# Properties

 Label 441.4.e.c 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} - 21 q^{8} +O(q^{10})$$ q - 3*z * q^2 + (z - 1) * q^4 + 3*z * q^5 - 21 * q^8 $$q - 3 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} - 21 q^{8} + ( - 9 \zeta_{6} + 9) q^{10} + (15 \zeta_{6} - 15) q^{11} + 64 q^{13} + 71 \zeta_{6} q^{16} + (84 \zeta_{6} - 84) q^{17} - 16 \zeta_{6} q^{19} - 3 q^{20} + 45 q^{22} - 84 \zeta_{6} q^{23} + ( - 116 \zeta_{6} + 116) q^{25} - 192 \zeta_{6} q^{26} + 297 q^{29} + (253 \zeta_{6} - 253) q^{31} + ( - 45 \zeta_{6} + 45) q^{32} + 252 q^{34} + 316 \zeta_{6} q^{37} + (48 \zeta_{6} - 48) q^{38} - 63 \zeta_{6} q^{40} + 360 q^{41} + 26 q^{43} - 15 \zeta_{6} q^{44} + (252 \zeta_{6} - 252) q^{46} + 30 \zeta_{6} q^{47} - 348 q^{50} + (64 \zeta_{6} - 64) q^{52} + ( - 363 \zeta_{6} + 363) q^{53} - 45 q^{55} - 891 \zeta_{6} q^{58} + ( - 15 \zeta_{6} + 15) q^{59} - 118 \zeta_{6} q^{61} + 759 q^{62} + 433 q^{64} + 192 \zeta_{6} q^{65} + ( - 370 \zeta_{6} + 370) q^{67} - 84 \zeta_{6} q^{68} + 342 q^{71} + ( - 362 \zeta_{6} + 362) q^{73} + ( - 948 \zeta_{6} + 948) q^{74} + 16 q^{76} - 467 \zeta_{6} q^{79} + (213 \zeta_{6} - 213) q^{80} - 1080 \zeta_{6} q^{82} + 477 q^{83} - 252 q^{85} - 78 \zeta_{6} q^{86} + ( - 315 \zeta_{6} + 315) q^{88} - 906 \zeta_{6} q^{89} + 84 q^{92} + ( - 90 \zeta_{6} + 90) q^{94} + ( - 48 \zeta_{6} + 48) q^{95} - 503 q^{97} +O(q^{100})$$ q - 3*z * q^2 + (z - 1) * q^4 + 3*z * q^5 - 21 * q^8 + (-9*z + 9) * q^10 + (15*z - 15) * q^11 + 64 * q^13 + 71*z * q^16 + (84*z - 84) * q^17 - 16*z * q^19 - 3 * q^20 + 45 * q^22 - 84*z * q^23 + (-116*z + 116) * q^25 - 192*z * q^26 + 297 * q^29 + (253*z - 253) * q^31 + (-45*z + 45) * q^32 + 252 * q^34 + 316*z * q^37 + (48*z - 48) * q^38 - 63*z * q^40 + 360 * q^41 + 26 * q^43 - 15*z * q^44 + (252*z - 252) * q^46 + 30*z * q^47 - 348 * q^50 + (64*z - 64) * q^52 + (-363*z + 363) * q^53 - 45 * q^55 - 891*z * q^58 + (-15*z + 15) * q^59 - 118*z * q^61 + 759 * q^62 + 433 * q^64 + 192*z * q^65 + (-370*z + 370) * q^67 - 84*z * q^68 + 342 * q^71 + (-362*z + 362) * q^73 + (-948*z + 948) * q^74 + 16 * q^76 - 467*z * q^79 + (213*z - 213) * q^80 - 1080*z * q^82 + 477 * q^83 - 252 * q^85 - 78*z * q^86 + (-315*z + 315) * q^88 - 906*z * q^89 + 84 * q^92 + (-90*z + 90) * q^94 + (-48*z + 48) * q^95 - 503 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} + 3 q^{5} - 42 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 - q^4 + 3 * q^5 - 42 * q^8 $$2 q - 3 q^{2} - q^{4} + 3 q^{5} - 42 q^{8} + 9 q^{10} - 15 q^{11} + 128 q^{13} + 71 q^{16} - 84 q^{17} - 16 q^{19} - 6 q^{20} + 90 q^{22} - 84 q^{23} + 116 q^{25} - 192 q^{26} + 594 q^{29} - 253 q^{31} + 45 q^{32} + 504 q^{34} + 316 q^{37} - 48 q^{38} - 63 q^{40} + 720 q^{41} + 52 q^{43} - 15 q^{44} - 252 q^{46} + 30 q^{47} - 696 q^{50} - 64 q^{52} + 363 q^{53} - 90 q^{55} - 891 q^{58} + 15 q^{59} - 118 q^{61} + 1518 q^{62} + 866 q^{64} + 