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Properties

Label 9248.2.a.bu
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $1$
Dimension $12$
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 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);
 
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")
 
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 = [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,-24,0,0,0,-12,0,0, 0,6,0,0,0,-30,0,0,0,-18,0,0,0,-54,0,0,0,12,0,0,0,6,0,0,0,18,0,0,0,-36, 0,0,0,-36,0,0,0,-90,0,0,0,18,0,0,0,-42,0,0,0,42,0,0,0,0,0,0,0,0,0,0,0, -18,0,0,0,-48,0,0,0,-78,0,0,0,42,0,0,0,-18,0,0,0,-24,0,0,0,-54,0,0,0,-6, 0,0,0,48,0,0,0,-24,0,0,0,-60,0,0,0,-48,0,0,0,24,0,0,0,42,0,0,0,24] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 207x^{8} - 806x^{6} + 1500x^{4} - 1224x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{11} - \beta_1) q^{7} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{9} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{11} + ( - \beta_{4} - 2 \beta_{2}) q^{13}+ \cdots + ( - \beta_{9} + 2 \beta_{5} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{9} - 24 q^{21} - 12 q^{25} + 6 q^{29} - 30 q^{33} - 18 q^{37} - 54 q^{41} + 12 q^{45} + 6 q^{49} + 18 q^{53} - 36 q^{57} - 36 q^{61} - 90 q^{65} + 18 q^{69} - 42 q^{73} + 42 q^{77} - 18 q^{89}+ \cdots - 78 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 24x^{10} + 207x^{8} - 806x^{6} + 1500x^{4} - 1224x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{10} + 439\nu^{8} - 3525\nu^{6} + 11795\nu^{4} - 15376\nu^{2} + 4692 ) / 289 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\nu^{11} - 1627\nu^{9} + 11213\nu^{7} - 28742\nu^{5} + 25808\nu^{3} - 4641\nu ) / 578 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67\nu^{10} - 1472\nu^{8} + 10894\nu^{6} - 32208\nu^{4} + 37413\nu^{2} - 11237 ) / 578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67\nu^{11} - 1472\nu^{9} + 10894\nu^{7} - 32208\nu^{5} + 37413\nu^{3} - 11237\nu ) / 578 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 88\nu^{10} - 1942\nu^{8} + 14425\nu^{6} - 42385\nu^{4} + 46422\nu^{2} - 10795 ) / 578 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -115\nu^{10} + 2505\nu^{8} - 18263\nu^{6} + 52332\nu^{4} - 56560\nu^{2} + 14025 ) / 578 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 155\nu^{10} - 3414\nu^{8} + 25319\nu^{6} - 74593\nu^{4} + 84413\nu^{2} - 23766 ) / 578 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -182\nu^{11} + 3977\nu^{9} - 29157\nu^{7} + 84540\nu^{5} - 93973\nu^{3} + 25262\nu ) / 578 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -86\nu^{11} + 1911\nu^{9} - 14419\nu^{7} + 44003\nu^{5} - 53078\nu^{3} + 17663\nu ) / 289 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 203\nu^{11} - 4447\nu^{9} + 32688\nu^{7} - 94717\nu^{5} + 102982\nu^{3} - 24242\nu ) / 578 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} - \beta_{5} + \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{8} - 2\beta_{7} - 10\beta_{6} - 14\beta_{4} - \beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{10} + 7\beta_{9} - 17\beta_{5} + 9\beta_{3} + 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 90\beta_{8} - 34\beta_{7} - 100\beta_{6} - 146\beta_{4} - 19\beta_{2} + 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{11} - 90\beta_{10} + 27\beta_{9} - 209\beta_{5} + 71\beta_{3} + 408\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 805\beta_{8} - 440\beta_{7} - 1007\beta_{6} - 1435\beta_{4} - 247\beta_{2} + 1081 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -202\beta_{11} - 805\beta_{10} - 84\beta_{9} - 2324\beta_{5} + 558\beta_{3} + 3698\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7301\beta_{8} - 5100\beta_{7} - 10113\beta_{6} - 13951\beta_{4} - 2818\beta_{2} + 9602 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2812\beta_{11} - 7301\beta_{10} - 3429\beta_{9} - 24681\beta_{5} + 4483\beta_{3} + 34317\beta_1 \) 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
