LMFDB - Newform orbit 2880.2.t.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{5} q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+O(q^{10}) $ q + b5 * q^5 + (-b15 + b13 - b12 + b11 - b10 - b8 + b7 + b4 - b1) * q^7	$ q + \beta_{5} q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 1) q^{97}+O(q^{100}) $ q + b5 * q^5 + (-b15 + b13 - b12 + b11 - b10 - b8 + b7 + b4 - b1) * q^7 + (b10 - b9 + b5 - 1) * q^11 + (-b15 - b14 + b13 - 2b12 + b11 - b10 + b9 - b8 + b7 + b6 - b5 - b1 + 1) q^13 + (-b14 + b13 - b12 - b7 + b6 - b5 - b3 - b2 + 1) * q^17 + (-b13 + b12 + b8 - b4 + b3 + b2 - b1) * q^19 + (-b13 + b12 - b11 + 3b8 - b7 + b5 - b4 + b1) q^23 - b8 * q^25 + (2b15 - 2b13 + 2b12 + b10 - b9 + 2b8 - b7 - b6 + b5 - b4 + b2 + 2b1) q^29 + (b13 + b12 - b7 - b6 + b3 + b1) * q^31 + (b12 + b3) * q^35 + (b14 + b11 + b8 - 2b7 + 2b6 - b4 - b2 - 1) * q^37 + (-2b12 - b11 - b8 + b4) q^41 + (b10 - b9 + b6 - 4b5 - 1) q^43 + (-b15 - b14 + b13 - b12 + b9 + b6 - 3b5 + b2 + b1 - 2) q^47 + (2b15 + 3b14 - 2b13 + 2b12 - 2b9 - 2b6 + b2 + 4b1 - 4) q^49 + (2b14 + 2b11 - 2b10 + b9 - b6 + b5 + b4 + b2 - 2) q^53 + (-b8 + b6 - b5 - b4 + b3) * q^55 + (2b14 + 2b11 - b8 + b7 - b6 - 2b5 + b4 + b2 - 2) q^59 + (2b15 + 3b14 + 2b12 - 3b11 + 3b8 - b4 + 2b3 + b2 + 2b1 - 1) q^61 + (b15 + b14 + b12 - b10 - b6 + b5 + b3 + b2 + b1 - 1) * q^65 + (2b15 + 2b14 - b13 + 4b12 - 2b11 + b10 - b9 - b7 - b6 + b5 - 2b4 + 2b3 + 2b2 + 7b1 - 5) * q^67 + (-2b15 + 2b13 - 4b12 + 2b11 - 2b10 - 2b8 + 2b7 - 2b6 - 2b5 + 2b4 - 2b3 - 6b1) * q^71 + (b15 - b12 + b11 + b10 - b8 + b6 - b5 - b4 + b3 + b1) * q^73 + (b15 + 2b14 - 3b13 - 2b11 - b10 + b9 + b7 + b6 - b5 - b4 + 2b3 + b2 + 3b1 - 2) q^77 + (2b14 + b13 - b12 - b7 + b6 + 2b5 - b3 - 3b1 - 2) q^79 + (2b14 - b13 + 2b12 - 2b11 + b10 - b9 + 4b8 - b7 - b6 + b5 + 3b1 + 1) q^83 + (-b9 - b6 + b5 + b4 + b2 - 2) * q^85 + (-2b15 + 2b13 - 2b12 + 4b11 - 2b10 - 4b8 + 2b7 + 2b5 + 2b4) q^89 + (4b14 + 4b11 - 3b10 - b9 - b8 + 3b7 - 3b6 + 7b5 + b4 + b2 - 5) * q^91 + (b13 - b12 - b10 + b9 + b6 - b2 - b1 + 3) * q^95 + (b15 - b14 + 2b13 + b12 - 3b10 + 2b9 - b6 + b5 + 3b3 + b2 + b1 + 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4*x^15 + 4*x^14 + 7*x^12 - 8*x^11 - 28*x^10 + 28*x^9 + 17*x^8 + 56*x^7 - 112*x^6 - 64*x^5 + 112*x^4 + 256*x^2 - 512*x + 256 :

