LMFDB - Newform orbit 1248.2.bv.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(673,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

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

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

chi := DirichletCharacter("1248.673");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.bv (of order $6$, degree $2$, 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:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
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} - 6 x^{11} + 31 x^{10} - 100 x^{9} + 268 x^{8} - 538 x^{7} + 891 x^{6} - 1144 x^{5} + 1178 x^{4} + \cdots + 37 $ x^12 - 6x^11 + 31x^10 - 100x^9 + 268x^8 - 538x^7 + 891x^6 - 1144x^5 + 1178x^4 - 910x^3 + 517x^2 - 188*x + 37
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{4} q^{3} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{4} - 1) q^{9}+O(q^{10}) $ q + b4 * q^3 + (b8 - b7 - b6 - b5 + b2 - 1) * q^5 - b5 * q^7 + (b4 - 1) * q^9	$ q + \beta_{4} q^{3} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{8} + \cdots - \beta_{5}) q^{99}+O(q^{100}) $ q + b4 * q^3 + (b8 - b7 - b6 - b5 + b2 - 1) * q^5 - b5 * q^7 + (b4 - 1) * q^9 + (b10 - b8 + b7 + b5) * q^11 + (b11 + b10 - b9 + b8 + b7) * q^13 + (-b10 - b7 - b6 - b5 + b2 - 1) * q^15 + (-2b9 - b5 - b4 + 2b2 + b1 - 1) * q^17 + (2b10 + 2b8 - b7 + b6) * q^19 + (-b5 + b2 - 1) * q^21 + (-b10 - b9 - b8 + b7 - b5 - 2b4 + 2b1) * q^23 + (-b10 - b8 - 2b6 - b5 + b3 + b2 - 1) q^25 - q^27 + (-b10 + b7 - b6 - 3b5 + b2 - 1) q^29 + (2b11 + b10 + b8 + b5 - b3 - b2 + 1) q^31 + (b7 - b6) * q^33 + (-b11 - b10 + 2b9 - b7 - b6 - b5 + b2 - b1 - 1) q^35 + (-2b10 + b9 - b8 - 2b7 - 3b6 - 2b5 + 3b2 - 3) q^37 + (2b8 - b7 - b6 - b5 + b3 + b2 - b1 - 1) q^39 + (b11 + b10 - b6 + b4 + b3 + 2b2 - 4) q^41 + (-b11 + b10 - 2b8 + 2b5 - b2 + 1) * q^43 + (-b10 - b8) * q^45 + (2b11 - b10 - 2b9 - b8 + 4b7 + 2b6 - 4b4 - b3 - 2b2 + 2b1 + 4) q^47 + (-b11 + b8 - b7 - b6 - 3b5 - 3b4 + b3 + 2b2 - 2) q^49 + (-b9 + b5 + b2 - b1) * q^51 + (-2b10 + b9 - 3b8 + b7 - 2b6 - b3 + b2 + b1 + 1) q^53 + (b11 + 2b10 - b8 - b7 + 3b6 + 3b5 + 4b4 - b3 - b2 + 1) * q^55 + (b10 + b8 - 2b7 - b6 - b5 + 2b2 - 2) * q^57 + (-b11 + b10 + 2b8 - 2b6 + b5 + 2b3 + b2 - 3b1 - 1) * q^59 + (b11 + 2b10 - 2b9 - b8 + 2b4 + 3b2 + b1 - 5) * q^61 + (b2 - 1) * q^63 + (-2b8 - b6 - b5 + 3b4 + b3 + b2 - b1 - 5) * q^65 + (-b11 - b10 + 2b9 - 2b8 - b6 - b3 - 2b2 + 2) q^67 + (-2b9 + b7 + b6 - 2b4 + b1 + 2) * q^69 + (-2b10 - 2b8 - b7 + b6 + 2b5 + 2b4 + b1 + 2) * q^71 + (-2b10 + 2b7 - 4b5 - 2b4 + 2b2 - 1) q^73 + (-b11 - 2b10 - b8 + b7 - b6 - b5 + b3) q^75 - 2 * q^77 + (-3b10 + 2b9 - b8 - 2b7 - 3b5 - b3 - b2 + 2b1 - 3) q^79 - b4 * q^81 + (-2b11 - 3b10 + b9 - b8 + 2b7 - 2b5 + b3 - b1) * q^83 + (-b10 - b8 + b7 - b6 - 2b5 - 2b4 + 3b1 - 2) q^85 + (-b10 + b8 - 2b5 + 2b2 - 2) * q^87 + (b11 + 3b10 - b9 + 2b7 + b6 + 2b5 + b3 - b2 + 1) q^89 + (-b11 + 3b10 - 2b9 + 2b8 + b7 + 2b6 + 2b5 + b3 - 2b2 - 2b1 + 2) q^91 + (b11 + b8 - b7 - b6 - b5 + b3) * q^93 + (b11 - b10 + 2b8 - 3b7 - 3b6 - 3b5 + 6b4 + 3b2 - 9) * q^95 + (b11 + 3b10 + 2b8 + b7 + b6 + 4b5 + 4b4 - 2b3 - b2 - 4b1 + 5) * q^97 + (-b10 + b8 - b6 - b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9	$ 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9} - 2 q^{13} - 8 q^{23} + 4 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{39} - 24 q^{41} + 6 q^{43} - 12 q^{49} + 24 q^{53} + 20 q^{55} - 12 q^{59} - 18 q^{61} - 6 q^{63} - 20 q^{65} + 18 q^{67} + 8 q^{69} + 24 q^{71} + 2 q^{75} - 24 q^{77} - 28 q^{79} - 6 q^{81} - 24 q^{85} - 8 q^{87} - 2 q^{91} + 6 q^{93} - 36 q^{95} + 54 q^{97}+O(q^{100}) $ 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9 - 2 * q^13 - 8 * q^23 + 4 * q^25 - 12 * q^27 + 8 * q^29 - 4 * q^39 - 24 * q^41 + 6 * q^43 - 12 * q^49 + 24 * q^53 + 20 * q^55 - 12 * q^59 - 18 * q^61 - 6 * q^63 - 20 * q^65 + 18 * q^67 + 8 * q^69 + 24 * q^71 + 2 * q^75 - 24 * q^77 - 28 * q^79 - 6 * q^81 - 24 * q^85 - 8 * q^87 - 2 * q^91 + 6 * q^93 - 36 * q^95 + 54 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 31 x^{10} - 100 x^{9} + 268 x^{8} - 538 x^{7} + 891 x^{6} - 1144 x^{5} + 1178 x^{4} + \cdots + 37 $ x^12 - 6*x^11 + 31*x^10 - 100*x^9 + 268*x^8 - 538*x^7 + 891*x^6 - 1144*x^5 + 1178*x^4 - 910*x^3 + 517*x^2 - 188*x + 37 :

