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Properties

Label 6776.2.a.bp
Level $6776$
Weight $2$
Character orbit 6776.a
Self dual yes
Analytic conductor $54.107$
Analytic rank $0$
Dimension $10$
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 178x^{6} - 266x^{5} - 515x^{4} + 700x^{3} + 433x^{2} - 624x + 132 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} + q^{7} + (\beta_{7} - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{6} q^{5} + q^{7} + (\beta_{7} - \beta_{6} + 2) q^{9} + ( - \beta_{4} + \beta_{3} - 1) q^{13} + ( - \beta_{6} + \beta_{3} + \cdots + \beta_1) q^{15}+ \cdots + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9} - 12 q^{13} + 10 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 8 q^{25} + 14 q^{27} - 16 q^{29} + 14 q^{31} + 4 q^{35} + 2 q^{37} - 4 q^{39} + 18 q^{41} + 16 q^{43} + 52 q^{45} - 4 q^{47} + 10 q^{49} + 16 q^{51} + 32 q^{53} - 24 q^{57} + 4 q^{59} + 16 q^{61} + 20 q^{63} - 4 q^{65} + 10 q^{67} - 14 q^{69} - 10 q^{71} - 22 q^{73} + 48 q^{75} + 36 q^{79} + 50 q^{81} + 2 q^{83} - 4 q^{85} - 2 q^{87} + 36 q^{89} - 12 q^{91} - 2 q^{93} + 52 q^{95} + 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 178x^{6} - 266x^{5} - 515x^{4} + 700x^{3} + 433x^{2} - 624x + 132 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 10 \nu^{8} + 57 \nu^{7} - 416 \nu^{6} - 1642 \nu^{5} + 7722 \nu^{4} + 12355 \nu^{3} + \cdots + 13530 ) / 2574 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{9} + 10 \nu^{8} + 21 \nu^{7} - 208 \nu^{6} - 152 \nu^{5} + 1326 \nu^{4} + 437 \nu^{3} + \cdots + 744 ) / 234 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{9} + 70 \nu^{8} + 30 \nu^{7} - 1378 \nu^{6} + 1198 \nu^{5} + 8580 \nu^{4} - 10123 \nu^{3} + \cdots + 1815 ) / 1287 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{9} + \nu^{8} + 84\nu^{7} - 13\nu^{6} - 530\nu^{5} + 39\nu^{4} + 929\nu^{3} + 25\nu^{2} - 60\nu - 66 ) / 117 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47 \nu^{9} - 41 \nu^{8} - 1182 \nu^{7} + 611 \nu^{6} + 9913 \nu^{5} - 2574 \nu^{4} - 30211 \nu^{3} + \cdots - 8877 ) / 1287 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47 \nu^{9} - 41 \nu^{8} - 1182 \nu^{7} + 611 \nu^{6} + 9913 \nu^{5} - 2574 \nu^{4} - 30211 \nu^{3} + \cdots - 15312 ) / 1287 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 95 \nu^{9} - 92 \nu^{8} - 2307 \nu^{7} + 1664 \nu^{6} + 19900 \nu^{5} - 10296 \nu^{4} - 72091 \nu^{3} + \cdots - 35112 ) / 2574 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{9} - 6 \nu^{8} - 111 \nu^{7} + 98 \nu^{6} + 810 \nu^{5} - 462 \nu^{4} - 2113 \nu^{3} + \cdots - 462 ) / 66 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + 10\beta_{7} - 11\beta_{6} + 2\beta_{4} - 2\beta_{3} + 2\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{9} - 12 \beta_{8} + 14 \beta_{7} - 16 \beta_{6} + 14 \beta_{5} + \beta_{4} + 13 \beta_{3} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{9} - 16\beta_{8} + 101\beta_{7} - 115\beta_{6} + 28\beta_{4} - 28\beta_{3} - 5\beta_{2} + 41\beta _1 + 371 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 162 \beta_{9} - 124 \beta_{8} + 173 \beta_{7} - 215 \beta_{6} + 158 \beta_{5} + 15 \beta_{4} + \cdots + 368 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 187 \beta_{9} - 204 \beta_{8} + 1041 \beta_{7} - 1204 \beta_{6} + 4 \beta_{5} + 320 \beta_{4} + \cdots + 3655 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1787 \beta_{9} - 1255 \beta_{8} + 2046 \beta_{7} - 2668 \beta_{6} + 1667 \beta_{5} + 191 \beta_{4} + \cdots + 4893 \) 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
