# Properties

 Label 2160.4.a.ba 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: $$\Q(\sqrt{69})$$ Defining polynomial: $$x^{2} - x - 17$$ x^2 - x - 17 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 $$\beta = 3\sqrt{69}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + ( - \beta - 5) q^{7}+O(q^{10})$$ q + 5 * q^5 + (-b - 5) * q^7 $$q + 5 q^{5} + ( - \beta - 5) q^{7} + (\beta - 9) q^{11} + (\beta + 17) q^{13} + (\beta + 12) q^{17} + (\beta - 14) q^{19} + ( - 4 \beta - 15) q^{23} + 25 q^{25} + ( - 5 \beta - 21) q^{29} + (7 \beta - 128) q^{31} + ( - 5 \beta - 25) q^{35} + 74 q^{37} + ( - 9 \beta - 111) q^{41} + (5 \beta - 209) q^{43} + (10 \beta + 6) q^{47} + (10 \beta + 303) q^{49} + (17 \beta - 24) q^{53} + (5 \beta - 45) q^{55} + ( - 7 \beta + 177) q^{59} + (6 \beta + 419) q^{61} + (5 \beta + 85) q^{65} + ( - 6 \beta - 620) q^{67} + (27 \beta - 231) q^{71} + (21 \beta + 443) q^{73} + (4 \beta - 576) q^{77} + (3 \beta - 578) q^{79} + ( - 28 \beta - 573) q^{83} + (5 \beta + 60) q^{85} + ( - 3 \beta + 75) q^{89} + ( - 22 \beta - 706) q^{91} + (5 \beta - 70) q^{95} + ( - 16 \beta + 392) q^{97}+O(q^{100})$$ q + 5 * q^5 + (-b - 5) * q^7 + (b - 9) * q^11 + (b + 17) * q^13 + (b + 12) * q^17 + (b - 14) * q^19 + (-4*b - 15) * q^23 + 25 * q^25 + (-5*b - 21) * q^29 + (7*b - 128) * q^31 + (-5*b - 25) * q^35 + 74 * q^37 + (-9*b - 111) * q^41 + (5*b - 209) * q^43 + (10*b + 6) * q^47 + (10*b + 303) * q^49 + (17*b - 24) * q^53 + (5*b - 45) * q^55 + (-7*b + 177) * q^59 + (6*b + 419) * q^61 + (5*b + 85) * q^65 + (-6*b - 620) * q^67 + (27*b - 231) * q^71 + (21*b + 443) * q^73 + (4*b - 576) * q^77 + (3*b - 578) * q^79 + (-28*b - 573) * q^83 + (5*b + 60) * q^85 + (-3*b + 75) * q^89 + (-22*b - 706) * q^91 + (5*b - 70) * q^95 + (-16*b + 392) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} - 10 q^{7}+O(q^{10})$$ 2 * q + 10 * q^5 - 10 * q^7 $$2 q + 10 q^{5} - 10 q^{7} - 18 q^{11} + 34 q^{13} + 24 q^{17} - 28 q^{19} - 30 q^{23} + 50 q^{25} - 42 q^{29} - 256 q^{31} - 50 q^{35} + 148 q^{37} - 222 q^{41} - 418 q^{43} + 12 q^{47} + 606 q^{49} - 48 q^{53} - 90 q^{55} + 354 q^{59} + 838 q^{61} + 170 q^{65} - 1240 q^{67} - 462 q^{71} + 886 q^{73} - 1152 q^{77} - 1156 q^{79} - 1146 q^{83} + 120 q^{85} + 150 q^{89} - 1412 q^{91} - 140 q^{95} + 784 q^{97}+O(q^{100})$$ 2 * q + 10 * q^5 - 10 * q^7 - 18 * q^11 + 34 * q^13 + 24 * q^17 - 28 * q^19 - 30 * q^23 + 50 * q^25 - 42 * q^29 - 256 * q^31 - 50 * q^35 + 148 * q^37 - 222 * q^41 - 418 * q^43 + 12 * q^47 + 606 * q^49 - 48 * q^53 - 90 * q^55 + 354 * q^59 + 838 * q^61 + 170 * q^65 - 1240 * q^67 - 462 * q^71 + 886 * q^73 - 1152 * q^77 - 1156 * q^79 - 1146 * q^83 + 120 * q^85 + 150 * q^89 - 1412 * q^91 - 140 * q^95 + 784 * 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
 4.65331 −3.65331
0 0 0 5.00000 0 −29.9199 0 0 0
1.2 0 0 0 5.00000 0 19.9199 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.ba 2
3.b odd 2 1 2160.4.a.v 2
4.b odd 2 1 540.4.a.i yes 2
12.b even 2 1 540.4.a.f 2
36.f odd 6 2 1620.4.i.n 4
36.h even 6 2 1620.4.i.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.f 2 12.b even 2 1
540.4.a.i yes 2 4.b odd 2 1
1620.4.i.n 4 36.f odd 6 2
1620.4.i.q 4 36.h even 6 2
2160.4.a.v 2 3.b odd 2 1
2160.4.a.ba 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{2} + 10T_{7} - 596$$ T7^2 + 10*T7 - 596 $$T_{11}^{2} + 18T_{11} - 540$$ T11^2 + 18*T11 - 540

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} + 10T - 596$$
$11$ $$T^{2} + 18T - 540$$
$13$ $$T^{2} - 34T - 332$$
$17$ $$T^{2} - 24T - 477$$
$19$ $$T^{2} + 28T - 425$$
$23$ $$T^{2} + 30T - 9711$$
$29$ $$T^{2} + 42T - 15084$$
$31$ $$T^{2} + 256T - 14045$$
$37$ $$(T - 74)^{2}$$
$41$ $$T^{2} + 222T - 37980$$
$43$ $$T^{2} + 418T + 28156$$
$47$ $$T^{2} - 12T - 62064$$
$53$ $$T^{2} + 48T - 178893$$
$59$ $$T^{2} - 354T + 900$$
$61$ $$T^{2} - 838T + 153205$$
$67$ $$T^{2} + 1240 T + 362044$$
$71$ $$T^{2} + 462T - 399348$$
$73$ $$T^{2} - 886T - 77612$$
$79$ $$T^{2} + 1156 T + 328495$$
$83$ $$T^{2} + 1146 T - 158535$$
$89$ $$T^{2} - 150T + 36$$
$97$ $$T^{2} - 784T - 5312$$