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Properties

Label 9248.2.a.bw
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $0$
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 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 = [12,0,0,0,12,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,36,0,0, 0,18,0,0,0,-30,0,0,0,30,0,0,0,-6,0,0,0,72,0,0,0,6,0,0,0,-6,0,0,0,12,0, 0,0,36,0,0,0,6,0,0,0,-6,0,0,0,30,0,0,0,-18,0,0,0,48,0,0,0,0,0,0,0,30,0, 0,0,96,0,0,0,42,0,0,0,6,0,0,0,-66,0,0,0,108,0,0,0,-30,0,0,0,6,0,0,0,0, 0,0,0,96,0,0,0,36,0,0,0,108,0,0,0,0,0,0,0,-30,0,0,0,0] 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: \(0\)
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: \( x^{12} - 24x^{10} + 195x^{8} - 666x^{6} + 948x^{4} - 420x^{2} + 17 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} - \beta_{2} + 1) q^{5} + \beta_{8} q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} + 1) q^{9} + ( - \beta_{10} + \beta_{6} + \beta_1) q^{11} + \beta_{7} q^{13} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_1) q^{15}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} + 12 q^{9} - 12 q^{21} + 36 q^{25} + 18 q^{29} - 30 q^{33} + 30 q^{37} - 6 q^{41} + 72 q^{45} + 6 q^{49} - 6 q^{53} + 12 q^{57} + 36 q^{61} + 6 q^{65} - 6 q^{69} + 30 q^{73} - 18 q^{77}+ \cdots + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 24x^{10} + 195x^{8} - 666x^{6} + 948x^{4} - 420x^{2} + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -18\nu^{10} + 583\nu^{8} - 6593\nu^{6} + 29890\nu^{4} - 45319\nu^{2} + 9644 ) / 5006 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\nu^{11} - 583\nu^{9} + 6593\nu^{7} - 29890\nu^{5} + 45319\nu^{3} - 14650\nu ) / 5006 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\nu^{10} - 545\nu^{8} + 1329\nu^{6} + 5838\nu^{4} - 3393\nu^{2} - 22110 ) / 5006 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 70\nu^{10} - 1711\nu^{8} + 14515\nu^{6} - 53942\nu^{4} + 82239\nu^{2} - 21374 ) / 5006 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 177\nu^{11} - 3647\nu^{9} + 21863\nu^{7} - 37778\nu^{5} + 520\nu^{3} + 9459\nu ) / 5006 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 195\nu^{10} - 4230\nu^{8} + 28456\nu^{6} - 67668\nu^{4} + 45839\nu^{2} - 185 ) / 5006 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 327\nu^{11} - 7671\nu^{9} + 60118\nu^{7} - 195919\nu^{5} + 272218\nu^{3} - 131814\nu ) / 5006 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 363\nu^{11} - 8837\nu^{9} + 73304\nu^{7} - 255699\nu^{5} + 357850\nu^{3} - 121066\nu ) / 5006 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 470\nu^{11} - 10773\nu^{9} + 80652\nu^{7} - 239535\nu^{5} + 276131\nu^{3} - 90233\nu ) / 5006 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -525\nu^{10} + 11581\nu^{8} - 80078\nu^{6} + 199319\nu^{4} - 152486\nu^{2} + 12628 ) / 5006 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 2\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + 2\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 6\beta_{7} - 13\beta_{5} + 10\beta_{4} - 25\beta_{2} + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{10} - 11\beta_{9} + 12\beta_{8} + 5\beta_{6} + 33\beta_{3} + 83\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{11} + 104\beta_{7} - 145\beta_{5} + 106\beta_{4} - 287\beta_{2} + 315 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -69\beta_{10} - 109\beta_{9} + 142\beta_{8} + 101\beta_{6} + 427\beta_{3} + 918\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 496\beta_{11} + 1408\beta_{7} - 1604\beta_{5} + 1169\beta_{4} - 3243\beta_{2} + 3363 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1053\beta_{10} - 1108\beta_{9} + 1665\beta_{8} + 1469\beta_{6} + 5147\beta_{3} + 10352\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6200\beta_{11} + 17474\beta_{7} - 17911\beta_{5} + 13125\beta_{4} - 36672\beta_{2} + 37149 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -13814\beta_{10} - 11711\beta_{9} + 19325\beta_{8} + 18888\beta_{6} + 60346\beta_{3} + 117545\beta_1 \) 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
