LMFDB - Newform orbit 624.2.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(337,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.c (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:	$4.98266508613$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
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 $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{3} + \beta q^{5} - 2 \beta q^{7} + q^{9} + 3 \beta q^{11} + ( - \beta + 3) q^{13} - \beta q^{15} - 2 q^{17} + 2 \beta q^{21} + 8 q^{23} + q^{25} - q^{27} + 2 q^{29} + 4 \beta q^{31} - 3 \beta q^{33} + \cdots + 3 \beta q^{99} +O(q^{100}) $ q - q^3 + b * q^5 - 2b q^7 + q^9 + 3b q^11 + (-b + 3) * q^13 - b * q^15 - 2 * q^17 + 2b q^21 + 8 * q^23 + q^25 - q^27 + 2 * q^29 + 4b q^31 - 3b q^33 + 8 * q^35 + 4b q^37 + (b - 3) * q^39 - b * q^41 + 8 * q^43 + b * q^45 + 3b q^47 - 9 * q^49 + 2 * q^51 + 6 * q^53 - 12 * q^55 + b * q^59 + 2 * q^61 - 2b q^63 + (3b + 4) q^65 - 2b q^67 - 8 * q^69 - 3b q^71 - 2b q^73 - q^75 + 24 * q^77 + q^81 - 7b q^83 - 2b q^85 - 2 * q^87 + 3b q^89 + (-6b - 8) q^91 - 4b q^93 - 6b q^97 + 3b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} + 16 q^{35} - 6 q^{39} + 16 q^{43} - 18 q^{49} + 4 q^{51} + 12 q^{53} - 24 q^{55} + 4 q^{61} + 8 q^{65} - 16 q^{69}+ \cdots - 16 q^{91}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 16 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 + 16 * q^35 - 6 * q^39 + 16 * q^43 - 18 * q^49 + 4 * q^51 + 12 * q^53 - 24 * q^55 + 4 * q^61 + 8 * q^65 - 16 * q^69 - 2 * q^75 + 48 * q^77 + 2 * q^81 - 4 * q^87 - 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

337.1

	−	1.00000i
		1.00000i

−1.00000

−

2.00000i

4.00000i

1.00000

337.2

−1.00000

2.00000i

−

4.00000i

1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.2.c.d		2
3.b	odd	2	1		1872.2.c.h		2
4.b	odd	2	1		156.2.b.b	✓	2
8.b	even	2	1		2496.2.c.i		2
8.d	odd	2	1		2496.2.c.b		2
12.b	even	2	1		468.2.b.c		2
13.b	even	2	1	inner	624.2.c.d		2
13.d	odd	4	1		8112.2.a.d		1
13.d	odd	4	1		8112.2.a.l		1
20.d	odd	2	1		3900.2.c.a		2
20.e	even	4	1		3900.2.j.b		2
20.e	even	4	1		3900.2.j.e		2
28.d	even	2	1		7644.2.e.b		2
39.d	odd	2	1		1872.2.c.h		2
52.b	odd	2	1		156.2.b.b	✓	2
52.f	even	4	1		2028.2.a.d		1
52.f	even	4	1		2028.2.a.f		1
52.i	odd	6	2		2028.2.q.e		4
52.j	odd	6	2		2028.2.q.e		4
52.l	even	12	2		2028.2.i.a		2
52.l	even	12	2		2028.2.i.d		2
104.e	even	2	1		2496.2.c.i		2
104.h	odd	2	1		2496.2.c.b		2
156.h	even	2	1		468.2.b.c		2
156.l	odd	4	1		6084.2.a.d		1
156.l	odd	4	1		6084.2.a.n		1
260.g	odd	2	1		3900.2.c.a		2
260.p	even	4	1		3900.2.j.b		2
260.p	even	4	1		3900.2.j.e		2
364.h	even	2	1		7644.2.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.b.b	✓	2	4.b	odd	2	1
156.2.b.b	✓	2	52.b	odd	2	1
468.2.b.c		2	12.b	even	2	1
468.2.b.c		2	156.h	even	2	1
624.2.c.d		2	1.a	even	1	1	trivial
624.2.c.d		2	13.b	even	2	1	inner
1872.2.c.h		2	3.b	odd	2	1
1872.2.c.h		2	39.d	odd	2	1
2028.2.a.d		1	52.f	even	4	1
2028.2.a.f		1	52.f	even	4	1
2028.2.i.a		2	52.l	even	12	2
2028.2.i.d		2	52.l	even	12	2
2028.2.q.e		4	52.i	odd	6	2
2028.2.q.e		4	52.j	odd	6	2
2496.2.c.b		2	8.d	odd	2	1
2496.2.c.b		2	104.h	odd	2	1
2496.2.c.i		2	8.b	even	2	1
2496.2.c.i		2	104.e	even	2	1
3900.2.c.a		2	20.d	odd	2	1
3900.2.c.a		2	260.g	odd	2	1
3900.2.j.b		2	20.e	even	4	1
3900.2.j.b		2	260.p	even	4	1
3900.2.j.e		2	20.e	even	4	1
3900.2.j.e		2	260.p	even	4	1
6084.2.a.d		1	156.l	odd	4	1
6084.2.a.n		1	156.l	odd	4	1
7644.2.e.b		2	28.d	even	2	1
7644.2.e.b		2	364.h	even	2	1
8112.2.a.d		1	13.d	odd	4	1
8112.2.a.l		1	13.d	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(624, [\chi])$:

$ T_{5}^{2} + 4 $

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

$ T_{11}^{2} + 36 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ (T + 1)^{2} $ (T + 1)^2
$5$	$ T^{2} + 4 $ T^2 + 4
$7$	$ T^{2} + 16 $ T^2 + 16
$11$	$ T^{2} + 36 $ T^2 + 36
$13$	$ T^{2} - 6T + 13 $ T^2 - 6*T + 13
$17$	$ (T + 2)^{2} $ (T + 2)^2
$19$	$ T^{2} $ T^2
$23$	$ (T - 8)^{2} $ (T - 8)^2
$29$	$ (T - 2)^{2} $ (T - 2)^2
$31$	$ T^{2} + 64 $ T^2 + 64
$37$	$ T^{2} + 64 $ T^2 + 64
$41$	$ T^{2} + 4 $ T^2 + 4
$43$	$ (T - 8)^{2} $ (T - 8)^2
$47$	$ T^{2} + 36 $ T^2 + 36
$53$	$ (T - 6)^{2} $ (T - 6)^2
$59$	$ T^{2} + 4 $ T^2 + 4
$61$	$ (T - 2)^{2} $ (T - 2)^2
$67$	$ T^{2} + 16 $ T^2 + 16
$71$	$ T^{2} + 36 $ T^2 + 36
$73$	$ T^{2} + 16 $ T^2 + 16
$79$	$ T^{2} $ T^2
$83$	$ T^{2} + 196 $ T^2 + 196
$89$	$ T^{2} + 36 $ T^2 + 36
$97$	$ T^{2} + 144 $ T^2 + 144
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Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - q^{3} + \beta q^{5} - 2 \beta q^{7} + q^{9} + 3 \beta q^{11} + ( - \beta + 3) q^{13} - \beta q^{15} - 2 q^{17} + 2 \beta q^{21} + 8 q^{23} + q^{25} - q^{27} + 2 q^{29} + 4 \beta q^{31} - 3 \beta q^{33} + \cdots + 3 \beta q^{99} +O(q^{100}) \) q - q^3 + b * q^5 - 2b q^7 + q^9 + 3b q^11 + (-b + 3) * q^13 - b * q^15 - 2 * q^17 + 2b q^21 + 8 * q^23 + q^25 - q^27 + 2 * q^29 + 4b q^31 - 3b q^33 + 8 * q^35 + 4b q^37 + (b - 3) * q^39 - b * q^41 + 8 * q^43 + b * q^45 + 3b q^47 - 9 * q^49 + 2 * q^51 + 6 * q^53 - 12 * q^55 + b * q^59 + 2 * q^61 - 2b q^63 + (3b + 4) q^65 - 2b q^67 - 8 * q^69 - 3b q^71 - 2b q^73 - q^75 + 24 * q^77 + q^81 - 7b q^83 - 2b q^85 - 2 * q^87 + 3b q^89 + (-6b - 8) q^91 - 4b q^93 - 6b q^97 + 3b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} + 16 q^{35} - 6 q^{39} + 16 q^{43} - 18 q^{49} + 4 q^{51} + 12 q^{53} - 24 q^{55} + 4 q^{61} + 8 q^{65} - 16 q^{69}+ \cdots - 16 q^{91}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 16 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 + 16 * q^35 - 6 * q^39 + 16 * q^43 - 18 * q^49 + 4 * q^51 + 12 * q^53 - 24 * q^55 + 4 * q^61 + 8 * q^65 - 16 * q^69 - 2 * q^75 + 48 * q^77 + 2 * q^81 - 4 * q^87 - 16 * q^91

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