LMFDB - Newform orbit 1470.2.n.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,2,Mod(79,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1470.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1470.n (of order $6$, 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:	$11.7380090971$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.7652750400000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} + \cdots + 64 $ x^12 - 6x^11 + 18x^10 - 14x^9 + 21x^8 - 108x^7 + 368x^6 - 216x^5 + 84x^4 - 112x^3 + 288x^2 - 192*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{3} q^{2} - \beta_1 q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{8} + 1) q^{9}+O(q^{10}) $ q + b3 * q^2 - b1 * q^3 + b8 * q^4 + b5 * q^5 - q^6 + (b3 - b1) * q^8 + (-b8 + 1) * q^9	$ q + \beta_{3} q^{2} - \beta_1 q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{8} + 1) q^{9} + ( - \beta_{11} - \beta_{4}) q^{10} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) $ q + b3 * q^2 - b1 * q^3 + b8 * q^4 + b5 * q^5 - q^6 + (b3 - b1) * q^8 + (-b8 + 1) * q^9 + (-b11 - b4) * q^10 + (-b11 - b8 + b7 - b6 - b4 + b2) * q^11 - b3 * q^12 + (-b10 + b7 + b6 + b5 - b3 + b1) * q^13 + b11 * q^15 + (b8 - 1) * q^16 + (2b11 + 2b4 + 2b2 + 2b1) * q^17 + b1 * q^18 + (b9 + b8 + b4 - b2 - 1) * q^19 + (b6 + b5) * q^20 + (-b11 - b10 - b9 + b7 + b6 + b5 - b3 + b1) * q^22 + (b10 - b5 - 3b3) q^23 - b8 * q^24 + (2b7 + b2) q^25 + (-b9 - b8 - b4 + b2 + 1) * q^26 + (b3 - b1) * q^27 + (-b10 + b7 - b6 - b5 - 4) * q^29 - b5 * q^30 + (-b11 - b4 + b2) * q^31 - b1 * q^32 + (b10 + b9 - b5 - b4 + b3 - b2) * q^33 + (-2b10 + 2b7 - 2b6 - 2b5 + 2) * q^34 + q^36 + (-b10 + b5 + 3b3) q^37 + (-b7 - b6 - b1) * q^38 + (b11 + b8 + b4 - b2) * q^39 - b4 * q^40 + (b11 - 2b10 - b9 + 2b7 - 2b6 - 2b5 + 3) * q^41 + (-2b3 + 2b1) * q^43 + (b10 - b9 - b8 + b5 - b4 + b2 + 1) * q^44 - b6 * q^45 + (b11 - 3b8 + b4 - b2) q^46 + (3b10 - 3b5 + b3) * q^47 + (-b3 + b1) * q^48 + (-b10 - 2b9 + b7) q^50 + (2b10 + 2b8 + 2b5 - 2) q^51 + (b7 + b6 + b1) * q^52 + (2b11 + 2b7 + 2b6 + 2b4 + 2b2 - 3b1) * q^53 + (b8 - 1) * q^54 + (-3b10 - b9 + 3b7 - b6 - b5 + 5b3 - 5b1 - 5) * q^55 + (-b10 + b7 + b6 + b5 - b3 + b1) * q^57 + (-b9 + b4 - 4b3 + b2) q^58 + (-2b11 + 4b8 - b7 + b6 - 2b4 + 2b2) * q^59 + (b11 + b4) * q^60 + (2b10 + 2b9 - 2b8 + 2b5 + 2b4 - 2b2 + 2) * q^61 + (-b10 + b7 + b6 + b5) * q^62 - q^64 + (2b10 - b9 + 5b8 + b4 + b2 - 5) * q^65 + (b11 + b8 - b7 + b6 + b4 - b2) * q^66 + (-2b11 + 2b7 + 2b6 - 2b4 - 2b2 + 2b1) * q^67 + (-2b9 + 2b4 + 2b3 + 2b2) * q^68 + (-b11 + b9 + 3) * q^69 + 6 * q^71 + b3 * q^72 - 4b1 q^73 + (-b11 + 3b8 - b4 + b2) q^74 + (b10 + 2b9 - 2b2) * q^75 + (-b11 + b9 - 1) * q^76 + (b10 - b7 - b6 - b5 + b3 - b1) * q^78 + (b9 - 8b8 + b4 - b2 + 8) q^79 + b6 * q^80 - b8 * q^81 + (b10 - 2b9 - b5 + 2b4 + 3b3 + 2b2) * q^82 + (-2b11 - b10 - 2b9 + b7 + b6 + b5 - 4b3 + 4b1) * q^83 + (-2b11 + 2b10 + 4b9 - 2b7 + 10b3 - 10b1) * q^85 + (-2b8 + 2) q^86 + (-b11 - b4 - b2 + 4b1) q^87 + (-b11 + b7 + b6 - b4 - b2 + b1) * q^88 + (2b10 - 2b9 - 2b8 + 2b5 - 2b4 + 2b2 + 2) * q^89 - b11 * q^90 + (b10 - b7 - b6 - b5 - 3b3 + 3b1) * q^92 + (b10 - b5) * q^93 + (3b11 + b8 + 3b4 - 3b2) q^94 + (-b7 + b6 + 2b2 + 5b1) * q^95 + (-b8 + 1) * q^96 + (b11 - 2b10 + b9 + 2b7 + 2b6 + 2b5 - 8b3 + 8b1) * q^97 + (-b11 - b10 + b9 + b7 - b6 - b5 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{4} - 12 q^{6} + 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^4 - 12 * q^6 + 6 * q^9	$ 12 q + 6 q^{4} - 12 q^{6} + 6 q^{9} - 6 q^{11} - 6 q^{16} - 6 q^{19} - 6 q^{24} + 6 q^{26} - 48 q^{29} + 24 q^{34} + 12 q^{36} + 6 q^{39} + 36 q^{41} + 6 q^{44} - 18 q^{46} - 12 q^{51} - 6 q^{54} - 60 q^{55} + 24 q^{59} + 12 q^{61} - 12 q^{64} - 30 q^{65} + 6 q^{66} + 36 q^{69} + 72 q^{71} + 18 q^{74} - 12 q^{76} + 48 q^{79} - 6 q^{81} + 12 q^{86} + 12 q^{89} + 6 q^{94} + 6 q^{96} - 12 q^{99}+O(q^{100}) $ 12 * q + 6 * q^4 - 12 * q^6 + 6 * q^9 - 6 * q^11 - 6 * q^16 - 6 * q^19 - 6 * q^24 + 6 * q^26 - 48 * q^29 + 24 * q^34 + 12 * q^36 + 6 * q^39 + 36 * q^41 + 6 * q^44 - 18 * q^46 - 12 * q^51 - 6 * q^54 - 60 * q^55 + 24 * q^59 + 12 * q^61 - 12 * q^64 - 30 * q^65 + 6 * q^66 + 36 * q^69 + 72 * q^71 + 18 * q^74 - 12 * q^76 + 48 * q^79 - 6 * q^81 + 12 * q^86 + 12 * q^89 + 6 * q^94 + 6 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} + \cdots + 64 $ x^12 - 6*x^11 + 18*x^10 - 14*x^9 + 21*x^8 - 108*x^7 + 368*x^6 - 216*x^5 + 84*x^4 - 112*x^3 + 288*x^2 - 192*x + 64 :

