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Properties

Label 7536.2.a.bj
Level $7536$
Weight $2$
Character orbit 7536.a
Self dual yes
Analytic conductor $60.175$
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] = [7536,2,Mod(1,7536)] 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("7536.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7536, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7536 = 2^{4} \cdot 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7536.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,9,0,8,0,2] 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: \(60.1752629632\)
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: \( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 39x^{5} - 53x^{4} - 49x^{3} + 45x^{2} + 14x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 471)
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{2} + 1) q^{5} + ( - \beta_{7} + \beta_{3}) q^{7} + q^{9} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{5} + \beta_{2} + \beta_1) q^{13} + (\beta_{2} + 1) q^{15} + ( - \beta_{8} + \beta_{7} - \beta_{2} + \cdots + 2) q^{17}+ \cdots + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} + 8 q^{5} + 2 q^{7} + 9 q^{9} - 7 q^{11} - q^{13} + 8 q^{15} + 20 q^{17} - 2 q^{19} + 2 q^{21} - 8 q^{23} + 17 q^{25} + 9 q^{27} + 19 q^{29} - 5 q^{31} - 7 q^{33} - 24 q^{35} - 8 q^{37} - q^{39}+ \cdots - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 39x^{5} - 53x^{4} - 49x^{3} + 45x^{2} + 14x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6\nu^{8} - 16\nu^{7} - 75\nu^{6} + 164\nu^{5} + 341\nu^{4} - 506\nu^{3} - 645\nu^{2} + 464\nu + 345 ) / 59 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{8} + 41\nu^{7} + 41\nu^{6} - 376\nu^{5} + 37\nu^{4} + 950\nu^{3} - 143\nu^{2} - 599\nu - 106 ) / 59 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{8} - 24\nu^{7} - 83\nu^{6} + 187\nu^{5} + 246\nu^{4} - 287\nu^{3} - 289\nu^{2} - 130\nu + 75 ) / 59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{8} - 23\nu^{7} - 200\nu^{6} + 221\nu^{5} + 811\nu^{4} - 661\nu^{3} - 1012\nu^{2} + 667\nu + 35 ) / 59 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{8} - 31\nu^{7} - 208\nu^{6} + 244\nu^{5} + 716\nu^{4} - 442\nu^{3} - 774\nu^{2} + 191\nu + 60 ) / 59 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{8} - 31\nu^{7} - 208\nu^{6} + 244\nu^{5} + 716\nu^{4} - 442\nu^{3} - 774\nu^{2} + 73\nu + 119 ) / 59 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -22\nu^{8} + 39\nu^{7} + 216\nu^{6} - 326\nu^{5} - 621\nu^{4} + 754\nu^{3} + 536\nu^{2} - 541\nu - 85 ) / 59 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\nu^{8} - 47\nu^{7} - 283\nu^{6} + 467\nu^{5} + 998\nu^{4} - 1361\nu^{3} - 1065\nu^{2} + 1127\nu - 8 ) / 59 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{3} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 5\beta_{6} + 4\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 5\beta_{6} + 3\beta_{4} + 5\beta_{3} + \beta_{2} - 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{8} - 2\beta_{7} - 31\beta_{6} + 18\beta_{5} - 7\beta_{4} + 18\beta_{3} + 9\beta_{2} + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{8} - 22\beta_{7} - 77\beta_{6} + 33\beta_{4} + 75\beta_{3} + 20\beta_{2} - 19\beta _1 + 156 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 64\beta_{8} - 32\beta_{7} - 209\beta_{6} + 87\beta_{5} - 42\beta_{4} + 142\beta_{3} + 74\beta_{2} + 273 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17\beta_{8} - 96\beta_{7} - 278\beta_{6} + 87\beta_{4} + 266\beta_{3} + 86\beta_{2} - 49\beta _1 + 476 \) 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
−1.34370
0.0604903
1.06833
2.31676
−0.316645
−2.29794
2.67810
1.51659
−1.68197
0 1.00000 0 −3.48167 0 4.92854 0 1.00000 0
1.2 0 1.00000 0 −1.41604 0 3.07838 0 1.00000 0
1.3 0 1.00000 0 −0.921169 0 −4.41562 0 1.00000 0
1.4 0 1.00000 0 −0.520771 0 −2.00784 0 1.00000 0
1.5 0 1.00000 0 1.69099 0 −0.220418 0 1.00000 0
1.6 0 1.00000 0 1.95196 0 3.21152 0 1.00000 0
1.7 0 1.00000 0 2.42967 0 1.48907 0 1.00000 0
1.8 0 1.00000 0 4.06961 0 0.404124 0 1.00000 0
1.9 0 1.00000 0 4.19743 0 −4.46776 0 1.00000 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
\(2\) \( -1 \)
\(3\) \( -1 \)
\(157\) \( -1 \)

