# Properties

 Label 5070.2.a.z Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.30278 −1.30278
−1.00000 −1.00000 1.00000 −1.00000 1.00000 −4.60555 −1.00000 1.00000 1.00000
1.2 −1.00000 −1.00000 1.00000 −1.00000 1.00000 2.60555 −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.z 2
13.b even 2 1 5070.2.a.bf 2
13.d odd 4 2 390.2.b.c 4
39.f even 4 2 1170.2.b.d 4
52.f even 4 2 3120.2.g.q 4
65.f even 4 2 1950.2.f.n 4
65.g odd 4 2 1950.2.b.k 4
65.k even 4 2 1950.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 13.d odd 4 2
1170.2.b.d 4 39.f even 4 2
1950.2.b.k 4 65.g odd 4 2
1950.2.f.m 4 65.k even 4 2
1950.2.f.n 4 65.f even 4 2
3120.2.g.q 4 52.f even 4 2
5070.2.a.z 2 1.a even 1 1 trivial
5070.2.a.bf 2 13.b even 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 2 T_{7} - 12$$ $$T_{11}$$ $$T_{17}^{2} + 2 T_{17} - 12$$ $$T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-12 + 2 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$-12 + 2 T + T^{2}$$
$19$ $$-12 - 2 T + T^{2}$$
$23$ $$12 - 10 T + T^{2}$$
$29$ $$-12 - 2 T + T^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$-48 + 4 T + T^{2}$$
$41$ $$-36 - 8 T + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-48 + 4 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-48 + 4 T + T^{2}$$
$61$ $$-36 + 8 T + T^{2}$$
$67$ $$-36 + 8 T + T^{2}$$
$71$ $$-48 + 4 T + T^{2}$$
$73$ $$12 + 10 T + T^{2}$$
$79$ $$-208 + T^{2}$$
$83$ $$48 + 20 T + T^{2}$$
$89$ $$12 - 16 T + T^{2}$$
$97$ $$12 + 10 T + T^{2}$$