LMFDB - Newform orbit 1026.2.e.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1026,2,Mod(343,1026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1026, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1026.343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1026 = 2 \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1026.e (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:	$8.19265124738$
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} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 $ x^12 + 8x^10 + 55x^8 - 2x^7 + 70x^6 - 32x^5 + 73x^4 - 18x^3 + 13x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 342)
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_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{7} - q^{8}+O(q^{10}) $ q + b4 * q^2 + (b4 - 1) * q^4 + b7 * q^5 + (-b9 + b6 - b4) * q^7 - q^8	$ q + \beta_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{7} - q^{8} + (\beta_{7} + \beta_{5}) q^{10} + (\beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{10} - \beta_{8} + \beta_{6} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots - 3) q^{98}+O(q^{100}) $ q + b4 * q^2 + (b4 - 1) * q^4 + b7 * q^5 + (-b9 + b6 - b4) * q^7 - q^8 + (b7 + b5) * q^10 + (b4 + b3 - b1) * q^11 + (-b10 - b8 + b6 + b4 + b1 - 1) * q^13 + (b6 - b4 + 1) * q^14 - b4 * q^16 + (-b11 + b10 - b9 - b3) * q^17 - q^19 + b5 * q^20 + (-b11 + b10 + b4 - b1 - 1) * q^22 + (-b11 + b10 - b8 + b7 - 2b4 - b1 + 2) q^23 + (-5b4 + 2b3) * q^25 + (-b10 + b9 - b8 + b2 - 1) * q^26 + (b9 + 1) * q^28 + (-b4 + b2) * q^29 + (-b11 - b10 + b7 + b4 + b1 - 1) * q^31 + (-b4 + 1) * q^32 + (-b9 + b6 - b3 + b1) * q^34 + (b11 + 2b9 - 2b8 + b3 + 2b2 - 2) q^35 + (-2b8 + b7 + b5 + 2b2 + 1) * q^37 - b4 * q^38 - b7 * q^40 + (-b10 - b7 + b4 + b1 - 1) * q^41 + (-2b9 + 2b6 + b5 - 2b4 + b3) q^43 + (-b11 + b10 - b3 - 1) * q^44 + (-b11 + b10 - b8 + b7 + b5 - b3 + b2 + 2) * q^46 + (-2b9 + 2b6 + b5 + b4 - b3) * q^47 + (-b8 + 2b7 - b6 + 3b4 - 3) * q^49 + (-2b11 - 5b4 + 5) * q^50 + (b9 - b6 - b4 + b2 - b1) * q^52 + (-b10 + b9 + b8 - b2 + 3) * q^53 + (-2b10 - 2b8 + 2b2 + 4) q^55 + (b9 - b6 + b4) * q^56 + (b8 - b4 + 1) * q^58 + (-b11 + 2b10 + b8 + b7 - b6 + 2b4 - 2b1 - 2) q^59 + (b5 - 2b4 - b3 + 2b1) * q^61 + (-b11 - b10 + b7 + b5 - b3 - 1) * q^62 + q^64 + (2b9 - 2b6 + 2b5 + b4 - 2b3 + 5b1) q^65 + (-2b11 + 2b10 - b8 + b7 + 2b6 + 3b4 - 2b1 - 3) q^67 + (b11 - b10 + b6 + b1) * q^68 + (2b9 - 2b6 - 2b4 + b3 + 2b2) * q^70 + (-b10 + 2b8 - b7 - b5 - 2b2 - 3) * q^71 + (b11 + b10 + b8 - 2b7 - 2b5 + b3 - b2 + 4) * q^73 + (b5 + b4 + 2b2) q^74 + (-b4 + 1) * q^76 + (3b11 - 3b10 + 2b8 - 3b7 - b4 + 3b1 + 1) q^77 + (-2b5 - 3b4 + 2b3 - 2b2 - 3b1) q^79 + (-b7 - b5) * q^80 + (-b10 - b7 - b5 - 1) * q^82 + (b5 - 2b4 - b3) q^83 + (b11 + 2b10 + 2b7 + 2b6 + 6b4 - 2b1 - 6) q^85 + (-b11 - b7 + 2b6 - 2b4 + 2) * q^86 + (-b4 - b3 + b1) * q^88 + (b11 + b10 + b7 + b5 + b3 - 1) * q^89 + (b9 + 2b8 + 2b7 + 2b5 - 2b2 - 2) * q^91 + (b5 + 2b4 - b3 + b2 + b1) q^92 + (b11 - b7 + 2b6 + b4 - 1) q^94 - b7 * q^95 + (3b4 - b3 - 2b2 - 3b1) q^97 + (-b9 - b8 + 2b7 + 2b5 + b2 - 3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q + 6 * q^2 - 6 * q^4 - 6 * q^7 - 12 * q^8	$ 12 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 12 q^{8} + 6 q^{11} - 6 q^{13} + 6 q^{14} - 6 q^{16} - 12 q^{19} - 6 q^{22} + 12 q^{23} - 30 q^{25} - 12 q^{26} + 12 q^{28} - 6 q^{29} - 6 q^{31} + 6 q^{32} - 24 q^{35} + 12 q^{37} - 6 q^{38} - 6 q^{41} - 12 q^{43} - 12 q^{44} + 24 q^{46} + 6 q^{47} - 18 q^{49} + 30 q^{50} - 6 q^{52} + 36 q^{53} + 48 q^{55} + 6 q^{56} + 6 q^{58} - 12 q^{59} - 12 q^{61} - 12 q^{62} + 12 q^{64} + 6 q^{65} - 18 q^{67} - 12 q^{70} - 36 q^{71} + 48 q^{73} + 6 q^{74} + 6 q^{76} + 6 q^{77} - 18 q^{79} - 12 q^{82} - 12 q^{83} - 36 q^{85} + 12 q^{86} - 6 q^{88} - 12 q^{89} - 24 q^{91} + 12 q^{92} - 6 q^{94} + 18 q^{97} - 36 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 - 6 * q^4 - 6 * q^7 - 12 * q^8 + 6 * q^11 - 6 * q^13 + 6 * q^14 - 6 * q^16 - 12 * q^19 - 6 * q^22 + 12 * q^23 - 30 * q^25 - 12 * q^26 + 12 * q^28 - 6 * q^29 - 6 * q^31 + 6 * q^32 - 24 * q^35 + 12 * q^37 - 6 * q^38 - 6 * q^41 - 12 * q^43 - 12 * q^44 + 24 * q^46 + 6 * q^47 - 18 * q^49 + 30 * q^50 - 6 * q^52 + 36 * q^53 + 48 * q^55 + 6 * q^56 + 6 * q^58 - 12 * q^59 - 12 * q^61 - 12 * q^62 + 12 * q^64 + 6 * q^65 - 18 * q^67 - 12 * q^70 - 36 * q^71 + 48 * q^73 + 6 * q^74 + 6 * q^76 + 6 * q^77 - 18 * q^79 - 12 * q^82 - 12 * q^83 - 36 * q^85 + 12 * q^86 - 6 * q^88 - 12 * q^89 - 24 * q^91 + 12 * q^92 - 6 * q^94 + 18 * q^97 - 36 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 $ x^12 + 8*x^10 + 55*x^8 - 2*x^7 + 70*x^6 - 32*x^5 + 73*x^4 - 18*x^3 + 13*x^2 + 2*x + 1 :

