LMFDB - Newform orbit 1617.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1617,4,Mod(1,1617)] 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("1617.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1617, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-3,3,1,4] 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:	$95.4060884793$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
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 - 3 q^{2} + 3 q^{3} + q^{4} + 4 q^{5} - 9 q^{6} + 21 q^{8} + 9 q^{9} - 12 q^{10} + 11 q^{11} + 3 q^{12} - 50 q^{13} + 12 q^{15} - 71 q^{16} + 28 q^{17} - 27 q^{18} - 30 q^{19} + 4 q^{20} - 33 q^{22}+ \cdots + 99 q^{99}+O(q^{100}) $

q - 3 * q^2 + 3 * q^3 + q^4 + 4 * q^5 - 9 * q^6 + 21 * q^8 + 9 * q^9 - 12 * q^10 + 11 * q^11 + 3 * q^12 - 50 * q^13 + 12 * q^15 - 71 * q^16 + 28 * q^17 - 27 * q^18 - 30 * q^19 + 4 * q^20 - 33 * q^22 + 112 * q^23 + 63 * q^24 - 109 * q^25 + 150 * q^26 + 27 * q^27 + 130 * q^29 - 36 * q^30 + 146 * q^31 + 45 * q^32 + 33 * q^33 - 84 * q^34 + 9 * q^36 - 302 * q^37 + 90 * q^38 - 150 * q^39 + 84 * q^40 - 4 * q^41 - 548 * q^43 + 11 * q^44 + 36 * q^45 - 336 * q^46 - 86 * q^47 - 213 * q^48 + 327 * q^50 + 84 * q^51 - 50 * q^52 - 246 * q^53 - 81 * q^54 + 44 * q^55 - 90 * q^57 - 390 * q^58 - 120 * q^59 + 12 * q^60 + 638 * q^61 - 438 * q^62 + 433 * q^64 - 200 * q^65 - 99 * q^66 - 132 * q^67 + 28 * q^68 + 336 * q^69 - 692 * q^71 + 189 * q^72 + 152 * q^73 + 906 * q^74 - 327 * q^75 - 30 * q^76 + 450 * q^78 + 768 * q^79 - 284 * q^80 + 81 * q^81 + 12 * q^82 - 1098 * q^83 + 112 * q^85 + 1644 * q^86 + 390 * q^87 + 231 * q^88 + 1158 * q^89 - 108 * q^90 + 112 * q^92 + 438 * q^93 + 258 * q^94 - 120 * q^95 + 135 * q^96 - 1618 * q^97 + 99 * 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

−3.00000

3.00000

1.00000

4.00000

−9.00000

21.0000

9.00000

−12.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ -1 $
$11$	$ -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	1617.4.a.b		1
7.b	odd	2	1		231.4.a.a	✓	1
21.c	even	2	1		693.4.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.4.a.a	✓	1	7.b	odd	2	1
693.4.a.f		1	21.c	even	2	1
1617.4.a.b		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1617))$:

$ T_{2} + 3 $

T2 + 3

$ T_{5} - 4 $

T5 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 3 $ T + 3
$3$	$ T - 3 $ T - 3
$5$	$ T - 4 $ T - 4
$7$	$ T $ T
$11$	$ T - 11 $ T - 11
$13$	$ T + 50 $ T + 50
$17$	$ T - 28 $ T - 28
$19$	$ T + 30 $ T + 30
$23$	$ T - 112 $ T - 112
$29$	$ T - 130 $ T - 130
$31$	$ T - 146 $ T - 146
$37$	$ T + 302 $ T + 302
$41$	$ T + 4 $ T + 4
$43$	$ T + 548 $ T + 548
$47$	$ T + 86 $ T + 86
$53$	$ T + 246 $ T + 246
$59$	$ T + 120 $ T + 120
$61$	$ T - 638 $ T - 638
$67$	$ T + 132 $ T + 132
$71$	$ T + 692 $ T + 692
$73$	$ T - 152 $ T - 152
$79$	$ T - 768 $ T - 768
$83$	$ T + 1098 $ T + 1098
$89$	$ T - 1158 $ T - 1158
$97$	$ T + 1618 $ T + 1618
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Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

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