LMFDB - Newform orbit 768.4.c.v

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.c (of order $2$, degree $1$, 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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 35x^{12} + 612x^{8} - 8960x^{4} + 65536 $ x^16 - 35x^12 + 612x^8 - 8960*x^4 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 24)
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 $1,\beta_1,\ldots,\beta_{15}$ 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_{9} q^{5} + \beta_{6} q^{7} + ( - 3 \beta_{7} + 6) q^{9}+O(q^{10}) $ q - b4 * q^3 + b9 * q^5 + b6 * q^7 + (-3b7 + 6) q^9	\( q - \beta_{4} q^{3} + \beta_{9} q^{5} + \beta_{6} q^{7} + ( - 3 \beta_{7} + 6) q^{9} + ( - \beta_{10} + \beta_{4} - \beta_{3}) q^{11} + \beta_{2} q^{13} - \beta_{13} q^{15} + (\beta_{11} - \beta_{8} - \beta_{7}) q^{17} + ( - 4 \beta_{4} - 4 \beta_{3} - 7 \beta_1) q^{19} + \beta_{14} q^{21} - \beta_{5} q^{23} + (\beta_{8} - \beta_{7} - 37) q^{25} + ( - 9 \beta_{10} - 3 \beta_{4} - 9 \beta_1) q^{27} + (\beta_{15} + \beta_{14} + \beta_{9}) q^{29} + (\beta_{13} + \beta_{12} + \beta_{6}) q^{31} + (\beta_{11} + 7 \beta_{8} + 2 \beta_{7} - 33) q^{33} + ( - 21 \beta_{10} + 2 \beta_{4} - 2 \beta_{3}) q^{35} + (\beta_{15} - \beta_{14} + 3 \beta_{2}) q^{37} + ( - \beta_{13} - 3 \beta_{6} + 3 \beta_{5}) q^{39} + (7 \beta_{11} + 16 \beta_{8} + 16 \beta_{7}) q^{41} + ( - 22 \beta_{4} - 22 \beta_{3} - 17 \beta_1) q^{43} + ( - 3 \beta_{15} + 6 \beta_{9} - 3 \beta_{2}) q^{45} + (5 \beta_{13} - 5 \beta_{12} - \beta_{5}) q^{47} + (23 \beta_{8} - 23 \beta_{7} - 119) q^{49} + (3 \beta_{10} + 8 \beta_{4} - 24 \beta_1) q^{51} + ( - 2 \beta_{15} - 2 \beta_{14} - 3 \beta_{9}) q^{53} + (\beta_{13} + \beta_{12} - 8 \beta_{6}) q^{55} + (7 \beta_{11} - 14 \beta_{8} + \cdots - 147) q^{57}+ \cdots + (12 \beta_{10} + 54 \beta_{4} + \cdots - 15 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + b9 * q^5 + b6 * q^7 + (-3b7 + 6) q^9 + (-b10 + b4 - b3) * q^11 + b2 * q^13 - b13 * q^15 + (b11 - b8 - b7) * q^17 + (-4b4 - 4b3 - 7b1) q^19 + b14 * q^21 - b5 * q^23 + (b8 - b7 - 37) * q^25 + (-9b10 - 3b4 - 9b1) q^27 + (b15 + b14 + b9) * q^29 + (b13 + b12 + b6) * q^31 + (b11 + 7b8 + 2b7 - 33) * q^33 + (-21b10 + 2b4 - 2b3) q^35 + (b15 - b14 + 3b2) q^37 + (-b13 - 3b6 + 3b5) * q^39 + (7b11 + 16b8 + 16b7) q^41 + (-22b4 - 22b3 - 17b1) q^43 + (-3b15 + 6b9 - 3b2) q^45 + (5b13 - 5b12 - b5) * q^47 + (23b8 - 23b7 - 119) * q^49 + (3b10 + 8b4 - 24b1) q^51 + (-2b15 - 2b14 - 3b9) q^53 + (b13 + b12 - 8b6) q^55 + (7b11 - 14b8 - 19b7 - 147) q^57 + (-4b10 - 11b4 + 11b3) q^59 + (3b15 - 3b14 - b2) * q^61 + (-3b13 + 9b12 - 3b6 + 3b5) * q^63 + (19b11 - 18b8 - 18b7) q^65 + (-29b4 - 29b3 + 20b1) q^67 + (b14 - 9b9 - 9b2) * q^69 + (5b13 - 5b12 - 6b5) q^71 + (74b8 - 74b7 - 134) * q^73 + (-3b10 + 40b4 + 9b3 - 3b1) * q^75 + (-2b15 - 2b14 - 22b9) q^77 + (-7b13 - 7b12 - 7b6) q^79 + (18b11 + 45b8 - 27b7 - 63) q^81 + (31b10 - 21b4 + 21b3) q^83 - 8b2 q^85 + (-4b13 + 9b12 + 24b6 + 3b5) * q^87 + (28b11 - 53b8 - 53b7) q^89 + (23b4 + 23b3 + 93b1) q^91 + (3b15 + b14 - 33b9 + 3b2) q^93 + (-4b13 + 4b12 - 7b5) q^95 + (45b8 - 45b7 - 484) * q^97 + (12b10 + 54b4 + 72b3 - 15b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 96 q^{9}+O(q^{10}) $ 16 * q + 96 * q^9	$ 16 q + 96 q^{9} - 592 q^{25} - 528 q^{33} - 1904 q^{49} - 2352 q^{57} - 2144 q^{73} - 1008 q^{81} - 7744 q^{97}+O(q^{100}) $ 16 * q + 96 * q^9 - 592 * q^25 - 528 * q^33 - 1904 * q^49 - 2352 * q^57 - 2144 * q^73 - 1008 * q^81 - 7744 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 35x^{12} + 612x^{8} - 8960x^{4} + 65536 $ x^16 - 35*x^12 + 612*x^8 - 8960*x^4 + 65536 :

