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Properties

Label 5776.2.a.ca
Level $5776$
Weight $2$
Character orbit 5776.a
Self dual yes
Analytic conductor $46.122$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [5776,2,Mod(1,5776)] 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("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-6,0,-2,0,2,0,2,0,-12,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.1215922075\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.10564000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2888)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{7} + (\beta_{7} + \beta_{6} - \beta_1 + 1) q^{9} + ( - \beta_{5} + \beta_{2} + 2 \beta_1 - 2) q^{11}+ \cdots + ( - 3 \beta_{7} - 5 \beta_{6} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 12 q^{11} + 6 q^{13} - 10 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 12 q^{27} + 18 q^{29} - 14 q^{31} + 40 q^{33} - 18 q^{35} + 16 q^{37} - 28 q^{39} + 12 q^{41}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 12\nu^{3} + 16\nu^{2} - 16\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 8\nu^{5} + 12\nu^{4} + 16\nu^{3} - 16\nu^{2} - 7\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 9\nu^{2} - 13\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 4\nu^{5} + 26\nu^{4} - 4\nu^{3} - 36\nu^{2} + 11\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} - 12\nu^{5} + 28\nu^{4} + 8\nu^{3} - 30\nu^{2} + 12\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} + 12\nu^{5} - 28\nu^{4} - 8\nu^{3} + 32\nu^{2} - 14\nu - 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} + 8\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} + 9\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} + 14\beta_{6} + 8\beta_{5} + 10\beta_{4} - 10\beta_{3} - 3\beta_{2} + 34\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 62\beta_{7} + 64\beta_{6} + 12\beta_{5} + 24\beta_{4} - 16\beta_{3} - 20\beta_{2} + 80\beta _1 + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 132\beta_{7} + 144\beta_{6} + 60\beta_{5} + 88\beta_{4} - 82\beta_{3} - 40\beta_{2} + 267\beta _1 + 208 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.11762
−1.32561
−1.26814
−0.0583903
0.650108
1.49959
1.67642
2.94364
0 −3.11762 0 0.354750 0 −0.325225 0 6.71958 0
1.2 0 −2.32561 0 −4.27353 0 4.02384 0 2.40845 0
1.3 0 −2.26814 0 2.03239 0 −4.86584 0 2.14447 0
1.4 0 −1.05839 0 3.24714 0 2.81724 0 −1.87981 0
1.5 0 −0.349892 0 −2.14090 0 3.07223 0 −2.87758 0
1.6 0 0.499591 0 0.989826 0 0.882762 0 −2.75041 0
1.7 0 0.676424 0 −1.78749 0 −3.10132 0 −2.54245 0
1.8 0 1.94364 0 −0.422186 0 −0.503690 0 0.777742 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5776.2.a.ca 8
4.b odd 2 1 2888.2.a.w yes 8
19.b odd 2 1 5776.2.a.cc 8
76.d even 2 1 2888.2.a.v ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2888.2.a.v ✓ 8 76.d even 2 1
2888.2.a.w yes 8 4.b odd 2 1
5776.2.a.ca 8 1.a even 1 1 trivial
5776.2.a.cc 8 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5776))\):

\( T_{3}^{8} + 6T_{3}^{7} + 5T_{3}^{6} - 26T_{3}^{5} - 41T_{3}^{4} + 14T_{3}^{3} + 30T_{3}^{2} - 4T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 19T_{5}^{6} - 24T_{5}^{5} + 89T_{5}^{4} + 68T_{5}^{3} - 116T_{5}^{2} - 16T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} - 32T_{7}^{6} + 76T_{7}^{5} + 225T_{7}^{4} - 574T_{7}^{3} - 102T_{7}^{2} + 268T_{7} + 76 \) Copy content Toggle raw display
\( T_{11}^{8} + 12T_{11}^{7} + 21T_{11}^{6} - 204T_{11}^{5} - 621T_{11}^{4} + 918T_{11}^{3} + 3234T_{11}^{2} - 996T_{11} - 1604 \) Copy content Toggle raw display
\( T_{13}^{8} - 6T_{13}^{7} - 25T_{13}^{6} + 136T_{13}^{5} + 64T_{13}^{4} - 694T_{13}^{3} + 475T_{13}^{2} + 284T_{13} - 239 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 76 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots - 1604 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + \cdots - 239 \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} + \cdots - 4724 \) Copy content Toggle raw display
$29$ \( T^{8} - 18 T^{7} + \cdots - 138119 \) Copy content Toggle raw display
$31$ \( T^{8} + 14 T^{7} + \cdots - 285444 \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + \cdots - 121999 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots - 1573619 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 8396 \) Copy content Toggle raw display
$47$ \( T^{8} + 24 T^{7} + \cdots - 59204 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots - 155439 \) Copy content Toggle raw display
$59$ \( T^{8} - 16 T^{7} + \cdots - 675044 \) Copy content Toggle raw display
$61$ \( T^{8} - 273 T^{6} + \cdots + 72041 \) Copy content Toggle raw display
$67$ \( T^{8} + 42 T^{7} + \cdots - 39124 \) Copy content Toggle raw display
$71$ \( T^{8} + 36 T^{7} + \cdots - 3765204 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + \cdots + 841525 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} + \cdots + 11324416 \) Copy content Toggle raw display
$83$ \( T^{8} + 12 T^{7} + \cdots - 64 \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 12007301 \) Copy content Toggle raw display
$97$ \( T^{8} - 28 T^{7} + \cdots - 5120839 \) Copy content Toggle raw display
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