LMFDB - Newform orbit 704.4.e.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(703,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.703");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.e (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:	$41.5373446440$
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} + 6x^{14} - 44x^{12} - 176x^{10} + 2176x^{8} - 11264x^{6} - 180224x^{4} + 1572864x^{2} + 16777216 $ x^16 + 6x^14 - 44x^12 - 176x^10 + 2176x^8 - 11264x^6 - 180224x^4 + 1572864*x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 44)
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_1 q^{3} + \beta_{2} q^{5} - \beta_{8} q^{7} + (\beta_{4} - 8) q^{9} - \beta_{6} q^{11} - \beta_{3} q^{13} + ( - \beta_{12} - \beta_1) q^{15} + \beta_{13} q^{17} - \beta_{11} q^{19} + ( - \beta_{15} + \beta_{13}) q^{21}+ \cdots + (5 \beta_{14} - 6 \beta_{12} + \cdots + 33 \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + b2 * q^5 - b8 * q^7 + (b4 - 8) * q^9 - b6 * q^11 - b3 * q^13 + (-b12 - b1) * q^15 + b13 * q^17 - b11 * q^19 + (-b15 + b13) * q^21 + (b12 - b9 - b6 - b5 - 2b1) q^23 + (-b7 + 23) * q^25 + (-b9 - b6 - b5 - 4b1) q^27 + (2b15 + b13 - b10) q^29 + (b12 - b6 - b5 + 3b1) q^31 + (2b13 - b10 + b7 - 2b4 - 2b3 - 8) q^33 + (b14 + 3b11 - 3b8 + 3b6 - 3b5) * q^35 + (-b7 + 3b4 - 3b2 + 3) * q^37 + (b14 - 2b11 + 2b8 + 2b6 - 2b5) * q^39 + (b15 - 3b13 - b10 - 5b3) * q^41 + (-b14 - 2b11 - 5b8 + b6 - b5) * q^43 + (-3b7 + b4 - 10b2 + 59) * q^45 + (4b9 - b6 - b5 + 12b1) * q^47 + (-3b7 + 3b4 + 2b2 + 32) q^49 + (b11 + 10b8 + 5b6 - 5b5) q^51 + (b7 + 7b4 + 2b2 - 83) * q^53 + (-b14 - b12 + 2b11 - 3b9 + 8b8 - 6b5) * q^55 + (-5b15 + b13 - b10 + 11b3) * q^57 + (-6b12 + 3b9 - 2b6 - 2b5 - 4b1) q^59 + (-4b15 - b13 - 3b10 + 2b3) q^61 + (-b14 + 4b11 + b8 + 6b6 - 6b5) q^63 + (2b15 + 3b13 + 2b10 - 8b3) * q^65 + (6b12 + 4b9 - 6b6 - 6b5 - 7b1) q^67 + (6b7 - 16b4 + 31b2 + 22) q^69 + (-b12 + 3b9 - 7b6 - 7b5 - 26b1) * q^71 + (3b15 - b13 + b10 + 9b3) * q^73 + (6b12 - 2b9 - 7b6 - 7b5 + 28b1) q^75 + (5b13 + 3b10 - 3b7 - 5b4 - 5b3 - 22b2 + 79) * q^77 + (-5b14 - 15b8 + 4b6 - 4b5) * q^79 + (3b7 + 11b4 - 6b2 - 100) q^81 + (-6b14 - 4b11 + 16b8 + 9b6 - 9b5) q^83 + (3b15 - 9b13 + 4b10 - 8b3) * q^85 + (5b14 - 2b11 - 35b8 + 12b6 - 12b5) q^87 + (7b7 - 20b4 - 20b2 + 14) q^89 + (-7b9 - 8b6 - 8b5 + 17b1) * q^91 + (5b7 - 3b4 + 37b2 - 145) q^93 + (b14 - 4b11 + 42b8 + 8b6 - 8b5) * q^95 + (25b4 + 4b2 - 245) * q^97 + (5b14 - 6b12 + b11 + 4b9 + 15b8 + 5b6 - 14b5 + 33b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{5} - 132 q^{9} + 364 q^{25} - 116 q^{33} + 20 q^{37} + 888 q^{45} + 496 q^{49} - 1344 q^{53} + 564 q^{69} + 1184 q^{77} - 1656 q^{81} + 252 q^{89} - 2140 q^{93} - 4004 q^{97}+O(q^{100}) $ 16 * q + 4 * q^5 - 132 * q^9 + 364 * q^25 - 116 * q^33 + 20 * q^37 + 888 * q^45 + 496 * q^49 - 1344 * q^53 + 564 * q^69 + 1184 * q^77 - 1656 * q^81 + 252 * q^89 - 2140 * q^93 - 4004 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 6x^{14} - 44x^{12} - 176x^{10} + 2176x^{8} - 11264x^{6} - 180224x^{4} + 1572864x^{2} + 16777216 $ x^16 + 6*x^14 - 44*x^12 - 176*x^10 + 2176*x^8 - 11264*x^6 - 180224*x^4 + 1572864*x^2 + 16777216 :

