LMFDB - Newform orbit 1176.2.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(589,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("1176.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.c (of order $2$, degree $1$, 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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.84396412309504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 $ x^12 - 6x^11 + 18x^10 - 32x^9 + 39x^8 - 28x^7 + 13x^6 + 10x^5 + 6x^4 + 14x^3 + 17x^2 + 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{6} q^{2} + \beta_{7} q^{3} - \beta_{11} q^{4} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{5}+ \cdots - q^{9}+O(q^{10}) $ q - b6 * q^2 + b7 * q^3 - b11 * q^4 + (-b8 + b7 - b5 - b2) * q^5 + b5 * q^6 + (b9 - b1) * q^8 - q^9	$ q - \beta_{6} q^{2} + \beta_{7} q^{3} - \beta_{11} q^{4} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) $ q - b6 * q^2 + b7 * q^3 - b11 * q^4 + (-b8 + b7 - b5 - b2) * q^5 + b5 * q^6 + (b9 - b1) * q^8 - q^9 + (-b10 - b7 + b5 + b3) * q^10 + (2b11 - 2b6 - b1 - 1) * q^11 - b2 * q^12 + (-b10 + b8 - b7 + 2b5) q^13 + (b1 - 1) * q^15 + (b9 + b1 - 2) * q^16 + (-b8 - b7 + b5 + 2b3 - b2) q^17 + b6 * q^18 + (-b10 - b8 + b7 - 2b2) q^19 + (2b8 + 2b7) * q^20 + (-2b11 - b9 + b6 + b4 + 2b1 - 1) * q^22 + (-b11 - b6 + b4) * q^23 + (-b8 - b5 + b3 - b2) * q^24 + (-b11 - b6 + b4 + b1) * q^25 + (4b7 - 2b5 - 2b2) q^26 - b7 * q^27 + (b11 + 2b9 - b6 + b4 - b1 - 1) q^29 + (-b9 + b6 - b4 + 1) * q^30 + (-b10 + b8 + b7 - 2b5 + 2b3 - 2b2) q^31 + (-b9 + 2b6 - 2b4 + b1) * q^32 + (-b8 - b7 + b5 + b2) * q^33 + (b10 + 2b8 - 3b7 + b5 + b3) * q^34 + b11 * q^36 + (b11 + 2b9 - b6 + b4 - b1 - 1) q^37 + (-2b10 + 2b7 + 2b3 - 2b2) * q^38 + (b11 + b6 - b4 - b1 + 1) * q^39 + (2b10 - 2b8 + 2b7) q^40 + (-2b10 - b8 - b7 - b5 + 2b3 + b2) * q^41 + (-2b9 - 2b6 - 2b4) q^43 + (2b6 - 2b1 + 6) * q^44 + (b8 - b7 + b5 + b2) * q^45 + (-2b11 + b9 + b6 + b4 + 3) q^46 + (-2b10 - 2b5 + 2b2) q^47 + (b8 - 2b7 + b5 + b3 + b2) q^48 + (-2b11 + b6 + 4) q^50 + (-2b9 + 2b6 + b1 + 1) * q^51 + (-2b8 + 2b5 + 2b3) q^52 + (b11 - 2b9 + 3b6 + b4 + b1 + 1) * q^53 - b5 * q^54 + (-b10 - 3b8 - 3b7 + 2b5 + 2b3 + 2b2) q^55 + (b11 + b6 - b4 + b1 - 1) * q^57 + (-2b11 + 2b6 + 4b1) q^58 + (-2b10 + 4b8 + 6b5 + 2b2) * q^59 + (2b11 - 2b6 - 2b1 - 2) q^60 + (b10 - 3b8 - 5b7 - 4b5 - 2b2) * q^61 + (2b10 + 2b7 + 2b3 + 2b2) * q^62 + (4b11 - b9 - 2b6 - 2b4 - 3b1 - 4) * q^64 + (2b11 + 2b6 - 2b4 - 4b1 + 4) * q^65 + (-b10 + 2b8 - b7 + b5 - b3) q^66 + (-2b11 + 2b9) * q^67 + (4b10 - 2b8 - 2b7 - 4b5) * q^68 + (-b10 + b5 - b2) * q^69 + (-b11 - b6 + b4 - 2b1 + 2) q^71 + (-b9 + b1) * q^72 + (2b10 + 2b5 - 2b2) q^73 + (-2b11 + 2b6 + 4b1) q^74 + (-b10 + b8 + 2b5) q^75 + (2b8 + 4b7 + 2b5 + 2b3) * q^76 + (2b11 - 2b6 - 4) * q^78 + (2b1 - 2) q^79 + (-8b7 + 4b5) * q^80 + q^81 + (-b10 + 6b8 + 3b7 + 3b5 - b3 + 4b2) * q^82 + (-2b10 - 4b7 + 2b5 - 2b2) * q^83 + (-3b11 - 2b9 + 7b6 + b4 + 3b1 + 3) * q^85 + (-2b6 - 4b1 - 4) * q^86 + (-b10 - b8 - b7 + 2b3) q^87 + (2b11 + 2b9 - 6b6 + 2b4 - 2) * q^88 + (2b10 - 3b8 - 3b7 + 5b5 - 2b3 + 3b2) * q^89 + (b10 + b7 - b5 - b3) * q^90 + (2b9 - 2b6 + 2) * q^92 + (3b11 - 2b9 - 3b6 - b4 - b1 - 1) q^93 + (-2b10 + 4b8 + 6b7 + 2b5 - 2b3 + 4b2) * q^94 + (-2b1 - 6) q^95 + (2b10 + b8 - b5 - b3 + b2) q^96 + (-2b10 - 2b5 + 2b2) q^97 + (-2b11 + 2b6 + b1 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - 6 q^{4} + 2 q^{8} - 12 q^{9}+O(q^{10}) $ 12 * q + 2 * q^2 - 6 * q^4 + 2 * q^8 - 12 * q^9	$ 12 q + 2 q^{2} - 6 q^{4} + 2 q^{8} - 12 q^{9} - 8 q^{15} - 14 q^{16} - 2 q^{18} - 28 q^{22} - 8 q^{23} - 4 q^{25} + 8 q^{30} + 2 q^{32} + 6 q^{36} + 16 q^{39} + 60 q^{44} + 24 q^{46} + 34 q^{50} - 16 q^{60} - 30 q^{64} + 48 q^{65} + 8 q^{71} - 2 q^{72} - 32 q^{78} - 16 q^{79} + 12 q^{81} - 60 q^{86} + 4 q^{88} + 40 q^{92} - 80 q^{95}+O(q^{100}) $ 12 * q + 2 * q^2 - 6 * q^4 + 2 * q^8 - 12 * q^9 - 8 * q^15 - 14 * q^16 - 2 * q^18 - 28 * q^22 - 8 * q^23 - 4 * q^25 + 8 * q^30 + 2 * q^32 + 6 * q^36 + 16 * q^39 + 60 * q^44 + 24 * q^46 + 34 * q^50 - 16 * q^60 - 30 * q^64 + 48 * q^65 + 8 * q^71 - 2 * q^72 - 32 * q^78 - 16 * q^79 + 12 * q^81 - 60 * q^86 + 4 * q^88 + 40 * q^92 - 80 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 $ x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 39*x^8 - 28*x^7 + 13*x^6 + 10*x^5 + 6*x^4 + 14*x^3 + 17*x^2 + 10*x + 2 :

