LMFDB - Embedded newform 2166.4.a.bj.1.4

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:	$1$
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			1.4
Root			$-9.08036$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.54827 q^{5} +6.00000 q^{6} +10.5206 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.54827 q^{5} +6.00000 q^{6} +10.5206 q^{7} -8.00000 q^{8} +9.00000 q^{9} +9.09654 q^{10} +25.5178 q^{11} -12.0000 q^{12} +0.908540 q^{13} -21.0412 q^{14} +13.6448 q^{15} +16.0000 q^{16} +97.6174 q^{17} -18.0000 q^{18} -18.1931 q^{20} -31.5618 q^{21} -51.0356 q^{22} -67.3081 q^{23} +24.0000 q^{24} -104.313 q^{25} -1.81708 q^{26} -27.0000 q^{27} +42.0824 q^{28} -176.591 q^{29} -27.2896 q^{30} -87.4990 q^{31} -32.0000 q^{32} -76.5534 q^{33} -195.235 q^{34} -47.8505 q^{35} +36.0000 q^{36} -57.3404 q^{37} -2.72562 q^{39} +36.3862 q^{40} +153.554 q^{41} +63.1236 q^{42} +313.914 q^{43} +102.071 q^{44} -40.9344 q^{45} +134.616 q^{46} -253.544 q^{47} -48.0000 q^{48} -232.317 q^{49} +208.626 q^{50} -292.852 q^{51} +3.63416 q^{52} -289.405 q^{53} +54.0000 q^{54} -116.062 q^{55} -84.1648 q^{56} +353.182 q^{58} +66.7954 q^{59} +54.5792 q^{60} +254.649 q^{61} +174.998 q^{62} +94.6854 q^{63} +64.0000 q^{64} -4.13229 q^{65} +153.107 q^{66} -610.525 q^{67} +390.470 q^{68} +201.924 q^{69} +95.7010 q^{70} +258.810 q^{71} -72.0000 q^{72} +627.304 q^{73} +114.681 q^{74} +312.940 q^{75} +268.462 q^{77} +5.45124 q^{78} +198.825 q^{79} -72.7723 q^{80} +81.0000 q^{81} -307.107 q^{82} +190.652 q^{83} -126.247 q^{84} -443.990 q^{85} -627.828 q^{86} +529.773 q^{87} -204.142 q^{88} +1246.20 q^{89} +81.8688 q^{90} +9.55839 q^{91} -269.232 q^{92} +262.497 q^{93} +507.089 q^{94} +96.0000 q^{96} -164.006 q^{97} +464.634 q^{98} +229.660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.54827	−0.406810	−0.203405	−	0.979095i	$-0.565201\pi$
$5$	−4.54827	−0.406810	−0.203405	+	0.979095i	$0.565201\pi$
$6$	6.00000	0.408248
$7$	10.5206	0.568059	0.284029	−	0.958816i	$-0.408329\pi$
$7$	10.5206	0.568059	0.284029	+	0.958816i	$0.408329\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	9.09654	0.287658
$11$	25.5178	0.699446	0.349723	−	0.936853i	$-0.386276\pi$
$11$	25.5178	0.699446	0.349723	+	0.936853i	$0.386276\pi$
$12$	−12.0000	−0.288675
$13$	0.908540	0.0193834	0.00969168	−	0.999953i	$-0.496915\pi$
$13$	0.908540	0.0193834	0.00969168	+	0.999953i	$0.496915\pi$
$14$	−21.0412	−0.401678
$15$	13.6448	0.234872
$16$	16.0000	0.250000
$17$	97.6174	1.39269	0.696344	−	0.717708i	$-0.254809\pi$
$17$	97.6174	1.39269	0.696344	+	0.717708i	$0.254809\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−18.1931	−0.203405
$21$	−31.5618	−0.327969
$22$	−51.0356	−0.494583
$23$	−67.3081	−0.610204	−0.305102	−	0.952320i	$-0.598691\pi$
$23$	−67.3081	−0.610204	−0.305102	+	0.952320i	$0.598691\pi$
$24$	24.0000	0.204124
$25$	−104.313	−0.834506
$26$	−1.81708	−0.0137061
$27$	−27.0000	−0.192450
$28$	42.0824	0.284029
$29$	−176.591	−1.13076	−0.565382	−	0.824829i	$-0.691271\pi$
$29$	−176.591	−1.13076	−0.565382	+	0.824829i	$0.691271\pi$
$30$	−27.2896	−0.166079
$31$	−87.4990	−0.506945	−0.253472	−	0.967343i	$-0.581573\pi$
$31$	−87.4990	−0.506945	−0.253472	+	0.967343i	$0.581573\pi$
$32$	−32.0000	−0.176777
$33$	−76.5534	−0.403825
$34$	−195.235	−0.984779
$35$	−47.8505	−0.231092
$36$	36.0000	0.166667
$37$	−57.3404	−0.254776	−0.127388	−	0.991853i	$-0.540659\pi$
$37$	−57.3404	−0.254776	−0.127388	+	0.991853i	$0.540659\pi$
$38$	0	0
$39$	−2.72562	−0.0111910
$40$	36.3862	0.143829
$41$	153.554	0.584904	0.292452	−	0.956280i	$-0.405529\pi$
$41$	153.554	0.584904	0.292452	+	0.956280i	$0.405529\pi$
$42$	63.1236	0.231909
$43$	313.914	1.11329	0.556644	−	0.830751i	$-0.312089\pi$
$43$	313.914	1.11329	0.556644	+	0.830751i	$0.312089\pi$
$44$	102.071	0.349723
$45$	−40.9344	−0.135603
$46$	134.616	0.431480
$47$	−253.544	−0.786878	−0.393439	−	0.919351i	$-0.628715\pi$
$47$	−253.544	−0.786878	−0.393439	+	0.919351i	$0.628715\pi$
$48$	−48.0000	−0.144338
$49$	−232.317	−0.677309
$50$	208.626	0.590085
$51$	−292.852	−0.804069
$52$	3.63416	0.00969168
$53$	−289.405	−0.750053	−0.375027	−	0.927014i	$-0.622366\pi$
$53$	−289.405	−0.750053	−0.375027	+	0.927014i	$0.622366\pi$
$54$	54.0000	0.136083
$55$	−116.062	−0.284541
$56$	−84.1648	−0.200839
$57$	0	0
$58$	353.182	0.799570
$59$	66.7954	0.147390	0.0736950	−	0.997281i	$-0.476521\pi$
$59$	66.7954	0.147390	0.0736950	+	0.997281i	$0.476521\pi$
$60$	54.5792	0.117436
$61$	254.649	0.534500	0.267250	−	0.963627i	$-0.413885\pi$
$61$	254.649	0.534500	0.267250	+	0.963627i	$0.413885\pi$
$62$	174.998	0.358464
$63$	94.6854	0.189353
$64$	64.0000	0.125000
$65$	−4.13229	−0.00788534
$66$	153.107	0.285548
$67$	−610.525	−1.11325	−0.556623	−	0.830765i	$-0.687903\pi$
$67$	−610.525	−1.11325	−0.556623	+	0.830765i	$0.687903\pi$
$68$	390.470	0.696344
$69$	201.924	0.352302
$70$	95.7010	0.163407
$71$	258.810	0.432608	0.216304	−	0.976326i	$-0.430600\pi$
$71$	258.810	0.432608	0.216304	+	0.976326i	$0.430600\pi$
$72$	−72.0000	−0.117851
$73$	627.304	1.00576	0.502880	−	0.864356i	$-0.332274\pi$
$73$	627.304	1.00576	0.502880	+	0.864356i	$0.332274\pi$
$74$	114.681	0.180154
$75$	312.940	0.481802
$76$	0	0
$77$	268.462	0.397326
$78$	5.45124	0.00791323
$79$	198.825	0.283159	0.141580	−	0.989927i	$-0.454782\pi$
$79$	198.825	0.283159	0.141580	+	0.989927i	$0.454782\pi$
$80$	−72.7723	−0.101702
$81$	81.0000	0.111111
$82$	−307.107	−0.413589
$83$	190.652	0.252129	0.126065	−	0.992022i	$-0.459765\pi$
