LMFDB - Newform orbit 2888.1.bm.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(11,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(114))

chi = DirichletCharacter(H, H._module([57, 57, 34]))

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

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

chi := DirichletCharacter("2888.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2888.bm (of order $114$, 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:	$1.44129975648$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{57})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} + \cdots + 1 $ x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{57}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{57} - \cdots)$

$q$-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_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} +O(q^{10}) $ q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9	$ q - \zeta_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} + \cdots + (\zeta_{114}^{46} + \cdots - \zeta_{114}^{7}) q^{99} +O(q^{100}) $ q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9 + (z^16 - z^11) * q^11 + (-z^31 + z^8) * q^12 - z^43 * q^16 + (z^40 - z^31) * q^17 + (-z^55 + z^44 - z^21) * q^18 + z^32 * q^19 + (-z^41 + z^36) * q^22 + (z^56 - z^33) * q^24 - z^29 * q^25 + (-z^45 - z^34 + z^11 + 1) * q^27 - z^11 * q^32 + (z^54 - z^49 - z^31 + z^26) * q^33 + (z^56 + z^8) * q^34 + (z^46 - z^23 + z^12) * q^36 + q^38 + (-z^39 - z^5) * q^41 + (z^52 + z^40) * q^43 + (-z^9 + z^4) * q^44 + (z^24 - z) * q^48 - z^33 * q^49 + z^54 * q^50 + (-z^55 + z^46 - z^21 + z^12) * q^51 + (z^36 - z^25 - z^13 + z^2) * q^54 + (-z^47 - z^13) * q^57 + (-z^35 - z^9) * q^59 + z^36 * q^64 + (z^56 - z^51 + z^22 - z^17) * q^66 + (z^52 - z^3) * q^67 + (-z^33 + z^24) * q^68 + (z^48 - z^37 + z^14) * q^72 + (z^42 - z^29) * q^73 + (z^44 + z^10) * q^75 - z^25 * q^76 + (z^49 + z^38 - z^26 + z^15 - z^3) * q^81 + (z^30 - z^7) * q^82 + (z^38 + z^28) * q^83 + (z^20 + z^8) * q^86 + (z^34 - z^29) * q^88 + (z^26 - z^17) * q^89 + (-z^49 + z^26) * q^96 + z * q^97 - z * q^98 + (z^46 - z^41 - z^35 + z^30 + z^12 - z^7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9}+O(q^{10}) $ 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9	$ 36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9} + 2 q^{11} + 2 q^{12} + q^{16} + 2 q^{17} + q^{19} - q^{22} - q^{24} + q^{25} + 36 q^{27} + q^{32} + q^{33} + 2 q^{34} + 36 q^{38} - q^{41} + 2 q^{43} - q^{44} - q^{48} - 2 q^{49} - 2 q^{50} - 2 q^{51} + q^{54} + 2 q^{57} - q^{59} - 2 q^{64} + q^{66} - q^{67} - 4 q^{68} - q^{73} + 2 q^{75} + q^{76} - 20 q^{81} - q^{82} - 17 q^{83} + 2 q^{86} + 2 q^{88} + 2 q^{89} + 2 q^{96} - q^{97} + q^{98}+O(q^{100}) $ 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9 + 2 * q^11 + 2 * q^12 + q^16 + 2 * q^17 + q^19 - q^22 - q^24 + q^25 + 36 * q^27 + q^32 + q^33 + 2 * q^34 + 36 * q^38 - q^41 + 2 * q^43 - q^44 - q^48 - 2 * q^49 - 2 * q^50 - 2 * q^51 + q^54 + 2 * q^57 - q^59 - 2 * q^64 + q^66 - q^67 - 4 * q^68 - q^73 + 2 * q^75 + q^76 - 20 * q^81 - q^82 - 17 * q^83 + 2 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^96 - q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times$.

