LMFDB - Embedded newform 6042.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.476452$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29654 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29654 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.29654 q^{10} +1.47645 q^{11} -1.00000 q^{12} -1.82009 q^{13} +2.00000 q^{14} +3.29654 q^{15} +1.00000 q^{16} +5.64018 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.29654 q^{20} -2.00000 q^{21} +1.47645 q^{22} -3.82009 q^{23} -1.00000 q^{24} +5.86719 q^{25} -1.82009 q^{26} -1.00000 q^{27} +2.00000 q^{28} +2.17991 q^{29} +3.29654 q^{30} -0.867185 q^{31} +1.00000 q^{32} -1.47645 q^{33} +5.64018 q^{34} -6.59308 q^{35} +1.00000 q^{36} +2.86719 q^{37} +1.00000 q^{38} +1.82009 q^{39} -3.29654 q^{40} -4.00000 q^{41} -2.00000 q^{42} -6.68727 q^{43} +1.47645 q^{44} -3.29654 q^{45} -3.82009 q^{46} +11.2965 q^{47} -1.00000 q^{48} -3.00000 q^{49} +5.86719 q^{50} -5.64018 q^{51} -1.82009 q^{52} +1.00000 q^{53} -1.00000 q^{54} -4.86719 q^{55} +2.00000 q^{56} -1.00000 q^{57} +2.17991 q^{58} +1.04710 q^{59} +3.29654 q^{60} -11.5460 q^{61} -0.867185 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -1.47645 q^{66} -9.54599 q^{67} +5.64018 q^{68} +3.82009 q^{69} -6.59308 q^{70} -0.867185 q^{71} +1.00000 q^{72} +10.5931 q^{73} +2.86719 q^{74} -5.86719 q^{75} +1.00000 q^{76} +2.95290 q^{77} +1.82009 q^{78} +12.5074 q^{79} -3.29654 q^{80} +1.00000 q^{81} -4.00000 q^{82} -7.64018 q^{83} -2.00000 q^{84} -18.5931 q^{85} -6.68727 q^{86} -2.17991 q^{87} +1.47645 q^{88} +14.6873 q^{89} -3.29654 q^{90} -3.64018 q^{91} -3.82009 q^{92} +0.867185 q^{93} +11.2965 q^{94} -3.29654 q^{95} -1.00000 q^{96} +8.59308 q^{97} -3.00000 q^{98} +1.47645 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{10} + 2 q^{11} - 3 q^{12} + 4 q^{13} + 6 q^{14} - 2 q^{15} + 3 q^{16} - 2 q^{17} + 3 q^{18} + 3 q^{19} + 2 q^{20} - 6 q^{21} + 2 q^{22} - 2 q^{23} - 3 q^{24} + 13 q^{25} + 4 q^{26} - 3 q^{27} + 6 q^{28} + 16 q^{29} - 2 q^{30} + 2 q^{31} + 3 q^{32} - 2 q^{33} - 2 q^{34} + 4 q^{35} + 3 q^{36} + 4 q^{37} + 3 q^{38} - 4 q^{39} + 2 q^{40} - 12 q^{41} - 6 q^{42} - 6 q^{43} + 2 q^{44} + 2 q^{45} - 2 q^{46} + 22 q^{47} - 3 q^{48} - 9 q^{49} + 13 q^{50} + 2 q^{51} + 4 q^{52} + 3 q^{53} - 3 q^{54} - 10 q^{55} + 6 q^{56} - 3 q^{57} + 16 q^{58} + 8 q^{59} - 2 q^{60} - 6 q^{61} + 2 q^{62} + 6 q^{63} + 3 q^{64} + 18 q^{65} - 2 q^{66} - 2 q^{68} + 2 q^{69} + 4 q^{70} + 2 q^{71} + 3 q^{72} + 8 q^{73} + 4 q^{74} - 13 q^{75} + 3 q^{76} + 4 q^{77} - 4 q^{78} + 14 q^{79} + 2 q^{80} + 3 q^{81} - 12 q^{82} - 4 q^{83} - 6 q^{84} - 32 q^{85} - 6 q^{86} - 16 q^{87} + 2 q^{88} + 30 q^{89} + 2 q^{90} + 8 q^{91} - 2 q^{92} - 2 q^{93} + 22 q^{94} + 2 q^{95} - 3 q^{96} + 2 q^{97} - 9 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^10 + 2 * q^11 - 3 * q^12 + 4 * q^13 + 6 * q^14 - 2 * q^15 + 3 * q^16 - 2 * q^17 + 3 * q^18 + 3 * q^19 + 2 * q^20 - 6 * q^21 + 2 * q^22 - 2 * q^23 - 3 * q^24 + 13 * q^25 + 4 * q^26 - 3 * q^27 + 6 * q^28 + 16 * q^29 - 2 * q^30 + 2 * q^31 + 3 * q^32 - 2 * q^33 - 2 * q^34 + 4 * q^35 + 3 * q^36 + 4 * q^37 + 3 * q^38 - 4 * q^39 + 2 * q^40 - 12 * q^41 - 6 * q^42 - 6 * q^43 + 2 * q^44 + 2 * q^45 - 2 * q^46 + 22 * q^47 - 3 * q^48 - 9 * q^49 + 13 * q^50 + 2 * q^51 + 4 * q^52 + 3 * q^53 - 3 * q^54 - 10 * q^55 + 6 * q^56 - 3 * q^57 + 16 * q^58 + 8 * q^59 - 2 * q^60 - 6 * q^61 + 2 * q^62 + 6 * q^63 + 3 * q^64 + 18 * q^65 - 2 * q^66 - 2 * q^68 + 2 * q^69 + 4 * q^70 + 2 * q^71 + 3 * q^72 + 8 * q^73 + 4 * q^74 - 13 * q^75 + 3 * q^76 + 4 * q^77 - 4 * q^78 + 14 * q^79 + 2 * q^80 + 3 * q^81 - 12 * q^82 - 4 * q^83 - 6 * q^84 - 32 * q^85 - 6 * q^86 - 16 * q^87 + 2 * q^88 + 30 * q^89 + 2 * q^90 + 8 * q^91 - 2 * q^92 - 2 * q^93 + 22 * q^94 + 2 * q^95 - 3 * q^96 + 2 * q^97 - 9 * q^98 + 2 * 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
$5$	−3.29654	−1.47426	−0.737129	−	0.675752i	$-0.763819\pi$
$5$	−3.29654	−1.47426	−0.737129	+	0.675752i	$0.763819\pi$
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.29654	−1.04246
$11$	1.47645	0.445167	0.222584	−	0.974914i	$-0.428551\pi$
$11$	1.47645	0.445167	0.222584	+	0.974914i	$0.428551\pi$
$12$	−1.00000	−0.288675
$13$	−1.82009	−0.504802	−0.252401	−	0.967623i	$-0.581220\pi$
$13$	−1.82009	−0.504802	−0.252401	+	0.967623i	$0.581220\pi$
$14$	2.00000	0.534522
$15$	3.29654	0.851163
$16$	1.00000	0.250000
$17$	5.64018	1.36794	0.683972	−	0.729508i	$-0.260251\pi$
$17$	5.64018	1.36794	0.683972	+	0.729508i	$0.260251\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.29654	−0.737129
$21$	−2.00000	−0.436436
$22$	1.47645	0.314781
$23$	−3.82009	−0.796544	−0.398272	−	0.917267i	$-0.630390\pi$
$23$	−3.82009	−0.796544	−0.398272	+	0.917267i	$0.630390\pi$
$24$	−1.00000	−0.204124
$25$	5.86719	1.17344
$26$	−1.82009	−0.356949
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	2.17991	0.404799	0.202400	−	0.979303i	$-0.435126\pi$
$29$	2.17991	0.404799	0.202400	+	0.979303i	$0.435126\pi$
$30$	3.29654	0.601863
$31$	−0.867185	−0.155751	−0.0778755	−	0.996963i	$-0.524814\pi$
$31$	−0.867185	−0.155751	−0.0778755	+	0.996963i	$0.524814\pi$
$32$	1.00000	0.176777
$33$	−1.47645	−0.257017
$34$	5.64018	0.967283
$35$	−6.59308	−1.11443
$36$	1.00000	0.166667
$37$	2.86719	0.471362	0.235681	−	0.971830i	$-0.424268\pi$
$37$	2.86719	0.471362	0.235681	+	0.971830i	$0.424268\pi$
$38$	1.00000	0.162221
$39$	1.82009	0.291448
$40$	−3.29654	−0.521229
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−2.00000	−0.308607
$43$	−6.68727	−1.01980	−0.509900	−	0.860234i	$-0.670318\pi$
$43$	−6.68727	−1.01980	−0.509900	+	0.860234i	$0.670318\pi$
$44$	1.47645	0.222584
$45$	−3.29654	−0.491419
$46$	−3.82009	−0.563241
$47$	11.2965	1.64777	0.823885	−	0.566757i	$-0.191802\pi$
$47$	11.2965	1.64777	0.823885	+	0.566757i	$0.191802\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	5.86719	0.829745
$51$	−5.64018	−0.789783
$52$	−1.82009	−0.252401
