# Properties

 Label 441.4.e.l Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ 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] = [441,4,Mod(226,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$26.0198423125$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 18 \zeta_{6} q^{5} +O(q^{10})$$ q + 4*z * q^2 + (8*z - 8) * q^4 - 18*z * q^5 $$q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 18 \zeta_{6} q^{5} + ( - 72 \zeta_{6} + 72) q^{10} + (50 \zeta_{6} - 50) q^{11} + 36 q^{13} + 64 \zeta_{6} q^{16} + (126 \zeta_{6} - 126) q^{17} - 72 \zeta_{6} q^{19} + 144 q^{20} - 200 q^{22} + 14 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} + 144 \zeta_{6} q^{26} - 158 q^{29} + (36 \zeta_{6} - 36) q^{31} + (256 \zeta_{6} - 256) q^{32} - 504 q^{34} + 162 \zeta_{6} q^{37} + ( - 288 \zeta_{6} + 288) q^{38} - 270 q^{41} - 324 q^{43} - 400 \zeta_{6} q^{44} + (56 \zeta_{6} - 56) q^{46} + 72 \zeta_{6} q^{47} - 796 q^{50} + (288 \zeta_{6} - 288) q^{52} + (22 \zeta_{6} - 22) q^{53} + 900 q^{55} - 632 \zeta_{6} q^{58} + (468 \zeta_{6} - 468) q^{59} + 792 \zeta_{6} q^{61} - 144 q^{62} - 512 q^{64} - 648 \zeta_{6} q^{65} + (232 \zeta_{6} - 232) q^{67} - 1008 \zeta_{6} q^{68} + 734 q^{71} + ( - 180 \zeta_{6} + 180) q^{73} + (648 \zeta_{6} - 648) q^{74} + 576 q^{76} - 236 \zeta_{6} q^{79} + ( - 1152 \zeta_{6} + 1152) q^{80} - 1080 \zeta_{6} q^{82} + 36 q^{83} + 2268 q^{85} - 1296 \zeta_{6} q^{86} - 234 \zeta_{6} q^{89} - 112 q^{92} + (288 \zeta_{6} - 288) q^{94} + (1296 \zeta_{6} - 1296) q^{95} - 468 q^{97} +O(q^{100})$$ q + 4*z * q^2 + (8*z - 8) * q^4 - 18*z * q^5 + (-72*z + 72) * q^10 + (50*z - 50) * q^11 + 36 * q^13 + 64*z * q^16 + (126*z - 126) * q^17 - 72*z * q^19 + 144 * q^20 - 200 * q^22 + 14*z * q^23 + (199*z - 199) * q^25 + 144*z * q^26 - 158 * q^29 + (36*z - 36) * q^31 + (256*z - 256) * q^32 - 504 * q^34 + 162*z * q^37 + (-288*z + 288) * q^38 - 270 * q^41 - 324 * q^43 - 400*z * q^44 + (56*z - 56) * q^46 + 72*z * q^47 - 796 * q^50 + (288*z - 288) * q^52 + (22*z - 22) * q^53 + 900 * q^55 - 632*z * q^58 + (468*z - 468) * q^59 + 792*z * q^61 - 144 * q^62 - 512 * q^64 - 648*z * q^65 + (232*z - 232) * q^67 - 1008*z * q^68 + 734 * q^71 + (-180*z + 180) * q^73 + (648*z - 648) * q^74 + 576 * q^76 - 236*z * q^79 + (-1152*z + 1152) * q^80 - 1080*z * q^82 + 36 * q^83 + 2268 * q^85 - 1296*z * q^86 - 234*z * q^89 - 112 * q^92 + (288*z - 288) * q^94 + (1296*z - 1296) * q^95 - 468 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 8 q^{4} - 18 q^{5}+O(q^{10})$$ 2 * q + 4 * q^2 - 8 * q^4 - 18 * q^5 $$2 q + 4 q^{2} - 8 q^{4} - 18 q^{5} + 72 q^{10} - 50 q^{11} + 72 q^{13} + 64 q^{16} - 126 q^{17} - 72 q^{19} + 288 q^{20} - 400 q^{22} + 14 q^{23} - 199 q^{25} + 144 q^{26} - 316 q^{29} - 36 q^{31} - 256 q^{32} - 1008 q^{34} + 162 q^{37} + 288 q^{38} - 540 q^{41} - 648 q^{43} - 400 q^{44} - 56 q^{46} + 72 q^{47} - 1592 q^{50} - 