LMFDB - Newform orbit 366.2.e.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [366,2,Mod(13,366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(366, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("366.13"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$366 = 2 \cdot 3 \cdot 61$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	366.e (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,3,-6,-3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$2.92252471398$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.65370672.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} + 10x^{4} - 13x^{3} + 92x^{2} - 99x + 121$ x^6 - x^5 + 10x^4 - 13x^3 + 92x^2 - 99x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/366\mathbb{Z}\right)^\times$ .

$n$	$245$	$307$
$\chi(n)$	$1$	$\beta_{4}$

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}

13.1

−1.54030	−	2.66788i
1.40489	+	2.43335i
0.635409	+	1.10056i
−1.54030	+	2.66788i
1.40489	−	2.43335i
0.635409	−	1.10056i

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.74506

3.02254i

−0.500000

0.866025i

2.04030

−

3.53391i

−1.00000

1.00000

1.74506

3.02254i

13.2

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

−0.947448

1.64103i

−0.500000

0.866025i

−0.904893

1.56732i

−1.00000

1.00000

0.947448

1.64103i

13.3

0.500000

−

0.866025i

−1.00000

−0.500000

−

0.866025i

2.19251

−

3.79754i

−0.500000

0.866025i

−0.135409

0.234536i

−1.00000

1.00000

−2.19251

−

3.79754i

169.1

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

−1.74506

−

3.02254i

−0.500000

−

0.866025i

2.04030

3.53391i

−1.00000

1.00000

1.74506

−

3.02254i

169.2

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

−0.947448

−

1.64103i

−0.500000

−

0.866025i

−0.904893

−

1.56732i

−1.00000

1.00000

0.947448

−

1.64103i

169.3

0.500000

0.866025i

−1.00000

−0.500000

0.866025i

2.19251

3.79754i

−0.500000

−

0.866025i

−0.135409

−

0.234536i

−1.00000

1.00000

−2.19251

3.79754i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	366.2.e.d	✓	6
3.b	odd	2	1		1098.2.f.f		6
61.c	even	3	1	inner	366.2.e.d	✓	6
183.k	odd	6	1		1098.2.f.f		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
366.2.e.d	✓	6	1.a	even	1	1	trivial
366.2.e.d	✓	6	61.c	even	3	1	inner
1098.2.f.f		6	3.b	odd	2	1
1098.2.f.f		6	183.k	odd	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{6} + T_{5}^{5} + 18T_{5}^{4} + 41T_{5}^{3} + 318T_{5}^{2} + 493T_{5} + 841$ T5^6 + T5^5 + 18*T5^4 + 41*T5^3 + 318*T5^2 + 493*T5 + 841 acting on $S_{2}^{\mathrm{new}}(366, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - T + 1)^{3}$ (T^2 - T + 1)^3
$3$	$(T + 1)^{6}$ (T + 1)^6
$5$	$T^{6} + T^{5} + \cdots + 841$ T^6 + T^5 + 18T^4 + 41T^3 + 318T^2 + 493T + 841
$7$	$T^{6} - 2 T^{5} + \cdots + 4$ T^6 - 2T^5 + 12T^4 + 20T^3 + 60T^2 + 16*T + 4
$11$	$(T^{3} - 2 T^{2} - 8 T - 2)^{2}$ (T^3 - 2T^2 - 8T - 2)^2
$13$	$T^{6} + 2 T^{5} + \cdots + 7744$ T^6 + 2T^5 + 40T^4 + 104T^3 + 1472T^2 + 3168*T + 7744
$17$	$T^{6} + 12 T^{5} + \cdots + 17424$ T^6 + 12T^5 + 128T^4 + 456T^3 + 1840T^2 - 2112*T + 17424
$19$	$(T^{2} - 2 T + 4)^{3}$ (T^2 - 2*T + 4)^3
$23$	$(T^{3} - 4 T^{2} - 10 T + 22)^{2}$ (T^3 - 4T^2 - 10T + 22)^2