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	−4.86719	−0.656291
$56$	2.00000	0.267261
$57$	−1.00000	−0.132453
$58$	2.17991	0.286236
$59$	1.04710	0.136320	0.0681601	−	0.997674i	$-0.478287\pi$
$59$	1.04710	0.136320	0.0681601	+	0.997674i	$0.478287\pi$
$60$	3.29654	0.425582
$61$	−11.5460	−1.47831	−0.739156	−	0.673534i	$-0.764775\pi$
$61$	−11.5460	−1.47831	−0.739156	+	0.673534i	$0.764775\pi$
$62$	−0.867185	−0.110133
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−1.47645	−0.181739
$67$	−9.54599	−1.16623	−0.583114	−	0.812390i	$-0.698166\pi$
$67$	−9.54599	−1.16623	−0.583114	+	0.812390i	$0.698166\pi$
$68$	5.64018	0.683972
$69$	3.82009	0.459885
$70$	−6.59308	−0.788024
$71$	−0.867185	−0.102916	−0.0514580	−	0.998675i	$-0.516387\pi$
$71$	−0.867185	−0.102916	−0.0514580	+	0.998675i	$0.516387\pi$
$72$	1.00000	0.117851
$73$	10.5931	1.23983	0.619913	−	0.784670i	$-0.287168\pi$
$73$	10.5931	1.23983	0.619913	+	0.784670i	$0.287168\pi$
$74$	2.86719	0.333304
$75$	−5.86719	−0.677484
$76$	1.00000	0.114708
$77$	2.95290	0.336515
$78$	1.82009	0.206085
$79$	12.5074	1.40719	0.703594	−	0.710602i	$-0.251577\pi$
$79$	12.5074	1.40719	0.703594	+	0.710602i	$0.251577\pi$
$80$	−3.29654	−0.368565
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	−7.64018	−0.838619	−0.419309	−	0.907843i	$-0.637728\pi$
$83$	−7.64018	−0.838619	−0.419309	+	0.907843i	$0.637728\pi$
$84$	−2.00000	−0.218218
$85$	−18.5931	−2.01670
$86$	−6.68727	−0.721107
$87$	−2.17991	−0.233711
$88$	1.47645	0.157390
$89$	14.6873	1.55685	0.778424	−	0.627739i	$-0.216019\pi$
$89$	14.6873	1.55685	0.778424	+	0.627739i	$0.216019\pi$
$90$	−3.29654	−0.347486
$91$	−3.64018	−0.381594
$92$	−3.82009	−0.398272
$93$	0.867185	0.0899229
$94$	11.2965	1.16515
$95$	−3.29654	−0.338218
$96$	−1.00000	−0.102062
$97$	8.59308	0.872495	0.436248	−	0.899827i	$-0.356307\pi$
$97$	8.59308	0.872495	0.436248	+	0.899827i	$0.356307\pi$
$98$	−3.00000	−0.303046
$99$	1.47645	0.148389
$100$	5.86719	0.586719
$101$	10.3436	1.02923	0.514615	−	0.857421i	$-0.327935\pi$
$101$	10.3436	1.02923	0.514615	+	0.857421i	$0.327935\pi$
$102$	−5.64018	−0.558461
$103$	−1.22701	−0.120901	−0.0604503	−	0.998171i	$-0.519254\pi$
$103$	−1.22701	−0.120901	−0.0604503	+	0.998171i	$0.519254\pi$
$104$	−1.82009	−0.178474
$105$	6.59308	0.643419
$106$	1.00000	0.0971286
$107$	16.3275	1.57843	0.789217	−	0.614114i	$-0.210487\pi$
$107$	16.3275	1.57843	0.789217	+	0.614114i	$0.210487\pi$
$108$	−1.00000	−0.0962250
$109$	8.24945	0.790154	0.395077	−	0.918648i	$-0.370718\pi$
$109$	8.24945	0.790154	0.395077	+	0.918648i	$0.370718\pi$
$110$	−4.86719	−0.464068
$111$	−2.86719	−0.272141
$112$	2.00000	0.188982
$113$	−18.4132	−1.73217	−0.866083	−	0.499901i	$-0.833370\pi$
$113$	−18.4132	−1.73217	−0.866083	+	0.499901i	$0.833370\pi$
$114$	−1.00000	−0.0936586
$115$	12.5931	1.17431
$116$	2.17991	0.202400
$117$	−1.82009	−0.168267
$118$	1.04710	0.0963930
$119$	11.2804	1.03407
$120$	3.29654	0.300932
$121$	−8.82009	−0.801826
$122$	−11.5460	−1.04532
$123$	4.00000	0.360668
$124$	−0.867185	−0.0778755
$125$	−2.85871	−0.255691
$126$	2.00000	0.178174
$127$	−13.7259	−1.21798	−0.608988	−	0.793179i	$-0.708424\pi$
$127$	−13.7259	−1.21798	−0.608988	+	0.793179i	$0.708424\pi$
$128$	1.00000	0.0883883
$129$	6.68727	0.588782
$130$	6.00000	0.526235
$131$	13.0224	1.13778	0.568888	−	0.822415i	$-0.307374\pi$
$131$	13.0224	1.13778	0.568888	+	0.822415i	$0.307374\pi$
$132$	−1.47645	−0.128509
$133$	2.00000	0.173422
$134$	−9.54599	−0.824648
$135$	3.29654	0.283721
$136$	5.64018	0.483641
$137$	12.4294	1.06191	0.530956	−	0.847400i	$-0.321833\pi$
$137$	12.4294	1.06191	0.530956	+	0.847400i	$0.321833\pi$
$138$	3.82009	0.325188
$139$	6.50736	0.551947	0.275974	−	0.961165i	$-0.411000\pi$
$139$	6.50736	0.551947	0.275974	+	0.961165i	$0.411000\pi$
$140$	−6.59308	−0.557217
$141$	−11.2965	−0.951340
$142$	−0.867185	−0.0727726
$143$	−2.68727	−0.224721
$144$	1.00000	0.0833333
$145$	−7.18617	−0.596779
$146$	10.5931	0.876690
$147$	3.00000	0.247436
$148$	2.86719	0.235681
$149$	1.82009	0.149108	0.0745538	−	0.997217i	$-0.476247\pi$
$149$	1.82009	0.149108	0.0745538	+	0.997217i	$0.476247\pi$
$150$	−5.86719	−0.479054
$151$	14.0534	1.14365	0.571823	−	0.820377i	$-0.306237\pi$
$151$	14.0534	1.14365	0.571823	+	0.820377i	$0.306237\pi$
$152$	1.00000	0.0811107
$153$	5.64018	0.455981
$154$	2.95290	0.237952
$155$	2.85871	0.229617
$156$	1.82009	0.145724
$157$	18.9529	1.51261	0.756303	−	0.654221i	$-0.227003\pi$
$157$	18.9529	1.51261	0.756303	+	0.654221i	$0.227003\pi$
$158$	12.5074	0.995032
$159$	−1.00000	−0.0793052
$160$	−3.29654	−0.260614
$161$	−7.64018	−0.602130
$162$	1.00000	0.0785674
$163$	5.54599	0.434395	0.217198	−	0.976128i	$-0.430308\pi$
$163$	5.54599	0.434395	0.217198	+	0.976128i	$0.430308\pi$
$164$	−4.00000	−0.312348
$165$	4.86719	0.378910
$166$	−7.64018	−0.592993
$167$	19.9592	1.54449	0.772243	−	0.635327i	$-0.219135\pi$
$167$	19.9592	1.54449	0.772243	+	0.635327i	$0.219135\pi$
$168$	−2.00000	−0.154303
$169$	−9.68727	−0.745175
$170$	−18.5931	−1.42602
$171$	1.00000	0.0764719
$172$	−6.68727	−0.509900
$173$	1.40692	0.106966	0.0534830	−	0.998569i	$-0.482968\pi$
$173$	1.40692	0.106966	0.0534830	+	0.998569i	$0.482968\pi$
$174$	−2.17991	−0.165259
$175$	11.7344	0.887035
$176$	1.47645	0.111292
$177$	−1.04710	−0.0787045
$178$	14.6873	1.10086
$179$	−7.91428	−0.591541	−0.295771	−	0.955259i	$-0.595576\pi$
$179$	−7.91428	−0.591541	−0.295771	+	0.955259i	$0.595576\pi$
$180$	−3.29654	−0.245710
$181$	−8.67109	−0.644517	−0.322259	−	0.946652i	$-0.604442\pi$
$181$	−8.67109	−0.644517	−0.322259	+	0.946652i	$0.604442\pi$
$182$	−3.64018	−0.269828
$183$	11.5460	0.853504
$184$	−3.82009	−0.281621
$185$	−9.45180	−0.694910
$186$	0.867185	0.0635851
$187$	8.32745	0.608964
$188$	11.2965	0.823885
$189$	−2.00000	−0.145479
$190$	−3.29654	−0.239156
$191$	−4.17991	−0.302448	−0.151224	−	0.988500i	$-0.548321\pi$
$191$	−4.17991	−0.302448	−0.151224	+	0.988500i	$0.548321\pi$
$192$	−1.00000	−0.0721688
