Properties

Label 3332.1.bc.a
Level $3332$
Weight $1$
Character orbit 3332.bc
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.962948.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{5} + \zeta_{12}^{3} q^{8} -\zeta_{12} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{5} + \zeta_{12}^{3} q^{8} -\zeta_{12} q^{9} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{10} -2 q^{13} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{5} q^{17} -\zeta_{12}^{2} q^{18} + ( -1 - \zeta_{12}^{3} ) q^{20} -\zeta_{12}^{5} q^{25} -2 \zeta_{12} q^{26} + ( -1 - \zeta_{12}^{3} ) q^{29} + \zeta_{12}^{5} q^{32} + q^{34} -\zeta_{12}^{3} q^{36} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{37} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{40} + ( -1 + \zeta_{12}^{3} ) q^{41} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{45} + q^{50} -2 \zeta_{12}^{2} q^{52} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{58} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{61} - q^{64} + ( 2 \zeta_{12} - 2 \zeta_{12}^{4} ) q^{65} + \zeta_{12} q^{68} -\zeta_{12}^{4} q^{72} + ( -\zeta_{12}^{2} - \zeta_{12}^{5} ) q^{73} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{74} + ( -\zeta_{12}^{2} - \zeta_{12}^{5} ) q^{80} + \zeta_{12}^{2} q^{81} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{82} + ( -1 + \zeta_{12}^{3} ) q^{85} + 2 \zeta_{12}^{4} q^{89} + ( 1 + \zeta_{12}^{3} ) q^{90} + ( 1 + \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{5} + O(q^{10}) \) \( 4 q + 2 q^{4} - 2 q^{5} - 2 q^{10} - 8 q^{13} - 2 q^{16} - 2 q^{18} - 4 q^{20} - 4 q^{29} + 4 q^{34} - 2 q^{37} + 2 q^{40} - 4 q^{41} + 2 q^{45} + 4 q^{50} - 4 q^{52} + 2 q^{58} + 2 q^{61} - 4 q^{64} + 4 q^{65} + 2 q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{80} + 2 q^{81} - 2 q^{82} - 4 q^{85} - 4 q^{89} + 4 q^{90} + 4 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
667.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0.366025 + 1.36603i 0 0 1.00000i 0.866025 + 0.500000i 0.366025 1.36603i
863.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.36603 + 0.366025i 0 0 1.00000i −0.866025 0.500000i −1.36603 0.366025i
2027.1 0.866025 0.500000i 0 0.500000 0.866025i −1.36603 0.366025i 0 0 1.00000i −0.866025 + 0.500000i −1.36603 + 0.366025i
2223.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.366025 1.36603i 0 0 1.00000i 0.866025 0.500000i 0.366025 + 1.36603i
\(n\): 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 \(\Q(\sqrt{-1}) \)
7.c even 3 1 inner
17.c even 4 1 inner
28.g odd 6 1 inner
68.f odd 4 1 inner
119.n even 12 1 inner
476.bb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bc.a 4
4.b odd 2 1 CM 3332.1.bc.a 4
7.b odd 2 1 3332.1.bc.d 4
7.c even 3 1 3332.1.m.c yes 2
7.c even 3 1 inner 3332.1.bc.a 4
7.d odd 6 1 3332.1.m.a 2
7.d odd 6 1 3332.1.bc.d 4
17.c even 4 1 inner 3332.1.bc.a 4
28.d even 2 1 3332.1.bc.d 4
28.f even 6 1 3332.1.m.a 2
28.f even 6 1 3332.1.bc.d 4
28.g odd 6 1 3332.1.m.c yes 2
28.g odd 6 1 inner 3332.1.bc.a 4
68.f odd 4 1 inner 3332.1.bc.a 4
119.f odd 4 1 3332.1.bc.d 4
119.m odd 12 1 3332.1.m.a 2
119.m odd 12 1 3332.1.bc.d 4
119.n even 12 1 3332.1.m.c yes 2
119.n even 12 1 inner 3332.1.bc.a 4
476.k even 4 1 3332.1.bc.d 4
476.z even 12 1 3332.1.m.a 2
476.z even 12 1 3332.1.bc.d 4
476.bb odd 12 1 3332.1.m.c yes 2
476.bb odd 12 1 inner 3332.1.bc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.m.a 2 7.d odd 6 1
3332.1.m.a 2 28.f even 6 1
3332.1.m.a 2 119.m odd 12 1
3332.1.m.a 2 476.z even 12 1
3332.1.m.c yes 2 7.c even 3 1
3332.1.m.c yes 2 28.g odd 6 1
3332.1.m.c yes 2 119.n even 12 1
3332.1.m.c yes 2 476.bb odd 12 1
3332.1.bc.a 4 1.a even 1 1 trivial
3332.1.bc.a 4 4.b odd 2 1 CM
3332.1.bc.a 4 7.c even 3 1 inner
3332.1.bc.a 4 17.c even 4 1 inner
3332.1.bc.a 4 28.g odd 6 1 inner
3332.1.bc.a 4 68.f odd 4 1 inner
3332.1.bc.a 4 119.n even 12 1 inner
3332.1.bc.a 4 476.bb odd 12 1 inner
3332.1.bc.d 4 7.b odd 2 1
3332.1.bc.d 4 7.d odd 6 1
3332.1.bc.d 4 28.d even 2 1
3332.1.bc.d 4 28.f even 6 1
3332.1.bc.d 4 119.f odd 4 1
3332.1.bc.d 4 119.m odd 12 1
3332.1.bc.d 4 476.k even 4 1
3332.1.bc.d 4 476.z even 12 1

Hecke kernels

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

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

Hecke characteristic polynomials

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