LMFDB - Newform orbit 790.2.m.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [790,2,Mod(21,790)] 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("790.21"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 18])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 790 = 2 \cdot 5 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	790.m (of order $13$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,1,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$6.30818175968$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{26})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

$q$-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 $\zeta_{26}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/790\mathbb{Z}\right)^\times$.

$n$	$161$	$317$
$\chi(n)$	$-\zeta_{26}^{7}$	$1$

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} $

21.1

0.748511	−	0.663123i
−0.568065	+	0.822984i
−0.885456	−	0.464723i
0.970942	+	0.239316i
0.748511	+	0.663123i
0.970942	−	0.239316i
−0.120537	+	0.992709i
−0.120537	−	0.992709i
0.354605	−	0.935016i
−0.568065	−	0.822984i
−0.885456	+	0.464723i
0.354605	+	0.935016i

0.748511

−

0.663123i

0.0773304

−

0.112032i

0.120537

−

0.992709i

−0.970942

0.239316i

−0.0164086

−

0.135137i

1.69459

−

2.45504i

−0.568065

−

0.822984i

1.05724

2.78772i

−0.568065

0.822984i

101.1

−0.568065

0.822984i

2.85640

−

0.704039i

−0.354605

−

0.935016i

−0.748511

−

0.663123i

−1.04321

2.75071i

−1.55961

0.384411i

0.970942

0.239316i

5.00697

−

2.62786i

0.970942

−

0.239316i

131.1

−0.885456

−

0.464723i

−0.0914785

0.753393i

0.568065

0.822984i

−0.354605

−

0.935016i

0.431119

−

0.624584i

0.508155

−

4.18504i

−0.120537

−

0.992709i

2.35359

0.580108i

−0.120537

0.992709i

141.1

0.970942

0.239316i

1.86905

−

1.65583i

0.885456

0.464723i

0.568065

−

0.822984i

2.21100

−

1.16042i

0.667796

−

0.591616i

0.748511

0.663123i

0.389950

−

3.21153i

0.748511

−

0.663123i

301.1

0.748511

0.663123i

0.0773304

0.112032i

0.120537

0.992709i

−0.970942

−

0.239316i

−0.0164086

0.135137i

1.69459

2.45504i

−0.568065

0.822984i

1.05724

−

2.78772i

−0.568065

−

0.822984i

381.1

0.970942

−

0.239316i

1.86905

1.65583i

0.885456

−

0.464723i

0.568065

0.822984i

2.21100

1.16042i

0.667796

0.591616i

0.748511

−

0.663123i

0.389950

3.21153i

0.748511

0.663123i

441.1

−0.120537

0.992709i

0.606094

1.59814i

−0.970942

−

0.239316i

0.885456

−

0.464723i

−1.65954

0.409041i

−1.72842

−

4.55748i

0.354605

−

0.935016i

0.0588344

−

0.0521227i

0.354605

0.935016i

541.1

−0.120537

−

0.992709i

0.606094

−

1.59814i

−0.970942

0.239316i

0.885456

0.464723i

−1.65954

−

0.409041i

−1.72842

4.55748i

0.354605

0.935016i

0.0588344

0.0521227i

0.354605

−

0.935016i

561.1

0.354605

−

0.935016i

0.682609

0.358261i

−0.748511

−

0.663123i

0.120537

−

0.992709i

0.577036

−

