LMFDB - Newform orbit 45.14.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,14,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.2539180284$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
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 + 76 q^{2} - 2416 q^{4} - 15625 q^{5} - 224160 q^{7} - 806208 q^{8} - 1187500 q^{10} - 2313836 q^{11} + 10537318 q^{13} - 17036160 q^{14} - 41479936 q^{16} + 186660598 q^{17} - 290440676 q^{19} + 37750000 q^{20}+ \cdots - 3544739165332 q^{98}+O(q^{100}) $

q + 76 * q^2 - 2416 * q^4 - 15625 * q^5 - 224160 * q^7 - 806208 * q^8 - 1187500 * q^10 - 2313836 * q^11 + 10537318 * q^13 - 17036160 * q^14 - 41479936 * q^16 + 186660598 * q^17 - 290440676 * q^19 + 37750000 * q^20 - 175851536 * q^22 + 866818560 * q^23 + 244140625 * q^25 + 800836168 * q^26 + 541570560 * q^28 + 1566981362 * q^29 + 1200623400 * q^31 + 3451980800 * q^32 + 14186205448 * q^34 + 3502500000 * q^35 - 12182249410 * q^37 - 22073491376 * q^38 + 12597000000 * q^40 + 29167834310 * q^41 + 49361767564 * q^43 + 5590227776 * q^44 + 65878210560 * q^46 + 11671527832 * q^47 - 46641304807 * q^49 + 18554687500 * q^50 - 25458160288 * q^52 - 100929409430 * q^53 + 36153687500 * q^55 + 180719585280 * q^56 + 119090583512 * q^58 + 265189749604 * q^59 - 566433594722 * q^61 + 91247378400 * q^62 + 602154176512 * q^64 - 164645593750 * q^65 + 1441180693572 * q^67 - 450972004768 * q^68 + 266190000000 * q^70 + 502944753848 * q^71 + 1574910852730 * q^73 - 925850955160 * q^74 + 701704673216 * q^76 + 518669477760 * q^77 + 338387056680 * q^79 + 648124000000 * q^80 + 2216755407560 * q^82 + 4771809968748 * q^83 - 2916571843750 * q^85 + 3751494334864 * q^86 + 1865433093888 * q^88 - 2746483865994 * q^89 - 2362045202880 * q^91 - 2094233640960 * q^92 + 887036115232 * q^94 + 4538135562500 * q^95 + 1979074481282 * q^97 - 3544739165332 * 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

76.0000

−2416.00

−15625.0

−224160.

−806208.

−1.18750e6

Atkin-Lehner signs

$ p $	Sign
$3$	$ -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	45.14.a.a		1
3.b	odd	2	1		15.14.a.a	✓	1
15.d	odd	2	1		75.14.a.b		1
15.e	even	4	2		75.14.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.14.a.a	✓	1	3.b	odd	2	1
45.14.a.a		1	1.a	even	1	1	trivial
75.14.a.b		1	15.d	odd	2	1
75.14.b.a		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 76 $ T2 - 76 acting on $S_{14}^{\mathrm{new}}(\Gamma_0(45))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 76 $ T - 76
$3$	$ T $ T
$5$	$ T + 15625 $ T + 15625
$7$	$ T + 224160 $ T + 224160
$11$	$ T + 2313836 $ T + 2313836
$13$	$ T - 10537318 $ T - 10537318
$17$	$ T - 186660598 $ T - 186660598
$19$	$ T + 290440676 $ T + 290440676
$23$	$ T - 866818560 $ T - 866818560
$29$	$ T - 1566981362 $ T - 1566981362
$31$	$ T - 1200623400 $ T - 1200623400
$37$	$ T + 12182249410 $ T + 12182249410
$41$	$ T - 29167834310 $ T - 29167834310
$43$	$ T - 49361767564 $ T - 49361767564
$47$	$ T - 11671527832 $ T - 11671527832
$53$	$ T + 100929409430 $ T + 100929409430
$59$	$ T - 265189749604 $ T - 265189749604
$61$	$ T + 566433594722 $ T + 566433594722
$67$	$ T - 1441180693572 $ T - 1441180693572
$71$	$ T - 502944753848 $ T - 502944753848
$73$	$ T - 1574910852730 $ T - 1574910852730
$79$	$ T - 338387056680 $ T - 338387056680
$83$	$ T - 4771809968748 $ T - 4771809968748
$89$	$ T + 2746483865994 $ T + 2746483865994
$97$	$ T - 1979074481282 $ T - 1979074481282
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Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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