# Properties

 Label 4400.2.a.i 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 11) 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 $$q - q^{3} - 2 q^{7} - 2 q^{9} - q^{11} - 4 q^{13} + 2 q^{17} + 2 q^{21} - q^{23} + 5 q^{27} - 7 q^{31} + q^{33} - 3 q^{37} + 4 q^{39} - 8 q^{41} - 6 q^{43} + 8 q^{47} - 3 q^{49} - 2 q^{51} + 6 q^{53} - 5 q^{59} + 12 q^{61} + 4 q^{63} - 7 q^{67} + q^{69} + 3 q^{71} - 4 q^{73} + 2 q^{77} + 10 q^{79} + q^{81} - 6 q^{83} + 15 q^{89} + 8 q^{91} + 7 q^{93} + 7 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^7 - 2 * q^9 - q^11 - 4 * q^13 + 2 * q^17 + 2 * q^21 - q^23 + 5 * q^27 - 7 * q^31 + q^33 - 3 * q^37 + 4 * q^39 - 8 * q^41 - 6 * q^43 + 8 * q^47 - 3 * q^49 - 2 * q^51 + 6 * q^53 - 5 * q^59 + 12 * q^61 + 4 * q^63 - 7 * q^67 + q^69 + 3 * q^71 - 4 * q^73 + 2 * q^77 + 10 * q^79 + q^81 - 6 * q^83 + 15 * q^89 + 8 * q^91 + 7 * 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.i 1
4.b odd 2 1 275.2.a.b 1
5.b even 2 1 176.2.a.b 1
5.c odd 4 2 4400.2.b.h 2
12.b even 2 1 2475.2.a.a 1
15.d odd 2 1 1584.2.a.g 1
20.d odd 2 1 11.2.a.a 1
20.e even 4 2 275.2.b.a 2
35.c odd 2 1 8624.2.a.j 1
40.e odd 2 1 704.2.a.h 1
40.f even 2 1 704.2.a.c 1
44.c even 2 1 3025.2.a.a 1
55.d odd 2 1 1936.2.a.i 1
60.h even 2 1 99.2.a.d 1
60.l odd 4 2 2475.2.c.a 2
80.k odd 4 2 2816.2.c.j 2
80.q even 4 2 2816.2.c.f 2
120.i odd 2 1 6336.2.a.bu 1
120.m even 2 1 6336.2.a.br 1
140.c even 2 1 539.2.a.a 1
140.p odd 6 2 539.2.e.h 2
140.s even 6 2 539.2.e.g 2
180.n even 6 2 891.2.e.b 2
180.p odd 6 2 891.2.e.k 2
220.g even 2 1 121.2.a.d 1
220.n odd 10 4 121.2.c.e 4
220.o even 10 4 121.2.c.a 4
260.g odd 2 1 1859.2.a.b 1
340.d odd 2 1 3179.2.a.a 1
380.d even 2 1 3971.2.a.b 1
420.o odd 2 1 4851.2.a.t 1
440.c even 2 1 7744.2.a.x 1
440.o odd 2 1 7744.2.a.k 1
460.g even 2 1 5819.2.a.a 1
580.e odd 2 1 9251.2.a.d 1
660.g odd 2 1 1089.2.a.b 1
1540.b odd 2 1 5929.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 20.d odd 2 1
99.2.a.d 1 60.h even 2 1
121.2.a.d 1 220.g even 2 1
121.2.c.a 4 220.o even 10 4
121.2.c.e 4 220.n odd 10 4
176.2.a.b 1 5.b even 2 1
275.2.a.b 1 4.b odd 2 1
275.2.b.a 2 20.e even 4 2
539.2.a.a 1 140.c even 2 1
539.2.e.g 2 140.s even 6 2
539.2.e.h 2 140.p odd 6 2
704.2.a.c 1 40.f even 2 1
704.2.a.h 1 40.e odd 2 1
891.2.e.b 2 180.n even 6 2
891.2.e.k 2 180.p odd 6 2
1089.2.a.b 1 660.g odd 2 1
1584.2.a.g 1 15.d odd 2 1
1859.2.a.b 1 260.g odd 2 1
1936.2.a.i 1 55.d odd 2 1
2475.2.a.a 1 12.b even 2 1
2475.2.c.a 2 60.l odd 4 2
2816.2.c.f 2 80.q even 4 2
2816.2.c.j 2 80.k odd 4 2
3025.2.a.a 1 44.c even 2 1
3179.2.a.a 1 340.d odd 2 1
3971.2.a.b 1 380.d even 2 1
4400.2.a.i 1 1.a even 1 1 trivial
4400.2.b.h 2 5.c odd 4 2
4851.2.a.t 1 420.o odd 2 1
5819.2.a.a 1 460.g even 2 1
5929.2.a.h 1 1540.b odd 2 1
6336.2.a.br 1 120.m even 2 1
6336.2.a.bu 1 120.i odd 2 1
7744.2.a.k 1 440.o odd 2 1
7744.2.a.x 1 440.c even 2 1
8624.2.a.j 1 35.c odd 2 1
9251.2.a.d 1 580.e odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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