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Properties

Label 396.2.j.d
Level $396$
Weight $2$
Character orbit 396.j
Analytic conductor $3.162$
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] = [396,2,Mod(37,396)] 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("396.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.16207592004\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.6
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} - 2 \beta_1) q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{7} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{11} + (3 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{13}+ \cdots + ( - 3 \beta_{5} + 9 \beta_{3} - 9 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{7} - 14 q^{13} - 16 q^{19} + 8 q^{31} + 30 q^{37} + 24 q^{43} + 12 q^{49} + 20 q^{55} - 38 q^{61} - 28 q^{67} - 40 q^{73} - 42 q^{79} - 50 q^{85} + 6 q^{91} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} + 75\beta_{6} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\)
\(\chi(n)\) \(-1 - \beta_{2} + \beta_{3} - \beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.46782 − 0.476925i
−1.46782 + 0.476925i
−1.26313 − 1.73855i
1.26313 + 1.73855i
1.46782 + 0.476925i
−1.46782 − 0.476925i
−1.26313 + 1.73855i
1.26313 − 1.73855i
0 0 0 −2.37499 + 1.72553i 0 0.309017 + 0.951057i 0 0 0
37.2 0 0 0 2.37499 − 1.72553i 0 0.309017 + 0.951057i 0 0 0
181.1 0 0 0 −0.780656 + 2.40261i 0 −0.809017 + 0.587785i 0 0 0
181.2 0 0 0 0.780656 − 2.40261i 0 −0.809017 + 0.587785i 0 0 0
289.1 0 0 0 −2.37499 − 1.72553i 0 0.309017 − 0.951057i 0 0 0
289.2 0 0 0 2.37499 + 1.72553i 0 0.309017 − 0.951057i 0 0 0
361.1 0 0 0 −0.780656 − 2.40261i 0 −0.809017 − 0.587785i 0 0 0
361.2 0 0 0 0.780656 + 2.40261i 0 −0.809017 − 0.587785i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.2.j.d ✓ 8
3.b odd 2 1 inner 396.2.j.d ✓ 8
11.c even 5 1 inner 396.2.j.d ✓ 8
11.c even 5 1 4356.2.a.z 4
11.d odd 10 1 4356.2.a.x 4
33.f even 10 1 4356.2.a.x 4
33.h odd 10 1 inner 396.2.j.d ✓ 8
33.h odd 10 1 4356.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.2.j.d ✓ 8 1.a even 1 1 trivial
396.2.j.d ✓ 8 3.b odd 2 1 inner
396.2.j.d ✓ 8 11.c even 5 1 inner
396.2.j.d ✓ 8 33.h odd 10 1 inner
4356.2.a.x 4 11.d odd 10 1
4356.2.a.x 4 33.f even 10 1
4356.2.a.z 4 11.c even 5 1
4356.2.a.z 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(396, [\chi])\):

\( T_{5}^{8} + 5T_{5}^{6} + 60T_{5}^{4} + 550T_{5}^{2} + 3025 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} + T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 31 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 7 T^{3} + 19 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 70 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 60 T^{2} + 55)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 140 T^{6} + \cdots + 774400 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 46 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 15 T^{3} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 25 T^{6} + \cdots + 1890625 \) Copy content Toggle raw display
$43$ \( (T - 3)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 55 T^{6} + \cdots + 44289025 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$59$ \( T^{8} - 70 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( (T^{4} + 19 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T - 49)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 185 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( (T^{4} + 20 T^{3} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 21 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 55 T^{6} + \cdots + 44289025 \) Copy content Toggle raw display
$89$ \( (T^{4} - 180 T^{2} + 6655)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 21 T^{3} + \cdots + 9801)^{2} \) Copy content Toggle raw display
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