# Properties

 Label 2790.2.a.bi Level $2790$ Weight $2$ Character orbit 2790.a Self dual yes Analytic conductor $22.278$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2790.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.2782621639$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 310) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - q^5 + (-b2 + b1) * q^7 - q^8 $$q - q^{2} + q^{4} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} - q^{8} + q^{10} + ( - \beta_{2} - 2 \beta_1 + 1) q^{11} + (\beta_1 - 3) q^{13} + (\beta_{2} - \beta_1) q^{14} + q^{16} + (\beta_{2} - \beta_1) q^{17} + (2 \beta_{2} + 2) q^{19} - q^{20} + (\beta_{2} + 2 \beta_1 - 1) q^{22} + (\beta_{2} + \beta_1) q^{23} + q^{25} + ( - \beta_1 + 3) q^{26} + ( - \beta_{2} + \beta_1) q^{28} + (3 \beta_{2} + 2 \beta_1 - 1) q^{29} + q^{31} - q^{32} + ( - \beta_{2} + \beta_1) q^{34} + (\beta_{2} - \beta_1) q^{35} + (2 \beta_{2} - \beta_1 - 3) q^{37} + ( - 2 \beta_{2} - 2) q^{38} + q^{40} + ( - 3 \beta_{2} - \beta_1 + 2) q^{41} + (2 \beta_{2} - 3 \beta_1 - 3) q^{43} + ( - \beta_{2} - 2 \beta_1 + 1) q^{44} + ( - \beta_{2} - \beta_1) q^{46} + (\beta_{2} + 3 \beta_1 - 8) q^{47} + ( - 4 \beta_1 + 5) q^{49} - q^{50} + (\beta_1 - 3) q^{52} + ( - 2 \beta_{2} + \beta_1 + 7) q^{53} + (\beta_{2} + 2 \beta_1 - 1) q^{55} + (\beta_{2} - \beta_1) q^{56} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{58} + ( - 2 \beta_1 - 6) q^{59} + ( - \beta_{2} - 2 \beta_1 - 5) q^{61} - q^{62} + q^{64} + ( - \beta_1 + 3) q^{65} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{67} + (\beta_{2} - \beta_1) q^{68} + ( - \beta_{2} + \beta_1) q^{70} + (2 \beta_{2} + 2 \beta_1 - 4) q^{71} + ( - 3 \beta_{2} + \beta_1 - 6) q^{73} + ( - 2 \beta_{2} + \beta_1 + 3) q^{74} + (2 \beta_{2} + 2) q^{76} - 4 \beta_{2} q^{77} + (6 \beta_1 - 2) q^{79} - q^{80} + (3 \beta_{2} + \beta_1 - 2) q^{82} + ( - 2 \beta_{2} + \beta_1 - 3) q^{83} + ( - \beta_{2} + \beta_1) q^{85} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{86} + (\beta_{2} + 2 \beta_1 - 1) q^{88} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{89} + (4 \beta_{2} - 4 \beta_1 + 4) q^{91} + (\beta_{2} + \beta_1) q^{92} + ( - \beta_{2} - 3 \beta_1 + 8) q^{94} + ( - 2 \beta_{2} - 2) q^{95} + ( - \beta_{2} + 3 \beta_1 - 4) q^{97} + (4 \beta_1 - 5) q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^5 + (-b2 + b1) * q^7 - q^8 + q^10 + (-b2 - 2*b1 + 1) * q^11 + (b1 - 3) * q^13 + (b2 - b1) * q^14 + q^16 + (b2 - b1) * q^17 + (2*b2 + 2) * q^19 - q^20 + (b2 + 2*b1 - 1) * q^22 + (b2 + b1) * q^23 + q^25 + (-b1 + 3) * q^26 + (-b2 + b1) * q^28 + (3*b2 + 2*b1 - 1) * q^29 + q^31 - q^32 + (-b2 + b1) * q^34 + (b2 - b1) * q^35 + (2*b2 - b1 - 3) * q^37 + (-2*b2 - 2) * q^38 + q^40 + (-3*b2 - b1 + 2) * q^41 + (2*b2 - 3*b1 - 3) * q^43 + (-b2 - 2*b1 + 1) * q^44 + (-b2 - b1) * q^46 + (b2 + 3*b1 - 8) * q^47 + (-4*b1 + 5) * q^49 - q^50 + (b1 - 3) * q^52 + (-2*b2 + b1 + 7) * q^53 + (b2 + 2*b1 - 1) * q^55 + (b2 - b1) * q^56 + (-3*b2 - 2*b1 + 1) * q^58 + (-2*b1 - 6) * q^59 + (-b2 - 2*b1 - 5) * q^61 - q^62 + q^64 + (-b1 + 3) * q^65 + (-4*b2 - 2*b1 - 2) * q^67 + (b2 - b1) * q^68 + (-b2 + b1) * q^70 + (2*b2 + 2*b1 - 4) * q^71 + (-3*b2 + b1 - 6) * q^73 + (-2*b2 + b1 + 3) * q^74 + (2*b2 + 2) * q^76 - 4*b2 * q^77 + (6*b1 - 2) * q^79 - q^80 + (3*b2 + b1 - 2) * q^82 + (-2*b2 + b1 - 3) * q^83 + (-b2 + b1) * q^85 + (-2*b2 + 3*b1 + 3) * q^86 + (b2 + 2*b1 - 1) * q^88 + (-2*b2 + 2*b1 - 6) * q^89 + (4*b2 - 4*b1 + 4) * q^91 + (b2 + b1) * q^92 + (-b2 - 3*b1 + 8) * q^94 + (-2*b2 - 2) * q^95 + (-b2 + 3*b1 - 4) * q^97 + (4*b1 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8} + 3 q^{10} - 8 q^{13} + 3 q^{16} + 8 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 8 q^{26} + 2 q^{29} + 3 q^{31} - 3 q^{32} - 8 q^{37} - 8 q^{38} + 3 q^{40} + 2 q^{41} - 10 q^{43} - 2 q^{46} - 20 q^{47} + 11 q^{49} - 3 q^{50} - 8 q^{52} + 20 q^{53} - 2 q^{58} - 20 q^{59} - 18 q^{61} - 3 q^{62} + 3 q^{64} + 8 q^{65} - 12 q^{67} - 8 q^{71} - 20 q^{73} + 8 q^{74} + 8 q^{76} - 4 q^{77} - 3 q^{80} - 2 q^{82} - 10 q^{83} + 10 q^{86} - 18 q^{89} + 12 q^{91} + 2 q^{92} + 20 q^{94} - 8 q^{95} - 10 q^{97} - 11 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8 + 3 * q^10 - 8 * q^13 + 3 * q^16 + 8 * q^19 - 3 * q^20 + 2 * q^23 + 3 * q^25 + 8 * q^26 + 2 * q^29 + 3 * q^31 - 3 * q^32 - 8 * q^37 - 8 * q^38 + 3 * q^40 + 2 * q^41 - 10 * q^43 - 2 * q^46 - 20 * q^47 + 11 * q^49 - 3 * q^50 - 8 * q^52 + 20 * q^53 - 2 * q^58 - 20 * q^59 - 18 * q^61 - 3 * q^62 + 3 * q^64 + 8 * q^65 - 12 * q^67 - 8 * q^71 - 20 * q^73 + 8 * q^74 + 8 * q^76 - 4 * q^77 - 3 * q^80 - 2 * q^82 - 10 * q^83 + 10 * q^86 - 18 * q^89 + 12 * q^91 + 2 * q^92 + 20 * q^94 - 8 * q^95 - 10 * q^97 - 11 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 0.311108 2.17009 −1.48119
−1.00000 0 1.00000 −1.00000 0 −4.42864 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.07838 −1.00000 0 1.00000
1.3 −1.00000 0 1.00000 −1.00000 0 3.35026 −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$31$$ $$-1$$

## 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 2790.2.a.bi 3
3.b odd 2 1 310.2.a.e 3
12.b even 2 1 2480.2.a.u 3
15.d odd 2 1 1550.2.a.k 3
15.e even 4 2 1550.2.b.j 6
24.f even 2 1 9920.2.a.bx 3
24.h odd 2 1 9920.2.a.bw 3
93.c even 2 1 9610.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.e 3 3.b odd 2 1
1550.2.a.k 3 15.d odd 2 1
1550.2.b.j 6 15.e even 4 2
2480.2.a.u 3 12.b even 2 1
2790.2.a.bi 3 1.a even 1 1 trivial
9610.2.a.u 3 93.c even 2 1
9920.2.a.bw 3 24.h odd 2 1
9920.2.a.bx 3 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7}^{3} - 16T_{7} + 16$$ T7^3 - 16*T7 + 16 $$T_{11}^{3} - 28T_{11} + 52$$ T11^3 - 28*T11 + 52 $$T_{13}^{3} + 8T_{13}^{2} + 16T_{13} + 4$$ T13^3 + 8*T13^2 + 16*T13 + 4 $$T_{17}^{3} - 16T_{17} - 16$$ T17^3 - 16*T17 - 16 $$T_{19}^{3} - 8T_{19}^{2} - 16T_{19} + 160$$ T19^3 - 8*T19^2 - 16*T19 + 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 16T + 16$$
$11$ $$T^{3} - 28T + 52$$
$13$ $$T^{3} + 8 T^{2} + 16 T + 4$$
$17$ $$T^{3} - 16T - 16$$
$19$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$23$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$29$ $$T^{3} - 2 T^{2} - 96 T + 260$$
$31$ $$(T - 1)^{3}$$
$37$ $$T^{3} + 8 T^{2} - 24 T - 92$$
$41$ $$T^{3} - 2 T^{2} - 84 T - 232$$
$43$ $$T^{3} + 10 T^{2} - 60 T - 604$$
$47$ $$T^{3} + 20 T^{2} + 80 T - 208$$
$53$ $$T^{3} - 20 T^{2} + 88 T - 4$$
$59$ $$T^{3} + 20 T^{2} + 112 T + 160$$
$61$ $$T^{3} + 18 T^{2} + 80 T + 100$$
$67$ $$T^{3} + 12 T^{2} - 112 T - 1184$$
$71$ $$T^{3} + 8 T^{2} - 32 T - 128$$
$73$ $$T^{3} + 20 T^{2} + 40 T - 464$$
$79$ $$T^{3} - 192T - 160$$
$83$ $$T^{3} + 10 T^{2} - 12 T - 124$$
$89$ $$T^{3} + 18 T^{2} + 44 T - 40$$
$97$ $$T^{3} + 10 T^{2} - 28 T - 8$$