LMFDB - Newform orbit 800.2.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(401,800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 200)
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 $\beta = \sqrt{-7}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta q^{3} - 4 q^{7} - 4 q^{9} +O(q^{10}) $ q - b * q^3 - 4 * q^7 - 4 * q^9	$ q - \beta q^{3} - 4 q^{7} - 4 q^{9} + \beta q^{11} - 3 q^{17} + \beta q^{19} + 4 \beta q^{21} - 4 q^{23} + \beta q^{27} - 4 q^{31} + 7 q^{33} + 4 \beta q^{37} - 5 q^{41} - 2 \beta q^{43} + 8 q^{47} + 9 q^{49} + 3 \beta q^{51} - 4 \beta q^{53} + 7 q^{57} + 2 \beta q^{59} - 4 \beta q^{61} + 16 q^{63} + 3 \beta q^{67} + 4 \beta q^{69} - 8 q^{71} - 7 q^{73} - 4 \beta q^{77} - 4 q^{79} - 5 q^{81} + 3 \beta q^{83} - q^{89} + 4 \beta q^{93} - 2 q^{97} - 4 \beta q^{99} +O(q^{100}) $ q - b * q^3 - 4 * q^7 - 4 * q^9 + b * q^11 - 3 * q^17 + b * q^19 + 4b q^21 - 4 * q^23 + b * q^27 - 4 * q^31 + 7 * q^33 + 4b q^37 - 5 * q^41 - 2b q^43 + 8 * q^47 + 9 * q^49 + 3b q^51 - 4b q^53 + 7 * q^57 + 2b q^59 - 4b q^61 + 16 * q^63 + 3b q^67 + 4b q^69 - 8 * q^71 - 7 * q^73 - 4b q^77 - 4 * q^79 - 5 * q^81 + 3b q^83 - q^89 + 4b q^93 - 2 * q^97 - 4b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 - 8 * q^9	$ 2 q - 8 q^{7} - 8 q^{9} - 6 q^{17} - 8 q^{23} - 8 q^{31} + 14 q^{33} - 10 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{57} + 32 q^{63} - 16 q^{71} - 14 q^{73} - 8 q^{79} - 10 q^{81} - 2 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 8 * q^9 - 6 * q^17 - 8 * q^23 - 8 * q^31 + 14 * q^33 - 10 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^57 + 32 * q^63 - 16 * q^71 - 14 * q^73 - 8 * q^79 - 10 * q^81 - 2 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$1$

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} $

401.1

0.500000	+	1.32288i
0.500000	−	1.32288i

−

2.64575i

−4.00000

401.2

2.64575i

−4.00000

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	800.2.d.a		2
3.b	odd	2	1		7200.2.k.b		2
4.b	odd	2	1		200.2.d.c	yes	2
5.b	even	2	1		800.2.d.d		2
5.c	odd	4	2		800.2.f.d		4
8.b	even	2	1	inner	800.2.d.a		2
8.d	odd	2	1		200.2.d.c	yes	2
12.b	even	2	1		1800.2.k.d		2
15.d	odd	2	1		7200.2.k.i		2
15.e	even	4	2		7200.2.d.m		4
16.e	even	4	2		6400.2.a.cb		2
16.f	odd	4	2		6400.2.a.bg		2
20.d	odd	2	1		200.2.d.b	✓	2
20.e	even	4	2		200.2.f.d		4
24.f	even	2	1		1800.2.k.d		2
24.h	odd	2	1		7200.2.k.b		2
40.e	odd	2	1		200.2.d.b	✓	2
40.f	even	2	1		800.2.d.d		2
40.i	odd	4	2		800.2.f.d		4
40.k	even	4	2		200.2.f.d		4
60.h	even	2	1		1800.2.k.f		2
60.l	odd	4	2		1800.2.d.m		4
80.k	odd	4	2		6400.2.a.cc		2
80.q	even	4	2		6400.2.a.bh		2
120.i	odd	2	1		7200.2.k.i		2
120.m	even	2	1		1800.2.k.f		2
120.q	odd	4	2		1800.2.d.m		4
120.w	even	4	2		7200.2.d.m		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.d.b	✓	2	20.d	odd	2	1
200.2.d.b	✓	2	40.e	odd	2	1
200.2.d.c	yes	2	4.b	odd	2	1
200.2.d.c	yes	2	8.d	odd	2	1
200.2.f.d		4	20.e	even	4	2
200.2.f.d		4	40.k	even	4	2
800.2.d.a		2	1.a	even	1	1	trivial
800.2.d.a		2	8.b	even	2	1	inner
800.2.d.d		2	5.b	even	2	1
800.2.d.d		2	40.f	even	2	1
800.2.f.d		4	5.c	odd	4	2
800.2.f.d		4	40.i	odd	4	2
1800.2.d.m		4	60.l	odd	4	2
1800.2.d.m		4	120.q	odd	4	2
1800.2.k.d		2	12.b	even	2	1
1800.2.k.d		2	24.f	even	2	1
1800.2.k.f		2	60.h	even	2	1
1800.2.k.f		2	120.m	even	2	1
6400.2.a.bg		2	16.f	odd	4	2
6400.2.a.bh		2	80.q	even	4	2
6400.2.a.cb		2	16.e	even	4	2
6400.2.a.cc		2	80.k	odd	4	2
7200.2.d.m		4	15.e	even	4	2
7200.2.d.m		4	120.w	even	4	2
7200.2.k.b		2	3.b	odd	2	1
7200.2.k.b		2	24.h	odd	2	1
7200.2.k.i		2	15.d	odd	2	1
7200.2.k.i		2	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(800, [\chi])$:

