LMFDB - Newform orbit 2919.1.cp.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2919,1,Mod(26,2919)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2919, base_ring=CyclotomicField(138))

chi = DirichletCharacter(H, H._module([69, 115, 65]))

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

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

chi := DirichletCharacter("2919.26");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2919 = 3 \cdot 7 \cdot 139 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2919.cp (of order $138$, degree $44$, 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:	$1.45677077188$
Analytic rank:	$0$
Dimension:	$44$
Coefficient field:	$\Q(\zeta_{69})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{44} - x^{43} + x^{41} - x^{40} + x^{38} - x^{37} + x^{35} - x^{34} + x^{32} - x^{31} + x^{29} + \cdots + 1 $ x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{138}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{138} - \cdots)$

$q$-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_{138}^{32} q^{3} + \zeta_{138}^{47} q^{4} + \zeta_{138}^{7} q^{7} + \zeta_{138}^{64} q^{9}+O(q^{10}) $ q + z^32 * q^3 + z^47 * q^4 + z^7 * q^7 + z^64 * q^9	\( q + \zeta_{138}^{32} q^{3} + \zeta_{138}^{47} q^{4} + \zeta_{138}^{7} q^{7} + \zeta_{138}^{64} q^{9} - \zeta_{138}^{10} q^{12} + (\zeta_{138}^{36} - \zeta_{138}^{28}) q^{13} - \zeta_{138}^{25} q^{16} + (\zeta_{138}^{26} + \zeta_{138}^{12}) q^{19} + \zeta_{138}^{39} q^{21} + \zeta_{138}^{63} q^{25} - \zeta_{138}^{27} q^{27} + \zeta_{138}^{54} q^{28} + (\zeta_{138}^{45} + \zeta_{138}^{34}) q^{31} - \zeta_{138}^{42} q^{36} + (\zeta_{138}^{42} - \zeta_{138}^{15}) q^{37} + (\zeta_{138}^{68} - \zeta_{138}^{60}) q^{39} + ( - \zeta_{138}^{38} + \zeta_{138}^{8}) q^{43} - \zeta_{138}^{57} q^{48} + \zeta_{138}^{14} q^{49} + ( - \zeta_{138}^{14} + \zeta_{138}^{6}) q^{52} + (\zeta_{138}^{58} + \zeta_{138}^{44}) q^{57} + (\zeta_{138}^{67} - \zeta_{138}^{16}) q^{61} - \zeta_{138}^{2} q^{63} + \zeta_{138}^{3} q^{64} + (\zeta_{138}^{28} - \zeta_{138}^{11}) q^{67} + ( - \zeta_{138}^{39} - \zeta_{138}^{33}) q^{73} - \zeta_{138}^{26} q^{75} + (\zeta_{138}^{59} - \zeta_{138}^{4}) q^{76} + ( - \zeta_{138}^{19} + \zeta_{138}^{18}) q^{79} - \zeta_{138}^{59} q^{81} - \zeta_{138}^{17} q^{84} + (\zeta_{138}^{43} - \zeta_{138}^{35}) q^{91} + (\zeta_{138}^{66} - \zeta_{138}^{8}) q^{93} + ( - \zeta_{138}^{68} - \zeta_{138}^{24}) q^{97} +O(q^{100}) \) q + z^32 * q^3 + z^47 * q^4 + z^7 * q^7 + z^64 * q^9 - z^10 * q^12 + (z^36 - z^28) * q^13 - z^25 * q^16 + (z^26 + z^12) * q^19 + z^39 * q^21 + z^63 * q^25 - z^27 * q^27 + z^54 * q^28 + (z^45 + z^34) * q^31 - z^42 * q^36 + (z^42 - z^15) * q^37 + (z^68 - z^60) * q^39 + (-z^38 + z^8) * q^43 - z^57 * q^48 + z^14 * q^49 + (-z^14 + z^6) * q^52 + (z^58 + z^44) * q^57 + (z^67 - z^16) * q^61 - z^2 * q^63 + z^3 * q^64 + (z^28 - z^11) * q^67 + (-z^39 - z^33) * q^73 - z^26 * q^75 + (z^59 - z^4) * q^76 + (-z^19 + z^18) * q^79 - z^59 * q^81 - z^17 * q^84 + (z^43 - z^35) * q^91 + (z^66 - z^8) * q^93 + (-z^68 - z^24) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + q^{3} - q^{4} - q^{7} + q^{9}+O(q^{10}) $ 44 * q + q^3 - q^4 - q^7 + q^9	$ 44 q + q^{3} - q^{4} - q^{7} + q^{9} - q^{12} - 3 q^{13} + q^{16} - q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} + 3 q^{31} + 2 q^{36} - 4 q^{37} + 3 q^{39} - 2 q^{48} + q^{49} - 3 q^{52} + 2 q^{57} - 2 q^{61} - q^{63} + 2 q^{64} + 2 q^{67} - 4 q^{73} - q^{75} - 2 q^{76} - q^{79} + q^{81} + q^{84} - 3 q^{93} + q^{97}+O(q^{100}) $ 44 * q + q^3 - q^4 - q^7 + q^9 - q^12 - 3 * q^13 + q^16 - q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 + 3 * q^31 + 2 * q^36 - 4 * q^37 + 3 * q^39 - 2 * q^48 + q^49 - 3 * q^52 + 2 * q^57 - 2 * q^61 - q^63 + 2 * q^64 + 2 * q^67 - 4 * q^73 - q^75 - 2 * q^76 - q^79 + q^81 + q^84 - 3 * q^93 + q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2919\mathbb{Z}\right)^\times$.

