LMFDB - Newform orbit 1472.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-8,0,4,0,-4] 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:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
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 - 8 q^{3} + 4 q^{5} - 4 q^{7} + 37 q^{9} - 26 q^{11} - 70 q^{13} - 32 q^{15} + 94 q^{17} - 54 q^{19} + 32 q^{21} - 23 q^{23} - 109 q^{25} - 80 q^{27} + 86 q^{29} - 144 q^{31} + 208 q^{33} - 16 q^{35}+ \cdots - 962 q^{99}+O(q^{100}) $

q - 8 * q^3 + 4 * q^5 - 4 * q^7 + 37 * q^9 - 26 * q^11 - 70 * q^13 - 32 * q^15 + 94 * q^17 - 54 * q^19 + 32 * q^21 - 23 * q^23 - 109 * q^25 - 80 * q^27 + 86 * q^29 - 144 * q^31 + 208 * q^33 - 16 * q^35 + 172 * q^37 + 560 * q^39 - 42 * q^41 - 386 * q^43 + 148 * q^45 - 80 * q^47 - 327 * q^49 - 752 * q^51 + 108 * q^53 - 104 * q^55 + 432 * q^57 - 164 * q^59 + 400 * q^61 - 148 * q^63 - 280 * q^65 - 398 * q^67 + 184 * q^69 - 320 * q^71 - 810 * q^73 + 872 * q^75 + 104 * q^77 - 204 * q^79 - 359 * q^81 - 102 * q^83 + 376 * q^85 - 688 * q^87 + 1018 * q^89 + 280 * q^91 + 1152 * q^93 - 216 * q^95 - 1370 * q^97 - 962 * 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

−8.00000

4.00000

−4.00000

37.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$23$	$ +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	1472.4.a.b		1
4.b	odd	2	1		1472.4.a.i		1
8.b	even	2	1		184.4.a.b	✓	1
8.d	odd	2	1		368.4.a.a		1
24.h	odd	2	1		1656.4.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
184.4.a.b	✓	1	8.b	even	2	1
368.4.a.a		1	8.d	odd	2	1
1472.4.a.b		1	1.a	even	1	1	trivial
1472.4.a.i		1	4.b	odd	2	1
1656.4.a.c		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 8 $ T3 + 8 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1472))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 8 $ T + 8
$5$	$ T - 4 $ T - 4
$7$	$ T + 4 $ T + 4
$11$	$ T + 26 $ T + 26
$13$	$ T + 70 $ T + 70
$17$	$ T - 94 $ T - 94
$19$	$ T + 54 $ T + 54
$23$	$ T + 23 $ T + 23
$29$	$ T - 86 $ T - 86
$31$	$ T + 144 $ T + 144
$37$	$ T - 172 $ T - 172
$41$	$ T + 42 $ T + 42
$43$	$ T + 386 $ T + 386
$47$	$ T + 80 $ T + 80
$53$	$ T - 108 $ T - 108
$59$	$ T + 164 $ T + 164
$61$	$ T - 400 $ T - 400
$67$	$ T + 398 $ T + 398
$71$	$ T + 320 $ T + 320
$73$	$ T + 810 $ T + 810
$79$	$ T + 204 $ T + 204
$83$	$ T + 102 $ T + 102
$89$	$ T - 1018 $ T - 1018
$97$	$ T + 1370 $ T + 1370
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Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

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