LMFDB - Newform orbit 324.2.h.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,2,Mod(107,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.33418400425706520576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8*x^14 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*x^4 - 8*x^2 + 1 :

$\beta_{1}$	$=$	$ ( -3\nu^{14} + 104\nu^{12} - 768\nu^{10} + 4072\nu^{8} - 7808\nu^{6} + 10848\nu^{4} - 5011\nu^{2} + 1944 ) / 528 $ (-3v^14 + 104v^12 - 768v^10 + 4072v^8 - 7808v^6 + 10848v^4 - 5011*v^2 + 1944) / 528
$\beta_{2}$	$=$	$ ( -13\nu^{15} + 227\nu^{13} - 1568\nu^{11} + 6968\nu^{9} - 12392\nu^{7} + 16208\nu^{5} - 6637\nu^{3} + 2451\nu ) / 528 $ (-13v^15 + 227v^13 - 1568v^11 + 6968v^9 - 12392v^7 + 16208v^5 - 6637v^3 + 2451v) / 528
$\beta_{3}$	$=$	$ ( 140\nu^{15} - 1051\nu^{13} + 6272\nu^{11} - 10976\nu^{9} + 14104\nu^{7} - 3584\nu^{5} + 588\nu^{3} - 1323\nu ) / 1056 $ (140v^15 - 1051v^13 + 6272v^11 - 10976v^9 + 14104v^7 - 3584v^5 + 588v^3 - 1323v) / 1056
$\beta_{4}$	$=$	$ ( -5\nu^{14} + 48\nu^{12} - 304\nu^{10} + 872\nu^{8} - 1392\nu^{6} + 1312\nu^{4} - 453\nu^{2} + 96 ) / 48 $ (-5v^14 + 48v^12 - 304v^10 + 872v^8 - 1392v^6 + 1312v^4 - 453*v^2 + 96) / 48
$\beta_{5}$	$=$	$ ( 169\nu^{15} - 1191\nu^{13} + 7008\nu^{11} - 9800\nu^{9} + 10968\nu^{7} + 7008\nu^{5} - 7175\nu^{3} + 7209\nu ) / 1056 $ (169v^15 - 1191v^13 + 7008v^11 - 9800v^9 + 10968v^7 + 7008v^5 - 7175v^3 + 7209v) / 1056
$\beta_{6}$	$=$	$ ( - 160 \nu^{15} + 1385 \nu^{13} - 8576 \nu^{11} + 20992 \nu^{9} - 31720 \nu^{7} + 24512 \nu^{5} + \cdots + 3833 \nu ) / 1056 $ (-160v^15 + 1385v^13 - 8576v^11 + 20992v^9 - 31720v^7 + 24512v^5 - 5952v^3 + 3833v) / 1056
$\beta_{7}$	$=$	$ ( 153 \nu^{15} - 1399 \nu^{13} + 8896 \nu^{11} - 24456 \nu^{9} + 42424 \nu^{7} - 42496 \nu^{5} + \cdots - 5127 \nu ) / 1056 $ (153v^15 - 1399v^13 + 8896v^11 - 24456v^9 + 42424v^7 - 42496v^5 + 22185v^3 - 5127v) / 1056
