Properties

Label 3528.1.g.b
Level $3528$
Weight $1$
Character orbit 3528.g
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -7, -168, 24
Inner twists $8$

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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: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{6}, \sqrt{-7})\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.5489031744.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{8} + q^{16} -2 i q^{23} + q^{25} + 2 i q^{29} -i q^{32} + 2 q^{43} -2 q^{46} -i q^{50} -2 i q^{53} + 2 q^{58} - q^{64} + 2 q^{67} -2 i q^{71} -2 i q^{86} + 2 i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{16} + 2q^{25} + 4q^{43} - 4q^{46} + 4q^{58} - 2q^{64} + 4q^{67} + O(q^{100}) \)

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
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
883.2 1.00000i 0 −1.00000 0 0 0 1.00000i 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 \(\Q(\sqrt{-7}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
168.e odd 2 1 CM by \(\Q(\sqrt{-42}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.g.b 2
3.b odd 2 1 inner 3528.1.g.b 2
7.b odd 2 1 CM 3528.1.g.b 2
7.c even 3 2 3528.1.bx.c 4
7.d odd 6 2 3528.1.bx.c 4
8.d odd 2 1 inner 3528.1.g.b 2
21.c even 2 1 inner 3528.1.g.b 2
21.g even 6 2 3528.1.bx.c 4
21.h odd 6 2 3528.1.bx.c 4
24.f even 2 1 RM 3528.1.g.b 2
56.e even 2 1 inner 3528.1.g.b 2
56.k odd 6 2 3528.1.bx.c 4
56.m even 6 2 3528.1.bx.c 4
168.e odd 2 1 CM 3528.1.g.b 2
168.v even 6 2 3528.1.bx.c 4
168.be odd 6 2 3528.1.bx.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.g.b 2 1.a even 1 1 trivial
3528.1.g.b 2 3.b odd 2 1 inner
3528.1.g.b 2 7.b odd 2 1 CM
3528.1.g.b 2 8.d odd 2 1 inner
3528.1.g.b 2 21.c even 2 1 inner
3528.1.g.b 2 24.f even 2 1 RM
3528.1.g.b 2 56.e even 2 1 inner
3528.1.g.b 2 168.e odd 2 1 CM
3528.1.bx.c 4 7.c even 3 2
3528.1.bx.c 4 7.d odd 6 2
3528.1.bx.c 4 21.g even 6 2
3528.1.bx.c 4 21.h odd 6 2
3528.1.bx.c 4 56.k odd 6 2
3528.1.bx.c 4 56.m even 6 2
3528.1.bx.c 4 168.v even 6 2
3528.1.bx.c 4 168.be odd 6 2

Hecke kernels

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

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

Hecke characteristic polynomials

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