# Properties

 Label 5070.2.a.be Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.73205 −1.73205
1.00000 −1.00000 1.00000 −1.00000 −1.00000 3.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 3.00000 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.be 2
13.b even 2 1 5070.2.a.ba 2
13.d odd 4 2 5070.2.b.p 4
13.f odd 12 2 390.2.bb.a 4
39.k even 12 2 1170.2.bs.d 4
65.o even 12 2 1950.2.y.d 4
65.s odd 12 2 1950.2.bc.a 4
65.t even 12 2 1950.2.y.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 13.f odd 12 2
1170.2.bs.d 4 39.k even 12 2
1950.2.y.d 4 65.o even 12 2
1950.2.y.e 4 65.t even 12 2
1950.2.bc.a 4 65.s odd 12 2
5070.2.a.ba 2 13.b even 2 1
5070.2.a.be 2 1.a even 1 1 trivial
5070.2.b.p 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7} - 3$$ T7 - 3 $$T_{11}^{2} + 4T_{11} + 1$$ T11^2 + 4*T11 + 1 $$T_{17} + 4$$ T17 + 4 $$T_{31}^{2} - 4T_{31} - 44$$ T31^2 - 4*T31 - 44

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2} + 4T + 1$$
$13$ $$T^{2}$$
$17$ $$(T + 4)^{2}$$
$19$ $$T^{2} + 8T + 13$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} + 4T - 8$$
$31$ $$T^{2} - 4T - 44$$
$37$ $$T^{2} + 2T - 47$$
$41$ $$(T + 4)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 6T - 3$$
$53$ $$T^{2} + 4T + 1$$
$59$ $$T^{2} + 16T + 52$$
$61$ $$T^{2} + 8T + 4$$
$67$ $$T^{2} + 4T - 8$$
$71$ $$T^{2} - 12T - 12$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2} - 20T + 52$$
$83$ $$T^{2} + 12T + 24$$
$89$ $$T^{2} - 4T - 143$$
$97$ $$T^{2} - 4T - 104$$