LMFDB - Newform orbit 832.2.bu.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(63,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 0, 7]))

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

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

chi := DirichletCharacter("832.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.bu (of order $12$, degree $4$, 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:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	16.0.102930383934669717504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 $ x^16 - 4x^15 + 5x^14 - 2x^13 + 5x^12 - 8x^11 - 12x^10 + 32x^9 - 36x^8 + 64x^7 - 48x^6 - 64x^5 + 80x^4 - 64x^3 + 320x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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_{14} - \beta_{2}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14}) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_1 - 1) q^{9}+O(q^{10}) $ q + (-b14 - b2) * q^3 + (-b8 - b6 + b3) * q^5 + (-b15 - b14) * q^7 + (-b8 - b6 - b1 - 1) * q^9	$ q + ( - \beta_{14} - \beta_{2}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14}) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_1 - 1) q^{9} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{2}) q^{11}+ \cdots + (2 \beta_{14} + 2 \beta_{11} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) $ q + (-b14 - b2) * q^3 + (-b8 - b6 + b3) * q^5 + (-b15 - b14) * q^7 + (-b8 - b6 - b1 - 1) * q^9 + (-b15 - b11 + b7 + b2) * q^11 + (b8 + b6 + 2b5 + 2b4 - 2b3 - b1 + 2) q^13 + (b15 + b13 - b11 - b10 - b9 - b7 - 2b2) q^15 + (-b12 + b5 + b4 + b1 + 2) * q^17 + (b15 + b13 + b11 - 2b7 - b2) q^19 + (-2b12 - 2b8 - b6 - 2b5 - 2b4) * q^21 + (-b14 + 2b10 + 2b9 + 2b7 + b2) q^23 + (-b12 - 2b8 - b6 - b5 + b4 - b3 - b1 - 1) q^25 + (b13 - b11 - b10 - b9 - b7 + b2) * q^27 + (-2b12 - b8 - 2b6 - 4b5 + 2b1 + 1) * q^29 + (b15 + b11 - b10 - 2b9 - b7 + b2) q^31 + (-3b12 + b3 + 3b1 + 1) * q^33 + (b14 + b11 - b9) * q^35 + (b5 + 1) * q^37 + (b14 - b13 - b11 - b10 + b9 - b7 - b2) * q^39 + (-3b5 + 3) q^41 + (b15 + b14 - 3b13 + b11 + b10 + 2b9 + 2b7) q^43 + (-b12 - 3b8 + 3b5 + 2b4 + 2b3 - b1 - 2) * q^45 + (3b14 + 3b11 - b7) * q^47 + (-b12 - 4b8 - 2b6 - 3b4 + b3 - 2b1 - 1) * q^49 + (b15 + b14 - b10 + 2b9) q^51 + (-2b12 + 2b6 + b5 + b4 + b1 + 4) * q^53 + (-2b15 + 2b14 - b13 - b11 + b10 + 2b7 + 4b2) * q^55 + (2b12 - b6 + 2b5 - 4b1 - 2) q^57 + (-3b15 - 3b14 - 2b13 + 4b10 + 2b9 + 4b7 + 2b2) q^59 + (b12 - 2b6 + b5 + b4 - 3b1 - 2) * q^61 + (-b14 + b13 + 2b11 + b10 + 2b9 - b2) * q^63 + (-3b12 - 2b8 + b6 + b5 + 2b4 - 3b3 + 3b1 + 2) q^65 + (2b14 - b11 - 3b9 - b2) * q^67 + (2b12 - 3b6 - 2b4 + b1 - 1) q^69 + (-b15 - 3b13 - 3b11 - 2b9 + 2b7 + b2) * q^71 + (b12 + 3b8 + 2b6 - b5 + 4b4 - 3b3 + 5) * q^73 + (2b14 + b13 - b11 - 2b10 - b9 - b7 - 2b2) q^75 + (-4b8 - 4b5 + b4 - 5b3 - b1 - 2) q^77 + (-2b13 - 2b11 + 2b10 + 2b7 + 2b2) q^79 + (2b12 + 7b8 + 4b5 - 2b4 + 2b3 + 5) q^81 + (2b15 + 2b11 - 2b10 - 3b9 - 2b7 + b2) q^83 + (-3b12 + 2b8 + 2b5 + 2b3 + 3b1 + 2) q^85 + (2b15 + b14 + 2b13 + 2b11 - 2b10) * q^87 + (-3b8 - 3b6 - 4b5 + 3b4 - 4) * q^89 + (-b15 - 2b14 + b13 - 2b11 + b10 - 2b7 + b2) q^91 + (6b12 + 8b8 - b6 - 2b5 - b4 + 3b3 - 3b1 + 6) q^93 + (b15 - b14 + b13 - b10 - b9 - 2b7 - 4b2) * q^95 + (b12 - 5b8 + 5b5 - 2b4 + 5b3 + b1 - 5) * q^97 + (2b14 + 2b11 + 2b9 + 2b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{5} + 4 q^{9}+O(q^{10}) $ 16 * q + 12 * q^5 + 4 * q^9	$ 16 q + 12 q^{5} + 4 q^{9} + 12 q^{13} + 12 q^{17} + 28 q^{21} + 8 q^{29} - 20 q^{33} + 16 q^{37} + 48 q^{41} - 20 q^{45} + 60 q^{49} + 32 q^{53} + 12 q^{57} - 4 q^{61} - 8 q^{65} + 12 q^{69} + 20 q^{73} + 48 q^{81} - 20 q^{85} - 52 q^{89} + 92 q^{93} - 28 q^{97}+O(q^{100}) $ 16 * q + 12 * q^5 + 4 * q^9 + 12 * q^13 + 12 * q^17 + 28 * q^21 + 8 * q^29 - 20 * q^33 + 16 * q^37 + 48 * q^41 - 20 * q^45 + 60 * q^49 + 32 * q^53 + 12 * q^57 - 4 * q^61 - 8 * q^65 + 12 * q^69 + 20 * q^73 + 48 * q^81 - 20 * q^85 - 52 * q^89 + 92 * q^93 - 28 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 $ x^16 - 4*x^15 + 5*x^14 - 2*x^13 + 5*x^12 - 8*x^11 - 12*x^10 + 32*x^9 - 36*x^8 + 64*x^7 - 48*x^6 - 64*x^5 + 80*x^4 - 64*x^3 + 320*x^2 - 512*x + 256 :