$\beta_{1}$	$=$	$ ( - 5 \nu^{11} + 30 \nu^{10} - 91 \nu^{9} + 166 \nu^{8} - 210 \nu^{7} + 154 \nu^{6} - 59 \nu^{5} + \cdots - 48 ) / 16 $ (-5v^11 + 30v^10 - 91v^9 + 166v^8 - 210v^7 + 154v^6 - 59v^5 - 80v^4 + 3v^3 - 102v^2 - 78*v - 48) / 16
$\beta_{2}$	$=$	$ ( 11 \nu^{11} - 75 \nu^{10} + 259 \nu^{9} - 561 \nu^{8} + 876 \nu^{7} - 992 \nu^{6} + 887 \nu^{5} + \cdots + 26 ) / 32 $ (11v^11 - 75v^10 + 259v^9 - 561v^8 + 876v^7 - 992v^6 + 887v^5 - 503v^4 + 315v^3 + 41v^2 + 100*v + 26) / 32
$\beta_{3}$	$=$	$ ( 16 \nu^{11} - 105 \nu^{10} + 348 \nu^{9} - 709 \nu^{8} + 1010 \nu^{7} - 954 \nu^{6} + 622 \nu^{5} + \cdots + 54 ) / 16 $ (16v^11 - 105v^10 + 348v^9 - 709v^8 + 1010v^7 - 954v^6 + 622v^5 - 55v^4 + 58v^3 + 201v^2 + 218*v + 54) / 16
$\beta_{4}$	$=$	$ ( 57 \nu^{11} - 379 \nu^{10} + 1269 \nu^{9} - 2617 \nu^{8} + 3792 \nu^{7} - 3748 \nu^{6} + 2721 \nu^{5} + \cdots + 242 ) / 32 $ (57v^11 - 379v^10 + 1269v^9 - 2617v^8 + 3792v^7 - 3748v^6 + 2721v^5 - 763v^4 + 577v^3 + 545v^2 + 608*v + 242) / 32
$\beta_{5}$	$=$	$ ( - 10 \nu^{11} + 64 \nu^{10} - 206 \nu^{9} + 405 \nu^{8} - 560 \nu^{7} + 518 \nu^{6} - 352 \nu^{5} + \cdots - 50 ) / 4 $ (-10v^11 + 64v^10 - 206v^9 + 405v^8 - 560v^7 + 518v^6 - 352v^5 + 48v^4 - 78v^3 - 117v^2 - 122*v - 50) / 4
$\beta_{6}$	$=$	$ ( - 81 \nu^{11} + 519 \nu^{10} - 1673 \nu^{9} + 3297 \nu^{8} - 4576 \nu^{7} + 4268 \nu^{6} - 2953 \nu^{5} + \cdots - 402 ) / 32 $ (-81v^11 + 519v^10 - 1673v^9 + 3297v^8 - 4576v^7 + 4268v^6 - 2953v^5 + 511v^4 - 757v^3 - 817v^2 - 1048*v - 402) / 32
$\beta_{7}$	$=$	$ ( 85 \nu^{11} - 545 \nu^{10} + 1753 \nu^{9} - 3431 \nu^{8} + 4692 \nu^{7} - 4248 \nu^{6} + 2793 \nu^{5} + \cdots + 422 ) / 32 $ (85v^11 - 545v^10 + 1753v^9 - 3431v^8 + 4692v^7 - 4248v^6 + 2793v^5 - 285v^4 + 657v^3 + 887v^2 + 1092*v + 422) / 32
$\beta_{8}$	$=$	$ ( - 56 \nu^{11} + 361 \nu^{10} - 1168 \nu^{9} + 2309 \nu^{8} - 3210 \nu^{7} + 3010 \nu^{6} - 2102 \nu^{5} + \cdots - 246 ) / 16 $ (-56v^11 + 361v^10 - 1168v^9 + 2309v^8 - 3210v^7 + 3010v^6 - 2102v^5 + 423v^4 - 550v^3 - 529v^2 - 690*v - 246) / 16
$\beta_{9}$	$=$	$ ( 31 \nu^{11} - 197 \nu^{10} + 628 \nu^{9} - 1216 \nu^{8} + 1646 \nu^{7} - 1468 \nu^{6} + 953 \nu^{5} + \cdots + 160 ) / 8 $ (31v^11 - 197v^10 + 628v^9 - 1216v^8 + 1646v^7 - 1468v^6 + 953v^5 - 61v^4 + 230v^3 + 338v^2 + 420*v + 160) / 8
$\beta_{10}$	$=$	$ ( 123 \nu^{11} - 793 \nu^{10} + 2567 \nu^{9} - 5067 \nu^{8} + 6992 \nu^{7} - 6396 \nu^{6} + 4179 \nu^{5} + \cdots + 566 ) / 32 $ (123v^11 - 793v^10 + 2567v^9 - 5067v^8 + 6992v^7 - 6396v^6 + 4179v^5 - 329v^4 + 683v^3 + 1459v^2 + 1520*v + 566) / 32
$\beta_{11}$	$=$	$ ( 88 \nu^{11} - 567 \nu^{10} + 1832 \nu^{9} - 3607 \nu^{8} + 4966 \nu^{7} - 4550 \nu^{6} + 3026 \nu^{5} + \cdots + 402 ) / 16 $ (88v^11 - 567v^10 + 1832v^9 - 3607v^8 + 4966v^7 - 4550v^6 + 3026v^5 - 361v^4 + 626v^3 + 955v^2 + 1046*v + 402) / 16

