\pi$
$138$	0	0
$139$	2604.90	1.58953	0.794764	−	0.606918i	$-0.207594\pi$
$139$	2604.90	1.58953	0.794764	+	0.606918i	$0.207594\pi$
$140$	0	0
$141$	−1247.41	−0.745043
$142$	0	0
$143$	−171.092	−0.100052
$144$	0	0
$145$	1696.78	0.971790
$146$	0	0
$147$	−500.880	−0.281033
$148$	0	0
$149$	2360.00	1.29758	0.648789	−	0.760968i	$-0.275276\pi$
$149$	2360.00	1.29758	0.648789	+	0.760968i	$0.275276\pi$
$150$	0	0
$151$	2579.82	1.39035	0.695174	−	0.718842i	$-0.255327\pi$
$151$	2579.82	1.39035	0.695174	+	0.718842i	$0.255327\pi$
$152$	0	0
$153$	−1127.47	−0.595756
$154$	0	0
$155$	502.797	0.260553
$156$	0	0
$157$	1051.22	0.534371	0.267185	−	0.963645i	$-0.413906\pi$
$157$	1051.22	0.534371	0.267185	+	0.963645i	$0.413906\pi$
$158$	0	0
$159$	−260.791	−0.130076
$160$	0	0
$161$	−2627.85	−1.28636
$162$	0	0
$163$	277.072	0.133141	0.0665703	−	0.997782i	$-0.478794\pi$
$163$	277.072	0.133141	0.0665703	+	0.997782i	$0.478794\pi$
$164$	0	0
$165$	257.116	0.121312
$166$	0	0
$167$	2486.70	1.15226	0.576128	−	0.817359i	$-0.304563\pi$
$167$	2486.70	1.15226	0.576128	+	0.817359i	$0.304563\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−296.558	−0.132622
$172$	0	0
$173$	−2381.25	−1.04649	−0.523245	−	0.852182i	$-0.675279\pi$
$173$	−2381.25	−1.04649	−0.523245	+	0.852182i	$0.675279\pi$
$174$	0	0
$175$	−1865.12	−0.805657
$176$	0	0
$177$	−1384.14	−0.587789
$178$	0	0
$179$	1712.08	0.714898	0.357449	−	0.933933i	$-0.383647\pi$
$179$	1712.08	0.714898	0.357449	+	0.933933i	$0.383647\pi$
$180$	0	0
$181$	−1549.09	−0.636151	−0.318075	−	0.948065i	$-0.603037\pi$
$181$	−1549.09	−0.636151	−0.318075	+	0.948065i	$0.603037\pi$
$182$	0	0
$183$	−507.689	−0.205079
$184$	0	0
$185$	399.936	0.158940
$186$	0	0
$187$	1648.72	0.644742
$188$	0	0
$189$	−609.722	−0.234660
$190$	0	0
$191$	3577.70	1.35536	0.677679	−	0.735358i	$-0.262986\pi$
$191$	3577.70	1.35536	0.677679	+	0.735358i	$0.262986\pi$
$192$	0	0
$193$	4139.79	1.54398	0.771990	−	0.635634i	$-0.219261\pi$
$193$	4139.79	1.54398	0.771990	+	0.635634i	$0.219261\pi$
$194$	0	0
$195$	−253.973	−0.0932686
$196$	0	0
$197$	−688.709	−0.249079	−0.124539	−	0.992215i	$-0.539745\pi$
$197$	−688.709	−0.249079	−0.124539	+	0.992215i	$0.539745\pi$
$198$	0	0
$199$	1245.05	0.443513	0.221757	−	0.975102i	$-0.428821\pi$
$199$	1245.05	0.443513	0.221757	+	0.975102i	$0.428821\pi$
$200$	0	0
$201$	−60.0341	−0.0210671
$202$	0	0
$203$	5883.96	2.03435
$204$	0	0
$205$	−114.157	−0.0388931
$206$	0	0
$207$	−1047.31	−0.351657
$208$	0	0
$209$	433.663	0.143527
$210$	0	0
$211$	2470.01	0.805889	0.402945	−	0.915224i	$-0.367987\pi$
$211$	2470.01	0.805889	0.402945	+	0.915224i	$0.367987\pi$
$212$	0	0
$213$	−1658.36	−0.533470
$214$	0	0
$215$	−674.647	−0.214003
$216$	0	0
$217$	1743.57	0.545442
$218$	0	0
$219$	−383.843	−0.118437
$220$	0	0
$221$	−1628.57	−0.495699
$222$	0	0
$223$	−6281.55	−1.88630	−0.943148	−	0.332373i	$-0.892151\pi$
$223$	−6281.55	−1.88630	−0.943148	+	0.332373i	$0.892151\pi$
$224$	0	0
$225$	−743.330	−0.220246
$226$	0	0
$227$	1321.57	0.386413	0.193206	−	0.981158i	$-0.438111\pi$
$227$	1321.57	0.386413	0.193206	+	0.981158i	$0.438111\pi$
$228$	0	0
$229$	−1499.85	−0.432809	−0.216404	−	0.976304i	$-0.569433\pi$
$229$	−1499.85	−0.432809	−0.216404	+	0.976304i	$0.569433\pi$
$230$	0	0
$231$	891.609	0.253955
$232$	0	0
$233$	−5601.84	−1.57506	−0.787530	−	0.616276i	$-0.788640\pi$
$233$	−5601.84	−1.57506	−0.787530	+	0.616276i	$0.788640\pi$
$234$	0	0
$235$	2707.77	0.751639
$236$	0	0
$237$	−1342.21	−0.367872
$238$	0	0
$239$	−445.036	−0.120448	−0.0602238	−	0.998185i	$-0.519181\pi$
$239$	−445.036	−0.120448	−0.0602238	+	0.998185i	$0.519181\pi$
$240$	0	0
$241$	2507.44	0.670202	0.335101	−	0.942182i	$-0.391230\pi$
$241$	2507.44	0.670202	0.335101	+	0.942182i	$0.391230\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1087.26	0.283522
