LMFDB - Newform orbit 1143.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1142,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("1143.1142");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.c (of order $2$, degree $1$, 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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128128 x^{10} + 274560 x^{8} + \cdots + 3969 $ x^20 + 40x^18 + 680x^16 + 6400x^14 + 36400x^12 + 128128x^10 + 274560x^8 + 337920x^6 + 211200x^4 + 51200*x^2 + 3969
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b2 - 2) * q^4 + (b3 - 2b1) q^8	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + (\beta_{3} - 2 \beta_1) q^{8} - \beta_{8} q^{11} + \beta_{10} q^{13} + (\beta_{4} - 2 \beta_{2} + 4) q^{16} - \beta_{7} q^{17} + \beta_{11} q^{19} + ( - \beta_{15} + \beta_{11} - \beta_{9}) q^{22} - 5 q^{25} + (\beta_{12} - \beta_{7} - \beta_{6}) q^{26} + ( - \beta_{16} + \beta_{4}) q^{31} + (\beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{32} + ( - \beta_{13} - \beta_{10} + \beta_{9}) q^{34} + (\beta_{16} + \beta_{4}) q^{37} + (\beta_{14} - \beta_{12} + \cdots + \beta_{5}) q^{38}+ \cdots + 7 \beta_1 q^{98}+O(q^{100}) $ q + b1 * q^2 + (b2 - 2) * q^4 + (b3 - 2b1) q^8 - b8 * q^11 + b10 * q^13 + (b4 - 2b2 + 4) q^16 - b7 * q^17 + b11 * q^19 + (-b15 + b11 - b9) * q^22 - 5 * q^25 + (b12 - b7 - b6) * q^26 + (-b16 + b4) * q^31 + (b5 - 2b3 + 4b1) * q^32 + (-b13 - b10 + b9) * q^34 + (b16 + b4) * q^37 + (b14 - b12 + b8 + b5) * q^38 + (b17 + b3) * q^41 + (-b14 + 2b8 - b7 + b6 + b5) q^44 + (-b17 + b3) * q^47 + 7 * q^49 - 5b1 q^50 + (b15 - b13 + b11 - 2b10) q^52 + (-2b15 + b11) q^61 + (-b17 + 2b14 + b5 - 2b3) * q^62 + (b15 - 2b11 - 2b4 + 4b2 - 8) q^64 + (-b12 - 2b8 + 4b7 + b6) * q^68 + (-b7 - 2b6) q^71 + (-b18 + b2) * q^73 + (b17 - 2b14 + b5 - 2b3) * q^74 + (b16 - 2b11 + 2b10 + b9 - 2b4) q^76 + (b18 + b2) * q^79 + (b18 - 2b16 + b4 - 2b2) * q^82 + (-b16 + 2b15 - b13 - 2b11 + 2b9 - 2b4) * q^88 + (-b18 + 2b16 + b4 - 2b2) * q^94 + 7b1 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4}+O(q^{10}) $ 20 * q - 40 * q^4	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 140 q^{49} - 160 q^{64}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 140 * q^49 - 160 * q^64

