LMFDB - Newform orbit 74.2.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(3,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([13]))

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

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

chi := DirichletCharacter("74.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	74.h (of order $18$, degree $6$, 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:	$0.590892974957$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-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 $\zeta_{36}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$.

$n$	$39$
$\chi(n)$	$\zeta_{36}^{2}$

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} $

3.1

−0.642788	+	0.766044i
0.642788	−	0.766044i
−0.342020	+	0.939693i
0.342020	−	0.939693i
−0.642788	−	0.766044i
0.642788	+	0.766044i
−0.984808	−	0.173648i
0.984808	+	0.173648i
−0.984808	+	0.173648i
0.984808	−	0.173648i
−0.342020	−	0.939693i
0.342020	+	0.939693i

−0.642788

0.766044i

1.43969

−

1.20805i

−0.173648

−

0.984808i

0.273629

−

0.751790i

1.87939i

−0.138449

−

0.0503913i

0.866025

0.500000i

0.0923963

−

0.524005i

0.400019

0.692853i

3.2

0.642788

−

0.766044i

1.43969

−

1.20805i

−0.173648

−

0.984808i

−1.45842

4.00698i

−

1.87939i

−3.39364

−

1.23518i

−0.866025

−

0.500000i

0.0923963

−

0.524005i

2.13207

3.69285i

21.1

−0.342020

0.939693i

0.326352

−

0.118782i

−0.766044

−

0.642788i

2.57176

0.453471i

0.347296i

−0.361075

2.04776i

0.866025

−

0.500000i

−2.20574

1.85083i

−1.30572

2.26157i

21.2

0.342020

−

0.939693i

0.326352

−

0.118782i

−0.766044

−

0.642788i

0.839712

0.148064i

−

0.347296i

0.240460

−

1.36372i

−0.866025

0.500000i

−2.20574

1.85083i

0.426333

−

0.738430i

25.1

−0.642788

−

0.766044i

1.43969

1.20805i

−0.173648

0.984808i

0.273629

0.751790i

−

1.87939i

−0.138449

0.0503913i

0.866025

−

0.500000i

0.0923963

0.524005i

0.400019

−

0.692853i

25.2

0.642788

0.766044i

1.43969

1.20805i

−0.173648

0.984808i

−1.45842

−

4.00698i

1.87939i

−3.39364

1.23518i

−0.866025

0.500000i

0.0923963

0.524005i

2.13207

−

3.69285i

41.1

−0.984808

−

0.173648i

−0.266044

−

1.50881i

0.939693

0.342020i

−1.97937

−

2.35892i

1.53209i

0.153180

−

0.128533i

−0.866025

−

0.500000i

0.613341

−

0.223238i

1.53967

2.66679i

41.2

0.984808

0.173648i

−0.266044

−

1.50881i

0.939693

0.342020i

−0.247315

−

0.294739i

−

1.53209i

−2.50048

2.09815i

0.866025

0.500000i