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

$\nu$	$=$	$ ( -\beta_{11} + \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{3} + \beta _1 + 2 ) / 4 $ (-b11 + b9 - b8 + 2*b6 - b5 + b3 + b1 + 2) / 4
$\nu^{2}$	$=$	$ ( - 2 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots + 1 ) / 4 $ (-2b11 - b10 + b9 - 2b8 + b7 + 5b6 - 5b5 + b4 + b3 - 2b2 + 2b1 + 1) / 4
$\nu^{3}$	$=$	$ ( -2\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} + 3\beta_{6} - 4\beta_{5} + 2\beta_{4} - 4\beta_{2} + 2\beta _1 - 1 ) / 2 $ (-2b11 - b10 + b9 - 2b8 + 3b6 - 4b5 + 2b4 - 4b2 + 2*b1 - 1) / 2
$\nu^{4}$	$=$	$ ( - 5 \beta_{11} + 7 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 10 \beta_{4} + \cdots - 10 ) / 4 $ (-5b11 + 7b9 - 5b8 - 10b7 + 4b6 - 5b5 + 10b4 - 5b3 - 14b2 + 3b1 - 10) / 4
$\nu^{5}$	$=$	$ ( - 2 \beta_{11} + 7 \beta_{10} + 17 \beta_{9} - 4 \beta_{8} - 31 \beta_{7} - 3 \beta_{6} + 15 \beta_{5} + \cdots - 29 ) / 4 $ (-2b11 + 7b10 + 17b9 - 4b8 - 31b7 - 3b6 + 15b5 + 19b4 - 17b3 - 12b2 - 12*b1 - 29) / 4
$\nu^{6}$	$=$	$ ( 3 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} - 2 \beta_{8} - 27 \beta_{7} - 10 \beta_{6} + 33 \beta_{5} + \cdots - 30 ) / 2 $ (3b11 + 13b10 + 11b9 - 2b8 - 27b7 - 10b6 + 33b5 + 12b4 - 19b3 + 4b2 - 29*b1 - 30) / 2
$\nu^{7}$	$=$	$ ( 11 \beta_{11} + 78 \beta_{10} - 9 \beta_{9} - 5 \beta_{8} - 38 \beta_{7} - 60 \beta_{6} + 157 \beta_{5} + \cdots - 96 ) / 4 $ (11b11 + 78b10 - 9b9 - 5b8 - 38b7 - 60b6 + 157b5 + 2b4 - 73b3 + 68b2 - 143*b1 - 96) / 4
$\nu^{8}$	$=$	$ ( 193 \beta_{10} - 139 \beta_{9} + 32 \beta_{8} + 135 \beta_{7} - 157 \beta_{6} + 279 \beta_{5} + \cdots - 133 ) / 4 $ (193b10 - 139b9 + 32b8 + 135b7 - 157b6 + 279b5 - 97b4 - 127b3 + 222b2 - 236b1 - 133) / 4
$\nu^{9}$	$=$	$ ( - 4 \beta_{11} + 180 \beta_{10} - 234 \beta_{9} + 126 \beta_{8} + 343 \beta_{7} - 204 \beta_{6} + \cdots - 84 ) / 2 $ (-4b11 + 180b10 - 234b9 + 126b8 + 343b7 - 204b6 + 191b5 - 181b4 - 83b3 + 286b2 - 112*b1 - 84) / 2
$\nu^{10}$	$=$	$ ( 133 \beta_{11} + 392 \beta_{10} - 1099 \beta_{9} + 957 \beta_{8} + 1842 \beta_{7} - 1016 \beta_{6} + \cdots - 94 ) / 4 $ (133b11 + 392b10 - 1099b9 + 957b8 + 1842b7 - 1016b6 + 321b5 - 938b4 - 3b3 + 1242b2 + 145*b1 - 94) / 4
$\nu^{11}$	$=$	$ ( 824 \beta_{11} - 331 \beta_{10} - 1983 \beta_{9} + 2486 \beta_{8} + 3487 \beta_{7} - 2137 \beta_{6} + \cdots + 673 ) / 4 $ (824b11 - 331b10 - 1983b9 + 2486b8 + 3487b7 - 2137b6 - 309b5 - 2007b4 + 927b3 + 2204b2 + 1358*b1 + 673) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

