Properties

Label 3332.1.o.e
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -4, -119, 476
Inner twists $16$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{119})\)
Artin image: $C_3\times D_4:C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + 2 \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + 2 \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} + 2 \zeta_{12} q^{10} -\zeta_{12}^{2} q^{16} + \zeta_{12} q^{17} -\zeta_{12}^{4} q^{18} -2 \zeta_{12}^{3} q^{20} -3 \zeta_{12}^{4} q^{25} + \zeta_{12}^{4} q^{32} -\zeta_{12}^{3} q^{34} - q^{36} + 2 \zeta_{12}^{5} q^{40} + 2 \zeta_{12}^{3} q^{41} -2 \zeta_{12} q^{45} -3 q^{50} + 2 \zeta_{12}^{4} q^{53} + 2 \zeta_{12}^{5} q^{61} + q^{64} + \zeta_{12}^{5} q^{68} + \zeta_{12}^{2} q^{72} -2 \zeta_{12} q^{73} + 2 \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} -2 \zeta_{12}^{5} q^{82} -2 q^{85} + 2 \zeta_{12}^{3} q^{90} -2 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} - 2 q^{16} + 2 q^{18} + 6 q^{25} - 2 q^{32} - 4 q^{36} - 12 q^{50} - 4 q^{53} + 4 q^{64} + 2 q^{72} - 2 q^{81} - 8 q^{85} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.73205 + 1.00000i 0 0 1.00000 0.500000 + 0.866025i 1.73205 + 1.00000i
67.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.73205 1.00000i 0 0 1.00000 0.500000 + 0.866025i −1.73205 1.00000i
2039.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.73205 1.00000i 0 0 1.00000 0.500000 0.866025i 1.73205 1.00000i
2039.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.73205 + 1.00000i 0 0 1.00000 0.500000 0.866025i −1.73205 + 1.00000i
\(n\): 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 \(\Q(\sqrt{-1}) \)
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
476.e even 2 1 RM by \(\Q(\sqrt{119}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
68.d odd 2 1 inner
119.h odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner
476.q even 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.e 4
4.b odd 2 1 CM 3332.1.o.e 4
7.b odd 2 1 inner 3332.1.o.e 4
7.c even 3 1 3332.1.g.h 2
7.c even 3 1 inner 3332.1.o.e 4
7.d odd 6 1 3332.1.g.h 2
7.d odd 6 1 inner 3332.1.o.e 4
17.b even 2 1 inner 3332.1.o.e 4
28.d even 2 1 inner 3332.1.o.e 4
28.f even 6 1 3332.1.g.h 2
28.f even 6 1 inner 3332.1.o.e 4
28.g odd 6 1 3332.1.g.h 2
28.g odd 6 1 inner 3332.1.o.e 4
68.d odd 2 1 inner 3332.1.o.e 4
119.d odd 2 1 CM 3332.1.o.e 4
119.h odd 6 1 3332.1.g.h 2
119.h odd 6 1 inner 3332.1.o.e 4
119.j even 6 1 3332.1.g.h 2
119.j even 6 1 inner 3332.1.o.e 4
476.e even 2 1 RM 3332.1.o.e 4
476.o odd 6 1 3332.1.g.h 2
476.o odd 6 1 inner 3332.1.o.e 4
476.q even 6 1 3332.1.g.h 2
476.q even 6 1 inner 3332.1.o.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.h 2 7.c even 3 1
3332.1.g.h 2 7.d odd 6 1
3332.1.g.h 2 28.f even 6 1
3332.1.g.h 2 28.g odd 6 1
3332.1.g.h 2 119.h odd 6 1
3332.1.g.h 2 119.j even 6 1
3332.1.g.h 2 476.o odd 6 1
3332.1.g.h 2 476.q even 6 1
3332.1.o.e 4 1.a even 1 1 trivial
3332.1.o.e 4 4.b odd 2 1 CM
3332.1.o.e 4 7.b odd 2 1 inner
3332.1.o.e 4 7.c even 3 1 inner
3332.1.o.e 4 7.d odd 6 1 inner
3332.1.o.e 4 17.b even 2 1 inner
3332.1.o.e 4 28.d even 2 1 inner
3332.1.o.e 4 28.f even 6 1 inner
3332.1.o.e 4 28.g odd 6 1 inner
3332.1.o.e 4 68.d odd 2 1 inner
3332.1.o.e 4 119.d odd 2 1 CM
3332.1.o.e 4 119.h odd 6 1 inner
3332.1.o.e 4 119.j even 6 1 inner
3332.1.o.e 4 476.e even 2 1 RM
3332.1.o.e 4 476.o odd 6 1 inner
3332.1.o.e 4 476.q even 6 1 inner

Hecke kernels

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

\( T_{3} \)
\( T_{5}^{4} - 4 T_{5}^{2} + 16 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 16 - 4 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( 1 - T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 4 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 4 + 2 T + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( 16 - 4 T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 16 - 4 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 4 + T^{2} )^{2} \)
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