# Properties

 Label 490.2.i.f Level $490$ Weight $2$ Character orbit 490.i Analytic conductor $3.913$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,2,Mod(79,490)]

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([3, 2]))

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

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

chi := DirichletCharacter("490.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.i (of order $$6$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + 3 \beta_{2} q^{9}+O(q^{10})$$ q + (b4 - b1) * q^2 + (-b6 - b5) * q^3 + (-b2 + 1) * q^4 + (-b7 + b4 + b2 - b1) * q^5 + (b7 - b5 + b3) * q^6 + b4 * q^8 + 3*b2 * q^9 $$q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - 6 \beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{99}+O(q^{100})$$ q + (b4 - b1) * q^2 + (-b6 - b5) * q^3 + (-b2 + 1) * q^4 + (-b7 + b4 + b2 - b1) * q^5 + (b7 - b5 + b3) * q^6 + b4 * q^8 + 3*b2 * q^9 + (b6 - b2 - b1 + 1) * q^10 + (-2*b6 + 2*b5) * q^11 + (-b7 - b6 + b3) * q^12 + (b7 - b5 + 2*b4 - b3) * q^13 + (2*b7 - 2*b5 + 3*b4 + 3) * q^15 - b2 * q^16 + 2*b1 * q^17 - 3*b1 * q^18 + (-b7 + b6 - b3 - 4*b2) * q^19 + (-b7 + b5 + b4 + 1) * q^20 + (2*b7 - 2*b5 - 2*b3) * q^22 + (2*b7 + 2*b6 - 2*b4 - 2*b3 + 2*b1) * q^23 + (b6 - b5) * q^24 + (2*b6 - 2*b5 + b1) * q^25 + (b7 - b6 + b3 - 2*b2) * q^26 + (2*b7 - 2*b5 + 2*b3 - 2) * q^29 + (-2*b6 + 3*b4 + 2*b3 - 3*b2 - 3*b1) * q^30 + (-2*b6 + 2*b5 - 4*b2 + 4) * q^31 + b1 * q^32 + (-12*b4 + 12*b1) * q^33 - 2 * q^34 + 3 * q^36 + (-2*b4 + 2*b1) * q^37 + (b6 + b5 + 4*b1) * q^38 + (2*b6 - 2*b5 + 6*b2 - 6) * q^39 + (b6 + b4 - b3 - b2 - b1) * q^40 + (2*b7 - 2*b5 + 2*b3 + 6) * q^41 + (2*b7 - 2*b5 - 4*b4 - 2*b3) * q^43 + (2*b7 - 2*b6 + 2*b3) * q^44 + (-3*b5 + 3*b2 - 3*b1 - 3) * q^45 + (-2*b6 + 2*b5 + 2*b2 - 2) * q^46 + (2*b7 + 2*b6 + 4*b4 - 2*b3 - 4*b1) * q^47 + (-b7 + b5 + b3) * q^48 + (-2*b7 + 2*b5 + 2*b3 - 1) * q^50 + (-2*b7 + 2*b6 - 2*b3) * q^51 + (-b6 - b5 + 2*b1) * q^52 + (2*b6 + 2*b5 - 6*b1) * q^53 + (-6*b4 - 4*b3 + 6) * q^55 + (-4*b7 + 4*b5 + 6*b4 + 4*b3) * q^57 + (-2*b7 - 2*b6 - 2*b4 + 2*b3 + 2*b1) * q^58 + (b6 - b5 - 4*b2 + 4) * q^59 + (-2*b5 - 3*b2 + 3*b1 + 3) * q^60 + (b7 - b6 + b3 + 6*b2) * q^61 + (2*b7 - 2*b5 + 4*b4 - 2*b3) * q^62 - q^64 + (2*b7 + 2*b6 + 5*b4 - 2*b3 + b2 - 5*b1) * q^65 + (12*b2 - 12) * q^66 + 8*b1 * q^67 + (-2*b4 + 2*b1) * q^68 + (-2*b7 + 2*b5 - 2*b3 - 12) * q^69 + (-2*b7 + 2*b5 - 2*b3 - 6) * q^71 + (3*b4 - 3*b1) * q^72 + (2*b6 + 2*b5 + 2*b1) * q^73 + (2*b2 - 2) * q^74 + (-b7 + b6 + 12*b4 - b3 - 12*b1) * q^75 + (-b7 + b5 - b3 - 4) * q^76 + (-2*b7 + 2*b5 - 6*b4 + 2*b3) * q^78 + (-2*b7 + 2*b6 - 2*b3 + 2*b2) * q^79 + (b5 - b2 + b1 + 1) * q^80 + (-9*b2 + 9) * q^81 + (-2*b7 - 2*b6 + 6*b4 + 2*b3 - 6*b1) * q^82 + (-b7 + b5 + b3) * q^83 + (2*b4 - 2*b3 - 2) * q^85 + (2*b7 - 2*b6 + 2*b3 + 4*b2) * q^86 + (2*b6 + 2*b5 - 12*b1) * q^87 + (-2*b6 - 2*b5) * q^88 + 10*b2 * q^89 + (-3*b4 + 3*b3 + 3) * q^90 + (2*b7 - 2*b5 - 2*b4 - 2*b3) * q^92 + (-4*b7 - 4*b6 - 12*b4 + 4*b3 + 12*b1) * q^93 + (-2*b6 + 2*b5 - 4*b2 + 4) * q^94 + (2*b6 + 4*b5 - b2 + 7*b1 + 1) * q^95 + (-b7 + b6 - b3) * q^96 + (-4*b7 + 4*b5 - 6*b4 + 4*b3) * q^97 + (-6*b7 + 6*b5 - 6*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10})$$ 8 * q + 4 * q^4 + 4 * q^5 + 12 * q^9 $$8 q + 4 q^{4} + 4 q^{5} + 12 q^{9} + 4 q^{10} + 24 q^{15} - 4 q^{16} - 16 q^{19} + 8 q^{20} - 8 q^{26} - 16 q^{29} - 12 q^{30} + 16 q^{31} - 16 q^{34} + 24 q^{36} - 24 q^{39} - 4 q^{40} + 48 q^{41} - 12 q^{45} - 8 q^{46} - 8 q^{50} + 48 q^{55} + 16 q^{59} + 12 q^{60} + 24 q^{61} - 8 q^{64} + 4 q^{65} - 48 q^{66} - 96 q^{69} - 48 q^{71} - 8 q^{74} - 32 q^{76} + 8 q^{79} + 4 q^{80} + 36 q^{81} - 16 q^{85} + 16 q^{86} + 40 q^{89} + 24 q^{90} + 16 q^{94} + 4 q^{95}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^5 + 12 * q^9 + 4 * q^10 + 24 * q^15 - 4 * q^16 - 16 * q^19 + 8 * q^20 - 8 * q^26 - 16 * q^29 - 12 * q^30 + 16 * q^31 - 16 * q^34 + 24 * q^36 - 24 * q^39 - 4 * q^40 + 48 * q^41 - 12 * q^45 - 8 * q^46 - 8 * q^50 + 48 * q^55 + 16 * q^59 + 12 * q^60 + 24 * q^61 - 8 * q^64 + 4 * q^65 - 48 * q^66 - 96 * q^69 - 48 * q^71 - 8 * q^74 - 32 * q^76 + 8 * q^79 + 4 * q^80 + 36 * q^81 - 16 * q^85 + 16 * q^86 + 40 * q^89 + 24 * q^90 + 16 * q^94 + 4 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{5} + \zeta_{24}$$ v^5 + v $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{5}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}^{3}$$ v^7 + v^3 $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{5} + 2\zeta_{24}$$ -v^5 + 2*v $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{3}$$ -v^7 + 2*v^3
 $$\zeta_{24}$$ $$=$$ $$( \beta_{6} + \beta_{3} ) / 3$$ (b6 + b3) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{5} ) / 3$$ (b7 + b5) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{6} + 2\beta_{3} ) / 3$$ (-b6 + 2*b3) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{4}$$ b4 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} ) / 3$$ (-b7 + 2*b5) / 3

