LMFDB - Newform orbit 209.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,2,Mod(18,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7, 5]))

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

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

chi := DirichletCharacter("209.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$209 = 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	209.k (of order $10$ , degree $4$ , minimal)

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:	no
Analytic conductor:	$1.66887340224$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.484000000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121$ x^8 + x^6 + 16x^4 + 66x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/209\mathbb{Z}\right)^\times$ .

$n$	$78$	$134$
$\chi(n)$	$-1$	$1 + \beta_{2} + \beta_{3} - \beta_{5}$

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}

18.1

0.476925	+	1.46782i
−0.476925	−	1.46782i
−1.73855	+	1.26313i
1.73855	−	1.26313i
0.476925	−	1.46782i
−0.476925	+	1.46782i
−1.73855	−	1.26313i
1.73855	+	1.26313i

−2.02029

−

1.46782i

1.30902

4.02874i

0.500000

−

0.363271i

−1.80902

0.587785i

1.72553

−

5.31064i

−2.42705

−

1.76336i

−1.54336

18.2

2.02029

1.46782i

1.30902

4.02874i

0.500000

−

0.363271i

−1.80902

0.587785i

−1.72553

5.31064i

−2.42705

−

1.76336i

1.54336

94.1

−0.410415

−

1.26313i

0.190983

−

0.138757i

0.500000

−

1.53884i

−0.690983

−

0.951057i

−2.40261

−

1.74560i

0.927051

2.85317i

−2.14896

94.2

0.410415

1.26313i

0.190983

−

0.138757i

0.500000

−

1.53884i

−0.690983

−

0.951057i

2.40261

1.74560i

0.927051

2.85317i

2.14896

151.1

−2.02029

1.46782i

1.30902

−

4.02874i

0.500000

0.363271i

−1.80902

−

0.587785i

1.72553

5.31064i

−2.42705

1.76336i

−1.54336

151.2

2.02029

−

1.46782i

1.30902

−

4.02874i

0.500000

0.363271i

−1.80902

−

0.587785i

−1.72553

−

5.31064i

−2.42705

1.76336i

1.54336

189.1

−0.410415

1.26313i

0.190983

0.138757i

0.500000

1.53884i

−0.690983

0.951057i

−2.40261

1.74560i

0.927051

−

2.85317i

−2.14896

189.2

0.410415

−

1.26313i

0.190983

0.138757i

0.500000

1.53884i

−0.690983

0.951057i

2.40261

−

1.74560i

0.927051

−

2.85317i

2.14896

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
19.b	odd	2	1	inner
209.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	209.2.k.b	✓	8
11.d	odd	10	1	inner	209.2.k.b	✓	8
19.b	odd	2	1	inner	209.2.k.b	✓	8
209.k	even	10	1	inner	209.2.k.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.2.k.b	✓	8	1.a	even	1	1	trivial
209.2.k.b	✓	8	11.d	odd	10	1	inner
209.2.k.b	✓	8	19.b	odd	2	1	inner
209.2.k.b	✓	8	209.k	even	10	1	inner

