# Properties

 Label 405.2.j.c Level $405$ Weight $2$ Character orbit 405.j Analytic conductor $3.234$ Analytic rank $0$ Dimension $4$ CM discriminant -15 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 3*b2 * q^4 + (-b3 + b1) * q^5 + b3 * q^8 $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + \beta_{3} q^{8} + 5 q^{10} + ( - \beta_{2} + 1) q^{16} + 2 \beta_{3} q^{17} - 4 q^{19} + 3 \beta_1 q^{20} + (4 \beta_{3} - 4 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} - 8 \beta_{2} q^{31} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + (10 \beta_{2} - 10) q^{34} - 4 \beta_1 q^{38} + 5 \beta_{2} q^{40} - 20 q^{46} + 4 \beta_1 q^{47} - 7 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{50} + 2 \beta_{3} q^{53} + (2 \beta_{2} - 2) q^{61} - 8 \beta_{3} q^{62} + 13 q^{64} + (6 \beta_{3} - 6 \beta_1) q^{68} - 12 \beta_{2} q^{76} + ( - 16 \beta_{2} + 16) q^{79} - \beta_{3} q^{80} - 8 \beta_1 q^{83} + 10 \beta_{2} q^{85} - 12 \beta_1 q^{92} + 20 \beta_{2} q^{94} + (4 \beta_{3} - 4 \beta_1) q^{95} - 7 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 3*b2 * q^4 + (-b3 + b1) * q^5 + b3 * q^8 + 5 * q^10 + (-b2 + 1) * q^16 + 2*b3 * q^17 - 4 * q^19 + 3*b1 * q^20 + (4*b3 - 4*b1) * q^23 + (-5*b2 + 5) * q^25 - 8*b2 * q^31 + (-3*b3 + 3*b1) * q^32 + (10*b2 - 10) * q^34 - 4*b1 * q^38 + 5*b2 * q^40 - 20 * q^46 + 4*b1 * q^47 - 7*b2 * q^49 + (-5*b3 + 5*b1) * q^50 + 2*b3 * q^53 + (2*b2 - 2) * q^61 - 8*b3 * q^62 + 13 * q^64 + (6*b3 - 6*b1) * q^68 - 12*b2 * q^76 + (-16*b2 + 16) * q^79 - b3 * q^80 - 8*b1 * q^83 + 10*b2 * q^85 - 12*b1 * q^92 + 20*b2 * q^94 + (4*b3 - 4*b1) * q^95 - 7*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4}+O(q^{10})$$ 4 * q + 6 * q^4 $$4 q + 6 q^{4} + 20 q^{10} + 2 q^{16} - 16 q^{19} + 10 q^{25} - 16 q^{31} - 20 q^{34} + 10 q^{40} - 80 q^{46} - 14 q^{49} - 4 q^{61} + 52 q^{64} - 24 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100})$$ 4 * q + 6 * q^4 + 20 * q^10 + 2 * q^16 - 16 * q^19 + 10 * q^25 - 16 * q^31 - 20 * q^34 + 10 * q^40 - 80 * q^46 - 14 * q^49 - 4 * q^61 + 52 * q^64 - 24 * q^76 + 32 * q^79 + 20 * q^85 + 40 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## 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
 −1.93649 + 1.11803i 1.93649 − 1.11803i −1.93649 − 1.11803i 1.93649 + 1.11803i
−1.93649 + 1.11803i 0 1.50000 2.59808i −1.93649 1.11803i 0 0 2.23607i 0 5.00000
109.2 1.93649 1.11803i 0 1.50000 2.59808i 1.93649 + 1.11803i 0 0 2.23607i 0 5.00000
379.1 −1.93649 1.11803i 0 1.50000 + 2.59808i −1.93649 + 1.11803i 0 0 2.23607i 0 5.00000
379.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 1.93649 1.11803i 0 0 2.23607i 0 5.00000
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 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.c 4
3.b odd 2 1 inner 405.2.j.c 4
5.b even 2 1 inner 405.2.j.c 4
9.c even 3 1 45.2.b.a 2
9.c even 3 1 inner 405.2.j.c 4
9.d odd 6 1 45.2.b.a 2
9.d odd 6 1 inner 405.2.j.c 4
15.d odd 2 1 CM 405.2.j.c 4
36.f odd 6 1 720.2.f.d 2
36.h even 6 1 720.2.f.d 2
45.h odd 6 1 45.2.b.a 2
45.h odd 6 1 inner 405.2.j.c 4
45.j even 6 1 45.2.b.a 2
45.j even 6 1 inner 405.2.j.c 4
45.k odd 12 2 225.2.a.f 2
45.l even 12 2 225.2.a.f 2
63.l odd 6 1 2205.2.d.a 2
63.o even 6 1 2205.2.d.a 2
72.j odd 6 1 2880.2.f.k 2
72.l even 6 1 2880.2.f.j 2
72.n even 6 1 2880.2.f.k 2
72.p odd 6 1 2880.2.f.j 2
180.n even 6 1 720.2.f.d 2
180.p odd 6 1 720.2.f.d 2
180.v odd 12 2 3600.2.a.bs 2
180.x even 12 2 3600.2.a.bs 2
315.z even 6 1 2205.2.d.a 2
315.bg odd 6 1 2205.2.d.a 2
360.z odd 6 1 2880.2.f.j 2
360.bd even 6 1 2880.2.f.j 2
360.bh odd 6 1 2880.2.f.k 2
360.bk even 6 1 2880.2.f.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 9.c even 3 1
45.2.b.a 2 9.d odd 6 1
45.2.b.a 2 45.h odd 6 1
45.2.b.a 2 45.j even 6 1
225.2.a.f 2 45.k odd 12 2
225.2.a.f 2 45.l even 12 2
405.2.j.c 4 1.a even 1 1 trivial
405.2.j.c 4 3.b odd 2 1 inner
405.2.j.c 4 5.b even 2 1 inner
405.2.j.c 4 9.c even 3 1 inner
405.2.j.c 4 9.d odd 6 1 inner
405.2.j.c 4 15.d odd 2 1 CM
405.2.j.c 4 45.h odd 6 1 inner
405.2.j.c 4 45.j even 6 1 inner
720.2.f.d 2 36.f odd 6 1
720.2.f.d 2 36.h even 6 1
720.2.f.d 2 180.n even 6 1
720.2.f.d 2 180.p odd 6 1
2205.2.d.a 2 63.l odd 6 1
2205.2.d.a 2 63.o even 6 1
2205.2.d.a 2 315.z even 6 1
2205.2.d.a 2 315.bg odd 6 1
2880.2.f.j 2 72.l even 6 1
2880.2.f.j 2 72.p odd 6 1
2880.2.f.j 2 360.z odd 6 1
2880.2.f.j 2 360.bd even 6 1
2880.2.f.k 2 72.j odd 6 1
2880.2.f.k 2 72.n even 6 1
2880.2.f.k 2 360.bh odd 6 1
2880.2.f.k 2 360.bk even 6 1
3600.2.a.bs 2 180.v odd 12 2
3600.2.a.bs 2 180.x even 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5T^{2} + 25$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$(T + 4)^{4}$$
$23$ $$T^{4} - 80T^{2} + 6400$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 8 T + 64)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} - 80T^{2} + 6400$$
$53$ $$(T^{2} + 20)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 2 T + 4)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 16 T + 256)^{2}$$
$83$ $$T^{4} - 320 T^{2} + 102400$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$