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Properties

Label 6936.2.a.bj
Level $6936$
Weight $2$
Character orbit 6936.a
Self dual yes
Analytic conductor $55.384$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6936,2,Mod(1,6936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.3842388420\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.50874368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 13x^{4} + 12x^{3} + 56x^{2} + 8x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 408)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{5} q^{5} + ( - \beta_{3} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{5} q^{5} + ( - \beta_{3} + 1) q^{7} + q^{9} + ( - \beta_{4} + \beta_1 + 1) q^{11} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{13} + \beta_{5} q^{15} + (\beta_{5} + \beta_{3} + \beta_{2} - 1) q^{19} + ( - \beta_{3} + 1) q^{21} + ( - \beta_{5} - \beta_{2} - \beta_1 + 2) q^{23} + ( - \beta_{4} + \beta_1 + 2) q^{25} + q^{27} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{29} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{31} + ( - \beta_{4} + \beta_1 + 1) q^{33} + ( - 2 \beta_{3} + \beta_1 + 4) q^{35} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{37}+ \cdots + ( - \beta_{4} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{19} + 4 q^{21} + 10 q^{23} + 12 q^{25} + 6 q^{27} - 8 q^{29} + 4 q^{31} + 6 q^{33} + 20 q^{35} + 12 q^{37} + 2 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} + 4 q^{47} + 18 q^{49} + 12 q^{53} - 10 q^{55} - 2 q^{57} + 16 q^{59} + 20 q^{61} + 4 q^{63} - 22 q^{65} - 28 q^{67} + 10 q^{69} + 36 q^{71} - 12 q^{73} + 12 q^{75} + 20 q^{77} + 16 q^{79} + 6 q^{81} + 20 q^{83} - 8 q^{87} + 4 q^{89} + 36 q^{91} + 4 q^{93} + 22 q^{95} - 32 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 13x^{4} + 12x^{3} + 56x^{2} + 8x - 34 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 6\nu^{3} - 14\nu^{2} - 10\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 8\nu^{3} + 20\nu^{2} + 18\nu - 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 10\nu^{3} - 22\nu^{2} - 30\nu + 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 5\nu^{4} + 6\nu^{3} - 34\nu^{2} - 18\nu + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 27\nu^{2} - 20\nu + 26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{5} + 7\beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} + 8\beta_{4} + 5\beta_{3} + 3\beta_{2} + 4\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -30\beta_{5} + 33\beta_{4} + 17\beta_{3} - 2\beta_{2} + 6\beta _1 + 108 \) 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.

comment: embeddings in the coefficient field
 
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
3.11578
−1.59250
3.29083
0.702165
−1.40373
−2.11254
0 1.00000 0 −3.36112 0 −1.25969 0 1.00000 0
1.2 0 1.00000 0 −2.92557 0 0.433936 0 1.00000 0
1.3 0 1.00000 0 0.423961 0 −1.64943 0 1.00000 0
1.4 0 1.00000 0 1.86974 0 4.94562 0 1.00000 0
1.5 0 1.00000 0 2.49138 0 −3.10014 0 1.00000 0
1.6 0 1.00000 0 3.50161 0 4.62971 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bj 6
17.b even 2 1 6936.2.a.bg 6
17.d even 8 2 408.2.v.c ✓ 12
51.g odd 8 2 1224.2.w.k 12
68.g odd 8 2 816.2.bd.f 12
204.p even 8 2 2448.2.be.y 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.v.c ✓ 12 17.d even 8 2
816.2.bd.f 12 68.g odd 8 2
1224.2.w.k 12 51.g odd 8 2
2448.2.be.y 12 204.p even 8 2
6936.2.a.bg 6 17.b even 2 1
6936.2.a.bj 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6936))\):

\( T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 40T_{5}^{3} + 80T_{5}^{2} - 200T_{5} + 68 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} - 22T_{7}^{4} + 48T_{7}^{3} + 176T_{7}^{2} + 64T_{7} - 64 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} - 39T_{11}^{4} + 240T_{11}^{3} + 160T_{11}^{2} - 1280T_{11} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 68 \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots - 128 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + \cdots - 128 \) Copy content Toggle raw display
$23$ \( T^{6} - 10 T^{5} + \cdots + 992 \) Copy content Toggle raw display
$29$ \( T^{6} + 8 T^{5} + \cdots - 2528 \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} + \cdots - 2176 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 712 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + \cdots + 71204 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} + \cdots - 3584 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + \cdots + 172544 \) Copy content Toggle raw display
$59$ \( T^{6} - 16 T^{5} + \cdots - 57088 \) Copy content Toggle raw display
$61$ \( T^{6} - 20 T^{5} + \cdots - 824 \) Copy content Toggle raw display
$67$ \( T^{6} + 28 T^{5} + \cdots - 11776 \) Copy content Toggle raw display
$71$ \( T^{6} - 36 T^{5} + \cdots + 353152 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + \cdots - 29296 \) Copy content Toggle raw display
$79$ \( T^{6} - 16 T^{5} + \cdots + 1158208 \) Copy content Toggle raw display
$83$ \( T^{6} - 20 T^{5} + \cdots + 8704 \) Copy content Toggle raw display
$89$ \( T^{6} - 4 T^{5} + \cdots - 1024 \) Copy content Toggle raw display
$97$ \( T^{6} + 32 T^{5} + \cdots + 42208 \) Copy content Toggle raw display
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