# Properties

 Label 3528.2.s.l 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -5 + 5 \zeta_{6} ) q^{11} -2 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} -3 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{37} -8 q^{41} -4 q^{43} + 4 \zeta_{6} q^{47} + ( -9 + 9 \zeta_{6} ) q^{53} + 5 q^{55} + ( -3 + 3 \zeta_{6} ) q^{59} -12 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + ( -2 + 2 \zeta_{6} ) q^{67} -8 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} -\zeta_{6} q^{79} -17 q^{83} -6 q^{85} + 18 \zeta_{6} q^{89} + ( 2 - 2 \zeta_{6} ) q^{95} -3 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + O(q^{10})$$ $$2q - q^{5} - 5q^{11} - 4q^{13} + 6q^{17} + 2q^{19} + 6q^{23} + 4q^{25} - 6q^{29} + 5q^{31} + 2q^{37} - 16q^{41} - 8q^{43} + 4q^{47} - 9q^{53} + 10q^{55} - 3q^{59} - 12q^{61} + 2q^{65} - 2q^{67} - 16q^{71} - 14q^{73} - q^{79} - 34q^{83} - 12q^{85} + 18q^{89} + 2q^{95} - 6q^{97} + O(q^{100})$$

## 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 −0.500000 0.866025i 0 0 0 0 0
3313.1 0 0 0 −0.500000 + 0.866025i 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.l 2
3.b odd 2 1 3528.2.s.r 2
7.b odd 2 1 504.2.s.f yes 2
7.c even 3 1 3528.2.a.s 1
7.c even 3 1 inner 3528.2.s.l 2
7.d odd 6 1 504.2.s.f yes 2
7.d odd 6 1 3528.2.a.l 1
21.c even 2 1 504.2.s.b 2
21.g even 6 1 504.2.s.b 2
21.g even 6 1 3528.2.a.o 1
21.h odd 6 1 3528.2.a.h 1
21.h odd 6 1 3528.2.s.r 2
28.d even 2 1 1008.2.s.l 2
28.f even 6 1 1008.2.s.l 2
28.f even 6 1 7056.2.a.r 1
28.g odd 6 1 7056.2.a.bh 1
84.h odd 2 1 1008.2.s.h 2
84.j odd 6 1 1008.2.s.h 2
84.j odd 6 1 7056.2.a.bm 1
84.n even 6 1 7056.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.b 2 21.c even 2 1
504.2.s.b 2 21.g even 6 1
504.2.s.f yes 2 7.b odd 2 1
504.2.s.f yes 2 7.d odd 6 1
1008.2.s.h 2 84.h odd 2 1
1008.2.s.h 2 84.j odd 6 1
1008.2.s.l 2 28.d even 2 1
1008.2.s.l 2 28.f even 6 1
3528.2.a.h 1 21.h odd 6 1
3528.2.a.l 1 7.d odd 6 1
3528.2.a.o 1 21.g even 6 1
3528.2.a.s 1 7.c even 3 1
3528.2.s.l 2 1.a even 1 1 trivial
3528.2.s.l 2 7.c even 3 1 inner
3528.2.s.r 2 3.b odd 2 1
3528.2.s.r 2 21.h odd 6 1
7056.2.a.r 1 28.f even 6 1
7056.2.a.v 1 84.n even 6 1
7056.2.a.bh 1 28.g odd 6 1
7056.2.a.bm 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13} + 2$$ $$T_{23}^{2} - 6 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$16 - 4 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$144 + 12 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$( 17 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$( 3 + T )^{2}$$
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