Properties

Label 2891.1.g.b
Level $2891$
Weight $1$
Character orbit 2891.g
Analytic conductor $1.443$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2891 = 7^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2891.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.44279695152\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 59)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.59.1
Artin image $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} +O(q^{10})\) \( q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{12} + q^{15} + \zeta_{6}^{2} q^{16} + 2 \zeta_{6} q^{17} + \zeta_{6}^{2} q^{19} + q^{20} - q^{27} - q^{29} + q^{41} + q^{48} -2 \zeta_{6}^{2} q^{51} + \zeta_{6} q^{53} + q^{57} + \zeta_{6} q^{59} -\zeta_{6} q^{60} + q^{64} -2 \zeta_{6}^{2} q^{68} + 2 q^{71} + q^{76} -\zeta_{6}^{2} q^{79} -\zeta_{6} q^{80} + \zeta_{6} q^{81} -2 q^{85} + \zeta_{6} q^{87} -\zeta_{6} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{4} - q^{5} + O(q^{10}) \) \( 2q - q^{3} - q^{4} - q^{5} - q^{12} + 2q^{15} - q^{16} + 2q^{17} - q^{19} + 2q^{20} - 2q^{27} - 2q^{29} + 2q^{41} + 2q^{48} + 2q^{51} + q^{53} + 2q^{57} + q^{59} - q^{60} + 2q^{64} + 2q^{68} + 4q^{71} + 2q^{76} + q^{79} - q^{80} + q^{81} - 4q^{85} + q^{87} - q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2891\mathbb{Z}\right)^\times\).

\(n\) \(591\) \(2598\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
471.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0 0 0 0
2713.1 0 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0 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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
7.c even 3 1 inner
413.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2891.1.g.b 2
7.b odd 2 1 2891.1.g.d 2
7.c even 3 1 2891.1.c.e 1
7.c even 3 1 inner 2891.1.g.b 2
7.d odd 6 1 59.1.b.a 1
7.d odd 6 1 2891.1.g.d 2
21.g even 6 1 531.1.c.a 1
28.f even 6 1 944.1.h.a 1
35.i odd 6 1 1475.1.c.b 1
35.k even 12 2 1475.1.d.a 2
56.j odd 6 1 3776.1.h.b 1
56.m even 6 1 3776.1.h.a 1
59.b odd 2 1 CM 2891.1.g.b 2
413.b even 2 1 2891.1.g.d 2
413.g odd 6 1 2891.1.c.e 1
413.g odd 6 1 inner 2891.1.g.b 2
413.h even 6 1 59.1.b.a 1
413.h even 6 1 2891.1.g.d 2
413.n even 174 28 3481.1.d.a 28
413.p odd 174 28 3481.1.d.a 28
1239.j odd 6 1 531.1.c.a 1
1652.k odd 6 1 944.1.h.a 1
2065.n even 6 1 1475.1.c.b 1
2065.x odd 12 2 1475.1.d.a 2
3304.r even 6 1 3776.1.h.b 1
3304.ba odd 6 1 3776.1.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 7.d odd 6 1
59.1.b.a 1 413.h even 6 1
531.1.c.a 1 21.g even 6 1
531.1.c.a 1 1239.j odd 6 1
944.1.h.a 1 28.f even 6 1
944.1.h.a 1 1652.k odd 6 1
1475.1.c.b 1 35.i odd 6 1
1475.1.c.b 1 2065.n even 6 1
1475.1.d.a 2 35.k even 12 2
1475.1.d.a 2 2065.x odd 12 2
2891.1.c.e 1 7.c even 3 1
2891.1.c.e 1 413.g odd 6 1
2891.1.g.b 2 1.a even 1 1 trivial
2891.1.g.b 2 7.c even 3 1 inner
2891.1.g.b 2 59.b odd 2 1 CM
2891.1.g.b 2 413.g odd 6 1 inner
2891.1.g.d 2 7.b odd 2 1
2891.1.g.d 2 7.d odd 6 1
2891.1.g.d 2 413.b even 2 1
2891.1.g.d 2 413.h even 6 1
3481.1.d.a 28 413.n even 174 28
3481.1.d.a 28 413.p odd 174 28
3776.1.h.a 1 56.m even 6 1
3776.1.h.a 1 3304.ba odd 6 1
3776.1.h.b 1 56.j odd 6 1
3776.1.h.b 1 3304.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2891, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)
\( T_{5}^{2} + T_{5} + 1 \)
\( T_{17}^{2} - 2 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 1 - T + T^{2} \)
$59$ \( 1 - T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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