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Properties

Label 1053.2.e.n
Level $1053$
Weight $2$
Character orbit 1053.e
Analytic conductor $8.408$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1053,2,Mod(352,1053)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1053.352"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4}) q^{7} + (2 \beta_{2} - 3) q^{8}+ \cdots + ( - 3 \beta_1 - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 3 q^{5} - 18 q^{8} - 18 q^{10} - 3 q^{13} + 9 q^{14} - 12 q^{16} + 12 q^{17} + 30 q^{19} - 30 q^{20} - 12 q^{22} + 3 q^{23} - 12 q^{25} + 48 q^{28} - 3 q^{31} + 18 q^{32} + 3 q^{34} + 48 q^{35}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{5} - 13\nu^{4} + 54\nu^{3} - 71\nu^{2} + 99\nu - 40 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} + 2\beta_{4} + 4\beta_{3} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20\beta_{5} + 14\beta_{4} + 25\beta_{3} + 13\beta_{2} - 25\beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{5} - 16\beta_{4} - 59\beta_{3} + 7\beta_{2} - 28\beta _1 + 121 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
352.1
0.500000 − 0.0585812i
0.500000 + 2.43956i
0.500000 − 1.51496i
0.500000 + 0.0585812i
0.500000 − 2.43956i
0.500000 + 1.51496i
−1.07255 − 1.85771i 0 −1.30073 + 2.25294i −0.800733 + 1.38691i 0 −0.300733 − 0.520884i 1.29021 0 3.43531
352.2 −0.261988 − 0.453777i 0 0.862724 − 1.49428i 1.36272 − 2.36031i 0 1.86272 + 3.22633i −1.95205 0 −1.42807
352.3 1.33454 + 2.31149i 0 −2.56199 + 4.43750i −2.06199 + 3.57147i 0 −1.56199 − 2.70545i −8.33816 0 −11.0072
703.1 −1.07255 + 1.85771i 0 −1.30073 − 2.25294i −0.800733 − 1.38691i 0 −0.300733 + 0.520884i 1.29021 0 3.43531
703.2 −0.261988 + 0.453777i 0 0.862724 + 1.49428i 1.36272 + 2.36031i 0 1.86272 − 3.22633i −1.95205 0 −1.42807
703.3 1.33454 − 2.31149i 0 −2.56199 − 4.43750i −2.06199 − 3.57147i 0 −1.56199 + 2.70545i −8.33816 0 −11.0072
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 352.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.2.e.n 6
3.b odd 2 1 1053.2.e.o 6
9.c even 3 1 1053.2.a.h yes 3
9.c even 3 1 inner 1053.2.e.n 6
9.d odd 6 1 1053.2.a.g ✓ 3
9.d odd 6 1 1053.2.e.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1053.2.a.g ✓ 3 9.d odd 6 1
1053.2.a.h yes 3 9.c even 3 1
1053.2.e.n 6 1.a even 1 1 trivial
1053.2.e.n 6 9.c even 3 1 inner
1053.2.e.o 6 3.b odd 2 1
1053.2.e.o 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

\( T_{2}^{6} + 6T_{2}^{4} + 6T_{2}^{3} + 36T_{2}^{2} + 18T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} + 18T_{5}^{4} + 9T_{5}^{3} + 135T_{5}^{2} + 162T_{5} + 324 \) Copy content Toggle raw display
\( T_{7}^{6} + 12T_{7}^{4} + 14T_{7}^{3} + 144T_{7}^{2} + 84T_{7} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( (T^{3} - 6 T^{2} + 3 T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 5)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 51 T + 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + \cdots + 2116 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( (T^{3} + 9 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 205209 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots + 1868689 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} + \cdots + 2155024 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} - 33 T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 12 T^{2} + \cdots + 652)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 6 T^{5} + \cdots + 27556 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} + \cdots + 194481 \) Copy content Toggle raw display
$89$ \( (T^{3} - 3 T^{2} + \cdots + 138)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 64)^{3} \) Copy content Toggle raw display
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