LMFDB - Newform orbit 1566.2.e.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1566,2,Mod(523,1566)] 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("1566.523"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1566, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1566 = 2 \cdot 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1566.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,6,0,-6,2] 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:	$12.5045729565$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 4x^11 + 6x^10 - 10x^9 + 22x^8 - 18x^7 - 3x^6 - 54x^5 + 198x^4 - 270x^3 + 486x^2 - 972*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 522)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{2} + ( - \beta_{7} - 1) q^{4} + ( - \beta_{9} + \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} - q^{8} + \beta_{4} q^{10} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{6} + 2 \beta_{2} + 1) q^{98}+O(q^{100}) $ q - b7 * q^2 + (-b7 - 1) * q^4 + (-b9 + b4) * q^5 + (-b10 - b7) * q^7 - q^8 + b4 * q^10 + (-b10 + b9 + b7 + b3 + b1) * q^11 + (-b10 + b9 + b8 + b6 - b4 + b2) * q^13 + (-b10 - b7 + b6 - 1) * q^14 + b7 * q^16 + (-b6 - 3) * q^17 + (b6 - b5 + b4 + b2 + 1) * q^19 + b9 * q^20 + (-b10 + b9 + b7 + b6 - b4 + b3 + 1) * q^22 + (-b10 + b7 + b6 + b3 + 1) * q^23 + (-b11 + 2b10 - b9 + b7) q^25 + (b6 - b4 + b2) * q^26 + (b6 - 1) * q^28 + b7 * q^29 + (b11 + b10 - b7 - b6 - b5 + b3 - 1) * q^31 + (b7 + 1) * q^32 + (-b10 + 3b7) q^34 + (-b6 + b5 + 3b4 + b2 + b1 - 2) q^35 + (-b5 - b4 + b2 + 1) * q^37 + (-b11 + b10 + b9 - b8 - b7) * q^38 + (b9 - b4) * q^40 + (2b9 + 2b7 - 2b4 + 2b3 + 2) * q^41 + (-b10 + b8 - b3 - b1) * q^43 + (b6 - b4 - b1 + 1) * q^44 + (b6 - b1 + 1) * q^46 + (b11 - b9 - 2b7 + b3 + b1) q^47 + (-4b10 + 2b8 + b7 + 4b6 + 2b2 + 1) * q^49 + (-b11 + 2b10 - b9 + b7 - 2b6 + b5 + b4 + 1) * q^50 + (b10 - b9 - b8) * q^52 + (-b6 + 2b4 + b2 - 4) q^53 + (-4b6 + 3b5 + b4 + b1 + 2) * q^55 + (b10 + b7) * q^56 + (b7 + 1) * q^58 + (-b10 + 2b8 + b7 + b6 + 2b2 + 1) * q^59 + (b11 + b9 + b8 + 5b7 + b3 + b1) q^61 + (-b6 - b5 - b1 - 1) * q^62 + q^64 + (3b11 - 2b10 + 2b9 - 5b7 + b3 + b1) * q^65 + (-b11 + 3b10 - b9 - 3b8 - 2b7 - 3b6 + b5 + b4 - 3b2 - 2) q^67 + (-b10 + 3b7 + b6 + 3) q^68 + (b11 - b10 + 3b9 - b8 + 2b7 + b3 + b1) * q^70 + (b5 + b4 + b2 - 1) * q^71 + (-b5 + b4 - b2 - 2b1 - 1) q^73 + (-b11 - b9 - b8 - b7) * q^74 + (-b11 + b10 + b9 - b8 - b7 - b6 + b5 - b4 - b2 - 1) * q^76 + (b11 - 3b10 + 3b9 + b8 + 3b6 - b5 - 3b4 + b3 + b2) * q^77 + (b11 + b10 - 2b8 - b7 + b3 + b1) q^79 - b4 * q^80 + (-2b4 - 2b1 + 2) * q^82 + (-b11 - b10 - b9 + b8 + b3 + b1) * q^83 + (-b11 + b10 + b9 + b8 - 2b7 - b6 + b5 - b4 - b3 + b2 - 2) q^85 + (-b10 + b8 + b6 - b3 + b2) * q^86 + (b10 - b9 - b7 - b3 - b1) * q^88 + (-4b6 + 2b4 + b1 - 4) * q^89 + (5b6 - 2b4 - b1 - 1) * q^91 + (b10 - b7 - b3 - b1) * q^92 + (b11 - b9 - 2b7 - b5 + b4 + b3 - 2) q^94 + (-b10 + b9 + 2b8 + 9b7 + b6 - b4 + 2b3 + 2b2 + 9) * q^95 + (-3b11 + 3b10 - b9 - 3b8 - 2b7 - b3 - b1) * q^97 + (4b6 + 2b2 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} - 6 q^{4} + 2 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} - 6 q^{13} - 8 q^{14} - 6 q^{16} - 32 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{23} - 14 q^{25} - 12 q^{26} - 16 q^{28} - 6 q^{29} - 8 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 - 6 * q^4 + 2 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 - 6 * q^13 - 8 * q^14 - 6 * q^16 - 32 * q^17 + 4 * q^19 + 2 * q^20 + 2 * q^23 - 14 * q^25 - 12 * q^26 - 16 * q^28 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 16 * q^34 - 4 * q^35 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 2 * q^43 + 4 * q^46 + 14 * q^47 - 6 * q^49 + 14 * q^50 - 6 * q^52 - 40 * q^53 + 60 * q^55 - 8 * q^56 + 6 * q^58 - 22 * q^61 - 16 * q^62 + 12 * q^64 + 46 * q^65 + 4 * q^67 + 16 * q^68 - 2 * q^70 - 8 * q^71 - 16 * q^73 - 2 * q^76 - 18 * q^77 + 4 * q^79 - 4 * q^80 + 8 * q^82 + 2 * q^83 - 10 * q^85 - 2 * q^86 - 20 * q^89 - 44 * q^91 + 2 * q^92 - 14 * q^94 + 42 * q^95 - 10 * q^97 - 12 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 4*x^11 + 6*x^10 - 10*x^9 + 22*x^8 - 18*x^7 - 3*x^6 - 54*x^5 + 198*x^4 - 270*x^3 + 486*x^2 - 972*x + 729 :