$\beta_{1}$	$=$	$ ( - 163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} + \cdots - 7296 ) / 2688 $ (-163v^15 + 58v^14 + 376v^13 + 568v^12 - 501v^11 - 2502v^10 - 632v^9 + 3284v^8 + 6101v^7 - 1962v^6 - 10212v^5 - 4496v^4 + 4544v^3 + 16768v^2 - 6656*v - 7296) / 2688
$\beta_{2}$	$=$	$ ( - 9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + \cdots - 12288 ) / 128 $ (-9v^15 - 80v^14 + 152v^13 + 168v^12 + 65v^11 - 740v^10 - 1008v^9 + 1260v^8 + 2359v^7 + 1444v^6 - 5092v^5 - 4728v^4 + 2944v^3 + 5856v^2 + 7168*v - 12288) / 128
$\beta_{3}$	$=$	$ ( 41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} + \cdots + 15488 ) / 384 $ (41v^15 + 80v^14 - 264v^13 - 312v^12 + 47v^11 + 1380v^10 + 1312v^9 - 2204v^8 - 4071v^7 - 1316v^6 + 8052v^5 + 6376v^4 - 4208v^3 - 9952v^2 - 7680*v + 15488) / 384
$\beta_{4}$	$=$	$ ( 303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} + \cdots - 209536 ) / 2688 $ (303v^15 - 1892v^14 + 1724v^13 + 1784v^12 + 3497v^11 - 6864v^10 - 19332v^9 + 11948v^8 + 27583v^7 + 39328v^6 - 63120v^5 - 81576v^4 + 32528v^3 + 58720v^2 + 166592*v - 209536) / 2688
$\beta_{5}$	$=$	$ ( 397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} + \cdots + 8832 ) / 2688 $ (397v^15 - 502v^14 - 772v^13 - 664v^12 + 2523v^11 + 5226v^10 - 3292v^9 - 9932v^8 - 10139v^7 + 16854v^6 + 24480v^5 - 4720v^4 - 21776v^3 - 40576v^2 + 42176*v + 8832) / 2688
$\beta_{6}$	$=$	$ ( - 655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} + \cdots - 123008 ) / 2688 $ (-655v^15 + 60v^14 + 2516v^13 + 2480v^12 - 3641v^11 - 14664v^10 - 3308v^9 + 27372v^8 + 36625v^7 - 15880v^6 - 84768v^5 - 33296v^4 + 64416v^3 + 114624v^2 - 6592*v - 123008) / 2688
$\beta_{7}$	$=$	$ ( - 715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + \cdots + 227968 ) / 2688 $ (-715v^15 + 2724v^14 - 1240v^13 - 1648v^12 - 7037v^11 + 2592v^10 + 27040v^9 - 2292v^8 - 22475v^7 - 67264v^6 + 45252v^5 + 102304v^4 - 7248v^3 - 26112v^2 - 239872*v + 227968) / 2688
$\beta_{8}$	$=$	$ ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} + \cdots - 83072 ) / 1344 $ (396v^15 - 1201v^14 + 256v^13 + 460v^12 + 3508v^11 + 489v^10 - 11700v^9 - 2228v^8 + 6320v^7 + 32015v^6 - 9900v^5 - 42864v^4 - 5840v^3 - 1504v^2 + 108736*v - 83072) / 1344
$\beta_{9}$	$=$	$ ( - 137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + \cdots + 32000 ) / 448 $ (-137v^15 + 466v^14 - 144v^13 - 264v^12 - 1303v^11 + 106v^10 + 4672v^9 + 492v^8 - 3289v^7 - 12506v^6 + 4972v^5 + 17552v^4 + 1864v^3 - 1184v^2 - 41984*v + 32000) / 448
$\beta_{10}$	$=$	$ ( - 935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + \cdots + 203136 ) / 2688 $ (-935v^15 + 3056v^14 - 676v^13 - 1552v^12 - 8961v^11 - 636v^10 + 30236v^9 + 5644v^8 - 17975v^7 - 81876v^6 + 25272v^5 + 108608v^4 + 16480v^3 + 1280v^2 - 270016*v + 203136) / 2688
$\beta_{11}$	$=$	$ ( - 1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + \cdots + 166912 ) / 2688 $ (-1011v^15 + 2684v^14 - 164v^13 - 632v^12 - 8357v^11 - 3480v^10 + 25260v^9 + 9028v^8 - 7939v^7 - 73048v^6 + 8664v^5 + 88680v^4 + 19696v^3 + 21344v^2 - 240704*v + 166912) / 2688
$\beta_{12}$	$=$	$ ( - 1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + \cdots + 176384 ) / 2688 $ (-1269v^15 + 3292v^14 + 236v^13 - 664v^12 - 11155v^11 - 6240v^10 + 31596v^9 + 17228v^8 - 5477v^7 - 95888v^6 - 4752v^5 + 107784v^4 + 43856v^3 + 44320v^2 - 296512*v + 176384) / 2688
$\beta_{13}$	$=$	$ ( - 1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + \cdots + 252160 ) / 2688 $ (-1583v^15 + 4222v^14 - 120v^13 - 1032v^12 - 13337v^11 - 5850v^10 + 40424v^9 + 15908v^8 - 12087v^7 - 117622v^6 + 9804v^5 + 141728v^4 + 39344v^3 + 38848v^2 - 383232*v + 252160) / 2688
$\beta_{14}$	$=$	$ ( 239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} + \cdots - 29952 ) / 384 $ (239v^15 - 554v^14 - 56v^13 + 16v^12 + 1833v^11 + 1254v^10 - 5096v^9 - 2764v^8 + 311v^7 + 15690v^6 + 1044v^5 - 17384v^4 - 6592v^3 - 8672v^2 + 50560*v - 29952) / 384
$\beta_{15}$	$=$	$ ( - 356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + \cdots + 65248 ) / 224 $ (-356v^15 + 1000v^14 - 139v^13 - 312v^12 - 3008v^11 - 856v^10 + 9619v^9 + 2572v^8 - 4104v^7 - 27068v^6 + 5701v^5 + 34644v^4 + 6408v^3 + 4944v^2 - 90864*v + 65248) / 224

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

$\nu$	$=$	$ ( - \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 3 ) / 4 $ (-b15 - 2b14 - b12 + b9 - b8 + b7 - b5 + b4 - b3 - b2 - 2b1 + 3) / 4
$\nu^{2}$	$=$	$ ( - \beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 4 ) / 4 $ (-b14 - b13 + b12 - 3b11 + b10 + b9 + 2b8 - 2b7 + 2b6 - 3b5 - 2b4 + b3 - 2b2 + 3b1 + 4) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} - 2\beta_{13} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1 ) / 2 $ (b15 - 2b13 + b12 - 2b11 + b10 - b9 - b6 + b2 + 1) / 2
$\nu^{4}$	$=$	$ ( 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} + \cdots + 2 ) / 4 $ (2b15 + b14 - 3b13 + b12 - 3b11 - b10 + b9 - 2b7 - 3b5 - 3b3 - 2b2 - 3b1 + 2) / 4
$\nu^{5}$	$=$	$ ( 3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + \cdots - 13 ) / 4 $ (3b15 + 2b14 - 6b13 + 9b12 - 8b11 + 2b10 - 3b9 + 5b8 + b7 - 4b6 - 3b5 - b4 + 3b3 + 3b2 + 12*b1 - 13) / 4
$\nu^{6}$	$=$	$ ( 2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} + \cdots - 6 ) / 2 $ (2b15 + 3b14 - 2b13 + b12 - b11 - 4b9 - 5b8 - 2b7 - 2b6 - 4b5 + b2 - 2*b1 - 6) / 2
$\nu^{7}$	$=$	$ ( 7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + \cdots - 19 ) / 4 $ (7b15 + 4b14 + 3b12 - 6b11 - 6b10 - 3b9 + 19b8 + 9b7 - 4b6 + 3b5 + b4 - 3b3 + 5b2 - 12*b1 - 19) / 4
$\nu^{8}$	$=$	$ ( - 2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} + \cdots - 32 ) / 4 $ (-2b15 + 13b14 + 3b13 + 13b12 + 13b11 - 17b10 - 3b9 - 16b8 + 8b7 - 14b6 - 11b5 + 2b4 + 11b3 + 14b2 + 11*b1 - 32) / 4
$\nu^{9}$	$=$	$ ( - 9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots - 13 ) / 2 $ (-9b15 - 6b14 + 23b13 - 16b12 + 6b11 - 7b10 - 3b9 - 16b8 + 11b7 - 2b5 + 10b4 - b3 - 3b2 - 17*b1 - 13) / 2
$\nu^{10}$	$=$	$ ( - 4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + \cdots + 8 ) / 4 $ (-4b15 - 13b14 + 17b13 - 3b12 + 13b11 - 13b10 + 3b9 + 36b8 - 10b7 + 10b6 + 7b5 - 30b4 + 3b3 + 4b2 - 51*b1 + 8) / 4
$\nu^{11}$	$=$	$ ( - 31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + \cdots + 23 ) / 4 $ (-31b15 + 4b14 + 84b13 - 43b12 + 30b11 - 56b10 + 35b9 - 41b8 + 69b7 - 26b6 - b5 + 37b4 + 9b3 + 31b2 - 20b1 + 23) / 4
$\nu^{12}$	$=$	$ ( - 46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + \cdots + 23 ) / 2 $ (-46b15 - 44b14 + 49b13 - 52b12 + 26b11 - 14b10 + 24b9 - 88b8 + b7 + 33b6 + 6b5 + b4 - 7b3 - 28b2 - 47*b1 + 23) / 2
$\nu^{13}$	$=$	$ ( - 5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} + \cdots + 139 ) / 4 $ (-5b15 - 38b14 + 76b13 - 17b12 - 84b11 + 88b10 - 3b9 + 139b8 - 75b7 + 36b6 + 55b5 - 75b4 - 37b3 - 33b2 - 6*b1 + 139) / 4
$\nu^{14}$	$=$	$ ( - 32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + \cdots + 68 ) / 4 $ (-32b15 + 9b14 - 23b13 - 81b12 + 27b11 - 9b10 + 95b9 - 178b8 + 50b7 - 18b6 - 61b5 - 14b4 - 33b3 + 66b2 - 131*b1 + 68) / 4
$\nu^{15}$	$=$	$ ( - 45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + \cdots + 63 ) / 2 $ (-45b15 + 4b14 + 82b13 - 85b12 - 58b11 + 11b10 + 77b9 - 108b8 + 125b6 + 24b5 - 4b4 - 32b3 - 89b2 + 32b1 + 63) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$-\beta_{8}$	$1$

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} $

721.1

−0.966675	−	1.03225i
1.38652	−	0.278517i
−1.39563	+	0.228522i
1.26868	−	0.624862i
−0.530822	+	1.31081i
−0.296075	−	1.38287i
1.32070	+	0.505727i
1.21331	−	0.726558i
1.26868	+	0.624862i
−1.39563	−	0.228522i
1.38652	+	0.278517i
−0.966675	+	1.03225i
1.21331	+	0.726558i
1.32070	−	0.505727i
−0.296075	+	1.38287i
−0.530822	−	1.31081i

−0.707107

0.707107i

−

1.73696i

721.2

−0.707107

0.707107i

−

0.982011i

721.3

−0.707107

0.707107i

0.690576i

721.4

−0.707107

0.707107i

4.02840i

721.5

0.707107

−

0.707107i

−

2.73482i

721.6

0.707107

−

0.707107i

−

2.66881i

721.7

0.707107

−

0.707107i

2.89402i

721.8

0.707107

−

0.707107i

4.50961i

2161.1

−0.707107

−

0.707107i

−

4.02840i

2161.2

−0.707107

−

0.707107i

−

0.690576i

2161.3

−0.707107

−

0.707107i

0.982011i

2161.4

−0.707107

−

0.707107i

1.73696i

2161.5

0.707107

0.707107i

−

4.50961i

2161.6

0.707107

0.707107i

−

2.89402i

2161.7

0.707107

0.707107i

2.66881i

2161.8

0.707107

0.707107i

2.73482i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2880.2.t.c		16
3.b	odd	2	1		320.2.l.a		16
4.b	odd	2	1		720.2.t.c		16
12.b	even	2	1		80.2.l.a	✓	16
15.d	odd	2	1		1600.2.l.i		16
15.e	even	4	1		1600.2.q.g		16
15.e	even	4	1		1600.2.q.h		16
16.e	even	4	1	inner	2880.2.t.c		16
16.f	odd	4	1		720.2.t.c		16
24.f	even	2	1		640.2.l.b		16
24.h	odd	2	1		640.2.l.a		16
48.i	odd	4	1		320.2.l.a		16
48.i	odd	4	1		640.2.l.a		16
48.k	even	4	1		80.2.l.a	✓	16
48.k	even	4	1		640.2.l.b		16
60.h	even	2	1		400.2.l.h		16
60.l	odd	4	1		400.2.q.g		16
60.l	odd	4	1		400.2.q.h		16
96.o	even	8	1		5120.2.a.s		8
96.o	even	8	1		5120.2.a.v		8
96.p	odd	8	1		5120.2.a.t		8
96.p	odd	8	1		5120.2.a.u		8
240.t	even	4	1		400.2.l.h		16
240.z	odd	4	1		400.2.q.g		16
240.bb	even	4	1		1600.2.q.h		16
240.bd	odd	4	1		400.2.q.h		16
240.bf	even	4	1		1600.2.q.g		16
240.bm	odd	4	1		1600.2.l.i		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.2.l.a	✓	16	12.b	even	2	1
80.2.l.a	✓	16	48.k	even	4	1
320.2.l.a		16	3.b	odd	2	1
320.2.l.a		16	48.i	odd	4	1
400.2.l.h		16	60.h	even	2	1
400.2.l.h		16	240.t	even	4	1
400.2.q.g		16	60.l	odd	4	1
400.2.q.g		16	240.z	odd	4	1
400.2.q.h		16	60.l	odd	4	1
400.2.q.h		16	240.bd	odd	4	1
640.2.l.a		16	24.h	odd	2	1
640.2.l.a		16	48.i	odd	4	1
640.2.l.b		16	24.f	even	2	1
640.2.l.b		16	48.k	even	4	1
720.2.t.c		16	4.b	odd	2	1
720.2.t.c		16	16.f	odd	4	1
1600.2.l.i		16	15.d	odd	2	1
1600.2.l.i		16	240.bm	odd	4	1
1600.2.q.g		16	15.e	even	4	1
1600.2.q.g		16	240.bf	even	4	1
1600.2.q.h		16	15.e	even	4	1
1600.2.q.h		16	240.bb	even	4	1
2880.2.t.c		16	1.a	even	1	1	trivial
2880.2.t.c		16	16.e	even	4	1	inner
5120.2.a.s		8	96.o	even	8	1
5120.2.a.t		8	96.p	odd	8	1
5120.2.a.u		8	96.p	odd	8	1
5120.2.a.v		8	96.o	even	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} + 64 T_{7}^{14} + 1616 T_{7}^{12} + 20736 T_{7}^{10} + 145224 T_{7}^{8} + 549632 T_{7}^{6} + \cdots + 204304 $ T7^16 + 64*T7^14 + 1616*T7^12 + 20736*T7^10 + 145224*T7^8 + 549632*T7^6 + 1033536*T7^4 + 811008*T7^2 + 204304 acting on $S_{2}^{\mathrm{new}}(2880, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$7$	$ T^{16} + 64 T^{14} + \cdots + 204304 $ T^16 + 64T^14 + 1616T^12 + 20736T^10 + 145224T^8 + 549632T^6 + 1033536T^4 + 811008*T^2 + 204304
$11$	$ T^{16} + 8 T^{15} + \cdots + 1290496 $ T^16 + 8T^15 + 32T^14 + 80T^13 + 1232T^12 + 9568T^11 + 40320T^10 + 82368T^9 + 139616T^8 + 446848T^7 + 1795584T^6 + 3673856T^5 + 3958016T^4 + 848384T^3 + 247808T^2 + 799744*T + 1290496
$13$	$ T^{16} + 128 T^{13} + \cdots + 20647936 $ T^16 + 128T^13 + 1600T^12 + 4352T^11 + 8192T^10 + 49152T^9 + 415872T^8 + 1400832T^7 + 2654208T^6 + 3137536T^5 + 9146368T^4 + 27869184T^3 + 52428800T^2 + 46530560*T + 20647936
$17$	$ (T^{8} - 72 T^{6} + \cdots + 13888)^{2} $ (T^8 - 72T^6 + 64T^5 + 1408T^4 - 1536T^3 - 7744T^2 + 5120T + 13888)^2
$19$	$ T^{16} - 8 T^{15} + \cdots + 614656 $ T^16 - 8T^15 + 32T^14 - 176T^13 + 3216T^12 - 27488T^11 + 132480T^10 - 363072T^9 + 731744T^8 - 1921408T^7 + 7595520T^6 - 19398912T^5 + 30308608T^4 - 26513920T^3 + 14108672T^2 - 4164608*T + 614656
$23$	$ T^{16} + 128 T^{14} + \cdots + 1731856 $ T^16 + 128T^14 + 4784T^12 + 77504T^10 + 622088T^8 + 2620928T^6 + 5719232T^4 + 5740288*T^2 + 1731856
$29$	$ T^{16} + \cdots + 3017085184 $ T^16 - 16T^15 + 128T^14 - 288T^13 + 1104T^12 - 18624T^11 + 198144T^10 - 351616T^9 - 199840T^8 + 2768128T^7 + 37230592T^6 - 1816064T^5 - 5714688T^4 + 456489984T^3 + 2708480000T^2 + 4042700800*T + 3017085184
$31$	$ (T^{8} - 96 T^{6} + \cdots - 20224)^{2} $ (T^8 - 96T^6 - 64T^5 + 2848T^4 + 4096T^3 - 26112T^2 - 58368T - 20224)^2
$37$	$ T^{16} + 16 T^{15} + \cdots + 18939904 $ T^16 + 16T^15 + 128T^14 + 704T^13 + 16320T^12 + 249088T^11 + 2144256T^10 + 9370624T^9 + 22554112T^8 + 40185856T^7 + 381124608T^6 + 1380728832T^5 + 2705047552T^4 + 2707357696T^3 + 1472724992T^2 + 236191744*T + 18939904
$41$	$ T^{16} + \cdots + 110660014336 $ T^16 + 384T^14 + 59088T^12 + 4686848T^10 + 206041952T^8 + 5024061440T^6 + 63304144128T^4 + 325786435584*T^2 + 110660014336
$43$	$ T^{16} + 8 T^{15} + \cdots + 53640976 $ T^16 + 8T^15 + 32T^14 - 440T^13 + 5328T^12 + 18816T^11 + 76832T^10 - 950192T^9 + 5444360T^8 + 1854880T^7 + 26364032T^6 - 280880992T^5 + 1191507520T^4 - 1500385792T^3 + 1021520000T^2 - 331044800*T + 53640976
$47$	$ (T^{8} + 20 T^{7} + \cdots + 575044)^{2} $ (T^8 + 20T^7 - 8T^6 - 2392T^5 - 14936T^4 + 584T^3 + 244768T^2 + 693376*T + 575044)^2
$53$	$ T^{16} + \cdots + 383725735936 $ T^16 + 16T^15 + 128T^14 + 448T^13 + 15552T^12 + 236416T^11 + 1892352T^10 + 6504448T^9 + 43316352T^8 + 498471936T^7 + 3861454848T^6 + 13942091776T^5 + 26109784064T^4 + 10552958976T^3 + 50384076800T^2 + 196640112640*T + 383725735936
$59$	$ T^{16} + \cdots + 12227051776 $ T^16 + 8T^15 + 32T^14 - 272T^13 + 8464T^12 + 65632T^11 + 291200T^10 - 2562240T^9 + 7403616T^8 + 20658560T^7 + 314344960T^6 - 3038184192T^5 + 12226302208T^4 - 27384356352T^3 + 37920376832T^2 - 30451745792*T + 12227051776
$61$	$ T^{16} + \cdots + 1393986371584 $ T^16 - 16T^15 + 128T^14 + 384T^13 + 11520T^12 - 176640T^11 + 1425408T^10 + 4136960T^9 + 17876992T^8 - 240730112T^7 + 2332164096T^6 + 9643622400T^5 + 27549499392T^4 - 95024578560T^3 + 428456542208T^2 + 1092943347712*T + 1393986371584
$67$	$ T^{16} + \cdots + 46120451769616 $ T^16 + 40T^15 + 800T^14 + 10680T^13 + 147664T^12 + 2604960T^11 + 43098400T^10 + 537576688T^9 + 4867387016T^8 + 31795927072T^7 + 148072661120T^6 + 475604544352T^5 + 1043974444608T^4 + 2087109786752T^3 + 7633293466752T^2 + 26534918246592*T + 46120451769616
$71$	$ T^{16} + \cdots + 3333516427264 $ T^16 + 640T^14 + 157440T^12 + 18817024T^10 + 1144522752T^8 + 33695596544T^6 + 408856297472T^4 + 2007385505792*T^2 + 3333516427264
$73$	$ T^{16} + \cdots + 15847788544 $ T^16 + 560T^14 + 108992T^12 + 9035648T^10 + 321195136T^8 + 4377603072T^6 + 17757564928T^4 + 28362989568*T^2 + 15847788544
$79$	$ (T^{8} + 8 T^{7} + \cdots + 4352)^{2} $ (T^8 + 8T^7 - 160T^6 - 352T^5 + 5856T^4 + 4992T^3 - 61952T^2 - 31232*T + 4352)^2
$83$	$ T^{16} + \cdots + 2050640656 $ T^16 - 40T^15 + 800T^14 - 7976T^13 + 37520T^12 + 36928T^11 + 315168T^10 - 9891472T^9 + 64272392T^8 + 379938272T^7 + 1135671424T^6 - 140621856T^5 + 1230069056T^4 + 5315507456T^3 + 8900981888T^2 + 6041972416*T + 2050640656
$89$	$ T^{16} + \cdots + 684153962496 $ T^16 + 416T^14 + 68032T^12 + 5576192T^10 + 244188672T^8 + 5734359040T^6 + 69045698560T^4 + 380947267584*T^2 + 684153962496
$97$	$ (T^{8} - 440 T^{6} + \cdots - 8549312)^{2} $ (T^8 - 440T^6 - 416T^5 + 47936T^4 - 78720T^3 - 1675968T^2 + 7621376T - 8549312)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+O(q^{10}) \) q + b5 * q^5 + (-b15 + b13 - b12 + b11 - b10 - b8 + b7 + b4 - b1) * q^7	\( q + \beta_{5} q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 1) q^{97}+O(q^{100}) \) q + b5 * q^5 + (-b15 + b13 - b12 + b11 - b10 - b8 + b7 + b4 - b1) * q^7 + (b10 - b9 + b5 - 1) * q^11 + (-b15 - b14 + b13 - 2b12 + b11 - b10 + b9 - b8 + b7 + b6 - b5 - b1 + 1) q^13 + (-b14 + b13 - b12 - b7 + b6 - b5 - b3 - b2 + 1) * q^17 + (-b13 + b12 + b8 - b4 + b3 + b2 - b1) * q^19 + (-b13 + b12 - b11 + 3b8 - b7 + b5 - b4 + b1) q^23 - b8 * q^25 + (2b15 - 2b13 + 2b12 + b10 - b9 + 2b8 - b7 - b6 + b5 - b4 + b2 + 2b1) q^29 + (b13 + b12 - b7 - b6 + b3 + b1) * q^31 + (b12 + b3) * q^35 + (b14 + b11 + b8 - 2b7 + 2b6 - b4 - b2 - 1) * q^37 + (-2b12 - b11 - b8 + b4) q^41 + (b10 - b9 + b6 - 4b5 - 1) q^43 + (-b15 - b14 + b13 - b12 + b9 + b6 - 3b5 + b2 + b1 - 2) q^47 + (2b15 + 3b14 - 2b13 + 2b12 - 2b9 - 2b6 + b2 + 4b1 - 4) q^49 + (2b14 + 2b11 - 2b10 + b9 - b6 + b5 + b4 + b2 - 2) q^53 + (-b8 + b6 - b5 - b4 + b3) * q^55 + (2b14 + 2b11 - b8 + b7 - b6 - 2b5 + b4 + b2 - 2) q^59 + (2b15 + 3b14 + 2b12 - 3b11 + 3b8 - b4 + 2b3 + b2 + 2b1 - 1) q^61 + (b15 + b14 + b12 - b10 - b6 + b5 + b3 + b2 + b1 - 1) * q^65 + (2b15 + 2b14 - b13 + 4b12 - 2b11 + b10 - b9 - b7 - b6 + b5 - 2b4 + 2b3 + 2b2 + 7b1 - 5) * q^67 + (-2b15 + 2b13 - 4b12 + 2b11 - 2b10 - 2b8 + 2b7 - 2b6 - 2b5 + 2b4 - 2b3 - 6b1) * q^71 + (b15 - b12 + b11 + b10 - b8 + b6 - b5 - b4 + b3 + b1) * q^73 + (b15 + 2b14 - 3b13 - 2b11 - b10 + b9 + b7 + b6 - b5 - b4 + 2b3 + b2 + 3b1 - 2) q^77 + (2b14 + b13 - b12 - b7 + b6 + 2b5 - b3 - 3b1 - 2) q^79 + (2b14 - b13 + 2b12 - 2b11 + b10 - b9 + 4b8 - b7 - b6 + b5 + 3b1 + 1) q^83 + (-b9 - b6 + b5 + b4 + b2 - 2) * q^85 + (-2b15 + 2b13 - 2b12 + 4b11 - 2b10 - 4b8 + 2b7 + 2b5 + 2b4) q^89 + (4b14 + 4b11 - 3b10 - b9 - b8 + 3b7 - 3b6 + 7b5 + b4 + b2 - 5) * q^91 + (b13 - b12 - b10 + b9 + b6 - b2 - b1 + 3) * q^95 + (b15 - b14 + 2b13 + b12 - 3b10 + 2b9 - b6 + b5 + 3b3 + b2 + b1 + 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2880.2.t.c

