LMFDB - Newform orbit 224.3.o.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(79,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("224.79");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.o (of order $6$, degree $2$, not 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:	$6.10355792167$
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} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 $ x^12 + 2x^10 - 12x^9 + 12x^8 - 12x^7 + 148x^6 - 48x^5 + 192x^4 - 768x^3 + 512*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 56)
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_{8} + \beta_{2} - 1) q^{3} + (\beta_{9} + \beta_{5} - \beta_{4} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{8} - \beta_{7} + \cdots - \beta_{2}) q^{9}+O(q^{10}) $ q + (-b8 + b2 - 1) * q^3 + (b9 + b5 - b4 + b1) * q^5 + (-b10 + b6 + b5 - b1) * q^7 + (b8 - b7 + b3 - b2) * q^9	$ q + ( - \beta_{8} + \beta_{2} - 1) q^{3} + (\beta_{9} + \beta_{5} - \beta_{4} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (5 \beta_{11} - 21 \beta_{7} + \cdots - 71) q^{99}+O(q^{100}) $ q + (-b8 + b2 - 1) * q^3 + (b9 + b5 - b4 + b1) * q^5 + (-b10 + b6 + b5 - b1) * q^7 + (b8 - b7 + b3 - b2) * q^9 + (-b11 + 2b8 - 2b2 + 2) * q^11 + (-b9 - b6 + b1) * q^13 + (4b9 - 6b6 + b5 - 4b1) q^15 + (-b11 + 3b8 + 14b2 - 14) * q^17 + (-2b8 + 2b7 + b3 + 16b2) q^19 + (-5b10 - b6 - 2b5 + b4 + b1) * q^21 + (-3b10 - b9 + 3b6 - b5 + b4 + 5b1) q^23 + (-4b11 - 6b8 - 18b2 + 18) q^25 + (-3b11 - 4b7 - 3b3 + 4) q^27 + (9b9 - 5b6 + 4b5 - 9b1) * q^29 + (-2b10 + b9 - 5b4) * q^31 + (-6b8 + 6b7 - 4b3 + 23b2) * q^33 + (-b11 + 7b7 + 4b3 - 22b2 - 11) q^35 + (-5b10 + b9 + 5b6 + b5 - b4 - 6b1) q^37 + (b10 - b9 + 2b4) q^39 + (3b11 + 3b7 + 3b3 + 11) q^41 + (4b11 - 6b7 + 4b3 - 2) q^43 + (11b10 - 7b9 + 4b4) q^45 + (6b10 + b9 - 6b6 + b5 - b4 + 12b1) q^47 + (5b11 + 14b8 - 7b7 + b3 - 16b2 - 8) * q^49 + (10b8 - 10b7 - 5b3 + 16b2) * q^51 + (9b10 - 9b9 - 5b4) q^53 + (-7b9 + 17b6 + 3b5 + 7b1) * q^55 + (-12b7 - 31) q^57 + (4b11 + 11b8 + 9b2 - 9) q^59 + (-11b10 - 3b9 + 11b6 - 3b5 + 3b4 + 32b1) * q^61 + (9b10 - 14b9 - 7b6 - 2b5 + 2b4 - 7b1) * q^63 + (-7b8 + 7b7 - b3 - 11b2) q^65 + (10b11 - 9b8 - 33b2 + 33) q^67 + (11b9 - 6b6 - 7b5 - 11b1) * q^69 + (22b9 + 8b6 - 8b5 - 22b1) * q^71 + (6b11 + 8b8 - 11b2 + 11) q^73 + (-34b8 + 34b7 - 2b3 - 24b2) * q^75 + (12b10 - 21b9 + 3b6 + 2b5 + b4 + 11b1) q^77 + (3b10 - 5b9 - 3b6 - 5b5 + 5b4 - 17b1) * q^79 + (-7b11 - 7b8 - 32b2 + 32) q^81 + (-14b11 - 6b7 - 14b3 + 28) q^83 + (6b9 + 23b6 - 15b5 - 6b1) * q^85 + (-b10 - 29b9 + 14b4) * q^87 + (-4b8 + 4b7 - 8b3 - 7b2) * q^89 + (b11 + 7b7 + 3b3 + 22b2 - 3) q^91 + (25b10 + b9 - 25b6 + b5 - b4 - 4b1) q^93 + (-7b10 + 30b9 - 7b4) q^95 + (13b11 - b7 + 13b3 - 11) * q^97 + (5b11 - 21b7 + 5b3 - 71) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 - 8 * q^9	$ 12 q - 6 q^{3} - 8 q^{9} + 14 q^{11} - 82 q^{17} + 94 q^{19} + 116 q^{25} + 60 q^{27} + 146 q^{33} - 270 q^{35} + 120 q^{41} - 40 q^{43} - 204 q^{49} + 106 q^{51} - 372 q^{57} - 62 q^{59} - 64 q^{65} + 178 q^{67} + 54 q^{73} - 140 q^{75} + 206 q^{81} + 392 q^{83} - 26 q^{89} + 88 q^{91} - 184 q^{97} - 872 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 - 8 * q^9 + 14 * q^11 - 82 * q^17 + 94 * q^19 + 116 * q^25 + 60 * q^27 + 146 * q^33 - 270 * q^35 + 120 * q^41 - 40 * q^43 - 204 * q^49 + 106 * q^51 - 372 * q^57 - 62 * q^59 - 64 * q^65 + 178 * q^67 + 54 * q^73 - 140 * q^75 + 206 * q^81 + 392 * q^83 - 26 * q^89 + 88 * q^91 - 184 * q^97 - 872 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 $ x^12 + 2*x^10 - 12*x^9 + 12*x^8 - 12*x^7 + 148*x^6 - 48*x^5 + 192*x^4 - 768*x^3 + 512*x^2 + 4096 :

