# Properties

 Label 405.2.a.i Level $405$ Weight $2$ Character orbit 405.a Self dual yes Analytic conductor $3.234$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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, 0]))

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

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## 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: yes Analytic conductor: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_1 + 2) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 2) * q^4 - q^5 + (-b1 + 2) * q^7 + (-b2 - b1 - 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_1 + 2) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8} + \beta_1 q^{10} + (\beta_{2} + 1) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{2} - 2 \beta_1 + 4) q^{14} + (2 \beta_1 + 1) q^{16} + (\beta_{2} + 1) q^{17} + ( - \beta_{2} + 1) q^{19} + ( - \beta_{2} - 2) q^{20} + ( - \beta_{2} - 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + \beta_1 - 2) q^{23} + q^{25} + (\beta_{2} + 1) q^{26} + (\beta_{2} - 3 \beta_1 + 3) q^{28} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{29} + ( - \beta_{2} + 4 \beta_1 + 1) q^{31} + (\beta_1 - 6) q^{32} + ( - \beta_{2} - 2 \beta_1 - 1) q^{34} + (\beta_1 - 2) q^{35} + (\beta_{2} - 2 \beta_1 + 3) q^{37} + (\beta_{2} + 1) q^{38} + (\beta_{2} + \beta_1 + 1) q^{40} + (\beta_{2} + 2 \beta_1 + 4) q^{41} + (\beta_{2} + 2 \beta_1 + 3) q^{43} + (\beta_{2} + 2 \beta_1 + 7) q^{44} + (\beta_{2} + 4 \beta_1 - 2) q^{46} + (\beta_{2} + 3 \beta_1 - 5) q^{47} + (\beta_{2} - 4 \beta_1 + 1) q^{49} - \beta_1 q^{50} + (\beta_{2} - 2 \beta_1 - 3) q^{52} + 2 \beta_1 q^{53} + ( - \beta_{2} - 1) q^{55} + 3 q^{56} + (\beta_1 - 6) q^{58} + 2 \beta_1 q^{59} + (\beta_{2} + 2 \beta_1) q^{61} + ( - 3 \beta_{2} - 15) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} - 1) q^{65} + (3 \beta_{2} - \beta_1 + 5) q^{67} + (\beta_{2} + 2 \beta_1 + 7) q^{68} + ( - \beta_{2} + 2 \beta_1 - 4) q^{70} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{71} + (4 \beta_1 - 4) q^{73} + (\beta_{2} - 4 \beta_1 + 7) q^{74} + (\beta_{2} - 2 \beta_1 - 3) q^{76} + (\beta_{2} - 2 \beta_1 + 1) q^{77} + ( - 4 \beta_1 + 2) q^{79} + ( - 2 \beta_1 - 1) q^{80} + ( - 3 \beta_{2} - 5 \beta_1 - 9) q^{82} + ( - 3 \beta_1 + 6) q^{83} + ( - \beta_{2} - 1) q^{85} + ( - 3 \beta_{2} - 4 \beta_1 - 9) q^{86} + ( - \beta_{2} - 4 \beta_1 - 7) q^{88} + 3 q^{89} + ( - \beta_{2} + 3) q^{91} + ( - \beta_{2} - \beta_1 - 13) q^{92} + ( - 4 \beta_{2} + 4 \beta_1 - 13) q^{94} + (\beta_{2} - 1) q^{95} + ( - 4 \beta_{2} + 2 \beta_1 - 8) q^{97} + (3 \beta_{2} - 2 \beta_1 + 15) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 2) * q^4 - q^5 + (-b1 + 