$\beta_{1}$	$=$	$ ( - 1870 \nu^{11} + 93796 \nu^{10} - 206030 \nu^{9} + 644380 \nu^{8} - 1616819 \nu^{7} + \cdots + 772695 ) / 895247 $ (-1870v^11 + 93796v^10 - 206030v^9 + 644380v^8 - 1616819v^7 + 4386085v^6 - 10641620v^5 + 1785263v^4 - 15006200v^3 + 9400655v^2 - 9228995*v + 772695) / 895247
$\beta_{2}$	$=$	$ ( - 1870 \nu^{11} + 93796 \nu^{10} - 206030 \nu^{9} + 644380 \nu^{8} - 1616819 \nu^{7} + \cdots + 772695 ) / 895247 $ (-1870v^11 + 93796v^10 - 206030v^9 + 644380v^8 - 1616819v^7 + 4386085v^6 - 10641620v^5 + 1785263v^4 - 15006200v^3 + 9400655v^2 - 6543254*v + 772695) / 895247
$\beta_{3}$	$=$	$ ( 24850 \nu^{11} + 285540 \nu^{10} - 421810 \nu^{9} + 1969300 \nu^{8} - 3499994 \nu^{7} + \cdots + 2370615 ) / 2685741 $ (24850v^11 + 285540v^10 - 421810v^9 + 1969300v^8 - 3499994v^7 + 13334085v^6 - 32369200v^5 + 4837781v^4 - 45989060v^3 + 28848695v^2 - 40790853*v + 2370615) / 2685741
$\beta_{4}$	$=$	$ ( - 475714 \nu^{11} + 219777 \nu^{10} - 3753182 \nu^{9} + 1727756 \nu^{8} - 25748182 \nu^{7} + \cdots + 1930593 ) / 2685741 $ (-475714v^11 + 219777v^10 - 3753182v^9 + 1727756v^8 - 25748182v^7 + 12842883v^6 - 30886544v^5 + 29151715v^4 - 38258716v^3 + 23369953v^2 - 5787570*v + 1930593) / 2685741
$\beta_{5}$	$=$	$ ( 619464 \nu^{11} - 153157 \nu^{10} + 4582497 \nu^{9} - 1233806 \nu^{8} + 31130660 \nu^{7} + \cdots - 1294728 ) / 2685741 $ (619464v^11 - 153157v^10 + 4582497v^9 - 1233806v^8 + 31130660v^7 - 9814648v^6 + 23455119v^5 - 31167245v^4 + 26145931v^3 - 15597658v^2 - 15860327*v - 1294728) / 2685741
$\beta_{6}$	$=$	$ ( - 295111 \nu^{11} + 579804 \nu^{10} - 2128552 \nu^{9} + 4583810 \nu^{8} - 14488844 \nu^{7} + \cdots - 48968 ) / 895247 $ (-295111v^11 + 579804v^10 - 2128552v^9 + 4583810v^8 - 14488844v^7 + 31938188v^6 - 9964895v^5 + 45814932v^4 - 30004460v^3 + 31484733v^2 - 2637508*v - 48968) / 895247
$\beta_{7}$	$=$	$ ( 419533 \nu^{11} + 219777 \nu^{10} + 3408794 \nu^{9} + 1727756 \nu^{8} + 23490403 \nu^{7} + \cdots + 1035346 ) / 895247 $ (419533v^11 + 219777v^10 + 3408794v^9 + 1727756v^8 + 23490403v^7 + 11052389v^6 + 31780746v^5 + 503811v^4 + 27094315v^3 + 6360260v^2 + 5850641*v + 1035346) / 895247
$\beta_{8}$	$=$	$ ( - 1525245 \nu^{11} - 363090 \nu^{10} - 12641764 \nu^{9} - 2877555 \nu^{8} - 87002900 \nu^{7} + \cdots - 3463016 ) / 2685741 $ (-1525245v^11 - 363090v^10 - 12641764v^9 - 2877555v^8 - 87002900v^7 - 16506934v^6 - 127110133v^5 + 27208266v^4 - 108729704v^3 + 25001872v^2 - 21057244*v - 3463016) / 2685741
$\beta_{9}$	$=$	$ ( 1543101 \nu^{11} + 945818 \nu^{10} + 12186642 \nu^{9} + 6888274 \nu^{8} + 83348183 \nu^{7} + \cdots - 7654161 ) / 2685741 $ (1543101v^11 + 945818v^10 + 12186642v^9 + 6888274v^8 + 83348183v^7 + 43559705v^6 + 95489559v^5 - 19531004v^4 + 59698732v^3 + 3723014v^2 + 1522372*v - 7654161) / 2685741
$\beta_{10}$	$=$	$ ( 2178966 \nu^{11} + 802068 \nu^{10} + 17206942 \nu^{9} + 6058959 \nu^{8} + 117826808 \nu^{7} + \cdots + 7208243 ) / 2685741 $ (2178966v^11 + 802068v^10 + 17206942v^9 + 6058959v^8 + 117826808v^7 + 36905497v^6 + 136971874v^5 - 32447259v^4 + 108132407v^3 + 4390229v^2 + 2016322*v + 7208243) / 2685741
$\beta_{11}$	$=$	$ ( 3253001 \nu^{11} + 779178 \nu^{10} + 26267532 \nu^{9} + 6170099 \nu^{8} + 180784077 \nu^{7} + \cdots + 7216198 ) / 2685741 $ (3253001v^11 + 779178v^10 + 26267532v^9 + 6170099v^8 + 180784077v^7 + 35638592v^6 + 238985409v^5 - 54943010v^4 + 225471231v^3 - 22474906v^2 + 43788575*v + 7216198) / 2685741

