LMFDB - Newform orbit 1890.2.l.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1890,2,Mod(361,1890)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1890, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 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("1890.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1890.l (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:	$15.0917259820$
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} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 $ x^12 - 3x^11 + 7x^10 - 3x^9 - 2x^8 + 24x^7 - 21x^6 + 72x^5 - 18x^4 - 81x^3 + 567x^2 - 729*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 630)
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_{7} q^{2} + (\beta_{7} - 1) q^{4} - q^{5} + ( - \beta_{7} - \beta_{3} + 1) q^{7} - q^{8}+O(q^{10}) $ q + b7 * q^2 + (b7 - 1) * q^4 - q^5 + (-b7 - b3 + 1) * q^7 - q^8	$ q + \beta_{7} q^{2} + (\beta_{7} - 1) q^{4} - q^{5} + ( - \beta_{7} - \beta_{3} + 1) q^{7} - q^{8} - \beta_{7} q^{10} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \cdots - 1) q^{11}+ \cdots + (3 \beta_{10} + 2 \beta_{7} + \cdots + \beta_1) q^{98}+O(q^{100}) $ q + b7 * q^2 + (b7 - 1) * q^4 - q^5 + (-b7 - b3 + 1) * q^7 - q^8 - b7 * q^10 + (-b9 - b8 + b7 + b4 + b3 + b2 + b1 - 1) * q^11 + (-b7 + b6 - b4 + b3 + b2 - 1) * q^13 + (b9 - b3 + 1) * q^14 - b7 * q^16 + (-b11 - b8 - b7 + b6) * q^17 + (-b10 - b9 - 2b7 - b5 - b4 + b3 + 1) q^19 + (-b7 + 1) * q^20 + (-b11 + b10 - b9 - b8 + b7 + b4 + b2 + 2b1 - 1) q^22 + (b10 + b9 + b8 + b2 + 2) * q^23 + q^25 + (-b10 - b9 + b5 - b4 + b3 - 1) * q^26 + (b9 + b7) * q^28 + (-b10 - b9 + 3b7 + b5 - b4 + b3 + b1 - 4) q^29 + (b11 - b10 - 2b7 + b3 + b1 + 2) q^31 + (-b7 + 1) * q^32 + (-b11 + b5) * q^34 + (b7 + b3 - 1) * q^35 + (-b11 - 2b9 - 4b7 + b5 - 2b4 + b1 + 2) q^37 + (-b10 - b9 - b7 + b6 - b5 - b2 + 2) * q^38 + q^40 + (b10 - b9 - b7 + b6 + b4 + b1) * q^41 + (-b9 + 3b7 + b5 - b4 + 3b1 - 4) * q^43 + (-b11 + b10 - b3 + b1) * q^44 + (b11 + b8 + 2b7 - b4 + b3 + b2 - 1) q^46 + (b10 - b9 + 3b7 + 3b6 + b3 + 2b2 + 2b1 - 2) * q^47 + (2b10 - 2b9 + 3b7 + 3b4 + 3b1 + 1) q^49 + b7 * q^50 + (-b10 - b9 + b7 - b6 + b5 - b2) * q^52 + (-b6 - b4 + b3 + b2 - 1) * q^53 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 1) * q^55 + (b7 + b3 - 1) * q^56 + (-b10 - b9 + b7 - b6 + b5 - 2b2 - 2) q^58 + (b11 - 2b10 - b9 + b7 - b5 - b4 + 2b3 + b1 - 2) * q^59 + (-3b11 - b10 + b9 - 3b8 - b7 + 2b6 - b4 + 3b2 + 2b1 - 3) q^61 + (-b9 - b8 + b7 + b4 + b3 - b2 + b1 + 3) * q^62 + q^64 + (b7 - b6 + b4 - b3 - b2 + 1) * q^65 + (-b11 - b9 + 2b7 + 2b5 - b4 + 5b1 - 3) q^67 + (b8 + b7 - b6 + b5) * q^68 + (-b9 + b3 - 1) * q^70 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 4) * q^71 + (-b11 + 2b10 - 2b9 - b8 + 4b7 + b6 + 2b4 + b2 + 3b1 - 1) q^73 + (-2b10 + b8 - b7 - b6 + b5 - 2b4 - 2b3 - 3b2 - 2b1 + 5) q^74 + (b7 + b6 + b4 - b3 - b2 + 1) * q^76 + (-3b10 - b9 + b7 + 3b6 - b5 - b4 - 2b2 - 2) q^77 + (5b7 + b2 + b1 - 1) q^79 + b7 * q^80 + (b10 - b7 + b5 - b3 + 1) * q^82 + (b11 + b10 + b9 - b7 - 3b5 + b4 - b3 - 2b1 + 2) * q^83 + (b11 + b8 + b7 - b6) * q^85 + (-b10 - b6 + b5 - b4 - b3 - 4b2 - b1) q^86 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 1) * q^88 + (-b11 + 3b10 + b9 - 3b7 + 2b5 + b4 - 3b3 + b1 + 4) * q^89 + (-b11 + b9 - 8b7 + 2b6 - b5 - 2b4 + b3 + 3b2 - b1 + 3) * q^91 + (b11 - b10 - b9 + 2b7 - b4 + b3 - 3) q^92 + (-b9 + 5b7 + 3b5 - b4 + b1 - 6) * q^94 + (b10 + b9 + 2b7 + b5 + b4 - b3 - 1) q^95 + (-2b11 - 3b10 - 3b9 - 3b4 + 3b3 + b1 - 3) q^97 + (3b10 + 2b7 + b4 - 2b3 + b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} - 6 q^{4} - 12 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q + 6 * q^2 - 6 * q^4 - 12 * q^5 + 4 * q^7 - 12 * q^8	$ 12 q + 6 q^{2} - 6 q^{4} - 12 q^{5} + 4 q^{7} - 12 q^{8} - 6 q^{10} + 6 q^{11} - 2 q^{13} + 8 q^{14} - 6 q^{16} - q^{17} + 8 q^{19} + 6 q^{20} + 3 q^{22} + 22 q^{23} + 12 q^{25} + 2 q^{26} + 4 q^{28} - 13 q^{29} + 21 q^{31} + 6 q^{32} + q^{34} - 4 q^{35} + 18 q^{37} + 16 q^{38} + 12 q^{40} - 5 q^{41} - 11 q^{43} - 3 q^{44} + 11 q^{46} + 23 q^{47} + 24 q^{49} + 6 q^{50} + 4 q^{52} - 2 q^{53} - 6 q^{55} - 4 q^{56} - 26 q^{58} - q^{59} - q^{61} + 42 q^{62} + 12 q^{64} + 2 q^{65} + 2 q^{67} + 2 q^{68} - 8 q^{70} + 30 q^{71} + 22 q^{73} + 36 q^{74} + 8 q^{76} - 11 q^{77} + 27 q^{79} + 6 q^{80} + 5 q^{82} - 6 q^{83} + q^{85} - 22 q^{86} - 6 q^{88} + 18 q^{89} + 14 q^{91} - 11 q^{92} - 23 q^{94} - 8 q^{95} - 4 q^{97}+O(q^{100}) $ 12 * q + 6 * q^2 - 6 * q^4 - 12 * q^5 + 4 * q^7 - 12 * q^8 - 6 * q^10 + 6 * q^11 - 2 * q^13 + 8 * q^14 - 6 * q^16 - q^17 + 8 * q^19 + 6 * q^20 + 3 * q^22 + 22 * q^23 + 12 * q^25 + 2 * q^26 + 4 * q^28 - 13 * q^29 + 21 * q^31 + 6 * q^32 + q^34 - 4 * q^35 + 18 * q^37 + 16 * q^38 + 12 * q^40 - 5 * q^41 - 11 * q^43 - 3 * q^44 + 11 * q^46 + 23 * q^47 + 24 * q^49 + 6 * q^50 + 4 * q^52 - 2 * q^53 - 6 * q^55 - 4 * q^56 - 26 * q^58 - q^59 - q^61 + 42 * q^62 + 12 * q^64 + 2 * q^65 + 2 * q^67 + 2 * q^68 - 8 * q^70 + 30 * q^71 + 22 * q^73 + 36 * q^74 + 8 * q^76 - 11 * q^77 + 27 * q^79 + 6 * q^80 + 5 * q^82 - 6 * q^83 + q^85 - 22 * q^86 - 6 * q^88 + 18 * q^89 + 14 * q^91 - 11 * q^92 - 23 * q^94 - 8 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 $ x^12 - 3*x^11 + 7*x^10 - 3*x^9 - 2*x^8 + 24*x^7 - 21*x^6 + 72*x^5 - 18*x^4 - 81*x^3 + 567*x^2 - 729*x + 729 :

