LMFDB - Embedded newform 2166.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,2,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, 2, names="a")

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

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$17.2955970778$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.38197 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.38197 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.38197 q^{10} +0.381966 q^{11} -1.00000 q^{12} -4.47214 q^{13} -2.61803 q^{14} -1.38197 q^{15} +1.00000 q^{16} +0.763932 q^{17} -1.00000 q^{18} +1.38197 q^{20} -2.61803 q^{21} -0.381966 q^{22} +2.47214 q^{23} +1.00000 q^{24} -3.09017 q^{25} +4.47214 q^{26} -1.00000 q^{27} +2.61803 q^{28} +7.61803 q^{29} +1.38197 q^{30} +10.4721 q^{31} -1.00000 q^{32} -0.381966 q^{33} -0.763932 q^{34} +3.61803 q^{35} +1.00000 q^{36} +10.9443 q^{37} +4.47214 q^{39} -1.38197 q^{40} -1.70820 q^{41} +2.61803 q^{42} +3.52786 q^{43} +0.381966 q^{44} +1.38197 q^{45} -2.47214 q^{46} -10.0000 q^{47} -1.00000 q^{48} -0.145898 q^{49} +3.09017 q^{50} -0.763932 q^{51} -4.47214 q^{52} -13.3262 q^{53} +1.00000 q^{54} +0.527864 q^{55} -2.61803 q^{56} -7.61803 q^{58} -5.14590 q^{59} -1.38197 q^{60} +4.47214 q^{61} -10.4721 q^{62} +2.61803 q^{63} +1.00000 q^{64} -6.18034 q^{65} +0.381966 q^{66} +8.94427 q^{67} +0.763932 q^{68} -2.47214 q^{69} -3.61803 q^{70} -14.9443 q^{71} -1.00000 q^{72} +8.32624 q^{73} -10.9443 q^{74} +3.09017 q^{75} +1.00000 q^{77} -4.47214 q^{78} +6.38197 q^{79} +1.38197 q^{80} +1.00000 q^{81} +1.70820 q^{82} +7.09017 q^{83} -2.61803 q^{84} +1.05573 q^{85} -3.52786 q^{86} -7.61803 q^{87} -0.381966 q^{88} +18.6525 q^{89} -1.38197 q^{90} -11.7082 q^{91} +2.47214 q^{92} -10.4721 q^{93} +10.0000 q^{94} +1.00000 q^{96} -4.47214 q^{97} +0.145898 q^{98} +0.381966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 5 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 3 q^{14} - 5 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 5 q^{20} - 3 q^{21} - 3 q^{22} - 4 q^{23} + 2 q^{24} + 5 q^{25} - 2 q^{27} + 3 q^{28} + 13 q^{29} + 5 q^{30} + 12 q^{31} - 2 q^{32} - 3 q^{33} - 6 q^{34} + 5 q^{35} + 2 q^{36} + 4 q^{37} - 5 q^{40} + 10 q^{41} + 3 q^{42} + 16 q^{43} + 3 q^{44} + 5 q^{45} + 4 q^{46} - 20 q^{47} - 2 q^{48} - 7 q^{49} - 5 q^{50} - 6 q^{51} - 11 q^{53} + 2 q^{54} + 10 q^{55} - 3 q^{56} - 13 q^{58} - 17 q^{59} - 5 q^{60} - 12 q^{62} + 3 q^{63} + 2 q^{64} + 10 q^{65} + 3 q^{66} + 6 q^{68} + 4 q^{69} - 5 q^{70} - 12 q^{71} - 2 q^{72} + q^{73} - 4 q^{74} - 5 q^{75} + 2 q^{77} + 15 q^{79} + 5 q^{80} + 2 q^{81} - 10 q^{82} + 3 q^{83} - 3 q^{84} + 20 q^{85} - 16 q^{86} - 13 q^{87} - 3 q^{88} + 6 q^{89} - 5 q^{90} - 10 q^{91} - 4 q^{92} - 12 q^{93} + 20 q^{94} + 2 q^{96} + 7 q^{98} + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 5 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - 5 * q^10 + 3 * q^11 - 2 * q^12 - 3 * q^14 - 5 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^18 + 5 * q^20 - 3 * q^21 - 3 * q^22 - 4 * q^23 + 2 * q^24 + 5 * q^25 - 2 * q^27 + 3 * q^28 + 13 * q^29 + 5 * q^30 + 12 * q^31 - 2 * q^32 - 3 * q^33 - 6 * q^34 + 5 * q^35 + 2 * q^36 + 4 * q^37 - 5 * q^40 + 10 * q^41 + 3 * q^42 + 16 * q^43 + 3 * q^44 + 5 * q^45 + 4 * q^46 - 20 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 6 * q^51 - 11 * q^53 + 2 * q^54 + 10 * q^55 - 3 * q^56 - 13 * q^58 - 17 * q^59 - 5 * q^60 - 12 * q^62 + 3 * q^63 + 2 * q^64 + 10 * q^65 + 3 * q^66 + 6 * q^68 + 4 * q^69 - 5 * q^70 - 12 * q^71 - 2 * q^72 + q^73 - 4 * q^74 - 5 * q^75 + 2 * q^77 + 15 * q^79 + 5 * q^80 + 2 * q^81 - 10 * q^82 + 3 * q^83 - 3 * q^84 + 20 * q^85 - 16 * q^86 - 13 * q^87 - 3 * q^88 + 6 * q^89 - 5 * q^90 - 10 * q^91 - 4 * q^92 - 12 * q^93 + 20 * q^94 + 2 * q^96 + 7 * q^98 + 3 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.38197	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	1.38197	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	1.00000	0.408248
$7$	2.61803	0.989524	0.494762	−	0.869029i	$-0.335255\pi$
$7$	2.61803	0.989524	0.494762	+	0.869029i	$0.335255\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.38197	−0.437016
$11$	0.381966	0.115167	0.0575835	−	0.998341i	$-0.481660\pi$
$11$	0.381966	0.115167	0.0575835	+	0.998341i	$0.481660\pi$
$12$	−1.00000	−0.288675
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	−2.61803	−0.699699
$15$	−1.38197	−0.356822
$16$	1.00000	0.250000
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	−1.00000	−0.235702
$19$	0	0
$19$	0	0
$20$	1.38197	0.309017
$21$	−2.61803	−0.571302
$22$	−0.381966	−0.0814354
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	1.00000	0.204124
$25$	−3.09017	−0.618034
$26$	4.47214	0.877058
$27$	−1.00000	−0.192450
$28$	2.61803	0.494762
$29$	7.61803	1.41463	0.707317	−	0.706897i	$-0.249905\pi$
$29$	7.61803	1.41463	0.707317	+	0.706897i	$0.249905\pi$
$30$	1.38197	0.252311
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	−1.00000	−0.176777
$33$	−0.381966	−0.0664917
$34$	−0.763932	−0.131013
$35$	3.61803	0.611559
$36$	1.00000	0.166667
$37$	10.9443	1.79923	0.899614	−	0.436687i	$-0.143848\pi$
$37$	10.9443	1.79923	0.899614	+	0.436687i	$0.143848\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	−1.38197	−0.218508
$41$	−1.70820	−0.266777	−0.133388	−	0.991064i	$-0.542586\pi$
$41$	−1.70820	−0.266777	−0.133388	+	0.991064i	$0.542586\pi$
$42$	2.61803	0.403971
$43$	3.52786	0.537994	0.268997	−	0.963141i	$-0.413308\pi$
$43$	3.52786	0.537994	0.268997	+	0.963141i	$0.413308\pi$
$44$	0.381966	0.0575835
$45$	1.38197	0.206011
$46$	−2.47214	−0.364497
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−1.00000	−0.144338
$49$	−0.145898	−0.0208426
$50$	3.09017	0.437016
$51$	−0.763932	−0.106972
$52$	−4.47214	−0.620174
$53$	−13.3262	−1.83050	−0.915250	−	0.402887i	$-0.868007\pi$
