# Properties

 Label 441.4.a.q Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 11 q^{4} -2 \beta q^{5} + 3 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 11 q^{4} -2 \beta q^{5} + 3 \beta q^{8} -38 q^{10} -10 \beta q^{11} -82 q^{13} -31 q^{16} + 18 \beta q^{17} + 20 q^{19} -22 \beta q^{20} -190 q^{22} -30 \beta q^{23} -49 q^{25} -82 \beta q^{26} + 56 \beta q^{29} -156 q^{31} -55 \beta q^{32} + 342 q^{34} + 186 q^{37} + 20 \beta q^{38} -114 q^{40} + 38 \beta q^{41} + 164 q^{43} -110 \beta q^{44} -570 q^{46} -108 \beta q^{47} -49 \beta q^{50} -902 q^{52} -36 \beta q^{53} + 380 q^{55} + 1064 q^{58} -36 \beta q^{59} -790 q^{61} -156 \beta q^{62} -797 q^{64} + 164 \beta q^{65} -44 q^{67} + 198 \beta q^{68} + 102 \beta q^{71} -126 q^{73} + 186 \beta q^{74} + 220 q^{76} -712 q^{79} + 62 \beta q^{80} + 722 q^{82} + 336 \beta q^{83} -684 q^{85} + 164 \beta q^{86} -570 q^{88} + 334 \beta q^{89} -330 \beta q^{92} -2052 q^{94} -40 \beta q^{95} -798 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 22 q^{4} + O(q^{10})$$ $$2 q + 22 q^{4} - 76 q^{10} - 164 q^{13} - 62 q^{16} + 40 q^{19} - 380 q^{22} - 98 q^{25} - 312 q^{31} + 684 q^{34} + 372 q^{37} - 228 q^{40} + 328 q^{43} - 1140 q^{46} - 1804 q^{52} + 760 q^{55} + 2128 q^{58} - 1580 q^{61} - 1594 q^{64} - 88 q^{67} - 252 q^{73} + 440 q^{76} - 1424 q^{79} + 1444 q^{82} - 1368 q^{85} - 1140 q^{88} - 4104 q^{94} - 1596 q^{97} + O(q^{100})$$

## 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
 −4.35890 4.35890
−4.35890 0 11.0000 8.71780 0 0 −13.0767 0 −38.0000
1.2 4.35890 0 11.0000 −8.71780 0 0 13.0767 0 −38.0000
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.q 2
3.b odd 2 1 inner 441.4.a.q 2
7.b odd 2 1 63.4.a.d 2
7.c even 3 2 441.4.e.s 4
7.d odd 6 2 441.4.e.r 4
21.c even 2 1 63.4.a.d 2
21.g even 6 2 441.4.e.r 4
21.h odd 6 2 441.4.e.s 4
28.d even 2 1 1008.4.a.be 2
35.c odd 2 1 1575.4.a.t 2
84.h odd 2 1 1008.4.a.be 2
105.g even 2 1 1575.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 7.b odd 2 1
63.4.a.d 2 21.c even 2 1
441.4.a.q 2 1.a even 1 1 trivial
441.4.a.q 2 3.b odd 2 1 inner
441.4.e.r 4 7.d odd 6 2
441.4.e.r 4 21.g even 6 2
441.4.e.s 4 7.c even 3 2
441.4.e.s 4 21.h odd 6 2
1008.4.a.be 2 28.d even 2 1
1008.4.a.be 2 84.h odd 2 1
1575.4.a.t 2 35.c odd 2 1
1575.4.a.t 2 105.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 19$$ $$T_{5}^{2} - 76$$ $$T_{13} + 82$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-19 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-76 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1900 + T^{2}$$
$13$ $$( 82 + T )^{2}$$
$17$ $$-6156 + T^{2}$$
$19$ $$( -20 + T )^{2}$$
$23$ $$-17100 + T^{2}$$
$29$ $$-59584 + T^{2}$$
$31$ $$( 156 + T )^{2}$$
$37$ $$( -186 + T )^{2}$$
$41$ $$-27436 + T^{2}$$
$43$ $$( -164 + T )^{2}$$
$47$ $$-221616 + T^{2}$$
$53$ $$-24624 + T^{2}$$
$59$ $$-24624 + T^{2}$$
$61$ $$( 790 + T )^{2}$$
$67$ $$( 44 + T )^{2}$$
$71$ $$-197676 + T^{2}$$
$73$ $$( 126 + T )^{2}$$
$79$ $$( 712 + T )^{2}$$
$83$ $$-2145024 + T^{2}$$
$89$ $$-2119564 + T^{2}$$
$97$ $$( 798 + T )^{2}$$