# Properties

 Label 405.2.b.b Level $405$ Weight $2$ Character orbit 405.b 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(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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{3} + \beta_1 - 2) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q - b2 * q^2 + (b3 + b1 - 2) * q^4 + (-b1 + 1) * q^5 + (-b3 - b2 + b1) * q^7 + (b3 + 2*b2 - b1) * q^8 $$q - \beta_{2} q^{2} + (\beta_{3} + \beta_1 - 2) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 2 \beta_{2}) q^{10} + (\beta_{3} + \beta_1 + 1) q^{11} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1 - 4) q^{14} + ( - \beta_{3} - \beta_1 + 4) q^{16} - \beta_{2} q^{17} + (\beta_{3} + \beta_1 - 3) q^{19} + (2 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{20} + (\beta_{3} + \beta_{2} - \beta_1) q^{22} + 2 \beta_{2} q^{23} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{3} + \beta_1 - 8) q^{26} + 6 \beta_{2} q^{28} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{29} + (\beta_{3} + \beta_1 + 3) q^{31} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{32} + (\beta_{3} + \beta_1 - 4) q^{34} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{35} - 3 \beta_{2} q^{37} + (\beta_{3} + 5 \beta_{2} - \beta_1) q^{38} + (5 \beta_{2} - \beta_1 - 4) q^{40} + ( - \beta_{3} - \beta_1 - 1) q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{43} + 6 q^{44} + ( - 2 \beta_{3} - 2 \beta_1 + 8) q^{46} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{47} - 5 q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{50}+ \cdots + 5 \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 + (b3 + b1 - 2) * q^4 + (-b1 + 1) * q^5 + (-b3 - b2 + b1) * q^7 + (b3 + 2*b2 - b1) * q^8 + (-b3 - 2*b2) * q^10 + (b3 + b1 + 1) * q^11 + (b3 - 2*b2 - b1) * q^13 + (2*b3 + 2*b1 - 4) * q^14 + (-b3 - b1 + 4) * q^16 - b2 * q^17 + (b3 + b1 - 3) * q^19 + (2*b3 - b2 + b1 - 6) * q^20 + (b3 + b2 - b1) * q^22 + 2*b2 * q^23 + (-2*b3 + b2 - b1 + 1) * q^25 + (b3 + b1 - 8) * q^26 + 6*b2 * q^28 + (-2*b3 - 2*b1 + 1) * q^29 + (b3 + b1 + 3) * q^31 + (b3 - 2*b2 - b1) * q^32 + (b3 + b1 - 4) * q^34 + (b3 - 3*b2 + b1 + 4) * q^35 - 3*b2 * q^37 + (b3 + 5*b2 - b1) * q^38 + (5*b2 - b1 - 4) * q^40 + (-b3 - b1 - 1) * q^41 + (-b3 - b2 + b1) * q^43 + 6 * q^44 + (-2*b3 - 2*b1 + 8) * q^46 + (b3 + 3*b2 - b1) * q^47 - 5 * q^49 + (-2*b3 - 4*b2 + b1 + 4) * q^50 + (3*b3 + 6*b2 - 3*b1) * q^52 + (-3*b3 - b2 + 3*b1) * q^53 + (2*b3 - b2 - 2*b1 - 3) * q^55 + (-2*b3 - 2*b1 + 16) * q^56 + (-2*b3 - 5*b2 + 2*b1) * q^58 + (-b3 - b1 + 5) * q^59 + (b3 + b1 + 6) * q^61 + (b3 - b2 - b1) * q^62 + (-b3 - b1) * q^64 + (-4*b3 - 3*b2 - b1 - 4) * q^65 + (-2*b3 + 4*b2 + 2*b1) * q^67 + (b3 + 4*b2 - b1) * q^68 + (4*b3 - 2*b2 + 2*b1 - 12) * q^70 + (-3*b3 - 3*b1 - 3) * q^71 + 3*b2 * q^73 + (3*b3 + 3*b1 - 12) * q^74 + (-4*b3 - 4*b1 + 14) * q^76 + (-3*b3 + 3*b2 + 3*b1) * q^77 + (-2*b3 - 2*b1) * q^79 + (-2*b3 + b2 - 3*b1 + 8) * q^80 + (-b3 - b2 + b1) * q^82 + (-3*b3 - b2 + 3*b1) * q^83 + (-b3 - 2*b2) * q^85 + (2*b3 + 2*b1 - 4) * q^86 + (2*b3 - 4*b2 - 2*b1) * q^88 - 3 * q^89 + (6*b3 + 6*b1) * q^91 + (-2*b3 - 8*b2 + 2*b1) * q^92 + (-4*b3 - 4*b1 + 12) * q^94 + (2*b3 - b2 + 2*b1 - 7) * q^95 + (-b3 - 7*b2 + b1) * q^97 + 5*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4} + 3 q^{5}+O(q^{10})$$ 4 * q - 6 * q^4 + 3 * q^5 $$4 q - 6 q^{4} + 3 q^{5} - q^{10} + 6 q^{11} - 12 q^{14} + 14 q^{16} - 10 q^{19} - 21 q^{20} + q^{25} - 30 q^{26} + 14 q^{31} - 14 q^{34} + 18 q^{35} - 17 q^{40} - 6 q^{41} + 24 q^{44} + 28 q^{46} - 20 q^{49} + 15 q^{50} - 12 q^{55} + 60 q^{56} + 18 q^{59} + 26 q^{61} - 2 q^{64} - 21 q^{65} - 42 q^{70} - 18 q^{71} - 42 q^{74} + 48 q^{76} - 4 q^{79} + 27 q^{80} - q^{85} - 12 q^{86} - 12 q^{89} + 12 q^{91} + 40 q^{94} - 24 q^{95}+O(q^{100})$$ 4 * q - 6 * q^4 + 3 * q^5 - q^10 + 6 * q^11 - 12 * q^14 + 14 * q^16 - 10 * q^19 - 21 * q^20 + q^25 - 30 * q^26 + 14 * q^31 - 14 * q^34 + 18 * q^35 - 17 * q^40 - 6 * q^41 + 24 * q^44 + 28 * q^46 - 20 * q^49 + 15 * q^50 - 12 * q^55 + 60 * q^56 + 18 * q^59 + 26 * q^61 - 2 * q^64 - 21 * q^65 - 42 * q^70 - 18 * q^71 - 42 * q^74 + 48 * q^76 - 4 * q^79 + 27 * q^80 - q^85 - 12 * q^86 - 12 * q^89 + 12 * q^91 + 40 * q^94 - 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3$$ (v^3 + 2*v^2 + v - 6) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} + \nu - 3 ) / 3$$ (v^3 - v^2 + v - 3) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3$$ (-2*v^3 - v^2 + 4*v + 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{2} + \beta _1 + 1$$ -b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2$$ (-b3 + 3*b2 + b1 + 8) / 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i 0 −4.37228 2.18614 + 0.469882i 0 3.46410i 5.98844i 0 1.18614 5.51856i
244.2 0.792287i 0 1.37228 −0.686141 2.12819i 0 3.46410i 2.67181i 0 −1.68614 + 0.543620i
244.3 0.792287i 0 1.37228 −0.686141 + 2.12819i 0 3.46410i 2.67181i 0 −1.68614 0.543620i
244.4 2.52434i 0 −4.37228 2.18614 0.469882i 0 3.46410i 5.98844i 0 1.18614 + 5.51856i
 $$n$$: 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.b yes 4
3.b odd 2 1 405.2.b.a 4
5.b even 2 1 inner 405.2.b.b yes 4
5.c odd 4 2 2025.2.a.x 4
9.c even 3 1 405.2.j.a 4
9.c even 3 1 405.2.j.d 4
9.d odd 6 1 405.2.j.b 4
9.d odd 6 1 405.2.j.e 4
15.d odd 2 1 405.2.b.a 4
15.e even 4 2 2025.2.a.w 4
45.h odd 6 1 405.2.j.b 4
45.h odd 6 1 405.2.j.e 4
45.j even 6 1 405.2.j.a 4
45.j even 6 1 405.2.j.d 4