$n$	$1445$	$2167$	$2529$
$\chi(n)$	$-1$	$-1$	$\zeta_{114}^{50}$

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} $

11.1

−0.754107	+	0.656752i
−0.592235	+	0.805765i
0.851919	−	0.523673i
0.975796	+	0.218681i
−0.926494	+	0.376309i
−0.191711	−	0.981451i
0.975796	−	0.218681i
−0.821778	+	0.569808i
−0.998482	+	0.0550878i
0.851919	+	0.523673i
0.993931	+	0.110008i
0.137354	−	0.990522i
−0.962268	−	0.272103i
−0.962268	+	0.272103i
0.904357	+	0.426776i
0.635724	+	0.771917i
−0.821778	−	0.569808i
0.451533	−	0.892254i
0.716783	+	0.697297i
−0.998482	−	0.0550878i

−0.592235

−

0.805765i

−0.254515

−

1.83543i

−0.298515

0.954405i

−1.32819

1.29208i

0.945817

−

0.324699i

−2.34174

0.662181i

83.1

0.137354

−

0.990522i

−0.582579

1.86261i

−0.962268

−

0.272103i

1.76494

0.832894i

−0.401695

0.915773i

−2.30814

−

1.60043i

163.1

0.350638

−

0.936511i

−0.901695

−

1.78180i

−0.754107

−

0.656752i

−1.98484

0.219682i

−0.879474

0.475947i

−1.76952

2.40751i

235.1

0.716783

−

0.697297i

−1.48636

0.701431i

0.0275543

−

0.999620i

−0.576292

1.53921i

−0.677282

−

0.735724i

1.08154

−

1.31324i

315.1

0.975796

−

0.218681i

−1.37947

−

1.34197i

0.904357

−

0.426776i

−1.63955

−

1.00783i

0.789141

−

0.614213i

0.0745025

2.70281i

387.1

0.993931

−

0.110008i

−0.254515

−

0.103375i

0.975796

−

0.218681i

−0.264342

−

0.0747488i

0.945817

−

0.324699i

−0.662691

−

0.644676i

467.1

0.716783

0.697297i

−1.48636

−

0.701431i

0.0275543

0.999620i

−0.576292

−

1.53921i

−0.677282

0.735724i

1.08154

1.31324i

539.1

0.851919

0.523673i

0.445817

1.19072i

0.451533

0.892254i

−0.243750

1.24786i

−0.0825793

0.996584i

−0.464966

0.404939i

619.1

−0.191711

0.981451i

−1.17728

−

0.130301i

−0.926494

−

0.376309i

0.353582

−

1.13046i

0.546948

−

0.837166i

0.393217

0.0881220i

691.1

0.350638

0.936511i

−0.901695

1.78180i

−0.754107

0.656752i

−1.98484

−

0.219682i

−0.879474

−

0.475947i

−1.76952

−

2.40751i

771.1

−0.926494

0.376309i

−0.582579

0.130559i

0.716783

−

0.697297i

0.490626

−

0.340192i

−0.401695

0.915773i

−0.582004

0.274654i

843.1

−0.298515

0.954405i

−1.37947

0.390078i

−0.821778

−

0.569808i

0.0395009

−

1.43302i

0.789141

−

0.614213i

0.898868

−

0.552532i

923.1

−0.821778

−

0.569808i

0.0469482

−

0.0288589i

0.350638

0.936511i

−0.0550250

−

0.00303581i

0.245485

−

0.969400i

−0.450162

0.889544i

995.1

−0.821778

0.569808i

0.0469482

0.0288589i

0.350638

−

0.936511i

−0.0550250

0.00303581i

0.245485

0.969400i

−0.450162

−

0.889544i

1075.1

0.0275543

−

0.999620i

0.445817

−

0.541326i

−0.998482

−

0.0550878i

−0.528836

−

0.460564i

−0.0825793

0.996584i

0.0974299

0.498787i

1147.1

−0.998482

−

0.0550878i

0.289141

1.48024i

0.993931

0.110008i

−0.207158

−

1.49392i

−0.986361

