# Properties

 Label 490.2.e.d Level $490$ Weight $2$ Character orbit 490.e Analytic conductor $3.913$ 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] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^4 + z * q^5 + q^8 + 3*z * q^9 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + (4 \zeta_{6} - 4) q^{11} - 6 q^{13} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} - q^{20} + 4 q^{22} + (\zeta_{6} - 1) q^{25} + 6 \zeta_{6} q^{26} + 6 q^{29} + (8 \zeta_{6} - 8) q^{31} + (\zeta_{6} - 1) q^{32} + 2 q^{34} - 3 q^{36} + 10 \zeta_{6} q^{37} + \zeta_{6} q^{40} + 2 q^{41} + 4 q^{43} - 4 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{45} - 8 \zeta_{6} q^{47} + q^{50} + ( - 6 \zeta_{6} + 6) q^{52} + ( - 2 \zeta_{6} + 2) q^{53} - 4 q^{55} - 6 \zeta_{6} q^{58} + ( - 8 \zeta_{6} + 8) q^{59} + 14 \zeta_{6} q^{61} + 8 q^{62} + q^{64} - 6 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} - 2 \zeta_{6} q^{68} - 16 q^{71} + 3 \zeta_{6} q^{72} + (2 \zeta_{6} - 2) q^{73} + ( - 10 \zeta_{6} + 10) q^{74} + 8 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (9 \zeta_{6} - 9) q^{81} - 2 \zeta_{6} q^{82} + 8 q^{83} - 2 q^{85} - 4 \zeta_{6} q^{86} + (4 \zeta_{6} - 4) q^{88} - 10 \zeta_{6} q^{89} + 3 q^{90} + (8 \zeta_{6} - 8) q^{94} + 2 q^{97} - 12 q^{99} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^4 + z * q^5 + q^8 + 3*z * q^9 + (-z + 1) * q^10 + (4*z - 4) * q^11 - 6 * q^13 - z * q^16 + (2*z - 2) * q^17 + (-3*z + 3) * q^18 - q^20 + 4 * q^22 + (z - 1) * q^25 + 6*z * q^26 + 6 * q^29 + (8*z - 8) * q^31 + (z - 1) * q^32 + 2 * q^34 - 3 * q^36 + 10*z * q^37 + z * q^40 + 2 * q^41 + 4 * q^43 - 4*z * q^44 + (3*z - 3) * q^45 - 8*z * q^47 + q^50 + (-6*z + 6) * q^52 + (-2*z + 2) * q^53 - 4 * q^55 - 6*z * q^58 + (-8*z + 8) * q^59 + 14*z * q^61 + 8 * q^62 + q^64 - 6*z * q^65 + (-12*z + 12) * q^67 - 2*z * q^68 - 16 * q^71 + 3*z * q^72 + (2*z - 2) * q^73 + (-10*z + 10) * q^74 + 8*z * q^79 + (-z + 1) * q^80 + (9*z - 9) * q^81 - 2*z * q^82 + 8 * q^83 - 2 * q^85 - 4*z * q^86 + (4*z - 4) * q^88 - 10*z * q^89 + 3 * q^90 + (8*z - 8) * q^94 + 2 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 + q^5 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} + q^{5} + 2 q^{8} + 3 q^{9} + q^{10} - 4 q^{11} - 12 q^{13} - q^{16} - 2 q^{17} + 3 q^{18} - 2 q^{20} + 8 q^{22} - q^{25} + 6 q^{26} + 12 q^{29} - 8 q^{31} - q^{32} + 4 q^{34} - 6 q^{36} + 10 q^{37} + q^{40} + 4 q^{41} + 8 q^{43} - 4 q^{44} - 3 q^{45} - 8 q^{47} + 2 q^{50} + 6 q^{52} + 