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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 41.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 57.41
Dual form 57.2.j.a.32.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.592396 - 1.62760i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.47178 + 4.28125i) q^{7} +(-2.29813 + 1.92836i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(1.08512 - 2.98135i) q^{13} +(-3.75877 + 1.36808i) q^{16} +(0.500000 + 4.33013i) q^{19} +(5.50387 - 6.55926i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.57398 - 6.35532i) q^{28} +(-4.66772 + 2.69491i) q^{31} +(4.59627 + 3.85673i) q^{36} -11.7352i q^{37} -5.49525 q^{39} +(-0.0957998 + 0.543308i) q^{43} +(4.45336 + 5.30731i) q^{48} +(-8.71941 + 15.1025i) q^{49} +(-6.24897 - 1.10186i) q^{52} +(6.75150 - 3.37895i) q^{57} +(0.762641 + 4.32515i) q^{61} +(-13.9363 - 5.07239i) q^{63} +(4.00000 + 6.92820i) q^{64} +(-1.42514 - 1.69842i) q^{67} +(14.4547 - 5.26108i) q^{73} +8.66025i q^{75} +(8.35504 - 2.48865i) q^{76} +(-5.09879 - 14.0088i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-14.8307 - 8.56250i) q^{84} +(15.4461 - 2.72356i) q^{91} +(7.15136 + 6.00070i) q^{93} +(-3.34002 + 3.98048i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{18}\right)\)

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 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(3\) −0.592396 − 1.62760i −0.342020 − 0.939693i
\(4\) −0.347296 − 1.96962i −0.173648 − 0.984808i
\(5\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(6\) 0 0
\(7\) 2.47178 + 4.28125i 0.934246 + 1.61816i 0.775974 + 0.630765i \(0.217259\pi\)
0.158272 + 0.987396i \(0.449408\pi\)
\(8\) 0 0
\(9\) −2.29813 + 1.92836i −0.766044 + 0.642788i
\(10\) 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
\(13\) 1.08512 − 2.98135i 0.300959 − 0.826877i −0.693375 − 0.720577i \(-0.743877\pi\)
0.994334 − 0.106301i \(-0.0339006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.75877 + 1.36808i −0.939693 + 0.342020i
\(17\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(18\) 0 0
\(19\) 0.500000 + 4.33013i 0.114708 + 0.993399i
\(20\) 0 0
\(21\) 5.50387 − 6.55926i 1.20104 − 1.43135i
\(22\) 0 0
\(23\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(24\) 0 0
\(25\) −4.69846 − 1.71010i −0.939693 − 0.342020i
\(26\) 0 0
\(27\) 4.50000 + 2.59808i 0.866025 + 0.500000i
\(28\) 7.57398 − 6.35532i 1.43135 − 1.20104i
\(29\) 0 0 −0.642788 − 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(30\) 0 0
\(31\) −4.66772 + 2.69491i −0.838347 + 0.484020i −0.856702 − 0.515812i \(-0.827490\pi\)
0.0183550 + 0.999832i \(0.494157\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 4.59627 + 3.85673i 0.766044 + 0.642788i
\(37\) − 11.7352i − 1.92925i −0.263620 − 0.964626i \(-0.584917\pi\)
0.263620 − 0.964626i \(-0.415083\pi\)
\(38\) 0 0
\(39\) −5.49525 −0.879945
\(40\) 0 0
\(41\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(42\) 0 0
\(43\) −0.0957998 + 0.543308i −0.0146093 + 0.0828537i −0.991241 − 0.132068i \(-0.957838\pi\)
0.976631 + 0.214921i \(0.0689495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(48\) 4.45336 + 5.30731i 0.642788 + 0.766044i
\(49\) −8.71941 + 15.1025i −1.24563 + 2.15749i
\(50\) 0 0
\(51\) 0 0
\(52\) −6.24897 − 1.10186i −0.866576 − 0.152801i
\(53\) 0 0 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.75150 − 3.37895i 0.894258 − 0.447553i
\(58\) 0 0
\(59\) 0 0 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(60\) 0 0
\(61\) 0.762641 + 4.32515i 0.0976462 + 0.553779i 0.993904 + 0.110246i \(0.0351639\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −13.9363 − 5.07239i −1.75581 − 0.639062i
\(64\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.42514 − 1.69842i −0.174109 − 0.207495i 0.671932 − 0.740613i \(-0.265465\pi\)
−0.846041 + 0.533118i \(0.821020\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(72\) 0 0
\(73\) 14.4547 − 5.26108i 1.69180 − 0.615763i 0.696946 − 0.717124i \(-0.254542\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 8.35504 − 2.48865i 0.958388 − 0.285467i
\(77\) 0 0
\(78\) 0 0
\(79\) −5.09879 − 14.0088i −0.573659 − 1.57612i −0.798677 − 0.601760i \(-0.794466\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(80\) 0 0
\(81\) 1.56283 − 8.86327i 0.173648 − 0.984808i
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) −14.8307 − 8.56250i −1.61816 − 0.934246i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.342020 − 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(90\) 0 0
\(91\) 15.4461 − 2.72356i 1.61919 − 0.285507i
\(92\) 0 0
\(93\) 7.15136 + 6.00070i 0.741561 + 0.622244i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.34002 + 3.98048i −0.339128 + 0.404157i −0.908474 − 0.417941i \(-0.862752\pi\)
0.569346 + 0.822098i \(0.307196\pi\)
\(98\) 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.41.1 yes 6
3.2 odd 2 CM 57.2.j.a.41.1 yes 6
4.3 odd 2 912.2.cc.a.497.1 6
12.11 even 2 912.2.cc.a.497.1 6
19.5 even 9 1083.2.d.a.1082.4 6
19.13 odd 18 inner 57.2.j.a.32.1 ✓ 6
19.14 odd 18 1083.2.d.a.1082.1 6
57.5 odd 18 1083.2.d.a.1082.4 6
57.14 even 18 1083.2.d.a.1082.1 6
57.32 even 18 inner 57.2.j.a.32.1 ✓ 6
76.51 even 18 912.2.cc.a.545.1 6
228.203 odd 18 912.2.cc.a.545.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.j.a.32.1 ✓ 6 19.13 odd 18 inner
57.2.j.a.32.1 ✓ 6 57.32 even 18 inner
57.2.j.a.41.1 yes 6 1.1 even 1 trivial
57.2.j.a.41.1 yes 6 3.2 odd 2 CM
912.2.cc.a.497.1 6 4.3 odd 2
912.2.cc.a.497.1 6 12.11 even 2
912.2.cc.a.545.1 6 76.51 even 18
912.2.cc.a.545.1 6 228.203 odd 18
1083.2.d.a.1082.1 6 19.14 odd 18
1083.2.d.a.1082.1 6 57.14 even 18
1083.2.d.a.1082.4 6 19.5 even 9
1083.2.d.a.1082.4 6 57.5 odd 18