# Properties

 Label 810.2.i.h Level $810$ Weight $2$ Character orbit 810.i Analytic conductor $6.468$ Analytic rank $0$ Dimension $8$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(109,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("810.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 810.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: $$6.46788256372$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.2702336256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ x^8 + 9*x^6 + 56*x^4 + 225*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{6} q^{8}+O(q^{10})$$ q + b4 * q^2 - b2 * q^4 - b1 * q^5 + (2*b5 + 2*b3 + b2 + 1) * q^7 - b6 * q^8 $$q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{6} q^{8} + (\beta_{5} + 1) q^{10} + ( - 2 \beta_{7} - \beta_{4}) q^{11} + ( - \beta_{6} - \beta_{4} - 2 \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + 4 \beta_{6} q^{17} - 6 q^{19} + (\beta_{7} + \beta_{4}) q^{20} + (2 \beta_{3} + \beta_{2}) q^{22} + ( - 2 \beta_{6} - 2 \beta_{4}) q^{23} + (\beta_{5} + \beta_{3} - 4 \beta_{2} - 4) q^{25} + (2 \beta_{5} + 1) q^{28} + 7 \beta_{2} q^{31} + ( - \beta_{6} - \beta_{4}) q^{32} + (4 \beta_{2} + 4) q^{34} + ( - \beta_{7} - 10 \beta_{6} + \cdots - \beta_1) q^{35}+ \cdots - 12 \beta_{6} q^{98}+O(q^{100})$$ q + b4 * q^2 - b2 * q^4 - b1 * q^5 + (2*b5 + 2*b3 + b2 + 1) * q^7 - b6 * q^8 + (b5 + 1) * q^10 + (-2*b7 - b4) * q^11 + (-b6 - b4 - 2*b1) * q^14 + (-b2 - 1) * q^16 + 4*b6 * q^17 - 6 * q^19 + (b7 + b4) * q^20 + (2*b3 + b2) * q^22 + (-2*b6 - 2*b4) * q^23 + (b5 + b3 - 4*b2 - 4) * q^25 + (2*b5 + 1) * q^28 + 7*b2 * q^31 + (-b6 - b4) * q^32 + (4*b2 + 4) * q^34 + (-b7 - 10*b6 - b4 - b1) * q^35 + (4*b5 + 2) * q^37 - 6*b4 * q^38 + (-b3 - b2) * q^40 + (2*b6 + 2*b4 + 4*b1) * q^41 + (4*b5 + 4*b3 + 2*b2 + 2) * q^43 + (-2*b7 - b6 - 2*b4 - 2*b1) * q^44 - 2 * q^46 + 2*b4 * q^47 - 12*b2 * q^49 + (-5*b6 - 5*b4 - b1) * q^50 + 3*b6 * q^53 + (-b5 + 9) * q^55 + (2*b7 + b4) * q^56 + (2*b6 + 2*b4 + 4*b1) * q^59 + (4*b2 + 4) * q^61 + 7*b6 * q^62 - q^64 + (-4*b3 - 2*b2) * q^67 + (4*b6 + 4*b4) * q^68 + (b5 + b3 - 9*b2 - 9) * q^70 + (2*b5 + 1) * q^73 + (4*b7 + 2*b4) * q^74 + 6*b2 * q^76 + (19*b6 + 19*b4) * q^77 + (b7 + b4 + b1) * q^80 + (-4*b5 - 2) * q^82 + 5*b4 * q^83 + (4*b3 + 4*b2) * q^85 + (-2*b6 - 2*b4 - 4*b1) * q^86 + (2*b5 + 2*b3 + b2 + 1) * q^88 + (4*b7 + 2*b6 + 4*b4 + 4*b1) * q^89 - 2*b4 * q^92 - 2*b2 * q^94 + 6*b1 * q^95 + (-2*b5 - 2*b3 - b2 - 1) * q^97 - 12*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4}+O(q^{10})$$ 8 * q + 4 * q^4 $$8 q + 4 q^{4} + 4 q^{10} - 4 q^{16} - 48 q^{19} - 18 q^{25} - 28 q^{31} + 16 q^{34} + 2 q^{40} - 16 q^{46} + 48 q^{49} + 76 q^{55} + 16 q^{61} - 8 q^{64} - 38 q^{70} - 24 q^{76} - 8 q^{85} + 8 q^{94}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^10 - 4 * q^16 - 48 * q^19 - 18 * q^25 - 28 * q^31 + 16 * q^34 + 2 * q^40 - 16 * q^46 + 48 * q^49 + 76 * q^55 + 16 * q^61 - 8 * q^64 - 38 * q^70 - 24 * q^76 - 8 * q^85 + 8 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -9\nu^{6} - 56\nu^{4} - 504\nu^{2} - 2025 ) / 1400$$ (-9*v^6 - 56*v^4 - 504*v^2 - 2025) / 1400 $$\beta_{3}$$ $$=$$ $$( -11\nu^{6} - 224\nu^{4} - 616\nu^{2} - 2475 ) / 1400$$ (-11*v^6 - 224*v^4 - 616*v^2 - 2475) / 1400 $$\beta_{4}$$ $$=$$ $$( 11\nu^{7} + 224\nu^{5} + 616\nu^{3} + 2475\nu ) / 7000$$ (11*v^7 + 224*v^5 + 616*v^3 + 2475*v) / 7000 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} - 1 ) / 56$$ (-v^6 - 1) / 56 $$\beta_{6}$$ $$=$$ $$( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750$$ (-9*v^7 - 56*v^5 - 154*v^3 - 625*v) / 1750 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} - 9\nu^{5} - 56\nu^{3} - 225\nu ) / 125$$ (-v^7 - 9*v^5 - 56*v^3 - 225*v) / 125
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3} - 4\beta_{2} - 4$$ b5 + b3 - 4*b2 - 4 $$\nu^{3}$$ $$=$$ $$-4\beta_{7} + 5\beta_{6} - 4\beta_{4} - 4\beta_1$$ -4*b7 + 5*b6 - 4*b4 - 4*b1 $$\nu^{4}$$ $$=$$ $$-9\beta_{3} + 11\beta_{2}$$ -9*b3 + 11*b2 $$\nu^{5}$$ $$=$$ $$11\beta_{7} + 56\beta_{4}$$ 11*b7 + 56*b4 $$\nu^{6}$$ $$=$$ $$-56\beta_{5} - 1$$ -56*b5 - 1 $$\nu^{7}$$ $$=$$ $$-280\beta_{6} - 280\beta_{4} - \beta_1$$ -280*b6 - 280*b4 - b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$$.