$\beta_{8}$	$=$	$ ( 196\nu^{14} - 1533\nu^{12} + 9344\nu^{10} - 18816\nu^{8} + 28616\nu^{6} - 16352\nu^{4} + 8708\nu^{2} - 365 ) / 1056 $ (196v^14 - 1533v^12 + 9344v^10 - 18816v^8 + 28616v^6 - 16352v^4 + 8708*v^2 - 365) / 1056
$\beta_{9}$	$=$	$ ( 216 \nu^{14} - 1867 \nu^{12} + 11648 \nu^{10} - 28832 \nu^{8} + 46232 \nu^{6} - 37280 \nu^{4} + \cdots - 1819 ) / 1056 $ (216v^14 - 1867v^12 + 11648v^10 - 28832v^8 + 46232v^6 - 37280v^4 + 14072*v^2 - 1819) / 1056
$\beta_{10}$	$=$	$ ( - 365 \nu^{15} + 2724 \nu^{13} - 16352 \nu^{11} + 28616 \nu^{9} - 39584 \nu^{7} + 9344 \nu^{5} + \cdots - 4732 \nu ) / 1056 $ (-365v^15 + 2724v^13 - 16352v^11 + 28616v^9 - 39584v^7 + 9344v^5 - 1533v^3 - 4732v) / 1056
$\beta_{11}$	$=$	$ ( - 307 \nu^{14} + 2576 \nu^{12} - 15936 \nu^{10} + 37304 \nu^{8} - 58528 \nu^{6} + 45312 \nu^{4} + \cdots + 2960 ) / 1056 $ (-307v^14 + 2576v^12 - 15936v^10 + 37304v^8 - 58528v^6 + 45312v^4 - 18115*v^2 + 2960) / 1056
$\beta_{12}$	$=$	$ ( 161\nu^{14} - 1295\nu^{12} + 7952\nu^{10} - 17128\nu^{8} + 26696\nu^{6} - 17744\nu^{4} + 7505\nu^{2} - 15 ) / 528 $ (161v^14 - 1295v^12 + 7952v^10 - 17128v^8 + 26696v^6 - 17744v^4 + 7505*v^2 - 15) / 528
$\beta_{13}$	$=$	$ ( -5\nu^{14} + 40\nu^{12} - 244\nu^{10} + 512\nu^{8} - 752\nu^{6} + 424\nu^{4} - 129\nu^{2} + 4 ) / 12 $ (-5v^14 + 40v^12 - 244v^10 + 512v^8 - 752v^6 + 424v^4 - 129*v^2 + 4) / 12
$\beta_{14}$	$=$	$ ( - 593 \nu^{15} + 4919 \nu^{13} - 30368 \nu^{11} + 69512 \nu^{9} - 108600 \nu^{7} + 79808 \nu^{5} + \cdots + 5479 \nu ) / 1056 $ (-593v^15 + 4919v^13 - 30368v^11 + 69512v^9 - 108600v^7 + 79808v^5 - 33537v^3 + 5479v) / 1056
$\beta_{15}$	$=$	$ ( - 721 \nu^{15} + 5642 \nu^{13} - 34272 \nu^{11} + 68424 \nu^{9} - 99920 \nu^{7} + 50208 \nu^{5} + \cdots - 1590 \nu ) / 1056 $ (-721v^15 + 5642v^13 - 34272v^11 + 68424v^9 - 99920v^7 + 50208v^5 - 16193v^3 - 1590v) / 1056

