Properties

Label 3332.1.bp.c
Level $3332$
Weight $1$
Character orbit 3332.bp
Analytic conductor $1.663$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
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.bp (of order \(24\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.3089659810545728.4

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( \zeta_{24}^{7} + \zeta_{24}^{10} ) q^{5} + \zeta_{24}^{9} q^{8} -\zeta_{24} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( \zeta_{24}^{7} + \zeta_{24}^{10} ) q^{5} + \zeta_{24}^{9} q^{8} -\zeta_{24} q^{9} + ( \zeta_{24}^{2} + \zeta_{24}^{5} ) q^{10} + \zeta_{24}^{4} q^{16} -\zeta_{24}^{11} q^{17} + \zeta_{24}^{8} q^{18} + ( 1 - \zeta_{24}^{9} ) q^{20} + ( -\zeta_{24}^{2} - \zeta_{24}^{5} - \zeta_{24}^{8} ) q^{25} + ( -\zeta_{24}^{3} - \zeta_{24}^{6} ) q^{29} -\zeta_{24}^{11} q^{32} -\zeta_{24}^{6} q^{34} + \zeta_{24}^{3} q^{36} + ( -\zeta_{24} - \zeta_{24}^{4} ) q^{37} + ( -\zeta_{24}^{4} - \zeta_{24}^{7} ) q^{40} + ( 1 - \zeta_{24}^{3} ) q^{41} + ( -\zeta_{24}^{8} - \zeta_{24}^{11} ) q^{45} + ( -1 - \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{50} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{53} + ( -\zeta_{24} + \zeta_{24}^{10} ) q^{58} + ( -\zeta_{24}^{4} + \zeta_{24}^{7} ) q^{61} -\zeta_{24}^{6} q^{64} -\zeta_{24} q^{68} -\zeta_{24}^{10} q^{72} + ( -\zeta_{24}^{2} + \zeta_{24}^{11} ) q^{73} + ( \zeta_{24}^{8} + \zeta_{24}^{11} ) q^{74} + ( -\zeta_{24}^{2} + \zeta_{24}^{11} ) q^{80} + \zeta_{24}^{2} q^{81} + ( -\zeta_{24}^{7} + \zeta_{24}^{10} ) q^{82} + ( \zeta_{24}^{6} + \zeta_{24}^{9} ) q^{85} -2 \zeta_{24}^{10} q^{89} + ( -\zeta_{24}^{3} - \zeta_{24}^{6} ) q^{90} + ( -\zeta_{24}^{3} - \zeta_{24}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 4 q^{16} - 4 q^{18} + 8 q^{20} + 4 q^{25} - 4 q^{37} - 4 q^{40} + 8 q^{41} + 4 q^{45} - 8 q^{50} + 4 q^{53} - 4 q^{61} - 4 q^{74} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{24}^{9}\) \(-\zeta_{24}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.0999004 0.758819i 0 0 0.707107 + 0.707107i −0.258819 + 0.965926i 0.0999004 0.758819i
655.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 1.83195 0.241181i 0 0 −0.707107 0.707107i 0.258819 0.965926i −1.83195 0.241181i
1243.1 0.258819 0.965926i 0 −0.866025 0.500000i −1.12484 + 1.46593i 0 0 −0.707107 + 0.707107i −0.965926 0.258819i 1.12484 + 1.46593i
1647.1 0.965926 0.258819i 0 0.866025 0.500000i −0.0999004 + 0.758819i 0 0 0.707107 0.707107i −0.258819 0.965926i 0.0999004 + 0.758819i
2235.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −0.607206 + 0.465926i 0 0 0.707107 + 0.707107i 0.965926 0.258819i 0.607206 + 0.465926i
2627.1 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −1.12484 1.46593i 0 0 −0.707107 0.707107i −0.965926 + 0.258819i 1.12484 1.46593i
