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Properties

Label 6760.2.a.bg
Level $6760$
Weight $2$
Character orbit 6760.a
Self dual yes
Analytic conductor $53.979$
Analytic rank $1$
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] = [6760,2,Mod(1,6760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6760, 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("6760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6760.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: \(53.9788717664\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 520)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{7} + (\beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{7} + (\beta_{3} - \beta_{2}) q^{9} + (\beta_{5} + \beta_1) q^{11} + \beta_1 q^{15} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{3} - 3 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{5} - 4 q^{7} - 2 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 18 q^{21} - 2 q^{23} + 6 q^{25} - 6 q^{27} - 12 q^{29} - 10 q^{31} - 12 q^{33} + 4 q^{35} + 10 q^{37} + 8 q^{41} - 18 q^{43} + 2 q^{45} - 6 q^{47} + 24 q^{49} - 34 q^{53} - 2 q^{55} + 16 q^{57} + 40 q^{59} - 18 q^{61} - 2 q^{63} - 10 q^{67} - 8 q^{69} - 4 q^{71} + 18 q^{73} - 32 q^{77} - 6 q^{79} - 14 q^{81} + 14 q^{83} + 8 q^{85} - 18 q^{87} + 4 q^{89} - 8 q^{93} - 10 q^{95} + 28 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - 5\nu^{3} + 2\nu^{2} + 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 3\nu - 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 7\beta_{4} - 5\beta_{3} + 2\beta_{2} + 10\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{5} + 21\beta_{4} - 17\beta_{3} + 12\beta_{2} + 16\beta _1 + 35 ) / 2 \) 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
−1.70320
2.38595
−1.10939
1.82356
−0.276564
0.879640
0 −2.60407 0 −1.00000 0 −4.96132 0 3.78120 0
1.2 0 −1.30682 0 −1.00000 0 2.90491 0 −1.29223 0
1.3 0 −0.340131 0 −1.00000 0 −4.71117 0 −2.88431 0
1.4 0 0.498200 0 −1.00000 0 −1.33854 0 −2.75180 0
1.5 0 1.64695 0 −1.00000 0 1.53831 0 −0.287562 0
1.6 0 2.10587 0 −1.00000 0 2.56781 0 1.43470 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
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 6760.2.a.bg 6
13.b even 2 1 6760.2.a.bj 6
13.f odd 12 2 520.2.bu.a ✓ 12
52.l even 12 2 1040.2.da.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bu.a ✓ 12 13.f odd 12 2
1040.2.da.e 12 52.l even 12 2
6760.2.a.bg 6 1.a even 1 1 trivial
6760.2.a.bj 6 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):

\( T_{3}^{6} - 8T_{3}^{4} + 2T_{3}^{3} + 13T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 25T_{7}^{4} - 60T_{7}^{3} + 231T_{7}^{2} + 80T_{7} - 359 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} - 17T_{11}^{4} + 20T_{11}^{3} + 67T_{11}^{2} - 42T_{11} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 8 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots - 359 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots - 11 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 8 T^{5} + \cdots - 194 \) Copy content Toggle raw display
$19$ \( T^{6} - 10 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{6} + 2 T^{5} + \cdots - 17096 \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} + \cdots - 3656 \) Copy content Toggle raw display
$31$ \( T^{6} + 10 T^{5} + \cdots - 4736 \) Copy content Toggle raw display
$37$ \( T^{6} - 10 T^{5} + \cdots + 6361 \) Copy content Toggle raw display
$41$ \( T^{6} - 8 T^{5} + \cdots - 355052 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{5} + \cdots - 29916 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + \cdots - 15984 \) Copy content Toggle raw display
$53$ \( T^{6} + 34 T^{5} + \cdots + 19792 \) Copy content Toggle raw display
$59$ \( T^{6} - 40 T^{5} + \cdots - 15404 \) Copy content Toggle raw display
$61$ \( T^{6} + 18 T^{5} + \cdots + 3478 \) Copy content Toggle raw display
$67$ \( T^{6} + 10 T^{5} + \cdots + 3904 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} + \cdots + 20452 \) Copy content Toggle raw display
$73$ \( T^{6} - 18 T^{5} + \cdots - 231968 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + \cdots + 50272 \) Copy content Toggle raw display
$83$ \( T^{6} - 14 T^{5} + \cdots + 469888 \) Copy content Toggle raw display
$89$ \( T^{6} - 4 T^{5} + \cdots - 48863 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + \cdots + 91702 \) Copy content Toggle raw display
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