# Properties

 Label 5070.2.b.f Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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 2.00000i 1.00000i 1.00000 1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 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.f 2
13.b even 2 1 inner 5070.2.b.f 2
13.d odd 4 1 390.2.a.b 1
13.d odd 4 1 5070.2.a.n 1
39.f even 4 1 1170.2.a.j 1
52.f even 4 1 3120.2.a.y 1
65.f even 4 1 1950.2.e.m 2
65.g odd 4 1 1950.2.a.ba 1
65.k even 4 1 1950.2.e.m 2
156.l odd 4 1 9360.2.a.v 1
195.j odd 4 1 5850.2.e.h 2
195.n even 4 1 5850.2.a.s 1
195.u odd 4 1 5850.2.e.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.b 1 13.d odd 4 1
1170.2.a.j 1 39.f even 4 1
1950.2.a.ba 1 65.g odd 4 1
1950.2.e.m 2 65.f even 4 1
1950.2.e.m 2 65.k even 4 1
3120.2.a.y 1 52.f even 4 1
5070.2.a.n 1 13.d odd 4 1
5070.2.b.f 2 1.a even 1 1 trivial
5070.2.b.f 2 13.b even 2 1 inner
5850.2.a.s 1 195.n even 4 1
5850.2.e.h 2 195.j odd 4 1
5850.2.e.h 2 195.u odd 4 1
9360.2.a.v 1 156.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 16$$ $$T_{17} + 4$$ $$T_{31}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$( 2 + T )^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$100 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$100 + T^{2}$$
$97$ $$64 + T^{2}$$