$\beta_{1}$	$=$	$ ( -\nu^{14} + 35\nu^{10} - 612\nu^{6} + 4864\nu^{2} ) / 2048 $ (-v^14 + 35v^10 - 612v^6 + 4864*v^2) / 2048
$\beta_{2}$	$=$	$ ( -\nu^{14} + 19\nu^{10} + 204\nu^{6} - 5696\nu^{2} ) / 640 $ (-v^14 + 19v^10 + 204v^6 - 5696*v^2) / 640
$\beta_{3}$	$=$	$ ( - 29 \nu^{15} - 132 \nu^{14} + 64 \nu^{13} + 631 \nu^{11} + 2572 \nu^{10} - 1216 \nu^{9} + \cdots - 331776 \nu ) / 81920 $ (-29v^15 - 132v^14 + 64v^13 + 631v^11 + 2572v^10 - 1216v^9 - 8404v^7 - 41872v^6 + 27904v^5 + 102656v^3 + 535552v^2 - 331776v) / 81920
$\beta_{4}$	$=$	$ ( 29 \nu^{15} - 132 \nu^{14} - 64 \nu^{13} - 631 \nu^{11} + 2572 \nu^{10} + 1216 \nu^{9} + \cdots + 331776 \nu ) / 81920 $ (29v^15 - 132v^14 - 64v^13 - 631v^11 + 2572v^10 + 1216v^9 + 8404v^7 - 41872v^6 - 27904v^5 - 102656v^3 + 535552v^2 + 331776v) / 81920
$\beta_{5}$	$=$	$ ( \nu^{15} + 4\nu^{13} + 61\nu^{11} - 396\nu^{9} - 1724\nu^{7} + 5264\nu^{5} + 9856\nu^{3} + 6144\nu ) / 2560 $ (v^15 + 4v^13 + 61v^11 - 396v^9 - 1724v^7 + 5264v^5 + 9856v^3 + 6144*v) / 2560
$\beta_{6}$	$=$	$ ( 25\nu^{12} - 491\nu^{8} + 8260\nu^{4} - 106496 ) / 640 $ (25v^12 - 491v^8 + 8260*v^4 - 106496) / 640
$\beta_{7}$	$=$	$ ( - 33 \nu^{15} - 112 \nu^{13} - 64 \nu^{12} + 643 \nu^{11} + 1872 \nu^{9} + 2240 \nu^{8} + \cdots + 286720 ) / 40960 $ (-33v^15 - 112v^13 - 64v^12 + 643v^11 + 1872v^9 + 2240v^8 - 10468v^7 - 29632v^5 - 22784v^4 + 154368v^3 + 356352*v + 286720) / 40960
$\beta_{8}$	$=$	$ ( - 33 \nu^{15} - 112 \nu^{13} + 64 \nu^{12} + 643 \nu^{11} + 1872 \nu^{9} - 2240 \nu^{8} + \cdots - 286720 ) / 40960 $ (-33v^15 - 112v^13 + 64v^12 + 643v^11 + 1872v^9 - 2240v^8 - 10468v^7 - 29632v^5 + 22784v^4 + 154368v^3 + 356352*v - 286720) / 40960
$\beta_{9}$	$=$	$ ( 21 \nu^{15} + 120 \nu^{13} - 415 \nu^{11} - 2664 \nu^{9} + 7796 \nu^{7} + 40160 \nu^{5} + \cdots - 507904 \nu ) / 20480 $ (21v^15 + 120v^13 - 415v^11 - 2664v^9 + 7796v^7 + 40160v^5 - 84480v^3 - 507904v) / 20480
$\beta_{10}$	$=$	$ ( - 33 \nu^{15} + 112 \nu^{13} + 643 \nu^{11} - 1872 \nu^{9} - 10468 \nu^{7} + 29632 \nu^{5} + \cdots - 356352 \nu ) / 20480 $ (-33v^15 + 112v^13 + 643v^11 - 1872v^9 - 10468v^7 + 29632v^5 + 154368v^3 - 356352v) / 20480
$\beta_{11}$	$=$	$ ( - 29 \nu^{15} - 64 \nu^{13} + 631 \nu^{11} + 1216 \nu^{9} - 8404 \nu^{7} - 27904 \nu^{5} + \cdots + 331776 \nu ) / 10240 $ (-29v^15 - 64v^13 + 631v^11 + 1216v^9 - 8404v^7 - 27904v^5 + 102656v^3 + 331776v) / 10240
$\beta_{12}$	$=$	$ ( - 11 \nu^{15} + 58 \nu^{13} - 140 \nu^{12} + 177 \nu^{11} - 1134 \nu^{9} + 2340 \nu^{8} + \cdots + 471040 ) / 2560 $ (-11v^15 + 58v^13 - 140v^12 + 177v^11 - 1134v^9 + 2340v^8 - 3036v^7 + 17448v^5 - 31920v^4 + 37312v^3 - 257024*v + 471040) / 2560
$\beta_{13}$	$=$	$ ( 11 \nu^{15} - 58 \nu^{13} - 140 \nu^{12} - 177 \nu^{11} + 1134 \nu^{9} + 2340 \nu^{8} + 3036 \nu^{7} + \cdots + 471040 ) / 2560 $ (11v^15 - 58v^13 - 140v^12 - 177v^11 + 1134v^9 + 2340v^8 + 3036v^7 - 17448v^5 - 31920v^4 - 37312v^3 + 257024*v + 471040) / 2560
$\beta_{14}$	$=$	$ ( 113 \nu^{15} + 112 \nu^{14} + 568 \nu^{13} - 1587 \nu^{11} - 3408 \nu^{10} - 10152 \nu^{9} + \cdots - 2588672 \nu ) / 20480 $ (113v^15 + 112v^14 + 568v^13 - 1587v^11 - 3408v^10 - 10152v^9 + 25188v^7 + 42432v^6 + 158688v^5 - 343552v^3 - 862208v^2 - 2588672v) / 20480
$\beta_{15}$	$=$	$ ( 113 \nu^{15} - 112 \nu^{14} + 568 \nu^{13} - 1587 \nu^{11} + 3408 \nu^{10} - 10152 \nu^{9} + \cdots - 2588672 \nu ) / 20480 $ (113v^15 - 112v^14 + 568v^13 - 1587v^11 + 3408v^10 - 10152v^9 + 25188v^7 - 42432v^6 + 158688v^5 - 343552v^3 + 862208v^2 - 2588672v) / 20480