$\beta_{1}$	$=$	$ ( - 11 \nu^{11} + 48 \nu^{10} + 298 \nu^{9} + 420 \nu^{8} + 1212 \nu^{7} - 1980 \nu^{6} + \cdots - 49152 ) / 93184 $ (-11v^11 + 48v^10 + 298v^9 + 420v^8 + 1212v^7 - 1980v^6 + 7012v^5 - 4656v^4 + 30912v^3 - 18432v^2 + 57856*v - 49152) / 93184
$\beta_{2}$	$=$	$ ( 6 \nu^{11} - 51 \nu^{10} + 36 \nu^{9} - 14 \nu^{8} + 828 \nu^{7} - 12 \nu^{6} + 444 \nu^{5} + \cdots + 52224 ) / 46592 $ (6v^11 - 51v^10 + 36v^9 - 14v^8 + 828v^7 - 12v^6 + 444v^5 - 3516v^4 + 1008v^3 + 2112v^2 + 28800*v + 52224) / 46592
$\beta_{3}$	$=$	$ ( - 10 \nu^{11} - 97 \nu^{10} - 60 \nu^{9} - 826 \nu^{8} + 804 \nu^{7} + 748 \nu^{6} + 7268 \nu^{5} + \cdots + 99328 ) / 46592 $ (-10v^11 - 97v^10 - 60v^9 - 826v^8 + 804v^7 + 748v^6 + 7268v^5 - 3604v^4 - 10416v^3 + 13952v^2 + 21888*v + 99328) / 46592
$\beta_{4}$	$=$	$ ( 16 \nu^{11} + 319 \nu^{10} + 96 \nu^{9} - 98 \nu^{8} - 340 \nu^{7} - 396 \nu^{6} + 1548 \nu^{5} + \cdots - 512 ) / 46592 $ (16v^11 + 319v^10 + 96v^9 - 98v^8 - 340v^7 - 396v^6 + 1548v^5 + 11372v^4 + 4144v^3 - 192v^2 - 28032*v - 512) / 46592
$\beta_{5}$	$=$	$ ( 3 \nu^{11} + 20 \nu^{10} - 86 \nu^{9} + 84 \nu^{8} - 132 \nu^{7} + 956 \nu^{6} - 324 \nu^{5} + \cdots + 46080 ) / 6656 $ (3v^11 + 20v^10 - 86v^9 + 84v^8 - 132v^7 + 956v^6 - 324v^5 + 2064v^4 - 9584v^3 + 3968v^2 - 3072*v + 46080) / 6656
$\beta_{6}$	$=$	$ ( - 8 \nu^{11} - 23 \nu^{10} + 43 \nu^{9} - 42 \nu^{8} + 170 \nu^{7} + 16 \nu^{6} + 136 \nu^{5} + \cdots + 11904 ) / 11648 $ (-8v^11 - 23v^10 + 43v^9 - 42v^8 + 170v^7 + 16v^6 + 136v^5 - 2592v^4 - 2436v^3 - 2816v^2 + 8192*v + 11904) / 11648
$\beta_{7}$	$=$	$ ( - \nu^{11} + 6 \nu^{9} + 12 \nu^{8} + 4 \nu^{7} - 84 \nu^{6} - 52 \nu^{5} - 48 \nu^{4} + 480 \nu^{3} + \cdots - 3072 ) / 1024 $ (-v^11 + 6v^9 + 12v^8 + 4v^7 - 84v^6 - 52v^5 - 48v^4 + 480v^3 + 384v^2 + 1024*v - 3072) / 1024
$\beta_{8}$	$=$	$ ( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} + \cdots - 115200 ) / 46592 $ (51v^11 - 24v^10 + 306v^9 - 756v^8 + 668v^7 - 3924v^6 + 7596v^5 - 4224v^4 + 23856v^3 - 43200v^2 + 64256*v - 115200) / 46592
$\beta_{9}$	$=$	$ ( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} + \cdots - 115200 ) / 46592 $ (51v^11 - 24v^10 + 306v^9 - 756v^8 + 668v^7 - 3924v^6 + 7596v^5 - 4224v^4 + 23856v^3 - 43200v^2 - 28928*v - 115200) / 46592
$\beta_{10}$	$=$	$ ( 75 \nu^{11} - 46 \nu^{10} + 450 \nu^{9} - 448 \nu^{8} + 1796 \nu^{7} - 1788 \nu^{6} + 7188 \nu^{5} + \cdots + 512 ) / 46592 $ (75v^11 - 46v^10 + 450v^9 - 448v^8 + 1796v^7 - 1788v^6 + 7188v^5 - 14648v^4 + 19152v^3 + 192v^2 + 86272*v + 512) / 46592
$\beta_{11}$	$=$	$ ( - 81 \nu^{11} - 267 \nu^{10} - 486 \nu^{9} - 266 \nu^{8} + 1744 \nu^{7} - 2568 \nu^{6} + \cdots - 192512 ) / 46592 $ (-81v^11 - 267v^10 - 486v^9 - 266v^8 + 1744v^7 - 2568v^6 - 3264v^5 - 35708v^4 - 2688v^3 - 72192v^2 + 71296*v - 192512) / 46592

