LMFDB - Embedded newform 69.4.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [69,4,Mod(1,69)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(69, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("69.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	69.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.07113179040$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2009704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 27x^{2} - 6x + 112 $ x^4 - 27x^2 - 6x + 112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.45983$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.45983 q^{2} +3.00000 q^{3} +3.97043 q^{4} +7.89053 q^{5} +10.3795 q^{6} +4.57766 q^{7} -13.9416 q^{8} +9.00000 q^{9} +27.2999 q^{10} -7.16805 q^{11} +11.9113 q^{12} +9.66367 q^{13} +15.8379 q^{14} +23.6716 q^{15} -79.9991 q^{16} -69.1871 q^{17} +31.1385 q^{18} -21.2382 q^{19} +31.3288 q^{20} +13.7330 q^{21} -24.8002 q^{22} +23.0000 q^{23} -41.8249 q^{24} -62.7395 q^{25} +33.4347 q^{26} +27.0000 q^{27} +18.1753 q^{28} +39.5338 q^{29} +81.8997 q^{30} +28.5625 q^{31} -165.251 q^{32} -21.5041 q^{33} -239.376 q^{34} +36.1202 q^{35} +35.7339 q^{36} -170.923 q^{37} -73.4807 q^{38} +28.9910 q^{39} -110.007 q^{40} +395.474 q^{41} +47.5138 q^{42} -214.499 q^{43} -28.4603 q^{44} +71.0148 q^{45} +79.5761 q^{46} +387.800 q^{47} -239.997 q^{48} -322.045 q^{49} -217.068 q^{50} -207.561 q^{51} +38.3690 q^{52} +268.151 q^{53} +93.4155 q^{54} -56.5597 q^{55} -63.8200 q^{56} -63.7147 q^{57} +136.780 q^{58} +552.468 q^{59} +93.9865 q^{60} +354.991 q^{61} +98.8214 q^{62} +41.1989 q^{63} +68.2540 q^{64} +76.2515 q^{65} -74.4007 q^{66} +293.680 q^{67} -274.703 q^{68} +69.0000 q^{69} +124.970 q^{70} +505.691 q^{71} -125.475 q^{72} +1048.26 q^{73} -591.365 q^{74} -188.219 q^{75} -84.3250 q^{76} -32.8129 q^{77} +100.304 q^{78} -38.4080 q^{79} -631.236 q^{80} +81.0000 q^{81} +1368.27 q^{82} -111.501 q^{83} +54.5259 q^{84} -545.923 q^{85} -742.130 q^{86} +118.601 q^{87} +99.9342 q^{88} -1479.10 q^{89} +245.699 q^{90} +44.2370 q^{91} +91.3200 q^{92} +85.6875 q^{93} +1341.72 q^{94} -167.581 q^{95} -495.752 q^{96} -1274.61 q^{97} -1114.22 q^{98} -64.5124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 4 q^{5} + 12 q^{6} - 14 q^{7} + 84 q^{8} + 36 q^{9} - 100 q^{10} + 70 q^{11} + 78 q^{12} - 12 q^{13} - 18 q^{14} + 12 q^{15} + 130 q^{16} + 178 q^{17} + 36 q^{18} + 96 q^{19}+ \cdots + 630 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 12 * q^3 + 26 * q^4 + 4 * q^5 + 12 * q^6 - 14 * q^7 + 84 * q^8 + 36 * q^9 - 100 * q^10 + 70 * q^11 + 78 * q^12 - 12 * q^13 - 18 * q^14 + 12 * q^15 + 130 * q^16 + 178 * q^17 + 36 * q^18 + 96 * q^19 - 296 * q^20 - 42 * q^21 - 326 * q^22 + 92 * q^23 + 252 * q^24 - 80 * q^25 - 464 * q^26 + 108 * q^27 - 774 * q^28 - 24 * q^29 - 300 * q^30 - 400 * q^31 + 636 * q^32 + 210 * q^33 - 18 * q^34 + 224 * q^35 + 234 * q^36 - 358 * q^37 - 136 * q^38 - 36 * q^39 - 1268 * q^40 + 152 * q^41 - 54 * q^42 - 196 * q^43 + 70 * q^44 + 36 * q^45 + 92 * q^46 + 260 * q^47 + 390 * q^48 + 476 * q^49 + 876 * q^50 + 534 * q^51 - 876 * q^52 + 968 * q^53 + 108 * q^54 + 684 * q^55 - 1234 * q^56 + 288 * q^57 + 904 * q^58 + 1236 * q^59 - 888 * q^60 - 482 * q^61 - 1304 * q^62 - 126 * q^63 + 1658 * q^64 + 1424 * q^65 - 978 * q^66 + 620 * q^67 + 838 * q^68 + 276 * q^69 + 2636 * q^70 - 88 * q^71 + 756 * q^72 - 948 * q^73 + 710 * q^74 - 240 * q^75 + 3344 * q^76 - 892 * q^77 - 1392 * q^78 + 254 * q^79 - 5144 * q^80 + 324 * q^81 + 1568 * q^82 - 550 * q^83 - 2322 * q^84 - 1208 * q^85 - 1236 * q^86 - 72 * q^87 + 202 * q^88 + 202 * q^89 - 900 * q^90 - 356 * q^91 + 598 * q^92 - 1200 * q^93 - 1328 * q^94 - 460 * q^95 + 1908 * q^96 - 1800 * q^97 - 2784 * q^98 + 630 * 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 $

$2$	3.45983	1.22324	0.611618	−	0.791154i	$-0.290519\pi$
$2$	3.45983	1.22324	0.611618	+	0.791154i	$0.290519\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.97043	0.496304
$5$	7.89053	0.705751	0.352875	−	0.935670i	$-0.385204\pi$
$5$	7.89053	0.705751	0.352875	+	0.935670i	$0.385204\pi$
$6$	10.3795	0.706235
$7$	4.57766	0.247170	0.123585	−	0.992334i	$-0.460561\pi$
$7$	4.57766	0.247170	0.123585	+	0.992334i	$0.460561\pi$
$8$	−13.9416	−0.616138
$9$	9.00000	0.333333
$10$	27.2999	0.863299
$11$	−7.16805	−0.196477	−0.0982386	−	0.995163i	$-0.531321\pi$
$11$	−7.16805	−0.196477	−0.0982386	+	0.995163i	$0.531321\pi$
$12$	11.9113	0.286541
$13$	9.66367	0.206171	0.103085	−	0.994673i	$-0.467129\pi$
$13$	9.66367	0.206171	0.103085	+	0.994673i	$0.467129\pi$
$14$	15.8379	0.302348
$15$	23.6716	0.407465
$16$	−79.9991	−1.24999
$17$	−69.1871	−0.987078	−0.493539	−	0.869724i	$-0.664297\pi$
$17$	−69.1871	−0.987078	−0.493539	+	0.869724i	$0.664297\pi$
$18$	31.1385	0.407745
$19$	−21.2382	−0.256441	−0.128221	−	0.991746i	$-0.540927\pi$
$19$	−21.2382	−0.256441	−0.128221	+	0.991746i	$0.540927\pi$
$20$	31.3288	0.350267
$21$	13.7330	0.142704
$22$	−24.8002	−0.240338
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−41.8249	−0.355728
$25$	−62.7395	−0.501916
$26$	33.4347	0.252195
$27$	27.0000	0.192450
$28$	18.1753	0.122672
$29$	39.5338	0.253146	0.126573	−	0.991957i	$-0.459602\pi$
$29$	39.5338	0.253146	0.126573	+	0.991957i	$0.459602\pi$
$30$	81.8997	0.498426
$31$	28.5625	0.165483	0.0827415	−	0.996571i	$-0.473632\pi$
$31$	28.5625	0.165483	0.0827415	+	0.996571i	$0.473632\pi$
$32$	−165.251	−0.912889
$33$	−21.5041	−0.113436
$34$	−239.376	−1.20743
$35$	36.1202	0.174441
$36$	35.7339	0.165435
$37$	−170.923	−0.759448	−0.379724	−	0.925100i	$-0.623981\pi$
$37$	−170.923	−0.759448	−0.379724	+	0.925100i	$0.623981\pi$
$38$	−73.4807	−0.313688
$39$	28.9910	0.119033
$40$	−110.007	−0.434840
$41$	395.474	1.50641	0.753203	−	0.657788i	$-0.228508\pi$
$41$	395.474	1.50641	0.753203	+	0.657788i	$0.228508\pi$
$42$	47.5138	0.174560
$43$	−214.499	−0.760716	−0.380358	−	0.924839i	$-0.624199\pi$
$43$	−214.499	−0.760716	−0.380358	+	0.924839i	$0.624199\pi$
$44$	−28.4603	−0.0975124