$246$	0	0
$247$	−428.361	−0.110348
$248$	0	0
$249$	−1704.05	−0.433694
$250$	0	0
$251$	−3702.60	−0.931099	−0.465549	−	0.885022i	$-0.654143\pi$
$251$	−3702.60	−0.931099	−0.465549	+	0.885022i	$0.654143\pi$
$252$	0	0
$253$	1531.50	0.380572
$254$	0	0
$255$	2447.41	0.601031
$256$	0	0
$257$	612.839	0.148747	0.0743733	−	0.997230i	$-0.476304\pi$
$257$	612.839	0.148747	0.0743733	+	0.997230i	$0.476304\pi$
$258$	0	0
$259$	1386.87	0.332726
$260$	0	0
$261$	2345.01	0.556139
$262$	0	0
$263$	1552.09	0.363901	0.181950	−	0.983308i	$-0.441759\pi$
$263$	1552.09	0.363901	0.181950	+	0.983308i	$0.441759\pi$
$264$	0	0
$265$	566.102	0.131228
$266$	0	0
$267$	−385.103	−0.0882694
$268$	0	0
$269$	−1668.83	−0.378254	−0.189127	−	0.981953i	$-0.560566\pi$
$269$	−1668.83	−0.378254	−0.189127	+	0.981953i	$0.560566\pi$
$270$	0	0
$271$	−452.859	−0.101510	−0.0507550	−	0.998711i	$-0.516163\pi$
$271$	−452.859	−0.101510	−0.0507550	+	0.998711i	$0.516163\pi$
$272$	0	0
$273$	−880.709	−0.195249
$274$	0	0
$275$	1086.99	0.238356
$276$	0	0
$277$	5103.35	1.10697	0.553484	−	0.832860i	$-0.313298\pi$
$277$	5103.35	1.10697	0.553484	+	0.832860i	$0.313298\pi$
$278$	0	0
$279$	694.885	0.149110
$280$	0	0
$281$	5683.36	1.20655	0.603275	−	0.797533i	$-0.293862\pi$
$281$	5683.36	1.20655	0.603275	+	0.797533i	$0.293862\pi$
$282$	0	0
$283$	−3307.47	−0.694730	−0.347365	−	0.937730i	$-0.612923\pi$
$283$	−3307.47	−0.694730	−0.347365	+	0.937730i	$0.612923\pi$
$284$	0	0
$285$	643.741	0.133796
$286$	0	0
$287$	−395.867	−0.0814191
$288$	0	0
$289$	10780.7	2.19433
$290$	0	0
$291$	2375.81	0.478599
$292$	0	0
$293$	2748.84	0.548085	0.274043	−	0.961718i	$-0.411639\pi$
$293$	2748.84	0.548085	0.274043	+	0.961718i	$0.411639\pi$
$294$	0	0
$295$	3004.57	0.592993
$296$	0	0
$297$	355.344	0.0694247
$298$	0	0
$299$	−1512.78	−0.292596
$300$	0	0
$301$	−2339.50	−0.447994
$302$	0	0
$303$	−3400.16	−0.644666
$304$	0	0
$305$	1102.04	0.206895
$306$	0	0
$307$	7534.52	1.40071	0.700355	−	0.713795i	$-0.253025\pi$
$307$	7534.52	1.40071	0.700355	+	0.713795i	$0.253025\pi$
$308$	0	0
$309$	3268.62	0.601765
$310$	0	0
$311$	6582.52	1.20019	0.600097	−	0.799927i	$-0.295128\pi$
$311$	6582.52	1.20019	0.600097	+	0.799927i	$0.295128\pi$
$312$	0	0
$313$	1766.54	0.319012	0.159506	−	0.987197i	$-0.449010\pi$
$313$	1766.54	0.319012	0.159506	+	0.987197i	$0.449010\pi$
$314$	0	0
$315$	1323.53	0.236738
$316$	0	0
$317$	5966.06	1.05706	0.528529	−	0.848915i	$-0.322744\pi$
$317$	5966.06	1.05706	0.528529	+	0.848915i	$0.322744\pi$
$318$	0	0
$319$	−3429.15	−0.601868
$320$	0	0
$321$	−4951.77	−0.860999
$322$	0	0
$323$	4127.91	0.711093
$324$	0	0
$325$	−1073.70	−0.183256
$326$	0	0
$327$	3039.50	0.514021
$328$	0	0
$329$	9389.81	1.57349
$330$	0	0
$331$	8087.52	1.34299	0.671496	−	0.741008i	$-0.265652\pi$
$331$	8087.52	1.34299	0.671496	+	0.741008i	$0.265652\pi$
$332$	0	0
$333$	552.726	0.0909586
$334$	0	0
$335$	130.317	0.0212536
$336$	0	0
$337$	−9553.64	−1.54427	−0.772137	−	0.635457i	$-0.780812\pi$
$337$	−9553.64	−1.54427	−0.772137	+	0.635457i	$0.780812\pi$
$338$	0	0
$339$	−926.753	−0.148479
$340$	0	0
$341$	−1016.14	−0.161370
$342$	0	0
$343$	−3975.39	−0.625804
$344$	0	0
$345$	2273.40	0.354770
$346$	0	0
$347$	3345.46	0.517561	0.258781	−	0.965936i	$-0.416679\pi$
$347$	3345.46	0.517561	0.258781	+	0.965936i	$0.416679\pi$
$348$	0	0
$349$	331.054	0.0507763	0.0253881	−	0.999678i	$-0.491918\pi$
$349$	331.054	0.0507763	0.0253881	+	0.999678i	$0.491918\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	1412.78	0.213016	0.106508	−	0.994312i	$-0.466033\pi$
$353$	1412.78	0.213016	0.106508	+	0.994312i	$0.466033\pi$
$354$	0	0
$355$	3599.82	0.538193
$356$	0	0
$357$	8486.97	1.25820
$358$	0	0
$359$	−5722.37	−0.841267	−0.420634	−	0.907231i	$-0.638192\pi$
