# Properties

 Label 4356.2.a.d Level 4356 Weight 2 Character orbit 4356.a Self dual yes Analytic conductor 34.783 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4356 = 2^{2} \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4356.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.7828351205$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 132) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{5} + 2q^{7} + O(q^{10})$$ $$q - 2q^{5} + 2q^{7} + 2q^{13} + 4q^{17} + 6q^{19} - q^{25} - 8q^{29} - 8q^{31} - 4q^{35} + 10q^{37} + 8q^{41} + 2q^{43} + 8q^{47} - 3q^{49} + 2q^{53} - 12q^{59} - 10q^{61} - 4q^{65} + 12q^{67} - 8q^{71} - 6q^{73} + 2q^{79} + 16q^{83} - 8q^{85} + 14q^{89} + 4q^{91} - 12q^{95} - 2q^{97} + O(q^{100})$$

## 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 −2.00000 0 2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 4356.2.a.d 1
3.b odd 2 1 1452.2.a.f 1
11.b odd 2 1 396.2.a.a 1
12.b even 2 1 5808.2.a.m 1
33.d even 2 1 132.2.a.b 1
33.f even 10 4 1452.2.i.e 4
33.h odd 10 4 1452.2.i.d 4
44.c even 2 1 1584.2.a.e 1
55.d odd 2 1 9900.2.a.w 1
55.e even 4 2 9900.2.c.f 2
88.b odd 2 1 6336.2.a.ca 1
88.g even 2 1 6336.2.a.cg 1
99.g even 6 2 3564.2.i.d 2
99.h odd 6 2 3564.2.i.i 2
132.d odd 2 1 528.2.a.e 1
165.d even 2 1 3300.2.a.f 1
165.l odd 4 2 3300.2.c.j 2
231.h odd 2 1 6468.2.a.b 1
264.m even 2 1 2112.2.a.c 1
264.p odd 2 1 2112.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.b 1 33.d even 2 1
396.2.a.a 1 11.b odd 2 1
528.2.a.e 1 132.d odd 2 1
1452.2.a.f 1 3.b odd 2 1
1452.2.i.d 4 33.h odd 10 4
1452.2.i.e 4 33.f even 10 4
1584.2.a.e 1 44.c even 2 1
2112.2.a.c 1 264.m even 2 1
2112.2.a.u 1 264.p odd 2 1
3300.2.a.f 1 165.d even 2 1
3300.2.c.j 2 165.l odd 4 2
3564.2.i.d 2 99.g even 6 2
3564.2.i.i 2 99.h odd 6 2
4356.2.a.d 1 1.a even 1 1 trivial
5808.2.a.m 1 12.b even 2 1
6336.2.a.ca 1 88.b odd 2 1
6336.2.a.cg 1 88.g even 2 1
6468.2.a.b 1 231.h odd 2 1
9900.2.a.w 1 55.d odd 2 1
9900.2.c.f 2 55.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4356))$$:

 $$T_{5} + 2$$ $$T_{7} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 + 2 T + 5 T^{2}$$
$7$ $$1 - 2 T + 7 T^{2}$$
$11$ 
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 - 4 T + 17 T^{2}$$
$19$ $$1 - 6 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 8 T + 29 T^{2}$$
$31$ $$1 + 8 T + 31 T^{2}$$
$37$ $$1 - 10 T + 37 T^{2}$$
$41$ $$1 - 8 T + 41 T^{2}$$
$43$ $$1 - 2 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 - 2 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 + 10 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 + 8 T + 71 T^{2}$$
$73$ $$1 + 6 T + 73 T^{2}$$
$79$ $$1 - 2 T + 79 T^{2}$$
$83$ $$1 - 16 T + 83 T^{2}$$
$89$ $$1 - 14 T + 89 T^{2}$$
$97$ $$1 + 2 T + 97 T^{2}$$