# Properties

 Label 4400.2.a.e Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 220) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - 4 q^{7} + q^{9} + O(q^{10})$$ $$q - 2 q^{3} - 4 q^{7} + q^{9} + q^{11} + 4 q^{13} + 4 q^{19} + 8 q^{21} - 6 q^{23} + 4 q^{27} - 6 q^{29} - 8 q^{31} - 2 q^{33} - 2 q^{37} - 8 q^{39} + 6 q^{41} + 8 q^{43} + 6 q^{47} + 9 q^{49} + 6 q^{53} - 8 q^{57} + 12 q^{59} + 2 q^{61} - 4 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} + 16 q^{73} - 4 q^{77} - 8 q^{79} - 11 q^{81} + 12 q^{87} + 6 q^{89} - 16 q^{91} + 16 q^{93} - 14 q^{97} + q^{99} + O(q^{100})$$

## 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 −2.00000 0 0 0 −4.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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.e 1
4.b odd 2 1 1100.2.a.e 1
5.b even 2 1 880.2.a.j 1
5.c odd 4 2 4400.2.b.f 2
12.b even 2 1 9900.2.a.bd 1
15.d odd 2 1 7920.2.a.o 1
20.d odd 2 1 220.2.a.a 1
20.e even 4 2 1100.2.b.a 2
40.e odd 2 1 3520.2.a.bd 1
40.f even 2 1 3520.2.a.d 1
55.d odd 2 1 9680.2.a.bb 1
60.h even 2 1 1980.2.a.a 1
60.l odd 4 2 9900.2.c.m 2
220.g even 2 1 2420.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 20.d odd 2 1
880.2.a.j 1 5.b even 2 1
1100.2.a.e 1 4.b odd 2 1
1100.2.b.a 2 20.e even 4 2
1980.2.a.a 1 60.h even 2 1
2420.2.a.b 1 220.g even 2 1
3520.2.a.d 1 40.f even 2 1
3520.2.a.bd 1 40.e odd 2 1
4400.2.a.e 1 1.a even 1 1 trivial
4400.2.b.f 2 5.c odd 4 2
7920.2.a.o 1 15.d odd 2 1
9680.2.a.bb 1 55.d odd 2 1
9900.2.a.bd 1 12.b even 2 1
9900.2.c.m 2 60.l odd 4 2

## Hecke kernels

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

 $$T_{3} + 2$$ $$T_{7} + 4$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

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