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Properties

Label 6664.2.a.be
Level $6664$
Weight $2$
Character orbit 6664.a
Self dual yes
Analytic conductor $53.212$
Analytic rank $0$
Dimension $12$
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] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.2123079070\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 16 x^{10} + 64 x^{9} + 96 x^{8} - 352 x^{7} - 250 x^{6} + 772 x^{5} + 166 x^{4} + \cdots - 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - \beta_{3} q^{5} + (\beta_{11} - \beta_{10} + \beta_{7} + \cdots + 1) q^{9} + ( - \beta_{8} - \beta_{6}) q^{11} - \beta_{8} q^{13} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{15}+ \cdots + (2 \beta_{10} - \beta_{9} - \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 4 q^{5} + 16 q^{9} - 8 q^{11} - 4 q^{13} - 12 q^{17} + 24 q^{19} - 8 q^{23} + 16 q^{25} + 32 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{37} + 8 q^{39} + 16 q^{41} + 4 q^{43} + 8 q^{45} + 28 q^{47}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 16 x^{10} + 64 x^{9} + 96 x^{8} - 352 x^{7} - 250 x^{6} + 772 x^{5} + 166 x^{4} + \cdots - 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 935 \nu^{11} - 7279 \nu^{10} - 4118 \nu^{9} + 121617 \nu^{8} - 61677 \nu^{7} - 730990 \nu^{6} + \cdots + 120820 ) / 46018 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1599 \nu^{11} - 4229 \nu^{10} - 31085 \nu^{9} + 57331 \nu^{8} + 232078 \nu^{7} - 191204 \nu^{6} + \cdots - 97048 ) / 23009 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3179 \nu^{11} + 15545 \nu^{10} + 41612 \nu^{9} - 247833 \nu^{8} - 186053 \nu^{7} + 1380934 \nu^{6} + \cdots - 548842 ) / 46018 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5610 \nu^{11} + 20665 \nu^{10} + 93735 \nu^{9} - 315540 \nu^{8} - 619325 \nu^{7} + 1601851 \nu^{6} + \cdots - 471821 ) / 46018 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8905 \nu^{11} + 32905 \nu^{10} + 148236 \nu^{9} - 510837 \nu^{8} - 944713 \nu^{7} + 2654514 \nu^{6} + \cdots - 789936 ) / 46018 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9629 \nu^{11} - 32906 \nu^{10} - 174729 \nu^{9} + 522521 \nu^{8} + 1239924 \nu^{7} - 2770083 \nu^{6} + \cdots + 944643 ) / 46018 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12938 \nu^{11} + 43557 \nu^{10} + 238323 \nu^{9} - 697196 \nu^{8} - 1720579 \nu^{7} + \cdots - 1491581 ) / 46018 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14898 \nu^{11} - 51187 \nu^{10} - 268857 \nu^{9} + 801286 \nu^{8} + 1921535 \nu^{7} - 4158123 \nu^{6} + \cdots + 1222165 ) / 46018 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17824 \nu^{11} + 64221 \nu^{10} + 305565 \nu^{9} - 1001052 \nu^{8} - 2044693 \nu^{7} + \cdots - 1624623 ) / 46018 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9274 \nu^{11} + 32111 \nu^{10} + 166029 \nu^{9} - 506368 \nu^{8} - 1169952 \nu^{7} + \cdots - 958692 ) / 23009 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{11} - 9 \beta_{10} - \beta_{9} - \beta_{8} + 12 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 17 \beta_{10} - 3 \beta_{9} - 10 \beta_{8} + 29 \beta_{7} + 20 \beta_{6} - 19 \beta_{5} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 133 \beta_{11} - 90 \beta_{10} - 21 \beta_{9} - 18 \beta_{8} + 159 \beta_{7} + 111 \beta_{6} + \cdots + 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 444 \beta_{11} - 237 \beta_{10} - 72 \beta_{9} - 102 \beta_{8} + 502 \beta_{7} + 309 \beta_{6} + \cdots + 376 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1698 \beta_{11} - 1007 \beta_{10} - 343 \beta_{9} - 259 \beta_{8} + 2168 \beta_{7} + 1345 \beta_{6} + \cdots + 1601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5971 \beta_{11} - 3140 \beta_{10} - 1238 \beta_{9} - 1142 \beta_{8} + 7511 \beta_{7} + 4356 \beta_{6} + \cdots + 4389 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22173 \beta_{11} - 12153 \beta_{10} - 5124 \beta_{9} - 3488 \beta_{8} + 29573 \beta_{7} + 17121 \beta_{6} + \cdots + 16786 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 79609 \beta_{11} - 41154 \beta_{10} - 18809 \beta_{9} - 13746 \beta_{8} + 106009 \beta_{7} + \cdots + 51849 \) 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
3.64645
2.87390
2.83452
1.07734
0.904453
0.532539
0.430647
−0.601810
−1.35122
−1.64088
−2.28793
−2.41801
0 −2.64645 0 0.782845 0 0 0 4.00370 0
1.2 0 −1.87390 0 −0.643029 0 0 0 0.511496 0
1.3 0 −1.83452 0 1.46820 0 0 0 0.365474 0
1.4 0 −0.0773396 0 4.04559 0 0 0 −2.99402 0
1.5 0 0.0955473 0 −3.42183 0 0 0 −2.99087 0
1.6 0 0.467461 0 −0.799673 0 0 0 −2.78148 0
1.7 0 0.569353 0 1.10359 0 0 0 −2.67584 0
1.8 0 1.60181 0 1.24041 0 0 0 −0.434206 0
1.9 0 2.35122 0 −3.32934 0 0 0 2.52826 0
1.10 0 2.64088 0 2.35452 0 0 0 3.97427 0
1.11 0 3.28793 0 4.06343 0 0 0 7.81046 0
1.12 0 3.41801 0 −2.86472 0 0 0 8.68276 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

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 6664.2.a.be yes 12
7.b odd 2 1 6664.2.a.bb ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6664.2.a.bb ✓ 12 7.b odd 2 1
6664.2.a.be yes 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6664))\):

\( T_{3}^{12} - 8 T_{3}^{11} + 6 T_{3}^{10} + 96 T_{3}^{9} - 213 T_{3}^{8} - 272 T_{3}^{7} + 1066 T_{3}^{6} + \cdots + 2 \) Copy content Toggle raw display
\( T_{5}^{12} - 4 T_{5}^{11} - 30 T_{5}^{10} + 124 T_{5}^{9} + 271 T_{5}^{8} - 1304 T_{5}^{7} - 394 T_{5}^{6} + \cdots - 1022 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{11} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots - 1022 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + \cdots - 49376 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 448 \) Copy content Toggle raw display
$17$ \( (T + 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 24 T^{11} + \cdots - 46144 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} + \cdots + 14709632 \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{11} + \cdots - 9184 \) Copy content Toggle raw display
$31$ \( T^{12} - 12 T^{11} + \cdots + 415816 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots + 27115744 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots - 40831672 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 354016736 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 878523968 \) Copy content Toggle raw display
$53$ \( T^{12} - 202 T^{10} + \cdots + 6154528 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 9049294912 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 185306254 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 1193586544 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 873774592 \) Copy content Toggle raw display
$73$ \( T^{12} + 16 T^{11} + \cdots + 46752272 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 9609634816 \) Copy content Toggle raw display
$83$ \( T^{12} - 36 T^{11} + \cdots - 30507968 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 5561253824 \) Copy content Toggle raw display
$97$ \( T^{12} + 36 T^{11} + \cdots + 91886536 \) Copy content Toggle raw display
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