−3.04642
−2.69826
−1.76789
−1.45987
0.291788
0.689053
1.53960
1.83824
3.27764
3.33612
0 −3.04642 0 1.52053 0 1.00000 0 6.28070 0
1.2 0 −2.69826 0 2.04005 0 1.00000 0 4.28062 0
1.3 0 −1.76789 0 −2.55889 0 1.00000 0 0.125418 0
1.4 0 −1.45987 0 −0.909528 0 1.00000 0 −0.868786 0
1.5 0 0.291788 0 0.737206 0 1.00000 0 −2.91486 0
1.6 0 0.689053 0 −3.44569 0 1.00000 0 −2.52521 0
1.7 0 1.53960 0 3.69314 0 1.00000 0 −0.629646 0
1.8 0 1.83824 0 −0.648759 0 1.00000 0 0.379111 0
1.9 0 3.27764 0 −0.590511 0 1.00000 0 7.74294 0
1.10 0 3.33612 0 4.16244 0 1.00000 0 8.12971 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bp yes 10
11.b odd 2 1 6776.2.a.bo ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6776.2.a.bo ✓ 10 11.b odd 2 1
6776.2.a.bp yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6776))\):

\( T_{3}^{10} - 2 T_{3}^{9} - 23 T_{3}^{8} + 40 T_{3}^{7} + 178 T_{3}^{6} - 266 T_{3}^{5} - 515 T_{3}^{4} + \cdots + 132 \) Copy content Toggle raw display
\( T_{5}^{10} - 4 T_{5}^{9} - 21 T_{5}^{8} + 80 T_{5}^{7} + 127 T_{5}^{6} - 432 T_{5}^{5} - 231 T_{5}^{4} + \cdots - 108 \) Copy content Toggle raw display
\( T_{13}^{10} + 12 T_{13}^{9} - 27 T_{13}^{8} - 772 T_{13}^{7} - 1467 T_{13}^{6} + 12072 T_{13}^{5} + \cdots + 17424 \) Copy content Toggle raw display
\( T_{17}^{10} - 2 T_{17}^{9} - 90 T_{17}^{8} + 246 T_{17}^{7} + 1902 T_{17}^{6} - 7416 T_{17}^{5} + \cdots - 416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 2 T^{9} + \cdots + 132 \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots - 108 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 12 T^{9} + \cdots + 17424 \) Copy content Toggle raw display
$17$ \( T^{10} - 2 T^{9} + \cdots - 416 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 150544 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 17424 \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} + \cdots - 1395648 \) Copy content Toggle raw display
$31$ \( T^{10} - 14 T^{9} + \cdots + 1480704 \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + \cdots + 527616 \) Copy content Toggle raw display
$41$ \( T^{10} - 18 T^{9} + \cdots + 14786064 \) Copy content Toggle raw display
$43$ \( T^{10} - 16 T^{9} + \cdots - 773888 \) Copy content Toggle raw display
$47$ \( T^{10} + 4 T^{9} + \cdots + 1004544 \) Copy content Toggle raw display
$53$ \( T^{10} - 32 T^{9} + \cdots - 14100512 \) Copy content Toggle raw display
$59$ \( T^{10} - 4 T^{9} + \cdots + 30023136 \) Copy content Toggle raw display
$61$ \( T^{10} - 16 T^{9} + \cdots - 28036224 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 258306048 \) Copy content Toggle raw display
$71$ \( T^{10} + 10 T^{9} + \cdots + 82570752 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 193189888 \) Copy content Toggle raw display
$79$ \( T^{10} - 36 T^{9} + \cdots + 1126368 \) Copy content Toggle raw display
$83$ \( T^{10} - 2 T^{9} + \cdots + 3562596 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 268423632 \) Copy content Toggle raw display
$97$ \( T^{10} - 44 T^{9} + \cdots - 47190144 \) Copy content Toggle raw display
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