LMFDB - Embedded newform 8030.2.a.be.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 24 x^{13} + 64 x^{12} + 237 x^{11} - 524 x^{10} - 1225 x^{9} + 2074 x^{8} + \cdots - 52 $ x^15 - 3x^14 - 24x^13 + 64x^12 + 237x^11 - 524x^10 - 1225x^9 + 2074x^8 + 3463x^7 - 4142x^6 - 5157x^5 + 3892x^4 + 3622x^3 - 1239x^2 - 907x - 52
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.42670$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.42670 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.42670 q^{6} +2.68339 q^{7} -1.00000 q^{8} +8.74228 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.42670 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.42670 q^{6} +2.68339 q^{7} -1.00000 q^{8} +8.74228 q^{9} -1.00000 q^{10} -1.00000 q^{11} +3.42670 q^{12} -5.61839 q^{13} -2.68339 q^{14} +3.42670 q^{15} +1.00000 q^{16} +6.29070 q^{17} -8.74228 q^{18} +2.23415 q^{19} +1.00000 q^{20} +9.19519 q^{21} +1.00000 q^{22} -7.18545 q^{23} -3.42670 q^{24} +1.00000 q^{25} +5.61839 q^{26} +19.6771 q^{27} +2.68339 q^{28} +6.59746 q^{29} -3.42670 q^{30} -2.97871 q^{31} -1.00000 q^{32} -3.42670 q^{33} -6.29070 q^{34} +2.68339 q^{35} +8.74228 q^{36} -7.39516 q^{37} -2.23415 q^{38} -19.2526 q^{39} -1.00000 q^{40} +7.44464 q^{41} -9.19519 q^{42} +2.76062 q^{43} -1.00000 q^{44} +8.74228 q^{45} +7.18545 q^{46} +3.02411 q^{47} +3.42670 q^{48} +0.200606 q^{49} -1.00000 q^{50} +21.5563 q^{51} -5.61839 q^{52} +5.79295 q^{53} -19.6771 q^{54} -1.00000 q^{55} -2.68339 q^{56} +7.65576 q^{57} -6.59746 q^{58} -9.03913 q^{59} +3.42670 q^{60} +14.1637 q^{61} +2.97871 q^{62} +23.4590 q^{63} +1.00000 q^{64} -5.61839 q^{65} +3.42670 q^{66} +15.4711 q^{67} +6.29070 q^{68} -24.6224 q^{69} -2.68339 q^{70} -15.9039 q^{71} -8.74228 q^{72} -1.00000 q^{73} +7.39516 q^{74} +3.42670 q^{75} +2.23415 q^{76} -2.68339 q^{77} +19.2526 q^{78} -0.450595 q^{79} +1.00000 q^{80} +41.2006 q^{81} -7.44464 q^{82} -7.40753 q^{83} +9.19519 q^{84} +6.29070 q^{85} -2.76062 q^{86} +22.6075 q^{87} +1.00000 q^{88} -9.29525 q^{89} -8.74228 q^{90} -15.0764 q^{91} -7.18545 q^{92} -10.2072 q^{93} -3.02411 q^{94} +2.23415 q^{95} -3.42670 q^{96} +10.6138 q^{97} -0.200606 q^{98} -8.74228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 3 * q^3 + 15 * q^4 + 15 * q^5 - 3 * q^6 + 7 * q^7 - 15 * q^8 + 12 * q^9	\( 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9} - 15 q^{10} - 15 q^{11} + 3 q^{12} + 13 q^{13} - 7 q^{14} + 3 q^{15} + 15 q^{16} + 20 q^{17} - 12 q^{18} + 3 q^{19} + 15 q^{20} + 22 q^{21} + 15 q^{22} + 2 q^{23} - 3 q^{24} + 15 q^{25} - 13 q^{26} + 33 q^{27} + 7 q^{28} + 11 q^{29} - 3 q^{30} - 3 q^{31} - 15 q^{32} - 3 q^{33} - 20 q^{34} + 7 q^{35} + 12 q^{36} + 9 q^{37} - 3 q^{38} + 11 q^{39} - 15 q^{40} + 17 q^{41} - 22 q^{42} + 29 q^{43} - 15 q^{44} + 12 q^{45} - 2 q^{46} - 2 q^{47} + 3 q^{48} + 20 q^{49} - 15 q^{50} + 7 q^{51} + 13 q^{52} + 3 q^{53} - 33 q^{54} - 15 q^{55} - 7 q^{56} + 13 q^{57} - 11 q^{58} - 32 q^{59} + 3 q^{60} + 61 q^{61} + 3 q^{62} + 20 q^{63} + 15 q^{64} + 13 q^{65} + 3 q^{66} + 7 q^{67} + 20 q^{68} - 23 q^{69} - 7 q^{70} - 6 q^{71} - 12 q^{72} - 15 q^{73} - 9 q^{74} + 3 q^{75} + 3 q^{76} - 7 q^{77} - 11 q^{78} + 12 q^{79} + 15 q^{80} + 3 q^{81} - 17 q^{82} + 17 q^{83} + 22 q^{84} + 20 q^{85} - 29 q^{86} + 23 q^{87} + 15 q^{88} - 18 q^{89} - 12 q^{90} - 15 q^{91} + 2 q^{92} + 32 q^{93} + 2 q^{94} + 3 q^{95} - 3 q^{96} + 36 q^{97} - 20 q^{98} - 12 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 3 * q^3 + 15 * q^4 + 15 * q^5 - 3 * q^6 + 7 * q^7 - 15 * q^8 + 12 * q^9 - 15 * q^10 - 15 * q^11 + 3 * q^12 + 13 * q^13 - 7 * q^14 + 3 * q^15 + 15 * q^16 + 20 * q^17 - 12 * q^18 + 3 * q^19 + 15 * q^20 + 22 * q^21 + 15 * q^22 + 2 * q^23 - 3 * q^24 + 15 * q^25 - 13 * q^26 + 33 * q^27 + 7 * q^28 + 11 * q^29 - 3 * q^30 - 3 * q^31 - 15 * q^32 - 3 * q^33 - 20 * q^34 + 7 * q^35 + 12 * q^36 + 9 * q^37 - 3 * q^38 + 11 * q^39 - 15 * q^40 + 17 * q^41 - 22 * q^42 + 29 * q^43 - 15 * q^44 + 12 * q^45 - 2 * q^46 - 2 * q^47 + 3 * q^48 + 20 * q^49 - 15 * q^50 + 7 * q^51 + 13 * q^52 + 3 * q^53 - 33 * q^54 - 15 * q^55 - 7 * q^56 + 13 * q^57 - 11 * q^58 - 32 * q^59 + 3 * q^60 + 61 * q^61 + 3 * q^62 + 20 * q^63 + 15 * q^64 + 13 * q^65 + 3 * q^66 + 7 * q^67 + 20 * q^68 - 23 * q^69 - 7 * q^70 - 6 * q^71 - 12 * q^72 - 15 * q^73 - 9 * q^74 + 3 * q^75 + 3 * q^76 - 7 * q^77 - 11 * q^78 + 12 * q^79 + 15 * q^80 + 3 * q^81 - 17 * q^82 + 17 * q^83 + 22 * q^84 + 20 * q^85 - 29 * q^86 + 23 * q^87 + 15 * q^88 - 18 * q^89 - 12 * q^90 - 15 * q^91 + 2 * q^92 + 32 * q^93 + 2 * q^94 + 3 * q^95 - 3 * q^96 + 36 * q^97 - 20 * q^98 - 12 * 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
$3$	3.42670	1.97841	0.989203	−	0.146550i	$-0.0468168\pi$
$3$	3.42670	1.97841	0.989203	+	0.146550i	$0.0468168\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.42670	−1.39894
$7$	2.68339	1.01423	0.507114	−	0.861879i	$-0.330712\pi$
$7$	2.68339	1.01423	0.507114	+	0.861879i	$0.330712\pi$
$8$	−1.00000	−0.353553
$9$	8.74228	2.91409
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.42670	0.989203
$13$	−5.61839	−1.55826	−0.779131	−	0.626861i	$-0.784339\pi$
$13$	−5.61839	−1.55826	−0.779131	+	0.626861i	$0.784339\pi$
$14$	−2.68339	−0.717167
$15$	3.42670	0.884770
$16$	1.00000	0.250000
$17$	6.29070	1.52572	0.762859	−	0.646565i	$-0.223795\pi$
$17$	6.29070	1.52572	0.762859	+	0.646565i	$0.223795\pi$
$18$	−8.74228	−2.06057
$19$	2.23415	0.512549	0.256274	−	0.966604i	$-0.417505\pi$
$19$	2.23415	0.512549	0.256274	+	0.966604i	$0.417505\pi$
$20$	1.00000	0.223607
$21$	9.19519	2.00655
$22$	1.00000	0.213201
$23$	−7.18545	−1.49827	−0.749134	−	0.662418i	$-0.769530\pi$
$23$	−7.18545	−1.49827	−0.749134	+	0.662418i	$0.769530\pi$
$24$	−3.42670	−0.699472
$25$	1.00000	0.200000
$26$	5.61839	1.10186
$27$	19.6771	3.78685
$28$	2.68339	0.507114
$29$	6.59746	1.22512	0.612559	−	0.790425i	$-0.290140\pi$
$29$	6.59746	1.22512	0.612559	+	0.790425i	$0.290140\pi$
$30$	−3.42670	−0.625627
$31$	−2.97871	−0.534992	−0.267496	−	0.963559i	$-0.586196\pi$
$31$	−2.97871	−0.534992	−0.267496	+	0.963559i	$0.586196\pi$
$32$	−1.00000	−0.176777
$33$	−3.42670	−0.596512
$34$	−6.29070	−1.07885
