Properties

Label 3528.2.s.t
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 56)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} +O(q^{10})\) \( q + 2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{11} + 2 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{25} -6 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{37} -2 q^{41} -4 q^{43} -8 \zeta_{6} q^{47} + ( 6 - 6 \zeta_{6} ) q^{53} -8 q^{55} + 6 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} + 8 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -16 \zeta_{6} q^{79} -8 q^{83} -12 q^{85} -6 \zeta_{6} q^{89} + ( 16 - 16 \zeta_{6} ) q^{95} -6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} - 4q^{11} + 4q^{13} - 6q^{17} - 8q^{19} + q^{25} - 12q^{29} - 8q^{31} + 2q^{37} - 4q^{41} - 8q^{43} - 8q^{47} + 6q^{53} - 16q^{55} + 6q^{61} + 4q^{65} + 4q^{67} + 16q^{71} - 10q^{73} - 16q^{79} - 16q^{83} - 24q^{85} - 6q^{89} + 16q^{95} - 12q^{97} + O(q^{100}) \)

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.t 2
3.b odd 2 1 392.2.i.c 2
7.b odd 2 1 3528.2.s.e 2
7.c even 3 1 504.2.a.c 1
7.c even 3 1 inner 3528.2.s.t 2
7.d odd 6 1 3528.2.a.x 1
7.d odd 6 1 3528.2.s.e 2
12.b even 2 1 784.2.i.e 2
21.c even 2 1 392.2.i.d 2
21.g even 6 1 392.2.a.d 1
21.g even 6 1 392.2.i.d 2
21.h odd 6 1 56.2.a.a 1
21.h odd 6 1 392.2.i.c 2
28.f even 6 1 7056.2.a.bo 1
28.g odd 6 1 1008.2.a.d 1
56.k odd 6 1 4032.2.a.bk 1
56.p even 6 1 4032.2.a.bb 1
84.h odd 2 1 784.2.i.g 2
84.j odd 6 1 784.2.a.e 1
84.j odd 6 1 784.2.i.g 2
84.n even 6 1 112.2.a.b 1
84.n even 6 1 784.2.i.e 2
105.o odd 6 1 1400.2.a.g 1
105.p even 6 1 9800.2.a.u 1
105.x even 12 2 1400.2.g.g 2
168.s odd 6 1 448.2.a.d 1
168.v even 6 1 448.2.a.e 1
168.ba even 6 1 3136.2.a.q 1
168.be odd 6 1 3136.2.a.p 1
231.l even 6 1 6776.2.a.g 1
273.w odd 6 1 9464.2.a.c 1
336.bt odd 12 2 1792.2.b.i 2
336.bu even 12 2 1792.2.b.d 2
420.ba even 6 1 2800.2.a.p 1
420.bp odd 12 2 2800.2.g.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 21.h odd 6 1
112.2.a.b 1 84.n even 6 1
392.2.a.d 1 21.g even 6 1
392.2.i.c 2 3.b odd 2 1
392.2.i.c 2 21.h odd 6 1
392.2.i.d 2 21.c even 2 1
392.2.i.d 2 21.g even 6 1
448.2.a.d 1 168.s odd 6 1
448.2.a.e 1 168.v even 6 1
504.2.a.c 1 7.c even 3 1
784.2.a.e 1 84.j odd 6 1
784.2.i.e 2 12.b even 2 1
784.2.i.e 2 84.n even 6 1
784.2.i.g 2 84.h odd 2 1
784.2.i.g 2 84.j odd 6 1
1008.2.a.d 1 28.g odd 6 1
1400.2.a.g 1 105.o odd 6 1
1400.2.g.g 2 105.x even 12 2
1792.2.b.d 2 336.bu even 12 2
1792.2.b.i 2 336.bt odd 12 2
2800.2.a.p 1 420.ba even 6 1
2800.2.g.p 2 420.bp odd 12 2
3136.2.a.p 1 168.be odd 6 1
3136.2.a.q 1 168.ba even 6 1
3528.2.a.x 1 7.d odd 6 1
3528.2.s.e 2 7.b odd 2 1
3528.2.s.e 2 7.d odd 6 1
3528.2.s.t 2 1.a even 1 1 trivial
3528.2.s.t 2 7.c even 3 1 inner
4032.2.a.bb 1 56.p even 6 1
4032.2.a.bk 1 56.k odd 6 1
6776.2.a.g 1 231.l even 6 1
7056.2.a.bo 1 28.f even 6 1
9464.2.a.c 1 273.w odd 6 1
9800.2.a.u 1 105.p even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 64 + 8 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 36 - 6 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 100 + 10 T + T^{2} \)
$79$ \( 256 + 16 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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