2 q^{53} - 8 q^{55} - 6 q^{58} + 8 q^{59} + 14 q^{61} + 16 q^{62} + 2 q^{64} - 6 q^{65} + 12 q^{67} - 2 q^{68} - 32 q^{71} + 3 q^{72} - 2 q^{73} + 10 q^{74} + 8 q^{79} + q^{80} - 9 q^{81} - 2 q^{82} + 16 q^{83} - 4 q^{85} - 4 q^{86} - 4 q^{88} - 10 q^{89} + 6 q^{90} - 8 q^{94} + 4 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 + q^5 + 2 * q^8 + 3 * q^9 + q^10 - 4 * q^11 - 12 * q^13 - q^16 - 2 * q^17 + 3 * q^18 - 2 * q^20 + 8 * q^22 - q^25 + 6 * q^26 + 12 * q^29 - 8 * q^31 - q^32 + 4 * q^34 - 6 * q^36 + 10 * q^37 + q^40 + 4 * q^41 + 8 * q^43 - 4 * q^44 - 3 * q^45 - 8 * q^47 + 2 * q^50 + 6 * q^52 + 2 * q^53 - 8 * q^55 - 6 * q^58 + 8 * q^59 + 14 * q^61 + 16 * q^62 + 2 * q^64 - 6 * q^65 + 12 * q^67 - 2 * q^68 - 32 * q^71 + 3 * q^72 - 2 * q^73 + 10 * q^74 + 8 * q^79 + q^80 - 9 * q^81 - 2 * q^82 + 16 * q^83 - 4 * q^85 - 4 * q^86 - 4 * q^88 - 10 * q^89 + 6 * q^90 - 8 * q^94 + 4 * q^97 - 24 * 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 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
471.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 0 1.00000 1.50000 2.59808i 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.d 2
7.b odd 2 1 490.2.e.c 2
7.c even 3 1 70.2.a.a 1
7.c even 3 1 inner 490.2.e.d 2
7.d odd 6 1 490.2.a.h 1
7.d odd 6 1 490.2.e.c 2
21.g even 6 1 4410.2.a.b 1
21.h odd 6 1 630.2.a.d 1
28.f even 6 1 3920.2.a.t 1
28.g odd 6 1 560.2.a.d 1
35.i odd 6 1 2450.2.a.l 1
35.j even 6 1 350.2.a.b 1
35.k even 12 2 2450.2.c.k 2
35.l odd 12 2 350.2.c.b 2
56.k odd 6 1 2240.2.a.q 1
56.p even 6 1 2240.2.a.n 1
77.h odd 6 1 8470.2.a.j 1
84.n even 6 1 5040.2.a.bm 1
105.o odd 6 1 3150.2.a.bj 1
105.x even 12 2 3150.2.g.c 2
140.p odd 6 1 2800.2.a.m 1
140.w even 12 2 2800.2.g.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 7.c even 3 1
350.2.a.b 1 35.j even 6 1
350.2.c.b 2 35.l odd 12 2
490.2.a.h 1 7.d odd 6 1
490.2.e.c 2 7.b odd 2 1
490.2.e.c 2 7.d odd 6 1
490.2.e.d 2 1.a even 1 1 trivial
490.2.e.d 2 7.c even 3 1 inner
560.2.a.d 1 28.g odd 6 1
630.2.a.d 1 21.h odd 6 1
2240.2.a.n 1 56.p even 6 1
2240.2.a.q 1 56.k odd 6 1
2450.2.a.l 1 35.i odd 6 1
2450.2.c.k 2 35.k even 12 2
2800.2.a.m 1 140.p odd 6 1
2800.2.g.n 2 140.w even 12 2
3150.2.a.bj 1 105.o odd 6 1
3150.2.g.c 2 105.x even 12 2
3920.2.a.t 1 28.f even 6 1
4410.2.a.b 1 21.g even 6 1
5040.2.a.bm 1 84.n even 6 1
8470.2.a.j 1 77.h odd 6 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13} + 6$$ T13 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$(T - 2)^{2}$$