# Properties

 Label 405.2.b.d Level $405$ Weight $2$ Character orbit 405.b Analytic conductor $3.234$ Analytic rank $0$ Dimension $4$ 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(244,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.b (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + (-b3 - b2 + b1) * q^5 + (-2*b3 + b1) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{10} + ( - \beta_{2} + 3) q^{11} + ( - \beta_{3} - \beta_1) q^{13} + \beta_{2} q^{14} + (2 \beta_{2} - 1) q^{16} + ( - 3 \beta_{3} + \beta_1) q^{17} + (\beta_{2} - 1) q^{19} + ( - \beta_{3} - \beta_1 - 3) q^{20} + ( - \beta_{3} + 5 \beta_1) q^{22} + \beta_1 q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{2} + 3) q^{26} - 3 \beta_{3} q^{28} + (2 \beta_{2} - 3) q^{29} + (\beta_{2} - 1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} + (\beta_{2} + 1) q^{34} + (3 \beta_{3} - \beta_{2} - 3) q^{35} + (3 \beta_{3} - 3 \beta_1) q^{37} + (\beta_{3} - 3 \beta_1) q^{38} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{40} + (\beta_{2} + 6) q^{41} + (\beta_{3} - 5 \beta_1) q^{43} + (3 \beta_{2} - 3) q^{44} + (\beta_{2} - 2) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} + 1) q^{49} + (2 \beta_{2} + \beta_1 - 6) q^{50} + ( - 3 \beta_{3} + 3 \beta_1) q^{52} - 2 \beta_1 q^{53} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{55}+ \cdots + ( - 3 \beta_{3} + 7 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + b2 * q^4 + (-b3 - b2 + b1) * q^5 + (-2*b3 + b1) * q^7 + b3 * q^8 + (-b3 + b2 + 2*b1 - 1) * q^10 + (-b2 + 3) * q^11 + (-b3 - b1) * q^13 + b2 * q^14 + (2*b2 - 1) * q^16 + (-3*b3 + b1) * q^17 + (b2 - 1) * q^19 + (-b3 - b1 - 3) * q^20 + (-b3 + 5*b1) * q^22 + b1 * q^23 + (2*b3 + 2*b1 + 1) * q^25 + (-b2 + 3) * q^26 - 3*b3 * q^28 + (2*b2 - 3) * q^29 + (b2 - 1) * q^31 + (4*b3 - 5*b1) * q^32 + (b2 + 1) * q^34 + (3*b3 - b2 - 3) * q^35 + (3*b3 - 3*b1) * q^37 + (b3 - 3*b1) * q^38 + (-2*b3 + b2 + b1 + 1) * q^40 + (b2 + 6) * q^41 + (b3 - 5*b1) * q^43 + (3*b2 - 3) * q^44 + (b2 - 2) * q^46 + (b3 + 2*b1) * q^47 + (-3*b2 + 1) * q^49 + (2*b2 + b1 - 6) * q^50 + (-3*b3 + 3*b1) * q^52 - 2*b1 * q^53 + (-2*b3 - 3*b2 + 4*b1 + 3) * q^55 + (2*b2 + 3) * q^56 + (2*b3 - 7*b1) * q^58 + (-2*b2 - 6) * q^59 + (-5*b2 + 2) * q^61 + (b3 - 3*b1) * q^62 + (-b2 + 4) * q^64 + (3*b3 - 2*b2 - 3*b1) * q^65 + (5*b3 - 4*b1) * q^67 + (-5*b3 + b1) * q^68 + (-b3 - b1 - 3) * q^70 + (3*b2 + 9) * q^71 + (6*b3 - 6*b1) * q^73 + (-3*b2 + 3) * q^74 + (-b2 + 3) * q^76 + (-3*b3 + 3*b1) * q^77 + (-2*b2 - 4) * q^79 + (-b3 + b2 - 3*b1 - 6) * q^80 + (b3 + 4*b1) * q^82 + (-6*b3 + 7*b1) * q^83 + (5*b3 - 2*b2 - b1 - 4) * q^85 + (-5*b2 + 9) * q^86 + (b3 + b1) * q^88 + (-6*b2 - 3) * q^89 + (-3*b2 - 3) * q^91 + (b3 - 2*b1) * q^92 + (2*b2 - 5) * q^94 + (b2 - 2*b1 - 3) * q^95 + (10*b3 - 8*b1) * q^97 + (-3*b3 + 7*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{10} + 12 q^{11} - 4 q^{16} - 4 q^{19} - 12 q^{20} + 4 q^{25} + 12 q^{26} - 12 q^{29} - 4 q^{31} + 4 q^{34} - 12 q^{35} + 4 q^{40} + 24 q^{41} - 12 q^{44} - 8 q^{46} + 4 q^{49} - 24 q^{50} + 12 q^{55} + 12 q^{56} - 24 q^{59} + 8 q^{61} + 16 q^{64} - 12 q^{70} + 36 q^{71} + 12 q^{74} + 12 q^{76} - 16 q^{79} - 24 q^{80} - 16 q^{85} + 36 q^{86} - 12 q^{89} - 12 q^{91} - 20 q^{94} - 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^10 + 12 * q^11 - 4 * q^16 - 4 * q^19 - 12 * q^20 + 4 * q^25 + 12 * q^26 - 12 * q^29 - 4 * q^31 + 4 * q^34 - 12 * q^35 + 4 * q^40 + 24 * q^41 - 12 * q^44 - 8 * q^46 + 4 * q^49 - 24 * q^50 + 12 * q^55 + 12 * q^56 - 24 * q^59 + 8 * q^61 + 16 * q^64 - 12 * q^70 + 36 * q^71 + 12 * q^74 + 12 * q^76 - 16 * q^79 - 24 * q^80 - 16 * q^85 + 36 * q^86 - 12 * q^89 - 12 * q^91 - 20 * q^94 - 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0 −1.73205 1.73205 1.41421i 0 0.896575i 0.517638i 0 −2.73205 3.34607i
244.2 0.517638i 0 1.73205 −1.73205 + 1.41421i 0 3.34607i 1.93185i 0 0.732051 + 0.896575i
244.3 0.517638i 0 1.73205 −1.73205 1.41421i 0 3.34607i 1.93185i 0 0.732051 0.896575i
244.4 1.93185i 0 −1.73205 1.73205 + 1.41421i 0 0.896575i 0.517638i 0 −2.73205 + 3.34607i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.b.d 4
3.b odd 2 1 405.2.b.c 4
5.b even 2 1 inner 405.2.b.d 4
5.c odd 4 2 2025.2.a.t 4
9.c even 3 2 45.2.j.a 8
9.d odd 6 2 135.2.j.a 8
15.d odd 2 1 405.2.b.c 4
15.e even 4 2 2025.2.a.r 4
36.f odd 6 2 720.2.by.d 8
36.h even 6 2 2160.2.by.c 8
45.h odd 6 2 135.2.j.a 8
45.j even 6 2 45.2.j.a 8
45.k odd 12 4 225.2.e.d 8
45.l even 12 4 675.2.e.d 8
180.n even 6 2 2160.2.by.c 8
180.p odd 6 2 720.2.by.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 9.c even 3 2
45.2.j.a 8 45.j even 6 2
135.2.j.a 8 9.d odd 6 2
135.2.j.a 8 45.h odd 6 2
225.2.e.d 8 45.k odd 12 4
405.2.b.c 4 3.b odd 2 1
405.2.b.c 4 15.d odd 2 1
405.2.b.d 4 1.a even 1 1 trivial
405.2.b.d 4 5.b even 2 1 inner
675.2.e.d 8 45.l even 12 4
720.2.by.d 8 36.f odd 6 2
720.2.by.d 8 180.p odd 6 2
2025.2.a.r 4 15.e even 4 2
2025.2.a.t 4 5.c odd 4 2
2160.2.by.c 8 36.h even 6 2
2160.2.by.c 8 180.n even 6 2

