# Properties

 Label 441.4.a.g Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} - 12 q^{5} - 15 q^{8}+O(q^{10})$$ q + q^2 - 7 * q^4 - 12 * q^5 - 15 * q^8 $$q + q^{2} - 7 q^{4} - 12 q^{5} - 15 q^{8} - 12 q^{10} - 20 q^{11} - 84 q^{13} + 41 q^{16} + 96 q^{17} + 12 q^{19} + 84 q^{20} - 20 q^{22} + 176 q^{23} + 19 q^{25} - 84 q^{26} - 58 q^{29} - 264 q^{31} + 161 q^{32} + 96 q^{34} + 258 q^{37} + 12 q^{38} + 180 q^{40} + 156 q^{43} + 140 q^{44} + 176 q^{46} + 408 q^{47} + 19 q^{50} + 588 q^{52} + 722 q^{53} + 240 q^{55} - 58 q^{58} - 492 q^{59} - 492 q^{61} - 264 q^{62} - 167 q^{64} + 1008 q^{65} + 412 q^{67} - 672 q^{68} - 296 q^{71} + 240 q^{73} + 258 q^{74} - 84 q^{76} + 776 q^{79} - 492 q^{80} - 924 q^{83} - 1152 q^{85} + 156 q^{86} + 300 q^{88} + 744 q^{89} - 1232 q^{92} + 408 q^{94} - 144 q^{95} - 168 q^{97}+O(q^{100})$$ q + q^2 - 7 * q^4 - 12 * q^5 - 15 * q^8 - 12 * q^10 - 20 * q^11 - 84 * q^13 + 41 * q^16 + 96 * q^17 + 12 * q^19 + 84 * q^20 - 20 * q^22 + 176 * q^23 + 19 * q^25 - 84 * q^26 - 58 * q^29 - 264 * q^31 + 161 * q^32 + 96 * q^34 + 258 * q^37 + 12 * q^38 + 180 * q^40 + 156 * q^43 + 140 * q^44 + 176 * q^46 + 408 * q^47 + 19 * q^50 + 588 * q^52 + 722 * q^53 + 240 * q^55 - 58 * q^58 - 492 * q^59 - 492 * q^61 - 264 * q^62 - 167 * q^64 + 1008 * q^65 + 412 * q^67 - 672 * q^68 - 296 * q^71 + 240 * q^73 + 258 * q^74 - 84 * q^76 + 776 * q^79 - 492 * q^80 - 924 * q^83 - 1152 * q^85 + 156 * q^86 + 300 * q^88 + 744 * q^89 - 1232 * q^92 + 408 * q^94 - 144 * q^95 - 168 * 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
 0
1.00000 0 −7.00000 −12.0000 0 0 −15.0000 0 −12.0000
 $$n$$: 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 441.4.a.g 1
3.b odd 2 1 147.4.a.d 1
7.b odd 2 1 441.4.a.h 1
7.c even 3 2 441.4.e.g 2
7.d odd 6 2 441.4.e.f 2
12.b even 2 1 2352.4.a.bi 1
21.c even 2 1 147.4.a.e yes 1
21.g even 6 2 147.4.e.e 2
21.h odd 6 2 147.4.e.f 2
84.h odd 2 1 2352.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 3.b odd 2 1
147.4.a.e yes 1 21.c even 2 1
147.4.e.e 2 21.g even 6 2
147.4.e.f 2 21.h odd 6 2
441.4.a.g 1 1.a even 1 1 trivial
441.4.a.h 1 7.b odd 2 1
441.4.e.f 2 7.d odd 6 2
441.4.e.g 2 7.c even 3 2
2352.4.a.b 1 84.h odd 2 1
2352.4.a.bi 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} + 12$$ T5 + 12 $$T_{13} + 84$$ T13 + 84

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 12$$
$7$ $$T$$
$11$ $$T + 20$$
$13$ $$T + 84$$
$17$ $$T - 96$$
$19$ $$T - 12$$
$23$ $$T - 176$$
$29$ $$T + 58$$
$31$ $$T + 264$$
$37$ $$T - 258$$
$41$ $$T$$
$43$ $$T - 156$$
$47$ $$T - 408$$
$53$ $$T - 722$$
$59$ $$T + 492$$
$61$ $$T + 492$$
$67$ $$T - 412$$
$71$ $$T + 296$$
$73$ $$T - 240$$
$79$ $$T - 776$$
$83$ $$T + 924$$
$89$ $$T - 744$$
$97$ $$T + 168$$