Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
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.

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

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{8}^{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0.707107 0.292893i 0 0 0.707107 0.707107i −0.707107 + 0.707107i −0.707107 0.292893i
1079.1 −0.707107 + 0.707107i 0 1.00000i 0.707107 + 0.292893i 0 0 0.707107 + 0.707107i −0.707107 0.707107i −0.707107 + 0.292893i
1471.1 0.707107 0.707107i 0 1.00000i −0.707107 + 1.70711i 0 0 −0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 1.70711i
2059.1 0.707107 + 0.707107i 0 1.00000i −0.707107 1.70711i 0 0 −0.707107 + 0.707107i 0.707107 0.707107i 0.707107 1.70711i
\(n\): 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 \(\Q(\sqrt{-1}) \)
17.d even 8 1 inner
68.g odd 8 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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