Properties

 Label 4410.2.a.g 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + q^{11} + 7q^{13} + q^{16} + 4q^{17} + q^{19} - q^{20} - q^{22} - q^{23} + q^{25} - 7q^{26} + 8q^{29} + 6q^{31} - q^{32} - 4q^{34} - 3q^{37} - q^{38} + q^{40} - 9q^{41} - 4q^{43} + q^{44} + q^{46} + 3q^{47} - q^{50} + 7q^{52} + q^{53} - q^{55} - 8q^{58} - 12q^{59} - 4q^{61} - 6q^{62} + q^{64} - 7q^{65} + 12q^{67} + 4q^{68} + 14q^{71} - 14q^{73} + 3q^{74} + q^{76} + 4q^{79} - q^{80} + 9q^{82} - 12q^{83} - 4q^{85} + 4q^{86} - q^{88} + 2q^{89} - q^{92} - 3q^{94} - q^{95} - 16q^{97} + O(q^{100})$$

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.g 1
3.b odd 2 1 1470.2.a.r 1
7.b odd 2 1 4410.2.a.q 1
7.c even 3 2 630.2.k.h 2
15.d odd 2 1 7350.2.a.j 1
21.c even 2 1 1470.2.a.k 1
21.g even 6 2 1470.2.i.i 2
21.h odd 6 2 210.2.i.a 2
84.n even 6 2 1680.2.bg.k 2
105.g even 2 1 7350.2.a.ba 1
105.o odd 6 2 1050.2.i.s 2
105.x even 12 4 1050.2.o.j 4

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

 $$T_{11} - 1$$ $$T_{13} - 7$$ $$T_{17} - 4$$ $$T_{19} - 1$$ $$T_{29} - 8$$ $$T_{31} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$-7 + T$$
$17$ $$-4 + T$$
$19$ $$-1 + T$$
$23$ $$1 + T$$
$29$ $$-8 + T$$
$31$ $$-6 + T$$
$37$ $$3 + T$$
$41$ $$9 + T$$
$43$ $$4 + T$$
$47$ $$-3 + T$$
$53$ $$-1 + T$$
$59$ $$12 + T$$
$61$ $$4 + T$$
$67$ $$-12 + T$$
$71$ $$-14 + T$$
$73$ $$14 + T$$
$79$ $$-4 + T$$
$83$ $$12 + T$$
$89$ $$-2 + T$$
$97$ $$16 + T$$