$\beta_{1}$	$=$	$ ( 2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 3 \nu^{12} - 6 \nu^{11} + 9 \nu^{10} + 34 \nu^{9} - 2 \nu^{8} + \cdots - 416 ) / 352 $ (2v^15 - v^14 - 4v^13 - 3v^12 - 6v^11 + 9v^10 + 34v^9 - 2v^8 - 48v^7 - 96v^6 + 48v^5 + 136v^4 + 144v^3 - 32v^2 - 352v - 416) / 352
$\beta_{2}$	$=$	$ ( - 4 \nu^{15} + 19 \nu^{14} - 54 \nu^{13} - 5 \nu^{12} - 8 \nu^{11} + 135 \nu^{10} + 66 \nu^{9} + \cdots + 4416 ) / 704 $ (-4v^15 + 19v^14 - 54v^13 - 5v^12 - 8v^11 + 135v^10 + 66v^9 - 328v^8 + 40v^7 - 428v^6 + 792v^5 + 864v^4 - 608v^3 - 464v^2 - 2912*v + 4416) / 704
$\beta_{3}$	$=$	$ ( 25 \nu^{15} - 6 \nu^{14} - 31 \nu^{13} - 76 \nu^{12} + 61 \nu^{11} + 182 \nu^{10} - 40 \nu^{8} + \cdots + 384 ) / 1408 $ (25v^15 - 6v^14 - 31v^13 - 76v^12 + 61v^11 + 182v^10 - 40v^8 - 580v^7 - 152v^6 + 704v^5 + 672v^4 + 16v^3 - 2336v^2 + 512v + 384) / 1408
$\beta_{4}$	$=$	$ ( - 3 \nu^{15} + 62 \nu^{14} - 27 \nu^{13} - 56 \nu^{12} - 167 \nu^{11} + 146 \nu^{10} + 400 \nu^{9} + \cdots + 2560 ) / 1408 $ (-3v^15 + 62v^14 - 27v^13 - 56v^12 - 167v^11 + 146v^10 + 400v^9 - 96v^8 + 28v^7 - 1352v^6 - 160v^5 + 1600v^4 + 1104v^3 + 224v^2 - 7040*v + 2560) / 1408
$\beta_{5}$	$=$	$ ( 25 \nu^{15} - 72 \nu^{14} + 13 \nu^{13} + 34 \nu^{12} + 149 \nu^{11} + 28 \nu^{10} - 396 \nu^{9} + \cdots - 2432 ) / 1408 $ (25v^15 - 72v^14 + 13v^13 + 34v^12 + 149v^11 + 28v^10 - 396v^9 + 136v^8 - 404v^7 + 992v^6 + 880v^5 - 1440v^4 - 688v^3 - 1984v^2 + 5440*v - 2432) / 1408
$\beta_{6}$	$=$	$ ( 17 \nu^{15} - 42 \nu^{14} + 17 \nu^{13} - 20 \nu^{12} + 101 \nu^{11} + 6 \nu^{10} - 208 \nu^{9} + \cdots - 3072 ) / 704 $ (17v^15 - 42v^14 + 17v^13 - 20v^12 + 101v^11 + 6v^10 - 208v^9 + 172v^8 - 308v^7 + 640v^6 + 48v^5 - 640v^4 + 48v^3 - 1152v^2 + 3968*v - 3072) / 704
$\beta_{7}$	$=$	$ ( - 15 \nu^{15} + 48 \nu^{14} + 13 \nu^{13} + 6 \nu^{12} - 123 \nu^{11} - 88 \nu^{10} + 292 \nu^{9} + \cdots + 384 ) / 704 $ (-15v^15 + 48v^14 + 13v^13 + 6v^12 - 123v^11 - 88v^10 + 292v^9 - 4v^8 + 444v^7 - 616v^6 - 768v^5 + 576v^4 + 720v^3 + 2528v^2 - 3904*v + 384) / 704
$\beta_{8}$	$=$	$ ( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + \cdots - 9408 ) / 704 $ (53v^15 - 139v^14 + 61v^13 + 3v^12 + 271v^11 - 59v^10 - 748v^9 + 628v^8 - 948v^7 + 2140v^6 + 528v^5 - 2912v^4 - 304v^3 - 3312v^2 + 12800*v - 9408) / 704
$\beta_{9}$	$=$	$ ( 37 \nu^{15} - 147 \nu^{14} + 119 \nu^{13} + 27 \nu^{12} + 233 \nu^{11} - 231 \nu^{10} - 754 \nu^{9} + \cdots - 12352 ) / 704 $ (37v^15 - 147v^14 + 119v^13 + 27v^12 + 233v^11 - 231v^10 - 754v^9 + 884v^8 - 708v^7 + 2420v^6 - 552v^5 - 3920v^4 + 688v^3 - 1648v^2 + 13408*v - 12352) / 704
$\beta_{10}$	$=$	$ ( - 54 \nu^{15} + 153 \nu^{14} - 72 \nu^{13} - 7 \nu^{12} - 302 \nu^{11} + 33 \nu^{10} + 818 \nu^{9} + \cdots + 9408 ) / 704 $ (-54v^15 + 153v^14 - 72v^13 - 7v^12 - 302v^11 + 33v^10 + 818v^9 - 604v^8 + 1088v^7 - 2420v^6 - 600v^5 + 3024v^4 + 128v^3 + 3504v^2 - 12576*v + 9408) / 704
$\beta_{11}$	$=$	$ ( 119 \nu^{15} - 308 \nu^{14} + 183 \nu^{13} - 30 \nu^{12} + 519 \nu^{11} - 128 \nu^{10} - 1600 \nu^{9} + \cdots - 21888 ) / 1408 $ (119v^15 - 308v^14 + 183v^13 - 30v^12 + 519v^11 - 128v^10 - 1600v^9 + 1656v^8 - 2172v^7 + 4416v^6 + 640v^5 - 6368v^4 + 1200v^3 - 6656v^2 + 27648*v - 21888) / 1408
$\beta_{12}$	$=$	$ ( 135 \nu^{15} - 254 \nu^{14} + 31 \nu^{13} - 120 \nu^{12} + 587 \nu^{11} + 326 \nu^{10} - 1344 \nu^{9} + \cdots - 11648 ) / 1408 $ (135v^15 - 254v^14 + 31v^13 - 120v^12 + 587v^11 + 326v^10 - 1344v^9 + 776v^8 - 2636v^7 + 3240v^6 + 2720v^5 - 3456v^4 - 16v^3 - 11488v^2 + 21376*v - 11648) / 1408
$\beta_{13}$	$=$	$ ( - 125 \nu^{15} + 402 \nu^{14} - 169 \nu^{13} - 16 \nu^{12} - 797 \nu^{11} + 62 \nu^{10} + \cdots + 18944 ) / 1408 $ (-125v^15 + 402v^14 - 169v^13 - 16v^12 - 797v^11 + 62v^10 + 2236v^9 - 1288v^8 + 2596v^7 - 6264v^6 - 2320v^5 + 7936v^4 + 1904v^3 + 10976v^2 - 33856*v + 18944) / 1408
$\beta_{14}$	$=$	$ ( - 163 \nu^{15} + 434 \nu^{14} - 143 \nu^{13} - 36 \nu^{12} - 851 \nu^{11} + 30 \nu^{10} + \cdots + 22528 ) / 1408 $ (-163v^15 + 434v^14 - 143v^13 - 36v^12 - 851v^11 + 30v^10 + 2412v^9 - 1456v^8 + 2844v^7 - 6392v^6 - 2672v^5 + 8928v^4 + 1936v^3 + 10528v^2 - 37056*v + 22528) / 1408
$\beta_{15}$	$=$	$ ( - 195 \nu^{15} + 606 \nu^{14} - 415 \nu^{13} - 32 \nu^{12} - 1099 \nu^{11} + 674 \nu^{10} + \cdots + 49152 ) / 1408 $ (-195v^15 + 606v^14 - 415v^13 - 32v^12 - 1099v^11 + 674v^10 + 3196v^9 - 3544v^8 + 3740v^7 - 9448v^6 + 816v^5 + 13760v^4 - 1648v^3 + 11040v^2 - 57152*v + 49152) / 1408

