# Properties

 Label 4056.2.a.a Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.3873230598$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 312) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{5} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^5 + q^9 $$q - q^{3} - 4 q^{5} + q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 11 q^{25} - q^{27} - 6 q^{29} + 4 q^{31} - 2 q^{33} - 6 q^{37} + 12 q^{41} + 4 q^{43} - 4 q^{45} + 6 q^{47} - 7 q^{49} - 2 q^{51} - 2 q^{53} - 8 q^{55} + 8 q^{57} + 14 q^{59} + 10 q^{61} + 4 q^{67} - 4 q^{69} - 2 q^{71} + 2 q^{73} - 11 q^{75} - 8 q^{79} + q^{81} - 14 q^{83} - 8 q^{85} + 6 q^{87} - 4 q^{93} + 32 q^{95} + 10 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^3 - 4 * q^5 + q^9 + 2 * q^11 + 4 * q^15 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 11 * q^25 - q^27 - 6 * q^29 + 4 * q^31 - 2 * q^33 - 6 * q^37 + 12 * q^41 + 4 * q^43 - 4 * q^45 + 6 * q^47 - 7 * q^49 - 2 * q^51 - 2 * q^53 - 8 * q^55 + 8 * q^57 + 14 * q^59 + 10 * q^61 + 4 * q^67 - 4 * q^69 - 2 * q^71 + 2 * q^73 - 11 * q^75 - 8 * q^79 + q^81 - 14 * q^83 - 8 * q^85 + 6 * q^87 - 4 * q^93 + 32 * q^95 + 10 * q^97 + 2 * q^99

## 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 −1.00000 0 −4.00000 0 0 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$13$$ $$1$$

## 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 4056.2.a.a 1
4.b odd 2 1 8112.2.a.q 1
13.b even 2 1 312.2.a.c 1
13.d odd 4 2 4056.2.c.d 2
39.d odd 2 1 936.2.a.a 1
52.b odd 2 1 624.2.a.j 1
65.d even 2 1 7800.2.a.s 1
104.e even 2 1 2496.2.a.p 1
104.h odd 2 1 2496.2.a.a 1
156.h even 2 1 1872.2.a.b 1
312.b odd 2 1 7488.2.a.cb 1
312.h even 2 1 7488.2.a.cc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.c 1 13.b even 2 1
624.2.a.j 1 52.b odd 2 1
936.2.a.a 1 39.d odd 2 1
1872.2.a.b 1 156.h even 2 1
2496.2.a.a 1 104.h odd 2 1
2496.2.a.p 1 104.e even 2 1
4056.2.a.a 1 1.a even 1 1 trivial
4056.2.c.d 2 13.d odd 4 2
7488.2.a.cb 1 312.b odd 2 1
7488.2.a.cc 1 312.h even 2 1
7800.2.a.s 1 65.d even 2 1
8112.2.a.q 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7}$$ T7

## Hecke characteristic polynomials

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