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

$\nu$	$=$	$ ( -\beta_{9} + \beta_{8} ) / 2 $ (-b9 + b8) / 2
$\nu^{2}$	$=$	$ ( \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 ) / 2 $ (b10 - b8 + b7 - b6 + b3 - b2 - b1) / 2
$\nu^{3}$	$=$	$ -\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta _1 + 3 $ -b9 - b7 - b6 - b5 + b1 + 3
$\nu^{4}$	$=$	$ -\beta_{11} - 3\beta_{10} + 2\beta_{8} + 7\beta_{2} - 7 $ -b11 - 3b10 + 2b8 + 7*b2 - 7
$\nu^{5}$	$=$	$ \beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} - 9\beta_{2} + 7\beta_1 $ b10 + 2b9 - b8 + b7 - b6 + 2b5 - 2b4 + 3b3 - 9b2 + 7b1
$\nu^{6}$	$=$	$ -2\beta_{11} - 10\beta_{9} - 14\beta_{7} + 4\beta_{6} - 6\beta_{5} - 2\beta_{3} + 10\beta _1 - 28 $ -2b11 - 10b9 - 14b7 + 4b6 - 6b5 - 2b3 + 10*b1 - 28
$\nu^{7}$	$=$	$ -12\beta_{10} + 14\beta_{9} - 10\beta_{8} + 12\beta_{4} + 84\beta_{2} - 84 $ -12b10 + 14b9 - 10b8 + 12b4 + 84*b2 - 84
$\nu^{8}$	$=$	$ 6 \beta_{10} + 24 \beta_{9} - 18 \beta_{8} + 18 \beta_{7} - 6 \beta_{6} + 24 \beta_{5} - 24 \beta_{4} + \cdots + 54 \beta_1 $ 6b10 + 24b9 - 18b8 + 18b7 - 6b6 + 24b5 - 24b4 - 18b3 - 22b2 + 54b1
$\nu^{9}$	$=$	$ -36\beta_{11} + 8\beta_{9} - 8\beta_{7} + 120\beta_{6} - 12\beta_{5} - 36\beta_{3} - 8\beta _1 - 120 $ -36b11 + 8b9 - 8b7 + 120b6 - 12b5 - 36b3 - 8*b1 - 120
$\nu^{10}$	$=$	$ 44\beta_{11} + 76\beta_{10} - 40\beta_{9} - 40\beta_{8} + 168\beta_{4} - 124\beta_{2} + 124 $ 44b11 + 76b10 - 40b9 - 40b8 + 168b4 - 124b2 + 124
$\nu^{11}$	$=$	$ 308 \beta_{10} + 136 \beta_{9} + 20 \beta_{8} - 20 \beta_{7} - 308 \beta_{6} + 136 \beta_{5} + \cdots - 316 \beta_1 $ 308b10 + 136b9 + 20b8 - 20b7 - 308b6 + 136b5 - 136b4 - 228b3 + 12b2 - 316b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-\beta_{2}$	$-1$

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} $

79.1

1.71059	+	1.03628i
0.0421483	+	1.99956i
1.83041	−	0.805972i
−1.61320	+	1.18220i
−0.685878	−	1.87872i
−1.28408	−	1.53335i
1.71059	−	1.03628i
0.0421483	−	1.99956i
1.83041	+	0.805972i
−1.61320	−	1.18220i
−0.685878	+	1.87872i
−1.28408	+	1.53335i

