# Properties

 Label 4410.2.a.bz 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.bz 2
3.b odd 2 1 1470.2.a.t yes 2
7.b odd 2 1 4410.2.a.bw 2
15.d odd 2 1 7350.2.a.dh 2
21.c even 2 1 1470.2.a.s 2
21.g even 6 2 1470.2.i.x 4
21.h odd 6 2 1470.2.i.w 4
105.g even 2 1 7350.2.a.dl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 21.c even 2 1
1470.2.a.t yes 2 3.b odd 2 1
1470.2.i.w 4 21.h odd 6 2
1470.2.i.x 4 21.g even 6 2
4410.2.a.bw 2 7.b odd 2 1
4410.2.a.bz 2 1.a even 1 1 trivial
7350.2.a.dh 2 15.d odd 2 1
7350.2.a.dl 2 105.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ 1
$5$ $$( 1 - T )^{2}$$
$7$ 1
$11$ $$1 - 4 T + 8 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T^{2} + 169 T^{4}$$
$17$ $$1 - 8 T + 48 T^{2} - 136 T^{3} + 289 T^{4}$$
$19$ $$1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$1 + 12 T + 74 T^{2} + 276 T^{3} + 529 T^{4}$$
$29$ $$1 + 8 T + 72 T^{2} + 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4}$$
$37$ $$1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 4 T + 14 T^{2} - 164 T^{3} + 1681 T^{4}$$
$43$ $$1 - 4 T + 40 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 4 T + 80 T^{2} - 188 T^{3} + 2209 T^{4}$$
$53$ $$1 + 12 T + 110 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 146 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 126 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T + 88 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 16 T + 222 T^{2} - 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 - 4 T + 174 T^{2} - 356 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T + 126 T^{2} + 388 T^{3} + 9409 T^{4}$$