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{13} + 2\beta_{11} - 2\beta_{7} - 2\beta_{4} - 2\beta_{2} ) / 4 $ (b15 + b13 + 2b11 - 2b7 - 2b4 - 2b2) / 4
$\nu^{2}$	$=$	$ ( -\beta_{14} + \beta_{13} - 2\beta_{12} + 2\beta_{11} - \beta_{10} - 2\beta_{8} - 2\beta_{4} + 4\beta_{3} + \beta_{2} ) / 4 $ (-b14 + b13 - 2b12 + 2b11 - b10 - 2b8 - 2b4 + 4*b3 + b2) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + \beta_{14} + 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 4 $ (b15 + b14 + 2b13 + 4b12 + 2b11 - b10 + 2b9 - 2b8 - 2b6 - 2b5 - 4b4 - 4*b3 + b2 - 2) / 4
$\nu^{4}$	$=$	$ ( - \beta_{15} + 2 \beta_{14} - \beta_{13} + 2 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + 2 \beta_{8} + \cdots + 2 ) / 4 $ (-b15 + 2b14 - b13 + 2b11 - 2b10 - 6b9 + 2b8 + 2b6 - 4b5 - 8b3 + 2*b1 + 2) / 4
$\nu^{5}$	$=$	$ ( 5 \beta_{14} - 3 \beta_{13} + 4 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} + \cdots - 10 ) / 4 $ (5b14 - 3b13 + 4b11 + 3b10 + 2b9 + 2b8 + 6b7 - 6b6 + 6b5 - 6b4 + 8b3 + 7b2 - 10) / 4
$\nu^{6}$	$=$	$ ( 7 \beta_{15} + 9 \beta_{14} + 2 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 20 \beta_{8} + 2 \beta_{6} + \cdots + 10 ) / 4 $ (7b15 + 9b14 + 2b12 - 7b10 + 2b9 + 20b8 + 2b6 - 8b5 + 2b4 - 12b3 + b2 - 2*b1 + 10) / 4
$\nu^{7}$	$=$	$ ( \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} + 12 \beta_{8} + \cdots - 4 \beta_1 ) / 4 $ (b15 + 6b14 - 5b13 + 4b12 - 2b11 + 10b10 + 12b8 + 10b7 + 20b6 + 12b5 + 10b4 - 20b3 + 8b2 - 4*b1) / 4
$\nu^{8}$	$=$	$ ( - 6 \beta_{15} + 19 \beta_{14} - \beta_{13} - 10 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} + 30 \beta_{8} + \cdots + 40 ) / 4 $ (-6b15 + 19b14 - b13 - 10b12 - 2b11 + 3b10 + 30b8 + 4b7 + 8b6 + 24b5 + 18b4 + 4b3 + 25b2 - 8*b1 + 40) / 4
$\nu^{9}$	$=$	$ ( 17 \beta_{15} + 19 \beta_{14} - 8 \beta_{12} - 2 \beta_{11} + \beta_{10} + 18 \beta_{9} + 34 \beta_{8} + \cdots + 62 ) / 4 $ (17b15 + 19b14 - 8b12 - 2b11 + b10 + 18b9 + 34b8 - 8b7 + 22b6 + 46b5 + 20b3 - 17b2 + 16b1 + 62) / 4
$\nu^{10}$	$=$	$ ( 21 \beta_{15} + 8 \beta_{14} + 27 \beta_{13} - 24 \beta_{12} + 30 \beta_{11} - 24 \beta_{10} + 6 \beta_{9} + \cdots + 6 ) / 4 $ (21b15 + 8b14 + 27b13 - 24b12 + 30b11 - 24b10 + 6b9 + 6b8 - 48b7 + 54b6 + 64b5 + 24b4 - 26b2 - 30b1 + 6) / 4
$\nu^{11}$	$=$	$ ( - 38 \beta_{15} + \beta_{14} + 7 \beta_{13} - 32 \beta_{11} + 7 \beta_{10} + 14 \beta_{9} - 66 \beta_{8} + \cdots + 2 ) / 4 $ (-38b15 + b14 + 7b13 - 32b11 + 7b10 + 14b9 - 66b8 + 38b7 + 22b6 + 110b5 - 22b4 + 88b3 + 39b2 - 20*b1 + 2) / 4
$\nu^{12}$	$=$	$ ( 7 \beta_{15} + 7 \beta_{14} + 62 \beta_{13} + 18 \beta_{12} + 8 \beta_{11} - 69 \beta_{10} - 34 \beta_{9} + \cdots + 62 ) / 4 $ (7b15 + 7b14 + 62b13 + 18b12 + 8b11 - 69b10 - 34b9 - 76b7 - 18b6 + 76b5 - 2b4 - 92b3 - 85b2 - 2b1 + 62) / 4
$\nu^{13}$	$=$	$ ( - 21 \beta_{15} - 20 \beta_{14} - 53 \beta_{13} - 32 \beta_{12} + 46 \beta_{11} - 52 \beta_{9} + \cdots - 132 ) / 4 $ (-21b15 - 20b14 - 53b13 - 32b12 + 46b11 - 52b9 - 172b8 + 42b7 + 172b5 - 26b4 + 132b3 - 98b2 - 32*b1 - 132) / 4
$\nu^{14}$	$=$	$ ( 23 \beta_{14} + 81 \beta_{13} + 78 \beta_{12} + 18 \beta_{11} - 65 \beta_{10} + 16 \beta_{9} + \cdots - 328 ) / 4 $ (23b14 + 81b13 + 78b12 + 18b11 - 65b10 + 16b9 - 10b8 + 16b7 - 208b6 - 130b4 + 108b3 - 23b2 - 208*b1 - 328) / 4
$\nu^{15}$	$=$	$ ( - 3 \beta_{15} - 123 \beta_{14} - 6 \beta_{13} + 108 \beta_{12} - 126 \beta_{11} + 3 \beta_{10} + \cdots + 190 ) / 4 $ (-3b15 - 123b14 - 6b13 + 108b12 - 126b11 + 3b10 - 126b9 + 54b8 + 192b7 + 54b6 - 82b5 + 108b4 - 244b3 - 123b2 + 190) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$1$	$-1$	$-\beta_{5}$

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} $

63.1

1.41121	+	0.0921725i
−0.00757716	+	1.41419i
−0.713659	−	1.22094i
1.17605	−	0.785427i
−0.873468	−	1.11223i
1.08916	+	0.902074i
−1.39427	+	0.236640i
1.31256	−	0.526485i
1.41121	−	0.0921725i
−0.00757716	−	1.41419i
−0.713659	+	1.22094i
1.17605	+	0.785427i
−0.873468	+	1.11223i
1.08916	−	0.902074i
−1.39427	−	0.236640i
1.31256	+	0.526485i

