Properties

Label 3528.2.s.bf
Level $3528$
Weight $2$
Character orbit 3528.s
Analytic conductor $28.171$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} +O(q^{10})\) \( q + \beta_{1} q^{5} + 2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + 2 \beta_{1} q^{19} + ( -6 - 6 \beta_{2} ) q^{23} + 3 \beta_{2} q^{25} -4 q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 2 + 2 \beta_{2} ) q^{37} -\beta_{3} q^{41} -4 q^{43} -4 \beta_{1} q^{47} + 12 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{59} + 2 \beta_{1} q^{61} + ( -16 - 16 \beta_{2} ) q^{65} -12 \beta_{2} q^{67} -6 q^{71} + ( -8 - 8 \beta_{2} ) q^{79} + 4 \beta_{3} q^{83} + 8 q^{85} + 3 \beta_{1} q^{89} + 16 \beta_{2} q^{95} -4 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} - 12q^{23} - 6q^{25} - 16q^{29} + 4q^{37} - 16q^{43} - 24q^{53} - 32q^{65} + 24q^{67} - 24q^{71} - 16q^{79} + 32q^{85} - 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 0 0 −1.41421 2.44949i 0 0 0 0 0
361.2 0 0 0 1.41421 + 2.44949i 0 0 0 0 0
3313.1 0 0 0 −1.41421 + 2.44949i 0 0 0 0 0
3313.2 0 0 0 1.41421 2.44949i 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.s.bf 4
3.b odd 2 1 3528.2.s.bi 4
7.b odd 2 1 inner 3528.2.s.bf 4
7.c even 3 1 3528.2.a.bi yes 2
7.c even 3 1 inner 3528.2.s.bf 4
7.d odd 6 1 3528.2.a.bi yes 2
7.d odd 6 1 inner 3528.2.s.bf 4
21.c even 2 1 3528.2.s.bi 4
21.g even 6 1 3528.2.a.bf 2
21.g even 6 1 3528.2.s.bi 4
21.h odd 6 1 3528.2.a.bf 2
21.h odd 6 1 3528.2.s.bi 4
28.f even 6 1 7056.2.a.ck 2
28.g odd 6 1 7056.2.a.ck 2
84.j odd 6 1 7056.2.a.cq 2
84.n even 6 1 7056.2.a.cq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.2.a.bf 2 21.g even 6 1
3528.2.a.bf 2 21.h odd 6 1
3528.2.a.bi yes 2 7.c even 3 1
3528.2.a.bi yes 2 7.d odd 6 1
3528.2.s.bf 4 1.a even 1 1 trivial
3528.2.s.bf 4 7.b odd 2 1 inner
3528.2.s.bf 4 7.c even 3 1 inner
3528.2.s.bf 4 7.d odd 6 1 inner
3528.2.s.bi 4 3.b odd 2 1
3528.2.s.bi 4 21.c even 2 1
3528.2.s.bi 4 21.g even 6 1
3528.2.s.bi 4 21.h odd 6 1
7056.2.a.ck 2 28.f even 6 1
7056.2.a.ck 2 28.g odd 6 1
7056.2.a.cq 2 84.j odd 6 1
7056.2.a.cq 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{4} + 8 T_{5}^{2} + 64 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13}^{2} - 32 \)
\( T_{23}^{2} + 6 T_{23} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 64 + 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 + 2 T + T^{2} )^{2} \)
$13$ \( ( -32 + T^{2} )^{2} \)
$17$ \( 64 + 8 T^{2} + T^{4} \)
$19$ \( 1024 + 32 T^{2} + T^{4} \)
$23$ \( ( 36 + 6 T + T^{2} )^{2} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( 1024 + 32 T^{2} + T^{4} \)
$37$ \( ( 4 - 2 T + T^{2} )^{2} \)
$41$ \( ( -8 + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 16384 + 128 T^{2} + T^{4} \)
$53$ \( ( 144 + 12 T + T^{2} )^{2} \)
$59$ \( 16384 + 128 T^{2} + T^{4} \)
$61$ \( 1024 + 32 T^{2} + T^{4} \)
$67$ \( ( 144 - 12 T + T^{2} )^{2} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 64 + 8 T + T^{2} )^{2} \)
$83$ \( ( -128 + T^{2} )^{2} \)
$89$ \( 5184 + 72 T^{2} + T^{4} \)
$97$ \( ( -128 + T^{2} )^{2} \)
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