$35$	2.68339	0.453576
$36$	8.74228	1.45705
$37$	−7.39516	−1.21576	−0.607879	−	0.794030i	$-0.707979\pi$
$37$	−7.39516	−1.21576	−0.607879	+	0.794030i	$0.707979\pi$
$38$	−2.23415	−0.362427
$39$	−19.2526	−3.08288
$40$	−1.00000	−0.158114
$41$	7.44464	1.16266	0.581329	−	0.813669i	$-0.302533\pi$
$41$	7.44464	1.16266	0.581329	+	0.813669i	$0.302533\pi$
$42$	−9.19519	−1.41885
$43$	2.76062	0.420990	0.210495	−	0.977595i	$-0.432492\pi$
$43$	2.76062	0.420990	0.210495	+	0.977595i	$0.432492\pi$
$44$	−1.00000	−0.150756
$45$	8.74228	1.30322
$46$	7.18545	1.05944
$47$	3.02411	0.441111	0.220556	−	0.975374i	$-0.429213\pi$
$47$	3.02411	0.441111	0.220556	+	0.975374i	$0.429213\pi$
$48$	3.42670	0.494602
$49$	0.200606	0.0286580
$50$	−1.00000	−0.141421
$51$	21.5563	3.01849
$52$	−5.61839	−0.779131
$53$	5.79295	0.795723	0.397861	−	0.917446i	$-0.369753\pi$
$53$	5.79295	0.795723	0.397861	+	0.917446i	$0.369753\pi$
$54$	−19.6771	−2.67771
$55$	−1.00000	−0.134840
$56$	−2.68339	−0.358584
$57$	7.65576	1.01403
$58$	−6.59746	−0.866289
$59$	−9.03913	−1.17680	−0.588398	−	0.808572i	$-0.700241\pi$
$59$	−9.03913	−1.17680	−0.588398	+	0.808572i	$0.700241\pi$
$60$	3.42670	0.442385
$61$	14.1637	1.81347	0.906736	−	0.421699i	$-0.138566\pi$
$61$	14.1637	1.81347	0.906736	+	0.421699i	$0.138566\pi$
$62$	2.97871	0.378297
$63$	23.4590	2.95555
$64$	1.00000	0.125000
$65$	−5.61839	−0.696876
$66$	3.42670	0.421798
$67$	15.4711	1.89010	0.945048	−	0.326932i	$-0.106015\pi$
$67$	15.4711	1.89010	0.945048	+	0.326932i	$0.106015\pi$
$68$	6.29070	0.762859
$69$	−24.6224	−2.96419
$70$	−2.68339	−0.320727
$71$	−15.9039	−1.88744	−0.943722	−	0.330739i	$-0.892702\pi$
$71$	−15.9039	−1.88744	−0.943722	+	0.330739i	$0.892702\pi$
$72$	−8.74228	−1.03029
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	7.39516	0.859670
$75$	3.42670	0.395681
$76$	2.23415	0.256274
$77$	−2.68339	−0.305801
$78$	19.2526	2.17992
$79$	−0.450595	−0.0506959	−0.0253480	−	0.999679i	$-0.508069\pi$
$79$	−0.450595	−0.0506959	−0.0253480	+	0.999679i	$0.508069\pi$
$80$	1.00000	0.111803
$81$	41.2006	4.57784
$82$	−7.44464	−0.822123
$83$	−7.40753	−0.813083	−0.406541	−	0.913632i	$-0.633265\pi$
$83$	−7.40753	−0.813083	−0.406541	+	0.913632i	$0.633265\pi$
$84$	9.19519	1.00328
$85$	6.29070	0.682322
$86$	−2.76062	−0.297685
$87$	22.6075	2.42378
$88$	1.00000	0.106600
$89$	−9.29525	−0.985294	−0.492647	−	0.870229i	$-0.663971\pi$
$89$	−9.29525	−0.985294	−0.492647	+	0.870229i	$0.663971\pi$
$90$	−8.74228	−0.921517
$91$	−15.0764	−1.58043
$92$	−7.18545	−0.749134
$93$	−10.2072	−1.05843
$94$	−3.02411	−0.311913
$95$	2.23415	0.229219
$96$	−3.42670	−0.349736
$97$	10.6138	1.07766	0.538832	−	0.842413i	$-0.318866\pi$
$97$	10.6138	1.07766	0.538832	+	0.842413i	$0.318866\pi$
$98$	−0.200606	−0.0202642
$99$	−8.74228	−0.878632
$100$	1.00000	0.100000
$101$	0.362692	0.0360892	0.0180446	−	0.999837i	$-0.494256\pi$
$101$	0.362692	0.0360892	0.0180446	+	0.999837i	$0.494256\pi$
$102$	−21.5563	−2.13440
$103$	−14.2983	−1.40886	−0.704428	−	0.709775i	$-0.748797\pi$
$103$	−14.2983	−1.40886	−0.704428	+	0.709775i	$0.748797\pi$
$104$	5.61839	0.550929
$105$	9.19519	0.897359
$106$	−5.79295	−0.562661
$107$	11.4447	1.10641	0.553203	−	0.833047i	$-0.313405\pi$
$107$	11.4447	1.10641	0.553203	+	0.833047i	$0.313405\pi$
$108$	19.6771	1.89343
$109$	−7.41131	−0.709875	−0.354937	−	0.934890i	$-0.615498\pi$
$109$	−7.41131	−0.709875	−0.354937	+	0.934890i	$0.615498\pi$
$110$	1.00000	0.0953463
$111$	−25.3410	−2.40526
$112$	2.68339	0.253557
$113$	9.45541	0.889490	0.444745	−	0.895657i	$-0.353294\pi$
$113$	9.45541	0.889490	0.444745	+	0.895657i	$0.353294\pi$
$114$	−7.65576	−0.717028
$115$	−7.18545	−0.670046
$116$	6.59746	0.612559
$117$	−49.1176	−4.54092
$118$	9.03913	0.832120
$119$	16.8804	1.54743
$120$	−3.42670	−0.312814
$121$	1.00000	0.0909091
$122$	−14.1637	−1.28232
$123$	25.5106	2.30021
$124$	−2.97871	−0.267496
$125$	1.00000	0.0894427
$126$	−23.4590	−2.08989
$127$	−3.65038	−0.323919	−0.161960	−	0.986797i	$-0.551781\pi$
$127$	−3.65038	−0.323919	−0.161960	+	0.986797i	$0.551781\pi$
$128$	−1.00000	−0.0883883
$129$	9.45981	0.832890
$130$	5.61839	0.492766
$131$	0.0953352	0.00832947	0.00416474	−	0.999991i	$-0.498674\pi$
$131$	0.0953352	0.00832947	0.00416474	+	0.999991i	$0.498674\pi$
$132$	−3.42670	−0.298256
$133$	5.99510	0.519841
$134$	−15.4711	−1.33650
$135$	19.6771	1.69353
$136$	−6.29070	−0.539423
$137$	20.4611	1.74811	0.874056	−	0.485825i	$-0.161481\pi$
$137$	20.4611	1.74811	0.874056	+	0.485825i	$0.161481\pi$
$138$	24.6224	2.09600
$139$	4.42742	0.375529	0.187765	−	0.982214i	$-0.439876\pi$
$139$	4.42742	0.375529	0.187765	+	0.982214i	$0.439876\pi$
$140$	2.68339	0.226788
$141$	10.3627	0.872697
$142$	15.9039	1.33462
$143$	5.61839	0.469834
$144$	8.74228	0.728523
$145$	6.59746	0.547889
$146$	1.00000	0.0827606
$147$	0.687416	0.0566971
$148$	−7.39516	−0.607879
$149$	9.99656	0.818950	0.409475	−	0.912321i	$-0.365712\pi$
$149$	9.99656	0.818950	0.409475	+	0.912321i	$0.365712\pi$
$150$	−3.42670	−0.279789
$151$	−4.49638	−0.365910	−0.182955	−	0.983121i	$-0.558566\pi$
$151$	−4.49638	−0.365910	−0.182955	+	0.983121i	$0.558566\pi$
$152$	−2.23415	−0.181213
$153$	54.9950	4.44608
$154$	2.68339	0.216234
$155$	−2.97871	−0.239256
$156$	−19.2526	−1.54144
$157$	−12.1885	−0.972746	−0.486373	−	0.873751i	$-0.661680\pi$
$157$	−12.1885	−0.972746	−0.486373	+	0.873751i	$0.661680\pi$
$158$	0.450595	0.0358474
$159$	19.8507	1.57426
$160$	−1.00000	−0.0790569
$161$	−19.2814	−1.51959
$162$	−41.2006	−3.23702
$163$	0.304628	0.0238603	0.0119301	−	0.999929i	$-0.496202\pi$
$163$	0.304628	0.0238603	0.0119301	+	0.999929i	$0.496202\pi$
$164$	7.44464	0.581329
$165$	−3.42670	−0.266768
$166$	7.40753	0.574936
$167$	−6.60955	−0.511463	−0.255731	−	0.966748i	$-0.582316\pi$
$167$	−6.60955	−0.511463	−0.255731	+	0.966748i	$0.582316\pi$
$168$	−9.19519	−0.709424
$169$	18.5664	1.42818
$170$	−6.29070	−0.482475
$171$	19.5316	1.49362
$172$	2.76062	0.210495
$173$	9.83572	0.747796	0.373898	−	0.927470i	$-0.378021\pi$
$173$	9.83572	0.747796	0.373898	+	0.927470i	$0.378021\pi$
$174$	−22.6075	−1.71387
$175$	2.68339	0.202846
$176$	−1.00000	−0.0753778
$177$	−30.9744	−2.32818
