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Properties

Label 656.2.u.g
Level $656$
Weight $2$
Character orbit 656.u
Analytic conductor $5.238$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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] = [656,2,Mod(305,656)] 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("656.305"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(656, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 656.u (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.23818637260\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 19 x^{13} + 65 x^{12} - 61 x^{11} + 374 x^{10} + 255 x^{9} + 1627 x^{8} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{9} + \beta_{7} + \cdots + \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{13} + \beta_{12} + \cdots - 1) q^{7} + ( - \beta_{14} - \beta_{10} - \beta_{9} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{14} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - 4 q^{5} + 18 q^{9} - q^{11} - q^{17} - 9 q^{19} - 15 q^{21} + 18 q^{23} - 22 q^{25} - 10 q^{27} + 4 q^{29} - 3 q^{31} - 15 q^{33} - q^{35} + 4 q^{37} - 4 q^{39} - 36 q^{41} + 4 q^{43}+ \cdots + 98 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 19 x^{13} + 65 x^{12} - 61 x^{11} + 374 x^{10} + 255 x^{9} + 1627 x^{8} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!47 \nu^{15} + \cdots + 50\!\cdots\!32 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!39 \nu^{15} + \cdots - 12\!\cdots\!48 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!01 \nu^{15} + \cdots + 66\!\cdots\!36 ) / 32\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!44 \nu^{15} + \cdots - 75\!\cdots\!12 ) / 40\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!42 \nu^{15} + \cdots + 75\!\cdots\!96 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70\!\cdots\!97 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 65\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!93 \nu^{15} + \cdots + 24\!\cdots\!04 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!43 \nu^{15} + \cdots + 86\!\cdots\!40 ) / 65\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!21 \nu^{15} + \cdots + 15\!\cdots\!16 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!89 \nu^{15} + \cdots - 51\!\cdots\!12 ) / 81\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{15} + \cdots + 88\!\cdots\!00 ) / 65\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!11 \nu^{15} + \cdots - 71\!\cdots\!04 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!43 \nu^{15} + \cdots + 96\!\cdots\!12 ) / 65\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35\!\cdots\!05 \nu^{15} + \cdots - 70\!\cdots\!96 ) / 40\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{5} + 4\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{12} + \beta_{11} + 2\beta_{9} + \beta_{8} - 2\beta_{6} + 7\beta_{5} + 2\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{12} - 3 \beta_{10} + 4 \beta_{9} + 9 \beta_{8} - 2 \beta_{7} - 24 \beta_{6} + \cdots + 12 \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{15} - 3\beta_{14} - 16\beta_{10} - 33\beta_{9} - 18\beta_{7} - 33\beta_{6} + 58\beta_{2} + 16\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 76 \beta_{15} - 2 \beta_{14} + 28 \beta_{13} - 6 \beta_{12} + 2 \beta_{11} + 45 \beta_{10} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 149 \beta_{14} + 152 \beta_{13} - 91 \beta_{11} + 513 \beta_{10} + 145 \beta_{9} - 149 \beta_{8} + \cdots - 336 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 665 \beta_{15} + 665 \beta_{14} + 116 \beta_{12} - 616 \beta_{11} + 665 \beta_{10} + 2833 \beta_{9} + \cdots - 2168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1461 \beta_{15} - 1330 \beta_{13} + 98 \beta_{12} - 1461 \beta_{11} - 3249 \beta_{10} + 4848 \beta_{9} + \cdots - 1330 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 797 \beta_{15} - 6040 \beta_{14} - 2922 \beta_{13} - 2922 \beta_{12} - 11805 \beta_{10} + \cdots + 17490 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9314 \beta_{15} - 13930 \beta_{14} + 1594 \beta_{13} - 12080 \beta_{12} + 13930 \beta_{11} + \cdots + 47220 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10883 \beta_{14} + 18628 \beta_{13} + 45494 \beta_{11} + 84895 \beta_{10} - 16977 \beta_{9} + \cdots - 46388 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 103523 \beta_{15} + 103523 \beta_{14} + 112754 \beta_{12} + 28215 \beta_{11} + 103523 \beta_{10} + \cdots - 452334 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 135059 \beta_{15} - 207046 \beta_{13} + 263476 \beta_{12} - 135059 \beta_{11} - 769352 \beta_{10} + \cdots - 207046 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1246053 \beta_{15} - 1111457 \beta_{14} - 270118 \beta_{13} - 270118 \beta_{12} - 2555679 \beta_{10} + \cdots + 4419696 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/656\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(165\) \(575\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.978017 + 3.01003i
