# Properties

 Label 441.4.a.d Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 4q^{4} - 7q^{5} + 24q^{8} + O(q^{10})$$ $$q - 2q^{2} - 4q^{4} - 7q^{5} + 24q^{8} + 14q^{10} + 5q^{11} - 14q^{13} - 16q^{16} + 21q^{17} + 49q^{19} + 28q^{20} - 10q^{22} + 159q^{23} - 76q^{25} + 28q^{26} - 58q^{29} + 147q^{31} - 160q^{32} - 42q^{34} + 219q^{37} - 98q^{38} - 168q^{40} - 350q^{41} - 124q^{43} - 20q^{44} - 318q^{46} - 525q^{47} + 152q^{50} + 56q^{52} - 303q^{53} - 35q^{55} + 116q^{58} + 105q^{59} - 413q^{61} - 294q^{62} + 448q^{64} + 98q^{65} + 415q^{67} - 84q^{68} + 432q^{71} - 1113q^{73} - 438q^{74} - 196q^{76} - 103q^{79} + 112q^{80} + 700q^{82} - 1092q^{83} - 147q^{85} + 248q^{86} + 120q^{88} + 329q^{89} - 636q^{92} + 1050q^{94} - 343q^{95} - 882q^{97} + O(q^{100})$$

## 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
−2.00000 0 −4.00000 −7.00000 0 0 24.0000 0 14.0000
 $$n$$: 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.d 1
3.b odd 2 1 49.4.a.d 1
7.b odd 2 1 441.4.a.e 1
7.c even 3 2 63.4.e.b 2
7.d odd 6 2 441.4.e.k 2
12.b even 2 1 784.4.a.b 1
15.d odd 2 1 1225.4.a.c 1
21.c even 2 1 49.4.a.c 1
21.g even 6 2 49.4.c.a 2
21.h odd 6 2 7.4.c.a 2
84.h odd 2 1 784.4.a.r 1
84.n even 6 2 112.4.i.c 2
105.g even 2 1 1225.4.a.d 1
105.o odd 6 2 175.4.e.a 2
105.x even 12 4 175.4.k.a 4
168.s odd 6 2 448.4.i.f 2
168.v even 6 2 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 21.h odd 6 2
49.4.a.c 1 21.c even 2 1
49.4.a.d 1 3.b odd 2 1
49.4.c.a 2 21.g even 6 2
63.4.e.b 2 7.c even 3 2
112.4.i.c 2 84.n even 6 2
175.4.e.a 2 105.o odd 6 2
175.4.k.a 4 105.x even 12 4
441.4.a.d 1 1.a even 1 1 trivial
441.4.a.e 1 7.b odd 2 1
441.4.e.k 2 7.d odd 6 2
448.4.i.a 2 168.v even 6 2
448.4.i.f 2 168.s odd 6 2
784.4.a.b 1 12.b even 2 1
784.4.a.r 1 84.h odd 2 1
1225.4.a.c 1 15.d odd 2 1
1225.4.a.d 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} + 2$$ $$T_{5} + 7$$ $$T_{13} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$7 + T$$
$7$ $$T$$
$11$ $$-5 + T$$
$13$ $$14 + T$$
$17$ $$-21 + T$$
$19$ $$-49 + T$$
$23$ $$-159 + T$$
$29$ $$58 + T$$
$31$ $$-147 + T$$
$37$ $$-219 + T$$
$41$ $$350 + T$$
$43$ $$124 + T$$
$47$ $$525 + T$$
$53$ $$303 + T$$
$59$ $$-105 + T$$
$61$ $$413 + T$$
$67$ $$-415 + T$$
$71$ $$-432 + T$$
$73$ $$1113 + T$$
$79$ $$103 + T$$
$83$ $$1092 + T$$
$89$ $$-329 + T$$
$97$ $$882 + T$$