$53$	−13.3262	−1.83050	−0.915250	+	0.402887i	$0.868007\pi$
$54$	1.00000	0.136083
$55$	0.527864	0.0711772
$56$	−2.61803	−0.349850
$57$	0	0
$58$	−7.61803	−1.00030
$59$	−5.14590	−0.669939	−0.334969	−	0.942229i	$-0.608726\pi$
$59$	−5.14590	−0.669939	−0.334969	+	0.942229i	$0.608726\pi$
$60$	−1.38197	−0.178411
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−10.4721	−1.32996
$63$	2.61803	0.329841
$64$	1.00000	0.125000
$65$	−6.18034	−0.766577
$66$	0.381966	0.0470168
$67$	8.94427	1.09272	0.546358	−	0.837552i	$-0.316014\pi$
$67$	8.94427	1.09272	0.546358	+	0.837552i	$0.316014\pi$
$68$	0.763932	0.0926404
$69$	−2.47214	−0.297610
$70$	−3.61803	−0.432438
$71$	−14.9443	−1.77356	−0.886779	−	0.462193i	$-0.847063\pi$
$71$	−14.9443	−1.77356	−0.886779	+	0.462193i	$0.847063\pi$
$72$	−1.00000	−0.117851
$73$	8.32624	0.974512	0.487256	−	0.873259i	$-0.337998\pi$
$73$	8.32624	0.974512	0.487256	+	0.873259i	$0.337998\pi$
$74$	−10.9443	−1.27225
$75$	3.09017	0.356822
$76$	0	0
$77$	1.00000	0.113961
$78$	−4.47214	−0.506370
$79$	6.38197	0.718027	0.359014	−	0.933332i	$-0.383113\pi$
$79$	6.38197	0.718027	0.359014	+	0.933332i	$0.383113\pi$
$80$	1.38197	0.154508
$81$	1.00000	0.111111
$82$	1.70820	0.188640
$83$	7.09017	0.778247	0.389124	−	0.921186i	$-0.372778\pi$
$83$	7.09017	0.778247	0.389124	+	0.921186i	$0.372778\pi$
$84$	−2.61803	−0.285651
$85$	1.05573	0.114510
$86$	−3.52786	−0.380419
$87$	−7.61803	−0.816739
$88$	−0.381966	−0.0407177
$89$	18.6525	1.97716	0.988579	−	0.150702i	$-0.0481534\pi$
$89$	18.6525	1.97716	0.988579	+	0.150702i	$0.0481534\pi$
$90$	−1.38197	−0.145672
$91$	−11.7082	−1.22735
$92$	2.47214	0.257738
$93$	−10.4721	−1.08591
$94$	10.0000	1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	−4.47214	−0.454077	−0.227038	−	0.973886i	$-0.572904\pi$
$97$	−4.47214	−0.454077	−0.227038	+	0.973886i	$0.572904\pi$
$98$	0.145898	0.0147379
$99$	0.381966	0.0383890
$100$	−3.09017	−0.309017
$101$	6.94427	0.690981	0.345490	−	0.938422i	$-0.387713\pi$
$101$	6.94427	0.690981	0.345490	+	0.938422i	$0.387713\pi$
$102$	0.763932	0.0756405
$103$	3.14590	0.309975	0.154987	−	0.987916i	$-0.450466\pi$
$103$	3.14590	0.309975	0.154987	+	0.987916i	$0.450466\pi$
$104$	4.47214	0.438529
$105$	−3.61803	−0.353084
$106$	13.3262	1.29436
$107$	−0.909830	−0.0879566	−0.0439783	−	0.999032i	$-0.514003\pi$
$107$	−0.909830	−0.0879566	−0.0439783	+	0.999032i	$0.514003\pi$
$108$	−1.00000	−0.0962250
$109$	8.76393	0.839432	0.419716	−	0.907655i	$-0.362130\pi$
$109$	8.76393	0.839432	0.419716	+	0.907655i	$0.362130\pi$
$110$	−0.527864	−0.0503299
$111$	−10.9443	−1.03878
$112$	2.61803	0.247381
$113$	12.1803	1.14583	0.572915	−	0.819615i	$-0.305813\pi$
$113$	12.1803	1.14583	0.572915	+	0.819615i	$0.305813\pi$
$114$	0	0
$115$	3.41641	0.318582
$116$	7.61803	0.707317
$117$	−4.47214	−0.413449
$118$	5.14590	0.473718
$119$	2.00000	0.183340
$120$	1.38197	0.126156
$121$	−10.8541	−0.986737
$122$	−4.47214	−0.404888
$123$	1.70820	0.154024
$124$	10.4721	0.940426
$125$	−11.1803	−1.00000
$126$	−2.61803	−0.233233
$127$	−16.0902	−1.42777	−0.713886	−	0.700262i	$-0.753066\pi$
$127$	−16.0902	−1.42777	−0.713886	+	0.700262i	$0.753066\pi$
$128$	−1.00000	−0.0883883
$129$	−3.52786	−0.310611
$130$	6.18034	0.542052
$131$	−8.61803	−0.752961	−0.376481	−	0.926425i	$-0.622866\pi$
$131$	−8.61803	−0.752961	−0.376481	+	0.926425i	$0.622866\pi$
$132$	−0.381966	−0.0332459
$133$	0	0
$134$	−8.94427	−0.772667
$135$	−1.38197	−0.118941
$136$	−0.763932	−0.0655066
$137$	8.76393	0.748753	0.374377	−	0.927277i	$-0.377857\pi$
$137$	8.76393	0.748753	0.374377	+	0.927277i	$0.377857\pi$
$138$	2.47214	0.210442
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	3.61803	0.305780
$141$	10.0000	0.842152
$142$	14.9443	1.25410
$143$	−1.70820	−0.142847
$144$	1.00000	0.0833333
$145$	10.5279	0.874292
$146$	−8.32624	−0.689084
$147$	0.145898	0.0120335
$148$	10.9443	0.899614
$149$	10.3262	0.845958	0.422979	−	0.906139i	$-0.360984\pi$
$149$	10.3262	0.845958	0.422979	+	0.906139i	$0.360984\pi$
$150$	−3.09017	−0.252311
$151$	13.6180	1.10822	0.554110	−	0.832443i	$-0.313059\pi$
$151$	13.6180	1.10822	0.554110	+	0.832443i	$0.313059\pi$
$152$	0	0
$153$	0.763932	0.0617602
$154$	−1.00000	−0.0805823
$155$	14.4721	1.16243
$156$	4.47214	0.358057
$157$	0.763932	0.0609684	0.0304842	−	0.999535i	$-0.490295\pi$
$157$	0.763932	0.0609684	0.0304842	+	0.999535i	$0.490295\pi$
$158$	−6.38197	−0.507722
$159$	13.3262	1.05684
$160$	−1.38197	−0.109254
$161$	6.47214	0.510076
$162$	−1.00000	−0.0785674
$163$	−1.70820	−0.133797	−0.0668984	−	0.997760i	$-0.521310\pi$
$163$	−1.70820	−0.133797	−0.0668984	+	0.997760i	$0.521310\pi$
$164$	−1.70820	−0.133388
$165$	−0.527864	−0.0410942
$166$	−7.09017	−0.550304
$167$	−12.4721	−0.965123	−0.482561	−	0.875862i	$-0.660293\pi$
$167$	−12.4721	−0.965123	−0.482561	+	0.875862i	$0.660293\pi$
$168$	2.61803	0.201986
$169$	7.00000	0.538462
$170$	−1.05573	−0.0809706
$171$	0	0
$172$	3.52786	0.268997
$173$	6.32624	0.480975	0.240487	−	0.970652i	$-0.422693\pi$
$173$	6.32624	0.480975	0.240487	+	0.970652i	$0.422693\pi$
$174$	7.61803	0.577522
$175$	−8.09017	−0.611559
$176$	0.381966	0.0287918
$177$	5.14590	0.386789
$178$	−18.6525	−1.39806
$179$	3.61803	0.270425	0.135212	−	0.990817i	$-0.456828\pi$
$179$	3.61803	0.270425	0.135212	+	0.990817i	$0.456828\pi$
$180$	1.38197	0.103006
$181$	20.9443	1.55678	0.778388	−	0.627784i	$-0.216038\pi$
$181$	20.9443	1.55678	0.778388	+	0.627784i	$0.216038\pi$
$182$	11.7082	0.867870
$183$	−4.47214	−0.330590
$184$	−2.47214	−0.182248
$185$	15.1246	1.11198
$186$	10.4721	0.767854
$187$	0.291796	0.0213382
$188$	−10.0000	−0.729325
$189$	−2.61803	−0.190434
$190$	0	0
$191$	6.94427	0.502470	0.251235	−	0.967926i	$-0.419163\pi$
$191$	6.94427	0.502470	0.251235	+	0.967926i	$0.419163\pi$
$192$	−1.00000	−0.0721688
$193$	−4.61803	−0.332413	−0.166207	−	0.986091i	$-0.553152\pi$