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

$\nu$	$=$	$ ( \beta_{2} - \beta_1 ) / 3 $ (b2 - b1) / 3
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{10} - 2\beta_{7} - \beta_{6} + 8\beta_{4} - \beta _1 - 8 ) / 3 $ (b11 + b10 - 2b7 - b6 + 8b4 - b1 - 8) / 3
$\nu^{3}$	$=$	$ ( -3\beta_{11} + 8\beta_{10} + 5\beta_{8} - 3\beta_{3} - 5\beta_{2} ) / 3 $ (-3b11 + 8b10 + 5b8 - 3b3 - 5*b2) / 3
$\nu^{4}$	$=$	$ ( -5\beta_{9} + 5\beta_{6} - 16\beta_{5} - 46\beta_{4} + 8\beta_{3} - \beta_{2} + 6\beta_1 ) / 3 $ (-5b9 + 5b6 - 16b5 - 46b4 + 8b3 - b2 + 6b1) / 3
$\nu^{5}$	$=$	$ ( 23\beta_{11} - 56\beta_{10} - 31\beta_{8} - \beta_{7} + \beta_{6} - 5\beta_{4} + 56\beta _1 + 5 ) / 3 $ (23b11 - 56b10 - 31b8 - b7 + b6 - 5b4 + 56*b1 + 5) / 3
$\nu^{6}$	$=$	$ ( - 55 \beta_{11} - 41 \beta_{10} + 31 \beta_{9} - 10 \beta_{8} + 110 \beta_{7} + 110 \beta_{5} + \cdots + 299 ) / 3 $ (-55b11 - 41b10 + 31b9 - 10b8 + 110b7 + 110b5 - 55b3 + 10b2 + 299) / 3
$\nu^{7}$	$=$	$ ( 10\beta_{9} - 10\beta_{6} - 4\beta_{5} + 56\beta_{4} + 155\beta_{3} + 204\beta_{2} - 379\beta_1 ) / 3 $ (10b9 - 10b6 - 4b5 + 56b4 + 155b3 + 204b2 - 379*b1) / 3
$\nu^{8}$	$=$	$ 121\beta_{11} + 97\beta_{10} + 27\beta_{8} - 246\beta_{7} - 68\beta_{6} + 662\beta_{4} - 97\beta _1 - 662 $ 121b11 + 97b10 + 27b8 - 246b7 - 68b6 + 662b4 - 97*b1 - 662
$\nu^{9}$	$=$	$ ( - 1020 \beta_{11} + 2548 \beta_{10} - 81 \beta_{9} + 1360 \beta_{8} - 9 \beta_{7} - 9 \beta_{5} + \cdots - 495 ) / 3 $ (-1020b11 + 2548b10 - 81b9 + 1360b8 - 9b7 - 9b5 - 1020b3 - 1360b2 - 495) / 3
$\nu^{10}$	$=$	$ ( -1360\beta_{9} + 1360\beta_{6} - 4928\beta_{5} - 13253\beta_{4} + 2371\beta_{3} - 621\beta_{2} + 2077\beta_1 ) / 3 $ (-1360b9 + 1360b6 - 4928b5 - 13253b4 + 2371b3 - 621b2 + 2077*b1) / 3
$\nu^{11}$	$=$	$ ( 6678 \beta_{11} - 17111 \beta_{10} - 9095 \beta_{8} + 327 \beta_{7} + 621 \beta_{6} - 4059 \beta_{4} + \cdots + 4059 ) / 3 $ (6678b11 - 17111b10 - 9095b8 + 327b7 + 621b6 - 4059b4 + 17111*b1 + 4059) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1026\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-1 + \beta_{4}$	$1$

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} $

343.1

0.289933	−	0.502178i
0.425485	−	0.736962i
−0.621041	+	1.07567i
−0.122364	+	0.211941i
1.30509	−	2.26048i
−1.27710	+	2.21200i
0.289933	+	0.502178i
0.425485	+	0.736962i
−0.621041	−	1.07567i
−0.122364	−	0.211941i
1.30509	+	2.26048i
−1.27710	−	2.21200i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−2.19415

