LMFDB - Newform orbit 3300.2.c.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1849,3300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3300.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3300.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 660)
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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - i q^{3} + 2 i q^{7} - q^{9} + q^{11} - 2 i q^{13} - 2 q^{19} + 2 q^{21} + i q^{27} + 8 q^{31} - i q^{33} + 2 i q^{37} - 2 q^{39} - 2 i q^{43} + 3 q^{49} - 6 i q^{53} + 2 i q^{57} + 12 q^{59} + 2 q^{61} + \cdots - q^{99} +O(q^{100})$ q - i * q^3 + 2i q^7 - q^9 + q^11 - 2i q^13 - 2 * q^19 + 2 * q^21 + i * q^27 + 8 * q^31 - i * q^33 + 2i q^37 - 2 * q^39 - 2i q^43 + 3 * q^49 - 6i q^53 + 2i q^57 + 12 * q^59 + 2 * q^61 - 2i q^63 - 4i q^67 - 2i q^73 + 2i q^77 + 10 * q^79 + q^81 + 12i q^83 + 6 * q^89 + 4 * q^91 - 8i q^93 + 14i q^97 - q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{9} + 2 q^{11} - 4 q^{19} + 4 q^{21} + 16 q^{31} - 4 q^{39} + 6 q^{49} + 24 q^{59} + 4 q^{61} + 20 q^{79} + 2 q^{81} + 12 q^{89} + 8 q^{91} - 2 q^{99}+O(q^{100})$ 2 * q - 2 * q^9 + 2 * q^11 - 4 * q^19 + 4 * q^21 + 16 * q^31 - 4 * q^39 + 6 * q^49 + 24 * q^59 + 4 * q^61 + 20 * q^79 + 2 * q^81 + 12 * q^89 + 8 * q^91 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$ .

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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}

1849.1

		1.00000i
	−	1.00000i

−

1.00000i

2.00000i

−1.00000

1849.2

1.00000i

−

2.00000i

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3300.2.c.i		2
3.b	odd	2	1		9900.2.c.d		2
5.b	even	2	1	inner	3300.2.c.i		2
5.c	odd	4	1		660.2.a.d	✓	1
5.c	odd	4	1		3300.2.a.b		1
15.d	odd	2	1		9900.2.c.d		2
15.e	even	4	1		1980.2.a.f		1
15.e	even	4	1		9900.2.a.e		1
20.e	even	4	1		2640.2.a.b		1
55.e	even	4	1		7260.2.a.l		1
60.l	odd	4	1		7920.2.a.y		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
660.2.a.d	✓	1	5.c	odd	4	1
1980.2.a.f		1	15.e	even	4	1
2640.2.a.b		1	20.e	even	4	1
3300.2.a.b		1	5.c	odd	4	1
3300.2.c.i		2	1.a	even	1	1	trivial
3300.2.c.i		2	5.b	even	2	1	inner
7260.2.a.l		1	55.e	even	4	1
7920.2.a.y		1	60.l	odd	4	1
9900.2.a.e		1	15.e	even	4	1
9900.2.c.d		2	3.b	odd	2	1
9900.2.c.d		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3300, [\chi])$ :

T_{7}^{2} + 4

T7^2 + 4

T_{13}^{2} + 4

T13^2 + 4

T_{17}

T17

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 1$ T^2 + 1
$5$	$T^{2}$ T^2
$7$	$T^{2} + 4$ T^2 + 4
$11$	$(T - 1)^{2}$ (T - 1)^2
$13$	$T^{2} + 4$ T^2 + 4
$17$	$T^{2}$ T^2
$19$	$(T + 2)^{2}$ (T + 2)^2
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$(T - 8)^{2}$ (T - 8)^2
$37$	$T^{2} + 4$ T^2 + 4
$41$	$T^{2}$ T^2
$43$	$T^{2} + 4$ T^2 + 4
$47$	$T^{2}$ T^2
$53$	$T^{2} + 36$ T^2 + 36
$59$	$(T - 12)^{2}$ (T - 12)^2
$61$	$(T - 2)^{2}$ (T - 2)^2
$67$	$T^{2} + 16$ T^2 + 16
$71$	$T^{2}$ T^2
$73$	$T^{2} + 4$ T^2 + 4
$79$	$(T - 10)^{2}$ (T - 10)^2
$83$	$T^{2} + 144$ T^2 + 144
$89$	$(T - 6)^{2}$ (T - 6)^2
$97$	$T^{2} + 196$ T^2 + 196
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