$45$	71.0148	0.235250
$46$	79.5761	0.255062
$47$	387.800	1.20354	0.601771	−	0.798669i	$-0.294462\pi$
$47$	387.800	1.20354	0.601771	+	0.798669i	$0.294462\pi$
$48$	−239.997	−0.721680
$49$	−322.045	−0.938907
$50$	−217.068	−0.613961
$51$	−207.561	−0.569890
$52$	38.3690	0.102323
$53$	268.151	0.694969	0.347484	−	0.937686i	$-0.387036\pi$
$53$	268.151	0.694969	0.347484	+	0.937686i	$0.387036\pi$
$54$	93.4155	0.235412
$55$	−56.5597	−0.138664
$56$	−63.8200	−0.152291
$57$	−63.7147	−0.148056
$58$	136.780	0.309658
$59$	552.468	1.21907	0.609535	−	0.792759i	$-0.291356\pi$
$59$	552.468	1.21907	0.609535	+	0.792759i	$0.291356\pi$
$60$	93.9865	0.202227
$61$	354.991	0.745113	0.372557	−	0.928009i	$-0.378481\pi$
$61$	354.991	0.745113	0.372557	+	0.928009i	$0.378481\pi$
$62$	98.8214	0.202425
$63$	41.1989	0.0823902
$64$	68.2540	0.133309
$65$	76.2515	0.145505
$66$	−74.4007	−0.138759
$67$	293.680	0.535504	0.267752	−	0.963488i	$-0.413719\pi$
$67$	293.680	0.535504	0.267752	+	0.963488i	$0.413719\pi$
$68$	−274.703	−0.489891
$69$	69.0000	0.120386
$70$	124.970	0.213382
$71$	505.691	0.845274	0.422637	−	0.906299i	$-0.361105\pi$
$71$	505.691	0.845274	0.422637	+	0.906299i	$0.361105\pi$
$72$	−125.475	−0.205379
$73$	1048.26	1.68069	0.840343	−	0.542055i	$-0.182354\pi$
$73$	1048.26	1.68069	0.840343	+	0.542055i	$0.182354\pi$
$74$	−591.365	−0.928984
$75$	−188.219	−0.289781
$76$	−84.3250	−0.127273
$77$	−32.8129	−0.0485633
$78$	100.304	0.145605
$79$	−38.4080	−0.0546992	−0.0273496	−	0.999626i	$-0.508707\pi$
$79$	−38.4080	−0.0546992	−0.0273496	+	0.999626i	$0.508707\pi$
$80$	−631.236	−0.882179
$81$	81.0000	0.111111
$82$	1368.27	1.84269
$83$	−111.501	−0.147455	−0.0737276	−	0.997278i	$-0.523490\pi$
$83$	−111.501	−0.147455	−0.0737276	+	0.997278i	$0.523490\pi$
$84$	54.5259	0.0708246
$85$	−545.923	−0.696631
$86$	−742.130	−0.930534
$87$	118.601	0.146154
$88$	99.9342	0.121057
$89$	−1479.10	−1.76163	−0.880813	−	0.473464i	$-0.843004\pi$
$89$	−1479.10	−1.76163	−0.880813	+	0.473464i	$0.843004\pi$
$90$	245.699	0.287766
$91$	44.2370	0.0509593
$92$	91.3200	0.103487
$93$	85.6875	0.0955417
$94$	1341.72	1.47222
$95$	−167.581	−0.180984
$96$	−495.752	−0.527057
$97$	−1274.61	−1.33420	−0.667099	−	0.744969i	$-0.732464\pi$
$97$	−1274.61	−1.33420	−0.667099	+	0.744969i	$0.732464\pi$
$98$	−1114.22	−1.14850
$99$	−64.5124	−0.0654924
$100$	−249.103	−0.249103
$101$	−1030.08	−1.01482	−0.507412	−	0.861704i	$-0.669398\pi$
$101$	−1030.08	−1.01482	−0.507412	+	0.861704i	$0.669398\pi$
$102$	−718.127	−0.697109
$103$	−1261.15	−1.20646	−0.603229	−	0.797568i	$-0.706119\pi$
$103$	−1261.15	−1.20646	−0.603229	+	0.797568i	$0.706119\pi$
$104$	−134.727	−0.127030
$105$	108.361	0.100713
$106$	927.757	0.850110
$107$	314.435	0.284089	0.142045	−	0.989860i	$-0.454632\pi$
$107$	314.435	0.284089	0.142045	+	0.989860i	$0.454632\pi$
$108$	107.202	0.0955138
$109$	−594.442	−0.522360	−0.261180	−	0.965290i	$-0.584112\pi$
$109$	−594.442	−0.522360	−0.261180	+	0.965290i	$0.584112\pi$
$110$	−195.687	−0.169618
$111$	−512.769	−0.438468
$112$	−366.209	−0.308960
$113$	705.117	0.587008	0.293504	−	0.955958i	$-0.405179\pi$
$113$	705.117	0.587008	0.293504	+	0.955958i	$0.405179\pi$
$114$	−220.442	−0.181108
$115$	181.482	0.147159
$116$	156.966	0.125638
$117$	86.9730	0.0687236
$118$	1911.45	1.49121
$119$	−316.715	−0.243977
$120$	−330.020	−0.251055
$121$	−1279.62	−0.961397
$122$	1228.21	0.911449
$123$	1186.42	0.869724
$124$	113.405	0.0821300
$125$	−1481.36	−1.05998
$126$	142.541	0.100783
$127$	365.035	0.255052	0.127526	−	0.991835i	$-0.459296\pi$
$127$	365.035	0.255052	0.127526	+	0.991835i	$0.459296\pi$
$128$	1558.15	1.07596
$129$	−643.497	−0.439199
$130$	263.817	0.177987
$131$	841.502	0.561239	0.280620	−	0.959819i	$-0.409460\pi$
$131$	841.502	0.561239	0.280620	+	0.959819i	$0.409460\pi$
$132$	−85.3808	−0.0562988
$133$	−97.2214	−0.0633847
$134$	1016.08	0.655047
$135$	213.044	0.135822
$136$	964.580	0.608177
$137$	2767.12	1.72563	0.862814	−	0.505521i	$-0.168700\pi$
$137$	2767.12	1.72563	0.862814	+	0.505521i	$0.168700\pi$
$138$	238.728	0.147260
$139$	−1201.78	−0.733336	−0.366668	−	0.930352i	$-0.619502\pi$
$139$	−1201.78	−0.733336	−0.366668	+	0.930352i	$0.619502\pi$
$140$	143.413	0.0865757
$141$	1163.40	0.694865
$142$	1749.61	1.03397
$143$	−69.2696	−0.0405078
$144$	−719.992	−0.416662
$145$	311.943	0.178658
$146$	3626.82	2.05587
$147$	−966.135	−0.542078
$148$	−678.639	−0.376917
$149$	−2411.55	−1.32592	−0.662961	−	0.748654i	$-0.730700\pi$
$149$	−2411.55	−1.32592	−0.662961	+	0.748654i	$0.730700\pi$
$150$	−651.204	−0.354471
$151$	2352.41	1.26779	0.633897	−	0.773418i	$-0.281454\pi$
$151$	2352.41	1.26779	0.633897	+	0.773418i	$0.281454\pi$
$152$	296.095	0.158003
$153$	−622.684	−0.329026
$154$	−113.527	−0.0594044
$155$	225.373	0.116790
$156$	115.107	0.0590764
$157$	−2720.30	−1.38283	−0.691414	−	0.722459i	$-0.743012\pi$
$157$	−2720.30	−1.38283	−0.691414	+	0.722459i	$0.743012\pi$
$158$	−132.885	−0.0669099
$159$	804.452	0.401240
$160$	−1303.91	−0.644272
$161$	105.286	0.0515386
$162$	280.246	0.135915
$163$	910.107	0.437332	0.218666	−	0.975800i	$-0.429830\pi$
$163$	910.107	0.437332	0.218666	+	0.975800i	$0.429830\pi$
$164$	1570.20	0.747636
$165$	−169.679	−0.0800576
$166$	−385.773	−0.180372
$167$	370.185	0.171532	0.0857658	−	0.996315i	$-0.472666\pi$
$167$	370.185	0.171532	0.0857658	+	0.996315i	$0.472666\pi$
$168$	−191.460	−0.0879254
$169$	−2103.61	−0.957494
$170$	−1888.80	−0.852144
$171$	−191.144	−0.0854804
$172$	−851.654	−0.377546
$173$	−1674.46	−0.735879	−0.367940	−	0.929850i	$-0.619937\pi$
$173$	−1674.46	−0.735879	−0.367940	+	0.929850i	$0.619937\pi$
$174$	410.341	0.178781
$175$	−287.200	−0.124059
$176$	573.438	0.245594
$177$	1657.40	0.703831
$178$	−5117.45	−2.15488
$179$	−1890.08	−0.789225	−0.394613	−	0.918848i	$-0.629121\pi$
$179$	−1890.08	−0.789225	−0.394613	+	0.918848i	$0.629121\pi$
$180$	281.960	0.116756
$181$	−1249.68	−0.513191	−0.256596	−	0.966519i	$-0.582601\pi$
$181$	−1249.68	−0.513191	−0.256596	+	0.966519i	$0.582601\pi$
