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Properties

Label 9216.2.a.g.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $2$
CM discriminant -4
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] = [9216,2,Mod(1,9216)] 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("9216.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,26,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,-14,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(67)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.24264 q^{5} +1.41421 q^{13} -8.00000 q^{17} +13.0000 q^{25} +9.89949 q^{29} +7.07107 q^{37} -8.00000 q^{41} -7.00000 q^{49} +7.07107 q^{53} -1.41421 q^{61} -6.00000 q^{65} +6.00000 q^{73} +33.9411 q^{85} +10.0000 q^{89} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{17} + 26 q^{25} - 16 q^{41} - 14 q^{49} - 12 q^{65} + 12 q^{73} + 20 q^{89} + 16 q^{97}+O(q^{100}) \) 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
\(5\) −4.24264 −1.89737 −0.948683 − 0.316228i \(-0.897584\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 − 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 13.0000 2.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.89949 1.83829 0.919145 − 0.393919i \(-0.128881\pi\)
0.919145 + 0.393919i \(0.128881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.07107 1.16248 0.581238 − 0.813733i \(-0.302568\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.07107 0.971286 0.485643 − 0.874157i \(-0.338586\pi\)
0.485643 + 0.874157i \(0.338586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.41421 −0.181071 −0.0905357 − 0.995893i \(-0.528858\pi\)
−0.0905357 + 0.995893i \(0.528858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 33.9411 3.68143
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 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 9216.2.a.g.1.1 2
3.2 odd 2 1024.2.a.d.1.2 2
4.3 odd 2 CM 9216.2.a.g.1.1 2
8.3 odd 2 inner 9216.2.a.g.1.2 2
8.5 even 2 inner 9216.2.a.g.1.2 2
12.11 even 2 1024.2.a.d.1.2 2
24.5 odd 2 1024.2.a.d.1.1 2
24.11 even 2 1024.2.a.d.1.1 2
32.3 odd 8 4608.2.k.w.1153.1 2
32.5 even 8 4608.2.k.b.3457.1 2
32.11 odd 8 4608.2.k.w.3457.1 2
32.13 even 8 4608.2.k.b.1153.1 2
32.19 odd 8 4608.2.k.b.1153.1 2
32.21 even 8 4608.2.k.w.3457.1 2
32.27 odd 8 4608.2.k.b.3457.1 2
32.29 even 8 4608.2.k.w.1153.1 2
48.5 odd 4 1024.2.b.d.513.2 2
48.11 even 4 1024.2.b.d.513.2 2
48.29 odd 4 1024.2.b.d.513.1 2
48.35 even 4 1024.2.b.d.513.1 2
96.5 odd 8 512.2.e.f.385.1 yes 2
96.11 even 8 512.2.e.c.385.1 yes 2
96.29 odd 8 512.2.e.c.129.1 ✓ 2
96.35 even 8 512.2.e.c.129.1 ✓ 2
96.53 odd 8 512.2.e.c.385.1 yes 2
96.59 even 8 512.2.e.f.385.1 yes 2
96.77 odd 8 512.2.e.f.129.1 yes 2
96.83 even 8 512.2.e.f.129.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.c.129.1 ✓ 2 96.29 odd 8
512.2.e.c.129.1 ✓ 2 96.35 even 8
512.2.e.c.385.1 yes 2 96.11 even 8
512.2.e.c.385.1 yes 2 96.53 odd 8
512.2.e.f.129.1 yes 2 96.77 odd 8
512.2.e.f.129.1 yes 2 96.83 even 8
512.2.e.f.385.1 yes 2 96.5 odd 8
512.2.e.f.385.1 yes 2 96.59 even 8
1024.2.a.d.1.1 2 24.5 odd 2
1024.2.a.d.1.1 2 24.11 even 2
1024.2.a.d.1.2 2 3.2 odd 2
1024.2.a.d.1.2 2 12.11 even 2
1024.2.b.d.513.1 2 48.29 odd 4
1024.2.b.d.513.1 2 48.35 even 4
1024.2.b.d.513.2 2 48.5 odd 4
1024.2.b.d.513.2 2 48.11 even 4
4608.2.k.b.1153.1 2 32.13 even 8
4608.2.k.b.1153.1 2 32.19 odd 8
4608.2.k.b.3457.1 2 32.5 even 8
4608.2.k.b.3457.1 2 32.27 odd 8
4608.2.k.w.1153.1 2 32.3 odd 8
4608.2.k.w.1153.1 2 32.29 even 8
4608.2.k.w.3457.1 2 32.11 odd 8
4608.2.k.w.3457.1 2 32.21 even 8
9216.2.a.g.1.1 2 1.1 even 1 trivial
9216.2.a.g.1.1 2 4.3 odd 2 CM
9216.2.a.g.1.2 2 8.3 odd 2 inner
9216.2.a.g.1.2 2 8.5 even 2 inner