Properties

Label 3332.1.g.b
Level $3332$
Weight $1$
Character orbit 3332.g
Self dual yes
Analytic conductor $1.663$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -68
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.3332.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.3332.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} - q^{11} - q^{12} - q^{13} + q^{16} + q^{17} - q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26} + q^{27} + 2 q^{31} + q^{32} + q^{33} + q^{34} + q^{39} - q^{44} + 2 q^{46} - q^{48} + q^{50} - q^{51} - q^{52} - q^{53} + q^{54} + 2 q^{62} + q^{64} + q^{66} + q^{68} - 2 q^{69} - q^{71} - q^{75} + q^{78} - q^{79} - q^{81} - q^{88} - q^{89} + 2 q^{92} - 2 q^{93} - q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 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
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.b 1
4.b odd 2 1 3332.1.g.e 1
7.b odd 2 1 3332.1.g.d 1
7.c even 3 2 476.1.o.b yes 2
7.d odd 6 2 3332.1.o.a 2
17.b even 2 1 3332.1.g.e 1
28.d even 2 1 3332.1.g.c 1
28.f even 6 2 3332.1.o.b 2
28.g odd 6 2 476.1.o.a 2
68.d odd 2 1 CM 3332.1.g.b 1
119.d odd 2 1 3332.1.g.c 1
119.h odd 6 2 3332.1.o.b 2
119.j even 6 2 476.1.o.a 2
476.e even 2 1 3332.1.g.d 1
476.o odd 6 2 476.1.o.b yes 2
476.q even 6 2 3332.1.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.a 2 28.g odd 6 2
476.1.o.a 2 119.j even 6 2
476.1.o.b yes 2 7.c even 3 2
476.1.o.b yes 2 476.o odd 6 2
3332.1.g.b 1 1.a even 1 1 trivial
3332.1.g.b 1 68.d odd 2 1 CM
3332.1.g.c 1 28.d even 2 1
3332.1.g.c 1 119.d odd 2 1
3332.1.g.d 1 7.b odd 2 1
3332.1.g.d 1 476.e even 2 1
3332.1.g.e 1 4.b odd 2 1
3332.1.g.e 1 17.b even 2 1
3332.1.o.a 2 7.d odd 6 2
3332.1.o.a 2 476.q even 6 2
3332.1.o.b 2 28.f even 6 2
3332.1.o.b 2 119.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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