$83$	190.652	0.252129	0.126065	+	0.992022i	$0.459765\pi$
$84$	−126.247	−0.163984
$85$	−443.990	−0.566559
$86$	−627.828	−0.787214
$87$	529.773	0.652847
$88$	−204.142	−0.247292
$89$	1246.20	1.48424	0.742119	−	0.670268i	$-0.233821\pi$
$89$	1246.20	1.48424	0.742119	+	0.670268i	$0.233821\pi$
$90$	81.8688	0.0958859
$91$	9.55839	0.0110109
$92$	−269.232	−0.305102
$93$	262.497	0.292685
$94$	507.089	0.556407
$95$	0	0
$96$	96.0000	0.102062
$97$	−164.006	−0.171674	−0.0858368	−	0.996309i	$-0.527356\pi$
$97$	−164.006	−0.171674	−0.0858368	+	0.996309i	$0.527356\pi$
$98$	464.634	0.478930
$99$	229.660	0.233149
$100$	−417.253	−0.417253
$101$	−1428.13	−1.40697	−0.703484	−	0.710711i	$-0.748373\pi$
$101$	−1428.13	−1.40697	−0.703484	+	0.710711i	$0.748373\pi$
$102$	585.704	0.568562
$103$	−1745.46	−1.66976	−0.834880	−	0.550433i	$-0.814463\pi$
$103$	−1745.46	−1.66976	−0.834880	+	0.550433i	$0.814463\pi$
$104$	−7.26832	−0.00685305
$105$	143.552	0.133421
$106$	578.810	0.530368
$107$	−405.354	−0.366234	−0.183117	−	0.983091i	$-0.558619\pi$
$107$	−405.354	−0.366234	−0.183117	+	0.983091i	$0.558619\pi$
$108$	−108.000	−0.0962250
$109$	−397.489	−0.349290	−0.174645	−	0.984632i	$-0.555878\pi$
$109$	−397.489	−0.349290	−0.174645	+	0.984632i	$0.555878\pi$
$110$	232.124	0.201201
$111$	172.021	0.147095
$112$	168.330	0.142015
$113$	−277.608	−0.231108	−0.115554	−	0.993301i	$-0.536864\pi$
$113$	−277.608	−0.231108	−0.115554	+	0.993301i	$0.536864\pi$
$114$	0	0
$115$	306.135	0.248237
$116$	−706.364	−0.565382
$117$	8.17686	0.00646112
$118$	−133.591	−0.104221
$119$	1026.99	0.791129
$120$	−109.158	−0.0830397
$121$	−679.842	−0.510775
$122$	−509.299	−0.377949
$123$	−460.661	−0.337694
$124$	−349.996	−0.253472
$125$	1042.98	0.746295
$126$	−189.371	−0.133893
$127$	−654.796	−0.457510	−0.228755	−	0.973484i	$-0.573465\pi$
$127$	−654.796	−0.457510	−0.228755	+	0.973484i	$0.573465\pi$
$128$	−128.000	−0.0883883
$129$	−941.742	−0.642757
$130$	8.26457	0.00557578
$131$	1011.62	0.674697	0.337348	−	0.941380i	$-0.390470\pi$
$131$	1011.62	0.674697	0.337348	+	0.941380i	$0.390470\pi$
$132$	−306.214	−0.201913
$133$	0	0
$134$	1221.05	0.787184
$135$	122.803	0.0782905
$136$	−780.939	−0.492390
$137$	2711.88	1.69118	0.845590	−	0.533833i	$-0.179249\pi$
$137$	2711.88	1.69118	0.845590	+	0.533833i	$0.179249\pi$
$138$	−403.848	−0.249115
$139$	−1461.49	−0.891815	−0.445907	−	0.895079i	$-0.647119\pi$
$139$	−1461.49	−0.891815	−0.445907	+	0.895079i	$0.647119\pi$
$140$	−191.402	−0.115546
$141$	760.633	0.454304
$142$	−517.621	−0.305900
$143$	23.1840	0.0135576
$144$	144.000	0.0833333
$145$	803.184	0.460005
$146$	−1254.61	−0.711179
$147$	696.951	0.391045
$148$	−229.361	−0.127388
$149$	2740.91	1.50701	0.753505	−	0.657443i	$-0.228362\pi$
$149$	2740.91	1.50701	0.753505	+	0.657443i	$0.228362\pi$
$150$	−625.879	−0.340686
$151$	860.653	0.463834	0.231917	−	0.972736i	$-0.425500\pi$
$151$	860.653	0.463834	0.231917	+	0.972736i	$0.425500\pi$
$152$	0	0
$153$	878.556	0.464229
$154$	−536.925	−0.280952
$155$	397.969	0.206230
$156$	−10.9025	−0.00559550
$157$	1361.10	0.691895	0.345947	−	0.938254i	$-0.387558\pi$
$157$	1361.10	0.691895	0.345947	+	0.938254i	$0.387558\pi$
$158$	−397.651	−0.200224
$159$	868.215	0.433044
$160$	145.545	0.0719144
$161$	−708.121	−0.346632
$162$	−162.000	−0.0785674
$163$	2919.34	1.40282	0.701411	−	0.712757i	$-0.252554\pi$
$163$	2919.34	1.40282	0.701411	+	0.712757i	$0.252554\pi$
$164$	614.215	0.292452
$165$	348.185	0.164280
$166$	−381.303	−0.178282
$167$	1898.52	0.879711	0.439855	−	0.898069i	$-0.355030\pi$
$167$	1898.52	0.879711	0.439855	+	0.898069i	$0.355030\pi$
$168$	252.494	0.115955
$169$	−2196.17	−0.999624
$170$	887.980	0.400618
$171$	0	0
$172$	1255.66	0.556644
$173$	−2975.53	−1.30766	−0.653830	−	0.756641i	$-0.726839\pi$
$173$	−2975.53	−1.30766	−0.653830	+	0.756641i	$0.726839\pi$
$174$	−1059.55	−0.461632
$175$	−1097.44	−0.474048
$176$	408.285	0.174862
$177$	−200.386	−0.0850957
$178$	−2492.40	−1.04951
$179$	1455.70	0.607846	0.303923	−	0.952697i	$-0.401703\pi$
$179$	1455.70	0.607846	0.303923	+	0.952697i	$0.401703\pi$
$180$	−163.738	−0.0678016
$181$	−1106.89	−0.454554	−0.227277	−	0.973830i	$-0.572982\pi$
$181$	−1106.89	−0.454554	−0.227277	+	0.973830i	$0.572982\pi$
$182$	−19.1168	−0.00778588
$183$	−763.948	−0.308594
$184$	538.465	0.215740
$185$	260.799	0.103645
$186$	−524.994	−0.206959
$187$	2490.98	0.974110
$188$	−1014.18	−0.393439
$189$	−284.056	−0.109323
$190$	0	0
$191$	−75.3231	−0.0285350	−0.0142675	−	0.999898i	$-0.504542\pi$
$191$	−75.3231	−0.0285350	−0.0142675	+	0.999898i	$0.504542\pi$
$192$	−192.000	−0.0721688
$193$	914.928	0.341233	0.170616	−	0.985338i	$-0.445424\pi$
$193$	914.928	0.341233	0.170616	+	0.985338i	$0.445424\pi$
$194$	328.013	0.121392
$195$	12.3969	0.00455260
$196$	−929.268	−0.338655
$197$	−288.375	−0.104294	−0.0521469	−	0.998639i	$-0.516606\pi$
$197$	−288.375	−0.104294	−0.0521469	+	0.998639i	$0.516606\pi$
$198$	−459.320	−0.164861
$199$	−3771.16	−1.34337	−0.671684	−	0.740837i	$-0.734429\pi$
$199$	−3771.16	−1.34337	−0.671684	+	0.740837i	$0.734429\pi$
$200$	834.506	0.295042
$201$	1831.58	0.642733
$202$	2856.25	0.994877
$203$	−1857.84	−0.642340
$204$	−1171.41	−0.402034
$205$	−698.403	−0.237944
$206$	3490.92	1.18070
$207$	−605.773	−0.203401
$208$	14.5366	0.00484584
$209$	0	0
$210$	−287.103	−0.0943428
$211$	−5640.66	−1.84037	−0.920187	−	0.391479i	$-0.871964\pi$
$211$	−5640.66	−1.84037	−0.920187	+	0.391479i	$0.871964\pi$
