LMFDB - Embedded newform 2166.4.a.q.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{30}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 30 $ x^2 - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.47723$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.9545 q^{5} +6.00000 q^{6} -1.95445 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.9545 q^{5} +6.00000 q^{6} -1.95445 q^{7} +8.00000 q^{8} +9.00000 q^{9} +29.9089 q^{10} -0.954451 q^{11} +12.0000 q^{12} +48.8178 q^{13} -3.90890 q^{14} +44.8634 q^{15} +16.0000 q^{16} +12.0000 q^{17} +18.0000 q^{18} +59.8178 q^{20} -5.86335 q^{21} -1.90890 q^{22} +50.9545 q^{23} +24.0000 q^{24} +98.6356 q^{25} +97.6356 q^{26} +27.0000 q^{27} -7.81780 q^{28} +53.9089 q^{29} +89.7267 q^{30} -11.4990 q^{31} +32.0000 q^{32} -2.86335 q^{33} +24.0000 q^{34} -29.2277 q^{35} +36.0000 q^{36} -176.089 q^{37} +146.453 q^{39} +119.636 q^{40} +253.727 q^{41} -11.7267 q^{42} +488.497 q^{43} -3.81780 q^{44} +134.590 q^{45} +101.909 q^{46} -351.996 q^{47} +48.0000 q^{48} -339.180 q^{49} +197.271 q^{50} +36.0000 q^{51} +195.271 q^{52} -184.863 q^{53} +54.0000 q^{54} -14.2733 q^{55} -15.6356 q^{56} +107.818 q^{58} +280.954 q^{59} +179.453 q^{60} +563.998 q^{61} -22.9979 q^{62} -17.5901 q^{63} +64.0000 q^{64} +730.043 q^{65} -5.72671 q^{66} -188.406 q^{67} +48.0000 q^{68} +152.863 q^{69} -58.4555 q^{70} +106.451 q^{71} +72.0000 q^{72} -982.723 q^{73} -352.178 q^{74} +295.907 q^{75} +1.86543 q^{77} +292.907 q^{78} +804.319 q^{79} +239.271 q^{80} +81.0000 q^{81} +507.453 q^{82} -1043.45 q^{83} -23.4534 q^{84} +179.453 q^{85} +976.994 q^{86} +161.727 q^{87} -7.63561 q^{88} +873.493 q^{89} +269.180 q^{90} -95.4120 q^{91} +203.818 q^{92} -34.4969 q^{93} -703.992 q^{94} +96.0000 q^{96} +122.824 q^{97} -678.360 q^{98} -8.59006 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 8 q^{5} + 12 q^{6} + 18 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 8 * q^5 + 12 * q^6 + 18 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 8 q^{5} + 12 q^{6} + 18 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 20 q^{11} + 24 q^{12} + 10 q^{13} + 36 q^{14} + 24 q^{15} + 32 q^{16} + 24 q^{17} + 36 q^{18} + 32 q^{20} + 54 q^{21} + 40 q^{22} + 80 q^{23} + 48 q^{24} + 22 q^{25} + 20 q^{26} + 54 q^{27} + 72 q^{28} + 64 q^{29} + 48 q^{30} + 218 q^{31} + 64 q^{32} + 60 q^{33} + 48 q^{34} - 168 q^{35} + 72 q^{36} + 86 q^{37} + 30 q^{39} + 64 q^{40} + 376 q^{41} + 108 q^{42} + 254 q^{43} + 80 q^{44} + 72 q^{45} + 160 q^{46} + 260 q^{47} + 96 q^{48} - 284 q^{49} + 44 q^{50} + 72 q^{51} + 40 q^{52} - 304 q^{53} + 108 q^{54} - 160 q^{55} + 144 q^{56} + 128 q^{58} + 540 q^{59} + 96 q^{60} + 646 q^{61} + 436 q^{62} + 162 q^{63} + 128 q^{64} + 1000 q^{65} + 120 q^{66} + 390 q^{67} + 96 q^{68} + 240 q^{69} - 336 q^{70} - 532 q^{71} + 144 q^{72} - 870 q^{73} + 172 q^{74} + 66 q^{75} + 420 q^{77} + 60 q^{78} + 1762 q^{79} + 128 q^{80} + 162 q^{81} + 752 q^{82} - 1824 q^{83} + 216 q^{84} + 96 q^{85} + 508 q^{86} + 192 q^{87} + 160 q^{88} + 60 q^{89} + 144 q^{90} - 870 q^{91} + 320 q^{92} + 654 q^{93} + 520 q^{94} + 192 q^{96} + 1604 q^{97} - 568 q^{98} + 180 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 8 * q^5 + 12 * q^6 + 18 * q^7 + 16 * q^8 + 18 * q^9 + 16 * q^10 + 20 * q^11 + 24 * q^12 + 10 * q^13 + 36 * q^14 + 24 * q^15 + 32 * q^16 + 24 * q^17 + 36 * q^18 + 32 * q^20 + 54 * q^21 + 40 * q^22 + 80 * q^23 + 48 * q^24 + 22 * q^25 + 20 * q^26 + 54 * q^27 + 72 * q^28 + 64 * q^29 + 48 * q^30 + 218 * q^31 + 64 * q^32 + 60 * q^33 + 48 * q^34 - 168 * q^35 + 72 * q^36 + 86 * q^37 + 30 * q^39 + 64 * q^40 + 376 * q^41 + 108 * q^42 + 254 * q^43 + 80 * q^44 + 72 * q^45 + 160 * q^46 + 260 * q^47 + 96 * q^48 - 284 * q^49 + 44 * q^50 + 72 * q^51 + 40 * q^52 - 304 * q^53 + 108 * q^54 - 160 * q^55 + 144 * q^56 + 128 * q^58 + 540 * q^59 + 96 * q^60 + 646 * q^61 + 436 * q^62 + 162 * q^63 + 128 * q^64 + 1000 * q^65 + 120 * q^66 + 390 * q^67 + 96 * q^68 + 240 * q^69 - 336 * q^70 - 532 * q^71 + 144 * q^72 - 870 * q^73 + 172 * q^74 + 66 * q^75 + 420 * q^77 + 60 * q^78 + 1762 * q^79 + 128 * q^80 + 162 * q^81 + 752 * q^82 - 1824 * q^83 + 216 * q^84 + 96 * q^85 + 508 * q^86 + 192 * q^87 + 160 * q^88 + 60 * q^89 + 144 * q^90 - 870 * q^91 + 320 * q^92 + 654 * q^93 + 520 * q^94 + 192 * q^96 + 1604 * q^97 - 568 * q^98 + 180 * 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
$5$	14.9545	1.33757	0.668783	−	0.743457i	$-0.266815\pi$
$5$	14.9545	1.33757	0.668783	+	0.743457i	$0.266815\pi$
$6$	6.00000	0.408248
$7$	−1.95445	−0.105530	−0.0527652	−	0.998607i	$-0.516803\pi$
$7$	−1.95445	−0.105530	−0.0527652	+	0.998607i	$0.516803\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	29.9089	0.945803
$11$	−0.954451	−0.0261616	−0.0130808	−	0.999914i	$-0.504164\pi$
$11$	−0.954451	−0.0261616	−0.0130808	+	0.999914i	$0.504164\pi$
$12$	12.0000	0.288675
$13$	48.8178	1.04151	0.520755	−	0.853706i	$-0.325651\pi$
$13$	48.8178	1.04151	0.520755	+	0.853706i	$0.325651\pi$
$14$	−3.90890	−0.0746213
$15$	44.8634	0.772245
$16$	16.0000	0.250000
$17$	12.0000	0.171202	0.0856008	−	0.996330i	$-0.472719\pi$
$17$	12.0000	0.171202	0.0856008	+	0.996330i	$0.472719\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	59.8178	0.668783
$21$	−5.86335	−0.0609280
$22$	−1.90890	−0.0184991
$23$	50.9545	0.461945	0.230973	−	0.972960i	$-0.425809\pi$
$23$	50.9545	0.461945	0.230973	+	0.972960i	$0.425809\pi$
$24$	24.0000	0.204124
$25$	98.6356	0.789085
$26$	97.6356	0.736458
$27$	27.0000	0.192450
$28$	−7.81780	−0.0527652
$29$	53.9089	0.345194	0.172597	−	0.984993i	$-0.444784\pi$
$29$	53.9089	0.345194	0.172597	+	0.984993i	$0.444784\pi$
$30$	89.7267	0.546059
$31$	−11.4990	−0.0666218	−0.0333109	−	0.999445i	$-0.510605\pi$
$31$	−11.4990	−0.0666218	−0.0333109	+	0.999445i	$0.510605\pi$
$32$	32.0000	0.176777
$33$	−2.86335	−0.0151044
$34$	24.0000	0.121058
$35$	−29.2277	−0.141154
$36$	36.0000	0.166667
$37$	−176.089	−0.782402	−0.391201	−	0.920305i	$-0.627940\pi$
$37$	−176.089	−0.782402	−0.391201	+	0.920305i	$0.627940\pi$
$38$	0	0
$39$	146.453	0.601316
$40$	119.636	0.472901
