LMFDB - Embedded newform 2166.4.a.y.1.1

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:	$4$
Coefficient field:	4.4.8742025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 299x^{2} + 300x + 21780 $ x^4 - 2x^3 - 299x^2 + 300*x + 21780
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.6093$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -12.2274 q^{5} -6.00000 q^{6} +1.48084 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -12.2274 q^{5} -6.00000 q^{6} +1.48084 q^{7} -8.00000 q^{8} +9.00000 q^{9} +24.4547 q^{10} -15.6728 q^{11} +12.0000 q^{12} -55.5419 q^{13} -2.96168 q^{14} -36.6821 q^{15} +16.0000 q^{16} -46.4056 q^{17} -18.0000 q^{18} -48.9095 q^{20} +4.44252 q^{21} +31.3457 q^{22} -115.903 q^{23} -24.0000 q^{24} +24.5084 q^{25} +111.084 q^{26} +27.0000 q^{27} +5.92336 q^{28} +14.6836 q^{29} +73.3642 q^{30} -65.4837 q^{31} -32.0000 q^{32} -47.0185 q^{33} +92.8112 q^{34} -18.1068 q^{35} +36.0000 q^{36} -361.936 q^{37} -166.626 q^{39} +97.8189 q^{40} +329.442 q^{41} -8.88505 q^{42} -208.665 q^{43} -62.6913 q^{44} -110.046 q^{45} +231.806 q^{46} +435.867 q^{47} +48.0000 q^{48} -340.807 q^{49} -49.0168 q^{50} -139.217 q^{51} -222.168 q^{52} +114.582 q^{53} -54.0000 q^{54} +191.637 q^{55} -11.8467 q^{56} -29.3671 q^{58} +270.392 q^{59} -146.728 q^{60} +266.447 q^{61} +130.967 q^{62} +13.3276 q^{63} +64.0000 q^{64} +679.131 q^{65} +94.0370 q^{66} -811.655 q^{67} -185.622 q^{68} -347.708 q^{69} +36.2136 q^{70} +748.881 q^{71} -72.0000 q^{72} +57.8492 q^{73} +723.872 q^{74} +73.5252 q^{75} -23.2090 q^{77} +333.251 q^{78} -267.860 q^{79} -195.638 q^{80} +81.0000 q^{81} -658.884 q^{82} -494.100 q^{83} +17.7701 q^{84} +567.418 q^{85} +417.330 q^{86} +44.0507 q^{87} +125.383 q^{88} -508.926 q^{89} +220.093 q^{90} -82.2488 q^{91} -463.611 q^{92} -196.451 q^{93} -871.733 q^{94} -96.0000 q^{96} +103.253 q^{97} +681.614 q^{98} -141.055 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} - 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 + 12 * q^3 + 16 * q^4 - 24 * q^6 + 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} - 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9} + 83 q^{11} + 48 q^{12} + 5 q^{13} - 56 q^{14} + 64 q^{16} + 108 q^{17} - 72 q^{18} + 84 q^{21} - 166 q^{22} + 112 q^{23} - 96 q^{24} + 106 q^{25} - 10 q^{26} + 108 q^{27} + 112 q^{28} - 387 q^{29} + 419 q^{31} - 128 q^{32} + 249 q^{33} - 216 q^{34} + 576 q^{35} + 144 q^{36} + 68 q^{37} + 15 q^{39} - 47 q^{41} - 168 q^{42} - 244 q^{43} + 332 q^{44} - 224 q^{46} + 1019 q^{47} + 192 q^{48} - 450 q^{49} - 212 q^{50} + 324 q^{51} + 20 q^{52} + 111 q^{53} - 216 q^{54} + 263 q^{55} - 224 q^{56} + 774 q^{58} + 279 q^{59} + 206 q^{61} - 838 q^{62} + 252 q^{63} + 256 q^{64} + 3 q^{65} - 498 q^{66} - 587 q^{67} + 432 q^{68} + 336 q^{69} - 1152 q^{70} + 1161 q^{71} - 288 q^{72} + 445 q^{73} - 136 q^{74} + 318 q^{75} + 349 q^{77} - 30 q^{78} + 629 q^{79} + 324 q^{81} + 94 q^{82} + 403 q^{83} + 336 q^{84} - 725 q^{85} + 488 q^{86} - 1161 q^{87} - 664 q^{88} - 1598 q^{89} - 337 q^{91} + 448 q^{92} + 1257 q^{93} - 2038 q^{94} - 384 q^{96} - 1158 q^{97} + 900 q^{98} + 747 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 12 * q^3 + 16 * q^4 - 24 * q^6 + 28 * q^7 - 32 * q^8 + 36 * q^9 + 83 * q^11 + 48 * q^12 + 5 * q^13 - 56 * q^14 + 64 * q^16 + 108 * q^17 - 72 * q^18 + 84 * q^21 - 166 * q^22 + 112 * q^23 - 96 * q^24 + 106 * q^25 - 10 * q^26 + 108 * q^27 + 112 * q^28 - 387 * q^29 + 419 * q^31 - 128 * q^32 + 249 * q^33 - 216 * q^34 + 576 * q^35 + 144 * q^36 + 68 * q^37 + 15 * q^39 - 47 * q^41 - 168 * q^42 - 244 * q^43 + 332 * q^44 - 224 * q^46 + 1019 * q^47 + 192 * q^48 - 450 * q^49 - 212 * q^50 + 324 * q^51 + 20 * q^52 + 111 * q^53 - 216 * q^54 + 263 * q^55 - 224 * q^56 + 774 * q^58 + 279 * q^59 + 206 * q^61 - 838 * q^62 + 252 * q^63 + 256 * q^64 + 3 * q^65 - 498 * q^66 - 587 * q^67 + 432 * q^68 + 336 * q^69 - 1152 * q^70 + 1161 * q^71 - 288 * q^72 + 445 * q^73 - 136 * q^74 + 318 * q^75 + 349 * q^77 - 30 * q^78 + 629 * q^79 + 324 * q^81 + 94 * q^82 + 403 * q^83 + 336 * q^84 - 725 * q^85 + 488 * q^86 - 1161 * q^87 - 664 * q^88 - 1598 * q^89 - 337 * q^91 + 448 * q^92 + 1257 * q^93 - 2038 * q^94 - 384 * q^96 - 1158 * q^97 + 900 * q^98 + 747 * 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
$5$	−12.2274	−1.09365	−0.546824	−	0.837247i	$-0.684163\pi$
$5$	−12.2274	−1.09365	−0.546824	+	0.837247i	$0.684163\pi$
$6$	−6.00000	−0.408248
$7$	1.48084	0.0799579	0.0399790	−	0.999201i	$-0.487271\pi$
$7$	1.48084	0.0799579	0.0399790	+	0.999201i	$0.487271\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	24.4547	0.773326
$11$	−15.6728	−0.429594	−0.214797	−	0.976659i	$-0.568909\pi$
$11$	−15.6728	−0.429594	−0.214797	+	0.976659i	$0.568909\pi$
$12$	12.0000	0.288675
$13$	−55.5419	−1.18497	−0.592483	−	0.805583i	$-0.701852\pi$
$13$	−55.5419	−1.18497	−0.592483	+	0.805583i	$0.701852\pi$
$14$	−2.96168	−0.0565388
$15$	−36.6821	−0.631418
$16$	16.0000	0.250000
$17$	−46.4056	−0.662060	−0.331030	−	0.943620i	$-0.607396\pi$
$17$	−46.4056	−0.662060	−0.331030	+	0.943620i	$0.607396\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−48.9095	−0.546824
$21$	4.44252	0.0461637
$22$	31.3457	0.303769
$23$	−115.903	−1.05076	−0.525378	−	0.850869i	$-0.676076\pi$
$23$	−115.903	−1.05076	−0.525378	+	0.850869i	$0.676076\pi$
$24$	−24.0000	−0.204124
$25$	24.5084	0.196067
$26$	111.084	0.837897
$27$	27.0000	0.192450
$28$	5.92336	0.0399790
$29$	14.6836	0.0940230	0.0470115	−	0.998894i	$-0.485030\pi$
$29$	14.6836	0.0940230	0.0470115	+	0.998894i	$0.485030\pi$
$30$	73.3642	0.446480
$31$	−65.4837	−0.379394	−0.189697	−	0.981843i	$-0.560751\pi$
$31$	−65.4837	−0.379394	−0.189697	+	0.981843i	$0.560751\pi$
$32$	−32.0000	−0.176777
$33$	−47.0185	−0.248026
$34$	92.8112	0.468147
$35$	−18.1068	−0.0874459
$36$	36.0000	0.166667
$37$	−361.936	−1.60816	−0.804080	−	0.594521i	$-0.797342\pi$
$37$	−361.936	−1.60816	−0.804080	+	0.594521i	$0.797342\pi$
$38$	0	0
$39$	−166.626	−0.684140
$40$	97.8189	0.386663
