LMFDB - Embedded newform 2166.4.a.bm.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 603 x^{7} - 764 x^{6} + 123192 x^{5} + 325506 x^{4} - 10023031 x^{3} - 37119420 x^{2} + \cdots + 1077539768 $ x^9 - 603x^7 - 764x^6 + 123192x^5 + 325506x^4 - 10023031x^3 - 37119420x^2 + 258857220*x + 1077539768
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3\cdot 19 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$14.0660$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +15.1867 q^{5} +6.00000 q^{6} +0.939071 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +15.1867 q^{5} +6.00000 q^{6} +0.939071 q^{7} +8.00000 q^{8} +9.00000 q^{9} +30.3733 q^{10} +50.0831 q^{11} +12.0000 q^{12} -30.5682 q^{13} +1.87814 q^{14} +45.5600 q^{15} +16.0000 q^{16} -36.9485 q^{17} +18.0000 q^{18} +60.7466 q^{20} +2.81721 q^{21} +100.166 q^{22} -24.4301 q^{23} +24.0000 q^{24} +105.634 q^{25} -61.1363 q^{26} +27.0000 q^{27} +3.75629 q^{28} +142.916 q^{29} +91.1199 q^{30} +141.256 q^{31} +32.0000 q^{32} +150.249 q^{33} -73.8970 q^{34} +14.2614 q^{35} +36.0000 q^{36} +126.952 q^{37} -91.7045 q^{39} +121.493 q^{40} +85.8312 q^{41} +5.63443 q^{42} -131.215 q^{43} +200.332 q^{44} +136.680 q^{45} -48.8602 q^{46} -138.119 q^{47} +48.0000 q^{48} -342.118 q^{49} +211.269 q^{50} -110.846 q^{51} -122.273 q^{52} +708.191 q^{53} +54.0000 q^{54} +760.595 q^{55} +7.51257 q^{56} +285.833 q^{58} -203.139 q^{59} +182.240 q^{60} +38.6668 q^{61} +282.512 q^{62} +8.45164 q^{63} +64.0000 q^{64} -464.228 q^{65} +300.499 q^{66} +582.240 q^{67} -147.794 q^{68} -73.2903 q^{69} +28.5227 q^{70} +996.713 q^{71} +72.0000 q^{72} -46.0794 q^{73} +253.904 q^{74} +316.903 q^{75} +47.0316 q^{77} -183.409 q^{78} +1173.01 q^{79} +242.986 q^{80} +81.0000 q^{81} +171.662 q^{82} -1492.61 q^{83} +11.2689 q^{84} -561.124 q^{85} -262.431 q^{86} +428.749 q^{87} +400.665 q^{88} +894.513 q^{89} +273.360 q^{90} -28.7057 q^{91} -97.7204 q^{92} +423.768 q^{93} -276.239 q^{94} +96.0000 q^{96} -1826.06 q^{97} -684.236 q^{98} +450.748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 18 q^{2} + 27 q^{3} + 36 q^{4} + 27 q^{5} + 54 q^{6} + 72 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 18 * q^2 + 27 * q^3 + 36 * q^4 + 27 * q^5 + 54 * q^6 + 72 * q^8 + 81 * q^9	\( 9 q + 18 q^{2} + 27 q^{3} + 36 q^{4} + 27 q^{5} + 54 q^{6} + 72 q^{8} + 81 q^{9} + 54 q^{10} + 39 q^{11} + 108 q^{12} + 99 q^{13} + 81 q^{15} + 144 q^{16} + 57 q^{17} + 162 q^{18} + 108 q^{20} + 78 q^{22} + 228 q^{23} + 216 q^{24} + 174 q^{25} + 198 q^{26} + 243 q^{27} + 459 q^{29} + 162 q^{30} + 243 q^{31} + 288 q^{32} + 117 q^{33} + 114 q^{34} + 324 q^{35} + 324 q^{36} + 711 q^{37} + 297 q^{39} + 216 q^{40} + 459 q^{41} + 252 q^{43} + 156 q^{44} + 243 q^{45} + 456 q^{46} - 66 q^{47} + 432 q^{48} + 2229 q^{49} + 348 q^{50} + 171 q^{51} + 396 q^{52} + 1197 q^{53} + 486 q^{54} + 762 q^{55} + 918 q^{58} + 1221 q^{59} + 324 q^{60} - 780 q^{61} + 486 q^{62} + 576 q^{64} - 237 q^{65} + 234 q^{66} + 1596 q^{67} + 228 q^{68} + 684 q^{69} + 648 q^{70} + 2538 q^{71} + 648 q^{72} + 225 q^{73} + 1422 q^{74} + 522 q^{75} - 135 q^{77} + 594 q^{78} + 834 q^{79} + 432 q^{80} + 729 q^{81} + 918 q^{82} + 2490 q^{83} - 1653 q^{85} + 504 q^{86} + 1377 q^{87} + 312 q^{88} - 507 q^{89} + 486 q^{90} + 6423 q^{91} + 912 q^{92} + 729 q^{93} - 132 q^{94} + 864 q^{96} + 2529 q^{97} + 4458 q^{98} + 351 q^{99}+O(q^{100}) \) 9 * q + 18 * q^2 + 27 * q^3 + 36 * q^4 + 27 * q^5 + 54 * q^6 + 72 * q^8 + 81 * q^9 + 54 * q^10 + 39 * q^11 + 108 * q^12 + 99 * q^13 + 81 * q^15 + 144 * q^16 + 57 * q^17 + 162 * q^18 + 108 * q^20 + 78 * q^22 + 228 * q^23 + 216 * q^24 + 174 * q^25 + 198 * q^26 + 243 * q^27 + 459 * q^29 + 162 * q^30 + 243 * q^31 + 288 * q^32 + 117 * q^33 + 114 * q^34 + 324 * q^35 + 324 * q^36 + 711 * q^37 + 297 * q^39 + 216 * q^40 + 459 * q^41 + 252 * q^43 + 156 * q^44 + 243 * q^45 + 456 * q^46 - 66 * q^47 + 432 * q^48 + 2229 * q^49 + 348 * q^50 + 171 * q^51 + 396 * q^52 + 1197 * q^53 + 486 * q^54 + 762 * q^55 + 918 * q^58 + 1221 * q^59 + 324 * q^60 - 780 * q^61 + 486 * q^62 + 576 * q^64 - 237 * q^65 + 234 * q^66 + 1596 * q^67 + 228 * q^68 + 684 * q^69 + 648 * q^70 + 2538 * q^71 + 648 * q^72 + 225 * q^73 + 1422 * q^74 + 522 * q^75 - 135 * q^77 + 594 * q^78 + 834 * q^79 + 432 * q^80 + 729 * q^81 + 918 * q^82 + 2490 * q^83 - 1653 * q^85 + 504 * q^86 + 1377 * q^87 + 312 * q^88 - 507 * q^89 + 486 * q^90 + 6423 * q^91 + 912 * q^92 + 729 * q^93 - 132 * q^94 + 864 * q^96 + 2529 * q^97 + 4458 * q^98 + 351 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	15.1867	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$5$	15.1867	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$6$	6.00000	0.408248
$7$	0.939071	0.0507051	0.0253525	−	0.999679i	$-0.491929\pi$
$7$	0.939071	0.0507051	0.0253525	+	0.999679i	$0.491929\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	30.3733	0.960488
$11$	50.0831	1.37278	0.686392	−	0.727232i	$-0.259193\pi$
$11$	50.0831	1.37278	0.686392	+	0.727232i	$0.259193\pi$
$12$	12.0000	0.288675
$13$	−30.5682	−0.652160	−0.326080	−	0.945342i	$-0.605728\pi$
$13$	−30.5682	−0.652160	−0.326080	+	0.945342i	$0.605728\pi$
$14$	1.87814	0.0358539
$15$	45.5600	0.784235
$16$	16.0000	0.250000
$17$	−36.9485	−0.527137	−0.263568	−	0.964641i	$-0.584899\pi$
$17$	−36.9485	−0.527137	−0.263568	+	0.964641i	$0.584899\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	60.7466	0.679168
$21$	2.81721	0.0292746
$22$	100.166	0.970705
$23$	−24.4301	−0.221479	−0.110740	−	0.993849i	$-0.535322\pi$
$23$	−24.4301	−0.221479	−0.110740	+	0.993849i	$0.535322\pi$
$24$	24.0000	0.204124
$25$	105.634	0.845075
$26$	−61.1363	−0.461147
$27$	27.0000	0.192450
$28$	3.75629	0.0253525
$29$	142.916	0.915135	0.457568	−	0.889175i	$-0.348721\pi$
$29$	142.916	0.915135	0.457568	+	0.889175i	$0.348721\pi$
$30$	91.1199	0.554538
$31$	141.256	0.818398	0.409199	−	0.912445i	$-0.365808\pi$
$31$	141.256	0.818398	0.409199	+	0.912445i	$0.365808\pi$
$32$	32.0000	0.176777
$33$	150.249	0.792577
$34$	−73.8970	−0.372742
$35$	14.2614	0.0688745