−

3.80037i

1.88373

−

3.26272i

−1.00000

−4.38829

343.2

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.72549

−

2.98863i

−1.90190

3.29418i

−1.00000

−3.45098

343.3

0.500000

−

0.866025i

−0.500000

−

0.866025i

−0.326066

−

0.564763i

−2.10656

3.64868i

−1.00000

−0.652132

343.4

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.573032

0.992520i

0.817447

−

1.41586i

−1.00000

1.14606

343.5

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.71495

2.97037i

−0.0414111

0.0717262i

−1.00000

3.42989

343.6

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.95772

3.39088i

−1.65131

2.86015i

−1.00000

3.91545

685.1

0.500000

0.866025i

−0.500000

0.866025i

−2.19415

3.80037i

1.88373

3.26272i

−1.00000

−4.38829

685.2

0.500000

0.866025i

−0.500000

0.866025i

−1.72549

2.98863i

−1.90190

−

3.29418i

−1.00000

−3.45098

685.3

0.500000

0.866025i

−0.500000

0.866025i

−0.326066

0.564763i

−2.10656

−

3.64868i

−1.00000

−0.652132

685.4

0.500000

0.866025i

−0.500000

0.866025i

0.573032

−

0.992520i

0.817447

1.41586i

−1.00000

1.14606

685.5

0.500000

0.866025i

−0.500000

0.866025i

1.71495

−

2.97037i

−0.0414111

−

0.0717262i

−1.00000

3.42989

685.6

0.500000

0.866025i

−0.500000

0.866025i

1.95772

−

3.39088i

−1.65131

−

2.86015i

−1.00000

3.91545

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	1026.2.e.d		12
3.b	odd	2	1		342.2.e.c	✓	12
9.c	even	3	1	inner	1026.2.e.d		12
9.c	even	3	1		3078.2.a.v		6
9.d	odd	6	1		342.2.e.c	✓	12
9.d	odd	6	1		3078.2.a.x		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
342.2.e.c	✓	12	3.b	odd	2	1
342.2.e.c	✓	12	9.d	odd	6	1
1026.2.e.d		12	1.a	even	1	1	trivial
1026.2.e.d		12	9.c	even	3	1	inner
3078.2.a.v		6	9.c	even	3	1
3078.2.a.x		6	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} + 30 T_{5}^{10} - 16 T_{5}^{9} + 672 T_{5}^{8} - 336 T_{5}^{7} + 6600 T_{5}^{6} + \cdots + 23104 $ T5^12 + 30*T5^10 - 16*T5^9 + 672*T5^8 - 336*T5^7 + 6600*T5^6 - 3936*T5^5 + 48192*T5^4 - 19456*T5^3 + 43872*T5^2 + 14592*T5 + 23104 acting on $S_{2}^{\mathrm{new}}(1026, [\chi])$.

