LMFDB - Newform orbit 2667.1.ej.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2667,1,Mod(44,2667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2667, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 42, 86]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2667.44");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2667.ej (of order $126$, degree $36$, 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:	$1.33100638869$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{126}^{20} q^{3} + \zeta_{126}^{24} q^{4} + \zeta_{126}^{56} q^{7} + \zeta_{126}^{40} q^{9}+O(q^{10}) $ q + z^20 * q^3 + z^24 * q^4 + z^56 * q^7 + z^40 * q^9	\( q + \zeta_{126}^{20} q^{3} + \zeta_{126}^{24} q^{4} + \zeta_{126}^{56} q^{7} + \zeta_{126}^{40} q^{9} + \zeta_{126}^{44} q^{12} + (\zeta_{126}^{54} - \zeta_{126}^{31}) q^{13} + \zeta_{126}^{48} q^{16} + (\zeta_{126}^{58} - \zeta_{126}^{5}) q^{19} - \zeta_{126}^{13} q^{21} - \zeta_{126}^{57} q^{25} + \zeta_{126}^{60} q^{27} - \zeta_{126}^{17} q^{28} + (\zeta_{126}^{46} + \zeta_{126}^{30}) q^{31} - \zeta_{126} q^{36} + ( - \zeta_{126}^{59} + \zeta_{126}^{18}) q^{37} + ( - \zeta_{126}^{51} - \zeta_{126}^{11}) q^{39} + (\zeta_{126}^{46} - \zeta_{126}^{43}) q^{43} - \zeta_{126}^{5} q^{48} - \zeta_{126}^{49} q^{49} + ( - \zeta_{126}^{55} - \zeta_{126}^{15}) q^{52} + ( - \zeta_{126}^{25} - \zeta_{126}^{15}) q^{57} + (\zeta_{126}^{10} + \zeta_{126}^{2}) q^{61} - \zeta_{126}^{33} q^{63} - \zeta_{126}^{9} q^{64} + ( - \zeta_{126}^{13} + \zeta_{126}^{6}) q^{67} + (\zeta_{126}^{32} + \zeta_{126}^{28}) q^{73} + \zeta_{126}^{14} q^{75} + ( - \zeta_{126}^{29} - \zeta_{126}^{19}) q^{76} + (\zeta_{126}^{40} + \zeta_{126}^{12}) q^{79} - \zeta_{126}^{17} q^{81} - \zeta_{126}^{37} q^{84} + ( - \zeta_{126}^{47} + \zeta_{126}^{24}) q^{91} + (\zeta_{126}^{50} - \zeta_{126}^{3}) q^{93} + (\zeta_{126}^{14} + \zeta_{126}^{2}) q^{97} +O(q^{100}) \) q + z^20 * q^3 + z^24 * q^4 + z^56 * q^7 + z^40 * q^9 + z^44 * q^12 + (z^54 - z^31) * q^13 + z^48 * q^16 + (z^58 - z^5) * q^19 - z^13 * q^21 - z^57 * q^25 + z^60 * q^27 - z^17 * q^28 + (z^46 + z^30) * q^31 - z * q^36 + (-z^59 + z^18) * q^37 + (-z^51 - z^11) * q^39 + (z^46 - z^43) * q^43 - z^5 * q^48 - z^49 * q^49 + (-z^55 - z^15) * q^52 + (-z^25 - z^15) * q^57 + (z^10 + z^2) * q^61 - z^33 * q^63 - z^9 * q^64 + (-z^13 + z^6) * q^67 + (z^32 + z^28) * q^73 + z^14 * q^75 + (-z^29 - z^19) * q^76 + (z^40 + z^12) * q^79 - z^17 * q^81 - z^37 * q^84 + (-z^47 + z^24) * q^91 + (z^50 - z^3) * q^93 + (z^14 + z^2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{4}+O(q^{10}) $ 36 * q + 3 * q^4	$ 36 q + 3 q^{4} - 6 q^{13} + 3 q^{16} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} + 3 q^{67} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) $ 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 + 3 * q^25 + 3 * q^27 + 3 * q^31 - 6 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 + 3 * q^67 + 3 * q^79 + 3 * q^91 + 3 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2667\mathbb{Z}\right)^\times$.

