LMFDB - Newform orbit 980.2.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(81,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("980.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.bg (of order $21$, degree $12$, 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:	$7.82533939809$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Character values

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

$n$	$101$	$197$	$491$
$\chi(n)$	$\zeta_{21}^{2}$	$1$	$1$

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} $

81.1

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

1.71171

0.527991i

−0.733052

0.680173i

−2.62053

0.364415i

0.172446

0.117572i

121.1

1.71171

−

0.527991i

−0.733052

−

0.680173i

−2.62053

−

0.364415i

0.172446

−

0.117572i

221.1

0.654431

1.66746i

−0.988831

−

0.149042i

−2.61152

−

0.424191i

−0.152997

0.141960i

261.1

0.133863

1.78628i

0.826239

−

0.563320i

−0.559228

2.58597i

−0.206381

0.0311069i

401.1

2.76011

0.416020i

0.365341

0.930874i

2.53695

0.750915i

4.57842

1.41226i

501.1

2.76011

−

0.416020i

0.365341

−

0.930874i

2.53695

−

0.750915i

4.57842

−

1.41226i

541.1

0.654431

−

1.66746i

−0.988831

0.149042i

−2.61152

0.424191i

−0.152997

−

0.141960i

641.1

2.04616

1.89856i

0.955573

−

0.294755i

−1.34897

2.27603i

0.358053

4.77789i

681.1

−2.30627

−

1.57239i

0.0747301

−

0.997204i

−2.39670

−

1.12064i

1.75045

4.46008i

781.1

0.133863

−

1.78628i

0.826239

0.563320i

−0.559228

−

2.58597i

−0.206381

−

0.0311069i

821.1

2.04616

−

1.89856i

0.955573

0.294755i

−1.34897

−

2.27603i

0.358053

−

4.77789i

921.1

−2.30627

1.57239i

0.0747301

0.997204i

−2.39670

1.12064i

1.75045

−

4.46008i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.bg.a	✓	12
49.g	even	21	1	inner	980.2.bg.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.2.bg.a	✓	12	1.a	even	1	1	trivial
980.2.bg.a	✓	12	49.g	even	21	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 10 T_{3}^{11} + 42 T_{3}^{10} - 92 T_{3}^{9} + 122 T_{3}^{8} - 315 T_{3}^{7} + 1681 T_{3}^{6} + \cdots + 15625 $ T3^12 - 10*T3^11 + 42*T3^10 - 92*T3^9 + 122*T3^8 - 315*T3^7 + 1681*T3^6 - 6405*T3^5 + 16700*T3^4 - 30500*T3^3 + 39375*T3^2 - 34375*T3 + 15625 acting on $S_{2}^{\mathrm{new}}(980, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 10 T^{11} + \cdots + 15625 $ T^12 - 10T^11 + 42T^10 - 92T^9 + 122T^8 - 315T^7 + 1681T^6 - 6405T^5 + 16700T^4 - 30500T^3 + 39375T^2 - 34375*T + 15625
