# Properties

 Label 441.4.e.s Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{19})$$ Defining polynomial: $$x^{4} + 19 x^{2} + 361$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 11 \beta_{2} q^{4} -2 \beta_{1} q^{5} + 3 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 11 \beta_{2} q^{4} -2 \beta_{1} q^{5} + 3 \beta_{3} q^{8} -38 \beta_{2} q^{10} + ( 10 \beta_{1} + 10 \beta_{3} ) q^{11} -82 q^{13} + ( 31 + 31 \beta_{2} ) q^{16} + ( -18 \beta_{1} - 18 \beta_{3} ) q^{17} + ( -20 - 20 \beta_{2} ) q^{19} -22 \beta_{3} q^{20} -190 q^{22} -30 \beta_{1} q^{23} -49 \beta_{2} q^{25} -82 \beta_{1} q^{26} + 56 \beta_{3} q^{29} -156 \beta_{2} q^{31} + ( 55 \beta_{1} + 55 \beta_{3} ) q^{32} + 342 q^{34} + ( -186 - 186 \beta_{2} ) q^{37} + ( -20 \beta_{1} - 20 \beta_{3} ) q^{38} + ( 114 + 114 \beta_{2} ) q^{40} + 38 \beta_{3} q^{41} + 164 q^{43} -110 \beta_{1} q^{44} -570 \beta_{2} q^{46} -108 \beta_{1} q^{47} -49 \beta_{3} q^{50} -902 \beta_{2} q^{52} + ( 36 \beta_{1} + 36 \beta_{3} ) q^{53} + 380 q^{55} + ( -1064 - 1064 \beta_{2} ) q^{58} + ( 36 \beta_{1} + 36 \beta_{3} ) q^{59} + ( 790 + 790 \beta_{2} ) q^{61} -156 \beta_{3} q^{62} -797 q^{64} + 164 \beta_{1} q^{65} -44 \beta_{2} q^{67} + 198 \beta_{1} q^{68} + 102 \beta_{3} q^{71} -126 \beta_{2} q^{73} + ( -186 \beta_{1} - 186 \beta_{3} ) q^{74} + 220 q^{76} + ( 712 + 712 \beta_{2} ) q^{79} + ( -62 \beta_{1} - 62 \beta_{3} ) q^{80} + ( -722 - 722 \beta_{2} ) q^{82} + 336 \beta_{3} q^{83} -684 q^{85} + 164 \beta_{1} q^{86} -570 \beta_{2} q^{88} + 334 \beta_{1} q^{89} -330 \beta_{3} q^{92} -2052 \beta_{2} q^{94} + ( 40 \beta_{1} + 40 \beta_{3} ) q^{95} -798 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 22q^{4} + O(q^{10})$$ $$4q - 22q^{4} + 76q^{10} - 328q^{13} + 62q^{16} - 40q^{19} - 760q^{22} + 98q^{25} + 312q^{31} + 1368q^{34} - 372q^{37} + 228q^{40} + 656q^{43} + 1140q^{46} + 1804q^{52} + 1520q^{55} - 2128q^{58} + 1580q^{61} - 3188q^{64} + 88q^{67} + 252q^{73} + 880q^{76} + 1424q^{79} - 1444q^{82} - 2736q^{85} + 1140q^{88} + 4104q^{94} - 3192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 19 x^{2} + 361$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/19$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$19 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$19 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −2.17945 + 3.77492i 2.17945 − 3.77492i −2.17945 − 3.77492i 2.17945 + 3.77492i
−2.17945 + 3.77492i 0 −5.50000 9.52628i 4.35890 7.54983i 0 0 13.0767 0 19.0000 + 32.9090i
226.2 2.17945 3.77492i 0 −5.50000 9.52628i −4.35890 + 7.54983i 0 0 −13.0767 0 19.0000 + 32.9090i
361.1 −2.17945 3.77492i 0 −5.50000 + 9.52628i 4.35890 + 7.54983i 0 0 13.0767 0 19.0000 32.9090i
361.2 2.17945 + 3.77492i 0 −5.50000 + 9.52628i −4.35890 7.54983i 0 0 −13.0767 0 19.0000 32.9090i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 19 T_{2}^{2} + 361$$ $$T_{5}^{4} + 76 T_{5}^{2} + 5776$$ $$T_{13} + 82$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$361 + 19 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$5776 + 76 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$3610000 + 1900 T^{2} + T^{4}$$
$13$ $$( 82 + T )^{4}$$
$17$ $$37896336 + 6156 T^{2} + T^{4}$$
$19$ $$( 400 + 20 T + T^{2} )^{2}$$
$23$ $$292410000 + 17100 T^{2} + T^{4}$$
$29$ $$( -59584 + T^{2} )^{2}$$
$31$ $$( 24336 - 156 T + T^{2} )^{2}$$
$37$ $$( 34596 + 186 T + T^{2} )^{2}$$
$41$ $$( -27436 + T^{2} )^{2}$$
$43$ $$( -164 + T )^{4}$$
$47$ $$49113651456 + 221616 T^{2} + T^{4}$$
$53$ $$606341376 + 24624 T^{2} + T^{4}$$
$59$ $$606341376 + 24624 T^{2} + T^{4}$$
$61$ $$( 624100 - 790 T + T^{2} )^{2}$$
$67$ $$( 1936 - 44 T + T^{2} )^{2}$$
$71$ $$( -197676 + T^{2} )^{2}$$
$73$ $$( 15876 - 126 T + T^{2} )^{2}$$
$79$ $$( 506944 - 712 T + T^{2} )^{2}$$
$83$ $$( -2145024 + T^{2} )^{2}$$
$89$ $$4492551550096 + 2119564 T^{2} + T^{4}$$
$97$ $$( 798 + T )^{4}$$