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Properties

Label 6936.2.a.bp
Level $6936$
Weight $2$
Character orbit 6936.a
Self dual yes
Analytic conductor $55.384$
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] = [6936,2,Mod(1,6936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6936.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6936.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,-9,0,-3,0,0,0,9,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.3842388420\)
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: \( x^{9} - 3x^{8} - 27x^{7} + 47x^{6} + 285x^{5} - 45x^{4} - 1141x^{3} - 1344x^{2} - 468x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_1 q^{5} - \beta_{7} q^{7} + q^{9} + ( - \beta_{8} - \beta_{5} - \beta_{3}) q^{11} + (\beta_{8} + \beta_{4}) q^{13} + \beta_1 q^{15} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{19}+ \cdots + ( - \beta_{8} - \beta_{5} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 3 q^{5} + 9 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{15} + 21 q^{19} - 21 q^{23} + 18 q^{25} - 9 q^{27} - 18 q^{29} + 3 q^{33} + 18 q^{35} + 3 q^{37} - 3 q^{39} + 12 q^{41} - 18 q^{43} - 3 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 27x^{7} + 47x^{6} + 285x^{5} - 45x^{4} - 1141x^{3} - 1344x^{2} - 468x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 195 \nu^{8} + 1475 \nu^{7} + 823 \nu^{6} - 23799 \nu^{5} + 16119 \nu^{4} + 103525 \nu^{3} + \cdots - 19352 ) / 11164 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 201 \nu^{8} + 1091 \nu^{7} + 3167 \nu^{6} - 17747 \nu^{5} - 21085 \nu^{4} + 61625 \nu^{3} + \cdots + 22304 ) / 11164 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 445 \nu^{8} + 2221 \nu^{7} + 7317 \nu^{6} - 35847 \nu^{5} - 47375 \nu^{4} + 130907 \nu^{3} + \cdots + 780 ) / 5582 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 689 \nu^{8} - 3351 \nu^{7} - 11467 \nu^{6} + 53947 \nu^{5} + 73665 \nu^{4} - 194607 \nu^{3} + \cdots - 1584 ) / 5582 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1389 \nu^{8} - 5998 \nu^{7} - 29092 \nu^{6} + 100520 \nu^{5} + 256472 \nu^{4} - 347750 \nu^{3} + \cdots - 42324 ) / 5582 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3143 \nu^{8} - 13755 \nu^{7} - 65879 \nu^{6} + 235711 \nu^{5} + 575949 \nu^{4} - 880833 \nu^{3} + \cdots + 19704 ) / 11164 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3149 \nu^{8} - 13371 \nu^{7} - 68223 \nu^{6} + 229659 \nu^{5} + 613153 \nu^{4} - 838933 \nu^{3} + \cdots - 88936 ) / 11164 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{3} - \beta_{2} + 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} + 12\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{8} - 14\beta_{7} + 2\beta_{6} + \beta_{5} + 4\beta_{4} + 19\beta_{3} - 18\beta_{2} + 36\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 27\beta_{8} - 27\beta_{7} + 2\beta_{6} + 22\beta_{5} + 47\beta_{4} + 9\beta_{3} - 39\beta_{2} + 170\beta _1 + 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 190 \beta_{8} - 212 \beta_{7} + 45 \beta_{6} + 53 \beta_{5} + 144 \beta_{4} + 313 \beta_{3} + \cdots + 1056 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 538 \beta_{8} - 569 \beta_{7} + 99 \beta_{6} + 441 \beta_{5} + 959 \beta_{4} + 575 \beta_{3} + \cdots + 3278 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2980 \beta_{8} - 3390 \beta_{7} + 860 \beta_{6} + 1488 \beta_{5} + 3518 \beta_{4} + 5410 \beta_{3} + \cdots + 15936 \) 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
4.20390
3.80482
3.55667
−0.0180114
−0.717796
−1.07057
−1.73425
−1.83888
−3.18589
0 −1.00000 0 −4.20390 0 −0.398181 0 1.00000 0
1.2 0 −1.00000 0 −3.80482 0 −1.97677 0 1.00000 0
1.3 0 −1.00000 0 −3.55667 0 1.16939 0 1.00000 0
1.4 0 −1.00000 0 0.0180114 0 −3.60324 0 1.00000 0
1.5 0 −1.00000 0 0.717796 0 −4.32432 0 1.00000 0
1.6 0 −1.00000 0 1.07057 0 4.75098 0 1.00000 0
1.7 0 −1.00000 0 1.73425 0 −1.24212 0 1.00000 0
1.8 0 −1.00000 0 1.83888 0 3.50222 0 1.00000 0
1.9 0 −1.00000 0 3.18589 0 2.12204 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
\(2\) \( -1 \)
\(3\) \( +1 \)
\(17\) \( +1 \)