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 8\beta_{10} + 2\beta_{9} - 8\beta_{8} - 8\beta_{7} + \beta_{5} ) / 128 $ (-b15 - b14 + b13 - b12 + 8b10 + 2b9 - 8b8 - 8b7 + b5) / 128
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} - 2\beta_{2} - 16\beta_1 ) / 64 $ (b15 - b14 - 2b2 - 16b1) / 64
$\nu^{3}$	$=$	$ ( \beta_{15} + \beta_{14} + 5 \beta_{13} - 5 \beta_{12} - 8 \beta_{11} + 40 \beta_{10} + \cdots - 32 \beta_{3} ) / 128 $ (b15 + b14 + 5b13 - 5b12 - 8b11 + 40b10 + 30b9 + 40b8 + 40b7 - 3b5 + 32b4 - 32b3) / 128
$\nu^{4}$	$=$	$ ( 3\beta_{13} + 3\beta_{12} - 20\beta_{8} + 20\beta_{7} + 10\beta_{6} + 280 ) / 32 $ (3b13 + 3b12 - 20b8 + 20b7 + 10*b6 + 280) / 32
$\nu^{5}$	$=$	$ ( - 11 \beta_{15} - 11 \beta_{14} + 15 \beta_{13} - 15 \beta_{12} - 72 \beta_{11} + \cdots + 288 \beta_{3} ) / 128 $ (-11b15 - 11b14 + 15b13 - 15b12 - 72b11 + 40b10 - 10b9 - 40b8 - 40b7 + 7b5 - 288b4 + 288b3) / 128
$\nu^{6}$	$=$	$ ( 19\beta_{15} - 19\beta_{14} - 40\beta_{4} - 40\beta_{3} + 42\beta_{2} - 296\beta_1 ) / 64 $ (19b15 - 19b14 - 40b4 - 40b3 + 42b2 - 296b1) / 64
$\nu^{7}$	$=$	$ ( - 9 \beta_{15} - 9 \beta_{14} + 91 \beta_{13} - 91 \beta_{12} + 120 \beta_{11} + 248 \beta_{10} + \cdots + 480 \beta_{3} ) / 128 $ (-9b15 - 9b14 + 91b13 - 91b12 + 120b11 + 248b10 + 818b9 + 248b8 + 248b7 - 109b5 - 480b4 + 480b3) / 128
$\nu^{8}$	$=$	$ ( 5\beta_{13} + 5\beta_{12} - 700\beta_{8} + 700\beta_{7} + 70\beta_{6} + 8 ) / 32 $ (5b13 + 5b12 - 700b8 + 700b7 + 70*b6 + 8) / 32
$\nu^{9}$	$=$	$ ( 51 \beta_{15} + 51 \beta_{14} + 249 \beta_{13} - 249 \beta_{12} - 920 \beta_{11} + \cdots + 3680 \beta_{3} ) / 128 $ (51b15 + 51b14 + 249b13 - 249b12 - 920b11 + 1112b10 - 2502b9 - 1112b8 - 1112b7 - 351b5 - 3680b4 + 3680b3) / 128
$\nu^{10}$	$=$	$ ( 309\beta_{15} - 309\beta_{14} - 2040\beta_{4} - 2040\beta_{3} + 902\beta_{2} + 3656\beta_1 ) / 64 $ (309b15 - 309b14 - 2040b4 - 2040b3 + 902b2 + 3656b1) / 64
$\nu^{11}$	$=$	$ ( 609 \beta_{15} + 609 \beta_{14} + 1405 \beta_{13} - 1405 \beta_{12} + 4488 \beta_{11} + \cdots + 17952 \beta_{3} ) / 128 $ (609b15 + 609b14 + 1405b13 - 1405b12 + 4488b11 - 440b10 + 5150b9 - 440b8 - 440b7 - 187b5 - 17952b4 + 17952b3) / 128
$\nu^{12}$	$=$	$ ( -893\beta_{13} - 893\beta_{12} - 7140\beta_{8} + 7140\beta_{7} - 1110\beta_{6} + 43960 ) / 32 $ (-893b13 - 893b12 - 7140b8 + 7140b7 - 1110*b6 + 43960) / 32
$\nu^{13}$	$=$	$ ( 581 \beta_{15} + 581 \beta_{14} + 3375 \beta_{13} - 3375 \beta_{12} + 3672 \beta_{11} + \cdots - 14688 \beta_{3} ) / 128 $ (581b15 + 581b14 + 3375b13 - 3375b12 + 3672b11 + 45160b10 - 32810b9 - 45160b8 - 45160b7 - 4537b5 + 14688b4 - 14688b3) / 128
$\nu^{14}$	$=$	$ ( 4051\beta_{15} - 4051\beta_{14} - 46920\beta_{4} - 46920\beta_{3} - 3862\beta_{2} + 100216\beta_1 ) / 64 $ (4051b15 - 4051b14 - 46920b4 - 46920b3 - 3862b2 + 100216b1) / 64
$\nu^{15}$	$=$	$ ( 19399 \beta_{15} + 19399 \beta_{14} + 21899 \beta_{13} - 21899 \beta_{12} + 11960 \beta_{11} + \cdots + 47840 \beta_{3} ) / 128 $ (19399b15 + 19399b14 + 21899b13 - 21899b12 + 11960b11 + 60152b10 - 18798b9 + 60152b8 + 60152b7 + 16899b5 - 47840b4 + 47840b3) / 128

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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.