2) * q^7 + (-b2 - b1 - 1) * q^8 + b1 * q^10 + (b2 + 1) * q^11 + (-b2 + 1) * q^13 + (b2 - 2*b1 + 4) * q^14 + (2*b1 + 1) * q^16 + (b2 + 1) * q^17 + (-b2 + 1) * q^19 + (-b2 - 2) * q^20 + (-b2 - 2*b1 - 1) * q^22 + (-2*b2 + b1 - 2) * q^23 + q^25 + (b2 + 1) * q^26 + (b2 - 3*b1 + 3) * q^28 + (-2*b2 + 2*b1 + 1) * q^29 + (-b2 + 4*b1 + 1) * q^31 + (b1 - 6) * q^32 + (-b2 - 2*b1 - 1) * q^34 + (b1 - 2) * q^35 + (b2 - 2*b1 + 3) * q^37 + (b2 + 1) * q^38 + (b2 + b1 + 1) * q^40 + (b2 + 2*b1 + 4) * q^41 + (b2 + 2*b1 + 3) * q^43 + (b2 + 2*b1 + 7) * q^44 + (b2 + 4*b1 - 2) * q^46 + (b2 + 3*b1 - 5) * q^47 + (b2 - 4*b1 + 1) * q^49 - b1 * q^50 + (b2 - 2*b1 - 3) * q^52 + 2*b1 * q^53 + (-b2 - 1) * q^55 + 3 * q^56 + (b1 - 6) * q^58 + 2*b1 * q^59 + (b2 + 2*b1) * q^61 + (-3*b2 - 15) * q^62 + (-b2 + 2*b1 - 6) * q^64 + (b2 - 1) * q^65 + (3*b2 - b1 + 5) * q^67 + (b2 + 2*b1 + 7) * q^68 + (-b2 + 2*b1 - 4) * q^70 + (-3*b2 - 2*b1 + 3) * q^71 + (4*b1 - 4) * q^73 + (b2 - 4*b1 + 7) * q^74 + (b2 - 2*b1 - 3) * q^76 + (b2 - 2*b1 + 1) * q^77 + (-4*b1 + 2) * q^79 + (-2*b1 - 1) * q^80 + (-3*b2 - 5*b1 - 9) * q^82 + (-3*b1 + 6) * q^83 + (-b2 - 1) * q^85 + (-3*b2 - 4*b1 - 9) * q^86 + (-b2 - 4*b1 - 7) * q^88 + 3 * q^89 + (-b2 + 3) * q^91 + (-b2 - b1 - 13) * q^92 + (-4*b2 + 4*b1 - 13) * q^94 + (b2 - 1) * q^95 + (-4*b2 + 2*b1 - 8) * q^97 + (3*b2 - 2*b1 + 15) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29} + 8 q^{31} - 17 q^{32} - 4 q^{34} - 5 q^{35} + 6 q^{37} + 2 q^{38} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 22 q^{44} - 3 q^{46} - 13 q^{47} - 2 q^{49} - q^{50} - 12 q^{52} + 2 q^{53} - 2 q^{55} + 9 q^{56} - 17 q^{58} + 2 q^{59} + q^{61} - 42 q^{62} - 15 q^{64} - 4 q^{65} + 11 q^{67} + 22 q^{68} - 9 q^{70} + 10 q^{71} - 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} - 5 q^{80} - 29 q^{82} + 15 q^{83} - 2 q^{85} - 28 q^{86} - 24 q^{88} + 9 q^{89} + 10 q^{91} - 39 q^{92} - 31 q^{94} - 4 q^{95} - 18 q^{97} + 40 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 + q^10 + 2 * q^11 + 4 * q^13 + 9 * q^14 + 5 * q^16 + 2 * q^17 + 4 * q^19 - 5 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 2 * q^26 + 5 * q^28 + 7 * q^29 + 8 * q^31 - 17 * q^32 - 4 * q^34 - 5 * q^35 + 6 * q^37 + 2 * q^38 + 3 * q^40 + 13 * q^41 + 10 * q^43 + 22 * q^44 - 3 * q^46 - 13 * q^47 - 2 * q^49 - q^50 - 12 * q^52 + 2 * q^53 - 2 * q^55 + 9 * q^56 - 17 * q^58 + 2 * q^59 + q^61 - 42 * q^62 - 15 * q^64 - 4 * q^65 + 11 * q^67 + 22 * q^68 - 9 * q^70 + 10 * q^71 - 8 * q^73 + 16 * q^74 - 12 * q^76 + 2 * q^79 - 5 * q^80 - 29 * q^82 + 15 * q^83 - 2 * q^85 - 28 * q^86 - 24 * q^88 + 9 * q^89 + 10 * q^91 - 39 * q^92 - 31 * q^94 - 4 * q^95 - 18 * q^97 + 40 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x + 3$$ :