$41$	253.727	0.966474	0.483237	−	0.875489i	$-0.339461\pi$
$41$	253.727	0.966474	0.483237	+	0.875489i	$0.339461\pi$
$42$	−11.7267	−0.0430826
$43$	488.497	1.73244	0.866222	−	0.499660i	$-0.166542\pi$
$43$	488.497	1.73244	0.866222	+	0.499660i	$0.166542\pi$
$44$	−3.81780	−0.0130808
$45$	134.590	0.445856
$46$	101.909	0.326645
$47$	−351.996	−1.09242	−0.546211	−	0.837647i	$-0.683930\pi$
$47$	−351.996	−1.09242	−0.546211	+	0.837647i	$0.683930\pi$
$48$	48.0000	0.144338
$49$	−339.180	−0.988863
$50$	197.271	0.557967
$51$	36.0000	0.0988433
$52$	195.271	0.520755
$53$	−184.863	−0.479112	−0.239556	−	0.970883i	$-0.577002\pi$
$53$	−184.863	−0.479112	−0.239556	+	0.970883i	$0.577002\pi$
$54$	54.0000	0.136083
$55$	−14.2733	−0.0349929
$56$	−15.6356	−0.0373106
$57$	0	0
$58$	107.818	0.244089
$59$	280.954	0.619952	0.309976	−	0.950744i	$-0.399679\pi$
$59$	280.954	0.619952	0.309976	+	0.950744i	$0.399679\pi$
$60$	179.453	0.386122
$61$	563.998	1.18381	0.591906	−	0.806007i	$-0.298376\pi$
$61$	563.998	1.18381	0.591906	+	0.806007i	$0.298376\pi$
$62$	−22.9979	−0.0471087
$63$	−17.5901	−0.0351768
$64$	64.0000	0.125000
$65$	730.043	1.39309
$66$	−5.72671	−0.0106804
$67$	−188.406	−0.343544	−0.171772	−	0.985137i	$-0.554949\pi$
$67$	−188.406	−0.343544	−0.171772	+	0.985137i	$0.554949\pi$
$68$	48.0000	0.0856008
$69$	152.863	0.266704
$70$	−58.4555	−0.0998110
$71$	106.451	0.177936	0.0889680	−	0.996034i	$-0.471643\pi$
$71$	106.451	0.177936	0.0889680	+	0.996034i	$0.471643\pi$
$72$	72.0000	0.117851
$73$	−982.723	−1.57560	−0.787801	−	0.615930i	$-0.788781\pi$
$73$	−982.723	−1.57560	−0.787801	+	0.615930i	$0.788781\pi$
$74$	−352.178	−0.553241
$75$	295.907	0.455578
$76$	0	0
$77$	1.86543	0.00276085
$78$	292.907	0.425194
$79$	804.319	1.14548	0.572740	−	0.819737i	$-0.305880\pi$
$79$	804.319	1.14548	0.572740	+	0.819737i	$0.305880\pi$
$80$	239.271	0.334392
$81$	81.0000	0.111111
$82$	507.453	0.683401
$83$	−1043.45	−1.37993	−0.689963	−	0.723844i	$-0.742373\pi$
$83$	−1043.45	−1.37993	−0.689963	+	0.723844i	$0.742373\pi$
$84$	−23.4534	−0.0304640
$85$	179.453	0.228994
$86$	976.994	1.22502
$87$	161.727	0.199298
$88$	−7.63561	−0.00924953
$89$	873.493	1.04034	0.520169	−	0.854063i	$-0.325869\pi$
$89$	873.493	1.04034	0.520169	+	0.854063i	$0.325869\pi$
$90$	269.180	0.315268
$91$	−95.4120	−0.109911
$92$	203.818	0.230973
$93$	−34.4969	−0.0384641
$94$	−703.992	−0.772460
$95$	0	0
$96$	96.0000	0.102062
$97$	122.824	0.128566	0.0642829	−	0.997932i	$-0.479524\pi$
$97$	122.824	0.128566	0.0642829	+	0.997932i	$0.479524\pi$
$98$	−678.360	−0.699232
$99$	−8.59006	−0.00872054
$100$	394.542	0.394542
$101$	1220.91	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$101$	1220.91	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$102$	72.0000	0.0698928
$103$	1972.67	1.88712	0.943560	−	0.331201i	$-0.107454\pi$
$103$	1972.67	1.88712	0.943560	+	0.331201i	$0.107454\pi$
$104$	390.542	0.368229
$105$	−87.6832	−0.0814953
$106$	−369.727	−0.338783
$107$	−1004.72	−0.907761	−0.453880	−	0.891063i	$-0.649961\pi$
$107$	−1004.72	−0.907761	−0.453880	+	0.891063i	$0.649961\pi$
$108$	108.000	0.0962250
$109$	−1576.91	−1.38569	−0.692845	−	0.721086i	$-0.743643\pi$
$109$	−1576.91	−1.38569	−0.692845	+	0.721086i	$0.743643\pi$
$110$	−28.5466	−0.0247437
$111$	−528.267	−0.451720
$112$	−31.2712	−0.0263826
$113$	475.315	0.395698	0.197849	−	0.980233i	$-0.436604\pi$
$113$	475.315	0.395698	0.197849	+	0.980233i	$0.436604\pi$
$114$	0	0
$115$	761.996	0.617882
$116$	215.636	0.172597
$117$	439.360	0.347170
$118$	561.909	0.438372
$119$	−23.4534	−0.0180670
$120$	358.907	0.273030
$121$	−1330.09	−0.999316
$122$	1128.00	0.837082
$123$	761.180	0.557994
$124$	−45.9959	−0.0333109
$125$	−394.265	−0.282113
$126$	−35.1801	−0.0248738
$127$	−1222.81	−0.854386	−0.427193	−	0.904161i	$-0.640497\pi$
$127$	−1222.81	−0.854386	−0.427193	+	0.904161i	$0.640497\pi$
$128$	128.000	0.0883883
$129$	1465.49	1.00023
$130$	1460.09	0.985062
$131$	2128.62	1.41968	0.709842	−	0.704361i	$-0.248767\pi$
$131$	2128.62	1.41968	0.709842	+	0.704361i	$0.248767\pi$
$132$	−11.4534	−0.00755221
$133$	0	0
$134$	−376.812	−0.242922
$135$	403.770	0.257415
$136$	96.0000	0.0605289
$137$	−1673.53	−1.04365	−0.521823	−	0.853054i	$-0.674748\pi$
$137$	−1673.53	−1.04365	−0.521823	+	0.853054i	$0.674748\pi$
$138$	305.727	0.188588
$139$	265.967	0.162295	0.0811475	−	0.996702i	$-0.474142\pi$
$139$	265.967	0.162295	0.0811475	+	0.996702i	$0.474142\pi$
$140$	−116.911	−0.0705770
$141$	−1055.99	−0.630711
$142$	212.903	0.125820
$143$	−46.5942	−0.0272476
$144$	144.000	0.0833333
$145$	806.178	0.461720
$146$	−1965.45	−1.11412
$147$	−1017.54	−0.570921
$148$	−704.356	−0.391201
$149$	3511.31	1.93059	0.965293	−	0.261168i	$-0.0841076\pi$
$149$	3511.31	1.93059	0.965293	+	0.261168i	$0.0841076\pi$
$150$	591.814	0.322143
$151$	2008.43	1.08241	0.541206	−	0.840890i	$-0.317968\pi$
$151$	2008.43	1.08241	0.541206	+	0.840890i	$0.317968\pi$
$152$	0	0
$153$	108.000	0.0570672
$154$	3.73086	0.00195221
$155$	−171.961	−0.0891111
$156$	585.814	0.300658
$157$	−906.545	−0.460829	−0.230414	−	0.973093i	$-0.574008\pi$
$157$	−906.545	−0.460829	−0.230414	+	0.973093i	$0.574008\pi$
$158$	1608.64	0.809977
$159$	−554.590	−0.276615
$160$	478.542	0.236451
$161$	−99.5880	−0.0487493
$162$	162.000	0.0785674
$163$	1594.02	0.765973	0.382987	−	0.923754i	$-0.374896\pi$
$163$	1594.02	0.765973	0.382987	+	0.923754i	$0.374896\pi$
$164$	1014.91	0.483237
$165$	−42.8199	−0.0202032
$166$	−2086.91	−0.975755
$167$	−2608.38	−1.20864	−0.604320	−	0.796742i	$-0.706555\pi$
$167$	−2608.38	−1.20864	−0.604320	+	0.796742i	$0.706555\pi$
$168$	−46.9068	−0.0215413
$169$	186.178	0.0847419
$170$	358.907	0.161923
$171$	0	0
$172$	1953.99	0.866222
$173$	−1157.89	−0.508861	−0.254430	−	0.967091i	$-0.581888\pi$
$173$	−1157.89	−0.508861	−0.254430	+	0.967091i	$0.581888\pi$
$174$	323.453	0.140925
$175$	−192.778	−0.0832725
$176$	−15.2712	−0.00654041
$177$	842.863	0.357929
$178$	1746.99	0.735630
$179$	684.238	0.285711	0.142856	−	0.989744i	$-0.454371\pi$
$179$	684.238	0.285711	0.142856	+	0.989744i	$0.454371\pi$
$180$	538.360	0.222928