$212$	−1157.62	−0.375027
$213$	−776.431	−0.249766
$214$	810.707	0.258966
$215$	−1427.76	−0.452896
$216$	216.000	0.0680414
$217$	−920.541	−0.287974
$218$	794.979	0.246985
$219$	−1881.91	−0.580675
$220$	−464.247	−0.142271
$221$	88.6893	0.0269950
$222$	−344.042	−0.104012
$223$	−814.436	−0.244568	−0.122284	−	0.992495i	$-0.539022\pi$
$223$	−814.436	−0.244568	−0.122284	+	0.992495i	$0.539022\pi$
$224$	−336.659	−0.100420
$225$	−938.819	−0.278169
$226$	555.216	0.163418
$227$	177.938	0.0520273	0.0260136	−	0.999662i	$-0.491719\pi$
$227$	177.938	0.0520273	0.0260136	+	0.999662i	$0.491719\pi$
$228$	0	0
$229$	−6490.11	−1.87283	−0.936416	−	0.350892i	$-0.885879\pi$
$229$	−6490.11	−1.87283	−0.936416	+	0.350892i	$0.885879\pi$
$230$	−612.270	−0.175530
$231$	−805.387	−0.229397
$232$	1412.73	0.399785
$233$	1361.43	0.382790	0.191395	−	0.981513i	$-0.438699\pi$
$233$	1361.43	0.382790	0.191395	+	0.981513i	$0.438699\pi$
$234$	−16.3537	−0.00456870
$235$	1153.19	0.320109
$236$	267.181	0.0736950
$237$	−596.476	−0.163482
$238$	−2053.99	−0.559412
$239$	7221.94	1.95460	0.977298	−	0.211871i	$-0.0679556\pi$
$239$	7221.94	1.95460	0.977298	+	0.211871i	$0.0679556\pi$
$240$	218.317	0.0587179
$241$	3909.80	1.04503	0.522516	−	0.852630i	$-0.324994\pi$
$241$	3909.80	1.04503	0.522516	+	0.852630i	$0.324994\pi$
$242$	1359.68	0.361173
$243$	−243.000	−0.0641500
$244$	1018.60	0.267250
$245$	1056.64	0.275536
$246$	921.322	0.238786
$247$	0	0
$248$	699.992	0.179232
$249$	−571.955	−0.145567
$250$	−2085.96	−0.527710
$251$	5643.55	1.41919	0.709597	−	0.704607i	$-0.248877\pi$
$251$	5643.55	1.41919	0.709597	+	0.704607i	$0.248877\pi$
$252$	378.741	0.0946765
$253$	−1717.55	−0.426805
$254$	1309.59	0.323509
$255$	1331.97	0.327103
$256$	256.000	0.0625000
$257$	−5872.23	−1.42529	−0.712645	−	0.701525i	$-0.752503\pi$
$257$	−5872.23	−1.42529	−0.712645	+	0.701525i	$0.752503\pi$
$258$	1883.48	0.454498
$259$	−603.255	−0.144728
$260$	−16.5291	−0.00394267
$261$	−1589.32	−0.376921
$262$	−2023.23	−0.477083
$263$	−4469.08	−1.04782	−0.523908	−	0.851775i	$-0.675526\pi$
$263$	−4469.08	−1.04782	−0.523908	+	0.851775i	$0.675526\pi$
$264$	612.427	0.142774
$265$	1316.29	0.305129
$266$	0	0
$267$	−3738.61	−0.856925
$268$	−2442.10	−0.556623
$269$	−6067.70	−1.37529	−0.687647	−	0.726045i	$-0.741356\pi$
$269$	−6067.70	−1.37529	−0.687647	+	0.726045i	$0.741356\pi$
$270$	−245.607	−0.0553598
$271$	308.049	0.0690503	0.0345252	−	0.999404i	$-0.489008\pi$
$271$	308.049	0.0690503	0.0345252	+	0.999404i	$0.489008\pi$
$272$	1561.88	0.348172
$273$	−28.6752	−0.00635714
$274$	−5423.76	−1.19585
$275$	−2661.84	−0.583692
$276$	807.697	0.176151
$277$	−1360.83	−0.295178	−0.147589	−	0.989049i	$-0.547151\pi$
$277$	−1360.83	−0.295178	−0.147589	+	0.989049i	$0.547151\pi$
$278$	2922.99	0.630608
$279$	−787.491	−0.168982
$280$	382.804	0.0817033
$281$	3181.80	0.675481	0.337741	−	0.941239i	$-0.390337\pi$
$281$	3181.80	0.675481	0.337741	+	0.941239i	$0.390337\pi$
$282$	−1521.27	−0.321241
$283$	−4873.49	−1.02367	−0.511836	−	0.859083i	$-0.671034\pi$
$283$	−4873.49	−1.02367	−0.511836	+	0.859083i	$0.671034\pi$
$284$	1035.24	0.216304
$285$	0	0
$286$	−46.3679	−0.00958668
$287$	1615.48	0.332260
$288$	−288.000	−0.0589256
$289$	4616.15	0.939579
$290$	−1606.37	−0.325273
$291$	492.019	0.0991158
$292$	2509.22	0.502880
$293$	−6803.04	−1.35644	−0.678221	−	0.734858i	$-0.737249\pi$
$293$	−6803.04	−1.35644	−0.678221	+	0.734858i	$0.737249\pi$
$294$	−1393.90	−0.276510
$295$	−303.803	−0.0599597
$296$	458.723	0.0900768
$297$	−688.981	−0.134608
$298$	−5481.83	−1.06562
$299$	−61.1521	−0.0118278
$300$	1251.76	0.240901
$301$	3302.56	0.632413
$302$	−1721.31	−0.327980
$303$	4284.38	0.812314
$304$	0	0
$305$	−1158.21	−0.217440
$306$	−1757.11	−0.328260
$307$	−7354.25	−1.36720	−0.683598	−	0.729858i	$-0.739586\pi$
$307$	−7354.25	−1.36720	−0.683598	+	0.729858i	$0.739586\pi$
$308$	1073.85	0.198663
$309$	5236.38	0.964036
$310$	−795.938	−0.145827
$311$	−9334.39	−1.70194	−0.850972	−	0.525211i	$-0.823986\pi$
$311$	−9334.39	−1.70194	−0.850972	+	0.525211i	$0.823986\pi$
$312$	21.8050	0.00395661
$313$	−9106.58	−1.64452	−0.822260	−	0.569113i	$-0.807287\pi$
$313$	−9106.58	−1.64452	−0.822260	+	0.569113i	$0.807287\pi$
$314$	−2722.20	−0.489244
$315$	−430.655	−0.0770306
$316$	795.301	0.141580
$317$	−759.056	−0.134488	−0.0672442	−	0.997737i	$-0.521421\pi$
$317$	−759.056	−0.134488	−0.0672442	+	0.997737i	$0.521421\pi$
$318$	−1736.43	−0.306208
$319$	−4506.22	−0.790908
$320$	−291.089	−0.0508512
$321$	1216.06	0.211445
$322$	1416.24	0.245106
$323$	0	0
$324$	324.000	0.0555556
$325$	−94.7728	−0.0161755
$326$	−5838.67	−0.991945
$327$	1192.47	0.201662
$328$	−1228.43	−0.206795
$329$	−2667.44	−0.446993
$330$	−696.371	−0.116164
$331$	11805.9	1.96046	0.980229	−	0.197868i	$-0.0634018\pi$
$331$	11805.9	1.96046	0.980229	+	0.197868i	$0.0634018\pi$
$332$	762.607	0.126065
$333$	−516.063	−0.0849252
$334$	−3797.04	−0.622049
$335$	2776.83	0.452879
$336$	−504.989	−0.0819922
$337$	−3312.04	−0.535367	−0.267683	−	0.963507i	$-0.586258\pi$
$337$	−3312.04	−0.535367	−0.267683	+	0.963507i	$0.586258\pi$
$338$	4392.35	0.706841
$339$	832.825	0.133430
$340$	−1775.96	−0.283279
$341$	−2232.78	−0.354580
$342$	0	0
$343$	−6052.68	−0.952810
$344$	−2511.31	−0.393607
$345$	−918.406	−0.143320
$346$	5951.06	0.924656
$347$	3506.57	0.542485	0.271243	−	0.962511i	$-0.412565\pi$
$347$	3506.57	0.542485	0.271243	+	0.962511i	$0.412565\pi$
$348$	2119.09	0.326423