$359$	−5722.37	−0.841267	−0.420634	+	0.907231i	$0.638192\pi$
$360$	0	0
$361$	−5773.24	−0.841703
$362$	0	0
$363$	3473.37	0.502217
$364$	0	0
$365$	833.210	0.119486
$366$	0	0
$367$	12313.9	1.75144	0.875721	−	0.482818i	$-0.160387\pi$
$367$	12313.9	1.75144	0.875721	+	0.482818i	$0.160387\pi$
$368$	0	0
$369$	−157.770	−0.0222579
$370$	0	0
$371$	1963.09	0.274713
$372$	0	0
$373$	1198.51	0.166371	0.0831855	−	0.996534i	$-0.473491\pi$
$373$	1198.51	0.166371	0.0831855	+	0.996534i	$0.473491\pi$
$374$	0	0
$375$	4055.60	0.558481
$376$	0	0
$377$	3387.23	0.462736
$378$	0	0
$379$	−8328.75	−1.12881	−0.564405	−	0.825498i	$-0.690894\pi$
$379$	−8328.75	−1.12881	−0.564405	+	0.825498i	$0.690894\pi$
$380$	0	0
$381$	4742.87	0.637755
$382$	0	0
$383$	−8173.12	−1.09041	−0.545205	−	0.838303i	$-0.683548\pi$
$383$	−8173.12	−1.09041	−0.545205	+	0.838303i	$0.683548\pi$
$384$	0	0
$385$	−1935.42	−0.256203
$386$	0	0
$387$	−932.388	−0.122470
$388$	0	0
$389$	13705.5	1.78637	0.893183	−	0.449694i	$-0.148467\pi$
$389$	13705.5	1.78637	0.893183	+	0.449694i	$0.148467\pi$
$390$	0	0
$391$	14577.9	1.88551
$392$	0	0
$393$	−3581.74	−0.459733
$394$	0	0
$395$	2913.54	0.371129
$396$	0	0
$397$	−12455.3	−1.57459	−0.787294	−	0.616578i	$-0.788519\pi$
$397$	−12455.3	−1.57459	−0.787294	+	0.616578i	$0.788519\pi$
$398$	0	0
$399$	2232.32	0.280090
$400$	0	0
$401$	−11270.2	−1.40351	−0.701755	−	0.712418i	$-0.747600\pi$
$401$	−11270.2	−1.40351	−0.701755	+	0.712418i	$0.747600\pi$
$402$	0	0
$403$	1003.72	0.124067
$404$	0	0
$405$	527.482	0.0647180
$406$	0	0
$407$	−808.263	−0.0984377
$408$	0	0
$409$	−10185.5	−1.23139	−0.615697	−	0.787983i	$-0.711126\pi$
$409$	−10185.5	−1.23139	−0.615697	+	0.787983i	$0.711126\pi$
$410$	0	0
$411$	5153.20	0.618464
$412$	0	0
$413$	10419.1	1.24138
$414$	0	0
$415$	3699.00	0.437534
$416$	0	0
$417$	−7814.69	−0.917715
$418$	0	0
$419$	2008.02	0.234124	0.117062	−	0.993125i	$-0.462652\pi$
$419$	2008.02	0.234124	0.117062	+	0.993125i	$0.462652\pi$
$420$	0	0
$421$	−11699.1	−1.35435	−0.677175	−	0.735822i	$-0.736796\pi$
$421$	−11699.1	−1.35435	−0.677175	+	0.735822i	$0.736796\pi$
$422$	0	0
$423$	3742.23	0.430151
$424$	0	0
$425$	10346.7	1.18092
$426$	0	0
$427$	3821.59	0.433114
$428$	0	0
$429$	513.275	0.0577649
$430$	0	0
$431$	15665.6	1.75078	0.875388	−	0.483421i	$-0.160606\pi$
$431$	15665.6	1.75078	0.875388	+	0.483421i	$0.160606\pi$
$432$	0	0
$433$	12046.0	1.33694	0.668470	−	0.743739i	$-0.266949\pi$
$433$	12046.0	1.33694	0.668470	+	0.743739i	$0.266949\pi$
$434$	0	0
$435$	−5090.33	−0.561063
$436$	0	0
$437$	3834.41	0.419737
$438$	0	0
$439$	15371.0	1.67112	0.835558	−	0.549403i	$-0.185145\pi$
$439$	15371.0	1.67112	0.835558	+	0.549403i	$0.185145\pi$
$440$	0	0
$441$	1502.64	0.162255
$442$	0	0
$443$	−16694.2	−1.79044	−0.895221	−	0.445622i	$-0.852982\pi$
$443$	−16694.2	−1.79044	−0.895221	+	0.445622i	$0.852982\pi$
$444$	0	0
$445$	835.947	0.0890510
$446$	0	0
$447$	−7080.01	−0.749157
$448$	0	0
$449$	−14300.7	−1.50310	−0.751551	−	0.659674i	$-0.770694\pi$
$449$	−14300.7	−1.50310	−0.751551	+	0.659674i	$0.770694\pi$
$450$	0	0
$451$	230.710	0.0240880
$452$	0	0
$453$	−7739.45	−0.802717
$454$	0	0
$455$	1911.76	0.196978
$456$	0	0
$457$	15350.3	1.57124	0.785620	−	0.618709i	$-0.212344\pi$
$457$	15350.3	1.57124	0.785620	+	0.618709i	$0.212344\pi$
$458$	0	0
$459$	3382.41	0.343960
$460$	0	0
$461$	12323.5	1.24504	0.622520	−	0.782604i	$-0.286109\pi$
$461$	12323.5	1.24504	0.622520	+	0.782604i	$0.286109\pi$
$462$	0	0
$463$	8439.91	0.847162	0.423581	−	0.905858i	$-0.360773\pi$
$463$	8439.91	0.847162	0.423581	+	0.905858i	$0.360773\pi$
$464$	0	0
$465$	−1508.39	−0.150430
$466$	0	0
$467$	3609.28	0.357639	0.178820	−	0.983882i	$-0.442772\pi$
$467$	3609.28	0.357639	0.178820	+	0.983882i	$0.442772\pi$
$468$	0	0
$469$	451.903	0.0444924
$470$	0	0
