LMFDB - Newform orbit 325.2.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$325 = 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	325.b (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:	$2.59513806569$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 65)
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.

$f(q)$	$=$	$q + i q^{2} - 2 i q^{3} + q^{4} + 2 q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} + 2 q^{11} - 2 i q^{12} - i q^{13} - 4 q^{14} - q^{16} - 2 i q^{17} - i q^{18} + 6 q^{19} + 8 q^{21} + 2 i q^{22} - 6 i q^{23} + \cdots - 2 q^{99} +O(q^{100})$ q + i * q^2 - 2i q^3 + q^4 + 2 * q^6 + 4i q^7 + 3i q^8 - q^9 + 2 * q^11 - 2i q^12 - i * q^13 - 4 * q^14 - q^16 - 2i q^17 - i * q^18 + 6 * q^19 + 8 * q^21 + 2i q^22 - 6i q^23 + 6 * q^24 + q^26 - 4i q^27 + 4i q^28 - 2 * q^29 - 10 * q^31 + 5i q^32 - 4i q^33 + 2 * q^34 - q^36 + 2i q^37 + 6i q^38 - 2 * q^39 - 6 * q^41 + 8i q^42 + 10i q^43 + 2 * q^44 + 6 * q^46 - 4i q^47 + 2i q^48 - 9 * q^49 - 4 * q^51 - i * q^52 + 2i q^53 + 4 * q^54 - 12 * q^56 - 12i q^57 - 2i q^58 - 6 * q^59 + 2 * q^61 - 10i q^62 - 4i q^63 - 7 * q^64 + 4 * q^66 + 4i q^67 - 2i q^68 - 12 * q^69 + 6 * q^71 - 3i q^72 - 6i q^73 - 2 * q^74 + 6 * q^76 + 8i q^77 - 2i q^78 + 12 * q^79 - 11 * q^81 - 6i q^82 - 16i q^83 + 8 * q^84 - 10 * q^86 + 4i q^87 + 6i q^88 - 2 * q^89 + 4 * q^91 - 6i q^92 + 20i q^93 + 4 * q^94 + 10 * q^96 + 2i q^97 - 9i q^98 - 2 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} - 8 q^{14} - 2 q^{16} + 12 q^{19} + 16 q^{21} + 12 q^{24} + 2 q^{26} - 4 q^{29} - 20 q^{31} + 4 q^{34} - 2 q^{36} - 4 q^{39} - 12 q^{41} + 4 q^{44} + 12 q^{46}+ \cdots - 4 q^{99}+O(q^{100})$ 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^11 - 8 * q^14 - 2 * q^16 + 12 * q^19 + 16 * q^21 + 12 * q^24 + 2 * q^26 - 4 * q^29 - 20 * q^31 + 4 * q^34 - 2 * q^36 - 4 * q^39 - 12 * q^41 + 4 * q^44 + 12 * q^46 - 18 * q^49 - 8 * q^51 + 8 * q^54 - 24 * q^56 - 12 * q^59 + 4 * q^61 - 14 * q^64 + 8 * q^66 - 24 * q^69 + 12 * q^71 - 4 * q^74 + 12 * q^76 + 24 * q^79 - 22 * q^81 + 16 * q^84 - 20 * q^86 - 4 * q^89 + 8 * q^91 + 8 * q^94 + 20 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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}

274.1

	−	1.00000i
		1.00000i

−

1.00000i

2.00000i

1.00000

2.00000

−

4.00000i

−

3.00000i

−1.00000

274.2

1.00000i

−

2.00000i

1.00000

2.00000

4.00000i

3.00000i

−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	325.2.b.b		2
3.b	odd	2	1		2925.2.c.h		2
5.b	even	2	1	inner	325.2.b.b		2
5.c	odd	4	1		65.2.a.a	✓	1
5.c	odd	4	1		325.2.a.d		1
15.d	odd	2	1		2925.2.c.h		2
15.e	even	4	1		585.2.a.h		1
15.e	even	4	1		2925.2.a.f		1
20.e	even	4	1		1040.2.a.f		1
20.e	even	4	1		5200.2.a.d		1
35.f	even	4	1		3185.2.a.e		1
40.i	odd	4	1		4160.2.a.q		1
40.k	even	4	1		4160.2.a.f		1
55.e	even	4	1		7865.2.a.c		1
60.l	odd	4	1		9360.2.a.ca		1
65.f	even	4	1		845.2.c.a		2
65.h	odd	4	1		845.2.a.a		1
65.h	odd	4	1		4225.2.a.g		1
65.k	even	4	1		845.2.c.a		2
65.o	even	12	2		845.2.m.b		4
65.q	odd	12	2		845.2.e.b		2
65.r	odd	12	2		845.2.e.a		2
65.t	even	12	2		845.2.m.b		4
195.s	even	4	1		7605.2.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.a	✓	1	5.c	odd	4	1
325.2.a.d		1	5.c	odd	4	1
325.2.b.b		2	1.a	even	1	1	trivial
325.2.b.b		2	5.b	even	2	1	inner
585.2.a.h		1	15.e	even	4	1
845.2.a.a		1	65.h	odd	4	1
845.2.c.a		2	65.f	even	4	1
845.2.c.a		2	65.k	even	4	1
845.2.e.a		2	65.r	odd	12	2
845.2.e.b		2	65.q	odd	12	2
845.2.m.b		4	65.o	even	12	2
845.2.m.b		4	65.t	even	12	2
1040.2.a.f		1	20.e	even	4	1
2925.2.a.f		1	15.e	even	4	1
2925.2.c.h		2	3.b	odd	2	1
2925.2.c.h		2	15.d	odd	2	1
3185.2.a.e		1	35.f	even	4	1
4160.2.a.f		1	40.k	even	4	1
4160.2.a.q		1	40.i	odd	4	1
4225.2.a.g		1	65.h	odd	4	1
5200.2.a.d		1	20.e	even	4	1
7605.2.a.f		1	195.s	even	4	1
7865.2.a.c		1	55.e	even	4	1
9360.2.a.ca		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(325, [\chi])$ :

T_{2}^{2} + 1

T_{3}^{2} + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 1$ T^2 + 1
$3$	$T^{2} + 4$ T^2 + 4
$5$	$T^{2}$ T^2
$7$	$T^{2} + 16$ T^2 + 16
$11$	$(T - 2)^{2}$ (T - 2)^2
$13$	$T^{2} + 1$ T^2 + 1
$17$	$T^{2} + 4$ T^2 + 4
$19$	$(T - 6)^{2}$ (T - 6)^2
$23$	$T^{2} + 36$ T^2 + 36
$29$	$(T + 2)^{2}$ (T + 2)^2
$31$	$(T + 10)^{2}$ (T + 10)^2
$37$	$T^{2} + 4$ T^2 + 4
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$T^{2} + 100$ T^2 + 100
$47$	$T^{2} + 16$ T^2 + 16
$53$	$T^{2} + 4$ T^2 + 4
$59$	$(T + 6)^{2}$ (T + 6)^2
$61$	$(T - 2)^{2}$ (T - 2)^2
$67$	$T^{2} + 16$ T^2 + 16
$71$	$(T - 6)^{2}$ (T - 6)^2
$73$	$T^{2} + 36$ T^2 + 36
$79$	$(T - 12)^{2}$ (T - 12)^2
$83$	$T^{2} + 256$ T^2 + 256
$89$	$(T + 2)^{2}$ (T + 2)^2
$97$	$T^{2} + 4$ T^2 + 4
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