LMFDB - Newform orbit 644.1.r.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [644,1,Mod(83,644)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(644, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 21])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("644.83"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$644 = 2^{2} \cdot 7 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	644.r (of order $22$ , degree $10$ , minimal)

Newform invariants

comment:select newform

sage:traces = [10,-1,0,-1,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$0.321397868136$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + \zeta_{22}^{6} q^{2} - \zeta_{22} q^{4} - \zeta_{22}^{9} q^{7} - \zeta_{22}^{7} q^{8} + \zeta_{22}^{3} q^{9} + ( - \zeta_{22}^{10} - \zeta_{22}^{2}) q^{11} + \zeta_{22}^{4} q^{14} + \zeta_{22}^{2} q^{16} + \cdots + ( - \zeta_{22}^{5} + \zeta_{22}^{2}) q^{99} +O(q^{100})$ q + z^6 * q^2 - z * q^4 - z^9 * q^7 - z^7 * q^8 + z^3 * q^9 + (-z^10 - z^2) * q^11 + z^4 * q^14 + z^2 * q^16 + z^9 * q^18 + (-z^8 + z^5) * q^22 + z^8 * q^23 + z^5 * q^25 + z^10 * q^28 + (z^7 - z^6) * q^29 + z^8 * q^32 - z^4 * q^36 + (-z^5 + z) * q^37 + (z^10 + z^4) * q^43 + (z^3 - 1) * q^44 - z^3 * q^46 - z^7 * q^49 - q^50 + (-z^4 - z^3) * q^53 - z^5 * q^56 + (-z^2 + z) * q^58 + z * q^63 - z^3 * q^64 + (-z^6 - z^4) * q^67 + (z^9 - z) * q^71 - z^10 * q^72 + (z^7 + 1) * q^74 + (-z^8 - 1) * q^77 + (-z^8 + z^7) * q^79 + z^6 * q^81 + (z^10 - z^5) * q^86 + (z^9 - z^6) * q^88 - z^9 * q^92 + z^2 * q^98 + (-z^5 + z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q - q^{2} - q^{4} - q^{7} - q^{8} + q^{9} + 2 q^{11} - q^{14} - q^{16} + q^{18} + 2 q^{22} - q^{23} + q^{25} - q^{28} + 2 q^{29} - q^{32} + q^{36} - 2 q^{43} - 9 q^{44} - q^{46} - q^{49}+ \cdots - 2 q^{99}+O(q^{100})$ 10 * q - q^2 - q^4 - q^7 - q^8 + q^9 + 2 * q^11 - q^14 - q^16 + q^18 + 2 * q^22 - q^23 + q^25 - q^28 + 2 * q^29 - q^32 + q^36 - 2 * q^43 - 9 * q^44 - q^46 - q^49 - 10 * q^50 - q^56 + 2 * q^58 + q^63 - q^64 + 2 * q^67 + q^72 + 11 * q^74 - 9 * q^77 + 2 * q^79 - q^81 - 2 * q^86 + 2 * q^88 - q^92 - q^98 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/644\mathbb{Z}\right)^\times$ .

$n$	$185$	$281$	$323$
$\chi(n)$	$-1$	$\zeta_{22}^{5}$	$-1$

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}

83.1

−0.841254	−	0.540641i
0.654861	+	0.755750i
−0.415415	−	0.909632i
−0.415415	+	0.909632i
−0.841254	+	0.540641i
0.142315	+	0.989821i
0.959493	−	0.281733i
0.959493	+	0.281733i
0.142315	−	0.989821i
0.654861	−	0.755750i