−

0.164595i

−1.18101

0.479684i

1227.1

0.851919

−

0.523673i

0.445817

−

1.19072i

0.451533

−

0.892254i

−0.243750

−

1.24786i

−0.0825793

−

0.996584i

−0.464966

−

0.404939i

1299.1

−0.754107

−

0.656752i

−1.17728

1.60175i

0.137354

0.990522i

1.93975

−

0.434708i

0.546948

−

0.837166i

−0.881094

−

2.81701i

1379.1

0.904357

0.426776i

0.0469482

−

1.70319i

0.635724

0.771917i

0.769340

−

1.52026i

0.245485

0.969400i

−1.90018

−

0.104836i

1451.1

−0.191711

−

0.981451i

−1.17728

0.130301i

−0.926494

0.376309i

0.353582

1.13046i

0.546948

0.837166i

0.393217

−

0.0881220i

See all 36 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
361.i	even	57	1	inner
2888.bm	odd	114	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2888.1.bm.a	✓	36
8.d	odd	2	1	CM	2888.1.bm.a	✓	36
361.i	even	57	1	inner	2888.1.bm.a	✓	36
2888.bm	odd	114	1	inner	2888.1.bm.a	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2888.1.bm.a	✓	36	1.a	even	1	1	trivial
2888.1.bm.a	✓	36	8.d	odd	2	1	CM
2888.1.bm.a	✓	36	361.i	even	57	1	inner
2888.1.bm.a	✓	36	2888.bm	odd	114	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2888, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{36} - T^{35} + \cdots + 1 $ T^36 - T^35 + T^33 - T^32 + T^30 - T^29 + T^27 - T^26 + T^24 - T^23 + T^21 - T^20 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$3$	$ T^{36} + 20 T^{35} + \cdots + 1 $ T^36 + 20T^35 + 209T^34 + 1500T^33 + 8245T^32 + 36784T^31 + 138036T^30 + 446216T^29 + 1263614T^28 + 3173360T^27 + 7132010T^26 + 14442584T^25 + 26485146T^24 + 44141656T^23 + 67023184T^22 + 92835576T^21 + 117350915T^20 + 135312908T^19 + 142153251T^18 + 135837194T^17 + 117875182T^16 + 92835728T^15 + 66501007T^14 + 43623203T^13 + 26477242T^12 + 14898869T^11 + 7545811T^10 + 3206458T^9 + 1067667T^8 + 309549T^7 + 121734T^6 + 56373T^5 + 17004T^4 + 2298T^3 - 18T + 1
$5$	$ T^{36} $ T^36
$7$	$ T^{36} $ T^36
$11$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 3T^34 + 15T^33 - 33T^32 + 51T^31 + 159T^30 - 369T^29 + 579T^28 + 902T^27 - 2383T^26 + 3826T^25 + 3908T^24 - 11642T^23 + 19148T^22 + 7546T^21 - 34240T^20 + 68572T^19 - 9006T^18 - 50558T^17 + 142612T^16 - 66801T^15 - 8630T^14 + 24135T^13 + 171545T^12 - 369885T^11 + 387725T^10 - 269829T^9 + 167703T^8 - 69187T^7 + 30559T^6 - 82T^5 + 2456T^4 + 357T^3 + 345T^2 + 36*T + 1
$13$	$ T^{36} $ T^36
$17$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 8T^33 + 3T^32 - 38T^31 + 64T^30 + 24T^29 - 57T^28 - 134T^27 + 496T^26 - 456T^25 + 638T^24 - 459T^23 - 1615T^22 + 5066T^21 + 4897T^20 - 4693T^19 - 4749T^18 + 28272T^17 - 15166T^16 - 87T^15 + 23655T^14 - 48330T^13 + 43194T^12 - 7144T^11 + 18763T^10 - 19913T^9 - 13357T^8 + 1676T^7 + 1702T^6 + 1349T^5 + 1628T^4 + 335T^3 + 38T^2 + 9T + 1
