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Properties

Label 6776.2.a.bl
Level $6776$
Weight $2$
Character orbit 6776.a
Self dual yes
Analytic conductor $54.107$
Analytic rank $1$
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] = [6776,2,Mod(1,6776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6776, 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("6776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6776.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: \(54.1066324096\)
Analytic rank: \(1\)
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: \( x^{8} - x^{7} - 13x^{6} + 13x^{5} + 34x^{4} - 14x^{3} - 23x^{2} + 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{2} - 1) q^{5} + q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{2} - 1) q^{5} + q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{9}+ \cdots + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 7 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 7 q^{5} + 8 q^{7} + 12 q^{9} - 8 q^{13} - 14 q^{15} - 3 q^{17} - 15 q^{19} + 2 q^{21} + 7 q^{23} - q^{25} + 2 q^{27} - 2 q^{29} + 11 q^{31} - 7 q^{35} - 27 q^{37} - 9 q^{39} - 12 q^{41} - 18 q^{43} - 35 q^{45} + 3 q^{47} + 8 q^{49} - 37 q^{51} - 22 q^{53} + 2 q^{57} + 6 q^{59} - 20 q^{61} + 12 q^{63} + 13 q^{65} - 10 q^{67} - 30 q^{69} + 14 q^{71} + 14 q^{73} + 44 q^{75} - 27 q^{79} + 20 q^{81} - 4 q^{83} - 4 q^{85} + 4 q^{87} - 8 q^{89} - 8 q^{91} - 25 q^{93} - 22 q^{95} + 23 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 13x^{6} + 13x^{5} + 34x^{4} - 14x^{3} - 23x^{2} + 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 7\nu^{6} + 3\nu^{5} - 79\nu^{4} + 88\nu^{3} + 78\nu^{2} - 77\nu + 10 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 9\nu^{6} + 19\nu^{5} + 113\nu^{4} - 120\nu^{3} - 290\nu^{2} + 67\nu + 122 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 3\nu^{6} - 63\nu^{5} + 43\nu^{4} + 152\nu^{3} - 22\nu^{2} - 47\nu - 2 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} + 29\nu^{6} + 113\nu^{5} - 357\nu^{4} - 8\nu^{3} + 522\nu^{2} - 127\nu - 194 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 15\nu^{6} - 107\nu^{5} + 183\nu^{4} + 184\nu^{3} - 190\nu^{2} - 91\nu + 22 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -35\nu^{7} + 37\nu^{6} + 441\nu^{5} - 461\nu^{4} - 1000\nu^{3} + 346\nu^{2} + 361\nu - 82 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} + 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 9\beta_{5} + 11\beta_{4} + 11\beta_{3} - 9\beta_{2} - 4\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 20\beta_{6} - 16\beta_{5} - 13\beta_{4} - 27\beta_{3} + 24\beta_{2} + 60\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} - 18\beta_{6} + 82\beta_{5} + 109\beta_{4} + 106\beta_{3} - 86\beta_{2} - 70\beta _1 + 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 107\beta_{7} + 189\beta_{6} - 195\beta_{5} - 155\beta_{4} - 306\beta_{3} + 263\beta_{2} + 545\beta _1 - 185 \) 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
0.586008
0.795423
−0.816154
−3.21575
−0.433956
2.75643
2.44567
−1.11767
0 −2.85959 0 −0.967327 0 1.00000 0 5.17726 0
1.2 0 −2.54974 0 −0.169222 0 1.00000 0 3.50117 0
1.3 0 −1.06131 0 1.69568 0 1.00000 0 −1.87361 0
1.4 0 0.205771 0 −3.79959 0 1.00000 0 −2.95766 0
1.5 0 1.16769 0 2.00711 0 1.00000 0 −1.63650 0
1.6 0 1.31114 0 −0.280283 0 1.00000 0 −1.28092 0
1.7 0 2.48613 0 −1.85823 0 1.00000 0 3.18083 0
1.8 0 3.29991 0 −3.62814 0 1.00000 0 7.88943 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
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 6776.2.a.bl 8
11.b odd 2 1 6776.2.a.bk 8
11.c even 5 2 616.2.r.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.r.d ✓ 16 11.c even 5 2
6776.2.a.bk 8 11.b odd 2 1
6776.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6776))\):

\( T_{3}^{8} - 2T_{3}^{7} - 16T_{3}^{6} + 30T_{3}^{5} + 68T_{3}^{4} - 128T_{3}^{3} - 39T_{3}^{2} + 110T_{3} - 20 \) Copy content Toggle raw display
\( T_{5}^{8} + 7T_{5}^{7} + 5T_{5}^{6} - 47T_{5}^{5} - 66T_{5}^{4} + 64T_{5}^{3} + 121T_{5}^{2} + 42T_{5} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} + 8T_{13}^{7} - 41T_{13}^{6} - 461T_{13}^{5} - 183T_{13}^{4} + 6035T_{13}^{3} + 10433T_{13}^{2} - 13552T_{13} - 26704 \) Copy content Toggle raw display
\( T_{17}^{8} + 3T_{17}^{7} - 52T_{17}^{6} - 180T_{17}^{5} + 732T_{17}^{4} + 3309T_{17}^{3} - 1011T_{17}^{2} - 16902T_{17} - 17404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots - 20 \) Copy content Toggle raw display
$5$ \( T^{8} + 7 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots - 26704 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots - 17404 \) Copy content Toggle raw display
$19$ \( T^{8} + 15 T^{7} + \cdots - 764 \) Copy content Toggle raw display
$23$ \( T^{8} - 7 T^{7} + \cdots - 1709 \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 1159 \) Copy content Toggle raw display
$31$ \( T^{8} - 11 T^{7} + \cdots + 284224 \) Copy content Toggle raw display
$37$ \( T^{8} + 27 T^{7} + \cdots - 1481251 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots - 6976 \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 400711 \) Copy content Toggle raw display
$47$ \( T^{8} - 3 T^{7} + \cdots - 104044 \) Copy content Toggle raw display
$53$ \( T^{8} + 22 T^{7} + \cdots - 6760745 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots - 1285324 \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + \cdots + 40220 \) Copy content Toggle raw display
$67$ \( T^{8} + 10 T^{7} + \cdots - 89669 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots - 3091 \) Copy content Toggle raw display
$73$ \( T^{8} - 14 T^{7} + \cdots + 1385764 \) Copy content Toggle raw display
$79$ \( T^{8} + 27 T^{7} + \cdots + 407401 \) Copy content Toggle raw display
$83$ \( T^{8} + 4 T^{7} + \cdots - 1495964 \) Copy content Toggle raw display
$89$ \( T^{8} + 8 T^{7} + \cdots - 361220 \) Copy content Toggle raw display
$97$ \( T^{8} - 23 T^{7} + \cdots + 562000 \) Copy content Toggle raw display
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