# Properties

 Label 5070.2.a.ba 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: $$x^{2} - 3$$ 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.

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

## 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.ba 2
13.b even 2 1 5070.2.a.be 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.e 4
65.s odd 12 2 1950.2.bc.a 4
65.t even 12 2 1950.2.y.d 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.t even 12 2
1950.2.y.e 4 65.o even 12 2
1950.2.bc.a 4 65.s odd 12 2
5070.2.a.ba 2 1.a even 1 1 trivial
5070.2.a.be 2 13.b even 2 1
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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7} + 3$$ $$T_{11}^{2} - 4 T_{11} + 1$$ $$T_{17} + 4$$ $$T_{31}^{2} + 4 T_{31} - 44$$

## Hecke characteristic polynomials

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