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} $

767.1

1.86600	−	0.719760i
−0.719760	+	1.86600i
−0.719760	−	1.86600i
1.86600	+	0.719760i
−0.0465478	+	1.99946i
−1.99946	+	0.0465478i
−1.99946	−	0.0465478i
−0.0465478	−	1.99946i
1.99946	−	0.0465478i
0.0465478	−	1.99946i
0.0465478	+	1.99946i
1.99946	+	0.0465478i
0.719760	−	1.86600i
−1.86600	+	0.719760i
−1.86600	−	0.719760i
0.719760	+	1.86600i

−5.01167

−

1.37228i

−

12.2683i

−

14.0624i

23.2337

13.7548i

767.2

−5.01167

−

1.37228i

12.2683i

14.0624i

23.2337

13.7548i

767.3

−5.01167

1.37228i

−

12.2683i

−

14.0624i

23.2337

−

13.7548i

767.4

−5.01167

1.37228i

12.2683i

14.0624i

23.2337

−

13.7548i

767.5

−2.80770

−

4.37228i

−

13.1715i

26.9490i

−11.2337

24.5521i

767.6

−2.80770

−

4.37228i

13.1715i

−

26.9490i

−11.2337

24.5521i

767.7

−2.80770

4.37228i

−

13.1715i

26.9490i

−11.2337

−

24.5521i

767.8

−2.80770

4.37228i

13.1715i

−

26.9490i

−11.2337

−

24.5521i

767.9

2.80770

−

4.37228i

−

13.1715i

−

26.9490i

−11.2337

−

24.5521i

767.10

2.80770

−

4.37228i

13.1715i

26.9490i

−11.2337

−

24.5521i

767.11

2.80770

4.37228i

−

13.1715i

−

26.9490i

−11.2337

24.5521i

767.12

2.80770

4.37228i

13.1715i

26.9490i

−11.2337

24.5521i

767.13

5.01167

−

1.37228i

−

12.2683i

14.0624i

23.2337

−

13.7548i

767.14

5.01167

−

1.37228i

12.2683i

−

14.0624i

23.2337

−

13.7548i

767.15

5.01167

1.37228i

−

12.2683i

14.0624i

23.2337

13.7548i

767.16

5.01167

1.37228i

12.2683i

−

14.0624i

23.2337

13.7548i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.4.c.v		16
3.b	odd	2	1	inner	768.4.c.v		16
4.b	odd	2	1	inner	768.4.c.v		16
8.b	even	2	1	inner	768.4.c.v		16
8.d	odd	2	1	inner	768.4.c.v		16
12.b	even	2	1	inner	768.4.c.v		16
16.e	even	4	1		24.4.f.b	✓	8
16.e	even	4	1		96.4.f.b		8
16.f	odd	4	1		24.4.f.b	✓	8
16.f	odd	4	1		96.4.f.b		8
24.f	even	2	1	inner	768.4.c.v		16
24.h	odd	2	1	inner	768.4.c.v		16
48.i	odd	4	1		24.4.f.b	✓	8
48.i	odd	4	1		96.4.f.b		8
48.k	even	4	1		24.4.f.b	✓	8
48.k	even	4	1		96.4.f.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.4.f.b	✓	8	16.e	even	4	1
24.4.f.b	✓	8	16.f	odd	4	1
24.4.f.b	✓	8	48.i	odd	4	1
24.4.f.b	✓	8	48.k	even	4	1
96.4.f.b		8	16.e	even	4	1
96.4.f.b		8	16.f	odd	4	1
96.4.f.b		8	48.i	odd	4	1
96.4.f.b		8	48.k	even	4	1
768.4.c.v		16	1.a	even	1	1	trivial
768.4.c.v		16	3.b	odd	2	1	inner
768.4.c.v		16	4.b	odd	2	1	inner
768.4.c.v		16	8.b	even	2	1	inner
768.4.c.v		16	8.d	odd	2	1	inner
768.4.c.v		16	12.b	even	2	1	inner
768.4.c.v		16	24.f	even	2	1	inner
768.4.c.v		16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(768, [\chi])$:

$ T_{5}^{4} + 324T_{5}^{2} + 26112 $

$ T_{7}^{4} + 924T_{7}^{2} + 143616 $

$ T_{11}^{4} - 484T_{11}^{2} + 352 $

$ T_{13}^{4} - 5808T_{13}^{2} + 574464 $

$ T_{23}^{4} - 17472T_{23}^{2} + 1671168 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 24 T^{6} + \cdots + 531441)^{2} $ (T^8 - 24T^6 + 414T^4 - 17496*T^2 + 531441)^2
$5$	$ (T^{4} + 324 T^{2} + 26112)^{4} $ (T^4 + 324*T^2 + 26112)^4
$7$	$ (T^{4} + 924 T^{2} + 143616)^{4} $ (T^4 + 924*T^2 + 143616)^4
$11$	$ (T^{4} - 484 T^{2} + 352)^{4} $ (T^4 - 484*T^2 + 352)^4
$13$	$ (T^{4} - 5808 T^{2} + 574464)^{4} $ (T^4 - 5808*T^2 + 574464)^4
$17$	$ (T^{4} + 2464 T^{2} + 90112)^{4} $ (T^4 + 2464*T^2 + 90112)^4
$19$	$ (T^{4} + 9044 T^{2} + 11999296)^{4} $ (T^4 + 9044*T^2 + 11999296)^4
$23$	$ (T^{4} - 17472 T^{2} + 1671168)^{4} $ (T^4 - 17472*T^2 + 1671168)^4
$29$	$ (T^{4} + 43620 T^{2} + 448108032)^{4} $ (T^4 + 43620*T^2 + 448108032)^4
$31$	$ (T^{4} + 20988 T^{2} + 41505024)^{4} $ (T^4 + 20988*T^2 + 41505024)^4
$37$	$ (T^{4} - 77616 T^{2} + 1494180864)^{4} $ (T^4 - 77616*T^2 + 1494180864)^4
$41$	$ (T^{4} + 193600 T^{2} + 3151126528)^{4} $ (T^4 + 193600*T^2 + 3151126528)^4
$43$	$ (T^{4} + 101108 T^{2} + 2483228224)^{4} $ (T^4 + 101108*T^2 + 2483228224)^4
$47$	$ (T^{4} - 321792 T^{2} + 427819008)^{4} $ (T^4 - 321792*T^2 + 427819008)^4
$53$	$ (T^{4} + 177156 T^{2} + 7033554432)^{4} $ (T^4 + 177156*T^2 + 7033554432)^4
$59$	$ (T^{4} - 21604 T^{2} + 296032)^{4} $ (T^4 - 21604*T^2 + 296032)^4
$61$	$ (T^{4} - 550704 T^{2} + 18820015104)^{4} $ (T^4 - 550704*T^2 + 18820015104)^4
$67$	$ (T^{4} + 75284 T^{2} + 1043031616)^{4} $ (T^4 + 75284*T^2 + 1043031616)^4
$71$	$ (T^{4} - 654912 T^{2} + 103614087168)^{4} $ (T^4 - 654912*T^2 + 103614087168)^4
$73$	$ (T^{2} + 268 T - 704876)^{8} $ (T^2 + 268*T - 704876)^8
$79$	$ (T^{4} + 1028412 T^{2} + 99653562624)^{4} $ (T^4 + 1028412*T^2 + 99653562624)^4
$83$	$ (T^{4} - 396484 T^{2} + 2128431712)^{4} $ (T^4 - 396484*T^2 + 2128431712)^4
$89$	$ (T^{4} + 2644576 T^{2} + 147293673472)^{4} $ (T^4 + 2644576*T^2 + 147293673472)^4
$97$	$ (T^{2} + 968 T - 33044)^{8} $ (T^2 + 968*T - 33044)^8
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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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + \beta_{9} q^{5} + \beta_{6} q^{7} + ( - 3 \beta_{7} + 6) q^{9}+O(q^{10}) \) q - b4 * q^3 + b9 * q^5 + b6 * q^7 + (-3b7 + 6) q^9	\( q - \beta_{4} q^{3} + \beta_{9} q^{5} + \beta_{6} q^{7} + ( - 3 \beta_{7} + 6) q^{9} + ( - \beta_{10} + \beta_{4} - \beta_{3}) q^{11} + \beta_{2} q^{13} - \beta_{13} q^{15} + (\beta_{11} - \beta_{8} - \beta_{7}) q^{17} + ( - 4 \beta_{4} - 4 \beta_{3} - 7 \beta_1) q^{19} + \beta_{14} q^{21} - \beta_{5} q^{23} + (\beta_{8} - \beta_{7} - 37) q^{25} + ( - 9 \beta_{10} - 3 \beta_{4} - 9 \beta_1) q^{27} + (\beta_{15} + \beta_{14} + \beta_{9}) q^{29} + (\beta_{13} + \beta_{12} + \beta_{6}) q^{31} + (\beta_{11} + 7 \beta_{8} + 2 \beta_{7} - 33) q^{33} + ( - 21 \beta_{10} + 2 \beta_{4} - 2 \beta_{3}) q^{35} + (\beta_{15} - \beta_{14} + 3 \beta_{2}) q^{37} + ( - \beta_{13} - 3 \beta_{6} + 3 \beta_{5}) q^{39} + (7 \beta_{11} + 16 \beta_{8} + 16 \beta_{7}) q^{41} + ( - 22 \beta_{4} - 22 \beta_{3} - 17 \beta_1) q^{43} + ( - 3 \beta_{15} + 6 \beta_{9} - 3 \beta_{2}) q^{45} + (5 \beta_{13} - 5 \beta_{12} - \beta_{5}) q^{47} + (23 \beta_{8} - 23 \beta_{7} - 119) q^{49} + (3 \beta_{10} + 8 \beta_{4} - 24 \beta_1) q^{51} + ( - 2 \beta_{15} - 2 \beta_{14} - 3 \beta_{9}) q^{53} + (\beta_{13} + \beta_{12} - 8 \beta_{6}) q^{55} + (7 \beta_{11} - 14 \beta_{8} + \cdots - 147) q^{57}+ \cdots + (12 \beta_{10} + 54 \beta_{4} + \cdots - 15 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + b9 * q^5 + b6 * q^7 + (-3b7 + 6) q^9 + (-b10 + b4 - b3) * q^11 + b2 * q^13 - b13 * q^15 + (b11 - b8 - b7) * q^17 + (-4b4 - 4b3 - 7b1) q^19 + b14 * q^21 - b5 * q^23 + (b8 - b7 - 37) * q^25 + (-9b10 - 3b4 - 9b1) q^27 + (b15 + b14 + b9) * q^29 + (b13 + b12 + b6) * q^31 + (b11 + 7b8 + 2b7 - 33) * q^33 + (-21b10 + 2b4 - 2b3) q^35 + (b15 - b14 + 3b2) q^37 + (-b13 - 3b6 + 3b5) * q^39 + (7b11 + 16b8 + 16b7) q^41 + (-22b4 - 22b3 - 17b1) q^43 + (-3b15 + 6b9 - 3b2) q^45 + (5b13 - 5b12 - b5) * q^47 + (23b8 - 23b7 - 119) * q^49 + (3b10 + 8b4 - 24b1) q^51 + (-2b15 - 2b14 - 3b9) q^53 + (b13 + b12 - 8b6) q^55 + (7b11 - 14b8 - 19b7 - 147) q^57 + (-4b10 - 11b4 + 11b3) q^59 + (3b15 - 3b14 - b2) * q^61 + (-3b13 + 9b12 - 3b6 + 3b5) * q^63 + (19b11 - 18b8 - 18b7) q^65 + (-29b4 - 29b3 + 20b1) q^67 + (b14 - 9b9 - 9b2) * q^69 + (5b13 - 5b12 - 6b5) q^71 + (74b8 - 74b7 - 134) * q^73 + (-3b10 + 40b4 + 9b3 - 3b1) * q^75 + (-2b15 - 2b14 - 22b9) q^77 + (-7b13 - 7b12 - 7b6) q^79 + (18b11 + 45b8 - 27b7 - 63) q^81 + (31b10 - 21b4 + 21b3) q^83 - 8b2 q^85 + (-4b13 + 9b12 + 24b6 + 3b5) * q^87 + (28b11 - 53b8 - 53b7) q^89 + (23b4 + 23b3 + 93b1) q^91 + (3b15 + b14 - 33b9 + 3b2) q^93 + (-4b13 + 4b12 - 7b5) q^95 + (45b8 - 45b7 - 484) * q^97 + (12b10 + 54b4 + 72b3 - 15b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 96 q^{9}+O(q^{10}) \) 16 * q + 96 * q^9	\( 16 q + 96 q^{9} - 592 q^{25} - 528 q^{33} - 1904 q^{49} - 2352 q^{57} - 2144 q^{73} - 1008 q^{81} - 7744 q^{97}+O(q^{100}) \) 16 * q + 96 * q^9 - 592 * q^25 - 528 * q^33 - 1904 * q^49 - 2352 * q^57 - 2144 * q^73 - 1008 * q^81 - 7744 * q^97

