Properties

Label 3249.1.p.b
Level $3249$
Weight $1$
Character orbit 3249.p
Analytic conductor $1.621$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3249.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.62146222604\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.22284891.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{4} + q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{4} + q^{7} + ( 1 - \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{25} -\zeta_{6} q^{28} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{31} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{37} -\zeta_{6}^{2} q^{43} + ( -1 - \zeta_{6} ) q^{52} -\zeta_{6} q^{61} + q^{64} + ( -1 + \zeta_{6}^{2} ) q^{67} -\zeta_{6}^{2} q^{73} + ( 1 + \zeta_{6} ) q^{79} + ( 1 - \zeta_{6}^{2} ) q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{4} + 2q^{7} + O(q^{10}) \) \( 2q - q^{4} + 2q^{7} + 3q^{13} - q^{16} + q^{25} - q^{28} + q^{43} - 3q^{52} - q^{61} + 2q^{64} - 3q^{67} + q^{73} + 3q^{79} + 3q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3249\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(2890\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1513.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −0.500000 + 0.866025i 0 0 1.00000 0 0 0
2098.1 0 0 −0.500000 0.866025i 0 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.1.p.b 2
3.b odd 2 1 CM 3249.1.p.b 2
19.b odd 2 1 171.1.p.a 2
19.c even 3 1 171.1.p.a 2
19.c even 3 1 3249.1.c.a 2
19.d odd 6 1 3249.1.c.a 2
19.d odd 6 1 inner 3249.1.p.b 2
19.e even 9 3 3249.1.ba.a 6
19.e even 9 3 3249.1.ba.b 6
19.f odd 18 3 3249.1.ba.a 6
19.f odd 18 3 3249.1.ba.b 6
57.d even 2 1 171.1.p.a 2
57.f even 6 1 3249.1.c.a 2
57.f even 6 1 inner 3249.1.p.b 2
57.h odd 6 1 171.1.p.a 2
57.h odd 6 1 3249.1.c.a 2
57.j even 18 3 3249.1.ba.a 6
57.j even 18 3 3249.1.ba.b 6
57.l odd 18 3 3249.1.ba.a 6
57.l odd 18 3 3249.1.ba.b 6
76.d even 2 1 2736.1.cd.a 2
76.g odd 6 1 2736.1.cd.a 2
171.g even 3 1 1539.1.i.a 2
171.h even 3 1 1539.1.s.a 2
171.j odd 6 1 1539.1.s.a 2
171.l even 6 1 1539.1.i.a 2
171.l even 6 1 1539.1.s.a 2
171.n odd 6 1 1539.1.i.a 2
171.o odd 6 1 1539.1.i.a 2
171.o odd 6 1 1539.1.s.a 2
228.b odd 2 1 2736.1.cd.a 2
228.m even 6 1 2736.1.cd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.p.a 2 19.b odd 2 1
171.1.p.a 2 19.c even 3 1
171.1.p.a 2 57.d even 2 1
171.1.p.a 2 57.h odd 6 1
1539.1.i.a 2 171.g even 3 1
1539.1.i.a 2 171.l even 6 1
1539.1.i.a 2 171.n odd 6 1
1539.1.i.a 2 171.o odd 6 1
1539.1.s.a 2 171.h even 3 1
1539.1.s.a 2 171.j odd 6 1
1539.1.s.a 2 171.l even 6 1
1539.1.s.a 2 171.o odd 6 1
2736.1.cd.a 2 76.d even 2 1
2736.1.cd.a 2 76.g odd 6 1
2736.1.cd.a 2 228.b odd 2 1
2736.1.cd.a 2 228.m even 6 1
3249.1.c.a 2 19.c even 3 1
3249.1.c.a 2 19.d odd 6 1
3249.1.c.a 2 57.f even 6 1
3249.1.c.a 2 57.h odd 6 1
3249.1.p.b 2 1.a even 1 1 trivial
3249.1.p.b 2 3.b odd 2 1 CM
3249.1.p.b 2 19.d odd 6 1 inner
3249.1.p.b 2 57.f even 6 1 inner
3249.1.ba.a 6 19.e even 9 3
3249.1.ba.a 6 19.f odd 18 3
3249.1.ba.a 6 57.j even 18 3
3249.1.ba.a 6 57.l odd 18 3
3249.1.ba.b 6 19.e even 9 3
3249.1.ba.b 6 19.f odd 18 3
3249.1.ba.b 6 57.j even 18 3
3249.1.ba.b 6 57.l odd 18 3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3249, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 3 - 3 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( 3 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 3 + 3 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 3 - 3 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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