$\beta_{1}$	$=$	$ ( - 2 \nu^{11} - 65 \nu^{10} + 52 \nu^{9} - 122 \nu^{8} - 191 \nu^{7} - 131 \nu^{6} - 261 \nu^{5} + \cdots - 5589 ) / 3807 $ (-2v^11 - 65v^10 + 52v^9 - 122v^8 - 191v^7 - 131v^6 - 261v^5 - 174v^4 - 1206v^3 - 2043v^2 + 3537*v - 5589) / 3807
$\beta_{2}$	$=$	$ ( - \nu^{11} + 3 \nu^{10} - 7 \nu^{9} + 3 \nu^{8} + 2 \nu^{7} - 24 \nu^{6} + 21 \nu^{5} - 72 \nu^{4} + \cdots + 729 ) / 243 $ (-v^11 + 3v^10 - 7v^9 + 3v^8 + 2v^7 - 24v^6 + 21v^5 - 72v^4 + 18v^3 + 81v^2 - 324v + 729) / 243
$\beta_{3}$	$=$	$ ( 52 \nu^{11} + 1032 \nu^{10} - 1634 \nu^{9} + 1950 \nu^{8} + 5107 \nu^{7} - 777 \nu^{6} + \cdots + 248103 ) / 11421 $ (52v^11 + 1032v^10 - 1634v^9 + 1950v^8 + 5107v^7 - 777v^6 + 9465v^5 - 2385v^4 + 41508v^3 + 49734v^2 - 138915*v + 248103) / 11421
$\beta_{4}$	$=$	$ ( - 83 \nu^{11} + 804 \nu^{10} - 1085 \nu^{9} + 624 \nu^{8} + 2578 \nu^{7} - 2076 \nu^{6} + \cdots + 127818 ) / 11421 $ (-83v^11 + 804v^10 - 1085v^9 + 624v^8 + 2578v^7 - 2076v^6 + 6018v^5 - 5670v^4 + 15939v^3 + 40635v^2 - 100035*v + 127818) / 11421
$\beta_{5}$	$=$	$ ( 91 \nu^{11} + 537 \nu^{10} - 956 \nu^{9} + 1509 \nu^{8} + 2275 \nu^{7} - 408 \nu^{6} + 4191 \nu^{5} + \cdots + 145800 ) / 11421 $ (91v^11 + 537v^10 - 956v^9 + 1509v^8 + 2275v^7 - 408v^6 + 4191v^5 - 1953v^4 + 26955v^3 + 12798v^2 - 80352*v + 145800) / 11421
$\beta_{6}$	$=$	$ ( - 44 \nu^{11} - 67 \nu^{10} + 157 \nu^{9} - 334 \nu^{8} - 536 \nu^{7} - 109 \nu^{6} - 807 \nu^{5} + \cdots - 27783 ) / 3807 $ (-44v^11 - 67v^10 + 157v^9 - 334v^8 - 536v^7 - 109v^6 - 807v^5 - 303v^4 - 7497v^3 - 954v^2 + 6750*v - 27783) / 3807
$\beta_{7}$	$=$	$ ( 51 \nu^{11} - 199 \nu^{10} + 225 \nu^{9} + 56 \nu^{8} - 558 \nu^{7} + 497 \nu^{6} - 1311 \nu^{5} + \cdots - 19278 ) / 3807 $ (51v^11 - 199v^10 + 225v^9 + 56v^8 - 558v^7 + 497v^6 - 1311v^5 + 2604v^4 - 1395v^3 - 12834v^2 + 25920*v - 19278) / 3807
$\beta_{8}$	$=$	$ ( - 4 \nu^{11} + 9 \nu^{10} - 19 \nu^{9} - 9 \nu^{8} + 17 \nu^{7} - 90 \nu^{6} + 12 \nu^{5} + \cdots + 1215 ) / 243 $ (-4v^11 + 9v^10 - 19v^9 - 9v^8 + 17v^7 - 90v^6 + 12v^5 - 225v^4 - 144v^3 + 621v^2 - 1782*v + 1215) / 243
$\beta_{9}$	$=$	$ ( - 71 \nu^{11} + 536 \nu^{10} - 833 \nu^{9} + 557 \nu^{8} + 2173 \nu^{7} - 1243 \nu^{6} + \cdots + 115668 ) / 3807 $ (-71v^11 + 536v^10 - 833v^9 + 557v^8 + 2173v^7 - 1243v^6 + 4764v^5 - 3498v^4 + 15561v^3 + 30474v^2 - 73035*v + 115668) / 3807
$\beta_{10}$	$=$	$ ( 542 \nu^{11} - 480 \nu^{10} + 572 \nu^{9} + 3264 \nu^{8} + 1283 \nu^{7} + 6126 \nu^{6} + \cdots + 86994 ) / 11421 $ (542v^11 - 480v^10 + 572v^9 + 3264v^8 + 1283v^7 + 6126v^6 + 2487v^5 + 22761v^4 + 43839v^3 - 41931v^2 + 93474*v + 86994) / 11421
$\beta_{11}$	$=$	$ ( - 256 \nu^{11} + 657 \nu^{10} - 958 \nu^{9} - 576 \nu^{8} + 1355 \nu^{7} - 2574 \nu^{6} + \cdots + 76464 ) / 3807 $ (-256v^11 + 657v^10 - 958v^9 - 576v^8 + 1355v^7 - 2574v^6 + 3816v^5 - 11133v^4 + 27v^3 + 46440v^2 - 96741*v + 76464) / 3807