$n$	$890$	$1144$	$2416$
$\chi(n)$	$-1$	$-\zeta_{126}^{21}$	$-\zeta_{126}^{19}$

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} $

44.1

0.456211	−	0.889872i
−0.583744	−	0.811938i
−0.998757	−	0.0498459i
0.270840	+	0.962624i
−0.318487	−	0.947927i
0.878222	−	0.478254i
−0.661686	−	0.749781i
−0.583744	+	0.811938i
−0.124344	+	0.992239i
0.542546	−	0.840026i
−0.0249307	−	0.999689i
0.921476	+	0.388435i
0.698237	−	0.715867i
0.995031	+	0.0995678i
0.456211	+	0.889872i
−0.853291	−	0.521435i
0.980172	−	0.198146i
−0.124344	−	0.992239i
−0.797133	+	0.603804i
−0.661686	+	0.749781i

−0.998757

−

0.0498459i

0.365341

−

0.930874i

0.173648

0.984808i

0.995031

0.0995678i

158.1

0.995031

0.0995678i

−0.733052

−

0.680173i

−0.939693

0.342020i

0.980172

0.198146i

242.1

0.542546

0.840026i

0.365341

0.930874i

−0.939693

0.342020i

−0.411287

0.911506i

275.1

0.698237

0.715867i

0.955573

−

0.294755i

−0.939693

−

0.342020i

−0.0249307

0.999689i

284.1

0.980172

−

0.198146i

0.0747301

−

0.997204i

0.766044

0.642788i

0.921476

−

0.388435i

296.1

−0.853291

0.521435i

0.826239

0.563320i

−0.939693

−

0.342020i

0.456211

−

0.889872i

338.1

−0.318487

−

0.947927i

0.0747301

0.997204i

−0.939693

−

0.342020i

−0.797133

0.603804i

422.1

0.995031

−

0.0995678i

−0.733052

0.680173i

−0.939693

−

0.342020i

0.980172

−

0.198146i

452.1

−0.797133

0.603804i

−0.988831

0.149042i

0.766044

0.642788i

0.270840

−

0.962624i

494.1

0.456211

−

0.889872i

0.365341

0.930874i

0.766044

0.642788i

−0.583744

−

0.811938i

578.1

0.878222

−

0.478254i

0.826239

−

0.563320i

0.173648

−

0.984808i

0.542546

−

0.840026i

590.1

−0.124344

0.992239i

−0.988831

−

0.149042i

−0.939693

−

0.342020i

−0.969077

−

0.246757i

653.1

−0.969077

0.246757i

0.955573

−

0.294755i

0.766044

−

0.642788i

0.878222

−

0.478254i

779.1

−0.411287

0.911506i

−0.733052

0.680173i

0.766044

−

0.642788i

−0.661686

−

0.749781i

788.1

−0.998757

0.0498459i

0.365341

0.930874i

0.173648

−

0.984808i

0.995031

−

0.0995678i

1052.1

−0.0249307

−

0.999689i

0.826239

0.563320i

0.766044

−

0.642788i

−0.998757

0.0498459i

1241.1

−0.661686

0.749781i

0.0747301

0.997204i

0.173648

0.984808i

−0.124344

−

0.992239i

1304.1

−0.797133

−

0.603804i

−0.988831

−

0.149042i

0.766044

−

0.642788i

0.270840

0.962624i

1535.1

0.921476

−

0.388435i

−0.988831

−

0.149042i

0.173648

0.984808i

0.698237

−

0.715867i

1586.1

−0.318487

0.947927i

0.0747301

−

0.997204i

−0.939693

0.342020i

−0.797133

−

0.603804i

See all 36 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
889.cc	even	63	1	inner
2667.ej	odd	126	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2667.1.ej.a	✓	36
3.b	odd	2	1	CM	2667.1.ej.a	✓	36
7.c	even	3	1		2667.1.em.a	yes	36
21.h	odd	6	1		2667.1.em.a	yes	36
127.k	even	63	1		2667.1.em.a	yes	36
381.w	odd	126	1		2667.1.em.a	yes	36
889.cc	even	63	1	inner	2667.1.ej.a	✓	36
2667.ej	odd	126	1	inner	2667.1.ej.a	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2667.1.ej.a	✓	36	1.a	even	1	1	trivial
2667.1.ej.a	✓	36	3.b	odd	2	1	CM
2667.1.ej.a	✓	36	889.cc	even	63	1	inner