$471$	−3153.65	−0.308519
$472$	0	0
$473$	1363.45	0.132540
$474$	0	0
$475$	2721.49	0.262885
$476$	0	0
$477$	782.374	0.0750995
$478$	0	0
$479$	12736.9	1.21496	0.607478	−	0.794336i	$-0.292181\pi$
$479$	12736.9	1.21496	0.607478	+	0.794336i	$0.292181\pi$
$480$	0	0
$481$	798.383	0.0756822
$482$	0	0
$483$	7883.54	0.742678
$484$	0	0
$485$	−5157.18	−0.482836
$486$	0	0
$487$	6441.08	0.599329	0.299664	−	0.954045i	$-0.403125\pi$
$487$	6441.08	0.599329	0.299664	+	0.954045i	$0.403125\pi$
$488$	0	0
$489$	−831.215	−0.0768688
$490$	0	0
$491$	4176.71	0.383895	0.191947	−	0.981405i	$-0.438520\pi$
$491$	4176.71	0.383895	0.191947	+	0.981405i	$0.438520\pi$
$492$	0	0
$493$	−32641.1	−2.98191
$494$	0	0
$495$	−771.348	−0.0700394
$496$	0	0
$497$	12483.2	1.12666
$498$	0	0
$499$	−11691.3	−1.04885	−0.524423	−	0.851458i	$-0.675719\pi$
$499$	−11691.3	−1.04885	−0.524423	+	0.851458i	$0.675719\pi$
$500$	0	0
$501$	−7460.11	−0.665256
$502$	0	0
$503$	6431.16	0.570082	0.285041	−	0.958515i	$-0.407993\pi$
$503$	6431.16	0.570082	0.285041	+	0.958515i	$0.407993\pi$
$504$	0	0
$505$	7380.75	0.650374
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	3233.43	0.281570	0.140785	−	0.990040i	$-0.455037\pi$
$509$	3233.43	0.281570	0.140785	+	0.990040i	$0.455037\pi$
$510$	0	0
$511$	2889.35	0.250132
$512$	0	0
$513$	889.674	0.0765693
$514$	0	0
$515$	−7095.22	−0.607093
$516$	0	0
$517$	−5472.35	−0.465519
$518$	0	0
$519$	7143.74	0.604191
$520$	0	0
$521$	−17045.9	−1.43338	−0.716691	−	0.697390i	$-0.754344\pi$
$521$	−17045.9	−1.43338	−0.716691	+	0.697390i	$0.754344\pi$
$522$	0	0
$523$	9386.30	0.784769	0.392384	−	0.919801i	$-0.371650\pi$
$523$	9386.30	0.784769	0.392384	+	0.919801i	$0.371650\pi$
$524$	0	0
$525$	5595.37	0.465146
$526$	0	0
$527$	−9672.38	−0.799498
$528$	0	0
$529$	1374.42	0.112963
$530$	0	0
$531$	4152.43	0.339360
$532$	0	0
$533$	−227.890	−0.0185197
$534$	0	0
$535$	10748.8	0.868623
$536$	0	0
$537$	−5136.24	−0.412747
$538$	0	0
$539$	−2197.34	−0.175596
$540$	0	0
$541$	−6637.81	−0.527508	−0.263754	−	0.964590i	$-0.584961\pi$
$541$	−6637.81	−0.527508	−0.263754	+	0.964590i	$0.584961\pi$
$542$	0	0
$543$	4647.28	0.367282
$544$	0	0
$545$	−6597.87	−0.518572
$546$	0	0
$547$	−16415.4	−1.28313	−0.641565	−	0.767069i	$-0.721714\pi$
$547$	−16415.4	−1.28313	−0.641565	+	0.767069i	$0.721714\pi$
$548$	0	0
$549$	1523.07	0.118402
$550$	0	0
$551$	−8585.56	−0.663806
$552$	0	0
$553$	10103.4	0.776924
$554$	0	0
$555$	−1199.81	−0.0917640
$556$	0	0
$557$	−13742.1	−1.04537	−0.522686	−	0.852525i	$-0.675070\pi$
$557$	−13742.1	−1.04537	−0.522686	+	0.852525i	$0.675070\pi$
$558$	0	0
$559$	−1346.78	−0.101901
$560$	0	0
$561$	−4946.17	−0.372242
$562$	0	0
$563$	10957.2	0.820233	0.410117	−	0.912033i	$-0.365488\pi$
$563$	10957.2	0.820233	0.410117	+	0.912033i	$0.365488\pi$
$564$	0	0
$565$	2011.71	0.149793
$566$	0	0
$567$	1829.17	0.135481
$568$	0	0
$569$	10525.2	0.775466	0.387733	−	0.921772i	$-0.373258\pi$
$569$	10525.2	0.775466	0.387733	+	0.921772i	$0.373258\pi$
$570$	0	0
$571$	7781.17	0.570284	0.285142	−	0.958485i	$-0.407959\pi$
$571$	7781.17	0.570284	0.285142	+	0.958485i	$0.407959\pi$
$572$	0	0
$573$	−10733.1	−0.782516
$574$	0	0
$575$	9611.06	0.697059
$576$	0	0
$577$	−2079.07	−0.150005	−0.0750025	−	0.997183i	$-0.523896\pi$
$577$	−2079.07	−0.150005	−0.0750025	+	0.997183i	$0.523896\pi$
$578$	0	0
$579$	−12419.4	−0.891418
$580$	0	0
$581$	12827.1	0.915937
$582$	0	0
$583$	−1144.08	−0.0812745
$584$	0	0
$585$	761.919	0.0538486
$586$	0	0
$587$	−14632.4	−1.02887	−0.514434	−	0.857530i	$-0.671998\pi$
$587$	−14632.4	−1.02887	−0.514434	+	0.857530i	$0.671998\pi$
$588$	0	0
$589$	−2544.12	−0.177977
$590$	0	0
$591$	2066.13	0.143806
$592$	0	0
$593$	1047.10	0.0725114	0.0362557	−	0.999343i	$-0.488457\pi$
$593$	1047.10	0.0725114	0.0362557	+	0.999343i	$0.488457\pi$