192 q^{65} + 370 q^{67} - 84 q^{68} + 684 q^{71} + 362 q^{73} + 948 q^{74} + 32 q^{76} - 467 q^{79} - 213 q^{80} - 1080 q^{82} + 954 q^{83} - 504 q^{85} - 78 q^{86} + 315 q^{88} - 906 q^{89} + 168 q^{92} + 90 q^{94} + 48 q^{95} - 1006 q^{97}+O(q^{100})$$ 2 * q - 3 * q^2 - q^4 + 3 * q^5 - 42 * q^8 + 9 * q^10 - 15 * q^11 + 128 * q^13 + 71 * q^16 - 84 * q^17 - 16 * q^19 - 6 * q^20 + 90 * q^22 - 84 * q^23 + 116 * q^25 - 192 * q^26 + 594 * q^29 - 253 * q^31 + 45 * q^32 + 504 * q^34 + 316 * q^37 - 48 * q^38 - 63 * q^40 + 720 * q^41 + 52 * q^43 - 15 * q^44 - 252 * q^46 + 30 * q^47 - 696 * q^50 - 64 * q^52 + 363 * q^53 - 90 * q^55 - 891 * q^58 + 15 * q^59 - 118 * q^61 + 1518 * q^62 + 866 * q^64 + 192 * q^65 + 370 * q^67 - 84 * q^68 + 684 * q^71 + 362 * q^73 + 948 * q^74 + 32 * q^76 - 467 * q^79 - 213 * q^80 - 1080 * q^82 + 954 * q^83 - 504 * q^85 - 78 * q^86 + 315 * q^88 - 906 * q^89 + 168 * q^92 + 90 * q^94 + 48 * q^95 - 1006 * 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
−1.50000 + 2.59808i 0 −0.500000 0.866025i 1.50000 2.59808i 0 0 −21.0000 0 4.50000 + 7.79423i
361.1 −1.50000 2.59808i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 0 −21.0000 0 4.50000 7.79423i
 $$n$$: 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.c 2
3.b odd 2 1 147.4.e.h 2
7.b odd 2 1 63.4.e.a 2
7.c even 3 1 441.4.a.k 1
7.c even 3 1 inner 441.4.e.c 2
7.d odd 6 1 63.4.e.a 2
7.d odd 6 1 441.4.a.l 1
21.c even 2 1 21.4.e.a 2
21.g even 6 1 21.4.e.a 2
21.g even 6 1 147.4.a.b 1
21.h odd 6 1 147.4.a.a 1
21.h odd 6 1 147.4.e.h 2
84.h odd 2 1 336.4.q.e 2
84.j odd 6 1 336.4.q.e 2
84.j odd 6 1 2352.4.a.i 1
84.n even 6 1 2352.4.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 21.c even 2 1
21.4.e.a 2 21.g even 6 1
63.4.e.a 2 7.b odd 2 1
63.4.e.a 2 7.d odd 6 1
147.4.a.a 1 21.h odd 6 1
147.4.a.b 1 21.g even 6 1
147.4.e.h 2 3.b odd 2 1
147.4.e.h 2 21.h odd 6 1
336.4.q.e 2 84.h odd 2 1
336.4.q.e 2 84.j odd 6 1
441.4.a.k 1 7.c even 3 1
441.4.a.l 1 7.d odd 6 1
441.4.e.c 2 1.a even 1 1 trivial
441.4.e.c 2 7.c even 3 1 inner
2352.4.a.i 1 84.j odd 6 1
2352.4.a.bd 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 $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + 3T_{2} + 9$$ T2^2 + 3*T2 + 9 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{13} - 64$$ T13 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 15T + 225$$
$13$ $$(T - 64)^{2}$$
$17$ $$T^{2} + 84T + 7056$$
$19$ $$T^{2} + 16T + 256$$
$23$ $$T^{2} + 84T + 7056$$
$29$ $$(T - 297)^{2}$$
$31$ $$T^{2} + 253T + 64009$$
$37$ $$T^{2} - 316T + 99856$$
$41$ $$(T - 360)^{2}$$
$43$ $$(T - 26)^{2}$$
$47$ $$T^{2} - 30T + 900$$
$53$ $$T^{2} - 363T + 131769$$
$59$ $$T^{2} - 15T + 225$$
$61$ $$T^{2} + 118T + 13924$$
$67$ $$T^{2} - 370T + 136900$$
$71$ $$(T - 342)^{2}$$
$73$ $$T^{2} - 362T + 131044$$
$79$ $$T^{2} + 467T + 218089$$
$83$ $$(T - 477)^{2}$$
$89$ $$T^{2} + 906T + 820836$$
$97$ $$(T + 503)^{2}$$