# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(136,405)]

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([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, 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 $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 4x^{2} + 3x + 9$$ x^4 - x^3 + 4*x^2 + 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} - 3 q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + b2 + b1 - 1) * q^4 + b2 * q^5 + (-2*b2 + 2*b1 - 2) * q^7 - 3 * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} - 3 q^{8} + (\beta_{3} - 1) q^{10} - 2 \beta_1 q^{11} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + 6 \beta_{2} q^{14} + (2 \beta_{2} - \beta_1 + 2) q^{16} + ( - 2 \beta_{3} - 1) q^{17} + ( - 2 \beta_{3} + 1) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{20} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 2) q^{22} - 3 \beta_{2} q^{23} + ( - \beta_{2} - 1) q^{25} + (4 \beta_{3} - 10) q^{26} + (2 \beta_{3} - 6) q^{28} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{29} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{32} + (6 \beta_{2} - \beta_1 + 6) q^{34} + 2 \beta_{3} q^{35} + 2 q^{37} + (6 \beta_{2} + \beta_1 + 6) q^{38} - 3 \beta_{2} q^{40} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{41} + (4 \beta_{2} - 2 \beta_1 + 4) q^{43} + ( - 4 \beta_{3} + 10) q^{44} + ( - 3 \beta_{3} + 3) q^{46} + 4 \beta_1 q^{47} + ( - 4 \beta_{3} + 9 \beta_{2} - 4 \beta_1 + 4) q^{49} + ( - \beta_{3} - \beta_1 + 1) q^{50} + ( - 8 \beta_{2} - 6 \beta_1 - 8) q^{52} + ( - 2 \beta_{3} - 1) q^{53} + ( - 2 \beta_{3} + 2) q^{55} + (6 \beta_{2} - 6 \beta_1 + 6) q^{56} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{58} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{59} + ( - 5 \beta_{2} + 4 \beta_1 - 5) q^{61} + ( - 3 \beta_{3} + 9) q^{62} + (6 \beta_{3} - 5) q^{64} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{65} + (4 \beta_{3} - 10 \beta_{2} + 4 \beta_1 - 4) q^{67} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{68} + ( - 6 \beta_{2} - 6) q^{70} + (2 \beta_{3} + 10) q^{71} + ( - 2 \beta_{3} + 10) q^{73} + 2 \beta_1 q^{74} + (3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 3) q^{76} - 12 \beta_{2} q^{77} + (7 \beta_{2} + 2 \beta_1 + 7) q^{79} + ( - \beta_{3} - 1) q^{80} + ( - 2 \beta_{3} + 8) q^{82} + ( - 3 \beta_{2} - 3) q^{83} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{85} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{86} + 6 \beta_1 q^{88} + (6 \beta_{3} - 6) q^{89} + (4 \beta_{3} - 12) q^{91} + (3 \beta_{2} + 3 \beta_1 + 3) q^{92} + (4 \beta_{3} + 12 \beta_{2} + 4 \beta_1 - 4) q^{94} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{95} + ( - 8 \beta_{2} - 8) q^{97} + (5 \beta_{3} + 7) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + b2 + b1 - 1) * q^4 + b2 * q^5 + (-2*b2 + 2*b1 - 2) * q^7 - 3 * q^8 + (b3 - 1) * q^10 - 2*b1 * q^11 + (2*b3 + 2*b2 + 2*b1 - 2) * q^13 + 6*b2 * q^14 + (2*b2 - b1 + 2) * q^16 + (-2*b3 - 1) * q^17 + (-2*b3 + 1) * q^19 + (-b2 - b1 - 1) * q^20 + (-2*b3 - 