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Properties

Label 832.2.bu.k
Level $832$
Weight $2$
Character orbit 832.bu
Analytic conductor $6.644$
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] = [832,2,Mod(63,832)] 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("832.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 832.bu (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,6,0,-4,0,-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: \(6.64355344817\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.22581504.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 416)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} + 1) q^{3} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{5} + ( - \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{7} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - 4 \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 6 q^{11} - 24 q^{13} - 16 q^{15} + 12 q^{17} - 6 q^{19} + 4 q^{21} - 6 q^{23} - 12 q^{31} + 12 q^{33} - 36 q^{35} - 22 q^{39} - 4 q^{41} + 22 q^{43} - 44 q^{45}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
0.665665 + 1.24775i
1.20036 − 0.747754i
−1.27597 + 0.609843i
1.40994 − 0.109843i
0.665665 − 1.24775i
1.20036 + 0.747754i
−1.27597 − 0.609843i
1.40994 + 0.109843i
0 −0.0820885 + 0.0473938i 0 0.0947876 + 0.0947876i 0 −2.54290 + 0.681368i 0 −1.49551 + 2.59030i 0
63.2 0 2.44811 − 1.41342i 0 −2.82684 − 2.82684i 0 2.90893 − 0.779445i 0 2.49551 − 4.32235i 0
319.1 0 −1.38581 − 0.800098i 0 −1.60020 − 1.60020i 0 −0.419589 + 1.56593i 0 −0.219687 − 0.380509i 0
319.2 0 2.01978 + 1.16612i 0 2.33225 + 2.33225i 0 −0.946436 + 3.53215i 0 1.21969 + 2.11256i 0
383.1 0 −0.0820885 − 0.0473938i 0 0.0947876 − 0.0947876i 0 −2.54290 − 0.681368i 0 −1.49551 − 2.59030i 0
383.2 0 2.44811 + 1.41342i 0 −2.82684 + 2.82684i 0 2.90893 + 0.779445i 0 2.49551 + 4.32235i 0
639.1 0 −1.38581 + 0.800098i 0 −1.60020 + 1.60020i 0 −0.419589 − 1.56593i 0 −0.219687 + 0.380509i 0
639.2 0 2.01978 − 1.16612i 0 2.33225 − 2.33225i 0 −0.946436 − 3.53215i 0 1.21969 − 2.11256i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 63.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.bu.k 8
4.b odd 2 1 832.2.bu.i 8
8.b even 2 1 416.2.bu.c ✓ 8
8.d odd 2 1 416.2.bu.d yes 8
13.f odd 12 1 832.2.bu.i 8
52.l even 12 1 inner 832.2.bu.k 8
104.u even 12 1 416.2.bu.c ✓ 8
104.x odd 12 1 416.2.bu.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.bu.c ✓ 8 8.b even 2 1
416.2.bu.c ✓ 8 104.u even 12 1
416.2.bu.d yes 8 8.d odd 2 1
416.2.bu.d yes 8 104.x odd 12 1
832.2.bu.i 8 4.b odd 2 1
832.2.bu.i 8 13.f odd 12 1
832.2.bu.k 8 1.a even 1 1 trivial
832.2.bu.k 8 52.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(832, [\chi])\):

\( T_{3}^{8} - 6T_{3}^{7} + 10T_{3}^{6} + 12T_{3}^{5} - 33T_{3}^{4} - 36T_{3}^{3} + 106T_{3}^{2} + 18T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} - 8T_{5}^{5} + 136T_{5}^{4} + 464T_{5}^{3} + 800T_{5}^{2} - 160T_{5} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 2209 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 6889 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 84681 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 567009 \) Copy content Toggle raw display
$29$ \( T^{8} + 78 T^{6} + \cdots + 962361 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{8} + 90 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 52441 \) Copy content Toggle raw display
$43$ \( T^{8} - 22 T^{7} + \cdots + 113569 \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 24 T^{3} + \cdots - 1584)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 22 T^{7} + \cdots + 279841 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 4068289 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + \cdots + 37249 \) Copy content Toggle raw display
$71$ \( T^{8} + 14 T^{7} + \cdots + 37249 \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} + \cdots + 12873744 \) Copy content Toggle raw display
$79$ \( T^{8} + 408 T^{6} + \cdots + 15872256 \) Copy content Toggle raw display
$83$ \( T^{8} + 44 T^{7} + \cdots + 30958096 \) Copy content Toggle raw display
$89$ \( T^{8} + 32 T^{7} + \cdots + 8862529 \) Copy content Toggle raw display
$97$ \( T^{8} - 68 T^{7} + \cdots + 322669369 \) Copy content Toggle raw display
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