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 6936.2.a.bp ✓ 9
17.b even 2 1 6936.2.a.bq yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6936.2.a.bp ✓ 9 1.a even 1 1 trivial
6936.2.a.bq yes 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6936))\):

\( T_{5}^{9} + 3T_{5}^{8} - 27T_{5}^{7} - 47T_{5}^{6} + 285T_{5}^{5} + 45T_{5}^{4} - 1141T_{5}^{3} + 1344T_{5}^{2} - 468T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{9} - 39T_{7}^{7} - 8T_{7}^{6} + 456T_{7}^{5} + 144T_{7}^{4} - 1684T_{7}^{3} - 669T_{7}^{2} + 1578T_{7} + 629 \) Copy content Toggle raw display
\( T_{11}^{9} + 3 T_{11}^{8} - 87 T_{11}^{7} - 227 T_{11}^{6} + 2721 T_{11}^{5} + 5847 T_{11}^{4} + \cdots + 204744 \) 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
$5$ \( T^{9} + 3 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{9} - 39 T^{7} + \cdots + 629 \) Copy content Toggle raw display
$11$ \( T^{9} + 3 T^{8} + \cdots + 204744 \) Copy content Toggle raw display
$13$ \( T^{9} - 3 T^{8} + \cdots + 25848 \) Copy content Toggle raw display
$17$ \( T^{9} \) Copy content Toggle raw display
$19$ \( T^{9} - 21 T^{8} + \cdots - 129144 \) Copy content Toggle raw display
$23$ \( T^{9} + 21 T^{8} + \cdots + 62016 \) Copy content Toggle raw display
$29$ \( T^{9} + 18 T^{8} + \cdots - 754344 \) Copy content Toggle raw display
$31$ \( T^{9} - 147 T^{7} + \cdots + 12527 \) Copy content Toggle raw display
$37$ \( T^{9} - 3 T^{8} + \cdots + 1081272 \) Copy content Toggle raw display
$41$ \( T^{9} - 12 T^{8} + \cdots - 248256 \) Copy content Toggle raw display
$43$ \( T^{9} + 18 T^{8} + \cdots + 1088 \) Copy content Toggle raw display
$47$ \( T^{9} + 6 T^{8} + \cdots + 1707072 \) Copy content Toggle raw display
$53$ \( T^{9} - 30 T^{8} + \cdots + 154632 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots + 139468312 \) Copy content Toggle raw display
$61$ \( T^{9} - 15 T^{8} + \cdots - 32264 \) Copy content Toggle raw display
$67$ \( T^{9} - 30 T^{8} + \cdots + 120768 \) Copy content Toggle raw display
$71$ \( T^{9} + 33 T^{8} + \cdots + 262656 \) Copy content Toggle raw display
$73$ \( T^{9} - 30 T^{8} + \cdots - 8792027 \) Copy content Toggle raw display
$79$ \( T^{9} - 12 T^{8} + \cdots - 112537 \) Copy content Toggle raw display
$83$ \( T^{9} + 33 T^{8} + \cdots - 874072 \) Copy content Toggle raw display
$89$ \( T^{9} - 12 T^{8} + \cdots + 10029504 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} + \cdots + 63669 \) Copy content Toggle raw display
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