−2.25274

−

3.90186i

−6.07099

−

3.50509i

−2.51181

−

6.53382i

−5.64968

9.78553i

79.2

−2.25274

−

3.90186i

6.07099

3.50509i

2.51181

6.53382i

−5.64968

9.78553i

79.3

−0.717214

−

1.24225i

−2.27256

−

1.31206i

5.39122

4.46483i

3.47121

−

6.01231i

79.4

−0.717214

−

1.24225i

2.27256

1.31206i

−5.39122

−

4.46483i

3.47121

−

6.01231i

79.5

1.46995

2.54604i

−7.59793

−

4.38667i

3.55324

−

6.03112i

0.178469

−

0.309118i

79.6

1.46995

2.54604i

7.59793

4.38667i

−3.55324

6.03112i

0.178469

−

0.309118i

207.1

−2.25274

3.90186i

−6.07099

3.50509i

−2.51181

6.53382i

−5.64968

−

9.78553i

207.2

−2.25274

3.90186i

6.07099

−

3.50509i

2.51181

−

6.53382i

−5.64968

−

9.78553i

207.3

−0.717214

1.24225i

−2.27256

1.31206i

5.39122

−

4.46483i

3.47121

6.01231i

207.4

−0.717214

1.24225i

2.27256

−

1.31206i

−5.39122

4.46483i

3.47121

6.01231i

207.5

1.46995

−

2.54604i

−7.59793

4.38667i

3.55324

6.03112i

0.178469

0.309118i

207.6

1.46995

−

2.54604i

7.59793

−

4.38667i

−3.55324

−

6.03112i

0.178469

0.309118i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
8.d	odd	2	1	inner
56.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.3.o.c		12
4.b	odd	2	1		56.3.k.c	✓	12
7.c	even	3	1	inner	224.3.o.c		12
7.c	even	3	1		1568.3.g.k		6
7.d	odd	6	1		1568.3.g.i		6
8.b	even	2	1		56.3.k.c	✓	12
8.d	odd	2	1	inner	224.3.o.c		12
28.d	even	2	1		392.3.k.k		12
28.f	even	6	1		392.3.g.l		6
28.f	even	6	1		392.3.k.k		12
28.g	odd	6	1		56.3.k.c	✓	12
28.g	odd	6	1		392.3.g.k		6
56.h	odd	2	1		392.3.k.k		12
56.j	odd	6	1		392.3.g.l		6
56.j	odd	6	1		392.3.k.k		12
56.k	odd	6	1	inner	224.3.o.c		12
56.k	odd	6	1		1568.3.g.k		6
56.m	even	6	1		1568.3.g.i		6
56.p	even	6	1		56.3.k.c	✓	12
56.p	even	6	1		392.3.g.k		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.k.c	✓	12	4.b	odd	2	1
56.3.k.c	✓	12	8.b	even	2	1
56.3.k.c	✓	12	28.g	odd	6	1
56.3.k.c	✓	12	56.p	even	6	1
224.3.o.c		12	1.a	even	1	1	trivial
224.3.o.c		12	7.c	even	3	1	inner
224.3.o.c		12	8.d	odd	2	1	inner
224.3.o.c		12	56.k	odd	6	1	inner
392.3.g.k		6	28.g	odd	6	1
392.3.g.k		6	56.p	even	6	1
392.3.g.l		6	28.f	even	6	1
392.3.g.l		6	56.j	odd	6	1
392.3.k.k		12	28.d	even	2	1
392.3.k.k		12	28.f	even	6	1
392.3.k.k		12	56.h	odd	2	1
392.3.k.k		12	56.j	odd	6	1
1568.3.g.i		6	7.d	odd	6	1
1568.3.g.i		6	56.m	even	6	1
1568.3.g.k		6	7.c	even	3	1
1568.3.g.k		6	56.k	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(224, [\chi])$:

$ T_{3}^{6} + 3T_{3}^{5} + 20T_{3}^{4} + 5T_{3}^{3} + 178T_{3}^{2} + 209T_{3} + 361 $

$ T_{5}^{12} - 133T_{5}^{10} + 13038T_{5}^{8} - 566489T_{5}^{6} + 18167550T_{5}^{4} - 121144597T_{5}^{2} + 678446209 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{6} + 3 T^{5} + \cdots + 361)^{2} $ (T^6 + 3T^5 + 20T^4 + 5T^3 + 178T^2 + 209*T + 361)^2
$5$	$ T^{12} + \cdots + 678446209 $ T^12 - 133T^10 + 13038T^8 - 566489T^6 + 18167550T^4 - 121144597*T^2 + 678446209
$7$	$ T^{12} + \cdots + 13841287201 $ T^12 + 102T^10 + 8463T^8 + 426692T^6 + 20319663T^4 + 588009702*T^2 + 13841287201
$11$	$ (T^{6} - 7 T^{5} + \cdots + 82369)^{2} $ (T^6 - 7T^5 + 170T^4 + 1421T^3 + 12632T^2 + 34727*T + 82369)^2
$13$	$ (T^{6} + 64 T^{4} + \cdots + 5488)^{2} $ (T^6 + 64T^4 + 1120T^2 + 5488)^2
$17$	$ (T^{6} + 41 T^{5} + \cdots + 2627641)^{2} $ (T^6 + 41T^5 + 1338T^4 + 17305T^3 + 184110T^2 - 556003*T + 2627641)^2
$19$	$ (T^{6} - 47 T^{5} + \cdots + 5139289)^{2} $ (T^6 - 47T^5 + 1570T^4 - 25499T^3 + 301772T^2 - 1448613*T + 5139289)^2
$23$	$ T^{12} + \cdots + 282099132781249 $ T^12 - 793T^10 + 426438T^8 - 126920309T^6 + 27651137970T^4 - 3399656090677*T^2 + 282099132781249
$29$	$ (T^{6} + 3208 T^{4} + \cdots + 747251568)^{2} $ (T^6 + 3208T^4 + 2906064T^2 + 747251568)^2
$31$	$ T^{12} + \cdots + 27\!\cdots\!09 $ T^12 - 2437T^10 + 3973602T^8 - 3739418485T^6 + 2583022025350T^4 - 1031995436549049*T^2 + 275719977530659809
$37$	$ T^{12} + \cdots + 50\!\cdots\!49 $ T^12 - 1813T^10 + 2589990T^8 - 1120889441T^6 + 356391821382T^4 - 49741121169397*T^2 + 5093212006428049
$41$	$ (T^{3} - 30 T^{2} + \cdots + 364)^{4} $ (T^3 - 30T^2 - 288T + 364)^4
$43$	$ (T^{3} + 10 T^{2} + \cdots + 16408)^{4} $ (T^3 + 10T^2 - 1692T + 16408)^4
$47$	$ T^{12} + \cdots + 7867032061329 $ T^12 - 4941T^10 + 19546754T^8 - 24040888461T^6 + 23671173062086T^4 - 13650307824321*T^2 + 7867032061329
$53$	$ T^{12} + \cdots + 18\!\cdots\!49 $ T^12 - 7581T^10 + 39500070T^8 - 109023176185T^6 + 219802017458598T^4 - 244580284856387613*T^2 + 185214367764856222849
$59$	$ (T^{6} + 31 T^{5} + \cdots + 19158129)^{2} $ (T^6 + 31T^5 + 2876T^4 - 68119T^3 + 3531538T^2 - 8381955*T + 19158129)^2
$61$	$ T^{12} + \cdots + 31\!\cdots\!69 $ T^12 - 22829T^10 + 375290886T^8 - 2976172636169T^6 + 17238611856745798T^4 - 25815567192061648365*T^2 + 31319682727146367431969
$67$	$ (T^{6} - 89 T^{5} + \cdots + 102323854161)^{2} $ (T^6 - 89T^5 + 13448T^4 - 147859T^3 + 59017138T^2 - 1767982287*T + 102323854161)^2
$71$	$ (T^{6} + 22416 T^{4} + \cdots + 203138627328)^{2} $ (T^6 + 22416T^4 + 143235904T^2 + 203138627328)^2
$73$	$ (T^{6} - 27 T^{5} + \cdots + 16072081)^{2} $ (T^6 - 27T^5 + 3110T^4 + 72305T^3 + 5560918T^2 + 9545429*T + 16072081)^2
$79$	$ T^{12} + \cdots + 73\!\cdots\!09 $ T^12 - 9961T^10 + 69363606T^8 - 243144641621T^6 + 621203105646258T^4 - 810195265404614005*T^2 + 736309573447307373409
$83$	$ (T^{3} - 98 T^{2} + \cdots + 823528)^{4} $ (T^3 - 98T^2 - 8484T + 823528)^4
$89$	$ (T^{6} + 13 T^{5} + \cdots + 86434209)^{2} $ (T^6 + 13T^5 + 4582T^4 - 38775T^3 + 19595430T^2 + 41027661*T + 86434209)^2
$97$	$ (T^{3} + 46 T^{2} + \cdots - 407276)^{4} $ (T^3 + 46T^2 - 9804T - 407276)^4
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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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{8} + \beta_{2} - 1) q^{3} + (\beta_{9} + \beta_{5} - \beta_{4} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{8} - \beta_{7} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) q + (-b8 + b2 - 1) * q^3 + (b9 + b5 - b4 + b1) * q^5 + (-b10 + b6 + b5 - b1) * q^7 + (b8 - b7 + b3 - b2) * q^9	\( q + ( - \beta_{8} + \beta_{2} - 1) q^{3} + (\beta_{9} + \beta_{5} - \beta_{4} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (5 \beta_{11} - 21 \beta_{7} + \cdots - 71) q^{99}+O(q^{100}) \) q + (-b8 + b2 - 1) * q^3 + (b9 + b5 - b4 + b1) * q^5 + (-b10 + b6 + b5 - b1) * q^7 + (b8 - b7 + b3 - b2) * q^9 + (-b11 + 2b8 - 2b2 + 2) * q^11 + (-b9 - b6 + b1) * q^13 + (4b9 - 6b6 + b5 - 4b1) q^15 + (-b11 + 3b8 + 14b2 - 14) * q^17 + (-2b8 + 2b7 + b3 + 16b2) q^19 + (-5b10 - b6 - 2b5 + b4 + b1) * q^21 + (-3b10 - b9 + 3b6 - b5 + b4 + 5b1) q^23 + (-4b11 - 6b8 - 18b2 + 18) q^25 + (-3b11 - 4b7 - 3b3 + 4) q^27 + (9b9 - 5b6 + 4b5 - 9b1) * q^29 + (-2b10 + b9 - 5b4) * q^31 + (-6b8 + 6b7 - 4b3 + 23b2) * q^33 + (-b11 + 7b7 + 4b3 - 22b2 - 11) q^35 + (-5b10 + b9 + 5b6 + b5 - b4 - 6b1) q^37 + (b10 - b9 + 2b4) q^39 + (3b11 + 3b7 + 3b3 + 11) q^41 + (4b11 - 6b7 + 4b3 - 2) q^43 + (11b10 - 7b9 + 4b4) q^45 + (6b10 + b9 - 6b6 + b5 - b4 + 12b1) q^47 + (5b11 + 14b8 - 7b7 + b3 - 16b2 - 8) * q^49 + (10b8 - 10b7 - 5b3 + 16b2) * q^51 + (9b10 - 9b9 - 5b4) q^53 + (-7b9 + 17b6 + 3b5 + 7b1) * q^55 + (-12b7 - 31) q^57 + (4b11 + 11b8 + 9b2 - 9) q^59 + (-11b10 - 3b9 + 11b6 - 3b5 + 3b4 + 32b1) * q^61 + (9b10 - 14b9 - 7b6 - 2b5 + 2b4 - 7b1) * q^63 + (-7b8 + 7b7 - b3 - 11b2) q^65 + (10b11 - 9b8 - 33b2 + 33) q^67 + (11b9 - 6b6 - 7b5 - 11b1) * q^69 + (22b9 + 8b6 - 8b5 - 22b1) * q^71 + (6b11 + 8b8 - 11b2 + 11) q^73 + (-34b8 + 34b7 - 2b3 - 24b2) * q^75 + (12b10 - 21b9 + 3b6 + 2b5 + b4 + 11b1) q^77 + (3b10 - 5b9 - 3b6 - 5b5 + 5b4 - 17b1) * q^79 + (-7b11 - 7b8 - 32b2 + 32) q^81 + (-14b11 - 6b7 - 14b3 + 28) q^83 + (6b9 + 23b6 - 15b5 - 6b1) * q^85 + (-b10 - 29b9 + 14b4) * q^87 + (-4b8 + 4b7 - 8b3 - 7b2) * q^89 + (b11 + 7b7 + 3b3 + 22b2 - 3) q^91 + (25b10 + b9 - 25b6 + b5 - b4 - 4b1) q^93 + (-7b10 + 30b9 - 7b4) q^95 + (13b11 - b7 + 13b3 - 11) * q^97 + (5b11 - 21b7 + 5b3 - 71) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 - 8 * q^9	\( 12 q - 6 q^{3} - 8 q^{9} + 14 q^{11} - 82 q^{17} + 94 q^{19} + 116 q^{25} + 60 q^{27} + 146 q^{33} - 270 q^{35} + 120 q^{41} - 40 q^{43} - 204 q^{49} + 106 q^{51} - 372 q^{57} - 62 q^{59} - 64 q^{65} + 178 q^{67} + 54 q^{73} - 140 q^{75} + 206 q^{81} + 392 q^{83} - 26 q^{89} + 88 q^{91} - 184 q^{97} - 872 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 - 8 * q^9 + 14 * q^11 - 82 * q^17 + 94 * q^19 + 116 * q^25 + 60 * q^27 + 146 * q^33 - 270 * q^35 + 120 * q^41 - 40 * q^43 - 204 * q^49 + 106 * q^51 - 372 * q^57 - 62 * q^59 - 64 * q^65 + 178 * q^67 + 54 * q^73 - 140 * q^75 + 206 * q^81 + 392 * q^83 - 26 * q^89 + 88 * q^91 - 184 * q^97 - 872 * 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	224.3.o.c