0.511209i

−1.58251

−

0.830564i

−0.885456

0.464723i

−1.36659

−

1.97985i

−0.885456

−

0.464723i

571.1

−0.568065

−

0.822984i

2.85640

0.704039i

−0.354605

0.935016i

−0.748511

0.663123i

−1.04321

−

2.75071i

−1.55961

−

0.384411i

0.970942

−

0.239316i

5.00697

2.62786i

0.970942

0.239316i

591.1

−0.885456

0.464723i

−0.0914785

−

0.753393i

0.568065

−

0.822984i

−0.354605

0.935016i

0.431119

0.624584i

0.508155

4.18504i

−0.120537

0.992709i

2.35359

−

0.580108i

−0.120537

−

0.992709i

721.1

0.354605

0.935016i

0.682609

−

0.358261i

−0.748511

0.663123i

0.120537

0.992709i

0.577036

0.511209i

−1.58251

0.830564i

−0.885456

−

0.464723i

−1.36659

1.97985i

−0.885456

0.464723i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
79.e	even	13	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	790.2.m.b	✓	12
79.e	even	13	1	inner	790.2.m.b	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
790.2.m.b	✓	12	1.a	even	1	1	trivial
790.2.m.b	✓	12	79.e	even	13	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 12 T_{3}^{11} + 66 T_{3}^{10} - 220 T_{3}^{9} + 495 T_{3}^{8} - 792 T_{3}^{7} + 924 T_{3}^{6} + \cdots + 1 $ T3^12 - 12*T3^11 + 66*T3^10 - 220*T3^9 + 495*T3^8 - 792*T3^7 + 924*T3^6 - 779*T3^5 + 521*T3^4 - 272*T3^3 + 92*T3^2 - 12*T3 + 1 acting on $S_{2}^{\mathrm{new}}(790, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{12} - 12 T^{11} + \cdots + 1 $ T^12 - 12T^11 + 66T^10 - 220T^9 + 495T^8 - 792T^7 + 924T^6 - 779T^5 + 521T^4 - 272T^3 + 92T^2 - 12*T + 1
$5$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$7$	$ T^{12} + 4 T^{11} + \cdots + 24649 $ T^12 + 4T^11 + 42T^10 + 116T^9 + 594T^8 + 1570T^7 + 3134T^6 + 11535T^5 + 25574T^4 + 13974T^3 - 12822T^2 + 5652*T + 24649
$11$	$ T^{12} - 14 T^{11} + \cdots + 2985984 $ T^12 - 14T^11 + 92T^10 - 456T^9 + 2432T^8 - 12416T^7 + 49024T^6 - 155520T^5 + 440064T^4 - 1009152T^3 + 1907712T^2 - 2985984*T + 2985984
$13$	$ T^{12} + 13 T^{11} + \cdots + 169 $ T^12 + 13T^11 + 104T^10 + 637T^9 + 3848T^8 + 17030T^7 + 28249T^6 - 113061T^5 - 414557T^4 + 1271387T^3 + 5943054T^2 - 7605*T + 169
$17$	$ T^{12} - 13 T^{10} + \cdots + 28561 $ T^12 - 13T^10 - 91T^9 - 338T^8 - 169T^7 + 9971T^6 + 59319T^5 + 208715T^4 + 415233T^3 + 485537T^2 + 142805T + 28561
$19$	$ T^{12} - 7 T^{11} + \cdots + 96721 $ T^12 - 7T^11 + 101T^10 - 499T^9 + 2609T^8 - 9865T^7 + 37543T^6 - 40709T^5 - 137160T^4 + 660067T^3 + 1277917T^2 + 526212*T + 96721
$23$	$ (T^{6} + 12 T^{5} + \cdots - 443)^{2} $ (T^6 + 12T^5 - 5T^4 - 438T^3 - 1515T^2 - 1537*T - 443)^2
$29$	$ T^{12} - 20 T^{11} + \cdots + 4096 $ T^12 - 20T^11 + 296T^10 - 2592T^9 + 18560T^8 - 87488T^7 + 328704T^6 - 695168T^5 + 664576T^4 - 59392T^3 - 36864T^2 - 8192*T + 4096
$31$	$ T^{12} + \cdots + 347561449 $ T^12 - 20T^11 + 88T^10 + 450T^9 - 719T^8 - 45576T^7 + 291147T^6 - 1606663T^5 + 28501203T^4 - 138559507T^3 + 499833494T^2 - 196832794*T + 347561449