$36$	36.0000	0.166667
$37$	126.952	0.564076	0.282038	−	0.959403i	$-0.408990\pi$
$37$	126.952	0.564076	0.282038	+	0.959403i	$0.408990\pi$
$38$	0	0
$39$	−91.7045	−0.376525
$40$	121.493	0.480244
$41$	85.8312	0.326941	0.163470	−	0.986548i	$-0.447731\pi$
$41$	85.8312	0.326941	0.163470	+	0.986548i	$0.447731\pi$
$42$	5.63443	0.0207003
$43$	−131.215	−0.465352	−0.232676	−	0.972554i	$-0.574748\pi$
$43$	−131.215	−0.465352	−0.232676	+	0.972554i	$0.574748\pi$
$44$	200.332	0.686392
$45$	136.680	0.452779
$46$	−48.8602	−0.156610
$47$	−138.119	−0.428655	−0.214328	−	0.976762i	$-0.568756\pi$
$47$	−138.119	−0.428655	−0.214328	+	0.976762i	$0.568756\pi$
$48$	48.0000	0.144338
$49$	−342.118	−0.997429
$50$	211.269	0.597559
$51$	−110.846	−0.304343
$52$	−122.273	−0.326080
$53$	708.191	1.83543	0.917713	−	0.397244i	$-0.130033\pi$
$53$	708.191	1.83543	0.917713	+	0.397244i	$0.130033\pi$
$54$	54.0000	0.136083
$55$	760.595	1.86470
$56$	7.51257	0.0179270
$57$	0	0
$58$	285.833	0.647098
$59$	−203.139	−0.448244	−0.224122	−	0.974561i	$-0.571951\pi$
$59$	−203.139	−0.448244	−0.224122	+	0.974561i	$0.571951\pi$
$60$	182.240	0.392118
$61$	38.6668	0.0811602	0.0405801	−	0.999176i	$-0.487079\pi$
$61$	38.6668	0.0811602	0.0405801	+	0.999176i	$0.487079\pi$
$62$	282.512	0.578695
$63$	8.45164	0.0169017
$64$	64.0000	0.125000
$65$	−464.228	−0.885853
$66$	300.499	0.560437
$67$	582.240	1.06167	0.530835	−	0.847475i	$-0.321878\pi$
$67$	582.240	1.06167	0.530835	+	0.847475i	$0.321878\pi$
$68$	−147.794	−0.263568
$69$	−73.2903	−0.127871
$70$	28.5227	0.0487016
$71$	996.713	1.66603	0.833015	−	0.553251i	$-0.186613\pi$
$71$	996.713	1.66603	0.833015	+	0.553251i	$0.186613\pi$
$72$	72.0000	0.117851
$73$	−46.0794	−0.0738793	−0.0369397	−	0.999317i	$-0.511761\pi$
$73$	−46.0794	−0.0738793	−0.0369397	+	0.999317i	$0.511761\pi$
$74$	253.904	0.398862
$75$	316.903	0.487904
$76$	0	0
$77$	47.0316	0.0696072
$78$	−183.409	−0.266243
$79$	1173.01	1.67056	0.835279	−	0.549827i	$-0.185306\pi$
$79$	1173.01	1.67056	0.835279	+	0.549827i	$0.185306\pi$
$80$	242.986	0.339584
$81$	81.0000	0.111111
$82$	171.662	0.231182
$83$	−1492.61	−1.97392	−0.986960	−	0.160965i	$-0.948539\pi$
$83$	−1492.61	−1.97392	−0.986960	+	0.160965i	$0.948539\pi$
$84$	11.2689	0.0146373
$85$	−561.124	−0.716029
$86$	−262.431	−0.329054
$87$	428.749	0.528354
$88$	400.665	0.485353
$89$	894.513	1.06537	0.532686	−	0.846313i	$-0.321182\pi$
$89$	894.513	1.06537	0.532686	+	0.846313i	$0.321182\pi$
$90$	273.360	0.320163
$91$	−28.7057	−0.0330678
$92$	−97.7204	−0.110740
$93$	423.768	0.472502
$94$	−276.239	−0.303105
$95$	0	0
$96$	96.0000	0.102062
$97$	−1826.06	−1.91142	−0.955712	−	0.294302i	$-0.904913\pi$
$97$	−1826.06	−1.91142	−0.955712	+	0.294302i	$0.904913\pi$
$98$	−684.236	−0.705289
$99$	450.748	0.457595
$100$	422.538	0.422538
$101$	142.374	0.140265	0.0701325	−	0.997538i	$-0.477658\pi$
$101$	142.374	0.140265	0.0701325	+	0.997538i	$0.477658\pi$
$102$	−221.691	−0.215203
$103$	−1090.23	−1.04294	−0.521471	−	0.853269i	$-0.674617\pi$
$103$	−1090.23	−1.04294	−0.521471	+	0.853269i	$0.674617\pi$
$104$	−244.545	−0.230574
$105$	42.7841	0.0397647
$106$	1416.38	1.29784
$107$	−1493.61	−1.34947	−0.674734	−	0.738061i	$-0.735742\pi$
$107$	−1493.61	−1.34947	−0.674734	+	0.738061i	$0.735742\pi$
$108$	108.000	0.0962250
$109$	1934.84	1.70022	0.850112	−	0.526602i	$-0.176534\pi$
$109$	1934.84	1.70022	0.850112	+	0.526602i	$0.176534\pi$
$110$	1521.19	1.31854
$111$	380.856	0.325669
$112$	15.0251	0.0126763
$113$	−1407.18	−1.17147	−0.585735	−	0.810503i	$-0.699194\pi$
$113$	−1407.18	−1.17147	−0.585735	+	0.810503i	$0.699194\pi$
$114$	0	0
$115$	−371.011	−0.300843
$116$	571.666	0.457568
$117$	−275.114	−0.217387
$118$	−406.278	−0.316957
$119$	−34.6973	−0.0267285
$120$	364.480	0.277269
$121$	1177.32	0.884537
$122$	77.3336	0.0573889
$123$	257.493	0.188759
$124$	565.024	0.409199
$125$	−294.098	−0.210440
$126$	16.9033	0.0119513
$127$	1635.39	1.14266	0.571330	−	0.820721i	$-0.306428\pi$
$127$	1635.39	1.14266	0.571330	+	0.820721i	$0.306428\pi$
$128$	128.000	0.0883883
$129$	−393.646	−0.268671
$130$	−928.456	−0.626392
$131$	−1940.96	−1.29452	−0.647260	−	0.762269i	$-0.724085\pi$
$131$	−1940.96	−1.29452	−0.647260	+	0.762269i	$0.724085\pi$
$132$	600.997	0.396289
$133$	0	0
$134$	1164.48	0.750715
$135$	410.040	0.261412
$136$	−295.588	−0.186371
$137$	−1351.46	−0.842794	−0.421397	−	0.906876i	$-0.638460\pi$
$137$	−1351.46	−0.842794	−0.421397	+	0.906876i	$0.638460\pi$
$138$	−146.581	−0.0904186
$139$	1198.42	0.731288	0.365644	−	0.930755i	$-0.380849\pi$
$139$	1198.42	0.731288	0.365644	+	0.930755i	$0.380849\pi$
$140$	57.0454	0.0344373
$141$	−414.358	−0.247484
$142$	1993.43	1.17806
$143$	−1530.95	−0.895276
$144$	144.000	0.0833333
$145$	2170.42	1.24306
$146$	−92.1589	−0.0522406
$147$	−1026.35	−0.575866
$148$	507.809	0.282038
$149$	−119.876	−0.0659103	−0.0329551	−	0.999457i	$-0.510492\pi$
$149$	−119.876	−0.0659103	−0.0329551	+	0.999457i	$0.510492\pi$
$150$	633.807	0.345001
$151$	−1558.55	−0.839952	−0.419976	−	0.907535i	$-0.637962\pi$
$151$	−1558.55	−0.839952	−0.419976	+	0.907535i	$0.637962\pi$
$152$	0	0
$153$	−332.537	−0.175712
$154$	94.0632	0.0492197
$155$	2145.21	1.11166
$156$	−366.818	−0.188262
$157$	−68.2354	−0.0346865	−0.0173432	−	0.999850i	$-0.505521\pi$
$157$	−68.2354	−0.0346865	−0.0173432	+	0.999850i	$0.505521\pi$
$158$	2346.02	1.18126
$159$	2124.57	1.05968
$160$	485.973	0.240122
$161$	−22.9416	−0.0112301
$162$	162.000	0.0785674
$163$	3677.16	1.76698	0.883490	−	0.468451i	$-0.155188\pi$
$163$	3677.16	1.76698	0.883490	+	0.468451i	$0.155188\pi$
$164$	343.325	0.163470
$165$	2281.78	1.07659
$166$	−2985.22	−1.39577
$167$	−867.550	−0.401994	−0.200997	−	0.979592i	$-0.564418\pi$
$167$	−867.550	−0.401994	−0.200997	+	0.979592i	$0.564418\pi$
$168$	22.5377	0.0103501
$169$	−1262.59	−0.574687
$170$	−1122.25	−0.506309
$171$	0	0
$172$	−524.861	−0.232676
$173$	2346.34	1.03115	0.515575	−	0.856844i	$-0.327578\pi$
$173$	2346.34	1.03115	0.515575	+	0.856844i	$0.327578\pi$
$174$	857.499	0.373602
$175$	99.1983	0.0428496
$176$	801.330	0.343196
$177$	−609.416	−0.258794
$178$	1789.03	0.753332
