# Properties

 Label 405.4.e.e 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: $$\Z[a_1, \ldots, a_{5}]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} - 24 q^{8}+O(q^{10})$$ q + (2*z - 2) * q^2 + 4*z * q^4 + 5*z * q^5 - 24 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} - 24 q^{8} - 10 q^{10} + ( - 10 \zeta_{6} + 10) q^{11} + 80 \zeta_{6} q^{13} + ( - 16 \zeta_{6} + 16) q^{16} - 7 q^{17} - 113 q^{19} + (20 \zeta_{6} - 20) q^{20} + 20 \zeta_{6} q^{22} - 81 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 160 q^{26} + (220 \zeta_{6} - 220) q^{29} + 189 \zeta_{6} q^{31} - 160 \zeta_{6} q^{32} + ( - 14 \zeta_{6} + 14) q^{34} + 170 q^{37} + ( - 226 \zeta_{6} + 226) q^{38} - 120 \zeta_{6} q^{40} - 130 \zeta_{6} q^{41} + (10 \zeta_{6} - 10) q^{43} + 40 q^{44} + 162 q^{46} + ( - 160 \zeta_{6} + 160) q^{47} + 343 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (320 \zeta_{6} - 320) q^{52} - 631 q^{53} + 50 q^{55} - 440 \zeta_{6} q^{58} - 560 \zeta_{6} q^{59} + (229 \zeta_{6} - 229) q^{61} - 378 q^{62} + 448 q^{64} + (400 \zeta_{6} - 400) q^{65} - 750 \zeta_{6} q^{67} - 28 \zeta_{6} q^{68} - 890 q^{71} - 890 q^{73} + (340 \zeta_{6} - 340) q^{74} - 452 \zeta_{6} q^{76} + ( - 27 \zeta_{6} + 27) q^{79} + 80 q^{80} + 260 q^{82} + ( - 429 \zeta_{6} + 429) q^{83} - 35 \zeta_{6} q^{85} - 20 \zeta_{6} q^{86} + (240 \zeta_{6} - 240) q^{88} + 750 q^{89} + ( - 324 \zeta_{6} + 324) q^{92} + 320 \zeta_{6} q^{94} - 565 \zeta_{6} q^{95} + ( - 1480 \zeta_{6} + 1480) q^{97} - 686 q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 + 4*z * q^4 + 5*z * q^5 - 24 * q^8 - 10 * q^10 + (-10*z + 10) * q^11 + 80*z * q^13 + (-16*z + 16) * q^16 - 7 * q^17 - 113 * q^19 + (20*z - 20) * q^20 + 20*z * q^22 - 81*z * q^23 + (25*z - 25) * q^25 - 160 * q^26 + (220*z - 220) * q^29 + 189*z * q^31 - 160*z * q^32 + (-14*z + 14) * q^34 + 170 * q^37 + (-226*z + 226) * q^38 - 120*z * q^40 - 130*z * q^41 + (10*z - 10) * q^43 + 40 * q^44 + 162 * q^46 + (-160*z + 160) * q^47 + 343*z * q^49 - 50*z * q^50 + (320*z - 320) * q^52 - 631 * q^53 + 50 * q^55 - 440*z * q^58 - 560*z * q^59 + (229*z - 229) * q^61 - 378 * q^62 + 448 * q^64 + (400*z - 400) * q^65 - 750*z * q^67 - 28*z * q^68 - 890 * q^71 - 890 * q^73 + (340*z - 340) * q^74 - 452*z * q^76 + (-27*z + 27) * q^79 + 80 * q^80 + 260 * q^82 + (-429*z + 429) * q^83 - 35*z * q^85 - 20*z * q^86 + (240*z - 240) * q^88 + 750 * q^89 + (-324*z + 324) * q^92 + 320*z * q^94 - 565*z * q^95 + (-1480*z + 1480) * q^97 - 686 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 48 * q^8 $$2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8} - 20 q^{10} + 10 q^{11} + 80 q^{13} + 16 q^{16} - 14 q^{17} - 226 q^{19} - 20 q^{20} + 20 