# Properties

 Label 441.6.a.y Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## 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: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{19}, \sqrt{69})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 71x^{2} + 72x - 15$$ x^4 - 2*x^3 - 71*x^2 + 72*x - 15 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 + 44 * q^4 + b2 * q^5 + 12*b1 * q^8 $$q + \beta_1 q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_1 q^{8} - \beta_{3} q^{10} + 43 \beta_1 q^{11} - \beta_{3} q^{13} - 496 q^{16} - 11 \beta_{2} q^{17} + \beta_{3} q^{19} + 44 \beta_{2} q^{20} + 3268 q^{22} + 341 \beta_1 q^{23} + 6811 q^{25} + 76 \beta_{2} q^{26} + 518 \beta_1 q^{29} + 11 \beta_{3} q^{31} - 880 \beta_1 q^{32} + 11 \beta_{3} q^{34} - 5466 q^{37} - 76 \beta_{2} q^{38} - 12 \beta_{3} q^{40} + 99 \beta_{2} q^{41} + 12540 q^{43} + 1892 \beta_1 q^{44} + 25916 q^{46} - 100 \beta_{2} q^{47} + 6811 \beta_1 q^{50} - 44 \beta_{3} q^{52} - 1738 \beta_1 q^{53} - 43 \beta_{3} q^{55} + 39368 q^{58} + 428 \beta_{2} q^{59} + 55 \beta_{3} q^{61} - 836 \beta_{2} q^{62} - 51008 q^{64} + 9936 \beta_1 q^{65} - 29996 q^{67} - 484 \beta_{2} q^{68} + 7051 \beta_1 q^{71} + 56 \beta_{3} q^{73} - 5466 \beta_1 q^{74} + 44 \beta_{3} q^{76} + 80168 q^{79} - 496 \beta_{2} q^{80} - 99 \beta_{3} q^{82} + 308 \beta_{2} q^{83} - 109296 q^{85} + 12540 \beta_1 q^{86} + 39216 q^{88} - 209 \beta_{2} q^{89} + 15004 \beta_1 q^{92} + 100 \beta_{3} q^{94} - 9936 \beta_1 q^{95} + 154 \beta_{3} q^{97}+O(q^{100})$$ q + b1 * q^2 + 44 * q^4 + b2 * q^5 + 12*b1 * q^8 - b3 * q^10 + 43*b1 * q^11 - b3 * q^13 - 496 * q^16 - 11*b2 * q^17 + b3 * q^19 + 44*b2 * q^20 + 3268 * q^22 + 341*b1 * q^23 + 6811 * q^25 + 76*b2 * q^26 + 518*b1 * q^29 + 11*b3 * q^31 - 880*b1 * q^32 + 11*b3 * q^34 - 5466 * q^37 - 76*b2 * q^38 - 12*b3 * q^40 + 99*b2 * q^41 + 12540 * q^43 + 1892*b1 * q^44 + 25916 * q^46 - 100*b2 * q^47 + 6811*b1 * q^50 - 44*b3 * q^52 - 1738*b1 * q^53 - 43*b3 * q^55 + 39368 * q^58 + 428*b2 * q^59 + 55*b3 * q^61 - 836*b2 * q^62 - 51008 * q^64 + 9936*b1 * q^65 - 29996 * q^67 - 484*b2 * q^68 + 7051*b1 * q^71 + 56*b3 * q^73 - 5466*b1 * q^74 + 44*b3 * q^76 + 80168 * q^79 - 496*b2 * q^80 - 99*b3 * q^82 + 308*b2 * q^83 - 109296 * q^85 + 12540*b1 * q^86 + 39216 * q^88 - 209*b2 * q^89 + 15004*b1 * q^92 + 100*b3 * q^94 - 9936*b1 * q^95 + 154*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 176 q^{4}+O(q^{10})$$ 4 * q + 176 * q^4 $$4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88}+O(q^{100})$$ 4 * q + 176 * q^4 - 1984 * q^16 + 13072 * q^22 + 27244 * q^25 - 21864 * q^37 + 50160 * q^43 + 103664 * q^46 + 157472 * q^58 - 204032 * q^64 - 119984 * q^67 + 320672 * q^79 - 437184 * q^85 + 156864 * q^88

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 71x^{2} + 72x - 15$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{3} - 6\nu^{2} - 294\nu + 148 ) / 7$$ (4*v^3 - 6*v^2 - 294*v + 148) / 7 $$\beta_{2}$$ $$=$$ $$( 48\nu^{3} - 72\nu^{2} - 3360\nu + 1692 ) / 7$$ (48*v^3 - 72*v^2 - 3360*v + 1692) / 7 $$\beta_{3}$$ $$=$$ $$24\nu^{2} - 24\nu - 864$$ 24*v^2 - 24*v - 864
 $$\nu$$ $$=$$ $$( \beta_{2} - 12\beta _1 + 12 ) / 24$$ (b2 - 12*b1 + 12) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} - 12\beta _1 + 876 ) / 24$$ (b3 + b2 - 12*b1 + 876) / 24 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 50\beta_{2} - 572\beta _1 + 872 ) / 16$$ (b3 + 50*b2 - 572*b1 + 872) / 16

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.705587 9.01221 −8.01221 0.294413
−8.71780 0 44.0000 −99.6795 0 0 −104.614 0 868.986
1.2 −8.71780 0 44.0000 99.6795 0 0 −104.614 0 −868.986
1.3 8.71780 0 44.0000 −99.6795 0 0 104.614 0 −868.986
1.4 8.71780 0 44.0000 99.6795 0 0 104.614 0 868.986
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.y 4
3.b odd 2 1 inner 441.6.a.y 4
7.b odd 2 1 inner 441.6.a.y 4
21.c even 2 1 inner 441.6.a.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.y 4 1.a even 1 1 trivial
441.6.a.y 4 3.b odd 2 1 inner
441.6.a.y 4 7.b odd 2 1 inner
441.6.a.y 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 76$$ T2^2 - 76 $$T_{5}^{2} - 9936$$ T5^2 - 9936 $$T_{13}^{2} - 755136$$ T13^2 - 755136

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 76)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 9936)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 140524)^{2}$$
$13$ $$(T^{2} - 755136)^{2}$$
$17$ $$(T^{2} - 1202256)^{2}$$
$19$ $$(T^{2} - 755136)^{2}$$
$23$ $$(T^{2} - 8837356)^{2}$$
$29$ $$(T^{2} - 20392624)^{2}$$
$31$ $$(T^{2} - 91371456)^{2}$$
$37$ $$(T + 5466)^{4}$$
$41$ $$(T^{2} - 97382736)^{2}$$
$43$ $$(T - 12540)^{4}$$
$47$ $$(T^{2} - 99360000)^{2}$$
$53$ $$(T^{2} - 229568944)^{2}$$
$59$ $$(T^{2} - 1820116224)^{2}$$
$61$ $$(T^{2} - 2284286400)^{2}$$
$67$ $$(T + 29996)^{4}$$
$71$ $$(T^{2} - 3778461676)^{2}$$
$73$ $$(T^{2} - 2368106496)^{2}$$
$79$ $$(T - 80168)^{4}$$
$83$ $$(T^{2} - 942568704)^{2}$$
$89$ $$(T^{2} - 434014416)^{2}$$
$97$ $$(T^{2} - 17908805376)^{2}$$