# Properties

 Label 405.2.e.e Level $405$ Weight $2$ Character orbit 405.e Analytic conductor $3.234$ 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] = [405,2,Mod(136,405)]

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([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$3.23394128186$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7}+O(q^{10})$$ q + 2*z * q^4 + z * q^5 + (2*z - 2) * q^7 $$q + 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + 4 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} - 6 q^{17} - q^{19} + (2 \zeta_{6} - 2) q^{20} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 4 q^{28} + ( - 9 \zeta_{6} + 9) q^{29} + \zeta_{6} q^{31} - 2 q^{35} + 8 q^{37} - 3 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + 6 q^{44} + (12 \zeta_{6} - 12) q^{47} + 3 \zeta_{6} q^{49} + (8 \zeta_{6} - 8) q^{52} + 6 q^{53} + 3 q^{55} - 3 \zeta_{6} q^{59} + ( - 10 \zeta_{6} + 10) q^{61} - 8 q^{64} + (4 \zeta_{6} - 4) q^{65} - 14 \zeta_{6} q^{67} - 12 \zeta_{6} q^{68} - 3 q^{71} + 2 q^{73} - 2 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + ( - 16 \zeta_{6} + 16) q^{79} - 4 q^{80} + ( - 12 \zeta_{6} + 12) q^{83} - 6 \zeta_{6} q^{85} + 15 q^{89} - 8 q^{91} + (12 \zeta_{6} - 12) q^{92} - \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 4) q^{97} +O(q^{100})$$ q + 2*z * q^4 + z * q^5 + (2*z - 2) * q^7 + (-3*z + 3) * q^11 + 4*z * q^13 + (4*z - 4) * q^16 - 6 * q^17 - q^19 + (2*z - 2) * q^20 + 6*z * q^23 + (z - 1) * q^25 - 4 * q^28 + (-9*z + 9) * q^29 + z * q^31 - 2 * q^35 + 8 * q^37 - 3*z * q^41 + (-4*z + 4) * q^43 + 6 * q^44 + (12*z - 12) * q^47 + 3*z * q^49 + (8*z - 8) * q^52 + 6 * q^53 + 3 * q^55 - 3*z * q^59 + (-10*z + 10) * q^61 - 8 * q^64 + (4*z - 4) * q^65 - 14*z * q^67 - 12*z * q^68 - 3 * q^71 + 2 * q^73 - 2*z * q^76 + 6*z * q^77 + (-16*z + 16) * q^79 - 4 * q^80 + (-12*z + 12) * q^83 - 6*z * q^85 + 15 * q^89 - 8 * q^91 + (12*z - 12) * q^92 - z * q^95 + (-4*z + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + q^5 - 2 * q^7 $$2 q + 2 q^{4} + q^{5} - 2 q^{7} + 3 q^{11} + 4 q^{13} - 4 q^{16} - 12 q^{17} - 2 q^{19} - 2 q^{20} + 6 q^{23} - q^{25} - 8 q^{28} + 9 q^{29} + q^{31} - 4 q^{35} + 16 q^{37} - 3 q^{41} + 4 q^{43} + 12 q^{44} - 12 q^{47} + 3 q^{49} - 8 q^{52} + 12 q^{53} + 6 q^{55} - 3 q^{59} + 10 q^{61} - 16 q^{64} - 4 q^{65} - 14 q^{67} - 12 q^{68} - 6 q^{71} + 4 q^{73} - 2 q^{76} + 6 q^{77} + 16 q^{79} - 8 q^{80} + 12 q^{83} - 6 q^{85} + 30 q^{89} - 16 q^{91} - 12 q^{92} - q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + q^5 - 2 * q^7 + 3 * q^11 + 4 * q^13 - 4 * q^16 - 12 * q^17 - 2 * q^19 - 2 * q^20 + 6 * q^23 - q^25 - 8 * q^28 + 9 * q^29 + q^31 - 4 * q^35 + 16 * q^37 - 3 * q^41 + 4 * q^43 + 12 * q^44 - 12 * q^47 + 3 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 3 * q^59 + 10 * q^61 - 16 * q^64 - 4 * q^65 - 14 * q^67 - 12 * q^68 - 6 * q^71 + 4 * q^73 - 2 * q^76 + 6 * q^77 + 16 * q^79 - 8 * q^80 + 12 * q^83 - 6 * q^85 + 30 * q^89 - 16 * q^91 - 12 * q^92 - q^95 + 4 * q^97

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 1.00000 + 1.73205i 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0
271.1 0 0 1.00000 1.73205i 0.500000 0.866025i 0 −1.00000 1.73205i 0 0 0
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.e.e 2
3.b odd 2 1 405.2.e.d 2
9.c even 3 1 405.2.a.c 1
9.c even 3 1 inner 405.2.e.e 2
9.d odd 6 1 405.2.a.d yes 1
9.d odd 6 1 405.2.e.d 2
36.f odd 6 1 6480.2.a.c 1
36.h even 6 1 6480.2.a.o 1
45.h odd 6 1 2025.2.a.d 1
45.j even 6 1 2025.2.a.c 1
45.k odd 12 2 2025.2.b.e 2
45.l even 12 2 2025.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.c 1 9.c even 3 1
405.2.a.d yes 1 9.d odd 6 1
405.2.e.d 2 3.b odd 2 1
405.2.e.d 2 9.d odd 6 1
405.2.e.e 2 1.a even 1 1 trivial
405.2.e.e 2 9.c even 3 1 inner
2025.2.a.c 1 45.j even 6 1
2025.2.a.d 1 45.h odd 6 1
2025.2.b.e 2 45.k odd 12 2
2025.2.b.f 2 45.l even 12 2
6480.2.a.c 1 36.f odd 6 1
6480.2.a.o 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} - T + 1$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} - 4T + 16$$