LMFDB - Newform orbit 3900.2.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,2,Mod(3301,3900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3900, 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("3900.3301");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.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:	$31.1416567883$
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:	$\Z[a_1, \ldots, a_{11}]$
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} + 2 \beta q^{7} + q^{9} + 3 \beta q^{11} + ( - \beta - 3) q^{13} + 2 q^{17} - 2 \beta q^{21} + 8 q^{23} - q^{27} + 2 q^{29} + 4 \beta q^{31} - 3 \beta q^{33} + 4 \beta q^{37} + (\beta + 3) q^{39} + \cdots + 3 \beta q^{99} +O(q^{100}) $ q - q^3 + 2b q^7 + q^9 + 3b q^11 + (-b - 3) * q^13 + 2 * q^17 - 2b q^21 + 8 * q^23 - q^27 + 2 * q^29 + 4b q^31 - 3b q^33 + 4b q^37 + (b + 3) * q^39 + b * q^41 + 8 * q^43 - 3b q^47 - 9 * q^49 - 2 * q^51 - 6 * q^53 + b * q^59 + 2 * q^61 + 2b q^63 + 2b q^67 - 8 * q^69 - 3b q^71 - 2b q^73 - 24 * q^77 + q^81 + 7b q^83 - 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^{27} + 4 q^{29} + 6 q^{39} + 16 q^{43} - 18 q^{49} - 4 q^{51} - 12 q^{53} + 4 q^{61} - 16 q^{69} - 48 q^{77} + 2 q^{81} - 4 q^{87} + 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^27 + 4 * q^29 + 6 * q^39 + 16 * q^43 - 18 * q^49 - 4 * q^51 - 12 * q^53 + 4 * q^61 - 16 * q^69 - 48 * q^77 + 2 * q^81 - 4 * q^87 + 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$-1$	$1$	$1$	$1$

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} $

3301.1

	−	1.00000i
		1.00000i

−1.00000

−

4.00000i

1.00000

3301.2

−1.00000

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	3900.2.c.a		2
5.b	even	2	1		156.2.b.b	✓	2
5.c	odd	4	1		3900.2.j.b		2
5.c	odd	4	1		3900.2.j.e		2
13.b	even	2	1	inner	3900.2.c.a		2
15.d	odd	2	1		468.2.b.c		2
20.d	odd	2	1		624.2.c.d		2
35.c	odd	2	1		7644.2.e.b		2
40.e	odd	2	1		2496.2.c.i		2
40.f	even	2	1		2496.2.c.b		2
60.h	even	2	1		1872.2.c.h		2
65.d	even	2	1		156.2.b.b	✓	2
65.g	odd	4	1		2028.2.a.d		1
65.g	odd	4	1		2028.2.a.f		1
65.h	odd	4	1		3900.2.j.b		2
65.h	odd	4	1		3900.2.j.e		2
65.l	even	6	2		2028.2.q.e		4
65.n	even	6	2		2028.2.q.e		4
65.s	odd	12	2		2028.2.i.a		2
65.s	odd	12	2		2028.2.i.d		2
195.e	odd	2	1		468.2.b.c		2
195.n	even	4	1		6084.2.a.d		1
195.n	even	4	1		6084.2.a.n		1
260.g	odd	2	1		624.2.c.d		2
260.u	even	4	1		8112.2.a.d		1
260.u	even	4	1		8112.2.a.l		1
455.h	odd	2	1		7644.2.e.b		2
520.b	odd	2	1		2496.2.c.i		2
520.p	even	2	1		2496.2.c.b		2
780.d	even	2	1		1872.2.c.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.b.b	✓	2	5.b	even	2	1
156.2.b.b	✓	2	65.d	even	2	1
468.2.b.c		2	15.d	odd	2	1
468.2.b.c		2	195.e	odd	2	1
624.2.c.d		2	20.d	odd	2	1
624.2.c.d		2	260.g	odd	2	1
1872.2.c.h		2	60.h	even	2	1
1872.2.c.h		2	780.d	even	2	1
2028.2.a.d		1	65.g	odd	4	1
2028.2.a.f		1	65.g	odd	4	1
2028.2.i.a		2	65.s	odd	12	2
2028.2.i.d		2	65.s	odd	12	2
2028.2.q.e		4	65.l	even	6	2
2028.2.q.e		4	65.n	even	6	2
2496.2.c.b		2	40.f	even	2	1
2496.2.c.b		2	520.p	even	2	1
2496.2.c.i		2	40.e	odd	2	1
2496.2.c.i		2	520.b	odd	2	1
3900.2.c.a		2	1.a	even	1	1	trivial
3900.2.c.a		2	13.b	even	2	1	inner
3900.2.j.b		2	5.c	odd	4	1
3900.2.j.b		2	65.h	odd	4	1
3900.2.j.e		2	5.c	odd	4	1
3900.2.j.e		2	65.h	odd	4	1
6084.2.a.d		1	195.n	even	4	1
6084.2.a.n		1	195.n	even	4	1
7644.2.e.b		2	35.c	odd	2	1
7644.2.e.b		2	455.h	odd	2	1
8112.2.a.d		1	260.u	even	4	1
8112.2.a.l		1	260.u	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3900, [\chi])$:

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

T7^2 + 16

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

T11^2 + 36

$ T_{17} - 2 $

T17 - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ (T + 1)^{2} $ (T + 1)^2
$5$	$ T^{2} $ T^2
$7$	$ 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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.c (of order \(2\), degree \(1\), minimal)

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

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