LMFDB - Newform orbit 3150.3.c.f

Newspace parameters

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8}+O(q^{10}) $ q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2b6 q^8	\( q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8} + (\beta_{12} + 2 \beta_{7} - \beta_{5}) q^{11} + (\beta_{14} + 2 \beta_{4} + 4 \beta_{2}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + (\beta_{13} + 2 \beta_{11} + 6 \beta_{6} - \beta_1) q^{17} + ( - \beta_{15} + \beta_{9} + 10) q^{19} + ( - 2 \beta_{10} + \beta_{4} - 2 \beta_{2}) q^{22} + (3 \beta_{11} + 2 \beta_{6} + \beta_1) q^{23} + ( - 4 \beta_{12} - 4 \beta_{5} - \beta_{3}) q^{26} + 2 \beta_{2} q^{28} + ( - 2 \beta_{12} + 2 \beta_{7} + 13 \beta_{5} + 2 \beta_{3}) q^{29} + (2 \beta_{15} + 3 \beta_{9} - 2 \beta_{8} + 14) q^{31} - 4 \beta_{6} q^{32} + ( - \beta_{15} + 2 \beta_{9} + 2 \beta_{8} - 12) q^{34} + ( - 4 \beta_{14} + 3 \beta_{10} + 8 \beta_{4} + 10 \beta_{2}) q^{37} + (2 \beta_{13} - 10 \beta_{6} - \beta_1) q^{38} + ( - 6 \beta_{12} - 6 \beta_{7} + 20 \beta_{5} - \beta_{3}) q^{41} + (6 \beta_{14} + 2 \beta_{10} + 4 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{12} + 4 \beta_{7} - 2 \beta_{5}) q^{44} + ( - 2 \beta_{9} + 3 \beta_{8} - 4) q^{46} + ( - 3 \beta_{13} - 2 \beta_{11} + 5 \beta_1) q^{47} - 7 q^{49} + (2 \beta_{14} + 4 \beta_{4} + 8 \beta_{2}) q^{52} + (2 \beta_{13} - 3 \beta_{11} + 15 \beta_{6} - 7 \beta_1) q^{53} - 2 \beta_{12} q^{56} + ( - 4 \beta_{14} - 2 \beta_{10} - 13 \beta_{4} + 4 \beta_{2}) q^{58} + ( - 12 \beta_{12} - 9 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{59} + (2 \beta_{15} - 6 \beta_{9} + 6 \beta_{8} - 18) q^{61} + ( - 4 \beta_{13} - 4 \beta_{11} - 14 \beta_{6} - 3 \beta_1) q^{62} + 8 q^{64} + ( - 4 \beta_{14} - \beta_{10} - 4 \beta_{4} + 14 \beta_{2}) q^{67} + (2 \beta_{13} + 4 \beta_{11} + 12 \beta_{6} - 2 \beta_1) q^{68} + ( - 13 \beta_{12} + 4 \beta_{7} + 7 \beta_{5}) q^{71} + (3 \beta_{14} + 4 \beta_{10} + 24 \beta_{2}) q^{73} + ( - 10 \beta_{12} - 6 \beta_{7} - 16 \beta_{5} + 4 \beta_{3}) q^{74} + ( - 2 \beta_{15} + 2 \beta_{9} + 20) q^{76} + ( - 2 \beta_{13} - \beta_{11} - 7 \beta_{6}) q^{77} + ( - 2 \beta_{15} - 14 \beta_{9} - 12 \beta_{8} + 8) q^{79} + (2 \beta_{14} + 6 \beta_{10} - 20 \beta_{4} + 12 \beta_{2}) q^{82} + ( - \beta_{13} + 10 \beta_{11} - 16 \beta_{6} + 23 \beta_1) q^{83} + (2 \beta_{12} - 4 \beta_{7} - 8 \beta_{5} - 6 \beta_{3}) q^{86} + ( - 4 \beta_{10} + 2 \beta_{4} - 4 \beta_{2}) q^{88} + (\beta_{7} - 26 \beta_{5} - 4 \beta_{3}) q^{89} + ( - 7 \beta_{9} - 2 \beta_{8} - 28) q^{91} + (6 \beta_{11} + 4 \beta_{6} + 2 \beta_1) q^{92} + (3 \beta_{15} - 10 \beta_{9} - 2 \beta_{8}) q^{94} + ( - 5 \beta_{14} - 2 \beta_{10} - 2 \beta_{4} + 32 \beta_{2}) q^{97} + 7 \beta_{6} q^{98}+O(q^{100}) \) q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2b6 q^8 + (b12 + 2b7 - b5) q^11 + (b14 + 2b4 + 4b2) * q^13 - b12 * q^14 + 4 * q^16 + (b13 + 2b11 + 6b6 - b1) * q^17 + (-b15 + b9 + 10) * q^19 + (-2b10 + b4 - 2b2) * q^22 + (3b11 + 2b6 + b1) * q^23 + (-4b12 - 4b5 - b3) * q^26 + 2b2 q^28 + (-2b12 + 2b7 + 13b5 + 2b3) * q^29 + (2b15 + 3b9 - 2b8 + 14) q^31 - 4b6 q^32 + (-b15 + 2b9 + 2b8 - 12) * q^34 + (-4b14 + 3b10 + 8b4 + 10b2) * q^37 + (2b13 - 10b6 - b1) * q^38 + (-6b12 - 6b7 + 20b5 - b3) q^41 + (6b14 + 2b10 + 4b4 - 2b2) * q^43 + (2b12 + 4b7 - 2b5) q^44 + (-2b9 + 3b8 - 4) * q^46 + (-3b13 - 2b11 + 5b1) q^47 - 7 * q^49 + (2b14 + 4b4 + 8b2) q^52 + (2b13 - 3b11 + 15b6 - 7b1) * q^53 - 2b12 q^56 + (-4b14 - 2b10 - 13b4 + 4b2) * q^58 + (-12b12 - 9b7 + 2b5 + 2b3) * q^59 + (2b15 - 6b9 + 6b8 - 18) q^61 + (-4b13 - 4b11 - 14b6 - 3b1) * q^62 + 8 * q^64 + (-4b14 - b10 - 4b4 + 14b2) q^67 + (2b13 + 4b11 + 12b6 - 2b1) * q^68 + (-13b12 + 4b7 + 7b5) q^71 + (3b14 + 4b10 + 24b2) q^73 + (-10b12 - 6b7 - 16b5 + 4b3) * q^74 + (-2b15 + 2b9 + 20) * q^76 + (-2b13 - b11 - 7b6) * q^77 + (-2b15 - 14b9 - 12b8 + 8) q^79 + (2b14 + 6b10 - 20b4 + 12b2) * q^82 + (-b13 + 10b11 - 16b6 + 23b1) q^83 + (2b12 - 4b7 - 8b5 - 6b3) * q^86 + (-4b10 + 2b4 - 4b2) q^88 + (b7 - 26b5 - 4b3) * q^89 + (-7b9 - 2b8 - 28) * q^91 + (6b11 + 4b6 + 2b1) q^92 + (3b15 - 10b9 - 2b8) q^94 + (-5b14 - 2b10 - 2b4 + 32b2) * q^97 + 7b6 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7*x^12 + 753*x^8 + 112*x^4 + 256 :

