LMFDB - Newform orbit 7840.2.a.bf

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7840,2,Mod(1,7840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("7840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7840 = 2^{5} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,10,0,0,0,4,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.6027151847$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
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)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{3} - q^{5} + 5 q^{9} + 2 \beta q^{11} + 2 q^{13} - \beta q^{15} - 2 q^{17} - \beta q^{23} + q^{25} + 2 \beta q^{27} + 6 q^{29} + 2 \beta q^{31} + 16 q^{33} - 10 q^{37} + 2 \beta q^{39} - 2 q^{41} + \cdots + 10 \beta q^{99} +O(q^{100}) $ q + b * q^3 - q^5 + 5 * q^9 + 2b q^11 + 2 * q^13 - b * q^15 - 2 * q^17 - b * q^23 + q^25 + 2b q^27 + 6 * q^29 + 2b q^31 + 16 * q^33 - 10 * q^37 + 2b q^39 - 2 * q^41 + 3b q^43 - 5 * q^45 - b * q^47 - 2b q^51 + 6 * q^53 - 2b q^55 + 4b q^59 + 2 * q^61 - 2 * q^65 + b * q^67 - 8 * q^69 - 2b q^71 + 6 * q^73 + b * q^75 - 4b q^79 + q^81 + b * q^83 + 2 * q^85 + 6b q^87 - 10 * q^89 + 16 * q^93 - 2 * q^97 + 10b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 10 q^{9} + 4 q^{13} - 4 q^{17} + 2 q^{25} + 12 q^{29} + 32 q^{33} - 20 q^{37} - 4 q^{41} - 10 q^{45} + 12 q^{53} + 4 q^{61} - 4 q^{65} - 16 q^{69} + 12 q^{73} + 2 q^{81} + 4 q^{85} - 20 q^{89}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 10 * q^9 + 4 * q^13 - 4 * q^17 + 2 * q^25 + 12 * q^29 + 32 * q^33 - 20 * q^37 - 4 * q^41 - 10 * q^45 + 12 * q^53 + 4 * q^61 - 4 * q^65 - 16 * q^69 + 12 * q^73 + 2 * q^81 + 4 * q^85 - 20 * q^89 + 32 * q^93 - 4 * q^97

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

−1.41421
1.41421

−2.82843

−1.00000

5.00000

1.2

2.82843

−1.00000

5.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$5$	$ +1 $
$7$	$ -1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7840.2.a.bf		2
4.b	odd	2	1	inner	7840.2.a.bf		2
7.b	odd	2	1		160.2.a.c	✓	2
21.c	even	2	1		1440.2.a.o		2
28.d	even	2	1		160.2.a.c	✓	2
35.c	odd	2	1		800.2.a.m		2
35.f	even	4	2		800.2.c.f		4
56.e	even	2	1		320.2.a.g		2
56.h	odd	2	1		320.2.a.g		2
84.h	odd	2	1		1440.2.a.o		2
105.g	even	2	1		7200.2.a.cm		2
105.k	odd	4	2		7200.2.f.bh		4
112.j	even	4	2		1280.2.d.l		4
112.l	odd	4	2		1280.2.d.l		4
140.c	even	2	1		800.2.a.m		2
140.j	odd	4	2		800.2.c.f		4
168.e	odd	2	1		2880.2.a.bk		2
168.i	even	2	1		2880.2.a.bk		2
280.c	odd	2	1		1600.2.a.bc		2
280.n	even	2	1		1600.2.a.bc		2
280.s	even	4	2		1600.2.c.n		4
280.y	odd	4	2		1600.2.c.n		4
420.o	odd	2	1		7200.2.a.cm		2
420.w	even	4	2		7200.2.f.bh		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.a.c	✓	2	7.b	odd	2	1
160.2.a.c	✓	2	28.d	even	2	1
320.2.a.g		2	56.e	even	2	1
320.2.a.g		2	56.h	odd	2	1
800.2.a.m		2	35.c	odd	2	1
800.2.a.m		2	140.c	even	2	1
800.2.c.f		4	35.f	even	4	2
800.2.c.f		4	140.j	odd	4	2
1280.2.d.l		4	112.j	even	4	2
1280.2.d.l		4	112.l	odd	4	2
1440.2.a.o		2	21.c	even	2	1
1440.2.a.o		2	84.h	odd	2	1
1600.2.a.bc		2	280.c	odd	2	1
1600.2.a.bc		2	280.n	even	2	1
1600.2.c.n		4	280.s	even	4	2
1600.2.c.n		4	280.y	odd	4	2
2880.2.a.bk		2	168.e	odd	2	1
2880.2.a.bk		2	168.i	even	2	1
7200.2.a.cm		2	105.g	even	2	1
7200.2.a.cm		2	420.o	odd	2	1
7200.2.f.bh		4	105.k	odd	4	2
7200.2.f.bh		4	420.w	even	4	2
7840.2.a.bf		2	1.a	even	1	1	trivial
7840.2.a.bf		2	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7840))$:

$ T_{3}^{2} - 8 $

T3^2 - 8

$ T_{11}^{2} - 32 $

T11^2 - 32

$ T_{13} - 2 $

T13 - 2

$ T_{19} $

T19

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} - 8 $ T^2 - 8
$5$	$ (T + 1)^{2} $ (T + 1)^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 32 $ T^2 - 32
$13$	$ (T - 2)^{2} $ (T - 2)^2
$17$	$ (T + 2)^{2} $ (T + 2)^2
$19$	$ T^{2} $ T^2
$23$	$ T^{2} - 8 $ T^2 - 8
$29$	$ (T - 6)^{2} $ (T - 6)^2
$31$	$ T^{2} - 32 $ T^2 - 32
$37$	$ (T + 10)^{2} $ (T + 10)^2
$41$	$ (T + 2)^{2} $ (T + 2)^2
$43$	$ T^{2} - 72 $ T^2 - 72
$47$	$ T^{2} - 8 $ T^2 - 8
$53$	$ (T - 6)^{2} $ (T - 6)^2
$59$	$ T^{2} - 128 $ T^2 - 128
$61$	$ (T - 2)^{2} $ (T - 2)^2
$67$	$ T^{2} - 8 $ T^2 - 8
$71$	$ T^{2} - 32 $ T^2 - 32
$73$	$ (T - 6)^{2} $ (T - 6)^2
$79$	$ T^{2} - 128 $ T^2 - 128
$83$	$ T^{2} - 8 $ T^2 - 8
$89$	$ (T + 10)^{2} $ (T + 10)^2
$97$	$ (T + 2)^{2} $ (T + 2)^2
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Level:	\( N \)	\(=\)	\( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7840.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{3} - q^{5} + 5 q^{9} + 2 \beta q^{11} + 2 q^{13} - \beta q^{15} - 2 q^{17} - \beta q^{23} + q^{25} + 2 \beta q^{27} + 6 q^{29} + 2 \beta q^{31} + 16 q^{33} - 10 q^{37} + 2 \beta q^{39} - 2 q^{41} + \cdots + 10 \beta q^{99} +O(q^{100}) \) q + b * q^3 - q^5 + 5 * q^9 + 2b q^11 + 2 * q^13 - b * q^15 - 2 * q^17 - b * q^23 + q^25 + 2b q^27 + 6 * q^29 + 2b q^31 + 16 * q^33 - 10 * q^37 + 2b q^39 - 2 * q^41 + 3b q^43 - 5 * q^45 - b * q^47 - 2b q^51 + 6 * q^53 - 2b q^55 + 4b q^59 + 2 * q^61 - 2 * q^65 + b * q^67 - 8 * q^69 - 2b q^71 + 6 * q^73 + b * q^75 - 4b q^79 + q^81 + b * q^83 + 2 * q^85 + 6b q^87 - 10 * q^89 + 16 * q^93 - 2 * q^97 + 10b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 10 q^{9} + 4 q^{13} - 4 q^{17} + 2 q^{25} + 12 q^{29} + 32 q^{33} - 20 q^{37} - 4 q^{41} - 10 q^{45} + 12 q^{53} + 4 q^{61} - 4 q^{65} - 16 q^{69} + 12 q^{73} + 2 q^{81} + 4 q^{85} - 20 q^{89}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 10 * q^9 + 4 * q^13 - 4 * q^17 + 2 * q^25 + 12 * q^29 + 32 * q^33 - 20 * q^37 - 4 * q^41 - 10 * q^45 + 12 * q^53 + 4 * q^61 - 4 * q^65 - 16 * q^69 + 12 * q^73 + 2 * q^81 + 4 * q^85 - 20 * q^89 + 32 * q^93 - 4 * q^97

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