Properties

Label 588.6.a.m.1.2
Level $588$
Weight $6$
Character 588.1
Self dual yes
Analytic conductor $94.306$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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] = [588,6,Mod(1,588)] 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("588.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-36,0,0,0,0,0,324,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{177 +28 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 87x^{2} + 88x + 1838 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.85863\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -33.4054 q^{5} +81.0000 q^{9} -707.705 q^{11} -83.0563 q^{13} +300.648 q^{15} -512.294 q^{17} -1034.68 q^{19} +565.691 q^{23} -2009.08 q^{25} -729.000 q^{27} +81.1930 q^{29} +2271.84 q^{31} +6369.35 q^{33} -10158.9 q^{37} +747.506 q^{39} -4261.21 q^{41} -18345.8 q^{43} -2705.84 q^{45} +22503.9 q^{47} +4610.65 q^{51} -3595.71 q^{53} +23641.2 q^{55} +9312.13 q^{57} -20014.3 q^{59} -32936.0 q^{61} +2774.53 q^{65} -53990.4 q^{67} -5091.22 q^{69} -130.028 q^{71} -31453.2 q^{73} +18081.7 q^{75} +46268.6 q^{79} +6561.00 q^{81} +85560.9 q^{83} +17113.4 q^{85} -730.737 q^{87} +20643.2 q^{89} -20446.6 q^{93} +34563.9 q^{95} +60446.8 q^{97} -57324.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9} + 1872 q^{17} + 1728 q^{19} - 3648 q^{23} - 3996 q^{25} - 2916 q^{27} - 1248 q^{29} + 3888 q^{31} - 12032 q^{37} - 9072 q^{41} - 2128 q^{43} + 19872 q^{47} - 16848 q^{51} - 22248 q^{53}+ \cdots + 320544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −33.4054 −0.597574 −0.298787 − 0.954320i \(-0.596582\pi\)
−0.298787 + 0.954320i \(0.596582\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −707.705 −1.76348 −0.881740 − 0.471735i \(-0.843628\pi\)
−0.881740 + 0.471735i \(0.843628\pi\)
\(12\) 0 0
\(13\) −83.0563 −0.136306 −0.0681529 − 0.997675i \(-0.521711\pi\)
−0.0681529 + 0.997675i \(0.521711\pi\)
\(14\) 0 0
\(15\) 300.648 0.345009
\(16\) 0 0
\(17\) −512.294 −0.429929 −0.214965 − 0.976622i \(-0.568964\pi\)
−0.214965 + 0.976622i \(0.568964\pi\)
\(18\) 0 0
\(19\) −1034.68 −0.657540 −0.328770 − 0.944410i \(-0.606634\pi\)
−0.328770 + 0.944410i \(0.606634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 565.691 0.222977 0.111488 − 0.993766i \(-0.464438\pi\)
0.111488 + 0.993766i \(0.464438\pi\)
\(24\) 0 0
\(25\) −2009.08 −0.642906
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 81.1930 0.0179277 0.00896383 − 0.999960i \(-0.497147\pi\)
0.00896383 + 0.999960i \(0.497147\pi\)
\(30\) 0 0
\(31\) 2271.84 0.424594 0.212297 − 0.977205i \(-0.431906\pi\)
0.212297 + 0.977205i \(0.431906\pi\)
\(32\) 0 0
\(33\) 6369.35 1.01815
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10158.9 −1.21995 −0.609975 − 0.792421i \(-0.708821\pi\)
−0.609975 + 0.792421i \(0.708821\pi\)
\(38\) 0 0
\(39\) 747.506 0.0786961
\(40\) 0 0
\(41\) −4261.21 −0.395888 −0.197944 − 0.980213i \(-0.563426\pi\)
−0.197944 + 0.980213i \(0.563426\pi\)
\(42\) 0 0
\(43\) −18345.8 −1.51310 −0.756548 − 0.653938i \(-0.773116\pi\)
−0.756548 + 0.653938i \(0.773116\pi\)
\(44\) 0 0
\(45\) −2705.84 −0.199191
\(46\) 0 0
\(47\) 22503.9 1.48598 0.742991 − 0.669302i \(-0.233407\pi\)
0.742991 + 0.669302i \(0.233407\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4610.65 0.248220
\(52\) 0 0
\(53\) −3595.71 −0.175831 −0.0879154 − 0.996128i \(-0.528021\pi\)
−0.0879154 + 0.996128i \(0.528021\pi\)
\(54\) 0 0
\(55\) 23641.2 1.05381
\(56\) 0 0
\(57\) 9312.13 0.379631
\(58\) 0 0
\(59\) −20014.3 −0.748530 −0.374265 − 0.927322i \(-0.622105\pi\)
−0.374265 + 0.927322i \(0.622105\pi\)
\(60\) 0 0
\(61\) −32936.0 −1.13330 −0.566652 − 0.823957i \(-0.691762\pi\)
−0.566652 + 0.823957i \(0.691762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2774.53 0.0814527
\(66\) 0 0
\(67\) −53990.4 −1.46937 −0.734683 − 0.678411i \(-0.762669\pi\)
−0.734683 + 0.678411i \(0.762669\pi\)
\(68\) 0 0
\(69\) −5091.22 −0.128736
\(70\) 0 0
\(71\) −130.028 −0.00306119 −0.00153060 − 0.999999i \(-0.500487\pi\)
−0.00153060 + 0.999999i \(0.500487\pi\)
\(72\) 0 0
\(73\) −31453.2 −0.690808 −0.345404 − 0.938454i \(-0.612258\pi\)
−0.345404 + 0.938454i \(0.612258\pi\)
\(74\) 0 0
\(75\) 18081.7 0.371182
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 46268.6 0.834100 0.417050 − 0.908884i \(-0.363064\pi\)
0.417050 + 0.908884i \(0.363064\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 85560.9 1.36326 0.681632 − 0.731695i \(-0.261270\pi\)
0.681632 + 0.731695i \(0.261270\pi\)
\(84\) 0 0
\(85\) 17113.4 0.256914
\(86\) 0 0
\(87\) −730.737 −0.0103505
\(88\) 0 0
\(89\) 20643.2 0.276250 0.138125 − 0.990415i \(-0.455892\pi\)
0.138125 + 0.990415i \(0.455892\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −20446.6 −0.245139
\(94\) 0 0
\(95\) 34563.9 0.392929
\(96\) 0 0
\(97\) 60446.8 0.652295 0.326147 − 0.945319i \(-0.394249\pi\)
0.326147 + 0.945319i \(0.394249\pi\)
\(98\) 0 0
\(99\) −57324.1 −0.587827
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 588.6.a.m.1.2 ✓ 4
7.2 even 3 588.6.i.q.361.3 8
7.3 odd 6 588.6.i.p.373.2 8
7.4 even 3 588.6.i.q.373.3 8
7.5 odd 6 588.6.i.p.361.2 8
7.6 odd 2 588.6.a.o.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.m.1.2 ✓ 4 1.1 even 1 trivial
588.6.a.o.1.3 yes 4 7.6 odd 2
588.6.i.p.361.2 8 7.5 odd 6
588.6.i.p.373.2 8 7.3 odd 6
588.6.i.q.361.3 8 7.2 even 3
588.6.i.q.373.3 8 7.4 even 3