# Properties

 Label 484.4.a.a Level $484$ Weight $4$ Character orbit 484.a Self dual yes Analytic conductor $28.557$ 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] = [484,4,Mod(1,484)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(484, 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("484.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$484 = 2^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 484.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: $$28.5569244428$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{3} - 7 q^{5} + 26 q^{7} - 2 q^{9}+O(q^{10})$$ q - 5 * q^3 - 7 * q^5 + 26 * q^7 - 2 * q^9 $$q - 5 q^{3} - 7 q^{5} + 26 q^{7} - 2 q^{9} - 52 q^{13} + 35 q^{15} - 46 q^{17} + 96 q^{19} - 130 q^{21} + 27 q^{23} - 76 q^{25} + 145 q^{27} - 16 q^{29} - 293 q^{31} - 182 q^{35} - 29 q^{37} + 260 q^{39} + 472 q^{41} + 110 q^{43} + 14 q^{45} - 224 q^{47} + 333 q^{49} + 230 q^{51} + 754 q^{53} - 480 q^{57} + 825 q^{59} + 548 q^{61} - 52 q^{63} + 364 q^{65} - 123 q^{67} - 135 q^{69} + 1001 q^{71} + 1020 q^{73} + 380 q^{75} - 526 q^{79} - 671 q^{81} + 158 q^{83} + 322 q^{85} + 80 q^{87} - 1217 q^{89} - 1352 q^{91} + 1465 q^{93} - 672 q^{95} - 263 q^{97}+O(q^{100})$$ q - 5 * q^3 - 7 * q^5 + 26 * q^7 - 2 * q^9 - 52 * q^13 + 35 * q^15 - 46 * q^17 + 96 * q^19 - 130 * q^21 + 27 * q^23 - 76 * q^25 + 145 * q^27 - 16 * q^29 - 293 * q^31 - 182 * q^35 - 29 * q^37 + 260 * q^39 + 472 * q^41 + 110 * q^43 + 14 * q^45 - 224 * q^47 + 333 * q^49 + 230 * q^51 + 754 * q^53 - 480 * q^57 + 825 * q^59 + 548 * q^61 - 52 * q^63 + 364 * q^65 - 123 * q^67 - 135 * q^69 + 1001 * q^71 + 1020 * q^73 + 380 * q^75 - 526 * q^79 - 671 * q^81 + 158 * q^83 + 322 * q^85 + 80 * q^87 - 1217 * q^89 - 1352 * q^91 + 1465 * q^93 - 672 * q^95 - 263 * 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
0 −5.00000 0 −7.00000 0 26.0000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$

## 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 484.4.a.a 1
4.b odd 2 1 1936.4.a.m 1
11.b odd 2 1 44.4.a.a 1
33.d even 2 1 396.4.a.e 1
44.c even 2 1 176.4.a.e 1
55.d odd 2 1 1100.4.a.d 1
55.e even 4 2 1100.4.b.c 2
77.b even 2 1 2156.4.a.b 1
88.b odd 2 1 704.4.a.j 1
88.g even 2 1 704.4.a.c 1
132.d odd 2 1 1584.4.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 11.b odd 2 1
176.4.a.e 1 44.c even 2 1
396.4.a.e 1 33.d even 2 1
484.4.a.a 1 1.a even 1 1 trivial
704.4.a.c 1 88.g even 2 1
704.4.a.j 1 88.b odd 2 1
1100.4.a.d 1 55.d odd 2 1
1100.4.b.c 2 55.e even 4 2
1584.4.a.p 1 132.d odd 2 1
1936.4.a.m 1 4.b odd 2 1
2156.4.a.b 1 77.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(484))$$:

 $$T_{3} + 5$$ T3 + 5 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 5$$
$5$ $$T + 7$$
$7$ $$T - 26$$
$11$ $$T$$
$13$ $$T + 52$$
$17$ $$T + 46$$
$19$ $$T - 96$$
$23$ $$T - 27$$
$29$ $$T + 16$$
$31$ $$T + 293$$
$37$ $$T + 29$$
$41$ $$T - 472$$
$43$ $$T - 110$$
$47$ $$T + 224$$
$53$ $$T - 754$$
$59$ $$T - 825$$
$61$ $$T - 548$$
$67$ $$T + 123$$
$71$ $$T - 1001$$
$73$ $$T - 1020$$
$79$ $$T + 526$$
$83$ $$T - 158$$
$89$ $$T + 1217$$
$97$ $$T + 263$$