LMFDB - Newform orbit 483.2.q.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [483,2,Mod(64,483)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("483.64"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	483.q (of order $11$, degree $10$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,5,-1,-1,-5] 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:	$3.85677441763$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$.

$n$	$323$	$346$	$442$
$\chi(n)$	$1$	$1$	$\zeta_{22}^{4}$

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} $

64.1

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

2.30075

0.675560i

−0.654861

0.755750i

3.15455

2.02730i

−0.195368

1.35881i

−2.01722

1.29639i

0.415415

0.909632i

2.74769

3.17101i

−0.142315

−

0.989821i

−1.36745

2.99430i

85.1

1.01255

−

1.16854i

0.841254

−

0.540641i

−0.0556075

−

0.386758i

0.996114

2.18119i

0.220047

−

1.53046i

−0.959493

0.281733i

2.09325

1.34525i

0.415415

−

0.909632i

3.55742

1.04455i

127.1

0.0741615

−

0.0476607i

−0.142315

0.989821i

−0.827602

1.81219i

−1.48357

0.435615i

0.0366213

0.0801894i

−0.654861

−

0.755750i

0.0500861

0.348356i

−0.959493

−

0.281733i

−0.0892619

0.103014i

169.1

−0.317178

2.20602i

0.415415

0.909632i

−2.84695

−

0.835939i

0.0577299

0.0666238i

−2.13843

0.627899i

0.841254

0.540641i

0.895412

−

1.96068i

−0.654861

0.755750i

−0.165284

0.106222i

190.1

−0.570276

1.24873i

−0.959493

−

0.281733i

0.0756100

0.0872586i

−1.87491

1.20493i

0.898983

−

1.03748i

−0.142315

0.989821i

−2.78644

0.818172i

0.841254

0.540641i

−0.435418

−

3.02840i

211.1

−0.570276

−

1.24873i

−0.959493

0.281733i

0.0756100

−

0.0872586i

−1.87491

−

1.20493i

0.898983

1.03748i

−0.142315

−

0.989821i

−2.78644

−

0.818172i

0.841254

−

0.540641i

−0.435418

3.02840i

232.1

0.0741615

0.0476607i

−0.142315

−

0.989821i

−0.827602

−

1.81219i

−1.48357

−

0.435615i

0.0366213

−

0.0801894i

−0.654861

0.755750i

0.0500861

−

0.348356i

−0.959493

0.281733i

−0.0892619

−

0.103014i

358.1

1.01255

1.16854i

0.841254

0.540641i

−0.0556075

0.386758i

0.996114

−

2.18119i

0.220047

1.53046i

−0.959493

−

0.281733i

2.09325

−

1.34525i

0.415415

0.909632i

3.55742