$349$	−2781.35	−0.426596	−0.213298	−	0.976987i	$-0.568421\pi$
$349$	−2781.35	−0.426596	−0.213298	+	0.976987i	$0.568421\pi$
$350$	2194.88	0.335203
$351$	−24.5306	−0.00373033
$352$	−816.570	−0.123646
$353$	9160.14	1.38115	0.690574	−	0.723262i	$-0.257358\pi$
$353$	9160.14	1.38115	0.690574	+	0.723262i	$0.257358\pi$
$354$	400.772	0.0601718
$355$	−1177.14	−0.175989
$356$	4984.81	0.742119
$357$	−3080.98	−0.456758
$358$	−2911.41	−0.429812
$359$	−8307.71	−1.22135	−0.610674	−	0.791882i	$-0.709102\pi$
$359$	−8307.71	−1.22135	−0.610674	+	0.791882i	$0.709102\pi$
$360$	327.475	0.0479430
$361$	0	0
$362$	2213.77	0.321418
$363$	2039.53	0.294896
$364$	38.2335	0.00550545
$365$	−2853.15	−0.409152
$366$	1527.90	0.218209
$367$	−10662.8	−1.51660	−0.758300	−	0.651905i	$-0.773970\pi$
$367$	−10662.8	−1.51660	−0.758300	+	0.651905i	$0.773970\pi$
$368$	−1076.93	−0.152551
$369$	1381.98	0.194968
$370$	−521.599	−0.0732882
$371$	−3044.71	−0.426074
$372$	1049.99	0.146342
$373$	11368.5	1.57812	0.789060	−	0.614316i	$-0.210568\pi$
$373$	11368.5	1.57812	0.789060	+	0.614316i	$0.210568\pi$
$374$	−4981.96	−0.688800
$375$	−3128.94	−0.430873
$376$	2028.36	0.278203
$377$	−160.440	−0.0219180
$378$	568.112	0.0773030
$379$	2395.24	0.324631	0.162316	−	0.986739i	$-0.448104\pi$
$379$	2395.24	0.324631	0.162316	+	0.986739i	$0.448104\pi$
$380$	0	0
$381$	1964.39	0.264144
$382$	150.646	0.0201773
$383$	5366.38	0.715950	0.357975	−	0.933731i	$-0.383467\pi$
$383$	5366.38	0.715950	0.357975	+	0.933731i	$0.383467\pi$
$384$	384.000	0.0510310
$385$	−1221.04	−0.161636
$386$	−1829.86	−0.241288
$387$	2825.22	0.371096
$388$	−656.026	−0.0858368
$389$	5220.97	0.680498	0.340249	−	0.940335i	$-0.389489\pi$
$389$	5220.97	0.680498	0.340249	+	0.940335i	$0.389489\pi$
$390$	−24.7937	−0.00321918
$391$	−6570.44	−0.849824
$392$	1858.54	0.239465
$393$	−3034.85	−0.389536
$394$	576.750	0.0737468
$395$	−904.311	−0.115192
$396$	918.641	0.116574
$397$	11464.2	1.44930	0.724648	−	0.689120i	$-0.242003\pi$
$397$	11464.2	1.44930	0.724648	+	0.689120i	$0.242003\pi$
$398$	7542.32	0.949905
$399$	0	0
$400$	−1669.01	−0.208626
$401$	−1696.04	−0.211212	−0.105606	−	0.994408i	$-0.533678\pi$
$401$	−1696.04	−0.211212	−0.105606	+	0.994408i	$0.533678\pi$
$402$	−3663.15	−0.454481
$403$	−79.4964	−0.00982629
$404$	−5712.50	−0.703484
$405$	−368.410	−0.0452011
$406$	3715.69	0.454203
$407$	−1463.20	−0.178202
$408$	2342.82	0.284281
$409$	−8653.64	−1.04620	−0.523099	−	0.852272i	$-0.675224\pi$
$409$	−8653.64	−1.04620	−0.523099	+	0.852272i	$0.675224\pi$
$410$	1396.81	0.168252
$411$	−8135.65	−0.976403
$412$	−6981.84	−0.834880
$413$	702.727	0.0837262
$414$	1211.55	0.143827
$415$	−867.135	−0.102569
$416$	−29.0733	−0.00342653
$417$	4384.48	0.514889
$418$	0	0
$419$	−8564.91	−0.998624	−0.499312	−	0.866422i	$-0.666414\pi$
$419$	−8564.91	−0.998624	−0.499312	+	0.866422i	$0.666414\pi$
$420$	574.206	0.0667104
$421$	−5578.66	−0.645813	−0.322907	−	0.946431i	$-0.604660\pi$
$421$	−5578.66	−0.645813	−0.322907	+	0.946431i	$0.604660\pi$
$422$	11281.3	1.30134
$423$	−2281.90	−0.262293
$424$	2315.24	0.265184
$425$	−10182.8	−1.16221
$426$	1552.86	0.176611
$427$	2679.06	0.303628
$428$	−1621.41	−0.183117
$429$	−69.5519	−0.00782749
$430$	2855.53	0.320246
$431$	53.5465	0.00598433	0.00299216	−	0.999996i	$-0.499048\pi$
$431$	53.5465	0.00598433	0.00299216	+	0.999996i	$0.499048\pi$
$432$	−432.000	−0.0481125
$433$	7503.47	0.832781	0.416390	−	0.909186i	$-0.363295\pi$
$433$	7503.47	0.832781	0.416390	+	0.909186i	$0.363295\pi$
$434$	1841.08	0.203629
$435$	−2409.55	−0.265584
$436$	−1589.96	−0.174645
$437$	0	0
$438$	3763.83	0.410599
$439$	−15556.3	−1.69126	−0.845628	−	0.533772i	$-0.820774\pi$
$439$	−15556.3	−1.69126	−0.845628	+	0.533772i	$0.820774\pi$
$440$	928.495	0.100601
$441$	−2090.85	−0.225770
$442$	−177.379	−0.0190883
$443$	−13593.7	−1.45791	−0.728957	−	0.684560i	$-0.759995\pi$
$443$	−13593.7	−1.45791	−0.728957	+	0.684560i	$0.759995\pi$
$444$	688.084	0.0735474
$445$	−5668.06	−0.603802
$446$	1628.87	0.172936
$447$	−8222.74	−0.870072
$448$	673.318	0.0710074
$449$	−14468.5	−1.52074	−0.760368	−	0.649492i	$-0.774982\pi$
$449$	−14468.5	−1.52074	−0.760368	+	0.649492i	$0.774982\pi$
$450$	1877.64	0.196695
$451$	3918.35	0.409109
$452$	−1110.43	−0.115554
$453$	−2581.96	−0.267795
$454$	−355.877	−0.0367888
$455$	−43.4741	−0.00447934
$456$	0	0
$457$	−10002.9	−1.02388	−0.511941	−	0.859021i	$-0.671073\pi$
$457$	−10002.9	−1.02388	−0.511941	+	0.859021i	$0.671073\pi$
$458$	12980.2	1.32429
$459$	−2635.67	−0.268023
$460$	1224.54	0.124118
$461$	−16929.5	−1.71038	−0.855188	−	0.518318i	$-0.826558\pi$
$461$	−16929.5	−1.71038	−0.855188	+	0.518318i	$0.826558\pi$
$462$	1610.77	0.162208
$463$	−12017.2	−1.20624	−0.603118	−	0.797652i	$-0.706075\pi$
$463$	−12017.2	−1.20624	−0.603118	+	0.797652i	$0.706075\pi$
$464$	−2825.46	−0.282691
$465$	−1193.91	−0.119067
$466$	−2722.86	−0.270674
$467$	−6345.20	−0.628738	−0.314369	−	0.949301i	$-0.601793\pi$
$467$	−6345.20	−0.628738	−0.314369	+	0.949301i	$0.601793\pi$
$468$	32.7075	0.00323056
$469$	−6423.09	−0.632389
$470$	−2306.38	−0.226351
$471$	−4083.30	−0.399466
$472$	−534.363	−0.0521103
$473$	8010.39	0.778685
$474$	1192.95	0.115599
$475$	0	0
$476$	4107.97	0.395564
$477$	−2604.64	−0.250018
$478$	−14443.9	−1.38211
$479$	8464.62	0.807429	0.403715	−	0.914885i	$-0.367719\pi$
$479$	8464.62	0.807429	0.403715	+	0.914885i	$0.367719\pi$
$480$	−436.634	−0.0415198
$481$	−52.0960	−0.00493841
$482$	−7819.61	−0.738949
$483$	2124.36	0.200128
