# Properties

 Label 441.4.a.m Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5q^{2} + 17q^{4} + 45q^{8} + O(q^{10})$$ $$q + 5q^{2} + 17q^{4} + 45q^{8} + 68q^{11} + 89q^{16} + 340q^{22} + 40q^{23} - 125q^{25} + 166q^{29} + 85q^{32} + 450q^{37} - 180q^{43} + 1156q^{44} + 200q^{46} - 625q^{50} - 590q^{53} + 830q^{58} - 287q^{64} - 740q^{67} - 688q^{71} + 2250q^{74} - 1384q^{79} - 900q^{86} + 3060q^{88} + 680q^{92} + O(q^{100})$$

## 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
5.00000 0 17.0000 0 0 0 45.0000 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 $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.m 1
3.b odd 2 1 49.4.a.a 1
7.b odd 2 1 CM 441.4.a.m 1
7.c even 3 2 441.4.e.a 2
7.d odd 6 2 441.4.e.a 2
12.b even 2 1 784.4.a.k 1
15.d odd 2 1 1225.4.a.l 1
21.c even 2 1 49.4.a.a 1
21.g even 6 2 49.4.c.d 2
21.h odd 6 2 49.4.c.d 2
84.h odd 2 1 784.4.a.k 1
105.g even 2 1 1225.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 3.b odd 2 1
49.4.a.a 1 21.c even 2 1
49.4.c.d 2 21.g even 6 2
49.4.c.d 2 21.h odd 6 2
441.4.a.m 1 1.a even 1 1 trivial
441.4.a.m 1 7.b odd 2 1 CM
441.4.e.a 2 7.c even 3 2
441.4.e.a 2 7.d odd 6 2
784.4.a.k 1 12.b even 2 1
784.4.a.k 1 84.h odd 2 1
1225.4.a.l 1 15.d odd 2 1
1225.4.a.l 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-68 + T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$-40 + T$$
$29$ $$-166 + T$$
$31$ $$T$$
$37$ $$-450 + T$$
$41$ $$T$$
$43$ $$180 + T$$
$47$ $$T$$
$53$ $$590 + T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$740 + T$$
$71$ $$688 + T$$
$73$ $$T$$
$79$ $$1384 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$
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