# Properties

 Label 441.6.a.u Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 39$$ x^2 - 39 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 49) 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 $$\beta = 12\sqrt{39}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 28 q^{4} - \beta q^{5} - 120 q^{8} +O(q^{10})$$ q + 2 * q^2 - 28 * q^4 - b * q^5 - 120 * q^8 $$q + 2 q^{2} - 28 q^{4} - \beta q^{5} - 120 q^{8} - 2 \beta q^{10} + 284 q^{11} + 7 \beta q^{13} + 656 q^{16} + 2 \beta q^{17} + 29 \beta q^{19} + 28 \beta q^{20} + 568 q^{22} - 1496 q^{23} + 2491 q^{25} + 14 \beta q^{26} + 4366 q^{29} - 86 \beta q^{31} + 5152 q^{32} + 4 \beta q^{34} - 12630 q^{37} + 58 \beta q^{38} + 120 \beta q^{40} + 126 \beta q^{41} - 1356 q^{43} - 7952 q^{44} - 2992 q^{46} - 134 \beta q^{47} + 4982 q^{50} - 196 \beta q^{52} - 14150 q^{53} - 284 \beta q^{55} + 8732 q^{58} + 499 \beta q^{59} - 475 \beta q^{61} - 172 \beta q^{62} - 10688 q^{64} - 39312 q^{65} - 3644 q^{67} - 56 \beta q^{68} - 35632 q^{71} + 544 \beta q^{73} - 25260 q^{74} - 812 \beta q^{76} - 54616 q^{79} - 656 \beta q^{80} + 252 \beta q^{82} + 7 \beta q^{83} - 11232 q^{85} - 2712 q^{86} - 34080 q^{88} + 272 \beta q^{89} + 41888 q^{92} - 268 \beta q^{94} - 162864 q^{95} + 2450 \beta q^{97} +O(q^{100})$$ q + 2 * q^2 - 28 * q^4 - b * q^5 - 120 * q^8 - 2*b * q^10 + 284 * q^11 + 7*b * q^13 + 656 * q^16 + 2*b * q^17 + 29*b * q^19 + 28*b * q^20 + 568 * q^22 - 1496 * q^23 + 2491 * q^25 + 14*b * q^26 + 4366 * q^29 - 86*b * q^31 + 5152 * q^32 + 4*b * q^34 - 12630 * q^37 + 58*b * q^38 + 120*b * q^40 + 126*b * q^41 - 1356 * q^43 - 7952 * q^44 - 2992 * q^46 - 134*b * q^47 + 4982 * q^50 - 196*b * q^52 - 14150 * q^53 - 284*b * q^55 + 8732 * q^58 + 499*b * q^59 - 475*b * q^61 - 172*b * q^62 - 10688 * q^64 - 39312 * q^65 - 3644 * q^67 - 56*b * q^68 - 35632 * q^71 + 544*b * q^73 - 25260 * q^74 - 812*b * q^76 - 54616 * q^79 - 656*b * q^80 + 252*b * q^82 + 7*b * q^83 - 11232 * q^85 - 2712 * q^86 - 34080 * q^88 + 272*b * q^89 + 41888 * q^92 - 268*b * q^94 - 162864 * q^95 + 2450*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 56 q^{4} - 240 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8 $$2 q + 4 q^{2} - 56 q^{4} - 240 q^{8} + 568 q^{11} + 1312 q^{16} + 1136 q^{22} - 2992 q^{23} + 4982 q^{25} + 8732 q^{29} + 10304 q^{32} - 25260 q^{37} - 2712 q^{43} - 15904 q^{44} - 5984 q^{46} + 9964 q^{50} - 28300 q^{53} + 17464 q^{58} - 21376 q^{64} - 78624 q^{65} - 7288 q^{67} - 71264 q^{71} - 50520 q^{74} - 109232 q^{79} - 22464 q^{85} - 5424 q^{86} - 68160 q^{88} + 83776 q^{92} - 325728 q^{95}+O(q^{100})$$ 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8 + 568 * q^11 + 1312 * q^16 + 1136 * q^22 - 2992 * q^23 + 4982 * q^25 + 8732 * q^29 + 10304 * q^32 - 25260 * q^37 - 2712 * q^43 - 15904 * q^44 - 5984 * q^46 + 9964 * q^50 - 28300 * q^53 + 17464 * q^58 - 21376 * q^64 - 78624 * q^65 - 7288 * q^67 - 71264 * q^71 - 50520 * q^74 - 109232 * q^79 - 22464 * q^85 - 5424 * q^86 - 68160 * q^88 + 83776 * q^92 - 325728 * q^95

## 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
 6.24500 −6.24500
2.00000 0 −28.0000 −74.9400 0 0 −120.000 0 −149.880
1.2 2.00000 0 −28.0000 74.9400 0 0 −120.000 0 149.880
 $$n$$: 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
7.b odd 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.u 2
3.b odd 2 1 49.6.a.c 2
7.b odd 2 1 inner 441.6.a.u 2
12.b even 2 1 784.6.a.z 2
21.c even 2 1 49.6.a.c 2
21.g even 6 2 49.6.c.g 4
21.h odd 6 2 49.6.c.g 4
84.h odd 2 1 784.6.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 3.b odd 2 1
49.6.a.c 2 21.c even 2 1
49.6.c.g 4 21.g even 6 2
49.6.c.g 4 21.h odd 6 2
441.6.a.u 2 1.a even 1 1 trivial
441.6.a.u 2 7.b odd 2 1 inner
784.6.a.z 2 12.b even 2 1
784.6.a.z 2 84.h odd 2 1

## Hecke kernels

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

 $$T_{2} - 2$$ T2 - 2 $$T_{5}^{2} - 5616$$ T5^2 - 5616 $$T_{13}^{2} - 275184$$ T13^2 - 275184

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5616$$
$7$ $$T^{2}$$
$11$ $$(T - 284)^{2}$$
$13$ $$T^{2} - 275184$$
$17$ $$T^{2} - 22464$$
$19$ $$T^{2} - 4723056$$
$23$ $$(T + 1496)^{2}$$
$29$ $$(T - 4366)^{2}$$
$31$ $$T^{2} - 41535936$$
$37$ $$(T + 12630)^{2}$$
$41$ $$T^{2} - 89159616$$
$43$ $$(T + 1356)^{2}$$
$47$ $$T^{2} - 100840896$$
$53$ $$(T + 14150)^{2}$$
$59$ $$T^{2} - 1398389616$$
$61$ $$T^{2} - 1267110000$$
$67$ $$(T + 3644)^{2}$$
$71$ $$(T + 35632)^{2}$$
$73$ $$T^{2} - 1661976576$$
$79$ $$(T + 54616)^{2}$$
$83$ $$T^{2} - 275184$$
$89$ $$T^{2} - 415494144$$
$97$ $$T^{2} - 33710040000$$