$\beta_{1}$	$=$	$ ( 3\nu^{14} - 6\nu^{12} + 108\nu^{10} + 784\nu^{8} + 6144\nu^{6} - 14336\nu^{4} + 40960\nu^{2} + 2490368 ) / 1441792 $ (3v^14 - 6v^12 + 108v^10 + 784v^8 + 6144v^6 - 14336v^4 + 40960*v^2 + 2490368) / 1441792
$\beta_{2}$	$=$	$ ( - 19 \nu^{14} - 2 \nu^{12} + 740 \nu^{10} - 6192 \nu^{8} + 21888 \nu^{6} + 429056 \nu^{4} + \cdots - 29622272 ) / 2883584 $ (-19v^14 - 2v^12 + 740v^10 - 6192v^8 + 21888v^6 + 429056v^4 + 557056*v^2 - 29622272) / 2883584
$\beta_{3}$	$=$	$ ( - 17 \nu^{15} + 250 \nu^{13} + 1836 \nu^{11} + 5936 \nu^{9} + 290176 \nu^{7} + \cdots - 3145728 \nu ) / 11534336 $ (-17v^15 + 250v^13 + 1836v^11 + 5936v^9 + 290176v^7 + 711680v^5 - 7716864v^3 - 3145728v) / 11534336
$\beta_{4}$	$=$	$ ( - 21 \nu^{14} + 146 \nu^{12} + 2300 \nu^{10} - 9808 \nu^{8} - 65920 \nu^{6} - 142336 \nu^{4} + \cdots - 49283072 ) / 2883584 $ (-21v^14 + 146v^12 + 2300v^10 - 9808v^8 - 65920v^6 - 142336v^4 - 1736704*v^2 - 49283072) / 2883584
$\beta_{5}$	$=$	$ ( - 29 \nu^{15} + 12 \nu^{14} + 274 \nu^{13} + 328 \nu^{12} + 1404 \nu^{11} - 3088 \nu^{10} + \cdots + 108003328 ) / 11534336 $ (-29v^15 + 12v^14 + 274v^13 + 328v^12 + 1404v^11 - 3088v^10 + 2800v^9 + 43968v^8 - 94848v^7 + 390656v^6 + 48128v^5 - 2985984v^4 - 6438912v^3 - 7405568v^2 - 65011712*v + 108003328) / 11534336
$\beta_{6}$	$=$	$ ( 29 \nu^{15} + 12 \nu^{14} - 274 \nu^{13} + 328 \nu^{12} - 1404 \nu^{11} - 3088 \nu^{10} + \cdots + 108003328 ) / 11534336 $ (29v^15 + 12v^14 - 274v^13 + 328v^12 - 1404v^11 - 3088v^10 - 2800v^9 + 43968v^8 + 94848v^7 + 390656v^6 - 48128v^5 - 2985984v^4 + 6438912v^3 - 7405568v^2 + 65011712*v + 108003328) / 11534336
$\beta_{7}$	$=$	$ ( - 71 \nu^{14} - 378 \nu^{12} + 4692 \nu^{10} + 18064 \nu^{8} - 301184 \nu^{6} - 9216 \nu^{4} + \cdots - 90963968 ) / 2883584 $ (-71v^14 - 378v^12 + 4692v^10 + 18064v^8 - 301184v^6 - 9216v^4 + 36077568*v^2 - 90963968) / 2883584
$\beta_{8}$	$=$	$ ( 43 \nu^{15} - 30 \nu^{13} - 1572 \nu^{11} + 9200 \nu^{9} + 70528 \nu^{7} - 1766400 \nu^{5} + \cdots + 79691776 \nu ) / 11534336 $ (43v^15 - 30v^13 - 1572v^11 + 9200v^9 + 70528v^7 - 1766400v^5 + 1654784v^3 + 79691776v) / 11534336
$\beta_{9}$	$=$	$ ( 41 \nu^{14} + 270 \nu^{12} - 2044 \nu^{10} - 8528 \nu^{8} + 89600 \nu^{6} - 481280 \nu^{4} + \cdots + 66715648 ) / 1441792 $ (41v^14 + 270v^12 - 2044v^10 - 8528v^8 + 89600v^6 - 481280v^4 + 3563520*v^2 + 66715648) / 1441792
$\beta_{10}$	$=$	$ ( - 35 \nu^{15} - 562 \nu^{13} + 6596 \nu^{11} + 72848 \nu^{9} - 214912 \nu^{7} + \cdots - 370147328 \nu ) / 11534336 $ (-35v^15 - 562v^13 + 6596v^11 + 72848v^9 - 214912v^7 + 2069504v^5 + 32030720v^3 - 370147328v) / 11534336
$\beta_{11}$	$=$	$ ( 5 \nu^{15} - 2 \nu^{13} + 612 \nu^{11} - 1520 \nu^{9} - 12160 \nu^{7} - 76800 \nu^{5} + \cdots - 8388608 \nu ) / 1048576 $ (5v^15 - 2v^13 + 612v^11 - 1520v^9 - 12160v^7 - 76800v^5 - 1851392v^3 - 8388608v) / 1048576
$\beta_{12}$	$=$	$ ( 53 \nu^{14} - 194 \nu^{12} - 2844 \nu^{10} - 112 \nu^{8} + 84608 \nu^{6} - 2014208 \nu^{4} + \cdots + 115605504 ) / 1441792 $ (53v^14 - 194v^12 - 2844v^10 - 112v^8 + 84608v^6 - 2014208v^4 + 933888*v^2 + 115605504) / 1441792
$\beta_{13}$	$=$	$ ( 101 \nu^{15} - 450 \nu^{13} - 6684 \nu^{11} + 45072 \nu^{9} + 110720 \nu^{7} + \cdots + 208666624 \nu ) / 11534336 $ (101v^15 - 450v^13 - 6684v^11 + 45072v^9 + 110720v^7 - 2092032v^5 - 15089664v^3 + 208666624v) / 11534336
$\beta_{14}$	$=$	$ ( - 133 \nu^{15} - 1086 \nu^{13} + 6172 \nu^{11} + 40176 \nu^{9} - 312448 \nu^{7} + \cdots + 171966464 \nu ) / 11534336 $ (-133v^15 - 1086v^13 + 6172v^11 + 40176v^9 - 312448v^7 + 216064v^5 + 33374208v^3 + 171966464v) / 11534336
$\beta_{15}$	$=$	$ ( - 79 \nu^{15} - 122 \nu^{13} + 1492 \nu^{11} - 17968 \nu^{9} + 61056 \nu^{7} + \cdots - 139460608 \nu ) / 5767168 $ (-79v^15 - 122v^13 + 1492v^11 - 17968v^9 + 61056v^7 + 312320v^5 - 4014080v^3 - 139460608v) / 5767168