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	768.4.c.v

Level	$768$
Weight	$4$
Character orbit	768.c
Analytic conductor	$45.313$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(45.3134668844\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 35x^{12} + 612x^{8} - 8960x^{4} + 65536 \) x^16 - 35x^12 + 612x^8 - 8960*x^4 + 65536
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{50} \)
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} + 35\nu^{10} - 612\nu^{6} + 4864\nu^{2} ) / 2048 \) (-v^14 + 35v^10 - 612v^6 + 4864*v^2) / 2048
\(\beta_{2}\)	\(=\)	\( ( -\nu^{14} + 19\nu^{10} + 204\nu^{6} - 5696\nu^{2} ) / 640 \) (-v^14 + 19v^10 + 204v^6 - 5696*v^2) / 640
\(\beta_{3}\)	\(=\)	\( ( - 29 \nu^{15} - 132 \nu^{14} + 64 \nu^{13} + 631 \nu^{11} + 2572 \nu^{10} - 1216 \nu^{9} + \cdots - 331776 \nu ) / 81920 \) (-29v^15 - 132v^14 + 64v^13 + 631v^11 + 2572v^10 - 1216v^9 - 8404v^7 - 41872v^6 + 27904v^5 + 102656v^3 + 535552v^2 - 331776v) / 81920
\(\beta_{4}\)	\(=\)	\( ( 29 \nu^{15} - 132 \nu^{14} - 64 \nu^{13} - 631 \nu^{11} + 2572 \nu^{10} + 1216 \nu^{9} + \cdots + 331776 \nu ) / 81920 \) (29v^15 - 132v^14 - 64v^13 - 631v^11 + 2572v^10 + 1216v^9 + 8404v^7 - 41872v^6 - 27904v^5 - 102656v^3 + 535552v^2 + 331776v) / 81920
\(\beta_{5}\)	\(=\)	\( ( \nu^{15} + 4\nu^{13} + 61\nu^{11} - 396\nu^{9} - 1724\nu^{7} + 5264\nu^{5} + 9856\nu^{3} + 6144\nu ) / 2560 \) (v^15 + 4v^13 + 61v^11 - 396v^9 - 1724v^7 + 5264v^5 + 9856v^3 + 6144*v) / 2560
\(\beta_{6}\)	\(=\)	\( ( 25\nu^{12} - 491\nu^{8} + 8260\nu^{4} - 106496 ) / 640 \) (25v^12 - 491v^8 + 8260*v^4 - 106496) / 640
\(\beta_{7}\)	\(=\)	\( ( - 33 \nu^{15} - 112 \nu^{13} - 64 \nu^{12} + 643 \nu^{11} + 1872 \nu^{9} + 2240 \nu^{8} + \cdots + 286720 ) / 40960 \) (-33v^15 - 112v^13 - 64v^12 + 643v^11 + 1872v^9 + 2240v^8 - 10468v^7 - 29632v^5 - 22784v^4 + 154368v^3 + 356352*v + 286720) / 40960
\(\beta_{8}\)	\(=\)	\( ( - 33 \nu^{15} - 112 \nu^{13} + 64 \nu^{12} + 643 \nu^{11} + 1872 \nu^{9} - 2240 \nu^{8} + \cdots - 286720 ) / 40960 \) (-33v^15 - 112v^13 + 64v^12 + 643v^11 + 1872v^9 - 2240v^8 - 10468v^7 - 29632v^5 + 22784v^4 + 154368v^3 + 356352*v - 286720) / 40960
\(\beta_{9}\)	\(=\)	\( ( 21 \nu^{15} + 120 \nu^{13} - 415 \nu^{11} - 2664 \nu^{9} + 7796 \nu^{7} + 40160 \nu^{5} + \cdots - 507904 \nu ) / 20480 \) (21v^15 + 120v^13 - 415v^11 - 2664v^9 + 7796v^7 + 40160v^5 - 84480v^3 - 507904v) / 20480
\(\beta_{10}\)	\(=\)	\( ( - 33 \nu^{15} + 112 \nu^{13} + 643 \nu^{11} - 1872 \nu^{9} - 10468 \nu^{7} + 29632 \nu^{5} + \cdots - 356352 \nu ) / 20480 \) (-33v^15 + 112v^13 + 643v^11 - 1872v^9 - 10468v^7 + 29632v^5 + 154368v^3 - 356352v) / 20480
\(\beta_{11}\)	\(=\)	\( ( - 29 \nu^{15} - 64 \nu^{13} + 631 \nu^{11} + 1216 \nu^{9} - 8404 \nu^{7} - 27904 \nu^{5} + \cdots + 331776 \nu ) / 10240 \) (-29v^15 - 64v^13 + 631v^11 + 1216v^9 - 8404v^7 - 27904v^5 + 102656v^3 + 331776v) / 10240
\(\beta_{12}\)	\(=\)	\( ( - 11 \nu^{15} + 58 \nu^{13} - 140 \nu^{12} + 177 \nu^{11} - 1134 \nu^{9} + 2340 \nu^{8} + \cdots + 471040 ) / 2560 \) (-11v^15 + 58v^13 - 140v^12 + 177v^11 - 1134v^9 + 2340v^8 - 3036v^7 + 17448v^5 - 31920v^4 + 37312v^3 - 257024*v + 471040) / 2560
\(\beta_{13}\)	\(=\)	\( ( 11 \nu^{15} - 58 \nu^{13} - 140 \nu^{12} - 177 \nu^{11} + 1134 \nu^{9} + 2340 \nu^{8} + 3036 \nu^{7} + \cdots + 471040 ) / 2560 \) (11v^15 - 58v^13 - 140v^12 - 177v^11 + 1134v^9 + 2340v^8 + 3036v^7 - 17448v^5 - 31920v^4 - 37312v^3 + 257024*v + 471040) / 2560
\(\beta_{14}\)	\(=\)	\( ( 113 \nu^{15} + 112 \nu^{14} + 568 \nu^{13} - 1587 \nu^{11} - 3408 \nu^{10} - 10152 \nu^{9} + \cdots - 2588672 \nu ) / 20480 \) (113v^15 + 112v^14 + 568v^13 - 1587v^11 - 3408v^10 - 10152v^9 + 25188v^7 + 42432v^6 + 158688v^5 - 343552v^3 - 862208v^2 - 2588672v) / 20480
\(\beta_{15}\)	\(=\)	\( ( 113 \nu^{15} - 112 \nu^{14} + 568 \nu^{13} - 1587 \nu^{11} + 3408 \nu^{10} - 10152 \nu^{9} + \cdots - 2588672 \nu ) / 20480 \) (113v^15 - 112v^14 + 568v^13 - 1587v^11 + 3408v^10 - 10152v^9 + 25188v^7 - 42432v^6 + 158688v^5 - 343552v^3 + 862208v^2 - 2588672v) / 20480