$193$	8.66262	0.623549	0.311774	−	0.950156i	$-0.399077\pi$
$193$	8.66262	0.623549	0.311774	+	0.950156i	$0.399077\pi$
$194$	8.59308	0.616947
$195$	−6.00000	−0.429669
$196$	−3.00000	−0.214286
$197$	−16.0534	−1.14375	−0.571877	−	0.820339i	$-0.693785\pi$
$197$	−16.0534	−1.14375	−0.571877	+	0.820339i	$0.693785\pi$
$198$	1.47645	0.104927
$199$	25.8734	1.83412	0.917060	−	0.398750i	$-0.130556\pi$
$199$	25.8734	1.83412	0.917060	+	0.398750i	$0.130556\pi$
$200$	5.86719	0.414873
$201$	9.54599	0.673322
$202$	10.3436	0.727776
$203$	4.35982	0.305999
$204$	−5.64018	−0.394891
$205$	13.1862	0.920962
$206$	−1.22701	−0.0854896
$207$	−3.82009	−0.265515
$208$	−1.82009	−0.126200
$209$	1.47645	0.102128
$210$	6.59308	0.454966
$211$	−11.4765	−0.790072	−0.395036	−	0.918666i	$-0.629268\pi$
$211$	−11.4765	−0.790072	−0.395036	+	0.918666i	$0.629268\pi$
$212$	1.00000	0.0686803
$213$	0.867185	0.0594186
$214$	16.3275	1.11612
$215$	22.0449	1.50345
$216$	−1.00000	−0.0680414
$217$	−1.73437	−0.117737
$218$	8.24945	0.558723
$219$	−10.5931	−0.715814
$220$	−4.86719	−0.328146
$221$	−10.2656	−0.690541
$222$	−2.86719	−0.192433
$223$	4.34364	0.290871	0.145436	−	0.989368i	$-0.453542\pi$
$223$	4.34364	0.290871	0.145436	+	0.989368i	$0.453542\pi$
$224$	2.00000	0.133631
$225$	5.86719	0.391146
$226$	−18.4132	−1.22483
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	−1.00000	−0.0662266
$229$	5.82009	0.384602	0.192301	−	0.981336i	$-0.438405\pi$
$229$	5.82009	0.384602	0.192301	+	0.981336i	$0.438405\pi$
$230$	12.5931	0.830363
$231$	−2.95290	−0.194287
$232$	2.17991	0.143118
$233$	7.38226	0.483628	0.241814	−	0.970323i	$-0.422258\pi$
$233$	7.38226	0.483628	0.241814	+	0.970323i	$0.422258\pi$
$234$	−1.82009	−0.118983
$235$	−37.2395	−2.42924
$236$	1.04710	0.0681601
$237$	−12.5074	−0.812441
$238$	11.2804	0.731197
$239$	15.7877	1.02122	0.510611	−	0.859812i	$-0.329419\pi$
$239$	15.7877	1.02122	0.510611	+	0.859812i	$0.329419\pi$
$240$	3.29654	0.212791
$241$	−2.49889	−0.160968	−0.0804839	−	0.996756i	$-0.525647\pi$
$241$	−2.49889	−0.160968	−0.0804839	+	0.996756i	$0.525647\pi$
$242$	−8.82009	−0.566977
$243$	−1.00000	−0.0641500
$244$	−11.5460	−0.739156
$245$	9.88962	0.631825
$246$	4.00000	0.255031
$247$	−1.82009	−0.115810
$248$	−0.867185	−0.0550663
$249$	7.64018	0.484177
$250$	−2.85871	−0.180801
$251$	14.5931	0.921107	0.460554	−	0.887632i	$-0.347651\pi$
$251$	14.5931	0.921107	0.460554	+	0.887632i	$0.347651\pi$
$252$	2.00000	0.125988
$253$	−5.64018	−0.354595
$254$	−13.7259	−0.861239
$255$	18.5931	1.16434
$256$	1.00000	0.0625000
$257$	23.1862	1.44631	0.723157	−	0.690684i	$-0.242690\pi$
$257$	23.1862	1.44631	0.723157	+	0.690684i	$0.242690\pi$
$258$	6.68727	0.416332
$259$	5.73437	0.356316
$260$	6.00000	0.372104
$261$	2.17991	0.134933
$262$	13.0224	0.804529
$263$	28.1475	1.73565	0.867826	−	0.496868i	$-0.165517\pi$
$263$	28.1475	1.73565	0.867826	+	0.496868i	$0.165517\pi$
$264$	−1.47645	−0.0908693
$265$	−3.29654	−0.202505
$266$	2.00000	0.122628
$267$	−14.6873	−0.898847
$268$	−9.54599	−0.583114
$269$	11.5460	0.703971	0.351986	−	0.936005i	$-0.385507\pi$
$269$	11.5460	0.703971	0.351986	+	0.936005i	$0.385507\pi$
$270$	3.29654	0.200621
$271$	−15.9676	−0.969965	−0.484982	−	0.874524i	$-0.661174\pi$
$271$	−15.9676	−0.969965	−0.484982	+	0.874524i	$0.661174\pi$
$272$	5.64018	0.341986
$273$	3.64018	0.220314
$274$	12.4294	0.750885
$275$	8.66262	0.522376
$276$	3.82009	0.229942
$277$	−3.54599	−0.213058	−0.106529	−	0.994310i	$-0.533974\pi$
$277$	−3.54599	−0.213058	−0.106529	+	0.994310i	$0.533974\pi$
$278$	6.50736	0.390286
$279$	−0.867185	−0.0519170
$280$	−6.59308	−0.394012
$281$	20.4047	1.21724	0.608621	−	0.793461i	$-0.291723\pi$
$281$	20.4047	1.21724	0.608621	+	0.793461i	$0.291723\pi$
$282$	−11.2965	−0.672699
$283$	−26.5607	−1.57887	−0.789435	−	0.613834i	$-0.789627\pi$
$283$	−26.5607	−1.57887	−0.789435	+	0.613834i	$0.789627\pi$
$284$	−0.867185	−0.0514580
$285$	3.29654	0.195270
$286$	−2.68727	−0.158902
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	14.8116	0.871272
$290$	−7.18617	−0.421986
$291$	−8.59308	−0.503735
$292$	10.5931	0.619913
$293$	5.99153	0.350029	0.175014	−	0.984566i	$-0.444003\pi$
$293$	5.99153	0.350029	0.175014	+	0.984566i	$0.444003\pi$
$294$	3.00000	0.174964
$295$	−3.45180	−0.200971
$296$	2.86719	0.166652
$297$	−1.47645	−0.0856724
$298$	1.82009	0.105435
$299$	6.95290	0.402097
$300$	−5.86719	−0.338742
$301$	−13.3745	−0.770896
$302$	14.0534	0.808679
$303$	−10.3436	−0.594226
$304$	1.00000	0.0573539
$305$	38.0618	2.17941
$306$	5.64018	0.322428
$307$	24.0695	1.37372	0.686860	−	0.726789i	$-0.258988\pi$
$307$	24.0695	1.37372	0.686860	+	0.726789i	$0.258988\pi$
$308$	2.95290	0.168257
$309$	1.22701	0.0698020
$310$	2.85871	0.162364
$311$	−2.34364	−0.132895	−0.0664477	−	0.997790i	$-0.521167\pi$
$311$	−2.34364	−0.132895	−0.0664477	+	0.997790i	$0.521167\pi$
$312$	1.82009	0.103042
$313$	−16.4989	−0.932572	−0.466286	−	0.884634i	$-0.654408\pi$
$313$	−16.4989	−0.932572	−0.466286	+	0.884634i	$0.654408\pi$
$314$	18.9529	1.06957
$315$	−6.59308	−0.371478
$316$	12.5074	0.703594
$317$	16.7815	0.942541	0.471271	−	0.881989i	$-0.343796\pi$
$317$	16.7815	0.942541	0.471271	+	0.881989i	$0.343796\pi$
$318$	−1.00000	−0.0560772
$319$	3.21853	0.180203
$320$	−3.29654	−0.184282
$321$	−16.3275	−0.911310
$322$	−7.64018	−0.425771
$323$	5.64018	0.313828
$324$	1.00000	0.0555556
$325$	−10.6788	−0.592353
$326$	5.54599	0.307164
$327$	−8.24945	−0.456195
$328$	−4.00000	−0.220863
$329$	22.5931	1.24560
$330$	4.86719	0.267930
$331$	4.61774	0.253814	0.126907	−	0.991915i	$-0.459495\pi$
$331$	4.61774	0.253814	0.126907	+	0.991915i	$0.459495\pi$
$332$	−7.64018	−0.419309
$333$	2.86719	0.157121
$334$	19.9592	1.09212
$335$	31.4687	1.71932
$336$	−2.00000	−0.109109
$337$	−29.5832	−1.61150	−0.805749	−	0.592258i	$-0.798237\pi$
$337$	−29.5832	−1.61150	−0.805749	+	0.592258i	$0.798237\pi$
$338$	−9.68727	−0.526918
$339$	18.4132	1.00007
$340$	−18.5931	−1.00835