$179$	−3678.74	−1.53610	−0.768049	−	0.640391i	$-0.778772\pi$
$179$	−3678.74	−1.53610	−0.768049	+	0.640391i	$0.778772\pi$
$180$	546.719	0.226389
$181$	588.750	0.241776	0.120888	−	0.992666i	$-0.461426\pi$
$181$	588.750	0.241776	0.120888	+	0.992666i	$0.461426\pi$
$182$	−57.4114	−0.0233825
$183$	116.000	0.0468579
$184$	−195.441	−0.0783048
$185$	1927.98	0.766204
$186$	847.536	0.334110
$187$	−1850.50	−0.723645
$188$	−552.478	−0.214328
$189$	25.3549	0.00975820
$190$	0	0
$191$	1237.70	0.468884	0.234442	−	0.972130i	$-0.424674\pi$
$191$	1237.70	0.468884	0.234442	+	0.972130i	$0.424674\pi$
$192$	192.000	0.0721688
$193$	−1339.51	−0.499586	−0.249793	−	0.968299i	$-0.580363\pi$
$193$	−1339.51	−0.499586	−0.249793	+	0.968299i	$0.580363\pi$
$194$	−3652.12	−1.35158
$195$	−1392.68	−0.511447
$196$	−1368.47	−0.498714
$197$	4184.16	1.51325	0.756623	−	0.653852i	$-0.226848\pi$
$197$	4184.16	1.51325	0.756623	+	0.653852i	$0.226848\pi$
$198$	901.496	0.323568
$199$	513.171	0.182803	0.0914014	−	0.995814i	$-0.470865\pi$
$199$	513.171	0.182803	0.0914014	+	0.995814i	$0.470865\pi$
$200$	845.075	0.298779
$201$	1746.72	0.612956
$202$	284.748	0.0991823
$203$	134.209	0.0464020
$204$	−443.382	−0.152171
$205$	1303.49	0.444095
$206$	−2180.45	−0.737472
$207$	−219.871	−0.0738265
$208$	−489.091	−0.163040
$209$	0	0
$210$	85.5681	0.0281179
$211$	−3663.00	−1.19512	−0.597562	−	0.801823i	$-0.703864\pi$
$211$	−3663.00	−1.19512	−0.597562	+	0.801823i	$0.703864\pi$
$212$	2832.77	0.917713
$213$	2990.14	0.961882
$214$	−2987.23	−0.954219
$215$	−1992.72	−0.632104
$216$	216.000	0.0680414
$217$	132.650	0.0414969
$218$	3869.69	1.20224
$219$	−138.238	−0.0426542
$220$	3042.38	0.932351
$221$	1129.45	0.343778
$222$	761.713	0.230283
$223$	2490.93	0.748004	0.374002	−	0.927428i	$-0.377985\pi$
$223$	2490.93	0.748004	0.374002	+	0.927428i	$0.377985\pi$
$224$	30.0503	0.00896348
$225$	950.710	0.281692
$226$	−2814.35	−0.828354
$227$	−3980.30	−1.16380	−0.581898	−	0.813262i	$-0.697690\pi$
$227$	−3980.30	−1.16380	−0.581898	+	0.813262i	$0.697690\pi$
$228$	0	0
$229$	−1653.20	−0.477060	−0.238530	−	0.971135i	$-0.576666\pi$
$229$	−1653.20	−0.477060	−0.238530	+	0.971135i	$0.576666\pi$
$230$	−742.023	−0.212728
$231$	141.095	0.0401877
$232$	1143.33	0.323549
$233$	5611.46	1.57776	0.788882	−	0.614544i	$-0.210660\pi$
$233$	5611.46	1.57776	0.788882	+	0.614544i	$0.210660\pi$
$234$	−550.227	−0.153716
$235$	−2097.57	−0.582258
$236$	−812.555	−0.224122
$237$	3519.03	0.964497
$238$	−69.3946	−0.0188999
$239$	2398.42	0.649125	0.324563	−	0.945864i	$-0.394783\pi$
$239$	2398.42	0.649125	0.324563	+	0.945864i	$0.394783\pi$
$240$	728.959	0.196059
$241$	6379.74	1.70521	0.852604	−	0.522558i	$-0.175022\pi$
$241$	6379.74	1.70521	0.852604	+	0.522558i	$0.175022\pi$
$242$	2354.64	0.625462
$243$	243.000	0.0641500
$244$	154.667	0.0405801
$245$	−5195.63	−1.35484
$246$	514.987	0.133473
$247$	0	0
$248$	1130.05	0.289347
$249$	−4477.83	−1.13964
$250$	−588.197	−0.148803
$251$	−485.880	−0.122185	−0.0610926	−	0.998132i	$-0.519458\pi$
$251$	−485.880	−0.122185	−0.0610926	+	0.998132i	$0.519458\pi$
$252$	33.8066	0.00845085
$253$	−1223.54	−0.304044
$254$	3270.79	0.807982
$255$	−1683.37	−0.413399
$256$	256.000	0.0625000
$257$	−4651.99	−1.12912	−0.564558	−	0.825393i	$-0.690954\pi$
$257$	−4651.99	−1.12912	−0.564558	+	0.825393i	$0.690954\pi$
$258$	−787.292	−0.189979
$259$	119.217	0.0286015
$260$	−1856.91	−0.442926
$261$	1286.25	0.305045
$262$	−3881.91	−0.915364
$263$	−4747.21	−1.11302	−0.556512	−	0.830840i	$-0.687861\pi$
$263$	−4747.21	−1.11302	−0.556512	+	0.830840i	$0.687861\pi$
$264$	1201.99	0.280218
$265$	10755.1	2.49312
$266$	0	0
$267$	2683.54	0.615093
$268$	2328.96	0.530835
$269$	−3568.21	−0.808764	−0.404382	−	0.914590i	$-0.632513\pi$
$269$	−3568.21	−0.808764	−0.404382	+	0.914590i	$0.632513\pi$
$270$	820.079	0.184846
$271$	−4110.01	−0.921274	−0.460637	−	0.887589i	$-0.652379\pi$
$271$	−4110.01	−0.921274	−0.460637	+	0.887589i	$0.652379\pi$
$272$	−591.176	−0.131784
$273$	−86.1171	−0.0190917
$274$	−2702.91	−0.595945
$275$	5290.50	1.16011
$276$	−293.161	−0.0639356
$277$	1809.82	0.392569	0.196285	−	0.980547i	$-0.437112\pi$
$277$	1809.82	0.392569	0.196285	+	0.980547i	$0.437112\pi$
$278$	2396.85	0.517099
$279$	1271.30	0.272799
$280$	114.091	0.0243508
$281$	656.871	0.139451	0.0697253	−	0.997566i	$-0.477788\pi$
$281$	656.871	0.139451	0.0697253	+	0.997566i	$0.477788\pi$
$282$	−828.717	−0.174998
$283$	991.543	0.208272	0.104136	−	0.994563i	$-0.466792\pi$
$283$	991.543	0.208272	0.104136	+	0.994563i	$0.466792\pi$
$284$	3986.85	0.833015
$285$	0	0
$286$	−3061.90	−0.633055
$287$	80.6016	0.0165776
$288$	288.000	0.0589256
$289$	−3547.81	−0.722127
$290$	4340.85	0.878977
$291$	−5478.18	−1.10356
$292$	−184.318	−0.0369397
$293$	−7187.18	−1.43304	−0.716518	−	0.697569i	$-0.754265\pi$
$293$	−7187.18	−1.43304	−0.716518	+	0.697569i	$0.754265\pi$
$294$	−2052.71	−0.407199
$295$	−3085.00	−0.608866
$296$	1015.62	0.199431
$297$	1352.24	0.264192
$298$	−239.752	−0.0466056
$299$	746.783	0.144440
$300$	1267.61	0.243952
$301$	−123.220	−0.0235957
$302$	−3117.10	−0.593936
$303$	427.123	0.0809820
$304$	0	0
$305$	587.219	0.110243
$306$	−665.073	−0.124247
$307$	391.509	0.0727837	0.0363918	−	0.999338i	$-0.488414\pi$
$307$	391.509	0.0727837	0.0363918	+	0.999338i	$0.488414\pi$
$308$	188.126	0.0348036
$309$	−3270.68	−0.602143
$310$	4290.41	0.786062
$311$	−9768.06	−1.78101	−0.890507	−	0.454969i	$-0.849650\pi$
$311$	−9768.06	−1.78101	−0.890507	+	0.454969i	$0.849650\pi$
$312$	−733.636	−0.133122
$313$	−4443.28	−0.802393	−0.401196	−	0.915992i	$-0.631406\pi$
$313$	−4443.28	−0.802393	−0.401196	+	0.915992i	$0.631406\pi$
$314$	−136.471	−0.0245270
$315$	128.352	0.0229582
$316$	4692.04	0.835279
$317$	1685.86	0.298698	0.149349	−	0.988785i	$-0.452282\pi$
$317$	1685.86	0.298698	0.149349	+	0.988785i	$0.452282\pi$
$318$	4249.15	0.749309
$319$	7157.70	1.25628
$320$	971.946	0.169792
$321$	−4480.84	−0.779116
$322$	−45.8832	−0.00794090
$323$	0	0
$324$	324.000	0.0555556
$325$	−3229.05	−0.551125
$326$	7354.33	1.24944
$327$	5804.53	0.981625