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{14} - \beta_{10} - \beta_{8} + \beta_{3} ) / 64 $ (-b15 + b14 - b10 - b8 + b3) / 64
$\nu^{2}$	$=$	$ ( \beta_{9} + \beta_{7} - \beta_{4} + 2\beta_{2} + \beta _1 - 13 ) / 16 $ (b9 + b7 - b4 + 2*b2 + b1 - 13) / 16
$\nu^{3}$	$=$	$ ( -\beta_{15} - 4\beta_{13} - 2\beta_{11} + \beta_{10} + 6\beta_{8} + 2\beta_{6} - 2\beta_{5} - \beta_{3} ) / 16 $ (-b15 - 4b13 - 2b11 + b10 + 6b8 + 2b6 - 2*b5 - b3) / 16
$\nu^{4}$	$=$	$ ( -4\beta_{12} + \beta_{9} - \beta_{7} - 4\beta_{6} - 4\beta_{5} - 11\beta_{4} + 2\beta_{2} + 17\beta _1 + 121 ) / 8 $ (-4b12 + b9 - b7 - 4b6 - 4b5 - 11b4 + 2b2 + 17b1 + 121) / 8
$\nu^{5}$	$=$	$ ( - 19 \beta_{15} - 3 \beta_{14} + 8 \beta_{13} - 4 \beta_{11} + \beta_{10} - 113 \beta_{8} + \cdots + 15 \beta_{3} ) / 16 $ (-19b15 - 3b14 + 8b13 - 4b11 + b10 - 113b8 + 20b6 - 20b5 + 15b3) / 16
$\nu^{6}$	$=$	$ ( 2\beta_{12} + 2\beta_{9} - \beta_{7} + 10\beta_{6} + 10\beta_{5} - 9\beta_{4} + 60\beta_{2} + 74\beta _1 - 137 ) / 2 $ (2b12 + 2b9 - b7 + 10b6 + 10b5 - 9b4 + 60b2 + 74*b1 - 137) / 2
$\nu^{7}$	$=$	$ ( 45 \beta_{15} - 17 \beta_{14} - 16 \beta_{13} - 16 \beta_{11} + 13 \beta_{10} + 129 \beta_{8} + \cdots + 195 \beta_{3} ) / 8 $ (45b15 - 17b14 - 16b13 - 16b11 + 13b10 + 129b8 + 96b6 - 96b5 + 195*b3) / 8
$\nu^{8}$	$=$	$ ( - 56 \beta_{12} - 27 \beta_{9} + 17 \beta_{7} + 88 \beta_{6} + 88 \beta_{5} - 89 \beta_{4} - 294 \beta_{2} + \cdots - 309 ) / 2 $ (-56b12 - 27b9 + 17b7 + 88b6 + 88b5 - 89b4 - 294b2 + 229b1 - 309) / 2
$\nu^{9}$	$=$	$ ( - 62 \beta_{15} + 37 \beta_{14} + 172 \beta_{13} - 74 \beta_{11} + 108 \beta_{10} + \cdots + 180 \beta_{3} ) / 2 $ (-62b15 + 37b14 + 172b13 - 74b11 + 108b10 + 25b8 - 214b6 + 214b5 + 180*b3) / 2
$\nu^{10}$	$=$	$ - 4 \beta_{12} - 53 \beta_{9} + 25 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 603 \beta_{4} + 310 \beta_{2} + \cdots + 9895 $ -4b12 - 53b9 + 25b7 - 4b6 - 4b5 + 603b4 + 310b2 + 4187b1 + 9895
$\nu^{11}$	$=$	$ ( - 527 \beta_{15} + 773 \beta_{14} - 968 \beta_{13} + 1748 \beta_{11} + 45 \beta_{10} + \cdots + 1891 \beta_{3} ) / 2 $ (-527b15 + 773b14 - 968b13 + 1748b11 + 45b10 + 1023b8 + 188b6 - 188b5 + 1891*b3) / 2
$\nu^{12}$	$=$	$ - 1992 \beta_{12} + 2380 \beta_{9} + 88 \beta_{7} + 1880 \beta_{6} + 1880 \beta_{5} + 3088 \beta_{4} + \cdots + 17952 $ -1992b12 + 2380b9 + 88b7 + 1880b6 + 1880b5 + 3088b4 - 6280b2 - 7252b1 + 17952
$\nu^{13}$	$=$	$ - 4849 \beta_{15} - 1727 \beta_{14} - 5760 \beta_{13} - 3992 \beta_{11} - 745 \beta_{10} + \cdots + 3673 \beta_{3} $ -4849b15 - 1727b14 - 5760b13 - 3992b11 - 745b10 + 7879b8 - 8568b6 + 8568b5 + 3673*b3
$\nu^{14}$	$=$	$ - 960 \beta_{12} + 7892 \beta_{9} - 3372 \beta_{7} - 20224 \beta_{6} - 20224 \beta_{5} - 404 \beta_{4} + \cdots - 886404 $ -960b12 + 7892b9 - 3372b7 - 20224b6 - 20224b5 - 404b4 - 47256b2 + 218964b1 - 886404
$\nu^{15}$	$=$	$ - 33028 \beta_{15} - 24208 \beta_{14} - 6672 \beta_{13} + 34904 \beta_{11} + 15204 \beta_{10} + \cdots - 10148 \beta_{3} $ -33028b15 - 24208b14 - 6672b13 + 34904b11 + 15204b10 - 12280b8 + 47208b6 - 47208b5 - 10148*b3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$-1$

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} $

703.1

−2.78722	+	0.481028i
2.78722	−	0.481028i
−0.337894	+	2.80817i
0.337894	−	2.80817i
−1.06306	−	2.62105i
1.06306	+	2.62105i
2.34246	+	1.58521i
−2.34246	−	1.58521i
2.34246	−	1.58521i
−2.34246	+	1.58521i
−1.06306	+	2.62105i
1.06306	−	2.62105i
−0.337894	−	2.80817i
0.337894	+	2.80817i
−2.78722	−	0.481028i
2.78722	+	0.481028i

−

8.15553i

7.84961

−6.08360

−39.5127

703.2

−

8.15553i

7.84961

6.08360

−39.5127

703.3

−

7.66734i

−15.8563

−24.1286

−31.7881

703.4

−

7.66734i

−15.8563

24.1286

−31.7881

703.5

−

3.80457i

15.3983

−27.1053

12.5252

703.6

−

3.80457i

15.3983

27.1053

12.5252

703.7

−

1.10653i

−6.39165

−11.9207

25.7756

703.8

−

1.10653i

−6.39165

11.9207

25.7756

703.9

1.10653i

−6.39165

−11.9207

25.7756

703.10

1.10653i

−6.39165

11.9207

25.7756

703.11

3.80457i

15.3983

−27.1053

12.5252

703.12

3.80457i

15.3983

27.1053

12.5252

703.13

7.66734i

−15.8563

−24.1286

−31.7881

703.14

7.66734i

−15.8563

24.1286

−31.7881

703.15

8.15553i

7.84961

−6.08360

−39.5127

703.16

8.15553i

7.84961

6.08360

−39.5127

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.4.e.f		16
4.b	odd	2	1	inner	704.4.e.f		16
8.b	even	2	1		44.4.c.a	✓	16
8.d	odd	2	1		44.4.c.a	✓	16
11.b	odd	2	1	inner	704.4.e.f		16
24.f	even	2	1		396.4.h.b		16
24.h	odd	2	1		396.4.h.b		16
44.c	even	2	1	inner	704.4.e.f		16
88.b	odd	2	1		44.4.c.a	✓	16
88.g	even	2	1		44.4.c.a	✓	16
264.m	even	2	1		396.4.h.b		16
264.p	odd	2	1		396.4.h.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.4.c.a	✓	16	8.b	even	2	1
44.4.c.a	✓	16	8.d	odd	2	1
44.4.c.a	✓	16	88.b	odd	2	1
44.4.c.a	✓	16	88.g	even	2	1
396.4.h.b		16	24.f	even	2	1
396.4.h.b		16	24.h	odd	2	1
396.4.h.b		16	264.m	even	2	1
396.4.h.b		16	264.p	odd	2	1
704.4.e.f		16	1.a	even	1	1	trivial
704.4.e.f		16	4.b	odd	2	1	inner
704.4.e.f		16	11.b	odd	2	1	inner
704.4.e.f		16	44.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(704, [\chi])$:

$ T_{3}^{8} + 141T_{3}^{6} + 5895T_{3}^{4} + 63607T_{3}^{2} + 69300 $

$ T_{7}^{8} - 1496T_{7}^{6} + 668864T_{7}^{4} - 83538432T_{7}^{2} + 2249555968 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 141 T^{6} + \cdots + 69300)^{2} $ (T^8 + 141T^6 + 5895T^4 + 63607*T^2 + 69300)^2
$5$	$ (T^{4} - T^{3} + \cdots + 12250)^{4} $ (T^4 - T^3 - 295T^2 + 333T + 12250)^4
$7$	$ (T^{8} - 1496 T^{6} + \cdots + 2249555968)^{2} $ (T^8 - 1496T^6 + 668864T^4 - 83538432*T^2 + 2249555968)^2
$11$	$ T^{16} + \cdots + 98\!\cdots\!41 $ T^16 - 632T^14 - 215204T^12 + 1800843000T^10 - 1590036874778T^8 + 3190303225923000T^6 - 675402340383866084T^4 - 3513867742127090295992*T^2 + 9849732675807611094711841
$13$	$ (T^{8} + 7384 T^{6} + \cdots + 192669474816)^{2} $ (T^8 + 7384T^6 + 16227136T^4 + 10535019008*T^2 + 192669474816)^2
$17$	$ (T^{8} + \cdots + 19002976763904)^{2} $ (T^8 + 14664T^6 + 63108288T^4 + 76738901504*T^2 + 19002976763904)^2
$19$	$ (T^{8} + \cdots + 550422481100800)^{2} $ (T^8 - 28040T^6 + 259573568T^4 - 860543866368*T^2 + 550422481100800)^2
$23$	$ (T^{8} + \cdots + 86\!\cdots\!00)^{2} $ (T^8 + 47069T^6 + 708614695T^4 + 4254382018183*T^2 + 8648373903592500)^2
$29$	$ (T^{8} + \cdots + 58\!\cdots\!00)^{2} $ (T^8 + 177744T^6 + 10117480448T^4 + 182429620895744*T^2 + 58932964255334400)^2
$31$	$ (T^{8} + \cdots + 171617000277492)^{2} $ (T^8 + 33125T^6 + 244820055T^4 + 473645172367*T^2 + 171617000277492)^2
$37$	$ (T^{4} - 5 T^{3} + \cdots - 29264670)^{4} $ (T^4 - 5T^3 - 29607T^2 + 2270537*T - 29264670)^4
$41$	$ (T^{8} + \cdots + 22\!\cdots\!00)^{2} $ (T^8 + 383240T^6 + 47273291456T^4 + 2061418050143744*T^2 + 22075399279580774400)^2
$43$	$ (T^{8} + \cdots + 66\!\cdots\!08)^{2} $ (T^8 - 177936T^6 + 9038700032T^4 - 152463379447808*T^2 + 660655015980433408)^2
$47$	$ (T^{8} + \cdots + 28\!\cdots\!00)^{2} $ (T^8 + 419072T^6 + 46548181152T^4 + 1971991569545728*T^2 + 28662079431048940800)^2
$53$	$ (T^{4} + 336 T^{3} + \cdots + 45417040)^{4} $ (T^4 + 336T^3 - 68536T^2 - 19046144*T + 45417040)^4
$59$	$ (T^{8} + \cdots + 95\!\cdots\!00)^{2} $ (T^8 + 880189T^6 + 262684963655T^4 + 30022424168816743*T^2 + 956317918004204332500)^2
$61$	$ (T^{8} + \cdots + 18\!\cdots\!00)^{2} $ (T^8 + 745840T^6 + 184944047616T^4 + 15995475483508736*T^2 + 182369385760201113600)^2
$67$	$ (T^{8} + \cdots + 15\!\cdots\!00)^{2} $ (T^8 + 1730469T^6 + 1016268506615T^4 + 227415738188260303*T^2 + 15090615901295639327700)^2
$71$	$ (T^{8} + \cdots + 30\!\cdots\!00)^{2} $ (T^8 + 762149T^6 + 87594176439T^4 + 3064923794257807*T^2 + 30101205363464812500)^2
$73$	$ (T^{8} + \cdots + 38\!\cdots\!96)^{2} $ (T^8 + 1001736T^6 + 375179607744T^4 + 62274241009468928*T^2 + 3865785935284366442496)^2
$79$	$ (T^{8} + \cdots + 15\!\cdots\!00)^{2} $ (T^8 - 1692800T^6 + 460716720128T^4 - 22720961386119168*T^2 + 155512676150149120000)^2
$83$	$ (T^{8} + \cdots + 88\!\cdots\!28)^{2} $ (T^8 - 3153584T^6 + 2947498508288T^4 - 815944256165249024*T^2 + 8825233202433100873728)^2
$89$	$ (T^{4} - 63 T^{3} + \cdots - 26023111450)^{4} $ (T^4 - 63T^3 - 1503131T^2 + 438306187*T - 26023111450)^4
$97$	$ (T^{4} + 1001 T^{3} + \cdots + 171117066190)^{4} $ (T^4 + 1001T^3 - 627235T^2 - 456830013*T + 171117066190)^4
show more
show less