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128128 x^{10} + 274560 x^{8} + \cdots + 3969 $ x^20 + 40*x^18 + 680*x^16 + 6400*x^14 + 36400*x^12 + 128128*x^10 + 274560*x^8 + 337920*x^6 + 211200*x^4 + 51200*x^2 + 3969 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ \nu^{3} + 6\nu $ v^3 + 6*v
$\beta_{4}$	$=$	$ \nu^{4} + 8\nu^{2} + 8 $ v^4 + 8*v^2 + 8
$\beta_{5}$	$=$	$ \nu^{5} + 10\nu^{3} + 20\nu $ v^5 + 10v^3 + 20v
$\beta_{6}$	$=$	$ ( \nu^{11} + 24\nu^{9} + 212\nu^{7} + 832\nu^{5} + 1360\nu^{3} + 640\nu ) / 21 $ (v^11 + 24v^9 + 212v^7 + 832v^5 + 1360v^3 + 640*v) / 21
$\beta_{7}$	$=$	$ ( 5\nu^{11} + 99\nu^{9} + 682\nu^{7} + 1892\nu^{5} + 1760\nu^{3} + 176\nu ) / 21 $ (5v^11 + 99v^9 + 682v^7 + 1892v^5 + 1760v^3 + 176v) / 21
$\beta_{8}$	$=$	$ ( \nu^{13} + 26\nu^{11} + 260\nu^{9} + 1235\nu^{7} + 2730\nu^{5} + 2184\nu^{3} + 104\nu ) / 21 $ (v^13 + 26v^11 + 260v^9 + 1235v^7 + 2730v^5 + 2184v^3 + 104v) / 21
$\beta_{9}$	$=$	$ ( -2\nu^{12} - 48\nu^{10} - 445\nu^{8} - 2000\nu^{6} - 4400\nu^{4} - 3968\nu^{2} - 672 ) / 21 $ (-2v^12 - 48v^10 - 445v^8 - 2000v^6 - 4400v^4 - 3968v^2 - 672) / 21
$\beta_{10}$	$=$	$ ( \nu^{12} + 24\nu^{10} + 219\nu^{8} + 944\nu^{6} + 1920\nu^{4} + 1536\nu^{2} + 224 ) / 7 $ (v^12 + 24v^10 + 219v^8 + 944v^6 + 1920v^4 + 1536*v^2 + 224) / 7
$\beta_{11}$	$=$	$ ( \nu^{14} + 28\nu^{12} + 308\nu^{10} + 1680\nu^{8} + 4709\nu^{6} + 6332\nu^{4} + 3316\nu^{2} + 336 ) / 21 $ (v^14 + 28v^12 + 308v^10 + 1680v^8 + 4709v^6 + 6332v^4 + 3316v^2 + 336) / 21
$\beta_{12}$	$=$	$ ( \nu^{13} + 26\nu^{11} + 260\nu^{9} + 1242\nu^{7} + 2828\nu^{5} + 2576\nu^{3} + 496\nu ) / 7 $ (v^13 + 26v^11 + 260v^9 + 1242v^7 + 2828v^5 + 2576v^3 + 496v) / 7
$\beta_{13}$	$=$	$ -\nu^{10} - 20\nu^{8} - 140\nu^{6} - 400\nu^{4} - 400\nu^{2} - 64 $ -v^10 - 20v^8 - 140v^6 - 400v^4 - 400v^2 - 64
$\beta_{14}$	$=$	$ ( \nu^{15} + 30\nu^{13} + 360\nu^{11} + 2200\nu^{9} + 7200\nu^{7} + 12065\nu^{5} + 8650\nu^{3} + 1300\nu ) / 21 $ (v^15 + 30v^13 + 360v^11 + 2200v^9 + 7200v^7 + 12065v^5 + 8650v^3 + 1300*v) / 21
$\beta_{15}$	$=$	$ ( 2\nu^{14} + 56\nu^{12} + 616\nu^{10} + 3360\nu^{8} + 9439\nu^{6} + 12916\nu^{4} + 7388\nu^{2} + 1008 ) / 21 $ (2v^14 + 56v^12 + 616v^10 + 3360v^8 + 9439v^6 + 12916v^4 + 7388*v^2 + 1008) / 21
$\beta_{16}$	$=$	$ ( \nu^{16} + 32 \nu^{14} + 416 \nu^{12} + 2816 \nu^{10} + 10560 \nu^{8} + 21504 \nu^{6} + 21566 \nu^{4} + \cdots + 1008 ) / 21 $ (v^16 + 32v^14 + 416v^12 + 2816v^10 + 10560v^8 + 21504v^6 + 21566v^4 + 8688*v^2 + 1008) / 21
$\beta_{17}$	$=$	$ ( \nu^{17} + 34 \nu^{15} + 476 \nu^{13} + 3536 \nu^{11} + 14960 \nu^{9} + 35904 \nu^{7} + 45696 \nu^{5} + \cdots + 3608 \nu ) / 21 $ (v^17 + 34v^15 + 476v^13 + 3536v^11 + 14960v^9 + 35904v^7 + 45696v^5 + 25988v^3 + 3608v) / 21
$\beta_{18}$	$=$	$ ( \nu^{18} + 36 \nu^{16} + 540 \nu^{14} + 4368 \nu^{12} + 20592 \nu^{10} + 57024 \nu^{8} + 88704 \nu^{6} + \cdots + 2016 ) / 21 $ (v^18 + 36v^16 + 540v^14 + 4368v^12 + 20592v^10 + 57024v^8 + 88704v^6 + 69120v^4 + 20984v^2 + 2016) / 21
$\beta_{19}$	$=$	$ ( \nu^{19} + 38 \nu^{17} + 608 \nu^{15} + 5320 \nu^{13} + 27664 \nu^{11} + 86944 \nu^{9} + \cdots + 9253 \nu ) / 21 $ (v^19 + 38v^17 + 608v^15 + 5320v^13 + 27664v^11 + 86944v^9 + 160512v^7 + 160512v^5 + 72960v^3 + 9253*v) / 21

