LMFDB - Newform orbit 18.16.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [18,16,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	18.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-128,0,16384,263040] 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:	$25.6848309180$
Analytic rank:	$0$
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 - 128 q^{2} + 16384 q^{4} + 263040 q^{5} + 3585764 q^{7} - 2097152 q^{8} - 33669120 q^{10} + 65754624 q^{11} - 215571382 q^{13} - 458977792 q^{14} + 268435456 q^{16} + 1166982912 q^{17} - 5076345256 q^{19}+ \cdots - 10\!\cdots\!84 q^{98}+O(q^{100}) $

q - 128 * q^2 + 16384 * q^4 + 263040 * q^5 + 3585764 * q^7 - 2097152 * q^8 - 33669120 * q^10 + 65754624 * q^11 - 215571382 * q^13 - 458977792 * q^14 + 268435456 * q^16 + 1166982912 * q^17 - 5076345256 * q^19 + 4309647360 * q^20 - 8416591872 * q^22 + 897561600 * q^23 + 38672463475 * q^25 + 27593136896 * q^26 + 58749157376 * q^28 + 177789823104 * q^29 - 59934035644 * q^31 - 34359738368 * q^32 - 149373812736 * q^34 + 943199362560 * q^35 + 492717849086 * q^37 + 649772192768 * q^38 - 551634862080 * q^40 - 768992044800 * q^41 - 2375218814056 * q^43 + 1077323759616 * q^44 - 114887884800 * q^46 + 3705502540800 * q^47 + 8110141953753 * q^49 - 4950075324800 * q^50 - 3531921522688 * q^52 - 15767028761472 * q^53 + 17296096296960 * q^55 - 7519892144128 * q^56 - 22757097357312 * q^58 - 10870153227264 * q^59 - 21391088815258 * q^61 + 7671556562432 * q^62 + 4398046511104 * q^64 - 56703896321280 * q^65 + 65214454605392 * q^67 + 19119848030208 * q^68 - 120729518407680 * q^70 + 93219847483392 * q^71 + 73536094422470 * q^73 - 63067884683008 * q^74 - 83170840674304 * q^76 + 235780563572736 * q^77 + 151007275893764 * q^79 + 70609262346240 * q^80 + 98430981734400 * q^82 + 47723528999424 * q^83 + 306963185172480 * q^85 + 304028008199168 * q^86 - 137897441230848 * q^88 + 254772905098752 * q^89 - 772988101005848 * q^91 + 14705649254400 * q^92 - 474304325222400 * q^94 - 1335281856138240 * q^95 + 54230408468174 * q^97 - 1038098170080384 * 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

−128.000

16384.0

263040.

3.58576e6

−2.09715e6

−3.36691e7

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +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	18.16.a.c	✓	1
3.b	odd	2	1		18.16.a.d	yes	1
4.b	odd	2	1		144.16.a.n		1
12.b	even	2	1		144.16.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.16.a.c	✓	1	1.a	even	1	1	trivial
18.16.a.d	yes	1	3.b	odd	2	1
144.16.a.c		1	12.b	even	2	1
144.16.a.n		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 263040 $ T5 - 263040 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(18))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 128 $ T + 128
$3$	$ T $ T
$5$	$ T - 263040 $ T - 263040
$7$	$ T - 3585764 $ T - 3585764
$11$	$ T - 65754624 $ T - 65754624
$13$	$ T + 215571382 $ T + 215571382
$17$	$ T - 1166982912 $ T - 1166982912
$19$	$ T + 5076345256 $ T + 5076345256
$23$	$ T - 897561600 $ T - 897561600
$29$	$ T - 177789823104 $ T - 177789823104
$31$	$ T + 59934035644 $ T + 59934035644
$37$	$ T - 492717849086 $ T - 492717849086
$41$	$ T + 768992044800 $ T + 768992044800
$43$	$ T + 2375218814056 $ T + 2375218814056
$47$	$ T - 3705502540800 $ T - 3705502540800
$53$	$ T + 15767028761472 $ T + 15767028761472
$59$	$ T + 10870153227264 $ T + 10870153227264
$61$	$ T + 21391088815258 $ T + 21391088815258
$67$	$ T - 65214454605392 $ T - 65214454605392
$71$	$ T - 93219847483392 $ T - 93219847483392
$73$	$ T - 73536094422470 $ T - 73536094422470
$79$	$ T - 151007275893764 $ T - 151007275893764
$83$	$ T - 47723528999424 $ T - 47723528999424
$89$	$ T - 254772905098752 $ T - 254772905098752
$97$	$ T - 54230408468174 $ T - 54230408468174
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Classical	Maass
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Elliptic curves over $\Q$ over number fields
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

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