$19$	$ T^{36} - T^{35} + \cdots + 1 $ T^36 - T^35 + T^33 - T^32 + T^30 - T^29 + T^27 - T^26 + T^24 - T^23 + T^21 - T^20 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$23$	$ T^{36} $ T^36
$29$	$ T^{36} $ T^36
$31$	$ T^{36} $ T^36
$37$	$ T^{36} $ T^36
$41$	$ T^{36} + T^{35} + \cdots + 1 $ T^36 + T^35 - T^33 - T^32 + T^30 - 189T^29 - 190T^28 - T^27 + 417T^26 + 513T^25 + 96T^24 - 303T^23 + 9500T^22 + 9822T^21 + 284T^20 - 31122T^19 - 40412T^18 - 9291T^17 + 72275T^16 - 23124T^15 - 88179T^14 - 87437T^13 + 114818T^12 + 250629T^11 + 142765T^10 - 25195T^9 + 9291T^8 + 44461T^7 + 10071T^6 + 2793T^5 + 4046T^4 - 989T^3 + 399T^2 - 37T + 1
$43$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 8T^33 - 16T^32 + 64T^30 - 71T^29 + 19T^28 + 208T^27 - 492T^26 + 152T^25 + 1664T^24 - 3917T^23 + 4161T^22 - 1375T^21 + 2294T^20 - 1482T^19 - 5813T^18 + 17556T^17 - 9694T^16 - 38201T^15 + 11571T^14 + 11254T^13 + 881T^12 - 2375T^11 + 1815T^10 + 9936T^9 + 32756T^8 + 30480T^7 + 19201T^6 + 7543T^5 + 2046T^4 + 202T^3 + 19T^2 + 9*T + 1
$47$	$ T^{36} $ T^36
$53$	$ T^{36} $ T^36
$59$	$ T^{36} + T^{35} + \cdots + 1 $ T^36 + T^35 - T^33 - 39T^32 - 38T^31 + T^30 + 39T^29 + 627T^28 + 664T^27 + 37T^26 - 627T^25 - 5452T^24 - 5832T^23 - 361T^22 + 5471T^21 + 27682T^20 + 22667T^19 + 381T^18 - 22287T^17 - 78262T^16 - 25290T^15 + 121923T^14 + 145788T^13 + 97509T^12 - 129865T^11 - 28463T^10 + 94429T^9 + 79781T^8 + 5207T^7 + 14593T^6 + 16112T^5 + 8948T^4 + 1880T^3 + 152T^2 - 18*T + 1
$61$	$ T^{36} $ T^36
$67$	$ T^{36} + T^{35} + \cdots + 1 $ T^36 + T^35 - T^33 - T^32 + T^30 + 58T^29 - 209T^28 - 248T^27 - 39T^26 + 209T^25 + 248T^24 + T^23 + 2736T^22 - 4276T^21 + 10430T^20 + 12312T^19 + 2224T^18 - 10089T^17 - 13396T^16 + 16909T^15 + 130302T^14 + 51301T^13 - 208600T^12 - 271814T^11 - 55386T^10 + 221976T^9 + 337307T^8 + 251067T^7 + 124945T^6 + 41743T^5 + 10677T^4 + 2602T^3 + 532T^2 + 39T + 1
$71$	$ T^{36} $ T^36
$73$	$ T^{36} + T^{35} + \cdots + 1 $ T^36 + T^35 - T^33 - T^32 - 76T^31 - 75T^30 + T^29 + 95T^28 + 94T^27 + 2089T^26 + 1995T^25 - 75T^24 - 2469T^23 - 2394T^22 - 25119T^21 - 22402T^20 + 2033T^19 + 18317T^18 + 16283T^17 + 126482T^16 + 123917T^15 + 2128T^14 - 24319T^13 - 25326T^12 - 216524T^11 - 98421T^10 + 124259T^9 + 123044T^8 + 12180T^7 - 19170T^6 - 6840T^5 + 1823T^4 - 362T^3 + 304T^2 + 20T + 1
$79$	$ T^{36} $ T^36
$83$	$ T^{36} + 17 T^{35} + \cdots + 1 $ T^36 + 17T^35 + 155T^34 + 984T^33 + 4831T^32 + 19393T^31 + 65880T^30 + 193811T^29 + 501932T^28 + 1158287T^27 + 2403758T^26 + 4518131T^25 + 7734552T^24 + 12111859T^23 + 17408404T^22 + 23025381T^21 + 28079357T^20 + 31613474T^19 + 32884953T^18 + 31613474T^17 + 28079338T^16 + 23025077T^15 + 17408062T^14 + 12116723T^13 + 7748422T^12 + 4512659T^11 + 2350444T^10 + 1119223T^9 + 540996T^8 + 247125T^7 + 71352T^6 + 5523T^5 - 33T^4 + 1326T^3 + 459T^2 + 36*T + 1
