# Properties

 Label 132.4.a.b Level $132$ Weight $4$ Character orbit 132.a Self dual yes Analytic conductor $7.788$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 2 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 2 * q^7 + 9 * q^9 $$q - 3 q^{3} + 2 q^{7} + 9 q^{9} - 11 q^{11} - 88 q^{13} - 66 q^{17} - 40 q^{19} - 6 q^{21} + 6 q^{23} - 125 q^{25} - 27 q^{27} - 54 q^{29} + 8 q^{31} + 33 q^{33} - 106 q^{37} + 264 q^{39} + 354 q^{41} - 124 q^{43} + 546 q^{47} - 339 q^{49} + 198 q^{51} - 408 q^{53} + 120 q^{57} + 552 q^{59} + 404 q^{61} + 18 q^{63} - 4 q^{67} - 18 q^{69} + 126 q^{71} - 166 q^{73} + 375 q^{75} - 22 q^{77} - 874 q^{79} + 81 q^{81} + 444 q^{83} + 162 q^{87} + 1002 q^{89} - 176 q^{91} - 24 q^{93} - 802 q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 + 2 * q^7 + 9 * q^9 - 11 * q^11 - 88 * q^13 - 66 * q^17 - 40 * q^19 - 6 * q^21 + 6 * q^23 - 125 * q^25 - 27 * q^27 - 54 * q^29 + 8 * q^31 + 33 * q^33 - 106 * q^37 + 264 * q^39 + 354 * q^41 - 124 * q^43 + 546 * q^47 - 339 * q^49 + 198 * q^51 - 408 * q^53 + 120 * q^57 + 552 * q^59 + 404 * q^61 + 18 * q^63 - 4 * q^67 - 18 * q^69 + 126 * q^71 - 166 * q^73 + 375 * q^75 - 22 * q^77 - 874 * q^79 + 81 * q^81 + 444 * q^83 + 162 * q^87 + 1002 * q^89 - 176 * q^91 - 24 * q^93 - 802 * 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 0 0 2.00000 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.b 1
3.b odd 2 1 396.4.a.d 1
4.b odd 2 1 528.4.a.i 1
8.b even 2 1 2112.4.a.t 1
8.d odd 2 1 2112.4.a.f 1
11.b odd 2 1 1452.4.a.b 1
12.b even 2 1 1584.4.a.j 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T + 11$$
$13$ $$T + 88$$
$17$ $$T + 66$$
$19$ $$T + 40$$
$23$ $$T - 6$$
$29$ $$T + 54$$
$31$ $$T - 8$$
$37$ $$T + 106$$
$41$ $$T - 354$$
$43$ $$T + 124$$
$47$ $$T - 546$$
$53$ $$T + 408$$
$59$ $$T - 552$$
$61$ $$T - 404$$
$67$ $$T + 4$$
$71$ $$T - 126$$
$73$ $$T + 166$$
$79$ $$T + 874$$
$83$ $$T - 444$$
$89$ $$T - 1002$$
$97$ $$T + 802$$
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