−

1.04455i

400.1

2.30075

−

0.675560i

−0.654861

−

0.755750i

3.15455

−

2.02730i

−0.195368

−

1.35881i

−2.01722

−

1.29639i

0.415415

−

0.909632i

2.74769

−

3.17101i

−0.142315

0.989821i

−1.36745

−

2.99430i

463.1

−0.317178

−

2.20602i

0.415415

−

0.909632i

−2.84695

0.835939i

0.0577299

−

0.0666238i

−2.13843

−

0.627899i

0.841254

−

0.540641i

0.895412

1.96068i

−0.654861

−

0.755750i

−0.165284

−

0.106222i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	483.2.q.b	✓	10
23.c	even	11	1	inner	483.2.q.b	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
483.2.q.b	✓	10	1.a	even	1	1	trivial
483.2.q.b	✓	10	23.c	even	11	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 5T_{2}^{9} + 14T_{2}^{8} - 37T_{2}^{7} + 75T_{2}^{6} - 100T_{2}^{5} + 126T_{2}^{4} - 135T_{2}^{3} + 147T_{2}^{2} - 20T_{2} + 1 $ T2^10 - 5*T2^9 + 14*T2^8 - 37*T2^7 + 75*T2^6 - 100*T2^5 + 126*T2^4 - 135*T2^3 + 147*T2^2 - 20*T2 + 1 acting on $S_{2}^{\mathrm{new}}(483, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 5 T^{9} + \cdots + 1 $ T^10 - 5T^9 + 14T^8 - 37T^7 + 75T^6 - 100T^5 + 126T^4 - 135T^3 + 147T^2 - 20*T + 1
$3$	$ 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
$5$	$ T^{10} + 5 T^{9} + \cdots + 1 $ T^10 + 5T^9 + 14T^8 + 37T^7 + 97T^6 + 177T^5 + 225T^4 + 212T^3 + 103T^2 - 13*T + 1
$7$	$ 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} - 9 T^{9} + \cdots + 1 $ T^10 - 9T^9 + 59T^8 - 223T^7 + 632T^6 - 1244T^5 + 1758T^4 - 1456T^3 + 542T^2 - 38*T + 1
$13$	$ T^{10} + 15 T^{9} + \cdots + 529 $ T^10 + 15T^9 + 104T^8 + 416T^7 + 1147T^6 + 2300T^5 + 3491T^4 + 3976T^3 + 3298T^2 + 1840*T + 529
$17$	$ 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
$19$	$ T^{10} + 19 T^{9} + \cdots + 1849 $ T^10 + 19T^9 + 185T^8 + 1018T^7 + 3546T^6 + 8392T^5 + 13676T^4 + 14885T^3 + 10851T^2 + 5375*T + 1849
$23$	$ T^{10} + 21 T^{9} + \cdots + 6436343 $ T^10 + 21T^9 + 243T^8 + 1913T^7 + 11771T^6 + 60543T^5 + 270733T^4 + 1011977T^3 + 2956581T^2 + 5876661*T + 6436343
$29$	$ T^{10} + 22 T^{8} + \cdots + 121 $ T^10 + 22T^8 - 649T^7 + 517T^6 + 6512T^5 + 56144T^4 - 34969T^3 + 7986T^2 - 242T + 121
$31$	$ T^{10} - 2 T^{9} + \cdots + 1024 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 64T^4 - 128T^3 + 256T^2 - 512*T + 1024
