# Properties

 Label 5070.2.a.bp Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + (\beta_{2} + \beta_1 - 2) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 + (b2 + b1 - 2) * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + (\beta_{2} + \beta_1 - 2) q^{7} - q^{8} + q^{9} - q^{10} + (\beta_{2} - 2 \beta_1 + 1) q^{11} + q^{12} + ( - \beta_{2} - \beta_1 + 2) q^{14} + q^{15} + q^{16} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{17} - q^{18} + ( - 3 \beta_{2} + \beta_1 - 1) q^{19} + q^{20} + (\beta_{2} + \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + \beta_1) q^{23} - q^{24} + q^{25} + q^{27} + (\beta_{2} + \beta_1 - 2) q^{28} + (5 \beta_{2} - 7 \beta_1 + 4) q^{29} - q^{30} + ( - \beta_{2} - 2 \beta_1 - 5) q^{31} - q^{32} + (\beta_{2} - 2 \beta_1 + 1) q^{33} + (2 \beta_{2} - 3 \beta_1 + 4) q^{34} + (\beta_{2} + \beta_1 - 2) q^{35} + q^{36} + (3 \beta_{2} - 5 \beta_1 - 4) q^{37} + (3 \beta_{2} - \beta_1 + 1) q^{38} - q^{40} + ( - \beta_{2} + 3 \beta_1 - 7) q^{41} + ( - \beta_{2} - \beta_1 + 2) q^{42} + ( - \beta_{2} - 4 \beta_1 + 4) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{44} + q^{45} + (2 \beta_{2} - \beta_1) q^{46} + ( - 2 \beta_{2} + 6 \beta_1 - 5) q^{47} + q^{48} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{49} - q^{50} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{51} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{53} - q^{54} + (\beta_{2} - 2 \beta_1 + 1) q^{55} + ( - \beta_{2} - \beta_1 + 2) q^{56} + ( - 3 \beta_{2} + \beta_1 - 1) q^{57} + ( - 5 \beta_{2} + 7 \beta_1 - 4) q^{58} + ( - \beta_{2} + 4 \beta_1 - 4) q^{59} + q^{60} + (\beta_{2} + 2) q^{61} + (\beta_{2} + 2 \beta_1 + 5) q^{62} + (\beta_{2} + \beta_1 - 2) q^{63} + q^{64} + ( - \beta_{2} + 2 \beta_1 - 1) q^{66} + (2 \beta_{2} - 3 \beta_1 - 4) q^{67} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{68} + ( - 2 \beta_{2} + \beta_1) q^{69} + ( - \beta_{2} - \beta_1 + 2) q^{70} + ( - 2 \beta_{2} - \beta_1 + 2) q^{71} - q^{72} + ( - 8 \beta_{2} + 7 \beta_1 - 6) q^{73} + ( - 3 \beta_{2} + 5 \beta_1 + 4) q^{74} + q^{75} + ( - 3 \beta_{2} + \beta_1 - 1) q^{76} + ( - 5 \beta_{2} + 6 \beta_1 - 6) q^{77} + (7 \beta_{2} - 3 \beta_1 + 9) q^{79} + q^{80} + q^{81} + (\beta_{2} - 3 \beta_1 + 7) q^{82} + (7 \beta_{2} - 8 \beta_1 + 3) q^{83} + (\beta_{2} + \beta_1 - 2) q^{84} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{85} + (\beta_{2} + 4 \beta_1 - 4) q^{86} + (5 \beta_{2} - 7 \beta_1 + 4) q^{87} + ( - \beta_{2} + 2 \beta_1 - 1) q^{88} + ( - 6 \beta_{2} - 3 \beta_1) q^{89} - q^{90} + ( - 2 \beta_{2} + \beta_1) q^{92} + ( - \beta_{2} - 2 \beta_1 - 5) q^{93} + (2 \beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 3 \beta_{2} + \beta_1 - 1) q^{95} - q^{96} + (5 \beta_{2} - \beta_1 - 6) q^{97} + (2 \beta_{2} + 3 \beta_1 - 2) q^{98} + (\beta_{2} - 2 \beta_1 + 1) q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 + (b2 + b1 - 2) * q^7 - q^8 + q^9 - q^10 + (b2 - 2*b1 + 1) * q^11 + q^12 + (-b2 - b1 + 2) * q^14 + q^15 + q^16 + (-2*b2 + 3*b1 - 4) * q^17 - q^18 + (-3*b2 + b1 - 1) * q^19 + q^20 + (b2 + b1 - 2) * q^21 + (-b2 + 2*b1 - 1) * q^22 + (-2*b2 + b1) * q^23 - q^24 + q^25 + q^27 + (b2 + b1 - 2) * q^28 + (5*b2 - 7*b1 + 4) * q^29 - q^30 + (-b2 - 2*b1 - 5) * q^31 - q^32 + (b2 - 2*b1 + 1) * q^33 + (2*b2 - 3*b1 + 4) * q^34 + (b2 + b1 - 2) * q^35 + q^36 + (3*b2 - 5*b1 - 4) * q^37 + (3*b2 - b1 + 1) * q^38 - q^40 + (-b2 + 3*b1 - 7) * q^41 + (-b2 - b1 + 2) * q^42 + (-b2 - 4*b1 + 4) * q^43 + (b2 - 2*b1 + 1) * q^44 + q^45 + (2*b2 - b1) * q^46 + (-2*b2 + 6*b1 - 5) * q^47 + q^48 + (-2*b2 - 3*b1 + 2) * q^49 - q^50 + (-2*b2 + 3*b1 - 4) * q^51 + (-3*b2 + 3*b1 - 1) * q^53 - q^54 + (b2 - 2*b1 + 1) * q^55 + (-b2 - b1 + 2) * q^56 + (-3*b2 + b1 - 1) * q^57 + (-5*b2 + 7*b1 - 4) * q^58 + (-b2 + 4*b1 - 4) * q^59 + q^60 + (b2 + 2) * q^61 + (b2 + 2*b1 + 5) * q^62 + (b2 + b1 - 2) * q^63 + q^64 + (-b2 + 2*b1 - 1) * q^66 + (2*b2 - 3*b1 - 4) * q^67 + (-2*b2 + 3*b1 - 4) * q^68 + (-2*b2 + b1) * q^69 + (-b2 - b1 + 2) * q^70 + (-2*b2 - b1 + 2) * q^71 - q^72 + (-8*b2 + 7*b1 - 6) * q^73 + (-3*b2 + 5*b1 + 4) * q^74 + q^75 + (-3*b2 + b1 - 1) * q^76 + (-5*b2 + 6*b1 - 6) * q^77 + (7*b2 - 3*b1 + 9) * q^79 + q^80 + q^81 + (b2 - 3*b1 + 7) * q^82 + (7*b2 - 8*b1 + 3) * q^83 + (b2 + b1 - 2) * q^84 + (-2*b2 + 3*b1 - 4) * q^85 + (b2 + 4*b1 - 4) * q^86 + (5*b2 - 7*b1 + 4) * q^87 + (-b2 + 2*b1 - 1) * q^88 + (-6*b2 - 3*b1) * q^89 - q^90 + (-2*b2 + b1) * q^92 + (-b2 - 2*b1 - 5) * q^93 + (2*b2 - 6*b1 + 5) * q^94 + (-3*b2 + b1 - 1) * q^95 - q^96 + (5*b2 - b1 - 6) * q^97 + (2*b2 + 3*b1 - 2) * q^98 + (b2 - 2*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} - 6 q^{28} - 3 q^{30} - 16 q^{31} - 3 q^{32} + 7 q^{34} - 6 q^{35} + 3 q^{36} - 20 q^{37} - q^{38} - 3 q^{40} - 17 q^{41} + 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} - 7 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} + 6 q^{56} + q^{57} - 7 q^{59} + 3 q^{60} + 5 q^{61} + 16 q^{62} - 6 q^{63} + 3 q^{64} - 17 q^{67} - 7 q^{68} + 3 q^{69} + 6 q^{70} + 7 q^{71} - 3 q^{72} - 3 q^{73} + 20 q^{74} + 3 q^{75} + q^{76} - 7 q^{77} + 17 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} - 6 q^{83} - 6 