# Properties

 Label 441.4.e.f Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 12 \zeta_{6} q^{5} - 15 q^{8} +O(q^{10})$$ q - z * q^2 + (-7*z + 7) * q^4 - 12*z * q^5 - 15 * q^8 $$q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 12 \zeta_{6} q^{5} - 15 q^{8} + (12 \zeta_{6} - 12) q^{10} + ( - 20 \zeta_{6} + 20) q^{11} + 84 q^{13} - 41 \zeta_{6} q^{16} + ( - 96 \zeta_{6} + 96) q^{17} + 12 \zeta_{6} q^{19} - 84 q^{20} - 20 q^{22} - 176 \zeta_{6} q^{23} + (19 \zeta_{6} - 19) q^{25} - 84 \zeta_{6} q^{26} - 58 q^{29} + (264 \zeta_{6} - 264) q^{31} + (161 \zeta_{6} - 161) q^{32} - 96 q^{34} - 258 \zeta_{6} q^{37} + ( - 12 \zeta_{6} + 12) q^{38} + 180 \zeta_{6} q^{40} + 156 q^{43} - 140 \zeta_{6} q^{44} + (176 \zeta_{6} - 176) q^{46} + 408 \zeta_{6} q^{47} + 19 q^{50} + ( - 588 \zeta_{6} + 588) q^{52} + (722 \zeta_{6} - 722) q^{53} - 240 q^{55} + 58 \zeta_{6} q^{58} + (492 \zeta_{6} - 492) q^{59} - 492 \zeta_{6} q^{61} + 264 q^{62} - 167 q^{64} - 1008 \zeta_{6} q^{65} + (412 \zeta_{6} - 412) q^{67} - 672 \zeta_{6} q^{68} - 296 q^{71} + ( - 240 \zeta_{6} + 240) q^{73} + (258 \zeta_{6} - 258) q^{74} + 84 q^{76} - 776 \zeta_{6} q^{79} + (492 \zeta_{6} - 492) q^{80} + 924 q^{83} - 1152 q^{85} - 156 \zeta_{6} q^{86} + (300 \zeta_{6} - 300) q^{88} + 744 \zeta_{6} q^{89} - 1232 q^{92} + ( - 408 \zeta_{6} + 408) q^{94} + ( - 144 \zeta_{6} + 144) q^{95} + 168 q^{97} +O(q^{100})$$ q - z * q^2 + (-7*z + 7) * q^4 - 12*z * q^5 - 15 * q^8 + (12*z - 12) * q^10 + (-20*z + 20) * q^11 + 84 * q^13 - 41*z * q^16 + (-96*z + 96) * q^17 + 12*z * q^19 - 84 * q^20 - 20 * q^22 - 176*z * q^23 + (19*z - 19) * q^25 - 84*z * q^26 - 58 * q^29 + (264*z - 264) * q^31 + (161*z - 161) * q^32 - 96 * q^34 - 258*z * q^37 + (-12*z + 12) * q^38 + 180*z * q^40 + 156 * q^43 - 140*z * q^44 + (176*z - 176) * q^46 + 408*z * q^47 + 19 * q^50 + (-588*z + 588) * q^52 + (722*z - 722) * q^53 - 240 * q^55 + 58*z * q^58 + (492*z - 492) * q^59 - 492*z * q^61 + 264 * q^62 - 167 * q^64 - 1008*z * q^65 + (412*z - 412) * q^67 - 672*z * q^68 - 296 * q^71 + (-240*z + 240) * q^73 + (258*z - 258) * q^74 + 84 * q^76 - 776*z * q^79 + (492*z - 492) * q^80 + 924 * q^83 - 1152 * q^85 - 156*z * q^86 + (300*z - 300) * q^88 + 744*z * q^89 - 1232 * q^92 + (-408*z + 408) * q^94 + (-144*z + 144) * q^95 + 168 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 12 q^{5} - 30 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 12 * q^5 - 30 * q^8 $$2 q - q^{2} + 7 q^{4} - 12 q^{5} - 30 q^{8} - 12 q^{10} + 20 q^{11} + 168 q^{13} - 41 q^{16} + 96 q^{17} + 12 q^{19} - 168 q^{20} - 40 q^{22} - 176 q^{23} - 19 q^{25} - 84 q^{26} - 116 q^{29} - 264 q^{31} - 161 q^{32} - 192 q^{34} - 258 q^{37} + 12 q^{38} + 180 q^{40} + 312 q^{43} - 140 q^{44} - 176 q^{46} + 408 q^{47} + 38 q^{50} + 588 q^{52} - 722 q^{53} - 480 q^{55} + 58 q^{58} - 492 q^{59} - 492 q^{61} + 528 q^{62} - 