# Properties

 Label 490.2.e.a Level $490$ Weight $2$ Character orbit 490.e Analytic conductor $3.913$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,2,Mod(361,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("490.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.e (of order $$3$$, degree $$2$$, 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: $$3.91266969904$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 2 q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - z * q^5 + 2 * q^6 + q^8 - z * q^9 $$q - \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 2 q^{6} + q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (3 \zeta_{6} - 3) q^{11} - 2 \zeta_{6} q^{12} - 5 q^{13} + 2 q^{15} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + (\zeta_{6} - 1) q^{18} - \zeta_{6} q^{19} + q^{20} + 3 q^{22} - 3 \zeta_{6} q^{23} + (2 \zeta_{6} - 2) q^{24} + (\zeta_{6} - 1) q^{25} + 5 \zeta_{6} q^{26} - 4 q^{27} - 6 q^{29} - 2 \zeta_{6} q^{30} + (4 \zeta_{6} - 4) q^{31} + (\zeta_{6} - 1) q^{32} - 6 \zeta_{6} q^{33} - 6 q^{34} + q^{36} - 11 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} + ( - 10 \zeta_{6} + 10) q^{39} - \zeta_{6} q^{40} - 3 q^{41} - 10 q^{43} - 3 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + (3 \zeta_{6} - 3) q^{46} + 3 \zeta_{6} q^{47} + 2 q^{48} + q^{50} + 12 \zeta_{6} q^{51} + ( - 5 \zeta_{6} + 5) q^{52} + (3 \zeta_{6} - 3) q^{53} + 4 \zeta_{6} q^{54} + 3 q^{55} + 2 q^{57} + 6 \zeta_{6} q^{58} + (2 \zeta_{6} - 2) q^{60} - 4 \zeta_{6} q^{61} + 4 q^{62} + q^{64} + 5 \zeta_{6} q^{65} + (6 \zeta_{6} - 6) q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + 6 q^{69} + 12 q^{71} - \zeta_{6} q^{72} + (4 \zeta_{6} - 4) q^{73} + (11 \zeta_{6} - 11) q^{74} - 2 \zeta_{6} q^{75} + q^{76} - 10 q^{78} + 10 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 3 \zeta_{6} q^{82} + 12 q^{83} - 6 q^{85} + 10 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{87} + (3 \zeta_{6} - 3) q^{88} + 6 \zeta_{6} q^{89} + q^{90} + 3 q^{92} - 8 \zeta_{6} q^{93} + ( - 3 \zeta_{6} + 3) q^{94} + (\zeta_{6} - 1) q^{95} - 2 \zeta_{6} q^{96} - 14 q^{97} + 3 q^{99} +O(q^{100})$$ q - z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - z * q^5 + 2 * q^6 + q^8 - z * q^9 + (z - 1) * q^10 + (3*z - 3) * q^11 - 2*z * q^12 - 5 * q^13 + 2 * q^15 - z * q^16 + (-6*z + 6) * q^17 + (z - 1) * q^18 - z * q^19 + q^20 + 3 * q^22 - 3*z * q^23 + (2*z - 2) * q^24 + (z - 1) * q^25 + 5*z * q^26 - 4 * q^27 - 6 * q^29 - 2*z * q^30 + (4*z - 4) * q^31 + (z - 1) * q^32 - 6*z * q^33 - 6 * q^34 + q^36 - 11*z * q^37 + (z - 1) * q^38 + (-10*z + 10) * q^39 - z * q^40 - 3 * q^41 - 10 * q^43 - 3*z * q^44 + (z - 1) * q^45 + (3*z - 3) * q^46 + 3*z * q^47 + 2 * q^48 + q^50 + 12*z * q^51 + (-5*z + 5) * q^52 + (3*z - 3) * q^53 + 4*z * q^54 + 3 * q^55 + 2 * q^57 + 6*z * q^58 + (2*z - 2) * q^60 - 4*z * q^61 + 4 * q^62 + q^64 + 5*z * q^65 + (6*z - 6) * q^66 + (-4*z + 4) * q^67 + 6*z * q^68 + 6 * q^69 + 12 * q^71 - z * q^72 + (4*z - 4) * q^73 + (11*z - 11) * q^74 - 2*z * q^75 + q^76 - 10 * q^78 + 10*z * q^79 + (z - 1) * q^80 + (-11*z + 11) * q^81 + 3*z * q^82 + 12 * q^83 - 6 * q^85 + 10*z * q^86 + (-12*z + 12) * q^87 + (3*z - 3) * q^88 + 6*z * q^89 + q^90 + 3 * q^92 - 8*z * q^93 + (-3*z + 3) * q^94 + (z - 1) * q^95 - 2*z * q^96 - 14 * q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 4 