LMFDB - Newform orbit 583.2.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [583,2,Mod(1,583)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(583, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("583.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 583 = 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	583.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$4.65527843784$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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^{2} + (\beta - 1) q^{3} + ( - \beta - 3) q^{5} + ( - \beta + 2) q^{6} + (\beta + 2) q^{7} - 2 \beta q^{8} - 2 \beta q^{9} + ( - 3 \beta - 2) q^{10} - q^{11} - \beta q^{13} + (2 \beta + 2) q^{14} + \cdots + 2 \beta q^{99} +O(q^{100}) $ q + b * q^2 + (b - 1) * q^3 + (-b - 3) * q^5 + (-b + 2) * q^6 + (b + 2) * q^7 - 2b q^8 - 2b q^9 + (-3b - 2) q^10 - q^11 - b * q^13 + (2b + 2) q^14 + (-2b + 1) q^15 - 4 * q^16 + (-4b - 2) q^17 - 4 * q^18 + (-2b + 4) q^19 + b * q^21 - b * q^22 + (b - 3) * q^23 + (2b - 4) q^24 + (6b + 6) q^25 - 2 * q^26 + (-b - 1) * q^27 + (3b + 4) q^29 + (b - 4) * q^30 + (-3b - 1) q^31 + (-b + 1) * q^33 + (-2b - 8) q^34 + (-5b - 8) q^35 + (2b - 9) q^37 + (4b - 4) q^38 + (b - 2) * q^39 + (6b + 4) q^40 + (3b - 2) q^41 + 2 * q^42 + (-b + 6) * q^43 + (6b + 4) q^45 + (-3b + 2) q^46 - 10 * q^47 + (-4b + 4) q^48 + (4b - 1) q^49 + (6b + 12) q^50 + (2b - 6) q^51 - q^53 + (-b - 2) * q^54 + (b + 3) * q^55 + (-4b - 4) q^56 + (6b - 8) q^57 + (4b + 6) q^58 + (2b - 3) q^59 + (-b + 10) * q^61 + (-b - 6) * q^62 + (-4b - 4) q^63 + 8 * q^64 + (3b + 2) q^65 + (b - 2) * q^66 + (-3b - 3) q^67 + (-4b + 5) q^69 + (-8b - 10) q^70 + (-5b - 1) q^71 + 8 * q^72 + (4b - 4) q^73 + (-9b + 4) q^74 + 6 * q^75 + (-b - 2) * q^77 + (-2b + 2) q^78 + (-2b + 10) q^79 + (4b + 12) q^80 + (6b - 1) q^81 + (-2b + 6) q^82 + (6b + 6) q^83 + (14b + 14) q^85 + (6b - 2) q^86 + (b + 2) * q^87 + 2b q^88 + (2b - 7) q^89 + (4b + 12) q^90 + (-2b - 2) q^91 + (2b - 5) q^93 - 10b q^94 + (2b - 8) q^95 + (-6b - 7) q^97 + (-b + 8) * q^98 + 2b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 6 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{10} - 2 q^{11} + 4 q^{14} + 2 q^{15} - 8 q^{16} - 4 q^{17} - 8 q^{18} + 8 q^{19} - 6 q^{23} - 8 q^{24} + 12 q^{25} - 4 q^{26} - 2 q^{27} + 8 q^{29} - 8 q^{30}+ \cdots + 16 q^{98}+O(q^{100}) $ 2 * q - 2 * q^3 - 6 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^10 - 2 * q^11 + 4 * q^14 + 2 * q^15 - 8 * q^16 - 4 * q^17 - 8 * q^18 + 8 * q^19 - 6 * q^23 - 8 * q^24 + 12 * q^25 - 4 * q^26 - 2 * q^27 + 8 * q^29 - 8 * q^30 - 2 * q^31 + 2 * q^33 - 16 * q^34 - 16 * q^35 - 18 * q^37 - 8 * q^38 - 4 * q^39 + 8 * q^40 - 4 * q^41 + 4 * q^42 + 12 * q^43 + 8 * q^45 + 4 * q^46 - 20 * q^47 + 8 * q^48 - 2 * q^49 + 24 * q^50 - 12 * q^51 - 2 * q^53 - 4 * q^54 + 6 * q^55 - 8 * q^56 - 16 * q^57 + 12 * q^58 - 6 * q^59 + 20 * q^61 - 12 * q^62 - 8 * q^63 + 16 * q^64 + 4 * q^65 - 4 * q^66 - 6 * q^67 + 10 * q^69 - 20 * q^70 - 2 * q^71 + 16 * q^72 - 8 * q^73 + 8 * q^74 + 12 * q^75 - 4 * q^77 + 4 * q^78 + 20 * q^79 + 24 * q^80 - 2 * q^81 + 12 * q^82 + 12 * q^83 + 28 * q^85 - 4 * q^86 + 4 * q^87 - 14 * q^89 + 24 * q^90 - 4 * q^91 - 10 * q^93 - 16 * q^95 - 14 * q^97 + 16 * q^98