$\beta_{1}$	$=$	$ ( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 9396 $ (-v^12 + 105v^8 + 15v^4 + 42056) / 9396
$\beta_{2}$	$=$	$ ( -14\nu^{12} - 96\nu^{8} - 10752\nu^{4} - 815 ) / 2349 $ (-14v^12 - 96v^8 - 10752*v^4 - 815) / 2349
$\beta_{3}$	$=$	$ ( -\nu^{12} - 7\nu^{8} - 737\nu^{4} - 56 ) / 36 $ (-v^12 - 7v^8 - 737v^4 - 56) / 36
$\beta_{4}$	$=$	$ ( 33\nu^{14} + 247\nu^{10} + 25025\nu^{6} + 18304\nu^{2} ) / 8352 $ (33v^14 + 247v^10 + 25025v^6 + 18304v^2) / 8352
$\beta_{5}$	$=$	$ ( 245 \nu^{15} - 374 \nu^{13} + 1419 \nu^{11} - 2490 \nu^{9} + 182157 \nu^{7} - 278358 \nu^{5} - 185272 \nu^{3} + 39712 \nu ) / 150336 $ (245v^15 - 374v^13 + 1419v^11 - 2490v^9 + 182157v^7 - 278358v^5 - 185272v^3 + 39712v) / 150336
$\beta_{6}$	$=$	$ ( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336 $ (245v^15 + 374v^13 + 1419v^11 + 2490v^9 + 182157v^7 + 278358v^5 - 185272v^3 - 39712v) / 150336
$\beta_{7}$	$=$	$ ( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 2592 $ (23v^14 + 177v^10 + 17367v^6 + 12704v^2) / 2592
$\beta_{8}$	$=$	$ ( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592 $ (31v^14 + 201v^10 + 23295v^6 - 10112v^2) / 2592
$\beta_{9}$	$=$	$ ( 91 \nu^{15} + 137 \nu^{13} + 537 \nu^{11} + 927 \nu^{9} + 67887 \nu^{7} + 103737 \nu^{5} - 69044 \nu^{3} - 14800 \nu ) / 25056 $ (91v^15 + 137v^13 + 537v^11 + 927v^9 + 67887v^7 + 103737v^5 - 69044v^3 - 14800v) / 25056
$\beta_{10}$	$=$	$ ( 91 \nu^{15} - 137 \nu^{13} + 537 \nu^{11} - 927 \nu^{9} + 67887 \nu^{7} - 103737 \nu^{5} - 69044 \nu^{3} + 14800 \nu ) / 12528 $ (91v^15 - 137v^13 + 537v^11 - 927v^9 + 67887v^7 - 103737v^5 - 69044v^3 + 14800v) / 12528
$\beta_{11}$	$=$	$ ( 403 \nu^{15} - 122 \nu^{13} + 2925 \nu^{11} - 1110 \nu^{9} + 303675 \nu^{7} - 92826 \nu^{5} + 115960 \nu^{3} - 149024 \nu ) / 50112 $ (403v^15 - 122v^13 + 2925v^11 - 1110v^9 + 303675v^7 - 92826v^5 + 115960v^3 - 149024v) / 50112
$\beta_{12}$	$=$	$ ( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112 $ (403v^15 + 122v^13 + 2925v^11 + 1110v^9 + 303675v^7 + 92826v^5 + 115960v^3 + 149024v) / 50112
$\beta_{13}$	$=$	$ ( 25\nu^{14} + 159\nu^{10} + 18649\nu^{6} - 8096\nu^{2} ) / 928 $ (25v^14 + 159v^10 + 18649v^6 - 8096v^2) / 928
$\beta_{14}$	$=$	$ ( 1349 \nu^{15} + 407 \nu^{13} + 9735 \nu^{11} + 3201 \nu^{9} + 1018545 \nu^{7} + 311271 \nu^{5} + 388916 \nu^{3} + 499664 \nu ) / 75168 $ (1349v^15 + 407v^13 + 9735v^11 + 3201v^9 + 1018545v^7 + 311271v^5 + 388916v^3 + 499664v) / 75168
$\beta_{15}$	$=$	$ ( - 1349 \nu^{15} + 407 \nu^{13} - 9735 \nu^{11} + 3201 \nu^{9} - 1018545 \nu^{7} + 311271 \nu^{5} - 388916 \nu^{3} + 499664 \nu ) / 37584 $ (-1349v^15 + 407v^13 - 9735v^11 + 3201v^9 - 1018545v^7 + 311271v^5 - 388916v^3 + 499664v) / 37584

