Properties

 Label 5070.2.b.l 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + q^{3} - q^{4} - i q^{5} - i q^{6} + 5 i q^{7} + i q^{8} + q^{9} +O(q^{10})$$ q - i * q^2 + q^3 - q^4 - i * q^5 - i * q^6 + 5*i * q^7 + i * q^8 + q^9 $$q - i q^{2} + q^{3} - q^{4} - i q^{5} - i q^{6} + 5 i q^{7} + i q^{8} + q^{9} - q^{10} + 3 i q^{11} - q^{12} + 5 q^{14} - i q^{15} + q^{16} + 8 q^{17} - i q^{18} - 5 i q^{19} + i q^{20} + 5 i q^{21} + 3 q^{22} + 4 q^{23} + i q^{24} - q^{25} + q^{27} - 5 i q^{28} - 4 q^{29} - q^{30} - 2 i q^{31} - i q^{32} + 3 i q^{33} - 8 i q^{34} + 5 q^{35} - q^{36} + 7 i q^{37} - 5 q^{38} + q^{40} + 6 i q^{41} + 5 q^{42} - 6 q^{43} - 3 i q^{44} - i q^{45} - 4 i q^{46} + 3 i q^{47} + q^{48} - 18 q^{49} + i q^{50} + 8 q^{51} + q^{53} - i q^{54} + 3 q^{55} - 5 q^{56} - 5 i q^{57} + 4 i q^{58} - 12 i q^{59} + i q^{60} + 2 q^{61} - 2 q^{62} + 5 i q^{63} - q^{64} + 3 q^{66} + 8 i q^{67} - 8 q^{68} + 4 q^{69} - 5 i q^{70} + 2 i q^{71} + i q^{72} + 7 q^{74} - q^{75} + 5 i q^{76} - 15 q^{77} - 2 q^{79} - i q^{80} + q^{81} + 6 q^{82} + 8 i q^{83} - 5 i q^{84} - 8 i q^{85} + 6 i q^{86} - 4 q^{87} - 3 q^{88} + 11 i q^{89} - q^{90} - 4 q^{92} - 2 i q^{93} + 3 q^{94} - 5 q^{95} - i q^{96} + 18 i q^{98} + 3 i q^{99} +O(q^{100})$$ q - i * q^2 + q^3 - q^4 - i * q^5 - i * q^6 + 5*i * q^7 + i * q^8 + q^9 - q^10 + 3*i * q^11 - q^12 + 5 * q^14 - i * q^15 + q^16 + 8 * q^17 - i * q^18 - 5*i * q^19 + i * q^20 + 5*i * q^21 + 3 * q^22 + 4 * q^23 + i * q^24 - q^25 + q^27 - 5*i * q^28 - 4 * q^29 - q^30 - 2*i * q^31 - i * q^32 + 3*i * q^33 - 8*i * q^34 + 5 * q^35 - q^36 + 7*i * q^37 - 5 * q^38 + q^40 + 6*i * q^41 + 5 * q^42 - 6 * q^43 - 3*i * q^44 - i * q^45 - 4*i * q^46 + 3*i * q^47 + q^48 - 18 * q^49 + i * q^50 + 8 * q^51 + q^53 - i * q^54 + 3 * q^55 - 5 * q^56 - 5*i * q^57 + 4*i * q^58 - 12*i * q^59 + i * q^60 + 2 * q^61 - 2 * q^62 + 5*i * q^63 - q^64 + 3 * q^66 + 8*i * q^67 - 8 * q^68 + 4 * q^69 - 5*i * q^70 + 2*i * q^71 + i * q^72 + 7 * q^74 - q^75 + 5*i * q^76 - 15 * q^77 - 2 * q^79 - i * q^80 + q^81 + 6 * q^82 + 8*i * q^83 - 5*i * q^84 - 8*i * q^85 + 6*i * q^86 - 4 * q^87 - 3 * q^88 + 11*i * q^89 - q^90 - 4 * q^92 - 2*i * q^93 + 3 * q^94 - 5 * q^95 - i * q^96 + 18*i * q^98 + 3*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 $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 10 q^{14} + 2 q^{16} + 16 q^{17} + 6 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{27} - 8 q^{29} - 2 q^{30} + 10 q^{35} - 2 q^{36} - 10 q^{38} + 2 q^{40} + 10 q^{42} - 12 q^{43} + 2 q^{48} - 36 q^{49} + 16 q^{51} + 2 q^{53} + 6 q^{55} - 10 q^{56} + 4 q^{61} - 4 q^{62} - 2 q^{64} + 6 q^{66} - 16 q^{68} + 8 q^{69} + 14 q^{74} - 2 q^{75} - 30 q^{77} - 4 q^{79} + 2 q^{81} + 12 q^{82} - 8 q^{87} - 6 q^{88} - 2 q^{90} - 8 q^{92} + 6 q^{94} - 10 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 10 * q^14 + 2 * q^16 + 16 * q^17 + 6 * q^22 + 8 * q^23 - 2 * q^25 + 2 * q^27 - 8 * q^29 - 2 * q^30 + 10 * q^35 - 2 * q^36 - 10 * q^38 + 2 * q^40 + 10 * q^42 - 12 * q^43 + 2 * q^48 - 36 * q^49 + 16 * q^51 + 2 * q^53 + 6 * q^55 - 10 * q^56 + 4 * q^61 - 4 * q^62 - 2 * q^64 + 6 * q^66 - 16 * q^68 + 8 * q^69 + 14 * q^74 - 2 * q^75 - 30 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^82 - 8 * q^87 - 6 * q^88 - 2 * q^90 - 8 * q^92 + 6 * q^94 - 10 * 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 5.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.00000i 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.l 2
13.b even 2 1 inner 5070.2.b.l 2
13.d odd 4 1 5070.2.a.i 1
13.d odd 4 1 5070.2.a.x 1
13.f odd 12 2 390.2.i.d 2
39.k even 12 2 1170.2.i.g 2
65.o even 12 2 1950.2.z.j 4
65.s odd 12 2 1950.2.i.i 2
65.t even 12 2 1950.2.z.j 4

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

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11}^{2} + 9$$ T11^2 + 9 $$T_{17} - 8$$ T17 - 8 $$T_{31}^{2} + 4$$ T31^2 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 25$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2}$$
$17$ $$(T - 8)^{2}$$
$19$ $$T^{2} + 25$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2} + 36$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$(T - 1)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2} + 4$$
$73$ $$T^{2}$$
$79$ $$(T + 2)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$T^{2} + 121$$
$97$ $$T^{2}$$