Properties

Label 3332.1.bc.d
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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

Hecke kernels

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

\( T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} - 4T_{5} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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