# Properties

 Label 2842.2.a.x Level $2842$ Weight $2$ Character orbit 2842.a Self dual yes Analytic conductor $22.693$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# 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: $$5$$ Coefficient field: 5.5.1019601.1 Defining polynomial: $$x^{5} - 10x^{3} - x^{2} + 24x + 7$$ x^5 - 10*x^3 - x^2 + 24*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 406) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_{3} - 1) q^{3} + q^{4} + (\beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{6} + q^{8} + (\beta_{4} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + q^2 + (-b3 - 1) * q^3 + q^4 + (b3 - b1 - 1) * q^5 + (-b3 - 1) * q^6 + q^8 + (b4 - b2 + 2) * q^9 $$q + q^{2} + ( - \beta_{3} - 1) q^{3} + q^{4} + (\beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{6} + q^{8} + (\beta_{4} - \beta_{2} + 2) q^{9} + (\beta_{3} - \beta_1 - 1) q^{10} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} - 1) q^{12} + ( - \beta_{3} - \beta_{2} - 2) q^{13} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{15} + q^{16} + (\beta_{2} + \beta_1 - 2) q^{17} + (\beta_{4} - \beta_{2} + 2) q^{18} + (\beta_{3} + 2 \beta_1) q^{19} + (\beta_{3} - \beta_1 - 1) q^{20} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{22} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{3} - 1) q^{24} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{25} + ( - \beta_{3} - \beta_{2} - 2) q^{26} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{27} - q^{29} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{30} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 3) q^{31} + q^{32} + (3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{33} + (\beta_{2} + \beta_1 - 2) q^{34} + (\beta_{4} - \beta_{2} + 2) q^{36} + (2 \beta_{4} + \beta_{2} - 2) q^{37} + (\beta_{3} + 2 \beta_1) q^{38} + (\beta_{4} - \beta_{2} + \beta_1 + 4) q^{39} + (\beta_{3} - \beta_1 - 1) q^{40} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{41} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{43} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{44} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{45} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{46} + (3 \beta_{3} - 2 \beta_1 - 2) q^{47} + ( - \beta_{3} - 1) q^{48} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{50} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{51} + ( - \beta_{3} - \beta_{2} - 2) q^{52} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{53} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{54} + (4 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 2) q^{55} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{57} - q^{58} + (2 \beta_{3} - \beta_{2} + 3) q^{59} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{60} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{61} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 