$n$	$974$	$1114$	$1669$
$\chi(n)$	$-1$	$\zeta_{138}$	$\zeta_{138}^{23}$

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} $

26.1

0.983462	−	0.181116i
0.898128	+	0.439735i
−0.829885	−	0.557934i
0.377419	−	0.926043i
0.0227632	−	0.999741i
−0.247808	−	0.968809i
0.995857	−	0.0909349i
−0.715110	−	0.699012i
0.377419	+	0.926043i
0.291646	−	0.956526i
−0.877183	+	0.480157i
−0.829885	+	0.557934i
−0.998964	−	0.0455146i
−0.974199	−	0.225690i
0.538899	−	0.842371i
−0.949640	−	0.313344i
0.613267	+	0.789876i
0.613267	−	0.789876i
0.983462	+	0.181116i
−0.648582	+	0.761145i

0.898128

0.439735i

0.648582

0.761145i

−0.291646

0.956526i

0.613267

0.789876i

227.1

−0.419177

0.907905i

0.829885

−

0.557934i

0.998964

0.0455146i

−0.648582

−

0.761145i

269.1

0.995857

0.0909349i

−0.898128

0.439735i

−0.538899

−

0.842371i

0.983462

0.181116i

290.1

0.983462

−

0.181116i

−0.613267

−

0.789876i

0.419177

0.907905i

0.934394

−

0.356242i

467.1

0.746184

0.665740i

0.877183

−

0.480157i

0.158683

−

0.987330i

0.113580

0.993529i

509.1

−0.158683

−

0.987330i

0.715110

−

0.699012i

−0.983462

0.181116i

−0.949640

0.313344i

521.1

−0.974199

−

0.225690i

0.419177

−

0.907905i

−0.803631

0.595128i

0.898128

0.439735i

551.1

0.934394

−

0.356242i

0.247808

−

0.968809i

0.648582

−

0.761145i

0.746184

−

0.665740i

614.1

0.983462

0.181116i

−0.613267

0.789876i

0.419177

−

0.907905i

0.934394

0.356242i

698.1

−0.998964

−

0.0455146i

0.974199

−

0.225690i

0.877183

0.480157i

0.995857

0.0909349i

710.1

−0.949640

0.313344i

−0.0227632

0.999741i

−0.934394

0.356242i

0.803631

−

0.595128i

803.1

0.995857

−

0.0909349i

−0.898128

−

0.439735i

−0.538899

0.842371i

0.983462

−

0.181116i

836.1

0.113580

0.993529i

−0.538899

0.842371i

0.949640

0.313344i

−0.974199

0.225690i

866.1

0.538899

0.842371i

−0.291646

−

0.956526i

−0.0227632

0.999741i

−0.419177

0.907905i

887.1

0.803631

−

0.595128i

0.998964

0.0455146i

−0.746184

0.665740i

0.291646

−

0.956526i

962.1

−0.715110

−

0.699012i

−0.746184

0.665740i

−0.613267

0.789876i

0.0227632

0.999741i

1013.1

−0.648582

−

0.761145i

−0.377419

0.926043i

−0.995857

−

0.0909349i

−0.158683

0.987330i

1046.1

−0.648582

0.761145i

−0.377419

−

0.926043i

−0.995857

0.0909349i

−0.158683

−

0.987330i

1235.1

0.898128

−

0.439735i

0.648582

−

0.761145i

−0.291646

−

0.956526i

0.613267

−

0.789876i

1244.1

−0.829885

−

0.557934i

−0.983462

−

0.181116i

0.974199

0.225690i

0.377419

0.926043i

See all 44 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
973.bi	even	138	1	inner
2919.cp	odd	138	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2919.1.cp.a	yes	44
3.b	odd	2	1	CM	2919.1.cp.a	yes	44
7.d	odd	6	1		2919.1.cc.a	✓	44
21.g	even	6	1		2919.1.cc.a	✓	44
139.h	odd	138	1		2919.1.cc.a	✓	44
417.p	even	138	1		2919.1.cc.a	✓	44
973.bi	even	138	1	inner	2919.1.cp.a	yes	44
2919.cp	odd	138	1	inner	2919.1.cp.a	yes	44