$594$	0	0
$595$	−18422.7	−1.26934
$596$	0	0
$597$	−3735.15	−0.256063
$598$	0	0
$599$	8729.16	0.595432	0.297716	−	0.954655i	$-0.403775\pi$
$599$	8729.16	0.595432	0.297716	+	0.954655i	$0.403775\pi$
$600$	0	0
$601$	758.735	0.0514966	0.0257483	−	0.999668i	$-0.491803\pi$
$601$	758.735	0.0514966	0.0257483	+	0.999668i	$0.491803\pi$
$602$	0	0
$603$	180.102	0.0121631
$604$	0	0
$605$	−7539.68	−0.506664
$606$	0	0
$607$	−7230.78	−0.483506	−0.241753	−	0.970338i	$-0.577722\pi$
$607$	−7230.78	−0.483506	−0.241753	+	0.970338i	$0.577722\pi$
$608$	0	0
$609$	−17651.9	−1.17453
$610$	0	0
$611$	5405.45	0.357907
$612$	0	0
$613$	10545.4	0.694822	0.347411	−	0.937713i	$-0.387061\pi$
$613$	10545.4	0.694822	0.347411	+	0.937713i	$0.387061\pi$
$614$	0	0
$615$	342.472	0.0224550
$616$	0	0
$617$	16441.2	1.07277	0.536384	−	0.843974i	$-0.319790\pi$
$617$	16441.2	1.07277	0.536384	+	0.843974i	$0.319790\pi$
$618$	0	0
$619$	−10865.8	−0.705550	−0.352775	−	0.935708i	$-0.614762\pi$
$619$	−10865.8	−0.705550	−0.352775	+	0.935708i	$0.614762\pi$
$620$	0	0
$621$	3141.92	0.203029
$622$	0	0
$623$	2898.84	0.186420
$624$	0	0
$625$	1520.51	0.0973124
$626$	0	0
$627$	−1300.99	−0.0828652
$628$	0	0
$629$	−7693.62	−0.487702
$630$	0	0
$631$	13405.8	0.845762	0.422881	−	0.906185i	$-0.361019\pi$
$631$	13405.8	0.845762	0.422881	+	0.906185i	$0.361019\pi$
$632$	0	0
$633$	−7410.04	−0.465281
$634$	0	0
$635$	−10295.4	−0.643402
$636$	0	0
$637$	2170.48	0.135004
$638$	0	0
$639$	4975.09	0.307999
$640$	0	0
$641$	1407.18	0.0867085	0.0433542	−	0.999060i	$-0.486196\pi$
$641$	1407.18	0.0867085	0.0433542	+	0.999060i	$0.486196\pi$
$642$	0	0
$643$	13866.2	0.850432	0.425216	−	0.905092i	$-0.360198\pi$
$643$	13866.2	0.850432	0.425216	+	0.905092i	$0.360198\pi$
$644$	0	0
$645$	2023.94	0.123554
$646$	0	0
$647$	1556.99	0.0946086	0.0473043	−	0.998881i	$-0.484937\pi$
$647$	1556.99	0.0946086	0.0473043	+	0.998881i	$0.484937\pi$
$648$	0	0
$649$	−6072.19	−0.367264
$650$	0	0
$651$	−5230.70	−0.314911
$652$	0	0
$653$	11808.5	0.707659	0.353830	−	0.935310i	$-0.384879\pi$
$653$	11808.5	0.707659	0.353830	+	0.935310i	$0.384879\pi$
$654$	0	0
$655$	7774.91	0.463803
$656$	0	0
$657$	1151.53	0.0683796
$658$	0	0
$659$	−21323.7	−1.26048	−0.630238	−	0.776402i	$-0.717043\pi$
$659$	−21323.7	−1.26048	−0.630238	+	0.776402i	$0.717043\pi$
$660$	0	0
$661$	−10085.6	−0.593471	−0.296735	−	0.954960i	$-0.595898\pi$
$661$	−10085.6	−0.593471	−0.296735	+	0.954960i	$0.595898\pi$
$662$	0	0
$663$	4885.71	0.286192
$664$	0	0
$665$	−4845.71	−0.282570
$666$	0	0
$667$	−30320.3	−1.76013
$668$	0	0
$669$	18844.7	1.08905
$670$	0	0
$671$	−2227.21	−0.128138
$672$	0	0
$673$	18383.1	1.05292	0.526461	−	0.850199i	$-0.323518\pi$
$673$	18383.1	1.05292	0.526461	+	0.850199i	$0.323518\pi$
$674$	0	0
$675$	2229.99	0.127159
$676$	0	0
$677$	15609.9	0.886170	0.443085	−	0.896480i	$-0.353884\pi$
$677$	15609.9	0.886170	0.443085	+	0.896480i	$0.353884\pi$
$678$	0	0
$679$	−17883.7	−1.01077
$680$	0	0
$681$	−3964.71	−0.223095
$682$	0	0
$683$	−22909.0	−1.28344	−0.641720	−	0.766939i	$-0.721779\pi$
$683$	−22909.0	−1.28344	−0.641720	+	0.766939i	$0.721779\pi$
$684$	0	0
$685$	−11186.1	−0.623940
$686$	0	0
$687$	4499.56	0.249882
$688$	0	0
$689$	1130.10	0.0624866
$690$	0	0
$691$	32473.9	1.78780	0.893898	−	0.448271i	$-0.147960\pi$
$691$	32473.9	1.78780	0.893898	+	0.448271i	$0.147960\pi$
$692$	0	0
$693$	−2674.83	−0.146621
$694$	0	0
$695$	16963.4	0.925840
$696$	0	0
$697$	2196.06	0.119342
$698$	0	0
$699$	16805.5	0.909361
$700$	0	0
$701$	23789.4	1.28176	0.640881	−	0.767640i	$-0.278569\pi$
$701$	23789.4	1.28176	0.640881	+	0.767640i	$0.278569\pi$
$702$	0	0
$703$	−2023.65	−0.108568
$704$	0	0
$705$	−8123.30	−0.433959
$706$	0	0
$707$	25594.5	1.36150
$708$	0	0
$709$	−21638.2	−1.14618	−0.573088	−	0.819494i	$-0.694255\pi$
