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Properties

Label 9408.2.a.dj
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [9408,2,Mod(1,9408)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9408.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-1,0,0,0,2,0,1,0,5,0,1,0,8,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - \beta q^{5} + q^{9} + \beta q^{11} + ( - \beta + 3) q^{13} + \beta q^{15} + 4 q^{17} + (\beta - 3) q^{19} + 4 q^{23} + (\beta + 9) q^{25} - q^{27} + (\beta - 2) q^{29} + q^{31} - \beta q^{33} + \cdots + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{5} + 2 q^{9} + q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 5 q^{19} + 8 q^{23} + 19 q^{25} - 2 q^{27} - 3 q^{29} + 2 q^{31} - q^{33} - 3 q^{37} - 5 q^{39} - 6 q^{41} + 7 q^{43} - q^{45}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.27492
−3.27492
0 −1.00000 0 −4.27492 0 0 0 1.00000 0
1.2 0 −1.00000 0 3.27492 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.dj 2
4.b odd 2 1 9408.2.a.dw 2
7.b odd 2 1 9408.2.a.ec 2
7.d odd 6 2 1344.2.q.w 4
8.b even 2 1 1176.2.a.n 2
8.d odd 2 1 2352.2.a.ba 2
24.f even 2 1 7056.2.a.ch 2
24.h odd 2 1 3528.2.a.bd 2
28.d even 2 1 9408.2.a.dp 2
28.f even 6 2 1344.2.q.x 4
56.e even 2 1 2352.2.a.bf 2
56.h odd 2 1 1176.2.a.k 2
56.j odd 6 2 168.2.q.c ✓ 4
56.k odd 6 2 2352.2.q.bf 4
56.m even 6 2 336.2.q.g 4
56.p even 6 2 1176.2.q.l 4
168.e odd 2 1 7056.2.a.cu 2
168.i even 2 1 3528.2.a.bk 2
168.s odd 6 2 3528.2.s.bk 4
168.ba even 6 2 504.2.s.i 4
168.be odd 6 2 1008.2.s.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c ✓ 4 56.j odd 6 2
336.2.q.g 4 56.m even 6 2
504.2.s.i 4 168.ba even 6 2
1008.2.s.r 4 168.be odd 6 2
1176.2.a.k 2 56.h odd 2 1
1176.2.a.n 2 8.b even 2 1
1176.2.q.l 4 56.p even 6 2
1344.2.q.w 4 7.d odd 6 2
1344.2.q.x 4 28.f even 6 2
2352.2.a.ba 2 8.d odd 2 1
2352.2.a.bf 2 56.e even 2 1
2352.2.q.bf 4 56.k odd 6 2
3528.2.a.bd 2 24.h odd 2 1
3528.2.a.bk 2 168.i even 2 1
3528.2.s.bk 4 168.s odd 6 2
7056.2.a.ch 2 24.f even 2 1
7056.2.a.cu 2 168.e odd 2 1
9408.2.a.dj 2 1.a even 1 1 trivial
9408.2.a.dp 2 28.d even 2 1
9408.2.a.dw 2 4.b odd 2 1
9408.2.a.ec 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} + T_{5} - 14 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 14 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} - 8 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} + 5T_{19} - 8 \) Copy content Toggle raw display
\( T_{31} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 14 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 8 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 48 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T - 2 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 11T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 5T - 8 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7T - 2 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T - 14 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 41 \) Copy content Toggle raw display
$83$ \( T^{2} - 7T - 2 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 25T + 142 \) Copy content Toggle raw display
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