LMFDB - Newform orbit 2667.1.en.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2667,1,Mod(173,2667)]

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, 105, 67]))

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

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

chi := DirichletCharacter("2667.173");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2667.en (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:	$D_{126}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{126} - \cdots)$

$q$-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}^{58} q^{3} - \zeta_{126}^{15} q^{4} - \zeta_{126}^{61} q^{7} - \zeta_{126}^{53} q^{9} +O(q^{10}) $ q - z^58 * q^3 - z^15 * q^4 - z^61 * q^7 - z^53 * q^9	\( q - \zeta_{126}^{58} q^{3} - \zeta_{126}^{15} q^{4} - \zeta_{126}^{61} q^{7} - \zeta_{126}^{53} q^{9} - \zeta_{126}^{10} q^{12} + (\zeta_{126}^{29} + \zeta_{126}^{18}) q^{13} + \zeta_{126}^{30} q^{16} + (\zeta_{126}^{17} + \zeta_{126}^{4}) q^{19} - \zeta_{126}^{56} q^{21} + \zeta_{126}^{54} q^{25} - \zeta_{126}^{48} q^{27} - \zeta_{126}^{13} q^{28} + (\zeta_{126}^{62} + \zeta_{126}^{45}) q^{31} - \zeta_{126}^{5} q^{36} + ( - \zeta_{126}^{22} - \zeta_{126}^{6}) q^{37} + (\zeta_{126}^{24} + \zeta_{126}^{13}) q^{39} + ( - \zeta_{126}^{41} - \zeta_{126}^{26}) q^{43} + \zeta_{126}^{25} q^{48} - \zeta_{126}^{59} q^{49} + ( - \zeta_{126}^{44} - \zeta_{126}^{33}) q^{52} + ( - \zeta_{126}^{62} + \zeta_{126}^{12}) q^{57} + ( - \zeta_{126}^{52} - \zeta_{126}^{29}) q^{61} - \zeta_{126}^{51} q^{63} - \zeta_{126}^{45} q^{64} + (\zeta_{126}^{51} - \zeta_{126}^{23}) q^{67} + (\zeta_{126}^{55} - \zeta_{126}^{35}) q^{73} + \zeta_{126}^{49} q^{75} + ( - \zeta_{126}^{32} - \zeta_{126}^{19}) q^{76} + (\zeta_{126}^{53} + \zeta_{126}^{39}) q^{79} - \zeta_{126}^{43} q^{81} - \zeta_{126}^{8} q^{84} + (\zeta_{126}^{27} + \zeta_{126}^{16}) q^{91} + (\zeta_{126}^{57} + \zeta_{126}^{40}) q^{93} + (\zeta_{126}^{10} - \zeta_{126}^{7}) q^{97} +O(q^{100}) \) q - z^58 * q^3 - z^15 * q^4 - z^61 * q^7 - z^53 * q^9 - z^10 * q^12 + (z^29 + z^18) * q^13 + z^30 * q^16 + (z^17 + z^4) * q^19 - z^56 * q^21 + z^54 * q^25 - z^48 * q^27 - z^13 * q^28 + (z^62 + z^45) * q^31 - z^5 * q^36 + (-z^22 - z^6) * q^37 + (z^24 + z^13) * q^39 + (-z^41 - z^26) * q^43 + z^25 * q^48 - z^59 * q^49 + (-z^44 - z^33) * q^52 + (-z^62 + z^12) * q^57 + (-z^52 - z^29) * q^61 - z^51 * q^63 - z^45 * q^64 + (z^51 - z^23) * q^67 + (z^55 - z^35) * q^73 + z^49 * q^75 + (-z^32 - z^19) * q^76 + (z^53 + z^39) * q^79 - z^43 * q^81 - z^8 * q^84 + (z^27 + z^16) * q^91 + (z^57 + z^40) * q^93 + (z^10 - z^7) * 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} - 6 q^{25} - 3 q^{27} + 6 q^{31} - 3 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} - 3 q^{67} - 3 q^{79} + 6 q^{91} - 3 q^{93}+O(q^{100}) $ 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 - 6 * q^25 - 3 * q^27 + 6 * q^31 - 3 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 - 3 * q^67 - 3 * q^79 + 6 * q^91 - 3 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$890$	$1144$	$2416$
$\chi(n)$	$-1$	$\zeta_{126}^{21}$	$-\zeta_{126}^{32}$

