LMFDB - Newform orbit 63.2.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,2,Mod(62,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("63.62");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	63.c (of order $2$, degree $1$, 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:	$0.503057532734$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b2 - 2) * q^4 - b2 * q^7 + (b3 - 2b1) q^8	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{3} + 2 \beta_1) q^{14} + ( - 2 \beta_{2} + 3) q^{16} + (\beta_{2} + 5) q^{22} + (\beta_{3} - 3 \beta_1) q^{23} - 5 q^{25} + (2 \beta_{2} - 7) q^{28} + (\beta_{3} + 3 \beta_1) q^{29} + 3 \beta_1 q^{32} + 4 \beta_{2} q^{37} - 2 \beta_{2} q^{43} + ( - \beta_{3} + \beta_1) q^{44} + ( - 5 \beta_{2} + 11) q^{46} + 7 q^{49} - 5 \beta_1 q^{50} + ( - \beta_{3} + 5 \beta_1) q^{53} - 7 \beta_1 q^{56} + (\beta_{2} - 13) q^{58} + ( - \beta_{2} - 6) q^{64} - 4 q^{67} + ( - 3 \beta_{3} + \beta_1) q^{71} + (4 \beta_{3} - 8 \beta_1) q^{74} + (3 \beta_{3} + \beta_1) q^{77} + 8 q^{79} + ( - 2 \beta_{3} + 4 \beta_1) q^{86} + (5 \beta_{2} + 7) q^{88} + ( - 3 \beta_{3} + 15 \beta_1) q^{92} + 7 \beta_1 q^{98}+O(q^{100}) $ q + b1 * q^2 + (b2 - 2) * q^4 - b2 * q^7 + (b3 - 2b1) q^8 + (-b3 - b1) * q^11 + (-b3 + 2b1) q^14 + (-2b2 + 3) q^16 + (b2 + 5) * q^22 + (b3 - 3b1) q^23 - 5 * q^25 + (2b2 - 7) q^28 + (b3 + 3b1) q^29 + 3b1 q^32 + 4b2 q^37 - 2b2 q^43 + (-b3 + b1) * q^44 + (-5b2 + 11) q^46 + 7 * q^49 - 5b1 q^50 + (-b3 + 5b1) q^53 - 7b1 q^56 + (b2 - 13) * q^58 + (-b2 - 6) * q^64 - 4 * q^67 + (-3b3 + b1) q^71 + (4b3 - 8b1) * q^74 + (3b3 + b1) q^77 + 8 * q^79 + (-2b3 + 4b1) * q^86 + (5b2 + 7) q^88 + (-3b3 + 15b1) * q^92 + 7b1 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^16 + 20 * q^22 - 20 * q^25 - 28 * q^28 + 44 * q^46 + 28 * q^49 - 52 * q^58 - 24 * q^64 - 16 * q^67 + 32 * q^79 + 28 * q^88

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ \nu^{3} + 6\nu $ v^3 + 6*v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ \beta_{3} - 6\beta_1 $ b3 - 6*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1$	$-1$