$328$	686.649	0.115591
$329$	−129.704	−0.0217350
$330$	4563.57	0.761261
$331$	−2676.13	−0.444390	−0.222195	−	0.975002i	$-0.571322\pi$
$331$	−2676.13	−0.444390	−0.222195	+	0.975002i	$0.571322\pi$
$332$	−5970.45	−0.986960
$333$	1142.57	0.188025
$334$	−1735.10	−0.284253
$335$	8842.28	1.44211
$336$	45.0754	0.00731865
$337$	−1274.51	−0.206015	−0.103008	−	0.994681i	$-0.532847\pi$
$337$	−1274.51	−0.206015	−0.103008	+	0.994681i	$0.532847\pi$
$338$	−2525.17	−0.406365
$339$	−4221.53	−0.676348
$340$	−2244.50	−0.358014
$341$	7074.54	1.12348
$342$	0	0
$343$	−643.375	−0.101280
$344$	−1049.72	−0.164527
$345$	−1113.03	−0.173692
$346$	4692.68	0.729133
$347$	4124.95	0.638152	0.319076	−	0.947729i	$-0.396627\pi$
$347$	4124.95	0.638152	0.319076	+	0.947729i	$0.396627\pi$
$348$	1715.00	0.264177
$349$	−7835.73	−1.20182	−0.600912	−	0.799315i	$-0.705196\pi$
$349$	−7835.73	−1.20182	−0.600912	+	0.799315i	$0.705196\pi$
$350$	198.397	0.0302993
$351$	−825.341	−0.125508
$352$	1602.66	0.242676
$353$	−10960.4	−1.65259	−0.826297	−	0.563235i	$-0.809557\pi$
$353$	−10960.4	−1.65259	−0.826297	+	0.563235i	$0.809557\pi$
$354$	−1218.83	−0.182995
$355$	15136.7	2.26303
$356$	3578.05	0.532686
$357$	−104.092	−0.0154317
$358$	−7357.47	−1.08619
$359$	1299.65	0.191067	0.0955336	−	0.995426i	$-0.469544\pi$
$359$	1299.65	0.191067	0.0955336	+	0.995426i	$0.469544\pi$
$360$	1093.44	0.160081
$361$	0	0
$362$	1177.50	0.170961
$363$	3531.96	0.510688
$364$	−114.823	−0.0165339
$365$	−699.792	−0.100353
$366$	232.001	0.0331335
$367$	2642.39	0.375835	0.187918	−	0.982185i	$-0.439826\pi$
$367$	2642.39	0.375835	0.187918	+	0.982185i	$0.439826\pi$
$368$	−390.882	−0.0553699
$369$	772.480	0.108980
$370$	3855.96	0.541788
$371$	665.042	0.0930654
$372$	1695.07	0.236251
$373$	9868.11	1.36984	0.684921	−	0.728617i	$-0.259837\pi$
$373$	9868.11	1.36984	0.684921	+	0.728617i	$0.259837\pi$
$374$	−3700.99	−0.511695
$375$	−882.295	−0.121497
$376$	−1104.96	−0.151552
$377$	−4368.69	−0.596815
$378$	50.7099	0.00690009
$379$	4390.48	0.595050	0.297525	−	0.954714i	$-0.403839\pi$
$379$	4390.48	0.595050	0.297525	+	0.954714i	$0.403839\pi$
$380$	0	0
$381$	4906.18	0.659715
$382$	2475.40	0.331551
$383$	3884.72	0.518277	0.259139	−	0.965840i	$-0.416561\pi$
$383$	3884.72	0.518277	0.259139	+	0.965840i	$0.416561\pi$
$384$	384.000	0.0510310
$385$	714.253	0.0945499
$386$	−2679.02	−0.353260
$387$	−1180.94	−0.155117
$388$	−7304.24	−0.955712
$389$	10710.0	1.39593	0.697967	−	0.716130i	$-0.254088\pi$
$389$	10710.0	1.39593	0.697967	+	0.716130i	$0.254088\pi$
$390$	−2785.37	−0.361648
$391$	902.656	0.116750
$392$	−2736.95	−0.352644
$393$	−5822.87	−0.747391
$394$	8368.33	1.07003
$395$	17814.1	2.26918
$396$	1802.99	0.228797
$397$	−14137.2	−1.78722	−0.893611	−	0.448842i	$-0.851837\pi$
$397$	−14137.2	−1.78722	−0.893611	+	0.448842i	$0.851837\pi$
$398$	1026.34	0.129261
$399$	0	0
$400$	1690.15	0.211269
$401$	−10480.3	−1.30514	−0.652569	−	0.757729i	$-0.726309\pi$
$401$	−10480.3	−1.30514	−0.652569	+	0.757729i	$0.726309\pi$
$402$	3493.44	0.433425
$403$	−4317.94	−0.533727
$404$	569.497	0.0701325
$405$	1230.12	0.150926
$406$	268.417	0.0328112
$407$	6358.16	0.774354
$408$	−886.764	−0.107601
$409$	11573.8	1.39923	0.699617	−	0.714519i	$-0.253354\pi$
$409$	11573.8	1.39923	0.699617	+	0.714519i	$0.253354\pi$
$410$	2606.98	0.314023
$411$	−4054.37	−0.486587
$412$	−4360.90	−0.521471
$413$	−190.762	−0.0227283
$414$	−439.742	−0.0522032
$415$	−22667.8	−2.68125
$416$	−978.181	−0.115287
$417$	3595.27	0.422210
$418$	0	0
$419$	16115.5	1.87899	0.939494	−	0.342566i	$-0.111296\pi$
$419$	16115.5	1.87899	0.939494	+	0.342566i	$0.111296\pi$
$420$	171.136	0.0198824
$421$	10958.6	1.26862	0.634312	−	0.773077i	$-0.281283\pi$
$421$	10958.6	1.26862	0.634312	+	0.773077i	$0.281283\pi$
$422$	−7326.00	−0.845080
$423$	−1243.08	−0.142885
$424$	5665.53	0.648921
$425$	−3903.03	−0.445470
$426$	5980.28	0.680154
$427$	36.3109	0.00411524
$428$	−5974.46	−0.674734
$429$	−4592.85	−0.516888
$430$	−3985.44	−0.446965
$431$	1254.09	0.140156	0.0700782	−	0.997541i	$-0.477675\pi$
$431$	1254.09	0.140156	0.0700782	+	0.997541i	$0.477675\pi$
$432$	432.000	0.0481125
$433$	3422.08	0.379803	0.189902	−	0.981803i	$-0.439183\pi$
$433$	3422.08	0.379803	0.189902	+	0.981803i	$0.439183\pi$
$434$	265.299	0.0293428
$435$	6511.27	0.717681
$436$	7739.38	0.850112
$437$	0	0
$438$	−276.477	−0.0301611
$439$	−2705.05	−0.294089	−0.147044	−	0.989130i	$-0.546976\pi$
$439$	−2705.05	−0.294089	−0.147044	+	0.989130i	$0.546976\pi$
$440$	6084.76	0.659272
$441$	−3079.06	−0.332476
$442$	2258.90	0.243088
$443$	−4631.11	−0.496683	−0.248342	−	0.968673i	$-0.579886\pi$
$443$	−4631.11	−0.496683	−0.248342	+	0.968673i	$0.579886\pi$
$444$	1523.43	0.162835
$445$	13584.7	1.44713
$446$	4981.86	0.528919
$447$	−359.628	−0.0380533
$448$	60.1006	0.00633814
$449$	16563.6	1.74094	0.870471	−	0.492219i	$-0.163814\pi$
$449$	16563.6	1.74094	0.870471	+	0.492219i	$0.163814\pi$
$450$	1901.42	0.199186
$451$	4298.69	0.448819
$452$	−5628.71	−0.585735
$453$	−4675.64	−0.484947
$454$	−7960.59	−0.822928
$455$	−435.943	−0.0449172
$456$	0	0
$457$	−4329.83	−0.443197	−0.221598	−	0.975138i	$-0.571127\pi$
$457$	−4329.83	−0.443197	−0.221598	+	0.975138i	$0.571127\pi$
$458$	−3306.41	−0.337333
$459$	−997.610	−0.101448
$460$	−1484.05	−0.150422
$461$	−6490.45	−0.655727	−0.327864	−	0.944725i	$-0.606329\pi$
$461$	−6490.45	−0.655727	−0.327864	+	0.944725i	$0.606329\pi$
$462$	282.190	0.0284170
$463$	4222.34	0.423820	0.211910	−	0.977289i	$-0.432032\pi$
$463$	4222.34	0.423820	0.211910	+	0.977289i	$0.432032\pi$
$464$	2286.66	0.228784
$465$	6435.62	0.641817
$466$	11222.9	1.11565
$467$	6193.87	0.613743	0.306872	−	0.951751i	$-0.400718\pi$
$467$	6193.87	0.613743	0.306872	+	0.951751i	$0.400718\pi$
$468$	−1100.45	−0.108693
$469$	546.765	0.0538321
$470$	−4195.14	−0.411718
$471$	−204.706	−0.0200262
$472$	−1625.11	−0.158478
$473$	−6571.67	−0.638828
$474$	7038.07	0.682002
$475$	0	0
$476$	−138.789	−0.0133643
$477$	6373.72	0.611809
$478$	4796.84	0.459001
$479$	14996.1	1.43046	0.715229	−	0.698890i	$-0.246322\pi$
$479$	14996.1	1.43046	0.715229	+	0.698890i	$0.246322\pi$