$341$	−1.28036	−0.0693352
$342$	1.00000	0.0540738
$343$	−20.0000	−1.07990
$344$	−6.68727	−0.360554
$345$	−12.5931	−0.677989
$346$	1.40692	0.0756363
$347$	−31.9430	−1.71479	−0.857394	−	0.514660i	$-0.827918\pi$
$347$	−31.9430	−1.71479	−0.857394	+	0.514660i	$0.827918\pi$
$348$	−2.17991	−0.116855
$349$	−29.8734	−1.59909	−0.799544	−	0.600607i	$-0.794926\pi$
$349$	−29.8734	−1.59909	−0.799544	+	0.600607i	$0.794926\pi$
$350$	11.7344	0.627228
$351$	1.82009	0.0971492
$352$	1.47645	0.0786952
$353$	0.0695349	0.00370097	0.00185048	−	0.999998i	$-0.499411\pi$
$353$	0.0695349	0.00370097	0.00185048	+	0.999998i	$0.499411\pi$
$354$	−1.04710	−0.0556525
$355$	2.85871	0.151725
$356$	14.6873	0.778424
$357$	−11.2804	−0.597020
$358$	−7.91428	−0.418283
$359$	−10.5846	−0.558634	−0.279317	−	0.960199i	$-0.590108\pi$
$359$	−10.5846	−0.558634	−0.279317	+	0.960199i	$0.590108\pi$
$360$	−3.29654	−0.173743
$361$	1.00000	0.0526316
$362$	−8.67109	−0.455742
$363$	8.82009	0.462935
$364$	−3.64018	−0.190797
$365$	−34.9205	−1.82782
$366$	11.5460	0.603518
$367$	4.68727	0.244674	0.122337	−	0.992489i	$-0.460961\pi$
$367$	4.68727	0.244674	0.122337	+	0.992489i	$0.460961\pi$
$368$	−3.82009	−0.199136
$369$	−4.00000	−0.208232
$370$	−9.45180	−0.491375
$371$	2.00000	0.103835
$372$	0.867185	0.0449615
$373$	−16.0780	−0.832488	−0.416244	−	0.909253i	$-0.636654\pi$
$373$	−16.0780	−0.832488	−0.416244	+	0.909253i	$0.636654\pi$
$374$	8.32745	0.430602
$375$	2.85871	0.147623
$376$	11.2965	0.582575
$377$	−3.96763	−0.204343
$378$	−2.00000	−0.102869
$379$	35.0920	1.80255	0.901277	−	0.433244i	$-0.142631\pi$
$379$	35.0920	1.80255	0.901277	+	0.433244i	$0.142631\pi$
$380$	−3.29654	−0.169109
$381$	13.7259	0.703199
$382$	−4.17991	−0.213863
$383$	23.9592	1.22426	0.612128	−	0.790759i	$-0.290314\pi$
$383$	23.9592	1.22426	0.612128	+	0.790759i	$0.290314\pi$
$384$	−1.00000	−0.0510310
$385$	−9.73437	−0.496109
$386$	8.66262	0.440916
$387$	−6.68727	−0.339933
$388$	8.59308	0.436248
$389$	−16.6093	−0.842123	−0.421062	−	0.907032i	$-0.638342\pi$
$389$	−16.6093	−0.842123	−0.421062	+	0.907032i	$0.638342\pi$
$390$	−6.00000	−0.303822
$391$	−21.5460	−1.08963
$392$	−3.00000	−0.151523
$393$	−13.0224	−0.656895
$394$	−16.0534	−0.808756
$395$	−41.2310	−2.07456
$396$	1.47645	0.0741945
$397$	23.7792	1.19345	0.596723	−	0.802447i	$-0.296469\pi$
$397$	23.7792	1.19345	0.596723	+	0.802447i	$0.296469\pi$
$398$	25.8734	1.29692
$399$	−2.00000	−0.100125
$400$	5.86719	0.293359
$401$	3.64018	0.181782	0.0908909	−	0.995861i	$-0.471029\pi$
$401$	3.64018	0.181782	0.0908909	+	0.995861i	$0.471029\pi$
$402$	9.54599	0.476111
$403$	1.57835	0.0786234
$404$	10.3436	0.514615
$405$	−3.29654	−0.163806
$406$	4.35982	0.216374
$407$	4.23326	0.209835
$408$	−5.64018	−0.279230
$409$	7.76674	0.384040	0.192020	−	0.981391i	$-0.438496\pi$
$409$	7.76674	0.384040	0.192020	+	0.981391i	$0.438496\pi$
$410$	13.1862	0.651218
$411$	−12.4294	−0.613095
$412$	−1.22701	−0.0604503
$413$	2.09419	0.103048
$414$	−3.82009	−0.187747
$415$	25.1862	1.23634
$416$	−1.82009	−0.0892372
$417$	−6.50736	−0.318667
$418$	1.47645	0.0722156
$419$	−3.67255	−0.179416	−0.0897078	−	0.995968i	$-0.528593\pi$
$419$	−3.67255	−0.179416	−0.0897078	+	0.995968i	$0.528593\pi$
$420$	6.59308	0.321710
$421$	−2.42088	−0.117987	−0.0589933	−	0.998258i	$-0.518789\pi$
$421$	−2.42088	−0.117987	−0.0589933	+	0.998258i	$0.518789\pi$
$422$	−11.4765	−0.558665
$423$	11.2965	0.549257
$424$	1.00000	0.0485643
$425$	33.0920	1.60520
$426$	0.867185	0.0420153
$427$	−23.0920	−1.11750
$428$	16.3275	0.789217
$429$	2.68727	0.129743
$430$	22.0449	1.06310
$431$	31.0147	1.49393	0.746963	−	0.664865i	$-0.231511\pi$
$431$	31.0147	1.49393	0.746963	+	0.664865i	$0.231511\pi$
$432$	−1.00000	−0.0481125
$433$	31.1538	1.49716	0.748578	−	0.663047i	$-0.230737\pi$
$433$	31.1538	1.49716	0.748578	+	0.663047i	$0.230737\pi$
$434$	−1.73437	−0.0832525
$435$	7.18617	0.344550
$436$	8.24945	0.395077
$437$	−3.82009	−0.182740
$438$	−10.5931	−0.506157
$439$	23.7182	1.13201	0.566004	−	0.824403i	$-0.308489\pi$
$439$	23.7182	1.13201	0.566004	+	0.824403i	$0.308489\pi$
$440$	−4.86719	−0.232034
$441$	−3.00000	−0.142857
$442$	−10.2656	−0.488286
$443$	−23.6078	−1.12164	−0.560820	−	0.827937i	$-0.689514\pi$
$443$	−23.6078	−1.12164	−0.560820	+	0.827937i	$0.689514\pi$
$444$	−2.86719	−0.136071
$445$	−48.4172	−2.29520
$446$	4.34364	0.205677
$447$	−1.82009	−0.0860873
$448$	2.00000	0.0944911
$449$	6.99374	0.330055	0.165028	−	0.986289i	$-0.447229\pi$
$449$	6.99374	0.330055	0.165028	+	0.986289i	$0.447229\pi$
$450$	5.86719	0.276582
$451$	−5.90581	−0.278094
$452$	−18.4132	−0.866083
$453$	−14.0534	−0.660284
$454$	2.00000	0.0938647
$455$	12.0000	0.562569
$456$	−1.00000	−0.0468293
$457$	−9.31273	−0.435631	−0.217815	−	0.975990i	$-0.569893\pi$
$457$	−9.31273	−0.435631	−0.217815	+	0.975990i	$0.569893\pi$
$458$	5.82009	0.271955
$459$	−5.64018	−0.263261
$460$	12.5931	0.587156
$461$	19.8650	0.925204	0.462602	−	0.886566i	$-0.346916\pi$
$461$	19.8650	0.925204	0.462602	+	0.886566i	$0.346916\pi$
$462$	−2.95290	−0.137382
$463$	−1.76674	−0.0821073	−0.0410536	−	0.999157i	$-0.513071\pi$
$463$	−1.76674	−0.0821073	−0.0410536	+	0.999157i	$0.513071\pi$
$464$	2.17991	0.101200
$465$	−2.85871	−0.132570
$466$	7.38226	0.341977
$467$	−3.02244	−0.139862	−0.0699309	−	0.997552i	$-0.522278\pi$
$467$	−3.02244	−0.139862	−0.0699309	+	0.997552i	$0.522278\pi$
$468$	−1.82009	−0.0841337
$469$	−19.0920	−0.881585
$470$	−37.2395	−1.71773
$471$	−18.9529	−0.873304
$472$	1.04710	0.0481965
$473$	−9.87344	−0.453981
$474$	−12.5074	−0.574482
$475$	5.86719	0.269205
$476$	11.2804	0.517034
$477$	1.00000	0.0457869
$478$	15.7877	0.722114
$479$	−9.58683	−0.438033	−0.219017	−	0.975721i	$-0.570285\pi$
$479$	−9.58683	−0.438033	−0.219017	+	0.975721i	$0.570285\pi$
$480$	3.29654	0.150466
$481$	−5.21853	−0.237945
$482$	−2.49889	−0.113821
$483$	7.64018	0.347640
$484$	−8.82009	−0.400913
$485$	−28.3275	−1.28628
$486$	−1.00000	−0.0453609