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

$\nu$	$=$	$ ( -\beta_{6} - \beta_{5} + \beta_{2} - \beta _1 + 1 ) / 3 $ (-b6 - b5 + b2 - b1 + 1) / 3
$\nu^{2}$	$=$	$ ( \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 $ (b11 + 2b10 - b9 + 2b8 - b7 - b6 - b5 + b4 - b3 - 2*b2 + 2) / 3
$\nu^{3}$	$=$	$ ( 3\beta_{9} - 4\beta_{6} + 2\beta_{5} - 6\beta_{3} - 5\beta_{2} - \beta _1 - 2 ) / 3 $ (3b9 - 4b6 + 2b5 - 6b3 - 5*b2 - b1 - 2) / 3
$\nu^{4}$	$=$	$ ( \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 14 \beta_{7} + 2 \beta_{6} - \beta_{5} + 7 \beta_{4} + \cdots - 7 ) / 3 $ (b11 - b10 + 2b9 - b8 + 14b7 + 2b6 - b5 + 7b4 - b3 - 8b2 - 3b1 - 7) / 3
$\nu^{5}$	$=$	$ ( -6\beta_{9} - 9\beta_{8} + 11\beta_{6} - 4\beta_{5} + 9\beta_{4} + 12\beta_{3} + 19\beta_{2} + 23\beta _1 - 26 ) / 3 $ (-6b9 - 9b8 + 11b6 - 4b5 + 9b4 + 12b3 + 19b2 + 23b1 - 26) / 3
$\nu^{6}$	$=$	$ ( 7 \beta_{11} - \beta_{10} - 10 \beta_{9} - 10 \beta_{8} + 5 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + \cdots - 4 ) / 3 $ (7b11 - b10 - 10b9 - 10b8 + 5b7 + 8b6 + 2b5 - 17b4 + 20b3 + 10b2 - 6b1 - 4) / 3
$\nu^{7}$	$=$	$ ( - 18 \beta_{11} - 18 \beta_{10} + 12 \beta_{9} - 27 \beta_{7} + 2 \beta_{6} - 16 \beta_{5} - 45 \beta_{4} + \cdots - 17 ) / 3 $ (-18b11 - 18b10 + 12b9 - 27b7 + 2b6 - 16b5 - 45b4 + 30b3 + 22b2 - 13b1 - 17) / 3
$\nu^{8}$	$=$	$ ( - 8 \beta_{11} + 29 \beta_{10} + 8 \beta_{9} + 38 \beta_{8} - 73 \beta_{7} + 35 \beta_{6} + 17 \beta_{5} + \cdots - 61 ) / 3 $ (-8b11 + 29b10 + 8b9 + 38b8 - 73b7 + 35b6 + 17b5 - 44b4 - 19b3 + 34b2 - 27*b1 - 61) / 3
$\nu^{9}$	$=$	$ ( - 54 \beta_{11} + 9 \beta_{10} + 66 \beta_{9} + 27 \beta_{8} - 198 \beta_{7} + 5 \beta_{6} + 65 \beta_{5} + \cdots + 331 ) / 3 $ (-54b11 + 9b10 + 66b9 + 27b8 - 198b7 + 5b6 + 65b5 - 9b4 - 132b3 - 86b2 + 17*b1 + 331) / 3
$\nu^{10}$	$=$	$ ( - 26 \beta_{11} - 55 \beta_{10} + 29 \beta_{9} - 55 \beta_{8} + 41 \beta_{7} - 79 \beta_{6} + \cdots + 317 ) / 3 $ (-26b11 - 55b10 + 29b9 - 55b8 + 41b7 - 79b6 + 26b5 + 142b4 - 109b3 - 62b2 - 192*b1 + 317) / 3
$\nu^{11}$	$=$	$ ( 81 \beta_{11} + 81 \beta_{10} - 384 \beta_{9} + 45 \beta_{8} + 27 \beta_{7} - 97 \beta_{6} - 139 \beta_{5} + \cdots + 460 ) / 3 $ (81b11 + 81b10 - 384b9 + 45b8 + 27b7 - 97b6 - 139b5 + 441b4 + 255b3 - 8b2 + 347*b1 + 460) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times$.

$n$	$757$	$1081$	$1541$
$\chi(n)$	$1$	$-1 + \beta_{7}$	$-\beta_{7}$

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} $

361.1

−1.67391	+	0.444996i
0.702501	−	1.58319i
1.22323	+	1.22626i
−0.778860	−	1.54705i
1.39898	−	1.02120i
0.628063	+	1.61417i
−1.67391	−	0.444996i
0.702501	+	1.58319i
1.22323	−	1.22626i
−0.778860	+	1.54705i
1.39898	+	1.02120i
0.628063	−	1.61417i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

