LMFDB - Newform orbit 90.12.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,12,Mod(1,90)] 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("90.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-32,0,1024,-3125,0,-5152] 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:	$69.1508862504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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 - 32 q^{2} + 1024 q^{4} - 3125 q^{5} - 5152 q^{7} - 32768 q^{8} + 100000 q^{10} + 93180 q^{11} + 203894 q^{13} + 164864 q^{14} + 1048576 q^{16} - 4432338 q^{17} + 17715404 q^{19} - 3200000 q^{20} - 2981760 q^{22}+ \cdots + 62425076448 q^{98}+O(q^{100}) $

q - 32 * q^2 + 1024 * q^4 - 3125 * q^5 - 5152 * q^7 - 32768 * q^8 + 100000 * q^10 + 93180 * q^11 + 203894 * q^13 + 164864 * q^14 + 1048576 * q^16 - 4432338 * q^17 + 17715404 * q^19 - 3200000 * q^20 - 2981760 * q^22 - 38596800 * q^23 + 9765625 * q^25 - 6524608 * q^26 - 5275648 * q^28 - 4076646 * q^29 + 73231568 * q^31 - 33554432 * q^32 + 141834816 * q^34 + 16100000 * q^35 + 195272606 * q^37 - 566892928 * q^38 + 102400000 * q^40 - 46765434 * q^41 + 1436617292 * q^43 + 95416320 * q^44 + 1235097600 * q^46 + 278996520 * q^47 - 1950783639 * q^49 - 312500000 * q^50 + 208787456 * q^52 + 2179465362 * q^53 - 291187500 * q^55 + 168820736 * q^56 + 130452672 * q^58 + 7925830620 * q^59 - 2219791450 * q^61 - 2343410176 * q^62 + 1073741824 * q^64 - 637168750 * q^65 - 12104306812 * q^67 - 4538714112 * q^68 - 515200000 * q^70 - 18129117960 * q^71 - 24090734086 * q^73 - 6248723392 * q^74 + 18140573696 * q^76 - 480063360 * q^77 - 12116137648 * q^79 - 3276800000 * q^80 + 1496493888 * q^82 - 44049478884 * q^83 + 13851056250 * q^85 - 45971753344 * q^86 - 3053322240 * q^88 - 31204135242 * q^89 - 1050461888 * q^91 - 39523123200 * q^92 - 8927888640 * q^94 - 55360637500 * q^95 - 28332695614 * q^97 + 62425076448 * q^98

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

−32.0000

1024.00

−3125.00

−5152.00

−32768.0

100000.

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	90.12.a.a		1
3.b	odd	2	1		30.12.a.e	✓	1
12.b	even	2	1		240.12.a.f		1
15.d	odd	2	1		150.12.a.d		1
15.e	even	4	2		150.12.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.12.a.e	✓	1	3.b	odd	2	1
90.12.a.a		1	1.a	even	1	1	trivial
150.12.a.d		1	15.d	odd	2	1
150.12.c.c		2	15.e	even	4	2
240.12.a.f		1	12.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 $S_{12}^{\mathrm{new}}(\Gamma_0(90))$:

$ T_{7} + 5152 $

T7 + 5152

$ T_{11} - 93180 $

T11 - 93180

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T $ T
$5$	$ T + 3125 $ T + 3125
$7$	$ T + 5152 $ T + 5152
$11$	$ T - 93180 $ T - 93180
$13$	$ T - 203894 $ T - 203894
$17$	$ T + 4432338 $ T + 4432338
$19$	$ T - 17715404 $ T - 17715404
$23$	$ T + 38596800 $ T + 38596800
$29$	$ T + 4076646 $ T + 4076646
$31$	$ T - 73231568 $ T - 73231568
$37$	$ T - 195272606 $ T - 195272606
$41$	$ T + 46765434 $ T + 46765434
$43$	$ T - 1436617292 $ T - 1436617292
$47$	$ T - 278996520 $ T - 278996520
$53$	$ T - 2179465362 $ T - 2179465362
$59$	$ T - 7925830620 $ T - 7925830620
$61$	$ T + 2219791450 $ T + 2219791450
$67$	$ T + 12104306812 $ T + 12104306812
$71$	$ T + 18129117960 $ T + 18129117960
$73$	$ T + 24090734086 $ T + 24090734086
$79$	$ T + 12116137648 $ T + 12116137648
$83$	$ T + 44049478884 $ T + 44049478884
$89$	$ T + 31204135242 $ T + 31204135242
$97$	$ T + 28332695614 $ T + 28332695614
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Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

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