$178$	9.29525	0.696708
$179$	−21.6972	−1.62172	−0.810862	−	0.585237i	$-0.801001\pi$
$179$	−21.6972	−1.62172	−0.810862	+	0.585237i	$0.801001\pi$
$180$	8.74228	0.651611
$181$	9.35029	0.695001	0.347501	−	0.937680i	$-0.387030\pi$
$181$	9.35029	0.695001	0.347501	+	0.937680i	$0.387030\pi$
$182$	15.0764	1.11753
$183$	48.5346	3.58778
$184$	7.18545	0.529718
$185$	−7.39516	−0.543703
$186$	10.2072	0.748425
$187$	−6.29070	−0.460021
$188$	3.02411	0.220556
$189$	52.8013	3.84073
$190$	−2.23415	−0.162082
$191$	11.6107	0.840122	0.420061	−	0.907496i	$-0.362009\pi$
$191$	11.6107	0.840122	0.420061	+	0.907496i	$0.362009\pi$
$192$	3.42670	0.247301
$193$	−12.7929	−0.920854	−0.460427	−	0.887698i	$-0.652304\pi$
$193$	−12.7929	−0.920854	−0.460427	+	0.887698i	$0.652304\pi$
$194$	−10.6138	−0.762023
$195$	−19.2526	−1.37870
$196$	0.200606	0.0143290
$197$	10.1657	0.724274	0.362137	−	0.932125i	$-0.382047\pi$
$197$	10.1657	0.724274	0.362137	+	0.932125i	$0.382047\pi$
$198$	8.74228	0.621287
$199$	9.30565	0.659660	0.329830	−	0.944040i	$-0.393009\pi$
$199$	9.30565	0.659660	0.329830	+	0.944040i	$0.393009\pi$
$200$	−1.00000	−0.0707107
$201$	53.0148	3.73938
$202$	−0.362692	−0.0255189
$203$	17.7036	1.24255
$204$	21.5563	1.50925
$205$	7.44464	0.519956
$206$	14.2983	0.996212
$207$	−62.8172	−4.36609
$208$	−5.61839	−0.389566
$209$	−2.23415	−0.154539
$210$	−9.19519	−0.634528
$211$	−11.2669	−0.775647	−0.387823	−	0.921734i	$-0.626773\pi$
$211$	−11.2669	−0.775647	−0.387823	+	0.921734i	$0.626773\pi$
$212$	5.79295	0.397861
$213$	−54.4979	−3.73413
$214$	−11.4447	−0.782347
$215$	2.76062	0.188273
$216$	−19.6771	−1.33885
$217$	−7.99306	−0.542604
$218$	7.41131	0.501957
$219$	−3.42670	−0.231555
$220$	−1.00000	−0.0674200
$221$	−35.3436	−2.37747
$222$	25.3410	1.70078
$223$	−22.9623	−1.53767	−0.768835	−	0.639448i	$-0.779163\pi$
$223$	−22.9623	−1.53767	−0.768835	+	0.639448i	$0.779163\pi$
$224$	−2.68339	−0.179292
$225$	8.74228	0.582819
$226$	−9.45541	−0.628965
$227$	−26.6925	−1.77165	−0.885823	−	0.464024i	$-0.846405\pi$
$227$	−26.6925	−1.77165	−0.885823	+	0.464024i	$0.846405\pi$
$228$	7.65576	0.507015
$229$	−9.20117	−0.608031	−0.304015	−	0.952667i	$-0.598327\pi$
$229$	−9.20117	−0.608031	−0.304015	+	0.952667i	$0.598327\pi$
$230$	7.18545	0.473794
$231$	−9.19519	−0.604999
$232$	−6.59746	−0.433145
$233$	−10.1449	−0.664613	−0.332307	−	0.943171i	$-0.607827\pi$
$233$	−10.1449	−0.664613	−0.332307	+	0.943171i	$0.607827\pi$
$234$	49.1176	3.21092
$235$	3.02411	0.197271
$236$	−9.03913	−0.588398
$237$	−1.54406	−0.100297
$238$	−16.8804	−1.09420
$239$	−0.186998	−0.0120959	−0.00604794	−	0.999982i	$-0.501925\pi$
$239$	−0.186998	−0.0120959	−0.00604794	+	0.999982i	$0.501925\pi$
$240$	3.42670	0.221193
$241$	−3.10803	−0.200206	−0.100103	−	0.994977i	$-0.531917\pi$
$241$	−3.10803	−0.200206	−0.100103	+	0.994977i	$0.531917\pi$
$242$	−1.00000	−0.0642824
$243$	82.1509	5.26998
$244$	14.1637	0.906736
$245$	0.200606	0.0128162
$246$	−25.5106	−1.62649
$247$	−12.5523	−0.798686
$248$	2.97871	0.189148
$249$	−25.3834	−1.60861
$250$	−1.00000	−0.0632456
$251$	−20.5781	−1.29888	−0.649440	−	0.760413i	$-0.724997\pi$
$251$	−20.5781	−1.29888	−0.649440	+	0.760413i	$0.724997\pi$
$252$	23.4590	1.47778
$253$	7.18545	0.451745
$254$	3.65038	0.229045
$255$	21.5563	1.34991
$256$	1.00000	0.0625000
$257$	18.6108	1.16091	0.580455	−	0.814293i	$-0.302875\pi$
$257$	18.6108	1.16091	0.580455	+	0.814293i	$0.302875\pi$
$258$	−9.45981	−0.588942
$259$	−19.8441	−1.23305
$260$	−5.61839	−0.348438
$261$	57.6768	3.57011
$262$	−0.0953352	−0.00588983
$263$	−15.2398	−0.939727	−0.469864	−	0.882739i	$-0.655697\pi$
$263$	−15.2398	−0.939727	−0.469864	+	0.882739i	$0.655697\pi$
$264$	3.42670	0.210899
$265$	5.79295	0.355858
$266$	−5.99510	−0.367583
$267$	−31.8520	−1.94931
$268$	15.4711	0.945048
$269$	−0.510953	−0.0311534	−0.0155767	−	0.999879i	$-0.504958\pi$
$269$	−0.510953	−0.0311534	−0.0155767	+	0.999879i	$0.504958\pi$
$270$	−19.6771	−1.19751
$271$	15.0429	0.913789	0.456895	−	0.889521i	$-0.348962\pi$
$271$	15.0429	0.913789	0.456895	+	0.889521i	$0.348962\pi$
$272$	6.29070	0.381430
$273$	−51.6622	−3.12674
$274$	−20.4611	−1.23610
$275$	−1.00000	−0.0603023
$276$	−24.6224	−1.48209
$277$	4.76430	0.286259	0.143130	−	0.989704i	$-0.454283\pi$
$277$	4.76430	0.286259	0.143130	+	0.989704i	$0.454283\pi$
$278$	−4.42742	−0.265539
$279$	−26.0407	−1.55902
$280$	−2.68339	−0.160363
$281$	−15.3690	−0.916838	−0.458419	−	0.888736i	$-0.651584\pi$
$281$	−15.3690	−0.916838	−0.458419	+	0.888736i	$0.651584\pi$
$282$	−10.3627	−0.617090
$283$	30.1171	1.79028	0.895138	−	0.445788i	$-0.147077\pi$
$283$	30.1171	1.79028	0.895138	+	0.445788i	$0.147077\pi$
$284$	−15.9039	−0.943722
$285$	7.65576	0.453488
$286$	−5.61839	−0.332223
$287$	19.9769	1.17920
$288$	−8.74228	−0.515144
$289$	22.5729	1.32782
$290$	−6.59746	−0.387416
$291$	36.3702	2.13206
$292$	−1.00000	−0.0585206
$293$	24.0722	1.40631	0.703156	−	0.711035i	$-0.251773\pi$
$293$	24.0722	1.40631	0.703156	+	0.711035i	$0.251773\pi$
$294$	−0.687416	−0.0400909
$295$	−9.03913	−0.526279
$296$	7.39516	0.429835
$297$	−19.6771	−1.14178
$298$	−9.99656	−0.579085
$299$	40.3707	2.33470
$300$	3.42670	0.197841
$301$	7.40783	0.426980
$302$	4.49638	0.258737
$303$	1.24284	0.0713991
$304$	2.23415	0.128137
$305$	14.1637	0.811009
$306$	−54.9950	−3.14386
$307$	20.7530	1.18443	0.592217	−	0.805778i	$-0.298253\pi$
$307$	20.7530	1.18443	0.592217	+	0.805778i	$0.298253\pi$
$308$	−2.68339	−0.152901
$309$	−48.9961	−2.78729
$310$	2.97871	0.169179
$311$	−32.1504	−1.82308	−0.911542	−	0.411207i	$-0.865107\pi$
$311$	−32.1504	−1.82308	−0.911542	+	0.411207i	$0.865107\pi$
$312$	19.2526	1.08996
$313$	−0.136315	−0.00770497	−0.00385248	−	0.999993i	$-0.501226\pi$
$313$	−0.136315	−0.00770497	−0.00385248	+	0.999993i	$0.501226\pi$
$314$	12.1885	0.687835
$315$	23.4590	1.32176
$316$	−0.450595	−0.0253480
$317$	−12.9121	−0.725218	−0.362609	−	0.931941i	$-0.618114\pi$
$317$	−12.9121	−0.725218	−0.362609	+	0.931941i	$0.618114\pi$
$318$	−19.8507	−1.11317
$319$	−6.59746	−0.369387
$320$	1.00000	0.0559017
$321$	39.2177	2.18892
$322$	19.2814	1.07451
$323$	14.0544	0.782005
$324$	41.2006	2.28892
$325$	−5.61839	−0.311652
