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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 529.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 592.529
Dual form 592.2.w.d.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)
 
\(f(q)\) \(=\) \(q+(1.36603 - 2.36603i) q^{3} +(-1.50000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(-2.23205 - 3.86603i) q^{9} -1.26795 q^{11} +(5.19615 + 3.00000i) q^{13} +(-4.09808 + 2.36603i) q^{15} +(-1.50000 + 0.866025i) q^{17} +(-4.09808 - 2.36603i) q^{19} +(-5.46410 - 9.46410i) q^{21} +4.73205i q^{23} +(-1.00000 - 1.73205i) q^{25} -4.00000 q^{27} +7.73205i q^{29} -4.73205i q^{31} +(-1.73205 + 3.00000i) q^{33} +(-6.00000 + 3.46410i) q^{35} +(-0.500000 - 6.06218i) q^{37} +(14.1962 - 8.19615i) q^{39} +(3.23205 - 5.59808i) q^{41} +2.53590i q^{43} +7.73205i q^{45} +5.66025 q^{47} +(-4.50000 - 7.79423i) q^{49} +4.73205i q^{51} +(4.73205 + 8.19615i) q^{53} +(1.90192 + 1.09808i) q^{55} +(-11.1962 + 6.46410i) q^{57} +(-8.19615 + 4.73205i) q^{59} +(2.30385 + 1.33013i) q^{61} -17.8564 q^{63} +(-5.19615 - 9.00000i) q^{65} +(-2.09808 + 3.63397i) q^{67} +(11.1962 + 6.46410i) q^{69} +(4.73205 - 8.19615i) q^{71} +8.39230 q^{73} -5.46410 q^{75} +(-2.53590 + 4.39230i) q^{77} +(1.90192 + 1.09808i) q^{79} +(1.23205 - 2.13397i) q^{81} +(-2.83013 - 4.90192i) q^{83} +3.00000 q^{85} +(18.2942 + 10.5622i) q^{87} +(4.50000 - 2.59808i) q^{89} +(20.7846 - 12.0000i) q^{91} +(-11.1962 - 6.46410i) q^{93} +(4.09808 + 7.09808i) q^{95} +5.19615i q^{97} +(2.83013 + 4.90192i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{5} + 8 q^{7} - 2 q^{9} - 12 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 8 q^{21} - 4 q^{25} - 16 q^{27} - 24 q^{35} - 2 q^{37} + 36 q^{39} + 6 q^{41} - 12 q^{47} - 18 q^{49} + 12 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) 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 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 2.00000 − 3.46410i 0.755929 − 1.30931i −0.188982 − 0.981981i \(-0.560519\pi\)
0.944911 − 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) −2.23205 − 3.86603i −0.744017 − 1.28868i
\(10\) 0 0
\(11\) −1.26795 −0.382301 −0.191151 − 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) 0 0
\(13\) 5.19615 + 3.00000i 1.44115 + 0.832050i 0.997927 − 0.0643593i \(-0.0205004\pi\)
0.443227 + 0.896410i \(0.353834\pi\)
\(14\) 0 0
\(15\) −4.09808 + 2.36603i −1.05812 + 0.610905i
\(16\) 0 0
\(17\) −1.50000 + 0.866025i −0.363803 + 0.210042i −0.670748 − 0.741685i \(-0.734027\pi\)
0.306944 + 0.951727i \(0.400693\pi\)
\(18\) 0 0
\(19\) −4.09808 − 2.36603i −0.940163 − 0.542803i −0.0501517 − 0.998742i \(-0.515970\pi\)
−0.890011 + 0.455938i \(0.849304\pi\)
\(20\) 0 0
\(21\) −5.46410 − 9.46410i −1.19236 − 2.06524i
\(22\) 0 0
\(23\) 4.73205i 0.986701i 0.869831 + 0.493350i \(0.164228\pi\)
−0.869831 + 0.493350i \(0.835772\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 7.73205i 1.43581i 0.696143 + 0.717903i \(0.254898\pi\)
−0.696143 + 0.717903i \(0.745102\pi\)
\(30\) 0 0
\(31\) − 4.73205i − 0.849901i −0.905216 − 0.424951i \(-0.860291\pi\)
0.905216 − 0.424951i \(-0.139709\pi\)
\(32\) 0 0
