LMFDB - Embedded newform 38.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,4,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	38.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:	$2.24207258022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +7.15207 q^{3} +4.00000 q^{4} -8.30413 q^{5} -14.3041 q^{6} +35.1521 q^{7} -8.00000 q^{8} +24.1521 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +7.15207 q^{3} +4.00000 q^{4} -8.30413 q^{5} -14.3041 q^{6} +35.1521 q^{7} -8.00000 q^{8} +24.1521 q^{9} +16.6083 q^{10} +18.3041 q^{11} +28.6083 q^{12} -40.0645 q^{13} -70.3041 q^{14} -59.3917 q^{15} +16.0000 q^{16} -125.281 q^{17} -48.3041 q^{18} -19.0000 q^{19} -33.2165 q^{20} +251.410 q^{21} -36.6083 q^{22} +8.97688 q^{23} -57.2165 q^{24} -56.0413 q^{25} +80.1289 q^{26} -20.3686 q^{27} +140.608 q^{28} +153.410 q^{29} +118.783 q^{30} -114.433 q^{31} -32.0000 q^{32} +130.912 q^{33} +250.562 q^{34} -291.908 q^{35} +96.6083 q^{36} +83.5669 q^{37} +38.0000 q^{38} -286.544 q^{39} +66.4331 q^{40} -355.088 q^{41} -502.820 q^{42} +467.299 q^{43} +73.2165 q^{44} -200.562 q^{45} -17.9538 q^{46} +166.083 q^{47} +114.433 q^{48} +892.668 q^{49} +112.083 q^{50} -896.018 q^{51} -160.258 q^{52} +258.369 q^{53} +40.7372 q^{54} -152.000 q^{55} -281.217 q^{56} -135.889 q^{57} -306.820 q^{58} -371.797 q^{59} -237.567 q^{60} -47.3090 q^{61} +228.866 q^{62} +848.995 q^{63} +64.0000 q^{64} +332.701 q^{65} -261.825 q^{66} -755.539 q^{67} -501.124 q^{68} +64.2032 q^{69} +583.815 q^{70} +349.345 q^{71} -193.217 q^{72} +54.8479 q^{73} -167.134 q^{74} -400.811 q^{75} -76.0000 q^{76} +643.428 q^{77} +573.088 q^{78} +438.820 q^{79} -132.866 q^{80} -797.783 q^{81} +710.175 q^{82} +1073.09 q^{83} +1005.64 q^{84} +1040.35 q^{85} -934.598 q^{86} +1097.20 q^{87} -146.433 q^{88} -501.521 q^{89} +401.124 q^{90} -1408.35 q^{91} +35.9075 q^{92} -818.433 q^{93} -332.165 q^{94} +157.779 q^{95} -228.866 q^{96} -1437.56 q^{97} -1785.34 q^{98} +442.083 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} + 13 q^{13} - 114 q^{14} - 172 q^{15} + 32 q^{16} - 51 q^{17} - 70 q^{18} - 38 q^{19} + 40 q^{20} + 117 q^{21} - 20 q^{22} - 155 q^{23} - 8 q^{24} + 154 q^{25} - 26 q^{26} + 79 q^{27} + 228 q^{28} - 79 q^{29} + 344 q^{30} - 16 q^{31} - 64 q^{32} + 182 q^{33} + 102 q^{34} + 108 q^{35} + 140 q^{36} + 380 q^{37} + 76 q^{38} - 613 q^{39} - 80 q^{40} - 790 q^{41} - 234 q^{42} + 296 q^{43} + 40 q^{44} - 2 q^{45} + 310 q^{46} - 200 q^{47} + 16 q^{48} + 1027 q^{49} - 308 q^{50} - 1353 q^{51} + 52 q^{52} + 397 q^{53} - 158 q^{54} - 304 q^{55} - 456 q^{56} - 19 q^{57} + 158 q^{58} + 201 q^{59} - 688 q^{60} - 680 q^{61} + 32 q^{62} + 1086 q^{63} + 128 q^{64} + 1304 q^{65} - 364 q^{66} - 939 q^{67} - 204 q^{68} + 1073 q^{69} - 216 q^{70} + 406 q^{71} - 280 q^{72} + 123 q^{73} - 760 q^{74} - 1693 q^{75} - 152 q^{76} + 462 q^{77} + 1226 q^{78} + 106 q^{79} + 160 q^{80} - 1702 q^{81} + 1580 q^{82} + 2226 q^{83} + 468 q^{84} + 2400 q^{85} - 592 q^{86} + 2527 q^{87} - 80 q^{88} - 870 q^{89} + 4 q^{90} - 249 q^{91} - 620 q^{92} - 1424 q^{93} + 400 q^{94} - 190 q^{95} - 32 q^{96} - 1864 q^{97} - 2054 q^{98} + 352 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 + 13 * q^13 - 114 * q^14 - 172 * q^15 + 32 * q^16 - 51 * q^17 - 70 * q^18 - 38 * q^19 + 40 * q^20 + 117 * q^21 - 20 * q^22 - 155 * q^23 - 8 * q^24 + 154 * q^25 - 26 * q^26 + 79 * q^27 + 228 * q^28 - 79 * q^29 + 344 * q^30 - 16 * q^31 - 64 * q^32 + 182 * q^33 + 102 * q^34 + 108 * q^35 + 140 * q^36 + 380 * q^37 + 76 * q^38 - 613 * q^39 - 80 * q^40 - 790 * q^41 - 234 * q^42 + 296 * q^43 + 40 * q^44 - 2 * q^45 + 310 * q^46 - 200 * q^47 + 16 * q^48 + 1027 * q^49 - 308 * q^50 - 1353 * q^51 + 52 * q^52 + 397 * q^53 - 158 * q^54 - 304 * q^55 - 456 * q^56 - 19 * q^57 + 158 * q^58 + 201 * q^59 - 688 * q^60 - 680 * q^61 + 32 * q^62 + 1086 * q^63 + 128 * q^64 + 1304 * q^65 - 364 * q^66 - 939 * q^67 - 204 * q^68 + 1073 * q^69 - 216 * q^70 + 406 * q^71 - 280 * q^72 + 123 * q^73 - 760 * q^74 - 1693 * q^75 - 152 * q^76 + 462 * q^77 + 1226 * q^78 + 106 * q^79 + 160 * q^80 - 1702 * q^81 + 1580 * q^82 + 2226 * q^83 + 468 * q^84 + 2400 * q^85 - 592 * q^86 + 2527 * q^87 - 80 * q^88 - 870 * q^89 + 4 * q^90 - 249 * q^91 - 620 * q^92 - 1424 * q^93 + 400 * q^94 - 190 * q^95 - 32 * q^96 - 1864 * q^97 - 2054 * q^98 + 352 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	7.15207	1.37642	0.688208	−	0.725513i	$-0.258398\pi$
$3$	7.15207	1.37642	0.688208	+	0.725513i	$0.258398\pi$
$4$	4.00000	0.500000
$5$	−8.30413	−0.742744	−0.371372	−	0.928484i	$-0.621113\pi$
$5$	−8.30413	−0.742744	−0.371372	+	0.928484i	$0.621113\pi$
$6$	−14.3041	−0.973273
$7$	35.1521	1.89803	0.949017	−	0.315226i	$-0.102080\pi$
$7$	35.1521	1.89803	0.949017	+	0.315226i	$0.102080\pi$
$8$	−8.00000	−0.353553
$9$	24.1521	0.894521
$10$	16.6083	0.525200
$11$	18.3041	0.501719	0.250859	−	0.968024i	$-0.419287\pi$
$11$	18.3041	0.501719	0.250859	+	0.968024i	$0.419287\pi$
$12$	28.6083	0.688208
$13$	−40.0645	−0.854760	−0.427380	−	0.904072i	$-0.640563\pi$
$13$	−40.0645	−0.854760	−0.427380	+	0.904072i	$0.640563\pi$
$14$	−70.3041	−1.34211
$15$	−59.3917	−1.02233
$16$	16.0000	0.250000
$17$	−125.281	−1.78736	−0.893680	−	0.448706i	$-0.851885\pi$
$17$	−125.281	−1.78736	−0.893680	+	0.448706i	$0.851885\pi$
$18$	−48.3041	−0.632522
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−33.2165	−0.371372
$21$	251.410	2.61248
$22$	−36.6083	−0.354769
$23$	8.97688	0.0813830	0.0406915	−	0.999172i	$-0.487044\pi$
$23$	8.97688	0.0813830	0.0406915	+	0.999172i	$0.487044\pi$
$24$	−57.2165	−0.486637
$25$	−56.0413	−0.448331
$26$	80.1289	0.604407
$27$	−20.3686	−0.145183
$28$	140.608	0.949017
$29$	153.410	0.982328	0.491164	−	0.871067i	$-0.336572\pi$
$29$	153.410	0.982328	0.491164	+	0.871067i	$0.336572\pi$
$30$	118.783	0.722893
$31$	−114.433	−0.662993	−0.331497	−	0.943456i	$-0.607554\pi$
$31$	−114.433	−0.662993	−0.331497	+	0.943456i	$0.607554\pi$
$32$	−32.0000	−0.176777
$33$	130.912	0.690573
$34$	250.562	1.26385
$35$	−291.908	−1.40975
$36$	96.6083	0.447261
$37$	83.5669	0.371306	0.185653	−	0.982615i	$-0.440560\pi$
$37$	83.5669	0.371306	0.185653	+	0.982615i	$0.440560\pi$
