Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,8,Mod(1,336)] 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("336.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 336.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: \(104.961368563\)
Analytic rank: \(0\)
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 1.1
Root \(30.7035\) of defining polynomial
Character \(\chi\) \(=\) 336.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -230.442 q^{5} -343.000 q^{7} +729.000 q^{9} +5027.09 q^{11} -13772.6 q^{13} -6221.93 q^{15} +32474.7 q^{17} -9651.95 q^{19} -9261.00 q^{21} -32034.5 q^{23} -25021.6 q^{25} +19683.0 q^{27} +103672. q^{29} -241936. q^{31} +135731. q^{33} +79041.5 q^{35} -127147. q^{37} -371860. q^{39} +607138. q^{41} -443862. q^{43} -167992. q^{45} -691778. q^{47} +117649. q^{49} +876817. q^{51} +672824. q^{53} -1.15845e6 q^{55} -260603. q^{57} +2.58646e6 q^{59} +1.53593e6 q^{61} -250047. q^{63} +3.17378e6 q^{65} +4.20823e6 q^{67} -864931. q^{69} -1.51772e6 q^{71} +5.47779e6 q^{73} -675584. q^{75} -1.72429e6 q^{77} +6.75855e6 q^{79} +531441. q^{81} -8.36612e6 q^{83} -7.48353e6 q^{85} +2.79914e6 q^{87} -5.17007e6 q^{89} +4.72400e6 q^{91} -6.53227e6 q^{93} +2.22421e6 q^{95} -1.06286e7 q^{97} +3.66475e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{3} + 264 q^{5} - 686 q^{7} + 1458 q^{9} + 4980 q^{11} - 10148 q^{13} + 7128 q^{15} + 17832 q^{17} - 6256 q^{19} - 18522 q^{21} - 14052 q^{23} + 141326 q^{25} + 39366 q^{27} + 243588 q^{29} - 470824 q^{31}+ \cdots + 3630420 q^{99}+O(q^{100}) \) 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 \(-0.635249\pi\)
−0.412227 + 0.911081i \(0.635249\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 5027.09 1.13879 0.569393 − 0.822065i \(-0.307178\pi\)
0.569393 + 0.822065i \(0.307178\pi\)
\(12\) 0 0
\(13\) −13772.6 −1.73866 −0.869329 − 0.494234i \(-0.835449\pi\)
−0.869329 + 0.494234i \(0.835449\pi\)
\(14\) 0 0
\(15\) −6221.93 −0.475998
\(16\) 0 0
\(17\) 32474.7 1.60315 0.801575 − 0.597894i \(-0.203996\pi\)
0.801575 + 0.597894i \(0.203996\pi\)
\(18\) 0 0
\(19\) −9651.95 −0.322833 −0.161416 − 0.986886i \(-0.551606\pi\)
−0.161416 + 0.986886i \(0.551606\pi\)
\(20\) 0 0
\(21\) −9261.00 −0.218218
\(22\) 0 0
\(23\) −32034.5 −0.548997 −0.274499 − 0.961587i \(-0.588512\pi\)
−0.274499 + 0.961587i \(0.588512\pi\)
\(24\) 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
\(35\) 79041.5 0.311614
\(36\) 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 \(-0.258541\pi\)
0.687882 + 0.725823i \(0.258541\pi\)
\(42\) 0 0
\(43\) −443862. −0.851350 −0.425675 − 0.904876i \(-0.639963\pi\)
−0.425675 + 0.904876i \(0.639963\pi\)
\(44\) 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
\(49\) 117649. 0.142857
\(50\) 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 \(-0.357385\pi\)
0.433198 + 0.901299i \(0.357385\pi\)
\(62\) 0 0
\(63\) −250047. −0.125988
\(64\) 0 0
\(65\) 3.17378e6 1.43344
\(66\) 0 0
\(67\) 4.20823e6 1.70937 0.854687 − 0.519143i \(-0.173749\pi\)
0.854687 + 0.519143i \(0.173749\pi\)
\(68\) 0 0
\(69\) −864931. −0.316964
\(70\) 0 0
\(71\) −1.51772e6 −0.503254 −0.251627 − 0.967824i \(-0.580966\pi\)
−0.251627 + 0.967824i \(0.580966\pi\)
\(72\) 0 0
\(73\) 5.47779e6 1.64807 0.824034 − 0.566540i \(-0.191718\pi\)
0.824034 + 0.566540i \(0.191718\pi\)
\(74\) 0 0
\(75\) −675584. −0.184912
\(76\) 0 0
\(77\) −1.72429e6 −0.430421
\(78\) 0 0
\(79\) 6.75855e6 1.54226 0.771132 − 0.636675i \(-0.219691\pi\)
0.771132 + 0.636675i \(0.219691\pi\)
\(80\) 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 \(-0.627072\pi\)
−0.388688 + 0.921369i \(0.627072\pi\)
\(90\) 0 0
\(91\) 4.72400e6 0.657151
\(92\) 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 \(-0.701350\pi\)
−0.591212 + 0.806516i \(0.701350\pi\)
\(98\) 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 336.8.a.q.1.1 2
4.3 odd 2 84.8.a.c.1.1 ✓ 2
12.11 even 2 252.8.a.c.1.2 2
28.3 even 6 588.8.i.j.373.1 4
28.11 odd 6 588.8.i.k.373.2 4
28.19 even 6 588.8.i.j.361.1 4
28.23 odd 6 588.8.i.k.361.2 4
28.27 even 2 588.8.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.1 ✓ 2 4.3 odd 2
252.8.a.c.1.2 2 12.11 even 2
336.8.a.q.1.1 2 1.1 even 1 trivial
588.8.a.f.1.2 2 28.27 even 2
588.8.i.j.361.1 4 28.19 even 6
588.8.i.j.373.1 4 28.3 even 6
588.8.i.k.361.2 4 28.23 odd 6
588.8.i.k.373.2 4 28.11 odd 6