LMFDB - Newform orbit 3528.2.s.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,2,Mod(361,3528)] 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("3528.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0,0,0,0,0,0,6,0,0,0,4,0,7,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 4 \zeta_{6} q^{5} + 3 q^{13} + ( - 4 \zeta_{6} + 4) q^{17} + 7 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + 8 q^{29} + (5 \zeta_{6} - 5) q^{31} - 3 \zeta_{6} q^{37} + 8 q^{41} + \cdots + 2 q^{97} +O(q^{100}) $ q - 4z q^5 + 3 * q^13 + (-4z + 4) q^17 + 7z q^19 + 4z q^23 + (11z - 11) q^25 + 8 * q^29 + (5z - 5) q^31 - 3z q^37 + 8 * q^41 + 11 * q^43 - 4z q^47 + (-4z + 4) q^53 + (12z - 12) q^59 - 2z q^61 - 12z q^65 + (-3z + 3) q^67 - 12 * q^71 + (-z + 1) * q^73 - z * q^79 + 12 * q^83 - 16 * q^85 - 8z q^89 + (-28z + 28) q^95 + 2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 6 q^{13} + 4 q^{17} + 7 q^{19} + 4 q^{23} - 11 q^{25} + 16 q^{29} - 5 q^{31} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 4 q^{53} - 12 q^{59} - 2 q^{61} - 12 q^{65} + 3 q^{67} - 24 q^{71}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^13 + 4 * q^17 + 7 * q^19 + 4 * q^23 - 11 * q^25 + 16 * q^29 - 5 * q^31 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 4 * q^53 - 12 * q^59 - 2 * q^61 - 12 * q^65 + 3 * q^67 - 24 * q^71 + q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 + 28 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$	$1$

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} $

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.00000

−

3.46410i

3313.1

−2.00000

3.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3528.2.s.b		2
3.b	odd	2	1		3528.2.s.bb		2
7.b	odd	2	1		504.2.s.h	yes	2
7.c	even	3	1		3528.2.a.ba		1
7.c	even	3	1	inner	3528.2.s.b		2
7.d	odd	6	1		504.2.s.h	yes	2
7.d	odd	6	1		3528.2.a.a		1
21.c	even	2	1		504.2.s.a	✓	2
21.g	even	6	1		504.2.s.a	✓	2
21.g	even	6	1		3528.2.a.z		1
21.h	odd	6	1		3528.2.a.c		1
21.h	odd	6	1		3528.2.s.bb		2
28.d	even	2	1		1008.2.s.q		2
28.f	even	6	1		1008.2.s.q		2
28.f	even	6	1		7056.2.a.b		1
28.g	odd	6	1		7056.2.a.cc		1
84.h	odd	2	1		1008.2.s.a		2
84.j	odd	6	1		1008.2.s.a		2
84.j	odd	6	1		7056.2.a.cb		1
84.n	even	6	1		7056.2.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.s.a	✓	2	21.c	even	2	1
504.2.s.a	✓	2	21.g	even	6	1
504.2.s.h	yes	2	7.b	odd	2	1
504.2.s.h	yes	2	7.d	odd	6	1
1008.2.s.a		2	84.h	odd	2	1
1008.2.s.a		2	84.j	odd	6	1
1008.2.s.q		2	28.d	even	2	1
1008.2.s.q		2	28.f	even	6	1
3528.2.a.a		1	7.d	odd	6	1
3528.2.a.c		1	21.h	odd	6	1
3528.2.a.z		1	21.g	even	6	1
3528.2.a.ba		1	7.c	even	3	1
3528.2.s.b		2	1.a	even	1	1	trivial
3528.2.s.b		2	7.c	even	3	1	inner
3528.2.s.bb		2	3.b	odd	2	1
3528.2.s.bb		2	21.h	odd	6	1
7056.2.a.b		1	28.f	even	6	1
7056.2.a.d		1	84.n	even	6	1
7056.2.a.cb		1	84.j	odd	6	1
7056.2.a.cc		1	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3528, [\chi])$:

$ T_{5}^{2} + 4T_{5} + 16 $

T5^2 + 4*T5 + 16

$ T_{11} $

T11

$ T_{13} - 3 $

T13 - 3

$ T_{23}^{2} - 4T_{23} + 16 $

T23^2 - 4*T23 + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 4T + 16 $ T^2 + 4*T + 16
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ (T - 3)^{2} $ (T - 3)^2
$17$	$ T^{2} - 4T + 16 $ T^2 - 4*T + 16
$19$	$ T^{2} - 7T + 49 $ T^2 - 7*T + 49
$23$	$ T^{2} - 4T + 16 $ T^2 - 4*T + 16
$29$	$ (T - 8)^{2} $ (T - 8)^2
$31$	$ T^{2} + 5T + 25 $ T^2 + 5*T + 25
$37$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
$41$	$ (T - 8)^{2} $ (T - 8)^2
$43$	$ (T - 11)^{2} $ (T - 11)^2
$47$	$ T^{2} + 4T + 16 $ T^2 + 4*T + 16
$53$	$ T^{2} - 4T + 16 $ T^2 - 4*T + 16
$59$	$ T^{2} + 12T + 144 $ T^2 + 12*T + 144
$61$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$67$	$ T^{2} - 3T + 9 $ T^2 - 3*T + 9
$71$	$ (T + 12)^{2} $ (T + 12)^2
$73$	$ T^{2} - T + 1 $ T^2 - T + 1
$79$	$ T^{2} + T + 1 $ T^2 + T + 1
$83$	$ (T - 12)^{2} $ (T - 12)^2
$89$	$ T^{2} + 8T + 64 $ T^2 + 8*T + 64
$97$	$ (T - 2)^{2} $ (T - 2)^2
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Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 4 \zeta_{6} q^{5} + 3 q^{13} + ( - 4 \zeta_{6} + 4) q^{17} + 7 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + 8 q^{29} + (5 \zeta_{6} - 5) q^{31} - 3 \zeta_{6} q^{37} + 8 q^{41} + \cdots + 2 q^{97} +O(q^{100}) \) q - 4z q^5 + 3 * q^13 + (-4z + 4) q^17 + 7z q^19 + 4z q^23 + (11z - 11) q^25 + 8 * q^29 + (5z - 5) q^31 - 3z q^37 + 8 * q^41 + 11 * q^43 - 4z q^47 + (-4z + 4) q^53 + (12z - 12) q^59 - 2z q^61 - 12z q^65 + (-3z + 3) q^67 - 12 * q^71 + (-z + 1) * q^73 - z * q^79 + 12 * q^83 - 16 * q^85 - 8z q^89 + (-28z + 28) q^95 + 2 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 6 q^{13} + 4 q^{17} + 7 q^{19} + 4 q^{23} - 11 q^{25} + 16 q^{29} - 5 q^{31} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 4 q^{53} - 12 q^{59} - 2 q^{61} - 12 q^{65} + 3 q^{67} - 24 q^{71}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^13 + 4 * q^17 + 7 * q^19 + 4 * q^23 - 11 * q^25 + 16 * q^29 - 5 * q^31 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 4 * q^53 - 12 * q^59 - 2 * q^61 - 12 * q^65 + 3 * q^67 - 24 * q^71 + q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 + 28 * q^95 + 4 * q^97

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