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} $

173.1

0.921476	+	0.388435i
0.921476	−	0.388435i
−0.411287	−	0.911506i
0.995031	+	0.0995678i
−0.583744	−	0.811938i
0.878222	−	0.478254i
0.698237	−	0.715867i
−0.969077	+	0.246757i
0.980172	+	0.198146i
−0.318487	−	0.947927i
0.878222	+	0.478254i
−0.0249307	−	0.999689i
0.270840	−	0.962624i
−0.661686	−	0.749781i
−0.411287	+	0.911506i
−0.583744	+	0.811938i
0.980172	−	0.198146i
−0.0249307	+	0.999689i
−0.797133	−	0.603804i
−0.969077	−	0.246757i

0.411287

0.911506i

0.955573

−

0.294755i

0.698237

−

0.715867i

−0.661686

0.749781i

185.1

0.411287

−

0.911506i

0.955573

0.294755i

0.698237

0.715867i

−0.661686

−

0.749781i

194.1

0.853291

0.521435i

0.0747301

0.997204i

−0.661686

−

0.749781i

0.456211

0.889872i

257.1

−0.878222

0.478254i

0.0747301

0.997204i

0.980172

−

0.198146i

0.542546

−

0.840026i

332.1

0.0249307

0.999689i

0.0747301

−

0.997204i

−0.318487

−

0.947927i

−0.998757

0.0498459i

404.1

0.797133

−

0.603804i

0.365341

−

0.930874i

0.542546

0.840026i

0.270840

−

0.962624i

446.1

0.661686

0.749781i

0.826239

0.563320i

−0.0249307

0.999689i

−0.124344

0.992239i

563.1

0.318487

0.947927i

0.826239

−

0.563320i

0.878222

0.478254i

−0.797133

0.603804i

626.1

−0.542546

0.840026i

−0.988831

0.149042i

0.921476

−

0.388435i

−0.411287

−

0.911506i

815.1

0.998757

0.0498459i

−0.988831

0.149042i

−0.797133

−

0.603804i

0.995031

0.0995678i

878.1

0.797133

0.603804i

0.365341

0.930874i

0.542546

−

0.840026i

0.270840

0.962624i

1109.1

0.124344

−

0.992239i

0.365341

0.930874i

−0.998757

−

0.0498459i

−0.969077

−

0.246757i

1172.1

−0.980172

−

0.198146i

0.826239

−

0.563320i

−0.853291

0.521435i

0.921476

0.388435i

1244.1

−0.456211

0.889872i

−0.988831

−

0.149042i

−0.124344

−

0.992239i

−0.583744

−

0.811938i

1361.1

0.853291

−

0.521435i

0.0747301

−

0.997204i

−0.661686

0.749781i

0.456211

−

0.889872i

1454.1

0.0249307

−

0.999689i

0.0747301

0.997204i

−0.318487

0.947927i

−0.998757

−

0.0498459i

1538.1

−0.542546

−

0.840026i

−0.988831

−

0.149042i

0.921476

0.388435i

−0.411287

0.911506i

1580.1

0.124344

0.992239i

0.365341

−

0.930874i

−0.998757

0.0498459i

−0.969077

0.246757i

1634.1

−0.995031

0.0995678i

0.955573

0.294755i

0.270840

−

0.962624i

0.980172

−

0.198146i

1748.1

0.318487

−

0.947927i

0.826239

0.563320i

0.878222

−

0.478254i

−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.cg	even	126	1	inner
2667.en	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.en.a	yes	36
3.b	odd	2	1	CM	2667.1.en.a	yes	36
7.d	odd	6	1		2667.1.ei.a	✓	36
21.g	even	6	1		2667.1.ei.a	✓	36
127.l	odd	126	1		2667.1.ei.a	✓	36
381.x	even	126	1		2667.1.ei.a	✓	36
889.cg	even	126	1	inner	2667.1.en.a	yes	36
2667.en	odd	126	1	inner	2667.1.en.a	yes	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2667.1.ei.a	✓	36	7.d	odd	6	1
2667.1.ei.a	✓	36	21.g	even	6	1
2667.1.ei.a	✓	36	127.l	odd	126	1
2667.1.ei.a	✓	36	381.x	even	126	1
2667.1.en.a	yes	36	1.a	even	1	1	trivial
2667.1.en.a	yes	36	3.b	odd	2	1	CM
2667.1.en.a	yes	36	889.cg	even	126	1	inner
2667.1.en.a	yes	36	2667.en	odd	126	1	inner