$182$	153.053	0.0623352
$183$	1064.97	0.430191
$184$	−320.657	−0.128474
$185$	−1348.67	−0.535981
$186$	296.464	0.116870
$187$	495.936	0.193938
$188$	1539.74	0.597323
$189$	123.597	0.0475680
$190$	−579.802	−0.221386
$191$	4304.02	1.63051	0.815257	−	0.579100i	$-0.196596\pi$
$191$	4304.02	1.63051	0.815257	+	0.579100i	$0.196596\pi$
$192$	204.762	0.0769657
$193$	−2955.56	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$193$	−2955.56	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$194$	−4409.94	−1.63204
$195$	228.754	0.0840074
$196$	−1278.66	−0.465983
$197$	2937.68	1.06244	0.531222	−	0.847233i	$-0.321733\pi$
$197$	2937.68	1.06244	0.531222	+	0.847233i	$0.321733\pi$
$198$	−223.202	−0.0801126
$199$	4241.65	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$199$	4241.65	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$200$	874.690	0.309250
$201$	881.040	0.309173
$202$	−3563.92	−1.24137
$203$	180.972	0.0625703
$204$	−824.108	−0.282839
$205$	3120.50	1.06315
$206$	−4363.38	−1.47578
$207$	207.000	0.0695048
$208$	−773.085	−0.257711
$209$	152.237	0.0503848
$210$	374.909	0.123196
$211$	5021.48	1.63835	0.819177	−	0.573541i	$-0.194431\pi$
$211$	5021.48	1.63835	0.819177	+	0.573541i	$0.194431\pi$
$212$	1064.68	0.344916
$213$	1517.07	0.488019
$214$	1087.89	0.347508
$215$	−1692.51	−0.536875
$216$	−376.424	−0.118576
$217$	130.749	0.0409025
$218$	−2056.67	−0.638969
$219$	3144.79	0.970344
$220$	−224.567	−0.0688195
$221$	−668.601	−0.203507
$222$	−1774.10	−0.536349
$223$	−1211.64	−0.363843	−0.181922	−	0.983313i	$-0.558232\pi$
$223$	−1211.64	−0.363843	−0.181922	+	0.983313i	$0.558232\pi$
$224$	−756.461	−0.225639
$225$	−564.656	−0.167305
$226$	2439.59	0.718048
$227$	4524.63	1.32295	0.661476	−	0.749966i	$-0.269930\pi$
$227$	4524.63	1.32295	0.661476	+	0.749966i	$0.269930\pi$
$228$	−252.975	−0.0734811
$229$	−6614.12	−1.90862	−0.954309	−	0.298821i	$-0.903407\pi$
$229$	−6614.12	−1.90862	−0.954309	+	0.298821i	$0.903407\pi$
$230$	627.898	0.180010
$231$	−98.4387	−0.0280381
$232$	−551.166	−0.155973
$233$	−2017.22	−0.567179	−0.283589	−	0.958946i	$-0.591525\pi$
$233$	−2017.22	−0.567179	−0.283589	+	0.958946i	$0.591525\pi$
$234$	300.912	0.0840651
$235$	3059.95	0.849401
$236$	2193.54	0.605030
$237$	−115.224	−0.0315806
$238$	−1095.78	−0.298441
$239$	−2191.56	−0.593140	−0.296570	−	0.955011i	$-0.595843\pi$
$239$	−2191.56	−0.593140	−0.296570	+	0.955011i	$0.595843\pi$
$240$	−1893.71	−0.509326
$241$	4000.34	1.06923	0.534615	−	0.845096i	$-0.320457\pi$
$241$	4000.34	1.06923	0.534615	+	0.845096i	$0.320457\pi$
$242$	−4427.27	−1.17601
$243$	243.000	0.0641500
$244$	1409.47	0.369803
$245$	−2541.11	−0.662634
$246$	4104.82	1.06388
$247$	−205.239	−0.0528707
$248$	−398.207	−0.101960
$249$	−334.502	−0.0851333
$250$	−5125.27	−1.29660
$251$	−2219.23	−0.558075	−0.279037	−	0.960280i	$-0.590015\pi$
$251$	−2219.23	−0.558075	−0.279037	+	0.960280i	$0.590015\pi$
$252$	163.578	0.0408906
$253$	−164.865	−0.0409683
$254$	1262.96	0.311989
$255$	−1637.77	−0.402200
$256$	4844.91	1.18284
$257$	−548.420	−0.133111	−0.0665554	−	0.997783i	$-0.521201\pi$
$257$	−548.420	−0.133111	−0.0665554	+	0.997783i	$0.521201\pi$
$258$	−2226.39	−0.537244
$259$	−782.428	−0.187713
$260$	302.751	0.0722148
$261$	355.804	0.0843822
$262$	2911.46	0.686528
$263$	−4180.82	−0.980230	−0.490115	−	0.871658i	$-0.663045\pi$
$263$	−4180.82	−0.980230	−0.490115	+	0.871658i	$0.663045\pi$
$264$	299.803	0.0698923
$265$	2115.85	0.490475
$266$	−336.370	−0.0775344
$267$	−4437.31	−1.01708
$268$	1166.04	0.265773
$269$	4339.45	0.983573	0.491787	−	0.870716i	$-0.336344\pi$
$269$	4339.45	0.983573	0.491787	+	0.870716i	$0.336344\pi$
$270$	737.098	0.166142
$271$	−1512.65	−0.339066	−0.169533	−	0.985524i	$-0.554226\pi$
$271$	−1512.65	−0.339066	−0.169533	+	0.985524i	$0.554226\pi$
$272$	5534.91	1.23383
$273$	132.711	0.0294214
$274$	9573.77	2.11085
$275$	449.720	0.0986150
$276$	273.960	0.0597480
$277$	537.755	0.116645	0.0583223	−	0.998298i	$-0.481425\pi$
$277$	537.755	0.116645	0.0583223	+	0.998298i	$0.481425\pi$
$278$	−4157.96	−0.897042
$279$	257.062	0.0551610
$280$	−503.574	−0.107480
$281$	8146.34	1.72943	0.864715	−	0.502262i	$-0.167499\pi$
$281$	8146.34	1.72943	0.864715	+	0.502262i	$0.167499\pi$
$282$	4025.17	0.849984
$283$	5231.91	1.09896	0.549479	−	0.835508i	$-0.314826\pi$
$283$	5231.91	1.09896	0.549479	+	0.835508i	$0.314826\pi$
$284$	2007.81	0.419513
$285$	−502.743	−0.104491
$286$	−239.661	−0.0495506
$287$	1810.35	0.372339
$288$	−1487.25	−0.304296
$289$	−126.148	−0.0256764
$290$	1079.27	0.218541
$291$	−3823.84	−0.770300
$292$	4162.06	0.834131
$293$	−1540.05	−0.307067	−0.153534	−	0.988143i	$-0.549065\pi$
$293$	−1540.05	−0.307067	−0.153534	+	0.988143i	$0.549065\pi$
$294$	−3342.66	−0.663089
$295$	4359.26	0.860360
$296$	2382.95	0.467925
$297$	−193.537	−0.0378120
$298$	−8343.57	−1.62191
$299$	222.264	0.0429896
$300$	−747.309	−0.143820
$301$	−981.903	−0.188026
$302$	8138.96	1.55081
$303$	−3090.25	−0.585909
$304$	1699.04	0.320548
$305$	2801.07	0.525864
$306$	−2154.38	−0.402476
$307$	−8549.38	−1.58938	−0.794689	−	0.607017i	$-0.792366\pi$
$307$	−8549.38	−1.58938	−0.794689	+	0.607017i	$0.792366\pi$
$308$	−130.281	−0.0241022
$309$	−3783.46	−0.696549
$310$	779.753	0.142861
$311$	−3678.75	−0.670749	−0.335374	−	0.942085i	$-0.608863\pi$
$311$	−3678.75	−0.670749	−0.335374	+	0.942085i	$0.608863\pi$
$312$	−404.181	−0.0733406
$313$	−4382.07	−0.791339	−0.395669	−	0.918393i	$-0.629487\pi$
$313$	−4382.07	−0.791339	−0.395669	+	0.918393i	$0.629487\pi$
$314$	−9411.80	−1.69152
$315$	325.082	0.0581469
$316$	−152.496	−0.0271474
$317$	−3015.16	−0.534221	−0.267111	−	0.963666i	$-0.586069\pi$
$317$	−3015.16	−0.534221	−0.267111	+	0.963666i	$0.586069\pi$
$318$	2783.27	0.490811
$319$	−283.380	−0.0497375
$320$	538.560	0.0940826
$321$	943.305	0.164019
$322$	364.273	0.0630438
$323$	1469.41	0.253128
$324$	321.605	0.0551449
$325$	−606.294	−0.103480
$326$	3148.82	0.534959
$327$	−1783.33	−0.301584
$328$	−5513.55	−0.928155
$329$	1775.22	0.297480