$\beta_{1}$	$=$	$ ( 10967 \nu^{11} + 32246 \nu^{10} - 320538 \nu^{9} + 1200782 \nu^{8} + 26475 \nu^{7} + \cdots + 11668096 ) / 6321728 $ (10967v^11 + 32246v^10 - 320538v^9 + 1200782v^8 + 26475v^7 + 192316v^6 - 5356872v^5 + 25802216v^4 + 4101028v^3 - 2644000v^2 - 7541936*v + 11668096) / 6321728
$\beta_{2}$	$=$	$ ( 71537 \nu^{11} - 395262 \nu^{10} + 1115946 \nu^{9} - 559878 \nu^{8} + 1522973 \nu^{7} + \cdots + 376704 ) / 6321728 $ (71537v^11 - 395262v^10 + 1115946v^9 - 559878v^8 + 1522973v^7 - 7374100v^6 + 23734728v^5 - 5540672v^4 + 7241340v^3 - 10074032v^2 + 29588496*v + 376704) / 6321728
$\beta_{3}$	$=$	$ ( 38425 \nu^{11} - 170111 \nu^{10} + 367550 \nu^{9} + 334360 \nu^{8} + 552915 \nu^{7} + \cdots + 2629264 ) / 3160864 $ (38425v^11 - 170111v^10 + 367550v^9 + 334360v^8 + 552915v^7 - 3125365v^6 + 8245842v^5 + 9778420v^4 + 2618720v^3 - 1196060v^2 + 4633520*v + 2629264) / 3160864
$\beta_{4}$	$=$	$ ( - 87643 \nu^{11} + 504878 \nu^{10} - 1490814 \nu^{9} + 1056954 \nu^{8} - 2126039 \nu^{7} + \cdots + 21343872 ) / 6321728 $ (-87643v^11 + 504878v^10 - 1490814v^9 + 1056954v^8 - 2126039v^7 + 9354400v^6 - 31186560v^5 + 15651488v^4 - 14480484v^3 + 11904288v^2 - 29339888*v + 21343872) / 6321728
$\beta_{5}$	$=$	$ ( 44272 \nu^{11} - 237241 \nu^{10} + 629192 \nu^{9} - 106362 \nu^{8} + 510126 \nu^{7} + \cdots - 1612208 ) / 3160864 $ (44272v^11 - 237241v^10 + 629192v^9 - 106362v^8 + 510126v^7 - 4079825v^6 + 13520382v^5 + 801716v^4 - 1987220v^3 - 690028v^2 + 15880448*v - 1612208) / 3160864
$\beta_{6}$	$=$	$ ( 112417 \nu^{11} - 805130 \nu^{10} + 2628966 \nu^{9} - 2969054 \nu^{8} + 1600117 \nu^{7} + \cdots - 33288832 ) / 6321728 $ (112417v^11 - 805130v^10 + 2628966v^9 - 2969054v^8 + 1600117v^7 - 14110440v^6 + 52377004v^5 - 54894792v^4 - 17165628v^3 - 21185216v^2 + 32776656*v - 33288832) / 6321728
$\beta_{7}$	$=$	$ ( 131935 \nu^{11} - 773178 \nu^{10} + 2298830 \nu^{9} - 1687650 \nu^{8} + 2890131 \nu^{7} + \cdots - 20220800 ) / 6321728 $ (131935v^11 - 773178v^10 + 2298830v^9 - 1687650v^8 + 2890131v^7 - 13565876v^6 + 46490256v^5 - 24737320v^4 + 15014452v^3 - 6351360v^2 + 31227728*v - 20220800) / 6321728
$\beta_{8}$	$=$	$ ( 227 \nu^{11} - 1290 \nu^{10} + 3622 \nu^{9} - 1778 \nu^{8} + 3655 \nu^{7} - 23804 \nu^{6} + \cdots - 16768 ) / 8768 $ (227v^11 - 1290v^10 + 3622v^9 - 1778v^8 + 3655v^7 - 23804v^6 + 75424v^5 - 20464v^4 + 228v^3 - 36720v^2 + 54704*v - 16768) / 8768
$\beta_{9}$	$=$	$ ( - 43824 \nu^{11} + 276857 \nu^{10} - 829457 \nu^{9} + 642506 \nu^{8} - 534986 \nu^{7} + \cdots + 9604288 ) / 1580432 $ (-43824v^11 + 276857v^10 - 829457v^9 + 642506v^8 - 534986v^7 + 4941719v^6 - 16904415v^5 + 10669054v^4 + 5302254v^3 + 7880932v^2 - 12722696*v + 9604288) / 1580432
$\beta_{10}$	$=$	$ ( - 128463 \nu^{11} + 645645 \nu^{10} - 1597836 \nu^{9} - 220844 \nu^{8} - 1660469 \nu^{7} + \cdots - 2470736 ) / 3160864 $ (-128463v^11 + 645645v^10 - 1597836v^9 - 220844v^8 - 1660469v^7 + 11871603v^6 - 34275032v^5 - 14186852v^4 + 1994316v^3 + 14559756v^2 - 17921792*v - 2470736) / 3160864
$\beta_{11}$	$=$	$ ( - 80946 \nu^{11} + 432381 \nu^{10} - 1148873 \nu^{9} + 259810 \nu^{8} - 1234124 \nu^{7} + \cdots + 782560 ) / 1580432 $ (-80946v^11 + 432381v^10 - 1148873v^9 + 259810v^8 - 1234124v^7 + 7919495v^6 - 24050043v^5 - 800286v^4 - 1154946v^3 + 10439620v^2 - 13207560*v + 782560) / 1580432

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

$\nu$	$=$	$ ( -2\beta_{9} - \beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 $ (-2b9 - b8 - 2b6 - b5 + b3 + b2 - 2*b1 + 2) / 3
$\nu^{2}$	$=$	$ ( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \cdots + 1 ) / 3 $ (-2b11 + 2b10 - 2b9 - 2b8 - 4b7 - 2b6 - 4b5 - b4 + 12b3 + b2 - 12*b1 + 1) / 3
$\nu^{3}$	$=$	$ - 5 \beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} - 4 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 3 $ -5b11 + 5b10 - b9 - 5b8 - 4b7 + b6 - b5 - b4 + 8b3 + 2b2 - 3*b1 - 3
$\nu^{4}$	$=$	$ ( - 20 \beta_{11} + 32 \beta_{10} + 20 \beta_{9} - 10 \beta_{7} + 32 \beta_{6} + 10 \beta_{5} + 23 \beta_{4} + \cdots - 99 ) / 3 $ (-20b11 + 32b10 + 20b9 - 10b7 + 32b6 + 10b5 + 23b4 + 28b3 + 23b2 + 28b1 - 99) / 3
$\nu^{5}$	$=$	$ ( 45 \beta_{11} + 45 \beta_{10} + 145 \beta_{9} + 191 \beta_{8} + 60 \beta_{7} + 145 \beta_{6} + \cdots - 289 ) / 3 $ (45b11 + 45b10 + 145b9 + 191b8 + 60b7 + 145b6 + 125b5 + 105b4 - 98b3 - 20b2 + 289*b1 - 289) / 3
$\nu^{6}$	$=$	$ 138 \beta_{11} - 68 \beta_{10} + 138 \beta_{9} + 290 \beta_{8} + 160 \beta_{7} + 68 \beta_{6} + \cdots - 145 $ 138b11 - 68b10 + 138b9 + 290b8 + 160b7 + 68b6 + 160b5 + 105b4 - 338b3 - 105b2 + 338*b1 - 145
$\nu^{7}$	$=$	$ ( 1579 \beta_{11} - 1579 \beta_{10} + 591 \beta_{9} + 2393 \beta_{8} + 1388 \beta_{7} - 591 \beta_{6} + \cdots + 1073 ) / 3 $ (1579b11 - 1579b10 + 591b9 + 2393b8 + 1388b7 - 591b6 + 849b5 + 191b4 - 3466b3 - 1440b2 + 1073*b1 + 1073) / 3
$\nu^{8}$	$=$	$ ( 2224 \beta_{11} - 5104 \beta_{10} - 2224 \beta_{9} + 1034 \beta_{7} - 5104 \beta_{6} - 1034 \beta_{5} + \cdots + 11361 ) / 3 $ (2224b11 - 5104b10 - 2224b9 + 1034b7 - 5104b6 - 1034b5 - 3373b4 - 5768b3 - 3373b2 - 5768b1 + 11361) / 3
$\nu^{9}$	$=$	$ - 2481 \beta_{11} - 2481 \beta_{10} - 6069 \beta_{9} - 9855 \beta_{8} - 3624 \beta_{7} - 6069 \beta_{6} + \cdots + 13889 $ -2481b11 - 2481b10 - 6069b9 - 9855b8 - 3624b7 - 6069b6 - 5361b5 - 6105b4 + 4034b3 + 708b2 - 13889*b1 + 13889
$\nu^{10}$	$=$	$ ( - 62050 \beta_{11} + 25420 \beta_{10} - 62050 \beta_{9} - 144778 \beta_{8} - 66620 \beta_{7} + \cdots + 72389 ) / 3 $ (-62050b11 + 25420b10 - 62050b9 - 144778b8 - 66620b7 - 25420b6 - 66620b5 - 50645b4 + 132702b3 + 50645b2 - 132702*b1 + 72389) / 3
$\nu^{11}$	$=$	$ ( - 215513 \beta_{11} + 215513 \beta_{10} - 91845 \beta_{9} - 361723 \beta_{8} - 190504 \beta_{7} + \cdots - 140359 ) / 3 $ (-215513b11 + 215513b10 - 91845b9 - 361723b8 - 190504b7 + 91845b6 - 134439b5 - 25009b4 + 502082b3 + 226284b2 - 140359*b1 - 140359) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1 + \beta_{8}$	$-1$

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} $

79.1

−0.685661	+	0.685661i
2.45845	−	2.45845i
0.593239	−	0.593239i
1.68566	+	1.68566i
−1.45845	−	1.45845i
0.406761	+	0.406761i
−0.685661	−	0.685661i
2.45845	+	2.45845i
0.593239	+	0.593239i
1.68566	−	1.68566i
−1.45845	+	1.45845i
0.406761	−	0.406761i

