# Properties

 Label 441.2.a.h Level 441 Weight 2 Character orbit 441.a Self dual yes Analytic conductor 3.521 Analytic rank 0 Dimension 2 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8} -2 \beta q^{11} + 11 q^{16} -14 q^{22} + 2 \beta q^{23} -5 q^{25} -4 \beta q^{29} + 5 \beta q^{32} + 6 q^{37} + 12 q^{43} -10 \beta q^{44} + 14 q^{46} -5 \beta q^{50} -4 \beta q^{53} -28 q^{58} + 13 q^{64} + 4 q^{67} -2 \beta q^{71} + 6 \beta q^{74} + 8 q^{79} + 12 \beta q^{86} -42 q^{88} + 10 \beta q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{4} + O(q^{10})$$ $$2q + 10q^{4} + 22q^{16} - 28q^{22} - 10q^{25} + 12q^{37} + 24q^{43} + 28q^{46} - 56q^{58} + 26q^{64} + 8q^{67} + 16q^{79} - 84q^{88} + O(q^{100})$$

## 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
 −2.64575 2.64575
−2.64575 0 5.00000 0 0 0 −7.93725 0 0
1.2 2.64575 0 5.00000 0 0 0 7.93725 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.h 2
3.b odd 2 1 inner 441.2.a.h 2
4.b odd 2 1 7056.2.a.co 2
7.b odd 2 1 CM 441.2.a.h 2
7.c even 3 2 441.2.e.h 4
7.d odd 6 2 441.2.e.h 4
12.b even 2 1 7056.2.a.co 2
21.c even 2 1 inner 441.2.a.h 2
21.g even 6 2 441.2.e.h 4
21.h odd 6 2 441.2.e.h 4
28.d even 2 1 7056.2.a.co 2
84.h odd 2 1 7056.2.a.co 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 1.a even 1 1 trivial
441.2.a.h 2 3.b odd 2 1 inner
441.2.a.h 2 7.b odd 2 1 CM
441.2.a.h 2 21.c even 2 1 inner
441.2.e.h 4 7.c even 3 2
441.2.e.h 4 7.d odd 6 2
441.2.e.h 4 21.g even 6 2
441.2.e.h 4 21.h odd 6 2
7056.2.a.co 2 4.b odd 2 1
7056.2.a.co 2 12.b even 2 1
7056.2.a.co 2 28.d even 2 1
7056.2.a.co 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 7$$ $$T_{5}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 6 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$1 - 54 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 12 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{2}$$
$71$ $$1 + 114 T^{2} + 5041 T^{4}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{2}$$