# Properties

 Label 4400.2.a.w Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 5 q^{7} - 2 q^{9} + O(q^{10})$$ $$q + q^{3} + 5 q^{7} - 2 q^{9} - q^{11} - 2 q^{13} - 3 q^{17} + 7 q^{19} + 5 q^{21} - 6 q^{23} - 5 q^{27} - 3 q^{29} + 7 q^{31} - q^{33} + 7 q^{37} - 2 q^{39} + 6 q^{41} + 8 q^{43} + 6 q^{47} + 18 q^{49} - 3 q^{51} + 3 q^{53} + 7 q^{57} + 6 q^{59} - q^{61} - 10 q^{63} + 8 q^{67} - 6 q^{69} - 3 q^{71} - 2 q^{73} - 5 q^{77} + 10 q^{79} + q^{81} - 6 q^{83} - 3 q^{87} + 9 q^{89} - 10 q^{91} + 7 q^{93} + 4 q^{97} + 2 q^{99} + O(q^{100})$$

## 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 1.00000 0 0 0 5.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 4400.2.a.w 1
4.b odd 2 1 550.2.a.i 1
5.b even 2 1 880.2.a.c 1
5.c odd 4 2 4400.2.b.g 2
12.b even 2 1 4950.2.a.a 1
15.d odd 2 1 7920.2.a.s 1
20.d odd 2 1 110.2.a.a 1
20.e even 4 2 550.2.b.b 2
40.e odd 2 1 3520.2.a.l 1
40.f even 2 1 3520.2.a.z 1
44.c even 2 1 6050.2.a.i 1
55.d odd 2 1 9680.2.a.j 1
60.h even 2 1 990.2.a.l 1
60.l odd 4 2 4950.2.c.a 2
140.c even 2 1 5390.2.a.h 1
220.g even 2 1 1210.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 20.d odd 2 1
550.2.a.i 1 4.b odd 2 1
550.2.b.b 2 20.e even 4 2
880.2.a.c 1 5.b even 2 1
990.2.a.l 1 60.h even 2 1
1210.2.a.k 1 220.g even 2 1
3520.2.a.l 1 40.e odd 2 1
3520.2.a.z 1 40.f even 2 1
4400.2.a.w 1 1.a even 1 1 trivial
4400.2.b.g 2 5.c odd 4 2
4950.2.a.a 1 12.b even 2 1
4950.2.c.a 2 60.l odd 4 2
5390.2.a.h 1 140.c even 2 1
6050.2.a.i 1 44.c even 2 1
7920.2.a.s 1 15.d odd 2 1
9680.2.a.j 1 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3} - 1$$ $$T_{7} - 5$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-5 + T$$
$11$ $$1 + T$$
$13$ $$2 + T$$
$17$ $$3 + T$$
$19$ $$-7 + T$$
$23$ $$6 + T$$
$29$ $$3 + T$$
$31$ $$-7 + T$$
$37$ $$-7 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$-6 + T$$
$53$ $$-3 + T$$
$59$ $$-6 + T$$
$61$ $$1 + T$$
$67$ $$-8 + T$$
$71$ $$3 + T$$
$73$ $$2 + T$$
$79$ $$-10 + T$$
$83$ $$6 + T$$
$89$ $$-9 + T$$
$97$ $$-4 + T$$