6*b2 - 2*b1 + 2) * q^22 - 3*b2 * q^23 + (-b2 - 1) * q^25 + (4*b3 - 10) * q^26 + (2*b3 - 6) * q^28 + (-6*b2 + 2*b1 - 6) * q^29 + (-2*b3 - b2 - 2*b1 + 2) * q^31 + (b3 + 3*b2 + b1 - 1) * q^32 + (6*b2 - b1 + 6) * q^34 + 2*b3 * q^35 + 2 * q^37 + (6*b2 + b1 + 6) * q^38 - 3*b2 * q^40 + (-2*b3 - 2*b1 + 2) * q^41 + (4*b2 - 2*b1 + 4) * q^43 + (-4*b3 + 10) * q^44 + (-3*b3 + 3) * q^46 + 4*b1 * q^47 + (-4*b3 + 9*b2 - 4*b1 + 4) * q^49 + (-b3 - b1 + 1) * q^50 + (-8*b2 - 6*b1 - 8) * q^52 + (-2*b3 - 1) * q^53 + (-2*b3 + 2) * q^55 + (6*b2 - 6*b1 + 6) * q^56 + (-4*b3 + 6*b2 - 4*b1 + 4) * q^58 + (2*b3 - 6*b2 + 2*b1 - 2) * q^59 + (-5*b2 + 4*b1 - 5) * q^61 + (-3*b3 + 9) * q^62 + (6*b3 - 5) * q^64 + (-2*b2 - 2*b1 - 2) * q^65 + (4*b3 - 10*b2 + 4*b1 - 4) * q^67 + (b3 + 3*b2 + b1 - 1) * q^68 + (-6*b2 - 6) * q^70 + (2*b3 + 10) * q^71 + (-2*b3 + 10) * q^73 + 2*b1 * q^74 + (3*b3 + 5*b2 + 3*b1 - 3) * q^76 - 12*b2 * q^77 + (7*b2 + 2*b1 + 7) * q^79 + (-b3 - 1) * q^80 + (-2*b3 + 8) * q^82 + (-3*b2 - 3) * q^83 + (2*b3 - 3*b2 + 2*b1 - 2) * q^85 + (2*b3 - 6*b2 + 2*b1 - 2) * q^86 + 6*b1 * q^88 + (6*b3 - 6) * q^89 + (4*b3 - 12) * q^91 + (3*b2 + 3*b1 + 3) * q^92 + (4*b3 + 12*b2 + 4*b1 - 4) * q^94 + (2*b3 - b2 + 2*b1 - 2) * q^95 + (-8*b2 - 8) * q^97 + (5*b3 + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 3 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + q^2 - 3 * q^4 - 2 * q^5 - 2 * q^7 - 12 * q^8 $$4 q + q^{2} - 3 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} - 2 q^{10} - 2 q^{11} - 6 q^{13} - 12 q^{14} + 3 q^{16} - 8 q^{17} - 3 q^{20} + 14 q^{22} + 6 q^{23} - 2 q^{25} - 32 q^{26} - 20 q^{28} - 10 q^{29} + 4 q^{31} - 7 q^{32} + 11 q^{34} + 4 q^{35} + 8 q^{37} + 13 q^{38} + 6 q^{40} + 2 q^{41} + 6 q^{43} + 32 q^{44} + 6 q^{46} + 4 q^{47} - 14 q^{49} + q^{50} - 22 q^{52} - 8 q^{53} + 4 q^{55} + 6 q^{56} - 8 q^{58} + 10 q^{59} - 6 q^{61} + 30 q^{62} - 8 q^{64} - 6 q^{65} + 16 q^{67} - 7 q^{68} - 12 q^{70} + 44 q^{71} + 36 q^{73} + 2 q^{74} - 13 q^{76} + 24 q^{77} + 16 q^{79} - 6 q^{80} + 28 q^{82} - 6 q^{83} + 4 q^{85} + 10 q^{86} + 6 q^{88} - 12 q^{89} - 40 q^{91} + 9 q^{92} - 28 q^{94} - 16 q^{97} + 38 q^{98}+O(q^{100})$$ 4 * q + q^2 - 3 * q^4 - 2 * q^5 - 2 * q^7 - 12 * q^8 - 2 * q^10 - 2 * q^11 - 6 * q^13 - 12 * q^14 + 3 * q^16 - 8 * q^17 - 3 * q^20 + 14 * q^22 + 6 * q^23 - 2 * q^25 - 32 * q^26 - 20 * q^28 - 10 * q^29 + 4 * q^31 - 7 * q^32 + 11 * q^34 + 4 * q^35 + 8 * q^37 + 13 * q^38 + 6 * q^40 + 2 * q^41 + 6 * q^43 + 32 * q^44 + 6 * q^46 + 4 * q^47 - 14 * q^49 + q^50 - 22 * q^52 - 8 * q^53 + 4 * q^55 + 6 * q^56 - 8 * q^58 + 10 * q^59 - 6 * q^61 + 30 * q^62 - 8 * q^64 - 6 * q^65 + 16 * q^67 - 7 * q^68 - 12 * q^70 + 44 * q^71 + 36 * q^73 + 2 * q^74 - 13 * q^76 + 24 * q^77 + 16 * q^79 - 6 * q^80 + 28 * q^82 - 6 * q^83 + 4 * q^85 + 10 * q^86 + 6 * q^88 - 12 * q^89 - 40 * q^91 + 9 * q^92 - 28 * q^94 - 16 * q^97 + 38 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12$$ (-v^3 + 4*v^2 - 4*v - 3) / 12 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 7 ) / 4$$ (v^3 + 7) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2} + \beta _1 - 1$$ b3 + 3*b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$4\beta_{3} - 7$$ 4*b3 - 7

