LMFDB - Newform orbit 600.2.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.f (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:	$4.79102412128$
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:	no (minimal twist has level 120)
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 i q^{7} - q^{9} - 6 i q^{13} + 2 i q^{17} - 4 q^{19} + 4 q^{21} - 8 i q^{23} - i q^{27} + 6 q^{29} + 6 i q^{37} + 6 q^{39} + 10 q^{41} - 4 i q^{43} - 8 i q^{47} - 9 q^{49} - 2 q^{51} + 10 i q^{53} + \cdots - 2 i q^{97} +O(q^{100}) $ q + i * q^3 - 4i q^7 - q^9 - 6i q^13 + 2i q^17 - 4 * q^19 + 4 * q^21 - 8i q^23 - i * q^27 + 6 * q^29 + 6i q^37 + 6 * q^39 + 10 * q^41 - 4i q^43 - 8i q^47 - 9 * q^49 - 2 * q^51 + 10i q^53 - 4i q^57 + 6 * q^61 + 4i q^63 + 4i q^67 + 8 * q^69 - 14i q^73 - 16 * q^79 + q^81 + 12i q^83 + 6i q^87 - 2 * q^89 - 24 * q^91 - 2i q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 8 q^{19} + 8 q^{21} + 12 q^{29} + 12 q^{39} + 20 q^{41} - 18 q^{49} - 4 q^{51} + 12 q^{61} + 16 q^{69} - 32 q^{79} + 2 q^{81} - 4 q^{89} - 48 q^{91}+O(q^{100}) $ 2 * q - 2 * q^9 - 8 * q^19 + 8 * q^21 + 12 * q^29 + 12 * q^39 + 20 * q^41 - 18 * q^49 - 4 * q^51 + 12 * q^61 + 16 * q^69 - 32 * q^79 + 2 * q^81 - 4 * q^89 - 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

49.1

	−	1.00000i
		1.00000i

−

1.00000i

4.00000i

−1.00000

49.2

1.00000i

−

4.00000i

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.2.f.c		2
3.b	odd	2	1		1800.2.f.g		2
4.b	odd	2	1		1200.2.f.f		2
5.b	even	2	1	inner	600.2.f.c		2
5.c	odd	4	1		120.2.a.a	✓	1
5.c	odd	4	1		600.2.a.a		1
8.b	even	2	1		4800.2.f.u		2
8.d	odd	2	1		4800.2.f.n		2
12.b	even	2	1		3600.2.f.l		2
15.d	odd	2	1		1800.2.f.g		2
15.e	even	4	1		360.2.a.e		1
15.e	even	4	1		1800.2.a.c		1
20.d	odd	2	1		1200.2.f.f		2
20.e	even	4	1		240.2.a.a		1
20.e	even	4	1		1200.2.a.r		1
35.f	even	4	1		5880.2.a.p		1
40.e	odd	2	1		4800.2.f.n		2
40.f	even	2	1		4800.2.f.u		2
40.i	odd	4	1		960.2.a.g		1
40.i	odd	4	1		4800.2.a.bl		1
40.k	even	4	1		960.2.a.n		1
40.k	even	4	1		4800.2.a.bh		1
45.k	odd	12	2		3240.2.q.m		2
45.l	even	12	2		3240.2.q.a		2
60.h	even	2	1		3600.2.f.l		2
60.l	odd	4	1		720.2.a.f		1
60.l	odd	4	1		3600.2.a.bo		1
80.i	odd	4	1		3840.2.k.a		2
80.j	even	4	1		3840.2.k.z		2
80.s	even	4	1		3840.2.k.z		2
80.t	odd	4	1		3840.2.k.a		2
120.q	odd	4	1		2880.2.a.b		1
120.w	even	4	1		2880.2.a.r		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.a.a	✓	1	5.c	odd	4	1
240.2.a.a		1	20.e	even	4	1
360.2.a.e		1	15.e	even	4	1
600.2.a.a		1	5.c	odd	4	1
600.2.f.c		2	1.a	even	1	1	trivial
600.2.f.c		2	5.b	even	2	1	inner
720.2.a.f		1	60.l	odd	4	1
960.2.a.g		1	40.i	odd	4	1
960.2.a.n		1	40.k	even	4	1
1200.2.a.r		1	20.e	even	4	1
1200.2.f.f		2	4.b	odd	2	1
1200.2.f.f		2	20.d	odd	2	1
1800.2.a.c		1	15.e	even	4	1
1800.2.f.g		2	3.b	odd	2	1
1800.2.f.g		2	15.d	odd	2	1
2880.2.a.b		1	120.q	odd	4	1
2880.2.a.r		1	120.w	even	4	1
3240.2.q.a		2	45.l	even	12	2
3240.2.q.m		2	45.k	odd	12	2
3600.2.a.bo		1	60.l	odd	4	1
3600.2.f.l		2	12.b	even	2	1
3600.2.f.l		2	60.h	even	2	1
3840.2.k.a		2	80.i	odd	4	1
3840.2.k.a		2	80.t	odd	4	1
3840.2.k.z		2	80.j	even	4	1
3840.2.k.z		2	80.s	even	4	1
4800.2.a.bh		1	40.k	even	4	1
4800.2.a.bl		1	40.i	odd	4	1
4800.2.f.n		2	8.d	odd	2	1
4800.2.f.n		2	40.e	odd	2	1
4800.2.f.u		2	8.b	even	2	1
4800.2.f.u		2	40.f	even	2	1
5880.2.a.p		1	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(600, [\chi])$:

$ T_{7}^{2} + 16 $

$ T_{11} $

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^{2} + 16 $ T^2 + 16
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 36 $ T^2 + 36
$17$	$ T^{2} + 4 $ T^2 + 4
$19$	$ (T + 4)^{2} $ (T + 4)^2
$23$	$ T^{2} + 64 $ T^2 + 64
$29$	$ (T - 6)^{2} $ (T - 6)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 36 $ T^2 + 36
$41$	$ (T - 10)^{2} $ (T - 10)^2
$43$	$ T^{2} + 16 $ T^2 + 16
$47$	$ T^{2} + 64 $ T^2 + 64
$53$	$ T^{2} + 100 $ T^2 + 100
$59$	$ T^{2} $ T^2
$61$	$ (T - 6)^{2} $ (T - 6)^2
$67$	$ T^{2} + 16 $ T^2 + 16
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 196 $ T^2 + 196
$79$	$ (T + 16)^{2} $ (T + 16)^2
$83$	$ T^{2} + 144 $ T^2 + 144
$89$	$ (T + 2)^{2} $ (T + 2)^2
$97$	$ T^{2} + 4 $ T^2 + 4
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + i q^{3} - 4 i q^{7} - q^{9} - 6 i q^{13} + 2 i q^{17} - 4 q^{19} + 4 q^{21} - 8 i q^{23} - i q^{27} + 6 q^{29} + 6 i q^{37} + 6 q^{39} + 10 q^{41} - 4 i q^{43} - 8 i q^{47} - 9 q^{49} - 2 q^{51} + 10 i q^{53} + \cdots - 2 i q^{97} +O(q^{100}) \) q + i * q^3 - 4i q^7 - q^9 - 6i q^13 + 2i q^17 - 4 * q^19 + 4 * q^21 - 8i q^23 - i * q^27 + 6 * q^29 + 6i q^37 + 6 * q^39 + 10 * q^41 - 4i q^43 - 8i q^47 - 9 * q^49 - 2 * q^51 + 10i q^53 - 4i q^57 + 6 * q^61 + 4i q^63 + 4i q^67 + 8 * q^69 - 14i q^73 - 16 * q^79 + q^81 + 12i q^83 + 6i q^87 - 2 * q^89 - 24 * q^91 - 2i q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 8 q^{19} + 8 q^{21} + 12 q^{29} + 12 q^{39} + 20 q^{41} - 18 q^{49} - 4 q^{51} + 12 q^{61} + 16 q^{69} - 32 q^{79} + 2 q^{81} - 4 q^{89} - 48 q^{91}+O(q^{100}) \) 2 * q - 2 * q^9 - 8 * q^19 + 8 * q^21 + 12 * q^29 + 12 * q^39 + 20 * q^41 - 18 * q^49 - 4 * q^51 + 12 * q^61 + 16 * q^69 - 32 * q^79 + 2 * q^81 - 4 * q^89 - 48 * q^91

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