$330$	−587.061	−0.0979293
$331$	8098.73	1.34485	0.672426	−	0.740164i	$-0.265252\pi$
$331$	8098.73	1.34485	0.672426	+	0.740164i	$0.265252\pi$
$332$	−442.706	−0.0731827
$333$	−1538.31	−0.253149
$334$	1280.78	0.209824
$335$	2317.29	0.377932
$336$	−1098.63	−0.178378
$337$	−1279.84	−0.206876	−0.103438	−	0.994636i	$-0.532984\pi$
$337$	−1279.84	−0.206876	−0.103438	+	0.994636i	$0.532984\pi$
$338$	−7278.15	−1.17124
$339$	2115.35	0.338909
$340$	−2167.55	−0.345741
$341$	−204.737	−0.0325136
$342$	−661.326	−0.104563
$343$	−3044.35	−0.479240
$344$	2990.46	0.468706
$345$	544.447	0.0849624
$346$	−5793.36	−0.900153
$347$	−6965.46	−1.07760	−0.538798	−	0.842435i	$-0.681121\pi$
$347$	−6965.46	−1.07760	−0.538798	+	0.842435i	$0.681121\pi$
$348$	470.899	0.0725370
$349$	11580.9	1.77625	0.888123	−	0.459606i	$-0.152009\pi$
$349$	11580.9	1.77625	0.888123	+	0.459606i	$0.152009\pi$
$350$	−993.664	−0.151753
$351$	260.919	0.0396776
$352$	1184.52	0.179362
$353$	−1359.33	−0.204958	−0.102479	−	0.994735i	$-0.532677\pi$
$353$	−1359.33	−0.204958	−0.102479	+	0.994735i	$0.532677\pi$
$354$	5734.34	0.860951
$355$	3990.17	0.596553
$356$	−5872.69	−0.874303
$357$	−950.145	−0.140860
$358$	−6539.36	−0.965408
$359$	−11315.1	−1.66348	−0.831738	−	0.555168i	$-0.812654\pi$
$359$	−11315.1	−1.66348	−0.831738	+	0.555168i	$0.812654\pi$
$360$	−990.061	−0.144947
$361$	−6407.94	−0.934238
$362$	−4323.67	−0.627754
$363$	−3838.86	−0.555063
$364$	175.640	0.0252913
$365$	8271.36	1.18614
$366$	3684.62	0.526225
$367$	−9260.84	−1.31720	−0.658600	−	0.752494i	$-0.728851\pi$
$367$	−9260.84	−1.31720	−0.658600	+	0.752494i	$0.728851\pi$
$368$	−1839.98	−0.260640
$369$	3559.27	0.502136
$370$	−4666.19	−0.655631
$371$	1227.50	0.171776
$372$	340.216	0.0474177
$373$	7448.35	1.03394	0.516972	−	0.856002i	$-0.327059\pi$
$373$	7448.35	1.03394	0.516972	+	0.856002i	$0.327059\pi$
$374$	1715.86	0.237232
$375$	−4444.09	−0.611979
$376$	−5406.56	−0.741549
$377$	382.042	0.0521914
$378$	427.624	0.0581868
$379$	13428.8	1.82003	0.910017	−	0.414570i	$-0.136068\pi$
$379$	13428.8	1.82003	0.910017	+	0.414570i	$0.136068\pi$
$380$	−665.369	−0.0898229
$381$	1095.11	0.147255
$382$	14891.2	1.99450
$383$	−12530.8	−1.67178	−0.835891	−	0.548896i	$-0.815048\pi$
$383$	−12530.8	−1.67178	−0.835891	+	0.548896i	$0.815048\pi$
$384$	4674.45	0.621204
$385$	−258.911	−0.0342736
$386$	−10225.8	−1.34839
$387$	−1930.49	−0.253572
$388$	−5060.76	−0.662168
$389$	−2037.90	−0.265619	−0.132809	−	0.991142i	$-0.542400\pi$
$389$	−2037.90	−0.265619	−0.132809	+	0.991142i	$0.542400\pi$
$390$	791.452	0.102761
$391$	−1591.30	−0.205820
$392$	4489.83	0.578496
$393$	2524.51	0.324032
$394$	10163.9	1.29962
$395$	−303.059	−0.0386040
$396$	−256.142	−0.0325041
$397$	−12445.6	−1.57337	−0.786683	−	0.617357i	$-0.788203\pi$
$397$	−12445.6	−1.57337	−0.786683	+	0.617357i	$0.788203\pi$
$398$	14675.4	1.84827
$399$	−291.664	−0.0365952
$400$	5019.11	0.627388
$401$	7515.90	0.935976	0.467988	−	0.883735i	$-0.344979\pi$
$401$	7515.90	0.935976	0.467988	+	0.883735i	$0.344979\pi$
$402$	3048.25	0.378191
$403$	276.018	0.0341178
$404$	−4089.88	−0.503661
$405$	639.133	0.0784167
$406$	626.134	0.0765382
$407$	1225.19	0.149214
$408$	2893.74	0.351131
$409$	4458.77	0.539051	0.269526	−	0.962993i	$-0.413133\pi$
$409$	4458.77	0.539051	0.269526	+	0.962993i	$0.413133\pi$
$410$	10796.4	1.30048
$411$	8301.36	0.996292
$412$	−5007.33	−0.598770
$413$	2529.01	0.301318
$414$	716.185	0.0850207
$415$	−879.799	−0.104067
$416$	−1596.93	−0.188211
$417$	−3605.34	−0.423392
$418$	526.713	0.0616325
$419$	4506.59	0.525445	0.262722	−	0.964871i	$-0.415380\pi$
$419$	4506.59	0.525445	0.262722	+	0.964871i	$0.415380\pi$
$420$	430.238	0.0499845
$421$	4391.61	0.508395	0.254197	−	0.967152i	$-0.418189\pi$
$421$	4391.61	0.508395	0.254197	+	0.967152i	$0.418189\pi$
$422$	17373.5	2.00409
$423$	3490.20	0.401181
$424$	−3738.46	−0.428197
$425$	4340.76	0.495431
$426$	5248.82	0.596962
$427$	1625.03	0.184170
$428$	1248.44	0.140995
$429$	−207.809	−0.0233872
$430$	−5855.80	−0.656725
$431$	−4034.85	−0.450932	−0.225466	−	0.974251i	$-0.572390\pi$
$431$	−4034.85	−0.450932	−0.225466	+	0.974251i	$0.572390\pi$
$432$	−2159.98	−0.240560
$433$	−2378.67	−0.263999	−0.131999	−	0.991250i	$-0.542140\pi$
$433$	−2378.67	−0.263999	−0.131999	+	0.991250i	$0.542140\pi$
$434$	452.371	0.0500334
$435$	935.829	0.103148
$436$	−2360.19	−0.259249
$437$	−488.479	−0.0534717
$438$	10880.5	1.18696
$439$	6767.22	0.735722	0.367861	−	0.929881i	$-0.380090\pi$
$439$	6767.22	0.735722	0.367861	+	0.929881i	$0.380090\pi$
$440$	788.534	0.0854361
$441$	−2898.41	−0.312969
$442$	−2313.25	−0.248936
$443$	2767.13	0.296773	0.148386	−	0.988929i	$-0.452592\pi$
$443$	2767.13	0.296773	0.148386	+	0.988929i	$0.452592\pi$
$444$	−2035.92	−0.217613
$445$	−11670.9	−1.24327
$446$	−4192.05	−0.445066
$447$	−7234.66	−0.765521
$448$	312.443	0.0329499
$449$	3616.87	0.380158	0.190079	−	0.981769i	$-0.439126\pi$
$449$	3616.87	0.380158	0.190079	+	0.981769i	$0.439126\pi$
$450$	−1953.61	−0.204654
$451$	−2834.78	−0.295974
$452$	2799.62	0.291334
$453$	7057.24	0.731961
$454$	15654.5	1.61828
$455$	349.053	0.0359646
$456$	888.286	0.0912233
$457$	−5912.55	−0.605202	−0.302601	−	0.953117i	$-0.597855\pi$
$457$	−5912.55	−0.605202	−0.302601	+	0.953117i	$0.597855\pi$
$458$	−22883.8	−2.33469
$459$	−1868.05	−0.189963
$460$	720.563	0.0730357
$461$	3865.90	0.390570	0.195285	−	0.980747i	$-0.437437\pi$
$461$	3865.90	0.390570	0.195285	+	0.980747i	$0.437437\pi$
$462$	−340.581	−0.0342971
$463$	−11789.4	−1.18337	−0.591687	−	0.806168i	$-0.701538\pi$
$463$	−11789.4	−1.18337	−0.591687	+	0.806168i	$0.701538\pi$
$464$	−3162.67	−0.316430
$465$	676.120	0.0674286
$466$	−6979.25	−0.693793
$467$	3867.34	0.383210	0.191605	−	0.981472i	$-0.438631\pi$
$467$	3867.34	0.383210	0.191605	+	0.981472i	$0.438631\pi$
$468$	345.321	0.0341078
$469$	1344.37	0.132361
$470$	10586.9	1.03902
$471$	−8160.91	−0.798376
$472$	−7702.30	−0.751116
$473$	1537.54	0.149463
$474$	−398.655	−0.0386305
$475$	1332.48	0.128712
