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Properties

Label 8325.2.a.cr
Level $8325$
Weight $2$
Character orbit 8325.a
Self dual yes
Analytic conductor $66.475$
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] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,5,0,11,0,0,-8,15,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
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: \( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 185)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{8} + ( - \beta_{8} - \beta_{3}) q^{11} + ( - \beta_{8} - \beta_{6} + \beta_{4} + \cdots - 1) q^{13}+ \cdots + ( - 3 \beta_{8} + 2 \beta_{7} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 14\nu^{2} - 4\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 8\nu^{2} + 14\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 5\nu^{6} + 25\nu^{4} - 11\nu^{3} - 34\nu^{2} + 8\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 5\nu^{7} + 26\nu^{5} - 13\nu^{4} - 39\nu^{3} + 14\nu^{2} + 10\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} + 5\nu^{7} + \nu^{6} - 29\nu^{5} + 8\nu^{4} + 52\nu^{3} - 4\nu^{2} - 22\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 3\beta_{3} + 9\beta_{2} + 9\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} - 2\beta_{4} + 13\beta_{3} + 24\beta_{2} + 31\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{8} + 2\beta_{7} + 17\beta_{5} - 11\beta_{4} + 41\beta_{3} + 81\beta_{2} + 75\beta _1 + 164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{8} + 10\beta_{7} + 2\beta_{6} + 60\beta_{5} - 30\beta_{4} + 141\beta_{3} + 236\beta_{2} + 231\beta _1 + 460 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50 \beta_{8} + 52 \beta_{7} + 10 \beta_{6} + 209 \beta_{5} - 111 \beta_{4} + 445 \beta_{3} + 737 \beta_{2} + \cdots + 1428 \) 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
3.05104
2.70607
1.97415
0.753109
0.296889
−0.356037
−1.15179
−1.58855
−1.68489
−2.05104 0 2.20678 0 0 −0.536422 −0.424122 0 0
1.2 −1.70607 0 0.910675 0 0 0.663740 1.85846 0 0
1.3 −0.974151 0 −1.05103 0 0 −4.15169 2.97216 0 0
1.4 0.246891 0 −1.93904 0 0 −2.66723 −0.972515 0 0
1.5 0.703111 0 −1.50564 0 0 −0.501390 −2.46485 0 0
1.6 1.35604 0 −0.161165 0 0 −0.748140 −2.93062 0 0
1.7 2.15179 0 2.63020 0 0 −4.19034 1.35605 0 0
1.8 2.58855 0 4.70058 0 0 0.648627 6.99059 0 0
1.9 2.68489 0 5.20864 0 0 3.48285 8.61484 0 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
\(3\) \( -1 \)
\(5\) \( -1 \)
\(37\) \( -1 \)

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 8325.2.a.cr 9
3.b odd 2 1 925.2.a.l 9
5.b even 2 1 8325.2.a.cq 9
5.c odd 4 2 1665.2.c.e 18
15.d odd 2 1 925.2.a.m 9
15.e even 4 2 185.2.b.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.b.a ✓ 18 15.e even 4 2
925.2.a.l 9 3.b odd 2 1
925.2.a.m 9 15.d odd 2 1
1665.2.c.e 18 5.c odd 4 2
8325.2.a.cq 9 5.b even 2 1
8325.2.a.cr 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{9} - 5T_{2}^{8} - 2T_{2}^{7} + 40T_{2}^{6} - 29T_{2}^{5} - 91T_{2}^{4} + 98T_{2}^{3} + 44T_{2}^{2} - 64T_{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{9} + 8T_{7}^{8} + 4T_{7}^{7} - 98T_{7}^{6} - 209T_{7}^{5} + 4T_{7}^{4} + 178T_{7}^{3} + 48T_{7}^{2} - 38T_{7} - 14 \) Copy content Toggle raw display
\( T_{11}^{9} - 42T_{11}^{7} + 24T_{11}^{6} + 481T_{11}^{5} - 456T_{11}^{4} - 1528T_{11}^{3} + 1760T_{11}^{2} + 896T_{11} - 1152 \) Copy content Toggle raw display
\( T_{13}^{9} + 6 T_{13}^{8} - 40 T_{13}^{7} - 252 T_{13}^{6} + 432 T_{13}^{5} + 2996 T_{13}^{4} + \cdots - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 5 T^{8} + \cdots + 12 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 8 T^{8} + \cdots - 14 \) Copy content Toggle raw display
$11$ \( T^{9} - 42 T^{7} + \cdots - 1152 \) Copy content Toggle raw display
$13$ \( T^{9} + 6 T^{8} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{9} - 18 T^{8} + \cdots - 7728 \) Copy content Toggle raw display
$19$ \( T^{9} + 4 T^{8} + \cdots + 63768 \) Copy content Toggle raw display
$23$ \( T^{9} - 16 T^{8} + \cdots + 3904 \) Copy content Toggle raw display
$29$ \( T^{9} - 2 T^{8} + \cdots - 1152 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} + \cdots + 2168344 \) Copy content Toggle raw display
$37$ \( (T - 1)^{9} \) Copy content Toggle raw display
$41$ \( T^{9} + 2 T^{8} + \cdots - 8368 \) Copy content Toggle raw display
$43$ \( T^{9} + 14 T^{8} + \cdots + 1852736 \) Copy content Toggle raw display
$47$ \( T^{9} - 34 T^{8} + \cdots + 11786326 \) Copy content Toggle raw display
$53$ \( T^{9} + 4 T^{8} + \cdots + 13034336 \) Copy content Toggle raw display
$59$ \( T^{9} + 10 T^{8} + \cdots - 689664 \) Copy content Toggle raw display
$61$ \( T^{9} - 8 T^{8} + \cdots + 6156288 \) Copy content Toggle raw display
$67$ \( T^{9} + 16 T^{8} + \cdots - 6502304 \) Copy content Toggle raw display
$71$ \( T^{9} + 8 T^{8} + \cdots + 5516928 \) Copy content Toggle raw display
$73$ \( T^{9} + 8 T^{8} + \cdots + 1036576 \) Copy content Toggle raw display
$79$ \( T^{9} + 26 T^{8} + \cdots - 70088 \) Copy content Toggle raw display
$83$ \( T^{9} - 70 T^{8} + \cdots + 37115022 \) Copy content Toggle raw display
$89$ \( T^{9} - 8 T^{8} + \cdots + 27504768 \) Copy content Toggle raw display
$97$ \( T^{9} - 2 T^{8} + \cdots + 106138304 \) Copy content Toggle raw display
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