$480$	1457.92	0.138635
$481$	−3880.69	−0.367868
$482$	12759.5	1.20576
$483$	−68.8248	−0.00648372
$484$	4709.28	0.442269
$485$	−27731.7	−2.59636
$486$	486.000	0.0453609
$487$	−2983.40	−0.277599	−0.138800	−	0.990320i	$-0.544324\pi$
$487$	−2983.40	−0.277599	−0.138800	+	0.990320i	$0.544324\pi$
$488$	309.334	0.0286945
$489$	11031.5	1.02017
$490$	−10391.3	−0.958019
$491$	18097.2	1.66337	0.831686	−	0.555246i	$-0.187376\pi$
$491$	18097.2	1.66337	0.831686	+	0.555246i	$0.187376\pi$
$492$	1029.97	0.0943797
$493$	−5280.55	−0.482402
$494$	0	0
$495$	6845.35	0.621567
$496$	2260.10	0.204599
$497$	935.985	0.0844762
$498$	−8955.67	−0.805850
$499$	20446.8	1.83432	0.917160	−	0.398519i	$-0.130476\pi$
$499$	20446.8	1.83432	0.917160	+	0.398519i	$0.130476\pi$
$500$	−1176.39	−0.105220
$501$	−2602.65	−0.232091
$502$	−971.760	−0.0863980
$503$	−15757.5	−1.39681	−0.698403	−	0.715705i	$-0.746105\pi$
$503$	−15757.5	−1.39681	−0.698403	+	0.715705i	$0.746105\pi$
$504$	67.6131	0.00597565
$505$	2162.19	0.190527
$506$	−2447.07	−0.214991
$507$	−3787.76	−0.331796
$508$	6541.58	0.571330
$509$	−3522.80	−0.306769	−0.153384	−	0.988167i	$-0.549017\pi$
$509$	−3522.80	−0.306769	−0.153384	+	0.988167i	$0.549017\pi$
$510$	−3366.74	−0.292318
$511$	−43.2719	−0.00374606
$512$	512.000	0.0441942
$513$	0	0
$514$	−9303.98	−0.798406
$515$	−16556.9	−1.41667
$516$	−1574.58	−0.134336
$517$	−6917.45	−0.588451
$518$	238.434	0.0202243
$519$	7039.02	0.595335
$520$	−3713.83	−0.313196
$521$	−14.7844	−0.00124321	−0.000621607	−	1.00000i	$-0.500198\pi$
$521$	−14.7844	−0.00124321	−0.000621607		1.00000i	$0.500198\pi$
$522$	2572.50	0.215699
$523$	−4224.24	−0.353180	−0.176590	−	0.984284i	$-0.556507\pi$
$523$	−4224.24	−0.353180	−0.176590	+	0.984284i	$0.556507\pi$
$524$	−7763.82	−0.647260
$525$	297.595	0.0247392
$526$	−9494.41	−0.787027
$527$	−5219.20	−0.431408
$528$	2403.99	0.198144
$529$	−11570.2	−0.950947
$530$	21510.1	1.76290
$531$	−1828.25	−0.149415
$532$	0	0
$533$	−2623.70	−0.213218
$534$	5367.08	0.434937
$535$	−22683.0	−1.83303
$536$	4657.92	0.375357
$537$	−11036.2	−0.886867
$538$	−7136.42	−0.571883
$539$	−17134.3	−1.36926
$540$	1640.16	0.130706
$541$	−22452.1	−1.78427	−0.892137	−	0.451765i	$-0.850795\pi$
$541$	−22452.1	−1.78427	−0.892137	+	0.451765i	$0.850795\pi$
$542$	−8220.02	−0.651439
$543$	1766.25	0.139589
$544$	−1182.35	−0.0931855
$545$	29383.8	2.30948
$546$	−172.234	−0.0134999
$547$	−12776.7	−0.998709	−0.499354	−	0.866398i	$-0.666429\pi$
$547$	−12776.7	−0.998709	−0.499354	+	0.866398i	$0.666429\pi$
$548$	−5405.83	−0.421397
$549$	348.001	0.0270534
$550$	10581.0	0.820319
$551$	0	0
$552$	−586.322	−0.0452093
$553$	1101.54	0.0847058
$554$	3619.65	0.277589
$555$	5783.93	0.442368
$556$	4793.70	0.365644
$557$	−15271.2	−1.16169	−0.580846	−	0.814013i	$-0.697278\pi$
$557$	−15271.2	−1.16169	−0.580846	+	0.814013i	$0.697278\pi$
$558$	2542.61	0.192898
$559$	4011.01	0.303484
$560$	228.182	0.0172186
$561$	−5551.49	−0.417797
$562$	1313.74	0.0986065
$563$	−11895.7	−0.890484	−0.445242	−	0.895410i	$-0.646882\pi$
$563$	−11895.7	−0.890484	−0.445242	+	0.895410i	$0.646882\pi$
$564$	−1657.43	−0.123742
$565$	−21370.3	−1.59125
$566$	1983.09	0.147271
$567$	76.0648	0.00563390
$568$	7973.70	0.589030
$569$	−572.397	−0.0421725	−0.0210862	−	0.999778i	$-0.506712\pi$
$569$	−572.397	−0.0421725	−0.0210862	+	0.999778i	$0.506712\pi$
$570$	0	0
$571$	20507.9	1.50303	0.751514	−	0.659718i	$-0.229324\pi$
$571$	20507.9	1.50303	0.751514	+	0.659718i	$0.229324\pi$
$572$	−6123.80	−0.447638
$573$	3713.10	0.270710
$574$	161.203	0.0117221
$575$	−2580.66	−0.187167
$576$	576.000	0.0416667
$577$	11277.1	0.813642	0.406821	−	0.913508i	$-0.366637\pi$
$577$	11277.1	0.813642	0.406821	+	0.913508i	$0.366637\pi$
$578$	−7095.62	−0.510621
$579$	−4018.53	−0.288436
$580$	8681.69	0.621530
$581$	−1401.67	−0.100088
$582$	−10956.4	−0.780336
$583$	35468.4	2.51964
$584$	−368.635	−0.0261203
$585$	−4178.05	−0.295284
$586$	−14374.4	−1.01331
$587$	−3154.54	−0.221809	−0.110904	−	0.993831i	$-0.535375\pi$
$587$	−3154.54	−0.221809	−0.110904	+	0.993831i	$0.535375\pi$
$588$	−4105.42	−0.287933
$589$	0	0
$590$	−6170.00	−0.430533
$591$	12552.5	0.873673
$592$	2031.23	0.141019
$593$	23915.2	1.65612	0.828061	−	0.560638i	$-0.189444\pi$
$593$	23915.2	1.65612	0.828061	+	0.560638i	$0.189444\pi$
$594$	2704.49	0.186812
$595$	−526.936	−0.0363063
$596$	−479.504	−0.0329551
$597$	1539.51	0.105541
$598$	1493.57	0.102135
$599$	−4138.05	−0.282264	−0.141132	−	0.989991i	$-0.545074\pi$
$599$	−4138.05	−0.282264	−0.141132	+	0.989991i	$0.545074\pi$
$600$	2535.23	0.172500
$601$	−21492.8	−1.45875	−0.729374	−	0.684115i	$-0.760189\pi$
$601$	−21492.8	−1.45875	−0.729374	+	0.684115i	$0.760189\pi$
$602$	−246.441	−0.0166847
$603$	5240.16	0.353890
$604$	−6234.19	−0.419976
$605$	17879.5	1.20150
$606$	854.245	0.0572629
$607$	−22000.3	−1.47111	−0.735555	−	0.677465i	$-0.763078\pi$
$607$	−22000.3	−1.47111	−0.735555	+	0.677465i	$0.763078\pi$
$608$	0	0
$609$	402.626	0.0267902
$610$	1174.44	0.0779534
$611$	4222.06	0.279552
$612$	−1330.15	−0.0878562
$613$	−4395.74	−0.289628	−0.144814	−	0.989459i	$-0.546258\pi$
$613$	−4395.74	−0.289628	−0.144814	+	0.989459i	$0.546258\pi$
$614$	783.018	0.0514658
$615$	3910.46	0.256399
$616$	376.253	0.0246098
$617$	14750.5	0.962453	0.481226	−	0.876596i	$-0.340192\pi$
$617$	14750.5	0.962453	0.481226	+	0.876596i	$0.340192\pi$
$618$	−6541.35	−0.425780
$619$	10408.7	0.675864	0.337932	−	0.941171i	$-0.390273\pi$
$619$	10408.7	0.675864	0.337932	+	0.941171i	$0.390273\pi$
$620$	8580.83	0.555829
$621$	−659.613	−0.0426237
$622$	−19536.1	−1.25937
$623$	840.012	0.0540198
$624$	−1467.27	−0.0941312
$625$	−17670.7	−1.13092
$626$	−8886.56	−0.567377
$627$	0	0
$628$	−272.942	−0.0173432
$629$	−4690.69	−0.297345
$630$	256.704	0.0162339
$631$	−14899.4	−0.939990	−0.469995	−	0.882669i	$-0.655744\pi$
$631$	−14899.4	−0.939990	−0.469995	+	0.882669i	$0.655744\pi$
$632$	9384.09	0.590631
$633$	−10989.0	−0.690005
$634$	3371.72	0.211212
$635$	24836.2	1.55212
$636$	8498.30	0.529842
$637$	10457.9	0.650484
$638$	14315.4	0.888326
$639$	8970.42	0.555343
