Show commands: Magma / Pari/GP / SageMath

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 29.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 57.29
Dual form 57.2.j.a.2.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.11334 - 1.32683i) q^{3} +(1.87939 - 0.684040i) q^{4} +(-0.418748 + 0.725293i) q^{7} +(-0.520945 + 2.95442i) q^{9} +(-3.00000 - 1.73205i) q^{12} +(-4.62449 + 5.51125i) q^{13} +(3.06418 - 2.57115i) q^{16} +(0.500000 - 4.33013i) q^{19} +(1.42855 - 0.251892i) q^{21} +(3.83022 + 3.21394i) q^{25} +(4.50000 - 2.59808i) q^{27} +(-0.290859 + 1.64955i) q^{28} +(-4.97431 - 2.87192i) q^{31} +(1.04189 + 5.90885i) q^{36} -8.64501i q^{37} +12.4611 q^{39} +(-10.9226 - 3.97551i) q^{43} +(-6.82295 - 1.20307i) q^{48} +(3.14930 + 5.45475i) q^{49} +(-4.92127 + 13.5211i) q^{52} +(-6.30200 + 4.15749i) q^{57} +(-10.1356 + 3.68907i) q^{61} +(-1.92468 - 1.61500i) q^{63} +(4.00000 - 6.92820i) q^{64} +(12.7417 + 2.24670i) q^{67} +(10.8289 - 9.08651i) q^{73} -8.66025i q^{75} +(-2.02229 - 8.48000i) q^{76} +(10.1814 + 12.1337i) q^{79} +(-8.45723 - 3.07818i) q^{81} +(2.51249 - 1.45059i) q^{84} +(-2.06077 - 5.66193i) q^{91} +(1.72756 + 9.79747i) q^{93} +(5.11721 - 0.902302i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(e\left(\frac{17}{18}\right)\)

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 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(3\) −1.11334 − 1.32683i −0.642788 − 0.766044i
\(4\) 1.87939 − 0.684040i 0.939693 − 0.342020i
\(5\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(6\) 0 0
\(7\) −0.418748 + 0.725293i −0.158272 + 0.274135i −0.934246 − 0.356630i \(-0.883926\pi\)
0.775974 + 0.630765i \(0.217259\pi\)
\(8\) 0 0
\(9\) −0.520945 + 2.95442i −0.173648 + 0.984808i
\(10\) 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
\(13\) −4.62449 + 5.51125i −1.28260 + 1.52854i −0.589226 + 0.807968i \(0.700567\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.06418 − 2.57115i 0.766044 − 0.642788i
\(17\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(18\) 0 0
\(19\) 0.500000 − 4.33013i 0.114708 − 0.993399i
\(20\) 0 0
\(21\) 1.42855 − 0.251892i 0.311735 − 0.0549673i
\(22\) 0 0
\(23\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(24\) 0 0
\(25\) 3.83022 + 3.21394i 0.766044 + 0.642788i
\(26\) 0 0
\(27\) 4.50000 − 2.59808i 0.866025 − 0.500000i
\(28\) −0.290859 + 1.64955i −0.0549673 + 0.311735i
\(29\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(30\) 0 0
\(31\) −4.97431 − 2.87192i −0.893412 − 0.515812i −0.0183550 − 0.999832i \(-0.505843\pi\)
−0.875057 + 0.484020i \(0.839176\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.04189 + 5.90885i 0.173648 + 0.984808i
\(37\) − 8.64501i − 1.42123i −0.703581 − 0.710615i \(-0.748417\pi\)
0.703581 − 0.710615i \(-0.251583\pi\)
\(38\) 0 0
\(39\) 12.4611 1.99537
\(40\) 0 0
\(41\) 0 0 −0.642788 − 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(42\) 0 0
\(43\) −10.9226 − 3.97551i −1.66568 − 0.606259i −0.674443 − 0.738327i \(-0.735616\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(48\) −6.82295 − 1.20307i −0.984808 − 0.173648i
\(49\) 3.14930 + 5.45475i 0.449900 + 0.779250i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.92127 + 13.5211i −0.682458 + 1.87504i
\(53\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.30200 + 4.15749i −0.834721 + 0.550673i
\(58\) 0 0
\(59\) 0 0 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(60\) 0 0
\(61\) −10.1356 + 3.68907i −1.29773 + 0.472337i −0.896258 − 0.443533i \(-0.853725\pi\)
−0.401476 + 0.915869i \(0.631503\pi\)
\(62\) 0 0
\(63\) −1.92468 − 1.61500i −0.242487 − 0.203470i
\(64\) 4.00000 − 6.92820i 0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 12.7417 + 2.24670i 1.55665 + 0.274479i 0.884714 − 0.466134i \(-0.154354\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.342020 − 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(72\) 0 0
\(73\) 10.8289 − 9.08651i 1.26742 − 1.06350i 0.272575 − 0.962135i \(-0.412125\pi\)
0.994850 − 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) − 8.66025i − 1.00000i
\(76\) −2.02229 − 8.48000i −0.231972 − 0.972722i
\(77\) 0 0
\(78\) 0 0
\(79\) 10.1814 + 12.1337i 1.14550 + 1.36515i 0.920478 + 0.390794i \(0.127800\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −8.45723 − 3.07818i −0.939693 − 0.342020i
\(82\) 0 0
\(83\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 2.51249 − 1.45059i 0.274135 − 0.158272i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(90\) 0 0
\(91\) −2.06077 − 5.66193i −0.216028 − 0.593532i
\(92\) 0 0
\(93\) 1.72756 + 9.79747i 0.179140 + 1.01595i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.11721 − 0.902302i 0.519574 − 0.0916149i 0.0922897 − 0.995732i \(-0.470581\pi\)
0.427284 + 0.904117i \(0.359470\pi\)
\(98\) 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.29.1 yes 6
3.2 odd 2 CM 57.2.j.a.29.1 yes 6
4.3 odd 2 912.2.cc.a.257.1 6
12.11 even 2 912.2.cc.a.257.1 6
19.2 odd 18 inner 57.2.j.a.2.1 ✓ 6
19.6 even 9 1083.2.d.a.1082.2 6
19.13 odd 18 1083.2.d.a.1082.5 6
57.2 even 18 inner 57.2.j.a.2.1 ✓ 6
57.32 even 18 1083.2.d.a.1082.5 6
57.44 odd 18 1083.2.d.a.1082.2 6
76.59 even 18 912.2.cc.a.401.1 6
228.59 odd 18 912.2.cc.a.401.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.j.a.2.1 ✓ 6 19.2 odd 18 inner
57.2.j.a.2.1 ✓ 6 57.2 even 18 inner
57.2.j.a.29.1 yes 6 1.1 even 1 trivial
57.2.j.a.29.1 yes 6 3.2 odd 2 CM
912.2.cc.a.257.1 6 4.3 odd 2
912.2.cc.a.257.1 6 12.11 even 2
912.2.cc.a.401.1 6 76.59 even 18
912.2.cc.a.401.1 6 228.59 odd 18
1083.2.d.a.1082.2 6 19.6 even 9
1083.2.d.a.1082.2 6 57.44 odd 18
1083.2.d.a.1082.5 6 19.13 odd 18
1083.2.d.a.1082.5 6 57.32 even 18