Properties

 Label 4410.2.a.a 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 630) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4q^{11} - 6q^{13} + q^{16} - 4q^{17} - 6q^{19} - q^{20} + 4q^{22} + q^{25} + 6q^{26} - 6q^{29} + 4q^{31} - q^{32} + 4q^{34} + 8q^{37} + 6q^{38} + q^{40} - 10q^{41} - 2q^{43} - 4q^{44} - 10q^{47} - q^{50} - 6q^{52} + 14q^{53} + 4q^{55} + 6q^{58} + 4q^{59} + 8q^{61} - 4q^{62} + q^{64} + 6q^{65} + 6q^{67} - 4q^{68} - 2q^{71} + 10q^{73} - 8q^{74} - 6q^{76} + 16q^{79} - q^{80} + 10q^{82} + 8q^{83} + 4q^{85} + 2q^{86} + 4q^{88} - 2q^{89} + 10q^{94} + 6q^{95} - 2q^{97} + O(q^{100})$$

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.a 1
3.b odd 2 1 4410.2.a.bl 1
7.b odd 2 1 630.2.a.e 1
21.c even 2 1 630.2.a.g yes 1
28.d even 2 1 5040.2.a.bp 1
35.c odd 2 1 3150.2.a.bh 1
35.f even 4 2 3150.2.g.b 2
84.h odd 2 1 5040.2.a.n 1
105.g even 2 1 3150.2.a.s 1
105.k odd 4 2 3150.2.g.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.a.e 1 7.b odd 2 1
630.2.a.g yes 1 21.c even 2 1
3150.2.a.s 1 105.g even 2 1
3150.2.a.bh 1 35.c odd 2 1
3150.2.g.b 2 35.f even 4 2
3150.2.g.s 2 105.k odd 4 2
4410.2.a.a 1 1.a even 1 1 trivial
4410.2.a.bl 1 3.b odd 2 1
5040.2.a.n 1 84.h odd 2 1
5040.2.a.bp 1 28.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$6 + T$$
$17$ $$4 + T$$
$19$ $$6 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-4 + T$$
$37$ $$-8 + T$$
$41$ $$10 + T$$
$43$ $$2 + T$$
$47$ $$10 + T$$
$53$ $$-14 + T$$
$59$ $$-4 + T$$
$61$ $$-8 + T$$
$67$ $$-6 + T$$
$71$ $$2 + T$$
$73$ $$-10 + T$$
$79$ $$-16 + T$$
$83$ $$-8 + T$$
$89$ $$2 + T$$
$97$ $$2 + T$$