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Properties

Label 588.8.a.a.1.1
Level $588$
Weight $8$
Character 588.1
Self dual yes
Analytic conductor $183.682$
Analytic rank $0$
Dimension $1$
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] = [588,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27,0,-270] 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: yes
Analytic conductor: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -270.000 q^{5} +729.000 q^{9} -5724.00 q^{11} +4570.00 q^{13} +7290.00 q^{15} +36558.0 q^{17} -51740.0 q^{19} +22248.0 q^{23} -5225.00 q^{25} -19683.0 q^{27} -157194. q^{29} +103936. q^{31} +154548. q^{33} -94834.0 q^{37} -123390. q^{39} -659610. q^{41} -75772.0 q^{43} -196830. q^{45} -405648. q^{47} -987066. q^{51} -1.34627e6 q^{53} +1.54548e6 q^{55} +1.39698e6 q^{57} +1.30388e6 q^{59} -1.83378e6 q^{61} -1.23390e6 q^{65} +1.36939e6 q^{67} -600696. q^{69} +2.71404e6 q^{71} -2.86879e6 q^{73} +141075. q^{75} -1.12965e6 q^{79} +531441. q^{81} -5.91203e6 q^{83} -9.87066e6 q^{85} +4.24424e6 q^{87} +897750. q^{89} -2.80627e6 q^{93} +1.39698e7 q^{95} -1.37191e7 q^{97} -4.17280e6 q^{99} +O(q^{100})\)

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
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −270.000 −0.965981 −0.482991 − 0.875625i \(-0.660450\pi\)
−0.482991 + 0.875625i \(0.660450\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −5724.00 −1.29666 −0.648329 − 0.761361i \(-0.724532\pi\)
−0.648329 + 0.761361i \(0.724532\pi\)
\(12\) 0 0
\(13\) 4570.00 0.576919 0.288459 − 0.957492i \(-0.406857\pi\)
0.288459 + 0.957492i \(0.406857\pi\)
\(14\) 0 0
\(15\) 7290.00 0.557710
\(16\) 0 0
\(17\) 36558.0 1.80473 0.902363 − 0.430977i \(-0.141831\pi\)
0.902363 + 0.430977i \(0.141831\pi\)
\(18\) 0 0
\(19\) −51740.0 −1.73057 −0.865284 − 0.501281i \(-0.832862\pi\)
−0.865284 + 0.501281i \(0.832862\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 22248.0 0.381280 0.190640 − 0.981660i \(-0.438944\pi\)
0.190640 + 0.981660i \(0.438944\pi\)
\(24\) 0 0
\(25\) −5225.00 −0.0668800
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −157194. −1.19686 −0.598429 − 0.801175i \(-0.704208\pi\)
−0.598429 + 0.801175i \(0.704208\pi\)
\(30\) 0 0
\(31\) 103936. 0.626614 0.313307 − 0.949652i \(-0.398563\pi\)
0.313307 + 0.949652i \(0.398563\pi\)
\(32\) 0 0
\(33\) 154548. 0.748625
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −94834.0 −0.307793 −0.153896 − 0.988087i \(-0.549182\pi\)
−0.153896 + 0.988087i \(0.549182\pi\)
\(38\) 0 0
\(39\) −123390. −0.333084
\(40\) 0 0
\(41\) −659610. −1.49466 −0.747332 − 0.664451i \(-0.768666\pi\)
−0.747332 + 0.664451i \(0.768666\pi\)
\(42\) 0 0
\(43\) −75772.0 −0.145335 −0.0726673 − 0.997356i \(-0.523151\pi\)
−0.0726673 + 0.997356i \(0.523151\pi\)
\(44\) 0 0
\(45\) −196830. −0.321994
\(46\) 0 0
\(47\) −405648. −0.569911 −0.284955 − 0.958541i \(-0.591979\pi\)
−0.284955 + 0.958541i \(0.591979\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −987066. −1.04196