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

$\nu$	$=$	$ ( \beta_{15} + 2\beta_{14} - 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{6} - 2\beta_{5} ) / 16 $ (b15 + 2b14 - 2b12 + 2b11 + b10 - 2b9 + 2b6 - 2b5) / 16
$\nu^{2}$	$=$	$ ( \beta_{13} - 3\beta_{8} - 3\beta_{7} + 9\beta_{4} ) / 8 $ (b13 - 3b8 - 3b7 + 9*b4) / 8
$\nu^{3}$	$=$	$ ( -\beta_{15} + 2\beta_{14} - 4\beta_{12} - 4\beta_{11} - 5\beta_{10} - 10\beta_{9} + 20\beta_{6} + 20\beta_{5} ) / 8 $ (-b15 + 2b14 - 4b12 - 4b11 - 5b10 - 10b9 + 20b6 + 20*b5) / 8
$\nu^{4}$	$=$	$ ( 9\beta_{3} - 42\beta_{2} + 3\beta _1 - 14 ) / 8 $ (9b3 - 42b2 + 3*b1 - 14) / 8
$\nu^{5}$	$=$	$ ( 5 \beta_{15} + 10 \beta_{14} - 22 \beta_{12} + 22 \beta_{11} - 55 \beta_{10} + 110 \beta_{9} - 242 \beta_{6} + 242 \beta_{5} ) / 16 $ (5b15 + 10b14 - 22b12 + 22b11 - 55b10 + 110b9 - 242b6 + 242b5) / 16
$\nu^{6}$	$=$	$ ( -20\beta_{13} + 45\beta_{8} - 36\beta_{7} + 81\beta_{4} ) / 4 $ (-20b13 + 45b8 - 36b7 + 81b4) / 4
$\nu^{7}$	$=$	$ ( - 91 \beta_{15} + 182 \beta_{14} - 406 \beta_{12} - 406 \beta_{11} + 169 \beta_{10} + 338 \beta_{9} - 754 \beta_{6} - 754 \beta_{5} ) / 16 $ (-91b15 + 182b14 - 406b12 - 406b11 + 169b10 + 338b9 - 754b6 - 754b5) / 16
$\nu^{8}$	$=$	$ ( -63\beta_{3} + 282\beta_{2} + 651\beta _1 - 2914 ) / 8 $ (-63b3 + 282b2 + 651*b1 - 2914) / 8
$\nu^{9}$	$=$	$ ( - 289 \beta_{15} - 578 \beta_{14} + 1292 \beta_{12} - 1292 \beta_{11} - 85 \beta_{10} + 170 \beta_{9} - 380 \beta_{6} + 380 \beta_{5} ) / 8 $ (-289b15 - 578b14 + 1292b12 - 1292b11 - 85b10 + 170b9 - 380b6 + 380b5) / 8
$\nu^{10}$	$=$	$ ( -605\beta_{13} + 1353\beta_{8} + 3135\beta_{7} - 7011\beta_{4} ) / 8 $ (-605b13 + 1353b8 + 3135b7 - 7011b4) / 8
$\nu^{11}$	$=$	$ ( 2047 \beta_{15} - 4094 \beta_{14} + 9154 \beta_{12} + 9154 \beta_{11} + 5963 \beta_{10} + 11926 \beta_{9} - 26666 \beta_{6} - 26666 \beta_{5} ) / 16 $ (2047b15 - 4094b14 + 9154b12 + 9154b11 + 5963b10 + 11926b9 - 26666b6 - 26666b5) / 16
$\nu^{12}$	$=$	$ ( -1620\beta_{3} + 7245\beta_{2} - 1692\beta _1 + 7567 ) / 2 $ (-1620b3 + 7245b2 - 1692*b1 + 7567) / 2
$\nu^{13}$	$=$	$ ( 233 \beta_{15} + 466 \beta_{14} - 1042 \beta_{12} + 1042 \beta_{11} + 42173 \beta_{10} - 84346 \beta_{9} + 188602 \beta_{6} - 188602 \beta_{5} ) / 16 $ (233b15 + 466b14 - 1042b12 + 1042b11 + 42173b10 - 84346b9 + 188602b6 - 188602b5) / 16
$\nu^{14}$	$=$	$ ( 34307\beta_{13} - 76713\beta_{8} + 32799\beta_{7} - 73341\beta_{4} ) / 8 $ (34307b13 - 76713b8 + 32799b7 - 73341b4) / 8
$\nu^{15}$	$=$	$ ( 27145 \beta_{15} - 54290 \beta_{14} + 121396 \beta_{12} + 121396 \beta_{11} - 83875 \beta_{10} - 167750 \beta_{9} + 375100 \beta_{6} + 375100 \beta_{5} ) / 8 $ (27145b15 - 54290b14 + 121396b12 + 121396b11 - 83875b10 - 167750b9 + 375100b6 + 375100b5) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

449.1

2.08559	+	0.941471i
0.359610	−	0.796626i
−0.796626	−	0.359610i
−0.941471	+	2.08559i
−0.941471	−	2.08559i
−0.796626	+	0.359610i
0.359610	+	0.796626i
2.08559	−	0.941471i
−0.359610	+	0.796626i
−2.08559	−	0.941471i
0.941471	−	2.08559i
0.796626	+	0.359610i
0.796626	−	0.359610i
0.941471	+	2.08559i
−2.08559	+	0.941471i
−0.359610	−	0.796626i

