LMFDB - Newform orbit 3600.3.l.s

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(1601,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.l (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:	$98.0928951697$
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} - 2x^{3} + 9x^{2} - 8x + 2 $ x^4 - 2x^3 + 9x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
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 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^{7}+O(q^{10}) $ q - b1 * q^7	$ q - \beta_1 q^{7} - \beta_{2} q^{11} + \beta_1 q^{13} - 2 \beta_{3} q^{17} + 20 q^{19} + \beta_{3} q^{23} + 2 \beta_{2} q^{29} - 26 q^{31} - 3 \beta_1 q^{37} - 13 \beta_{2} q^{41} - 2 \beta_1 q^{43} - 4 \beta_{3} q^{47} + 77 q^{49} - 16 \beta_{3} q^{53} + 11 \beta_{2} q^{59} - 22 q^{61} - 8 \beta_1 q^{67} + 12 \beta_{2} q^{71} + 6 \beta_1 q^{73} - 9 \beta_{3} q^{77} + 14 q^{79} - 14 \beta_{3} q^{83} - 21 \beta_{2} q^{89} - 126 q^{91} - 2 \beta_1 q^{97}+O(q^{100}) $ q - b1 * q^7 - b2 * q^11 + b1 * q^13 - 2b3 q^17 + 20 * q^19 + b3 * q^23 + 2b2 q^29 - 26 * q^31 - 3b1 q^37 - 13b2 q^41 - 2b1 q^43 - 4b3 q^47 + 77 * q^49 - 16b3 q^53 + 11b2 q^59 - 22 * q^61 - 8b1 q^67 + 12b2 q^71 + 6b1 q^73 - 9b3 q^77 + 14 * q^79 - 14b3 q^83 - 21b2 q^89 - 126 * q^91 - 2b1 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 80 q^{19} - 104 q^{31} + 308 q^{49} - 88 q^{61} + 56 q^{79} - 504 q^{91}+O(q^{100}) $ 4 * q + 80 * q^19 - 104 * q^31 + 308 * q^49 - 88 * q^61 + 56 * q^79 - 504 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} - 2x^{3} + 9x^{2} - 8x + 2 $ x^4 - 2*x^3 + 9*x^2 - 8*x + 2 :

$\beta_{1}$	$=$	$ 3\nu^{2} - 3\nu + 12 $ 3v^2 - 3v + 12
$\beta_{2}$	$=$	$ 6\nu^{3} - 9\nu^{2} + 51\nu - 24 $ 6v^3 - 9v^2 + 51*v - 24
$\beta_{3}$	$=$	$ -8\nu^{3} + 12\nu^{2} - 64\nu + 30 $ -8v^3 + 12v^2 - 64*v + 30

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

$\nu$	$=$	$ ( 3\beta_{3} + 4\beta_{2} + 6 ) / 12 $ (3b3 + 4b2 + 6) / 12
$\nu^{2}$	$=$	$ ( 3\beta_{3} + 4\beta_{2} + 4\beta _1 - 42 ) / 12 $ (3b3 + 4b2 + 4*b1 - 42) / 12
$\nu^{3}$	$=$	$ ( -21\beta_{3} - 26\beta_{2} + 6\beta _1 - 66 ) / 12 $ (-21b3 - 26b2 + 6*b1 - 66) / 12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$1$	$1$	$-1$	$1$

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} $

1601.1

0.500000	+	0.0913379i
0.500000	−	0.0913379i
0.500000	+	2.73709i
0.500000	−	2.73709i

−11.2250

1601.2

−11.2250

1601.3

11.2250

1601.4

11.2250

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3600.3.l.s		4
3.b	odd	2	1	inner	3600.3.l.s		4
4.b	odd	2	1		225.3.c.d		4
5.b	even	2	1	inner	3600.3.l.s		4
5.c	odd	4	2		720.3.c.a		4
12.b	even	2	1		225.3.c.d		4
15.d	odd	2	1	inner	3600.3.l.s		4
15.e	even	4	2		720.3.c.a		4
20.d	odd	2	1		225.3.c.d		4
20.e	even	4	2		45.3.d.a	✓	4
40.i	odd	4	2		2880.3.c.g		4
40.k	even	4	2		2880.3.c.b		4
60.h	even	2	1		225.3.c.d		4
60.l	odd	4	2		45.3.d.a	✓	4
120.q	odd	4	2		2880.3.c.b		4
120.w	even	4	2		2880.3.c.g		4
180.v	odd	12	4		405.3.h.j		8
180.x	even	12	4		405.3.h.j		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.3.d.a	✓	4	20.e	even	4	2
45.3.d.a	✓	4	60.l	odd	4	2
225.3.c.d		4	4.b	odd	2	1
225.3.c.d		4	12.b	even	2	1
225.3.c.d		4	20.d	odd	2	1
225.3.c.d		4	60.h	even	2	1
405.3.h.j		8	180.v	odd	12	4
405.3.h.j		8	180.x	even	12	4
720.3.c.a		4	5.c	odd	4	2
720.3.c.a		4	15.e	even	4	2
2880.3.c.b		4	40.k	even	4	2
2880.3.c.b		4	120.q	odd	4	2
2880.3.c.g		4	40.i	odd	4	2
2880.3.c.g		4	120.w	even	4	2
3600.3.l.s		4	1.a	even	1	1	trivial
3600.3.l.s		4	3.b	odd	2	1	inner
3600.3.l.s		4	5.b	even	2	1	inner
3600.3.l.s		4	15.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(3600, [\chi])$:

$ T_{7}^{2} - 126 $

$ T_{11}^{2} + 18 $

$ T_{13}^{2} - 126 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 126)^{2} $ (T^2 - 126)^2
$11$	$ (T^{2} + 18)^{2} $ (T^2 + 18)^2
$13$	$ (T^{2} - 126)^{2} $ (T^2 - 126)^2
$17$	$ (T^{2} + 112)^{2} $ (T^2 + 112)^2
$19$	$ (T - 20)^{4} $ (T - 20)^4
$23$	$ (T^{2} + 28)^{2} $ (T^2 + 28)^2
$29$	$ (T^{2} + 72)^{2} $ (T^2 + 72)^2
$31$	$ (T + 26)^{4} $ (T + 26)^4
$37$	$ (T^{2} - 1134)^{2} $ (T^2 - 1134)^2
$41$	$ (T^{2} + 3042)^{2} $ (T^2 + 3042)^2
$43$	$ (T^{2} - 504)^{2} $ (T^2 - 504)^2
$47$	$ (T^{2} + 448)^{2} $ (T^2 + 448)^2
$53$	$ (T^{2} + 7168)^{2} $ (T^2 + 7168)^2
$59$	$ (T^{2} + 2178)^{2} $ (T^2 + 2178)^2
$61$	$ (T + 22)^{4} $ (T + 22)^4
$67$	$ (T^{2} - 8064)^{2} $ (T^2 - 8064)^2
$71$	$ (T^{2} + 2592)^{2} $ (T^2 + 2592)^2
$73$	$ (T^{2} - 4536)^{2} $ (T^2 - 4536)^2
$79$	$ (T - 14)^{4} $ (T - 14)^4
$83$	$ (T^{2} + 5488)^{2} $ (T^2 + 5488)^2
$89$	$ (T^{2} + 7938)^{2} $ (T^2 + 7938)^2
$97$	$ (T^{2} - 504)^{2} $ (T^2 - 504)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{7}+O(q^{10}) \) q - b1 * q^7	\( q - \beta_1 q^{7} - \beta_{2} q^{11} + \beta_1 q^{13} - 2 \beta_{3} q^{17} + 20 q^{19} + \beta_{3} q^{23} + 2 \beta_{2} q^{29} - 26 q^{31} - 3 \beta_1 q^{37} - 13 \beta_{2} q^{41} - 2 \beta_1 q^{43} - 4 \beta_{3} q^{47} + 77 q^{49} - 16 \beta_{3} q^{53} + 11 \beta_{2} q^{59} - 22 q^{61} - 8 \beta_1 q^{67} + 12 \beta_{2} q^{71} + 6 \beta_1 q^{73} - 9 \beta_{3} q^{77} + 14 q^{79} - 14 \beta_{3} q^{83} - 21 \beta_{2} q^{89} - 126 q^{91} - 2 \beta_1 q^{97}+O(q^{100}) \) q - b1 * q^7 - b2 * q^11 + b1 * q^13 - 2b3 q^17 + 20 * q^19 + b3 * q^23 + 2b2 q^29 - 26 * q^31 - 3b1 q^37 - 13b2 q^41 - 2b1 q^43 - 4b3 q^47 + 77 * q^49 - 16b3 q^53 + 11b2 q^59 - 22 * q^61 - 8b1 q^67 + 12b2 q^71 + 6b1 q^73 - 9b3 q^77 + 14 * q^79 - 14b3 q^83 - 21b2 q^89 - 126 * q^91 - 2b1 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 80 q^{19} - 104 q^{31} + 308 q^{49} - 88 q^{61} + 56 q^{79} - 504 q^{91}+O(q^{100}) \) 4 * q + 80 * q^19 - 104 * q^31 + 308 * q^49 - 88 * q^61 + 56 * q^79 - 504 * q^91

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