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{8} q^{7} + (\beta_{4} - 8) q^{9} - \beta_{6} q^{11} - \beta_{3} q^{13} + ( - \beta_{12} - \beta_1) q^{15} + \beta_{13} q^{17} - \beta_{11} q^{19} + ( - \beta_{15} + \beta_{13}) q^{21}+ \cdots + (5 \beta_{14} - 6 \beta_{12} + \cdots + 33 \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + b2 * q^5 - b8 * q^7 + (b4 - 8) * q^9 - b6 * q^11 - b3 * q^13 + (-b12 - b1) * q^15 + b13 * q^17 - b11 * q^19 + (-b15 + b13) * q^21 + (b12 - b9 - b6 - b5 - 2b1) q^23 + (-b7 + 23) * q^25 + (-b9 - b6 - b5 - 4b1) q^27 + (2b15 + b13 - b10) q^29 + (b12 - b6 - b5 + 3b1) q^31 + (2b13 - b10 + b7 - 2b4 - 2b3 - 8) q^33 + (b14 + 3b11 - 3b8 + 3b6 - 3b5) * q^35 + (-b7 + 3b4 - 3b2 + 3) * q^37 + (b14 - 2b11 + 2b8 + 2b6 - 2b5) * q^39 + (b15 - 3b13 - b10 - 5b3) * q^41 + (-b14 - 2b11 - 5b8 + b6 - b5) * q^43 + (-3b7 + b4 - 10b2 + 59) * q^45 + (4b9 - b6 - b5 + 12b1) * q^47 + (-3b7 + 3b4 + 2b2 + 32) q^49 + (b11 + 10b8 + 5b6 - 5b5) q^51 + (b7 + 7b4 + 2b2 - 83) * q^53 + (-b14 - b12 + 2b11 - 3b9 + 8b8 - 6b5) * q^55 + (-5b15 + b13 - b10 + 11b3) * q^57 + (-6b12 + 3b9 - 2b6 - 2b5 - 4b1) q^59 + (-4b15 - b13 - 3b10 + 2b3) q^61 + (-b14 + 4b11 + b8 + 6b6 - 6b5) q^63 + (2b15 + 3b13 + 2b10 - 8b3) * q^65 + (6b12 + 4b9 - 6b6 - 6b5 - 7b1) q^67 + (6b7 - 16b4 + 31b2 + 22) q^69 + (-b12 + 3b9 - 7b6 - 7b5 - 26b1) * q^71 + (3b15 - b13 + b10 + 9b3) * q^73 + (6b12 - 2b9 - 7b6 - 7b5 + 28b1) q^75 + (5b13 + 3b10 - 3b7 - 5b4 - 5b3 - 22b2 + 79) * q^77 + (-5b14 - 15b8 + 4b6 - 4b5) * q^79 + (3b7 + 11b4 - 6b2 - 100) q^81 + (-6b14 - 4b11 + 16b8 + 9b6 - 9b5) q^83 + (3b15 - 9b13 + 4b10 - 8b3) * q^85 + (5b14 - 2b11 - 35b8 + 12b6 - 12b5) q^87 + (7b7 - 20b4 - 20b2 + 14) q^89 + (-7b9 - 8b6 - 8b5 + 17b1) * q^91 + (5b7 - 3b4 + 37b2 - 145) q^93 + (b14 - 4b11 + 42b8 + 8b6 - 8b5) * q^95 + (25b4 + 4b2 - 245) * q^97 + (5b14 - 6b12 + b11 + 4b9 + 15b8 + 5b6 - 14b5 + 33b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{5} - 132 q^{9} + 364 q^{25} - 116 q^{33} + 20 q^{37} + 888 q^{45} + 496 q^{49} - 1344 q^{53} + 564 q^{69} + 1184 q^{77} - 1656 q^{81} + 252 q^{89} - 2140 q^{93} - 4004 q^{97}+O(q^{100}) \) 16 * q + 4 * q^5 - 132 * q^9 + 364 * q^25 - 116 * q^33 + 20 * q^37 + 888 * q^45 + 496 * q^49 - 1344 * q^53 + 564 * q^69 + 1184 * q^77 - 1656 * q^81 + 252 * q^89 - 2140 * q^93 - 4004 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	704.4.e.f

