# Properties

 Label 28.4.a.a Level $28$ Weight $4$ Character orbit 28.a Self dual yes Analytic conductor $1.652$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 28.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.65205348016$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 10q^{3} - 8q^{5} - 7q^{7} + 73q^{9} + O(q^{10})$$ $$q - 10q^{3} - 8q^{5} - 7q^{7} + 73q^{9} - 40q^{11} - 12q^{13} + 80q^{15} - 58q^{17} + 26q^{19} + 70q^{21} - 64q^{23} - 61q^{25} - 460q^{27} - 62q^{29} + 252q^{31} + 400q^{33} + 56q^{35} + 26q^{37} + 120q^{39} + 6q^{41} + 416q^{43} - 584q^{45} - 396q^{47} + 49q^{49} + 580q^{51} - 450q^{53} + 320q^{55} - 260q^{57} + 274q^{59} - 576q^{61} - 511q^{63} + 96q^{65} - 476q^{67} + 640q^{69} - 448q^{71} - 158q^{73} + 610q^{75} + 280q^{77} - 936q^{79} + 2629q^{81} + 530q^{83} + 464q^{85} + 620q^{87} - 390q^{89} + 84q^{91} - 2520q^{93} - 208q^{95} + 214q^{97} - 2920q^{99} + O(q^{100})$$

## 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 −10.0000 0 −8.00000 0 −7.00000 0 73.0000 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
$$2$$ $$-1$$
$$7$$ $$1$$

## 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 28.4.a.a 1
3.b odd 2 1 252.4.a.d 1
4.b odd 2 1 112.4.a.g 1
5.b even 2 1 700.4.a.n 1
5.c odd 4 2 700.4.e.a 2
7.b odd 2 1 196.4.a.d 1
7.c even 3 2 196.4.e.f 2
7.d odd 6 2 196.4.e.a 2
8.b even 2 1 448.4.a.p 1
8.d odd 2 1 448.4.a.a 1
12.b even 2 1 1008.4.a.o 1
21.c even 2 1 1764.4.a.c 1
21.g even 6 2 1764.4.k.m 2
21.h odd 6 2 1764.4.k.d 2
28.d even 2 1 784.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 1.a even 1 1 trivial
112.4.a.g 1 4.b odd 2 1
196.4.a.d 1 7.b odd 2 1
196.4.e.a 2 7.d odd 6 2
196.4.e.f 2 7.c even 3 2
252.4.a.d 1 3.b odd 2 1
448.4.a.a 1 8.d odd 2 1
448.4.a.p 1 8.b even 2 1
700.4.a.n 1 5.b even 2 1
700.4.e.a 2 5.c odd 4 2
784.4.a.a 1 28.d even 2 1
1008.4.a.o 1 12.b even 2 1
1764.4.a.c 1 21.c even 2 1
1764.4.k.d 2 21.h odd 6 2
1764.4.k.m 2 21.g even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 10$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(28))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$10 + T$$
$5$ $$8 + T$$
$7$ $$7 + T$$
$11$ $$40 + T$$
$13$ $$12 + T$$
$17$ $$58 + T$$
$19$ $$-26 + T$$
$23$ $$64 + T$$
$29$ $$62 + T$$
$31$ $$-252 + T$$
$37$ $$-26 + T$$
$41$ $$-6 + T$$
$43$ $$-416 + T$$
$47$ $$396 + T$$
$53$ $$450 + T$$
$59$ $$-274 + T$$
$61$ $$576 + T$$
$67$ $$476 + T$$
$71$ $$448 + T$$
$73$ $$158 + T$$
$79$ $$936 + T$$
$83$ $$-530 + T$$
$89$ $$390 + T$$
$97$ $$-214 + T$$