# Properties

 Label 2790.2.a.bk Level $2790$ Weight $2$ Character orbit 2790.a Self dual yes Analytic conductor $22.278$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + ( - \beta_{2} + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 + (-b2 + 1) * q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} + ( - \beta_{2} + 1) q^{7} + q^{8} + q^{10} + (\beta_{2} + 1) q^{11} + ( - \beta_1 + 2) q^{13} + ( - \beta_{2} + 1) q^{14} + q^{16} + (\beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + q^{20} + (\beta_{2} + 1) q^{22} + ( - \beta_{3} + \beta_{2}) q^{23} + q^{25} + ( - \beta_1 + 2) q^{26} + ( - \beta_{2} + 1) q^{28} + 2 \beta_1 q^{29} + q^{31} + q^{32} + (\beta_{3} - \beta_1 + 1) q^{34} + ( - \beta_{2} + 1) q^{35} + ( - 2 \beta_{3} + \beta_1) q^{37} + ( - \beta_{2} + \beta_1 + 1) q^{38} + q^{40} + 2 \beta_1 q^{41} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{43} + (\beta_{2} + 1) q^{44} + ( - \beta_{3} + \beta_{2}) q^{46} + ( - 2 \beta_{3} - 2) q^{47} + ( - 3 \beta_{2} + \beta_1) q^{49} + q^{50} + ( - \beta_1 + 2) q^{52} + ( - \beta_{2} - \beta_1 - 1) q^{53} + (\beta_{2} + 1) q^{55} + ( - \beta_{2} + 1) q^{56} + 2 \beta_1 q^{58} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{59} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{61} + q^{62} + q^{64} + ( - \beta_1 + 2) q^{65} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{67} + (\beta_{3} - \beta_1 + 1) q^{68} + ( - \beta_{2} + 1) q^{70} + (\beta_{2} - 2 \beta_1 + 1) q^{71} + (3 \beta_{2} - \beta_1 + 5) q^{73} + ( - 2 \beta_{3} + \beta_1) q^{74} + ( - \beta_{2} + \beta_1 + 1) q^{76} + (\beta_{2} - \beta_1 - 5) q^{77} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{79} + q^{80} + 2 \beta_1 q^{82} + (2 \beta_{3} - 2 \beta_{2}) q^{83} + (\beta_{3} - \beta_1 + 1) q^{85} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{86} + (\beta_{2} + 1) q^{88} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{89} + (2 \beta_{3} - 2 \beta_1 + 6) q^{91} + ( - \beta_{3} + \beta_{2}) q^{92} + ( - 2 \beta_{3} - 2) q^{94} + ( - \beta_{2} + \beta_1 + 1) q^{95} + (2 \beta_{2} - 2 \beta_1 + 4) q^{97} + ( - 3 \beta_{2} + \beta_1) q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 + (-b2 + 1) * q^7 + q^8 + q^10 + (b2 + 1) * q^11 + (-b1 + 2) * q^13 + (-b2 + 1) * q^14 + q^16 + (b3 - b1 + 1) * q^17 + (-b2 + b1 + 1) * q^19 + q^20 + (b2 + 1) * q^22 + (-b3 + b2) * q^23 + q^25 + (-b1 + 2) * q^26 + (-b2 + 1) * q^28 + 2*b1 * q^29 + q^31 + q^32 + (b3 - b1 + 1) * q^34 + (-b2 + 1) * q^35 + (-2*b3 + b1) * q^37 + (-b2 + b1 + 1) * q^38 + q^40 + 2*b1 * q^41 + (2*b3 + b2 - b1 + 5) * q^43 + (b2 + 1) * q^44 + (-b3 + b2) * q^46 + (-2*b3 - 2) * q^47 + (-3*b2 + b1) * q^49 + q^50 + (-b1 + 2) * q^52 + (-b2 - b1 - 1) * q^53 + (b2 + 1) * q^55 + (-b2 + 1) * q^56 + 2*b1 * q^58 + (2*b3 + 2*b2 - b1 + 4) * q^59 + (-b3 + 4*b2 + b1 - 1) * q^61 + q^62 + q^64 + (-b1 + 2) * q^65 + (2*b3 - 