# Properties

 Label 5070.2.b.e 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: $$x^{2} + 1$$ 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 $$q - i q^{2} - q^{3} - q^{4} + i q^{5} + i q^{6} + 2 i q^{7} + i q^{8} + q^{9} + q^{10} + 4 i q^{11} + q^{12} + 2 q^{14} - i q^{15} + q^{16} - 8 q^{17} - i q^{18} + 6 i q^{19} - i q^{20} - 2 i q^{21} + 4 q^{22} - 6 q^{23} - i q^{24} - q^{25} - q^{27} - 2 i q^{28} - 4 q^{29} - q^{30} - i q^{32} - 4 i q^{33} + 8 i q^{34} - 2 q^{35} - q^{36} - 2 i q^{37} + 6 q^{38} - q^{40} + 2 i q^{41} - 2 q^{42} + 4 q^{43} - 4 i q^{44} + i q^{45} + 6 i q^{46} - q^{48} + 3 q^{49} + i q^{50} + 8 q^{51} - 10 q^{53} + i q^{54} - 4 q^{55} - 2 q^{56} - 6 i q^{57} + 4 i q^{58} + 4 i q^{59} + i q^{60} - 10 q^{61} + 2 i q^{63} - q^{64} - 4 q^{66} - 12 i q^{67} + 8 q^{68} + 6 q^{69} + 2 i q^{70} + 8 i q^{71} + i q^{72} - 8 i q^{73} - 2 q^{74} + q^{75} - 6 i q^{76} - 8 q^{77} + 8 q^{79} + i q^{80} + q^{81} + 2 q^{82} - 12 i q^{83} + 2 i q^{84} - 8 i q^{85} - 4 i q^{86} + 4 q^{87} - 4 q^{88} - 14 i q^{89} + q^{90} + 6 q^{92} - 6 q^{95} + i q^{96} + 16 i q^{97} - 3 i q^{98} + 4 i q^{99} +O(q^{100})$$ 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 + 4*i * q^11 + q^12 + 2 * q^14 - i * q^15 + q^16 - 8 * q^17 - i * q^18 + 6*i * q^19 - i * q^20 - 2*i * q^21 + 4 * q^22 - 6 * q^23 - i * q^24 - q^25 - q^27 - 2*i * q^28 - 4 * q^29 - q^30 - i * q^32 - 4*i * q^33 + 8*i * q^34 - 2 * q^35 - q^36 - 2*i * q^37 + 6 * q^38 - q^40 + 2*i * q^41 - 2 * q^42 + 4 * q^43 - 4*i * q^44 + i * q^45 + 6*i * q^46 - q^48 + 3 * q^49 + i * q^50 + 8 * q^51 - 10 * q^53 + i * q^54 - 4 * q^55 - 2 * q^56 - 6*i * q^57 + 4*i * q^58 + 4*i * q^59 + i * q^60 - 10 * q^61 + 2*i * q^63 - q^64 - 4 * q^66 - 12*i * q^67 + 8 * q^68 + 6 * q^69 + 2*i * q^70 + 8*i * q^71 + i * q^72 - 8*i * q^73 - 2 * q^74 + q^75 - 6*i * q^76 - 8 * q^77 + 8 * q^79 + i * q^80 + q^81 + 2 * q^82 - 12*i * q^83 + 2*i * q^84 - 8*i * q^85 - 4*i * q^86 + 4 * q^87 - 4 * q^88 - 14*i * q^89 + q^90 + 6 * q^92 - 6 * q^95 + i * q^96 + 16*i * q^97 - 3*i * q^98 + 4*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 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 16 q^{17} + 8 q^{22} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 8 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 12 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} - 2 q^{48} + 6 q^{49} + 16 q^{51} - 20 q^{53} - 8 q^{55} - 4 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{68} + 12 q^{69} - 4 q^{74} + 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{82} + 8 q^{87} - 8 q^{88} + 2 q^{90} + 12 q^{92} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 16 * q^17 + 8 * q^22 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 8 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 + 12 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 - 2 * q^48 + 6 * q^49 + 16 * q^51 - 20 * q^53 - 8 * q^55 - 4 * q^56 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^68 + 12 * q^69 - 4 * q^74 + 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 4 * q^82 + 8 * q^87 - 8 * q^88 + 2 * q^90 + 12 * q^92 - 12 * 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.e 2
13.b even 2 1 inner 5070.2.b.e 2
13.d odd 4 1 390.2.a.e 1
13.d odd 4 1 5070.2.a.e 1
39.f even 4 1 1170.2.a.e 1
52.f even 4 1 3120.2.a.o 1
65.f even 4 1 1950.2.e.f 2
65.g odd 4 1 1950.2.a.h 1
65.k even 4 1 1950.2.e.f 2
156.l odd 4 1 9360.2.a.bh 1
195.j odd 4 1 5850.2.e.i 2
195.n even 4 1 5850.2.a.bi 1
195.u odd 4 1 5850.2.e.i 2

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{17} + 8$$ T17 + 8 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2}$$
$17$ $$(T + 8)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 4$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 64$$
$73$ $$T^{2} + 64$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 256$$