LMFDB - Newform orbit 945.2.i.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(316,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 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("945.316");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.i (of order $3$, 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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + \cdots + 729 $ x^12 - x^11 + 2x^10 - x^9 - 4x^8 + 20x^7 - 38x^6 + 60x^5 - 36x^4 - 27x^3 + 162x^2 - 243*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{9} + 2 \beta_{5} - \beta_{2} - 2) q^{4} + (\beta_{5} - 1) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{8}+O(q^{10}) $ q + (-b5 + b4 - b1 + 1) * q^2 + (-b9 + 2b5 - b2 - 2) q^4 + (b5 - 1) * q^5 - b5 * q^7 + (-2b4 - b3 - 1) q^8	$ q + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{9} + 2 \beta_{5} - \beta_{2} - 2) q^{4} + (\beta_{5} - 1) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{8} - \beta_{4} q^{10} + ( - \beta_{11} - \beta_{6} - \beta_{5} + \cdots - 1) q^{11}+ \cdots - \beta_{4} q^{98}+O(q^{100}) $ q + (-b5 + b4 - b1 + 1) * q^2 + (-b9 + 2b5 - b2 - 2) q^4 + (b5 - 1) * q^5 - b5 * q^7 + (-2b4 - b3 - 1) q^8 - b4 * q^10 + (-b11 - b6 - b5 - b4 + b2 + b1 - 1) * q^11 + (-b11 - b9 - b8 + 2b5 + b1 - 2) q^13 + (b5 + b1 - 1) * q^14 + (b11 + 2b9 - 2b7 - 2b6 - 2b5 - b4 + 2b2 + b1 - 1) q^16 + (-b8 - b6 - b4 - 2) * q^17 + (-b7 + b6 + b4 - b3 + 4) * q^19 + (b9 - b7 - b6 - 2b5 + b2) q^20 + (b11 + b9 + b8 + 2b1) q^22 + (b10 + 2b5 - b3 + b2 - b1 - 1) q^23 - b5 * q^25 + (b8 + b7 - 3b4 - b3 + 1) q^26 + (b7 + b6 + 2) * q^28 + (-b9 + b7 + 2b5) q^29 + (b11 + b10 + b9 + b8 + b5 - b3 + b2 - 2b1) q^31 + (-b11 - b10 + 2b9 - b8 - 2b5 + b3 + b2 + 3b1 + 1) q^32 + (-b11 - 2b10 + 3b9 - 3b7 - 2b5 - 2b4 + 2b1 - 2) * q^34 + q^35 + (b8 + 2b7 - 2b6 - b4 + 2) * q^37 + (b11 + b10 - b9 + b7 - 3b6 - b5 + 2b4 + 3b2 - 2b1 + 2) * q^38 + (-b10 + b3 + 2b1 - 1) q^40 + (b11 + b10 + b9 + b8 - b3 + 1) * q^41 + (-b11 + b10 - 2b9 + 2b7 - 2b5) q^43 + (b8 + 2b7 + b6 + 2b4 + b3 + 7) * q^44 + (-b8 - 2b7 - 4b6 - 2b4 + b3 - 5) q^46 + (b11 - b9 + b7 - 2b5 + b4 - b1 + 1) q^47 + (b5 - 1) * q^49 + (b5 + b1 - 1) * q^50 + (b10 + 3b9 - 3b7 - 4b6 - 6b5 + b4 + 4b2 - b1 + 1) q^52 + (b8 + 3b7 - b6 - 2b4 + b3 - 4) * q^53 + (-b8 + b6 + b4 + 2) * q^55 + (b10 + 2b4 - 2b1 + 2) * q^56 + (b9 - b5 - b2 - 4b1 + 1) q^58 + (-b11 - b10 - b8 + 2b5 + b3 - 2b1 - 3) * q^59 + (-b10 - 2b6 - b5 - b4 + 2b2 + b1 - 1) * q^61 + (-2b8 - 3b7 - 4b6 + 2b3 - 6) * q^62 + (-b7 + 2b6 + 6b4 + 9) * q^64 + (b11 + b9 - b7 - 2b5 + b4 - b1 + 1) q^65 + (2b10 + 2b9 + b5 - 2b3 + b2 - 2b1 + 1) * q^67 + (b11 + b10 + 2b9 + b8 - 4b5 - b3 + 7b2 + 6b1 + 5) * q^68 + (-b5 + b4 - b1 + 1) * q^70 + (-b8 + 2b6 + 3b4 + 7) * q^71 + (b8 + 3b7 - b6 - 4b4 - b3 - 1) * q^73 + (b11 - b10 + 4b9 - 4b7 + 3b6 - b5 + 6b4 - 3b2 - 6b1 + 6) * q^74 + (-2b11 - 2b10 - b9 - 2b8 + 6b5 + 2b3 - 6b2 - 8) * q^76 + (b11 + b8 + b5 - b2 - b1 - 1) * q^77 + (2b10 - 2b9 + 2b7 - b6 + 4b5 + b2) * q^79 + (b8 + 2b7 + 2b6 + b4 + 3) * q^80 + (-2b8 - 2b7 - b6 + 3b4 + b3 + 4) q^82 + (-b10 - b9 + b7 - 3b4 + 3b1 - 3) * q^83 + (b11 + b8 - b5 + b2 + b1 + 1) * q^85 + (b10 - b9 + 5b5 - b3 - 4b2 - b1 - 4) * q^86 + (2b11 + 2b10 - 2b9 + 2b7 + 5b6 + 2b5 + 7b4 - 5b2 - 7b1 + 7) q^88 + (-b7 + 3b6 + 1) q^89 + (b8 + b7 - b4 + 1) * q^91 + (-2b11 - 3b10 + 3b9 - 3b7 + b5 - 7b4 + 7b1 - 7) * q^92 + (-b11 - b10 + b9 - b8 + 5b5 + b3 - 2b2 - b1 - 6) * q^94 + (-b10 + b9 + 2b5 + b3 - b2 - b1 - 3) q^95 + (-b10 + 2b9 - 2b7 - 6b5 + b4 - b1 + 1) q^97 - b4 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 11 q^{4} - 6 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 - 11 * q^4 - 6 * q^5 - 6 * q^7 - 6 * q^8	$ 12 q - 3 q^{2} - 11 q^{4} - 6 q^{5} - 6 q^{7} - 6 q^{8} + 6 q^{10} - 7 q^{11} - 10 q^{13} - 3 q^{14} - 13 q^{16} - 14 q^{17} + 30 q^{19} - 11 q^{20} + 7 q^{22} - 14 q^{23} - 6 q^{25} + 26 q^{26} + 22 q^{28} + 13 q^{29} - 10 q^{31} + 18 q^{32} - 15 q^{34} + 12 q^{35} + 42 q^{37} + 4 q^{38} + 3 q^{40} + 4 q^{41} - 13 q^{43} + 78 q^{44} - 30 q^{46} - 8 q^{47} - 6 q^{49} - 3 q^{50} - 31 q^{52} - 20 q^{53} + 14 q^{55} + 3 q^{56} - 3 q^{58} - 21 q^{59} - 2 q^{61} - 50 q^{62} + 62 q^{64} - 10 q^{65} - 6 q^{67} + 33 q^{68} - 3 q^{70} + 58 q^{71} + 16 q^{73} + 5 q^{74} - 31 q^{76} - 7 q^{77} + 22 q^{79} + 26 q^{80} + 36 q^{82} - 5 q^{83} + 7 q^{85} - 23 q^{86} + 19 q^{88} - 2 q^{89} + 20 q^{91} - 9 q^{92} - 31 q^{94} - 15 q^{95} - 32 q^{97} + 6 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 - 11 * q^4 - 6 * q^5 - 6 * q^7 - 6 * q^8 + 6 * q^10 - 7 * q^11 - 10 * q^13 - 3 * q^14 - 13 * q^16 - 14 * q^17 + 30 * q^19 - 11 * q^20 + 7 * q^22 - 14 * q^23 - 6 * q^25 + 26 * q^26 + 22 * q^28 + 13 * q^29 - 10 * q^31 + 18 * q^32 - 15 * q^34 + 12 * q^35 + 42 * q^37 + 4 * q^38 + 3 * q^40 + 4 * q^41 - 13 * q^43 + 78 * q^44 - 30 * q^46 - 8 * q^47 - 6 * q^49 - 3 * q^50 - 31 * q^52 - 20 * q^53 + 14 * q^55 + 3 * q^56 - 3 * q^58 - 21 * q^59 - 2 * q^61 - 50 * q^62 + 62 * q^64 - 10 * q^65 - 6 * q^67 + 33 * q^68 - 3 * q^70 + 58 * q^71 + 16 * q^73 + 5 * q^74 - 31 * q^76 - 7 * q^77 + 22 * q^79 + 26 * q^80 + 36 * q^82 - 5 * q^83 + 7 * q^85 - 23 * q^86 + 19 * q^88 - 2 * q^89 + 20 * q^91 - 9 * q^92 - 31 * q^94 - 15 * q^95 - 32 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + \cdots + 729 $ x^12 - x^11 + 2*x^10 - x^9 - 4*x^8 + 20*x^7 - 38*x^6 + 60*x^5 - 36*x^4 - 27*x^3 + 162*x^2 - 243*x + 729 :

