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Properties

Label 6664.2.a.ba
Level $6664$
Weight $2$
Character orbit 6664.a
Self dual yes
Analytic conductor $53.212$
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] = [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 = [10,0,1,0,1] 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: \(10\)
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: \( x^{10} - x^{9} - 24x^{8} + 19x^{7} + 197x^{6} - 103x^{5} - 640x^{4} + 97x^{3} + 676x^{2} + 216x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{7} q^{5} + (\beta_{2} + 2) q^{9} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{11} - \beta_{9} q^{13} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{15} + q^{17}+ \cdots + ( - 2 \beta_{9} - 3 \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + q^{5} + 19 q^{9} + 2 q^{11} + 4 q^{13} + 8 q^{15} + 10 q^{17} - 4 q^{19} + 13 q^{23} + 31 q^{25} + 10 q^{27} + 13 q^{29} + 12 q^{31} - 11 q^{33} + 20 q^{37} + 4 q^{39} + 16 q^{41} - 14 q^{43}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 24x^{8} + 19x^{7} + 197x^{6} - 103x^{5} - 640x^{4} + 97x^{3} + 676x^{2} + 216x - 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 559 \nu^{9} - 2883 \nu^{8} - 12543 \nu^{7} + 55359 \nu^{6} + 98727 \nu^{5} - 307059 \nu^{4} + \cdots + 77430 ) / 37649 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 928 \nu^{9} - 543 \nu^{8} - 17994 \nu^{7} + 13169 \nu^{6} + 92169 \nu^{5} - 101202 \nu^{4} + \cdots - 39363 ) / 37649 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1035 \nu^{9} - 1701 \nu^{8} - 20799 \nu^{7} + 35013 \nu^{6} + 126692 \nu^{5} - 239855 \nu^{4} + \cdots - 296369 ) / 37649 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3449 \nu^{9} - 1085 \nu^{8} - 78804 \nu^{7} + 23263 \nu^{6} + 588531 \nu^{5} - 178713 \nu^{4} + \cdots - 172464 ) / 75298 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4567 \nu^{9} + 6851 \nu^{8} + 103890 \nu^{7} - 133981 \nu^{6} - 785985 \nu^{5} + 792831 \nu^{4} + \cdots - 57694 ) / 75298 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5177 \nu^{9} - 17675 \nu^{8} - 122696 \nu^{7} + 361959 \nu^{6} + 991243 \nu^{5} - 2265187 \nu^{4} + \cdots - 489830 ) / 75298 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4896 \nu^{9} + 9356 \nu^{8} + 111811 \nu^{7} - 194109 \nu^{6} - 852375 \nu^{5} + 1255750 \nu^{4} + \cdots + 284270 ) / 37649 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{3} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} + 2\beta_{3} + 11\beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{9} - 4 \beta_{8} + 15 \beta_{7} + 11 \beta_{6} - \beta_{5} - 2 \beta_{4} + 16 \beta_{3} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{9} - 18 \beta_{8} + 23 \beta_{7} - \beta_{6} - 33 \beta_{5} + 16 \beta_{4} + 31 \beta_{3} + \cdots + 320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 78 \beta_{9} - 80 \beta_{8} + 183 \beta_{7} + 101 \beta_{6} - 16 \beta_{5} - 32 \beta_{4} + 206 \beta_{3} + \cdots + 67 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 309 \beta_{9} - 252 \beta_{8} + 349 \beta_{7} - 13 \beta_{6} - 426 \beta_{5} + 197 \beta_{4} + \cdots + 2861 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1107 \beta_{9} - 1155 \beta_{8} + 2078 \beta_{7} + 905 \beta_{6} - 210 \beta_{5} - 384 \beta_{4} + \cdots + 676 \) 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
−3.15394
−2.71873
−1.92223
−0.774185
−0.624306
0.0988820
1.59660
2.53492
2.65458
3.30842
0 −3.15394 0 −1.15329 0 0 0 6.94734 0
1.2 0 −2.71873 0 −4.07378 0 0 0 4.39147 0
1.3 0 −1.92223 0 4.22586 0 0 0 0.694976 0
1.4 0 −0.774185 0 0.527476 0 0 0 −2.40064 0
1.5 0 −0.624306 0 3.34286 0 0 0 −2.61024 0
1.6 0 0.0988820 0 −3.40663 0 0 0 −2.99022 0
1.7 0 1.59660 0 2.08287 0 0 0 −0.450881 0
1.8 0 2.53492 0 −1.98012 0 0 0 3.42581 0
1.9 0 2.65458 0 −1.81934 0 0 0 4.04678 0
1.10 0 3.30842 0 3.25409 0 0 0 7.94561 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
\(2\) \( +1 \)
\(7\) \( +1 \)
\(17\) \( -1 \)

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.ba 10
7.b odd 2 1 6664.2.a.z 10
7.c even 3 2 952.2.q.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.q.f ✓ 20 7.c even 3 2
6664.2.a.z 10 7.b odd 2 1
6664.2.a.ba 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6664))\):

\( T_{3}^{10} - T_{3}^{9} - 24T_{3}^{8} + 19T_{3}^{7} + 197T_{3}^{6} - 103T_{3}^{5} - 640T_{3}^{4} + 97T_{3}^{3} + 676T_{3}^{2} + 216T_{3} - 28 \) Copy content Toggle raw display
\( T_{5}^{10} - T_{5}^{9} - 40 T_{5}^{8} + 29 T_{5}^{7} + 558 T_{5}^{6} - 214 T_{5}^{5} - 3251 T_{5}^{4} + \cdots - 2912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots - 28 \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} + \cdots - 2912 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots + 303868 \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + \cdots - 56756 \) Copy content Toggle raw display
$17$ \( (T - 1)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + 4 T^{9} + \cdots + 3124208 \) Copy content Toggle raw display
$23$ \( T^{10} - 13 T^{9} + \cdots - 416 \) Copy content Toggle raw display
$29$ \( T^{10} - 13 T^{9} + \cdots + 3677056 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} + \cdots + 68368 \) Copy content Toggle raw display
$37$ \( T^{10} - 20 T^{9} + \cdots + 112832 \) Copy content Toggle raw display
$41$ \( T^{10} - 16 T^{9} + \cdots - 26083712 \) Copy content Toggle raw display
$43$ \( T^{10} + 14 T^{9} + \cdots + 19511824 \) Copy content Toggle raw display
$47$ \( T^{10} + 21 T^{9} + \cdots + 170048 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 545854328 \) Copy content Toggle raw display
$59$ \( T^{10} + 4 T^{9} + \cdots + 25450960 \) Copy content Toggle raw display
$61$ \( T^{10} - 327 T^{8} + \cdots + 68956928 \) Copy content Toggle raw display
$67$ \( T^{10} + 19 T^{9} + \cdots + 19679472 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 11239499907 \) Copy content Toggle raw display
$73$ \( T^{10} + 17 T^{9} + \cdots + 14772096 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 334809436 \) Copy content Toggle raw display
$83$ \( T^{10} - 16 T^{9} + \cdots - 93184 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 956799808 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 6561065024 \) Copy content Toggle raw display
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