0.613341

−

0.223238i

−0.192377

−

0.333207i

65.1

−0.984808

0.173648i

−0.266044

1.50881i

0.939693

−

0.342020i

−1.97937

2.35892i

−

1.53209i

0.153180

0.128533i

−0.866025

0.500000i

0.613341

0.223238i

1.53967

−

2.66679i

65.2

0.984808

−

0.173648i

−0.266044

1.50881i

0.939693

−

0.342020i

−0.247315

0.294739i

1.53209i

−2.50048

−

2.09815i

0.866025

−

0.500000i

0.613341

0.223238i

−0.192377

0.333207i

67.1

−0.342020

−

0.939693i

0.326352

0.118782i

−0.766044

0.642788i

2.57176

−

0.453471i

−

0.347296i

−0.361075

−

2.04776i

0.866025

0.500000i

−2.20574

−

1.85083i

−1.30572

−

2.26157i

67.2

0.342020

0.939693i

0.326352

0.118782i

−0.766044

0.642788i

0.839712

−

0.148064i

0.347296i

0.240460

1.36372i

−0.866025

−

0.500000i

−2.20574

−

1.85083i

0.426333

0.738430i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.h	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	74.2.h.a	✓	12
3.b	odd	2	1		666.2.bj.c		12
4.b	odd	2	1		592.2.bq.b		12
37.h	even	18	1	inner	74.2.h.a	✓	12
37.i	odd	36	1		2738.2.a.r		6
37.i	odd	36	1		2738.2.a.s		6
111.n	odd	18	1		666.2.bj.c		12
148.o	odd	18	1		592.2.bq.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
74.2.h.a	✓	12	1.a	even	1	1	trivial
74.2.h.a	✓	12	37.h	even	18	1	inner
592.2.bq.b		12	4.b	odd	2	1
592.2.bq.b		12	148.o	odd	18	1
666.2.bj.c		12	3.b	odd	2	1
666.2.bj.c		12	111.n	odd	18	1
2738.2.a.r		6	37.i	odd	36	1
2738.2.a.s		6	37.i	odd	36	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(74, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{6} + 1 $ T^12 - T^6 + 1
$3$	$ (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} $ (T^6 - 3T^5 + 6T^4 - 8T^3 + 12T^2 - 6*T + 1)^2
$5$	$ T^{12} + 9 T^{10} + \cdots + 81 $ T^12 + 9T^10 - 72T^9 + 36T^8 - 162T^7 + 1476T^6 - 2268T^5 + 1701T^4 - 810T^3 + 81*T^2 + 81
$7$	$ T^{12} + 12 T^{11} + \cdots + 1 $ T^12 + 12T^11 + 66T^10 + 218T^9 + 504T^8 + 882T^7 + 1164T^6 + 990T^5 + 1098T^4 - 52T^3 - 21T^2 + 6*T + 1
$11$	$ T^{12} + 6 T^{11} + \cdots + 408321 $ T^12 + 6T^11 + 63T^10 + 138T^9 + 1431T^8 + 2322T^7 + 22680T^6 + 1458T^5 + 119151T^4 - 63396T^3 + 546426T^2 - 414072*T + 408321
$13$	$ T^{12} - 6 T^{11} + \cdots + 288369 $ T^12 - 6T^11 + 30T^10 - 174T^9 + 882T^8 - 2880T^7 + 7530T^6 - 33174T^5 + 112788T^4 - 178128T^3 + 177255T^2 - 289980*T + 288369
$17$	$ T^{12} + 36 T^{10} + \cdots + 81 $ T^12 + 36T^10 + 234T^9 + 576T^8 + 3240T^7 + 14355T^6 + 61560T^5 + 186624T^4 + 184518T^3 + 103356T^2 - 5832T + 81