$41$	329.442	1.25488	0.627441	−	0.778664i	$-0.284102\pi$
$41$	329.442	1.25488	0.627441	+	0.778664i	$0.284102\pi$
$42$	−8.88505	−0.0326427
$43$	−208.665	−0.740025	−0.370013	−	0.929027i	$-0.620647\pi$
$43$	−208.665	−0.740025	−0.370013	+	0.929027i	$0.620647\pi$
$44$	−62.6913	−0.214797
$45$	−110.046	−0.364550
$46$	231.806	0.742997
$47$	435.867	1.35272	0.676358	−	0.736573i	$-0.263557\pi$
$47$	435.867	1.35272	0.676358	+	0.736573i	$0.263557\pi$
$48$	48.0000	0.144338
$49$	−340.807	−0.993607
$50$	−49.0168	−0.138640
$51$	−139.217	−0.382240
$52$	−222.168	−0.592483
$53$	114.582	0.296964	0.148482	−	0.988915i	$-0.452561\pi$
$53$	114.582	0.296964	0.148482	+	0.988915i	$0.452561\pi$
$54$	−54.0000	−0.136083
$55$	191.637	0.469825
$56$	−11.8467	−0.0282694
$57$	0	0
$58$	−29.3671	−0.0664843
$59$	270.392	0.596644	0.298322	−	0.954465i	$-0.403573\pi$
$59$	270.392	0.596644	0.298322	+	0.954465i	$0.403573\pi$
$60$	−146.728	−0.315709
$61$	266.447	0.559264	0.279632	−	0.960107i	$-0.409788\pi$
$61$	266.447	0.559264	0.279632	+	0.960107i	$0.409788\pi$
$62$	130.967	0.268272
$63$	13.3276	0.0266526
$64$	64.0000	0.125000
$65$	679.131	1.29594
$66$	94.0370	0.175381
$67$	−811.655	−1.47999	−0.739996	−	0.672612i	$-0.765172\pi$
$67$	−811.655	−1.47999	−0.739996	+	0.672612i	$0.765172\pi$
$68$	−185.622	−0.331030
$69$	−347.708	−0.606654
$70$	36.2136	0.0618336
$71$	748.881	1.25177	0.625886	−	0.779914i	$-0.284737\pi$
$71$	748.881	1.25177	0.625886	+	0.779914i	$0.284737\pi$
$72$	−72.0000	−0.117851
$73$	57.8492	0.0927498	0.0463749	−	0.998924i	$-0.485233\pi$
$73$	57.8492	0.0927498	0.0463749	+	0.998924i	$0.485233\pi$
$74$	723.872	1.13714
$75$	73.5252	0.113199
$76$	0	0
$77$	−23.2090	−0.0343495
$78$	333.251	0.483760
$79$	−267.860	−0.381475	−0.190738	−	0.981641i	$-0.561088\pi$
$79$	−267.860	−0.381475	−0.190738	+	0.981641i	$0.561088\pi$
$80$	−195.638	−0.273412
$81$	81.0000	0.111111
$82$	−658.884	−0.887336
$83$	−494.100	−0.653428	−0.326714	−	0.945123i	$-0.605941\pi$
$83$	−494.100	−0.653428	−0.326714	+	0.945123i	$0.605941\pi$
$84$	17.7701	0.0230819
$85$	567.418	0.724061
$86$	417.330	0.523277
$87$	44.0507	0.0542842
$88$	125.383	0.151885
$89$	−508.926	−0.606135	−0.303068	−	0.952969i	$-0.598011\pi$
$89$	−508.926	−0.606135	−0.303068	+	0.952969i	$0.598011\pi$
$90$	220.093	0.257775
$91$	−82.2488	−0.0947474
$92$	−463.611	−0.525378
$93$	−196.451	−0.219043
$94$	−871.733	−0.956515
$95$	0	0
$96$	−96.0000	−0.102062
$97$	103.253	0.108080	0.0540399	−	0.998539i	$-0.482790\pi$
$97$	103.253	0.108080	0.0540399	+	0.998539i	$0.482790\pi$
$98$	681.614	0.702586
$99$	−141.055	−0.143198
$100$	98.0336	0.0980336
$101$	−569.310	−0.560876	−0.280438	−	0.959872i	$-0.590480\pi$
$101$	−569.310	−0.560876	−0.280438	+	0.959872i	$0.590480\pi$
$102$	278.434	0.270285
$103$	1295.77	1.23957	0.619787	−	0.784770i	$-0.287219\pi$
$103$	1295.77	1.23957	0.619787	+	0.784770i	$0.287219\pi$
$104$	444.335	0.418949
$105$	−54.3203	−0.0504869
$106$	−229.165	−0.209985
$107$	−1119.61	−1.01156	−0.505780	−	0.862662i	$-0.668795\pi$
$107$	−1119.61	−1.01156	−0.505780	+	0.862662i	$0.668795\pi$
$108$	108.000	0.0962250
$109$	−111.183	−0.0977008	−0.0488504	−	0.998806i	$-0.515556\pi$
$109$	−111.183	−0.0977008	−0.0488504	+	0.998806i	$0.515556\pi$
$110$	−383.275	−0.332217
$111$	−1085.81	−0.928472
$112$	23.6935	0.0199895
$113$	−73.1513	−0.0608982	−0.0304491	−	0.999536i	$-0.509694\pi$
$113$	−73.1513	−0.0608982	−0.0304491	+	0.999536i	$0.509694\pi$
$114$	0	0
$115$	1417.19	1.14916
$116$	58.7342	0.0470115
$117$	−499.877	−0.394989
$118$	−540.784	−0.421891
$119$	−68.7193	−0.0529369
$120$	293.457	0.223240
$121$	−1085.36	−0.815449
$122$	−532.895	−0.395459
$123$	988.326	0.724507
$124$	−261.935	−0.189697
$125$	1228.75	0.879220
$126$	−26.6551	−0.0188463
$127$	−1364.13	−0.953128	−0.476564	−	0.879140i	$-0.658118\pi$
$127$	−1364.13	−0.953128	−0.476564	+	0.879140i	$0.658118\pi$
$128$	−128.000	−0.0883883
$129$	−625.994	−0.427254
$130$	−1358.26	−0.916365
$131$	2080.94	1.38788	0.693940	−	0.720033i	$-0.255873\pi$
$131$	2080.94	1.38788	0.693940	+	0.720033i	$0.255873\pi$
$132$	−188.074	−0.124013
$133$	0	0
$134$	1623.31	1.04651
$135$	−330.139	−0.210473
$136$	371.245	0.234073
$137$	−741.651	−0.462508	−0.231254	−	0.972893i	$-0.574283\pi$
$137$	−741.651	−0.462508	−0.231254	+	0.972893i	$0.574283\pi$
$138$	695.417	0.428969
$139$	2254.22	1.37554	0.687771	−	0.725928i	$-0.258589\pi$
$139$	2254.22	1.37554	0.687771	+	0.725928i	$0.258589\pi$
$140$	−72.4271	−0.0437229
$141$	1307.60	0.780991
$142$	−1497.76	−0.885137
$143$	870.499	0.509054
$144$	144.000	0.0833333
$145$	−179.541	−0.102828
$146$	−115.698	−0.0655840
$147$	−1022.42	−0.573659
$148$	−1447.74	−0.804080
$149$	2973.55	1.63492	0.817458	−	0.575989i	$-0.195383\pi$
$149$	2973.55	1.63492	0.817458	+	0.575989i	$0.195383\pi$
$150$	−147.050	−0.0800441
$151$	−3252.13	−1.75268	−0.876341	−	0.481691i	$-0.840023\pi$
$151$	−3252.13	−1.75268	−0.876341	+	0.481691i	$0.840023\pi$
$152$	0	0
$153$	−417.650	−0.220687
$154$	46.4180	0.0242887
$155$	800.693	0.414924
$156$	−666.503	−0.342070
$157$	3456.25	1.75693	0.878467	−	0.477804i	$-0.158567\pi$
$157$	3456.25	1.75693	0.878467	+	0.477804i	$0.158567\pi$
$158$	535.719	0.269744
$159$	343.747	0.171452
$160$	391.276	0.193332
$161$	−171.634	−0.0840163
$162$	−162.000	−0.0785674
$163$	1749.28	0.840576	0.420288	−	0.907391i	$-0.361929\pi$
$163$	1749.28	0.840576	0.420288	+	0.907391i	$0.361929\pi$
$164$	1317.77	0.627441
$165$	574.912	0.271254
$166$	988.201	0.462044
$167$	−1518.46	−0.703605	−0.351802	−	0.936074i	$-0.614431\pi$
$167$	−1518.46	−0.703605	−0.351802	+	0.936074i	$0.614431\pi$
$168$	−35.5402	−0.0163213
$169$	887.904	0.404144
$170$	−1134.84	−0.511988
$171$	0	0
$172$	−834.659	−0.370013
$173$	−4232.55	−1.86009	−0.930044	−	0.367449i	$-0.880231\pi$
$173$	−4232.55	−1.86009	−0.930044	+	0.367449i	$0.880231\pi$
$174$	−88.1013	−0.0383847
$175$	36.2930	0.0156771
$176$	−250.765	−0.107399
$177$	811.175	0.344473
$178$	1017.85	0.428602
$179$	3913.11	1.63396	0.816982	−	0.576664i	$-0.195646\pi$
$179$	3913.11	1.63396	0.816982	+	0.576664i	$0.195646\pi$
