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Properties

Label 3225.2.a.be
Level $3225$
Weight $2$
Character orbit 3225.a
Self dual yes
Analytic conductor $25.752$
Analytic rank $1$
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] = [3225,2,Mod(1,3225)] 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("3225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3225 = 3 \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-3,-9,13,0,3,-8,-6,9,0,1,-13,-3] 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: \(25.7517546519\)
Analytic rank: \(1\)
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: \( x^{9} - 3x^{8} - 11x^{7} + 36x^{6} + 29x^{5} - 120x^{4} - 13x^{3} + 127x^{2} - 4x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + (\beta_{8} + \beta_{5} - 1) q^{7} + ( - \beta_{8} - \beta_{7} - \beta_{5} + \cdots - 1) q^{8} + q^{9} - \beta_{6} q^{11} + ( - \beta_{2} - 1) q^{12}+ \cdots - \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 9 q^{3} + 13 q^{4} + 3 q^{6} - 8 q^{7} - 6 q^{8} + 9 q^{9} + q^{11} - 13 q^{12} - 3 q^{13} + 3 q^{14} + 21 q^{16} - 11 q^{17} - 3 q^{18} + 7 q^{19} + 8 q^{21} - 15 q^{22} - 30 q^{23} + 6 q^{24}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 11x^{7} + 36x^{6} + 29x^{5} - 120x^{4} - 13x^{3} + 127x^{2} - 4x - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 3\nu^{7} - 11\nu^{6} + 32\nu^{5} + 29\nu^{4} - 80\nu^{3} - 13\nu^{2} + 39\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 15\nu^{6} - 32\nu^{5} - 69\nu^{4} + 80\nu^{3} + 101\nu^{2} - 35\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 13\nu^{6} - 10\nu^{5} - 51\nu^{4} + 20\nu^{3} + 69\nu^{2} - 3\nu - 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{8} + \nu^{7} + 37\nu^{6} - 12\nu^{5} - 131\nu^{4} + 32\nu^{3} + 143\nu^{2} - 13\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{8} - \nu^{7} - 37\nu^{6} + 12\nu^{5} + 135\nu^{4} - 32\nu^{3} - 175\nu^{2} + 13\nu + 56 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 13\nu^{6} - 12\nu^{5} - 51\nu^{4} + 38\nu^{3} + 69\nu^{2} - 31\nu - 22 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{8} + 9\beta_{7} + 10\beta_{5} - 9\beta_{4} + 31\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} + 10\beta_{6} + \beta_{4} + \beta_{3} + 58\beta_{2} - \beta _1 + 102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 58\beta_{8} + 68\beta_{7} - \beta_{6} + 79\beta_{5} - 68\beta_{4} - \beta_{3} - 2\beta_{2} + 208\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{8} + 77\beta_{7} + 78\beta_{6} - 3\beta_{5} + 15\beta_{4} + 12\beta_{3} + 413\beta_{2} - 18\beta _1 + 691 \) 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
2.63207
2.61101
1.64847
1.52360
0.658266
−0.557056
−1.07358
−1.74674
−2.69605
−2.63207 −1.00000 4.92780 0 2.63207 −3.08178 −7.70618 1.00000 0
1.2 −2.61101 −1.00000 4.81738 0 2.61101 1.30127 −7.35622 1.00000 0
1.3 −1.64847 −1.00000 0.717463 0 1.64847 −3.70598 2.11423 1.00000 0
1.4 −1.52360 −1.00000 0.321366 0 1.52360 2.34301 2.55757 1.00000 0
1.5 −0.658266 −1.00000 −1.56669 0 0.658266 −4.87941 2.34783 1.00000 0
1.6 0.557056 −1.00000 −1.68969 0 −0.557056 −0.0899122 −2.05536 1.00000 0
1.7 1.07358 −1.00000 −0.847434 0 −1.07358 4.32861 −3.05694 1.00000 0
1.8 1.74674 −1.00000 1.05111 0 −1.74674 0.733330 −1.65746 1.00000 0
1.9 2.69605 −1.00000 5.26868 0 −2.69605 −4.94914 8.81254 1.00000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(43\) \( -1 \)