$19$	$ T^{12} + 18 T^{11} + \cdots + 10439361 $ T^12 + 18T^11 + 135T^10 + 342T^9 - 1827T^8 - 16146T^7 - 45180T^6 + 19440T^5 + 637794T^4 + 1808244T^3 + 3360285T^2 + 1919214*T + 10439361
$23$	$ T^{12} - 63 T^{10} + \cdots + 431649 $ T^12 - 63T^10 + 3303T^8 + 4482T^7 - 38916T^6 - 47952T^5 + 400869T^4 + 35964T^3 - 436590T^2 - 35478*T + 431649
$29$	$ T^{12} - 18 T^{11} + \cdots + 110889 $ T^12 - 18T^11 + 126T^10 - 324T^9 - 423T^8 + 2592T^7 + 14112T^6 - 124578T^5 + 400707T^4 - 711504T^3 + 742689T^2 - 431568*T + 110889
$31$	$ T^{12} + \cdots + 317445489 $ T^12 + 210T^10 + 16929T^8 + 655566T^6 + 12463938T^4 + 106546320*T^2 + 317445489
$37$	$ T^{12} + \cdots + 2565726409 $ T^12 - 30T^11 + 417T^10 - 3838T^9 + 27450T^8 - 164430T^7 + 953241T^6 - 6083910T^5 + 37579050T^4 - 194406214T^3 + 781525137T^2 - 2080318710*T + 2565726409
$41$	$ T^{12} - 24 T^{11} + \cdots + 331776 $ T^12 - 24T^11 + 216T^10 - 336T^9 - 5184T^8 + 19008T^7 + 29376T^6 - 145152T^5 + 2654208T^4 - 691200T^3 + 5142528T^2 + 1990656*T + 331776
$43$	$ T^{12} + 156 T^{10} + \cdots + 2277081 $ T^12 + 156T^10 + 8910T^8 + 238146T^6 + 3015693T^4 + 14710302*T^2 + 2277081
$47$	$ T^{12} + \cdots + 2027430729 $ T^12 - 6T^11 + 153T^10 - 498T^9 + 13338T^8 - 35478T^7 + 656073T^6 - 328698T^5 + 17956242T^4 + 3894210T^3 + 338937777T^2 + 571392630*T + 2027430729
$53$	$ T^{12} + \cdots + 45041148441 $ T^12 + 12T^11 + 162T^10 + 798T^9 + 1188T^8 - 235818T^7 - 1286712T^6 + 6328530T^5 + 304555950T^4 + 2947900932T^3 + 20285966511T^2 + 51376820778*T + 45041148441
$59$	$ T^{12} + \cdots + 17477104401 $ T^12 + 1404T^9 + 9108T^8 + 89046T^7 - 31518T^6 - 4378536T^5 + 17490168T^4 - 155749392T^3 - 8372727T^2 - 349803846*T + 17477104401
$61$	$ T^{12} + 36 T^{11} + \cdots + 331776 $ T^12 + 36T^11 + 756T^10 + 10368T^9 + 101664T^8 + 711936T^7 + 3508992T^6 + 11425536T^5 + 18786816T^4 + 2488320T^3 - 663552T^2 + 995328*T + 331776
$67$	$ T^{12} + \cdots + 59697637561 $ T^12 + 30T^11 + 669T^10 + 9578T^9 + 101016T^8 + 764046T^7 + 5658987T^6 + 48777696T^5 + 525509181T^4 + 4827632402T^3 + 26000383650T^2 + 64028404536*T + 59697637561
$71$	$ T^{12} + \cdots + 4131885551616 $ T^12 - 12T^11 + 216T^10 - 528T^9 - 30672T^8 + 162432T^7 + 1615680T^6 - 23576832T^5 + 774779904T^4 + 5570297856T^3 - 21649959936T^2 - 270463463424*T + 4131885551616
$73$	$ (T^{6} - 366 T^{4} + \cdots + 94609)^{2} $ (T^6 - 366T^4 + 322T^3 + 28305T^2 - 115410T + 94609)^2
$79$	$ T^{12} - 6 T^{11} + \cdots + 47961 $ T^12 - 6T^11 + 12T^10 - 174T^9 + 1674T^8 - 7560T^7 + 21543T^6 - 44784T^5 + 71694T^4 - 95022T^3 + 107784T^2 - 90666*T + 47961