$193$	−4.61803	−0.332413	−0.166207	+	0.986091i	$0.553152\pi$
$194$	4.47214	0.321081
$195$	6.18034	0.442583
$196$	−0.145898	−0.0104213
$197$	3.61803	0.257774	0.128887	−	0.991659i	$-0.458860\pi$
$197$	3.61803	0.257774	0.128887	+	0.991659i	$0.458860\pi$
$198$	−0.381966	−0.0271451
$199$	7.90983	0.560713	0.280356	−	0.959896i	$-0.409547\pi$
$199$	7.90983	0.560713	0.280356	+	0.959896i	$0.409547\pi$
$200$	3.09017	0.218508
$201$	−8.94427	−0.630880
$202$	−6.94427	−0.488597
$203$	19.9443	1.39981
$204$	−0.763932	−0.0534859
$205$	−2.36068	−0.164877
$206$	−3.14590	−0.219185
$207$	2.47214	0.171825
$208$	−4.47214	−0.310087
$209$	0	0
$210$	3.61803	0.249668
$211$	25.1246	1.72965	0.864825	−	0.502074i	$-0.167429\pi$
$211$	25.1246	1.72965	0.864825	+	0.502074i	$0.167429\pi$
$212$	−13.3262	−0.915250
$213$	14.9443	1.02396
$214$	0.909830	0.0621947
$215$	4.87539	0.332499
$216$	1.00000	0.0680414
$217$	27.4164	1.86115
$218$	−8.76393	−0.593568
$219$	−8.32624	−0.562635
$220$	0.527864	0.0355886
$221$	−3.41641	−0.229812
$222$	10.9443	0.734531
$223$	−8.79837	−0.589183	−0.294591	−	0.955623i	$-0.595184\pi$
$223$	−8.79837	−0.589183	−0.294591	+	0.955623i	$0.595184\pi$
$224$	−2.61803	−0.174925
$225$	−3.09017	−0.206011
$226$	−12.1803	−0.810224
$227$	−26.5066	−1.75930	−0.879652	−	0.475618i	$-0.842224\pi$
$227$	−26.5066	−1.75930	−0.879652	+	0.475618i	$0.842224\pi$
$228$	0	0
$229$	13.5279	0.893946	0.446973	−	0.894547i	$-0.352502\pi$
$229$	13.5279	0.893946	0.446973	+	0.894547i	$0.352502\pi$
$230$	−3.41641	−0.225271
$231$	−1.00000	−0.0657952
$232$	−7.61803	−0.500148
$233$	18.4721	1.21015	0.605075	−	0.796169i	$-0.293143\pi$
$233$	18.4721	1.21015	0.605075	+	0.796169i	$0.293143\pi$
$234$	4.47214	0.292353
$235$	−13.8197	−0.901495
$236$	−5.14590	−0.334969
$237$	−6.38197	−0.414553
$238$	−2.00000	−0.129641
$239$	−23.8885	−1.54522	−0.772611	−	0.634880i	$-0.781050\pi$
$239$	−23.8885	−1.54522	−0.772611	+	0.634880i	$0.781050\pi$
$240$	−1.38197	−0.0892055
$241$	−9.03444	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$241$	−9.03444	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$242$	10.8541	0.697728
$243$	−1.00000	−0.0641500
$244$	4.47214	0.286299
$245$	−0.201626	−0.0128814
$246$	−1.70820	−0.108911
$247$	0	0
$248$	−10.4721	−0.664981
$249$	−7.09017	−0.449321
$250$	11.1803	0.707107
$251$	−9.32624	−0.588667	−0.294333	−	0.955703i	$-0.595098\pi$
$251$	−9.32624	−0.588667	−0.294333	+	0.955703i	$0.595098\pi$
$252$	2.61803	0.164921
$253$	0.944272	0.0593659
$254$	16.0902	1.00959
$255$	−1.05573	−0.0661123
$256$	1.00000	0.0625000
$257$	−8.65248	−0.539727	−0.269863	−	0.962899i	$-0.586979\pi$
$257$	−8.65248	−0.539727	−0.269863	+	0.962899i	$0.586979\pi$
$258$	3.52786	0.219635
$259$	28.6525	1.78038
$260$	−6.18034	−0.383288
$261$	7.61803	0.471544
$262$	8.61803	0.532424
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0.381966	0.0235084
$265$	−18.4164	−1.13131
$266$	0	0
$267$	−18.6525	−1.14151
$268$	8.94427	0.546358
$269$	11.5623	0.704966	0.352483	−	0.935818i	$-0.385337\pi$
$269$	11.5623	0.704966	0.352483	+	0.935818i	$0.385337\pi$
$270$	1.38197	0.0841038
$271$	−11.5623	−0.702360	−0.351180	−	0.936308i	$-0.614219\pi$
$271$	−11.5623	−0.702360	−0.351180	+	0.936308i	$0.614219\pi$
$272$	0.763932	0.0463202
$273$	11.7082	0.708613
$274$	−8.76393	−0.529448
$275$	−1.18034	−0.0711772
$276$	−2.47214	−0.148805
$277$	24.8328	1.49206	0.746030	−	0.665913i	$-0.231958\pi$
$277$	24.8328	1.49206	0.746030	+	0.665913i	$0.231958\pi$
$278$	−16.0000	−0.959616
$279$	10.4721	0.626950
$280$	−3.61803	−0.216219
$281$	13.4164	0.800356	0.400178	−	0.916437i	$-0.368948\pi$
$281$	13.4164	0.800356	0.400178	+	0.916437i	$0.368948\pi$
$282$	−10.0000	−0.595491
$283$	12.4721	0.741392	0.370696	−	0.928754i	$-0.379119\pi$
$283$	12.4721	0.741392	0.370696	+	0.928754i	$0.379119\pi$
$284$	−14.9443	−0.886779
$285$	0	0
$286$	1.70820	0.101008
$287$	−4.47214	−0.263982
$288$	−1.00000	−0.0589256
$289$	−16.4164	−0.965671
$290$	−10.5279	−0.618217
$291$	4.47214	0.262161
$292$	8.32624	0.487256
$293$	−20.7984	−1.21505	−0.607527	−	0.794299i	$-0.707838\pi$
$293$	−20.7984	−1.21505	−0.607527	+	0.794299i	$0.707838\pi$
$294$	−0.145898	−0.00850895
$295$	−7.11146	−0.414045
$296$	−10.9443	−0.636123
$297$	−0.381966	−0.0221639
$298$	−10.3262	−0.598183
$299$	−11.0557	−0.639369
$300$	3.09017	0.178411
$301$	9.23607	0.532358
$302$	−13.6180	−0.783630
$303$	−6.94427	−0.398938
$304$	0	0
$305$	6.18034	0.353885
$306$	−0.763932	−0.0436711
$307$	−1.70820	−0.0974923	−0.0487462	−	0.998811i	$-0.515523\pi$
$307$	−1.70820	−0.0974923	−0.0487462	+	0.998811i	$0.515523\pi$
$308$	1.00000	0.0569803
$309$	−3.14590	−0.178964
$310$	−14.4721	−0.821962
$311$	−28.3607	−1.60819	−0.804093	−	0.594503i	$-0.797349\pi$
$311$	−28.3607	−1.60819	−0.804093	+	0.594503i	$0.797349\pi$
$312$	−4.47214	−0.253185
$313$	−18.1459	−1.02567	−0.512833	−	0.858488i	$-0.671404\pi$
$313$	−18.1459	−1.02567	−0.512833	+	0.858488i	$0.671404\pi$
$314$	−0.763932	−0.0431112
$315$	3.61803	0.203853
$316$	6.38197	0.359014
$317$	31.9787	1.79610	0.898052	−	0.439890i	$-0.144983\pi$
$317$	31.9787	1.79610	0.898052	+	0.439890i	$0.144983\pi$
$318$	−13.3262	−0.747298
$319$	2.90983	0.162919
$320$	1.38197	0.0772542
$321$	0.909830	0.0507818
$322$	−6.47214	−0.360678
$323$	0	0
$324$	1.00000	0.0555556
$325$	13.8197	0.766577
$326$	1.70820	0.0946087
$327$	−8.76393	−0.484647
$328$	1.70820	0.0943198
$329$	−26.1803	−1.44337
$330$	0.527864	0.0290580
$331$	11.8885	0.653453	0.326727	−	0.945119i	$-0.394054\pi$
$331$	11.8885	0.653453	0.326727	+	0.945119i	$0.394054\pi$
$332$	7.09017	0.389124
$333$	10.9443	0.599742
$334$	12.4721	0.682445
$335$	12.3607	0.675336
$336$	−2.61803	−0.142825
$337$	29.4164	1.60241	0.801207	−	0.598387i	$-0.204192\pi$
$337$	29.4164	1.60241	0.801207	+	0.598387i	$0.204192\pi$
$338$	−7.00000	−0.380750