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^{36} - T^{33} + \cdots + 1 $ T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$11$	$ T^{36} $ T^36
$13$	$ T^{36} + 6 T^{35} + \cdots + 1 $ T^36 + 6T^35 + 21T^34 + 57T^33 + 114T^32 + 171T^31 + 186T^30 + 84T^29 - 3T^28 + 405T^27 + 2088T^26 + 5850T^25 + 10220T^24 + 9849T^23 - 5271T^22 - 40606T^21 - 82086T^20 - 96969T^19 - 49768T^18 + 64392T^17 + 180162T^16 + 198429T^15 + 173241T^14 + 187107T^13 + 281518T^12 + 376641T^11 + 294666T^10 + 79724T^9 - 42267T^8 - 30894T^7 + 3669T^6 + 5316T^5 + 1308T^4 - 541T^3 + 45T^2 + 3*T + 1
$17$	$ T^{36} $ T^36
$19$	$ T^{36} - 18 T^{34} + \cdots + 1 $ T^36 - 18T^34 + 189T^32 + 9T^31 - 1339T^30 - 135T^29 + 7125T^28 + 1215T^27 - 29241T^26 - 7218T^25 + 95219T^24 + 31941T^23 - 245715T^22 - 106029T^21 + 504243T^20 + 273069T^19 - 803577T^18 - 533790T^17 + 977535T^16 + 797283T^15 - 842292T^14 - 855504T^13 + 483306T^12 + 657693T^11 - 112626T^10 - 299862T^9 - 9666T^8 + 88623T^7 + 26260T^6 - 4887T^5 - 1992T^4 + 324T^3 + 90T^2 - 18*T + 1
$23$	$ T^{36} $ T^36
$29$	$ T^{36} $ T^36
$31$	$ T^{36} - 6 T^{35} + \cdots + 1 $ T^36 - 6T^35 + 21T^34 - 57T^33 + 114T^32 - 171T^31 + 186T^30 - 84T^29 - 3T^28 - 405T^27 + 2088T^26 - 5850T^25 + 10220T^24 - 9849T^23 - 5271T^22 + 40606T^21 - 82086T^20 + 96969T^19 - 49768T^18 - 64392T^17 + 180162T^16 - 198429T^15 + 173241T^14 - 187107T^13 + 281518T^12 - 376641T^11 + 294666T^10 - 79724T^9 - 42267T^8 + 30894T^7 + 3669T^6 - 5316T^5 + 1308T^4 + 541T^3 + 45T^2 - 3*T + 1
$37$	$ T^{36} + 3 T^{35} + \cdots + 1 $ T^36 + 3T^35 + 3T^34 - T^33 + 18T^31 + 97T^30 + 171T^29 + 75T^28 - 78T^27 + 339T^26 + 1158T^25 + 4048T^24 + 8460T^23 + 9504T^22 + 5723T^21 + 11889T^20 + 23685T^19 + 44790T^18 + 55722T^17 + 24411T^16 - 15232T^15 + 50265T^14 + 45432T^13 + 69904T^12 + 27270T^11 - 44574T^10 - 15331T^9 + 29964T^8 + 23853T^7 + 28496T^6 + 19359T^5 + 8271T^4 + 1189T^3 + 468T^2 + 39*T + 1
$41$	$ T^{36} $ T^36
$43$	$ T^{36} - 9 T^{33} + \cdots + 1 $ T^36 - 9T^33 + 33T^30 - 348T^27 + 2192T^24 + 2702T^21 + 16766T^18 - 4662T^15 + 33559T^12 - 14947T^9 + 1944T^6 - 61*T^3 + 1
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} $ T^36
$61$	$ T^{36} - 6 T^{34} + \cdots + 1 $ T^36 - 6T^34 + 27T^32 - 21T^31 - 111T^30 + 126T^29 + 441T^28 - 371T^27 - 1413T^26 + 1155T^25 + 4842T^24 - 3717T^23 - 11478T^22 + 10766T^21 + 29529T^20 - 34587T^19 - 52824T^18 + 50841T^17 + 97356T^16 - 26061T^15 - 173493T^14 - 2814T^13 + 226318T^12 + 75096T^11 - 76842T^10 - 45941T^9 + 9693T^8 + 32760T^7 - 7706T^6 - 11025T^5 + 7245T^4 - 1897T^3 + 270T^2 - 21*T + 1
$67$	$ T^{36} + 3 T^{35} + \cdots + 729 $ T^36 + 3T^35 + 3T^34 - 3T^33 - 9T^32 + 9T^31 + 60T^30 + 207T^29 + 306T^28 + 45T^27 - 432T^26 + 243T^25 + 3411T^24 + 5562T^23 + 9450T^22 + 15309T^21 + 23895T^20 + 35802T^19 + 45657T^18 + 48843T^17 + 59049T^16 + 66339T^15 + 66096T^14 + 47142T^13 + 5913T^12 - 15066T^11 + 25029T^10 + 1944T^9 + 5832T^8 + 14580T^7 + 2673T^6 + 729T^5 - 1458T^4 + 1458*T^3 + 729
$71$	$ T^{36} $ T^36
$73$	$ T^{36} - 6 T^{34} + \cdots + 1 $ T^36 - 6T^34 + 7T^33 + 27T^32 - 42T^31 - 83T^30 + 189T^29 + 252T^28 - 700T^27 - 846T^26 + 1848T^25 + 3218T^24 - 5355T^23 - 7152T^22 + 17864T^21 + 16866T^20 - 35742T^19 - 46125T^18 + 59913T^17 + 133245T^16 - 91000T^15 - 224208T^14 + 5019T^13 + 283459T^12 + 63504T^11 - 126528T^10 - 90706T^9 + 39177T^8 + 13503T^7 + 9661T^6 - 4788T^5 + 357T^4 + 378T^3 + 144T^2 + 21T + 1
$79$	$ 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
$83$	$ 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
show more
show less