$326$	−0.304628	−0.0168718
$327$	−25.3963	−1.40442
$328$	−7.44464	−0.411062
$329$	8.11487	0.447387
$330$	3.42670	0.188634
$331$	2.81263	0.154596	0.0772981	−	0.997008i	$-0.475371\pi$
$331$	2.81263	0.154596	0.0772981	+	0.997008i	$0.475371\pi$
$332$	−7.40753	−0.406541
$333$	−64.6506	−3.54283
$334$	6.60955	0.361659
$335$	15.4711	0.845276
$336$	9.19519	0.501639
$337$	6.16686	0.335930	0.167965	−	0.985793i	$-0.446280\pi$
$337$	6.16686	0.335930	0.167965	+	0.985793i	$0.446280\pi$
$338$	−18.5664	−1.00988
$339$	32.4009	1.75977
$340$	6.29070	0.341161
$341$	2.97871	0.161306
$342$	−19.5316	−1.05615
$343$	−18.2455	−0.985162
$344$	−2.76062	−0.148843
$345$	−24.6224	−1.32562
$346$	−9.83572	−0.528772
$347$	7.28404	0.391028	0.195514	−	0.980701i	$-0.437363\pi$
$347$	7.28404	0.391028	0.195514	+	0.980701i	$0.437363\pi$
$348$	22.6075	1.21189
$349$	18.1432	0.971185	0.485592	−	0.874185i	$-0.338604\pi$
$349$	18.1432	0.971185	0.485592	+	0.874185i	$0.338604\pi$
$350$	−2.68339	−0.143433
$351$	−110.554	−5.90091
$352$	1.00000	0.0533002
$353$	−24.8361	−1.32189	−0.660946	−	0.750433i	$-0.729845\pi$
$353$	−24.8361	−1.32189	−0.660946	+	0.750433i	$0.729845\pi$
$354$	30.9744	1.64627
$355$	−15.9039	−0.844091
$356$	−9.29525	−0.492647
$357$	57.8442	3.06144
$358$	21.6972	1.14673
$359$	−20.8661	−1.10127	−0.550634	−	0.834747i	$-0.685614\pi$
$359$	−20.8661	−1.10127	−0.550634	+	0.834747i	$0.685614\pi$
$360$	−8.74228	−0.460759
$361$	−14.0086	−0.737294
$362$	−9.35029	−0.491440
$363$	3.42670	0.179855
$364$	−15.0764	−0.790217
$365$	−1.00000	−0.0523424
$366$	−48.5346	−2.53695
$367$	15.9704	0.833648	0.416824	−	0.908987i	$-0.363143\pi$
$367$	15.9704	0.833648	0.416824	+	0.908987i	$0.363143\pi$
$368$	−7.18545	−0.374567
$369$	65.0831	3.38809
$370$	7.39516	0.384456
$371$	15.5448	0.807044
$372$	−10.2072	−0.529216
$373$	13.6655	0.707571	0.353786	−	0.935327i	$-0.384894\pi$
$373$	13.6655	0.707571	0.353786	+	0.935327i	$0.384894\pi$
$374$	6.29070	0.325284
$375$	3.42670	0.176954
$376$	−3.02411	−0.155956
$377$	−37.0671	−1.90906
$378$	−52.8013	−2.71581
$379$	−15.3923	−0.790650	−0.395325	−	0.918541i	$-0.629368\pi$
$379$	−15.3923	−0.790650	−0.395325	+	0.918541i	$0.629368\pi$
$380$	2.23415	0.114609
$381$	−12.5088	−0.640844
$382$	−11.6107	−0.594056
$383$	−38.7109	−1.97803	−0.989016	−	0.147806i	$-0.952779\pi$
$383$	−38.7109	−1.97803	−0.989016	+	0.147806i	$0.952779\pi$
$384$	−3.42670	−0.174868
$385$	−2.68339	−0.136758
$386$	12.7929	0.651142
$387$	24.1341	1.22680
$388$	10.6138	0.538832
$389$	−37.4642	−1.89951	−0.949755	−	0.312995i	$-0.898668\pi$
$389$	−37.4642	−1.89951	−0.949755	+	0.312995i	$0.898668\pi$
$390$	19.2526	0.974891
$391$	−45.2015	−2.28594
$392$	−0.200606	−0.0101321
$393$	0.326685	0.0164791
$394$	−10.1657	−0.512139
$395$	−0.450595	−0.0226719
$396$	−8.74228	−0.439316
$397$	24.3011	1.21964	0.609818	−	0.792541i	$-0.291242\pi$
$397$	24.3011	1.21964	0.609818	+	0.792541i	$0.291242\pi$
$398$	−9.30565	−0.466450
$399$	20.5434	1.02846
$400$	1.00000	0.0500000
$401$	1.67058	0.0834246	0.0417123	−	0.999130i	$-0.486719\pi$
$401$	1.67058	0.0834246	0.0417123	+	0.999130i	$0.486719\pi$
$402$	−53.0148	−2.64414
$403$	16.7356	0.833659
$404$	0.362692	0.0180446
$405$	41.2006	2.04727
$406$	−17.7036	−0.878615
$407$	7.39516	0.366565
$408$	−21.5563	−1.06720
$409$	25.0327	1.23779	0.618895	−	0.785474i	$-0.287581\pi$
$409$	25.0327	1.23779	0.618895	+	0.785474i	$0.287581\pi$
$410$	−7.44464	−0.367665
$411$	70.1142	3.45848
$412$	−14.2983	−0.704428
$413$	−24.2556	−1.19354
$414$	62.8172	3.08729
$415$	−7.40753	−0.363622
$416$	5.61839	0.275464
$417$	15.1715	0.742949
$418$	2.23415	0.109276
$419$	15.1263	0.738968	0.369484	−	0.929237i	$-0.379535\pi$
$419$	15.1263	0.738968	0.369484	+	0.929237i	$0.379535\pi$
$420$	9.19519	0.448679
$421$	−12.4413	−0.606352	−0.303176	−	0.952935i	$-0.598047\pi$
$421$	−12.4413	−0.606352	−0.303176	+	0.952935i	$0.598047\pi$
$422$	11.2669	0.548465
$423$	26.4376	1.28544
$424$	−5.79295	−0.281330
$425$	6.29070	0.305144
$426$	54.4979	2.64043
$427$	38.0067	1.83927
$428$	11.4447	0.553203
$429$	19.2526	0.929522
$430$	−2.76062	−0.133129
$431$	12.3964	0.597112	0.298556	−	0.954392i	$-0.403495\pi$
$431$	12.3964	0.597112	0.298556	+	0.954392i	$0.403495\pi$
$432$	19.6771	0.946713
$433$	−29.3303	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$433$	−29.3303	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$434$	7.99306	0.383679
$435$	22.6075	1.08395
$436$	−7.41131	−0.354937
$437$	−16.0534	−0.767936
$438$	3.42670	0.163734
$439$	4.10463	0.195903	0.0979517	−	0.995191i	$-0.468771\pi$
$439$	4.10463	0.195903	0.0979517	+	0.995191i	$0.468771\pi$
$440$	1.00000	0.0476731
$441$	1.75375	0.0835120
$442$	35.3436	1.68112
$443$	3.98628	0.189394	0.0946969	−	0.995506i	$-0.469812\pi$
$443$	3.98628	0.189394	0.0946969	+	0.995506i	$0.469812\pi$
$444$	−25.3410	−1.20263
$445$	−9.29525	−0.440637
$446$	22.9623	1.08730
$447$	34.2552	1.62022
$448$	2.68339	0.126778
$449$	11.0820	0.522992	0.261496	−	0.965205i	$-0.415784\pi$
$449$	11.0820	0.522992	0.261496	+	0.965205i	$0.415784\pi$
$450$	−8.74228	−0.412115
$451$	−7.44464	−0.350555
$452$	9.45541	0.444745
$453$	−15.4077	−0.723919
$454$	26.6925	1.25274
$455$	−15.0764	−0.706791
$456$	−7.65576	−0.358514
$457$	−16.7197	−0.782114	−0.391057	−	0.920366i	$-0.627891\pi$
$457$	−16.7197	−0.782114	−0.391057	+	0.920366i	$0.627891\pi$
$458$	9.20117	0.429943
$459$	123.782	5.77767
$460$	−7.18545	−0.335023
$461$	16.9832	0.790985	0.395492	−	0.918469i	$-0.370574\pi$
$461$	16.9832	0.790985	0.395492	+	0.918469i	$0.370574\pi$
$462$	9.19519	0.427799
$463$	19.9916	0.929088	0.464544	−	0.885550i	$-0.346218\pi$
$463$	19.9916	0.929088	0.464544	+	0.885550i	$0.346218\pi$
$464$	6.59746	0.306279
$465$	−10.2072	−0.473345
$466$	10.1449	0.469953
$467$	4.47054	0.206872	0.103436	−	0.994636i	$-0.467016\pi$
$467$	4.47054	0.206872	0.103436	+	0.994636i	$0.467016\pi$
$468$	−49.1176	−2.27046
$469$	41.5151	1.91699
$470$	−3.02411	−0.139492
$471$	−41.7663	−1.92449
$472$	9.03913	0.416060
$473$	−2.76062	−0.126933
$474$	1.54406	0.0709208
$475$	2.23415	0.102510
$476$	16.8804	0.773713
$477$	50.6436	2.31881
$478$	0.186998	0.00855307
$479$	−19.3770	−0.885356	−0.442678	−	0.896681i	$-0.645971\pi$
