Properties

Label 3528.1.g.a
Level $3528$
Weight $1$
Character orbit 3528.g
Self dual yes
Analytic conductor $1.761$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -7, -8, 56
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.24696.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} + 2q^{11} + q^{16} - 2q^{22} + q^{25} - q^{32} - 2q^{43} + 2q^{44} - q^{50} + q^{64} + 2q^{67} + 2q^{86} - 2q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
−1.00000 0 1.00000 0 0 0 −1.00000 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
56.e even 2 1 RM by \(\Q(\sqrt{14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.g.a 1
3.b odd 2 1 392.1.g.a 1
7.b odd 2 1 CM 3528.1.g.a 1
7.c even 3 2 3528.1.bx.a 2
7.d odd 6 2 3528.1.bx.a 2
8.d odd 2 1 CM 3528.1.g.a 1
12.b even 2 1 1568.1.g.a 1
21.c even 2 1 392.1.g.a 1
21.g even 6 2 392.1.k.a 2
21.h odd 6 2 392.1.k.a 2
24.f even 2 1 392.1.g.a 1
24.h odd 2 1 1568.1.g.a 1
56.e even 2 1 RM 3528.1.g.a 1
56.k odd 6 2 3528.1.bx.a 2
56.m even 6 2 3528.1.bx.a 2
84.h odd 2 1 1568.1.g.a 1
84.j odd 6 2 1568.1.o.a 2
84.n even 6 2 1568.1.o.a 2
168.e odd 2 1 392.1.g.a 1
168.i even 2 1 1568.1.g.a 1
168.s odd 6 2 1568.1.o.a 2
168.v even 6 2 392.1.k.a 2
168.ba even 6 2 1568.1.o.a 2
168.be odd 6 2 392.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.a 1 3.b odd 2 1
392.1.g.a 1 21.c even 2 1
392.1.g.a 1 24.f even 2 1
392.1.g.a 1 168.e odd 2 1
392.1.k.a 2 21.g even 6 2
392.1.k.a 2 21.h odd 6 2
392.1.k.a 2 168.v even 6 2
392.1.k.a 2 168.be odd 6 2
1568.1.g.a 1 12.b even 2 1
1568.1.g.a 1 24.h odd 2 1
1568.1.g.a 1 84.h odd 2 1
1568.1.g.a 1 168.i even 2 1
1568.1.o.a 2 84.j odd 6 2
1568.1.o.a 2 84.n even 6 2
1568.1.o.a 2 168.s odd 6 2
1568.1.o.a 2 168.ba even 6 2
3528.1.g.a 1 1.a even 1 1 trivial
3528.1.g.a 1 7.b odd 2 1 CM
3528.1.g.a 1 8.d odd 2 1 CM
3528.1.g.a 1 56.e even 2 1 RM
3528.1.bx.a 2 7.c even 3 2
3528.1.bx.a 2 7.d odd 6 2
3528.1.bx.a 2 56.k odd 6 2
3528.1.bx.a 2 56.m even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{11} - 2 \)
\( T_{17} \)

Hecke characteristic polynomials

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