LMFDB - Newform orbit 1225.2.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1225 = 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1225.a (trivial)

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:	yes
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $\beta = \sqrt{2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta - 1) q^{2} + (\beta + 1) q^{3} + ( - 2 \beta + 1) q^{4} + q^{6} + (\beta - 3) q^{8} + 2 \beta q^{9} + (2 \beta + 2) q^{11} + ( - \beta - 3) q^{12} + (2 \beta - 2) q^{13} + 3 q^{16} + ( - 2 \beta + 2) q^{17}+ \cdots + (4 \beta + 8) q^{99}+O(q^{100})$ q + (b - 1) * q^2 + (b + 1) * q^3 + (-2b + 1) q^4 + q^6 + (b - 3) * q^8 + 2b q^9 + (2b + 2) q^11 + (-b - 3) * q^12 + (2b - 2) q^13 + 3 * q^16 + (-2b + 2) q^17 + (-2b + 4) q^18 + 2b q^19 + 2 * q^22 + (b + 1) * q^23 + (-2b - 1) q^24 + (-4b + 6) q^26 + (-b + 1) * q^27 - q^29 + 6 * q^31 + (b + 3) * q^32 + (4b + 6) q^33 + (4b - 6) q^34 + (2b - 8) q^36 + (-2b + 4) q^38 + 2 * q^39 + (-2b + 5) q^41 + (-b - 5) * q^43 + (-2b - 6) q^44 + q^46 + 2 * q^47 + (3b + 3) q^48 - 2 * q^51 + (6b - 10) q^52 + (2b + 4) q^53 + (2b - 3) q^54 + (2b + 4) q^57 + (-b + 1) * q^58 + (6b + 4) q^59 + (6b + 3) q^61 + (6b - 6) q^62 + (2b - 7) q^64 + (2b + 2) q^66 + (-b - 11) * q^67 + (-6b + 10) q^68 + (2b + 3) q^69 + (-6b - 4) q^71 + (-6b + 4) q^72 + (2b + 2) q^73 + (2b - 8) q^76 + (2b - 2) q^78 + (-2b + 12) q^79 + (-6b - 1) q^81 + (7b - 9) q^82 + (-9b + 1) q^83 + (-4b + 3) q^86 + (-b - 1) * q^87 + (-4b - 2) q^88 + (-4b + 3) q^89 + (-b - 3) * q^92 + (6b + 6) q^93 + (2b - 2) q^94 + (4b + 5) q^96 + (-4b + 6) q^97 + (4b + 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 6 q^{16} + 4 q^{17} + 8 q^{18} + 4 q^{22} + 2 q^{23} - 2 q^{24} + 12 q^{26} + 2 q^{27} - 2 q^{29} + 12 q^{31} + 6 q^{32}+ \cdots + 16 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^11 - 6 * q^12 - 4 * q^13 + 6 * q^16 + 4 * q^17 + 8 * q^18 + 4 * q^22 + 2 * q^23 - 2 * q^24 + 12 * q^26 + 2 * q^27 - 2 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^33 - 12 * q^34 - 16 * q^36 + 8 * q^38 + 4 * q^39 + 10 * q^41 - 10 * q^43 - 12 * q^44 + 2 * q^46 + 4 * q^47 + 6 * q^48 - 4 * q^51 - 20 * q^52 + 8 * q^53 - 6 * q^54 + 8 * q^57 + 2 * q^58 + 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^66 - 22 * q^67 + 20 * q^68 + 6 * q^69 - 8 * q^71 + 8 * q^72 + 4 * q^73 - 16 * q^76 - 4 * q^78 + 24 * q^79 - 2 * q^81 - 18 * q^82 + 2 * q^83 + 6 * q^86 - 2 * q^87 - 4 * q^88 + 6 * q^89 - 6 * q^92 + 12 * q^93 - 4 * q^94 + 10 * q^96 + 12 * q^97 + 16 * q^99

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}

1.1

−1.41421
1.41421

−2.41421

−0.414214

3.82843

1.00000

−4.41421

−2.82843

1.2

0.414214

2.41421

−1.82843

1.00000

−1.58579

2.82843

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1225.2.a.m		2
5.b	even	2	1		245.2.a.g		2
5.c	odd	4	2		1225.2.b.h		4
7.b	odd	2	1		1225.2.a.k		2
7.d	odd	6	2		175.2.e.c		4
15.d	odd	2	1		2205.2.a.q		2
20.d	odd	2	1		3920.2.a.bv		2
35.c	odd	2	1		245.2.a.h		2
35.f	even	4	2		1225.2.b.g		4
35.i	odd	6	2		35.2.e.a	✓	4
35.j	even	6	2		245.2.e.e		4
35.k	even	12	4		175.2.k.a		8
105.g	even	2	1		2205.2.a.n		2
105.p	even	6	2		315.2.j.e		4
140.c	even	2	1		3920.2.a.bq		2
140.s	even	6	2		560.2.q.k		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.e.a	✓	4	35.i	odd	6	2
175.2.e.c		4	7.d	odd	6	2
175.2.k.a		8	35.k	even	12	4
245.2.a.g		2	5.b	even	2	1
245.2.a.h		2	35.c	odd	2	1
245.2.e.e		4	35.j	even	6	2
315.2.j.e		4	105.p	even	6	2
560.2.q.k		4	140.s	even	6	2
1225.2.a.k		2	7.b	odd	2	1
1225.2.a.m		2	1.a	even	1	1	trivial
1225.2.b.g		4	35.f	even	4	2
1225.2.b.h		4	5.c	odd	4	2
2205.2.a.n		2	105.g	even	2	1
2205.2.a.q		2	15.d	odd	2	1
3920.2.a.bq		2	140.c	even	2	1
3920.2.a.bv		2	20.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 $S_{2}^{\mathrm{new}}(\Gamma_0(1225))$ :

T_{2}^{2} + 2T_{2} - 1

T_{3}^{2} - 2T_{3} - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2T - 1$ T^2 + 2*T - 1
$3$	$T^{2} - 2T - 1$ T^2 - 2*T - 1
$5$	$T^{2}$ T^2
$7$	$T^{2}$ T^2
$11$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$13$	$T^{2} + 4T - 4$ T^2 + 4*T - 4
$17$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$19$	$T^{2} - 8$ T^2 - 8
$23$	$T^{2} - 2T - 1$ T^2 - 2*T - 1
$29$	$(T + 1)^{2}$ (T + 1)^2
$31$	$(T - 6)^{2}$ (T - 6)^2
$37$	$T^{2}$ T^2
$41$	$T^{2} - 10T + 17$ T^2 - 10*T + 17
$43$	$T^{2} + 10T + 23$ T^2 + 10*T + 23
$47$	$(T - 2)^{2}$ (T - 2)^2
$53$	$T^{2} - 8T + 8$ T^2 - 8*T + 8
$59$	$T^{2} - 8T - 56$ T^2 - 8*T - 56
$61$	$T^{2} - 6T - 63$ T^2 - 6*T - 63
$67$	$T^{2} + 22T + 119$ T^2 + 22*T + 119
$71$	$T^{2} + 8T - 56$ T^2 + 8*T - 56
$73$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$79$	$T^{2} - 24T + 136$ T^2 - 24*T + 136
$83$	$T^{2} - 2T - 161$ T^2 - 2*T - 161
$89$	$T^{2} - 6T - 23$ T^2 - 6*T - 23
$97$	$T^{2} - 12T + 4$ T^2 - 12*T + 4
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