LMFDB - Embedded newform 2166.4.a.t.1.3

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:	$3$
Coefficient field:	3.3.14457.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 32x - 3 $ x^3 - x^2 - 32*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.12716$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.2543 q^{5} -6.00000 q^{6} +13.2543 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.2543 q^{5} -6.00000 q^{6} +13.2543 q^{7} -8.00000 q^{8} +9.00000 q^{9} -28.5086 q^{10} -65.9138 q^{11} +12.0000 q^{12} +68.5259 q^{13} -26.5086 q^{14} +42.7629 q^{15} +16.0000 q^{16} +99.6940 q^{17} -18.0000 q^{18} +57.0172 q^{20} +39.7629 q^{21} +131.828 q^{22} +3.53886 q^{23} -24.0000 q^{24} +78.1854 q^{25} -137.052 q^{26} +27.0000 q^{27} +53.0172 q^{28} +81.9008 q^{29} -85.5259 q^{30} +247.190 q^{31} -32.0000 q^{32} -197.741 q^{33} -199.388 q^{34} +188.931 q^{35} +36.0000 q^{36} -421.065 q^{37} +205.578 q^{39} -114.034 q^{40} +345.259 q^{41} -79.5259 q^{42} -366.168 q^{43} -263.655 q^{44} +128.289 q^{45} -7.07773 q^{46} -90.9399 q^{47} +48.0000 q^{48} -167.323 q^{49} -156.371 q^{50} +299.082 q^{51} +274.103 q^{52} +688.491 q^{53} -54.0000 q^{54} -939.556 q^{55} -106.034 q^{56} -163.802 q^{58} -182.289 q^{59} +171.052 q^{60} -0.517370 q^{61} -494.379 q^{62} +119.289 q^{63} +64.0000 q^{64} +976.789 q^{65} +395.483 q^{66} +159.862 q^{67} +398.776 q^{68} +10.6166 q^{69} -377.862 q^{70} +791.884 q^{71} -72.0000 q^{72} +322.129 q^{73} +842.130 q^{74} +234.556 q^{75} -873.642 q^{77} -411.155 q^{78} -318.371 q^{79} +228.069 q^{80} +81.0000 q^{81} -690.517 q^{82} +684.160 q^{83} +159.052 q^{84} +1421.07 q^{85} +732.336 q^{86} +245.703 q^{87} +527.311 q^{88} -720.406 q^{89} -256.578 q^{90} +908.263 q^{91} +14.1555 q^{92} +741.569 q^{93} +181.880 q^{94} -96.0000 q^{96} -415.392 q^{97} +334.646 q^{98} -593.224 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} - 20 q^{10} - 44 q^{11} + 36 q^{12} + 9 q^{13} - 14 q^{14} + 30 q^{15} + 48 q^{16} - 84 q^{17} - 54 q^{18} + 40 q^{20} + 21 q^{21} + 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} + 81 q^{27} + 28 q^{28} - 92 q^{29} - 60 q^{30} + 109 q^{31} - 96 q^{32} - 132 q^{33} + 168 q^{34} + 282 q^{35} + 108 q^{36} - 245 q^{37} + 27 q^{39} - 80 q^{40} + 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} + 4 q^{46} + 322 q^{47} + 144 q^{48} - 754 q^{49} + 166 q^{50} - 252 q^{51} + 36 q^{52} + 1322 q^{53} - 162 q^{54} - 248 q^{55} - 56 q^{56} + 184 q^{58} - 252 q^{59} + 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} + 1582 q^{65} + 264 q^{66} + 719 q^{67} - 336 q^{68} - 6 q^{69} - 564 q^{70} + 62 q^{71} - 216 q^{72} - 581 q^{73} + 490 q^{74} - 249 q^{75} - 204 q^{77} - 54 q^{78} + 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} + 84 q^{84} + 1632 q^{85} + 206 q^{86} - 276 q^{87} + 352 q^{88} - 1584 q^{89} - 180 q^{90} + 1573 q^{91} - 8 q^{92} + 327 q^{93} - 644 q^{94} - 288 q^{96} - 974 q^{97} + 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9 - 20 * q^10 - 44 * q^11 + 36 * q^12 + 9 * q^13 - 14 * q^14 + 30 * q^15 + 48 * q^16 - 84 * q^17 - 54 * q^18 + 40 * q^20 + 21 * q^21 + 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 + 81 * q^27 + 28 * q^28 - 92 * q^29 - 60 * q^30 + 109 * q^31 - 96 * q^32 - 132 * q^33 + 168 * q^34 + 282 * q^35 + 108 * q^36 - 245 * q^37 + 27 * q^39 - 80 * q^40 + 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 + 4 * q^46 + 322 * q^47 + 144 * q^48 - 754 * q^49 + 166 * q^50 - 252 * q^51 + 36 * q^52 + 1322 * q^53 - 162 * q^54 - 248 * q^55 - 56 * q^56 + 184 * q^58 - 252 * q^59 + 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 + 1582 * q^65 + 264 * q^66 + 719 * q^67 - 336 * q^68 - 6 * q^69 - 564 * q^70 + 62 * q^71 - 216 * q^72 - 581 * q^73 + 490 * q^74 - 249 * q^75 - 204 * q^77 - 54 * q^78 + 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 + 84 * q^84 + 1632 * q^85 + 206 * q^86 - 276 * q^87 + 352 * q^88 - 1584 * q^89 - 180 * q^90 + 1573 * q^91 - 8 * q^92 + 327 * q^93 - 644 * q^94 - 288 * q^96 - 974 * q^97 + 1508 * q^98 - 396 * 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
$5$	14.2543	1.27494	0.637472	−	0.770473i	$-0.279980\pi$
$5$	14.2543	1.27494	0.637472	+	0.770473i	$0.279980\pi$
$6$	−6.00000	−0.408248
$7$	13.2543	0.715666	0.357833	−	0.933786i	$-0.383516\pi$
$7$	13.2543	0.715666	0.357833	+	0.933786i	$0.383516\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−28.5086	−0.901522
$11$	−65.9138	−1.80671	−0.903353	−	0.428898i	$-0.858902\pi$
$11$	−65.9138	−1.80671	−0.903353	+	0.428898i	$0.858902\pi$
$12$	12.0000	0.288675
$13$	68.5259	1.46197	0.730987	−	0.682392i	$-0.239060\pi$
$13$	68.5259	1.46197	0.730987	+	0.682392i	$0.239060\pi$
$14$	−26.5086	−0.506052
$15$	42.7629	0.736089
$16$	16.0000	0.250000
$17$	99.6940	1.42231	0.711157	−	0.703033i	$-0.248171\pi$
$17$	99.6940	1.42231	0.711157	+	0.703033i	$0.248171\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	57.0172	0.637472
$21$	39.7629	0.413190
$22$	131.828	1.27753
$23$	3.53886	0.0320828	0.0160414	−	0.999871i	$-0.494894\pi$
$23$	3.53886	0.0320828	0.0160414	+	0.999871i	$0.494894\pi$
$24$	−24.0000	−0.204124
$25$	78.1854	0.625483
$26$	−137.052	−1.03377
$27$	27.0000	0.192450
$28$	53.0172	0.357833
$29$	81.9008	0.524435	0.262217	−	0.965009i	$-0.415546\pi$
$29$	81.9008	0.524435	0.262217	+	0.965009i	$0.415546\pi$
$30$	−85.5259	−0.520494
$31$	247.190	1.43215	0.716074	−	0.698025i	$-0.245937\pi$
$31$	247.190	1.43215	0.716074	+	0.698025i	$0.245937\pi$
$32$	−32.0000	−0.176777
$33$	−197.741	−1.04310
$34$	−199.388	−1.00573
$35$	188.931	0.912434
$36$	36.0000	0.166667
$37$	−421.065	−1.87088	−0.935441	−	0.353483i	$-0.884997\pi$
$37$	−421.065	−1.87088	−0.935441	+	0.353483i	$0.884997\pi$
$38$	0	0
$39$	205.578	0.844071
$40$	−114.034	−0.450761
$41$	345.259	1.31513	0.657565	−	0.753398i	$-0.271587\pi$
$41$	345.259	1.31513	0.657565	+	0.753398i	$0.271587\pi$
$42$	−79.5259	−0.292169
$43$	−366.168	−1.29861	−0.649304	−	0.760529i	$-0.724940\pi$