$38$	38.0000	0.162221
$39$	−286.544	−1.17651
$40$	66.4331	0.262600
$41$	−355.088	−1.35257	−0.676285	−	0.736640i	$-0.736411\pi$
$41$	−355.088	−1.35257	−0.676285	+	0.736640i	$0.736411\pi$
$42$	−502.820	−1.84730
$43$	467.299	1.65727	0.828633	−	0.559792i	$-0.189119\pi$
$43$	467.299	1.65727	0.828633	+	0.559792i	$0.189119\pi$
$44$	73.2165	0.250859
$45$	−200.562	−0.664400
$46$	−17.9538	−0.0575464
$47$	166.083	0.515439	0.257720	−	0.966220i	$-0.417029\pi$
$47$	166.083	0.515439	0.257720	+	0.966220i	$0.417029\pi$
$48$	114.433	0.344104
$49$	892.668	2.60253
$50$	112.083	0.317018
$51$	−896.018	−2.46015
$52$	−160.258	−0.427380
$53$	258.369	0.669616	0.334808	−	0.942286i	$-0.391329\pi$
$53$	258.369	0.669616	0.334808	+	0.942286i	$0.391329\pi$
$54$	40.7372	0.102660
$55$	−152.000	−0.372649
$56$	−281.217	−0.671056
$57$	−135.889	−0.315771
$58$	−306.820	−0.694611
$59$	−371.797	−0.820404	−0.410202	−	0.911995i	$-0.634542\pi$
$59$	−371.797	−0.820404	−0.410202	+	0.911995i	$0.634542\pi$
$60$	−237.567	−0.511163
$61$	−47.3090	−0.0993000	−0.0496500	−	0.998767i	$-0.515811\pi$
$61$	−47.3090	−0.0993000	−0.0496500	+	0.998767i	$0.515811\pi$
$62$	228.866	0.468807
$63$	848.995	1.69783
$64$	64.0000	0.125000
$65$	332.701	0.634868
$66$	−261.825	−0.488309
$67$	−755.539	−1.37767	−0.688834	−	0.724919i	$-0.741877\pi$
$67$	−755.539	−1.37767	−0.688834	+	0.724919i	$0.741877\pi$
$68$	−501.124	−0.893680
$69$	64.2032	0.112017
$70$	583.815	0.996846
$71$	349.345	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$71$	349.345	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$72$	−193.217	−0.316261
$73$	54.8479	0.0879379	0.0439689	−	0.999033i	$-0.486000\pi$
$73$	54.8479	0.0879379	0.0439689	+	0.999033i	$0.486000\pi$
$74$	−167.134	−0.262553
$75$	−400.811	−0.617090
$76$	−76.0000	−0.114708
$77$	643.428	0.952279
$78$	573.088	0.831915
$79$	438.820	0.624951	0.312475	−	0.949926i	$-0.398842\pi$
$79$	438.820	0.624951	0.312475	+	0.949926i	$0.398842\pi$
$80$	−132.866	−0.185686
$81$	−797.783	−1.09435
$82$	710.175	0.956411
$83$	1073.09	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$83$	1073.09	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$84$	1005.64	1.30624
$85$	1040.35	1.32755
$86$	−934.598	−1.17186
$87$	1097.20	1.35209
$88$	−146.433	−0.177384
$89$	−501.521	−0.597316	−0.298658	−	0.954360i	$-0.596539\pi$
$89$	−501.521	−0.597316	−0.298658	+	0.954360i	$0.596539\pi$
$90$	401.124	0.469802
$91$	−1408.35	−1.62236
$92$	35.9075	0.0406915
$93$	−818.433	−0.912554
$94$	−332.165	−0.364471
$95$	157.779	0.170397
$96$	−228.866	−0.243318
$97$	−1437.56	−1.50476	−0.752380	−	0.658729i	$-0.771094\pi$
$97$	−1437.56	−1.50476	−0.752380	+	0.658729i	$0.771094\pi$
$98$	−1785.34	−1.84027
$99$	442.083	0.448798
$100$	−224.165	−0.224165
$101$	395.124	0.389270	0.194635	−	0.980876i	$-0.437648\pi$
$101$	395.124	0.389270	0.194635	+	0.980876i	$0.437648\pi$
$102$	1792.04	1.73959
$103$	1285.68	1.22992	0.614958	−	0.788559i	$-0.289173\pi$
$103$	1285.68	1.22992	0.614958	+	0.788559i	$0.289173\pi$
$104$	320.516	0.302203
$105$	−2087.74	−1.94041
$106$	−516.737	−0.473490
$107$	1203.10	1.08699	0.543494	−	0.839413i	$-0.317101\pi$
$107$	1203.10	1.08699	0.543494	+	0.839413i	$0.317101\pi$
$108$	−81.4744	−0.0725915
$109$	−1333.74	−1.17201	−0.586005	−	0.810307i	$-0.699300\pi$
$109$	−1333.74	−1.17201	−0.586005	+	0.810307i	$0.699300\pi$
$110$	304.000	0.263502
$111$	597.676	0.511071
$112$	562.433	0.474508
$113$	836.506	0.696388	0.348194	−	0.937422i	$-0.386795\pi$
$113$	836.506	0.696388	0.348194	+	0.937422i	$0.386795\pi$
$114$	271.779	0.223284
$115$	−74.5452	−0.0604467
$116$	613.640	0.491164
$117$	−967.640	−0.764601
$118$	743.594	0.580113
$119$	−4403.89	−3.39247
$120$	475.134	0.361447
$121$	−995.959	−0.748278
$122$	94.6181	0.0702157
$123$	−2539.61	−1.86170
$124$	−457.732	−0.331497
$125$	1503.39	1.07574
$126$	−1697.99	−1.20055
$127$	−385.954	−0.269668	−0.134834	−	0.990868i	$-0.543050\pi$
$127$	−385.954	−0.269668	−0.134834	+	0.990868i	$0.543050\pi$
$128$	−128.000	−0.0883883
$129$	3342.16	2.28109
$130$	−665.402	−0.448920
$131$	1737.90	1.15909	0.579545	−	0.814940i	$-0.303230\pi$
$131$	1737.90	1.15909	0.579545	+	0.814940i	$0.303230\pi$
$132$	523.650	0.345287
$133$	−667.889	−0.435439
$134$	1511.08	0.974159
$135$	169.144	0.107834
$136$	1002.25	0.631927
$137$	41.7603	0.0260425	0.0130213	−	0.999915i	$-0.495855\pi$
$137$	41.7603	0.0260425	0.0130213	+	0.999915i	$0.495855\pi$
$138$	−128.406	−0.0792078
$139$	1536.79	0.937763	0.468882	−	0.883261i	$-0.344657\pi$
$139$	1536.79	0.937763	0.468882	+	0.883261i	$0.344657\pi$
$140$	−1167.63	−0.704877
$141$	1187.83	0.709459
$142$	−698.691	−0.412907
$143$	−733.345	−0.428849
$144$	386.433	0.223630
$145$	−1273.94	−0.729619
$146$	−109.696	−0.0621815
$147$	6384.42	3.58216
$148$	334.268	0.185653
$149$	1656.65	0.910862	0.455431	−	0.890271i	$-0.349485\pi$
$149$	1656.65	0.910862	0.455431	+	0.890271i	$0.349485\pi$
$150$	801.623	0.436348
$151$	−714.985	−0.385329	−0.192664	−	0.981265i	$-0.561713\pi$
$151$	−714.985	−0.385329	−0.192664	+	0.981265i	$0.561713\pi$
$152$	152.000	0.0811107
$153$	−3025.80	−1.59883
$154$	−1286.86	−0.673363
$155$	950.268	0.492434
$156$	−1146.18	−0.588253
$157$	1684.23	0.856153	0.428077	−	0.903742i	$-0.359191\pi$
$157$	1684.23	0.856153	0.428077	+	0.903742i	$0.359191\pi$
$158$	−877.640	−0.441907
$159$	1847.87	0.921670
$160$	265.732	0.131300
$161$	315.556	0.154468
$162$	1595.57	0.773825
$163$	702.175	0.337415	0.168707	−	0.985666i	$-0.446041\pi$
$163$	702.175	0.337415	0.168707	+	0.985666i	$0.446041\pi$
$164$	−1420.35	−0.676285
$165$	−1087.11	−0.512920
$166$	−2146.18	−1.00347
$167$	−282.506	−0.130904	−0.0654520	−	0.997856i	$-0.520849\pi$
$167$	−282.506	−0.130904	−0.0654520	+	0.997856i	$0.520849\pi$
$168$	−2011.28	−0.923652
$169$	−591.838	−0.269385
$170$	−2080.70	−0.938720
$171$	−458.889	−0.205217
$172$	1869.20	0.828633
$173$	2183.44	0.959558	0.479779	−	0.877389i	$-0.340717\pi$
$173$	2183.44	0.959558	0.479779	+	0.877389i	$0.340717\pi$
$174$	−2194.40	−0.956073
$175$	−1969.97	−0.850947
$176$	292.866	0.125430
$177$	−2659.12	−1.12922
$178$	1003.04	0.422366
$179$	−3198.51	−1.33557	−0.667786	−	0.744353i	$-0.732758\pi$
$179$	−3198.51	−1.33557	−0.667786	+	0.744353i	$0.732758\pi$