−0.866025

−

0.500000i

0.866025

−

0.500000i

0.500000

0.866025i

−2.20942

−

0.344208i

−1.00000

−

1.00000i

0.500000

−

0.866025i

1.74131

1.40280i

79.2

−0.866025

−

0.500000i

0.866025

−

0.500000i

0.500000

0.866025i

0.806615

2.08551i

−1.00000

−

1.00000i

0.500000

−

0.866025i

0.344208

−

2.20942i

79.3

−0.866025

−

0.500000i

0.866025

−

0.500000i

0.500000

0.866025i

1.40280

−

1.74131i

−1.00000

−

1.00000i

0.500000

−

0.866025i

−2.08551

0.806615i

79.4

0.866025

0.500000i

−0.866025

0.500000i

0.500000

0.866025i

−2.20942

0.344208i

−1.00000

1.00000i

0.500000

−

0.866025i

−2.08551

−

0.806615i

79.5

0.866025

0.500000i

−0.866025

0.500000i

0.500000

0.866025i

0.806615

−

2.08551i

−1.00000

1.00000i

0.500000

−

0.866025i

1.74131

−

1.40280i

79.6

0.866025

0.500000i

−0.866025

0.500000i

0.500000

0.866025i

1.40280

1.74131i

−1.00000

1.00000i

0.500000

−

0.866025i

0.344208

2.20942i

949.1

−0.866025

0.500000i

0.866025

0.500000i

0.500000

−

0.866025i

−2.20942

0.344208i

−1.00000

1.00000i

0.500000

0.866025i

1.74131

−

1.40280i

949.2

−0.866025

0.500000i

0.866025

0.500000i

0.500000

−

0.866025i

0.806615

−

2.08551i

−1.00000

1.00000i

0.500000

0.866025i

0.344208

2.20942i

949.3

−0.866025

0.500000i

0.866025

0.500000i

0.500000

−

0.866025i

1.40280

1.74131i

−1.00000

1.00000i

0.500000

0.866025i

−2.08551

−

0.806615i

949.4

0.866025

−

0.500000i

−0.866025

−

0.500000i

0.500000

−

0.866025i

−2.20942

−

0.344208i

−1.00000

−

1.00000i

0.500000

0.866025i

−2.08551

0.806615i

949.5

0.866025

−

0.500000i

−0.866025

−

0.500000i

0.500000

−

0.866025i

0.806615

2.08551i

−1.00000

−

1.00000i

0.500000

0.866025i

1.74131

1.40280i

949.6

0.866025

−

0.500000i

−0.866025

−

0.500000i

0.500000

−

0.866025i

1.40280

−

1.74131i

−1.00000

−

1.00000i

0.500000

0.866025i

0.344208

−

2.20942i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1470.2.n.j		12
5.b	even	2	1	inner	1470.2.n.j		12
7.b	odd	2	1		210.2.n.b	✓	12
7.c	even	3	1		1470.2.g.h		6
7.c	even	3	1	inner	1470.2.n.j		12
7.d	odd	6	1		210.2.n.b	✓	12
7.d	odd	6	1		1470.2.g.i		6
21.c	even	2	1		630.2.u.f		12
21.g	even	6	1		630.2.u.f		12
28.d	even	2	1		1680.2.di.c		12
28.f	even	6	1		1680.2.di.c		12
35.c	odd	2	1		210.2.n.b	✓	12
35.f	even	4	1		1050.2.i.u		6
35.f	even	4	1		1050.2.i.v		6
35.i	odd	6	1		210.2.n.b	✓	12
35.i	odd	6	1		1470.2.g.i		6
35.j	even	6	1		1470.2.g.h		6
35.j	even	6	1	inner	1470.2.n.j		12
35.k	even	12	1		1050.2.i.u		6
35.k	even	12	1		1050.2.i.v		6
35.k	even	12	1		7350.2.a.dn		3
35.k	even	12	1		7350.2.a.dq		3
35.l	odd	12	1		7350.2.a.do		3
35.l	odd	12	1		7350.2.a.dp		3
105.g	even	2	1		630.2.u.f		12
105.p	even	6	1		630.2.u.f		12
140.c	even	2	1		1680.2.di.c		12
140.s	even	6	1		1680.2.di.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.2.n.b	✓	12	7.b	odd	2	1
210.2.n.b	✓	12	7.d	odd	6	1
210.2.n.b	✓	12	35.c	odd	2	1
210.2.n.b	✓	12	35.i	odd	6	1
630.2.u.f		12	21.c	even	2	1
630.2.u.f		12	21.g	even	6	1
630.2.u.f		12	105.g	even	2	1
630.2.u.f		12	105.p	even	6	1
1050.2.i.u		6	35.f	even	4	1
1050.2.i.u		6	35.k	even	12	1
1050.2.i.v		6	35.f	even	4	1
1050.2.i.v		6	35.k	even	12	1
1470.2.g.h		6	7.c	even	3	1
1470.2.g.h		6	35.j	even	6	1
1470.2.g.i		6	7.d	odd	6	1
1470.2.g.i		6	35.i	odd	6	1
1470.2.n.j		12	1.a	even	1	1	trivial
1470.2.n.j		12	5.b	even	2	1	inner
1470.2.n.j		12	7.c	even	3	1	inner
1470.2.n.j		12	35.j	even	6	1	inner
1680.2.di.c		12	28.d	even	2	1
1680.2.di.c		12	28.f	even	6	1
1680.2.di.c		12	140.c	even	2	1
1680.2.di.c		12	140.s	even	6	1
7350.2.a.dn		3	35.k	even	12	1
7350.2.a.do		3	35.l	odd	12	1
7350.2.a.dp		3	35.l	odd	12	1
7350.2.a.dq		3	35.k	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1470, [\chi])$:

$ T_{11}^{6} + 3T_{11}^{5} + 36T_{11}^{4} + 17T_{11}^{3} + 876T_{11}^{2} + 1323T_{11} + 2401 $

$ T_{17}^{12} - 132T_{17}^{10} + 11856T_{17}^{8} - 587008T_{17}^{6} + 21236736T_{17}^{4} - 411942912T_{17}^{2} + 5473632256 $

