# Properties

 Label 3528.2.s.bj 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: $$1$$ Twist minimal: no (minimal twist has level 392) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{1} q^{5} +O(q^{10})$$ $$q + 2 \beta_{1} q^{5} -6 \beta_{2} q^{11} + 4 \beta_{3} q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + 3 \beta_{1} q^{19} + ( 4 + 4 \beta_{2} ) q^{23} + 3 \beta_{2} q^{25} + 6 q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( -2 - 2 \beta_{2} ) q^{37} -\beta_{3} q^{41} + 10 q^{43} -2 \beta_{1} q^{47} + 2 \beta_{2} q^{53} -12 \beta_{3} q^{55} + ( -\beta_{1} - \beta_{3} ) q^{59} + 6 \beta_{1} q^{61} + ( -16 - 16 \beta_{2} ) q^{65} + 4 \beta_{2} q^{67} + 12 q^{71} + ( -7 \beta_{1} - 7 \beta_{3} ) q^{73} + ( 4 + 4 \beta_{2} ) q^{79} + \beta_{3} q^{83} + 4 q^{85} -3 \beta_{1} q^{89} + 12 \beta_{2} q^{95} -9 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 12q^{11} + 8q^{23} - 6q^{25} + 24q^{29} - 4q^{37} + 40q^{43} - 4q^{53} - 32q^{65} - 8q^{67} + 48q^{71} + 8q^{79} + 16q^{85} - 24q^{95} + O(q^{100})$$

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

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

## Character values

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

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

## 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.bj 4
3.b odd 2 1 392.2.i.h 4
7.b odd 2 1 inner 3528.2.s.bj 4
7.c even 3 1 3528.2.a.be 2
7.c even 3 1 inner 3528.2.s.bj 4
7.d odd 6 1 3528.2.a.be 2
7.d odd 6 1 inner 3528.2.s.bj 4
12.b even 2 1 784.2.i.n 4
21.c even 2 1 392.2.i.h 4
21.g even 6 1 392.2.a.g 2
21.g even 6 1 392.2.i.h 4
21.h odd 6 1 392.2.a.g 2
21.h odd 6 1 392.2.i.h 4
28.f even 6 1 7056.2.a.ct 2
28.g odd 6 1 7056.2.a.ct 2
84.h odd 2 1 784.2.i.n 4
84.j odd 6 1 784.2.a.k 2
84.j odd 6 1 784.2.i.n 4
84.n even 6 1 784.2.a.k 2
84.n even 6 1 784.2.i.n 4
105.o odd 6 1 9800.2.a.bv 2
105.p even 6 1 9800.2.a.bv 2
168.s odd 6 1 3136.2.a.bk 2
168.v even 6 1 3136.2.a.bp 2
168.ba even 6 1 3136.2.a.bk 2
168.be odd 6 1 3136.2.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 21.g even 6 1
392.2.a.g 2 21.h odd 6 1
392.2.i.h 4 3.b odd 2 1
392.2.i.h 4 21.c even 2 1
392.2.i.h 4 21.g even 6 1
392.2.i.h 4 21.h odd 6 1
784.2.a.k 2 84.j odd 6 1
784.2.a.k 2 84.n even 6 1
784.2.i.n 4 12.b even 2 1
784.2.i.n 4 84.h odd 2 1
784.2.i.n 4 84.j odd 6 1
784.2.i.n 4 84.n even 6 1
3136.2.a.bk 2 168.s odd 6 1
3136.2.a.bk 2 168.ba even 6 1
3136.2.a.bp 2 168.v even 6 1
3136.2.a.bp 2 168.be odd 6 1
3528.2.a.be 2 7.c even 3 1
3528.2.a.be 2 7.d odd 6 1
3528.2.s.bj 4 1.a even 1 1 trivial
3528.2.s.bj 4 7.b odd 2 1 inner
3528.2.s.bj 4 7.c even 3 1 inner
3528.2.s.bj 4 7.d odd 6 1 inner
7056.2.a.ct 2 28.f even 6 1
7056.2.a.ct 2 28.g odd 6 1
9800.2.a.bv 2 105.o odd 6 1
9800.2.a.bv 2 105.p even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 8 T_{5}^{2} + 64$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13}^{2} - 32$$ $$T_{23}^{2} - 4 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )$$
$7$ 1
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8}$$
$19$ $$1 - 20 T^{2} + 39 T^{4} - 7220 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{4}$$
$31$ $$1 - 54 T^{2} + 1955 T^{4} - 51894 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 80 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{4}$$
$47$ $$1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 50 T^{2} - 1221 T^{4} - 186050 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{4}$$
$73$ $$1 - 48 T^{2} - 3025 T^{4} - 255792 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 164 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 160 T^{2} + 17679 T^{4} - 1267360 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 32 T^{2} + 9409 T^{4} )^{2}$$