$640$	1943.89	0.120061
$641$	−14725.0	−0.907339	−0.453669	−	0.891170i	$-0.649885\pi$
$641$	−14725.0	−0.907339	−0.453669	+	0.891170i	$0.649885\pi$
$642$	−8961.69	−0.550918
$643$	550.623	0.0337706	0.0168853	−	0.999857i	$-0.494625\pi$
$643$	550.623	0.0337706	0.0168853	+	0.999857i	$0.494625\pi$
$644$	−91.7664	−0.00561507
$645$	−5978.16	−0.364946
$646$	0	0
$647$	−9960.87	−0.605258	−0.302629	−	0.953108i	$-0.597864\pi$
$647$	−9960.87	−0.605258	−0.302629	+	0.953108i	$0.597864\pi$
$648$	648.000	0.0392837
$649$	−10173.8	−0.615343
$650$	−6458.10	−0.389704
$651$	397.949	0.0239583
$652$	14708.7	0.883490
$653$	−11009.2	−0.659759	−0.329880	−	0.944023i	$-0.607008\pi$
$653$	−11009.2	−0.659759	−0.329880	+	0.944023i	$0.607008\pi$
$654$	11609.1	0.694114
$655$	−29476.6	−1.75839
$656$	1373.30	0.0817352
$657$	−414.715	−0.0246264
$658$	−259.408	−0.0153690
$659$	−7491.20	−0.442816	−0.221408	−	0.975181i	$-0.571065\pi$
$659$	−7491.20	−0.442816	−0.221408	+	0.975181i	$0.571065\pi$
$660$	9127.14	0.538293
$661$	−3360.53	−0.197745	−0.0988725	−	0.995100i	$-0.531524\pi$
$661$	−3360.53	−0.197745	−0.0988725	+	0.995100i	$0.531524\pi$
$662$	−5352.25	−0.314231
$663$	3388.34	0.198480
$664$	−11940.9	−0.697886
$665$	0	0
$666$	2285.14	0.132954
$667$	−3491.46	−0.202684
$668$	−3470.20	−0.200997
$669$	7472.79	0.431860
$670$	17684.6	1.01972
$671$	1936.55	0.111416
$672$	90.1508	0.00517507
$673$	−24288.0	−1.39114	−0.695568	−	0.718460i	$-0.744847\pi$
$673$	−24288.0	−1.39114	−0.695568	+	0.718460i	$0.744847\pi$
$674$	−2549.03	−0.145675
$675$	2852.13	0.162635
$676$	−5050.35	−0.287343
$677$	−10604.3	−0.602003	−0.301001	−	0.953624i	$-0.597321\pi$
$677$	−10604.3	−0.602003	−0.301001	+	0.953624i	$0.597321\pi$
$678$	−8443.06	−0.478250
$679$	−1714.80	−0.0969190
$680$	−4488.99	−0.253154
$681$	−11940.9	−0.671918
$682$	14149.1	0.794423
$683$	−3972.25	−0.222539	−0.111269	−	0.993790i	$-0.535492\pi$
$683$	−3972.25	−0.222539	−0.111269	+	0.993790i	$0.535492\pi$
$684$	0	0
$685$	−20524.1	−1.14480
$686$	−1286.75	−0.0716156
$687$	−4959.61	−0.275431
$688$	−2099.44	−0.116338
$689$	−21648.1	−1.19699
$690$	−2226.07	−0.122819
$691$	−13684.4	−0.753368	−0.376684	−	0.926342i	$-0.622936\pi$
$691$	−13684.4	−0.753368	−0.376684	+	0.926342i	$0.622936\pi$
$692$	9385.36	0.515575
$693$	423.285	0.0232024
$694$	8249.89	0.451242
$695$	18200.1	0.993335
$696$	3429.99	0.186801
$697$	−3171.33	−0.172343
$698$	−15671.5	−0.849818
$699$	16834.4	0.910923
$700$	396.793	0.0214248
$701$	−10717.2	−0.577435	−0.288717	−	0.957414i	$-0.593229\pi$
$701$	−10717.2	−0.577435	−0.288717	+	0.957414i	$0.593229\pi$
$702$	−1650.68	−0.0887478
$703$	0	0
$704$	3205.32	0.171598
$705$	−6292.72	−0.336167
$706$	−21920.9	−1.16856
$707$	133.700	0.00711215
$708$	−2437.67	−0.129397
$709$	−8653.19	−0.458360	−0.229180	−	0.973384i	$-0.573604\pi$
$709$	−8653.19	−0.458360	−0.229180	+	0.973384i	$0.573604\pi$
$710$	30273.5	1.60020
$711$	10557.1	0.556852
$712$	7156.10	0.376666
$713$	−3450.90	−0.181258
$714$	−208.184	−0.0109119
$715$	−23250.0	−1.21608
$716$	−14714.9	−0.768049
$717$	7195.26	0.374773
$718$	2599.31	0.135105
$719$	26820.5	1.39115	0.695574	−	0.718455i	$-0.255150\pi$
$719$	26820.5	1.39115	0.695574	+	0.718455i	$0.255150\pi$
$720$	2186.88	0.113195
$721$	−1023.80	−0.0528825
$722$	0	0
$723$	19139.2	0.984502
$724$	2355.00	0.120888
$725$	15096.9	0.773358
$726$	7063.91	0.361111
$727$	−23235.5	−1.18536	−0.592681	−	0.805437i	$-0.701931\pi$
$727$	−23235.5	−1.18536	−0.592681	+	0.805437i	$0.701931\pi$
$728$	−229.646	−0.0116912
$729$	729.000	0.0370370
$730$	−1399.58	−0.0709602
$731$	4848.21	0.245304
$732$	464.001	0.0234289
$733$	−30607.0	−1.54228	−0.771142	−	0.636663i	$-0.780314\pi$
$733$	−30607.0	−1.54228	−0.771142	+	0.636663i	$0.780314\pi$
$734$	5284.78	0.265756
$735$	−15586.9	−0.782219
$736$	−781.763	−0.0391524
$737$	29160.4	1.45745
$738$	1544.96	0.0770607
$739$	−32445.6	−1.61506	−0.807532	−	0.589824i	$-0.799197\pi$
$739$	−32445.6	−1.61506	−0.807532	+	0.589824i	$0.799197\pi$
$740$	7711.91	0.383102
$741$	0	0
$742$	1330.08	0.0658072
$743$	−2725.64	−0.134581	−0.0672906	−	0.997733i	$-0.521435\pi$
$743$	−2725.64	−0.134581	−0.0672906	+	0.997733i	$0.521435\pi$
$744$	3390.15	0.167055
$745$	−1820.52	−0.0895282
$746$	19736.2	0.968625
$747$	−13433.5	−0.657973
$748$	−7401.99	−0.361823
$749$	−1402.61	−0.0684249
$750$	−1764.59	−0.0859116
$751$	39564.3	1.92240	0.961200	−	0.275853i	$-0.0889605\pi$
$751$	39564.3	1.92240	0.961200	+	0.275853i	$0.0889605\pi$
$752$	−2209.91	−0.107164
$753$	−1457.64	−0.0705436
$754$	−8737.39	−0.422012
$755$	−23669.1	−1.14094
$756$	101.420	0.00487910
$757$	30446.9	1.46184	0.730919	−	0.682464i	$-0.239092\pi$
$757$	30446.9	1.46184	0.730919	+	0.682464i	$0.239092\pi$
$758$	8780.97	0.420764
$759$	−3670.61	−0.175540
$760$	0	0
$761$	−17440.9	−0.830792	−0.415396	−	0.909641i	$-0.636357\pi$
$761$	−17440.9	−0.830792	−0.415396	+	0.909641i	$0.636357\pi$
$762$	9812.37	0.466489
$763$	1816.96	0.0862100
$764$	4950.80	0.234442
$765$	−5050.12	−0.238676
$766$	7769.45	0.366477
$767$	6209.58	0.292327
$768$	768.000	0.0360844
$769$	−18028.8	−0.845429	−0.422714	−	0.906263i	$-0.638923\pi$
$769$	−18028.8	−0.845429	−0.422714	+	0.906263i	$0.638923\pi$
$770$	1428.51	0.0668569
$771$	−13956.0	−0.651896
$772$	−5358.04	−0.249793
$773$	−33051.0	−1.53786	−0.768928	−	0.639335i	$-0.779210\pi$
$773$	−33051.0	−1.53786	−0.768928	+	0.639335i	$0.779210\pi$
$774$	−2361.87	−0.109685
$775$	14921.5	0.691608
$776$	−14608.5	−0.675791
$777$	357.651	0.0165131
$778$	21420.0	0.987074
$779$	0	0
$780$	−5570.74	−0.255724
$781$	49918.5	2.28710
$782$	1805.31	0.0825547
$783$	3858.74	0.176118
$784$	−5473.89	−0.249357
$785$	−1036.27	−0.0471159
$786$	−11645.7	−0.528485
$787$	−21904.2	−0.992122	−0.496061	−	0.868288i	$-0.665221\pi$
$787$	−21904.2	−0.992122	−0.496061	+	0.868288i	$0.665221\pi$
$788$	16736.7	0.756623
$789$	−14241.6	−0.642605
$790$	35628.2	1.60455
$791$	−1321.44	−0.0593995
$792$	3605.98	0.161784
$793$	−1181.97	−0.0529295
$794$	−28274.5	−1.26376
$795$	32265.2	1.43941
$796$	2052.69	0.0914014
$797$	−28429.8	−1.26353	−0.631766	−	0.775159i	$-0.717670\pi$
