LMFDB - Newform orbit 144.16.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,16,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	144.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,42010,0,-385728] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$205.478647344$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
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 + 42010 q^{5} - 385728 q^{7} + 20444300 q^{11} + 75221302 q^{13} - 639707746 q^{17} - 267006356 q^{19} + 571925576 q^{23} - 28752738025 q^{25} + 62437913154 q^{29} + 1243357000 q^{31} - 16204433280 q^{35}+ \cdots - 12\!\cdots\!26 q^{97}+O(q^{100}) $

q + 42010 * q^5 - 385728 * q^7 + 20444300 * q^11 + 75221302 * q^13 - 639707746 * q^17 - 267006356 * q^19 + 571925576 * q^23 - 28752738025 * q^25 + 62437913154 * q^29 + 1243357000 * q^31 - 16204433280 * q^35 - 408247831602 * q^37 + 1376815491990 * q^41 + 1923521494772 * q^43 - 4743411679104 * q^47 - 4598775419959 * q^49 + 3260491936570 * q^53 + 858865043000 * q^55 + 6734040423500 * q^59 - 10498750519274 * q^61 + 3160046897020 * q^65 - 42186307399892 * q^67 + 81526680652600 * q^71 - 37550447155142 * q^73 - 7885938950400 * q^77 - 169301409686344 * q^79 + 357833604131476 * q^83 - 26874122409460 * q^85 + 157672146871542 * q^89 - 29014962377856 * q^91 - 11216937015560 * q^95 - 1220442820436926 * 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

42010.0

−385728.

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -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	144.16.a.i		1
3.b	odd	2	1		48.16.a.e		1
4.b	odd	2	1		72.16.a.b		1
12.b	even	2	1		24.16.a.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.16.a.a	✓	1	12.b	even	2	1
48.16.a.e		1	3.b	odd	2	1
72.16.a.b		1	4.b	odd	2	1
144.16.a.i		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 42010 $ T5 - 42010 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(144))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T - 42010 $ T - 42010
$7$	$ T + 385728 $ T + 385728
$11$	$ T - 20444300 $ T - 20444300
$13$	$ T - 75221302 $ T - 75221302
$17$	$ T + 639707746 $ T + 639707746
$19$	$ T + 267006356 $ T + 267006356
$23$	$ T - 571925576 $ T - 571925576
$29$	$ T - 62437913154 $ T - 62437913154
$31$	$ T - 1243357000 $ T - 1243357000
$37$	$ T + 408247831602 $ T + 408247831602
$41$	$ T - 1376815491990 $ T - 1376815491990
$43$	$ T - 1923521494772 $ T - 1923521494772
$47$	$ T + 4743411679104 $ T + 4743411679104
$53$	$ T - 3260491936570 $ T - 3260491936570
$59$	$ T - 6734040423500 $ T - 6734040423500
$61$	$ T + 10498750519274 $ T + 10498750519274
$67$	$ T + 42186307399892 $ T + 42186307399892
$71$	$ T - 81526680652600 $ T - 81526680652600
$73$	$ T + 37550447155142 $ T + 37550447155142
$79$	$ T + 169301409686344 $ T + 169301409686344
$83$	$ T - 357833604131476 $ T - 357833604131476
$89$	$ T - 157672146871542 $ T - 157672146871542
$97$	$ T + 1220442820436926 $ T + 1220442820436926
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Classical	Maass
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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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