3007.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −0.607206 0.465926i 0 0 0.707107 0.707107i 0.965926 + 0.258819i 0.607206 0.465926i
3215.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 1.83195 + 0.241181i 0 0 −0.707107 + 0.707107i 0.258819 + 0.965926i −1.83195 + 0.241181i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3215.1
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.d even 8 1 inner
28.g odd 6 1 inner
68.g odd 8 1 inner
119.q even 24 1 inner
476.bg odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bp.c 8
4.b odd 2 1 CM 3332.1.bp.c 8
7.b odd 2 1 3332.1.bp.b 8
7.c even 3 1 3332.1.w.c yes 4
7.c even 3 1 inner 3332.1.bp.c 8
7.d odd 6 1 3332.1.w.b 4
7.d odd 6 1 3332.1.bp.b 8
17.d even 8 1 inner 3332.1.bp.c 8
28.d even 2 1 3332.1.bp.b 8
28.f even 6 1 3332.1.w.b 4
28.f even 6 1 3332.1.bp.b 8
28.g odd 6 1 3332.1.w.c yes 4
28.g odd 6 1 inner 3332.1.bp.c 8
68.g odd 8 1 inner 3332.1.bp.c 8
119.l odd 8 1 3332.1.bp.b 8
119.q even 24 1 3332.1.w.c yes 4
119.q even 24 1 inner 3332.1.bp.c 8
119.r odd 24 1 3332.1.w.b 4
119.r odd 24 1 3332.1.bp.b 8
476.w even 8 1 3332.1.bp.b 8
476.bg odd 24 1 3332.1.w.c yes 4
476.bg odd 24 1 inner 3332.1.bp.c 8
476.bj even 24 1 3332.1.w.b 4
476.bj even 24 1 3332.1.bp.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.w.b 4 7.d odd 6 1
3332.1.w.b 4 28.f even 6 1
3332.1.w.b 4 119.r odd 24 1
3332.1.w.b 4 476.bj even 24 1
3332.1.w.c yes 4 7.c even 3 1
3332.1.w.c yes 4 28.g odd 6 1
3332.1.w.c yes 4 119.q even 24 1
3332.1.w.c yes 4 476.bg odd 24 1
3332.1.bp.b 8 7.b odd 2 1
3332.1.bp.b 8 7.d odd 6 1
3332.1.bp.b 8 28.d even 2 1
3332.1.bp.b 8 28.f even 6 1
3332.1.bp.b 8 119.l odd 8 1
3332.1.bp.b 8 119.r odd 24 1
3332.1.bp.b 8 476.w even 8 1
3332.1.bp.b 8 476.bj even 24 1
3332.1.bp.c 8 1.a even 1 1 trivial
3332.1.bp.c 8 4.b odd 2 1 CM
3332.1.bp.c 8 7.c even 3 1 inner
3332.1.bp.c 8 17.d even 8 1 inner
3332.1.bp.c 8 28.g odd 6 1 inner
3332.1.bp.c 8 68.g odd 8 1 inner
3332.1.bp.c 8 119.q even 24 1 inner
3332.1.bp.c 8 476.bg odd 24 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2 T_{5}^{6} - 8 T_{5}^{5} + 2 T_{5}^{4} + 8 T_{5}^{3} + 12 T_{5}^{2} + 8 T_{5} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 4 + 8 T + 12 T^{2} + 8 T^{3} + 2 T^{4} - 8 T^{5} - 2 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( 1 - T^{4} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( 4 + 8 T + 4 T^{2} + 8 T^{3} + 18 T^{4} + 16 T^{5} + 10 T^{6} + 4 T^{7} + T^{8} \)
$41$ \( ( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( 4 + 8 T + 4 T^{2} + 8 T^{3} + 18 T^{4} + 16 T^{5} + 10 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 4 - 8 T + 12 T^{2} - 8 T^{3} + 2 T^{4} + 8 T^{5} - 2 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$97$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{2} \)
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