# Properties

 Label 441.4.e.m 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 21) 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} +O(q^{10})$$ q + 4*z * q^2 + (8*z - 8) * q^4 - 4*z * q^5 $$q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{10} + ( - 62 \zeta_{6} + 62) q^{11} - 62 q^{13} + 64 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} - 100 \zeta_{6} q^{19} + 32 q^{20} + 248 q^{22} - 42 \zeta_{6} q^{23} + ( - 109 \zeta_{6} + 109) q^{25} - 248 \zeta_{6} q^{26} + 10 q^{29} + ( - 48 \zeta_{6} + 48) q^{31} + (256 \zeta_{6} - 256) q^{32} + 336 q^{34} + 246 \zeta_{6} q^{37} + ( - 400 \zeta_{6} + 400) q^{38} + 248 q^{41} + 68 q^{43} + 496 \zeta_{6} q^{44} + ( - 168 \zeta_{6} + 168) q^{46} + 324 \zeta_{6} q^{47} + 436 q^{50} + ( - 496 \zeta_{6} + 496) q^{52} + ( - 258 \zeta_{6} + 258) q^{53} - 248 q^{55} + 40 \zeta_{6} q^{58} + ( - 120 \zeta_{6} + 120) q^{59} - 622 \zeta_{6} q^{61} + 192 q^{62} - 512 q^{64} + 248 \zeta_{6} q^{65} + (904 \zeta_{6} - 904) q^{67} + 672 \zeta_{6} q^{68} + 678 q^{71} + ( - 642 \zeta_{6} + 642) q^{73} + (984 \zeta_{6} - 984) q^{74} + 800 q^{76} - 740 \zeta_{6} q^{79} + ( - 256 \zeta_{6} + 256) q^{80} + 992 \zeta_{6} q^{82} - 468 q^{83} - 336 q^{85} + 272 \zeta_{6} q^{86} + 200 \zeta_{6} q^{89} + 336 q^{92} + (1296 \zeta_{6} - 1296) q^{94} + (400 \zeta_{6} - 400) q^{95} - 1266 q^{97} +O(q^{100})$$ q + 4*z * q^2 + (8*z - 8) * q^4 - 4*z * q^5 + (-16*z + 16) * q^10 + (-62*z + 62) * q^11 - 62 * q^13 + 64*z * q^16 + (-84*z + 84) * q^17 - 100*z * q^19 + 32 * q^20 + 248 * q^22 - 42*z * q^23 + (-109*z + 109) * q^25 - 248*z * q^26 + 10 * q^29 + (-48*z + 48) * q^31 + (256*z - 256) * q^32 + 336 * q^34 + 246*z * q^37 + (-400*z + 400) * q^38 + 248 * q^41 + 68 * q^43 + 496*z * q^44 + (-168*z + 168) * q^46 + 324*z * q^47 + 436 * q^50 + (-496*z + 496) * q^52 + (-258*z + 258) * q^53 - 248 * q^55 + 40*z * q^58 + (-120*z + 120) * q^59 - 622*z * q^61 + 192 * q^62 - 512 * q^64 + 248*z * q^65 + (904*z - 904) * q^67 + 672*z * q^68 + 678 * q^71 + (-642*z + 642) * q^73 + (984*z - 984) * q^74 + 800 * q^76 - 740*z * q^79 + (-256*z + 256) * q^80 + 992*z * q^82 - 468 * q^83 - 336 * q^85 + 272*z * q^86 + 200*z * q^89 + 336 * q^92 + (1296*z - 1296) * q^94 + (400*z - 400) * q^95 - 1266 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 8 q^{4} - 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^2 - 8 * q^4 - 4 * q^5 $$2 q + 4 q^{2} - 8 q^{4} - 4 q^{5} + 16 q^{10} + 62 q^{11} - 124 q^{13} + 64 q^{16} + 84 q^{17} - 100 q^{19} + 64 q^{20} + 496 q^{22} - 42 q^{23} + 109 q^{25} - 248 q^{26} + 20 q^{29} + 48 q^{31} - 256 q^{32} + 672 q^{34} + 246 q^{37} + 400 q^{38} + 496 q^{41} + 136 q^{43} + 496 q^{44} + 168 q^{46} + 324 q^{47} + 872 q^{50} + 496 q^{52} + 258 q^{53} - 496 q^{55} + 40 q^{58} + 120 q^{59} - 622 q^{61} + 384 q^{62} - 1024 q^{64} + 248 q^{65} - 904 q^{67} + 672 q^{68} + 1356 q^{71} + 642 q^{73} - 984 q^{74} + 1600 q^{76} - 740 q^{79} + 