$484$	−2719.37	−0.255388
$485$	745.946	0.0698384
$486$	486.000	0.0453609
$487$	9621.23	0.895235	0.447618	−	0.894225i	$-0.352273\pi$
$487$	9621.23	0.895235	0.447618	+	0.894225i	$0.352273\pi$
$488$	−2037.19	−0.188974
$489$	−8758.01	−0.809920
$490$	−2113.28	−0.194833
$491$	1381.93	0.127018	0.0635090	−	0.997981i	$-0.479771\pi$
$491$	1381.93	0.127018	0.0635090	+	0.997981i	$0.479771\pi$
$492$	−1842.64	−0.168847
$493$	−17238.4	−1.57480
$494$	0	0
$495$	−1044.56	−0.0948471
$496$	−1399.98	−0.126736
$497$	2722.84	0.245747
$498$	1143.91	0.102931
$499$	−4911.96	−0.440660	−0.220330	−	0.975425i	$-0.570713\pi$
$499$	−4911.96	−0.440660	−0.220330	+	0.975425i	$0.570713\pi$
$500$	4171.91	0.373147
$501$	−5695.55	−0.507901
$502$	−11287.1	−1.00352
$503$	879.596	0.0779707	0.0389854	−	0.999240i	$-0.487587\pi$
$503$	879.596	0.0779707	0.0389854	+	0.999240i	$0.487587\pi$
$504$	−757.483	−0.0669464
$505$	6495.50	0.572368
$506$	3435.11	0.301797
$507$	6588.52	0.577133
$508$	−2619.19	−0.228755
$509$	9387.05	0.817433	0.408717	−	0.912661i	$-0.365976\pi$
$509$	9387.05	0.817433	0.408717	+	0.912661i	$0.365976\pi$
$510$	−2663.94	−0.231297
$511$	6599.62	0.571330
$512$	−512.000	−0.0441942
$513$	0	0
$514$	11744.5	1.00783
$515$	7938.82	0.679274
$516$	−3766.97	−0.321379
$517$	−6469.90	−0.550378
$518$	1206.51	0.102338
$519$	8926.59	0.754978
$520$	33.0583	0.00278789
$521$	23680.9	1.99132	0.995659	−	0.0930734i	$-0.0296691\pi$
$521$	23680.9	1.99132	0.995659	+	0.0930734i	$0.0296691\pi$
$522$	3178.64	0.266523
$523$	3670.77	0.306906	0.153453	−	0.988156i	$-0.450961\pi$
$523$	3670.77	0.306906	0.153453	+	0.988156i	$0.450961\pi$
$524$	4046.46	0.337348
$525$	3292.31	0.273692
$526$	8938.16	0.740917
$527$	−8541.42	−0.706015
$528$	−1224.85	−0.100956
$529$	−7636.62	−0.627651
$530$	−2632.58	−0.215759
$531$	601.158	0.0491300
$532$	0	0
$533$	139.510	0.0113374
$534$	7477.21	0.605937
$535$	1843.66	0.148987
$536$	4884.20	0.393592
$537$	−4367.11	−0.350940
$538$	12135.4	0.972479
$539$	−5928.22	−0.473741
$540$	491.213	0.0391453
$541$	−24694.9	−1.96250	−0.981252	−	0.192729i	$-0.938266\pi$
$541$	−24694.9	−1.96250	−0.981252	+	0.192729i	$0.938266\pi$
$542$	−616.098	−0.0488260
$543$	3320.66	0.262437
$544$	−3123.76	−0.246195
$545$	1807.89	0.142094
$546$	57.3503	0.00449518
$547$	−341.298	−0.0266779	−0.0133390	−	0.999911i	$-0.504246\pi$
$547$	−341.298	−0.0266779	−0.0133390	+	0.999911i	$0.504246\pi$
$548$	10847.5	0.845590
$549$	2291.84	0.178167
$550$	5323.69	0.412733
$551$	0	0
$552$	−1615.39	−0.124557
$553$	2091.76	0.160851
$554$	2721.66	0.208722
$555$	−782.398	−0.0598395
$556$	−5845.97	−0.445907
$557$	−17411.7	−1.32452	−0.662260	−	0.749274i	$-0.730403\pi$
$557$	−17411.7	−1.32452	−0.662260	+	0.749274i	$0.730403\pi$
$558$	1574.98	0.119488
$559$	285.203	0.0215793
$560$	−765.608	−0.0577729
$561$	−7472.94	−0.562403
$562$	−6363.60	−0.477637
$563$	−13139.5	−0.983597	−0.491798	−	0.870709i	$-0.663660\pi$
$563$	−13139.5	−0.983597	−0.491798	+	0.870709i	$0.663660\pi$
$564$	3042.53	0.227152
$565$	1262.64	0.0940169
$566$	9746.98	0.723845
$567$	852.168	0.0631176
$568$	−2070.48	−0.152950
$569$	−13006.1	−0.958253	−0.479126	−	0.877746i	$-0.659046\pi$
$569$	−13006.1	−0.958253	−0.479126	+	0.877746i	$0.659046\pi$
$570$	0	0
$571$	−6042.41	−0.442849	−0.221425	−	0.975177i	$-0.571071\pi$
$571$	−6042.41	−0.442849	−0.221425	+	0.975177i	$0.571071\pi$
$572$	92.7358	0.00677881
$573$	225.969	0.0164747
$574$	−3230.95	−0.234943
$575$	7021.12	0.509219
$576$	576.000	0.0416667
$577$	18981.8	1.36953	0.684767	−	0.728762i	$-0.259904\pi$
$577$	18981.8	1.36953	0.684767	+	0.728762i	$0.259904\pi$
$578$	−9232.31	−0.664383
$579$	−2744.78	−0.197011
$580$	3212.73	0.230003
$581$	2005.77	0.143224
$582$	−984.039	−0.0700854
$583$	−7384.98	−0.524622
$584$	−5018.43	−0.355590
$585$	−37.1906	−0.00262845
$586$	13606.1	0.959150
$587$	−5226.77	−0.367516	−0.183758	−	0.982972i	$-0.558826\pi$
$587$	−5226.77	−0.367516	−0.183758	+	0.982972i	$0.558826\pi$
$588$	2787.80	0.195522
$589$	0	0
$590$	607.607	0.0423979
$591$	865.125	0.0602140
$592$	−917.446	−0.0636939
$593$	−2692.03	−0.186422	−0.0932110	−	0.995646i	$-0.529713\pi$
$593$	−2692.03	−0.186422	−0.0932110	+	0.995646i	$0.529713\pi$
$594$	1377.96	0.0951826
$595$	−4671.04	−0.321839
$596$	10963.7	0.753505
$597$	11313.5	0.775594
$598$	122.304	0.00836353
$599$	−2003.22	−0.136643	−0.0683216	−	0.997663i	$-0.521764\pi$
$599$	−2003.22	−0.136643	−0.0683216	+	0.997663i	$0.521764\pi$
$600$	−2503.52	−0.170343
$601$	4881.17	0.331293	0.165646	−	0.986185i	$-0.447029\pi$
$601$	4881.17	0.331293	0.165646	+	0.986185i	$0.447029\pi$
$602$	−6605.12	−0.447184
$603$	−5494.73	−0.371082
$604$	3442.61	0.231917
$605$	3092.10	0.207788
$606$	−8568.75	−0.574393
$607$	−11010.1	−0.736219	−0.368110	−	0.929782i	$-0.619995\pi$
$607$	−11010.1	−0.736219	−0.368110	+	0.929782i	$0.619995\pi$
$608$	0	0
$609$	5573.53	0.370855
$610$	2316.43	0.153753
$611$	−230.355	−0.0152523
$612$	3514.23	0.232115
$613$	7281.20	0.479747	0.239873	−	0.970804i	$-0.422894\pi$
$613$	7281.20	0.479747	0.239873	+	0.970804i	$0.422894\pi$
$614$	14708.5	0.966754
$615$	2095.21	0.137377
$616$	−2147.70	−0.140476
$617$	17226.4	1.12400	0.562000	−	0.827137i	$-0.310032\pi$
$617$	17226.4	1.12400	0.562000	+	0.827137i	$0.310032\pi$
$618$	−10472.8	−0.681676
$619$	−27081.0	−1.75844	−0.879221	−	0.476414i	$-0.841936\pi$
$619$	−27081.0	−1.75844	−0.879221	+	0.476414i	$0.841936\pi$
$620$	1591.88	0.103115
$621$	1817.32	0.117434
$622$	18668.8	1.20346