$83$	$ T^{12} + 48 T^{11} + \cdots + 110889 $ T^12 + 48T^11 + 1008T^10 + 10968T^9 + 68364T^8 + 295434T^7 + 1072386T^6 + 9542124T^5 + 318723984T^4 + 201295908T^3 + 892811565T^2 - 18719262*T + 110889
$89$	$ T^{12} + \cdots + 687331089 $ T^12 + 18T^11 + 9T^10 + 306T^9 + 216T^8 - 102870T^7 + 1249155T^6 - 18528264T^5 + 122173353T^4 - 299077110T^3 + 325925694T^2 + 509658480*T + 687331089
$97$	$ T^{12} + \cdots + 27455164416 $ T^12 - 36T^11 + 336T^10 + 3456T^9 - 52272T^8 - 1091520T^7 + 32966016T^6 - 384618240T^5 + 2599135488T^4 - 10960137216T^3 + 28495328256T^2 - 41898553344*T + 27455164416
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{36} q^{2} + ( - \zeta_{36}^{6} - \zeta_{36}^{4} + 1) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} - \zeta_{36}^{8} + \cdots - \zeta_{36}) q^{5} + ( - \zeta_{36}^{7} - \zeta_{36}^{5} + \zeta_{36}) q^{6}+ \cdots + (2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} + \cdots + 3) q^{99}+O(q^{100}) \) q + z * q^2 + (-z^6 - z^4 + 1) * q^3 + z^2 * q^4 + (-z^9 - z^8 + z^7 + z^5 + z^3 - z^2 - z) * q^5 + (-z^7 - z^5 + z) * q^6 + (2z^11 + z^10 - z^5 - 1) q^7 + z^3 * q^8 + (-z^10 + z^8 - z^6 + z^4) * q^9 + (-z^10 - z^9 + z^8 + z^6 + z^4 - z^3 - z^2) * q^10 + (-z^10 + 4z^9 - z^8 + z^6 - 2z^3 + z^2 - 1) * q^11 + (-z^8 - z^6 + z^2) * q^12 + (-2z^11 + z^10 + 2z^8 + z^6 - z^4 + z^3 - z^2 - z) * q^13 + (z^11 + z^6 - z - 2) * q^14 + (-2z^11 - 2z^9 + z^8 + z^7 + 2z^6 + 2z^5 - 2z^2 - z - 1) q^15 + z^4 * q^16 + (-2z^11 + z^10 - 2z^7 + z^4 - 2z^3) q^17 + (-z^11 + z^9 - z^7 + z^5) * q^18 + (2z^11 + 2z^10 - z^8 + 2z^7 - z^6 - 2z^5 - z^4 - 2z^3 + 2z^2 - 2z - 1) q^19 + (-z^11 - z^10 + z^9 + z^7 + z^5 - z^4 - z^3) * q^20 + (z^11 - z^9 - z^8 + z^6 + z^5 + 2z^4 + 2z^3 + z^2 - 1) * q^21 + (-z^11 + 4z^10 - z^9 + z^7 - 2z^4 + z^3 - z) * q^22 + (-3z^11 - 2z^10 + z^9 - z^8 - 2z^7 + 2z^5 + z^4 - z^3 + 2z^2 + 3z) * q^23 + (-z^9 - z^7 + z^3) * q^24 + (2z^11 + z^10 - 2z^9 - z^8 - 4z^7 - z^6 - 4z^5 + 4z^3 + 2z^2 + 2z - 1) q^25 + (z^11 + 2z^9 + z^7 - 2z^6 - z^5 + z^4 - z^3 - z^2 + 2) * q^26 + (-3z^10 - z^6 - 3z^2) * q^27 + (z^7 + z^6 - z^2 - 2z - 1) q^28 + (-2z^10 + 2z^8 + z^6 + z^5 + z^4 - z^3 - z^2 + z + 1) * q^29 + (-2z^10 + z^9 + z^8 + 2z^7 - 2z^3 - z^2 - z + 2) q^30 + (2z^11 - 3z^9 - z^8 + 4z^7 + 2z^5 - z^4 + z^2 - 2z) q^31 + z^5 * q^32 + (-z^10 + 2z^9 - 2z^7 + z^6 + 2z^3 - z^2 + 4z - 1) * q^33 + (z^11 - 2z^8 - 2z^6 + z^5 - 2z^4 + 2) q^34 + (z^10 + 2z^9 + 3z^8 - 3z^6 - 2z^5 - 2z^4 - 2z^3 + 4z + 3) q^35 + (z^10 - z^8 + 1) * q^36 + (5z^11 + 