Properties

 Label 405.2.j.d Level $405$ Weight $2$ Character orbit 405.j 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(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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (2 \beta_{2} + 2) q^{7} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b3 + b2 - 2*b1 + 1) * q^4 + (-b2 + b1 + 1) * q^5 + (2*b2 + 2) * q^7 + (b3 + 5*b2 - b1 - 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (2 \beta_{2} + 2) q^{7} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 1) q^{10}+ \cdots + (5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b3 + b2 - 2*b1 + 1) * q^4 + (-b2 + b1 + 1) * q^5 + (2*b2 + 2) * q^7 + (b3 + 5*b2 - b1 - 2) * q^8 + (-2*b3 - 4*b2 + b1 + 1) * q^10 + (-2*b3 + b2 + b1 - 2) * q^11 + (3*b3 + b2 + 1) * q^13 + (2*b3 + 2*b2 - 4*b1 + 2) * q^14 + (2*b3 + 4*b2 - b1 - 3) * q^16 + (-b3 - b2 + b1) * q^17 + (b3 + b2 + b1 - 3) * q^19 + (-3*b3 - 7*b2 + 5) * q^20 + (-2*b2 + 4) * q^22 + (-2*b3 - 2*b2 + 2) * q^23 + (3*b3 + 2*b2) * q^25 + (b3 + b2 + b1 - 8) * q^26 + (6*b3 + 6*b2 - 6*b1) * q^28 + (4*b3 + b2 - 2*b1 + 1) * q^29 + (b3 - 4*b2 - 2*b1 + 1) * q^31 + (3*b3 + b2 + 1) * q^32 + (-2*b3 - 4*b2 + b1 + 3) * q^34 + (-2*b3 - 2*b2 + 4*b1 + 4) * q^35 + (-3*b3 - 3*b2 + 3*b1) * q^37 + (-2*b2 + 4*b1 - 6) * q^38 + (-5*b3 - b2 + b1 + 6) * q^40 + (-b3 + 2*b2 + 2*b1 - 1) * q^41 + (2*b2 + 2) * q^43 + 6 * q^44 + (-2*b3 - 2*b2 - 2*b1 + 8) * q^46 + (-2*b2 + 2*b1 - 4) * q^47 + 5*b2 * q^49 + (2*b3 + 2*b2 + b1 - 9) * q^50 + (-6*b2 + 3*b1 - 9) * q^52 + (2*b3 - 10*b2 - 2*b1 + 6) * q^53 + (-3*b3 + 5*b2 + 3*b1 - 7) * q^55 + (4*b3 + 16*b2 - 2*b1 - 14) * q^56 + (3*b3 + 7*b2 - 11) * q^58 + (-b3 - 4*b2 + 2*b1 - 1) * q^59 + (-2*b3 + 6*b2 + b1 - 7) * q^61 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^62 + (-b3 - b2 - b1) * q^64 + (5*b3 - b2 - 4*b1 + 11) * q^65 + (-6*b3 - 2*b2 - 2) * q^67 + (-3*b3 - 5*b2 + 7) * q^68 + (-6*b3 - 14*b2 + 10) * q^70 + (-3*b3 - 3*b2 - 3*b1 - 3) * q^71 + (3*b3 + 3*b2 - 3*b1) * q^73 + (-6*b3 - 12*b2 + 3*b1 + 9) * q^74 + (-4*b3 - 10*b2 + 8*b1 - 4) * q^76 + (-6*b3 - 6) * q^77 + (4*b3 - 2*b1 + 2) * q^79 + (-2*b3 - 3*b1 + 7) * q^80 + (-4*b2 + 2) * q^82 + (6*b2 + 2*b1 + 4) * q^83 + (b3 + 3*b2 + b1 - 4) * q^85 + (2*b3 + 2*b2 - 4*b1 + 2) * q^86 + (-4*b2 - 6*b1 + 2) * q^88 - 3 * q^89 + (6*b3 + 6*b2 + 6*b1) * q^91 + (4*b2 - 6*b1 + 10) * q^92 + (-4*b3 - 8*b2 + 8*b1 - 4) * q^94 + (3*b3 + 6*b2 - 4*b1 + 1) * q^95 + (2*b2 - 6*b1 + 8) * q^97 + (5*b3 + 5*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 12 q^{7}+O(q^{10})$$ 4 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 12 * q^7 $$4 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 12 q^{7} - q^{10} - 3 q^{11} + 3 q^{13} + 6 q^{14} - 7 q^{16} - 10 q^{19} + 9 q^{20} + 12 q^{22} + 6 q^{23} + q^{25} - 30 q^{26} - 7 q^{31} + 3 q^{32} + 7 q^{34} + 18 q^{35} - 24 q^{38} + 28 q^{40} + 3 q^{41} + 12 q^{43} + 24 q^{44} + 28 q^{46} - 18 q^{47} + 10 q^{49} - 33 q^{50} - 45 q^{52} - 12 q^{55} - 30 q^{56} - 33 q^{58} - 9 q^{59} - 13 q^{61} - 2 q^{64} + 33 q^{65} - 6 q^{67} + 21 q^{68} + 18 q^{70} - 18 q^{71} + 21 q^{74} - 24 q^{76} - 18 q^{77} + 2 q^{79} + 27 q^{80} + 30 q^{83} - 10 q^{85} + 6 q^{86} - 6 q^{88} - 12 q^{89} + 12 q^{91} + 42 q^{92} - 20 q^{94} + 9 q^{95} + 30 q^{97}+O(q^{100})$$ 4 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 12 * q^7 - q^10 - 3 * q^11 + 3 * q^13 + 6 * q^14 - 7 * q^16 - 10 * q^19 + 9 * q^20 + 12 * q^22 + 6 * q^23 + q^25 - 30 * q^26 - 7 * q^31 + 3 * q^32 + 7 * q^34 + 18 * q^35 - 24 * q^38 + 28 * q^40 + 3 * q^41 + 12 * q^43 + 24 * q^44 + 28 * q^46 - 18 * q^47 + 10 * q^49 - 33 * q^50 - 45 * q^52 - 12 * q^55 - 30 * q^56 - 33 * q^58 - 9 * q^59 - 13 * q^61 - 2 * q^64 + 33 * q^65 - 6 * q^67 + 21 * q^68 + 18 * q^70 - 18 * q^71 + 21 * q^74 - 24 * q^76 - 18 * q^77 + 2 * q^79 + 27 * q^80 + 30 * q^83 - 10 * q^85 + 6 * q^86 - 6 * q^88 - 12 * q^89 + 12 * q^91 + 42 * q^92 - 20 * q^94 + 9 * q^95 + 30 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2}$$ b3 + 3*b2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 2\beta _1 + 3$$ -2*b3 + 2*b1 + 3

Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

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
 1.68614 − 0.396143i −1.18614 + 1.26217i 1.68614 + 0.396143i −1.18614 − 1.26217i
−0.686141 + 0.396143i 0 −0.686141 + 1.18843i 2.18614 + 0.469882i 0 3.00000 1.73205i 2.67181i 0 −1.68614 + 0.543620i
109.2 2.18614 1.26217i 0 2.18614 3.78651i −0.686141 + 2.12819i 0 3.00000 1.73205i 5.98844i 0 1.18614 + 5.51856i
379.1 −0.686141 0.396143i 0 −0.686141 1.18843i 2.18614 0.469882i 0 3.00000 + 1.73205i 2.67181i 0 −1.68614 0.543620i
379.2 2.18614 + 1.26217i 0 2.18614 + 3.78651i −0.686141 2.12819i 0 3.00000 + 1.73205i 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
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.d 4
3.b odd 2 1 405.2.j.b 4
5.b even 2 1 405.2.j.a 4
9.c even 3 1 405.2.b.b yes 4
9.c even 3 1 405.2.j.a 4
9.d odd 6 1 405.2.b.a 4
9.d odd 6 1 405.2.j.e 4
15.d odd 2 1 405.2.j.e 4
45.h odd 6 1 405.2.b.a 4
45.h odd 6 1 405.2.j.b 4
45.j even 6 1 405.2.b.b yes 4
45.j even 6 1 inner 405.2.j.d 4
45.k odd 12 2 2025.2.a.x 4
45.l even 12 2 2025.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.b.a 4 9.d odd 6 1
405.2.b.a 4 45.h odd 6 1
405.2.b.b yes 4 9.c even 3 1
405.2.b.b yes 4 45.j even 6 1
405.2.j.a 4 5.b even 2 1
405.2.j.a 4 9.c even 3 1
405.2.j.b 4 3.b odd 2 1
405.2.j.b 4 45.h odd 6 1
405.2.j.d 4 1.a even 1 1 trivial
405.2.j.d 4 45.j even 6 1 inner
405.2.j.e 4 9.d odd 6 1
405.2.j.e 4 15.d odd 2 1
2025.2.a.w 4 45.l even 12 2
2025.2.a.x 4 45.k odd 12 2

Hecke kernels

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

 $$T_{2}^{4} - 3T_{2}^{3} + T_{2}^{2} + 6T_{2} + 4$$ T2^4 - 3*T2^3 + T2^2 + 6*T2 + 4 $$T_{7}^{2} - 6T_{7} + 12$$ T7^2 - 6*T7 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + \cdots + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} - 6 T + 12)^{2}$$
$11$ $$T^{4} + 3 T^{3} + \cdots + 36$$
$13$ $$T^{4} - 3 T^{3} + \cdots + 576$$
$17$ $$T^{4} + 7T^{2} + 4$$
$19$ $$(T^{2} + 5 T - 2)^{2}$$
$23$ $$T^{4} - 6 T^{3} + \cdots + 64$$
$29$ $$T^{4} + 33T^{2} + 1089$$
$31$ $$T^{4} + 7 T^{3} + \cdots + 16$$
$37$ $$T^{4} + 63T^{2} + 324$$
$41$ $$T^{4} - 3 T^{3} + \cdots + 36$$
$43$ $$(T^{2} - 6 T + 12)^{2}$$
$47$ $$T^{4} + 18 T^{3} + \cdots + 256$$
$53$ $$T^{4} + 172T^{2} + 4096$$
$59$ $$T^{4} + 9 T^{3} + \cdots + 144$$
$61$ $$T^{4} + 13 T^{3} + \cdots + 1156$$
$67$ $$T^{4} + 6 T^{3} + \cdots + 9216$$
$71$ $$(T^{2} + 9 T - 54)^{2}$$
$73$ $$T^{4} + 63T^{2} + 324$$
$79$ $$T^{4} - 2 T^{3} + \cdots + 1024$$
$83$ $$T^{4} - 30 T^{3} + \cdots + 4096$$
$89$ $$(T + 3)^{4}$$
$97$ $$T^{4} - 30 T^{3} + \cdots + 576$$