−1.41421

2.00000

−

2.64575i

−2.82843

449.2

−1.41421

2.00000

−

2.64575i

−2.82843

449.3

−1.41421

2.00000

−

2.64575i

−2.82843

449.4

−1.41421

2.00000

−

2.64575i

−2.82843

449.5

−1.41421

2.00000

2.64575i

−2.82843

449.6

−1.41421

2.00000

2.64575i

−2.82843

449.7

−1.41421

2.00000

2.64575i

−2.82843

449.8

−1.41421

2.00000

2.64575i

−2.82843

449.9

1.41421

2.00000

−

2.64575i

2.82843

449.10

1.41421

2.00000

−

2.64575i

2.82843

449.11

1.41421

2.00000

−

2.64575i

2.82843

449.12

1.41421

2.00000

−

2.64575i

2.82843

449.13

1.41421

2.00000

2.64575i

2.82843

449.14

1.41421

2.00000

2.64575i

2.82843

449.15

1.41421

2.00000

2.64575i

2.82843

449.16

1.41421

2.00000

2.64575i

2.82843

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3150.3.c.f		16
3.b	odd	2	1	inner	3150.3.c.f		16
5.b	even	2	1	inner	3150.3.c.f		16
5.c	odd	4	1		630.3.e.b	✓	8
5.c	odd	4	1		3150.3.e.f		8
15.d	odd	2	1	inner	3150.3.c.f		16
15.e	even	4	1		630.3.e.b	✓	8
15.e	even	4	1		3150.3.e.f		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.3.e.b	✓	8	5.c	odd	4	1
630.3.e.b	✓	8	15.e	even	4	1
3150.3.c.f		16	1.a	even	1	1	trivial
3150.3.c.f		16	3.b	odd	2	1	inner
3150.3.c.f		16	5.b	even	2	1	inner
3150.3.c.f		16	15.d	odd	2	1	inner
3150.3.e.f		8	5.c	odd	4	1
3150.3.e.f		8	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{8} + 384T_{11}^{6} + 44832T_{11}^{4} + 1673216T_{11}^{2} + 15872256 $ T11^8 + 384*T11^6 + 44832*T11^4 + 1673216*T11^2 + 15872256 acting on $S_{3}^{\mathrm{new}}(3150, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{2} + 7)^{8} $ (T^2 + 7)^8
$11$	$ (T^{8} + 384 T^{6} + 44832 T^{4} + \cdots + 15872256)^{2} $ (T^8 + 384T^6 + 44832T^4 + 1673216*T^2 + 15872256)^2
$13$	$ (T^{8} + 792 T^{6} + 149208 T^{4} + \cdots + 14470416)^{2} $ (T^8 + 792T^6 + 149208T^4 + 5015392*T^2 + 14470416)^2
$17$	$ (T^{8} - 1152 T^{6} + 117888 T^{4} + \cdots + 331776)^{2} $ (T^8 - 1152T^6 + 117888T^4 - 2957312*T^2 + 331776)^2
$19$	$ (T^{4} - 40 T^{3} + 20 T^{2} + 7600 T + 24900)^{4} $ (T^4 - 40T^3 + 20T^2 + 7600*T + 24900)^4
$23$	$ (T^{8} - 616 T^{6} + 112792 T^{4} + \cdots + 80353296)^{2} $ (T^8 - 616T^6 + 112792T^4 - 6812064*T^2 + 80353296)^2
$29$	$ (T^{8} + 4136 T^{6} + \cdots + 182633150736)^{2} $ (T^8 + 4136T^6 + 5285592T^4 + 2064468384*T^2 + 182633150736)^2
$31$	$ (T^{4} - 56 T^{3} - 1468 T^{2} + \cdots - 53724)^{4} $ (T^4 - 56T^3 - 1468T^2 + 89936*T - 53724)^4
$37$	$ (T^{8} + 9744 T^{6} + \cdots + 17685591019776)^{2} $ (T^8 + 9744T^6 + 32528736T^4 + 42572556544*T^2 + 17685591019776)^2
$41$	$ (T^{8} + 8656 T^{6} + \cdots + 11776317808896)^{2} $ (T^8 + 8656T^6 + 24770656T^4 + 28820802816*T^2 + 11776317808896)^2
$43$	$ (T^{8} + 11088 T^{6} + \cdots + 23646201405696)^{2} $ (T^8 + 11088T^6 + 42181728T^4 + 61368149248*T^2 + 23646201405696)^2
$47$	$ (T^{8} - 7264 T^{6} + \cdots + 147161235456)^{2} $ (T^8 - 7264T^6 + 13958656T^4 - 5044506624*T^2 + 147161235456)^2
$53$	$ (T^{8} - 8464 T^{6} + \cdots + 8432658441216)^{2} $ (T^8 - 8464T^6 + 23391616T^4 - 25129465344*T^2 + 8432658441216)^2
$59$	$ (T^{8} + 16816 T^{6} + \cdots + 7297000079616)^{2} $ (T^8 + 16816T^6 + 57550432T^4 + 41795230464*T^2 + 7297000079616)^2
$61$	$ (T^{4} + 72 T^{3} - 3032 T^{2} + \cdots + 5776)^{4} $ (T^4 + 72T^3 - 3032T^2 + 5472*T + 5776)^4
$67$	$ (T^{8} + 10384 T^{6} + \cdots + 913966592256)^{2} $ (T^8 + 10384T^6 + 29058912T^4 + 11129446656*T^2 + 913966592256)^2
$71$	$ (T^{8} + 11136 T^{6} + \cdots + 13463498301696)^{2} $ (T^8 + 11136T^6 + 38341152T^4 + 45609580544*T^2 + 13463498301696)^2
$73$	$ (T^{8} + 21208 T^{6} + \cdots + 36190379032336)^{2} $ (T^8 + 21208T^6 + 124476504T^4 + 231628973152*T^2 + 36190379032336)^2
$79$	$ (T^{4} - 32 T^{3} - 13840 T^{2} + \cdots - 14801856)^{4} $ (T^4 - 32T^3 - 13840T^2 + 978176*T - 14801856)^4
$83$	$ (T^{8} - 50528 T^{6} + \cdots + 36\!\cdots\!96)^{2} $ (T^8 - 50528T^6 + 825306624T^4 - 4565309755392*T^2 + 3665777429200896)^2
$89$	$ (T^{8} + 14448 T^{6} + \cdots + 416365629696)^{2} $ (T^8 + 14448T^6 + 53476704T^4 + 13199228672*T^2 + 416365629696)^2