Level	$704$
Weight	$4$
Character orbit	704.e
Analytic conductor	$41.537$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(41.5373446440\)
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:	\( x^{16} + 6x^{14} - 44x^{12} - 176x^{10} + 2176x^{8} - 11264x^{6} - 180224x^{4} + 1572864x^{2} + 16777216 \) x^16 + 6x^14 - 44x^12 - 176x^10 + 2176x^8 - 11264x^6 - 180224x^4 + 1572864*x^2 + 16777216
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{42}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 3\nu^{14} - 6\nu^{12} + 108\nu^{10} + 784\nu^{8} + 6144\nu^{6} - 14336\nu^{4} + 40960\nu^{2} + 2490368 ) / 1441792 \) (3v^14 - 6v^12 + 108v^10 + 784v^8 + 6144v^6 - 14336v^4 + 40960*v^2 + 2490368) / 1441792
\(\beta_{2}\)	\(=\)	\( ( - 19 \nu^{14} - 2 \nu^{12} + 740 \nu^{10} - 6192 \nu^{8} + 21888 \nu^{6} + 429056 \nu^{4} + \cdots - 29622272 ) / 2883584 \) (-19v^14 - 2v^12 + 740v^10 - 6192v^8 + 21888v^6 + 429056v^4 + 557056*v^2 - 29622272) / 2883584
\(\beta_{3}\)	\(=\)	\( ( - 17 \nu^{15} + 250 \nu^{13} + 1836 \nu^{11} + 5936 \nu^{9} + 290176 \nu^{7} + \cdots - 3145728 \nu ) / 11534336 \) (-17v^15 + 250v^13 + 1836v^11 + 5936v^9 + 290176v^7 + 711680v^5 - 7716864v^3 - 3145728v) / 11534336
\(\beta_{4}\)	\(=\)	\( ( - 21 \nu^{14} + 146 \nu^{12} + 2300 \nu^{10} - 9808 \nu^{8} - 65920 \nu^{6} - 142336 \nu^{4} + \cdots - 49283072 ) / 2883584 \) (-21v^14 + 146v^12 + 2300v^10 - 9808v^8 - 65920v^6 - 142336v^4 - 1736704*v^2 - 49283072) / 2883584
\(\beta_{5}\)	\(=\)	\( ( - 29 \nu^{15} + 12 \nu^{14} + 274 \nu^{13} + 328 \nu^{12} + 1404 \nu^{11} - 3088 \nu^{10} + \cdots + 108003328 ) / 11534336 \) (-29v^15 + 12v^14 + 274v^13 + 328v^12 + 1404v^11 - 3088v^10 + 2800v^9 + 43968v^8 - 94848v^7 + 390656v^6 + 48128v^5 - 2985984v^4 - 6438912v^3 - 7405568v^2 - 65011712*v + 108003328) / 11534336
\(\beta_{6}\)	\(=\)	\( ( 29 \nu^{15} + 12 \nu^{14} - 274 \nu^{13} + 328 \nu^{12} - 1404 \nu^{11} - 3088 \nu^{10} + \cdots + 108003328 ) / 11534336 \) (29v^15 + 12v^14 - 274v^13 + 328v^12 - 1404v^11 - 3088v^10 - 2800v^9 + 43968v^8 + 94848v^7 + 390656v^6 - 48128v^5 - 2985984v^4 + 6438912v^3 - 7405568v^2 + 65011712*v + 108003328) / 11534336
\(\beta_{7}\)	\(=\)	\( ( - 71 \nu^{14} - 378 \nu^{12} + 4692 \nu^{10} + 18064 \nu^{8} - 301184 \nu^{6} - 9216 \nu^{4} + \cdots - 90963968 ) / 2883584 \) (-71v^14 - 378v^12 + 4692v^10 + 18064v^8 - 301184v^6 - 9216v^4 + 36077568*v^2 - 90963968) / 2883584
\(\beta_{8}\)	\(=\)	\( ( 43 \nu^{15} - 30 \nu^{13} - 1572 \nu^{11} + 9200 \nu^{9} + 70528 \nu^{7} - 1766400 \nu^{5} + \cdots + 79691776 \nu ) / 11534336 \) (43v^15 - 30v^13 - 1572v^11 + 9200v^9 + 70528v^7 - 1766400v^5 + 1654784v^3 + 79691776v) / 11534336
\(\beta_{9}\)	\(=\)	\( ( 41 \nu^{14} + 270 \nu^{12} - 2044 \nu^{10} - 8528 \nu^{8} + 89600 \nu^{6} - 481280 \nu^{4} + \cdots + 66715648 ) / 1441792 \) (41v^14 + 270v^12 - 2044v^10 - 8528v^8 + 89600v^6 - 481280v^4 + 3563520*v^2 + 66715648) / 1441792
\(\beta_{10}\)	\(=\)	\( ( - 35 \nu^{15} - 562 \nu^{13} + 6596 \nu^{11} + 72848 \nu^{9} - 214912 \nu^{7} + \cdots - 370147328 \nu ) / 11534336 \) (-35v^15 - 562v^13 + 6596v^11 + 72848v^9 - 214912v^7 + 2069504v^5 + 32030720v^3 - 370147328v) / 11534336
\(\beta_{11}\)	\(=\)	\( ( 5 \nu^{15} - 2 \nu^{13} + 612 \nu^{11} - 1520 \nu^{9} - 12160 \nu^{7} - 76800 \nu^{5} + \cdots - 8388608 \nu ) / 1048576 \) (5v^15 - 2v^13 + 612v^11 - 1520v^9 - 12160v^7 - 76800v^5 - 1851392v^3 - 8388608v) / 1048576
\(\beta_{12}\)	\(=\)	\( ( 53 \nu^{14} - 194 \nu^{12} - 2844 \nu^{10} - 112 \nu^{8} + 84608 \nu^{6} - 2014208 \nu^{4} + \cdots + 115605504 ) / 1441792 \) (53v^14 - 194v^12 - 2844v^10 - 112v^8 + 84608v^6 - 2014208v^4 + 933888*v^2 + 115605504) / 1441792
\(\beta_{13}\)	\(=\)	\( ( 101 \nu^{15} - 450 \nu^{13} - 6684 \nu^{11} + 45072 \nu^{9} + 110720 \nu^{7} + \cdots + 208666624 \nu ) / 11534336 \) (101v^15 - 450v^13 - 6684v^11 + 45072v^9 + 110720v^7 - 2092032v^5 - 15089664v^3 + 208666624v) / 11534336
\(\beta_{14}\)	\(=\)	\( ( - 133 \nu^{15} - 1086 \nu^{13} + 6172 \nu^{11} + 40176 \nu^{9} - 312448 \nu^{7} + \cdots + 171966464 \nu ) / 11534336 \) (-133v^15 - 1086v^13 + 6172v^11 + 40176v^9 - 312448v^7 + 216064v^5 + 33374208v^3 + 171966464v) / 11534336
\(\beta_{15}\)	\(=\)	\( ( - 79 \nu^{15} - 122 \nu^{13} + 1492 \nu^{11} - 17968 \nu^{9} + 61056 \nu^{7} + \cdots - 139460608 \nu ) / 5767168 \) (-79v^15 - 122v^13 + 1492v^11 - 17968v^9 + 61056v^7 + 312320v^5 - 4014080v^3 - 139460608v) / 5767168

