LMFDB - Newform orbit 8112.2.a.bl

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,-2,0,2,0,2,0,0,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
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 $\beta = \sqrt{13}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{3} - q^{5} + (\beta - 1) q^{7} + q^{9} + ( - \beta + 1) q^{11} + q^{15} + (\beta - 2) q^{17} + (\beta + 3) q^{19} + ( - \beta + 1) q^{21} + ( - \beta + 1) q^{23} - 4 q^{25} - q^{27} + 3 q^{29}+ \cdots + ( - \beta + 1) q^{99}+O(q^{100}) $ q - q^3 - q^5 + (b - 1) * q^7 + q^9 + (-b + 1) * q^11 + q^15 + (b - 2) * q^17 + (b + 3) * q^19 + (-b + 1) * q^21 + (-b + 1) * q^23 - 4 * q^25 - q^27 + 3 * q^29 + (-2b + 2) q^31 + (b - 1) * q^33 + (-b + 1) * q^35 + (b - 6) * q^37 + (-b - 8) * q^41 + (b - 1) * q^43 - q^45 + (-b + 5) * q^47 + (-2b + 7) q^49 + (-b + 2) * q^51 + 3 * q^53 + (b - 1) * q^55 + (-b - 3) * q^57 + (-2b - 2) q^59 + (-b - 4) * q^61 + (b - 1) * q^63 + (b + 7) * q^67 + (b - 1) * q^69 + (-b + 13) * q^71 + 7 * q^73 + 4 * q^75 + (2b - 14) q^77 - 12 * q^79 + q^81 + (3b + 1) q^83 + (-b + 2) * q^85 - 3 * q^87 + (-4b - 2) q^89 + (2b - 2) q^93 + (-b - 3) * q^95 + (2b + 4) q^97 + (-b + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{33} + 2 q^{35} - 12 q^{37} - 16 q^{41} - 2 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 6 * q^29 + 4 * q^31 - 2 * q^33 + 2 * q^35 - 12 * q^37 - 16 * q^41 - 2 * q^43 - 2 * q^45 + 10 * q^47 + 14 * q^49 + 4 * q^51 + 6 * q^53 - 2 * q^55 - 6 * q^57 - 4 * q^59 - 8 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 26 * q^71 + 14 * q^73 + 8 * q^75 - 28 * q^77 - 24 * q^79 + 2 * q^81 + 2 * q^83 + 4 * q^85 - 6 * q^87 - 4 * q^89 - 4 * q^93 - 6 * q^95 + 8 * q^97 + 2 * 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

−1.30278
2.30278

−1.00000

−4.60555

1.00000

1.2

−1.00000

2.60555

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$13$	$ +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	8112.2.a.bl		2
4.b	odd	2	1		4056.2.a.v		2
13.b	even	2	1		8112.2.a.bn		2
13.c	even	3	2		624.2.q.i		4
39.i	odd	6	2		1872.2.t.q		4
52.b	odd	2	1		4056.2.a.w		2
52.f	even	4	2		4056.2.c.l		4
52.j	odd	6	2		312.2.q.d	✓	4
156.p	even	6	2		936.2.t.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.q.d	✓	4	52.j	odd	6	2
624.2.q.i		4	13.c	even	3	2
936.2.t.e		4	156.p	even	6	2
1872.2.t.q		4	39.i	odd	6	2
4056.2.a.v		2	4.b	odd	2	1
4056.2.a.w		2	52.b	odd	2	1
4056.2.c.l		4	52.f	even	4	2
8112.2.a.bl		2	1.a	even	1	1	trivial
8112.2.a.bn		2	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8112))$:

$ T_{5} + 1 $

T5 + 1

$ T_{7}^{2} + 2T_{7} - 12 $

T7^2 + 2*T7 - 12

$ T_{11}^{2} - 2T_{11} - 12 $

T11^2 - 2*T11 - 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ (T + 1)^{2} $ (T + 1)^2
$5$	$ (T + 1)^{2} $ (T + 1)^2
$7$	$ T^{2} + 2T - 12 $ T^2 + 2*T - 12
$11$	$ T^{2} - 2T - 12 $ T^2 - 2*T - 12
$13$	$ T^{2} $ T^2
$17$	$ T^{2} + 4T - 9 $ T^2 + 4*T - 9
$19$	$ T^{2} - 6T - 4 $ T^2 - 6*T - 4
$23$	$ T^{2} - 2T - 12 $ T^2 - 2*T - 12
$29$	$ (T - 3)^{2} $ (T - 3)^2
$31$	$ T^{2} - 4T - 48 $ T^2 - 4*T - 48
$37$	$ T^{2} + 12T + 23 $ T^2 + 12*T + 23
$41$	$ T^{2} + 16T + 51 $ T^2 + 16*T + 51
$43$	$ T^{2} + 2T - 12 $ T^2 + 2*T - 12
$47$	$ T^{2} - 10T + 12 $ T^2 - 10*T + 12
$53$	$ (T - 3)^{2} $ (T - 3)^2
$59$	$ T^{2} + 4T - 48 $ T^2 + 4*T - 48
$61$	$ T^{2} + 8T + 3 $ T^2 + 8*T + 3
$67$	$ T^{2} - 14T + 36 $ T^2 - 14*T + 36
$71$	$ T^{2} - 26T + 156 $ T^2 - 26*T + 156
$73$	$ (T - 7)^{2} $ (T - 7)^2
$79$	$ (T + 12)^{2} $ (T + 12)^2
$83$	$ T^{2} - 2T - 116 $ T^2 - 2*T - 116
$89$	$ T^{2} + 4T - 204 $ T^2 + 4*T - 204
$97$	$ T^{2} - 8T - 36 $ T^2 - 8*T - 36
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Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{3} - q^{5} + (\beta - 1) q^{7} + q^{9} + ( - \beta + 1) q^{11} + q^{15} + (\beta - 2) q^{17} + (\beta + 3) q^{19} + ( - \beta + 1) q^{21} + ( - \beta + 1) q^{23} - 4 q^{25} - q^{27} + 3 q^{29}+ \cdots + ( - \beta + 1) q^{99}+O(q^{100}) \) q - q^3 - q^5 + (b - 1) * q^7 + q^9 + (-b + 1) * q^11 + q^15 + (b - 2) * q^17 + (b + 3) * q^19 + (-b + 1) * q^21 + (-b + 1) * q^23 - 4 * q^25 - q^27 + 3 * q^29 + (-2b + 2) q^31 + (b - 1) * q^33 + (-b + 1) * q^35 + (b - 6) * q^37 + (-b - 8) * q^41 + (b - 1) * q^43 - q^45 + (-b + 5) * q^47 + (-2b + 7) q^49 + (-b + 2) * q^51 + 3 * q^53 + (b - 1) * q^55 + (-b - 3) * q^57 + (-2b - 2) q^59 + (-b - 4) * q^61 + (b - 1) * q^63 + (b + 7) * q^67 + (b - 1) * q^69 + (-b + 13) * q^71 + 7 * q^73 + 4 * q^75 + (2b - 14) q^77 - 12 * q^79 + q^81 + (3b + 1) q^83 + (-b + 2) * q^85 - 3 * q^87 + (-4b - 2) q^89 + (2b - 2) q^93 + (-b - 3) * q^95 + (2b + 4) q^97 + (-b + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{33} + 2 q^{35} - 12 q^{37} - 16 q^{41} - 2 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 6 * q^29 + 4 * q^31 - 2 * q^33 + 2 * q^35 - 12 * q^37 - 16 * q^41 - 2 * q^43 - 2 * q^45 + 10 * q^47 + 14 * q^49 + 4 * q^51 + 6 * q^53 - 2 * q^55 - 6 * q^57 - 4 * q^59 - 8 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 26 * q^71 + 14 * q^73 + 8 * q^75 - 28 * q^77 - 24 * q^79 + 2 * q^81 + 2 * q^83 + 4 * q^85 - 6 * q^87 - 4 * q^89 - 4 * q^93 - 6 * q^95 + 8 * q^97 + 2 * q^99

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