$709$	−21638.2	−1.14618	−0.573088	+	0.819494i	$0.694255\pi$
$710$	0	0
$711$	4026.62	0.212391
$712$	0	0
$713$	−8984.68	−0.471920
$714$	0	0
$715$	−1114.17	−0.0582763
$716$	0	0
$717$	1335.11	0.0695405
$718$	0	0
$719$	3482.57	0.180637	0.0903185	−	0.995913i	$-0.471211\pi$
$719$	3482.57	0.180637	0.0903185	+	0.995913i	$0.471211\pi$
$720$	0	0
$721$	−24604.3	−1.27089
$722$	0	0
$723$	−7522.33	−0.386941
$724$	0	0
$725$	−21519.9	−1.10239
$726$	0	0
$727$	−13591.7	−0.693381	−0.346690	−	0.937980i	$-0.612695\pi$
$727$	−13591.7	−0.693381	−0.346690	+	0.937980i	$0.612695\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	12978.3	0.656661
$732$	0	0
$733$	−18531.9	−0.933820	−0.466910	−	0.884305i	$-0.654633\pi$
$733$	−18531.9	−0.933820	−0.466910	+	0.884305i	$0.654633\pi$
$734$	0	0
$735$	−3261.79	−0.163691
$736$	0	0
$737$	−263.367	−0.0131632
$738$	0	0
$739$	5161.20	0.256912	0.128456	−	0.991715i	$-0.458998\pi$
$739$	5161.20	0.256912	0.128456	+	0.991715i	$0.458998\pi$
$740$	0	0
$741$	1285.08	0.0637095
$742$	0	0
$743$	−16650.3	−0.822128	−0.411064	−	0.911607i	$-0.634843\pi$
$743$	−16650.3	−0.822128	−0.411064	+	0.911607i	$0.634843\pi$
$744$	0	0
$745$	15368.6	0.755790
$746$	0	0
$747$	5112.16	0.250394
$748$	0	0
$749$	37274.1	1.81838
$750$	0	0
$751$	−9917.76	−0.481896	−0.240948	−	0.970538i	$-0.577458\pi$
$751$	−9917.76	−0.481896	−0.240948	+	0.970538i	$0.577458\pi$
$752$	0	0
$753$	11107.8	0.537570
$754$	0	0
$755$	16800.1	0.809825
$756$	0	0
$757$	7455.62	0.357965	0.178982	−	0.983852i	$-0.442720\pi$
$757$	7455.62	0.357965	0.178982	+	0.983852i	$0.442720\pi$
$758$	0	0
$759$	−4594.50	−0.219723
$760$	0	0
$761$	−6527.00	−0.310912	−0.155456	−	0.987843i	$-0.549685\pi$
$761$	−6527.00	−0.310912	−0.155456	+	0.987843i	$0.549685\pi$
$762$	0	0
$763$	−22879.6	−1.08558
$764$	0	0
$765$	−7342.24	−0.347005
$766$	0	0
$767$	5997.96	0.282365
$768$	0	0
$769$	−15895.1	−0.745371	−0.372685	−	0.927958i	$-0.621563\pi$
$769$	−15895.1	−0.745371	−0.372685	+	0.927958i	$0.621563\pi$
$770$	0	0
$771$	−1838.52	−0.0858788
$772$	0	0
$773$	19104.6	0.888932	0.444466	−	0.895796i	$-0.353394\pi$
$773$	19104.6	0.888932	0.444466	+	0.895796i	$0.353394\pi$
$774$	0	0
$775$	−6376.90	−0.295568
$776$	0	0
$777$	−4160.61	−0.192099
$778$	0	0
$779$	577.628	0.0265670
$780$	0	0
$781$	−7275.17	−0.333324
$782$	0	0
$783$	−7035.02	−0.321087
$784$	0	0
$785$	6845.65	0.311251
$786$	0	0
$787$	42737.1	1.93572	0.967861	−	0.251484i	$-0.0809185\pi$
$787$	42737.1	1.93572	0.967861	+	0.251484i	$0.0809185\pi$
$788$	0	0
$789$	−4656.27	−0.210098
$790$	0	0
$791$	6976.07	0.313579
$792$	0	0
$793$	2199.99	0.0985167
$794$	0	0
$795$	−1698.31	−0.0757644
$796$	0	0
$797$	38280.1	1.70132	0.850660	−	0.525716i	$-0.176203\pi$
$797$	38280.1	1.70132	0.850660	+	0.525716i	$0.176203\pi$
$798$	0	0
$799$	−52089.7	−2.30638
$800$	0	0
$801$	1155.31	0.0509624
$802$	0	0
$803$	−1683.90	−0.0740020
$804$	0	0
$805$	−17112.9	−0.749254
$806$	0	0
$807$	5006.48	0.218385
$808$	0	0
$809$	15897.5	0.690885	0.345443	−	0.938440i	$-0.387729\pi$
$809$	15897.5	0.690885	0.345443	+	0.938440i	$0.387729\pi$
$810$	0	0
$811$	−32091.5	−1.38950	−0.694750	−	0.719251i	$-0.744485\pi$
$811$	−32091.5	−1.38950	−0.694750	+	0.719251i	$0.744485\pi$
$812$	0	0
$813$	1358.58	0.0586068
$814$	0	0
$815$	1804.33	0.0775494
$816$	0	0
$817$	3413.67	0.146180
$818$	0	0
$819$	2642.13	0.112727
$820$	0	0
$821$	25323.7	1.07650	0.538248	−	0.842786i	$-0.319086\pi$
$821$	25323.7	1.07650	0.538248	+	0.842786i	$0.319086\pi$
$822$	0	0
$823$	45147.5	1.91220	0.956102	−	0.293033i	$-0.0946645\pi$
$823$	45147.5	1.91220	0.956102	+	0.293033i	$0.0946645\pi$
$824$	0	0
$825$	−3260.96	−0.137615
$826$	0	0
$827$	−45428.3	−1.91016	−0.955078	−	0.296356i	$-0.904229\pi$
$827$	−45428.3	−1.91016	−0.955078	+	0.296356i	$0.904229\pi$