$\beta_{1}$	$=$	$ ( - 57 \nu^{11} + 229 \nu^{10} - 1061 \nu^{9} + 2381 \nu^{8} - 4875 \nu^{7} + 6005 \nu^{6} - 5619 \nu^{5} + \cdots + 669 ) / 169 $ (-57v^11 + 229v^10 - 1061v^9 + 2381v^8 - 4875v^7 + 6005v^6 - 5619v^5 + 2995v^4 - 161v^3 + 667v^2 - 659*v + 669) / 169
$\beta_{2}$	$=$	$ ( 66 \nu^{11} - 363 \nu^{10} + 1780 \nu^{9} - 5034 \nu^{8} + 11960 \nu^{7} - 18543 \nu^{6} + 22748 \nu^{5} + \cdots + 5932 ) / 338 $ (66v^11 - 363v^10 + 1780v^9 - 5034v^8 + 11960v^7 - 18543v^6 + 22748v^5 - 13430v^4 + 222v^3 + 14411v^2 - 13006*v + 5932) / 338
$\beta_{3}$	$=$	$ ( 140 \nu^{11} - 601 \nu^{10} + 2772 \nu^{9} - 6530 \nu^{8} + 13468 \nu^{7} - 16703 \nu^{6} + \cdots + 4850 ) / 338 $ (140v^11 - 601v^10 + 2772v^9 - 6530v^8 + 13468v^7 - 16703v^6 + 14566v^5 - 1284v^4 - 11482v^3 + 17945v^2 - 11006*v + 4850) / 338
$\beta_{4}$	$=$	$ ( 72 \nu^{11} - 396 \nu^{10} + 2034 \nu^{9} - 6183 \nu^{8} + 16120 \nu^{7} - 30338 \nu^{6} + 47462 \nu^{5} + \cdots - 2455 ) / 338 $ (72v^11 - 396v^10 + 2034v^9 - 6183v^8 + 16120v^7 - 30338v^6 + 47462v^5 - 55257v^4 + 50512v^3 - 32152v^2 + 13374*v - 2455) / 338
$\beta_{5}$	$=$	$ ( - 66 \nu^{11} + 363 \nu^{10} - 1780 \nu^{9} + 5541 \nu^{8} - 13988 \nu^{7} + 28007 \nu^{6} + \cdots + 6405 ) / 338 $ (-66v^11 + 363v^10 - 1780v^9 + 5541v^8 - 13988v^7 + 28007v^6 - 44042v^5 + 58215v^4 - 56668v^3 + 45415v^2 - 21808*v + 6405) / 338
$\beta_{6}$	$=$	$ ( 206 \nu^{11} - 964 \nu^{10} + 4552 \nu^{9} - 11733 \nu^{8} + 26104 \nu^{7} - 38626 \nu^{6} + \cdots + 2163 ) / 338 $ (206v^11 - 964v^10 + 4552v^9 - 11733v^8 + 26104v^7 - 38626v^6 + 45088v^5 - 32797v^4 + 12738v^3 + 2950v^2 - 5422*v + 2163) / 338
$\beta_{7}$	$=$	$ ( - 2 \nu^{10} + 10 \nu^{9} - 47 \nu^{8} + 128 \nu^{7} - 290 \nu^{6} + 464 \nu^{5} - 583 \nu^{4} + \cdots - 21 ) / 2 $ (-2v^10 + 10v^9 - 47v^8 + 128v^7 - 290v^6 + 464v^5 - 583v^4 + 522v^3 - 324v^2 + 120v - 21) / 2
$\beta_{8}$	$=$	$ ( - 103 \nu^{11} + 482 \nu^{10} - 2276 \nu^{9} + 5951 \nu^{8} - 13390 \nu^{7} + 20834 \nu^{6} + \cdots + 355 ) / 169 $ (-103v^11 + 482v^10 - 2276v^9 + 5951v^8 - 13390v^7 + 20834v^6 - 25924v^5 + 23074v^4 - 14481v^3 + 6299v^2 - 1683*v + 355) / 169
$\beta_{9}$	$=$	$ ( 57 \nu^{11} - 398 \nu^{10} + 1906 \nu^{9} - 6268 \nu^{8} + 15353 \nu^{7} - 28820 \nu^{6} + 40940 \nu^{5} + \cdots + 514 ) / 169 $ (57v^11 - 398v^10 + 1906v^9 - 6268v^8 + 15353v^7 - 28820v^6 + 40940v^5 - 43386v^4 + 32609v^3 - 14694v^2 + 2856*v + 514) / 169
$\beta_{10}$	$=$	$ ( 66 \nu^{11} - 701 \nu^{10} + 3470 \nu^{9} - 13484 \nu^{8} + 35620 \nu^{7} - 77017 \nu^{6} + \cdots - 10292 ) / 338 $ (66v^11 - 701v^10 + 3470v^9 - 13484v^8 + 35620v^7 - 77017v^6 + 122458v^5 - 156742v^4 + 144886v^3 - 100171v^2 + 42764*v - 10292) / 338
$\beta_{11}$	$=$	$ ( - 258 \nu^{11} + 1757 \nu^{10} - 8894 \nu^{9} + 29803 \nu^{8} - 78156 \nu^{7} + 155665 \nu^{6} + \cdots + 17571 ) / 338 $ (-258v^11 + 1757v^10 - 8894v^9 + 29803v^8 - 78156v^7 + 155665v^6 - 245756v^5 + 297503v^4 - 275416v^3 + 182417v^2 - 78090*v + 17571) / 338

