LMFDB - Newform orbit 5800.2.a.x

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5800,2,Mod(1,5800)] 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("5800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5800 = 2^{3} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,0,0,-4,0,5,0,-1] 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:	$46.3132331723$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.16196689.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 11x^{2} - 7x + 1 $ x^6 - x^5 - 8x^4 + 5x^3 + 11x^2 - 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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 a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} + ( - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{9} + (\beta_{5} + \beta_{2}) q^{11} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + (3 \beta_{2} - 3 \beta_1 + 4) q^{99}+O(q^{100}) $ q + b2 * q^3 + (-b3 + b1 - 1) * q^7 + (b5 - b4 + b3 + 1) * q^9 + (b5 + b2) * q^11 + (-b4 + b3 - b2) * q^13 + (-b5 + b4 + b3 - b2 + 1) * q^17 + (-2b5 + b4 - b3 - 2) q^19 + (-2b5 + b4 - 2b2 + b1 - 2) * q^21 + (b5 + b4 - b2 - 2) * q^23 + (b5 + b4 + b2 - 3b1 + 2) q^27 - q^29 + (-3b3 - b2 - b1 - 4) q^31 + (-b4 + b3 - b1 + 3) * q^33 + (b5 - 2b4 - 2b3 - b2 - b1 + 4) * q^37 + (b5 + 2b4 - b3 + 3b2 - 2b1 - 1) q^39 + (-4b5 - 3b2 - b1 - 3) * q^41 + (-3b5 - 3b4 - 2b3 - 2b2 - 4b1 + 1) q^43 + (2b4 + b2 - 2b1 + 1) * q^47 + (-b5 + b3 + b2 - 4b1 + 2) q^49 + (-b3 + 3b1 - 4) q^51 + (3b5 + 3b4 + 2b3 + b2 - 3b1 + 1) * q^53 + (-b4 - 5b2 + 4b1 - 1) * q^57 + (-b5 + b4 + 2b2 + 3b1) * q^59 + (3b5 + 2b4 - b2 - 3) * q^61 + (-2b5 + 2b4 + b3 - 4b2 + 2b1 - 6) * q^63 + (-b5 + 4b3 - 3b2 + 3b1 - 1) q^67 + (-3b5 - b3 - 4b2 + b1 - 7) * q^69 + (2b5 - 2b4 + 2b3 - 2b2 + 5b1 - 5) q^71 + (2b5 + 4b3 + 4b2 + 3b1) * q^73 + (-2b5 + 2b4 + b3 - 3b2 - 1) q^77 + (-3b5 - 2b4 - b3 + b1 + 1) * q^79 + (-b5 - 2b4 - 2b3 - 2b1 + 1) q^81 + (5b5 + b4 + 3b3 + b2 + 4b1 + 1) q^83 - b2 * q^87 + (b5 + b4 + 4b3 + b2 + 2b1 - 3) * q^89 + (b3 - b2 + b1 - 4) * q^91 + (-3b5 - b3 - 7b2 - b1 - 6) * q^93 + (2b5 - 2b3 - b2 - 4b1 + 7) q^97 + (3b2 - 3b1 + 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - 4 q^{7} + 5 q^{9} - q^{11} + 6 q^{17} - 11 q^{19} - 9 q^{21} - 11 q^{23} + 8 q^{27} - 6 q^{29} - 21 q^{31} + 16 q^{33} + 26 q^{37} - 10 q^{39} - 16 q^{41} + 6 q^{43} + 3 q^{47} + 6 q^{49}+ \cdots + 18 q^{99}+O(q^{100}) $ 6 * q - q^3 - 4 * q^7 + 5 * q^9 - q^11 + 6 * q^17 - 11 * q^19 - 9 * q^21 - 11 * q^23 + 8 * q^27 - 6 * q^29 - 21 * q^31 + 16 * q^33 + 26 * q^37 - 10 * q^39 - 16 * q^41 + 6 * q^43 + 3 * q^47 + 6 * q^49 - 20 * q^51 + 3 * q^57 + q^59 - 17 * q^61 - 31 * q^63 - 4 * q^67 - 36 * q^69 - 25 * q^71 - 5 * q^73 - 4 * q^77 + 8 * q^79 + 6 * q^81 + 6 * q^83 + q^87 - 21 * q^89 - 23 * q^91 - 29 * q^93 + 41 * q^97 + 18 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 11x^{2} - 7x + 1 $ x^6 - x^5 - 8*x^4 + 5*x^3 + 11*x^2 - 7*x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ -\nu^{5} + \nu^{4} + 8\nu^{3} - 4\nu^{2} - 12\nu + 3 $ -v^5 + v^4 + 8v^3 - 4v^2 - 12*v + 3
$\beta_{3}$	$=$	$ \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 11\nu - 6 $ v^5 - v^4 - 8v^3 + 5v^2 + 11*v - 6
$\beta_{4}$	$=$	$ 3\nu^{5} - 2\nu^{4} - 24\nu^{3} + 6\nu^{2} + 32\nu - 8 $ 3v^5 - 2v^4 - 24v^3 + 6v^2 + 32*v - 8
$\beta_{5}$	$=$	$ 3\nu^{5} - 2\nu^{4} - 25\nu^{3} + 7\nu^{2} + 37\nu - 10 $ 3v^5 - 2v^4 - 25v^3 + 7v^2 + 37*v - 10

