LMFDB - Newform orbit 6525.2.a.bu

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6525 = 3^{2} \cdot 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14$ x^7 - 2x^6 - 10x^5 + 19x^4 + 24x^3 - 44x^2 - 3x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2175)
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 $1,\beta_1,\ldots,\beta_{6}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{6} - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{11} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{13}+ \cdots + ( - \beta_{6} + \beta_{4} + 2 \beta_{3} + \cdots + 8) q^{98}+O(q^{100})$ q - b1 * q^2 + (b2 + 1) * q^4 + (b6 - b2 + b1) * q^7 + (-b3 - b1) * q^8 + (b4 + b3 - b2) * q^11 + (b6 + b4 - b3 + b1) * q^13 + (b5 - b2 + b1 - 2) * q^14 + (b5 - b4 + b2 + 1) * q^16 + (-2b5 - b4 + b3 + b2 - 2) q^17 + (-b6 + b5 - b4 + b2 - 2b1 + 2) q^19 + (-b6 - 2b5 - b4 + 2b3 - b2 + b1) * q^22 + (-2b6 - 2) q^23 + (-b6 + b5 - 3b4 + 2b2 - 2b1 - 2) q^26 + (-b6 + 2b1 - 4) q^28 + q^29 + (-b6 - b5 + b4 + b2 + 2b1) q^31 + (2b6 + b5 + 2b4 - b3 - 2b2 - 1) q^32 + (-b6 + 3b4 - b2 + b1 + 2) q^34 - 2b2 q^37 + (2b6 + 2b4 - 2b3 - 2b1 + 4) * q^38 + (b6 + b5 + b4 - 2b3 + b2 - 3) q^41 + (-b6 - b5 - b4 + b2 + 2b1 - 4) q^43 + (-b6 - 2b5 + 2b4 + b3 - 2b2 + 4b1 - 2) * q^44 + (-2b5 + 2b3 + 4b1 - 2) q^46 + (-2b6 + b5 - b4 + 2b2 - b1 - 4) * q^47 + (b6 - b5 + b3 - 2b2 - 2b1 + 3) * q^49 + (2b6 + 2b5 + 4b4 - 3b3 - 2b2 + 4) q^52 + (2b6 - 2b4 - 2) * q^53 + (-3b5 + b3 + 3b1 - 3) * q^56 - b1 * q^58 + (4b1 - 2) q^59 + (-b6 + 3b5 + b4 - b2 + 4) q^61 + (-2b6 - 2b5 - 2b4 + 2b3 - 2b1 - 6) q^62 + (-b6 - b5 - 3b4 + b3 + b2 + b1 - 1) q^64 + (-3b6 + b5 - 2b4 + b3 - 2b1 - 2) q^67 + (-3b6 - 4b4 + 3b3 - 2b1) * q^68 + (-2b6 + 2b3 - 4) * q^71 + (-2b6 + 2b5 - 2b4 - 2b3 + 2b2 - 2b1 - 4) * q^73 + (2b3 + 4b1) * q^74 + (-4b4 + 6b2 - 4b1 + 4) q^76 + (b6 + 6b5 - 2b3 - b2 - 3b1 + 1) q^77 + (-2b6 - 2b4 - 2b1 + 2) q^79 + (2b5 - 4b4 - 2b3 + 4b2 - b1) * q^82 + (2b6 + 2b1 - 2) * q^83 + (2b4 - 2b2 + 4b1 - 6) q^86 + (-2b6 - b4 + 3b3 + 3b1 - 11) q^88 + (-2b6 + b5 - 2b4 - b3 + 3b2 + b1 - 4) q^89 + (-2b6 - 3b5 + b3 + b2 + b1 + 1) * q^91 + (2b6 - 2b5 + 2b4 + 2b3 - 6b2 + 2b1 - 6) * q^92 + (2b6 - b5 + 2b4 - 2b3 - b2 + 3b1) * q^94 + (3b6 + 3b5 + b4 - 2b3 - 3b2 - 2b1 - 2) q^97 + (-b6 + b4 + 2b3 + b2 + 8) q^98
$\operatorname{Tr}(f)(q)$	$=$	$7 q - 2 q^{2} + 10 q^{4} - q^{7} - 3 q^{8} - 4 q^{11} - q^{13} - 15 q^{14} + 12 q^{16} - 8 q^{17} + 15 q^{19} + 3 q^{22} - 14 q^{23} - 6 q^{26} - 24 q^{28} + 7 q^{29} + 5 q^{31} - 18 q^{32} + 7 q^{34}+ \cdots + 59 q^{98}+O(q^{100})$ 7 * q - 2 * q^2 + 10 * q^4 - q^7 - 3 * q^8 - 4 * q^11 - q^13 - 15 * q^14 + 12 * q^16 - 8 * q^17 + 15 * q^19 + 3 * q^22 - 14 * q^23 - 6 * q^26 - 24 * q^28 + 7 * q^29 + 5 * q^31 - 18 * q^32 + 7 * q^34 - 6 * q^37 + 18 * q^38 - 22 * q^41 - 19 * q^43 - 15 * q^44 - 4 * q^46 - 22 * q^47 + 12 * q^49 + 11 * q^52 - 10 * q^53 - 14 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 - 40 * q^62 + 5 * q^64 - 13 * q^67 + 7 * q^68 - 26 * q^71 - 24 * q^73 + 10 * q^74 + 46 * q^76 - 4 * q^77 + 14 * q^79 + 16 * q^82 - 10 * q^83 - 44 * q^86 - 66 * q^88 - 14 * q^89 + 13 * q^91 - 58 * q^92 - 3 * q^94 - 31 * q^97 + 59 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14$ x^7 - 2*x^6 - 10*x^5 + 19*x^4 + 24*x^3 - 44*x^2 - 3*x + 14 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 3$ v^2 - 3
$\beta_{3}$	$=$	$\nu^{3} - 5\nu$ v^3 - 5*v
$\beta_{4}$	$=$	$( -2\nu^{6} + \nu^{5} + 19\nu^{4} - 7\nu^{3} - 41\nu^{2} + 9\nu + 7 ) / 5$ (-2v^6 + v^5 + 19v^4 - 7v^3 - 41v^2 + 9*v + 7) / 5
$\beta_{5}$	$=$	$( -2\nu^{6} + \nu^{5} + 24\nu^{4} - 7\nu^{3} - 76\nu^{2} + 9\nu + 37 ) / 5$ (-2v^6 + v^5 + 24v^4 - 7v^3 - 76v^2 + 9*v + 37) / 5
$\beta_{6}$	$=$	$( 3\nu^{6} - 4\nu^{5} - 31\nu^{4} + 33\nu^{3} + 84\nu^{2} - 56\nu - 38 ) / 5$ (3v^6 - 4v^5 - 31v^4 + 33v^3 + 84v^2 - 56v - 38) / 5