$ T_{19}^{6} + 3T_{19}^{5} + 21T_{19}^{4} + 12T_{19}^{3} + 216T_{19}^{2} + 288T_{19} + 576 $

$ T_{31}^{6} + 15T_{31}^{4} + 20T_{31}^{3} + 225T_{31}^{2} + 150T_{31} + 100 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$3$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$5$	$ (T^{6} + 20 T^{3} + 125)^{2} $ (T^6 + 20*T^3 + 125)^2
$7$	$ T^{12} $ T^12
$11$	$ (T^{6} + 3 T^{5} + \cdots + 2401)^{2} $ (T^6 + 3T^5 + 36T^4 + 17T^3 + 876T^2 + 1323*T + 2401)^2
$13$	$ (T^{6} + 33 T^{4} + \cdots + 576)^{2} $ (T^6 + 33T^4 + 288T^2 + 576)^2
$17$	$ T^{12} + \cdots + 5473632256 $ T^12 - 132T^10 + 11856T^8 - 587008T^6 + 21236736T^4 - 411942912*T^2 + 5473632256
$19$	$ (T^{6} + 3 T^{5} + \cdots + 576)^{2} $ (T^6 + 3T^5 + 21T^4 + 12T^3 + 216T^2 + 288*T + 576)^2
$23$	$ T^{12} - 57 T^{10} + \cdots + 4096 $ T^12 - 57T^10 + 2961T^8 - 16288T^6 + 79296T^4 - 18432*T^2 + 4096
$29$	$ (T^{3} + 12 T^{2} + \cdots + 24)^{4} $ (T^3 + 12T^2 + 33T + 24)^4
$31$	$ (T^{6} + 15 T^{4} + \cdots + 100)^{2} $ (T^6 + 15T^4 + 20T^3 + 225T^2 + 150T + 100)^2
$37$	$ T^{12} - 57 T^{10} + \cdots + 4096 $ T^12 - 57T^10 + 2961T^8 - 16288T^6 + 79296T^4 - 18432*T^2 + 4096
$41$	$ (T^{3} - 9 T^{2} + \cdots + 128)^{4} $ (T^3 - 9T^2 - 48T + 128)^4
$43$	$ (T^{2} + 4)^{6} $ (T^2 + 4)^6
$47$	$ T^{12} + \cdots + 26639462656 $ T^12 - 273T^10 + 54681T^8 - 5092072T^6 + 349385136T^4 - 3239511168*T^2 + 26639462656
$53$	$ T^{12} + \cdots + 436880018961 $ T^12 - 267T^10 + 48006T^8 - 4894623T^6 + 365619366T^4 - 15389341227*T^2 + 436880018961
$59$	$ (T^{6} - 12 T^{5} + \cdots + 190096)^{2} $ (T^6 - 12T^5 + 171T^4 - 548T^3 + 5961T^2 - 11772*T + 190096)^2
$61$	$ (T^{6} - 6 T^{5} + \cdots + 506944)^{2} $ (T^6 - 6T^5 + 144T^4 - 776T^3 + 15936T^2 - 76896*T + 506944)^2
$67$	$ T^{12} + \cdots + 23612624896 $ T^12 - 252T^10 + 47136T^8 - 3817408T^6 + 229188096T^4 - 2515172352*T^2 + 23612624896
$71$	$ (T - 6)^{12} $ (T - 6)^12
$73$	$ (T^{4} - 16 T^{2} + 256)^{3} $ (T^4 - 16*T^2 + 256)^3
$79$	$ (T^{6} - 24 T^{5} + \cdots + 161604)^{2} $ (T^6 - 24T^5 + 399T^4 - 3444T^3 + 21681T^2 - 71154*T + 161604)^2
$83$	$ (T^{6} + 198 T^{4} + \cdots + 7396)^{2} $ (T^6 + 198T^4 + 2793T^2 + 7396)^2
$89$	$ (T^{6} - 6 T^{5} + \cdots + 153664)^{2} $ (T^6 - 6T^5 + 144T^4 - 136T^3 + 14016T^2 - 42336*T + 153664)^2
$97$	$ (T^{6} + 342 T^{4} + \cdots + 12544)^{2} $ (T^6 + 342T^4 + 8313T^2 + 12544)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} - \beta_1 q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{8} + 1) q^{9}+O(q^{10}) \) q + b3 * q^2 - b1 * q^3 + b8 * q^4 + b5 * q^5 - q^6 + (b3 - b1) * q^8 + (-b8 + 1) * q^9	\( q + \beta_{3} q^{2} - \beta_1 q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{8} + 1) q^{9} + ( - \beta_{11} - \beta_{4}) q^{10} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) q + b3 * q^2 - b1 * q^3 + b8 * q^4 + b5 * q^5 - q^6 + (b3 - b1) * q^8 + (-b8 + 1) * q^9 + (-b11 - b4) * q^10 + (-b11 - b8 + b7 - b6 - b4 + b2) * q^11 - b3 * q^12 + (-b10 + b7 + b6 + b5 - b3 + b1) * q^13 + b11 * q^15 + (b8 - 1) * q^16 + (2b11 + 2b4 + 2b2 + 2b1) * q^17 + b1 * q^18 + (b9 + b8 + b4 - b2 - 1) * q^19 + (b6 + b5) * q^20 + (-b11 - b10 - b9 + b7 + b6 + b5 - b3 + b1) * q^22 + (b10 - b5 - 3b3) q^23 - b8 * q^24 + (2b7 + b2) q^25 + (-b9 - b8 - b4 + b2 + 1) * q^26 + (b3 - b1) * q^27 + (-b10 + b7 - b6 - b5 - 4) * q^29 - b5 * q^30 + (-b11 - b4 + b2) * q^31 - b1 * q^32 + (b10 + b9 - b5 - b4 + b3 - b2) * q^33 + (-2b10 + 2b7 - 2b6 - 2b5 + 2) * q^34 + q^36 + (-b10 + b5 + 3b3) q^37 + (-b7 - b6 - b1) * q^38 + (b11 + b8 + b4 - b2) * q^39 - b4 * q^40 + (b11 - 2b10 - b9 + 2b7 - 2b6 - 2b5 + 3) * q^41 + (-2b3 + 2b1) * q^43 + (b10 - b9 - b8 + b5 - b4 + b2 + 1) * q^44 - b6 * q^45 + (b11 - 3b8 + b4 - b2) q^46 + (3b10 - 3b5 + b3) * q^47 + (-b3 + b1) * q^48 + (-b10 - 2b9 + b7) q^50 + (2b10 + 2b8 + 2b5 - 2) q^51 + (b7 + b6 + b1) * q^52 + (2b11 + 2b7 + 2b6 + 2b4 + 2b2 - 3b1) * q^53 + (b8 - 1) * q^54 + (-3b10 - b9 + 3b7 - b6 - b5 + 5b3 - 5b1 - 5) * q^55 + (-b10 + b7 + b6 + b5 - b3 + b1) * q^57 + (-b9 + b4 - 4b3 + b2) q^58 + (-2b11 + 4b8 - b7 + b6 - 2b4 + 2b2) * q^59 + (b11 + b4) * q^60 + (2b10 + 2b9 - 2b8 + 2b5 + 2b4 - 2b2 + 2) * q^61 + (-b10 + b7 + b6 + b5) * q^62 - q^64 + (2b10 - b9 + 5b8 + b4 + b2 - 5) * q^65 + (b11 + b8 - b7 + b6 + b4 - b2) * q^66 + (-2b11 + 2b7 + 2b6 - 2b4 - 2b2 + 2b1) * q^67 + (-2b9 + 2b4 + 2b3 + 2b2) * q^68 + (-b11 + b9 + 3) * q^69 + 6 * q^71 + b3 * q^72 - 4b1 q^73 + (-b11 + 3b8 - b4 + b2) q^74 + (b10 + 2b9 - 2b2) * q^75 + (-b11 + b9 - 1) * q^76 + (b10 - b7 - b6 - b5 + b3 - b1) * q^78 + (b9 - 8b8 + b4 - b2 + 8) q^79 + b6 * q^80 - b8 * q^81 + (b10 - 2b9 - b5 + 2b4 + 3b3 + 2b2) * q^82 + (-2b11 - b10 - 2b9 + b7 + b6 + b5 - 4b3 + 4b1) * q^83 + (-2b11 + 2b10 + 4b9 - 2b7 + 10b3 - 10b1) * q^85 + (-2b8 + 2) q^86 + (-b11 - b4 - b2 + 4b1) q^87 + (-b11 + b7 + b6 - b4 - b2 + b1) * q^88 + (2b10 - 2b9 - 2b8 + 2b5 - 2b4 + 2b2 + 2) * q^89 - b11 * q^90 + (b10 - b7 - b6 - b5 - 3b3 + 3b1) * q^92 + (b10 - b5) * q^93 + (3b11 + b8 + 3b4 - 3b2) q^94 + (-b7 + b6 + 2b2 + 5b1) * q^95 + (-b8 + 1) * q^96 + (b11 - 2b10 + b9 + 2b7 + 2b6 + 2b5 - 8b3 + 8b1) * q^97 + (-b11 - b10 + b9 + b7 - b6 - b5 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{4} - 12 q^{6} + 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^4 - 12 * q^6 + 6 * q^9	\( 12 q + 6 q^{4} - 12 q^{6} + 6 q^{9} - 6 q^{11} - 6 q^{16} - 6 q^{19} - 6 q^{24} + 6 q^{26} - 48 q^{29} + 24 q^{34} + 12 q^{36} + 6 q^{39} + 36 q^{41} + 6 q^{44} - 18 q^{46} - 12 q^{51} - 6 q^{54} - 60 q^{55} + 24 q^{59} + 12 q^{61} - 12 q^{64} - 30 q^{65} + 6 q^{66} + 36 q^{69} + 72 q^{71} + 18 q^{74} - 12 q^{76} + 48 q^{79} - 6 q^{81} + 12 q^{86} + 12 q^{89} + 6 q^{94} + 6 q^{96} - 12 q^{99}+O(q^{100}) \) 12 * q + 6 * q^4 - 12 * q^6 + 6 * q^9 - 6 * q^11 - 6 * q^16 - 6 * q^19 - 6 * q^24 + 6 * q^26 - 48 * q^29 + 24 * q^34 + 12 * q^36 + 6 * q^39 + 36 * q^41 + 6 * q^44 - 18 * q^46 - 12 * q^51 - 6 * q^54 - 60 * q^55 + 24 * q^59 + 12 * q^61 - 12 * q^64 - 30 * q^65 + 6 * q^66 + 36 * q^69 + 72 * q^71 + 18 * q^74 - 12 * q^76 + 48 * q^79 - 6 * q^81 + 12 * q^86 + 12 * q^89 + 6 * q^94 + 6 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1470.2.n.j