$37$	$ T^{10} + 27 T^{9} + \cdots + 17161 $ T^10 + 27T^9 + 366T^8 + 3095T^7 + 17730T^6 + 68025T^5 + 170351T^4 + 251177T^3 + 166687T^2 + 10349*T + 17161
$41$	$ T^{10} - 10 T^{9} + \cdots + 58081 $ T^10 - 10T^9 + 155T^8 - 934T^7 + 20516T^6 - 55362T^5 + 211663T^4 - 299133T^3 + 2663T^2 + 132309*T + 58081
$43$	$ T^{10} - 15 T^{9} + \cdots + 1849 $ T^10 - 15T^9 + 71T^8 - 207T^7 + 1246T^6 - 1002T^5 + 4514T^4 - 4262T^3 + 12032T^2 + 8170*T + 1849
$47$	$ (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} $ (T^5 + 9T^4 - 5T^3 - 9T^2 + 4T + 1)^2
$53$	$ T^{10} - 19 T^{9} + \cdots + 6285049 $ T^10 - 19T^9 + 251T^8 - 1722T^7 + 15305T^6 - 87647T^5 + 450101T^4 - 1478721T^3 + 3543820T^2 + 10993195*T + 6285049
$59$	$ T^{10} + \cdots + 549011761 $ T^10 + 13T^9 + 15T^8 + 382T^7 + 8651T^6 + 8975T^5 + 11295T^4 + 170045T^3 + 34946090T^2 - 42902161*T + 549011761
$61$	$ T^{10} + \cdots + 389825536 $ T^10 + 18T^9 + 412T^8 + 5392T^7 + 60448T^6 + 319648T^5 + 863680T^4 - 5040128T^3 - 30119168T^2 - 8845312*T + 389825536
$67$	$ T^{10} + 18 T^{9} + \cdots + 529 $ T^10 + 18T^9 + 115T^8 - 108T^7 - 162T^6 - 9186T^5 + 29011T^4 - 20245T^3 + 9733T^2 - 2093*T + 529
$71$	$ T^{10} - 38 T^{9} + \cdots + 12538681 $ T^10 - 38T^9 + 718T^8 - 7418T^7 + 48079T^6 - 273725T^5 + 1612792T^4 - 6689906T^3 + 14414668T^2 - 10173293*T + 12538681
$73$	$ T^{10} - 14 T^{9} + \cdots + 51739249 $ T^10 - 14T^9 + 273T^8 - 3327T^7 + 37866T^6 - 350109T^5 + 2618322T^4 - 13485437T^3 + 43270914T^2 - 74109479*T + 51739249
$79$	$ T^{10} - 25 T^{9} + \cdots + 9765625 $ T^10 - 25T^9 + 350T^8 - 4625T^7 + 60625T^6 - 553125T^5 + 3515625T^4 - 16562500T^3 + 40234375T^2 + 25390625*T + 9765625
$83$	$ T^{10} + 19 T^{9} + \cdots + 69169 $ T^10 + 19T^9 + 185T^8 + 1029T^7 + 4338T^6 + 17984T^5 + 70700T^4 + 138448T^3 + 119014T^2 + 70484*T + 69169
$89$	$ T^{10} + \cdots + 570875449 $ T^10 + 17T^9 + 113T^8 - 4151T^7 - 67839T^6 - 212917T^5 + 9060397T^4 + 116895589T^3 + 584187773T^2 + 278902989*T + 570875449
$97$	$ T^{10} + \cdots + 2220577129 $ T^10 + 49T^9 + 1147T^8 + 16251T^7 + 166197T^6 + 1333949T^5 + 9201758T^4 + 53103823T^3 + 253613188T^2 + 987556711*T + 2220577129
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.q (of order \(11\), degree \(10\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	483.2.q.b

