# Properties

 Label 441.2.a.j Level $441$ Weight $2$ Character orbit 441.a Self dual yes Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,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, 2, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + ( - \beta + 2) q^{5} + (\beta + 3) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + (-b + 2) * q^5 + (b + 3) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + ( - \beta + 2) q^{5} + (\beta + 3) q^{8} + \beta q^{10} + 2 q^{11} + ( - \beta - 4) q^{13} + 3 q^{16} + (3 \beta + 2) q^{17} - 2 \beta q^{19} + (3 \beta - 2) q^{20} + (2 \beta + 2) q^{22} + ( - 4 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} + ( - 5 \beta - 6) q^{26} + ( - 2 \beta + 4) q^{29} + (2 \beta + 4) q^{31} + (\beta - 3) q^{32} + (5 \beta + 8) q^{34} - 4 q^{37} + ( - 2 \beta - 4) q^{38} + ( - \beta + 4) q^{40} + ( - 3 \beta + 2) q^{41} - 4 \beta q^{43} + (4 \beta + 2) q^{44} + ( - 2 \beta - 6) q^{46} + 2 \beta q^{47} + ( - 3 \beta - 7) q^{50} + ( - 9 \beta - 8) q^{52} + 2 q^{53} + ( - 2 \beta + 4) q^{55} + 2 \beta q^{58} + ( - 2 \beta - 4) q^{59} + (3 \beta - 8) q^{61} + (6 \beta + 8) q^{62} + ( - 2 \beta - 7) q^{64} + (2 \beta - 6) q^{65} + 4 \beta q^{67} + (7 \beta + 14) q^{68} + (8 \beta + 2) q^{71} + (7 \beta - 4) q^{73} + ( - 4 \beta - 4) q^{74} + ( - 2 \beta - 8) q^{76} + ( - 4 \beta + 8) q^{79} + ( - 3 \beta + 6) q^{80} + ( - \beta - 4) q^{82} + ( - 8 \beta - 4) q^{83} + (4 \beta - 2) q^{85} + ( - 4 \beta - 8) q^{86} + (2 \beta + 6) q^{88} + (3 \beta - 10) q^{89} - 14 q^{92} + (2 \beta + 4) q^{94} + ( - 4 \beta + 4) q^{95} + ( - \beta - 4) q^{97} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + (-b + 2) * q^5 + (b + 3) * q^8 + b * q^10 + 2 * q^11 + (-b - 4) * q^13 + 3 * q^16 + (3*b + 2) * q^17 - 2*b * q^19 + (3*b - 2) * q^20 + (2*b + 2) * q^22 + (-4*b + 2) * q^23 + (-4*b + 1) * q^25 + (-5*b - 6) * q^26 + (-2*b + 4) * q^29 + (2*b + 4) * q^31 + (b - 3) * q^32 + (5*b + 8) * q^34 - 4 * q^37 + (-2*b - 4) * q^38 + (-b + 4) * q^40 + (-3*b + 2) * q^41 - 4*b * q^43 + (4*b + 2) * q^44 + (-2*b - 6) * q^46 + 2*b * q^47 + (-3*b - 7) * q^50 + (-9*b - 8) * q^52 + 2 * q^53 + (-2*b + 4) * q^55 + 2*b * q^58 + (-2*b - 4) * q^59 + (3*b - 8) * q^61 + (6*b + 8) * q^62 + (-2*b - 7) * q^64 + (2*b - 6) * q^65 + 4*b * q^67 + (7*b + 14) * q^68 + (8*b + 2) * q^71 + (7*b - 4) * q^73 + (-4*b - 4) * q^74 + (-2*b - 8) * q^76 + (-4*b + 8) * q^79 + (-3*b + 6) * q^80 + (-b - 4) * q^82 + (-8*b - 4) * q^83 + (4*b - 2) * q^85 + (-4*b - 8) * q^86 + (2*b + 6) * q^88 + (3*b - 10) * q^89 - 14 * q^92 + (2*b + 4) * q^94 + (-4*b + 4) * q^95 + (-b - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{11} - 8 q^{13} + 6 q^{16} + 4 q^{17} - 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 6 q^{32} + 16 q^{34} - 8 q^{37} - 8 q^{38} + 8 q^{40} + 4 q^{41} + 4 q^{44} - 12 q^{46} - 14 q^{50} - 16 