$37$	$ T^{12} + 11 T^{11} + \cdots + 17530969 $ T^12 + 11T^11 + 95T^10 + 525T^9 + 2954T^8 + 20846T^7 + 190293T^6 + 1182287T^5 + 4765276T^4 + 8250952T^3 + 8697894T^2 + 15918974*T + 17530969
$41$	$ T^{12} - 20 T^{11} + \cdots + 6241 $ T^12 - 20T^11 + 166T^10 - 603T^9 + 3051T^8 - 29482T^7 + 139996T^6 + 34067T^5 + 690147T^4 - 280080T^3 + 280180T^2 - 77183*T + 6241
$43$	$ T^{12} + \cdots + 131767441 $ T^12 - 26T^11 + 364T^10 - 3614T^9 + 28340T^8 - 184366T^7 + 1012791T^6 - 4687384T^5 + 17963348T^4 - 54735213T^3 + 123161285T^2 - 180564670*T + 131767441
$47$	$ T^{12} - 13 T^{11} + \cdots + 1792921 $ T^12 - 13T^11 + 130T^10 - 1014T^9 + 5551T^8 - 17602T^7 + 20761T^6 + 39884T^5 - 101062T^4 - 257049T^3 + 672282T^2 + 748501*T + 1792921
$53$	$ T^{12} - 7 T^{11} + \cdots + 4214809 $ T^12 - 7T^11 + 127T^10 - 928T^9 + 4468T^8 - 4639T^7 - 17135T^6 + 137417T^5 - 40778T^4 - 627115T^3 + 491521T^2 + 2245982*T + 4214809
$59$	$ T^{12} + \cdots + 5239298689 $ T^12 + 35T^11 + 536T^10 + 5513T^9 + 52724T^8 + 493405T^7 + 4076632T^6 + 26946669T^5 + 136386877T^4 + 517812946T^3 + 1557879852T^2 + 3519985290*T + 5239298689
$61$	$ T^{12} - 12 T^{11} + \cdots + 271441 $ T^12 - 12T^11 + 430T^10 - 2144T^9 + 49700T^8 - 47709T^7 + 4317288T^6 - 25206037T^5 + 419249312T^4 - 1046989540T^3 + 754542400T^2 - 25127309*T + 271441
$67$	$ T^{12} - 16 T^{11} + \cdots + 1079521 $ T^12 - 16T^11 + 269T^10 - 5305T^9 + 49286T^8 - 440488T^7 + 6267158T^6 - 48936735T^5 + 189983179T^4 - 131268801T^3 + 45637595T^2 - 9094367*T + 1079521
$71$	$ T^{12} + \cdots + 110901961 $ T^12 - 7T^11 + 10T^10 - 304T^9 + 21784T^8 - 273440T^7 + 1661672T^6 - 5239474T^5 + 34632055T^4 - 95789741T^3 + 141759505T^2 - 163072535*T + 110901961
$73$	$ T^{12} + \cdots + 1349754121 $ T^12 + 37T^11 + 797T^10 + 13993T^9 + 186384T^8 + 1670338T^7 + 10016421T^6 + 42995643T^5 + 141814240T^4 + 533491902T^3 + 3019269016T^2 + 2978577686*T + 1349754121
$79$	$ T^{12} + \cdots + 243087455521 $ T^12 + 64T^11 + 2107T^10 + 46422T^9 + 755418T^8 + 9511241T^7 + 94733575T^6 + 751388039T^5 + 4714563738T^4 + 22887856458T^3 + 82067820667T^2 + 196931609536*T + 243087455521
$83$	$ T^{12} + \cdots + 119361267169 $ T^12 + T^11 + 469T^10 + 3641T^9 + 111073T^8 + 1780039T^7 + 19925569T^6 + 212680664T^5 + 2169724844T^4 + 15263203138T^3 + 63833805101T^2 + 138184435390T + 119361267169
$89$	$ T^{12} - 36 T^{11} + \cdots + 11607649 $ T^12 - 36T^11 + 594T^10 - 6278T^9 + 50495T^8 - 339148T^7 + 1949650T^6 - 8831859T^5 + 28281249T^4 - 56752084T^3 + 60281575T^2 - 18217229*T + 11607649
$97$	$ T^{12} + \cdots + 2538849769 $ T^12 + 11T^11 + 173T^10 - 242T^9 + 237T^8 - 234279T^7 + 1324205T^6 - 3408741T^5 + 45097698T^4 - 269238741T^3 + 1039870257T^2 - 2335487837*T + 2538849769
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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:	\( N \)	\(=\)	\( 790 = 2 \cdot 5 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	790.m (of order \(13\), degree \(12\), minimal)

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

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	790.2.m.b

Level	$790$
Weight	$2$
Character orbit	790.m
Analytic conductor	$6.308$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.30818175968\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{26})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 21.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{12} - 12 T^{11} + \cdots + 1 \) T^12 - 12T^11 + 66T^10 - 220T^9 + 495T^8 - 792T^7 + 924T^6 - 779T^5 + 521T^4 - 272T^3 + 92T^2 - 12*T + 1
$5$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$7$	\( T^{12} + 4 T^{11} + \cdots + 24649 \) T^12 + 4T^11 + 42T^10 + 116T^9 + 594T^8 + 1570T^7 + 3134T^6 + 11535T^5 + 25574T^4 + 13974T^3 - 12822T^2 + 5652*T + 24649
$11$	\( T^{12} - 14 T^{11} + \cdots + 2985984 \) T^12 - 14T^11 + 92T^10 - 456T^9 + 2432T^8 - 12416T^7 + 49024T^6 - 155520T^5 + 440064T^4 - 1009152T^3 + 1907712T^2 - 2985984*T + 2985984
$13$	\( T^{12} + 13 T^{11} + \cdots + 169 \) T^12 + 13T^11 + 104T^10 + 637T^9 + 3848T^8 + 17030T^7 + 28249T^6 - 113061T^5 - 414557T^4 + 1271387T^3 + 5943054T^2 - 7605*T + 169
$17$	\( T^{12} - 13 T^{10} + \cdots + 28561 \) T^12 - 13T^10 - 91T^9 - 338T^8 - 169T^7 + 9971T^6 + 59319T^5 + 208715T^4 + 415233T^3 + 485537T^2 + 142805T + 28561
$19$	\( T^{12} - 7 T^{11} + \cdots + 96721 \) T^12 - 7T^11 + 101T^10 - 499T^9 + 2609T^8 - 9865T^7 + 37543T^6 - 40709T^5 - 137160T^4 + 660067T^3 + 1277917T^2 + 526212*T + 96721
$23$	\( (T^{6} + 12 T^{5} + \cdots - 443)^{2} \) (T^6 + 12T^5 - 5T^4 - 438T^3 - 1515T^2 - 1537*T - 443)^2
$29$	\( T^{12} - 20 T^{11} + \cdots + 4096 \) T^12 - 20T^11 + 296T^10 - 2592T^9 + 18560T^8 - 87488T^7 + 328704T^6 - 695168T^5 + 664576T^4 - 59392T^3 - 36864T^2 - 8192*T + 4096
$31$	\( T^{12} + \cdots + 347561449 \) T^12 - 20T^11 + 88T^10 + 450T^9 - 719T^8 - 45576T^7 + 291147T^6 - 1606663T^5 + 28501203T^4 - 138559507T^3 + 499833494T^2 - 196832794*T + 347561449
$37$	\( T^{12} + 11 T^{11} + \cdots + 17530969 \) T^12 + 11T^11 + 95T^10 + 525T^9 + 2954T^8 + 20846T^7 + 190293T^6 + 1182287T^5 + 4765276T^4 + 8250952T^3 + 8697894T^2 + 15918974*T + 17530969
$41$	\( T^{12} - 20 T^{11} + \cdots + 6241 \) T^12 - 20T^11 + 166T^10 - 603T^9 + 3051T^8 - 29482T^7 + 139996T^6 + 34067T^5 + 690147T^4 - 280080T^3 + 280180T^2 - 77183*T + 6241
$43$	\( T^{12} + \cdots + 131767441 \) T^12 - 26T^11 + 364T^10 - 3614T^9 + 28340T^8 - 184366T^7 + 1012791T^6 - 4687384T^5 + 17963348T^4 - 54735213T^3 + 123161285T^2 - 180564670*T + 131767441
$47$	\( T^{12} - 13 T^{11} + \cdots + 1792921 \) T^12 - 13T^11 + 130T^10 - 1014T^9 + 5551T^8 - 17602T^7 + 20761T^6 + 39884T^5 - 101062T^4 - 257049T^3 + 672282T^2 + 748501*T + 1792921
$53$	\( T^{12} - 7 T^{11} + \cdots + 4214809 \) T^12 - 7T^11 + 127T^10 - 928T^9 + 4468T^8 - 4639T^7 - 17135T^6 + 137417T^5 - 40778T^4 - 627115T^3 + 491521T^2 + 2245982*T + 4214809
$59$	\( T^{12} + \cdots + 5239298689 \) T^12 + 35T^11 + 536T^10 + 5513T^9 + 52724T^8 + 493405T^7 + 4076632T^6 + 26946669T^5 + 136386877T^4 + 517812946T^3 + 1557879852T^2 + 3519985290*T + 5239298689
$61$	\( T^{12} - 12 T^{11} + \cdots + 271441 \) T^12 - 12T^11 + 430T^10 - 2144T^9 + 49700T^8 - 47709T^7 + 4317288T^6 - 25206037T^5 + 419249312T^4 - 1046989540T^3 + 754542400T^2 - 25127309*T + 271441
$67$	\( T^{12} - 16 T^{11} + \cdots + 1079521 \) T^12 - 16T^11 + 269T^10 - 5305T^9 + 49286T^8 - 440488T^7 + 6267158T^6 - 48936735T^5 + 189983179T^4 - 131268801T^3 + 45637595T^2 - 9094367*T + 1079521
$71$	\( T^{12} + \cdots + 110901961 \) T^12 - 7T^11 + 10T^10 - 304T^9 + 21784T^8 - 273440T^7 + 1661672T^6 - 5239474T^5 + 34632055T^4 - 95789741T^3 + 141759505T^2 - 163072535*T + 110901961
$73$	\( T^{12} + \cdots + 1349754121 \) T^12 + 37T^11 + 797T^10 + 13993T^9 + 186384T^8 + 1670338T^7 + 10016421T^6 + 42995643T^5 + 141814240T^4 + 533491902T^3 + 3019269016T^2 + 2978577686*T + 1349754121
$79$	\( T^{12} + \cdots + 243087455521 \) T^12 + 64T^11 + 2107T^10 + 46422T^9 + 755418T^8 + 9511241T^7 + 94733575T^6 + 751388039T^5 + 4714563738T^4 + 22887856458T^3 + 82067820667T^2 + 196931609536*T + 243087455521
$83$	\( T^{12} + \cdots + 119361267169 \) T^12 + T^11 + 469T^10 + 3641T^9 + 111073T^8 + 1780039T^7 + 19925569T^6 + 212680664T^5 + 2169724844T^4 + 15263203138T^3 + 63833805101T^2 + 138184435390T + 119361267169
$89$	\( T^{12} - 36 T^{11} + \cdots + 11607649 \) T^12 - 36T^11 + 594T^10 - 6278T^9 + 50495T^8 - 339148T^7 + 1949650T^6 - 8831859T^5 + 28281249T^4 - 56752084T^3 + 60281575T^2 - 18217229*T + 11607649
$97$	\( T^{12} + \cdots + 2538849769 \) T^12 + 11T^11 + 173T^10 - 242T^9 + 237T^8 - 234279T^7 + 1324205T^6 - 3408741T^5 + 45097698T^4 - 269238741T^3 + 1039870257T^2 - 2335487837*T + 2538849769
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\(n\)	\(161\)	\(317\)
\(\chi(n)\)	\(-\zeta_{26}^{7}\)	\(1\)