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Properties

Label 4576.2.a.w
Level $4576$
Weight $2$
Character orbit 4576.a
Self dual yes
Analytic conductor $36.540$
Analytic rank $0$
Dimension $9$
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] = [4576,2,Mod(1,4576)] 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("4576.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4576 = 2^{5} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4576.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,-2,0,2,0,5] 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: \(36.5395439649\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 20x^{7} + 36x^{6} + 114x^{5} - 166x^{4} - 183x^{3} + 168x^{2} + 100x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{8} + 1) q^{7} + (\beta_{5} - \beta_{4} + 2) q^{9} + q^{11} - q^{13} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots + 1) q^{15} + (\beta_{7} + \beta_{6} - \beta_{2} + \beta_1) q^{17}+ \cdots + (\beta_{5} - \beta_{4} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 2 q^{3} + 2 q^{5} + 5 q^{7} + 17 q^{9} + 9 q^{11} - 9 q^{13} + 14 q^{15} + q^{17} + 11 q^{21} + 7 q^{23} + 15 q^{25} - 8 q^{27} + 24 q^{29} + 17 q^{31} - 2 q^{33} - 13 q^{35} + 8 q^{37} + 2 q^{39}+ \cdots + 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 20x^{7} + 36x^{6} + 114x^{5} - 166x^{4} - 183x^{3} + 168x^{2} + 100x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{8} - 4\nu^{7} + 52\nu^{6} + 44\nu^{5} - 262\nu^{4} - 10\nu^{3} + 529\nu^{2} - 202\nu - 280 ) / 88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 8\nu^{7} - 120\nu^{5} - 138\nu^{4} + 394\nu^{3} + 329\nu^{2} - 210\nu - 120 ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 4\nu^{7} - 28\nu^{6} - 92\nu^{5} + 226\nu^{4} + 582\nu^{3} - 467\nu^{2} - 642\nu + 184 ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 4\nu^{7} - 28\nu^{6} - 92\nu^{5} + 226\nu^{4} + 582\nu^{3} - 379\nu^{2} - 642\nu - 256 ) / 88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{8} - 128\nu^{6} - 8\nu^{5} + 610\nu^{4} + 198\nu^{3} - 481\nu^{2} - 742\nu - 280 ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 18\nu^{6} + 74\nu^{5} + 74\nu^{4} - 362\nu^{3} + 9\nu^{2} + 386\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{8} + 4\nu^{7} + 41\nu^{6} - 65\nu^{5} - 235\nu^{4} + 241\nu^{3} + 319\nu^{2} - 137\nu - 70 ) / 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - 2\beta_{7} - \beta_{6} + 10\beta_{5} - 12\beta_{4} + \beta_{3} + 2\beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - 11\beta_{7} + 13\beta_{6} - 12\beta_{5} - 13\beta_{4} - 15\beta_{3} - 3\beta_{2} + 90\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{8} - 28\beta_{7} - 15\beta_{6} + 102\beta_{5} - 131\beta_{4} + 21\beta_{3} + 5\beta_{2} + 28\beta _1 + 472 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{8} - 110 \beta_{7} + 152 \beta_{6} - 134 \beta_{5} - 136 \beta_{4} - 186 \beta_{3} - 56 \beta_{2} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 372 \beta_{8} - 322 \beta_{7} - 188 \beta_{6} + 1077 \beta_{5} - 1405 \beta_{4} + 308 \beta_{3} + \cdots + 4989 \) 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
3.24401
3.21164
1.70009
1.21018
0.134539
−0.640259
−1.05378
−2.44979
−3.35663
0 −3.24401 0 2.62946 0 −2.16008 0 7.52360 0
1.2 0 −3.21164 0 −4.15198 0 0.804420 0 7.31461 0
1.3 0 −1.70009 0 −1.98413 0 4.11003 0 −0.109680 0
1.4 0 −1.21018 0 0.918726 0 −4.48011 0 −1.53546 0
1.5 0 −0.134539 0 3.38306 0 4.73411 0 −2.98190 0
1.6 0 0.640259 0 −0.269583 0 −1.51477 0 −2.59007 0
1.7 0 1.05378 0 −2.54729 0 1.51817 0 −1.88954 0
1.8 0 2.44979 0 3.58766 0 −1.62652 0 3.00145 0
1.9 0 3.35663 0 0.434073 0 3.61476 0 8.26698 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( -1 \)
\(13\) \( +1 \)

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 4576.2.a.w ✓ 9
4.b odd 2 1 4576.2.a.y yes 9
8.b even 2 1 9152.2.a.db 9
8.d odd 2 1 9152.2.a.cz 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4576.2.a.w ✓ 9 1.a even 1 1 trivial
4576.2.a.y yes 9 4.b odd 2 1
9152.2.a.cz 9 8.d odd 2 1
9152.2.a.db 9 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4576))\):

\( T_{3}^{9} + 2T_{3}^{8} - 20T_{3}^{7} - 36T_{3}^{6} + 114T_{3}^{5} + 166T_{3}^{4} - 183T_{3}^{3} - 168T_{3}^{2} + 100T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{9} - 2T_{5}^{8} - 28T_{5}^{7} + 56T_{5}^{6} + 221T_{5}^{5} - 394T_{5}^{4} - 526T_{5}^{3} + 748T_{5}^{2} - 36T_{5} - 72 \) Copy content Toggle raw display
\( T_{7}^{9} - 5T_{7}^{8} - 30T_{7}^{7} + 152T_{7}^{6} + 246T_{7}^{5} - 1176T_{7}^{4} - 1017T_{7}^{3} + 2853T_{7}^{2} + 1376T_{7} - 2048 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{9} - 2 T^{8} + \cdots - 72 \) Copy content Toggle raw display
$7$ \( T^{9} - 5 T^{8} + \cdots - 2048 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( (T + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} + \cdots + 12928 \) Copy content Toggle raw display
$19$ \( T^{9} - 61 T^{7} + \cdots + 2048 \) Copy content Toggle raw display
$23$ \( T^{9} - 7 T^{8} + \cdots - 784 \) Copy content Toggle raw display
$29$ \( T^{9} - 24 T^{8} + \cdots + 170496 \) Copy content Toggle raw display
$31$ \( T^{9} - 17 T^{8} + \cdots - 16384 \) Copy content Toggle raw display
$37$ \( T^{9} - 8 T^{8} + \cdots - 224768 \) Copy content Toggle raw display
$41$ \( T^{9} - 10 T^{8} + \cdots - 682576 \) Copy content Toggle raw display
$43$ \( T^{9} + 27 T^{8} + \cdots - 24224 \) Copy content Toggle raw display
$47$ \( T^{9} - 15 T^{8} + \cdots + 472064 \) Copy content Toggle raw display
$53$ \( T^{9} - 28 T^{8} + \cdots - 61557728 \) Copy content Toggle raw display
$59$ \( T^{9} + T^{8} + \cdots - 1936512 \) Copy content Toggle raw display
$61$ \( T^{9} - 20 T^{8} + \cdots - 14143232 \) Copy content Toggle raw display
$67$ \( T^{9} + 3 T^{8} + \cdots + 17216 \) Copy content Toggle raw display
$71$ \( T^{9} + 8 T^{8} + \cdots + 683264 \) Copy content Toggle raw display
$73$ \( T^{9} - 2 T^{8} + \cdots - 1071184 \) Copy content Toggle raw display
$79$ \( T^{9} - 24 T^{8} + \cdots + 10616832 \) Copy content Toggle raw display
$83$ \( T^{9} + 4 T^{8} + \cdots - 89940992 \) Copy content Toggle raw display
$89$ \( T^{9} - 17 T^{8} + \cdots - 28017664 \) Copy content Toggle raw display
$97$ \( T^{9} - 3 T^{8} + \cdots - 2080768 \) Copy content Toggle raw display
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