−3.13779
−2.65056
−1.65054
−1.46946
−1.36537
−0.617237
0.617237
1.36537
1.46946
1.65054
2.65056
3.13779
0 −3.13779 0 −0.899091 0 −0.178036 0 6.84570 0
1.2 0 −2.65056 0 2.74951 0 2.42587 0 4.02547 0
1.3 0 −1.65054 0 1.38735 0 1.00535 0 −0.275718 0
1.4 0 −1.46946 0 −1.21742 0 −2.45988 0 −0.840680 0
1.5 0 −1.36537 0 −3.26673 0 4.21284 0 −1.13576 0
1.6 0 −0.617237 0 1.24639 0 3.77806 0 −2.61902 0
1.7 0 0.617237 0 1.24639 0 −3.77806 0 −2.61902 0
1.8 0 1.36537 0 −3.26673 0 −4.21284 0 −1.13576 0
1.9 0 1.46946 0 −1.21742 0 2.45988 0 −0.840680 0
1.10 0 1.65054 0 1.38735 0 −1.00535 0 −0.275718 0
1.11 0 2.65056 0 2.74951 0 −2.42587 0 4.02547 0
1.12 0 3.13779 0 −0.899091 0 0.178036 0 6.84570 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
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
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 9248.2.a.bu yes 12
4.b odd 2 1 inner 9248.2.a.bu yes 12
17.b even 2 1 9248.2.a.bt ✓ 12
68.d odd 2 1 9248.2.a.bt ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9248.2.a.bt ✓ 12 17.b even 2 1
9248.2.a.bt ✓ 12 68.d odd 2 1
9248.2.a.bu yes 12 1.a even 1 1 trivial
9248.2.a.bu yes 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{12} - 24T_{3}^{10} + 207T_{3}^{8} - 806T_{3}^{6} + 1500T_{3}^{4} - 1224T_{3}^{2} + 289 \) Copy content Toggle raw display
\( T_{5}^{6} - 12T_{5}^{4} + 4T_{5}^{3} + 27T_{5}^{2} - 6T_{5} - 17 \) Copy content Toggle raw display
\( T_{7}^{12} - 45T_{7}^{10} + 717T_{7}^{8} - 4865T_{7}^{6} + 13383T_{7}^{4} - 9537T_{7}^{2} + 289 \) Copy content Toggle raw display
\( T_{19}^{12} - 96T_{19}^{10} + 3024T_{19}^{8} - 37454T_{19}^{6} + 181203T_{19}^{4} - 280908T_{19}^{2} + 83521 \) Copy content Toggle raw display
\( T_{43}^{12} - 330T_{43}^{10} + 40743T_{43}^{8} - 2256512T_{43}^{6} + 50346690T_{43}^{4} - 208969542T_{43}^{2} + 24137569 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 24 T^{10} + \cdots + 289 \) Copy content Toggle raw display
$5$ \( (T^{6} - 12 T^{4} + \cdots - 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 45 T^{10} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{12} - 90 T^{10} + \cdots + 6677056 \) Copy content Toggle raw display
$13$ \( (T^{6} - 48 T^{4} + \cdots - 593)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 96 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
$23$ \( T^{12} - 141 T^{10} + \cdots + 30151081 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} + \cdots + 1819)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 360582121 \) Copy content Toggle raw display
$37$ \( (T^{6} + 9 T^{5} + \cdots + 7633)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 27 T^{5} + \cdots - 4607)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 330 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2442237561 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9 T^{5} + \cdots - 3672)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 421029361 \) Copy content Toggle raw display
$61$ \( (T^{6} + 18 T^{5} + \cdots + 40375)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 33254804881 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 172685881 \) Copy content Toggle raw display
$73$ \( (T^{6} + 21 T^{5} + \cdots + 110007)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 248985032256 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 6975757441 \) Copy content Toggle raw display
$89$ \( (T^{6} + 9 T^{5} + 3 T^{4} + \cdots - 3)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 39 T^{5} + \cdots - 294848)^{2} \) Copy content Toggle raw display
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