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 = [2,0,0,0,0,0,0,0,0,0,6,0,8] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1295.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.1295
Dual form 1296.2.c.d.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)
 
\(f(q)\) \(=\) \(q-3.46410i q^{5} +3.46410i q^{7} +3.00000 q^{11} +4.00000 q^{13} -1.73205i q^{17} +1.73205i q^{19} -7.00000 q^{25} +3.46410i q^{29} +12.0000 q^{35} +2.00000 q^{37} -5.19615i q^{41} -5.19615i q^{43} +12.0000 q^{47} -5.00000 q^{49} -10.3923i q^{55} +15.0000 q^{59} +8.00000 q^{61} -13.8564i q^{65} -8.66025i q^{67} -6.00000 q^{71} -11.0000 q^{73} +10.3923i q^{77} -3.46410i q^{79} -12.0000 q^{83} -6.00000 q^{85} -13.8564i q^{89} +13.8564i q^{91} +6.00000 q^{95} +13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{11} + 8 q^{13} - 14 q^{25} + 24 q^{35} + 4 q^{37} + 24 q^{47} - 10 q^{49} + 30 q^{59} + 16 q^{61} - 12 q^{71} - 22 q^{73} - 24 q^{83} - 12 q^{85} + 12 q^{95} + 26 q^{97}+O(q^{100}) \) 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
\(5\) − 3.46410i − 1.54919i −0.632456 − 0.774597i \(-0.717953\pi\)
0.632456 − 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.73205i − 0.420084i −0.977692 − 0.210042i \(-0.932640\pi\)
0.977692 − 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.46410i 0.643268i 0.946864 + 0.321634i \(0.104232\pi\)
−0.946864 + 0.321634i \(0.895768\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.19615i − 0.811503i −0.913984 − 0.405751i \(-0.867010\pi\)
0.913984 − 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) − 5.19615i − 0.792406i −0.918163 − 0.396203i \(-0.870328\pi\)
0.918163 − 0.396203i \(-0.129672\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) − 10.3923i − 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.0000 1.95283 0.976417 − 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 13.8564i − 1.71868i
\(66\) 0 0
\(67\) − 8.66025i − 1.05802i −0.848616 − 0.529009i \(-0.822564\pi\)
0.848616 − 0.529009i \(-0.177436\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923i 1.18431i
\(78\) 0 0
\(79\) − 3.46410i − 0.389742i −0.980829 − 0.194871i \(-0.937571\pi\)
0.980829 − 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 13.8564i − 1.46878i −0.678730 − 0.734388i \(-0.737469\pi\)
0.678730 − 0.734388i \(-0.262531\pi\)
\(90\) 0 0
\(91\) 13.8564i 1.45255i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 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.d.1295.1 2
3.2 odd 2 1296.2.c.b.1295.2 2
4.3 odd 2 1296.2.c.b.1295.1 2
8.3 odd 2 5184.2.c.c.5183.2 2
8.5 even 2 5184.2.c.a.5183.2 2
9.2 odd 6 144.2.s.d.95.1 yes 2
9.4 even 3 144.2.s.a.47.1 ✓ 2
9.5 odd 6 432.2.s.c.143.1 2
9.7 even 3 432.2.s.d.287.1 2
12.11 even 2 inner 1296.2.c.d.1295.2 2
24.5 odd 2 5184.2.c.c.5183.1 2
24.11 even 2 5184.2.c.a.5183.1 2
36.7 odd 6 432.2.s.c.287.1 2
36.11 even 6 144.2.s.a.95.1 yes 2
36.23 even 6 432.2.s.d.143.1 2
36.31 odd 6 144.2.s.d.47.1 yes 2
72.5 odd 6 1728.2.s.a.575.1 2
72.11 even 6 576.2.s.d.383.1 2
72.13 even 6 576.2.s.d.191.1 2
72.29 odd 6 576.2.s.a.383.1 2
72.43 odd 6 1728.2.s.a.1151.1 2
72.59 even 6 1728.2.s.b.575.1 2
72.61 even 6 1728.2.s.b.1151.1 2
72.67 odd 6 576.2.s.a.191.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.a.47.1 ✓ 2 9.4 even 3
144.2.s.a.95.1 yes 2 36.11 even 6
144.2.s.d.47.1 yes 2 36.31 odd 6
144.2.s.d.95.1 yes 2 9.2 odd 6
432.2.s.c.143.1 2 9.5 odd 6
432.2.s.c.287.1 2 36.7 odd 6
432.2.s.d.143.1 2 36.23 even 6
432.2.s.d.287.1 2 9.7 even 3
576.2.s.a.191.1 2 72.67 odd 6
576.2.s.a.383.1 2 72.29 odd 6
576.2.s.d.191.1 2 72.13 even 6
576.2.s.d.383.1 2 72.11 even 6
1296.2.c.b.1295.1 2 4.3 odd 2
1296.2.c.b.1295.2 2 3.2 odd 2
1296.2.c.d.1295.1 2 1.1 even 1 trivial
1296.2.c.d.1295.2 2 12.11 even 2 inner
1728.2.s.a.575.1 2 72.5 odd 6
1728.2.s.a.1151.1 2 72.43 odd 6
1728.2.s.b.575.1 2 72.59 even 6
1728.2.s.b.1151.1 2 72.61 even 6
5184.2.c.a.5183.1 2 24.11 even 2
5184.2.c.a.5183.2 2 8.5 even 2
5184.2.c.c.5183.1 2 24.5 odd 2
5184.2.c.c.5183.2 2 8.3 odd 2