$29$	$T^{6} - 7 T^{5} + \cdots + 121$ T^6 - 7T^5 + 70T^4 + 125T^3 + 518T^2 - 231*T + 121
$31$	$(T^{2} - 2 T + 4)^{3}$ (T^2 - 2*T + 4)^3
$37$	$(T^{3} + 7 T^{2} + T - 27)^{2}$ (T^3 + 7*T^2 + T - 27)^2
$41$	$(T^{3} - 11 T^{2} + \cdots - 23)^{2}$ (T^3 - 11T^2 + 31T - 23)^2
$43$	$T^{6} - 12 T^{5} + \cdots + 14884$ T^6 - 12T^5 + 134T^4 - 364T^3 + 1564T^2 + 1220*T + 14884
$47$	$T^{6} + 10 T^{5} + \cdots + 948676$ T^6 + 10T^5 + 198T^4 + 968T^3 + 19344T^2 + 95452*T + 948676
$53$	$(T^{3} + 11 T^{2} + \cdots - 723)^{2}$ (T^3 + 11T^2 - 85T - 723)^2
$59$	$T^{6} + 4 T^{5} + \cdots + 9216$ T^6 + 4T^5 + 80T^4 - 448T^3 + 3712T^2 - 6144*T + 9216
$61$	$T^{6} - 9 T^{5} + \cdots + 226981$ T^6 - 9T^5 + 72T^4 - 605T^3 + 4392T^2 - 33489*T + 226981
$67$	$T^{6} + 2 T^{5} + \cdots + 36$ T^6 + 2T^5 + 18T^4 - 16T^3 + 208T^2 + 84*T + 36
$71$	$T^{6} - 10 T^{5} + \cdots + 576$ T^6 - 10T^5 + 104T^4 - 8T^3 + 256T^2 - 96*T + 576
$73$	$(T^{2} - 5 T + 25)^{3}$ (T^2 - 5*T + 25)^3
$79$	$T^{6} + 6 T^{5} + \cdots + 627264$ T^6 + 6T^5 + 152T^4 + 888T^3 + 18208T^2 + 91872*T + 627264
$83$	$T^{6} + 14 T^{5} + \cdots + 4356$ T^6 + 14T^5 + 140T^4 + 652T^3 + 2212T^2 + 3696*T + 4356
$89$	$(T^{3} - 13 T^{2} + \cdots + 3309)^{2}$ (T^3 - 13T^2 - 251T + 3309)^2
$97$	$T^{6} - 27 T^{5} + \cdots + 259081$ T^6 - 27T^5 + 518T^4 - 4679T^3 + 30778T^2 - 107399*T + 259081
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + (\beta_{4} + 1) q^{2} - q^{3} + \beta_{4} q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{5} + ( - \beta_{4} - 1) q^{6} + (\beta_{4} - \beta_{2} + \beta_1 + 1) q^{7} - q^{8} + q^{9}+ \cdots + ( - \beta_{2} + 1) q^{99}+O(q^{100})$ q + (b4 + 1) * q^2 - q^3 + b4 * q^4 + (b5 - b4 - b3 + b2 - b1 - 1) * q^5 + (-b4 - 1) * q^6 + (b4 - b2 + b1 + 1) * q^7 - q^8 + q^9 + (b5 - b4 - b1) * q^10 + (-b2 + 1) * q^11 - b4 * q^12 + (-2b2 + 2b1) * q^13 + (b4 + b1) * q^14 + (-b5 + b4 + b3 - b2 + b1 + 1) * q^15 + (-b4 - 1) * q^16 + (b5 + 4b4 + b1) q^17 + (b4 + 1) * q^18 - 2b4 q^19 + (b3 - b2 + 1) * q^20 + (-b4 + b2 - b1 - 1) * q^21 + (b4 - b2 + b1 + 1) * q^22 + (-b3 + 1) * q^23 + q^24 + (2b5 + 6b4) * q^25 + 2b1 q^26 - q^27 + (b2 - 1) * q^28 + (-3b4 - 2b1) * q^29 + (-b5 + b4 + b1) * q^30 - 2b4 q^31 - b4 * q^32 + (b2 - 1) * q^33 + (b3 + b2 - 4) * q^34 + (-5b4 - 3b1) * q^35 + b4 * q^36 + (b3 - 2) * q^37 + 2 * q^38 + (2b2 - 2b1) * q^39 + (-b5 + b4 + b3 - b2 + b1 + 1) * q^40 + (-b2 + 4) * q^41 + (-b4 - b1) * q^42 + (b5 + 3b4 - b3 + 2b2 - 2b1 + 3) q^43 + (b4 + b1) * q^44 + (b5 - b4 - b3 + b2 - b1 - 1) * q^45 + (b5 + b4 - b3 + 1) * q^46 + (-3b5 + 5b4 + 2b1) q^47 + (b4 + 1) * q^48 + (-b5 + b4 + 3b1) q^49 + (2b3 - 6) q^50 + (-b5 - 4b4 - b1) q^51 + 2b2 q^52 + (-3b3 + b2 - 5) q^53 + (-b4 - 1) * q^54 + (-5b4 + 3b2 - 3b1 - 5) q^55 + (-b4 + b2 - b1 - 1) * q^56 + 2b4 q^57 + (-2b2 + 3) q^58 + (-2b5 + 2b3 - 2b2 + 2b1) * q^59 + (-b3 + b2 - 1) * q^60 + (b5 + 2b4 - 2b3 - b2 + 2b1 + 2) q^61 + 2 * q^62 + (b4 - b2 + b1 + 1) * q^63 + q^64 + (-2b5 - 8b4 - 4b1) q^65 + (-b4 + b2 - b1 - 1) * q^66 + (b5 - b4 - b3 - 1) * q^67 + (-b5 - 4b4 + b3 + b2 - b1 - 4) q^68 + (b3 - 1) * q^69 + (-3b2 + 5) q^70 + (-4b4 - 2b1) * q^71 - q^72 - 5b4 q^73 + (-b5 - 2b4 + b3 - 2) q^74 + (-2b5 - 6b4) * q^75 + (2b4 + 2) q^76 + (-b5 + 8b4 + b3 - 3b2 + 3b1 + 8) q^77 - 2b1 q^78 + (-2b5 - 2b4 + 2b3 + 2b2 - 2b1 - 2) q^79 + (-b5 + b4 + b1) * q^80 + q^81 + (4b4 - b2 + b1 + 4) q^82 + (-5b4 + b2 - b1 - 5) q^83 + (-b2 + 1) * q^84 + (5b3 - 9b2 + 2) * q^85 + (b5 + 3b4 - 2b1) * q^86 + (3b4 + 2b1) * q^87 + (b2 - 1) * q^88 + (-b3 + 6b2 + 2) q^89 + (b5 - b4 - b1) * q^90 + (-2b5 + 14b4 + 4b1) q^91 + (b5 + b4) * q^92 + 2b4 q^93 + (-3b3 + 2b2 - 5) * q^94 + (-2b3 + 2b2 - 2) * q^95 + b4 * q^96 + (b5 - 9b4 + b1) q^97 + (-b3 + 3b2 - 1) q^98 + (-b2 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{2} - 6 q^{3} - 3 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9} + q^{10} + 4 q^{11} + 3 q^{12} - 2 q^{13} - 2 q^{14} + q^{15} - 3 q^{16} - 12 q^{17} + 3 q^{18} + 6 q^{19} + 2 q^{20}+ \cdots + 4 q^{99}+O(q^{100})$ 6 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 - q^5 - 3 * q^6 + 2 * q^7 - 6 * q^8 + 6 * q^9 + q^10 + 4 * q^11 + 3 * q^12 - 2 * q^13 - 2 * q^14 + q^15 - 3 * q^16 - 12 * q^17 + 3 * q^18 + 6 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^22 + 8 * q^23 + 6 * q^24 - 20 * q^25 + 2 * q^26 - 6 * q^27 - 4 * q^28 + 7 * q^29 - q^30 + 6 * q^31 + 3 * q^32 - 4 * q^33 - 24 * q^34 + 12 * q^35 - 3 * q^36 - 14 * q^37 + 12 * q^38 + 2 * q^39 + q^40 + 22 * q^41 + 2 * q^42 + 12 * q^43 - 2 * q^44 - q^45 + 4 * q^46 - 10 * q^47 + 3 * q^48 + q^49 - 40 * q^50 + 12 * q^51 + 4 * q^52 - 22 * q^53 - 3 * q^54 - 12 * q^55 - 2 * q^56 - 6 * q^57 + 14 * q^58 - 4 * q^59 - 2 * q^60 + 9 * q^61 + 12 * q^62 + 2 * q^63 + 6 * q^64 + 22 * q^65 - 2 * q^66 - 2 * q^67 - 12 * q^68 - 8 * q^69 + 24 * q^70 + 10 * q^71 - 6 * q^72 + 15 * q^73 - 7 * q^74 + 20 * q^75 + 6 * q^76 + 20 * q^77 - 2 * q^78 - 6 * q^79 - q^80 + 6 * q^81 + 11 * q^82 - 14 * q^83 + 4 * q^84 - 16 * q^85 - 12 * q^86 - 7 * q^87 - 4 * q^88 + 26 * q^89 + q^90 - 36 * q^91 - 4 * q^92 - 6 * q^93 - 20 * q^94 - 4 * q^95 - 3 * q^96 + 27 * q^97 + 2 * q^98 + 4 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 92\nu^{2} - 99\nu + 990 ) / 821$ (-v^5 + 10v^4 - 100v^3 + 92v^2 - 99v + 990) / 821
$\beta_{3}$	$=$	$( 8\nu^{5} - 80\nu^{4} - 21\nu^{3} - 736\nu^{2} + 792\nu - 4636 ) / 821$ (8v^5 - 80v^4 - 21v^3 - 736v^2 + 792*v - 4636) / 821
$\beta_{4}$	$=$	$( -90\nu^{5} + 79\nu^{4} - 790\nu^{3} + 70\nu^{2} - 7268\nu - 1210 ) / 9031$ (-90v^5 + 79v^4 - 790v^3 + 70v^2 - 7268*v - 1210) / 9031
$\beta_{5}$	$=$	$( -531\nu^{5} - 437\nu^{4} - 4661\nu^{3} + 413\nu^{2} - 32044\nu - 7139 ) / 9031$ (-531v^5 - 437v^4 - 4661v^3 + 413v^2 - 32044*v - 7139) / 9031

Introduction

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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	366.2.e.d

Level	$366$
Weight	$2$
Character orbit	366.e
Analytic conductor	$2.923$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{5} - 7\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 7$ b5 - 7*b4 - b3 + b2 - b1 - 7
$\nu^{3}$	$=$	$-\beta_{3} - 8\beta_{2} + 4$ -b3 - 8*b2 + 4
$\nu^{4}$	$=$	$-10\beta_{5} + 59\beta_{4} + 12\beta_1$ -10b5 + 59b4 + 12*b1
$\nu^{5}$	$=$	$-8\beta_{5} - 54\beta_{4} + 8\beta_{3} + 71\beta_{2} - 71\beta _1 - 54$ -8b5 - 54b4 + 8b3 + 71b2 - 71*b1 - 54

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.3
Significant digits:
Format: