# Properties

 Label 2160.4.a.x Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 540) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{21}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta + 1) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b + 1) * q^7 $$q - 5 q^{5} + (\beta + 1) q^{7} + (\beta - 21) q^{11} + (5 \beta - 7) q^{13} + (7 \beta + 12) q^{17} + ( - \beta + 70) q^{19} + (8 \beta - 63) q^{23} + 25 q^{25} + ( - 11 \beta - 63) q^{29} + (17 \beta + 28) q^{31} + ( - 5 \beta - 5) q^{35} - 286 q^{37} + (21 \beta - 33) q^{41} + (7 \beta + 265) q^{43} + ( - 14 \beta + 198) q^{47} + (2 \beta - 153) q^{49} + ( - 25 \beta - 252) q^{53} + ( - 5 \beta + 105) q^{55} + ( - 7 \beta - 219) q^{59} + (42 \beta - 301) q^{61} + ( - 25 \beta + 35) q^{65} + ( - 42 \beta + 460) q^{67} + (15 \beta - 399) q^{71} + ( - 27 \beta - 385) q^{73} + ( - 20 \beta + 168) q^{77} + (21 \beta + 862) q^{79} + ( - 28 \beta + 243) q^{83} + ( - 35 \beta - 60) q^{85} + (63 \beta - 147) q^{89} + ( - 2 \beta + 938) q^{91} + (5 \beta - 350) q^{95} + ( - 80 \beta + 56) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b + 1) * q^7 + (b - 21) * q^11 + (5*b - 7) * q^13 + (7*b + 12) * q^17 + (-b + 70) * q^19 + (8*b - 63) * q^23 + 25 * q^25 + (-11*b - 63) * q^29 + (17*b + 28) * q^31 + (-5*b - 5) * q^35 - 286 * q^37 + (21*b - 33) * q^41 + (7*b + 265) * q^43 + (-14*b + 198) * q^47 + (2*b - 153) * q^49 + (-25*b - 252) * q^53 + (-5*b + 105) * q^55 + (-7*b - 219) * q^59 + (42*b - 301) * q^61 + (-25*b + 35) * q^65 + (-42*b + 460) * q^67 + (15*b - 399) * q^71 + (-27*b - 385) * q^73 + (-20*b + 168) * q^77 + (21*b + 862) * q^79 + (-28*b + 243) * q^83 + (-35*b - 60) * q^85 + (63*b - 147) * q^89 + (-2*b + 938) * q^91 + (5*b - 350) * q^95 + (-80*b + 56) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 10 * q^5 + 2 * q^7 $$2 q - 10 q^{5} + 2 q^{7} - 42 q^{11} - 14 q^{13} + 24 q^{17} + 140 q^{19} - 126 q^{23} + 50 q^{25} - 126 q^{29} + 56 q^{31} - 10 q^{35} - 572 q^{37} - 66 q^{41} + 530 q^{43} + 396 q^{47} - 306 q^{49} - 504 q^{53} + 210 q^{55} - 438 q^{59} - 602 q^{61} + 70 q^{65} + 920 q^{67} - 798 q^{71} - 770 q^{73} + 336 q^{77} + 1724 q^{79} + 486 q^{83} - 120 q^{85} - 294 q^{89} + 1876 q^{91} - 700 q^{95} + 112 q^{97}+O(q^{100})$$ 2 * q - 10 * q^5 + 2 * q^7 - 42 * q^11 - 14 * q^13 + 24 * q^17 + 140 * q^19 - 126 * q^23 + 50 * q^25 - 126 * q^29 + 56 * q^31 - 10 * q^35 - 572 * q^37 - 66 * q^41 + 530 * q^43 + 396 * q^47 - 306 * q^49 - 504 * q^53 + 210 * q^55 - 438 * q^59 - 602 * q^61 + 70 * q^65 + 920 * q^67 - 798 * q^71 - 770 * q^73 + 336 * q^77 + 1724 * q^79 + 486 * q^83 - 120 * q^85 - 294 * q^89 + 1876 * q^91 - 700 * q^95 + 112 * q^97

## 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
 −1.79129 2.79129
0 0 0 −5.00000 0 −12.7477 0 0 0
1.2 0 0 0 −5.00000 0 14.7477 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 2160.4.a.x 2
3.b odd 2 1 2160.4.a.bc 2
4.b odd 2 1 540.4.a.e 2
12.b even 2 1 540.4.a.h yes 2
36.f odd 6 2 1620.4.i.r 4
36.h even 6 2 1620.4.i.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.e 2 4.b odd 2 1
540.4.a.h yes 2 12.b even 2 1
1620.4.i.o 4 36.h even 6 2
1620.4.i.r 4 36.f odd 6 2
2160.4.a.x 2 1.a even 1 1 trivial
2160.4.a.bc 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{2} - 2T_{7} - 188$$ T7^2 - 2*T7 - 188 $$T_{11}^{2} + 42T_{11} + 252$$ T11^2 + 42*T11 + 252

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} - 2T - 188$$
$11$ $$T^{2} + 42T + 252$$
$13$ $$T^{2} + 14T - 4676$$
$17$ $$T^{2} - 24T - 9117$$
$19$ $$T^{2} - 140T + 4711$$
$23$ $$T^{2} + 126T - 8127$$
$29$ $$T^{2} + 126T - 18900$$
$31$ $$T^{2} - 56T - 53837$$
$37$ $$(T + 286)^{2}$$
$41$ $$T^{2} + 66T - 82260$$
$43$ $$T^{2} - 530T + 60964$$
$47$ $$T^{2} - 396T + 2160$$
$53$ $$T^{2} + 504T - 54621$$
$59$ $$T^{2} + 438T + 38700$$
$61$ $$T^{2} + 602T - 242795$$
$67$ $$T^{2} - 920T - 121796$$
$71$ $$T^{2} + 798T + 116676$$
$73$ $$T^{2} + 770T + 10444$$
$79$ $$T^{2} - 1724 T + 659695$$
$83$ $$T^{2} - 486T - 89127$$
$89$ $$T^{2} + 294T - 728532$$
$97$ $$T^{2} - 112 T - 1206464$$