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Properties

Label 6760.2.a.bl
Level $6760$
Weight $2$
Character orbit 6760.a
Self dual yes
Analytic conductor $53.979$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 27x^{4} - 14x^{3} - 22x^{2} + 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{5} + 1) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{5} + 1) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{9}+ \cdots + (5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{5} + 6 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{5} + 6 q^{7} + 16 q^{9} + 6 q^{11} + 4 q^{15} + 4 q^{17} + 10 q^{19} + 10 q^{21} + 6 q^{23} + 8 q^{25} + 22 q^{27} + 16 q^{29} + 18 q^{31} + 6 q^{35} - 12 q^{37} - 8 q^{41} + 6 q^{43} + 16 q^{45} + 8 q^{47} - 4 q^{49} + 20 q^{51} + 2 q^{53} + 6 q^{55} - 8 q^{57} + 2 q^{61} + 44 q^{63} + 2 q^{67} + 52 q^{69} + 24 q^{71} - 14 q^{73} + 4 q^{75} - 16 q^{77} + 18 q^{79} + 28 q^{81} - 6 q^{83} + 4 q^{85} + 22 q^{87} - 28 q^{89} - 36 q^{93} + 10 q^{95} - 16 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 27x^{4} - 14x^{3} - 22x^{2} + 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 12\nu^{4} + 25\nu^{3} - 12\nu^{2} - 12\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 12\nu^{4} + 27\nu^{3} - 14\nu^{2} - 18\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 12\nu^{4} + 27\nu^{3} - 14\nu^{2} - 22\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 2\nu^{6} - 8\nu^{5} + 10\nu^{4} + 20\nu^{3} - 5\nu^{2} - 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 18\nu^{4} + 10\nu^{3} - 25\nu^{2} - 2\nu + 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 3\beta_{4} - \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 9\beta_{4} - 2\beta_{3} + 2\beta_{2} + 12\beta _1 + 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{6} - 23\beta_{5} + 37\beta_{4} - 18\beta_{3} + 4\beta_{2} + 20\beta _1 + 79 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + 3\beta_{6} - 24\beta_{5} + 36\beta_{4} - 16\beta_{3} + 12\beta_{2} + 38\beta _1 + 125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{7} + 30\beta_{6} - 143\beta_{5} + 243\beta_{4} - 148\beta_{3} + 60\beta_{2} + 162\beta _1 + 585 ) / 2 \) 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
−1.97402
2.79253
2.44974
−1.44614
−0.897932
−0.432338
0.956612
0.551543
0 −2.87076 0 1.00000 0 −0.0131629 0 5.24124 0
1.2 0 −2.00568 0 1.00000 0 1.71620 0 1.02274 0
1.3 0 −0.551495 0 1.00000 0 1.81641 0 −2.69585 0
1.4 0 −0.537453 0 1.00000 0 −0.382994 0 −2.71114 0
1.5 0 1.29579 0 1.00000 0 −1.22734 0 −1.32094 0
1.6 0 2.38075 0 1.00000 0 −3.62601 0 2.66795 0
1.7 0 3.04151 0 1.00000 0 3.09071 0 6.25076 0
1.8 0 3.24734 0 1.00000 0 4.62619 0 7.54524 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bl 8
13.b even 2 1 6760.2.a.bk 8
13.f odd 12 2 520.2.bu.b ✓ 16
52.l even 12 2 1040.2.da.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bu.b ✓ 16 13.f odd 12 2
1040.2.da.f 16 52.l even 12 2
6760.2.a.bk 8 13.b even 2 1
6760.2.a.bl 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):

\( T_{3}^{8} - 4T_{3}^{7} - 12T_{3}^{6} + 54T_{3}^{5} + 33T_{3}^{4} - 190T_{3}^{3} - 22T_{3}^{2} + 140T_{3} + 52 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{7} - 8T_{7}^{6} + 90T_{7}^{5} - 82T_{7}^{4} - 162T_{7}^{3} + 152T_{7}^{2} + 78T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} - 64T_{11}^{6} + 490T_{11}^{5} + 354T_{11}^{4} - 9386T_{11}^{3} + 24120T_{11}^{2} - 21530T_{11} + 6229 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 52 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 6229 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots - 7964 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 2497 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 35152 \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$31$ \( T^{8} - 18 T^{7} + \cdots - 196352 \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + \cdots - 4643 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots - 37232 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots - 378608 \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{7} + \cdots - 1143536 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots - 663344 \) Copy content Toggle raw display
$59$ \( T^{8} - 270 T^{6} + \cdots - 1939184 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + \cdots + 256036 \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{7} + \cdots - 822272 \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + \cdots + 817168 \) Copy content Toggle raw display
$73$ \( T^{8} + 14 T^{7} + \cdots + 360256 \) Copy content Toggle raw display
$79$ \( T^{8} - 18 T^{7} + \cdots + 659776 \) Copy content Toggle raw display
$83$ \( T^{8} + 6 T^{7} + \cdots + 9472 \) Copy content Toggle raw display
$89$ \( T^{8} + 28 T^{7} + \cdots - 2346227 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots - 1633532 \) Copy content Toggle raw display
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