−2.56238

0.658939i

−1.00000

−0.500000

0.866025i

361.2

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

−2.56238

0.658939i

−1.00000

−0.500000

0.866025i

361.3

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

1.23855

2.33795i

−1.00000

−0.500000

0.866025i

361.4

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

1.23855

2.33795i

−1.00000

−0.500000

0.866025i

361.5

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

2.32383

−

1.26483i

−1.00000

−0.500000

0.866025i

361.6

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.00000

2.32383

−

1.26483i

−1.00000

−0.500000

0.866025i

1801.1

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

−2.56238

−

0.658939i

−1.00000

−0.500000

−

0.866025i

1801.2

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

−2.56238

−

0.658939i

−1.00000

−0.500000

−

0.866025i

1801.3

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

1.23855

−

2.33795i

−1.00000

−0.500000

−

0.866025i

1801.4

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

1.23855

−

2.33795i

−1.00000

−0.500000

−

0.866025i

1801.5

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

2.32383

1.26483i

−1.00000

−0.500000

−

0.866025i

1801.6

0.500000

0.866025i

−0.500000

0.866025i

−1.00000

2.32383

1.26483i

−1.00000

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1890.2.l.g		12
3.b	odd	2	1		630.2.l.g	yes	12
7.c	even	3	1		1890.2.i.g		12
9.c	even	3	1		1890.2.i.g		12
9.d	odd	6	1		630.2.i.g	✓	12
21.h	odd	6	1		630.2.i.g	✓	12
63.g	even	3	1	inner	1890.2.l.g		12
63.n	odd	6	1		630.2.l.g	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.i.g	✓	12	9.d	odd	6	1
630.2.i.g	✓	12	21.h	odd	6	1
630.2.l.g	yes	12	3.b	odd	2	1
630.2.l.g	yes	12	63.n	odd	6	1
1890.2.i.g		12	7.c	even	3	1
1890.2.i.g		12	9.c	even	3	1
1890.2.l.g		12	1.a	even	1	1	trivial
1890.2.l.g		12	63.g	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1890, [\chi])$:

$ T_{11}^{6} - 3T_{11}^{5} - 44T_{11}^{4} + 36T_{11}^{3} + 550T_{11}^{2} + 750T_{11} + 279 $

$ T_{13}^{12} + 2 T_{13}^{11} + 38 T_{13}^{10} + 24 T_{13}^{9} + 1108 T_{13}^{8} + 1129 T_{13}^{7} + \cdots + 15625 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ (T + 1)^{12} $ (T + 1)^12
$7$	$ (T^{6} - 2 T^{5} + \cdots + 343)^{2} $ (T^6 - 2T^5 - 4T^4 + 31T^3 - 28T^2 - 98*T + 343)^2
$11$	$ (T^{6} - 3 T^{5} + \cdots + 279)^{2} $ (T^6 - 3T^5 - 44T^4 + 36T^3 + 550T^2 + 750*T + 279)^2
$13$	$ T^{12} + 2 T^{11} + \cdots + 15625 $ T^12 + 2T^11 + 38T^10 + 24T^9 + 1108T^8 + 1129T^7 + 6376T^6 + 1810T^5 + 21100T^4 + 6000T^3 + 33125T^2 - 15625*T + 15625
$17$	$ T^{12} + T^{11} + \cdots + 9801 $ T^12 + T^11 + 52T^10 - 143T^9 + 2311T^8 - 2813T^7 + 14341T^6 + 13267T^5 + 53521T^4 + 14232T^3 + 24597T^2 - 2079T + 9801
$19$	$ T^{12} - 8 T^{11} + \cdots + 5929 $ T^12 - 8T^11 + 98T^10 - 240T^9 + 3142T^8 - 8209T^7 + 63514T^6 - 17308T^5 + 128458T^4 + 8610T^3 + 251783T^2 + 38269*T + 5929
$23$	$ (T^{6} - 11 T^{5} + \cdots + 5949)^{2} $ (T^6 - 11T^5 - 15T^4 + 406T^3 - 659T^2 - 2598*T + 5949)^2
$29$	$ T^{12} + 13 T^{11} + \cdots + 729 $ T^12 + 13T^11 + 141T^10 + 466T^9 + 1507T^8 + 177T^7 + 3750T^6 + 891T^5 + 5139T^4 + 54T^3 + 3645T^2 + 1215*T + 729
$31$	$ T^{12} + \cdots + 1198336689 $ T^12 - 21T^11 + 341T^10 - 3168T^9 + 25872T^8 - 141426T^7 + 837200T^6 - 3376329T^5 + 17802724T^4 - 40743936T^3 + 161902086T^2 + 28039770*T + 1198336689
$37$	$ T^{12} + \cdots + 313396209 $ T^12 - 18T^11 + 341T^10 - 3342T^9 + 43926T^8 - 396447T^7 + 3531623T^6 - 18589272T^5 + 73508290T^4 - 189779961T^3 + 362897766T^2 - 416746323*T + 313396209
$41$	$ T^{12} + 5 T^{11} + \cdots + 531441 $ T^12 + 5T^11 + 57T^10 + 146T^9 + 1654T^8 + 4518T^7 + 24327T^6 + 45198T^5 + 190269T^4 + 354294T^3 + 846369T^2 + 708588*T + 531441
$43$	$ T^{12} + \cdots + 3018293721 $ T^12 + 11T^11 + 211T^10 + 1556T^9 + 22888T^8 + 155366T^7 + 1400971T^6 + 5838314T^5 + 35509333T^4 + 121406784T^3 + 596599791T^2 + 1292494914*T + 3018293721
$47$	$ T^{12} + \cdots + 3525153129 $ T^12 - 23T^11 + 447T^10 - 4340T^9 + 41566T^8 - 218370T^7 + 1835583T^6 - 7124994T^5 + 56659941T^4 - 50239764T^3 + 600987357T^2 - 856039914*T + 3525153129
$53$	$ T^{12} + 2 T^{11} + \cdots + 88209 $ T^12 + 2T^11 + 63T^10 + 2T^9 + 3040T^8 + 1071T^7 + 36555T^6 - 60804T^5 + 283698T^4 - 161865T^3 + 217242T^2 + 66825*T + 88209
$59$	$ T^{12} + \cdots + 422179209 $ T^12 + T^11 + 174T^10 + 43T^9 + 23491T^8 + 14403T^7 + 1094217T^6 + 2321091T^5 + 40247073T^4 + 53238654T^3 + 212134383T^2 - 181039617T + 422179209
$61$	$ T^{12} + \cdots + 9181664041 $ T^12 + T^11 + 383T^10 - 1350T^9 + 110092T^8 - 407674T^7 + 14080924T^6 - 98992507T^5 + 1359696286T^4 - 5468832048T^3 + 19744287392T^2 - 14573415890T + 9181664041
$67$	$ T^{12} + \cdots + 7010210529 $ T^12 - 2T^11 + 285T^10 + 1300T^9 + 56431T^8 + 275214T^7 + 6260973T^6 + 39534840T^5 + 482491710T^4 + 1776517218T^3 + 8940682233T^2 - 7062958539*T + 7010210529
$71$	$ (T^{6} - 15 T^{5} + \cdots + 4887)^{2} $ (T^6 - 15T^5 + 46T^4 + 222T^3 - 1097T^2 - 513*T + 4887)^2
$73$	$ T^{12} - 22 T^{11} + \cdots + 30614089 $ T^12 - 22T^11 + 392T^10 - 3168T^9 + 23506T^8 - 61559T^7 + 431704T^6 - 417998T^5 + 8982460T^4 + 8494446T^3 + 49973075T^2 - 33369523*T + 30614089
$79$	$ T^{12} - 27 T^{11} + \cdots + 3932289 $ T^12 - 27T^11 + 440T^10 - 4719T^9 + 37692T^8 - 224304T^7 + 1029083T^6 - 3518943T^5 + 9012706T^4 - 15681648T^3 + 18679731T^2 - 10303668*T + 3932289
$83$	$ T^{12} + \cdots + 265070961 $ T^12 + 6T^11 + 168T^10 + 252T^9 + 17478T^8 + 47439T^7 + 618516T^6 + 2575314T^5 + 19699038T^4 + 64617102T^3 + 189238923T^2 + 251883351*T + 265070961
$89$	$ T^{12} + \cdots + 18815334561 $ T^12 - 18T^11 + 419T^10 - 4458T^9 + 76398T^8 - 746079T^7 + 8188745T^6 - 44007936T^5 + 234216028T^4 - 536450709T^3 + 2308328118T^2 - 3580522407*T + 18815334561
$97$	$ T^{12} + \cdots + 1888367924041 $ T^12 + 4T^11 + 440T^10 + 276T^9 + 137905T^8 + 113420T^7 + 17401894T^6 + 1800542T^5 + 1577372065T^4 + 124967952T^3 + 66786650261T^2 - 85028699804*T + 1888367924041
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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:	\( N \)	\(=\)	\( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1890.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} + (\beta_{7} - 1) q^{4} - q^{5} + ( - \beta_{7} - \beta_{3} + 1) q^{7} - q^{8}+O(q^{10}) \) q + b7 * q^2 + (b7 - 1) * q^4 - q^5 + (-b7 - b3 + 1) * q^7 - q^8	\( q + \beta_{7} q^{2} + (\beta_{7} - 1) q^{4} - q^{5} + ( - \beta_{7} - \beta_{3} + 1) q^{7} - q^{8} - \beta_{7} q^{10} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \cdots - 1) q^{11}+ \cdots + (3 \beta_{10} + 2 \beta_{7} + \cdots + \beta_1) q^{98}+O(q^{100}) \) q + b7 * q^2 + (b7 - 1) * q^4 - q^5 + (-b7 - b3 + 1) * q^7 - q^8 - b7 * q^10 + (-b9 - b8 + b7 + b4 + b3 + b2 + b1 - 1) * q^11 + (-b7 + b6 - b4 + b3 + b2 - 1) * q^13 + (b9 - b3 + 1) * q^14 - b7 * q^16 + (-b11 - b8 - b7 + b6) * q^17 + (-b10 - b9 - 2b7 - b5 - b4 + b3 + 1) q^19 + (-b7 + 1) * q^20 + (-b11 + b10 - b9 - b8 + b7 + b4 + b2 + 2b1 - 1) q^22 + (b10 + b9 + b8 + b2 + 2) * q^23 + q^25 + (-b10 - b9 + b5 - b4 + b3 - 1) * q^26 + (b9 + b7) * q^28 + (-b10 - b9 + 3b7 + b5 - b4 + b3 + b1 - 4) q^29 + (b11 - b10 - 2b7 + b3 + b1 + 2) q^31 + (-b7 + 1) * q^32 + (-b11 + b5) * q^34 + (b7 + b3 - 1) * q^35 + (-b11 - 2b9 - 4b7 + b5 - 2b4 + b1 + 2) q^37 + (-b10 - b9 - b7 + b6 - b5 - b2 + 2) * q^38 + q^40 + (b10 - b9 - b7 + b6 + b4 + b1) * q^41 + (-b9 + 3b7 + b5 - b4 + 3b1 - 4) * q^43 + (-b11 + b10 - b3 + b1) * q^44 + (b11 + b8 + 2b7 - b4 + b3 + b2 - 1) q^46 + (b10 - b9 + 3b7 + 3b6 + b3 + 2b2 + 2b1 - 2) * q^47 + (2b10 - 2b9 + 3b7 + 3b4 + 3b1 + 1) q^49 + b7 * q^50 + (-b10 - b9 + b7 - b6 + b5 - b2) * q^52 + (-b6 - b4 + b3 + b2 - 1) * q^53 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 1) * q^55 + (b7 + b3 - 1) * q^56 + (-b10 - b9 + b7 - b6 + b5 - 2b2 - 2) q^58 + (b11 - 2b10 - b9 + b7 - b5 - b4 + 2b3 + b1 - 2) * q^59 + (-3b11 - b10 + b9 - 3b8 - b7 + 2b6 - b4 + 3b2 + 2b1 - 3) q^61 + (-b9 - b8 + b7 + b4 + b3 - b2 + b1 + 3) * q^62 + q^64 + (b7 - b6 + b4 - b3 - b2 + 1) * q^65 + (-b11 - b9 + 2b7 + 2b5 - b4 + 5b1 - 3) q^67 + (b8 + b7 - b6 + b5) * q^68 + (-b9 + b3 - 1) * q^70 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 4) * q^71 + (-b11 + 2b10 - 2b9 - b8 + 4b7 + b6 + 2b4 + b2 + 3b1 - 1) q^73 + (-2b10 + b8 - b7 - b6 + b5 - 2b4 - 2b3 - 3b2 - 2b1 + 5) q^74 + (b7 + b6 + b4 - b3 - b2 + 1) * q^76 + (-3b10 - b9 + b7 + 3b6 - b5 - b4 - 2b2 - 2) q^77 + (5b7 + b2 + b1 - 1) q^79 + b7 * q^80 + (b10 - b7 + b5 - b3 + 1) * q^82 + (b11 + b10 + b9 - b7 - 3b5 + b4 - b3 - 2b1 + 2) * q^83 + (b11 + b8 + b7 - b6) * q^85 + (-b10 - b6 + b5 - b4 - b3 - 4b2 - b1) q^86 + (b9 + b8 - b7 - b4 - b3 - b2 - b1 + 1) * q^88 + (-b11 + 3b10 + b9 - 3b7 + 2b5 + b4 - 3b3 + b1 + 4) * q^89 + (-b11 + b9 - 8b7 + 2b6 - b5 - 2b4 + b3 + 3b2 - b1 + 3) * q^91 + (b11 - b10 - b9 + 2b7 - b4 + b3 - 3) q^92 + (-b9 + 5b7 + 3b5 - b4 + b1 - 6) * q^94 + (b10 + b9 + 2b7 + b5 + b4 - b3 - 1) q^95 + (-2b11 - 3b10 - 3b9 - 3b4 + 3b3 + b1 - 3) q^97 + (3b10 + 2b7 + b4 - 2b3 + b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} - 6 q^{4} - 12 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q + 6 * q^2 - 6 * q^4 - 12 * q^5 + 4 * q^7 - 12 * q^8	\( 12 q + 6 q^{2} - 6 q^{4} - 12 q^{5} + 4 q^{7} - 12 q^{8} - 6 q^{10} + 6 q^{11} - 2 q^{13} + 8 q^{14} - 6 q^{16} - q^{17} + 8 q^{19} + 6 q^{20} + 3 q^{22} + 22 q^{23} + 12 q^{25} + 2 q^{26} + 4 q^{28} - 13 q^{29} + 21 q^{31} + 6 q^{32} + q^{34} - 4 q^{35} + 18 q^{37} + 16 q^{38} + 12 q^{40} - 5 q^{41} - 11 q^{43} - 3 q^{44} + 11 q^{46} + 23 q^{47} + 24 q^{49} + 6 q^{50} + 4 q^{52} - 2 q^{53} - 6 q^{55} - 4 q^{56} - 26 q^{58} - q^{59} - q^{61} + 42 q^{62} + 12 q^{64} + 2 q^{65} + 2 q^{67} + 2 q^{68} - 8 q^{70} + 30 q^{71} + 22 q^{73} + 36 q^{74} + 8 q^{76} - 11 q^{77} + 27 q^{79} + 6 q^{80} + 5 q^{82} - 6 q^{83} + q^{85} - 22 q^{86} - 6 q^{88} + 18 q^{89} + 14 q^{91} - 11 q^{92} - 23 q^{94} - 8 q^{95} - 4 q^{97}+O(q^{100}) \) 12 * q + 6 * q^2 - 6 * q^4 - 12 * q^5 + 4 * q^7 - 12 * q^8 - 6 * q^10 + 6 * q^11 - 2 * q^13 + 8 * q^14 - 6 * q^16 - q^17 + 8 * q^19 + 6 * q^20 + 3 * q^22 + 22 * q^23 + 12 * q^25 + 2 * q^26 + 4 * q^28 - 13 * q^29 + 21 * q^31 + 6 * q^32 + q^34 - 4 * q^35 + 18 * q^37 + 16 * q^38 + 12 * q^40 - 5 * q^41 - 11 * q^43 - 3 * q^44 + 11 * q^46 + 23 * q^47 + 24 * q^49 + 6 * q^50 + 4 * q^52 - 2 * q^53 - 6 * q^55 - 4 * q^56 - 26 * q^58 - q^59 - q^61 + 42 * q^62 + 12 * q^64 + 2 * q^65 + 2 * q^67 + 2 * q^68 - 8 * q^70 + 30 * q^71 + 22 * q^73 + 36 * q^74 + 8 * q^76 - 11 * q^77 + 27 * q^79 + 6 * q^80 + 5 * q^82 - 6 * q^83 + q^85 - 22 * q^86 - 6 * q^88 + 18 * q^89 + 14 * q^91 - 11 * q^92 - 23 * q^94 - 8 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1890.2.l.g