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 3225.2.a.be ✓ 9
3.b odd 2 1 9675.2.a.ct 9
5.b even 2 1 3225.2.a.bf yes 9
15.d odd 2 1 9675.2.a.cs 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3225.2.a.be ✓ 9 1.a even 1 1 trivial
3225.2.a.bf yes 9 5.b even 2 1
9675.2.a.cs 9 15.d odd 2 1
9675.2.a.ct 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3225))\):

\( T_{2}^{9} + 3T_{2}^{8} - 11T_{2}^{7} - 36T_{2}^{6} + 29T_{2}^{5} + 120T_{2}^{4} - 13T_{2}^{3} - 127T_{2}^{2} - 4T_{2} + 32 \) Copy content Toggle raw display
\( T_{7}^{9} + 8T_{7}^{8} - 17T_{7}^{7} - 233T_{7}^{6} - 65T_{7}^{5} + 1807T_{7}^{4} + 500T_{7}^{3} - 4744T_{7}^{2} + 2240T_{7} + 240 \) Copy content Toggle raw display
\( T_{11}^{9} - T_{11}^{8} - 54 T_{11}^{7} + 4 T_{11}^{6} + 1067 T_{11}^{5} + 754 T_{11}^{4} - 8635 T_{11}^{3} + \cdots + 33400 \) Copy content Toggle raw display
\( T_{13}^{9} + 3 T_{13}^{8} - 74 T_{13}^{7} - 219 T_{13}^{6} + 1537 T_{13}^{5} + 4090 T_{13}^{4} + \cdots + 48618 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 8 T^{8} + \cdots + 240 \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} + \cdots + 33400 \) Copy content Toggle raw display
$13$ \( T^{9} + 3 T^{8} + \cdots + 48618 \) Copy content Toggle raw display
$17$ \( T^{9} + 11 T^{8} + \cdots + 972 \) Copy content Toggle raw display
$19$ \( T^{9} - 7 T^{8} + \cdots + 7008 \) Copy content Toggle raw display
$23$ \( T^{9} + 30 T^{8} + \cdots - 359344 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots + 386816 \) Copy content Toggle raw display
$31$ \( T^{9} - 13 T^{8} + \cdots - 55696 \) Copy content Toggle raw display
$37$ \( T^{9} + 5 T^{8} + \cdots + 458048 \) Copy content Toggle raw display
$41$ \( T^{9} - 2 T^{8} + \cdots + 2942 \) Copy content Toggle raw display
$43$ \( (T - 1)^{9} \) Copy content Toggle raw display
$47$ \( T^{9} + 24 T^{8} + \cdots + 523 \) Copy content Toggle raw display
$53$ \( T^{9} + 16 T^{8} + \cdots - 211534 \) Copy content Toggle raw display
$59$ \( T^{9} + 12 T^{8} + \cdots + 435415 \) Copy content Toggle raw display
$61$ \( T^{9} - 6 T^{8} + \cdots + 11926336 \) Copy content Toggle raw display
$67$ \( T^{9} + 23 T^{8} + \cdots + 45616 \) Copy content Toggle raw display
$71$ \( T^{9} + 30 T^{8} + \cdots - 16726176 \) Copy content Toggle raw display
$73$ \( T^{9} - 4 T^{8} + \cdots - 26310656 \) Copy content Toggle raw display
$79$ \( T^{9} - 9 T^{8} + \cdots + 44368733 \) Copy content Toggle raw display
$83$ \( T^{9} - 7 T^{8} + \cdots - 245532920 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 2560664448 \) Copy content Toggle raw display
$97$ \( T^{9} + 19 T^{8} + \cdots + 383574 \) Copy content Toggle raw display
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