$43$	−366.168	−1.29861	−0.649304	+	0.760529i	$0.724940\pi$
$44$	−263.655	−0.903353
$45$	128.289	0.424981
$46$	−7.07773	−0.0226860
$47$	−90.9399	−0.282233	−0.141116	−	0.989993i	$-0.545069\pi$
$47$	−90.9399	−0.282233	−0.141116	+	0.989993i	$0.545069\pi$
$48$	48.0000	0.144338
$49$	−167.323	−0.487823
$50$	−156.371	−0.442283
$51$	299.082	0.821174
$52$	274.103	0.730987
$53$	688.491	1.78437	0.892185	−	0.451671i	$-0.149172\pi$
$53$	688.491	1.78437	0.892185	+	0.451671i	$0.149172\pi$
$54$	−54.0000	−0.136083
$55$	−939.556	−2.30345
$56$	−106.034	−0.253026
$57$	0	0
$58$	−163.802	−0.370831
$59$	−182.289	−0.402237	−0.201118	−	0.979567i	$-0.564458\pi$
$59$	−182.289	−0.402237	−0.201118	+	0.979567i	$0.564458\pi$
$60$	171.052	0.368045
$61$	−0.517370	−0.00108594	−0.000542971	−	1.00000i	$-0.500173\pi$
$61$	−0.517370	−0.00108594	−0.000542971		1.00000i	$0.500173\pi$
$62$	−494.379	−1.01268
$63$	119.289	0.238555
$64$	64.0000	0.125000
$65$	976.789	1.86393
$66$	395.483	0.737585
$67$	159.862	0.291496	0.145748	−	0.989322i	$-0.453441\pi$
$67$	159.862	0.291496	0.145748	+	0.989322i	$0.453441\pi$
$68$	398.776	0.711157
$69$	10.6166	0.0185230
$70$	−377.862	−0.645188
$71$	791.884	1.32365	0.661826	−	0.749657i	$-0.269782\pi$
$71$	791.884	1.32365	0.661826	+	0.749657i	$0.269782\pi$
$72$	−72.0000	−0.117851
$73$	322.129	0.516471	0.258235	−	0.966082i	$-0.416859\pi$
$73$	322.129	0.516471	0.258235	+	0.966082i	$0.416859\pi$
$74$	842.130	1.32291
$75$	234.556	0.361123
$76$	0	0
$77$	−873.642	−1.29300
$78$	−411.155	−0.596848
$79$	−318.371	−0.453412	−0.226706	−	0.973963i	$-0.572796\pi$
$79$	−318.371	−0.453412	−0.226706	+	0.973963i	$0.572796\pi$
$80$	228.069	0.318736
$81$	81.0000	0.111111
$82$	−690.517	−0.929937
$83$	684.160	0.904774	0.452387	−	0.891822i	$-0.350573\pi$
$83$	684.160	0.904774	0.452387	+	0.891822i	$0.350573\pi$
$84$	159.052	0.206595
$85$	1421.07	1.81337
$86$	732.336	0.918254
$87$	245.703	0.302782
$88$	527.311	0.638767
$89$	−720.406	−0.858009	−0.429005	−	0.903302i	$-0.641136\pi$
$89$	−720.406	−0.858009	−0.429005	+	0.903302i	$0.641136\pi$
$90$	−256.578	−0.300507
$91$	908.263	1.04628
$92$	14.1555	0.0160414
$93$	741.569	0.826851
$94$	181.880	0.199569
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−415.392	−0.434811	−0.217406	−	0.976081i	$-0.569759\pi$
$97$	−415.392	−0.434811	−0.217406	+	0.976081i	$0.569759\pi$
$98$	334.646	0.344943
$99$	−593.224	−0.602235
$100$	312.742	0.312742
$101$	337.341	0.332343	0.166172	−	0.986097i	$-0.446859\pi$
$101$	337.341	0.332343	0.166172	+	0.986097i	$0.446859\pi$
$102$	−598.164	−0.580658
$103$	1467.05	1.40342	0.701711	−	0.712462i	$-0.252420\pi$
$103$	1467.05	1.40342	0.701711	+	0.712462i	$0.252420\pi$
$104$	−548.207	−0.516886
$105$	566.793	0.526794
$106$	−1376.98	−1.26174
$107$	−382.014	−0.345146	−0.172573	−	0.984997i	$-0.555208\pi$
$107$	−382.014	−0.345146	−0.172573	+	0.984997i	$0.555208\pi$
$108$	108.000	0.0962250
$109$	132.857	0.116747	0.0583736	−	0.998295i	$-0.481409\pi$
$109$	132.857	0.116747	0.0583736	+	0.998295i	$0.481409\pi$
$110$	1879.11	1.62879
$111$	−1263.19	−1.08015
$112$	212.069	0.178916
$113$	1137.94	0.947331	0.473665	−	0.880705i	$-0.342931\pi$
$113$	1137.94	0.947331	0.473665	+	0.880705i	$0.342931\pi$
$114$	0	0
$115$	50.4441	0.0409038
$116$	327.603	0.262217
$117$	616.733	0.487325
$118$	364.578	0.284424
$119$	1321.38	1.01790
$120$	−342.103	−0.260247
$121$	3013.63	2.26419
$122$	1.03474	0.000767877 0
$123$	1035.78	0.759291
$124$	988.759	0.716074
$125$	−667.310	−0.477488
$126$	−238.578	−0.168684
$127$	1509.97	1.05503	0.527514	−	0.849546i	$-0.323124\pi$
$127$	1509.97	1.05503	0.527514	+	0.849546i	$0.323124\pi$
$128$	−128.000	−0.0883883
$129$	−1098.50	−0.749751
$130$	−1953.58	−1.31800
$131$	−370.944	−0.247401	−0.123701	−	0.992320i	$-0.539476\pi$
$131$	−370.944	−0.247401	−0.123701	+	0.992320i	$0.539476\pi$
$132$	−790.966	−0.521551
$133$	0	0
$134$	−319.724	−0.206119
$135$	384.866	0.245363
$136$	−797.552	−0.502864
$137$	−1690.37	−1.05415	−0.527075	−	0.849819i	$-0.676711\pi$
$137$	−1690.37	−1.05415	−0.527075	+	0.849819i	$0.676711\pi$
$138$	−21.2332	−0.0130977
$139$	−2711.52	−1.65459	−0.827296	−	0.561766i	$-0.810122\pi$
$139$	−2711.52	−1.65459	−0.827296	+	0.561766i	$0.810122\pi$
$140$	755.724	0.456217
$141$	−272.820	−0.162947
$142$	−1583.77	−0.935964
$143$	−4516.80	−2.64136
$144$	144.000	0.0833333
$145$	1167.44	0.668625
$146$	−644.258	−0.365200
$147$	−501.970	−0.281645
$148$	−1684.26	−0.935441
$149$	2598.90	1.42893	0.714464	−	0.699672i	$-0.246671\pi$
$149$	2598.90	1.42893	0.714464	+	0.699672i	$0.246671\pi$
$150$	−469.112	−0.255352
$151$	−594.863	−0.320591	−0.160296	−	0.987069i	$-0.551245\pi$
$151$	−594.863	−0.320591	−0.160296	+	0.987069i	$0.551245\pi$
$152$	0	0
$153$	897.246	0.474105
$154$	1747.28	0.914287
$155$	3523.52	1.82591
$156$	822.310	0.422035
$157$	−58.2549	−0.0296130	−0.0148065	−	0.999890i	$-0.504713\pi$
$157$	−58.2549	−0.0296130	−0.0148065	+	0.999890i	$0.504713\pi$
$158$	636.742	0.320610
$159$	2065.47	1.03021
$160$	−456.138	−0.225380
$161$	46.9052	0.0229605
$162$	−162.000	−0.0785674
$163$	1152.53	0.553825	0.276912	−	0.960895i	$-0.410689\pi$
$163$	1152.53	0.553825	0.276912	+	0.960895i	$0.410689\pi$
$164$	1381.03	0.657565
$165$	−2818.67	−1.32990
$166$	−1368.32	−0.639772
$167$	−1259.11	−0.583430	−0.291715	−	0.956505i	$-0.594226\pi$
$167$	−1259.11	−0.583430	−0.291715	+	0.956505i	$0.594226\pi$
$168$	−318.103	−0.146085
$169$	2498.79	1.13737
$170$	−2842.14	−1.28225
$171$	0	0
$172$	−1464.67	−0.649304
$173$	1274.10	0.559931	0.279965	−	0.960010i	$-0.409677\pi$
$173$	1274.10	0.559931	0.279965	+	0.960010i	$0.409677\pi$
$174$	−491.405	−0.214100
$175$	1036.29	0.447637
$176$	−1054.62	−0.451677
$177$	−546.866	−0.232232
$178$	1440.81	0.606704
$179$	−2034.50	−0.849528	−0.424764	−	0.905304i	$-0.639643\pi$
$179$	−2034.50	−0.849528	−0.424764	+	0.905304i	$0.639643\pi$
$180$	513.155	0.212491
$181$	−1033.31	−0.424340	−0.212170	−	0.977233i	$-0.568053\pi$
$181$	−1033.31	−0.424340	−0.212170	+	0.977233i	$0.568053\pi$
$182$	−1816.53	−0.739835
