Properties

 Label 405.4.e.a Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $2$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + (5 \zeta_{6} - 5) q^{2} - 17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (9 \zeta_{6} - 9) q^{7} + 45 q^{8} +O(q^{10})$$ q + (5*z - 5) * q^2 - 17*z * q^4 + 5*z * q^5 + (9*z - 9) * q^7 + 45 * q^8 $$q + (5 \zeta_{6} - 5) q^{2} - 17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (9 \zeta_{6} - 9) q^{7} + 45 q^{8} - 25 q^{10} + (8 \zeta_{6} - 8) q^{11} - 43 \zeta_{6} q^{13} - 45 \zeta_{6} q^{14} + (89 \zeta_{6} - 89) q^{16} + 122 q^{17} - 59 q^{19} + ( - 85 \zeta_{6} + 85) q^{20} - 40 \zeta_{6} q^{22} - 213 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 215 q^{26} + 153 q^{28} + ( - 224 \zeta_{6} + 224) q^{29} + 36 \zeta_{6} q^{31} - 85 \zeta_{6} q^{32} + (610 \zeta_{6} - 610) q^{34} - 45 q^{35} + 206 q^{37} + ( - 295 \zeta_{6} + 295) q^{38} + 225 \zeta_{6} q^{40} + 413 \zeta_{6} q^{41} + ( - 392 \zeta_{6} + 392) q^{43} + 136 q^{44} + 1065 q^{46} + (311 \zeta_{6} - 311) q^{47} + 262 \zeta_{6} q^{49} - 125 \zeta_{6} q^{50} + (731 \zeta_{6} - 731) q^{52} + 377 q^{53} - 40 q^{55} + (405 \zeta_{6} - 405) q^{56} + 1120 \zeta_{6} q^{58} + 337 \zeta_{6} q^{59} + (40 \zeta_{6} - 40) q^{61} - 180 q^{62} - 287 q^{64} + ( - 215 \zeta_{6} + 215) q^{65} - 348 \zeta_{6} q^{67} - 2074 \zeta_{6} q^{68} + ( - 225 \zeta_{6} + 225) q^{70} - 62 q^{71} - 1214 q^{73} + (1030 \zeta_{6} - 1030) q^{74} + 1003 \zeta_{6} q^{76} - 72 \zeta_{6} q^{77} + ( - 294 \zeta_{6} + 294) q^{79} - 445 q^{80} - 2065 q^{82} + ( - 534 \zeta_{6} + 534) q^{83} + 610 \zeta_{6} q^{85} + 1960 \zeta_{6} q^{86} + (360 \zeta_{6} - 360) q^{88} + 810 q^{89} + 387 q^{91} + (3621 \zeta_{6} - 3621) q^{92} - 1555 \zeta_{6} q^{94} - 295 \zeta_{6} q^{95} + ( - 928 \zeta_{6} + 928) q^{97} - 1310 q^{98} +O(q^{100})$$ q + (5*z - 5) * q^2 - 17*z * q^4 + 5*z * q^5 + (9*z - 9) * q^7 + 45 * q^8 - 25 * q^10 + (8*z - 8) * q^11 - 43*z * q^13 - 45*z * q^14 + (89*z - 89) * q^16 + 122 * q^17 - 59 * q^19 + (-85*z + 85) * q^20 - 40*z * q^22 - 213*z * q^23 + (25*z - 25) * q^25 + 215 * q^26 + 153 * q^28 + (-224*z + 224) * q^29 + 36*z * q^31 - 85*z * q^32 + (610*z - 610) * q^34 - 45 * q^35 + 206 * q^37 + (-295*z + 295) * q^38 + 225*z * q^40 + 413*z * q^41 + (-392*z + 392) * q^43 + 136 * q^44 + 1065 * q^46 + (311*z - 311) * q^47 + 262*z * q^49 - 125*z * q^50 + (731*z - 731) * q^52 + 377 * q^53 - 40 * q^55 + (405*z - 405) * q^56 + 1120*z * q^58 + 337*z * q^59 + (40*z - 40) * q^61 - 180 * q^62 - 287 * q^64 + (-215*z + 215) * q^65 - 348*z * q^67 - 2074*z * q^68 + (-225*z + 225) * q^70 - 62 * q^71 - 1214 * q^73 + (1030*z - 1030) * q^74 + 1003*z * q^76 - 72*z * q^77 + (-294*z + 294) * q^79 - 445 * q^80 - 2065 * q^82 + (-534*z + 534) * q^83 + 610*z * q^85 + 1960*z * q^86 + (360*z - 360) * q^88 + 810 * q^89 + 387 * q^91 + (3621*z - 3621) * q^92 - 1555*z * q^94 - 295*z * q^95 + (-928*z + 928) * q^97 - 1310 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} - 9 q^{7} + 90 q^{8}+O(q^{10})$$ 2 * q - 5 * q^2 - 17 * q^4 + 5 * q^5 - 9 * q^7 + 90 * q^8 $$2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} - 9 q^{7} + 90 q^{8} - 50 q^{10} - 8 q^{11} - 43 q^{13} - 45 q^{14} - 89 q^{16} + 244 q^{17} - 118 q^{19} + 85 q^{20} - 40 q^{22} - 213 q^{23} - 25 q^{25} + 430 q^{26} + 306 q^{28} + 224 q^{29} + 36 q^{31} - 85 q^{32} - 610 q^{34} - 90 q^{35} + 412 q^{37} + 295 q^{38} + 225 q^{40} + 413 q^{41} + 392 q^{43} + 272 q^{44} + 2130 q^{46} - 311 q^{47} + 262 q^{49} - 125 q^{50} - 731 q^{52} + 754 q^{53} - 80 q^{55} - 405 q^{56} + 1120 q^{58} + 337 q^{59} - 40 q^{61} - 360 q^{62} - 574 q^{64} + 215 q^{65} - 348 q^{67} - 2074 q^{68} + 225 q^{70} - 124 q^{71} - 2428 q^{73} - 1030 q^{74} + 1003 q^{76} - 72 q^{77} + 294 q^{79} - 890 q^{80} - 4130 q^{82} + 534 q^{83} + 610 q^{85} + 1960 q^{86} - 360 q^{88} + 1620 q^{89} + 774 q^{91} - 3621 q^{92} - 1555 q^{94} - 295 q^{95} + 928 q^{97} - 2620 q^{98}+O(q^{100})$$ 2 * q - 5 * q^2 - 17 * q^4 + 5 * q^5 - 9 * q^7 + 90 * q^8 - 50 * q^10 - 8 * q^11 - 43 * q^13 - 45 * q^14 - 89 * q^16 + 244 * q^17 - 118 * q^19 + 85 * q^20 - 40 * q^22 - 213 * q^23 - 25 * q^25 + 430 * q^26 + 306 * q^28 + 224 * q^29 + 36 * q^31 - 85 * q^32 - 610 * q^34 - 90 * q^35 + 412 * q^37 + 295 * q^38 + 225 * q^40 + 413 * q^41 + 392 * q^43 + 272 * q^44 + 2130 * q^46 - 311 * q^47 + 262 * q^49 - 125 * q^50 - 731 * q^52 + 754 * q^53 - 80 * q^55 - 405 * q^56 + 1120 * q^58 + 337 * q^59 - 40 * q^61 - 360 * q^62 - 574 * q^64 + 215 * q^65 - 348 * q^67 - 2074 * q^68 + 225 * q^70 - 124 * q^71 - 2428 * q^73 - 1030 * q^74 + 1003 * q^76 - 72 * q^77 + 294 * q^79 - 890 * q^80 - 4130 * q^82 + 534 * q^83 + 610 * q^85 + 1960 * q^86 - 360 * q^88 + 1620 * q^89 + 774 * q^91 - 3621 * q^92 - 1555 * q^94 - 295 * q^95 + 928 * q^97 - 2620 * 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 −4.50000 + 7.79423i 45.0000 0 −25.0000
271.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 2.50000 4.33013i 0 −4.50000 7.79423i 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.a 2
3.b odd 2 1 405.4.e.m 2
9.c even 3 1 405.4.a.b yes 1
9.c even 3 1 inner 405.4.e.a 2
9.d odd 6 1 405.4.a.a 1
9.d odd 6 1 405.4.e.m 2
45.h odd 6 1 2025.4.a.f 1
45.j even 6 1 2025.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.a 1 9.d odd 6 1
405.4.a.b yes 1 9.c even 3 1
405.4.e.a 2 1.a even 1 1 trivial
405.4.e.a 2 9.c even 3 1 inner
405.4.e.m 2 3.b odd 2 1
405.4.e.m 2 9.d odd 6 1
2025.4.a.a 1 45.j even 6 1
2025.4.a.f 1 45.h odd 6 1