$\beta_{1}$	$=$	$ ( \nu^{11} - 3 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 15 \nu^{7} + 6 \nu^{6} - 8 \nu^{5} - 81 \nu^{4} + \cdots - 567 ) / 81 $ (v^11 - 3v^10 + 3v^9 - 8v^8 + 15v^7 + 6v^6 - 8v^5 - 81v^4 + 102v^3 - 135v^2 + 540v - 567) / 81
$\beta_{2}$	$=$	$ ( - 7 \nu^{11} + 16 \nu^{10} - 15 \nu^{9} + 55 \nu^{8} - 70 \nu^{7} - 18 \nu^{6} - 9 \nu^{5} + \cdots + 2916 ) / 243 $ (-7v^11 + 16v^10 - 15v^9 + 55v^8 - 70v^7 - 18v^6 - 9v^5 + 405v^4 - 666v^3 + 675v^2 - 2835*v + 2916) / 243
$\beta_{3}$	$=$	$ ( - 8 \nu^{11} + 17 \nu^{10} - 12 \nu^{9} + 59 \nu^{8} - 98 \nu^{7} - 45 \nu^{6} + 9 \nu^{5} + \cdots + 3645 ) / 243 $ (-8v^11 + 17v^10 - 12v^9 + 59v^8 - 98v^7 - 45v^6 + 9v^5 + 486v^4 - 603v^3 + 621v^2 - 2835*v + 3645) / 243
$\beta_{4}$	$=$	$ ( 7 \nu^{11} - 22 \nu^{10} + 39 \nu^{9} - 91 \nu^{8} + 103 \nu^{7} - 6 \nu^{6} + 36 \nu^{5} + \cdots - 4617 ) / 243 $ (7v^11 - 22v^10 + 39v^9 - 91v^8 + 103v^7 - 6v^6 + 36v^5 - 441v^4 + 882v^3 - 1701v^2 + 4374*v - 4617) / 243
$\beta_{5}$	$=$	$ ( 13 \nu^{11} - 31 \nu^{10} + 15 \nu^{9} - 61 \nu^{8} + 139 \nu^{7} - 90 \nu^{5} - 702 \nu^{4} + \cdots - 5103 ) / 243 $ (13v^11 - 31v^10 + 15v^9 - 61v^8 + 139v^7 - 90v^5 - 702v^4 + 1476v^3 - 918v^2 + 2916v - 5103) / 243
$\beta_{6}$	$=$	$ ( 13 \nu^{11} - 31 \nu^{10} + 18 \nu^{9} - 73 \nu^{8} + 148 \nu^{7} + 6 \nu^{6} - 51 \nu^{5} + \cdots - 5589 ) / 243 $ (13v^11 - 31v^10 + 18v^9 - 73v^8 + 148v^7 + 6v^6 - 51v^5 - 774v^4 + 1350v^3 - 864v^2 + 3645*v - 5589) / 243
$\beta_{7}$	$=$	$ ( - 16 \nu^{11} + 34 \nu^{10} - 30 \nu^{9} + 106 \nu^{8} - 178 \nu^{7} - 21 \nu^{6} + 30 \nu^{5} + \cdots + 6804 ) / 243 $ (-16v^11 + 34v^10 - 30v^9 + 106v^8 - 178v^7 - 21v^6 + 30v^5 + 918v^4 - 1494v^3 + 1350v^2 - 5184*v + 6804) / 243
$\beta_{8}$	$=$	$ ( 32 \nu^{11} - 71 \nu^{10} + 69 \nu^{9} - 200 \nu^{8} + 305 \nu^{7} + 24 \nu^{6} - 9 \nu^{5} + \cdots - 11664 ) / 243 $ (32v^11 - 71v^10 + 69v^9 - 200v^8 + 305v^7 + 24v^6 - 9v^5 - 1764v^4 + 3087v^3 - 3375v^2 + 10044*v - 11664) / 243
$\beta_{9}$	$=$	$ ( - 13 \nu^{11} + 29 \nu^{10} - 23 \nu^{9} + 74 \nu^{8} - 137 \nu^{7} - 13 \nu^{6} + 29 \nu^{5} + \cdots + 5184 ) / 81 $ (-13v^11 + 29v^10 - 23v^9 + 74v^8 - 137v^7 - 13v^6 + 29v^5 + 738v^4 - 1317v^3 + 1089v^2 - 3699*v + 5184) / 81
$\beta_{10}$	$=$	$ ( - 20 \nu^{11} + 45 \nu^{10} - 42 \nu^{9} + 133 \nu^{8} - 225 \nu^{7} - 21 \nu^{6} + 25 \nu^{5} + \cdots + 8748 ) / 81 $ (-20v^11 + 45v^10 - 42v^9 + 133v^8 - 225v^7 - 21v^6 + 25v^5 + 1137v^4 - 1986v^3 + 1953v^2 - 6588*v + 8748) / 81
$\beta_{11}$	$=$	$ ( - 83 \nu^{11} + 197 \nu^{10} - 195 \nu^{9} + 527 \nu^{8} - 881 \nu^{7} - 33 \nu^{6} + 93 \nu^{5} + \cdots + 34020 ) / 243 $ (-83v^11 + 197v^10 - 195v^9 + 527v^8 - 881v^7 - 33v^6 + 93v^5 + 4599v^4 - 8613v^3 + 9045v^2 - 26649*v + 34020) / 243