$183$	−1.55211	−0.000626969 0
$184$	−28.3109	−0.0113430
$185$	−6001.99	−2.38527
$186$	−1483.14	−0.584672
$187$	−6571.21	−2.56970
$188$	−363.760	−0.141116
$189$	357.866	0.137730
$190$	0	0
$191$	−1135.46	−0.430154	−0.215077	−	0.976597i	$-0.569000\pi$
$191$	−1135.46	−0.430154	−0.215077	+	0.976597i	$0.569000\pi$
$192$	192.000	0.0721688
$193$	−1091.07	−0.406928	−0.203464	−	0.979082i	$-0.565220\pi$
$193$	−1091.07	−0.406928	−0.203464	+	0.979082i	$0.565220\pi$
$194$	830.784	0.307458
$195$	2930.37	1.07614
$196$	−669.293	−0.243911
$197$	4138.08	1.49658	0.748289	−	0.663373i	$-0.230876\pi$
$197$	4138.08	1.49658	0.748289	+	0.663373i	$0.230876\pi$
$198$	1186.45	0.425845
$199$	−2734.23	−0.973993	−0.486997	−	0.873404i	$-0.661908\pi$
$199$	−2734.23	−0.973993	−0.486997	+	0.873404i	$0.661908\pi$
$200$	−625.483	−0.221142
$201$	479.586	0.168296
$202$	−674.682	−0.235002
$203$	1085.54	0.375320
$204$	1196.33	0.410587
$205$	4921.42	1.67672
$206$	−2934.10	−0.992369
$207$	31.8498	0.0106943
$208$	1096.41	0.365493
$209$	0	0
$210$	−1133.59	−0.372500
$211$	−3058.01	−0.997734	−0.498867	−	0.866679i	$-0.666250\pi$
$211$	−3058.01	−0.997734	−0.498867	+	0.866679i	$0.666250\pi$
$212$	2753.97	0.892185
$213$	2375.65	0.764211
$214$	764.027	0.244055
$215$	−5219.48	−1.65565
$216$	−216.000	−0.0680414
$217$	3276.33	1.02494
$218$	−265.715	−0.0825527
$219$	966.388	0.298185
$220$	−3758.23	−1.15172
$221$	6831.62	2.07939
$222$	2526.39	0.763784
$223$	−1618.27	−0.485951	−0.242976	−	0.970032i	$-0.578124\pi$
$223$	−1618.27	−0.485951	−0.242976	+	0.970032i	$0.578124\pi$
$224$	−424.138	−0.126513
$225$	703.668	0.208494
$226$	−2275.88	−0.669864
$227$	6065.43	1.77347	0.886733	−	0.462282i	$-0.152969\pi$
$227$	6065.43	1.77347	0.886733	+	0.462282i	$0.152969\pi$
$228$	0	0
$229$	−1916.36	−0.552999	−0.276500	−	0.961014i	$-0.589174\pi$
$229$	−1916.36	−0.552999	−0.276500	+	0.961014i	$0.589174\pi$
$230$	−100.888	−0.0289233
$231$	−2620.93	−0.746512
$232$	−655.207	−0.185416
$233$	−4008.54	−1.12707	−0.563537	−	0.826091i	$-0.690560\pi$
$233$	−4008.54	−1.12707	−0.563537	+	0.826091i	$0.690560\pi$
$234$	−1233.47	−0.344590
$235$	−1296.29	−0.359831
$236$	−729.155	−0.201118
$237$	−955.112	−0.261777
$238$	−2642.75	−0.719765
$239$	530.292	0.143522	0.0717610	−	0.997422i	$-0.477138\pi$
$239$	530.292	0.143522	0.0717610	+	0.997422i	$0.477138\pi$
$240$	684.207	0.184022
$241$	−1271.23	−0.339779	−0.169890	−	0.985463i	$-0.554341\pi$
$241$	−1271.23	−0.339779	−0.169890	+	0.985463i	$0.554341\pi$
$242$	−6027.27	−1.60102
$243$	243.000	0.0641500
$244$	−2.06948	−0.000542971 0
$245$	−2385.08	−0.621947
$246$	−2071.55	−0.536900
$247$	0	0
$248$	−1977.52	−0.506341
$249$	2052.48	0.522372
$250$	1334.62	0.337635
$251$	551.989	0.138810	0.0694048	−	0.997589i	$-0.477890\pi$
$251$	551.989	0.138810	0.0694048	+	0.997589i	$0.477890\pi$
$252$	477.155	0.119278
$253$	−233.260	−0.0579642
$254$	−3019.95	−0.746018
$255$	4263.21	1.04695
$256$	256.000	0.0625000
$257$	5235.84	1.27083	0.635414	−	0.772172i	$-0.280830\pi$
$257$	5235.84	1.27083	0.635414	+	0.772172i	$0.280830\pi$
$258$	2197.01	0.530154
$259$	−5580.92	−1.33893
$260$	3907.16	0.931967
$261$	737.108	0.174812
$262$	741.888	0.174939
$263$	−1814.26	−0.425368	−0.212684	−	0.977121i	$-0.568221\pi$
$263$	−1814.26	−0.425368	−0.212684	+	0.977121i	$0.568221\pi$
$264$	1581.93	0.368792
$265$	9813.97	2.27497
$266$	0	0
$267$	−2161.22	−0.495372
$268$	639.448	0.145748
$269$	−1013.40	−0.229695	−0.114848	−	0.993383i	$-0.536638\pi$
$269$	−1013.40	−0.229695	−0.114848	+	0.993383i	$0.536638\pi$
$270$	−769.733	−0.173498
$271$	−1918.52	−0.430043	−0.215021	−	0.976609i	$-0.568982\pi$
$271$	−1918.52	−0.430043	−0.215021	+	0.976609i	$0.568982\pi$
$272$	1595.10	0.355579
$273$	2724.79	0.604072
$274$	3380.75	0.745396
$275$	−5153.50	−1.13006
$276$	42.4664	0.00926150
$277$	6283.45	1.36295	0.681473	−	0.731844i	$-0.261340\pi$
$277$	6283.45	1.36295	0.681473	+	0.731844i	$0.261340\pi$
$278$	5423.04	1.16997
$279$	2224.71	0.477382
$280$	−1511.45	−0.322594
$281$	8057.36	1.71054	0.855270	−	0.518182i	$-0.173391\pi$
$281$	8057.36	1.71054	0.855270	+	0.518182i	$0.173391\pi$
$282$	545.639	0.115221
$283$	−193.417	−0.0406269	−0.0203135	−	0.999794i	$-0.506466\pi$
$283$	−193.417	−0.0406269	−0.0203135	+	0.999794i	$0.506466\pi$
$284$	3167.54	0.661826
$285$	0	0
$286$	9033.61	1.86772
$287$	4576.16	0.941193
$288$	−288.000	−0.0589256
$289$	5025.90	1.02298
$290$	−2334.88	−0.472789
$291$	−1246.18	−0.251038
$292$	1288.52	0.258235
$293$	4735.47	0.944194	0.472097	−	0.881547i	$-0.343497\pi$
$293$	4735.47	0.944194	0.472097	+	0.881547i	$0.343497\pi$
$294$	1003.94	0.199153
$295$	−2598.40	−0.512830
$296$	3368.52	0.661457
$297$	−1779.67	−0.347701
$298$	−5197.80	−1.01040
$299$	242.504	0.0469042
$300$	938.225	0.180561
$301$	−4853.31	−0.929369
$302$	1189.73	0.226692
$303$	1012.02	0.191879
$304$	0	0
$305$	−7.37475	−0.00138451
$306$	−1794.49	−0.335243
$307$	28.7122	0.00533775	0.00266888	−	0.999996i	$-0.499150\pi$
$307$	28.7122	0.00533775	0.00266888	+	0.999996i	$0.499150\pi$
$308$	−3494.57	−0.646499
$309$	4401.14	0.810266
$310$	−7047.04	−1.29111
$311$	−1165.69	−0.212540	−0.106270	−	0.994337i	$-0.533891\pi$
$311$	−1165.69	−0.212540	−0.106270	+	0.994337i	$0.533891\pi$
$312$	−1644.62	−0.298424
$313$	2512.75	0.453767	0.226884	−	0.973922i	$-0.427146\pi$
$313$	2512.75	0.453767	0.226884	+	0.973922i	$0.427146\pi$
$314$	116.510	0.0209396
$315$	1700.38	0.304145
$316$	−1273.48	−0.226706
$317$	1727.57	0.306087	0.153044	−	0.988219i	$-0.451092\pi$
$317$	1727.57	0.306087	0.153044	+	0.988219i	$0.451092\pi$
$318$	−4130.95	−0.728466
$319$	−5398.40	−0.947499
$320$	912.276	0.159368
$321$	−1146.04	−0.199270
$322$	−93.8104	−0.0162356
$323$	0	0
$324$	324.000	0.0555556
$325$	5357.72	0.914440
$326$	−2305.07	−0.391613
$327$	398.572	0.0674040
$328$	−2762.07	−0.464969
$329$	−1205.35	−0.201984
$330$	5637.34	0.940379
$331$	6816.13	1.13187	0.565934	−	0.824450i	$-0.308516\pi$
$331$	6816.13	1.13187	0.565934	+	0.824450i	$0.308516\pi$
$332$	2736.64	0.452387
