# Properties

 Label 3080.2.a.m Level $3080$ Weight $2$ Character orbit 3080.a Self dual yes Analytic conductor $24.594$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3080.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.5939238226$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 - q^5 - q^7 + (b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} - q^{11} + (\beta_1 - 1) q^{13} + (\beta_1 - 1) q^{15} + ( - 2 \beta_{2} + \beta_1 - 1) q^{17} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{2} + \beta_1) q^{23} + q^{25} + 2 \beta_{2} q^{27} - 2 q^{29} + (\beta_{2} + 2 \beta_1 - 3) q^{31} + (\beta_1 - 1) q^{33} + q^{35} + (\beta_{2} + \beta_1 - 2) q^{37} + ( - \beta_{2} + \beta_1 - 4) q^{39} + ( - \beta_{2} - 2 \beta_1 - 3) q^{41} + ( - 3 \beta_{2} - \beta_1 - 2) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{45} + (2 \beta_{2} + \beta_1 + 3) q^{47} + q^{49} + ( - 3 \beta_{2} + 5 \beta_1 - 2) q^{51} + ( - \beta_{2} + 3 \beta_1 + 2) q^{53} + q^{55} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{57} + (5 \beta_{2} - 1) q^{59} + (3 \beta_{2} - 4 \beta_1 - 7) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{63} + ( - \beta_1 + 1) q^{65} + (3 \beta_{2} - 3 \beta_1 - 2) q^{67} + ( - 2 \beta_1 - 4) q^{69} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{71} + (2 \beta_{2} - \beta_1 - 7) q^{73} + ( - \beta_1 + 1) q^{75} + q^{77} + (\beta_{2} - 3 \beta_1 - 10) q^{79} + ( - \beta_{2} - \beta_1 - 5) q^{81} + (2 \beta_{2} + 4) q^{83} + (2 \beta_{2} - \beta_1 + 1) q^{85} + (2 \beta_1 - 2) q^{87} + ( - 2 \beta_1 + 2) q^{89} + ( - \beta_1 + 1) q^{91} + ( - \beta_{2} + \beta_1 - 10) q^{93} + (2 \beta_{2} - 2 \beta_1 - 2) q^{95} + (4 \beta_{2} - 6 \beta_1 + 2) q^{97} + ( - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 - q^5 - q^7 + (b2 - b1 + 1) * q^9 - q^11 + (b1 - 1) * q^13 + (b1 - 1) * q^15 + (-2*b2 + b1 - 1) * q^17 + (-2*b2 + 2*b1 + 2) * q^19 + (b1 - 1) * q^21 + (b2 + b1) * q^23 + q^25 + 2*b2 * q^27 - 2 * q^29 + (b2 + 2*b1 - 3) * q^31 + (b1 - 1) * q^33 + q^35 + (b2 + b1 - 2) * q^37 + (-b2 + b1 - 4) * q^39 + (-b2 - 2*b1 - 3) * q^41 + (-3*b2 - b1 - 2) * q^43 + (-b2 + b1 - 1) * q^45 + (2*b2 + b1 + 3) * q^47 + q^49 + (-3*b2 + 5*b1 - 2) * q^51 + (-b2 + 3*b1 + 2) * q^53 + q^55 + (-4*b2 + 2*b1 - 2) * q^57 + (5*b2 - 1) * q^59 + (3*b2 - 4*b1 - 7) * q^61 + (-b2 + b1 - 1) * q^63 + (-b1 + 1) * q^65 + (3*b2 - 3*b1 - 2) * q^67 + (-2*b1 - 4) * q^69 + (-4*b2 + 4*b1 - 4) * q^71 + (2*b2 - b1 - 7) * q^73 + (-b1 + 1) * q^75 + q^77 + (b2 - 3*b1 - 10) * q^79 + (-b2 - b1 - 5) * q^81 + (2*b2 + 4) * q^83 + (2*b2 - b1 + 1) * q^85 + (2*b1 - 2) * q^87 + (-2*b1 + 2) * q^89 + (-b1 + 1) * q^91 + (-b2 + b1 - 10) * q^93 + (2*b2 - 2*b1 - 2) * q^95 + (4*b2 - 6*b1 + 2) * q^97 + (-b2 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 2 q^{23} + 3 q^{25} + 2 q^{27} - 6 q^{29} - 6 q^{31} - 2 q^{33} + 3 q^{35} - 4 q^{37} - 12 q^{39} - 12 q^{41} - 10 q^{43} - 3 q^{45} + 12 q^{47} + 3 q^{49} - 4 q^{51} + 8 q^{53} + 3 q^{55} - 8 q^{57} + 2 q^{59} - 22 q^{61} - 3 q^{63} + 2 q^{65} - 6 q^{67} - 14 q^{69} - 12 q^{71} - 20 q^{73} + 2 q^{75} + 3 q^{77} - 32 q^{79} - 17 q^{81} + 14 q^{83} + 4 q^{85} - 4 q^{87} + 4 q^{89} + 2 q^{91} - 30 q^{93} - 6 q^{95} + 4 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 2 * q^15 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 2 * q^23 + 3 * q^25 + 2 * q^27 - 6 * q^29 - 6 * q^31 - 2 * q^33 + 3 * q^35 - 4 * q^37 - 12 * q^39 - 12 * q^41 - 10 * q^43 - 3 * q^45 + 12 * q^47 + 3 * q^49 - 4 * q^51 + 8 * q^53 + 3 * q^55 - 8 * q^57 + 2 * q^59 - 22 * q^61 - 3 * q^63 + 2 * q^65 - 6 * q^67 - 14 * q^69 - 12 * q^71 - 20 * q^73 + 2 * q^75 + 3 * q^77 - 32 * q^79 - 17 * q^81 + 14 * q^83 + 4 * q^85 - 4 * q^87 + 4 * q^89 + 2 * q^91 - 30 * q^93 - 6 * q^95 + 4 * q^97 - 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 2.86620 −0.210756 −1.65544
0 −1.86620 0 −1.00000 0 −1.00000 0 0.482696 0
1.2 0 1.21076 0 −1.00000 0 −1.00000 0 −1.53407 0
1.3 0 2.65544 0 −1.00000 0 −1.00000 0 4.05137 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 3080.2.a.m 3
4.b odd 2 1 6160.2.a.bd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.m 3 1.a even 1 1 trivial
6160.2.a.bd 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3080))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 6$$ T3^3 - 2*T3^2 - 4*T3 + 6 $$T_{13}^{3} + 2T_{13}^{2} - 4T_{13} - 6$$ T13^3 + 2*T13^2 - 4*T13 - 6 $$T_{17}^{3} + 4T_{17}^{2} - 20T_{17} - 66$$ T17^3 + 4*T17^2 - 20*T17 - 66

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} - 4 T + 6$$
$5$ $$(T + 1)^{3}$$
$7$ $$(T + 1)^{3}$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} + 2 T^{2} - 4 T - 6$$
$17$ $$T^{3} + 4 T^{2} - 20 T - 66$$
$19$ $$T^{3} - 6 T^{2} - 20 T + 88$$
$23$ $$T^{3} - 2 T^{2} - 16 T - 4$$
$29$ $$(T + 2)^{3}$$
$31$ $$T^{3} + 6 T^{2} - 26 T - 154$$
$37$ $$T^{3} + 4 T^{2} - 12 T - 36$$
$41$ $$T^{3} + 12 T^{2} + 10 T - 2$$
$43$ $$T^{3} + 10 T^{2} - 52 T - 348$$
$47$ $$T^{3} - 12 T^{2} + 4 T + 118$$
$53$ $$T^{3} - 8 T^{2} - 20 T + 148$$
$59$ $$T^{3} - 2 T^{2} - 182 T + 946$$
$61$ $$T^{3} + 22 T^{2} + 66 T - 626$$
$67$ $$T^{3} + 6 T^{2} - 60 T - 244$$
$71$ $$T^{3} + 12 T^{2} - 80 T - 192$$
$73$ $$T^{3} + 20 T^{2} + 108 T + 162$$
$79$ $$T^{3} + 32 T^{2} + 300 T + 716$$
$83$ $$T^{3} - 14 T^{2} + 36 T + 88$$
$89$ $$T^{3} - 4 T^{2} - 16 T + 48$$
$97$ $$T^{3} - 4 T^{2} - 192 T - 784$$