LMFDB - Newform orbit 297.2.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,2,Mod(82,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("297.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.f (of order $5$, degree $4$, 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:	$2.37155694003$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{5})$
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} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 $ x^16 - 9x^14 + 51x^12 - 249x^10 + 1476x^8 - 2875x^6 + 2335x^4 + 125*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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_{12} q^{2} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{4} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{5} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots + 1) q^{7} + ( - \beta_{15} - 2 \beta_{14} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{15} + 10 \beta_{13} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) $ q + b12 * q^2 + (b9 - b7 + b6 - b5 - b2) * q^4 + (-b15 + b13 + b11 + b10 + b8 + b1) * q^5 + (b6 - b5 + b3 - b2 + 1) * q^7 + (-b15 - 2b14 - 2b13 - 3b12 - 2b11 + 2b10 - 3b8 - 2b1) q^8 + (b4 - b3 + b2) * q^10 + (b15 - 2b13 + b11 - 2b10) * q^11 + (b6 - 3b4 + 1) q^13 + (-3b15 - 2b14 - 2b13 - 5b12 - 2b11 + 3b10 - 3b8 - b1) q^14 + (-b9 + b6 + b5 - b4 + b3 + b2) * q^16 + (-2b14 - 2b12 + b11 + 3b10 - 2b8 - b1) * q^17 + (-b9 + 2b7 - 3b6 + 3b4 - 3b3 + 2b2 - 3) q^19 + (2b12 + b11 - 2b10 + 2b1) q^20 + (-b7 + b6 - b4 + 4b3 - 2b2) * q^22 + (2b15 - b10 - b8) q^23 + (2b9 - b7 - 2b6 - b4 - 2b3 - b2 + 1) q^25 + (2b14 - b13 - b8 - b1) q^26 + (-b9 - 4b6 + 3b5 + 4b4 + b2) q^28 + (-b14 + 2b13 + 2b8 + 2b1) q^29 + (-b9 + 2b7 - 3b6 + b5 + b4 - 3) * q^31 + (-2b15 + b13 + b12 + b10 + 2b8 + b1) * q^32 + (2b7 + 2b5 - b4 + b3 + 2b2 - 4) q^34 + (b12 - 2b11 + 4b10 - 4b1) q^35 + (-2b3 - 2) q^37 + (-4b15 + 7b14 + 4b13 + 7b12 + 2b11 - 4b10 + 10b8 + 7b1) * q^38 + 5b6 q^40 + (6b15 - b14 - b13 + 5b12 - 4b10 - b8 - 5b1) * q^41 + (-b7 - b5 + 2b4 - 2b3 + 1) * q^43 + (2b15 - 3b14 - 5b12 - 2b11 + b10 - 4b8 - 2b1) * q^44 + (2b7 - b6 - 3b4 - 1) * q^46 + (-b15 + 3b14 + 3b13 + 2b12 + 4b11 - 4b10 + 5b8 + 3b1) q^47 + (-2b9 - 3b6 + b5 + 3b4 + 2b2) * q^49 + (3b15 + b14 - 3b13 + b12 - b11 - b10 - 3b8 - 3b1) * q^50 + (-2b9 - b7 - 2b6 + 3b4 - 2b3 - b2 - 3) * q^52 + (-3b15 - 3b14 - 2b12) q^53 + (2b9 - 3b7 + b6 - 2b5 - 5b4 + 3b3 - b2 + 2) q^55 + (3b15 + 4b13 + 4b12 - b10 + 3b8 + 4b1) q^56 + (b9 + 2b7 + 3b6 - 9b4 + 3b3 + 2b2 + 9) q^58 + (3b14 - 2b11 + b10 - 2b1) q^59 + (b9 - 2b6 - 3b5 + 2b4 + 3b3 - b2) * q^61 + (7b14 + 5b13 + b11 - b10 + 6b8 + 7b1) * q^62 + (-b9 - 2b7 + 6b6 + b5 + 6) * q^64 + (2b15 + b13 + b12 + b11 - 2b10 - b8) * q^65 + (-3b4 + 3b3 - 1) * q^67 + (5b15 + 5b14 + 2b12 + 2b11 - 4b10 + 4b1) * q^68 + (b9 - b7 + b6 - b5 - 4b3 - b2 - 4) q^70 + (5b15 - 2b14 - 5b13 - 2b12 - 3b11 - b10 - 7b8 - 6b1) q^71 + (8b6 - 2b5 + 4b3 - 2b2 + 4) * q^73 + (-2b15 - 2b12) * q^74 + (-3b7 - 3b5 + 11b4 - 11b3 - b2) * q^76 + (-b15 - 2b14 + 3b13 - b12 + 5b11 - b10 + 5b8 + 3b1) q^77 + (b7 + 5b6 + 5) q^79 + (-b14 - b13 - b12 - 4b11 - 3b8 - 5b1) q^80 + (6b9 + 6b6 - 5b5 - 6b4 + 7b3 - 6b2) * q^82 + (5b15 - 2b14 - 5b13 - 2b12 - 4b10 - b8 - 3b1) * q^83 + (5b9 - 2b7 - 8b6 + 8b4 - 8b3 - 2b2 - 8) * q^85 + (-5b15 - 5b14 - b11 + 2b10 - b8 - b1) q^86 + (3b7 - 5b6 + b5 - 2b4 + 4b3 + b2 + 4) * q^88 + (-4b15 - b13 - b12 + b11 + 2b10 + b8 - 2b1) q^89 + (-b9 - 3b7 - b6 - b4 - b3 - 3b2 + 1) * q^91 + (6b14 - b13 - 2b11 + b8 - b1) * q^92 + (-b9 - 11b6 - 2b5 + 11b4 - 3b3 + b2) * q^94 + (4b14 + 2b13 - 4b11 + b10 + 4b8) * q^95 + (-2b9 - b7 + 9b6 + 2b5 - 7b4 + 9) * q^97 + (-b15 + 10b13 + 10b12 + 3b11 - b10 + 9b8 + 7b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{4} + 12 q^{10} - 10 q^{16} - 2 q^{19} - 36 q^{22} + 32 q^{25} + 42 q^{28} - 26 q^{31} - 48 q^{34} - 24 q^{37} - 20 q^{40} + 24 q^{43} - 16 q^{46} + 24 q^{49} - 40 q^{52} - 16 q^{55} + 106 q^{58}+ \cdots + 72 q^{97}+O(q^{100}) $ 16 * q - 10 * q^4 + 12 * q^10 - 10 * q^16 - 2 * q^19 - 36 * q^22 + 32 * q^25 + 42 * q^28 - 26 * q^31 - 48 * q^34 - 24 * q^37 - 20 * q^40 + 24 * q^43 - 16 * q^46 + 24 * q^49 - 40 * q^52 - 16 * q^55 + 106 * q^58 - 6 * q^61 + 62 * q^64 - 40 * q^67 - 58 * q^70 + 60 * q^76 + 64 * q^79 - 84 * q^82 - 18 * q^85 + 80 * q^88 - 10 * q^91 + 90 * q^94 + 72 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 $ x^16 - 9*x^14 + 51*x^12 - 249*x^10 + 1476*x^8 - 2875*x^6 + 2335*x^4 + 125*x^2 + 25 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 16594091 \nu^{14} + 132889182 \nu^{12} - 669375495 \nu^{10} + 3344289682 \nu^{8} + \cdots + 109896768237 ) / 48030072872 $ (-16594091v^14 + 132889182v^12 - 669375495v^10 + 3344289682v^8 - 21196587086v^6 + 27818686175v^4 + 1584015460*v^2 + 109896768237) / 48030072872
