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Properties

Label 6776.2.a.bn
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 / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6776,2,Mod(1,6776)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-3,0,5,0,10,0,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.1066324096\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 19x^{8} + 52x^{7} + 128x^{6} - 304x^{5} - 349x^{4} + 691x^{3} + 279x^{2} - 460x + 100 \) 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} + 1) q^{5} + q^{7} + (\beta_{9} + \beta_{8} - \beta_{5} + \cdots + 2) q^{9} + ( - \beta_{9} - \beta_{6} + \cdots - \beta_1) q^{13} + ( - 2 \beta_{9} - \beta_{8} - \beta_{7} + \cdots - 2) q^{15}+ \cdots + ( - 2 \beta_{9} + 2 \beta_{8} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} + 5 q^{5} + 10 q^{7} + 17 q^{9} + 4 q^{13} - 6 q^{15} + 10 q^{17} + 12 q^{19} - 3 q^{21} - 3 q^{23} + 9 q^{25} - 24 q^{27} + 2 q^{29} + q^{31} + 5 q^{35} + 25 q^{37} + 29 q^{39} + 11 q^{41}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 19x^{8} + 52x^{7} + 128x^{6} - 304x^{5} - 349x^{4} + 691x^{3} + 279x^{2} - 460x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{9} - 134 \nu^{8} + 413 \nu^{7} + 946 \nu^{6} - 7356 \nu^{5} + 3898 \nu^{4} + 31143 \nu^{3} + \cdots + 21190 ) / 1660 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{9} + 27 \nu^{8} - 38 \nu^{7} - 369 \nu^{6} + 755 \nu^{5} + 1437 \nu^{4} - 3061 \nu^{3} + \cdots - 770 ) / 166 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17 \nu^{9} - 144 \nu^{8} + 1088 \nu^{7} + 1636 \nu^{6} - 13046 \nu^{5} - 3182 \nu^{4} + \cdots + 16280 ) / 1660 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33 \nu^{9} - 11 \nu^{8} + 867 \nu^{7} + 344 \nu^{6} - 7504 \nu^{5} - 2808 \nu^{4} + 22907 \nu^{3} + \cdots + 5970 ) / 1660 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{9} + 132 \nu^{8} - 29 \nu^{7} - 1638 \nu^{6} + 2898 \nu^{5} + 5061 \nu^{4} - 13849 \nu^{3} + \cdots - 4410 ) / 830 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74 \nu^{9} - 252 \nu^{8} - 1001 \nu^{7} + 3278 \nu^{6} + 3522 \nu^{5} - 10756 \nu^{4} + 294 \nu^{3} + \cdots + 5250 ) / 1660 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77 \nu^{9} + 251 \nu^{8} + 1193 \nu^{7} - 3624 \nu^{6} - 6166 \nu^{5} + 15858 \nu^{4} + \cdots + 5630 ) / 1660 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 81 \nu^{9} - 388 \nu^{8} - 1034 \nu^{7} + 6022 \nu^{6} + 4158 \nu^{5} - 29854 \nu^{4} - 10199 \nu^{3} + \cdots - 16540 ) / 1660 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{9} + 8\beta_{8} + 6\beta_{6} - 8\beta_{5} + 12\beta_{4} + 12\beta_{3} + 4\beta_{2} + 3\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{9} - 11 \beta_{8} - 14 \beta_{7} + 30 \beta_{6} - 13 \beta_{5} + 29 \beta_{4} + 19 \beta_{3} + \cdots + 51 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 127 \beta_{9} + 51 \beta_{8} - 11 \beta_{7} + 94 \beta_{6} - 65 \beta_{5} + 134 \beta_{4} + 133 \beta_{3} + \cdots + 359 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 258 \beta_{9} - 122 \beta_{8} - 170 \beta_{7} + 360 \beta_{6} - 138 \beta_{5} + 360 \beta_{4} + \cdots + 652 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1321 \beta_{9} + 229 \beta_{8} - 260 \beta_{7} + 1164 \beta_{6} - 553 \beta_{5} + 1477 \beta_{4} + \cdots + 3433 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3145 \beta_{9} - 1493 \beta_{8} - 2005 \beta_{7} + 4106 \beta_{6} - 1357 \beta_{5} + 4266 \beta_{4} + \cdots + 7643 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
