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Properties

Label 6897.2.a.bc
Level $6897$
Weight $2$
Character orbit 6897.a
Self dual yes
Analytic conductor $55.073$
Analytic rank $1$
Dimension $8$
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] = [6897,2,Mod(1,6897)] 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("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,-8,9,-2,1,-6,-6,8,-3,0,-9,-1] 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: \(55.0728222741\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 9x^{5} + 40x^{4} - 22x^{3} - 28x^{2} + 12x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{7} - \beta_{6} + 1) q^{4} - \beta_{4} q^{5} + \beta_1 q^{6} + ( - \beta_{7} + \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{8} + q^{9}+ \cdots + ( - 4 \beta_{7} + 2 \beta_{6} + \cdots - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 12x^{6} + 9x^{5} + 40x^{4} - 22x^{3} - 28x^{2} + 12x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 3\nu^{6} + 46\nu^{5} - 19\nu^{4} - 140\nu^{3} + 20\nu^{2} + 62\nu - 5 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} - 3\nu^{6} - 46\nu^{5} + 19\nu^{4} + 151\nu^{3} - 20\nu^{2} - 128\nu + 5 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - \nu^{6} - 63\nu^{5} - \nu^{4} + 219\nu^{3} + 30\nu^{2} - 160\nu - 2 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + \nu^{6} + 12\nu^{5} - 8\nu^{4} - 41\nu^{3} + 15\nu^{2} + 32\nu - 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -16\nu^{7} + 12\nu^{6} + 195\nu^{5} - 98\nu^{4} - 659\nu^{3} + 201\nu^{2} + 468\nu - 75 ) / 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -16\nu^{7} + 12\nu^{6} + 195\nu^{5} - 98\nu^{4} - 659\nu^{3} + 212\nu^{2} + 468\nu - 108 ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} - 7\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 11\beta_{3} + 7\beta_{2} + 38\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{7} - 47\beta_{6} + 13\beta_{5} - 9\beta_{4} + 15\beta_{3} + 12\beta_{2} + 28\beta _1 + 95 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15\beta_{7} - 30\beta_{6} + 28\beta_{5} - 25\beta_{4} + 98\beta_{3} + 47\beta_{2} + 254\beta _1 + 55 \) 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
2.82309
2.17304
1.07029
0.267052
0.118442
−0.962830
−2.05203
−2.43706
−2.82309 −1.00000 5.96985 0.754028 2.82309 2.16126 −11.2072 1.00000 −2.12869
1.2 −2.17304 −1.00000 2.72212 −0.272062 2.17304 −5.04465 −1.56920 1.00000 0.591202
1.3 −1.07029 −1.00000 −0.854469 −4.21558 1.07029 −3.04581 3.05512 1.00000 4.51192
1.4 −0.267052 −1.00000 −1.92868 3.50077 0.267052 −4.33946 1.04916 1.00000 −0.934889
1.5 −0.118442 −1.00000 −1.98597 1.83342 0.118442 2.90053 0.472107 1.00000 −0.217154
1.6 0.962830 −1.00000 −1.07296 −2.82062 −0.962830 3.66641 −2.95874 1.00000 −2.71578
1.7 2.05203 −1.00000 2.21083 0.534513 −2.05203 −0.350409 0.432637 1.00000 1.09684
1.8 2.43706 −1.00000 3.93928 −1.31447 −2.43706 −1.94788 4.72615 1.00000 −3.20345
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( -1 \)
\(19\) \( -1 \)

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 6897.2.a.bc ✓ 8
11.b odd 2 1 6897.2.a.bd yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6897.2.a.bc ✓ 8 1.a even 1 1 trivial
6897.2.a.bd yes 8 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):

\( T_{2}^{8} + T_{2}^{7} - 12T_{2}^{6} - 9T_{2}^{5} + 40T_{2}^{4} + 22T_{2}^{3} - 28T_{2}^{2} - 12T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 20T_{5}^{6} - 26T_{5}^{5} + 96T_{5}^{4} + 43T_{5}^{3} - 104T_{5}^{2} + 11T_{5} + 11 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 24T_{7}^{6} - 162T_{7}^{5} + 160T_{7}^{4} + 1321T_{7}^{3} - 131T_{7}^{2} - 3184T_{7} - 1046 \) Copy content Toggle raw display
\( T_{13}^{8} + T_{13}^{7} - 12T_{13}^{6} - 9T_{13}^{5} + 40T_{13}^{4} + 22T_{13}^{3} - 28T_{13}^{2} - 12T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 12 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 11 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots - 1046 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} - 12 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 15 T^{7} + \cdots - 30491 \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 5 T^{7} + \cdots - 456932 \) Copy content Toggle raw display
$29$ \( T^{8} + 17 T^{7} + \cdots + 683 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + \cdots + 4423766 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots - 174347 \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots + 495667 \) Copy content Toggle raw display
$43$ \( T^{8} + 10 T^{7} + \cdots - 616036 \) Copy content Toggle raw display
$47$ \( T^{8} - 5 T^{7} + \cdots + 79002 \) Copy content Toggle raw display
$53$ \( T^{8} + 19 T^{7} + \cdots + 3465779 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots - 27194 \) Copy content Toggle raw display
$61$ \( T^{8} - 5 T^{7} + \cdots + 110368 \) Copy content Toggle raw display
$67$ \( T^{8} + 12 T^{7} + \cdots - 1431892 \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{7} + \cdots - 12466 \) Copy content Toggle raw display
$73$ \( T^{8} - 12 T^{7} + \cdots - 9136 \) Copy content Toggle raw display
$79$ \( T^{8} + 14 T^{7} + \cdots + 710206 \) Copy content Toggle raw display
$83$ \( T^{8} + 17 T^{7} + \cdots - 62705152 \) Copy content Toggle raw display
$89$ \( T^{8} + 9 T^{7} + \cdots + 16071583 \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} + \cdots + 6097261 \) Copy content Toggle raw display
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