$\beta_{1}$	$=$	$ ( - \nu^{11} + 40 \nu^{10} + 274 \nu^{9} - 209 \nu^{8} + 487 \nu^{7} - 1085 \nu^{6} + 341 \nu^{5} + \cdots + 17253 ) / 17496 $ (-v^11 + 40v^10 + 274v^9 - 209v^8 + 487v^7 - 1085v^6 + 341v^5 + 1329v^4 - 2655v^3 + 12204v^2 - 10530v + 17253) / 17496
$\beta_{2}$	$=$	$ ( \nu^{10} - 3 \nu^{9} + 7 \nu^{8} - 8 \nu^{7} + 13 \nu^{6} - 2 \nu^{5} - 27 \nu^{4} + 88 \nu^{3} + \cdots - 243 ) / 108 $ (v^10 - 3v^9 + 7v^8 - 8v^7 + 13v^6 - 2v^5 - 27v^4 + 88v^3 - 153v^2 + 279*v - 243) / 108
$\beta_{3}$	$=$	$ ( \nu^{11} - 121 \nu^{10} + 131 \nu^{9} - 34 \nu^{8} - \nu^{7} + 842 \nu^{6} - 2771 \nu^{5} + \cdots - 10692 ) / 8748 $ (v^11 - 121v^10 + 131v^9 - 34v^8 - v^7 + 842v^6 - 2771v^5 + 2154v^4 - 2367v^3 + 4563v^2 + 5427*v - 10692) / 8748
$\beta_{4}$	$=$	$ ( - 13 \nu^{11} + 25 \nu^{10} + 61 \nu^{9} + 46 \nu^{8} + 49 \nu^{7} - 110 \nu^{6} + 311 \nu^{5} + \cdots - 3888 ) / 8748 $ (-13v^11 + 25v^10 + 61v^9 + 46v^8 + 49v^7 - 110v^6 + 311v^5 - 822v^4 - 1305v^3 - 351v^2 + 2349*v - 3888) / 8748
$\beta_{5}$	$=$	$ ( 5 \nu^{11} - 17 \nu^{10} + 49 \nu^{9} - 110 \nu^{8} + 181 \nu^{7} - 68 \nu^{6} - 79 \nu^{5} + \cdots + 4374 ) / 2916 $ (5v^11 - 17v^10 + 49v^9 - 110v^8 + 181v^7 - 68v^6 - 79v^5 + 702v^4 - 1629v^3 + 2673v^2 - 2997*v + 4374) / 2916
$\beta_{6}$	$=$	$ ( \nu^{11} + \nu^{10} - 3 \nu^{9} + 6 \nu^{8} + 15 \nu^{7} - 12 \nu^{6} - 13 \nu^{5} - 22 \nu^{4} + \cdots + 162 ) / 486 $ (v^11 + v^10 - 3v^9 + 6v^8 + 15v^7 - 12v^6 - 13v^5 - 22v^4 + 9v^3 + 99v^2 - 405*v + 162) / 486
$\beta_{7}$	$=$	$ ( - \nu^{11} + \nu^{10} - 2 \nu^{9} + \nu^{8} + 4 \nu^{7} - 20 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + \cdots + 243 ) / 243 $ (-v^11 + v^10 - 2v^9 + v^8 + 4v^7 - 20v^6 + 38v^5 - 60v^4 + 36v^3 + 27v^2 + 81v + 243) / 243
$\beta_{8}$	$=$	$ ( 37 \nu^{11} - 49 \nu^{10} + 95 \nu^{9} + 146 \nu^{8} - 253 \nu^{7} + 1130 \nu^{6} - 1979 \nu^{5} + \cdots - 10692 ) / 8748 $ (37v^11 - 49v^10 + 95v^9 + 146v^8 - 253v^7 + 1130v^6 - 1979v^5 + 210v^4 + 2817v^3 - 9045v^2 + 8343*v - 10692) / 8748
$\beta_{9}$	$=$	$ ( 3 \nu^{11} - 10 \nu^{10} + 28 \nu^{9} - 59 \nu^{8} + 79 \nu^{7} - 89 \nu^{6} - 71 \nu^{5} + \cdots + 1863 ) / 486 $ (3v^11 - 10v^10 + 28v^9 - 59v^8 + 79v^7 - 89v^6 - 71v^5 + 341v^4 - 855v^3 + 1422v^2 - 1620*v + 1863) / 486
$\beta_{10}$	$=$	$ ( 115 \nu^{11} - 640 \nu^{10} + 1358 \nu^{9} - 1957 \nu^{8} + 2999 \nu^{7} - 1765 \nu^{6} + \cdots + 42525 ) / 17496 $ (115v^11 - 640v^10 + 1358v^9 - 1957v^8 + 2999v^7 - 1765v^6 - 6239v^5 + 21117v^4 - 35199v^3 + 50328v^2 - 60426*v + 42525) / 17496
$\beta_{11}$	$=$	$ ( 161 \nu^{11} - 284 \nu^{10} + 598 \nu^{9} - 1019 \nu^{8} + 1621 \nu^{7} - 923 \nu^{6} + 827 \nu^{5} + \cdots + 74115 ) / 17496 $ (161v^11 - 284v^10 + 598v^9 - 1019v^8 + 1621v^7 - 923v^6 + 827v^5 + 8619v^4 - 17397v^3 + 41364v^2 - 39690*v + 74115) / 17496