$479$	−19.3770	−0.885356	−0.442678	+	0.896681i	$0.645971\pi$
$480$	−3.42670	−0.156407
$481$	41.5489	1.89447
$482$	3.10803	0.141567
$483$	−66.0715	−3.00636
$484$	1.00000	0.0454545
$485$	10.6138	0.481946
$486$	−82.1509	−3.72644
$487$	−4.61079	−0.208935	−0.104467	−	0.994528i	$-0.533314\pi$
$487$	−4.61079	−0.208935	−0.104467	+	0.994528i	$0.533314\pi$
$488$	−14.1637	−0.641159
$489$	1.04387	0.0472053
$490$	−0.200606	−0.00906245
$491$	−2.84640	−0.128456	−0.0642280	−	0.997935i	$-0.520458\pi$
$491$	−2.84640	−0.128456	−0.0642280	+	0.997935i	$0.520458\pi$
$492$	25.5106	1.15010
$493$	41.5026	1.86919
$494$	12.5523	0.564756
$495$	−8.74228	−0.392936
$496$	−2.97871	−0.133748
$497$	−42.6764	−1.91430
$498$	25.3834	1.13746
$499$	−23.3086	−1.04343	−0.521717	−	0.853119i	$-0.674708\pi$
$499$	−23.3086	−1.04343	−0.521717	+	0.853119i	$0.674708\pi$
$500$	1.00000	0.0447214
$501$	−22.6490	−1.01188
$502$	20.5781	0.918447
$503$	−32.2068	−1.43603	−0.718015	−	0.696027i	$-0.754949\pi$
$503$	−32.2068	−1.43603	−0.718015	+	0.696027i	$0.754949\pi$
$504$	−23.4590	−1.04495
$505$	0.362692	0.0161396
$506$	−7.18545	−0.319432
$507$	63.6214	2.82552
$508$	−3.65038	−0.161960
$509$	−14.4446	−0.640244	−0.320122	−	0.947376i	$-0.603724\pi$
$509$	−14.4446	−0.640244	−0.320122	+	0.947376i	$0.603724\pi$
$510$	−21.5563	−0.954531
$511$	−2.68339	−0.118706
$512$	−1.00000	−0.0441942
$513$	43.9615	1.94095
$514$	−18.6108	−0.820887
$515$	−14.2983	−0.630060
$516$	9.45981	0.416445
$517$	−3.02411	−0.133000
$518$	19.8441	0.871901
$519$	33.7041	1.47944
$520$	5.61839	0.246383
$521$	−12.9151	−0.565820	−0.282910	−	0.959146i	$-0.591300\pi$
$521$	−12.9151	−0.565820	−0.282910	+	0.959146i	$0.591300\pi$
$522$	−57.6768	−2.52445
$523$	−41.7277	−1.82462	−0.912312	−	0.409495i	$-0.865705\pi$
$523$	−41.7277	−1.82462	−0.912312	+	0.409495i	$0.865705\pi$
$524$	0.0953352	0.00416474
$525$	9.19519	0.401311
$526$	15.2398	0.664488
$527$	−18.7382	−0.816248
$528$	−3.42670	−0.149128
$529$	28.6306	1.24481
$530$	−5.79295	−0.251630
$531$	−79.0226	−3.42929
$532$	5.99510	0.259921
$533$	−41.8269	−1.81173
$534$	31.8520	1.37837
$535$	11.4447	0.494800
$536$	−15.4711	−0.668250
$537$	−74.3498	−3.20843
$538$	0.510953	0.0220288
$539$	−0.200606	−0.00864070
$540$	19.6771	0.846766
$541$	−28.0044	−1.20400	−0.602001	−	0.798495i	$-0.705630\pi$
$541$	−28.0044	−1.20400	−0.602001	+	0.798495i	$0.705630\pi$
$542$	−15.0429	−0.646146
$543$	32.0406	1.37500
$544$	−6.29070	−0.269711
$545$	−7.41131	−0.317466
$546$	51.6622	2.21094
$547$	−24.3025	−1.03910	−0.519550	−	0.854440i	$-0.673900\pi$
$547$	−24.3025	−1.03910	−0.519550	+	0.854440i	$0.673900\pi$
$548$	20.4611	0.874056
$549$	123.823	5.28462
$550$	1.00000	0.0426401
$551$	14.7397	0.627933
$552$	24.6224	1.04800
$553$	−1.20912	−0.0514172
$554$	−4.76430	−0.202416
$555$	−25.3410	−1.07567
$556$	4.42742	0.187765
$557$	−7.36632	−0.312121	−0.156060	−	0.987748i	$-0.549879\pi$
$557$	−7.36632	−0.312121	−0.156060	+	0.987748i	$0.549879\pi$
$558$	26.0407	1.10239
$559$	−15.5102	−0.656013
$560$	2.68339	0.113394
$561$	−21.5563	−0.910109
$562$	15.3690	0.648302
$563$	−38.2414	−1.61168	−0.805842	−	0.592131i	$-0.798287\pi$
$563$	−38.2414	−1.61168	−0.805842	+	0.592131i	$0.798287\pi$
$564$	10.3627	0.436349
$565$	9.45541	0.397792
$566$	−30.1171	−1.26592
$567$	110.557	4.64298
$568$	15.9039	0.667312
$569$	−14.8033	−0.620585	−0.310293	−	0.950641i	$-0.600427\pi$
$569$	−14.8033	−0.620585	−0.310293	+	0.950641i	$0.600427\pi$
$570$	−7.65576	−0.320665
$571$	17.4311	0.729471	0.364735	−	0.931111i	$-0.381159\pi$
$571$	17.4311	0.729471	0.364735	+	0.931111i	$0.381159\pi$
$572$	5.61839	0.234917
$573$	39.7865	1.66210
$574$	−19.9769	−0.833820
$575$	−7.18545	−0.299654
$576$	8.74228	0.364262
$577$	−30.8477	−1.28421	−0.642104	−	0.766617i	$-0.721938\pi$
$577$	−30.8477	−1.28421	−0.642104	+	0.766617i	$0.721938\pi$
$578$	−22.5729	−0.938908
$579$	−43.8375	−1.82182
$580$	6.59746	0.273945
$581$	−19.8773	−0.824651
$582$	−36.3702	−1.50759
$583$	−5.79295	−0.239919
$584$	1.00000	0.0413803
$585$	−49.1176	−2.03076
$586$	−24.0722	−0.994413
$587$	1.58698	0.0655015	0.0327508	−	0.999464i	$-0.489573\pi$
$587$	1.58698	0.0655015	0.0327508	+	0.999464i	$0.489573\pi$
$588$	0.687416	0.0283486
$589$	−6.65489	−0.274210
$590$	9.03913	0.372135
$591$	34.8347	1.43291
$592$	−7.39516	−0.303939
$593$	16.6052	0.681896	0.340948	−	0.940082i	$-0.389252\pi$
$593$	16.6052	0.681896	0.340948	+	0.940082i	$0.389252\pi$
$594$	19.6771	0.807360
$595$	16.8804	0.692030
$596$	9.99656	0.409475
$597$	31.8877	1.30508
$598$	−40.3707	−1.65088
$599$	−23.2396	−0.949545	−0.474773	−	0.880108i	$-0.657470\pi$
$599$	−23.2396	−0.949545	−0.474773	+	0.880108i	$0.657470\pi$
$600$	−3.42670	−0.139894
$601$	14.7176	0.600344	0.300172	−	0.953885i	$-0.402956\pi$
$601$	14.7176	0.600344	0.300172	+	0.953885i	$0.402956\pi$
$602$	−7.40783	−0.301920
$603$	135.253	5.50791
$604$	−4.49638	−0.182955
$605$	1.00000	0.0406558
$606$	−1.24284	−0.0504868
$607$	46.7910	1.89919	0.949594	−	0.313484i	$-0.101496\pi$
$607$	46.7910	1.89919	0.949594	+	0.313484i	$0.101496\pi$
$608$	−2.23415	−0.0906067
$609$	60.6649	2.45827
$610$	−14.1637	−0.573470
$611$	−16.9906	−0.687367
$612$	54.9950	2.22304
$613$	−18.8704	−0.762168	−0.381084	−	0.924540i	$-0.624449\pi$
$613$	−18.8704	−0.762168	−0.381084	+	0.924540i	$0.624449\pi$
$614$	−20.7530	−0.837522
$615$	25.5106	1.02869
$616$	2.68339	0.108117
$617$	−21.2804	−0.856717	−0.428358	−	0.903609i	$-0.640908\pi$
$617$	−21.2804	−0.856717	−0.428358	+	0.903609i	$0.640908\pi$
$618$	48.9961	1.97091
$619$	−11.4220	−0.459087	−0.229544	−	0.973298i	$-0.573723\pi$
$619$	−11.4220	−0.459087	−0.229544	+	0.973298i	$0.573723\pi$
$620$	−2.97871	−0.119628
$621$	−141.388	−5.67372
$622$	32.1504	1.28912
$623$	−24.9428	−0.999313
$624$	−19.2526	−0.770719
$625$	1.00000	0.0400000
$626$	0.136315	0.00544823
$627$	−7.65576	−0.305742
$628$	−12.1885	−0.486373
$629$	−46.5207	−1.85490
$630$	−23.4590	−0.934628
$631$	7.27980	0.289804	0.144902	−	0.989446i	$-0.453713\pi$
$631$	7.27980	0.289804	0.144902	+	0.989446i	$0.453713\pi$
$632$	0.450595	0.0179237
$633$	−38.6084	−1.53454
$634$	12.9121	0.512807
$635$	−3.65038	−0.144861