−0.959493

−

0.281733i

0.841254

0.540641i

0.415415

−

0.909632i

−0.654861

−

0.755750i

0.142315

−

0.989821i

111.1

0.415415

−

0.909632i

−0.654861

−

0.755750i

−0.142315

−

0.989821i

−0.959493

0.281733i

−0.841254

0.540641i

195.1

0.841254

0.540641i

0.415415

0.909632i

−0.654861

−

0.755750i

−0.142315

0.989821i

0.959493

0.281733i

251.1

0.841254

−

0.540641i

0.415415

−

0.909632i

−0.654861

0.755750i

−0.142315

−

0.989821i

0.959493

−

0.281733i

419.1

−0.959493

0.281733i

0.841254

−

0.540641i

0.415415

0.909632i

−0.654861

0.755750i

0.142315

0.989821i

447.1

−0.654861

0.755750i

−0.142315

−

0.989821i

−0.959493

−

0.281733i

0.841254

0.540641i

−0.415415

−

0.909632i

475.1

−0.142315

−

0.989821i

−0.959493

0.281733i

0.841254

0.540641i

0.415415

0.909632i

0.654861

−

0.755750i

503.1

−0.142315

0.989821i

−0.959493

−

0.281733i

0.841254

−

0.540641i

0.415415

−

0.909632i

0.654861

0.755750i

559.1

−0.654861

−

0.755750i

−0.142315

0.989821i

−0.959493

0.281733i

0.841254

−

0.540641i

−0.415415

0.909632i

615.1

0.415415

0.909632i

−0.654861

0.755750i

−0.142315

0.989821i

−0.959493

−

0.281733i

−0.841254

−

0.540641i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
92.h	even	22	1	inner
644.r	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.1.r.a	✓	10
4.b	odd	2	1		644.1.r.b	yes	10
7.b	odd	2	1	CM	644.1.r.a	✓	10
23.d	odd	22	1		644.1.r.b	yes	10
28.d	even	2	1		644.1.r.b	yes	10
92.h	even	22	1	inner	644.1.r.a	✓	10
161.k	even	22	1		644.1.r.b	yes	10
644.r	odd	22	1	inner	644.1.r.a	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.1.r.a	✓	10	1.a	even	1	1	trivial
644.1.r.a	✓	10	7.b	odd	2	1	CM
644.1.r.a	✓	10	92.h	even	22	1	inner
644.1.r.a	✓	10	644.r	odd	22	1	inner
644.1.r.b	yes	10	4.b	odd	2	1
644.1.r.b	yes	10	23.d	odd	22	1
644.1.r.b	yes	10	28.d	even	2	1
644.1.r.b	yes	10	161.k	even	22	1

This newform subspace can be constructed as the kernel of the linear operator $T_{11}^{10} - 2T_{11}^{9} + 4T_{11}^{8} + 3T_{11}^{7} - 6T_{11}^{6} + 12T_{11}^{5} + 9T_{11}^{4} - 7T_{11}^{3} + 14T_{11}^{2} - 6T_{11} + 1$ T11^10 - 2*T11^9 + 4*T11^8 + 3*T11^7 - 6*T11^6 + 12*T11^5 + 9*T11^4 - 7*T11^3 + 14*T11^2 - 6*T11 + 1 acting on $S_{1}^{\mathrm{new}}(644, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$T^{10}$ T^10
$5$	$T^{10}$ T^10
$7$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$13$	$T^{10}$ T^10
$17$	$T^{10}$ T^10
$19$	$T^{10}$ T^10
$23$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$29$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$31$	$T^{10}$ T^10
$37$	$T^{10} + 11 T^{7} + \cdots + 11$ T^10 + 11T^7 + 33T^4 - 11T^3 + 22T + 11
$41$	$T^{10}$ T^10
$43$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$47$	$T^{10}$ T^10
$53$	$T^{10} - 11 T^{7} + \cdots + 11$ T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$59$	$T^{10}$ T^10
$61$	$T^{10}$ T^10
$67$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$71$	$T^{10} + 11 T^{7} + \cdots + 11$ T^10 + 11T^7 + 33T^4 - 11T^3 + 22T + 11
$73$	$T^{10}$ T^10
$79$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$83$	$T^{10}$ T^10
$89$	$T^{10}$ T^10
$97$	$T^{10}$ T^10
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Elliptic curves over $\Q$
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$n$ :		e.g. 2-40 or 990-1000
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Introduction

L-functions

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	644.1.r.a

Level	$644$
Weight	$1$
Character orbit	644.r
Analytic conductor	$0.321$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-7
Inner twists	$4$