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Properties

Label 7225.2.a.bk
Level $7225$
Weight $2$
Character orbit 7225.a
Self dual yes
Analytic conductor $57.692$
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] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8,0,6,0] 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: yes
Analytic conductor: \(57.6919154604\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.4217732978944.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 40x^{4} - 47x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1445)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{5} + 1) q^{6} + \beta_1 q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{9} + ( - \beta_{6} + 1) q^{11}+ \cdots + ( - \beta_{6} - \beta_{5} - 3 \beta_{2} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 6 q^{6} + 8 q^{9} + 10 q^{11} + 24 q^{14} + 16 q^{16} - 4 q^{19} + 6 q^{21} + 12 q^{24} + 24 q^{26} - 4 q^{29} + 24 q^{31} - 18 q^{36} + 26 q^{39} + 22 q^{44} + 8 q^{46} - 32 q^{49} + 28 q^{54}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 40x^{4} - 47x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 10\nu^{5} + 20\nu^{3} - 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 10\nu^{4} + 20\nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 11\nu^{4} - 29\nu^{2} + 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 11\nu^{5} - 29\nu^{3} + 19\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + 9\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{4} + 9\beta_{3} + 33\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{6} + 11\beta_{5} + 70\beta_{2} + 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{7} + 22\beta_{4} + 70\beta_{3} + 237\beta_1 \) 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
−2.72535
−1.58320
−1.22010
−0.759813
0.759813
1.22010
1.58320
2.72535
−2.72535 −0.502135 5.42752 0 1.36849 −2.72535 −9.34117 −2.74786 0
1.2 −1.58320 −3.12562 0.506523 0 4.94849 −1.58320 2.36447 6.76953 0
1.3 −1.22010 2.38602 −0.511354 0 −2.91119 −1.22010 3.06410 2.69310 0
1.4 −0.759813 0.534070 −1.42268 0 −0.405793 −0.759813 2.60060 −2.71477 0
1.5 0.759813 −0.534070 −1.42268 0 −0.405793 0.759813 −2.60060 −2.71477 0
1.6 1.22010 −2.38602 −0.511354 0 −2.91119 1.22010 −3.06410 2.69310 0
1.7 1.58320 3.12562 0.506523 0 4.94849 1.58320 −2.36447 6.76953 0
1.8 2.72535 0.502135 5.42752 0 1.36849 2.72535 9.34117 −2.74786 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
\(5\) \( -1 \)
\(17\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7225.2.a.bk 8
5.b even 2 1 inner 7225.2.a.bk 8
5.c odd 4 2 1445.2.b.c ✓ 8
17.b even 2 1 7225.2.a.bj 8
85.c even 2 1 7225.2.a.bj 8
85.g odd 4 2 1445.2.b.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1445.2.b.c ✓ 8 5.c odd 4 2
1445.2.b.d yes 8 85.g odd 4 2
7225.2.a.bj 8 17.b even 2 1
7225.2.a.bj 8 85.c even 2 1
7225.2.a.bk 8 1.a even 1 1 trivial
7225.2.a.bk 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):

\( T_{2}^{8} - 12T_{2}^{6} + 40T_{2}^{4} - 47T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{8} - 16T_{3}^{6} + 64T_{3}^{4} - 31T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 12T_{7}^{6} + 40T_{7}^{4} - 47T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 5T_{11}^{3} - 11T_{11}^{2} + 65T_{11} - 58 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 12 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 5 T^{3} - 11 T^{2} + \cdots - 58)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 53 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots - 278)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 104 T^{6} + \cdots + 1444 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots + 514)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + \cdots - 304)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 124 T^{6} + \cdots + 473344 \) Copy content Toggle raw display
$41$ \( (T^{4} - 33 T^{2} - 28 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 198 T^{6} + \cdots + 4227136 \) Copy content Toggle raw display
$47$ \( T^{8} - 218 T^{6} + \cdots + 3984016 \) Copy content Toggle raw display
$53$ \( T^{8} - 53 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{4} + 11 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3 T^{3} - 83 T^{2} + \cdots + 38)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 215 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$71$ \( (T^{4} - T^{3} - 90 T^{2} + \cdots + 1184)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 482 T^{6} + \cdots + 152078224 \) Copy content Toggle raw display
$79$ \( (T^{4} + 15 T^{3} + 68 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 308 T^{6} + \cdots + 4605316 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 505 T^{6} + \cdots + 26790976 \) Copy content Toggle raw display
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