LMFDB - Newform orbit 2480.2.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2480,2,Mod(1,2480)] 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("2480.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2480 = 2^{4} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-2,0,0,0,3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.8028997013$
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:	no (minimal twist has level 1240)
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} - q^{5} + (\beta + 1) q^{9} + ( - 2 \beta + 2) q^{11} - 2 q^{13} - \beta q^{15} + ( - \beta - 4) q^{17} + ( - \beta - 4) q^{19} + ( - 2 \beta + 2) q^{23} + q^{25} + ( - \beta + 4) q^{27} + \cdots + ( - 2 \beta - 6) q^{99} +O(q^{100}) $ q + b * q^3 - q^5 + (b + 1) * q^9 + (-2b + 2) q^11 - 2 * q^13 - b * q^15 + (-b - 4) * q^17 + (-b - 4) * q^19 + (-2b + 2) q^23 + q^25 + (-b + 4) * q^27 + (2b - 4) q^29 - q^31 - 8 * q^33 + (-3b + 6) q^37 - 2b q^39 + (-b - 6) * q^41 - b * q^43 + (-b - 1) * q^45 + (2b + 4) q^47 - 7 * q^49 + (-5b - 4) q^51 + (b + 6) * q^53 + (2b - 2) q^55 + (-5b - 4) q^57 + (3b + 4) q^59 + 2 * q^65 + (2b + 4) q^67 - 8 * q^69 + (5b - 4) q^71 + (-3b - 4) q^73 + b * q^75 + (2b + 4) q^79 - 7 * q^81 + (3b - 4) q^83 + (b + 4) * q^85 + (-2b + 8) q^87 + (6b - 2) q^89 - b * q^93 + (b + 4) * q^95 - 10 * q^97 + (-2b - 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} + 3 q^{9} + 2 q^{11} - 4 q^{13} - q^{15} - 9 q^{17} - 9 q^{19} + 2 q^{23} + 2 q^{25} + 7 q^{27} - 6 q^{29} - 2 q^{31} - 16 q^{33} + 9 q^{37} - 2 q^{39} - 13 q^{41} - q^{43} - 3 q^{45}+ \cdots - 14 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 + 3 * q^9 + 2 * q^11 - 4 * q^13 - q^15 - 9 * q^17 - 9 * q^19 + 2 * q^23 + 2 * q^25 + 7 * q^27 - 6 * q^29 - 2 * q^31 - 16 * q^33 + 9 * q^37 - 2 * q^39 - 13 * q^41 - q^43 - 3 * q^45 + 10 * q^47 - 14 * q^49 - 13 * q^51 + 13 * q^53 - 2 * q^55 - 13 * q^57 + 11 * q^59 + 4 * q^65 + 10 * q^67 - 16 * q^69 - 3 * q^71 - 11 * q^73 + q^75 + 10 * q^79 - 14 * q^81 - 5 * q^83 + 9 * q^85 + 14 * q^87 + 2 * q^89 - q^93 + 9 * q^95 - 20 * q^97 - 14 * 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

−1.56155
2.56155

−1.56155

−1.00000

−0.561553

1.2

2.56155

−1.00000

3.56155

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$5$	$ +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	2480.2.a.r		2
4.b	odd	2	1		1240.2.a.h	✓	2
8.b	even	2	1		9920.2.a.bm		2
8.d	odd	2	1		9920.2.a.br		2
20.d	odd	2	1		6200.2.a.o		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1240.2.a.h	✓	2	4.b	odd	2	1
2480.2.a.r		2	1.a	even	1	1	trivial
6200.2.a.o		2	20.d	odd	2	1
9920.2.a.bm		2	8.b	even	2	1
9920.2.a.br		2	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2480))$:

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

T3^2 - T3 - 4

$ T_{7} $

T7

$ T_{11}^{2} - 2T_{11} - 16 $

T11^2 - 2*T11 - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} - T - 4 $ T^2 - T - 4
$5$	$ (T + 1)^{2} $ (T + 1)^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 2T - 16 $ T^2 - 2*T - 16
$13$	$ (T + 2)^{2} $ (T + 2)^2
$17$	$ T^{2} + 9T + 16 $ T^2 + 9*T + 16
$19$	$ T^{2} + 9T + 16 $ T^2 + 9*T + 16
$23$	$ T^{2} - 2T - 16 $ T^2 - 2*T - 16
$29$	$ T^{2} + 6T - 8 $ T^2 + 6*T - 8
$31$	$ (T + 1)^{2} $ (T + 1)^2
$37$	$ T^{2} - 9T - 18 $ T^2 - 9*T - 18
$41$	$ T^{2} + 13T + 38 $ T^2 + 13*T + 38
$43$	$ T^{2} + T - 4 $ T^2 + T - 4
$47$	$ T^{2} - 10T + 8 $ T^2 - 10*T + 8
$53$	$ T^{2} - 13T + 38 $ T^2 - 13*T + 38
$59$	$ T^{2} - 11T - 8 $ T^2 - 11*T - 8
$61$	$ T^{2} $ T^2
$67$	$ T^{2} - 10T + 8 $ T^2 - 10*T + 8
$71$	$ T^{2} + 3T - 104 $ T^2 + 3*T - 104
$73$	$ T^{2} + 11T - 8 $ T^2 + 11*T - 8
$79$	$ T^{2} - 10T + 8 $ T^2 - 10*T + 8
$83$	$ T^{2} + 5T - 32 $ T^2 + 5*T - 32
$89$	$ T^{2} - 2T - 152 $ T^2 - 2*T - 152
$97$	$ (T + 10)^{2} $ (T + 10)^2
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Level:	\( N \)	\(=\)	\( 2480 = 2^{4} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2480.a (trivial)

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

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