0.289148 + 0.889907i
−0.360378 − 1.10913i
−0.715804 − 2.20302i
0.978017 − 3.01003i
0.289148 − 0.889907i
−0.360378 + 1.10913i
−0.715804 + 2.20302i
−1.57854 + 1.14687i
−0.724335 + 0.526260i
1.29198 − 0.938678i
2.31991 − 1.68551i
−1.57854 − 1.14687i
−0.724335 − 0.526260i
1.29198 + 0.938678i
2.31991 + 1.68551i
0 −3.16493 0 −1.45054 + 1.05388i 0 0.779454 − 2.39891i 0 7.01678 0
305.2 0 −0.935703 0 1.16958 − 0.849749i 0 −0.783712 + 2.41202i 0 −2.12446 0
305.3 0 1.16621 0 −1.63186 + 1.18561i 0 1.27567 − 3.92610i 0 −1.63996 0
305.4 0 2.31639 0 −0.205216 + 0.149098i 0 −1.27141 + 3.91299i 0 2.36567 0
385.1 0 −3.16493 0 −1.45054 − 1.05388i 0 0.779454 + 2.39891i 0 7.01678 0
385.2 0 −0.935703 0 1.16958 + 0.849749i 0 −0.783712 − 2.41202i 0 −2.12446 0
385.3 0 1.16621 0 −1.63186 − 1.18561i 0 1.27567 + 3.92610i 0 −1.63996 0
385.4 0 2.31639 0 −0.205216 − 0.149098i 0 −1.27141 − 3.91299i 0 2.36567 0
529.1 0 −1.95118 0 −0.450567 + 1.38670i 0 −1.37545 − 0.999326i 0 0.807096 0
529.2 0 −0.895327 0 1.28616 − 3.95838i 0 1.67625 + 1.21787i 0 −2.19839 0
529.3 0 1.59697 0 −0.953882 + 2.93575i 0 3.64351 + 2.64716i 0 −0.449675 0
529.4 0 2.86757 0 0.236327 − 0.727340i 0 −3.94431 − 2.86571i 0 5.22293 0
625.1 0 −1.95118 0 −0.450567 − 1.38670i 0 −1.37545 + 0.999326i 0 0.807096 0
625.2 0 −0.895327 0 1.28616 + 3.95838i 0 1.67625 − 1.21787i 0 −2.19839 0
625.3 0 1.59697 0 −0.953882 − 2.93575i 0 3.64351 − 2.64716i 0 −0.449675 0
625.4 0 2.86757 0 0.236327 + 0.727340i 0 −3.94431 + 2.86571i 0 5.22293 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 656.2.u.g 16
4.b odd 2 1 164.2.g.a ✓ 16
12.b even 2 1 1476.2.n.f 16
41.d even 5 1 inner 656.2.u.g 16
164.j odd 10 1 164.2.g.a ✓ 16
164.j odd 10 1 6724.2.a.f 8
164.l odd 10 1 6724.2.a.g 8
492.w even 10 1 1476.2.n.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.g.a ✓ 16 4.b odd 2 1
164.2.g.a ✓ 16 164.j odd 10 1
656.2.u.g 16 1.a even 1 1 trivial
656.2.u.g 16 41.d even 5 1 inner
1476.2.n.f 16 12.b even 2 1
1476.2.n.f 16 492.w even 10 1
6724.2.a.f 8 164.j odd 10 1
6724.2.a.g 8 164.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 16T_{3}^{5} + 73T_{3}^{4} - 58T_{3}^{3} - 116T_{3}^{2} + 48T_{3} + 64 \) acting on \(S_{2}^{\mathrm{new}}(656, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{7} - 16 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} + \cdots + 361 \) Copy content Toggle raw display
$7$ \( T^{16} + 22 T^{14} + \cdots + 70627216 \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots + 56610576 \) Copy content Toggle raw display
$13$ \( T^{16} + 55 T^{14} + \cdots + 51854401 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 571162201 \) Copy content Toggle raw display
$19$ \( T^{16} + 9 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{16} - 18 T^{15} + \cdots + 2085136 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 496576656 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 149426176 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12245414281 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4846320462096 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 163989361936 \) Copy content Toggle raw display
$53$ \( T^{16} + 19 T^{15} + \cdots + 1437601 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5102466570496 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 31680796081 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 145057936 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4570171737616 \) Copy content Toggle raw display
$73$ \( (T^{8} - 21 T^{7} + \cdots - 44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 9 T^{7} + \cdots + 25344)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 26 T^{7} + \cdots - 99584)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 680080651662736 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2868566481 \) Copy content Toggle raw display
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