$180$	−440.185	−0.182275
$181$	2252.25	0.924910	0.462455	−	0.886643i	$-0.346969\pi$
$181$	2252.25	0.924910	0.462455	+	0.886643i	$0.346969\pi$
$182$	164.498	0.0669965
$183$	799.342	0.322891
$184$	927.222	0.371498
$185$	4425.53	1.75876
$186$	392.902	0.154887
$187$	727.307	0.284417
$188$	1743.47	0.676358
$189$	39.9827	0.0153879
$190$	0	0
$191$	3077.97	1.16604	0.583021	−	0.812457i	$-0.301871\pi$
$191$	3077.97	1.16604	0.583021	+	0.812457i	$0.301871\pi$
$192$	192.000	0.0721688
$193$	119.488	0.0445644	0.0222822	−	0.999752i	$-0.492907\pi$
$193$	119.488	0.0445644	0.0222822	+	0.999752i	$0.492907\pi$
$194$	−206.506	−0.0764239
$195$	2037.39	0.748209
$196$	−1363.23	−0.496803
$197$	−1399.03	−0.505973	−0.252986	−	0.967470i	$-0.581413\pi$
$197$	−1399.03	−0.505973	−0.252986	+	0.967470i	$0.581413\pi$
$198$	282.111	0.101256
$199$	1441.45	0.513477	0.256739	−	0.966481i	$-0.417352\pi$
$199$	1441.45	0.513477	0.256739	+	0.966481i	$0.417352\pi$
$200$	−196.067	−0.0693202
$201$	−2434.96	−0.854473
$202$	1138.62	0.396599
$203$	21.7440	0.00751788
$204$	−556.867	−0.191120
$205$	−4028.21	−1.37240
$206$	−2591.54	−0.876511
$207$	−1043.12	−0.350252
$208$	−888.671	−0.296241
$209$	0	0
$210$	108.641	0.0356996
$211$	−1151.19	−0.375599	−0.187799	−	0.982207i	$-0.560136\pi$
$211$	−1151.19	−0.375599	−0.187799	+	0.982207i	$0.560136\pi$
$212$	458.329	0.148482
$213$	2246.64	0.722711
$214$	2239.22	0.715281
$215$	2551.42	0.809327
$216$	−216.000	−0.0680414
$217$	−96.9710	−0.0303356
$218$	222.366	0.0690849
$219$	173.548	0.0535491
$220$	766.550	0.234913
$221$	2577.46	0.784518
$222$	2171.62	0.656529
$223$	−4751.00	−1.42668	−0.713342	−	0.700816i	$-0.752820\pi$
$223$	−4751.00	−1.42668	−0.713342	+	0.700816i	$0.752820\pi$
$224$	−47.3869	−0.0141347
$225$	220.576	0.0653557
$226$	146.303	0.0430615
$227$	2329.07	0.680996	0.340498	−	0.940245i	$-0.389404\pi$
$227$	2329.07	0.680996	0.340498	+	0.940245i	$0.389404\pi$
$228$	0	0
$229$	−4645.41	−1.34051	−0.670256	−	0.742130i	$-0.733816\pi$
$229$	−4645.41	−1.34051	−0.670256	+	0.742130i	$0.733816\pi$
$230$	−2834.37	−0.812578
$231$	−69.6269	−0.0198317
$232$	−117.468	−0.0332422
$233$	6376.47	1.79286	0.896431	−	0.443184i	$-0.146151\pi$
$233$	6376.47	1.79286	0.896431	+	0.443184i	$0.146151\pi$
$234$	999.754	0.279299
$235$	−5329.50	−1.47940
$236$	1081.57	0.298322
$237$	−803.579	−0.220245
$238$	137.439	0.0374320
$239$	−1015.92	−0.274957	−0.137479	−	0.990505i	$-0.543900\pi$
$239$	−1015.92	−0.274957	−0.137479	+	0.990505i	$0.543900\pi$
$240$	−586.913	−0.157855
$241$	−1254.41	−0.335284	−0.167642	−	0.985848i	$-0.553615\pi$
$241$	−1254.41	−0.335284	−0.167642	+	0.985848i	$0.553615\pi$
$242$	2170.72	0.576609
$243$	243.000	0.0641500
$244$	1065.79	0.279632
$245$	4167.17	1.08666
$246$	−1976.65	−0.512304
$247$	0	0
$248$	523.870	0.134136
$249$	−1482.30	−0.377257
$250$	−2457.49	−0.621702
$251$	214.111	0.0538428	0.0269214	−	0.999638i	$-0.491430\pi$
$251$	214.111	0.0538428	0.0269214	+	0.999638i	$0.491430\pi$
$252$	53.3103	0.0133263
$253$	1816.52	0.451399
$254$	2728.27	0.673963
$255$	1702.25	0.418037
$256$	256.000	0.0625000
$257$	−6548.46	−1.58942	−0.794711	−	0.606988i	$-0.792378\pi$
$257$	−6548.46	−1.58942	−0.794711	+	0.606988i	$0.792378\pi$
$258$	1251.99	0.302114
$259$	−535.970	−0.128585
$260$	2716.52	0.647968
$261$	132.152	0.0313410
$262$	−4161.87	−0.981379
$263$	66.5987	0.0156146	0.00780732	−	0.999970i	$-0.497515\pi$
$263$	66.5987	0.0156146	0.00780732	+	0.999970i	$0.497515\pi$
$264$	376.148	0.0876906
$265$	−1401.04	−0.324774
$266$	0	0
$267$	−1526.78	−0.349952
$268$	−3246.62	−0.739996
$269$	5371.52	1.21750	0.608749	−	0.793363i	$-0.291672\pi$
$269$	5371.52	1.21750	0.608749	+	0.793363i	$0.291672\pi$
$270$	660.278	0.148827
$271$	4741.74	1.06288	0.531439	−	0.847097i	$-0.321651\pi$
$271$	4741.74	1.06288	0.531439	+	0.847097i	$0.321651\pi$
$272$	−742.490	−0.165515
$273$	−246.746	−0.0547024
$274$	1483.30	0.327042
$275$	−384.116	−0.0842293
$276$	−1390.83	−0.303327
$277$	6593.42	1.43018	0.715090	−	0.699032i	$-0.246386\pi$
$277$	6593.42	1.43018	0.715090	+	0.699032i	$0.246386\pi$
$278$	−4508.44	−0.972655
$279$	−589.353	−0.126465
$280$	144.854	0.0309168
$281$	−4174.03	−0.886127	−0.443064	−	0.896490i	$-0.646108\pi$
$281$	−4174.03	−0.886127	−0.443064	+	0.896490i	$0.646108\pi$
$282$	−2615.20	−0.552244
$283$	1186.00	0.249118	0.124559	−	0.992212i	$-0.460248\pi$
$283$	1186.00	0.249118	0.124559	+	0.992212i	$0.460248\pi$
$284$	2995.52	0.625886
$285$	0	0
$286$	−1741.00	−0.359956
$287$	487.851	0.100338
$288$	−288.000	−0.0589256
$289$	−2759.52	−0.561677
$290$	359.082	0.0727105
$291$	309.758	0.0623999
$292$	231.397	0.0463749
$293$	1051.74	0.209705	0.104852	−	0.994488i	$-0.466563\pi$
$293$	1051.74	0.209705	0.104852	+	0.994488i	$0.466563\pi$
$294$	2044.84	0.405638
$295$	−3306.18	−0.652519
$296$	2895.49	0.568571
$297$	−423.166	−0.0826755
$298$	−5947.09	−1.15606
$299$	6437.46	1.24511
$300$	294.101	0.0565997
$301$	−308.999	−0.0591709
$302$	6504.27	1.23933
$303$	−1707.93	−0.323822
$304$	0	0
$305$	−3257.95	−0.611638
$306$	835.301	0.156049
$307$	9728.81	1.80864	0.904320	−	0.426855i	$-0.140378\pi$
$307$	9728.81	1.80864	0.904320	+	0.426855i	$0.140378\pi$
$308$	−92.8359	−0.0171747
$309$	3887.31	0.715669
$310$	−1601.39	−0.293396
$311$	3860.94	0.703967	0.351984	−	0.936006i	$-0.385507\pi$
$311$	3860.94	0.703967	0.351984	+	0.936006i	$0.385507\pi$
$312$	1333.01	0.241880
$313$	−6056.57	−1.09373	−0.546865	−	0.837221i	$-0.684179\pi$
$313$	−6056.57	−1.09373	−0.546865	+	0.837221i	$0.684179\pi$
$314$	−6912.49	−1.24234
$315$	−162.961	−0.0291486
$316$	−1071.44	−0.190738
$317$	−5324.38	−0.943365	−0.471683	−	0.881768i	$-0.656353\pi$
$317$	−5324.38	−0.943365	−0.471683	+	0.881768i	$0.656353\pi$
$318$	−687.494	−0.121235
$319$	−230.133	−0.0403917
$320$	−782.551	−0.136706
$321$	−3358.83	−0.584024
$322$	343.267	0.0594085
$323$	0	0
$324$	324.000	0.0555556
$325$	−1361.24	−0.232333
$326$	−3498.55	−0.594377
$327$	−333.549	−0.0564076
$328$	−2635.54	−0.443668
$329$	645.449	0.108160
$330$	−1149.82	−0.191805
$331$	7210.00	1.19727	0.598636	−	0.801021i	$-0.295709\pi$
$331$	7210.00	1.19727	0.598636	+	0.801021i	$0.295709\pi$
$332$	−1976.40	−0.326714
$333$	−3257.43	−0.536054
$334$	3036.92	0.497524
$335$	9924.40	1.61859
$336$	71.0804	0.0115409
$337$	−6731.15	−1.08804	−0.544019	−	0.839073i	$-0.683098\pi$
$337$	−6731.15	−1.08804	−0.544019	+	0.839073i	$0.683098\pi$
$338$	−1775.81	−0.285773
$339$	−219.454	−0.0351596
$340$	2269.67	0.362030
$341$	1026.32	0.162986
$342$	0	0
$343$	−1012.61	−0.159405
$344$	1669.32	0.261638
$345$	4251.56	0.663467
$346$	8465.11	1.31528
$347$	−145.796	−0.0225555	−0.0112777	−	0.999936i	$-0.503590\pi$
$347$	−145.796	−0.0225555	−0.0112777	+	0.999936i	$0.503590\pi$
$348$	176.203	0.0271421
$349$	11163.5	1.71224	0.856118	−	0.516780i	$-0.172870\pi$
$349$	11163.5	1.71224	0.856118	+	0.516780i	$0.172870\pi$
$350$	−72.5861	−0.0110854
$351$	−1499.63	−0.228047
$352$	501.531	0.0759423
$353$	−6837.99	−1.03102	−0.515509	−	0.856884i	$-0.672397\pi$
$353$	−6837.99	−1.03102	−0.515509	+	0.856884i	$0.672397\pi$
$354$	−1622.35	−0.243579
$355$	−9156.84	−1.36900
$356$	−2035.70	−0.303068
$357$	−206.158	−0.0305631
$358$	−7826.22	−1.15539
$359$	6573.70	0.966425	0.483213	−	0.875503i	$-0.339470\pi$
$359$	6573.70	0.966425	0.483213	+	0.875503i	$0.339470\pi$
$360$	880.370	0.128888
$361$	0	0
$362$	−4504.51	−0.654010
$363$	−3256.09	−0.470800
$364$	−328.995	−0.0473737
$365$	−707.343	−0.101436
$366$	−1598.68	−0.228318
$367$	4684.90	0.666348	0.333174	−	0.942865i	$-0.391880\pi$
$367$	4684.90	0.666348	0.333174	+	0.942865i	$0.391880\pi$
$368$	−1854.44	−0.262689
$369$	2964.98	0.418294
$370$	−8851.05	−1.24363
$371$	169.678	0.0237446
$372$	−785.805	−0.109522
$373$	−6792.85	−0.942950	−0.471475	−	0.881880i	$-0.656278\pi$
$373$	−6792.85	−0.942950	−0.471475	+	0.881880i	$0.656278\pi$
$374$	−1454.61	−0.201113
$375$	3686.24	0.507618
$376$	−3486.93	−0.478258
$377$	−815.553	−0.111414
$378$	−79.9654	−0.0108809
$379$	−9449.62	−1.28072	−0.640362	−	0.768073i	$-0.721216\pi$
$379$	−9449.62	−1.28072	−0.640362	+	0.768073i	$0.721216\pi$
$380$	0	0
$381$	−4092.40	−0.550289
$382$	−6155.94	−0.824516
$383$	8933.34	1.19183	0.595917	−	0.803046i	$-0.296789\pi$
$383$	8933.34	1.19183	0.595917	+	0.803046i	$0.296789\pi$
$384$	−384.000	−0.0510310
$385$	283.785	0.0375662
$386$	−238.976	−0.0315118
$387$	−1877.98	−0.246675
$388$	413.011	0.0540399
$389$	−5252.23	−0.684573	−0.342286	−	0.939596i	$-0.611201\pi$
$389$	−5252.23	−0.684573	−0.342286	+	0.939596i	$0.611201\pi$
$390$	−4074.79	−0.529064
$391$	5378.54	0.695663
$392$	2726.46	0.351293
$393$	6242.81	0.801293
$394$	2798.06	0.357777
$395$	3275.22	0.417200
$396$	−564.222	−0.0715990
$397$	5441.21	0.687876	0.343938	−	0.938992i	$-0.388239\pi$
$397$	5441.21	0.687876	0.343938	+	0.938992i	$0.388239\pi$
$398$	−2882.91	−0.363083
$399$	0	0
$400$	392.134	0.0490168
$401$	4500.69	0.560483	0.280242	−	0.959930i	$-0.409585\pi$
$401$	4500.69	0.560483	0.280242	+	0.959930i	$0.409585\pi$
$402$	4869.93	0.604204
$403$	3637.09	0.449569
$404$	−2277.24	−0.280438
$405$	−990.416	−0.121517
$406$	−43.4880	−0.00531595
$407$	5672.57	0.690857
$408$	1113.73	0.135142
$409$	826.947	0.0999753	0.0499877	−	0.998750i	$-0.484082\pi$
$409$	826.947	0.0999753	0.0499877	+	0.998750i	$0.484082\pi$
$410$	8056.41	0.970434
$411$	−2224.95	−0.267029
$412$	5183.09	0.619787
$413$	400.407	0.0477064
$414$	2086.25	0.247666
$415$	6041.54	0.714621
$416$	1777.34	0.209474
$417$	6762.66	0.794169
$418$	0	0
$419$	−10791.6	−1.25825	−0.629124	−	0.777305i	$-0.716586\pi$
$419$	−10791.6	−1.25825	−0.629124	+	0.777305i	$0.716586\pi$
$420$	−217.281	−0.0252434
$421$	15959.3	1.84753	0.923764	−	0.382962i	$-0.125096\pi$
$421$	15959.3	1.84753	0.923764	+	0.382962i	$0.125096\pi$
$422$	2302.39	0.265589
$423$	3922.80	0.450906
$424$	−916.659	−0.104993
$425$	−1137.33	−0.129808
$426$	−4493.29	−0.511034
$427$	394.566	0.0447175
$428$	−4478.45	−0.505780
$429$	2611.50	0.293903
$430$	−5102.84	−0.572281
$431$	−8182.27	−0.914446	−0.457223	−	0.889352i	$-0.651156\pi$
$431$	−8182.27	−0.914446	−0.457223	+	0.889352i	$0.651156\pi$
$432$	432.000	0.0481125
$433$	15570.4	1.72810	0.864049	−	0.503407i	$-0.167920\pi$
$433$	15570.4	1.72810	0.864049	+	0.503407i	$0.167920\pi$
$434$	193.942	0.0214505
$435$	−538.623	−0.0593679
$436$	−444.731	−0.0488504
$437$	0	0
$438$	−347.095	−0.0378649
$439$	6609.99	0.718628	0.359314	−	0.933217i	$-0.383011\pi$
$439$	6609.99	0.718628	0.359314	+	0.933217i	$0.383011\pi$
$440$	−1533.10	−0.166108
$441$	−3067.26	−0.331202
$442$	−5154.91	−0.554738
$443$	9396.43	1.00776	0.503880	−	0.863774i	$-0.331905\pi$
$443$	9396.43	1.00776	0.503880	+	0.863774i	$0.331905\pi$
$444$	−4343.23	−0.464236
$445$	6222.82	0.662899
$446$	9502.01	1.00882
$447$	8920.64	0.943919
$448$	94.7738	0.00999474
$449$	10534.5	1.10724	0.553621	−	0.832769i	$-0.313246\pi$
$449$	10534.5	1.10724	0.553621	+	0.832769i	$0.313246\pi$
$450$	−441.151	−0.0462135
$451$	−5163.29	−0.539090
$452$	−292.605	−0.0304491
$453$	−9756.40	−1.01191
$454$	−4658.15	−0.481537
$455$	1005.69	0.103620
$456$	0	0
$457$	−4427.98	−0.453244	−0.226622	−	0.973983i	$-0.572768\pi$
$457$	−4427.98	−0.453244	−0.226622	+	0.973983i	$0.572768\pi$
$458$	9290.81	0.947885
$459$	−1252.95	−0.127413
$460$	5668.74	0.574579
$461$	15108.4	1.52640	0.763200	−	0.646162i	$-0.223627\pi$
$461$	15108.4	1.52640	0.763200	+	0.646162i	$0.223627\pi$
$462$	139.254	0.0140231
$463$	2088.43	0.209628	0.104814	−	0.994492i	$-0.466575\pi$
$463$	2088.43	0.209628	0.104814	+	0.994492i	$0.466575\pi$
$464$	234.937	0.0235058
$465$	2402.08	0.239556
$466$	−12752.9	−1.26774
$467$	−1755.64	−0.173964	−0.0869822	−	0.996210i	$-0.527722\pi$
$467$	−1755.64	−0.173964	−0.0869822	+	0.996210i	$0.527722\pi$
$468$	−1999.51	−0.197494
$469$	−1201.93	−0.118337
$470$	10659.0	1.04609
$471$	10368.7	1.01437
$472$	−2163.13	−0.210946
$473$	3270.37	0.317911
$474$	1607.16	0.155737
$475$	0	0
$476$	−274.877	−0.0264685
$477$	1031.24	0.0989880
$478$	2031.85	0.194424
$479$	8453.24	0.806343	0.403172	−	0.915124i	$-0.367908\pi$
$479$	8453.24	0.806343	0.403172	+	0.915124i	$0.367908\pi$
$480$	1173.83	0.111620
$481$	20102.6	1.90562
$482$	2508.82	0.237082
$483$	−514.901	−0.0485068
$484$	−4341.45	−0.407724
$485$	−1262.51	−0.118201
$486$	−486.000	−0.0453609
$487$	13797.8	1.28385	0.641926	−	0.766767i	$-0.278136\pi$
$487$	13797.8	1.28385	0.641926	+	0.766767i	$0.278136\pi$
$488$	−2131.58	−0.197730
$489$	5247.83	0.485307
$490$	−8334.34	−0.768382
$491$	−2503.98	−0.230148	−0.115074	−	0.993357i	$-0.536711\pi$
$491$	−2503.98	−0.230148	−0.115074	+	0.993357i	$0.536711\pi$
$492$	3953.30	0.362253
$493$	−681.399	−0.0622488
$494$	0	0
$495$	1724.74	0.156608
$496$	−1047.74	−0.0948486
$497$	1108.97	0.100089
$498$	2964.60	0.266761
$499$	−10265.7	−0.920956	−0.460478	−	0.887671i	$-0.652322\pi$
$499$	−10265.7	−0.920956	−0.460478	+	0.887671i	$0.652322\pi$
$500$	4914.99	0.439610
$501$	−4555.38	−0.406226
$502$	−428.221	−0.0380726
$503$	20702.9	1.83518	0.917590	−	0.397527i	$-0.130132\pi$
$503$	20702.9	1.83518	0.917590	+	0.397527i	$0.130132\pi$
$504$	−106.621	−0.00942313
$505$	6961.16	0.613401
$506$	−3633.05	−0.319187
$507$	2663.71	0.233333
$508$	−5456.53	−0.476564
$509$	1920.35	0.167226	0.0836130	−	0.996498i	$-0.473354\pi$
$509$	1920.35	0.167226	0.0836130	+	0.996498i	$0.473354\pi$
$510$	−3404.51	−0.295596
$511$	85.6655	0.00741608
$512$	−512.000	−0.0441942
$513$	0	0
$514$	13096.9	1.12389
$515$	−15843.9	−1.35566
$516$	−2503.98	−0.213627
$517$	−6831.27	−0.581119
$518$	1071.94	0.0909234
$519$	−12697.7	−1.07392
$520$	−5433.05	−0.458183
$521$	−21087.3	−1.77322	−0.886612	−	0.462514i	$-0.846947\pi$
$521$	−21087.3	−1.77322	−0.886612	+	0.462514i	$0.846947\pi$
$522$	−264.304	−0.0221614
$523$	12496.6	1.04482	0.522408	−	0.852696i	$-0.325034\pi$
$523$	12496.6	1.04482	0.522408	+	0.852696i	$0.325034\pi$
$524$	8323.75	0.693940
$525$	108.879	0.00905119
$526$	−133.197	−0.0110412
$527$	3038.81	0.251182
$528$	−752.296	−0.0620066
$529$	1266.45	0.104089
$530$	2802.08	0.229650
$531$	2433.53	0.198881
$532$	0	0
$533$	−18297.8	−1.48699
$534$	3053.56	0.247454
$535$	13689.9	1.10629
$536$	6493.24	0.523256
$537$	11739.3	0.943369
$538$	−10743.0	−0.860901
$539$	5341.41	0.426848
$540$	−1320.56	−0.105236
$541$	−8373.02	−0.665405	−0.332703	−	0.943032i	$-0.607961\pi$
$541$	−8373.02	−0.665405	−0.332703	+	0.943032i	$0.607961\pi$
$542$	−9483.47	−0.751568
$543$	6756.76	0.533997
$544$	1484.98	0.117037
$545$	1359.47	0.106850
$546$	493.493	0.0386805
$547$	−4295.65	−0.335774	−0.167887	−	0.985806i	$-0.553694\pi$
$547$	−4295.65	−0.335774	−0.167887	+	0.985806i	$0.553694\pi$
$548$	−2966.61	−0.231254
$549$	2398.03	0.186421
$550$	768.232	0.0595591
$551$	0	0
$552$	2781.67	0.214485
$553$	−396.658	−0.0305020
$554$	−13186.8	−1.01129
$555$	13276.6	1.01542
$556$	9016.87	0.687771
$557$	25616.4	1.94865	0.974327	−	0.225136i	$-0.0722825\pi$
$557$	25616.4	1.94865	0.974327	+	0.225136i	$0.0722825\pi$
$558$	1178.71	0.0894241
$559$	11589.6	0.876904
$560$	−289.709	−0.0218615
$561$	2181.92	0.164208
$562$	8348.06	0.626587
$563$	21503.6	1.60971	0.804855	−	0.593471i	$-0.202243\pi$
$563$	21503.6	1.60971	0.804855	+	0.593471i	$0.202243\pi$
$564$	5230.40	0.390496
$565$	894.447	0.0666012
$566$	−2372.00	−0.176153
$567$	119.948	0.00888421
$568$	−5991.05	−0.442568
$569$	1149.78	0.0847123	0.0423562	−	0.999103i	$-0.486514\pi$
$569$	1149.78	0.0847123	0.0423562	+	0.999103i	$0.486514\pi$
$570$	0	0
$571$	16807.2	1.23180	0.615902	−	0.787823i	$-0.288792\pi$
$571$	16807.2	1.23180	0.615902	+	0.787823i	$0.288792\pi$
$572$	3482.00	0.254527
$573$	9233.91	0.673215
$574$	−975.702	−0.0709495
$575$	−2840.59	−0.206019
$576$	576.000	0.0416667
$577$	−1755.53	−0.126661	−0.0633307	−	0.997993i	$-0.520172\pi$
$577$	−1755.53	−0.126661	−0.0633307	+	0.997993i	$0.520172\pi$
$578$	5519.04	0.397166
$579$	358.464	0.0257293
$580$	−718.165	−0.0514141
$581$	−731.684	−0.0522468
$582$	−619.517	−0.0441234
$583$	−1795.83	−0.127574
$584$	−462.793	−0.0327920
$585$	6112.18	0.431979
$586$	−2103.49	−0.148284
$587$	−25597.7	−1.79988	−0.899939	−	0.436015i	$-0.856390\pi$
$587$	−25597.7	−1.79988	−0.899939	+	0.436015i	$0.856390\pi$
$588$	−4089.69	−0.286830
$589$	0	0
$590$	6612.36	0.461401
$591$	−4197.09	−0.292124
$592$	−5790.98	−0.402040
$593$	3782.07	0.261907	0.130953	−	0.991389i	$-0.458196\pi$
$593$	3782.07	0.261907	0.130953	+	0.991389i	$0.458196\pi$
$594$	846.333	0.0584604
$595$	840.256	0.0578944
$596$	11894.2	0.817458
$597$	4324.36	0.296456
$598$	−12874.9	−0.880426
$599$	−1776.50	−0.121178	−0.0605892	−	0.998163i	$-0.519298\pi$
$599$	−1776.50	−0.121178	−0.0605892	+	0.998163i	$0.519298\pi$
$600$	−588.202	−0.0400220
$601$	−7805.84	−0.529795	−0.264897	−	0.964277i	$-0.585338\pi$
$601$	−7805.84	−0.529795	−0.264897	+	0.964277i	$0.585338\pi$
$602$	617.999	0.0418401
$603$	−7304.89	−0.493330
$604$	−13008.5	−0.876341
$605$	13271.1	0.891814
$606$	3415.86	0.228976
$607$	−10926.2	−0.730610	−0.365305	−	0.930888i	$-0.619035\pi$
$607$	−10926.2	−0.730610	−0.365305	+	0.930888i	$0.619035\pi$
$608$	0	0
$609$	65.2320	0.00434045
$610$	6515.89	0.432493
$611$	−24208.9	−1.60292
$612$	−1670.60	−0.110343
$613$	−22866.5	−1.50664	−0.753320	−	0.657654i	$-0.771549\pi$
$613$	−22866.5	−1.50664	−0.753320	+	0.657654i	$0.771549\pi$
$614$	−19457.6	−1.27890
$615$	−12084.6	−0.792356
$616$	185.672	0.0121444
$617$	14830.6	0.967678	0.483839	−	0.875157i	$-0.339242\pi$
$617$	14830.6	0.967678	0.483839	+	0.875157i	$0.339242\pi$
$618$	−7774.63	−0.506054
$619$	21096.2	1.36984	0.684919	−	0.728619i	$-0.259838\pi$
$619$	21096.2	1.36984	0.684919	+	0.728619i	$0.259838\pi$
$620$	3202.77	0.207462
$621$	−3129.37	−0.202218
$622$	−7721.88	−0.497780
$623$	−753.639	−0.0484653
$624$	−2666.01	−0.171035
$625$	−18087.9	−1.15762
$626$	12113.1	0.773384
$627$	0	0
$628$	13825.0	0.878467
$629$	16795.9	1.06470
$630$	325.922	0.0206112
$631$	1069.86	0.0674965	0.0337483	−	0.999430i	$-0.489256\pi$
$631$	1069.86	0.0674965	0.0337483	+	0.999430i	$0.489256\pi$
$632$	2142.88	0.134872
$633$	−3453.58	−0.216852
$634$	10648.8	0.667060
$635$	16679.8	1.04239
$636$	1374.99	0.0857262
$637$	18929.1	1.17739
$638$	460.266	0.0285613
$639$	6739.93	0.417257
$640$	1565.10	0.0966658
$641$	−6912.37	−0.425932	−0.212966	−	0.977060i	$-0.568312\pi$
$641$	−6912.37	−0.425932	−0.212966	+	0.977060i	$0.568312\pi$
$642$	6717.67	0.412968
$643$	−1813.07	−0.111198	−0.0555992	−	0.998453i	$-0.517707\pi$
$643$	−1813.07	−0.111198	−0.0555992	+	0.998453i	$0.517707\pi$
$644$	−686.534	−0.0420081
$645$	7654.26	0.467265
$646$	0	0
$647$	13986.2	0.849854	0.424927	−	0.905228i	$-0.360300\pi$
$647$	13986.2	0.849854	0.424927	+	0.905228i	$0.360300\pi$
$648$	−648.000	−0.0392837
$649$	−4237.81	−0.256315
$650$	2722.49	0.164284
$651$	−290.913	−0.0175142
$652$	6997.11	0.420288
$653$	−18234.3	−1.09275	−0.546374	−	0.837541i	$-0.683992\pi$
$653$	−18234.3	−1.09275	−0.546374	+	0.837541i	$0.683992\pi$
$654$	667.097	0.0398862
$655$	−25444.4	−1.51785
$656$	5271.07	0.313721
$657$	520.643	0.0309166
$658$	−1290.90	−0.0764810
$659$	18752.7	1.10850	0.554249	−	0.832351i	$-0.313005\pi$
$659$	18752.7	1.10850	0.554249	+	0.832351i	$0.313005\pi$
$660$	2299.65	0.135627
$661$	−19059.7	−1.12154	−0.560768	−	0.827973i	$-0.689494\pi$
$661$	−19059.7	−1.12154	−0.560768	+	0.827973i	$0.689494\pi$
$662$	−14420.0	−0.846600
$663$	7732.37	0.452942
$664$	3952.80	0.231022
$665$	0	0
$666$	6514.85	0.379047
$667$	−1701.86	−0.0987953
$668$	−6073.84	−0.351802
$669$	−14253.0	−0.823697
$670$	−19848.8	−1.14452
$671$	−4175.98	−0.240256
$672$	−142.161	−0.00816067
$673$	−31202.8	−1.78719	−0.893594	−	0.448875i	$-0.851825\pi$
$673$	−31202.8	−1.78719	−0.893594	+	0.448875i	$0.851825\pi$
$674$	13462.3	0.769360
$675$	661.727	0.0377331
$676$	3551.62	0.202072
$677$	29253.3	1.66071	0.830353	−	0.557238i	$-0.188139\pi$
$677$	29253.3	1.66071	0.830353	+	0.557238i	$0.188139\pi$
$678$	438.908	0.0248616
$679$	152.901	0.00864183
$680$	−4539.35	−0.255994
$681$	6987.22	0.393173
$682$	−2052.63	−0.115248
$683$	−13862.3	−0.776614	−0.388307	−	0.921530i	$-0.626940\pi$
$683$	−13862.3	−0.776614	−0.388307	+	0.921530i	$0.626940\pi$
$684$	0	0
$685$	9068.44	0.505821
$686$	2025.22	0.112716
$687$	−13936.2	−0.773944
$688$	−3338.64	−0.185006
$689$	−6364.12	−0.351892
$690$	−8503.11	−0.469142
$691$	−8334.09	−0.458818	−0.229409	−	0.973330i	$-0.573679\pi$
$691$	−8334.09	−0.458818	−0.229409	+	0.973330i	$0.573679\pi$
$692$	−16930.2	−0.930044
$693$	−208.881	−0.0114498
$694$	291.592	0.0159491
$695$	−27563.1	−1.50436
$696$	−352.405	−0.0191924
$697$	−15288.0	−0.830807
$698$	−22327.1	−1.21073
$699$	19129.4	1.03511
$700$	145.172	0.00783856
$701$	22201.0	1.19618	0.598089	−	0.801430i	$-0.295927\pi$
$701$	22201.0	1.19618	0.598089	+	0.801430i	$0.295927\pi$
$702$	2999.26	0.161253
$703$	0	0
$704$	−1003.06	−0.0536993
$705$	−15988.5	−0.854130
$706$	13676.0	0.729040
$707$	−843.057	−0.0448464
$708$	3244.70	0.172236
$709$	21170.7	1.12141	0.560706	−	0.828015i	$-0.310530\pi$
$709$	21170.7	1.12141	0.560706	+	0.828015i	$0.310530\pi$
$710$	18313.7	0.968028
$711$	−2410.74	−0.127158
$712$	4071.41	0.214301
$713$	7589.74	0.398651
$714$	412.316	0.0216114
$715$	−10643.9	−0.556727
$716$	15652.4	0.816982
$717$	−3047.77	−0.158747
$718$	−13147.4	−0.683366
$719$	12898.8	0.669048	0.334524	−	0.942387i	$-0.391424\pi$
$719$	12898.8	0.669048	0.334524	+	0.942387i	$0.391424\pi$
$720$	−1760.74	−0.0911374
$721$	1918.83	0.0991138
$722$	0	0
$723$	−3763.22	−0.193576
$724$	9009.02	0.462455
$725$	359.870	0.0184348
$726$	6512.17	0.332906
$727$	−19872.7	−1.01380	−0.506902	−	0.862003i	$-0.669210\pi$
$727$	−19872.7	−1.01380	−0.506902	+	0.862003i	$0.669210\pi$
$728$	657.990	0.0334983
$729$	729.000	0.0370370
$730$	1414.69	0.0717259
$731$	9683.22	0.489941
$732$	3197.37	0.161445
$733$	−17941.9	−0.904092	−0.452046	−	0.891995i	$-0.649306\pi$
$733$	−17941.9	−0.904092	−0.452046	+	0.891995i	$0.649306\pi$
$734$	−9369.80	−0.471179
$735$	12501.5	0.627381
$736$	3708.89	0.185749
$737$	12720.9	0.635796
$738$	−5929.95	−0.295779
$739$	−25466.4	−1.26766	−0.633828	−	0.773474i	$-0.718517\pi$
$739$	−25466.4	−1.26766	−0.633828	+	0.773474i	$0.718517\pi$
$740$	17702.1	0.879381
$741$	0	0
$742$	−339.357	−0.0167900
$743$	−8569.65	−0.423136	−0.211568	−	0.977363i	$-0.567857\pi$
$743$	−8569.65	−0.423136	−0.211568	+	0.977363i	$0.567857\pi$
$744$	1571.61	0.0774435
$745$	−36358.6	−1.78802
$746$	13585.7	0.666766
$747$	−4446.90	−0.217809
$748$	2909.23	0.142208
$749$	−1657.97	−0.0808822
$750$	−7372.48	−0.358940
$751$	20228.3	0.982876	0.491438	−	0.870912i	$-0.336471\pi$
$751$	20228.3	0.982876	0.491438	+	0.870912i	$0.336471\pi$
$752$	6973.87	0.338179
$753$	642.332	0.0310861
$754$	1631.11	0.0787816
$755$	39765.0	1.91682
$756$	159.931	0.00769395
$757$	9587.32	0.460313	0.230157	−	0.973154i	$-0.426076\pi$
$757$	9587.32	0.460313	0.230157	+	0.973154i	$0.426076\pi$
$758$	18899.2	0.905609
$759$	5449.57	0.260615
$760$	0	0
$761$	−23969.6	−1.14179	−0.570893	−	0.821025i	$-0.693403\pi$
$761$	−23969.6	−1.14179	−0.570893	+	0.821025i	$0.693403\pi$
$762$	8184.80	0.389113
$763$	−164.644	−0.00781195
$764$	12311.9	0.583021
$765$	5106.76	0.241354
$766$	−17866.7	−0.842754
$767$	−15018.1	−0.707003
$768$	768.000	0.0360844
$769$	2945.14	0.138107	0.0690536	−	0.997613i	$-0.478002\pi$
$769$	2945.14	0.138107	0.0690536	+	0.997613i	$0.478002\pi$
$770$	−567.569	−0.0265633
$771$	−19645.4	−0.917654
$772$	477.952	0.0222822
$773$	6646.64	0.309266	0.154633	−	0.987972i	$-0.450580\pi$
$773$	6646.64	0.309266	0.154633	+	0.987972i	$0.450580\pi$
$774$	3755.97	0.174426
$775$	−1604.90	−0.0743868
$776$	−826.023	−0.0382120
$777$	−1607.91	−0.0742387
$778$	10504.5	0.484066
$779$	0	0
$780$	8149.57	0.374105
$781$	−11737.1	−0.537754
$782$	−10757.1	−0.491908
$783$	396.456	0.0180947
$784$	−5452.91	−0.248402
$785$	−42260.8	−1.92147
$786$	−12485.6	−0.566600
$787$	33502.0	1.51743	0.758716	−	0.651422i	$-0.225827\pi$
$787$	33502.0	1.51743	0.758716	+	0.651422i	$0.225827\pi$
$788$	−5596.11	−0.252986
$789$	199.796	0.00901512
$790$	−6550.43	−0.295005
$791$	−108.325	−0.00486929
$792$	1128.44	0.0506282
$793$	−14799.0	−0.662708
$794$	−10882.4	−0.486401
$795$	−4203.12	−0.187509
$796$	5765.81	0.256739
