Properties

Label 3364.1.b.b
Level $3364$
Weight $1$
Character orbit 3364.b
Self dual yes
Analytic conductor $1.679$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3364 = 2^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3364.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.67885470250\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} -\beta_{1} q^{5} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{4} -\beta_{1} q^{5} - q^{8} + q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{1} - \beta_{2} ) q^{13} + q^{16} -\beta_{2} q^{17} - q^{18} -\beta_{1} q^{20} + ( 1 + \beta_{2} ) q^{25} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} - q^{32} + \beta_{2} q^{34} + q^{36} + ( 1 - \beta_{1} + \beta_{2} ) q^{37} + \beta_{1} q^{40} + ( 1 - \beta_{1} + \beta_{2} ) q^{41} -\beta_{1} q^{45} + q^{49} + ( -1 - \beta_{2} ) q^{50} + ( -1 + \beta_{1} - \beta_{2} ) q^{52} + \beta_{2} q^{53} -\beta_{2} q^{61} + q^{64} + ( -1 + \beta_{1} ) q^{65} -\beta_{2} q^{68} - q^{72} -\beta_{2} q^{73} + ( -1 + \beta_{1} - \beta_{2} ) q^{74} -\beta_{1} q^{80} + q^{81} + ( -1 + \beta_{1} - \beta_{2} ) q^{82} + ( 1 + \beta_{2} ) q^{85} + \beta_{1} q^{89} + \beta_{1} q^{90} + ( 1 - \beta_{1} + \beta_{2} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - q^{5} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - q^{5} - 3q^{8} + 3q^{9} + q^{10} - q^{13} + 3q^{16} + q^{17} - 3q^{18} - q^{20} + 2q^{25} + q^{26} - 3q^{32} - q^{34} + 3q^{36} + q^{37} + q^{40} + q^{41} - q^{45} + 3q^{49} - 2q^{50} - q^{52} - q^{53} + q^{61} + 3q^{64} - 2q^{65} + q^{68} - 3q^{72} + q^{73} - q^{74} - q^{80} + 3q^{81} - q^{82} + 2q^{85} + q^{89} + q^{90} + q^{97} - 3q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times\).

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1683.1
1.80194
0.445042
−1.24698
−1.00000 0 1.00000 −1.80194 0 0 −1.00000 1.00000 1.80194
1683.2 −1.00000 0 1.00000 −0.445042 0 0 −1.00000 1.00000 0.445042
1683.3 −1.00000 0 1.00000 1.24698 0 0 −1.00000 1.00000 −1.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.b.b 3
4.b odd 2 1 CM 3364.1.b.b 3
29.b even 2 1 3364.1.b.c 3
29.c odd 4 2 3364.1.d.a 6
29.d even 7 2 3364.1.j.c 6
29.d even 7 2 3364.1.j.d 6
29.d even 7 2 3364.1.j.e 6
29.e even 14 2 116.1.j.a 6
29.e even 14 2 3364.1.j.a 6
29.e even 14 2 3364.1.j.b 6
29.f odd 28 4 3364.1.h.c 12
29.f odd 28 4 3364.1.h.d 12
29.f odd 28 4 3364.1.h.e 12
87.h odd 14 2 1044.1.bb.a 6
116.d odd 2 1 3364.1.b.c 3
116.e even 4 2 3364.1.d.a 6
116.h odd 14 2 116.1.j.a 6
116.h odd 14 2 3364.1.j.a 6
116.h odd 14 2 3364.1.j.b 6
116.j odd 14 2 3364.1.j.c 6
116.j odd 14 2 3364.1.j.d 6
116.j odd 14 2 3364.1.j.e 6
116.l even 28 4 3364.1.h.c 12
116.l even 28 4 3364.1.h.d 12
116.l even 28 4 3364.1.h.e 12
145.l even 14 2 2900.1.bj.a 6
145.q odd 28 4 2900.1.bd.a 12
232.o even 14 2 1856.1.bh.a 6
232.t odd 14 2 1856.1.bh.a 6
348.t even 14 2 1044.1.bb.a 6
580.y odd 14 2 2900.1.bj.a 6
580.bh even 28 4 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 29.e even 14 2
116.1.j.a 6 116.h odd 14 2
1044.1.bb.a 6 87.h odd 14 2
1044.1.bb.a 6 348.t even 14 2
1856.1.bh.a 6 232.o even 14 2
1856.1.bh.a 6 232.t odd 14 2
2900.1.bd.a 12 145.q odd 28 4
2900.1.bd.a 12 580.bh even 28 4
2900.1.bj.a 6 145.l even 14 2
2900.1.bj.a 6 580.y odd 14 2
3364.1.b.b 3 1.a even 1 1 trivial
3364.1.b.b 3 4.b odd 2 1 CM
3364.1.b.c 3 29.b even 2 1
3364.1.b.c 3 116.d odd 2 1
3364.1.d.a 6 29.c odd 4 2
3364.1.d.a 6 116.e even 4 2
3364.1.h.c 12 29.f odd 28 4
3364.1.h.c 12 116.l even 28 4
3364.1.h.d 12 29.f odd 28 4
3364.1.h.d 12 116.l even 28 4
3364.1.h.e 12 29.f odd 28 4
3364.1.h.e 12 116.l even 28 4
3364.1.j.a 6 29.e even 14 2
3364.1.j.a 6 116.h odd 14 2
3364.1.j.b 6 29.e even 14 2
3364.1.j.b 6 116.h odd 14 2
3364.1.j.c 6 29.d even 7 2
3364.1.j.c 6 116.j odd 14 2
3364.1.j.d 6 29.d even 7 2
3364.1.j.d 6 116.j odd 14 2
3364.1.j.e 6 29.d even 7 2
3364.1.j.e 6 116.j odd 14 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3364, [\chi])\):

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

Hecke characteristic polynomials

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