LMFDB - Newform orbit 616.2.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,2,Mod(1,616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$616 = 2^{3} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	616.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:	$4.91878476451$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 4$ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
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 $\beta = \frac{1}{2}(1 + \sqrt{17})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{3} + (\beta - 2) q^{5} - q^{7} + (\beta + 1) q^{9} + q^{11} + (2 \beta - 2) q^{13} + (\beta - 4) q^{15} - 2 q^{17} - 2 \beta q^{19} + \beta q^{21} + (\beta - 4) q^{23} + ( - 3 \beta + 3) q^{25} + \cdots + (\beta + 1) q^{99} +O(q^{100})$ q - b * q^3 + (b - 2) * q^5 - q^7 + (b + 1) * q^9 + q^11 + (2b - 2) q^13 + (b - 4) * q^15 - 2 * q^17 - 2b q^19 + b * q^21 + (b - 4) * q^23 + (-3b + 3) q^25 + (b - 4) * q^27 - 2 * q^29 + (-b - 8) * q^31 - b * q^33 + (-b + 2) * q^35 + (b + 2) * q^37 - 8 * q^39 - 10 * q^41 + 4 * q^43 + 2 * q^45 + (-4b + 4) q^47 + q^49 + 2b q^51 + (-4b + 6) q^53 + (b - 2) * q^55 + (2b + 8) q^57 + (b - 8) * q^59 + (-4b + 6) q^61 + (-b - 1) * q^63 + (-4b + 12) q^65 - b * q^67 + (3b - 4) q^69 + (3b - 4) q^71 + (-4b + 6) q^73 + 12 * q^75 - q^77 + 2b q^79 - 7 * q^81 - 8 * q^83 + (-2b + 4) q^85 + 2b q^87 + (-b - 10) * q^89 + (-2b + 2) q^91 + (9b + 4) q^93 + (2b - 8) q^95 + (b + 6) * q^97 + (b + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 7 q^{15} - 4 q^{17} - 2 q^{19} + q^{21} - 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} - q^{33} + 3 q^{35} + 5 q^{37} - 16 q^{39}+ \cdots + 3 q^{99}+O(q^{100})$ 2 * q - q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - 7 * q^15 - 4 * q^17 - 2 * q^19 + q^21 - 7 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 17 * q^31 - q^33 + 3 * q^35 + 5 * q^37 - 16 * q^39 - 20 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 + 2 * q^49 + 2 * q^51 + 8 * q^53 - 3 * q^55 + 18 * q^57 - 15 * q^59 + 8 * q^61 - 3 * q^63 + 20 * q^65 - q^67 - 5 * q^69 - 5 * q^71 + 8 * q^73 + 24 * q^75 - 2 * q^77 + 2 * q^79 - 14 * q^81 - 16 * q^83 + 6 * q^85 + 2 * q^87 - 21 * q^89 + 2 * q^91 + 17 * q^93 - 14 * q^95 + 13 * q^97 + 3 * 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

2.56155
−1.56155

−2.56155

0.561553

−1.00000

3.56155

1.2

1.56155

−3.56155

−1.00000

−0.561553

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$+1$
$11$	$-1$

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	616.2.a.f	✓	2
3.b	odd	2	1		5544.2.a.bf		2
4.b	odd	2	1		1232.2.a.o		2
7.b	odd	2	1		4312.2.a.t		2
8.b	even	2	1		4928.2.a.bs		2
8.d	odd	2	1		4928.2.a.bo		2
11.b	odd	2	1		6776.2.a.l		2
28.d	even	2	1		8624.2.a.bi		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
616.2.a.f	✓	2	1.a	even	1	1	trivial
1232.2.a.o		2	4.b	odd	2	1
4312.2.a.t		2	7.b	odd	2	1
4928.2.a.bo		2	8.d	odd	2	1
4928.2.a.bs		2	8.b	even	2	1
5544.2.a.bf		2	3.b	odd	2	1
6776.2.a.l		2	11.b	odd	2	1
8624.2.a.bi		2	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(616))$ :

T_{3}^{2} + T_{3} - 4

T_{5}^{2} + 3T_{5} - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + T - 4$ T^2 + T - 4
$5$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$7$	$(T + 1)^{2}$ (T + 1)^2
$11$	$(T - 1)^{2}$ (T - 1)^2
$13$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$17$	$(T + 2)^{2}$ (T + 2)^2
$19$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$23$	$T^{2} + 7T + 8$ T^2 + 7*T + 8
$29$	$(T + 2)^{2}$ (T + 2)^2
$31$	$T^{2} + 17T + 68$ T^2 + 17*T + 68
$37$	$T^{2} - 5T + 2$ T^2 - 5*T + 2
$41$	$(T + 10)^{2}$ (T + 10)^2
$43$	$(T - 4)^{2}$ (T - 4)^2
$47$	$T^{2} - 4T - 64$ T^2 - 4*T - 64
$53$	$T^{2} - 8T - 52$ T^2 - 8*T - 52
$59$	$T^{2} + 15T + 52$ T^2 + 15*T + 52
$61$	$T^{2} - 8T - 52$ T^2 - 8*T - 52
$67$	$T^{2} + T - 4$ T^2 + T - 4
$71$	$T^{2} + 5T - 32$ T^2 + 5*T - 32
$73$	$T^{2} - 8T - 52$ T^2 - 8*T - 52
$79$	$T^{2} - 2T - 16$ T^2 - 2*T - 16
$83$	$(T + 8)^{2}$ (T + 8)^2
$89$	$T^{2} + 21T + 106$ T^2 + 21*T + 106
$97$	$T^{2} - 13T + 38$ T^2 - 13*T + 38
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