LMFDB - Newform orbit 168.3.e.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [168,3,Mod(83,168)] 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("168.83"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	168.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.57766844125$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9} + 12 q^{12} + 2 q^{13} - 14 q^{14} + 16 q^{16} - 22 q^{17} - 18 q^{18} + 21 q^{21} + 38 q^{23} - 24 q^{24} + 25 q^{25} - 4 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) $

q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 + 7 * q^7 - 8 * q^8 + 9 * q^9 + 12 * q^12 + 2 * q^13 - 14 * q^14 + 16 * q^16 - 22 * q^17 - 18 * q^18 + 21 * q^21 + 38 * q^23 - 24 * q^24 + 25 * q^25 - 4 * q^26 + 27 * q^27 + 28 * q^28 + 26 * q^29 - 34 * q^31 - 32 * q^32 + 44 * q^34 + 36 * q^36 + 6 * q^39 + 26 * q^41 - 42 * q^42 - 82 * q^43 - 76 * q^46 + 48 * q^48 + 49 * q^49 - 50 * q^50 - 66 * q^51 + 8 * q^52 - 22 * q^53 - 54 * q^54 - 56 * q^56 - 52 * q^58 - 106 * q^59 - 94 * q^61 + 68 * q^62 + 63 * q^63 + 64 * q^64 - 34 * q^67 - 88 * q^68 + 114 * q^69 - 58 * q^71 - 72 * q^72 + 75 * q^75 - 12 * q^78 + 81 * q^81 - 52 * q^82 - 58 * q^83 + 84 * q^84 + 164 * q^86 + 78 * q^87 + 122 * q^89 + 14 * q^91 + 152 * q^92 - 102 * q^93 - 96 * q^96 - 98 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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} $

83.1

−2.00000

3.00000

4.00000

−6.00000

7.00000

−8.00000

9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
168.e	odd	2	1	CM by $\Q(\sqrt{-42}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.3.e.b	yes	1
3.b	odd	2	1		168.3.e.c	yes	1
4.b	odd	2	1		672.3.e.a		1
7.b	odd	2	1		168.3.e.a	✓	1
8.b	even	2	1		672.3.e.b		1
8.d	odd	2	1		168.3.e.d	yes	1
12.b	even	2	1		672.3.e.c		1
21.c	even	2	1		168.3.e.d	yes	1
24.f	even	2	1		168.3.e.a	✓	1
24.h	odd	2	1		672.3.e.d		1
28.d	even	2	1		672.3.e.d		1
56.e	even	2	1		168.3.e.c	yes	1
56.h	odd	2	1		672.3.e.c		1
84.h	odd	2	1		672.3.e.b		1
168.e	odd	2	1	CM	168.3.e.b	yes	1
168.i	even	2	1		672.3.e.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.3.e.a	✓	1	7.b	odd	2	1
168.3.e.a	✓	1	24.f	even	2	1
168.3.e.b	yes	1	1.a	even	1	1	trivial
168.3.e.b	yes	1	168.e	odd	2	1	CM
168.3.e.c	yes	1	3.b	odd	2	1
168.3.e.c	yes	1	56.e	even	2	1
168.3.e.d	yes	1	8.d	odd	2	1
168.3.e.d	yes	1	21.c	even	2	1
672.3.e.a		1	4.b	odd	2	1
672.3.e.a		1	168.i	even	2	1
672.3.e.b		1	8.b	even	2	1
672.3.e.b		1	84.h	odd	2	1
672.3.e.c		1	12.b	even	2	1
672.3.e.c		1	56.h	odd	2	1
672.3.e.d		1	24.h	odd	2	1
672.3.e.d		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(168, [\chi])$:

$ T_{5} $

T5

$ T_{13} - 2 $

T13 - 2

$ T_{17} + 22 $

T17 + 22

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T - 3 $ T - 3
$5$	$ T $ T
$7$	$ T - 7 $ T - 7
$11$	$ T $ T
$13$	$ T - 2 $ T - 2
$17$	$ T + 22 $ T + 22
$19$	$ T $ T
$23$	$ T - 38 $ T - 38
$29$	$ T - 26 $ T - 26
$31$	$ T + 34 $ T + 34
$37$	$ T $ T
$41$	$ T - 26 $ T - 26
$43$	$ T + 82 $ T + 82
$47$	$ T $ T
$53$	$ T + 22 $ T + 22
$59$	$ T + 106 $ T + 106
$61$	$ T + 94 $ T + 94
$67$	$ T + 34 $ T + 34
$71$	$ T + 58 $ T + 58
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T + 58 $ T + 58
$89$	$ T - 122 $ T - 122
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	168.e (of order \(2\), degree \(1\), minimal)

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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