# Properties

 Label 4400.2.a.d 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 550) 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 $$q - 2 q^{3} - 4 q^{7} + q^{9} + q^{11} - 5 q^{13} + 7 q^{19} + 8 q^{21} + 3 q^{23} + 4 q^{27} + 3 q^{29} - 5 q^{31} - 2 q^{33} + 4 q^{37} + 10 q^{39} + 12 q^{41} + 5 q^{43} + 9 q^{49} - 6 q^{53} - 14 q^{57} - 12 q^{59} - 10 q^{61} - 4 q^{63} + 14 q^{67} - 6 q^{69} - 3 q^{71} - 8 q^{73} - 4 q^{77} + 4 q^{79} - 11 q^{81} - 15 q^{83} - 6 q^{87} + 3 q^{89} + 20 q^{91} + 10 q^{93} + 13 q^{97} + q^{99}+O(q^{100})$$ q - 2 * q^3 - 4 * q^7 + q^9 + q^11 - 5 * q^13 + 7 * q^19 + 8 * q^21 + 3 * q^23 + 4 * q^27 + 3 * q^29 - 5 * q^31 - 2 * q^33 + 4 * q^37 + 10 * q^39 + 12 * q^41 + 5 * q^43 + 9 * q^49 - 6 * q^53 - 14 * q^57 - 12 * q^59 - 10 * q^61 - 4 * q^63 + 14 * q^67 - 6 * q^69 - 3 * q^71 - 8 * q^73 - 4 * q^77 + 4 * q^79 - 11 * q^81 - 15 * q^83 - 6 * q^87 + 3 * q^89 + 20 * q^91 + 10 * q^93 + 13 * q^97 + 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 −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.d 1
4.b odd 2 1 550.2.a.m yes 1
5.b even 2 1 4400.2.a.bc 1
5.c odd 4 2 4400.2.b.e 2
12.b even 2 1 4950.2.a.u 1
20.d odd 2 1 550.2.a.a 1
20.e even 4 2 550.2.b.d 2
44.c even 2 1 6050.2.a.n 1
60.h even 2 1 4950.2.a.y 1
60.l odd 4 2 4950.2.c.ba 2
220.g even 2 1 6050.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 20.d odd 2 1
550.2.a.m yes 1 4.b odd 2 1
550.2.b.d 2 20.e even 4 2
4400.2.a.d 1 1.a even 1 1 trivial
4400.2.a.bc 1 5.b even 2 1
4400.2.b.e 2 5.c odd 4 2
4950.2.a.u 1 12.b even 2 1
4950.2.a.y 1 60.h even 2 1
4950.2.c.ba 2 60.l odd 4 2
6050.2.a.n 1 44.c even 2 1
6050.2.a.bb 1 220.g even 2 1

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 4$$ T7 + 4 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 1$$
$13$ $$T + 5$$
$17$ $$T$$
$19$ $$T - 7$$
$23$ $$T - 3$$
$29$ $$T - 3$$
$31$ $$T + 5$$
$37$ $$T - 4$$
$41$ $$T - 12$$
$43$ $$T - 5$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T - 14$$
$71$ $$T + 3$$
$73$ $$T + 8$$
$79$ $$T - 4$$
$83$ $$T + 15$$
$89$ $$T - 3$$
$97$ $$T - 13$$