q^{52} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 16 q^{61} + 16 q^{62} - 14 q^{64} - 12 q^{65} + 28 q^{68} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 16 q^{76} + 16 q^{79} + 12 q^{80} - 8 q^{82} - 8 q^{83} - 4 q^{85} - 16 q^{86} + 12 q^{88} - 20 q^{89} - 28 q^{92} + 8 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^11 - 8 * q^13 + 6 * q^16 + 4 * q^17 - 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 - 12 * q^26 + 8 * q^29 + 8 * q^31 - 6 * q^32 + 16 * q^34 - 8 * q^37 - 8 * q^38 + 8 * q^40 + 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 - 16 * q^52 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 16 * q^61 + 16 * q^62 - 14 * q^64 - 12 * q^65 + 28 * q^68 + 4 * q^71 - 8 * q^73 - 8 * q^74 - 16 * q^76 + 16 * q^79 + 12 * q^80 - 8 * q^82 - 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 - 20 * q^89 - 28 * q^92 + 8 * q^94 + 8 * q^95 - 8 * q^97

## 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
 −1.41421 1.41421
−0.414214 0 −1.82843 3.41421 0 0 1.58579 0 −1.41421
1.2 2.41421 0 3.82843 0.585786 0 0 4.41421 0 1.41421
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.j 2
3.b odd 2 1 147.2.a.d 2
4.b odd 2 1 7056.2.a.cv 2
7.b odd 2 1 441.2.a.i 2
7.c even 3 2 441.2.e.f 4
7.d odd 6 2 441.2.e.g 4
12.b even 2 1 2352.2.a.be 2
15.d odd 2 1 3675.2.a.bf 2
21.c even 2 1 147.2.a.e yes 2
21.g even 6 2 147.2.e.d 4
21.h odd 6 2 147.2.e.e 4
24.f even 2 1 9408.2.a.dq 2
24.h odd 2 1 9408.2.a.ef 2
28.d even 2 1 7056.2.a.cf 2
84.h odd 2 1 2352.2.a.bc 2
84.j odd 6 2 2352.2.q.bd 4
84.n even 6 2 2352.2.q.bb 4
105.g even 2 1 3675.2.a.bd 2
168.e odd 2 1 9408.2.a.dt 2
168.i even 2 1 9408.2.a.di 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 3.b odd 2 1
147.2.a.e yes 2 21.c even 2 1
147.2.e.d 4 21.g even 6 2
147.2.e.e 4 21.h odd 6 2
441.2.a.i 2 7.b odd 2 1
441.2.a.j 2 1.a even 1 1 trivial
441.2.e.f 4 7.c even 3 2
441.2.e.g 4 7.d odd 6 2
2352.2.a.bc 2 84.h odd 2 1
2352.2.a.be 2 12.b even 2 1
2352.2.q.bb 4 84.n even 6 2
2352.2.q.bd 4 84.j odd 6 2
3675.2.a.bd 2 105.g even 2 1
3675.2.a.bf 2 15.d odd 2 1
7056.2.a.cf 2 28.d even 2 1
7056.2.a.cv 2 4.b odd 2 1
9408.2.a.di 2 168.i even 2 1
9408.2.a.dq 2 24.f even 2 1
9408.2.a.dt 2 168.e odd 2 1
9408.2.a.ef 2 24.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{13}^{2} + 8T_{13} + 14$$ T13^2 + 8*T13 + 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 8T + 14$$
$17$ $$T^{2} - 4T - 14$$
$19$ $$T^{2} - 8$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$T^{2} - 8T + 8$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 4T - 14$$
$43$ $$T^{2} - 32$$
$47$ $$T^{2} - 8$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} + 16T + 46$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} - 4T - 124$$
$73$ $$T^{2} + 8T - 82$$
$79$ $$T^{2} - 16T + 32$$
$83$ $$T^{2} + 8T - 112$$
$89$ $$T^{2} + 20T + 82$$
$97$ $$T^{2} + 8T + 14$$