Level	$1470$
Weight	$2$
Character orbit	1470.n
Analytic conductor	$11.738$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.7380090971\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.7652750400000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} + \cdots + 64 \) x^12 - 6x^11 + 18x^10 - 14x^9 + 21x^8 - 108x^7 + 368x^6 - 216x^5 + 84x^4 - 112x^3 + 288x^2 - 192*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 10967 \nu^{11} + 32246 \nu^{10} - 320538 \nu^{9} + 1200782 \nu^{8} + 26475 \nu^{7} + \cdots + 11668096 ) / 6321728 \) (10967v^11 + 32246v^10 - 320538v^9 + 1200782v^8 + 26475v^7 + 192316v^6 - 5356872v^5 + 25802216v^4 + 4101028v^3 - 2644000v^2 - 7541936*v + 11668096) / 6321728
\(\beta_{2}\)	\(=\)	\( ( 71537 \nu^{11} - 395262 \nu^{10} + 1115946 \nu^{9} - 559878 \nu^{8} + 1522973 \nu^{7} + \cdots + 376704 ) / 6321728 \) (71537v^11 - 395262v^10 + 1115946v^9 - 559878v^8 + 1522973v^7 - 7374100v^6 + 23734728v^5 - 5540672v^4 + 7241340v^3 - 10074032v^2 + 29588496*v + 376704) / 6321728
\(\beta_{3}\)	\(=\)	\( ( 38425 \nu^{11} - 170111 \nu^{10} + 367550 \nu^{9} + 334360 \nu^{8} + 552915 \nu^{7} + \cdots + 2629264 ) / 3160864 \) (38425v^11 - 170111v^10 + 367550v^9 + 334360v^8 + 552915v^7 - 3125365v^6 + 8245842v^5 + 9778420v^4 + 2618720v^3 - 1196060v^2 + 4633520*v + 2629264) / 3160864
\(\beta_{4}\)	\(=\)	\( ( - 87643 \nu^{11} + 504878 \nu^{10} - 1490814 \nu^{9} + 1056954 \nu^{8} - 2126039 \nu^{7} + \cdots + 21343872 ) / 6321728 \) (-87643v^11 + 504878v^10 - 1490814v^9 + 1056954v^8 - 2126039v^7 + 9354400v^6 - 31186560v^5 + 15651488v^4 - 14480484v^3 + 11904288v^2 - 29339888*v + 21343872) / 6321728
\(\beta_{5}\)	\(=\)	\( ( 44272 \nu^{11} - 237241 \nu^{10} + 629192 \nu^{9} - 106362 \nu^{8} + 510126 \nu^{7} + \cdots - 1612208 ) / 3160864 \) (44272v^11 - 237241v^10 + 629192v^9 - 106362v^8 + 510126v^7 - 4079825v^6 + 13520382v^5 + 801716v^4 - 1987220v^3 - 690028v^2 + 15880448*v - 1612208) / 3160864
\(\beta_{6}\)	\(=\)	\( ( 112417 \nu^{11} - 805130 \nu^{10} + 2628966 \nu^{9} - 2969054 \nu^{8} + 1600117 \nu^{7} + \cdots - 33288832 ) / 6321728 \) (112417v^11 - 805130v^10 + 2628966v^9 - 2969054v^8 + 1600117v^7 - 14110440v^6 + 52377004v^5 - 54894792v^4 - 17165628v^3 - 21185216v^2 + 32776656*v - 33288832) / 6321728
\(\beta_{7}\)	\(=\)	\( ( 131935 \nu^{11} - 773178 \nu^{10} + 2298830 \nu^{9} - 1687650 \nu^{8} + 2890131 \nu^{7} + \cdots - 20220800 ) / 6321728 \) (131935v^11 - 773178v^10 + 2298830v^9 - 1687650v^8 + 2890131v^7 - 13565876v^6 + 46490256v^5 - 24737320v^4 + 15014452v^3 - 6351360v^2 + 31227728*v - 20220800) / 6321728
\(\beta_{8}\)	\(=\)	\( ( 227 \nu^{11} - 1290 \nu^{10} + 3622 \nu^{9} - 1778 \nu^{8} + 3655 \nu^{7} - 23804 \nu^{6} + \cdots - 16768 ) / 8768 \) (227v^11 - 1290v^10 + 3622v^9 - 1778v^8 + 3655v^7 - 23804v^6 + 75424v^5 - 20464v^4 + 228v^3 - 36720v^2 + 54704*v - 16768) / 8768
\(\beta_{9}\)	\(=\)	\( ( - 43824 \nu^{11} + 276857 \nu^{10} - 829457 \nu^{9} + 642506 \nu^{8} - 534986 \nu^{7} + \cdots + 9604288 ) / 1580432 \) (-43824v^11 + 276857v^10 - 829457v^9 + 642506v^8 - 534986v^7 + 4941719v^6 - 16904415v^5 + 10669054v^4 + 5302254v^3 + 7880932v^2 - 12722696*v + 9604288) / 1580432
\(\beta_{10}\)	\(=\)	\( ( - 128463 \nu^{11} + 645645 \nu^{10} - 1597836 \nu^{9} - 220844 \nu^{8} - 1660469 \nu^{7} + \cdots - 2470736 ) / 3160864 \) (-128463v^11 + 645645v^10 - 1597836v^9 - 220844v^8 - 1660469v^7 + 11871603v^6 - 34275032v^5 - 14186852v^4 + 1994316v^3 + 14559756v^2 - 17921792*v - 2470736) / 3160864
\(\beta_{11}\)	\(=\)	\( ( - 80946 \nu^{11} + 432381 \nu^{10} - 1148873 \nu^{9} + 259810 \nu^{8} - 1234124 \nu^{7} + \cdots + 782560 ) / 1580432 \) (-80946v^11 + 432381v^10 - 1148873v^9 + 259810v^8 - 1234124v^7 + 7919495v^6 - 24050043v^5 - 800286v^4 - 1154946v^3 + 10439620v^2 - 13207560*v + 782560) / 1580432

