# Properties

 Label 1053.2.e.b Level $1053$ Weight $2$ Character orbit 1053.e Analytic conductor $8.408$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - 3 q^{8} +O(q^{10})$$ q + (z - 1) * q^2 + z * q^4 - 2*z * q^5 + (-4*z + 4) * q^7 - 3 * q^8 $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - 3 q^{8} + 2 q^{10} + (4 \zeta_{6} - 4) q^{11} - \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} + 2 q^{17} + ( - 2 \zeta_{6} + 2) q^{20} - 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{25} + q^{26} + 4 q^{28} + ( - 10 \zeta_{6} + 10) q^{29} - 4 \zeta_{6} q^{31} - 5 \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{34} - 8 q^{35} - 2 q^{37} + 6 \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} + ( - 12 \zeta_{6} + 12) q^{43} - 4 q^{44} - 9 \zeta_{6} q^{49} + \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} + 6 q^{53} + 8 q^{55} + (12 \zeta_{6} - 12) q^{56} + 10 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 4 q^{62} + 7 q^{64} + (2 \zeta_{6} - 2) q^{65} + 8 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( - 8 \zeta_{6} + 8) q^{70} + 2 q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 16 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} - 2 q^{80} + 6 q^{82} + (4 \zeta_{6} - 4) q^{83} - 4 \zeta_{6} q^{85} + 12 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{88} - 2 q^{89} - 4 q^{91} + (10 \zeta_{6} - 10) q^{97} + 9 q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + z * q^4 - 2*z * q^5 + (-4*z + 4) * q^7 - 3 * q^8 + 2 * q^10 + (4*z - 4) * q^11 - z * q^13 + 4*z * q^14 + (-z + 1) * q^16 + 2 * q^17 + (-2*z + 2) * q^20 - 4*z * q^22 + (-z + 1) * q^25 + q^26 + 4 * q^28 + (-10*z + 10) * q^29 - 4*z * q^31 - 5*z * q^32 + (2*z - 2) * q^34 - 8 * q^35 - 2 * q^37 + 6*z * q^40 - 6*z * q^41 + (-12*z + 12) * q^43 - 4 * q^44 - 9*z * q^49 + z * q^50 + (-z + 1) * q^52 + 6 * q^53 + 8 * q^55 + (12*z - 12) * q^56 + 10*z * q^58 - 12*z * q^59 + (-2*z + 2) * q^61 + 4 * q^62 + 7 * q^64 + (2*z - 2) * q^65 + 8*z * q^67 + 2*z * q^68 + (-8*z + 8) * q^70 + 2 * q^73 + (-2*z + 2) * q^74 + 16*z * q^77 + (8*z - 8) * q^79 - 2 * q^80 + 6 * q^82 + (4*z - 4) * q^83 - 4*z * q^85 + 12*z * q^86 + (-12*z + 12) * q^88 - 2 * q^89 - 4 * q^91 + (10*z - 10) * q^97 + 9 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 $$2 q - q^{2} + q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{10} - 4 q^{11} - q^{13} + 4 q^{14} + q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + q^{25} + 2 q^{26} + 8 q^{28} + 10 q^{29} - 4 q^{31} - 5 q^{32} - 2 q^{34} - 16 q^{35} - 4 q^{37} + 6 q^{40} - 6 q^{41} + 12 q^{43} - 8 q^{44} - 9 q^{49} + q^{50} + q^{52} + 12 q^{53} + 16 q^{55} - 12 q^{56} + 10 q^{58} - 12 q^{59} + 2 q^{61} + 8 q^{62} + 14 q^{64} - 2 q^{65} + 8 q^{67} + 2 q^{68} + 8 q^{70} + 4 q^{73} + 2 q^{74} + 16 q^{77} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 4 q^{85} + 12 q^{86} + 12 q^{88} - 4 q^{89} - 8 q^{91} - 10 q^{97} + 18 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^10 - 4 * q^11 - q^13 + 4 * q^14 + q^16 + 4 * q^17 + 2 * q^20 - 4 * q^22 + q^25 + 2 * q^26 + 8 * q^28 + 10 * q^29 - 4 * q^31 - 5 * q^32 - 2 * q^34 - 16 * q^35 - 4 * q^37 + 6 * q^40 - 6 * q^41 + 12 * q^43 - 8 * q^44 - 9 * q^49 + q^50 + q^52 + 12 * q^53 + 16 * q^55 - 12 * q^56 + 10 * q^58 - 12 * q^59 + 2 * q^61 + 8 * q^62 + 14 * q^64 - 2 * q^65 + 8 * q^67 + 2 * q^68 + 8 * q^70 + 4 * q^73 + 2 * q^74 + 16 * q^77 - 8 * q^79 - 4 * q^80 + 12 * q^82 - 4 * q^83 - 4 * q^85 + 12 * q^86 + 12 * q^88 - 4 * q^89 - 8 * q^91 - 10 * q^97 + 18 * q^98