$89$	$ T^{36} - 2 T^{35} + \cdots + 1 $ T^36 - 2T^35 + 8T^33 + 3T^32 - 38T^31 + 64T^30 + 24T^29 - 57T^28 - 134T^27 + 496T^26 - 456T^25 + 638T^24 - 459T^23 - 1615T^22 + 5066T^21 + 4897T^20 - 4693T^19 - 4749T^18 + 28272T^17 - 15166T^16 - 87T^15 + 23655T^14 - 48330T^13 + 43194T^12 - 7144T^11 + 18763T^10 - 19913T^9 - 13357T^8 + 1676T^7 + 1702T^6 + 1349T^5 + 1628T^4 + 335T^3 + 38T^2 + 9T + 1
$97$	$ T^{36} + T^{35} + \cdots + 1 $ T^36 + T^35 - T^33 - T^32 + T^30 + T^29 - T^27 - T^26 + T^24 + T^23 - T^21 - T^20 + T^18 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2888.bm (of order \(114\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} +O(q^{10}) \) q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9	\( q - \zeta_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} + \cdots + (\zeta_{114}^{46} + \cdots - \zeta_{114}^{7}) q^{99} +O(q^{100}) \) q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9 + (z^16 - z^11) * q^11 + (-z^31 + z^8) * q^12 - z^43 * q^16 + (z^40 - z^31) * q^17 + (-z^55 + z^44 - z^21) * q^18 + z^32 * q^19 + (-z^41 + z^36) * q^22 + (z^56 - z^33) * q^24 - z^29 * q^25 + (-z^45 - z^34 + z^11 + 1) * q^27 - z^11 * q^32 + (z^54 - z^49 - z^31 + z^26) * q^33 + (z^56 + z^8) * q^34 + (z^46 - z^23 + z^12) * q^36 + q^38 + (-z^39 - z^5) * q^41 + (z^52 + z^40) * q^43 + (-z^9 + z^4) * q^44 + (z^24 - z) * q^48 - z^33 * q^49 + z^54 * q^50 + (-z^55 + z^46 - z^21 + z^12) * q^51 + (z^36 - z^25 - z^13 + z^2) * q^54 + (-z^47 - z^13) * q^57 + (-z^35 - z^9) * q^59 + z^36 * q^64 + (z^56 - z^51 + z^22 - z^17) * q^66 + (z^52 - z^3) * q^67 + (-z^33 + z^24) * q^68 + (z^48 - z^37 + z^14) * q^72 + (z^42 - z^29) * q^73 + (z^44 + z^10) * q^75 - z^25 * q^76 + (z^49 + z^38 - z^26 + z^15 - z^3) * q^81 + (z^30 - z^7) * q^82 + (z^38 + z^28) * q^83 + (z^20 + z^8) * q^86 + (z^34 - z^29) * q^88 + (z^26 - z^17) * q^89 + (-z^49 + z^26) * q^96 + z * q^97 - z * q^98 + (z^46 - z^41 - z^35 + z^30 + z^12 - z^7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9}+O(q^{10}) \) 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9	\( 36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9} + 2 q^{11} + 2 q^{12} + q^{16} + 2 q^{17} + q^{19} - q^{22} - q^{24} + q^{25} + 36 q^{27} + q^{32} + q^{33} + 2 q^{34} + 36 q^{38} - q^{41} + 2 q^{43} - q^{44} - q^{48} - 2 q^{49} - 2 q^{50} - 2 q^{51} + q^{54} + 2 q^{57} - q^{59} - 2 q^{64} + q^{66} - q^{67} - 4 q^{68} - q^{73} + 2 q^{75} + q^{76} - 20 q^{81} - q^{82} - 17 q^{83} + 2 q^{86} + 2 q^{88} + 2 q^{89} + 2 q^{96} - q^{97} + q^{98}+O(q^{100}) \) 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9 + 2 * q^11 + 2 * q^12 + q^16 + 2 * q^17 + q^19 - q^22 - q^24 + q^25 + 36 * q^27 + q^32 + q^33 + 2 * q^34 + 36 * q^38 - q^41 + 2 * q^43 - q^44 - q^48 - 2 * q^49 - 2 * q^50 - 2 * q^51 + q^54 + 2 * q^57 - q^59 - 2 * q^64 + q^66 - q^67 - 4 * q^68 - q^73 + 2 * q^75 + q^76 - 20 * q^81 - q^82 - 17 * q^83 + 2 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^96 - q^97 + q^98

