# Properties

 Label 405.2.e.f Level $405$ Weight $2$ Character orbit 405.e Analytic conductor $3.234$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + 3 q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 - z * q^5 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + 3 q^{8} - q^{10} + ( - 4 \zeta_{6} + 4) q^{11} + 2 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{16} + 2 q^{17} + 4 q^{19} + ( - \zeta_{6} + 1) q^{20} - 4 \zeta_{6} q^{22} + (\zeta_{6} - 1) q^{25} + 2 q^{26} + ( - 2 \zeta_{6} + 2) q^{29} + 5 \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{34} - 10 q^{37} + ( - 4 \zeta_{6} + 4) q^{38} - 3 \zeta_{6} q^{40} - 10 \zeta_{6} q^{41} + (4 \zeta_{6} - 4) q^{43} + 4 q^{44} + (8 \zeta_{6} - 8) q^{47} + 7 \zeta_{6} q^{49} + \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} - 10 q^{53} - 4 q^{55} - 2 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 7 q^{64} + ( - 2 \zeta_{6} + 2) q^{65} - 12 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} - 8 q^{71} + 10 q^{73} + (10 \zeta_{6} - 10) q^{74} + 4 \zeta_{6} q^{76} - q^{80} - 10 q^{82} + (12 \zeta_{6} - 12) q^{83} - 2 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{88} - 6 q^{89} + 8 \zeta_{6} q^{94} - 4 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} + 7 q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 - z * q^5 + 3 * q^8 - q^10 + (-4*z + 4) * q^11 + 2*z * q^13 + (-z + 1) * q^16 + 2 * q^17 + 4 * q^19 + (-z + 1) * q^20 - 4*z * q^22 + (z - 1) * q^25 + 2 * q^26 + (-2*z + 2) * q^29 + 5*z * q^32 + (-2*z + 2) * q^34 - 10 * q^37 + (-4*z + 4) * q^38 - 3*z * q^40 - 10*z * q^41 + (4*z - 4) * q^43 + 4 * q^44 + (8*z - 8) * q^47 + 7*z * q^49 + z * q^50 + (2*z - 2) * q^52 - 10 * q^53 - 4 * q^55 - 2*z * q^58 + 4*z * q^59 + (-2*z + 2) * q^61 + 7 * q^64 + (-2*z + 2) * q^65 - 12*z * q^67 + 2*z * q^68 - 8 * q^71 + 10 * q^73 + (10*z - 10) * q^74 + 4*z * q^76 - q^80 - 10 * q^82 + (12*z - 12) * q^83 - 2*z * q^85 + 4*z * q^86 + (-12*z + 12) * q^88 - 6 * q^89 + 8*z * q^94 - 4*z * q^95 + (2*z - 2) * q^97 + 7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - q^5 + 6 * q^8 $$2 q + q^{2} + q^{4} - q^{5} + 6 q^{8} - 2 q^{10} + 4 q^{11} + 2 q^{13} + q^{16} + 4 q^{17} + 8 q^{19} + q^{20} - 4 q^{22} - q^{25} + 4 q^{26} + 2 q^{29} + 5 q^{32} + 2 q^{34} - 20 q^{37} + 4 q^{38} - 3 q^{40} - 10 q^{41} - 4 q^{43} + 8 q^{44} - 8 q^{47} + 7 q^{49} + q^{50} - 2 q^{52} - 20 q^{53} - 8 q^{55} - 2 q^{58} + 4 q^{59} + 2 q^{61} + 14 q^{64} + 2 q^{65} - 12 q^{67} + 2 q^{68} - 16 q^{71} + 20 q^{73} - 10 q^{74} + 4 q^{76} - 2 q^{80} - 20 q^{82} - 12 q^{83} - 2 q^{85} + 4 q^{86} + 12 q^{88} - 12 q^{89} + 8 q^{94} - 4 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 - q^5 + 6 * q^8 - 2 * q^10 + 4 * q^11 + 2 * q^13 + q^16 + 4 * q^17 + 8 * q^19 + q^20 - 4 * q^22 - q^25 + 4 * q^26 + 2 * q^29 + 5 * q^32 + 2 * q^34 - 20 * q^37 + 4 * q^38 - 3 * q^40 - 10 * q^41 - 4 * q^43 + 8 * q^44 - 8 * q^47 + 7 * q^49 + q^50 - 2 * q^52 - 20 * q^53 - 8 * q^55 - 2 * q^58 + 4 * q^59 + 2 * q^61 + 14 * q^64 + 2 * q^65 - 12 * q^67 + 2 * q^68 - 16 * q^71 + 20 * q^73 - 10 * q^74 + 4 * q^76 - 2 * q^80 - 20 * q^82 - 12 * q^83 - 2 * q^85 + 4 * q^86 + 12 * q^88 - 12 * q^89 + 8 * q^94 - 4 * q^95 - 2 * q^97 + 14 * q^98