$181$	−1865.54	−0.766102	−0.383051	−	0.923727i	$-0.625127\pi$
$181$	−1865.54	−0.766102	−0.383051	+	0.923727i	$0.625127\pi$
$182$	−190.824	−0.0777188
$183$	1691.99	0.683474
$184$	407.636	0.163322
$185$	−2633.31	−1.04651
$186$	−68.9938	−0.0271982
$187$	−11.4534	−0.00447891
$188$	−1407.98	−0.546211
$189$	−52.7702	−0.0203093
$190$	0	0
$191$	−1224.93	−0.464048	−0.232024	−	0.972710i	$-0.574535\pi$
$191$	−1224.93	−0.464048	−0.232024	+	0.972710i	$0.574535\pi$
$192$	192.000	0.0721688
$193$	3165.82	1.18073	0.590365	−	0.807137i	$-0.298984\pi$
$193$	3165.82	1.18073	0.590365	+	0.807137i	$0.298984\pi$
$194$	245.648	0.0909098
$195$	2190.13	0.804300
$196$	−1356.72	−0.494432
$197$	−5375.48	−1.94410	−0.972049	−	0.234780i	$-0.924563\pi$
$197$	−5375.48	−1.94410	−0.972049	+	0.234780i	$0.924563\pi$
$198$	−17.1801	−0.00616635
$199$	2156.94	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$199$	2156.94	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$200$	789.085	0.278984
$201$	−565.217	−0.198345
$202$	2441.81	0.850522
$203$	−105.362	−0.0364285
$204$	144.000	0.0494217
$205$	3794.34	1.29272
$206$	3945.35	1.33440
$207$	458.590	0.153982
$208$	781.085	0.260377
$209$	0	0
$210$	−175.366	−0.0576259
$211$	5334.55	1.74050	0.870250	−	0.492610i	$-0.163957\pi$
$211$	5334.55	1.74050	0.870250	+	0.492610i	$0.163957\pi$
$212$	−739.453	−0.239556
$213$	319.354	0.102731
$214$	−2009.45	−0.641884
$215$	7305.20	2.31726
$216$	216.000	0.0680414
$217$	22.4742	0.00703062
$218$	−3153.81	−0.979831
$219$	−2948.17	−0.909674
$220$	−57.0932	−0.0174965
$221$	585.814	0.178308
$222$	−1056.53	−0.319414
$223$	−5777.57	−1.73495	−0.867477	−	0.497477i	$-0.834260\pi$
$223$	−5777.57	−1.73495	−0.867477	+	0.497477i	$0.834260\pi$
$224$	−62.5424	−0.0186553
$225$	887.720	0.263028
$226$	950.629	0.279801
$227$	−294.304	−0.0860514	−0.0430257	−	0.999074i	$-0.513700\pi$
$227$	−294.304	−0.0860514	−0.0430257	+	0.999074i	$0.513700\pi$
$228$	0	0
$229$	−393.161	−0.113453	−0.0567267	−	0.998390i	$-0.518066\pi$
$229$	−393.161	−0.113453	−0.0567267	+	0.998390i	$0.518066\pi$
$230$	1523.99	0.436909
$231$	5.59628	0.00159398
$232$	431.271	0.122045
$233$	−5767.04	−1.62151	−0.810754	−	0.585386i	$-0.800943\pi$
$233$	−5767.04	−1.62151	−0.810754	+	0.585386i	$0.800943\pi$
$234$	878.720	0.245486
$235$	−5263.90	−1.46119
$236$	1123.82	0.309976
$237$	2412.96	0.661343
$238$	−46.9068	−0.0127753
$239$	−1992.21	−0.539186	−0.269593	−	0.962974i	$-0.586889\pi$
$239$	−1992.21	−0.539186	−0.269593	+	0.962974i	$0.586889\pi$
$240$	717.814	0.193061
$241$	2287.27	0.611354	0.305677	−	0.952135i	$-0.401117\pi$
$241$	2287.27	0.611354	0.305677	+	0.952135i	$0.401117\pi$
$242$	−2660.18	−0.706623
$243$	243.000	0.0641500
$244$	2255.99	0.591906
$245$	−5072.25	−1.32267
$246$	1522.36	0.394562
$247$	0	0
$248$	−91.9917	−0.0235544
$249$	−3130.36	−0.796701
$250$	−788.530	−0.199484
$251$	−2662.29	−0.669492	−0.334746	−	0.942308i	$-0.608651\pi$
$251$	−2662.29	−0.669492	−0.334746	+	0.942308i	$0.608651\pi$
$252$	−70.3602	−0.0175884
$253$	−48.6335	−0.0120852
$254$	−2445.62	−0.604142
$255$	538.360	0.132210
$256$	256.000	0.0625000
$257$	−5747.30	−1.39497	−0.697484	−	0.716600i	$-0.745697\pi$
$257$	−5747.30	−1.39497	−0.697484	+	0.716600i	$0.745697\pi$
$258$	2930.98	0.707267
$259$	344.157	0.0825672
$260$	2920.17	0.696544
$261$	485.180	0.115065
$262$	4257.24	1.00387
$263$	1715.21	0.402145	0.201072	−	0.979576i	$-0.435557\pi$
$263$	1715.21	0.402145	0.201072	+	0.979576i	$0.435557\pi$
$264$	−22.9068	−0.00534022
$265$	−2764.53	−0.640844
$266$	0	0
$267$	2620.48	0.600639
$268$	−753.623	−0.171772
$269$	−3447.55	−0.781416	−0.390708	−	0.920515i	$-0.627770\pi$
$269$	−3447.55	−0.781416	−0.390708	+	0.920515i	$0.627770\pi$
$270$	807.540	0.182020
$271$	611.818	0.137141	0.0685706	−	0.997646i	$-0.478156\pi$
$271$	611.818	0.137141	0.0685706	+	0.997646i	$0.478156\pi$
$272$	192.000	0.0428004
$273$	−286.236	−0.0634571
$274$	−3347.06	−0.737969
$275$	−94.1429	−0.0206437
$276$	611.453	0.133352
$277$	2750.63	0.596641	0.298320	−	0.954466i	$-0.403574\pi$
$277$	2750.63	0.596641	0.298320	+	0.954466i	$0.403574\pi$
$278$	531.934	0.114760
$279$	−103.491	−0.0222073
$280$	−233.822	−0.0499055
$281$	1812.80	0.384849	0.192425	−	0.981312i	$-0.438365\pi$
$281$	1812.80	0.384849	0.192425	+	0.981312i	$0.438365\pi$
$282$	−2111.98	−0.445980
$283$	−2640.55	−0.554644	−0.277322	−	0.960777i	$-0.589447\pi$
$283$	−2640.55	−0.554644	−0.277322	+	0.960777i	$0.589447\pi$
$284$	425.805	0.0889680
$285$	0	0
$286$	−93.1884	−0.0192669
$287$	−495.896	−0.101992
$288$	288.000	0.0589256
$289$	−4769.00	−0.970690
$290$	1612.36	0.326486
$291$	368.472	0.0742275
$292$	−3930.89	−0.787801
$293$	8459.94	1.68681	0.843404	−	0.537280i	$-0.180548\pi$
$293$	8459.94	1.68681	0.843404	+	0.537280i	$0.180548\pi$
$294$	−2035.08	−0.403702
$295$	4201.52	0.829227
$296$	−1408.71	−0.276621
$297$	−25.7702	−0.00503481
$298$	7022.61	1.36513
$299$	2487.48	0.481120
$300$	1183.63	0.227789
$301$	−954.743	−0.182826
$302$	4016.87	0.765380
$303$	3662.72	0.694448
$304$	0	0
$305$	8434.28	1.58343
$306$	216.000	0.0403526
$307$	2728.61	0.507264	0.253632	−	0.967301i	$-0.418375\pi$
$307$	2728.61	0.507264	0.253632	+	0.967301i	$0.418375\pi$
$308$	7.46171	0.00138042
$309$	5918.02	1.08953
$310$	−343.921	−0.0630110
$311$	−861.716	−0.157117	−0.0785586	−	0.996909i	$-0.525032\pi$
$311$	−861.716	−0.157117	−0.0785586	+	0.996909i	$0.525032\pi$
$312$	1171.63	0.212597
$313$	−1828.34	−0.330173	−0.165086	−	0.986279i	$-0.552790\pi$
$313$	−1828.34	−0.330173	−0.165086	+	0.986279i	$0.552790\pi$
$314$	−1813.09	−0.325855
$315$	−263.050	−0.0470513
$316$	3217.28	0.572740
$317$	1263.61	0.223885	0.111943	−	0.993715i	$-0.464293\pi$
$317$	1263.61	0.223885	0.111943	+	0.993715i	$0.464293\pi$
$318$	−1109.18	−0.195597
$319$	−51.4534	−0.00903084
$320$	957.085	0.167196
$321$	−3014.17	−0.524096
$322$	−199.176	−0.0344709
$323$	0	0
$324$	324.000	0.0555556
$325$	4815.17	0.821839
$326$	3188.05	0.541625
$327$	−4730.72	−0.800029
$328$	2029.81	0.341700
$329$	687.959	0.115284
$330$	−85.6398	−0.0142858
$331$	9805.95	1.62835	0.814175	−	0.580620i	$-0.197190\pi$
$331$	9805.95	1.62835	0.814175	+	0.580620i	$0.197190\pi$
