# Properties

 Label 1053.2.e.e Level $1053$ Weight $2$ Character orbit 1053.e Analytic conductor $8.408$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1053 = 3^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1053.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.40824733284$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( 3 + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( 3 + \beta_{3} ) q^{8} + ( -4 - 2 \beta_{3} ) q^{10} + ( -2 - 2 \beta_{2} ) q^{11} -\beta_{2} q^{13} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -3 - 3 \beta_{2} ) q^{16} + ( -2 + 4 \beta_{3} ) q^{17} + 2 \beta_{3} q^{19} + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{20} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{22} + 4 \beta_{2} q^{23} + ( -3 - 3 \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{26} + ( -8 - 2 \beta_{3} ) q^{28} + ( 2 + 2 \beta_{2} ) q^{29} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{34} + 8 q^{35} + ( -2 + 4 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{38} + ( 6 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{40} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 2 + 4 \beta_{3} ) q^{44} + ( 4 + 4 \beta_{3} ) q^{46} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{47} + \beta_{2} q^{49} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{50} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{52} + 2 q^{53} -4 \beta_{3} q^{55} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{56} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{59} + ( -2 - 8 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -8 - 6 \beta_{3} ) q^{62} + ( -7 - 2 \beta_{3} ) q^{64} + 2 \beta_{1} q^{65} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + 14 \beta_{2} q^{68} + ( -8 + 8 \beta_{1} - 8 \beta_{2} ) q^{70} -2 q^{71} + ( 6 + 4 \beta_{3} ) q^{73} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{74} + ( -2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{77} + 8 \beta_{1} q^{79} -6 \beta_{3} q^{80} + ( -12 - 10 \beta_{3} ) q^{82} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{85} + ( -8 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{86} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{88} + ( -12 + 2 \beta_{3} ) q^{89} + 2 \beta_{3} q^{91} + ( -4 + 8 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{94} -8 \beta_{2} q^{95} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 1 + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 12q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 12q^{8} - 16q^{10} - 4q^{11} + 2q^{13} + 8q^{14} - 6q^{16} - 8q^{17} + 16q^{20} - 4q^{22} - 8q^{23} - 6q^{25} - 4q^{26} - 32q^{28} + 4q^{29} + 8q^{31} + 6q^{32} - 12q^{34} + 32q^{35} - 8q^{37} - 8q^{38} + 8q^{40} + 16q^{41} - 8q^{43} + 8q^{44} + 16q^{46} - 12q^{47} - 2q^{49} - 6q^{50} + 2q^{52} + 8q^{53} + 8q^{56} + 4q^{58} + 4q^{59} - 4q^{61} - 32q^{62} - 28q^{64} - 8q^{67} - 28q^{68} - 16q^{70} - 8q^{71} + 24q^{73} - 12q^{74} - 16q^{76} - 48q^{82} - 4q^{83} + 32q^{85} - 24q^{86} - 12q^{88} - 48q^{89} - 8q^{92} + 4q^{94} + 16q^{95} + 4q^{97} + 4q^{98} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$730$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
352.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.20711 2.09077i 0 −1.91421 + 3.31552i 1.41421 2.44949i 0 1.41421 + 2.44949i 4.41421 0 −6.82843
352.2 0.207107 + 0.358719i 0 0.914214 1.58346i −1.41421 + 2.44949i 0 −1.41421 2.44949i 1.58579 0 −1.17157
703.1 −1.20711 + 2.09077i 0 −1.91421 3.31552i 1.41421 + 2.44949i 0 1.41421 2.44949i 4.41421 0 −6.82843
703.2 0.207107 0.358719i 0 0.914214 + 1.58346i −1.41421 2.44949i 0 −1.41421 + 2.44949i 1.58579 0 −1.17157
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.2.e.e 4
3.b odd 2 1 1053.2.e.m 4
9.c even 3 1 117.2.a.c 2
9.c even 3 1 inner 1053.2.e.e 4
9.d odd 6 1 39.2.a.b 2
9.d odd 6 1 1053.2.e.m 4
36.f odd 6 1 1872.2.a.w 2
36.h even 6 1 624.2.a.k 2
45.h odd 6 1 975.2.a.l 2
45.j even 6 1 2925.2.a.v 2
45.k odd 12 2 2925.2.c.u 4
45.l even 12 2 975.2.c.h 4
63.l odd 6 1 5733.2.a.u 2
63.o even 6 1 1911.2.a.h 2
72.j odd 6 1 2496.2.a.bf 2
72.l even 6 1 2496.2.a.bi 2
72.n even 6 1 7488.2.a.cl 2
72.p odd 6 1 7488.2.a.co 2
99.g even 6 1 4719.2.a.p 2
117.k odd 6 1 507.2.e.h 4
117.m odd 6 1 507.2.e.d 4
117.n odd 6 1 507.2.a.h 2
117.t even 6 1 1521.2.a.f 2
117.u odd 6 1 507.2.e.h 4
117.v odd 6 1 507.2.e.d 4
117.x even 12 2 507.2.j.f 8
117.y odd 12 2 1521.2.b.j 4
117.z even 12 2 507.2.b.e 4
117.bc even 12 2 507.2.j.f 8
468.x even 6 1 8112.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 9.d odd 6 1
117.2.a.c 2 9.c even 3 1
507.2.a.h 2 117.n odd 6 1
507.2.b.e 4 117.z even 12 2
507.2.e.d 4 117.m odd 6 1
507.2.e.d 4 117.v odd 6 1
507.2.e.h 4 117.k odd 6 1
507.2.e.h 4 117.u odd 6 1
507.2.j.f 8 117.x even 12 2
507.2.j.f 8 117.bc even 12 2
624.2.a.k 2 36.h even 6 1
975.2.a.l 2 45.h odd 6 1
975.2.c.h 4 45.l even 12 2
1053.2.e.e 4 1.a even 1 1 trivial
1053.2.e.e 4 9.c even 3 1 inner
1053.2.e.m 4 3.b odd 2 1
1053.2.e.m 4 9.d odd 6 1
1521.2.a.f 2 117.t even 6 1
1521.2.b.j 4 117.y odd 12 2
1872.2.a.w 2 36.f odd 6 1
1911.2.a.h 2 63.o even 6 1
2496.2.a.bf 2 72.j odd 6 1
2496.2.a.bi 2 72.l even 6 1
2925.2.a.v 2 45.j even 6 1
2925.2.c.u 4 45.k odd 12 2
4719.2.a.p 2 99.g even 6 1
5733.2.a.u 2 63.l odd 6 1
7488.2.a.cl 2 72.n even 6 1
7488.2.a.co 2 72.p odd 6 1
8112.2.a.bm 2 468.x even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1053, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{5}^{4} + 8 T_{5}^{2} + 64$$ $$T_{7}^{4} + 8 T_{7}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$64 + 8 T^{2} + T^{4}$$
$7$ $$64 + 8 T^{2} + T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$( -28 + 4 T + T^{2} )^{2}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$( 16 + 4 T + T^{2} )^{2}$$
$29$ $$( 4 - 2 T + T^{2} )^{2}$$
$31$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$( -28 + 4 T + T^{2} )^{2}$$
$41$ $$3136 - 896 T + 200 T^{2} - 16 T^{3} + T^{4}$$
$43$ $$256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$( -2 + T )^{4}$$
$59$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$( 2 + T )^{4}$$
$73$ $$( 4 - 12 T + T^{2} )^{2}$$
$79$ $$16384 + 128 T^{2} + T^{4}$$
$83$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$( 136 + 24 T + T^{2} )^{2}$$
$97$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$