−1.81380

1.04720i

−0.894007

−

0.894007i

−4.37156

1.17136i

0.693255

−

1.20075i

63.2

−1.40004

0.808315i

1.52798

1.52798i

1.97429

−

0.529008i

−0.193255

0.334727i

63.3

1.40004

−

0.808315i

1.52798

1.52798i

−1.97429

0.529008i

−0.193255

0.334727i

63.4

1.81380

−

1.04720i

−0.894007

−

0.894007i

4.37156

−

1.17136i

0.693255

−

1.20075i

319.1

−2.16981

−

1.25274i

2.19962

2.19962i

−0.152604

0.569525i

1.63871

2.83834i

319.2

−0.736159

−

0.425021i

0.166404

0.166404i

−0.684384

2.55416i

−1.13871

−

1.97231i

319.3

0.736159

0.425021i

0.166404

0.166404i

0.684384

−

2.55416i

−1.13871

−

1.97231i

319.4

2.16981

1.25274i

2.19962

2.19962i

0.152604

−

0.569525i

1.63871

2.83834i

383.1

−1.81380

−

1.04720i

−0.894007

0.894007i

−4.37156

−

1.17136i

0.693255

1.20075i

383.2

−1.40004

−

0.808315i

1.52798

−

1.52798i

1.97429

0.529008i

−0.193255

−

0.334727i

383.3

1.40004

0.808315i

1.52798

−

1.52798i

−1.97429

−

0.529008i

−0.193255

−

0.334727i

383.4

1.81380

1.04720i

−0.894007

0.894007i

4.37156

1.17136i

0.693255

1.20075i

639.1

−2.16981

1.25274i

2.19962

−

2.19962i

−0.152604

−

0.569525i

1.63871

−

2.83834i

639.2

−0.736159

0.425021i

0.166404

−

0.166404i

−0.684384

−

2.55416i

−1.13871

1.97231i

639.3

0.736159

−

0.425021i

0.166404

−

0.166404i

0.684384

2.55416i

−1.13871

1.97231i

639.4

2.16981

−

1.25274i

2.19962

−

2.19962i

0.152604

0.569525i

1.63871

−

2.83834i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.f	odd	12	1	inner
52.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	832.2.bu.n		16
4.b	odd	2	1	inner	832.2.bu.n		16
8.b	even	2	1		52.2.l.b	✓	16
8.d	odd	2	1		52.2.l.b	✓	16
13.f	odd	12	1	inner	832.2.bu.n		16
24.f	even	2	1		468.2.cb.f		16
24.h	odd	2	1		468.2.cb.f		16
52.l	even	12	1	inner	832.2.bu.n		16
104.e	even	2	1		676.2.l.k		16
104.h	odd	2	1		676.2.l.k		16
104.j	odd	4	1		676.2.l.i		16
104.j	odd	4	1		676.2.l.m		16
104.m	even	4	1		676.2.l.i		16
104.m	even	4	1		676.2.l.m		16
104.n	odd	6	1		676.2.f.h		16
104.n	odd	6	1		676.2.l.m		16
104.p	odd	6	1		676.2.f.i		16
104.p	odd	6	1		676.2.l.i		16
104.r	even	6	1		676.2.f.h		16
104.r	even	6	1		676.2.l.m		16
104.s	even	6	1		676.2.f.i		16
104.s	even	6	1		676.2.l.i		16
104.u	even	12	1		52.2.l.b	✓	16
104.u	even	12	1		676.2.f.h		16
104.u	even	12	1		676.2.f.i		16
104.u	even	12	1		676.2.l.k		16
104.x	odd	12	1		52.2.l.b	✓	16
104.x	odd	12	1		676.2.f.h		16
104.x	odd	12	1		676.2.f.i		16
104.x	odd	12	1		676.2.l.k		16
312.bo	even	12	1		468.2.cb.f		16
312.bq	odd	12	1		468.2.cb.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.l.b	✓	16	8.b	even	2	1
52.2.l.b	✓	16	8.d	odd	2	1
52.2.l.b	✓	16	104.u	even	12	1
52.2.l.b	✓	16	104.x	odd	12	1
468.2.cb.f		16	24.f	even	2	1
468.2.cb.f		16	24.h	odd	2	1
468.2.cb.f		16	312.bo	even	12	1
468.2.cb.f		16	312.bq	odd	12	1
676.2.f.h		16	104.n	odd	6	1
676.2.f.h		16	104.r	even	6	1
676.2.f.h		16	104.u	even	12	1
676.2.f.h		16	104.x	odd	12	1
676.2.f.i		16	104.p	odd	6	1
676.2.f.i		16	104.s	even	6	1
676.2.f.i		16	104.u	even	12	1
676.2.f.i		16	104.x	odd	12	1
676.2.l.i		16	104.j	odd	4	1
676.2.l.i		16	104.m	even	4	1
676.2.l.i		16	104.p	odd	6	1
676.2.l.i		16	104.s	even	6	1
676.2.l.k		16	104.e	even	2	1
676.2.l.k		16	104.h	odd	2	1
676.2.l.k		16	104.u	even	12	1
676.2.l.k		16	104.x	odd	12	1
676.2.l.m		16	104.j	odd	4	1
676.2.l.m		16	104.m	even	4	1
676.2.l.m		16	104.n	odd	6	1
676.2.l.m		16	104.r	even	6	1
832.2.bu.n		16	1.a	even	1	1	trivial
832.2.bu.n		16	4.b	odd	2	1	inner
832.2.bu.n		16	13.f	odd	12	1	inner
832.2.bu.n		16	52.l	even	12	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(832, [\chi])$:

$ T_{3}^{16} - 14T_{3}^{14} + 131T_{3}^{12} - 686T_{3}^{10} + 2605T_{3}^{8} - 5824T_{3}^{6} + 9164T_{3}^{4} - 5824T_{3}^{2} + 2704 $

$ T_{5}^{8} - 6T_{5}^{7} + 18T_{5}^{6} - 18T_{5}^{5} + 5T_{5}^{4} + 72T_{5}^{2} - 24T_{5} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} - 14 T^{14} + \cdots + 2704 $ T^16 - 14T^14 + 131T^12 - 686T^10 + 2605T^8 - 5824T^6 + 9164T^4 - 5824*T^2 + 2704
$5$	$ (T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 4)^{2} $ (T^8 - 6T^7 + 18T^6 - 18T^5 + 5T^4 + 72T^2 - 24T + 4)^2
$7$	$ T^{16} - 30 T^{14} + \cdots + 43264 $ T^16 - 30T^14 + 207T^12 + 2790T^10 - 1399T^8 - 91512T^6 + 303408T^4 + 204672*T^2 + 43264
$11$	$ T^{16} + 18 T^{14} + \cdots + 692224 $ T^16 + 18T^14 - 57T^12 - 2970T^10 + 25241T^8 + 31680T^6 - 124992T^4 - 159744*T^2 + 692224
$13$	$ (T^{8} - 6 T^{7} + \cdots + 28561)^{2} $ (T^8 - 6T^7 + 33T^6 - 102T^5 + 428T^4 - 1326T^3 + 5577T^2 - 13182*T + 28561)^2
$17$	$ (T^{8} - 6 T^{7} + 2 T^{6} + \cdots + 1)^{2} $ (T^8 - 6T^7 + 2T^6 + 60T^5 + 39T^4 - 300T^3 + 290T^2 + 30*T + 1)^2
$19$	$ T^{16} - 54 T^{14} + \cdots + 77228944 $ T^16 - 54T^14 + 891T^12 + 4374T^10 - 72643T^8 - 316872T^6 + 4389420T^4 + 34378656*T^2 + 77228944
$23$	$ T^{16} + 106 T^{14} + \cdots + 77228944 $ T^16 + 106T^14 + 8843T^12 + 232858T^10 + 4615261T^8 + 23024144T^6 + 87130316T^4 + 91395200*T^2 + 77228944
$29$	$ (T^{8} - 4 T^{7} + \cdots + 51529)^{2} $ (T^8 - 4T^7 + 62T^6 - 304T^5 + 3319T^4 - 13040T^3 + 49094T^2 - 55388*T + 51529)^2
$31$	$ T^{16} + \cdots + 1235663104 $ T^16 + 9072T^12 + 18310880T^8 + 9894433536*T^4 + 1235663104
$37$	$ (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{4} $ (T^4 - 4T^3 + 5T^2 - 2*T + 1)^4
$41$	$ (T^{4} - 12 T^{3} + \cdots + 81)^{4} $ (T^4 - 12T^3 + 45T^2 - 54*T + 81)^4
$43$	$ T^{16} + \cdots + 5671027857664 $ T^16 + 266T^14 + 48143T^12 + 4728770T^10 + 337890073T^8 + 13276514728T^6 + 359783787440T^4 + 1531577976448*T^2 + 5671027857664
$47$	$ T^{16} + \cdots + 5671027857664 $ T^16 + 20976T^12 + 70290656T^8 + 39057328896*T^4 + 5671027857664
$53$	$ (T^{4} - 8 T^{3} + \cdots - 128)^{4} $ (T^4 - 8T^3 - 3T^2 + 112*T - 128)^4
$59$	$ T^{16} + \cdots + 171720267307264 $ T^16 - 6T^14 - 8121T^12 + 48798T^10 + 53073017T^8 + 128241144T^6 - 106493647056T^4 - 206627151744*T^2 + 171720267307264
$61$	$ (T^{8} + 2 T^{7} + \cdots + 6889)^{2} $ (T^8 + 2T^7 + 62T^6 + 200T^5 + 3763T^4 + 9496T^3 + 20150T^2 + 13114*T + 6889)^2
$67$	$ T^{16} + \cdots + 2205735869584 $ T^16 - 126T^14 + 1587T^12 + 466830T^10 - 3260179T^8 - 1367500680T^6 + 39908056812T^4 + 548171044512*T^2 + 2205735869584
$71$	$ T^{16} + \cdots + 5067731344 $ T^16 - 126T^14 - 141T^12 + 684558T^10 + 28929197T^8 - 66891096T^6 - 336235956T^4 + 876466656*T^2 + 5067731344
$73$	$ (T^{8} - 10 T^{7} + \cdots + 676)^{2} $ (T^8 - 10T^7 + 50T^6 - 6T^5 + 3533T^4 - 35424T^3 + 177608T^2 - 15496*T + 676)^2
$79$	$ (T^{8} + 160 T^{6} + \cdots + 53248)^{2} $ (T^8 + 160T^6 + 7040T^4 + 51200*T^2 + 53248)^2
$83$	$ T^{16} + \cdots + 538409280507904 $ T^16 + 61704T^12 + 891937424T^8 + 3819952817664*T^4 + 538409280507904
$89$	$ (T^{8} + 26 T^{7} + \cdots + 2896804)^{2} $ (T^8 + 26T^7 + 215T^6 + 1736T^5 + 19699T^4 + 5950T^3 + 479246T^2 - 2474708*T + 2896804)^2
$97$	$ (T^{8} + 14 T^{7} + \cdots + 2116)^{2} $ (T^8 + 14T^7 + 323T^6 + 1812T^5 + 9371T^4 + 23070T^3 + 25694T^2 + 12236*T + 2116)^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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.bu (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{14} - \beta_{2}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14}) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_1 - 1) q^{9}+O(q^{10}) \) q + (-b14 - b2) * q^3 + (-b8 - b6 + b3) * q^5 + (-b15 - b14) * q^7 + (-b8 - b6 - b1 - 1) * q^9	\( q + ( - \beta_{14} - \beta_{2}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14}) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_1 - 1) q^{9} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{2}) q^{11}+ \cdots + (2 \beta_{14} + 2 \beta_{11} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) q + (-b14 - b2) * q^3 + (-b8 - b6 + b3) * q^5 + (-b15 - b14) * q^7 + (-b8 - b6 - b1 - 1) * q^9 + (-b15 - b11 + b7 + b2) * q^11 + (b8 + b6 + 2b5 + 2b4 - 2b3 - b1 + 2) q^13 + (b15 + b13 - b11 - b10 - b9 - b7 - 2b2) q^15 + (-b12 + b5 + b4 + b1 + 2) * q^17 + (b15 + b13 + b11 - 2b7 - b2) q^19 + (-2b12 - 2b8 - b6 - 2b5 - 2b4) * q^21 + (-b14 + 2b10 + 2b9 + 2b7 + b2) q^23 + (-b12 - 2b8 - b6 - b5 + b4 - b3 - b1 - 1) q^25 + (b13 - b11 - b10 - b9 - b7 + b2) * q^27 + (-2b12 - b8 - 2b6 - 4b5 + 2b1 + 1) * q^29 + (b15 + b11 - b10 - 2b9 - b7 + b2) q^31 + (-3b12 + b3 + 3b1 + 1) * q^33 + (b14 + b11 - b9) * q^35 + (b5 + 1) * q^37 + (b14 - b13 - b11 - b10 + b9 - b7 - b2) * q^39 + (-3b5 + 3) q^41 + (b15 + b14 - 3b13 + b11 + b10 + 2b9 + 2b7) q^43 + (-b12 - 3b8 + 3b5 + 2b4 + 2b3 - b1 - 2) * q^45 + (3b14 + 3b11 - b7) * q^47 + (-b12 - 4b8 - 2b6 - 3b4 + b3 - 2b1 - 1) * q^49 + (b15 + b14 - b10 + 2b9) q^51 + (-2b12 + 2b6 + b5 + b4 + b1 + 4) * q^53 + (-2b15 + 2b14 - b13 - b11 + b10 + 2b7 + 4b2) * q^55 + (2b12 - b6 + 2b5 - 4b1 - 2) q^57 + (-3b15 - 3b14 - 2b13 + 4b10 + 2b9 + 4b7 + 2b2) q^59 + (b12 - 2b6 + b5 + b4 - 3b1 - 2) * q^61 + (-b14 + b13 + 2b11 + b10 + 2b9 - b2) * q^63 + (-3b12 - 2b8 + b6 + b5 + 2b4 - 3b3 + 3b1 + 2) q^65 + (2b14 - b11 - 3b9 - b2) * q^67 + (2b12 - 3b6 - 2b4 + b1 - 1) q^69 + (-b15 - 3b13 - 3b11 - 2b9 + 2b7 + b2) * q^71 + (b12 + 3b8 + 2b6 - b5 + 4b4 - 3b3 + 5) * q^73 + (2b14 + b13 - b11 - 2b10 - b9 - b7 - 2b2) q^75 + (-4b8 - 4b5 + b4 - 5b3 - b1 - 2) q^77 + (-2b13 - 2b11 + 2b10 + 2b7 + 2b2) q^79 + (2b12 + 7b8 + 4b5 - 2b4 + 2b3 + 5) q^81 + (2b15 + 2b11 - 2b10 - 3b9 - 2b7 + b2) q^83 + (-3b12 + 2b8 + 2b5 + 2b3 + 3b1 + 2) q^85 + (2b15 + b14 + 2b13 + 2b11 - 2b10) * q^87 + (-3b8 - 3b6 - 4b5 + 3b4 - 4) * q^89 + (-b15 - 2b14 + b13 - 2b11 + b10 - 2b7 + b2) q^91 + (6b12 + 8b8 - b6 - 2b5 - b4 + 3b3 - 3b1 + 6) q^93 + (b15 - b14 + b13 - b10 - b9 - 2b7 - 4b2) * q^95 + (b12 - 5b8 + 5b5 - 2b4 + 5b3 + b1 - 5) * q^97 + (2b14 + 2b11 + 2b9 + 2b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{5} + 4 q^{9}+O(q^{10}) \) 16 * q + 12 * q^5 + 4 * q^9	\( 16 q + 12 q^{5} + 4 q^{9} + 12 q^{13} + 12 q^{17} + 28 q^{21} + 8 q^{29} - 20 q^{33} + 16 q^{37} + 48 q^{41} - 20 q^{45} + 60 q^{49} + 32 q^{53} + 12 q^{57} - 4 q^{61} - 8 q^{65} + 12 q^{69} + 20 q^{73} + 48 q^{81} - 20 q^{85} - 52 q^{89} + 92 q^{93} - 28 q^{97}+O(q^{100}) \) 16 * q + 12 * q^5 + 4 * q^9 + 12 * q^13 + 12 * q^17 + 28 * q^21 + 8 * q^29 - 20 * q^33 + 16 * q^37 + 48 * q^41 - 20 * q^45 + 60 * q^49 + 32 * q^53 + 12 * q^57 - 4 * q^61 - 8 * q^65 + 12 * q^69 + 20 * q^73 + 48 * q^81 - 20 * q^85 - 52 * q^89 + 92 * q^93 - 28 * q^97

