LMFDB - Newform orbit 2925.2.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2925,2,Mod(2224,2925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2925.2224"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2925 = 3^{2} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2925.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,0,0,0,0,-10,0,0,-4,0,-8,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$23.3562425912$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 195)
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.

$f(q)$	$=$	$q + 2 i q^{2} - 2 q^{4} + i q^{7} - 5 q^{11} - i q^{13} - 2 q^{14} - 4 q^{16} - 7 i q^{17} + 6 q^{19} - 10 i q^{22} - 3 i q^{23} + 2 q^{26} - 2 i q^{28} + 2 q^{29} + 2 q^{31} - 8 i q^{32} + 14 q^{34} - 7 i q^{37} + \cdots + 12 i q^{98} +O(q^{100})$ q + 2i q^2 - 2 * q^4 + i * q^7 - 5 * q^11 - i * q^13 - 2 * q^14 - 4 * q^16 - 7i q^17 + 6 * q^19 - 10i q^22 - 3i q^23 + 2 * q^26 - 2i q^28 + 2 * q^29 + 2 * q^31 - 8i q^32 + 14 * q^34 - 7i q^37 + 12i q^38 - 9 * q^41 - 8i q^43 + 10 * q^44 + 6 * q^46 + 10i q^47 + 6 * q^49 + 2i q^52 - 5i q^53 + 4i q^58 + 5 * q^61 + 4i q^62 + 8 * q^64 + 4i q^67 + 14i q^68 - 9 * q^71 - 6i q^73 + 14 * q^74 - 12 * q^76 - 5i q^77 + 3 * q^79 - 18i q^82 + 4i q^83 + 16 * q^86 + 11 * q^89 + q^91 + 6i q^92 - 20 * q^94 + 11i q^97 + 12i q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{4} - 10 q^{11} - 4 q^{14} - 8 q^{16} + 12 q^{19} + 4 q^{26} + 4 q^{29} + 4 q^{31} + 28 q^{34} - 18 q^{41} + 20 q^{44} + 12 q^{46} + 12 q^{49} + 10 q^{61} + 16 q^{64} - 18 q^{71} + 28 q^{74} - 24 q^{76}+ \cdots - 40 q^{94}+O(q^{100})$ 2 * q - 4 * q^4 - 10 * q^11 - 4 * q^14 - 8 * q^16 + 12 * q^19 + 4 * q^26 + 4 * q^29 + 4 * q^31 + 28 * q^34 - 18 * q^41 + 20 * q^44 + 12 * q^46 + 12 * q^49 + 10 * q^61 + 16 * q^64 - 18 * q^71 + 28 * q^74 - 24 * q^76 + 6 * q^79 + 32 * q^86 + 22 * q^89 + 2 * q^91 - 40 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$ .

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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}

2224.1

	−	1.00000i
		1.00000i

−

2.00000i

−2.00000

−

1.00000i

2224.2

2.00000i

−2.00000

1.00000i

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	2925.2.c.a		2
3.b	odd	2	1		975.2.c.c		2
5.b	even	2	1	inner	2925.2.c.a		2
5.c	odd	4	1		585.2.a.c		1
5.c	odd	4	1		2925.2.a.s		1
15.d	odd	2	1		975.2.c.c		2
15.e	even	4	1		195.2.a.c	✓	1
15.e	even	4	1		975.2.a.a		1
20.e	even	4	1		9360.2.a.bv		1
60.l	odd	4	1		3120.2.a.d		1
65.h	odd	4	1		7605.2.a.t		1
105.k	odd	4	1		9555.2.a.u		1
195.s	even	4	1		2535.2.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.a.c	✓	1	15.e	even	4	1
585.2.a.c		1	5.c	odd	4	1
975.2.a.a		1	15.e	even	4	1
975.2.c.c		2	3.b	odd	2	1
975.2.c.c		2	15.d	odd	2	1
2535.2.a.d		1	195.s	even	4	1
2925.2.a.s		1	5.c	odd	4	1
2925.2.c.a		2	1.a	even	1	1	trivial
2925.2.c.a		2	5.b	even	2	1	inner
3120.2.a.d		1	60.l	odd	4	1
7605.2.a.t		1	65.h	odd	4	1
9360.2.a.bv		1	20.e	even	4	1
9555.2.a.u		1	105.k	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2925, [\chi])$ :

T_{2}^{2} + 4

T2^2 + 4

T_{7}^{2} + 1

T7^2 + 1

T_{11} + 5

T11 + 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$ T^2 + 4
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1$ T^2 + 1
$11$	$(T + 5)^{2}$ (T + 5)^2
$13$	$T^{2} + 1$ T^2 + 1
$17$	$T^{2} + 49$ T^2 + 49
$19$	$(T - 6)^{2}$ (T - 6)^2
$23$	$T^{2} + 9$ T^2 + 9
$29$	$(T - 2)^{2}$ (T - 2)^2
$31$	$(T - 2)^{2}$ (T - 2)^2
$37$	$T^{2} + 49$ T^2 + 49
$41$	$(T + 9)^{2}$ (T + 9)^2
$43$	$T^{2} + 64$ T^2 + 64
$47$	$T^{2} + 100$ T^2 + 100
$53$	$T^{2} + 25$ T^2 + 25
$59$	$T^{2}$ T^2
$61$	$(T - 5)^{2}$ (T - 5)^2
$67$	$T^{2} + 16$ T^2 + 16
$71$	$(T + 9)^{2}$ (T + 9)^2
$73$	$T^{2} + 36$ T^2 + 36
$79$	$(T - 3)^{2}$ (T - 3)^2
$83$	$T^{2} + 16$ T^2 + 16
$89$	$(T - 11)^{2}$ (T - 11)^2
$97$	$T^{2} + 121$ T^2 + 121
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