$487$	−14.3436	−0.649972	−0.324986	−	0.945719i	$-0.605360\pi$
$487$	−14.3436	−0.649972	−0.324986	+	0.945719i	$0.605360\pi$
$488$	−11.5460	−0.522662
$489$	−5.54599	−0.250798
$490$	9.88962	0.446768
$491$	42.0449	1.89746	0.948729	−	0.316089i	$-0.102370\pi$
$491$	42.0449	1.89746	0.948729	+	0.316089i	$0.102370\pi$
$492$	4.00000	0.180334
$493$	12.2951	0.553743
$494$	−1.82009	−0.0818897
$495$	−4.86719	−0.218764
$496$	−0.867185	−0.0389378
$497$	−1.73437	−0.0777972
$498$	7.64018	0.342365
$499$	−25.2719	−1.13132	−0.565662	−	0.824637i	$-0.691379\pi$
$499$	−25.2719	−1.13132	−0.565662	+	0.824637i	$0.691379\pi$
$500$	−2.85871	−0.127845
$501$	−19.9592	−0.891709
$502$	14.5931	0.651321
$503$	1.03862	0.0463099	0.0231550	−	0.999732i	$-0.492629\pi$
$503$	1.03862	0.0463099	0.0231550	+	0.999732i	$0.492629\pi$
$504$	2.00000	0.0890871
$505$	−34.0982	−1.51735
$506$	−5.64018	−0.250737
$507$	9.68727	0.430227
$508$	−13.7259	−0.608988
$509$	−10.5931	−0.469530	−0.234765	−	0.972052i	$-0.575432\pi$
$509$	−10.5931	−0.469530	−0.234765	+	0.972052i	$0.575432\pi$
$510$	18.5931	0.823316
$511$	21.1862	0.937221
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	23.1862	1.02270
$515$	4.04488	0.178239
$516$	6.68727	0.294391
$517$	16.6788	0.733533
$518$	5.73437	0.251954
$519$	−1.40692	−0.0617568
$520$	6.00000	0.263117
$521$	12.1799	0.533612	0.266806	−	0.963750i	$-0.414032\pi$
$521$	12.1799	0.533612	0.266806	+	0.963750i	$0.414032\pi$
$522$	2.17991	0.0954121
$523$	24.2086	1.05857	0.529284	−	0.848445i	$-0.322461\pi$
$523$	24.2086	1.05857	0.529284	+	0.848445i	$0.322461\pi$
$524$	13.0224	0.568888
$525$	−11.7344	−0.512130
$526$	28.1475	1.22729
$527$	−4.89108	−0.213059
$528$	−1.47645	−0.0642543
$529$	−8.40692	−0.365518
$530$	−3.29654	−0.143193
$531$	1.04710	0.0454401
$532$	2.00000	0.0867110
$533$	7.28036	0.315347
$534$	−14.6873	−0.635581
$535$	−53.8241	−2.32702
$536$	−9.54599	−0.412324
$537$	7.91428	0.341526
$538$	11.5460	0.497783
$539$	−4.42936	−0.190786
$540$	3.29654	0.141861
$541$	33.2480	1.42944	0.714721	−	0.699409i	$-0.246554\pi$
$541$	33.2480	1.42944	0.714721	+	0.699409i	$0.246554\pi$
$542$	−15.9676	−0.685869
$543$	8.67109	0.372112
$544$	5.64018	0.241821
$545$	−27.1946	−1.16489
$546$	3.64018	0.155785
$547$	−2.94519	−0.125927	−0.0629637	−	0.998016i	$-0.520055\pi$
$547$	−2.94519	−0.125927	−0.0629637	+	0.998016i	$0.520055\pi$
$548$	12.4294	0.530956
$549$	−11.5460	−0.492771
$550$	8.66262	0.369375
$551$	2.17991	0.0928673
$552$	3.82009	0.162594
$553$	25.0147	1.06373
$554$	−3.54599	−0.150655
$555$	9.45180	0.401206
$556$	6.50736	0.275974
$557$	7.70124	0.326312	0.163156	−	0.986600i	$-0.447833\pi$
$557$	7.70124	0.326312	0.163156	+	0.986600i	$0.447833\pi$
$558$	−0.867185	−0.0367109
$559$	12.1714	0.514797
$560$	−6.59308	−0.278609
$561$	−8.32745	−0.351585
$562$	20.4047	0.860720
$563$	−12.4132	−0.523153	−0.261576	−	0.965183i	$-0.584242\pi$
$563$	−12.4132	−0.523153	−0.261576	+	0.965183i	$0.584242\pi$
$564$	−11.2965	−0.475670
$565$	60.6998	2.55366
$566$	−26.5607	−1.11643
$567$	2.00000	0.0839921
$568$	−0.867185	−0.0363863
$569$	−12.0772	−0.506304	−0.253152	−	0.967426i	$-0.581467\pi$
$569$	−12.0772	−0.506304	−0.253152	+	0.967426i	$0.581467\pi$
$570$	3.29654	0.138077
$571$	−14.1799	−0.593411	−0.296705	−	0.954969i	$-0.595888\pi$
$571$	−14.1799	−0.593411	−0.296705	+	0.954969i	$0.595888\pi$
$572$	−2.68727	−0.112361
$573$	4.17991	0.174618
$574$	−8.00000	−0.333914
$575$	−22.4132	−0.934694
$576$	1.00000	0.0416667
$577$	−9.12434	−0.379851	−0.189926	−	0.981798i	$-0.560825\pi$
$577$	−9.12434	−0.379851	−0.189926	+	0.981798i	$0.560825\pi$
$578$	14.8116	0.616082
$579$	−8.66262	−0.360006
$580$	−7.18617	−0.298389
$581$	−15.2804	−0.633936
$582$	−8.59308	−0.356195
$583$	1.47645	0.0611484
$584$	10.5931	0.438345
$585$	6.00000	0.248069
$586$	5.99153	0.247508
$587$	16.1637	0.667148	0.333574	−	0.942724i	$-0.391745\pi$
$587$	16.1637	0.667148	0.333574	+	0.942724i	$0.391745\pi$
$588$	3.00000	0.123718
$589$	−0.867185	−0.0357317
$590$	−3.45180	−0.142108
$591$	16.0534	0.660347
$592$	2.86719	0.117841
$593$	−2.18838	−0.0898661	−0.0449331	−	0.998990i	$-0.514307\pi$
$593$	−2.18838	−0.0898661	−0.0449331	+	0.998990i	$0.514307\pi$
$594$	−1.47645	−0.0605796
$595$	−37.1862	−1.52448
$596$	1.82009	0.0745538
$597$	−25.8734	−1.05893
$598$	6.95290	0.284325
$599$	−27.3576	−1.11780	−0.558901	−	0.829235i	$-0.688777\pi$
$599$	−27.3576	−1.11780	−0.558901	+	0.829235i	$0.688777\pi$
$600$	−5.86719	−0.239527
$601$	−19.6648	−0.802145	−0.401073	−	0.916046i	$-0.631363\pi$
$601$	−19.6648	−0.802145	−0.401073	+	0.916046i	$0.631363\pi$
$602$	−13.3745	−0.545106
$603$	−9.54599	−0.388743
$604$	14.0534	0.571823
$605$	29.0758	1.18210
$606$	−10.3436	−0.420182
$607$	41.4974	1.68433	0.842164	−	0.539221i	$-0.181281\pi$
$607$	41.4974	1.68433	0.842164	+	0.539221i	$0.181281\pi$
$608$	1.00000	0.0405554
$609$	−4.35982	−0.176669
$610$	38.0618	1.54108
$611$	−20.5607	−0.831797
$612$	5.64018	0.227991
$613$	20.6703	0.834867	0.417433	−	0.908708i	$-0.362930\pi$
$613$	20.6703	0.834867	0.417433	+	0.908708i	$0.362930\pi$
$614$	24.0695	0.971367
$615$	−13.1862	−0.531718
$616$	2.95290	0.118976
$617$	17.8812	0.719868	0.359934	−	0.932978i	$-0.382799\pi$
$617$	17.8812	0.719868	0.359934	+	0.932978i	$0.382799\pi$
$618$	1.22701	0.0493574
$619$	−3.73437	−0.150097	−0.0750485	−	0.997180i	$-0.523911\pi$
$619$	−3.73437	−0.150097	−0.0750485	+	0.997180i	$0.523911\pi$
$620$	2.85871	0.114809
$621$	3.82009	0.153295
$622$	−2.34364	−0.0939713
$623$	29.3745	1.17687
$624$	1.82009	0.0728619
$625$	−19.9121	−0.796483
$626$	−16.4989	−0.659428
$627$	−1.47645	−0.0589638
$628$	18.9529	0.756303
$629$	16.1714	0.644797
$630$	−6.59308	−0.262675
$631$	−6.60156	−0.262804	−0.131402	−	0.991329i	$-0.541948\pi$
$631$	−6.60156	−0.262804	−0.131402	+	0.991329i	$0.541948\pi$
$632$	12.5074	0.497516
$633$	11.4765	0.456148
$634$	16.7815	0.666477
$635$	45.2480	1.79561
$636$	−1.00000	−0.0396526
$637$	5.46027	0.216344
