# Properties

 Label 4400.2.a.v 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 44) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + 2 * q^7 - 2 * q^9 $$q + q^{3} + 2 q^{7} - 2 q^{9} + q^{11} + 4 q^{13} - 6 q^{17} - 8 q^{19} + 2 q^{21} - 3 q^{23} - 5 q^{27} - 5 q^{31} + q^{33} + q^{37} + 4 q^{39} - 10 q^{43} - 3 q^{49} - 6 q^{51} + 6 q^{53} - 8 q^{57} - 3 q^{59} - 4 q^{61} - 4 q^{63} - q^{67} - 3 q^{69} - 15 q^{71} + 4 q^{73} + 2 q^{77} - 2 q^{79} + q^{81} + 6 q^{83} - 9 q^{89} + 8 q^{91} - 5 q^{93} + 7 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 + 2 * q^7 - 2 * q^9 + q^11 + 4 * q^13 - 6 * q^17 - 8 * q^19 + 2 * q^21 - 3 * q^23 - 5 * q^27 - 5 * q^31 + q^33 + q^37 + 4 * q^39 - 10 * q^43 - 3 * q^49 - 6 * q^51 + 6 * q^53 - 8 * q^57 - 3 * q^59 - 4 * q^61 - 4 * q^63 - q^67 - 3 * q^69 - 15 * q^71 + 4 * q^73 + 2 * q^77 - 2 * q^79 + q^81 + 6 * q^83 - 9 * q^89 + 8 * q^91 - 5 * q^93 + 7 * 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 0 0 2.00000 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.v 1
4.b odd 2 1 1100.2.a.b 1
5.b even 2 1 176.2.a.a 1
5.c odd 4 2 4400.2.b.k 2
12.b even 2 1 9900.2.a.h 1
15.d odd 2 1 1584.2.a.p 1
20.d odd 2 1 44.2.a.a 1
20.e even 4 2 1100.2.b.c 2
35.c odd 2 1 8624.2.a.w 1
40.e odd 2 1 704.2.a.f 1
40.f even 2 1 704.2.a.i 1
55.d odd 2 1 1936.2.a.c 1
60.h even 2 1 396.2.a.c 1
60.l odd 4 2 9900.2.c.g 2
80.k odd 4 2 2816.2.c.e 2
80.q even 4 2 2816.2.c.k 2
120.i odd 2 1 6336.2.a.i 1
120.m even 2 1 6336.2.a.j 1
140.c even 2 1 2156.2.a.a 1
140.p odd 6 2 2156.2.i.b 2
140.s even 6 2 2156.2.i.c 2
180.n even 6 2 3564.2.i.a 2
180.p odd 6 2 3564.2.i.j 2
220.g even 2 1 484.2.a.a 1
220.n odd 10 4 484.2.e.a 4
220.o even 10 4 484.2.e.b 4
260.g odd 2 1 7436.2.a.d 1
440.c even 2 1 7744.2.a.m 1
440.o odd 2 1 7744.2.a.bc 1
660.g odd 2 1 4356.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 20.d odd 2 1
176.2.a.a 1 5.b even 2 1
396.2.a.c 1 60.h even 2 1
484.2.a.a 1 220.g even 2 1
484.2.e.a 4 220.n odd 10 4
484.2.e.b 4 220.o even 10 4
704.2.a.f 1 40.e odd 2 1
704.2.a.i 1 40.f even 2 1
1100.2.a.b 1 4.b odd 2 1
1100.2.b.c 2 20.e even 4 2
1584.2.a.p 1 15.d odd 2 1
1936.2.a.c 1 55.d odd 2 1
2156.2.a.a 1 140.c even 2 1
2156.2.i.b 2 140.p odd 6 2
2156.2.i.c 2 140.s even 6 2
2816.2.c.e 2 80.k odd 4 2
2816.2.c.k 2 80.q even 4 2
3564.2.i.a 2 180.n even 6 2
3564.2.i.j 2 180.p odd 6 2
4356.2.a.j 1 660.g odd 2 1
4400.2.a.v 1 1.a even 1 1 trivial
4400.2.b.k 2 5.c odd 4 2
6336.2.a.i 1 120.i odd 2 1
6336.2.a.j 1 120.m even 2 1
7436.2.a.d 1 260.g odd 2 1
7744.2.a.m 1 440.c even 2 1
7744.2.a.bc 1 440.o odd 2 1
8624.2.a.w 1 35.c odd 2 1
9900.2.a.h 1 12.b even 2 1
9900.2.c.g 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 2$$ T7 - 2 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T - 1$$
$13$ $$T - 4$$
$17$ $$T + 6$$
$19$ $$T + 8$$
$23$ $$T + 3$$
$29$ $$T$$
$31$ $$T + 5$$
$37$ $$T - 1$$
$41$ $$T$$
$43$ $$T + 10$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 3$$
$61$ $$T + 4$$
$67$ $$T + 1$$
$71$ $$T + 15$$
$73$ $$T - 4$$
$79$ $$T + 2$$
$83$ $$T - 6$$
$89$ $$T + 9$$
$97$ $$T - 7$$