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 = [2,0,54,0,-264] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-29.7035\) 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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +230.442 q^{5} +729.000 q^{9} -5027.09 q^{11} +13772.6 q^{13} +6221.93 q^{15} -32474.7 q^{17} -9651.95 q^{19} +32034.5 q^{23} -25021.6 q^{25} +19683.0 q^{27} +103672. q^{29} -241936. q^{31} -135731. q^{33} -127147. q^{37} +371860. q^{39} -607138. q^{41} +443862. q^{43} +167992. q^{45} -691778. q^{47} -876817. q^{51} +672824. q^{53} -1.15845e6 q^{55} -260603. q^{57} +2.58646e6 q^{59} -1.53593e6 q^{61} +3.17378e6 q^{65} -4.20823e6 q^{67} +864931. q^{69} +1.51772e6 q^{71} -5.47779e6 q^{73} -675584. q^{75} -6.75855e6 q^{79} +531441. q^{81} -8.36612e6 q^{83} -7.48353e6 q^{85} +2.79914e6 q^{87} +5.17007e6 q^{89} -6.53227e6 q^{93} -2.22421e6 q^{95} +1.06286e7 q^{97} -3.66475e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{3} - 264 q^{5} + 1458 q^{9} - 4980 q^{11} + 10148 q^{13} - 7128 q^{15} - 17832 q^{17} - 6256 q^{19} + 14052 q^{23} + 141326 q^{25} + 39366 q^{27} + 243588 q^{29} - 470824 q^{31} - 134460 q^{33}+ \cdots - 3630420 q^{99}+O(q^{100}) \) 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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 230.442 0.824453 0.412227 − 0.911081i \(-0.364751\pi\)
0.412227 + 0.911081i \(0.364751\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −5027.09 −1.13879 −0.569393 − 0.822065i \(-0.692822\pi\)
−0.569393 + 0.822065i \(0.692822\pi\)
\(12\) 0 0
\(13\) 13772.6 1.73866 0.869329 − 0.494234i \(-0.164551\pi\)
0.869329 + 0.494234i \(0.164551\pi\)
\(14\) 0 0
\(15\) 6221.93 0.475998
\(16\) 0 0
\(17\) −32474.7 −1.60315 −0.801575 − 0.597894i \(-0.796004\pi\)
−0.801575 + 0.597894i \(0.796004\pi\)
\(18\) 0 0
\(19\) −9651.95 −0.322833 −0.161416 − 0.986886i \(-0.551606\pi\)
−0.161416 + 0.986886i \(0.551606\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 32034.5 0.548997 0.274499 − 0.961587i \(-0.411488\pi\)
0.274499 + 0.961587i \(0.411488\pi\)
\(24\) 0 0
\(25\) −25021.6 −0.320277
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 103672. 0.789347 0.394674 − 0.918821i \(-0.370858\pi\)
0.394674 + 0.918821i \(0.370858\pi\)
\(30\) 0 0
\(31\) −241936. −1.45859 −0.729297 − 0.684197i \(-0.760153\pi\)
−0.729297 + 0.684197i \(0.760153\pi\)
\(32\) 0 0
\(33\) −135731. −0.657479
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −127147. −0.412666 −0.206333 − 0.978482i \(-0.566153\pi\)
−0.206333 + 0.978482i \(0.566153\pi\)
\(38\) 0 0
\(39\) 371860. 1.00381
\(40\) 0 0
\(41\) −607138. −1.37576 −0.687882 − 0.725823i \(-0.741459\pi\)
−0.687882 + 0.725823i \(0.741459\pi\)
\(42\) 0 0
\(43\) 443862. 0.851350 0.425675 − 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(44\) 0 0
\(45\) 167992. 0.274818
\(46\) 0 0
\(47\) −691778. −0.971905 −0.485953 − 0.873985i \(-0.661527\pi\)
−0.485953 + 0.873985i \(0.661527\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −876817. −0.925579
\(52\) 0 0
\(53\) 672824. 0.620778 0.310389 − 0.950610i \(-0.399541\pi\)
0.310389 + 0.950610i \(0.399541\pi\)
\(54\) 0 0
\(55\) −1.15845e6 −0.938877
\(56\) 0 0
\(57\) −260603. −0.186388
\(58\) 0 0
\(59\) 2.58646e6 1.63955 0.819775 − 0.572686i \(-0.194099\pi\)
0.819775 + 0.572686i \(0.194099\pi\)
\(60\) 0 0
\(61\) −1.53593e6 −0.866397 −0.433198 − 0.901299i \(-0.642615\pi\)
−0.433198 + 0.901299i \(0.642615\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.17378e6 1.43344
\(66\) 0 0
\(67\) −4.20823e6 −1.70937 −0.854687 − 0.519143i \(-0.826251\pi\)
−0.854687 + 0.519143i \(0.826251\pi\)
\(68\) 0 0
\(69\) 864931. 0.316964
\(70\) 0 0
\(71\) 1.51772e6 0.503254 0.251627 − 0.967824i \(-0.419034\pi\)
0.251627 + 0.967824i \(0.419034\pi\)
\(72\) 0 0
\(73\) −5.47779e6 −1.64807 −0.824034 − 0.566540i \(-0.808282\pi\)
−0.824034 + 0.566540i \(0.808282\pi\)
\(74\) 0 0
\(75\) −675584. −0.184912
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.75855e6 −1.54226 −0.771132 − 0.636675i \(-0.780309\pi\)
−0.771132 + 0.636675i \(0.780309\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −8.36612e6 −1.60602 −0.803009 − 0.595966i \(-0.796769\pi\)
−0.803009 + 0.595966i \(0.796769\pi\)
\(84\) 0 0
\(85\) −7.48353e6 −1.32172
\(86\) 0 0
\(87\) 2.79914e6 0.455730
\(88\) 0 0
\(89\) 5.17007e6 0.777377 0.388688 − 0.921369i \(-0.372928\pi\)
0.388688 + 0.921369i \(0.372928\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.53227e6 −0.842120
\(94\) 0 0
\(95\) −2.22421e6 −0.266160
\(96\) 0 0
\(97\) 1.06286e7 1.18242 0.591212 − 0.806516i \(-0.298650\pi\)
0.591212 + 0.806516i \(0.298650\pi\)
\(98\) 0 0
\(99\) −3.66475e6 −0.379596
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.f.1.2 2
7.2 even 3 588.8.i.j.361.1 4
7.3 odd 6 588.8.i.k.373.2 4
7.4 even 3 588.8.i.j.373.1 4
7.5 odd 6 588.8.i.k.361.2 4
7.6 odd 2 84.8.a.c.1.1 ✓ 2
21.20 even 2 252.8.a.c.1.2 2
28.27 even 2 336.8.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.1 ✓ 2 7.6 odd 2
252.8.a.c.1.2 2 21.20 even 2
336.8.a.q.1.1 2 28.27 even 2
588.8.a.f.1.2 2 1.1 even 1 trivial
588.8.i.j.361.1 4 7.2 even 3
588.8.i.j.373.1 4 7.4 even 3
588.8.i.k.361.2 4 7.5 odd 6
588.8.i.k.373.2 4 7.3 odd 6