\(\nu\)	\(=\)	\( ( -2\beta_{9} - \beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) (-2b9 - b8 - 2b6 - b5 + b3 + b2 - 2*b1 + 2) / 3
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \cdots + 1 ) / 3 \) (-2b11 + 2b10 - 2b9 - 2b8 - 4b7 - 2b6 - 4b5 - b4 + 12b3 + b2 - 12*b1 + 1) / 3
\(\nu^{3}\)	\(=\)	\( - 5 \beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} - 4 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 3 \) -5b11 + 5b10 - b9 - 5b8 - 4b7 + b6 - b5 - b4 + 8b3 + 2b2 - 3*b1 - 3
\(\nu^{4}\)	\(=\)	\( ( - 20 \beta_{11} + 32 \beta_{10} + 20 \beta_{9} - 10 \beta_{7} + 32 \beta_{6} + 10 \beta_{5} + 23 \beta_{4} + \cdots - 99 ) / 3 \) (-20b11 + 32b10 + 20b9 - 10b7 + 32b6 + 10b5 + 23b4 + 28b3 + 23b2 + 28b1 - 99) / 3
\(\nu^{5}\)	\(=\)	\( ( 45 \beta_{11} + 45 \beta_{10} + 145 \beta_{9} + 191 \beta_{8} + 60 \beta_{7} + 145 \beta_{6} + \cdots - 289 ) / 3 \) (45b11 + 45b10 + 145b9 + 191b8 + 60b7 + 145b6 + 125b5 + 105b4 - 98b3 - 20b2 + 289*b1 - 289) / 3
\(\nu^{6}\)	\(=\)	\( 138 \beta_{11} - 68 \beta_{10} + 138 \beta_{9} + 290 \beta_{8} + 160 \beta_{7} + 68 \beta_{6} + \cdots - 145 \) 138b11 - 68b10 + 138b9 + 290b8 + 160b7 + 68b6 + 160b5 + 105b4 - 338b3 - 105b2 + 338*b1 - 145
\(\nu^{7}\)	\(=\)	\( ( 1579 \beta_{11} - 1579 \beta_{10} + 591 \beta_{9} + 2393 \beta_{8} + 1388 \beta_{7} - 591 \beta_{6} + \cdots + 1073 ) / 3 \) (1579b11 - 1579b10 + 591b9 + 2393b8 + 1388b7 - 591b6 + 849b5 + 191b4 - 3466b3 - 1440b2 + 1073*b1 + 1073) / 3
\(\nu^{8}\)	\(=\)	\( ( 2224 \beta_{11} - 5104 \beta_{10} - 2224 \beta_{9} + 1034 \beta_{7} - 5104 \beta_{6} - 1034 \beta_{5} + \cdots + 11361 ) / 3 \) (2224b11 - 5104b10 - 2224b9 + 1034b7 - 5104b6 - 1034b5 - 3373b4 - 5768b3 - 3373b2 - 5768b1 + 11361) / 3
\(\nu^{9}\)	\(=\)	\( - 2481 \beta_{11} - 2481 \beta_{10} - 6069 \beta_{9} - 9855 \beta_{8} - 3624 \beta_{7} - 6069 \beta_{6} + \cdots + 13889 \) -2481b11 - 2481b10 - 6069b9 - 9855b8 - 3624b7 - 6069b6 - 5361b5 - 6105b4 + 4034b3 + 708b2 - 13889*b1 + 13889
\(\nu^{10}\)	\(=\)	\( ( - 62050 \beta_{11} + 25420 \beta_{10} - 62050 \beta_{9} - 144778 \beta_{8} - 66620 \beta_{7} + \cdots + 72389 ) / 3 \) (-62050b11 + 25420b10 - 62050b9 - 144778b8 - 66620b7 - 25420b6 - 66620b5 - 50645b4 + 132702b3 + 50645b2 - 132702*b1 + 72389) / 3
\(\nu^{11}\)	\(=\)	\( ( - 215513 \beta_{11} + 215513 \beta_{10} - 91845 \beta_{9} - 361723 \beta_{8} - 190504 \beta_{7} + \cdots - 140359 ) / 3 \) (-215513b11 + 215513b10 - 91845b9 - 361723b8 - 190504b7 + 91845b6 - 134439b5 - 25009b4 + 502082b3 + 226284b2 - 140359*b1 - 140359) / 3