$5$	$ 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} + 14 T^{11} + \cdots + 117649 $ T^12 + 14T^11 + 84T^10 + 259T^9 + 245T^8 - 1323T^7 - 6125T^6 - 9261T^5 + 12005T^4 + 88837T^3 + 201684T^2 + 235298*T + 117649
$11$	$ T^{12} + 9 T^{11} + \cdots + 729 $ T^12 + 9T^11 + 21T^10 - 162T^9 - 1332T^8 - 3591T^7 + 4509T^6 + 61803T^5 + 197802T^4 + 303750T^3 + 193914T^2 + 2916*T + 729
$13$	$ T^{12} - 2 T^{11} + \cdots + 15625 $ T^12 - 2T^11 + 24T^10 + 24T^9 + 432T^8 + 673T^7 + 4711T^6 + 6560T^5 + 13725T^4 + 5750T^3 - 14375T^2 - 9375*T + 15625
$17$	$ T^{12} - 12 T^{11} + \cdots + 729 $ T^12 - 12T^11 + 84T^10 - 414T^9 + 1440T^8 - 3591T^7 + 6021T^6 - 3969T^5 - 648T^4 + 972T^3 + 1701T^2 - 2187*T + 729
$19$	$ T^{12} + 7 T^{11} + \cdots + 49 $ T^12 + 7T^11 + 49T^10 + 210T^9 + 1015T^8 + 3703T^7 + 12558T^6 + 29449T^5 + 55615T^4 + 62230T^3 + 49049T^2 - 1519*T + 49
$23$	$ T^{12} + 9 T^{11} + \cdots + 5022081 $ T^12 + 9T^11 + 42T^10 + 90T^9 - 261T^8 + 1134T^7 + 17739T^6 + 11907T^5 + 161514T^4 + 63909T^3 - 207522T^2 + 3146364*T + 5022081
$29$	$ T^{12} + 6 T^{11} + \cdots + 729 $ T^12 + 6T^11 + 27T^10 + 108T^9 + 342T^8 + 702T^7 + 1323T^6 + 1782T^5 + 2754T^4 + 486T^3 + 1944T^2 + 2187*T + 729
$31$	$ T^{12} + 16 T^{11} + \cdots + 16129 $ T^12 + 16T^11 + 175T^10 + 1140T^9 + 5731T^8 + 18928T^7 + 52452T^6 + 93520T^5 + 224185T^4 + 300376T^3 + 639842T^2 - 97282*T + 16129
$37$	$ T^{12} - 22 T^{11} + \cdots + 3568321 $ T^12 - 22T^11 + 308T^10 - 3163T^9 + 31940T^8 - 243271T^7 + 1431774T^6 - 6676432T^5 + 22027921T^4 - 44803205T^3 + 50071056T^2 - 22900347*T + 3568321
$41$	$ T^{12} - 24 T^{11} + \cdots + 32455809 $ T^12 - 24T^11 + 327T^10 - 2376T^9 + 10755T^8 - 18036T^7 - 1323T^6 + 158436T^5 - 16038T^4 - 2534976T^3 + 10712169T^2 - 21380841*T + 32455809
$43$	$ T^{12} + 37 T^{11} + \cdots + 175561 $ T^12 + 37T^11 + 612T^10 + 5765T^9 + 33171T^8 + 113742T^7 + 207277T^6 + 176406T^5 + 370851T^4 + 30853T^3 + 547354T^2 + 95951*T + 175561
$47$	$ T^{12} + \cdots + 104714289 $ T^12 + 15T^11 - 162T^9 + 5508T^8 - 38745T^7 - 193563T^6 - 533547T^5 + 10122489T^4 - 17235261T^3 + 96257889T^2 - 156380706T + 104714289
$53$	$ T^{12} + 3 T^{11} + \cdots + 5022081 $ T^12 + 3T^11 + 126T^10 - 531T^9 + 2754T^8 - 36288T^7 + 14715T^6 + 526743T^5 + 1146150T^4 - 3018546T^3 + 2369493T^2 + 10286190*T + 5022081
