LMFDB - Newform orbit 160.4.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(49,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("160.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.f (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:	$9.44030560092$
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} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 $ x^16 - 2x^14 + 44x^12 + 400x^10 - 3200x^8 + 25600x^6 + 180224x^4 - 524288*x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 40)
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_{8} q^{3} + \beta_{6} q^{5} - \beta_{9} q^{7} + ( - \beta_1 + 6) q^{9}+O(q^{10}) $ q + b8 * q^3 + b6 * q^5 - b9 * q^7 + (-b1 + 6) * q^9	$ q + \beta_{8} q^{3} + \beta_{6} q^{5} - \beta_{9} q^{7} + ( - \beta_1 + 6) q^{9} + \beta_{3} q^{11} + ( - \beta_{11} + \beta_{8}) q^{13} + ( - \beta_{13} + 4) q^{15} + ( - \beta_{15} - \beta_{13} + \beta_{9}) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{19} + (\beta_{4} + \beta_{3}) q^{21} + ( - 2 \beta_{15} - \beta_{14} + \cdots - 1) q^{23}+ \cdots + (59 \beta_{7} + 59 \beta_{6} + \cdots - 4 \beta_{3}) q^{99}+O(q^{100}) $ q + b8 * q^3 + b6 * q^5 - b9 * q^7 + (-b1 + 6) * q^9 + b3 * q^11 + (-b11 + b8) * q^13 + (-b13 + 4) * q^15 + (-b15 - b13 + b9) * q^17 + (-b7 - b6 + b5) * q^19 + (b4 + b3) * q^21 + (-2b15 - b14 - b13 - b10 + b9 - 1) q^23 + (-b15 - 2b14 - b13 - b10 - b9 - b2 + b1 - 2) q^25 + (b12 - 2b11 + 3b8 - b7 + b6) * q^27 + (-3b7 - 3b6 + 2b5 - b4 + b3) q^29 + (-b14 - b13 - 2b2 + 7) q^31 + (b15 + 2b14 - b13 - 5b9 + 2) * q^33 + (-b12 - 2b11 + 4b7 + 2b6 - b5 + 2b4) * q^35 + (2b12 + 10b8 + b7 - b6) * q^37 + (2b14 + 2b13 + 2b2 - 8b1 + 42) * q^39 + (-2b14 - 2b13 + 2b2 + b1 + 15) q^41 + (-b12 + 2b11 - 2b8 - 7b7 + 7b6) * q^43 + (-2b12 + b11 + 5b8 + 3b7 + 6b6 - 2b5 - b4 + 5b3) * q^45 + (b14 - b13 + 4b10 + 3b9 + 1) * q^47 + (4b14 + 4b13 - 3b1 - 14) q^49 + (13b7 + 13b6 - b5 - 4b4 + b3) q^51 + (-2b12 + b11 - 11b8) * q^53 + (2b15 - b14 + 2b13 - 3b10 - 3b9 + 2b2 + 8b1 - 21) * q^55 + (3b15 - 2b14 + 5b13 + 6b10 - 7b9 - 2) q^57 + (-17b7 - 17b6 + b5 - 4b3) q^59 + (-b7 - b6 - 2b5 + 4b4 - 14b3) q^61 + (6b15 + b14 + 5b13 - b10 - 3b9 + 1) q^63 + (2b15 + 4b14 - 3b10 + 12b9 - 3b2 + 3b1 - 33) * q^65 + (7b12 + 10b11 - 4b8 + 25b7 - 25b6) q^67 + (-2b5 - 3b4 - 5b3) q^69 + (b14 + b13 - 4b2 + 8b1 - 127) * q^71 + (-b15 - 6b14 + 5b13 - 4b10 + 29b9 - 6) * q^73 + (-6b12 + 8b11 - 15b8 - 31b7 + 5b6 - b5 + 2b4 - 5b3) q^75 + (10b12 + b11 - 47b8 + 11b7 - 11b6) * q^77 + (b14 + b13 + 2b2 + 24b1 - 175) * q^79 + (6b14 + 6b13 + 6b2 + 3b1 - 44) * q^81 + (-5b12 - 6b11 - 12b8 + 29b7 - 29b6) q^83 + (-8b12 - b11 - 55b8 - 3b7 + 3b6 + 2b5 - 4b4 - 10b3) q^85 + (2b15 - 2b14 + 4b13 + 13b10 + 6b9 - 2) q^87 + (-6b14 - 6b13 - 2b2 + 6b1 - 10) * q^89 + (-35b7 - 35b6 - b5 - 4b4 + 3b3) * q^91 + (-6b12 + 10b11 + 24b8 - 8b7 + 8b6) q^93 + (-6b15 - 2b14 + b13 - 11b10 + 9b9 + 4b2 - 24b1 + 50) * q^95 + (5b15 + 8b14 - 3b13 + 18b10 + 19b9 + 8) q^97 + (59b7 + 59b6 - 7b5 + 4b4 - 4b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 104 q^{9}+O(q^{10}) $ 16 * q + 104 * q^9	$ 16 q + 104 q^{9} + 56 q^{15} - 24 q^{25} + 112 q^{31} + 736 q^{39} + 232 q^{41} - 200 q^{49} - 392 q^{55} - 600 q^{65} - 2096 q^{71} - 2992 q^{79} - 728 q^{81} - 208 q^{89} + 1064 q^{95}+O(q^{100}) $ 16 * q + 104 * q^9 + 56 * q^15 - 24 * q^25 + 112 * q^31 + 736 * q^39 + 232 * q^41 - 200 * q^49 - 392 * q^55 - 600 * q^65 - 2096 * q^71 - 2992 * q^79 - 728 * q^81 - 208 * q^89 + 1064 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 $ x^16 - 2*x^14 + 44*x^12 + 400*x^10 - 3200*x^8 + 25600*x^6 + 180224*x^4 - 524288*x^2 + 16777216 :

