# Properties

 Label 3528.2.a.bg Level $3528$ Weight $2$ Character orbit 3528.a Self dual yes Analytic conductor $28.171$ Analytic rank $1$ 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1712218331$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} -\beta q^{13} -\beta q^{17} -4 q^{23} -3 q^{25} -6 q^{29} -4 \beta q^{31} -4 q^{37} -5 \beta q^{41} + 4 q^{43} + 8 \beta q^{47} -4 q^{53} -4 \beta q^{59} -3 \beta q^{61} -2 q^{65} + 4 q^{67} -12 q^{71} + 5 \beta q^{73} + 8 q^{79} -4 \beta q^{83} -2 q^{85} + 9 \beta q^{89} -3 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 8q^{23} - 6q^{25} - 12q^{29} - 8q^{37} + 8q^{43} - 8q^{53} - 4q^{65} + 8q^{67} - 24q^{71} + 16q^{79} - 4q^{85} + O(q^{100})$$

## 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 −1.41421 0 0 0 0 0
1.2 0 0 0 1.41421 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.bg 2
3.b odd 2 1 3528.2.a.bh yes 2
4.b odd 2 1 7056.2.a.cp 2
7.b odd 2 1 inner 3528.2.a.bg 2
7.c even 3 2 3528.2.s.bh 4
7.d odd 6 2 3528.2.s.bh 4
12.b even 2 1 7056.2.a.cn 2
21.c even 2 1 3528.2.a.bh yes 2
21.g even 6 2 3528.2.s.bg 4
21.h odd 6 2 3528.2.s.bg 4
28.d even 2 1 7056.2.a.cp 2
84.h odd 2 1 7056.2.a.cn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.2.a.bg 2 1.a even 1 1 trivial
3528.2.a.bg 2 7.b odd 2 1 inner
3528.2.a.bh yes 2 3.b odd 2 1
3528.2.a.bh yes 2 21.c even 2 1
3528.2.s.bg 4 21.g even 6 2
3528.2.s.bg 4 21.h odd 6 2
3528.2.s.bh 4 7.c even 3 2
3528.2.s.bh 4 7.d odd 6 2
7056.2.a.cn 2 12.b even 2 1
7056.2.a.cn 2 84.h odd 2 1
7056.2.a.cp 2 4.b odd 2 1
7056.2.a.cp 2 28.d even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 + 24 T^{2} + 169 T^{4}$$
$17$ $$1 + 32 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 30 T^{2} + 961 T^{4}$$
$37$ $$( 1 + 4 T + 37 T^{2} )^{2}$$
$41$ $$1 + 32 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 34 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 + 86 T^{2} + 3481 T^{4}$$
$61$ $$1 + 104 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 + 96 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 134 T^{2} + 6889 T^{4}$$
$89$ $$1 + 16 T^{2} + 7921 T^{4}$$
$97$ $$1 + 176 T^{2} + 9409 T^{4}$$