Level	$224$
Weight	$3$
Character orbit	224.o
Analytic conductor	$6.104$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.10355792167\)
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:	\( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) x^12 + 2x^10 - 12x^9 + 12x^8 - 12x^7 + 148x^6 - 48x^5 + 192x^4 - 768x^3 + 512*x^2 + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 11 \nu^{11} + 48 \nu^{10} + 298 \nu^{9} + 420 \nu^{8} + 1212 \nu^{7} - 1980 \nu^{6} + \cdots - 49152 ) / 93184 \) (-11v^11 + 48v^10 + 298v^9 + 420v^8 + 1212v^7 - 1980v^6 + 7012v^5 - 4656v^4 + 30912v^3 - 18432v^2 + 57856*v - 49152) / 93184
\(\beta_{2}\)	\(=\)	\( ( 6 \nu^{11} - 51 \nu^{10} + 36 \nu^{9} - 14 \nu^{8} + 828 \nu^{7} - 12 \nu^{6} + 444 \nu^{5} + \cdots + 52224 ) / 46592 \) (6v^11 - 51v^10 + 36v^9 - 14v^8 + 828v^7 - 12v^6 + 444v^5 - 3516v^4 + 1008v^3 + 2112v^2 + 28800*v + 52224) / 46592
\(\beta_{3}\)	\(=\)	\( ( - 10 \nu^{11} - 97 \nu^{10} - 60 \nu^{9} - 826 \nu^{8} + 804 \nu^{7} + 748 \nu^{6} + 7268 \nu^{5} + \cdots + 99328 ) / 46592 \) (-10v^11 - 97v^10 - 60v^9 - 826v^8 + 804v^7 + 748v^6 + 7268v^5 - 3604v^4 - 10416v^3 + 13952v^2 + 21888*v + 99328) / 46592
\(\beta_{4}\)	\(=\)	\( ( 16 \nu^{11} + 319 \nu^{10} + 96 \nu^{9} - 98 \nu^{8} - 340 \nu^{7} - 396 \nu^{6} + 1548 \nu^{5} + \cdots - 512 ) / 46592 \) (16v^11 + 319v^10 + 96v^9 - 98v^8 - 340v^7 - 396v^6 + 1548v^5 + 11372v^4 + 4144v^3 - 192v^2 - 28032*v - 512) / 46592
\(\beta_{5}\)	\(=\)	\( ( 3 \nu^{11} + 20 \nu^{10} - 86 \nu^{9} + 84 \nu^{8} - 132 \nu^{7} + 956 \nu^{6} - 324 \nu^{5} + \cdots + 46080 ) / 6656 \) (3v^11 + 20v^10 - 86v^9 + 84v^8 - 132v^7 + 956v^6 - 324v^5 + 2064v^4 - 9584v^3 + 3968v^2 - 3072*v + 46080) / 6656
\(\beta_{6}\)	\(=\)	\( ( - 8 \nu^{11} - 23 \nu^{10} + 43 \nu^{9} - 42 \nu^{8} + 170 \nu^{7} + 16 \nu^{6} + 136 \nu^{5} + \cdots + 11904 ) / 11648 \) (-8v^11 - 23v^10 + 43v^9 - 42v^8 + 170v^7 + 16v^6 + 136v^5 - 2592v^4 - 2436v^3 - 2816v^2 + 8192*v + 11904) / 11648
\(\beta_{7}\)	\(=\)	\( ( - \nu^{11} + 6 \nu^{9} + 12 \nu^{8} + 4 \nu^{7} - 84 \nu^{6} - 52 \nu^{5} - 48 \nu^{4} + 480 \nu^{3} + \cdots - 3072 ) / 1024 \) (-v^11 + 6v^9 + 12v^8 + 4v^7 - 84v^6 - 52v^5 - 48v^4 + 480v^3 + 384v^2 + 1024*v - 3072) / 1024
\(\beta_{8}\)	\(=\)	\( ( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} + \cdots - 115200 ) / 46592 \) (51v^11 - 24v^10 + 306v^9 - 756v^8 + 668v^7 - 3924v^6 + 7596v^5 - 4224v^4 + 23856v^3 - 43200v^2 + 64256*v - 115200) / 46592
\(\beta_{9}\)	\(=\)	\( ( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} + \cdots - 115200 ) / 46592 \) (51v^11 - 24v^10 + 306v^9 - 756v^8 + 668v^7 - 3924v^6 + 7596v^5 - 4224v^4 + 23856v^3 - 43200v^2 - 28928*v - 115200) / 46592
\(\beta_{10}\)	\(=\)	\( ( 75 \nu^{11} - 46 \nu^{10} + 450 \nu^{9} - 448 \nu^{8} + 1796 \nu^{7} - 1788 \nu^{6} + 7188 \nu^{5} + \cdots + 512 ) / 46592 \) (75v^11 - 46v^10 + 450v^9 - 448v^8 + 1796v^7 - 1788v^6 + 7188v^5 - 14648v^4 + 19152v^3 + 192v^2 + 86272*v + 512) / 46592
\(\beta_{11}\)	\(=\)	\( ( - 81 \nu^{11} - 267 \nu^{10} - 486 \nu^{9} - 266 \nu^{8} + 1744 \nu^{7} - 2568 \nu^{6} + \cdots - 192512 ) / 46592 \) (-81v^11 - 267v^10 - 486v^9 - 266v^8 + 1744v^7 - 2568v^6 - 3264v^5 - 35708v^4 - 2688v^3 - 72192v^2 + 71296*v - 192512) / 46592