$332$	−4173.81	−0.689963
$333$	−1584.80	−0.260801
$334$	−5216.77	−0.854637
$335$	−2817.51	−0.459513
$336$	−93.8137	−0.0152320
$337$	7781.30	1.25779	0.628894	−	0.777491i	$-0.283508\pi$
$337$	7781.30	1.25779	0.628894	+	0.777491i	$0.283508\pi$
$338$	372.356	0.0599216
$339$	1425.94	0.228456
$340$	717.814	0.114497
$341$	10.9752	0.00174293
$342$	0	0
$343$	1333.29	0.209886
$344$	3907.98	0.612511
$345$	2285.99	0.356735
$346$	−2315.78	−0.359819
$347$	7124.35	1.10218	0.551088	−	0.834447i	$-0.314213\pi$
$347$	7124.35	1.10218	0.551088	+	0.834447i	$0.314213\pi$
$348$	646.907	0.0996490
$349$	−9423.36	−1.44533	−0.722666	−	0.691197i	$-0.757084\pi$
$349$	−9423.36	−1.44533	−0.722666	+	0.691197i	$0.757084\pi$
$350$	−385.557	−0.0588825
$351$	1318.08	0.200439
$352$	−30.5424	−0.00462477
$353$	4051.75	0.610914	0.305457	−	0.952206i	$-0.401191\pi$
$353$	4051.75	0.610914	0.305457	+	0.952206i	$0.401191\pi$
$354$	1685.73	0.253094
$355$	1591.92	0.238001
$356$	3493.97	0.520169
$357$	−70.3602	−0.0104310
$358$	1368.48	0.202029
$359$	−10737.2	−1.57852	−0.789262	−	0.614057i	$-0.789537\pi$
$359$	−10737.2	−1.57852	−0.789262	+	0.614057i	$0.789537\pi$
$360$	1076.72	0.157634
$361$	0	0
$362$	−3731.08	−0.541716
$363$	−3990.27	−0.576955
$364$	−381.648	−0.0549555
$365$	−14696.1	−2.10747
$366$	3383.99	0.483289
$367$	5839.35	0.830550	0.415275	−	0.909696i	$-0.363685\pi$
$367$	5839.35	0.830550	0.415275	+	0.909696i	$0.363685\pi$
$368$	815.271	0.115486
$369$	2283.54	0.322158
$370$	−5266.63	−0.739997
$371$	361.306	0.0505609
$372$	−137.988	−0.0192320
$373$	1924.80	0.267192	0.133596	−	0.991036i	$-0.457348\pi$
$373$	1924.80	0.267192	0.133596	+	0.991036i	$0.457348\pi$
$374$	−22.9068	−0.00316707
$375$	−1182.79	−0.162878
$376$	−2815.97	−0.386230
$377$	2631.71	0.359523
$378$	−105.540	−0.0143609
$379$	−3816.49	−0.517256	−0.258628	−	0.965977i	$-0.583270\pi$
$379$	−3816.49	−0.517256	−0.258628	+	0.965977i	$0.583270\pi$
$380$	0	0
$381$	−3668.43	−0.493280
$382$	−2449.87	−0.328131
$383$	−6796.20	−0.906709	−0.453355	−	0.891330i	$-0.649773\pi$
$383$	−6796.20	−0.906709	−0.453355	+	0.891330i	$0.649773\pi$
$384$	384.000	0.0510310
$385$	27.8965	0.00369282
$386$	6331.64	0.834901
$387$	4396.47	0.577481
$388$	491.296	0.0642829
$389$	−12087.0	−1.57541	−0.787707	−	0.616050i	$-0.788732\pi$
$389$	−12087.0	−1.57541	−0.787707	+	0.616050i	$0.788732\pi$
$390$	4380.26	0.568726
$391$	611.453	0.0790858
$392$	−2713.44	−0.349616
$393$	6385.86	0.819654
$394$	−10751.0	−1.37468
$395$	12028.1	1.53216
$396$	−34.3602	−0.00436027
$397$	−9724.80	−1.22940	−0.614702	−	0.788759i	$-0.710724\pi$
$397$	−9724.80	−1.22940	−0.614702	+	0.788759i	$0.710724\pi$
$398$	4313.89	0.543306
$399$	0	0
$400$	1578.17	0.197271
$401$	4518.90	0.562750	0.281375	−	0.959598i	$-0.409209\pi$
$401$	4518.90	0.562750	0.281375	+	0.959598i	$0.409209\pi$
$402$	−1130.43	−0.140251
$403$	−561.354	−0.0693872
$404$	4883.63	0.601410
$405$	1211.31	0.148619
$406$	−210.725	−0.0257588
$407$	168.068	0.0204689
$408$	288.000	0.0349464
$409$	7803.20	0.943382	0.471691	−	0.881764i	$-0.343644\pi$
$409$	7803.20	0.943382	0.471691	+	0.881764i	$0.343644\pi$
$410$	7588.69	0.914094
$411$	−5020.60	−0.602549
$412$	7890.70	0.943560
$413$	−549.112	−0.0654238
$414$	917.180	0.108882
$415$	−15604.3	−1.84574
$416$	1562.17	0.184115
$417$	797.901	0.0937011
$418$	0	0
$419$	4967.51	0.579186	0.289593	−	0.957150i	$-0.406480\pi$
$419$	4967.51	0.579186	0.289593	+	0.957150i	$0.406480\pi$
$420$	−350.733	−0.0407477
$421$	1066.31	0.123442	0.0617209	−	0.998093i	$-0.480341\pi$
$421$	1066.31	0.123442	0.0617209	+	0.998093i	$0.480341\pi$
$422$	10669.1	1.23072
$423$	−3167.96	−0.364141
$424$	−1478.91	−0.169392
$425$	1183.63	0.135093
$426$	638.708	0.0726420
$427$	−1102.31	−0.124928
$428$	−4018.90	−0.453880
$429$	−139.783	−0.0157314
$430$	14610.4	1.63855
$431$	11228.4	1.25488	0.627442	−	0.778663i	$-0.284102\pi$
$431$	11228.4	1.25488	0.627442	+	0.778663i	$0.284102\pi$
$432$	432.000	0.0481125
$433$	380.482	0.0422282	0.0211141	−	0.999777i	$-0.493279\pi$
$433$	380.482	0.0422282	0.0211141	+	0.999777i	$0.493279\pi$
$434$	44.9483	0.00497140
$435$	2418.53	0.266574
$436$	−6307.63	−0.692845
$437$	0	0
$438$	−5896.34	−0.643237
$439$	13436.6	1.46080	0.730402	−	0.683018i	$-0.239333\pi$
$439$	13436.6	1.46080	0.730402	+	0.683018i	$0.239333\pi$
$440$	−114.186	−0.0123719
$441$	−3052.62	−0.329621
$442$	1171.63	0.126083
$443$	17797.3	1.90875	0.954375	−	0.298611i	$-0.0965232\pi$
$443$	17797.3	1.90875	0.954375	+	0.298611i	$0.0965232\pi$
$444$	−2113.07	−0.225860
$445$	13062.6	1.39152
$446$	−11555.1	−1.22680
$447$	10533.9	1.11462
$448$	−125.085	−0.0131913
$449$	−4491.05	−0.472040	−0.236020	−	0.971748i	$-0.575843\pi$
$449$	−4491.05	−0.472040	−0.236020	+	0.971748i	$0.575843\pi$
$450$	1775.44	0.185989
$451$	−242.170	−0.0252845
$452$	1901.26	0.197849
$453$	6025.30	0.624930
$454$	−588.609	−0.0608475
$455$	−1426.83	−0.147013
$456$	0	0
$457$	−18138.6	−1.85664	−0.928322	−	0.371777i	$-0.878749\pi$
$457$	−18138.6	−1.85664	−0.928322	+	0.371777i	$0.878749\pi$
$458$	−786.323	−0.0802237
$459$	324.000	0.0329478
$460$	3047.98	0.308941
$461$	2873.82	0.290340	0.145170	−	0.989407i	$-0.453627\pi$
$461$	2873.82	0.290340	0.145170	+	0.989407i	$0.453627\pi$
$462$	11.1926	0.00112711
$463$	6302.53	0.632621	0.316310	−	0.948656i	$-0.397556\pi$
$463$	6302.53	0.632621	0.316310	+	0.948656i	$0.397556\pi$
$464$	862.542	0.0862986
$465$	−515.882	−0.0514483
$466$	−11534.1	−1.14658
$467$	−14867.8	−1.47323	−0.736617	−	0.676310i	$-0.763578\pi$
$467$	−14867.8	−1.47323	−0.736617	+	0.676310i	$0.763578\pi$
$468$	1757.44	0.173585
$469$	368.230	0.0362543
$470$	−10527.8	−1.03322
$471$	−2719.63	−0.266060
$472$	2247.64	0.219186
$473$	−466.246	−0.0453235
$474$	4825.91	0.467640
$475$	0	0
$476$	−93.8137	−0.00903349
$477$	−1663.77	−0.159704
$478$	−3984.43	−0.381262
$479$	−3077.25	−0.293535	−0.146767	−	0.989171i	$-0.546887\pi$
$479$	−3077.25	−0.293535	−0.146767	+	0.989171i	$0.546887\pi$
$480$	1435.63	0.136515
$481$	−8596.28	−0.814879
$482$	4574.55	0.432292
$483$	−298.764	−0.0281454
$484$	−5320.36	−0.499658
$485$	1836.77	0.171965
$486$	486.000	0.0453609
$487$	7954.58	0.740157	0.370078	−	0.929001i	$-0.379331\pi$
$487$	7954.58	0.740157	0.370078	+	0.929001i	$0.379331\pi$
$488$	4511.98	0.418541
$489$	4782.07	0.442235
$490$	−10144.5	−0.935269
$491$	136.008	0.0125009	0.00625047	−	0.999980i	$-0.498010\pi$
$491$	136.008	0.0125009	0.00625047	+	0.999980i	$0.498010\pi$
$492$	3044.72	0.278997
$493$	646.907	0.0590978
$494$	0	0
$495$	−128.460	−0.0116643
$496$	−183.983	−0.0166554
$497$	−208.054	−0.0187777
$498$	−6260.72	−0.563353
$499$	−8319.59	−0.746364	−0.373182	−	0.927758i	$-0.621733\pi$
$499$	−8319.59	−0.746364	−0.373182	+	0.927758i	$0.621733\pi$
$500$	−1577.06	−0.141057
$501$	−7825.15	−0.697808
$502$	−5324.59	−0.473402
$503$	19145.2	1.69710	0.848551	−	0.529114i	$-0.177476\pi$
$503$	19145.2	1.69710	0.848551	+	0.529114i	$0.177476\pi$
$504$	−140.720	−0.0124369
$505$	18258.0	1.60885
$506$	−97.2671	−0.00854555
$507$	558.534	0.0489258
$508$	−4891.25	−0.427193
$509$	20673.3	1.80025	0.900124	−	0.435634i	$-0.143476\pi$
$509$	20673.3	1.80025	0.900124	+	0.435634i	$0.143476\pi$
$510$	1076.72	0.0934862
$511$	1920.68	0.166274
$512$	512.000	0.0441942
$513$	0	0
$514$	−11494.6	−0.986392
$515$	29500.3	2.52415
$516$	5861.96	0.500113
$517$	335.963	0.0285796
$518$	688.315	0.0583838
$519$	−3473.68	−0.293791
$520$	5840.35	0.492531
$521$	−18437.8	−1.55043	−0.775216	−	0.631696i	$-0.782359\pi$
$521$	−18437.8	−1.55043	−0.775216	+	0.631696i	$0.782359\pi$
$522$	970.360	0.0813631
$523$	−13666.8	−1.14265	−0.571324	−	0.820724i	$-0.693570\pi$
$523$	−13666.8	−1.14265	−0.571324	+	0.820724i	$0.693570\pi$
$524$	8514.48	0.709842
$525$	−578.335	−0.0480774
$526$	3430.41	0.284359
$527$	−137.988	−0.0114058
$528$	−45.8137	−0.00377610
$529$	−9570.64	−0.786607
$530$	−5529.06	−0.453145
$531$	2528.59	0.206651
$532$	0	0
$533$	12386.4	1.00659
$534$	5240.96	0.424716
$535$	−15025.1	−1.21419
$536$	−1507.25	−0.121461
$537$	2052.71	0.164956
$538$	−6895.10	−0.552544
$539$	323.731	0.0258703
$540$	1615.08	0.128707
$541$	−194.292	−0.0154404	−0.00772020	−	0.999970i	$-0.502457\pi$
$541$	−194.292	−0.0154404	−0.00772020	+	0.999970i	$0.502457\pi$
$542$	1223.64	0.0969735
$543$	−5596.62	−0.442309
$544$	384.000	0.0302645
$545$	−23581.8	−1.85345
$546$	−572.472	−0.0448710
$547$	−23137.5	−1.80857	−0.904287	−	0.426925i	$-0.859597\pi$
$547$	−23137.5	−1.80857	−0.904287	+	0.426925i	$0.859597\pi$
$548$	−6694.13	−0.521823
$549$	5075.98	0.394604
$550$	−188.286	−0.0145973
$551$	0	0
$552$	1222.91	0.0942942
$553$	−1572.00	−0.120883
$554$	5501.27	0.421889
$555$	−7899.94	−0.604205
$556$	1063.87	0.0811475
$557$	−7809.50	−0.594074	−0.297037	−	0.954866i	$-0.595998\pi$
$557$	−7809.50	−0.594074	−0.297037	+	0.954866i	$0.595998\pi$
$558$	−206.981	−0.0157029
$559$	23847.3	1.80436
$560$	−467.644	−0.0352885
$561$	−34.3602	−0.00258590
$562$	3625.60	0.272130
$563$	−14420.0	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$563$	−14420.0	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$564$	−4223.95	−0.315355
$565$	7108.07	0.529272
$566$	−5281.09	−0.392192
$567$	−158.311	−0.0117256
$568$	851.611	0.0629098
$569$	20606.6	1.51823	0.759114	−	0.650958i	$-0.225632\pi$
$569$	20606.6	1.51823	0.759114	+	0.650958i	$0.225632\pi$
$570$	0	0
$571$	−17355.7	−1.27200	−0.636001	−	0.771688i	$-0.719413\pi$
$571$	−17355.7	−1.27200	−0.636001	+	0.771688i	$0.719413\pi$
$572$	−186.377	−0.0136238
$573$	−3674.80	−0.267918
$574$	−991.793	−0.0721196
$575$	5025.92	0.364514
$576$	576.000	0.0416667
$577$	−10518.5	−0.758910	−0.379455	−	0.925210i	$-0.623888\pi$
$577$	−10518.5	−0.758910	−0.379455	+	0.925210i	$0.623888\pi$
$578$	−9538.00	−0.686381
$579$	9497.46	0.681694
$580$	3224.71	0.230860
$581$	2039.38	0.145624
$582$	736.944	0.0524868
$583$	176.443	0.0125343
$584$	−7861.78	−0.557060
$585$	6570.39	0.464363
$586$	16919.9	1.19275
$587$	21497.9	1.51161	0.755804	−	0.654798i	$-0.227246\pi$
$587$	21497.9	1.51161	0.755804	+	0.654798i	$0.227246\pi$
$588$	−4070.16	−0.285460
$589$	0	0
$590$	8403.04	0.586352
$591$	−16126.4	−1.12242
$592$	−2817.42	−0.195600
$593$	−8101.68	−0.561039	−0.280520	−	0.959848i	$-0.590507\pi$
$593$	−8101.68	−0.561039	−0.280520	+	0.959848i	$0.590507\pi$
$594$	−51.5404	−0.00356015
$595$	−350.733	−0.0241658
$596$	14045.2	0.965293
$597$	6470.83	0.443607
$598$	4974.97	0.340203
$599$	−15285.2	−1.04263	−0.521317	−	0.853363i	$-0.674559\pi$
$599$	−15285.2	−1.04263	−0.521317	+	0.853363i	$0.674559\pi$
$600$	2367.25	0.161071
$601$	−15368.7	−1.04310	−0.521548	−	0.853222i	$-0.674645\pi$
$601$	−15368.7	−1.04310	−0.521548	+	0.853222i	$0.674645\pi$
$602$	−1909.49	−0.129277
$603$	−1695.65	−0.114515
$604$	8033.74	0.541206
$605$	−19890.8	−1.33665
$606$	7325.44	0.491049
$607$	8777.56	0.586936	0.293468	−	0.955969i	$-0.405191\pi$
$607$	8777.56	0.586936	0.293468	+	0.955969i	$0.405191\pi$
$608$	0	0
$609$	−316.087	−0.0210320
$610$	16868.6	1.11965
$611$	−17183.7	−1.13777
$612$	432.000	0.0285336
$613$	−20991.8	−1.38312	−0.691559	−	0.722320i	$-0.743076\pi$
$613$	−20991.8	−1.38312	−0.691559	+	0.722320i	$0.743076\pi$
$614$	5457.22	0.358690
$615$	11383.0	0.746355
$616$	14.9234	0.000976107 0
$617$	−9879.90	−0.644651	−0.322326	−	0.946629i	$-0.604465\pi$
$617$	−9879.90	−0.644651	−0.322326	+	0.946629i	$0.604465\pi$
$618$	11836.0	0.770414
$619$	−25209.7	−1.63694	−0.818468	−	0.574552i	$-0.805176\pi$
$619$	−25209.7	−1.63694	−0.818468	+	0.574552i	$0.805176\pi$
$620$	−687.843	−0.0445555
$621$	1375.77	0.0889014
$622$	−1723.43	−0.111099
$623$	−1707.20	−0.109787
$624$	2343.25	0.150329
$625$	−18225.5	−1.16643
$626$	−3656.69	−0.233467
$627$	0	0
$628$	−3626.18	−0.230414
$629$	−2113.07	−0.133948
$630$	−526.099	−0.0332703
$631$	4793.74	0.302434	0.151217	−	0.988501i	$-0.451681\pi$
$631$	4793.74	0.302434	0.151217	+	0.988501i	$0.451681\pi$
$632$	6434.55	0.404988
$633$	16003.7	1.00488
$634$	2527.23	0.158311
$635$	−18286.5	−1.14280
$636$	−2218.36	−0.138308
$637$	−16558.0	−1.02991
$638$	−102.907	−0.00638577
$639$	958.062	0.0593120
$640$	1914.17	0.118225
$641$	−308.588	−0.0190148	−0.00950740	−	0.999955i	$-0.503026\pi$
$641$	−308.588	−0.0190148	−0.00950740	+	0.999955i	$0.503026\pi$
$642$	−6028.35	−0.370592