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

$\nu$	$=$	$ ( \beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{2} ) / 3 $ (b9 + b7 - b6 + 2*b2) / 3
$\nu^{2}$	$=$	$ ( \beta_{11} - \beta_{10} - \beta_{8} - \beta_{4} + 2\beta_{3} + 2\beta _1 - 3 ) / 3 $ (b11 - b10 - b8 - b4 + 2b3 + 2b1 - 3) / 3
$\nu^{3}$	$=$	$ ( -2\beta_{9} + 3\beta_{8} + 4\beta_{7} - \beta_{6} + 3\beta_{5} - 9\beta_{4} - \beta_{2} + 6\beta _1 - 9 ) / 3 $ (-2b9 + 3b8 + 4b7 - b6 + 3b5 - 9b4 - b2 + 6b1 - 9) / 3
$\nu^{4}$	$=$	$ ( - 2 \beta_{11} + 5 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \cdots + 3 ) / 3 $ (-2b11 + 5b10 - 3b9 - 4b8 - 3b7 - 3b6 + 3b5 - 7b4 - 4b3 + 6b2 + 5*b1 + 3) / 3
$\nu^{5}$	$=$	$ ( 6 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} + \cdots - 18 ) / 3 $ (6b11 + 3b10 - 8b9 + 4b7 - 7b6 + 6b5 + 3b4 - 6b3 - b2 - 3*b1 - 18) / 3
$\nu^{6}$	$=$	$ ( 4 \beta_{11} - 7 \beta_{10} - 18 \beta_{9} + 2 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} + 51 \beta_{5} + \cdots - 9 ) / 3 $ (4b11 - 7b10 - 18b9 + 2b8 - 9b7 - 3b6 + 51b5 + 5b4 + 8b3 - 12b2 - b1 - 9) / 3
$\nu^{7}$	$=$	$ ( - 9 \beta_{11} + 9 \beta_{10} - 14 \beta_{9} + 18 \beta_{8} + 40 \beta_{7} + 23 \beta_{6} + 81 \beta_{5} + \cdots - 54 ) / 3 $ (-9b11 + 9b10 - 14b9 + 18b8 + 40b7 + 23b6 + 81b5 - 9b4 - 9b3 + 17b2 - 54) / 3
$\nu^{8}$	$=$	$ ( 19 \beta_{11} + 35 \beta_{10} - 36 \beta_{9} - 19 \beta_{8} + 54 \beta_{6} - 63 \beta_{5} + 71 \beta_{4} + \cdots + 78 ) / 3 $ (19b11 + 35b10 - 36b9 - 19b8 + 54b6 - 63b5 + 71b4 + 2b3 + 9b2 + 2b1 + 78) / 3
$\nu^{9}$	$=$	$ ( 18 \beta_{11} + 9 \beta_{10} - 38 \beta_{9} + 66 \beta_{8} + 13 \beta_{7} - 37 \beta_{6} - 15 \beta_{5} + \cdots - 36 ) / 3 $ (18b11 + 9b10 - 38b9 + 66b8 + 13b7 - 37b6 - 15b5 + 72b4 - 9b3 - 46b2 + 105*b1 - 36) / 3
$\nu^{10}$	$=$	$ ( - 92 \beta_{11} - 67 \beta_{10} + 60 \beta_{9} - 76 \beta_{8} - 228 \beta_{7} + 6 \beta_{6} + \cdots + 138 ) / 3 $ (-92b11 - 67b10 + 60b9 - 76b8 - 228b7 + 6b6 + 210b5 + 38b4 - 31b3 + 105b2 + 221*b1 + 138) / 3
$\nu^{11}$	$=$	$ ( 150 \beta_{11} - 87 \beta_{10} + 289 \beta_{9} + 126 \beta_{8} - 86 \beta_{7} + 83 \beta_{6} + \cdots - 288 ) / 3 $ (150b11 - 87b10 + 289b9 + 126b8 - 86b7 + 83b6 - 363b5 + 12b4 - 141b3 + 242b2 - 111*b1 - 288) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/945\mathbb{Z}\right)^\times$.

$n$	$136$	$596$	$757$
$\chi(n)$	$1$	$-1 + \beta_{5}$	$1$

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} $

316.1

−0.382799	+	1.68922i
1.65373	−	0.514941i
−1.73190	−	0.0231100i
−0.764584	−	1.55416i
0.478182	−	1.66473i
1.24736	+	1.20170i
−0.382799	−	1.68922i
1.65373	+	0.514941i
−1.73190	+	0.0231100i
−0.764584	+	1.55416i
0.478182	+	1.66473i
1.24736	−	1.20170i