$476$	−1257.50	−0.121087
$477$	2413.36	0.231656
$478$	−7582.44	−0.725549
$479$	−17656.8	−1.68425	−0.842127	−	0.539279i	$-0.818697\pi$
$479$	−17656.8	−1.68425	−0.842127	+	0.539279i	$0.818697\pi$
$480$	−3911.74	−0.371971
$481$	−1651.74	−0.156576
$482$	13840.5	1.30792
$483$	315.859	0.0297558
$484$	−5080.64	−0.477145
$485$	−10057.4	−0.941611
$486$	840.739	0.0784706
$487$	−5750.71	−0.535092	−0.267546	−	0.963545i	$-0.586213\pi$
$487$	−5750.71	−0.535092	−0.267546	+	0.963545i	$0.586213\pi$
$488$	−4949.14	−0.459093
$489$	2730.32	0.252493
$490$	−8791.80	−0.810557
$491$	12090.0	1.11123	0.555616	−	0.831439i	$-0.312483\pi$
$491$	12090.0	1.11123	0.555616	+	0.831439i	$0.312483\pi$
$492$	4710.61	0.431648
$493$	−2735.23	−0.249875
$494$	−710.093	−0.0646733
$495$	−509.037	−0.0462213
$496$	−2284.97	−0.206852
$497$	2314.88	0.208927
$498$	−1157.32	−0.104138
$499$	14759.4	1.32410	0.662048	−	0.749462i	$-0.269688\pi$
$499$	14759.4	1.32410	0.662048	+	0.749462i	$0.269688\pi$
$500$	−5881.66	−0.526072
$501$	1110.56	0.0990339
$502$	−7678.17	−0.682657
$503$	15790.7	1.39975	0.699875	−	0.714265i	$-0.253239\pi$
$503$	15790.7	1.39975	0.699875	+	0.714265i	$0.253239\pi$
$504$	−574.380	−0.0507637
$505$	−8127.91	−0.716212
$506$	−570.406	−0.0501139
$507$	−6310.84	−0.552809
$508$	1449.35	0.126584
$509$	−4800.46	−0.418029	−0.209014	−	0.977913i	$-0.567026\pi$
$509$	−4800.46	−0.418029	−0.209014	+	0.977913i	$0.567026\pi$
$510$	−5666.40	−0.491985
$511$	4798.60	0.415416
$512$	4297.36	0.370934
$513$	−573.432	−0.0493522
$514$	−1897.44	−0.162826
$515$	−9951.17	−0.851458
$516$	−2554.96	−0.217976
$517$	−2779.77	−0.236469
$518$	−2707.07	−0.229617
$519$	−5023.39	−0.424860
$520$	−1063.07	−0.0896513
$521$	−5068.16	−0.426180	−0.213090	−	0.977033i	$-0.568353\pi$
$521$	−5068.16	−0.426180	−0.213090	+	0.977033i	$0.568353\pi$
$522$	1231.02	0.103219
$523$	−4316.59	−0.360901	−0.180451	−	0.983584i	$-0.557756\pi$
$523$	−4316.59	−0.360901	−0.180451	+	0.983584i	$0.557756\pi$
$524$	3341.13	0.278546
$525$	−861.601	−0.0716254
$526$	−14464.9	−1.19905
$527$	−1976.16	−0.163345
$528$	1720.31	0.141794
$529$	529.000	0.0434783
$530$	7320.49	0.599966
$531$	4972.21	0.406357
$532$	−386.011	−0.0314581
$533$	3821.73	0.310577
$534$	−15352.4	−1.24412
$535$	2481.06	0.200496
$536$	−4094.38	−0.329944
$537$	−5670.25	−0.455660
$538$	15013.8	1.20314
$539$	2308.43	0.184474
$540$	845.879	0.0674089
$541$	19439.9	1.54490	0.772448	−	0.635079i	$-0.219032\pi$
$541$	19439.9	1.54490	0.772448	+	0.635079i	$0.219032\pi$
$542$	−5233.51	−0.414758
$543$	−3749.03	−0.296291
$544$	11433.2	0.901093
$545$	−4690.46	−0.368656
$546$	459.158	0.0359893
$547$	17162.2	1.34150	0.670752	−	0.741682i	$-0.265972\pi$
$547$	17162.2	1.34150	0.670752	+	0.741682i	$0.265972\pi$
$548$	10986.7	0.856437
$549$	3194.92	0.248371
$550$	1555.96	0.120629
$551$	−839.629	−0.0649172
$552$	−961.972	−0.0741743
$553$	−175.819	−0.0135200
$554$	1860.54	0.142684
$555$	−4046.02	−0.309449
$556$	−4771.59	−0.363958
$557$	9533.83	0.725245	0.362622	−	0.931936i	$-0.381881\pi$
$557$	9533.83	0.725245	0.362622	+	0.931936i	$0.381881\pi$
$558$	889.393	0.0674749
$559$	−2072.85	−0.156837
$560$	−2889.58	−0.218048
$561$	1487.81	0.111970
$562$	28185.0	2.11550
$563$	−14499.5	−1.08540	−0.542702	−	0.839925i	$-0.682599\pi$
$563$	−14499.5	−1.08540	−0.542702	+	0.839925i	$0.682599\pi$
$564$	4619.21	0.344865
$565$	5563.75	0.414281
$566$	18101.5	1.34428
$567$	370.791	0.0274634
$568$	−7050.15	−0.520806
$569$	−3689.09	−0.271801	−0.135900	−	0.990723i	$-0.543393\pi$
$569$	−3689.09	−0.271801	−0.135900	+	0.990723i	$0.543393\pi$
$570$	−1739.41	−0.127817
$571$	−18230.6	−1.33612	−0.668061	−	0.744106i	$-0.732876\pi$
$571$	−18230.6	−1.33612	−0.668061	+	0.744106i	$0.732876\pi$
$572$	−275.031	−0.0201042
$573$	12912.1	0.941377
$574$	6263.49	0.455458
$575$	−1443.01	−0.104657
$576$	614.286	0.0444362
$577$	20164.4	1.45486	0.727432	−	0.686180i	$-0.240714\pi$
$577$	20164.4	1.45486	0.727432	+	0.686180i	$0.240714\pi$
$578$	−436.451	−0.0314083
$579$	−8866.69	−0.636420
$580$	1238.55	0.0886689
$581$	−510.412	−0.0364466
$582$	−13229.8	−0.942258
$583$	−1922.12	−0.136545
$584$	−14614.5	−1.03553
$585$	686.263	0.0485017
$586$	−5328.32	−0.375616
$587$	3604.51	0.253448	0.126724	−	0.991938i	$-0.459554\pi$
$587$	3604.51	0.253448	0.126724	+	0.991938i	$0.459554\pi$
$588$	−3835.98	−0.269036
$589$	−606.617	−0.0424367
$590$	15082.3	1.05242
$591$	8813.05	0.613402
$592$	13673.7	0.949300
$593$	21451.6	1.48551	0.742757	−	0.669561i	$-0.233518\pi$
$593$	21451.6	1.48551	0.742757	+	0.669561i	$0.233518\pi$
$594$	−669.607	−0.0462530
$595$	−2499.05	−0.172187
$596$	−9574.92	−0.658060
$597$	12724.9	0.872357
$598$	768.997	0.0525863
$599$	−14824.4	−1.01120	−0.505599	−	0.862769i	$-0.668728\pi$
$599$	−14824.4	−1.01120	−0.505599	+	0.862769i	$0.668728\pi$
$600$	2624.07	0.178545
$601$	−18786.5	−1.27507	−0.637535	−	0.770422i	$-0.720046\pi$
$601$	−18786.5	−1.27507	−0.637535	+	0.770422i	$0.720046\pi$
$602$	−3397.22	−0.230001
$603$	2643.12	0.178501
$604$	9340.11	0.629211
$605$	−10096.9	−0.678506
$606$	−10691.8	−0.716704
$607$	−2160.73	−0.144484	−0.0722418	−	0.997387i	$-0.523015\pi$
$607$	−2160.73	−0.144484	−0.0722418	+	0.997387i	$0.523015\pi$
$608$	3509.63	0.234102
$609$	542.917	0.0361250
$610$	9691.21	0.643255
$611$	3747.57	0.248135
$612$	−2472.32	−0.163297
$613$	20705.6	1.36426	0.682131	−	0.731230i	$-0.261053\pi$
$613$	20705.6	1.36426	0.682131	+	0.731230i	$0.261053\pi$
$614$	−29579.4	−1.94418
$615$	9361.50	0.613808
$616$	457.465	0.0299217
$617$	−19079.1	−1.24488	−0.622442	−	0.782666i	$-0.713860\pi$
$617$	−19079.1	−1.24488	−0.622442	+	0.782666i	$0.713860\pi$
$618$	−13090.1	−0.852043
$619$	18244.3	1.18465	0.592325	−	0.805699i	$-0.298210\pi$
$619$	18244.3	1.18465	0.592325	+	0.805699i	$0.298210\pi$
$620$	894.830	0.0579633
$621$	621.000	0.0401286
$622$	−12727.9	−0.820484
$623$	−6770.84	−0.435422
$624$	−2319.25	−0.148789
$625$	−3846.31	−0.246164
$626$	−15161.2	−0.967993
$627$	456.710	0.0290897
$628$	−10800.8	−0.686303