Level	$2880$
Weight	$2$
Character orbit	2880.t
Analytic conductor	$22.997$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(22.9969157821\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} + \cdots - 7296 ) / 2688 \) (-163v^15 + 58v^14 + 376v^13 + 568v^12 - 501v^11 - 2502v^10 - 632v^9 + 3284v^8 + 6101v^7 - 1962v^6 - 10212v^5 - 4496v^4 + 4544v^3 + 16768v^2 - 6656*v - 7296) / 2688
\(\beta_{2}\)	\(=\)	\( ( - 9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + \cdots - 12288 ) / 128 \) (-9v^15 - 80v^14 + 152v^13 + 168v^12 + 65v^11 - 740v^10 - 1008v^9 + 1260v^8 + 2359v^7 + 1444v^6 - 5092v^5 - 4728v^4 + 2944v^3 + 5856v^2 + 7168*v - 12288) / 128
\(\beta_{3}\)	\(=\)	\( ( 41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} + \cdots + 15488 ) / 384 \) (41v^15 + 80v^14 - 264v^13 - 312v^12 + 47v^11 + 1380v^10 + 1312v^9 - 2204v^8 - 4071v^7 - 1316v^6 + 8052v^5 + 6376v^4 - 4208v^3 - 9952v^2 - 7680*v + 15488) / 384
\(\beta_{4}\)	\(=\)	\( ( 303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} + \cdots - 209536 ) / 2688 \) (303v^15 - 1892v^14 + 1724v^13 + 1784v^12 + 3497v^11 - 6864v^10 - 19332v^9 + 11948v^8 + 27583v^7 + 39328v^6 - 63120v^5 - 81576v^4 + 32528v^3 + 58720v^2 + 166592*v - 209536) / 2688
\(\beta_{5}\)	\(=\)	\( ( 397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} + \cdots + 8832 ) / 2688 \) (397v^15 - 502v^14 - 772v^13 - 664v^12 + 2523v^11 + 5226v^10 - 3292v^9 - 9932v^8 - 10139v^7 + 16854v^6 + 24480v^5 - 4720v^4 - 21776v^3 - 40576v^2 + 42176*v + 8832) / 2688
\(\beta_{6}\)	\(=\)	\( ( - 655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} + \cdots - 123008 ) / 2688 \) (-655v^15 + 60v^14 + 2516v^13 + 2480v^12 - 3641v^11 - 14664v^10 - 3308v^9 + 27372v^8 + 36625v^7 - 15880v^6 - 84768v^5 - 33296v^4 + 64416v^3 + 114624v^2 - 6592*v - 123008) / 2688
\(\beta_{7}\)	\(=\)	\( ( - 715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + \cdots + 227968 ) / 2688 \) (-715v^15 + 2724v^14 - 1240v^13 - 1648v^12 - 7037v^11 + 2592v^10 + 27040v^9 - 2292v^8 - 22475v^7 - 67264v^6 + 45252v^5 + 102304v^4 - 7248v^3 - 26112v^2 - 239872*v + 227968) / 2688
\(\beta_{8}\)	\(=\)	\( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} + \cdots - 83072 ) / 1344 \) (396v^15 - 1201v^14 + 256v^13 + 460v^12 + 3508v^11 + 489v^10 - 11700v^9 - 2228v^8 + 6320v^7 + 32015v^6 - 9900v^5 - 42864v^4 - 5840v^3 - 1504v^2 + 108736*v - 83072) / 1344
\(\beta_{9}\)	\(=\)	\( ( - 137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + \cdots + 32000 ) / 448 \) (-137v^15 + 466v^14 - 144v^13 - 264v^12 - 1303v^11 + 106v^10 + 4672v^9 + 492v^8 - 3289v^7 - 12506v^6 + 4972v^5 + 17552v^4 + 1864v^3 - 1184v^2 - 41984*v + 32000) / 448
\(\beta_{10}\)	\(=\)	\( ( - 935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + \cdots + 203136 ) / 2688 \) (-935v^15 + 3056v^14 - 676v^13 - 1552v^12 - 8961v^11 - 636v^10 + 30236v^9 + 5644v^8 - 17975v^7 - 81876v^6 + 25272v^5 + 108608v^4 + 16480v^3 + 1280v^2 - 270016*v + 203136) / 2688
\(\beta_{11}\)	\(=\)	\( ( - 1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + \cdots + 166912 ) / 2688 \) (-1011v^15 + 2684v^14 - 164v^13 - 632v^12 - 8357v^11 - 3480v^10 + 25260v^9 + 9028v^8 - 7939v^7 - 73048v^6 + 8664v^5 + 88680v^4 + 19696v^3 + 21344v^2 - 240704*v + 166912) / 2688
\(\beta_{12}\)	\(=\)	\( ( - 1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + \cdots + 176384 ) / 2688 \) (-1269v^15 + 3292v^14 + 236v^13 - 664v^12 - 11155v^11 - 6240v^10 + 31596v^9 + 17228v^8 - 5477v^7 - 95888v^6 - 4752v^5 + 107784v^4 + 43856v^3 + 44320v^2 - 296512*v + 176384) / 2688
\(\beta_{13}\)	\(=\)	\( ( - 1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + \cdots + 252160 ) / 2688 \) (-1583v^15 + 4222v^14 - 120v^13 - 1032v^12 - 13337v^11 - 5850v^10 + 40424v^9 + 15908v^8 - 12087v^7 - 117622v^6 + 9804v^5 + 141728v^4 + 39344v^3 + 38848v^2 - 383232*v + 252160) / 2688
\(\beta_{14}\)	\(=\)	\( ( 239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} + \cdots - 29952 ) / 384 \) (239v^15 - 554v^14 - 56v^13 + 16v^12 + 1833v^11 + 1254v^10 - 5096v^9 - 2764v^8 + 311v^7 + 15690v^6 + 1044v^5 - 17384v^4 - 6592v^3 - 8672v^2 + 50560*v - 29952) / 384
\(\beta_{15}\)	\(=\)	\( ( - 356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + \cdots + 65248 ) / 224 \) (-356v^15 + 1000v^14 - 139v^13 - 312v^12 - 3008v^11 - 856v^10 + 9619v^9 + 2572v^8 - 4104v^7 - 27068v^6 + 5701v^5 + 34644v^4 + 6408v^3 + 4944v^2 - 90864*v + 65248) / 224

\(\nu\)	\(=\)	\( ( - \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 3 ) / 4 \) (-b15 - 2b14 - b12 + b9 - b8 + b7 - b5 + b4 - b3 - b2 - 2b1 + 3) / 4
\(\nu^{2}\)	\(=\)	\( ( - \beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 4 ) / 4 \) (-b14 - b13 + b12 - 3b11 + b10 + b9 + 2b8 - 2b7 + 2b6 - 3b5 - 2b4 + b3 - 2b2 + 3b1 + 4) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} - 2\beta_{13} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1 ) / 2 \) (b15 - 2b13 + b12 - 2b11 + b10 - b9 - b6 + b2 + 1) / 2
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} + \cdots + 2 ) / 4 \) (2b15 + b14 - 3b13 + b12 - 3b11 - b10 + b9 - 2b7 - 3b5 - 3b3 - 2b2 - 3b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( 3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + \cdots - 13 ) / 4 \) (3b15 + 2b14 - 6b13 + 9b12 - 8b11 + 2b10 - 3b9 + 5b8 + b7 - 4b6 - 3b5 - b4 + 3b3 + 3b2 + 12*b1 - 13) / 4
\(\nu^{6}\)	\(=\)	\( ( 2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} + \cdots - 6 ) / 2 \) (2b15 + 3b14 - 2b13 + b12 - b11 - 4b9 - 5b8 - 2b7 - 2b6 - 4b5 + b2 - 2*b1 - 6) / 2
\(\nu^{7}\)	\(=\)	\( ( 7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + \cdots - 19 ) / 4 \) (7b15 + 4b14 + 3b12 - 6b11 - 6b10 - 3b9 + 19b8 + 9b7 - 4b6 + 3b5 + b4 - 3b3 + 5b2 - 12*b1 - 19) / 4
\(\nu^{8}\)	\(=\)	\( ( - 2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} + \cdots - 32 ) / 4 \) (-2b15 + 13b14 + 3b13 + 13b12 + 13b11 - 17b10 - 3b9 - 16b8 + 8b7 - 14b6 - 11b5 + 2b4 + 11b3 + 14b2 + 11*b1 - 32) / 4
\(\nu^{9}\)	\(=\)	\( ( - 9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots - 13 ) / 2 \) (-9b15 - 6b14 + 23b13 - 16b12 + 6b11 - 7b10 - 3b9 - 16b8 + 11b7 - 2b5 + 10b4 - b3 - 3b2 - 17*b1 - 13) / 2
\(\nu^{10}\)	\(=\)	\( ( - 4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + \cdots + 8 ) / 4 \) (-4b15 - 13b14 + 17b13 - 3b12 + 13b11 - 13b10 + 3b9 + 36b8 - 10b7 + 10b6 + 7b5 - 30b4 + 3b3 + 4b2 - 51*b1 + 8) / 4
\(\nu^{11}\)	\(=\)	\( ( - 31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + \cdots + 23 ) / 4 \) (-31b15 + 4b14 + 84b13 - 43b12 + 30b11 - 56b10 + 35b9 - 41b8 + 69b7 - 26b6 - b5 + 37b4 + 9b3 + 31b2 - 20b1 + 23) / 4
\(\nu^{12}\)	\(=\)	\( ( - 46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + \cdots + 23 ) / 2 \) (-46b15 - 44b14 + 49b13 - 52b12 + 26b11 - 14b10 + 24b9 - 88b8 + b7 + 33b6 + 6b5 + b4 - 7b3 - 28b2 - 47*b1 + 23) / 2
\(\nu^{13}\)	\(=\)	\( ( - 5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} + \cdots + 139 ) / 4 \) (-5b15 - 38b14 + 76b13 - 17b12 - 84b11 + 88b10 - 3b9 + 139b8 - 75b7 + 36b6 + 55b5 - 75b4 - 37b3 - 33b2 - 6*b1 + 139) / 4
\(\nu^{14}\)	\(=\)	\( ( - 32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + \cdots + 68 ) / 4 \) (-32b15 + 9b14 - 23b13 - 81b12 + 27b11 - 9b10 + 95b9 - 178b8 + 50b7 - 18b6 - 61b5 - 14b4 - 33b3 + 66b2 - 131*b1 + 68) / 4
\(\nu^{15}\)	\(=\)	\( ( - 45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + \cdots + 63 ) / 2 \) (-45b15 + 4b14 + 82b13 - 85b12 - 58b11 + 11b10 + 77b9 - 108b8 + 125b6 + 24b5 - 4b4 - 32b3 - 89b2 + 32b1 + 63) / 2