$\beta_{3}$	$=$	$ ( 399459369 \nu^{14} - 3872461014 \nu^{12} + 22604705541 \nu^{10} - 111418189614 \nu^{8} + \cdots - 163815397855 ) / 240150364360 $ (399459369v^14 - 3872461014v^12 + 22604705541v^10 - 111418189614v^8 + 645839076946v^6 - 1502229627733v^4 + 1396478211400*v^2 - 163815397855) / 240150364360
$\beta_{4}$	$=$	$ ( 641538645 \nu^{14} - 5845087188 \nu^{12} + 33077583727 \nu^{10} - 161239139198 \nu^{8} + \cdots + 220551349875 ) / 240150364360 $ (641538645v^14 - 5845087188v^12 + 33077583727v^10 - 161239139198v^8 + 953682741102v^6 - 1904544632553v^4 + 1373579575990*v^2 + 220551349875) / 240150364360
$\beta_{5}$	$=$	$ ( 420560697 \nu^{14} - 4175189607 \nu^{12} + 24562062708 \nu^{10} - 121476379662 \nu^{8} + \cdots - 186816081040 ) / 120075182180 $ (420560697v^14 - 4175189607v^12 + 24562062708v^10 - 121476379662v^8 + 702524611788v^6 - 1706165602009v^4 + 1593672822325*v^2 - 186816081040) / 120075182180
$\beta_{6}$	$=$	$ ( 1070034741 \nu^{14} - 9471203848 \nu^{12} + 53263591527 \nu^{10} - 259312649798 \nu^{8} + \cdots + 118775784515 ) / 240150364360 $ (1070034741v^14 - 9471203848v^12 + 53263591527v^10 - 259312649798v^8 + 1546271776542v^6 - 2874489151649v^4 + 2235309546290*v^2 + 118775784515) / 240150364360
$\beta_{7}$	$=$	$ ( - 724570675 \nu^{14} + 6672080139 \nu^{12} - 37857976976 \nu^{10} + 184639030474 \nu^{8} + \cdots - 251290182000 ) / 120075182180 $ (-724570675v^14 + 6672080139v^12 - 37857976976v^10 + 184639030474v^8 - 1088328501456v^6 + 2209378603819v^4 - 1565036862845*v^2 - 251290182000) / 120075182180
$\beta_{8}$	$=$	$ ( 399459369 \nu^{15} - 3872461014 \nu^{13} + 22604705541 \nu^{11} - 111418189614 \nu^{9} + \cdots - 163815397855 \nu ) / 240150364360 $ (399459369v^15 - 3872461014v^13 + 22604705541v^11 - 111418189614v^9 + 645839076946v^7 - 1502229627733v^5 + 1396478211400v^3 - 163815397855v) / 240150364360
$\beta_{9}$	$=$	$ ( 921163875 \nu^{14} - 8325322917 \nu^{12} + 47647278148 \nu^{10} - 233617980722 \nu^{8} + \cdots - 28361391040 ) / 120075182180 $ (921163875v^14 - 8325322917v^12 + 47647278148v^10 - 233617980722v^8 + 1380800798028v^6 - 2755988903967v^4 + 2580447302375*v^2 - 28361391040) / 120075182180
$\beta_{10}$	$=$	$ ( 827955465 \nu^{15} - 7498577674 \nu^{13} + 42790713341 \nu^{11} - 209491700214 \nu^{9} + \cdots - 25440598855 \nu ) / 240150364360 $ (827955465v^15 - 7498577674v^13 + 42790713341v^11 - 209491700214v^9 + 1238428112386v^7 - 2472174146829v^5 + 2258208181700v^3 - 25440598855v) / 240150364360
$\beta_{11}$	$=$	$ ( 1070034741 \nu^{15} - 9471203848 \nu^{13} + 53263591527 \nu^{11} - 259312649798 \nu^{9} + \cdots + 118775784515 \nu ) / 240150364360 $ (1070034741v^15 - 9471203848v^13 + 53263591527v^11 - 259312649798v^9 + 1546271776542v^7 - 2874489151649v^5 + 2235309546290v^3 + 118775784515v) / 240150364360
$\beta_{12}$	$=$	$ ( - 914980841 \nu^{15} + 8770447689 \nu^{13} - 51196668716 \nu^{11} + 253062739159 \nu^{9} + \cdots + 962789857750 \nu ) / 150093977725 $ (-914980841v^15 + 8770447689v^13 - 51196668716v^11 + 253062739159v^9 - 1473103609566v^7 + 3371306212175v^5 - 3348910912605v^3 + 962789857750v) / 150093977725
$\beta_{13}$	$=$	$ ( - 12695929441 \nu^{15} + 111086725194 \nu^{13} - 620796945061 \nu^{11} + \cdots - 5798667404425 \nu ) / 1200751821800 $ (-12695929441v^15 + 111086725194v^13 - 620796945061v^11 + 3015391643614v^9 - 18031800047586v^7 + 32228335033405v^5 - 23090825524980v^3 - 5798667404425v) / 1200751821800
$\beta_{14}$	$=$	$ ( - 13576283373 \nu^{15} + 121399649892 \nu^{13} - 682891277823 \nu^{11} + \cdots - 8793919655675 \nu ) / 1200751821800 $ (-13576283373v^15 + 121399649892v^13 - 682891277823v^11 + 3319835423102v^9 - 19730608058398v^7 + 37341837633425v^5 - 26201048561390v^3 - 8793919655675v) / 1200751821800
$\beta_{15}$	$=$	$ ( 24923473777 \nu^{15} - 228729353598 \nu^{13} + 1310185729437 \nu^{11} + \cdots - 3637970480775 \nu ) / 1200751821800 $ (24923473777v^15 - 228729353598v^13 + 1310185729437v^11 - 6423697280038v^9 + 37842347738762v^7 - 77962619135165v^5 + 69730609456160v^3 - 3637970480775v) / 1200751821800