$97$	$ (T^{8} + 36376 T^{6} + \cdots + 615175117820176)^{2} $ (T^8 + 36376T^6 + 391877592T^4 + 1149277896544*T^2 + 615175117820176)^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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8}+O(q^{10}) \) q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2b6 q^8	\( q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8} + (\beta_{12} + 2 \beta_{7} - \beta_{5}) q^{11} + (\beta_{14} + 2 \beta_{4} + 4 \beta_{2}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + (\beta_{13} + 2 \beta_{11} + 6 \beta_{6} - \beta_1) q^{17} + ( - \beta_{15} + \beta_{9} + 10) q^{19} + ( - 2 \beta_{10} + \beta_{4} - 2 \beta_{2}) q^{22} + (3 \beta_{11} + 2 \beta_{6} + \beta_1) q^{23} + ( - 4 \beta_{12} - 4 \beta_{5} - \beta_{3}) q^{26} + 2 \beta_{2} q^{28} + ( - 2 \beta_{12} + 2 \beta_{7} + 13 \beta_{5} + 2 \beta_{3}) q^{29} + (2 \beta_{15} + 3 \beta_{9} - 2 \beta_{8} + 14) q^{31} - 4 \beta_{6} q^{32} + ( - \beta_{15} + 2 \beta_{9} + 2 \beta_{8} - 12) q^{34} + ( - 4 \beta_{14} + 3 \beta_{10} + 8 \beta_{4} + 10 \beta_{2}) q^{37} + (2 \beta_{13} - 10 \beta_{6} - \beta_1) q^{38} + ( - 6 \beta_{12} - 6 \beta_{7} + 20 \beta_{5} - \beta_{3}) q^{41} + (6 \beta_{14} + 2 \beta_{10} + 4 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{12} + 4 \beta_{7} - 2 \beta_{5}) q^{44} + ( - 2 \beta_{9} + 3 \beta_{8} - 4) q^{46} + ( - 3 \beta_{13} - 2 \beta_{11} + 5 \beta_1) q^{47} - 7 q^{49} + (2 \beta_{14} + 4 \beta_{4} + 8 \beta_{2}) q^{52} + (2 \beta_{13} - 3 \beta_{11} + 15 \beta_{6} - 7 \beta_1) q^{53} - 2 \beta_{12} q^{56} + ( - 4 \beta_{14} - 2 \beta_{10} - 13 \beta_{4} + 4 \beta_{2}) q^{58} + ( - 12 \beta_{12} - 9 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{59} + (2 \beta_{15} - 6 \beta_{9} + 6 \beta_{8} - 18) q^{61} + ( - 4 \beta_{13} - 4 \beta_{11} - 14 \beta_{6} - 3 \beta_1) q^{62} + 8 q^{64} + ( - 4 \beta_{14} - \beta_{10} - 4 \beta_{4} + 14 \beta_{2}) q^{67} + (2 \beta_{13} + 4 \beta_{11} + 12 \beta_{6} - 2 \beta_1) q^{68} + ( - 13 \beta_{12} + 4 \beta_{7} + 7 \beta_{5}) q^{71} + (3 \beta_{14} + 4 \beta_{10} + 24 \beta_{2}) q^{73} + ( - 10 \beta_{12} - 6 \beta_{7} - 16 \beta_{5} + 4 \beta_{3}) q^{74} + ( - 2 \beta_{15} + 2 \beta_{9} + 20) q^{76} + ( - 2 \beta_{13} - \beta_{11} - 7 \beta_{6}) q^{77} + ( - 2 \beta_{15} - 14 \beta_{9} - 12 \beta_{8} + 8) q^{79} + (2 \beta_{14} + 6 \beta_{10} - 20 \beta_{4} + 12 \beta_{2}) q^{82} + ( - \beta_{13} + 10 \beta_{11} - 16 \beta_{6} + 23 \beta_1) q^{83} + (2 \beta_{12} - 4 \beta_{7} - 8 \beta_{5} - 6 \beta_{3}) q^{86} + ( - 4 \beta_{10} + 2 \beta_{4} - 4 \beta_{2}) q^{88} + (\beta_{7} - 26 \beta_{5} - 4 \beta_{3}) q^{89} + ( - 7 \beta_{9} - 2 \beta_{8} - 28) q^{91} + (6 \beta_{11} + 4 \beta_{6} + 2 \beta_1) q^{92} + (3 \beta_{15} - 10 \beta_{9} - 2 \beta_{8}) q^{94} + ( - 5 \beta_{14} - 2 \beta_{10} - 2 \beta_{4} + 32 \beta_{2}) q^{97} + 7 \beta_{6} q^{98}+O(q^{100}) \) q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2b6 q^8 + (b12 + 2b7 - b5) q^11 + (b14 + 2b4 + 4b2) * q^13 - b12 * q^14 + 4 * q^16 + (b13 + 2b11 + 6b6 - b1) * q^17 + (-b15 + b9 + 10) * q^19 + (-2b10 + b4 - 2b2) * q^22 + (3b11 + 2b6 + b1) * q^23 + (-4b12 - 4b5 - b3) * q^26 + 2b2 q^28 + (-2b12 + 2b7 + 13b5 + 2b3) * q^29 + (2b15 + 3b9 - 2b8 + 14) q^31 - 4b6 q^32 + (-b15 + 2b9 + 2b8 - 12) * q^34 + (-4b14 + 3b10 + 8b4 + 10b2) * q^37 + (2b13 - 10b6 - b1) * q^38 + (-6b12 - 6b7 + 20b5 - b3) q^41 + (6b14 + 2b10 + 4b4 - 2b2) * q^43 + (2b12 + 4b7 - 2b5) q^44 + (-2b9 + 3b8 - 4) * q^46 + (-3b13 - 2b11 + 5b1) q^47 - 7 * q^49 + (2b14 + 4b4 + 8b2) q^52 + (2b13 - 3b11 + 15b6 - 7b1) * q^53 - 2b12 q^56 + (-4b14 - 2b10 - 13b4 + 4b2) * q^58 + (-12b12 - 9b7 + 2b5 + 2b3) * q^59 + (2b15 - 6b9 + 6b8 - 18) q^61 + (-4b13 - 4b11 - 14b6 - 3b1) * q^62 + 8 * q^64 + (-4b14 - b10 - 4b4 + 14b2) q^67 + (2b13 + 4b11 + 12b6 - 2b1) * q^68 + (-13b12 + 4b7 + 7b5) q^71 + (3b14 + 4b10 + 24b2) q^73 + (-10b12 - 6b7 - 16b5 + 4b3) * q^74 + (-2b15 + 2b9 + 20) * q^76 + (-2b13 - b11 - 7b6) * q^77 + (-2b15 - 14b9 - 12b8 + 8) q^79 + (2b14 + 6b10 - 20b4 + 12b2) * q^82 + (-b13 + 10b11 - 16b6 + 23b1) q^83 + (2b12 - 4b7 - 8b5 - 6b3) * q^86 + (-4b10 + 2b4 - 4b2) q^88 + (b7 - 26b5 - 4b3) * q^89 + (-7b9 - 2b8 - 28) * q^91 + (6b11 + 4b6 + 2b1) q^92 + (3b15 - 10b9 - 2b8) q^94 + (-5b14 - 2b10 - 2b4 + 32b2) * q^97 + 7b6 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

