# Properties

 Label 441.4.a.k Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## 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 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{2} + q^{4} - 3q^{5} - 21q^{8} + O(q^{10})$$ $$q + 3q^{2} + q^{4} - 3q^{5} - 21q^{8} - 9q^{10} + 15q^{11} + 64q^{13} - 71q^{16} + 84q^{17} + 16q^{19} - 3q^{20} + 45q^{22} + 84q^{23} - 116q^{25} + 192q^{26} + 297q^{29} + 253q^{31} - 45q^{32} + 252q^{34} - 316q^{37} + 48q^{38} + 63q^{40} + 360q^{41} + 26q^{43} + 15q^{44} + 252q^{46} - 30q^{47} - 348q^{50} + 64q^{52} - 363q^{53} - 45q^{55} + 891q^{58} - 15q^{59} + 118q^{61} + 759q^{62} + 433q^{64} - 192q^{65} - 370q^{67} + 84q^{68} + 342q^{71} - 362q^{73} - 948q^{74} + 16q^{76} + 467q^{79} + 213q^{80} + 1080q^{82} + 477q^{83} - 252q^{85} + 78q^{86} - 315q^{88} + 906q^{89} + 84q^{92} - 90q^{94} - 48q^{95} - 503q^{97} + O(q^{100})$$

## 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
 0
3.00000 0 1.00000 −3.00000 0 0 −21.0000 0 −9.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 441.4.a.k 1
3.b odd 2 1 147.4.a.a 1
7.b odd 2 1 441.4.a.l 1
7.c even 3 2 441.4.e.c 2
7.d odd 6 2 63.4.e.a 2
12.b even 2 1 2352.4.a.bd 1
21.c even 2 1 147.4.a.b 1
21.g even 6 2 21.4.e.a 2
21.h odd 6 2 147.4.e.h 2
84.h odd 2 1 2352.4.a.i 1
84.j odd 6 2 336.4.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 21.g even 6 2
63.4.e.a 2 7.d odd 6 2
147.4.a.a 1 3.b odd 2 1
147.4.a.b 1 21.c even 2 1
147.4.e.h 2 21.h odd 6 2
336.4.q.e 2 84.j odd 6 2
441.4.a.k 1 1.a even 1 1 trivial
441.4.a.l 1 7.b odd 2 1
441.4.e.c 2 7.c even 3 2
2352.4.a.i 1 84.h odd 2 1
2352.4.a.bd 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 3$$ $$T_{5} + 3$$ $$T_{13} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T$$
$3$ $$T$$
$5$ $$3 + T$$
$7$ $$T$$
$11$ $$-15 + T$$
$13$ $$-64 + T$$
$17$ $$-84 + T$$
$19$ $$-16 + T$$
$23$ $$-84 + T$$
$29$ $$-297 + T$$
$31$ $$-253 + T$$
$37$ $$316 + T$$
$41$ $$-360 + T$$
$43$ $$-26 + T$$
$47$ $$30 + T$$
$53$ $$363 + T$$
$59$ $$15 + T$$
$61$ $$-118 + T$$
$67$ $$370 + T$$
$71$ $$-342 + T$$
$73$ $$362 + T$$
$79$ $$-467 + T$$
$83$ $$-477 + T$$
$89$ $$-906 + T$$
$97$ $$503 + T$$