Properties

Label 3332.1.g.h
Level $3332$
Weight $1$
Character orbit 3332.g
Analytic conductor $1.663$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -119, 476
Inner twists $8$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{119})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.8704143616.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - i q^{5} + q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - i q^{5} + q^{8} - q^{9} - 2 i q^{10} + q^{16} - i q^{17} - q^{18} - 2 i q^{20} - 3 q^{25} + q^{32} - i q^{34} - q^{36} - 2 i q^{40} + i q^{41} + 2 i q^{45} - 3 q^{50} + q^{53} - i q^{61} + q^{64} - i q^{68} - q^{72} + i q^{73} - 2 i q^{80} + q^{81} + 2 i q^{82} - 2 q^{85} + 2 i q^{90} - i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{16} - 2 q^{18} - 6 q^{25} + 2 q^{32} - 2 q^{36} - 6 q^{50} + 4 q^{53} + 2 q^{64} - 2 q^{72} + 2 q^{81} - 4 q^{85}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000 0 1.00000 2.00000i 0 0 1.00000 −1.00000 2.00000i
883.2 1.00000 0 1.00000 2.00000i 0 0 1.00000 −1.00000 2.00000i
\(n\): 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 \(\Q(\sqrt{-1}) \)
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
476.e even 2 1 RM by \(\Q(\sqrt{119}) \)
7.b odd 2 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
68.d odd 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.h 2
4.b odd 2 1 CM 3332.1.g.h 2
7.b odd 2 1 inner 3332.1.g.h 2
7.c even 3 2 3332.1.o.e 4
7.d odd 6 2 3332.1.o.e 4
17.b even 2 1 inner 3332.1.g.h 2
28.d even 2 1 inner 3332.1.g.h 2
28.f even 6 2 3332.1.o.e 4
28.g odd 6 2 3332.1.o.e 4
68.d odd 2 1 inner 3332.1.g.h 2
119.d odd 2 1 CM 3332.1.g.h 2
119.h odd 6 2 3332.1.o.e 4
119.j even 6 2 3332.1.o.e 4
476.e even 2 1 RM 3332.1.g.h 2
476.o odd 6 2 3332.1.o.e 4
476.q even 6 2 3332.1.o.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.h 2 1.a even 1 1 trivial
3332.1.g.h 2 4.b odd 2 1 CM
3332.1.g.h 2 7.b odd 2 1 inner
3332.1.g.h 2 17.b even 2 1 inner
3332.1.g.h 2 28.d even 2 1 inner
3332.1.g.h 2 68.d odd 2 1 inner
3332.1.g.h 2 119.d odd 2 1 CM
3332.1.g.h 2 476.e even 2 1 RM
3332.1.o.e 4 7.c even 3 2
3332.1.o.e 4 7.d odd 6 2
3332.1.o.e 4 28.f even 6 2
3332.1.o.e 4 28.g odd 6 2
3332.1.o.e 4 119.h odd 6 2
3332.1.o.e 4 119.j even 6 2
3332.1.o.e 4 476.o odd 6 2
3332.1.o.e 4 476.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 \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

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

Hecke characteristic polynomials

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