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2919.1.cc.a	✓	44	7.d	odd	6	1
2919.1.cc.a	✓	44	21.g	even	6	1
2919.1.cc.a	✓	44	139.h	odd	138	1
2919.1.cc.a	✓	44	417.p	even	138	1
2919.1.cp.a	yes	44	1.a	even	1	1	trivial
2919.1.cp.a	yes	44	3.b	odd	2	1	CM
2919.1.cp.a	yes	44	973.bi	even	138	1	inner
2919.1.cp.a	yes	44	2919.cp	odd	138	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2919, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{44} $ T^44
$3$	$ T^{44} - T^{43} + \cdots + 1 $ T^44 - T^43 + T^41 - T^40 + T^38 - T^37 + T^35 - T^34 + T^32 - T^31 + T^29 - T^28 + T^26 - T^25 + T^23 - T^22 + T^21 - T^19 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$5$	$ T^{44} $ T^44
$7$	$ T^{44} + T^{43} + \cdots + 1 $ T^44 + T^43 - T^41 - T^40 + T^38 + T^37 - T^35 - T^34 + T^32 + T^31 - T^29 - T^28 + T^26 + T^25 - T^23 - T^22 - T^21 + T^19 + T^18 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$11$	$ T^{44} $ T^44
$13$	$ T^{44} + 3 T^{43} + \cdots + 1 $ T^44 + 3T^43 + 6T^42 + 32T^41 + 78T^40 + 138T^39 + 502T^38 + 1092T^37 + 1770T^36 + 4955T^35 + 9555T^34 + 13800T^33 + 32331T^32 + 55570T^31 + 69717T^30 + 138880T^29 + 208570T^28 + 209070T^27 + 364210T^26 + 480968T^25 + 350274T^24 + 613110T^23 + 737701T^22 + 373770T^21 + 998286T^20 + 1251444T^19 + 759934T^18 + 1532835T^17 + 1942055T^16 + 1221312T^15 + 916022T^14 + 1059416T^13 + 501712T^12 + 141358T^11 + 139341T^10 - 87586T^9 + 42891T^8 + 1029T^7 - 28250T^6 + 207T^5 + 3905T^4 + 790T^3 + 85T^2 + T + 1
$17$	$ T^{44} $ T^44
$19$	$ T^{44} + T^{43} + \cdots + 1 $ T^44 + T^43 - T^41 - T^40 + T^38 + T^37 - 322T^36 - 323T^35 - T^34 + 1012T^33 + 1013T^32 + 139T^31 - 276T^30 - 415T^29 + 28680T^28 + 28727T^27 + 70T^26 - 139011T^25 - 139058T^24 - 22196T^23 + 496661T^22 + 518856T^21 - 539603T^20 - 1411831T^19 - 856887T^18 + 1601306T^17 + 2685134T^16 + 1304421T^15 + 479780T^14 - 905946T^13 + 550943T^12 + 496225T^11 + 16375T^10 + 658558T^9 + 513245T^8 - 15110T^7 - 31624T^6 + 18538T^5 + 620T^4 - 2623T^3 + 575T^2 - 22*T + 1
$23$	$ T^{44} $ T^44
$29$	$ T^{44} $ T^44
