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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 14.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 57.14
Dual form 57.2.j.a.53.1

$q$-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 \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{18}\right)\)

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 \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(3\) 1.70574 + 0.300767i 0.984808 + 0.173648i
\(4\) −1.53209 + 1.28558i −0.766044 + 0.642788i
\(5\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(6\) 0 0
\(7\) −2.05303 − 3.55596i −0.775974 − 1.34403i −0.934246 − 0.356630i \(-0.883926\pi\)
0.158272 − 0.987396i \(-0.449408\pi\)
\(8\) 0 0
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −3.00000 + 1.73205i −0.866025 + 0.500000i
\(13\) −3.96064 + 0.698367i −1.09848 + 0.193692i −0.693375 − 0.720577i \(-0.743877\pi\)
−0.405108 + 0.914269i \(0.632766\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.694593 − 3.93923i 0.173648 − 0.984808i
\(17\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 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 \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(24\) 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 \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(30\) 0 0
\(31\) 9.64203 − 5.56683i 1.73176 − 0.999832i 0.856702 − 0.515812i \(-0.172510\pi\)
0.875057 − 0.484020i \(-0.160824\pi\)
\(32\) 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 \(0.0817500\pi\)
−0.967201 + 0.254011i \(0.918250\pi\)
\(38\) 0 0
\(39\) −6.96585 −1.11543
\(40\) 0 0
\(41\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(42\) 0 0
\(43\) −8.48158 − 7.11689i −1.29343 − 1.08532i −0.991241 − 0.132068i \(-0.957838\pi\)
−0.302188 − 0.953248i \(-0.597717\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(48\) 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 \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(54\) 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 \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(60\) 0 0
\(61\) −11.6270 + 9.75622i −1.48869 + 1.24916i −0.592428 + 0.805623i \(0.701831\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 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 −0.0386729 − 0.999252i \(-0.512313\pi\)
0.671932 − 0.740613i \(-0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(72\) 0 0
\(73\) 0.216415 − 1.22735i 0.0253294 − 0.143650i −0.969520 − 0.245011i \(-0.921208\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 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 0.225018 − 0.974355i \(-0.427756\pi\)
−0.121802 + 0.992554i \(0.538867\pi\)
\(80\) 0 0
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 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 \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(90\) 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 0.816185 − 0.577791i \(-0.196085\pi\)
−0.996631 + 0.0820195i \(0.973863\pi\)
\(98\) 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.14.1 ✓ 6
3.2 odd 2 CM 57.2.j.a.14.1 ✓ 6
4.3 odd 2 912.2.cc.a.641.1 6
12.11 even 2 912.2.cc.a.641.1 6
19.2 odd 18 1083.2.d.a.1082.3 6
19.15 odd 18 inner 57.2.j.a.53.1 yes 6
19.17 even 9 1083.2.d.a.1082.6 6
57.2 even 18 1083.2.d.a.1082.3 6
57.17 odd 18 1083.2.d.a.1082.6 6
57.53 even 18 inner 57.2.j.a.53.1 yes 6
76.15 even 18 912.2.cc.a.737.1 6
228.167 odd 18 912.2.cc.a.737.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.j.a.14.1 ✓ 6 1.1 even 1 trivial
57.2.j.a.14.1 ✓ 6 3.2 odd 2 CM
57.2.j.a.53.1 yes 6 19.15 odd 18 inner
57.2.j.a.53.1 yes 6 57.53 even 18 inner
912.2.cc.a.641.1 6 4.3 odd 2
912.2.cc.a.641.1 6 12.11 even 2
912.2.cc.a.737.1 6 76.15 even 18
912.2.cc.a.737.1 6 228.167 odd 18
1083.2.d.a.1082.3 6 19.2 odd 18
1083.2.d.a.1082.3 6 57.2 even 18
1083.2.d.a.1082.6 6 19.17 even 9
1083.2.d.a.1082.6 6 57.17 odd 18