# Properties

 Label 441.4.e.k 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( - 4 \zeta_{6} + 4) q^{4} - 7 \zeta_{6} q^{5} + 24 q^{8} +O(q^{10})$$ q + 2*z * q^2 + (-4*z + 4) * q^4 - 7*z * q^5 + 24 * q^8 $$q + 2 \zeta_{6} q^{2} + ( - 4 \zeta_{6} + 4) q^{4} - 7 \zeta_{6} q^{5} + 24 q^{8} + ( - 14 \zeta_{6} + 14) q^{10} + (5 \zeta_{6} - 5) q^{11} + 14 q^{13} + 16 \zeta_{6} q^{16} + ( - 21 \zeta_{6} + 21) q^{17} + 49 \zeta_{6} q^{19} - 28 q^{20} - 10 q^{22} - 159 \zeta_{6} q^{23} + ( - 76 \zeta_{6} + 76) q^{25} + 28 \zeta_{6} q^{26} - 58 q^{29} + ( - 147 \zeta_{6} + 147) q^{31} + ( - 160 \zeta_{6} + 160) q^{32} + 42 q^{34} - 219 \zeta_{6} q^{37} + (98 \zeta_{6} - 98) q^{38} - 168 \zeta_{6} q^{40} + 350 q^{41} - 124 q^{43} + 20 \zeta_{6} q^{44} + ( - 318 \zeta_{6} + 318) q^{46} - 525 \zeta_{6} q^{47} + 152 q^{50} + ( - 56 \zeta_{6} + 56) q^{52} + ( - 303 \zeta_{6} + 303) q^{53} + 35 q^{55} - 116 \zeta_{6} q^{58} + ( - 105 \zeta_{6} + 105) q^{59} - 413 \zeta_{6} q^{61} + 294 q^{62} + 448 q^{64} - 98 \zeta_{6} q^{65} + (415 \zeta_{6} - 415) q^{67} - 84 \zeta_{6} q^{68} + 432 q^{71} + (1113 \zeta_{6} - 1113) q^{73} + ( - 438 \zeta_{6} + 438) q^{74} + 196 q^{76} + 103 \zeta_{6} q^{79} + ( - 112 \zeta_{6} + 112) q^{80} + 700 \zeta_{6} q^{82} + 1092 q^{83} - 147 q^{85} - 248 \zeta_{6} q^{86} + (120 \zeta_{6} - 120) q^{88} + 329 \zeta_{6} q^{89} - 636 q^{92} + ( - 1050 \zeta_{6} + 1050) q^{94} + ( - 343 \zeta_{6} + 343) q^{95} + 882 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (-4*z + 4) * q^4 - 7*z * q^5 + 24 * q^8 + (-14*z + 14) * q^10 + (5*z - 5) * q^11 + 14 * q^13 + 16*z * q^16 + (-21*z + 21) * q^17 + 49*z * q^19 - 28 * q^20 - 10 * q^22 - 159*z * q^23 + (-76*z + 76) * q^25 + 28*z * q^26 - 58 * q^29 + (-147*z + 147) * q^31 + (-160*z + 160) * q^32 + 42 * q^34 - 219*z * q^37 + (98*z - 98) * q^38 - 168*z * q^40 + 350 * q^41 - 124 * q^43 + 20*z * q^44 + (-318*z + 318) * q^46 - 525*z * q^47 + 152 * q^50 + (-56*z + 56) * q^52 + (-303*z + 303) * q^53 + 35 * q^55 - 116*z * q^58 + (-105*z + 105) * q^59 - 413*z * q^61 + 294 * q^62 + 448 * q^64 - 98*z * q^65 + (415*z - 415) * q^67 - 84*z * q^68 + 432 * q^71 + (1113*z - 1113) * q^73 + (-438*z + 438) * q^74 + 196 * q^76 + 103*z * q^79 + (-112*z + 112) * q^80 + 700*z * q^82 + 1092 * q^83 - 147 * q^85 - 248*z * q^86 + (120*z - 120) * q^88 + 329*z * q^89 - 636 * q^92 + (-1050*z + 1050) * q^94 + (-343*z + 343) * q^95 + 882 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{4} - 7 q^{5} + 48 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^4 - 7 * q^5 + 48 * q^8 $$2 q + 2 q^{2} + 4 q^{4} - 7 q^{5} + 48 q^{8} + 14 q^{10} - 5 q^{11} + 28 q^{13} + 16 q^{16} + 21 q^{17} + 49 q^{19} - 56 q^{20} - 20 q^{22} - 159 q^{23} + 76 q^{25} + 28 q^{26} - 116 q^{29} + 147 q^{31} + 160 q^{32} + 84 q^{34} - 219 q^{37} - 98 q^{38} - 168 q^{40} + 700 q^{41} - 248 q^{43} + 20 q^{44} + 318 q^{46} - 525 q^{47} + 304 q^{50} + 56 q^{52} + 303 q^{53} + 70 q^{55} - 116 q^{58} + 105 q^{59} - 413 q^{61} + 588 q^{62} + 896 q^{64} - 98 q^{65} - 415 q^{67} - 84 q^{68} + 864 q^{71} - 1113 q^{73} + 438 q^{74} + 392 q^{76} + 103 q^{79} + 112 q^{80} + 700 q^{82} + 2184 q^{83} - 294 q^{85} - 248 q^{86} - 120 q^{88} + 329 