LMFDB - Newform orbit 60.8.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [60,8,Mod(1,60)] 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("60.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	60.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,27,0,125] 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:	$18.7431015290$
Analytic rank:	$1$
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 + 27 q^{3} + 125 q^{5} - 1408 q^{7} + 729 q^{9} - 4044 q^{11} - 5890 q^{13} + 3375 q^{15} + 31002 q^{17} - 40300 q^{19} - 38016 q^{21} - 78912 q^{23} + 15625 q^{25} + 19683 q^{27} - 157194 q^{29} + 114824 q^{31}+ \cdots - 2948076 q^{99}+O(q^{100}) $

q + 27 * q^3 + 125 * q^5 - 1408 * q^7 + 729 * q^9 - 4044 * q^11 - 5890 * q^13 + 3375 * q^15 + 31002 * q^17 - 40300 * q^19 - 38016 * q^21 - 78912 * q^23 + 15625 * q^25 + 19683 * q^27 - 157194 * q^29 + 114824 * q^31 - 109188 * q^33 - 176000 * q^35 - 471994 * q^37 - 159030 * q^39 - 404310 * q^41 - 253852 * q^43 + 91125 * q^45 + 437688 * q^47 + 1158921 * q^49 + 837054 * q^51 + 334926 * q^53 - 505500 * q^55 - 1088100 * q^57 + 562596 * q^59 + 3246662 * q^61 - 1026432 * q^63 - 736250 * q^65 + 3895148 * q^67 - 2130624 * q^69 - 2345160 * q^71 + 5726954 * q^73 + 421875 * q^75 + 5693952 * q^77 - 5222008 * q^79 + 531441 * q^81 - 2928132 * q^83 + 3875250 * q^85 - 4244238 * q^87 - 3160230 * q^89 + 8293120 * q^91 + 3100248 * q^93 - 5037500 * q^95 - 1898686 * q^97 - 2948076 * 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

125.000

−1408.00

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	60.8.a.d	✓	1
3.b	odd	2	1		180.8.a.a		1
4.b	odd	2	1		240.8.a.g		1
5.b	even	2	1		300.8.a.d		1
5.c	odd	4	2		300.8.d.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.8.a.d	✓	1	1.a	even	1	1	trivial
180.8.a.a		1	3.b	odd	2	1
240.8.a.g		1	4.b	odd	2	1
300.8.a.d		1	5.b	even	2	1
300.8.d.b		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 1408 $ T7 + 1408 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(60))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 27 $ T - 27
$5$	$ T - 125 $ T - 125
$7$	$ T + 1408 $ T + 1408
$11$	$ T + 4044 $ T + 4044
$13$	$ T + 5890 $ T + 5890
$17$	$ T - 31002 $ T - 31002
$19$	$ T + 40300 $ T + 40300
$23$	$ T + 78912 $ T + 78912
$29$	$ T + 157194 $ T + 157194
$31$	$ T - 114824 $ T - 114824
$37$	$ T + 471994 $ T + 471994
$41$	$ T + 404310 $ T + 404310
$43$	$ T + 253852 $ T + 253852
$47$	$ T - 437688 $ T - 437688
$53$	$ T - 334926 $ T - 334926
$59$	$ T - 562596 $ T - 562596
$61$	$ T - 3246662 $ T - 3246662
$67$	$ T - 3895148 $ T - 3895148
$71$	$ T + 2345160 $ T + 2345160
$73$	$ T - 5726954 $ T - 5726954
$79$	$ T + 5222008 $ T + 5222008
$83$	$ T + 2928132 $ T + 2928132
$89$	$ T + 3160230 $ T + 3160230
$97$	$ T + 1898686 $ T + 1898686
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Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

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