LMFDB - Newform orbit 132.4.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [132,4,Mod(1,132)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 132 = 2^{2} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	132.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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} + 22 q^{5} - 20 q^{7} + 9 q^{9} + 11 q^{11} + 22 q^{13} - 66 q^{15} + 110 q^{17} + 48 q^{19} + 60 q^{21} + 72 q^{23} + 359 q^{25} - 27 q^{27} - 142 q^{29} + 184 q^{31} - 33 q^{33} - 440 q^{35}+ \cdots + 99 q^{99}+O(q^{100}) $

q - 3 * q^3 + 22 * q^5 - 20 * q^7 + 9 * q^9 + 11 * q^11 + 22 * q^13 - 66 * q^15 + 110 * q^17 + 48 * q^19 + 60 * q^21 + 72 * q^23 + 359 * q^25 - 27 * q^27 - 142 * q^29 + 184 * q^31 - 33 * q^33 - 440 * q^35 - 194 * q^37 - 66 * q^39 - 482 * q^41 - 80 * q^43 + 198 * q^45 + 392 * q^47 + 57 * q^49 - 330 * q^51 - 34 * q^53 + 242 * q^55 - 144 * q^57 - 108 * q^59 + 382 * q^61 - 180 * q^63 + 484 * q^65 + 84 * q^67 - 216 * q^69 - 1040 * q^71 - 606 * q^73 - 1077 * q^75 - 220 * q^77 - 1292 * q^79 + 81 * q^81 + 356 * q^83 + 2420 * q^85 + 426 * q^87 - 406 * q^89 - 440 * q^91 - 552 * q^93 + 1056 * q^95 + 1090 * 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

−3.00000

22.0000

−20.0000

9.00000

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.c	✓	1
3.b	odd	2	1		396.4.a.a		1
4.b	odd	2	1		528.4.a.l		1
8.b	even	2	1		2112.4.a.n		1
8.d	odd	2	1		2112.4.a.a		1
11.b	odd	2	1		1452.4.a.c		1
12.b	even	2	1		1584.4.a.a		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 22 $ T5 - 22 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(132))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 3 $ T + 3
$5$	$ T - 22 $ T - 22
$7$	$ T + 20 $ T + 20
$11$	$ T - 11 $ T - 11
$13$	$ T - 22 $ T - 22
$17$	$ T - 110 $ T - 110
$19$	$ T - 48 $ T - 48
$23$	$ T - 72 $ T - 72
$29$	$ T + 142 $ T + 142
$31$	$ T - 184 $ T - 184
$37$	$ T + 194 $ T + 194
$41$	$ T + 482 $ T + 482
$43$	$ T + 80 $ T + 80
$47$	$ T - 392 $ T - 392
$53$	$ T + 34 $ T + 34
$59$	$ T + 108 $ T + 108
$61$	$ T - 382 $ T - 382
$67$	$ T - 84 $ T - 84
$71$	$ T + 1040 $ T + 1040
$73$	$ T + 606 $ T + 606
$79$	$ T + 1292 $ T + 1292
$83$	$ T - 356 $ T - 356
$89$	$ T + 406 $ T + 406
$97$	$ T - 1090 $ T - 1090
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Level:	\( N \)	\(=\)	\( 132 = 2^{2} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	132.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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