Properties

 Label 5070.2.b.n 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.

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

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.n 2
13.b even 2 1 inner 5070.2.b.n 2
13.d odd 4 1 390.2.a.g 1
13.d odd 4 1 5070.2.a.k 1
39.f even 4 1 1170.2.a.g 1
52.f even 4 1 3120.2.a.b 1
65.f even 4 1 1950.2.e.k 2
65.g odd 4 1 1950.2.a.b 1
65.k even 4 1 1950.2.e.k 2
156.l odd 4 1 9360.2.a.bg 1
195.j odd 4 1 5850.2.e.r 2
195.n even 4 1 5850.2.a.bk 1
195.u odd 4 1 5850.2.e.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.g 1 13.d odd 4 1
1170.2.a.g 1 39.f even 4 1
1950.2.a.b 1 65.g odd 4 1
1950.2.e.k 2 65.f even 4 1
1950.2.e.k 2 65.k even 4 1
3120.2.a.b 1 52.f even 4 1
5070.2.a.k 1 13.d odd 4 1
5070.2.b.n 2 1.a even 1 1 trivial
5070.2.b.n 2 13.b even 2 1 inner
5850.2.a.bk 1 195.n even 4 1
5850.2.e.r 2 195.j odd 4 1
5850.2.e.r 2 195.u odd 4 1
9360.2.a.bg 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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$64 + T^{2}$$