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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 545.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 592.545
Dual form 592.2.w.e.529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.366025 - 0.633975i) q^{3} +(1.50000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(1.23205 - 2.13397i) q^{9} +1.26795 q^{11} +(-3.00000 + 1.73205i) q^{13} +(-1.09808 - 0.633975i) q^{15} +(-3.69615 - 2.13397i) q^{17} +(4.09808 - 2.36603i) q^{19} +(-0.732051 + 1.26795i) q^{21} -1.26795i q^{23} +(-1.00000 + 1.73205i) q^{25} -4.00000 q^{27} -8.66025i q^{29} +4.73205i q^{31} +(-0.464102 - 0.803848i) q^{33} +(-3.00000 - 1.73205i) q^{35} +(4.69615 - 3.86603i) q^{37} +(2.19615 + 1.26795i) q^{39} +(1.96410 + 3.40192i) q^{41} -12.9282i q^{43} -4.26795i q^{45} -1.26795 q^{47} +(1.50000 - 2.59808i) q^{49} +3.12436i q^{51} +(-4.73205 + 8.19615i) q^{53} +(1.90192 - 1.09808i) q^{55} +(-3.00000 - 1.73205i) q^{57} +(8.19615 + 4.73205i) q^{59} +(1.50000 - 0.866025i) q^{61} -4.92820 q^{63} +(-3.00000 + 5.19615i) q^{65} +(0.0980762 + 0.169873i) q^{67} +(-0.803848 + 0.464102i) q^{69} +(-1.73205 - 3.00000i) q^{71} +4.00000 q^{73} +1.46410 q^{75} +(-1.26795 - 2.19615i) q^{77} +(-14.4904 + 8.36603i) q^{79} +(-2.23205 - 3.86603i) q^{81} +(5.83013 - 10.0981i) q^{83} -7.39230 q^{85} +(-5.49038 + 3.16987i) q^{87} +(14.8923 + 8.59808i) q^{89} +(6.00000 + 3.46410i) q^{91} +(3.00000 - 1.73205i) q^{93} +(4.09808 - 7.09808i) q^{95} +4.26795i q^{97} +(1.56218 - 2.70577i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} - 2 q^{9} + 12 q^{11} - 12 q^{13} + 6 q^{15} + 6 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{25} - 16 q^{27} + 12 q^{33} - 12 q^{35} - 2 q^{37} - 12 q^{39} - 6 q^{41} - 12 q^{47}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −0.366025 − 0.633975i −0.211325 − 0.366025i 0.740805 − 0.671721i \(-0.234444\pi\)
−0.952129 + 0.305695i \(0.901111\pi\)
\(4\) 0 0
\(5\) 1.50000 − 0.866025i 0.670820 − 0.387298i −0.125567 − 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) −1.00000 − 1.73205i −0.377964 − 0.654654i 0.612801 − 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 1.23205 − 2.13397i 0.410684 − 0.711325i
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) −3.00000 + 1.73205i −0.832050 + 0.480384i −0.854554 − 0.519362i \(-0.826170\pi\)
0.0225039 + 0.999747i \(0.492836\pi\)
\(14\) 0 0
\(15\) −1.09808 − 0.633975i −0.283522 − 0.163692i
\(16\) 0 0
\(17\) −3.69615 − 2.13397i −0.896449 − 0.517565i −0.0204023 − 0.999792i \(-0.506495\pi\)
−0.876046 + 0.482227i \(0.839828\pi\)
\(18\) 0 0
\(19\) 4.09808 − 2.36603i 0.940163 − 0.542803i 0.0501517 − 0.998742i \(-0.484030\pi\)
0.890011 + 0.455938i \(0.150696\pi\)
\(20\) 0 0
\(21\) −0.732051 + 1.26795i −0.159747 + 0.276689i
\(22\) 0 0
\(23\) − 1.26795i − 0.264386i −0.991224 − 0.132193i \(-0.957798\pi\)
0.991224 − 0.132193i \(-0.0422018\pi\)
\(24\) 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 \(-0.702656\pi\)
0.594515 − 0.804084i \(-0.297344\pi\)
\(30\) 0 0
\(31\) 4.73205i 0.849901i 0.905216 + 0.424951i \(0.139709\pi\)
−0.905216 + 0.424951i \(0.860291\pi\)
\(32\) 0 0