Introduction

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Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Hecke kernels

Hecke characteristic polynomials

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

Label	832.2.bu.n

Level	$832$
Weight	$2$
Character orbit	832.bu
Analytic conductor	$6.644$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.64355344817\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	16.0.102930383934669717504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 \) x^16 - 4x^15 + 5x^14 - 2x^13 + 5x^12 - 8x^11 - 12x^10 + 32x^9 - 36x^8 + 64x^7 - 48x^6 - 64x^5 + 80x^4 - 64x^3 + 320x^2 - 512*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(\beta_{1}\)	\(=\)	\( ( 2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 3 \nu^{12} - 6 \nu^{11} + 9 \nu^{10} + 34 \nu^{9} - 2 \nu^{8} + \cdots - 416 ) / 352 \) (2v^15 - v^14 - 4v^13 - 3v^12 - 6v^11 + 9v^10 + 34v^9 - 2v^8 - 48v^7 - 96v^6 + 48v^5 + 136v^4 + 144v^3 - 32v^2 - 352v - 416) / 352
\(\beta_{2}\)	\(=\)	\( ( - 4 \nu^{15} + 19 \nu^{14} - 54 \nu^{13} - 5 \nu^{12} - 8 \nu^{11} + 135 \nu^{10} + 66 \nu^{9} + \cdots + 4416 ) / 704 \) (-4v^15 + 19v^14 - 54v^13 - 5v^12 - 8v^11 + 135v^10 + 66v^9 - 328v^8 + 40v^7 - 428v^6 + 792v^5 + 864v^4 - 608v^3 - 464v^2 - 2912*v + 4416) / 704
\(\beta_{3}\)	\(=\)	\( ( 25 \nu^{15} - 6 \nu^{14} - 31 \nu^{13} - 76 \nu^{12} + 61 \nu^{11} + 182 \nu^{10} - 40 \nu^{8} + \cdots + 384 ) / 1408 \) (25v^15 - 6v^14 - 31v^13 - 76v^12 + 61v^11 + 182v^10 - 40v^8 - 580v^7 - 152v^6 + 704v^5 + 672v^4 + 16v^3 - 2336v^2 + 512v + 384) / 1408
\(\beta_{4}\)	\(=\)	\( ( - 3 \nu^{15} + 62 \nu^{14} - 27 \nu^{13} - 56 \nu^{12} - 167 \nu^{11} + 146 \nu^{10} + 400 \nu^{9} + \cdots + 2560 ) / 1408 \) (-3v^15 + 62v^14 - 27v^13 - 56v^12 - 167v^11 + 146v^10 + 400v^9 - 96v^8 + 28v^7 - 1352v^6 - 160v^5 + 1600v^4 + 1104v^3 + 224v^2 - 7040*v + 2560) / 1408
\(\beta_{5}\)	\(=\)	\( ( 25 \nu^{15} - 72 \nu^{14} + 13 \nu^{13} + 34 \nu^{12} + 149 \nu^{11} + 28 \nu^{10} - 396 \nu^{9} + \cdots - 2432 ) / 1408 \) (25v^15 - 72v^14 + 13v^13 + 34v^12 + 149v^11 + 28v^10 - 396v^9 + 136v^8 - 404v^7 + 992v^6 + 880v^5 - 1440v^4 - 688v^3 - 1984v^2 + 5440*v - 2432) / 1408
\(\beta_{6}\)	\(=\)	\( ( 17 \nu^{15} - 42 \nu^{14} + 17 \nu^{13} - 20 \nu^{12} + 101 \nu^{11} + 6 \nu^{10} - 208 \nu^{9} + \cdots - 3072 ) / 704 \) (17v^15 - 42v^14 + 17v^13 - 20v^12 + 101v^11 + 6v^10 - 208v^9 + 172v^8 - 308v^7 + 640v^6 + 48v^5 - 640v^4 + 48v^3 - 1152v^2 + 3968*v - 3072) / 704
\(\beta_{7}\)	\(=\)	\( ( - 15 \nu^{15} + 48 \nu^{14} + 13 \nu^{13} + 6 \nu^{12} - 123 \nu^{11} - 88 \nu^{10} + 292 \nu^{9} + \cdots + 384 ) / 704 \) (-15v^15 + 48v^14 + 13v^13 + 6v^12 - 123v^11 - 88v^10 + 292v^9 - 4v^8 + 444v^7 - 616v^6 - 768v^5 + 576v^4 + 720v^3 + 2528v^2 - 3904*v + 384) / 704
\(\beta_{8}\)	\(=\)	\( ( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + \cdots - 9408 ) / 704 \) (53v^15 - 139v^14 + 61v^13 + 3v^12 + 271v^11 - 59v^10 - 748v^9 + 628v^8 - 948v^7 + 2140v^6 + 528v^5 - 2912v^4 - 304v^3 - 3312v^2 + 12800*v - 9408) / 704
\(\beta_{9}\)	\(=\)	\( ( 37 \nu^{15} - 147 \nu^{14} + 119 \nu^{13} + 27 \nu^{12} + 233 \nu^{11} - 231 \nu^{10} - 754 \nu^{9} + \cdots - 12352 ) / 704 \) (37v^15 - 147v^14 + 119v^13 + 27v^12 + 233v^11 - 231v^10 - 754v^9 + 884v^8 - 708v^7 + 2420v^6 - 552v^5 - 3920v^4 + 688v^3 - 1648v^2 + 13408*v - 12352) / 704
\(\beta_{10}\)	\(=\)	\( ( - 54 \nu^{15} + 153 \nu^{14} - 72 \nu^{13} - 7 \nu^{12} - 302 \nu^{11} + 33 \nu^{10} + 818 \nu^{9} + \cdots + 9408 ) / 704 \) (-54v^15 + 153v^14 - 72v^13 - 7v^12 - 302v^11 + 33v^10 + 818v^9 - 604v^8 + 1088v^7 - 2420v^6 - 600v^5 + 3024v^4 + 128v^3 + 3504v^2 - 12576*v + 9408) / 704
\(\beta_{11}\)	\(=\)	\( ( 119 \nu^{15} - 308 \nu^{14} + 183 \nu^{13} - 30 \nu^{12} + 519 \nu^{11} - 128 \nu^{10} - 1600 \nu^{9} + \cdots - 21888 ) / 1408 \) (119v^15 - 308v^14 + 183v^13 - 30v^12 + 519v^11 - 128v^10 - 1600v^9 + 1656v^8 - 2172v^7 + 4416v^6 + 640v^5 - 6368v^4 + 1200v^3 - 6656v^2 + 27648*v - 21888) / 1408
\(\beta_{12}\)	\(=\)	\( ( 135 \nu^{15} - 254 \nu^{14} + 31 \nu^{13} - 120 \nu^{12} + 587 \nu^{11} + 326 \nu^{10} - 1344 \nu^{9} + \cdots - 11648 ) / 1408 \) (135v^15 - 254v^14 + 31v^13 - 120v^12 + 587v^11 + 326v^10 - 1344v^9 + 776v^8 - 2636v^7 + 3240v^6 + 2720v^5 - 3456v^4 - 16v^3 - 11488v^2 + 21376*v - 11648) / 1408
\(\beta_{13}\)	\(=\)	\( ( - 125 \nu^{15} + 402 \nu^{14} - 169 \nu^{13} - 16 \nu^{12} - 797 \nu^{11} + 62 \nu^{10} + \cdots + 18944 ) / 1408 \) (-125v^15 + 402v^14 - 169v^13 - 16v^12 - 797v^11 + 62v^10 + 2236v^9 - 1288v^8 + 2596v^7 - 6264v^6 - 2320v^5 + 7936v^4 + 1904v^3 + 10976v^2 - 33856*v + 18944) / 1408
\(\beta_{14}\)	\(=\)	\( ( - 163 \nu^{15} + 434 \nu^{14} - 143 \nu^{13} - 36 \nu^{12} - 851 \nu^{11} + 30 \nu^{10} + \cdots + 22528 ) / 1408 \) (-163v^15 + 434v^14 - 143v^13 - 36v^12 - 851v^11 + 30v^10 + 2412v^9 - 1456v^8 + 2844v^7 - 6392v^6 - 2672v^5 + 8928v^4 + 1936v^3 + 10528v^2 - 37056*v + 22528) / 1408
\(\beta_{15}\)	\(=\)	\( ( - 195 \nu^{15} + 606 \nu^{14} - 415 \nu^{13} - 32 \nu^{12} - 1099 \nu^{11} + 674 \nu^{10} + \cdots + 49152 ) / 1408 \) (-195v^15 + 606v^14 - 415v^13 - 32v^12 - 1099v^11 + 674v^10 + 3196v^9 - 3544v^8 + 3740v^7 - 9448v^6 + 816v^5 + 13760v^4 - 1648v^3 + 11040v^2 - 57152*v + 49152) / 1408

