LMFDB - Newform orbit 150.16.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,16,Mod(1,150)] 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("150.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,128,2187,16384,0,279936,511994] 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:	$214.040257650$
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 + 128 q^{2} + 2187 q^{3} + 16384 q^{4} + 279936 q^{6} + 511994 q^{7} + 2097152 q^{8} + 4782969 q^{9} - 19947598 q^{11} + 35831808 q^{12} - 180100962 q^{13} + 65535232 q^{14} + 268435456 q^{16} + 565654514 q^{17}+ \cdots - 95408742858462 q^{99}+O(q^{100}) $

q + 128 * q^2 + 2187 * q^3 + 16384 * q^4 + 279936 * q^6 + 511994 * q^7 + 2097152 * q^8 + 4782969 * q^9 - 19947598 * q^11 + 35831808 * q^12 - 180100962 * q^13 + 65535232 * q^14 + 268435456 * q^16 + 565654514 * q^17 + 612220032 * q^18 - 2169230520 * q^19 + 1119730878 * q^21 - 2553292544 * q^22 - 3443072572 * q^23 + 4586471424 * q^24 - 23052923136 * q^26 + 10460353203 * q^27 + 8388509696 * q^28 + 58843361520 * q^29 - 131248934648 * q^31 + 34359738368 * q^32 - 43625396826 * q^33 + 72403777792 * q^34 + 78364164096 * q^36 - 1018926148246 * q^37 - 277661506560 * q^38 - 393880803894 * q^39 - 678311212798 * q^41 + 143325552384 * q^42 + 1869778918508 * q^43 - 326821445632 * q^44 - 440713289216 * q^46 - 2655546152576 * q^47 + 587068342272 * q^48 - 4485423653907 * q^49 + 1237086422118 * q^51 - 2950774161408 * q^52 + 7733409097998 * q^53 + 1338925209984 * q^54 + 1073729241088 * q^56 - 4744107147240 * q^57 + 7531950274560 * q^58 + 12384358030090 * q^59 - 28742040118198 * q^61 - 16799863634944 * q^62 + 2448851430186 * q^63 + 4398046511104 * q^64 - 5584050793728 * q^66 - 62114558153336 * q^67 + 9267683557376 * q^68 - 7529999714964 * q^69 + 17045715067452 * q^71 + 10030613004288 * q^72 + 94423385896028 * q^73 - 130422546975488 * q^74 - 35540672839680 * q^76 - 10213050490412 * q^77 - 50416742898432 * q^78 + 5162001412320 * q^79 + 22876792454961 * q^81 - 86823835238144 * q^82 - 388824931818532 * q^83 + 18345670705152 * q^84 + 239331701569024 * q^86 + 128690431644240 * q^87 - 41833145040896 * q^88 + 64970078898710 * q^89 - 92210611938228 * q^91 - 56411301019648 * q^92 - 287041420075176 * q^93 - 339909907529728 * q^94 + 75144747810816 * q^96 - 424522131387176 * q^97 - 574134227700096 * q^98 - 95408742858462 * 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

128.000

2187.00

16384.0

279936.

511994.

2.09715e6

4.78297e6

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	150.16.a.i		1
5.b	even	2	1		150.16.a.b		1
5.c	odd	4	2		30.16.c.a	✓	2
15.e	even	4	2		90.16.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.16.c.a	✓	2	5.c	odd	4	2
90.16.c.a		2	15.e	even	4	2
150.16.a.b		1	5.b	even	2	1
150.16.a.i		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 511994 $ T7 - 511994 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(150))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 128 $ T - 128
$3$	$ T - 2187 $ T - 2187
$5$	$ T $ T
$7$	$ T - 511994 $ T - 511994
$11$	$ T + 19947598 $ T + 19947598
$13$	$ T + 180100962 $ T + 180100962
$17$	$ T - 565654514 $ T - 565654514
$19$	$ T + 2169230520 $ T + 2169230520
$23$	$ T + 3443072572 $ T + 3443072572
$29$	$ T - 58843361520 $ T - 58843361520
$31$	$ T + 131248934648 $ T + 131248934648
$37$	$ T + 1018926148246 $ T + 1018926148246
$41$	$ T + 678311212798 $ T + 678311212798
$43$	$ T - 1869778918508 $ T - 1869778918508
$47$	$ T + 2655546152576 $ T + 2655546152576
$53$	$ T - 7733409097998 $ T - 7733409097998
$59$	$ T - 12384358030090 $ T - 12384358030090
$61$	$ T + 28742040118198 $ T + 28742040118198
$67$	$ T + 62114558153336 $ T + 62114558153336
$71$	$ T - 17045715067452 $ T - 17045715067452
$73$	$ T - 94423385896028 $ T - 94423385896028
$79$	$ T - 5162001412320 $ T - 5162001412320
$83$	$ T + 388824931818532 $ T + 388824931818532
$89$	$ T - 64970078898710 $ T - 64970078898710
$97$	$ T + 424522131387176 $ T + 424522131387176
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

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