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Properties

Label 592.2.i.g.417.1
Level $592$
Weight $2$
Character 592.417
Analytic conductor $4.727$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(417,592)] 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("592.417"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.1
Root \(1.08504 + 1.87935i\) of defining polynomial
Character \(\chi\) \(=\) 592.417
Dual form 592.2.i.g.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.854638 - 1.48028i) q^{3} +(1.35464 + 2.34630i) q^{5} +(0.315449 + 0.546373i) q^{7} +(0.0391889 - 0.0678771i) q^{9} +(1.00000 + 1.73205i) q^{13} +(2.31545 - 4.01047i) q^{15} +(-2.13090 + 3.69082i) q^{17} +(2.63090 + 4.55685i) q^{19} +(0.539189 - 0.933903i) q^{21} +4.04945 q^{23} +(-1.17009 + 2.02665i) q^{25} -5.26180 q^{27} +7.04945 q^{29} +0.581449 q^{31} +(-0.854638 + 1.48028i) q^{35} +(6.06391 - 0.478496i) q^{37} +(1.70928 - 2.96055i) q^{39} +(-2.67009 - 4.62473i) q^{41} +7.86603 q^{43} +0.212347 q^{45} -3.41855 q^{47} +(3.30098 - 5.71747i) q^{49} +7.28458 q^{51} +(0.460811 - 0.798148i) q^{53} +(4.49693 - 7.78891i) q^{57} +(-2.63090 + 4.55685i) q^{59} +(-1.47528 - 2.55525i) q^{61} +0.0494483 q^{63} +(-2.70928 + 4.69260i) q^{65} +(5.41855 + 9.38521i) q^{67} +(-3.46081 - 5.99430i) q^{69} +(-4.65562 - 8.06377i) q^{71} -5.23513 q^{73} +4.00000 q^{75} +(1.55252 + 2.68904i) q^{79} +(4.37936 + 7.58528i) q^{81} +(-5.64229 + 9.77273i) q^{83} -11.5464 q^{85} +(-6.02472 - 10.4351i) q^{87} +(5.71953 - 9.90652i) q^{89} +(-0.630898 + 1.09275i) q^{91} +(-0.496928 - 0.860705i) q^{93} +(-7.12783 + 12.3458i) q^{95} -10.1773 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + q^{5} - 2 q^{7} - 3 q^{9} + 6 q^{13} + 10 q^{15} - 5 q^{17} + 8 q^{19} - 12 q^{23} + 4 q^{25} - 16 q^{27} + 6 q^{29} + 32 q^{31} + 2 q^{35} + 15 q^{37} - 4 q^{39} - 5 q^{41} + 20 q^{43}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −0.854638 − 1.48028i −0.493425 − 0.854638i 0.506546 − 0.862213i \(-0.330922\pi\)
−0.999971 + 0.00757525i \(0.997589\pi\)
\(4\) 0 0
\(5\) 1.35464 + 2.34630i 0.605812 + 1.04930i 0.991923 + 0.126845i \(0.0404851\pi\)
−0.386110 + 0.922453i \(0.626182\pi\)
\(6\) 0 0
\(7\) 0.315449 + 0.546373i 0.119228 + 0.206510i 0.919462 − 0.393179i \(-0.128625\pi\)
−0.800234 + 0.599688i \(0.795291\pi\)
\(8\) 0 0
\(9\) 0.0391889 − 0.0678771i 0.0130630 − 0.0226257i
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 − 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 2.31545 − 4.01047i 0.597846 − 1.03550i
\(16\) 0 0
\(17\) −2.13090 + 3.69082i −0.516819 + 0.895156i 0.482991 + 0.875625i \(0.339550\pi\)
−0.999809 + 0.0195306i \(0.993783\pi\)
\(18\) 0 0
\(19\) 2.63090 + 4.55685i 0.603569 + 1.04541i 0.992276 + 0.124051i \(0.0395887\pi\)
−0.388707 + 0.921362i \(0.627078\pi\)
\(20\) 0 0
\(21\) 0.539189 − 0.933903i 0.117661 − 0.203794i
\(22\) 0 0
\(23\) 4.04945 0.844368 0.422184 − 0.906510i \(-0.361264\pi\)
0.422184 + 0.906510i \(0.361264\pi\)
\(24\) 0 0
\(25\) −1.17009 + 2.02665i −0.234017 + 0.405330i
\(26\) 0 0
\(27\) −5.26180 −1.01263
\(28\) 0 0
\(29\) 7.04945 1.30905 0.654525 − 0.756040i \(-0.272869\pi\)
0.654525 + 0.756040i \(0.272869\pi\)
\(30\) 0 0