−1.27897

2.21523i

−2.27151

−

3.93437i

−0.500000

−

0.866025i

−0.500000

0.866025i

6.50486

2.55793

316.2

−1.09108

1.88981i

−1.38091

−

2.39181i

−0.500000

−

0.866025i

−0.500000

0.866025i

1.66244

2.18216

316.3

−0.746337

1.29269i

−0.114038

−

0.197519i

−0.500000

−

0.866025i

−0.500000

0.866025i

−2.64491

1.49267

316.4

−0.368623

0.638475i

0.728233

1.26134i

−0.500000

−

0.866025i

−0.500000

0.866025i

−2.54827

0.737247

316.5

0.631422

−

1.09366i

0.202612

0.350934i

−0.500000

−

0.866025i

−0.500000

0.866025i

3.03742

−1.26284

316.6

1.35359

−

2.34448i

−2.66438

−

4.61485i

−0.500000

−

0.866025i

−0.500000

0.866025i

−9.01155

−2.70717

631.1

−1.27897

−

2.21523i

−2.27151

3.93437i

−0.500000

0.866025i

−0.500000

−

0.866025i

6.50486

2.55793

631.2

−1.09108

−

1.88981i

−1.38091

2.39181i

−0.500000

0.866025i

−0.500000

−

0.866025i

1.66244

2.18216

631.3

−0.746337

−

1.29269i

−0.114038

0.197519i

−0.500000

0.866025i

−0.500000

−

0.866025i

−2.64491

1.49267

631.4

−0.368623

−

0.638475i

0.728233

−

1.26134i

−0.500000

0.866025i

−0.500000

−

0.866025i

−2.54827

0.737247

631.5

0.631422

1.09366i

0.202612

−

0.350934i

−0.500000

0.866025i

−0.500000

−

0.866025i

3.03742

−1.26284

631.6

1.35359

2.34448i

−2.66438

4.61485i

−0.500000

0.866025i

−0.500000

−

0.866025i

−9.01155

−2.70717

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	945.2.i.e		12
3.b	odd	2	1		315.2.i.e	✓	12
9.c	even	3	1	inner	945.2.i.e		12
9.c	even	3	1		2835.2.a.v		6
9.d	odd	6	1		315.2.i.e	✓	12
9.d	odd	6	1		2835.2.a.u		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.i.e	✓	12	3.b	odd	2	1
315.2.i.e	✓	12	9.d	odd	6	1
945.2.i.e		12	1.a	even	1	1	trivial
945.2.i.e		12	9.c	even	3	1	inner
2835.2.a.u		6	9.d	odd	6	1
2835.2.a.v		6	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} + 3 T_{2}^{11} + 16 T_{2}^{10} + 33 T_{2}^{9} + 135 T_{2}^{8} + 255 T_{2}^{7} + 628 T_{2}^{6} + \cdots + 441 $ T2^12 + 3*T2^11 + 16*T2^10 + 33*T2^9 + 135*T2^8 + 255*T2^7 + 628*T2^6 + 702*T2^5 + 1144*T2^4 + 954*T2^3 + 1401*T2^2 + 756*T2 + 441 acting on $S_{2}^{\mathrm{new}}(945, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} + 3 T^{11} + \cdots + 441 $ T^12 + 3T^11 + 16T^10 + 33T^9 + 135T^8 + 255T^7 + 628T^6 + 702T^5 + 1144T^4 + 954T^3 + 1401T^2 + 756*T + 441
$3$	$ T^{12} $ T^12
$5$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$7$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$11$	$ T^{12} + 7 T^{11} + \cdots + 99225 $ T^12 + 7T^11 + 68T^10 + 409T^9 + 3043T^8 + 15269T^7 + 63986T^6 + 177470T^5 + 374470T^4 + 512220T^3 + 509625T^2 + 274050*T + 99225
$13$	$ T^{12} + 10 T^{11} + \cdots + 677329 $ T^12 + 10T^11 + 91T^10 + 380T^9 + 1822T^8 + 4864T^7 + 21800T^6 + 44143T^5 + 127879T^4 + 137111T^3 + 361294T^2 + 287227*T + 677329
$17$	$ (T^{6} + 7 T^{5} + \cdots - 501)^{2} $ (T^6 + 7T^5 - 51T^4 - 401T^3 + 133T^2 + 2964*T - 501)^2
$19$	$ (T^{6} - 15 T^{5} + \cdots + 400)^{2} $ (T^6 - 15T^5 + 29T^4 + 433T^3 - 2015T^2 + 2200*T + 400)^2
$23$	$ T^{12} + 14 T^{11} + \cdots + 46656 $ T^12 + 14T^11 + 172T^10 + 1244T^9 + 9397T^8 + 54452T^7 + 316252T^6 + 1292342T^5 + 4414321T^4 + 8855352T^3 + 12951144T^2 + 793152*T + 46656
$29$	$ T^{12} - 13 T^{11} + \cdots + 3600 $ T^12 - 13T^11 + 115T^10 - 586T^9 + 2257T^8 - 5422T^7 + 10714T^6 - 13150T^5 + 22705T^4 - 24060T^3 + 26700T^2 - 10800*T + 3600
$31$	$ T^{12} + 10 T^{11} + \cdots + 19360000 $ T^12 + 10T^11 + 154T^10 + 1442T^9 + 17101T^8 + 131614T^7 + 816431T^6 + 3393325T^5 + 10704625T^4 + 22914200T^3 + 35950000T^2 + 32560000*T + 19360000