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{10} + \beta_{9} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 3 $ (-b10 + b9 + b7 - b4 + b3 - b2 + b1 + 1) / 3
$\nu^{2}$	$=$	$ ( -\beta_{11} + 2\beta_{10} - \beta_{9} - 2\beta_{8} - 4\beta_{7} - \beta_{5} - 2\beta_{2} - \beta_1 ) / 3 $ (-b11 + 2b10 - b9 - 2b8 - 4b7 - b5 - 2b2 - b1) / 3
$\nu^{3}$	$=$	$ ( \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 5 ) / 3 $ (b11 - b10 - 3b9 - b8 + 3b7 - 3b6 + b5 + b4 + 2b3 - 3*b2 + 5) / 3
$\nu^{4}$	$=$	$ ( - 3 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} - 4 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + \cdots + 2 ) / 3 $ (-3b11 - 2b10 + 6b9 - 4b8 - 4b6 + 3b5 - 5b4 - b3 - 6b2 - 3*b1 + 2) / 3
$\nu^{5}$	$=$	$ ( 2 \beta_{11} - 10 \beta_{10} - \beta_{9} + \beta_{8} + 29 \beta_{7} + 9 \beta_{6} - 10 \beta_{5} + \cdots + 15 ) / 3 $ (2b11 - 10b10 - b9 + b8 + 29b7 + 9b6 - 10b5 - 3b4 - 2b2 - 7b1 + 15) / 3
$\nu^{6}$	$=$	$ ( - 2 \beta_{11} + 8 \beta_{10} - 18 \beta_{9} - 10 \beta_{8} + 3 \beta_{7} - 24 \beta_{6} + 4 \beta_{5} + \cdots - 4 ) / 3 $ (-2b11 + 8b10 - 18b9 - 10b8 + 3b7 - 24b6 + 4b5 + 4b4 - 16b3 - 12b2 - 4) / 3
$\nu^{7}$	$=$	$ ( - 23 \beta_{10} + 12 \beta_{9} - 7 \beta_{8} + 30 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 11 \beta_{4} + \cdots + 119 ) / 3 $ (-23b10 + 12b9 - 7b8 + 30b7 + 5b6 - 6b5 - 11b4 - b3 + 18b2 + 119) / 3
$\nu^{8}$	$=$	$ ( - 4 \beta_{11} - 46 \beta_{10} + 29 \beta_{9} - 2 \beta_{8} + 74 \beta_{7} - 19 \beta_{5} - 51 \beta_{4} + \cdots - 6 ) / 3 $ (-4b11 - 46b10 + 29b9 - 2b8 + 74b7 - 19b5 - 51b4 + 33b3 - 2b2 + 47b1 - 6) / 3
$\nu^{9}$	$=$	$ ( - 50 \beta_{11} + 116 \beta_{10} - 78 \beta_{9} - 82 \beta_{8} - 222 \beta_{7} - 63 \beta_{6} + \cdots - 67 ) / 3 $ (-50b11 + 116b10 - 78b9 - 82b8 - 222b7 - 63b6 - 47b5 + 25b4 - 16b3 - 3b2 - 18*b1 - 67) / 3
$\nu^{10}$	$=$	$ ( 66 \beta_{11} - 86 \beta_{10} - 69 \beta_{9} + 20 \beta_{8} - 172 \beta_{6} + 21 \beta_{5} + 4 \beta_{4} + \cdots + 80 ) / 3 $ (66b11 - 86b10 - 69b9 + 20b8 - 172b6 + 21b5 + 4b4 + 131b3 + 30b2 + 60b1 + 80) / 3
$\nu^{11}$	$=$	$ ( - 136 \beta_{11} - 7 \beta_{10} + 296 \beta_{9} - 29 \beta_{8} - 364 \beta_{7} - 204 \beta_{6} + \cdots - 435 ) / 3 $ (-136b11 - 7b10 + 296b9 - 29b8 - 364b7 - 204b6 + 44b5 - 321b4 - 9b3 - 41b2 - 121*b1 - 435) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1566\mathbb{Z}\right)^\times$.

$n$	$379$	$407$
$\chi(n)$	$1$	$-1 - \beta_{7}$

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} $

523.1

−0.452567	−	1.67188i
1.73173	+	0.0331860i
1.16678	+	1.28009i
1.64965	−	0.527869i
−1.55661	−	0.759577i
−0.538988	+	1.64605i
−0.452567	+	1.67188i
1.73173	−	0.0331860i
1.16678	−	1.28009i
1.64965	+	0.527869i
−1.55661	+	0.759577i
−0.538988	−	1.64605i

0.500000

0.866025i

−0.500000

0.866025i

−1.58186

2.73987i

0.567358

0.982693i

−1.00000

−3.16372

523.2

0.500000

0.866025i

−0.500000

0.866025i

−1.26263

2.18694i

2.51863

4.36239i

−1.00000

−2.52526

523.3

0.500000

0.866025i

−0.500000

0.866025i

−0.353519

0.612313i

−0.134930

−

0.233706i

−1.00000

−0.707038

523.4

0.500000

0.866025i

−0.500000

0.866025i

0.851534

−

1.47490i

−1.18704

−

2.05602i

−1.00000

1.70307

523.5

0.500000

0.866025i

−0.500000

0.866025i

1.19682

−

2.07295i

0.279091

0.483399i

−1.00000

2.39363

523.6

0.500000

0.866025i

−0.500000

0.866025i

2.14966

−

3.72332i

1.95690

3.38944i

−1.00000

4.29932

1045.1

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.58186

−

2.73987i

0.567358

−

0.982693i

−1.00000

−3.16372

1045.2

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.26263

−

2.18694i

2.51863

−

4.36239i

−1.00000

−2.52526

1045.3

0.500000

−

0.866025i

−0.500000

−

0.866025i

−0.353519

−

0.612313i

−0.134930

0.233706i

−1.00000

−0.707038

1045.4

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.851534

1.47490i

−1.18704

2.05602i

−1.00000

1.70307

1045.5

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.19682

2.07295i

0.279091

−

0.483399i

−1.00000

2.39363

1045.6

0.500000

−

0.866025i

−0.500000

−

0.866025i

2.14966

3.72332i

1.95690

−

3.38944i

−1.00000

4.29932

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1566.2.e.h		12
3.b	odd	2	1		522.2.e.h	✓	12
9.c	even	3	1	inner	1566.2.e.h		12
9.c	even	3	1		4698.2.a.be		6
9.d	odd	6	1		522.2.e.h	✓	12
9.d	odd	6	1		4698.2.a.bh		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
522.2.e.h	✓	12	3.b	odd	2	1
522.2.e.h	✓	12	9.d	odd	6	1
1566.2.e.h		12	1.a	even	1	1	trivial
1566.2.e.h		12	9.c	even	3	1	inner
4698.2.a.be		6	9.c	even	3	1
4698.2.a.bh		6	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1566, [\chi])$:

$ T_{5}^{12} - 2 T_{5}^{11} + 24 T_{5}^{10} - 12 T_{5}^{9} + 351 T_{5}^{8} - 204 T_{5}^{7} + 2322 T_{5}^{6} + \cdots + 9801 $

T5^12 - 2*T5^11 + 24*T5^10 - 12*T5^9 + 351*T5^8 - 204*T5^7 + 2322*T5^6 - 696*T5^5 + 10509*T5^4 - 3740*T5^3 + 17743*T5^2 + 8712*T5 + 9801

$ T_{7}^{12} - 8 T_{7}^{11} + 56 T_{7}^{10} - 160 T_{7}^{9} + 516 T_{7}^{8} - 696 T_{7}^{7} + 2928 T_{7}^{6} + \cdots + 64 $

T7^12 - 8*T7^11 + 56*T7^10 - 160*T7^9 + 516*T7^8 - 696*T7^7 + 2928*T7^6 - 3456*T7^5 + 4176*T7^4 - 1312*T7^3 + 608*T7^2 + 64*T7 + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ T^{12} - 2 T^{11} + \cdots + 9801 $ T^12 - 2T^11 + 24T^10 - 12T^9 + 351T^8 - 204T^7 + 2322T^6 - 696T^5 + 10509T^4 - 3740T^3 + 17743T^2 + 8712*T + 9801
$7$	$ T^{12} - 8 T^{11} + \cdots + 64 $ T^12 - 8T^11 + 56T^10 - 160T^9 + 516T^8 - 696T^7 + 2928T^6 - 3456T^5 + 4176T^4 - 1312T^3 + 608T^2 + 64*T + 64
$11$	$ T^{12} + 48 T^{10} + \cdots + 6561 $ T^12 + 48T^10 + 36T^9 + 1809T^8 + 1512T^7 + 23922T^6 + 53298T^5 + 252801T^4 + 317844T^3 + 459999T^2 - 52488T + 6561
$13$	$ T^{12} + 6 T^{11} + \cdots + 121 $ T^12 + 6T^11 + 72T^10 + 244T^9 + 2553T^8 + 8058T^7 + 49782T^6 + 61932T^5 + 303153T^4 + 149182T^3 + 1573869T^2 - 13794*T + 121
$17$	$ (T^{6} + 16 T^{5} + \cdots - 24)^{2} $ (T^6 + 16T^5 + 88T^4 + 176T^3 - 4T^2 - 280*T - 24)^2
$19$	$ (T^{6} - 2 T^{5} + \cdots - 1233)^{2} $ (T^6 - 2T^5 - 63T^4 + 60T^3 + 735T^2 + 174*T - 1233)^2
$23$	$ T^{12} - 2 T^{11} + \cdots + 788544 $ T^12 - 2T^11 + 42T^10 - 60T^9 + 1236T^8 - 1584T^7 + 15168T^6 - 3408T^5 + 110160T^4 - 8576T^3 + 446848T^2 + 333888*T + 788544
$29$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$31$	$ T^{12} + 8 T^{11} + \cdots + 531441 $ T^12 + 8T^11 + 148T^10 + 816T^9 + 12207T^8 + 60414T^7 + 533490T^6 + 1201536T^5 + 8566461T^4 + 7513182T^3 + 115802685T^2 - 7825086*T + 531441
$37$	$ (T^{6} - 90 T^{4} + \cdots - 344)^{2} $ (T^6 - 90T^4 + 140T^3 + 1416T^2 - 2856T - 344)^2
$41$	$ T^{12} + \cdots + 6410244096 $ T^12 - 4T^11 + 204T^10 - 432T^9 + 29808T^8 - 60096T^7 + 1628160T^6 + 475392T^5 + 54544128T^4 - 305152T^3 + 777594880T^2 + 963330048*T + 6410244096
$43$	$ T^{12} - 2 T^{11} + \cdots + 63001 $ T^12 - 2T^11 + 68T^10 - 472T^9 + 5055T^8 - 20736T^7 + 67326T^6 - 121002T^5 + 174945T^4 - 114700T^3 + 100109T^2 - 25100*T + 63001
$47$	$ T^{12} - 14 T^{11} + \cdots + 463761 $ T^12 - 14T^11 + 204T^10 - 996T^9 + 8445T^8 - 23832T^7 + 273954T^6 - 367116T^5 + 1437777T^4 + 1942048T^3 + 3184375T^2 + 1293900*T + 463761
$53$	$ (T^{6} + 20 T^{5} + \cdots + 66189)^{2} $ (T^6 + 20T^5 + 34T^4 - 1496T^3 - 9133T^2 - 734*T + 66189)^2
$59$	$ T^{12} + 96 T^{10} + \cdots + 5645376 $ T^12 + 96T^10 - 336T^9 + 7044T^8 - 22680T^7 + 241488T^6 - 893088T^5 + 6046416T^4 - 15029280T^3 + 37768032T^2 - 15567552T + 5645376
$61$	$ T^{12} + 22 T^{11} + \cdots + 7744 $ T^12 + 22T^11 + 392T^10 + 3440T^9 + 28716T^8 + 136704T^7 + 1017168T^6 + 4027008T^5 + 19109040T^4 + 18130496T^3 + 14829728T^2 + 343552*T + 7744
$67$	$ T^{12} + \cdots + 158703437376 $ T^12 - 4T^11 + 250T^10 - 360T^9 + 39984T^8 - 37536T^7 + 3587064T^6 + 1231488T^5 + 224369280T^4 + 85390272T^3 + 7533234720T^2 + 9886098816*T + 158703437376
$71$	$ (T^{6} + 4 T^{5} + \cdots - 8232)^{2} $ (T^6 + 4T^5 - 98T^4 - 172T^3 + 2072T^2 + 2744*T - 8232)^2
$73$	$ (T^{6} + 8 T^{5} + \cdots + 14696)^{2} $ (T^6 + 8T^5 - 178T^4 - 420T^3 + 9208T^2 - 25448*T + 14696)^2
$79$	$ T^{12} - 4 T^{11} + \cdots + 81 $ T^12 - 4T^11 + 196T^10 - 672T^9 + 29235T^8 - 92622T^7 + 1495482T^6 - 1235700T^5 + 45783045T^4 - 88829838T^3 + 223077897T^2 + 134406*T + 81
$83$	$ T^{12} + \cdots + 153958464 $ T^12 - 2T^11 + 192T^10 + 864T^9 + 26100T^8 + 109704T^7 + 1738464T^6 + 8720160T^5 + 81363984T^4 + 246187744T^3 + 938545312T^2 - 357449664*T + 153958464
