# Properties

 Label 405.4.e.n Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ 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] = [405,4,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, 4, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$23.8957735523$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + ( - 5 \zeta_{6} + 5) q^{2} - 17 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 30 \zeta_{6} + 30) q^{7} - 45 q^{8} +O(q^{10})$$ q + (-5*z + 5) * q^2 - 17*z * q^4 - 5*z * q^5 + (-30*z + 30) * q^7 - 45 * q^8 $$q + ( - 5 \zeta_{6} + 5) q^{2} - 17 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 30 \zeta_{6} + 30) q^{7} - 45 q^{8} - 25 q^{10} + ( - 50 \zeta_{6} + 50) q^{11} + 20 \zeta_{6} q^{13} - 150 \zeta_{6} q^{14} + (89 \zeta_{6} - 89) q^{16} + 10 q^{17} - 44 q^{19} + (85 \zeta_{6} - 85) q^{20} - 250 \zeta_{6} q^{22} + 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 100 q^{26} - 510 q^{28} + (50 \zeta_{6} - 50) q^{29} - 108 \zeta_{6} q^{31} + 85 \zeta_{6} q^{32} + ( - 50 \zeta_{6} + 50) q^{34} - 150 q^{35} - 40 q^{37} + (220 \zeta_{6} - 220) q^{38} + 225 \zeta_{6} q^{40} + 400 \zeta_{6} q^{41} + (280 \zeta_{6} - 280) q^{43} - 850 q^{44} + 600 q^{46} + (280 \zeta_{6} - 280) q^{47} - 557 \zeta_{6} q^{49} + 125 \zeta_{6} q^{50} + ( - 340 \zeta_{6} + 340) q^{52} + 610 q^{53} - 250 q^{55} + (1350 \zeta_{6} - 1350) q^{56} + 250 \zeta_{6} q^{58} + 50 \zeta_{6} q^{59} + ( - 518 \zeta_{6} + 518) q^{61} - 540 q^{62} - 287 q^{64} + ( - 100 \zeta_{6} + 100) q^{65} + 180 \zeta_{6} q^{67} - 170 \zeta_{6} q^{68} + (750 \zeta_{6} - 750) q^{70} - 700 q^{71} - 410 q^{73} + (200 \zeta_{6} - 200) q^{74} + 748 \zeta_{6} q^{76} - 1500 \zeta_{6} q^{77} + ( - 516 \zeta_{6} + 516) q^{79} + 445 q^{80} + 2000 q^{82} + ( - 660 \zeta_{6} + 660) q^{83} - 50 \zeta_{6} q^{85} + 1400 \zeta_{6} q^{86} + (2250 \zeta_{6} - 2250) q^{88} + 1500 q^{89} + 600 q^{91} + ( - 2040 \zeta_{6} + 2040) q^{92} + 1400 \zeta_{6} q^{94} + 220 \zeta_{6} q^{95} + ( - 1630 \zeta_{6} + 1630) q^{97} - 2785 q^{98} +O(q^{100})$$ q + (-5*z + 5) * q^2 - 17*z * q^4 - 5*z * q^5 + (-30*z + 30) * q^7 - 45 * q^8 - 25 * q^10 + (-50*z + 50) * q^11 + 20*z * q^13 - 150*z * q^14 + (89*z - 89) * q^16 + 10 * q^17 - 44 * q^19 + (85*z - 85) * q^20 - 250*z * q^22 + 120*z * q^23 + (25*z - 25) * q^25 + 100 * q^26 - 510 * q^28 + (50*z - 50) * q^29 - 108*z * q^31 + 85*z * q^32 + (-50*z + 50) * q^34 - 150 * q^35 - 40 * q^37 + (220*z - 220) * q^38 + 225*z * q^40 + 400*z * q^41 + (280*z - 280) * q^43 - 850 * q^44 + 600 * q^46 + (280*z - 280) * q^47 - 557*z * q^49 + 125*z * q^50 + (-340*z + 340) * q^52 + 610 * q^53 - 250 * q^55 + (1350*z - 1350) * q^56 + 250*z * q^58 + 50*z * q^59 + (-518*z + 518) * q^61 - 540 * q^62 - 287 * q^64 + (-100*z + 100) * q^65 + 180*z * q^67 - 170*z * q^68 + (750*z - 750) * q^70 - 700 * q^71 - 410 * q^73 + (200*z - 200) * q^74 + 748*z * q^76 - 1500*z * q^77 + (-516*z + 516) * q^79 + 445 * q^80 + 2000 * q^82 + (-660*z + 660) * q^83 - 50*z * q^85 + 1400*z * q^86 + (2250*z - 2250) * q^88 + 1500 * q^89 + 600 * q^91 + (-2040*z + 2040) * q^92 + 1400*z * q^94 + 220*z * q^95 + (-1630*z + 1630) * q^97 - 2785 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} + 30 q^{7} - 90 q^{8}+O(q^{10})$$ 2 * q + 5 * q^2 - 17 * q^4 - 5 * q^5 + 30 * q^7 - 90 * q^8 $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} + 30 q^{7} - 90 q^{8} - 50 q^{10} + 50 q^{11} + 20 q^{13} - 150 q^{14} - 89 q^{16} + 20 q^{17} - 88 q^{19} - 