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 8\beta_{10} + 2\beta_{9} - 8\beta_{8} - 8\beta_{7} + \beta_{5} ) / 128 \) (-b15 - b14 + b13 - b12 + 8b10 + 2b9 - 8b8 - 8b7 + b5) / 128
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} - 2\beta_{2} - 16\beta_1 ) / 64 \) (b15 - b14 - 2b2 - 16b1) / 64
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + \beta_{14} + 5 \beta_{13} - 5 \beta_{12} - 8 \beta_{11} + 40 \beta_{10} + \cdots - 32 \beta_{3} ) / 128 \) (b15 + b14 + 5b13 - 5b12 - 8b11 + 40b10 + 30b9 + 40b8 + 40b7 - 3b5 + 32b4 - 32b3) / 128
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{13} + 3\beta_{12} - 20\beta_{8} + 20\beta_{7} + 10\beta_{6} + 280 ) / 32 \) (3b13 + 3b12 - 20b8 + 20b7 + 10*b6 + 280) / 32
\(\nu^{5}\)	\(=\)	\( ( - 11 \beta_{15} - 11 \beta_{14} + 15 \beta_{13} - 15 \beta_{12} - 72 \beta_{11} + \cdots + 288 \beta_{3} ) / 128 \) (-11b15 - 11b14 + 15b13 - 15b12 - 72b11 + 40b10 - 10b9 - 40b8 - 40b7 + 7b5 - 288b4 + 288b3) / 128
\(\nu^{6}\)	\(=\)	\( ( 19\beta_{15} - 19\beta_{14} - 40\beta_{4} - 40\beta_{3} + 42\beta_{2} - 296\beta_1 ) / 64 \) (19b15 - 19b14 - 40b4 - 40b3 + 42b2 - 296b1) / 64
\(\nu^{7}\)	\(=\)	\( ( - 9 \beta_{15} - 9 \beta_{14} + 91 \beta_{13} - 91 \beta_{12} + 120 \beta_{11} + 248 \beta_{10} + \cdots + 480 \beta_{3} ) / 128 \) (-9b15 - 9b14 + 91b13 - 91b12 + 120b11 + 248b10 + 818b9 + 248b8 + 248b7 - 109b5 - 480b4 + 480b3) / 128
\(\nu^{8}\)	\(=\)	\( ( 5\beta_{13} + 5\beta_{12} - 700\beta_{8} + 700\beta_{7} + 70\beta_{6} + 8 ) / 32 \) (5b13 + 5b12 - 700b8 + 700b7 + 70*b6 + 8) / 32
\(\nu^{9}\)	\(=\)	\( ( 51 \beta_{15} + 51 \beta_{14} + 249 \beta_{13} - 249 \beta_{12} - 920 \beta_{11} + \cdots + 3680 \beta_{3} ) / 128 \) (51b15 + 51b14 + 249b13 - 249b12 - 920b11 + 1112b10 - 2502b9 - 1112b8 - 1112b7 - 351b5 - 3680b4 + 3680b3) / 128
\(\nu^{10}\)	\(=\)	\( ( 309\beta_{15} - 309\beta_{14} - 2040\beta_{4} - 2040\beta_{3} + 902\beta_{2} + 3656\beta_1 ) / 64 \) (309b15 - 309b14 - 2040b4 - 2040b3 + 902b2 + 3656b1) / 64
\(\nu^{11}\)	\(=\)	\( ( 609 \beta_{15} + 609 \beta_{14} + 1405 \beta_{13} - 1405 \beta_{12} + 4488 \beta_{11} + \cdots + 17952 \beta_{3} ) / 128 \) (609b15 + 609b14 + 1405b13 - 1405b12 + 4488b11 - 440b10 + 5150b9 - 440b8 - 440b7 - 187b5 - 17952b4 + 17952b3) / 128
\(\nu^{12}\)	\(=\)	\( ( -893\beta_{13} - 893\beta_{12} - 7140\beta_{8} + 7140\beta_{7} - 1110\beta_{6} + 43960 ) / 32 \) (-893b13 - 893b12 - 7140b8 + 7140b7 - 1110*b6 + 43960) / 32
\(\nu^{13}\)	\(=\)	\( ( 581 \beta_{15} + 581 \beta_{14} + 3375 \beta_{13} - 3375 \beta_{12} + 3672 \beta_{11} + \cdots - 14688 \beta_{3} ) / 128 \) (581b15 + 581b14 + 3375b13 - 3375b12 + 3672b11 + 45160b10 - 32810b9 - 45160b8 - 45160b7 - 4537b5 + 14688b4 - 14688b3) / 128
\(\nu^{14}\)	\(=\)	\( ( 4051\beta_{15} - 4051\beta_{14} - 46920\beta_{4} - 46920\beta_{3} - 3862\beta_{2} + 100216\beta_1 ) / 64 \) (4051b15 - 4051b14 - 46920b4 - 46920b3 - 3862b2 + 100216b1) / 64
\(\nu^{15}\)	\(=\)	\( ( 19399 \beta_{15} + 19399 \beta_{14} + 21899 \beta_{13} - 21899 \beta_{12} + 11960 \beta_{11} + \cdots + 47840 \beta_{3} ) / 128 \) (19399b15 + 19399b14 + 21899b13 - 21899b12 + 11960b11 + 60152b10 - 18798b9 + 60152b8 + 60152b7 + 16899b5 - 47840b4 + 47840b3) / 128

