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Properties

Label 6664.2.a.bd
Level $6664$
Weight $2$
Character orbit 6664.a
Self dual yes
Analytic conductor $53.212$
Analytic rank $1$
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 = [12,0,4,0,-8] 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: \(1\)
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: \( x^{12} - 4 x^{11} - 14 x^{10} + 64 x^{9} + 59 x^{8} - 348 x^{7} - 74 x^{6} + 760 x^{5} + 27 x^{4} + \cdots - 14 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{10} - 1) q^{5} + (\beta_{2} + \beta_1) q^{9} + \beta_{9} q^{11} + (\beta_{7} - \beta_{6} - \beta_{3} + \cdots - \beta_1) q^{13} + (\beta_{8} - \beta_{4} - 2 \beta_1) q^{15}+ \cdots + (\beta_{11} - 2 \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 8 q^{5} + 8 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} - 12 q^{17} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 4 q^{31} - 16 q^{33} + 12 q^{37} - 16 q^{39} - 24 q^{41} - 4 q^{43} - 28 q^{45} - 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 14 x^{10} + 64 x^{9} + 59 x^{8} - 348 x^{7} - 74 x^{6} + 760 x^{5} + 27 x^{4} + \cdots - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2441 \nu^{11} + 3265 \nu^{10} + 46834 \nu^{9} - 34413 \nu^{8} - 330686 \nu^{7} + 26959 \nu^{6} + \cdots + 208664 ) / 203870 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2949 \nu^{11} + 16810 \nu^{10} - 148911 \nu^{9} - 185263 \nu^{8} + 1803419 \nu^{7} + 164619 \nu^{6} + \cdots + 2461194 ) / 203870 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6013 \nu^{11} - 21030 \nu^{10} - 88767 \nu^{9} + 335829 \nu^{8} + 380333 \nu^{7} - 1813377 \nu^{6} + \cdots + 513108 ) / 203870 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6499 \nu^{11} - 12660 \nu^{10} - 121811 \nu^{9} + 186667 \nu^{8} + 822509 \nu^{7} - 889261 \nu^{6} + \cdots + 34174 ) / 203870 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5062 \nu^{11} + 3430 \nu^{10} + 121008 \nu^{9} - 66436 \nu^{8} - 1040337 \nu^{7} + 478053 \nu^{6} + \cdots - 760647 ) / 101935 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11771 \nu^{11} + 80430 \nu^{10} + 2889 \nu^{9} - 1099773 \nu^{8} + 1755439 \nu^{7} + \cdots + 2944614 ) / 203870 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25467 \nu^{11} - 105055 \nu^{10} - 286838 \nu^{9} + 1534061 \nu^{8} + 374882 \nu^{7} + \cdots - 701548 ) / 203870 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3453 \nu^{11} - 15284 \nu^{10} - 41980 \nu^{9} + 236172 \nu^{8} + 112276 \nu^{7} - 1206442 \nu^{6} + \cdots + 112325 ) / 20387 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35228 \nu^{11} - 152145 \nu^{10} - 460127 \nu^{9} + 2422674 \nu^{8} + 1608063 \nu^{7} + \cdots + 1463138 ) / 203870 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{10} - 12 \beta_{9} - 12 \beta_{8} - 15 \beta_{7} - \beta_{6} + 12 \beta_{5} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{11} + 13 \beta_{10} - 18 \beta_{9} - 18 \beta_{8} - 37 \beta_{7} - 27 \beta_{6} + 28 \beta_{5} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32 \beta_{11} + 32 \beta_{10} - 126 \beta_{9} - 128 \beta_{8} - 181 \beta_{7} - 27 \beta_{6} + \cdots - 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 139 \beta_{11} + 137 \beta_{10} - 238 \beta_{9} - 245 \beta_{8} - 495 \beta_{7} - 297 \beta_{6} + \cdots + 512 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 385 \beta_{11} + 378 \beta_{10} - 1280 \beta_{9} - 1333 \beta_{8} - 2031 \beta_{7} - 444 \beta_{6} + \cdots - 386 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1430 \beta_{11} + 1377 \beta_{10} - 2835 \beta_{9} - 3012 \beta_{8} - 5905 \beta_{7} - 3106 \beta_{6} + \cdots + 3196 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4238 \beta_{11} + 4061 \beta_{10} - 12955 \beta_{9} - 13839 \beta_{8} - 22134 \beta_{7} - 5991 \beta_{6} + \cdots - 3322 \) 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
−2.66272
−2.21792
−1.51257
−0.836454
−0.726451
0.166977
0.291629
1.38408
2.01001
2.26807
2.57346
3.26188
0 −2.66272 0 −1.73730 0 0 0 4.09005 0
1.2 0 −2.21792 0 2.31274 0 0 0 1.91918 0
1.3 0 −1.51257 0 −4.17827 0 0 0 −0.712117 0
1.4 0 −0.836454 0 1.79195 0 0 0 −2.30034 0
1.5 0 −0.726451 0 −2.90303 0 0 0 −2.47227 0
1.6 0 0.166977 0 2.77381 0 0 0 −2.97212 0
1.7 0 0.291629 0 −0.918965 0 0 0 −2.91495 0
1.8 0 1.38408 0 1.37442 0 0 0 −1.08431 0
1.9 0 2.01001 0 −1.57380 0 0 0 1.04012 0
1.10 0 2.26807 0 −3.33749 0 0 0 2.14415 0
1.11 0 2.57346 0 0.811450 0 0 0 3.62272 0
1.12 0 3.26188 0 −2.41552 0 0 0 7.63989 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.bd yes 12
7.b odd 2 1 6664.2.a.bc ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6664.2.a.bc ✓ 12 7.b odd 2 1
6664.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6664))\):

\( T_{3}^{12} - 4 T_{3}^{11} - 14 T_{3}^{10} + 64 T_{3}^{9} + 59 T_{3}^{8} - 348 T_{3}^{7} - 74 T_{3}^{6} + \cdots - 14 \) Copy content Toggle raw display
\( T_{5}^{12} + 8 T_{5}^{11} - 2 T_{5}^{10} - 148 T_{5}^{9} - 193 T_{5}^{8} + 964 T_{5}^{7} + 1794 T_{5}^{6} + \cdots - 3150 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{11} + \cdots - 3150 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots - 2272 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 18368 \) Copy content Toggle raw display
$17$ \( (T + 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 98 T^{10} + \cdots + 13248 \) Copy content Toggle raw display
$23$ \( T^{12} - 162 T^{10} + \cdots + 19308416 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + \cdots - 1282784 \) Copy content Toggle raw display
$31$ \( T^{12} + 4 T^{11} + \cdots + 119304 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots - 19338016 \) Copy content Toggle raw display
$41$ \( T^{12} + 24 T^{11} + \cdots - 56535928 \) Copy content Toggle raw display
$43$ \( T^{12} + 4 T^{11} + \cdots + 2842912 \) Copy content Toggle raw display
$47$ \( T^{12} + 28 T^{11} + \cdots + 68586944 \) Copy content Toggle raw display
$53$ \( T^{12} - 234 T^{10} + \cdots - 2495968 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 130783168 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 134701442 \) Copy content Toggle raw display
$67$ \( T^{12} - 314 T^{10} + \cdots - 13678448 \) Copy content Toggle raw display
$71$ \( T^{12} - 310 T^{10} + \cdots - 1958400 \) Copy content Toggle raw display
$73$ \( T^{12} + 40 T^{11} + \cdots - 31753584 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 642522112 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2506193984 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 30372724672 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 4283108552 \) Copy content Toggle raw display
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