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

$\nu$	$=$	$ ( -\beta_{15} + 2\beta_{10} + \beta_{6} + \beta_{5} ) / 3 $ (-b15 + 2*b10 + b6 + b5) / 3
$\nu^{2}$	$=$	$ ( \beta_{13} - 4\beta_{12} - 2\beta_{11} + 3\beta_{9} + 3\beta_{8} + \beta_{4} + 2\beta _1 + 2 ) / 3 $ (b13 - 4b12 - 2b11 + 3b9 + 3b8 + b4 + 2*b1 + 2) / 3
$\nu^{3}$	$=$	$ -4\beta_{15} + 2\beta_{14} + 3\beta_{10} + 2\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} $ -4b15 + 2b14 + 3b10 + 2b6 - b5 - b3 - b2
$\nu^{4}$	$=$	$ ( 11\beta_{13} - 11\beta_{12} - 10\beta_{11} + 15\beta_{9} + 9\beta_{8} - 4\beta_{4} + 22\beta _1 - 20 ) / 3 $ (11b13 - 11b12 - 10b11 + 15b9 + 9b8 - 4b4 + 22*b1 - 20) / 3
$\nu^{5}$	$=$	$ ( -35\beta_{15} + 39\beta_{14} - 14\beta_{10} + 6\beta_{7} + 14\beta_{6} - 28\beta_{5} - 15\beta_{3} - 36\beta_{2} ) / 3 $ (-35b15 + 39b14 - 14b10 + 6b7 + 14b6 - 28b5 - 15b3 - 36b2) / 3
$\nu^{6}$	$=$	$ 13\beta_{13} + 20\beta_{12} + 6\beta_{11} - 8\beta_{9} - 27\beta_{4} + 20\beta _1 - 54 $ 13b13 + 20b12 + 6b11 - 8b9 - 27b4 + 20b1 - 54
$\nu^{7}$	$=$	$ ( 197\beta_{15} - 373\beta_{10} - 74\beta_{6} - 74\beta_{5} + 24\beta_{3} - 123\beta_{2} ) / 3 $ (197b15 - 373b10 - 74b6 - 74b5 + 24b3 - 123b2) / 3
$\nu^{8}$	$=$	$ ( -116\beta_{13} + 662\beta_{12} + 454\beta_{11} - 570\beta_{9} - 222\beta_{8} - 362\beta_{4} - 331\beta _1 - 331 ) / 3 $ (-116b13 + 662b12 + 454b11 - 570b9 - 222b8 - 362b4 - 331*b1 - 331) / 3
$\nu^{9}$	$=$	$ 732 \beta_{15} - 421 \beta_{14} - 556 \beta_{10} - 90 \beta_{7} - 270 \beta_{6} + 135 \beta_{5} + \cdots + 141 \beta_{2} $ 732b15 - 421b14 - 556b10 - 90b7 - 270b6 + 135b5 + 190b3 + 141b2
$\nu^{10}$	$=$	$ ( - 1834 \beta_{13} + 1834 \beta_{12} + 2030 \beta_{11} - 2331 \beta_{9} - 1215 \beta_{8} + 497 \beta_{4} + \cdots + 3049 ) / 3 $ (-1834b13 + 1834b12 + 2030b11 - 2331b9 - 1215b8 + 497b4 - 3668*b1 + 3049) / 3
$\nu^{11}$	$=$	$ ( 6103 \beta_{15} - 7035 \beta_{14} + 2239 \beta_{10} - 1533 \beta_{7} - 2239 \beta_{6} + \cdots + 6195 \beta_{2} ) / 3 $ (6103b15 - 7035b14 + 2239b10 - 1533b7 - 2239b6 + 4478b5 + 2331b3 + 6195b2) / 3
$\nu^{12}$	$=$	$ -2200\beta_{13} - 3392\beta_{12} - 912\beta_{11} + 1568\beta_{9} + 4680\beta_{4} - 3392\beta _1 + 9023 $ -2200b13 - 3392b12 - 912b11 + 1568b9 + 4680b4 - 3392b1 + 9023
$\nu^{13}$	$=$	$ ( -33895\beta_{15} + 63926\beta_{10} + 12415\beta_{6} + 12415\beta_{5} - 4704\beta_{3} + 21480\beta_{2} ) / 3 $ (-33895b15 + 63926b10 + 12415b6 + 12415b5 - 4704b3 + 21480b2) / 3
$\nu^{14}$	$=$	$ ( 19855 \beta_{13} - 112972 \beta_{12} - 77966 \beta_{11} + 97821 \beta_{9} + 37245 \beta_{8} + \cdots + 56486 ) / 3 $ (19855b13 - 112972b12 - 77966b11 + 97821b9 + 37245b8 + 62815b4 + 56486*b1 + 56486) / 3
$\nu^{15}$	$=$	$ - 125468 \beta_{15} + 72374 \beta_{14} + 95341 \beta_{10} + 15888 \beta_{7} + 45934 \beta_{6} + \cdots - 23879 \beta_{2} $ -125468b15 + 72374b14 + 95341b10 + 15888b7 + 45934b6 - 22967b5 - 32607b3 - 23879b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$1 - \beta_{8}$