589.1

−0.414092	−	0.0185074i
1.26951	+	1.38830i
1.26951	−	1.38830i
−0.414092	+	0.0185074i
1.97872	+	1.02084i
0.410511	+	1.28550i
0.410511	−	1.28550i
1.97872	−	1.02084i
−0.349377	−	0.423297i
0.104733	−	1.09476i
0.104733	+	1.09476i
−0.349377	+	0.423297i

−0.906803

−

1.08522i

−

1.00000i

−0.355416

1.96817i

−

3.10278i

−1.08522

0.906803i

2.45819

−

1.39903i

−1.00000

−3.36720

2.81361i

589.2

−0.906803

−

1.08522i

1.00000i

−0.355416

1.96817i

3.10278i

1.08522

−

0.906803i

2.45819

−

1.39903i

−1.00000

3.36720

−

2.81361i

589.3

−0.906803

1.08522i

−

1.00000i

−0.355416

−

1.96817i

−

3.10278i

1.08522

0.906803i

2.45819

1.39903i

−1.00000

3.36720

2.81361i

589.4

−0.906803

1.08522i

1.00000i

−0.355416

−

1.96817i

3.10278i

−1.08522

−

0.906803i

2.45819

1.39903i

−1.00000

−3.36720

−

2.81361i

589.5

0.235342

−

1.39449i

−

1.00000i

−1.88923

−

0.656365i

2.24914i

−1.39449

−

0.235342i

−1.35991

2.48005i

−1.00000

3.13641

0.529317i

589.6

0.235342

−

1.39449i

1.00000i

−1.88923

−

0.656365i

−

2.24914i

1.39449

0.235342i

−1.35991

2.48005i

−1.00000

−3.13641

−

0.529317i

589.7

0.235342

1.39449i

−

1.00000i

−1.88923

0.656365i

2.24914i

1.39449

−

0.235342i

−1.35991

−

2.48005i

−1.00000

−3.13641

0.529317i

589.8

0.235342

1.39449i

1.00000i

−1.88923

0.656365i

−

2.24914i

−1.39449

0.235342i

−1.35991

−

2.48005i

−1.00000

3.13641

−

0.529317i

589.9

1.17146

−

0.792261i

−

1.00000i

0.744644

−

1.85621i

−

1.14637i

−0.792261

−

1.17146i

−0.598279

−

2.76443i

−1.00000

−0.908221

−

1.34292i

589.10

1.17146

−

0.792261i

1.00000i

0.744644

−

1.85621i

1.14637i

0.792261

1.17146i

−0.598279

−

2.76443i

−1.00000

0.908221

1.34292i

589.11

1.17146

0.792261i

−

1.00000i

0.744644

1.85621i

−

1.14637i

0.792261

−

1.17146i

−0.598279

2.76443i

−1.00000

0.908221

−

1.34292i

589.12

1.17146

0.792261i

1.00000i

0.744644

1.85621i

1.14637i

−0.792261

1.17146i

−0.598279

2.76443i

−1.00000

−0.908221

1.34292i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.b	even	2	1	inner
56.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1176.2.c.d	✓	12
4.b	odd	2	1		4704.2.c.d		12
7.b	odd	2	1	inner	1176.2.c.d	✓	12
8.b	even	2	1	inner	1176.2.c.d	✓	12
8.d	odd	2	1		4704.2.c.d		12
28.d	even	2	1		4704.2.c.d		12
56.e	even	2	1		4704.2.c.d		12
56.h	odd	2	1	inner	1176.2.c.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.2.c.d	✓	12	1.a	even	1	1	trivial
1176.2.c.d	✓	12	7.b	odd	2	1	inner
1176.2.c.d	✓	12	8.b	even	2	1	inner
1176.2.c.d	✓	12	56.h	odd	2	1	inner
4704.2.c.d		12	4.b	odd	2	1
4704.2.c.d		12	8.d	odd	2	1
4704.2.c.d		12	28.d	even	2	1
4704.2.c.d		12	56.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1176, [\chi])$:

$ T_{5}^{6} + 16T_{5}^{4} + 68T_{5}^{2} + 64 $

$ T_{17}^{6} - 80T_{17}^{4} + 1604T_{17}^{2} - 5888 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{5} + 2 T^{4} + \cdots + 8)^{2} $ (T^6 - T^5 + 2T^4 - 2T^3 + 4T^2 - 4T + 8)^2
$3$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$5$	$ (T^{6} + 16 T^{4} + \cdots + 64)^{2} $ (T^6 + 16T^4 + 68T^2 + 64)^2
$7$	$ T^{12} $ T^12
$11$	$ (T^{6} + 44 T^{4} + \cdots + 368)^{2} $ (T^6 + 44T^4 + 260T^2 + 368)^2
$13$	$ (T^{6} + 40 T^{4} + \cdots + 256)^{2} $ (T^6 + 40T^4 + 272T^2 + 256)^2
$17$	$ (T^{6} - 80 T^{4} + \cdots - 5888)^{2} $ (T^6 - 80T^4 + 1604T^2 - 5888)^2
$19$	$ (T^{6} + 56 T^{4} + \cdots + 256)^{2} $ (T^6 + 56T^4 + 784T^2 + 256)^2
$23$	$ (T^{3} + 2 T^{2} - 14 T - 32)^{4} $ (T^3 + 2T^2 - 14T - 32)^4
$29$	$ (T^{6} + 88 T^{4} + \cdots + 5888)^{2} $ (T^6 + 88T^4 + 2064T^2 + 5888)^2
$31$	$ (T^{6} - 152 T^{4} + \cdots - 94208)^{2} $ (T^6 - 152T^4 + 6800T^2 - 94208)^2
$37$	$ (T^{6} + 88 T^{4} + \cdots + 5888)^{2} $ (T^6 + 88T^4 + 2064T^2 + 5888)^2
$41$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9956T^2 - 94208)^2
$43$	$ (T^{6} + 148 T^{4} + \cdots + 1472)^{2} $ (T^6 + 148T^4 + 1072T^2 + 1472)^2
$47$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
$53$	$ (T^{6} + 168 T^{4} + \cdots + 5888)^{2} $ (T^6 + 168T^4 + 2192T^2 + 5888)^2
$59$	$ (T^{6} + 272 T^{4} + \cdots + 369664)^{2} $ (T^6 + 272T^4 + 18496T^2 + 369664)^2
$61$	$ (T^{6} + 216 T^{4} + \cdots + 135424)^{2} $ (T^6 + 216T^4 + 12176T^2 + 135424)^2
$67$	$ (T^{6} + 100 T^{4} + \cdots + 1472)^{2} $ (T^6 + 100T^4 + 1264T^2 + 1472)^2
$71$	$ (T^{3} - 2 T^{2} + \cdots + 176)^{4} $ (T^3 - 2T^2 - 54T + 176)^4
$73$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
$79$	$ (T^{3} + 4 T^{2} - 24 T - 64)^{4} $ (T^3 + 4T^2 - 24T - 64)^4
$83$	$ (T^{6} + 208 T^{4} + \cdots + 123904)^{2} $ (T^6 + 208T^4 + 11840T^2 + 123904)^2
$89$	$ (T^{6} - 544 T^{4} + \cdots - 1701632)^{2} $ (T^6 - 544T^4 + 80356T^2 - 1701632)^2
$97$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
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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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + \beta_{7} q^{3} - \beta_{11} q^{4} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{5}+ \cdots - q^{9}+O(q^{10}) \) q - b6 * q^2 + b7 * q^3 - b11 * q^4 + (-b8 + b7 - b5 - b2) * q^5 + b5 * q^6 + (b9 - b1) * q^8 - q^9	\( q - \beta_{6} q^{2} + \beta_{7} q^{3} - \beta_{11} q^{4} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) \) q - b6 * q^2 + b7 * q^3 - b11 * q^4 + (-b8 + b7 - b5 - b2) * q^5 + b5 * q^6 + (b9 - b1) * q^8 - q^9 + (-b10 - b7 + b5 + b3) * q^10 + (2b11 - 2b6 - b1 - 1) * q^11 - b2 * q^12 + (-b10 + b8 - b7 + 2b5) q^13 + (b1 - 1) * q^15 + (b9 + b1 - 2) * q^16 + (-b8 - b7 + b5 + 2b3 - b2) q^17 + b6 * q^18 + (-b10 - b8 + b7 - 2b2) q^19 + (2b8 + 2b7) * q^20 + (-2b11 - b9 + b6 + b4 + 2b1 - 1) * q^22 + (-b11 - b6 + b4) * q^23 + (-b8 - b5 + b3 - b2) * q^24 + (-b11 - b6 + b4 + b1) * q^25 + (4b7 - 2b5 - 2b2) q^26 - b7 * q^27 + (b11 + 2b9 - b6 + b4 - b1 - 1) q^29 + (-b9 + b6 - b4 + 1) * q^30 + (-b10 + b8 + b7 - 2b5 + 2b3 - 2b2) q^31 + (-b9 + 2b6 - 2b4 + b1) * q^32 + (-b8 - b7 + b5 + b2) * q^33 + (b10 + 2b8 - 3b7 + b5 + b3) * q^34 + b11 * q^36 + (b11 + 2b9 - b6 + b4 - b1 - 1) q^37 + (-2b10 + 2b7 + 2b3 - 2b2) * q^38 + (b11 + b6 - b4 - b1 + 1) * q^39 + (2b10 - 2b8 + 2b7) q^40 + (-2b10 - b8 - b7 - b5 + 2b3 + b2) * q^41 + (-2b9 - 2b6 - 2b4) q^43 + (2b6 - 2b1 + 6) * q^44 + (b8 - b7 + b5 + b2) * q^45 + (-2b11 + b9 + b6 + b4 + 3) q^46 + (-2b10 - 2b5 + 2b2) q^47 + (b8 - 2b7 + b5 + b3 + b2) q^48 + (-2b11 + b6 + 4) q^50 + (-2b9 + 2b6 + b1 + 1) * q^51 + (-2b8 + 2b5 + 2b3) q^52 + (b11 - 2b9 + 3b6 + b4 + b1 + 1) * q^53 - b5 * q^54 + (-b10 - 3b8 - 3b7 + 2b5 + 2b3 + 2b2) q^55 + (b11 + b6 - b4 + b1 - 1) * q^57 + (-2b11 + 2b6 + 4b1) q^58 + (-2b10 + 4b8 + 6b5 + 2b2) * q^59 + (2b11 - 2b6 - 2b1 - 2) q^60 + (b10 - 3b8 - 5b7 - 4b5 - 2b2) * q^61 + (2b10 + 2b7 + 2b3 + 2b2) * q^62 + (4b11 - b9 - 2b6 - 2b4 - 3b1 - 4) * q^64 + (2b11 + 2b6 - 2b4 - 4b1 + 4) * q^65 + (-b10 + 2b8 - b7 + b5 - b3) q^66 + (-2b11 + 2b9) * q^67 + (4b10 - 2b8 - 2b7 - 4b5) * q^68 + (-b10 + b5 - b2) * q^69 + (-b11 - b6 + b4 - 2b1 + 2) q^71 + (-b9 + b1) * q^72 + (2b10 + 2b5 - 2b2) q^73 + (-2b11 + 2b6 + 4b1) q^74 + (-b10 + b8 + 2b5) q^75 + (2b8 + 4b7 + 2b5 + 2b3) * q^76 + (2b11 - 2b6 - 4) * q^78 + (2b1 - 2) q^79 + (-8b7 + 4b5) * q^80 + q^81 + (-b10 + 6b8 + 3b7 + 3b5 - b3 + 4b2) * q^82 + (-2b10 - 4b7 + 2b5 - 2b2) * q^83 + (-3b11 - 2b9 + 7b6 + b4 + 3b1 + 3) * q^85 + (-2b6 - 4b1 - 4) * q^86 + (-b10 - b8 - b7 + 2b3) q^87 + (2b11 + 2b9 - 6b6 + 2b4 - 2) * q^88 + (2b10 - 3b8 - 3b7 + 5b5 - 2b3 + 3b2) * q^89 + (b10 + b7 - b5 - b3) * q^90 + (2b9 - 2b6 + 2) * q^92 + (3b11 - 2b9 - 3b6 - b4 - b1 - 1) q^93 + (-2b10 + 4b8 + 6b7 + 2b5 - 2b3 + 4b2) * q^94 + (-2b1 - 6) q^95 + (2b10 + b8 - b5 - b3 + b2) q^96 + (-2b10 - 2b5 + 2b2) q^97 + (-2b11 + 2b6 + b1 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 6 q^{4} + 2 q^{8} - 12 q^{9}+O(q^{10}) \) 12 * q + 2 * q^2 - 6 * q^4 + 2 * q^8 - 12 * q^9	\( 12 q + 2 q^{2} - 6 q^{4} + 2 q^{8} - 12 q^{9} - 8 q^{15} - 14 q^{16} - 2 q^{18} - 28 q^{22} - 8 q^{23} - 4 q^{25} + 8 q^{30} + 2 q^{32} + 6 q^{36} + 16 q^{39} + 60 q^{44} + 24 q^{46} + 34 q^{50} - 16 q^{60} - 30 q^{64} + 48 q^{65} + 8 q^{71} - 2 q^{72} - 32 q^{78} - 16 q^{79} + 12 q^{81} - 60 q^{86} + 4 q^{88} + 40 q^{92} - 80 q^{95}+O(q^{100}) \) 12 * q + 2 * q^2 - 6 * q^4 + 2 * q^8 - 12 * q^9 - 8 * q^15 - 14 * q^16 - 2 * q^18 - 28 * q^22 - 8 * q^23 - 4 * q^25 + 8 * q^30 + 2 * q^32 + 6 * q^36 + 16 * q^39 + 60 * q^44 + 24 * q^46 + 34 * q^50 - 16 * q^60 - 30 * q^64 + 48 * q^65 + 8 * q^71 - 2 * q^72 - 32 * q^78 - 16 * q^79 + 12 * q^81 - 60 * q^86 + 4 * q^88 + 40 * q^92 - 80 * q^95

