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: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{4} - 5T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{4} + 3T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

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