$828$	0	0
$829$	−4979.31	−0.208611	−0.104306	−	0.994545i	$-0.533262\pi$
$829$	−4979.31	−0.208611	−0.104306	+	0.994545i	$0.533262\pi$
$830$	0	0
$831$	−15310.0	−0.639109
$832$	0	0
$833$	−20915.9	−0.869978
$834$	0	0
$835$	16193.7	0.671146
$836$	0	0
$837$	−2084.65	−0.0860887
$838$	0	0
$839$	22087.3	0.908865	0.454432	−	0.890781i	$-0.349842\pi$
$839$	22087.3	0.908865	0.454432	+	0.890781i	$0.349842\pi$
$840$	0	0
$841$	43500.7	1.78362
$842$	0	0
$843$	−17050.1	−0.696602
$844$	0	0
$845$	1100.55	0.0448048
$846$	0	0
$847$	−26145.6	−1.06065
$848$	0	0
$849$	9922.40	0.401102
$850$	0	0
$851$	−7146.60	−0.287876
$852$	0	0
$853$	−27493.8	−1.10360	−0.551799	−	0.833977i	$-0.686058\pi$
$853$	−27493.8	−1.10360	−0.551799	+	0.833977i	$0.686058\pi$
$854$	0	0
$855$	−1931.22	−0.0772473
$856$	0	0
$857$	22693.2	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$857$	22693.2	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$858$	0	0
$859$	4842.18	0.192332	0.0961660	−	0.995365i	$-0.469342\pi$
$859$	4842.18	0.192332	0.0961660	+	0.995365i	$0.469342\pi$
$860$	0	0
$861$	1187.60	0.0470073
$862$	0	0
$863$	−18433.4	−0.727094	−0.363547	−	0.931576i	$-0.618434\pi$
$863$	−18433.4	−0.727094	−0.363547	+	0.931576i	$0.618434\pi$
$864$	0	0
$865$	−15507.0	−0.609541
$866$	0	0
$867$	−32342.2	−1.26690
$868$	0	0
$869$	−5888.21	−0.229855
$870$	0	0
$871$	260.148	0.0101203
$872$	0	0
$873$	−7127.42	−0.276319
$874$	0	0
$875$	−30528.2	−1.17948
$876$	0	0
$877$	−37558.1	−1.44612	−0.723059	−	0.690786i	$-0.757264\pi$
$877$	−37558.1	−1.44612	−0.723059	+	0.690786i	$0.757264\pi$
$878$	0	0
$879$	−8246.52	−0.316437
$880$	0	0
$881$	20119.9	0.769418	0.384709	−	0.923038i	$-0.374302\pi$
$881$	20119.9	0.769418	0.384709	+	0.923038i	$0.374302\pi$
$882$	0	0
$883$	−33843.1	−1.28982	−0.644911	−	0.764258i	$-0.723105\pi$
$883$	−33843.1	−1.28982	−0.644911	+	0.764258i	$0.723105\pi$
$884$	0	0
$885$	−9013.72	−0.342365
$886$	0	0
$887$	−42015.1	−1.59045	−0.795225	−	0.606315i	$-0.792647\pi$
$887$	−42015.1	−1.59045	−0.795225	+	0.606315i	$0.792647\pi$
$888$	0	0
$889$	−35701.6	−1.34690
$890$	0	0
$891$	−1066.03	−0.0400824
$892$	0	0
$893$	−13701.1	−0.513427
$894$	0	0
$895$	11149.3	0.416401
$896$	0	0
$897$	4538.34	0.168930
$898$	0	0
$899$	20117.4	0.746333
$900$	0	0
$901$	−10890.2	−0.402669
$902$	0	0
$903$	7018.49	0.258650
$904$	0	0
$905$	−10087.9	−0.370534
$906$	0	0
$907$	11281.8	0.413017	0.206508	−	0.978445i	$-0.433790\pi$
$907$	11281.8	0.413017	0.206508	+	0.978445i	$0.433790\pi$
$908$	0	0
$909$	10200.5	0.372198
$910$	0	0
$911$	−9299.64	−0.338212	−0.169106	−	0.985598i	$-0.554088\pi$
$911$	−9299.64	−0.338212	−0.169106	+	0.985598i	$0.554088\pi$
$912$	0	0
$913$	−7475.61	−0.270982
$914$	0	0
$915$	−3306.13	−0.119451
$916$	0	0
$917$	26961.3	0.970928
$918$	0	0
$919$	−52240.0	−1.87512	−0.937562	−	0.347818i	$-0.886923\pi$
$919$	−52240.0	−1.87512	−0.937562	+	0.347818i	$0.886923\pi$
$920$	0	0
$921$	−22603.6	−0.808700
$922$	0	0
$923$	7186.23	0.256271
$924$	0	0
$925$	−5072.32	−0.180299
$926$	0	0
$927$	−9805.86	−0.347429
$928$	0	0
$929$	26925.4	0.950907	0.475453	−	0.879741i	$-0.342284\pi$
$929$	26925.4	0.950907	0.475453	+	0.879741i	$0.342284\pi$
$930$	0	0
$931$	−5501.48	−0.193667
$932$	0	0
$933$	−19747.6	−0.692933
$934$	0	0
$935$	10736.7	0.375538
$936$	0	0
$937$	−27026.2	−0.942272	−0.471136	−	0.882061i	$-0.656156\pi$
$937$	−27026.2	−0.942272	−0.471136	+	0.882061i	$0.656156\pi$
$938$	0	0
$939$	−5299.61	−0.184181
$940$	0	0
$941$	−30867.1	−1.06933	−0.534665	−	0.845064i	$-0.679562\pi$
$941$	−30867.1	−1.06933	−0.534665	+	0.845064i	$0.679562\pi$
$942$	0	0
$943$	2039.92	0.0704442
$944$	0	0
$945$	−3970.59	−0.136681
$946$	0	0
$947$	32287.0	1.10791	0.553953	−	0.832548i	$-0.313119\pi$
$947$	32287.0	1.10791	0.553953	+	0.832548i	$0.313119\pi$