3.37904
2.50884
1.73538
1.59475
0.829540
0.211852
−0.211852
−0.829540
−1.59475
−1.73538
−2.50884
−3.37904
0 −3.37904 0 3.80290 0 2.38844 0 8.41793 0
1.2 0 −2.50884 0 3.69982 0 0.702390 0 3.29427 0
1.3 0 −1.73538 0 −2.55381 0 1.08964 0 0.0115556 0
1.4 0 −1.59475 0 −2.04712 0 −1.24713 0 −0.456777 0
1.5 0 −0.829540 0 0.0764835 0 −3.20558 0 −2.31186 0
1.6 0 −0.211852 0 3.02172 0 −5.07775 0 −2.95512 0
1.7 0 0.211852 0 3.02172 0 5.07775 0 −2.95512 0
1.8 0 0.829540 0 0.0764835 0 3.20558 0 −2.31186 0
1.9 0 1.59475 0 −2.04712 0 1.24713 0 −0.456777 0
1.10 0 1.73538 0 −2.55381 0 −1.08964 0 0.0115556 0
1.11 0 2.50884 0 3.69982 0 −0.702390 0 3.29427 0
1.12 0 3.37904 0 3.80290 0 −2.38844 0 8.41793 0
\(n\): e.g. 2-40 or 990-1000
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.bw yes 12
4.b odd 2 1 inner 9248.2.a.bw yes 12
17.b even 2 1 9248.2.a.bs ✓ 12
68.d odd 2 1 9248.2.a.bs ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9248.2.a.bs ✓ 12 17.b even 2 1
9248.2.a.bs ✓ 12 68.d odd 2 1
9248.2.a.bw yes 12 1.a even 1 1 trivial
9248.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{12} - 24T_{3}^{10} + 195T_{3}^{8} - 666T_{3}^{6} + 948T_{3}^{4} - 420T_{3}^{2} + 17 \) Copy content Toggle raw display
\( T_{5}^{6} - 6T_{5}^{5} - 6T_{5}^{4} + 72T_{5}^{3} - 9T_{5}^{2} - 222T_{5} + 17 \) Copy content Toggle raw display
\( T_{7}^{12} - 45T_{7}^{10} + 609T_{7}^{8} - 3169T_{7}^{6} + 6435T_{7}^{4} - 5265T_{7}^{2} + 1377 \) Copy content Toggle raw display
\( T_{19}^{12} - 168T_{19}^{10} + 10512T_{19}^{8} - 305262T_{19}^{6} + 4258995T_{19}^{4} - 26772756T_{19}^{2} + 56248937 \) Copy content Toggle raw display
\( T_{43}^{12} - 342T_{43}^{10} + 38607T_{43}^{8} - 1781380T_{43}^{6} + 34228782T_{43}^{4} - 221889474T_{43}^{2} + 350242857 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 24 T^{10} + \cdots + 17 \) Copy content Toggle raw display
$5$ \( (T^{6} - 6 T^{5} - 6 T^{4} + \cdots + 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 45 T^{10} + \cdots + 1377 \) Copy content Toggle raw display
$11$ \( T^{12} - 66 T^{10} + \cdots + 392768 \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 168 T^{10} + \cdots + 56248937 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 383401017 \) Copy content Toggle raw display
$29$ \( (T^{6} - 9 T^{5} - 33 T^{4} + \cdots - 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 201 T^{10} + \cdots + 194633 \) Copy content Toggle raw display
$37$ \( (T^{6} - 15 T^{5} + \cdots - 17)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 3 T^{5} + \cdots + 8721)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 350242857 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 143661033 \) Copy content Toggle raw display
$53$ \( (T^{6} + 3 T^{5} + \cdots + 33048)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 6351374097 \) Copy content Toggle raw display
$61$ \( (T^{6} - 18 T^{5} + \cdots + 5457)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 189 T^{10} + \cdots + 1419857 \) Copy content Toggle raw display
$71$ \( T^{12} - 312 T^{10} + \cdots + 87367913 \) Copy content Toggle raw display
$73$ \( (T^{6} - 15 T^{5} + \cdots + 106539)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 21073255488 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17866856537 \) Copy content Toggle raw display
$89$ \( (T^{6} - 15 T^{5} + \cdots + 29637)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 21 T^{5} + \cdots + 177344)^{2} \) Copy content Toggle raw display
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