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 767.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 24 T^{6} + \cdots + 531441)^{2} \) (T^8 - 24T^6 + 414T^4 - 17496*T^2 + 531441)^2
$5$	\( (T^{4} + 324 T^{2} + 26112)^{4} \) (T^4 + 324*T^2 + 26112)^4
$7$	\( (T^{4} + 924 T^{2} + 143616)^{4} \) (T^4 + 924*T^2 + 143616)^4
$11$	\( (T^{4} - 484 T^{2} + 352)^{4} \) (T^4 - 484*T^2 + 352)^4
$13$	\( (T^{4} - 5808 T^{2} + 574464)^{4} \) (T^4 - 5808*T^2 + 574464)^4
$17$	\( (T^{4} + 2464 T^{2} + 90112)^{4} \) (T^4 + 2464*T^2 + 90112)^4
$19$	\( (T^{4} + 9044 T^{2} + 11999296)^{4} \) (T^4 + 9044*T^2 + 11999296)^4
$23$	\( (T^{4} - 17472 T^{2} + 1671168)^{4} \) (T^4 - 17472*T^2 + 1671168)^4
$29$	\( (T^{4} + 43620 T^{2} + 448108032)^{4} \) (T^4 + 43620*T^2 + 448108032)^4
$31$	\( (T^{4} + 20988 T^{2} + 41505024)^{4} \) (T^4 + 20988*T^2 + 41505024)^4
$37$	\( (T^{4} - 77616 T^{2} + 1494180864)^{4} \) (T^4 - 77616*T^2 + 1494180864)^4
$41$	\( (T^{4} + 193600 T^{2} + 3151126528)^{4} \) (T^4 + 193600*T^2 + 3151126528)^4
$43$	\( (T^{4} + 101108 T^{2} + 2483228224)^{4} \) (T^4 + 101108*T^2 + 2483228224)^4
$47$	\( (T^{4} - 321792 T^{2} + 427819008)^{4} \) (T^4 - 321792*T^2 + 427819008)^4
$53$	\( (T^{4} + 177156 T^{2} + 7033554432)^{4} \) (T^4 + 177156*T^2 + 7033554432)^4
$59$	\( (T^{4} - 21604 T^{2} + 296032)^{4} \) (T^4 - 21604*T^2 + 296032)^4
$61$	\( (T^{4} - 550704 T^{2} + 18820015104)^{4} \) (T^4 - 550704*T^2 + 18820015104)^4
$67$	\( (T^{4} + 75284 T^{2} + 1043031616)^{4} \) (T^4 + 75284*T^2 + 1043031616)^4
$71$	\( (T^{4} - 654912 T^{2} + 103614087168)^{4} \) (T^4 - 654912*T^2 + 103614087168)^4
$73$	\( (T^{2} + 268 T - 704876)^{8} \) (T^2 + 268*T - 704876)^8
$79$	\( (T^{4} + 1028412 T^{2} + 99653562624)^{4} \) (T^4 + 1028412*T^2 + 99653562624)^4
$83$	\( (T^{4} - 396484 T^{2} + 2128431712)^{4} \) (T^4 - 396484*T^2 + 2128431712)^4
$89$	\( (T^{4} + 2644576 T^{2} + 147293673472)^{4} \) (T^4 + 2644576*T^2 + 147293673472)^4
$97$	\( (T^{2} + 968 T - 33044)^{8} \) (T^2 + 968*T - 33044)^8
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\(n\)	\(257\)	\(511\)	\(517\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)