2667.1.ej.a	✓	36	2667.ej	odd	126	1	inner
2667.1.em.a	yes	36	7.c	even	3	1
2667.1.em.a	yes	36	21.h	odd	6	1
2667.1.em.a	yes	36	127.k	even	63	1
2667.1.em.a	yes	36	381.w	odd	126	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2667, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{36} $ T^36
$3$	$ T^{36} - T^{33} + \cdots + 1 $ T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$5$	$ T^{36} $ T^36
$7$	$ (T^{6} + T^{3} + 1)^{6} $ (T^6 + T^3 + 1)^6
$11$	$ T^{36} $ T^36
$13$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 55T^33 + 138T^32 + 333T^31 + 738T^30 + 1446T^29 + 2715T^28 + 4985T^27 + 8664T^26 + 13512T^25 + 19164T^24 + 23799T^23 + 20349T^22 - 54T^21 - 38244T^20 - 83631T^19 - 121730T^18 - 143004T^17 - 138378T^16 - 109203T^15 + 14991T^14 + 190053T^13 + 298416T^12 + 263403T^11 + 144756T^10 + 7426T^9 - 40923T^8 - 24132T^7 + 4465T^6 + 13446T^5 + 2604T^4 - 397T^3 + 417T^2 - 39*T + 1
$17$	$ T^{36} $ T^36
$19$	$ (T^{18} - 18 T^{16} + \cdots + 1)^{2} $ (T^18 - 18T^16 - T^15 + 135T^14 + 15T^13 - 546T^12 - 90T^11 + 1287T^10 + 276T^9 - 1782T^8 - 459T^7 + 1385T^6 + 405T^5 - 534T^4 - 170T^3 + 72T^2 + 24*T + 1)^2
$23$	$ T^{36} $ T^36
$29$	$ T^{36} $ T^36
$31$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 + 12T^32 - 36T^31 + 9T^30 + 69T^29 + 168T^28 - 631T^27 + 429T^26 - 258T^25 + 3126T^24 - 4767T^23 - 3915T^22 + 11151T^21 + 5010T^20 - 11910T^19 - 11894T^18 + 32487T^17 - 9660T^16 + 30927T^15 - 26445T^14 - 20349T^13 + 9759T^12 - 40140T^11 + 137385T^10 - 154817T^9 + 99720T^8 - 48648T^7 + 10864T^6 - 351T^5 - 1257T^4 + 1628T^3 + 444T^2 + 15T + 1
$37$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 367T^30 + 711T^29 + 1434T^28 + 2724T^27 + 4443T^26 + 5196T^25 + 4912T^24 + 6894T^23 + 19845T^22 + 57241T^21 + 128979T^20 + 212070T^19 + 261114T^18 + 239910T^17 + 155685T^16 + 81463T^15 + 95211T^14 + 128250T^13 + 119125T^12 + 50904T^11 - 35025T^10 - 46607T^9 + 7104T^8 + 14991T^7 + 5249T^6 + 1665T^5 + 4725T^4 - 2350T^3 + 207T^2 + 24*T + 1
$41$	$ T^{36} $ T^36
$43$	$ T^{36} + T^{33} + \cdots + 1 $ T^36 + T^33 + 21T^30 + 314T^27 + 1014T^24 + 14028T^21 + 117454T^18 + 349356T^15 + 277815T^12 - 372737T^9 + 90622T^6 - 223T^3 + 1
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} $ T^36
$61$	$ T^{36} - 3 T^{34} + \cdots + 1 $ T^36 - 3T^34 - 2T^33 + 30T^31 + 30T^30 - 63T^29 - 141T^28 + 57T^27 + 504T^26 + 558T^25 - 198T^24 - 2583T^23 + 4176T^22 + 3808T^21 - 7758T^20 + 3528T^19 + 29533T^18 - 36513T^17 - 29694T^16 + 38189T^15 - 56241T^14 - 969T^13 + 138471T^12 - 78903T^11 + 31950T^10 + 10052T^9 - 60111T^8 + 6999T^7 + 12175T^6 - 5292T^5 + 3834T^4 + 1908T^3 + 252T^2 - 24*T + 1
$67$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 18T^31 + 27T^30 - 150T^29 + 432T^28 - 559T^27 + 150T^26 - 3T^25 + 1808T^24 - 3483T^23 + 9042T^22 - 19674T^21 + 27051T^20 - 19569T^19 + 14410T^18 - 52593T^17 + 78654T^16 - 145467T^15 + 256233T^14 - 304551T^13 + 131840T^12 + 106878T^11 + 40980T^10 + 8891T^9 - 171T^8 - 1236T^7 - 540T^6 - 207T^5 - 45T^4 + T^3 + 6T^2 + 3T + 1
$71$	$ T^{36} $ T^36
$73$	$ T^{36} - 3 T^{34} + \cdots + 1 $ T^36 - 3T^34 + 5T^33 - 12T^31 + 44T^30 - 162T^28 + 92T^27 + 126T^26 + 264T^25 + 1013T^24 - 1386T^23 + 3567T^22 - 448T^21 - 17397T^20 + 5775T^19 + 24486T^18 + 2547T^17 + 294T^16 - 55555T^15 - 18189T^14 - 1158T^13 + 128608T^12 - 15840T^11 + 11643T^10 - 59822T^9 + 12843T^8 - 16899T^7 + 30501T^6 - 7938T^5 - 4377T^4 + 1173T^3 + 441T^2 + 18T + 1
$79$	$ T^{36} - 3 T^{35} + \cdots + 1 $ T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$83$	$ T^{36} $ T^36
$89$	$ T^{36} $ T^36
$97$	$ T^{36} + 8 T^{33} + \cdots + 1 $ T^36 + 8T^33 + 35T^30 + 188T^27 + 2288T^24 - 3577T^21 - 41985T^18 - 111958T^15 + 590386T^12 - 55511T^9 + 90048T^6 - 573*T^3 + 1
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2667.ej (of order \(126\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{126}^{20} q^{3} + \zeta_{126}^{24} q^{4} + \zeta_{126}^{56} q^{7} + \zeta_{126}^{40} q^{9}+O(q^{10}) \) q + z^20 * q^3 + z^24 * q^4 + z^56 * q^7 + z^40 * q^9	\( q + \zeta_{126}^{20} q^{3} + \zeta_{126}^{24} q^{4} + \zeta_{126}^{56} q^{7} + \zeta_{126}^{40} q^{9} + \zeta_{126}^{44} q^{12} + (\zeta_{126}^{54} - \zeta_{126}^{31}) q^{13} + \zeta_{126}^{48} q^{16} + (\zeta_{126}^{58} - \zeta_{126}^{5}) q^{19} - \zeta_{126}^{13} q^{21} - \zeta_{126}^{57} q^{25} + \zeta_{126}^{60} q^{27} - \zeta_{126}^{17} q^{28} + (\zeta_{126}^{46} + \zeta_{126}^{30}) q^{31} - \zeta_{126} q^{36} + ( - \zeta_{126}^{59} + \zeta_{126}^{18}) q^{37} + ( - \zeta_{126}^{51} - \zeta_{126}^{11}) q^{39} + (\zeta_{126}^{46} - \zeta_{126}^{43}) q^{43} - \zeta_{126}^{5} q^{48} - \zeta_{126}^{49} q^{49} + ( - \zeta_{126}^{55} - \zeta_{126}^{15}) q^{52} + ( - \zeta_{126}^{25} - \zeta_{126}^{15}) q^{57} + (\zeta_{126}^{10} + \zeta_{126}^{2}) q^{61} - \zeta_{126}^{33} q^{63} - \zeta_{126}^{9} q^{64} + ( - \zeta_{126}^{13} + \zeta_{126}^{6}) q^{67} + (\zeta_{126}^{32} + \zeta_{126}^{28}) q^{73} + \zeta_{126}^{14} q^{75} + ( - \zeta_{126}^{29} - \zeta_{126}^{19}) q^{76} + (\zeta_{126}^{40} + \zeta_{126}^{12}) q^{79} - \zeta_{126}^{17} q^{81} - \zeta_{126}^{37} q^{84} + ( - \zeta_{126}^{47} + \zeta_{126}^{24}) q^{91} + (\zeta_{126}^{50} - \zeta_{126}^{3}) q^{93} + (\zeta_{126}^{14} + \zeta_{126}^{2}) q^{97} +O(q^{100}) \) q + z^20 * q^3 + z^24 * q^4 + z^56 * q^7 + z^40 * q^9 + z^44 * q^12 + (z^54 - z^31) * q^13 + z^48 * q^16 + (z^58 - z^5) * q^19 - z^13 * q^21 - z^57 * q^25 + z^60 * q^27 - z^17 * q^28 + (z^46 + z^30) * q^31 - z * q^36 + (-z^59 + z^18) * q^37 + (-z^51 - z^11) * q^39 + (z^46 - z^43) * q^43 - z^5 * q^48 - z^49 * q^49 + (-z^55 - z^15) * q^52 + (-z^25 - z^15) * q^57 + (z^10 + z^2) * q^61 - z^33 * q^63 - z^9 * q^64 + (-z^13 + z^6) * q^67 + (z^32 + z^28) * q^73 + z^14 * q^75 + (-z^29 - z^19) * q^76 + (z^40 + z^12) * q^79 - z^17 * q^81 - z^37 * q^84 + (-z^47 + z^24) * q^91 + (z^50 - z^3) * q^93 + (z^14 + z^2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{4}+O(q^{10}) \) 36 * q + 3 * q^4	\( 36 q + 3 q^{4} - 6 q^{13} + 3 q^{16} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} + 3 q^{67} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) \) 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 + 3 * q^25 + 3 * q^27 + 3 * q^31 - 6 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 + 3 * q^67 + 3 * q^79 + 3 * q^91 + 3 * q^93