$31$	$ T^{44} - 3 T^{43} + \cdots + 1 $ T^44 - 3T^43 + 6T^42 - 9T^41 + 9T^40 - 27T^38 + 81T^37 - 162T^36 + 243T^35 - 243T^34 + 706T^32 - 3130T^31 + 3983T^30 + 7308T^29 - 983T^28 - 2231T^27 - 1306T^26 + 1227T^25 - 1074T^24 - 229T^23 + 3955T^22 - 11175T^21 + 22189T^20 - 21404T^19 + 178034T^18 + 740462T^17 + 2691096T^16 + 5900661T^15 + 9525957T^14 + 11488533T^13 + 10891686T^12 + 8251365T^11 + 5103408T^10 + 2597944T^9 + 1094060T^8 + 388660T^7 + 119226T^6 + 25921T^5 + 2548T^4 + 291T^3 + 154T^2 + 22*T + 1
$37$	$ (T^{22} + 2 T^{21} + \cdots + 1)^{2} $ (T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 64T^16 + 128T^15 + 256T^14 + 512T^13 + 1024T^12 + 2048T^11 + 4073T^10 + 7640T^9 + 11991T^8 + 14115T^7 + 11785T^6 + 6826T^5 + 2704T^4 + 716T^3 + 121T^2 + 12*T + 1)^2
$41$	$ T^{44} $ T^44
$43$	$ T^{44} - 23 T^{42} + \cdots + 529 $ T^44 - 23T^42 + 299T^40 - 2668T^38 + 18055T^36 - 96646T^34 + 421245T^32 - 1516689T^30 + 4557519T^28 - 11467961T^26 + 24199128T^24 - 42662286T^22 + 62532561T^20 - 75392022T^18 + 73935156T^16 - 57768387T^14 + 35301228T^12 - 16195335T^10 + 5446584T^8 - 1210352T^6 + 180389T^4 - 11638*T^2 + 529
$47$	$ T^{44} $ T^44
$53$	$ T^{44} $ T^44
$59$	$ T^{44} $ T^44
$61$	$ T^{44} + 2 T^{43} + \cdots + 1 $ T^44 + 2T^43 - 8T^41 + 7T^40 + 46T^39 + 64T^38 - 56T^37 - 23T^36 + 224T^35 + 540T^34 + 184T^33 + 1244T^32 + 1545T^31 - 1886T^30 - 9952T^29 + 7052T^28 + 14375T^27 + 1209T^26 - 55082T^25 - 36547T^24 + 17905T^23 + 166680T^22 + 261738T^21 + 83697T^20 + 81106T^19 + 324155T^18 + 327106T^17 - 289036T^16 - 26003T^15 - 34270T^14 - 14047T^13 + 488762T^12 - 12650T^11 - 43679T^10 - 169399T^9 - 49703T^8 + 16392T^7 + 11003T^6 + 7429T^5 + 4150T^4 + 664T^3 + 69T^2 + 12T + 1
$67$	$ T^{44} - 2 T^{43} + \cdots + 1 $ T^44 - 2T^43 + 3T^42 - 4T^41 + 5T^40 - 6T^39 + 7T^38 - 8T^37 - 37T^36 + 404T^35 - 495T^34 + 540T^33 - 585T^32 + 630T^31 - 675T^30 + 720T^29 - 52T^28 - 7056T^27 + 109223T^26 - 303390T^25 + 565407T^24 - 833565T^23 + 1102298T^22 - 1371029T^21 + 1635528T^20 - 1892621T^19 + 2849581T^18 - 3247917T^17 + 1714345T^16 - 1303311T^15 + 1174763T^14 - 903431T^13 + 642472T^12 - 293423T^11 + 668000T^10 + 771893T^9 + 866465T^8 + 502542T^7 + 218530T^6 + 65682T^5 + 11850T^4 + 2503T^3 + 647T^2 + 44*T + 1
$71$	$ T^{44} $ T^44
$73$	$ (T^{22} + 2 T^{21} + \cdots + 1)^{2} $ (T^22 + 2T^21 + 4T^20 + 8T^19 - 7T^18 - 14T^17 - 28T^16 - 56T^15 + 72T^14 + 121T^13 + 242T^12 + 484T^11 + 370T^10 + 4T^9 + 8T^8 + 16T^7 + 722T^6 - 5T^5 + 59T^4 + 118T^3 + 75T^2 + 12*T + 1)^2
$79$	$ T^{44} + T^{43} + \cdots + 1 $ T^44 + T^43 - T^41 - T^40 + 46T^39 + 47T^38 + T^37 - 46T^36 - 47T^35 + 1540T^34 + 1357T^33 - 183T^32 - 1540T^31 - 1334T^30 + 23298T^29 + 20331T^28 - 2944T^27 - 23275T^26 - 21619T^25 + 254702T^24 + 433595T^23 + 191106T^22 - 242490T^21 - 465336T^20 + 708194T^19 + 2120969T^18 + 1273648T^17 - 849391T^16 - 2095531T^15 + 1378229T^14 + 1668812T^13 + 1198347T^12 - 499997T^11 - 1303595T^10 - 330672T^9 + 18078T^8 + 120176T^7 + 147155T^6 + 68172T^5 + 21481T^4 + 3564T^3 + 184T^2 - 22T + 1
$83$	$ T^{44} $ T^44
$89$	$ T^{44} $ T^44
$97$	$ T^{44} - T^{43} + \cdots + 1 $ T^44 - T^43 + 23T^42 - 22T^41 + 298T^40 - 276T^39 + 2646T^38 - 2370T^37 + 17779T^36 - 15409T^35 + 94276T^34 - 78867T^33 + 405836T^32 - 326969T^31 + 1437822T^30 - 1110853T^29 + 4230550T^28 - 3119697T^27 + 10357108T^26 - 7237411T^25 + 21079431T^24 - 13842020T^23 + 35424875T^22 - 21582854T^21 + 48690540T^20 - 27107640T^19 + 53809122T^18 - 26701252T^17 + 46827286T^16 - 20127345T^15 + 31068446T^14 - 10950485T^13 + 15183267T^12 - 4243730T^11 + 5255798T^10 - 995324T^9 + 1186110T^8 - 157896T^7 + 182138T^6 - 14375T^5 + 12626T^4 - 1540T^3 + 552T^2 - 24T + 1
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2919 = 3 \cdot 7 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2919.cp (of order \(138\), degree \(44\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{138}^{32} q^{3} + \zeta_{138}^{47} q^{4} + \zeta_{138}^{7} q^{7} + \zeta_{138}^{64} q^{9}+O(q^{10}) \) q + z^32 * q^3 + z^47 * q^4 + z^7 * q^7 + z^64 * q^9	\( q + \zeta_{138}^{32} q^{3} + \zeta_{138}^{47} q^{4} + \zeta_{138}^{7} q^{7} + \zeta_{138}^{64} q^{9} - \zeta_{138}^{10} q^{12} + (\zeta_{138}^{36} - \zeta_{138}^{28}) q^{13} - \zeta_{138}^{25} q^{16} + (\zeta_{138}^{26} + \zeta_{138}^{12}) q^{19} + \zeta_{138}^{39} q^{21} + \zeta_{138}^{63} q^{25} - \zeta_{138}^{27} q^{27} + \zeta_{138}^{54} q^{28} + (\zeta_{138}^{45} + \zeta_{138}^{34}) q^{31} - \zeta_{138}^{42} q^{36} + (\zeta_{138}^{42} - \zeta_{138}^{15}) q^{37} + (\zeta_{138}^{68} - \zeta_{138}^{60}) q^{39} + ( - \zeta_{138}^{38} + \zeta_{138}^{8}) q^{43} - \zeta_{138}^{57} q^{48} + \zeta_{138}^{14} q^{49} + ( - \zeta_{138}^{14} + \zeta_{138}^{6}) q^{52} + (\zeta_{138}^{58} + \zeta_{138}^{44}) q^{57} + (\zeta_{138}^{67} - \zeta_{138}^{16}) q^{61} - \zeta_{138}^{2} q^{63} + \zeta_{138}^{3} q^{64} + (\zeta_{138}^{28} - \zeta_{138}^{11}) q^{67} + ( - \zeta_{138}^{39} - \zeta_{138}^{33}) q^{73} - \zeta_{138}^{26} q^{75} + (\zeta_{138}^{59} - \zeta_{138}^{4}) q^{76} + ( - \zeta_{138}^{19} + \zeta_{138}^{18}) q^{79} - \zeta_{138}^{59} q^{81} - \zeta_{138}^{17} q^{84} + (\zeta_{138}^{43} - \zeta_{138}^{35}) q^{91} + (\zeta_{138}^{66} - \zeta_{138}^{8}) q^{93} + ( - \zeta_{138}^{68} - \zeta_{138}^{24}) q^{97} +O(q^{100}) \) q + z^32 * q^3 + z^47 * q^4 + z^7 * q^7 + z^64 * q^9 - z^10 * q^12 + (z^36 - z^28) * q^13 - z^25 * q^16 + (z^26 + z^12) * q^19 + z^39 * q^21 + z^63 * q^25 - z^27 * q^27 + z^54 * q^28 + (z^45 + z^34) * q^31 - z^42 * q^36 + (z^42 - z^15) * q^37 + (z^68 - z^60) * q^39 + (-z^38 + z^8) * q^43 - z^57 * q^48 + z^14 * q^49 + (-z^14 + z^6) * q^52 + (z^58 + z^44) * q^57 + (z^67 - z^16) * q^61 - z^2 * q^63 + z^3 * q^64 + (z^28 - z^11) * q^67 + (-z^39 - z^33) * q^73 - z^26 * q^75 + (z^59 - z^4) * q^76 + (-z^19 + z^18) * q^79 - z^59 * q^81 - z^17 * q^84 + (z^43 - z^35) * q^91 + (z^66 - z^8) * q^93 + (-z^68 - z^24) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{3} - q^{4} - q^{7} + q^{9}+O(q^{10}) \) 44 * q + q^3 - q^4 - q^7 + q^9	\( 44 q + q^{3} - q^{4} - q^{7} + q^{9} - q^{12} - 3 q^{13} + q^{16} - q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} + 3 q^{31} + 2 q^{36} - 4 q^{37} + 3 q^{39} - 2 q^{48} + q^{49} - 3 q^{52} + 2 q^{57} - 2 q^{61} - q^{63} + 2 q^{64} + 2 q^{67} - 4 q^{73} - q^{75} - 2 q^{76} - q^{79} + q^{81} + q^{84} - 3 q^{93} + q^{97}+O(q^{100}) \) 44 * q + q^3 - q^4 - q^7 + q^9 - q^12 - 3 * q^13 + q^16 - q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 + 3 * q^31 + 2 * q^36 - 4 * q^37 + 3 * q^39 - 2 * q^48 + q^49 - 3 * q^52 + 2 * q^57 - 2 * q^61 - q^63 + 2 * q^64 + 2 * q^67 - 4 * q^73 - q^75 - 2 * q^76 - q^79 + q^81 + q^84 - 3 * q^93 + q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2919.1.cp.a

Level	$2919$
Weight	$1$
Character orbit	2919.cp
Analytic conductor	$1.457$
Analytic rank	$0$
Dimension	$44$
Projective image	$D_{138}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.45677077188\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Coefficient field:	\(\Q(\zeta_{69})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{44} - x^{43} + x^{41} - x^{40} + x^{38} - x^{37} + x^{35} - x^{34} + x^{32} - x^{31} + x^{29} + \cdots + 1 \) x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{138}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{138} - \cdots)\)

\(n\)	\(974\)	\(1114\)	\(1669\)
\(\chi(n)\)	\(-1\)	\(\zeta_{138}\)	\(\zeta_{138}^{23}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 26.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
973.bi	even	138	1	inner
2919.cp	odd	138	1	inner

$p$	$F_p(T)$
$2$	\( T^{44} \) T^44
$3$	\( T^{44} - T^{43} + \cdots + 1 \) T^44 - T^43 + T^41 - T^40 + T^38 - T^37 + T^35 - T^34 + T^32 - T^31 + T^29 - T^28 + T^26 - T^25 + T^23 - T^22 + T^21 - T^19 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$5$	\( T^{44} \) T^44
$7$	\( T^{44} + T^{43} + \cdots + 1 \) T^44 + T^43 - T^41 - T^40 + T^38 + T^37 - T^35 - T^34 + T^32 + T^31 - T^29 - T^28 + T^26 + T^25 - T^23 - T^22 - T^21 + T^19 + T^18 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$11$	\( T^{44} \) T^44
$13$	\( T^{44} + 3 T^{43} + \cdots + 1 \) T^44 + 3T^43 + 6T^42 + 32T^41 + 78T^40 + 138T^39 + 502T^38 + 1092T^37 + 1770T^36 + 4955T^35 + 9555T^34 + 13800T^33 + 32331T^32 + 55570T^31 + 69717T^30 + 138880T^29 + 208570T^28 + 209070T^27 + 364210T^26 + 480968T^25 + 350274T^24 + 613110T^23 + 737701T^22 + 373770T^21 + 998286T^20 + 1251444T^19 + 759934T^18 + 1532835T^17 + 1942055T^16 + 1221312T^15 + 916022T^14 + 1059416T^13 + 501712T^12 + 141358T^11 + 139341T^10 - 87586T^9 + 42891T^8 + 1029T^7 - 28250T^6 + 207T^5 + 3905T^4 + 790T^3 + 85T^2 + T + 1
$17$	\( T^{44} \) T^44
$19$	\( T^{44} + T^{43} + \cdots + 1 \) T^44 + T^43 - T^41 - T^40 + T^38 + T^37 - 322T^36 - 323T^35 - T^34 + 1012T^33 + 1013T^32 + 139T^31 - 276T^30 - 415T^29 + 28680T^28 + 28727T^27 + 70T^26 - 139011T^25 - 139058T^24 - 22196T^23 + 496661T^22 + 518856T^21 - 539603T^20 - 1411831T^19 - 856887T^18 + 1601306T^17 + 2685134T^16 + 1304421T^15 + 479780T^14 - 905946T^13 + 550943T^12 + 496225T^11 + 16375T^10 + 658558T^9 + 513245T^8 - 15110T^7 - 31624T^6 + 18538T^5 + 620T^4 - 2623T^3 + 575T^2 - 22*T + 1
$23$	\( T^{44} \) T^44
$29$	\( T^{44} \) T^44
$31$	\( T^{44} - 3 T^{43} + \cdots + 1 \) T^44 - 3T^43 + 6T^42 - 9T^41 + 9T^40 - 27T^38 + 81T^37 - 162T^36 + 243T^35 - 243T^34 + 706T^32 - 3130T^31 + 3983T^30 + 7308T^29 - 983T^28 - 2231T^27 - 1306T^26 + 1227T^25 - 1074T^24 - 229T^23 + 3955T^22 - 11175T^21 + 22189T^20 - 21404T^19 + 178034T^18 + 740462T^17 + 2691096T^16 + 5900661T^15 + 9525957T^14 + 11488533T^13 + 10891686T^12 + 8251365T^11 + 5103408T^10 + 2597944T^9 + 1094060T^8 + 388660T^7 + 119226T^6 + 25921T^5 + 2548T^4 + 291T^3 + 154T^2 + 22*T + 1