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{9} + 3\beta_{6} - 4\beta_{4} + 3\beta_{3} + 4 $ -b9 + 3b6 - 4b4 + 3*b3 + 4
$\nu^{3}$	$=$	$ -2\beta_{15} - \beta_{14} - 2\beta_{12} - \beta_{11} + 7\beta_{10} - \beta_{8} $ -2b15 - b14 - 2b12 - b11 + 7*b10 - b8
$\nu^{4}$	$=$	$ -\beta_{9} - 6\beta_{7} + 6\beta_{6} + \beta_{5} - 22\beta_{4} + 6 $ -b9 - 6b7 + 6b6 + b5 - 22*b4 + 6
$\nu^{5}$	$=$	$ -8\beta_{15} - 14\beta_{14} + \beta_{13} - 15\beta_{12} - 34\beta_{11} + 43\beta_{10} - 42\beta_{8} - 33\beta_1 $ -8b15 - 14b14 + b13 - 15b12 - 34b11 + 43b10 - 42b8 - 33*b1
$\nu^{6}$	$=$	$ -9\beta_{9} + 39\beta_{6} + 43\beta_{5} - 39\beta_{4} - 89\beta_{3} + 9\beta_{2} $ -9b9 + 39b6 + 43b5 - 39b4 - 89b3 + 9b2
$\nu^{7}$	$=$	$ -9\beta_{15} - 52\beta_{14} + 52\beta_{13} - 43\beta_{12} + 66\beta_{10} - 205\beta_{8} - 14\beta_1 $ -9b15 - 52b14 + 52b13 - 43b12 + 66b10 - 205b8 - 14*b1
$\nu^{8}$	$=$	$ -191\beta_{9} + 191\beta_{7} + 501\beta_{6} + 257\beta_{5} - 256\beta_{3} + 257\beta_{2} - 256 $ -191b9 + 191b7 + 501b6 + 257b5 - 256b3 + 257b2 - 256
$\nu^{9}$	$=$	$ 66\beta_{15} - 66\beta_{14} + 514\beta_{13} + 257\beta_{12} + 1074\beta_{11} + 60\beta_{8} + 60\beta_1 $ 66b15 - 66b14 + 514b13 + 257b12 + 1074b11 + 60b8 + 60*b1
$\nu^{10}$	$=$	$ 454\beta_{7} + 454\beta_{5} + 1664\beta_{4} - 1664\beta_{3} + 1528\beta_{2} - 4503 $ 454b7 + 454b5 + 1664b4 - 1664b3 + 1528*b2 - 4503
$\nu^{11}$	$=$	$ 1982 \beta_{15} + 454 \beta_{14} + 1982 \beta_{13} + 3510 \beta_{12} + 3026 \beta_{11} + \cdots - 4079 \beta_1 $ 1982b15 + 454b14 + 1982b13 + 3510b12 + 3026b11 - 3026b10 + 1982b8 - 4079b1
$\nu^{12}$	$=$	$ 6061\beta_{9} + 3026\beta_{7} - 16171\beta_{6} + 26862\beta_{4} - 16171\beta_{3} + 3026\beta_{2} - 26862 $ 6061b9 + 3026b7 - 16171b6 + 26862b4 - 16171b3 + 3026b2 - 26862
$\nu^{13}$	$=$	$ 18174 \beta_{15} + 12113 \beta_{14} + 3026 \beta_{13} + 24226 \beta_{12} + 19769 \beta_{11} + \cdots + 3026 \beta_1 $ 18174b15 + 12113b14 + 3026b13 + 24226b12 + 19769b11 - 54123b10 + 22795b8 + 3026b1
$\nu^{14}$	$=$	$ 19769\beta_{9} + 34354\beta_{7} - 68004\beta_{6} - 19769\beta_{5} + 160548\beta_{4} - 68004 $ 19769b9 + 34354b7 - 68004b6 - 19769b5 + 160548*b4 - 68004
$\nu^{15}$	$=$	$ 73892 \beta_{15} + 108246 \beta_{14} - 19769 \beta_{13} + 128015 \beta_{12} + 195606 \beta_{11} + \cdots + 175837 \beta_1 $ 73892b15 + 108246b14 - 19769b13 + 128015b12 + 195606b11 - 322917b10 + 303148b8 + 175837b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$1$	$\beta_{6}$