3) q^{62} + q^{64} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{65} + (3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{66} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 4 \beta_1) q^{67} + (\beta_{2} + \beta_1 - 2) q^{68} + (2 \beta_{4} + 3 \beta_{2} + 3 \beta_1 - 4) q^{69} + (\beta_{3} - 4 \beta_1 - 4) q^{71} + (\beta_{4} - \beta_{2} + 2) q^{72} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - \beta_1) q^{73} + (2 \beta_{4} + \beta_{2} - 2) q^{74} + (3 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + \beta_1 + 6) q^{75} + (\beta_{3} + 2 \beta_1) q^{76} + (\beta_{4} - \beta_{2} + \beta_1 + 4) q^{78} + ( - 2 \beta_{4} + \beta_{2} - 3) q^{79} + (\beta_{3} - \beta_1 - 1) q^{80} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 3) q^{81} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{82} + (\beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 6) q^{83} + (2 \beta_{4} - 4 \beta_{3} + \beta_1 - 3) q^{85} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{86} + (\beta_{3} + 1) q^{87} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{88} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 3) q^{89} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{90} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{92} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 5) q^{93} + (3 \beta_{3} - 2 \beta_1 - 2) q^{94} + (2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{95} + ( - \beta_{3} - 1) q^{96} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 8) q^{97} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 8) q^{99}+O(q^{100})$$ q + q^2 + (-b3 - 1) * q^3 + q^4 + (b3 - b1 - 1) * q^5 + (-b3 - 1) * q^6 + q^8 + (b4 - b2 + 2) * q^9 + (b3 - b1 - 1) * q^10 + (-b4 + b3 + b2 + b1) * q^11 + (-b3 - 1) * q^12 + (-b3 - b2 - 2) * q^13 + (2*b3 + 2*b2 + b1 - 2) * q^15 + q^16 + (b2 + b1 - 2) * q^17 + (b4 - b2 + 2) * q^18 + (b3 + 2*b1) * q^19 + (b3 - b1 - 1) * q^20 + (-b4 + b3 + b2 + b1) * q^22 + (-b4 + b3 - b2 - b1 + 1) * q^23 + (-b3 - 1) * q^24 + (-b4 - 3*b3 - 2*b2 + 2*b1 + 2) * q^25 + (-b3 - b2 - 2) * q^26 + (-2*b4 - b3 - b2 - 3) * q^27 - q^29 + (2*b3 + 2*b2 + b1 - 2) * q^30 + (2*b4 - b3 + b2 - 3) * q^31 + q^32 + (3*b3 + b2 - b1 - 1) * q^33 + (b2 + b1 - 2) * q^34 + (b4 - b2 + 2) * q^36 + (2*b4 + b2 - 2) * q^37 + (b3 + 2*b1) * q^38 + (b4 - b2 + b1 + 4) * q^39 + (b3 - b1 - 1) * q^40 + (-b4 + b3 + 2*b2 + 2*b1 - 5) * q^41 + (3*b4 - 2*b3 - b2 - 1) * q^43 + (-b4 + b3 + b2 + b1) * q^44 + (-3*b4 + 3*b3 + b2) * q^45 + (-b4 + b3 - b2 - b1 + 1) * q^46 + (3*b3 - 2*b1 - 2) * q^47 + (-b3 - 1) * q^48 + (-b4 - 3*b3 - 2*b2 + 2*b1 + 2) * q^50 + (-b4 + 3*b3 - b2 - 2*b1 + 3) * q^51 + (-b3 - b2 - 2) * q^52 + (b3 + 2*b2 - 2*b1 + 1) * q^53 + (-2*b4 - b3 - b2 - 3) * q^54 + (4*b4 - 5*b3 - 3*b2 - 3*b1 - 2) * q^55 + (-3*b4 + b3 - b2 - 2*b1 - 6) * q^57 - q^58 + (2*b3 - b2 + 3) * q^59 + (2*b3 + 2*b2 + b1 - 2) * q^60 + (2*b3 + 