$\beta_{1}$	$=$	$ ( 3\nu^{14} - 22\nu^{12} - 92\nu^{10} + 1008\nu^{8} - 10880\nu^{6} + 58368\nu^{4} + 753664\nu^{2} - 4325376 ) / 131072 $ (3v^14 - 22v^12 - 92v^10 + 1008v^8 - 10880v^6 + 58368v^4 + 753664*v^2 - 4325376) / 131072
$\beta_{2}$	$=$	$ ( -\nu^{14} + 2\nu^{12} - 44\nu^{10} - 400\nu^{8} + 3200\nu^{6} - 25600\nu^{4} + 81920\nu^{2} + 393216 ) / 32768 $ (-v^14 + 2v^12 - 44v^10 - 400v^8 + 3200v^6 - 25600v^4 + 81920v^2 + 393216) / 32768
$\beta_{3}$	$=$	$ ( -3\nu^{14} - 58\nu^{12} - 4\nu^{10} + 80\nu^{8} - 24192\nu^{6} + 46080\nu^{4} - 802816\nu^{2} - 16515072 ) / 131072 $ (-3v^14 - 58v^12 - 4v^10 + 80v^8 - 24192v^6 + 46080v^4 - 802816*v^2 - 16515072) / 131072
$\beta_{4}$	$=$	$ ( - 3 \nu^{14} - 586 \nu^{12} + 1308 \nu^{10} + 13200 \nu^{8} - 281472 \nu^{6} + 1264640 \nu^{4} + \cdots - 164102144 ) / 917504 $ (-3v^14 - 586v^12 + 1308v^10 + 13200v^8 - 281472v^6 + 1264640v^4 + 4653056*v^2 - 164102144) / 917504
$\beta_{5}$	$=$	$ ( 29 \nu^{14} + 662 \nu^{12} - 1444 \nu^{10} + 13072 \nu^{8} + 122496 \nu^{6} - 402432 \nu^{4} + \cdots + 134217728 ) / 917504 $ (29v^14 + 662v^12 - 1444v^10 + 13072v^8 + 122496v^6 - 402432v^4 + 12746752*v^2 + 134217728) / 917504
$\beta_{6}$	$=$	$ ( - 39 \nu^{15} - 80 \nu^{14} - 114 \nu^{13} - 992 \nu^{12} + 204 \nu^{11} + 832 \nu^{10} + \cdots - 270532608 ) / 14680064 $ (-39v^15 - 80v^14 - 114v^13 - 992v^12 + 204v^11 + 832v^10 - 10736v^9 + 11520v^8 - 48256v^7 - 180224v^6 + 427008v^5 + 1802240v^4 - 8208384v^3 - 16908288v^2 - 45088768*v - 270532608) / 14680064
$\beta_{7}$	$=$	$ ( 39 \nu^{15} - 80 \nu^{14} + 114 \nu^{13} - 992 \nu^{12} - 204 \nu^{11} + 832 \nu^{10} + \cdots - 270532608 ) / 14680064 $ (39v^15 - 80v^14 + 114v^13 - 992v^12 - 204v^11 + 832v^10 + 10736v^9 + 11520v^8 + 48256v^7 - 180224v^6 - 427008v^5 + 1802240v^4 + 8208384v^3 - 16908288v^2 + 45088768*v - 270532608) / 14680064
$\beta_{8}$	$=$	$ ( - 69 \nu^{15} - 150 \nu^{13} + 1636 \nu^{11} - 23888 \nu^{9} + 40064 \nu^{7} + \cdots - 34603008 \nu ) / 14680064 $ (-69v^15 - 150v^13 + 1636v^11 - 23888v^9 + 40064v^7 + 171008v^5 - 18104320v^3 - 34603008v) / 14680064
$\beta_{9}$	$=$	$ ( 11 \nu^{15} - 182 \nu^{13} - 220 \nu^{11} + 7600 \nu^{9} - 95104 \nu^{7} + 89088 \nu^{5} + \cdots - 50331648 \nu ) / 2097152 $ (11v^15 - 182v^13 - 220v^11 + 7600v^9 - 95104v^7 + 89088v^5 + 3129344v^3 - 50331648v) / 2097152
$\beta_{10}$	$=$	$ ( \nu^{15} - 2\nu^{13} + 44\nu^{11} + 400\nu^{9} - 3200\nu^{7} + 25600\nu^{5} + 180224\nu^{3} + 1572864\nu ) / 131072 $ (v^15 - 2v^13 + 44v^11 + 400v^9 - 3200v^7 + 25600v^5 + 180224v^3 + 1572864*v) / 131072
$\beta_{11}$	$=$	$ ( 17 \nu^{15} - 66 \nu^{13} - 212 \nu^{11} - 752 \nu^{9} - 30336 \nu^{7} + 160768 \nu^{5} + \cdots - 3145728 \nu ) / 2097152 $ (17v^15 - 66v^13 - 212v^11 - 752v^9 - 30336v^7 + 160768v^5 + 5128192v^3 - 3145728v) / 2097152
$\beta_{12}$	$=$	$ ( - 121 \nu^{15} + 146 \nu^{13} - 6156 \nu^{11} - 42384 \nu^{9} + 221824 \nu^{7} + \cdots + 238026752 \nu ) / 14680064 $ (-121v^15 + 146v^13 - 6156v^11 - 42384v^9 + 221824v^7 - 2184192v^5 - 18497536v^3 + 238026752v) / 14680064
$\beta_{13}$	$=$	$ ( - 15 \nu^{15} - 176 \nu^{14} + 510 \nu^{13} + 608 \nu^{12} + 428 \nu^{11} + 4032 \nu^{10} + \cdots + 171966464 ) / 2097152 $ (-15v^15 - 176v^14 + 510v^13 + 608v^12 + 428v^11 + 4032v^10 - 13552v^9 - 83712v^8 + 182656v^7 + 419840v^6 - 740352v^5 + 114688v^4 - 1818624v^3 - 50855936v^2 + 130023424*v + 171966464) / 2097152
$\beta_{14}$	$=$	$ ( 15 \nu^{15} - 176 \nu^{14} - 510 \nu^{13} + 608 \nu^{12} - 428 \nu^{11} + 4032 \nu^{10} + \cdots + 169869312 ) / 2097152 $ (15v^15 - 176v^14 - 510v^13 + 608v^12 - 428v^11 + 4032v^10 + 13552v^9 - 83712v^8 - 182656v^7 + 419840v^6 + 740352v^5 + 114688v^4 + 1818624v^3 - 50855936v^2 - 130023424*v + 169869312) / 2097152
$\beta_{15}$	$=$	$ ( - \nu^{15} + 11 \nu^{14} + 14 \nu^{13} - 38 \nu^{12} - 4 \nu^{11} - 252 \nu^{10} - 1024 \nu^{9} + \cdots - 10747904 ) / 131072 $ (-v^15 + 11v^14 + 14v^13 - 38v^12 - 4v^11 - 252v^10 - 1024v^9 + 5232v^8 + 4672v^7 - 26240v^6 - 34304v^5 - 7168v^4 - 520192v^3 + 3178496v^2 + 2883584v - 10747904) / 131072

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

$\nu$	$=$	$ ( \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} ) / 32 $ (b12 + b10 + b8 + b7 - b6) / 32
$\nu^{2}$	$=$	$ ( -\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 4 ) / 16 $ (-b7 - b6 + b4 - b3 + b2 + 4) / 16
$\nu^{3}$	$=$	$ ( -2\beta_{15} - 2\beta_{14} + 2\beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - 2 ) / 8 $ (-2b15 - 2b14 + 2b11 - b10 + 2b9 + 2b8 - 2b7 + 2*b6 - 2) / 8
$\nu^{4}$	$=$	$ ( 2 \beta_{14} + 2 \beta_{13} + 9 \beta_{7} + 9 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 78 ) / 8 $ (2b14 + 2b13 + 9b7 + 9b6 + 2b5 + b4 - 3b3 - b2 + 12*b1 - 78) / 8
$\nu^{5}$	$=$	$ ( - 4 \beta_{15} + 4 \beta_{14} - 8 \beta_{13} - 5 \beta_{12} + 4 \beta_{11} + 5 \beta_{10} - 44 \beta_{9} + \cdots + 4 ) / 8 $ (-4b15 + 4b14 - 8b13 - 5b12 + 4b11 + 5b10 - 44b9 - 65b8 - 57b7 + 57b6 + 4) / 8
$\nu^{6}$	$=$	$ ( - \beta_{14} - \beta_{13} + 34 \beta_{7} + 34 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - 16 \beta_{3} + \cdots - 363 ) / 2 $ (-b14 - b13 + 34b7 + 34b6 - 5b5 - 3b4 - 16b3 + 2b2 + 2*b1 - 363) / 2
$\nu^{7}$	$=$	$ ( - 40 \beta_{15} - 40 \beta_{13} - 7 \beta_{12} + 8 \beta_{11} - 55 \beta_{10} - 136 \beta_{9} + \cdots - 193 \beta_{6} ) / 4 $ (-40b15 - 40b13 - 7b12 + 8b11 - 55b10 - 136b9 + 129b8 + 193b7 - 193*b6) / 4
$\nu^{8}$	$=$	$ ( - 24 \beta_{14} - 24 \beta_{13} + 291 \beta_{7} + 291 \beta_{6} + 32 \beta_{5} - 19 \beta_{4} + \cdots + 3132 ) / 2 $ (-24b14 - 24b13 + 291b7 + 291b6 + 32b5 - 19b4 + 3b3 - 35b2 - 128*b1 + 3132) / 2
$\nu^{9}$	$=$	$ - 58 \beta_{15} - 18 \beta_{14} - 40 \beta_{13} + 28 \beta_{12} - 214 \beta_{11} + 31 \beta_{10} + \cdots - 18 $ -58b15 - 18b14 - 40b13 + 28b12 - 214b11 + 31b10 - 22b9 - 154b8 + 98b7 - 98b6 - 18
$\nu^{10}$	$=$	$ - 38 \beta_{14} - 38 \beta_{13} - 339 \beta_{7} - 339 \beta_{6} - 150 \beta_{5} + 261 \beta_{4} + \cdots - 118 $ -38b14 - 38b13 - 339b7 - 339b6 - 150b5 + 261b4 - 335b3 - 85b2 - 644*b1 - 118
$\nu^{11}$	$=$	$ - 276 \beta_{15} - 524 \beta_{14} + 248 \beta_{13} - 801 \beta_{12} + 84 \beta_{11} + 369 \beta_{10} + \cdots - 524 $ -276b15 - 524b14 + 248b13 - 801b12 + 84b11 + 369b10 + 1380b9 + 5235b8 + 2011b7 - 2011b6 - 524
$\nu^{12}$	$=$	$ 428 \beta_{14} + 428 \beta_{13} - 3560 \beta_{7} - 3560 \beta_{6} + 1820 \beta_{5} + 340 \beta_{4} + \cdots - 172380 $ 428b14 + 428b13 - 3560b7 - 3560b6 + 1820b5 + 340b4 + 1840b3 - 776b2 - 216*b1 - 172380
$\nu^{13}$	$=$	$ 2608 \beta_{15} - 448 \beta_{14} + 3056 \beta_{13} - 7718 \beta_{12} - 4208 \beta_{11} - 1702 \beta_{10} + \cdots - 448 $ 2608b15 - 448b14 + 3056b13 - 7718b12 - 4208b11 - 1702b10 + 1520b9 - 39830b8 - 31382b7 + 31382b6 - 448
$\nu^{14}$	$=$	$ - 672 \beta_{14} - 672 \beta_{13} - 29924 \beta_{7} - 29924 \beta_{6} - 10560 \beta_{5} - 9884 \beta_{4} + \cdots - 883472 $ -672b14 - 672b13 - 29924b7 - 29924b6 - 10560b5 - 9884b4 - 3300b3 - 12060b2 + 18304*b1 - 883472
$\nu^{15}$	$=$	$ 66416 \beta_{15} + 61616 \beta_{14} + 4800 \beta_{13} - 30144 \beta_{12} + 22032 \beta_{11} + \cdots + 61616 $ 66416b15 + 61616b14 + 4800b13 - 30144b12 + 22032b11 + 12408b10 - 61936b9 - 31408b8 + 142256b7 - 142256b6 + 61616

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$-1$

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} $

49.1

−2.15123	−	1.83636i
−2.15123	+	1.83636i
2.07496	−	1.92212i
2.07496	+	1.92212i
0.407618	+	2.79890i
0.407618	−	2.79890i
2.72041	−	0.774197i
2.72041	+	0.774197i
−2.72041	+	0.774197i
−2.72041	−	0.774197i
−0.407618	−	2.79890i
−0.407618	+	2.79890i
−2.07496	+	1.92212i
−2.07496	−	1.92212i
2.15123	+	1.83636i
2.15123	−	1.83636i