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{3} + \beta_{2} + \beta _1 + 3 $ b3 + b2 + b1 + 3
$\nu^{3}$	$=$	$ -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 1 $ -b5 + b4 + b3 + b2 + 6*b1 + 1
$\nu^{4}$	$=$	$ \beta_{4} + 6\beta_{3} + 9\beta_{2} + 10\beta _1 + 17 $ b4 + 6b3 + 9b2 + 10*b1 + 17
$\nu^{5}$	$=$	$ -8\beta_{5} + 9\beta_{4} + 10\beta_{3} + 12\beta_{2} + 42\beta _1 + 16 $ -8b5 + 9b4 + 10b3 + 12b2 + 42*b1 + 16

Display $\beta_i$ in terms of $\nu^j$

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

1.28248
0.359514
−1.42864
0.236718
2.82229
−2.27236

−2.85798

0.0622207

5.16806

1.2

−1.44873

1.14105

−0.901181

1.3

−1.23031

−4.12861

−1.48634

1.4

0.0437611

2.46116

−2.99808

1.5

1.49734

1.17661

−0.757966

1.6

2.99592

−4.71243

5.97551

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ -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	5800.2.a.x	✓	6
5.b	even	2	1		5800.2.a.y	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5800.2.a.x	✓	6	1.a	even	1	1	trivial
5800.2.a.y	yes	6	5.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 $S_{2}^{\mathrm{new}}(\Gamma_0(5800))$:

$ T_{3}^{6} + T_{3}^{5} - 11T_{3}^{4} - 12T_{3}^{3} + 20T_{3}^{2} + 22T_{3} - 1 $

T3^6 + T3^5 - 11*T3^4 - 12*T3^3 + 20*T3^2 + 22*T3 - 1

$ T_{7}^{6} + 4T_{7}^{5} - 16T_{7}^{4} - 33T_{7}^{3} + 110T_{7}^{2} - 71T_{7} + 4 $

T7^6 + 4*T7^5 - 16*T7^4 - 33*T7^3 + 110*T7^2 - 71*T7 + 4

$ T_{11}^{6} + T_{11}^{5} - 13T_{11}^{4} - 15T_{11}^{3} + 26T_{11}^{2} + 31T_{11} + 1 $

T11^6 + T11^5 - 13*T11^4 - 15*T11^3 + 26*T11^2 + 31*T11 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + T^{5} - 11 T^{4} + \cdots - 1 $ T^6 + T^5 - 11T^4 - 12T^3 + 20T^2 + 22T - 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 4 T^{5} + \cdots + 4 $ T^6 + 4T^5 - 16T^4 - 33T^3 + 110T^2 - 71*T + 4
$11$	$ T^{6} + T^{5} - 13 T^{4} + \cdots + 1 $ T^6 + T^5 - 13T^4 - 15T^3 + 26T^2 + 31T + 1
$13$	$ T^{6} - 30 T^{4} + \cdots - 100 $ T^6 - 30T^4 - 43T^3 + 110T^2 + 121T - 100
$17$	$ T^{6} - 6 T^{5} + \cdots + 13 $ T^6 - 6T^5 - 28T^4 + 80T^3 + 77T^2 - 111*T + 13
$19$	$ T^{6} + 11 T^{5} + \cdots - 1625 $ T^6 + 11T^5 - 7T^4 - 262T^3 + 136T^2 + 1502*T - 1625
$23$	$ T^{6} + 11 T^{5} + \cdots - 130 $ T^6 + 11T^5 - 2T^4 - 321T^3 - 967T^2 - 691*T - 130
$29$	$ (T + 1)^{6} $ (T + 1)^6
$31$	$ T^{6} + 21 T^{5} + \cdots + 24194 $ T^6 + 21T^5 + 84T^4 - 675T^3 - 4183T^2 + 2205*T + 24194
$37$	$ T^{6} - 26 T^{5} + \cdots + 70802 $ T^6 - 26T^5 + 163T^4 + 635T^3 - 7365T^2 - 21*T + 70802
$41$	$ T^{6} + 16 T^{5} + \cdots + 4187 $ T^6 + 16T^5 - 50T^4 - 1531T^3 - 3717T^2 + 4992*T + 4187
$43$	$ T^{6} - 6 T^{5} + \cdots + 431344 $ T^6 - 6T^5 - 277T^4 + 2174T^3 + 16484T^2 - 188505*T + 431344
$47$	$ T^{6} - 3 T^{5} + \cdots - 274 $ T^6 - 3T^5 - 87T^4 + 414T^3 + 101T^2 - 1089*T - 274
$53$	$ T^{6} - 271 T^{4} + \cdots - 419796 $ T^6 - 271T^4 - 245T^3 + 20589T^2 + 17937T - 419796
$59$	$ T^{6} - T^{5} + \cdots + 2404 $ T^6 - T^5 - 128T^4 - 661T^3 - 649T^2 + 1615T + 2404
$61$	$ T^{6} + 17 T^{5} + \cdots - 44318 $ T^6 + 17T^5 - 77T^4 - 3049T^3 - 20734T^2 - 52447*T - 44318
$67$	$ T^{6} + 4 T^{5} + \cdots - 438271 $ T^6 + 4T^5 - 258T^4 - 595T^3 + 20223T^2 + 12272*T - 438271
$71$	$ T^{6} + 25 T^{5} + \cdots + 357326 $ T^6 + 25T^5 - 74T^4 - 6027T^3 - 40873T^2 + 9721*T + 357326
$73$	$ T^{6} + 5 T^{5} + \cdots + 349939 $ T^6 + 5T^5 - 300T^4 - 1809T^3 + 18187T^2 + 169041*T + 349939
$79$	$ T^{6} - 8 T^{5} + \cdots + 8224 $ T^6 - 8T^5 - 112T^4 + 1351T^3 - 3890T^2 + 127*T + 8224
$83$	$ T^{6} - 6 T^{5} + \cdots - 241331 $ T^6 - 6T^5 - 278T^4 + 1548T^3 + 17307T^2 - 71581*T - 241331