2*b2 + 2*b1 - 3) * q^61 + (2*b4 - b3 + b2 - 3) * q^62 + q^64 + (-b4 + 2*b3 + 2*b2 + 2*b1 + 1) * q^65 + (3*b3 + b2 - b1 - 1) * q^66 + (-b4 + b3 - b2 - 4*b1) * q^67 + (b2 + b1 - 2) * q^68 + (2*b4 + 3*b2 + 3*b1 - 4) * q^69 + (b3 - 4*b1 - 4) * q^71 + (b4 - b2 + 2) * q^72 + (3*b4 - 3*b3 - 3*b2 - b1) * q^73 + (2*b4 + b2 - 2) * q^74 + (3*b4 - 6*b3 - 4*b2 + b1 + 6) * q^75 + (b3 + 2*b1) * q^76 + (b4 - b2 + b1 + 4) * q^78 + (-2*b4 + b2 - 3) * q^79 + (b3 - b1 - 1) * q^80 + (2*b4 + 3*b3 + 4*b2 + 3*b1 + 3) * q^81 + (-b4 + b3 + 2*b2 + 2*b1 - 5) * q^82 + (b4 - 3*b3 - 2*b2 - b1 - 6) * q^83 + (2*b4 - 4*b3 + b1 - 3) * q^85 + (3*b4 - 2*b3 - b2 - 1) * q^86 + (b3 + 1) * q^87 + (-b4 + b3 + b2 + b1) * q^88 + (-b4 + 3*b3 + 2*b2 - 5*b1 - 3) * q^89 + (-3*b4 + 3*b3 + b2) * q^90 + (-b4 + b3 - b2 - b1 + 1) * q^92 + (-3*b4 + b3 - 3*b2 - 3*b1 + 5) * q^93 + (3*b3 - 2*b1 - 2) * q^94 + (2*b4 - 5*b3 - 2*b2 - 2*b1 - 3) * q^95 + (-b3 - 1) * q^96 + (-3*b4 + 2*b3 + b2 - 8) * q^97 + (b4 + 2*b3 + b2 - 3*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 - 3 * q^3 + 5 * q^4 - 7 * q^5 - 3 * q^6 + 5 * q^8 + 8 * q^9 $$5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9} - 7 q^{10} - 3 q^{12} - 10 q^{13} - 10 q^{15} + 5 q^{16} - 8 q^{17} + 8 q^{18} - 2 q^{19} - 7 q^{20} + q^{23} - 3 q^{24} + 12 q^{25} - 10 q^{26} - 15 q^{27} - 5 q^{29} - 10 q^{30} - 11 q^{31} + 5 q^{32} - 9 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 2 q^{38} + 18 q^{39} - 7 q^{40} - 23 q^{41} - 3 q^{43} - 4 q^{45} + q^{46} - 16 q^{47} - 3 q^{48} + 12 q^{50} + 7 q^{51} - 10 q^{52} + 7 q^{53} - 15 q^{54} - 6 q^{55} - 34 q^{57} - 5 q^{58} + 9 q^{59} - 10 q^{60} - 15 q^{61} - 11 q^{62} + 5 q^{64} + 5 q^{65} - 9 q^{66} - 4 q^{67} - 8 q^{68} - 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} + 34 q^{75} - 2 q^{76} + 18 q^{78} - 13 q^{79} - 7 q^{80} + 17 q^{81} - 23 q^{82} - 28 q^{83} - 7 q^{85} - 3 q^{86} + 3 q^{87} - 17 q^{89} - 4 q^{90} + q^{92} + 17 q^{93} - 16 q^{94} - 9 q^{95} - 3 q^{96} - 42 q^{97} - 42 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 - 3 * q^3 + 5 * q^4 - 7 * q^5 - 3 * q^6 + 5 * q^8 + 8 * q^9 - 7 * q^10 - 3 * q^12 - 10 * q^13 - 10 * q^15 + 5 * q^16 - 8 * q^17 + 8 * q^18 - 2 * q^19 - 7 * q^20 + q^23 - 3 * q^24 + 12 * q^25 - 10 * q^26 - 15 * q^27 - 5 * q^29 - 10 * q^30 - 11 * q^31 + 5 * q^32 - 9 * q^33 - 8 * q^34 + 8 * q^36 - 8 * q^37 - 2 * q^38 + 18 * q^39 - 7 * q^40 - 23 * q^41 - 3 * q^43 - 4 * q^45 + q^46 - 16 * q^47 - 3 * q^48 + 12 * q^50 + 7 * q^51 - 10 * q^52 + 7 * q^53 - 15 * q^54 - 6 * q^55 - 34 * q^57 - 5 * q^58 + 9 * q^59 - 10 * q^60 - 15 * q^61 - 11 * q^62 + 5 * q^64 + 5 * q^65 - 9 * q^66 - 4 * q^67 - 8 * q^68 - 14 * q^69 - 22 * q^71 + 8 * q^72 - 8 * q^74 + 34 * q^75 - 2 * q^76 + 18 * q^78 - 13 * q^79 - 7 * q^80 + 17 * q^81 - 23 * q^82 - 28 * q^83 - 7 * q^85 - 3 * q^86 + 3 * q^87 - 17 * q^89 - 4 * q^90 + q^92 + 17 * q^93 - 16 * q^94 - 9 * q^95 - 3 * q^96 - 42 * q^97 - 42 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 10x^{3} - x^{2} + 24x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5\nu + 1$$ -v^3 + 5*v + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + \nu^{2} - 5\nu - 5$$ v^3 + v^2 - 5*v - 5 $$\beta_{4}$$ $$=$$ $$\nu^{4} + 2\nu^{3} - 5\nu^{2} - 10\nu - 2$$ v^4 + 2*v^3 - 5*v^2 - 10*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4$$ b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 5\beta _1 + 1$$ -b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{3} + 7\beta_{2} + 20$$ b4 + 5*b3 + 7*b2 + 20

## 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
 2.37936 −1.80092 2.20194 −2.48141 −0.298978
1.00000 −3.23497 1.00000 −1.14439 −3.23497 0 1.00000 7.46501 −1.14439
1.2 1.00000 −2.40697 1.00000 2.20789 −2.40697 0 1.00000 2.79350 2.20789
1.3 1.00000 −0.515089 1.00000 −3.68685 −0.515089 0 1.00000 −2.73468 −3.68685
1.4 1.00000 0.714579 1.00000 −0.233169 0.714579 0 1.00000 −2.48938 −0.233169
1.5 1.00000 2.44245 1.00000 −4.14347 2.44245 0 1.00000 2.96555 −4.14347
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 2842.2.a.x 5
7.b odd 2 1 2842.2.a.z 5
7.d odd 6 2 406.2.e.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.a 10 7.d odd 6 2
2842.2.a.x 5 1.a even 1 1 trivial
2842.2.a.z 5 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2842))$$:

 $$T_{3}^{5} + 3T_{3}^{4} - 7T_{3}^{3} - 19T_{3}^{2} + 6T_{3} + 7$$ T3^5 + 3*T3^4 - 7*T3^3 - 19*T3^2 + 6*T3 + 7 $$T_{5}^{5} + 7T_{5}^{4} + 6T_{5}^{3} - 35T_{5}^{2} - 47T_{5} - 9$$ T5^5 + 7*T5^4 + 6*T5^3 - 35*T5^2 - 47*T5 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{5}$$
$3$ $$T^{5} + 3 T^{4} - 7 T^{3} - 19 T^{2} + \cdots + 7$$
$5$ $$T^{5} + 7 T^{4} + 6 T^{3} - 35 T^{2} + \cdots - 9$$
$7$ $$T^{5}$$
$11$ $$T^{5} - 33 T^{3} - 54 T^{2} - 12 T + 9$$
$13$ $$T^{5} + 10 T^{4} + 28 T^{3} + \cdots - 103$$
$17$ $$T^{5} + 8 T^{4} + 9 T^{3} - 22 T^{2} + \cdots - 3$$
$19$ $$T^{5} + 2 T^{4} - 63 T^{3} + \cdots + 1623$$
$23$ $$T^{5} - T^{4} - 39 T^{3} - 97 T^{2} + \cdots + 33$$
$29$ $$(T + 1)^{5}$$
$31$ $$T^{5} + 11 T^{4} - 24 T^{3} + \cdots - 387$$
$37$ $$T^{5} + 8 T^{4} - 47 T^{3} - 422 T^{2} + \cdots + 349$$
$41$ $$T^{5} + 23 T^{4} + 132 T^{3} + \cdots - 7113$$
$43$ $$T^{5} + 3 T^{4} - 163 T^{3} + \cdots + 16843$$
$47$ $$T^{5} + 16 T^{4} + 9 T^{3} - 584 T^{2} + \cdots - 63$$
$53$ $$T^{5} - 7 T^{4} - 57 T^{3} - 49 T^{2} + \cdots + 183$$
$59$ $$T^{5} - 9 T^{4} - 39 T^{3} + 291 T^{2} + \cdots - 657$$
$61$ $$T^{5} + 15 T^{4} - 10 T^{3} + \cdots + 5327$$
$67$ $$T^{5} + 4 T^{4} - 174 T^{3} + \cdots + 11481$$
$71$ $$T^{5} + 22 T^{4} + 51 T^{3} + \cdots + 18189$$
$73$ $$T^{5} - 247 T^{3} + 226 T^{2} + \cdots + 341$$
$79$ $$T^{5} + 13 T^{4} - 33 T^{3} + \cdots + 1563$$
$83$ $$T^{5} + 28 T^{4} + 213 T^{3} + \cdots - 25851$$
$89$ $$T^{5} + 17 T^{4} - 195 T^{3} + \cdots + 162771$$
$97$ $$T^{5} + 42 T^{4} + 539 T^{3} + \cdots - 15007$$