# Properties

 Label 441.4.a.j 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

 $$f(q)$$ $$=$$ $$q + 3q^{2} + q^{4} - 18q^{5} - 21q^{8} + O(q^{10})$$ $$q + 3q^{2} + q^{4} - 18q^{5} - 21q^{8} - 54q^{10} + 36q^{11} + 34q^{13} - 71q^{16} + 42q^{17} + 124q^{19} - 18q^{20} + 108q^{22} + 199q^{25} + 102q^{26} - 102q^{29} + 160q^{31} - 45q^{32} + 126q^{34} + 398q^{37} + 372q^{38} + 378q^{40} - 318q^{41} - 268q^{43} + 36q^{44} + 240q^{47} + 597q^{50} + 34q^{52} + 498q^{53} - 648q^{55} - 306q^{58} - 132q^{59} - 398q^{61} + 480q^{62} + 433q^{64} - 612q^{65} + 92q^{67} + 42q^{68} + 720q^{71} + 502q^{73} + 1194q^{74} + 124q^{76} - 1024q^{79} + 1278q^{80} - 954q^{82} - 204q^{83} - 756q^{85} - 804q^{86} - 756q^{88} + 354q^{89} + 720q^{94} - 2232q^{95} + 286q^{97} + O(q^{100})$$

## 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 −18.0000 0 0 −21.0000 0 −54.0000
 $$n$$: 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.j 1
3.b odd 2 1 147.4.a.c 1
7.b odd 2 1 63.4.a.c 1
7.c even 3 2 441.4.e.d 2
7.d odd 6 2 441.4.e.b 2
12.b even 2 1 2352.4.a.r 1
21.c even 2 1 21.4.a.a 1
21.g even 6 2 147.4.e.i 2
21.h odd 6 2 147.4.e.g 2
28.d even 2 1 1008.4.a.v 1
35.c odd 2 1 1575.4.a.b 1
84.h odd 2 1 336.4.a.f 1
105.g even 2 1 525.4.a.g 1
105.k odd 4 2 525.4.d.c 2
168.e odd 2 1 1344.4.a.n 1
168.i even 2 1 1344.4.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 21.c even 2 1
63.4.a.c 1 7.b odd 2 1
147.4.a.c 1 3.b odd 2 1
147.4.e.g 2 21.h odd 6 2
147.4.e.i 2 21.g even 6 2
336.4.a.f 1 84.h odd 2 1
441.4.a.j 1 1.a even 1 1 trivial
441.4.e.b 2 7.d odd 6 2
441.4.e.d 2 7.c even 3 2
525.4.a.g 1 105.g even 2 1
525.4.d.c 2 105.k odd 4 2
1008.4.a.v 1 28.d even 2 1
1344.4.a.n 1 168.e odd 2 1
1344.4.a.ba 1 168.i even 2 1
1575.4.a.b 1 35.c odd 2 1
2352.4.a.r 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 3$$ $$T_{5} + 18$$ $$T_{13} - 34$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T$$
$3$ $$T$$
$5$ $$18 + T$$
$7$ $$T$$
$11$ $$-36 + T$$
$13$ $$-34 + T$$
$17$ $$-42 + T$$
$19$ $$-124 + T$$
$23$ $$T$$
$29$ $$102 + T$$
$31$ $$-160 + T$$
$37$ $$-398 + T$$
$41$ $$318 + T$$
$43$ $$268 + T$$
$47$ $$-240 + T$$
$53$ $$-498 + T$$
$59$ $$132 + T$$
$61$ $$398 + T$$
$67$ $$-92 + T$$
$71$ $$-720 + T$$
$73$ $$-502 + T$$
$79$ $$1024 + T$$
$83$ $$204 + T$$
$89$ $$-354 + T$$
$97$ $$-286 + T$$