LMFDB - Newform orbit 90.10.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,10,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, 10); 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, 10, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,16,0,256,-625,0,-7168] 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:	$46.3532252547$
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 + 16 q^{2} + 256 q^{4} - 625 q^{5} - 7168 q^{7} + 4096 q^{8} - 10000 q^{10} + 83748 q^{11} + 128126 q^{13} - 114688 q^{14} + 65536 q^{16} - 560802 q^{17} - 577660 q^{19} - 160000 q^{20} + 1339968 q^{22}+ \cdots + 176425872 q^{98}+O(q^{100}) $

q + 16 * q^2 + 256 * q^4 - 625 * q^5 - 7168 * q^7 + 4096 * q^8 - 10000 * q^10 + 83748 * q^11 + 128126 * q^13 - 114688 * q^14 + 65536 * q^16 - 560802 * q^17 - 577660 * q^19 - 160000 * q^20 + 1339968 * q^22 - 2431296 * q^23 + 390625 * q^25 + 2050016 * q^26 - 1835008 * q^28 - 5791710 * q^29 + 4145312 * q^31 + 1048576 * q^32 - 8972832 * q^34 + 4480000 * q^35 - 7011658 * q^37 - 9242560 * q^38 - 2560000 * q^40 + 8881398 * q^41 - 15730684 * q^43 + 21439488 * q^44 - 38900736 * q^46 - 60552072 * q^47 + 11026617 * q^49 + 6250000 * q^50 + 32800256 * q^52 - 30273366 * q^53 - 52342500 * q^55 - 29360128 * q^56 - 92667360 * q^58 - 45957660 * q^59 + 37595102 * q^61 + 66324992 * q^62 + 16777216 * q^64 - 80078750 * q^65 + 196784012 * q^67 - 143565312 * q^68 + 71680000 * q^70 - 56047992 * q^71 - 159688054 * q^73 - 112186528 * q^74 - 147880960 * q^76 - 600305664 * q^77 + 201923360 * q^79 - 40960000 * q^80 + 142102368 * q^82 + 362955444 * q^83 + 350501250 * q^85 - 251690944 * q^86 + 343031808 * q^88 + 272479110 * q^89 - 918407168 * q^91 - 622411776 * q^92 - 968833152 * q^94 + 361037500 * q^95 - 600852478 * q^97 + 176425872 * 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

16.0000

256.000

−625.000

−7168.00

4096.00

−10000.0

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	90.10.a.f		1
3.b	odd	2	1		30.10.a.b	✓	1
12.b	even	2	1		240.10.a.i		1
15.d	odd	2	1		150.10.a.k		1
15.e	even	4	2		150.10.c.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.10.a.b	✓	1	3.b	odd	2	1
90.10.a.f		1	1.a	even	1	1	trivial
150.10.a.k		1	15.d	odd	2	1
150.10.c.f		2	15.e	even	4	2
240.10.a.i		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 $S_{10}^{\mathrm{new}}(\Gamma_0(90))$:

$ T_{7} + 7168 $

T7 + 7168

$ T_{11} - 83748 $

T11 - 83748

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 16 $ T - 16
$3$	$ T $ T
$5$	$ T + 625 $ T + 625
$7$	$ T + 7168 $ T + 7168
$11$	$ T - 83748 $ T - 83748
$13$	$ T - 128126 $ T - 128126
$17$	$ T + 560802 $ T + 560802
$19$	$ T + 577660 $ T + 577660
$23$	$ T + 2431296 $ T + 2431296
$29$	$ T + 5791710 $ T + 5791710
$31$	$ T - 4145312 $ T - 4145312
$37$	$ T + 7011658 $ T + 7011658
$41$	$ T - 8881398 $ T - 8881398
$43$	$ T + 15730684 $ T + 15730684
$47$	$ T + 60552072 $ T + 60552072
$53$	$ T + 30273366 $ T + 30273366
$59$	$ T + 45957660 $ T + 45957660
$61$	$ T - 37595102 $ T - 37595102
$67$	$ T - 196784012 $ T - 196784012
$71$	$ T + 56047992 $ T + 56047992
$73$	$ T + 159688054 $ T + 159688054
$79$	$ T - 201923360 $ T - 201923360
$83$	$ T - 362955444 $ T - 362955444
$89$	$ T - 272479110 $ T - 272479110
$97$	$ T + 600852478 $ T + 600852478
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Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

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