# Properties

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

# Learn more about

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 8q^{4} + 18q^{5} + O(q^{10})$$ $$q - 4q^{2} + 8q^{4} + 18q^{5} - 72q^{10} + 50q^{11} + 36q^{13} - 64q^{16} + 126q^{17} + 72q^{19} + 144q^{20} - 200q^{22} - 14q^{23} + 199q^{25} - 144q^{26} - 158q^{29} + 36q^{31} + 256q^{32} - 504q^{34} - 162q^{37} - 288q^{38} - 270q^{41} - 324q^{43} + 400q^{44} + 56q^{46} - 72q^{47} - 796q^{50} + 288q^{52} + 22q^{53} + 900q^{55} + 632q^{58} + 468q^{59} - 792q^{61} - 144q^{62} - 512q^{64} + 648q^{65} + 232q^{67} + 1008q^{68} + 734q^{71} - 180q^{73} + 648q^{74} + 576q^{76} + 236q^{79} - 1152q^{80} + 1080q^{82} + 36q^{83} + 2268q^{85} + 1296q^{86} + 234q^{89} - 112q^{92} + 288q^{94} + 1296q^{95} - 468q^{97} + O(q^{100})$$

## 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
−4.00000 0 8.00000 18.0000 0 0 0 0 −72.0000
 $$n$$: 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.c 1
3.b odd 2 1 147.4.a.f 1
7.b odd 2 1 441.4.a.a 1
7.c even 3 2 441.4.e.l 2
7.d odd 6 2 441.4.e.o 2
12.b even 2 1 2352.4.a.t 1
21.c even 2 1 147.4.a.h yes 1
21.g even 6 2 147.4.e.a 2
21.h odd 6 2 147.4.e.d 2
84.h odd 2 1 2352.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 3.b odd 2 1
147.4.a.h yes 1 21.c even 2 1
147.4.e.a 2 21.g even 6 2
147.4.e.d 2 21.h odd 6 2
441.4.a.a 1 7.b odd 2 1
441.4.a.c 1 1.a even 1 1 trivial
441.4.e.l 2 7.c even 3 2
441.4.e.o 2 7.d odd 6 2
2352.4.a.s 1 84.h odd 2 1
2352.4.a.t 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} + 4$$ $$T_{5} - 18$$ $$T_{13} - 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$-18 + T$$
$7$ $$T$$
$11$ $$-50 + T$$
$13$ $$-36 + T$$
$17$ $$-126 + T$$
$19$ $$-72 + T$$
$23$ $$14 + T$$
$29$ $$158 + T$$
$31$ $$-36 + T$$
$37$ $$162 + T$$
$41$ $$270 + T$$
$43$ $$324 + T$$
$47$ $$72 + T$$
$53$ $$-22 + T$$
$59$ $$-468 + T$$
$61$ $$792 + T$$
$67$ $$-232 + T$$
$71$ $$-734 + T$$
$73$ $$180 + T$$
$79$ $$-236 + T$$
$83$ $$-36 + T$$
$89$ $$-234 + T$$
$97$ $$468 + T$$
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