$643$	−544.779	−0.0334121	−0.0167060	−	0.999860i	$-0.505318\pi$
$643$	−544.779	−0.0334121	−0.0167060	+	0.999860i	$0.505318\pi$
$644$	−398.352	−0.0243746
$645$	21915.6	1.33787
$646$	0	0
$647$	26288.6	1.59739	0.798695	−	0.601737i	$-0.205524\pi$
$647$	26288.6	1.59739	0.798695	+	0.601737i	$0.205524\pi$
$648$	648.000	0.0392837
$649$	−268.157	−0.0162189
$650$	9630.35	0.581128
$651$	67.4225	0.00405913
$652$	6376.10	0.382987
$653$	−9560.11	−0.572919	−0.286459	−	0.958092i	$-0.592478\pi$
$653$	−9560.11	−0.572919	−0.286459	+	0.958092i	$0.592478\pi$
$654$	−9461.44	−0.565706
$655$	31832.4	1.89892
$656$	4059.63	0.241619
$657$	−8844.50	−0.525201
$658$	1375.92	0.0815180
$659$	−25006.7	−1.47818	−0.739091	−	0.673605i	$-0.764745\pi$
$659$	−25006.7	−1.47818	−0.739091	+	0.673605i	$0.764745\pi$
$660$	−171.280	−0.0101016
$661$	3566.41	0.209860	0.104930	−	0.994480i	$-0.466538\pi$
$661$	3566.41	0.209860	0.104930	+	0.994480i	$0.466538\pi$
$662$	19611.9	1.15142
$663$	1757.44	0.102946
$664$	−8347.63	−0.487878
$665$	0	0
$666$	−3169.60	−0.184414
$667$	2746.90	0.159461
$668$	−10433.5	−0.604320
$669$	−17332.7	−1.00168
$670$	−5635.01	−0.324925
$671$	−538.308	−0.0309704
$672$	−187.627	−0.0107707
$673$	−10280.6	−0.588838	−0.294419	−	0.955676i	$-0.595126\pi$
$673$	−10280.6	−0.588838	−0.294419	+	0.955676i	$0.595126\pi$
$674$	15562.6	0.889390
$675$	2663.16	0.151859
$676$	744.712	0.0423710
$677$	−19616.7	−1.11363	−0.556817	−	0.830635i	$-0.687978\pi$
$677$	−19616.7	−1.11363	−0.556817	+	0.830635i	$0.687978\pi$
$678$	2851.89	0.161543
$679$	−240.054	−0.0135676
$680$	1435.63	0.0809615
$681$	−882.913	−0.0496818
$682$	21.9504	0.00123244
$683$	8069.93	0.452104	0.226052	−	0.974115i	$-0.427418\pi$
$683$	8069.93	0.452104	0.226052	+	0.974115i	$0.427418\pi$
$684$	0	0
$685$	−25026.8	−1.39595
$686$	2666.58	0.148412
$687$	−1179.48	−0.0655024
$688$	7815.95	0.433111
$689$	−9024.62	−0.499000
$690$	4571.98	0.252249
$691$	−7892.46	−0.434505	−0.217253	−	0.976115i	$-0.569710\pi$
$691$	−7892.46	−0.434505	−0.217253	+	0.976115i	$0.569710\pi$
$692$	−4631.57	−0.254430
$693$	16.7889	0.000920283 0
$694$	14248.7	0.779357
$695$	3977.39	0.217081
$696$	1293.81	0.0704625
$697$	3044.72	0.165462
$698$	−18846.7	−1.02200
$699$	−17301.1	−0.936179
$700$	−771.114	−0.0416362
$701$	−320.114	−0.0172475	−0.00862377	−	0.999963i	$-0.502745\pi$
$701$	−320.114	−0.0172475	−0.00862377	+	0.999963i	$0.502745\pi$
$702$	2636.16	0.141731
$703$	0	0
$704$	−61.0849	−0.00327020
$705$	−15791.7	−0.843618
$706$	8103.49	0.431981
$707$	−2386.20	−0.126934
$708$	3371.45	0.178965
$709$	−21212.9	−1.12365	−0.561826	−	0.827256i	$-0.689901\pi$
$709$	−21212.9	−1.12365	−0.561826	+	0.827256i	$0.689901\pi$
$710$	3183.84	0.168292
$711$	7238.87	0.381827
$712$	6987.94	0.367815
$713$	−585.923	−0.0307756
$714$	−140.720	−0.00737581
$715$	−696.791	−0.0364455
$716$	2736.95	0.142856
$717$	−5976.64	−0.311299
$718$	−21474.5	−1.11618
$719$	−12263.6	−0.636097	−0.318048	−	0.948075i	$-0.603027\pi$
$719$	−12263.6	−0.636097	−0.318048	+	0.948075i	$0.603027\pi$
$720$	2153.44	0.111464
$721$	−3855.50	−0.199149
$722$	0	0
$723$	6861.82	0.352965
$724$	−7462.16	−0.383051
$725$	5317.34	0.272388
$726$	−7980.53	−0.407969
$727$	−7283.24	−0.371555	−0.185778	−	0.982592i	$-0.559480\pi$
$727$	−7283.24	−0.371555	−0.185778	+	0.982592i	$0.559480\pi$
$728$	−763.296	−0.0388594
$729$	729.000	0.0370370
$730$	−29392.2	−1.49021
$731$	5861.96	0.296597
$732$	6767.98	0.341737
$733$	−1212.49	−0.0610973	−0.0305486	−	0.999533i	$-0.509725\pi$
$733$	−1212.49	−0.0610973	−0.0305486	+	0.999533i	$0.509725\pi$
$734$	11678.7	0.587287
$735$	−15216.8	−0.763644
$736$	1630.54	0.0816611
$737$	179.824	0.00898766
$738$	4567.08	0.227800
$739$	−13510.6	−0.672526	−0.336263	−	0.941768i	$-0.609163\pi$
$739$	−13510.6	−0.672526	−0.336263	+	0.941768i	$0.609163\pi$
$740$	−10533.3	−0.523257
$741$	0	0
$742$	722.613	0.0357520
$743$	−21833.5	−1.07805	−0.539027	−	0.842288i	$-0.681208\pi$
$743$	−21833.5	−1.07805	−0.539027	+	0.842288i	$0.681208\pi$
$744$	−275.975	−0.0135991
$745$	52509.7	2.58229
$746$	3849.61	0.188933
$747$	−9391.08	−0.459975
$748$	−45.8137	−0.00223946
$749$	1963.69	0.0957964
$750$	−2365.59	−0.115172
$751$	−8294.60	−0.403029	−0.201514	−	0.979486i	$-0.564586\pi$
$751$	−8294.60	−0.403029	−0.201514	+	0.979486i	$0.564586\pi$
$752$	−5631.93	−0.273106
$753$	−7986.88	−0.386531
$754$	5263.43	0.254221
$755$	30035.0	1.44780
$756$	−211.081	−0.0101547
$757$	8477.06	0.407007	0.203503	−	0.979074i	$-0.434767\pi$
$757$	8477.06	0.407007	0.203503	+	0.979074i	$0.434767\pi$
$758$	−7632.99	−0.365755
$759$	−145.901	−0.00697741
$760$	0	0
$761$	10485.1	0.499455	0.249727	−	0.968316i	$-0.419659\pi$
$761$	10485.1	0.499455	0.249727	+	0.968316i	$0.419659\pi$
$762$	−7336.87	−0.348801
$763$	3081.99	0.146233
$764$	−4899.73	−0.232024
$765$	1615.08	0.0763312
$766$	−13592.4	−0.641140
$767$	13715.6	0.645686
$768$	768.000	0.0360844
$769$	5998.42	0.281286	0.140643	−	0.990060i	$-0.455083\pi$
$769$	5998.42	0.281286	0.140643	+	0.990060i	$0.455083\pi$
$770$	55.7929	0.00261122
$771$	−17241.9	−0.805385
$772$	12663.3	0.590365
$773$	34841.8	1.62118	0.810590	−	0.585615i	$-0.199147\pi$
$773$	34841.8	1.62118	0.810590	+	0.585615i	$0.199147\pi$
$774$	8792.94	0.408341
$775$	−1134.21	−0.0525702
$776$	982.592	0.0454549
$777$	1032.47	0.0476702
$778$	−24174.0	−1.11399
$779$	0	0
$780$	8760.52	0.402150
$781$	−101.603	−0.00465509
$782$	1222.91	0.0559221
$783$	1455.54	0.0664327
$784$	−5426.88	−0.247216
$785$	−13556.9	−0.616389
$786$	12771.7	0.579583
$787$	−765.743	−0.0346834	−0.0173417	−	0.999850i	$-0.505520\pi$
$787$	−765.743	−0.0346834	−0.0173417	+	0.999850i	$0.505520\pi$
$788$	−21501.9	−0.972049
$789$	5145.62	0.232178
$790$	24056.3	1.08340
$791$	−928.979	−0.0417582
$792$	−68.7205	−0.00308318
$793$	27533.1	1.23295
$794$	−19449.6	−0.869321
$795$	−8293.59	−0.369992
$796$	8627.78	0.384175
$797$	391.549	0.0174020	0.00870099	−	0.999962i	$-0.497230\pi$
$797$	391.549	0.0174020	0.00870099	+	0.999962i	$0.497230\pi$
$798$	0	0
$799$	−4223.95	−0.187025
$800$	3156.34	0.139492
$801$	7861.43	0.346779
$802$	9037.79	0.397925