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{7} + \beta_{5} + 1 ) / 2 $ (b10 - b7 + b5 + 1) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{10} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 4 ) / 2 $ (2b10 + 2b8 - 2*b7 + b6 + b5 + b3 - 4) / 2
$\nu^{3}$	$=$	$ ( \beta_{11} - 2 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 6 ) / 2 $ (b11 - 2b10 + b9 + 3b8 + 3b7 + 2b6 - b5 + 5b4 + b3 - 3b2 - b1 - 6) / 2
$\nu^{4}$	$=$	$ \beta_{11} - 6 \beta_{10} - 5 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} - \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + \cdots + 6 $ b11 - 6b10 - 5b8 + 8b7 - 3b6 - b5 + 5b4 - 2b3 - 2b2 - 2b1 + 6
$\nu^{5}$	$=$	$ ( - 6 \beta_{11} - 15 \beta_{9} - 33 \beta_{8} + 2 \beta_{7} - 20 \beta_{6} + 3 \beta_{5} - 29 \beta_{4} + \cdots + 38 ) / 2 $ (-6b11 - 15b9 - 33b8 + 2b7 - 20b6 + 3b5 - 29b4 - 7b3 + 20b2 + 5b1 + 38) / 2
$\nu^{6}$	$=$	$ ( - 23 \beta_{11} + 69 \beta_{10} - 22 \beta_{9} + 26 \beta_{8} - 96 \beta_{7} + 25 \beta_{6} + 2 \beta_{5} + \cdots - 33 ) / 2 $ (-23b11 + 69b10 - 22b9 + 26b8 - 96b7 + 25b6 + 2b5 - 112b4 + 19b3 + 49b2 + 48*b1 - 33) / 2
$\nu^{7}$	$=$	$ ( 16 \beta_{11} + 55 \beta_{10} + 102 \beta_{9} + 246 \beta_{8} - 117 \beta_{7} + 161 \beta_{6} + \cdots - 243 ) / 2 $ (16b11 + 55b10 + 102b9 + 246b8 - 117b7 + 161b6 - 30b5 + 82b4 + 55b3 - 98b2 + 24*b1 - 243) / 2
$\nu^{8}$	$=$	$ 88 \beta_{11} - 199 \beta_{10} + 150 \beta_{9} + 47 \beta_{8} + 248 \beta_{7} - 14 \beta_{6} - 7 \beta_{5} + \cdots + 2 $ 88b11 - 199b10 + 150b9 + 47b8 + 248b7 - 14b6 - 7b5 + 437b4 - 41b3 - 223b2 - 174*b1 + 2
$\nu^{9}$	$=$	$ ( 74 \beta_{11} - 707 \beta_{10} - 423 \beta_{9} - 1539 \beta_{8} + 1273 \beta_{7} - 1136 \beta_{6} + \cdots + 1551 ) / 2 $ (74b11 - 707b10 - 423b9 - 1539b8 + 1273b7 - 1136b6 + 262b5 + 349b4 - 433b3 + 248b2 - 591*b1 + 1551) / 2
$\nu^{10}$	$=$	$ ( - 1121 \beta_{11} + 2045 \beta_{10} - 2666 \beta_{9} - 2212 \beta_{8} - 2014 \beta_{7} - 874 \beta_{6} + \cdots + 1333 ) / 2 $ (-1121b11 + 2045b10 - 2666b9 - 2212b8 - 2014b7 - 874b6 + 345b5 - 5644b4 + 168b3 + 3367b2 + 1864*b1 + 1333) / 2
$\nu^{11}$	$=$	$ ( - 1627 \beta_{11} + 6703 \beta_{10} + 129 \beta_{9} + 8129 \beta_{8} - 10552 \beta_{7} + 6991 \beta_{6} + \cdots - 9175 ) / 2 $ (-1627b11 + 6703b10 + 129b9 + 8129b8 - 10552b7 + 6991b6 - 1697b5 - 8105b4 + 3096b3 + 1651b2 + 6251*b1 - 9175) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1 - \beta_{4}$	$1$	$1$

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} $

673.1

0.500000	+	1.38157i
0.500000	−	1.77042i
0.500000	−	2.57244i
0.500000	+	0.418260i
0.500000	−	0.675156i
0.500000	+	1.48614i
0.500000	−	1.48614i
0.500000	+	0.675156i
0.500000	−	0.418260i
0.500000	+	2.57244i
0.500000	+	1.77042i
0.500000	−	1.38157i

0.500000

0.866025i

−

3.05170i

0.282067

0.162852i

−0.500000

0.866025i

673.2

0.500000

0.866025i

−

1.18206i

2.61613

1.51042i

−0.500000

0.866025i

673.3

0.500000

0.866025i

−

0.911864i

−0.371524

−

0.214499i

−0.500000

0.866025i

673.4

0.500000

0.866025i

−

0.350612i

−1.84229

−

1.06365i

−0.500000

0.866025i

673.5

0.500000

0.866025i

1.96357i

3.32151

1.91767i

−0.500000

0.866025i

673.6

0.500000

0.866025i

3.53268i

−1.00589

−

0.580753i

−0.500000

0.866025i

1057.1

0.500000

−

0.866025i

−

3.53268i

−1.00589

0.580753i

−0.500000

−

0.866025i

1057.2

0.500000

−

0.866025i

−

1.96357i

3.32151

−

1.91767i

−0.500000

−

0.866025i

1057.3

0.500000

−

0.866025i

0.350612i

−1.84229

1.06365i

−0.500000

−

0.866025i

1057.4

0.500000

−

0.866025i

0.911864i

−0.371524

0.214499i

−0.500000

−

0.866025i

1057.5

0.500000

−

0.866025i

1.18206i

2.61613

−

1.51042i

−0.500000

−

0.866025i

1057.6

0.500000

−

0.866025i

3.05170i

0.282067

−

0.162852i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1248.2.bv.b	yes	12
4.b	odd	2	1		1248.2.bv.a	✓	12
13.e	even	6	1	inner	1248.2.bv.b	yes	12
52.i	odd	6	1		1248.2.bv.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1248.2.bv.a	✓	12	4.b	odd	2	1
1248.2.bv.a	✓	12	52.i	odd	6	1
1248.2.bv.b	yes	12	1.a	even	1	1	trivial
1248.2.bv.b	yes	12	13.e	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1248, [\chi])$:

$ T_{5}^{12} + 28T_{5}^{10} + 262T_{5}^{8} + 956T_{5}^{6} + 1345T_{5}^{4} + 672T_{5}^{2} + 64 $

$ T_{7}^{12} - 6 T_{7}^{11} + 3 T_{7}^{10} + 54 T_{7}^{9} - 23 T_{7}^{8} - 324 T_{7}^{7} + 152 T_{7}^{6} + \cdots + 16 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$5$	$ T^{12} + 28 T^{10} + \cdots + 64 $ T^12 + 28T^10 + 262T^8 + 956T^6 + 1345T^4 + 672*T^2 + 64
$7$	$ T^{12} - 6 T^{11} + \cdots + 16 $ T^12 - 6T^11 + 3T^10 + 54T^9 - 23T^8 - 324T^7 + 152T^6 + 1128T^5 + 988T^4 - 128*T^2 + 16
$11$	$ T^{12} - 32 T^{10} + \cdots + 256 $ T^12 - 32T^10 + 988T^8 - 2256T^7 + 608T^6 + 2592T^5 - 368T^4 - 1728T^3 + 192T^2 + 768*T + 256
$13$	$ T^{12} + 2 T^{11} + \cdots + 4826809 $ T^12 + 2T^11 + 13T^10 + 86T^9 + 206T^8 + 434T^7 + 4697T^6 + 5642T^5 + 34814T^4 + 188942T^3 + 371293T^2 + 742586*T + 4826809
$17$	$ T^{12} + 54 T^{10} + \cdots + 35344 $ T^12 + 54T^10 - 176T^9 + 2583T^8 - 5040T^7 + 25350T^6 - 1800T^5 + 126081T^4 - 62816T^3 + 145548T^2 + 54144T + 35344
$19$	$ T^{12} - 48 T^{10} + \cdots + 256 $ T^12 - 48T^10 + 1900T^8 - 1872T^7 - 19168T^6 + 9696T^5 + 168208T^4 + 290880T^3 + 166336T^2 - 11520*T + 256
$23$	$ T^{12} + 8 T^{11} + \cdots + 1024 $ T^12 + 8T^11 + 76T^10 + 432T^9 + 3052T^8 + 15152T^7 + 66096T^6 + 191840T^5 + 434704T^4 + 581696T^3 + 540032T^2 + 24064*T + 1024
$29$	$ T^{12} + \cdots + 119946304 $ T^12 - 8T^11 + 114T^10 - 312T^9 + 4243T^8 - 4188T^7 + 127538T^6 + 74196T^5 + 2121633T^4 + 3302684T^3 + 28650584T^2 + 44552736*T + 119946304
$31$	$ T^{12} + \cdots + 100962304 $ T^12 + 214T^10 + 16857T^8 + 620064T^6 + 10843520T^4 + 76900352*T^2 + 100962304
$37$	$ T^{12} - 94 T^{10} + \cdots + 38142976 $ T^12 - 94T^10 + 6647T^8 - 17016T^7 - 180486T^6 + 735504T^5 + 3740649T^4 - 31889352T^3 + 84261472T^2 - 89971968*T + 38142976
$41$	$ T^{12} + 24 T^{11} + \cdots + 746496 $ T^12 + 24T^11 + 162T^10 - 720T^9 - 9657T^8 + 70200T^7 + 1506330T^6 + 9731016T^5 + 32622345T^4 + 59340600T^3 + 51360480T^2 + 10264320*T + 746496
$43$	$ T^{12} - 6 T^{11} + \cdots + 17073424 $ T^12 - 6T^11 + 119T^10 - 118T^9 + 7473T^8 - 15420T^7 + 177896T^6 - 469464T^5 + 3488044T^4 - 8895648T^3 + 20017728T^2 - 20759168*T + 17073424
$47$	$ T^{12} + \cdots + 20565854464 $ T^12 + 384T^10 + 56872T^8 + 4169312T^6 + 158639248T^4 + 2936798080*T^2 + 20565854464
$53$	$ (T^{6} - 12 T^{5} + \cdots + 17488)^{2} $ (T^6 - 12T^5 - 74T^4 + 1196T^3 - 2639T^2 - 5960*T + 17488)^2
$59$	$ T^{12} + \cdots + 4294967296 $ T^12 + 12T^11 - 104T^10 - 1824T^9 + 12864T^8 + 116736T^7 - 761856T^6 - 4718592T^5 + 41418752T^4 - 469762048*T^2 + 4294967296
$61$	$ T^{12} + \cdots + 20054241769 $ T^12 + 18T^11 + 341T^10 + 3238T^9 + 38670T^8 + 293058T^7 + 2782433T^6 + 15042570T^5 + 96690574T^4 + 310737582T^3 + 1787076981T^2 + 4173901562*T + 20054241769
$67$	$ T^{12} - 18 T^{11} + \cdots + 21904 $ T^12 - 18T^11 + 27T^10 + 1458T^9 - 503T^8 - 97692T^7 + 543464T^6 - 627624T^5 - 1600532T^4 + 2166528T^3 + 6433216T^2 + 646464*T + 21904
$71$	$ T^{12} - 24 T^{11} + \cdots + 2262016 $ T^12 - 24T^11 + 108T^10 + 2016T^9 - 10388T^8 - 185136T^7 + 1751600T^6 - 1850976T^5 - 4344560T^4 + 5081664T^3 + 14782336T^2 + 9601536*T + 2262016
$73$	$ T^{12} + \cdots + 147042338521 $ T^12 + 466T^10 + 87943T^8 + 8614556T^6 + 462609343T^4 + 12928803186*T^2 + 147042338521
$79$	$ (T^{6} + 14 T^{5} + \cdots + 251200)^{2} $ (T^6 + 14T^5 - 167T^4 - 3080T^3 - 6032T^2 + 70080*T + 251200)^2
$83$	$ T^{12} + \cdots + 31946557696 $ T^12 + 496T^10 + 82504T^8 + 6376544T^6 + 247268368T^4 + 4583936640*T^2 + 31946557696
$89$	$ T^{12} - 132 T^{10} + \cdots + 11075584 $ T^12 - 132T^10 + 15376T^8 - 62592T^7 - 186880T^6 + 983040T^5 + 3632128T^4 - 1572864T^3 - 6619136T^2 + 2555904*T + 11075584