$333$	−3789.58	−0.623627
$334$	2518.22	0.412547
$335$	2278.72	0.371642
$336$	636.207	0.103297
$337$	10408.4	1.68244	0.841218	−	0.540696i	$-0.181839\pi$
$337$	10408.4	1.68244	0.841218	+	0.540696i	$0.181839\pi$
$338$	−4997.59	−0.804240
$339$	3413.82	0.546942
$340$	5684.28	0.906686
$341$	−16293.2	−2.58747
$342$	0	0
$343$	−6763.98	−1.06478
$344$	2929.35	0.459127
$345$	151.332	0.0236158
$346$	−2548.20	−0.395931
$347$	−5584.03	−0.863881	−0.431940	−	0.901902i	$-0.642171\pi$
$347$	−5584.03	−0.863881	−0.431940	+	0.901902i	$0.642171\pi$
$348$	982.810	0.151391
$349$	5505.82	0.844470	0.422235	−	0.906486i	$-0.361246\pi$
$349$	5505.82	0.844470	0.422235	+	0.906486i	$0.361246\pi$
$350$	−2072.59	−0.316527
$351$	1850.20	0.281357
$352$	2109.24	0.319384
$353$	−5782.46	−0.871868	−0.435934	−	0.899979i	$-0.643582\pi$
$353$	−5782.46	−0.871868	−0.435934	+	0.899979i	$0.643582\pi$
$354$	1093.73	0.164213
$355$	11287.8	1.68758
$356$	−2881.62	−0.429005
$357$	3964.13	0.587686
$358$	4068.99	0.600707
$359$	−1903.64	−0.279861	−0.139931	−	0.990161i	$-0.544688\pi$
$359$	−1903.64	−0.279861	−0.139931	+	0.990161i	$0.544688\pi$
$360$	−1026.31	−0.150254
$361$	0	0
$362$	2066.63	0.300054
$363$	9040.90	1.30723
$364$	3633.05	0.523142
$365$	4591.73	0.658472
$366$	3.10422	0.000443334 0
$367$	−12748.0	−1.81319	−0.906597	−	0.421998i	$-0.861329\pi$
$367$	−12748.0	−1.81319	−0.906597	+	0.421998i	$0.861329\pi$
$368$	56.6218	0.00802070
$369$	3107.33	0.438377
$370$	12004.0	1.68664
$371$	9125.48	1.27701
$372$	2966.28	0.413425
$373$	736.660	0.102260	0.0511298	−	0.998692i	$-0.483718\pi$
$373$	736.660	0.102260	0.0511298	+	0.998692i	$0.483718\pi$
$374$	13142.4	1.81706
$375$	−2001.93	−0.275678
$376$	727.519	0.0997844
$377$	5612.33	0.766710
$378$	−715.733	−0.0973897
$379$	−1543.57	−0.209202	−0.104601	−	0.994514i	$-0.533357\pi$
$379$	−1543.57	−0.209202	−0.104601	+	0.994514i	$0.533357\pi$
$380$	0	0
$381$	4529.92	0.609121
$382$	2270.93	0.304165
$383$	−6564.36	−0.875779	−0.437889	−	0.899029i	$-0.644274\pi$
$383$	−6564.36	−0.875779	−0.437889	+	0.899029i	$0.644274\pi$
$384$	−384.000	−0.0510310
$385$	−12453.2	−1.64850
$386$	2182.15	0.287742
$387$	−3295.51	−0.432869
$388$	−1661.57	−0.217406
$389$	−3498.27	−0.455962	−0.227981	−	0.973666i	$-0.573212\pi$
$389$	−3498.27	−0.455962	−0.227981	+	0.973666i	$0.573212\pi$
$390$	−5860.73	−0.760948
$391$	352.803	0.0456318
$392$	1338.59	0.172471
$393$	−1112.83	−0.142837
$394$	−8276.16	−1.05824
$395$	−4538.16	−0.578075
$396$	−2372.90	−0.301118
$397$	8610.60	1.08855	0.544274	−	0.838908i	$-0.316805\pi$
$397$	8610.60	1.08855	0.544274	+	0.838908i	$0.316805\pi$
$398$	5468.47	0.688717
$399$	0	0
$400$	1250.97	0.156371
$401$	−1765.74	−0.219893	−0.109946	−	0.993938i	$-0.535068\pi$
$401$	−1765.74	−0.219893	−0.109946	+	0.993938i	$0.535068\pi$
$402$	−959.173	−0.119003
$403$	16938.9	2.09376
$404$	1349.36	0.166172
$405$	1154.60	0.141660
$406$	−2171.08	−0.265391
$407$	27754.0	3.38013
$408$	−2392.66	−0.290329
$409$	−13395.7	−1.61950	−0.809751	−	0.586774i	$-0.800398\pi$
$409$	−13395.7	−1.61950	−0.809751	+	0.586774i	$0.800398\pi$
$410$	−9842.85	−1.18562
$411$	−5071.12	−0.608613
$412$	5868.19	0.701711
$413$	−2416.11	−0.287867
$414$	−63.6995	−0.00756198
$415$	9752.22	1.15354
$416$	−2192.83	−0.258443
$417$	−8134.57	−0.955279
$418$	0	0
$419$	9117.71	1.06308	0.531539	−	0.847034i	$-0.321614\pi$
$419$	9117.71	1.06308	0.531539	+	0.847034i	$0.321614\pi$
$420$	2267.17	0.263397
$421$	−8204.45	−0.949788	−0.474894	−	0.880043i	$-0.657514\pi$
$421$	−8204.45	−0.949788	−0.474894	+	0.880043i	$0.657514\pi$
$422$	6116.01	0.705504
$423$	−818.459	−0.0940776
$424$	−5507.93	−0.630870
$425$	7794.61	0.889634
$426$	−4751.30	−0.540379
$427$	−6.85738	−0.000777171 0
$428$	−1528.05	−0.172573
$429$	−13550.4	−1.52499
$430$	10439.0	1.17072
$431$	1271.34	0.142085	0.0710423	−	0.997473i	$-0.477367\pi$
$431$	1271.34	0.142085	0.0710423	+	0.997473i	$0.477367\pi$
$432$	432.000	0.0481125
$433$	3967.59	0.440347	0.220173	−	0.975461i	$-0.429338\pi$
$433$	3967.59	0.440347	0.220173	+	0.975461i	$0.429338\pi$
$434$	−6552.66	−0.724741
$435$	3502.32	0.386031
$436$	531.430	0.0583736
$437$	0	0
$438$	−1932.78	−0.210848
$439$	−209.038	−0.0227263	−0.0113632	−	0.999935i	$-0.503617\pi$
$439$	−209.038	−0.0227263	−0.0113632	+	0.999935i	$0.503617\pi$
$440$	7516.45	0.814393
$441$	−1505.91	−0.162608
$442$	−13663.2	−1.47035
$443$	5734.04	0.614972	0.307486	−	0.951553i	$-0.400512\pi$
$443$	5734.04	0.614972	0.307486	+	0.951553i	$0.400512\pi$
$444$	−5052.78	−0.540077
$445$	−10268.9	−1.09391
$446$	3236.53	0.343619
$447$	7796.70	0.824992
$448$	848.276	0.0894582
$449$	−9741.13	−1.02386	−0.511929	−	0.859028i	$-0.671069\pi$
$449$	−9741.13	−1.02386	−0.511929	+	0.859028i	$0.671069\pi$
$450$	−1407.34	−0.147428
$451$	−22757.3	−2.37605
$452$	4551.76	0.473665
$453$	−1784.59	−0.185093
$454$	−12130.9	−1.25403
$455$	12946.7	1.33395
$456$	0	0
$457$	−17233.8	−1.76403	−0.882016	−	0.471220i	$-0.843814\pi$
$457$	−17233.8	−1.76403	−0.882016	+	0.471220i	$0.843814\pi$
$458$	3832.73	0.391029
$459$	2691.74	0.273725
$460$	201.776	0.0204519
$461$	1665.23	0.168238	0.0841190	−	0.996456i	$-0.473192\pi$
$461$	1665.23	0.168238	0.0841190	+	0.996456i	$0.473192\pi$
$462$	5241.85	0.527864
$463$	−10694.0	−1.07342	−0.536710	−	0.843767i	$-0.680333\pi$
$463$	−10694.0	−1.07342	−0.536710	+	0.843767i	$0.680333\pi$
$464$	1310.41	0.131109
$465$	10570.6	1.05419
$466$	8017.08	0.796961
$467$	13212.7	1.30923	0.654614	−	0.755963i	$-0.272831\pi$
$467$	13212.7	1.30923	0.654614	+	0.755963i	$0.272831\pi$
$468$	2466.93	0.243662
$469$	2118.86	0.208614
$470$	2592.57	0.254439
$471$	−174.765	−0.0170971
$472$	1458.31	0.142212
$473$	24135.5	2.34620
$474$	1910.22	0.185105
$475$	0	0
$476$	5285.50	0.508951
$477$	6196.42	0.594790
$478$	−1060.58	−0.101485
$479$	5409.53	0.516008	0.258004	−	0.966144i	$-0.416935\pi$
$479$	5409.53	0.516008	0.258004	+	0.966144i	$0.416935\pi$
$480$	−1368.41	−0.130123
$481$	−28853.8	−2.73518
$482$	2542.45	0.240260
$483$	140.716	0.0132563
$484$	12054.5	1.13209
