# Properties

 Label 405.4.e.l 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 5) 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.

 $$f(q)$$ $$=$$ $$q + ( - 4 \zeta_{6} + 4) q^{2} - 8 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (6 \zeta_{6} - 6) q^{7} +O(q^{10})$$ q + (-4*z + 4) * q^2 - 8*z * q^4 + 5*z * q^5 + (6*z - 6) * q^7 $$q + ( - 4 \zeta_{6} + 4) q^{2} - 8 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (6 \zeta_{6} - 6) q^{7} + 20 q^{10} + (32 \zeta_{6} - 32) q^{11} + 38 \zeta_{6} q^{13} + 24 \zeta_{6} q^{14} + ( - 64 \zeta_{6} + 64) q^{16} + 26 q^{17} + 100 q^{19} + ( - 40 \zeta_{6} + 40) q^{20} + 128 \zeta_{6} q^{22} + 78 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 152 q^{26} + 48 q^{28} + ( - 50 \zeta_{6} + 50) q^{29} + 108 \zeta_{6} q^{31} - 256 \zeta_{6} q^{32} + ( - 104 \zeta_{6} + 104) q^{34} - 30 q^{35} + 266 q^{37} + ( - 400 \zeta_{6} + 400) q^{38} - 22 \zeta_{6} q^{41} + (442 \zeta_{6} - 442) q^{43} + 256 q^{44} + 312 q^{46} + ( - 514 \zeta_{6} + 514) q^{47} + 307 \zeta_{6} q^{49} + 100 \zeta_{6} q^{50} + ( - 304 \zeta_{6} + 304) q^{52} + 2 q^{53} - 160 q^{55} - 200 \zeta_{6} q^{58} - 500 \zeta_{6} q^{59} + ( - 518 \zeta_{6} + 518) q^{61} + 432 q^{62} - 512 q^{64} + (190 \zeta_{6} - 190) q^{65} - 126 \zeta_{6} q^{67} - 208 \zeta_{6} q^{68} + (120 \zeta_{6} - 120) q^{70} + 412 q^{71} - 878 q^{73} + ( - 1064 \zeta_{6} + 1064) q^{74} - 800 \zeta_{6} q^{76} - 192 \zeta_{6} q^{77} + (600 \zeta_{6} - 600) q^{79} + 320 q^{80} - 88 q^{82} + (282 \zeta_{6} - 282) q^{83} + 130 \zeta_{6} q^{85} + 1768 \zeta_{6} q^{86} - 150 q^{89} - 228 q^{91} + ( - 624 \zeta_{6} + 624) q^{92} - 2056 \zeta_{6} q^{94} + 500 \zeta_{6} q^{95} + (386 \zeta_{6} - 386) q^{97} + 1228 q^{98} +O(q^{100})$$ q + (-4*z + 4) * q^2 - 8*z * q^4 + 5*z * q^5 + (6*z - 6) * q^7 + 20 * q^10 + (32*z - 32) * q^11 + 38*z * q^13 + 24*z * q^14 + (-64*z + 64) * q^16 + 26 * q^17 + 100 * q^19 + (-40*z + 40) * q^20 + 128*z * q^22 + 78*z * q^23 + (25*z - 25) * q^25 + 152 * q^26 + 48 * q^28 + (-50*z + 50) * q^29 + 108*z * q^31 - 256*z * q^32 + (-104*z + 104) * q^34 - 30 * q^35 + 266 * q^37 + (-400*z + 400) * q^38 - 22*z * q^41 + (442*z - 442) * q^43 + 256 * q^44 + 312 * q^46 + (-514*z + 514) * q^47 + 307*z * q^49 + 100*z * q^50 + (-304*z + 304) * q^52 + 2 * q^53 - 160 * q^55 - 200*z * q^58 - 500*z * q^59 + (-518*z + 518) * q^61 + 432 * q^62 - 512 * q^64 + (190*z - 190) * q^65 - 126*z * q^67 - 208*z * q^68 + (120*z - 120) * q^70 + 412 * q^71 - 878 * q^73 + (-1064*z + 1064) * q^74 - 800*z * q^76 - 192*z * q^77 + (600*z - 600) * q^79 + 320 * q^80 - 88 * q^82 + (282*z - 282) * q^83 + 130*z * q^85 + 1768*z * q^86 - 150 * q^89 - 228 * q^91 + (-624*z + 624) * q^92 - 2056*z * q^94 + 500*z * q^95 + (386*z - 386) * q^97 + 1228 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7}+O(q^{10})$$ 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7 $$2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7} + 40 q^{10} - 32 q^{11} + 38 q^{13} + 24 q^{14} + 64 q^{16} + 52 q^{17} + 200 q^{19} + 40 q^{20} + 128 q^{22} + 78 q^{23} - 25 q^{25} + 304 q^{26} + 96 q^{28} + 50 q^{29} + 108 q^{31} - 256 q^{32} + 104 q^{34} - 60 q^{35} + 532 q^{37} + 400 q^{38} - 22 q^{41} - 442 q^{43} + 512 q^{44} + 624 q^{46} + 514 q^{47} + 307 q^{49} + 100 q^{50} + 304 q^{52} + 4 q^{53} - 320 q^{55} - 200 q^{58} - 500 q^{59} + 518 q^{61} + 864 q^{62} - 1024 q^{64} - 190 q^{65} - 126 q^{67} - 208 q^{68} - 120 q^{70} + 824 q^{71} - 1756 q^{73} + 1064 q^{74} - 800 q^{76} - 192 q^{77} - 600 q^{79} + 640 q^{80} - 176 q^{82} - 282 q^{83} + 130 q^{85} + 1768 q^{86} - 300 q^{89} - 456 q^{91} + 624 