$89$	$ (T^{6} + 10 T^{5} + \cdots + 227064)^{2} $ (T^6 + 10T^5 - 266T^4 - 3064T^3 + 5696T^2 + 111920*T + 227064)^2
$97$	$ T^{12} + \cdots + 309750128704 $ T^12 + 10T^11 + 452T^10 + 2312T^9 + 125076T^8 + 576936T^7 + 16139088T^6 - 9552672T^5 + 908951664T^4 - 159742432T^3 + 27734927072T^2 - 61367649728*T + 309750128704
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1566 = 2 \cdot 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1566.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + ( - \beta_{7} - 1) q^{4} + ( - \beta_{9} + \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} - q^{8} + \beta_{4} q^{10} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{6} + 2 \beta_{2} + 1) q^{98}+O(q^{100}) \) q - b7 * q^2 + (-b7 - 1) * q^4 + (-b9 + b4) * q^5 + (-b10 - b7) * q^7 - q^8 + b4 * q^10 + (-b10 + b9 + b7 + b3 + b1) * q^11 + (-b10 + b9 + b8 + b6 - b4 + b2) * q^13 + (-b10 - b7 + b6 - 1) * q^14 + b7 * q^16 + (-b6 - 3) * q^17 + (b6 - b5 + b4 + b2 + 1) * q^19 + b9 * q^20 + (-b10 + b9 + b7 + b6 - b4 + b3 + 1) * q^22 + (-b10 + b7 + b6 + b3 + 1) * q^23 + (-b11 + 2b10 - b9 + b7) q^25 + (b6 - b4 + b2) * q^26 + (b6 - 1) * q^28 + b7 * q^29 + (b11 + b10 - b7 - b6 - b5 + b3 - 1) * q^31 + (b7 + 1) * q^32 + (-b10 + 3b7) q^34 + (-b6 + b5 + 3b4 + b2 + b1 - 2) q^35 + (-b5 - b4 + b2 + 1) * q^37 + (-b11 + b10 + b9 - b8 - b7) * q^38 + (b9 - b4) * q^40 + (2b9 + 2b7 - 2b4 + 2b3 + 2) * q^41 + (-b10 + b8 - b3 - b1) * q^43 + (b6 - b4 - b1 + 1) * q^44 + (b6 - b1 + 1) * q^46 + (b11 - b9 - 2b7 + b3 + b1) q^47 + (-4b10 + 2b8 + b7 + 4b6 + 2b2 + 1) * q^49 + (-b11 + 2b10 - b9 + b7 - 2b6 + b5 + b4 + 1) * q^50 + (b10 - b9 - b8) * q^52 + (-b6 + 2b4 + b2 - 4) q^53 + (-4b6 + 3b5 + b4 + b1 + 2) * q^55 + (b10 + b7) * q^56 + (b7 + 1) * q^58 + (-b10 + 2b8 + b7 + b6 + 2b2 + 1) * q^59 + (b11 + b9 + b8 + 5b7 + b3 + b1) q^61 + (-b6 - b5 - b1 - 1) * q^62 + q^64 + (3b11 - 2b10 + 2b9 - 5b7 + b3 + b1) * q^65 + (-b11 + 3b10 - b9 - 3b8 - 2b7 - 3b6 + b5 + b4 - 3b2 - 2) q^67 + (-b10 + 3b7 + b6 + 3) q^68 + (b11 - b10 + 3b9 - b8 + 2b7 + b3 + b1) * q^70 + (b5 + b4 + b2 - 1) * q^71 + (-b5 + b4 - b2 - 2b1 - 1) q^73 + (-b11 - b9 - b8 - b7) * q^74 + (-b11 + b10 + b9 - b8 - b7 - b6 + b5 - b4 - b2 - 1) * q^76 + (b11 - 3b10 + 3b9 + b8 + 3b6 - b5 - 3b4 + b3 + b2) * q^77 + (b11 + b10 - 2b8 - b7 + b3 + b1) q^79 - b4 * q^80 + (-2b4 - 2b1 + 2) * q^82 + (-b11 - b10 - b9 + b8 + b3 + b1) * q^83 + (-b11 + b10 + b9 + b8 - 2b7 - b6 + b5 - b4 - b3 + b2 - 2) q^85 + (-b10 + b8 + b6 - b3 + b2) * q^86 + (b10 - b9 - b7 - b3 - b1) * q^88 + (-4b6 + 2b4 + b1 - 4) * q^89 + (5b6 - 2b4 - b1 - 1) * q^91 + (b10 - b7 - b3 - b1) * q^92 + (b11 - b9 - 2b7 - b5 + b4 + b3 - 2) q^94 + (-b10 + b9 + 2b8 + 9b7 + b6 - b4 + 2b3 + 2b2 + 9) * q^95 + (-3b11 + 3b10 - b9 - 3b8 - 2b7 - b3 - b1) * q^97 + (4b6 + 2b2 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} - 6 q^{4} + 2 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} - 6 q^{13} - 8 q^{14} - 6 q^{16} - 32 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{23} - 14 q^{25} - 12 q^{26} - 16 q^{28} - 6 q^{29} - 8 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 - 6 * q^4 + 2 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 - 6 * q^13 - 8 * q^14 - 6 * q^16 - 32 * q^17 + 4 * q^19 + 2 * q^20 + 2 * q^23 - 14 * q^25 - 12 * q^26 - 16 * q^28 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 16 * q^34 - 4 * q^35 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 2 * q^43 + 4 * q^46 + 14 * q^47 - 6 * q^49 + 14 * q^50 - 6 * q^52 - 40 * q^53 + 60 * q^55 - 8 * q^56 + 6 * q^58 - 22 * q^61 - 16 * q^62 + 12 * q^64 + 46 * q^65 + 4 * q^67 + 16 * q^68 - 2 * q^70 - 8 * q^71 - 16 * q^73 - 2 * q^76 - 18 * q^77 + 4 * q^79 - 4 * q^80 + 8 * q^82 + 2 * q^83 - 10 * q^85 - 2 * q^86 - 20 * q^89 - 44 * q^91 + 2 * q^92 - 14 * q^94 + 42 * q^95 - 10 * q^97 - 12 * q^98