$636$	19.8507	0.787131
$637$	−1.12708	−0.0446566
$638$	6.59746	0.261196
$639$	−139.036	−5.50019
$640$	−1.00000	−0.0395285
$641$	10.5623	0.417186	0.208593	−	0.978002i	$-0.433112\pi$
$641$	10.5623	0.417186	0.208593	+	0.978002i	$0.433112\pi$
$642$	−39.2177	−1.54780
$643$	−28.0458	−1.10602	−0.553008	−	0.833176i	$-0.686520\pi$
$643$	−28.0458	−1.10602	−0.553008	+	0.833176i	$0.686520\pi$
$644$	−19.2814	−0.759793
$645$	9.45981	0.372480
$646$	−14.0544	−0.552961
$647$	9.09434	0.357535	0.178768	−	0.983891i	$-0.442789\pi$
$647$	9.09434	0.357535	0.178768	+	0.983891i	$0.442789\pi$
$648$	−41.2006	−1.61851
$649$	9.03913	0.354817
$650$	5.61839	0.220372
$651$	−27.3898	−1.07349
$652$	0.304628	0.0119301
$653$	−29.7498	−1.16420	−0.582101	−	0.813117i	$-0.697769\pi$
$653$	−29.7498	−1.16420	−0.582101	+	0.813117i	$0.697769\pi$
$654$	25.3963	0.993076
$655$	0.0953352	0.00372505
$656$	7.44464	0.290664
$657$	−8.74228	−0.341069
$658$	−8.11487	−0.316351
$659$	−6.75428	−0.263109	−0.131555	−	0.991309i	$-0.541997\pi$
$659$	−6.75428	−0.263109	−0.131555	+	0.991309i	$0.541997\pi$
$660$	−3.42670	−0.133384
$661$	16.1211	0.627039	0.313520	−	0.949582i	$-0.398492\pi$
$661$	16.1211	0.627039	0.313520	+	0.949582i	$0.398492\pi$
$662$	−2.81263	−0.109316
$663$	−121.112	−4.70360
$664$	7.40753	0.287468
$665$	5.99510	0.232480
$666$	64.6506	2.50516
$667$	−47.4057	−1.83556
$668$	−6.60955	−0.255731
$669$	−78.6849	−3.04213
$670$	−15.4711	−0.597701
$671$	−14.1637	−0.546782
$672$	−9.19519	−0.354712
$673$	20.1950	0.778459	0.389229	−	0.921141i	$-0.372741\pi$
$673$	20.1950	0.778459	0.389229	+	0.921141i	$0.372741\pi$
$674$	−6.16686	−0.237539
$675$	19.6771	0.757371
$676$	18.5664	0.714091
$677$	−45.3538	−1.74309	−0.871544	−	0.490317i	$-0.836881\pi$
$677$	−45.3538	−1.74309	−0.871544	+	0.490317i	$0.836881\pi$
$678$	−32.4009	−1.24435
$679$	28.4809	1.09300
$680$	−6.29070	−0.241237
$681$	−91.4673	−3.50504
$682$	−2.97871	−0.114061
$683$	43.8508	1.67790	0.838951	−	0.544206i	$-0.183169\pi$
$683$	43.8508	1.67790	0.838951	+	0.544206i	$0.183169\pi$
$684$	19.5316	0.746808
$685$	20.4611	0.781779
$686$	18.2455	0.696615
$687$	−31.5297	−1.20293
$688$	2.76062	0.105248
$689$	−32.5471	−1.23994
$690$	24.6224	0.937358
$691$	32.4122	1.23302	0.616510	−	0.787347i	$-0.288546\pi$
$691$	32.4122	1.23302	0.616510	+	0.787347i	$0.288546\pi$
$692$	9.83572	0.373898
$693$	−23.4590	−0.891133
$694$	−7.28404	−0.276498
$695$	4.42742	0.167942
$696$	−22.6075	−0.856936
$697$	46.8320	1.77389
$698$	−18.1432	−0.686731
$699$	−34.7635	−1.31488
$700$	2.68339	0.101423
$701$	−24.2442	−0.915692	−0.457846	−	0.889032i	$-0.651379\pi$
$701$	−24.2442	−0.915692	−0.457846	+	0.889032i	$0.651379\pi$
$702$	110.554	4.17257
$703$	−16.5219	−0.623135
$704$	−1.00000	−0.0376889
$705$	10.3627	0.390282
$706$	24.8361	0.934719
$707$	0.973246	0.0366027
$708$	−30.9744	−1.16409
$709$	2.03183	0.0763068	0.0381534	−	0.999272i	$-0.487852\pi$
$709$	2.03183	0.0763068	0.0381534	+	0.999272i	$0.487852\pi$
$710$	15.9039	0.596862
$711$	−3.93923	−0.147733
$712$	9.29525	0.348354
$713$	21.4034	0.801563
$714$	−57.8442	−2.16476
$715$	5.61839	0.210116
$716$	−21.6972	−0.810862
$717$	−0.640785	−0.0239306
$718$	20.8661	0.778714
$719$	13.5369	0.504841	0.252420	−	0.967618i	$-0.418773\pi$
$719$	13.5369	0.504841	0.252420	+	0.967618i	$0.418773\pi$
$720$	8.74228	0.325805
$721$	−38.3681	−1.42890
$722$	14.0086	0.521345
$723$	−10.6503	−0.396089
$724$	9.35029	0.347501
$725$	6.59746	0.245024
$726$	−3.42670	−0.127177
$727$	31.1202	1.15418	0.577092	−	0.816679i	$-0.304187\pi$
$727$	31.1202	1.15418	0.577092	+	0.816679i	$0.304187\pi$
$728$	15.0764	0.558767
$729$	157.905	5.84832
$730$	1.00000	0.0370117
$731$	17.3662	0.642313
$732$	48.5346	1.79389
$733$	5.00186	0.184748	0.0923739	−	0.995724i	$-0.470555\pi$
$733$	5.00186	0.184748	0.0923739	+	0.995724i	$0.470555\pi$
$734$	−15.9704	−0.589478
$735$	0.687416	0.0253557
$736$	7.18545	0.264859
$737$	−15.4711	−0.569885
$738$	−65.0831	−2.39574
$739$	3.87234	0.142446	0.0712232	−	0.997460i	$-0.477310\pi$
$739$	3.87234	0.142446	0.0712232	+	0.997460i	$0.477310\pi$
$740$	−7.39516	−0.271852
$741$	−43.0131	−1.58013
$742$	−15.5448	−0.570666
$743$	−5.89912	−0.216418	−0.108209	−	0.994128i	$-0.534512\pi$
$743$	−5.89912	−0.216418	−0.108209	+	0.994128i	$0.534512\pi$
$744$	10.2072	0.374212
$745$	9.99656	0.366245
$746$	−13.6655	−0.500329
$747$	−64.7587	−2.36940
$748$	−6.29070	−0.230011
$749$	30.7108	1.12215
$750$	−3.42670	−0.125125
$751$	51.1414	1.86617	0.933087	−	0.359650i	$-0.117104\pi$
$751$	51.1414	1.86617	0.933087	+	0.359650i	$0.117104\pi$
$752$	3.02411	0.110278
$753$	−70.5151	−2.56971
$754$	37.0671	1.34991
$755$	−4.49638	−0.163640
$756$	52.8013	1.92037
$757$	13.8990	0.505168	0.252584	−	0.967575i	$-0.418720\pi$
$757$	13.8990	0.505168	0.252584	+	0.967575i	$0.418720\pi$
$758$	15.3923	0.559074
$759$	24.6224	0.893735
$760$	−2.23415	−0.0810411
$761$	−22.2287	−0.805789	−0.402894	−	0.915246i	$-0.631996\pi$
$761$	−22.2287	−0.805789	−0.402894	+	0.915246i	$0.631996\pi$
$762$	12.5088	0.453145
$763$	−19.8875	−0.719975
$764$	11.6107	0.420061
$765$	54.9950	1.98835
$766$	38.7109	1.39868
$767$	50.7854	1.83376
$768$	3.42670	0.123650
$769$	−4.99590	−0.180157	−0.0900785	−	0.995935i	$-0.528712\pi$
$769$	−4.99590	−0.180157	−0.0900785	+	0.995935i	$0.528712\pi$
$770$	2.68339	0.0967028
$771$	63.7736	2.29675
$772$	−12.7929	−0.460427
$773$	34.6397	1.24590	0.622952	−	0.782260i	$-0.285933\pi$
$773$	34.6397	1.24590	0.622952	+	0.782260i	$0.285933\pi$
$774$	−24.1341	−0.867482
$775$	−2.97871	−0.106998
$776$	−10.6138	−0.381012
$777$	−67.9999	−2.43948
$778$	37.4642	1.34316
$779$	16.6324	0.595919
$780$	−19.2526	−0.689352
$781$	15.9039	0.569086
$782$	45.2015	1.61640
$783$	129.819	4.63934
$784$	0.200606	0.00716449
$785$	−12.1885	−0.435025
$786$	−0.326685	−0.0116525
$787$	−45.1942	−1.61100	−0.805500	−	0.592595i	$-0.798103\pi$
$787$	−45.1942	−1.61100	−0.805500	+	0.592595i	$0.798103\pi$
$788$	10.1657	0.362137
$789$	−52.2223	−1.85916
$790$	0.450595	0.0160315
$791$	25.3726	0.902146
$792$	8.74228	0.310643
$793$	−79.5771	−2.82586
$794$	−24.3011	−0.862413
$795$	19.8507	0.704032
$796$	9.30565	0.329830