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^{12} + 30 T^{10} + \cdots + 23104 $ T^12 + 30T^10 - 16T^9 + 672T^8 - 336T^7 + 6600T^6 - 3936T^5 + 48192T^4 - 19456T^3 + 43872T^2 + 14592T + 23104
$7$	$ T^{12} + 6 T^{11} + \cdots + 729 $ T^12 + 6T^11 + 48T^10 + 144T^9 + 825T^8 + 2016T^7 + 9378T^6 + 11502T^5 + 36405T^4 - 4860T^3 + 105867T^2 + 8748*T + 729
$11$	$ T^{12} - 6 T^{11} + \cdots + 760384 $ T^12 - 6T^11 + 60T^10 - 88T^9 + 996T^8 + 96T^7 + 15744T^6 + 16512T^5 + 105360T^4 + 83072T^3 + 427296T^2 + 376704*T + 760384
$13$	$ T^{12} + 6 T^{11} + \cdots + 19035769 $ T^12 + 6T^11 + 84T^10 + 352T^9 + 3903T^8 + 15420T^7 + 103062T^6 + 299010T^5 + 1564305T^4 + 4048072T^3 + 13903221T^2 + 17068056*T + 19035769
$17$	$ (T^{6} - 66 T^{4} + \cdots + 405)^{2} $ (T^6 - 66T^4 + 18T^3 + 549T^2 - 918T + 405)^2
$19$	$ (T + 1)^{12} $ (T + 1)^12
$23$	$ T^{12} - 12 T^{11} + \cdots + 66049 $ T^12 - 12T^11 + 138T^10 - 868T^9 + 6513T^8 - 36264T^7 + 194160T^6 - 699978T^5 + 2030487T^4 - 3490000T^3 + 4280427T^2 - 558204*T + 66049
$29$	$ T^{12} + 6 T^{11} + \cdots + 72361 $ T^12 + 6T^11 + 66T^10 + 52T^9 + 1365T^8 + 858T^7 + 18948T^6 - 19410T^5 + 62691T^4 - 27758T^3 + 84639T^2 - 40350*T + 72361
$31$	$ T^{12} + \cdots + 202094656 $ T^12 + 6T^11 + 138T^10 + 644T^9 + 12084T^8 + 49776T^7 + 514272T^6 + 801744T^5 + 9677040T^4 + 4644992T^3 + 145805184T^2 - 153191616*T + 202094656
$37$	$ (T^{6} - 6 T^{5} + \cdots - 90329)^{2} $ (T^6 - 6T^5 - 135T^4 + 628T^3 + 5967T^2 - 15702*T - 90329)^2
$41$	$ T^{12} + 6 T^{11} + \cdots + 1600 $ T^12 + 6T^11 + 66T^10 + 172T^9 + 1908T^8 + 4896T^7 + 31344T^6 + 3312T^5 + 37296T^4 + 23296T^3 + 34944T^2 + 7680*T + 1600
$43$	$ T^{12} + 12 T^{11} + \cdots + 72863296 $ T^12 + 12T^11 + 210T^10 + 1088T^9 + 17028T^8 + 100944T^7 + 742848T^6 + 2136384T^5 + 7667280T^4 + 8397248T^3 + 42088128T^2 + 46913856*T + 72863296
$47$	$ T^{12} + \cdots + 2953161649 $ T^12 - 6T^11 + 195T^10 - 1078T^9 + 26970T^8 - 144606T^7 + 1625979T^6 - 7279374T^5 + 64583322T^4 - 268838998T^3 + 1052625315T^2 - 1953413478*T + 2953161649
$53$	$ (T^{6} - 18 T^{5} + \cdots + 1927)^{2} $ (T^6 - 18T^5 + 36T^4 + 664T^3 - 1791T^2 - 5112*T + 1927)^2
$59$	$ T^{12} + 12 T^{11} + \cdots + 20331081 $ T^12 + 12T^11 + 234T^10 + 2028T^9 + 30903T^8 + 244368T^7 + 1992528T^6 + 7372818T^5 + 25110387T^4 - 5880168T^3 + 41659839T^2 + 21589092*T + 20331081
$61$	$ T^{12} + 12 T^{11} + \cdots + 4494400 $ T^12 + 12T^11 + 174T^10 + 976T^9 + 9852T^8 + 46272T^7 + 368688T^6 + 825888T^5 + 3329520T^4 - 6359168T^3 + 12213504T^2 - 7988160*T + 4494400
$67$	$ T^{12} + \cdots + 275017385241 $ T^12 + 18T^11 + 426T^10 + 4764T^9 + 78315T^8 + 758556T^7 + 8990640T^6 + 61100118T^5 + 507276063T^4 + 2477647440T^3 + 17817460731T^2 + 60602090760*T + 275017385241
$71$	$ (T^{6} + 18 T^{5} + \cdots - 50840)^{2} $ (T^6 + 18T^5 - 30T^4 - 1232T^3 + 1992T^2 + 20688*T - 50840)^2
$73$	$ (T^{6} - 24 T^{5} + \cdots - 59967)^{2} $ (T^6 - 24T^5 + 54T^4 + 2622T^3 - 24927T^2 + 76878*T - 59967)^2
$79$	$ T^{12} + \cdots + 1126273600 $ T^12 + 18T^11 + 468T^10 + 3832T^9 + 82644T^8 + 657768T^7 + 8797872T^6 + 26342688T^5 + 165856560T^4 - 411710816T^3 + 2159765664T^2 - 1608463680*T + 1126273600
$83$	$ T^{12} + 12 T^{11} + \cdots + 10497600 $ T^12 + 12T^11 + 138T^10 + 720T^9 + 4644T^8 + 16848T^7 + 97632T^6 + 254016T^5 + 988848T^4 + 1010880T^3 + 4618944T^2 + 4898880*T + 10497600
$89$	$ (T^{6} + 6 T^{5} + \cdots - 5400)^{2} $ (T^6 + 6T^5 - 102T^4 - 252T^3 + 1632T^2 + 2088*T - 5400)^2
$97$	$ T^{12} + \cdots + 5833925283904 $ T^12 - 18T^11 + 732T^10 - 8968T^9 + 283092T^8 - 3161280T^7 + 67102464T^6 - 548139648T^5 + 9400729488T^4 - 67168201984T^3 + 773710769184T^2 - 2222394359424*T + 5833925283904