$629$	11825.7	0.749635
$630$	1124.73	0.0711273
$631$	4623.60	0.291700	0.145850	−	0.989307i	$-0.453408\pi$
$631$	4623.60	0.291700	0.145850	+	0.989307i	$0.453408\pi$
$632$	535.469	0.0337022
$633$	15064.4	0.945904
$634$	−10431.9	−0.653478
$635$	2880.32	0.180003
$636$	3194.03	0.199137
$637$	−3112.14	−0.193575
$638$	−980.449	−0.0608406
$639$	4551.22	0.281758
$640$	12294.6	0.759357
$641$	21059.1	1.29763	0.648816	−	0.760945i	$-0.275264\pi$
$641$	21059.1	1.29763	0.648816	+	0.760945i	$0.275264\pi$
$642$	3263.68	0.200634
$643$	18828.5	1.15478	0.577390	−	0.816468i	$-0.304071\pi$
$643$	18828.5	1.15478	0.577390	+	0.816468i	$0.304071\pi$
$644$	418.032	0.0255788
$645$	−5077.53	−0.309965
$646$	5083.92	0.309635
$647$	−18686.1	−1.13543	−0.567716	−	0.823224i	$-0.692173\pi$
$647$	−18686.1	−1.13543	−0.567716	+	0.823224i	$0.692173\pi$
$648$	−1129.27	−0.0684598
$649$	−3960.12	−0.239520
$650$	−2097.67	−0.126581
$651$	392.248	0.0236151
$652$	3613.52	0.217050
$653$	−13267.2	−0.795078	−0.397539	−	0.917585i	$-0.630136\pi$
$653$	−13267.2	−0.795078	−0.397539	+	0.917585i	$0.630136\pi$
$654$	−6170.01	−0.368909
$655$	6639.90	0.396095
$656$	−31637.6	−1.88299
$657$	9434.38	0.560228
$658$	6141.96	0.363888
$659$	9430.56	0.557455	0.278727	−	0.960370i	$-0.410087\pi$
$659$	9430.56	0.557455	0.278727	+	0.960370i	$0.410087\pi$
$660$	−673.700	−0.0397329
$661$	−8433.69	−0.496267	−0.248134	−	0.968726i	$-0.579817\pi$
$661$	−8433.69	−0.496267	−0.248134	+	0.968726i	$0.579817\pi$
$662$	28020.2	1.64507
$663$	−2005.80	−0.117495
$664$	1554.50	0.0908528
$665$	−767.129	−0.0447338
$666$	−5322.29	−0.309661
$667$	909.278	0.0527847
$668$	1469.80	0.0851319
$669$	−3634.91	−0.210065
$670$	8017.44	0.462300
$671$	−2544.59	−0.146398
$672$	−2269.38	−0.130273
$673$	1486.31	0.0851310	0.0425655	−	0.999094i	$-0.486447\pi$
$673$	1486.31	0.0851310	0.0425655	+	0.999094i	$0.486447\pi$
$674$	−4428.03	−0.253058
$675$	−1693.97	−0.0965938
$676$	−8352.26	−0.475208
$677$	3996.72	0.226893	0.113446	−	0.993544i	$-0.463811\pi$
$677$	3996.72	0.226893	0.113446	+	0.993544i	$0.463811\pi$
$678$	7318.76	0.414565
$679$	−5834.74	−0.329774
$680$	7611.05	0.429221
$681$	13573.9	0.763807
$682$	−708.357	−0.0397718
$683$	−4484.96	−0.251262	−0.125631	−	0.992077i	$-0.540096\pi$
$683$	−4484.96	−0.251262	−0.125631	+	0.992077i	$0.540096\pi$
$684$	−758.925	−0.0424243
$685$	21834.0	1.21786
$686$	−10532.9	−0.586224
$687$	−19842.4	−1.10194
$688$	17159.7	0.950884
$689$	2591.32	0.143282
$690$	1883.69	0.103929
$691$	8534.58	0.469856	0.234928	−	0.972013i	$-0.424515\pi$
$691$	8534.58	0.469856	0.234928	+	0.972013i	$0.424515\pi$
$692$	−6648.35	−0.365220
$693$	−295.316	−0.0161878
$694$	−24099.3	−1.31815
$695$	−9482.69	−0.517552
$696$	−1653.50	−0.0900512
$697$	−27361.7	−1.48694
$698$	40067.9	2.17277
$699$	−6051.67	−0.327461
$700$	−1140.31	−0.0615709
$701$	5562.07	0.299681	0.149841	−	0.988710i	$-0.452124\pi$
$701$	5562.07	0.299681	0.149841	+	0.988710i	$0.452124\pi$
$702$	902.736	0.0485350
$703$	3630.11	0.194754
$704$	−489.248	−0.0261921
$705$	9179.85	0.490402
$706$	−4703.07	−0.250711
$707$	−4715.38	−0.250834
$708$	6580.61	0.349314
$709$	8050.95	0.426460	0.213230	−	0.977002i	$-0.431602\pi$
$709$	8050.95	0.426460	0.213230	+	0.977002i	$0.431602\pi$
$710$	13805.3	0.729724
$711$	−345.672	−0.0182331
$712$	20621.1	1.08541
$713$	656.937	0.0345056
$714$	−3287.34	−0.172305
$715$	−546.574	−0.0285884
$716$	−7504.45	−0.391696
$717$	−6574.69	−0.342449
$718$	−39148.3	−2.03482
$719$	5103.08	0.264691	0.132345	−	0.991204i	$-0.457749\pi$
$719$	5103.08	0.264691	0.132345	+	0.991204i	$0.457749\pi$
$720$	−5681.12	−0.294060
$721$	−5773.13	−0.298201
$722$	−22170.4	−1.14279
$723$	12001.0	0.617320
$724$	−4961.75	−0.254699
$725$	−2480.33	−0.127058
$726$	−13281.8	−0.678972
$727$	−19611.5	−1.00048	−0.500240	−	0.865887i	$-0.666755\pi$
$727$	−19611.5	−1.00048	−0.500240	+	0.865887i	$0.666755\pi$
$728$	−616.735	−0.0313980
$729$	729.000	0.0370370
$730$	28617.5	1.45093
$731$	14840.5	0.750886
$732$	4228.40	0.213506
$733$	−13236.7	−0.666998	−0.333499	−	0.942750i	$-0.608229\pi$
$733$	−13236.7	−0.666998	−0.333499	+	0.942750i	$0.608229\pi$
$734$	−32041.0	−1.61124
$735$	−7623.32	−0.382572
$736$	−3800.76	−0.190350
$737$	−2105.11	−0.105214
$738$	12314.5	0.614230
$739$	−25308.9	−1.25981	−0.629906	−	0.776671i	$-0.716907\pi$
$739$	−25308.9	−1.25981	−0.629906	+	0.776671i	$0.716907\pi$
$740$	−5354.82	−0.266010
$741$	−615.718	−0.0305249
$742$	4246.95	0.210122
$743$	15214.4	0.751230	0.375615	−	0.926776i	$-0.377432\pi$
$743$	15214.4	0.751230	0.375615	+	0.926776i	$0.377432\pi$
$744$	−1194.62	−0.0588669
$745$	−19028.4	−0.935770
$746$	25770.0	1.26476
$747$	−1003.51	−0.0491517
$748$	1969.08	0.0962524
$749$	1439.38	0.0702185
$750$	−15375.8	−0.748594
$751$	−2600.26	−0.126345	−0.0631724	−	0.998003i	$-0.520122\pi$
$751$	−2600.26	−0.126345	−0.0631724	+	0.998003i	$0.520122\pi$
$752$	−31023.7	−1.50441
$753$	−6657.70	−0.322205
$754$	1321.80	0.0638423
$755$	18561.8	0.894746
$756$	490.733	0.0236082
$757$	−24500.2	−1.17632	−0.588160	−	0.808745i	$-0.700147\pi$
$757$	−24500.2	−1.17632	−0.588160	+	0.808745i	$0.700147\pi$
$758$	46461.5	2.22633
$759$	−494.595	−0.0236531
$760$	2336.35	0.111511
$761$	7901.08	0.376365	0.188183	−	0.982134i	$-0.439740\pi$
$761$	7901.08	0.376365	0.188183	+	0.982134i	$0.439740\pi$
$762$	3788.88	0.180127
$763$	−2721.15	−0.129112
$764$	17088.8	0.809231
$765$	−4913.31	−0.232210
$766$	−43354.3	−2.04498
$767$	5338.86	0.251337
$768$	14534.7	0.682913
$769$	−33754.8	−1.58287	−0.791436	−	0.611252i	$-0.790666\pi$
$769$	−33754.8	−1.58287	−0.791436	+	0.611252i	$0.790666\pi$
$770$	−895.789	−0.0419247
$771$	−1645.26	−0.0768516
$772$	−11734.9	−0.547082
$773$	9500.18	0.442041	0.221021	−	0.975269i	$-0.429061\pi$
$773$	9500.18	0.442041	0.221021	+	0.975269i	$0.429061\pi$
$774$	−6679.17	−0.310178
$775$	−1792.00	−0.0830586
$776$	17770.2	0.822051
$777$	−2347.28	−0.108376
$778$	−7050.79	−0.324914
$779$	−8399.17	−0.386305
$780$	908.254	0.0416932
$781$	−3624.82	−0.166077
