↑

Properties

Label 9248.2.a.y
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

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 = [4,0,-2,0,2,0,-6,0,6,0,0,0,2,0,8,0,0,0,-6,0,12,0,-6,0,2,0,-8, 0,2,0,0,0,-10,0,-14,0,0,0,-26,0,-16,0,8,0,-20,0,8,0,14,0,0,0,-6,0,-22, 0,6,0,-16,0,20,0,-24,0,4,0,0,0,-4,0,0,0,4,0,-8,0,-2,0,-24,0,32,0,26,0, 0,0,-10,0,0,0,-32,0,26,0,32,0,-22,0,50,0,-4,0,-32,0,-14,0,-32,0,28,0,32, 0,10,0,10,0,2,0,0,0,-8,0,-8,0,36,0,-8,0,-46,0,-16,0,4,0,38,0,24,0,-22, 0,2,0,-2,0,-20] 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: \(4\)
Coefficient field: 4.4.14272.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 2x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{9} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - \beta_{2}) q^{13}+ \cdots + ( - 5 \beta_{3} - 4 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 6 q^{9} + 2 q^{13} + 8 q^{15} - 6 q^{19} + 12 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} - 10 q^{33} - 14 q^{35} - 26 q^{39} - 16 q^{41} + 8 q^{43} - 20 q^{45}+ \cdots + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 5x^{2} + 2x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 2\nu + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 3\beta_{2} + 8\beta _1 + 3 \) 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
−1.43700
3.25919
0.894176
−0.716360
0 −3.28831 0 −1.93899 0 −3.78633 0 7.81299 0
1.2 0 −0.765213 0 −1.10393 0 2.59790 0 −2.41445 0
1.3 0 −0.472066 0 3.98880 0 −4.56669 0 −2.77715 0
1.4 0 2.52559 0 1.05411 0 −0.244878 0 3.37861 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 9248.2.a.y yes 4
4.b odd 2 1 9248.2.a.bk yes 4
17.b even 2 1 9248.2.a.bj yes 4
68.d odd 2 1 9248.2.a.x ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9248.2.a.x ✓ 4 68.d odd 2 1
9248.2.a.y yes 4 1.a even 1 1 trivial
9248.2.a.bj yes 4 17.b even 2 1
9248.2.a.bk yes 4 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 10T_{3} - 3 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} - 9T_{5}^{2} + 2T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 6T_{7}^{3} - 3T_{7}^{2} - 46T_{7} - 11 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 47T_{19}^{2} - 230T_{19} + 293 \) Copy content Toggle raw display
\( T_{43}^{4} - 8T_{43}^{3} - 42T_{43}^{2} + 288T_{43} + 373 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 41 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 293 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - 34 T^{2} + \cdots + 253 \) Copy content Toggle raw display
$37$ \( T^{4} - 56 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + \cdots - 1088 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + \cdots + 373 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots - 1131 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots - 1791 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} + \cdots - 1088 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 152 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$71$ \( T^{4} - 64 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 1977 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots - 1472 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + \cdots - 507 \) Copy content Toggle raw display
$89$ \( T^{4} - 128 T^{2} + \cdots - 832 \) Copy content Toggle raw display
$97$ \( T^{4} + 22 T^{3} + \cdots - 263 \) Copy content Toggle raw display
show more
show less