# Properties

 Label 2160.4.a.bd Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $1$ 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: $$1$$ 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.bd 2
3.b odd 2 1 2160.4.a.y 2
4.b odd 2 1 1080.4.a.b yes 2
12.b even 2 1 1080.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.a 2 12.b even 2 1
1080.4.a.b yes 2 4.b odd 2 1
2160.4.a.y 2 3.b odd 2 1
2160.4.a.bd 2 1.a even 1 1 trivial

## 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

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