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

−2.41421

−1.58579

3.41421

0.585786

2.82843

2.24264

1.2

1.41421

0.414214

−4.41421

0.585786

3.41421

−2.82843

−6.24264

Atkin-Lehner signs

$ p $	Sign
$11$	$ +1 $
$53$	$ +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	583.2.a.e	✓	2
3.b	odd	2	1		5247.2.a.e		2
4.b	odd	2	1		9328.2.a.q		2
11.b	odd	2	1		6413.2.a.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
583.2.a.e	✓	2	1.a	even	1	1	trivial
5247.2.a.e		2	3.b	odd	2	1
6413.2.a.m		2	11.b	odd	2	1
9328.2.a.q		2	4.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 $S_{2}^{\mathrm{new}}(\Gamma_0(583))$:

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

T2^2 - 2

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

T3^2 + 2*T3 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 2 $ T^2 - 2
$3$	$ T^{2} + 2T - 1 $ T^2 + 2*T - 1
$5$	$ T^{2} + 6T + 7 $ T^2 + 6*T + 7
$7$	$ T^{2} - 4T + 2 $ T^2 - 4*T + 2
$11$	$ (T + 1)^{2} $ (T + 1)^2
$13$	$ T^{2} - 2 $ T^2 - 2
$17$	$ T^{2} + 4T - 28 $ T^2 + 4*T - 28
$19$	$ T^{2} - 8T + 8 $ T^2 - 8*T + 8
$23$	$ T^{2} + 6T + 7 $ T^2 + 6*T + 7
$29$	$ T^{2} - 8T - 2 $ T^2 - 8*T - 2
$31$	$ T^{2} + 2T - 17 $ T^2 + 2*T - 17
$37$	$ T^{2} + 18T + 73 $ T^2 + 18*T + 73
$41$	$ T^{2} + 4T - 14 $ T^2 + 4*T - 14
$43$	$ T^{2} - 12T + 34 $ T^2 - 12*T + 34
$47$	$ (T + 10)^{2} $ (T + 10)^2
$53$	$ (T + 1)^{2} $ (T + 1)^2
$59$	$ T^{2} + 6T + 1 $ T^2 + 6*T + 1
$61$	$ T^{2} - 20T + 98 $ T^2 - 20*T + 98
$67$	$ T^{2} + 6T - 9 $ T^2 + 6*T - 9
$71$	$ T^{2} + 2T - 49 $ T^2 + 2*T - 49
$73$	$ T^{2} + 8T - 16 $ T^2 + 8*T - 16
$79$	$ T^{2} - 20T + 92 $ T^2 - 20*T + 92
$83$	$ T^{2} - 12T - 36 $ T^2 - 12*T - 36
$89$	$ T^{2} + 14T + 41 $ T^2 + 14*T + 41
$97$	$ T^{2} + 14T - 23 $ T^2 + 14*T - 23
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Level:	\( N \)	\(=\)	\( 583 = 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	583.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (\beta - 1) q^{3} + ( - \beta - 3) q^{5} + ( - \beta + 2) q^{6} + (\beta + 2) q^{7} - 2 \beta q^{8} - 2 \beta q^{9} + ( - 3 \beta - 2) q^{10} - q^{11} - \beta q^{13} + (2 \beta + 2) q^{14} + \cdots + 2 \beta q^{99} +O(q^{100}) \) q + b * q^2 + (b - 1) * q^3 + (-b - 3) * q^5 + (-b + 2) * q^6 + (b + 2) * q^7 - 2b q^8 - 2b q^9 + (-3b - 2) q^10 - q^11 - b * q^13 + (2b + 2) q^14 + (-2b + 1) q^15 - 4 * q^16 + (-4b - 2) q^17 - 4 * q^18 + (-2b + 4) q^19 + b * q^21 - b * q^22 + (b - 3) * q^23 + (2b - 4) q^24 + (6b + 6) q^25 - 2 * q^26 + (-b - 1) * q^27 + (3b + 4) q^29 + (b - 4) * q^30 + (-3b - 1) q^31 + (-b + 1) * q^33 + (-2b - 8) q^34 + (-5b - 8) q^35 + (2b - 9) q^37 + (4b - 4) q^38 + (b - 2) * q^39 + (6b + 4) q^40 + (3b - 2) q^41 + 2 * q^42 + (-b + 6) * q^43 + (6b + 4) q^45 + (-3b + 2) q^46 - 10 * q^47 + (-4b + 4) q^48 + (4b - 1) q^49 + (6b + 12) q^50 + (2b - 6) q^51 - q^53 + (-b - 2) * q^54 + (b + 3) * q^55 + (-4b - 4) q^56 + (6b - 8) q^57 + (4b + 6) q^58 + (2b - 3) q^59 + (-b + 10) * q^61 + (-b - 6) * q^62 + (-4b - 4) q^63 + 8 * q^64 + (3b + 2) q^65 + (b - 2) * q^66 + (-3b - 3) q^67 + (-4b + 5) q^69 + (-8b - 10) q^70 + (-5b - 1) q^71 + 8 * q^72 + (4b - 4) q^73 + (-9b + 4) q^74 + 6 * q^75 + (-b - 2) * q^77 + (-2b + 2) q^78 + (-2b + 10) q^79 + (4b + 12) q^80 + (6b - 1) q^81 + (-2b + 6) q^82 + (6b + 6) q^83 + (14b + 14) q^85 + (6b - 2) q^86 + (b + 2) * q^87 + 2b q^88 + (2b - 7) q^89 + (4b + 12) q^90 + (-2b - 2) q^91 + (2b - 5) q^93 - 10b q^94 + (2b - 8) q^95 + (-6b - 7) q^97 + (-b + 8) * q^98 + 2b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 6 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{10} - 2 q^{11} + 4 q^{14} + 2 q^{15} - 8 q^{16} - 4 q^{17} - 8 q^{18} + 8 q^{19} - 6 q^{23} - 8 q^{24} + 12 q^{25} - 4 q^{26} - 2 q^{27} + 8 q^{29} - 8 q^{30}+ \cdots + 16 q^{98}+O(q^{100}) \) 2 * q - 2 * q^3 - 6 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^10 - 2 * q^11 + 4 * q^14 + 2 * q^15 - 8 * q^16 - 4 * q^17 - 8 * q^18 + 8 * q^19 - 6 * q^23 - 8 * q^24 + 12 * q^25 - 4 * q^26 - 2 * q^27 + 8 * q^29 - 8 * q^30 - 2 * q^31 + 2 * q^33 - 16 * q^34 - 16 * q^35 - 18 * q^37 - 8 * q^38 - 4 * q^39 + 8 * q^40 - 4 * q^41 + 4 * q^42 + 12 * q^43 + 8 * q^45 + 4 * q^46 - 20 * q^47 + 8 * q^48 - 2 * q^49 + 24 * q^50 - 12 * q^51 - 2 * q^53 - 4 * q^54 + 6 * q^55 - 8 * q^56 - 16 * q^57 + 12 * q^58 - 6 * q^59 + 20 * q^61 - 12 * q^62 - 8 * q^63 + 16 * q^64 + 4 * q^65 - 4 * q^66 - 6 * q^67 + 10 * q^69 - 20 * q^70 - 2 * q^71 + 16 * q^72 - 8 * q^73 + 8 * q^74 + 12 * q^75 - 4 * q^77 + 4 * q^78 + 20 * q^79 + 24 * q^80 - 2 * q^81 + 12 * q^82 + 12 * q^83 + 28 * q^85 - 4 * q^86 + 4 * q^87 - 14 * q^89 + 24 * q^90 - 4 * q^91 - 10 * q^93 - 16 * q^95 - 14 * q^97 + 16 * q^98

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