# Properties

 Label 405.2.a.g Level $405$ Weight $2$ Character orbit 405.a Self dual yes Analytic conductor $3.234$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta - 3) q^{7} + (2 \beta - 6) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 2) * q^4 + q^5 + (-b - 3) * q^7 + (2*b - 6) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta - 3) q^{7} + (2 \beta - 6) q^{8} + (\beta - 1) q^{10} + ( - \beta - 4) q^{11} + (2 \beta - 2) q^{13} - 2 \beta q^{14} + ( - 4 \beta + 8) q^{16} + ( - \beta - 1) q^{17} + (2 \beta + 1) q^{19} + ( - 2 \beta + 2) q^{20} + ( - 3 \beta + 1) q^{22} - 2 \beta q^{23} + q^{25} + ( - 4 \beta + 8) q^{26} + 4 \beta q^{28} + (3 \beta - 2) q^{29} - 3 q^{31} + (8 \beta - 8) q^{32} - 2 q^{34} + ( - \beta - 3) q^{35} + ( - \beta - 1) q^{37} + ( - \beta + 5) q^{38} + (2 \beta - 6) q^{40} + ( - 3 \beta - 2) q^{41} + (3 \beta - 5) q^{43} + (6 \beta - 2) q^{44} + (2 \beta - 6) q^{46} + ( - \beta - 7) q^{47} + (6 \beta + 5) q^{49} + (\beta - 1) q^{50} + (8 \beta - 16) q^{52} + (\beta + 5) q^{53} + ( - \beta - 4) q^{55} + 12 q^{56} + ( - 5 \beta + 11) q^{58} + (\beta - 10) q^{59} + 4 q^{61} + ( - 3 \beta + 3) q^{62} + ( - 8 \beta + 16) q^{64} + (2 \beta - 2) q^{65} + 2 \beta q^{67} + 4 q^{68} - 2 \beta q^{70} + ( - \beta - 2) q^{71} + ( - 5 \beta + 1) q^{73} - 2 q^{74} + (2 \beta - 10) q^{76} + (7 \beta + 15) q^{77} + (2 \beta + 12) q^{79} + ( - 4 \beta + 8) q^{80} + (\beta - 7) q^{82} + (3 \beta - 3) q^{83} + ( - \beta - 1) q^{85} + ( - 8 \beta + 14) q^{86} + ( - 2 \beta + 18) q^{88} + 3 \beta q^{89} - 4 \beta q^{91} + ( - 4 \beta + 12) q^{92} + ( - 6 \beta + 4) q^{94} + (2 \beta + 1) q^{95} + ( - 5 \beta - 1) q^{97} + ( - \beta + 13) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 2) * q^4 + q^5 + (-b - 3) * q^7 + (2*b - 6) * q^8 + (b - 1) * q^10 + (-b - 4) * q^11 + (2*b - 2) * q^13 - 2*b * q^14 + (-4*b + 8) * q^16 + (-b - 1) * q^17 + (2*b + 1) * q^19 + (-2*b + 2) * q^20 + (-3*b + 1) * q^22 - 2*b * q^23 + q^25 + (-4*b + 8) * q^26 + 4*b * q^28 + (3*b - 2) * q^29 - 3 * q^31 + (8*b - 8) * q^32 - 2 * q^34 + (-b - 3) * q^35 + (-b - 1) * q^37 + (-b + 5) * q^38 + (2*b - 6) * q^40 + (-3*b - 2) * q^41 + (3*b - 5) * q^43 + (6*b - 2) * q^44 + (2*b - 6) * q^46 + (-b - 7) * q^47 + (6*b + 5) * q^49 + (b - 1) * q^50 + (8*b - 16) * q^52 + (b + 5) * q^53 + (-b - 4) * q^55 + 12 * q^56 + (-5*b + 11) * q^58 + (b - 10) * q^59 + 4 * q^61 + (-3*b + 3) * q^62 + (-8*b + 16) * q^64 + (2*b - 2) * q^65 + 2*b * q^67 + 4 * q^68 - 2*b * q^70 + (-b - 2) * q^71 + (-5*b + 1) * q^73 - 2 * q^74 + (2*b - 10) * q^76 + (7*b + 15) * q^77 + (2*b + 12) * q^79 + (-4*b + 8) * q^80 + (b - 7) * q^82 + (3*b - 3) * q^83 + (-b - 1) * q^85 + (-8*b + 14) * q^86 + (-2*b + 18) * q^88 + 3*b * q^89 - 4*b * q^91 + (-4*b + 12) * q^92 + (-6*b + 4) * q^94 + (2*b + 1) * q^95 + (-5*b - 1) * q^97 + (-b + 13) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 6 * q^7 - 12 * q^8 $$2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8} - 2 q^{10} - 8 q^{11} - 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{25} + 16 q^{26} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} - 6 q^{35} - 2 q^{37} + 10 q^{38} - 12 q^{40} - 4 q^{41} - 10 q^{43} - 4 q^{44} - 12 q^{46} - 14 q^{47} + 10 q^{49} - 2 q^{50} - 32 q^{52} + 10 q^{53} - 8 q^{55} + 24 q^{56} + 22 q^{58} - 20 q^{59} + 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{65} + 8 q^{68} - 4 q^{71} + 2 q^{73} - 4 q^{74} - 20 q^{76} + 30 q^{77} + 24 q^{79} + 16 q^{80} - 14 q^{82} - 6 q^{83} - 2 q^{85} + 28 q^{86} + 36 q^{88} + 24 q^{92} + 8 q^{94} + 2 q^{95} - 2 q^{97} + 26 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 6 * q^7 - 12 * q^8 - 2 * q^10 - 8 * q^11 - 4 * q^13 + 16 * q^16 - 2 * q^17 + 2 * q^19 + 4 * q^20 + 2 * q^22 + 2 * q^25 + 16 * q^26 - 4 * q^29 - 6 * q^31 - 16 * q^32 - 4 * q^34 - 6 * q^35 - 2 * q^37 + 10 * q^38 - 12 * q^40 - 4 * q^41 - 10 * q^43 - 4 * q^44 - 12 * q^46 - 14 * q^47 + 10 * q^49 - 2 * q^50 - 32 * q^52 + 10 * q^53 - 8 * q^55 + 24 * q^56 + 22 * q^58 - 20 * q^59 + 8 * q^61 + 6 * q^62 + 32 * q^64 - 4 * q^65 + 8 * q^68 - 4 * q^71 + 2 * q^73 - 4 * q^74 - 20 * q^76 + 30 * q^77 + 24 * q^79 + 16 * q^80 - 14 * q^82 - 6 * q^83 - 2 * q^85 + 28 * q^86 + 36 * q^88 + 24 * q^92 + 8 * q^94 + 2 * q^95 - 2 * q^97 + 26 * q^98

## 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
 −1.73205 1.73205
−2.73205 0 5.46410 1.00000 0 −1.26795 −9.46410 0 −2.73205
1.2 0.732051 0 −1.46410 1.00000 0 −4.73205 −2.53590 0 0.732051
 $$n$$: 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.g 2
3.b odd 2 1 405.2.a.h yes 2
4.b odd 2 1 6480.2.a.br 2
5.b even 2 1 2025.2.a.m 2
5.c odd 4 2 2025.2.b.g 4
9.c even 3 2 405.2.e.l 4
9.d odd 6 2 405.2.e.i 4
12.b even 2 1 6480.2.a.bi 2
15.d odd 2 1 2025.2.a.g 2
15.e even 4 2 2025.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 1.a even 1 1 trivial
405.2.a.h yes 2 3.b odd 2 1
405.2.e.i 4 9.d odd 6 2
405.2.e.l 4 9.c even 3 2
2025.2.a.g 2 15.d odd 2 1
2025.2.a.m 2 5.b even 2 1
2025.2.b.g 4 5.c odd 4 2
2025.2.b.h 4 15.e even 4 2
6480.2.a.bi 2 12.b even 2 1
6480.2.a.br 2 4.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 2T_{2} - 2$$ T2^2 + 2*T2 - 2 $$T_{11}^{2} + 8T_{11} + 13$$ T11^2 + 8*T11 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 2$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 6T + 6$$
$11$ $$T^{2} + 8T + 13$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$T^{2} + 2T - 2$$
$19$ $$T^{2} - 2T - 11$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} + 4T - 23$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 2T - 2$$
$41$ $$T^{2} + 4T - 23$$
$43$ $$T^{2} + 10T - 2$$
$47$ $$T^{2} + 14T + 46$$
$53$ $$T^{2} - 10T + 22$$
$59$ $$T^{2} + 20T + 97$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} - 12$$
$71$ $$T^{2} + 4T + 1$$
$73$ $$T^{2} - 2T - 74$$
$79$ $$T^{2} - 24T + 132$$
$83$ $$T^{2} + 6T - 18$$
$89$ $$T^{2} - 27$$
$97$ $$T^{2} + 2T - 74$$