\(\nu\)	\(=\)	\( ( -\beta_{9} + \beta_{8} ) / 2 \) (-b9 + b8) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) (b10 - b8 + b7 - b6 + b3 - b2 - b1) / 2
\(\nu^{3}\)	\(=\)	\( -\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta _1 + 3 \) -b9 - b7 - b6 - b5 + b1 + 3
\(\nu^{4}\)	\(=\)	\( -\beta_{11} - 3\beta_{10} + 2\beta_{8} + 7\beta_{2} - 7 \) -b11 - 3b10 + 2b8 + 7*b2 - 7
\(\nu^{5}\)	\(=\)	\( \beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} - 9\beta_{2} + 7\beta_1 \) b10 + 2b9 - b8 + b7 - b6 + 2b5 - 2b4 + 3b3 - 9b2 + 7b1
\(\nu^{6}\)	\(=\)	\( -2\beta_{11} - 10\beta_{9} - 14\beta_{7} + 4\beta_{6} - 6\beta_{5} - 2\beta_{3} + 10\beta _1 - 28 \) -2b11 - 10b9 - 14b7 + 4b6 - 6b5 - 2b3 + 10*b1 - 28
\(\nu^{7}\)	\(=\)	\( -12\beta_{10} + 14\beta_{9} - 10\beta_{8} + 12\beta_{4} + 84\beta_{2} - 84 \) -12b10 + 14b9 - 10b8 + 12b4 + 84*b2 - 84
\(\nu^{8}\)	\(=\)	\( 6 \beta_{10} + 24 \beta_{9} - 18 \beta_{8} + 18 \beta_{7} - 6 \beta_{6} + 24 \beta_{5} - 24 \beta_{4} + \cdots + 54 \beta_1 \) 6b10 + 24b9 - 18b8 + 18b7 - 6b6 + 24b5 - 24b4 - 18b3 - 22b2 + 54b1
\(\nu^{9}\)	\(=\)	\( -36\beta_{11} + 8\beta_{9} - 8\beta_{7} + 120\beta_{6} - 12\beta_{5} - 36\beta_{3} - 8\beta _1 - 120 \) -36b11 + 8b9 - 8b7 + 120b6 - 12b5 - 36b3 - 8*b1 - 120
\(\nu^{10}\)	\(=\)	\( 44\beta_{11} + 76\beta_{10} - 40\beta_{9} - 40\beta_{8} + 168\beta_{4} - 124\beta_{2} + 124 \) 44b11 + 76b10 - 40b9 - 40b8 + 168b4 - 124b2 + 124
\(\nu^{11}\)	\(=\)	\( 308 \beta_{10} + 136 \beta_{9} + 20 \beta_{8} - 20 \beta_{7} - 308 \beta_{6} + 136 \beta_{5} + \cdots - 316 \beta_1 \) 308b10 + 136b9 + 20b8 - 20b7 - 308b6 + 136b5 - 136b4 - 228b3 + 12b2 - 316b1