$948$	0	0
$949$	1663.32	0.0568952
$950$	0	0
$951$	−17898.2	−0.610293
$952$	0	0
$953$	11157.2	0.379240	0.189620	−	0.981858i	$-0.439274\pi$
$953$	11157.2	0.379240	0.189620	+	0.981858i	$0.439274\pi$
$954$	0	0
$955$	23298.4	0.789444
$956$	0	0
$957$	10287.5	0.347488
$958$	0	0
$959$	−38790.4	−1.30616
$960$	0	0
$961$	−23829.7	−0.799896
$962$	0	0
$963$	14855.3	0.497098
$964$	0	0
$965$	26958.8	0.899310
$966$	0	0
$967$	−3135.64	−0.104277	−0.0521383	−	0.998640i	$-0.516604\pi$
$967$	−3135.64	−0.104277	−0.0521383	+	0.998640i	$0.516604\pi$
$968$	0	0
$969$	−12383.7	−0.410550
$970$	0	0
$971$	−9327.09	−0.308260	−0.154130	−	0.988051i	$-0.549257\pi$
$971$	−9327.09	−0.308260	−0.154130	+	0.988051i	$0.549257\pi$
$972$	0	0
$973$	58824.6	1.93816
$974$	0	0
$975$	3221.10	0.105803
$976$	0	0
$977$	364.050	0.0119212	0.00596058	−	0.999982i	$-0.498103\pi$
$977$	364.050	0.0119212	0.00596058	+	0.999982i	$0.498103\pi$
$978$	0	0
$979$	−1689.43	−0.0551527
$980$	0	0
$981$	−9118.51	−0.296770
$982$	0	0
$983$	−7203.60	−0.233733	−0.116866	−	0.993148i	$-0.537285\pi$
$983$	−7203.60	−0.233733	−0.116866	+	0.993148i	$0.537285\pi$
$984$	0	0
$985$	−4484.96	−0.145079
$986$	0	0
$987$	−28169.4	−0.908452
$988$	0	0
$989$	12055.5	0.387607
$990$	0	0
$991$	−19971.0	−0.640163	−0.320081	−	0.947390i	$-0.603710\pi$
$991$	−19971.0	−0.640163	−0.320081	+	0.947390i	$0.603710\pi$
$992$	0	0
$993$	−24262.6	−0.775377
$994$	0	0
$995$	8107.91	0.258330
$996$	0	0
$997$	8654.50	0.274915	0.137458	−	0.990508i	$-0.456107\pi$
$997$	8654.50	0.274915	0.137458	+	0.990508i	$0.456107\pi$
$998$	0	0
$999$	−1658.18	−0.0525150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bt.1.3		4
4.3	odd	2		2496.4.a.bv.1.3		4
8.3	odd	2		1248.4.a.c.1.2	✓	4
8.5	even	2		1248.4.a.e.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.4.a.c.1.2	✓	4	8.3	odd	2
1248.4.a.e.1.2	yes	4	8.5	even	2
2496.4.a.bt.1.3		4	1.1	even	1	trivial
2496.4.a.bv.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +6.51212 q^{5} +22.5823 q^{7} +9.00000 q^{9} -13.1609 q^{11} +13.0000 q^{13} -19.5364 q^{15} -125.275 q^{17} -32.9509 q^{19} -67.7469 q^{21} -116.368 q^{23} -82.5922 q^{25} -27.0000 q^{27} +260.556 q^{29} +77.2094 q^{31} +39.4827 q^{33} +147.059 q^{35} +61.4140 q^{37} -39.0000 q^{39} -17.5300 q^{41} -103.599 q^{43} +58.6091 q^{45} +415.804 q^{47} +166.960 q^{49} +375.824 q^{51} +86.9305 q^{53} -85.7053 q^{55} +98.8526 q^{57} +461.382 q^{59} +169.230 q^{61} +203.241 q^{63} +84.6576 q^{65} +20.0114 q^{67} +349.103 q^{69} +552.787 q^{71} +127.948 q^{73} +247.777 q^{75} -297.203 q^{77} +447.402 q^{79} +81.0000 q^{81} +568.017 q^{83} -815.804 q^{85} -781.669 q^{87} +128.368 q^{89} +293.570 q^{91} -231.628 q^{93} -214.580 q^{95} -791.936 q^{97} -118.448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 2 q^{5} + 10 q^{7} + 36 q^{9} + 56 q^{11} + 52 q^{13} - 6 q^{15} + 36 q^{17} + 186 q^{19} - 30 q^{21} + 52 q^{23} - 32 q^{25} - 108 q^{27} + 316 q^{29} + 130 q^{31} - 168 q^{33} + 348 q^{35}+ \cdots + 504 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 2 * q^5 + 10 * q^7 + 36 * q^9 + 56 * q^11 + 52 * q^13 - 6 * q^15 + 36 * q^17 + 186 * q^19 - 30 * q^21 + 52 * q^23 - 32 * q^25 - 108 * q^27 + 316 * q^29 + 130 * q^31 - 168 * q^33 + 348 * q^35 + 404 * q^37 - 156 * q^39 - 382 * q^41 + 268 * q^43 + 18 * q^45 - 64 * q^47 - 600 * q^49 - 108 * q^51 - 156 * q^53 - 436 * q^55 - 558 * q^57 + 1132 * q^59 + 100 * q^61 + 90 * q^63 + 26 * q^65 + 534 * q^67 - 156 * q^69 - 60 * q^71 - 388 * q^73 + 96 * q^75 + 204 * q^77 + 888 * q^79 + 324 * q^81 + 1716 * q^83 - 964 * q^85 - 948 * q^87 - 2554 * q^89 + 130 * q^91 - 390 * q^93 - 2284 * q^95 - 1084 * q^97 + 504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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