LMFDB - Newform orbit 980.1.bq.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,1,Mod(39,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 21, 34])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.39"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$980 = 2^{2} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	980.bq (of order $42$ , degree $12$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$0.489083712380$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} + \cdots)$

$q$ -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.

$f(q)$	$=$	$q - \zeta_{42}^{4} q^{2} + (\zeta_{42}^{7} + \zeta_{42}^{3}) q^{3} + \zeta_{42}^{8} q^{4} - \zeta_{42}^{19} q^{5} + ( - \zeta_{42}^{11} - \zeta_{42}^{7}) q^{6} - \zeta_{42}^{8} q^{7} - \zeta_{42}^{12} q^{8} + \cdots - \zeta_{42}^{20} q^{98} +O(q^{100})$ q - z^4 * q^2 + (z^7 + z^3) * q^3 + z^8 * q^4 - z^19 * q^5 + (-z^11 - z^7) * q^6 - z^8 * q^7 - z^12 * q^8 + (z^14 + z^10 + z^6) * q^9 - z^2 * q^10 + (z^15 + z^11) * q^12 + z^12 * q^14 + (z^5 + z) * q^15 + z^16 * q^16 + (-z^18 - z^14 - z^10) * q^18 + z^6 * q^20 + (-z^15 - z^11) * q^21 + (z^17 - z^6) * q^23 + (-z^19 - z^15) * q^24 - z^17 * q^25 + (z^17 + z^13 + z^9 - 1) * q^27 - z^16 * q^28 + (-z^7 - z^5) * q^29 + (-z^9 - z^5) * q^30 - z^20 * q^32 - z^6 * q^35 + (z^18 + z^14 - z) * q^36 - z^10 * q^40 + (z^2 - z) * q^41 + (z^19 + z^15) * q^42 + (-z^20 + z^19) * q^43 + (z^12 + z^8 + z^4) * q^45 + (z^10 + 1) * q^46 + (-z^16 + z^13) * q^47 + (z^19 - z^2) * q^48 + z^16 * q^49 - q^50 + (-z^17 - z^13 + z^4 + 1) * q^54 + z^20 * q^56 + (z^11 + z^9) * q^58 + (z^13 + z^9) * q^60 + (z^18 + z^8) * q^61 + (-z^18 - z^14 + z) * q^63 - z^3 * q^64 + z^14 * q^67 + (z^20 - z^13 - z^9 - z^3) * q^69 + z^10 * q^70 + (-z^18 + z^5 + z) * q^72 + (-z^20 + z^3) * q^75 + z^14 * q^80 + (z^20 + z^16 + z^12 - z^7 - z^3) * q^81 + (-z^6 + z^5) * q^82 + (-z^16 + z^11) * q^83 + (-z^19 + z^2) * q^84 + (-z^3 + z^2) * q^86 + (-z^14 - z^12 - z^10 - z^8) * q^87 + (-z^11 - z^9) * q^89 + (-z^16 - z^12 - z^8) * q^90 + (-z^14 - z^4) * q^92 + (z^20 - z^17) * q^94 + (z^6 + z^2) * q^96 - z^20 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$12 q - q^{2} + 8 q^{3} + q^{4} + q^{5} - 5 q^{6} - q^{7} + 2 q^{8} - 7 q^{9} - q^{10} + q^{12} - 2 q^{14} - 2 q^{15} + q^{16} + 7 q^{18} - 2 q^{20} - q^{21} + q^{23} - q^{24} + q^{25} - 12 q^{27}+ \cdots - q^{98}+O(q^{100})$ 12 * q - q^2 + 8 * q^3 + q^4 + q^5 - 5 * q^6 - q^7 + 2 * q^8 - 7 * q^9 - q^10 + q^12 - 2 * q^14 - 2 * q^15 + q^16 + 7 * q^18 - 2 * q^20 - q^21 + q^23 - q^24 + q^25 - 12 * q^27 - q^28 - 5 * q^29 - q^30 - q^32 + 2 * q^35 - 7 * q^36 - q^40 + 2 * q^41 + q^42 - 2 * q^43 + 13 * q^46 - 2 * q^47 - 2 * q^48 + q^49 - 12 * q^50 + 15 * q^54 + q^56 + q^58 + q^60 - q^61 + 7 * q^63 - 2 * q^64 - 6 * q^67 - 2 * q^69 + q^70 + q^75 - 6 * q^80 - 8 * q^81 + q^82 - 2 * q^83 + 2 * q^84 - q^86 + 6 * q^87 - q^89 + 5 * q^92 + 2 * q^94 - q^96 - q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$ .