$180$	−802.248	−0.332200
$181$	−2151.05	−0.883350	−0.441675	−	0.897175i	$-0.645616\pi$
$181$	−2151.05	−0.883350	−0.441675	+	0.897175i	$0.645616\pi$
$182$	2816.70	1.14718
$183$	−338.357	−0.136678
$184$	−71.8150	−0.0287732
$185$	−693.951	−0.275785
$186$	1636.87	0.645273
$187$	−2293.16	−0.896751
$188$	664.331	0.257720
$189$	−715.999	−0.275562
$190$	−315.557	−0.120489
$191$	−4435.52	−1.68033	−0.840165	−	0.542331i	$-0.817542\pi$
$191$	−4435.52	−1.68033	−0.840165	+	0.542331i	$0.817542\pi$
$192$	457.732	0.172052
$193$	2720.60	1.01468	0.507339	−	0.861746i	$-0.330629\pi$
$193$	2720.60	1.01468	0.507339	+	0.861746i	$0.330629\pi$
$194$	2875.11	1.06403
$195$	2379.50	0.873843
$196$	3570.67	1.30127
$197$	1254.08	0.453549	0.226775	−	0.973947i	$-0.427182\pi$
$197$	1254.08	0.453549	0.226775	+	0.973947i	$0.427182\pi$
$198$	−884.165	−0.317348
$199$	4155.19	1.48017	0.740084	−	0.672515i	$-0.234786\pi$
$199$	4155.19	1.48017	0.740084	+	0.672515i	$0.234786\pi$
$200$	448.331	0.158509
$201$	−5403.67	−1.89624
$202$	−790.248	−0.275256
$203$	5392.68	1.86449
$204$	−3584.07	−1.23007
$205$	2948.70	1.00461
$206$	−2571.35	−0.869683
$207$	216.810	0.0727988
$208$	−641.032	−0.213690
$209$	−347.779	−0.115102
$210$	4175.48	1.37208
$211$	−633.437	−0.206671	−0.103335	−	0.994647i	$-0.532952\pi$
$211$	−633.437	−0.206671	−0.103335	+	0.994647i	$0.532952\pi$
$212$	1033.47	0.334808
$213$	2498.54	0.803743
$214$	−2406.19	−0.768616
$215$	−3880.52	−1.23093
$216$	162.949	0.0513299
$217$	−4022.56	−1.25838
$218$	2667.48	0.828737
$219$	392.276	0.121039
$220$	−608.000	−0.186324
$221$	5019.32	1.52776
$222$	−1195.35	−0.361382
$223$	−4798.34	−1.44090	−0.720449	−	0.693507i	$-0.756064\pi$
$223$	−4798.34	−1.44090	−0.720449	+	0.693507i	$0.756064\pi$
$224$	−1124.87	−0.335528
$225$	−1353.51	−0.401041
$226$	−1673.01	−0.492421
$227$	641.770	0.187647	0.0938233	−	0.995589i	$-0.470091\pi$
$227$	641.770	0.187647	0.0938233	+	0.995589i	$0.470091\pi$
$228$	−543.557	−0.157886
$229$	−1231.34	−0.355325	−0.177662	−	0.984091i	$-0.556854\pi$
$229$	−1231.34	−0.355325	−0.177662	+	0.984091i	$0.556854\pi$
$230$	149.090	0.0427423
$231$	4601.84	1.31073
$232$	−1227.28	−0.347305
$233$	226.701	0.0637410	0.0318705	−	0.999492i	$-0.489854\pi$
$233$	226.701	0.0637410	0.0318705	+	0.999492i	$0.489854\pi$
$234$	1935.28	0.540655
$235$	−1379.17	−0.382840
$236$	−1487.19	−0.410202
$237$	3138.47	0.860192
$238$	8807.77	2.39884
$239$	2433.27	0.658557	0.329278	−	0.944233i	$-0.393195\pi$
$239$	2433.27	0.658557	0.329278	+	0.944233i	$0.393195\pi$
$240$	−950.268	−0.255581
$241$	−901.336	−0.240913	−0.120457	−	0.992719i	$-0.538436\pi$
$241$	−901.336	−0.240913	−0.120457	+	0.992719i	$0.538436\pi$
$242$	1991.92	0.529113
$243$	−5155.85	−1.36110
$244$	−189.236	−0.0496500
$245$	−7412.83	−1.93301
$246$	5079.22	1.31642
$247$	761.225	0.196095
$248$	915.465	0.234403
$249$	7674.79	1.95329
$250$	−3006.78	−0.760663
$251$	−5675.92	−1.42734	−0.713668	−	0.700484i	$-0.752967\pi$
$251$	−5675.92	−1.42734	−0.713668	+	0.700484i	$0.752967\pi$
$252$	3395.98	0.848915
$253$	164.314	0.0408313
$254$	771.908	0.190684
$255$	7440.66	1.82726
$256$	256.000	0.0625000
$257$	2035.88	0.494143	0.247072	−	0.968997i	$-0.420532\pi$
$257$	2035.88	0.494143	0.247072	+	0.968997i	$0.420532\pi$
$258$	−6684.31	−1.61297
$259$	2937.55	0.704751
$260$	1330.80	0.317434
$261$	3705.17	0.878713
$262$	−3475.80	−0.819601
$263$	1924.20	0.451147	0.225573	−	0.974226i	$-0.427574\pi$
$263$	1924.20	0.451147	0.225573	+	0.974226i	$0.427574\pi$
$264$	−1047.30	−0.244155
$265$	−2145.53	−0.497354
$266$	1335.78	0.307902
$267$	−3586.91	−0.822155
$268$	−3022.16	−0.688834
$269$	−829.280	−0.187963	−0.0939815	−	0.995574i	$-0.529959\pi$
$269$	−829.280	−0.187963	−0.0939815	+	0.995574i	$0.529959\pi$
$270$	−338.287	−0.0762500
$271$	−1223.13	−0.274169	−0.137085	−	0.990559i	$-0.543773\pi$
$271$	−1223.13	−0.274169	−0.137085	+	0.990559i	$0.543773\pi$
$272$	−2004.50	−0.446840
$273$	−10072.6	−2.23305
$274$	−83.5207	−0.0184148
$275$	−1025.79	−0.224936
$276$	256.813	0.0560084
$277$	1622.81	0.352004	0.176002	−	0.984390i	$-0.443683\pi$
$277$	1622.81	0.352004	0.176002	+	0.984390i	$0.443683\pi$
$278$	−3073.59	−0.663099
$279$	−2763.80	−0.593061
$280$	2335.26	0.498423
$281$	3618.96	0.768288	0.384144	−	0.923273i	$-0.374497\pi$
$281$	3618.96	0.768288	0.384144	+	0.923273i	$0.374497\pi$
$282$	−2375.67	−0.501663
$283$	2102.43	0.441613	0.220807	−	0.975318i	$-0.429131\pi$
$283$	2102.43	0.441613	0.220807	+	0.975318i	$0.429131\pi$
$284$	1397.38	0.291970
$285$	1128.44	0.234537
$286$	1466.69	0.303242
$287$	−12482.1	−2.56722
$288$	−772.866	−0.158130
$289$	10782.3	2.19465
$290$	2547.87	0.515918
$291$	−10281.5	−2.07118
$292$	219.392	0.0439689
$293$	−5383.99	−1.07350	−0.536751	−	0.843741i	$-0.680348\pi$
$293$	−5383.99	−1.07350	−0.536751	+	0.843741i	$0.680348\pi$
$294$	−12768.8	−2.53297
$295$	3087.45	0.609350
$296$	−668.535	−0.131276
$297$	−372.830	−0.0728410
$298$	−3313.31	−0.644077
$299$	−359.654	−0.0695629
$300$	−1603.25	−0.308545
$301$	16426.5	3.14555
$302$	1429.97	0.272469
$303$	2825.95	0.535798
$304$	−304.000	−0.0573539
$305$	392.861	0.0737545
$306$	6051.59	1.13054
$307$	3692.85	0.686521	0.343260	−	0.939240i	$-0.388469\pi$
$307$	3692.85	0.686521	0.343260	+	0.939240i	$0.388469\pi$
$308$	2573.71	0.476139
$309$	9195.24	1.69288
$310$	−1900.54	−0.348204
$311$	−4427.67	−0.807299	−0.403649	−	0.914914i	$-0.632258\pi$
$311$	−4427.67	−0.807299	−0.403649	+	0.914914i	$0.632258\pi$
$312$	2292.35	0.415958
$313$	−7356.47	−1.32847	−0.664236	−	0.747523i	$-0.731243\pi$
$313$	−7356.47	−1.32847	−0.664236	+	0.747523i	$0.731243\pi$
$314$	−3368.46	−0.605392
$315$	−7050.17	−1.26105
$316$	1755.28	0.312475
$317$	1612.77	0.285747	0.142874	−	0.989741i	$-0.454366\pi$
$317$	1612.77	0.285747	0.142874	+	0.989741i	$0.454366\pi$
$318$	−3695.74	−0.651719
$319$	2808.04	0.492852
$320$	−531.465	−0.0928430
$321$	8604.62	1.49615
$322$	−631.111	−0.109225
$323$	2380.34	0.410048
$324$	−3191.13	−0.547177
$325$	2245.27	0.383215
$326$	−1404.35	−0.238588
$327$	−9539.00	−1.61317
$328$	2840.70	0.478206
$329$	5838.15	0.978321
$330$	2174.23	0.362689
$331$	−2262.78	−0.375752	−0.187876	−	0.982193i	$-0.560160\pi$
$331$	−2262.78	−0.375752	−0.187876	+	0.982193i	$0.560160\pi$
$332$	4292.35	0.709558
$333$	2018.31	0.332141
$334$	565.012	0.0925631
$335$	6274.10	1.02326
$336$	4022.56	0.653121
$337$	−2037.63	−0.329367	−0.164683	−	0.986346i	$-0.552660\pi$
$337$	−2037.63	−0.329367	−0.164683	+	0.986346i	$0.552660\pi$
$338$	1183.68	0.190484
$339$	5982.75	0.958520
$340$	4161.40	0.663776
$341$	−2094.60	−0.332636
$342$	917.779	0.145110
$343$	19322.0	3.04166
$344$	−3738.39	−0.585932
$345$	−533.152	−0.0831998
$346$	−4366.87	−0.678510
$347$	−1844.33	−0.285328	−0.142664	−	0.989771i	$-0.545567\pi$
$347$	−1844.33	−0.285328	−0.142664	+	0.989771i	$0.545567\pi$
$348$	4388.79	0.676046
$349$	−10156.9	−1.55784	−0.778918	−	0.627125i	$-0.784231\pi$
$349$	−10156.9	−1.55784	−0.778918	+	0.627125i	$0.784231\pi$
$350$	3939.94	0.601710
$351$	816.057	0.124097
$352$	−585.732	−0.0886922
$353$	1905.43	0.287296	0.143648	−	0.989629i	$-0.454117\pi$
$353$	1905.43	0.287296	0.143648	+	0.989629i	$0.454117\pi$
$354$	5318.23	0.798477
$355$	−2901.01	−0.433718
$356$	−2006.08	−0.298658
$357$	−31496.9	−4.66945
$358$	6397.01	0.944393
$359$	−2496.50	−0.367020	−0.183510	−	0.983018i	$-0.558746\pi$
$359$	−2496.50	−0.367020	−0.183510	+	0.983018i	$0.558746\pi$
$360$	1604.50	0.234901
$361$	361.000	0.0526316
$362$	4302.10	0.624623
$363$	−7123.16	−1.02994
$364$	−5633.40	−0.811182
$365$	−455.465	−0.0653154
$366$	676.715	0.0966460
$367$	−8748.42	−1.24432	−0.622158	−	0.782892i	$-0.713744\pi$
$367$	−8748.42	−1.24432	−0.622158	+	0.782892i	$0.713744\pi$
$368$	143.630	0.0203457
$369$	−8576.10	−1.20990
$370$	1387.90	0.195010
$371$	9082.19	1.27095
$372$	−3273.73	−0.456277
$373$	1460.62	0.202756	0.101378	−	0.994848i	$-0.467675\pi$
$373$	1460.62	0.202756	0.101378	+	0.994848i	$0.467675\pi$
$374$	4586.32	0.634099
$375$	10752.4	1.48067
$376$	−1328.66	−0.182235
$377$	−6146.29	−0.839655
$378$	1432.00	0.194852
$379$	10581.0	1.43406	0.717029	−	0.697043i	$-0.245501\pi$
$379$	10581.0	1.43406	0.717029	+	0.697043i	$0.245501\pi$
$380$	631.114	0.0851986
$381$	−2760.37	−0.371176
$382$	8871.04	1.18817
$383$	−2932.54	−0.391242	−0.195621	−	0.980680i	$-0.562672\pi$
$383$	−2932.54	−0.391242	−0.195621	+	0.980680i	$0.562672\pi$
$384$	−915.465	−0.121659
$385$	−5343.11	−0.707300
$386$	−5441.20	−0.717486
$387$	11286.2	1.48246
$388$	−5750.23	−0.752380
$389$	−3631.19	−0.473287	−0.236644	−	0.971597i	$-0.576047\pi$
$389$	−3631.19	−0.473287	−0.236644	+	0.971597i	$0.576047\pi$
$390$	−4759.00	−0.617900
$391$	−1124.63	−0.145461
$392$	−7141.34	−0.920133
$393$	12429.6	1.59539
$394$	−2508.15	−0.320708
$395$	−3644.02	−0.464179
$396$	1768.33	0.224399
$397$	−2005.32	−0.253512	−0.126756	−	0.991934i	$-0.540456\pi$
$397$	−2005.32	−0.253512	−0.126756	+	0.991934i	$0.540456\pi$
$398$	−8310.37	−1.04664
$399$	−4776.79	−0.599345
$400$	−896.662	−0.112083
$401$	−8187.30	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$401$	−8187.30	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$402$	10807.3	1.34085
$403$	4584.70	0.566700
$404$	1580.50	0.194635
$405$	6624.90	0.812825
$406$	−10785.4	−1.31839
$407$	1529.62	0.186291
$408$	7168.15	0.869794
$409$	−15565.0	−1.88175	−0.940877	−	0.338747i	$-0.889997\pi$
$409$	−15565.0	−1.88175	−0.940877	+	0.338747i	$0.889997\pi$
$410$	−5897.39	−0.710369
$411$	298.673	0.0358454
$412$	5142.71	0.614958
$413$	−13069.4	−1.55715
$414$	−433.620	−0.0514765
$415$	−8911.06	−1.05404
$416$	1282.06	0.151102
$417$	10991.2	1.29075
$418$	695.557	0.0813895
$419$	−1839.83	−0.214514	−0.107257	−	0.994231i	$-0.534207\pi$
$419$	−1839.83	−0.214514	−0.107257	+	0.994231i	$0.534207\pi$
$420$	−8350.97	−0.970204
$421$	4658.28	0.539266	0.269633	−	0.962963i	$-0.413098\pi$
$421$	4658.28	0.539266	0.269633	+	0.962963i	$0.413098\pi$
$422$	1266.87	0.146138
$423$	4011.24	0.461071
$424$	−2066.95	−0.236745
$425$	7020.92	0.801328
$426$	−4997.08	−0.568332
$427$	−1663.01	−0.188475
$428$	4812.38	0.543494
$429$	−5244.94	−0.590275
$430$	7761.03	0.870396
$431$	2820.47	0.315214	0.157607	−	0.987502i	$-0.449622\pi$
$431$	2820.47	0.315214	0.157607	+	0.987502i	$0.449622\pi$
$432$	−325.898	−0.0362957
$433$	8981.39	0.996809	0.498404	−	0.866945i	$-0.333920\pi$
$433$	8981.39	0.996809	0.498404	+	0.866945i	$0.333920\pi$
$434$	8045.12	0.889811
$435$	−9111.28	−1.00426
$436$	−5334.96	−0.586005
$437$	−170.561	−0.0186705
$438$	−784.552	−0.0855876
$439$	14131.1	1.53631	0.768157	−	0.640261i	$-0.221174\pi$
$439$	14131.1	1.53631	0.768157	+	0.640261i	$0.221174\pi$
$440$	1216.00	0.131751
$441$	21559.8	2.32802
$442$	−10038.6	−1.08029
$443$	1659.59	0.177990	0.0889951	−	0.996032i	$-0.471634\pi$
$443$	1659.59	0.177990	0.0889951	+	0.996032i	$0.471634\pi$
$444$	2390.71	0.255536
$445$	4164.70	0.443653
$446$	9596.68	1.01887
$447$	11848.5	1.25372
$448$	2249.73	0.237254
$449$	−13732.6	−1.44338	−0.721692	−	0.692214i	$-0.756635\pi$
$449$	−13732.6	−1.44338	−0.721692	+	0.692214i	$0.756635\pi$
$450$	2707.03	0.283579
$451$	−6499.57	−0.678609
$452$	3346.02	0.348194
$453$	−5113.62	−0.530373
$454$	−1283.54	−0.132686
$455$	11695.1	1.20500
$456$	1087.11	0.111642
$457$	−5273.37	−0.539776	−0.269888	−	0.962892i	$-0.586987\pi$
$457$	−5273.37	−0.539776	−0.269888	+	0.962892i	$0.586987\pi$
$458$	2462.69	0.251253
$459$	2551.80	0.259494
$460$	−298.181	−0.0302234
$461$	16402.4	1.65713	0.828565	−	0.559893i	$-0.189158\pi$
$461$	16402.4	1.65713	0.828565	+	0.559893i	$0.189158\pi$
$462$	−9203.68	−0.926827
$463$	−5296.03	−0.531593	−0.265796	−	0.964029i	$-0.585635\pi$
$463$	−5296.03	−0.531593	−0.265796	+	0.964029i	$0.585635\pi$
$464$	2454.56	0.245582
$465$	6796.38	0.677795
$466$	−453.402	−0.0450717
$467$	3470.94	0.343931	0.171966	−	0.985103i	$-0.444988\pi$
$467$	3470.94	0.343931	0.171966	+	0.985103i	$0.444988\pi$
$468$	−3870.56	−0.382301
$469$	−26558.8	−2.61486
$470$	2758.35	0.270709
$471$	12045.7	1.17842
$472$	2974.37	0.290057
$473$	8553.51	0.831481
$474$	−6276.94	−0.608248
$475$	1064.79	0.102854
$476$	−17615.5	−1.69623
$477$	6240.14	0.598986
$478$	−4866.54	−0.465670
$479$	−16294.7	−1.55433	−0.777163	−	0.629300i	$-0.783342\pi$
$479$	−16294.7	−1.55433	−0.777163	+	0.629300i	$0.783342\pi$
$480$	1900.54	0.180723
$481$	−3348.06	−0.317378
$482$	1802.67	0.170352
$483$	2256.88	0.212612
$484$	−3983.83	−0.374139