$339$	−12.1803	−0.661545
$340$	1.05573	0.0572549
$341$	4.00000	0.216612
$342$	0	0
$343$	−18.7082	−1.01015
$344$	−3.52786	−0.190210
$345$	−3.41641	−0.183933
$346$	−6.32624	−0.340101
$347$	9.50658	0.510340	0.255170	−	0.966896i	$-0.417869\pi$
$347$	9.50658	0.510340	0.255170	+	0.966896i	$0.417869\pi$
$348$	−7.61803	−0.408369
$349$	−25.1246	−1.34489	−0.672445	−	0.740147i	$-0.734756\pi$
$349$	−25.1246	−1.34489	−0.672445	+	0.740147i	$0.734756\pi$
$350$	8.09017	0.432438
$351$	4.47214	0.238705
$352$	−0.381966	−0.0203589
$353$	23.4164	1.24633	0.623165	−	0.782091i	$-0.285847\pi$
$353$	23.4164	1.24633	0.623165	+	0.782091i	$0.285847\pi$
$354$	−5.14590	−0.273501
$355$	−20.6525	−1.09612
$356$	18.6525	0.988579
$357$	−2.00000	−0.105851
$358$	−3.61803	−0.191219
$359$	35.5967	1.87872	0.939362	−	0.342926i	$-0.111418\pi$
$359$	35.5967	1.87872	0.939362	+	0.342926i	$0.111418\pi$
$360$	−1.38197	−0.0728360
$361$	0	0
$362$	−20.9443	−1.10081
$363$	10.8541	0.569693
$364$	−11.7082	−0.613677
$365$	11.5066	0.602282
$366$	4.47214	0.233762
$367$	−13.5279	−0.706149	−0.353074	−	0.935595i	$-0.614864\pi$
$367$	−13.5279	−0.706149	−0.353074	+	0.935595i	$0.614864\pi$
$368$	2.47214	0.128869
$369$	−1.70820	−0.0889255
$370$	−15.1246	−0.786291
$371$	−34.8885	−1.81132
$372$	−10.4721	−0.542955
$373$	0.763932	0.0395549	0.0197775	−	0.999804i	$-0.493704\pi$
$373$	0.763932	0.0395549	0.0197775	+	0.999804i	$0.493704\pi$
$374$	−0.291796	−0.0150884
$375$	11.1803	0.577350
$376$	10.0000	0.515711
$377$	−34.0689	−1.75464
$378$	2.61803	0.134657
$379$	21.4164	1.10009	0.550043	−	0.835136i	$-0.314611\pi$
$379$	21.4164	1.10009	0.550043	+	0.835136i	$0.314611\pi$
$380$	0	0
$381$	16.0902	0.824324
$382$	−6.94427	−0.355300
$383$	1.41641	0.0723751	0.0361875	−	0.999345i	$-0.488479\pi$
$383$	1.41641	0.0723751	0.0361875	+	0.999345i	$0.488479\pi$
$384$	1.00000	0.0510310
$385$	1.38197	0.0704315
$386$	4.61803	0.235052
$387$	3.52786	0.179331
$388$	−4.47214	−0.227038
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−6.18034	−0.312954
$391$	1.88854	0.0955078
$392$	0.145898	0.00736896
$393$	8.61803	0.434722
$394$	−3.61803	−0.182274
$395$	8.81966	0.443765
$396$	0.381966	0.0191945
$397$	12.3607	0.620365	0.310182	−	0.950677i	$-0.399610\pi$
$397$	12.3607	0.620365	0.310182	+	0.950677i	$0.399610\pi$
$398$	−7.90983	−0.396484
$399$	0	0
$400$	−3.09017	−0.154508
$401$	−6.18034	−0.308631	−0.154316	−	0.988022i	$-0.549317\pi$
$401$	−6.18034	−0.308631	−0.154316	+	0.988022i	$0.549317\pi$
$402$	8.94427	0.446100
$403$	−46.8328	−2.33291
$404$	6.94427	0.345490
$405$	1.38197	0.0686704
$406$	−19.9443	−0.989818
$407$	4.18034	0.207212
$408$	0.763932	0.0378203
$409$	2.72949	0.134965	0.0674823	−	0.997720i	$-0.478503\pi$
$409$	2.72949	0.134965	0.0674823	+	0.997720i	$0.478503\pi$
$410$	2.36068	0.116586
$411$	−8.76393	−0.432293
$412$	3.14590	0.154987
$413$	−13.4721	−0.662920
$414$	−2.47214	−0.121499
$415$	9.79837	0.480983
$416$	4.47214	0.219265
$417$	−16.0000	−0.783523
$418$	0	0
$419$	28.9787	1.41570	0.707851	−	0.706361i	$-0.249665\pi$
$419$	28.9787	1.41570	0.707851	+	0.706361i	$0.249665\pi$
$420$	−3.61803	−0.176542
$421$	−24.1803	−1.17848	−0.589239	−	0.807959i	$-0.700572\pi$
$421$	−24.1803	−1.17848	−0.589239	+	0.807959i	$0.700572\pi$
$422$	−25.1246	−1.22305
$423$	−10.0000	−0.486217
$424$	13.3262	0.647179
$425$	−2.36068	−0.114510
$426$	−14.9443	−0.724052
$427$	11.7082	0.566600
$428$	−0.909830	−0.0439783
$429$	1.70820	0.0824729
$430$	−4.87539	−0.235112
$431$	−31.2361	−1.50459	−0.752294	−	0.658827i	$-0.771053\pi$
$431$	−31.2361	−1.50459	−0.752294	+	0.658827i	$0.771053\pi$
$432$	−1.00000	−0.0481125
$433$	20.0902	0.965472	0.482736	−	0.875766i	$-0.339643\pi$
$433$	20.0902	0.965472	0.482736	+	0.875766i	$0.339643\pi$
$434$	−27.4164	−1.31603
$435$	−10.5279	−0.504772
$436$	8.76393	0.419716
$437$	0	0
$438$	8.32624	0.397843
$439$	−28.8541	−1.37713	−0.688566	−	0.725174i	$-0.741759\pi$
$439$	−28.8541	−1.37713	−0.688566	+	0.725174i	$0.741759\pi$
$440$	−0.527864	−0.0251649
$441$	−0.145898	−0.00694753
$442$	3.41641	0.162502
$443$	1.27051	0.0603637	0.0301819	−	0.999544i	$-0.490391\pi$
$443$	1.27051	0.0603637	0.0301819	+	0.999544i	$0.490391\pi$
$444$	−10.9443	−0.519392
$445$	25.7771	1.22195
$446$	8.79837	0.416615
$447$	−10.3262	−0.488414
$448$	2.61803	0.123690
$449$	−19.7082	−0.930088	−0.465044	−	0.885288i	$-0.653961\pi$
$449$	−19.7082	−0.930088	−0.465044	+	0.885288i	$0.653961\pi$
$450$	3.09017	0.145672
$451$	−0.652476	−0.0307239
$452$	12.1803	0.572915
$453$	−13.6180	−0.639831
$454$	26.5066	1.24402
$455$	−16.1803	−0.758546
$456$	0	0
$457$	−23.5279	−1.10059	−0.550294	−	0.834971i	$-0.685484\pi$
$457$	−23.5279	−1.10059	−0.550294	+	0.834971i	$0.685484\pi$
$458$	−13.5279	−0.632116
$459$	−0.763932	−0.0356573
$460$	3.41641	0.159291
$461$	−14.0344	−0.653649	−0.326825	−	0.945085i	$-0.605979\pi$
$461$	−14.0344	−0.653649	−0.326825	+	0.945085i	$0.605979\pi$
$462$	1.00000	0.0465242
$463$	7.50658	0.348860	0.174430	−	0.984670i	$-0.444192\pi$
$463$	7.50658	0.348860	0.174430	+	0.984670i	$0.444192\pi$
$464$	7.61803	0.353658
$465$	−14.4721	−0.671129
$466$	−18.4721	−0.855705
$467$	−29.5623	−1.36798	−0.683990	−	0.729491i	$-0.739757\pi$
$467$	−29.5623	−1.36798	−0.683990	+	0.729491i	$0.739757\pi$
$468$	−4.47214	−0.206725
$469$	23.4164	1.08127
$470$	13.8197	0.637453
$471$	−0.763932	−0.0352001
$472$	5.14590	0.236859
$473$	1.34752	0.0619592
$474$	6.38197	0.293133
$475$	0	0
$476$	2.00000	0.0916698
$477$	−13.3262	−0.610167
$478$	23.8885	1.09264
$479$	21.1246	0.965208	0.482604	−	0.875839i	$-0.339691\pi$
$479$	21.1246	0.965208	0.482604	+	0.875839i	$0.339691\pi$
$480$	1.38197	0.0630778
$481$	−48.9443	−2.23167
$482$	9.03444	0.411508
$483$	−6.47214	−0.294492
$484$	−10.8541	−0.493368
$485$	−6.18034	−0.280635
$486$	1.00000	0.0453609
$487$	−36.2148	−1.64105	−0.820524	−	0.571612i	$-0.806318\pi$
$487$	−36.2148	−1.64105	−0.820524	+	0.571612i	$0.806318\pi$
$488$	−4.47214	−0.202444
$489$	1.70820	0.0772477
$490$	0.201626	0.00910854
$491$	−25.2705	−1.14044	−0.570221	−	0.821491i	$-0.693142\pi$
$491$	−25.2705	−1.14044	−0.570221	+	0.821491i	$0.693142\pi$
$492$	1.70820	0.0770118
$493$	5.81966	0.262104
$494$	0	0
$495$	0.527864	0.0237257
$496$	10.4721	0.470213
$497$	−39.1246	−1.75498
$498$	7.09017	0.317718
$499$	7.05573	0.315858	0.157929	−	0.987450i	$-0.449518\pi$
$499$	7.05573	0.315858	0.157929	+	0.987450i	$0.449518\pi$
$500$	−11.1803	−0.500000
$501$	12.4721	0.557214
$502$	9.32624	0.416250
$503$	−16.6525	−0.742497	−0.371249	−	0.928534i	$-0.621070\pi$
$503$	−16.6525	−0.742497	−0.371249	+	0.928534i	$0.621070\pi$
$504$	−2.61803	−0.116617
$505$	9.59675	0.427050
$506$	−0.944272	−0.0419780
$507$	−7.00000	−0.310881
$508$	−16.0902	−0.713886
$509$	22.0344	0.976659	0.488330	−	0.872659i	$-0.337606\pi$
$509$	22.0344	0.976659	0.488330	+	0.872659i	$0.337606\pi$
$510$	1.05573	0.0467484
$511$	21.7984	0.964303
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.65248	0.381644
$515$	4.34752	0.191575
$516$	−3.52786	−0.155306
$517$	−3.81966	−0.167988
$518$	−28.6525	−1.25892
$519$	−6.32624	−0.277691
$520$	6.18034	0.271026
$521$	5.70820	0.250081	0.125040	−	0.992152i	$-0.460094\pi$
$521$	5.70820	0.250081	0.125040	+	0.992152i	$0.460094\pi$
$522$	−7.61803	−0.333432
$523$	−13.5967	−0.594544	−0.297272	−	0.954793i	$-0.596077\pi$
$523$	−13.5967	−0.594544	−0.297272	+	0.954793i	$0.596077\pi$
$524$	−8.61803	−0.376481
$525$	8.09017	0.353084
$526$	−4.00000	−0.174408
$527$	8.00000	0.348485
$528$	−0.381966	−0.0166229
$529$	−16.8885	−0.734285
$530$	18.4164	0.799958
$531$	−5.14590	−0.223313
$532$	0	0
$533$	7.63932	0.330896
$534$	18.6525	0.807172
$535$	−1.25735	−0.0543602
$536$	−8.94427	−0.386334
$537$	−3.61803	−0.156130
$538$	−11.5623	−0.498486
$539$	−0.0557281	−0.00240038
$540$	−1.38197	−0.0594703
$541$	−5.88854	−0.253168	−0.126584	−	0.991956i	$-0.540401\pi$
$541$	−5.88854	−0.253168	−0.126584	+	0.991956i	$0.540401\pi$
$542$	11.5623	0.496644
$543$	−20.9443	−0.898805
$544$	−0.763932	−0.0327533
$545$	12.1115	0.518798
$546$	−11.7082	−0.501065
$547$	−8.11146	−0.346821	−0.173410	−	0.984850i	$-0.555479\pi$
$547$	−8.11146	−0.346821	−0.173410	+	0.984850i	$0.555479\pi$
$548$	8.76393	0.374377
$549$	4.47214	0.190866
$550$	1.18034	0.0503299
$551$	0	0
$552$	2.47214	0.105221
$553$	16.7082	0.710505
$554$	−24.8328	−1.05505
$555$	−15.1246	−0.642004
$556$	16.0000	0.678551
$557$	18.2705	0.774146	0.387073	−	0.922049i	$-0.373486\pi$
$557$	18.2705	0.774146	0.387073	+	0.922049i	$0.373486\pi$
$558$	−10.4721	−0.443321
$559$	−15.7771	−0.667300
$560$	3.61803	0.152890
$561$	−0.291796	−0.0123196
$562$	−13.4164	−0.565937
$563$	−31.7984	−1.34014	−0.670071	−	0.742297i	$-0.733736\pi$
$563$	−31.7984	−1.34014	−0.670071	+	0.742297i	$0.733736\pi$
$564$	10.0000	0.421076
$565$	16.8328	0.708162
$566$	−12.4721	−0.524243
$567$	2.61803	0.109947
$568$	14.9443	0.627048
$569$	28.7639	1.20585	0.602923	−	0.797799i	$-0.294002\pi$
$569$	28.7639	1.20585	0.602923	+	0.797799i	$0.294002\pi$
$570$	0	0
$571$	−16.3607	−0.684673	−0.342337	−	0.939577i	$-0.611218\pi$
$571$	−16.3607	−0.684673	−0.342337	+	0.939577i	$0.611218\pi$
$572$	−1.70820	−0.0714236
$573$	−6.94427	−0.290101
$574$	4.47214	0.186663
$575$	−7.63932	−0.318582
$576$	1.00000	0.0416667
$577$	20.4721	0.852266	0.426133	−	0.904660i	$-0.359876\pi$
$577$	20.4721	0.852266	0.426133	+	0.904660i	$0.359876\pi$
$578$	16.4164	0.682833
$579$	4.61803	0.191919
$580$	10.5279	0.437146
$581$	18.5623	0.770094
$582$	−4.47214	−0.185376
$583$	−5.09017	−0.210813
$584$	−8.32624	−0.344542
$585$	−6.18034	−0.255526
$586$	20.7984	0.859173
$587$	13.5279	0.558355	0.279177	−	0.960240i	$-0.409938\pi$
$587$	13.5279	0.558355	0.279177	+	0.960240i	$0.409938\pi$
$588$	0.145898	0.00601673
$589$	0	0
$590$	7.11146	0.292774
$591$	−3.61803	−0.148826
$592$	10.9443	0.449807
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	0.381966	0.0156723
$595$	2.76393	0.113310
$596$	10.3262	0.422979
$597$	−7.90983	−0.323728
$598$	11.0557	0.452102
$599$	−1.88854	−0.0771638	−0.0385819	−	0.999255i	$-0.512284\pi$
$599$	−1.88854	−0.0771638	−0.0385819	+	0.999255i	$0.512284\pi$
$600$	−3.09017	−0.126156
$601$	12.9787	0.529413	0.264706	−	0.964329i	$-0.414725\pi$
$601$	12.9787	0.529413	0.264706	+	0.964329i	$0.414725\pi$
$602$	−9.23607	−0.376434
$603$	8.94427	0.364239
$604$	13.6180	0.554110
$605$	−15.0000	−0.609837
$606$	6.94427	0.282092
$607$	−16.1459	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$607$	−16.1459	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$608$	0	0
$609$	−19.9443	−0.808183
$610$	−6.18034	−0.250235
$611$	44.7214	1.80923
$612$	0.763932	0.0308801
$613$	−35.8885	−1.44952	−0.724762	−	0.688999i	$-0.758050\pi$
$613$	−35.8885	−1.44952	−0.724762	+	0.688999i	$0.758050\pi$
$614$	1.70820	0.0689375
$615$	2.36068	0.0951918
$616$	−1.00000	−0.0402911
$617$	−0.875388	−0.0352418	−0.0176209	−	0.999845i	$-0.505609\pi$
$617$	−0.875388	−0.0352418	−0.0176209	+	0.999845i	$0.505609\pi$
$618$	3.14590	0.126547
$619$	−17.8885	−0.719001	−0.359501	−	0.933145i	$-0.617053\pi$
$619$	−17.8885	−0.719001	−0.359501	+	0.933145i	$0.617053\pi$
$620$	14.4721	0.581215
$621$	−2.47214	−0.0992034
$622$	28.3607	1.13716
$623$	48.8328	1.95645
$624$	4.47214	0.179029
$625$	0	0
$626$	18.1459	0.725256
$627$	0	0
$628$	0.763932	0.0304842
$629$	8.36068	0.333362
$630$	−3.61803	−0.144146
$631$	1.96556	0.0782476	0.0391238	−	0.999234i	$-0.487543\pi$
$631$	1.96556	0.0782476	0.0391238	+	0.999234i	$0.487543\pi$
$632$	−6.38197	−0.253861
$633$	−25.1246	−0.998614
$634$	−31.9787	−1.27004
$635$	−22.2361	−0.882411
$636$	13.3262	0.528420
$637$	0.652476	0.0258520
$638$	−2.90983	−0.115201
$639$	−14.9443	−0.591186
$640$	−1.38197	−0.0546270