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	3150.3.c.f

Level	$3150$
Weight	$3$
Character orbit	3150.c
Analytic conductor	$85.831$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(85.8312832735\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.9671731157401600000000.1
Defining polynomial:	\( x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 \) x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{28} \)
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 9396 \) (-v^12 + 105v^8 + 15v^4 + 42056) / 9396
\(\beta_{2}\)	\(=\)	\( ( -14\nu^{12} - 96\nu^{8} - 10752\nu^{4} - 815 ) / 2349 \) (-14v^12 - 96v^8 - 10752*v^4 - 815) / 2349
\(\beta_{3}\)	\(=\)	\( ( -\nu^{12} - 7\nu^{8} - 737\nu^{4} - 56 ) / 36 \) (-v^12 - 7v^8 - 737v^4 - 56) / 36
\(\beta_{4}\)	\(=\)	\( ( 33\nu^{14} + 247\nu^{10} + 25025\nu^{6} + 18304\nu^{2} ) / 8352 \) (33v^14 + 247v^10 + 25025v^6 + 18304v^2) / 8352
\(\beta_{5}\)	\(=\)	\( ( 245 \nu^{15} - 374 \nu^{13} + 1419 \nu^{11} - 2490 \nu^{9} + 182157 \nu^{7} - 278358 \nu^{5} - 185272 \nu^{3} + 39712 \nu ) / 150336 \) (245v^15 - 374v^13 + 1419v^11 - 2490v^9 + 182157v^7 - 278358v^5 - 185272v^3 + 39712v) / 150336
\(\beta_{6}\)	\(=\)	\( ( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336 \) (245v^15 + 374v^13 + 1419v^11 + 2490v^9 + 182157v^7 + 278358v^5 - 185272v^3 - 39712v) / 150336
\(\beta_{7}\)	\(=\)	\( ( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 2592 \) (23v^14 + 177v^10 + 17367v^6 + 12704v^2) / 2592
\(\beta_{8}\)	\(=\)	\( ( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592 \) (31v^14 + 201v^10 + 23295v^6 - 10112v^2) / 2592
\(\beta_{9}\)	\(=\)	\( ( 91 \nu^{15} + 137 \nu^{13} + 537 \nu^{11} + 927 \nu^{9} + 67887 \nu^{7} + 103737 \nu^{5} - 69044 \nu^{3} - 14800 \nu ) / 25056 \) (91v^15 + 137v^13 + 537v^11 + 927v^9 + 67887v^7 + 103737v^5 - 69044v^3 - 14800v) / 25056
\(\beta_{10}\)	\(=\)	\( ( 91 \nu^{15} - 137 \nu^{13} + 537 \nu^{11} - 927 \nu^{9} + 67887 \nu^{7} - 103737 \nu^{5} - 69044 \nu^{3} + 14800 \nu ) / 12528 \) (91v^15 - 137v^13 + 537v^11 - 927v^9 + 67887v^7 - 103737v^5 - 69044v^3 + 14800v) / 12528
\(\beta_{11}\)	\(=\)	\( ( 403 \nu^{15} - 122 \nu^{13} + 2925 \nu^{11} - 1110 \nu^{9} + 303675 \nu^{7} - 92826 \nu^{5} + 115960 \nu^{3} - 149024 \nu ) / 50112 \) (403v^15 - 122v^13 + 2925v^11 - 1110v^9 + 303675v^7 - 92826v^5 + 115960v^3 - 149024v) / 50112
\(\beta_{12}\)	\(=\)	\( ( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112 \) (403v^15 + 122v^13 + 2925v^11 + 1110v^9 + 303675v^7 + 92826v^5 + 115960v^3 + 149024v) / 50112
\(\beta_{13}\)	\(=\)	\( ( 25\nu^{14} + 159\nu^{10} + 18649\nu^{6} - 8096\nu^{2} ) / 928 \) (25v^14 + 159v^10 + 18649v^6 - 8096v^2) / 928
\(\beta_{14}\)	\(=\)	\( ( 1349 \nu^{15} + 407 \nu^{13} + 9735 \nu^{11} + 3201 \nu^{9} + 1018545 \nu^{7} + 311271 \nu^{5} + 388916 \nu^{3} + 499664 \nu ) / 75168 \) (1349v^15 + 407v^13 + 9735v^11 + 3201v^9 + 1018545v^7 + 311271v^5 + 388916v^3 + 499664v) / 75168
\(\beta_{15}\)	\(=\)	\( ( - 1349 \nu^{15} + 407 \nu^{13} - 9735 \nu^{11} + 3201 \nu^{9} - 1018545 \nu^{7} + 311271 \nu^{5} - 388916 \nu^{3} + 499664 \nu ) / 37584 \) (-1349v^15 + 407v^13 - 9735v^11 + 3201v^9 - 1018545v^7 + 311271v^5 - 388916v^3 + 499664v) / 37584

