# Properties

 Label 405.2.e.g 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + z * q^5 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + 2 q^{10} + ( - 5 \zeta_{6} + 5) q^{11} - 4 \zeta_{6} q^{13} + ( - 4 \zeta_{6} + 4) q^{16} + 4 q^{17} - 5 q^{19} + ( - 2 \zeta_{6} + 2) q^{20} - 10 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 8 q^{26} + (5 \zeta_{6} - 5) q^{29} + 9 \zeta_{6} q^{31} - 8 \zeta_{6} q^{32} + ( - 8 \zeta_{6} + 8) q^{34} - 10 q^{37} + (10 \zeta_{6} - 10) q^{38} + 7 \zeta_{6} q^{41} + ( - 2 \zeta_{6} + 2) q^{43} - 10 q^{44} + 12 q^{46} + ( - 2 \zeta_{6} + 2) q^{47} + 7 \zeta_{6} q^{49} + 2 \zeta_{6} q^{50} + (8 \zeta_{6} - 8) q^{52} - 8 q^{53} + 5 q^{55} + 10 \zeta_{6} q^{58} - \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 18 q^{62} - 8 q^{64} + ( - 4 \zeta_{6} + 4) q^{65} - 6 \zeta_{6} q^{67} - 8 \zeta_{6} q^{68} - q^{71} - 8 q^{73} + (20 \zeta_{6} - 20) q^{74} + 10 \zeta_{6} q^{76} + (12 \zeta_{6} - 12) q^{79} + 4 q^{80} + 14 q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + 4 \zeta_{6} q^{85} - 4 \zeta_{6} q^{86} + 9 q^{89} + ( - 12 \zeta_{6} + 12) q^{92} - 4 \zeta_{6} q^{94} - 5 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} + 14 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + z * q^5 + 2 * q^10 + (-5*z + 5) * q^11 - 4*z * q^13 + (-4*z + 4) * q^16 + 4 * q^17 - 5 * q^19 + (-2*z + 2) * q^20 - 10*z * q^22 + 6*z * q^23 + (z - 1) * q^25 - 8 * q^26 + (5*z - 5) * q^29 + 9*z * q^31 - 8*z * q^32 + (-8*z + 8) * q^34 - 10 * q^37 + (10*z - 10) * q^38 + 7*z * q^41 + (-2*z + 2) * q^43 - 10 * q^44 + 12 * q^46 + (-2*z + 2) * q^47 + 7*z * q^49 + 2*z * q^50 + (8*z - 8) * q^52 - 8 * q^53 + 5 * q^55 + 10*z * q^58 - z * q^59 + (-2*z + 2) * q^61 + 18 * q^62 - 8 * q^64 + (-4*z + 4) * q^65 - 6*z * q^67 - 8*z * q^68 - q^71 - 8 * q^73 + (20*z - 20) * q^74 + 10*z * q^76 + (12*z - 12) * q^79 + 4 * q^80 + 14 * q^82 + (-6*z + 6) * q^83 + 4*z * q^85 - 4*z * q^86 + 9 * q^89 + (-12*z + 12) * q^92 - 4*z * q^94 - 5*z * q^95 + (14*z - 14) * q^97 + 14 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} + q^{5}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 + q^5 $$2 q + 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{10} + 5 q^{11} - 4 q^{13} + 4 q^{16} + 8 q^{17} - 10 q^{19} + 2 q^{20} - 10 q^{22} + 6 q^{23} - q^{25} - 16 q^{26} - 5 q^{29} + 9 q^{31} - 8 q^{32} + 8 q^{34} - 20 q^{37} - 10 q^{38} + 7 q^{41} + 2 q^{43} - 20 q^{44} + 24 q^{46} + 2 q^{47} + 7 q^{49} + 2 q^{50} - 8 q^{52} - 16 q^{53} + 10 q^{55} + 10 q^{58} - q^{59} + 2 q^{61} + 36 q^{62} - 16 q^{64} + 4 q^{65} - 6 q^{67} - 8 q^{68} - 2 q^{71} - 16 q^{73} - 20 q^{74} + 10 q^{76} - 12 q^{79} + 8 q^{80} + 28 q^{82} + 6 q^{83} + 4 q^{85} - 4 q^{86} + 18 q^{89} + 12 q^{92} - 4 q^{94} - 5 q^{95} - 14 q^{97} + 28 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 4 * q^10 + 5 * q^11 - 4 * q^13 + 4 * q^16 + 8 * q^17 - 10 * q^19 + 2 * q^20 - 10 * q^22 + 6 * q^23 - q^25 - 16 * q^26 - 5 * q^29 + 9 * q^31 - 8 * q^32 + 8 * q^34 - 20 * q^37 - 10 * q^38 + 7 * q^41 + 2 * q^43 - 20 * q^44 + 24 * q^46 + 2 * q^47 + 7 * q^49 + 2 * q^50 - 8 * q^52 - 16 * q^53 + 10 * q^55 + 10 * q^58 - q^59 + 2 * q^61 + 36 * q^62 - 16 * q^64 + 4 * q^65 - 6 * q^67 - 8 * q^68 - 2 * q^71 - 16 * q^73 - 20 * q^74 + 10 * q^76 - 12 * q^79 + 8 * q^80 + 28 * q^82 + 6 * q^83 + 4 * q^85 - 4 * q^86 + 18 * q^89 + 12 * q^92 - 4 * q^94 - 5 * q^95 - 14 * q^97 + 28 * q^98

## 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
1.00000 1.73205i 0 −1.00000 1.73205i 0.500000 + 0.866025i 0 0 0 0 2.00000
271.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0.500000 0.866025i 0 0 0 0 2.00000
 $$n$$: 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.g 2
3.b odd 2 1 405.2.e.a 2
9.c even 3 1 405.2.a.a 1
9.c even 3 1 inner 405.2.e.g 2
9.d odd 6 1 405.2.a.f yes 1
9.d odd 6 1 405.2.e.a 2
36.f odd 6 1 6480.2.a.f 1
36.h even 6 1 6480.2.a.r 1
45.h odd 6 1 2025.2.a.a 1
45.j even 6 1 2025.2.a.f 1
45.k odd 12 2 2025.2.b.a 2
45.l even 12 2 2025.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 9.c even 3 1
405.2.a.f yes 1 9.d odd 6 1
405.2.e.a 2 3.b odd 2 1
405.2.e.a 2 9.d odd 6 1
405.2.e.g 2 1.a even 1 1 trivial
405.2.e.g 2 9.c even 3 1 inner
2025.2.a.a 1 45.h odd 6 1
2025.2.a.f 1 45.j even 6 1
2025.2.b.a 2 45.k odd 12 2
2025.2.b.b 2 45.l even 12 2
6480.2.a.f 1 36.f odd 6 1
6480.2.a.r 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 $$S_{2}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{7}$$ T7 $$T_{11}^{2} - 5T_{11} + 25$$ T11^2 - 5*T11 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$(T - 4)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 5T + 25$$
$31$ $$T^{2} - 9T + 81$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 7T + 49$$
$43$ $$T^{2} - 2T + 4$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T + 8)^{2}$$
$59$ $$T^{2} + T + 1$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 6T + 36$$
$71$ $$(T + 1)^{2}$$
$73$ $$(T + 8)^{2}$$
$79$ $$T^{2} + 12T + 144$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} + 14T + 196$$