# Properties

 Label 3528.2.a.bj Level $3528$ Weight $2$ Character orbit 3528.a Self dual yes 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1712218331$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 392) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} +O(q^{10})$$ $$q + 2 \beta q^{5} + 4 q^{11} + 2 \beta q^{13} -4 \beta q^{17} + 2 \beta q^{19} + 3 q^{25} -2 q^{29} + 4 \beta q^{31} + 10 q^{37} + 4 \beta q^{41} -4 q^{43} + 4 \beta q^{47} -6 q^{53} + 8 \beta q^{55} -2 \beta q^{59} -10 \beta q^{61} + 8 q^{65} + 12 q^{67} + 8 q^{79} -10 \beta q^{83} -16 q^{85} + 8 q^{95} -4 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 8q^{11} + 6q^{25} - 4q^{29} + 20q^{37} - 8q^{43} - 12q^{53} + 16q^{65} + 24q^{67} + 16q^{79} - 32q^{85} + 16q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 0 0 −2.82843 0 0 0 0 0
1.2 0 0 0 2.82843 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.a.bj 2
3.b odd 2 1 392.2.a.h 2
4.b odd 2 1 7056.2.a.cj 2
7.b odd 2 1 inner 3528.2.a.bj 2
7.c even 3 2 3528.2.s.be 4
7.d odd 6 2 3528.2.s.be 4
12.b even 2 1 784.2.a.n 2
15.d odd 2 1 9800.2.a.bw 2
21.c even 2 1 392.2.a.h 2
21.g even 6 2 392.2.i.g 4
21.h odd 6 2 392.2.i.g 4
24.f even 2 1 3136.2.a.bq 2
24.h odd 2 1 3136.2.a.bt 2
28.d even 2 1 7056.2.a.cj 2
84.h odd 2 1 784.2.a.n 2
84.j odd 6 2 784.2.i.k 4
84.n even 6 2 784.2.i.k 4
105.g even 2 1 9800.2.a.bw 2
168.e odd 2 1 3136.2.a.bq 2
168.i even 2 1 3136.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h 2 3.b odd 2 1
392.2.a.h 2 21.c even 2 1
392.2.i.g 4 21.g even 6 2
392.2.i.g 4 21.h odd 6 2
784.2.a.n 2 12.b even 2 1
784.2.a.n 2 84.h odd 2 1
784.2.i.k 4 84.j odd 6 2
784.2.i.k 4 84.n even 6 2
3136.2.a.bq 2 24.f even 2 1
3136.2.a.bq 2 168.e odd 2 1
3136.2.a.bt 2 24.h odd 2 1
3136.2.a.bt 2 168.i even 2 1
3528.2.a.bj 2 1.a even 1 1 trivial
3528.2.a.bj 2 7.b odd 2 1 inner
3528.2.s.be 4 7.c even 3 2
3528.2.s.be 4 7.d odd 6 2
7056.2.a.cj 2 4.b odd 2 1
7056.2.a.cj 2 28.d even 2 1
9800.2.a.bw 2 15.d odd 2 1
9800.2.a.bw 2 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3528))$$:

 $$T_{5}^{2} - 8$$ $$T_{11} - 4$$ $$T_{13}^{2} - 8$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$1 + 18 T^{2} + 169 T^{4}$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$1 + 30 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 10 T + 37 T^{2} )^{2}$$
$41$ $$1 + 50 T^{2} + 1681 T^{4}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 62 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$1 + 110 T^{2} + 3481 T^{4}$$
$61$ $$1 - 78 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 12 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 34 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 + 162 T^{2} + 9409 T^{4}$$
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