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 1.52274 − 1.63746i −0.656712 + 2.13746i 0.656712 − 2.13746i −1.52274 + 1.63746i 1.52274 + 1.63746i −0.656712 − 2.13746i 0.656712 + 2.13746i −1.52274 − 1.63746i
−0.866025 + 0.500000i 0 0.500000 0.866025i −1.52274 + 1.63746i 0 3.77492 2.17945i 1.00000i 0 0.500000 2.17945i
109.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.656712 2.13746i 0 −3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
109.3 0.866025 0.500000i 0 0.500000 0.866025i −0.656712 + 2.13746i 0 −3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
109.4 0.866025 0.500000i 0 0.500000 0.866025i 1.52274 1.63746i 0 3.77492 2.17945i 1.00000i 0 0.500000 2.17945i
379.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.52274 1.63746i 0 3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
379.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.656712 + 2.13746i 0 −3.77492 2.17945i 1.00000i 0 0.500000 2.17945i
379.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.656712 2.13746i 0 −3.77492 2.17945i 1.00000i 0 0.500000 2.17945i
379.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.52274 + 1.63746i 0 3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.i.h 8
3.b odd 2 1 inner 810.2.i.h 8
5.b even 2 1 inner 810.2.i.h 8
9.c even 3 1 270.2.c.c 4
9.c even 3 1 inner 810.2.i.h 8
9.d odd 6 1 270.2.c.c 4
9.d odd 6 1 inner 810.2.i.h 8
15.d odd 2 1 inner 810.2.i.h 8
36.f odd 6 1 2160.2.f.m 4
36.h even 6 1 2160.2.f.m 4
45.h odd 6 1 270.2.c.c 4
45.h odd 6 1 inner 810.2.i.h 8
45.j even 6 1 270.2.c.c 4
45.j even 6 1 inner 810.2.i.h 8
45.k odd 12 1 1350.2.a.w 2
45.k odd 12 1 1350.2.a.x 2
45.l even 12 1 1350.2.a.w 2
45.l even 12 1 1350.2.a.x 2
180.n even 6 1 2160.2.f.m 4
180.p odd 6 1 2160.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 9.c even 3 1
270.2.c.c 4 9.d odd 6 1
270.2.c.c 4 45.h odd 6 1
270.2.c.c 4 45.j even 6 1
810.2.i.h 8 1.a even 1 1 trivial
810.2.i.h 8 3.b odd 2 1 inner
810.2.i.h 8 5.b even 2 1 inner
810.2.i.h 8 9.c even 3 1 inner
810.2.i.h 8 9.d odd 6 1 inner
810.2.i.h 8 15.d odd 2 1 inner
810.2.i.h 8 45.h odd 6 1 inner
810.2.i.h 8 45.j even 6 1 inner
1350.2.a.w 2 45.k odd 12 1
1350.2.a.w 2 45.l even 12 1
1350.2.a.x 2 45.k odd 12 1
1350.2.a.x 2 45.l even 12 1
2160.2.f.m 4 36.f odd 6 1
2160.2.f.m 4 36.h even 6 1
2160.2.f.m 4 180.n even 6 1
2160.2.f.m 4 180.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{4} - 19T_{7}^{2} + 361$$ T7^4 - 19*T7^2 + 361 $$T_{11}^{4} + 19T_{11}^{2} + 361$$ T11^4 + 19*T11^2 + 361

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 9 T^{6} + \cdots + 625$$
$7$ $$(T^{4} - 19 T^{2} + 361)^{2}$$
$11$ $$(T^{4} + 19 T^{2} + 361)^{2}$$
$13$ $$T^{8}$$
$17$ $$(T^{2} + 16)^{4}$$
$19$ $$(T + 6)^{8}$$
$23$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$29$ $$T^{8}$$
$31$ $$(T^{2} + 7 T + 49)^{4}$$
$37$ $$(T^{2} + 76)^{4}$$
$41$ $$(T^{4} + 76 T^{2} + 5776)^{2}$$
$43$ $$(T^{4} - 76 T^{2} + 5776)^{2}$$
$47$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$53$ $$(T^{2} + 9)^{4}$$
$59$ $$(T^{4} + 76 T^{2} + 5776)^{2}$$
$61$ $$(T^{2} - 4 T + 16)^{4}$$
$67$ $$(T^{4} - 76 T^{2} + 5776)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} + 19)^{4}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} - 25 T^{2} + 625)^{2}$$
$89$ $$(T^{2} - 76)^{4}$$
$97$ $$(T^{4} - 19 T^{2} + 361)^{2}$$