$485$	−5921.13	−0.554360
$486$	−486.000	−0.0453609
$487$	−14297.0	−1.33031	−0.665154	−	0.746706i	$-0.731634\pi$
$487$	−14297.0	−1.33031	−0.665154	+	0.746706i	$0.731634\pi$
$488$	4.13896	0.000383938 0
$489$	3457.60	0.319751
$490$	4770.15	0.439783
$491$	2802.42	0.257580	0.128790	−	0.991672i	$-0.458891\pi$
$491$	2802.42	0.257580	0.128790	+	0.991672i	$0.458891\pi$
$492$	4143.10	0.379645
$493$	8165.02	0.745911
$494$	0	0
$495$	−8456.01	−0.767817
$496$	3955.03	0.358037
$497$	10495.9	0.947293
$498$	−4104.96	−0.369373
$499$	14178.3	1.27196	0.635978	−	0.771708i	$-0.280597\pi$
$499$	14178.3	1.27196	0.635978	+	0.771708i	$0.280597\pi$
$500$	−2669.24	−0.238744
$501$	−3777.33	−0.336843
$502$	−1103.98	−0.0981533
$503$	−5537.85	−0.490896	−0.245448	−	0.969410i	$-0.578935\pi$
$503$	−5537.85	−0.490896	−0.245448	+	0.969410i	$0.578935\pi$
$504$	−954.310	−0.0843420
$505$	4808.56	0.423719
$506$	466.520	0.0409869
$507$	7496.38	0.656659
$508$	6039.90	0.527514
$509$	−9115.09	−0.793751	−0.396876	−	0.917872i	$-0.629906\pi$
$509$	−9115.09	−0.793751	−0.396876	+	0.917872i	$0.629906\pi$
$510$	−8526.42	−0.740306
$511$	4269.60	0.369620
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−10471.7	−0.898610
$515$	20911.8	1.78929
$516$	−4394.02	−0.374876
$517$	5994.20	0.509912
$518$	11161.8	0.946763
$519$	3822.30	0.323276
$520$	−7814.31	−0.659001
$521$	2705.46	0.227502	0.113751	−	0.993509i	$-0.463713\pi$
$521$	2705.46	0.227502	0.113751	+	0.993509i	$0.463713\pi$
$522$	−1474.22	−0.123610
$523$	23128.3	1.93371	0.966856	−	0.255323i	$-0.0821820\pi$
$523$	23128.3	1.93371	0.966856	+	0.255323i	$0.0821820\pi$
$524$	−1483.78	−0.123701
$525$	3108.88	0.258443
$526$	3628.52	0.300781
$527$	24643.3	2.03696
$528$	−3163.86	−0.260776
$529$	−12154.5	−0.998971
$530$	−19627.9	−1.60865
$531$	−1640.60	−0.134079
$532$	0	0
$533$	23659.1	1.92269
$534$	4322.43	0.350281
$535$	−5445.34	−0.440042
$536$	−1278.90	−0.103060
$537$	−6103.49	−0.490475
$538$	2026.80	0.162419
$539$	11028.9	0.881353
$540$	1539.47	0.122682
$541$	−9641.17	−0.766185	−0.383093	−	0.923710i	$-0.625141\pi$
$541$	−9641.17	−0.766185	−0.383093	+	0.923710i	$0.625141\pi$
$542$	3837.03	0.304086
$543$	−3099.94	−0.244993
$544$	−3190.21	−0.251432
$545$	1893.79	0.148846
$546$	−5449.58	−0.427144
$547$	17377.0	1.35829	0.679147	−	0.734002i	$-0.262350\pi$
$547$	17377.0	1.35829	0.679147	+	0.734002i	$0.262350\pi$
$548$	−6761.50	−0.527075
$549$	−4.65633	−0.000361980 0
$550$	10307.0	0.799076
$551$	0	0
$552$	−84.9327	−0.00654887
$553$	−4219.79	−0.324491
$554$	−12566.9	−0.963748
$555$	−18006.0	−1.37714
$556$	−10846.1	−0.827296
$557$	18744.3	1.42589	0.712945	−	0.701220i	$-0.247361\pi$
$557$	18744.3	1.42589	0.712945	+	0.701220i	$0.247361\pi$
$558$	−4449.41	−0.337560
$559$	−25092.0	−1.89853
$560$	3022.90	0.228108
$561$	−19713.6	−1.48362
$562$	−16114.7	−1.20953
$563$	7618.70	0.570320	0.285160	−	0.958480i	$-0.407953\pi$
$563$	7618.70	0.570320	0.285160	+	0.958480i	$0.407953\pi$
$564$	−1091.28	−0.0814736
$565$	16220.6	1.20779
$566$	386.833	0.0287276
$567$	1073.60	0.0795184
$568$	−6335.07	−0.467982
$569$	13081.6	0.963816	0.481908	−	0.876222i	$-0.339944\pi$
$569$	13081.6	0.963816	0.481908	+	0.876222i	$0.339944\pi$
$570$	0	0
$571$	8643.49	0.633483	0.316741	−	0.948512i	$-0.397411\pi$
$571$	8643.49	0.633483	0.316741	+	0.948512i	$0.397411\pi$
$572$	−18067.2	−1.32068
$573$	−3406.39	−0.248349
$574$	−9152.33	−0.665524
$575$	276.687	0.0200672
$576$	576.000	0.0416667
$577$	9841.26	0.710047	0.355024	−	0.934857i	$-0.384473\pi$
$577$	9841.26	0.710047	0.355024	+	0.934857i	$0.384473\pi$
$578$	−10051.8	−0.723355
$579$	−3273.22	−0.234940
$580$	4669.76	0.334312
$581$	9068.06	0.647516
$582$	2492.35	0.177511
$583$	−45381.1	−3.22383
$584$	−2577.03	−0.182600
$585$	8791.10	0.621312
$586$	−9470.93	−0.667646
$587$	19898.5	1.39915	0.699573	−	0.714562i	$-0.253374\pi$
$587$	19898.5	1.39915	0.699573	+	0.714562i	$0.253374\pi$
$588$	−2007.88	−0.140822
$589$	0	0
$590$	5196.80	0.362625
$591$	12414.2	0.864050
$592$	−6737.04	−0.467720
$593$	22195.0	1.53699	0.768497	−	0.639853i	$-0.221005\pi$
$593$	22195.0	1.53699	0.768497	+	0.639853i	$0.221005\pi$
$594$	3559.35	0.245862
$595$	18835.3	1.29777
$596$	10395.6	0.714464
$597$	−8202.70	−0.562335
$598$	−485.007	−0.0331663
$599$	−19696.2	−1.34352	−0.671758	−	0.740771i	$-0.734460\pi$
$599$	−19696.2	−1.34352	−0.671758	+	0.740771i	$0.734460\pi$
$600$	−1876.45	−0.127676
$601$	14626.5	0.992724	0.496362	−	0.868116i	$-0.334669\pi$
$601$	14626.5	0.992724	0.496362	+	0.868116i	$0.334669\pi$
$602$	9706.61	0.657163
$603$	1438.76	0.0971655
$604$	−2379.45	−0.160296
$605$	42957.3	2.88671
$606$	−2024.05	−0.135679
$607$	16312.0	1.09075	0.545375	−	0.838192i	$-0.316387\pi$
$607$	16312.0	1.09075	0.545375	+	0.838192i	$0.316387\pi$
$608$	0	0
$609$	3256.62	0.216691
$610$	14.7495	0.000979000 0
$611$	−6231.73	−0.412617
$612$	3588.98	0.237052
$613$	17748.7	1.16944	0.584718	−	0.811237i	$-0.301205\pi$
$613$	17748.7	1.16944	0.584718	+	0.811237i	$0.301205\pi$
$614$	−57.4243	−0.00377436
$615$	14764.3	0.968053
$616$	6989.14	0.457144
$617$	3729.80	0.243365	0.121682	−	0.992569i	$-0.461171\pi$
$617$	3729.80	0.243365	0.121682	+	0.992569i	$0.461171\pi$
$618$	−8802.29	−0.572945
$619$	−1003.67	−0.0651714	−0.0325857	−	0.999469i	$-0.510374\pi$
$619$	−1003.67	−0.0651714	−0.0325857	+	0.999469i	$0.510374\pi$
$620$	14094.1	0.912954
$621$	95.5493	0.00617433
$622$	2331.37	0.150289
$623$	−9548.48	−0.614048
$624$	3289.24	0.211018
$625$	−19285.2	−1.23425
$626$	−5025.51	−0.320862
$627$	0	0
$628$	−233.019	−0.0148065
$629$	−41977.6	−2.66098
$630$	−3400.76	−0.215063
$631$	16846.8	1.06285	0.531426	−	0.847105i	$-0.321656\pi$
$631$	16846.8	1.06285	0.531426	+	0.847105i	$0.321656\pi$
$632$	2546.97	0.160305
$633$	−9174.02	−0.576042
$634$	−3455.13	−0.216437
$635$	21523.7	1.34510
$636$	8261.90	0.515103
$637$	−11466.0	−0.713184
$638$	10796.8	0.669983
$639$	7126.96	0.441218
$640$	−1824.55	−0.112690
$641$	161.995	0.00998196	0.00499098	−	0.999988i	$-0.498411\pi$