\(52\) 0 0
\(53\) −1.34627e6 −1.24213 −0.621066 − 0.783758i \(-0.713300\pi\)
−0.621066 + 0.783758i \(0.713300\pi\)
\(54\) 0 0
\(55\) 1.54548e6 1.25255
\(56\) 0 0
\(57\) 1.39698e6 0.999144
\(58\) 0 0
\(59\) 1.30388e6 0.826527 0.413263 − 0.910612i \(-0.364389\pi\)
0.413263 + 0.910612i \(0.364389\pi\)
\(60\) 0 0
\(61\) −1.83378e6 −1.03441 −0.517206 − 0.855861i \(-0.673028\pi\)
−0.517206 + 0.855861i \(0.673028\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.23390e6 −0.557293
\(66\) 0 0
\(67\) 1.36939e6 0.556243 0.278122 − 0.960546i \(-0.410288\pi\)
0.278122 + 0.960546i \(0.410288\pi\)
\(68\) 0 0
\(69\) −600696. −0.220132
\(70\) 0 0
\(71\) 2.71404e6 0.899937 0.449968 − 0.893044i \(-0.351435\pi\)
0.449968 + 0.893044i \(0.351435\pi\)
\(72\) 0 0
\(73\) −2.86879e6 −0.863116 −0.431558 − 0.902085i \(-0.642036\pi\)
−0.431558 + 0.902085i \(0.642036\pi\)
\(74\) 0 0
\(75\) 141075. 0.0386132
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.12965e6 −0.257779 −0.128890 − 0.991659i \(-0.541141\pi\)
−0.128890 + 0.991659i \(0.541141\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −5.91203e6 −1.13491 −0.567457 − 0.823403i \(-0.692073\pi\)
−0.567457 + 0.823403i \(0.692073\pi\)
\(84\) 0 0
\(85\) −9.87066e6 −1.74333
\(86\) 0 0
\(87\) 4.24424e6 0.691007
\(88\) 0 0
\(89\) 897750. 0.134987 0.0674933 − 0.997720i \(-0.478500\pi\)
0.0674933 + 0.997720i \(0.478500\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.80627e6 −0.361776
\(94\) 0 0
\(95\) 1.39698e7 1.67170
\(96\) 0 0
\(97\) −1.37191e7 −1.52624 −0.763122 − 0.646255i \(-0.776334\pi\)
−0.763122 + 0.646255i \(0.776334\pi\)
\(98\) 0 0
\(99\) −4.17280e6 −0.432219
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 588.8.a.a.1.1 1
7.2 even 3 588.8.i.g.361.1 2
7.3 odd 6 588.8.i.b.373.1 2
7.4 even 3 588.8.i.g.373.1 2
7.5 odd 6 588.8.i.b.361.1 2
7.6 odd 2 12.8.a.b.1.1 ✓ 1
21.20 even 2 36.8.a.a.1.1 1
28.27 even 2 48.8.a.d.1.1 1
35.13 even 4 300.8.d.a.49.2 2
35.27 even 4 300.8.d.a.49.1 2
35.34 odd 2 300.8.a.a.1.1 1
56.13 odd 2 192.8.a.b.1.1 1
56.27 even 2 192.8.a.j.1.1 1
63.13 odd 6 324.8.e.b.217.1 2
63.20 even 6 324.8.e.e.109.1 2
63.34 odd 6 324.8.e.b.109.1 2
63.41 even 6 324.8.e.e.217.1 2
84.83 odd 2 144.8.a.c.1.1 1
168.83 odd 2 576.8.a.u.1.1 1
168.125 even 2 576.8.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.b.1.1 ✓ 1 7.6 odd 2
36.8.a.a.1.1 1 21.20 even 2
48.8.a.d.1.1 1 28.27 even 2
144.8.a.c.1.1 1 84.83 odd 2
192.8.a.b.1.1 1 56.13 odd 2
192.8.a.j.1.1 1 56.27 even 2
300.8.a.a.1.1 1 35.34 odd 2
300.8.d.a.49.1 2 35.27 even 4
300.8.d.a.49.2 2 35.13 even 4
324.8.e.b.109.1 2 63.34 odd 6
324.8.e.b.217.1 2 63.13 odd 6
324.8.e.e.109.1 2 63.20 even 6
324.8.e.e.217.1 2 63.41 even 6
576.8.a.u.1.1 1 168.83 odd 2
576.8.a.v.1.1 1 168.125 even 2
588.8.a.a.1.1 1 1.1 even 1 trivial
588.8.i.b.361.1 2 7.5 odd 6
588.8.i.b.373.1 2 7.3 odd 6
588.8.i.g.361.1 2 7.2 even 3
588.8.i.g.373.1 2 7.4 even 3