q^{89} - 1272 q^{92} + 1050 q^{94} + 343 q^{95} + 1764 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^4 - 7 * q^5 + 48 * q^8 + 14 * q^10 - 5 * q^11 + 28 * q^13 + 16 * q^16 + 21 * q^17 + 49 * q^19 - 56 * q^20 - 20 * q^22 - 159 * q^23 + 76 * q^25 + 28 * q^26 - 116 * q^29 + 147 * q^31 + 160 * q^32 + 84 * q^34 - 219 * q^37 - 98 * q^38 - 168 * q^40 + 700 * q^41 - 248 * q^43 + 20 * q^44 + 318 * q^46 - 525 * q^47 + 304 * q^50 + 56 * q^52 + 303 * q^53 + 70 * q^55 - 116 * q^58 + 105 * q^59 - 413 * q^61 + 588 * q^62 + 896 * q^64 - 98 * q^65 - 415 * q^67 - 84 * q^68 + 864 * q^71 - 1113 * q^73 + 438 * q^74 + 392 * q^76 + 103 * q^79 + 112 * q^80 + 700 * q^82 + 2184 * q^83 - 294 * q^85 - 248 * q^86 - 120 * q^88 + 329 * q^89 - 1272 * q^92 + 1050 * q^94 + 343 * q^95 + 1764 * 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
1.00000 1.73205i 0 2.00000 + 3.46410i −3.50000 + 6.06218i 0 0 24.0000 0 7.00000 + 12.1244i
361.1 1.00000 + 1.73205i 0 2.00000 3.46410i −3.50000 6.06218i 0 0 24.0000 0 7.00000 12.1244i
 $$n$$: 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.k 2
3.b odd 2 1 49.4.c.a 2
7.b odd 2 1 63.4.e.b 2
7.c even 3 1 441.4.a.e 1
7.c even 3 1 inner 441.4.e.k 2
7.d odd 6 1 63.4.e.b 2
7.d odd 6 1 441.4.a.d 1
21.c even 2 1 7.4.c.a 2
21.g even 6 1 7.4.c.a 2
21.g even 6 1 49.4.a.d 1
21.h odd 6 1 49.4.a.c 1
21.h odd 6 1 49.4.c.a 2
84.h odd 2 1 112.4.i.c 2
84.j odd 6 1 112.4.i.c 2
84.j odd 6 1 784.4.a.b 1
84.n even 6 1 784.4.a.r 1
105.g even 2 1 175.4.e.a 2
105.k odd 4 2 175.4.k.a 4
105.o odd 6 1 1225.4.a.d 1
105.p even 6 1 175.4.e.a 2
105.p even 6 1 1225.4.a.c 1
105.w odd 12 2 175.4.k.a 4
168.e odd 2 1 448.4.i.a 2
168.i even 2 1 448.4.i.f 2
168.ba even 6 1 448.4.i.f 2
168.be odd 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 21.c even 2 1
7.4.c.a 2 21.g even 6 1
49.4.a.c 1 21.h odd 6 1
49.4.a.d 1 21.g even 6 1
49.4.c.a 2 3.b odd 2 1
49.4.c.a 2 21.h odd 6 1
63.4.e.b 2 7.b odd 2 1
63.4.e.b 2 7.d odd 6 1
112.4.i.c 2 84.h odd 2 1
112.4.i.c 2 84.j odd 6 1
175.4.e.a 2 105.g even 2 1
175.4.e.a 2 105.p even 6 1
175.4.k.a 4 105.k odd 4 2
175.4.k.a 4 105.w odd 12 2
441.4.a.d 1 7.d odd 6 1
441.4.a.e 1 7.c even 3 1
441.4.e.k 2 1.a even 1 1 trivial
441.4.e.k 2 7.c even 3 1 inner
448.4.i.a 2 168.e odd 2 1
448.4.i.a 2 168.be odd 6 1
448.4.i.f 2 168.i even 2 1
448.4.i.f 2 168.ba even 6 1
784.4.a.b 1 84.j odd 6 1
784.4.a.r 1 84.n even 6 1
1225.4.a.c 1 105.p even 6 1
1225.4.a.d 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{5}^{2} + 7T_{5} + 49$$ T5^2 + 7*T5 + 49 $$T_{13} - 14$$ T13 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 7T + 49$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$(T - 14)^{2}$$
$17$ $$T^{2} - 21T + 441$$
$19$ $$T^{2} - 49T + 2401$$
$23$ $$T^{2} + 159T + 25281$$
$29$ $$(T + 58)^{2}$$
$31$ $$T^{2} - 147T + 21609$$
$37$ $$T^{2} + 219T + 47961$$
$41$ $$(T - 350)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} + 525T + 275625$$
$53$ $$T^{2} - 303T + 91809$$
$59$ $$T^{2} - 105T + 11025$$
$61$ $$T^{2} + 413T + 170569$$
$67$ $$T^{2} + 415T + 172225$$
$71$ $$(T - 432)^{2}$$
$73$ $$T^{2} + 1113 T + 1238769$$
$79$ $$T^{2} - 103T + 10609$$
$83$ $$(T - 1092)^{2}$$
$89$ $$T^{2} - 329T + 108241$$
$97$ $$(T - 882)^{2}$$