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

\(f(q)\)	\(=\)	\( q - \zeta_{126}^{58} q^{3} - \zeta_{126}^{15} q^{4} - \zeta_{126}^{61} q^{7} - \zeta_{126}^{53} q^{9} +O(q^{10}) \) q - z^58 * q^3 - z^15 * q^4 - z^61 * q^7 - z^53 * q^9	\( q - \zeta_{126}^{58} q^{3} - \zeta_{126}^{15} q^{4} - \zeta_{126}^{61} q^{7} - \zeta_{126}^{53} q^{9} - \zeta_{126}^{10} q^{12} + (\zeta_{126}^{29} + \zeta_{126}^{18}) q^{13} + \zeta_{126}^{30} q^{16} + (\zeta_{126}^{17} + \zeta_{126}^{4}) q^{19} - \zeta_{126}^{56} q^{21} + \zeta_{126}^{54} q^{25} - \zeta_{126}^{48} q^{27} - \zeta_{126}^{13} q^{28} + (\zeta_{126}^{62} + \zeta_{126}^{45}) q^{31} - \zeta_{126}^{5} q^{36} + ( - \zeta_{126}^{22} - \zeta_{126}^{6}) q^{37} + (\zeta_{126}^{24} + \zeta_{126}^{13}) q^{39} + ( - \zeta_{126}^{41} - \zeta_{126}^{26}) q^{43} + \zeta_{126}^{25} q^{48} - \zeta_{126}^{59} q^{49} + ( - \zeta_{126}^{44} - \zeta_{126}^{33}) q^{52} + ( - \zeta_{126}^{62} + \zeta_{126}^{12}) q^{57} + ( - \zeta_{126}^{52} - \zeta_{126}^{29}) q^{61} - \zeta_{126}^{51} q^{63} - \zeta_{126}^{45} q^{64} + (\zeta_{126}^{51} - \zeta_{126}^{23}) q^{67} + (\zeta_{126}^{55} - \zeta_{126}^{35}) q^{73} + \zeta_{126}^{49} q^{75} + ( - \zeta_{126}^{32} - \zeta_{126}^{19}) q^{76} + (\zeta_{126}^{53} + \zeta_{126}^{39}) q^{79} - \zeta_{126}^{43} q^{81} - \zeta_{126}^{8} q^{84} + (\zeta_{126}^{27} + \zeta_{126}^{16}) q^{91} + (\zeta_{126}^{57} + \zeta_{126}^{40}) q^{93} + (\zeta_{126}^{10} - \zeta_{126}^{7}) q^{97} +O(q^{100}) \) q - z^58 * q^3 - z^15 * q^4 - z^61 * q^7 - z^53 * q^9 - z^10 * q^12 + (z^29 + z^18) * q^13 + z^30 * q^16 + (z^17 + z^4) * q^19 - z^56 * q^21 + z^54 * q^25 - z^48 * q^27 - z^13 * q^28 + (z^62 + z^45) * q^31 - z^5 * q^36 + (-z^22 - z^6) * q^37 + (z^24 + z^13) * q^39 + (-z^41 - z^26) * q^43 + z^25 * q^48 - z^59 * q^49 + (-z^44 - z^33) * q^52 + (-z^62 + z^12) * q^57 + (-z^52 - z^29) * q^61 - z^51 * q^63 - z^45 * q^64 + (z^51 - z^23) * q^67 + (z^55 - z^35) * q^73 + z^49 * q^75 + (-z^32 - z^19) * q^76 + (z^53 + z^39) * q^79 - z^43 * q^81 - z^8 * q^84 + (z^27 + z^16) * q^91 + (z^57 + z^40) * q^93 + (z^10 - z^7) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{4}+O(q^{10}) \) 36 * q + 3 * q^4	\( 36 q + 3 q^{4} - 6 q^{13} + 3 q^{16} - 6 q^{25} - 3 q^{27} + 6 q^{31} - 3 q^{37} + 3 q^{39} + 3 q^{52} + 3 q^{57} + 3 q^{63} - 6 q^{64} - 3 q^{67} - 3 q^{79} + 6 q^{91} - 3 q^{93}+O(q^{100}) \) 36 * q + 3 * q^4 - 6 * q^13 + 3 * q^16 - 6 * q^25 - 3 * q^27 + 6 * q^31 - 3 * q^37 + 3 * q^39 + 3 * q^52 + 3 * q^57 + 3 * q^63 - 6 * q^64 - 3 * q^67 - 3 * q^79 + 6 * q^91 - 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	2667.1.en.a

Level	$2667$
Weight	$1$
Character orbit	2667.en
Analytic conductor	$1.331$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{126}$
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:	\(D_{126}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{126} - \cdots)\)

\(n\)	\(890\)	\(1144\)	\(2416\)
\(\chi(n)\)	\(-1\)	\(\zeta_{126}^{21}\)	\(-\zeta_{126}^{32}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 173.1
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.cg	even	126	1	inner
2667.en	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
$7$	\( T^{36} - T^{33} + \cdots + 1 \) T^36 - T^33 + T^27 - T^24 + T^18 - T^12 + T^9 - T^3 + 1
$11$	\( T^{36} \) T^36
$13$	\( T^{36} + 6 T^{35} + \cdots + 1 \) T^36 + 6T^35 + 21T^34 + 57T^33 + 114T^32 + 171T^31 + 186T^30 + 84T^29 - 3T^28 + 405T^27 + 2088T^26 + 5850T^25 + 10220T^24 + 9849T^23 - 5271T^22 - 40606T^21 - 82086T^20 - 96969T^19 - 49768T^18 + 64392T^17 + 180162T^16 + 198429T^15 + 173241T^14 + 187107T^13 + 281518T^12 + 376641T^11 + 294666T^10 + 79724T^9 - 42267T^8 - 30894T^7 + 3669T^6 + 5316T^5 + 1308T^4 - 541T^3 + 45T^2 + 3*T + 1
$17$	\( T^{36} \) T^36
$19$	\( T^{36} - 18 T^{34} + \cdots + 1 \) T^36 - 18T^34 + 189T^32 + 9T^31 - 1339T^30 - 135T^29 + 7125T^28 + 1215T^27 - 29241T^26 - 7218T^25 + 95219T^24 + 31941T^23 - 245715T^22 - 106029T^21 + 504243T^20 + 273069T^19 - 803577T^18 - 533790T^17 + 977535T^16 + 797283T^15 - 842292T^14 - 855504T^13 + 483306T^12 + 657693T^11 - 112626T^10 - 299862T^9 - 9666T^8 + 88623T^7 + 26260T^6 - 4887T^5 - 1992T^4 + 324T^3 + 90T^2 - 18*T + 1
$23$	\( T^{36} \) T^36
$29$	\( T^{36} \) T^36
$31$	\( T^{36} - 6 T^{35} + \cdots + 1 \) T^36 - 6T^35 + 21T^34 - 57T^33 + 114T^32 - 171T^31 + 186T^30 - 84T^29 - 3T^28 - 405T^27 + 2088T^26 - 5850T^25 + 10220T^24 - 9849T^23 - 5271T^22 + 40606T^21 - 82086T^20 + 96969T^19 - 49768T^18 - 64392T^17 + 180162T^16 - 198429T^15 + 173241T^14 - 187107T^13 + 281518T^12 - 376641T^11 + 294666T^10 - 79724T^9 - 42267T^8 + 30894T^7 + 3669T^6 - 5316T^5 + 1308T^4 + 541T^3 + 45T^2 - 3*T + 1
$37$	\( T^{36} + 3 T^{35} + \cdots + 1 \) T^36 + 3T^35 + 3T^34 - T^33 + 18T^31 + 97T^30 + 171T^29 + 75T^28 - 78T^27 + 339T^26 + 1158T^25 + 4048T^24 + 8460T^23 + 9504T^22 + 5723T^21 + 11889T^20 + 23685T^19 + 44790T^18 + 55722T^17 + 24411T^16 - 15232T^15 + 50265T^14 + 45432T^13 + 69904T^12 + 27270T^11 - 44574T^10 - 15331T^9 + 29964T^8 + 23853T^7 + 28496T^6 + 19359T^5 + 8271T^4 + 1189T^3 + 468T^2 + 39*T + 1
$41$	\( T^{36} \) T^36
$43$	\( T^{36} - 9 T^{33} + \cdots + 1 \) T^36 - 9T^33 + 33T^30 - 348T^27 + 2192T^24 + 2702T^21 + 16766T^18 - 4662T^15 + 33559T^12 - 14947T^9 + 1944T^6 - 61*T^3 + 1
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} \) T^36
$61$	\( T^{36} - 6 T^{34} + \cdots + 1 \) T^36 - 6T^34 + 27T^32 - 21T^31 - 111T^30 + 126T^29 + 441T^28 - 371T^27 - 1413T^26 + 1155T^25 + 4842T^24 - 3717T^23 - 11478T^22 + 10766T^21 + 29529T^20 - 34587T^19 - 52824T^18 + 50841T^17 + 97356T^16 - 26061T^15 - 173493T^14 - 2814T^13 + 226318T^12 + 75096T^11 - 76842T^10 - 45941T^9 + 9693T^8 + 32760T^7 - 7706T^6 - 11025T^5 + 7245T^4 - 1897T^3 + 270T^2 - 21*T + 1
$67$	\( T^{36} + 3 T^{35} + \cdots + 729 \) T^36 + 3T^35 + 3T^34 - 3T^33 - 9T^32 + 9T^31 + 60T^30 + 207T^29 + 306T^28 + 45T^27 - 432T^26 + 243T^25 + 3411T^24 + 5562T^23 + 9450T^22 + 15309T^21 + 23895T^20 + 35802T^19 + 45657T^18 + 48843T^17 + 59049T^16 + 66339T^15 + 66096T^14 + 47142T^13 + 5913T^12 - 15066T^11 + 25029T^10 + 1944T^9 + 5832T^8 + 14580T^7 + 2673T^6 + 729T^5 - 1458T^4 + 1458*T^3 + 729
$71$	\( T^{36} \) T^36
$73$	\( T^{36} - 6 T^{34} + \cdots + 1 \) T^36 - 6T^34 + 7T^33 + 27T^32 - 42T^31 - 83T^30 + 189T^29 + 252T^28 - 700T^27 - 846T^26 + 1848T^25 + 3218T^24 - 5355T^23 - 7152T^22 + 17864T^21 + 16866T^20 - 35742T^19 - 46125T^18 + 59913T^17 + 133245T^16 - 91000T^15 - 224208T^14 + 5019T^13 + 283459T^12 + 63504T^11 - 126528T^10 - 90706T^9 + 39177T^8 + 13503T^7 + 9661T^6 - 4788T^5 + 357T^4 + 378T^3 + 144T^2 + 21T + 1
$79$	\( T^{36} + 3 T^{35} + \cdots + 1 \) 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
$83$	\( T^{36} \) 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
show more
show less