$623$	13110.8	0.843134
$624$	−43.6099	−0.00279775
$625$	8295.41	0.530906
$626$	18213.2	1.16285
$627$	0	0
$628$	5444.39	0.345947
$629$	−5597.42	−0.354823
$630$	861.309	0.0544688
$631$	−24550.4	−1.54887	−0.774435	−	0.632653i	$-0.781966\pi$
$631$	−24550.4	−1.54887	−0.774435	+	0.632653i	$0.781966\pi$
$632$	−1590.60	−0.100112
$633$	16922.0	1.06254
$634$	1518.11	0.0950977
$635$	2978.19	0.186119
$636$	3472.86	0.216522
$637$	−211.069	−0.0131285
$638$	9012.43	0.559256
$639$	2329.29	0.144203
$640$	582.178	0.0359572
$641$	−10000.0	−0.616188	−0.308094	−	0.951356i	$-0.599691\pi$
$641$	−10000.0	−0.616188	−0.308094	+	0.951356i	$0.599691\pi$
$642$	−2432.12	−0.149514
$643$	24537.9	1.50494	0.752472	−	0.658625i	$-0.228861\pi$
$643$	24537.9	1.50494	0.752472	+	0.658625i	$0.228861\pi$
$644$	−2832.48	−0.173316
$645$	4283.29	0.261480
$646$	0	0
$647$	8904.57	0.541074	0.270537	−	0.962710i	$-0.412799\pi$
$647$	8904.57	0.541074	0.270537	+	0.962710i	$0.412799\pi$
$648$	−648.000	−0.0392837
$649$	1704.47	0.103091
$650$	189.546	0.0114378
$651$	2761.62	0.166262
$652$	11677.3	0.701411
$653$	3189.13	0.191118	0.0955592	−	0.995424i	$-0.469536\pi$
$653$	3189.13	0.191118	0.0955592	+	0.995424i	$0.469536\pi$
$654$	−2384.94	−0.142597
$655$	−4601.10	−0.274473
$656$	2456.86	0.146226
$657$	5645.74	0.335253
$658$	5334.88	0.316072
$659$	−4475.48	−0.264552	−0.132276	−	0.991213i	$-0.542229\pi$
$659$	−4475.48	−0.264552	−0.132276	+	0.991213i	$0.542229\pi$
$660$	1392.74	0.0821400
$661$	−15805.6	−0.930054	−0.465027	−	0.885297i	$-0.653955\pi$
$661$	−15805.6	−0.930054	−0.465027	+	0.885297i	$0.653955\pi$
$662$	−23611.8	−1.38625
$663$	−266.068	−0.0155856
$664$	−1525.21	−0.0891412
$665$	0	0
$666$	1032.13	0.0600512
$667$	11886.0	0.689997
$668$	7594.07	0.439855
$669$	2443.31	0.141201
$670$	−5553.66	−0.320234
$671$	6498.09	0.373854
$672$	1009.98	0.0579773
$673$	−5856.55	−0.335443	−0.167722	−	0.985834i	$-0.553641\pi$
$673$	−5856.55	−0.335443	−0.167722	+	0.985834i	$0.553641\pi$
$674$	6624.09	0.378561
$675$	2816.46	0.160601
$676$	−8784.70	−0.499812
$677$	−8243.45	−0.467979	−0.233989	−	0.972239i	$-0.575178\pi$
$677$	−8243.45	−0.467979	−0.233989	+	0.972239i	$0.575178\pi$
$678$	−1665.65	−0.0943494
$679$	−1725.45	−0.0975207
$680$	3551.92	0.200309
$681$	−533.815	−0.0300380
$682$	4465.56	0.250726
$683$	9076.91	0.508519	0.254259	−	0.967136i	$-0.418168\pi$
$683$	9076.91	0.508519	0.254259	+	0.967136i	$0.418168\pi$
$684$	0	0
$685$	−12334.4	−0.687988
$686$	12105.4	0.673739
$687$	19470.3	1.08128
$688$	5022.62	0.278322
$689$	−262.936	−0.0145386
$690$	1836.81	0.101342
$691$	14600.7	0.803816	0.401908	−	0.915680i	$-0.368347\pi$
$691$	14600.7	0.803816	0.401908	+	0.915680i	$0.368347\pi$
$692$	−11902.1	−0.653830
$693$	2416.16	0.132442
$694$	−7013.13	−0.383595
$695$	6647.26	0.362799
$696$	−4238.19	−0.230816
$697$	14989.5	0.814588
$698$	5562.69	0.301649
$699$	−4084.29	−0.221004
$700$	−4389.75	−0.237024
$701$	−5264.76	−0.283662	−0.141831	−	0.989891i	$-0.545299\pi$
$701$	−5264.76	−0.283662	−0.141831	+	0.989891i	$0.545299\pi$
$702$	49.0612	0.00263774
$703$	0	0
$704$	1633.14	0.0874308
$705$	−3459.56	−0.184815
$706$	−18320.3	−0.976619
$707$	−15024.7	−0.799241
$708$	−801.544	−0.0425479
$709$	−30920.1	−1.63784	−0.818920	−	0.573908i	$-0.805427\pi$
$709$	−30920.1	−1.63784	−0.818920	+	0.573908i	$0.805427\pi$
$710$	2354.28	0.124443
$711$	1789.43	0.0943865
$712$	−9969.62	−0.524757
$713$	5889.39	0.309340
$714$	6161.96	0.322977
$715$	−105.447	−0.00551537
$716$	5822.82	0.303923
$717$	−21665.8	−1.12849
$718$	16615.4	0.863624
$719$	−2815.48	−0.146036	−0.0730179	−	0.997331i	$-0.523263\pi$
$719$	−2815.48	−0.146036	−0.0730179	+	0.997331i	$0.523263\pi$
$720$	−654.951	−0.0339008
$721$	−18363.3	−0.948521
$722$	0	0
$723$	−11729.4	−0.603349
$724$	−4427.55	−0.227277
$725$	18420.8	0.943629
$726$	−4079.05	−0.208523
$727$	13905.3	0.709380	0.354690	−	0.934984i	$-0.384586\pi$
$727$	13905.3	0.709380	0.354690	+	0.934984i	$0.384586\pi$
$728$	−76.4671	−0.00389294
$729$	729.000	0.0370370
$730$	5706.30	0.289314
$731$	30643.4	1.55046
$732$	−3055.79	−0.154297
$733$	−30724.3	−1.54820	−0.774099	−	0.633064i	$-0.781797\pi$
$733$	−30724.3	−1.54820	−0.774099	+	0.633064i	$0.781797\pi$
$734$	21325.6	1.07240
$735$	−3169.92	−0.159081
$736$	2153.86	0.107870
$737$	−15579.3	−0.778656
$738$	−2763.97	−0.137863
$739$	33564.2	1.67074	0.835372	−	0.549685i	$-0.185252\pi$
$739$	33564.2	1.67074	0.835372	+	0.549685i	$0.185252\pi$
$740$	1043.20	0.0518226
$741$	0	0
$742$	6089.43	0.301280
$743$	−20279.4	−1.00132	−0.500659	−	0.865645i	$-0.666909\pi$
$743$	−20279.4	−1.00132	−0.500659	+	0.865645i	$0.666909\pi$
$744$	−2099.98	−0.103480
$745$	−12466.4	−0.613066
$746$	−22737.0	−1.11590
$747$	1715.87	0.0840431
$748$	9963.92	0.487055
$749$	−4264.56	−0.208042
$750$	6257.87	0.304673
$751$	38424.3	1.86701	0.933504	−	0.358568i	$-0.116735\pi$
$751$	38424.3	1.86701	0.933504	+	0.358568i	$0.116735\pi$
$752$	−4056.71	−0.196719
$753$	−16930.7	−0.819372
$754$	320.880	0.0154984
$755$	−3914.48	−0.188692
$756$	−1136.22	−0.0546615
$757$	2345.54	0.112616	0.0563080	−	0.998413i	$-0.482067\pi$
$757$	2345.54	0.112616	0.0563080	+	0.998413i	$0.482067\pi$
$758$	−4790.48	−0.229549
$759$	5152.66	0.246416
$760$	0	0
$761$	16257.3	0.774409	0.387205	−	0.921994i	$-0.373441\pi$
$761$	16257.3	0.774409	0.387205	+	0.921994i	$0.373441\pi$
$762$	−3928.78	−0.186778
$763$	−4181.82	−0.198417
$764$	−301.292	−0.0142675
$765$	−3995.91	−0.188853
$766$	−10732.8	−0.506253
