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Properties

Label 9248.2.a.bl
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-18,0,0,0,0, 0,0,0,-24,0,0,0,0,0,0,0,0,0,-24,0,0,0,-24,0,30,0,0,0,0,0,-24,0,0,0,0,0, 0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0,0,0,0,0,0,-18,0,-48,0,0,0,-24,0,-24,0, 0,0,48,0,0,0,0,0,0,0,48,0,-24,0,0,0,0,0,0,0,-48,0,0,0,24,0,-48,0,0,0,6, 0,-48,0,0,0,24,0,0,0,0,0,0,0,24,0,-24,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.8456517893\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.3359232.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 12x^{4} + 36x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 544)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{5} q^{7} + (\beta_{3} + 1) q^{9} + ( - 2 \beta_{2} - \beta_1) q^{11} - \beta_{3} q^{13} + (\beta_{4} + \beta_{3}) q^{15} + (\beta_{4} + \beta_{3}) q^{19} + ( - 2 \beta_{4} - \beta_{3}) q^{21}+ \cdots + ( - 2 \beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} - 18 q^{25} - 24 q^{33} - 24 q^{43} - 24 q^{47} + 30 q^{49} - 24 q^{55} - 24 q^{67} - 18 q^{81} - 48 q^{83} - 24 q^{87} - 24 q^{89} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 12x^{4} + 36x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 8\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 10\nu^{3} + 22\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 8\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 20\beta_{2} + 38\beta_1 \) 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
−2.65785
−2.16670
−0.491151
0.491151
2.16670
2.65785
0 −2.65785 0 −1.41421 0 1.67555 0 4.06418 0
1.2 0 −2.16670 0 1.41421 0 −3.14900 0 1.69459 0
1.3 0 −0.491151 0 1.41421 0 4.82455 0 −2.75877 0
1.4 0 0.491151 0 −1.41421 0 −4.82455 0 −2.75877 0
1.5 0 2.16670 0 −1.41421 0 3.14900 0 1.69459 0
1.6 0 2.65785 0 1.41421 0 −1.67555 0 4.06418 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(17\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9248.2.a.bl 6
4.b odd 2 1 9248.2.a.bm 6
17.b even 2 1 inner 9248.2.a.bl 6
17.d even 8 2 544.2.o.h yes 6
68.d odd 2 1 9248.2.a.bm 6
68.g odd 8 2 544.2.o.g ✓ 6
136.o even 8 2 1088.2.o.u 6
136.p odd 8 2 1088.2.o.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.o.g ✓ 6 68.g odd 8 2
544.2.o.h yes 6 17.d even 8 2
1088.2.o.u 6 136.o even 8 2
1088.2.o.v 6 136.p odd 8 2
9248.2.a.bl 6 1.a even 1 1 trivial
9248.2.a.bl 6 17.b even 2 1 inner
9248.2.a.bm 6 4.b odd 2 1
9248.2.a.bm 6 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{6} - 12T_{3}^{4} + 36T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} - 36T_{7}^{4} + 324T_{7}^{2} - 648 \) Copy content Toggle raw display
\( T_{19}^{3} - 12T_{19} - 8 \) Copy content Toggle raw display
\( T_{43}^{3} + 12T_{43}^{2} + 12T_{43} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 12 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 36 T^{4} + \cdots - 648 \) Copy content Toggle raw display
$11$ \( T^{6} - 36 T^{4} + \cdots - 72 \) Copy content Toggle raw display
$13$ \( (T^{3} - 12 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} - 12 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 60 T^{4} + \cdots - 2888 \) Copy content Toggle raw display
$29$ \( T^{6} - 54 T^{4} + \cdots - 2888 \) Copy content Toggle raw display
$31$ \( T^{6} - 108 T^{4} + \cdots - 20808 \) Copy content Toggle raw display
$37$ \( T^{6} - 198 T^{4} + \cdots - 98568 \) Copy content Toggle raw display
$41$ \( T^{6} - 198 T^{4} + \cdots - 2312 \) Copy content Toggle raw display
$43$ \( (T^{3} + 12 T^{2} + 12 T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 12 T^{2} - 192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 144 T - 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 84 T + 136)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 198 T^{4} + \cdots - 98568 \) Copy content Toggle raw display
$67$ \( (T + 4)^{6} \) Copy content Toggle raw display
$71$ \( T^{6} - 84 T^{4} + \cdots - 10952 \) Copy content Toggle raw display
$73$ \( T^{6} - 198 T^{4} + \cdots - 98568 \) Copy content Toggle raw display
$79$ \( T^{6} - 300 T^{4} + \cdots - 129032 \) Copy content Toggle raw display
$83$ \( (T^{3} + 24 T^{2} + \cdots - 456)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} + \cdots - 712)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 198 T^{4} + \cdots - 648 \) Copy content Toggle raw display
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