$n$	$101$	$197$	$491$
$\chi(n)$	$\zeta_{42}^{10}$	$-1$	$-1$

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}

39.1

−0.733052	+	0.680173i
0.365341	−	0.930874i
0.365341	+	0.930874i
0.0747301	+	0.997204i
0.955573	+	0.294755i
0.826239	−	0.563320i
0.826239	+	0.563320i
0.0747301	−	0.997204i
−0.988831	−	0.149042i
−0.733052	−	0.680173i
0.955573	−	0.294755i
−0.988831	+	0.149042i

0.988831

0.149042i

−0.123490

0.0841939i

0.955573

0.294755i

0.0747301

0.997204i

−0.134659

0.0648483i

−0.955573

−

0.294755i

0.900969

0.433884i

−0.357180

0.910080i

−0.0747301

0.997204i

179.1

−0.0747301

−

0.997204i

1.40097

0.432142i

−0.988831

0.149042i

−0.733052

0.680173i

0.326239

−

1.42935i

0.988831

−

0.149042i

0.222521

0.974928i

0.949729

0.647514i

0.733052

0.680173i

219.1

−0.0747301

0.997204i

1.40097

−

0.432142i

−0.988831

−

0.149042i

−0.733052

−

0.680173i

0.326239

1.42935i

0.988831

0.149042i

0.222521

−

0.974928i

0.949729

−

0.647514i

0.733052

−

0.680173i

319.1

−0.955573

0.294755i

0.722521

1.84095i

0.826239

−

0.563320i

−0.988831

−

0.149042i

−1.23305

−

1.54620i

−0.826239

0.563320i

−0.623490

0.781831i

−2.13402

1.98008i

0.988831

−

0.149042i

359.1

−0.365341

−

0.930874i

−0.123490

−

1.64786i

−0.733052

0.680173i

0.826239

−

0.563320i

−1.48883

0.716983i

0.733052

−

0.680173i

0.900969

0.433884i

−1.71135

0.257945i

−0.826239

−

0.563320i

499.1

0.733052

0.680173i

0.722521

0.108903i

0.0747301

0.997204i

0.365341

0.930874i

0.455573

0.571270i

−0.0747301

−

0.997204i

−0.623490

0.781831i

−0.445396

−

0.137386i

−0.365341

0.930874i

599.1

0.733052

−

0.680173i

0.722521

−

0.108903i

0.0747301

−

0.997204i

0.365341

−

0.930874i

0.455573

−

0.571270i

−0.0747301

0.997204i

−0.623490

−

0.781831i

−0.445396

0.137386i

−0.365341

−

0.930874i

639.1

−0.955573

−

0.294755i

0.722521

−

1.84095i

0.826239

0.563320i

−0.988831

0.149042i

−1.23305

1.54620i

−0.826239

−

0.563320i

−0.623490

−

0.781831i

−2.13402

−

1.98008i

0.988831

0.149042i

739.1

−0.826239

−

0.563320i

1.40097

1.29991i

0.365341

0.930874i

0.955573

−

0.294755i

−0.425270

−

1.86323i

−0.365341

−

0.930874i

0.222521

−

0.974928i

0.198220

2.64506i

−0.955573

−

0.294755i

779.1

0.988831

−

0.149042i

−0.123490

−

0.0841939i

0.955573

−

0.294755i

0.0747301

−

0.997204i

−0.134659

−

0.0648483i

−0.955573

0.294755i

0.900969

−

0.433884i

−0.357180

−

0.910080i

−0.0747301

−

0.997204i

879.1

−0.365341

0.930874i

−0.123490

1.64786i

−0.733052

−

0.680173i

0.826239

0.563320i

−1.48883

−

0.716983i

0.733052

0.680173i

0.900969

−

0.433884i

−1.71135

−

0.257945i

−0.826239

0.563320i

919.1

−0.826239

0.563320i

1.40097

−

1.29991i

0.365341

−

0.930874i

0.955573

0.294755i

−0.425270

1.86323i

−0.365341

0.930874i

0.222521

0.974928i

0.198220

−

2.64506i

−0.955573