Introduction

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Newspace parameters

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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	1176.2.c.d

Level	$1176$
Weight	$2$
Character orbit	1176.c
Analytic conductor	$9.390$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.39040727770\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.84396412309504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 \) x^12 - 6x^11 + 18x^10 - 32x^9 + 39x^8 - 28x^7 + 13x^6 + 10x^5 + 6x^4 + 14x^3 + 17x^2 + 10*x + 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 5 \nu^{11} + 30 \nu^{10} - 91 \nu^{9} + 166 \nu^{8} - 210 \nu^{7} + 154 \nu^{6} - 59 \nu^{5} + \cdots - 48 ) / 16 \) (-5v^11 + 30v^10 - 91v^9 + 166v^8 - 210v^7 + 154v^6 - 59v^5 - 80v^4 + 3v^3 - 102v^2 - 78*v - 48) / 16
\(\beta_{2}\)	\(=\)	\( ( 11 \nu^{11} - 75 \nu^{10} + 259 \nu^{9} - 561 \nu^{8} + 876 \nu^{7} - 992 \nu^{6} + 887 \nu^{5} + \cdots + 26 ) / 32 \) (11v^11 - 75v^10 + 259v^9 - 561v^8 + 876v^7 - 992v^6 + 887v^5 - 503v^4 + 315v^3 + 41v^2 + 100*v + 26) / 32
\(\beta_{3}\)	\(=\)	\( ( 16 \nu^{11} - 105 \nu^{10} + 348 \nu^{9} - 709 \nu^{8} + 1010 \nu^{7} - 954 \nu^{6} + 622 \nu^{5} + \cdots + 54 ) / 16 \) (16v^11 - 105v^10 + 348v^9 - 709v^8 + 1010v^7 - 954v^6 + 622v^5 - 55v^4 + 58v^3 + 201v^2 + 218*v + 54) / 16
\(\beta_{4}\)	\(=\)	\( ( 57 \nu^{11} - 379 \nu^{10} + 1269 \nu^{9} - 2617 \nu^{8} + 3792 \nu^{7} - 3748 \nu^{6} + 2721 \nu^{5} + \cdots + 242 ) / 32 \) (57v^11 - 379v^10 + 1269v^9 - 2617v^8 + 3792v^7 - 3748v^6 + 2721v^5 - 763v^4 + 577v^3 + 545v^2 + 608*v + 242) / 32
\(\beta_{5}\)	\(=\)	\( ( - 10 \nu^{11} + 64 \nu^{10} - 206 \nu^{9} + 405 \nu^{8} - 560 \nu^{7} + 518 \nu^{6} - 352 \nu^{5} + \cdots - 50 ) / 4 \) (-10v^11 + 64v^10 - 206v^9 + 405v^8 - 560v^7 + 518v^6 - 352v^5 + 48v^4 - 78v^3 - 117v^2 - 122*v - 50) / 4
\(\beta_{6}\)	\(=\)	\( ( - 81 \nu^{11} + 519 \nu^{10} - 1673 \nu^{9} + 3297 \nu^{8} - 4576 \nu^{7} + 4268 \nu^{6} - 2953 \nu^{5} + \cdots - 402 ) / 32 \) (-81v^11 + 519v^10 - 1673v^9 + 3297v^8 - 4576v^7 + 4268v^6 - 2953v^5 + 511v^4 - 757v^3 - 817v^2 - 1048*v - 402) / 32
\(\beta_{7}\)	\(=\)	\( ( 85 \nu^{11} - 545 \nu^{10} + 1753 \nu^{9} - 3431 \nu^{8} + 4692 \nu^{7} - 4248 \nu^{6} + 2793 \nu^{5} + \cdots + 422 ) / 32 \) (85v^11 - 545v^10 + 1753v^9 - 3431v^8 + 4692v^7 - 4248v^6 + 2793v^5 - 285v^4 + 657v^3 + 887v^2 + 1092*v + 422) / 32
\(\beta_{8}\)	\(=\)	\( ( - 56 \nu^{11} + 361 \nu^{10} - 1168 \nu^{9} + 2309 \nu^{8} - 3210 \nu^{7} + 3010 \nu^{6} - 2102 \nu^{5} + \cdots - 246 ) / 16 \) (-56v^11 + 361v^10 - 1168v^9 + 2309v^8 - 3210v^7 + 3010v^6 - 2102v^5 + 423v^4 - 550v^3 - 529v^2 - 690*v - 246) / 16
\(\beta_{9}\)	\(=\)	\( ( 31 \nu^{11} - 197 \nu^{10} + 628 \nu^{9} - 1216 \nu^{8} + 1646 \nu^{7} - 1468 \nu^{6} + 953 \nu^{5} + \cdots + 160 ) / 8 \) (31v^11 - 197v^10 + 628v^9 - 1216v^8 + 1646v^7 - 1468v^6 + 953v^5 - 61v^4 + 230v^3 + 338v^2 + 420*v + 160) / 8
\(\beta_{10}\)	\(=\)	\( ( 123 \nu^{11} - 793 \nu^{10} + 2567 \nu^{9} - 5067 \nu^{8} + 6992 \nu^{7} - 6396 \nu^{6} + 4179 \nu^{5} + \cdots + 566 ) / 32 \) (123v^11 - 793v^10 + 2567v^9 - 5067v^8 + 6992v^7 - 6396v^6 + 4179v^5 - 329v^4 + 683v^3 + 1459v^2 + 1520*v + 566) / 32
\(\beta_{11}\)	\(=\)	\( ( 88 \nu^{11} - 567 \nu^{10} + 1832 \nu^{9} - 3607 \nu^{8} + 4966 \nu^{7} - 4550 \nu^{6} + 3026 \nu^{5} + \cdots + 402 ) / 16 \) (88v^11 - 567v^10 + 1832v^9 - 3607v^8 + 4966v^7 - 4550v^6 + 3026v^5 - 361v^4 + 626v^3 + 955v^2 + 1046*v + 402) / 16

