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Properties

Label 1056.2.y.c
Level $1056$
Weight $2$
Character orbit 1056.y
Analytic conductor $8.432$
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] = [1056,2,Mod(97,1056)] 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("1056.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1056, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1056.y (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,0,2,0,8,0,-2,0,0] 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: \(8.43220245345\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{6} - \beta_{2}) q^{5} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{7} - \beta_{5} q^{9} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{11} + \beta_{5} q^{13}+ \cdots + ( - \beta_{7} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{5} + 8 q^{7} - 2 q^{9} + 2 q^{13} + 2 q^{15} - 10 q^{17} - 6 q^{19} - 12 q^{21} - 8 q^{23} - 4 q^{25} - 2 q^{27} - 20 q^{29} + 10 q^{31} + 26 q^{37} + 2 q^{39} + 6 q^{41} + 12 q^{43}+ \cdots + 30 q^{97}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1056\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(353\) \(673\) \(991\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2} - \beta_{3} + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1.73855 + 1.26313i
−1.73855 − 1.26313i
0.476925 − 1.46782i
−0.476925 + 1.46782i
1.73855 − 1.26313i
−1.73855 + 1.26313i
0.476925 + 1.46782i
−0.476925 − 1.46782i
0 −0.809017 + 0.587785i 0 −0.973083 − 2.99484i 0 3.85658 + 2.80197i 0 0.309017 − 0.951057i 0
97.2 0 −0.809017 + 0.587785i 0 0.355049 + 1.09273i 0 0.379488 + 0.275714i 0 0.309017 − 0.951057i 0
289.1 0 0.309017 + 0.951057i 0 −0.439589 − 0.319380i 0 0.358891 − 1.10455i 0 −0.809017 + 0.587785i 0
289.2 0 0.309017 + 0.951057i 0 2.05762 + 1.49495i 0 −0.594959 + 1.83110i 0 −0.809017 + 0.587785i 0
577.1 0 −0.809017 − 0.587785i 0 −0.973083 + 2.99484i 0 3.85658 − 2.80197i 0 0.309017 + 0.951057i 0
577.2 0 −0.809017 − 0.587785i 0 0.355049 − 1.09273i 0 0.379488 − 0.275714i 0 0.309017 + 0.951057i 0
961.1 0 0.309017 − 0.951057i 0 −0.439589 + 0.319380i 0 0.358891 + 1.10455i 0 −0.809017 − 0.587785i 0
961.2 0 0.309017 − 0.951057i 0 2.05762 − 1.49495i 0 −0.594959 − 1.83110i 0 −0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1056.2.y.c ✓ 8
4.b odd 2 1 1056.2.y.d yes 8
11.c even 5 1 inner 1056.2.y.c ✓ 8
44.h odd 10 1 1056.2.y.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.y.c ✓ 8 1.a even 1 1 trivial
1056.2.y.c ✓ 8 11.c even 5 1 inner
1056.2.y.d yes 8 4.b odd 2 1
1056.2.y.d yes 8 44.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 \(S_{2}^{\mathrm{new}}(1056, [\chi])\):

\( T_{5}^{8} - 2T_{5}^{7} + 9T_{5}^{6} - 28T_{5}^{5} + 66T_{5}^{4} - 10T_{5}^{3} + 40T_{5}^{2} + 50T_{5} + 25 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{7} + 29T_{7}^{6} - 42T_{7}^{5} + 131T_{7}^{4} - 150T_{7}^{3} + 185T_{7}^{2} - 100T_{7} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} + 31 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 12 T^{2} + \cdots - 41)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 20 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 942841 \) Copy content Toggle raw display
$37$ \( T^{8} - 26 T^{7} + \cdots + 1234321 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 1500625 \) Copy content Toggle raw display
$43$ \( (T^{4} - 6 T^{3} - 72 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 1874161 \) Copy content Toggle raw display
$53$ \( T^{8} - 14 T^{7} + \cdots + 483025 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots + 600625 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + \cdots + 1500625 \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} + \cdots - 409)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + \cdots + 60025 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 101761 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + \cdots + 3798601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 18 T^{3} + \cdots - 13955)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 30 T^{7} + \cdots + 410881 \) Copy content Toggle raw display
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