85 q^{20} - 250 q^{22} + 120 q^{23} - 25 q^{25} + 200 q^{26} - 1020 q^{28} - 50 q^{29} - 108 q^{31} + 85 q^{32} + 50 q^{34} - 300 q^{35} - 80 q^{37} - 220 q^{38} + 225 q^{40} + 400 q^{41} - 280 q^{43} - 1700 q^{44} + 1200 q^{46} - 280 q^{47} - 557 q^{49} + 125 q^{50} + 340 q^{52} + 1220 q^{53} - 500 q^{55} - 1350 q^{56} + 250 q^{58} + 50 q^{59} + 518 q^{61} - 1080 q^{62} - 574 q^{64} + 100 q^{65} + 180 q^{67} - 170 q^{68} - 750 q^{70} - 1400 q^{71} - 820 q^{73} - 200 q^{74} + 748 q^{76} - 1500 q^{77} + 516 q^{79} + 890 q^{80} + 4000 q^{82} + 660 q^{83} - 50 q^{85} + 1400 q^{86} - 2250 q^{88} + 3000 q^{89} + 1200 q^{91} + 2040 q^{92} + 1400 q^{94} + 220 q^{95} + 1630 q^{97} - 5570 q^{98}+O(q^{100})$$ 2 * q + 5 * q^2 - 17 * q^4 - 5 * q^5 + 30 * q^7 - 90 * q^8 - 50 * q^10 + 50 * q^11 + 20 * q^13 - 150 * q^14 - 89 * q^16 + 20 * q^17 - 88 * q^19 - 85 * q^20 - 250 * q^22 + 120 * q^23 - 25 * q^25 + 200 * q^26 - 1020 * q^28 - 50 * q^29 - 108 * q^31 + 85 * q^32 + 50 * q^34 - 300 * q^35 - 80 * q^37 - 220 * q^38 + 225 * q^40 + 400 * q^41 - 280 * q^43 - 1700 * q^44 + 1200 * q^46 - 280 * q^47 - 557 * q^49 + 125 * q^50 + 340 * q^52 + 1220 * q^53 - 500 * q^55 - 1350 * q^56 + 250 * q^58 + 50 * q^59 + 518 * q^61 - 1080 * q^62 - 574 * q^64 + 100 * q^65 + 180 * q^67 - 170 * q^68 - 750 * q^70 - 1400 * q^71 - 820 * q^73 - 200 * q^74 + 748 * q^76 - 1500 * q^77 + 516 * q^79 + 890 * q^80 + 4000 * q^82 + 660 * q^83 - 50 * q^85 + 1400 * q^86 - 2250 * q^88 + 3000 * q^89 + 1200 * q^91 + 2040 * q^92 + 1400 * q^94 + 220 * q^95 + 1630 * q^97 - 5570 * 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
2.50000 4.33013i 0 −8.50000 14.7224i −2.50000 4.33013i 0 15.0000 25.9808i −45.0000 0 −25.0000
271.1 2.50000 + 4.33013i 0 −8.50000 + 14.7224i −2.50000 + 4.33013i 0 15.0000 + 25.9808i −45.0000 0 −25.0000
 $$n$$: 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 405.4.e.n 2
3.b odd 2 1 405.4.e.b 2
9.c even 3 1 45.4.a.a 1
9.c even 3 1 inner 405.4.e.n 2
9.d odd 6 1 45.4.a.e yes 1
9.d odd 6 1 405.4.e.b 2
36.f odd 6 1 720.4.a.bc 1
36.h even 6 1 720.4.a.o 1
45.h odd 6 1 225.4.a.a 1
45.j even 6 1 225.4.a.h 1
45.k odd 12 2 225.4.b.a 2
45.l even 12 2 225.4.b.b 2
63.l odd 6 1 2205.4.a.a 1
63.o even 6 1 2205.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 9.c even 3 1
45.4.a.e yes 1 9.d odd 6 1
225.4.a.a 1 45.h odd 6 1
225.4.a.h 1 45.j even 6 1
225.4.b.a 2 45.k odd 12 2
225.4.b.b 2 45.l even 12 2
405.4.e.b 2 3.b odd 2 1
405.4.e.b 2 9.d odd 6 1
405.4.e.n 2 1.a even 1 1 trivial
405.4.e.n 2 9.c even 3 1 inner
720.4.a.o 1 36.h even 6 1
720.4.a.bc 1 36.f odd 6 1
2205.4.a.a 1 63.l odd 6 1
2205.4.a.t 1 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - 5T_{2} + 25$$ T2^2 - 5*T2 + 25 $$T_{7}^{2} - 30T_{7} + 900$$ T7^2 - 30*T7 + 900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5T + 25$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 30T + 900$$
$11$ $$T^{2} - 50T + 2500$$
$13$ $$T^{2} - 20T + 400$$
$17$ $$(T - 10)^{2}$$
$19$ $$(T + 44)^{2}$$
$23$ $$T^{2} - 120T + 14400$$
$29$ $$T^{2} + 50T + 2500$$
$31$ $$T^{2} + 108T + 11664$$
$37$ $$(T + 40)^{2}$$
$41$ $$T^{2} - 400T + 160000$$
$43$ $$T^{2} + 280T + 78400$$
$47$ $$T^{2} + 280T + 78400$$
$53$ $$(T - 610)^{2}$$
$59$ $$T^{2} - 50T + 2500$$
$61$ $$T^{2} - 518T + 268324$$
$67$ $$T^{2} - 180T + 32400$$
$71$ $$(T + 700)^{2}$$
$73$ $$(T + 410)^{2}$$
$79$ $$T^{2} - 516T + 266256$$
$83$ $$T^{2} - 660T + 435600$$
$89$ $$(T - 1500)^{2}$$
$97$ $$T^{2} - 1630 T + 2656900$$