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 7536.2.a.bj 9
4.b odd 2 1 471.2.a.d ✓ 9
12.b even 2 1 1413.2.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.a.d ✓ 9 4.b odd 2 1
1413.2.a.f 9 12.b even 2 1
7536.2.a.bj 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7536))\):

\( T_{5}^{9} - 8T_{5}^{8} + T_{5}^{7} + 122T_{5}^{6} - 214T_{5}^{5} - 330T_{5}^{4} + 728T_{5}^{3} + 380T_{5}^{2} - 648T_{5} - 324 \) Copy content Toggle raw display
\( T_{7}^{9} - 2T_{7}^{8} - 43T_{7}^{7} + 88T_{7}^{6} + 528T_{7}^{5} - 1152T_{7}^{4} - 1472T_{7}^{3} + 3264T_{7}^{2} - 384T_{7} - 256 \) Copy content Toggle raw display
\( T_{11}^{9} + 7 T_{11}^{8} - 37 T_{11}^{7} - 328 T_{11}^{6} + 64 T_{11}^{5} + 3696 T_{11}^{4} + \cdots - 5632 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 8 T^{8} + \cdots - 324 \) Copy content Toggle raw display
$7$ \( T^{9} - 2 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$11$ \( T^{9} + 7 T^{8} + \cdots - 5632 \) Copy content Toggle raw display
$13$ \( T^{9} + T^{8} + \cdots + 17536 \) Copy content Toggle raw display
$17$ \( T^{9} - 20 T^{8} + \cdots + 896 \) Copy content Toggle raw display
$19$ \( T^{9} + 2 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots - 44716 \) Copy content Toggle raw display
$29$ \( T^{9} - 19 T^{8} + \cdots + 183636 \) Copy content Toggle raw display
$31$ \( T^{9} + 5 T^{8} + \cdots + 1193456 \) Copy content Toggle raw display
$37$ \( T^{9} + 8 T^{8} + \cdots - 4672 \) Copy content Toggle raw display
$41$ \( T^{9} - 42 T^{8} + \cdots + 3700116 \) Copy content Toggle raw display
$43$ \( T^{9} - 34 T^{8} + \cdots - 22791168 \) Copy content Toggle raw display
$47$ \( T^{9} + 3 T^{8} + \cdots - 34068992 \) Copy content Toggle raw display
$53$ \( T^{9} - 16 T^{8} + \cdots - 1940144 \) Copy content Toggle raw display
$59$ \( T^{9} - 6 T^{8} + \cdots - 3271216 \) Copy content Toggle raw display
$61$ \( T^{9} - 14 T^{8} + \cdots + 253103744 \) Copy content Toggle raw display
$67$ \( T^{9} - 37 T^{8} + \cdots + 931072 \) Copy content Toggle raw display
$71$ \( T^{9} - 21 T^{8} + \cdots + 4146176 \) Copy content Toggle raw display
$73$ \( T^{9} - 23 T^{8} + \cdots - 628096 \) Copy content Toggle raw display
$79$ \( T^{9} - 36 T^{8} + \cdots - 1985792 \) Copy content Toggle raw display
$83$ \( T^{9} - 3 T^{8} + \cdots + 19772404 \) Copy content Toggle raw display
$89$ \( T^{9} - 27 T^{8} + \cdots - 53488512 \) Copy content Toggle raw display
$97$ \( T^{9} + 2 T^{8} + \cdots + 6125184 \) Copy content Toggle raw display
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