Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,2,Mod(1,1296)] 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("1296.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,0,0,4] 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: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 648)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1296.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.267949 q^{5} +3.46410 q^{7} +2.00000 q^{11} +4.46410 q^{13} -5.73205 q^{17} +0.535898 q^{19} +8.92820 q^{23} -4.92820 q^{25} -7.73205 q^{29} -2.92820 q^{31} -0.928203 q^{35} +6.46410 q^{37} +6.92820 q^{41} +11.4641 q^{43} -6.92820 q^{47} +5.00000 q^{49} -2.92820 q^{53} -0.535898 q^{55} +8.00000 q^{59} +3.53590 q^{61} -1.19615 q^{65} +7.46410 q^{67} -2.00000 q^{71} +1.00000 q^{73} +6.92820 q^{77} -7.46410 q^{79} +10.9282 q^{83} +1.53590 q^{85} -5.19615 q^{89} +15.4641 q^{91} -0.143594 q^{95} +15.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{11} + 2 q^{13} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} - 12 q^{29} + 8 q^{31} + 12 q^{35} + 6 q^{37} + 16 q^{43} + 10 q^{49} + 8 q^{53} - 8 q^{55} + 16 q^{59} + 14 q^{61} + 8 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.267949 −0.119831 −0.0599153 − 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 4.46410 1.23812 0.619060 − 0.785344i \(-0.287514\pi\)
0.619060 + 0.785344i \(0.287514\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.73205 −1.39023 −0.695113 − 0.718900i \(-0.744646\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) 0.535898 0.122944 0.0614718 − 0.998109i \(-0.480421\pi\)
0.0614718 + 0.998109i \(0.480421\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.92820 1.86166 0.930830 − 0.365454i \(-0.119086\pi\)
0.930830 + 0.365454i \(0.119086\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.73205 −1.43581 −0.717903 − 0.696143i \(-0.754898\pi\)
−0.717903 + 0.696143i \(0.754898\pi\)
\(30\) 0 0
\(31\) −2.92820 −0.525921 −0.262960 − 0.964807i \(-0.584699\pi\)
−0.262960 + 0.964807i \(0.584699\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.928203 −0.156895
\(36\) 0 0
\(37\) 6.46410 1.06269 0.531346 − 0.847155i \(-0.321686\pi\)
0.531346 + 0.847155i \(0.321686\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 − 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 11.4641 1.74826 0.874130 − 0.485693i \(-0.161433\pi\)
0.874130 + 0.485693i \(0.161433\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 − 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.92820 −0.402220 −0.201110 − 0.979569i \(-0.564455\pi\)
−0.201110 + 0.979569i \(0.564455\pi\)
\(54\) 0 0
\(55\) −0.535898 −0.0722605
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 3.53590 0.452725 0.226363 − 0.974043i \(-0.427317\pi\)
0.226363 + 0.974043i \(0.427317\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.19615 −0.148364
\(66\) 0 0
\(67\) 7.46410 0.911885 0.455943 − 0.890009i \(-0.349302\pi\)
0.455943 + 0.890009i \(0.349302\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) −7.46410 −0.839777 −0.419889 − 0.907576i \(-0.637931\pi\)
−0.419889 + 0.907576i \(0.637931\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.9282 1.19953 0.599763 − 0.800178i \(-0.295261\pi\)
0.599763 + 0.800178i \(0.295261\pi\)
\(84\) 0 0
\(85\) 1.53590 0.166592
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 −0.550791 −0.275396 − 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 0 0
\(91\) 15.4641 1.62108
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.143594 −0.0147324
\(96\) 0 0
\(97\) 15.8564 1.60997 0.804987 − 0.593292i \(-0.202172\pi\)
0.804987 + 0.593292i \(0.202172\pi\)
\(98\) 0 0
\(99\) 0 0
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 1296.2.a.m.1.2 2
3.2 odd 2 1296.2.a.q.1.1 2
4.3 odd 2 648.2.a.e.1.2 ✓ 2
8.3 odd 2 5184.2.a.cb.1.1 2
8.5 even 2 5184.2.a.bz.1.1 2
9.2 odd 6 1296.2.i.r.433.2 4
9.4 even 3 1296.2.i.t.865.1 4
9.5 odd 6 1296.2.i.r.865.2 4
9.7 even 3 1296.2.i.t.433.1 4
12.11 even 2 648.2.a.h.1.1 yes 2
24.5 odd 2 5184.2.a.bi.1.2 2
24.11 even 2 5184.2.a.bg.1.2 2
36.7 odd 6 648.2.i.j.433.1 4
36.11 even 6 648.2.i.i.433.2 4
36.23 even 6 648.2.i.i.217.2 4
36.31 odd 6 648.2.i.j.217.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.2.a.e.1.2 ✓ 2 4.3 odd 2
648.2.a.h.1.1 yes 2 12.11 even 2
648.2.i.i.217.2 4 36.23 even 6
648.2.i.i.433.2 4 36.11 even 6
648.2.i.j.217.1 4 36.31 odd 6
648.2.i.j.433.1 4 36.7 odd 6
1296.2.a.m.1.2 2 1.1 even 1 trivial
1296.2.a.q.1.1 2 3.2 odd 2
1296.2.i.r.433.2 4 9.2 odd 6
1296.2.i.r.865.2 4 9.5 odd 6
1296.2.i.t.433.1 4 9.7 even 3
1296.2.i.t.865.1 4 9.4 even 3
5184.2.a.bg.1.2 2 24.11 even 2
5184.2.a.bi.1.2 2 24.5 odd 2
5184.2.a.bz.1.1 2 8.5 even 2
5184.2.a.cb.1.1 2 8.3 odd 2