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{8} - T_{2}^{6} + 31T_{2}^{4} + 99T_{2}^{2} + 121$ T2^8 - T2^6 + 31*T2^4 + 99*T2^2 + 121 acting on $S_{2}^{\mathrm{new}}(209, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} - T^{6} + \cdots + 121$ T^8 - T^6 + 31T^4 + 99T^2 + 121
$3$	$T^{8}$ T^8
$5$	$(T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2}$ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$7$	$(T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2}$ (T^4 + 5T^3 + 10T^2 + 10*T + 5)^2
$11$	$(T^{4} - 9 T^{3} + \cdots + 121)^{2}$ (T^4 - 9T^3 + 41T^2 - 99*T + 121)^2
$13$	$T^{8} - 4 T^{6} + \cdots + 30976$ T^8 - 4T^6 + 496T^4 + 6336*T^2 + 30976
$17$	$(T^{4} + 15 T^{3} + \cdots + 405)^{2}$ (T^4 + 15T^3 + 90T^2 + 270*T + 405)^2
$19$	$T^{8} + 10 T^{7} + \cdots + 130321$ T^8 + 10T^7 + 39T^6 - 40T^5 - 699T^4 - 760T^3 + 14079T^2 + 68590*T + 130321
$23$	$(T^{2} + 3 T + 1)^{4}$ (T^2 + 3*T + 1)^4
$29$	$T^{8} - 24 T^{6} + \cdots + 30976$ T^8 - 24T^6 + 2176T^4 + 13376*T^2 + 30976
$31$	$T^{8} - 20 T^{6} + \cdots + 774400$ T^8 - 20T^6 + 960T^4 - 35200*T^2 + 774400
$37$	$T^{8} + 20 T^{6} + \cdots + 774400$ T^8 + 20T^6 + 3760T^4 - 88000*T^2 + 774400
$41$	$T^{8} + 20 T^{6} + \cdots + 19360000$ T^8 + 20T^6 + 6400T^4 + 528000*T^2 + 19360000
$43$	$(T^{4} + 145 T^{2} + 4805)^{2}$ (T^4 + 145*T^2 + 4805)^2
$47$	$(T^{4} - 8 T^{3} + \cdots + 121)^{2}$ (T^4 - 8T^3 + 34T^2 - 77*T + 121)^2
$53$	$T^{8} - 280 T^{6} + \cdots + 774400$ T^8 - 280T^6 + 29760T^4 + 88000*T^2 + 774400
$59$	$T^{8} - 140 T^{6} + \cdots + 774400$ T^8 - 140T^6 + 7360T^4 + 35200*T^2 + 774400
$61$	$(T^{4} - 15 T^{3} + \cdots + 125)^{2}$ (T^4 - 15T^3 + 150T^2 - 250*T + 125)^2
$67$	$(T^{4} + 180 T^{2} + 880)^{2}$ (T^4 + 180*T^2 + 880)^2
$71$	$T^{8} - 100 T^{6} + \cdots + 484000000$ T^8 - 100T^6 + 24000T^4 - 4400000*T^2 + 484000000
$73$	$(T^{4} + 320 T + 1280)^{2}$ (T^4 + 320*T + 1280)^2
$79$	$T^{8} - 20 T^{6} + \cdots + 19360000$ T^8 - 20T^6 + 12400T^4 + 792000*T^2 + 19360000
$83$	$(T^{4} + 20 T^{3} + \cdots + 125)^{2}$ (T^4 + 20T^3 + 100T^2 - 125*T + 125)^2
$89$	$(T^{4} + 60 T^{2} + 880)^{2}$ (T^4 + 60*T^2 + 880)^2
$97$	$T^{8} + 40 T^{6} + \cdots + 774400$ T^8 + 40T^6 + 7360T^4 - 123200*T^2 + 774400
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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

