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Properties

Label 1296.2.c.g.1295.7
Level $1296$
Weight $2$
Character 1296.1295
Analytic conductor $10.349$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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 = [8,0,0,0,0,0,0,0,0,0,0,0,8] 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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1295.7
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1296.1295
Dual form 1296.2.c.g.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)
 
\(f(q)\) \(=\) \(q+3.34607i q^{5} -4.73205i q^{7} -4.24264 q^{11} +1.00000 q^{13} -3.34607i q^{17} -1.26795i q^{19} -7.34847 q^{23} -6.19615 q^{25} +4.00240i q^{29} -6.00000i q^{31} +15.8338 q^{35} -9.19615 q^{37} -7.34847i q^{41} +2.19615i q^{43} +3.10583 q^{47} -15.3923 q^{49} -7.34847i q^{53} -14.1962i q^{55} -3.10583 q^{59} +7.19615 q^{61} +3.34607i q^{65} -7.26795i q^{67} +15.8338 q^{71} -1.19615 q^{73} +20.0764i q^{77} -1.26795i q^{79} -8.48528 q^{83} +11.1962 q^{85} -9.38186i q^{89} -4.73205i q^{91} +4.24264 q^{95} -6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} - 8 q^{25} - 32 q^{37} - 40 q^{49} + 16 q^{61} + 32 q^{73} + 48 q^{85} + 32 q^{97}+O(q^{100}) \) 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
\(5\) 3.34607i 1.49641i 0.663470 + 0.748203i \(0.269083\pi\)
−0.663470 + 0.748203i \(0.730917\pi\)
\(6\) 0 0
\(7\) − 4.73205i − 1.78855i −0.447521 − 0.894274i \(-0.647693\pi\)
0.447521 − 0.894274i \(-0.352307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.24264 −1.27920 −0.639602 − 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 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
\(17\) − 3.34607i − 0.811540i −0.913975 − 0.405770i \(-0.867003\pi\)
0.913975 − 0.405770i \(-0.132997\pi\)
\(18\) 0 0
\(19\) − 1.26795i − 0.290887i −0.989367 − 0.145444i \(-0.953539\pi\)
0.989367 − 0.145444i \(-0.0464610\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.34847 −1.53226 −0.766131 − 0.642685i \(-0.777821\pi\)
−0.766131 + 0.642685i \(0.777821\pi\)
\(24\) 0 0
\(25\) −6.19615 −1.23923
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00240i 0.743228i 0.928387 + 0.371614i \(0.121195\pi\)
−0.928387 + 0.371614i \(0.878805\pi\)
\(30\) 0 0
\(31\) − 6.00000i − 1.07763i −0.842424 − 0.538816i \(-0.818872\pi\)
0.842424 − 0.538816i \(-0.181128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.8338 2.67639
\(36\) 0 0
\(37\) −9.19615 −1.51184 −0.755919 − 0.654665i \(-0.772810\pi\)
−0.755919 + 0.654665i \(0.772810\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.34847i − 1.14764i −0.818982 − 0.573819i \(-0.805461\pi\)
0.818982 − 0.573819i \(-0.194539\pi\)
\(42\) 0 0
\(43\) 2.19615i 0.334910i 0.985880 + 0.167455i \(0.0535549\pi\)
−0.985880 + 0.167455i \(0.946445\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.10583 0.453032 0.226516 − 0.974007i \(-0.427266\pi\)
0.226516 + 0.974007i \(0.427266\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.34847i − 1.00939i −0.863298 − 0.504695i \(-0.831605\pi\)
0.863298 − 0.504695i \(-0.168395\pi\)
\(54\) 0 0
\(55\) − 14.1962i − 1.91421i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.10583 −0.404344 −0.202172 − 0.979350i \(-0.564800\pi\)
−0.202172 + 0.979350i \(0.564800\pi\)
\(60\) 0 0
\(61\) 7.19615 0.921373 0.460686 − 0.887563i \(-0.347603\pi\)
0.460686 + 0.887563i \(0.347603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.34607i 0.415028i
\(66\) 0 0
\(67\) − 7.26795i − 0.887921i −0.896046 − 0.443961i \(-0.853573\pi\)
0.896046 − 0.443961i \(-0.146427\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.8338 1.87912 0.939560 − 0.342384i \(-0.111234\pi\)
0.939560 + 0.342384i \(0.111234\pi\)
\(72\) 0 0
\(73\) −1.19615 −0.139999 −0.0699995 − 0.997547i \(-0.522300\pi\)
−0.0699995 + 0.997547i \(0.522300\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 20.0764i 2.28792i
\(78\) 0 0
\(79\) − 1.26795i − 0.142655i −0.997453 − 0.0713277i \(-0.977276\pi\)
0.997453 − 0.0713277i \(-0.0227236\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.48528 −0.931381 −0.465690 − 0.884948i \(-0.654194\pi\)
−0.465690 + 0.884948i \(0.654194\pi\)
\(84\) 0 0
\(85\) 11.1962 1.21439
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.38186i − 0.994475i −0.867615 − 0.497237i \(-0.834348\pi\)
0.867615 − 0.497237i \(-0.165652\pi\)
\(90\) 0 0
\(91\) − 4.73205i − 0.496054i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.24264 0.435286
\(96\) 0 0
\(97\) −6.39230 −0.649040 −0.324520 − 0.945879i \(-0.605203\pi\)
−0.324520 + 0.945879i \(0.605203\pi\)
\(98\) 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.g.1295.7 yes 8
3.2 odd 2 inner 1296.2.c.g.1295.1 ✓ 8
4.3 odd 2 inner 1296.2.c.g.1295.8 yes 8
8.3 odd 2 5184.2.c.h.5183.2 8
8.5 even 2 5184.2.c.h.5183.1 8
9.2 odd 6 1296.2.s.j.863.4 8
9.4 even 3 1296.2.s.l.431.4 8
9.5 odd 6 1296.2.s.l.431.1 8
9.7 even 3 1296.2.s.j.863.1 8
12.11 even 2 inner 1296.2.c.g.1295.2 yes 8
24.5 odd 2 5184.2.c.h.5183.7 8
24.11 even 2 5184.2.c.h.5183.8 8
36.7 odd 6 1296.2.s.l.863.1 8
36.11 even 6 1296.2.s.l.863.4 8
36.23 even 6 1296.2.s.j.431.1 8
36.31 odd 6 1296.2.s.j.431.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.1 ✓ 8 3.2 odd 2 inner
1296.2.c.g.1295.2 yes 8 12.11 even 2 inner
1296.2.c.g.1295.7 yes 8 1.1 even 1 trivial
1296.2.c.g.1295.8 yes 8 4.3 odd 2 inner
1296.2.s.j.431.1 8 36.23 even 6
1296.2.s.j.431.4 8 36.31 odd 6
1296.2.s.j.863.1 8 9.7 even 3
1296.2.s.j.863.4 8 9.2 odd 6
1296.2.s.l.431.1 8 9.5 odd 6
1296.2.s.l.431.4 8 9.4 even 3
1296.2.s.l.863.1 8 36.7 odd 6
1296.2.s.l.863.4 8 36.11 even 6
5184.2.c.h.5183.1 8 8.5 even 2
5184.2.c.h.5183.2 8 8.3 odd 2
5184.2.c.h.5183.7 8 24.5 odd 2
5184.2.c.h.5183.8 8 24.11 even 2