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} $

107.1

−1.11871	+	0.645885i
0.367543	−	0.212201i
2.04058	−	1.17813i
0.670418	−	0.387066i
−0.670418	+	0.387066i
−2.04058	+	1.17813i
−0.367543	+	0.212201i
1.11871	−	0.645885i
−1.11871	−	0.645885i
0.367543	+	0.212201i
2.04058	+	1.17813i
0.670418	+	0.387066i
−0.670418	−	0.387066i
−2.04058	−	1.17813i
−0.367543	−	0.212201i
1.11871	+	0.645885i

−1.35299

0.411599i

1.66117

−

1.11378i

−0.448288

0.258819i

−2.53020

−

1.46081i

−1.78912

2.19067i

0.500000

−

0.534695i

107.2

−0.919308

1.07465i

−0.309746

−

1.97587i

−1.67303

0.965926i

3.40559

1.96622i

2.40812

1.48356i

0.500000

−

2.68591i

107.3

−0.471020

1.33347i

−1.55628

−

1.25618i

1.67303

−

0.965926i

−3.40559

−

1.96622i

2.40812

−

1.48356i

0.500000

2.68591i

107.4

−0.320041

−

1.37752i

−1.79515

0.881728i

−0.448288

0.258819i

2.53020

1.46081i

1.78912

2.19067i

0.500000

0.534695i

107.5

0.320041

1.37752i

−1.79515

0.881728i

0.448288

−

0.258819i

2.53020

1.46081i

−1.78912

−

2.19067i

0.500000

0.534695i

107.6

0.471020

−

1.33347i

−1.55628

−

1.25618i

−1.67303

0.965926i

−3.40559

−

1.96622i

−2.40812

1.48356i

0.500000

2.68591i

107.7

0.919308

−

1.07465i

−0.309746

−

1.97587i

1.67303

−

0.965926i

3.40559

1.96622i

−2.40812

−

1.48356i

0.500000

−

2.68591i

107.8

1.35299

−

0.411599i

1.66117

−

1.11378i

0.448288

−

0.258819i

−2.53020

−

1.46081i

1.78912

−

2.19067i

0.500000

−

0.534695i

215.1

−1.35299

−

0.411599i

1.66117

1.11378i

−0.448288

−

0.258819i

−2.53020

1.46081i

−1.78912

−

2.19067i

0.500000

0.534695i

215.2

−0.919308

−

1.07465i

−0.309746

1.97587i

−1.67303

−

0.965926i

3.40559

−

1.96622i

2.40812

−

1.48356i

0.500000

2.68591i

215.3

−0.471020

−

1.33347i

−1.55628

1.25618i

1.67303

0.965926i

−3.40559

1.96622i

2.40812

1.48356i

0.500000

−

2.68591i

215.4

−0.320041

1.37752i

−1.79515

−

0.881728i

−0.448288

−

0.258819i

2.53020

−

1.46081i

1.78912

−

2.19067i

0.500000

−

0.534695i

215.5

0.320041

−

1.37752i

−1.79515

−

0.881728i

0.448288

0.258819i

2.53020

−

1.46081i

−1.78912

2.19067i

0.500000

−

0.534695i

215.6

0.471020

1.33347i

−1.55628

1.25618i

−1.67303

−

0.965926i

−3.40559

1.96622i

−2.40812

−

1.48356i

0.500000

−

2.68591i

215.7

0.919308

1.07465i

−0.309746

1.97587i

1.67303

0.965926i

3.40559

−

1.96622i

−2.40812

1.48356i

0.500000

2.68591i

215.8

1.35299

0.411599i

1.66117

1.11378i

0.448288

0.258819i

−2.53020

1.46081i

1.78912

2.19067i

0.500000

0.534695i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
12.b	even	2	1	inner
36.f	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.2.h.f		16
3.b	odd	2	1	inner	324.2.h.f		16
4.b	odd	2	1	inner	324.2.h.f		16
9.c	even	3	1		324.2.b.c	✓	8
9.c	even	3	1	inner	324.2.h.f		16
9.d	odd	6	1		324.2.b.c	✓	8
9.d	odd	6	1	inner	324.2.h.f		16
12.b	even	2	1	inner	324.2.h.f		16
36.f	odd	6	1		324.2.b.c	✓	8
36.f	odd	6	1	inner	324.2.h.f		16
36.h	even	6	1		324.2.b.c	✓	8
36.h	even	6	1	inner	324.2.h.f		16
72.j	odd	6	1		5184.2.c.k		8
72.l	even	6	1		5184.2.c.k		8
72.n	even	6	1		5184.2.c.k		8
72.p	odd	6	1		5184.2.c.k		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.2.b.c	✓	8	9.c	even	3	1
324.2.b.c	✓	8	9.d	odd	6	1
324.2.b.c	✓	8	36.f	odd	6	1
324.2.b.c	✓	8	36.h	even	6	1
324.2.h.f		16	1.a	even	1	1	trivial
324.2.h.f		16	3.b	odd	2	1	inner
324.2.h.f		16	4.b	odd	2	1	inner
324.2.h.f		16	9.c	even	3	1	inner
324.2.h.f		16	9.d	odd	6	1	inner
324.2.h.f		16	12.b	even	2	1	inner
324.2.h.f		16	36.f	odd	6	1	inner
324.2.h.f		16	36.h	even	6	1	inner
5184.2.c.k		8	72.j	odd	6	1
5184.2.c.k		8	72.l	even	6	1
5184.2.c.k		8	72.n	even	6	1
5184.2.c.k		8	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(324, [\chi])$:

$ T_{5}^{8} - 4T_{5}^{6} + 15T_{5}^{4} - 4T_{5}^{2} + 1 $

T5^8 - 4*T5^6 + 15*T5^4 - 4*T5^2 + 1

$ T_{7}^{8} - 24T_{7}^{6} + 444T_{7}^{4} - 3168T_{7}^{2} + 17424 $

T7^8 - 24*T7^6 + 444*T7^4 - 3168*T7^2 + 17424

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 4 T^{14} + \cdots + 256 $ T^16 + 4T^14 + 7T^12 + 4T^10 + T^8 + 16T^6 + 112T^4 + 256T^2 + 256
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} $ (T^8 - 4T^6 + 15T^4 - 4*T^2 + 1)^2
$7$	$ (T^{8} - 24 T^{6} + \cdots + 17424)^{2} $ (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$11$	$ (T^{8} + 36 T^{6} + \cdots + 17424)^{2} $ (T^8 + 36T^6 + 1164T^4 + 4752*T^2 + 17424)^2
$13$	$ (T^{4} - 2 T^{3} + \cdots + 121)^{4} $ (T^4 - 2T^3 + 15T^2 + 22*T + 121)^4
$17$	$ (T^{4} + 28 T^{2} + 121)^{4} $ (T^4 + 28*T^2 + 121)^4
$19$	$ (T^{4} + 72 T^{2} + 1188)^{4} $ (T^4 + 72*T^2 + 1188)^4
$23$	$ (T^{8} + 60 T^{6} + \cdots + 17424)^{2} $ (T^8 + 60T^6 + 3468T^4 + 7920*T^2 + 17424)^2
$29$	$ (T^{8} - 28 T^{6} + \cdots + 14641)^{2} $ (T^8 - 28T^6 + 663T^4 - 3388*T^2 + 14641)^2
$31$	$ (T^{8} - 72 T^{6} + \cdots + 278784)^{2} $ (T^8 - 72T^6 + 4656T^4 - 38016*T^2 + 278784)^2
$37$	$ (T^{2} - 4 T + 1)^{8} $ (T^2 - 4*T + 1)^8
$41$	$ (T^{8} - 76 T^{6} + \cdots + 456976)^{2} $ (T^8 - 76T^6 + 5100T^4 - 51376*T^2 + 456976)^2
$43$	$ (T^{8} - 24 T^{6} + \cdots + 17424)^{2} $ (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$47$	$ (T^{8} + 48 T^{6} + \cdots + 278784)^{2} $ (T^8 + 48T^6 + 1776T^4 + 25344*T^2 + 278784)^2
$53$	$ (T^{4} + 124 T^{2} + 2116)^{4} $ (T^4 + 124*T^2 + 2116)^4
$59$	$ (T^{8} + 48 T^{6} + \cdots + 278784)^{2} $ (T^8 + 48T^6 + 1776T^4 + 25344*T^2 + 278784)^2
$61$	$ (T^{4} - 8 T^{3} + \cdots + 169)^{4} $ (T^4 - 8T^3 + 51T^2 - 104*T + 169)^4
$67$	$ (T^{8} - 24 T^{6} + \cdots + 17424)^{2} $ (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$71$	$ (T^{4} - 108 T^{2} + 1188)^{4} $ (T^4 - 108*T^2 + 1188)^4
$73$	$ (T^{2} + 8 T - 59)^{8} $ (T^2 + 8*T - 59)^8
$79$	$ (T^{8} - 120 T^{6} + \cdots + 17424)^{2} $ (T^8 - 120T^6 + 14268T^4 - 15840*T^2 + 17424)^2
$83$	$ (T^{8} + 144 T^{6} + \cdots + 4460544)^{2} $ (T^8 + 144T^6 + 18624T^4 + 304128*T^2 + 4460544)^2
$89$	$ (T^{4} + 196 T^{2} + 2401)^{4} $ (T^4 + 196*T^2 + 2401)^4
$97$	$ (T^{4} - 8 T^{3} + 60 T^{2} + \cdots + 16)^{4} $ (T^4 - 8T^3 + 60T^2 - 32*T + 16)^4
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.h (of order \(6\), degree \(2\), not minimal)

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

Introduction

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Modular forms

Varieties

Fields

Representations

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Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	324.2.h.f

Level	$324$
Weight	$2$
Character orbit	324.h
Analytic conductor	$2.587$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.33418400425706520576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( -3\nu^{14} + 104\nu^{12} - 768\nu^{10} + 4072\nu^{8} - 7808\nu^{6} + 10848\nu^{4} - 5011\nu^{2} + 1944 ) / 528 \) (-3v^14 + 104v^12 - 768v^10 + 4072v^8 - 7808v^6 + 10848v^4 - 5011*v^2 + 1944) / 528
\(\beta_{2}\)	\(=\)	\( ( -13\nu^{15} + 227\nu^{13} - 1568\nu^{11} + 6968\nu^{9} - 12392\nu^{7} + 16208\nu^{5} - 6637\nu^{3} + 2451\nu ) / 528 \) (-13v^15 + 227v^13 - 1568v^11 + 6968v^9 - 12392v^7 + 16208v^5 - 6637v^3 + 2451v) / 528
\(\beta_{3}\)	\(=\)	\( ( 140\nu^{15} - 1051\nu^{13} + 6272\nu^{11} - 10976\nu^{9} + 14104\nu^{7} - 3584\nu^{5} + 588\nu^{3} - 1323\nu ) / 1056 \) (140v^15 - 1051v^13 + 6272v^11 - 10976v^9 + 14104v^7 - 3584v^5 + 588v^3 - 1323v) / 1056
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{14} + 48\nu^{12} - 304\nu^{10} + 872\nu^{8} - 1392\nu^{6} + 1312\nu^{4} - 453\nu^{2} + 96 ) / 48 \) (-5v^14 + 48v^12 - 304v^10 + 872v^8 - 1392v^6 + 1312v^4 - 453*v^2 + 96) / 48
\(\beta_{5}\)	\(=\)	\( ( 169\nu^{15} - 1191\nu^{13} + 7008\nu^{11} - 9800\nu^{9} + 10968\nu^{7} + 7008\nu^{5} - 7175\nu^{3} + 7209\nu ) / 1056 \) (169v^15 - 1191v^13 + 7008v^11 - 9800v^9 + 10968v^7 + 7008v^5 - 7175v^3 + 7209v) / 1056
\(\beta_{6}\)	\(=\)	\( ( - 160 \nu^{15} + 1385 \nu^{13} - 8576 \nu^{11} + 20992 \nu^{9} - 31720 \nu^{7} + 24512 \nu^{5} + \cdots + 3833 \nu ) / 1056 \) (-160v^15 + 1385v^13 - 8576v^11 + 20992v^9 - 31720v^7 + 24512v^5 - 5952v^3 + 3833v) / 1056
\(\beta_{7}\)	\(=\)	\( ( 153 \nu^{15} - 1399 \nu^{13} + 8896 \nu^{11} - 24456 \nu^{9} + 42424 \nu^{7} - 42496 \nu^{5} + \cdots - 5127 \nu ) / 1056 \) (153v^15 - 1399v^13 + 8896v^11 - 24456v^9 + 42424v^7 - 42496v^5 + 22185v^3 - 5127v) / 1056
\(\beta_{8}\)	\(=\)	\( ( 196\nu^{14} - 1533\nu^{12} + 9344\nu^{10} - 18816\nu^{8} + 28616\nu^{6} - 16352\nu^{4} + 8708\nu^{2} - 365 ) / 1056 \) (196v^14 - 1533v^12 + 9344v^10 - 18816v^8 + 28616v^6 - 16352v^4 + 8708*v^2 - 365) / 1056
\(\beta_{9}\)	\(=\)	\( ( 216 \nu^{14} - 1867 \nu^{12} + 11648 \nu^{10} - 28832 \nu^{8} + 46232 \nu^{6} - 37280 \nu^{4} + \cdots - 1819 ) / 1056 \) (216v^14 - 1867v^12 + 11648v^10 - 28832v^8 + 46232v^6 - 37280v^4 + 14072*v^2 - 1819) / 1056
\(\beta_{10}\)	\(=\)	\( ( - 365 \nu^{15} + 2724 \nu^{13} - 16352 \nu^{11} + 28616 \nu^{9} - 39584 \nu^{7} + 9344 \nu^{5} + \cdots - 4732 \nu ) / 1056 \) (-365v^15 + 2724v^13 - 16352v^11 + 28616v^9 - 39584v^7 + 9344v^5 - 1533v^3 - 4732v) / 1056
\(\beta_{11}\)	\(=\)	\( ( - 307 \nu^{14} + 2576 \nu^{12} - 15936 \nu^{10} + 37304 \nu^{8} - 58528 \nu^{6} + 45312 \nu^{4} + \cdots + 2960 ) / 1056 \) (-307v^14 + 2576v^12 - 15936v^10 + 37304v^8 - 58528v^6 + 45312v^4 - 18115*v^2 + 2960) / 1056
\(\beta_{12}\)	\(=\)	\( ( 161\nu^{14} - 1295\nu^{12} + 7952\nu^{10} - 17128\nu^{8} + 26696\nu^{6} - 17744\nu^{4} + 7505\nu^{2} - 15 ) / 528 \) (161v^14 - 1295v^12 + 7952v^10 - 17128v^8 + 26696v^6 - 17744v^4 + 7505*v^2 - 15) / 528
\(\beta_{13}\)	\(=\)	\( ( -5\nu^{14} + 40\nu^{12} - 244\nu^{10} + 512\nu^{8} - 752\nu^{6} + 424\nu^{4} - 129\nu^{2} + 4 ) / 12 \) (-5v^14 + 40v^12 - 244v^10 + 512v^8 - 752v^6 + 424v^4 - 129*v^2 + 4) / 12
\(\beta_{14}\)	\(=\)	\( ( - 593 \nu^{15} + 4919 \nu^{13} - 30368 \nu^{11} + 69512 \nu^{9} - 108600 \nu^{7} + 79808 \nu^{5} + \cdots + 5479 \nu ) / 1056 \) (-593v^15 + 4919v^13 - 30368v^11 + 69512v^9 - 108600v^7 + 79808v^5 - 33537v^3 + 5479v) / 1056
\(\beta_{15}\)	\(=\)	\( ( - 721 \nu^{15} + 5642 \nu^{13} - 34272 \nu^{11} + 68424 \nu^{9} - 99920 \nu^{7} + 50208 \nu^{5} + \cdots - 1590 \nu ) / 1056 \) (-721v^15 + 5642v^13 - 34272v^11 + 68424v^9 - 99920v^7 + 50208v^5 - 16193v^3 - 1590v) / 1056