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ \beta_{3} - 6\beta_1 $ b3 - 6*b1
$\nu^{4}$	$=$	$ \beta_{4} - 8\beta_{2} + 24 $ b4 - 8*b2 + 24
$\nu^{5}$	$=$	$ \beta_{5} - 10\beta_{3} + 40\beta_1 $ b5 - 10b3 + 40b1
$\nu^{6}$	$=$	$ \beta_{15} - 2\beta_{11} - 12\beta_{4} + 60\beta_{2} - 160 $ b15 - 2b11 - 12b4 + 60*b2 - 160
$\nu^{7}$	$=$	$ \beta_{12} - 3\beta_{8} - 14\beta_{5} + 84\beta_{3} - 280\beta_1 $ b12 - 3b8 - 14b5 + 84b3 - 280b1
$\nu^{8}$	$=$	$ -16\beta_{15} + 32\beta_{11} - 2\beta_{10} - 3\beta_{9} + 112\beta_{4} - 448\beta_{2} + 1120 $ -16b15 + 32b11 - 2b10 - 3b9 + 112b4 - 448b2 + 1120
$\nu^{9}$	$=$	$ -18\beta_{12} + 54\beta_{8} - \beta_{7} + 5\beta_{6} + 144\beta_{5} - 672\beta_{3} + 2016\beta_1 $ -18b12 + 54b8 - b7 + 5b6 + 144b5 - 672b3 + 2016b1
$\nu^{10}$	$=$	$ 180\beta_{15} - \beta_{13} - 360\beta_{11} + 40\beta_{10} + 60\beta_{9} - 960\beta_{4} + 3360\beta_{2} - 8064 $ 180b15 - b13 - 360b11 + 40b10 + 60b9 - 960b4 + 3360b2 - 8064
$\nu^{11}$	$=$	$ 220\beta_{12} - 660\beta_{8} + 24\beta_{7} - 99\beta_{6} - 1320\beta_{5} + 5280\beta_{3} - 14784\beta_1 $ 220b12 - 660b8 + 24b7 - 99b6 - 1320b5 + 5280b3 - 14784*b1
$\nu^{12}$	$=$	$ - 1760 \beta_{15} + 24 \beta_{13} + 3520 \beta_{11} - 515 \beta_{10} - 783 \beta_{9} + 7920 \beta_{4} + \cdots + 59136 $ -1760b15 + 24b13 + 3520b11 - 515b10 - 783b9 + 7920b4 - 25344*b2 + 59136
$\nu^{13}$	$=$	$ -2275\beta_{12} + 6846\beta_{8} - 364\beta_{7} + 1274\beta_{6} + 11440\beta_{5} - 41184\beta_{3} + 109824\beta_1 $ -2275b12 + 6846b8 - 364b7 + 1274b6 + 11440b5 - 41184b3 + 109824*b1
$\nu^{14}$	$=$	$ 16011 \beta_{15} - 364 \beta_{13} - 32001 \beta_{11} + 5460 \beta_{10} + 8484 \beta_{9} - 64064 \beta_{4} + \cdots - 439296 $ 16011b15 - 364b13 - 32001b11 + 5460b10 + 8484b9 - 64064b4 + 192192*b2 - 439296
$\nu^{15}$	$=$	$ 21 \beta_{14} + 21450 \beta_{12} - 64980 \beta_{8} + 4480 \beta_{7} - 13580 \beta_{6} + \cdots - 823680 \beta_1 $ 21b14 + 21450b12 - 64980b8 + 4480b7 - 13580b6 - 96065b5 + 320320b3 - 823680b1
$\nu^{16}$	$=$	$ 21 \beta_{16} - 139616 \beta_{15} + 4480 \beta_{13} + 278560 \beta_{11} - 52000 \beta_{10} + \cdots + 3294720 $ 21b16 - 139616b15 + 4480b13 + 278560b11 - 52000b10 - 83040b9 + 512450b4 - 1464320b2 + 3294720
$\nu^{17}$	$=$	$ 21 \beta_{17} - 714 \beta_{14} - 190944 \beta_{12} + 584256 \beta_{8} - 48960 \beta_{7} + \cdots + 6223360 \beta_1 $ 21b17 - 714b14 - 190944b12 + 584256b8 - 48960b7 + 130560b6 + 791010b5 - 2489220b3 + 6223360*b1
$\nu^{18}$	$=$	$ 21 \beta_{18} - 756 \beta_{16} + 1185036 \beta_{15} - 48960 \beta_{13} - 2357220 \beta_{11} + \cdots - 24893440 $ 21b18 - 756b16 + 1185036b15 - 48960b13 - 2357220b11 + 463488b10 + 763776b9 - 4071240b4 + 11201800*b2 - 24893440
$\nu^{19}$	$=$	$ 21 \beta_{19} - 798 \beta_{17} + 14364 \beta_{14} + 1635672 \beta_{12} - 5069808 \beta_{8} + \cdots - 47297061 \beta_1 $ 21b19 - 798b17 + 14364b14 + 1635672b12 - 5069808b8 + 496128b7 - 1178304b6 - 6428460b5 + 19344280b3 - 47297061b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$-1$

