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Properties

Label 9280.2.a.cu
Level $9280$
Weight $2$
Character orbit 9280.a
Self dual yes
Analytic conductor $74.101$
Analytic rank $0$
Dimension $10$
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] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,10,0,0,0,16,0,0,0,-6,0,0,0,18,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.1011730757\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 23x^{8} + 169x^{6} - 440x^{4} + 308x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4640)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} - \beta_{9} q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + ( - \beta_{5} + \beta_{4} - 1) q^{13} + \beta_1 q^{15} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 2) q^{17}+ \cdots + (\beta_{7} - 2 \beta_{6} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{5} + 16 q^{9} - 6 q^{13} + 18 q^{17} + 10 q^{25} + 10 q^{29} + 32 q^{33} + 4 q^{37} + 16 q^{45} + 24 q^{49} + 14 q^{53} + 12 q^{57} - 6 q^{61} - 6 q^{65} - 6 q^{69} + 42 q^{73} - 4 q^{77} + 58 q^{81}+ \cdots + 62 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 23x^{8} + 169x^{6} - 440x^{4} + 308x^{2} - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} + 113\nu^{6} - 795\nu^{4} + 1818\nu^{2} - 656 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{8} + 67\nu^{6} - 465\nu^{4} + 1054\nu^{2} - 388 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{8} - 113\nu^{6} + 799\nu^{4} - 1862\nu^{2} + 704 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{9} - 203\nu^{7} + 1429\nu^{5} - 3300\nu^{3} + 1236\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{9} + 135\nu^{7} - 947\nu^{5} + 2175\nu^{3} - 794\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{9} + 293\nu^{7} - 2063\nu^{5} + 4790\nu^{3} - 1888\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{9} - 248\nu^{7} + 1746\nu^{5} - 4041\nu^{3} + 1526\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{3} + 11\beta_{2} + 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{9} + 12\beta_{8} + \beta_{7} - 18\beta_{6} + 89\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{5} - 5\beta_{4} + 36\beta_{3} + 119\beta_{2} + 422 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 180\beta_{9} + 129\beta_{8} + 5\beta_{7} - 247\beta_{6} + 928\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 180\beta_{5} - 113\beta_{4} + 494\beta_{3} + 1304\beta_{2} + 4387 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2045\beta_{9} + 1371\beta_{8} - 46\beta_{7} - 3079\beta_{6} + 9963\beta_1 \) 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
−3.34879
−2.67216
−1.92812
−0.687730
−0.674199
0.674199
0.687730
1.92812
2.67216
3.34879
0 −3.34879 0 1.00000 0 1.39029 0 8.21441 0
1.2 0 −2.67216 0 1.00000 0 0.473071 0 4.14044 0
1.3 0 −1.92812 0 1.00000 0 −3.41774 0 0.717631 0
1.4 0 −0.687730 0 1.00000 0 −3.50513 0 −2.52703 0
1.5 0 −0.674199 0 1.00000 0 4.56907 0 −2.54546 0
1.6 0 0.674199 0 1.00000 0 −4.56907 0 −2.54546 0
1.7 0 0.687730 0 1.00000 0 3.50513 0 −2.52703 0
1.8 0 1.92812 0 1.00000 0 3.41774 0 0.717631 0
1.9 0 2.67216 0 1.00000 0 −0.473071 0 4.14044 0
1.10 0 3.34879 0 1.00000 0 −1.39029 0 8.21441 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9280.2.a.cu 10
4.b odd 2 1 inner 9280.2.a.cu 10
8.b even 2 1 4640.2.a.z ✓ 10
8.d odd 2 1 4640.2.a.z ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4640.2.a.z ✓ 10 8.b even 2 1
4640.2.a.z ✓ 10 8.d odd 2 1
9280.2.a.cu 10 1.a even 1 1 trivial
9280.2.a.cu 10 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9280))\):

\( T_{3}^{10} - 23T_{3}^{8} + 169T_{3}^{6} - 440T_{3}^{4} + 308T_{3}^{2} - 64 \) Copy content Toggle raw display
\( T_{7}^{10} - 47T_{7}^{8} + 741T_{7}^{6} - 4404T_{7}^{4} + 6740T_{7}^{2} - 1296 \) Copy content Toggle raw display
\( T_{11}^{10} - 68T_{11}^{8} + 1460T_{11}^{6} - 10240T_{11}^{4} + 12800T_{11}^{2} - 256 \) Copy content Toggle raw display
\( T_{13}^{5} + 3T_{13}^{4} - 41T_{13}^{3} - 78T_{13}^{2} + 368T_{13} + 448 \) Copy content Toggle raw display
\( T_{19}^{10} - 92T_{19}^{8} + 2196T_{19}^{6} - 14400T_{19}^{4} + 32000T_{19}^{2} - 20736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 23 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 47 T^{8} + \cdots - 1296 \) Copy content Toggle raw display
$11$ \( T^{10} - 68 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( (T^{5} + 3 T^{4} + \cdots + 448)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 9 T^{4} + \cdots - 164)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 92 T^{8} + \cdots - 20736 \) Copy content Toggle raw display
$23$ \( T^{10} - 147 T^{8} + \cdots - 309136 \) Copy content Toggle raw display
$29$ \( (T - 1)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 203 T^{8} + \cdots - 71824 \) Copy content Toggle raw display
$37$ \( (T^{5} - 2 T^{4} + \cdots - 1648)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 96 T^{3} + \cdots - 4032)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 160782400 \) Copy content Toggle raw display
$47$ \( T^{10} - 268 T^{8} + \cdots - 24920064 \) Copy content Toggle raw display
$53$ \( (T^{5} - 7 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 1864857856 \) Copy content Toggle raw display
$61$ \( (T^{5} + 3 T^{4} - 123 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 320 T^{8} + \cdots - 17305600 \) Copy content Toggle raw display
$71$ \( T^{10} - 336 T^{8} + \cdots - 14745600 \) Copy content Toggle raw display
$73$ \( (T^{5} - 21 T^{4} + \cdots - 217276)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 215 T^{8} + \cdots - 4112784 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 5026242816 \) Copy content Toggle raw display
$89$ \( (T^{5} - 4 T^{4} + \cdots + 3744)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 31 T^{4} + \cdots + 10452)^{2} \) Copy content Toggle raw display
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