# Properties

 Label 2790.2.a.bg Level $2790$ Weight $2$ Character orbit 2790.a Self dual yes Analytic conductor $22.278$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.44949 2.44949
1.00000 0 1.00000 −1.00000 0 −2.00000 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −2.00000 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.bg 2
3.b odd 2 1 310.2.a.d 2
12.b even 2 1 2480.2.a.q 2
15.d odd 2 1 1550.2.a.i 2
15.e even 4 2 1550.2.b.g 4
24.f even 2 1 9920.2.a.bq 2
24.h odd 2 1 9920.2.a.bo 2
93.c even 2 1 9610.2.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.d 2 3.b odd 2 1
1550.2.a.i 2 15.d odd 2 1
1550.2.b.g 4 15.e even 4 2
2480.2.a.q 2 12.b even 2 1
2790.2.a.bg 2 1.a even 1 1 trivial
9610.2.a.f 2 93.c even 2 1
9920.2.a.bo 2 24.h odd 2 1
9920.2.a.bq 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} + 4T_{11} - 2$$ T11^2 + 4*T11 - 2 $$T_{13}^{2} - 4T_{13} - 2$$ T13^2 - 4*T13 - 2 $$T_{17}^{2} - 24$$ T17^2 - 24 $$T_{19}^{2} - 24$$ T19^2 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 4T - 2$$
$13$ $$T^{2} - 4T - 2$$
$17$ $$T^{2} - 24$$
$19$ $$T^{2} - 24$$
$23$ $$(T + 2)^{2}$$
$29$ $$T^{2} + 16T + 58$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} - 4T - 2$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 16T + 58$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} - 4T - 50$$
$59$ $$T^{2} + 8T - 8$$
$61$ $$T^{2} + 8T + 10$$
$67$ $$T^{2} + 16T + 40$$
$71$ $$T^{2} - 96$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 24$$
$83$ $$T^{2} + 8T - 134$$
$89$ $$T^{2} + 12T - 60$$
$97$ $$T^{2} - 8T - 8$$