Properties

Label 3528.2.s.i
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: \(\Z[a_1, \ldots, a_{25}]\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{5} +O(q^{10})\) \( q -2 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{11} -2 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( -4 + 4 \zeta_{6} ) q^{31} -10 \zeta_{6} q^{37} + 2 q^{41} -4 q^{43} -4 \zeta_{6} q^{47} + ( -12 + 12 \zeta_{6} ) q^{53} -4 q^{55} + ( -12 + 12 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} + 14 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{79} -16 q^{83} + 12 q^{85} + 6 \zeta_{6} q^{89} + ( -8 + 8 \zeta_{6} ) q^{95} + 18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{11} - 4q^{13} - 6q^{17} - 4q^{19} + 6q^{23} + q^{25} - 4q^{31} - 10q^{37} + 4q^{41} - 8q^{43} - 4q^{47} - 12q^{53} - 8q^{55} - 12q^{59} + 6q^{61} + 4q^{65} + 4q^{67} + 28q^{71} - 2q^{73} + 8q^{79} - 32q^{83} + 24q^{85} + 6q^{89} - 8q^{95} + 36q^{97} + O(q^{100}) \)

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 −1.00000 1.73205i 0 0 0 0 0
3313.1 0 0 0 −1.00000 + 1.73205i 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.i 2
3.b odd 2 1 3528.2.s.u 2
7.b odd 2 1 3528.2.s.x 2
7.c even 3 1 3528.2.a.u 1
7.c even 3 1 inner 3528.2.s.i 2
7.d odd 6 1 504.2.a.a 1
7.d odd 6 1 3528.2.s.x 2
21.c even 2 1 3528.2.s.f 2
21.g even 6 1 504.2.a.f yes 1
21.g even 6 1 3528.2.s.f 2
21.h odd 6 1 3528.2.a.e 1
21.h odd 6 1 3528.2.s.u 2
28.f even 6 1 1008.2.a.f 1
28.g odd 6 1 7056.2.a.bt 1
56.j odd 6 1 4032.2.a.bf 1
56.m even 6 1 4032.2.a.bg 1
84.j odd 6 1 1008.2.a.k 1
84.n even 6 1 7056.2.a.l 1
168.ba even 6 1 4032.2.a.g 1
168.be odd 6 1 4032.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.a.a 1 7.d odd 6 1
504.2.a.f yes 1 21.g even 6 1
1008.2.a.f 1 28.f even 6 1
1008.2.a.k 1 84.j odd 6 1
3528.2.a.e 1 21.h odd 6 1
3528.2.a.u 1 7.c even 3 1
3528.2.s.f 2 21.c even 2 1
3528.2.s.f 2 21.g even 6 1
3528.2.s.i 2 1.a even 1 1 trivial
3528.2.s.i 2 7.c even 3 1 inner
3528.2.s.u 2 3.b odd 2 1
3528.2.s.u 2 21.h odd 6 1
3528.2.s.x 2 7.b odd 2 1
3528.2.s.x 2 7.d odd 6 1
4032.2.a.g 1 168.ba even 6 1
4032.2.a.l 1 168.be odd 6 1
4032.2.a.bf 1 56.j odd 6 1
4032.2.a.bg 1 56.m even 6 1
7056.2.a.l 1 84.n even 6 1
7056.2.a.bt 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 \(S_{2}^{\mathrm{new}}(3528, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 16 + 4 T + T^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 36 - 6 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( ( -14 + T )^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 64 - 8 T + T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -18 + T )^{2} \)
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