# Properties

 Label 4410.2.a.bg 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

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

## 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

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