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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:	\( N \)	\(=\)	\( 1026 = 2 \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1026.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{7} - q^{8}+O(q^{10}) \) q + b4 * q^2 + (b4 - 1) * q^4 + b7 * q^5 + (-b9 + b6 - b4) * q^7 - q^8	\( q + \beta_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{7} - q^{8} + (\beta_{7} + \beta_{5}) q^{10} + (\beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{10} - \beta_{8} + \beta_{6} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots - 3) q^{98}+O(q^{100}) \) q + b4 * q^2 + (b4 - 1) * q^4 + b7 * q^5 + (-b9 + b6 - b4) * q^7 - q^8 + (b7 + b5) * q^10 + (b4 + b3 - b1) * q^11 + (-b10 - b8 + b6 + b4 + b1 - 1) * q^13 + (b6 - b4 + 1) * q^14 - b4 * q^16 + (-b11 + b10 - b9 - b3) * q^17 - q^19 + b5 * q^20 + (-b11 + b10 + b4 - b1 - 1) * q^22 + (-b11 + b10 - b8 + b7 - 2b4 - b1 + 2) q^23 + (-5b4 + 2b3) * q^25 + (-b10 + b9 - b8 + b2 - 1) * q^26 + (b9 + 1) * q^28 + (-b4 + b2) * q^29 + (-b11 - b10 + b7 + b4 + b1 - 1) * q^31 + (-b4 + 1) * q^32 + (-b9 + b6 - b3 + b1) * q^34 + (b11 + 2b9 - 2b8 + b3 + 2b2 - 2) q^35 + (-2b8 + b7 + b5 + 2b2 + 1) * q^37 - b4 * q^38 - b7 * q^40 + (-b10 - b7 + b4 + b1 - 1) * q^41 + (-2b9 + 2b6 + b5 - 2b4 + b3) q^43 + (-b11 + b10 - b3 - 1) * q^44 + (-b11 + b10 - b8 + b7 + b5 - b3 + b2 + 2) * q^46 + (-2b9 + 2b6 + b5 + b4 - b3) * q^47 + (-b8 + 2b7 - b6 + 3b4 - 3) * q^49 + (-2b11 - 5b4 + 5) * q^50 + (b9 - b6 - b4 + b2 - b1) * q^52 + (-b10 + b9 + b8 - b2 + 3) * q^53 + (-2b10 - 2b8 + 2b2 + 4) q^55 + (b9 - b6 + b4) * q^56 + (b8 - b4 + 1) * q^58 + (-b11 + 2b10 + b8 + b7 - b6 + 2b4 - 2b1 - 2) q^59 + (b5 - 2b4 - b3 + 2b1) * q^61 + (-b11 - b10 + b7 + b5 - b3 - 1) * q^62 + q^64 + (2b9 - 2b6 + 2b5 + b4 - 2b3 + 5b1) q^65 + (-2b11 + 2b10 - b8 + b7 + 2b6 + 3b4 - 2b1 - 3) q^67 + (b11 - b10 + b6 + b1) * q^68 + (2b9 - 2b6 - 2b4 + b3 + 2b2) * q^70 + (-b10 + 2b8 - b7 - b5 - 2b2 - 3) * q^71 + (b11 + b10 + b8 - 2b7 - 2b5 + b3 - b2 + 4) * q^73 + (b5 + b4 + 2b2) q^74 + (-b4 + 1) * q^76 + (3b11 - 3b10 + 2b8 - 3b7 - b4 + 3b1 + 1) q^77 + (-2b5 - 3b4 + 2b3 - 2b2 - 3b1) q^79 + (-b7 - b5) * q^80 + (-b10 - b7 - b5 - 1) * q^82 + (b5 - 2b4 - b3) q^83 + (b11 + 2b10 + 2b7 + 2b6 + 6b4 - 2b1 - 6) q^85 + (-b11 - b7 + 2b6 - 2b4 + 2) * q^86 + (-b4 - b3 + b1) * q^88 + (b11 + b10 + b7 + b5 + b3 - 1) * q^89 + (b9 + 2b8 + 2b7 + 2b5 - 2b2 - 2) * q^91 + (b5 + 2b4 - b3 + b2 + b1) q^92 + (b11 - b7 + 2b6 + b4 - 1) q^94 - b7 * q^95 + (3b4 - b3 - 2b2 - 3b1) q^97 + (-b9 - b8 + 2b7 + 2b5 + b2 - 3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q + 6 * q^2 - 6 * q^4 - 6 * q^7 - 12 * q^8	\( 12 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 12 q^{8} + 6 q^{11} - 6 q^{13} + 6 q^{14} - 6 q^{16} - 12 q^{19} - 6 q^{22} + 12 q^{23} - 30 q^{25} - 12 q^{26} + 12 q^{28} - 6 q^{29} - 6 q^{31} + 6 q^{32} - 24 q^{35} + 12 q^{37} - 6 q^{38} - 6 q^{41} - 12 q^{43} - 12 q^{44} + 24 q^{46} + 6 q^{47} - 18 q^{49} + 30 q^{50} - 6 q^{52} + 36 q^{53} + 48 q^{55} + 6 q^{56} + 6 q^{58} - 12 q^{59} - 12 q^{61} - 12 q^{62} + 12 q^{64} + 6 q^{65} - 18 q^{67} - 12 q^{70} - 36 q^{71} + 48 q^{73} + 6 q^{74} + 6 q^{76} + 6 q^{77} - 18 q^{79} - 12 q^{82} - 12 q^{83} - 36 q^{85} + 12 q^{86} - 6 q^{88} - 12 q^{89} - 24 q^{91} + 12 q^{92} - 6 q^{94} + 18 q^{97} - 36 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 - 6 * q^4 - 6 * q^7 - 12 * q^8 + 6 * q^11 - 6 * q^13 + 6 * q^14 - 6 * q^16 - 12 * q^19 - 6 * q^22 + 12 * q^23 - 30 * q^25 - 12 * q^26 + 12 * q^28 - 6 * q^29 - 6 * q^31 + 6 * q^32 - 24 * q^35 + 12 * q^37 - 6 * q^38 - 6 * q^41 - 12 * q^43 - 12 * q^44 + 24 * q^46 + 6 * q^47 - 18 * q^49 + 30 * q^50 - 6 * q^52 + 36 * q^53 + 48 * q^55 + 6 * q^56 + 6 * q^58 - 12 * q^59 - 12 * q^61 - 12 * q^62 + 12 * q^64 + 6 * q^65 - 18 * q^67 - 12 * q^70 - 36 * q^71 + 48 * q^73 + 6 * q^74 + 6 * q^76 + 6 * q^77 - 18 * q^79 - 12 * q^82 - 12 * q^83 - 36 * q^85 + 12 * q^86 - 6 * q^88 - 12 * q^89 - 24 * q^91 + 12 * q^92 - 6 * q^94 + 18 * q^97 - 36 * q^98