$37$	\( (T^{22} + 2 T^{21} + \cdots + 1)^{2} \) (T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 64T^16 + 128T^15 + 256T^14 + 512T^13 + 1024T^12 + 2048T^11 + 4073T^10 + 7640T^9 + 11991T^8 + 14115T^7 + 11785T^6 + 6826T^5 + 2704T^4 + 716T^3 + 121T^2 + 12*T + 1)^2
$41$	\( T^{44} \) T^44
$43$	\( T^{44} - 23 T^{42} + \cdots + 529 \) T^44 - 23T^42 + 299T^40 - 2668T^38 + 18055T^36 - 96646T^34 + 421245T^32 - 1516689T^30 + 4557519T^28 - 11467961T^26 + 24199128T^24 - 42662286T^22 + 62532561T^20 - 75392022T^18 + 73935156T^16 - 57768387T^14 + 35301228T^12 - 16195335T^10 + 5446584T^8 - 1210352T^6 + 180389T^4 - 11638*T^2 + 529
$47$	\( T^{44} \) T^44
$53$	\( T^{44} \) T^44
$59$	\( T^{44} \) T^44
$61$	\( T^{44} + 2 T^{43} + \cdots + 1 \) T^44 + 2T^43 - 8T^41 + 7T^40 + 46T^39 + 64T^38 - 56T^37 - 23T^36 + 224T^35 + 540T^34 + 184T^33 + 1244T^32 + 1545T^31 - 1886T^30 - 9952T^29 + 7052T^28 + 14375T^27 + 1209T^26 - 55082T^25 - 36547T^24 + 17905T^23 + 166680T^22 + 261738T^21 + 83697T^20 + 81106T^19 + 324155T^18 + 327106T^17 - 289036T^16 - 26003T^15 - 34270T^14 - 14047T^13 + 488762T^12 - 12650T^11 - 43679T^10 - 169399T^9 - 49703T^8 + 16392T^7 + 11003T^6 + 7429T^5 + 4150T^4 + 664T^3 + 69T^2 + 12T + 1
$67$	\( T^{44} - 2 T^{43} + \cdots + 1 \) T^44 - 2T^43 + 3T^42 - 4T^41 + 5T^40 - 6T^39 + 7T^38 - 8T^37 - 37T^36 + 404T^35 - 495T^34 + 540T^33 - 585T^32 + 630T^31 - 675T^30 + 720T^29 - 52T^28 - 7056T^27 + 109223T^26 - 303390T^25 + 565407T^24 - 833565T^23 + 1102298T^22 - 1371029T^21 + 1635528T^20 - 1892621T^19 + 2849581T^18 - 3247917T^17 + 1714345T^16 - 1303311T^15 + 1174763T^14 - 903431T^13 + 642472T^12 - 293423T^11 + 668000T^10 + 771893T^9 + 866465T^8 + 502542T^7 + 218530T^6 + 65682T^5 + 11850T^4 + 2503T^3 + 647T^2 + 44*T + 1
$71$	\( T^{44} \) T^44
$73$	\( (T^{22} + 2 T^{21} + \cdots + 1)^{2} \) (T^22 + 2T^21 + 4T^20 + 8T^19 - 7T^18 - 14T^17 - 28T^16 - 56T^15 + 72T^14 + 121T^13 + 242T^12 + 484T^11 + 370T^10 + 4T^9 + 8T^8 + 16T^7 + 722T^6 - 5T^5 + 59T^4 + 118T^3 + 75T^2 + 12*T + 1)^2
$79$	\( T^{44} + T^{43} + \cdots + 1 \) T^44 + T^43 - T^41 - T^40 + 46T^39 + 47T^38 + T^37 - 46T^36 - 47T^35 + 1540T^34 + 1357T^33 - 183T^32 - 1540T^31 - 1334T^30 + 23298T^29 + 20331T^28 - 2944T^27 - 23275T^26 - 21619T^25 + 254702T^24 + 433595T^23 + 191106T^22 - 242490T^21 - 465336T^20 + 708194T^19 + 2120969T^18 + 1273648T^17 - 849391T^16 - 2095531T^15 + 1378229T^14 + 1668812T^13 + 1198347T^12 - 499997T^11 - 1303595T^10 - 330672T^9 + 18078T^8 + 120176T^7 + 147155T^6 + 68172T^5 + 21481T^4 + 3564T^3 + 184T^2 - 22T + 1
$83$	\( T^{44} \) T^44
$89$	\( T^{44} \) T^44
$97$	\( T^{44} - T^{43} + \cdots + 1 \) T^44 - T^43 + 23T^42 - 22T^41 + 298T^40 - 276T^39 + 2646T^38 - 2370T^37 + 17779T^36 - 15409T^35 + 94276T^34 - 78867T^33 + 405836T^32 - 326969T^31 + 1437822T^30 - 1110853T^29 + 4230550T^28 - 3119697T^27 + 10357108T^26 - 7237411T^25 + 21079431T^24 - 13842020T^23 + 35424875T^22 - 21582854T^21 + 48690540T^20 - 27107640T^19 + 53809122T^18 - 26701252T^17 + 46827286T^16 - 20127345T^15 + 31068446T^14 - 10950485T^13 + 15183267T^12 - 4243730T^11 + 5255798T^10 - 995324T^9 + 1186110T^8 - 157896T^7 + 182138T^6 - 14375T^5 + 12626T^4 - 1540T^3 + 552T^2 - 24T + 1
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