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Properties

Label 57.2.e.b
Level $57$
Weight $2$
Character orbit 57.e
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
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] = [57,2,Mod(7,57)] 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("57.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{2} - \beta_{3} q^{3} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - \beta_{5} q^{6} + \beta_{2} q^{7} + (\beta_{4} - \beta_{2} - 1) q^{8}+ \cdots + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 3 q^{3} - 5 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - 3 q^{9} + 4 q^{10} + 10 q^{12} + q^{13} + 3 q^{14} - 2 q^{15} - 5 q^{16} - 2 q^{18} + 4 q^{19} + 44 q^{20} + q^{21} - 18 q^{22}+ \cdots + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.71903 − 0.211943i
−1.62241 − 0.606458i
0.403374 + 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 − 1.68443i
−1.04307 − 1.80664i −0.500000 − 0.866025i −1.17597 + 2.03684i −0.675970 − 1.17081i −1.04307 + 1.80664i 0.351939 0.734191 −0.500000 + 0.866025i −1.41016 + 2.44247i
7.2 0.285997 + 0.495361i −0.500000 − 0.866025i 0.836412 − 1.44871i 1.33641 + 2.31473i 0.285997 − 0.495361i −3.67282 2.10083 −0.500000 + 0.866025i −0.764419 + 1.32401i
7.3 1.25707 + 2.17731i −0.500000 − 0.866025i −2.16044 + 3.74200i −1.66044 − 2.87597i 1.25707 − 2.17731i 2.32088 −5.83502 −0.500000 + 0.866025i 4.17458 − 7.23058i
49.1 −1.04307 + 1.80664i −0.500000 + 0.866025i −1.17597 − 2.03684i −0.675970 + 1.17081i −1.04307 − 1.80664i 0.351939 0.734191 −0.500000 − 0.866025i −1.41016 − 2.44247i
49.2 0.285997 − 0.495361i −0.500000 + 0.866025i 0.836412 + 1.44871i 1.33641 − 2.31473i 0.285997 + 0.495361i −3.67282 2.10083 −0.500000 − 0.866025i −0.764419 − 1.32401i
49.3 1.25707 − 2.17731i −0.500000 + 0.866025i −2.16044 − 3.74200i −1.66044 + 2.87597i 1.25707 + 2.17731i 2.32088 −5.83502 −0.500000 − 0.866025i 4.17458 + 7.23058i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.e.b ✓ 6
3.b odd 2 1 171.2.f.b 6
4.b odd 2 1 912.2.q.l 6
12.b even 2 1 2736.2.s.z 6
19.c even 3 1 inner 57.2.e.b ✓ 6
19.c even 3 1 1083.2.a.l 3
19.d odd 6 1 1083.2.a.o 3
57.f even 6 1 3249.2.a.t 3
57.h odd 6 1 171.2.f.b 6
57.h odd 6 1 3249.2.a.y 3
76.g odd 6 1 912.2.q.l 6
228.m even 6 1 2736.2.s.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b ✓ 6 1.a even 1 1 trivial
57.2.e.b ✓ 6 19.c even 3 1 inner
171.2.f.b 6 3.b odd 2 1
171.2.f.b 6 57.h odd 6 1
912.2.q.l 6 4.b odd 2 1
912.2.q.l 6 76.g odd 6 1
1083.2.a.l 3 19.c even 3 1
1083.2.a.o 3 19.d odd 6 1
2736.2.s.z 6 12.b even 2 1
2736.2.s.z 6 228.m even 6 1
3249.2.a.t 3 57.f even 6 1
3249.2.a.y 3 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{3} + T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 24 T - 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + 22 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 14 T^{5} + \cdots + 24336 \) Copy content Toggle raw display
$29$ \( T^{6} + 4 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + \cdots + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + \cdots + 104976 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{6} + 13 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 292681 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + \cdots + 419904 \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( T^{6} + 11 T^{5} + \cdots + 29241 \) Copy content Toggle raw display
$83$ \( (T^{3} + 4 T^{2} + \cdots - 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 16 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} + \cdots + 2096704 \) Copy content Toggle raw display
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