\(31\) 0.581449 0.104431 0.0522157 − 0.998636i \(-0.483372\pi\)
0.0522157 + 0.998636i \(0.483372\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.854638 + 1.48028i −0.144460 + 0.250212i
\(36\) 0 0
\(37\) 6.06391 − 0.478496i 0.996901 − 0.0786643i
\(38\) 0 0
\(39\) 1.70928 − 2.96055i 0.273703 − 0.474068i
\(40\) 0 0
\(41\) −2.67009 − 4.62473i −0.416997 − 0.722261i 0.578638 − 0.815584i \(-0.303584\pi\)
−0.995636 + 0.0933234i \(0.970251\pi\)
\(42\) 0 0
\(43\) 7.86603 1.19956 0.599779 − 0.800166i \(-0.295255\pi\)
0.599779 + 0.800166i \(0.295255\pi\)
\(44\) 0 0
\(45\) 0.212347 0.0316548
\(46\) 0 0
\(47\) −3.41855 −0.498647 −0.249323 − 0.968420i \(-0.580208\pi\)
−0.249323 + 0.968420i \(0.580208\pi\)
\(48\) 0 0
\(49\) 3.30098 − 5.71747i 0.471569 − 0.816782i
\(50\) 0 0
\(51\) 7.28458 1.02005
\(52\) 0 0
\(53\) 0.460811 − 0.798148i 0.0632973 − 0.109634i −0.832640 − 0.553814i \(-0.813172\pi\)
0.895937 + 0.444180i \(0.146505\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.49693 − 7.78891i 0.595633 − 1.03167i
\(58\) 0 0
\(59\) −2.63090 + 4.55685i −0.342514 + 0.593251i −0.984899 − 0.173131i \(-0.944612\pi\)
0.642385 + 0.766382i \(0.277945\pi\)
\(60\) 0 0
\(61\) −1.47528 − 2.55525i −0.188890 − 0.327167i 0.755991 − 0.654582i \(-0.227156\pi\)
−0.944880 + 0.327416i \(0.893822\pi\)
\(62\) 0 0
\(63\) 0.0494483 0.00622990
\(64\) 0 0
\(65\) −2.70928 + 4.69260i −0.336044 + 0.582046i
\(66\) 0 0
\(67\) 5.41855 + 9.38521i 0.661981 + 1.14659i 0.980094 + 0.198532i \(0.0636175\pi\)
−0.318113 + 0.948053i \(0.603049\pi\)
\(68\) 0 0
\(69\) −3.46081 − 5.99430i −0.416633 − 0.721629i
\(70\) 0 0
\(71\) −4.65562 − 8.06377i −0.552521 − 0.956994i −0.998092 − 0.0617473i \(-0.980333\pi\)
0.445571 − 0.895247i \(-0.353001\pi\)
\(72\) 0 0
\(73\) −5.23513 −0.612726 −0.306363 − 0.951915i \(-0.599112\pi\)
−0.306363 + 0.951915i \(0.599112\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.55252 + 2.68904i 0.174672 + 0.302541i 0.940048 − 0.341043i \(-0.110780\pi\)
−0.765376 + 0.643584i \(0.777447\pi\)
\(80\) 0 0
\(81\) 4.37936 + 7.58528i 0.486596 + 0.842809i
\(82\) 0 0
\(83\) −5.64229 + 9.77273i −0.619322 + 1.07270i 0.370288 + 0.928917i \(0.379259\pi\)
−0.989610 + 0.143780i \(0.954074\pi\)
\(84\) 0 0
\(85\) −11.5464 −1.25238
\(86\) 0 0
\(87\) −6.02472 − 10.4351i −0.645918 − 1.11876i
\(88\) 0 0
\(89\) 5.71953 − 9.90652i 0.606269 − 1.05009i −0.385580 − 0.922674i \(-0.625999\pi\)
0.991849 − 0.127415i \(-0.0406680\pi\)
\(90\) 0 0
\(91\) −0.630898 + 1.09275i −0.0661360 + 0.114551i
\(92\) 0 0
\(93\) −0.496928 − 0.860705i −0.0515291 − 0.0892510i
\(94\) 0 0
\(95\) −7.12783 + 12.3458i −0.731300 + 1.26665i
\(96\) 0 0
\(97\) −10.1773 −1.03335 −0.516673 − 0.856183i \(-0.672830\pi\)
−0.516673 + 0.856183i \(0.672830\pi\)
\(98\) 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 592.2.i.g.417.1 6
4.3 odd 2 296.2.i.b.121.3 ✓ 6
12.11 even 2 2664.2.r.i.1009.1 6
37.26 even 3 inner 592.2.i.g.433.1 6
148.63 odd 6 296.2.i.b.137.3 yes 6
444.359 even 6 2664.2.r.i.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.3 ✓ 6 4.3 odd 2
296.2.i.b.137.3 yes 6 148.63 odd 6
592.2.i.g.417.1 6 1.1 even 1 trivial
592.2.i.g.433.1 6 37.26 even 3 inner
2664.2.r.i.433.1 6 444.359 even 6
2664.2.r.i.1009.1 6 12.11 even 2