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 0 3.00000 0 −1.00000
271.1 0.500000 + 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 0 3.00000 0 −1.00000
 $$n$$: 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.f 2
3.b odd 2 1 405.2.e.c 2
9.c even 3 1 15.2.a.a 1
9.c even 3 1 inner 405.2.e.f 2
9.d odd 6 1 45.2.a.a 1
9.d odd 6 1 405.2.e.c 2
36.f odd 6 1 240.2.a.d 1
36.h even 6 1 720.2.a.c 1
45.h odd 6 1 225.2.a.b 1
45.j even 6 1 75.2.a.b 1
45.k odd 12 2 75.2.b.b 2
45.l even 12 2 225.2.b.b 2
63.g even 3 1 735.2.i.e 2
63.h even 3 1 735.2.i.e 2
63.k odd 6 1 735.2.i.d 2
63.l odd 6 1 735.2.a.c 1
63.o even 6 1 2205.2.a.i 1
63.t odd 6 1 735.2.i.d 2
72.j odd 6 1 2880.2.a.y 1
72.l even 6 1 2880.2.a.bc 1
72.n even 6 1 960.2.a.l 1
72.p odd 6 1 960.2.a.a 1
99.g even 6 1 5445.2.a.c 1
99.h odd 6 1 1815.2.a.d 1
117.n odd 6 1 7605.2.a.g 1
117.t even 6 1 2535.2.a.j 1
144.v odd 12 2 3840.2.k.r 2
144.x even 12 2 3840.2.k.m 2
153.h even 6 1 4335.2.a.c 1
171.o odd 6 1 5415.2.a.j 1
180.n even 6 1 3600.2.a.u 1
180.p odd 6 1 1200.2.a.e 1
180.v odd 12 2 3600.2.f.e 2
180.x even 12 2 1200.2.f.h 2
207.f odd 6 1 7935.2.a.d 1
315.bg odd 6 1 3675.2.a.j 1
360.z odd 6 1 4800.2.a.bz 1
360.bk even 6 1 4800.2.a.t 1
360.bo even 12 2 4800.2.f.c 2
360.bu odd 12 2 4800.2.f.bf 2
495.o odd 6 1 9075.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 9.c even 3 1
45.2.a.a 1 9.d odd 6 1
75.2.a.b 1 45.j even 6 1
75.2.b.b 2 45.k odd 12 2
225.2.a.b 1 45.h odd 6 1
225.2.b.b 2 45.l even 12 2
240.2.a.d 1 36.f odd 6 1
405.2.e.c 2 3.b odd 2 1
405.2.e.c 2 9.d odd 6 1
405.2.e.f 2 1.a even 1 1 trivial
405.2.e.f 2 9.c even 3 1 inner
720.2.a.c 1 36.h even 6 1
735.2.a.c 1 63.l odd 6 1
735.2.i.d 2 63.k odd 6 1
735.2.i.d 2 63.t odd 6 1
735.2.i.e 2 63.g even 3 1
735.2.i.e 2 63.h even 3 1
960.2.a.a 1 72.p odd 6 1
960.2.a.l 1 72.n even 6 1
1200.2.a.e 1 180.p odd 6 1
1200.2.f.h 2 180.x even 12 2
1815.2.a.d 1 99.h odd 6 1
2205.2.a.i 1 63.o even 6 1
2535.2.a.j 1 117.t even 6 1
2880.2.a.y 1 72.j odd 6 1
2880.2.a.bc 1 72.l even 6 1
3600.2.a.u 1 180.n even 6 1
3600.2.f.e 2 180.v odd 12 2
3675.2.a.j 1 315.bg odd 6 1
3840.2.k.m 2 144.x even 12 2
3840.2.k.r 2 144.v odd 12 2
4335.2.a.c 1 153.h even 6 1
4800.2.a.t 1 360.bk even 6 1
4800.2.a.bz 1 360.z odd 6 1
4800.2.f.c 2 360.bo even 12 2
4800.2.f.bf 2 360.bu odd 12 2
5415.2.a.j 1 171.o odd 6 1
5445.2.a.c 1 99.g even 6 1
7605.2.a.g 1 117.n odd 6 1
7935.2.a.d 1 207.f odd 6 1
9075.2.a.g 1 495.o odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{7}$$ T7 $$T_{11}^{2} - 4T_{11} + 16$$ T11^2 - 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 10T + 100$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 2T + 4$$