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	2667.1.ej.a

Level	$2667$
Weight	$1$
Character orbit	2667.ej
Analytic conductor	$1.331$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{63}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.33100638869\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Coefficient field:	\(\Q(\zeta_{63})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 \) x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{63}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{63} - \cdots)\)

\(n\)	\(890\)	\(1144\)	\(2416\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{126}^{21}\)	\(-\zeta_{126}^{19}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 44.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
889.cc	even	63	1	inner
2667.ej	odd	126	1	inner

$p$	$F_p(T)$
$2$	\( T^{36} \) T^36
$3$	\( T^{36} - T^{33} + \cdots + 1 \) T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$5$	\( T^{36} \) T^36
$7$	\( (T^{6} + T^{3} + 1)^{6} \) (T^6 + T^3 + 1)^6
$11$	\( T^{36} \) T^36
$13$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 55T^33 + 138T^32 + 333T^31 + 738T^30 + 1446T^29 + 2715T^28 + 4985T^27 + 8664T^26 + 13512T^25 + 19164T^24 + 23799T^23 + 20349T^22 - 54T^21 - 38244T^20 - 83631T^19 - 121730T^18 - 143004T^17 - 138378T^16 - 109203T^15 + 14991T^14 + 190053T^13 + 298416T^12 + 263403T^11 + 144756T^10 + 7426T^9 - 40923T^8 - 24132T^7 + 4465T^6 + 13446T^5 + 2604T^4 - 397T^3 + 417T^2 - 39*T + 1
$17$	\( T^{36} \) T^36
$19$	\( (T^{18} - 18 T^{16} + \cdots + 1)^{2} \) (T^18 - 18T^16 - T^15 + 135T^14 + 15T^13 - 546T^12 - 90T^11 + 1287T^10 + 276T^9 - 1782T^8 - 459T^7 + 1385T^6 + 405T^5 - 534T^4 - 170T^3 + 72T^2 + 24*T + 1)^2
$23$	\( T^{36} \) T^36
$29$	\( T^{36} \) T^36
$31$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 + 12T^32 - 36T^31 + 9T^30 + 69T^29 + 168T^28 - 631T^27 + 429T^26 - 258T^25 + 3126T^24 - 4767T^23 - 3915T^22 + 11151T^21 + 5010T^20 - 11910T^19 - 11894T^18 + 32487T^17 - 9660T^16 + 30927T^15 - 26445T^14 - 20349T^13 + 9759T^12 - 40140T^11 + 137385T^10 - 154817T^9 + 99720T^8 - 48648T^7 + 10864T^6 - 351T^5 - 1257T^4 + 1628T^3 + 444T^2 + 15T + 1