\(\nu\)	\(=\)	\( ( \beta_{15} + 2\beta_{14} - 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{6} - 2\beta_{5} ) / 16 \) (b15 + 2b14 - 2b12 + 2b11 + b10 - 2b9 + 2b6 - 2b5) / 16
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - 3\beta_{8} - 3\beta_{7} + 9\beta_{4} ) / 8 \) (b13 - 3b8 - 3b7 + 9*b4) / 8
\(\nu^{3}\)	\(=\)	\( ( -\beta_{15} + 2\beta_{14} - 4\beta_{12} - 4\beta_{11} - 5\beta_{10} - 10\beta_{9} + 20\beta_{6} + 20\beta_{5} ) / 8 \) (-b15 + 2b14 - 4b12 - 4b11 - 5b10 - 10b9 + 20b6 + 20*b5) / 8
\(\nu^{4}\)	\(=\)	\( ( 9\beta_{3} - 42\beta_{2} + 3\beta _1 - 14 ) / 8 \) (9b3 - 42b2 + 3*b1 - 14) / 8
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{15} + 10 \beta_{14} - 22 \beta_{12} + 22 \beta_{11} - 55 \beta_{10} + 110 \beta_{9} - 242 \beta_{6} + 242 \beta_{5} ) / 16 \) (5b15 + 10b14 - 22b12 + 22b11 - 55b10 + 110b9 - 242b6 + 242b5) / 16
\(\nu^{6}\)	\(=\)	\( ( -20\beta_{13} + 45\beta_{8} - 36\beta_{7} + 81\beta_{4} ) / 4 \) (-20b13 + 45b8 - 36b7 + 81b4) / 4
\(\nu^{7}\)	\(=\)	\( ( - 91 \beta_{15} + 182 \beta_{14} - 406 \beta_{12} - 406 \beta_{11} + 169 \beta_{10} + 338 \beta_{9} - 754 \beta_{6} - 754 \beta_{5} ) / 16 \) (-91b15 + 182b14 - 406b12 - 406b11 + 169b10 + 338b9 - 754b6 - 754b5) / 16
\(\nu^{8}\)	\(=\)	\( ( -63\beta_{3} + 282\beta_{2} + 651\beta _1 - 2914 ) / 8 \) (-63b3 + 282b2 + 651*b1 - 2914) / 8
\(\nu^{9}\)	\(=\)	\( ( - 289 \beta_{15} - 578 \beta_{14} + 1292 \beta_{12} - 1292 \beta_{11} - 85 \beta_{10} + 170 \beta_{9} - 380 \beta_{6} + 380 \beta_{5} ) / 8 \) (-289b15 - 578b14 + 1292b12 - 1292b11 - 85b10 + 170b9 - 380b6 + 380b5) / 8
\(\nu^{10}\)	\(=\)	\( ( -605\beta_{13} + 1353\beta_{8} + 3135\beta_{7} - 7011\beta_{4} ) / 8 \) (-605b13 + 1353b8 + 3135b7 - 7011b4) / 8
\(\nu^{11}\)	\(=\)	\( ( 2047 \beta_{15} - 4094 \beta_{14} + 9154 \beta_{12} + 9154 \beta_{11} + 5963 \beta_{10} + 11926 \beta_{9} - 26666 \beta_{6} - 26666 \beta_{5} ) / 16 \) (2047b15 - 4094b14 + 9154b12 + 9154b11 + 5963b10 + 11926b9 - 26666b6 - 26666b5) / 16
\(\nu^{12}\)	\(=\)	\( ( -1620\beta_{3} + 7245\beta_{2} - 1692\beta _1 + 7567 ) / 2 \) (-1620b3 + 7245b2 - 1692*b1 + 7567) / 2
\(\nu^{13}\)	\(=\)	\( ( 233 \beta_{15} + 466 \beta_{14} - 1042 \beta_{12} + 1042 \beta_{11} + 42173 \beta_{10} - 84346 \beta_{9} + 188602 \beta_{6} - 188602 \beta_{5} ) / 16 \) (233b15 + 466b14 - 1042b12 + 1042b11 + 42173b10 - 84346b9 + 188602b6 - 188602b5) / 16
\(\nu^{14}\)	\(=\)	\( ( 34307\beta_{13} - 76713\beta_{8} + 32799\beta_{7} - 73341\beta_{4} ) / 8 \) (34307b13 - 76713b8 + 32799b7 - 73341b4) / 8
\(\nu^{15}\)	\(=\)	\( ( 27145 \beta_{15} - 54290 \beta_{14} + 121396 \beta_{12} + 121396 \beta_{11} - 83875 \beta_{10} - 167750 \beta_{9} + 375100 \beta_{6} + 375100 \beta_{5} ) / 8 \) (27145b15 - 54290b14 + 121396b12 + 121396b11 - 83875b10 - 167750b9 + 375100b6 + 375100b5) / 8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{2} + 7)^{8} \) (T^2 + 7)^8
