# Properties

 Label 5070.2.b.q Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.82843i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.82843i 1.00000i 1.00000 −1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.82843i 1.00000i 1.00000 −1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.82843i 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.q 4
13.b even 2 1 inner 5070.2.b.q 4
13.d odd 4 1 390.2.a.h 2
13.d odd 4 1 5070.2.a.bc 2
39.f even 4 1 1170.2.a.o 2
52.f even 4 1 3120.2.a.bc 2
65.f even 4 1 1950.2.e.o 4
65.g odd 4 1 1950.2.a.bd 2
65.k even 4 1 1950.2.e.o 4
156.l odd 4 1 9360.2.a.ch 2
195.j odd 4 1 5850.2.e.bk 4
195.n even 4 1 5850.2.a.cl 2
195.u odd 4 1 5850.2.e.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.h 2 13.d odd 4 1
1170.2.a.o 2 39.f even 4 1
1950.2.a.bd 2 65.g odd 4 1
1950.2.e.o 4 65.f even 4 1
1950.2.e.o 4 65.k even 4 1
3120.2.a.bc 2 52.f even 4 1
5070.2.a.bc 2 13.d odd 4 1
5070.2.b.q 4 1.a even 1 1 trivial
5070.2.b.q 4 13.b even 2 1 inner
5850.2.a.cl 2 195.n even 4 1
5850.2.e.bk 4 195.j odd 4 1
5850.2.e.bk 4 195.u odd 4 1
9360.2.a.ch 2 156.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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{2} + 8$$ $$T_{11}^{2} + 32$$ $$T_{17}^{2} - 4 T_{17} - 4$$ $$T_{31}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -4 - 4 T + T^{2} )^{2}$$
$19$ $$( 8 + T^{2} )^{2}$$
$23$ $$( -72 + T^{2} )^{2}$$
$29$ $$( 28 + 12 T + T^{2} )^{2}$$
$31$ $$( 16 + T^{2} )^{2}$$
$37$ $$16 + 136 T^{2} + T^{4}$$
$41$ $$784 + 72 T^{2} + T^{4}$$
$43$ $$( -16 + 8 T + T^{2} )^{2}$$
$47$ $$( 64 + T^{2} )^{2}$$
$53$ $$( -124 - 4 T + T^{2} )^{2}$$
$59$ $$1024 + 192 T^{2} + T^{4}$$
$61$ $$( -6 + T )^{4}$$
$67$ $$( 32 + T^{2} )^{2}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$1296 + 216 T^{2} + T^{4}$$
$79$ $$( 32 - 16 T + T^{2} )^{2}$$
$83$ $$12544 + 352 T^{2} + T^{4}$$
$89$ $$4624 + 264 T^{2} + T^{4}$$
$97$ $$784 + 88 T^{2} + T^{4}$$