$485$	11937.7	1.11765
$486$	10311.7	0.962445
$487$	−2245.82	−0.208969	−0.104485	−	0.994527i	$-0.533319\pi$
$487$	−2245.82	−0.208969	−0.104485	+	0.994527i	$0.533319\pi$
$488$	378.472	0.0351079
$489$	5022.00	0.464423
$490$	14825.7	1.36685
$491$	−7578.64	−0.696577	−0.348288	−	0.937387i	$-0.613237\pi$
$491$	−7578.64	−0.696577	−0.348288	+	0.937387i	$0.613237\pi$
$492$	−10158.4	−0.930849
$493$	−19219.4	−1.75577
$494$	−1522.45	−0.138660
$495$	−3671.11	−0.333342
$496$	−1830.93	−0.165748
$497$	12280.2	1.10834
$498$	−15349.6	−1.38119
$499$	−10558.4	−0.947209	−0.473604	−	0.880738i	$-0.657047\pi$
$499$	−10558.4	−0.947209	−0.473604	+	0.880738i	$0.657047\pi$
$500$	6013.57	0.537870
$501$	−2020.50	−0.180178
$502$	11351.8	1.00928
$503$	−5816.42	−0.515590	−0.257795	−	0.966200i	$-0.582996\pi$
$503$	−5816.42	−0.515590	−0.257795	+	0.966200i	$0.582996\pi$
$504$	−6791.96	−0.600274
$505$	−3281.16	−0.289128
$506$	−328.628	−0.0288721
$507$	−4232.87	−0.370785
$508$	−1543.82	−0.134834
$509$	18051.0	1.57190	0.785950	−	0.618290i	$-0.212174\pi$
$509$	18051.0	1.57190	0.785950	+	0.618290i	$0.212174\pi$
$510$	−14881.3	−1.29207
$511$	1928.02	0.166909
$512$	−512.000	−0.0441942
$513$	387.004	0.0333073
$514$	−4071.76	−0.349412
$515$	−10676.4	−0.913514
$516$	13368.6	1.14054
$517$	3040.00	0.258606
$518$	−5875.10	−0.498334
$519$	15616.1	1.32075
$520$	−2661.61	−0.224460
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5	−	1.00000i	$-0.500009\pi$
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5		1.00000i	$0.500009\pi$
$522$	−7410.34	−0.621344
$523$	16912.1	1.41399	0.706994	−	0.707219i	$-0.250051\pi$
$523$	16912.1	1.41399	0.706994	+	0.707219i	$0.250051\pi$
$524$	6951.59	0.579545
$525$	−14089.4	−1.17126
$526$	−3848.41	−0.319009
$527$	14336.3	1.18501
$528$	2094.60	0.172643
$529$	−12086.4	−0.993377
$530$	4291.06	0.351682
$531$	−8979.66	−0.733868
$532$	−2671.56	−0.217719
$533$	14226.4	1.15612
$534$	7173.82	0.581351
$535$	−9990.67	−0.807354
$536$	6044.31	0.487079
$537$	−22875.9	−1.83830
$538$	1658.56	0.132910
$539$	16339.5	1.30574
$540$	676.575	0.0539169
$541$	−8893.75	−0.706788	−0.353394	−	0.935475i	$-0.614972\pi$
$541$	−8893.75	−0.706788	−0.353394	+	0.935475i	$0.614972\pi$
$542$	2446.26	0.193867
$543$	−15384.5	−1.21586
$544$	4008.99	0.315963
$545$	11075.6	0.870505
$546$	20145.2	1.57900
$547$	16972.8	1.32670	0.663351	−	0.748308i	$-0.269134\pi$
$547$	16972.8	1.32670	0.663351	+	0.748308i	$0.269134\pi$
$548$	167.041	0.0130213
$549$	−1142.61	−0.0888260
$550$	2051.58	0.159054
$551$	−2914.79	−0.225362
$552$	−513.626	−0.0396039
$553$	15425.4	1.18618
$554$	−3245.62	−0.248905
$555$	−4963.18	−0.379595
$556$	6147.17	0.468882
$557$	−17045.8	−1.29669	−0.648343	−	0.761348i	$-0.724538\pi$
$557$	−17045.8	−1.29669	−0.648343	+	0.761348i	$0.724538\pi$
$558$	5527.59	0.419358
$559$	−18722.1	−1.41657
$560$	−4670.52	−0.352438
$561$	−16400.8	−1.23430
$562$	−7237.91	−0.543261
$563$	−6973.83	−0.522046	−0.261023	−	0.965333i	$-0.584060\pi$
$563$	−6973.83	−0.522046	−0.261023	+	0.965333i	$0.584060\pi$
$564$	4751.34	0.354730
$565$	−6946.46	−0.517238
$566$	−4204.86	−0.312268
$567$	−28043.7	−2.07712
$568$	−2794.76	−0.206454
$569$	−21644.0	−1.59466	−0.797332	−	0.603541i	$-0.793756\pi$
$569$	−21644.0	−1.59466	−0.797332	+	0.603541i	$0.793756\pi$
$570$	−2256.89	−0.165843
$571$	−13891.0	−1.01807	−0.509035	−	0.860746i	$-0.669998\pi$
$571$	−13891.0	−1.01807	−0.509035	+	0.860746i	$0.669998\pi$
$572$	−2933.38	−0.214425
$573$	−31723.1	−2.31283
$574$	24964.1	1.81530
$575$	−503.076	−0.0364865
$576$	1545.73	0.111815
$577$	−2386.14	−0.172160	−0.0860798	−	0.996288i	$-0.527434\pi$
$577$	−2386.14	−0.172160	−0.0860798	+	0.996288i	$0.527434\pi$
$578$	−21564.7	−1.55185
$579$	19457.9	1.39662
$580$	−5095.75	−0.364809
$581$	37721.2	2.69353
$582$	20563.0	1.46454
$583$	4729.21	0.335959
$584$	−438.783	−0.0310907
$585$	8035.41	0.567903
$586$	10768.0	0.759080
$587$	−826.851	−0.0581393	−0.0290697	−	0.999577i	$-0.509254\pi$
$587$	−826.851	−0.0581393	−0.0290697	+	0.999577i	$0.509254\pi$
$588$	25537.7	1.79108
$589$	2174.23	0.152101
$590$	−6174.90	−0.430876
$591$	8969.23	0.624272
$592$	1337.07	0.0928265
$593$	9789.50	0.677920	0.338960	−	0.940801i	$-0.389925\pi$
$593$	9789.50	0.677920	0.338960	+	0.940801i	$0.389925\pi$
$594$	745.659	0.0515064
$595$	36570.5	2.51974
$596$	6626.62	0.455431
$597$	29718.2	2.03733
$598$	719.308	0.0491884
$599$	−23555.7	−1.60678	−0.803389	−	0.595455i	$-0.796972\pi$
$599$	−23555.7	−1.60678	−0.803389	+	0.595455i	$0.796972\pi$
$600$	3206.49	0.218174
$601$	12263.4	0.832334	0.416167	−	0.909288i	$-0.363373\pi$
$601$	12263.4	0.832334	0.416167	+	0.909288i	$0.363373\pi$
$602$	−32853.1	−2.22424
$603$	−18247.8	−1.23235
$604$	−2859.94	−0.192664
$605$	8270.57	0.555780
$606$	−5651.91	−0.378866
$607$	464.109	0.0310340	0.0155170	−	0.999880i	$-0.495061\pi$
$607$	464.109	0.0310340	0.0155170	+	0.999880i	$0.495061\pi$
$608$	608.000	0.0405554
$609$	38568.8	2.56632
$610$	−785.721	−0.0521523
$611$	−6654.02	−0.440577
$612$	−12103.2	−0.799415
$613$	11526.0	0.759427	0.379714	−	0.925104i	$-0.376023\pi$
$613$	11526.0	0.759427	0.379714	+	0.925104i	$0.376023\pi$
$614$	−7385.69	−0.485444
$615$	21089.3	1.38277
$616$	−5147.43	−0.336681
$617$	17632.5	1.15050	0.575249	−	0.817978i	$-0.304905\pi$
$617$	17632.5	1.15050	0.575249	+	0.817978i	$0.304905\pi$
$618$	−18390.5	−1.19705
$619$	16538.0	1.07386	0.536928	−	0.843628i	$-0.319585\pi$
$619$	16538.0	1.07386	0.536928	+	0.843628i	$0.319585\pi$
$620$	3801.07	0.246217
$621$	−182.846	−0.0118154
$622$	8855.33	0.570846
$623$	−17629.5	−1.13372
$624$	−4584.70	−0.294126
$625$	−5479.20	−0.350669
$626$	14712.9	0.939372
$627$	−2487.34	−0.158428
$628$	6736.91	0.428077
$629$	−10469.3	−0.663657
$630$	14100.3	0.891700
$631$	−8881.84	−0.560349	−0.280175	−	0.959949i	$-0.590392\pi$
$631$	−8881.84	−0.560349	−0.280175	+	0.959949i	$0.590392\pi$
$632$	−3510.56	−0.220953
$633$	−4530.38	−0.284465
$634$	−3225.53	−0.202054
$635$	3205.01	0.200295
$636$	7391.48	0.460835
$637$	−35764.3	−2.22454
$638$	−5616.07	−0.348499
$639$	8437.42	0.522346
$640$	1062.93	0.0656499
$641$	22459.3	1.38391	0.691956	−	0.721939i	$-0.256749\pi$
$641$	22459.3	1.38391	0.691956	+	0.721939i	$0.256749\pi$