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 −0.651388 + 1.12824i 1.15139 − 1.99426i −0.651388 − 1.12824i 1.15139 + 1.99426i
−0.651388 + 1.12824i 0 0.151388 + 0.262211i −0.500000 0.866025i 0 −2.30278 + 3.98852i −3.00000 0 1.30278
136.2 1.15139 1.99426i 0 −1.65139 2.86029i −0.500000 0.866025i 0 1.30278 2.25647i −3.00000 0 −2.30278
271.1 −0.651388 1.12824i 0 0.151388 0.262211i −0.500000 + 0.866025i 0 −2.30278 3.98852i −3.00000 0 1.30278
271.2 1.15139 + 1.99426i 0 −1.65139 + 2.86029i −0.500000 + 0.866025i 0 1.30278 + 2.25647i −3.00000 0 −2.30278
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.e.k 4
3.b odd 2 1 405.2.e.j 4
9.c even 3 1 135.2.a.c 2
9.c even 3 1 inner 405.2.e.k 4
9.d odd 6 1 135.2.a.d yes 2
9.d odd 6 1 405.2.e.j 4
36.f odd 6 1 2160.2.a.ba 2
36.h even 6 1 2160.2.a.y 2
45.h odd 6 1 675.2.a.k 2
45.j even 6 1 675.2.a.p 2
45.k odd 12 2 675.2.b.i 4
45.l even 12 2 675.2.b.h 4
63.l odd 6 1 6615.2.a.p 2
63.o even 6 1 6615.2.a.v 2
72.j odd 6 1 8640.2.a.df 2
72.l even 6 1 8640.2.a.cy 2
72.n even 6 1 8640.2.a.cr 2
72.p odd 6 1 8640.2.a.ck 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 9.c even 3 1
135.2.a.d yes 2 9.d odd 6 1
405.2.e.j 4 3.b odd 2 1
405.2.e.j 4 9.d odd 6 1
405.2.e.k 4 1.a even 1 1 trivial
405.2.e.k 4 9.c even 3 1 inner
675.2.a.k 2 45.h odd 6 1
675.2.a.p 2 45.j even 6 1
675.2.b.h 4 45.l even 12 2
675.2.b.i 4 45.k odd 12 2
2160.2.a.y 2 36.h even 6 1
2160.2.a.ba 2 36.f odd 6 1
6615.2.a.p 2 63.l odd 6 1
6615.2.a.v 2 63.o even 6 1
8640.2.a.ck 2 72.p odd 6 1
8640.2.a.cr 2 72.n even 6 1
8640.2.a.cy 2 72.l even 6 1
8640.2.a.df 2 72.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9$$ T2^4 - T2^3 + 4*T2^2 + 3*T2 + 9 $$T_{7}^{4} + 2T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 144$$ T7^4 + 2*T7^3 + 16*T7^2 - 24*T7 + 144 $$T_{11}^{4} + 2T_{11}^{3} + 16T_{11}^{2} - 24T_{11} + 144$$ T11^4 + 2*T11^3 + 16*T11^2 - 24*T11 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 4 T^{2} + 3 T + 9$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 2 T^{3} + 16 T^{2} - 24 T + 144$$
$11$ $$T^{4} + 2 T^{3} + 16 T^{2} - 24 T + 144$$
$13$ $$T^{4} + 6 T^{3} + 40 T^{2} - 24 T + 16$$
$17$ $$(T^{2} + 4 T - 9)^{2}$$
$19$ $$(T^{2} - 13)^{2}$$
$23$ $$(T^{2} - 3 T + 9)^{2}$$
$29$ $$T^{4} + 10 T^{3} + 88 T^{2} + \cdots + 144$$
$31$ $$T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81$$
$37$ $$(T - 2)^{4}$$
$41$ $$T^{4} - 2 T^{3} + 16 T^{2} + 24 T + 144$$
$43$ $$T^{4} - 6 T^{3} + 40 T^{2} + 24 T + 16$$
$47$ $$T^{4} - 4 T^{3} + 64 T^{2} + \cdots + 2304$$
$53$ $$(T^{2} + 4 T - 9)^{2}$$
$59$ $$T^{4} - 10 T^{3} + 88 T^{2} + \cdots + 144$$
$61$ $$T^{4} + 6 T^{3} + 79 T^{2} + \cdots + 1849$$
$67$ $$T^{4} - 16 T^{3} + 244 T^{2} + \cdots + 144$$
$71$ $$(T^{2} - 22 T + 108)^{2}$$
$73$ $$(T^{2} - 18 T + 68)^{2}$$
$79$ $$T^{4} - 16 T^{3} + 205 T^{2} + \cdots + 2601$$
$83$ $$(T^{2} + 3 T + 9)^{2}$$
$89$ $$(T^{2} + 6 T - 108)^{2}$$
$97$ $$(T^{2} + 8 T + 64)^{2}$$