\(33\) −1.73205 + 3.00000i −0.301511 + 0.522233i
\(34\) 0 0
\(35\) −6.00000 + 3.46410i −1.01419 + 0.585540i
\(36\) 0 0
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) 0 0
\(39\) 14.1962 − 8.19615i 2.27320 − 1.31243i
\(40\) 0 0
\(41\) 3.23205 − 5.59808i 0.504762 − 0.874273i −0.495223 − 0.868766i \(-0.664914\pi\)
0.999985 − 0.00550690i \(-0.00175291\pi\)
\(42\) 0 0
\(43\) 2.53590i 0.386721i 0.981128 + 0.193360i \(0.0619387\pi\)
−0.981128 + 0.193360i \(0.938061\pi\)
\(44\) 0 0
\(45\) 7.73205i 1.15263i
\(46\) 0 0
\(47\) 5.66025 0.825633 0.412816 − 0.910814i \(-0.364545\pi\)
0.412816 + 0.910814i \(0.364545\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 0 0
\(51\) 4.73205i 0.662620i
\(52\) 0 0
\(53\) 4.73205 + 8.19615i 0.649997 + 1.12583i 0.983123 + 0.182946i \(0.0585633\pi\)
−0.333126 + 0.942882i \(0.608103\pi\)
\(54\) 0 0
\(55\) 1.90192 + 1.09808i 0.256455 + 0.148065i
\(56\) 0 0
\(57\) −11.1962 + 6.46410i −1.48297 + 0.856191i
\(58\) 0 0
\(59\) −8.19615 + 4.73205i −1.06705 + 0.616061i −0.927373 − 0.374137i \(-0.877939\pi\)
−0.139675 + 0.990197i \(0.544606\pi\)
\(60\) 0 0
\(61\) 2.30385 + 1.33013i 0.294977 + 0.170305i 0.640184 − 0.768221i \(-0.278858\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) −17.8564 −2.24970
\(64\) 0 0
\(65\) −5.19615 − 9.00000i −0.644503 − 1.11631i
\(66\) 0 0
\(67\) −2.09808 + 3.63397i −0.256321 + 0.443961i −0.965253 − 0.261316i \(-0.915844\pi\)
0.708933 + 0.705276i \(0.249177\pi\)
\(68\) 0 0
\(69\) 11.1962 + 6.46410i 1.34786 + 0.778186i
\(70\) 0 0
\(71\) 4.73205 − 8.19615i 0.561591 − 0.972704i −0.435767 − 0.900060i \(-0.643523\pi\)
0.997358 − 0.0726447i \(-0.0231439\pi\)
\(72\) 0 0
\(73\) 8.39230 0.982245 0.491122 − 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(74\) 0 0
\(75\) −5.46410 −0.630940
\(76\) 0 0
\(77\) −2.53590 + 4.39230i −0.288992 + 0.500550i
\(78\) 0 0
\(79\) 1.90192 + 1.09808i 0.213983 + 0.123543i 0.603161 − 0.797619i \(-0.293908\pi\)
−0.389178 + 0.921163i \(0.627241\pi\)
\(80\) 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
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 18.2942 + 10.5622i 1.96135 + 1.13238i
\(88\) 0 0
\(89\) 4.50000 − 2.59808i 0.476999 − 0.275396i −0.242166 − 0.970235i \(-0.577858\pi\)
0.719165 + 0.694839i \(0.244525\pi\)
\(90\) 0 0
\(91\) 20.7846 − 12.0000i 2.17882 − 1.25794i
\(92\) 0 0
\(93\) −11.1962 − 6.46410i −1.16099 − 0.670296i
\(94\) 0 0
\(95\) 4.09808 + 7.09808i 0.420454 + 0.728247i
\(96\) 0 0
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 2.83013 + 4.90192i 0.284438 + 0.492662i
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.d.529.2 4
4.3 odd 2 74.2.e.a.11.1 ✓ 4
12.11 even 2 666.2.s.e.307.2 4
37.27 even 6 inner 592.2.w.d.545.2 4
148.27 odd 6 74.2.e.a.27.1 yes 4
148.103 even 12 2738.2.a.j.1.1 2
148.119 even 12 2738.2.a.f.1.1 2
444.323 even 6 666.2.s.e.397.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.1 ✓ 4 4.3 odd 2
74.2.e.a.27.1 yes 4 148.27 odd 6
592.2.w.d.529.2 4 1.1 even 1 trivial
592.2.w.d.545.2 4 37.27 even 6 inner
666.2.s.e.307.2 4 12.11 even 2
666.2.s.e.397.2 4 444.323 even 6
2738.2.a.f.1.1 2 148.119 even 12
2738.2.a.j.1.1 2 148.103 even 12