# Properties

 Label 5070.2.b.h 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: $$x^{2} + 1$$ 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 $$q - i q^{2} - q^{3} - q^{4} + i q^{5} + i q^{6} + 2 i q^{7} + i q^{8} + q^{9} + q^{10} + 5 i q^{11} + q^{12} + 2 q^{14} - i q^{15} + q^{16} + 2 q^{17} - i q^{18} - 2 i q^{19} - i q^{20} - 2 i q^{21} + 5 q^{22} + q^{23} - i q^{24} - q^{25} - q^{27} - 2 i q^{28} + 5 q^{29} - q^{30} - 11 i q^{31} - i q^{32} - 5 i q^{33} - 2 i q^{34} - 2 q^{35} - q^{36} - 3 i q^{37} - 2 q^{38} - q^{40} - 2 i q^{41} - 2 q^{42} + 11 q^{43} - 5 i q^{44} + i q^{45} - i q^{46} - 9 i q^{47} - q^{48} + 3 q^{49} + i q^{50} - 2 q^{51} + 6 q^{53} + i q^{54} - 5 q^{55} - 2 q^{56} + 2 i q^{57} - 5 i q^{58} + 15 i q^{59} + i q^{60} + 10 q^{61} - 11 q^{62} + 2 i q^{63} - q^{64} - 5 q^{66} + 16 i q^{67} - 2 q^{68} - q^{69} + 2 i q^{70} + i q^{72} + 6 i q^{73} - 3 q^{74} + q^{75} + 2 i q^{76} - 10 q^{77} - 11 q^{79} + i q^{80} + q^{81} - 2 q^{82} + 6 i q^{83} + 2 i q^{84} + 2 i q^{85} - 11 i q^{86} - 5 q^{87} - 5 q^{88} - 2 i q^{89} + q^{90} - q^{92} + 11 i q^{93} - 9 q^{94} + 2 q^{95} + i q^{96} - 2 i q^{97} - 3 i q^{98} + 5 i q^{99} +O(q^{100})$$ 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 + 5*i * q^11 + q^12 + 2 * q^14 - i * q^15 + q^16 + 2 * q^17 - i * q^18 - 2*i * q^19 - i * q^20 - 2*i * q^21 + 5 * q^22 + q^23 - i * q^24 - q^25 - q^27 - 2*i * q^28 + 5 * q^29 - q^30 - 11*i * q^31 - i * q^32 - 5*i * q^33 - 2*i * q^34 - 2 * q^35 - q^36 - 3*i * q^37 - 2 * q^38 - q^40 - 2*i * q^41 - 2 * q^42 + 11 * q^43 - 5*i * q^44 + i * q^45 - i * q^46 - 9*i * q^47 - q^48 + 3 * q^49 + i * q^50 - 2 * q^51 + 6 * q^53 + i * q^54 - 5 * q^55 - 2 * q^56 + 2*i * q^57 - 5*i * q^58 + 15*i * q^59 + i * q^60 + 10 * q^61 - 11 * q^62 + 2*i * q^63 - q^64 - 5 * q^66 + 16*i * q^67 - 2 * q^68 - q^69 + 2*i * q^70 + i * q^72 + 6*i * q^73 - 3 * q^74 + q^75 + 2*i * q^76 - 10 * q^77 - 11 * q^79 + i * q^80 + q^81 - 2 * q^82 + 6*i * q^83 + 2*i * q^84 + 2*i * q^85 - 11*i * q^86 - 5 * q^87 - 5 * q^88 - 2*i * q^89 + q^90 - q^92 + 11*i * q^93 - 9 * q^94 + 2 * q^95 + i * q^96 - 2*i * q^97 - 3*i * q^98 + 5*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 10 q^{22} + 2 q^{23} - 2 q^{25} - 2 q^{27} + 10 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 4 q^{38} - 2 q^{40} - 4 q^{42} + 22 q^{43} - 2 q^{48} + 6 q^{49} - 4 q^{51} + 12 q^{53} - 10 q^{55} - 4 q^{56} + 20 q^{61} - 22 q^{62} - 2 q^{64} - 10 q^{66} - 4 q^{68} - 2 q^{69} - 6 q^{74} + 2 q^{75} - 20 q^{77} - 22 q^{79} + 2 q^{81} - 4 q^{82} - 10 q^{87} - 10 q^{88} + 2 q^{90} - 2 q^{92} - 18 q^{94} + 4 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 10 * q^22 + 2 * q^23 - 2 * q^25 - 2 * q^27 + 10 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 4 * q^38 - 2 * q^40 - 4 * q^42 + 22 * q^43 - 2 * q^48 + 6 * q^49 - 4 * q^51 + 12 * q^53 - 10 * q^55 - 4 * q^56 + 20 * q^61 - 22 * q^62 - 2 * q^64 - 10 * q^66 - 4 * q^68 - 2 * q^69 - 6 * q^74 + 2 * q^75 - 20 * q^77 - 22 * q^79 + 2 * q^81 - 4 * q^82 - 10 * q^87 - 10 * q^88 + 2 * q^90 - 2 * q^92 - 18 * q^94 + 4 * q^95

## 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.h 2
13.b even 2 1 inner 5070.2.b.h 2
13.d odd 4 1 5070.2.a.d 1
13.d odd 4 1 5070.2.a.p 1
13.f odd 12 2 390.2.i.f 2
39.k even 12 2 1170.2.i.a 2
65.o even 12 2 1950.2.z.e 4
65.s odd 12 2 1950.2.i.d 2
65.t even 12 2 1950.2.z.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.f 2 13.f odd 12 2
1170.2.i.a 2 39.k even 12 2
1950.2.i.d 2 65.s odd 12 2
1950.2.z.e 4 65.o even 12 2
1950.2.z.e 4 65.t even 12 2
5070.2.a.d 1 13.d odd 4 1
5070.2.a.p 1 13.d odd 4 1
5070.2.b.h 2 1.a even 1 1 trivial
5070.2.b.h 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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} + 25$$ T11^2 + 25 $$T_{17} - 2$$ T17 - 2 $$T_{31}^{2} + 121$$ T31^2 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 25$$
$13$ $$T^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T - 1)^{2}$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2} + 121$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2} + 4$$
$43$ $$(T - 11)^{2}$$
$47$ $$T^{2} + 81$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 225$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 4$$