# Properties

 Label 405.2.j.f Level $405$ Weight $2$ Character orbit 405.j Analytic conductor $3.234$ 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] = [405,2,Mod(109,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.j (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: $$3.23394128186$$ 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: $$3^{4}$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} - \beta_{4} q^{5} - \beta_1 q^{7} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8}+O(q^{10})$$ q + b6 * q^2 - b4 * q^5 - b1 * q^7 + (-2*b6 + 2*b5) * q^8 $$q + \beta_{6} q^{2} - \beta_{4} q^{5} - \beta_1 q^{7} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8} + ( - \beta_{3} - 1) q^{10} + (2 \beta_{7} - \beta_{6} + \cdots - 2 \beta_{4}) q^{11}+ \cdots + (2 \beta_{6} - 2 \beta_{5}) q^{98}+O(q^{100})$$ q + b6 * q^2 - b4 * q^5 - b1 * q^7 + (-2*b6 + 2*b5) * q^8 + (-b3 - 1) * q^10 + (2*b7 - b6 + 2*b5 - 2*b4) * q^11 + (b3 - b1) * q^13 + (b5 - 2*b4) * q^14 + (-4*b2 + 4) * q^16 + (2*b6 - 2*b5) * q^17 - q^19 + (-2*b3 + 2*b1) * q^22 + 5*b5 * q^23 + (4*b2 + b1 - 4) * q^25 + (-2*b7 + b6 - b5) * q^26 + (2*b7 - b6 + 2*b5 - 2*b4) * q^29 - 2*b2 * q^31 + (4*b2 - 4) * q^34 + (b7 + 4*b6 - 4*b5) * q^35 + 3*b3 * q^37 - b6 * q^38 + (2*b3 + 2*b2 - 2*b1) * q^40 + (b5 - 2*b4) * q^41 + 2*b1 * q^43 + 10 * q^46 - 2*b6 * q^47 + 2*b2 * q^49 + (-5*b5 + 2*b4) * q^50 + (-7*b6 + 7*b5) * q^53 + (b3 - 9) * q^55 + (-4*b7 + 2*b6 - 4*b5 + 4*b4) * q^56 + (-2*b3 + 2*b1) * q^58 + (-2*b5 + 4*b4) * q^59 + (-13*b2 + 13) * q^61 + (-2*b6 + 2*b5) * q^62 - 8 * q^64 + (b7 + 4*b6 + b5 - b4) * q^65 + (b3 - b1) * q^67 + (9*b2 + b1 - 9) * q^70 + (6*b7 - 3*b6 + 3*b5) * q^71 - 3*b3 * q^73 + (-6*b7 + 3*b6 - 6*b5 + 6*b4) * q^74 - 9*b5 * q^77 + (5*b2 - 5) * q^79 + (-4*b7 + 4*b6 - 4*b5) * q^80 - 2*b3 * q^82 + b6 * q^83 + (-2*b3 - 2*b2 + 2*b1) * q^85 + (-2*b5 + 4*b4) * q^86 - 4*b1 * q^88 + 9 * q^91 - 4*b2 * q^94 + b4 * q^95 - b1 * q^97 + (2*b6 - 2*b5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{10} + 16 q^{16} - 8 q^{19} - 16 q^{25} - 8 q^{31} - 16 q^{34} + 8 q^{40} + 80 q^{46} + 8 q^{49} - 72 q^{55} + 52 q^{61} - 64 q^{64} - 36 q^{70} - 20 q^{79} - 8 q^{85} + 72 q^{91} - 16 q^{94}+O(q^{100})$$ 8 * q - 8 * q^10 + 16 * q^16 - 8 * q^19 - 16 * q^25 - 8 * q^31 - 16 * q^34 + 8 * q^40 + 80 * q^46 + 8 * q^49 - 72 * q^55 + 52 * q^61 - 64 * q^64 - 36 * q^70 - 20 * q^79 - 8 * q^85 + 72 * q^91 - 16 * q^94

Basis of coefficient ring

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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
 −0.258819 − 0.965926i −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
−1.22474 + 0.707107i 0 0 −0.448288 + 2.19067i 0 2.59808 1.50000i 2.82843i 0 −1.00000 3.00000i
109.2 −1.22474 + 0.707107i 0 0 1.67303 1.48356i 0 −2.59808 + 1.50000i 2.82843i 0 −1.00000 + 3.00000i
109.3 1.22474 0.707107i 0 0 −1.67303 + 1.48356i 0 −2.59808 + 1.50000i 2.82843i 0 −1.00000 + 3.00000i
109.4 1.22474 0.707107i 0 0 0.448288 2.19067i 0 2.59808 1.50000i 2.82843i 0 −1.00000 3.00000i
379.1 −1.22474 0.707107i 0 0 −0.448288 2.19067i 0 2.59808 + 1.50000i 2.82843i 0 −1.00000 + 3.00000i
379.2 −1.22474 0.707107i 0 0 1.67303 + 1.48356i 0 −2.59808 1.50000i 2.82843i 0 −1.00000 3.00000i
379.3 1.22474 + 0.707107i 0 0 −1.67303 1.48356i 0 −2.59808 1.50000i 2.82843i 0 −1.00000 3.00000i
379.4 1.22474 + 0.707107i 0 0 0.448288 + 2.19067i 0 2.59808 + 1.50000i 2.82843i 0 −1.00000 + 3.00000i
 $$n$$: 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 405.2.j.f 8
3.b odd 2 1 inner 405.2.j.f 8
5.b even 2 1 inner 405.2.j.f 8
9.c even 3 1 135.2.b.b 4
9.c even 3 1 inner 405.2.j.f 8
9.d odd 6 1 135.2.b.b 4
9.d odd 6 1 inner 405.2.j.f 8
15.d odd 2 1 inner 405.2.j.f 8
36.f odd 6 1 2160.2.f.k 4
36.h even 6 1 2160.2.f.k 4
45.h odd 6 1 135.2.b.b 4
45.h odd 6 1 inner 405.2.j.f 8
45.j even 6 1 135.2.b.b 4
45.j even 6 1 inner 405.2.j.f 8
45.k odd 12 1 675.2.a.l 2
45.k odd 12 1 675.2.a.m 2
45.l even 12 1 675.2.a.l 2
45.l even 12 1 675.2.a.m 2
180.n even 6 1 2160.2.f.k 4
180.p odd 6 1 2160.2.f.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b 4 9.c even 3 1
135.2.b.b 4 9.d odd 6 1
135.2.b.b 4 45.h odd 6 1
135.2.b.b 4 45.j even 6 1
405.2.j.f 8 1.a even 1 1 trivial
405.2.j.f 8 3.b odd 2 1 inner
405.2.j.f 8 5.b even 2 1 inner
405.2.j.f 8 9.c even 3 1 inner
405.2.j.f 8 9.d odd 6 1 inner
405.2.j.f 8 15.d odd 2 1 inner
405.2.j.f 8 45.h odd 6 1 inner
405.2.j.f 8 45.j even 6 1 inner
675.2.a.l 2 45.k odd 12 1
675.2.a.l 2 45.l even 12 1
675.2.a.m 2 45.k odd 12 1
675.2.a.m 2 45.l even 12 1
2160.2.f.k 4 36.f odd 6 1
2160.2.f.k 4 36.h even 6 1
2160.2.f.k 4 180.n even 6 1
2160.2.f.k 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 $$S_{2}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{4} - 2T_{2}^{2} + 4$$ T2^4 - 2*T2^2 + 4 $$T_{7}^{4} - 9T_{7}^{2} + 81$$ T7^4 - 9*T7^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 8 T^{6} + \cdots + 625$$
$7$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$11$ $$(T^{4} + 18 T^{2} + 324)^{2}$$
$13$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$17$ $$(T^{2} + 8)^{4}$$
$19$ $$(T + 1)^{8}$$
$23$ $$(T^{4} - 50 T^{2} + 2500)^{2}$$
$29$ $$(T^{4} + 18 T^{2} + 324)^{2}$$
$31$ $$(T^{2} + 2 T + 4)^{4}$$
$37$ $$(T^{2} + 81)^{4}$$
$41$ $$(T^{4} + 18 T^{2} + 324)^{2}$$
$43$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$47$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 98)^{4}$$
$59$ $$(T^{4} + 72 T^{2} + 5184)^{2}$$
$61$ $$(T^{2} - 13 T + 169)^{4}$$
$67$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$71$ $$(T^{2} - 162)^{4}$$
$73$ $$(T^{2} + 81)^{4}$$
$79$ $$(T^{2} + 5 T + 25)^{4}$$
$83$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$89$ $$T^{8}$$
$97$ $$(T^{4} - 9 T^{2} + 81)^{2}$$