# Properties

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

# Related objects

## 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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.x 2
3.b odd 2 1 3528.2.s.f 2
7.b odd 2 1 3528.2.s.i 2
7.c even 3 1 504.2.a.a 1
7.c even 3 1 inner 3528.2.s.x 2
7.d odd 6 1 3528.2.a.u 1
7.d odd 6 1 3528.2.s.i 2
21.c even 2 1 3528.2.s.u 2
21.g even 6 1 3528.2.a.e 1
21.g even 6 1 3528.2.s.u 2
21.h odd 6 1 504.2.a.f yes 1
21.h odd 6 1 3528.2.s.f 2
28.f even 6 1 7056.2.a.bt 1
28.g odd 6 1 1008.2.a.f 1
56.k odd 6 1 4032.2.a.bg 1
56.p even 6 1 4032.2.a.bf 1
84.j odd 6 1 7056.2.a.l 1
84.n even 6 1 1008.2.a.k 1
168.s odd 6 1 4032.2.a.g 1
168.v even 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.c even 3 1
504.2.a.f yes 1 21.h odd 6 1
1008.2.a.f 1 28.g odd 6 1
1008.2.a.k 1 84.n even 6 1
3528.2.a.e 1 21.g even 6 1
3528.2.a.u 1 7.d odd 6 1
3528.2.s.f 2 3.b odd 2 1
3528.2.s.f 2 21.h odd 6 1
3528.2.s.i 2 7.b odd 2 1
3528.2.s.i 2 7.d odd 6 1
3528.2.s.u 2 21.c even 2 1
3528.2.s.u 2 21.g even 6 1
3528.2.s.x 2 1.a even 1 1 trivial
3528.2.s.x 2 7.c even 3 1 inner
4032.2.a.g 1 168.s odd 6 1
4032.2.a.l 1 168.v even 6 1
4032.2.a.bf 1 56.p even 6 1
4032.2.a.bg 1 56.k odd 6 1
7056.2.a.l 1 84.j odd 6 1
7056.2.a.bt 1 28.f even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} - 2$$ $$T_{23}^{2} - 6 T_{23} + 36$$

## Hecke characteristic polynomials

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