# Properties

 Label 3528.2.s.g Level $3528$ Weight $2$ Character orbit 3528.s Analytic conductor $28.171$ Analytic rank $0$ 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.s (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.1712218331$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{5} +O(q^{10})$$ $$q -2 \zeta_{6} q^{5} -6 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 10 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{37} -2 q^{41} -4 q^{43} -8 \zeta_{6} q^{47} + ( -10 + 10 \zeta_{6} ) q^{53} + ( -12 + 12 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + 12 \zeta_{6} q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} + 12 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{79} + 12 q^{83} -4 q^{85} + 2 \zeta_{6} q^{89} + ( 8 - 8 \zeta_{6} ) q^{95} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + O(q^{10})$$ $$2q - 2q^{5} - 12q^{13} + 2q^{17} + 4q^{19} - 4q^{23} + q^{25} + 20q^{29} - 8q^{31} - 6q^{37} - 4q^{41} - 8q^{43} - 8q^{47} - 10q^{53} - 12q^{59} - 2q^{61} + 12q^{65} - 12q^{67} + 24q^{71} - 14q^{73} + 8q^{79} + 24q^{83} - 8q^{85} + 2q^{89} + 8q^{95} - 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 1.73205i 0 0 0 0 0
3313.1 0 0 0 −1.00000 + 1.73205i 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.g 2
3.b odd 2 1 1176.2.q.d 2
7.b odd 2 1 3528.2.s.w 2
7.c even 3 1 3528.2.a.v 1
7.c even 3 1 inner 3528.2.s.g 2
7.d odd 6 1 504.2.a.e 1
7.d odd 6 1 3528.2.s.w 2
12.b even 2 1 2352.2.q.w 2
21.c even 2 1 1176.2.q.f 2
21.g even 6 1 168.2.a.a 1
21.g even 6 1 1176.2.q.f 2
21.h odd 6 1 1176.2.a.f 1
21.h odd 6 1 1176.2.q.d 2
28.f even 6 1 1008.2.a.b 1
28.g odd 6 1 7056.2.a.bq 1
56.j odd 6 1 4032.2.a.bh 1
56.m even 6 1 4032.2.a.bc 1
84.h odd 2 1 2352.2.q.d 2
84.j odd 6 1 336.2.a.e 1
84.j odd 6 1 2352.2.q.d 2
84.n even 6 1 2352.2.a.c 1
84.n even 6 1 2352.2.q.w 2
105.p even 6 1 4200.2.a.t 1
105.w odd 12 2 4200.2.t.j 2
168.s odd 6 1 9408.2.a.be 1
168.v even 6 1 9408.2.a.da 1
168.ba even 6 1 1344.2.a.m 1
168.be odd 6 1 1344.2.a.b 1
336.bo even 12 2 5376.2.c.d 2
336.br odd 12 2 5376.2.c.bb 2
420.be odd 6 1 8400.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 21.g even 6 1
336.2.a.e 1 84.j odd 6 1
504.2.a.e 1 7.d odd 6 1
1008.2.a.b 1 28.f even 6 1
1176.2.a.f 1 21.h odd 6 1
1176.2.q.d 2 3.b odd 2 1
1176.2.q.d 2 21.h odd 6 1
1176.2.q.f 2 21.c even 2 1
1176.2.q.f 2 21.g even 6 1
1344.2.a.b 1 168.be odd 6 1
1344.2.a.m 1 168.ba even 6 1
2352.2.a.c 1 84.n even 6 1
2352.2.q.d 2 84.h odd 2 1
2352.2.q.d 2 84.j odd 6 1
2352.2.q.w 2 12.b even 2 1
2352.2.q.w 2 84.n even 6 1
3528.2.a.v 1 7.c even 3 1
3528.2.s.g 2 1.a even 1 1 trivial
3528.2.s.g 2 7.c even 3 1 inner
3528.2.s.w 2 7.b odd 2 1
3528.2.s.w 2 7.d odd 6 1
4032.2.a.bc 1 56.m even 6 1
4032.2.a.bh 1 56.j odd 6 1
4200.2.a.t 1 105.p even 6 1
4200.2.t.j 2 105.w odd 12 2
5376.2.c.d 2 336.bo even 12 2
5376.2.c.bb 2 336.br odd 12 2
7056.2.a.bq 1 28.g odd 6 1
8400.2.a.y 1 420.be odd 6 1
9408.2.a.be 1 168.s odd 6 1
9408.2.a.da 1 168.v even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}$$ $$T_{13} + 6$$ $$T_{23}^{2} + 4 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -10 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$64 + 8 T + T^{2}$$
$53$ $$100 + 10 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$4 - 2 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$