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{14} - \beta_{10} - \beta_{8} + \beta_{3} ) / 64 \) (-b15 + b14 - b10 - b8 + b3) / 64
\(\nu^{2}\)	\(=\)	\( ( \beta_{9} + \beta_{7} - \beta_{4} + 2\beta_{2} + \beta _1 - 13 ) / 16 \) (b9 + b7 - b4 + 2*b2 + b1 - 13) / 16
\(\nu^{3}\)	\(=\)	\( ( -\beta_{15} - 4\beta_{13} - 2\beta_{11} + \beta_{10} + 6\beta_{8} + 2\beta_{6} - 2\beta_{5} - \beta_{3} ) / 16 \) (-b15 - 4b13 - 2b11 + b10 + 6b8 + 2b6 - 2*b5 - b3) / 16
\(\nu^{4}\)	\(=\)	\( ( -4\beta_{12} + \beta_{9} - \beta_{7} - 4\beta_{6} - 4\beta_{5} - 11\beta_{4} + 2\beta_{2} + 17\beta _1 + 121 ) / 8 \) (-4b12 + b9 - b7 - 4b6 - 4b5 - 11b4 + 2b2 + 17b1 + 121) / 8
\(\nu^{5}\)	\(=\)	\( ( - 19 \beta_{15} - 3 \beta_{14} + 8 \beta_{13} - 4 \beta_{11} + \beta_{10} - 113 \beta_{8} + \cdots + 15 \beta_{3} ) / 16 \) (-19b15 - 3b14 + 8b13 - 4b11 + b10 - 113b8 + 20b6 - 20b5 + 15b3) / 16
\(\nu^{6}\)	\(=\)	\( ( 2\beta_{12} + 2\beta_{9} - \beta_{7} + 10\beta_{6} + 10\beta_{5} - 9\beta_{4} + 60\beta_{2} + 74\beta _1 - 137 ) / 2 \) (2b12 + 2b9 - b7 + 10b6 + 10b5 - 9b4 + 60b2 + 74*b1 - 137) / 2
\(\nu^{7}\)	\(=\)	\( ( 45 \beta_{15} - 17 \beta_{14} - 16 \beta_{13} - 16 \beta_{11} + 13 \beta_{10} + 129 \beta_{8} + \cdots + 195 \beta_{3} ) / 8 \) (45b15 - 17b14 - 16b13 - 16b11 + 13b10 + 129b8 + 96b6 - 96b5 + 195*b3) / 8
\(\nu^{8}\)	\(=\)	\( ( - 56 \beta_{12} - 27 \beta_{9} + 17 \beta_{7} + 88 \beta_{6} + 88 \beta_{5} - 89 \beta_{4} - 294 \beta_{2} + \cdots - 309 ) / 2 \) (-56b12 - 27b9 + 17b7 + 88b6 + 88b5 - 89b4 - 294b2 + 229b1 - 309) / 2
\(\nu^{9}\)	\(=\)	\( ( - 62 \beta_{15} + 37 \beta_{14} + 172 \beta_{13} - 74 \beta_{11} + 108 \beta_{10} + \cdots + 180 \beta_{3} ) / 2 \) (-62b15 + 37b14 + 172b13 - 74b11 + 108b10 + 25b8 - 214b6 + 214b5 + 180*b3) / 2
\(\nu^{10}\)	\(=\)	\( - 4 \beta_{12} - 53 \beta_{9} + 25 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 603 \beta_{4} + 310 \beta_{2} + \cdots + 9895 \) -4b12 - 53b9 + 25b7 - 4b6 - 4b5 + 603b4 + 310b2 + 4187b1 + 9895
\(\nu^{11}\)	\(=\)	\( ( - 527 \beta_{15} + 773 \beta_{14} - 968 \beta_{13} + 1748 \beta_{11} + 45 \beta_{10} + \cdots + 1891 \beta_{3} ) / 2 \) (-527b15 + 773b14 - 968b13 + 1748b11 + 45b10 + 1023b8 + 188b6 - 188b5 + 1891*b3) / 2
\(\nu^{12}\)	\(=\)	\( - 1992 \beta_{12} + 2380 \beta_{9} + 88 \beta_{7} + 1880 \beta_{6} + 1880 \beta_{5} + 3088 \beta_{4} + \cdots + 17952 \) -1992b12 + 2380b9 + 88b7 + 1880b6 + 1880b5 + 3088b4 - 6280b2 - 7252b1 + 17952
\(\nu^{13}\)	\(=\)	\( - 4849 \beta_{15} - 1727 \beta_{14} - 5760 \beta_{13} - 3992 \beta_{11} - 745 \beta_{10} + \cdots + 3673 \beta_{3} \) -4849b15 - 1727b14 - 5760b13 - 3992b11 - 745b10 + 7879b8 - 8568b6 + 8568b5 + 3673*b3
\(\nu^{14}\)	\(=\)	\( - 960 \beta_{12} + 7892 \beta_{9} - 3372 \beta_{7} - 20224 \beta_{6} - 20224 \beta_{5} - 404 \beta_{4} + \cdots - 886404 \) -960b12 + 7892b9 - 3372b7 - 20224b6 - 20224b5 - 404b4 - 47256b2 + 218964b1 - 886404
\(\nu^{15}\)	\(=\)	\( - 33028 \beta_{15} - 24208 \beta_{14} - 6672 \beta_{13} + 34904 \beta_{11} + 15204 \beta_{10} + \cdots - 10148 \beta_{3} \) -33028b15 - 24208b14 - 6672b13 + 34904b11 + 15204b10 - 12280b8 + 47208b6 - 47208b5 - 10148*b3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 703.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 141 T^{6} + \cdots + 69300)^{2} \) (T^8 + 141T^6 + 5895T^4 + 63607*T^2 + 69300)^2