$641$	161.995	0.00998196	0.00499098	+	0.999988i	$0.498411\pi$
$642$	2292.08	0.140905
$643$	6290.91	0.385831	0.192915	−	0.981215i	$-0.438206\pi$
$643$	6290.91	0.385831	0.192915	+	0.981215i	$0.438206\pi$
$644$	187.621	0.0114803
$645$	−15658.4	−0.955891
$646$	0	0
$647$	28249.8	1.71656	0.858280	−	0.513181i	$-0.171533\pi$
$647$	28249.8	1.71656	0.858280	+	0.513181i	$0.171533\pi$
$648$	−648.000	−0.0392837
$649$	12015.4	0.726724
$650$	−10715.4	−0.646607
$651$	9828.99	0.591749
$652$	4610.14	0.276912
$653$	11423.5	0.684587	0.342293	−	0.939593i	$-0.388796\pi$
$653$	11423.5	0.684587	0.342293	+	0.939593i	$0.388796\pi$
$654$	−797.145	−0.0476618
$655$	−5287.55	−0.315423
$656$	5524.14	0.328782
$657$	2899.16	0.172157
$658$	2410.69	0.142824
$659$	−200.138	−0.0118304	−0.00591522	−	0.999983i	$-0.501883\pi$
$659$	−200.138	−0.0118304	−0.00591522	+	0.999983i	$0.501883\pi$
$660$	−11274.7	−0.664949
$661$	2607.01	0.153405	0.0767026	−	0.997054i	$-0.475561\pi$
$661$	2607.01	0.153405	0.0767026	+	0.997054i	$0.475561\pi$
$662$	−13632.3	−0.800352
$663$	20494.9	1.20053
$664$	−5473.28	−0.319886
$665$	0	0
$666$	7579.17	0.440971
$667$	289.836	0.0168253
$668$	−5036.44	−0.291715
$669$	−4854.80	−0.280564
$670$	−4557.45	−0.262790
$671$	34.1018	0.00196198
$672$	−1272.41	−0.0730423
$673$	−14561.2	−0.834015	−0.417007	−	0.908903i	$-0.636921\pi$
$673$	−14561.2	−0.834015	−0.417007	+	0.908903i	$0.636921\pi$
$674$	−20816.8	−1.18966
$675$	2111.01	0.120374
$676$	9995.18	0.568683
$677$	−12684.6	−0.720100	−0.360050	−	0.932933i	$-0.617240\pi$
$677$	−12684.6	−0.720100	−0.360050	+	0.932933i	$0.617240\pi$
$678$	−6827.64	−0.386746
$679$	−5505.74	−0.311179
$680$	−11368.6	−0.641124
$681$	18196.3	1.02391
$682$	32586.4	1.82962
$683$	−12054.5	−0.675332	−0.337666	−	0.941266i	$-0.609637\pi$
$683$	−12054.5	−0.675332	−0.337666	+	0.941266i	$0.609637\pi$
$684$	0	0
$685$	−24095.1	−1.34398
$686$	13528.0	0.752916
$687$	−5749.09	−0.319274
$688$	−5858.69	−0.324652
$689$	47179.5	2.60870
$690$	−302.664	−0.0166989
$691$	3338.42	0.183791	0.0918953	−	0.995769i	$-0.470707\pi$
$691$	3338.42	0.183791	0.0918953	+	0.995769i	$0.470707\pi$
$692$	5096.40	0.279965
$693$	−7862.78	−0.430999
$694$	11168.1	0.610856
$695$	−38650.9	−2.10951
$696$	−1965.62	−0.107050
$697$	34420.2	1.87053
$698$	−11011.6	−0.597130
$699$	−12025.6	−0.650716
$700$	4145.17	0.223818
$701$	−23782.8	−1.28140	−0.640702	−	0.767790i	$-0.721356\pi$
$701$	−23782.8	−1.28140	−0.640702	+	0.767790i	$0.721356\pi$
$702$	−3700.40	−0.198949
$703$	0	0
$704$	−4218.49	−0.225838
$705$	−3888.86	−0.207749
$706$	11564.9	0.616504
$707$	4471.22	0.237847
$708$	−2187.47	−0.116116
$709$	32185.1	1.70485	0.852424	−	0.522851i	$-0.175132\pi$
$709$	32185.1	1.70485	0.852424	+	0.522851i	$0.175132\pi$
$710$	−22575.5	−1.19330
$711$	−2865.34	−0.151137
$712$	5763.24	0.303352
$713$	874.770	0.0459473
$714$	−7928.25	−0.415557
$715$	−64383.9	−3.36758
$716$	−8137.99	−0.424764
$717$	1590.88	0.0828624
$718$	3807.28	0.197892
$719$	−38319.4	−1.98758	−0.993791	−	0.111266i	$-0.964509\pi$
$719$	−38319.4	−1.98758	−0.993791	+	0.111266i	$0.964509\pi$
$720$	2052.62	0.106245
$721$	19444.7	1.00438
$722$	0	0
$723$	−3813.68	−0.196172
$724$	−4133.26	−0.212170
$725$	6403.45	0.328025
$726$	−18081.8	−0.924351
$727$	−36543.1	−1.86425	−0.932125	−	0.362138i	$-0.882047\pi$
$727$	−36543.1	−1.86425	−0.932125	+	0.362138i	$0.882047\pi$
$728$	−7266.11	−0.369917
$729$	729.000	0.0370370
$730$	−9183.46	−0.465610
$731$	−36504.8	−1.84703
$732$	−6.20844	−0.000313484 0
$733$	−15689.3	−0.790584	−0.395292	−	0.918555i	$-0.629357\pi$
$733$	−15689.3	−0.790584	−0.395292	+	0.918555i	$0.629357\pi$
$734$	25496.1	1.28212
$735$	−7155.23	−0.359081
$736$	−113.244	−0.00567149
$737$	−10537.1	−0.526648
$738$	−6214.65	−0.309979
$739$	−36445.7	−1.81418	−0.907088	−	0.420940i	$-0.861700\pi$
$739$	−36445.7	−1.81418	−0.907088	+	0.420940i	$0.861700\pi$
$740$	−24008.0	−1.19264
$741$	0	0
$742$	−18251.0	−0.902984
$743$	−1678.29	−0.0828673	−0.0414337	−	0.999141i	$-0.513193\pi$
$743$	−1678.29	−0.0828673	−0.0414337	+	0.999141i	$0.513193\pi$
$744$	−5932.55	−0.292336
$745$	37045.6	1.82180
$746$	−1473.32	−0.0723084
$747$	6157.44	0.301591
$748$	−26284.9	−1.28485
$749$	−5063.33	−0.247009
$750$	4003.86	0.194934
$751$	−8641.67	−0.419892	−0.209946	−	0.977713i	$-0.567329\pi$
$751$	−8641.67	−0.419892	−0.209946	+	0.977713i	$0.567329\pi$
$752$	−1455.04	−0.0705582
$753$	1655.97	0.0801418
$754$	−11224.7	−0.542145
$755$	−8479.36	−0.408736
$756$	1431.47	0.0688649
$757$	3586.79	0.172212	0.0861058	−	0.996286i	$-0.472558\pi$
$757$	3586.79	0.172212	0.0861058	+	0.996286i	$0.472558\pi$
$758$	3087.13	0.147928
$759$	−699.780	−0.0334656
$760$	0	0
$761$	−11615.4	−0.553297	−0.276649	−	0.960971i	$-0.589224\pi$
$761$	−11615.4	−0.553297	−0.276649	+	0.960971i	$0.589224\pi$
$762$	−9059.85	−0.430714
$763$	1760.93	0.0835519
$764$	−4541.86	−0.215077
$765$	12789.6	0.604457
$766$	13128.7	0.619269
$767$	−12491.5	−0.588060
$768$	768.000	0.0360844
$769$	6021.83	0.282383	0.141192	−	0.989982i	$-0.454907\pi$
$769$	6021.83	0.282383	0.141192	+	0.989982i	$0.454907\pi$
$770$	24906.3	1.16567
$771$	15707.5	0.733712
$772$	−4364.29	−0.203464
$773$	1508.99	0.0702128	0.0351064	−	0.999384i	$-0.488823\pi$
$773$	1508.99	0.0702128	0.0351064	+	0.999384i	$0.488823\pi$
$774$	6591.03	0.306085
$775$	19326.6	0.895784
$776$	3323.14	0.153729
$777$	−16742.8	−0.773029
$778$	6996.54	0.322414
$779$	0	0
$780$	11721.5	0.538072
$781$	−52196.1	−2.39145
$782$	−705.607	−0.0322666
$783$	2211.32	0.100927
$784$	−2677.17	−0.121956
$785$	−830.383	−0.0377550
$786$	2225.67	0.101001
$787$	−3255.84	−0.147469	−0.0737346	−	0.997278i	$-0.523492\pi$
$787$	−3255.84	−0.147469	−0.0737346	+	0.997278i	$0.523492\pi$
$788$	16552.3	0.748289
$789$	−5442.77	−0.245587
$790$	9076.31	0.408760
$791$	15082.6	0.677972
$792$	4745.80	0.212922
$793$	−35.4532	−0.00158762
$794$	−17221.2	−0.769720
$795$	29441.9	1.31346
$796$	−10936.9	−0.486997
$797$	−31732.8	−1.41033	−0.705164	−	0.709044i	$-0.749127\pi$
$797$	−31732.8	−1.41033	−0.705164	+	0.709044i	$0.749127\pi$