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	1566.2.e.h

Level	$1566$
Weight	$2$
Character orbit	1566.e
Analytic conductor	$12.505$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(12.5045729565\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 \) x^12 - 4x^11 + 6x^10 - 10x^9 + 22x^8 - 18x^7 - 3x^6 - 54x^5 + 198x^4 - 270x^3 + 486x^2 - 972*x + 729
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	no (minimal twist has level 522)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} - 3 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 15 \nu^{7} + 6 \nu^{6} - 8 \nu^{5} - 81 \nu^{4} + \cdots - 567 ) / 81 \) (v^11 - 3v^10 + 3v^9 - 8v^8 + 15v^7 + 6v^6 - 8v^5 - 81v^4 + 102v^3 - 135v^2 + 540v - 567) / 81
\(\beta_{2}\)	\(=\)	\( ( - 7 \nu^{11} + 16 \nu^{10} - 15 \nu^{9} + 55 \nu^{8} - 70 \nu^{7} - 18 \nu^{6} - 9 \nu^{5} + \cdots + 2916 ) / 243 \) (-7v^11 + 16v^10 - 15v^9 + 55v^8 - 70v^7 - 18v^6 - 9v^5 + 405v^4 - 666v^3 + 675v^2 - 2835*v + 2916) / 243
\(\beta_{3}\)	\(=\)	\( ( - 8 \nu^{11} + 17 \nu^{10} - 12 \nu^{9} + 59 \nu^{8} - 98 \nu^{7} - 45 \nu^{6} + 9 \nu^{5} + \cdots + 3645 ) / 243 \) (-8v^11 + 17v^10 - 12v^9 + 59v^8 - 98v^7 - 45v^6 + 9v^5 + 486v^4 - 603v^3 + 621v^2 - 2835*v + 3645) / 243
\(\beta_{4}\)	\(=\)	\( ( 7 \nu^{11} - 22 \nu^{10} + 39 \nu^{9} - 91 \nu^{8} + 103 \nu^{7} - 6 \nu^{6} + 36 \nu^{5} + \cdots - 4617 ) / 243 \) (7v^11 - 22v^10 + 39v^9 - 91v^8 + 103v^7 - 6v^6 + 36v^5 - 441v^4 + 882v^3 - 1701v^2 + 4374*v - 4617) / 243
\(\beta_{5}\)	\(=\)	\( ( 13 \nu^{11} - 31 \nu^{10} + 15 \nu^{9} - 61 \nu^{8} + 139 \nu^{7} - 90 \nu^{5} - 702 \nu^{4} + \cdots - 5103 ) / 243 \) (13v^11 - 31v^10 + 15v^9 - 61v^8 + 139v^7 - 90v^5 - 702v^4 + 1476v^3 - 918v^2 + 2916v - 5103) / 243
\(\beta_{6}\)	\(=\)	\( ( 13 \nu^{11} - 31 \nu^{10} + 18 \nu^{9} - 73 \nu^{8} + 148 \nu^{7} + 6 \nu^{6} - 51 \nu^{5} + \cdots - 5589 ) / 243 \) (13v^11 - 31v^10 + 18v^9 - 73v^8 + 148v^7 + 6v^6 - 51v^5 - 774v^4 + 1350v^3 - 864v^2 + 3645*v - 5589) / 243
\(\beta_{7}\)	\(=\)	\( ( - 16 \nu^{11} + 34 \nu^{10} - 30 \nu^{9} + 106 \nu^{8} - 178 \nu^{7} - 21 \nu^{6} + 30 \nu^{5} + \cdots + 6804 ) / 243 \) (-16v^11 + 34v^10 - 30v^9 + 106v^8 - 178v^7 - 21v^6 + 30v^5 + 918v^4 - 1494v^3 + 1350v^2 - 5184*v + 6804) / 243
\(\beta_{8}\)	\(=\)	\( ( 32 \nu^{11} - 71 \nu^{10} + 69 \nu^{9} - 200 \nu^{8} + 305 \nu^{7} + 24 \nu^{6} - 9 \nu^{5} + \cdots - 11664 ) / 243 \) (32v^11 - 71v^10 + 69v^9 - 200v^8 + 305v^7 + 24v^6 - 9v^5 - 1764v^4 + 3087v^3 - 3375v^2 + 10044*v - 11664) / 243
\(\beta_{9}\)	\(=\)	\( ( - 13 \nu^{11} + 29 \nu^{10} - 23 \nu^{9} + 74 \nu^{8} - 137 \nu^{7} - 13 \nu^{6} + 29 \nu^{5} + \cdots + 5184 ) / 81 \) (-13v^11 + 29v^10 - 23v^9 + 74v^8 - 137v^7 - 13v^6 + 29v^5 + 738v^4 - 1317v^3 + 1089v^2 - 3699*v + 5184) / 81
\(\beta_{10}\)	\(=\)	\( ( - 20 \nu^{11} + 45 \nu^{10} - 42 \nu^{9} + 133 \nu^{8} - 225 \nu^{7} - 21 \nu^{6} + 25 \nu^{5} + \cdots + 8748 ) / 81 \) (-20v^11 + 45v^10 - 42v^9 + 133v^8 - 225v^7 - 21v^6 + 25v^5 + 1137v^4 - 1986v^3 + 1953v^2 - 6588*v + 8748) / 81
\(\beta_{11}\)	\(=\)	\( ( - 83 \nu^{11} + 197 \nu^{10} - 195 \nu^{9} + 527 \nu^{8} - 881 \nu^{7} - 33 \nu^{6} + 93 \nu^{5} + \cdots + 34020 ) / 243 \) (-83v^11 + 197v^10 - 195v^9 + 527v^8 - 881v^7 - 33v^6 + 93v^5 + 4599v^4 - 8613v^3 + 9045v^2 - 26649*v + 34020) / 243