q^{6} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 - q^4 - q^5 + 4 * q^6 + 2 * q^8 - q^9 $$2 q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 4 q^{6} + 2 q^{8} - q^{9} - q^{10} - 3 q^{11} - 2 q^{12} - 10 q^{13} + 4 q^{15} - q^{16} + 6 q^{17} - q^{18} - q^{19} + 2 q^{20} + 6 q^{22} - 3 q^{23} - 2 q^{24} - q^{25} + 5 q^{26} - 8 q^{27} - 12 q^{29} - 2 q^{30} - 4 q^{31} - q^{32} - 6 q^{33} - 12 q^{34} + 2 q^{36} - 11 q^{37} - q^{38} + 10 q^{39} - q^{40} - 6 q^{41} - 20 q^{43} - 3 q^{44} - q^{45} - 3 q^{46} + 3 q^{47} + 4 q^{48} + 2 q^{50} + 12 q^{51} + 5 q^{52} - 3 q^{53} + 4 q^{54} + 6 q^{55} + 4 q^{57} + 6 q^{58} - 2 q^{60} - 4 q^{61} + 8 q^{62} + 2 q^{64} + 5 q^{65} - 6 q^{66} + 4 q^{67} + 6 q^{68} + 12 q^{69} + 24 q^{71} - q^{72} - 4 q^{73} - 11 q^{74} - 2 q^{75} + 2 q^{76} - 20 q^{78} + 10 q^{79} - q^{80} + 11 q^{81} + 3 q^{82} + 24 q^{83} - 12 q^{85} + 10 q^{86} + 12 q^{87} - 3 q^{88} + 6 q^{89} + 2 q^{90} + 6 q^{92} - 8 q^{93} + 3 q^{94} - q^{95} - 2 q^{96} - 28 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 - q^4 - q^5 + 4 * q^6 + 2 * q^8 - q^9 - q^10 - 3 * q^11 - 2 * q^12 - 10 * q^13 + 4 * q^15 - q^16 + 6 * q^17 - q^18 - q^19 + 2 * q^20 + 6 * q^22 - 3 * q^23 - 2 * q^24 - q^25 + 5 * q^26 - 8 * q^27 - 12 * q^29 - 2 * q^30 - 4 * q^31 - q^32 - 6 * q^33 - 12 * q^34 + 2 * q^36 - 11 * q^37 - q^38 + 10 * q^39 - q^40 - 6 * q^41 - 20 * q^43 - 3 * q^44 - q^45 - 3 * q^46 + 3 * q^47 + 4 * q^48 + 2 * q^50 + 12 * q^51 + 5 * q^52 - 3 * q^53 + 4 * q^54 + 6 * q^55 + 4 * q^57 + 6 * q^58 - 2 * q^60 - 4 * q^61 + 8 * q^62 + 2 * q^64 + 5 * q^65 - 6 * q^66 + 4 * q^67 + 6 * q^68 + 12 * q^69 + 24 * q^71 - q^72 - 4 * q^73 - 11 * q^74 - 2 * q^75 + 2 * q^76 - 20 * q^78 + 10 * q^79 - q^80 + 11 * q^81 + 3 * q^82 + 24 * q^83 - 12 * q^85 + 10 * q^86 + 12 * q^87 - 3 * q^88 + 6 * q^89 + 2 * q^90 + 6 * q^92 - 8 * q^93 + 3 * q^94 - q^95 - 2 * q^96 - 28 * q^97 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i −0.500000 0.866025i 2.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
471.1 −0.500000 + 0.866025i −1.00000 1.73205i −0.500000 0.866025i −0.500000 + 0.866025i 2.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.e.a 2
7.b odd 2 1 70.2.e.b 2
7.c even 3 1 490.2.a.j 1
7.c even 3 1 inner 490.2.e.a 2
7.d odd 6 1 70.2.e.b 2
7.d odd 6 1 490.2.a.g 1
21.c even 2 1 630.2.k.e 2
21.g even 6 1 630.2.k.e 2
21.g even 6 1 4410.2.a.m 1
21.h odd 6 1 4410.2.a.c 1
28.d even 2 1 560.2.q.d 2
28.f even 6 1 560.2.q.d 2
28.f even 6 1 3920.2.a.be 1
28.g odd 6 1 3920.2.a.g 1
35.c odd 2 1 350.2.e.h 2
35.f even 4 2 350.2.j.a 4
35.i odd 6 1 350.2.e.h 2
35.i odd 6 1 2450.2.a.p 1
35.j even 6 1 2450.2.a.f 1
35.k even 12 2 350.2.j.a 4
35.k even 12 2 2450.2.c.f 2
35.l odd 12 2 2450.2.c.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 7.b odd 2 1
70.2.e.b 2 7.d odd 6 1
350.2.e.h 2 35.c odd 2 1
350.2.e.h 2 35.i odd 6 1
350.2.j.a 4 35.f even 4 2
350.2.j.a 4 35.k even 12 2
490.2.a.g 1 7.d odd 6 1
490.2.a.j 1 7.c even 3 1
490.2.e.a 2 1.a even 1 1 trivial
490.2.e.a 2 7.c even 3 1 inner
560.2.q.d 2 28.d even 2 1
560.2.q.d 2 28.f even 6 1
630.2.k.e 2 21.c even 2 1
630.2.k.e 2 21.g even 6 1
2450.2.a.f 1 35.j even 6 1
2450.2.a.p 1 35.i odd 6 1
2450.2.c.f 2 35.k even 12 2
2450.2.c.p 2 35.l odd 12 2
3920.2.a.g 1 28.g odd 6 1
3920.2.a.be 1 28.f even 6 1
4410.2.a.c 1 21.h odd 6 1
4410.2.a.m 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 14)^{2}$$