Properties

Label 588.6.a.m.1.4
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 1.4
Root \(-5.36093\) 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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +77.8114 q^{5} +81.0000 q^{9} +58.2662 q^{11} -792.502 q^{13} -700.302 q^{15} +464.641 q^{17} +984.556 q^{19} -635.942 q^{23} +2929.61 q^{25} -729.000 q^{27} +6396.56 q^{29} +8428.62 q^{31} -524.396 q^{33} -1404.96 q^{37} +7132.52 q^{39} -19074.6 q^{41} +6023.14 q^{43} +6302.72 q^{45} -1880.94 q^{47} -4181.77 q^{51} +5255.39 q^{53} +4533.77 q^{55} -8861.01 q^{57} -3112.76 q^{59} +22094.5 q^{61} -61665.6 q^{65} -54711.8 q^{67} +5723.48 q^{69} -15211.7 q^{71} +50076.6 q^{73} -26366.5 q^{75} +1142.17 q^{79} +6561.00 q^{81} +122870. q^{83} +36154.3 q^{85} -57569.0 q^{87} -70492.6 q^{89} -75857.6 q^{93} +76609.7 q^{95} +132975. q^{97} +4719.56 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9} + 1872 q^{17} + 1728 q^{19} - 3648 q^{23} - 3996 q^{25} - 2916 q^{27} - 1248 q^{29} + 3888 q^{31} - 12032 q^{37} - 9072 q^{41} - 2128 q^{43} + 19872 q^{47} - 16848 q^{51} - 22248 q^{53}+ \cdots + 320544 q^{97}+O(q^{100}) \) 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
\(5\) 77.8114 1.39193 0.695966 − 0.718075i \(-0.254976\pi\)
0.695966 + 0.718075i \(0.254976\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 58.2662 0.145189 0.0725947 − 0.997362i \(-0.476872\pi\)
0.0725947 + 0.997362i \(0.476872\pi\)
\(12\) 0 0
\(13\) −792.502 −1.30059 −0.650297 − 0.759680i \(-0.725356\pi\)
−0.650297 + 0.759680i \(0.725356\pi\)
\(14\) 0 0
\(15\) −700.302 −0.803632
\(16\) 0 0
\(17\) 464.641 0.389937 0.194969 − 0.980809i \(-0.437539\pi\)
0.194969 + 0.980809i \(0.437539\pi\)
\(18\) 0 0
\(19\) 984.556 0.625686 0.312843 − 0.949805i \(-0.398719\pi\)
0.312843 + 0.949805i \(0.398719\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −635.942 −0.250667 −0.125334 − 0.992115i \(-0.540000\pi\)
−0.125334 + 0.992115i \(0.540000\pi\)
\(24\) 0 0
\(25\) 2929.61 0.937474
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 6396.56 1.41238 0.706189 − 0.708023i \(-0.250413\pi\)
0.706189 + 0.708023i \(0.250413\pi\)
\(30\) 0 0
\(31\) 8428.62 1.57526 0.787630 − 0.616148i \(-0.211308\pi\)
0.787630 + 0.616148i \(0.211308\pi\)
\(32\) 0 0
\(33\) −524.396 −0.0838251
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1404.96 −0.168717 −0.0843585 − 0.996435i \(-0.526884\pi\)
−0.0843585 + 0.996435i \(0.526884\pi\)
\(38\) 0 0
\(39\) 7132.52 0.750898
\(40\) 0 0
\(41\) −19074.6 −1.77213 −0.886066 − 0.463559i \(-0.846572\pi\)
−0.886066 + 0.463559i \(0.846572\pi\)
\(42\) 0 0
\(43\) 6023.14 0.496766 0.248383 − 0.968662i \(-0.420101\pi\)
0.248383 + 0.968662i \(0.420101\pi\)
\(44\) 0 0
\(45\) 6302.72 0.463977
\(46\) 0 0
\(47\) −1880.94 −0.124203 −0.0621013 − 0.998070i \(-0.519780\pi\)
−0.0621013 + 0.998070i \(0.519780\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4181.77 −0.225130
\(52\) 0 0
\(53\) 5255.39 0.256990 0.128495 − 0.991710i \(-0.458985\pi\)
0.128495 + 0.991710i \(0.458985\pi\)
\(54\) 0 0
\(55\) 4533.77 0.202094
\(56\) 0 0
\(57\) −8861.01 −0.361240
\(58\) 0 0
\(59\) −3112.76 −0.116417 −0.0582083 − 0.998304i \(-0.518539\pi\)
−0.0582083 + 0.998304i \(0.518539\pi\)
\(60\) 0 0
\(61\) 22094.5 0.760256 0.380128 − 0.924934i \(-0.375880\pi\)
0.380128 + 0.924934i \(0.375880\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −61665.6 −1.81034
\(66\) 0 0
\(67\) −54711.8 −1.48900 −0.744498 − 0.667624i \(-0.767311\pi\)
−0.744498 + 0.667624i \(0.767311\pi\)
\(68\) 0 0
\(69\) 5723.48 0.144723
\(70\) 0 0
\(71\) −15211.7 −0.358122 −0.179061 − 0.983838i \(-0.557306\pi\)
−0.179061 + 0.983838i \(0.557306\pi\)
\(72\) 0 0
\(73\) 50076.6 1.09983 0.549917 − 0.835219i \(-0.314659\pi\)
0.549917 + 0.835219i \(0.314659\pi\)
\(74\) 0 0
\(75\) −26366.5 −0.541251
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1142.17 0.0205902 0.0102951 − 0.999947i \(-0.496723\pi\)
0.0102951 + 0.999947i \(0.496723\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 122870. 1.95772 0.978859 − 0.204537i \(-0.0655689\pi\)
0.978859 + 0.204537i \(0.0655689\pi\)
\(84\) 0 0
\(85\) 36154.3 0.542766
\(86\) 0 0
\(87\) −57569.0 −0.815437
\(88\) 0 0
\(89\) −70492.6 −0.943341 −0.471670 − 0.881775i \(-0.656349\pi\)
−0.471670 + 0.881775i \(0.656349\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −75857.6 −0.909477
\(94\) 0 0
\(95\) 76609.7 0.870912
\(96\) 0 0
\(97\) 132975. 1.43497 0.717484 − 0.696575i \(-0.245294\pi\)
0.717484 + 0.696575i \(0.245294\pi\)
\(98\) 0 0
\(99\) 4719.56 0.0483965
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.1.4 ✓ 4
7.2 even 3 588.6.i.q.361.1 8
7.3 odd 6 588.6.i.p.373.4 8
7.4 even 3 588.6.i.q.373.1 8
7.5 odd 6 588.6.i.p.361.4 8
7.6 odd 2 588.6.a.o.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.m.1.4 ✓ 4 1.1 even 1 trivial
588.6.a.o.1.1 yes 4 7.6 odd 2
588.6.i.p.361.4 8 7.5 odd 6
588.6.i.p.373.4 8 7.3 odd 6
588.6.i.q.361.1 8 7.2 even 3
588.6.i.q.373.1 8 7.4 even 3