$89$	$ T^{6} + 21 T^{5} + \cdots + 559 $ T^6 + 21T^5 + 38T^4 - 677T^3 - 2917T^2 - 2637*T + 559
$97$	$ T^{6} - 41 T^{5} + \cdots - 849434 $ T^6 - 41T^5 + 495T^4 + 516T^3 - 53189T^2 + 380223*T - 849434
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5800.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} + ( - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{9} + (\beta_{5} + \beta_{2}) q^{11} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + (3 \beta_{2} - 3 \beta_1 + 4) q^{99}+O(q^{100}) \) q + b2 * q^3 + (-b3 + b1 - 1) * q^7 + (b5 - b4 + b3 + 1) * q^9 + (b5 + b2) * q^11 + (-b4 + b3 - b2) * q^13 + (-b5 + b4 + b3 - b2 + 1) * q^17 + (-2b5 + b4 - b3 - 2) q^19 + (-2b5 + b4 - 2b2 + b1 - 2) * q^21 + (b5 + b4 - b2 - 2) * q^23 + (b5 + b4 + b2 - 3b1 + 2) q^27 - q^29 + (-3b3 - b2 - b1 - 4) q^31 + (-b4 + b3 - b1 + 3) * q^33 + (b5 - 2b4 - 2b3 - b2 - b1 + 4) * q^37 + (b5 + 2b4 - b3 + 3b2 - 2b1 - 1) q^39 + (-4b5 - 3b2 - b1 - 3) * q^41 + (-3b5 - 3b4 - 2b3 - 2b2 - 4b1 + 1) q^43 + (2b4 + b2 - 2b1 + 1) * q^47 + (-b5 + b3 + b2 - 4b1 + 2) q^49 + (-b3 + 3b1 - 4) q^51 + (3b5 + 3b4 + 2b3 + b2 - 3b1 + 1) * q^53 + (-b4 - 5b2 + 4b1 - 1) * q^57 + (-b5 + b4 + 2b2 + 3b1) * q^59 + (3b5 + 2b4 - b2 - 3) * q^61 + (-2b5 + 2b4 + b3 - 4b2 + 2b1 - 6) * q^63 + (-b5 + 4b3 - 3b2 + 3b1 - 1) q^67 + (-3b5 - b3 - 4b2 + b1 - 7) * q^69 + (2b5 - 2b4 + 2b3 - 2b2 + 5b1 - 5) q^71 + (2b5 + 4b3 + 4b2 + 3b1) * q^73 + (-2b5 + 2b4 + b3 - 3b2 - 1) q^77 + (-3b5 - 2b4 - b3 + b1 + 1) * q^79 + (-b5 - 2b4 - 2b3 - 2b1 + 1) q^81 + (5b5 + b4 + 3b3 + b2 + 4b1 + 1) q^83 - b2 * q^87 + (b5 + b4 + 4b3 + b2 + 2b1 - 3) * q^89 + (b3 - b2 + b1 - 4) * q^91 + (-3b5 - b3 - 7b2 - b1 - 6) * q^93 + (2b5 - 2b3 - b2 - 4b1 + 7) q^97 + (3b2 - 3b1 + 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - 4 q^{7} + 5 q^{9} - q^{11} + 6 q^{17} - 11 q^{19} - 9 q^{21} - 11 q^{23} + 8 q^{27} - 6 q^{29} - 21 q^{31} + 16 q^{33} + 26 q^{37} - 10 q^{39} - 16 q^{41} + 6 q^{43} + 3 q^{47} + 6 q^{49}+ \cdots + 18 q^{99}+O(q^{100}) \) 6 * q - q^3 - 4 * q^7 + 5 * q^9 - q^11 + 6 * q^17 - 11 * q^19 - 9 * q^21 - 11 * q^23 + 8 * q^27 - 6 * q^29 - 21 * q^31 + 16 * q^33 + 26 * q^37 - 10 * q^39 - 16 * q^41 + 6 * q^43 + 3 * q^47 + 6 * q^49 - 20 * q^51 + 3 * q^57 + q^59 - 17 * q^61 - 31 * q^63 - 4 * q^67 - 36 * q^69 - 25 * q^71 - 5 * q^73 - 4 * q^77 + 8 * q^79 + 6 * q^81 + 6 * q^83 + q^87 - 21 * q^89 - 23 * q^91 - 29 * q^93 + 41 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more