\(\nu\)	\(=\)	\( ( -\beta_{11} + \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{3} + \beta _1 + 2 ) / 4 \) (-b11 + b9 - b8 + 2*b6 - b5 + b3 + b1 + 2) / 4
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots + 1 ) / 4 \) (-2b11 - b10 + b9 - 2b8 + b7 + 5b6 - 5b5 + b4 + b3 - 2b2 + 2b1 + 1) / 4
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} + 3\beta_{6} - 4\beta_{5} + 2\beta_{4} - 4\beta_{2} + 2\beta _1 - 1 ) / 2 \) (-2b11 - b10 + b9 - 2b8 + 3b6 - 4b5 + 2b4 - 4b2 + 2*b1 - 1) / 2
\(\nu^{4}\)	\(=\)	\( ( - 5 \beta_{11} + 7 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 10 \beta_{4} + \cdots - 10 ) / 4 \) (-5b11 + 7b9 - 5b8 - 10b7 + 4b6 - 5b5 + 10b4 - 5b3 - 14b2 + 3b1 - 10) / 4
\(\nu^{5}\)	\(=\)	\( ( - 2 \beta_{11} + 7 \beta_{10} + 17 \beta_{9} - 4 \beta_{8} - 31 \beta_{7} - 3 \beta_{6} + 15 \beta_{5} + \cdots - 29 ) / 4 \) (-2b11 + 7b10 + 17b9 - 4b8 - 31b7 - 3b6 + 15b5 + 19b4 - 17b3 - 12b2 - 12*b1 - 29) / 4
\(\nu^{6}\)	\(=\)	\( ( 3 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} - 2 \beta_{8} - 27 \beta_{7} - 10 \beta_{6} + 33 \beta_{5} + \cdots - 30 ) / 2 \) (3b11 + 13b10 + 11b9 - 2b8 - 27b7 - 10b6 + 33b5 + 12b4 - 19b3 + 4b2 - 29*b1 - 30) / 2
\(\nu^{7}\)	\(=\)	\( ( 11 \beta_{11} + 78 \beta_{10} - 9 \beta_{9} - 5 \beta_{8} - 38 \beta_{7} - 60 \beta_{6} + 157 \beta_{5} + \cdots - 96 ) / 4 \) (11b11 + 78b10 - 9b9 - 5b8 - 38b7 - 60b6 + 157b5 + 2b4 - 73b3 + 68b2 - 143*b1 - 96) / 4
\(\nu^{8}\)	\(=\)	\( ( 193 \beta_{10} - 139 \beta_{9} + 32 \beta_{8} + 135 \beta_{7} - 157 \beta_{6} + 279 \beta_{5} + \cdots - 133 ) / 4 \) (193b10 - 139b9 + 32b8 + 135b7 - 157b6 + 279b5 - 97b4 - 127b3 + 222b2 - 236b1 - 133) / 4
\(\nu^{9}\)	\(=\)	\( ( - 4 \beta_{11} + 180 \beta_{10} - 234 \beta_{9} + 126 \beta_{8} + 343 \beta_{7} - 204 \beta_{6} + \cdots - 84 ) / 2 \) (-4b11 + 180b10 - 234b9 + 126b8 + 343b7 - 204b6 + 191b5 - 181b4 - 83b3 + 286b2 - 112*b1 - 84) / 2
\(\nu^{10}\)	\(=\)	\( ( 133 \beta_{11} + 392 \beta_{10} - 1099 \beta_{9} + 957 \beta_{8} + 1842 \beta_{7} - 1016 \beta_{6} + \cdots - 94 ) / 4 \) (133b11 + 392b10 - 1099b9 + 957b8 + 1842b7 - 1016b6 + 321b5 - 938b4 - 3b3 + 1242b2 + 145*b1 - 94) / 4
\(\nu^{11}\)	\(=\)	\( ( 824 \beta_{11} - 331 \beta_{10} - 1983 \beta_{9} + 2486 \beta_{8} + 3487 \beta_{7} - 2137 \beta_{6} + \cdots + 673 ) / 4 \) (824b11 - 331b10 - 1983b9 + 2486b8 + 3487b7 - 2137b6 - 309b5 - 2007b4 + 927b3 + 2204b2 + 1358*b1 + 673) / 4

\(n\)	\(295\)	\(589\)	\(785\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 589.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{5} + 2 T^{4} + \cdots + 8)^{2} \) (T^6 - T^5 + 2T^4 - 2T^3 + 4T^2 - 4T + 8)^2
$3$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$5$	\( (T^{6} + 16 T^{4} + \cdots + 64)^{2} \) (T^6 + 16T^4 + 68T^2 + 64)^2
$7$	\( T^{12} \) T^12
$11$	\( (T^{6} + 44 T^{4} + \cdots + 368)^{2} \) (T^6 + 44T^4 + 260T^2 + 368)^2
$13$	\( (T^{6} + 40 T^{4} + \cdots + 256)^{2} \) (T^6 + 40T^4 + 272T^2 + 256)^2
$17$	\( (T^{6} - 80 T^{4} + \cdots - 5888)^{2} \) (T^6 - 80T^4 + 1604T^2 - 5888)^2
$19$	\( (T^{6} + 56 T^{4} + \cdots + 256)^{2} \) (T^6 + 56T^4 + 784T^2 + 256)^2
$23$	\( (T^{3} + 2 T^{2} - 14 T - 32)^{4} \) (T^3 + 2T^2 - 14T - 32)^4
$29$	\( (T^{6} + 88 T^{4} + \cdots + 5888)^{2} \) (T^6 + 88T^4 + 2064T^2 + 5888)^2
$31$	\( (T^{6} - 152 T^{4} + \cdots - 94208)^{2} \) (T^6 - 152T^4 + 6800T^2 - 94208)^2
$37$	\( (T^{6} + 88 T^{4} + \cdots + 5888)^{2} \) (T^6 + 88T^4 + 2064T^2 + 5888)^2
$41$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9956T^2 - 94208)^2
$43$	\( (T^{6} + 148 T^{4} + \cdots + 1472)^{2} \) (T^6 + 148T^4 + 1072T^2 + 1472)^2
$47$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
$53$	\( (T^{6} + 168 T^{4} + \cdots + 5888)^{2} \) (T^6 + 168T^4 + 2192T^2 + 5888)^2
$59$	\( (T^{6} + 272 T^{4} + \cdots + 369664)^{2} \) (T^6 + 272T^4 + 18496T^2 + 369664)^2
$61$	\( (T^{6} + 216 T^{4} + \cdots + 135424)^{2} \) (T^6 + 216T^4 + 12176T^2 + 135424)^2
$67$	\( (T^{6} + 100 T^{4} + \cdots + 1472)^{2} \) (T^6 + 100T^4 + 1264T^2 + 1472)^2
$71$	\( (T^{3} - 2 T^{2} + \cdots + 176)^{4} \) (T^3 - 2T^2 - 54T + 176)^4
$73$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
$79$	\( (T^{3} + 4 T^{2} - 24 T - 64)^{4} \) (T^3 + 4T^2 - 24T - 64)^4
$83$	\( (T^{6} + 208 T^{4} + \cdots + 123904)^{2} \) (T^6 + 208T^4 + 11840T^2 + 123904)^2
$89$	\( (T^{6} - 544 T^{4} + \cdots - 1701632)^{2} \) (T^6 - 544T^4 + 80356T^2 - 1701632)^2
$97$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
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