2z^10 - z^8 + z^7 + z^6 - 3z^5 - z^4 + 2z^3 + 2z^2 - z + 2) q^37 + (2z^11 - z^9 + 2z^8 - z^7 - z^5 - 2z^4 + 2z^3 - 2z^2 - z - 2) q^38 + (z^9 + z^8 - z^6 - z^5 - z^3 + 2z^2 - z + 3) q^39 + (-z^11 + z^10 + z^8 - z^5 - z^4 + 1) * q^40 + (-2z^9 + 2z^7 + 2z^4 + 4z^3 - 2z^2 + 2z + 2) * q^41 + (-z^10 - z^9 + z^7 + 2z^6 + 2z^5 + 2z^4 + z^3 - z - 1) q^42 + (2z^11 - z^10 - z^8 - 2z^7 - 4z^5 + 4z) * q^43 + (4z^11 - z^10 + z^8 - z^6 - 2z^5 + z^4 - z^2 + 1) * q^44 + (-2z^10 + 2z^8 - z^6 + z^5 + z^4 - z^3 - z^2 + z - 1) * q^45 + (-2z^11 + z^10 - z^9 - 2z^8 - z^6 + z^5 - z^4 + 2z^3 + 3z^2 + 3) * q^46 + (-z^11 + 4z^10 - 2z^9 - 3z^8 - z^7 + z^6 - z^5 - 3z^4 - 2z^3 + 4z^2 - z) * q^47 + (-z^10 - z^8 + z^4) * q^48 + (-4z^11 + 2z^10 - 2z^9 + 2z^5 - 2z^3 - z^2 + 1) q^49 + (z^11 - 2z^10 - z^9 - 4z^8 - z^7 - 2z^6 + 4z^4 + 2z^3 + 2z^2 - z - 2) * q^50 + (2z^11 - z^10 + 4z^9 - 2z^8 + 2z^7 - 2z^5 + 2z^4 - 4z^3 + z^2 - 2z) * q^51 + (2z^10 + z^8 - 2z^7 + z^5 - z^4 - z^3 + 2z - 1) q^52 + (-2z^11 + 2z^9 + z^8 - 3z^7 - 4z^6 + 4z^5 - 4z^4 + 2z^3 - 4z^2 - 3z + 1) q^53 + (-3z^11 - z^7 - 3z^3) * q^54 + (-6z^11 - z^10 + z^9 + z^8 - 2z^7 + 5z^6 + 6z^5 - 4z^4 + z^3 - 2z^2 + z - 4) * q^55 + (z^8 + z^7 - z^3 - 2z^2 - z) q^56 + (2z^10 + 2z^9 - 3z^8 + 4z^7 + 2z^5 + 2z^4 + 3z^2 - 3) q^57 + (-2z^11 + 2z^9 + z^7 + z^6 + z^5 - z^4 - z^3 + z^2 + z) * q^58 + (-6z^11 - 2z^10 - 6z^9 - z^8 + 2z^7 - 2z^6 + 2z^5 + z^4 + 4z^3 + 2z^2 - 2z + 1) q^59 + (-2z^11 + z^10 + z^9 + 2z^8 - 2z^4 - z^3 - z^2 + 2z) * q^60 + (2z^11 + 2z^10 - 2z^9 + 4z^8 - 2z^7 + 2z^6 + 2z^4 - 2z^2 + 2z - 4) q^61 + (-3z^10 - z^9 + 4z^8 + 4z^6 - z^5 + z^3 - 2z^2 - 2) * q^62 + (-z^11 + 2z^9 - z^7 + z^6 + 2z^5 - z^3 - z - 1) * q^63 + z^6 * q^64 + (-3z^10 - 4z^8 + 4z^7 + 2z^6 + 3z^4 + 2z^2 - 8z + 2) q^65 + (-z^11 + 2z^10 - 2z^8 + z^7 + 2z^4 - z^3 + 4z^2 - z) * q^66 + (4z^11 + 4z^9 + 3z^8 + 2z^7 - 5z^6 - 2z^5 + 5z^4 - 2z^3 - 3z^2 - 4z) * q^67 + (-2z^9 - 2z^7 + 2z^6 - 2z^5 + 2z - 1) q^68 + (-z^10 + 2z^9 - z^8 - 3z^7 - z^6 - 4z^5 - z^4 - 3z^3 - z^2 + 2z - 1) q^69 + (z^11 + 2z^10 + 3z^9 - 3z^7 - 2z^6 - 2z^5 - 2z^4 + 4z^2 + 3z) * q^70 + (4z^11 + 4z^10 - 2z^9 + 4z^8 - 4z^7 + 6z^6 + 4z^5 + 2z^4 - 2z^3 + 2z - 2) * q^71 + (z^11 - z^9 + z) * q^72 + (-2z^10 + 2z^9 + 5z^8 + 6z^7 - 6z^5 - 3z^4 - 4z^3 - 3z^2 - 6z) q^73 + (2z^11 - z^9 + z^8 + z^7 + 2z^6 - z^5 + 2z^4 + 2z^3 - z^2 + 2z - 5) q^74 + (8z^11 + z^10 - 2z^9 - 3z^8 - 4z^7 - 4z^5 + 2z^4 + 4z^3 + 2z^2 - 4z - 3) q^75 + (-z^10 + 2z^9 - z^8 + z^6 - 2z^5 + 2z^4 - 2z^3 - z^2 - 2z - 2) q^76 + (-z^11 + z^10 - 3z^9 - 5z^8 - z^5 - z^4 + 3z^3 - 3z + 1) * q^77 + (z^10 + z^9 - z^7 - z^6 - z^4 + 2z^3 - z^2 + 3z) * q^78 + (z^10 - z^8 + z^7 - z^6 + 2z^5 - z^4 + z^3 - z^2 + 1) q^79 + (z^11 + z^9 - 2z^6 - z^5 + z + 1) q^80 + (z^10 + 3z^8 + 6z^6 - 6z^4 - 3z^2 - 1) * q^81 + (-2z^10 + 2z^8 + 2z^5 + 4z^4 - 2z^3 + 2z^2 + 2z) q^82 + (-6z^11 - z^10 + 3z^8 + 2z^6 + z^4 + 6z^3 - 5z^2 - 5) q^83 + (-z^11 - z^10 + z^8 + 2z^7 + 2z^6 + 2z^5 + z^4 - z^2 - z) q^84 + (3z^11 - 4z^10 + 6z^9 - 4z^8 + 3z^7 - 3z^6 + 2z^4 - 3z^3 + 2z^2 - 3z + 3) * q^85 + (-z^11 - z^9 - 2z^8 - 2z^6 + 4z^2 - 2) q^86 + (-z^11 - 2z^10 + 2z^8 - 2z^6 - 2z^4 - z^3 - z^2 + z + 4) * q^87 + (-z^11 + z^9 - z^7 + 2z^6 + z^5 - z^3 + z - 4) q^88 + (-2z^10 + z^8 - z^6 + 6z^5 + z^4 - 6z^3 + z^2 - 1) q^89 + (-2z^11 + 2z^9 - z^7 + z^6 + z^5 - z^4 - z^3 + z^2 - z) * q^90 + (2z^11 + 2z^10 - 2z^9 - 2z^8 - 2z^7 - 3z^6 - 2z^5 + 2z^4 + 2z^3 - z^2 + 1) q^91 + (z^11 - z^10 - 2z^9 - z^7 - z^6 - z^5 + 2z^4 + 3z^3 + 3z + 2) * q^92 + (-6z^11 + z^10 - 4z^9 + 5z^7 + 6z^5 - z^4 - z^3 - z - 1) * q^93 + (4z^11 - 2z^10 - 3z^9 - z^8 + z^7 - 2z^6 - 3z^5 - 2z^4 + 4z^3 - z^2 + 1) q^94 + (-2z^11 + 2z^9 + z^8 + 4z^7 - 2z^6 + z^5 - 3z^4 - z^3 - 2z^2 + 4z + 1) q^95 + (-z^11 - z^9 + z^5) * q^96 + (-4z^11 - 4z^10 + 2z^9 - 6z^7 - 2z^6 + 6z^5 - 2z^3 + 4z^2 + 4z + 4) q^97 + (2z^11 - 2z^10 - 2z^6 - 2z^4 - z^3 + z + 4) * q^98 + (2z^11 + 2z^10 - 2z^9 - z^8 + 4z^7 - z^6 - 4z^5 + 4z^3 - 2z^2 - 2z + 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} - 12 q^{7} - 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 18 q^{14} - 18 q^{19} - 6 q^{21} - 18 q^{25} + 12 q^{26} - 6 q^{27} - 6 q^{28} + 18 q^{29} + 24 q^{30} - 6 q^{33} + 12 q^{34}+ \cdots + 30 q^{99}+O(q^{100}) \) 12 * q + 6 * q^3 - 12 * q^7 - 6 * q^9 + 6 * q^10 - 6 * q^11 - 6 * q^12 + 6 * q^13 - 18 * q^14 - 18 * q^19 - 6 * q^21 - 18 * q^25 + 12 * q^26 - 6 * q^27 - 6 * q^28 + 18 * q^29 + 24 * q^30 - 6 * q^33 + 12 * q^34 + 18 * q^35 + 12 * q^36 + 30 * q^37 - 24 * q^38 + 30 * q^39 + 12 * q^40 + 24 * q^41 + 6 * q^44 - 18 * q^45 + 30 * q^46 + 6 * q^47 + 12 * q^49 - 36 * q^50 - 12 * q^52 - 12 * q^53 - 18 * q^55 - 36 * q^57 + 6 * q^58 - 36 * q^61 - 6 * q^63 + 6 * q^64 + 36 * q^65 - 30 * q^67 - 18 * q^69 - 12 * q^70 + 12 * q^71 - 48 * q^74 - 36 * q^75 - 18 * q^76 + 12 * q^77 - 6 * q^78 + 6 * q^79 + 24 * q^81 - 48 * q^83 + 12 * q^84 + 18 * q^85 - 36 * q^86 + 36 * q^87 - 36 * q^88 - 18 * q^89 + 6 * q^90 - 6 * q^91 + 18 * q^92 - 12 * q^93 + 36 * q^97 + 36 * q^98 + 30 * 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	74.2.h.a