$641$	−38.0689	−1.50363	−0.751815	−	0.659374i	$-0.770821\pi$
$641$	−38.0689	−1.50363	−0.751815	+	0.659374i	$0.770821\pi$
$642$	−0.909830	−0.0359081
$643$	42.0689	1.65903	0.829517	−	0.558481i	$-0.188616\pi$
$643$	42.0689	1.65903	0.829517	+	0.558481i	$0.188616\pi$
$644$	6.47214	0.255038
$645$	−4.87539	−0.191968
$646$	0	0
$647$	−12.0689	−0.474477	−0.237238	−	0.971451i	$-0.576242\pi$
$647$	−12.0689	−0.474477	−0.237238	+	0.971451i	$0.576242\pi$
$648$	−1.00000	−0.0392837
$649$	−1.96556	−0.0771549
$650$	−13.8197	−0.542052
$651$	−27.4164	−1.07453
$652$	−1.70820	−0.0668984
$653$	−15.9230	−0.623115	−0.311557	−	0.950227i	$-0.600851\pi$
$653$	−15.9230	−0.623115	−0.311557	+	0.950227i	$0.600851\pi$
$654$	8.76393	0.342697
$655$	−11.9098	−0.465356
$656$	−1.70820	−0.0666942
$657$	8.32624	0.324837
$658$	26.1803	1.02062
$659$	−18.0902	−0.704693	−0.352346	−	0.935870i	$-0.614616\pi$
$659$	−18.0902	−0.704693	−0.352346	+	0.935870i	$0.614616\pi$
$660$	−0.527864	−0.0205471
$661$	46.0689	1.79187	0.895936	−	0.444183i	$-0.146506\pi$
$661$	46.0689	1.79187	0.895936	+	0.444183i	$0.146506\pi$
$662$	−11.8885	−0.462061
$663$	3.41641	0.132682
$664$	−7.09017	−0.275152
$665$	0	0
$666$	−10.9443	−0.424082
$667$	18.8328	0.729210
$668$	−12.4721	−0.482561
$669$	8.79837	0.340165
$670$	−12.3607	−0.477535
$671$	1.70820	0.0659445
$672$	2.61803	0.100993
$673$	−40.4721	−1.56009	−0.780043	−	0.625726i	$-0.784803\pi$
$673$	−40.4721	−1.56009	−0.780043	+	0.625726i	$0.784803\pi$
$674$	−29.4164	−1.13308
$675$	3.09017	0.118941
$676$	7.00000	0.269231
$677$	−43.9787	−1.69024	−0.845120	−	0.534577i	$-0.820471\pi$
$677$	−43.9787	−1.69024	−0.845120	+	0.534577i	$0.820471\pi$
$678$	12.1803	0.467783
$679$	−11.7082	−0.449320
$680$	−1.05573	−0.0404853
$681$	26.5066	1.01573
$682$	−4.00000	−0.153168
$683$	−33.2705	−1.27306	−0.636530	−	0.771252i	$-0.719631\pi$
$683$	−33.2705	−1.27306	−0.636530	+	0.771252i	$0.719631\pi$
$684$	0	0
$685$	12.1115	0.462755
$686$	18.7082	0.714283
$687$	−13.5279	−0.516120
$688$	3.52786	0.134499
$689$	59.5967	2.27046
$690$	3.41641	0.130060
$691$	−11.7082	−0.445401	−0.222701	−	0.974887i	$-0.571487\pi$
$691$	−11.7082	−0.445401	−0.222701	+	0.974887i	$0.571487\pi$
$692$	6.32624	0.240487
$693$	1.00000	0.0379869
$694$	−9.50658	−0.360865
$695$	22.1115	0.838735
$696$	7.61803	0.288761
$697$	−1.30495	−0.0494286
$698$	25.1246	0.950981
$699$	−18.4721	−0.698680
$700$	−8.09017	−0.305780
$701$	6.67376	0.252065	0.126032	−	0.992026i	$-0.459776\pi$
$701$	6.67376	0.252065	0.126032	+	0.992026i	$0.459776\pi$
$702$	−4.47214	−0.168790
$703$	0	0
$704$	0.381966	0.0143959
$705$	13.8197	0.520479
$706$	−23.4164	−0.881288
$707$	18.1803	0.683742
$708$	5.14590	0.193395
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	20.6525	0.775074
$711$	6.38197	0.239342
$712$	−18.6525	−0.699031
$713$	25.8885	0.969534
$714$	2.00000	0.0748481
$715$	−2.36068	−0.0882844
$716$	3.61803	0.135212
$717$	23.8885	0.892134
$718$	−35.5967	−1.32846
$719$	45.7082	1.70463	0.852314	−	0.523030i	$-0.175198\pi$
$719$	45.7082	1.70463	0.852314	+	0.523030i	$0.175198\pi$
$720$	1.38197	0.0515028
$721$	8.23607	0.306727
$722$	0	0
$723$	9.03444	0.335995
$724$	20.9443	0.778388
$725$	−23.5410	−0.874292
$726$	−10.8541	−0.402834
$727$	−45.6180	−1.69188	−0.845940	−	0.533279i	$-0.820960\pi$
$727$	−45.6180	−1.69188	−0.845940	+	0.533279i	$0.820960\pi$
$728$	11.7082	0.433935
$729$	1.00000	0.0370370
$730$	−11.5066	−0.425878
$731$	2.69505	0.0996800
$732$	−4.47214	−0.165295
$733$	−17.5967	−0.649951	−0.324975	−	0.945722i	$-0.605356\pi$
$733$	−17.5967	−0.649951	−0.324975	+	0.945722i	$0.605356\pi$
$734$	13.5279	0.499323
$735$	0.201626	0.00743709
$736$	−2.47214	−0.0911241
$737$	3.41641	0.125845
$738$	1.70820	0.0628799
$739$	−19.8197	−0.729078	−0.364539	−	0.931188i	$-0.618773\pi$
$739$	−19.8197	−0.729078	−0.364539	+	0.931188i	$0.618773\pi$
$740$	15.1246	0.555992
$741$	0	0
$742$	34.8885	1.28080
$743$	−27.7082	−1.01652	−0.508258	−	0.861205i	$-0.669710\pi$
$743$	−27.7082	−1.01652	−0.508258	+	0.861205i	$0.669710\pi$
$744$	10.4721	0.383927
$745$	14.2705	0.522831
$746$	−0.763932	−0.0279695
$747$	7.09017	0.259416
$748$	0.291796	0.0106691
$749$	−2.38197	−0.0870351
$750$	−11.1803	−0.408248
$751$	7.56231	0.275952	0.137976	−	0.990436i	$-0.455940\pi$
$751$	7.56231	0.275952	0.137976	+	0.990436i	$0.455940\pi$
$752$	−10.0000	−0.364662
$753$	9.32624	0.339867
$754$	34.0689	1.24072
$755$	18.8197	0.684918
$756$	−2.61803	−0.0952170
$757$	−32.4721	−1.18022	−0.590110	−	0.807323i	$-0.700916\pi$
$757$	−32.4721	−1.18022	−0.590110	+	0.807323i	$0.700916\pi$
$758$	−21.4164	−0.777879
$759$	−0.944272	−0.0342749
$760$	0	0
$761$	18.0689	0.654997	0.327498	−	0.944852i	$-0.393794\pi$
$761$	18.0689	0.654997	0.327498	+	0.944852i	$0.393794\pi$
$762$	−16.0902	−0.582885
$763$	22.9443	0.830638
$764$	6.94427	0.251235
$765$	1.05573	0.0381699
$766$	−1.41641	−0.0511769
$767$	23.0132	0.830957
$768$	−1.00000	−0.0360844
$769$	2.56231	0.0923991	0.0461996	−	0.998932i	$-0.485289\pi$
$769$	2.56231	0.0923991	0.0461996	+	0.998932i	$0.485289\pi$
$770$	−1.38197	−0.0498026
$771$	8.65248	0.311611
$772$	−4.61803	−0.166207
$773$	6.97871	0.251007	0.125503	−	0.992093i	$-0.459945\pi$
$773$	6.97871	0.251007	0.125503	+	0.992093i	$0.459945\pi$
$774$	−3.52786	−0.126806
$775$	−32.3607	−1.16243
$776$	4.47214	0.160540
$777$	−28.6525	−1.02790
$778$	2.00000	0.0717035
$779$	0	0
$780$	6.18034	0.221292
$781$	−5.70820	−0.204256
$782$	−1.88854	−0.0675342
$783$	−7.61803	−0.272246
$784$	−0.145898	−0.00521064
$785$	1.05573	0.0376806
$786$	−8.61803	−0.307395
$787$	−24.5410	−0.874793	−0.437396	−	0.899269i	$-0.644099\pi$
$787$	−24.5410	−0.874793	−0.437396	+	0.899269i	$0.644099\pi$
$788$	3.61803	0.128887
$789$	−4.00000	−0.142404
$790$	−8.81966	−0.313789
$791$	31.8885	1.13383
$792$	−0.381966	−0.0135726
$793$	−20.0000	−0.710221
$794$	−12.3607	−0.438664
