LMFDB - Newform orbit 150.12.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,12,Mod(1,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.1"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,32,243,1024,0,7776,57376] 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:	$115.251477084$
Analytic rank:	$0$
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} + 243 q^{3} + 1024 q^{4} + 7776 q^{6} + 57376 q^{7} + 32768 q^{8} + 59049 q^{9} + 954372 q^{11} + 248832 q^{12} - 978038 q^{13} + 1836032 q^{14} + 1048576 q^{16} + 4574766 q^{17} + 1889568 q^{18}+ \cdots + 56354712228 q^{99}+O(q^{100}) $

q + 32 * q^2 + 243 * q^3 + 1024 * q^4 + 7776 * q^6 + 57376 * q^7 + 32768 * q^8 + 59049 * q^9 + 954372 * q^11 + 248832 * q^12 - 978038 * q^13 + 1836032 * q^14 + 1048576 * q^16 + 4574766 * q^17 + 1889568 * q^18 + 429260 * q^19 + 13942368 * q^21 + 30539904 * q^22 + 25641792 * q^23 + 7962624 * q^24 - 31297216 * q^26 + 14348907 * q^27 + 58753024 * q^28 - 188685210 * q^29 + 34469072 * q^31 + 33554432 * q^32 + 231912396 * q^33 + 146392512 * q^34 + 60466176 * q^36 + 381698146 * q^37 + 13736320 * q^38 - 237663234 * q^39 - 1116342918 * q^41 + 446155776 * q^42 + 182578612 * q^43 + 977276928 * q^44 + 820537344 * q^46 - 2055898584 * q^47 + 254803968 * q^48 + 1314678633 * q^49 + 1111668138 * q^51 - 1001510912 * q^52 + 5352288402 * q^53 + 459165024 * q^54 + 1880096768 * q^56 + 104310180 * q^57 - 6037926720 * q^58 - 2306052060 * q^59 + 2262182822 * q^61 + 1103010304 * q^62 + 3387995424 * q^63 + 1073741824 * q^64 + 7421196672 * q^66 + 16091830396 * q^67 + 4684560384 * q^68 + 6230955456 * q^69 + 7283041032 * q^71 + 1934917632 * q^72 - 28423422458 * q^73 + 12214340672 * q^74 + 439562240 * q^76 + 54758047872 * q^77 - 7605223488 * q^78 - 385693360 * q^79 + 3486784401 * q^81 - 35722973376 * q^82 + 14785428252 * q^83 + 14276984832 * q^84 + 5842515584 * q^86 - 45850506030 * q^87 + 31272861696 * q^88 - 95789444790 * q^89 - 56115908288 * q^91 + 26257195008 * q^92 + 8375984496 * q^93 - 65788754688 * q^94 + 8153726976 * q^96 + 150483759166 * q^97 + 42069716256 * q^98 + 56354712228 * 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

32.0000

243.000

1024.00

7776.00

57376.0

32768.0

59049.0

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	150.12.a.i		1
5.b	even	2	1		30.12.a.b	✓	1
5.c	odd	4	2		150.12.c.i		2
15.d	odd	2	1		90.12.a.f		1
20.d	odd	2	1		240.12.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.12.a.b	✓	1	5.b	even	2	1
90.12.a.f		1	15.d	odd	2	1
150.12.a.i		1	1.a	even	1	1	trivial
150.12.c.i		2	5.c	odd	4	2
240.12.a.g		1	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 57376 $ T7 - 57376 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(150))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 32 $ T - 32
$3$	$ T - 243 $ T - 243
$5$	$ T $ T
$7$	$ T - 57376 $ T - 57376
$11$	$ T - 954372 $ T - 954372
$13$	$ T + 978038 $ T + 978038
$17$	$ T - 4574766 $ T - 4574766
$19$	$ T - 429260 $ T - 429260
$23$	$ T - 25641792 $ T - 25641792
$29$	$ T + 188685210 $ T + 188685210
$31$	$ T - 34469072 $ T - 34469072
$37$	$ T - 381698146 $ T - 381698146
$41$	$ T + 1116342918 $ T + 1116342918
$43$	$ T - 182578612 $ T - 182578612
$47$	$ T + 2055898584 $ T + 2055898584
$53$	$ T - 5352288402 $ T - 5352288402
$59$	$ T + 2306052060 $ T + 2306052060
$61$	$ T - 2262182822 $ T - 2262182822
$67$	$ T - 16091830396 $ T - 16091830396
$71$	$ T - 7283041032 $ T - 7283041032
$73$	$ T + 28423422458 $ T + 28423422458
$79$	$ T + 385693360 $ T + 385693360
$83$	$ T - 14785428252 $ T - 14785428252
$89$	$ T + 95789444790 $ T + 95789444790
$97$	$ T - 150483759166 $ T - 150483759166
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

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