$642$	−17209.2	−1.05794
$643$	19685.1	1.20731	0.603657	−	0.797244i	$-0.293710\pi$
$643$	19685.1	1.20731	0.603657	+	0.797244i	$0.293710\pi$
$644$	1262.22	0.0772338
$645$	−27753.7	−1.69427
$646$	−4760.68	−0.289948
$647$	−11755.6	−0.714312	−0.357156	−	0.934045i	$-0.616254\pi$
$647$	−11755.6	−0.714312	−0.357156	+	0.934045i	$0.616254\pi$
$648$	6382.27	0.386912
$649$	−6805.42	−0.411612
$650$	−4490.53	−0.270974
$651$	−28769.6	−1.73206
$652$	2808.70	0.168707
$653$	18350.7	1.09972	0.549862	−	0.835256i	$-0.314680\pi$
$653$	18350.7	1.09972	0.549862	+	0.835256i	$0.314680\pi$
$654$	19078.0	1.14069
$655$	−14431.7	−0.860908
$656$	−5681.40	−0.338142
$657$	1324.69	0.0786623
$658$	−11676.3	−0.691777
$659$	14617.0	0.864034	0.432017	−	0.901866i	$-0.357802\pi$
$659$	14617.0	0.864034	0.432017	+	0.901866i	$0.357802\pi$
$660$	−4348.46	−0.256460
$661$	−2932.29	−0.172546	−0.0862729	−	0.996272i	$-0.527496\pi$
$661$	−2932.29	−0.172546	−0.0862729	+	0.996272i	$0.527496\pi$
$662$	4525.56	0.265696
$663$	35898.5	2.10284
$664$	−8584.70	−0.501733
$665$	5546.24	0.323420
$666$	−4036.63	−0.234859
$667$	1377.14	0.0799448
$668$	−1130.02	−0.0654520
$669$	−34318.0	−1.98328
$670$	−12548.2	−0.723551
$671$	−865.951	−0.0498207
$672$	−8045.12	−0.461826
$673$	21271.4	1.21835	0.609177	−	0.793034i	$-0.291500\pi$
$673$	21271.4	1.21835	0.609177	+	0.793034i	$0.291500\pi$
$674$	4075.25	0.232898
$675$	1141.48	0.0650900
$676$	−2367.35	−0.134692
$677$	−32878.2	−1.86649	−0.933245	−	0.359241i	$-0.883036\pi$
$677$	−32878.2	−1.86649	−0.933245	+	0.359241i	$0.883036\pi$
$678$	−11965.5	−0.677776
$679$	−50533.1	−2.85609
$680$	−8322.80	−0.469360
$681$	4589.98	0.258280
$682$	4189.20	0.235209
$683$	−7771.01	−0.435358	−0.217679	−	0.976020i	$-0.569849\pi$
$683$	−7771.01	−0.435358	−0.217679	+	0.976020i	$0.569849\pi$
$684$	−1835.56	−0.102609
$685$	−346.783	−0.0193429
$686$	−38643.9	−2.15078
$687$	−8806.65	−0.489075
$688$	7476.79	0.414317
$689$	−10351.4	−0.572361
$690$	1066.30	0.0588312
$691$	3655.65	0.201255	0.100628	−	0.994924i	$-0.467915\pi$
$691$	3655.65	0.201255	0.100628	+	0.994924i	$0.467915\pi$
$692$	8733.74	0.479779
$693$	15540.1	0.851833
$694$	3688.66	0.201757
$695$	−12761.7	−0.696518
$696$	−8777.59	−0.478037
$697$	44485.7	2.41753
$698$	20313.8	1.10156
$699$	1621.38	0.0877342
$700$	−7879.88	−0.425473
$701$	−28207.0	−1.51977	−0.759887	−	0.650055i	$-0.774746\pi$
$701$	−28207.0	−1.51977	−0.759887	+	0.650055i	$0.774746\pi$
$702$	−1632.11	−0.0877496
$703$	−1587.77	−0.0851834
$704$	1171.46	0.0627148
$705$	−9863.94	−0.526947
$706$	−3810.85	−0.203149
$707$	13889.4	0.738848
$708$	−10636.5	−0.564608
$709$	36417.7	1.92905	0.964524	−	0.263997i	$-0.0850409\pi$
$709$	36417.7	1.92905	0.964524	+	0.263997i	$0.0850409\pi$
$710$	5802.02	0.306685
$711$	10598.4	0.559031
$712$	4012.17	0.211183
$713$	−1027.25	−0.0539563
$714$	62993.8	3.30180
$715$	6089.80	0.318525
$716$	−12794.0	−0.667786
$717$	17402.9	0.906448
$718$	4993.00	0.259522
$719$	27227.6	1.41227	0.706133	−	0.708079i	$-0.250438\pi$
$719$	27227.6	1.41227	0.706133	+	0.708079i	$0.250438\pi$
$720$	−3208.99	−0.166100
$721$	45194.2	2.33442
$722$	−722.000	−0.0372161
$723$	−6446.41	−0.331597
$724$	−8604.20	−0.441675
$725$	−8597.30	−0.440408
$726$	14246.3	0.728279
$727$	27647.2	1.41043	0.705213	−	0.708996i	$-0.250851\pi$
$727$	27647.2	1.41043	0.705213	+	0.708996i	$0.250851\pi$
$728$	11266.8	0.573592
$729$	−15334.8	−0.779090
$730$	910.929	0.0461849
$731$	−58543.7	−2.96213
$732$	−1353.43	−0.0683391
$733$	−32139.8	−1.61952	−0.809761	−	0.586760i	$-0.800403\pi$
$733$	−32139.8	−1.61952	−0.809761	+	0.586760i	$0.800403\pi$
$734$	17496.8	0.879864
$735$	−53017.1	−2.66063
$736$	−287.260	−0.0143866
$737$	−13829.5	−0.691202
$738$	17152.2	0.855530
$739$	16224.9	0.807635	0.403818	−	0.914839i	$-0.367683\pi$
$739$	16224.9	0.807635	0.403818	+	0.914839i	$0.367683\pi$
$740$	−2775.80	−0.137893
$741$	5444.33	0.269909
$742$	−18164.4	−0.898700
$743$	22463.2	1.10915	0.554573	−	0.832135i	$-0.312882\pi$
$743$	22463.2	1.10915	0.554573	+	0.832135i	$0.312882\pi$
$744$	6547.46	0.322637
$745$	−13757.1	−0.676538
$746$	−2921.24	−0.143370
$747$	25917.3	1.26943
$748$	−9172.64	−0.448376
$749$	42291.3	2.06314
$750$	−21504.7	−1.04699
$751$	9310.24	0.452377	0.226189	−	0.974083i	$-0.427373\pi$
$751$	9310.24	0.452377	0.226189	+	0.974083i	$0.427373\pi$
$752$	2657.32	0.128860
$753$	−40594.6	−1.96461
$754$	12292.6	0.593726
$755$	5937.33	0.286201
$756$	−2863.99	−0.137781
$757$	28707.1	1.37830	0.689152	−	0.724617i	$-0.257983\pi$
$757$	28707.1	1.37830	0.689152	+	0.724617i	$0.257983\pi$
$758$	−21161.9	−1.01403
$759$	1175.18	0.0562009
$760$	−1262.23	−0.0602445
$761$	−20820.3	−0.991768	−0.495884	−	0.868389i	$-0.665156\pi$
$761$	−20820.3	−0.991768	−0.495884	+	0.868389i	$0.665156\pi$
$762$	5520.73	0.262461
$763$	−46883.7	−2.22452
$764$	−17742.1	−0.840165
$765$	25126.6	1.18752
$766$	5865.08	0.276650
$767$	14895.8	0.701249
$768$	1830.93	0.0860260
$769$	28026.2	1.31424	0.657120	−	0.753786i	$-0.271775\pi$
$769$	28026.2	1.31424	0.657120	+	0.753786i	$0.271775\pi$
$770$	10686.2	0.500136
$771$	14560.8	0.680146
$772$	10882.4	0.507339
$773$	−16405.1	−0.763326	−0.381663	−	0.924302i	$-0.624648\pi$
$773$	−16405.1	−0.763326	−0.381663	+	0.924302i	$0.624648\pi$
$774$	−22572.5	−1.04826
$775$	6412.98	0.297240
$776$	11500.5	0.532013
$777$	21009.6	0.970030
$778$	7262.38	0.334665
$779$	6746.66	0.310301
$780$	9517.99	0.436922
$781$	6394.47	0.292973
$782$	2249.26	0.102856
$783$	−3124.75	−0.142617
$784$	14282.7	0.650633
$785$	−13986.1	−0.635903
$786$	−24859.1	−1.12811
$787$	−28468.8	−1.28946	−0.644730	−	0.764411i	$-0.723030\pi$
$787$	−28468.8	−1.28946	−0.644730	+	0.764411i	$0.723030\pi$
$788$	5016.30	0.226775
$789$	13762.0	0.620965
$790$	7288.04	0.328224
$791$	29404.9	1.32177
$792$	−3536.66	−0.158674
$793$	1895.41	0.0848777
$794$	4010.64	0.179260
$795$	−15345.0	−0.684566
$796$	16620.7	0.740084
$797$	7256.68	0.322515	0.161258	−	0.986912i	$-0.448445\pi$
$797$	7256.68	0.322515	0.161258	+	0.986912i	$0.448445\pi$
$798$	9553.58	0.423801
$799$	−20807.0	−0.921275
$800$	1793.32	0.0792544
$801$	−12112.8	−0.534311
$802$	16374.6	0.720957
$803$	1003.94	0.0441201
$804$	−21614.7	−0.948122
$805$	−2620.42	−0.114730