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{6} + 3 T^{5} + \cdots + 361)^{2} \) (T^6 + 3T^5 + 20T^4 + 5T^3 + 178T^2 + 209*T + 361)^2
$5$	\( T^{12} + \cdots + 678446209 \) T^12 - 133T^10 + 13038T^8 - 566489T^6 + 18167550T^4 - 121144597*T^2 + 678446209
$7$	\( T^{12} + \cdots + 13841287201 \) T^12 + 102T^10 + 8463T^8 + 426692T^6 + 20319663T^4 + 588009702*T^2 + 13841287201
$11$	\( (T^{6} - 7 T^{5} + \cdots + 82369)^{2} \) (T^6 - 7T^5 + 170T^4 + 1421T^3 + 12632T^2 + 34727*T + 82369)^2
$13$	\( (T^{6} + 64 T^{4} + \cdots + 5488)^{2} \) (T^6 + 64T^4 + 1120T^2 + 5488)^2
$17$	\( (T^{6} + 41 T^{5} + \cdots + 2627641)^{2} \) (T^6 + 41T^5 + 1338T^4 + 17305T^3 + 184110T^2 - 556003*T + 2627641)^2
$19$	\( (T^{6} - 47 T^{5} + \cdots + 5139289)^{2} \) (T^6 - 47T^5 + 1570T^4 - 25499T^3 + 301772T^2 - 1448613*T + 5139289)^2
$23$	\( T^{12} + \cdots + 282099132781249 \) T^12 - 793T^10 + 426438T^8 - 126920309T^6 + 27651137970T^4 - 3399656090677*T^2 + 282099132781249
$29$	\( (T^{6} + 3208 T^{4} + \cdots + 747251568)^{2} \) (T^6 + 3208T^4 + 2906064T^2 + 747251568)^2
$31$	\( T^{12} + \cdots + 27\!\cdots\!09 \) T^12 - 2437T^10 + 3973602T^8 - 3739418485T^6 + 2583022025350T^4 - 1031995436549049*T^2 + 275719977530659809
$37$	\( T^{12} + \cdots + 50\!\cdots\!49 \) T^12 - 1813T^10 + 2589990T^8 - 1120889441T^6 + 356391821382T^4 - 49741121169397*T^2 + 5093212006428049
$41$	\( (T^{3} - 30 T^{2} + \cdots + 364)^{4} \) (T^3 - 30T^2 - 288T + 364)^4
$43$	\( (T^{3} + 10 T^{2} + \cdots + 16408)^{4} \) (T^3 + 10T^2 - 1692T + 16408)^4
$47$	\( T^{12} + \cdots + 7867032061329 \) T^12 - 4941T^10 + 19546754T^8 - 24040888461T^6 + 23671173062086T^4 - 13650307824321*T^2 + 7867032061329
$53$	\( T^{12} + \cdots + 18\!\cdots\!49 \) T^12 - 7581T^10 + 39500070T^8 - 109023176185T^6 + 219802017458598T^4 - 244580284856387613*T^2 + 185214367764856222849
$59$	\( (T^{6} + 31 T^{5} + \cdots + 19158129)^{2} \) (T^6 + 31T^5 + 2876T^4 - 68119T^3 + 3531538T^2 - 8381955*T + 19158129)^2
$61$	\( T^{12} + \cdots + 31\!\cdots\!69 \) T^12 - 22829T^10 + 375290886T^8 - 2976172636169T^6 + 17238611856745798T^4 - 25815567192061648365*T^2 + 31319682727146367431969
$67$	\( (T^{6} - 89 T^{5} + \cdots + 102323854161)^{2} \) (T^6 - 89T^5 + 13448T^4 - 147859T^3 + 59017138T^2 - 1767982287*T + 102323854161)^2
$71$	\( (T^{6} + 22416 T^{4} + \cdots + 203138627328)^{2} \) (T^6 + 22416T^4 + 143235904T^2 + 203138627328)^2
$73$	\( (T^{6} - 27 T^{5} + \cdots + 16072081)^{2} \) (T^6 - 27T^5 + 3110T^4 + 72305T^3 + 5560918T^2 + 9545429*T + 16072081)^2
$79$	\( T^{12} + \cdots + 73\!\cdots\!09 \) T^12 - 9961T^10 + 69363606T^8 - 243144641621T^6 + 621203105646258T^4 - 810195265404614005*T^2 + 736309573447307373409
$83$	\( (T^{3} - 98 T^{2} + \cdots + 823528)^{4} \) (T^3 - 98T^2 - 8484T + 823528)^4
$89$	\( (T^{6} + 13 T^{5} + \cdots + 86434209)^{2} \) (T^6 + 13T^5 + 4582T^4 - 38775T^3 + 19595430T^2 + 41027661*T + 86434209)^2
$97$	\( (T^{3} + 46 T^{2} + \cdots - 407276)^{4} \) (T^3 + 46T^2 - 9804T - 407276)^4
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