# Properties

 Label 3528.2.a.a Level $3528$ Weight $2$ Character orbit 3528.a Self dual yes Analytic conductor $28.171$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{5} + O(q^{10})$$ $$q - 4q^{5} - 3q^{13} + 4q^{17} + 7q^{19} - 4q^{23} + 11q^{25} + 8q^{29} - 5q^{31} + 3q^{37} - 8q^{41} + 11q^{43} - 4q^{47} - 4q^{53} - 12q^{59} - 2q^{61} + 12q^{65} - 3q^{67} - 12q^{71} + q^{73} + q^{79} - 12q^{83} - 16q^{85} - 8q^{89} - 28q^{95} - 2q^{97} + O(q^{100})$$

## 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
 0
0 0 0 −4.00000 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.a 1
3.b odd 2 1 3528.2.a.z 1
4.b odd 2 1 7056.2.a.b 1
7.b odd 2 1 3528.2.a.ba 1
7.c even 3 2 504.2.s.h yes 2
7.d odd 6 2 3528.2.s.b 2
12.b even 2 1 7056.2.a.cb 1
21.c even 2 1 3528.2.a.c 1
21.g even 6 2 3528.2.s.bb 2
21.h odd 6 2 504.2.s.a 2
28.d even 2 1 7056.2.a.cc 1
28.g odd 6 2 1008.2.s.q 2
84.h odd 2 1 7056.2.a.d 1
84.n even 6 2 1008.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.a 2 21.h odd 6 2
504.2.s.h yes 2 7.c even 3 2
1008.2.s.a 2 84.n even 6 2
1008.2.s.q 2 28.g odd 6 2
3528.2.a.a 1 1.a even 1 1 trivial
3528.2.a.c 1 21.c even 2 1
3528.2.a.z 1 3.b odd 2 1
3528.2.a.ba 1 7.b odd 2 1
3528.2.s.b 2 7.d odd 6 2
3528.2.s.bb 2 21.g even 6 2
7056.2.a.b 1 4.b odd 2 1
7056.2.a.d 1 84.h odd 2 1
7056.2.a.cb 1 12.b even 2 1
7056.2.a.cc 1 28.d even 2 1

## Hecke kernels

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

 $$T_{5} + 4$$ $$T_{11}$$ $$T_{13} + 3$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T + 5 T^{2}$$
$7$ 1
$11$ $$1 + 11 T^{2}$$
$13$ $$1 + 3 T + 13 T^{2}$$
$17$ $$1 - 4 T + 17 T^{2}$$
$19$ $$1 - 7 T + 19 T^{2}$$
$23$ $$1 + 4 T + 23 T^{2}$$
$29$ $$1 - 8 T + 29 T^{2}$$
$31$ $$1 + 5 T + 31 T^{2}$$
$37$ $$1 - 3 T + 37 T^{2}$$
$41$ $$1 + 8 T + 41 T^{2}$$
$43$ $$1 - 11 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 + 4 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 + 2 T + 61 T^{2}$$
$67$ $$1 + 3 T + 67 T^{2}$$
$71$ $$1 + 12 T + 71 T^{2}$$
$73$ $$1 - T + 73 T^{2}$$
$79$ $$1 - T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 8 T + 89 T^{2}$$
$97$ $$1 + 2 T + 97 T^{2}$$