288 q^{52} - 22 q^{53} + 1800 q^{55} - 632 q^{58} - 468 q^{59} + 792 q^{61} - 288 q^{62} - 1024 q^{64} - 648 q^{65} - 232 q^{67} - 1008 q^{68} + 1468 q^{71} + 180 q^{73} - 648 q^{74} + 1152 q^{76} - 236 q^{79} + 1152 q^{80} - 1080 q^{82} + 72 q^{83} + 4536 q^{85} - 1296 q^{86} - 234 q^{89} - 224 q^{92} - 288 q^{94} - 1296 q^{95} - 936 q^{97}+O(q^{100})$$ 2 * q + 4 * q^2 - 8 * q^4 - 18 * q^5 + 72 * q^10 - 50 * q^11 + 72 * q^13 + 64 * q^16 - 126 * q^17 - 72 * q^19 + 288 * q^20 - 400 * q^22 + 14 * q^23 - 199 * q^25 + 144 * q^26 - 316 * q^29 - 36 * q^31 - 256 * q^32 - 1008 * q^34 + 162 * q^37 + 288 * q^38 - 540 * q^41 - 648 * q^43 - 400 * q^44 - 56 * q^46 + 72 * q^47 - 1592 * q^50 - 288 * q^52 - 22 * q^53 + 1800 * q^55 - 632 * q^58 - 468 * q^59 + 792 * q^61 - 288 * q^62 - 1024 * q^64 - 648 * q^65 - 232 * q^67 - 1008 * q^68 + 1468 * q^71 + 180 * q^73 - 648 * q^74 + 1152 * q^76 - 236 * q^79 + 1152 * q^80 - 1080 * q^82 + 72 * q^83 + 4536 * q^85 - 1296 * q^86 - 234 * q^89 - 224 * q^92 - 288 * q^94 - 1296 * q^95 - 936 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
2.00000 3.46410i 0 −4.00000 6.92820i −9.00000 + 15.5885i 0 0 0 0 36.0000 + 62.3538i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i −9.00000 15.5885i 0 0 0 0 36.0000 62.3538i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.l 2
3.b odd 2 1 147.4.e.d 2
7.b odd 2 1 441.4.e.o 2
7.c even 3 1 441.4.a.c 1
7.c even 3 1 inner 441.4.e.l 2
7.d odd 6 1 441.4.a.a 1
7.d odd 6 1 441.4.e.o 2
21.c even 2 1 147.4.e.a 2
21.g even 6 1 147.4.a.h yes 1
21.g even 6 1 147.4.e.a 2
21.h odd 6 1 147.4.a.f 1
21.h odd 6 1 147.4.e.d 2
84.j odd 6 1 2352.4.a.s 1
84.n even 6 1 2352.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 21.h odd 6 1
147.4.a.h yes 1 21.g even 6 1
147.4.e.a 2 21.c even 2 1
147.4.e.a 2 21.g even 6 1
147.4.e.d 2 3.b odd 2 1
147.4.e.d 2 21.h odd 6 1
441.4.a.a 1 7.d odd 6 1
441.4.a.c 1 7.c even 3 1
441.4.e.l 2 1.a even 1 1 trivial
441.4.e.l 2 7.c even 3 1 inner
441.4.e.o 2 7.b odd 2 1
441.4.e.o 2 7.d odd 6 1
2352.4.a.s 1 84.j odd 6 1
2352.4.a.t 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} - 4T_{2} + 16$$ T2^2 - 4*T2 + 16 $$T_{5}^{2} + 18T_{5} + 324$$ T5^2 + 18*T5 + 324 $$T_{13} - 36$$ T13 - 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 18T + 324$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 50T + 2500$$
$13$ $$(T - 36)^{2}$$
$17$ $$T^{2} + 126T + 15876$$
$19$ $$T^{2} + 72T + 5184$$
$23$ $$T^{2} - 14T + 196$$
$29$ $$(T + 158)^{2}$$
$31$ $$T^{2} + 36T + 1296$$
$37$ $$T^{2} - 162T + 26244$$
$41$ $$(T + 270)^{2}$$
$43$ $$(T + 324)^{2}$$
$47$ $$T^{2} - 72T + 5184$$
$53$ $$T^{2} + 22T + 484$$
$59$ $$T^{2} + 468T + 219024$$
$61$ $$T^{2} - 792T + 627264$$
$67$ $$T^{2} + 232T + 53824$$
$71$ $$(T - 734)^{2}$$
$73$ $$T^{2} - 180T + 32400$$
$79$ $$T^{2} + 236T + 55696$$
$83$ $$(T - 36)^{2}$$
$89$ $$T^{2} + 234T + 54756$$
$97$ $$(T + 468)^{2}$$