# Properties

 Label 3528.2.s.bd Level $3528$ Weight $2$ Character orbit 3528.s Analytic conductor $28.171$ Analytic rank $0$ Dimension $4$ 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.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: $$1$$ Twist minimal: no (minimal twist has level 1176) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} +O(q^{10})$$ $$q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{11} -\beta_{3} q^{13} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{25} + 6 \beta_{3} q^{29} + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{31} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{37} + ( -2 - \beta_{3} ) q^{41} -8 q^{43} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -2 - 8 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -8 - 6 \beta_{3} ) q^{55} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{59} + ( -7 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 8 + 8 \beta_{2} ) q^{67} + ( -2 - 2 \beta_{3} ) q^{71} + ( -4 - 5 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 4 - 8 \beta_{3} ) q^{83} + ( -10 - 8 \beta_{3} ) q^{85} + ( -9 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} ) q^{89} + ( 4 + 4 \beta_{2} ) q^{95} + ( 12 + 3 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + O(q^{10})$$ $$4q - 4q^{5} + 4q^{11} + 4q^{17} + 8q^{19} + 4q^{23} - 2q^{25} + 16q^{31} + 8q^{37} - 8q^{41} - 32q^{43} - 8q^{47} - 4q^{53} - 32q^{55} - 16q^{59} + 8q^{61} + 4q^{65} + 16q^{67} - 8q^{71} - 8q^{73} + 16q^{79} + 16q^{83} - 40q^{85} - 4q^{89} + 8q^{95} + 48q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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 −1.70711 2.95680i 0 0 0 0 0
361.2 0 0 0 −0.292893 0.507306i 0 0 0 0 0
3313.1 0 0 0 −1.70711 + 2.95680i 0 0 0 0 0
3313.2 0 0 0 −0.292893 + 0.507306i 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.bd 4
3.b odd 2 1 1176.2.q.o 4
7.b odd 2 1 3528.2.s.bm 4
7.c even 3 1 3528.2.a.bl 2
7.c even 3 1 inner 3528.2.s.bd 4
7.d odd 6 1 3528.2.a.bb 2
7.d odd 6 1 3528.2.s.bm 4
12.b even 2 1 2352.2.q.bc 4
21.c even 2 1 1176.2.q.k 4
21.g even 6 1 1176.2.a.o yes 2
21.g even 6 1 1176.2.q.k 4
21.h odd 6 1 1176.2.a.j 2
21.h odd 6 1 1176.2.q.o 4
28.f even 6 1 7056.2.a.cg 2
28.g odd 6 1 7056.2.a.cx 2
84.h odd 2 1 2352.2.q.be 4
84.j odd 6 1 2352.2.a.bb 2
84.j odd 6 1 2352.2.q.be 4
84.n even 6 1 2352.2.a.bd 2
84.n even 6 1 2352.2.q.bc 4
168.s odd 6 1 9408.2.a.ee 2
168.v even 6 1 9408.2.a.ds 2
168.ba even 6 1 9408.2.a.dg 2
168.be odd 6 1 9408.2.a.du 2

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

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{3} + 14 T_{5}^{2} + 8 T_{5} + 4$$ $$T_{11}^{4} - 4 T_{11}^{3} + 20 T_{11}^{2} + 16 T_{11} + 16$$ $$T_{13}^{2} - 2$$ $$T_{23}^{4} - 4 T_{23}^{3} + 20 T_{23}^{2} + 16 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$23$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -72 + T^{2} )^{2}$$
$31$ $$3136 - 896 T + 200 T^{2} - 16 T^{3} + T^{4}$$
$37$ $$256 + 128 T + 80 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$( 2 + 4 T + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4}$$
$61$ $$6724 + 656 T + 146 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$( 64 - 8 T + T^{2} )^{2}$$
$71$ $$( -4 + 4 T + T^{2} )^{2}$$
$73$ $$1156 - 272 T + 98 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$1024 - 512 T + 224 T^{2} - 16 T^{3} + T^{4}$$
$83$ $$( -112 - 8 T + T^{2} )^{2}$$
$89$ $$24964 - 632 T + 174 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$( 126 - 24 T + T^{2} )^{2}$$