$767$	60.6863	0.00285692
$768$	−768.000	−0.0360844
$769$	−8591.30	−0.402874	−0.201437	−	0.979501i	$-0.564561\pi$
$769$	−8591.30	−0.402874	−0.201437	+	0.979501i	$0.564561\pi$
$770$	2442.08	0.114294
$771$	17616.7	0.822891
$772$	3659.71	0.170616
$773$	28680.4	1.33449	0.667245	−	0.744838i	$-0.267473\pi$
$773$	28680.4	1.33449	0.667245	+	0.744838i	$0.267473\pi$
$774$	−5650.45	−0.262405
$775$	9127.30	0.423048
$776$	1312.05	0.0606958
$777$	1809.76	0.0835585
$778$	−10441.9	−0.481185
$779$	0	0
$780$	49.5874	0.00227630
$781$	6604.27	0.302586
$782$	13140.9	0.600917
$783$	4767.96	0.217616
$784$	−3717.07	−0.169327
$785$	−6190.64	−0.281469
$786$	6069.69	0.275444
$787$	8664.18	0.392433	0.196216	−	0.980561i	$-0.437135\pi$
$787$	8664.18	0.392433	0.196216	+	0.980561i	$0.437135\pi$
$788$	−1153.50	−0.0521469
$789$	13407.2	0.604956
$790$	1808.62	0.0814530
$791$	−2920.60	−0.131283
$792$	−1837.28	−0.0824305
$793$	231.359	0.0103604
$794$	−22928.3	−1.02481
$795$	−3948.87	−0.176166
$796$	−15084.6	−0.671684
$797$	−17346.7	−0.770955	−0.385477	−	0.922717i	$-0.625963\pi$
$797$	−17346.7	−0.770955	−0.385477	+	0.922717i	$0.625963\pi$
$798$	0	0
$799$	−24750.3	−1.09588
$800$	3338.02	0.147521
$801$	11215.8	0.494746
$802$	3392.07	0.149349
$803$	16007.4	0.703474
$804$	7326.30	0.321367
$805$	3220.72	0.141013
$806$	158.993	0.00694824
$807$	18203.1	0.794026
$808$	11425.0	0.497439
$809$	17662.9	0.767606	0.383803	−	0.923415i	$-0.374614\pi$
$809$	17662.9	0.767606	0.383803	+	0.923415i	$0.374614\pi$
$810$	736.820	0.0319620
$811$	−27124.8	−1.17445	−0.587227	−	0.809423i	$-0.699780\pi$
$811$	−27124.8	−1.17445	−0.587227	+	0.809423i	$0.699780\pi$
$812$	−7431.37	−0.321170
$813$	−924.147	−0.0398662
$814$	2926.40	0.126008
$815$	−13277.9	−0.570682
$816$	−4685.63	−0.201017
$817$	0	0
$818$	17307.3	0.739774
$819$	86.0255	0.00367030
$820$	−2793.61	−0.118972
$821$	−19164.8	−0.814685	−0.407342	−	0.913276i	$-0.633544\pi$
$821$	−19164.8	−0.814685	−0.407342	+	0.913276i	$0.633544\pi$
$822$	16271.3	0.690421
$823$	−2079.05	−0.0880574	−0.0440287	−	0.999030i	$-0.514019\pi$
$823$	−2079.05	−0.0880574	−0.0440287	+	0.999030i	$0.514019\pi$
$824$	13963.7	0.590349
$825$	7985.53	0.336995
$826$	−1405.45	−0.0592034
$827$	19618.9	0.824930	0.412465	−	0.910973i	$-0.364668\pi$
$827$	19618.9	0.824930	0.412465	+	0.910973i	$0.364668\pi$
$828$	−2423.09	−0.101701
$829$	−5102.68	−0.213780	−0.106890	−	0.994271i	$-0.534089\pi$
$829$	−5102.68	−0.213780	−0.106890	+	0.994271i	$0.534089\pi$
$830$	1734.27	0.0725270
$831$	4082.49	0.170421
$832$	58.1466	0.00242292
$833$	−22678.2	−0.943280
$834$	−8768.96	−0.364082
$835$	−8634.97	−0.357875
$836$	0	0
$837$	2362.47	0.0975615
$838$	17129.8	0.706134
$839$	26387.7	1.08582	0.542911	−	0.839790i	$-0.317322\pi$
$839$	26387.7	1.08582	0.542911	+	0.839790i	$0.317322\pi$
$840$	−1148.41	−0.0471714
$841$	6795.41	0.278626
$842$	11157.3	0.456659
$843$	−9545.40	−0.389989
$844$	−22562.6	−0.920187
$845$	9988.79	0.406657
$846$	4563.80	0.185469
$847$	−7152.34	−0.290150
$848$	−4630.48	−0.187513
$849$	14620.5	0.591017
$850$	20365.6	0.821804
$851$	3859.47	0.155465
$852$	−3105.72	−0.124883
$853$	−4304.97	−0.172801	−0.0864006	−	0.996260i	$-0.527536\pi$
$853$	−4304.97	−0.172801	−0.0864006	+	0.996260i	$0.527536\pi$
$854$	−5358.13	−0.214697
$855$	0	0
$856$	3242.83	0.129483
$857$	40813.9	1.62681	0.813405	−	0.581697i	$-0.197611\pi$
$857$	40813.9	1.62681	0.813405	+	0.581697i	$0.197611\pi$
$858$	139.104	0.00553487
$859$	−30512.0	−1.21194	−0.605970	−	0.795488i	$-0.707215\pi$
$859$	−30512.0	−1.21194	−0.605970	+	0.795488i	$0.707215\pi$
$860$	−5711.06	−0.226448
$861$	−4846.43	−0.191830
$862$	−107.093	−0.00423156
$863$	12681.1	0.500195	0.250098	−	0.968221i	$-0.419537\pi$
$863$	12681.1	0.500195	0.250098	+	0.968221i	$0.419537\pi$
$864$	864.000	0.0340207
$865$	13533.5	0.531969
$866$	−15006.9	−0.588865
$867$	−13848.5	−0.542466
$868$	−3682.17	−0.143987
$869$	5073.58	0.198055
$870$	4819.10	0.187796
$871$	−554.687	−0.0215785
$872$	3179.91	0.123493
$873$	−1476.06	−0.0572245
$874$	0	0
$875$	10972.8	0.423939
$876$	−7527.65	−0.290338
$877$	24071.3	0.926831	0.463416	−	0.886141i	$-0.346624\pi$
$877$	24071.3	0.926831	0.463416	+	0.886141i	$0.346624\pi$
$878$	31112.6	1.19590
$879$	20409.1	0.783142
$880$	−1856.99	−0.0711353
$881$	−12979.8	−0.496367	−0.248183	−	0.968713i	$-0.579834\pi$
$881$	−12979.8	−0.496367	−0.248183	+	0.968713i	$0.579834\pi$
$882$	4181.71	0.159643
$883$	−27248.1	−1.03847	−0.519237	−	0.854630i	$-0.673784\pi$
$883$	−27248.1	−1.03847	−0.519237	+	0.854630i	$0.673784\pi$
$884$	354.757	0.0134975
$885$	911.410	0.0346177
$886$	27187.4	1.03090
$887$	22913.4	0.867368	0.433684	−	0.901065i	$-0.357213\pi$
$887$	22913.4	0.867368	0.433684	+	0.901065i	$0.357213\pi$
$888$	−1376.17	−0.0520058
$889$	−6888.85	−0.259893
$890$	11336.1	0.426952
$891$	2066.94	0.0777162
$892$	−3257.74	−0.122284
$893$	0	0
$894$	16445.5	0.615234
$895$	−6620.93	−0.247278
$896$	−1346.64	−0.0502098
$897$	183.456	0.00682879
$898$	28937.0	1.07532
$899$	15451.5	0.573234
$900$	−3755.28	−0.139084
$901$	−28251.0	−1.04459
$902$	−7836.70	−0.289283
$903$	−9907.68	−0.365124
$904$	2220.87	0.0817090
$905$	5034.42	0.184917
$906$	5163.92	0.189360
$907$	53634.2	1.96350	0.981751	−	0.190173i	$-0.0609050\pi$
$907$	53634.2	1.96350	0.981751	+	0.190173i	$0.0609050\pi$
$908$	711.754	0.0260136
$909$	−12853.1	−0.468990
$910$	86.9482	0.00316737
$911$	19050.4	0.692831	0.346416	−	0.938081i	$-0.387399\pi$