\(33\) −0.464102 − 0.803848i −0.0807897 − 0.139932i
\(34\) 0 0
\(35\) −3.00000 − 1.73205i −0.507093 − 0.292770i
\(36\) 0 0
\(37\) 4.69615 − 3.86603i 0.772043 − 0.635571i
\(38\) 0 0
\(39\) 2.19615 + 1.26795i 0.351666 + 0.203034i
\(40\) 0 0
\(41\) 1.96410 + 3.40192i 0.306741 + 0.531291i 0.977647 − 0.210251i \(-0.0674281\pi\)
−0.670906 + 0.741542i \(0.734095\pi\)
\(42\) 0 0
\(43\) − 12.9282i − 1.97153i −0.168122 − 0.985766i \(-0.553770\pi\)
0.168122 − 0.985766i \(-0.446230\pi\)
\(44\) 0 0
\(45\) − 4.26795i − 0.636228i
\(46\) 0 0
\(47\) −1.26795 −0.184949 −0.0924747 − 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 1.50000 − 2.59808i 0.214286 − 0.371154i
\(50\) 0 0
\(51\) 3.12436i 0.437497i
\(52\) 0 0
\(53\) −4.73205 + 8.19615i −0.649997 + 1.12583i 0.333126 + 0.942882i \(0.391897\pi\)
−0.983123 + 0.182946i \(0.941437\pi\)
\(54\) 0 0
\(55\) 1.90192 − 1.09808i 0.256455 − 0.148065i
\(56\) 0 0
\(57\) −3.00000 − 1.73205i −0.397360 − 0.229416i
\(58\) 0 0
\(59\) 8.19615 + 4.73205i 1.06705 + 0.616061i 0.927373 − 0.374137i \(-0.122061\pi\)
0.139675 + 0.990197i \(0.455394\pi\)
\(60\) 0 0
\(61\) 1.50000 − 0.866025i 0.192055 − 0.110883i −0.400889 − 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) 0 0
\(63\) −4.92820 −0.620895
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) 0.0980762 + 0.169873i 0.0119819 + 0.0207533i 0.871954 − 0.489587i \(-0.162853\pi\)
−0.859972 + 0.510341i \(0.829519\pi\)
\(68\) 0 0
\(69\) −0.803848 + 0.464102i −0.0967719 + 0.0558713i
\(70\) 0 0
\(71\) −1.73205 − 3.00000i −0.205557 − 0.356034i 0.744753 − 0.667340i \(-0.232567\pi\)
−0.950310 + 0.311305i \(0.899234\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 1.46410 0.169060
\(76\) 0 0
\(77\) −1.26795 − 2.19615i −0.144496 − 0.250275i
\(78\) 0 0
\(79\) −14.4904 + 8.36603i −1.63030 + 0.941251i −0.646294 + 0.763088i \(0.723682\pi\)
−0.984001 + 0.178163i \(0.942985\pi\)
\(80\) 0 0
\(81\) −2.23205 − 3.86603i −0.248006 − 0.429558i
\(82\) 0 0
\(83\) 5.83013 − 10.0981i 0.639940 − 1.10841i −0.345506 − 0.938417i \(-0.612293\pi\)
0.985446 − 0.169991i \(-0.0543740\pi\)
\(84\) 0 0
\(85\) −7.39230 −0.801808
\(86\) 0 0
\(87\) −5.49038 + 3.16987i −0.588631 + 0.339846i
\(88\) 0 0
\(89\) 14.8923 + 8.59808i 1.57858 + 0.911394i 0.995058 + 0.0992979i \(0.0316597\pi\)
0.583523 + 0.812096i \(0.301674\pi\)
\(90\) 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
\(95\) 4.09808 − 7.09808i 0.420454 − 0.728247i
\(96\) 0 0
\(97\) 4.26795i 0.433345i 0.976244 + 0.216672i \(0.0695203\pi\)
−0.976244 + 0.216672i \(0.930480\pi\)
\(98\) 0 0
\(99\) 1.56218 − 2.70577i 0.157005 − 0.271940i
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.545.1 4
4.3 odd 2 74.2.e.b.27.1 yes 4
12.11 even 2 666.2.s.a.397.2 4
37.11 even 6 inner 592.2.w.e.529.1 4
148.11 odd 6 74.2.e.b.11.1 ✓ 4
148.23 even 12 2738.2.a.i.1.2 2
148.51 even 12 2738.2.a.e.1.2 2
444.11 even 6 666.2.s.a.307.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 148.11 odd 6
74.2.e.b.27.1 yes 4 4.3 odd 2
592.2.w.e.529.1 4 37.11 even 6 inner
592.2.w.e.545.1 4 1.1 even 1 trivial
666.2.s.a.307.2 4 444.11 even 6
666.2.s.a.397.2 4 12.11 even 2
2738.2.a.e.1.2 2 148.51 even 12
2738.2.a.i.1.2 2 148.23 even 12