# Properties

 Label 5070.2.a.bg Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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, 0]))

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

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

chi := DirichletCharacter("5070.1");

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.a (trivial)

## 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: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.73205 −1.73205
1.00000 −1.00000 1.00000 1.00000 −1.00000 2.00000 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 2.00000 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.a.bg 2
13.b even 2 1 5070.2.a.y 2
13.d odd 4 2 5070.2.b.o 4
13.f odd 12 2 390.2.bb.b 4
39.k even 12 2 1170.2.bs.e 4
65.o even 12 2 1950.2.y.c 4
65.s odd 12 2 1950.2.bc.b 4
65.t even 12 2 1950.2.y.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 13.f odd 12 2
1170.2.bs.e 4 39.k even 12 2
1950.2.y.c 4 65.o even 12 2
1950.2.y.f 4 65.t even 12 2
1950.2.bc.b 4 65.s odd 12 2
5070.2.a.y 2 13.b even 2 1
5070.2.a.bg 2 1.a even 1 1 trivial
5070.2.b.o 4 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} - 6T_{11} - 3$$ T11^2 - 6*T11 - 3 $$T_{17} - 4$$ T17 - 4 $$T_{31}^{2} - 3$$ T31^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} - 6T - 3$$
$13$ $$T^{2}$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 8T + 4$$
$23$ $$T^{2} + 4T + 1$$
$29$ $$T^{2} - 4T + 1$$
$31$ $$T^{2} - 3$$
$37$ $$T^{2} - 8T - 11$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 10T - 23$$
$47$ $$T^{2} - 14T + 37$$
$53$ $$T^{2} + 12T - 12$$
$59$ $$T^{2} + 10T + 13$$
$61$ $$T^{2} - 108$$
$67$ $$T^{2} - 16T + 52$$
$71$ $$T^{2} + 4T - 104$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 14T + 1$$
$83$ $$T^{2} - 4T - 44$$
$89$ $$T^{2} + 8T + 4$$
$97$ $$T^{2} - 8T + 4$$