$59$	$ T^{12} - 15 T^{11} + \cdots + 20820969 $ T^12 - 15T^11 - 42T^10 + 603T^9 + 7965T^8 + 22491T^7 + 70281T^6 - 221697T^5 - 192375T^4 - 1844856T^3 - 287469T^2 + 1601613*T + 20820969
$61$	$ T^{12} + \cdots + 353778481 $ T^12 - 7T^11 + 77T^10 + 77T^9 - 2926T^8 - 133084T^7 + 693315T^6 - 2612092T^5 + 116201638T^4 + 682696567T^3 + 617726193T^2 - 727174749*T + 353778481
$67$	$ T^{12} + \cdots + 123581074681 $ T^12 - 32T^11 + 805T^10 - 10716T^9 + 127312T^8 - 893417T^7 + 7950363T^6 - 40500656T^5 + 357251284T^4 - 943680695T^3 + 7361862284T^2 - 6317543311*T + 123581074681
$71$	$ T^{12} + \cdots + 129208689 $ T^12 - 12T^11 + 108T^10 - 549T^9 + 3141T^8 - 3375T^7 - 29295T^6 + 253287T^5 + 2170719T^4 - 2982339T^3 + 99189198T^2 + 138109050*T + 129208689
$73$	$ T^{12} - 35 T^{11} + \cdots + 7038409 $ T^12 - 35T^11 + 693T^10 - 11032T^9 + 153594T^8 - 1751799T^7 + 15851857T^6 - 118496553T^5 + 697778130T^4 - 2635304966T^3 + 4513439392T^2 - 293013238*T + 7038409
$79$	$ T^{12} - 11 T^{11} + \cdots + 29343889 $ T^12 - 11T^11 + 238T^10 + 99T^9 + 15130T^8 + 48706T^7 + 1005621T^6 + 4525801T^5 + 21647008T^4 + 37798090T^3 + 65509745T^2 - 33357886*T + 29343889
$83$	$ T^{12} + \cdots + 10511076242241 $ T^12 + 9T^11 + 297T^10 + 459T^9 + 26955T^8 + 137808T^7 + 5809104T^6 + 24567381T^5 + 199763982T^4 - 571929417T^3 + 65150181792T^2 + 950204723715*T + 10511076242241
$89$	$ T^{12} + 6 T^{11} + \cdots + 11758041 $ T^12 + 6T^11 + 21T^10 + 792T^9 + 6012T^8 + 6804T^7 + 170073T^6 + 2383668T^5 + 9218691T^4 + 8007336T^3 + 43110144T^2 - 47402496*T + 11758041
$97$	$ (T^{6} + 17 T^{5} + \cdots - 125)^{2} $ (T^6 + 17T^5 - 12T^4 - 981T^3 - 1375T^2 + 7600*T - 125)^2
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.bg (of order \(21\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{21}^{11} - \zeta_{21}^{10} + \cdots + 1) q^{3}+ \cdots + (2 \zeta_{21}^{11} - \zeta_{21}^{9} + \cdots - 1) q^{9}+O(q^{10}) \) q + (-z^11 - z^10 + z^6 + z^5 - z^4 - z^3 + z^2 - z + 1) * q^3 + z^8 * q^5 + (-z^10 + z^9 + z^8 - z^7 + z^5 - 2z^3 + 2z^2 - z - 2) * q^7 + (2z^11 - z^9 + z^8 - z^7 - 2z^6 + z^5 + 2z^4 - 2z^3 + 3z^2 - 1) q^9	\( q + ( - \zeta_{21}^{11} - \zeta_{21}^{10} + \cdots + 1) q^{3}+ \cdots + ( - 2 \zeta_{21}^{11} - 2 \zeta_{21}^{10} + \cdots - 5) q^{99}+O(q^{100}) \) q + (-z^11 - z^10 + z^6 + z^5 - z^4 - z^3 + z^2 - z + 1) * q^3 + z^8 * q^5 + (-z^10 + z^9 + z^8 - z^7 + z^5 - 2z^3 + 2z^2 - z - 2) * q^7 + (2z^11 - z^9 + z^8 - z^7 - 2z^6 + z^5 + 2z^4 - 2z^3 + 3z^2 - 1) q^9 + (3z^11 - z^10 - z^9 + 4z^8 - 2z^7 - 3z^6 + 3z^5 + 3z^4 - 2z^3 + z^2 + 2z - 4) * q^11 + (-z^11 + z^10 - z^9 + z^7 - z^5 + z^4 + z^3 - 3z^2 - z + 1) q^13 + (z^11 - z^9 + 2z^8 - 2z^7 - z^6 + 2z^5 + z^4 - z^3 + z^2 - 2) q^15 + (-z^11 + z^10 + z^9 + z^7 + z^4 + 2z^3 - z^2 + 2) q^17 + (z^11 - 2z^9 + z^8 - 2z^7 - 2z^6 + 2z^5 - 2z^3 + z^2 - 3) q^19 + (3z^11 + 2z^10 + z^9 - z^8 - 2z^7 - z^6 - 2z^5 + 4z^4 - z^3 + 3z^2 - z - 3) * q^21 + (z^11 + 2z^10 + 3z^9 - 2z^8 + z^7 + 2z^6 - z^5 + z^4 + 3z^3 + 2z^2 - 2z + 1) q^23 + (-z^9 - z^2) * q^25 - 5z^3 q^27 + (2z^11 + z^10 - 2z^9 + 2z^8 - 2z^6 + z^5 + 2z^4 + 2z - 2) * q^29 + (-z^11 + z^10 + 2z^9 - z^8 + 3z^7 + z^6 + z^5 + z^3 - z^2 - z + 1) * q^31 + (-z^11 + 5z^10 + 4z^9 + z^8 - 2z^4 + z^3 - z^2 + 2z) * q^33 + (-2z^9 - z^7 - z^6 + z^5 + z^4 - 2z^3 + z - 1) * q^35 + (2z^11 - 2z^10 + z^9 + 2z^8 - 5z^6 + 4z^5 + 2z^4 - 6z^3 + 3z^2 + 3z - 1) q^37 + (5z^11 - 5z^9 + 5z^8 - 5z^7 - 5z^6 + 5z^5 + 5z^4 - 5z^3 + 5z - 5) q^39 + (-z^11 + 2z^10 + 5z^9 - 6z^8 + z^7 + 4z^6 - 2z^5 - z^4 + 5z^3 - z^2 - 5z + 6) q^41 + (-5z^11 + z^10 + 4z^9 - 4z^8 - z^7 + 5z^6 - z^5 - 4z^4 + 4z^3 + z^2 - 5z) q^43 + (-z^11 + 3z^10 - z^9 + z^8 + z^7 - z^6 - z^5 + z + 1) q^45 + (-4z^10 + 3z^9 + 5z^8 - 3z^7 - 4z^6 + 2z^5 - 4z^4 - 3z^3 + 5z^2 + 3z - 4) * q^47 + (2z^10 - z^9 - 5z^8 + 4z^6 - 5z^5 + z^4 + 3z^3 - z^2 + 4) q^49 + (z^10 + z^9 + z^6 + 4z^5 + z^4 + z + 1) q^51 + (2z^11 - 3z^10 + 4z^9 + z^6 - 2z^5 - z^4 + z^3 + z + 1) * q^53 + (2z^11 - z^10 - 2z^9 + z^8 - 3z^6 + z^4 - 2z^3 - z^2 + 2z) q^55 + (-3z^11 + 6z^10 - 2z^9 - 2z^8 + 4z^7 - 3z^6 - 4z^5 - z^4 - 2z^2 + z + 1) * q^57 + (z^11 + 3z^10 - 4z^9 + z^8 - z^7 - z^6 + 2z^5 - 2z^4 + 2z^3 - 2z) * q^59 + (z^11 + 2z^10 + 2z^9 + 5z^8 - 6z^7 - 3z^5 - z^4 + 4z^3 + z^2 + 2z - 2) q^61 + (-5z^11 + z^10 + 4z^9 - 3z^8 + 3z^7 + 7z^6 - 8z^5 - 3z^2 - 2z + 5) * q^63 + (z^11 - z^10 - 3z^9 + z^8 + z^7 - z^6 + z^4 - z^2 + z - 1) q^65 + (4z^11 - z^10 - 6z^9 + 4z^8 - 6z^7 - 2z^6 - 3z^5 + 2z^4 - 2z^3 + 3z^2 + 3z - 3) * q^67 + (3z^11 + 3z^9 - z^8 + 5z^7 + 4z^6 - 2z^5 + 4z^4 + z^3 - 2z^2 - 4z + 5) * q^69 + (-3z^11 + 4z^10 - z^9 - 4z^8 + 8z^7 + 3z^6 - 3z^5 + 5z^3 - 5z^2 + z + 7) * q^71 + (3z^11 + 7z^9 - z^8 - z^7 + 3z^6 + 3z^5 - z^4 - z^3 + 7z^2 + 3) q^73 + (z^11 - 2z^9 + 2z^8 - z^7 - 2z^6 + z^5 - z^3 + 2z - 1) * q^75 + (-7z^11 + 2z^10 + 2z^9 - 5z^8 + z^7 + 3z^6 - 3z^5 - 8z^4 + z^3 - 2z^2 + 2z + 5) q^77 + (4z^11 - z^10 - 6z^9 + 3z^8 + z^7 - 7z^6 + 2z^5 + 5z^4 - 2z^3 - z^2 - z - 2) q^79 + (2z^11 - 2z^10 - 2z^9 - 2z^7 - 4z^4 - 4z^3 + 2z^2 - 4) q^81 + (-4z^11 - 6z^10 + 7z^9 - 7z^8 - 6z^7 - z^6 + 2z^5 + 2z^3 + z^2 - 3z - 1) * q^83 + (3z^11 - 2z^10 - 2z^9 + 3z^8 - 2z^6 + z^5 + z^4 - 3z^3 + z - 2) * q^85 + (-3z^11 + 4z^10 + 2z^8 + 4z^7 - z^6 - z^5 - z^4 + z^3 - 4z^2 + 3z + 1) * q^87 + (5z^11 + 2z^10 - 2z^9 - 3z^7 - 4z^6 + 2z^5 + 3z^4 - 2z^2 + 2z - 4) q^89 + (4z^11 - 4z^10 - 4z^9 + 9z^8 - 5z^7 - 6z^6 + 6z^5 + 3z^4 - 8z^3 + 5z^2 + 9z - 7) q^91 + (11z^11 - 2z^10 - z^9 + 2z^8 - z^7 - 6z^6 + 7z^5 + 8z^4 - 4z^3 + 4z^2 + 4z - 8) q^93 + (-z^11 + z^10 - z^6 + z^5 - z^4 + z^3 + z^2 + z + 1) * q^95 + (-6z^11 - z^9 - 6z^8 + z^7 + 5z^6 - 4z^4 + 5z^3 - 2z^2 - 4z + 1) q^97 + (-2z^11 - 2z^10 - 4z^9 - z^8 + z^6 + 4z^5 - 4z^2 - z - 5) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{3} + q^{5} - 14 q^{7} + 13 q^{9}+O(q^{10}) \) 12 * q + 10 * q^3 + q^5 - 14 * q^7 + 13 * q^9	\( 12 q + 10 q^{3} + q^{5} - 14 q^{7} + 13 q^{9} - 9 q^{11} + 2 q^{13} + q^{15} + 12 q^{17} - 7 q^{19} - 14 q^{21} - 9 q^{23} + q^{25} + 10 q^{27} - 6 q^{29} - 16 q^{31} - 6 q^{33} + 7 q^{35} + 22 q^{37} + 25 q^{39} + 24 q^{41} - 37 q^{43} + 13 q^{45} - 15 q^{47} + 28 q^{49} + 15 q^{51} - 3 q^{53} + 18 q^{55} - 7 q^{57} + 15 q^{59} + 7 q^{61} - 8 q^{65} + 32 q^{67} + 12 q^{69} + 12 q^{71} + 35 q^{73} + 10 q^{75} + 21 q^{77} + 11 q^{79} - 26 q^{81} - 9 q^{83} - 3 q^{85} - 12 q^{87} - 6 q^{89} + 14 q^{91} - 34 q^{93} + 14 q^{95} - 34 q^{97} - 60 q^{99}+O(q^{100}) \) 12 * q + 10 * q^3 + q^5 - 14 * q^7 + 13 * q^9 - 9 * q^11 + 2 * q^13 + q^15 + 12 * q^17 - 7 * q^19 - 14 * q^21 - 9 * q^23 + q^25 + 10 * q^27 - 6 * q^29 - 16 * q^31 - 6 * q^33 + 7 * q^35 + 22 * q^37 + 25 * q^39 + 24 * q^41 - 37 * q^43 + 13 * q^45 - 15 * q^47 + 28 * q^49 + 15 * q^51 - 3 * q^53 + 18 * q^55 - 7 * q^57 + 15 * q^59 + 7 * q^61 - 8 * q^65 + 32 * q^67 + 12 * q^69 + 12 * q^71 + 35 * q^73 + 10 * q^75 + 21 * q^77 + 11 * q^79 - 26 * q^81 - 9 * q^83 - 3 * q^85 - 12 * q^87 - 6 * q^89 + 14 * q^91 - 34 * q^93 + 14 * q^95 - 34 * q^97 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more