LMFDB - Newform orbit 1984.2.a.q

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1984,2,Mod(1,1984)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("1984.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,2,0,-2,0,4,0,4,0,0,0,-4,0,6] 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:	yes
Analytic conductor:	$15.8423197610$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 992)
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 q^{3} + q^{5} + (\beta + 1) q^{7} - q^{9} + ( - 2 \beta + 2) q^{11} + (\beta + 2) q^{13} + \beta q^{15} + (3 \beta - 2) q^{17} + ( - \beta + 3) q^{19} + (\beta + 2) q^{21} + ( - \beta + 4) q^{23}+ \cdots + (2 \beta - 2) q^{99}+O(q^{100}) $ q + b * q^3 + q^5 + (b + 1) * q^7 - q^9 + (-2b + 2) q^11 + (b + 2) * q^13 + b * q^15 + (3b - 2) q^17 + (-b + 3) * q^19 + (b + 2) * q^21 + (-b + 4) * q^23 - 4 * q^25 - 4b q^27 + 5b q^29 + q^31 + (2b - 4) q^33 + (b + 1) * q^35 + (-4b + 6) q^37 + (2b + 2) q^39 + (-4b - 1) q^41 + (7b - 2) q^43 - q^45 + (-2b + 4) q^47 + (2b - 4) q^49 + (-2b + 6) q^51 + 4 * q^53 + (-2b + 2) q^55 + (3b - 2) q^57 + (5b + 3) q^59 + (-3b + 6) q^61 + (-b - 1) * q^63 + (b + 2) * q^65 + 6 * q^67 + (4b - 2) q^69 + (-b + 5) * q^71 + (-4b - 6) q^73 - 4b q^75 - 2 * q^77 + (-b - 6) * q^79 - 5 * q^81 + (2b + 2) q^83 + (3b - 2) q^85 + 10 * q^87 + (-3b + 8) q^89 + (3b + 4) q^91 + b * q^93 + (-b + 3) * q^95 + (-6b - 1) q^97 + (2b - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7} - 2 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{17} + 6 q^{19} + 4 q^{21} + 8 q^{23} - 8 q^{25} + 2 q^{31} - 8 q^{33} + 2 q^{35} + 12 q^{37} + 4 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 - 2 * q^9 + 4 * q^11 + 4 * q^13 - 4 * q^17 + 6 * q^19 + 4 * q^21 + 8 * q^23 - 8 * q^25 + 2 * q^31 - 8 * q^33 + 2 * q^35 + 12 * q^37 + 4 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^45 + 8 * q^47 - 8 * q^49 + 12 * q^51 + 8 * q^53 + 4 * q^55 - 4 * q^57 + 6 * q^59 + 12 * q^61 - 2 * q^63 + 4 * q^65 + 12 * q^67 - 4 * q^69 + 10 * q^71 - 12 * q^73 - 4 * q^77 - 12 * q^79 - 10 * q^81 + 4 * q^83 - 4 * q^85 + 20 * q^87 + 16 * q^89 + 8 * q^91 + 6 * q^95 - 2 * q^97 - 4 * 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

−1.41421

1.00000

−0.414214

−1.00000

1.2

1.41421

1.00000

2.41421

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$31$	$ -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	1984.2.a.q		2
4.b	odd	2	1		1984.2.a.p		2
8.b	even	2	1		992.2.a.b	yes	2
8.d	odd	2	1		992.2.a.a	✓	2
24.f	even	2	1		8928.2.a.s		2
24.h	odd	2	1		8928.2.a.v		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
992.2.a.a	✓	2	8.d	odd	2	1
992.2.a.b	yes	2	8.b	even	2	1
1984.2.a.p		2	4.b	odd	2	1
1984.2.a.q		2	1.a	even	1	1	trivial
8928.2.a.s		2	24.f	even	2	1
8928.2.a.v		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1984))$:

$ T_{3}^{2} - 2 $

T3^2 - 2

$ T_{5} - 1 $

T5 - 1

$ T_{7}^{2} - 2T_{7} - 1 $

T7^2 - 2*T7 - 1

$ T_{19}^{2} - 6T_{19} + 7 $

T19^2 - 6*T19 + 7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} - 2 $ T^2 - 2
$5$	$ (T - 1)^{2} $ (T - 1)^2
$7$	$ T^{2} - 2T - 1 $ T^2 - 2*T - 1
$11$	$ T^{2} - 4T - 4 $ T^2 - 4*T - 4
$13$	$ T^{2} - 4T + 2 $ T^2 - 4*T + 2
$17$	$ T^{2} + 4T - 14 $ T^2 + 4*T - 14
$19$	$ T^{2} - 6T + 7 $ T^2 - 6*T + 7
$23$	$ T^{2} - 8T + 14 $ T^2 - 8*T + 14
$29$	$ T^{2} - 50 $ T^2 - 50
$31$	$ (T - 1)^{2} $ (T - 1)^2
$37$	$ T^{2} - 12T + 4 $ T^2 - 12*T + 4
$41$	$ T^{2} + 2T - 31 $ T^2 + 2*T - 31
$43$	$ T^{2} + 4T - 94 $ T^2 + 4*T - 94
$47$	$ T^{2} - 8T + 8 $ T^2 - 8*T + 8
$53$	$ (T - 4)^{2} $ (T - 4)^2
$59$	$ T^{2} - 6T - 41 $ T^2 - 6*T - 41
$61$	$ T^{2} - 12T + 18 $ T^2 - 12*T + 18
$67$	$ (T - 6)^{2} $ (T - 6)^2
$71$	$ T^{2} - 10T + 23 $ T^2 - 10*T + 23
$73$	$ T^{2} + 12T + 4 $ T^2 + 12*T + 4
$79$	$ T^{2} + 12T + 34 $ T^2 + 12*T + 34
$83$	$ T^{2} - 4T - 4 $ T^2 - 4*T - 4
$89$	$ T^{2} - 16T + 46 $ T^2 - 16*T + 46
$97$	$ T^{2} + 2T - 71 $ T^2 + 2*T - 71
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Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

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

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