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} $

82.1

−1.18970	−	0.386556i
−2.33991	−	0.760284i
2.33991	+	0.760284i
1.18970	+	0.386556i
−0.185814	−	0.255752i
−1.35089	−	1.85934i
1.35089	+	1.85934i
0.185814	+	0.255752i
−1.18970	+	0.386556i
−2.33991	+	0.760284i
2.33991	−	0.760284i
1.18970	−	0.386556i
−0.185814	+	0.255752i
−1.35089	+	1.85934i
1.35089	−	1.85934i
0.185814	−	0.255752i

−0.760284

−

2.33991i

−3.27913

2.38243i

−0.305860

0.941339i

−3.49672

2.54052i

4.08684

2.96926i

2.43519

82.2

−0.386556

−

1.18970i

0.352078

−

0.255800i

0.507211

−

1.56103i

2.37869

−

1.72822i

−2.46446

−

1.79053i

−2.05323

82.3

0.386556

1.18970i

0.352078

−

0.255800i

−0.507211

1.56103i

2.37869

−

1.72822i

2.46446

1.79053i

−2.05323

82.4

0.760284

2.33991i

−3.27913

2.38243i

0.305860

−

0.941339i

−3.49672

2.54052i

−4.08684

−

2.96926i

2.43519

136.1

−1.85934

−

1.35089i

1.01420

3.12140i

−1.37287

0.997447i

0.0641710

0.197498i

0.910502

−

2.80224i

3.90006

136.2

−0.255752

−

0.185814i

−0.587152

−

1.80707i

3.28092

−

2.38373i

1.05386

3.24346i

−0.380991

1.17257i

−1.28203

136.3

0.255752

0.185814i

−0.587152

−

1.80707i

−3.28092

2.38373i

1.05386

3.24346i

0.380991

−

1.17257i

−1.28203

136.4

1.85934

1.35089i

1.01420

3.12140i

1.37287

−

0.997447i

0.0641710

0.197498i

−0.910502

2.80224i

3.90006

163.1

−0.760284

2.33991i

−3.27913

−

2.38243i

−0.305860

−

0.941339i

−3.49672

−

2.54052i

4.08684

−

2.96926i

2.43519

163.2

−0.386556

1.18970i

0.352078

0.255800i

0.507211

1.56103i

2.37869

1.72822i

−2.46446

1.79053i

−2.05323

163.3

0.386556

−

1.18970i

0.352078

0.255800i

−0.507211

−

1.56103i

2.37869

1.72822i

2.46446

−

1.79053i

−2.05323

163.4

0.760284

−

2.33991i

−3.27913

−

2.38243i

0.305860

0.941339i

−3.49672

−

2.54052i

−4.08684

2.96926i

2.43519

190.1

−1.85934

1.35089i

1.01420

−

3.12140i

−1.37287

−

0.997447i

0.0641710

−

0.197498i

0.910502

2.80224i

3.90006

190.2

−0.255752

0.185814i

−0.587152

1.80707i

3.28092

2.38373i

1.05386

−

3.24346i

−0.380991

−

1.17257i

−1.28203

190.3

0.255752

−

0.185814i

−0.587152

1.80707i

−3.28092

−

2.38373i

1.05386

−

3.24346i

0.380991

1.17257i

−1.28203

190.4

1.85934

−

1.35089i

1.01420

−

3.12140i

1.37287

0.997447i

0.0641710

−

0.197498i

−0.910502

−

2.80224i

3.90006

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.c	even	5	1	inner
33.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.2.f.b	✓	16
3.b	odd	2	1	inner	297.2.f.b	✓	16
9.c	even	3	2		891.2.n.h		32
9.d	odd	6	2		891.2.n.h		32
11.c	even	5	1	inner	297.2.f.b	✓	16
11.c	even	5	1		3267.2.a.bj		8
11.d	odd	10	1		3267.2.a.bi		8
33.f	even	10	1		3267.2.a.bi		8
33.h	odd	10	1	inner	297.2.f.b	✓	16
33.h	odd	10	1		3267.2.a.bj		8
99.m	even	15	2		891.2.n.h		32
99.n	odd	30	2		891.2.n.h		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
297.2.f.b	✓	16	1.a	even	1	1	trivial
297.2.f.b	✓	16	3.b	odd	2	1	inner
297.2.f.b	✓	16	11.c	even	5	1	inner
297.2.f.b	✓	16	33.h	odd	10	1	inner
891.2.n.h		32	9.c	even	3	2
891.2.n.h		32	9.d	odd	6	2
891.2.n.h		32	99.m	even	15	2
891.2.n.h		32	99.n	odd	30	2
3267.2.a.bi		8	11.d	odd	10	1
3267.2.a.bi		8	33.f	even	10	1
3267.2.a.bj		8	11.c	even	5	1
3267.2.a.bj		8	33.h	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} + 9T_{2}^{14} + 51T_{2}^{12} + 249T_{2}^{10} + 1476T_{2}^{8} + 2875T_{2}^{6} + 2335T_{2}^{4} - 125T_{2}^{2} + 25 $ T2^16 + 9*T2^14 + 51*T2^12 + 249*T2^10 + 1476*T2^8 + 2875*T2^6 + 2335*T2^4 - 125*T2^2 + 25 acting on $S_{2}^{\mathrm{new}}(297, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 9 T^{14} + \cdots + 25 $ T^16 + 9T^14 + 51T^12 + 249T^10 + 1476T^8 + 2875T^6 + 2335T^4 - 125*T^2 + 25
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - 6 T^{14} + \cdots + 15625 $ T^16 - 6T^14 + 241T^12 + 1034T^10 + 3191T^8 + 9350T^6 + 27125T^4 + 31250*T^2 + 15625
$7$	$ (T^{8} + T^{6} + 10 T^{5} + \cdots + 81)^{2} $ (T^8 + T^6 + 10T^5 + 151T^4 - 690T^3 + 1971T^2 - 270*T + 81)^2
$11$	$ T^{16} + \cdots + 214358881 $ T^16 + 45T^14 + 956T^12 + 13185T^10 + 149631T^8 + 1595385T^6 + 13996796T^4 + 79720245*T^2 + 214358881
$13$	$ (T^{4} + 10 T^{2} + \cdots + 25)^{4} $ (T^4 + 10T^2 + 25T + 25)^4
$17$	$ T^{16} + \cdots + 25665642025 $ T^16 + 44T^14 + 1961T^12 + 82959T^10 + 3418726T^8 + 21381715T^6 + 527083285T^4 + 5401311575*T^2 + 25665642025
$19$	$ (T^{8} + T^{7} + \cdots + 450241)^{2} $ (T^8 + T^7 + 47T^6 + 273T^5 + 1330T^4 - 5403T^3 + 23277T^2 - 27511T + 450241)^2
$23$	$ (T^{8} - 42 T^{6} + \cdots + 405)^{2} $ (T^8 - 42T^6 + 586T^4 - 2745*T^2 + 405)^2
$29$	$ T^{16} - 84 T^{14} + \cdots + 25 $ T^16 - 84T^14 + 13711T^12 - 251014T^10 + 9178691T^8 + 4490990T^6 + 846035T^4 + 7400*T^2 + 25
$31$	$ (T^{8} + 13 T^{7} + \cdots + 44521)^{2} $ (T^8 + 13T^7 + 62T^6 - 135T^5 + 661T^4 - 1185T^3 + 16368T^2 + 12449*T + 44521)^2
$37$	$ (T^{4} + 6 T^{3} + 16 T^{2} + \cdots + 16)^{4} $ (T^4 + 6T^3 + 16T^2 + 16*T + 16)^4
$41$	$ T^{16} + \cdots + 43\!\cdots\!25 $ T^16 + 156T^14 + 26791T^12 + 4304066T^10 + 508079291T^8 + 33838073750T^6 + 1869638046875T^4 + 98226171875000*T^2 + 4316900634765625
$43$	$ (T^{4} - 6 T^{3} - 16 T^{2} + \cdots + 5)^{4} $ (T^4 - 6T^3 - 16T^2 + 5*T + 5)^4
$47$	$ T^{16} + \cdots + 980347515625 $ T^16 - 15T^14 + 25410T^12 + 2719125T^10 + 120083025T^8 + 388537375T^6 + 40987847500T^4 + 318547965625*T^2 + 980347515625
$53$	$ T^{16} + \cdots + 5358972025 $ T^16 + 156T^14 + 11001T^12 + 341301T^10 + 20122326T^8 + 582298585T^6 + 7801969285T^4 - 3675989075*T^2 + 5358972025
$59$	$ T^{16} + \cdots + 415257804025 $ T^16 + 176T^14 + 14456T^12 + 662916T^10 + 25649026T^8 + 962690920T^6 + 25816076035T^4 - 51281749900*T^2 + 415257804025
$61$	$ (T^{8} + 3 T^{7} + \cdots + 194481)^{2} $ (T^8 + 3T^7 + 118T^6 + 1051T^5 + 6805T^4 + 16821T^3 + 38808T^2 - 157437*T + 194481)^2
$67$	$ (T^{2} + 5 T - 5)^{8} $ (T^2 + 5*T - 5)^8
$71$	$ T^{16} + \cdots + 64072265625 $ T^16 - 131T^14 + 36466T^12 - 1312761T^10 + 166241671T^8 + 4793655375T^6 + 52496100000T^4 + 91979296875*T^2 + 64072265625
$73$	$ (T^{8} + 164 T^{6} + \cdots + 14622976)^{2} $ (T^8 + 164T^6 + 1440T^5 + 13776T^4 - 71040T^3 + 1605184T^2 - 7342080T + 14622976)^2
$79$	$ (T^{8} - 32 T^{7} + \cdots + 421201)^{2} $ (T^8 - 32T^7 + 497T^6 - 4540T^5 + 28141T^4 - 115180T^3 + 299033T^2 - 379016*T + 421201)^2
$83$	$ T^{16} + \cdots + 463495448025 $ T^16 + 169T^14 + 35551T^12 + 7277349T^10 + 1155940156T^8 + 18488958795T^6 + 113429470635T^4 - 83606258025*T^2 + 463495448025
$89$	$ (T^{8} - 357 T^{6} + \cdots + 25515405)^{2} $ (T^8 - 357T^6 + 41116T^4 - 1773540*T^2 + 25515405)^2
$97$	$ (T^{8} - 36 T^{7} + \cdots + 160801)^{2} $ (T^8 - 36T^7 + 893T^6 - 12915T^5 + 110436T^4 - 511815T^3 + 1058557T^2 + 193683*T + 160801)^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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.f (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{12} q^{2} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{4} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{5} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots + 1) q^{7} + ( - \beta_{15} - 2 \beta_{14} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{15} + 10 \beta_{13} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) q + b12 * q^2 + (b9 - b7 + b6 - b5 - b2) * q^4 + (-b15 + b13 + b11 + b10 + b8 + b1) * q^5 + (b6 - b5 + b3 - b2 + 1) * q^7 + (-b15 - 2b14 - 2b13 - 3b12 - 2b11 + 2b10 - 3b8 - 2b1) q^8 + (b4 - b3 + b2) * q^10 + (b15 - 2b13 + b11 - 2b10) * q^11 + (b6 - 3b4 + 1) q^13 + (-3b15 - 2b14 - 2b13 - 5b12 - 2b11 + 3b10 - 3b8 - b1) q^14 + (-b9 + b6 + b5 - b4 + b3 + b2) * q^16 + (-2b14 - 2b12 + b11 + 3b10 - 2b8 - b1) * q^17 + (-b9 + 2b7 - 3b6 + 3b4 - 3b3 + 2b2 - 3) q^19 + (2b12 + b11 - 2b10 + 2b1) q^20 + (-b7 + b6 - b4 + 4b3 - 2b2) * q^22 + (2b15 - b10 - b8) q^23 + (2b9 - b7 - 2b6 - b4 - 2b3 - b2 + 1) q^25 + (2b14 - b13 - b8 - b1) q^26 + (-b9 - 4b6 + 3b5 + 4b4 + b2) q^28 + (-b14 + 2b13 + 2b8 + 2b1) q^29 + (-b9 + 2b7 - 3b6 + b5 + b4 - 3) * q^31 + (-2b15 + b13 + b12 + b10 + 2b8 + b1) * q^32 + (2b7 + 2b5 - b4 + b3 + 2b2 - 4) q^34 + (b12 - 2b11 + 4b10 - 4b1) q^35 + (-2b3 - 2) q^37 + (-4b15 + 7b14 + 4b13 + 7b12 + 2b11 - 4b10 + 10b8 + 7b1) * q^38 + 5b6 q^40 + (6b15 - b14 - b13 + 5b12 - 4b10 - b8 - 5b1) * q^41 + (-b7 - b5 + 2b4 - 2b3 + 1) * q^43 + (2b15 - 3b14 - 5b12 - 2b11 + b10 - 4b8 - 2b1) * q^44 + (2b7 - b6 - 3b4 - 1) * q^46 + (-b15 + 3b14 + 3b13 + 2b12 + 4b11 - 4b10 + 5b8 + 3b1) q^47 + (-2b9 - 3b6 + b5 + 3b4 + 2b2) * q^49 + (3b15 + b14 - 3b13 + b12 - b11 - b10 - 3b8 - 3b1) * q^50 + (-2b9 - b7 - 2b6 + 3b4 - 2b3 - b2 - 3) * q^52 + (-3b15 - 3b14 - 2b12) q^53 + (2b9 - 3b7 + b6 - 2b5 - 5b4 + 3b3 - b2 + 2) q^55 + (3b15 + 4b13 + 4b12 - b10 + 3b8 + 4b1) q^56 + (b9 + 2b7 + 3b6 - 9b4 + 3b3 + 2b2 + 9) q^58 + (3b14 - 2b11 + b10 - 2b1) q^59 + (b9 - 2b6 - 3b5 + 2b4 + 3b3 - b2) * q^61 + (7b14 + 5b13 + b11 - b10 + 6b8 + 7b1) * q^62 + (-b9 - 2b7 + 6b6 + b5 + 6) * q^64 + (2b15 + b13 + b12 + b11 - 2b10 - b8) * q^65 + (-3b4 + 3b3 - 1) * q^67 + (5b15 + 5b14 + 2b12 + 2b11 - 4b10 + 4b1) * q^68 + (b9 - b7 + b6 - b5 - 4b3 - b2 - 4) q^70 + (5b15 - 2b14 - 5b13 - 2b12 - 3b11 - b10 - 7b8 - 6b1) q^71 + (8b6 - 2b5 + 4b3 - 2b2 + 4) * q^73 + (-2b15 - 2b12) * q^74 + (-3b7 - 3b5 + 11b4 - 11b3 - b2) * q^76 + (-b15 - 2b14 + 3b13 - b12 + 5b11 - b10 + 5b8 + 3b1) q^77 + (b7 + 5b6 + 5) q^79 + (-b14 - b13 - b12 - 4b11 - 3b8 - 5b1) q^80 + (6b9 + 6b6 - 5b5 - 6b4 + 7b3 - 6b2) * q^82 + (5b15 - 2b14 - 5b13 - 2b12 - 4b10 - b8 - 3b1) * q^83 + (5b9 - 2b7 - 8b6 + 8b4 - 8b3 - 2b2 - 8) * q^85 + (-5b15 - 5b14 - b11 + 2b10 - b8 - b1) q^86 + (3b7 - 5b6 + b5 - 2b4 + 4b3 + b2 + 4) * q^88 + (-4b15 - b13 - b12 + b11 + 2b10 + b8 - 2b1) q^89 + (-b9 - 3b7 - b6 - b4 - b3 - 3b2 + 1) * q^91 + (6b14 - b13 - 2b11 + b8 - b1) * q^92 + (-b9 - 11b6 - 2b5 + 11b4 - 3b3 + b2) * q^94 + (4b14 + 2b13 - 4b11 + b10 + 4b8) * q^95 + (-2b9 - b7 + 9b6 + 2b5 - 7b4 + 9) * q^97 + (-b15 + 10b13 + 10b12 + 3b11 - b10 + 9b8 + 7b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{4} + 12 q^{10} - 10 q^{16} - 2 q^{19} - 36 q^{22} + 32 q^{25} + 42 q^{28} - 26 q^{31} - 48 q^{34} - 24 q^{37} - 20 q^{40} + 24 q^{43} - 16 q^{46} + 24 q^{49} - 40 q^{52} - 16 q^{55} + 106 q^{58}+ \cdots + 72 q^{97}+O(q^{100}) \) 16 * q - 10 * q^4 + 12 * q^10 - 10 * q^16 - 2 * q^19 - 36 * q^22 + 32 * q^25 + 42 * q^28 - 26 * q^31 - 48 * q^34 - 24 * q^37 - 20 * q^40 + 24 * q^43 - 16 * q^46 + 24 * q^49 - 40 * q^52 - 16 * q^55 + 106 * q^58 - 6 * q^61 + 62 * q^64 - 40 * q^67 - 58 * q^70 + 60 * q^76 + 64 * q^79 - 84 * q^82 - 18 * q^85 + 80 * q^88 - 10 * q^91 + 90 * q^94 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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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	297.2.f.b

Level	$297$
Weight	$2$
Character orbit	297.f
Analytic conductor	$2.372$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 \) x^16 - 9x^14 + 51x^12 - 249x^10 + 1476x^8 - 2875x^6 + 2335x^4 + 125*x^2 + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 16594091 \nu^{14} + 132889182 \nu^{12} - 669375495 \nu^{10} + 3344289682 \nu^{8} + \cdots + 109896768237 ) / 48030072872 \) (-16594091v^14 + 132889182v^12 - 669375495v^10 + 3344289682v^8 - 21196587086v^6 + 27818686175v^4 + 1584015460*v^2 + 109896768237) / 48030072872
\(\beta_{3}\)	\(=\)	\( ( 399459369 \nu^{14} - 3872461014 \nu^{12} + 22604705541 \nu^{10} - 111418189614 \nu^{8} + \cdots - 163815397855 ) / 240150364360 \) (399459369v^14 - 3872461014v^12 + 22604705541v^10 - 111418189614v^8 + 645839076946v^6 - 1502229627733v^4 + 1396478211400*v^2 - 163815397855) / 240150364360
\(\beta_{4}\)	\(=\)	\( ( 641538645 \nu^{14} - 5845087188 \nu^{12} + 33077583727 \nu^{10} - 161239139198 \nu^{8} + \cdots + 220551349875 ) / 240150364360 \) (641538645v^14 - 5845087188v^12 + 33077583727v^10 - 161239139198v^8 + 953682741102v^6 - 1904544632553v^4 + 1373579575990*v^2 + 220551349875) / 240150364360
\(\beta_{5}\)	\(=\)	\( ( 420560697 \nu^{14} - 4175189607 \nu^{12} + 24562062708 \nu^{10} - 121476379662 \nu^{8} + \cdots - 186816081040 ) / 120075182180 \) (420560697v^14 - 4175189607v^12 + 24562062708v^10 - 121476379662v^8 + 702524611788v^6 - 1706165602009v^4 + 1593672822325*v^2 - 186816081040) / 120075182180
\(\beta_{6}\)	\(=\)	\( ( 1070034741 \nu^{14} - 9471203848 \nu^{12} + 53263591527 \nu^{10} - 259312649798 \nu^{8} + \cdots + 118775784515 ) / 240150364360 \) (1070034741v^14 - 9471203848v^12 + 53263591527v^10 - 259312649798v^8 + 1546271776542v^6 - 2874489151649v^4 + 2235309546290*v^2 + 118775784515) / 240150364360
\(\beta_{7}\)	\(=\)	\( ( - 724570675 \nu^{14} + 6672080139 \nu^{12} - 37857976976 \nu^{10} + 184639030474 \nu^{8} + \cdots - 251290182000 ) / 120075182180 \) (-724570675v^14 + 6672080139v^12 - 37857976976v^10 + 184639030474v^8 - 1088328501456v^6 + 2209378603819v^4 - 1565036862845*v^2 - 251290182000) / 120075182180
\(\beta_{8}\)	\(=\)	\( ( 399459369 \nu^{15} - 3872461014 \nu^{13} + 22604705541 \nu^{11} - 111418189614 \nu^{9} + \cdots - 163815397855 \nu ) / 240150364360 \) (399459369v^15 - 3872461014v^13 + 22604705541v^11 - 111418189614v^9 + 645839076946v^7 - 1502229627733v^5 + 1396478211400v^3 - 163815397855v) / 240150364360
\(\beta_{9}\)	\(=\)	\( ( 921163875 \nu^{14} - 8325322917 \nu^{12} + 47647278148 \nu^{10} - 233617980722 \nu^{8} + \cdots - 28361391040 ) / 120075182180 \) (921163875v^14 - 8325322917v^12 + 47647278148v^10 - 233617980722v^8 + 1380800798028v^6 - 2755988903967v^4 + 2580447302375*v^2 - 28361391040) / 120075182180
\(\beta_{10}\)	\(=\)	\( ( 827955465 \nu^{15} - 7498577674 \nu^{13} + 42790713341 \nu^{11} - 209491700214 \nu^{9} + \cdots - 25440598855 \nu ) / 240150364360 \) (827955465v^15 - 7498577674v^13 + 42790713341v^11 - 209491700214v^9 + 1238428112386v^7 - 2472174146829v^5 + 2258208181700v^3 - 25440598855v) / 240150364360
\(\beta_{11}\)	\(=\)	\( ( 1070034741 \nu^{15} - 9471203848 \nu^{13} + 53263591527 \nu^{11} - 259312649798 \nu^{9} + \cdots + 118775784515 \nu ) / 240150364360 \) (1070034741v^15 - 9471203848v^13 + 53263591527v^11 - 259312649798v^9 + 1546271776542v^7 - 2874489151649v^5 + 2235309546290v^3 + 118775784515v) / 240150364360
\(\beta_{12}\)	\(=\)	\( ( - 914980841 \nu^{15} + 8770447689 \nu^{13} - 51196668716 \nu^{11} + 253062739159 \nu^{9} + \cdots + 962789857750 \nu ) / 150093977725 \) (-914980841v^15 + 8770447689v^13 - 51196668716v^11 + 253062739159v^9 - 1473103609566v^7 + 3371306212175v^5 - 3348910912605v^3 + 962789857750v) / 150093977725
\(\beta_{13}\)	\(=\)	\( ( - 12695929441 \nu^{15} + 111086725194 \nu^{13} - 620796945061 \nu^{11} + \cdots - 5798667404425 \nu ) / 1200751821800 \) (-12695929441v^15 + 111086725194v^13 - 620796945061v^11 + 3015391643614v^9 - 18031800047586v^7 + 32228335033405v^5 - 23090825524980v^3 - 5798667404425v) / 1200751821800
\(\beta_{14}\)	\(=\)	\( ( - 13576283373 \nu^{15} + 121399649892 \nu^{13} - 682891277823 \nu^{11} + \cdots - 8793919655675 \nu ) / 1200751821800 \) (-13576283373v^15 + 121399649892v^13 - 682891277823v^11 + 3319835423102v^9 - 19730608058398v^7 + 37341837633425v^5 - 26201048561390v^3 - 8793919655675v) / 1200751821800
\(\beta_{15}\)	\(=\)	\( ( 24923473777 \nu^{15} - 228729353598 \nu^{13} + 1310185729437 \nu^{11} + \cdots - 3637970480775 \nu ) / 1200751821800 \) (24923473777v^15 - 228729353598v^13 + 1310185729437v^11 - 6423697280038v^9 + 37842347738762v^7 - 77962619135165v^5 + 69730609456160v^3 - 3637970480775v) / 1200751821800