−8.63004

2.63740

−

10.8648i

−

4.74133i

47.4777

49.2

−8.63004

2.63740

10.8648i

4.74133i

47.4777

49.3

−6.67494

−11.1461

−

0.874848i

29.8250i

17.5548

49.4

−6.67494

−11.1461

0.874848i

−

29.8250i

17.5548

49.5

−3.86846

9.66751

−

5.61599i

9.16068i

−12.0350

49.6

−3.86846

9.66751

5.61599i

−

9.16068i

−12.0350

49.7

−0.0507699

4.72745

−

10.1317i

−

20.6415i

−26.9974

49.8

−0.0507699

4.72745

10.1317i

20.6415i

−26.9974

49.9

0.0507699

−4.72745

−

10.1317i

20.6415i

−26.9974

49.10

0.0507699

−4.72745

10.1317i

−

20.6415i

−26.9974

49.11

3.86846

−9.66751

−

5.61599i

−

9.16068i

−12.0350

49.12

3.86846

−9.66751

5.61599i

9.16068i

−12.0350

49.13

6.67494

11.1461

−

0.874848i

−

29.8250i

17.5548

49.14

6.67494

11.1461

0.874848i

29.8250i

17.5548

49.15

8.63004

−2.63740

−

10.8648i

4.74133i

47.4777

49.16

8.63004

−2.63740

10.8648i

−

4.74133i

47.4777

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.4.f.a		16
3.b	odd	2	1		1440.4.d.d		16
4.b	odd	2	1		40.4.f.a	✓	16
5.b	even	2	1	inner	160.4.f.a		16
5.c	odd	4	2		800.4.d.e		16
8.b	even	2	1	inner	160.4.f.a		16
8.d	odd	2	1		40.4.f.a	✓	16
12.b	even	2	1		360.4.d.d		16
15.d	odd	2	1		1440.4.d.d		16
20.d	odd	2	1		40.4.f.a	✓	16
20.e	even	4	2		200.4.d.e		16
24.f	even	2	1		360.4.d.d		16
24.h	odd	2	1		1440.4.d.d		16
40.e	odd	2	1		40.4.f.a	✓	16
40.f	even	2	1	inner	160.4.f.a		16
40.i	odd	4	2		800.4.d.e		16
40.k	even	4	2		200.4.d.e		16
60.h	even	2	1		360.4.d.d		16
120.i	odd	2	1		1440.4.d.d		16
120.m	even	2	1		360.4.d.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.4.f.a	✓	16	4.b	odd	2	1
40.4.f.a	✓	16	8.d	odd	2	1
40.4.f.a	✓	16	20.d	odd	2	1
40.4.f.a	✓	16	40.e	odd	2	1
160.4.f.a		16	1.a	even	1	1	trivial
160.4.f.a		16	5.b	even	2	1	inner
160.4.f.a		16	8.b	even	2	1	inner
160.4.f.a		16	40.f	even	2	1	inner
200.4.d.e		16	20.e	even	4	2
200.4.d.e		16	40.k	even	4	2
360.4.d.d		16	12.b	even	2	1
360.4.d.d		16	24.f	even	2	1
360.4.d.d		16	60.h	even	2	1
360.4.d.d		16	120.m	even	2	1
800.4.d.e		16	5.c	odd	4	2
800.4.d.e		16	40.i	odd	4	2
1440.4.d.d		16	3.b	odd	2	1
1440.4.d.d		16	15.d	odd	2	1
1440.4.d.d		16	24.h	odd	2	1
1440.4.d.d		16	120.i	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(160, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 134 T^{6} + \cdots + 128)^{2} $ (T^8 - 134T^6 + 5100T^4 - 49672*T^2 + 128)^2
$5$	$ T^{16} + \cdots + 59\!\cdots\!25 $ T^16 + 12T^14 - 13164T^12 - 961900T^10 + 191573750T^8 - 15029687500T^6 - 3213867187500T^4 + 45776367187500*T^2 + 59604644775390625
$7$	$ (T^{8} + 1422 T^{6} + \cdots + 714987936)^{2} $ (T^8 + 1422T^6 + 520868T^4 + 42807208*T^2 + 714987936)^2
$11$	$ (T^{8} + \cdots + 2238493420800)^{2} $ (T^8 + 5120T^6 + 9493152T^4 + 7611914752*T^2 + 2238493420800)^2
$13$	$ (T^{8} + \cdots + 9784472371200)^{2} $ (T^8 - 9060T^6 + 28218368T^4 - 33454882816*T^2 + 9784472371200)^2
$17$	$ (T^{8} + \cdots + 12119907631104)^{2} $ (T^8 + 20808T^6 + 143506112T^4 + 330111030784*T^2 + 12119907631104)^2
$19$	$ (T^{8} + 28704 T^{6} + \cdots + 38178047232)^{2} $ (T^8 + 28704T^6 + 217354016T^4 + 168683868160*T^2 + 38178047232)^2
$23$	$ (T^{8} + \cdots + 23\!\cdots\!96)^{2} $ (T^8 + 46862T^6 + 692502564T^4 + 3478125501352*T^2 + 2374625191944096)^2
$29$	$ (T^{8} + \cdots + 58\!\cdots\!00)^{2} $ (T^8 + 106992T^6 + 3177830400T^4 + 31539931316224*T^2 + 58518211343155200)^2
$31$	$ (T^{4} - 28 T^{3} + \cdots + 400132096)^{4} $ (T^4 - 28T^3 - 50064T^2 + 1897920*T + 400132096)^4
$37$	$ (T^{8} + \cdots + 930111561566208)^{2} $ (T^8 - 77596T^6 + 1619216672T^4 - 4775065212160*T^2 + 930111561566208)^2
$41$	$ (T^{4} - 58 T^{3} + \cdots + 2763694080)^{4} $ (T^4 - 58T^3 - 150976T^2 + 14374016*T + 2763694080)^4
$43$	$ (T^{8} - 111062 T^{6} + \cdots + 542115358848)^{2} $ (T^8 - 111062T^6 + 3282346540T^4 - 14440319717704*T^2 + 542115358848)^2
$47$	$ (T^{8} + \cdots + 21\!\cdots\!64)^{2} $ (T^8 + 247326T^6 + 15278864228T^4 + 35827752956392*T^2 + 21277806044904864)^2
$53$	$ (T^{8} + \cdots + 27\!\cdots\!48)^{2} $ (T^8 - 95140T^6 + 2172259712T^4 - 14836063186944*T^2 + 27747256973262848)^2
$59$	$ (T^{8} + \cdots + 43\!\cdots\!48)^{2} $ (T^8 + 422240T^6 + 27600487968T^4 + 619747875576832*T^2 + 4364850634106197248)^2
$61$	$ (T^{8} + \cdots + 18\!\cdots\!00)^{2} $ (T^8 + 1274036T^6 + 546416266864T^4 + 83188858348299712*T^2 + 1879163420950839628800)^2
$67$	$ (T^{8} + \cdots + 13\!\cdots\!68)^{2} $ (T^8 - 1919030T^6 + 1168519372972T^4 - 228352813341226696*T^2 + 130877537747150371968)^2
$71$	$ (T^{4} + 524 T^{3} + \cdots - 30115911168)^{4} $ (T^4 + 524T^3 - 268592T^2 - 202218176*T - 30115911168)^4
$73$	$ (T^{8} + \cdots + 68\!\cdots\!04)^{2} $ (T^8 + 2125704T^6 + 1649904703680T^4 + 554636387446621696*T^2 + 68153875753273879855104)^2
$79$	$ (T^{4} + 748 T^{3} + \cdots + 45513395200)^{4} $ (T^4 + 748T^3 - 830928T^2 - 462088896*T + 45513395200)^4
$83$	$ (T^{8} + \cdots + 39\!\cdots\!48)^{2} $ (T^8 - 1687142T^6 + 686343675500T^4 - 68638149138819464*T^2 + 394987658083584107648)^2
$89$	$ (T^{4} + 52 T^{3} + \cdots - 4266435600)^{4} $ (T^4 + 52T^3 - 483008T^2 - 96195664*T - 4266435600)^4
$97$	$ (T^{8} + \cdots + 22\!\cdots\!44)^{2} $ (T^8 + 4783944T^6 + 5812981166784T^4 + 1095500072805045760*T^2 + 22160595202119257554944)^2
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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{3} + \beta_{6} q^{5} - \beta_{9} q^{7} + ( - \beta_1 + 6) q^{9}+O(q^{10}) \) q + b8 * q^3 + b6 * q^5 - b9 * q^7 + (-b1 + 6) * q^9	\( q + \beta_{8} q^{3} + \beta_{6} q^{5} - \beta_{9} q^{7} + ( - \beta_1 + 6) q^{9} + \beta_{3} q^{11} + ( - \beta_{11} + \beta_{8}) q^{13} + ( - \beta_{13} + 4) q^{15} + ( - \beta_{15} - \beta_{13} + \beta_{9}) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{19} + (\beta_{4} + \beta_{3}) q^{21} + ( - 2 \beta_{15} - \beta_{14} + \cdots - 1) q^{23}+ \cdots + (59 \beta_{7} + 59 \beta_{6} + \cdots - 4 \beta_{3}) q^{99}+O(q^{100}) \) q + b8 * q^3 + b6 * q^5 - b9 * q^7 + (-b1 + 6) * q^9 + b3 * q^11 + (-b11 + b8) * q^13 + (-b13 + 4) * q^15 + (-b15 - b13 + b9) * q^17 + (-b7 - b6 + b5) * q^19 + (b4 + b3) * q^21 + (-2b15 - b14 - b13 - b10 + b9 - 1) q^23 + (-b15 - 2b14 - b13 - b10 - b9 - b2 + b1 - 2) q^25 + (b12 - 2b11 + 3b8 - b7 + b6) * q^27 + (-3b7 - 3b6 + 2b5 - b4 + b3) q^29 + (-b14 - b13 - 2b2 + 7) q^31 + (b15 + 2b14 - b13 - 5b9 + 2) * q^33 + (-b12 - 2b11 + 4b7 + 2b6 - b5 + 2b4) * q^35 + (2b12 + 10b8 + b7 - b6) * q^37 + (2b14 + 2b13 + 2b2 - 8b1 + 42) * q^39 + (-2b14 - 2b13 + 2b2 + b1 + 15) q^41 + (-b12 + 2b11 - 2b8 - 7b7 + 7b6) * q^43 + (-2b12 + b11 + 5b8 + 3b7 + 6b6 - 2b5 - b4 + 5b3) * q^45 + (b14 - b13 + 4b10 + 3b9 + 1) * q^47 + (4b14 + 4b13 - 3b1 - 14) q^49 + (13b7 + 13b6 - b5 - 4b4 + b3) q^51 + (-2b12 + b11 - 11b8) * q^53 + (2b15 - b14 + 2b13 - 3b10 - 3b9 + 2b2 + 8b1 - 21) * q^55 + (3b15 - 2b14 + 5b13 + 6b10 - 7b9 - 2) q^57 + (-17b7 - 17b6 + b5 - 4b3) q^59 + (-b7 - b6 - 2b5 + 4b4 - 14b3) q^61 + (6b15 + b14 + 5b13 - b10 - 3b9 + 1) q^63 + (2b15 + 4b14 - 3b10 + 12b9 - 3b2 + 3b1 - 33) * q^65 + (7b12 + 10b11 - 4b8 + 25b7 - 25b6) q^67 + (-2b5 - 3b4 - 5b3) q^69 + (b14 + b13 - 4b2 + 8b1 - 127) * q^71 + (-b15 - 6b14 + 5b13 - 4b10 + 29b9 - 6) * q^73 + (-6b12 + 8b11 - 15b8 - 31b7 + 5b6 - b5 + 2b4 - 5b3) q^75 + (10b12 + b11 - 47b8 + 11b7 - 11b6) * q^77 + (b14 + b13 + 2b2 + 24b1 - 175) * q^79 + (6b14 + 6b13 + 6b2 + 3b1 - 44) * q^81 + (-5b12 - 6b11 - 12b8 + 29b7 - 29b6) q^83 + (-8b12 - b11 - 55b8 - 3b7 + 3b6 + 2b5 - 4b4 - 10b3) q^85 + (2b15 - 2b14 + 4b13 + 13b10 + 6b9 - 2) q^87 + (-6b14 - 6b13 - 2b2 + 6b1 - 10) * q^89 + (-35b7 - 35b6 - b5 - 4b4 + 3b3) * q^91 + (-6b12 + 10b11 + 24b8 - 8b7 + 8b6) q^93 + (-6b15 - 2b14 + b13 - 11b10 + 9b9 + 4b2 - 24b1 + 50) * q^95 + (5b15 + 8b14 - 3b13 + 18b10 + 19b9 + 8) q^97 + (59b7 + 59b6 - 7b5 + 4b4 - 4b3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 104 q^{9}+O(q^{10}) \) 16 * q + 104 * q^9	\( 16 q + 104 q^{9} + 56 q^{15} - 24 q^{25} + 112 q^{31} + 736 q^{39} + 232 q^{41} - 200 q^{49} - 392 q^{55} - 600 q^{65} - 2096 q^{71} - 2992 q^{79} - 728 q^{81} - 208 q^{89} + 1064 q^{95}+O(q^{100}) \) 16 * q + 104 * q^9 + 56 * q^15 - 24 * q^25 + 112 * q^31 + 736 * q^39 + 232 * q^41 - 200 * q^49 - 392 * q^55 - 600 * q^65 - 2096 * q^71 - 2992 * q^79 - 728 * q^81 - 208 * q^89 + 1064 * q^95

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	160.4.f.a

Level	$160$
Weight	$4$
Character orbit	160.f
Analytic conductor	$9.440$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.44030560092\)
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:	\( x^{16} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 \) x^16 - 2x^14 + 44x^12 + 400x^10 - 3200x^8 + 25600x^6 + 180224x^4 - 524288*x^2 + 16777216
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{40}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 3\nu^{14} - 22\nu^{12} - 92\nu^{10} + 1008\nu^{8} - 10880\nu^{6} + 58368\nu^{4} + 753664\nu^{2} - 4325376 ) / 131072 \) (3v^14 - 22v^12 - 92v^10 + 1008v^8 - 10880v^6 + 58368v^4 + 753664*v^2 - 4325376) / 131072
\(\beta_{2}\)	\(=\)	\( ( -\nu^{14} + 2\nu^{12} - 44\nu^{10} - 400\nu^{8} + 3200\nu^{6} - 25600\nu^{4} + 81920\nu^{2} + 393216 ) / 32768 \) (-v^14 + 2v^12 - 44v^10 - 400v^8 + 3200v^6 - 25600v^4 + 81920v^2 + 393216) / 32768
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{14} - 58\nu^{12} - 4\nu^{10} + 80\nu^{8} - 24192\nu^{6} + 46080\nu^{4} - 802816\nu^{2} - 16515072 ) / 131072 \) (-3v^14 - 58v^12 - 4v^10 + 80v^8 - 24192v^6 + 46080v^4 - 802816*v^2 - 16515072) / 131072
\(\beta_{4}\)	\(=\)	\( ( - 3 \nu^{14} - 586 \nu^{12} + 1308 \nu^{10} + 13200 \nu^{8} - 281472 \nu^{6} + 1264640 \nu^{4} + \cdots - 164102144 ) / 917504 \) (-3v^14 - 586v^12 + 1308v^10 + 13200v^8 - 281472v^6 + 1264640v^4 + 4653056*v^2 - 164102144) / 917504
\(\beta_{5}\)	\(=\)	\( ( 29 \nu^{14} + 662 \nu^{12} - 1444 \nu^{10} + 13072 \nu^{8} + 122496 \nu^{6} - 402432 \nu^{4} + \cdots + 134217728 ) / 917504 \) (29v^14 + 662v^12 - 1444v^10 + 13072v^8 + 122496v^6 - 402432v^4 + 12746752*v^2 + 134217728) / 917504
\(\beta_{6}\)	\(=\)	\( ( - 39 \nu^{15} - 80 \nu^{14} - 114 \nu^{13} - 992 \nu^{12} + 204 \nu^{11} + 832 \nu^{10} + \cdots - 270532608 ) / 14680064 \) (-39v^15 - 80v^14 - 114v^13 - 992v^12 + 204v^11 + 832v^10 - 10736v^9 + 11520v^8 - 48256v^7 - 180224v^6 + 427008v^5 + 1802240v^4 - 8208384v^3 - 16908288v^2 - 45088768*v - 270532608) / 14680064
\(\beta_{7}\)	\(=\)	\( ( 39 \nu^{15} - 80 \nu^{14} + 114 \nu^{13} - 992 \nu^{12} - 204 \nu^{11} + 832 \nu^{10} + \cdots - 270532608 ) / 14680064 \) (39v^15 - 80v^14 + 114v^13 - 992v^12 - 204v^11 + 832v^10 + 10736v^9 + 11520v^8 + 48256v^7 - 180224v^6 - 427008v^5 + 1802240v^4 + 8208384v^3 - 16908288v^2 + 45088768*v - 270532608) / 14680064
\(\beta_{8}\)	\(=\)	\( ( - 69 \nu^{15} - 150 \nu^{13} + 1636 \nu^{11} - 23888 \nu^{9} + 40064 \nu^{7} + \cdots - 34603008 \nu ) / 14680064 \) (-69v^15 - 150v^13 + 1636v^11 - 23888v^9 + 40064v^7 + 171008v^5 - 18104320v^3 - 34603008v) / 14680064
\(\beta_{9}\)	\(=\)	\( ( 11 \nu^{15} - 182 \nu^{13} - 220 \nu^{11} + 7600 \nu^{9} - 95104 \nu^{7} + 89088 \nu^{5} + \cdots - 50331648 \nu ) / 2097152 \) (11v^15 - 182v^13 - 220v^11 + 7600v^9 - 95104v^7 + 89088v^5 + 3129344v^3 - 50331648v) / 2097152
\(\beta_{10}\)	\(=\)	\( ( \nu^{15} - 2\nu^{13} + 44\nu^{11} + 400\nu^{9} - 3200\nu^{7} + 25600\nu^{5} + 180224\nu^{3} + 1572864\nu ) / 131072 \) (v^15 - 2v^13 + 44v^11 + 400v^9 - 3200v^7 + 25600v^5 + 180224v^3 + 1572864*v) / 131072
\(\beta_{11}\)	\(=\)	\( ( 17 \nu^{15} - 66 \nu^{13} - 212 \nu^{11} - 752 \nu^{9} - 30336 \nu^{7} + 160768 \nu^{5} + \cdots - 3145728 \nu ) / 2097152 \) (17v^15 - 66v^13 - 212v^11 - 752v^9 - 30336v^7 + 160768v^5 + 5128192v^3 - 3145728v) / 2097152
\(\beta_{12}\)	\(=\)	\( ( - 121 \nu^{15} + 146 \nu^{13} - 6156 \nu^{11} - 42384 \nu^{9} + 221824 \nu^{7} + \cdots + 238026752 \nu ) / 14680064 \) (-121v^15 + 146v^13 - 6156v^11 - 42384v^9 + 221824v^7 - 2184192v^5 - 18497536v^3 + 238026752v) / 14680064
\(\beta_{13}\)	\(=\)	\( ( - 15 \nu^{15} - 176 \nu^{14} + 510 \nu^{13} + 608 \nu^{12} + 428 \nu^{11} + 4032 \nu^{10} + \cdots + 171966464 ) / 2097152 \) (-15v^15 - 176v^14 + 510v^13 + 608v^12 + 428v^11 + 4032v^10 - 13552v^9 - 83712v^8 + 182656v^7 + 419840v^6 - 740352v^5 + 114688v^4 - 1818624v^3 - 50855936v^2 + 130023424*v + 171966464) / 2097152
\(\beta_{14}\)	\(=\)	\( ( 15 \nu^{15} - 176 \nu^{14} - 510 \nu^{13} + 608 \nu^{12} - 428 \nu^{11} + 4032 \nu^{10} + \cdots + 169869312 ) / 2097152 \) (15v^15 - 176v^14 - 510v^13 + 608v^12 - 428v^11 + 4032v^10 + 13552v^9 - 83712v^8 - 182656v^7 + 419840v^6 + 740352v^5 + 114688v^4 + 1818624v^3 - 50855936v^2 - 130023424*v + 169869312) / 2097152
\(\beta_{15}\)	\(=\)	\( ( - \nu^{15} + 11 \nu^{14} + 14 \nu^{13} - 38 \nu^{12} - 4 \nu^{11} - 252 \nu^{10} - 1024 \nu^{9} + \cdots - 10747904 ) / 131072 \) (-v^15 + 11v^14 + 14v^13 - 38v^12 - 4v^11 - 252v^10 - 1024v^9 + 5232v^8 + 4672v^7 - 26240v^6 - 34304v^5 - 7168v^4 - 520192v^3 + 3178496v^2 + 2883584v - 10747904) / 131072

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} ) / 32 \) (b12 + b10 + b8 + b7 - b6) / 32
\(\nu^{2}\)	\(=\)	\( ( -\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 4 ) / 16 \) (-b7 - b6 + b4 - b3 + b2 + 4) / 16
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{15} - 2\beta_{14} + 2\beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - 2 ) / 8 \) (-2b15 - 2b14 + 2b11 - b10 + 2b9 + 2b8 - 2b7 + 2*b6 - 2) / 8
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{14} + 2 \beta_{13} + 9 \beta_{7} + 9 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 78 ) / 8 \) (2b14 + 2b13 + 9b7 + 9b6 + 2b5 + b4 - 3b3 - b2 + 12*b1 - 78) / 8
\(\nu^{5}\)	\(=\)	\( ( - 4 \beta_{15} + 4 \beta_{14} - 8 \beta_{13} - 5 \beta_{12} + 4 \beta_{11} + 5 \beta_{10} - 44 \beta_{9} + \cdots + 4 ) / 8 \) (-4b15 + 4b14 - 8b13 - 5b12 + 4b11 + 5b10 - 44b9 - 65b8 - 57b7 + 57b6 + 4) / 8
\(\nu^{6}\)	\(=\)	\( ( - \beta_{14} - \beta_{13} + 34 \beta_{7} + 34 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - 16 \beta_{3} + \cdots - 363 ) / 2 \) (-b14 - b13 + 34b7 + 34b6 - 5b5 - 3b4 - 16b3 + 2b2 + 2*b1 - 363) / 2
\(\nu^{7}\)	\(=\)	\( ( - 40 \beta_{15} - 40 \beta_{13} - 7 \beta_{12} + 8 \beta_{11} - 55 \beta_{10} - 136 \beta_{9} + \cdots - 193 \beta_{6} ) / 4 \) (-40b15 - 40b13 - 7b12 + 8b11 - 55b10 - 136b9 + 129b8 + 193b7 - 193*b6) / 4
\(\nu^{8}\)	\(=\)	\( ( - 24 \beta_{14} - 24 \beta_{13} + 291 \beta_{7} + 291 \beta_{6} + 32 \beta_{5} - 19 \beta_{4} + \cdots + 3132 ) / 2 \) (-24b14 - 24b13 + 291b7 + 291b6 + 32b5 - 19b4 + 3b3 - 35b2 - 128*b1 + 3132) / 2
\(\nu^{9}\)	\(=\)	\( - 58 \beta_{15} - 18 \beta_{14} - 40 \beta_{13} + 28 \beta_{12} - 214 \beta_{11} + 31 \beta_{10} + \cdots - 18 \) -58b15 - 18b14 - 40b13 + 28b12 - 214b11 + 31b10 - 22b9 - 154b8 + 98b7 - 98b6 - 18
\(\nu^{10}\)	\(=\)	\( - 38 \beta_{14} - 38 \beta_{13} - 339 \beta_{7} - 339 \beta_{6} - 150 \beta_{5} + 261 \beta_{4} + \cdots - 118 \) -38b14 - 38b13 - 339b7 - 339b6 - 150b5 + 261b4 - 335b3 - 85b2 - 644*b1 - 118
\(\nu^{11}\)	\(=\)	\( - 276 \beta_{15} - 524 \beta_{14} + 248 \beta_{13} - 801 \beta_{12} + 84 \beta_{11} + 369 \beta_{10} + \cdots - 524 \) -276b15 - 524b14 + 248b13 - 801b12 + 84b11 + 369b10 + 1380b9 + 5235b8 + 2011b7 - 2011b6 - 524
\(\nu^{12}\)	\(=\)	\( 428 \beta_{14} + 428 \beta_{13} - 3560 \beta_{7} - 3560 \beta_{6} + 1820 \beta_{5} + 340 \beta_{4} + \cdots - 172380 \) 428b14 + 428b13 - 3560b7 - 3560b6 + 1820b5 + 340b4 + 1840b3 - 776b2 - 216*b1 - 172380
\(\nu^{13}\)	\(=\)	\( 2608 \beta_{15} - 448 \beta_{14} + 3056 \beta_{13} - 7718 \beta_{12} - 4208 \beta_{11} - 1702 \beta_{10} + \cdots - 448 \) 2608b15 - 448b14 + 3056b13 - 7718b12 - 4208b11 - 1702b10 + 1520b9 - 39830b8 - 31382b7 + 31382b6 - 448
\(\nu^{14}\)	\(=\)	\( - 672 \beta_{14} - 672 \beta_{13} - 29924 \beta_{7} - 29924 \beta_{6} - 10560 \beta_{5} - 9884 \beta_{4} + \cdots - 883472 \) -672b14 - 672b13 - 29924b7 - 29924b6 - 10560b5 - 9884b4 - 3300b3 - 12060b2 + 18304*b1 - 883472
\(\nu^{15}\)	\(=\)	\( 66416 \beta_{15} + 61616 \beta_{14} + 4800 \beta_{13} - 30144 \beta_{12} + 22032 \beta_{11} + \cdots + 61616 \) 66416b15 + 61616b14 + 4800b13 - 30144b12 + 22032b11 + 12408b10 - 61936b9 - 31408b8 + 142256b7 - 142256b6 + 61616