$638$	3.21853	0.127423
$639$	−0.867185	−0.0343053
$640$	−3.29654	−0.130307
$641$	−34.4989	−1.36262	−0.681312	−	0.731993i	$-0.738590\pi$
$641$	−34.4989	−1.36262	−0.681312	+	0.731993i	$0.738590\pi$
$642$	−16.3275	−0.644393
$643$	26.6998	1.05294	0.526468	−	0.850195i	$-0.323516\pi$
$643$	26.6998	1.05294	0.526468	+	0.850195i	$0.323516\pi$
$644$	−7.64018	−0.301065
$645$	−22.0449	−0.868016
$646$	5.64018	0.221910
$647$	−8.43783	−0.331725	−0.165863	−	0.986149i	$-0.553041\pi$
$647$	−8.43783	−0.331725	−0.165863	+	0.986149i	$0.553041\pi$
$648$	1.00000	0.0392837
$649$	1.54599	0.0606853
$650$	−10.6788	−0.418857
$651$	1.73437	0.0679753
$652$	5.54599	0.217198
$653$	0.102665	0.00401759	0.00200879	−	0.999998i	$-0.499361\pi$
$653$	0.102665	0.00401759	0.00200879	+	0.999998i	$0.499361\pi$
$654$	−8.24945	−0.322579
$655$	−42.9290	−1.67738
$656$	−4.00000	−0.156174
$657$	10.5931	0.413276
$658$	22.5931	0.880770
$659$	27.7259	1.08005	0.540024	−	0.841650i	$-0.318415\pi$
$659$	27.7259	1.08005	0.540024	+	0.841650i	$0.318415\pi$
$660$	4.86719	0.189455
$661$	−3.39844	−0.132184	−0.0660921	−	0.997814i	$-0.521053\pi$
$661$	−3.39844	−0.132184	−0.0660921	+	0.997814i	$0.521053\pi$
$662$	4.61774	0.179474
$663$	10.2656	0.398684
$664$	−7.64018	−0.296496
$665$	−6.59308	−0.255669
$666$	2.86719	0.111101
$667$	−8.32745	−0.322440
$668$	19.9592	0.772243
$669$	−4.34364	−0.167935
$670$	31.4687	1.21574
$671$	−17.0471	−0.658096
$672$	−2.00000	−0.0771517
$673$	16.7815	0.646878	0.323439	−	0.946249i	$-0.395161\pi$
$673$	16.7815	0.646878	0.323439	+	0.946249i	$0.395161\pi$
$674$	−29.5832	−1.13950
$675$	−5.86719	−0.225828
$676$	−9.68727	−0.372587
$677$	−39.3745	−1.51329	−0.756643	−	0.653828i	$-0.773162\pi$
$677$	−39.3745	−1.51329	−0.756643	+	0.653828i	$0.773162\pi$
$678$	18.4132	0.707154
$679$	17.1862	0.659545
$680$	−18.5931	−0.713012
$681$	−2.00000	−0.0766402
$682$	−1.28036	−0.0490274
$683$	−42.8263	−1.63870	−0.819352	−	0.573290i	$-0.805667\pi$
$683$	−42.8263	−1.63870	−0.819352	+	0.573290i	$0.805667\pi$
$684$	1.00000	0.0382360
$685$	−40.9739	−1.56553
$686$	−20.0000	−0.763604
$687$	−5.82009	−0.222050
$688$	−6.68727	−0.254950
$689$	−1.82009	−0.0693399
$690$	−12.5931	−0.479410
$691$	−40.3275	−1.53413	−0.767064	−	0.641570i	$-0.778283\pi$
$691$	−40.3275	−1.53413	−0.767064	+	0.641570i	$0.778283\pi$
$692$	1.40692	0.0534830
$693$	2.95290	0.112172
$694$	−31.9430	−1.21254
$695$	−21.4518	−0.813713
$696$	−2.17991	−0.0826293
$697$	−22.5607	−0.854548
$698$	−29.8734	−1.13073
$699$	−7.38226	−0.279223
$700$	11.7344	0.443518
$701$	−21.7012	−0.819645	−0.409822	−	0.912165i	$-0.634409\pi$
$701$	−21.7012	−0.819645	−0.409822	+	0.912165i	$0.634409\pi$
$702$	1.82009	0.0686948
$703$	2.86719	0.108138
$704$	1.47645	0.0556459
$705$	37.2395	1.40252
$706$	0.0695349	0.00261698
$707$	20.6873	0.778025
$708$	−1.04710	−0.0393523
$709$	−12.0942	−0.454207	−0.227103	−	0.973871i	$-0.572926\pi$
$709$	−12.0942	−0.454207	−0.227103	+	0.973871i	$0.572926\pi$
$710$	2.85871	0.107286
$711$	12.5074	0.469063
$712$	14.6873	0.550429
$713$	3.31273	0.124063
$714$	−11.2804	−0.422157
$715$	8.85871	0.331297
$716$	−7.91428	−0.295771
$717$	−15.7877	−0.589603
$718$	−10.5846	−0.395014
$719$	15.1328	0.564359	0.282179	−	0.959362i	$-0.408943\pi$
$719$	15.1328	0.564359	0.282179	+	0.959362i	$0.408943\pi$
$720$	−3.29654	−0.122855
$721$	−2.45401	−0.0913922
$722$	1.00000	0.0372161
$723$	2.49889	0.0929348
$724$	−8.67109	−0.322259
$725$	12.7899	0.475006
$726$	8.82009	0.327344
$727$	−47.6851	−1.76854	−0.884270	−	0.466975i	$-0.845344\pi$
$727$	−47.6851	−1.76854	−0.884270	+	0.466975i	$0.845344\pi$
$728$	−3.64018	−0.134914
$729$	1.00000	0.0370370
$730$	−34.9205	−1.29247
$731$	−37.7174	−1.39503
$732$	11.5460	0.426752
$733$	−27.1946	−1.00446	−0.502228	−	0.864735i	$-0.667486\pi$
$733$	−27.1946	−1.00446	−0.502228	+	0.864735i	$0.667486\pi$
$734$	4.68727	0.173010
$735$	−9.88962	−0.364784
$736$	−3.82009	−0.140810
$737$	−14.0942	−0.519166
$738$	−4.00000	−0.147242
$739$	−17.9058	−0.658676	−0.329338	−	0.944212i	$-0.606826\pi$
$739$	−17.9058	−0.658676	−0.329338	+	0.944212i	$0.606826\pi$
$740$	−9.45180	−0.347455
$741$	1.82009	0.0668627
$742$	2.00000	0.0734223
$743$	−14.6549	−0.537636	−0.268818	−	0.963191i	$-0.586633\pi$
$743$	−14.6549	−0.537636	−0.268818	+	0.963191i	$0.586633\pi$
$744$	0.867185	0.0317926
$745$	−6.00000	−0.219823
$746$	−16.0780	−0.588658
$747$	−7.64018	−0.279540
$748$	8.32745	0.304482
$749$	32.6549	1.19318
$750$	2.85871	0.104385
$751$	5.79543	0.211478	0.105739	−	0.994394i	$-0.466279\pi$
$751$	5.79543	0.211478	0.105739	+	0.994394i	$0.466279\pi$
$752$	11.2965	0.411942
$753$	−14.5931	−0.531802
$754$	−3.96763	−0.144493
$755$	−46.3275	−1.68603
$756$	−2.00000	−0.0727393
$757$	47.6851	1.73314	0.866571	−	0.499053i	$-0.166319\pi$
$757$	47.6851	1.73314	0.866571	+	0.499053i	$0.166319\pi$
$758$	35.0920	1.27460
$759$	5.64018	0.204726
$760$	−3.29654	−0.119578
$761$	−30.7075	−1.11315	−0.556573	−	0.830799i	$-0.687884\pi$
$761$	−30.7075	−1.11315	−0.556573	+	0.830799i	$0.687884\pi$
$762$	13.7259	0.497237
$763$	16.4989	0.597300
$764$	−4.17991	−0.151224
$765$	−18.5931	−0.672234
$766$	23.9592	0.865680
$767$	−1.90581	−0.0688147
$768$	−1.00000	−0.0360844
$769$	−21.5784	−0.778135	−0.389068	−	0.921209i	$-0.627203\pi$
$769$	−21.5784	−0.778135	−0.389068	+	0.921209i	$0.627203\pi$
$770$	−9.73437	−0.350802
$771$	−23.1862	−0.835029
$772$	8.66262	0.311774
$773$	−36.6998	−1.32000	−0.660000	−	0.751266i	$-0.729444\pi$
$773$	−36.6998	−1.32000	−0.660000	+	0.751266i	$0.729444\pi$
$774$	−6.68727	−0.240369
$775$	−5.08794	−0.182764
$776$	8.59308	0.308474
$777$	−5.73437	−0.205719
$778$	−16.6093	−0.595471
$779$	−4.00000	−0.143315
$780$	−6.00000	−0.214834
$781$	−1.28036	−0.0458148
$782$	−21.5460	−0.770483
$783$	−2.17991	−0.0779037
$784$	−3.00000	−0.107143
$785$	−62.4790	−2.22997
$786$	−13.0224	−0.464495
$787$	−32.9654	−1.17509	−0.587545	−	0.809191i	$-0.699906\pi$
$787$	−32.9654	−1.17509	−0.587545	+	0.809191i	$0.699906\pi$