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	1026.2.e.d

Level	$1026$
Weight	$2$
Character orbit	1026.e
Analytic conductor	$8.193$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.19265124738\)
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:	\( x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 \) x^12 + 8x^10 + 55x^8 - 2x^7 + 70x^6 - 32x^5 + 73x^4 - 18x^3 + 13x^2 + 2*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{6} \)
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 1870 \nu^{11} + 93796 \nu^{10} - 206030 \nu^{9} + 644380 \nu^{8} - 1616819 \nu^{7} + \cdots + 772695 ) / 895247 \) (-1870v^11 + 93796v^10 - 206030v^9 + 644380v^8 - 1616819v^7 + 4386085v^6 - 10641620v^5 + 1785263v^4 - 15006200v^3 + 9400655v^2 - 9228995*v + 772695) / 895247
\(\beta_{2}\)	\(=\)	\( ( - 1870 \nu^{11} + 93796 \nu^{10} - 206030 \nu^{9} + 644380 \nu^{8} - 1616819 \nu^{7} + \cdots + 772695 ) / 895247 \) (-1870v^11 + 93796v^10 - 206030v^9 + 644380v^8 - 1616819v^7 + 4386085v^6 - 10641620v^5 + 1785263v^4 - 15006200v^3 + 9400655v^2 - 6543254*v + 772695) / 895247
\(\beta_{3}\)	\(=\)	\( ( 24850 \nu^{11} + 285540 \nu^{10} - 421810 \nu^{9} + 1969300 \nu^{8} - 3499994 \nu^{7} + \cdots + 2370615 ) / 2685741 \) (24850v^11 + 285540v^10 - 421810v^9 + 1969300v^8 - 3499994v^7 + 13334085v^6 - 32369200v^5 + 4837781v^4 - 45989060v^3 + 28848695v^2 - 40790853*v + 2370615) / 2685741
\(\beta_{4}\)	\(=\)	\( ( - 475714 \nu^{11} + 219777 \nu^{10} - 3753182 \nu^{9} + 1727756 \nu^{8} - 25748182 \nu^{7} + \cdots + 1930593 ) / 2685741 \) (-475714v^11 + 219777v^10 - 3753182v^9 + 1727756v^8 - 25748182v^7 + 12842883v^6 - 30886544v^5 + 29151715v^4 - 38258716v^3 + 23369953v^2 - 5787570*v + 1930593) / 2685741
\(\beta_{5}\)	\(=\)	\( ( 619464 \nu^{11} - 153157 \nu^{10} + 4582497 \nu^{9} - 1233806 \nu^{8} + 31130660 \nu^{7} + \cdots - 1294728 ) / 2685741 \) (619464v^11 - 153157v^10 + 4582497v^9 - 1233806v^8 + 31130660v^7 - 9814648v^6 + 23455119v^5 - 31167245v^4 + 26145931v^3 - 15597658v^2 - 15860327*v - 1294728) / 2685741
\(\beta_{6}\)	\(=\)	\( ( - 295111 \nu^{11} + 579804 \nu^{10} - 2128552 \nu^{9} + 4583810 \nu^{8} - 14488844 \nu^{7} + \cdots - 48968 ) / 895247 \) (-295111v^11 + 579804v^10 - 2128552v^9 + 4583810v^8 - 14488844v^7 + 31938188v^6 - 9964895v^5 + 45814932v^4 - 30004460v^3 + 31484733v^2 - 2637508*v - 48968) / 895247
\(\beta_{7}\)	\(=\)	\( ( 419533 \nu^{11} + 219777 \nu^{10} + 3408794 \nu^{9} + 1727756 \nu^{8} + 23490403 \nu^{7} + \cdots + 1035346 ) / 895247 \) (419533v^11 + 219777v^10 + 3408794v^9 + 1727756v^8 + 23490403v^7 + 11052389v^6 + 31780746v^5 + 503811v^4 + 27094315v^3 + 6360260v^2 + 5850641*v + 1035346) / 895247
\(\beta_{8}\)	\(=\)	\( ( - 1525245 \nu^{11} - 363090 \nu^{10} - 12641764 \nu^{9} - 2877555 \nu^{8} - 87002900 \nu^{7} + \cdots - 3463016 ) / 2685741 \) (-1525245v^11 - 363090v^10 - 12641764v^9 - 2877555v^8 - 87002900v^7 - 16506934v^6 - 127110133v^5 + 27208266v^4 - 108729704v^3 + 25001872v^2 - 21057244*v - 3463016) / 2685741
\(\beta_{9}\)	\(=\)	\( ( 1543101 \nu^{11} + 945818 \nu^{10} + 12186642 \nu^{9} + 6888274 \nu^{8} + 83348183 \nu^{7} + \cdots - 7654161 ) / 2685741 \) (1543101v^11 + 945818v^10 + 12186642v^9 + 6888274v^8 + 83348183v^7 + 43559705v^6 + 95489559v^5 - 19531004v^4 + 59698732v^3 + 3723014v^2 + 1522372*v - 7654161) / 2685741
\(\beta_{10}\)	\(=\)	\( ( 2178966 \nu^{11} + 802068 \nu^{10} + 17206942 \nu^{9} + 6058959 \nu^{8} + 117826808 \nu^{7} + \cdots + 7208243 ) / 2685741 \) (2178966v^11 + 802068v^10 + 17206942v^9 + 6058959v^8 + 117826808v^7 + 36905497v^6 + 136971874v^5 - 32447259v^4 + 108132407v^3 + 4390229v^2 + 2016322*v + 7208243) / 2685741
\(\beta_{11}\)	\(=\)	\( ( 3253001 \nu^{11} + 779178 \nu^{10} + 26267532 \nu^{9} + 6170099 \nu^{8} + 180784077 \nu^{7} + \cdots + 7216198 ) / 2685741 \) (3253001v^11 + 779178v^10 + 26267532v^9 + 6170099v^8 + 180784077v^7 + 35638592v^6 + 238985409v^5 - 54943010v^4 + 225471231v^3 - 22474906v^2 + 43788575*v + 7216198) / 2685741

