# Properties

 Label 3528.2.a.bb Level $3528$ Weight $2$ Character orbit 3528.a Self dual yes Analytic conductor $28.171$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: no (minimal twist has level 1176) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta ) q^{5} +O(q^{10})$$ $$q + ( -2 + \beta ) q^{5} + ( -2 + 2 \beta ) q^{11} -\beta q^{13} + ( 2 - 3 \beta ) q^{17} + ( 4 + 2 \beta ) q^{19} + ( -2 - 2 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} -6 \beta q^{29} + ( 8 - 2 \beta ) q^{31} + ( -4 + 4 \beta ) q^{37} + ( 2 - \beta ) q^{41} -8 q^{43} + ( -4 - 2 \beta ) q^{47} + ( 2 + 8 \beta ) q^{53} + ( 8 - 6 \beta ) q^{55} + ( -8 + 2 \beta ) q^{59} + ( 4 + 7 \beta ) q^{61} + ( -2 + 2 \beta ) q^{65} -8 q^{67} + ( -2 + 2 \beta ) q^{71} + ( -4 - 5 \beta ) q^{73} + ( -8 + 4 \beta ) q^{79} + ( -4 - 8 \beta ) q^{83} + ( -10 + 8 \beta ) q^{85} + ( -2 + 9 \beta ) q^{89} -4 q^{95} + ( -12 + 3 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + O(q^{10})$$ $$2q - 4q^{5} - 4q^{11} + 4q^{17} + 8q^{19} - 4q^{23} + 2q^{25} + 16q^{31} - 8q^{37} + 4q^{41} - 16q^{43} - 8q^{47} + 4q^{53} + 16q^{55} - 16q^{59} + 8q^{61} - 4q^{65} - 16q^{67} - 4q^{71} - 8q^{73} - 16q^{79} - 8q^{83} - 20q^{85} - 4q^{89} - 8q^{95} - 24q^{97} + O(q^{100})$$

## 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 −3.41421 0 0 0 0 0
1.2 0 0 0 −0.585786 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.a.bb 2
3.b odd 2 1 1176.2.a.o yes 2
4.b odd 2 1 7056.2.a.cg 2
7.b odd 2 1 3528.2.a.bl 2
7.c even 3 2 3528.2.s.bm 4
7.d odd 6 2 3528.2.s.bd 4
12.b even 2 1 2352.2.a.bb 2
21.c even 2 1 1176.2.a.j 2
21.g even 6 2 1176.2.q.o 4
21.h odd 6 2 1176.2.q.k 4
24.f even 2 1 9408.2.a.du 2
24.h odd 2 1 9408.2.a.dg 2
28.d even 2 1 7056.2.a.cx 2
84.h odd 2 1 2352.2.a.bd 2
84.j odd 6 2 2352.2.q.bc 4
84.n even 6 2 2352.2.q.be 4
168.e odd 2 1 9408.2.a.ds 2
168.i even 2 1 9408.2.a.ee 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 21.c even 2 1
1176.2.a.o yes 2 3.b odd 2 1
1176.2.q.k 4 21.h odd 6 2
1176.2.q.o 4 21.g even 6 2
2352.2.a.bb 2 12.b even 2 1
2352.2.a.bd 2 84.h odd 2 1
2352.2.q.bc 4 84.j odd 6 2
2352.2.q.be 4 84.n even 6 2
3528.2.a.bb 2 1.a even 1 1 trivial
3528.2.a.bl 2 7.b odd 2 1
3528.2.s.bd 4 7.d odd 6 2
3528.2.s.bm 4 7.c even 3 2
7056.2.a.cg 2 4.b odd 2 1
7056.2.a.cx 2 28.d even 2 1
9408.2.a.dg 2 24.h odd 2 1
9408.2.a.ds 2 168.e odd 2 1
9408.2.a.du 2 24.f even 2 1
9408.2.a.ee 2 168.i even 2 1

## Hecke kernels

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

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