$806$	−9169.40	−0.400718
$807$	−5931.06	−0.258715
$808$	−3160.99	−0.137628
$809$	10647.1	0.462711	0.231356	−	0.972869i	$-0.425684\pi$
$809$	10647.1	0.462711	0.231356	+	0.972869i	$0.425684\pi$
$810$	−13249.8	−0.574754
$811$	−27076.9	−1.17238	−0.586190	−	0.810174i	$-0.699373\pi$
$811$	−27076.9	−1.17238	−0.586190	+	0.810174i	$0.699373\pi$
$812$	21570.7	0.932246
$813$	−8747.91	−0.377371
$814$	−3059.24	−0.131728
$815$	−5830.96	−0.250613
$816$	−14336.3	−0.615037
$817$	−8878.69	−0.380203
$818$	31129.9	1.33060
$819$	−34014.5	−1.45124
$820$	11794.8	0.502307
$821$	1880.29	0.0799301	0.0399650	−	0.999201i	$-0.487275\pi$
$821$	1880.29	0.0799301	0.0399650	+	0.999201i	$0.487275\pi$
$822$	−597.345	−0.0253465
$823$	36904.5	1.56307	0.781537	−	0.623859i	$-0.214436\pi$
$823$	36904.5	1.56307	0.781537	+	0.623859i	$0.214436\pi$
$824$	−10285.4	−0.434841
$825$	−7336.51	−0.309605
$826$	26138.9	1.10107
$827$	−23091.4	−0.970939	−0.485470	−	0.874254i	$-0.661351\pi$
$827$	−23091.4	−0.970939	−0.485470	+	0.874254i	$0.661351\pi$
$828$	867.240	0.0363994
$829$	−15527.1	−0.650515	−0.325258	−	0.945625i	$-0.605451\pi$
$829$	−15527.1	−0.650515	−0.325258	+	0.945625i	$0.605451\pi$
$830$	17822.1	0.745319
$831$	11606.4	0.484504
$832$	−2564.13	−0.106845
$833$	−111834.	−4.65166
$834$	−21982.5	−0.912700
$835$	2345.97	0.0972282
$836$	−1391.11	−0.0575511
$837$	2330.84	0.0962553
$838$	3679.66	0.151685
$839$	25695.1	1.05732	0.528661	−	0.848833i	$-0.322694\pi$
$839$	25695.1	1.05732	0.528661	+	0.848833i	$0.322694\pi$
$840$	16701.9	0.686038
$841$	−854.386	−0.0350316
$842$	−9316.57	−0.381319
$843$	25883.0	1.05748
$844$	−2533.75	−0.103335
$845$	4914.70	0.200084
$846$	−8022.48	−0.326027
$847$	−35010.0	−1.42026
$848$	4133.90	0.167404
$849$	15036.7	0.607843
$850$	−14041.8	−0.566625
$851$	750.170	0.0302180
$852$	9994.17	0.401872
$853$	−4534.78	−0.182026	−0.0910129	−	0.995850i	$-0.529010\pi$
$853$	−4534.78	−0.182026	−0.0910129	+	0.995850i	$0.529010\pi$
$854$	3326.02	0.133272
$855$	3810.68	0.152424
$856$	−9624.77	−0.384308
$857$	48754.1	1.94330	0.971651	−	0.236422i	$-0.0759747\pi$
$857$	48754.1	1.94330	0.971651	+	0.236422i	$0.0759747\pi$
$858$	10489.9	0.417387
$859$	−11730.5	−0.465937	−0.232968	−	0.972484i	$-0.574844\pi$
$859$	−11730.5	−0.465937	−0.232968	+	0.972484i	$0.574844\pi$
$860$	−15522.1	−0.615463
$861$	−89272.6	−3.53357
$862$	−5640.93	−0.222890
$863$	9907.21	0.390783	0.195391	−	0.980725i	$-0.437402\pi$
$863$	9907.21	0.390783	0.195391	+	0.980725i	$0.437402\pi$
$864$	651.795	0.0256650
$865$	−18131.5	−0.712706
$866$	−17962.8	−0.704850
$867$	77116.0	3.02076
$868$	−16090.2	−0.629192
$869$	8032.22	0.313549
$870$	18222.6	0.710118
$871$	30270.3	1.17758
$872$	10669.9	0.414368
$873$	−34720.0	−1.34604
$874$	341.121	0.0132021
$875$	52847.3	2.04179
$876$	1569.10	0.0605195
$877$	23741.6	0.914136	0.457068	−	0.889432i	$-0.348900\pi$
$877$	23741.6	0.914136	0.457068	+	0.889432i	$0.348900\pi$
$878$	−28262.3	−1.08634
$879$	−38506.7	−1.47758
$880$	−2432.00	−0.0931622
$881$	−24381.3	−0.932378	−0.466189	−	0.884685i	$-0.654373\pi$
$881$	−24381.3	−0.932378	−0.466189	+	0.884685i	$0.654373\pi$
$882$	−43119.5	−1.64616
$883$	343.893	0.0131064	0.00655319	−	0.999979i	$-0.497914\pi$
$883$	343.893	0.0131064	0.00655319	+	0.999979i	$0.497914\pi$
$884$	20077.3	0.763882
$885$	22081.7	0.838719
$886$	−3319.19	−0.125858
$887$	33805.3	1.27967	0.639837	−	0.768511i	$-0.279002\pi$
$887$	33805.3	1.27967	0.639837	+	0.768511i	$0.279002\pi$
$888$	−4781.41	−0.180691
$889$	−13567.1	−0.511839
$890$	−8329.39	−0.313710
$891$	−14602.7	−0.549057
$892$	−19193.4	−0.720449
$893$	−3155.57	−0.118250
$894$	−23697.0	−0.886517
$895$	26560.8	0.991989
$896$	−4499.46	−0.167764
$897$	−2572.27	−0.0957475
$898$	27465.1	1.02063
$899$	−17555.2	−0.651277
$900$	−5414.06	−0.200521
$901$	−32368.7	−1.19684
$902$	12999.1	0.479849
$903$	117484.	4.32958
$904$	−6692.05	−0.246210
$905$	17862.6	0.656103
$906$	10227.2	0.375030
$907$	−26456.3	−0.968540	−0.484270	−	0.874919i	$-0.660915\pi$
$907$	−26456.3	−0.968540	−0.484270	+	0.874919i	$0.660915\pi$
$908$	2567.08	0.0938233
$909$	9543.06	0.348211
$910$	−23390.2	−0.852065
$911$	−310.509	−0.0112927	−0.00564633	−	0.999984i	$-0.501797\pi$
$911$	−310.509	−0.0112927	−0.00564633	+	0.999984i	$0.501797\pi$
$912$	−2174.23	−0.0789429
$913$	19641.9	0.711997
$914$	10546.7	0.381679
$915$	2809.77	0.101517
$916$	−4925.37	−0.177662
$917$	61090.7	2.19999
$918$	−5103.60	−0.183490
$919$	−17316.8	−0.621575	−0.310787	−	0.950479i	$-0.600593\pi$
$919$	−17316.8	−0.621575	−0.310787	+	0.950479i	$0.600593\pi$
$920$	596.361	0.0213711
$921$	26411.5	0.944938
$922$	−32804.8	−1.17177
$923$	−13996.3	−0.499128
$924$	18407.4	0.655366
$925$	−4683.20	−0.166468
$926$	10592.1	0.375893
$927$	31051.7	1.10019
$928$	−4909.12	−0.173653
$929$	−36473.5	−1.28811	−0.644056	−	0.764978i	$-0.722750\pi$
$929$	−36473.5	−1.28811	−0.644056	+	0.764978i	$0.722750\pi$
$930$	−13592.8	−0.479273
$931$	−16960.7	−0.597061
$932$	906.803	0.0318705
$933$	−31667.0	−1.11118
$934$	−6941.88	−0.243196
$935$	19042.7	0.666057
$936$	7741.12	0.270327
$937$	4777.88	0.166581	0.0832905	−	0.996525i	$-0.473457\pi$
$937$	4777.88	0.166581	0.0832905	+	0.996525i	$0.473457\pi$
$938$	53117.5	1.84899
$939$	−52613.9	−1.82853
$940$	−5516.69	−0.191420
$941$	31366.5	1.08663	0.543315	−	0.839529i	$-0.317169\pi$
$941$	31366.5	1.08663	0.543315	+	0.839529i	$0.317169\pi$
$942$	−24091.4	−0.833271
$943$	−3187.58	−0.110076
$944$	−5948.75	−0.205101
$945$	5945.75	0.204672
$946$	−17107.0	−0.587946
$947$	26496.1	0.909195	0.454597	−	0.890697i	$-0.349783\pi$
$947$	26496.1	0.909195	0.454597	+	0.890697i	$0.349783\pi$
$948$	12553.9	0.430096
$949$	−2197.45	−0.0751658
$950$	−2129.57	−0.0727289
$951$	11534.6	0.393307
$952$	35231.1	1.19942
$953$	−11928.3	−0.405451	−0.202725	−	0.979236i	$-0.564980\pi$
$953$	−11928.3	−0.405451	−0.202725	+	0.979236i	$0.564980\pi$
$954$	−12480.3	−0.423547
$955$	36833.1	1.24806
$956$	9733.07	0.329278
$957$	20083.3	0.678370
$958$	32589.3	1.09907
$959$	1467.96	0.0494296
$960$	−3801.07	−0.127791
$961$	−16696.1	−0.560440
$962$	6696.13	0.224420
$963$	29057.3	0.972333
$964$	−3605.34	−0.120457
$965$	−22592.2	−0.753647
$966$	−4513.75	−0.150339
