# Properties

 Label 441.4.a.o Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -5 - 2 \beta ) q^{4} + ( 10 - 7 \beta ) q^{5} + ( 9 - 11 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -5 - 2 \beta ) q^{4} + ( 10 - 7 \beta ) q^{5} + ( 9 - 11 \beta ) q^{8} + ( -24 + 17 \beta ) q^{10} + ( 10 + 24 \beta ) q^{11} + ( -52 + 25 \beta ) q^{13} + ( 9 + 36 \beta ) q^{16} + ( 58 + 45 \beta ) q^{17} + ( -96 - 22 \beta ) q^{19} + ( -22 + 15 \beta ) q^{20} + ( 38 - 14 \beta ) q^{22} + ( -14 - 28 \beta ) q^{23} + ( 73 - 140 \beta ) q^{25} + ( 102 - 77 \beta ) q^{26} + ( -148 - 62 \beta ) q^{29} + ( 52 - 50 \beta ) q^{31} + ( -9 + 61 \beta ) q^{32} + ( 32 + 13 \beta ) q^{34} + ( -124 - 48 \beta ) q^{37} + ( 52 - 74 \beta ) q^{38} + ( 244 - 173 \beta ) q^{40} + ( 10 + 219 \beta ) q^{41} + ( -360 + 100 \beta ) q^{43} + ( -146 - 140 \beta ) q^{44} + ( -42 + 14 \beta ) q^{46} + ( -48 - 250 \beta ) q^{47} + ( -353 + 213 \beta ) q^{50} + ( 160 - 21 \beta ) q^{52} + ( -134 - 360 \beta ) q^{53} + ( -236 + 170 \beta ) q^{55} + ( 24 - 86 \beta ) q^{58} + ( -308 + 226 \beta ) q^{59} + ( -8 - 3 \beta ) q^{61} + ( -152 + 102 \beta ) q^{62} + ( 59 - 358 \beta ) q^{64} + ( -870 + 614 \beta ) q^{65} + ( -72 + 524 \beta ) q^{67} + ( -470 - 341 \beta ) q^{68} + ( -494 - 232 \beta ) q^{71} + ( -52 + 401 \beta ) q^{73} + ( 28 - 76 \beta ) q^{74} + ( 568 + 302 \beta ) q^{76} + ( -472 - 236 \beta ) q^{79} + ( -414 + 297 \beta ) q^{80} + ( 428 - 209 \beta ) q^{82} + ( 508 - 80 \beta ) q^{83} + ( -50 + 44 \beta ) q^{85} + ( 560 - 460 \beta ) q^{86} + ( -438 + 106 \beta ) q^{88} + ( -194 - 339 \beta ) q^{89} + ( 182 + 168 \beta ) q^{92} + ( -452 + 202 \beta ) q^{94} + ( -652 + 452 \beta ) q^{95} + ( -244 - 599 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 10q^{4} + 20q^{5} + 18q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 10q^{4} + 20q^{5} + 18q^{8} - 48q^{10} + 20q^{11} - 104q^{13} + 18q^{16} + 116q^{17} - 192q^{19} - 44q^{20} + 76q^{22} - 28q^{23} + 146q^{25} + 204q^{26} - 296q^{29} + 104q^{31} - 18q^{32} + 64q^{34} - 248q^{37} + 104q^{38} + 488q^{40} + 20q^{41} - 720q^{43} - 292q^{44} - 84q^{46} - 96q^{47} - 706q^{50} + 320q^{52} - 268q^{53} - 472q^{55} + 48q^{58} - 616q^{59} - 16q^{61} - 304q^{62} + 118q^{64} - 1740q^{65} - 144q^{67} - 940q^{68} - 988q^{71} - 104q^{73} + 56q^{74} + 1136q^{76} - 944q^{79} - 828q^{80} + 856q^{82} + 1016q^{83} - 100q^{85} + 1120q^{86} - 876q^{88} - 388q^{89} + 364q^{92} - 904q^{94} - 1304q^{95} - 488q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.41421 0 −2.17157 19.8995 0 0 24.5563 0 −48.0416
1.2 0.414214 0 −7.82843 0.100505 0 0 −6.55635 0 0.0416306
 $$n$$: 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.o 2
3.b odd 2 1 147.4.a.k yes 2
7.b odd 2 1 441.4.a.n 2
7.c even 3 2 441.4.e.u 4
7.d odd 6 2 441.4.e.v 4
12.b even 2 1 2352.4.a.bl 2
21.c even 2 1 147.4.a.j 2
21.g even 6 2 147.4.e.k 4
21.h odd 6 2 147.4.e.j 4
84.h odd 2 1 2352.4.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 21.c even 2 1
147.4.a.k yes 2 3.b odd 2 1
147.4.e.j 4 21.h odd 6 2
147.4.e.k 4 21.g even 6 2
441.4.a.n 2 7.b odd 2 1
441.4.a.o 2 1.a even 1 1 trivial
441.4.e.u 4 7.c even 3 2
441.4.e.v 4 7.d odd 6 2
2352.4.a.bl 2 12.b even 2 1
2352.4.a.cf 2 84.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{5}^{2} - 20 T_{5} + 2$$ $$T_{13}^{2} + 104 T_{13} + 1454$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 - 20 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1052 - 20 T + T^{2}$$
$13$ $$1454 + 104 T + T^{2}$$
$17$ $$-686 - 116 T + T^{2}$$
$19$ $$8248 + 192 T + T^{2}$$
$23$ $$-1372 + 28 T + T^{2}$$
$29$ $$14216 + 296 T + T^{2}$$
$31$ $$-2296 - 104 T + T^{2}$$
$37$ $$10768 + 248 T + T^{2}$$
$41$ $$-95822 - 20 T + T^{2}$$
$43$ $$109600 + 720 T + T^{2}$$
$47$ $$-122696 + 96 T + T^{2}$$
$53$ $$-241244 + 268 T + T^{2}$$
$59$ $$-7288 + 616 T + T^{2}$$
$61$ $$46 + 16 T + T^{2}$$
$67$ $$-543968 + 144 T + T^{2}$$
$71$ $$136388 + 988 T + T^{2}$$
$73$ $$-318898 + 104 T + T^{2}$$
$79$ $$111392 + 944 T + T^{2}$$
$83$ $$245264 - 1016 T + T^{2}$$
$89$ $$-192206 + 388 T + T^{2}$$
$97$ $$-658066 + 488 T + T^{2}$$