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Properties

Label 832.2.bu.j
Level $832$
Weight $2$
Character orbit 832.bu
Analytic conductor $6.644$
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] = [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 = [8,0,0,0,0,0,0] 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: 8.0.454201344.7
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 208)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{7} + (4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4) q^{9} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{11}+ \cdots + (4 \beta_{7} + 4 \beta_{6} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9} - 24 q^{13} - 12 q^{17} - 4 q^{21} + 12 q^{29} - 36 q^{33} - 8 q^{37} - 24 q^{41} + 72 q^{45} - 36 q^{49} + 52 q^{57} + 36 q^{61} - 84 q^{69} + 32 q^{73} - 28 q^{81} + 12 q^{85} - 24 q^{89}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 836 \nu^{7} - 2179 \nu^{6} - 23026 \nu^{5} + 99650 \nu^{4} - 1634 \nu^{3} - 349865 \nu^{2} + \cdots - 531206 ) / 284739 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2105\nu^{7} - 5146\nu^{6} - 8251\nu^{5} + 29525\nu^{4} - 17057\nu^{3} + 62854\nu^{2} - 30938\nu - 155675 ) / 284739 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2365\nu^{7} + 80\nu^{6} - 7692\nu^{5} + 19645\nu^{4} + 42834\nu^{3} - 11896\nu^{2} + 58801\nu + 328068 ) / 284739 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\nu^{7} + 84\nu^{6} - 716\nu^{5} + 773\nu^{4} + 2168\nu^{3} - 3968\nu^{2} + 6246\nu - 4576 ) / 5811 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3277 \nu^{7} - 19554 \nu^{6} + 10331 \nu^{5} + 50698 \nu^{4} - 157403 \nu^{3} + 98255 \nu^{2} + \cdots + 41938 ) / 284739 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4786 \nu^{7} + 17697 \nu^{6} + 40087 \nu^{5} - 168580 \nu^{4} + 95639 \nu^{3} + 434146 \nu^{2} + \cdots + 1041560 ) / 284739 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 856\nu^{7} - 1777\nu^{6} - 1080\nu^{5} + 11278\nu^{4} - 18930\nu^{3} + 16736\nu^{2} + 29692\nu - 34032 ) / 21903 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{4} + 3\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} - 2\beta_{5} - 6\beta_{4} + 13\beta_{3} + 7\beta_{2} - 2\beta _1 - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 2\beta_{5} - 5\beta_{4} + 4\beta_{3} + 8\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 4\beta_{6} - \beta_{5} - 29\beta_{4} + 60\beta_{3} - 29\beta_{2} + 3\beta _1 - 89 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{7} + 29\beta_{6} + 18\beta_{4} - 27\beta_{3} - 27\beta_{2} + 32\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50\beta_{7} + 31\beta_{6} + 81\beta_{5} + 111\beta_{4} + 111\beta_{3} - 322\beta_{2} - 31\beta _1 - 322 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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
2.02641 + 1.27503i
−2.39244 + 0.0909984i
1.02715 + 1.10132i
0.338876 − 1.46735i
2.02641 − 1.27503i
−2.39244 − 0.0909984i
1.02715 − 1.10132i
0.338876 + 1.46735i
0 −2.80144 + 1.61741i 0 1.73205 + 1.73205i 0 −1.61741 + 0.433385i 0 3.73205 − 6.46410i 0
63.2 0 2.80144 − 1.61741i 0 1.73205 + 1.73205i 0 1.61741 − 0.433385i 0 3.73205 − 6.46410i 0
319.1 0 −1.62847 − 0.940199i 0 −1.73205 − 1.73205i 0 0.940199 − 3.50887i 0 0.267949 + 0.464102i 0
319.2 0 1.62847 + 0.940199i 0 −1.73205 − 1.73205i 0 −0.940199 + 3.50887i 0 0.267949 + 0.464102i 0
383.1 0 −2.80144 − 1.61741i 0 1.73205 − 1.73205i 0 −1.61741 − 0.433385i 0 3.73205 + 6.46410i 0
383.2 0 2.80144 + 1.61741i 0 1.73205 − 1.73205i 0 1.61741 + 0.433385i 0 3.73205 + 6.46410i 0
639.1 0 −1.62847 + 0.940199i 0 −1.73205 + 1.73205i 0 0.940199 + 3.50887i 0 0.267949 − 0.464102i 0
639.2 0 1.62847 − 0.940199i 0 −1.73205 + 1.73205i 0 −0.940199 − 3.50887i 0 0.267949 − 0.464102i 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
4.b odd 2 1 inner
13.f odd 12 1 inner
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.j 8
4.b odd 2 1 inner 832.2.bu.j 8
8.b even 2 1 208.2.bm.f ✓ 8
8.d odd 2 1 208.2.bm.f ✓ 8
13.f odd 12 1 inner 832.2.bu.j 8
52.l even 12 1 inner 832.2.bu.j 8
104.u even 12 1 208.2.bm.f ✓ 8
104.x odd 12 1 208.2.bm.f ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.2.bm.f ✓ 8 8.b even 2 1
208.2.bm.f ✓ 8 8.d odd 2 1
208.2.bm.f ✓ 8 104.u even 12 1
208.2.bm.f ✓ 8 104.x odd 12 1
832.2.bu.j 8 1.a even 1 1 trivial
832.2.bu.j 8 4.b odd 2 1 inner
832.2.bu.j 8 13.f odd 12 1 inner
832.2.bu.j 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 \(S_{2}^{\mathrm{new}}(832, [\chi])\):

\( T_{3}^{8} - 14T_{3}^{6} + 159T_{3}^{4} - 518T_{3}^{2} + 1369 \) Copy content Toggle raw display
\( T_{5}^{4} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 14 T^{6} + \cdots + 1369 \) Copy content Toggle raw display
$5$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 18 T^{6} + \cdots + 1369 \) Copy content Toggle raw display
$11$ \( T^{8} - 54 T^{6} + \cdots + 110889 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 66 T^{6} + \cdots + 1369 \) Copy content Toggle raw display
$23$ \( T^{8} + 42 T^{6} + \cdots + 110889 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 488 T^{4} + 21904 \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 529)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 170 T^{6} + \cdots + 39100009 \) Copy content Toggle raw display
$47$ \( T^{8} + 4392 T^{4} + 1774224 \) Copy content Toggle raw display
$53$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 198 T^{6} + \cdots + 110889 \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{3} + \cdots + 4761)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 42 T^{6} + \cdots + 20043529 \) Copy content Toggle raw display
$71$ \( T^{8} - 54 T^{6} + \cdots + 110889 \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 39528 T^{4} + 143712144 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 3721)^{2} \) Copy content Toggle raw display
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