## Hecke kernels

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

 $$T_{2}^{4} + 4T_{2}^{2} + 1$$ T2^4 + 4*T2^2 + 1 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2T^{2} + 25$$
$7$ $$T^{4} + 12T^{2} + 9$$
$11$ $$(T^{2} - 6 T + 6)^{2}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$T^{4} + 28T^{2} + 4$$
$19$ $$(T^{2} + 2 T - 2)^{2}$$
$23$ $$T^{4} + 4T^{2} + 1$$
$29$ $$(T^{2} + 6 T - 3)^{2}$$
$31$ $$(T^{2} + 2 T - 2)^{2}$$
$37$ $$(T^{2} + 18)^{2}$$
$41$ $$(T^{2} - 12 T + 33)^{2}$$
$43$ $$T^{4} + 84T^{2} + 36$$
$47$ $$T^{4} + 28T^{2} + 169$$
$53$ $$T^{4} + 16T^{2} + 16$$
$59$ $$(T^{2} + 12 T + 24)^{2}$$
$61$ $$(T^{2} - 4 T - 71)^{2}$$
$67$ $$T^{4} + 84T^{2} + 1521$$
$71$ $$(T^{2} - 18 T + 54)^{2}$$
$73$ $$(T^{2} + 72)^{2}$$
$79$ $$(T^{2} + 8 T + 4)^{2}$$
$83$ $$T^{4} + 172T^{2} + 6889$$
$89$ $$(T^{2} + 6 T - 99)^{2}$$
$97$ $$T^{4} + 336 T^{2} + 24336$$