$37$	$ (T^{6} - 21 T^{5} + \cdots + 42472)^{2} $ (T^6 - 21T^5 + 35T^4 + 1693T^3 - 10577T^2 + 6688*T + 42472)^2
$41$	$ T^{12} - 4 T^{11} + \cdots + 4410000 $ T^12 - 4T^11 + 88T^10 - 670T^9 + 7675T^8 - 40288T^7 + 187441T^6 - 456625T^5 + 1056625T^4 - 1321800T^3 + 2647500T^2 - 2520000*T + 4410000
$43$	$ T^{12} + 13 T^{11} + \cdots + 41938576 $ T^12 + 13T^11 + 199T^10 + 938T^9 + 9469T^8 + 25006T^7 + 342422T^6 + 277420T^5 + 4274425T^4 - 8176516T^3 + 45620164T^2 - 43544624*T + 41938576
$47$	$ T^{12} + 8 T^{11} + \cdots + 86545809 $ T^12 + 8T^11 + 119T^10 + 206T^9 + 4342T^8 + 334T^7 + 132680T^6 - 171155T^5 + 2011267T^4 - 2410191T^3 + 19858182T^2 - 26429823*T + 86545809
$53$	$ (T^{6} + 10 T^{5} + \cdots - 21480)^{2} $ (T^6 + 10T^5 - 124T^4 - 1219T^3 + 2065T^2 + 17280*T - 21480)^2
$59$	$ T^{12} + \cdots + 5625000000 $ T^12 + 21T^11 + 385T^10 + 3408T^9 + 31647T^8 + 165654T^7 + 1364656T^6 + 5558700T^5 + 38295625T^4 + 91275000T^3 + 605625000T^2 + 1125000000*T + 5625000000
$61$	$ T^{12} + 2 T^{11} + \cdots + 21086464 $ T^12 + 2T^11 + 137T^10 + 594T^9 + 15652T^8 + 52186T^7 + 547849T^6 + 496450T^5 + 10612993T^4 + 15124728T^3 + 56652080T^2 - 30233728*T + 21086464
$67$	$ T^{12} + \cdots + 71742551104 $ T^12 + 6T^11 + 267T^10 + 526T^9 + 43434T^8 + 64224T^7 + 3808329T^6 - 2099988T^5 + 213421545T^4 - 21184304T^3 + 5177749560T^2 - 8395427712*T + 71742551104
$71$	$ (T^{6} - 29 T^{5} + \cdots + 6219)^{2} $ (T^6 - 29T^5 + 227T^4 + 59T^3 - 5969T^2 + 13152*T + 6219)^2
$73$	$ (T^{6} - 8 T^{5} + \cdots + 86137)^{2} $ (T^6 - 8T^5 - 233T^4 + 1891T^3 + 6161T^2 - 61557*T + 86137)^2
$79$	$ T^{12} + \cdots + 18108815761 $ T^12 - 22T^11 + 486T^10 - 4304T^9 + 53242T^8 - 293238T^7 + 3865894T^6 - 14925258T^5 + 140085138T^4 - 310007512T^3 + 3313740790T^2 - 6842833650*T + 18108815761
$83$	$ T^{12} + \cdots + 375003225 $ T^12 + 5T^11 + 137T^10 + 176T^9 + 11134T^8 + 15901T^7 + 424769T^6 + 316615T^5 + 11037700T^4 + 9098610T^3 + 114560475T^2 - 139137525*T + 375003225
$89$	$ (T^{6} + T^{5} + \cdots - 36492)^{2} $ (T^6 + T^5 - 219T^4 - 365T^3 + 12079T^2 + 24648T - 36492)^2
$97$	$ T^{12} + 32 T^{11} + \cdots + 95277121 $ T^12 + 32T^11 + 678T^10 + 8176T^9 + 72112T^8 + 414354T^7 + 1814518T^6 + 5331096T^5 + 12952026T^4 + 21291320T^3 + 42648952T^2 + 53705022*T + 95277121
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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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{9} + 2 \beta_{5} - \beta_{2} - 2) q^{4} + (\beta_{5} - 1) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{8}+O(q^{10}) \) q + (-b5 + b4 - b1 + 1) * q^2 + (-b9 + 2b5 - b2 - 2) q^4 + (b5 - 1) * q^5 - b5 * q^7 + (-2b4 - b3 - 1) q^8	\( q + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{9} + 2 \beta_{5} - \beta_{2} - 2) q^{4} + (\beta_{5} - 1) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{8} - \beta_{4} q^{10} + ( - \beta_{11} - \beta_{6} - \beta_{5} + \cdots - 1) q^{11}+ \cdots - \beta_{4} q^{98}+O(q^{100}) \) q + (-b5 + b4 - b1 + 1) * q^2 + (-b9 + 2b5 - b2 - 2) q^4 + (b5 - 1) * q^5 - b5 * q^7 + (-2b4 - b3 - 1) q^8 - b4 * q^10 + (-b11 - b6 - b5 - b4 + b2 + b1 - 1) * q^11 + (-b11 - b9 - b8 + 2b5 + b1 - 2) q^13 + (b5 + b1 - 1) * q^14 + (b11 + 2b9 - 2b7 - 2b6 - 2b5 - b4 + 2b2 + b1 - 1) q^16 + (-b8 - b6 - b4 - 2) * q^17 + (-b7 + b6 + b4 - b3 + 4) * q^19 + (b9 - b7 - b6 - 2b5 + b2) q^20 + (b11 + b9 + b8 + 2b1) q^22 + (b10 + 2b5 - b3 + b2 - b1 - 1) q^23 - b5 * q^25 + (b8 + b7 - 3b4 - b3 + 1) q^26 + (b7 + b6 + 2) * q^28 + (-b9 + b7 + 2b5) q^29 + (b11 + b10 + b9 + b8 + b5 - b3 + b2 - 2b1) q^31 + (-b11 - b10 + 2b9 - b8 - 2b5 + b3 + b2 + 3b1 + 1) q^32 + (-b11 - 2b10 + 3b9 - 3b7 - 2b5 - 2b4 + 2b1 - 2) * q^34 + q^35 + (b8 + 2b7 - 2b6 - b4 + 2) * q^37 + (b11 + b10 - b9 + b7 - 3b6 - b5 + 2b4 + 3b2 - 2b1 + 2) * q^38 + (-b10 + b3 + 2b1 - 1) q^40 + (b11 + b10 + b9 + b8 - b3 + 1) * q^41 + (-b11 + b10 - 2b9 + 2b7 - 2b5) q^43 + (b8 + 2b7 + b6 + 2b4 + b3 + 7) * q^44 + (-b8 - 2b7 - 4b6 - 2b4 + b3 - 5) q^46 + (b11 - b9 + b7 - 2b5 + b4 - b1 + 1) q^47 + (b5 - 1) * q^49 + (b5 + b1 - 1) * q^50 + (b10 + 3b9 - 3b7 - 4b6 - 6b5 + b4 + 4b2 - b1 + 1) q^52 + (b8 + 3b7 - b6 - 2b4 + b3 - 4) * q^53 + (-b8 + b6 + b4 + 2) * q^55 + (b10 + 2b4 - 2b1 + 2) * q^56 + (b9 - b5 - b2 - 4b1 + 1) q^58 + (-b11 - b10 - b8 + 2b5 + b3 - 2b1 - 3) * q^59 + (-b10 - 2b6 - b5 - b4 + 2b2 + b1 - 1) * q^61 + (-2b8 - 3b7 - 4b6 + 2b3 - 6) * q^62 + (-b7 + 2b6 + 6b4 + 9) * q^64 + (b11 + b9 - b7 - 2b5 + b4 - b1 + 1) q^65 + (2b10 + 2b9 + b5 - 2b3 + b2 - 2b1 + 1) * q^67 + (b11 + b10 + 2b9 + b8 - 4b5 - b3 + 7b2 + 6b1 + 5) * q^68 + (-b5 + b4 - b1 + 1) * q^70 + (-b8 + 2b6 + 3b4 + 7) * q^71 + (b8 + 3b7 - b6 - 4b4 - b3 - 1) * q^73 + (b11 - b10 + 4b9 - 4b7 + 3b6 - b5 + 6b4 - 3b2 - 6b1 + 6) * q^74 + (-2b11 - 2b10 - b9 - 2b8 + 6b5 + 2b3 - 6b2 - 8) * q^76 + (b11 + b8 + b5 - b2 - b1 - 1) * q^77 + (2b10 - 2b9 + 2b7 - b6 + 4b5 + b2) * q^79 + (b8 + 2b7 + 2b6 + b4 + 3) * q^80 + (-2b8 - 2b7 - b6 + 3b4 + b3 + 4) q^82 + (-b10 - b9 + b7 - 3b4 + 3b1 - 3) * q^83 + (b11 + b8 - b5 + b2 + b1 + 1) * q^85 + (b10 - b9 + 5b5 - b3 - 4b2 - b1 - 4) * q^86 + (2b11 + 2b10 - 2b9 + 2b7 + 5b6 + 2b5 + 7b4 - 5b2 - 7b1 + 7) q^88 + (-b7 + 3b6 + 1) q^89 + (b8 + b7 - b4 + 1) * q^91 + (-2b11 - 3b10 + 3b9 - 3b7 + b5 - 7b4 + 7b1 - 7) * q^92 + (-b11 - b10 + b9 - b8 + 5b5 + b3 - 2b2 - b1 - 6) * q^94 + (-b10 + b9 + 2b5 + b3 - b2 - b1 - 3) q^95 + (-b10 + 2b9 - 2b7 - 6b5 + b4 - b1 + 1) q^97 - b4 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 11 q^{4} - 6 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 - 11 * q^4 - 6 * q^5 - 6 * q^7 - 6 * q^8	\( 12 q - 3 q^{2} - 11 q^{4} - 6 q^{5} - 6 q^{7} - 6 q^{8} + 6 q^{10} - 7 q^{11} - 10 q^{13} - 3 q^{14} - 13 q^{16} - 14 q^{17} + 30 q^{19} - 11 q^{20} + 7 q^{22} - 14 q^{23} - 6 q^{25} + 26 q^{26} + 22 q^{28} + 13 q^{29} - 10 q^{31} + 18 q^{32} - 15 q^{34} + 12 q^{35} + 42 q^{37} + 4 q^{38} + 3 q^{40} + 4 q^{41} - 13 q^{43} + 78 q^{44} - 30 q^{46} - 8 q^{47} - 6 q^{49} - 3 q^{50} - 31 q^{52} - 20 q^{53} + 14 q^{55} + 3 q^{56} - 3 q^{58} - 21 q^{59} - 2 q^{61} - 50 q^{62} + 62 q^{64} - 10 q^{65} - 6 q^{67} + 33 q^{68} - 3 q^{70} + 58 q^{71} + 16 q^{73} + 5 q^{74} - 31 q^{76} - 7 q^{77} + 22 q^{79} + 26 q^{80} + 36 q^{82} - 5 q^{83} + 7 q^{85} - 23 q^{86} + 19 q^{88} - 2 q^{89} + 20 q^{91} - 9 q^{92} - 31 q^{94} - 15 q^{95} - 32 q^{97} + 6 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 - 11 * q^4 - 6 * q^5 - 6 * q^7 - 6 * q^8 + 6 * q^10 - 7 * q^11 - 10 * q^13 - 3 * q^14 - 13 * q^16 - 14 * q^17 + 30 * q^19 - 11 * q^20 + 7 * q^22 - 14 * q^23 - 6 * q^25 + 26 * q^26 + 22 * q^28 + 13 * q^29 - 10 * q^31 + 18 * q^32 - 15 * q^34 + 12 * q^35 + 42 * q^37 + 4 * q^38 + 3 * q^40 + 4 * q^41 - 13 * q^43 + 78 * q^44 - 30 * q^46 - 8 * q^47 - 6 * q^49 - 3 * q^50 - 31 * q^52 - 20 * q^53 + 14 * q^55 + 3 * q^56 - 3 * q^58 - 21 * q^59 - 2 * q^61 - 50 * q^62 + 62 * q^64 - 10 * q^65 - 6 * q^67 + 33 * q^68 - 3 * q^70 + 58 * q^71 + 16 * q^73 + 5 * q^74 - 31 * q^76 - 7 * q^77 + 22 * q^79 + 26 * q^80 + 36 * q^82 - 5 * q^83 + 7 * q^85 - 23 * q^86 + 19 * q^88 - 2 * q^89 + 20 * q^91 - 9 * q^92 - 31 * q^94 - 15 * q^95 - 32 * q^97 + 6 * q^98

