# Properties

 Label 5070.2.b.c Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} +O(q^{10})$$ q - i * q^2 - q^3 - q^4 - i * q^5 + i * q^6 + i * q^8 + q^9 $$q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} - q^{10} + q^{12} + i q^{15} + q^{16} + 6 q^{17} - i q^{18} + i q^{20} + 4 q^{23} - i q^{24} - q^{25} - q^{27} - 10 q^{29} + q^{30} - i q^{32} - 6 i q^{34} - q^{36} + 6 i q^{37} + q^{40} + 2 i q^{41} + 4 q^{43} - i q^{45} - 4 i q^{46} - q^{48} + 7 q^{49} + i q^{50} - 6 q^{51} - 6 q^{53} + i q^{54} + 10 i q^{58} - i q^{60} + 6 q^{61} - q^{64} + 4 i q^{67} - 6 q^{68} - 4 q^{69} + 16 i q^{71} + i q^{72} + 2 i q^{73} + 6 q^{74} + q^{75} - i q^{80} + q^{81} + 2 q^{82} + 4 i q^{83} - 6 i q^{85} - 4 i q^{86} + 10 q^{87} + 6 i q^{89} - q^{90} - 4 q^{92} + i q^{96} + 14 i q^{97} - 7 i q^{98} +O(q^{100})$$ q - i * q^2 - q^3 - q^4 - i * q^5 + i * q^6 + i * q^8 + q^9 - q^10 + q^12 + i * q^15 + q^16 + 6 * q^17 - i * q^18 + i * q^20 + 4 * q^23 - i * q^24 - q^25 - q^27 - 10 * q^29 + q^30 - i * q^32 - 6*i * q^34 - q^36 + 6*i * q^37 + q^40 + 2*i * q^41 + 4 * q^43 - i * q^45 - 4*i * q^46 - q^48 + 7 * q^49 + i * q^50 - 6 * q^51 - 6 * q^53 + i * q^54 + 10*i * q^58 - i * q^60 + 6 * q^61 - q^64 + 4*i * q^67 - 6 * q^68 - 4 * q^69 + 16*i * q^71 + i * q^72 + 2*i * q^73 + 6 * q^74 + q^75 - i * q^80 + q^81 + 2 * q^82 + 4*i * q^83 - 6*i * q^85 - 4*i * q^86 + 10 * q^87 + 6*i * q^89 - q^90 - 4 * q^92 + i * q^96 + 14*i * q^97 - 7*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 20 q^{29} + 2 q^{30} - 2 q^{36} + 2 q^{40} + 8 q^{43} - 2 q^{48} + 14 q^{49} - 12 q^{51} - 12 q^{53} + 12 q^{61} - 2 q^{64} - 12 q^{68} - 8 q^{69} + 12 q^{74} + 2 q^{75} + 2 q^{81} + 4 q^{82} + 20 q^{87} - 2 q^{90} - 8 q^{92}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^16 + 12 * q^17 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 20 * q^29 + 2 * q^30 - 2 * q^36 + 2 * q^40 + 8 * q^43 - 2 * q^48 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 12 * q^61 - 2 * q^64 - 12 * q^68 - 8 * q^69 + 12 * q^74 + 2 * q^75 + 2 * q^81 + 4 * q^82 + 20 * q^87 - 2 * q^90 - 8 * q^92

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 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.c 2
13.b even 2 1 inner 5070.2.b.c 2
13.d odd 4 1 390.2.a.a 1
13.d odd 4 1 5070.2.a.s 1
39.f even 4 1 1170.2.a.m 1
52.f even 4 1 3120.2.a.q 1
65.f even 4 1 1950.2.e.l 2
65.g odd 4 1 1950.2.a.y 1
65.k even 4 1 1950.2.e.l 2
156.l odd 4 1 9360.2.a.bn 1
195.j odd 4 1 5850.2.e.p 2
195.n even 4 1 5850.2.a.m 1
195.u odd 4 1 5850.2.e.p 2

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

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{17} - 6$$ T17 - 6 $$T_{31}$$ T31

## Hecke characteristic polynomials

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