$5$	\( (T^{4} - T^{3} + \cdots + 12250)^{4} \) (T^4 - T^3 - 295T^2 + 333T + 12250)^4
$7$	\( (T^{8} - 1496 T^{6} + \cdots + 2249555968)^{2} \) (T^8 - 1496T^6 + 668864T^4 - 83538432*T^2 + 2249555968)^2
$11$	\( T^{16} + \cdots + 98\!\cdots\!41 \) T^16 - 632T^14 - 215204T^12 + 1800843000T^10 - 1590036874778T^8 + 3190303225923000T^6 - 675402340383866084T^4 - 3513867742127090295992*T^2 + 9849732675807611094711841
$13$	\( (T^{8} + 7384 T^{6} + \cdots + 192669474816)^{2} \) (T^8 + 7384T^6 + 16227136T^4 + 10535019008*T^2 + 192669474816)^2
$17$	\( (T^{8} + \cdots + 19002976763904)^{2} \) (T^8 + 14664T^6 + 63108288T^4 + 76738901504*T^2 + 19002976763904)^2
$19$	\( (T^{8} + \cdots + 550422481100800)^{2} \) (T^8 - 28040T^6 + 259573568T^4 - 860543866368*T^2 + 550422481100800)^2
$23$	\( (T^{8} + \cdots + 86\!\cdots\!00)^{2} \) (T^8 + 47069T^6 + 708614695T^4 + 4254382018183*T^2 + 8648373903592500)^2
$29$	\( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) (T^8 + 177744T^6 + 10117480448T^4 + 182429620895744*T^2 + 58932964255334400)^2
$31$	\( (T^{8} + \cdots + 171617000277492)^{2} \) (T^8 + 33125T^6 + 244820055T^4 + 473645172367*T^2 + 171617000277492)^2
$37$	\( (T^{4} - 5 T^{3} + \cdots - 29264670)^{4} \) (T^4 - 5T^3 - 29607T^2 + 2270537*T - 29264670)^4
$41$	\( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) (T^8 + 383240T^6 + 47273291456T^4 + 2061418050143744*T^2 + 22075399279580774400)^2
$43$	\( (T^{8} + \cdots + 66\!\cdots\!08)^{2} \) (T^8 - 177936T^6 + 9038700032T^4 - 152463379447808*T^2 + 660655015980433408)^2
$47$	\( (T^{8} + \cdots + 28\!\cdots\!00)^{2} \) (T^8 + 419072T^6 + 46548181152T^4 + 1971991569545728*T^2 + 28662079431048940800)^2
$53$	\( (T^{4} + 336 T^{3} + \cdots + 45417040)^{4} \) (T^4 + 336T^3 - 68536T^2 - 19046144*T + 45417040)^4
$59$	\( (T^{8} + \cdots + 95\!\cdots\!00)^{2} \) (T^8 + 880189T^6 + 262684963655T^4 + 30022424168816743*T^2 + 956317918004204332500)^2
$61$	\( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) (T^8 + 745840T^6 + 184944047616T^4 + 15995475483508736*T^2 + 182369385760201113600)^2
$67$	\( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) (T^8 + 1730469T^6 + 1016268506615T^4 + 227415738188260303*T^2 + 15090615901295639327700)^2
$71$	\( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) (T^8 + 762149T^6 + 87594176439T^4 + 3064923794257807*T^2 + 30101205363464812500)^2
$73$	\( (T^{8} + \cdots + 38\!\cdots\!96)^{2} \) (T^8 + 1001736T^6 + 375179607744T^4 + 62274241009468928*T^2 + 3865785935284366442496)^2
$79$	\( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) (T^8 - 1692800T^6 + 460716720128T^4 - 22720961386119168*T^2 + 155512676150149120000)^2
$83$	\( (T^{8} + \cdots + 88\!\cdots\!28)^{2} \) (T^8 - 3153584T^6 + 2947498508288T^4 - 815944256165249024*T^2 + 8825233202433100873728)^2
$89$	\( (T^{4} - 63 T^{3} + \cdots - 26023111450)^{4} \) (T^4 - 63T^3 - 1503131T^2 + 438306187*T - 26023111450)^4
$97$	\( (T^{4} + 1001 T^{3} + \cdots + 171117066190)^{4} \) (T^4 + 1001T^3 - 627235T^2 - 456830013*T + 171117066190)^4
show more
show less

\(n\)	\(133\)	\(321\)	\(639\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)