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	2888.1.bm.a

Level	$2888$
Weight	$1$
Character orbit	2888.bm
Analytic conductor	$1.441$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{57}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.44129975648\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Coefficient field:	\(\Q(\zeta_{57})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} + \cdots + 1 \) x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + 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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{57}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{57} - \cdots)\)

\(n\)	\(1445\)	\(2167\)	\(2529\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{114}^{50}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
361.i	even	57	1	inner
2888.bm	odd	114	1	inner

$p$	$F_p(T)$
$2$	\( T^{36} - T^{35} + \cdots + 1 \) T^36 - T^35 + T^33 - T^32 + T^30 - T^29 + T^27 - T^26 + T^24 - T^23 + T^21 - T^20 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$3$	\( T^{36} + 20 T^{35} + \cdots + 1 \) T^36 + 20T^35 + 209T^34 + 1500T^33 + 8245T^32 + 36784T^31 + 138036T^30 + 446216T^29 + 1263614T^28 + 3173360T^27 + 7132010T^26 + 14442584T^25 + 26485146T^24 + 44141656T^23 + 67023184T^22 + 92835576T^21 + 117350915T^20 + 135312908T^19 + 142153251T^18 + 135837194T^17 + 117875182T^16 + 92835728T^15 + 66501007T^14 + 43623203T^13 + 26477242T^12 + 14898869T^11 + 7545811T^10 + 3206458T^9 + 1067667T^8 + 309549T^7 + 121734T^6 + 56373T^5 + 17004T^4 + 2298T^3 - 18T + 1
$5$	\( T^{36} \) T^36
$7$	\( T^{36} \) T^36
$11$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 3T^34 + 15T^33 - 33T^32 + 51T^31 + 159T^30 - 369T^29 + 579T^28 + 902T^27 - 2383T^26 + 3826T^25 + 3908T^24 - 11642T^23 + 19148T^22 + 7546T^21 - 34240T^20 + 68572T^19 - 9006T^18 - 50558T^17 + 142612T^16 - 66801T^15 - 8630T^14 + 24135T^13 + 171545T^12 - 369885T^11 + 387725T^10 - 269829T^9 + 167703T^8 - 69187T^7 + 30559T^6 - 82T^5 + 2456T^4 + 357T^3 + 345T^2 + 36*T + 1
$13$	\( T^{36} \) T^36
$17$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 8T^33 + 3T^32 - 38T^31 + 64T^30 + 24T^29 - 57T^28 - 134T^27 + 496T^26 - 456T^25 + 638T^24 - 459T^23 - 1615T^22 + 5066T^21 + 4897T^20 - 4693T^19 - 4749T^18 + 28272T^17 - 15166T^16 - 87T^15 + 23655T^14 - 48330T^13 + 43194T^12 - 7144T^11 + 18763T^10 - 19913T^9 - 13357T^8 + 1676T^7 + 1702T^6 + 1349T^5 + 1628T^4 + 335T^3 + 38T^2 + 9T + 1
$19$	\( T^{36} - T^{35} + \cdots + 1 \) T^36 - T^35 + T^33 - T^32 + T^30 - T^29 + T^27 - T^26 + T^24 - T^23 + T^21 - T^20 + T^18 - T^16 + T^15 - T^13 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$23$	\( T^{36} \) T^36
$29$	\( T^{36} \) T^36
$31$	\( T^{36} \) T^36
$37$	\( T^{36} \) T^36
$41$	\( T^{36} + T^{35} + \cdots + 1 \) T^36 + T^35 - T^33 - T^32 + T^30 - 189T^29 - 190T^28 - T^27 + 417T^26 + 513T^25 + 96T^24 - 303T^23 + 9500T^22 + 9822T^21 + 284T^20 - 31122T^19 - 40412T^18 - 9291T^17 + 72275T^16 - 23124T^15 - 88179T^14 - 87437T^13 + 114818T^12 + 250629T^11 + 142765T^10 - 25195T^9 + 9291T^8 + 44461T^7 + 10071T^6 + 2793T^5 + 4046T^4 - 989T^3 + 399T^2 - 37T + 1
