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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 529.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 592.529
Dual form 592.2.w.e.545.1

$q$-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 \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

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

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
\(3\) −0.366025 + 0.633975i −0.211325 + 0.366025i −0.952129 − 0.305695i \(-0.901111\pi\)
0.740805 + 0.671721i \(0.234444\pi\)
\(4\) 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
\(9\) 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 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) 0 0
\(15\) −1.09808 + 0.633975i −0.283522 + 0.163692i
\(16\) 0 0
\(17\) −3.69615 + 2.13397i −0.896449 + 0.517565i −0.876046 − 0.482227i \(-0.839828\pi\)
−0.0204023 + 0.999792i \(0.506495\pi\)
\(18\) 0 0
\(19\) 4.09808 + 2.36603i 0.940163 + 0.542803i 0.890011 − 0.455938i \(-0.150696\pi\)
0.0501517 + 0.998742i \(0.484030\pi\)
\(20\) 0 0
\(21\) −0.732051 − 1.26795i −0.159747 − 0.276689i
\(22\) 0 0
\(23\) 1.26795i 0.264386i 0.991224 + 0.132193i \(0.0422018\pi\)
−0.991224 + 0.132193i \(0.957798\pi\)
\(24\) 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
\(31\) − 4.73205i − 0.849901i −0.905216 − 0.424951i \(-0.860291\pi\)
0.905216 − 0.424951i \(-0.139709\pi\)
\(32\) 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 −0.670906 − 0.741542i \(-0.734095\pi\)
0.977647 + 0.210251i \(0.0674281\pi\)
\(42\) 0 0
\(43\) 12.9282i 1.97153i 0.168122 + 0.985766i \(0.446230\pi\)
−0.168122 + 0.985766i \(0.553770\pi\)
\(44\) 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 −0.983123 − 0.182946i \(-0.941437\pi\)
0.333126 − 0.942882i \(-0.391897\pi\)
\(54\) 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 0.139675 − 0.990197i \(-0.455394\pi\)
0.927373 + 0.374137i \(0.122061\pi\)
\(60\) 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
\(63\) −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 −0.859972 − 0.510341i \(-0.829519\pi\)
0.871954 + 0.489587i \(0.162853\pi\)
\(68\) 0 0
\(69\) −0.803848 − 0.464102i −0.0967719 − 0.0558713i
\(70\) 0 0
\(71\) −1.73205 + 3.00000i −0.205557 + 0.356034i −0.950310 − 0.311305i \(-0.899234\pi\)
0.744753 + 0.667340i \(0.232567\pi\)
\(72\) 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 −0.984001 − 0.178163i \(-0.942985\pi\)
−0.646294 − 0.763088i \(-0.723682\pi\)
\(80\) 0 0
\(81\) −2.23205 + 3.86603i −0.248006 + 0.429558i
\(82\) 0 0
\(83\) 5.83013 + 10.0981i 0.639940 + 1.10841i 0.985446 + 0.169991i \(0.0543740\pi\)
−0.345506 + 0.938417i \(0.612293\pi\)
\(84\) 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 0.583523 − 0.812096i \(-0.301674\pi\)
0.995058 − 0.0992979i \(-0.0316597\pi\)
\(90\) 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 \(-0.930480\pi\)
0.976244 − 0.216672i \(-0.0695203\pi\)
\(98\) 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.529.1 4
4.3 odd 2 74.2.e.b.11.1 ✓ 4
12.11 even 2 666.2.s.a.307.2 4
37.27 even 6 inner 592.2.w.e.545.1 4
148.27 odd 6 74.2.e.b.27.1 yes 4
148.103 even 12 2738.2.a.i.1.2 2
148.119 even 12 2738.2.a.e.1.2 2
444.323 even 6 666.2.s.a.397.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 4.3 odd 2
74.2.e.b.27.1 yes 4 148.27 odd 6
592.2.w.e.529.1 4 1.1 even 1 trivial
592.2.w.e.545.1 4 37.27 even 6 inner
666.2.s.a.307.2 4 12.11 even 2
666.2.s.a.397.2 4 444.323 even 6
2738.2.a.e.1.2 2 148.119 even 12
2738.2.a.i.1.2 2 148.103 even 12