# Properties

 Label 441.4.e.r Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 19x^{2} + 361$$ x^4 + 19*x^2 + 361 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 11 \beta_{2} q^{4} + 2 \beta_1 q^{5} + 3 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 11*b2 * q^4 + 2*b1 * q^5 + 3*b3 * q^8 $$q + \beta_1 q^{2} + 11 \beta_{2} q^{4} + 2 \beta_1 q^{5} + 3 \beta_{3} q^{8} + 38 \beta_{2} q^{10} + (10 \beta_{3} + 10 \beta_1) q^{11} + 82 q^{13} + (31 \beta_{2} + 31) q^{16} + (18 \beta_{3} + 18 \beta_1) q^{17} + (20 \beta_{2} + 20) q^{19} + 22 \beta_{3} q^{20} - 190 q^{22} - 30 \beta_1 q^{23} - 49 \beta_{2} q^{25} + 82 \beta_1 q^{26} + 56 \beta_{3} q^{29} + 156 \beta_{2} q^{31} + (55 \beta_{3} + 55 \beta_1) q^{32} - 342 q^{34} + ( - 186 \beta_{2} - 186) q^{37} + (20 \beta_{3} + 20 \beta_1) q^{38} + ( - 114 \beta_{2} - 114) q^{40} - 38 \beta_{3} q^{41} + 164 q^{43} - 110 \beta_1 q^{44} - 570 \beta_{2} q^{46} + 108 \beta_1 q^{47} - 49 \beta_{3} q^{50} + 902 \beta_{2} q^{52} + (36 \beta_{3} + 36 \beta_1) q^{53} - 380 q^{55} + ( - 1064 \beta_{2} - 1064) q^{58} + ( - 36 \beta_{3} - 36 \beta_1) q^{59} + ( - 790 \beta_{2} - 790) q^{61} + 156 \beta_{3} q^{62} - 797 q^{64} + 164 \beta_1 q^{65} - 44 \beta_{2} q^{67} - 198 \beta_1 q^{68} + 102 \beta_{3} q^{71} + 126 \beta_{2} q^{73} + ( - 186 \beta_{3} - 186 \beta_1) q^{74} - 220 q^{76} + (712 \beta_{2} + 712) q^{79} + (62 \beta_{3} + 62 \beta_1) q^{80} + (722 \beta_{2} + 722) q^{82} - 336 \beta_{3} q^{83} - 684 q^{85} + 164 \beta_1 q^{86} - 570 \beta_{2} q^{88} - 334 \beta_1 q^{89} - 330 \beta_{3} q^{92} + 2052 \beta_{2} q^{94} + (40 \beta_{3} + 40 \beta_1) q^{95} + 798 q^{97}+O(q^{100})$$ q + b1 * q^2 + 11*b2 * q^4 + 2*b1 * q^5 + 3*b3 * q^8 + 38*b2 * q^10 + (10*b3 + 10*b1) * q^11 + 82 * q^13 + (31*b2 + 31) * q^16 + (18*b3 + 18*b1) * q^17 + (20*b2 + 20) * q^19 + 22*b3 * q^20 - 190 * q^22 - 30*b1 * q^23 - 49*b2 * q^25 + 82*b1 * q^26 + 56*b3 * q^29 + 156*b2 * q^31 + (55*b3 + 55*b1) * q^32 - 342 * q^34 + (-186*b2 - 186) * q^37 + (20*b3 + 20*b1) * q^38 + (-114*b2 - 114) * q^40 - 38*b3 * q^41 + 164 * q^43 - 110*b1 * q^44 - 570*b2 * q^46 + 108*b1 * q^47 - 49*b3 * q^50 + 902*b2 * q^52 + (36*b3 + 36*b1) * q^53 - 380 * q^55 + (-1064*b2 - 1064) * q^58 + (-36*b3 - 36*b1) * q^59 + (-790*b2 - 790) * q^61 + 156*b3 * q^62 - 797 * q^64 + 164*b1 * q^65 - 44*b2 * q^67 - 198*b1 * q^68 + 102*b3 * q^71 + 126*b2 * q^73 + (-186*b3 - 186*b1) * q^74 - 220 * q^76 + (712*b2 + 712) * q^79 + (62*b3 + 62*b1) * q^80 + (722*b2 + 722) * q^82 - 336*b3 * q^83 - 684 * q^85 + 164*b1 * q^86 - 570*b2 * q^88 - 334*b1 * q^89 - 330*b3 * q^92 + 2052*b2 * q^94 + (40*b3 + 40*b1) * q^95 + 798 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 22 q^{4}+O(q^{10})$$ 4 * q - 22 * q^4 $$4 q - 22 q^{4} - 76 q^{10} + 328 q^{13} + 62 q^{16} + 40 q^{19} - 760 q^{22} + 98 q^{25} - 312 q^{31} - 1368 q^{34} - 372 q^{37} - 228 q^{40} + 656 q^{43} + 1140 q^{46} - 1804 q^{52} - 1520 q^{55} - 2128 q^{58} - 1580 q^{61} - 3188 q^{64} + 88 q^{67} - 252 q^{73} - 880 q^{76} + 1424 q^{79} + 1444 q^{82} - 2736 q^{85} + 1140 q^{88} - 4104 q^{94} + 3192 q^{97}+O(q^{100})$$ 4 * q - 22 * q^4 - 76 * q^10 + 328 * q^13 + 62 * q^16 + 40 * q^19 - 760 * q^22 + 98 * q^25 - 312 * q^31 - 