3.39984
3.06783
2.08894
1.81368
0.525921
0.316060
−1.24331
−2.07452
−2.15336
−2.74108
0 −3.39984 0 0.687479 0 1.00000 0 8.55894 0
1.2 0 −3.06783 0 3.86956 0 1.00000 0 6.41157 0
1.3 0 −2.08894 0 −2.11319 0 1.00000 0 1.36366 0
1.4 0 −1.81368 0 −2.26477 0 1.00000 0 0.289425 0
1.5 0 −0.525921 0 2.17642 0 1.00000 0 −2.72341 0
1.6 0 −0.316060 0 0.332826 0 1.00000 0 −2.90011 0
1.7 0 1.24331 0 3.35765 0 1.00000 0 −1.45419 0
1.8 0 2.07452 0 2.96067 0 1.00000 0 1.30365 0
1.9 0 2.15336 0 −2.72639 0 1.00000 0 1.63697 0
1.10 0 2.74108 0 −1.28025 0 1.00000 0 4.51350 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.bn 10
11.b odd 2 1 6776.2.a.bm 10
11.d odd 10 2 616.2.r.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.r.f ✓ 20 11.d odd 10 2
6776.2.a.bm 10 11.b odd 2 1
6776.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6776))\):

\( T_{3}^{10} + 3 T_{3}^{9} - 19 T_{3}^{8} - 52 T_{3}^{7} + 128 T_{3}^{6} + 304 T_{3}^{5} - 349 T_{3}^{4} + \cdots + 100 \) Copy content Toggle raw display
\( T_{5}^{10} - 5 T_{5}^{9} - 17 T_{5}^{8} + 97 T_{5}^{7} + 102 T_{5}^{6} - 652 T_{5}^{5} - 265 T_{5}^{4} + \cdots + 320 \) Copy content Toggle raw display
\( T_{13}^{10} - 4 T_{13}^{9} - 73 T_{13}^{8} + 253 T_{13}^{7} + 1855 T_{13}^{6} - 5475 T_{13}^{5} + \cdots + 74896 \) Copy content Toggle raw display
\( T_{17}^{10} - 10 T_{17}^{9} - 38 T_{17}^{8} + 501 T_{17}^{7} + 462 T_{17}^{6} - 8179 T_{17}^{5} + \cdots + 7180 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} + \cdots + 320 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + \cdots + 74896 \) Copy content Toggle raw display
$17$ \( T^{10} - 10 T^{9} + \cdots + 7180 \) Copy content Toggle raw display
$19$ \( T^{10} - 12 T^{9} + \cdots - 47924 \) Copy content Toggle raw display
$23$ \( T^{10} + 3 T^{9} + \cdots - 368404 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots - 798976 \) Copy content Toggle raw display
$31$ \( T^{10} - T^{9} + \cdots - 318704 \) Copy content Toggle raw display
$37$ \( T^{10} - 25 T^{9} + \cdots - 76396 \) Copy content Toggle raw display
$41$ \( T^{10} - 11 T^{9} + \cdots - 4544 \) Copy content Toggle raw display
$43$ \( T^{10} - 21 T^{9} + \cdots - 398345 \) Copy content Toggle raw display
$47$ \( T^{10} + 21 T^{9} + \cdots - 82736 \) Copy content Toggle raw display
$53$ \( T^{10} - 46 T^{9} + \cdots + 4145920 \) Copy content Toggle raw display
$59$ \( T^{10} + 7 T^{9} + \cdots + 154304 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 165574720 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} + \cdots + 78764149 \) Copy content Toggle raw display
$71$ \( T^{10} + 16 T^{9} + \cdots + 3469520 \) Copy content Toggle raw display
$73$ \( T^{10} - 37 T^{9} + \cdots - 11376784 \) Copy content Toggle raw display
$79$ \( T^{10} - 11 T^{9} + \cdots + 952436 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 395302976 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 455935796 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 4219135504 \) Copy content Toggle raw display
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