Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(529,592)] 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("592.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 529.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 592.529
Dual form 592.2.w.e.545.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36603 - 2.36603i) q^{3} +(1.50000 + 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-2.23205 - 3.86603i) q^{9} +4.73205 q^{11} +(-3.00000 - 1.73205i) q^{13} +(4.09808 - 2.36603i) q^{15} +(6.69615 - 3.86603i) q^{17} +(-1.09808 - 0.633975i) q^{19} +(2.73205 + 4.73205i) q^{21} -4.73205i q^{23} +(-1.00000 - 1.73205i) q^{25} -4.00000 q^{27} +8.66025i q^{29} +1.26795i q^{31} +(6.46410 - 11.1962i) q^{33} +(-3.00000 + 1.73205i) q^{35} +(-5.69615 - 2.13397i) q^{37} +(-8.19615 + 4.73205i) q^{39} +(-4.96410 + 8.59808i) q^{41} +0.928203i q^{43} -7.73205i q^{45} -4.73205 q^{47} +(1.50000 + 2.59808i) q^{49} -21.1244i q^{51} +(-1.26795 - 2.19615i) q^{53} +(7.09808 + 4.09808i) q^{55} +(-3.00000 + 1.73205i) q^{57} +(-2.19615 + 1.26795i) q^{59} +(1.50000 + 0.866025i) q^{61} +8.92820 q^{63} +(-3.00000 - 5.19615i) q^{65} +(-5.09808 + 8.83013i) q^{67} +(-11.1962 - 6.46410i) q^{69} +(1.73205 - 3.00000i) q^{71} +4.00000 q^{73} -5.46410 q^{75} +(-4.73205 + 8.19615i) q^{77} +(11.4904 + 6.63397i) q^{79} +(1.23205 - 2.13397i) q^{81} +(-2.83013 - 4.90192i) q^{83} +13.3923 q^{85} +(20.4904 + 11.8301i) q^{87} +(-5.89230 + 3.40192i) q^{89} +(6.00000 - 3.46410i) q^{91} +(3.00000 + 1.73205i) q^{93} +(-1.09808 - 1.90192i) q^{95} +7.73205i q^{97} +(-10.5622 - 18.2942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} - 2 q^{9} + 12 q^{11} - 12 q^{13} + 6 q^{15} + 6 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{25} - 16 q^{27} + 12 q^{33} - 12 q^{35} - 2 q^{37} - 12 q^{39} - 6 q^{41} - 12 q^{47}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 1.36603 − 2.36603i 0.788675 − 1.36603i −0.138104 − 0.990418i \(-0.544101\pi\)
0.926779 − 0.375608i \(-0.122566\pi\)
\(4\) 0 0
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 − 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −1.00000 + 1.73205i −0.377964 + 0.654654i −0.990766 − 0.135583i \(-0.956709\pi\)
0.612801 + 0.790237i \(0.290043\pi\)
\(8\) 0 0
\(9\) −2.23205 − 3.86603i −0.744017 − 1.28868i
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) −3.00000 − 1.73205i −0.832050 − 0.480384i 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) 0 0
\(15\) 4.09808 − 2.36603i 1.05812 − 0.610905i
\(16\) 0 0
\(17\) 6.69615 − 3.86603i 1.62406 − 0.937649i 0.638236 − 0.769841i \(-0.279664\pi\)
0.985820 − 0.167808i \(-0.0536689\pi\)
\(18\) 0 0
\(19\) −1.09808 − 0.633975i −0.251916 − 0.145444i 0.368725 − 0.929538i \(-0.379794\pi\)
−0.620641 + 0.784095i \(0.713128\pi\)
\(20\) 0 0
\(21\) 2.73205 + 4.73205i 0.596182 + 1.03262i
\(22\) 0 0
\(23\) − 4.73205i − 0.986701i −0.869831 − 0.493350i \(-0.835772\pi\)
0.869831 − 0.493350i \(-0.164228\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 8.66025i 1.60817i 0.594515 + 0.804084i \(0.297344\pi\)
−0.594515 + 0.804084i \(0.702656\pi\)
\(30\) 0 0
\(31\) 1.26795i 0.227730i 0.993496 + 0.113865i \(0.0363232\pi\)
−0.993496 + 0.113865i \(0.963677\pi\)
\(32\) 0 0