$795$	18.4164	0.653163
$796$	7.90983	0.280356
$797$	32.8328	1.16300	0.581499	−	0.813547i	$-0.302466\pi$
$797$	32.8328	1.16300	0.581499	+	0.813547i	$0.302466\pi$
$798$	0	0
$799$	−7.63932	−0.270260
$800$	3.09017	0.109254
$801$	18.6525	0.659053
$802$	6.18034	0.218235
$803$	3.18034	0.112232
$804$	−8.94427	−0.315440
$805$	8.94427	0.315244
$806$	46.8328	1.64962
$807$	−11.5623	−0.407012
$808$	−6.94427	−0.244299
$809$	−0.472136	−0.0165994	−0.00829971	−	0.999966i	$-0.502642\pi$
$809$	−0.472136	−0.0165994	−0.00829971	+	0.999966i	$0.502642\pi$
$810$	−1.38197	−0.0485573
$811$	−20.1803	−0.708628	−0.354314	−	0.935127i	$-0.615286\pi$
$811$	−20.1803	−0.708628	−0.354314	+	0.935127i	$0.615286\pi$
$812$	19.9443	0.699907
$813$	11.5623	0.405508
$814$	−4.18034	−0.146521
$815$	−2.36068	−0.0826910
$816$	−0.763932	−0.0267430
$817$	0	0
$818$	−2.72949	−0.0954344
$819$	−11.7082	−0.409118
$820$	−2.36068	−0.0824385
$821$	−14.2705	−0.498044	−0.249022	−	0.968498i	$-0.580109\pi$
$821$	−14.2705	−0.498044	−0.249022	+	0.968498i	$0.580109\pi$
$822$	8.76393	0.305677
$823$	40.7984	1.42214	0.711071	−	0.703120i	$-0.248210\pi$
$823$	40.7984	1.42214	0.711071	+	0.703120i	$0.248210\pi$
$824$	−3.14590	−0.109593
$825$	1.18034	0.0410942
$826$	13.4721	0.468756
$827$	−41.1459	−1.43078	−0.715392	−	0.698724i	$-0.753752\pi$
$827$	−41.1459	−1.43078	−0.715392	+	0.698724i	$0.753752\pi$
$828$	2.47214	0.0859127
$829$	−18.4721	−0.641564	−0.320782	−	0.947153i	$-0.603946\pi$
$829$	−18.4721	−0.641564	−0.320782	+	0.947153i	$0.603946\pi$
$830$	−9.79837	−0.340107
$831$	−24.8328	−0.861441
$832$	−4.47214	−0.155043
$833$	−0.111456	−0.00386173
$834$	16.0000	0.554035
$835$	−17.2361	−0.596479
$836$	0	0
$837$	−10.4721	−0.361970
$838$	−28.9787	−1.00105
$839$	46.0689	1.59047	0.795237	−	0.606298i	$-0.207346\pi$
$839$	46.0689	1.59047	0.795237	+	0.606298i	$0.207346\pi$
$840$	3.61803	0.124834
$841$	29.0344	1.00119
$842$	24.1803	0.833310
$843$	−13.4164	−0.462086
$844$	25.1246	0.864825
$845$	9.67376	0.332788
$846$	10.0000	0.343807
$847$	−28.4164	−0.976399
$848$	−13.3262	−0.457625
$849$	−12.4721	−0.428043
$850$	2.36068	0.0809706
$851$	27.0557	0.927458
$852$	14.9443	0.511982
$853$	−48.3607	−1.65584	−0.827919	−	0.560848i	$-0.810475\pi$
$853$	−48.3607	−1.65584	−0.827919	+	0.560848i	$0.810475\pi$
$854$	−11.7082	−0.400646
$855$	0	0
$856$	0.909830	0.0310974
$857$	−22.6525	−0.773794	−0.386897	−	0.922123i	$-0.626453\pi$
$857$	−22.6525	−0.773794	−0.386897	+	0.922123i	$0.626453\pi$
$858$	−1.70820	−0.0583171
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	4.87539	0.166249
$861$	4.47214	0.152410
$862$	31.2361	1.06390
$863$	−7.59675	−0.258596	−0.129298	−	0.991606i	$-0.541272\pi$
$863$	−7.59675	−0.258596	−0.129298	+	0.991606i	$0.541272\pi$
$864$	1.00000	0.0340207
$865$	8.74265	0.297259
$866$	−20.0902	−0.682692
$867$	16.4164	0.557530
$868$	27.4164	0.930574
$869$	2.43769	0.0826931
$870$	10.5279	0.356928
$871$	−40.0000	−1.35535
$872$	−8.76393	−0.296784
$873$	−4.47214	−0.151359
$874$	0	0
$875$	−29.2705	−0.989524
$876$	−8.32624	−0.281318
$877$	−14.3607	−0.484926	−0.242463	−	0.970161i	$-0.577955\pi$
$877$	−14.3607	−0.484926	−0.242463	+	0.970161i	$0.577955\pi$
$878$	28.8541	0.973779
$879$	20.7984	0.701512
$880$	0.527864	0.0177943
$881$	−13.5279	−0.455765	−0.227883	−	0.973689i	$-0.573180\pi$
$881$	−13.5279	−0.455765	−0.227883	+	0.973689i	$0.573180\pi$
$882$	0.145898	0.00491264
$883$	45.5967	1.53445	0.767226	−	0.641376i	$-0.221636\pi$
$883$	45.5967	1.53445	0.767226	+	0.641376i	$0.221636\pi$
$884$	−3.41641	−0.114906
$885$	7.11146	0.239049
$886$	−1.27051	−0.0426836
$887$	−36.4721	−1.22461	−0.612307	−	0.790620i	$-0.709758\pi$
$887$	−36.4721	−1.22461	−0.612307	+	0.790620i	$0.709758\pi$
$888$	10.9443	0.367266
$889$	−42.1246	−1.41281
$890$	−25.7771	−0.864050
$891$	0.381966	0.0127963
$892$	−8.79837	−0.294591
$893$	0	0
$894$	10.3262	0.345361
$895$	5.00000	0.167132
$896$	−2.61803	−0.0874624
$897$	11.0557	0.369140
$898$	19.7082	0.657671
$899$	79.7771	2.66071
$900$	−3.09017	−0.103006
$901$	−10.1803	−0.339156
$902$	0.652476	0.0217251
$903$	−9.23607	−0.307357
$904$	−12.1803	−0.405112
$905$	28.9443	0.962140
$906$	13.6180	0.452429
$907$	6.29180	0.208916	0.104458	−	0.994529i	$-0.466689\pi$
$907$	6.29180	0.208916	0.104458	+	0.994529i	$0.466689\pi$
$908$	−26.5066	−0.879652
$909$	6.94427	0.230327
$910$	16.1803	0.536373
$911$	24.1803	0.801130	0.400565	−	0.916268i	$-0.368814\pi$
$911$	24.1803	0.801130	0.400565	+	0.916268i	$0.368814\pi$
$912$	0	0
$913$	2.70820	0.0896285
$914$	23.5279	0.778233
$915$	−6.18034	−0.204316
$916$	13.5279	0.446973
$917$	−22.5623	−0.745073
$918$	0.763932	0.0252135
$919$	−10.7426	−0.354367	−0.177184	−	0.984178i	$-0.556699\pi$
$919$	−10.7426	−0.354367	−0.177184	+	0.984178i	$0.556699\pi$
$920$	−3.41641	−0.112636
$921$	1.70820	0.0562872
$922$	14.0344	0.462200
$923$	66.8328	2.19983
$924$	−1.00000	−0.0328976
$925$	−33.8197	−1.11198
$926$	−7.50658	−0.246681
$927$	3.14590	0.103325
$928$	−7.61803	−0.250074
$929$	−32.2492	−1.05806	−0.529031	−	0.848602i	$-0.677445\pi$
$929$	−32.2492	−1.05806	−0.529031	+	0.848602i	$0.677445\pi$
$930$	14.4721	0.474560
$931$	0	0
$932$	18.4721	0.605075
$933$	28.3607	0.928487
$934$	29.5623	0.967308
$935$	0.403252	0.0131878
$936$	4.47214	0.146176
$937$	−23.5066	−0.767926	−0.383963	−	0.923348i	$-0.625441\pi$
$937$	−23.5066	−0.767926	−0.383963	+	0.923348i	$0.625441\pi$
$938$	−23.4164	−0.764573
$939$	18.1459	0.592169
$940$	−13.8197	−0.450748
$941$	50.2705	1.63877	0.819386	−	0.573242i	$-0.194315\pi$
$941$	50.2705	1.63877	0.819386	+	0.573242i	$0.194315\pi$
$942$	0.763932	0.0248903
$943$	−4.22291	−0.137517
$944$	−5.14590	−0.167485
$945$	−3.61803	−0.117695
$946$	−1.34752	−0.0438118
$947$	3.38197	0.109899	0.0549496	−	0.998489i	$-0.482500\pi$
$947$	3.38197	0.109899	0.0549496	+	0.998489i	$0.482500\pi$
$948$	−6.38197	−0.207277
