# Properties

 Label 132.4.a.d Level $132$ Weight $4$ Character orbit 132.a Self dual yes Analytic conductor $7.788$ 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] = [132,4,Mod(1,132)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(132, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("132.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$132 = 2^{2} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 132.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: $$7.78825212076$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 10 * q^5 + 8 * q^7 + 9 * q^9 $$q + 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9} - 11 q^{11} + 18 q^{13} + 30 q^{15} + 46 q^{17} + 40 q^{19} + 24 q^{21} + 44 q^{23} - 25 q^{25} + 27 q^{27} + 186 q^{29} - 72 q^{31} - 33 q^{33} + 80 q^{35} - 114 q^{37} + 54 q^{39} + 174 q^{41} - 416 q^{43} + 90 q^{45} - 156 q^{47} - 279 q^{49} + 138 q^{51} - 62 q^{53} - 110 q^{55} + 120 q^{57} - 348 q^{59} - 446 q^{61} + 72 q^{63} + 180 q^{65} - 956 q^{67} + 132 q^{69} - 444 q^{71} + 306 q^{73} - 75 q^{75} - 88 q^{77} - 664 q^{79} + 81 q^{81} - 124 q^{83} + 460 q^{85} + 558 q^{87} + 602 q^{89} + 144 q^{91} - 216 q^{93} + 400 q^{95} + 1522 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + 10 * q^5 + 8 * q^7 + 9 * q^9 - 11 * q^11 + 18 * q^13 + 30 * q^15 + 46 * q^17 + 40 * q^19 + 24 * q^21 + 44 * q^23 - 25 * q^25 + 27 * q^27 + 186 * q^29 - 72 * q^31 - 33 * q^33 + 80 * q^35 - 114 * q^37 + 54 * q^39 + 174 * q^41 - 416 * q^43 + 90 * q^45 - 156 * q^47 - 279 * q^49 + 138 * q^51 - 62 * q^53 - 110 * q^55 + 120 * q^57 - 348 * q^59 - 446 * q^61 + 72 * q^63 + 180 * q^65 - 956 * q^67 + 132 * q^69 - 444 * q^71 + 306 * q^73 - 75 * q^75 - 88 * q^77 - 664 * q^79 + 81 * q^81 - 124 * q^83 + 460 * q^85 + 558 * q^87 + 602 * q^89 + 144 * q^91 - 216 * q^93 + 400 * q^95 + 1522 * q^97 - 99 * q^99

## 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 3.00000 0 10.0000 0 8.00000 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 132.4.a.d 1
3.b odd 2 1 396.4.a.c 1
4.b odd 2 1 528.4.a.e 1
8.b even 2 1 2112.4.a.e 1
8.d odd 2 1 2112.4.a.q 1
11.b odd 2 1 1452.4.a.i 1
12.b even 2 1 1584.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.d 1 1.a even 1 1 trivial
396.4.a.c 1 3.b odd 2 1
528.4.a.e 1 4.b odd 2 1
1452.4.a.i 1 11.b odd 2 1
1584.4.a.f 1 12.b even 2 1
2112.4.a.e 1 8.b even 2 1
2112.4.a.q 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 10$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(132))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 10$$
$7$ $$T - 8$$
$11$ $$T + 11$$
$13$ $$T - 18$$
$17$ $$T - 46$$
$19$ $$T - 40$$
$23$ $$T - 44$$
$29$ $$T - 186$$
$31$ $$T + 72$$
$37$ $$T + 114$$
$41$ $$T - 174$$
$43$ $$T + 416$$
$47$ $$T + 156$$
$53$ $$T + 62$$
$59$ $$T + 348$$
$61$ $$T + 446$$
$67$ $$T + 956$$
$71$ $$T + 444$$
$73$ $$T - 306$$
$79$ $$T + 664$$
$83$ $$T + 124$$
$89$ $$T - 602$$
$97$ $$T - 1522$$