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 3$ b2 + 3
$\nu^{3}$	$=$	$\beta_{3} + 5\beta_1$ b3 + 5*b1
$\nu^{4}$	$=$	$\beta_{5} - \beta_{4} + 7\beta_{2} + 15$ b5 - b4 + 7*b2 + 15
$\nu^{5}$	$=$	$-2\beta_{6} - \beta_{5} - 2\beta_{4} + 9\beta_{3} + 2\beta_{2} + 28\beta _1 + 1$ -2b6 - b5 - 2b4 + 9b3 + 2b2 + 28*b1 + 1
$\nu^{6}$	$=$	$-\beta_{6} + 9\beta_{5} - 13\beta_{4} + \beta_{3} + 47\beta_{2} + \beta _1 + 85$ -b6 + 9b5 - 13b4 + b3 + 47*b2 + b1 + 85

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

2.66356
2.26695
1.14889
0.792771
−0.560139
−1.83397
−2.47806

−2.66356

5.09453

−0.0170416

−8.24244

1.2

−2.26695

3.13907

0.159887

−2.58223

1.3

−1.14889

−0.680059

3.52165

3.07908

1.4

−0.792771

−1.37151

−2.01830

2.67284

1.5

0.560139

−1.68624

4.36313

−2.06481

1.6

1.83397

1.36343

−4.17416

−1.16745

1.7

2.47806

4.14079

−2.83516

5.30500

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$29$	$-1$

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	6525.2.a.bu		7
3.b	odd	2	1		2175.2.a.bb	yes	7
5.b	even	2	1		6525.2.a.bx		7
15.d	odd	2	1		2175.2.a.ba	✓	7
15.e	even	4	2		2175.2.c.o		14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2175.2.a.ba	✓	7	15.d	odd	2	1
2175.2.a.bb	yes	7	3.b	odd	2	1
2175.2.c.o		14	15.e	even	4	2
6525.2.a.bu		7	1.a	even	1	1	trivial
6525.2.a.bx		7	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 $S_{2}^{\mathrm{new}}(\Gamma_0(6525))$ :

