LMFDB - Newform orbit 8624.2.a.ch

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8624 = 2^{4} \cdot 7^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 77)
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 a basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_1 - 1) q^{3} + ( - \beta_{2} + \beta_1 + 2) q^{5} + (\beta_{2} + 2 \beta_1) q^{9} - q^{11} + ( - 2 \beta_{2} + \beta_1 + 1) q^{13} + ( - 2 \beta_1 - 3) q^{15} + (3 \beta_{2} - 4 \beta_1 - 1) q^{17}+ \cdots + ( - \beta_{2} - 2 \beta_1) q^{99}+O(q^{100})$ q + (-b1 - 1) * q^3 + (-b2 + b1 + 2) * q^5 + (b2 + 2b1) q^9 - q^11 + (-2b2 + b1 + 1) q^13 + (-2b1 - 3) q^15 + (3b2 - 4b1 - 1) * q^17 + (-b2 + 2b1 - 3) q^19 + (-3b2 - 2b1) * q^23 + (-4b2 + 3b1 + 1) * q^25 + (-3b2 - 2) q^27 + (-b2 - 3b1 - 1) q^29 + (-b2 + b1 - 3) * q^31 + (b1 + 1) * q^33 + (-2b2 - 2b1) * q^37 + (b2 - 1) * q^39 + (2b2 - 3b1 + 3) * q^41 + (-2b2 + b1) q^43 + (5b2 + 2b1 + 1) * q^45 + (-3b2 + 6b1 + 1) * q^47 + (b2 + 2b1 + 6) q^51 + (6b2 - 3b1 + 3) * q^53 + (b2 - b1 - 2) * q^55 + (-b2 + 2b1) q^57 + (6b2 - 5b1) * q^59 + (2b2 + 4b1 + 4) * q^61 + (-6b2 + 2b1 + 5) * q^65 - 2b1 q^67 + (5b2 + 5b1 + 7) * q^69 + (7b2 - 2b1 + 3) * q^71 + (-b1 + 2) * q^73 + (b2 - 3) * q^75 + (-7b2 + 5b1 + 1) * q^79 + (-b1 + 5) * q^81 + (-3b2 - 2b1 + 5) * q^83 + (6b2 - 5b1 - 9) * q^85 + (4b2 + 5b1 + 8) * q^87 + (-b2 + b1 - 5) * q^89 + (3b1 + 2) q^93 + (2b2 - b1 - 3) q^95 + (-b1 + 15) * q^97 + (-b2 - 2b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{3} + 6 q^{5} - 3 q^{11} + 3 q^{13} - 9 q^{15} - 3 q^{17} - 9 q^{19} + 3 q^{25} - 6 q^{27} - 3 q^{29} - 9 q^{31} + 3 q^{33} - 3 q^{39} + 9 q^{41} + 3 q^{45} + 3 q^{47} + 18 q^{51} + 9 q^{53} - 6 q^{55}+ \cdots + 45 q^{97}+O(q^{100})$ 3 * q - 3 * q^3 + 6 * q^5 - 3 * q^11 + 3 * q^13 - 9 * q^15 - 3 * q^17 - 9 * q^19 + 3 * q^25 - 6 * q^27 - 3 * q^29 - 9 * q^31 + 3 * q^33 - 3 * q^39 + 9 * q^41 + 3 * q^45 + 3 * q^47 + 18 * q^51 + 9 * q^53 - 6 * q^55 + 12 * q^61 + 15 * q^65 + 21 * q^69 + 9 * q^71 + 6 * q^73 - 9 * q^75 + 3 * q^79 + 15 * q^81 + 15 * q^83 - 27 * q^85 + 24 * q^87 - 15 * q^89 + 6 * q^93 - 9 * q^95 + 45 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{18} + \zeta_{18}^{-1}$ :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2

Display $\beta_i$ in terms of $\nu^j$

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

1.87939
−0.347296
−1.53209

−2.87939

2.34730

5.29086

1.2

−0.652704

3.53209

−2.57398

1.3

0.532089

0.120615

−2.71688

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	8624.2.a.ch		3
4.b	odd	2	1		539.2.a.j		3
7.b	odd	2	1		8624.2.a.co		3
7.c	even	3	2		1232.2.q.m		6
12.b	even	2	1		4851.2.a.bj		3
28.d	even	2	1		539.2.a.g		3
28.f	even	6	2		539.2.e.m		6
28.g	odd	6	2		77.2.e.a	✓	6
44.c	even	2	1		5929.2.a.x		3
84.h	odd	2	1		4851.2.a.bk		3
84.n	even	6	2		693.2.i.h		6
308.g	odd	2	1		5929.2.a.u		3
308.n	even	6	2		847.2.e.c		6
308.bb	odd	30	8		847.2.n.g		24
308.bc	even	30	8		847.2.n.f		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	28.g	odd	6	2
539.2.a.g		3	28.d	even	2	1
539.2.a.j		3	4.b	odd	2	1
539.2.e.m		6	28.f	even	6	2
693.2.i.h		6	84.n	even	6	2
847.2.e.c		6	308.n	even	6	2
847.2.n.f		24	308.bc	even	30	8
847.2.n.g		24	308.bb	odd	30	8
1232.2.q.m		6	7.c	even	3	2
4851.2.a.bj		3	12.b	even	2	1
4851.2.a.bk		3	84.h	odd	2	1
5929.2.a.u		3	308.g	odd	2	1
5929.2.a.x		3	44.c	even	2	1
8624.2.a.ch		3	1.a	even	1	1	trivial
8624.2.a.co		3	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8624))$ :

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

T_{5}^{3} - 6T_{5}^{2} + 9T_{5} - 1

T_{13}^{3} - 3T_{13}^{2} - 6T_{13} - 1

T_{17}^{3} + 3T_{17}^{2} - 36T_{17} - 127

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + 3T^{2} - 1$ T^3 + 3*T^2 - 1
$5$	$T^{3} - 6 T^{2} + \cdots - 1$ T^3 - 6T^2 + 9T - 1
$7$	$T^{3}$ T^3
$11$	$(T + 1)^{3}$ (T + 1)^3
$13$	$T^{3} - 3 T^{2} + \cdots - 1$ T^3 - 3T^2 - 6T - 1
$17$	$T^{3} + 3 T^{2} + \cdots - 127$ T^3 + 3T^2 - 36T - 127
$19$	$T^{3} + 9 T^{2} + \cdots + 9$ T^3 + 9T^2 + 18T + 9
$23$	$T^{3} - 57T + 107$ T^3 - 57*T + 107
$29$	$T^{3} + 3 T^{2} + \cdots + 51$ T^3 + 3T^2 - 36T + 51
$31$	$T^{3} + 9 T^{2} + \cdots + 19$ T^3 + 9T^2 + 24T + 19
$37$	$T^{3} - 36T + 72$ T^3 - 36*T + 72
$41$	$T^{3} - 9 T^{2} + \cdots - 1$ T^3 - 9T^2 + 6T - 1
$43$	$T^{3} - 9T - 9$ T^3 - 9*T - 9
$47$	$T^{3} - 3 T^{2} + \cdots + 323$ T^3 - 3T^2 - 78T + 323
$53$	$T^{3} - 9 T^{2} + \cdots + 459$ T^3 - 9T^2 - 54T + 459
$59$	$T^{3} - 93T - 19$ T^3 - 93*T - 19
$61$	$T^{3} - 12 T^{2} + \cdots - 24$ T^3 - 12T^2 - 36T - 24
$67$	$T^{3} - 12T + 8$ T^3 - 12*T + 8
$71$	$T^{3} - 9 T^{2} + \cdots + 801$ T^3 - 9T^2 - 90T + 801
$73$	$T^{3} - 6 T^{2} + \cdots - 1$ T^3 - 6T^2 + 9T - 1
$79$	$T^{3} - 3 T^{2} + \cdots - 37$ T^3 - 3T^2 - 114T - 37
$83$	$T^{3} - 15 T^{2} + \cdots + 267$ T^3 - 15T^2 + 18T + 267
$89$	$T^{3} + 15 T^{2} + \cdots + 111$ T^3 + 15T^2 + 72T + 111
$97$	$T^{3} - 45 T^{2} + \cdots - 3329$ T^3 - 45T^2 + 672T - 3329
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