1368 * q^34 - 372 * q^37 - 228 * q^40 + 656 * q^43 + 1140 * q^46 - 1804 * q^52 - 1520 * q^55 - 2128 * q^58 - 1580 * q^61 - 3188 * q^64 + 88 * q^67 - 252 * q^73 - 880 * q^76 + 1424 * q^79 + 1444 * q^82 - 2736 * q^85 + 1140 * q^88 - 4104 * q^94 + 3192 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 19x^{2} + 361$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 19$$ (v^2) / 19 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 19$$ (v^3) / 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$19\beta_{2}$$ 19*b2 $$\nu^{3}$$ $$=$$ $$19\beta_{3}$$ 19*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 −2.17945 + 3.77492i 2.17945 − 3.77492i −2.17945 − 3.77492i 2.17945 + 3.77492i
−2.17945 + 3.77492i 0 −5.50000 9.52628i −4.35890 + 7.54983i 0 0 13.0767 0 −19.0000 32.9090i
226.2 2.17945 3.77492i 0 −5.50000 9.52628i 4.35890 7.54983i 0 0 −13.0767 0 −19.0000 32.9090i
361.1 −2.17945 3.77492i 0 −5.50000 + 9.52628i −4.35890 7.54983i 0 0 13.0767 0 −19.0000 + 32.9090i
361.2 2.17945 + 3.77492i 0 −5.50000 + 9.52628i 4.35890 + 7.54983i 0 0 −13.0767 0 −19.0000 + 32.9090i
 $$n$$: 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.r 4
3.b odd 2 1 inner 441.4.e.r 4
7.b odd 2 1 441.4.e.s 4
7.c even 3 1 63.4.a.d 2
7.c even 3 1 inner 441.4.e.r 4
7.d odd 6 1 441.4.a.q 2
7.d odd 6 1 441.4.e.s 4
21.c even 2 1 441.4.e.s 4
21.g even 6 1 441.4.a.q 2
21.g even 6 1 441.4.e.s 4
21.h odd 6 1 63.4.a.d 2
21.h odd 6 1 inner 441.4.e.r 4
28.g odd 6 1 1008.4.a.be 2
35.j even 6 1 1575.4.a.t 2
84.n even 6 1 1008.4.a.be 2
105.o odd 6 1 1575.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 7.c even 3 1
63.4.a.d 2 21.h odd 6 1
441.4.a.q 2 7.d odd 6 1
441.4.a.q 2 21.g even 6 1
441.4.e.r 4 1.a even 1 1 trivial
441.4.e.r 4 3.b odd 2 1 inner
441.4.e.r 4 7.c even 3 1 inner
441.4.e.r 4 21.h odd 6 1 inner
441.4.e.s 4 7.b odd 2 1
441.4.e.s 4 7.d odd 6 1
441.4.e.s 4 21.c even 2 1
441.4.e.s 4 21.g even 6 1
1008.4.a.be 2 28.g odd 6 1
1008.4.a.be 2 84.n even 6 1
1575.4.a.t 2 35.j even 6 1
1575.4.a.t 2 105.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 $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{4} + 19T_{2}^{2} + 361$$ T2^4 + 19*T2^2 + 361 $$T_{5}^{4} + 76T_{5}^{2} + 5776$$ T5^4 + 76*T5^2 + 5776 $$T_{13} - 82$$ T13 - 82

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 19T^{2} + 361$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 76T^{2} + 5776$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 1900 T^{2} + \cdots + 3610000$$
$13$ $$(T - 82)^{4}$$
$17$ $$T^{4} + 6156 T^{2} + \cdots + 37896336$$
$19$ $$(T^{2} - 20 T + 400)^{2}$$
$23$ $$T^{4} + 17100 T^{2} + \cdots + 292410000$$
$29$ $$(T^{2} - 59584)^{2}$$
$31$ $$(T^{2} + 156 T + 24336)^{2}$$
$37$ $$(T^{2} + 186 T + 34596)^{2}$$
$41$ $$(T^{2} - 27436)^{2}$$
$43$ $$(T - 164)^{4}$$
$47$ $$T^{4} + 221616 T^{2} + \cdots + 49113651456$$
$53$ $$T^{4} + 24624 T^{2} + \cdots + 606341376$$
$59$ $$T^{4} + 24624 T^{2} + \cdots + 606341376$$
$61$ $$(T^{2} + 790 T + 624100)^{2}$$
$67$ $$(T^{2} - 44 T + 1936)^{2}$$
$71$ $$(T^{2} - 197676)^{2}$$
$73$ $$(T^{2} + 126 T + 15876)^{2}$$
$79$ $$(T^{2} - 712 T + 506944)^{2}$$
$83$ $$(T^{2} - 2145024)^{2}$$
$89$ $$T^{4} + 2119564 T^{2} + \cdots + 4492551550096$$
$97$ $$(T - 798)^{4}$$