# Properties

 Label 5070.2.b.k 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 30) 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.

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

## 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 4.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.00000i 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.k 2
13.b even 2 1 inner 5070.2.b.k 2
13.d odd 4 1 30.2.a.a 1
13.d odd 4 1 5070.2.a.w 1
39.f even 4 1 90.2.a.c 1
52.f even 4 1 240.2.a.b 1
65.f even 4 1 150.2.c.a 2
65.g odd 4 1 150.2.a.b 1
65.k even 4 1 150.2.c.a 2
91.i even 4 1 1470.2.a.d 1
91.z odd 12 2 1470.2.i.o 2
91.bb even 12 2 1470.2.i.q 2
104.j odd 4 1 960.2.a.e 1
104.m even 4 1 960.2.a.p 1
117.y odd 12 2 810.2.e.l 2
117.z even 12 2 810.2.e.b 2
143.g even 4 1 3630.2.a.w 1
156.l odd 4 1 720.2.a.j 1
195.j odd 4 1 450.2.c.b 2
195.n even 4 1 450.2.a.d 1
195.u odd 4 1 450.2.c.b 2
208.l even 4 1 3840.2.k.f 2
208.m odd 4 1 3840.2.k.y 2
208.r odd 4 1 3840.2.k.y 2
208.s even 4 1 3840.2.k.f 2
221.g odd 4 1 8670.2.a.g 1
260.l odd 4 1 1200.2.f.e 2
260.s odd 4 1 1200.2.f.e 2
260.u even 4 1 1200.2.a.k 1
273.o odd 4 1 4410.2.a.z 1
312.w odd 4 1 2880.2.a.q 1
312.y even 4 1 2880.2.a.a 1
455.u even 4 1 7350.2.a.ct 1
520.t even 4 1 4800.2.a.d 1
520.x odd 4 1 4800.2.f.w 2
520.y even 4 1 4800.2.f.p 2
520.bj even 4 1 4800.2.f.p 2
520.bk odd 4 1 4800.2.f.w 2
520.bo odd 4 1 4800.2.a.cq 1
780.u even 4 1 3600.2.f.i 2
780.bb odd 4 1 3600.2.a.f 1
780.bn even 4 1 3600.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 13.d odd 4 1
90.2.a.c 1 39.f even 4 1
150.2.a.b 1 65.g odd 4 1
150.2.c.a 2 65.f even 4 1
150.2.c.a 2 65.k even 4 1
240.2.a.b 1 52.f even 4 1
450.2.a.d 1 195.n even 4 1
450.2.c.b 2 195.j odd 4 1
450.2.c.b 2 195.u odd 4 1
720.2.a.j 1 156.l odd 4 1
810.2.e.b 2 117.z even 12 2
810.2.e.l 2 117.y odd 12 2
960.2.a.e 1 104.j odd 4 1
960.2.a.p 1 104.m even 4 1
1200.2.a.k 1 260.u even 4 1
1200.2.f.e 2 260.l odd 4 1
1200.2.f.e 2 260.s odd 4 1
1470.2.a.d 1 91.i even 4 1
1470.2.i.o 2 91.z odd 12 2
1470.2.i.q 2 91.bb even 12 2
2880.2.a.a 1 312.y even 4 1
2880.2.a.q 1 312.w odd 4 1
3600.2.a.f 1 780.bb odd 4 1
3600.2.f.i 2 780.u even 4 1
3600.2.f.i 2 780.bn even 4 1
3630.2.a.w 1 143.g even 4 1
3840.2.k.f 2 208.l even 4 1
3840.2.k.f 2 208.s even 4 1
3840.2.k.y 2 208.m odd 4 1
3840.2.k.y 2 208.r odd 4 1
4410.2.a.z 1 273.o odd 4 1
4800.2.a.d 1 520.t even 4 1
4800.2.a.cq 1 520.bo odd 4 1
4800.2.f.p 2 520.y even 4 1
4800.2.f.p 2 520.bj even 4 1
4800.2.f.w 2 520.x odd 4 1
4800.2.f.w 2 520.bk odd 4 1
5070.2.a.w 1 13.d odd 4 1
5070.2.b.k 2 1.a even 1 1 trivial
5070.2.b.k 2 13.b even 2 1 inner
7350.2.a.ct 1 455.u even 4 1
8670.2.a.g 1 221.g odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11}$$ $$T_{17} + 6$$ $$T_{31}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$64 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$324 + T^{2}$$
$97$ $$4 + T^{2}$$