# Properties

 Label 441.4.e.e 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} - 15 q^{8} +O(q^{10})$$ q - z * q^2 + (-7*z + 7) * q^4 - 16*z * q^5 - 15 * q^8 $$q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} - 15 q^{8} + (16 \zeta_{6} - 16) q^{10} + (8 \zeta_{6} - 8) q^{11} - 28 q^{13} - 41 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} - 110 \zeta_{6} q^{19} - 112 q^{20} + 8 q^{22} + 48 \zeta_{6} q^{23} + (131 \zeta_{6} - 131) q^{25} + 28 \zeta_{6} q^{26} + 110 q^{29} + ( - 12 \zeta_{6} + 12) q^{31} + (161 \zeta_{6} - 161) q^{32} + 54 q^{34} + 246 \zeta_{6} q^{37} + (110 \zeta_{6} - 110) q^{38} + 240 \zeta_{6} q^{40} + 182 q^{41} + 128 q^{43} + 56 \zeta_{6} q^{44} + ( - 48 \zeta_{6} + 48) q^{46} - 324 \zeta_{6} q^{47} + 131 q^{50} + (196 \zeta_{6} - 196) q^{52} + (162 \zeta_{6} - 162) q^{53} + 128 q^{55} - 110 \zeta_{6} q^{58} + (810 \zeta_{6} - 810) q^{59} - 488 \zeta_{6} q^{61} - 12 q^{62} - 167 q^{64} + 448 \zeta_{6} q^{65} + (244 \zeta_{6} - 244) q^{67} + 378 \zeta_{6} q^{68} + 768 q^{71} + (702 \zeta_{6} - 702) q^{73} + ( - 246 \zeta_{6} + 246) q^{74} - 770 q^{76} - 440 \zeta_{6} q^{79} + (656 \zeta_{6} - 656) q^{80} - 182 \zeta_{6} q^{82} - 1302 q^{83} + 864 q^{85} - 128 \zeta_{6} q^{86} + ( - 120 \zeta_{6} + 120) q^{88} - 730 \zeta_{6} q^{89} + 336 q^{92} + (324 \zeta_{6} - 324) q^{94} + (1760 \zeta_{6} - 1760) q^{95} - 294 q^{97} +O(q^{100})$$ q - z * q^2 + (-7*z + 7) * q^4 - 16*z * q^5 - 15 * q^8 + (16*z - 16) * q^10 + (8*z - 8) * q^11 - 28 * q^13 - 41*z * q^16 + (54*z - 54) * q^17 - 110*z * q^19 - 112 * q^20 + 8 * q^22 + 48*z * q^23 + (131*z - 131) * q^25 + 28*z * q^26 + 110 * q^29 + (-12*z + 12) * q^31 + (161*z - 161) * q^32 + 54 * q^34 + 246*z * q^37 + (110*z - 110) * q^38 + 240*z * q^40 + 182 * q^41 + 128 * q^43 + 56*z * q^44 + (-48*z + 48) * q^46 - 324*z * q^47 + 131 * q^50 + (196*z - 196) * q^52 + (162*z - 162) * q^53 + 128 * q^55 - 110*z * q^58 + (810*z - 810) * q^59 - 488*z * q^61 - 12 * q^62 - 167 * q^64 + 448*z * q^65 + (244*z - 244) * q^67 + 378*z * q^68 + 768 * q^71 + (702*z - 702) * q^73 + (-246*z + 246) * q^74 - 770 * q^76 - 440*z * q^79 + (656*z - 656) * q^80 - 182*z * q^82 - 1302 * q^83 + 864 * q^85 - 128*z * q^86 + (-120*z + 120) * q^88 - 730*z * q^89 + 336 * q^92 + (324*z - 324) * q^94 + (1760*z - 1760) * q^95 - 294 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 16 q^{5} - 30 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 16 * q^5 - 30 * q^8 $$2 q - q^{2} + 7 q^{4} - 16 q^{5} - 30 q^{8} - 16 q^{10} - 8 q^{11} - 56 q^{13} - 41 q^{16} - 54 q^{17} - 110 q^{19} - 224 q^{20} + 16 q^{22} + 48 q^{23} - 131 q^{25} + 28 q^{26} + 220 q^{29} + 12 q^{31} - 161 q^{32} + 108 q^{34} + 246 q^{37} - 110 q^{38} + 240 q^{40} + 364 q^{41} + 256 q^{43} + 56 q^{44} + 48 q^{46} - 324 q^{47} + 262 q^{50} - 196 q^{52} - 162 q^{53} + 256 q^{55} - 110 q^{58} - 810 q^{59} - 488 q^{61} - 24 q^{62} - 334 q^{64} + 448 q^{65} - 244 q^{67} + 378 q^{68} + 1536 q^{71} - 702 q^{73} + 246 q^{74} - 1540 q^{76} - 440 q^{79} - 656 q^{80} - 182 q^{82} - 2604 q^{83} + 1728 q^{85} - 128 q^{86} + 120 