Properties

Label 3332.1.o.a
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -68
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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

$q$-expansion

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

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

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0 1.00000 0 0
2039.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 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 \(\Q(\sqrt{-17}) \)
7.c even 3 1 inner
476.o odd 6 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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