$797$	−28429.8	−1.26353	−0.631766	+	0.775159i	$0.717670\pi$
$798$	0	0
$799$	5103.31	0.225960
$800$	3380.30	0.149390
$801$	8050.62	0.355124
$802$	−20960.6	−0.922872
$803$	−2307.80	−0.101420
$804$	6986.88	0.306478
$805$	−348.406	−0.0152543
$806$	−8635.88	−0.377402
$807$	−10704.6	−0.466940
$808$	1138.99	0.0495912
$809$	−32993.1	−1.43384	−0.716919	−	0.697157i	$-0.754448\pi$
$809$	−32993.1	−1.43384	−0.716919	+	0.697157i	$0.754448\pi$
$810$	2460.24	0.106721
$811$	13638.2	0.590507	0.295253	−	0.955419i	$-0.404596\pi$
$811$	13638.2	0.590507	0.295253	+	0.955419i	$0.404596\pi$
$812$	536.835	0.0232010
$813$	−12330.0	−0.531898
$814$	12716.3	0.547551
$815$	55843.8	2.40015
$816$	−1773.53	−0.0760857
$817$	0	0
$818$	23147.5	0.989407
$819$	−258.351	−0.0110226
$820$	5213.95	0.222048
$821$	15141.2	0.643643	0.321821	−	0.946800i	$-0.395705\pi$
$821$	15141.2	0.643643	0.321821	+	0.946800i	$0.395705\pi$
$822$	−8108.74	−0.344069
$823$	37901.6	1.60530	0.802652	−	0.596447i	$-0.203422\pi$
$823$	37901.6	1.60530	0.802652	+	0.596447i	$0.203422\pi$
$824$	−8721.80	−0.368736
$825$	15871.5	0.669788
$826$	−381.524	−0.0160713
$827$	−1944.52	−0.0817626	−0.0408813	−	0.999164i	$-0.513017\pi$
$827$	−1944.52	−0.0817626	−0.0408813	+	0.999164i	$0.513017\pi$
$828$	−879.484	−0.0369132
$829$	20602.7	0.863162	0.431581	−	0.902074i	$-0.357956\pi$
$829$	20602.7	0.863162	0.431581	+	0.902074i	$0.357956\pi$
$830$	−45335.5	−1.89593
$831$	5429.47	0.226650
$832$	−1956.36	−0.0815200
$833$	12640.8	0.525782
$834$	7190.55	0.298547
$835$	−13175.2	−0.546043
$836$	0	0
$837$	3813.91	0.157501
$838$	32231.1	1.32864
$839$	−15001.1	−0.617276	−0.308638	−	0.951180i	$-0.599873\pi$
$839$	−15001.1	−0.617276	−0.308638	+	0.951180i	$0.599873\pi$
$840$	342.272	0.0140590
$841$	−3963.89	−0.162528
$842$	21917.2	0.897053
$843$	1970.61	0.0805119
$844$	−14652.0	−0.597562
$845$	−19174.5	−0.780618
$846$	−2486.15	−0.101035
$847$	1105.59	0.0448505
$848$	11331.1	0.458856
$849$	2974.63	0.120246
$850$	−7806.07	−0.314995
$851$	−3101.45	−0.124931
$852$	11960.6	0.480941
$853$	23076.0	0.926270	0.463135	−	0.886288i	$-0.346725\pi$
$853$	23076.0	0.926270	0.463135	+	0.886288i	$0.346725\pi$
$854$	72.6217	0.00290991
$855$	0	0
$856$	−11948.9	−0.477109
$857$	37091.7	1.47845	0.739224	−	0.673460i	$-0.235193\pi$
$857$	37091.7	1.47845	0.739224	+	0.673460i	$0.235193\pi$
$858$	−9185.70	−0.365495
$859$	20110.2	0.798778	0.399389	−	0.916782i	$-0.369222\pi$
$859$	20110.2	0.798778	0.399389	+	0.916782i	$0.369222\pi$
$860$	−7970.88	−0.316052
$861$	241.805	0.00957106
$862$	2508.18	0.0991056
$863$	−3647.34	−0.143866	−0.0719332	−	0.997409i	$-0.522917\pi$
$863$	−3647.34	−0.143866	−0.0719332	+	0.997409i	$0.522917\pi$
$864$	864.000	0.0340207
$865$	35633.1	1.40065
$866$	6844.17	0.268562
$867$	−10643.4	−0.416920
$868$	530.598	0.0207485
$869$	58748.0	2.29332
$870$	13022.5	0.507477
$871$	−17798.0	−0.692380
$872$	15478.8	0.601120
$873$	−16434.5	−0.637142
$874$	0	0
$875$	−276.179	−0.0106704
$876$	−552.953	−0.0213271
$877$	−35833.7	−1.37972	−0.689861	−	0.723942i	$-0.742328\pi$
$877$	−35833.7	−1.37972	−0.689861	+	0.723942i	$0.742328\pi$
$878$	−5410.10	−0.207952
$879$	−21561.5	−0.827363
$880$	12169.5	0.466175
$881$	9364.52	0.358114	0.179057	−	0.983839i	$-0.442695\pi$
$881$	9364.52	0.358114	0.179057	+	0.983839i	$0.442695\pi$
$882$	−6158.13	−0.235096
$883$	31800.9	1.21199	0.605995	−	0.795469i	$-0.292775\pi$
$883$	31800.9	1.21199	0.605995	+	0.795469i	$0.292775\pi$
$884$	4517.79	0.171889
$885$	−9255.00	−0.351529
$886$	−9262.22	−0.351208
$887$	−44200.6	−1.67318	−0.836589	−	0.547831i	$-0.815454\pi$
$887$	−44200.6	−1.67318	−0.836589	+	0.547831i	$0.815454\pi$
$888$	3046.85	0.115141
$889$	1535.75	0.0579387
$890$	27169.3	1.02328
$891$	4056.73	0.152532
$892$	9963.72	0.374002
$893$	0	0
$894$	−719.256	−0.0269077
$895$	−55867.7	−2.08654
$896$	120.201	0.00448174
$897$	2240.35	0.0833925
$898$	33127.1	1.23103
$899$	20187.8	0.748945
$900$	3802.84	0.140846
$901$	−26166.6	−0.967521
$902$	8597.38	0.317363
$903$	−369.661	−0.0136230
$904$	−11257.4	−0.414177
$905$	8941.14	0.328413
$906$	−9351.29	−0.342909
$907$	−21282.3	−0.779127	−0.389564	−	0.921000i	$-0.627374\pi$
$907$	−21282.3	−0.779127	−0.389564	+	0.921000i	$0.627374\pi$
$908$	−15921.2	−0.581898
$909$	1281.37	0.0467550
$910$	−871.887	−0.0317613
$911$	989.352	0.0359810	0.0179905	−	0.999838i	$-0.494273\pi$
$911$	989.352	0.0359810	0.0179905	+	0.999838i	$0.494273\pi$
$912$	0	0
$913$	−74754.6	−2.70977
$914$	−8659.66	−0.313387
$915$	1761.66	0.0636487
$916$	−6612.82	−0.238530
$917$	−1822.70	−0.0656387
$918$	−1995.22	−0.0717343
$919$	−49987.5	−1.79427	−0.897135	−	0.441756i	$-0.854356\pi$
$919$	−49987.5	−1.79427	−0.897135	+	0.441756i	$0.854356\pi$
$920$	−2968.09	−0.106364
$921$	1174.53	0.0420217
$922$	−12980.9	−0.463669
$923$	−30467.7	−1.08652
$924$	564.379	0.0200939
$925$	13410.5	0.476687
$926$	8444.68	0.299686
$927$	−9812.03	−0.347648
$928$	4573.33	0.161775
$929$	22674.1	0.800769	0.400384	−	0.916347i	$-0.368877\pi$
$929$	22674.1	0.800769	0.400384	+	0.916347i	$0.368877\pi$
$930$	12871.2	0.453833
$931$	0	0
$932$	22445.9	0.788882
$933$	−29304.2	−1.02827
$934$	12387.7	0.433982
$935$	−28102.8	−0.982953
$936$	−2200.91	−0.0768578
$937$	−38569.5	−1.34473	−0.672364	−	0.740221i	$-0.734721\pi$
$937$	−38569.5	−1.34473	−0.672364	+	0.740221i	$0.734721\pi$
$938$	1093.53	0.0380650
$939$	−13329.8	−0.463262
$940$	−8390.29	−0.291129
$941$	−8558.54	−0.296494	−0.148247	−	0.988950i	$-0.547363\pi$
$941$	−8558.54	−0.296494	−0.148247	+	0.988950i	$0.547363\pi$
$942$	−409.412	−0.0141607
$943$	−2096.86	−0.0724107
$944$	−3250.22	−0.112061
$945$	385.056	0.0132549
$946$	−13143.3	−0.451720
$947$	19093.4	0.655178	0.327589	−	0.944820i	$-0.393764\pi$
$947$	19093.4	0.655178	0.327589	+	0.944820i	$0.393764\pi$
$948$	14076.1	0.482248
$949$	1408.56	0.0481812
$950$	0	0
$951$	5057.58	0.172453
$952$	−277.578	−0.00944996
$953$	20071.5	0.682246	0.341123	−	0.940019i	$-0.389193\pi$
$953$	20071.5	0.682246	0.341123	+	0.940019i	$0.389193\pi$
$954$	12747.4	0.432614
$955$	18796.5	0.636902
$956$	9593.68	0.324563