$f(q)$	$=$	$q + ( - \beta_{4} - \beta_1) q^{2} + (3 \beta_{5} - 3 \beta_{3} - \beta_{2} - 1) q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + ( - 6 \beta_{5} - 6 \beta_{3} + 3 \beta_{2}) q^{99}+O(q^{100})$ q + (-b4 - b1) * q^2 + (3b5 - 3b3 - b2 - 1) * q^4 + (b3 + b2 + 1) * q^5 + (b2 - 1) * q^7 + (-2b7 + b6 - 2b4 - b1) * q^8 - 3b5 q^9 - b4 * q^10 + (-2b5 + 3b3 + 2b2 + 4) q^11 + (-2b4 - 2b1) * q^13 + (-b7 + b4 + 2b1) q^14 + (-4b3 - b2 - 4) q^16 + (3b3 - 3) q^17 + (-3b6 + 3b4 + 3b1) q^18 + (2b7 - 2b5 + 2b4 + b3 + 2b2) * q^19 + (2b5 - b3 - b2) q^20 + (b7 - 2b6 - 2b4) * q^22 + (b5 - b2 - 2) * q^23 + (b5 - 4b3 - 1) q^25 + (10b5 - 10b3 - 6b2 - 6) q^26 + (-6b5 + 5b3 + 3b2 + 1) q^28 + (-2b6 + 2b4) * q^29 + (-2b7 + 2b6 - 2b4) q^31 + (-b7 + b1) * q^32 + (3b7 + 3b4 + 3b1) q^34 + (b5 - b3 - b2 - 2) * q^35 + (-6b5 + 9b3 + 6) * q^36 + (4b7 + 2b6 + 2b4 - 2b1) * q^37 + (-b7 - 2b6 + 2b4 + 6b3 + 2b2 + 4b1 + 6) q^38 + (2b6 - 2b4 - b1) * q^40 + (2b7 + 4b6 + 2b4 - 4b1) * q^41 + (-7b5 + 4b3 - 3b2 + 2) q^43 + (8b5 - 2b3 - 6b2 + 3) q^44 + (-3b5 + 3b2 + 3) * q^45 + (b7 + b6 + b4) * q^46 + (-3b5 + b3 + 3) q^47 + (b3 + 5b2 + 1) q^49 + (-4b7 + b6) q^50 + (-4b7 + 6b6 - 4b4 - 6b1) * q^52 + (2b7 - 2b6 + 4b4 - 2b1) * q^53 + (4b3 + 3b2 + 1) * q^55 + (2b7 - 2b6 + 3b4 + 4b1) * q^56 + (2b5 + 6b3 - 2) * q^58 + (-4b7 + 2b1) * q^59 + (8b5 - 7b3 - 4b2 - 1) q^61 + (-6b5 - 6b3 + 4b2 - 4) q^62 + (3b5 + 3) q^63 + (b5 - 6b3 - 6b2) * q^64 - 2b4 q^65 + (4b7 + 4b6 + 2b4) q^67 + (-3b5 + 9b3 - 3b2 + 6) q^68 + (b6 + b4) * q^70 + (-2b7 + 2b6 - 8b4 - 4b1) * q^71 + (3b7 - 6b6) * q^72 + (-4b5 - 4b3) * q^73 + (4b5 + 2b3 + 4b2 + 8) q^74 + (4b7 - 8b5 - 2b4 + 10b3 + 2b2 - 4b1 + 5) * q^76 + (5b5 - 4b3 - b2 - 5) * q^77 + (-4b7 - 4b6 + 2b4 + 6b1) * q^79 + (-b5 - 4b3 + 1) q^80 + (9b5 - 9b3 - 9b2 - 9) q^81 + (-2b3 + 14b2 - 2) * q^82 + (6b5 + b3 - 3b2 - 7) * q^83 + (-3b3 - 6b2 - 6) * q^85 + (7b7 - 7b6 + 5b4 + 2b1) * q^86 + (4b6 - 9b4 - 7b1) q^88 + (2b7 + 2b4 + 2b1) q^89 + (-3b7 - 3b6 + 3b1) q^90 + (-2b7 + 2b4 + 4b1) q^91 + (-b5 + b3) * q^92 + (b7 - 3b6) q^94 + (-2b6 + 2b4 + b2 - 1) * q^95 + (-2b7 + 2b6 - 2b1) q^97 + (-4b7 - b4 + 4b1) * q^98 + (-6b5 - 6b3 + 3b2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 6 q^{4} + 4 q^{5} - 10 q^{7} - 6 q^{9} + 18 q^{11} - 22 q^{16} - 30 q^{17} - 10 q^{19} + 8 q^{20} - 12 q^{23} + 2 q^{25} + 4 q^{26} - 20 q^{28} - 10 q^{35} + 18 q^{36} + 32 q^{38} + 56 q^{44} + 12 q^{45}+ \cdots - 6 q^{99}+O(q^{100})$ 8 * q + 6 * q^4 + 4 * q^5 - 10 * q^7 - 6 * q^9 + 18 * q^11 - 22 * q^16 - 30 * q^17 - 10 * q^19 + 8 * q^20 - 12 * q^23 + 2 * q^25 + 4 * q^26 - 20 * q^28 - 10 * q^35 + 18 * q^36 + 32 * q^38 + 56 * q^44 + 12 * q^45 + 16 * q^47 - 4 * q^49 - 6 * q^55 - 24 * q^58 + 30 * q^61 - 40 * q^62 + 30 * q^63 + 26 * q^64 + 30 * q^68 + 60 * q^74 - 20 * q^77 + 14 * q^80 - 18 * q^81 - 40 * q^82 - 40 * q^83 - 30 * q^85 - 4 * q^92 - 10 * q^95 - 6 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991$ (7v^6 - 37v^4 + 629*v^2 - 363) / 1991
$\beta_{3}$	$=$	$( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991$ (-28v^6 + 148v^4 - 525*v^2 - 539) / 1991
$\beta_{4}$	$=$	$( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991$ (-28v^7 + 148v^5 - 525v^3 - 539v) / 1991
$\beta_{5}$	$=$	$( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991$ (40v^6 + 73v^4 + 750*v^2 + 2761) / 1991
$\beta_{6}$	$=$	$( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991$ (-61v^7 + 38v^5 - 646v^3 - 1672v) / 1991
$\beta_{7}$	$=$	$( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991$ (68v^7 - 75v^5 + 1275v^3 + 3300v) / 1991

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	209.2.k.b

Level	$209$
Weight	$2$
Character orbit	209.k
Analytic conductor	$1.669$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} + 4\beta_{2} + 1$ b3 + 4*b2 + 1
$\nu^{3}$	$=$	$4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1$ 4b7 + 4b6 + b4 - 3*b1
$\nu^{4}$	$=$	$7\beta_{5} + 10\beta_{3} - 7$ 7b5 + 10b3 - 7
$\nu^{5}$	$=$	$7\beta_{7} + 17\beta_{4} - 7\beta_1$ 7b7 + 17b4 - 7*b1
$\nu^{6}$	$=$	$37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75$ 37b5 - 37b3 - 75*b2 - 75
$\nu^{7}$	$=$	$-38\beta_{7} - 75\beta_{6}$ -38b7 - 75b6

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 18.2
Significant digits:
Format: