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Properties

Label 9280.2.a.ct
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,-10,0,0,0,32,0,0,0,-18,0,0,0,18,0,0,0,-16] 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: \( x^{10} - 31x^{8} + 353x^{6} - 1760x^{4} + 3412x^{2} - 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + \beta_{7} q^{7} + (\beta_{2} + 3) q^{9} + \beta_{8} q^{11} + (\beta_{6} + \beta_{2} - 2) q^{13} - \beta_1 q^{15} + (\beta_{6} + \beta_{5} + 2) q^{17} + (\beta_{8} - \beta_{7} + \beta_{3}) q^{19}+ \cdots + (\beta_{8} + 4 \beta_{7} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{5} + 32 q^{9} - 18 q^{13} + 18 q^{17} - 16 q^{21} + 10 q^{25} - 10 q^{29} + 8 q^{33} - 4 q^{37} + 16 q^{41} - 32 q^{45} + 48 q^{49} + 2 q^{53} + 52 q^{57} + 22 q^{61} + 18 q^{65} + 14 q^{69}+ \cdots - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 31x^{8} + 353x^{6} - 1760x^{4} + 3412x^{2} - 1152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 17\nu^{5} - 71\nu^{3} + 18\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 17\nu^{5} + 83\nu^{3} - 114\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 25\nu^{6} + 195\nu^{4} - 478\nu^{2} + 144 ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 21\nu^{6} + 139\nu^{4} - 302\nu^{2} + 120 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 23\nu^{7} + 169\nu^{5} - 400\nu^{3} + 76\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 23\nu^{7} + 177\nu^{5} - 512\nu^{3} + 428\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{8} + 23\nu^{6} - 173\nu^{4} + 450\nu^{2} - 198 ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{6} - 2\beta_{5} + 10\beta_{2} + 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{8} - 3\beta_{7} + 14\beta_{4} + 14\beta_{3} + 68\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{9} - 11\beta_{6} - 34\beta_{5} + 96\beta_{2} + 428 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 51\beta_{8} - 51\beta_{7} + 167\beta_{4} + 155\beta_{3} + 606\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -155\beta_{9} - 80\beta_{6} - 436\beta_{5} + 928\beta_{2} + 3869 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 666\beta_{8} - 642\beta_{7} + 1875\beta_{4} + 1599\beta_{3} + 5570\beta_1 \) 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
−3.19336
−2.98091
−2.72980
−2.01067
−0.649617
0.649617
2.01067
2.72980
2.98091
3.19336
0 −3.19336 0 −1.00000 0 0.662809 0 7.19754 0
1.2 0 −2.98091 0 −1.00000 0 4.60019 0 5.88584 0
1.3 0 −2.72980 0 −1.00000 0 −5.23110 0 4.45183 0
1.4 0 −2.01067 0 −1.00000 0 2.64497 0 1.04278 0
1.5 0 −0.649617 0 −1.00000 0 1.74317 0 −2.57800 0
1.6 0 0.649617 0 −1.00000 0 −1.74317 0 −2.57800 0
1.7 0 2.01067 0 −1.00000 0 −2.64497 0 1.04278 0
1.8 0 2.72980 0 −1.00000 0 5.23110 0 4.45183 0
1.9 0 2.98091 0 −1.00000 0 −4.60019 0 5.88584 0
1.10 0 3.19336 0 −1.00000 0 −0.662809 0 7.19754 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.ct 10
4.b odd 2 1 inner 9280.2.a.ct 10
8.b even 2 1 4640.2.a.ba ✓ 10
8.d odd 2 1 4640.2.a.ba ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4640.2.a.ba ✓ 10 8.b even 2 1
4640.2.a.ba ✓ 10 8.d odd 2 1
9280.2.a.ct 10 1.a even 1 1 trivial
9280.2.a.ct 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9280))\):

\( T_{3}^{10} - 31T_{3}^{8} + 353T_{3}^{6} - 1760T_{3}^{4} + 3412T_{3}^{2} - 1152 \) Copy content Toggle raw display
\( T_{7}^{10} - 59T_{7}^{8} + 1113T_{7}^{6} - 7320T_{7}^{4} + 15316T_{7}^{2} - 5408 \) Copy content Toggle raw display
\( T_{11}^{10} - 104T_{11}^{8} + 3540T_{11}^{6} - 40080T_{11}^{4} + 28864T_{11}^{2} - 512 \) Copy content Toggle raw display
\( T_{13}^{5} + 9T_{13}^{4} - 17T_{13}^{3} - 278T_{13}^{2} - 384T_{13} + 112 \) Copy content Toggle raw display
\( T_{19}^{10} - 176T_{19}^{8} + 11316T_{19}^{6} - 314832T_{19}^{4} + 3202240T_{19}^{2} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 31 T^{8} + \cdots - 1152 \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 59 T^{8} + \cdots - 5408 \) Copy content Toggle raw display
$11$ \( T^{10} - 104 T^{8} + \cdots - 512 \) Copy content Toggle raw display
$13$ \( (T^{5} + 9 T^{4} + \cdots + 112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 9 T^{4} + \cdots - 564)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 176 T^{8} + \cdots - 512 \) Copy content Toggle raw display
$23$ \( T^{10} - 119 T^{8} + \cdots - 1451808 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 159 T^{8} + \cdots - 2592 \) Copy content Toggle raw display
$37$ \( (T^{5} + 2 T^{4} + \cdots + 1488)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 8 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} - 163 T^{8} + \cdots - 93312 \) Copy content Toggle raw display
$47$ \( T^{10} - 228 T^{8} + \cdots - 8388608 \) Copy content Toggle raw display
$53$ \( (T^{5} - T^{4} - 161 T^{3} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 271 T^{8} + \cdots - 3527168 \) Copy content Toggle raw display
$61$ \( (T^{5} - 11 T^{4} + \cdots - 3568)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 196 T^{8} + \cdots - 8192 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 289538048 \) Copy content Toggle raw display
$73$ \( (T^{5} + 15 T^{4} + \cdots - 9252)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 195 T^{8} + \cdots - 39926048 \) Copy content Toggle raw display
$83$ \( T^{10} - 312 T^{8} + \cdots - 4608 \) Copy content Toggle raw display
$89$ \( (T^{5} - 12 T^{4} + \cdots - 9376)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 5 T^{4} + \cdots + 38956)^{2} \) Copy content Toggle raw display
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