256 q^{80} + 992 q^{82} - 936 q^{83} - 672 q^{85} + 272 q^{86} + 200 q^{89} + 672 q^{92} - 1296 q^{94} - 400 q^{95} - 2532 q^{97}+O(q^{100})$$ 2 * q + 4 * q^2 - 8 * q^4 - 4 * q^5 + 16 * q^10 + 62 * q^11 - 124 * q^13 + 64 * q^16 + 84 * q^17 - 100 * q^19 + 64 * q^20 + 496 * q^22 - 42 * q^23 + 109 * q^25 - 248 * q^26 + 20 * q^29 + 48 * q^31 - 256 * q^32 + 672 * q^34 + 246 * q^37 + 400 * q^38 + 496 * q^41 + 136 * q^43 + 496 * q^44 + 168 * q^46 + 324 * q^47 + 872 * q^50 + 496 * q^52 + 258 * q^53 - 496 * q^55 + 40 * q^58 + 120 * q^59 - 622 * q^61 + 384 * q^62 - 1024 * q^64 + 248 * q^65 - 904 * q^67 + 672 * q^68 + 1356 * q^71 + 642 * q^73 - 984 * q^74 + 1600 * q^76 - 740 * q^79 + 256 * q^80 + 992 * q^82 - 936 * q^83 - 672 * q^85 + 272 * q^86 + 200 * q^89 + 672 * q^92 - 1296 * q^94 - 400 * q^95 - 2532 * 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 −2.00000 + 3.46410i 0 0 0 0 8.00000 + 13.8564i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i −2.00000 3.46410i 0 0 0 0 8.00000 13.8564i
 $$n$$: 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.m 2
3.b odd 2 1 147.4.e.c 2
7.b odd 2 1 441.4.e.n 2
7.c even 3 1 63.4.a.a 1
7.c even 3 1 inner 441.4.e.m 2
7.d odd 6 1 441.4.a.b 1
7.d odd 6 1 441.4.e.n 2
21.c even 2 1 147.4.e.b 2
21.g even 6 1 147.4.a.g 1
21.g even 6 1 147.4.e.b 2
21.h odd 6 1 21.4.a.b 1
21.h odd 6 1 147.4.e.c 2
28.g odd 6 1 1008.4.a.m 1
35.j even 6 1 1575.4.a.k 1
84.j odd 6 1 2352.4.a.l 1
84.n even 6 1 336.4.a.h 1
105.o odd 6 1 525.4.a.b 1
105.x even 12 2 525.4.d.b 2
168.s odd 6 1 1344.4.a.w 1
168.v even 6 1 1344.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 21.h odd 6 1
63.4.a.a 1 7.c even 3 1
147.4.a.g 1 21.g even 6 1
147.4.e.b 2 21.c even 2 1
147.4.e.b 2 21.g even 6 1
147.4.e.c 2 3.b odd 2 1
147.4.e.c 2 21.h odd 6 1
336.4.a.h 1 84.n even 6 1
441.4.a.b 1 7.d odd 6 1
441.4.e.m 2 1.a even 1 1 trivial
441.4.e.m 2 7.c even 3 1 inner
441.4.e.n 2 7.b odd 2 1
441.4.e.n 2 7.d odd 6 1
525.4.a.b 1 105.o odd 6 1
525.4.d.b 2 105.x even 12 2
1008.4.a.m 1 28.g odd 6 1
1344.4.a.i 1 168.v even 6 1
1344.4.a.w 1 168.s odd 6 1
1575.4.a.k 1 35.j even 6 1
2352.4.a.l 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 4T_{2} + 16$$ T2^2 - 4*T2 + 16 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16 $$T_{13} + 62$$ T13 + 62

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 62T + 3844$$
$13$ $$(T + 62)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} + 100T + 10000$$
$23$ $$T^{2} + 42T + 1764$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} - 48T + 2304$$
$37$ $$T^{2} - 246T + 60516$$
$41$ $$(T - 248)^{2}$$
$43$ $$(T - 68)^{2}$$
$47$ $$T^{2} - 324T + 104976$$
$53$ $$T^{2} - 258T + 66564$$
$59$ $$T^{2} - 120T + 14400$$
$61$ $$T^{2} + 622T + 386884$$
$67$ $$T^{2} + 904T + 817216$$
$71$ $$(T - 678)^{2}$$
$73$ $$T^{2} - 642T + 412164$$
$79$ $$T^{2} + 740T + 547600$$
$83$ $$(T + 468)^{2}$$
$89$ $$T^{2} - 200T + 40000$$
$97$ $$(T + 1266)^{2}$$