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Properties

Label 1547.2.a.g
Level $1547$
Weight $2$
Character orbit 1547.a
Self dual yes
Analytic conductor $12.353$
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] = [1547,2,Mod(1,1547)] 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("1547.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1547, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1547 = 7 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1547.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.3528571927\)
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: \( x^{9} - 3x^{8} - 6x^{7} + 21x^{6} + 6x^{5} - 34x^{4} - 2x^{3} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{7} + \beta_1 - 1) q^{5} + ( - \beta_{8} - \beta_{6} - \beta_{3}) q^{6} + q^{7} + (\beta_{7} - \beta_{6} - \beta_1) q^{8}+ \cdots + ( - 2 \beta_{8} - \beta_{6} + \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 4 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} + 9 q^{7} - 6 q^{8} + 5 q^{9} - 9 q^{10} - 7 q^{11} + 2 q^{12} - 9 q^{13} - 3 q^{14} - 5 q^{15} + 3 q^{16} + 9 q^{17} - 9 q^{18} - 6 q^{19} + 6 q^{20}+ \cdots - 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 6x^{7} + 21x^{6} + 6x^{5} - 34x^{4} - 2x^{3} + 13x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 13\nu^{4} + 11\nu^{3} - 17\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 5\nu^{6} - 19\nu^{5} + \nu^{4} + 22\nu^{3} - 9\nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 6\nu^{6} - 20\nu^{5} - 7\nu^{4} + 27\nu^{3} + 7\nu^{2} - 3\nu - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - 3\nu^{7} - 6\nu^{6} + 21\nu^{5} + 6\nu^{4} - 33\nu^{3} - 2\nu^{2} + 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 3\nu^{7} - 6\nu^{6} + 21\nu^{5} + 6\nu^{4} - 34\nu^{3} - 2\nu^{2} + 13\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 26\nu^{4} + 27\nu^{3} + 29\nu^{2} - 6\nu - 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 6\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 6\beta_{7} + 8\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 26\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{8} - \beta_{7} + 11\beta_{6} - 7\beta_{5} + 8\beta_{4} + 9\beta_{3} + 33\beta_{2} + 35\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{8} - 33\beta_{7} + 54\beta_{6} - \beta_{5} + 10\beta_{4} + 13\beta_{3} + 12\beta_{2} + 140\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 63\beta_{8} - 12\beta_{7} + 88\beta_{6} - 39\beta_{5} + 51\beta_{4} + 66\beta_{3} + 179\beta_{2} + 207\beta _1 + 181 \) 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
2.44309
2.16066
1.77588
0.615646
0.333078
−0.310487
−0.655374
−1.16158
−2.20091
−2.44309 1.20103 3.96867 1.03377 −2.93421 1.00000 −4.80962 −1.55754 −2.52558
1.2 −2.16066 −2.82658 2.66845 0.697839 6.10729 1.00000 −1.44430 4.98957 −1.50779
1.3 −1.77588 −0.523287 1.15374 0.212776 0.929293 1.00000 1.50285 −2.72617 −0.377864
1.4 −0.615646 2.73864 −1.62098 −2.00866 −1.68604 1.00000 2.22924 4.50017 1.23663
1.5 −0.333078 −1.38368 −1.88906 −3.66922 0.460874 1.00000 1.29536 −1.08543 1.22214
1.6 0.310487 −1.20477 −1.90360 1.91026 −0.374064 1.00000 −1.21202 −1.54854 0.593111
1.7 0.655374 −3.21526 −1.57048 −0.129529 −2.10720 1.00000 −2.34000 7.33787 −0.0848897
1.8 1.16158 1.02724 −0.650726 −1.30069 1.19323 1.00000 −3.07904 −1.94477 −1.51086
1.9 2.20091 0.186661 2.84398 −2.74655 0.410824 1.00000 1.85753 −2.96516 −6.04489
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(13\) \( +1 \)
\(17\) \( -1 \)

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 1547.2.a.g ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1547.2.a.g ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1547))\):

\( T_{2}^{9} + 3T_{2}^{8} - 6T_{2}^{7} - 21T_{2}^{6} + 6T_{2}^{5} + 34T_{2}^{4} - 2T_{2}^{3} - 13T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{9} + 4T_{3}^{8} - 8T_{3}^{7} - 41T_{3}^{6} + T_{3}^{5} + 90T_{3}^{4} + 24T_{3}^{3} - 60T_{3}^{2} - 17T_{3} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{9} + 4 T^{8} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( T^{9} + 6 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 7 T^{8} + \cdots - 48 \) Copy content Toggle raw display
$13$ \( (T + 1)^{9} \) Copy content Toggle raw display
$17$ \( (T - 1)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + 6 T^{8} + \cdots - 135472 \) Copy content Toggle raw display
$23$ \( T^{9} + 3 T^{8} + \cdots - 144 \) Copy content Toggle raw display
$29$ \( T^{9} + 4 T^{8} + \cdots + 7344 \) Copy content Toggle raw display
$31$ \( T^{9} - 240 T^{7} + \cdots + 2157937 \) Copy content Toggle raw display
$37$ \( T^{9} + 31 T^{8} + \cdots + 5357824 \) Copy content Toggle raw display
$41$ \( T^{9} - 3 T^{8} + \cdots + 1425 \) Copy content Toggle raw display
$43$ \( T^{9} + 24 T^{8} + \cdots - 854991 \) Copy content Toggle raw display
$47$ \( T^{9} - 7 T^{8} + \cdots + 101840 \) Copy content Toggle raw display
$53$ \( T^{9} + 18 T^{8} + \cdots + 73231 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots + 424240 \) Copy content Toggle raw display
$61$ \( T^{9} + 22 T^{8} + \cdots - 22985 \) Copy content Toggle raw display
$67$ \( T^{9} + 32 T^{8} + \cdots + 1109253 \) Copy content Toggle raw display
$71$ \( T^{9} - 18 T^{8} + \cdots + 2226128 \) Copy content Toggle raw display
$73$ \( T^{9} + 14 T^{8} + \cdots - 460535 \) Copy content Toggle raw display
$79$ \( T^{9} + 46 T^{8} + \cdots - 32764880 \) Copy content Toggle raw display
$83$ \( T^{9} + 16 T^{8} + \cdots - 1769808 \) Copy content Toggle raw display
$89$ \( T^{9} + 10 T^{8} + \cdots + 87712176 \) Copy content Toggle raw display
$97$ \( T^{9} + 15 T^{8} + \cdots - 193612229 \) Copy content Toggle raw display
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