q^{84} - 7 q^{85} - 9 q^{86} + 3 q^{89} - 3 q^{90} + 3 q^{92} - 16 q^{93} + 7 q^{94} + q^{95} - 3 q^{96} - 24 q^{97} - 5 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 + 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + q^19 + 3 * q^20 - 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 - 6 * q^28 - 3 * q^30 - 16 * q^31 - 3 * q^32 + 7 * q^34 - 6 * q^35 + 3 * q^36 - 20 * q^37 - q^38 - 3 * q^40 - 17 * q^41 + 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 - 7 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 + 6 * q^56 + q^57 - 7 * q^59 + 3 * q^60 + 5 * q^61 + 16 * q^62 - 6 * q^63 + 3 * q^64 - 17 * q^67 - 7 * q^68 + 3 * q^69 + 6 * q^70 + 7 * q^71 - 3 * q^72 - 3 * q^73 + 20 * q^74 + 3 * q^75 + q^76 - 7 * q^77 + 17 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 - 6 * q^83 - 6 * q^84 - 7 * q^85 - 9 * q^86 + 3 * q^89 - 3 * q^90 + 3 * q^92 - 16 * q^93 + 7 * q^94 + q^95 - 3 * q^96 - 24 * q^97 - 5 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 −1.24698 0.445042 1.80194
−1.00000 1.00000 1.00000 1.00000 −1.00000 −3.69202 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −3.35690 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 1.04892 −1.00000 1.00000 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 5070.2.a.bp 3
13.b even 2 1 5070.2.a.bw yes 3
13.d odd 4 2 5070.2.b.z 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bp 3 1.a even 1 1 trivial
5070.2.a.bw yes 3 13.b even 2 1
5070.2.b.z 6 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}^{3} + 6T_{7}^{2} + 5T_{7} - 13$$ T7^3 + 6*T7^2 + 5*T7 - 13 $$T_{11}^{3} - 7T_{11} - 7$$ T11^3 - 7*T11 - 7 $$T_{17}^{3} + 7T_{17}^{2} - 7$$ T17^3 + 7*T17^2 - 7 $$T_{31}^{3} + 16T_{31}^{2} + 69T_{31} + 83$$ T31^3 + 16*T31^2 + 69*T31 + 83

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 6 T^{2} + 5 T - 13$$
$11$ $$T^{3} - 7T - 7$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 7T^{2} - 7$$
$19$ $$T^{3} - T^{2} - 16 T - 13$$
$23$ $$T^{3} - 3 T^{2} - 4 T - 1$$
$29$ $$T^{3} - 91T - 203$$
$31$ $$T^{3} + 16 T^{2} + 69 T + 83$$
$37$ $$T^{3} + 20 T^{2} + 89 T - 97$$
$41$ $$T^{3} + 17 T^{2} + 80 T + 113$$
$43$ $$T^{3} - 9 T^{2} - 22 T + 169$$
$47$ $$T^{3} + 7 T^{2} - 49 T + 49$$
$53$ $$T^{3} - 3 T^{2} - 18 T + 13$$
$59$ $$T^{3} + 7 T^{2} - 14 T - 7$$
$61$ $$T^{3} - 5 T^{2} + 6 T - 1$$
$67$ $$T^{3} + 17 T^{2} + 80 T + 71$$
$71$ $$T^{3} - 7T^{2} + 49$$
$73$ $$T^{3} + 3 T^{2} - 130 T - 433$$
$79$ $$T^{3} - 17 T^{2} + 10 T + 587$$
$83$ $$T^{3} + 6 T^{2} - 121 T - 349$$
$89$ $$T^{3} - 3 T^{2} - 144 T + 783$$
$97$ $$T^{3} + 24 T^{2} + 143 T + 169$$