# Properties

 Label 4410.2.a.c 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 3q^{11} - 5q^{13} + q^{16} + 6q^{17} + q^{19} - q^{20} + 3q^{22} - 3q^{23} + q^{25} + 5q^{26} + 6q^{29} + 4q^{31} - q^{32} - 6q^{34} + 11q^{37} - q^{38} + q^{40} + 3q^{41} - 10q^{43} - 3q^{44} + 3q^{46} + 3q^{47} - q^{50} - 5q^{52} - 3q^{53} + 3q^{55} - 6q^{58} + 4q^{61} - 4q^{62} + q^{64} + 5q^{65} - 4q^{67} + 6q^{68} - 12q^{71} + 4q^{73} - 11q^{74} + q^{76} - 10q^{79} - q^{80} - 3q^{82} - 12q^{83} - 6q^{85} + 10q^{86} + 3q^{88} + 6q^{89} - 3q^{92} - 3q^{94} - q^{95} - 14q^{97} + O(q^{100})$$

## 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.c 1
3.b odd 2 1 490.2.a.j 1
7.b odd 2 1 4410.2.a.m 1
7.d odd 6 2 630.2.k.e 2
12.b even 2 1 3920.2.a.g 1
15.d odd 2 1 2450.2.a.f 1
15.e even 4 2 2450.2.c.p 2
21.c even 2 1 490.2.a.g 1
21.g even 6 2 70.2.e.b 2
21.h odd 6 2 490.2.e.a 2
84.h odd 2 1 3920.2.a.be 1
84.j odd 6 2 560.2.q.d 2
105.g even 2 1 2450.2.a.p 1
105.k odd 4 2 2450.2.c.f 2
105.p even 6 2 350.2.e.h 2
105.w odd 12 4 350.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 21.g even 6 2
350.2.e.h 2 105.p even 6 2
350.2.j.a 4 105.w odd 12 4
490.2.a.g 1 21.c even 2 1
490.2.a.j 1 3.b odd 2 1
490.2.e.a 2 21.h odd 6 2
560.2.q.d 2 84.j odd 6 2
630.2.k.e 2 7.d odd 6 2
2450.2.a.f 1 15.d odd 2 1
2450.2.a.p 1 105.g even 2 1
2450.2.c.f 2 105.k odd 4 2
2450.2.c.p 2 15.e even 4 2
3920.2.a.g 1 12.b even 2 1
3920.2.a.be 1 84.h odd 2 1
4410.2.a.c 1 1.a even 1 1 trivial
4410.2.a.m 1 7.b odd 2 1

## Hecke kernels

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

 $$T_{11} + 3$$ $$T_{13} + 5$$ $$T_{17} - 6$$ $$T_{19} - 1$$ $$T_{29} - 6$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$3 + T$$
$13$ $$5 + T$$
$17$ $$-6 + T$$
$19$ $$-1 + T$$
$23$ $$3 + T$$
$29$ $$-6 + T$$
$31$ $$-4 + T$$
$37$ $$-11 + T$$
$41$ $$-3 + T$$
$43$ $$10 + T$$
$47$ $$-3 + T$$
$53$ $$3 + T$$
$59$ $$T$$
$61$ $$-4 + T$$
$67$ $$4 + T$$
$71$ $$12 + T$$
$73$ $$-4 + T$$
$79$ $$10 + T$$
$83$ $$12 + T$$
$89$ $$-6 + T$$
$97$ $$14 + T$$