$43$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 8T^33 - 16T^32 + 64T^30 - 71T^29 + 19T^28 + 208T^27 - 492T^26 + 152T^25 + 1664T^24 - 3917T^23 + 4161T^22 - 1375T^21 + 2294T^20 - 1482T^19 - 5813T^18 + 17556T^17 - 9694T^16 - 38201T^15 + 11571T^14 + 11254T^13 + 881T^12 - 2375T^11 + 1815T^10 + 9936T^9 + 32756T^8 + 30480T^7 + 19201T^6 + 7543T^5 + 2046T^4 + 202T^3 + 19T^2 + 9*T + 1
$47$	\( T^{36} \) T^36
$53$	\( T^{36} \) T^36
$59$	\( T^{36} + T^{35} + \cdots + 1 \) T^36 + T^35 - T^33 - 39T^32 - 38T^31 + T^30 + 39T^29 + 627T^28 + 664T^27 + 37T^26 - 627T^25 - 5452T^24 - 5832T^23 - 361T^22 + 5471T^21 + 27682T^20 + 22667T^19 + 381T^18 - 22287T^17 - 78262T^16 - 25290T^15 + 121923T^14 + 145788T^13 + 97509T^12 - 129865T^11 - 28463T^10 + 94429T^9 + 79781T^8 + 5207T^7 + 14593T^6 + 16112T^5 + 8948T^4 + 1880T^3 + 152T^2 - 18*T + 1
$61$	\( T^{36} \) T^36
$67$	\( T^{36} + T^{35} + \cdots + 1 \) T^36 + T^35 - T^33 - T^32 + T^30 + 58T^29 - 209T^28 - 248T^27 - 39T^26 + 209T^25 + 248T^24 + T^23 + 2736T^22 - 4276T^21 + 10430T^20 + 12312T^19 + 2224T^18 - 10089T^17 - 13396T^16 + 16909T^15 + 130302T^14 + 51301T^13 - 208600T^12 - 271814T^11 - 55386T^10 + 221976T^9 + 337307T^8 + 251067T^7 + 124945T^6 + 41743T^5 + 10677T^4 + 2602T^3 + 532T^2 + 39T + 1
$71$	\( T^{36} \) T^36
$73$	\( T^{36} + T^{35} + \cdots + 1 \) T^36 + T^35 - T^33 - T^32 - 76T^31 - 75T^30 + T^29 + 95T^28 + 94T^27 + 2089T^26 + 1995T^25 - 75T^24 - 2469T^23 - 2394T^22 - 25119T^21 - 22402T^20 + 2033T^19 + 18317T^18 + 16283T^17 + 126482T^16 + 123917T^15 + 2128T^14 - 24319T^13 - 25326T^12 - 216524T^11 - 98421T^10 + 124259T^9 + 123044T^8 + 12180T^7 - 19170T^6 - 6840T^5 + 1823T^4 - 362T^3 + 304T^2 + 20T + 1
$79$	\( T^{36} \) T^36
$83$	\( T^{36} + 17 T^{35} + \cdots + 1 \) T^36 + 17T^35 + 155T^34 + 984T^33 + 4831T^32 + 19393T^31 + 65880T^30 + 193811T^29 + 501932T^28 + 1158287T^27 + 2403758T^26 + 4518131T^25 + 7734552T^24 + 12111859T^23 + 17408404T^22 + 23025381T^21 + 28079357T^20 + 31613474T^19 + 32884953T^18 + 31613474T^17 + 28079338T^16 + 23025077T^15 + 17408062T^14 + 12116723T^13 + 7748422T^12 + 4512659T^11 + 2350444T^10 + 1119223T^9 + 540996T^8 + 247125T^7 + 71352T^6 + 5523T^5 - 33T^4 + 1326T^3 + 459T^2 + 36*T + 1
$89$	\( T^{36} - 2 T^{35} + \cdots + 1 \) T^36 - 2T^35 + 8T^33 + 3T^32 - 38T^31 + 64T^30 + 24T^29 - 57T^28 - 134T^27 + 496T^26 - 456T^25 + 638T^24 - 459T^23 - 1615T^22 + 5066T^21 + 4897T^20 - 4693T^19 - 4749T^18 + 28272T^17 - 15166T^16 - 87T^15 + 23655T^14 - 48330T^13 + 43194T^12 - 7144T^11 + 18763T^10 - 19913T^9 - 13357T^8 + 1676T^7 + 1702T^6 + 1349T^5 + 1628T^4 + 335T^3 + 38T^2 + 9T + 1
$97$	\( T^{36} + T^{35} + \cdots + 1 \) T^36 + T^35 - T^33 - T^32 + T^30 + T^29 - T^27 - T^26 + T^24 + T^23 - T^21 - T^20 + T^18 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
show more
show less