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Properties

Label 1352.2.i.j
Level $1352$
Weight $2$
Character orbit 1352.i
Analytic conductor $10.796$
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] = [1352,2,Mod(529,1352)] 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("1352.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,4,0,8,0,-1] 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: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} + 1) q^{3} + ( - \beta_{3} + 1) q^{5} + (\beta_{4} + \beta_{3}) q^{7} + (2 \beta_{4} + 2 \beta_{3} + \cdots - \beta_1) q^{9} + ( - \beta_{5} + 3 \beta_{4} + 1) q^{11} + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots + 3) q^{15}+ \cdots + (2 \beta_{3} + 10 \beta_{2} - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 8 q^{5} - q^{7} - q^{9} + 6 q^{11} + 10 q^{15} + 2 q^{17} + 6 q^{19} - 12 q^{21} + 9 q^{23} - 10 q^{25} - 2 q^{27} - 3 q^{29} - 10 q^{31} - 22 q^{33} - 6 q^{35} - 12 q^{37} - 3 q^{41}+ \cdots - 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 + 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 − 1.07992i
0 −0.123490 + 0.213891i 0 −0.246980 0 0.623490 + 1.07992i 0 1.46950 + 2.54525i 0
529.2 0 0.722521 − 1.25144i 0 1.44504 0 −0.222521 − 0.385418i 0 0.455927 + 0.789689i 0
529.3 0 1.40097 − 2.42655i 0 2.80194 0 −0.900969 − 1.56052i 0 −2.42543 − 4.20096i 0
1329.1 0 −0.123490 − 0.213891i 0 −0.246980 0 0.623490 − 1.07992i 0 1.46950 − 2.54525i 0
1329.2 0 0.722521 + 1.25144i 0 1.44504 0 −0.222521 + 0.385418i 0 0.455927 − 0.789689i 0
1329.3 0 1.40097 + 2.42655i 0 2.80194 0 −0.900969 + 1.56052i 0 −2.42543 + 4.20096i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.i.j 6
13.b even 2 1 1352.2.i.i 6
13.c even 3 1 1352.2.a.j yes 3
13.c even 3 1 inner 1352.2.i.j 6
13.d odd 4 2 1352.2.o.g 12
13.e even 6 1 1352.2.a.i ✓ 3
13.e even 6 1 1352.2.i.i 6
13.f odd 12 2 1352.2.f.e 6
13.f odd 12 2 1352.2.o.g 12
52.i odd 6 1 2704.2.a.bb 3
52.j odd 6 1 2704.2.a.bc 3
52.l even 12 2 2704.2.f.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.2.a.i ✓ 3 13.e even 6 1
1352.2.a.j yes 3 13.c even 3 1
1352.2.f.e 6 13.f odd 12 2
1352.2.i.i 6 13.b even 2 1
1352.2.i.i 6 13.e even 6 1
1352.2.i.j 6 1.a even 1 1 trivial
1352.2.i.j 6 13.c even 3 1 inner
1352.2.o.g 12 13.d odd 4 2
1352.2.o.g 12 13.f odd 12 2
2704.2.a.bb 3 52.i odd 6 1
2704.2.a.bc 3 52.j odd 6 1
2704.2.f.p 6 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{6} - 4T_{3}^{5} + 13T_{3}^{4} - 14T_{3}^{3} + 13T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5}^{2} + 3T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + T_{7}^{5} + 3T_{7}^{4} + 5T_{7}^{2} + 2T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{3} - 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$19$ \( T^{6} - 6 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$23$ \( T^{6} - 9 T^{5} + \cdots + 12769 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 63001 \) Copy content Toggle raw display
$31$ \( (T^{3} + 5 T^{2} - 22 T - 97)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} + \cdots + 142129 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$43$ \( T^{6} - 17 T^{5} + \cdots + 12769 \) Copy content Toggle raw display
$47$ \( (T^{3} + 8 T^{2} - 23 T + 13)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 15 T^{2} + \cdots - 211)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 17 T^{5} + \cdots + 142129 \) Copy content Toggle raw display
$61$ \( T^{6} - 28 T^{5} + \cdots + 82369 \) Copy content Toggle raw display
$67$ \( T^{6} - 17 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$71$ \( T^{6} + 7 T^{5} + \cdots + 2798929 \) Copy content Toggle raw display
$73$ \( (T^{3} - 15 T^{2} + \cdots - 83)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 13 T^{2} + \cdots - 377)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 13 T^{2} + 26 T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 19 T^{5} + \cdots + 1692601 \) Copy content Toggle raw display
$97$ \( T^{6} + 5 T^{5} + \cdots + 841 \) Copy content Toggle raw display
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