Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 \zeta_{6} q^{5} +O(q^{10})\) \( q -4 \zeta_{6} q^{5} + 3 q^{13} + ( 4 - 4 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 8 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{37} + 8 q^{41} + 11 q^{43} -4 \zeta_{6} q^{47} + ( 4 - 4 \zeta_{6} ) q^{53} + ( -12 + 12 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} -12 \zeta_{6} q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} -12 q^{71} + ( 1 - \zeta_{6} ) q^{73} -\zeta_{6} q^{79} + 12 q^{83} -16 q^{85} -8 \zeta_{6} q^{89} + ( 28 - 28 \zeta_{6} ) q^{95} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} + 6q^{13} + 4q^{17} + 7q^{19} + 4q^{23} - 11q^{25} + 16q^{29} - 5q^{31} - 3q^{37} + 16q^{41} + 22q^{43} - 4q^{47} + 4q^{53} - 12q^{59} - 2q^{61} - 12q^{65} + 3q^{67} - 24q^{71} + q^{73} - q^{79} + 24q^{83} - 32q^{85} - 8q^{89} + 28q^{95} + 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 3.46410i 0 0 0 0 0
3313.1 0 0 0 −2.00000 + 3.46410i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.s.b 2
3.b odd 2 1 3528.2.s.bb 2
7.b odd 2 1 504.2.s.h yes 2
7.c even 3 1 3528.2.a.ba 1
7.c even 3 1 inner 3528.2.s.b 2
7.d odd 6 1 504.2.s.h yes 2
7.d odd 6 1 3528.2.a.a 1
21.c even 2 1 504.2.s.a 2
21.g even 6 1 504.2.s.a 2
21.g even 6 1 3528.2.a.z 1
21.h odd 6 1 3528.2.a.c 1
21.h odd 6 1 3528.2.s.bb 2
28.d even 2 1 1008.2.s.q 2
28.f even 6 1 1008.2.s.q 2
28.f even 6 1 7056.2.a.b 1
28.g odd 6 1 7056.2.a.cc 1
84.h odd 2 1 1008.2.s.a 2
84.j odd 6 1 1008.2.s.a 2
84.j odd 6 1 7056.2.a.cb 1
84.n even 6 1 7056.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.a 2 21.c even 2 1
504.2.s.a 2 21.g even 6 1
504.2.s.h yes 2 7.b odd 2 1
504.2.s.h yes 2 7.d odd 6 1
1008.2.s.a 2 84.h odd 2 1
1008.2.s.a 2 84.j odd 6 1
1008.2.s.q 2 28.d even 2 1
1008.2.s.q 2 28.f even 6 1
3528.2.a.a 1 7.d odd 6 1
3528.2.a.c 1 21.h odd 6 1
3528.2.a.z 1 21.g even 6 1
3528.2.a.ba 1 7.c even 3 1
3528.2.s.b 2 1.a even 1 1 trivial
3528.2.s.b 2 7.c even 3 1 inner
3528.2.s.bb 2 3.b odd 2 1
3528.2.s.bb 2 21.h odd 6 1
7056.2.a.b 1 28.f even 6 1
7056.2.a.d 1 84.n even 6 1
7056.2.a.cb 1 84.j odd 6 1
7056.2.a.cc 1 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{11} \)
\( T_{13} - 3 \)
\( T_{23}^{2} - 4 T_{23} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -3 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 9 + 3 T + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( ( -11 + T )^{2} \)
$47$ \( 16 + 4 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 64 + 8 T + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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