# Properties

 Label 3528.2.s.bf Level $3528$ Weight $2$ Character orbit 3528.s Analytic conductor $28.171$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} +O(q^{10})$$ $$q + \beta_{1} q^{5} + 2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + 2 \beta_{1} q^{19} + ( -6 - 6 \beta_{2} ) q^{23} + 3 \beta_{2} q^{25} -4 q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 2 + 2 \beta_{2} ) q^{37} -\beta_{3} q^{41} -4 q^{43} -4 \beta_{1} q^{47} + 12 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{59} + 2 \beta_{1} q^{61} + ( -16 - 16 \beta_{2} ) q^{65} -12 \beta_{2} q^{67} -6 q^{71} + ( -8 - 8 \beta_{2} ) q^{79} + 4 \beta_{3} q^{83} + 8 q^{85} + 3 \beta_{1} q^{89} + 16 \beta_{2} q^{95} -4 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{11} - 12q^{23} - 6q^{25} - 16q^{29} + 4q^{37} - 16q^{43} - 24q^{53} - 32q^{65} + 24q^{67} - 24q^{71} - 16q^{79} + 32q^{85} - 32q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 0 0 −1.41421 2.44949i 0 0 0 0 0
361.2 0 0 0 1.41421 + 2.44949i 0 0 0 0 0
3313.1 0 0 0 −1.41421 + 2.44949i 0 0 0 0 0
3313.2 0 0 0 1.41421 2.44949i 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.s.bf 4
3.b odd 2 1 3528.2.s.bi 4
7.b odd 2 1 inner 3528.2.s.bf 4
7.c even 3 1 3528.2.a.bi yes 2
7.c even 3 1 inner 3528.2.s.bf 4
7.d odd 6 1 3528.2.a.bi yes 2
7.d odd 6 1 inner 3528.2.s.bf 4
21.c even 2 1 3528.2.s.bi 4
21.g even 6 1 3528.2.a.bf 2
21.g even 6 1 3528.2.s.bi 4
21.h odd 6 1 3528.2.a.bf 2
21.h odd 6 1 3528.2.s.bi 4
28.f even 6 1 7056.2.a.ck 2
28.g odd 6 1 7056.2.a.ck 2
84.j odd 6 1 7056.2.a.cq 2
84.n even 6 1 7056.2.a.cq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.2.a.bf 2 21.g even 6 1
3528.2.a.bf 2 21.h odd 6 1
3528.2.a.bi yes 2 7.c even 3 1
3528.2.a.bi yes 2 7.d odd 6 1
3528.2.s.bf 4 1.a even 1 1 trivial
3528.2.s.bf 4 7.b odd 2 1 inner
3528.2.s.bf 4 7.c even 3 1 inner
3528.2.s.bf 4 7.d odd 6 1 inner
3528.2.s.bi 4 3.b odd 2 1
3528.2.s.bi 4 21.c even 2 1
3528.2.s.bi 4 21.g even 6 1
3528.2.s.bi 4 21.h odd 6 1
7056.2.a.ck 2 28.f even 6 1
7056.2.a.ck 2 28.g odd 6 1
7056.2.a.cq 2 84.j odd 6 1
7056.2.a.cq 2 84.n even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 8 T_{5}^{2} + 64$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13}^{2} - 32$$ $$T_{23}^{2} + 6 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$64 + 8 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$( -32 + T^{2} )^{2}$$
$17$ $$64 + 8 T^{2} + T^{4}$$
$19$ $$1024 + 32 T^{2} + T^{4}$$
$23$ $$( 36 + 6 T + T^{2} )^{2}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$1024 + 32 T^{2} + T^{4}$$
$37$ $$( 4 - 2 T + T^{2} )^{2}$$
$41$ $$( -8 + T^{2} )^{2}$$
$43$ $$( 4 + T )^{4}$$
$47$ $$16384 + 128 T^{2} + T^{4}$$
$53$ $$( 144 + 12 T + T^{2} )^{2}$$
$59$ $$16384 + 128 T^{2} + T^{4}$$
$61$ $$1024 + 32 T^{2} + T^{4}$$
$67$ $$( 144 - 12 T + T^{2} )^{2}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$T^{4}$$
$79$ $$( 64 + 8 T + T^{2} )^{2}$$
$83$ $$( -128 + T^{2} )^{2}$$
$89$ $$5184 + 72 T^{2} + T^{4}$$
$97$ $$( -128 + T^{2} )^{2}$$