q^{22} - 81 q^{23} - 25 q^{25} - 320 q^{26} - 220 q^{29} + 189 q^{31} - 160 q^{32} + 14 q^{34} + 340 q^{37} + 226 q^{38} - 120 q^{40} - 130 q^{41} - 10 q^{43} + 80 q^{44} + 324 q^{46} + 160 q^{47} + 343 q^{49} - 50 q^{50} - 320 q^{52} - 1262 q^{53} + 100 q^{55} - 440 q^{58} - 560 q^{59} - 229 q^{61} - 756 q^{62} + 896 q^{64} - 400 q^{65} - 750 q^{67} - 28 q^{68} - 1780 q^{71} - 1780 q^{73} - 340 q^{74} - 452 q^{76} + 27 q^{79} + 160 q^{80} + 520 q^{82} + 429 q^{83} - 35 q^{85} - 20 q^{86} - 240 q^{88} + 1500 q^{89} + 324 q^{92} + 320 q^{94} - 565 q^{95} + 1480 q^{97} - 1372 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 48 * q^8 - 20 * q^10 + 10 * q^11 + 80 * q^13 + 16 * q^16 - 14 * q^17 - 226 * q^19 - 20 * q^20 + 20 * q^22 - 81 * q^23 - 25 * q^25 - 320 * q^26 - 220 * q^29 + 189 * q^31 - 160 * q^32 + 14 * q^34 + 340 * q^37 + 226 * q^38 - 120 * q^40 - 130 * q^41 - 10 * q^43 + 80 * q^44 + 324 * q^46 + 160 * q^47 + 343 * q^49 - 50 * q^50 - 320 * q^52 - 1262 * q^53 + 100 * q^55 - 440 * q^58 - 560 * q^59 - 229 * q^61 - 756 * q^62 + 896 * q^64 - 400 * q^65 - 750 * q^67 - 28 * q^68 - 1780 * q^71 - 1780 * q^73 - 340 * q^74 - 452 * q^76 + 27 * q^79 + 160 * q^80 + 520 * q^82 + 429 * q^83 - 35 * q^85 - 20 * q^86 - 240 * q^88 + 1500 * q^89 + 324 * q^92 + 320 * q^94 - 565 * q^95 + 1480 * q^97 - 1372 * 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
−1.00000 + 1.73205i 0 2.00000 + 3.46410i 2.50000 + 4.33013i 0 0 −24.0000 0 −10.0000
271.1 −1.00000 1.73205i 0 2.00000 3.46410i 2.50000 4.33013i 0 0 −24.0000 0 −10.0000
 $$n$$: 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.e 2
3.b odd 2 1 405.4.e.j 2
9.c even 3 1 135.4.a.d yes 1
9.c even 3 1 inner 405.4.e.e 2
9.d odd 6 1 135.4.a.a 1
9.d odd 6 1 405.4.e.j 2
36.f odd 6 1 2160.4.a.d 1
36.h even 6 1 2160.4.a.n 1
45.h odd 6 1 675.4.a.i 1
45.j even 6 1 675.4.a.b 1
45.k odd 12 2 675.4.b.c 2
45.l even 12 2 675.4.b.d 2

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

## Hecke kernels

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

 $$T_{2}^{2} + 2T_{2} + 4$$ T2^2 + 2*T2 + 4 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 10T + 100$$
$13$ $$T^{2} - 80T + 6400$$
$17$ $$(T + 7)^{2}$$
$19$ $$(T + 113)^{2}$$
$23$ $$T^{2} + 81T + 6561$$
$29$ $$T^{2} + 220T + 48400$$
$31$ $$T^{2} - 189T + 35721$$
$37$ $$(T - 170)^{2}$$
$41$ $$T^{2} + 130T + 16900$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$T^{2} - 160T + 25600$$
$53$ $$(T + 631)^{2}$$
$59$ $$T^{2} + 560T + 313600$$
$61$ $$T^{2} + 229T + 52441$$
$67$ $$T^{2} + 750T + 562500$$
$71$ $$(T + 890)^{2}$$
$73$ $$(T + 890)^{2}$$
$79$ $$T^{2} - 27T + 729$$
$83$ $$T^{2} - 429T + 184041$$
$89$ $$(T - 750)^{2}$$
$97$ $$T^{2} - 1480 T + 2190400$$