Level	$1890$
Weight	$2$
Character orbit	1890.l
Analytic conductor	$15.092$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(15.0917259820\)
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:	\( x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 \) x^12 - 3x^11 + 7x^10 - 3x^9 - 2x^8 + 24x^7 - 21x^6 + 72x^5 - 18x^4 - 81x^3 + 567x^2 - 729*x + 729
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 2 \nu^{11} - 65 \nu^{10} + 52 \nu^{9} - 122 \nu^{8} - 191 \nu^{7} - 131 \nu^{6} - 261 \nu^{5} + \cdots - 5589 ) / 3807 \) (-2v^11 - 65v^10 + 52v^9 - 122v^8 - 191v^7 - 131v^6 - 261v^5 - 174v^4 - 1206v^3 - 2043v^2 + 3537*v - 5589) / 3807
\(\beta_{2}\)	\(=\)	\( ( - \nu^{11} + 3 \nu^{10} - 7 \nu^{9} + 3 \nu^{8} + 2 \nu^{7} - 24 \nu^{6} + 21 \nu^{5} - 72 \nu^{4} + \cdots + 729 ) / 243 \) (-v^11 + 3v^10 - 7v^9 + 3v^8 + 2v^7 - 24v^6 + 21v^5 - 72v^4 + 18v^3 + 81v^2 - 324v + 729) / 243
\(\beta_{3}\)	\(=\)	\( ( 52 \nu^{11} + 1032 \nu^{10} - 1634 \nu^{9} + 1950 \nu^{8} + 5107 \nu^{7} - 777 \nu^{6} + \cdots + 248103 ) / 11421 \) (52v^11 + 1032v^10 - 1634v^9 + 1950v^8 + 5107v^7 - 777v^6 + 9465v^5 - 2385v^4 + 41508v^3 + 49734v^2 - 138915*v + 248103) / 11421
\(\beta_{4}\)	\(=\)	\( ( - 83 \nu^{11} + 804 \nu^{10} - 1085 \nu^{9} + 624 \nu^{8} + 2578 \nu^{7} - 2076 \nu^{6} + \cdots + 127818 ) / 11421 \) (-83v^11 + 804v^10 - 1085v^9 + 624v^8 + 2578v^7 - 2076v^6 + 6018v^5 - 5670v^4 + 15939v^3 + 40635v^2 - 100035*v + 127818) / 11421
\(\beta_{5}\)	\(=\)	\( ( 91 \nu^{11} + 537 \nu^{10} - 956 \nu^{9} + 1509 \nu^{8} + 2275 \nu^{7} - 408 \nu^{6} + 4191 \nu^{5} + \cdots + 145800 ) / 11421 \) (91v^11 + 537v^10 - 956v^9 + 1509v^8 + 2275v^7 - 408v^6 + 4191v^5 - 1953v^4 + 26955v^3 + 12798v^2 - 80352*v + 145800) / 11421
\(\beta_{6}\)	\(=\)	\( ( - 44 \nu^{11} - 67 \nu^{10} + 157 \nu^{9} - 334 \nu^{8} - 536 \nu^{7} - 109 \nu^{6} - 807 \nu^{5} + \cdots - 27783 ) / 3807 \) (-44v^11 - 67v^10 + 157v^9 - 334v^8 - 536v^7 - 109v^6 - 807v^5 - 303v^4 - 7497v^3 - 954v^2 + 6750*v - 27783) / 3807
\(\beta_{7}\)	\(=\)	\( ( 51 \nu^{11} - 199 \nu^{10} + 225 \nu^{9} + 56 \nu^{8} - 558 \nu^{7} + 497 \nu^{6} - 1311 \nu^{5} + \cdots - 19278 ) / 3807 \) (51v^11 - 199v^10 + 225v^9 + 56v^8 - 558v^7 + 497v^6 - 1311v^5 + 2604v^4 - 1395v^3 - 12834v^2 + 25920*v - 19278) / 3807
\(\beta_{8}\)	\(=\)	\( ( - 4 \nu^{11} + 9 \nu^{10} - 19 \nu^{9} - 9 \nu^{8} + 17 \nu^{7} - 90 \nu^{6} + 12 \nu^{5} + \cdots + 1215 ) / 243 \) (-4v^11 + 9v^10 - 19v^9 - 9v^8 + 17v^7 - 90v^6 + 12v^5 - 225v^4 - 144v^3 + 621v^2 - 1782*v + 1215) / 243
\(\beta_{9}\)	\(=\)	\( ( - 71 \nu^{11} + 536 \nu^{10} - 833 \nu^{9} + 557 \nu^{8} + 2173 \nu^{7} - 1243 \nu^{6} + \cdots + 115668 ) / 3807 \) (-71v^11 + 536v^10 - 833v^9 + 557v^8 + 2173v^7 - 1243v^6 + 4764v^5 - 3498v^4 + 15561v^3 + 30474v^2 - 73035*v + 115668) / 3807
\(\beta_{10}\)	\(=\)	\( ( 542 \nu^{11} - 480 \nu^{10} + 572 \nu^{9} + 3264 \nu^{8} + 1283 \nu^{7} + 6126 \nu^{6} + \cdots + 86994 ) / 11421 \) (542v^11 - 480v^10 + 572v^9 + 3264v^8 + 1283v^7 + 6126v^6 + 2487v^5 + 22761v^4 + 43839v^3 - 41931v^2 + 93474*v + 86994) / 11421
\(\beta_{11}\)	\(=\)	\( ( - 256 \nu^{11} + 657 \nu^{10} - 958 \nu^{9} - 576 \nu^{8} + 1355 \nu^{7} - 2574 \nu^{6} + \cdots + 76464 ) / 3807 \) (-256v^11 + 657v^10 - 958v^9 - 576v^8 + 1355v^7 - 2574v^6 + 3816v^5 - 11133v^4 + 27v^3 + 46440v^2 - 96741*v + 76464) / 3807