## Character values

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

 $$n$$ $$326$$ $$730$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 + 3.46410i −3.00000 0 2.00000
703.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 3.46410i −3.00000 0 2.00000
 $$n$$: 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.b 2
3.b odd 2 1 1053.2.e.d 2
9.c even 3 1 39.2.a.a 1
9.c even 3 1 inner 1053.2.e.b 2
9.d odd 6 1 117.2.a.a 1
9.d odd 6 1 1053.2.e.d 2
36.f odd 6 1 624.2.a.i 1
36.h even 6 1 1872.2.a.h 1
45.h odd 6 1 2925.2.a.p 1
45.j even 6 1 975.2.a.f 1
45.k odd 12 2 975.2.c.f 2
45.l even 12 2 2925.2.c.e 2
63.l odd 6 1 1911.2.a.f 1
63.o even 6 1 5733.2.a.e 1
72.j odd 6 1 7488.2.a.bl 1
72.l even 6 1 7488.2.a.by 1
72.n even 6 1 2496.2.a.q 1
72.p odd 6 1 2496.2.a.e 1
99.h odd 6 1 4719.2.a.c 1
117.f even 3 1 507.2.e.a 2
117.h even 3 1 507.2.e.a 2
117.l even 6 1 507.2.e.b 2
117.n odd 6 1 1521.2.a.e 1
117.r even 6 1 507.2.e.b 2
117.t even 6 1 507.2.a.a 1
117.w odd 12 2 507.2.j.e 4
117.y odd 12 2 507.2.b.a 2
117.z even 12 2 1521.2.b.b 2
117.bb odd 12 2 507.2.j.e 4
468.bg odd 6 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 9.c even 3 1
117.2.a.a 1 9.d odd 6 1
507.2.a.a 1 117.t even 6 1
507.2.b.a 2 117.y odd 12 2
507.2.e.a 2 117.f even 3 1
507.2.e.a 2 117.h even 3 1
507.2.e.b 2 117.l even 6 1
507.2.e.b 2 117.r even 6 1
507.2.j.e 4 117.w odd 12 2
507.2.j.e 4 117.bb odd 12 2
624.2.a.i 1 36.f odd 6 1
975.2.a.f 1 45.j even 6 1
975.2.c.f 2 45.k odd 12 2
1053.2.e.b 2 1.a even 1 1 trivial
1053.2.e.b 2 9.c even 3 1 inner
1053.2.e.d 2 3.b odd 2 1
1053.2.e.d 2 9.d odd 6 1
1521.2.a.e 1 117.n odd 6 1
1521.2.b.b 2 117.z even 12 2
1872.2.a.h 1 36.h even 6 1
1911.2.a.f 1 63.l odd 6 1
2496.2.a.e 1 72.p odd 6 1
2496.2.a.q 1 72.n even 6 1
2925.2.a.p 1 45.h odd 6 1
2925.2.c.e 2 45.l even 12 2
4719.2.a.c 1 99.h odd 6 1
5733.2.a.e 1 63.o even 6 1
7488.2.a.bl 1 72.j odd 6 1
7488.2.a.by 1 72.l even 6 1
8112.2.a.s 1 468.bg odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 12T + 144$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} + 4T + 16$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + 10T + 100$$