$798$	0	0
$799$	−9066.16	−0.401424
$800$	−2501.93	−0.110571
$801$	−6483.65	−0.286003
$802$	3531.49	0.155488
$803$	−21232.8	−0.933111
$804$	1918.35	0.0841478
$805$	668.601	0.0292734
$806$	−33877.8	−1.48051
$807$	−3040.20	−0.132615
$808$	−2698.73	−0.117501
$809$	3853.96	0.167488	0.0837441	−	0.996487i	$-0.473312\pi$
$809$	3853.96	0.167488	0.0837441	+	0.996487i	$0.473312\pi$
$810$	−2309.20	−0.100169
$811$	−31871.8	−1.37999	−0.689995	−	0.723814i	$-0.742387\pi$
$811$	−31871.8	−1.37999	−0.689995	+	0.723814i	$0.742387\pi$
$812$	4342.16	0.187660
$813$	−5755.55	−0.248285
$814$	−55508.0	−2.39012
$815$	16428.6	0.706096
$816$	4785.31	0.205293
$817$	0	0
$818$	26791.4	1.14516
$819$	8174.37	0.348761
$820$	19685.7	0.838359
$821$	−39170.0	−1.66509	−0.832547	−	0.553954i	$-0.813118\pi$
$821$	−39170.0	−1.66509	−0.832547	+	0.553954i	$0.813118\pi$
$822$	10142.2	0.430355
$823$	−18838.8	−0.797910	−0.398955	−	0.916971i	$-0.630627\pi$
$823$	−18838.8	−0.797910	−0.398955	+	0.916971i	$0.630627\pi$
$824$	−11736.4	−0.496185
$825$	−15460.5	−0.652443
$826$	4832.22	0.203553
$827$	−887.307	−0.0373092	−0.0186546	−	0.999826i	$-0.505938\pi$
$827$	−887.307	−0.0373092	−0.0186546	+	0.999826i	$0.505938\pi$
$828$	127.399	0.00534713
$829$	−32877.0	−1.37740	−0.688700	−	0.725046i	$-0.741818\pi$
$829$	−32877.0	−1.37740	−0.688700	+	0.725046i	$0.741818\pi$
$830$	−19504.4	−0.815674
$831$	18850.3	0.786897
$832$	4385.66	0.182747
$833$	−16681.1	−0.693838
$834$	16269.1	0.675484
$835$	−17947.7	−0.743840
$836$	0	0
$837$	6674.12	0.275617
$838$	−18235.4	−0.751709
$839$	41512.7	1.70820	0.854099	−	0.520110i	$-0.174109\pi$
$839$	41512.7	1.70820	0.854099	+	0.520110i	$0.174109\pi$
$840$	−4534.35	−0.186250
$841$	−17681.3	−0.724968
$842$	16408.9	0.671601
$843$	24172.1	0.987581
$844$	−12232.0	−0.498867
$845$	35618.6	1.45008
$846$	1636.92	0.0665229
$847$	39943.6	1.62040
$848$	11015.9	0.446092
$849$	−580.250	−0.0234560
$850$	−15589.2	−0.629066
$851$	−1490.09	−0.0600231
$852$	9502.61	0.382106
$853$	1487.75	0.0597182	0.0298591	−	0.999554i	$-0.490494\pi$
$853$	1487.75	0.0597182	0.0298591	+	0.999554i	$0.490494\pi$
$854$	13.7148	0.000549543 0
$855$	0	0
$856$	3056.11	0.122028
$857$	−20276.1	−0.808188	−0.404094	−	0.914718i	$-0.632413\pi$
$857$	−20276.1	−0.808188	−0.404094	+	0.914718i	$0.632413\pi$
$858$	27100.8	1.07833
$859$	7436.86	0.295393	0.147696	−	0.989033i	$-0.452814\pi$
$859$	7436.86	0.295393	0.147696	+	0.989033i	$0.452814\pi$
$860$	−20877.9	−0.827826
$861$	13728.5	0.543398
$862$	−2542.69	−0.100469
$863$	−35570.4	−1.40305	−0.701524	−	0.712646i	$-0.747497\pi$
$863$	−35570.4	−1.40305	−0.701524	+	0.712646i	$0.747497\pi$
$864$	−864.000	−0.0340207
$865$	18161.4	0.713880
$866$	−7935.18	−0.311372
$867$	15077.7	0.590617
$868$	13105.3	0.512469
$869$	20985.0	0.819182
$870$	−7004.64	−0.272965
$871$	10954.7	0.426160
$872$	−1062.86	−0.0412764
$873$	−3738.53	−0.144937
$874$	0	0
$875$	−8844.74	−0.341722
$876$	3865.55	0.149092
$877$	−3868.42	−0.148948	−0.0744740	−	0.997223i	$-0.523728\pi$
$877$	−3868.42	−0.148948	−0.0744740	+	0.997223i	$0.523728\pi$
$878$	418.076	0.0160699
$879$	14206.4	0.545131
$880$	−15032.9	−0.575862
$881$	−20877.4	−0.798384	−0.399192	−	0.916867i	$-0.630709\pi$
$881$	−20877.4	−0.798384	−0.399192	+	0.916867i	$0.630709\pi$
$882$	3011.82	0.114981
$883$	−24206.8	−0.922565	−0.461283	−	0.887253i	$-0.652611\pi$
$883$	−24206.8	−0.922565	−0.461283	+	0.887253i	$0.652611\pi$
$884$	27326.5	1.03969
$885$	−7795.20	−0.296082
$886$	−11468.1	−0.434851
$887$	27117.5	1.02651	0.513256	−	0.858235i	$-0.328439\pi$
$887$	27117.5	1.02651	0.513256	+	0.858235i	$0.328439\pi$
$888$	10105.6	0.381892
$889$	20013.7	0.755047
$890$	20537.8	0.773514
$891$	−5339.02	−0.200745
$892$	−6473.07	−0.242976
$893$	0	0
$894$	−15593.4	−0.583357
$895$	−29000.4	−1.08310
$896$	−1696.55	−0.0632565
$897$	727.511	0.0270801
$898$	19482.3	0.723977
$899$	20245.0	0.751068
$900$	2814.67	0.104247
$901$	68638.5	2.53793
$902$	45514.6	1.68012
$903$	−14559.9	−0.536571
$904$	−9103.52	−0.334932
$905$	−14729.2	−0.541010
$906$	3569.18	0.130881
$907$	35594.9	1.30310	0.651548	−	0.758608i	$-0.274120\pi$
$907$	35594.9	1.30310	0.651548	+	0.758608i	$0.274120\pi$
$908$	24261.7	0.886733
$909$	3036.07	0.110781
$910$	−25893.3	−0.943248
$911$	−27909.5	−1.01502	−0.507510	−	0.861646i	$-0.669434\pi$
$911$	−27909.5	−1.01502	−0.507510	+	0.861646i	$0.669434\pi$
$912$	0	0
$913$	−45095.6	−1.63466
$914$	34467.6	1.24736
$915$	−22.1243	−0.000799350 0
$916$	−7665.45	−0.276500
$917$	−4916.61	−0.177056
$918$	−5383.48	−0.193553
$919$	−48960.0	−1.75739	−0.878696	−	0.477383i	$-0.841586\pi$
$919$	−48960.0	−1.75739	−0.878696	+	0.477383i	$0.841586\pi$
$920$	−403.552	−0.0144617
$921$	86.1365	0.00308175
$922$	−3330.47	−0.118962
$923$	54264.5	1.93515
$924$	−10483.7	−0.373256
$925$	−32921.1	−1.17020
$926$	21388.1	0.759023
$927$	13203.4	0.467807
$928$	−2620.83	−0.0927078
$929$	−11971.7	−0.422799	−0.211399	−	0.977400i	$-0.567802\pi$
$929$	−11971.7	−0.422799	−0.211399	+	0.977400i	$0.567802\pi$
$930$	−21141.1	−0.745424
$931$	0	0
$932$	−16034.2	−0.563537
$933$	−3497.06	−0.122710
$934$	−26425.4	−0.925765
$935$	−93668.1	−3.27623
$936$	−4933.86	−0.172295
$937$	−50434.2	−1.75839	−0.879197	−	0.476459i	$-0.841920\pi$
$937$	−50434.2	−1.75839	−0.879197	+	0.476459i	$0.841920\pi$
$938$	−4237.72	−0.147512
$939$	7538.26	0.261983
$940$	−5185.14	−0.179916
$941$	−12594.3	−0.436303	−0.218151	−	0.975915i	$-0.570003\pi$
$941$	−12594.3	−0.436303	−0.218151	+	0.975915i	$0.570003\pi$
$942$	349.529	0.0120895
$943$	1221.82	0.0421930
$944$	−2916.62	−0.100559
$945$	5101.14	0.175598
$946$	−48271.1	−1.65902
$947$	27632.6	0.948192	0.474096	−	0.880473i	$-0.342775\pi$
$947$	27632.6	0.948192	0.474096	+	0.880473i	$0.342775\pi$
$948$	−3820.45	−0.130889
$949$	22074.2	0.755067
$950$	0	0
$951$	5182.70	0.176720
$952$	−10571.0	−0.359883
$953$	−32021.7	−1.08844	−0.544221	−	0.838942i	$-0.683175\pi$
$953$	−32021.7	−1.08844	−0.544221	+	0.838942i	$0.683175\pi$
$954$	−12392.8	−0.420580
$955$	−16185.3	−0.548422