\(33\) 6.46410 − 11.1962i 1.12526 − 1.94900i
\(34\) 0 0
\(35\) −3.00000 + 1.73205i −0.507093 + 0.292770i
\(36\) 0 0
\(37\) −5.69615 − 2.13397i −0.936442 − 0.350823i
\(38\) 0 0
\(39\) −8.19615 + 4.73205i −1.31243 + 0.757735i
\(40\) 0 0
\(41\) −4.96410 + 8.59808i −0.775262 + 1.34279i 0.159384 + 0.987217i \(0.449049\pi\)
−0.934647 + 0.355577i \(0.884284\pi\)
\(42\) 0 0
\(43\) 0.928203i 0.141550i 0.997492 + 0.0707748i \(0.0225472\pi\)
−0.997492 + 0.0707748i \(0.977453\pi\)
\(44\) 0 0
\(45\) − 7.73205i − 1.15263i
\(46\) 0 0
\(47\) −4.73205 −0.690241 −0.345120 − 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) − 21.1244i − 2.95800i
\(52\) 0 0
\(53\) −1.26795 − 2.19615i −0.174166 − 0.301665i 0.765706 − 0.643191i \(-0.222390\pi\)
−0.939872 + 0.341526i \(0.889056\pi\)
\(54\) 0 0
\(55\) 7.09808 + 4.09808i 0.957104 + 0.552584i
\(56\) 0 0
\(57\) −3.00000 + 1.73205i −0.397360 + 0.229416i
\(58\) 0 0
\(59\) −2.19615 + 1.26795i −0.285915 + 0.165073i −0.636098 − 0.771608i \(-0.719453\pi\)
0.350183 + 0.936681i \(0.386119\pi\)
\(60\) 0 0
\(61\) 1.50000 + 0.866025i 0.192055 + 0.110883i 0.592944 − 0.805243i \(-0.297965\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(62\) 0 0
\(63\) 8.92820 1.12485
\(64\) 0 0
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) 0 0
\(67\) −5.09808 + 8.83013i −0.622829 + 1.07877i 0.366127 + 0.930565i \(0.380683\pi\)
−0.988956 + 0.148207i \(0.952650\pi\)
\(68\) 0 0
\(69\) −11.1962 − 6.46410i −1.34786 − 0.778186i
\(70\) 0 0
\(71\) 1.73205 − 3.00000i 0.205557 − 0.356034i −0.744753 − 0.667340i \(-0.767433\pi\)
0.950310 + 0.311305i \(0.100766\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) −5.46410 −0.630940
\(76\) 0 0
\(77\) −4.73205 + 8.19615i −0.539267 + 0.934038i
\(78\) 0 0
\(79\) 11.4904 + 6.63397i 1.29277 + 0.746380i 0.979144 − 0.203167i \(-0.0651233\pi\)
0.313625 + 0.949547i \(0.398457\pi\)
\(80\) 0 0
\(81\) 1.23205 − 2.13397i 0.136895 − 0.237108i
\(82\) 0 0
\(83\) −2.83013 − 4.90192i −0.310647 − 0.538056i 0.667856 − 0.744291i \(-0.267212\pi\)
−0.978503 + 0.206235i \(0.933879\pi\)
\(84\) 0 0
\(85\) 13.3923 1.45260
\(86\) 0 0
\(87\) 20.4904 + 11.8301i 2.19680 + 1.26832i
\(88\) 0 0
\(89\) −5.89230 + 3.40192i −0.624583 + 0.360603i −0.778651 − 0.627457i \(-0.784096\pi\)
0.154068 + 0.988060i \(0.450762\pi\)
\(90\) 0 0
\(91\) 6.00000 − 3.46410i 0.628971 − 0.363137i
\(92\) 0 0
\(93\) 3.00000 + 1.73205i 0.311086 + 0.179605i
\(94\) 0 0
\(95\) −1.09808 − 1.90192i −0.112660 − 0.195133i
\(96\) 0 0
\(97\) 7.73205i 0.785071i 0.919737 + 0.392535i \(0.128402\pi\)
−0.919737 + 0.392535i \(0.871598\pi\)
\(98\) 0 0
\(99\) −10.5622 − 18.2942i −1.06154 − 1.83864i
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 592.2.w.e.529.2 4
4.3 odd 2 74.2.e.b.11.2 ✓ 4
12.11 even 2 666.2.s.a.307.1 4
37.27 even 6 inner 592.2.w.e.545.2 4
148.27 odd 6 74.2.e.b.27.2 yes 4
148.103 even 12 2738.2.a.e.1.1 2
148.119 even 12 2738.2.a.i.1.1 2
444.323 even 6 666.2.s.a.397.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.2 ✓ 4 4.3 odd 2
74.2.e.b.27.2 yes 4 148.27 odd 6
592.2.w.e.529.2 4 1.1 even 1 trivial
592.2.w.e.545.2 4 37.27 even 6 inner
666.2.s.a.307.1 4 12.11 even 2
666.2.s.a.397.1 4 444.323 even 6
2738.2.a.e.1.1 2 148.103 even 12
2738.2.a.i.1.1 2 148.119 even 12