Level	$483$
Weight	$2$
Character orbit	483.q
Analytic conductor	$3.857$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.85677441763\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{22}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 64.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 5 T^{9} + \cdots + 1 \) T^10 - 5T^9 + 14T^8 - 37T^7 + 75T^6 - 100T^5 + 126T^4 - 135T^3 + 147T^2 - 20*T + 1
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} + 5 T^{9} + \cdots + 1 \) T^10 + 5T^9 + 14T^8 + 37T^7 + 97T^6 + 177T^5 + 225T^4 + 212T^3 + 103T^2 - 13*T + 1
$7$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	\( T^{10} - 9 T^{9} + \cdots + 1 \) T^10 - 9T^9 + 59T^8 - 223T^7 + 632T^6 - 1244T^5 + 1758T^4 - 1456T^3 + 542T^2 - 38*T + 1
$13$	\( T^{10} + 15 T^{9} + \cdots + 529 \) T^10 + 15T^9 + 104T^8 + 416T^7 + 1147T^6 + 2300T^5 + 3491T^4 + 3976T^3 + 3298T^2 + 1840*T + 529
$17$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$19$	\( T^{10} + 19 T^{9} + \cdots + 1849 \) T^10 + 19T^9 + 185T^8 + 1018T^7 + 3546T^6 + 8392T^5 + 13676T^4 + 14885T^3 + 10851T^2 + 5375*T + 1849
$23$	\( T^{10} + 21 T^{9} + \cdots + 6436343 \) T^10 + 21T^9 + 243T^8 + 1913T^7 + 11771T^6 + 60543T^5 + 270733T^4 + 1011977T^3 + 2956581T^2 + 5876661*T + 6436343
$29$	\( T^{10} + 22 T^{8} + \cdots + 121 \) T^10 + 22T^8 - 649T^7 + 517T^6 + 6512T^5 + 56144T^4 - 34969T^3 + 7986T^2 - 242T + 121
$31$	\( T^{10} - 2 T^{9} + \cdots + 1024 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 64T^4 - 128T^3 + 256T^2 - 512*T + 1024
$37$	\( T^{10} + 27 T^{9} + \cdots + 17161 \) T^10 + 27T^9 + 366T^8 + 3095T^7 + 17730T^6 + 68025T^5 + 170351T^4 + 251177T^3 + 166687T^2 + 10349*T + 17161
$41$	\( T^{10} - 10 T^{9} + \cdots + 58081 \) T^10 - 10T^9 + 155T^8 - 934T^7 + 20516T^6 - 55362T^5 + 211663T^4 - 299133T^3 + 2663T^2 + 132309*T + 58081
$43$	\( T^{10} - 15 T^{9} + \cdots + 1849 \) T^10 - 15T^9 + 71T^8 - 207T^7 + 1246T^6 - 1002T^5 + 4514T^4 - 4262T^3 + 12032T^2 + 8170*T + 1849
$47$	\( (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} \) (T^5 + 9T^4 - 5T^3 - 9T^2 + 4T + 1)^2
$53$	\( T^{10} - 19 T^{9} + \cdots + 6285049 \) T^10 - 19T^9 + 251T^8 - 1722T^7 + 15305T^6 - 87647T^5 + 450101T^4 - 1478721T^3 + 3543820T^2 + 10993195*T + 6285049
$59$	\( T^{10} + \cdots + 549011761 \) T^10 + 13T^9 + 15T^8 + 382T^7 + 8651T^6 + 8975T^5 + 11295T^4 + 170045T^3 + 34946090T^2 - 42902161*T + 549011761
$61$	\( T^{10} + \cdots + 389825536 \) T^10 + 18T^9 + 412T^8 + 5392T^7 + 60448T^6 + 319648T^5 + 863680T^4 - 5040128T^3 - 30119168T^2 - 8845312*T + 389825536
$67$	\( T^{10} + 18 T^{9} + \cdots + 529 \) T^10 + 18T^9 + 115T^8 - 108T^7 - 162T^6 - 9186T^5 + 29011T^4 - 20245T^3 + 9733T^2 - 2093*T + 529
$71$	\( T^{10} - 38 T^{9} + \cdots + 12538681 \) T^10 - 38T^9 + 718T^8 - 7418T^7 + 48079T^6 - 273725T^5 + 1612792T^4 - 6689906T^3 + 14414668T^2 - 10173293*T + 12538681
$73$	\( T^{10} - 14 T^{9} + \cdots + 51739249 \) T^10 - 14T^9 + 273T^8 - 3327T^7 + 37866T^6 - 350109T^5 + 2618322T^4 - 13485437T^3 + 43270914T^2 - 74109479*T + 51739249
$79$	\( T^{10} - 25 T^{9} + \cdots + 9765625 \) T^10 - 25T^9 + 350T^8 - 4625T^7 + 60625T^6 - 553125T^5 + 3515625T^4 - 16562500T^3 + 40234375T^2 + 25390625*T + 9765625
$83$	\( T^{10} + 19 T^{9} + \cdots + 69169 \) T^10 + 19T^9 + 185T^8 + 1029T^7 + 4338T^6 + 17984T^5 + 70700T^4 + 138448T^3 + 119014T^2 + 70484*T + 69169
$89$	\( T^{10} + \cdots + 570875449 \) T^10 + 17T^9 + 113T^8 - 4151T^7 - 67839T^6 - 212917T^5 + 9060397T^4 + 116895589T^3 + 584187773T^2 + 278902989*T + 570875449
$97$	\( T^{10} + \cdots + 2220577129 \) T^10 + 49T^9 + 1147T^8 + 16251T^7 + 166197T^6 + 1333949T^5 + 9201758T^4 + 53103823T^3 + 253613188T^2 + 987556711*T + 2220577129
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