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

## Hecke kernels

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

 $$T_{2}^{4} + 7T_{2}^{2} + 4$$ T2^4 + 7*T2^2 + 4 $$T_{11}^{2} - 3T_{11} - 6$$ T11^2 - 3*T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$(T^{2} - 3 T - 6)^{2}$$
$13$ $$T^{4} + 51T^{2} + 576$$
$17$ $$T^{4} + 7T^{2} + 4$$
$19$ $$(T^{2} + 5 T - 2)^{2}$$
$23$ $$T^{4} + 28T^{2} + 64$$
$29$ $$(T^{2} - 33)^{2}$$
$31$ $$(T^{2} - 7 T + 4)^{2}$$
$37$ $$T^{4} + 63T^{2} + 324$$
$41$ $$(T^{2} + 3 T - 6)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$T^{4} + 76T^{2} + 256$$
$53$ $$T^{4} + 172T^{2} + 4096$$
$59$ $$(T^{2} - 9 T + 12)^{2}$$
$61$ $$(T^{2} - 13 T + 34)^{2}$$
$67$ $$T^{4} + 204T^{2} + 9216$$
$71$ $$(T^{2} + 9 T - 54)^{2}$$
$73$ $$T^{4} + 63T^{2} + 324$$
$79$ $$(T^{2} + 2 T - 32)^{2}$$
$83$ $$T^{4} + 172T^{2} + 4096$$
$89$ $$(T + 3)^{4}$$
$97$ $$T^{4} + 348T^{2} + 576$$