# Properties

 Label 5070.2.b.m 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} - i q^{5} + i q^{6} + 2 i q^{7} - i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 + q^3 - q^4 - i * q^5 + i * q^6 + 2*i * q^7 - i * q^8 + q^9 $$q + i q^{2} + q^{3} - q^{4} - i q^{5} + i q^{6} + 2 i q^{7} - i q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{14} - i q^{15} + q^{16} + i q^{18} - 2 i q^{19} + i q^{20} + 2 i q^{21} + 6 q^{23} - i q^{24} - q^{25} + q^{27} - 2 i q^{28} + q^{30} - 8 i q^{31} + i q^{32} + 2 q^{35} - q^{36} + 2 i q^{37} + 2 q^{38} - q^{40} - 6 i q^{41} - 2 q^{42} + 4 q^{43} - i q^{45} + 6 i q^{46} + q^{48} + 3 q^{49} - i q^{50} - 6 q^{53} + i q^{54} + 2 q^{56} - 2 i q^{57} + i q^{60} + 14 q^{61} + 8 q^{62} + 2 i q^{63} - q^{64} + 4 i q^{67} + 6 q^{69} + 2 i q^{70} - i q^{72} - 4 i q^{73} - 2 q^{74} - q^{75} + 2 i q^{76} - 16 q^{79} - i q^{80} + q^{81} + 6 q^{82} + 12 i q^{83} - 2 i q^{84} + 4 i q^{86} - 6 i q^{89} + q^{90} - 6 q^{92} - 8 i q^{93} - 2 q^{95} + i q^{96} + 4 i q^{97} + 3 i q^{98} +O(q^{100})$$ 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 - q^12 - 2 * q^14 - i * q^15 + q^16 + i * q^18 - 2*i * q^19 + i * q^20 + 2*i * q^21 + 6 * q^23 - i * q^24 - q^25 + q^27 - 2*i * q^28 + q^30 - 8*i * q^31 + i * q^32 + 2 * q^35 - q^36 + 2*i * q^37 + 2 * q^38 - q^40 - 6*i * q^41 - 2 * q^42 + 4 * q^43 - i * q^45 + 6*i * q^46 + q^48 + 3 * q^49 - i * q^50 - 6 * q^53 + i * q^54 + 2 * q^56 - 2*i * q^57 + i * q^60 + 14 * q^61 + 8 * q^62 + 2*i * q^63 - q^64 + 4*i * q^67 + 6 * q^69 + 2*i * q^70 - i * q^72 - 4*i * q^73 - 2 * q^74 - q^75 + 2*i * q^76 - 16 * q^79 - i * q^80 + q^81 + 6 * q^82 + 12*i * q^83 - 2*i * q^84 + 4*i * q^86 - 6*i * q^89 + q^90 - 6 * q^92 - 8*i * q^93 - 2 * q^95 + i * q^96 + 4*i * q^97 + 3*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 $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 12 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{30} + 4 q^{35} - 2 q^{36} + 4 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{53} + 4 q^{56} + 28 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{69} - 4 q^{74} - 2 q^{75} - 32 q^{79} + 2 q^{81} + 12 q^{82} + 2 q^{90} - 12 q^{92} - 4 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 + 12 * q^23 - 2 * q^25 + 2 * q^27 + 2 * q^30 + 4 * q^35 - 2 * q^36 + 4 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 12 * q^53 + 4 * q^56 + 28 * q^61 + 16 * q^62 - 2 * q^64 + 12 * q^69 - 4 * q^74 - 2 * q^75 - 32 * q^79 + 2 * q^81 + 12 * q^82 + 2 * q^90 - 12 * q^92 - 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.

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 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.m 2
13.b even 2 1 inner 5070.2.b.m 2
13.d odd 4 1 390.2.a.d 1
13.d odd 4 1 5070.2.a.t 1
39.f even 4 1 1170.2.a.k 1
52.f even 4 1 3120.2.a.j 1
65.f even 4 1 1950.2.e.d 2
65.g odd 4 1 1950.2.a.o 1
65.k even 4 1 1950.2.e.d 2
156.l odd 4 1 9360.2.a.g 1
195.j odd 4 1 5850.2.e.o 2
195.n even 4 1 5850.2.a.g 1
195.u odd 4 1 5850.2.e.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.d 1 13.d odd 4 1
1170.2.a.k 1 39.f even 4 1
1950.2.a.o 1 65.g odd 4 1
1950.2.e.d 2 65.f even 4 1
1950.2.e.d 2 65.k even 4 1
3120.2.a.j 1 52.f even 4 1
5070.2.a.t 1 13.d odd 4 1
5070.2.b.m 2 1.a even 1 1 trivial
5070.2.b.m 2 13.b even 2 1 inner
5850.2.a.g 1 195.n even 4 1
5850.2.e.o 2 195.j odd 4 1
5850.2.e.o 2 195.u odd 4 1
9360.2.a.g 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 $$T_{11}$$ T11 $$T_{17}$$ T17 $$T_{31}^{2} + 64$$ T31^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 36$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 16$$