\(\nu\)	\(=\)	\( ( -\beta_{15} + 2\beta_{10} + \beta_{6} + \beta_{5} ) / 3 \) (-b15 + 2*b10 + b6 + b5) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - 4\beta_{12} - 2\beta_{11} + 3\beta_{9} + 3\beta_{8} + \beta_{4} + 2\beta _1 + 2 ) / 3 \) (b13 - 4b12 - 2b11 + 3b9 + 3b8 + b4 + 2*b1 + 2) / 3
\(\nu^{3}\)	\(=\)	\( -4\beta_{15} + 2\beta_{14} + 3\beta_{10} + 2\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} \) -4b15 + 2b14 + 3b10 + 2b6 - b5 - b3 - b2
\(\nu^{4}\)	\(=\)	\( ( 11\beta_{13} - 11\beta_{12} - 10\beta_{11} + 15\beta_{9} + 9\beta_{8} - 4\beta_{4} + 22\beta _1 - 20 ) / 3 \) (11b13 - 11b12 - 10b11 + 15b9 + 9b8 - 4b4 + 22*b1 - 20) / 3
\(\nu^{5}\)	\(=\)	\( ( -35\beta_{15} + 39\beta_{14} - 14\beta_{10} + 6\beta_{7} + 14\beta_{6} - 28\beta_{5} - 15\beta_{3} - 36\beta_{2} ) / 3 \) (-35b15 + 39b14 - 14b10 + 6b7 + 14b6 - 28b5 - 15b3 - 36b2) / 3
\(\nu^{6}\)	\(=\)	\( 13\beta_{13} + 20\beta_{12} + 6\beta_{11} - 8\beta_{9} - 27\beta_{4} + 20\beta _1 - 54 \) 13b13 + 20b12 + 6b11 - 8b9 - 27b4 + 20b1 - 54
\(\nu^{7}\)	\(=\)	\( ( 197\beta_{15} - 373\beta_{10} - 74\beta_{6} - 74\beta_{5} + 24\beta_{3} - 123\beta_{2} ) / 3 \) (197b15 - 373b10 - 74b6 - 74b5 + 24b3 - 123b2) / 3
\(\nu^{8}\)	\(=\)	\( ( -116\beta_{13} + 662\beta_{12} + 454\beta_{11} - 570\beta_{9} - 222\beta_{8} - 362\beta_{4} - 331\beta _1 - 331 ) / 3 \) (-116b13 + 662b12 + 454b11 - 570b9 - 222b8 - 362b4 - 331*b1 - 331) / 3
\(\nu^{9}\)	\(=\)	\( 732 \beta_{15} - 421 \beta_{14} - 556 \beta_{10} - 90 \beta_{7} - 270 \beta_{6} + 135 \beta_{5} + \cdots + 141 \beta_{2} \) 732b15 - 421b14 - 556b10 - 90b7 - 270b6 + 135b5 + 190b3 + 141b2
\(\nu^{10}\)	\(=\)	\( ( - 1834 \beta_{13} + 1834 \beta_{12} + 2030 \beta_{11} - 2331 \beta_{9} - 1215 \beta_{8} + 497 \beta_{4} + \cdots + 3049 ) / 3 \) (-1834b13 + 1834b12 + 2030b11 - 2331b9 - 1215b8 + 497b4 - 3668*b1 + 3049) / 3
\(\nu^{11}\)	\(=\)	\( ( 6103 \beta_{15} - 7035 \beta_{14} + 2239 \beta_{10} - 1533 \beta_{7} - 2239 \beta_{6} + \cdots + 6195 \beta_{2} ) / 3 \) (6103b15 - 7035b14 + 2239b10 - 1533b7 - 2239b6 + 4478b5 + 2331b3 + 6195b2) / 3
\(\nu^{12}\)	\(=\)	\( -2200\beta_{13} - 3392\beta_{12} - 912\beta_{11} + 1568\beta_{9} + 4680\beta_{4} - 3392\beta _1 + 9023 \) -2200b13 - 3392b12 - 912b11 + 1568b9 + 4680b4 - 3392b1 + 9023
\(\nu^{13}\)	\(=\)	\( ( -33895\beta_{15} + 63926\beta_{10} + 12415\beta_{6} + 12415\beta_{5} - 4704\beta_{3} + 21480\beta_{2} ) / 3 \) (-33895b15 + 63926b10 + 12415b6 + 12415b5 - 4704b3 + 21480b2) / 3
\(\nu^{14}\)	\(=\)	\( ( 19855 \beta_{13} - 112972 \beta_{12} - 77966 \beta_{11} + 97821 \beta_{9} + 37245 \beta_{8} + \cdots + 56486 ) / 3 \) (19855b13 - 112972b12 - 77966b11 + 97821b9 + 37245b8 + 62815b4 + 56486*b1 + 56486) / 3