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{13} + 2\beta_{11} - 2\beta_{7} - 2\beta_{4} - 2\beta_{2} ) / 4 \) (b15 + b13 + 2b11 - 2b7 - 2b4 - 2b2) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{14} + \beta_{13} - 2\beta_{12} + 2\beta_{11} - \beta_{10} - 2\beta_{8} - 2\beta_{4} + 4\beta_{3} + \beta_{2} ) / 4 \) (-b14 + b13 - 2b12 + 2b11 - b10 - 2b8 - 2b4 + 4*b3 + b2) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + \beta_{14} + 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 4 \) (b15 + b14 + 2b13 + 4b12 + 2b11 - b10 + 2b9 - 2b8 - 2b6 - 2b5 - 4b4 - 4*b3 + b2 - 2) / 4
\(\nu^{4}\)	\(=\)	\( ( - \beta_{15} + 2 \beta_{14} - \beta_{13} + 2 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + 2 \beta_{8} + \cdots + 2 ) / 4 \) (-b15 + 2b14 - b13 + 2b11 - 2b10 - 6b9 + 2b8 + 2b6 - 4b5 - 8b3 + 2*b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{14} - 3 \beta_{13} + 4 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} + \cdots - 10 ) / 4 \) (5b14 - 3b13 + 4b11 + 3b10 + 2b9 + 2b8 + 6b7 - 6b6 + 6b5 - 6b4 + 8b3 + 7b2 - 10) / 4
\(\nu^{6}\)	\(=\)	\( ( 7 \beta_{15} + 9 \beta_{14} + 2 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 20 \beta_{8} + 2 \beta_{6} + \cdots + 10 ) / 4 \) (7b15 + 9b14 + 2b12 - 7b10 + 2b9 + 20b8 + 2b6 - 8b5 + 2b4 - 12b3 + b2 - 2*b1 + 10) / 4
\(\nu^{7}\)	\(=\)	\( ( \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} + 12 \beta_{8} + \cdots - 4 \beta_1 ) / 4 \) (b15 + 6b14 - 5b13 + 4b12 - 2b11 + 10b10 + 12b8 + 10b7 + 20b6 + 12b5 + 10b4 - 20b3 + 8b2 - 4*b1) / 4
\(\nu^{8}\)	\(=\)	\( ( - 6 \beta_{15} + 19 \beta_{14} - \beta_{13} - 10 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} + 30 \beta_{8} + \cdots + 40 ) / 4 \) (-6b15 + 19b14 - b13 - 10b12 - 2b11 + 3b10 + 30b8 + 4b7 + 8b6 + 24b5 + 18b4 + 4b3 + 25b2 - 8*b1 + 40) / 4
\(\nu^{9}\)	\(=\)	\( ( 17 \beta_{15} + 19 \beta_{14} - 8 \beta_{12} - 2 \beta_{11} + \beta_{10} + 18 \beta_{9} + 34 \beta_{8} + \cdots + 62 ) / 4 \) (17b15 + 19b14 - 8b12 - 2b11 + b10 + 18b9 + 34b8 - 8b7 + 22b6 + 46b5 + 20b3 - 17b2 + 16b1 + 62) / 4
\(\nu^{10}\)	\(=\)	\( ( 21 \beta_{15} + 8 \beta_{14} + 27 \beta_{13} - 24 \beta_{12} + 30 \beta_{11} - 24 \beta_{10} + 6 \beta_{9} + \cdots + 6 ) / 4 \) (21b15 + 8b14 + 27b13 - 24b12 + 30b11 - 24b10 + 6b9 + 6b8 - 48b7 + 54b6 + 64b5 + 24b4 - 26b2 - 30b1 + 6) / 4
\(\nu^{11}\)	\(=\)	\( ( - 38 \beta_{15} + \beta_{14} + 7 \beta_{13} - 32 \beta_{11} + 7 \beta_{10} + 14 \beta_{9} - 66 \beta_{8} + \cdots + 2 ) / 4 \) (-38b15 + b14 + 7b13 - 32b11 + 7b10 + 14b9 - 66b8 + 38b7 + 22b6 + 110b5 - 22b4 + 88b3 + 39b2 - 20*b1 + 2) / 4
\(\nu^{12}\)	\(=\)	\( ( 7 \beta_{15} + 7 \beta_{14} + 62 \beta_{13} + 18 \beta_{12} + 8 \beta_{11} - 69 \beta_{10} - 34 \beta_{9} + \cdots + 62 ) / 4 \) (7b15 + 7b14 + 62b13 + 18b12 + 8b11 - 69b10 - 34b9 - 76b7 - 18b6 + 76b5 - 2b4 - 92b3 - 85b2 - 2b1 + 62) / 4
\(\nu^{13}\)	\(=\)	\( ( - 21 \beta_{15} - 20 \beta_{14} - 53 \beta_{13} - 32 \beta_{12} + 46 \beta_{11} - 52 \beta_{9} + \cdots - 132 ) / 4 \) (-21b15 - 20b14 - 53b13 - 32b12 + 46b11 - 52b9 - 172b8 + 42b7 + 172b5 - 26b4 + 132b3 - 98b2 - 32*b1 - 132) / 4
\(\nu^{14}\)	\(=\)	\( ( 23 \beta_{14} + 81 \beta_{13} + 78 \beta_{12} + 18 \beta_{11} - 65 \beta_{10} + 16 \beta_{9} + \cdots - 328 ) / 4 \) (23b14 + 81b13 + 78b12 + 18b11 - 65b10 + 16b9 - 10b8 + 16b7 - 208b6 - 130b4 + 108b3 - 23b2 - 208*b1 - 328) / 4
\(\nu^{15}\)	\(=\)	\( ( - 3 \beta_{15} - 123 \beta_{14} - 6 \beta_{13} + 108 \beta_{12} - 126 \beta_{11} + 3 \beta_{10} + \cdots + 190 ) / 4 \) (-3b15 - 123b14 - 6b13 + 108b12 - 126b11 + 3b10 - 126b9 + 54b8 + 192b7 + 54b6 - 82b5 + 108b4 - 244b3 - 123b2 + 190) / 4

