# Properties

 Label 405.4.e.f 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: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 6 \zeta_{6} + 6) q^{7} - 15 q^{8} +O(q^{10})$$ q + (z - 1) * q^2 + 7*z * q^4 - 5*z * q^5 + (-6*z + 6) * q^7 - 15 * q^8 $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 6 \zeta_{6} + 6) q^{7} - 15 q^{8} + 5 q^{10} + ( - 47 \zeta_{6} + 47) q^{11} + 5 \zeta_{6} q^{13} + 6 \zeta_{6} q^{14} + (41 \zeta_{6} - 41) q^{16} - 131 q^{17} - 56 q^{19} + ( - 35 \zeta_{6} + 35) q^{20} + 47 \zeta_{6} q^{22} - 3 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 5 q^{26} + 42 q^{28} + ( - 157 \zeta_{6} + 157) q^{29} - 225 \zeta_{6} q^{31} - 161 \zeta_{6} q^{32} + ( - 131 \zeta_{6} + 131) q^{34} - 30 q^{35} - 70 q^{37} + ( - 56 \zeta_{6} + 56) q^{38} + 75 \zeta_{6} q^{40} - 140 \zeta_{6} q^{41} + (397 \zeta_{6} - 397) q^{43} + 329 q^{44} + 3 q^{46} + ( - 347 \zeta_{6} + 347) q^{47} + 307 \zeta_{6} q^{49} - 25 \zeta_{6} q^{50} + (35 \zeta_{6} - 35) q^{52} + 4 q^{53} - 235 q^{55} + (90 \zeta_{6} - 90) q^{56} + 157 \zeta_{6} q^{58} - 748 \zeta_{6} q^{59} + ( - 338 \zeta_{6} + 338) q^{61} + 225 q^{62} - 167 q^{64} + ( - 25 \zeta_{6} + 25) q^{65} - 492 \zeta_{6} q^{67} - 917 \zeta_{6} q^{68} + ( - 30 \zeta_{6} + 30) q^{70} + 32 q^{71} + 970 q^{73} + ( - 70 \zeta_{6} + 70) q^{74} - 392 \zeta_{6} q^{76} - 282 \zeta_{6} q^{77} + ( - 1257 \zeta_{6} + 1257) q^{79} + 205 q^{80} + 140 q^{82} + ( - 102 \zeta_{6} + 102) q^{83} + 655 \zeta_{6} q^{85} - 397 \zeta_{6} q^{86} + (705 \zeta_{6} - 705) q^{88} - 1488 q^{89} + 30 q^{91} + ( - 21 \zeta_{6} + 21) q^{92} + 347 \zeta_{6} q^{94} + 280 \zeta_{6} q^{95} + (974 \zeta_{6} - 974) q^{97} - 307 q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + 7*z * q^4 - 5*z * q^5 + (-6*z + 6) * q^7 - 15 * q^8 + 5 * q^10 + (-47*z + 47) * q^11 + 5*z * q^13 + 6*z * q^14 + (41*z - 41) * q^16 - 131 * q^17 - 56 * q^19 + (-35*z + 35) * q^20 + 47*z * q^22 - 3*z * q^23 + (25*z - 25) * q^25 - 5 * q^26 + 42 * q^28 + (-157*z + 157) * q^29 - 225*z * q^31 - 161*z * q^32 + (-131*z + 131) * q^34 - 30 * q^35 - 70 * q^37 + (-56*z + 56) * q^38 + 75*z * q^40 - 140*z * q^41 + (397*z - 397) * q^43 + 329 * q^44 + 3 * q^46 + (-347*z + 347) * q^47 + 307*z * q^49 - 25*z * q^50 + (35*z - 35) * q^52 + 4 * q^53 - 235 * q^55 + (90*z - 90) * q^56 + 157*z * q^58 - 748*z * q^59 + (-338*z + 338) * q^61 + 225 * q^62 - 167 * q^64 + (-25*z + 25) * q^65 - 492*z * q^67 - 917*z * q^68 + (-30*z + 30) * q^70 + 32 * q^71 + 970 * q^73 + (-70*z + 70) * q^74 - 392*z * q^76 - 282*z * q^77 + (-1257*z + 1257) * q^79 + 205 * q^80 + 140 * q^82 + (-102*z + 102) * q^83 + 655*z * q^85 - 397*z * q^86 + (705*z - 705) * q^88 - 1488 * q^89 + 30 * q^91 + (-21*z + 21) * q^92 + 347*z * q^94 + 280*z * q^95 + (974*z - 974) * q^97 - 307 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 6 q^{7} - 30 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 6 * q^7 - 30 * q^8 $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 6 q^{7} - 30 q^{8} + 10 q^{10} + 47 q^{11} + 5 q^{13} + 6 q^{14} - 41 q^{16} - 262 q^{17} - 112 q^{19} + 35 q^{20} + 47 q^{22} - 3 q^{23} - 25 q^{25} - 10 q^{26} + 84 q^{28} + 