# Properties

 Label 405.2.e.l Level $405$ Weight $2$ Character orbit 405.e Analytic conductor $3.234$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + (\beta_1 - 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 6) q^{8}+O(q^{10})$$ q + (-b2 + b1) * q^2 + (2*b3 - 2*b2 + 2*b1 - 2) * q^4 + (b1 - 1) * q^5 + (b2 + 3*b1) * q^7 + (2*b3 - 6) * q^8 $$q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + (\beta_1 - 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 6) q^{8} + (\beta_{3} - 1) q^{10} + (\beta_{2} + 4 \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{13}+ \cdots + ( - \beta_{3} + 13) q^{98}+O(q^{100})$$ q + (-b2 + b1) * q^2 + (2*b3 - 2*b2 + 2*b1 - 2) * q^4 + (b1 - 1) * q^5 + (b2 + 3*b1) * q^7 + (2*b3 - 6) * q^8 + (b3 - 1) * q^10 + (b2 + 4*b1) * q^11 + (-2*b3 + 2*b2 - 2*b1 + 2) * q^13 + (2*b3 - 2*b2) * q^14 + (4*b2 - 8*b1) * q^16 + (-b3 - 1) * q^17 + (2*b3 + 1) * q^19 + (2*b2 - 2*b1) * q^20 + (3*b3 - 3*b2 + b1 - 1) * q^22 + (2*b3 - 2*b2) * q^23 - b1 * q^25 + (-4*b3 + 8) * q^26 + 4*b3 * q^28 + (-3*b2 + 2*b1) * q^29 + (-3*b1 + 3) * q^31 + (-8*b3 + 8*b2 - 8*b1 + 8) * q^32 + 2*b1 * q^34 + (-b3 - 3) * q^35 + (-b3 - 1) * q^37 + (b2 - 5*b1) * q^38 + (-2*b3 + 2*b2 - 6*b1 + 6) * q^40 + (3*b3 - 3*b2 - 2*b1 + 2) * q^41 + (-3*b2 + 5*b1) * q^43 + (6*b3 - 2) * q^44 + (2*b3 - 6) * q^46 + (b2 + 7*b1) * q^47 + (-6*b3 + 6*b2 + 5*b1 - 5) * q^49 + (-b3 + b2 - b1 + 1) * q^50 + (-8*b2 + 16*b1) * q^52 + (b3 + 5) * q^53 + (-b3 - 4) * q^55 - 12*b1 * q^56 + (5*b3 - 5*b2 + 11*b1 - 11) * q^58 + (-b3 + b2 - 10*b1 + 10) * q^59 - 4*b1 * q^61 + (-3*b3 + 3) * q^62 + (-8*b3 + 16) * q^64 + (-2*b2 + 2*b1) * q^65 + (-2*b3 + 2*b2) * q^67 + (4*b1 - 4) * q^68 + 2*b2 * q^70 + (-b3 - 2) * q^71 + (-5*b3 + 1) * q^73 + 2*b1 * q^74 + (-2*b3 + 2*b2 - 10*b1 + 10) * q^76 + (-7*b3 + 7*b2 + 15*b1 - 15) * q^77 + (-2*b2 - 12*b1) * q^79 + (-4*b3 + 8) * q^80 + (b3 - 7) * q^82 + (-3*b2 + 3*b1) * q^83 + (b3 - b2 - b1 + 1) * q^85 + (8*b3 - 8*b2 + 14*b1 - 14) * q^86 + (2*b2 - 18*b1) * q^88 + 3*b3 * q^89 - 4*b3 * q^91 + (4*b2 - 12*b1) * q^92 + (6*b3 - 6*b2 + 4*b1 - 4) * q^94 + (-2*b3 + 2*b2 + b1 - 1) * q^95 + (5*b2 + b1) * q^97 + (-b3 + 13) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 $$4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34} - 12 q^{35} - 4 q^{37} - 10 q^{38} + 12 q^{40} + 4 q^{41} + 10 q^{43} - 8 q^{44} - 24 q^{46} + 14 q^{47} - 10 q^{49} + 2 q^{50} + 32 q^{52} + 20 q^{53} - 16 q^{55} - 24 q^{56} - 22 q^{58} + 20 q^{59} - 8 q^{61} + 12 q^{62} + 64 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{71} + 4 q^{73} + 4 q^{74} + 20 q^{76} - 30 q^{77} - 24 q^{79} + 32 q^{80} - 28 q^{82} + 6 q^{83} + 2 q^{85} - 28 q^{86} - 36 q^{88} - 24 q^{92} - 8 q^{94} - 2 q^{95} + 2 q^{97} + 52 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 - 4 * q^10 + 8 * q^11 + 4 * q^13 - 16 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^25 + 32 * q^26 + 4 * q^29 + 6 * q^31 + 16 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 - 10 * q^38 + 12 * q^40 + 4 * q^41 + 10 * q^43 - 8 * q^44 - 24 * q^46 + 14 * q^47 - 10 * q^49 + 2 * q^50 + 32 * q^52 + 20 * q^53 - 16 * q^55 - 24 * q^56 - 22 * q^58 + 20 * q^59 - 8 * q^61 + 12 * q^62 + 64 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^71 + 4 * q^73 + 4 * q^74 + 20 * q^76 - 30 * q^77 - 24 * q^79 + 32 * q^80 - 28 * q^82 + 6 * q^83 + 2 * q^85 - 28 * q^86 - 36 * q^88 - 24 * q^92 - 8 * q^94 - 2 * q^95 + 2 * q^97 + 52 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.366025 + 0.633975i 0 0.732051 + 1.26795i −0.500000 0.866025i 0 2.36603 4.09808i −2.53590 0 0.732051
136.2 1.36603 2.36603i 0 −2.73205 4.73205i −0.500000 0.866025i 0 0.633975 1.09808i −9.46410 0 −2.73205
271.1 −0.366025 0.633975i 0 0.732051 1.26795i −0.500000 + 0.866025i 0 2.36603 + 4.09808i −2.53590 0 0.732051
271.2 1.36603 + 2.36603i 0 −2.73205 + 4.73205i −0.500000 + 0.866025i 0 0.633975 + 1.09808i −9.46410 0 −2.73205
 $$n$$: 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.l 4
3.b odd 2 1 405.2.e.i 4
9.c even 3 1 405.2.a.g 2
9.c even 3 1 inner 405.2.e.l 4
9.d odd 6 1 405.2.a.h yes 2
9.d odd 6 1 405.2.e.i 4
36.f odd 6 1 6480.2.a.br 2
36.h even 6 1 6480.2.a.bi 2
45.h odd 6 1 2025.2.a.g 2
45.j even 6 1 2025.2.a.m 2
45.k odd 12 2 2025.2.b.g 4
45.l even 12 2 2025.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 9.c even 3 1
405.2.a.h yes 2 9.d odd 6 1
405.2.e.i 4 3.b odd 2 1
405.2.e.i 4 9.d odd 6 1
405.2.e.l 4 1.a even 1 1 trivial
405.2.e.l 4 9.c even 3 1 inner
2025.2.a.g 2 45.h odd 6 1
2025.2.a.m 2 45.j even 6 1
2025.2.b.g 4 45.k odd 12 2
2025.2.b.h 4 45.l even 12 2
6480.2.a.bi 2 36.h even 6 1
6480.2.a.br 2 36.f odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4$$ T2^4 - 2*T2^3 + 6*T2^2 + 4*T2 + 4 $$T_{7}^{4} - 6T_{7}^{3} + 30T_{7}^{2} - 36T_{7} + 36$$ T7^4 - 6*T7^3 + 30*T7^2 - 36*T7 + 36 $$T_{11}^{4} - 8T_{11}^{3} + 51T_{11}^{2} - 104T_{11} + 169$$ T11^4 - 8*T11^3 + 51*T11^2 - 104*T11 + 169

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$11$ $$T^{4} - 8 T^{3} + \cdots + 169$$
$13$ $$T^{4} - 4 T^{3} + \cdots + 64$$
$17$ $$(T^{2} + 2 T - 2)^{2}$$
$19$ $$(T^{2} - 2 T - 11)^{2}$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4} - 4 T^{3} + \cdots + 529$$
$31$ $$(T^{2} - 3 T + 9)^{2}$$
$37$ $$(T^{2} + 2 T - 2)^{2}$$
$41$ $$T^{4} - 4 T^{3} + \cdots + 529$$
$43$ $$T^{4} - 10 T^{3} + \cdots + 4$$
$47$ $$T^{4} - 14 T^{3} + \cdots + 2116$$
$53$ $$(T^{2} - 10 T + 22)^{2}$$
$59$ $$T^{4} - 20 T^{3} + \cdots + 9409$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} + 12T^{2} + 144$$
$71$ $$(T^{2} + 4 T + 1)^{2}$$
$73$ $$(T^{2} - 2 T - 74)^{2}$$
$79$ $$T^{4} + 24 T^{3} + \cdots + 17424$$
$83$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$T^{4} - 2 T^{3} + \cdots + 5476$$