\(n\)	\(577\)	\(641\)	\(901\)	\(2431\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{8}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 721.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$7$	\( T^{16} + 64 T^{14} + \cdots + 204304 \) T^16 + 64T^14 + 1616T^12 + 20736T^10 + 145224T^8 + 549632T^6 + 1033536T^4 + 811008*T^2 + 204304
$11$	\( T^{16} + 8 T^{15} + \cdots + 1290496 \) T^16 + 8T^15 + 32T^14 + 80T^13 + 1232T^12 + 9568T^11 + 40320T^10 + 82368T^9 + 139616T^8 + 446848T^7 + 1795584T^6 + 3673856T^5 + 3958016T^4 + 848384T^3 + 247808T^2 + 799744*T + 1290496
$13$	\( T^{16} + 128 T^{13} + \cdots + 20647936 \) T^16 + 128T^13 + 1600T^12 + 4352T^11 + 8192T^10 + 49152T^9 + 415872T^8 + 1400832T^7 + 2654208T^6 + 3137536T^5 + 9146368T^4 + 27869184T^3 + 52428800T^2 + 46530560*T + 20647936
$17$	\( (T^{8} - 72 T^{6} + \cdots + 13888)^{2} \) (T^8 - 72T^6 + 64T^5 + 1408T^4 - 1536T^3 - 7744T^2 + 5120T + 13888)^2
$19$	\( T^{16} - 8 T^{15} + \cdots + 614656 \) T^16 - 8T^15 + 32T^14 - 176T^13 + 3216T^12 - 27488T^11 + 132480T^10 - 363072T^9 + 731744T^8 - 1921408T^7 + 7595520T^6 - 19398912T^5 + 30308608T^4 - 26513920T^3 + 14108672T^2 - 4164608*T + 614656
$23$	\( T^{16} + 128 T^{14} + \cdots + 1731856 \) T^16 + 128T^14 + 4784T^12 + 77504T^10 + 622088T^8 + 2620928T^6 + 5719232T^4 + 5740288*T^2 + 1731856
$29$	\( T^{16} + \cdots + 3017085184 \) T^16 - 16T^15 + 128T^14 - 288T^13 + 1104T^12 - 18624T^11 + 198144T^10 - 351616T^9 - 199840T^8 + 2768128T^7 + 37230592T^6 - 1816064T^5 - 5714688T^4 + 456489984T^3 + 2708480000T^2 + 4042700800*T + 3017085184
$31$	\( (T^{8} - 96 T^{6} + \cdots - 20224)^{2} \) (T^8 - 96T^6 - 64T^5 + 2848T^4 + 4096T^3 - 26112T^2 - 58368T - 20224)^2
$37$	\( T^{16} + 16 T^{15} + \cdots + 18939904 \) T^16 + 16T^15 + 128T^14 + 704T^13 + 16320T^12 + 249088T^11 + 2144256T^10 + 9370624T^9 + 22554112T^8 + 40185856T^7 + 381124608T^6 + 1380728832T^5 + 2705047552T^4 + 2707357696T^3 + 1472724992T^2 + 236191744*T + 18939904
$41$	\( T^{16} + \cdots + 110660014336 \) T^16 + 384T^14 + 59088T^12 + 4686848T^10 + 206041952T^8 + 5024061440T^6 + 63304144128T^4 + 325786435584*T^2 + 110660014336
$43$	\( T^{16} + 8 T^{15} + \cdots + 53640976 \) T^16 + 8T^15 + 32T^14 - 440T^13 + 5328T^12 + 18816T^11 + 76832T^10 - 950192T^9 + 5444360T^8 + 1854880T^7 + 26364032T^6 - 280880992T^5 + 1191507520T^4 - 1500385792T^3 + 1021520000T^2 - 331044800*T + 53640976
$47$	\( (T^{8} + 20 T^{7} + \cdots + 575044)^{2} \) (T^8 + 20T^7 - 8T^6 - 2392T^5 - 14936T^4 + 584T^3 + 244768T^2 + 693376*T + 575044)^2
$53$	\( T^{16} + \cdots + 383725735936 \) T^16 + 16T^15 + 128T^14 + 448T^13 + 15552T^12 + 236416T^11 + 1892352T^10 + 6504448T^9 + 43316352T^8 + 498471936T^7 + 3861454848T^6 + 13942091776T^5 + 26109784064T^4 + 10552958976T^3 + 50384076800T^2 + 196640112640*T + 383725735936
$59$	\( T^{16} + \cdots + 12227051776 \) T^16 + 8T^15 + 32T^14 - 272T^13 + 8464T^12 + 65632T^11 + 291200T^10 - 2562240T^9 + 7403616T^8 + 20658560T^7 + 314344960T^6 - 3038184192T^5 + 12226302208T^4 - 27384356352T^3 + 37920376832T^2 - 30451745792*T + 12227051776
$61$	\( T^{16} + \cdots + 1393986371584 \) T^16 - 16T^15 + 128T^14 + 384T^13 + 11520T^12 - 176640T^11 + 1425408T^10 + 4136960T^9 + 17876992T^8 - 240730112T^7 + 2332164096T^6 + 9643622400T^5 + 27549499392T^4 - 95024578560T^3 + 428456542208T^2 + 1092943347712*T + 1393986371584
$67$	\( T^{16} + \cdots + 46120451769616 \) T^16 + 40T^15 + 800T^14 + 10680T^13 + 147664T^12 + 2604960T^11 + 43098400T^10 + 537576688T^9 + 4867387016T^8 + 31795927072T^7 + 148072661120T^6 + 475604544352T^5 + 1043974444608T^4 + 2087109786752T^3 + 7633293466752T^2 + 26534918246592*T + 46120451769616
$71$	\( T^{16} + \cdots + 3333516427264 \) T^16 + 640T^14 + 157440T^12 + 18817024T^10 + 1144522752T^8 + 33695596544T^6 + 408856297472T^4 + 2007385505792*T^2 + 3333516427264
$73$	\( T^{16} + \cdots + 15847788544 \) T^16 + 560T^14 + 108992T^12 + 9035648T^10 + 321195136T^8 + 4377603072T^6 + 17757564928T^4 + 28362989568*T^2 + 15847788544
$79$	\( (T^{8} + 8 T^{7} + \cdots + 4352)^{2} \) (T^8 + 8T^7 - 160T^6 - 352T^5 + 5856T^4 + 4992T^3 - 61952T^2 - 31232*T + 4352)^2
$83$	\( T^{16} + \cdots + 2050640656 \) T^16 - 40T^15 + 800T^14 - 7976T^13 + 37520T^12 + 36928T^11 + 315168T^10 - 9891472T^9 + 64272392T^8 + 379938272T^7 + 1135671424T^6 - 140621856T^5 + 1230069056T^4 + 5315507456T^3 + 8900981888T^2 + 6041972416*T + 2050640656
$89$	\( T^{16} + \cdots + 684153962496 \) T^16 + 416T^14 + 68032T^12 + 5576192T^10 + 244188672T^8 + 5734359040T^6 + 69045698560T^4 + 380947267584*T^2 + 684153962496
$97$	\( (T^{8} - 440 T^{6} + \cdots - 8549312)^{2} \) (T^8 - 440T^6 - 416T^5 + 47936T^4 - 78720T^3 - 1675968T^2 + 7621376T - 8549312)^2
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