# Properties

 Label 4410.2.a.x Level 4410 Weight 2 Character orbit 4410.a Self dual yes Analytic conductor 35.214 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 3q^{11} - q^{13} + q^{16} + 6q^{17} - q^{19} - q^{20} - 3q^{22} - 9q^{23} + q^{25} - q^{26} - 6q^{29} + 8q^{31} + q^{32} + 6q^{34} - 7q^{37} - q^{38} - q^{40} - 3q^{41} + 2q^{43} - 3q^{44} - 9q^{46} - 9q^{47} + q^{50} - q^{52} - 9q^{53} + 3q^{55} - 6q^{58} + 8q^{61} + 8q^{62} + q^{64} + q^{65} + 8q^{67} + 6q^{68} - 4q^{73} - 7q^{74} - q^{76} - 10q^{79} - q^{80} - 3q^{82} - 6q^{85} + 2q^{86} - 3q^{88} - 6q^{89} - 9q^{92} - 9q^{94} + q^{95} - 10q^{97} + O(q^{100})$$

## 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
1.00000 0 1.00000 −1.00000 0 0 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 4410.2.a.x 1
3.b odd 2 1 490.2.a.a 1
7.b odd 2 1 4410.2.a.bg 1
7.c even 3 2 630.2.k.d 2
12.b even 2 1 3920.2.a.bh 1
15.d odd 2 1 2450.2.a.bf 1
15.e even 4 2 2450.2.c.q 2
21.c even 2 1 490.2.a.d 1
21.g even 6 2 490.2.e.g 2
21.h odd 6 2 70.2.e.d 2
84.h odd 2 1 3920.2.a.e 1
84.n even 6 2 560.2.q.b 2
105.g even 2 1 2450.2.a.v 1
105.k odd 4 2 2450.2.c.e 2
105.o odd 6 2 350.2.e.b 2
105.x even 12 4 350.2.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 21.h odd 6 2
350.2.e.b 2 105.o odd 6 2
350.2.j.d 4 105.x even 12 4
490.2.a.a 1 3.b odd 2 1
490.2.a.d 1 21.c even 2 1
490.2.e.g 2 21.g even 6 2
560.2.q.b 2 84.n even 6 2
630.2.k.d 2 7.c even 3 2
2450.2.a.v 1 105.g even 2 1
2450.2.a.bf 1 15.d odd 2 1
2450.2.c.e 2 105.k odd 4 2
2450.2.c.q 2 15.e even 4 2
3920.2.a.e 1 84.h odd 2 1
3920.2.a.bh 1 12.b even 2 1
4410.2.a.x 1 1.a even 1 1 trivial
4410.2.a.bg 1 7.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

 $$T_{11} + 3$$ $$T_{13} + 1$$ $$T_{17} - 6$$ $$T_{19} + 1$$ $$T_{29} + 6$$ $$T_{31} - 8$$

## Hecke Characteristic Polynomials

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