$957$	21473.1	0.725316
$958$	29992.2	1.01149
$959$	−1269.11	−0.0427339
$960$	2915.84	0.0980294
$961$	−9837.73	−0.330225
$962$	−7761.39	−0.260122
$963$	−13442.5	−0.449823
$964$	25519.0	0.852604
$965$	−20342.7	−0.678605
$966$	−137.650	−0.00458468
$967$	−20382.7	−0.677832	−0.338916	−	0.940817i	$-0.610060\pi$
$967$	−20382.7	−0.677832	−0.338916	+	0.940817i	$0.610060\pi$
$968$	9418.55	0.312731
$969$	0	0
$970$	−55463.5	−1.83590
$971$	33998.7	1.12366	0.561828	−	0.827254i	$-0.310098\pi$
$971$	33998.7	1.12366	0.561828	+	0.827254i	$0.310098\pi$
$972$	972.000	0.0320750
$973$	1125.41	0.0370800
$974$	−5966.81	−0.196292
$975$	−9687.15	−0.318192
$976$	618.669	0.0202901
$977$	−394.787	−0.0129277	−0.00646384	−	0.999979i	$-0.502058\pi$
$977$	−394.787	−0.0129277	−0.00646384	+	0.999979i	$0.502058\pi$
$978$	22063.0	0.721366
$979$	44800.0	1.46253
$980$	−20782.5	−0.677422
$981$	17413.6	0.566741
$982$	36194.4	1.17618
$983$	18810.8	0.610348	0.305174	−	0.952297i	$-0.401285\pi$
$983$	18810.8	0.610348	0.305174	+	0.952297i	$0.401285\pi$
$984$	2059.95	0.0667365
$985$	63543.5	2.05550
$986$	−10561.1	−0.341109
$987$	−389.112	−0.0125487
$988$	0	0
$989$	3205.60	0.103066
$990$	13690.7	0.439514
$991$	−48971.8	−1.56977	−0.784884	−	0.619643i	$-0.787277\pi$
$991$	−48971.8	−1.56977	−0.784884	+	0.619643i	$0.787277\pi$
$992$	4520.19	0.144674
$993$	−8028.38	−0.256569
$994$	1871.97	0.0597337
$995$	7793.36	0.248308
$996$	−17911.3	−0.569822
$997$	−39990.0	−1.27031	−0.635154	−	0.772385i	$-0.719063\pi$
$997$	−39990.0	−1.27031	−0.635154	+	0.772385i	$0.719063\pi$
$998$	40893.6	1.29706
$999$	3427.71	0.108556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bm.1.8		9
19.6	even	9		114.4.i.d.55.1	✓	18
19.16	even	9		114.4.i.d.85.1	yes	18
19.18	odd	2		2166.4.a.bj.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.d.55.1	✓	18	19.6	even	9
114.4.i.d.85.1	yes	18	19.16	even	9
2166.4.a.bj.1.8		9	19.18	odd	2
2166.4.a.bm.1.8		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +15.1867 q^{5} +6.00000 q^{6} +0.939071 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +15.1867 q^{5} +6.00000 q^{6} +0.939071 q^{7} +8.00000 q^{8} +9.00000 q^{9} +30.3733 q^{10} +50.0831 q^{11} +12.0000 q^{12} -30.5682 q^{13} +1.87814 q^{14} +45.5600 q^{15} +16.0000 q^{16} -36.9485 q^{17} +18.0000 q^{18} +60.7466 q^{20} +2.81721 q^{21} +100.166 q^{22} -24.4301 q^{23} +24.0000 q^{24} +105.634 q^{25} -61.1363 q^{26} +27.0000 q^{27} +3.75629 q^{28} +142.916 q^{29} +91.1199 q^{30} +141.256 q^{31} +32.0000 q^{32} +150.249 q^{33} -73.8970 q^{34} +14.2614 q^{35} +36.0000 q^{36} +126.952 q^{37} -91.7045 q^{39} +121.493 q^{40} +85.8312 q^{41} +5.63443 q^{42} -131.215 q^{43} +200.332 q^{44} +136.680 q^{45} -48.8602 q^{46} -138.119 q^{47} +48.0000 q^{48} -342.118 q^{49} +211.269 q^{50} -110.846 q^{51} -122.273 q^{52} +708.191 q^{53} +54.0000 q^{54} +760.595 q^{55} +7.51257 q^{56} +285.833 q^{58} -203.139 q^{59} +182.240 q^{60} +38.6668 q^{61} +282.512 q^{62} +8.45164 q^{63} +64.0000 q^{64} -464.228 q^{65} +300.499 q^{66} +582.240 q^{67} -147.794 q^{68} -73.2903 q^{69} +28.5227 q^{70} +996.713 q^{71} +72.0000 q^{72} -46.0794 q^{73} +253.904 q^{74} +316.903 q^{75} +47.0316 q^{77} -183.409 q^{78} +1173.01 q^{79} +242.986 q^{80} +81.0000 q^{81} +171.662 q^{82} -1492.61 q^{83} +11.2689 q^{84} -561.124 q^{85} -262.431 q^{86} +428.749 q^{87} +400.665 q^{88} +894.513 q^{89} +273.360 q^{90} -28.7057 q^{91} -97.7204 q^{92} +423.768 q^{93} -276.239 q^{94} +96.0000 q^{96} -1826.06 q^{97} -684.236 q^{98} +450.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 18 q^{2} + 27 q^{3} + 36 q^{4} + 27 q^{5} + 54 q^{6} + 72 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 18 * q^2 + 27 * q^3 + 36 * q^4 + 27 * q^5 + 54 * q^6 + 72 * q^8 + 81 * q^9	\( 9 q + 18 q^{2} + 27 q^{3} + 36 q^{4} + 27 q^{5} + 54 q^{6} + 72 q^{8} + 81 q^{9} + 54 q^{10} + 39 q^{11} + 108 q^{12} + 99 q^{13} + 81 q^{15} + 144 q^{16} + 57 q^{17} + 162 q^{18} + 108 q^{20} + 78 q^{22} + 228 q^{23} + 216 q^{24} + 174 q^{25} + 198 q^{26} + 243 q^{27} + 459 q^{29} + 162 q^{30} + 243 q^{31} + 288 q^{32} + 117 q^{33} + 114 q^{34} + 324 q^{35} + 324 q^{36} + 711 q^{37} + 297 q^{39} + 216 q^{40} + 459 q^{41} + 252 q^{43} + 156 q^{44} + 243 q^{45} + 456 q^{46} - 66 q^{47} + 432 q^{48} + 2229 q^{49} + 348 q^{50} + 171 q^{51} + 396 q^{52} + 1197 q^{53} + 486 q^{54} + 762 q^{55} + 918 q^{58} + 1221 q^{59} + 324 q^{60} - 780 q^{61} + 486 q^{62} + 576 q^{64} - 237 q^{65} + 234 q^{66} + 1596 q^{67} + 228 q^{68} + 684 q^{69} + 648 q^{70} + 2538 q^{71} + 648 q^{72} + 225 q^{73} + 1422 q^{74} + 522 q^{75} - 135 q^{77} + 594 q^{78} + 834 q^{79} + 432 q^{80} + 729 q^{81} + 918 q^{82} + 2490 q^{83} - 1653 q^{85} + 504 q^{86} + 1377 q^{87} + 312 q^{88} - 507 q^{89} + 486 q^{90} + 6423 q^{91} + 912 q^{92} + 729 q^{93} - 132 q^{94} + 864 q^{96} + 2529 q^{97} + 4458 q^{98} + 351 q^{99}+O(q^{100}) \) 9 * q + 18 * q^2 + 27 * q^3 + 36 * q^4 + 27 * q^5 + 54 * q^6 + 72 * q^8 + 81 * q^9 + 54 * q^10 + 39 * q^11 + 108 * q^12 + 99 * q^13 + 81 * q^15 + 144 * q^16 + 57 * q^17 + 162 * q^18 + 108 * q^20 + 78 * q^22 + 228 * q^23 + 216 * q^24 + 174 * q^25 + 198 * q^26 + 243 * q^27 + 459 * q^29 + 162 * q^30 + 243 * q^31 + 288 * q^32 + 117 * q^33 + 114 * q^34 + 324 * q^35 + 324 * q^36 + 711 * q^37 + 297 * q^39 + 216 * q^40 + 459 * q^41 + 252 * q^43 + 156 * q^44 + 243 * q^45 + 456 * q^46 - 66 * q^47 + 432 * q^48 + 2229 * q^49 + 348 * q^50 + 171 * q^51 + 396 * q^52 + 1197 * q^53 + 486 * q^54 + 762 * q^55 + 918 * q^58 + 1221 * q^59 + 324 * q^60 - 780 * q^61 + 486 * q^62 + 576 * q^64 - 237 * q^65 + 234 * q^66 + 1596 * q^67 + 228 * q^68 + 684 * q^69 + 648 * q^70 + 2538 * q^71 + 648 * q^72 + 225 * q^73 + 1422 * q^74 + 522 * q^75 - 135 * q^77 + 594 * q^78 + 834 * q^79 + 432 * q^80 + 729 * q^81 + 918 * q^82 + 2490 * q^83 - 1653 * q^85 + 504 * q^86 + 1377 * q^87 + 312 * q^88 - 507 * q^89 + 486 * q^90 + 6423 * q^91 + 912 * q^92 + 729 * q^93 - 132 * q^94 + 864 * q^96 + 2529 * q^97 + 4458 * q^98 + 351 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more