\(n\)	\(261\)	\(703\)	\(769\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\beta_{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 63.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} - 14 T^{14} + \cdots + 2704 \) T^16 - 14T^14 + 131T^12 - 686T^10 + 2605T^8 - 5824T^6 + 9164T^4 - 5824*T^2 + 2704
$5$	\( (T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 4)^{2} \) (T^8 - 6T^7 + 18T^6 - 18T^5 + 5T^4 + 72T^2 - 24T + 4)^2
$7$	\( T^{16} - 30 T^{14} + \cdots + 43264 \) T^16 - 30T^14 + 207T^12 + 2790T^10 - 1399T^8 - 91512T^6 + 303408T^4 + 204672*T^2 + 43264
$11$	\( T^{16} + 18 T^{14} + \cdots + 692224 \) T^16 + 18T^14 - 57T^12 - 2970T^10 + 25241T^8 + 31680T^6 - 124992T^4 - 159744*T^2 + 692224
$13$	\( (T^{8} - 6 T^{7} + \cdots + 28561)^{2} \) (T^8 - 6T^7 + 33T^6 - 102T^5 + 428T^4 - 1326T^3 + 5577T^2 - 13182*T + 28561)^2
$17$	\( (T^{8} - 6 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) (T^8 - 6T^7 + 2T^6 + 60T^5 + 39T^4 - 300T^3 + 290T^2 + 30*T + 1)^2
$19$	\( T^{16} - 54 T^{14} + \cdots + 77228944 \) T^16 - 54T^14 + 891T^12 + 4374T^10 - 72643T^8 - 316872T^6 + 4389420T^4 + 34378656*T^2 + 77228944
$23$	\( T^{16} + 106 T^{14} + \cdots + 77228944 \) T^16 + 106T^14 + 8843T^12 + 232858T^10 + 4615261T^8 + 23024144T^6 + 87130316T^4 + 91395200*T^2 + 77228944
$29$	\( (T^{8} - 4 T^{7} + \cdots + 51529)^{2} \) (T^8 - 4T^7 + 62T^6 - 304T^5 + 3319T^4 - 13040T^3 + 49094T^2 - 55388*T + 51529)^2
$31$	\( T^{16} + \cdots + 1235663104 \) T^16 + 9072T^12 + 18310880T^8 + 9894433536*T^4 + 1235663104
$37$	\( (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{4} \) (T^4 - 4T^3 + 5T^2 - 2*T + 1)^4
$41$	\( (T^{4} - 12 T^{3} + \cdots + 81)^{4} \) (T^4 - 12T^3 + 45T^2 - 54*T + 81)^4
$43$	\( T^{16} + \cdots + 5671027857664 \) T^16 + 266T^14 + 48143T^12 + 4728770T^10 + 337890073T^8 + 13276514728T^6 + 359783787440T^4 + 1531577976448*T^2 + 5671027857664
$47$	\( T^{16} + \cdots + 5671027857664 \) T^16 + 20976T^12 + 70290656T^8 + 39057328896*T^4 + 5671027857664
$53$	\( (T^{4} - 8 T^{3} + \cdots - 128)^{4} \) (T^4 - 8T^3 - 3T^2 + 112*T - 128)^4
$59$	\( T^{16} + \cdots + 171720267307264 \) T^16 - 6T^14 - 8121T^12 + 48798T^10 + 53073017T^8 + 128241144T^6 - 106493647056T^4 - 206627151744*T^2 + 171720267307264
$61$	\( (T^{8} + 2 T^{7} + \cdots + 6889)^{2} \) (T^8 + 2T^7 + 62T^6 + 200T^5 + 3763T^4 + 9496T^3 + 20150T^2 + 13114*T + 6889)^2
$67$	\( T^{16} + \cdots + 2205735869584 \) T^16 - 126T^14 + 1587T^12 + 466830T^10 - 3260179T^8 - 1367500680T^6 + 39908056812T^4 + 548171044512*T^2 + 2205735869584
$71$	\( T^{16} + \cdots + 5067731344 \) T^16 - 126T^14 - 141T^12 + 684558T^10 + 28929197T^8 - 66891096T^6 - 336235956T^4 + 876466656*T^2 + 5067731344
$73$	\( (T^{8} - 10 T^{7} + \cdots + 676)^{2} \) (T^8 - 10T^7 + 50T^6 - 6T^5 + 3533T^4 - 35424T^3 + 177608T^2 - 15496*T + 676)^2
$79$	\( (T^{8} + 160 T^{6} + \cdots + 53248)^{2} \) (T^8 + 160T^6 + 7040T^4 + 51200*T^2 + 53248)^2
$83$	\( T^{16} + \cdots + 538409280507904 \) T^16 + 61704T^12 + 891937424T^8 + 3819952817664*T^4 + 538409280507904
$89$	\( (T^{8} + 26 T^{7} + \cdots + 2896804)^{2} \) (T^8 + 26T^7 + 215T^6 + 1736T^5 + 19699T^4 + 5950T^3 + 479246T^2 - 2474708*T + 2896804)^2
$97$	\( (T^{8} + 14 T^{7} + \cdots + 2116)^{2} \) (T^8 + 14T^7 + 323T^6 + 1812T^5 + 9371T^4 + 23070T^3 + 25694T^2 + 12236*T + 2116)^2
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