\(\nu\)	\(=\)	\( ( -\beta_{10} + \beta_{9} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 3 \) (-b10 + b9 + b7 - b4 + b3 - b2 + b1 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} + 2\beta_{10} - \beta_{9} - 2\beta_{8} - 4\beta_{7} - \beta_{5} - 2\beta_{2} - \beta_1 ) / 3 \) (-b11 + 2b10 - b9 - 2b8 - 4b7 - b5 - 2b2 - b1) / 3
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 5 ) / 3 \) (b11 - b10 - 3b9 - b8 + 3b7 - 3b6 + b5 + b4 + 2b3 - 3*b2 + 5) / 3
\(\nu^{4}\)	\(=\)	\( ( - 3 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} - 4 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + \cdots + 2 ) / 3 \) (-3b11 - 2b10 + 6b9 - 4b8 - 4b6 + 3b5 - 5b4 - b3 - 6b2 - 3*b1 + 2) / 3
\(\nu^{5}\)	\(=\)	\( ( 2 \beta_{11} - 10 \beta_{10} - \beta_{9} + \beta_{8} + 29 \beta_{7} + 9 \beta_{6} - 10 \beta_{5} + \cdots + 15 ) / 3 \) (2b11 - 10b10 - b9 + b8 + 29b7 + 9b6 - 10b5 - 3b4 - 2b2 - 7b1 + 15) / 3
\(\nu^{6}\)	\(=\)	\( ( - 2 \beta_{11} + 8 \beta_{10} - 18 \beta_{9} - 10 \beta_{8} + 3 \beta_{7} - 24 \beta_{6} + 4 \beta_{5} + \cdots - 4 ) / 3 \) (-2b11 + 8b10 - 18b9 - 10b8 + 3b7 - 24b6 + 4b5 + 4b4 - 16b3 - 12b2 - 4) / 3
\(\nu^{7}\)	\(=\)	\( ( - 23 \beta_{10} + 12 \beta_{9} - 7 \beta_{8} + 30 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 11 \beta_{4} + \cdots + 119 ) / 3 \) (-23b10 + 12b9 - 7b8 + 30b7 + 5b6 - 6b5 - 11b4 - b3 + 18b2 + 119) / 3
\(\nu^{8}\)	\(=\)	\( ( - 4 \beta_{11} - 46 \beta_{10} + 29 \beta_{9} - 2 \beta_{8} + 74 \beta_{7} - 19 \beta_{5} - 51 \beta_{4} + \cdots - 6 ) / 3 \) (-4b11 - 46b10 + 29b9 - 2b8 + 74b7 - 19b5 - 51b4 + 33b3 - 2b2 + 47b1 - 6) / 3
\(\nu^{9}\)	\(=\)	\( ( - 50 \beta_{11} + 116 \beta_{10} - 78 \beta_{9} - 82 \beta_{8} - 222 \beta_{7} - 63 \beta_{6} + \cdots - 67 ) / 3 \) (-50b11 + 116b10 - 78b9 - 82b8 - 222b7 - 63b6 - 47b5 + 25b4 - 16b3 - 3b2 - 18*b1 - 67) / 3
\(\nu^{10}\)	\(=\)	\( ( 66 \beta_{11} - 86 \beta_{10} - 69 \beta_{9} + 20 \beta_{8} - 172 \beta_{6} + 21 \beta_{5} + 4 \beta_{4} + \cdots + 80 ) / 3 \) (66b11 - 86b10 - 69b9 + 20b8 - 172b6 + 21b5 + 4b4 + 131b3 + 30b2 + 60b1 + 80) / 3
\(\nu^{11}\)	\(=\)	\( ( - 136 \beta_{11} - 7 \beta_{10} + 296 \beta_{9} - 29 \beta_{8} - 364 \beta_{7} - 204 \beta_{6} + \cdots - 435 ) / 3 \) (-136b11 - 7b10 + 296b9 - 29b8 - 364b7 - 204b6 + 44b5 - 321b4 - 9b3 - 41b2 - 121*b1 - 435) / 3