334 q^{64} - 1008 q^{65} - 412 q^{67} - 672 q^{68} - 592 q^{71} + 240 q^{73} - 258 q^{74} + 168 q^{76} - 776 q^{79} - 492 q^{80} + 1848 q^{83} - 2304 q^{85} - 156 q^{86} - 300 q^{88} + 744 q^{89} - 2464 q^{92} + 408 q^{94} + 144 q^{95} + 336 q^{97}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 12 * q^5 - 30 * q^8 - 12 * q^10 + 20 * q^11 + 168 * q^13 - 41 * q^16 + 96 * q^17 + 12 * q^19 - 168 * q^20 - 40 * q^22 - 176 * q^23 - 19 * q^25 - 84 * q^26 - 116 * q^29 - 264 * q^31 - 161 * q^32 - 192 * q^34 - 258 * q^37 + 12 * q^38 + 180 * q^40 + 312 * q^43 - 140 * q^44 - 176 * q^46 + 408 * q^47 + 38 * q^50 + 588 * q^52 - 722 * q^53 - 480 * q^55 + 58 * q^58 - 492 * q^59 - 492 * q^61 + 528 * q^62 - 334 * q^64 - 1008 * q^65 - 412 * q^67 - 672 * q^68 - 592 * q^71 + 240 * q^73 - 258 * q^74 + 168 * q^76 - 776 * q^79 - 492 * q^80 + 1848 * q^83 - 2304 * q^85 - 156 * q^86 - 300 * q^88 + 744 * q^89 - 2464 * q^92 + 408 * q^94 + 144 * q^95 + 336 * 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
−0.500000 + 0.866025i 0 3.50000 + 6.06218i −6.00000 + 10.3923i 0 0 −15.0000 0 −6.00000 10.3923i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i −6.00000 10.3923i 0 0 −15.0000 0 −6.00000 + 10.3923i
 $$n$$: 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.f 2
3.b odd 2 1 147.4.e.e 2
7.b odd 2 1 441.4.e.g 2
7.c even 3 1 441.4.a.h 1
7.c even 3 1 inner 441.4.e.f 2
7.d odd 6 1 441.4.a.g 1
7.d odd 6 1 441.4.e.g 2
21.c even 2 1 147.4.e.f 2
21.g even 6 1 147.4.a.d 1
21.g even 6 1 147.4.e.f 2
21.h odd 6 1 147.4.a.e yes 1
21.h odd 6 1 147.4.e.e 2
84.j odd 6 1 2352.4.a.bi 1
84.n even 6 1 2352.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 21.g even 6 1
147.4.a.e yes 1 21.h odd 6 1
147.4.e.e 2 3.b odd 2 1
147.4.e.e 2 21.h odd 6 1
147.4.e.f 2 21.c even 2 1
147.4.e.f 2 21.g even 6 1
441.4.a.g 1 7.d odd 6 1
441.4.a.h 1 7.c even 3 1
441.4.e.f 2 1.a even 1 1 trivial
441.4.e.f 2 7.c even 3 1 inner
441.4.e.g 2 7.b odd 2 1
441.4.e.g 2 7.d odd 6 1
2352.4.a.b 1 84.n even 6 1
2352.4.a.bi 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 $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5}^{2} + 12T_{5} + 144$$ T5^2 + 12*T5 + 144 $$T_{13} - 84$$ T13 - 84

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 12T + 144$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 20T + 400$$
$13$ $$(T - 84)^{2}$$
$17$ $$T^{2} - 96T + 9216$$
$19$ $$T^{2} - 12T + 144$$
$23$ $$T^{2} + 176T + 30976$$
$29$ $$(T + 58)^{2}$$
$31$ $$T^{2} + 264T + 69696$$
$37$ $$T^{2} + 258T + 66564$$
$41$ $$T^{2}$$
$43$ $$(T - 156)^{2}$$
$47$ $$T^{2} - 408T + 166464$$
$53$ $$T^{2} + 722T + 521284$$
$59$ $$T^{2} + 492T + 242064$$
$61$ $$T^{2} + 492T + 242064$$
$67$ $$T^{2} + 412T + 169744$$
$71$ $$(T + 296)^{2}$$
$73$ $$T^{2} - 240T + 57600$$
$79$ $$T^{2} + 776T + 602176$$
$83$ $$(T - 924)^{2}$$
$89$ $$T^{2} - 744T + 553536$$
$97$ $$(T - 168)^{2}$$