q^{92} - 2056 q^{94} + 500 q^{95} - 386 q^{97} + 2456 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7 + 40 * q^10 - 32 * q^11 + 38 * q^13 + 24 * q^14 + 64 * q^16 + 52 * q^17 + 200 * q^19 + 40 * q^20 + 128 * q^22 + 78 * q^23 - 25 * q^25 + 304 * q^26 + 96 * q^28 + 50 * q^29 + 108 * q^31 - 256 * q^32 + 104 * q^34 - 60 * q^35 + 532 * q^37 + 400 * q^38 - 22 * q^41 - 442 * q^43 + 512 * q^44 + 624 * q^46 + 514 * q^47 + 307 * q^49 + 100 * q^50 + 304 * q^52 + 4 * q^53 - 320 * q^55 - 200 * q^58 - 500 * q^59 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 - 126 * q^67 - 208 * q^68 - 120 * q^70 + 824 * q^71 - 1756 * q^73 + 1064 * q^74 - 800 * q^76 - 192 * q^77 - 600 * q^79 + 640 * q^80 - 176 * q^82 - 282 * q^83 + 130 * q^85 + 1768 * q^86 - 300 * q^89 - 456 * q^91 + 624 * q^92 - 2056 * q^94 + 500 * q^95 - 386 * q^97 + 2456 * 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
2.00000 3.46410i 0 −4.00000 6.92820i 2.50000 + 4.33013i 0 −3.00000 + 5.19615i 0 0 20.0000
271.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i 2.50000 4.33013i 0 −3.00000 5.19615i 0 0 20.0000
 $$n$$: 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.l 2
3.b odd 2 1 405.4.e.c 2
9.c even 3 1 5.4.a.a 1
9.c even 3 1 inner 405.4.e.l 2
9.d odd 6 1 45.4.a.d 1
9.d odd 6 1 405.4.e.c 2
36.f odd 6 1 80.4.a.d 1
36.h even 6 1 720.4.a.u 1
45.h odd 6 1 225.4.a.b 1
45.j even 6 1 25.4.a.c 1
45.k odd 12 2 25.4.b.a 2
45.l even 12 2 225.4.b.c 2
63.g even 3 1 245.4.e.f 2
63.h even 3 1 245.4.e.f 2
63.k odd 6 1 245.4.e.g 2
63.l odd 6 1 245.4.a.a 1
63.o even 6 1 2205.4.a.q 1
63.t odd 6 1 245.4.e.g 2
72.n even 6 1 320.4.a.g 1
72.p odd 6 1 320.4.a.h 1
99.h odd 6 1 605.4.a.d 1
117.t even 6 1 845.4.a.b 1
144.v odd 12 2 1280.4.d.l 2
144.x even 12 2 1280.4.d.e 2
153.h even 6 1 1445.4.a.a 1
171.o odd 6 1 1805.4.a.h 1
180.p odd 6 1 400.4.a.m 1
180.x even 12 2 400.4.c.k 2
315.bg odd 6 1 1225.4.a.k 1
360.z odd 6 1 1600.4.a.s 1
360.bk even 6 1 1600.4.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 9.c even 3 1
25.4.a.c 1 45.j even 6 1
25.4.b.a 2 45.k odd 12 2
45.4.a.d 1 9.d odd 6 1
80.4.a.d 1 36.f odd 6 1
225.4.a.b 1 45.h odd 6 1
225.4.b.c 2 45.l even 12 2
245.4.a.a 1 63.l odd 6 1
245.4.e.f 2 63.g even 3 1
245.4.e.f 2 63.h even 3 1
245.4.e.g 2 63.k odd 6 1
245.4.e.g 2 63.t odd 6 1
320.4.a.g 1 72.n even 6 1
320.4.a.h 1 72.p odd 6 1
400.4.a.m 1 180.p odd 6 1
400.4.c.k 2 180.x even 12 2
405.4.e.c 2 3.b odd 2 1
405.4.e.c 2 9.d odd 6 1
405.4.e.l 2 1.a even 1 1 trivial
405.4.e.l 2 9.c even 3 1 inner
605.4.a.d 1 99.h odd 6 1
720.4.a.u 1 36.h even 6 1
845.4.a.b 1 117.t even 6 1
1225.4.a.k 1 315.bg odd 6 1
1280.4.d.e 2 144.x even 12 2
1280.4.d.l 2 144.v odd 12 2
1445.4.a.a 1 153.h even 6 1
1600.4.a.s 1 360.z odd 6 1
1600.4.a.bi 1 360.bk even 6 1
1805.4.a.h 1 171.o odd 6 1
2205.4.a.q 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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - 4T_{2} + 16$$ T2^2 - 4*T2 + 16 $$T_{7}^{2} + 6T_{7} + 36$$ T7^2 + 6*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} + 6T + 36$$
$11$ $$T^{2} + 32T + 1024$$
$13$ $$T^{2} - 38T + 1444$$
$17$ $$(T - 26)^{2}$$
$19$ $$(T - 100)^{2}$$
$23$ $$T^{2} - 78T + 6084$$
$29$ $$T^{2} - 50T + 2500$$
$31$ $$T^{2} - 108T + 11664$$
$37$ $$(T - 266)^{2}$$
$41$ $$T^{2} + 22T + 484$$
$43$ $$T^{2} + 442T + 195364$$
$47$ $$T^{2} - 514T + 264196$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} + 500T + 250000$$
$61$ $$T^{2} - 518T + 268324$$
$67$ $$T^{2} + 126T + 15876$$
$71$ $$(T - 412)^{2}$$
$73$ $$(T + 878)^{2}$$
$79$ $$T^{2} + 600T + 360000$$
$83$ $$T^{2} + 282T + 79524$$
$89$ $$(T + 150)^{2}$$
$97$ $$T^{2} + 386T + 148996$$