$949$	−37.2361	−1.20873
$950$	0	0
$951$	−31.9787	−1.03698
$952$	−2.00000	−0.0648204
$953$	10.6525	0.345068	0.172534	−	0.985004i	$-0.444805\pi$
$953$	10.6525	0.345068	0.172534	+	0.985004i	$0.444805\pi$
$954$	13.3262	0.431453
$955$	9.59675	0.310543
$956$	−23.8885	−0.772611
$957$	−2.90983	−0.0940614
$958$	−21.1246	−0.682505
$959$	22.9443	0.740909
$960$	−1.38197	−0.0446028
$961$	78.6656	2.53760
$962$	48.9443	1.57803
$963$	−0.909830	−0.0293189
$964$	−9.03444	−0.290980
$965$	−6.38197	−0.205443
$966$	6.47214	0.208238
$967$	−45.0344	−1.44821	−0.724105	−	0.689690i	$-0.757747\pi$
$967$	−45.0344	−1.44821	−0.724105	+	0.689690i	$0.757747\pi$
$968$	10.8541	0.348864
$969$	0	0
$970$	6.18034	0.198439
$971$	−24.9098	−0.799394	−0.399697	−	0.916647i	$-0.630885\pi$
$971$	−24.9098	−0.799394	−0.399697	+	0.916647i	$0.630885\pi$
$972$	−1.00000	−0.0320750
$973$	41.8885	1.34289
$974$	36.2148	1.16040
$975$	−13.8197	−0.442583
$976$	4.47214	0.143150
$977$	52.7214	1.68671	0.843353	−	0.537360i	$-0.180578\pi$
$977$	52.7214	1.68671	0.843353	+	0.537360i	$0.180578\pi$
$978$	−1.70820	−0.0546223
$979$	7.12461	0.227704
$980$	−0.201626	−0.00644071
$981$	8.76393	0.279811
$982$	25.2705	0.806414
$983$	−33.8197	−1.07868	−0.539340	−	0.842088i	$-0.681326\pi$
$983$	−33.8197	−1.07868	−0.539340	+	0.842088i	$0.681326\pi$
$984$	−1.70820	−0.0544556
$985$	5.00000	0.159313
$986$	−5.81966	−0.185336
$987$	26.1803	0.833329
$988$	0	0
$989$	8.72136	0.277323
$990$	−0.527864	−0.0167766
$991$	21.3820	0.679221	0.339610	−	0.940566i	$-0.389705\pi$
$991$	21.3820	0.679221	0.339610	+	0.940566i	$0.389705\pi$
$992$	−10.4721	−0.332491
$993$	−11.8885	−0.377272
$994$	39.1246	1.24096
$995$	10.9311	0.346540
$996$	−7.09017	−0.224661
$997$	0.360680	0.0114228	0.00571142	−	0.999984i	$-0.498182\pi$
$997$	0.360680	0.0114228	0.00571142	+	0.999984i	$0.498182\pi$
$998$	−7.05573	−0.223345
$999$	−10.9443	−0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.2.a.j.1.1	✓	2
3.2	odd	2		6498.2.a.bf.1.2		2
19.18	odd	2		2166.2.a.m.1.1	yes	2
57.56	even	2		6498.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.2.a.j.1.1	✓	2	1.1	even	1	trivial
2166.2.a.m.1.1	yes	2	19.18	odd	2
6498.2.a.z.1.2		2	57.56	even	2
6498.2.a.bf.1.2		2	3.2	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.38197 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.38197 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.38197 q^{10} +0.381966 q^{11} -1.00000 q^{12} -4.47214 q^{13} -2.61803 q^{14} -1.38197 q^{15} +1.00000 q^{16} +0.763932 q^{17} -1.00000 q^{18} +1.38197 q^{20} -2.61803 q^{21} -0.381966 q^{22} +2.47214 q^{23} +1.00000 q^{24} -3.09017 q^{25} +4.47214 q^{26} -1.00000 q^{27} +2.61803 q^{28} +7.61803 q^{29} +1.38197 q^{30} +10.4721 q^{31} -1.00000 q^{32} -0.381966 q^{33} -0.763932 q^{34} +3.61803 q^{35} +1.00000 q^{36} +10.9443 q^{37} +4.47214 q^{39} -1.38197 q^{40} -1.70820 q^{41} +2.61803 q^{42} +3.52786 q^{43} +0.381966 q^{44} +1.38197 q^{45} -2.47214 q^{46} -10.0000 q^{47} -1.00000 q^{48} -0.145898 q^{49} +3.09017 q^{50} -0.763932 q^{51} -4.47214 q^{52} -13.3262 q^{53} +1.00000 q^{54} +0.527864 q^{55} -2.61803 q^{56} -7.61803 q^{58} -5.14590 q^{59} -1.38197 q^{60} +4.47214 q^{61} -10.4721 q^{62} +2.61803 q^{63} +1.00000 q^{64} -6.18034 q^{65} +0.381966 q^{66} +8.94427 q^{67} +0.763932 q^{68} -2.47214 q^{69} -3.61803 q^{70} -14.9443 q^{71} -1.00000 q^{72} +8.32624 q^{73} -10.9443 q^{74} +3.09017 q^{75} +1.00000 q^{77} -4.47214 q^{78} +6.38197 q^{79} +1.38197 q^{80} +1.00000 q^{81} +1.70820 q^{82} +7.09017 q^{83} -2.61803 q^{84} +1.05573 q^{85} -3.52786 q^{86} -7.61803 q^{87} -0.381966 q^{88} +18.6525 q^{89} -1.38197 q^{90} -11.7082 q^{91} +2.47214 q^{92} -10.4721 q^{93} +10.0000 q^{94} +1.00000 q^{96} -4.47214 q^{97} +0.145898 q^{98} +0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 5 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 3 q^{14} - 5 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 5 q^{20} - 3 q^{21} - 3 q^{22} - 4 q^{23} + 2 q^{24} + 5 q^{25} - 2 q^{27} + 3 q^{28} + 13 q^{29} + 5 q^{30} + 12 q^{31} - 2 q^{32} - 3 q^{33} - 6 q^{34} + 5 q^{35} + 2 q^{36} + 4 q^{37} - 5 q^{40} + 10 q^{41} + 3 q^{42} + 16 q^{43} + 3 q^{44} + 5 q^{45} + 4 q^{46} - 20 q^{47} - 2 q^{48} - 7 q^{49} - 5 q^{50} - 6 q^{51} - 11 q^{53} + 2 q^{54} + 10 q^{55} - 3 q^{56} - 13 q^{58} - 17 q^{59} - 5 q^{60} - 12 q^{62} + 3 q^{63} + 2 q^{64} + 10 q^{65} + 3 q^{66} + 6 q^{68} + 4 q^{69} - 5 q^{70} - 12 q^{71} - 2 q^{72} + q^{73} - 4 q^{74} - 5 q^{75} + 2 q^{77} + 15 q^{79} + 5 q^{80} + 2 q^{81} - 10 q^{82} + 3 q^{83} - 3 q^{84} + 20 q^{85} - 16 q^{86} - 13 q^{87} - 3 q^{88} + 6 q^{89} - 5 q^{90} - 10 q^{91} - 4 q^{92} - 12 q^{93} + 20 q^{94} + 2 q^{96} + 7 q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 5 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - 5 * q^10 + 3 * q^11 - 2 * q^12 - 3 * q^14 - 5 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^18 + 5 * q^20 - 3 * q^21 - 3 * q^22 - 4 * q^23 + 2 * q^24 + 5 * q^25 - 2 * q^27 + 3 * q^28 + 13 * q^29 + 5 * q^30 + 12 * q^31 - 2 * q^32 - 3 * q^33 - 6 * q^34 + 5 * q^35 + 2 * q^36 + 4 * q^37 - 5 * q^40 + 10 * q^41 + 3 * q^42 + 16 * q^43 + 3 * q^44 + 5 * q^45 + 4 * q^46 - 20 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 6 * q^51 - 11 * q^53 + 2 * q^54 + 10 * q^55 - 3 * q^56 - 13 * q^58 - 17 * q^59 - 5 * q^60 - 12 * q^62 + 3 * q^63 + 2 * q^64 + 10 * q^65 + 3 * q^66 + 6 * q^68 + 4 * q^69 - 5 * q^70 - 12 * q^71 - 2 * q^72 + q^73 - 4 * q^74 - 5 * q^75 + 2 * q^77 + 15 * q^79 + 5 * q^80 + 2 * q^81 - 10 * q^82 + 3 * q^83 - 3 * q^84 + 20 * q^85 - 16 * q^86 - 13 * q^87 - 3 * q^88 + 6 * q^89 - 5 * q^90 - 10 * q^91 - 4 * q^92 - 12 * q^93 + 20 * q^94 + 2 * q^96 + 7 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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