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Properties

Label 3510.2.j.h
Level $3510$
Weight $2$
Character orbit 3510.j
Analytic conductor $28.027$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [3510,2,Mod(1171,3510)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3510, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 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("3510.1171"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3510 = 2 \cdot 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3510.j (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,0,-4,-4,0,7] 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: \(28.0274911095\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.856615824.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1170)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} - \beta_1 q^{5} + (\beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{7} - q^{8} - q^{10} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 3) q^{11} + \beta_1 q^{13} + (\beta_{4} - 2 \beta_1) q^{14}+ \cdots + (\beta_{3} + 4 \beta_{2} - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 4 q^{5} + 7 q^{7} - 8 q^{8} - 8 q^{10} - 11 q^{11} + 4 q^{13} - 7 q^{14} - 4 q^{16} + 8 q^{17} - 6 q^{19} - 4 q^{20} + 11 q^{22} + 15 q^{23} - 4 q^{25} + 8 q^{26} - 14 q^{28} - 4 q^{29}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} - 6\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 31\nu^{3} + 2\nu^{2} + 30\nu + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 10\nu^{5} + \nu^{4} - 28\nu^{3} + 6\nu^{2} - 18\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 9\nu^{4} + 21\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 2\nu^{6} + 32\nu^{5} + 18\nu^{4} + 95\nu^{3} + 42\nu^{2} + 50\nu + 14 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + \beta_{6} - 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 4\beta _1 - 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 2\beta_{6} + 10\beta_{5} + 8\beta_{4} + 5\beta_{3} - 4\beta_{2} - 8\beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} - 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{7} + 9\beta_{6} - 50\beta_{5} - 46\beta_{4} - 25\beta_{3} + 23\beta_{2} + 48\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 9\beta_{3} + 33\beta_{2} - 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 86\beta_{7} - 43\beta_{6} + 250\beta_{5} + 254\beta_{4} + 125\beta_{3} - 127\beta_{2} - 292\beta _1 + 146 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3510\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2081\) \(2107\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1171.1
− 0.385731i
− 1.07834i
2.06288i
− 2.33086i
0.385731i
1.07834i
− 2.06288i
2.33086i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.425606 − 0.737171i −1.00000 0 −1.00000
1171.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.0814062 + 0.141000i −1.00000 0 −1.00000
1171.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.62774 + 2.81933i −1.00000 0 −1.00000
1171.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 2.21646 + 3.83902i −1.00000 0 −1.00000
2341.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 −0.425606 + 0.737171i −1.00000 0 −1.00000
2341.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 0.0814062 − 0.141000i −1.00000 0 −1.00000
2341.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 1.62774 − 2.81933i −1.00000 0 −1.00000
2341.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 2.21646 − 3.83902i −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1171.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3510.2.j.h 8
3.b odd 2 1 1170.2.j.h ✓ 8
9.c even 3 1 inner 3510.2.j.h 8
9.d odd 6 1 1170.2.j.h ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.j.h ✓ 8 3.b odd 2 1
1170.2.j.h ✓ 8 9.d odd 6 1
3510.2.j.h 8 1.a even 1 1 trivial
3510.2.j.h 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3510, [\chi])\):

\( T_{7}^{8} - 7T_{7}^{7} + 40T_{7}^{6} - 85T_{7}^{5} + 160T_{7}^{4} + 71T_{7}^{3} + 139T_{7}^{2} - 22T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} + 11T_{11}^{7} + 85T_{11}^{6} + 332T_{11}^{5} + 940T_{11}^{4} + 1064T_{11}^{3} + 880T_{11}^{2} + 128T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 7 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 11 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} - 36 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{3} - 6 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + \cdots + 254016 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots + 4214809 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 404496 \) Copy content Toggle raw display
$37$ \( (T^{4} + 11 T^{3} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 11 T^{7} + \cdots + 5798464 \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{8} - 15 T^{7} + \cdots + 224676 \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + \cdots + 3856)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} + \cdots + 399424 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + \cdots + 5564881 \) Copy content Toggle raw display
$67$ \( T^{8} - 9 T^{7} + \cdots + 2643876 \) Copy content Toggle raw display
$71$ \( (T^{4} + 10 T^{3} + \cdots + 6016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 10 T^{3} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 2027776 \) Copy content Toggle raw display
$83$ \( T^{8} - 27 T^{7} + \cdots + 629909604 \) Copy content Toggle raw display
$89$ \( (T^{4} + 7 T^{3} + \cdots + 1486)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 7 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
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