LMFDB - Newform orbit 588.8.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [588,8,Mod(1,588)] 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("588.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	588.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,27,0,-100] 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:	$183.682394985$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
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 + 27 q^{3} - 100 q^{5} + 729 q^{9} + 2774 q^{11} + 3294 q^{13} - 2700 q^{15} - 5900 q^{17} - 6644 q^{19} + 1982 q^{23} - 68125 q^{25} + 19683 q^{27} - 208106 q^{29} + 117792 q^{31} + 74898 q^{33} - 335686 q^{37}+ \cdots + 2022246 q^{99}+O(q^{100}) $

q + 27 * q^3 - 100 * q^5 + 729 * q^9 + 2774 * q^11 + 3294 * q^13 - 2700 * q^15 - 5900 * q^17 - 6644 * q^19 + 1982 * q^23 - 68125 * q^25 + 19683 * q^27 - 208106 * q^29 + 117792 * q^31 + 74898 * q^33 - 335686 * q^37 + 88938 * q^39 + 265488 * q^41 - 93292 * q^43 - 72900 * q^45 + 657516 * q^47 - 159300 * q^51 - 608718 * q^53 - 277400 * q^55 - 179388 * q^57 + 536120 * q^59 + 1797090 * q^61 - 329400 * q^65 + 2123176 * q^67 + 53514 * q^69 - 1191214 * q^71 - 1056430 * q^73 - 1839375 * q^75 + 998484 * q^79 + 531441 * q^81 - 3898004 * q^83 + 590000 * q^85 - 5618862 * q^87 + 4622352 * q^89 + 3180384 * q^93 + 664400 * q^95 - 15287710 * q^97 + 2022246 * 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

27.0000

−100.000

729.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$7$	$ -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	588.8.a.c		1
7.b	odd	2	1		84.8.a.a	✓	1
7.c	even	3	2		588.8.i.c		2
7.d	odd	6	2		588.8.i.f		2
21.c	even	2	1		252.8.a.a		1
28.d	even	2	1		336.8.a.j		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.8.a.a	✓	1	7.b	odd	2	1
252.8.a.a		1	21.c	even	2	1
336.8.a.j		1	28.d	even	2	1
588.8.a.c		1	1.a	even	1	1	trivial
588.8.i.c		2	7.c	even	3	2
588.8.i.f		2	7.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 100 $ T5 + 100 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(588))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 27 $ T - 27
$5$	$ T + 100 $ T + 100
$7$	$ T $ T
$11$	$ T - 2774 $ T - 2774
$13$	$ T - 3294 $ T - 3294
$17$	$ T + 5900 $ T + 5900
$19$	$ T + 6644 $ T + 6644
$23$	$ T - 1982 $ T - 1982
$29$	$ T + 208106 $ T + 208106
$31$	$ T - 117792 $ T - 117792
$37$	$ T + 335686 $ T + 335686
$41$	$ T - 265488 $ T - 265488
$43$	$ T + 93292 $ T + 93292
$47$	$ T - 657516 $ T - 657516
$53$	$ T + 608718 $ T + 608718
$59$	$ T - 536120 $ T - 536120
$61$	$ T - 1797090 $ T - 1797090
$67$	$ T - 2123176 $ T - 2123176
$71$	$ T + 1191214 $ T + 1191214
$73$	$ T + 1056430 $ T + 1056430
$79$	$ T - 998484 $ T - 998484
$83$	$ T + 3898004 $ T + 3898004
$89$	$ T - 4622352 $ T - 4622352
$97$	$ T + 15287710 $ T + 15287710
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Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

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