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} $

1142.1

	−	2.80100i
	−	2.78534i
	−	2.54248i
	−	2.49702i
	−	2.03509i
	−	1.96429i
	−	1.32848i
	−	1.23927i
	−	0.491841i
	−	0.392948i
		0.392948i
		0.491841i
		1.23927i
		1.32848i
		1.96429i
		2.03509i
		2.49702i
		2.54248i
		2.78534i
		2.80100i

−

2.80100i

−5.84559

10.7715i

1142.2

−

2.78534i

−5.75809

10.4675i

1142.3

−

2.54248i

−4.46421

6.26519i

1142.4

−

2.49702i

−4.23513

5.58117i

1142.5

−

2.03509i

−2.14158

0.288121i

1142.6

−

1.96429i

−1.85842

−

0.278097i

1142.7

−

1.32848i

0.235130

−

2.96933i

1142.8

−

1.23927i

0.464206

−

3.05382i

1142.9

−

0.491841i

1.75809

−

1.84838i

1142.10

−

0.392948i

1.84559

−

1.51112i

1142.11

0.392948i

1.84559

1.51112i

1142.12

0.491841i

1.75809

1.84838i

1142.13

1.23927i

0.464206

3.05382i

1142.14

1.32848i

0.235130

2.96933i

1142.15

1.96429i

−1.85842

0.278097i

1142.16

2.03509i

−2.14158

−

0.288121i

1142.17

2.49702i

−4.23513

−

5.58117i

1142.18

2.54248i

−4.46421

−

6.26519i

1142.19

2.78534i

−5.75809

−

10.4675i

1142.20

2.80100i

−5.84559

−

10.7715i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
127.b	odd	2	1	CM by $\Q(\sqrt{-127}) $
3.b	odd	2	1	inner
381.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1143.2.c.b	✓	20
3.b	odd	2	1	inner	1143.2.c.b	✓	20
127.b	odd	2	1	CM	1143.2.c.b	✓	20
381.c	even	2	1	inner	1143.2.c.b	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1143.2.c.b	✓	20	1.a	even	1	1	trivial
1143.2.c.b	✓	20	3.b	odd	2	1	inner
1143.2.c.b	✓	20	127.b	odd	2	1	CM
1143.2.c.b	✓	20	381.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} + 40 T_{2}^{18} + 680 T_{2}^{16} + 6400 T_{2}^{14} + 36400 T_{2}^{12} + 128128 T_{2}^{10} + \cdots + 3969 $ T2^20 + 40*T2^18 + 680*T2^16 + 6400*T2^14 + 36400*T2^12 + 128128*T2^10 + 274560*T2^8 + 337920*T2^6 + 211200*T2^4 + 51200*T2^2 + 3969 acting on $S_{2}^{\mathrm{new}}(1143, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} + 40 T^{18} + \cdots + 3969 $ T^20 + 40T^18 + 680T^16 + 6400T^14 + 36400T^12 + 128128T^10 + 274560T^8 + 337920T^6 + 211200T^4 + 51200*T^2 + 3969
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + \cdots + 90118839204 $ T^20 + 220T^18 + 20570T^16 + 1064800T^14 + 33308275T^12 + 644848204T^10 + 7599996690T^8 + 51446131440T^6 + 176846076825T^4 + 235794769100*T^2 + 90118839204
$13$	$ (T^{5} - 65 T^{3} + \cdots - 62)^{4} $ (T^5 - 65T^3 + 845T - 62)^4
$17$	$ T^{20} + \cdots + 7048037374596 $ T^20 + 340T^18 + 49130T^16 + 3930400T^14 + 190010275T^12 + 5685107428T^10 + 103550171010T^8 + 1083294096720T^6 + 5754999888825T^4 + 11858787649700*T^2 + 7048037374596
$19$	$ (T^{10} - 190 T^{8} + \cdots - 8193532)^{2} $ (T^10 - 190T^8 + 12635T^6 - 342950T^4 + 3258025T^2 - 8193532)^2
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ (T^{5} - 155 T^{3} + \cdots - 7352)^{4} $ (T^5 - 155T^3 + 4805T - 7352)^4
$37$	$ (T^{5} - 185 T^{3} + \cdots - 8246)^{4} $ (T^5 - 185T^3 + 6845T - 8246)^4
