LMFDB - Newform orbit 300.8.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [300,8,Mod(1,300)] 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("300.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-27,0,0,0,-722,0,729,0,-3994] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$93.7155076452$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
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} - 722 q^{7} + 729 q^{9} - 3994 q^{11} + 3030 q^{13} - 20582 q^{17} - 25320 q^{19} + 19494 q^{21} + 66652 q^{23} - 19683 q^{27} - 152664 q^{29} - 123776 q^{31} + 107838 q^{33} - 337886 q^{37}+ \cdots - 2911626 q^{99}+O(q^{100}) $

q - 27 * q^3 - 722 * q^7 + 729 * q^9 - 3994 * q^11 + 3030 * q^13 - 20582 * q^17 - 25320 * q^19 + 19494 * q^21 + 66652 * q^23 - 19683 * q^27 - 152664 * q^29 - 123776 * q^31 + 107838 * q^33 - 337886 * q^37 - 81810 * q^39 + 396530 * q^41 + 442852 * q^43 + 170432 * q^47 - 302259 * q^49 + 555714 * q^51 - 1239426 * q^53 + 683640 * q^57 - 302354 * q^59 - 2830198 * q^61 - 526338 * q^63 + 3741272 * q^67 - 1799604 * q^69 - 1007580 * q^71 + 2404636 * q^73 + 2883668 * q^77 + 7517832 * q^79 + 531441 * q^81 - 5299628 * q^83 + 4121928 * q^87 - 7650250 * q^89 - 2187660 * q^91 + 3341952 * q^93 - 10055944 * q^97 - 2911626 * 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

−722.000

729.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ -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	300.8.a.b		1
5.b	even	2	1		300.8.a.f		1
5.c	odd	4	2		60.8.d.a	✓	2
15.e	even	4	2		180.8.d.a		2
20.e	even	4	2		240.8.f.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.8.d.a	✓	2	5.c	odd	4	2
180.8.d.a		2	15.e	even	4	2
240.8.f.b		2	20.e	even	4	2
300.8.a.b		1	1.a	even	1	1	trivial
300.8.a.f		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(300))$:

$ T_{7} + 722 $

T7 + 722

$ T_{11} + 3994 $

T11 + 3994

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 27 $ T + 27
$5$	$ T $ T
$7$	$ T + 722 $ T + 722
$11$	$ T + 3994 $ T + 3994
$13$	$ T - 3030 $ T - 3030
$17$	$ T + 20582 $ T + 20582
$19$	$ T + 25320 $ T + 25320
$23$	$ T - 66652 $ T - 66652
$29$	$ T + 152664 $ T + 152664
$31$	$ T + 123776 $ T + 123776
$37$	$ T + 337886 $ T + 337886
$41$	$ T - 396530 $ T - 396530
$43$	$ T - 442852 $ T - 442852
$47$	$ T - 170432 $ T - 170432
$53$	$ T + 1239426 $ T + 1239426
$59$	$ T + 302354 $ T + 302354
$61$	$ T + 2830198 $ T + 2830198
$67$	$ T - 3741272 $ T - 3741272
$71$	$ T + 1007580 $ T + 1007580
$73$	$ T - 2404636 $ T - 2404636
$79$	$ T - 7517832 $ T - 7517832
$83$	$ T + 5299628 $ T + 5299628
$89$	$ T + 7650250 $ T + 7650250
$97$	$ T + 10055944 $ T + 10055944
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Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)

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