$782$	−5505.64	−0.251766
$783$	1067.41	0.0487181
$784$	25763.3	1.17362
$785$	−21464.6	−0.975931
$786$	8734.37	0.396367
$787$	−37524.0	−1.69960	−0.849800	−	0.527105i	$-0.823277\pi$
$787$	−37524.0	−1.69960	−0.849800	+	0.527105i	$0.823277\pi$
$788$	11663.9	0.527295
$789$	−12542.5	−0.565936
$790$	−1048.53	−0.0472217
$791$	3227.79	0.145091
$792$	899.408	0.0403524
$793$	3430.51	0.153620
$794$	−43059.6	−1.92460
$795$	6347.56	0.283176
$796$	16841.2	0.749899
$797$	33330.8	1.48135	0.740675	−	0.671864i	$-0.234506\pi$
$797$	33330.8	1.48135	0.740675	+	0.671864i	$0.234506\pi$
$798$	−1009.11	−0.0447645
$799$	−26830.8	−1.18799
$800$	10367.7	0.458194
$801$	−13311.9	−0.587209
$802$	26003.8	1.14492
$803$	−7514.01	−0.330216
$804$	3498.11	0.153444
$805$	830.764	0.0363734
$806$	954.977	0.0417340
$807$	13018.4	0.567866
$808$	14361.0	0.625272
$809$	16696.0	0.725586	0.362793	−	0.931870i	$-0.381823\pi$
$809$	16696.0	0.725586	0.362793	+	0.931870i	$0.381823\pi$
$810$	2211.29	0.0959221
$811$	27187.8	1.17718	0.588589	−	0.808432i	$-0.299684\pi$
$811$	27187.8	1.17718	0.588589	+	0.808432i	$0.299684\pi$
$812$	718.539	0.0310539
$813$	−4537.95	−0.195760
$814$	4238.94	0.182524
$815$	7181.22	0.308647
$816$	16604.7	0.712355
$817$	4555.58	0.195079
$818$	15426.6	0.659387
$819$	398.133	0.0169864
$820$	12389.7	0.527645
$821$	−14087.5	−0.598852	−0.299426	−	0.954119i	$-0.596795\pi$
$821$	−14087.5	−0.598852	−0.299426	+	0.954119i	$0.596795\pi$
$822$	28721.3	1.21870
$823$	10841.3	0.459178	0.229589	−	0.973288i	$-0.426262\pi$
$823$	10841.3	0.459178	0.229589	+	0.973288i	$0.426262\pi$
$824$	17582.5	0.743345
$825$	1349.16	0.0569354
$826$	8749.95	0.368583
$827$	11420.3	0.480195	0.240097	−	0.970749i	$-0.422821\pi$
$827$	11420.3	0.480195	0.240097	+	0.970749i	$0.422821\pi$
$828$	821.880	0.0344955
$829$	20555.8	0.861196	0.430598	−	0.902544i	$-0.358303\pi$
$829$	20555.8	0.861196	0.430598	+	0.902544i	$0.358303\pi$
$830$	−3043.96	−0.127298
$831$	1613.27	0.0673448
$832$	659.583	0.0274843
$833$	22281.4	0.926775
$834$	−12473.9	−0.517908
$835$	2920.96	0.121059
$836$	604.446	0.0250062
$837$	771.187	0.0318472
$838$	15592.0	0.642742
$839$	−12.5459	−0.000516247 0	−0.000258123	−	1.00000i	$-0.500082\pi$
$839$	−12.5459	−0.000516247 0	−0.000258123		1.00000i	$0.500082\pi$
$840$	−1510.72	−0.0620534
$841$	−22826.1	−0.935917
$842$	15194.2	0.621886
$843$	24439.0	0.998487
$844$	19937.4	0.813122
$845$	−16598.6	−0.675752
$846$	12075.5	0.490738
$847$	−5857.66	−0.237629
$848$	−21451.8	−0.868701
$849$	15695.7	0.634483
$850$	15018.3	0.606028
$851$	−3931.23	−0.158356
$852$	6023.44	0.242206
$853$	−19060.7	−0.765096	−0.382548	−	0.923936i	$-0.624953\pi$
$853$	−19060.7	−0.765096	−0.382548	+	0.923936i	$0.624953\pi$
$854$	5622.32	0.225283
$855$	−1508.23	−0.0603279
$856$	−4383.73	−0.175038
$857$	−15239.1	−0.607418	−0.303709	−	0.952765i	$-0.598225\pi$
$857$	−15239.1	−0.607418	−0.303709	+	0.952765i	$0.598225\pi$
$858$	−718.984	−0.0286080
$859$	9970.27	0.396020	0.198010	−	0.980200i	$-0.436552\pi$
$859$	9970.27	0.396020	0.198010	+	0.980200i	$0.436552\pi$
$860$	−6720.00	−0.266454
$861$	5431.04	0.214970
$862$	−13959.9	−0.551596
$863$	35257.3	1.39070	0.695350	−	0.718671i	$-0.255249\pi$
$863$	35257.3	1.39070	0.695350	+	0.718671i	$0.255249\pi$
$864$	−4461.76	−0.175686
$865$	−13212.4	−0.519347
$866$	−8229.79	−0.322932
$867$	−378.444	−0.0148243
$868$	519.132	0.0203001
$869$	275.310	0.0107471
$870$	3237.81	0.126175
$871$	2838.03	0.110405
$872$	8287.48	0.321846
$873$	−11471.5	−0.444733
$874$	−1690.06	−0.0654085
$875$	−6781.18	−0.261995
$876$	12486.2	0.481586
$877$	−19721.8	−0.759358	−0.379679	−	0.925118i	$-0.623966\pi$
$877$	−19721.8	−0.759358	−0.379679	+	0.925118i	$0.623966\pi$
$878$	23413.4	0.899960
$879$	−4620.15	−0.177285
$880$	4524.73	0.173328
$881$	27621.7	1.05630	0.528150	−	0.849151i	$-0.322886\pi$
$881$	27621.7	1.05630	0.528150	+	0.849151i	$0.322886\pi$
$882$	−10028.0	−0.382835
$883$	35195.3	1.34135	0.670677	−	0.741750i	$-0.266004\pi$
$883$	35195.3	1.34135	0.670677	+	0.741750i	$0.266004\pi$
$884$	−2654.64	−0.101001
$885$	13077.8	0.496729
$886$	9573.80	0.363023
$887$	−37874.1	−1.43369	−0.716847	−	0.697231i	$-0.754415\pi$
$887$	−37874.1	−1.43369	−0.716847	+	0.697231i	$0.754415\pi$
$888$	7148.84	0.270157
$889$	1671.01	0.0630414
$890$	−40379.4	−1.52081
$891$	−580.612	−0.0218308
$892$	−4810.72	−0.180577
$893$	−8236.19	−0.308638
$894$	−25030.7	−0.936412
$895$	−14913.7	−0.556996
$896$	7132.69	0.265945
$897$	666.793	0.0248200
$898$	12513.8	0.465022
$899$	1129.18	0.0418915
$900$	−2241.93	−0.0830344
$901$	−18552.6	−0.685989
$902$	−9807.85	−0.362046
$903$	−2945.71	−0.108557
$904$	−9830.48	−0.361678
$905$	−9860.60	−0.362185
$906$	24416.9	0.895360
$907$	−23648.3	−0.865744	−0.432872	−	0.901455i	$-0.642500\pi$
$907$	−23648.3	−0.865744	−0.432872	+	0.901455i	$0.642500\pi$
$908$	17964.7	0.656587
$909$	−9270.76	−0.338275
$910$	1207.67	0.0439931
$911$	−24936.9	−0.906911	−0.453456	−	0.891279i	$-0.649809\pi$
$911$	−24936.9	−0.906911	−0.453456	+	0.891279i	$0.649809\pi$
$912$	5097.12	0.185069
$913$	799.242	0.0289716
$914$	−20456.4	−0.740304
$915$	8403.20	0.303608
$916$	−26260.9	−0.947255
$917$	3852.11	0.138722
$918$	−6463.14	−0.232370
$919$	38348.8	1.37651	0.688253	−	0.725470i	$-0.258378\pi$
$919$	38348.8	1.37651	0.688253	+	0.725470i	$0.258378\pi$
$920$	−2530.16	−0.0906704
$921$	−25648.1	−0.917628
$922$	13375.4	0.477759
$923$	4886.83	0.174271
$924$	−390.844	−0.0139154
$925$	10723.6	0.381179
$926$	−40789.5	−1.44754
$927$	−11350.4	−0.402153
$928$	−6532.99	−0.231095
$929$	−28584.5	−1.00950	−0.504751	−	0.863265i	$-0.668416\pi$
$929$	−28584.5	−1.00950	−0.504751	+	0.863265i	$0.668416\pi$
$930$	2339.26	0.0824810
$931$	6839.67	0.240774
$932$	−8009.25	−0.281493
$933$	−11036.3	−0.387257
$934$	13380.3	0.468756
$935$	3913.20	0.136872
$936$	−1212.54	−0.0423432
$937$	−6982.17	−0.243434	−0.121717	−	0.992565i	$-0.538840\pi$
$937$	−6982.17	−0.243434	−0.121717	+	0.992565i	$0.538840\pi$
$938$	4651.29	0.161908
$939$	−13146.2	−0.456880
