Properties

Label 3332.1.g.i
Level $3332$
Weight $1$
Character orbit 3332.g
Analytic conductor $1.663$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -119
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.205346735104.4

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{20}^{6} q^{2} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{3} -\zeta_{20}^{2} q^{4} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{5} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{6} + \zeta_{20}^{8} q^{8} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q -\zeta_{20}^{6} q^{2} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{3} -\zeta_{20}^{2} q^{4} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{5} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{6} + \zeta_{20}^{8} q^{8} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{9} + ( -\zeta_{20}^{3} + \zeta_{20}^{9} ) q^{10} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{12} + ( -\zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{15} + \zeta_{20}^{4} q^{16} + \zeta_{20}^{5} q^{17} + ( -\zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{18} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{20} + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{24} + ( -1 - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{25} + ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{27} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{30} + ( -\zeta_{20}^{3} + \zeta_{20}^{7} ) q^{31} + q^{32} + \zeta_{20} q^{34} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{36} + ( \zeta_{20} + \zeta_{20}^{5} ) q^{40} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{41} + ( \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{43} + ( -\zeta_{20} - \zeta_{20}^{3} - 2 \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{45} + ( \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{48} + ( -1 + \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{50} + ( \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{51} + ( -\zeta_{20}^{2} + \zeta_{20}^{8} ) q^{53} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{54} + ( -1 + \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{60} + ( \zeta_{20} + \zeta_{20}^{9} ) q^{61} + ( \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{62} -\zeta_{20}^{6} q^{64} + ( \zeta_{20}^{2} + \zeta_{20}^{8} ) q^{67} -\zeta_{20}^{7} q^{68} + ( -1 + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{72} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{73} + ( -\zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{75} + ( \zeta_{20} - \zeta_{20}^{7} ) q^{80} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{81} + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{82} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{85} + ( 1 + \zeta_{20}^{2} ) q^{86} + ( -2 \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{90} + ( -\zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{93} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{96} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 10 q^{30} + 8 q^{32} - 8 q^{36} - 4 q^{50} - 4 q^{53} - 10 q^{60} - 2 q^{64} - 8 q^{72} + 8 q^{81} + 4 q^{85} + 10 q^{86} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.809017 0.587785i −1.17557 0.309017 + 0.951057i 0.618034i 0.951057 + 0.690983i 0 0.309017 0.951057i 0.381966 −0.363271 + 0.500000i
883.2 −0.809017 0.587785i 1.17557 0.309017 + 0.951057i 0.618034i −0.951057 0.690983i 0 0.309017 0.951057i 0.381966 0.363271 0.500000i
883.3 −0.809017 + 0.587785i −1.17557 0.309017 0.951057i 0.618034i 0.951057 0.690983i 0 0.309017 + 0.951057i 0.381966 −0.363271 0.500000i
883.4 −0.809017 + 0.587785i 1.17557 0.309017 0.951057i 0.618034i −0.951057 + 0.690983i 0 0.309017 + 0.951057i 0.381966 0.363271 + 0.500000i
883.5 0.309017 0.951057i −1.90211 −0.809017 0.587785i 1.61803i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i 2.61803 1.53884 + 0.500000i
883.6 0.309017 0.951057i 1.90211 −0.809017 0.587785i 1.61803i 0.587785 1.80902i 0 −0.809017 + 0.587785i 2.61803 −1.53884 0.500000i
883.7 0.309017 + 0.951057i −1.90211 −0.809017 + 0.587785i 1.61803i −0.587785 1.80902i 0 −0.809017 0.587785i 2.61803 1.53884 0.500000i
883.8 0.309017 + 0.951057i 1.90211 −0.809017 + 0.587785i 1.61803i 0.587785 + 1.80902i 0 −0.809017 0.587785i 2.61803 −1.53884 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
68.d odd 2 1 inner
476.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.i 8
4.b odd 2 1 inner 3332.1.g.i 8
7.b odd 2 1 inner 3332.1.g.i 8
7.c even 3 2 3332.1.o.g 16
7.d odd 6 2 3332.1.o.g 16
17.b even 2 1 inner 3332.1.g.i 8
28.d even 2 1 inner 3332.1.g.i 8
28.f even 6 2 3332.1.o.g 16
28.g odd 6 2 3332.1.o.g 16
68.d odd 2 1 inner 3332.1.g.i 8
119.d odd 2 1 CM 3332.1.g.i 8
119.h odd 6 2 3332.1.o.g 16
119.j even 6 2 3332.1.o.g 16
476.e even 2 1 inner 3332.1.g.i 8
476.o odd 6 2 3332.1.o.g 16
476.q even 6 2 3332.1.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.i 8 1.a even 1 1 trivial
3332.1.g.i 8 4.b odd 2 1 inner
3332.1.g.i 8 7.b odd 2 1 inner
3332.1.g.i 8 17.b even 2 1 inner
3332.1.g.i 8 28.d even 2 1 inner
3332.1.g.i 8 68.d odd 2 1 inner
3332.1.g.i 8 119.d odd 2 1 CM
3332.1.g.i 8 476.e even 2 1 inner
3332.1.o.g 16 7.c even 3 2
3332.1.o.g 16 7.d odd 6 2
3332.1.o.g 16 28.f even 6 2
3332.1.o.g 16 28.g odd 6 2
3332.1.o.g 16 119.h odd 6 2
3332.1.o.g 16 119.j even 6 2
3332.1.o.g 16 476.o odd 6 2
3332.1.o.g 16 476.q even 6 2

Hecke kernels

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

\( T_{3}^{4} - 5 T_{3}^{2} + 5 \)
\( T_{5}^{4} + 3 T_{5}^{2} + 1 \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

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