$797$	−18882.6	−0.839216	−0.419608	−	0.907705i	$-0.637833\pi$
$797$	−18882.6	−0.839216	−0.419608	+	0.907705i	$0.637833\pi$
$798$	0	0
$799$	−20226.7	−0.895579
$800$	−784.269	−0.0346601
$801$	−4580.33	−0.202045
$802$	−9001.38	−0.396321
$803$	−906.661	−0.0398448
$804$	−9739.86	−0.427237
$805$	2098.63	0.0918843
$806$	−7274.18	−0.317893
$807$	16114.5	0.702923
$808$	4554.48	0.198299
$809$	−19583.4	−0.851068	−0.425534	−	0.904942i	$-0.639914\pi$
$809$	−19583.4	−0.851068	−0.425534	+	0.904942i	$0.639914\pi$
$810$	1980.83	0.0859251
$811$	−29726.8	−1.28711	−0.643557	−	0.765398i	$-0.722542\pi$
$811$	−29726.8	−1.28711	−0.643557	+	0.765398i	$0.722542\pi$
$812$	86.9760	0.00375894
$813$	14225.2	0.613653
$814$	−11345.1	−0.488509
$815$	−21389.0	−0.919295
$816$	−2227.47	−0.0955601
$817$	0	0
$818$	−1653.89	−0.0706932
$819$	−740.239	−0.0315825
$820$	−16112.8	−0.686200
$821$	−7951.37	−0.338008	−0.169004	−	0.985615i	$-0.554055\pi$
$821$	−7951.37	−0.338008	−0.169004	+	0.985615i	$0.554055\pi$
$822$	4449.91	0.188818
$823$	114.530	0.00485086	0.00242543	−	0.999997i	$-0.499228\pi$
$823$	114.530	0.00485086	0.00242543	+	0.999997i	$0.499228\pi$
$824$	−10366.2	−0.438256
$825$	−1152.35	−0.0486298
$826$	−800.815	−0.0337335
$827$	−9948.36	−0.418305	−0.209153	−	0.977883i	$-0.567071\pi$
$827$	−9948.36	−0.418305	−0.209153	+	0.977883i	$0.567071\pi$
$828$	−4172.50	−0.175126
$829$	−22652.3	−0.949030	−0.474515	−	0.880247i	$-0.657377\pi$
$829$	−22652.3	−0.949030	−0.474515	+	0.880247i	$0.657377\pi$
$830$	−12083.1	−0.505313
$831$	19780.2	0.825715
$832$	−3554.68	−0.148121
$833$	15815.4	0.657827
$834$	−13525.3	−0.561563
$835$	18566.8	0.769496
$836$	0	0
$837$	−1768.06	−0.0730145
$838$	21583.3	0.889716
$839$	−15033.9	−0.618626	−0.309313	−	0.950960i	$-0.600099\pi$
$839$	−15033.9	−0.618626	−0.309313	+	0.950960i	$0.600099\pi$
$840$	434.563	0.0178498
$841$	−24173.4	−0.991160
$842$	−31918.6	−1.30640
$843$	−12522.1	−0.511606
$844$	−4604.77	−0.187799
$845$	−10856.7	−0.441991
$846$	−7845.60	−0.318838
$847$	−1607.25	−0.0652016
$848$	1833.32	0.0742410
$849$	3558.00	0.143828
$850$	2274.65	0.0917882
$851$	41949.4	1.68979
$852$	8986.57	0.361356
$853$	−33056.8	−1.32690	−0.663448	−	0.748222i	$-0.730908\pi$
$853$	−33056.8	−1.32690	−0.663448	+	0.748222i	$0.730908\pi$
$854$	−789.132	−0.0316201
$855$	0	0
$856$	8956.89	0.357640
$857$	−1801.04	−0.0717880	−0.0358940	−	0.999356i	$-0.511428\pi$
$857$	−1801.04	−0.0717880	−0.0358940	+	0.999356i	$0.511428\pi$
$858$	−5222.99	−0.207821
$859$	−11756.3	−0.466962	−0.233481	−	0.972361i	$-0.575012\pi$
$859$	−11756.3	−0.466962	−0.233481	+	0.972361i	$0.575012\pi$
$860$	10205.7	0.404664
$861$	1463.55	0.0579300
$862$	16364.5	0.646611
$863$	−24141.9	−0.952258	−0.476129	−	0.879375i	$-0.657961\pi$
$863$	−24141.9	−0.952258	−0.476129	+	0.879375i	$0.657961\pi$
$864$	−864.000	−0.0340207
$865$	51753.0	2.03428
$866$	−31140.8	−1.22195
$867$	−8278.56	−0.324284
$868$	−387.884	−0.0151678
$869$	4198.12	0.163880
$870$	1077.25	0.0419794
$871$	45080.9	1.75374
$872$	889.463	0.0345424
$873$	929.275	0.0360266
$874$	0	0
$875$	1819.58	0.0703006
$876$	694.190	0.0267746
$877$	7778.93	0.299516	0.149758	−	0.988723i	$-0.452150\pi$
$877$	7778.93	0.299516	0.149758	+	0.988723i	$0.452150\pi$
$878$	−13220.0	−0.508147
$879$	3155.23	0.121073
$880$	3066.20	0.117456
$881$	5507.50	0.210616	0.105308	−	0.994440i	$-0.466417\pi$
$881$	5507.50	0.210616	0.105308	+	0.994440i	$0.466417\pi$
$882$	6134.53	0.234195
$883$	21807.2	0.831110	0.415555	−	0.909568i	$-0.363587\pi$
$883$	21807.2	0.831110	0.415555	+	0.909568i	$0.363587\pi$
$884$	10309.8	0.392259
$885$	−9918.54	−0.376732
$886$	−18792.9	−0.712594
$887$	4696.26	0.177773	0.0888866	−	0.996042i	$-0.471669\pi$
$887$	4696.26	0.177773	0.0888866	+	0.996042i	$0.471669\pi$
$888$	8686.47	0.328264
$889$	−2020.07	−0.0762101
$890$	−12445.6	−0.468740
$891$	−1269.50	−0.0477327
$892$	−19004.0	−0.713342
$893$	0	0
$894$	−17841.3	−0.667451
$895$	−47847.0	−1.78698
$896$	−189.548	−0.00706735
$897$	19312.4	0.718865
$898$	−21068.9	−0.782938
$899$	−961.534	−0.0356718
$900$	882.302	0.0326779
$901$	−5317.26	−0.196608
$902$	10326.6	0.381194
$903$	−926.998	−0.0341623
$904$	585.210	0.0215308
$905$	−27539.1	−1.01153
$906$	19512.8	0.715529
$907$	−44727.1	−1.63742	−0.818710	−	0.574207i	$-0.805310\pi$
$907$	−44727.1	−1.63742	−0.818710	+	0.574207i	$0.805310\pi$
$908$	9316.29	0.340498
$909$	−5123.79	−0.186959
$910$	−2011.37	−0.0732706
$911$	13419.5	0.488043	0.244021	−	0.969770i	$-0.421533\pi$
$911$	13419.5	0.488043	0.244021	+	0.969770i	$0.421533\pi$
$912$	0	0
$913$	7743.95	0.280709
$914$	8855.97	0.320492
$915$	−9773.84	−0.353129
$916$	−18581.6	−0.670256
$917$	3081.54	0.110972
$918$	2505.90	0.0900949
$919$	36503.7	1.31028	0.655139	−	0.755508i	$-0.272610\pi$
$919$	36503.7	1.31028	0.655139	+	0.755508i	$0.272610\pi$
$920$	−11337.5	−0.406289
$921$	29186.4	1.04422
$922$	−30216.9	−1.07933
$923$	−41594.3	−1.48331
$924$	−278.508	−0.00991583
$925$	−8870.48	−0.315308
$926$	−4176.86	−0.148229
$927$	11661.9	0.413191
$928$	−469.874	−0.0166211
$929$	−25619.7	−0.904797	−0.452399	−	0.891816i	$-0.649432\pi$
$929$	−25619.7	−0.904797	−0.452399	+	0.891816i	$0.649432\pi$
$930$	−4804.16	−0.169392
$931$	0	0
$932$	25505.9	0.896431
$933$	11582.8	0.406436
$934$	3511.28	0.123011
$935$	−8893.05	−0.311052
$936$	3999.02	0.139650
$937$	−18961.3	−0.661088	−0.330544	−	0.943791i	$-0.607232\pi$
$937$	−18961.3	−0.661088	−0.330544	+	0.943791i	$0.607232\pi$
$938$	2403.86	0.0836769
$939$	−18169.7	−0.631466
$940$	−21318.0	−0.739698
$941$	−14961.6	−0.518314	−0.259157	−	0.965835i	$-0.583445\pi$
$941$	−14961.6	−0.518314	−0.259157	+	0.965835i	$0.583445\pi$
$942$	−20737.5	−0.717265
$943$	−38183.2	−1.31858
$944$	4326.27	0.149161
$945$	−488.883	−0.0168290
$946$	−6540.74	−0.224797
$947$	33262.3	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$947$	33262.3	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$948$	−3214.32	−0.110122
$949$	−3213.05	−0.109905
$950$	0	0
$951$	−15973.1	−0.544652
$952$	549.755	0.0187160
$953$	4973.79	0.169063	0.0845313	−	0.996421i	$-0.473061\pi$