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 523.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} - 2 T^{11} + \cdots + 9801 \) T^12 - 2T^11 + 24T^10 - 12T^9 + 351T^8 - 204T^7 + 2322T^6 - 696T^5 + 10509T^4 - 3740T^3 + 17743T^2 + 8712*T + 9801
$7$	\( T^{12} - 8 T^{11} + \cdots + 64 \) T^12 - 8T^11 + 56T^10 - 160T^9 + 516T^8 - 696T^7 + 2928T^6 - 3456T^5 + 4176T^4 - 1312T^3 + 608T^2 + 64*T + 64
$11$	\( T^{12} + 48 T^{10} + \cdots + 6561 \) T^12 + 48T^10 + 36T^9 + 1809T^8 + 1512T^7 + 23922T^6 + 53298T^5 + 252801T^4 + 317844T^3 + 459999T^2 - 52488T + 6561
$13$	\( T^{12} + 6 T^{11} + \cdots + 121 \) T^12 + 6T^11 + 72T^10 + 244T^9 + 2553T^8 + 8058T^7 + 49782T^6 + 61932T^5 + 303153T^4 + 149182T^3 + 1573869T^2 - 13794*T + 121
$17$	\( (T^{6} + 16 T^{5} + \cdots - 24)^{2} \) (T^6 + 16T^5 + 88T^4 + 176T^3 - 4T^2 - 280*T - 24)^2
$19$	\( (T^{6} - 2 T^{5} + \cdots - 1233)^{2} \) (T^6 - 2T^5 - 63T^4 + 60T^3 + 735T^2 + 174*T - 1233)^2
$23$	\( T^{12} - 2 T^{11} + \cdots + 788544 \) T^12 - 2T^11 + 42T^10 - 60T^9 + 1236T^8 - 1584T^7 + 15168T^6 - 3408T^5 + 110160T^4 - 8576T^3 + 446848T^2 + 333888*T + 788544
$29$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$31$	\( T^{12} + 8 T^{11} + \cdots + 531441 \) T^12 + 8T^11 + 148T^10 + 816T^9 + 12207T^8 + 60414T^7 + 533490T^6 + 1201536T^5 + 8566461T^4 + 7513182T^3 + 115802685T^2 - 7825086*T + 531441
$37$	\( (T^{6} - 90 T^{4} + \cdots - 344)^{2} \) (T^6 - 90T^4 + 140T^3 + 1416T^2 - 2856T - 344)^2
$41$	\( T^{12} + \cdots + 6410244096 \) T^12 - 4T^11 + 204T^10 - 432T^9 + 29808T^8 - 60096T^7 + 1628160T^6 + 475392T^5 + 54544128T^4 - 305152T^3 + 777594880T^2 + 963330048*T + 6410244096
$43$	\( T^{12} - 2 T^{11} + \cdots + 63001 \) T^12 - 2T^11 + 68T^10 - 472T^9 + 5055T^8 - 20736T^7 + 67326T^6 - 121002T^5 + 174945T^4 - 114700T^3 + 100109T^2 - 25100*T + 63001
$47$	\( T^{12} - 14 T^{11} + \cdots + 463761 \) T^12 - 14T^11 + 204T^10 - 996T^9 + 8445T^8 - 23832T^7 + 273954T^6 - 367116T^5 + 1437777T^4 + 1942048T^3 + 3184375T^2 + 1293900*T + 463761
$53$	\( (T^{6} + 20 T^{5} + \cdots + 66189)^{2} \) (T^6 + 20T^5 + 34T^4 - 1496T^3 - 9133T^2 - 734*T + 66189)^2
$59$	\( T^{12} + 96 T^{10} + \cdots + 5645376 \) T^12 + 96T^10 - 336T^9 + 7044T^8 - 22680T^7 + 241488T^6 - 893088T^5 + 6046416T^4 - 15029280T^3 + 37768032T^2 - 15567552T + 5645376
$61$	\( T^{12} + 22 T^{11} + \cdots + 7744 \) T^12 + 22T^11 + 392T^10 + 3440T^9 + 28716T^8 + 136704T^7 + 1017168T^6 + 4027008T^5 + 19109040T^4 + 18130496T^3 + 14829728T^2 + 343552*T + 7744
$67$	\( T^{12} + \cdots + 158703437376 \) T^12 - 4T^11 + 250T^10 - 360T^9 + 39984T^8 - 37536T^7 + 3587064T^6 + 1231488T^5 + 224369280T^4 + 85390272T^3 + 7533234720T^2 + 9886098816*T + 158703437376
$71$	\( (T^{6} + 4 T^{5} + \cdots - 8232)^{2} \) (T^6 + 4T^5 - 98T^4 - 172T^3 + 2072T^2 + 2744*T - 8232)^2
$73$	\( (T^{6} + 8 T^{5} + \cdots + 14696)^{2} \) (T^6 + 8T^5 - 178T^4 - 420T^3 + 9208T^2 - 25448*T + 14696)^2
$79$	\( T^{12} - 4 T^{11} + \cdots + 81 \) T^12 - 4T^11 + 196T^10 - 672T^9 + 29235T^8 - 92622T^7 + 1495482T^6 - 1235700T^5 + 45783045T^4 - 88829838T^3 + 223077897T^2 + 134406*T + 81
$83$	\( T^{12} + \cdots + 153958464 \) T^12 - 2T^11 + 192T^10 + 864T^9 + 26100T^8 + 109704T^7 + 1738464T^6 + 8720160T^5 + 81363984T^4 + 246187744T^3 + 938545312T^2 - 357449664*T + 153958464
$89$	\( (T^{6} + 10 T^{5} + \cdots + 227064)^{2} \) (T^6 + 10T^5 - 266T^4 - 3064T^3 + 5696T^2 + 111920*T + 227064)^2
$97$	\( T^{12} + \cdots + 309750128704 \) T^12 + 10T^11 + 452T^10 + 2312T^9 + 125076T^8 + 576936T^7 + 16139088T^6 - 9552672T^5 + 908951664T^4 - 159742432T^3 + 27734927072T^2 - 61367649728*T + 309750128704
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\(n\)	\(379\)	\(407\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{7}\)