LMFDB - Newform orbit 174.2.h.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [174,2,Mod(13,174)] 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("174.13"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(174, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 9])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 174 = 2 \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	174.h (of order $14$, degree $6$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.38939699517$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-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 $\zeta_{28}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/174\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-\zeta_{28}^{4}$	$1$

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} $

13.1

−0.974928	−	0.222521i
0.974928	+	0.222521i
−0.974928	+	0.222521i
0.974928	−	0.222521i
0.781831	−	0.623490i
−0.781831	+	0.623490i
0.781831	+	0.623490i
−0.781831	−	0.623490i
−0.433884	+	0.900969i
0.433884	−	0.900969i
−0.433884	−	0.900969i
0.433884	+	0.900969i

−0.433884

0.900969i

0.781831

0.623490i

−0.623490

−

0.781831i

1.28967

0.621074i

−0.900969

0.433884i

−2.18522

2.74017i

0.974928

−

0.222521i

0.222521

0.974928i

−1.11914

0.892482i

13.2

0.433884

−

0.900969i

−0.781831

−

0.623490i

−0.623490

−

0.781831i

−3.09161

−

1.48884i

−0.900969

0.433884i

0.246215

−

0.308743i

−0.974928

0.222521i

0.222521

0.974928i

−2.68280

2.13946i

67.1

−0.433884

−

0.900969i

0.781831

−

0.623490i

−0.623490

0.781831i

1.28967

−

0.621074i

−0.900969

−

0.433884i

−2.18522

−

2.74017i

0.974928

0.222521i

0.222521

−

0.974928i

−1.11914

−

0.892482i

67.2

0.433884

0.900969i

−0.781831

0.623490i

−0.623490

0.781831i

−3.09161

1.48884i

−0.900969

−

0.433884i

0.246215

0.308743i

−0.974928

−

0.222521i

0.222521

−

0.974928i

−2.68280

−

2.13946i

91.1

−0.974928

−

0.222521i

0.433884

0.900969i

0.900969

0.433884i

0.404459

−

1.77205i

−0.222521

−

0.974928i

1.51662

−

0.730364i

−0.781831

−

0.623490i

−0.623490

0.781831i

−0.788637

1.63762i

91.2

0.974928

0.222521i

−0.433884

−

0.900969i

0.900969

0.433884i

−0.849501

3.72191i

−0.222521

−

0.974928i

4.33424

−

2.08726i

0.781831

0.623490i

−0.623490

0.781831i

−1.65640

3.43956i

109.1

−0.974928

0.222521i

0.433884

−

0.900969i

0.900969

−

0.433884i

0.404459

1.77205i

−0.222521

0.974928i

1.51662

0.730364i

−0.781831

0.623490i

−0.623490

−

0.781831i

−0.788637

−

1.63762i

109.2

0.974928

−

0.222521i

−0.433884

0.900969i

0.900969

−

0.433884i

−0.849501

−

3.72191i

−0.222521

0.974928i

4.33424

2.08726i

0.781831

−

0.623490i

−0.623490

−

0.781831i

−1.65640

−

3.43956i

121.1

−0.781831

0.623490i

−0.974928

0.222521i

0.222521

−

0.974928i

0.864277

1.08377i

0.623490

−

0.781831i

0.237169

1.03911i

0.433884

0.900969i