$956$	2121.17	0.0717610
$957$	−16195.2	−0.547039
$958$	−10819.1	−0.364873
$959$	−22404.8	−0.754418
$960$	2736.83	0.0920112
$961$	31311.7	1.05105
$962$	57707.7	1.93406
$963$	−3438.12	−0.115049
$964$	−5084.90	−0.169890
$965$	−15552.5	−0.518811
$966$	−281.431	−0.00937360
$967$	−12876.4	−0.428209	−0.214104	−	0.976811i	$-0.568683\pi$
$967$	−12876.4	−0.428209	−0.214104	+	0.976811i	$0.568683\pi$
$968$	−24109.1	−0.800511
$969$	0	0
$970$	11842.3	0.391992
$971$	−2414.51	−0.0797996	−0.0398998	−	0.999204i	$-0.512704\pi$
$971$	−2414.51	−0.0797996	−0.0398998	+	0.999204i	$0.512704\pi$
$972$	972.000	0.0320750
$973$	−35939.4	−1.18413
$974$	28594.0	0.940670
$975$	16073.2	0.527952
$976$	−8.27792	−0.000271485 0
$977$	14131.4	0.462746	0.231373	−	0.972865i	$-0.425678\pi$
$977$	14131.4	0.462746	0.231373	+	0.972865i	$0.425678\pi$
$978$	−6915.21	−0.226098
$979$	47484.7	1.55017
$980$	−9540.31	−0.310973
$981$	1195.72	0.0389157
$982$	−5604.85	−0.182136
$983$	20268.2	0.657634	0.328817	−	0.944394i	$-0.393350\pi$
$983$	20268.2	0.657634	0.328817	+	0.944394i	$0.393350\pi$
$984$	−8286.21	−0.268450
$985$	58985.5	1.90805
$986$	−16330.0	−0.527439
$987$	−3616.04	−0.116616
$988$	0	0
$989$	−1295.82	−0.0416629
$990$	16912.0	0.542928
$991$	−15550.2	−0.498453	−0.249226	−	0.968445i	$-0.580176\pi$
$991$	−15550.2	−0.498453	−0.249226	+	0.968445i	$0.580176\pi$
$992$	−7910.07	−0.253170
$993$	20448.4	0.653485
$994$	−20991.8	−0.669837
$995$	−38974.6	−1.24179
$996$	8209.92	0.261186
$997$	12715.8	0.403925	0.201962	−	0.979393i	$-0.435268\pi$
$997$	12715.8	0.403925	0.201962	+	0.979393i	$0.435268\pi$
$998$	−28356.5	−0.899408
$999$	−11368.7	−0.360051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.t.1.3		3
19.8	odd	6		114.4.e.d.7.1	✓	6
19.12	odd	6		114.4.e.d.49.1	yes	6
19.18	odd	2		2166.4.a.u.1.3		3
57.8	even	6		342.4.g.h.235.3		6
57.50	even	6		342.4.g.h.163.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.d.7.1	✓	6	19.8	odd	6
114.4.e.d.49.1	yes	6	19.12	odd	6
342.4.g.h.163.3		6	57.50	even	6
342.4.g.h.235.3		6	57.8	even	6
2166.4.a.t.1.3		3	1.1	even	1	trivial
2166.4.a.u.1.3		3	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.2543 q^{5} -6.00000 q^{6} +13.2543 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.2543 q^{5} -6.00000 q^{6} +13.2543 q^{7} -8.00000 q^{8} +9.00000 q^{9} -28.5086 q^{10} -65.9138 q^{11} +12.0000 q^{12} +68.5259 q^{13} -26.5086 q^{14} +42.7629 q^{15} +16.0000 q^{16} +99.6940 q^{17} -18.0000 q^{18} +57.0172 q^{20} +39.7629 q^{21} +131.828 q^{22} +3.53886 q^{23} -24.0000 q^{24} +78.1854 q^{25} -137.052 q^{26} +27.0000 q^{27} +53.0172 q^{28} +81.9008 q^{29} -85.5259 q^{30} +247.190 q^{31} -32.0000 q^{32} -197.741 q^{33} -199.388 q^{34} +188.931 q^{35} +36.0000 q^{36} -421.065 q^{37} +205.578 q^{39} -114.034 q^{40} +345.259 q^{41} -79.5259 q^{42} -366.168 q^{43} -263.655 q^{44} +128.289 q^{45} -7.07773 q^{46} -90.9399 q^{47} +48.0000 q^{48} -167.323 q^{49} -156.371 q^{50} +299.082 q^{51} +274.103 q^{52} +688.491 q^{53} -54.0000 q^{54} -939.556 q^{55} -106.034 q^{56} -163.802 q^{58} -182.289 q^{59} +171.052 q^{60} -0.517370 q^{61} -494.379 q^{62} +119.289 q^{63} +64.0000 q^{64} +976.789 q^{65} +395.483 q^{66} +159.862 q^{67} +398.776 q^{68} +10.6166 q^{69} -377.862 q^{70} +791.884 q^{71} -72.0000 q^{72} +322.129 q^{73} +842.130 q^{74} +234.556 q^{75} -873.642 q^{77} -411.155 q^{78} -318.371 q^{79} +228.069 q^{80} +81.0000 q^{81} -690.517 q^{82} +684.160 q^{83} +159.052 q^{84} +1421.07 q^{85} +732.336 q^{86} +245.703 q^{87} +527.311 q^{88} -720.406 q^{89} -256.578 q^{90} +908.263 q^{91} +14.1555 q^{92} +741.569 q^{93} +181.880 q^{94} -96.0000 q^{96} -415.392 q^{97} +334.646 q^{98} -593.224 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} - 20 q^{10} - 44 q^{11} + 36 q^{12} + 9 q^{13} - 14 q^{14} + 30 q^{15} + 48 q^{16} - 84 q^{17} - 54 q^{18} + 40 q^{20} + 21 q^{21} + 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} + 81 q^{27} + 28 q^{28} - 92 q^{29} - 60 q^{30} + 109 q^{31} - 96 q^{32} - 132 q^{33} + 168 q^{34} + 282 q^{35} + 108 q^{36} - 245 q^{37} + 27 q^{39} - 80 q^{40} + 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} + 4 q^{46} + 322 q^{47} + 144 q^{48} - 754 q^{49} + 166 q^{50} - 252 q^{51} + 36 q^{52} + 1322 q^{53} - 162 q^{54} - 248 q^{55} - 56 q^{56} + 184 q^{58} - 252 q^{59} + 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} + 1582 q^{65} + 264 q^{66} + 719 q^{67} - 336 q^{68} - 6 q^{69} - 564 q^{70} + 62 q^{71} - 216 q^{72} - 581 q^{73} + 490 q^{74} - 249 q^{75} - 204 q^{77} - 54 q^{78} + 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} + 84 q^{84} + 1632 q^{85} + 206 q^{86} - 276 q^{87} + 352 q^{88} - 1584 q^{89} - 180 q^{90} + 1573 q^{91} - 8 q^{92} + 327 q^{93} - 644 q^{94} - 288 q^{96} - 974 q^{97} + 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9 - 20 * q^10 - 44 * q^11 + 36 * q^12 + 9 * q^13 - 14 * q^14 + 30 * q^15 + 48 * q^16 - 84 * q^17 - 54 * q^18 + 40 * q^20 + 21 * q^21 + 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 + 81 * q^27 + 28 * q^28 - 92 * q^29 - 60 * q^30 + 109 * q^31 - 96 * q^32 - 132 * q^33 + 168 * q^34 + 282 * q^35 + 108 * q^36 - 245 * q^37 + 27 * q^39 - 80 * q^40 + 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 + 4 * q^46 + 322 * q^47 + 144 * q^48 - 754 * q^49 + 166 * q^50 - 252 * q^51 + 36 * q^52 + 1322 * q^53 - 162 * q^54 - 248 * q^55 - 56 * q^56 + 184 * q^58 - 252 * q^59 + 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 + 1582 * q^65 + 264 * q^66 + 719 * q^67 - 336 * q^68 - 6 * q^69 - 564 * q^70 + 62 * q^71 - 216 * q^72 - 581 * q^73 + 490 * q^74 - 249 * q^75 - 204 * q^77 - 54 * q^78 + 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 + 84 * q^84 + 1632 * q^85 + 206 * q^86 - 276 * q^87 + 352 * q^88 - 1584 * q^89 - 180 * q^90 + 1573 * q^91 - 8 * q^92 + 327 * q^93 - 644 * q^94 - 288 * q^96 - 974 * q^97 + 1508 * q^98 - 396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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