# Properties

 Label 405.2.b.e Level $405$ Weight $2$ Character orbit 405.b Analytic conductor $3.234$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{5} + \beta_{3} q^{7} - 2 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1) * q^5 + b3 * q^7 - 2*b1 * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{5} + \beta_{3} q^{7} - 2 \beta_1 q^{8} + ( - \beta_{3} + 2) q^{10} + \beta_{2} q^{11} + 2 \beta_{3} q^{13} + 2 \beta_{2} q^{14} - 4 q^{16} - \beta_1 q^{17} + 5 q^{19} - \beta_{3} q^{22} - 4 \beta_1 q^{23} + (2 \beta_{3} + 1) q^{25} + 4 \beta_{2} q^{26} + \beta_{2} q^{29} - 7 q^{31} - 2 q^{34} + ( - 2 \beta_{2} + 3 \beta_1) q^{35} - 3 \beta_{3} q^{37} - 5 \beta_1 q^{38} + ( - 2 \beta_{3} + 4) q^{40} - 7 \beta_{2} q^{41} - 5 \beta_{3} q^{43} - 8 q^{46} + 5 \beta_1 q^{47} + q^{49} + (4 \beta_{2} - \beta_1) q^{50} - \beta_1 q^{53} + (\beta_{3} + 3) q^{55} + 4 \beta_{2} q^{56} - \beta_{3} q^{58} - \beta_{2} q^{59} - 4 q^{61} + 7 \beta_1 q^{62} - 8 q^{64} + ( - 4 \beta_{2} + 6 \beta_1) q^{65} + 2 \beta_{3} q^{67} + (2 \beta_{3} + 6) q^{70} + 3 \beta_{2} q^{71} + 3 \beta_{3} q^{73} - 6 \beta_{2} q^{74} + 3 \beta_1 q^{77} + 14 q^{79} + ( - 4 \beta_{2} - 4 \beta_1) q^{80} + 7 \beta_{3} q^{82} - \beta_1 q^{83} + ( - \beta_{3} + 2) q^{85} - 10 \beta_{2} q^{86} - 2 \beta_{3} q^{88} - 9 \beta_{2} q^{89} - 12 q^{91} + 10 q^{94} + (5 \beta_{2} + 5 \beta_1) q^{95} + \beta_{3} q^{97} - \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1) * q^5 + b3 * q^7 - 2*b1 * q^8 + (-b3 + 2) * q^10 + b2 * q^11 + 2*b3 * q^13 + 2*b2 * q^14 - 4 * q^16 - b1 * q^17 + 5 * q^19 - b3 * q^22 - 4*b1 * q^23 + (2*b3 + 1) * q^25 + 4*b2 * q^26 + b2 * q^29 - 7 * q^31 - 2 * q^34 + (-2*b2 + 3*b1) * q^35 - 3*b3 * q^37 - 5*b1 * q^38 + (-2*b3 + 4) * q^40 - 7*b2 * q^41 - 5*b3 * q^43 - 8 * q^46 + 5*b1 * q^47 + q^49 + (4*b2 - b1) * q^50 - b1 * q^53 + (b3 + 3) * q^55 + 4*b2 * q^56 - b3 * q^58 - b2 * q^59 - 4 * q^61 + 7*b1 * q^62 - 8 * q^64 + (-4*b2 + 6*b1) * q^65 + 2*b3 * q^67 + (2*b3 + 6) * q^70 + 3*b2 * q^71 + 3*b3 * q^73 - 6*b2 * q^74 + 3*b1 * q^77 + 14 * q^79 + (-4*b2 - 4*b1) * q^80 + 7*b3 * q^82 - b1 * q^83 + (-b3 + 2) * q^85 - 10*b2 * q^86 - 2*b3 * q^88 - 9*b2 * q^89 - 12 * q^91 + 10 * q^94 + (5*b2 + 5*b1) * q^95 + b3 * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{10} - 16 q^{16} + 20 q^{19} + 4 q^{25} - 28 q^{31} - 8 q^{34} + 16 q^{40} - 32 q^{46} + 4 q^{49} + 12 q^{55} - 16 q^{61} - 32 q^{64} + 24 q^{70} + 56 q^{79} + 8 q^{85} - 48 q^{91} + 40 q^{94}+O(q^{100})$$ 4 * q + 8 * q^10 - 16 * q^16 + 20 * q^19 + 4 * q^25 - 28 * q^31 - 8 * q^34 + 16 * q^40 - 32 * q^46 + 4 * q^49 + 12 * q^55 - 16 * q^61 - 32 * q^64 + 24 * q^70 + 56 * q^79 + 8 * q^85 - 48 * q^91 + 40 * q^94

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

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

## 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 1.93185i − 0.517638i
1.41421i 0 0 −1.73205 + 1.41421i 0 2.44949i 2.82843i 0 2.00000 + 2.44949i
244.2 1.41421i 0 0 1.73205 + 1.41421i 0 2.44949i 2.82843i 0 2.00000 2.44949i
244.3 1.41421i 0 0 −1.73205 1.41421i 0 2.44949i 2.82843i 0 2.00000 2.44949i
244.4 1.41421i 0 0 1.73205 1.41421i 0 2.44949i 2.82843i 0 2.00000 + 2.44949i
 $$n$$: 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 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.e 4
3.b odd 2 1 inner 405.2.b.e 4
5.b even 2 1 inner 405.2.b.e 4
5.c odd 4 2 2025.2.a.s 4
9.c even 3 2 405.2.j.g 8
9.d odd 6 2 405.2.j.g 8
15.d odd 2 1 inner 405.2.b.e 4
15.e even 4 2 2025.2.a.s 4
45.h odd 6 2 405.2.j.g 8
45.j even 6 2 405.2.j.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.b.e 4 1.a even 1 1 trivial
405.2.b.e 4 3.b odd 2 1 inner
405.2.b.e 4 5.b even 2 1 inner
405.2.b.e 4 15.d odd 2 1 inner
405.2.j.g 8 9.c even 3 2
405.2.j.g 8 9.d odd 6 2
405.2.j.g 8 45.h odd 6 2
405.2.j.g 8 45.j even 6 2
2025.2.a.s 4 5.c odd 4 2
2025.2.a.s 4 15.e even 4 2

## Hecke kernels

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

 $$T_{2}^{2} + 2$$ T2^2 + 2 $$T_{11}^{2} - 3$$ T11^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2T^{2} + 25$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} - 3)^{2}$$
$13$ $$(T^{2} + 24)^{2}$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T - 5)^{4}$$
$23$ $$(T^{2} + 32)^{2}$$
$29$ $$(T^{2} - 3)^{2}$$
$31$ $$(T + 7)^{4}$$
$37$ $$(T^{2} + 54)^{2}$$
$41$ $$(T^{2} - 147)^{2}$$
$43$ $$(T^{2} + 150)^{2}$$
$47$ $$(T^{2} + 50)^{2}$$
$53$ $$(T^{2} + 2)^{2}$$
$59$ $$(T^{2} - 3)^{2}$$
$61$ $$(T + 4)^{4}$$
$67$ $$(T^{2} + 24)^{2}$$
$71$ $$(T^{2} - 27)^{2}$$
$73$ $$(T^{2} + 54)^{2}$$
$79$ $$(T - 14)^{4}$$
$83$ $$(T^{2} + 2)^{2}$$
$89$ $$(T^{2} - 243)^{2}$$
$97$ $$(T^{2} + 6)^{2}$$