0.900969

−

0.433884i

−1.35144

−

0.308457i

121.2

0.781831

−

0.623490i

0.974928

−

0.222521i

0.222521

−

0.974928i

0.382702

0.479894i

0.623490

−

0.781831i

−0.149023

−

0.652914i

−0.433884

−

0.900969i

0.900969

−

0.433884i

0.598418

0.136585i

151.1

−0.781831

−

0.623490i

−0.974928

−

0.222521i

0.222521

0.974928i

0.864277

−

1.08377i

0.623490

0.781831i

0.237169

−

1.03911i

0.433884

−

0.900969i

0.900969

0.433884i

−1.35144

0.308457i

151.2

0.781831

0.623490i

0.974928

0.222521i

0.222521

0.974928i

0.382702

−

0.479894i

0.623490

0.781831i

−0.149023

0.652914i

−0.433884

0.900969i

0.900969

0.433884i

0.598418

−

0.136585i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.e	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	174.2.h.a	✓	12
3.b	odd	2	1		522.2.n.b		12
29.e	even	14	1	inner	174.2.h.a	✓	12
29.f	odd	28	1		5046.2.a.bn		6
29.f	odd	28	1		5046.2.a.bp		6
87.h	odd	14	1		522.2.n.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
174.2.h.a	✓	12	1.a	even	1	1	trivial
174.2.h.a	✓	12	29.e	even	14	1	inner
522.2.n.b		12	3.b	odd	2	1
522.2.n.b		12	87.h	odd	14	1
5046.2.a.bn		6	29.f	odd	28	1
5046.2.a.bp		6	29.f	odd	28	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} + 2 T_{5}^{11} + 10 T_{5}^{10} + 4 T_{5}^{9} - 30 T_{5}^{8} + 6 T_{5}^{7} + 553 T_{5}^{6} + \cdots + 841 $ T5^12 + 2*T5^11 + 10*T5^10 + 4*T5^9 - 30*T5^8 + 6*T5^7 + 553*T5^6 - 2290*T5^5 + 5304*T5^4 - 7444*T5^3 + 6625*T5^2 - 3190*T5 + 841 acting on $S_{2}^{\mathrm{new}}(174, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	$ T^{12} + 2 T^{11} + \cdots + 841 $ T^12 + 2T^11 + 10T^10 + 4T^9 - 30T^8 + 6T^7 + 553T^6 - 2290T^5 + 5304T^4 - 7444T^3 + 6625T^2 - 3190*T + 841
$7$	$ T^{12} - 8 T^{11} + \cdots + 64 $ T^12 - 8T^11 + 20T^10 - 32T^9 + 328T^8 - 1104T^7 + 1904T^6 - 2208T^5 + 2096T^4 - 1152T^3 + 768T^2 - 256*T + 64
$11$	$ T^{12} - 112 T^{9} + \cdots + 3136 $ T^12 - 112T^9 + 28T^8 + 1120T^7 + 112T^6 - 3136T^5 - 1568T^4 + 12544T^3 + 47040T^2 + 18816*T + 3136
$13$	$ T^{12} - 6 T^{11} + \cdots + 12769 $ T^12 - 6T^11 + 34T^10 - 66T^9 + 454T^8 - 1220T^7 + 4529T^6 - 8950T^5 + 11224T^4 - 6030T^3 - 2795T^2 + 16272*T + 12769
$17$	$ T^{12} + 54 T^{10} + \cdots + 1 $ T^12 + 54T^10 + 473T^8 + 1378T^6 + 1178T^4 + 192*T^2 + 1
$19$	$ T^{12} + 14 T^{11} + \cdots + 1236544 $ T^12 + 14T^11 + 90T^10 + 224T^9 + 596T^8 + 3416T^7 + 26536T^6 + 106736T^5 + 226864T^4 + 575680T^3 + 1688224T^2 + 2428608*T + 1236544
$23$	$ T^{12} - 12 T^{11} + \cdots + 1236544 $ T^12 - 12T^11 + 80T^10 - 696T^9 + 5108T^8 - 17648T^7 + 84448T^6 - 634432T^5 + 2072096T^4 - 1722752T^3 + 10646016T^2 - 4501376*T + 1236544
$29$	$ T^{12} + \cdots + 594823321 $ T^12 + 18T^11 + 211T^10 + 1610T^9 + 9687T^8 + 46672T^7 + 241023T^6 + 1353488T^5 + 8146767T^4 + 39266290T^3 + 149236291T^2 + 369200682*T + 594823321
