LMFDB - Newform orbit 3200.2.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3200,2,Mod(1601,3200)] 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("3200.1601"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-8,0,4,0,0,0,0,0,0,0,-2,0,0,0,0,0,-8,0,0,0,0,0,0, 0,8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)

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

Self dual:	no
Analytic conductor:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + i q^{3} - 4 q^{7} + 2 q^{9} - 3 i q^{11} - q^{17} + 7 i q^{19} - 4 i q^{21} - 4 q^{23} + 5 i q^{27} - 8 i q^{29} + 4 q^{31} + 3 q^{33} + 4 i q^{37} + 3 q^{41} - 8 i q^{43} + 9 q^{49} - i q^{51} - 12 i q^{53} + \cdots - 6 i q^{99} +O(q^{100}) $ q + i * q^3 - 4 * q^7 + 2 * q^9 - 3i q^11 - q^17 + 7i q^19 - 4i q^21 - 4 * q^23 + 5i q^27 - 8i q^29 + 4 * q^31 + 3 * q^33 + 4i q^37 + 3 * q^41 - 8i q^43 + 9 * q^49 - i * q^51 - 12i q^53 - 7 * q^57 - 8i q^59 + 4i q^61 - 8 * q^63 - 9i q^67 - 4i q^69 - 16 * q^71 + 11 * q^73 + 12i q^77 + 4 * q^79 + q^81 - i * q^83 + 8 * q^87 + 13 * q^89 + 4i q^93 - 14 * q^97 - 6i q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} + 4 q^{9} - 2 q^{17} - 8 q^{23} + 8 q^{31} + 6 q^{33} + 6 q^{41} + 18 q^{49} - 14 q^{57} - 16 q^{63} - 32 q^{71} + 22 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 26 q^{89} - 28 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 + 4 * q^9 - 2 * q^17 - 8 * q^23 + 8 * q^31 + 6 * q^33 + 6 * q^41 + 18 * q^49 - 14 * q^57 - 16 * q^63 - 32 * q^71 + 22 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 26 * q^89 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$1$

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} $

1601.1

	−	1.00000i
		1.00000i

−

1.00000i

−4.00000

2.00000

1601.2

1.00000i

−4.00000

2.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3200.2.d.a	✓	2
4.b	odd	2	1		3200.2.d.g	yes	2
5.b	even	2	1		3200.2.d.h	yes	2
5.c	odd	4	1		3200.2.f.b		2
5.c	odd	4	1		3200.2.f.f		2
8.b	even	2	1	inner	3200.2.d.a	✓	2
8.d	odd	2	1		3200.2.d.g	yes	2
16.e	even	4	1		6400.2.a.h		1
16.e	even	4	1		6400.2.a.w		1
16.f	odd	4	1		6400.2.a.b		1
16.f	odd	4	1		6400.2.a.q		1
20.d	odd	2	1		3200.2.d.b	yes	2
20.e	even	4	1		3200.2.f.a		2
20.e	even	4	1		3200.2.f.e		2
40.e	odd	2	1		3200.2.d.b	yes	2
40.f	even	2	1		3200.2.d.h	yes	2
40.i	odd	4	1		3200.2.f.b		2
40.i	odd	4	1		3200.2.f.f		2
40.k	even	4	1		3200.2.f.a		2
40.k	even	4	1		3200.2.f.e		2
80.k	odd	4	1		6400.2.a.i		1
80.k	odd	4	1		6400.2.a.v		1
80.q	even	4	1		6400.2.a.c		1
80.q	even	4	1		6400.2.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3200.2.d.a	✓	2	1.a	even	1	1	trivial
3200.2.d.a	✓	2	8.b	even	2	1	inner
3200.2.d.b	yes	2	20.d	odd	2	1
3200.2.d.b	yes	2	40.e	odd	2	1
3200.2.d.g	yes	2	4.b	odd	2	1
3200.2.d.g	yes	2	8.d	odd	2	1
3200.2.d.h	yes	2	5.b	even	2	1
3200.2.d.h	yes	2	40.f	even	2	1
3200.2.f.a		2	20.e	even	4	1
3200.2.f.a		2	40.k	even	4	1
3200.2.f.b		2	5.c	odd	4	1
3200.2.f.b		2	40.i	odd	4	1
3200.2.f.e		2	20.e	even	4	1
3200.2.f.e		2	40.k	even	4	1
3200.2.f.f		2	5.c	odd	4	1
3200.2.f.f		2	40.i	odd	4	1
6400.2.a.b		1	16.f	odd	4	1
6400.2.a.c		1	80.q	even	4	1
6400.2.a.h		1	16.e	even	4	1
6400.2.a.i		1	80.k	odd	4	1
6400.2.a.p		1	80.q	even	4	1
6400.2.a.q		1	16.f	odd	4	1
6400.2.a.v		1	80.k	odd	4	1
6400.2.a.w		1	16.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3200, [\chi])$:

$ T_{3}^{2} + 1 $

T3^2 + 1

$ T_{7} + 4 $

T7 + 4

$ T_{11}^{2} + 9 $

T11^2 + 9

$ T_{13} $

T13

$ T_{17} + 1 $

T17 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 1 $ T^2 + 1
$5$	$ T^{2} $ T^2
$7$	$ (T + 4)^{2} $ (T + 4)^2
$11$	$ T^{2} + 9 $ T^2 + 9
$13$	$ T^{2} $ T^2
$17$	$ (T + 1)^{2} $ (T + 1)^2
$19$	$ T^{2} + 49 $ T^2 + 49
$23$	$ (T + 4)^{2} $ (T + 4)^2
$29$	$ T^{2} + 64 $ T^2 + 64
$31$	$ (T - 4)^{2} $ (T - 4)^2
$37$	$ T^{2} + 16 $ T^2 + 16
$41$	$ (T - 3)^{2} $ (T - 3)^2
$43$	$ T^{2} + 64 $ T^2 + 64
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + 144 $ T^2 + 144
$59$	$ T^{2} + 64 $ T^2 + 64
$61$	$ T^{2} + 16 $ T^2 + 16
$67$	$ T^{2} + 81 $ T^2 + 81
$71$	$ (T + 16)^{2} $ (T + 16)^2
$73$	$ (T - 11)^{2} $ (T - 11)^2
$79$	$ (T - 4)^{2} $ (T - 4)^2
$83$	$ T^{2} + 1 $ T^2 + 1
$89$	$ (T - 13)^{2} $ (T - 13)^2
$97$	$ (T + 14)^{2} $ (T + 14)^2
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Elliptic curves over $\Q$ over number fields
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Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + i q^{3} - 4 q^{7} + 2 q^{9} - 3 i q^{11} - q^{17} + 7 i q^{19} - 4 i q^{21} - 4 q^{23} + 5 i q^{27} - 8 i q^{29} + 4 q^{31} + 3 q^{33} + 4 i q^{37} + 3 q^{41} - 8 i q^{43} + 9 q^{49} - i q^{51} - 12 i q^{53} + \cdots - 6 i q^{99} +O(q^{100}) \) q + i * q^3 - 4 * q^7 + 2 * q^9 - 3i q^11 - q^17 + 7i q^19 - 4i q^21 - 4 * q^23 + 5i q^27 - 8i q^29 + 4 * q^31 + 3 * q^33 + 4i q^37 + 3 * q^41 - 8i q^43 + 9 * q^49 - i * q^51 - 12i q^53 - 7 * q^57 - 8i q^59 + 4i q^61 - 8 * q^63 - 9i q^67 - 4i q^69 - 16 * q^71 + 11 * q^73 + 12i q^77 + 4 * q^79 + q^81 - i * q^83 + 8 * q^87 + 13 * q^89 + 4i q^93 - 14 * q^97 - 6i q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} + 4 q^{9} - 2 q^{17} - 8 q^{23} + 8 q^{31} + 6 q^{33} + 6 q^{41} + 18 q^{49} - 14 q^{57} - 16 q^{63} - 32 q^{71} + 22 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 26 q^{89} - 28 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 + 4 * q^9 - 2 * q^17 - 8 * q^23 + 8 * q^31 + 6 * q^33 + 6 * q^41 + 18 * q^49 - 14 * q^57 - 16 * q^63 - 32 * q^71 + 22 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 26 * q^89 - 28 * q^97

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