LMFDB - Newform orbit 91.3.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [91,3,Mod(90,91)] 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("91.90"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	91.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,0,4,-3] 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:	$2.47957040568$
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 + 4 q^{4} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 13 q^{13} + 16 q^{16} + 25 q^{19} - 12 q^{20} - 45 q^{23} - 16 q^{25} + 28 q^{28} - 33 q^{29} - 55 q^{31} - 21 q^{35} + 36 q^{36} + 30 q^{41} - 5 q^{43} - 27 q^{45}+ \cdots - 131 q^{97}+O(q^{100}) $

q + 4 * q^4 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 13 * q^13 + 16 * q^16 + 25 * q^19 - 12 * q^20 - 45 * q^23 - 16 * q^25 + 28 * q^28 - 33 * q^29 - 55 * q^31 - 21 * q^35 + 36 * q^36 + 30 * q^41 - 5 * q^43 - 27 * q^45 + 81 * q^47 + 49 * q^49 - 52 * q^52 + 15 * q^53 - 90 * q^59 + 63 * q^63 + 64 * q^64 + 39 * q^65 + 29 * q^73 + 100 * q^76 + 67 * q^79 - 48 * q^80 + 81 * q^81 - 159 * q^83 + 165 * q^89 - 91 * q^91 - 180 * q^92 - 75 * q^95 - 131 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$.

$n$	$15$	$66$
$\chi(n)$	$-1$	$-1$

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} $

90.1

4.00000

−3.00000

7.00000

9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.b	odd	2	1	CM by $\Q(\sqrt{-91}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	91.3.b.a	✓	1
3.b	odd	2	1		819.3.d.b		1
7.b	odd	2	1		91.3.b.b	yes	1
13.b	even	2	1		91.3.b.b	yes	1
21.c	even	2	1		819.3.d.a		1
39.d	odd	2	1		819.3.d.a		1
91.b	odd	2	1	CM	91.3.b.a	✓	1
273.g	even	2	1		819.3.d.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.3.b.a	✓	1	1.a	even	1	1	trivial
91.3.b.a	✓	1	91.b	odd	2	1	CM
91.3.b.b	yes	1	7.b	odd	2	1
91.3.b.b	yes	1	13.b	even	2	1
819.3.d.a		1	21.c	even	2	1
819.3.d.a		1	39.d	odd	2	1
819.3.d.b		1	3.b	odd	2	1
819.3.d.b		1	273.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(91, [\chi])$:

$ T_{2} $

T2

$ T_{5} + 3 $

T5 + 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 3 $ T + 3
$7$	$ T - 7 $ T - 7
$11$	$ T $ T
$13$	$ T + 13 $ T + 13
$17$	$ T $ T
$19$	$ T - 25 $ T - 25
$23$	$ T + 45 $ T + 45
$29$	$ T + 33 $ T + 33
$31$	$ T + 55 $ T + 55
$37$	$ T $ T
$41$	$ T - 30 $ T - 30
$43$	$ T + 5 $ T + 5
$47$	$ T - 81 $ T - 81
$53$	$ T - 15 $ T - 15
$59$	$ T + 90 $ T + 90
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T - 29 $ T - 29
$79$	$ T - 67 $ T - 67
$83$	$ T + 159 $ T + 159
$89$	$ T - 165 $ T - 165
$97$	$ T + 131 $ T + 131
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Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	91.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-1\)	\(-1\)

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