# Properties

 Label 5070.2.b.a Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} - q^{4} + i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} + q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{3} - q^{4} + i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} + q^{9} - q^{10} -i q^{11} + q^{12} -3 q^{14} -i q^{15} + q^{16} + i q^{18} -5 i q^{19} -i q^{20} -3 i q^{21} + q^{22} + 4 q^{23} + i q^{24} - q^{25} - q^{27} -3 i q^{28} + q^{30} + 10 i q^{31} + i q^{32} + i q^{33} -3 q^{35} - q^{36} + i q^{37} + 5 q^{38} + q^{40} + 6 i q^{41} + 3 q^{42} + 2 q^{43} + i q^{44} + i q^{45} + 4 i q^{46} + 9 i q^{47} - q^{48} -2 q^{49} -i q^{50} -13 q^{53} -i q^{54} + q^{55} + 3 q^{56} + 5 i q^{57} -4 i q^{59} + i q^{60} -2 q^{61} -10 q^{62} + 3 i q^{63} - q^{64} - q^{66} -12 i q^{67} -4 q^{69} -3 i q^{70} -2 i q^{71} -i q^{72} + 16 i q^{73} - q^{74} + q^{75} + 5 i q^{76} + 3 q^{77} -10 q^{79} + i q^{80} + q^{81} -6 q^{82} + 12 i q^{83} + 3 i q^{84} + 2 i q^{86} - q^{88} -i q^{89} - q^{90} -4 q^{92} -10 i q^{93} -9 q^{94} + 5 q^{95} -i q^{96} + 12 i q^{97} -2 i q^{98} -i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} + 2q^{12} - 6q^{14} + 2q^{16} + 2q^{22} + 8q^{23} - 2q^{25} - 2q^{27} + 2q^{30} - 6q^{35} - 2q^{36} + 10q^{38} + 2q^{40} + 6q^{42} + 4q^{43} - 2q^{48} - 4q^{49} - 26q^{53} + 2q^{55} + 6q^{56} - 4q^{61} - 20q^{62} - 2q^{64} - 2q^{66} - 8q^{69} - 2q^{74} + 2q^{75} + 6q^{77} - 20q^{79} + 2q^{81} - 12q^{82} - 2q^{88} - 2q^{90} - 8q^{92} - 18q^{94} + 10q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.a 2
13.b even 2 1 inner 5070.2.b.a 2
13.d odd 4 1 5070.2.a.c 1
13.d odd 4 1 5070.2.a.q 1
13.f odd 12 2 390.2.i.b 2
39.k even 12 2 1170.2.i.j 2
65.o even 12 2 1950.2.z.i 4
65.s odd 12 2 1950.2.i.o 2
65.t even 12 2 1950.2.z.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.b 2 13.f odd 12 2
1170.2.i.j 2 39.k even 12 2
1950.2.i.o 2 65.s odd 12 2
1950.2.z.i 4 65.o even 12 2
1950.2.z.i 4 65.t even 12 2
5070.2.a.c 1 13.d odd 4 1
5070.2.a.q 1 13.d odd 4 1
5070.2.b.a 2 1.a even 1 1 trivial
5070.2.b.a 2 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ $$T_{11}^{2} + 1$$ $$T_{17}$$ $$T_{31}^{2} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$25 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$( 13 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$256 + T^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$1 + T^{2}$$
$97$ $$144 + T^{2}$$