Properties

 Label 2842.2.a.p Level $2842$ Weight $2$ Character orbit 2842.a Self dual yes Analytic conductor $22.693$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2842 = 2 \cdot 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2842.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.6934842544$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.9248.1 Defining polynomial: $$x^{4} - 5x^{2} + 2$$ x^4 - 5*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_{3} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q - q^2 - b3 * q^3 + q^4 - b1 * q^5 + b3 * q^6 - q^8 - b2 * q^9 $$q - q^{2} - \beta_{3} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9} + \beta_1 q^{10} + ( - \beta_{2} - 3) q^{11} - \beta_{3} q^{12} + (2 \beta_{3} + \beta_1) q^{13} + (\beta_{2} + 1) q^{15} + q^{16} + (\beta_{3} - 3 \beta_1) q^{17} + \beta_{2} q^{18} + ( - 2 \beta_{3} + 4 \beta_1) q^{19} - \beta_1 q^{20} + (\beta_{2} + 3) q^{22} + (4 \beta_{2} + 2) q^{23} + \beta_{3} q^{24} + (\beta_{2} - 2) q^{25} + ( - 2 \beta_{3} - \beta_1) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + q^{29} + ( - \beta_{2} - 1) q^{30} + (2 \beta_{3} + \beta_1) q^{31} - q^{32} + (\beta_{3} + 2 \beta_1) q^{33} + ( - \beta_{3} + 3 \beta_1) q^{34} - \beta_{2} q^{36} + ( - 2 \beta_{2} + 2) q^{37} + (2 \beta_{3} - 4 \beta_1) q^{38} + (\beta_{2} - 7) q^{39} + \beta_1 q^{40} + (\beta_{3} - 3 \beta_1) q^{41} + (3 \beta_{2} + 3) q^{43} + ( - \beta_{2} - 3) q^{44} + (\beta_{3} + \beta_1) q^{45} + ( - 4 \beta_{2} - 2) q^{46} - \beta_1 q^{47} - \beta_{3} q^{48} + ( - \beta_{2} + 2) q^{50} + 4 \beta_{2} q^{51} + (2 \beta_{3} + \beta_1) q^{52} + (\beta_{2} - 9) q^{53} + ( - \beta_{3} - 2 \beta_1) q^{54} + (\beta_{3} + 4 \beta_1) q^{55} + ( - 6 \beta_{2} + 2) q^{57} - q^{58} + ( - \beta_{3} + 3 \beta_1) q^{59} + (\beta_{2} + 1) q^{60} + 2 \beta_1 q^{61} + ( - 2 \beta_{3} - \beta_1) q^{62} + q^{64} + ( - 3 \beta_{2} - 5) q^{65} + ( - \beta_{3} - 2 \beta_1) q^{66} + ( - 2 \beta_{2} - 10) q^{67} + (\beta_{3} - 3 \beta_1) q^{68} + (6 \beta_{3} - 8 \beta_1) q^{69} + ( - 6 \beta_{2} - 4) q^{71} + \beta_{2} q^{72} + (5 \beta_{3} + 5 \beta_1) q^{73} + (2 \beta_{2} - 2) q^{74} + (4 \beta_{3} - 2 \beta_1) q^{75} + ( - 2 \beta_{3} + 4 \beta_1) q^{76} + ( - \beta_{2} + 7) q^{78} + ( - 5 \beta_{2} - 5) q^{79} - \beta_1 q^{80} + (2 \beta_{2} - 5) q^{81} + ( - \beta_{3} + 3 \beta_1) q^{82} + (3 \beta_{3} + 3 \beta_1) q^{83} + (2 \beta_{2} + 8) q^{85} + ( - 3 \beta_{2} - 3) q^{86} - \beta_{3} q^{87} + (\beta_{2} + 3) q^{88} + (3 \beta_{3} - 3 \beta_1) q^{89} + ( - \beta_{3} - \beta_1) q^{90} + (4 \beta_{2} + 2) q^{92} + (\beta_{2} - 7) q^{93} + \beta_1 q^{94} + ( - 2 \beta_{2} - 10) q^{95} + \beta_{3} q^{96} + (\beta_{3} - 7 \beta_1) q^{97} + (2 \beta_{2} + 4) q^{99}+O(q^{100})$$ q - q^2 - b3 * q^3 + q^4 - b1 * q^5 + b3 * q^6 - q^8 - b2 * q^9 + b1 * q^10 + (-b2 - 3) * q^11 - b3 * q^12 + (2*b3 + b1) * q^13 + (b2 + 1) * q^15 + q^16 + (b3 - 3*b1) * q^17 + b2 * q^18 + (-2*b3 + 4*b1) * q^19 - b1 * q^20 + (b2 + 3) * q^22 + (4*b2 + 2) * q^23 + b3 * q^24 + (b2 - 2) * q^25 + (-2*b3 - b1) * q^26 + (b3 + 2*b1) * q^27 + q^29 + (-b2 - 1) * q^30 + (2*b3 + b1) * q^31 - q^32 + (b3 + 2*b1) * q^33 + (-b3 + 3*b1) * q^34 - b2 * q^36 + (-2*b2 + 2) * q^37 + (2*b3 - 4*b1) * q^38 + (b2 - 7) * q^39 + b1 * q^40 + (b3 - 3*b1) * q^41 + (3*b2 + 3) * q^43 + (-b2 - 3) * q^44 + (b3 + b1) * q^45 + (-4*b2 - 2) * q^46 - b1 * q^47 - b3 * q^48 + (-b2 + 2) * q^50 + 4*b2 * q^51 + (2*b3 + b1) * q^52 + (b2 - 9) * q^53 + (-b3 - 2*b1) * q^54 + (b3 + 4*b1) * q^55 + (-6*b2 + 2) * q^57 - q^58 + (-b3 + 3*b1) * q^59 + (b2 + 1) * q^60 + 2*b1 * q^61 + (-2*b3 - b1) * q^62 + q^64 + (-3*b2 - 5) * q^65 + (-b3 - 2*b1) * q^66 + (-2*b2 - 10) * q^67 + (b3 - 3*b1) * q^68 + (6*b3 - 8*b1) * q^69 + (-6*b2 - 4) * q^71 + b2 * q^72 + (5*b3 + 5*b1) * q^73 + (2*b2 - 2) * q^74 + (4*b3 - 2*b1) * q^75 + (-2*b3 + 4*b1) * q^76 + (-b2 + 7) * q^78 + (-5*b2 - 5) * q^79 - b1 * q^80 + (2*b2 - 5) * q^81 + (-b3 + 3*b1) * q^82 + (3*b3 + 3*b1) * q^83 + (2*b2 + 8) * q^85 + (-3*b2 - 3) * q^86 - b3 * q^87 + (b2 + 3) * q^88 + (3*b3 - 3*b1) * q^89 + (-b3 - b1) * q^90 + (4*b2 + 2) * q^92 + (b2 - 7) * q^93 + b1 * q^94 + (-2*b2 - 10) * q^95 + b3 * q^96 + (b3 - 7*b1) * q^97 + (2*b2 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^16 - 2 * q^18 + 10 * q^22 - 10 * q^25 + 4 * q^29 - 2 * q^30 - 4 * q^32 + 2 * q^36 + 12 * q^37 - 30 * q^39 + 6 * q^43 - 10 * q^44 + 10 * q^50 - 8 * q^51 - 38 * q^53 + 20 * q^57 - 4 * q^58 + 2 * q^60 + 4 * q^64 - 14 * q^65 - 36 * q^67 - 4 * q^71 - 2 * q^72 - 12 * q^74 + 30 * q^78 - 10 * q^79 - 24 * q^81 + 28 * q^85 - 6 * q^86 + 10 * q^88 - 30 * q^93 - 36 * q^95 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1