Introduction

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Newspace parameters

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$q$-expansion

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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	945.2.i.e

Level	$945$
Weight	$2$
Character orbit	945.i
Analytic conductor	$7.546$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.54586299101\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + \cdots + 729 \) x^12 - x^11 + 2x^10 - x^9 - 4x^8 + 20x^7 - 38x^6 + 60x^5 - 36x^4 - 27x^3 + 162x^2 - 243*x + 729
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{11} + 40 \nu^{10} + 274 \nu^{9} - 209 \nu^{8} + 487 \nu^{7} - 1085 \nu^{6} + 341 \nu^{5} + \cdots + 17253 ) / 17496 \) (-v^11 + 40v^10 + 274v^9 - 209v^8 + 487v^7 - 1085v^6 + 341v^5 + 1329v^4 - 2655v^3 + 12204v^2 - 10530v + 17253) / 17496
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} - 3 \nu^{9} + 7 \nu^{8} - 8 \nu^{7} + 13 \nu^{6} - 2 \nu^{5} - 27 \nu^{4} + 88 \nu^{3} + \cdots - 243 ) / 108 \) (v^10 - 3v^9 + 7v^8 - 8v^7 + 13v^6 - 2v^5 - 27v^4 + 88v^3 - 153v^2 + 279*v - 243) / 108
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} - 121 \nu^{10} + 131 \nu^{9} - 34 \nu^{8} - \nu^{7} + 842 \nu^{6} - 2771 \nu^{5} + \cdots - 10692 ) / 8748 \) (v^11 - 121v^10 + 131v^9 - 34v^8 - v^7 + 842v^6 - 2771v^5 + 2154v^4 - 2367v^3 + 4563v^2 + 5427*v - 10692) / 8748
\(\beta_{4}\)	\(=\)	\( ( - 13 \nu^{11} + 25 \nu^{10} + 61 \nu^{9} + 46 \nu^{8} + 49 \nu^{7} - 110 \nu^{6} + 311 \nu^{5} + \cdots - 3888 ) / 8748 \) (-13v^11 + 25v^10 + 61v^9 + 46v^8 + 49v^7 - 110v^6 + 311v^5 - 822v^4 - 1305v^3 - 351v^2 + 2349*v - 3888) / 8748
\(\beta_{5}\)	\(=\)	\( ( 5 \nu^{11} - 17 \nu^{10} + 49 \nu^{9} - 110 \nu^{8} + 181 \nu^{7} - 68 \nu^{6} - 79 \nu^{5} + \cdots + 4374 ) / 2916 \) (5v^11 - 17v^10 + 49v^9 - 110v^8 + 181v^7 - 68v^6 - 79v^5 + 702v^4 - 1629v^3 + 2673v^2 - 2997*v + 4374) / 2916
\(\beta_{6}\)	\(=\)	\( ( \nu^{11} + \nu^{10} - 3 \nu^{9} + 6 \nu^{8} + 15 \nu^{7} - 12 \nu^{6} - 13 \nu^{5} - 22 \nu^{4} + \cdots + 162 ) / 486 \) (v^11 + v^10 - 3v^9 + 6v^8 + 15v^7 - 12v^6 - 13v^5 - 22v^4 + 9v^3 + 99v^2 - 405*v + 162) / 486
\(\beta_{7}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} - 2 \nu^{9} + \nu^{8} + 4 \nu^{7} - 20 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + \cdots + 243 ) / 243 \) (-v^11 + v^10 - 2v^9 + v^8 + 4v^7 - 20v^6 + 38v^5 - 60v^4 + 36v^3 + 27v^2 + 81v + 243) / 243
\(\beta_{8}\)	\(=\)	\( ( 37 \nu^{11} - 49 \nu^{10} + 95 \nu^{9} + 146 \nu^{8} - 253 \nu^{7} + 1130 \nu^{6} - 1979 \nu^{5} + \cdots - 10692 ) / 8748 \) (37v^11 - 49v^10 + 95v^9 + 146v^8 - 253v^7 + 1130v^6 - 1979v^5 + 210v^4 + 2817v^3 - 9045v^2 + 8343*v - 10692) / 8748
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{11} - 10 \nu^{10} + 28 \nu^{9} - 59 \nu^{8} + 79 \nu^{7} - 89 \nu^{6} - 71 \nu^{5} + \cdots + 1863 ) / 486 \) (3v^11 - 10v^10 + 28v^9 - 59v^8 + 79v^7 - 89v^6 - 71v^5 + 341v^4 - 855v^3 + 1422v^2 - 1620*v + 1863) / 486
\(\beta_{10}\)	\(=\)	\( ( 115 \nu^{11} - 640 \nu^{10} + 1358 \nu^{9} - 1957 \nu^{8} + 2999 \nu^{7} - 1765 \nu^{6} + \cdots + 42525 ) / 17496 \) (115v^11 - 640v^10 + 1358v^9 - 1957v^8 + 2999v^7 - 1765v^6 - 6239v^5 + 21117v^4 - 35199v^3 + 50328v^2 - 60426*v + 42525) / 17496
\(\beta_{11}\)	\(=\)	\( ( 161 \nu^{11} - 284 \nu^{10} + 598 \nu^{9} - 1019 \nu^{8} + 1621 \nu^{7} - 923 \nu^{6} + 827 \nu^{5} + \cdots + 74115 ) / 17496 \) (161v^11 - 284v^10 + 598v^9 - 1019v^8 + 1621v^7 - 923v^6 + 827v^5 + 8619v^4 - 17397v^3 + 41364v^2 - 39690*v + 74115) / 17496

\(\nu\)	\(=\)	\( ( \beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{2} ) / 3 \) (b9 + b7 - b6 + 2*b2) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} - \beta_{10} - \beta_{8} - \beta_{4} + 2\beta_{3} + 2\beta _1 - 3 ) / 3 \) (b11 - b10 - b8 - b4 + 2b3 + 2b1 - 3) / 3
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{9} + 3\beta_{8} + 4\beta_{7} - \beta_{6} + 3\beta_{5} - 9\beta_{4} - \beta_{2} + 6\beta _1 - 9 ) / 3 \) (-2b9 + 3b8 + 4b7 - b6 + 3b5 - 9b4 - b2 + 6b1 - 9) / 3
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{11} + 5 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \cdots + 3 ) / 3 \) (-2b11 + 5b10 - 3b9 - 4b8 - 3b7 - 3b6 + 3b5 - 7b4 - 4b3 + 6b2 + 5*b1 + 3) / 3
\(\nu^{5}\)	\(=\)	\( ( 6 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} + \cdots - 18 ) / 3 \) (6b11 + 3b10 - 8b9 + 4b7 - 7b6 + 6b5 + 3b4 - 6b3 - b2 - 3*b1 - 18) / 3
\(\nu^{6}\)	\(=\)	\( ( 4 \beta_{11} - 7 \beta_{10} - 18 \beta_{9} + 2 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} + 51 \beta_{5} + \cdots - 9 ) / 3 \) (4b11 - 7b10 - 18b9 + 2b8 - 9b7 - 3b6 + 51b5 + 5b4 + 8b3 - 12b2 - b1 - 9) / 3
\(\nu^{7}\)	\(=\)	\( ( - 9 \beta_{11} + 9 \beta_{10} - 14 \beta_{9} + 18 \beta_{8} + 40 \beta_{7} + 23 \beta_{6} + 81 \beta_{5} + \cdots - 54 ) / 3 \) (-9b11 + 9b10 - 14b9 + 18b8 + 40b7 + 23b6 + 81b5 - 9b4 - 9b3 + 17b2 - 54) / 3
\(\nu^{8}\)	\(=\)	\( ( 19 \beta_{11} + 35 \beta_{10} - 36 \beta_{9} - 19 \beta_{8} + 54 \beta_{6} - 63 \beta_{5} + 71 \beta_{4} + \cdots + 78 ) / 3 \) (19b11 + 35b10 - 36b9 - 19b8 + 54b6 - 63b5 + 71b4 + 2b3 + 9b2 + 2b1 + 78) / 3
\(\nu^{9}\)	\(=\)	\( ( 18 \beta_{11} + 9 \beta_{10} - 38 \beta_{9} + 66 \beta_{8} + 13 \beta_{7} - 37 \beta_{6} - 15 \beta_{5} + \cdots - 36 ) / 3 \) (18b11 + 9b10 - 38b9 + 66b8 + 13b7 - 37b6 - 15b5 + 72b4 - 9b3 - 46b2 + 105*b1 - 36) / 3
\(\nu^{10}\)	\(=\)	\( ( - 92 \beta_{11} - 67 \beta_{10} + 60 \beta_{9} - 76 \beta_{8} - 228 \beta_{7} + 6 \beta_{6} + \cdots + 138 ) / 3 \) (-92b11 - 67b10 + 60b9 - 76b8 - 228b7 + 6b6 + 210b5 + 38b4 - 31b3 + 105b2 + 221*b1 + 138) / 3
\(\nu^{11}\)	\(=\)	\( ( 150 \beta_{11} - 87 \beta_{10} + 289 \beta_{9} + 126 \beta_{8} - 86 \beta_{7} + 83 \beta_{6} + \cdots - 288 ) / 3 \) (150b11 - 87b10 + 289b9 + 126b8 - 86b7 + 83b6 - 363b5 + 12b4 - 141b3 + 242b2 - 111*b1 - 288) / 3

