Properties

 Label 4410.2.a.be 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 1470) 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 $$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 6 q^{11} - 6 q^{13} + q^{16} + 4 q^{19} + q^{20} - 6 q^{22} + q^{25} - 6 q^{26} + 8 q^{29} - 2 q^{31} + q^{32} + 4 q^{37} + 4 q^{38} + q^{40} - 10 q^{41} - 6 q^{43} - 6 q^{44} - 2 q^{47} + q^{50} - 6 q^{52} - 10 q^{53} - 6 q^{55} + 8 q^{58} - 4 q^{59} - 14 q^{61} - 2 q^{62} + q^{64} - 6 q^{65} + 14 q^{67} - 8 q^{71} - 6 q^{73} + 4 q^{74} + 4 q^{76} - 8 q^{79} + q^{80} - 10 q^{82} - 8 q^{83} - 6 q^{86} - 6 q^{88} - 18 q^{89} - 2 q^{94} + 4 q^{95} - 2 q^{97}+O(q^{100})$$ q + q^2 + q^4 + q^5 + q^8 + q^10 - 6 * q^11 - 6 * q^13 + q^16 + 4 * q^19 + q^20 - 6 * q^22 + q^25 - 6 * q^26 + 8 * q^29 - 2 * q^31 + q^32 + 4 * q^37 + 4 * q^38 + q^40 - 10 * q^41 - 6 * q^43 - 6 * q^44 - 2 * q^47 + q^50 - 6 * q^52 - 10 * q^53 - 6 * q^55 + 8 * q^58 - 4 * q^59 - 14 * q^61 - 2 * q^62 + q^64 - 6 * q^65 + 14 * q^67 - 8 * q^71 - 6 * q^73 + 4 * q^74 + 4 * q^76 - 8 * q^79 + q^80 - 10 * q^82 - 8 * q^83 - 6 * q^86 - 6 * q^88 - 18 * q^89 - 2 * q^94 + 4 * q^95 - 2 * q^97

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.be 1
3.b odd 2 1 1470.2.a.c 1
7.b odd 2 1 4410.2.a.v 1
15.d odd 2 1 7350.2.a.da 1
21.c even 2 1 1470.2.a.i yes 1
21.g even 6 2 1470.2.i.k 2
21.h odd 6 2 1470.2.i.r 2
105.g even 2 1 7350.2.a.cf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.c 1 3.b odd 2 1
1470.2.a.i yes 1 21.c even 2 1
1470.2.i.k 2 21.g even 6 2
1470.2.i.r 2 21.h odd 6 2
4410.2.a.v 1 7.b odd 2 1
4410.2.a.be 1 1.a even 1 1 trivial
7350.2.a.cf 1 105.g even 2 1
7350.2.a.da 1 15.d odd 2 1

Hecke kernels

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

 $$T_{11} + 6$$ T11 + 6 $$T_{13} + 6$$ T13 + 6 $$T_{17}$$ T17 $$T_{19} - 4$$ T19 - 4 $$T_{29} - 8$$ T29 - 8 $$T_{31} + 2$$ T31 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 6$$
$13$ $$T + 6$$
$17$ $$T$$
$19$ $$T - 4$$
$23$ $$T$$
$29$ $$T - 8$$
$31$ $$T + 2$$
$37$ $$T - 4$$
$41$ $$T + 10$$
$43$ $$T + 6$$
$47$ $$T + 2$$
$53$ $$T + 10$$
$59$ $$T + 4$$
$61$ $$T + 14$$
$67$ $$T - 14$$
$71$ $$T + 8$$
$73$ $$T + 6$$
$79$ $$T + 8$$
$83$ $$T + 8$$
$89$ $$T + 18$$
$97$ $$T + 2$$