$97$	$ T^{12} + \cdots + 394886560000 $ T^12 - 54T^11 + 975T^10 - 162T^9 - 169679T^8 + 135384T^7 + 46168760T^6 - 566386848T^5 + 2031098416T^4 + 4878343680T^3 - 27743657600T^2 - 62136192000*T + 394886560000
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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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.bv (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{4} - 1) q^{9}+O(q^{10}) \) q + b4 * q^3 + (b8 - b7 - b6 - b5 + b2 - 1) * q^5 - b5 * q^7 + (b4 - 1) * q^9	\( q + \beta_{4} q^{3} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{8} + \cdots - \beta_{5}) q^{99}+O(q^{100}) \) q + b4 * q^3 + (b8 - b7 - b6 - b5 + b2 - 1) * q^5 - b5 * q^7 + (b4 - 1) * q^9 + (b10 - b8 + b7 + b5) * q^11 + (b11 + b10 - b9 + b8 + b7) * q^13 + (-b10 - b7 - b6 - b5 + b2 - 1) * q^15 + (-2b9 - b5 - b4 + 2b2 + b1 - 1) * q^17 + (2b10 + 2b8 - b7 + b6) * q^19 + (-b5 + b2 - 1) * q^21 + (-b10 - b9 - b8 + b7 - b5 - 2b4 + 2b1) * q^23 + (-b10 - b8 - 2b6 - b5 + b3 + b2 - 1) q^25 - q^27 + (-b10 + b7 - b6 - 3b5 + b2 - 1) q^29 + (2b11 + b10 + b8 + b5 - b3 - b2 + 1) q^31 + (b7 - b6) * q^33 + (-b11 - b10 + 2b9 - b7 - b6 - b5 + b2 - b1 - 1) q^35 + (-2b10 + b9 - b8 - 2b7 - 3b6 - 2b5 + 3b2 - 3) q^37 + (2b8 - b7 - b6 - b5 + b3 + b2 - b1 - 1) q^39 + (b11 + b10 - b6 + b4 + b3 + 2b2 - 4) q^41 + (-b11 + b10 - 2b8 + 2b5 - b2 + 1) * q^43 + (-b10 - b8) * q^45 + (2b11 - b10 - 2b9 - b8 + 4b7 + 2b6 - 4b4 - b3 - 2b2 + 2b1 + 4) q^47 + (-b11 + b8 - b7 - b6 - 3b5 - 3b4 + b3 + 2b2 - 2) q^49 + (-b9 + b5 + b2 - b1) * q^51 + (-2b10 + b9 - 3b8 + b7 - 2b6 - b3 + b2 + b1 + 1) q^53 + (b11 + 2b10 - b8 - b7 + 3b6 + 3b5 + 4b4 - b3 - b2 + 1) * q^55 + (b10 + b8 - 2b7 - b6 - b5 + 2b2 - 2) * q^57 + (-b11 + b10 + 2b8 - 2b6 + b5 + 2b3 + b2 - 3b1 - 1) * q^59 + (b11 + 2b10 - 2b9 - b8 + 2b4 + 3b2 + b1 - 5) * q^61 + (b2 - 1) * q^63 + (-2b8 - b6 - b5 + 3b4 + b3 + b2 - b1 - 5) * q^65 + (-b11 - b10 + 2b9 - 2b8 - b6 - b3 - 2b2 + 2) q^67 + (-2b9 + b7 + b6 - 2b4 + b1 + 2) * q^69 + (-2b10 - 2b8 - b7 + b6 + 2b5 + 2b4 + b1 + 2) * q^71 + (-2b10 + 2b7 - 4b5 - 2b4 + 2b2 - 1) q^73 + (-b11 - 2b10 - b8 + b7 - b6 - b5 + b3) q^75 - 2 * q^77 + (-3b10 + 2b9 - b8 - 2b7 - 3b5 - b3 - b2 + 2b1 - 3) q^79 - b4 * q^81 + (-2b11 - 3b10 + b9 - b8 + 2b7 - 2b5 + b3 - b1) * q^83 + (-b10 - b8 + b7 - b6 - 2b5 - 2b4 + 3b1 - 2) q^85 + (-b10 + b8 - 2b5 + 2b2 - 2) * q^87 + (b11 + 3b10 - b9 + 2b7 + b6 + 2b5 + b3 - b2 + 1) q^89 + (-b11 + 3b10 - 2b9 + 2b8 + b7 + 2b6 + 2b5 + b3 - 2b2 - 2b1 + 2) q^91 + (b11 + b8 - b7 - b6 - b5 + b3) * q^93 + (b11 - b10 + 2b8 - 3b7 - 3b6 - 3b5 + 6b4 + 3b2 - 9) * q^95 + (b11 + 3b10 + 2b8 + b7 + b6 + 4b5 + 4b4 - 2b3 - b2 - 4b1 + 5) * q^97 + (-b10 + b8 - b6 - b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9	\( 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9} - 2 q^{13} - 8 q^{23} + 4 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{39} - 24 q^{41} + 6 q^{43} - 12 q^{49} + 24 q^{53} + 20 q^{55} - 12 q^{59} - 18 q^{61} - 6 q^{63} - 20 q^{65} + 18 q^{67} + 8 q^{69} + 24 q^{71} + 2 q^{75} - 24 q^{77} - 28 q^{79} - 6 q^{81} - 24 q^{85} - 8 q^{87} - 2 q^{91} + 6 q^{93} - 36 q^{95} + 54 q^{97}+O(q^{100}) \) 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9 - 2 * q^13 - 8 * q^23 + 4 * q^25 - 12 * q^27 + 8 * q^29 - 4 * q^39 - 24 * q^41 + 6 * q^43 - 12 * q^49 + 24 * q^53 + 20 * q^55 - 12 * q^59 - 18 * q^61 - 6 * q^63 - 20 * q^65 + 18 * q^67 + 8 * q^69 + 24 * q^71 + 2 * q^75 - 24 * q^77 - 28 * q^79 - 6 * q^81 - 24 * q^85 - 8 * q^87 - 2 * q^91 + 6 * q^93 - 36 * q^95 + 54 * q^97