$953$	4973.79	0.169063	0.0845313	+	0.996421i	$0.473061\pi$
$954$	−2062.48	−0.0699951
$955$	−37635.4	−1.27524
$956$	−4063.70	−0.137479
$957$	−690.399	−0.0233202
$958$	−16906.5	−0.570171
$959$	−1098.27	−0.0369812
$960$	−2347.65	−0.0789273
$961$	−25502.9	−0.856060
$962$	−40205.3	−1.34747
$963$	−10076.5	−0.337187
$964$	−5017.63	−0.167642
$965$	−1461.02	−0.0487378
$966$	1029.80	0.0342995
$967$	29720.9	0.988375	0.494187	−	0.869355i	$-0.335466\pi$
$967$	29720.9	0.988375	0.494187	+	0.869355i	$0.335466\pi$
$968$	8682.90	0.288305
$969$	0	0
$970$	2525.02	0.0835809
$971$	42672.6	1.41033	0.705165	−	0.709043i	$-0.250873\pi$
$971$	42672.6	1.41033	0.705165	+	0.709043i	$0.250873\pi$
$972$	972.000	0.0320750
$973$	3338.14	0.109985
$974$	−27595.5	−0.907820
$975$	−4083.73	−0.134137
$976$	4263.16	0.139816
$977$	−32516.3	−1.06478	−0.532389	−	0.846500i	$-0.678706\pi$
$977$	−32516.3	−1.06478	−0.532389	+	0.846500i	$0.678706\pi$
$978$	−10495.7	−0.343164
$979$	7976.31	0.260392
$980$	16668.7	0.543328
$981$	−1000.65	−0.0325669
$982$	5007.95	0.162740
$983$	4939.14	0.160259	0.0801293	−	0.996784i	$-0.474467\pi$
$983$	4939.14	0.160259	0.0801293	+	0.996784i	$0.474467\pi$
$984$	−7906.61	−0.256152
$985$	17106.4	0.553356
$986$	1362.80	0.0440166
$987$	1936.35	0.0624464
$988$	0	0
$989$	24184.8	0.777586
$990$	−3449.47	−0.110739
$991$	−48978.6	−1.56999	−0.784993	−	0.619504i	$-0.787334\pi$
$991$	−48978.6	−1.56999	−0.784993	+	0.619504i	$0.787334\pi$
$992$	2095.48	0.0670681
$993$	21630.0	0.691246
$994$	−2217.95	−0.0707737
$995$	−17625.2	−0.561563
$996$	−5929.20	−0.188629
$997$	−51504.8	−1.63608	−0.818042	−	0.575159i	$-0.804940\pi$
$997$	−51504.8	−1.63608	−0.818042	+	0.575159i	$0.804940\pi$
$998$	20531.5	0.651214
$999$	−9772.28	−0.309491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.y.1.1	✓	4
19.18	odd	2		2166.4.a.z.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.y.1.1	✓	4	1.1	even	1	trivial
2166.4.a.z.1.1	yes	4	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -12.2274 q^{5} -6.00000 q^{6} +1.48084 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -12.2274 q^{5} -6.00000 q^{6} +1.48084 q^{7} -8.00000 q^{8} +9.00000 q^{9} +24.4547 q^{10} -15.6728 q^{11} +12.0000 q^{12} -55.5419 q^{13} -2.96168 q^{14} -36.6821 q^{15} +16.0000 q^{16} -46.4056 q^{17} -18.0000 q^{18} -48.9095 q^{20} +4.44252 q^{21} +31.3457 q^{22} -115.903 q^{23} -24.0000 q^{24} +24.5084 q^{25} +111.084 q^{26} +27.0000 q^{27} +5.92336 q^{28} +14.6836 q^{29} +73.3642 q^{30} -65.4837 q^{31} -32.0000 q^{32} -47.0185 q^{33} +92.8112 q^{34} -18.1068 q^{35} +36.0000 q^{36} -361.936 q^{37} -166.626 q^{39} +97.8189 q^{40} +329.442 q^{41} -8.88505 q^{42} -208.665 q^{43} -62.6913 q^{44} -110.046 q^{45} +231.806 q^{46} +435.867 q^{47} +48.0000 q^{48} -340.807 q^{49} -49.0168 q^{50} -139.217 q^{51} -222.168 q^{52} +114.582 q^{53} -54.0000 q^{54} +191.637 q^{55} -11.8467 q^{56} -29.3671 q^{58} +270.392 q^{59} -146.728 q^{60} +266.447 q^{61} +130.967 q^{62} +13.3276 q^{63} +64.0000 q^{64} +679.131 q^{65} +94.0370 q^{66} -811.655 q^{67} -185.622 q^{68} -347.708 q^{69} +36.2136 q^{70} +748.881 q^{71} -72.0000 q^{72} +57.8492 q^{73} +723.872 q^{74} +73.5252 q^{75} -23.2090 q^{77} +333.251 q^{78} -267.860 q^{79} -195.638 q^{80} +81.0000 q^{81} -658.884 q^{82} -494.100 q^{83} +17.7701 q^{84} +567.418 q^{85} +417.330 q^{86} +44.0507 q^{87} +125.383 q^{88} -508.926 q^{89} +220.093 q^{90} -82.2488 q^{91} -463.611 q^{92} -196.451 q^{93} -871.733 q^{94} -96.0000 q^{96} +103.253 q^{97} +681.614 q^{98} -141.055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} - 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 + 12 * q^3 + 16 * q^4 - 24 * q^6 + 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} - 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9} + 83 q^{11} + 48 q^{12} + 5 q^{13} - 56 q^{14} + 64 q^{16} + 108 q^{17} - 72 q^{18} + 84 q^{21} - 166 q^{22} + 112 q^{23} - 96 q^{24} + 106 q^{25} - 10 q^{26} + 108 q^{27} + 112 q^{28} - 387 q^{29} + 419 q^{31} - 128 q^{32} + 249 q^{33} - 216 q^{34} + 576 q^{35} + 144 q^{36} + 68 q^{37} + 15 q^{39} - 47 q^{41} - 168 q^{42} - 244 q^{43} + 332 q^{44} - 224 q^{46} + 1019 q^{47} + 192 q^{48} - 450 q^{49} - 212 q^{50} + 324 q^{51} + 20 q^{52} + 111 q^{53} - 216 q^{54} + 263 q^{55} - 224 q^{56} + 774 q^{58} + 279 q^{59} + 206 q^{61} - 838 q^{62} + 252 q^{63} + 256 q^{64} + 3 q^{65} - 498 q^{66} - 587 q^{67} + 432 q^{68} + 336 q^{69} - 1152 q^{70} + 1161 q^{71} - 288 q^{72} + 445 q^{73} - 136 q^{74} + 318 q^{75} + 349 q^{77} - 30 q^{78} + 629 q^{79} + 324 q^{81} + 94 q^{82} + 403 q^{83} + 336 q^{84} - 725 q^{85} + 488 q^{86} - 1161 q^{87} - 664 q^{88} - 1598 q^{89} - 337 q^{91} + 448 q^{92} + 1257 q^{93} - 2038 q^{94} - 384 q^{96} - 1158 q^{97} + 900 q^{98} + 747 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 12 * q^3 + 16 * q^4 - 24 * q^6 + 28 * q^7 - 32 * q^8 + 36 * q^9 + 83 * q^11 + 48 * q^12 + 5 * q^13 - 56 * q^14 + 64 * q^16 + 108 * q^17 - 72 * q^18 + 84 * q^21 - 166 * q^22 + 112 * q^23 - 96 * q^24 + 106 * q^25 - 10 * q^26 + 108 * q^27 + 112 * q^28 - 387 * q^29 + 419 * q^31 - 128 * q^32 + 249 * q^33 - 216 * q^34 + 576 * q^35 + 144 * q^36 + 68 * q^37 + 15 * q^39 - 47 * q^41 - 168 * q^42 - 244 * q^43 + 332 * q^44 - 224 * q^46 + 1019 * q^47 + 192 * q^48 - 450 * q^49 - 212 * q^50 + 324 * q^51 + 20 * q^52 + 111 * q^53 - 216 * q^54 + 263 * q^55 - 224 * q^56 + 774 * q^58 + 279 * q^59 + 206 * q^61 - 838 * q^62 + 252 * q^63 + 256 * q^64 + 3 * q^65 - 498 * q^66 - 587 * q^67 + 432 * q^68 + 336 * q^69 - 1152 * q^70 + 1161 * q^71 - 288 * q^72 + 445 * q^73 - 136 * q^74 + 318 * q^75 + 349 * q^77 - 30 * q^78 + 629 * q^79 + 324 * q^81 + 94 * q^82 + 403 * q^83 + 336 * q^84 - 725 * q^85 + 488 * q^86 - 1161 * q^87 - 664 * q^88 - 1598 * q^89 - 337 * q^91 + 448 * q^92 + 1257 * q^93 - 2038 * q^94 - 384 * q^96 - 1158 * q^97 + 900 * q^98 + 747 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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