$967$	−58320.0	−1.93945	−0.969723	−	0.244209i	$-0.921472\pi$
$967$	−58320.0	−1.93945	−0.969723	+	0.244209i	$0.921472\pi$
$968$	7967.67	0.264556
$969$	17024.3	0.564397
$970$	−23875.3	−0.790300
$971$	−27526.8	−0.909760	−0.454880	−	0.890553i	$-0.650318\pi$
$971$	−27526.8	−0.909760	−0.454880	+	0.890553i	$0.650318\pi$
$972$	−20623.4	−0.680551
$973$	54021.5	1.77991
$974$	4491.64	0.147763
$975$	16058.3	0.527464
$976$	−756.945	−0.0248250
$977$	−36194.1	−1.18521	−0.592605	−	0.805493i	$-0.701901\pi$
$977$	−36194.1	−1.18521	−0.592605	+	0.805493i	$0.701901\pi$
$978$	−10044.0	−0.328397
$979$	−9179.90	−0.299684
$980$	−29651.3	−0.966507
$981$	−32212.6	−1.04839
$982$	15157.3	0.492554
$983$	−58081.3	−1.88454	−0.942271	−	0.334850i	$-0.891314\pi$
$983$	−58081.3	−1.88454	−0.942271	+	0.334850i	$0.891314\pi$
$984$	20316.9	0.658210
$985$	−10414.0	−0.336871
$986$	38438.7	1.24152
$987$	41754.8	1.34658
$988$	3044.90	0.0980477
$989$	4194.89	0.134873
$990$	7342.23	0.235708
$991$	54921.4	1.76048	0.880240	−	0.474529i	$-0.157382\pi$
$991$	54921.4	1.76048	0.880240	+	0.474529i	$0.157382\pi$
$992$	3661.86	0.117202
$993$	−16183.6	−0.517190
$994$	−24560.4	−0.783712
$995$	−34505.2	−1.09939
$996$	30699.2	0.976647
$997$	−33999.5	−1.08001	−0.540007	−	0.841661i	$-0.681578\pi$
$997$	−33999.5	−1.08001	−0.540007	+	0.841661i	$0.681578\pi$
$998$	21116.7	0.669778
$999$	−1702.14	−0.0539073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.a.b.1.2	✓	2
3.2	odd	2		342.4.a.k.1.2		2
4.3	odd	2		304.4.a.d.1.1		2
5.2	odd	4		950.4.b.g.799.1		4
5.3	odd	4		950.4.b.g.799.4		4
5.4	even	2		950.4.a.h.1.1		2
7.6	odd	2		1862.4.a.b.1.1		2
8.3	odd	2		1216.4.a.l.1.2		2
8.5	even	2		1216.4.a.j.1.1		2
19.18	odd	2		722.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.2	✓	2	1.1	even	1	trivial
304.4.a.d.1.1		2	4.3	odd	2
342.4.a.k.1.2		2	3.2	odd	2
722.4.a.i.1.1		2	19.18	odd	2
950.4.a.h.1.1		2	5.4	even	2
950.4.b.g.799.1		4	5.2	odd	4
950.4.b.g.799.4		4	5.3	odd	4
1216.4.a.j.1.1		2	8.5	even	2
1216.4.a.l.1.2		2	8.3	odd	2
1862.4.a.b.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +7.15207 q^{3} +4.00000 q^{4} -8.30413 q^{5} -14.3041 q^{6} +35.1521 q^{7} -8.00000 q^{8} +24.1521 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +7.15207 q^{3} +4.00000 q^{4} -8.30413 q^{5} -14.3041 q^{6} +35.1521 q^{7} -8.00000 q^{8} +24.1521 q^{9} +16.6083 q^{10} +18.3041 q^{11} +28.6083 q^{12} -40.0645 q^{13} -70.3041 q^{14} -59.3917 q^{15} +16.0000 q^{16} -125.281 q^{17} -48.3041 q^{18} -19.0000 q^{19} -33.2165 q^{20} +251.410 q^{21} -36.6083 q^{22} +8.97688 q^{23} -57.2165 q^{24} -56.0413 q^{25} +80.1289 q^{26} -20.3686 q^{27} +140.608 q^{28} +153.410 q^{29} +118.783 q^{30} -114.433 q^{31} -32.0000 q^{32} +130.912 q^{33} +250.562 q^{34} -291.908 q^{35} +96.6083 q^{36} +83.5669 q^{37} +38.0000 q^{38} -286.544 q^{39} +66.4331 q^{40} -355.088 q^{41} -502.820 q^{42} +467.299 q^{43} +73.2165 q^{44} -200.562 q^{45} -17.9538 q^{46} +166.083 q^{47} +114.433 q^{48} +892.668 q^{49} +112.083 q^{50} -896.018 q^{51} -160.258 q^{52} +258.369 q^{53} +40.7372 q^{54} -152.000 q^{55} -281.217 q^{56} -135.889 q^{57} -306.820 q^{58} -371.797 q^{59} -237.567 q^{60} -47.3090 q^{61} +228.866 q^{62} +848.995 q^{63} +64.0000 q^{64} +332.701 q^{65} -261.825 q^{66} -755.539 q^{67} -501.124 q^{68} +64.2032 q^{69} +583.815 q^{70} +349.345 q^{71} -193.217 q^{72} +54.8479 q^{73} -167.134 q^{74} -400.811 q^{75} -76.0000 q^{76} +643.428 q^{77} +573.088 q^{78} +438.820 q^{79} -132.866 q^{80} -797.783 q^{81} +710.175 q^{82} +1073.09 q^{83} +1005.64 q^{84} +1040.35 q^{85} -934.598 q^{86} +1097.20 q^{87} -146.433 q^{88} -501.521 q^{89} +401.124 q^{90} -1408.35 q^{91} +35.9075 q^{92} -818.433 q^{93} -332.165 q^{94} +157.779 q^{95} -228.866 q^{96} -1437.56 q^{97} -1785.34 q^{98} +442.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} + 13 q^{13} - 114 q^{14} - 172 q^{15} + 32 q^{16} - 51 q^{17} - 70 q^{18} - 38 q^{19} + 40 q^{20} + 117 q^{21} - 20 q^{22} - 155 q^{23} - 8 q^{24} + 154 q^{25} - 26 q^{26} + 79 q^{27} + 228 q^{28} - 79 q^{29} + 344 q^{30} - 16 q^{31} - 64 q^{32} + 182 q^{33} + 102 q^{34} + 108 q^{35} + 140 q^{36} + 380 q^{37} + 76 q^{38} - 613 q^{39} - 80 q^{40} - 790 q^{41} - 234 q^{42} + 296 q^{43} + 40 q^{44} - 2 q^{45} + 310 q^{46} - 200 q^{47} + 16 q^{48} + 1027 q^{49} - 308 q^{50} - 1353 q^{51} + 52 q^{52} + 397 q^{53} - 158 q^{54} - 304 q^{55} - 456 q^{56} - 19 q^{57} + 158 q^{58} + 201 q^{59} - 688 q^{60} - 680 q^{61} + 32 q^{62} + 1086 q^{63} + 128 q^{64} + 1304 q^{65} - 364 q^{66} - 939 q^{67} - 204 q^{68} + 1073 q^{69} - 216 q^{70} + 406 q^{71} - 280 q^{72} + 123 q^{73} - 760 q^{74} - 1693 q^{75} - 152 q^{76} + 462 q^{77} + 1226 q^{78} + 106 q^{79} + 160 q^{80} - 1702 q^{81} + 1580 q^{82} + 2226 q^{83} + 468 q^{84} + 2400 q^{85} - 592 q^{86} + 2527 q^{87} - 80 q^{88} - 870 q^{89} + 4 q^{90} - 249 q^{91} - 620 q^{92} - 1424 q^{93} + 400 q^{94} - 190 q^{95} - 32 q^{96} - 1864 q^{97} - 2054 q^{98} + 352 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 + 13 * q^13 - 114 * q^14 - 172 * q^15 + 32 * q^16 - 51 * q^17 - 70 * q^18 - 38 * q^19 + 40 * q^20 + 117 * q^21 - 20 * q^22 - 155 * q^23 - 8 * q^24 + 154 * q^25 - 26 * q^26 + 79 * q^27 + 228 * q^28 - 79 * q^29 + 344 * q^30 - 16 * q^31 - 64 * q^32 + 182 * q^33 + 102 * q^34 + 108 * q^35 + 140 * q^36 + 380 * q^37 + 76 * q^38 - 613 * q^39 - 80 * q^40 - 790 * q^41 - 234 * q^42 + 296 * q^43 + 40 * q^44 - 2 * q^45 + 310 * q^46 - 200 * q^47 + 16 * q^48 + 1027 * q^49 - 308 * q^50 - 1353 * q^51 + 52 * q^52 + 397 * q^53 - 158 * q^54 - 304 * q^55 - 456 * q^56 - 19 * q^57 + 158 * q^58 + 201 * q^59 - 688 * q^60 - 680 * q^61 + 32 * q^62 + 1086 * q^63 + 128 * q^64 + 1304 * q^65 - 364 * q^66 - 939 * q^67 - 204 * q^68 + 1073 * q^69 - 216 * q^70 + 406 * q^71 - 280 * q^72 + 123 * q^73 - 760 * q^74 - 1693 * q^75 - 152 * q^76 + 462 * q^77 + 1226 * q^78 + 106 * q^79 + 160 * q^80 - 1702 * q^81 + 1580 * q^82 + 2226 * q^83 + 468 * q^84 + 2400 * q^85 - 592 * q^86 + 2527 * q^87 - 80 * q^88 - 870 * q^89 + 4 * q^90 - 249 * q^91 - 620 * q^92 - 1424 * q^93 + 400 * q^94 - 190 * q^95 - 32 * q^96 - 1864 * q^97 - 2054 * q^98 + 352 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more