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Properties

Label 2736.2.s.y
Level $2736$
Weight $2$
Character orbit 2736.s
Analytic conductor $21.847$
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] = [2736,2,Mod(577,2736)] 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("2736.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,1,0,4,0,0,0,8,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{3} + \beta_{2} + 1) q^{7} + ( - \beta_{2} + 1) q^{11} + (\beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{13} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1 + 3) q^{17} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{19}+ \cdots + (\beta_{4} + 2 \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{5} + 4 q^{7} + 8 q^{11} + q^{13} + 11 q^{17} - q^{23} + 6 q^{25} - 3 q^{29} + 12 q^{31} - 12 q^{35} - 24 q^{37} - 19 q^{41} + 5 q^{43} - 17 q^{47} + 22 q^{49} - 5 q^{53} + 10 q^{55} - 13 q^{59}+ \cdots - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.906803 − 1.57063i
0.235342 + 0.407624i
1.17146 + 2.02903i
−0.906803 + 1.57063i
0.235342 − 0.407624i
1.17146 − 2.02903i
0 0 0 −0.906803 − 1.57063i 0 2.52444 0 0 0
577.2 0 0 0 0.235342 + 0.407624i 0 3.30777 0 0 0
577.3 0 0 0 1.17146 + 2.02903i 0 −3.83221 0 0 0
1873.1 0 0 0 −0.906803 + 1.57063i 0 2.52444 0 0 0
1873.2 0 0 0 0.235342 − 0.407624i 0 3.30777 0 0 0
1873.3 0 0 0 1.17146 − 2.02903i 0 −3.83221 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.3
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 2736.2.s.y 6
3.b odd 2 1 304.2.i.f 6
4.b odd 2 1 1368.2.s.k 6
12.b even 2 1 152.2.i.c ✓ 6
19.c even 3 1 inner 2736.2.s.y 6
24.f even 2 1 1216.2.i.n 6
24.h odd 2 1 1216.2.i.m 6
57.f even 6 1 5776.2.a.bq 3
57.h odd 6 1 304.2.i.f 6
57.h odd 6 1 5776.2.a.bk 3
76.g odd 6 1 1368.2.s.k 6
228.m even 6 1 152.2.i.c ✓ 6
228.m even 6 1 2888.2.a.r 3
228.n odd 6 1 2888.2.a.n 3
456.u even 6 1 1216.2.i.n 6
456.x odd 6 1 1216.2.i.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.c ✓ 6 12.b even 2 1
152.2.i.c ✓ 6 228.m even 6 1
304.2.i.f 6 3.b odd 2 1
304.2.i.f 6 57.h odd 6 1
1216.2.i.m 6 24.h odd 2 1
1216.2.i.m 6 456.x odd 6 1
1216.2.i.n 6 24.f even 2 1
1216.2.i.n 6 456.u even 6 1
1368.2.s.k 6 4.b odd 2 1
1368.2.s.k 6 76.g odd 6 1
2736.2.s.y 6 1.a even 1 1 trivial
2736.2.s.y 6 19.c even 3 1 inner
2888.2.a.n 3 228.n odd 6 1
2888.2.a.r 3 228.m even 6 1
5776.2.a.bk 3 57.h odd 6 1
5776.2.a.bq 3 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 5T_{5}^{4} + 18T_{5}^{2} - 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 14T_{7} + 32 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} + T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{6} - T_{13}^{5} + 33T_{13}^{4} - 120T_{13}^{3} + 1100T_{13}^{2} - 2432T_{13} + 5776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + 5 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} + T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( T^{6} - 11 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 18 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + 5 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + \cdots - 292)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 19 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$43$ \( T^{6} - 5 T^{5} + \cdots + 118336 \) Copy content Toggle raw display
$47$ \( T^{6} + 17 T^{5} + \cdots + 521284 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + \cdots + 1936 \) Copy content Toggle raw display
$59$ \( T^{6} + 13 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 58564 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{6} - 11 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$79$ \( T^{6} + 19 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{3} - 12 T^{2} + \cdots + 632)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 295936 \) Copy content Toggle raw display
$97$ \( T^{6} + T^{5} + 18 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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