2*b2 + b1 + 2) * q^67 + (b3 - b1 + 1) * q^68 + (-b2 + 1) * q^70 + (b2 - 2*b1 + 1) * q^71 + (3*b2 - b1 + 5) * q^73 + (-2*b3 + b1) * q^74 + (-b2 + b1 + 1) * q^76 + (b2 - b1 - 5) * q^77 + (-2*b3 + b2 - b1 + 1) * q^79 + q^80 + 2*b1 * q^82 + (2*b3 - 2*b2) * q^83 + (b3 - b1 + 1) * q^85 + (2*b3 + b2 - b1 + 5) * q^86 + (b2 + 1) * q^88 + (-2*b3 - 3*b2 + 2*b1 - 5) * q^89 + (2*b3 - 2*b1 + 6) * q^91 + (-b3 + b2) * q^92 + (-2*b3 - 2) * q^94 + (-b2 + b1 + 1) * q^95 + (2*b2 - 2*b1 + 4) * q^97 + (-3*b2 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 7$$ 2*v^2 - 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 7 ) / 2$$ (b3 + 7) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 1$$ b2 + 2*b1 + 1

## 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
 2.36865 −0.787711 −2.10710 1.52616
1.00000 0 1.00000 1.00000 0 −1.81471 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −0.662077 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 2.92682 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 4.54997 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.bk yes 4
3.b odd 2 1 2790.2.a.bj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2790.2.a.bj 4 3.b odd 2 1
2790.2.a.bk yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7}^{4} - 5T_{7}^{3} - 4T_{7}^{2} + 24T_{7} + 16$$ T7^4 - 5*T7^3 - 4*T7^2 + 24*T7 + 16 $$T_{11}^{4} - 3T_{11}^{3} - 10T_{11}^{2} + 20T_{11} + 24$$ T11^4 - 3*T11^3 - 10*T11^2 + 20*T11 + 24 $$T_{13}^{4} - 6T_{13}^{3} - 12T_{13}^{2} + 56T_{13} + 64$$ T13^4 - 6*T13^3 - 12*T13^2 + 56*T13 + 64 $$T_{17}^{4} - 40T_{17}^{2} - 80T_{17} + 48$$ T17^4 - 40*T17^2 - 80*T17 + 48 $$T_{19}^{4} - 7T_{19}^{3} - 12T_{19}^{2} + 48T_{19} + 64$$ T19^4 - 7*T19^3 - 12*T19^2 + 48*T19 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} - 5 T^{3} - 4 T^{2} + 24 T + 16$$
$11$ $$T^{4} - 3 T^{3} - 10 T^{2} + 20 T + 24$$
$13$ $$T^{4} - 6 T^{3} - 12 T^{2} + 56 T + 64$$
$17$ $$T^{4} - 40 T^{2} - 80 T + 48$$
$19$ $$T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64$$
$23$ $$T^{4} - T^{3} - 36 T^{2} - 80 T - 48$$
$29$ $$T^{4} - 4 T^{3} - 96 T^{2} + \cdots + 1536$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} - 6 T^{3} - 100 T^{2} + \cdots + 2272$$
$41$ $$T^{4} - 4 T^{3} - 96 T^{2} + \cdots + 1536$$
$43$ $$T^{4} - 13 T^{3} - 68 T^{2} + \cdots + 2624$$
$47$ $$T^{4} + 4 T^{3} - 112 T^{2} + \cdots + 1536$$
$53$ $$T^{4} + 5 T^{3} - 38 T^{2} - 68 T - 24$$
$59$ $$T^{4} - 8 T^{3} - 152 T^{2} + \cdots + 3072$$
$61$ $$T^{4} + 4 T^{3} - 256 T^{2} + \cdots + 15376$$
$67$ $$T^{4} - 8 T^{3} - 160 T^{2} + \cdots - 5552$$
$71$ $$T^{4} + T^{3} - 98 T^{2} - 28 T + 2136$$
$73$ $$T^{4} - 15 T^{3} - 36 T^{2} + \cdots - 3008$$
$79$ $$T^{4} - 5 T^{3} - 156 T^{2} - 208 T - 64$$
$83$ $$T^{4} + 2 T^{3} - 144 T^{2} + \cdots - 768$$
$89$ $$T^{4} + 9 T^{3} - 238 T^{2} + \cdots - 3768$$
$97$ $$T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128$$