$803$	937.961	0.0412203
$804$	−2260.87	−0.0991725
$805$	−1489.28	−0.0652054
$806$	−1122.71	−0.0490642
$807$	−10342.7	−0.451151
$808$	9767.25	0.425261
$809$	−24108.3	−1.04771	−0.523857	−	0.851806i	$-0.675508\pi$
$809$	−24108.3	−1.04771	−0.523857	+	0.851806i	$0.675508\pi$
$810$	2422.62	0.105089
$811$	9320.14	0.403544	0.201772	−	0.979432i	$-0.435330\pi$
$811$	9320.14	0.403544	0.201772	+	0.979432i	$0.435330\pi$
$812$	−421.449	−0.0182142
$813$	1835.45	0.0791786
$814$	336.137	0.0144737
$815$	23837.8	1.02454
$816$	576.000	0.0247108
$817$	0	0
$818$	15606.4	0.667072
$819$	−858.708	−0.0366370
$820$	15177.4	0.646362
$821$	13779.2	0.585745	0.292872	−	0.956152i	$-0.405389\pi$
$821$	13779.2	0.585745	0.292872	+	0.956152i	$0.405389\pi$
$822$	−10041.2	−0.426067
$823$	−29248.8	−1.23882	−0.619411	−	0.785067i	$-0.712628\pi$
$823$	−29248.8	−1.23882	−0.619411	+	0.785067i	$0.712628\pi$
$824$	15781.4	0.667198
$825$	−282.429	−0.0119187
$826$	−1098.22	−0.0462616
$827$	3553.49	0.149416	0.0747079	−	0.997205i	$-0.476198\pi$
$827$	3553.49	0.149416	0.0747079	+	0.997205i	$0.476198\pi$
$828$	1834.36	0.0769909
$829$	20223.3	0.847265	0.423633	−	0.905834i	$-0.360755\pi$
$829$	20223.3	0.847265	0.423633	+	0.905834i	$0.360755\pi$
$830$	−31208.5	−1.30514
$831$	8251.90	0.344471
$832$	3124.34	0.130189
$833$	−4070.16	−0.169295
$834$	1595.80	0.0662567
$835$	−39006.9	−1.61664
$836$	0	0
$837$	−310.472	−0.0128214
$838$	9935.03	0.409546
$839$	4427.75	0.182196	0.0910982	−	0.995842i	$-0.470962\pi$
$839$	4427.75	0.182196	0.0910982	+	0.995842i	$0.470962\pi$
$840$	−701.466	−0.0288129
$841$	−21482.8	−0.880841
$842$	2132.63	0.0872865
$843$	5438.40	0.222193
$844$	21338.2	0.870250
$845$	2784.19	0.113348
$846$	−6335.93	−0.257487
$847$	2599.59	0.105458
$848$	−2957.81	−0.119778
$849$	−7921.64	−0.320224
$850$	2367.25	0.0955249
$851$	−8972.52	−0.361427
$852$	1277.42	0.0513657
$853$	28949.0	1.16201	0.581006	−	0.813899i	$-0.302659\pi$
$853$	28949.0	1.16201	0.581006	+	0.813899i	$0.302659\pi$
$854$	−2204.61	−0.0883376
$855$	0	0
$856$	−8037.80	−0.320942
$857$	−32611.1	−1.29985	−0.649927	−	0.759997i	$-0.725200\pi$
$857$	−32611.1	−1.29985	−0.649927	+	0.759997i	$0.725200\pi$
$858$	−279.565	−0.0111238
$859$	5903.52	0.234488	0.117244	−	0.993103i	$-0.462594\pi$
$859$	5903.52	0.234488	0.117244	+	0.993103i	$0.462594\pi$
$860$	29220.8	1.15863
$861$	−1487.69	−0.0588854
$862$	22456.9	0.887337
$863$	−9437.15	−0.372241	−0.186121	−	0.982527i	$-0.559592\pi$
$863$	−9437.15	−0.372241	−0.186121	+	0.982527i	$0.559592\pi$
$864$	864.000	0.0340207
$865$	−17315.6	−0.680635
$866$	760.965	0.0298599
$867$	−14307.0	−0.560428
$868$	89.8966	0.00351531
$869$	−767.683	−0.0299676
$870$	4837.07	0.188497
$871$	−9197.56	−0.357804
$872$	−12615.3	−0.489916
$873$	1105.42	0.0428553
$874$	0	0
$875$	770.572	0.0297715
$876$	−11792.7	−0.454837
$877$	4367.12	0.168149	0.0840747	−	0.996459i	$-0.473207\pi$
$877$	4367.12	0.168149	0.0840747	+	0.996459i	$0.473207\pi$
$878$	26873.2	1.03294
$879$	25379.8	0.973879
$880$	−228.373	−0.00874823
$881$	−6456.93	−0.246923	−0.123462	−	0.992349i	$-0.539400\pi$
$881$	−6456.93	−0.246923	−0.123462	+	0.992349i	$0.539400\pi$
$882$	−6105.24	−0.233077
$883$	−31191.8	−1.18878	−0.594388	−	0.804179i	$-0.702605\pi$
$883$	−31191.8	−1.18878	−0.594388	+	0.804179i	$0.702605\pi$
$884$	2343.25	0.0891541
$885$	12604.6	0.478754
$886$	35594.6	1.34969
$887$	−20836.9	−0.788765	−0.394382	−	0.918946i	$-0.629041\pi$
$887$	−20836.9	−0.788765	−0.394382	+	0.918946i	$0.629041\pi$
$888$	−4226.14	−0.159707
$889$	2389.93	0.0901637
$890$	26125.2	0.983954
$891$	−77.3105	−0.00290685
$892$	−23110.3	−0.867477
$893$	0	0
$894$	21067.8	0.788159
$895$	10232.4	0.382158
$896$	−250.170	−0.00932766
$897$	7462.45	0.277775
$898$	−8982.10	−0.333782
$899$	−619.896	−0.0229974
$900$	3550.88	0.131514
$901$	−2218.36	−0.0820247
$902$	−484.339	−0.0178789
$903$	−2864.23	−0.105554
$904$	3802.52	0.139900
$905$	−27898.1	−1.02471
$906$	12050.6	0.441893
$907$	12288.4	0.449868	0.224934	−	0.974374i	$-0.427783\pi$
$907$	12288.4	0.449868	0.224934	+	0.974374i	$0.427783\pi$
$908$	−1177.22	−0.0430257
$909$	10988.2	0.400940
$910$	−2853.67	−0.103954
$911$	−8026.96	−0.291927	−0.145963	−	0.989290i	$-0.546628\pi$
$911$	−8026.96	−0.291927	−0.145963	+	0.989290i	$0.546628\pi$
$912$	0	0
$913$	995.925	0.0361011
$914$	−36277.1	−1.31285
$915$	25302.8	0.914192
$916$	−1572.65	−0.0567267
$917$	−4160.29	−0.149820
$918$	648.000	0.0232976
$919$	−30372.0	−1.09018	−0.545092	−	0.838376i	$-0.683505\pi$
$919$	−30372.0	−1.09018	−0.545092	+	0.838376i	$0.683505\pi$
$920$	6095.97	0.218454
$921$	8185.83	0.292869
$922$	5747.63	0.205302
$923$	5196.72	0.185322
$924$	22.3851	0.000796988 0
$925$	−17368.6	−0.617381
$926$	12605.1	0.447330
$927$	17754.1	0.629040
$928$	1725.08	0.0610223
$929$	18346.9	0.647946	0.323973	−	0.946066i	$-0.394981\pi$
$929$	18346.9	0.647946	0.323973	+	0.946066i	$0.394981\pi$
$930$	−1031.76	−0.0363794
$931$	0	0
$932$	−23068.2	−0.810754
$933$	−2585.15	−0.0907117
$934$	−29735.6	−1.04173
$935$	−171.280	−0.00599084
$936$	3514.88	0.122743
$937$	−50703.5	−1.76778	−0.883890	−	0.467694i	$-0.845085\pi$
$937$	−50703.5	−1.76778	−0.883890	+	0.467694i	$0.845085\pi$
$938$	736.460	0.0256357
$939$	−5485.03	−0.190625
$940$	−21055.6	−0.730594
$941$	40535.6	1.40428	0.702138	−	0.712041i	$-0.252229\pi$
$941$	40535.6	1.40428	0.702138	+	0.712041i	$0.252229\pi$
$942$	−5439.27	−0.188133
$943$	12928.5	0.446458
$944$	4495.27	0.154988
$945$	−789.149	−0.0271651
$946$	−932.493	−0.0320486
$947$	−6304.36	−0.216330	−0.108165	−	0.994133i	$-0.534497\pi$
$947$	−6304.36	−0.216330	−0.108165	+	0.994133i	$0.534497\pi$
$948$	9651.83	0.330672
$949$	−47974.4	−1.64100
$950$	0	0
$951$	3790.84	0.129260
$952$	−187.627	−0.00638764
$953$	43469.8	1.47757	0.738786	−	0.673941i	$-0.235400\pi$
$953$	43469.8	1.47757	0.738786	+	0.673941i	$0.235400\pi$
$954$	−3327.54	−0.112928
$955$	−18318.2	−0.620695
$956$	−7968.85	−0.269593
$957$	−154.360	−0.00521396
$958$	−6154.50	−0.207560
$959$	3270.84	0.110136
$960$	2871.25	0.0965306
$961$	−29658.8	−0.995562
$962$	−17192.6	−0.576206
$963$	−9042.52	−0.302587
$964$	9149.09	0.305677