$31$	$ T^{12} + 14 T^{11} + \cdots + 440896 $ T^12 + 14T^11 + 54T^10 + 420T^9 + 5352T^8 + 14000T^7 - 31592T^6 + 102928T^5 + 500320T^4 - 1953952T^3 + 3179648T^2 - 1784832*T + 440896
$37$	$ T^{12} - 14 T^{11} + \cdots + 27889 $ T^12 - 14T^11 + 54T^10 - 14T^9 - 234T^8 + 1260T^7 + 4745T^6 - 1834T^5 - 8160T^4 + 19306T^3 + 14913T^2 - 18704*T + 27889
$41$	$ T^{12} + 234 T^{10} + \cdots + 10374841 $ T^12 + 234T^10 + 18685T^8 + 605562T^6 + 7646706T^4 + 31432448*T^2 + 10374841
$43$	$ T^{12} + \cdots + 260370496 $ T^12 - 8T^10 + 448T^9 + 2108T^8 - 28896T^7 - 42288T^6 + 1146880T^5 - 444128T^4 - 19174400T^3 + 20191296T^2 + 34337408T + 260370496
$47$	$ T^{12} + \cdots + 155950144 $ T^12 - 14T^11 - 14T^10 + 504T^9 + 4620T^8 - 45976T^7 + 67928T^6 + 193648T^5 - 442960T^4 - 12418560T^3 + 65004576T^2 + 74828096*T + 155950144
$53$	$ T^{12} + 16 T^{11} + \cdots + 82791801 $ T^12 + 16T^11 + 73T^10 + 60T^9 + 3491T^8 - 17872T^7 + 84469T^6 - 394752T^5 + 1792593T^4 - 2480166T^3 + 18815247T^2 - 41928192*T + 82791801
$59$	$ (T^{6} + 30 T^{5} + \cdots + 55112)^{2} $ (T^6 + 30T^5 + 214T^4 - 972T^3 - 12864T^2 - 11952*T + 55112)^2
$61$	$ T^{12} + \cdots + 8260446769 $ T^12 + 42T^11 + 796T^10 + 7588T^9 + 24938T^8 - 239232T^7 - 3069599T^6 - 9327710T^5 + 49143516T^4 + 416876222T^3 + 701534179T^2 - 283749214*T + 8260446769
$67$	$ T^{12} + \cdots + 370793536 $ T^12 - 32T^11 + 740T^10 - 12072T^9 + 168416T^8 - 1751312T^7 + 13550992T^6 - 67995296T^5 + 221076624T^4 - 483184384T^3 + 737361344T^2 - 736349440*T + 370793536
$71$	$ T^{12} - 4 T^{11} + \cdots + 64 $ T^12 - 4T^11 - 16T^10 + 1564T^9 + 7024T^8 - 78984T^7 + 625576T^6 + 443200T^5 + 245440T^4 + 78368T^3 + 12704T^2 + 832*T + 64
$73$	$ T^{12} + \cdots + 9041917921 $ T^12 - 42T^11 + 607T^10 + 112T^9 - 93047T^8 + 815570T^7 + 525699T^6 - 49534618T^5 + 417511565T^4 - 1871328844T^3 + 5322342035T^2 - 8792879830*T + 9041917921
$79$	$ T^{12} + \cdots + 2980153121344 $ T^12 + 70T^11 + 2310T^10 + 45752T^9 + 580244T^8 + 4800600T^7 + 32320120T^6 + 389441808T^5 + 6066821712T^4 + 63009233280T^3 + 408999696224T^2 + 1607583165888*T + 2980153121344
$83$	$ T^{12} + \cdots + 11562270784 $ T^12 - 62T^11 + 2162T^10 - 50144T^9 + 844108T^8 - 10452360T^7 + 94620344T^6 - 607084944T^5 + 2643050672T^4 - 7340359680T^3 + 14460152544T^2 - 18416535616*T + 11562270784
$89$	$ T^{12} + \cdots + 634346938681 $ T^12 - 70T^11 + 2232T^10 - 42238T^9 + 562822T^8 - 6504162T^7 + 70804761T^6 - 619957898T^5 + 3726366372T^4 - 15288011564T^3 + 56731483731T^2 - 240068671780*T + 634346938681
$97$	$ T^{12} + \cdots + 134769017881 $ T^12 + 42T^11 + 811T^10 + 6720T^9 - 49979T^8 - 1841154T^7 - 17644229T^6 - 5305398T^5 + 1685823077T^4 + 16920125824T^3 + 77816630863T^2 + 153224688638*T + 134769017881
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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:	\( N \)	\(=\)	\( 174 = 2 \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	174.h (of order \(14\), degree \(6\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	174.2.h.a

