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(1295,1296)] 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("1296.1295"); 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1295.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1296.1295
Dual form 1296.2.c.f.1295.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{5} -3.00000i q^{7} -5.19615 q^{11} +1.00000 q^{13} +3.46410i q^{17} +6.00000i q^{19} +5.19615 q^{23} +2.00000 q^{25} +8.66025i q^{29} +3.00000i q^{31} +5.19615 q^{35} -4.00000 q^{37} +5.19615i q^{41} -3.00000i q^{43} +5.19615 q^{47} -2.00000 q^{49} +10.3923i q^{53} -9.00000i q^{55} -5.19615 q^{59} -7.00000 q^{61} +1.73205i q^{65} -9.00000i q^{67} -10.3923 q^{71} +4.00000 q^{73} +15.5885i q^{77} +15.0000i q^{79} +5.19615 q^{83} -6.00000 q^{85} -3.46410i q^{89} -3.00000i q^{91} -10.3923 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{13} + 8 q^{25} - 16 q^{37} - 8 q^{49} - 28 q^{61} + 16 q^{73} - 24 q^{85} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 − 0.566947i \(-0.808125\pi\)
0.823754 − 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.19615 −1.56670 −0.783349 − 0.621582i \(-0.786490\pi\)
−0.783349 + 0.621582i \(0.786490\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615 1.08347 0.541736 − 0.840548i \(-0.317767\pi\)
0.541736 + 0.840548i \(0.317767\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.66025i 1.60817i 0.594515 + 0.804084i \(0.297344\pi\)
−0.594515 + 0.804084i \(0.702656\pi\)
\(30\) 0 0
\(31\) 3.00000i 0.538816i 0.963026 + 0.269408i \(0.0868280\pi\)
−0.963026 + 0.269408i \(0.913172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.19615 0.878310
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615i 0.811503i 0.913984 + 0.405751i \(0.132990\pi\)
−0.913984 + 0.405751i \(0.867010\pi\)
\(42\) 0 0
\(43\) − 3.00000i − 0.457496i −0.973486 − 0.228748i \(-0.926537\pi\)
0.973486 − 0.228748i \(-0.0734631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.19615 0.757937 0.378968 − 0.925410i \(-0.376279\pi\)
0.378968 + 0.925410i \(0.376279\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3923i 1.42749i 0.700404 + 0.713746i \(0.253003\pi\)
−0.700404 + 0.713746i \(0.746997\pi\)
\(54\) 0 0
\(55\) − 9.00000i − 1.21356i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.19615 −0.676481 −0.338241 − 0.941060i \(-0.609832\pi\)
−0.338241 + 0.941060i \(0.609832\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.73205i 0.214834i
\(66\) 0 0
\(67\) − 9.00000i − 1.09952i −0.835321 − 0.549762i \(-0.814718\pi\)
0.835321 − 0.549762i \(-0.185282\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 − 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.5885i 1.77647i
\(78\) 0 0
\(79\) 15.0000i 1.68763i 0.536633 + 0.843816i \(0.319696\pi\)
−0.536633 + 0.843816i \(0.680304\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.19615 0.570352 0.285176 − 0.958475i \(-0.407948\pi\)
0.285176 + 0.958475i \(0.407948\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.46410i − 0.367194i −0.983002 − 0.183597i \(-0.941226\pi\)
0.983002 − 0.183597i \(-0.0587741\pi\)
\(90\) 0 0
\(91\) − 3.00000i − 0.314485i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.3923 −1.06623
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 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.c.f.1295.3 4
3.2 odd 2 inner 1296.2.c.f.1295.1 4
4.3 odd 2 inner 1296.2.c.f.1295.4 4
8.3 odd 2 5184.2.c.f.5183.2 4
8.5 even 2 5184.2.c.f.5183.1 4
9.2 odd 6 144.2.s.e.95.2 yes 4
9.4 even 3 144.2.s.e.47.1 ✓ 4
9.5 odd 6 432.2.s.e.143.2 4
9.7 even 3 432.2.s.e.287.1 4
12.11 even 2 inner 1296.2.c.f.1295.2 4
24.5 odd 2 5184.2.c.f.5183.3 4
24.11 even 2 5184.2.c.f.5183.4 4
36.7 odd 6 432.2.s.e.287.2 4
36.11 even 6 144.2.s.e.95.1 yes 4
36.23 even 6 432.2.s.e.143.1 4
36.31 odd 6 144.2.s.e.47.2 yes 4
72.5 odd 6 1728.2.s.e.575.2 4
72.11 even 6 576.2.s.e.383.2 4
72.13 even 6 576.2.s.e.191.2 4
72.29 odd 6 576.2.s.e.383.1 4
72.43 odd 6 1728.2.s.e.1151.2 4
72.59 even 6 1728.2.s.e.575.1 4
72.61 even 6 1728.2.s.e.1151.1 4
72.67 odd 6 576.2.s.e.191.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.e.47.1 ✓ 4 9.4 even 3
144.2.s.e.47.2 yes 4 36.31 odd 6
144.2.s.e.95.1 yes 4 36.11 even 6
144.2.s.e.95.2 yes 4 9.2 odd 6
432.2.s.e.143.1 4 36.23 even 6
432.2.s.e.143.2 4 9.5 odd 6
432.2.s.e.287.1 4 9.7 even 3
432.2.s.e.287.2 4 36.7 odd 6
576.2.s.e.191.1 4 72.67 odd 6
576.2.s.e.191.2 4 72.13 even 6
576.2.s.e.383.1 4 72.29 odd 6
576.2.s.e.383.2 4 72.11 even 6
1296.2.c.f.1295.1 4 3.2 odd 2 inner
1296.2.c.f.1295.2 4 12.11 even 2 inner
1296.2.c.f.1295.3 4 1.1 even 1 trivial
1296.2.c.f.1295.4 4 4.3 odd 2 inner
1728.2.s.e.575.1 4 72.59 even 6
1728.2.s.e.575.2 4 72.5 odd 6
1728.2.s.e.1151.1 4 72.61 even 6
1728.2.s.e.1151.2 4 72.43 odd 6
5184.2.c.f.5183.1 4 8.5 even 2
5184.2.c.f.5183.2 4 8.3 odd 2
5184.2.c.f.5183.3 4 24.5 odd 2
5184.2.c.f.5183.4 4 24.11 even 2