157 q^{29} - 225 q^{31} - 161 q^{32} + 131 q^{34} - 60 q^{35} - 140 q^{37} + 56 q^{38} + 75 q^{40} - 140 q^{41} - 397 q^{43} + 658 q^{44} + 6 q^{46} + 347 q^{47} + 307 q^{49} - 25 q^{50} - 35 q^{52} + 8 q^{53} - 470 q^{55} - 90 q^{56} + 157 q^{58} - 748 q^{59} + 338 q^{61} + 450 q^{62} - 334 q^{64} + 25 q^{65} - 492 q^{67} - 917 q^{68} + 30 q^{70} + 64 q^{71} + 1940 q^{73} + 70 q^{74} - 392 q^{76} - 282 q^{77} + 1257 q^{79} + 410 q^{80} + 280 q^{82} + 102 q^{83} + 655 q^{85} - 397 q^{86} - 705 q^{88} - 2976 q^{89} + 60 q^{91} + 21 q^{92} + 347 q^{94} + 280 q^{95} - 974 q^{97} - 614 q^{98}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 6 * q^7 - 30 * q^8 + 10 * q^10 + 47 * q^11 + 5 * q^13 + 6 * q^14 - 41 * q^16 - 262 * q^17 - 112 * q^19 + 35 * q^20 + 47 * q^22 - 3 * q^23 - 25 * q^25 - 10 * q^26 + 84 * q^28 + 157 * q^29 - 225 * q^31 - 161 * q^32 + 131 * q^34 - 60 * q^35 - 140 * q^37 + 56 * q^38 + 75 * q^40 - 140 * q^41 - 397 * q^43 + 658 * q^44 + 6 * q^46 + 347 * q^47 + 307 * q^49 - 25 * q^50 - 35 * q^52 + 8 * q^53 - 470 * q^55 - 90 * q^56 + 157 * q^58 - 748 * q^59 + 338 * q^61 + 450 * q^62 - 334 * q^64 + 25 * q^65 - 492 * q^67 - 917 * q^68 + 30 * q^70 + 64 * q^71 + 1940 * q^73 + 70 * q^74 - 392 * q^76 - 282 * q^77 + 1257 * q^79 + 410 * q^80 + 280 * q^82 + 102 * q^83 + 655 * q^85 - 397 * q^86 - 705 * q^88 - 2976 * q^89 + 60 * q^91 + 21 * q^92 + 347 * q^94 + 280 * q^95 - 974 * q^97 - 614 * 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 3.50000 + 6.06218i −2.50000 4.33013i 0 3.00000 5.19615i −15.0000 0 5.00000
271.1 −0.500000 0.866025i 0 3.50000 6.06218i −2.50000 + 4.33013i 0 3.00000 + 5.19615i −15.0000 0 5.00000
 $$n$$: 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 405.4.e.f 2
3.b odd 2 1 405.4.e.h 2
9.c even 3 1 135.4.a.c yes 1
9.c even 3 1 inner 405.4.e.f 2
9.d odd 6 1 135.4.a.b 1
9.d odd 6 1 405.4.e.h 2
36.f odd 6 1 2160.4.a.p 1
36.h even 6 1 2160.4.a.f 1
45.h odd 6 1 675.4.a.h 1
45.j even 6 1 675.4.a.c 1
45.k odd 12 2 675.4.b.e 2
45.l even 12 2 675.4.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.b 1 9.d odd 6 1
135.4.a.c yes 1 9.c even 3 1
405.4.e.f 2 1.a even 1 1 trivial
405.4.e.f 2 9.c even 3 1 inner
405.4.e.h 2 3.b odd 2 1
405.4.e.h 2 9.d odd 6 1
675.4.a.c 1 45.j even 6 1
675.4.a.h 1 45.h odd 6 1
675.4.b.e 2 45.k odd 12 2
675.4.b.f 2 45.l even 12 2
2160.4.a.f 1 36.h even 6 1
2160.4.a.p 1 36.f odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{7}^{2} - 6T_{7} + 36$$ T7^2 - 6*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 6T + 36$$
$11$ $$T^{2} - 47T + 2209$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 131)^{2}$$
$19$ $$(T + 56)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} - 157T + 24649$$
$31$ $$T^{2} + 225T + 50625$$
$37$ $$(T + 70)^{2}$$
$41$ $$T^{2} + 140T + 19600$$
$43$ $$T^{2} + 397T + 157609$$
$47$ $$T^{2} - 347T + 120409$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2} + 748T + 559504$$
$61$ $$T^{2} - 338T + 114244$$
$67$ $$T^{2} + 492T + 242064$$
$71$ $$(T - 32)^{2}$$
$73$ $$(T - 970)^{2}$$
$79$ $$T^{2} - 1257 T + 1580049$$
$83$ $$T^{2} - 102T + 10404$$
$89$ $$(T + 1488)^{2}$$
$97$ $$T^{2} + 974T + 948676$$