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 134 T^{6} + \cdots + 128)^{2} \) (T^8 - 134T^6 + 5100T^4 - 49672*T^2 + 128)^2
$5$	\( T^{16} + \cdots + 59\!\cdots\!25 \) T^16 + 12T^14 - 13164T^12 - 961900T^10 + 191573750T^8 - 15029687500T^6 - 3213867187500T^4 + 45776367187500*T^2 + 59604644775390625
$7$	\( (T^{8} + 1422 T^{6} + \cdots + 714987936)^{2} \) (T^8 + 1422T^6 + 520868T^4 + 42807208*T^2 + 714987936)^2
$11$	\( (T^{8} + \cdots + 2238493420800)^{2} \) (T^8 + 5120T^6 + 9493152T^4 + 7611914752*T^2 + 2238493420800)^2
$13$	\( (T^{8} + \cdots + 9784472371200)^{2} \) (T^8 - 9060T^6 + 28218368T^4 - 33454882816*T^2 + 9784472371200)^2
$17$	\( (T^{8} + \cdots + 12119907631104)^{2} \) (T^8 + 20808T^6 + 143506112T^4 + 330111030784*T^2 + 12119907631104)^2
$19$	\( (T^{8} + 28704 T^{6} + \cdots + 38178047232)^{2} \) (T^8 + 28704T^6 + 217354016T^4 + 168683868160*T^2 + 38178047232)^2
$23$	\( (T^{8} + \cdots + 23\!\cdots\!96)^{2} \) (T^8 + 46862T^6 + 692502564T^4 + 3478125501352*T^2 + 2374625191944096)^2
$29$	\( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) (T^8 + 106992T^6 + 3177830400T^4 + 31539931316224*T^2 + 58518211343155200)^2
$31$	\( (T^{4} - 28 T^{3} + \cdots + 400132096)^{4} \) (T^4 - 28T^3 - 50064T^2 + 1897920*T + 400132096)^4
$37$	\( (T^{8} + \cdots + 930111561566208)^{2} \) (T^8 - 77596T^6 + 1619216672T^4 - 4775065212160*T^2 + 930111561566208)^2
$41$	\( (T^{4} - 58 T^{3} + \cdots + 2763694080)^{4} \) (T^4 - 58T^3 - 150976T^2 + 14374016*T + 2763694080)^4
$43$	\( (T^{8} - 111062 T^{6} + \cdots + 542115358848)^{2} \) (T^8 - 111062T^6 + 3282346540T^4 - 14440319717704*T^2 + 542115358848)^2
$47$	\( (T^{8} + \cdots + 21\!\cdots\!64)^{2} \) (T^8 + 247326T^6 + 15278864228T^4 + 35827752956392*T^2 + 21277806044904864)^2
$53$	\( (T^{8} + \cdots + 27\!\cdots\!48)^{2} \) (T^8 - 95140T^6 + 2172259712T^4 - 14836063186944*T^2 + 27747256973262848)^2
$59$	\( (T^{8} + \cdots + 43\!\cdots\!48)^{2} \) (T^8 + 422240T^6 + 27600487968T^4 + 619747875576832*T^2 + 4364850634106197248)^2
$61$	\( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) (T^8 + 1274036T^6 + 546416266864T^4 + 83188858348299712*T^2 + 1879163420950839628800)^2
$67$	\( (T^{8} + \cdots + 13\!\cdots\!68)^{2} \) (T^8 - 1919030T^6 + 1168519372972T^4 - 228352813341226696*T^2 + 130877537747150371968)^2
$71$	\( (T^{4} + 524 T^{3} + \cdots - 30115911168)^{4} \) (T^4 + 524T^3 - 268592T^2 - 202218176*T - 30115911168)^4
$73$	\( (T^{8} + \cdots + 68\!\cdots\!04)^{2} \) (T^8 + 2125704T^6 + 1649904703680T^4 + 554636387446621696*T^2 + 68153875753273879855104)^2
$79$	\( (T^{4} + 748 T^{3} + \cdots + 45513395200)^{4} \) (T^4 + 748T^3 - 830928T^2 - 462088896*T + 45513395200)^4
$83$	\( (T^{8} + \cdots + 39\!\cdots\!48)^{2} \) (T^8 - 1687142T^6 + 686343675500T^4 - 68638149138819464*T^2 + 394987658083584107648)^2
$89$	\( (T^{4} + 52 T^{3} + \cdots - 4266435600)^{4} \) (T^4 + 52T^3 - 483008T^2 - 96195664*T - 4266435600)^4
$97$	\( (T^{8} + \cdots + 22\!\cdots\!44)^{2} \) (T^8 + 4783944T^6 + 5812981166784T^4 + 1095500072805045760*T^2 + 22160595202119257554944)^2
show more
show less

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)