# Properties

 Label 2160.4.a.y 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: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ x^2 - x - 60 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{241})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + ( - \beta + 3) q^{7}+O(q^{10})$$ q - 5 * q^5 + (-b + 3) * q^7 $$q - 5 q^{5} + ( - \beta + 3) q^{7} + ( - \beta - 2) q^{11} + ( - 6 \beta - 9) q^{13} + ( - 7 \beta + 24) q^{17} + ( - 11 \beta - 29) q^{19} + ( - 17 \beta - 42) q^{23} + 25 q^{25} + ( - 13 \beta + 58) q^{29} + (19 \beta + 44) q^{31} + (5 \beta - 15) q^{35} + (5 \beta - 131) q^{37} + ( - 10 \beta - 54) q^{41} + (25 \beta + 48) q^{43} + (7 \beta + 4) q^{47} + ( - 5 \beta - 274) q^{49} + (50 \beta + 258) q^{53} + (5 \beta + 10) q^{55} + (24 \beta + 252) q^{59} + ( - 5 \beta - 85) q^{61} + (30 \beta + 45) q^{65} + ( - 73 \beta + 137) q^{67} + ( - 46 \beta + 10) q^{71} + (79 \beta - 501) q^{73} + 54 q^{77} + ( - 30 \beta + 203) q^{79} + (36 \beta + 558) q^{83} + (35 \beta - 120) q^{85} + ( - 132 \beta - 8) q^{89} + ( - 3 \beta + 333) q^{91} + (55 \beta + 145) q^{95} + ( - 69 \beta - 637) q^{97}+O(q^{100})$$ q - 5 * q^5 + (-b + 3) * q^7 + (-b - 2) * q^11 + (-6*b - 9) * q^13 + (-7*b + 24) * q^17 + (-11*b - 29) * q^19 + (-17*b - 42) * q^23 + 25 * q^25 + (-13*b + 58) * q^29 + (19*b + 44) * q^31 + (5*b - 15) * q^35 + (5*b - 131) * q^37 + (-10*b - 54) * q^41 + (25*b + 48) * q^43 + (7*b + 4) * q^47 + (-5*b - 274) * q^49 + (50*b + 258) * q^53 + (5*b + 10) * q^55 + (24*b + 252) * q^59 + (-5*b - 85) * q^61 + (30*b + 45) * q^65 + (-73*b + 137) * q^67 + (-46*b + 10) * q^71 + (79*b - 501) * q^73 + 54 * q^77 + (-30*b + 203) * q^79 + (36*b + 558) * q^83 + (35*b - 120) * q^85 + (-132*b - 8) * q^89 + (-3*b + 333) * q^91 + (55*b + 145) * q^95 + (-69*b - 637) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{5} + 5 q^{7}+O(q^{10})$$ 2 * q - 10 * q^5 + 5 * q^7 $$2 q - 10 q^{5} + 5 q^{7} - 5 q^{11} - 24 q^{13} + 41 q^{17} - 69 q^{19} - 101 q^{23} + 50 q^{25} + 103 q^{29} + 107 q^{31} - 25 q^{35} - 257 q^{37} - 118 q^{41} + 121 q^{43} + 15 q^{47} - 553 q^{49} + 566 q^{53} + 25 q^{55} + 528 q^{59} - 175 q^{61} + 120 q^{65} + 201 q^{67} - 26 q^{71} - 923 q^{73} + 108 q^{77} + 376 q^{79} + 1152 q^{83} - 205 q^{85} - 148 q^{89} + 663 q^{91} + 345 q^{95} - 1343 q^{97}+O(q^{100})$$ 2 * q - 10 * q^5 + 5 * q^7 - 5 * q^11 - 24 * q^13 + 41 * q^17 - 69 * q^19 - 101 * q^23 + 50 * q^25 + 103 * q^29 + 107 * q^31 - 25 * q^35 - 257 * q^37 - 118 * q^41 + 121 * q^43 + 15 * q^47 - 553 * q^49 + 566 * q^53 + 25 * q^55 + 528 * q^59 - 175 * q^61 + 120 * q^65 + 201 * q^67 - 26 * q^71 - 923 * q^73 + 108 * q^77 + 376 * q^79 + 1152 * q^83 - 205 * q^85 - 148 * q^89 + 663 * q^91 + 345 * q^95 - 1343 * 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
 8.26209 −7.26209
0 0 0 −5.00000 0 −5.26209 0 0 0
1.2 0 0 0 −5.00000 0 10.2621 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.y 2
3.b odd 2 1 2160.4.a.bd 2
4.b odd 2 1 1080.4.a.a 2
12.b even 2 1 1080.4.a.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.a 2 4.b odd 2 1
1080.4.a.b yes 2 12.b even 2 1
2160.4.a.y 2 1.a even 1 1 trivial
2160.4.a.bd 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{2} - 5T_{7} - 54$$ T7^2 - 5*T7 - 54 $$T_{11}^{2} + 5T_{11} - 54$$ T11^2 + 5*T11 - 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} - 5T - 54$$
$11$ $$T^{2} + 5T - 54$$
$13$ $$T^{2} + 24T - 2025$$
$17$ $$T^{2} - 41T - 2532$$
$19$ $$T^{2} + 69T - 6100$$
$23$ $$T^{2} + 101T - 14862$$
$29$ $$T^{2} - 103T - 7530$$
$31$ $$T^{2} - 107T - 18888$$
$37$ $$T^{2} + 257T + 15006$$
$41$ $$T^{2} + 118T - 2544$$
$43$ $$T^{2} - 121T - 33996$$
$47$ $$T^{2} - 15T - 2896$$
$53$ $$T^{2} - 566T - 70536$$
$59$ $$T^{2} - 528T + 34992$$
$61$ $$T^{2} + 175T + 6150$$
$67$ $$T^{2} - 201T - 310972$$
$71$ $$T^{2} + 26T - 127320$$
$73$ $$T^{2} + 923T - 163038$$
$79$ $$T^{2} - 376T - 18881$$
$83$ $$T^{2} - 1152 T + 253692$$
$89$ $$T^{2} + 148 T - 1044320$$
$97$ $$T^{2} + 1343 T + 164062$$