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{9} + 3\beta_{6} - 4\beta_{4} + 3\beta_{3} + 4 \) -b9 + 3b6 - 4b4 + 3*b3 + 4
\(\nu^{3}\)	\(=\)	\( -2\beta_{15} - \beta_{14} - 2\beta_{12} - \beta_{11} + 7\beta_{10} - \beta_{8} \) -2b15 - b14 - 2b12 - b11 + 7*b10 - b8
\(\nu^{4}\)	\(=\)	\( -\beta_{9} - 6\beta_{7} + 6\beta_{6} + \beta_{5} - 22\beta_{4} + 6 \) -b9 - 6b7 + 6b6 + b5 - 22*b4 + 6
\(\nu^{5}\)	\(=\)	\( -8\beta_{15} - 14\beta_{14} + \beta_{13} - 15\beta_{12} - 34\beta_{11} + 43\beta_{10} - 42\beta_{8} - 33\beta_1 \) -8b15 - 14b14 + b13 - 15b12 - 34b11 + 43b10 - 42b8 - 33*b1
\(\nu^{6}\)	\(=\)	\( -9\beta_{9} + 39\beta_{6} + 43\beta_{5} - 39\beta_{4} - 89\beta_{3} + 9\beta_{2} \) -9b9 + 39b6 + 43b5 - 39b4 - 89b3 + 9b2
\(\nu^{7}\)	\(=\)	\( -9\beta_{15} - 52\beta_{14} + 52\beta_{13} - 43\beta_{12} + 66\beta_{10} - 205\beta_{8} - 14\beta_1 \) -9b15 - 52b14 + 52b13 - 43b12 + 66b10 - 205b8 - 14*b1
\(\nu^{8}\)	\(=\)	\( -191\beta_{9} + 191\beta_{7} + 501\beta_{6} + 257\beta_{5} - 256\beta_{3} + 257\beta_{2} - 256 \) -191b9 + 191b7 + 501b6 + 257b5 - 256b3 + 257b2 - 256
\(\nu^{9}\)	\(=\)	\( 66\beta_{15} - 66\beta_{14} + 514\beta_{13} + 257\beta_{12} + 1074\beta_{11} + 60\beta_{8} + 60\beta_1 \) 66b15 - 66b14 + 514b13 + 257b12 + 1074b11 + 60b8 + 60*b1
\(\nu^{10}\)	\(=\)	\( 454\beta_{7} + 454\beta_{5} + 1664\beta_{4} - 1664\beta_{3} + 1528\beta_{2} - 4503 \) 454b7 + 454b5 + 1664b4 - 1664b3 + 1528*b2 - 4503
\(\nu^{11}\)	\(=\)	\( 1982 \beta_{15} + 454 \beta_{14} + 1982 \beta_{13} + 3510 \beta_{12} + 3026 \beta_{11} + \cdots - 4079 \beta_1 \) 1982b15 + 454b14 + 1982b13 + 3510b12 + 3026b11 - 3026b10 + 1982b8 - 4079b1
\(\nu^{12}\)	\(=\)	\( 6061\beta_{9} + 3026\beta_{7} - 16171\beta_{6} + 26862\beta_{4} - 16171\beta_{3} + 3026\beta_{2} - 26862 \) 6061b9 + 3026b7 - 16171b6 + 26862b4 - 16171b3 + 3026b2 - 26862
\(\nu^{13}\)	\(=\)	\( 18174 \beta_{15} + 12113 \beta_{14} + 3026 \beta_{13} + 24226 \beta_{12} + 19769 \beta_{11} + \cdots + 3026 \beta_1 \) 18174b15 + 12113b14 + 3026b13 + 24226b12 + 19769b11 - 54123b10 + 22795b8 + 3026b1
\(\nu^{14}\)	\(=\)	\( 19769\beta_{9} + 34354\beta_{7} - 68004\beta_{6} - 19769\beta_{5} + 160548\beta_{4} - 68004 \) 19769b9 + 34354b7 - 68004b6 - 19769b5 + 160548*b4 - 68004
\(\nu^{15}\)	\(=\)	\( 73892 \beta_{15} + 108246 \beta_{14} - 19769 \beta_{13} + 128015 \beta_{12} + 195606 \beta_{11} + \cdots + 175837 \beta_1 \) 73892b15 + 108246b14 - 19769b13 + 128015b12 + 195606b11 - 322917b10 + 303148b8 + 175837b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 82.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 9 T^{14} + \cdots + 25 \) T^16 + 9T^14 + 51T^12 + 249T^10 + 1476T^8 + 2875T^6 + 2335T^4 - 125*T^2 + 25
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 6 T^{14} + \cdots + 15625 \) T^16 - 6T^14 + 241T^12 + 1034T^10 + 3191T^8 + 9350T^6 + 27125T^4 + 31250*T^2 + 15625
$7$	\( (T^{8} + T^{6} + 10 T^{5} + \cdots + 81)^{2} \) (T^8 + T^6 + 10T^5 + 151T^4 - 690T^3 + 1971T^2 - 270*T + 81)^2
$11$	\( T^{16} + \cdots + 214358881 \) T^16 + 45T^14 + 956T^12 + 13185T^10 + 149631T^8 + 1595385T^6 + 13996796T^4 + 79720245*T^2 + 214358881
$13$	\( (T^{4} + 10 T^{2} + \cdots + 25)^{4} \) (T^4 + 10T^2 + 25T + 25)^4
$17$	\( T^{16} + \cdots + 25665642025 \) T^16 + 44T^14 + 1961T^12 + 82959T^10 + 3418726T^8 + 21381715T^6 + 527083285T^4 + 5401311575*T^2 + 25665642025
$19$	\( (T^{8} + T^{7} + \cdots + 450241)^{2} \) (T^8 + T^7 + 47T^6 + 273T^5 + 1330T^4 - 5403T^3 + 23277T^2 - 27511T + 450241)^2
$23$	\( (T^{8} - 42 T^{6} + \cdots + 405)^{2} \) (T^8 - 42T^6 + 586T^4 - 2745*T^2 + 405)^2
$29$	\( T^{16} - 84 T^{14} + \cdots + 25 \) T^16 - 84T^14 + 13711T^12 - 251014T^10 + 9178691T^8 + 4490990T^6 + 846035T^4 + 7400*T^2 + 25
$31$	\( (T^{8} + 13 T^{7} + \cdots + 44521)^{2} \) (T^8 + 13T^7 + 62T^6 - 135T^5 + 661T^4 - 1185T^3 + 16368T^2 + 12449*T + 44521)^2
$37$	\( (T^{4} + 6 T^{3} + 16 T^{2} + \cdots + 16)^{4} \) (T^4 + 6T^3 + 16T^2 + 16*T + 16)^4
$41$	\( T^{16} + \cdots + 43\!\cdots\!25 \) T^16 + 156T^14 + 26791T^12 + 4304066T^10 + 508079291T^8 + 33838073750T^6 + 1869638046875T^4 + 98226171875000*T^2 + 4316900634765625
$43$	\( (T^{4} - 6 T^{3} - 16 T^{2} + \cdots + 5)^{4} \) (T^4 - 6T^3 - 16T^2 + 5*T + 5)^4
$47$	\( T^{16} + \cdots + 980347515625 \) T^16 - 15T^14 + 25410T^12 + 2719125T^10 + 120083025T^8 + 388537375T^6 + 40987847500T^4 + 318547965625*T^2 + 980347515625
$53$	\( T^{16} + \cdots + 5358972025 \) T^16 + 156T^14 + 11001T^12 + 341301T^10 + 20122326T^8 + 582298585T^6 + 7801969285T^4 - 3675989075*T^2 + 5358972025
$59$	\( T^{16} + \cdots + 415257804025 \) T^16 + 176T^14 + 14456T^12 + 662916T^10 + 25649026T^8 + 962690920T^6 + 25816076035T^4 - 51281749900*T^2 + 415257804025
$61$	\( (T^{8} + 3 T^{7} + \cdots + 194481)^{2} \) (T^8 + 3T^7 + 118T^6 + 1051T^5 + 6805T^4 + 16821T^3 + 38808T^2 - 157437*T + 194481)^2
$67$	\( (T^{2} + 5 T - 5)^{8} \) (T^2 + 5*T - 5)^8
$71$	\( T^{16} + \cdots + 64072265625 \) T^16 - 131T^14 + 36466T^12 - 1312761T^10 + 166241671T^8 + 4793655375T^6 + 52496100000T^4 + 91979296875*T^2 + 64072265625
$73$	\( (T^{8} + 164 T^{6} + \cdots + 14622976)^{2} \) (T^8 + 164T^6 + 1440T^5 + 13776T^4 - 71040T^3 + 1605184T^2 - 7342080T + 14622976)^2
$79$	\( (T^{8} - 32 T^{7} + \cdots + 421201)^{2} \) (T^8 - 32T^7 + 497T^6 - 4540T^5 + 28141T^4 - 115180T^3 + 299033T^2 - 379016*T + 421201)^2
$83$	\( T^{16} + \cdots + 463495448025 \) T^16 + 169T^14 + 35551T^12 + 7277349T^10 + 1155940156T^8 + 18488958795T^6 + 113429470635T^4 - 83606258025*T^2 + 463495448025
$89$	\( (T^{8} - 357 T^{6} + \cdots + 25515405)^{2} \) (T^8 - 357T^6 + 41116T^4 - 1773540*T^2 + 25515405)^2
$97$	\( (T^{8} - 36 T^{7} + \cdots + 160801)^{2} \) (T^8 - 36T^7 + 893T^6 - 12915T^5 + 110436T^4 - 511815T^3 + 1058557T^2 + 193683*T + 160801)^2
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\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(1\)	\(\beta_{6}\)