Level	$74$
Weight	$2$
Character orbit	74.h
Analytic conductor	$0.591$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.590892974957\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{36})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{6} + 1 \) x^12 - x^6 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{6} + 1 \) T^12 - T^6 + 1
$3$	\( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) (T^6 - 3T^5 + 6T^4 - 8T^3 + 12T^2 - 6*T + 1)^2
$5$	\( T^{12} + 9 T^{10} + \cdots + 81 \) T^12 + 9T^10 - 72T^9 + 36T^8 - 162T^7 + 1476T^6 - 2268T^5 + 1701T^4 - 810T^3 + 81*T^2 + 81
$7$	\( T^{12} + 12 T^{11} + \cdots + 1 \) T^12 + 12T^11 + 66T^10 + 218T^9 + 504T^8 + 882T^7 + 1164T^6 + 990T^5 + 1098T^4 - 52T^3 - 21T^2 + 6*T + 1
$11$	\( T^{12} + 6 T^{11} + \cdots + 408321 \) T^12 + 6T^11 + 63T^10 + 138T^9 + 1431T^8 + 2322T^7 + 22680T^6 + 1458T^5 + 119151T^4 - 63396T^3 + 546426T^2 - 414072*T + 408321
$13$	\( T^{12} - 6 T^{11} + \cdots + 288369 \) T^12 - 6T^11 + 30T^10 - 174T^9 + 882T^8 - 2880T^7 + 7530T^6 - 33174T^5 + 112788T^4 - 178128T^3 + 177255T^2 - 289980*T + 288369
$17$	\( T^{12} + 36 T^{10} + \cdots + 81 \) T^12 + 36T^10 + 234T^9 + 576T^8 + 3240T^7 + 14355T^6 + 61560T^5 + 186624T^4 + 184518T^3 + 103356T^2 - 5832T + 81
$19$	\( T^{12} + 18 T^{11} + \cdots + 10439361 \) T^12 + 18T^11 + 135T^10 + 342T^9 - 1827T^8 - 16146T^7 - 45180T^6 + 19440T^5 + 637794T^4 + 1808244T^3 + 3360285T^2 + 1919214*T + 10439361
$23$	\( T^{12} - 63 T^{10} + \cdots + 431649 \) T^12 - 63T^10 + 3303T^8 + 4482T^7 - 38916T^6 - 47952T^5 + 400869T^4 + 35964T^3 - 436590T^2 - 35478*T + 431649
$29$	\( T^{12} - 18 T^{11} + \cdots + 110889 \) T^12 - 18T^11 + 126T^10 - 324T^9 - 423T^8 + 2592T^7 + 14112T^6 - 124578T^5 + 400707T^4 - 711504T^3 + 742689T^2 - 431568*T + 110889
$31$	\( T^{12} + \cdots + 317445489 \) T^12 + 210T^10 + 16929T^8 + 655566T^6 + 12463938T^4 + 106546320*T^2 + 317445489
$37$	\( T^{12} + \cdots + 2565726409 \) T^12 - 30T^11 + 417T^10 - 3838T^9 + 27450T^8 - 164430T^7 + 953241T^6 - 6083910T^5 + 37579050T^4 - 194406214T^3 + 781525137T^2 - 2080318710*T + 2565726409
$41$	\( T^{12} - 24 T^{11} + \cdots + 331776 \) T^12 - 24T^11 + 216T^10 - 336T^9 - 5184T^8 + 19008T^7 + 29376T^6 - 145152T^5 + 2654208T^4 - 691200T^3 + 5142528T^2 + 1990656*T + 331776
