Properties

Label 6760.2.a.bi
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: 6.6.2249737.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 7x^{4} + 9x^{3} + 14x^{2} - 8x - 8 \) 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

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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + q^{5} + \beta_{2} q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + q^{5} + \beta_{2} q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} - \beta_1) q^{15} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{17} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{4} - \beta_{3} - 2) q^{21} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{23} + q^{25} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{27} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{29} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{31} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{33} + \beta_{2} q^{35} + (2 \beta_{5} - 3 \beta_{3} - 3) q^{37} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{41} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{43} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{45} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_1 - 3) q^{47} + (\beta_{4} - 5) q^{49} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{51} + (2 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1) q^{53} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{55} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 4) q^{57} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 2 \beta_1 + 5) q^{59} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{61} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{63} + (2 \beta_{5} - 6 \beta_{2} + \beta_1 - 1) q^{67} + ( - 3 \beta_{4} - \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 5) q^{69} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - 2) q^{71} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{73} + ( - \beta_{2} - \beta_1) q^{75} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{77} + (2 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{79} + (2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{81} + (4 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 2) q^{83} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{85} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} + 4 \beta_1) q^{87} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_1 - 7) q^{89} + ( - \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 12) q^{93} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{95} + (\beta_{5} - 5 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{97} + ( - 2 \beta_{5} - 8 \beta_{4} - \beta_{3} + 2 \beta_{2} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{17} + 2 q^{19} - 7 q^{21} + 3 q^{23} + 6 q^{25} - 11 q^{29} - 5 q^{31} - 20 q^{33} - 2 q^{35} - 7 q^{37} - 27 q^{41} - 4 q^{43} + 4 q^{45} - 19 q^{47} - 32 q^{49} + 5 q^{51} - 6 q^{53} - 5 q^{55} + 6 q^{57} + 19 q^{59} - 4 q^{61} - 6 q^{63} + 10 q^{67} - 15 q^{69} - 15 q^{71} - 14 q^{73} + 4 q^{77} + 7 q^{79} + 2 q^{81} + 12 q^{83} - 9 q^{85} + 21 q^{87} - 34 q^{89} - 55 q^{93} + 2 q^{95} + 2 q^{97} - 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 7x^{4} + 9x^{3} + 14x^{2} - 8x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 5\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 5\nu^{2} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - 3\nu^{3} + 19\nu^{2} - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 2\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 2\beta_{4} - \beta_{3} - 2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} - 4\beta_{4} - 7\beta_{3} - 2\beta_{2} + 10\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 23\beta_{5} - 18\beta_{4} - 12\beta_{3} - 14\beta_{2} + 36\beta _1 + 2 \) Copy content Toggle raw display

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
2.92965
1.06899
1.71866
−1.87093
−0.682674
−1.16370
0 −2.48461 0 1.00000 0 −0.445042 0 3.17330 0
1.2 0 −2.31597 0 1.00000 0 1.24698 0 2.36371 0
1.3 0 0.0832805 0 1.00000 0 −1.80194 0 −2.99306 0
1.4 0 0.623947 0 1.00000 0 1.24698 0 −2.61069 0
1.5 0 1.12772 0 1.00000 0 −0.445042 0 −1.72826 0
1.6 0 2.96564 0 1.00000 0 −1.80194 0 5.79500 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.bi yes 6
13.b even 2 1 6760.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6760.2.a.bh 6 13.b even 2 1
6760.2.a.bi yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):

\( T_{3}^{6} - 11T_{3}^{4} + 24T_{3}^{2} - 14T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 2T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 5T_{11}^{5} - 14T_{11}^{4} - 61T_{11}^{3} + 56T_{11}^{2} + 20T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 11 T^{4} + 24 T^{2} - 14 T + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} - 14 T^{4} - 61 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} + 16 T^{4} - 67 T^{3} + \cdots - 104 \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} - 35 T^{4} + 79 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} - 37 T^{4} + 121 T^{3} + \cdots + 13 \) Copy content Toggle raw display
$29$ \( T^{6} + 11 T^{5} - 5 T^{4} - 195 T^{3} + \cdots + 239 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} - 98 T^{4} + \cdots - 13448 \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{5} - 65 T^{4} + \cdots + 8128 \) Copy content Toggle raw display
$41$ \( T^{6} + 27 T^{5} + 237 T^{4} + \cdots + 7937 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} - 69 T^{4} + \cdots + 1373 \) Copy content Toggle raw display
$47$ \( T^{6} + 19 T^{5} + 25 T^{4} + \cdots - 3191 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} - 85 T^{4} + 19 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$59$ \( T^{6} - 19 T^{5} + 31 T^{4} + \cdots + 5144 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} - 111 T^{4} + \cdots - 14209 \) Copy content Toggle raw display
$67$ \( T^{6} - 10 T^{5} - 168 T^{4} + \cdots - 3753 \) Copy content Toggle raw display
$71$ \( T^{6} + 15 T^{5} + 4 T^{4} - 249 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$73$ \( T^{6} + 14 T^{5} - 60 T^{4} + \cdots + 80704 \) Copy content Toggle raw display
$79$ \( T^{6} - 7 T^{5} - 128 T^{4} + \cdots - 216 \) Copy content Toggle raw display
$83$ \( T^{6} - 12 T^{5} - 199 T^{4} + \cdots + 122851 \) Copy content Toggle raw display
$89$ \( T^{6} + 34 T^{5} + 386 T^{4} + \cdots - 839 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} - 361 T^{4} + \cdots - 644344 \) Copy content Toggle raw display
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