$788$	−16.0534	−0.571877
$789$	−28.1475	−1.00208
$790$	−41.2310	−1.46693
$791$	−36.8263	−1.30939
$792$	1.47645	0.0524634
$793$	21.0147	0.746255
$794$	23.7792	0.843894
$795$	3.29654	0.116916
$796$	25.8734	0.917060
$797$	45.1538	1.59943	0.799715	−	0.600380i	$-0.204984\pi$
$797$	45.1538	1.59943	0.799715	+	0.600380i	$0.204984\pi$
$798$	−2.00000	−0.0707992
$799$	63.7145	2.25406
$800$	5.86719	0.207436
$801$	14.6873	0.518949
$802$	3.64018	0.128539
$803$	15.6402	0.551930
$804$	9.54599	0.336661
$805$	25.1862	0.887696
$806$	1.57835	0.0555952
$807$	−11.5460	−0.406438
$808$	10.3436	0.363888
$809$	−28.8341	−1.01375	−0.506876	−	0.862019i	$-0.669200\pi$
$809$	−28.8341	−1.01375	−0.506876	+	0.862019i	$0.669200\pi$
$810$	−3.29654	−0.115829
$811$	−38.2256	−1.34228	−0.671140	−	0.741330i	$-0.734195\pi$
$811$	−38.2256	−1.34228	−0.671140	+	0.741330i	$0.734195\pi$
$812$	4.35982	0.153000
$813$	15.9676	0.560009
$814$	4.23326	0.148376
$815$	−18.2826	−0.640411
$816$	−5.64018	−0.197446
$817$	−6.68727	−0.233958
$818$	7.76674	0.271558
$819$	−3.64018	−0.127198
$820$	13.1862	0.460481
$821$	48.3561	1.68764	0.843821	−	0.536625i	$-0.180301\pi$
$821$	48.3561	1.68764	0.843821	+	0.536625i	$0.180301\pi$
$822$	−12.4294	−0.433524
$823$	−19.0471	−0.663940	−0.331970	−	0.943290i	$-0.607713\pi$
$823$	−19.0471	−0.663940	−0.331970	+	0.943290i	$0.607713\pi$
$824$	−1.22701	−0.0427448
$825$	−8.66262	−0.301594
$826$	2.09419	0.0728663
$827$	−57.0188	−1.98274	−0.991368	−	0.131106i	$-0.958147\pi$
$827$	−57.0188	−1.98274	−0.991368	+	0.131106i	$0.958147\pi$
$828$	−3.82009	−0.132757
$829$	14.1229	0.490508	0.245254	−	0.969459i	$-0.421129\pi$
$829$	14.1229	0.490508	0.245254	+	0.969459i	$0.421129\pi$
$830$	25.1862	0.874225
$831$	3.54599	0.123009
$832$	−1.82009	−0.0631002
$833$	−16.9205	−0.586262
$834$	−6.50736	−0.225332
$835$	−65.7962	−2.27697
$836$	1.47645	0.0510642
$837$	0.867185	0.0299743
$838$	−3.67255	−0.126866
$839$	−17.0920	−0.590080	−0.295040	−	0.955485i	$-0.595333\pi$
$839$	−17.0920	−0.590080	−0.295040	+	0.955485i	$0.595333\pi$
$840$	6.59308	0.227483
$841$	−24.2480	−0.836138
$842$	−2.42088	−0.0834291
$843$	−20.4047	−0.702775
$844$	−11.4765	−0.395036
$845$	31.9345	1.09858
$846$	11.2965	0.388383
$847$	−17.6402	−0.606124
$848$	1.00000	0.0343401
$849$	26.5607	0.911561
$850$	33.0920	1.13505
$851$	−10.9529	−0.375461
$852$	0.867185	0.0297093
$853$	50.1839	1.71827	0.859133	−	0.511753i	$-0.171004\pi$
$853$	50.1839	1.71827	0.859133	+	0.511753i	$0.171004\pi$
$854$	−23.0920	−0.790191
$855$	−3.29654	−0.112739
$856$	16.3275	0.558061
$857$	36.0982	1.23309	0.616546	−	0.787319i	$-0.288531\pi$
$857$	36.0982	1.23309	0.616546	+	0.787319i	$0.288531\pi$
$858$	2.68727	0.0917420
$859$	−42.3105	−1.44362	−0.721808	−	0.692093i	$-0.756689\pi$
$859$	−42.3105	−1.44362	−0.721808	+	0.692093i	$0.756689\pi$
$860$	22.0449	0.751724
$861$	8.00000	0.272639
$862$	31.0147	1.05637
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−1.00000	−0.0340207
$865$	−4.63796	−0.157695
$866$	31.1538	1.05865
$867$	−14.8116	−0.503029
$868$	−1.73437	−0.0588684
$869$	18.4665	0.626434
$870$	7.18617	0.243634
$871$	17.3745	0.588714
$872$	8.24945	0.279361
$873$	8.59308	0.290832
$874$	−3.82009	−0.129216
$875$	−5.71742	−0.193284
$876$	−10.5931	−0.357907
$877$	0.633924	0.0214061	0.0107030	−	0.999943i	$-0.496593\pi$
$877$	0.633924	0.0214061	0.0107030	+	0.999943i	$0.496593\pi$
$878$	23.7182	0.800450
$879$	−5.99153	−0.202089
$880$	−4.86719	−0.164073
$881$	−35.8488	−1.20778	−0.603888	−	0.797069i	$-0.706383\pi$
$881$	−35.8488	−1.20778	−0.603888	+	0.797069i	$0.706383\pi$
$882$	−3.00000	−0.101015
$883$	−13.1004	−0.440865	−0.220433	−	0.975402i	$-0.570747\pi$
$883$	−13.1004	−0.440865	−0.220433	+	0.975402i	$0.570747\pi$
$884$	−10.2656	−0.345270
$885$	3.45180	0.116031
$886$	−23.6078	−0.793120
$887$	−37.6766	−1.26506	−0.632528	−	0.774538i	$-0.717983\pi$
$887$	−37.6766	−1.26506	−0.632528	+	0.774538i	$0.717983\pi$
$888$	−2.86719	−0.0962164
$889$	−27.4518	−0.920704
$890$	−48.4172	−1.62295
$891$	1.47645	0.0494630
$892$	4.34364	0.145436
$893$	11.2965	0.378024
$894$	−1.82009	−0.0608729
$895$	26.0898	0.872084
$896$	2.00000	0.0668153
$897$	−6.95290	−0.232151
$898$	6.99374	0.233384
$899$	−1.89039	−0.0630479
$900$	5.86719	0.195573
$901$	5.64018	0.187902
$902$	−5.90581	−0.196642
$903$	13.3745	0.445077
$904$	−18.4132	−0.612413
$905$	28.5846	0.950185
$906$	−14.0534	−0.466891
$907$	16.8341	0.558966	0.279483	−	0.960151i	$-0.409837\pi$
$907$	16.8341	0.558966	0.279483	+	0.960151i	$0.409837\pi$
$908$	2.00000	0.0663723
$909$	10.3436	0.343077
$910$	12.0000	0.397796
$911$	14.8263	0.491219	0.245609	−	0.969369i	$-0.421012\pi$
$911$	14.8263	0.491219	0.245609	+	0.969369i	$0.421012\pi$
$912$	−1.00000	−0.0331133
$913$	−11.2804	−0.373325
$914$	−9.31273	−0.308038
$915$	−38.0618	−1.25829
$916$	5.82009	0.192301
$917$	26.0449	0.860078
$918$	−5.64018	−0.186154
$919$	−30.7491	−1.01432	−0.507160	−	0.861852i	$-0.669305\pi$
$919$	−30.7491	−1.01432	−0.507160	+	0.861852i	$0.669305\pi$
$920$	12.5931	0.415182
$921$	−24.0695	−0.793118
$922$	19.8650	0.654218
$923$	1.57835	0.0519522
$924$	−2.95290	−0.0971434
$925$	16.8223	0.553114
$926$	−1.76674	−0.0580586
$927$	−1.22701	−0.0403002
$928$	2.17991	0.0715591
$929$	6.17144	0.202478	0.101239	−	0.994862i	$-0.467719\pi$
$929$	6.17144	0.202478	0.101239	+	0.994862i	$0.467719\pi$
$930$	−2.85871	−0.0937409
$931$	−3.00000	−0.0983210
$932$	7.38226	0.241814
$933$	2.34364	0.0767272
$934$	−3.02244	−0.0988972
$935$	−27.4518	−0.897770
$936$	−1.82009	−0.0594915
$937$	−4.81383	−0.157261	−0.0786306	−	0.996904i	$-0.525055\pi$
$937$	−4.81383	−0.157261	−0.0786306	+	0.996904i	$0.525055\pi$
$938$	−19.0920	−0.623375
$939$	16.4989	0.538421
$940$	−37.2395	−1.21462
$941$	3.72590	0.121461	0.0607304	−	0.998154i	$-0.480657\pi$
$941$	3.72590	0.121461	0.0607304	+	0.998154i	$0.480657\pi$
$942$	−18.9529	−0.617519
$943$	15.2804	0.497597
$944$	1.04710	0.0340801