\(n\)	\(491\)	\(1081\)	\(1177\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{8}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$3$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$5$	\( (T^{6} + 20 T^{3} + 125)^{2} \) (T^6 + 20*T^3 + 125)^2
$7$	\( T^{12} \) T^12
$11$	\( (T^{6} + 3 T^{5} + \cdots + 2401)^{2} \) (T^6 + 3T^5 + 36T^4 + 17T^3 + 876T^2 + 1323*T + 2401)^2
$13$	\( (T^{6} + 33 T^{4} + \cdots + 576)^{2} \) (T^6 + 33T^4 + 288T^2 + 576)^2
$17$	\( T^{12} + \cdots + 5473632256 \) T^12 - 132T^10 + 11856T^8 - 587008T^6 + 21236736T^4 - 411942912*T^2 + 5473632256
$19$	\( (T^{6} + 3 T^{5} + \cdots + 576)^{2} \) (T^6 + 3T^5 + 21T^4 + 12T^3 + 216T^2 + 288*T + 576)^2
$23$	\( T^{12} - 57 T^{10} + \cdots + 4096 \) T^12 - 57T^10 + 2961T^8 - 16288T^6 + 79296T^4 - 18432*T^2 + 4096
$29$	\( (T^{3} + 12 T^{2} + \cdots + 24)^{4} \) (T^3 + 12T^2 + 33T + 24)^4
$31$	\( (T^{6} + 15 T^{4} + \cdots + 100)^{2} \) (T^6 + 15T^4 + 20T^3 + 225T^2 + 150T + 100)^2
$37$	\( T^{12} - 57 T^{10} + \cdots + 4096 \) T^12 - 57T^10 + 2961T^8 - 16288T^6 + 79296T^4 - 18432*T^2 + 4096
$41$	\( (T^{3} - 9 T^{2} + \cdots + 128)^{4} \) (T^3 - 9T^2 - 48T + 128)^4
$43$	\( (T^{2} + 4)^{6} \) (T^2 + 4)^6
$47$	\( T^{12} + \cdots + 26639462656 \) T^12 - 273T^10 + 54681T^8 - 5092072T^6 + 349385136T^4 - 3239511168*T^2 + 26639462656
$53$	\( T^{12} + \cdots + 436880018961 \) T^12 - 267T^10 + 48006T^8 - 4894623T^6 + 365619366T^4 - 15389341227*T^2 + 436880018961
$59$	\( (T^{6} - 12 T^{5} + \cdots + 190096)^{2} \) (T^6 - 12T^5 + 171T^4 - 548T^3 + 5961T^2 - 11772*T + 190096)^2
$61$	\( (T^{6} - 6 T^{5} + \cdots + 506944)^{2} \) (T^6 - 6T^5 + 144T^4 - 776T^3 + 15936T^2 - 76896*T + 506944)^2
$67$	\( T^{12} + \cdots + 23612624896 \) T^12 - 252T^10 + 47136T^8 - 3817408T^6 + 229188096T^4 - 2515172352*T^2 + 23612624896
$71$	\( (T - 6)^{12} \) (T - 6)^12
$73$	\( (T^{4} - 16 T^{2} + 256)^{3} \) (T^4 - 16*T^2 + 256)^3
$79$	\( (T^{6} - 24 T^{5} + \cdots + 161604)^{2} \) (T^6 - 24T^5 + 399T^4 - 3444T^3 + 21681T^2 - 71154*T + 161604)^2
$83$	\( (T^{6} + 198 T^{4} + \cdots + 7396)^{2} \) (T^6 + 198T^4 + 2793T^2 + 7396)^2
$89$	\( (T^{6} - 6 T^{5} + \cdots + 153664)^{2} \) (T^6 - 6T^5 + 144T^4 - 136T^3 + 14016T^2 - 42336*T + 153664)^2
$97$	\( (T^{6} + 342 T^{4} + \cdots + 12544)^{2} \) (T^6 + 342T^4 + 8313T^2 + 12544)^2
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