Properties

 Label 4410.2.a.bw Level $4410$ Weight $2$ Character orbit 4410.a Self dual yes Analytic conductor $35.214$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1470) 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.

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

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.bw 2
3.b odd 2 1 1470.2.a.s 2
7.b odd 2 1 4410.2.a.bz 2
15.d odd 2 1 7350.2.a.dl 2
21.c even 2 1 1470.2.a.t yes 2
21.g even 6 2 1470.2.i.w 4
21.h odd 6 2 1470.2.i.x 4
105.g even 2 1 7350.2.a.dh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 3.b odd 2 1
1470.2.a.t yes 2 21.c even 2 1
1470.2.i.w 4 21.g even 6 2
1470.2.i.x 4 21.h odd 6 2
4410.2.a.bw 2 1.a even 1 1 trivial
4410.2.a.bz 2 7.b odd 2 1
7350.2.a.dh 2 105.g even 2 1
7350.2.a.dl 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

 $$T_{11}^{2} - 4 T_{11} - 14$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} + 8 T_{17} + 14$$ $$T_{19}^{2} + 8 T_{19} + 8$$ $$T_{29}^{2} + 8 T_{29} + 14$$ $$T_{31}^{2} + 12 T_{31} + 18$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-14 - 4 T + T^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$14 + 8 T + T^{2}$$
$19$ $$8 + 8 T + T^{2}$$
$23$ $$28 + 12 T + T^{2}$$
$29$ $$14 + 8 T + T^{2}$$
$31$ $$18 + 12 T + T^{2}$$
$37$ $$-2 - 8 T + T^{2}$$
$41$ $$-68 + 4 T + T^{2}$$
$43$ $$-46 - 4 T + T^{2}$$
$47$ $$-14 + 4 T + T^{2}$$
$53$ $$4 + 12 T + T^{2}$$
$59$ $$28 - 12 T + T^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$-46 - 4 T + T^{2}$$
$71$ $$-56 + 8 T + T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$56 + 16 T + T^{2}$$
$89$ $$-4 + 4 T + T^{2}$$
$97$ $$-68 - 4 T + T^{2}$$