\(\nu\)	\(=\)	\( ( -\beta_{6} - \beta_{5} + \beta_{2} - \beta _1 + 1 ) / 3 \) (-b6 - b5 + b2 - b1 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 \) (b11 + 2b10 - b9 + 2b8 - b7 - b6 - b5 + b4 - b3 - 2*b2 + 2) / 3
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{9} - 4\beta_{6} + 2\beta_{5} - 6\beta_{3} - 5\beta_{2} - \beta _1 - 2 ) / 3 \) (3b9 - 4b6 + 2b5 - 6b3 - 5*b2 - b1 - 2) / 3
\(\nu^{4}\)	\(=\)	\( ( \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 14 \beta_{7} + 2 \beta_{6} - \beta_{5} + 7 \beta_{4} + \cdots - 7 ) / 3 \) (b11 - b10 + 2b9 - b8 + 14b7 + 2b6 - b5 + 7b4 - b3 - 8b2 - 3b1 - 7) / 3
\(\nu^{5}\)	\(=\)	\( ( -6\beta_{9} - 9\beta_{8} + 11\beta_{6} - 4\beta_{5} + 9\beta_{4} + 12\beta_{3} + 19\beta_{2} + 23\beta _1 - 26 ) / 3 \) (-6b9 - 9b8 + 11b6 - 4b5 + 9b4 + 12b3 + 19b2 + 23b1 - 26) / 3
\(\nu^{6}\)	\(=\)	\( ( 7 \beta_{11} - \beta_{10} - 10 \beta_{9} - 10 \beta_{8} + 5 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + \cdots - 4 ) / 3 \) (7b11 - b10 - 10b9 - 10b8 + 5b7 + 8b6 + 2b5 - 17b4 + 20b3 + 10b2 - 6b1 - 4) / 3
\(\nu^{7}\)	\(=\)	\( ( - 18 \beta_{11} - 18 \beta_{10} + 12 \beta_{9} - 27 \beta_{7} + 2 \beta_{6} - 16 \beta_{5} - 45 \beta_{4} + \cdots - 17 ) / 3 \) (-18b11 - 18b10 + 12b9 - 27b7 + 2b6 - 16b5 - 45b4 + 30b3 + 22b2 - 13b1 - 17) / 3
\(\nu^{8}\)	\(=\)	\( ( - 8 \beta_{11} + 29 \beta_{10} + 8 \beta_{9} + 38 \beta_{8} - 73 \beta_{7} + 35 \beta_{6} + 17 \beta_{5} + \cdots - 61 ) / 3 \) (-8b11 + 29b10 + 8b9 + 38b8 - 73b7 + 35b6 + 17b5 - 44b4 - 19b3 + 34b2 - 27*b1 - 61) / 3
\(\nu^{9}\)	\(=\)	\( ( - 54 \beta_{11} + 9 \beta_{10} + 66 \beta_{9} + 27 \beta_{8} - 198 \beta_{7} + 5 \beta_{6} + 65 \beta_{5} + \cdots + 331 ) / 3 \) (-54b11 + 9b10 + 66b9 + 27b8 - 198b7 + 5b6 + 65b5 - 9b4 - 132b3 - 86b2 + 17*b1 + 331) / 3
\(\nu^{10}\)	\(=\)	\( ( - 26 \beta_{11} - 55 \beta_{10} + 29 \beta_{9} - 55 \beta_{8} + 41 \beta_{7} - 79 \beta_{6} + \cdots + 317 ) / 3 \) (-26b11 - 55b10 + 29b9 - 55b8 + 41b7 - 79b6 + 26b5 + 142b4 - 109b3 - 62b2 - 192*b1 + 317) / 3
\(\nu^{11}\)	\(=\)	\( ( 81 \beta_{11} + 81 \beta_{10} - 384 \beta_{9} + 45 \beta_{8} + 27 \beta_{7} - 97 \beta_{6} - 139 \beta_{5} + \cdots + 460 ) / 3 \) (81b11 + 81b10 - 384b9 + 45b8 + 27b7 - 97b6 - 139b5 + 441b4 + 255b3 - 8b2 + 347*b1 + 460) / 3