Level	$174$
Weight	$2$
Character orbit	174.h
Analytic conductor	$1.389$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.38939699517\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{12} + 2 T^{11} + \cdots + 841 \) T^12 + 2T^11 + 10T^10 + 4T^9 - 30T^8 + 6T^7 + 553T^6 - 2290T^5 + 5304T^4 - 7444T^3 + 6625T^2 - 3190*T + 841
$7$	\( T^{12} - 8 T^{11} + \cdots + 64 \) T^12 - 8T^11 + 20T^10 - 32T^9 + 328T^8 - 1104T^7 + 1904T^6 - 2208T^5 + 2096T^4 - 1152T^3 + 768T^2 - 256*T + 64
$11$	\( T^{12} - 112 T^{9} + \cdots + 3136 \) T^12 - 112T^9 + 28T^8 + 1120T^7 + 112T^6 - 3136T^5 - 1568T^4 + 12544T^3 + 47040T^2 + 18816*T + 3136
$13$	\( T^{12} - 6 T^{11} + \cdots + 12769 \) T^12 - 6T^11 + 34T^10 - 66T^9 + 454T^8 - 1220T^7 + 4529T^6 - 8950T^5 + 11224T^4 - 6030T^3 - 2795T^2 + 16272*T + 12769
$17$	\( T^{12} + 54 T^{10} + \cdots + 1 \) T^12 + 54T^10 + 473T^8 + 1378T^6 + 1178T^4 + 192*T^2 + 1
$19$	\( T^{12} + 14 T^{11} + \cdots + 1236544 \) T^12 + 14T^11 + 90T^10 + 224T^9 + 596T^8 + 3416T^7 + 26536T^6 + 106736T^5 + 226864T^4 + 575680T^3 + 1688224T^2 + 2428608*T + 1236544
$23$	\( T^{12} - 12 T^{11} + \cdots + 1236544 \) T^12 - 12T^11 + 80T^10 - 696T^9 + 5108T^8 - 17648T^7 + 84448T^6 - 634432T^5 + 2072096T^4 - 1722752T^3 + 10646016T^2 - 4501376*T + 1236544
$29$	\( T^{12} + \cdots + 594823321 \) T^12 + 18T^11 + 211T^10 + 1610T^9 + 9687T^8 + 46672T^7 + 241023T^6 + 1353488T^5 + 8146767T^4 + 39266290T^3 + 149236291T^2 + 369200682*T + 594823321
$31$	\( T^{12} + 14 T^{11} + \cdots + 440896 \) T^12 + 14T^11 + 54T^10 + 420T^9 + 5352T^8 + 14000T^7 - 31592T^6 + 102928T^5 + 500320T^4 - 1953952T^3 + 3179648T^2 - 1784832*T + 440896
$37$	\( T^{12} - 14 T^{11} + \cdots + 27889 \) T^12 - 14T^11 + 54T^10 - 14T^9 - 234T^8 + 1260T^7 + 4745T^6 - 1834T^5 - 8160T^4 + 19306T^3 + 14913T^2 - 18704*T + 27889
$41$	\( T^{12} + 234 T^{10} + \cdots + 10374841 \) T^12 + 234T^10 + 18685T^8 + 605562T^6 + 7646706T^4 + 31432448*T^2 + 10374841
$43$	\( T^{12} + \cdots + 260370496 \) T^12 - 8T^10 + 448T^9 + 2108T^8 - 28896T^7 - 42288T^6 + 1146880T^5 - 444128T^4 - 19174400T^3 + 20191296T^2 + 34337408T + 260370496