Hecke kernels

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

 $$T_{2}^{2} + 5T_{2} + 25$$ T2^2 + 5*T2 + 25 $$T_{7}^{2} + 9T_{7} + 81$$ T7^2 + 9*T7 + 81

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 25$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} + 9T + 81$$
$11$ $$T^{2} + 8T + 64$$
$13$ $$T^{2} + 43T + 1849$$
$17$ $$(T - 122)^{2}$$
$19$ $$(T + 59)^{2}$$
$23$ $$T^{2} + 213T + 45369$$
$29$ $$T^{2} - 224T + 50176$$
$31$ $$T^{2} - 36T + 1296$$
$37$ $$(T - 206)^{2}$$
$41$ $$T^{2} - 413T + 170569$$
$43$ $$T^{2} - 392T + 153664$$
$47$ $$T^{2} + 311T + 96721$$
$53$ $$(T - 377)^{2}$$
$59$ $$T^{2} - 337T + 113569$$
$61$ $$T^{2} + 40T + 1600$$
$67$ $$T^{2} + 348T + 121104$$
$71$ $$(T + 62)^{2}$$
$73$ $$(T + 1214)^{2}$$
$79$ $$T^{2} - 294T + 86436$$
$83$ $$T^{2} - 534T + 285156$$
$89$ $$(T - 810)^{2}$$
$97$ $$T^{2} - 928T + 861184$$