\(n\)	\(136\)	\(596\)	\(757\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{5}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 316.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} + 3 T^{11} + \cdots + 441 \) T^12 + 3T^11 + 16T^10 + 33T^9 + 135T^8 + 255T^7 + 628T^6 + 702T^5 + 1144T^4 + 954T^3 + 1401T^2 + 756*T + 441
$3$	\( T^{12} \) T^12
$5$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$7$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$11$	\( T^{12} + 7 T^{11} + \cdots + 99225 \) T^12 + 7T^11 + 68T^10 + 409T^9 + 3043T^8 + 15269T^7 + 63986T^6 + 177470T^5 + 374470T^4 + 512220T^3 + 509625T^2 + 274050*T + 99225
$13$	\( T^{12} + 10 T^{11} + \cdots + 677329 \) T^12 + 10T^11 + 91T^10 + 380T^9 + 1822T^8 + 4864T^7 + 21800T^6 + 44143T^5 + 127879T^4 + 137111T^3 + 361294T^2 + 287227*T + 677329
$17$	\( (T^{6} + 7 T^{5} + \cdots - 501)^{2} \) (T^6 + 7T^5 - 51T^4 - 401T^3 + 133T^2 + 2964*T - 501)^2
$19$	\( (T^{6} - 15 T^{5} + \cdots + 400)^{2} \) (T^6 - 15T^5 + 29T^4 + 433T^3 - 2015T^2 + 2200*T + 400)^2
$23$	\( T^{12} + 14 T^{11} + \cdots + 46656 \) T^12 + 14T^11 + 172T^10 + 1244T^9 + 9397T^8 + 54452T^7 + 316252T^6 + 1292342T^5 + 4414321T^4 + 8855352T^3 + 12951144T^2 + 793152*T + 46656
$29$	\( T^{12} - 13 T^{11} + \cdots + 3600 \) T^12 - 13T^11 + 115T^10 - 586T^9 + 2257T^8 - 5422T^7 + 10714T^6 - 13150T^5 + 22705T^4 - 24060T^3 + 26700T^2 - 10800*T + 3600
$31$	\( T^{12} + 10 T^{11} + \cdots + 19360000 \) T^12 + 10T^11 + 154T^10 + 1442T^9 + 17101T^8 + 131614T^7 + 816431T^6 + 3393325T^5 + 10704625T^4 + 22914200T^3 + 35950000T^2 + 32560000*T + 19360000
$37$	\( (T^{6} - 21 T^{5} + \cdots + 42472)^{2} \) (T^6 - 21T^5 + 35T^4 + 1693T^3 - 10577T^2 + 6688*T + 42472)^2
$41$	\( T^{12} - 4 T^{11} + \cdots + 4410000 \) T^12 - 4T^11 + 88T^10 - 670T^9 + 7675T^8 - 40288T^7 + 187441T^6 - 456625T^5 + 1056625T^4 - 1321800T^3 + 2647500T^2 - 2520000*T + 4410000
$43$	\( T^{12} + 13 T^{11} + \cdots + 41938576 \) T^12 + 13T^11 + 199T^10 + 938T^9 + 9469T^8 + 25006T^7 + 342422T^6 + 277420T^5 + 4274425T^4 - 8176516T^3 + 45620164T^2 - 43544624*T + 41938576
$47$	\( T^{12} + 8 T^{11} + \cdots + 86545809 \) T^12 + 8T^11 + 119T^10 + 206T^9 + 4342T^8 + 334T^7 + 132680T^6 - 171155T^5 + 2011267T^4 - 2410191T^3 + 19858182T^2 - 26429823*T + 86545809
$53$	\( (T^{6} + 10 T^{5} + \cdots - 21480)^{2} \) (T^6 + 10T^5 - 124T^4 - 1219T^3 + 2065T^2 + 17280*T - 21480)^2
$59$	\( T^{12} + \cdots + 5625000000 \) T^12 + 21T^11 + 385T^10 + 3408T^9 + 31647T^8 + 165654T^7 + 1364656T^6 + 5558700T^5 + 38295625T^4 + 91275000T^3 + 605625000T^2 + 1125000000*T + 5625000000
$61$	\( T^{12} + 2 T^{11} + \cdots + 21086464 \) T^12 + 2T^11 + 137T^10 + 594T^9 + 15652T^8 + 52186T^7 + 547849T^6 + 496450T^5 + 10612993T^4 + 15124728T^3 + 56652080T^2 - 30233728*T + 21086464
$67$	\( T^{12} + \cdots + 71742551104 \) T^12 + 6T^11 + 267T^10 + 526T^9 + 43434T^8 + 64224T^7 + 3808329T^6 - 2099988T^5 + 213421545T^4 - 21184304T^3 + 5177749560T^2 - 8395427712*T + 71742551104
$71$	\( (T^{6} - 29 T^{5} + \cdots + 6219)^{2} \) (T^6 - 29T^5 + 227T^4 + 59T^3 - 5969T^2 + 13152*T + 6219)^2
$73$	\( (T^{6} - 8 T^{5} + \cdots + 86137)^{2} \) (T^6 - 8T^5 - 233T^4 + 1891T^3 + 6161T^2 - 61557*T + 86137)^2
$79$	\( T^{12} + \cdots + 18108815761 \) T^12 - 22T^11 + 486T^10 - 4304T^9 + 53242T^8 - 293238T^7 + 3865894T^6 - 14925258T^5 + 140085138T^4 - 310007512T^3 + 3313740790T^2 - 6842833650*T + 18108815761
$83$	\( T^{12} + \cdots + 375003225 \) T^12 + 5T^11 + 137T^10 + 176T^9 + 11134T^8 + 15901T^7 + 424769T^6 + 316615T^5 + 11037700T^4 + 9098610T^3 + 114560475T^2 - 139137525*T + 375003225
$89$	\( (T^{6} + T^{5} + \cdots - 36492)^{2} \) (T^6 + T^5 - 219T^4 - 365T^3 + 12079T^2 + 24648T - 36492)^2
$97$	\( T^{12} + 32 T^{11} + \cdots + 95277121 \) T^12 + 32T^11 + 678T^10 + 8176T^9 + 72112T^8 + 414354T^7 + 1814518T^6 + 5331096T^5 + 12952026T^4 + 21291320T^3 + 42648952T^2 + 53705022*T + 95277121
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