$37$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 55T^33 + 117T^32 + 207T^31 + 367T^30 + 711T^29 + 1434T^28 + 2724T^27 + 4443T^26 + 5196T^25 + 4912T^24 + 6894T^23 + 19845T^22 + 57241T^21 + 128979T^20 + 212070T^19 + 261114T^18 + 239910T^17 + 155685T^16 + 81463T^15 + 95211T^14 + 128250T^13 + 119125T^12 + 50904T^11 - 35025T^10 - 46607T^9 + 7104T^8 + 14991T^7 + 5249T^6 + 1665T^5 + 4725T^4 - 2350T^3 + 207T^2 + 24*T + 1
$41$	\( T^{36} \) T^36
$43$	\( T^{36} + T^{33} + \cdots + 1 \) T^36 + T^33 + 21T^30 + 314T^27 + 1014T^24 + 14028T^21 + 117454T^18 + 349356T^15 + 277815T^12 - 372737T^9 + 90622T^6 - 223T^3 + 1
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} \) T^36
$61$	\( T^{36} - 3 T^{34} + \cdots + 1 \) T^36 - 3T^34 - 2T^33 + 30T^31 + 30T^30 - 63T^29 - 141T^28 + 57T^27 + 504T^26 + 558T^25 - 198T^24 - 2583T^23 + 4176T^22 + 3808T^21 - 7758T^20 + 3528T^19 + 29533T^18 - 36513T^17 - 29694T^16 + 38189T^15 - 56241T^14 - 969T^13 + 138471T^12 - 78903T^11 + 31950T^10 + 10052T^9 - 60111T^8 + 6999T^7 + 12175T^6 - 5292T^5 + 3834T^4 + 1908T^3 + 252T^2 - 24*T + 1
$67$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 18T^31 + 27T^30 - 150T^29 + 432T^28 - 559T^27 + 150T^26 - 3T^25 + 1808T^24 - 3483T^23 + 9042T^22 - 19674T^21 + 27051T^20 - 19569T^19 + 14410T^18 - 52593T^17 + 78654T^16 - 145467T^15 + 256233T^14 - 304551T^13 + 131840T^12 + 106878T^11 + 40980T^10 + 8891T^9 - 171T^8 - 1236T^7 - 540T^6 - 207T^5 - 45T^4 + T^3 + 6T^2 + 3T + 1
$71$	\( T^{36} \) T^36
$73$	\( T^{36} - 3 T^{34} + \cdots + 1 \) T^36 - 3T^34 + 5T^33 - 12T^31 + 44T^30 - 162T^28 + 92T^27 + 126T^26 + 264T^25 + 1013T^24 - 1386T^23 + 3567T^22 - 448T^21 - 17397T^20 + 5775T^19 + 24486T^18 + 2547T^17 + 294T^16 - 55555T^15 - 18189T^14 - 1158T^13 + 128608T^12 - 15840T^11 + 11643T^10 - 59822T^9 + 12843T^8 - 16899T^7 + 30501T^6 - 7938T^5 - 4377T^4 + 1173T^3 + 441T^2 + 18T + 1
$79$	\( T^{36} - 3 T^{35} + \cdots + 1 \) T^36 - 3T^35 + 3T^34 + T^33 - 9T^32 + 27T^31 - 54T^30 + 111T^29 - 126T^28 - 127T^27 + 744T^26 - 1497T^25 + 1943T^24 - 630T^23 + 2490T^22 - 5607T^21 - 8157T^20 + 35592T^19 - 41021T^18 + 27741T^17 + 7203T^16 + 34245T^15 - 81024T^14 - 142380T^13 + 427463T^12 - 261816T^11 + 119325T^10 - 46324T^9 + 15804T^8 - 4674T^7 + 1323T^6 - 603T^5 + 234T^4 - 80T^3 + 24T^2 - 6T + 1
$83$	\( T^{36} \) T^36
$89$	\( T^{36} \) T^36
$97$	\( T^{36} + 8 T^{33} + \cdots + 1 \) T^36 + 8T^33 + 35T^30 + 188T^27 + 2288T^24 - 3577T^21 - 41985T^18 - 111958T^15 + 590386T^12 - 55511T^9 + 90048T^6 - 573*T^3 + 1
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