$41$	$ T^{20} + \cdots + 12\!\cdots\!36 $ T^20 + 820T^18 + 285770T^16 + 55136800T^14 + 6428606275T^12 + 463888228804T^10 + 20377947193890T^8 + 514151283045840T^6 + 6587563314024825T^4 + 32738193439396100*T^2 + 12815916773248836
$43$	$ T^{20} $ T^20
$47$	$ T^{20} + \cdots + 31\!\cdots\!84 $ T^20 + 940T^18 + 375530T^16 + 83058400T^14 + 11101274275T^12 + 918297408028T^10 + 46242833761410T^8 + 1337485038022320T^6 + 19644311495952825T^4 + 111913047310276700*T^2 + 3186250035946884
$53$	$ T^{20} $ T^20
$59$	$ T^{20} $ T^20
$61$	$ (T^{10} - 610 T^{8} + \cdots - 2445327088)^{2} $ (T^10 - 610T^8 + 130235T^6 - 11349050T^4 + 346146025T^2 - 2445327088)^2
$67$	$ T^{20} $ T^20
$71$	$ T^{20} + \cdots + 18\!\cdots\!96 $ T^20 + 1420T^18 + 856970T^16 + 286328800T^14 + 57811574275T^12 + 7224134321404T^10 + 549550218021090T^8 + 24011117218152240T^6 + 532746663277752825T^4 + 4584850071844903100*T^2 + 1881141409115091396
$73$	$ (T^{10} - 730 T^{8} + \cdots - 3114332608)^{2} $ (T^10 - 730T^8 + 186515T^6 - 19450850T^4 + 709956025T^2 - 3114332608)^2
$79$	$ (T^{10} - 790 T^{8} + \cdots - 373100092)^{2} $ (T^10 - 790T^8 + 218435T^6 - 24651950T^4 + 973752025T^2 - 373100092)^2
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} $ T^20
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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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b2 - 2) * q^4 + (b3 - 2b1) q^8	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + (\beta_{3} - 2 \beta_1) q^{8} - \beta_{8} q^{11} + \beta_{10} q^{13} + (\beta_{4} - 2 \beta_{2} + 4) q^{16} - \beta_{7} q^{17} + \beta_{11} q^{19} + ( - \beta_{15} + \beta_{11} - \beta_{9}) q^{22} - 5 q^{25} + (\beta_{12} - \beta_{7} - \beta_{6}) q^{26} + ( - \beta_{16} + \beta_{4}) q^{31} + (\beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{32} + ( - \beta_{13} - \beta_{10} + \beta_{9}) q^{34} + (\beta_{16} + \beta_{4}) q^{37} + (\beta_{14} - \beta_{12} + \cdots + \beta_{5}) q^{38}+ \cdots + 7 \beta_1 q^{98}+O(q^{100}) \) q + b1 * q^2 + (b2 - 2) * q^4 + (b3 - 2b1) q^8 - b8 * q^11 + b10 * q^13 + (b4 - 2b2 + 4) q^16 - b7 * q^17 + b11 * q^19 + (-b15 + b11 - b9) * q^22 - 5 * q^25 + (b12 - b7 - b6) * q^26 + (-b16 + b4) * q^31 + (b5 - 2b3 + 4b1) * q^32 + (-b13 - b10 + b9) * q^34 + (b16 + b4) * q^37 + (b14 - b12 + b8 + b5) * q^38 + (b17 + b3) * q^41 + (-b14 + 2b8 - b7 + b6 + b5) q^44 + (-b17 + b3) * q^47 + 7 * q^49 - 5b1 q^50 + (b15 - b13 + b11 - 2b10) q^52 + (-2b15 + b11) q^61 + (-b17 + 2b14 + b5 - 2b3) * q^62 + (b15 - 2b11 - 2b4 + 4b2 - 8) q^64 + (-b12 - 2b8 + 4b7 + b6) * q^68 + (-b7 - 2b6) q^71 + (-b18 + b2) * q^73 + (b17 - 2b14 + b5 - 2b3) * q^74 + (b16 - 2b11 + 2b10 + b9 - 2b4) q^76 + (b18 + b2) * q^79 + (b18 - 2b16 + b4 - 2b2) * q^82 + (-b16 + 2b15 - b13 - 2b11 + 2b9 - 2b4) * q^88 + (-b18 + 2b16 + b4 - 2b2) * q^94 + 7b1 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 20 * q - 40 * q^4	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 140 q^{49} - 160 q^{64}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 140 * q^49 - 160 * q^64

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