q^{88} - 730 q^{89} + 672 q^{92} - 324 q^{94} - 1760 q^{95} - 588 q^{97}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 16 * q^5 - 30 * q^8 - 16 * q^10 - 8 * q^11 - 56 * q^13 - 41 * q^16 - 54 * q^17 - 110 * q^19 - 224 * q^20 + 16 * q^22 + 48 * q^23 - 131 * q^25 + 28 * q^26 + 220 * q^29 + 12 * q^31 - 161 * q^32 + 108 * q^34 + 246 * q^37 - 110 * q^38 + 240 * q^40 + 364 * q^41 + 256 * q^43 + 56 * q^44 + 48 * q^46 - 324 * q^47 + 262 * q^50 - 196 * q^52 - 162 * q^53 + 256 * q^55 - 110 * q^58 - 810 * q^59 - 488 * q^61 - 24 * q^62 - 334 * q^64 + 448 * q^65 - 244 * q^67 + 378 * q^68 + 1536 * q^71 - 702 * q^73 + 246 * q^74 - 1540 * q^76 - 440 * q^79 - 656 * q^80 - 182 * q^82 - 2604 * q^83 + 1728 * q^85 - 128 * q^86 + 120 * q^88 - 730 * q^89 + 672 * q^92 - 324 * q^94 - 1760 * q^95 - 588 * 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 −8.00000 + 13.8564i 0 0 −15.0000 0 −8.00000 13.8564i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i −8.00000 13.8564i 0 0 −15.0000 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.e 2
3.b odd 2 1 49.4.c.b 2
7.b odd 2 1 441.4.e.h 2
7.c even 3 1 441.4.a.i 1
7.c even 3 1 inner 441.4.e.e 2
7.d odd 6 1 63.4.a.b 1
7.d odd 6 1 441.4.e.h 2
21.c even 2 1 49.4.c.c 2
21.g even 6 1 7.4.a.a 1
21.g even 6 1 49.4.c.c 2
21.h odd 6 1 49.4.a.b 1
21.h odd 6 1 49.4.c.b 2
28.f even 6 1 1008.4.a.c 1
35.i odd 6 1 1575.4.a.e 1
84.j odd 6 1 112.4.a.f 1
84.n even 6 1 784.4.a.g 1
105.o odd 6 1 1225.4.a.j 1
105.p even 6 1 175.4.a.b 1
105.w odd 12 2 175.4.b.b 2
168.ba even 6 1 448.4.a.i 1
168.be odd 6 1 448.4.a.e 1
231.k odd 6 1 847.4.a.b 1
273.ba even 6 1 1183.4.a.b 1
357.s even 6 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 21.g even 6 1
49.4.a.b 1 21.h odd 6 1
49.4.c.b 2 3.b odd 2 1
49.4.c.b 2 21.h odd 6 1
49.4.c.c 2 21.c even 2 1
49.4.c.c 2 21.g even 6 1
63.4.a.b 1 7.d odd 6 1
112.4.a.f 1 84.j odd 6 1
175.4.a.b 1 105.p even 6 1
175.4.b.b 2 105.w odd 12 2
441.4.a.i 1 7.c even 3 1
441.4.e.e 2 1.a even 1 1 trivial
441.4.e.e 2 7.c even 3 1 inner
441.4.e.h 2 7.b odd 2 1
441.4.e.h 2 7.d odd 6 1
448.4.a.e 1 168.be odd 6 1
448.4.a.i 1 168.ba even 6 1
784.4.a.g 1 84.n even 6 1
847.4.a.b 1 231.k odd 6 1
1008.4.a.c 1 28.f even 6 1
1183.4.a.b 1 273.ba even 6 1
1225.4.a.j 1 105.o odd 6 1
1575.4.a.e 1 35.i odd 6 1
2023.4.a.a 1 357.s even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5}^{2} + 16T_{5} + 256$$ T5^2 + 16*T5 + 256 $$T_{13} + 28$$ T13 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 16T + 256$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8T + 64$$
$13$ $$(T + 28)^{2}$$
$17$ $$T^{2} + 54T + 2916$$
$19$ $$T^{2} + 110T + 12100$$
$23$ $$T^{2} - 48T + 2304$$
$29$ $$(T - 110)^{2}$$
$31$ $$T^{2} - 12T + 144$$
$37$ $$T^{2} - 246T + 60516$$
$41$ $$(T - 182)^{2}$$
$43$ $$(T - 128)^{2}$$
$47$ $$T^{2} + 324T + 104976$$
$53$ $$T^{2} + 162T + 26244$$
$59$ $$T^{2} + 810T + 656100$$
$61$ $$T^{2} + 488T + 238144$$
$67$ $$T^{2} + 244T + 59536$$
$71$ $$(T - 768)^{2}$$
$73$ $$T^{2} + 702T + 492804$$
$79$ $$T^{2} + 440T + 193600$$
$83$ $$(T + 1302)^{2}$$
$89$ $$T^{2} + 730T + 532900$$
$97$ $$(T + 294)^{2}$$