LMFDB - Newform orbit 702.2.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [702,2,Mod(1,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("702.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 702 = 2 \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	702.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,1,0,-5] 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:	$5.60549822189$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - q^{2} + q^{4} + q^{5} - 5 q^{7} - q^{8} - q^{10} + 5 q^{11} - q^{13} + 5 q^{14} + q^{16} - 6 q^{17} - 6 q^{19} + q^{20} - 5 q^{22} + 4 q^{23} - 4 q^{25} + q^{26} - 5 q^{28} - 2 q^{29} + q^{31}+ \cdots - 18 q^{98}+O(q^{100}) $

q - q^2 + q^4 + q^5 - 5 * q^7 - q^8 - q^10 + 5 * q^11 - q^13 + 5 * q^14 + q^16 - 6 * q^17 - 6 * q^19 + q^20 - 5 * q^22 + 4 * q^23 - 4 * q^25 + q^26 - 5 * q^28 - 2 * q^29 + q^31 - q^32 + 6 * q^34 - 5 * q^35 - 6 * q^37 + 6 * q^38 - q^40 - 12 * q^41 + 4 * q^43 + 5 * q^44 - 4 * q^46 + 2 * q^47 + 18 * q^49 + 4 * q^50 - q^52 - 3 * q^53 + 5 * q^55 + 5 * q^56 + 2 * q^58 + 4 * q^59 - 14 * q^61 - q^62 + q^64 - q^65 - 8 * q^67 - 6 * q^68 + 5 * q^70 + 2 * q^71 - 13 * q^73 + 6 * q^74 - 6 * q^76 - 25 * q^77 + 8 * q^79 + q^80 + 12 * q^82 - 15 * q^83 - 6 * q^85 - 4 * q^86 - 5 * q^88 + 5 * q^91 + 4 * q^92 - 2 * q^94 - 6 * q^95 - 7 * q^97 - 18 * q^98

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

−1.00000

1.00000

−5.00000

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$13$	$ +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	702.2.a.e	✓	1
3.b	odd	2	1		702.2.a.l	yes	1
4.b	odd	2	1		5616.2.a.x		1
9.c	even	3	2		2106.2.e.u		2
9.d	odd	6	2		2106.2.e.i		2
12.b	even	2	1		5616.2.a.o		1
13.b	even	2	1		9126.2.a.x		1
39.d	odd	2	1		9126.2.a.o		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
702.2.a.e	✓	1	1.a	even	1	1	trivial
702.2.a.l	yes	1	3.b	odd	2	1
2106.2.e.i		2	9.d	odd	6	2
2106.2.e.u		2	9.c	even	3	2
5616.2.a.o		1	12.b	even	2	1
5616.2.a.x		1	4.b	odd	2	1
9126.2.a.o		1	39.d	odd	2	1
9126.2.a.x		1	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(702))$:

$ T_{5} - 1 $

T5 - 1

$ T_{7} + 5 $

T7 + 5

$ T_{11} - 5 $

T11 - 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T $ T
$5$	$ T - 1 $ T - 1
$7$	$ T + 5 $ T + 5
$11$	$ T - 5 $ T - 5
$13$	$ T + 1 $ T + 1
$17$	$ T + 6 $ T + 6
$19$	$ T + 6 $ T + 6
$23$	$ T - 4 $ T - 4
$29$	$ T + 2 $ T + 2
$31$	$ T - 1 $ T - 1
$37$	$ T + 6 $ T + 6
$41$	$ T + 12 $ T + 12
$43$	$ T - 4 $ T - 4
$47$	$ T - 2 $ T - 2
$53$	$ T + 3 $ T + 3
$59$	$ T - 4 $ T - 4
$61$	$ T + 14 $ T + 14
$67$	$ T + 8 $ T + 8
$71$	$ T - 2 $ T - 2
$73$	$ T + 13 $ T + 13
$79$	$ T - 8 $ T - 8
$83$	$ T + 15 $ T + 15
$89$	$ T $ T
$97$	$ T + 7 $ T + 7
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Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.a (trivial)

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