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
 −0.662153 2.13578 −2.13578 0.662153
−1.00000 −2.35829 1.00000 0.662153 2.35829 0 −1.00000 2.56155 −0.662153
1.2 −1.00000 −1.19935 1.00000 −2.13578 1.19935 0 −1.00000 −1.56155 2.13578
1.3 −1.00000 1.19935 1.00000 2.13578 −1.19935 0 −1.00000 −1.56155 −2.13578
1.4 −1.00000 2.35829 1.00000 −0.662153 −2.35829 0 −1.00000 2.56155 0.662153
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$29$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2842.2.a.p 4
7.b odd 2 1 inner 2842.2.a.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.p 4 1.a even 1 1 trivial
2842.2.a.p 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2842))$$:

 $$T_{3}^{4} - 7T_{3}^{2} + 8$$ T3^4 - 7*T3^2 + 8 $$T_{5}^{4} - 5T_{5}^{2} + 2$$ T5^4 - 5*T5^2 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4} - 7T^{2} + 8$$
$5$ $$T^{4} - 5T^{2} + 2$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 5 T + 2)^{2}$$
$13$ $$T^{4} - 37T^{2} + 338$$
$17$ $$T^{4} - 46T^{2} + 512$$
$19$ $$T^{4} - 92T^{2} + 2048$$
$23$ $$(T^{2} - 68)^{2}$$
$29$ $$(T - 1)^{4}$$
$31$ $$T^{4} - 37T^{2} + 338$$
$37$ $$(T^{2} - 6 T - 8)^{2}$$
$41$ $$T^{4} - 46T^{2} + 512$$
$43$ $$(T^{2} - 3 T - 36)^{2}$$
$47$ $$T^{4} - 5T^{2} + 2$$
$53$ $$(T^{2} + 19 T + 86)^{2}$$
$59$ $$T^{4} - 46T^{2} + 512$$
$61$ $$T^{4} - 20T^{2} + 32$$
$67$ $$(T^{2} + 18 T + 64)^{2}$$
$71$ $$(T^{2} + 2 T - 152)^{2}$$
$73$ $$T^{4} - 350 T^{2} + 20000$$
$79$ $$(T^{2} + 5 T - 100)^{2}$$
$83$ $$T^{4} - 126T^{2} + 2592$$
$89$ $$T^{4} - 90T^{2} + 648$$
$97$ $$T^{4} - 238T^{2} + 9248$$