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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [57,2,Mod(2,57)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("57.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 53.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 57.53
Dual form 57.2.j.a.14.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.70574 - 0.300767i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(-2.05303 + 3.55596i) q^{7} +(2.81908 - 1.02606i) q^{9} +(-3.00000 - 1.73205i) q^{12} +(-3.96064 - 0.698367i) q^{13} +(0.694593 + 3.93923i) q^{16} +(0.500000 - 4.33013i) q^{19} +(-2.43242 + 6.68302i) q^{21} +(0.868241 - 4.92404i) q^{25} +(4.50000 - 2.59808i) q^{27} +(7.71688 - 2.80872i) q^{28} +(9.64203 + 5.56683i) q^{31} +(-5.63816 - 2.05212i) q^{36} -3.09018i q^{37} -6.96585 q^{39} +(-8.48158 + 7.11689i) q^{43} +(2.36959 + 6.51038i) q^{48} +(-4.92989 - 8.53882i) q^{49} +(5.17024 + 6.16166i) q^{52} +(-0.449493 - 7.53644i) q^{57} +(-11.6270 - 9.75622i) q^{61} +(-2.13903 + 12.1311i) q^{63} +(4.00000 - 6.92820i) q^{64} +(5.18345 + 14.2414i) q^{67} +(0.216415 + 1.22735i) q^{73} -8.66025i q^{75} +(-6.33275 + 5.99135i) q^{76} +(0.917404 - 0.161763i) q^{79} +(6.89440 - 5.78509i) q^{81} +(12.3182 - 7.11192i) q^{84} +(10.6147 - 12.6501i) q^{91} +(18.1211 + 6.59553i) q^{93} +(-1.77719 + 4.88279i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.342020 − 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(3\) 1.70574 − 0.300767i 0.984808 − 0.173648i
\(4\) −1.53209 − 1.28558i −0.766044 − 0.642788i
\(5\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(6\) 0 0
\(7\) −2.05303 + 3.55596i −0.775974 + 1.34403i 0.158272 + 0.987396i \(0.449408\pi\)
−0.934246 + 0.356630i \(0.883926\pi\)
\(8\) 0 0
\(9\) 2.81908 − 1.02606i 0.939693 − 0.342020i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −3.00000 − 1.73205i −0.866025 − 0.500000i
\(13\) −3.96064 − 0.698367i −1.09848 − 0.193692i −0.405108 − 0.914269i \(-0.632766\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.694593 + 3.93923i 0.173648 + 0.984808i
\(17\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(18\) 0 0
\(19\) 0.500000 − 4.33013i 0.114708 − 0.993399i
\(20\) 0 0
\(21\) −2.43242 + 6.68302i −0.530797 + 1.45835i
\(22\) 0 0
\(23\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(24\) 0 0
\(25\) 0.868241 − 4.92404i 0.173648 − 0.984808i
\(26\) 0 0
\(27\) 4.50000 − 2.59808i 0.866025 − 0.500000i
\(28\) 7.71688 − 2.80872i 1.45835 − 0.530797i
\(29\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(30\) 0 0
\(31\) 9.64203 + 5.56683i 1.73176 + 0.999832i 0.875057 + 0.484020i \(0.160824\pi\)
0.856702 + 0.515812i \(0.172510\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −5.63816 − 2.05212i −0.939693 − 0.342020i
\(37\) − 3.09018i − 0.508022i −0.967201 − 0.254011i \(-0.918250\pi\)
0.967201 − 0.254011i \(-0.0817500\pi\)
\(38\) 0 0
\(39\) −6.96585 −1.11543
\(40\) 0 0
\(41\) 0 0 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(42\) 0 0
\(43\) −8.48158 + 7.11689i −1.29343 + 1.08532i −0.302188 + 0.953248i \(0.597717\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(48\) 2.36959 + 6.51038i 0.342020 + 0.939693i
\(49\) −4.92989 − 8.53882i −0.704270 − 1.21983i
\(50\) 0 0
\(51\) 0 0
\(52\) 5.17024 + 6.16166i 0.716984 + 0.854468i
\(53\) 0 0 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.449493 − 7.53644i −0.0595368 − 0.998226i
\(58\) 0 0
\(59\) 0 0 0.342020 − 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(60\) 0 0
\(61\) −11.6270 − 9.75622i −1.48869 − 1.24916i −0.896258 − 0.443533i \(-0.853725\pi\)
−0.592428 − 0.805623i \(-0.701831\pi\)
\(62\) 0 0
\(63\) −2.13903 + 12.1311i −0.269493 + 1.52837i
\(64\) 4.00000 − 6.92820i 0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 5.18345 + 14.2414i 0.633259 + 1.73986i 0.671932 + 0.740613i \(0.265465\pi\)
−0.0386729 + 0.999252i \(0.512313\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.642788 − 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(72\) 0 0
\(73\) 0.216415 + 1.22735i 0.0253294 + 0.143650i 0.994850 − 0.101361i \(-0.0323196\pi\)
−0.969520 + 0.245011i \(0.921208\pi\)
\(74\) 0 0
\(75\) − 8.66025i − 1.00000i
\(76\) −6.33275 + 5.99135i −0.726416 + 0.687255i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.917404 − 0.161763i 0.103216 − 0.0181998i −0.121802 − 0.992554i \(-0.538867\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 6.89440 − 5.78509i 0.766044 − 0.642788i
\(82\) 0 0
\(83\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 12.3182 − 7.11192i 1.34403 − 0.775974i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(90\) 0 0
\(91\) 10.6147 − 12.6501i 1.11272 − 1.32609i
\(92\) 0 0
\(93\) 18.1211 + 6.59553i 1.87907 + 0.683925i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.77719 + 4.88279i −0.180446 + 0.495772i −0.996631 − 0.0820195i \(-0.973863\pi\)
0.816185 + 0.577791i \(0.196085\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 57.2.j.a.53.1 yes 6
3.2 odd 2 CM 57.2.j.a.53.1 yes 6
4.3 odd 2 912.2.cc.a.737.1 6
12.11 even 2 912.2.cc.a.737.1 6
19.9 even 9 1083.2.d.a.1082.3 6
19.10 odd 18 1083.2.d.a.1082.6 6
19.14 odd 18 inner 57.2.j.a.14.1 ✓ 6
57.14 even 18 inner 57.2.j.a.14.1 ✓ 6
57.29 even 18 1083.2.d.a.1082.6 6
57.47 odd 18 1083.2.d.a.1082.3 6
76.71 even 18 912.2.cc.a.641.1 6
228.71 odd 18 912.2.cc.a.641.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.j.a.14.1 ✓ 6 19.14 odd 18 inner
57.2.j.a.14.1 ✓ 6 57.14 even 18 inner
57.2.j.a.53.1 yes 6 1.1 even 1 trivial
57.2.j.a.53.1 yes 6 3.2 odd 2 CM
912.2.cc.a.641.1 6 76.71 even 18
912.2.cc.a.641.1 6 228.71 odd 18
912.2.cc.a.737.1 6 4.3 odd 2
912.2.cc.a.737.1 6 12.11 even 2
1083.2.d.a.1082.3 6 19.9 even 9
1083.2.d.a.1082.3 6 57.47 odd 18
1083.2.d.a.1082.6 6 19.10 odd 18
1083.2.d.a.1082.6 6 57.29 even 18