# Properties

 Label 5070.2.a.bc 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −2.82843 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 2.82843 −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.bc 2
13.b even 2 1 390.2.a.h 2
13.d odd 4 2 5070.2.b.q 4
39.d odd 2 1 1170.2.a.o 2
52.b odd 2 1 3120.2.a.bc 2
65.d even 2 1 1950.2.a.bd 2
65.h odd 4 2 1950.2.e.o 4
156.h even 2 1 9360.2.a.ch 2
195.e odd 2 1 5850.2.a.cl 2
195.s even 4 2 5850.2.e.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.h 2 13.b even 2 1
1170.2.a.o 2 39.d odd 2 1
1950.2.a.bd 2 65.d even 2 1
1950.2.e.o 4 65.h odd 4 2
3120.2.a.bc 2 52.b odd 2 1
5070.2.a.bc 2 1.a even 1 1 trivial
5070.2.b.q 4 13.d odd 4 2
5850.2.a.cl 2 195.e odd 2 1
5850.2.e.bk 4 195.s even 4 2
9360.2.a.ch 2 156.h even 2 1

## Hecke kernels

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

 $$T_{7}^{2} - 8$$ $$T_{11}^{2} - 32$$ $$T_{17}^{2} + 4 T_{17} - 4$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-32 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-4 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-72 + T^{2}$$
$29$ $$28 + 12 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$4 - 12 T + T^{2}$$
$41$ $$-28 - 4 T + T^{2}$$
$43$ $$-16 - 8 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$-124 - 4 T + T^{2}$$
$59$ $$32 - 16 T + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$-36 - 12 T + T^{2}$$
$79$ $$32 - 16 T + T^{2}$$
$83$ $$112 + 24 T + T^{2}$$
$89$ $$68 - 20 T + T^{2}$$
$97$ $$28 - 12 T + T^{2}$$