$43$	\( T^{12} + 156 T^{10} + \cdots + 2277081 \) T^12 + 156T^10 + 8910T^8 + 238146T^6 + 3015693T^4 + 14710302*T^2 + 2277081
$47$	\( T^{12} + \cdots + 2027430729 \) T^12 - 6T^11 + 153T^10 - 498T^9 + 13338T^8 - 35478T^7 + 656073T^6 - 328698T^5 + 17956242T^4 + 3894210T^3 + 338937777T^2 + 571392630*T + 2027430729
$53$	\( T^{12} + \cdots + 45041148441 \) T^12 + 12T^11 + 162T^10 + 798T^9 + 1188T^8 - 235818T^7 - 1286712T^6 + 6328530T^5 + 304555950T^4 + 2947900932T^3 + 20285966511T^2 + 51376820778*T + 45041148441
$59$	\( T^{12} + \cdots + 17477104401 \) T^12 + 1404T^9 + 9108T^8 + 89046T^7 - 31518T^6 - 4378536T^5 + 17490168T^4 - 155749392T^3 - 8372727T^2 - 349803846*T + 17477104401
$61$	\( T^{12} + 36 T^{11} + \cdots + 331776 \) T^12 + 36T^11 + 756T^10 + 10368T^9 + 101664T^8 + 711936T^7 + 3508992T^6 + 11425536T^5 + 18786816T^4 + 2488320T^3 - 663552T^2 + 995328*T + 331776
$67$	\( T^{12} + \cdots + 59697637561 \) T^12 + 30T^11 + 669T^10 + 9578T^9 + 101016T^8 + 764046T^7 + 5658987T^6 + 48777696T^5 + 525509181T^4 + 4827632402T^3 + 26000383650T^2 + 64028404536*T + 59697637561
$71$	\( T^{12} + \cdots + 4131885551616 \) T^12 - 12T^11 + 216T^10 - 528T^9 - 30672T^8 + 162432T^7 + 1615680T^6 - 23576832T^5 + 774779904T^4 + 5570297856T^3 - 21649959936T^2 - 270463463424*T + 4131885551616
$73$	\( (T^{6} - 366 T^{4} + \cdots + 94609)^{2} \) (T^6 - 366T^4 + 322T^3 + 28305T^2 - 115410T + 94609)^2
$79$	\( T^{12} - 6 T^{11} + \cdots + 47961 \) T^12 - 6T^11 + 12T^10 - 174T^9 + 1674T^8 - 7560T^7 + 21543T^6 - 44784T^5 + 71694T^4 - 95022T^3 + 107784T^2 - 90666*T + 47961
$83$	\( T^{12} + 48 T^{11} + \cdots + 110889 \) T^12 + 48T^11 + 1008T^10 + 10968T^9 + 68364T^8 + 295434T^7 + 1072386T^6 + 9542124T^5 + 318723984T^4 + 201295908T^3 + 892811565T^2 - 18719262*T + 110889
$89$	\( T^{12} + \cdots + 687331089 \) T^12 + 18T^11 + 9T^10 + 306T^9 + 216T^8 - 102870T^7 + 1249155T^6 - 18528264T^5 + 122173353T^4 - 299077110T^3 + 325925694T^2 + 509658480*T + 687331089
$97$	\( T^{12} + \cdots + 27455164416 \) T^12 - 36T^11 + 336T^10 + 3456T^9 - 52272T^8 - 1091520T^7 + 32966016T^6 - 384618240T^5 + 2599135488T^4 - 10960137216T^3 + 28495328256T^2 - 41898553344*T + 27455164416
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\(n\)	\(39\)
\(\chi(n)\)	\(\zeta_{36}^{2}\)