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.51414 0.571993 −2.08613
−2.51414 0 4.32088 −1.00000 0 −0.514137 −5.83502 0 2.51414
1.2 −0.571993 0 −1.67282 −1.00000 0 1.42801 2.10083 0 0.571993
1.3 2.08613 0 2.35194 −1.00000 0 4.08613 0.734191 0 −2.08613
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.a.i 3
3.b odd 2 1 405.2.a.j 3
4.b odd 2 1 6480.2.a.bs 3
5.b even 2 1 2025.2.a.o 3
5.c odd 4 2 2025.2.b.m 6
9.c even 3 2 135.2.e.b 6
9.d odd 6 2 45.2.e.b 6
12.b even 2 1 6480.2.a.bv 3
15.d odd 2 1 2025.2.a.n 3
15.e even 4 2 2025.2.b.l 6
36.f odd 6 2 2160.2.q.k 6
36.h even 6 2 720.2.q.i 6
45.h odd 6 2 225.2.e.b 6
45.j even 6 2 675.2.e.b 6
45.k odd 12 4 675.2.k.b 12
45.l even 12 4 225.2.k.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 9.d odd 6 2
135.2.e.b 6 9.c even 3 2
225.2.e.b 6 45.h odd 6 2
225.2.k.b 12 45.l even 12 4
405.2.a.i 3 1.a even 1 1 trivial
405.2.a.j 3 3.b odd 2 1
675.2.e.b 6 45.j even 6 2
675.2.k.b 12 45.k odd 12 4
720.2.q.i 6 36.h even 6 2
2025.2.a.n 3 15.d odd 2 1
2025.2.a.o 3 5.b even 2 1
2025.2.b.l 6 15.e even 4 2
2025.2.b.m 6 5.c odd 4 2
2160.2.q.k 6 36.f odd 6 2
6480.2.a.bs 3 4.b odd 2 1
6480.2.a.bv 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(405))$$:

 $$T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3$$ T2^3 + T2^2 - 5*T2 - 3 $$T_{11}^{3} - 2T_{11}^{2} - 8T_{11} + 12$$ T11^3 - 2*T11^2 - 8*T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 5T - 3$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 5 T^{2} + 3 T + 3$$
$11$ $$T^{3} - 2 T^{2} - 8 T + 12$$
$13$ $$T^{3} - 4 T^{2} - 4 T + 4$$
$17$ $$T^{3} - 2 T^{2} - 8 T + 12$$
$19$ $$T^{3} - 4 T^{2} - 4 T + 4$$
$23$ $$T^{3} + 3 T^{2} - 33 T - 117$$
$29$ $$T^{3} - 7 T^{2} - 29 T + 51$$
$31$ $$T^{3} - 8 T^{2} - 60 T + 468$$
$37$ $$T^{3} - 6 T^{2} - 12 T + 4$$
$41$ $$T^{3} - 13 T^{2} + 19 T - 3$$
$43$ $$T^{3} - 10 T^{2} - 4 T + 4$$
$47$ $$T^{3} + 13 T^{2} - 11 T - 369$$
$53$ $$T^{3} - 2 T^{2} - 20 T + 24$$
$59$ $$T^{3} - 2 T^{2} - 20 T + 24$$
$61$ $$T^{3} - T^{2} - 37 T - 71$$
$67$ $$T^{3} - 11 T^{2} - 39 T + 507$$
$71$ $$T^{3} - 10 T^{2} - 92 T + 708$$
$73$ $$T^{3} + 8 T^{2} - 64 T - 128$$
$79$ $$T^{3} - 2 T^{2} - 84 T - 24$$
$83$ $$T^{3} - 15 T^{2} + 27 T + 81$$
$89$ $$(T - 3)^{3}$$
$97$ $$T^{3} + 18 T^{2} - 36 T - 1304$$