$ T_{3}^{2} + 7 $

$ T_{7} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 7 $ T^2 + 7
$5$	$ T^{2} $ T^2
$7$	$ (T + 4)^{2} $ (T + 4)^2
$11$	$ T^{2} + 7 $ T^2 + 7
$13$	$ T^{2} $ T^2
$17$	$ (T + 3)^{2} $ (T + 3)^2
$19$	$ T^{2} + 7 $ T^2 + 7
$23$	$ (T + 4)^{2} $ (T + 4)^2
$29$	$ T^{2} $ T^2
$31$	$ (T + 4)^{2} $ (T + 4)^2
$37$	$ T^{2} + 112 $ T^2 + 112
$41$	$ (T + 5)^{2} $ (T + 5)^2
$43$	$ T^{2} + 28 $ T^2 + 28
$47$	$ (T - 8)^{2} $ (T - 8)^2
$53$	$ T^{2} + 112 $ T^2 + 112
$59$	$ T^{2} + 28 $ T^2 + 28
$61$	$ T^{2} + 112 $ T^2 + 112
$67$	$ T^{2} + 63 $ T^2 + 63
$71$	$ (T + 8)^{2} $ (T + 8)^2
$73$	$ (T + 7)^{2} $ (T + 7)^2
$79$	$ (T + 4)^{2} $ (T + 4)^2
$83$	$ T^{2} + 63 $ T^2 + 63
$89$	$ (T + 1)^{2} $ (T + 1)^2
$97$	$ (T + 2)^{2} $ (T + 2)^2
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Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 4 q^{7} - 4 q^{9} +O(q^{10}) \) q - b * q^3 - 4 * q^7 - 4 * q^9	\( q - \beta q^{3} - 4 q^{7} - 4 q^{9} + \beta q^{11} - 3 q^{17} + \beta q^{19} + 4 \beta q^{21} - 4 q^{23} + \beta q^{27} - 4 q^{31} + 7 q^{33} + 4 \beta q^{37} - 5 q^{41} - 2 \beta q^{43} + 8 q^{47} + 9 q^{49} + 3 \beta q^{51} - 4 \beta q^{53} + 7 q^{57} + 2 \beta q^{59} - 4 \beta q^{61} + 16 q^{63} + 3 \beta q^{67} + 4 \beta q^{69} - 8 q^{71} - 7 q^{73} - 4 \beta q^{77} - 4 q^{79} - 5 q^{81} + 3 \beta q^{83} - q^{89} + 4 \beta q^{93} - 2 q^{97} - 4 \beta q^{99} +O(q^{100}) \) q - b * q^3 - 4 * q^7 - 4 * q^9 + b * q^11 - 3 * q^17 + b * q^19 + 4b q^21 - 4 * q^23 + b * q^27 - 4 * q^31 + 7 * q^33 + 4b q^37 - 5 * q^41 - 2b q^43 + 8 * q^47 + 9 * q^49 + 3b q^51 - 4b q^53 + 7 * q^57 + 2b q^59 - 4b q^61 + 16 * q^63 + 3b q^67 + 4b q^69 - 8 * q^71 - 7 * q^73 - 4b q^77 - 4 * q^79 - 5 * q^81 + 3b q^83 - q^89 + 4b q^93 - 2 * q^97 - 4b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 - 8 * q^9	\( 2 q - 8 q^{7} - 8 q^{9} - 6 q^{17} - 8 q^{23} - 8 q^{31} + 14 q^{33} - 10 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{57} + 32 q^{63} - 16 q^{71} - 14 q^{73} - 8 q^{79} - 10 q^{81} - 2 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 8 * q^9 - 6 * q^17 - 8 * q^23 - 8 * q^31 + 14 * q^33 - 10 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^57 + 32 * q^63 - 16 * q^71 - 14 * q^73 - 8 * q^79 - 10 * q^81 - 2 * q^89 - 4 * q^97

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