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} $

62.1

	−	2.57794i
	−	1.16372i
		1.16372i
		2.57794i

−

2.57794i

−4.64575

2.64575

6.82058i

62.2

−

1.16372i

0.645751

−2.64575

−

3.07892i

62.3

1.16372i

0.645751

−2.64575

3.07892i

62.4

2.57794i

−4.64575

2.64575

−

6.82058i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
3.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.2.c.a	✓	4
3.b	odd	2	1	inner	63.2.c.a	✓	4
4.b	odd	2	1		1008.2.k.a		4
5.b	even	2	1		1575.2.b.a		4
5.c	odd	4	2		1575.2.g.d		8
7.b	odd	2	1	CM	63.2.c.a	✓	4
7.c	even	3	2		441.2.p.b		8
7.d	odd	6	2		441.2.p.b		8
8.b	even	2	1		4032.2.k.c		4
8.d	odd	2	1		4032.2.k.b		4
9.c	even	3	2		567.2.o.f		8
9.d	odd	6	2		567.2.o.f		8
12.b	even	2	1		1008.2.k.a		4
15.d	odd	2	1		1575.2.b.a		4
15.e	even	4	2		1575.2.g.d		8
21.c	even	2	1	inner	63.2.c.a	✓	4
21.g	even	6	2		441.2.p.b		8
21.h	odd	6	2		441.2.p.b		8
24.f	even	2	1		4032.2.k.b		4
24.h	odd	2	1		4032.2.k.c		4
28.d	even	2	1		1008.2.k.a		4
35.c	odd	2	1		1575.2.b.a		4
35.f	even	4	2		1575.2.g.d		8
56.e	even	2	1		4032.2.k.b		4
56.h	odd	2	1		4032.2.k.c		4
63.l	odd	6	2		567.2.o.f		8
63.o	even	6	2		567.2.o.f		8
84.h	odd	2	1		1008.2.k.a		4
105.g	even	2	1		1575.2.b.a		4
105.k	odd	4	2		1575.2.g.d		8
168.e	odd	2	1		4032.2.k.b		4
168.i	even	2	1		4032.2.k.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.c.a	✓	4	1.a	even	1	1	trivial
63.2.c.a	✓	4	3.b	odd	2	1	inner
63.2.c.a	✓	4	7.b	odd	2	1	CM
63.2.c.a	✓	4	21.c	even	2	1	inner
441.2.p.b		8	7.c	even	3	2
441.2.p.b		8	7.d	odd	6	2
441.2.p.b		8	21.g	even	6	2
441.2.p.b		8	21.h	odd	6	2
567.2.o.f		8	9.c	even	3	2
567.2.o.f		8	9.d	odd	6	2
567.2.o.f		8	63.l	odd	6	2
567.2.o.f		8	63.o	even	6	2
1008.2.k.a		4	4.b	odd	2	1
1008.2.k.a		4	12.b	even	2	1
1008.2.k.a		4	28.d	even	2	1
1008.2.k.a		4	84.h	odd	2	1
1575.2.b.a		4	5.b	even	2	1
1575.2.b.a		4	15.d	odd	2	1
1575.2.b.a		4	35.c	odd	2	1
1575.2.b.a		4	105.g	even	2	1
1575.2.g.d		8	5.c	odd	4	2
1575.2.g.d		8	15.e	even	4	2
1575.2.g.d		8	35.f	even	4	2
1575.2.g.d		8	105.k	odd	4	2
4032.2.k.b		4	8.d	odd	2	1
4032.2.k.b		4	24.f	even	2	1
4032.2.k.b		4	56.e	even	2	1
4032.2.k.b		4	168.e	odd	2	1
4032.2.k.c		4	8.b	even	2	1
4032.2.k.c		4	24.h	odd	2	1
4032.2.k.c		4	56.h	odd	2	1
4032.2.k.c		4	168.i	even	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(63, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 8T^{2} + 9 $ T^4 + 8*T^2 + 9
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 7)^{2} $ (T^2 - 7)^2
$11$	$ T^{4} + 44T^{2} + 36 $ T^4 + 44*T^2 + 36
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} + 92T^{2} + 324 $ T^4 + 92*T^2 + 324
$29$	$ T^{4} + 116T^{2} + 2916 $ T^4 + 116*T^2 + 2916
$31$	$ T^{4} $ T^4
$37$	$ (T^{2} - 112)^{2} $ (T^2 - 112)^2
$41$	$ T^{4} $ T^4
$43$	$ (T^{2} - 28)^{2} $ (T^2 - 28)^2
$47$	$ T^{4} $ T^4
$53$	$ T^{4} + 212T^{2} + 36 $ T^4 + 212*T^2 + 36
$59$	$ T^{4} $ T^4
$61$	$ T^{4} $ T^4
$67$	$ (T + 4)^{4} $ (T + 4)^4
$71$	$ T^{4} + 284 T^{2} + 12996 $ T^4 + 284*T^2 + 12996
$73$	$ T^{4} $ T^4
$79$	$ (T - 8)^{4} $ (T - 8)^4
$83$	$ T^{4} $ T^4
$89$	$ T^{4} $ T^4
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b2 - 2) * q^4 - b2 * q^7 + (b3 - 2b1) q^8	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{3} + 2 \beta_1) q^{14} + ( - 2 \beta_{2} + 3) q^{16} + (\beta_{2} + 5) q^{22} + (\beta_{3} - 3 \beta_1) q^{23} - 5 q^{25} + (2 \beta_{2} - 7) q^{28} + (\beta_{3} + 3 \beta_1) q^{29} + 3 \beta_1 q^{32} + 4 \beta_{2} q^{37} - 2 \beta_{2} q^{43} + ( - \beta_{3} + \beta_1) q^{44} + ( - 5 \beta_{2} + 11) q^{46} + 7 q^{49} - 5 \beta_1 q^{50} + ( - \beta_{3} + 5 \beta_1) q^{53} - 7 \beta_1 q^{56} + (\beta_{2} - 13) q^{58} + ( - \beta_{2} - 6) q^{64} - 4 q^{67} + ( - 3 \beta_{3} + \beta_1) q^{71} + (4 \beta_{3} - 8 \beta_1) q^{74} + (3 \beta_{3} + \beta_1) q^{77} + 8 q^{79} + ( - 2 \beta_{3} + 4 \beta_1) q^{86} + (5 \beta_{2} + 7) q^{88} + ( - 3 \beta_{3} + 15 \beta_1) q^{92} + 7 \beta_1 q^{98}+O(q^{100}) \) q + b1 * q^2 + (b2 - 2) * q^4 - b2 * q^7 + (b3 - 2b1) q^8 + (-b3 - b1) * q^11 + (-b3 + 2b1) q^14 + (-2b2 + 3) q^16 + (b2 + 5) * q^22 + (b3 - 3b1) q^23 - 5 * q^25 + (2b2 - 7) q^28 + (b3 + 3b1) q^29 + 3b1 q^32 + 4b2 q^37 - 2b2 q^43 + (-b3 + b1) * q^44 + (-5b2 + 11) q^46 + 7 * q^49 - 5b1 q^50 + (-b3 + 5b1) q^53 - 7b1 q^56 + (b2 - 13) * q^58 + (-b2 - 6) * q^64 - 4 * q^67 + (-3b3 + b1) q^71 + (4b3 - 8b1) * q^74 + (3b3 + b1) q^77 + 8 * q^79 + (-2b3 + 4b1) * q^86 + (5b2 + 7) q^88 + (-3b3 + 15b1) * q^92 + 7b1 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^16 + 20 * q^22 - 20 * q^25 - 28 * q^28 + 44 * q^46 + 28 * q^49 - 52 * q^58 - 24 * q^64 - 16 * q^67 + 32 * q^79 + 28 * q^88

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