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	1248.2.bv.b

Level	$1248$
Weight	$2$
Character orbit	1248.bv
Analytic conductor	$9.965$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.96533017226\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 31 x^{10} - 100 x^{9} + 268 x^{8} - 538 x^{7} + 891 x^{6} - 1144 x^{5} + 1178 x^{4} + \cdots + 37 \) x^12 - 6x^11 + 31x^10 - 100x^9 + 268x^8 - 538x^7 + 891x^6 - 1144x^5 + 1178x^4 - 910x^3 + 517x^2 - 188*x + 37
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 57 \nu^{11} + 229 \nu^{10} - 1061 \nu^{9} + 2381 \nu^{8} - 4875 \nu^{7} + 6005 \nu^{6} - 5619 \nu^{5} + \cdots + 669 ) / 169 \) (-57v^11 + 229v^10 - 1061v^9 + 2381v^8 - 4875v^7 + 6005v^6 - 5619v^5 + 2995v^4 - 161v^3 + 667v^2 - 659*v + 669) / 169
\(\beta_{2}\)	\(=\)	\( ( 66 \nu^{11} - 363 \nu^{10} + 1780 \nu^{9} - 5034 \nu^{8} + 11960 \nu^{7} - 18543 \nu^{6} + 22748 \nu^{5} + \cdots + 5932 ) / 338 \) (66v^11 - 363v^10 + 1780v^9 - 5034v^8 + 11960v^7 - 18543v^6 + 22748v^5 - 13430v^4 + 222v^3 + 14411v^2 - 13006*v + 5932) / 338
\(\beta_{3}\)	\(=\)	\( ( 140 \nu^{11} - 601 \nu^{10} + 2772 \nu^{9} - 6530 \nu^{8} + 13468 \nu^{7} - 16703 \nu^{6} + \cdots + 4850 ) / 338 \) (140v^11 - 601v^10 + 2772v^9 - 6530v^8 + 13468v^7 - 16703v^6 + 14566v^5 - 1284v^4 - 11482v^3 + 17945v^2 - 11006*v + 4850) / 338
\(\beta_{4}\)	\(=\)	\( ( 72 \nu^{11} - 396 \nu^{10} + 2034 \nu^{9} - 6183 \nu^{8} + 16120 \nu^{7} - 30338 \nu^{6} + 47462 \nu^{5} + \cdots - 2455 ) / 338 \) (72v^11 - 396v^10 + 2034v^9 - 6183v^8 + 16120v^7 - 30338v^6 + 47462v^5 - 55257v^4 + 50512v^3 - 32152v^2 + 13374*v - 2455) / 338
\(\beta_{5}\)	\(=\)	\( ( - 66 \nu^{11} + 363 \nu^{10} - 1780 \nu^{9} + 5541 \nu^{8} - 13988 \nu^{7} + 28007 \nu^{6} + \cdots + 6405 ) / 338 \) (-66v^11 + 363v^10 - 1780v^9 + 5541v^8 - 13988v^7 + 28007v^6 - 44042v^5 + 58215v^4 - 56668v^3 + 45415v^2 - 21808*v + 6405) / 338
\(\beta_{6}\)	\(=\)	\( ( 206 \nu^{11} - 964 \nu^{10} + 4552 \nu^{9} - 11733 \nu^{8} + 26104 \nu^{7} - 38626 \nu^{6} + \cdots + 2163 ) / 338 \) (206v^11 - 964v^10 + 4552v^9 - 11733v^8 + 26104v^7 - 38626v^6 + 45088v^5 - 32797v^4 + 12738v^3 + 2950v^2 - 5422*v + 2163) / 338
\(\beta_{7}\)	\(=\)	\( ( - 2 \nu^{10} + 10 \nu^{9} - 47 \nu^{8} + 128 \nu^{7} - 290 \nu^{6} + 464 \nu^{5} - 583 \nu^{4} + \cdots - 21 ) / 2 \) (-2v^10 + 10v^9 - 47v^8 + 128v^7 - 290v^6 + 464v^5 - 583v^4 + 522v^3 - 324v^2 + 120v - 21) / 2
\(\beta_{8}\)	\(=\)	\( ( - 103 \nu^{11} + 482 \nu^{10} - 2276 \nu^{9} + 5951 \nu^{8} - 13390 \nu^{7} + 20834 \nu^{6} + \cdots + 355 ) / 169 \) (-103v^11 + 482v^10 - 2276v^9 + 5951v^8 - 13390v^7 + 20834v^6 - 25924v^5 + 23074v^4 - 14481v^3 + 6299v^2 - 1683*v + 355) / 169
\(\beta_{9}\)	\(=\)	\( ( 57 \nu^{11} - 398 \nu^{10} + 1906 \nu^{9} - 6268 \nu^{8} + 15353 \nu^{7} - 28820 \nu^{6} + 40940 \nu^{5} + \cdots + 514 ) / 169 \) (57v^11 - 398v^10 + 1906v^9 - 6268v^8 + 15353v^7 - 28820v^6 + 40940v^5 - 43386v^4 + 32609v^3 - 14694v^2 + 2856*v + 514) / 169
\(\beta_{10}\)	\(=\)	\( ( 66 \nu^{11} - 701 \nu^{10} + 3470 \nu^{9} - 13484 \nu^{8} + 35620 \nu^{7} - 77017 \nu^{6} + \cdots - 10292 ) / 338 \) (66v^11 - 701v^10 + 3470v^9 - 13484v^8 + 35620v^7 - 77017v^6 + 122458v^5 - 156742v^4 + 144886v^3 - 100171v^2 + 42764*v - 10292) / 338
\(\beta_{11}\)	\(=\)	\( ( - 258 \nu^{11} + 1757 \nu^{10} - 8894 \nu^{9} + 29803 \nu^{8} - 78156 \nu^{7} + 155665 \nu^{6} + \cdots + 17571 ) / 338 \) (-258v^11 + 1757v^10 - 8894v^9 + 29803v^8 - 78156v^7 + 155665v^6 - 245756v^5 + 297503v^4 - 275416v^3 + 182417v^2 - 78090*v + 17571) / 338

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{7} + \beta_{5} + 1 ) / 2 \) (b10 - b7 + b5 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{10} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 4 ) / 2 \) (2b10 + 2b8 - 2*b7 + b6 + b5 + b3 - 4) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} - 2 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 6 ) / 2 \) (b11 - 2b10 + b9 + 3b8 + 3b7 + 2b6 - b5 + 5b4 + b3 - 3b2 - b1 - 6) / 2
\(\nu^{4}\)	\(=\)	\( \beta_{11} - 6 \beta_{10} - 5 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} - \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + \cdots + 6 \) b11 - 6b10 - 5b8 + 8b7 - 3b6 - b5 + 5b4 - 2b3 - 2b2 - 2b1 + 6
\(\nu^{5}\)	\(=\)	\( ( - 6 \beta_{11} - 15 \beta_{9} - 33 \beta_{8} + 2 \beta_{7} - 20 \beta_{6} + 3 \beta_{5} - 29 \beta_{4} + \cdots + 38 ) / 2 \) (-6b11 - 15b9 - 33b8 + 2b7 - 20b6 + 3b5 - 29b4 - 7b3 + 20b2 + 5b1 + 38) / 2
\(\nu^{6}\)	\(=\)	\( ( - 23 \beta_{11} + 69 \beta_{10} - 22 \beta_{9} + 26 \beta_{8} - 96 \beta_{7} + 25 \beta_{6} + 2 \beta_{5} + \cdots - 33 ) / 2 \) (-23b11 + 69b10 - 22b9 + 26b8 - 96b7 + 25b6 + 2b5 - 112b4 + 19b3 + 49b2 + 48*b1 - 33) / 2
\(\nu^{7}\)	\(=\)	\( ( 16 \beta_{11} + 55 \beta_{10} + 102 \beta_{9} + 246 \beta_{8} - 117 \beta_{7} + 161 \beta_{6} + \cdots - 243 ) / 2 \) (16b11 + 55b10 + 102b9 + 246b8 - 117b7 + 161b6 - 30b5 + 82b4 + 55b3 - 98b2 + 24*b1 - 243) / 2
\(\nu^{8}\)	\(=\)	\( 88 \beta_{11} - 199 \beta_{10} + 150 \beta_{9} + 47 \beta_{8} + 248 \beta_{7} - 14 \beta_{6} - 7 \beta_{5} + \cdots + 2 \) 88b11 - 199b10 + 150b9 + 47b8 + 248b7 - 14b6 - 7b5 + 437b4 - 41b3 - 223b2 - 174*b1 + 2
\(\nu^{9}\)	\(=\)	\( ( 74 \beta_{11} - 707 \beta_{10} - 423 \beta_{9} - 1539 \beta_{8} + 1273 \beta_{7} - 1136 \beta_{6} + \cdots + 1551 ) / 2 \) (74b11 - 707b10 - 423b9 - 1539b8 + 1273b7 - 1136b6 + 262b5 + 349b4 - 433b3 + 248b2 - 591*b1 + 1551) / 2
\(\nu^{10}\)	\(=\)	\( ( - 1121 \beta_{11} + 2045 \beta_{10} - 2666 \beta_{9} - 2212 \beta_{8} - 2014 \beta_{7} - 874 \beta_{6} + \cdots + 1333 ) / 2 \) (-1121b11 + 2045b10 - 2666b9 - 2212b8 - 2014b7 - 874b6 + 345b5 - 5644b4 + 168b3 + 3367b2 + 1864*b1 + 1333) / 2
\(\nu^{11}\)	\(=\)	\( ( - 1627 \beta_{11} + 6703 \beta_{10} + 129 \beta_{9} + 8129 \beta_{8} - 10552 \beta_{7} + 6991 \beta_{6} + \cdots - 9175 ) / 2 \) (-1627b11 + 6703b10 + 129b9 + 8129b8 - 10552b7 + 6991b6 - 1697b5 - 8105b4 + 3096b3 + 1651b2 + 6251*b1 - 9175) / 2

\(n\)	\(703\)	\(769\)	\(833\)	\(1093\)
\(\chi(n)\)	\(1\)	\(1 - \beta_{4}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 673.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$5$	\( T^{12} + 28 T^{10} + \cdots + 64 \) T^12 + 28T^10 + 262T^8 + 956T^6 + 1345T^4 + 672*T^2 + 64
$7$	\( T^{12} - 6 T^{11} + \cdots + 16 \) T^12 - 6T^11 + 3T^10 + 54T^9 - 23T^8 - 324T^7 + 152T^6 + 1128T^5 + 988T^4 - 128*T^2 + 16
$11$	\( T^{12} - 32 T^{10} + \cdots + 256 \) T^12 - 32T^10 + 988T^8 - 2256T^7 + 608T^6 + 2592T^5 - 368T^4 - 1728T^3 + 192T^2 + 768*T + 256
$13$	\( T^{12} + 2 T^{11} + \cdots + 4826809 \) T^12 + 2T^11 + 13T^10 + 86T^9 + 206T^8 + 434T^7 + 4697T^6 + 5642T^5 + 34814T^4 + 188942T^3 + 371293T^2 + 742586*T + 4826809
$17$	\( T^{12} + 54 T^{10} + \cdots + 35344 \) T^12 + 54T^10 - 176T^9 + 2583T^8 - 5040T^7 + 25350T^6 - 1800T^5 + 126081T^4 - 62816T^3 + 145548T^2 + 54144T + 35344
$19$	\( T^{12} - 48 T^{10} + \cdots + 256 \) T^12 - 48T^10 + 1900T^8 - 1872T^7 - 19168T^6 + 9696T^5 + 168208T^4 + 290880T^3 + 166336T^2 - 11520*T + 256
$23$	\( T^{12} + 8 T^{11} + \cdots + 1024 \) T^12 + 8T^11 + 76T^10 + 432T^9 + 3052T^8 + 15152T^7 + 66096T^6 + 191840T^5 + 434704T^4 + 581696T^3 + 540032T^2 + 24064*T + 1024
$29$	\( T^{12} + \cdots + 119946304 \) T^12 - 8T^11 + 114T^10 - 312T^9 + 4243T^8 - 4188T^7 + 127538T^6 + 74196T^5 + 2121633T^4 + 3302684T^3 + 28650584T^2 + 44552736*T + 119946304
$31$	\( T^{12} + \cdots + 100962304 \) T^12 + 214T^10 + 16857T^8 + 620064T^6 + 10843520T^4 + 76900352*T^2 + 100962304
$37$	\( T^{12} - 94 T^{10} + \cdots + 38142976 \) T^12 - 94T^10 + 6647T^8 - 17016T^7 - 180486T^6 + 735504T^5 + 3740649T^4 - 31889352T^3 + 84261472T^2 - 89971968*T + 38142976
$41$	\( T^{12} + 24 T^{11} + \cdots + 746496 \) T^12 + 24T^11 + 162T^10 - 720T^9 - 9657T^8 + 70200T^7 + 1506330T^6 + 9731016T^5 + 32622345T^4 + 59340600T^3 + 51360480T^2 + 10264320*T + 746496
$43$	\( T^{12} - 6 T^{11} + \cdots + 17073424 \) T^12 - 6T^11 + 119T^10 - 118T^9 + 7473T^8 - 15420T^7 + 177896T^6 - 469464T^5 + 3488044T^4 - 8895648T^3 + 20017728T^2 - 20759168*T + 17073424
$47$	\( T^{12} + \cdots + 20565854464 \) T^12 + 384T^10 + 56872T^8 + 4169312T^6 + 158639248T^4 + 2936798080*T^2 + 20565854464
$53$	\( (T^{6} - 12 T^{5} + \cdots + 17488)^{2} \) (T^6 - 12T^5 - 74T^4 + 1196T^3 - 2639T^2 - 5960*T + 17488)^2
$59$	\( T^{12} + \cdots + 4294967296 \) T^12 + 12T^11 - 104T^10 - 1824T^9 + 12864T^8 + 116736T^7 - 761856T^6 - 4718592T^5 + 41418752T^4 - 469762048*T^2 + 4294967296
$61$	\( T^{12} + \cdots + 20054241769 \) T^12 + 18T^11 + 341T^10 + 3238T^9 + 38670T^8 + 293058T^7 + 2782433T^6 + 15042570T^5 + 96690574T^4 + 310737582T^3 + 1787076981T^2 + 4173901562*T + 20054241769
$67$	\( T^{12} - 18 T^{11} + \cdots + 21904 \) T^12 - 18T^11 + 27T^10 + 1458T^9 - 503T^8 - 97692T^7 + 543464T^6 - 627624T^5 - 1600532T^4 + 2166528T^3 + 6433216T^2 + 646464*T + 21904
$71$	\( T^{12} - 24 T^{11} + \cdots + 2262016 \) T^12 - 24T^11 + 108T^10 + 2016T^9 - 10388T^8 - 185136T^7 + 1751600T^6 - 1850976T^5 - 4344560T^4 + 5081664T^3 + 14782336T^2 + 9601536*T + 2262016
$73$	\( T^{12} + \cdots + 147042338521 \) T^12 + 466T^10 + 87943T^8 + 8614556T^6 + 462609343T^4 + 12928803186*T^2 + 147042338521
$79$	\( (T^{6} + 14 T^{5} + \cdots + 251200)^{2} \) (T^6 + 14T^5 - 167T^4 - 3080T^3 - 6032T^2 + 70080*T + 251200)^2
$83$	\( T^{12} + \cdots + 31946557696 \) T^12 + 496T^10 + 82504T^8 + 6376544T^6 + 247268368T^4 + 4583936640*T^2 + 31946557696
$89$	\( T^{12} - 132 T^{10} + \cdots + 11075584 \) T^12 - 132T^10 + 15376T^8 - 62592T^7 - 186880T^6 + 983040T^5 + 3632128T^4 - 1572864T^3 - 6619136T^2 + 2555904*T + 11075584
$97$	\( T^{12} + \cdots + 394886560000 \) T^12 - 54T^11 + 975T^10 - 162T^9 - 169679T^8 + 135384T^7 + 46168760T^6 - 566386848T^5 + 2031098416T^4 + 4878343680T^3 - 27743657600T^2 - 62136192000*T + 394886560000
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