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Properties

Label 1296.2.c.g.1295.5
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 1295.5
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1296.1295
Dual form 1296.2.c.g.1295.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.896575i q^{5} -1.26795i q^{7} -4.24264 q^{11} +1.00000 q^{13} -0.896575i q^{17} -4.73205i q^{19} +7.34847 q^{23} +4.19615 q^{25} -8.24504i q^{29} -6.00000i q^{31} +1.13681 q^{35} +1.19615 q^{37} +7.34847i q^{41} -8.19615i q^{43} -11.5911 q^{47} +5.39230 q^{49} +7.34847i q^{53} -3.80385i q^{55} +11.5911 q^{59} -3.19615 q^{61} +0.896575i q^{65} -10.7321i q^{67} +1.13681 q^{71} +9.19615 q^{73} +5.37945i q^{77} -4.73205i q^{79} -8.48528 q^{83} +0.803848 q^{85} -11.8313i q^{89} -1.26795i q^{91} +4.24264 q^{95} +14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} - 8 q^{25} - 32 q^{37} - 40 q^{49} + 16 q^{61} + 32 q^{73} + 48 q^{85} + 32 q^{97}+O(q^{100}) \) 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
\(5\) 0.896575i 0.400961i 0.979698 + 0.200480i \(0.0642503\pi\)
−0.979698 + 0.200480i \(0.935750\pi\)
\(6\) 0 0
\(7\) − 1.26795i − 0.479240i −0.970867 − 0.239620i \(-0.922977\pi\)
0.970867 − 0.239620i \(-0.0770228\pi\)
\(8\) 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
\(17\) − 0.896575i − 0.217451i −0.994072 − 0.108726i \(-0.965323\pi\)
0.994072 − 0.108726i \(-0.0346770\pi\)
\(18\) 0 0
\(19\) − 4.73205i − 1.08561i −0.839860 − 0.542803i \(-0.817363\pi\)
0.839860 − 0.542803i \(-0.182637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.34847 1.53226 0.766131 − 0.642685i \(-0.222179\pi\)
0.766131 + 0.642685i \(0.222179\pi\)
\(24\) 0 0
\(25\) 4.19615 0.839230
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.24504i − 1.53107i −0.643396 − 0.765533i \(-0.722475\pi\)
0.643396 − 0.765533i \(-0.277525\pi\)
\(30\) 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
\(35\) 1.13681 0.192156
\(36\) 0 0
\(37\) 1.19615 0.196646 0.0983231 − 0.995155i \(-0.468652\pi\)
0.0983231 + 0.995155i \(0.468652\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.34847i 1.14764i 0.818982 + 0.573819i \(0.194539\pi\)
−0.818982 + 0.573819i \(0.805461\pi\)
\(42\) 0 0
\(43\) − 8.19615i − 1.24990i −0.780664 − 0.624951i \(-0.785119\pi\)
0.780664 − 0.624951i \(-0.214881\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.5911 −1.69074 −0.845369 − 0.534183i \(-0.820619\pi\)
−0.845369 + 0.534183i \(0.820619\pi\)
\(48\) 0 0
\(49\) 5.39230 0.770329
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.34847i 1.00939i 0.863298 + 0.504695i \(0.168395\pi\)
−0.863298 + 0.504695i \(0.831605\pi\)
\(54\) 0 0
\(55\) − 3.80385i − 0.512911i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.5911 1.50903 0.754517 − 0.656281i \(-0.227871\pi\)
0.754517 + 0.656281i \(0.227871\pi\)
\(60\) 0 0
\(61\) −3.19615 −0.409225 −0.204613 − 0.978843i \(-0.565593\pi\)
−0.204613 + 0.978843i \(0.565593\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.896575i 0.111207i
\(66\) 0 0
\(67\) − 10.7321i − 1.31113i −0.755139 − 0.655564i \(-0.772431\pi\)
0.755139 − 0.655564i \(-0.227569\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.13681 0.134915 0.0674574 − 0.997722i \(-0.478511\pi\)
0.0674574 + 0.997722i \(0.478511\pi\)
\(72\) 0 0
\(73\) 9.19615 1.07633 0.538164 − 0.842840i \(-0.319118\pi\)
0.538164 + 0.842840i \(0.319118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.37945i 0.613046i
\(78\) 0 0
\(79\) − 4.73205i − 0.532397i −0.963918 − 0.266199i \(-0.914232\pi\)
0.963918 − 0.266199i \(-0.0857677\pi\)
\(80\) 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
\(85\) 0.803848 0.0871895
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 11.8313i − 1.25412i −0.778971 − 0.627060i \(-0.784258\pi\)
0.778971 − 0.627060i \(-0.215742\pi\)
\(90\) 0 0
\(91\) − 1.26795i − 0.132917i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.24264 0.435286
\(96\) 0 0
\(97\) 14.3923 1.46132 0.730659 − 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) 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.1295.5 yes 8
3.2 odd 2 inner 1296.2.c.g.1295.3 ✓ 8
4.3 odd 2 inner 1296.2.c.g.1295.6 yes 8
8.3 odd 2 5184.2.c.h.5183.4 8
8.5 even 2 5184.2.c.h.5183.3 8
9.2 odd 6 1296.2.s.l.863.3 8
9.4 even 3 1296.2.s.j.431.3 8
9.5 odd 6 1296.2.s.j.431.2 8
9.7 even 3 1296.2.s.l.863.2 8
12.11 even 2 inner 1296.2.c.g.1295.4 yes 8
24.5 odd 2 5184.2.c.h.5183.5 8
24.11 even 2 5184.2.c.h.5183.6 8
36.7 odd 6 1296.2.s.j.863.2 8
36.11 even 6 1296.2.s.j.863.3 8
36.23 even 6 1296.2.s.l.431.2 8
36.31 odd 6 1296.2.s.l.431.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.3 ✓ 8 3.2 odd 2 inner
1296.2.c.g.1295.4 yes 8 12.11 even 2 inner
1296.2.c.g.1295.5 yes 8 1.1 even 1 trivial
1296.2.c.g.1295.6 yes 8 4.3 odd 2 inner
1296.2.s.j.431.2 8 9.5 odd 6
1296.2.s.j.431.3 8 9.4 even 3
1296.2.s.j.863.2 8 36.7 odd 6
1296.2.s.j.863.3 8 36.11 even 6
1296.2.s.l.431.2 8 36.23 even 6
1296.2.s.l.431.3 8 36.31 odd 6
1296.2.s.l.863.2 8 9.7 even 3
1296.2.s.l.863.3 8 9.2 odd 6
5184.2.c.h.5183.3 8 8.5 even 2
5184.2.c.h.5183.4 8 8.3 odd 2
5184.2.c.h.5183.5 8 24.5 odd 2
5184.2.c.h.5183.6 8 24.11 even 2