# Properties

 Label 4410.2.a.bh Level $4410$ Weight $2$ Character orbit 4410.a Self dual yes Analytic conductor $35.214$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 3q^{11} + 5q^{13} + q^{16} + 5q^{19} + q^{20} - 3q^{22} + 9q^{23} + q^{25} + 5q^{26} - 10q^{31} + q^{32} - q^{37} + 5q^{38} + q^{40} - 9q^{41} + 8q^{43} - 3q^{44} + 9q^{46} - 3q^{47} + q^{50} + 5q^{52} + 3q^{53} - 3q^{55} - 12q^{59} + 8q^{61} - 10q^{62} + q^{64} + 5q^{65} + 8q^{67} + 6q^{71} + 2q^{73} - q^{74} + 5q^{76} + 8q^{79} + q^{80} - 9q^{82} + 8q^{86} - 3q^{88} - 6q^{89} + 9q^{92} - 3q^{94} + 5q^{95} + 8q^{97} + O(q^{100})$$

## 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

## Atkin-Lehner signs

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

## 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 4410.2.a.bh 1
3.b odd 2 1 1470.2.a.f 1
7.b odd 2 1 4410.2.a.w 1
7.c even 3 2 630.2.k.a 2
15.d odd 2 1 7350.2.a.cd 1
21.c even 2 1 1470.2.a.e 1
21.g even 6 2 1470.2.i.p 2
21.h odd 6 2 210.2.i.c 2
84.n even 6 2 1680.2.bg.n 2
105.g even 2 1 7350.2.a.cx 1
105.o odd 6 2 1050.2.i.i 2
105.x even 12 4 1050.2.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 21.h odd 6 2
630.2.k.a 2 7.c even 3 2
1050.2.i.i 2 105.o odd 6 2
1050.2.o.c 4 105.x even 12 4
1470.2.a.e 1 21.c even 2 1
1470.2.a.f 1 3.b odd 2 1
1470.2.i.p 2 21.g even 6 2
1680.2.bg.n 2 84.n even 6 2
4410.2.a.w 1 7.b odd 2 1
4410.2.a.bh 1 1.a even 1 1 trivial
7350.2.a.cd 1 15.d odd 2 1
7350.2.a.cx 1 105.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

 $$T_{11} + 3$$ $$T_{13} - 5$$ $$T_{17}$$ $$T_{19} - 5$$ $$T_{29}$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ 1
$5$ $$1 - T$$
$7$ 1
$11$ $$1 + 3 T + 11 T^{2}$$
$13$ $$1 - 5 T + 13 T^{2}$$
$17$ $$1 + 17 T^{2}$$
$19$ $$1 - 5 T + 19 T^{2}$$
$23$ $$1 - 9 T + 23 T^{2}$$
$29$ $$1 + 29 T^{2}$$
$31$ $$1 + 10 T + 31 T^{2}$$
$37$ $$1 + T + 37 T^{2}$$
$41$ $$1 + 9 T + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 + 3 T + 47 T^{2}$$
$53$ $$1 - 3 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 - 8 T + 61 T^{2}$$
$67$ $$1 - 8 T + 67 T^{2}$$
$71$ $$1 - 6 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 - 8 T + 79 T^{2}$$
$83$ $$1 + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 - 8 T + 97 T^{2}$$