$47$	\( T^{12} + \cdots + 155950144 \) T^12 - 14T^11 - 14T^10 + 504T^9 + 4620T^8 - 45976T^7 + 67928T^6 + 193648T^5 - 442960T^4 - 12418560T^3 + 65004576T^2 + 74828096*T + 155950144
$53$	\( T^{12} + 16 T^{11} + \cdots + 82791801 \) T^12 + 16T^11 + 73T^10 + 60T^9 + 3491T^8 - 17872T^7 + 84469T^6 - 394752T^5 + 1792593T^4 - 2480166T^3 + 18815247T^2 - 41928192*T + 82791801
$59$	\( (T^{6} + 30 T^{5} + \cdots + 55112)^{2} \) (T^6 + 30T^5 + 214T^4 - 972T^3 - 12864T^2 - 11952*T + 55112)^2
$61$	\( T^{12} + \cdots + 8260446769 \) T^12 + 42T^11 + 796T^10 + 7588T^9 + 24938T^8 - 239232T^7 - 3069599T^6 - 9327710T^5 + 49143516T^4 + 416876222T^3 + 701534179T^2 - 283749214*T + 8260446769
$67$	\( T^{12} + \cdots + 370793536 \) T^12 - 32T^11 + 740T^10 - 12072T^9 + 168416T^8 - 1751312T^7 + 13550992T^6 - 67995296T^5 + 221076624T^4 - 483184384T^3 + 737361344T^2 - 736349440*T + 370793536
$71$	\( T^{12} - 4 T^{11} + \cdots + 64 \) T^12 - 4T^11 - 16T^10 + 1564T^9 + 7024T^8 - 78984T^7 + 625576T^6 + 443200T^5 + 245440T^4 + 78368T^3 + 12704T^2 + 832*T + 64
$73$	\( T^{12} + \cdots + 9041917921 \) T^12 - 42T^11 + 607T^10 + 112T^9 - 93047T^8 + 815570T^7 + 525699T^6 - 49534618T^5 + 417511565T^4 - 1871328844T^3 + 5322342035T^2 - 8792879830*T + 9041917921
$79$	\( T^{12} + \cdots + 2980153121344 \) T^12 + 70T^11 + 2310T^10 + 45752T^9 + 580244T^8 + 4800600T^7 + 32320120T^6 + 389441808T^5 + 6066821712T^4 + 63009233280T^3 + 408999696224T^2 + 1607583165888*T + 2980153121344
$83$	\( T^{12} + \cdots + 11562270784 \) T^12 - 62T^11 + 2162T^10 - 50144T^9 + 844108T^8 - 10452360T^7 + 94620344T^6 - 607084944T^5 + 2643050672T^4 - 7340359680T^3 + 14460152544T^2 - 18416535616*T + 11562270784
$89$	\( T^{12} + \cdots + 634346938681 \) T^12 - 70T^11 + 2232T^10 - 42238T^9 + 562822T^8 - 6504162T^7 + 70804761T^6 - 619957898T^5 + 3726366372T^4 - 15288011564T^3 + 56731483731T^2 - 240068671780*T + 634346938681
$97$	\( T^{12} + \cdots + 134769017881 \) T^12 + 42T^11 + 811T^10 + 6720T^9 - 49979T^8 - 1841154T^7 - 17644229T^6 - 5305398T^5 + 1685823077T^4 + 16920125824T^3 + 77816630863T^2 + 153224688638*T + 134769017881
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\(n\)	\(31\)	\(59\)
\(\chi(n)\)	\(-\zeta_{28}^{4}\)	\(1\)