Properties

Label 3528.1.d.b
Level $3528$
Weight $1$
Character orbit 3528.d
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.21168.2
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.263473523712.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} -\beta q^{11} + q^{13} + q^{19} - q^{25} + q^{31} + q^{37} + \beta q^{41} - q^{43} -\beta q^{47} + 2 q^{55} + \beta q^{65} - q^{67} + \beta q^{71} - q^{73} + q^{79} + \beta q^{83} + \beta q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{13} + 2q^{19} - 2q^{25} + 2q^{31} + 2q^{37} - 2q^{43} + 4q^{55} - 2q^{67} - 2q^{73} + 2q^{79} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1961.1
1.41421i
1.41421i
0 0 0 1.41421i 0 0 0 0 0
1961.2 0 0 0 1.41421i 0 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.d.b 2
3.b odd 2 1 inner 3528.1.d.b 2
7.b odd 2 1 3528.1.d.a 2
7.c even 3 2 3528.1.cu.a 4
7.d odd 6 2 504.1.cu.a 4
21.c even 2 1 3528.1.d.a 2
21.g even 6 2 504.1.cu.a 4
21.h odd 6 2 3528.1.cu.a 4
28.f even 6 2 1008.1.dc.a 4
84.j odd 6 2 1008.1.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a 4 7.d odd 6 2
504.1.cu.a 4 21.g even 6 2
1008.1.dc.a 4 28.f even 6 2
1008.1.dc.a 4 84.j odd 6 2
3528.1.d.a 2 7.b odd 2 1
3528.1.d.a 2 21.c even 2 1
3528.1.d.b 2 1.a even 1 1 trivial
3528.1.d.b 2 3.b odd 2 1 inner
3528.1.cu.a 4 7.c even 3 2
3528.1.cu.a 4 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

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