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
−0.866025 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i −2.03906 0.917738i 2.44949 0 1.00000i 1.50000 2.59808i 1.30701 + 1.81431i
79.2 −0.866025 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.30701 1.81431i −2.44949 0 1.00000i 1.50000 2.59808i −2.03906 + 0.917738i
79.3 0.866025 + 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i 0.917738 2.03906i −2.44949 0 1.00000i 1.50000 2.59808i 1.81431 1.30701i
79.4 0.866025 + 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.81431 + 1.30701i 2.44949 0 1.00000i 1.50000 2.59808i 0.917738 + 2.03906i
459.1 −0.866025 + 0.500000i −2.12132 1.22474i 0.500000 0.866025i −2.03906 + 0.917738i 2.44949 0 1.00000i 1.50000 + 2.59808i 1.30701 1.81431i
459.2 −0.866025 + 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.30701 + 1.81431i −2.44949 0 1.00000i 1.50000 + 2.59808i −2.03906 0.917738i
459.3 0.866025 0.500000i −2.12132 1.22474i 0.500000 0.866025i 0.917738 + 2.03906i −2.44949 0 1.00000i 1.50000 + 2.59808i 1.81431 + 1.30701i
459.4 0.866025 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.81431 1.30701i 2.44949 0 1.00000i 1.50000 + 2.59808i 0.917738 2.03906i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.i.f 8
5.b even 2 1 inner 490.2.i.f 8
7.b odd 2 1 490.2.i.c 8
7.c even 3 1 490.2.c.e 4
7.c even 3 1 inner 490.2.i.f 8
7.d odd 6 1 70.2.c.a 4
7.d odd 6 1 490.2.i.c 8
21.g even 6 1 630.2.g.g 4
28.f even 6 1 560.2.g.e 4
35.c odd 2 1 490.2.i.c 8
35.i odd 6 1 70.2.c.a 4
35.i odd 6 1 490.2.i.c 8
35.j even 6 1 490.2.c.e 4
35.j even 6 1 inner 490.2.i.f 8
35.k even 12 1 350.2.a.g 2
35.k even 12 1 350.2.a.h 2
35.l odd 12 1 2450.2.a.bl 2
35.l odd 12 1 2450.2.a.bq 2
56.j odd 6 1 2240.2.g.j 4
56.m even 6 1 2240.2.g.i 4
84.j odd 6 1 5040.2.t.t 4
105.p even 6 1 630.2.g.g 4
105.w odd 12 1 3150.2.a.bs 2
105.w odd 12 1 3150.2.a.bt 2
140.s even 6 1 560.2.g.e 4
140.x odd 12 1 2800.2.a.bl 2
140.x odd 12 1 2800.2.a.bm 2
280.ba even 6 1 2240.2.g.i 4
280.bk odd 6 1 2240.2.g.j 4
420.be odd 6 1 5040.2.t.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 7.d odd 6 1
70.2.c.a 4 35.i odd 6 1
350.2.a.g 2 35.k even 12 1
350.2.a.h 2 35.k even 12 1
490.2.c.e 4 7.c even 3 1
490.2.c.e 4 35.j even 6 1
490.2.i.c 8 7.b odd 2 1
490.2.i.c 8 7.d odd 6 1
490.2.i.c 8 35.c odd 2 1
490.2.i.c 8 35.i odd 6 1
490.2.i.f 8 1.a even 1 1 trivial
490.2.i.f 8 5.b even 2 1 inner
490.2.i.f 8 7.c even 3 1 inner
490.2.i.f 8 35.j even 6 1 inner
560.2.g.e 4 28.f even 6 1
560.2.g.e 4 140.s even 6 1
630.2.g.g 4 21.g even 6 1
630.2.g.g 4 105.p even 6 1
2240.2.g.i 4 56.m even 6 1
2240.2.g.i 4 280.ba even 6 1
2240.2.g.j 4 56.j odd 6 1
2240.2.g.j 4 280.bk odd 6 1
2450.2.a.bl 2 35.l odd 12 1
2450.2.a.bq 2 35.l odd 12 1
2800.2.a.bl 2 140.x odd 12 1
2800.2.a.bm 2 140.x odd 12 1
3150.2.a.bs 2 105.w odd 12 1
3150.2.a.bt 2 105.w odd 12 1
5040.2.t.t 4 84.j odd 6 1
5040.2.t.t 4 420.be odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 6T_{3}^{2} + 36$$ T3^4 - 6*T3^2 + 36 $$T_{11}^{4} + 24T_{11}^{2} + 576$$ T11^4 + 24*T11^2 + 576 $$T_{19}^{4} + 8T_{19}^{3} + 54T_{19}^{2} + 80T_{19} + 100$$ T19^4 + 8*T19^3 + 54*T19^2 + 80*T19 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$(T^{4} - 6 T^{2} + 36)^{2}$$
$5$ $$T^{8} - 4 T^{7} + \cdots + 625$$
$7$ $$T^{8}$$
$11$ $$(T^{4} + 24 T^{2} + 576)^{2}$$
$13$ $$(T^{4} + 20 T^{2} + 4)^{2}$$
$17$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$19$ $$(T^{4} + 8 T^{3} + \cdots + 100)^{2}$$
$23$ $$T^{8} - 56 T^{6} + \cdots + 160000$$
$29$ $$(T^{2} + 4 T - 20)^{4}$$
$31$ $$(T^{4} - 8 T^{3} + 72 T^{2} + \cdots + 64)^{2}$$
$37$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$41$ $$(T^{2} - 12 T + 12)^{4}$$
$43$ $$(T^{4} + 80 T^{2} + 64)^{2}$$
$47$ $$T^{8} - 80 T^{6} + \cdots + 4096$$
$53$ $$T^{8} - 120 T^{6} + \cdots + 20736$$
$59$ $$(T^{4} - 8 T^{3} + \cdots + 100)^{2}$$
$61$ $$(T^{4} - 12 T^{3} + \cdots + 900)^{2}$$
$67$ $$(T^{4} - 64 T^{2} + 4096)^{2}$$
$71$ $$(T^{2} + 12 T + 12)^{4}$$
$73$ $$T^{8} - 56 T^{6} + \cdots + 160000$$
$79$ $$(T^{4} - 4 T^{3} + \cdots + 400)^{2}$$
$83$ $$(T^{2} + 6)^{4}$$
$89$ $$(T^{2} - 10 T + 100)^{4}$$
$97$ $$(T^{4} + 264 T^{2} + 3600)^{2}$$