0.294755i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5})$
49.g	even	21	1	inner
980.bq	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.1.bq.a	✓	12
4.b	odd	2	1		980.1.bq.b	yes	12
5.b	even	2	1		980.1.bq.b	yes	12
20.d	odd	2	1	CM	980.1.bq.a	✓	12
49.g	even	21	1	inner	980.1.bq.a	✓	12
196.o	odd	42	1		980.1.bq.b	yes	12
245.t	even	42	1		980.1.bq.b	yes	12
980.bq	odd	42	1	inner	980.1.bq.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.1.bq.a	✓	12	1.a	even	1	1	trivial
980.1.bq.a	✓	12	20.d	odd	2	1	CM
980.1.bq.a	✓	12	49.g	even	21	1	inner
980.1.bq.a	✓	12	980.bq	odd	42	1	inner
980.1.bq.b	yes	12	4.b	odd	2	1
980.1.bq.b	yes	12	5.b	even	2	1
980.1.bq.b	yes	12	196.o	odd	42	1
980.1.bq.b	yes	12	245.t	even	42	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{12} - 8 T_{3}^{11} + 35 T_{3}^{10} - 104 T_{3}^{9} + 230 T_{3}^{8} - 392 T_{3}^{7} + 519 T_{3}^{6} + \cdots + 1$ T3^12 - 8*T3^11 + 35*T3^10 - 104*T3^9 + 230*T3^8 - 392*T3^7 + 519*T3^6 - 518*T3^5 + 349*T3^4 - 118*T3^3 + 6*T3 + 1 acting on $S_{1}^{\mathrm{new}}(980, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$3$	$T^{12} - 8 T^{11} + \cdots + 1$ T^12 - 8T^11 + 35T^10 - 104T^9 + 230T^8 - 392T^7 + 519T^6 - 518T^5 + 349T^4 - 118T^3 + 6T + 1
$5$	$T^{12} - T^{11} + \cdots + 1$ T^12 - T^11 + T^9 - T^8 + T^6 - T^4 + T^3 - T + 1
$7$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$11$	$T^{12}$ T^12
$13$	$T^{12}$ T^12
$17$	$T^{12}$ T^12
$19$	$T^{12}$ T^12
$23$	$T^{12} - T^{11} + \cdots + 1$ T^12 - T^11 + T^9 + 6T^8 + 21T^7 - 20T^6 + 69T^4 + 29T^3 + 49T^2 + 13*T + 1
$29$	$T^{12} + 5 T^{11} + \cdots + 1$ T^12 + 5T^11 + 17T^10 + 38T^9 + 68T^8 + 92T^7 + 105T^6 + 92T^5 + 61T^4 + 10T^3 + 45T^2 + 12*T + 1
$31$	$T^{12}$ T^12
$37$	$T^{12}$ T^12
$41$	$T^{12} - 2 T^{11} + \cdots + 1$ T^12 - 2T^11 + 3T^10 - 4T^9 + 12T^8 - 6T^7 + 7T^6 - 6T^5 + 54T^4 + 94T^3 + 52T^2 + 5*T + 1
$43$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 3T^10 + 4T^9 + 12T^8 + 6T^7 + 7T^6 + 6T^5 + 54T^4 - 94T^3 + 52T^2 - 5*T + 1
$47$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 6T^9 + 12T^8 - 7T^7 + T^6 + 28T^5 + 3T^4 - 8T^3 + 7T^2 - 3*T + 1
$53$	$T^{12}$ T^12
$59$	$T^{12}$ T^12
$61$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 - T^9 - 15T^8 + 22T^6 + 21T^5 + 48T^4 - 71T^3 + 28T^2 + 8*T + 1
$67$	$(T^{2} + T + 1)^{6}$ (T^2 + T + 1)^6
$71$	$T^{12}$ T^12
$73$	$T^{12}$ T^12
$79$	$T^{12}$ T^12
$83$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 3T^10 + 4T^9 + 12T^8 + 6T^7 + 7T^6 + 6T^5 + 54T^4 - 94T^3 + 52T^2 - 5*T + 1
$89$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 - T^9 - 15T^8 + 22T^6 + 21T^5 + 48T^4 - 71T^3 + 28T^2 + 8*T + 1
$97$	$T^{12}$ T^12
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 39.1
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	980.1.bq.a

Level	$980$
Weight	$1$
Character orbit	980.bq
Analytic conductor	$0.489$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{21}$
CM discriminant	-20
Inner twists	$4$