LMFDB - Newform orbit 1856.2.a.u

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1856,2,Mod(1,1856)] 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("1856.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2,0,-4,0,0,0,2,0,-2,0,-2,0,0] 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:	$14.8202346151$
Analytic rank:	$1$
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 928)
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^{3} - q^{5} - 2 q^{7} + 2 \beta q^{9} + ( - \beta + 1) q^{11} + ( - 2 \beta - 1) q^{13} + ( - \beta - 1) q^{15} - 2 \beta q^{17} + ( - 4 \beta + 2) q^{19} + ( - 2 \beta - 2) q^{21} + ( - 2 \beta - 4) q^{23} + \cdots + (2 \beta - 4) q^{99} +O(q^{100}) $ q + (b + 1) * q^3 - q^5 - 2 * q^7 + 2b q^9 + (-b + 1) * q^11 + (-2b - 1) q^13 + (-b - 1) * q^15 - 2b q^17 + (-4b + 2) q^19 + (-2b - 2) q^21 + (-2b - 4) q^23 - 4 * q^25 + (-b + 1) * q^27 - q^29 + (b - 1) * q^31 - q^33 + 2 * q^35 + 4b q^37 + (-3b - 5) q^39 + (2b - 2) q^41 + (3b + 1) q^43 - 2b q^45 + (b - 3) * q^47 - 3 * q^49 + (-2b - 4) q^51 + (6b - 3) q^53 + (b - 1) * q^55 + (-2b - 6) q^57 + (6b + 4) q^59 + (6b - 4) q^61 - 4b q^63 + (2b + 1) q^65 + 4 * q^67 + (-6b - 8) q^69 + (2b - 6) q^71 - 8 * q^73 + (-4b - 4) q^75 + (2b - 2) q^77 + (7b - 1) q^79 + (-6b - 1) q^81 + (-6b + 4) q^83 + 2b q^85 + (-b - 1) * q^87 + (-2b - 6) q^89 + (4b + 2) q^91 + q^93 + (4b - 2) q^95 + (-2b - 6) q^97 + (2b - 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{19} - 4 q^{21} - 8 q^{23} - 8 q^{25} + 2 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{35} - 10 q^{39} - 4 q^{41} + 2 q^{43} - 6 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^19 - 4 * q^21 - 8 * q^23 - 8 * q^25 + 2 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^35 - 10 * q^39 - 4 * q^41 + 2 * q^43 - 6 * q^47 - 6 * q^49 - 8 * q^51 - 6 * q^53 - 2 * q^55 - 12 * q^57 + 8 * q^59 - 8 * q^61 + 2 * q^65 + 8 * q^67 - 16 * q^69 - 12 * q^71 - 16 * q^73 - 8 * q^75 - 4 * q^77 - 2 * q^79 - 2 * q^81 + 8 * q^83 - 2 * q^87 - 12 * q^89 + 4 * q^91 + 2 * q^93 - 4 * q^95 - 12 * q^97 - 8 * 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

−0.414214

−1.00000

−2.00000

−2.82843

1.2

2.41421

−1.00000

−2.00000

2.82843

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$29$	$ +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	1856.2.a.u		2
4.b	odd	2	1		1856.2.a.q		2
8.b	even	2	1		928.2.a.c	✓	2
8.d	odd	2	1		928.2.a.e	yes	2
24.f	even	2	1		8352.2.a.o		2
24.h	odd	2	1		8352.2.a.l		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
928.2.a.c	✓	2	8.b	even	2	1
928.2.a.e	yes	2	8.d	odd	2	1
1856.2.a.q		2	4.b	odd	2	1
1856.2.a.u		2	1.a	even	1	1	trivial
8352.2.a.l		2	24.h	odd	2	1
8352.2.a.o		2	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1856))$:

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

T3^2 - 2*T3 - 1

$ T_{5} + 1 $

T5 + 1

$ T_{17}^{2} - 8 $

T17^2 - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} - 2T - 1 $ T^2 - 2*T - 1
$5$	$ (T + 1)^{2} $ (T + 1)^2
$7$	$ (T + 2)^{2} $ (T + 2)^2
$11$	$ T^{2} - 2T - 1 $ T^2 - 2*T - 1
$13$	$ T^{2} + 2T - 7 $ T^2 + 2*T - 7
$17$	$ T^{2} - 8 $ T^2 - 8
$19$	$ T^{2} - 4T - 28 $ T^2 - 4*T - 28
$23$	$ T^{2} + 8T + 8 $ T^2 + 8*T + 8
$29$	$ (T + 1)^{2} $ (T + 1)^2
$31$	$ T^{2} + 2T - 1 $ T^2 + 2*T - 1
$37$	$ T^{2} - 32 $ T^2 - 32
$41$	$ T^{2} + 4T - 4 $ T^2 + 4*T - 4
$43$	$ T^{2} - 2T - 17 $ T^2 - 2*T - 17
$47$	$ T^{2} + 6T + 7 $ T^2 + 6*T + 7
$53$	$ T^{2} + 6T - 63 $ T^2 + 6*T - 63
$59$	$ T^{2} - 8T - 56 $ T^2 - 8*T - 56
$61$	$ T^{2} + 8T - 56 $ T^2 + 8*T - 56
$67$	$ (T - 4)^{2} $ (T - 4)^2
$71$	$ T^{2} + 12T + 28 $ T^2 + 12*T + 28
$73$	$ (T + 8)^{2} $ (T + 8)^2
$79$	$ T^{2} + 2T - 97 $ T^2 + 2*T - 97
$83$	$ T^{2} - 8T - 56 $ T^2 - 8*T - 56
$89$	$ T^{2} + 12T + 28 $ T^2 + 12*T + 28
$97$	$ T^{2} + 12T + 28 $ T^2 + 12*T + 28
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Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

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

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