$797$	8.43168	0.298665	0.149333	−	0.988787i	$-0.452287\pi$
$797$	8.43168	0.298665	0.149333	+	0.988787i	$0.452287\pi$
$798$	−20.5434	−0.727229
$799$	19.0237	0.673012
$800$	−1.00000	−0.0353553
$801$	−81.2616	−2.87124
$802$	−1.67058	−0.0589901
$803$	1.00000	0.0352892
$804$	53.0148	1.86969
$805$	−19.2814	−0.679579
$806$	−16.7356	−0.589486
$807$	−1.75088	−0.0616340
$808$	−0.362692	−0.0127595
$809$	19.1448	0.673097	0.336549	−	0.941666i	$-0.390740\pi$
$809$	19.1448	0.673097	0.336549	+	0.941666i	$0.390740\pi$
$810$	−41.2006	−1.44764
$811$	−27.2814	−0.957980	−0.478990	−	0.877820i	$-0.658997\pi$
$811$	−27.2814	−0.957980	−0.478990	+	0.877820i	$0.658997\pi$
$812$	17.7036	0.621274
$813$	51.5474	1.80785
$814$	−7.39516	−0.259200
$815$	0.304628	0.0106706
$816$	21.5563	0.754623
$817$	6.16763	0.215778
$818$	−25.0327	−0.875250
$819$	−131.802	−4.60553
$820$	7.44464	0.259978
$821$	−12.4183	−0.433400	−0.216700	−	0.976238i	$-0.569529\pi$
$821$	−12.4183	−0.433400	−0.216700	+	0.976238i	$0.569529\pi$
$822$	−70.1142	−2.44551
$823$	51.2681	1.78709	0.893546	−	0.448971i	$-0.148209\pi$
$823$	51.2681	1.78709	0.893546	+	0.448971i	$0.148209\pi$
$824$	14.2983	0.498106
$825$	−3.42670	−0.119302
$826$	24.2556	0.843959
$827$	−41.7205	−1.45076	−0.725382	−	0.688346i	$-0.758337\pi$
$827$	−41.7205	−1.45076	−0.725382	+	0.688346i	$0.758337\pi$
$828$	−62.8172	−2.18305
$829$	43.1989	1.50036	0.750181	−	0.661233i	$-0.229967\pi$
$829$	43.1989	1.50036	0.750181	+	0.661233i	$0.229967\pi$
$830$	7.40753	0.257119
$831$	16.3258	0.566337
$832$	−5.61839	−0.194783
$833$	1.26195	0.0437240
$834$	−15.1715	−0.525344
$835$	−6.60955	−0.228733
$836$	−2.23415	−0.0772697
$837$	−58.6123	−2.02594
$838$	−15.1263	−0.522529
$839$	−4.11719	−0.142141	−0.0710706	−	0.997471i	$-0.522642\pi$
$839$	−4.11719	−0.142141	−0.0710706	+	0.997471i	$0.522642\pi$
$840$	−9.19519	−0.317264
$841$	14.5265	0.500914
$842$	12.4413	0.428756
$843$	−52.6650	−1.81388
$844$	−11.2669	−0.387823
$845$	18.5664	0.638702
$846$	−26.4376	−0.908943
$847$	2.68339	0.0922025
$848$	5.79295	0.198931
$849$	103.202	3.54190
$850$	−6.29070	−0.215769
$851$	53.1375	1.82153
$852$	−54.4979	−1.86707
$853$	−1.56325	−0.0535245	−0.0267623	−	0.999642i	$-0.508520\pi$
$853$	−1.56325	−0.0535245	−0.0267623	+	0.999642i	$0.508520\pi$
$854$	−38.0067	−1.30056
$855$	19.5316	0.667965
$856$	−11.4447	−0.391173
$857$	−19.7459	−0.674507	−0.337253	−	0.941414i	$-0.609498\pi$
$857$	−19.7459	−0.674507	−0.337253	+	0.941414i	$0.609498\pi$
$858$	−19.2526	−0.657272
$859$	30.1993	1.03039	0.515193	−	0.857074i	$-0.327720\pi$
$859$	30.1993	1.03039	0.515193	+	0.857074i	$0.327720\pi$
$860$	2.76062	0.0941363
$861$	68.4549	2.33294
$862$	−12.3964	−0.422222
$863$	−24.4153	−0.831106	−0.415553	−	0.909569i	$-0.636412\pi$
$863$	−24.4153	−0.831106	−0.415553	+	0.909569i	$0.636412\pi$
$864$	−19.6771	−0.669427
$865$	9.83572	0.334425
$866$	29.3303	0.996683
$867$	77.3505	2.62696
$868$	−7.99306	−0.271302
$869$	0.450595	0.0152854
$870$	−22.6075	−0.766467
$871$	−86.9227	−2.94526
$872$	7.41131	0.250979
$873$	92.7884	3.14041
$874$	16.0534	0.543013
$875$	2.68339	0.0907153
$876$	−3.42670	−0.115777
$877$	−12.5230	−0.422870	−0.211435	−	0.977392i	$-0.567814\pi$
$877$	−12.5230	−0.422870	−0.211435	+	0.977392i	$0.567814\pi$
$878$	−4.10463	−0.138525
$879$	82.4882	2.78226
$880$	−1.00000	−0.0337100
$881$	−19.5733	−0.659440	−0.329720	−	0.944079i	$-0.606954\pi$
$881$	−19.5733	−0.659440	−0.329720	+	0.944079i	$0.606954\pi$
$882$	−1.75375	−0.0590519
$883$	−8.39302	−0.282448	−0.141224	−	0.989978i	$-0.545104\pi$
$883$	−8.39302	−0.282448	−0.141224	+	0.989978i	$0.545104\pi$
$884$	−35.3436	−1.18873
$885$	−30.9744	−1.04119
$886$	−3.98628	−0.133922
$887$	15.0718	0.506061	0.253031	−	0.967458i	$-0.418573\pi$
$887$	15.0718	0.506061	0.253031	+	0.967458i	$0.418573\pi$
$888$	25.3410	0.850388
$889$	−9.79542	−0.328528
$890$	9.29525	0.311577
$891$	−41.2006	−1.38027
$892$	−22.9623	−0.768835
$893$	6.75631	0.226091
$894$	−34.2552	−1.14567
$895$	−21.6972	−0.725257
$896$	−2.68339	−0.0896459
$897$	138.338	4.61898
$898$	−11.0820	−0.369811
$899$	−19.6519	−0.655429
$900$	8.74228	0.291409
$901$	36.4417	1.21405
$902$	7.44464	0.247879
$903$	25.3844	0.844740
$904$	−9.45541	−0.314482
$905$	9.35029	0.310814
$906$	15.4077	0.511888
$907$	−13.6531	−0.453344	−0.226672	−	0.973971i	$-0.572785\pi$
$907$	−13.6531	−0.453344	−0.226672	+	0.973971i	$0.572785\pi$
$908$	−26.6925	−0.885823
$909$	3.17075	0.105167
$910$	15.0764	0.499777
$911$	31.4450	1.04182	0.520910	−	0.853611i	$-0.325593\pi$
$911$	31.4450	1.04182	0.520910	+	0.853611i	$0.325593\pi$
$912$	7.65576	0.253508
$913$	7.40753	0.245154
$914$	16.7197	0.553038
$915$	48.5346	1.60451
$916$	−9.20117	−0.304015
$917$	0.255822	0.00844798
$918$	−123.782	−4.08543
$919$	−34.2997	−1.13144	−0.565721	−	0.824597i	$-0.691402\pi$
$919$	−34.2997	−1.13144	−0.565721	+	0.824597i	$0.691402\pi$
$920$	7.18545	0.236897
$921$	71.1142	2.34329
$922$	−16.9832	−0.559311
$923$	89.3543	2.94113
$924$	−9.19519	−0.302500
$925$	−7.39516	−0.243151
$926$	−19.9916	−0.656965
$927$	−125.000	−4.10554
$928$	−6.59746	−0.216572
$929$	14.6847	0.481791	0.240895	−	0.970551i	$-0.422559\pi$
$929$	14.6847	0.481791	0.240895	+	0.970551i	$0.422559\pi$
$930$	10.2072	0.334706
$931$	0.448183	0.0146886
$932$	−10.1449	−0.332307
$933$	−110.170	−3.60680
$934$	−4.47054	−0.146281
$935$	−6.29070	−0.205728
$936$	49.1176	1.60546
$937$	−11.8612	−0.387489	−0.193745	−	0.981052i	$-0.562063\pi$
$937$	−11.8612	−0.387489	−0.193745	+	0.981052i	$0.562063\pi$
$938$	−41.5151	−1.35551
$939$	−0.467110	−0.0152436
$940$	3.02411	0.0986355
$941$	28.8146	0.939329	0.469665	−	0.882845i	$-0.344375\pi$
$941$	28.8146	0.939329	0.469665	+	0.882845i	$0.344375\pi$
$942$	41.7663	1.36082
$943$	−53.4931	−1.74197
$944$	−9.03913	−0.294199
$945$	52.8013	1.71763
$946$	2.76062	0.0897554
$947$	21.4026	0.695490	0.347745	−	0.937589i	$-0.386947\pi$
$947$	21.4026	0.695490	0.347745	+	0.937589i	$0.386947\pi$
$948$	−1.54406	−0.0501486
$949$	5.61839	0.182381
$950$	−2.23415	−0.0724854
$951$	−44.2460	−1.43478
$952$	−16.8804	−0.547098
$953$	3.96489	0.128435	0.0642177	−	0.997936i	$-0.479545\pi$
$953$	3.96489	0.128435	0.0642177	+	0.997936i	$0.479545\pi$