\(\nu\)	\(=\)	\( ( \beta_{2} - \beta_1 ) / 3 \) (b2 - b1) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + \beta_{10} - 2\beta_{7} - \beta_{6} + 8\beta_{4} - \beta _1 - 8 ) / 3 \) (b11 + b10 - 2b7 - b6 + 8b4 - b1 - 8) / 3
\(\nu^{3}\)	\(=\)	\( ( -3\beta_{11} + 8\beta_{10} + 5\beta_{8} - 3\beta_{3} - 5\beta_{2} ) / 3 \) (-3b11 + 8b10 + 5b8 - 3b3 - 5*b2) / 3
\(\nu^{4}\)	\(=\)	\( ( -5\beta_{9} + 5\beta_{6} - 16\beta_{5} - 46\beta_{4} + 8\beta_{3} - \beta_{2} + 6\beta_1 ) / 3 \) (-5b9 + 5b6 - 16b5 - 46b4 + 8b3 - b2 + 6b1) / 3
\(\nu^{5}\)	\(=\)	\( ( 23\beta_{11} - 56\beta_{10} - 31\beta_{8} - \beta_{7} + \beta_{6} - 5\beta_{4} + 56\beta _1 + 5 ) / 3 \) (23b11 - 56b10 - 31b8 - b7 + b6 - 5b4 + 56*b1 + 5) / 3
\(\nu^{6}\)	\(=\)	\( ( - 55 \beta_{11} - 41 \beta_{10} + 31 \beta_{9} - 10 \beta_{8} + 110 \beta_{7} + 110 \beta_{5} + \cdots + 299 ) / 3 \) (-55b11 - 41b10 + 31b9 - 10b8 + 110b7 + 110b5 - 55b3 + 10b2 + 299) / 3
\(\nu^{7}\)	\(=\)	\( ( 10\beta_{9} - 10\beta_{6} - 4\beta_{5} + 56\beta_{4} + 155\beta_{3} + 204\beta_{2} - 379\beta_1 ) / 3 \) (10b9 - 10b6 - 4b5 + 56b4 + 155b3 + 204b2 - 379*b1) / 3
\(\nu^{8}\)	\(=\)	\( 121\beta_{11} + 97\beta_{10} + 27\beta_{8} - 246\beta_{7} - 68\beta_{6} + 662\beta_{4} - 97\beta _1 - 662 \) 121b11 + 97b10 + 27b8 - 246b7 - 68b6 + 662b4 - 97*b1 - 662
\(\nu^{9}\)	\(=\)	\( ( - 1020 \beta_{11} + 2548 \beta_{10} - 81 \beta_{9} + 1360 \beta_{8} - 9 \beta_{7} - 9 \beta_{5} + \cdots - 495 ) / 3 \) (-1020b11 + 2548b10 - 81b9 + 1360b8 - 9b7 - 9b5 - 1020b3 - 1360b2 - 495) / 3
\(\nu^{10}\)	\(=\)	\( ( -1360\beta_{9} + 1360\beta_{6} - 4928\beta_{5} - 13253\beta_{4} + 2371\beta_{3} - 621\beta_{2} + 2077\beta_1 ) / 3 \) (-1360b9 + 1360b6 - 4928b5 - 13253b4 + 2371b3 - 621b2 + 2077*b1) / 3
\(\nu^{11}\)	\(=\)	\( ( 6678 \beta_{11} - 17111 \beta_{10} - 9095 \beta_{8} + 327 \beta_{7} + 621 \beta_{6} - 4059 \beta_{4} + \cdots + 4059 ) / 3 \) (6678b11 - 17111b10 - 9095b8 + 327b7 + 621b6 - 4059b4 + 17111*b1 + 4059) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 343.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 30 T^{10} + \cdots + 23104 \) T^12 + 30T^10 - 16T^9 + 672T^8 - 336T^7 + 6600T^6 - 3936T^5 + 48192T^4 - 19456T^3 + 43872T^2 + 14592T + 23104
$7$	\( T^{12} + 6 T^{11} + \cdots + 729 \) T^12 + 6T^11 + 48T^10 + 144T^9 + 825T^8 + 2016T^7 + 9378T^6 + 11502T^5 + 36405T^4 - 4860T^3 + 105867T^2 + 8748*T + 729
$11$	\( T^{12} - 6 T^{11} + \cdots + 760384 \) T^12 - 6T^11 + 60T^10 - 88T^9 + 996T^8 + 96T^7 + 15744T^6 + 16512T^5 + 105360T^4 + 83072T^3 + 427296T^2 + 376704*T + 760384
$13$	\( T^{12} + 6 T^{11} + \cdots + 19035769 \) T^12 + 6T^11 + 84T^10 + 352T^9 + 3903T^8 + 15420T^7 + 103062T^6 + 299010T^5 + 1564305T^4 + 4048072T^3 + 13903221T^2 + 17068056*T + 19035769
$17$	\( (T^{6} - 66 T^{4} + \cdots + 405)^{2} \) (T^6 - 66T^4 + 18T^3 + 549T^2 - 918T + 405)^2
$19$	\( (T + 1)^{12} \) (T + 1)^12
$23$	\( T^{12} - 12 T^{11} + \cdots + 66049 \) T^12 - 12T^11 + 138T^10 - 868T^9 + 6513T^8 - 36264T^7 + 194160T^6 - 699978T^5 + 2030487T^4 - 3490000T^3 + 4280427T^2 - 558204*T + 66049
$29$	\( T^{12} + 6 T^{11} + \cdots + 72361 \) T^12 + 6T^11 + 66T^10 + 52T^9 + 1365T^8 + 858T^7 + 18948T^6 - 19410T^5 + 62691T^4 - 27758T^3 + 84639T^2 - 40350*T + 72361
$31$	\( T^{12} + \cdots + 202094656 \) T^12 + 6T^11 + 138T^10 + 644T^9 + 12084T^8 + 49776T^7 + 514272T^6 + 801744T^5 + 9677040T^4 + 4644992T^3 + 145805184T^2 - 153191616*T + 202094656
$37$	\( (T^{6} - 6 T^{5} + \cdots - 90329)^{2} \) (T^6 - 6T^5 - 135T^4 + 628T^3 + 5967T^2 - 15702*T - 90329)^2
$41$	\( T^{12} + 6 T^{11} + \cdots + 1600 \) T^12 + 6T^11 + 66T^10 + 172T^9 + 1908T^8 + 4896T^7 + 31344T^6 + 3312T^5 + 37296T^4 + 23296T^3 + 34944T^2 + 7680*T + 1600
$43$	\( T^{12} + 12 T^{11} + \cdots + 72863296 \) T^12 + 12T^11 + 210T^10 + 1088T^9 + 17028T^8 + 100944T^7 + 742848T^6 + 2136384T^5 + 7667280T^4 + 8397248T^3 + 42088128T^2 + 46913856*T + 72863296
$47$	\( T^{12} + \cdots + 2953161649 \) T^12 - 6T^11 + 195T^10 - 1078T^9 + 26970T^8 - 144606T^7 + 1625979T^6 - 7279374T^5 + 64583322T^4 - 268838998T^3 + 1052625315T^2 - 1953413478*T + 2953161649
$53$	\( (T^{6} - 18 T^{5} + \cdots + 1927)^{2} \) (T^6 - 18T^5 + 36T^4 + 664T^3 - 1791T^2 - 5112*T + 1927)^2
$59$	\( T^{12} + 12 T^{11} + \cdots + 20331081 \) T^12 + 12T^11 + 234T^10 + 2028T^9 + 30903T^8 + 244368T^7 + 1992528T^6 + 7372818T^5 + 25110387T^4 - 5880168T^3 + 41659839T^2 + 21589092*T + 20331081
$61$	\( T^{12} + 12 T^{11} + \cdots + 4494400 \) T^12 + 12T^11 + 174T^10 + 976T^9 + 9852T^8 + 46272T^7 + 368688T^6 + 825888T^5 + 3329520T^4 - 6359168T^3 + 12213504T^2 - 7988160*T + 4494400
$67$	\( T^{12} + \cdots + 275017385241 \) T^12 + 18T^11 + 426T^10 + 4764T^9 + 78315T^8 + 758556T^7 + 8990640T^6 + 61100118T^5 + 507276063T^4 + 2477647440T^3 + 17817460731T^2 + 60602090760*T + 275017385241
$71$	\( (T^{6} + 18 T^{5} + \cdots - 50840)^{2} \) (T^6 + 18T^5 - 30T^4 - 1232T^3 + 1992T^2 + 20688*T - 50840)^2
$73$	\( (T^{6} - 24 T^{5} + \cdots - 59967)^{2} \) (T^6 - 24T^5 + 54T^4 + 2622T^3 - 24927T^2 + 76878*T - 59967)^2
$79$	\( T^{12} + \cdots + 1126273600 \) T^12 + 18T^11 + 468T^10 + 3832T^9 + 82644T^8 + 657768T^7 + 8797872T^6 + 26342688T^5 + 165856560T^4 - 411710816T^3 + 2159765664T^2 - 1608463680*T + 1126273600
$83$	\( T^{12} + 12 T^{11} + \cdots + 10497600 \) T^12 + 12T^11 + 138T^10 + 720T^9 + 4644T^8 + 16848T^7 + 97632T^6 + 254016T^5 + 988848T^4 + 1010880T^3 + 4618944T^2 + 4898880*T + 10497600
$89$	\( (T^{6} + 6 T^{5} + \cdots - 5400)^{2} \) (T^6 + 6T^5 - 102T^4 - 252T^3 + 1632T^2 + 2088*T - 5400)^2
$97$	\( T^{12} + \cdots + 5833925283904 \) T^12 - 18T^11 + 732T^10 - 8968T^9 + 283092T^8 - 3161280T^7 + 67102464T^6 - 548139648T^5 + 9400729488T^4 - 67168201984T^3 + 773710769184T^2 - 2222394359424*T + 5833925283904
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\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(-1 + \beta_{4}\)	\(1\)