$945$	6.59308	0.214473
$946$	−9.87344	−0.321013
$947$	17.4610	0.567407	0.283704	−	0.958912i	$-0.408437\pi$
$947$	17.4610	0.567407	0.283704	+	0.958912i	$0.408437\pi$
$948$	−12.5074	−0.406220
$949$	−19.2804	−0.625867
$950$	5.86719	0.190357
$951$	−16.7815	−0.544176
$952$	11.2804	0.365598
$953$	16.0942	0.521342	0.260671	−	0.965428i	$-0.416056\pi$
$953$	16.0942	0.521342	0.260671	+	0.965428i	$0.416056\pi$
$954$	1.00000	0.0323762
$955$	13.7792	0.445886
$956$	15.7877	0.510611
$957$	−3.21853	−0.104040
$958$	−9.58683	−0.309736
$959$	24.8587	0.802730
$960$	3.29654	0.106395
$961$	−30.2480	−0.975742
$962$	−5.21853	−0.168252
$963$	16.3275	0.526145
$964$	−2.49889	−0.0804839
$965$	−28.5567	−0.919272
$966$	7.64018	0.245819
$967$	11.6078	0.373282	0.186641	−	0.982428i	$-0.440240\pi$
$967$	11.6078	0.373282	0.186641	+	0.982428i	$0.440240\pi$
$968$	−8.82009	−0.283488
$969$	−5.64018	−0.181189
$970$	−28.3275	−0.909540
$971$	−39.7020	−1.27410	−0.637049	−	0.770823i	$-0.719845\pi$
$971$	−39.7020	−1.27410	−0.637049	+	0.770823i	$0.719845\pi$
$972$	−1.00000	−0.0320750
$973$	13.0147	0.417233
$974$	−14.3436	−0.459600
$975$	10.6788	0.341995
$976$	−11.5460	−0.369578
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	−5.54599	−0.177341
$979$	21.6851	0.693057
$980$	9.88962	0.315912
$981$	8.24945	0.263385
$982$	42.0449	1.34171
$983$	−56.1685	−1.79150	−0.895749	−	0.444560i	$-0.853360\pi$
$983$	−56.1685	−1.79150	−0.895749	+	0.444560i	$0.853360\pi$
$984$	4.00000	0.127515
$985$	52.9205	1.68619
$986$	12.2951	0.391555
$987$	−22.5931	−0.719146
$988$	−1.82009	−0.0579048
$989$	25.5460	0.812315
$990$	−4.86719	−0.154689
$991$	3.37379	0.107172	0.0535859	−	0.998563i	$-0.482935\pi$
$991$	3.37379	0.107172	0.0535859	+	0.998563i	$0.482935\pi$
$992$	−0.867185	−0.0275332
$993$	−4.61774	−0.146540
$994$	−1.73437	−0.0550109
$995$	−85.2929	−2.70397
$996$	7.64018	0.242088
$997$	47.1777	1.49413	0.747066	−	0.664750i	$-0.231462\pi$
$997$	47.1777	1.49413	0.747066	+	0.664750i	$0.231462\pi$
$998$	−25.2719	−0.799968
$999$	−2.86719	−0.0907137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.r.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.r.1.1	✓	3	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29654 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29654 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.29654 q^{10} +1.47645 q^{11} -1.00000 q^{12} -1.82009 q^{13} +2.00000 q^{14} +3.29654 q^{15} +1.00000 q^{16} +5.64018 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.29654 q^{20} -2.00000 q^{21} +1.47645 q^{22} -3.82009 q^{23} -1.00000 q^{24} +5.86719 q^{25} -1.82009 q^{26} -1.00000 q^{27} +2.00000 q^{28} +2.17991 q^{29} +3.29654 q^{30} -0.867185 q^{31} +1.00000 q^{32} -1.47645 q^{33} +5.64018 q^{34} -6.59308 q^{35} +1.00000 q^{36} +2.86719 q^{37} +1.00000 q^{38} +1.82009 q^{39} -3.29654 q^{40} -4.00000 q^{41} -2.00000 q^{42} -6.68727 q^{43} +1.47645 q^{44} -3.29654 q^{45} -3.82009 q^{46} +11.2965 q^{47} -1.00000 q^{48} -3.00000 q^{49} +5.86719 q^{50} -5.64018 q^{51} -1.82009 q^{52} +1.00000 q^{53} -1.00000 q^{54} -4.86719 q^{55} +2.00000 q^{56} -1.00000 q^{57} +2.17991 q^{58} +1.04710 q^{59} +3.29654 q^{60} -11.5460 q^{61} -0.867185 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -1.47645 q^{66} -9.54599 q^{67} +5.64018 q^{68} +3.82009 q^{69} -6.59308 q^{70} -0.867185 q^{71} +1.00000 q^{72} +10.5931 q^{73} +2.86719 q^{74} -5.86719 q^{75} +1.00000 q^{76} +2.95290 q^{77} +1.82009 q^{78} +12.5074 q^{79} -3.29654 q^{80} +1.00000 q^{81} -4.00000 q^{82} -7.64018 q^{83} -2.00000 q^{84} -18.5931 q^{85} -6.68727 q^{86} -2.17991 q^{87} +1.47645 q^{88} +14.6873 q^{89} -3.29654 q^{90} -3.64018 q^{91} -3.82009 q^{92} +0.867185 q^{93} +11.2965 q^{94} -3.29654 q^{95} -1.00000 q^{96} +8.59308 q^{97} -3.00000 q^{98} +1.47645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{10} + 2 q^{11} - 3 q^{12} + 4 q^{13} + 6 q^{14} - 2 q^{15} + 3 q^{16} - 2 q^{17} + 3 q^{18} + 3 q^{19} + 2 q^{20} - 6 q^{21} + 2 q^{22} - 2 q^{23} - 3 q^{24} + 13 q^{25} + 4 q^{26} - 3 q^{27} + 6 q^{28} + 16 q^{29} - 2 q^{30} + 2 q^{31} + 3 q^{32} - 2 q^{33} - 2 q^{34} + 4 q^{35} + 3 q^{36} + 4 q^{37} + 3 q^{38} - 4 q^{39} + 2 q^{40} - 12 q^{41} - 6 q^{42} - 6 q^{43} + 2 q^{44} + 2 q^{45} - 2 q^{46} + 22 q^{47} - 3 q^{48} - 9 q^{49} + 13 q^{50} + 2 q^{51} + 4 q^{52} + 3 q^{53} - 3 q^{54} - 10 q^{55} + 6 q^{56} - 3 q^{57} + 16 q^{58} + 8 q^{59} - 2 q^{60} - 6 q^{61} + 2 q^{62} + 6 q^{63} + 3 q^{64} + 18 q^{65} - 2 q^{66} - 2 q^{68} + 2 q^{69} + 4 q^{70} + 2 q^{71} + 3 q^{72} + 8 q^{73} + 4 q^{74} - 13 q^{75} + 3 q^{76} + 4 q^{77} - 4 q^{78} + 14 q^{79} + 2 q^{80} + 3 q^{81} - 12 q^{82} - 4 q^{83} - 6 q^{84} - 32 q^{85} - 6 q^{86} - 16 q^{87} + 2 q^{88} + 30 q^{89} + 2 q^{90} + 8 q^{91} - 2 q^{92} - 2 q^{93} + 22 q^{94} + 2 q^{95} - 3 q^{96} + 2 q^{97} - 9 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^10 + 2 * q^11 - 3 * q^12 + 4 * q^13 + 6 * q^14 - 2 * q^15 + 3 * q^16 - 2 * q^17 + 3 * q^18 + 3 * q^19 + 2 * q^20 - 6 * q^21 + 2 * q^22 - 2 * q^23 - 3 * q^24 + 13 * q^25 + 4 * q^26 - 3 * q^27 + 6 * q^28 + 16 * q^29 - 2 * q^30 + 2 * q^31 + 3 * q^32 - 2 * q^33 - 2 * q^34 + 4 * q^35 + 3 * q^36 + 4 * q^37 + 3 * q^38 - 4 * q^39 + 2 * q^40 - 12 * q^41 - 6 * q^42 - 6 * q^43 + 2 * q^44 + 2 * q^45 - 2 * q^46 + 22 * q^47 - 3 * q^48 - 9 * q^49 + 13 * q^50 + 2 * q^51 + 4 * q^52 + 3 * q^53 - 3 * q^54 - 10 * q^55 + 6 * q^56 - 3 * q^57 + 16 * q^58 + 8 * q^59 - 2 * q^60 - 6 * q^61 + 2 * q^62 + 6 * q^63 + 3 * q^64 + 18 * q^65 - 2 * q^66 - 2 * q^68 + 2 * q^69 + 4 * q^70 + 2 * q^71 + 3 * q^72 + 8 * q^73 + 4 * q^74 - 13 * q^75 + 3 * q^76 + 4 * q^77 - 4 * q^78 + 14 * q^79 + 2 * q^80 + 3 * q^81 - 12 * q^82 - 4 * q^83 - 6 * q^84 - 32 * q^85 - 6 * q^86 - 16 * q^87 + 2 * q^88 + 30 * q^89 + 2 * q^90 + 8 * q^91 - 2 * q^92 - 2 * q^93 + 22 * q^94 + 2 * q^95 - 3 * q^96 + 2 * q^97 - 9 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more