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Properties

Label 930.2.i.n
Level $930$
Weight $2$
Character orbit 930.i
Analytic conductor $7.426$
Analytic rank $0$
Dimension $6$
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] = [930,2,Mod(211,930)] 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("930.211"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-6,3,6,-3,-3,-4,-6,-3,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3636603.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_1 - 1) q^{5} + \beta_1 q^{6} + ( - \beta_{4} + \beta_1) q^{7} - q^{8} + ( - \beta_1 - 1) q^{9} + (\beta_1 + 1) q^{10} + (\beta_{5} - \beta_{3} + 2 \beta_1 + 2) q^{11}+ \cdots + (\beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{6} - 4 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 5 q^{11} + 3 q^{12} - 2 q^{13} + 4 q^{14} - 6 q^{15} + 6 q^{16} - 11 q^{17} + 3 q^{18} - 2 q^{19} - 3 q^{20}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 2\nu^{2} + 8\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + \nu^{3} - 22\nu^{2} - 24\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} - 9\nu^{3} - 82\nu^{2} + 40\nu - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} + 3\nu^{4} + 7\nu^{3} + 38\nu^{2} + 120\nu - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 3\nu^{3} - 4\nu^{2} + 4\nu + 8 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - 2\beta_{2} - \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 4\beta_{2} + 36\beta _1 - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 2\beta_{4} - 3\beta_{3} + 10\beta_{2} + 3\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} + 20\beta_{4} + \beta_{3} + 58\beta_{2} - 75\beta _1 + 36 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
0.715814 + 1.86751i
−1.55951 − 1.25217i
1.34370 − 1.48137i
0.715814 − 1.86751i
−1.55951 + 1.25217i
1.34370 + 1.48137i
−1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i −2.25941 − 3.91341i −1.00000 −0.500000 + 0.866025i 0.500000 − 0.866025i
211.2 −1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.695350 − 1.20438i −1.00000 −0.500000 + 0.866025i 0.500000 − 0.866025i
211.3 −1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i 0.954758 + 1.65369i −1.00000 −0.500000 + 0.866025i 0.500000 − 0.866025i
811.1 −1.00000 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i −2.25941 + 3.91341i −1.00000 −0.500000 − 0.866025i 0.500000 + 0.866025i
811.2 −1.00000 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.695350 + 1.20438i −1.00000 −0.500000 − 0.866025i 0.500000 + 0.866025i
811.3 −1.00000 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i 0.954758 − 1.65369i −1.00000 −0.500000 − 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.n ✓ 6
31.c even 3 1 inner 930.2.i.n ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.n ✓ 6 1.a even 1 1 trivial
930.2.i.n ✓ 6 31.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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{6} + 4T_{7}^{5} + 21T_{7}^{4} + 4T_{7}^{3} + 73T_{7}^{2} + 60T_{7} + 144 \) Copy content Toggle raw display
\( T_{11}^{6} - 5T_{11}^{5} + 54T_{11}^{4} - 113T_{11}^{3} + 1486T_{11}^{2} - 3741T_{11} + 16641 \) Copy content Toggle raw display
\( T_{13}^{6} + 2T_{13}^{5} + 44T_{13}^{4} - 16T_{13}^{3} + 1664T_{13}^{2} + 1280T_{13} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} + \cdots + 16641 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{6} + 11 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( (T^{3} - 7 T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + \cdots + 146)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots + 29791 \) Copy content Toggle raw display
$37$ \( T^{6} - 17 T^{5} + \cdots + 15376 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + \cdots + 419904 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{3} - 3 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 2 T^{5} + \cdots + 8464 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} + \cdots - 248)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 17 T^{5} + \cdots + 15376 \) Copy content Toggle raw display
$71$ \( T^{6} - 10 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{6} + 177 T^{4} + \cdots + 44944 \) Copy content Toggle raw display
$89$ \( (T^{3} + 28 T^{2} + \cdots + 432)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 20 T^{2} + \cdots + 2766)^{2} \) Copy content Toggle raw display
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