Properties

Label 3332.1.g.f
Level $3332$
Weight $1$
Character orbit 3332.g
Self dual yes
Analytic conductor $1.663$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -68
Inner twists $4$

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: yes
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.188737808.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} - q^{8} + 2 q^{9} +O(q^{10})\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} - q^{8} + 2 q^{9} -\beta q^{11} -\beta q^{12} - q^{13} + q^{16} - q^{17} -2 q^{18} + \beta q^{22} + \beta q^{24} + q^{25} + q^{26} -\beta q^{27} - q^{32} + 3 q^{33} + q^{34} + 2 q^{36} + \beta q^{39} -\beta q^{44} -\beta q^{48} - q^{50} + \beta q^{51} - q^{52} - q^{53} + \beta q^{54} + q^{64} -3 q^{66} - q^{68} + \beta q^{71} -2 q^{72} -\beta q^{75} -\beta q^{78} -\beta q^{79} + q^{81} + \beta q^{88} - q^{89} + \beta q^{96} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} - 2 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{25} + 2 q^{26} - 2 q^{32} + 6 q^{33} + 2 q^{34} + 4 q^{36} - 2 q^{50} - 2 q^{52} - 2 q^{53} + 2 q^{64} - 6 q^{66} - 2 q^{68} - 4 q^{72} + 2 q^{81} - 2 q^{89} + O(q^{100}) \)

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
1.73205
−1.73205
−1.00000 −1.73205 1.00000 0 1.73205 0 −1.00000 2.00000 0
883.2 −1.00000 1.73205 1.00000 0 −1.73205 0 −1.00000 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
17.b 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.f 2
4.b odd 2 1 inner 3332.1.g.f 2
7.b odd 2 1 3332.1.g.g 2
7.c even 3 2 3332.1.o.f 4
7.d odd 6 2 476.1.o.c 4
17.b even 2 1 inner 3332.1.g.f 2
28.d even 2 1 3332.1.g.g 2
28.f even 6 2 476.1.o.c 4
28.g odd 6 2 3332.1.o.f 4
68.d odd 2 1 CM 3332.1.g.f 2
119.d odd 2 1 3332.1.g.g 2
119.h odd 6 2 476.1.o.c 4
119.j even 6 2 3332.1.o.f 4
476.e even 2 1 3332.1.g.g 2
476.o odd 6 2 3332.1.o.f 4
476.q even 6 2 476.1.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.c 4 7.d odd 6 2
476.1.o.c 4 28.f even 6 2
476.1.o.c 4 119.h odd 6 2
476.1.o.c 4 476.q even 6 2
3332.1.g.f 2 1.a even 1 1 trivial
3332.1.g.f 2 4.b odd 2 1 inner
3332.1.g.f 2 17.b even 2 1 inner
3332.1.g.f 2 68.d odd 2 1 CM
3332.1.g.g 2 7.b odd 2 1
3332.1.g.g 2 28.d even 2 1
3332.1.g.g 2 119.d odd 2 1
3332.1.g.g 2 476.e even 2 1
3332.1.o.f 4 7.c even 3 2
3332.1.o.f 4 28.g odd 6 2
3332.1.o.f 4 119.j even 6 2
3332.1.o.f 4 476.o odd 6 2

Hecke kernels

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

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

Hecke characteristic polynomials

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