$11$	\( (T^{8} + 384 T^{6} + 44832 T^{4} + \cdots + 15872256)^{2} \) (T^8 + 384T^6 + 44832T^4 + 1673216*T^2 + 15872256)^2
$13$	\( (T^{8} + 792 T^{6} + 149208 T^{4} + \cdots + 14470416)^{2} \) (T^8 + 792T^6 + 149208T^4 + 5015392*T^2 + 14470416)^2
$17$	\( (T^{8} - 1152 T^{6} + 117888 T^{4} + \cdots + 331776)^{2} \) (T^8 - 1152T^6 + 117888T^4 - 2957312*T^2 + 331776)^2
$19$	\( (T^{4} - 40 T^{3} + 20 T^{2} + 7600 T + 24900)^{4} \) (T^4 - 40T^3 + 20T^2 + 7600*T + 24900)^4
$23$	\( (T^{8} - 616 T^{6} + 112792 T^{4} + \cdots + 80353296)^{2} \) (T^8 - 616T^6 + 112792T^4 - 6812064*T^2 + 80353296)^2
$29$	\( (T^{8} + 4136 T^{6} + \cdots + 182633150736)^{2} \) (T^8 + 4136T^6 + 5285592T^4 + 2064468384*T^2 + 182633150736)^2
$31$	\( (T^{4} - 56 T^{3} - 1468 T^{2} + \cdots - 53724)^{4} \) (T^4 - 56T^3 - 1468T^2 + 89936*T - 53724)^4
$37$	\( (T^{8} + 9744 T^{6} + \cdots + 17685591019776)^{2} \) (T^8 + 9744T^6 + 32528736T^4 + 42572556544*T^2 + 17685591019776)^2
$41$	\( (T^{8} + 8656 T^{6} + \cdots + 11776317808896)^{2} \) (T^8 + 8656T^6 + 24770656T^4 + 28820802816*T^2 + 11776317808896)^2
$43$	\( (T^{8} + 11088 T^{6} + \cdots + 23646201405696)^{2} \) (T^8 + 11088T^6 + 42181728T^4 + 61368149248*T^2 + 23646201405696)^2
$47$	\( (T^{8} - 7264 T^{6} + \cdots + 147161235456)^{2} \) (T^8 - 7264T^6 + 13958656T^4 - 5044506624*T^2 + 147161235456)^2
$53$	\( (T^{8} - 8464 T^{6} + \cdots + 8432658441216)^{2} \) (T^8 - 8464T^6 + 23391616T^4 - 25129465344*T^2 + 8432658441216)^2
$59$	\( (T^{8} + 16816 T^{6} + \cdots + 7297000079616)^{2} \) (T^8 + 16816T^6 + 57550432T^4 + 41795230464*T^2 + 7297000079616)^2
$61$	\( (T^{4} + 72 T^{3} - 3032 T^{2} + \cdots + 5776)^{4} \) (T^4 + 72T^3 - 3032T^2 + 5472*T + 5776)^4
$67$	\( (T^{8} + 10384 T^{6} + \cdots + 913966592256)^{2} \) (T^8 + 10384T^6 + 29058912T^4 + 11129446656*T^2 + 913966592256)^2
$71$	\( (T^{8} + 11136 T^{6} + \cdots + 13463498301696)^{2} \) (T^8 + 11136T^6 + 38341152T^4 + 45609580544*T^2 + 13463498301696)^2
$73$	\( (T^{8} + 21208 T^{6} + \cdots + 36190379032336)^{2} \) (T^8 + 21208T^6 + 124476504T^4 + 231628973152*T^2 + 36190379032336)^2
$79$	\( (T^{4} - 32 T^{3} - 13840 T^{2} + \cdots - 14801856)^{4} \) (T^4 - 32T^3 - 13840T^2 + 978176*T - 14801856)^4
$83$	\( (T^{8} - 50528 T^{6} + \cdots + 36\!\cdots\!96)^{2} \) (T^8 - 50528T^6 + 825306624T^4 - 4565309755392*T^2 + 3665777429200896)^2
$89$	\( (T^{8} + 14448 T^{6} + \cdots + 416365629696)^{2} \) (T^8 + 14448T^6 + 53476704T^4 + 13199228672*T^2 + 416365629696)^2
$97$	\( (T^{8} + 36376 T^{6} + \cdots + 615175117820176)^{2} \) (T^8 + 36376T^6 + 391877592T^4 + 1149277896544*T^2 + 615175117820176)^2
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\(n\)	\(127\)	\(451\)	\(2801\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)