$965$	47343.1	1.57930
$966$	−597.528	−0.0199018
$967$	−20899.7	−0.695025	−0.347512	−	0.937675i	$-0.612973\pi$
$967$	−20899.7	−0.695025	−0.347512	+	0.937675i	$0.612973\pi$
$968$	−10640.7	−0.353311
$969$	0	0
$970$	3673.53	0.121598
$971$	−54625.8	−1.80538	−0.902691	−	0.430290i	$-0.858411\pi$
$971$	−54625.8	−1.80538	−0.902691	+	0.430290i	$0.858411\pi$
$972$	972.000	0.0320750
$973$	−519.819	−0.0171271
$974$	15909.2	0.523370
$975$	14445.5	0.474489
$976$	9023.97	0.295953
$977$	−4941.49	−0.161814	−0.0809070	−	0.996722i	$-0.525782\pi$
$977$	−4941.49	−0.161814	−0.0809070	+	0.996722i	$0.525782\pi$
$978$	9564.15	0.312707
$979$	−833.706	−0.0272169
$980$	−20289.0	−0.661335
$981$	−14192.2	−0.461897
$982$	272.016	0.00883950
$983$	39258.5	1.27381	0.636903	−	0.770944i	$-0.280215\pi$
$983$	39258.5	1.27381	0.636903	+	0.770944i	$0.280215\pi$
$984$	6089.44	0.197281
$985$	−80387.4	−2.60036
$986$	1293.81	0.0417885
$987$	2063.88	0.0665592
$988$	0	0
$989$	24891.1	0.800294
$990$	−256.919	−0.00824791
$991$	−10373.4	−0.332514	−0.166257	−	0.986082i	$-0.553168\pi$
$991$	−10373.4	−0.332514	−0.166257	+	0.986082i	$0.553168\pi$
$992$	−367.967	−0.0117772
$993$	29417.8	0.940128
$994$	−416.108	−0.0132778
$995$	32255.9	1.02772
$996$	−12521.4	−0.398350
$997$	44371.3	1.40948	0.704741	−	0.709465i	$-0.251063\pi$
$997$	44371.3	1.40948	0.704741	+	0.709465i	$0.251063\pi$
$998$	−16639.2	−0.527759
$999$	−4754.40	−0.150573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.q.1.2		2
19.7	even	3		114.4.e.c.49.1	yes	4
19.11	even	3		114.4.e.c.7.1	✓	4
19.18	odd	2		2166.4.a.k.1.2		2
57.11	odd	6		342.4.g.e.235.2		4
57.26	odd	6		342.4.g.e.163.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.c.7.1	✓	4	19.11	even	3
114.4.e.c.49.1	yes	4	19.7	even	3
342.4.g.e.163.2		4	57.26	odd	6
342.4.g.e.235.2		4	57.11	odd	6
2166.4.a.k.1.2		2	19.18	odd	2
2166.4.a.q.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.9545 q^{5} +6.00000 q^{6} -1.95445 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +14.9545 q^{5} +6.00000 q^{6} -1.95445 q^{7} +8.00000 q^{8} +9.00000 q^{9} +29.9089 q^{10} -0.954451 q^{11} +12.0000 q^{12} +48.8178 q^{13} -3.90890 q^{14} +44.8634 q^{15} +16.0000 q^{16} +12.0000 q^{17} +18.0000 q^{18} +59.8178 q^{20} -5.86335 q^{21} -1.90890 q^{22} +50.9545 q^{23} +24.0000 q^{24} +98.6356 q^{25} +97.6356 q^{26} +27.0000 q^{27} -7.81780 q^{28} +53.9089 q^{29} +89.7267 q^{30} -11.4990 q^{31} +32.0000 q^{32} -2.86335 q^{33} +24.0000 q^{34} -29.2277 q^{35} +36.0000 q^{36} -176.089 q^{37} +146.453 q^{39} +119.636 q^{40} +253.727 q^{41} -11.7267 q^{42} +488.497 q^{43} -3.81780 q^{44} +134.590 q^{45} +101.909 q^{46} -351.996 q^{47} +48.0000 q^{48} -339.180 q^{49} +197.271 q^{50} +36.0000 q^{51} +195.271 q^{52} -184.863 q^{53} +54.0000 q^{54} -14.2733 q^{55} -15.6356 q^{56} +107.818 q^{58} +280.954 q^{59} +179.453 q^{60} +563.998 q^{61} -22.9979 q^{62} -17.5901 q^{63} +64.0000 q^{64} +730.043 q^{65} -5.72671 q^{66} -188.406 q^{67} +48.0000 q^{68} +152.863 q^{69} -58.4555 q^{70} +106.451 q^{71} +72.0000 q^{72} -982.723 q^{73} -352.178 q^{74} +295.907 q^{75} +1.86543 q^{77} +292.907 q^{78} +804.319 q^{79} +239.271 q^{80} +81.0000 q^{81} +507.453 q^{82} -1043.45 q^{83} -23.4534 q^{84} +179.453 q^{85} +976.994 q^{86} +161.727 q^{87} -7.63561 q^{88} +873.493 q^{89} +269.180 q^{90} -95.4120 q^{91} +203.818 q^{92} -34.4969 q^{93} -703.992 q^{94} +96.0000 q^{96} +122.824 q^{97} -678.360 q^{98} -8.59006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 8 q^{5} + 12 q^{6} + 18 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 8 * q^5 + 12 * q^6 + 18 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 8 q^{5} + 12 q^{6} + 18 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 20 q^{11} + 24 q^{12} + 10 q^{13} + 36 q^{14} + 24 q^{15} + 32 q^{16} + 24 q^{17} + 36 q^{18} + 32 q^{20} + 54 q^{21} + 40 q^{22} + 80 q^{23} + 48 q^{24} + 22 q^{25} + 20 q^{26} + 54 q^{27} + 72 q^{28} + 64 q^{29} + 48 q^{30} + 218 q^{31} + 64 q^{32} + 60 q^{33} + 48 q^{34} - 168 q^{35} + 72 q^{36} + 86 q^{37} + 30 q^{39} + 64 q^{40} + 376 q^{41} + 108 q^{42} + 254 q^{43} + 80 q^{44} + 72 q^{45} + 160 q^{46} + 260 q^{47} + 96 q^{48} - 284 q^{49} + 44 q^{50} + 72 q^{51} + 40 q^{52} - 304 q^{53} + 108 q^{54} - 160 q^{55} + 144 q^{56} + 128 q^{58} + 540 q^{59} + 96 q^{60} + 646 q^{61} + 436 q^{62} + 162 q^{63} + 128 q^{64} + 1000 q^{65} + 120 q^{66} + 390 q^{67} + 96 q^{68} + 240 q^{69} - 336 q^{70} - 532 q^{71} + 144 q^{72} - 870 q^{73} + 172 q^{74} + 66 q^{75} + 420 q^{77} + 60 q^{78} + 1762 q^{79} + 128 q^{80} + 162 q^{81} + 752 q^{82} - 1824 q^{83} + 216 q^{84} + 96 q^{85} + 508 q^{86} + 192 q^{87} + 160 q^{88} + 60 q^{89} + 144 q^{90} - 870 q^{91} + 320 q^{92} + 654 q^{93} + 520 q^{94} + 192 q^{96} + 1604 q^{97} - 568 q^{98} + 180 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 8 * q^5 + 12 * q^6 + 18 * q^7 + 16 * q^8 + 18 * q^9 + 16 * q^10 + 20 * q^11 + 24 * q^12 + 10 * q^13 + 36 * q^14 + 24 * q^15 + 32 * q^16 + 24 * q^17 + 36 * q^18 + 32 * q^20 + 54 * q^21 + 40 * q^22 + 80 * q^23 + 48 * q^24 + 22 * q^25 + 20 * q^26 + 54 * q^27 + 72 * q^28 + 64 * q^29 + 48 * q^30 + 218 * q^31 + 64 * q^32 + 60 * q^33 + 48 * q^34 - 168 * q^35 + 72 * q^36 + 86 * q^37 + 30 * q^39 + 64 * q^40 + 376 * q^41 + 108 * q^42 + 254 * q^43 + 80 * q^44 + 72 * q^45 + 160 * q^46 + 260 * q^47 + 96 * q^48 - 284 * q^49 + 44 * q^50 + 72 * q^51 + 40 * q^52 - 304 * q^53 + 108 * q^54 - 160 * q^55 + 144 * q^56 + 128 * q^58 + 540 * q^59 + 96 * q^60 + 646 * q^61 + 436 * q^62 + 162 * q^63 + 128 * q^64 + 1000 * q^65 + 120 * q^66 + 390 * q^67 + 96 * q^68 + 240 * q^69 - 336 * q^70 - 532 * q^71 + 144 * q^72 - 870 * q^73 + 172 * q^74 + 66 * q^75 + 420 * q^77 + 60 * q^78 + 1762 * q^79 + 128 * q^80 + 162 * q^81 + 752 * q^82 - 1824 * q^83 + 216 * q^84 + 96 * q^85 + 508 * q^86 + 192 * q^87 + 160 * q^88 + 60 * q^89 + 144 * q^90 - 870 * q^91 + 320 * q^92 + 654 * q^93 + 520 * q^94 + 192 * q^96 + 1604 * q^97 - 568 * q^98 + 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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