$954$	−50.6436	−1.63965
$955$	11.6107	0.375714
$956$	−0.186998	−0.00604794
$957$	−22.6075	−0.730798
$958$	19.3770	0.626041
$959$	54.9053	1.77298
$960$	3.42670	0.110596
$961$	−22.1273	−0.713783
$962$	−41.5489	−1.33959
$963$	100.053	3.22417
$964$	−3.10803	−0.100103
$965$	−12.7929	−0.411818
$966$	66.0715	2.12582
$967$	42.6186	1.37052	0.685261	−	0.728297i	$-0.259688\pi$
$967$	42.6186	1.37052	0.685261	+	0.728297i	$0.259688\pi$
$968$	−1.00000	−0.0321412
$969$	48.1601	1.54712
$970$	−10.6138	−0.340787
$971$	−39.8335	−1.27832	−0.639158	−	0.769075i	$-0.720717\pi$
$971$	−39.8335	−1.27832	−0.639158	+	0.769075i	$0.720717\pi$
$972$	82.1509	2.63499
$973$	11.8805	0.380872
$974$	4.61079	0.147739
$975$	−19.2526	−0.616575
$976$	14.1637	0.453368
$977$	43.2764	1.38454	0.692268	−	0.721641i	$-0.256612\pi$
$977$	43.2764	1.38454	0.692268	+	0.721641i	$0.256612\pi$
$978$	−1.04387	−0.0333792
$979$	9.29525	0.297077
$980$	0.200606	0.00640812
$981$	−64.7917	−2.06864
$982$	2.84640	0.0908321
$983$	1.13015	0.0360462	0.0180231	−	0.999838i	$-0.494263\pi$
$983$	1.13015	0.0360462	0.0180231	+	0.999838i	$0.494263\pi$
$984$	−25.5106	−0.813247
$985$	10.1657	0.323905
$986$	−41.5026	−1.32171
$987$	27.8072	0.885114
$988$	−12.5523	−0.399343
$989$	−19.8363	−0.630757
$990$	8.74228	0.277848
$991$	−2.64461	−0.0840089	−0.0420044	−	0.999117i	$-0.513374\pi$
$991$	−2.64461	−0.0840089	−0.0420044	+	0.999117i	$0.513374\pi$
$992$	2.97871	0.0945742
$993$	9.63805	0.305854
$994$	42.6764	1.35361
$995$	9.30565	0.295009
$996$	−25.3834	−0.804304
$997$	28.0174	0.887320	0.443660	−	0.896195i	$-0.353680\pi$
$997$	28.0174	0.887320	0.443660	+	0.896195i	$0.353680\pi$
$998$	23.3086	0.737819
$999$	−145.515	−4.60389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.be.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.be.1.15	✓	15	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:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.42670 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.42670 q^{6} +2.68339 q^{7} -1.00000 q^{8} +8.74228 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.42670 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.42670 q^{6} +2.68339 q^{7} -1.00000 q^{8} +8.74228 q^{9} -1.00000 q^{10} -1.00000 q^{11} +3.42670 q^{12} -5.61839 q^{13} -2.68339 q^{14} +3.42670 q^{15} +1.00000 q^{16} +6.29070 q^{17} -8.74228 q^{18} +2.23415 q^{19} +1.00000 q^{20} +9.19519 q^{21} +1.00000 q^{22} -7.18545 q^{23} -3.42670 q^{24} +1.00000 q^{25} +5.61839 q^{26} +19.6771 q^{27} +2.68339 q^{28} +6.59746 q^{29} -3.42670 q^{30} -2.97871 q^{31} -1.00000 q^{32} -3.42670 q^{33} -6.29070 q^{34} +2.68339 q^{35} +8.74228 q^{36} -7.39516 q^{37} -2.23415 q^{38} -19.2526 q^{39} -1.00000 q^{40} +7.44464 q^{41} -9.19519 q^{42} +2.76062 q^{43} -1.00000 q^{44} +8.74228 q^{45} +7.18545 q^{46} +3.02411 q^{47} +3.42670 q^{48} +0.200606 q^{49} -1.00000 q^{50} +21.5563 q^{51} -5.61839 q^{52} +5.79295 q^{53} -19.6771 q^{54} -1.00000 q^{55} -2.68339 q^{56} +7.65576 q^{57} -6.59746 q^{58} -9.03913 q^{59} +3.42670 q^{60} +14.1637 q^{61} +2.97871 q^{62} +23.4590 q^{63} +1.00000 q^{64} -5.61839 q^{65} +3.42670 q^{66} +15.4711 q^{67} +6.29070 q^{68} -24.6224 q^{69} -2.68339 q^{70} -15.9039 q^{71} -8.74228 q^{72} -1.00000 q^{73} +7.39516 q^{74} +3.42670 q^{75} +2.23415 q^{76} -2.68339 q^{77} +19.2526 q^{78} -0.450595 q^{79} +1.00000 q^{80} +41.2006 q^{81} -7.44464 q^{82} -7.40753 q^{83} +9.19519 q^{84} +6.29070 q^{85} -2.76062 q^{86} +22.6075 q^{87} +1.00000 q^{88} -9.29525 q^{89} -8.74228 q^{90} -15.0764 q^{91} -7.18545 q^{92} -10.2072 q^{93} -3.02411 q^{94} +2.23415 q^{95} -3.42670 q^{96} +10.6138 q^{97} -0.200606 q^{98} -8.74228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 3 * q^3 + 15 * q^4 + 15 * q^5 - 3 * q^6 + 7 * q^7 - 15 * q^8 + 12 * q^9	\( 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9} - 15 q^{10} - 15 q^{11} + 3 q^{12} + 13 q^{13} - 7 q^{14} + 3 q^{15} + 15 q^{16} + 20 q^{17} - 12 q^{18} + 3 q^{19} + 15 q^{20} + 22 q^{21} + 15 q^{22} + 2 q^{23} - 3 q^{24} + 15 q^{25} - 13 q^{26} + 33 q^{27} + 7 q^{28} + 11 q^{29} - 3 q^{30} - 3 q^{31} - 15 q^{32} - 3 q^{33} - 20 q^{34} + 7 q^{35} + 12 q^{36} + 9 q^{37} - 3 q^{38} + 11 q^{39} - 15 q^{40} + 17 q^{41} - 22 q^{42} + 29 q^{43} - 15 q^{44} + 12 q^{45} - 2 q^{46} - 2 q^{47} + 3 q^{48} + 20 q^{49} - 15 q^{50} + 7 q^{51} + 13 q^{52} + 3 q^{53} - 33 q^{54} - 15 q^{55} - 7 q^{56} + 13 q^{57} - 11 q^{58} - 32 q^{59} + 3 q^{60} + 61 q^{61} + 3 q^{62} + 20 q^{63} + 15 q^{64} + 13 q^{65} + 3 q^{66} + 7 q^{67} + 20 q^{68} - 23 q^{69} - 7 q^{70} - 6 q^{71} - 12 q^{72} - 15 q^{73} - 9 q^{74} + 3 q^{75} + 3 q^{76} - 7 q^{77} - 11 q^{78} + 12 q^{79} + 15 q^{80} + 3 q^{81} - 17 q^{82} + 17 q^{83} + 22 q^{84} + 20 q^{85} - 29 q^{86} + 23 q^{87} + 15 q^{88} - 18 q^{89} - 12 q^{90} - 15 q^{91} + 2 q^{92} + 32 q^{93} + 2 q^{94} + 3 q^{95} - 3 q^{96} + 36 q^{97} - 20 q^{98} - 12 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 3 * q^3 + 15 * q^4 + 15 * q^5 - 3 * q^6 + 7 * q^7 - 15 * q^8 + 12 * q^9 - 15 * q^10 - 15 * q^11 + 3 * q^12 + 13 * q^13 - 7 * q^14 + 3 * q^15 + 15 * q^16 + 20 * q^17 - 12 * q^18 + 3 * q^19 + 15 * q^20 + 22 * q^21 + 15 * q^22 + 2 * q^23 - 3 * q^24 + 15 * q^25 - 13 * q^26 + 33 * q^27 + 7 * q^28 + 11 * q^29 - 3 * q^30 - 3 * q^31 - 15 * q^32 - 3 * q^33 - 20 * q^34 + 7 * q^35 + 12 * q^36 + 9 * q^37 - 3 * q^38 + 11 * q^39 - 15 * q^40 + 17 * q^41 - 22 * q^42 + 29 * q^43 - 15 * q^44 + 12 * q^45 - 2 * q^46 - 2 * q^47 + 3 * q^48 + 20 * q^49 - 15 * q^50 + 7 * q^51 + 13 * q^52 + 3 * q^53 - 33 * q^54 - 15 * q^55 - 7 * q^56 + 13 * q^57 - 11 * q^58 - 32 * q^59 + 3 * q^60 + 61 * q^61 + 3 * q^62 + 20 * q^63 + 15 * q^64 + 13 * q^65 + 3 * q^66 + 7 * q^67 + 20 * q^68 - 23 * q^69 - 7 * q^70 - 6 * q^71 - 12 * q^72 - 15 * q^73 - 9 * q^74 + 3 * q^75 + 3 * q^76 - 7 * q^77 - 11 * q^78 + 12 * q^79 + 15 * q^80 + 3 * q^81 - 17 * q^82 + 17 * q^83 + 22 * q^84 + 20 * q^85 - 29 * q^86 + 23 * q^87 + 15 * q^88 - 18 * q^89 - 12 * q^90 - 15 * q^91 + 2 * q^92 + 32 * q^93 + 2 * q^94 + 3 * q^95 - 3 * q^96 + 36 * q^97 - 20 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more