$911$	19050.4	0.692831	0.346416	+	0.938081i	$0.387399\pi$
$912$	0	0
$913$	4865.01	0.176351
$914$	20005.7	0.723994
$915$	3474.64	0.125539
$916$	−25960.4	−0.936416
$917$	10642.8	0.383267
$918$	5271.34	0.189521
$919$	−12749.2	−0.457626	−0.228813	−	0.973470i	$-0.573484\pi$
$919$	−12749.2	−0.457626	−0.228813	+	0.973470i	$0.573484\pi$
$920$	−2449.08	−0.0877650
$921$	22062.8	0.789351
$922$	33858.9	1.20942
$923$	235.140	0.00838539
$924$	−3221.55	−0.114698
$925$	5981.36	0.212612
$926$	24034.4	0.852937
$927$	−15709.1	−0.556586
$928$	5650.91	0.199893
$929$	−36570.8	−1.29155	−0.645775	−	0.763528i	$-0.723466\pi$
$929$	−36570.8	−1.29155	−0.645775	+	0.763528i	$0.723466\pi$
$930$	2387.81	0.0841930
$931$	0	0
$932$	5445.71	0.191395
$933$	28003.2	0.982618
$934$	12690.4	0.444585
$935$	−11329.7	−0.396277
$936$	−65.4149	−0.00228435
$937$	−14283.4	−0.497993	−0.248996	−	0.968504i	$-0.580101\pi$
$937$	−14283.4	−0.497993	−0.248996	+	0.968504i	$0.580101\pi$
$938$	12846.2	0.447167
$939$	27319.7	0.949464
$940$	4612.75	0.160055
$941$	−6526.01	−0.226081	−0.113040	−	0.993590i	$-0.536059\pi$
$941$	−6526.01	−0.226081	−0.113040	+	0.993590i	$0.536059\pi$
$942$	8166.59	0.282465
$943$	−10335.4	−0.356911
$944$	1068.73	0.0368475
$945$	1291.96	0.0444736
$946$	−16020.8	−0.550614
$947$	−18286.9	−0.627502	−0.313751	−	0.949505i	$-0.601586\pi$
$947$	−18286.9	−0.627502	−0.313751	+	0.949505i	$0.601586\pi$
$948$	−2385.90	−0.0817411
$949$	569.931	0.0194950
$950$	0	0
$951$	2277.17	0.0776469
$952$	−8215.95	−0.279706
$953$	16818.1	0.571661	0.285830	−	0.958280i	$-0.407731\pi$
$953$	16818.1	0.571661	0.285830	+	0.958280i	$0.407731\pi$
$954$	5209.29	0.176789
$955$	342.590	0.0116083
$956$	28887.7	0.977298
$957$	13518.6	0.456631
$958$	−16929.2	−0.570939
$959$	28530.6	0.960690
$960$	873.268	0.0293590
$961$	−22134.9	−0.743007
$962$	104.192	0.00349198
$963$	−3648.18	−0.122078
$964$	15639.2	0.522516
$965$	−4161.34	−0.138817
$966$	−4248.73	−0.141512
$967$	42110.1	1.40038	0.700191	−	0.713955i	$-0.253098\pi$
$967$	42110.1	1.40038	0.700191	+	0.713955i	$0.253098\pi$
$968$	5438.73	0.180586
$969$	0	0
$970$	−1491.89	−0.0493832
$971$	−15482.1	−0.511684	−0.255842	−	0.966719i	$-0.582353\pi$
$971$	−15482.1	−0.511684	−0.255842	+	0.966719i	$0.582353\pi$
$972$	−972.000	−0.0320750
$973$	−15375.8	−0.506603
$974$	−19242.5	−0.633027
$975$	284.318	0.00933895
$976$	4074.39	0.133625
$977$	19056.2	0.624013	0.312007	−	0.950080i	$-0.398999\pi$
$977$	19056.2	0.624013	0.312007	+	0.950080i	$0.398999\pi$
$978$	17516.0	0.572700
$979$	31800.3	1.03814
$980$	4226.56	0.137768
$981$	−3577.40	−0.116430
$982$	−2763.87	−0.0898153
$983$	50240.0	1.63012	0.815060	−	0.579377i	$-0.196704\pi$
$983$	50240.0	1.63012	0.815060	+	0.579377i	$0.196704\pi$
$984$	3685.29	0.119393
$985$	1311.61	0.0424277
$986$	34476.7	1.11355
$987$	8002.32	0.258071
$988$	0	0
$989$	−21128.9	−0.679334
$990$	2089.11	0.0670670
$991$	50971.3	1.63386	0.816931	−	0.576736i	$-0.195674\pi$
$991$	50971.3	1.63386	0.816931	+	0.576736i	$0.195674\pi$
$992$	2799.97	0.0896160
$993$	−35417.7	−1.13187
$994$	−5445.68	−0.173769
$995$	17152.2	0.546495
$996$	−2287.82	−0.0727835
$997$	−30368.2	−0.964665	−0.482332	−	0.875988i	$-0.660210\pi$
$997$	−30368.2	−0.964665	−0.482332	+	0.875988i	$0.660210\pi$
$998$	9823.91	0.311594
$999$	1548.19	0.0490316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bj.1.4		9
19.2	odd	18		114.4.i.d.61.2	yes	18
19.10	odd	18		114.4.i.d.43.2	✓	18
19.18	odd	2		2166.4.a.bm.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.d.43.2	✓	18	19.10	odd	18
114.4.i.d.61.2	yes	18	19.2	odd	18
2166.4.a.bj.1.4		9	1.1	even	1	trivial
2166.4.a.bm.1.4		9	19.18	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.54827 q^{5} +6.00000 q^{6} +10.5206 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.54827 q^{5} +6.00000 q^{6} +10.5206 q^{7} -8.00000 q^{8} +9.00000 q^{9} +9.09654 q^{10} +25.5178 q^{11} -12.0000 q^{12} +0.908540 q^{13} -21.0412 q^{14} +13.6448 q^{15} +16.0000 q^{16} +97.6174 q^{17} -18.0000 q^{18} -18.1931 q^{20} -31.5618 q^{21} -51.0356 q^{22} -67.3081 q^{23} +24.0000 q^{24} -104.313 q^{25} -1.81708 q^{26} -27.0000 q^{27} +42.0824 q^{28} -176.591 q^{29} -27.2896 q^{30} -87.4990 q^{31} -32.0000 q^{32} -76.5534 q^{33} -195.235 q^{34} -47.8505 q^{35} +36.0000 q^{36} -57.3404 q^{37} -2.72562 q^{39} +36.3862 q^{40} +153.554 q^{41} +63.1236 q^{42} +313.914 q^{43} +102.071 q^{44} -40.9344 q^{45} +134.616 q^{46} -253.544 q^{47} -48.0000 q^{48} -232.317 q^{49} +208.626 q^{50} -292.852 q^{51} +3.63416 q^{52} -289.405 q^{53} +54.0000 q^{54} -116.062 q^{55} -84.1648 q^{56} +353.182 q^{58} +66.7954 q^{59} +54.5792 q^{60} +254.649 q^{61} +174.998 q^{62} +94.6854 q^{63} +64.0000 q^{64} -4.13229 q^{65} +153.107 q^{66} -610.525 q^{67} +390.470 q^{68} +201.924 q^{69} +95.7010 q^{70} +258.810 q^{71} -72.0000 q^{72} +627.304 q^{73} +114.681 q^{74} +312.940 q^{75} +268.462 q^{77} +5.45124 q^{78} +198.825 q^{79} -72.7723 q^{80} +81.0000 q^{81} -307.107 q^{82} +190.652 q^{83} -126.247 q^{84} -443.990 q^{85} -627.828 q^{86} +529.773 q^{87} -204.142 q^{88} +1246.20 q^{89} +81.8688 q^{90} +9.55839 q^{91} -269.232 q^{92} +262.497 q^{93} +507.089 q^{94} +96.0000 q^{96} -164.006 q^{97} +464.634 q^{98} +229.660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 18 q^{2} - 27 q^{3} + 36 q^{4} + 27 q^{5} + 54 q^{6} - 72 q^{8} + 81 q^{9}+O(q^{10}) \) 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

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