# Properties

 Label 4410.2.a.by Level $4410$ Weight $2$ Character orbit 4410.a Self dual yes Analytic conductor $35.214$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 490) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + ( -2 + 2 \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} + q^{16} + ( 4 - \beta ) q^{17} + ( -2 + \beta ) q^{19} + q^{20} + ( -2 + 2 \beta ) q^{22} + ( 4 - 2 \beta ) q^{23} + q^{25} + ( 2 + 2 \beta ) q^{26} + ( 2 + 2 \beta ) q^{29} + 2 \beta q^{31} + q^{32} + ( 4 - \beta ) q^{34} + ( -2 - 4 \beta ) q^{37} + ( -2 + \beta ) q^{38} + q^{40} + ( 4 - 5 \beta ) q^{41} + ( -6 - 2 \beta ) q^{43} + ( -2 + 2 \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + ( 8 - 2 \beta ) q^{47} + q^{50} + ( 2 + 2 \beta ) q^{52} + ( 2 - 6 \beta ) q^{53} + ( -2 + 2 \beta ) q^{55} + ( 2 + 2 \beta ) q^{58} + ( 10 - \beta ) q^{59} + ( 2 - 8 \beta ) q^{61} + 2 \beta q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( -4 + 4 \beta ) q^{67} + ( 4 - \beta ) q^{68} + ( -4 + 6 \beta ) q^{71} + ( 8 + \beta ) q^{73} + ( -2 - 4 \beta ) q^{74} + ( -2 + \beta ) q^{76} + ( -4 - 2 \beta ) q^{79} + q^{80} + ( 4 - 5 \beta ) q^{82} + ( 2 - 3 \beta ) q^{83} + ( 4 - \beta ) q^{85} + ( -6 - 2 \beta ) q^{86} + ( -2 + 2 \beta ) q^{88} + 9 \beta q^{89} + ( 4 - 2 \beta ) q^{92} + ( 8 - 2 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( 12 - 3 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + 2q^{10} - 4q^{11} + 4q^{13} + 2q^{16} + 8q^{17} - 4q^{19} + 2q^{20} - 4q^{22} + 8q^{23} + 2q^{25} + 4q^{26} + 4q^{29} + 2q^{32} + 8q^{34} - 4q^{37} - 4q^{38} + 2q^{40} + 8q^{41} - 12q^{43} - 4q^{44} + 8q^{46} + 16q^{47} + 2q^{50} + 4q^{52} + 4q^{53} - 4q^{55} + 4q^{58} + 20q^{59} + 4q^{61} + 2q^{64} + 4q^{65} - 8q^{67} + 8q^{68} - 8q^{71} + 16q^{73} - 4q^{74} - 4q^{76} - 8q^{79} + 2q^{80} + 8q^{82} + 4q^{83} + 8q^{85} - 12q^{86} - 4q^{88} + 8q^{92} + 16q^{94} - 4q^{95} + 24q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
1.00000 0 1.00000 1.00000 0 0 1.00000 0 1.00000
1.2 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.by 2
3.b odd 2 1 490.2.a.l 2
7.b odd 2 1 4410.2.a.bt 2
12.b even 2 1 3920.2.a.ca 2
15.d odd 2 1 2450.2.a.bs 2
15.e even 4 2 2450.2.c.w 4
21.c even 2 1 490.2.a.m yes 2
21.g even 6 2 490.2.e.i 4
21.h odd 6 2 490.2.e.j 4
84.h odd 2 1 3920.2.a.bm 2
105.g even 2 1 2450.2.a.bn 2
105.k odd 4 2 2450.2.c.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 3.b odd 2 1
490.2.a.m yes 2 21.c even 2 1
490.2.e.i 4 21.g even 6 2
490.2.e.j 4 21.h odd 6 2
2450.2.a.bn 2 105.g even 2 1
2450.2.a.bs 2 15.d odd 2 1
2450.2.c.t 4 105.k odd 4 2
2450.2.c.w 4 15.e even 4 2
3920.2.a.bm 2 84.h odd 2 1
3920.2.a.ca 2 12.b even 2 1
4410.2.a.bt 2 7.b odd 2 1
4410.2.a.by 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

 $$T_{11}^{2} + 4 T_{11} - 4$$ $$T_{13}^{2} - 4 T_{13} - 4$$ $$T_{17}^{2} - 8 T_{17} + 14$$ $$T_{19}^{2} + 4 T_{19} + 2$$ $$T_{29}^{2} - 4 T_{29} - 4$$ $$T_{31}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 + 4 T + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$14 - 8 T + T^{2}$$
$19$ $$2 + 4 T + T^{2}$$
$23$ $$8 - 8 T + T^{2}$$
$29$ $$-4 - 4 T + T^{2}$$
$31$ $$-8 + T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$-34 - 8 T + T^{2}$$
$43$ $$28 + 12 T + T^{2}$$
$47$ $$56 - 16 T + T^{2}$$
$53$ $$-68 - 4 T + T^{2}$$
$59$ $$98 - 20 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$-16 + 8 T + T^{2}$$
$71$ $$-56 + 8 T + T^{2}$$
$73$ $$62 - 16 T + T^{2}$$
$79$ $$8 + 8 T + T^{2}$$
$83$ $$-14 - 4 T + T^{2}$$
$89$ $$-162 + T^{2}$$
$97$ $$126 - 24 T + T^{2}$$