$940$	12149.3	0.421561
$941$	−34182.7	−1.18419	−0.592095	−	0.805868i	$-0.701699\pi$
$941$	−34182.7	−1.18419	−0.592095	+	0.805868i	$0.701699\pi$
$942$	−28235.4	−0.976602
$943$	9095.90	0.314108
$944$	−44196.9	−1.52382
$945$	975.245	0.0335711
$946$	5319.62	0.182829
$947$	−44105.7	−1.51346	−0.756729	−	0.653729i	$-0.773204\pi$
$947$	−44105.7	−1.51346	−0.756729	+	0.653729i	$0.773204\pi$
$948$	−457.489	−0.0156736
$949$	10130.1	0.346508
$950$	4610.14	0.157445
$951$	−9045.47	−0.308433
$952$	4415.52	0.150323
$953$	22993.3	0.781560	0.390780	−	0.920484i	$-0.372205\pi$
$953$	22993.3	0.781560	0.390780	+	0.920484i	$0.372205\pi$
$954$	8349.81	0.283370
$955$	33961.0	1.15074
$956$	−8701.45	−0.294378
$957$	−850.141	−0.0287160
$958$	−61089.4	−2.06024
$959$	12666.9	0.426524
$960$	1615.68	0.0543186
$961$	−28975.2	−0.972615
$962$	−5714.76	−0.191529
$963$	2829.91	0.0946965
$964$	15883.1	0.530664
$965$	−23321.0	−0.777957
$966$	1092.82	0.0363984
$967$	30170.2	1.00332	0.501658	−	0.865066i	$-0.332724\pi$
$967$	30170.2	1.00332	0.501658	+	0.865066i	$0.332724\pi$
$968$	17840.0	0.592353
$969$	4408.23	0.146143
$970$	−34796.8	−1.15181
$971$	59927.1	1.98059	0.990295	−	0.138984i	$-0.0443838\pi$
$971$	59927.1	1.98059	0.990295	+	0.138984i	$0.0443838\pi$
$972$	964.816	0.0318379
$973$	−5501.34	−0.181259
$974$	−19896.5	−0.654543
$975$	−1818.88	−0.0597444
$976$	−28398.9	−0.931381
$977$	−22546.4	−0.738304	−0.369152	−	0.929369i	$-0.620352\pi$
$977$	−22546.4	−0.738304	−0.369152	+	0.929369i	$0.620352\pi$
$978$	9446.45	0.308859
$979$	10602.3	0.346119
$980$	−10089.3	−0.328868
$981$	−5349.98	−0.174120
$982$	41829.4	1.35930
$983$	1932.66	0.0627084	0.0313542	−	0.999508i	$-0.490018\pi$
$983$	1932.66	0.0627084	0.0313542	+	0.999508i	$0.490018\pi$
$984$	−16540.6	−0.535871
$985$	23179.9	0.749820
$986$	−9463.44	−0.305656
$987$	5325.66	0.171750
$988$	−814.889	−0.0262399
$989$	−4933.47	−0.158620
$990$	−1761.18	−0.0565395
$991$	−47353.0	−1.51788	−0.758939	−	0.651161i	$-0.774282\pi$
$991$	−47353.0	−1.51788	−0.758939	+	0.651161i	$0.774282\pi$
$992$	−4719.97	−0.151068
$993$	24296.2	0.776451
$994$	8009.10	0.255567
$995$	33468.8	1.06637
$996$	−1328.12	−0.0422520
$997$	20613.3	0.654795	0.327397	−	0.944887i	$-0.393828\pi$
$997$	20613.3	0.654795	0.327397	+	0.944887i	$0.393828\pi$
$998$	51065.2	1.61968
$999$	−4614.92	−0.146156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.a.d.1.3	✓	4
3.2	odd	2		207.4.a.d.1.2		4
4.3	odd	2		1104.4.a.t.1.3		4
5.4	even	2		1725.4.a.p.1.2		4
23.22	odd	2		1587.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.d.1.3	✓	4	1.1	even	1	trivial
207.4.a.d.1.2		4	3.2	odd	2
1104.4.a.t.1.3		4	4.3	odd	2
1587.4.a.g.1.3		4	23.22	odd	2
1725.4.a.p.1.2		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+3.45983 q^{2} +3.00000 q^{3} +3.97043 q^{4} +7.89053 q^{5} +10.3795 q^{6} +4.57766 q^{7} -13.9416 q^{8} +9.00000 q^{9} +27.2999 q^{10} -7.16805 q^{11} +11.9113 q^{12} +9.66367 q^{13} +15.8379 q^{14} +23.6716 q^{15} -79.9991 q^{16} -69.1871 q^{17} +31.1385 q^{18} -21.2382 q^{19} +31.3288 q^{20} +13.7330 q^{21} -24.8002 q^{22} +23.0000 q^{23} -41.8249 q^{24} -62.7395 q^{25} +33.4347 q^{26} +27.0000 q^{27} +18.1753 q^{28} +39.5338 q^{29} +81.8997 q^{30} +28.5625 q^{31} -165.251 q^{32} -21.5041 q^{33} -239.376 q^{34} +36.1202 q^{35} +35.7339 q^{36} -170.923 q^{37} -73.4807 q^{38} +28.9910 q^{39} -110.007 q^{40} +395.474 q^{41} +47.5138 q^{42} -214.499 q^{43} -28.4603 q^{44} +71.0148 q^{45} +79.5761 q^{46} +387.800 q^{47} -239.997 q^{48} -322.045 q^{49} -217.068 q^{50} -207.561 q^{51} +38.3690 q^{52} +268.151 q^{53} +93.4155 q^{54} -56.5597 q^{55} -63.8200 q^{56} -63.7147 q^{57} +136.780 q^{58} +552.468 q^{59} +93.9865 q^{60} +354.991 q^{61} +98.8214 q^{62} +41.1989 q^{63} +68.2540 q^{64} +76.2515 q^{65} -74.4007 q^{66} +293.680 q^{67} -274.703 q^{68} +69.0000 q^{69} +124.970 q^{70} +505.691 q^{71} -125.475 q^{72} +1048.26 q^{73} -591.365 q^{74} -188.219 q^{75} -84.3250 q^{76} -32.8129 q^{77} +100.304 q^{78} -38.4080 q^{79} -631.236 q^{80} +81.0000 q^{81} +1368.27 q^{82} -111.501 q^{83} +54.5259 q^{84} -545.923 q^{85} -742.130 q^{86} +118.601 q^{87} +99.9342 q^{88} -1479.10 q^{89} +245.699 q^{90} +44.2370 q^{91} +91.3200 q^{92} +85.6875 q^{93} +1341.72 q^{94} -167.581 q^{95} -495.752 q^{96} -1274.61 q^{97} -1114.22 q^{98} -64.5124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 4 q^{5} + 12 q^{6} - 14 q^{7} + 84 q^{8} + 36 q^{9} - 100 q^{10} + 70 q^{11} + 78 q^{12} - 12 q^{13} - 18 q^{14} + 12 q^{15} + 130 q^{16} + 178 q^{17} + 36 q^{18} + 96 q^{19}+ \cdots + 630 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 12 * q^3 + 26 * q^4 + 4 * q^5 + 12 * q^6 - 14 * q^7 + 84 * q^8 + 36 * q^9 - 100 * q^10 + 70 * q^11 + 78 * q^12 - 12 * q^13 - 18 * q^14 + 12 * q^15 + 130 * q^16 + 178 * q^17 + 36 * q^18 + 96 * q^19 - 296 * q^20 - 42 * q^21 - 326 * q^22 + 92 * q^23 + 252 * q^24 - 80 * q^25 - 464 * q^26 + 108 * q^27 - 774 * q^28 - 24 * q^29 - 300 * q^30 - 400 * q^31 + 636 * q^32 + 210 * q^33 - 18 * q^34 + 224 * q^35 + 234 * q^36 - 358 * q^37 - 136 * q^38 - 36 * q^39 - 1268 * q^40 + 152 * q^41 - 54 * q^42 - 196 * q^43 + 70 * q^44 + 36 * q^45 + 92 * q^46 + 260 * q^47 + 390 * q^48 + 476 * q^49 + 876 * q^50 + 534 * q^51 - 876 * q^52 + 968 * q^53 + 108 * q^54 + 684 * q^55 - 1234 * q^56 + 288 * q^57 + 904 * q^58 + 1236 * q^59 - 888 * q^60 - 482 * q^61 - 1304 * q^62 - 126 * q^63 + 1658 * q^64 + 1424 * q^65 - 978 * q^66 + 620 * q^67 + 838 * q^68 + 276 * q^69 + 2636 * q^70 - 88 * q^71 + 756 * q^72 - 948 * q^73 + 710 * q^74 - 240 * q^75 + 3344 * q^76 - 892 * q^77 - 1392 * q^78 + 254 * q^79 - 5144 * q^80 + 324 * q^81 + 1568 * q^82 - 550 * q^83 - 2322 * q^84 - 1208 * q^85 - 1236 * q^86 - 72 * q^87 + 202 * q^88 + 202 * q^89 - 900 * q^90 - 356 * q^91 + 598 * q^92 - 1200 * q^93 - 1328 * q^94 - 460 * q^95 + 1908 * q^96 - 1800 * q^97 - 2784 * q^98 + 630 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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