T_{2}^{7} + 2T_{2}^{6} - 10T_{2}^{5} - 19T_{2}^{4} + 24T_{2}^{3} + 44T_{2}^{2} - 3T_{2} - 14

T_{7}^{7} + T_{7}^{6} - 30T_{7}^{5} - 38T_{7}^{4} + 217T_{7}^{3} + 337T_{7}^{2} - 53T_{7} - 1

T_{11}^{7} + 4T_{11}^{6} - 60T_{11}^{5} - 190T_{11}^{4} + 1105T_{11}^{3} + 2448T_{11}^{2} - 5733T_{11} - 10746

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{7} + 2 T^{6} + \cdots - 14$ T^7 + 2T^6 - 10T^5 - 19T^4 + 24T^3 + 44T^2 - 3T - 14
$3$	$T^{7}$ T^7
$5$	$T^{7}$ T^7
$7$	$T^{7} + T^{6} - 30 T^{5} + \cdots - 1$ T^7 + T^6 - 30T^5 - 38T^4 + 217T^3 + 337T^2 - 53*T - 1
$11$	$T^{7} + 4 T^{6} + \cdots - 10746$ T^7 + 4T^6 - 60T^5 - 190T^4 + 1105T^3 + 2448T^2 - 5733T - 10746
$13$	$T^{7} + T^{6} + \cdots - 6803$ T^7 + T^6 - 58T^5 - 124T^4 + 755T^3 + 1783T^2 - 2801*T - 6803
$17$	$T^{7} + 8 T^{6} + \cdots - 86134$ T^7 + 8T^6 - 76T^5 - 580T^4 + 1973T^3 + 13002T^2 - 17443T - 86134
$19$	$T^{7} - 15 T^{6} + \cdots - 15680$ T^7 - 15T^6 + 28T^5 + 452T^4 - 1736T^3 - 2496T^2 + 16160T - 15680
$23$	$T^{7} + 14 T^{6} + \cdots + 27008$ T^7 + 14T^6 - 8T^5 - 936T^4 - 4560T^3 - 3648T^2 + 17344T + 27008
$29$	$(T - 1)^{7}$ (T - 1)^7
$31$	$T^{7} - 5 T^{6} + \cdots - 12096$ T^7 - 5T^6 - 120T^5 + 620T^4 + 1744T^3 - 13392T^2 + 22752T - 12096
$37$	$T^{7} + 6 T^{6} + \cdots - 13696$ T^7 + 6T^6 - 76T^5 - 336T^4 + 1968T^3 + 4960T^2 - 16384T - 13696
$41$	$T^{7} + 22 T^{6} + \cdots + 626632$ T^7 + 22T^6 + 39T^5 - 2084T^4 - 13729T^3 + 15750T^2 + 315001T + 626632
$43$	$T^{7} + 19 T^{6} + \cdots - 42304$ T^7 + 19T^6 + 46T^5 - 736T^4 - 2312T^3 + 9520T^2 + 20864T - 42304
$47$	$T^{7} + 22 T^{6} + \cdots + 2842$ T^7 + 22T^6 + 92T^5 - 740T^4 - 4681T^3 + 3036T^2 + 34131T + 2842
$53$	$T^{7} + 10 T^{6} + \cdots + 1229824$ T^7 + 10T^6 - 200T^5 - 1256T^4 + 17072T^3 + 25792T^2 - 568128T + 1229824
$59$	$T^{7} + 6 T^{6} + \cdots + 87680$ T^7 + 6T^6 - 172T^5 - 584T^4 + 8752T^3 + 6944T^2 - 93760T + 87680
$61$	$T^{7} - 23 T^{6} + \cdots + 51776$ T^7 - 23T^6 - 16T^5 + 3804T^4 - 27056T^3 + 16240T^2 + 194432T + 51776
$67$	$T^{7} + 13 T^{6} + \cdots + 37259$ T^7 + 13T^6 - 174T^5 - 2130T^4 + 3985T^3 + 45105T^2 - 95817T + 37259
$71$	$T^{7} + 26 T^{6} + \cdots + 34048$ T^7 + 26T^6 + 88T^5 - 1584T^4 - 5616T^3 + 17824T^2 + 71744T + 34048
$73$	$T^{7} + 24 T^{6} + \cdots + 1658752$ T^7 + 24T^6 - 52T^5 - 4584T^4 - 19024T^3 + 190304T^2 + 1305216T + 1658752
$79$	$T^{7} - 14 T^{6} + \cdots + 2560$ T^7 - 14T^6 - 104T^5 + 2096T^4 - 10112T^3 + 19712T^2 - 15040T + 2560
$83$	$T^{7} + 10 T^{6} + \cdots + 5248$ T^7 + 10T^6 - 80T^5 - 504T^4 + 2768T^3 - 1792T^2 - 5056T + 5248
$89$	$T^{7} + 14 T^{6} + \cdots - 231800$ T^7 + 14T^6 - 202T^5 - 2800T^4 + 6783T^3 + 99722T^2 + 109465T - 231800
$97$	$T^{7} + 31 T^{6} + \cdots + 6565312$ T^7 + 31T^6 - 70T^5 - 8440T^4 - 40128T^3 + 401248T^2 + 3399552T + 6565312
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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.7
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more