\(\nu^{15}\)	\(=\)	\( - 125468 \beta_{15} + 72374 \beta_{14} + 95341 \beta_{10} + 15888 \beta_{7} + 45934 \beta_{6} + \cdots - 23879 \beta_{2} \) -125468b15 + 72374b14 + 95341b10 + 15888b7 + 45934b6 - 22967b5 - 32607b3 - 23879b2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 4 T^{14} + \cdots + 256 \) T^16 + 4T^14 + 7T^12 + 4T^10 + T^8 + 16T^6 + 112T^4 + 256T^2 + 256
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) (T^8 - 4T^6 + 15T^4 - 4*T^2 + 1)^2
$7$	\( (T^{8} - 24 T^{6} + \cdots + 17424)^{2} \) (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$11$	\( (T^{8} + 36 T^{6} + \cdots + 17424)^{2} \) (T^8 + 36T^6 + 1164T^4 + 4752*T^2 + 17424)^2
$13$	\( (T^{4} - 2 T^{3} + \cdots + 121)^{4} \) (T^4 - 2T^3 + 15T^2 + 22*T + 121)^4
$17$	\( (T^{4} + 28 T^{2} + 121)^{4} \) (T^4 + 28*T^2 + 121)^4
$19$	\( (T^{4} + 72 T^{2} + 1188)^{4} \) (T^4 + 72*T^2 + 1188)^4
$23$	\( (T^{8} + 60 T^{6} + \cdots + 17424)^{2} \) (T^8 + 60T^6 + 3468T^4 + 7920*T^2 + 17424)^2
$29$	\( (T^{8} - 28 T^{6} + \cdots + 14641)^{2} \) (T^8 - 28T^6 + 663T^4 - 3388*T^2 + 14641)^2
$31$	\( (T^{8} - 72 T^{6} + \cdots + 278784)^{2} \) (T^8 - 72T^6 + 4656T^4 - 38016*T^2 + 278784)^2
$37$	\( (T^{2} - 4 T + 1)^{8} \) (T^2 - 4*T + 1)^8
$41$	\( (T^{8} - 76 T^{6} + \cdots + 456976)^{2} \) (T^8 - 76T^6 + 5100T^4 - 51376*T^2 + 456976)^2
$43$	\( (T^{8} - 24 T^{6} + \cdots + 17424)^{2} \) (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$47$	\( (T^{8} + 48 T^{6} + \cdots + 278784)^{2} \) (T^8 + 48T^6 + 1776T^4 + 25344*T^2 + 278784)^2
$53$	\( (T^{4} + 124 T^{2} + 2116)^{4} \) (T^4 + 124*T^2 + 2116)^4
$59$	\( (T^{8} + 48 T^{6} + \cdots + 278784)^{2} \) (T^8 + 48T^6 + 1776T^4 + 25344*T^2 + 278784)^2
$61$	\( (T^{4} - 8 T^{3} + \cdots + 169)^{4} \) (T^4 - 8T^3 + 51T^2 - 104*T + 169)^4
$67$	\( (T^{8} - 24 T^{6} + \cdots + 17424)^{2} \) (T^8 - 24T^6 + 444T^4 - 3168*T^2 + 17424)^2
$71$	\( (T^{4} - 108 T^{2} + 1188)^{4} \) (T^4 - 108*T^2 + 1188)^4
$73$	\( (T^{2} + 8 T - 59)^{8} \) (T^2 + 8*T - 59)^8
$79$	\( (T^{8} - 120 T^{6} + \cdots + 17424)^{2} \) (T^8 - 120T^6 + 14268T^4 - 15840*T^2 + 17424)^2
$83$	\( (T^{8} + 144 T^{6} + \cdots + 4460544)^{2} \) (T^8 + 144T^6 + 18624T^4 + 304128*T^2 + 4460544)^2
$89$	\( (T^{4} + 196 T^{2} + 2401)^{4} \) (T^4 + 196*T^2 + 2401)^4
$97$	\( (T^{4} - 8 T^{3} + 60 T^{2} + \cdots + 16)^{4} \) (T^4 - 8T^3 + 60T^2 - 32*T + 16)^4
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\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{8}\)