\(n\)	\(757\)	\(1081\)	\(1541\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{7}\)	\(-\beta_{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( (T + 1)^{12} \) (T + 1)^12
$7$	\( (T^{6} - 2 T^{5} + \cdots + 343)^{2} \) (T^6 - 2T^5 - 4T^4 + 31T^3 - 28T^2 - 98*T + 343)^2
$11$	\( (T^{6} - 3 T^{5} + \cdots + 279)^{2} \) (T^6 - 3T^5 - 44T^4 + 36T^3 + 550T^2 + 750*T + 279)^2
$13$	\( T^{12} + 2 T^{11} + \cdots + 15625 \) T^12 + 2T^11 + 38T^10 + 24T^9 + 1108T^8 + 1129T^7 + 6376T^6 + 1810T^5 + 21100T^4 + 6000T^3 + 33125T^2 - 15625*T + 15625
$17$	\( T^{12} + T^{11} + \cdots + 9801 \) T^12 + T^11 + 52T^10 - 143T^9 + 2311T^8 - 2813T^7 + 14341T^6 + 13267T^5 + 53521T^4 + 14232T^3 + 24597T^2 - 2079T + 9801
$19$	\( T^{12} - 8 T^{11} + \cdots + 5929 \) T^12 - 8T^11 + 98T^10 - 240T^9 + 3142T^8 - 8209T^7 + 63514T^6 - 17308T^5 + 128458T^4 + 8610T^3 + 251783T^2 + 38269*T + 5929
$23$	\( (T^{6} - 11 T^{5} + \cdots + 5949)^{2} \) (T^6 - 11T^5 - 15T^4 + 406T^3 - 659T^2 - 2598*T + 5949)^2
$29$	\( T^{12} + 13 T^{11} + \cdots + 729 \) T^12 + 13T^11 + 141T^10 + 466T^9 + 1507T^8 + 177T^7 + 3750T^6 + 891T^5 + 5139T^4 + 54T^3 + 3645T^2 + 1215*T + 729
$31$	\( T^{12} + \cdots + 1198336689 \) T^12 - 21T^11 + 341T^10 - 3168T^9 + 25872T^8 - 141426T^7 + 837200T^6 - 3376329T^5 + 17802724T^4 - 40743936T^3 + 161902086T^2 + 28039770*T + 1198336689
$37$	\( T^{12} + \cdots + 313396209 \) T^12 - 18T^11 + 341T^10 - 3342T^9 + 43926T^8 - 396447T^7 + 3531623T^6 - 18589272T^5 + 73508290T^4 - 189779961T^3 + 362897766T^2 - 416746323*T + 313396209
$41$	\( T^{12} + 5 T^{11} + \cdots + 531441 \) T^12 + 5T^11 + 57T^10 + 146T^9 + 1654T^8 + 4518T^7 + 24327T^6 + 45198T^5 + 190269T^4 + 354294T^3 + 846369T^2 + 708588*T + 531441
$43$	\( T^{12} + \cdots + 3018293721 \) T^12 + 11T^11 + 211T^10 + 1556T^9 + 22888T^8 + 155366T^7 + 1400971T^6 + 5838314T^5 + 35509333T^4 + 121406784T^3 + 596599791T^2 + 1292494914*T + 3018293721
$47$	\( T^{12} + \cdots + 3525153129 \) T^12 - 23T^11 + 447T^10 - 4340T^9 + 41566T^8 - 218370T^7 + 1835583T^6 - 7124994T^5 + 56659941T^4 - 50239764T^3 + 600987357T^2 - 856039914*T + 3525153129
$53$	\( T^{12} + 2 T^{11} + \cdots + 88209 \) T^12 + 2T^11 + 63T^10 + 2T^9 + 3040T^8 + 1071T^7 + 36555T^6 - 60804T^5 + 283698T^4 - 161865T^3 + 217242T^2 + 66825*T + 88209
$59$	\( T^{12} + \cdots + 422179209 \) T^12 + T^11 + 174T^10 + 43T^9 + 23491T^8 + 14403T^7 + 1094217T^6 + 2321091T^5 + 40247073T^4 + 53238654T^3 + 212134383T^2 - 181039617T + 422179209
$61$	\( T^{12} + \cdots + 9181664041 \) T^12 + T^11 + 383T^10 - 1350T^9 + 110092T^8 - 407674T^7 + 14080924T^6 - 98992507T^5 + 1359696286T^4 - 5468832048T^3 + 19744287392T^2 - 14573415890T + 9181664041
$67$	\( T^{12} + \cdots + 7010210529 \) T^12 - 2T^11 + 285T^10 + 1300T^9 + 56431T^8 + 275214T^7 + 6260973T^6 + 39534840T^5 + 482491710T^4 + 1776517218T^3 + 8940682233T^2 - 7062958539*T + 7010210529
$71$	\( (T^{6} - 15 T^{5} + \cdots + 4887)^{2} \) (T^6 - 15T^5 + 46T^4 + 222T^3 - 1097T^2 - 513*T + 4887)^2
$73$	\( T^{12} - 22 T^{11} + \cdots + 30614089 \) T^12 - 22T^11 + 392T^10 - 3168T^9 + 23506T^8 - 61559T^7 + 431704T^6 - 417998T^5 + 8982460T^4 + 8494446T^3 + 49973075T^2 - 33369523*T + 30614089
$79$	\( T^{12} - 27 T^{11} + \cdots + 3932289 \) T^12 - 27T^11 + 440T^10 - 4719T^9 + 37692T^8 - 224304T^7 + 1029083T^6 - 3518943T^5 + 9012706T^4 - 15681648T^3 + 18679731T^2 - 10303668*T + 3932289
$83$	\( T^{12} + \cdots + 265070961 \) T^12 + 6T^11 + 168T^10 + 252T^9 + 17478T^8 + 47439T^7 + 618516T^6 + 2575314T^5 + 19699038T^4 + 64617102T^3 + 189238923T^2 + 251883351*T + 265070961
$89$	\( T^{12} + \cdots + 18815334561 \) T^12 - 18T^11 + 419T^10 - 4458T^9 + 76398T^8 - 746079T^7 + 8188745T^6 - 44007936T^5 + 234216028T^4 - 536450709T^3 + 2308328118T^2 - 3580522407*T + 18815334561
$97$	\( T^{12} + \cdots + 1888367924041 \) T^12 + 4T^11 + 440T^10 + 276T^9 + 137905T^8 + 113420T^7 + 17401894T^6 + 1800542T^5 + 1577372065T^4 + 124967952T^3 + 66786650261T^2 - 85028699804*T + 1888367924041
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