LMFDB - Newform orbit 2768.1.w.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2768,1,Mod(15,2768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2768, base_ring=CyclotomicField(86))

chi = DirichletCharacter(H, H._module([43, 0, 33]))

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

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

chi := DirichletCharacter("2768.15");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2768 = 2^{4} \cdot 173 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2768.w (of order $86$, degree $42$, 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:	$1.38141195497$
Analytic rank:	$0$
Dimension:	$42$
Coefficient field:	$\Q(\zeta_{86})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{42} - x^{41} + x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} + \cdots + 1 $ x^42 - x^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{86}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{86} - \cdots)$

$q$-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_{86}^{23} + \zeta_{86}^{2}) q^{5} - \zeta_{86}^{14} q^{9} +O(q^{10}) $ q + (z^23 + z^2) * q^5 - z^14 * q^9	$ q + (\zeta_{86}^{23} + \zeta_{86}^{2}) q^{5} - \zeta_{86}^{14} q^{9} + (\zeta_{86}^{18} + \zeta_{86}^{8}) q^{13} + ( - \zeta_{86}^{42} - \zeta_{86}^{7}) q^{17} + (\zeta_{86}^{25} + \cdots - \zeta_{86}^{3}) q^{25} + \cdots + ( - \zeta_{86}^{6} - \zeta_{86}) q^{97} +O(q^{100}) $ q + (z^23 + z^2) * q^5 - z^14 * q^9 + (z^18 + z^8) * q^13 + (-z^42 - z^7) * q^17 + (z^25 + z^4 - z^3) * q^25 + (z^35 + z^11) * q^29 + (z^24 - z^17) * q^37 + (-z^33 - z^29) * q^41 + (-z^37 - z^16) * q^45 + z^19 * q^49 + (z^40 - z^34) * q^53 + (z^32 + z^5) * q^61 + (z^41 + z^31 + z^20 + z^10) * q^65 + (z^38 - z^21) * q^73 + z^28 * q^81 + (-z^30 + z^22 - z^9 + z) * q^85 + (z^30 + z^12) * q^89 + (-z^6 - z) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + q^{9}+O(q^{10}) $ 42 * q + q^9	$ 42 q + q^{9} - 2 q^{13} - q^{25} + 2 q^{29} - 2 q^{37} - 2 q^{41} + q^{49} - 2 q^{73} - q^{81} - 2 q^{89}+O(q^{100}) $ 42 * q + q^9 - 2 * q^13 - q^25 + 2 * q^29 - 2 * q^37 - 2 * q^41 + q^49 - 2 * q^73 - q^81 - 2 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2768\mathbb{Z}\right)^\times$.

$n$	$693$	$1039$	$1905$
$\chi(n)$	$1$	$-1$	$-\zeta_{86}^{26}$

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} $

15.1

0.872049	−	0.489418i
0.976076	−	0.217430i
−0.252933	+	0.967484i
0.581859	+	0.813290i
0.934016	−	0.357231i
−0.957601	−	0.288099i
−0.252933	−	0.967484i
0.694074	−	0.719903i
0.181637	+	0.983366i
0.581859	−	0.813290i
0.0365220	−	0.999333i
−0.833998	−	0.551768i
−0.639673	+	0.768647i
−0.957601	+	0.288099i
−0.989343	−	0.145601i
−0.989343	+	0.145601i
0.934016	+	0.357231i
0.0365220	+	0.999333i
0.322880	−	0.946440i
0.322880	+	0.946440i

1.21501

−

0.133690i

−0.639673

0.768647i

31.1

1.22833

0.521983i

0.997332

0.0729953i

159.1

−1.26315

−

1.40976i

−0.905448

−

0.424457i

207.1

−1.31222

1.09204i

−0.744772

−

0.667319i

303.1

0.223832

−

1.52091i

−0.391105

−

0.920346i

367.1

−0.0714504

0.127311i

0.581859

0.813290i

383.1

−1.26315

1.40976i

−0.905448

0.424457i

543.1

0.897494

−

0.642102i

−0.252933

−

0.967484i

559.1

−0.0619667

0.846649i

−0.833998

−

0.551768i

575.1

−1.31222

−

1.09204i

−0.744772

0.667319i

607.1

−1.74210

0.594323i

0.872049

0.489418i

943.1

−0.248568

0.151699i

0.322880

−

0.946440i

991.1

−0.434570

−

0.0158820i

−0.957601

−

0.288099i

1071.1

−0.0714504

−

0.127311i

0.581859

−

0.813290i

1087.1

1.93368

0.505529i

0.457242

−

0.889342i

1151.1

1.93368

−

0.505529i

0.457242

0.889342i

1215.1

0.223832

1.52091i

−0.391105

0.920346i

1327.1

−1.74210

−

0.594323i

0.872049

−

0.489418i

1439.1

−1.74910

−

0.323075i

−0.109371

−

0.994001i

1535.1

−1.74910

0.323075i

−0.109371

0.994001i

See all 42 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
173.e	even	86	1	inner
692.h	odd	86	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2768.1.w.a	✓	42
4.b	odd	2	1	CM	2768.1.w.a	✓	42
173.e	even	86	1	inner	2768.1.w.a	✓	42
692.h	odd	86	1	inner	2768.1.w.a	✓	42

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2768.1.w.a	✓	42	1.a	even	1	1	trivial
2768.1.w.a	✓	42	4.b	odd	2	1	CM
2768.1.w.a	✓	42	173.e	even	86	1	inner
2768.1.w.a	✓	42	692.h	odd	86	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2768, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{42} $ T^42
$3$	$ T^{42} $ T^42
$5$	$ T^{42} + 43 T^{31} + \cdots + 43 $ T^42 + 43T^31 + 2838T^30 + 2365T^29 + 43T^28 + 473T^20 - 66693T^19 + 614900T^18 - 953009T^17 + 334884T^16 - 26875T^15 + 344T^14 + 946T^9 + 30745T^8 + 171054T^7 + 333336T^6 + 304612T^5 + 150930T^4 + 43645T^3 + 7568T^2 + 774T + 43
$7$	$ T^{42} $ T^42
$11$	$ T^{42} $ T^42
$13$	$ T^{42} + 2 T^{41} + \cdots + 1 $ T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 - 87T^35 - 174T^34 - 348T^33 - 696T^32 - 1392T^31 - 2784T^30 - 5568T^29 + 2753T^28 + 5506T^27 + 9507T^26 + 19014T^25 + 38028T^24 + 75669T^23 + 151338T^22 + 27476T^21 + 54952T^20 - 24858T^19 - 49716T^18 - 99948T^17 - 225481T^16 - 450962T^15 + 167744T^14 + 335488T^13 + 11356T^12 + 22669T^11 + 71095T^10 + 65521T^9 + 130999T^8 + 18360T^7 + 36720T^6 + 1157T^5 + 2916T^4 + 586T^3 + 54T^2 - 21*T + 1
$17$	$ T^{42} + 43 T^{34} + \cdots + 43 $ T^42 + 43T^34 - 5676T^29 + 602T^26 + 20468T^24 + 172602T^21 - 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 - 349977T^13 + 560462T^11 + 3913T^10 - 688T^9 + 409790T^8 + 39947T^6 + 23779T^5 + 43T^4 - 8815T^3 + 301T^2 + 215T + 43
$19$	$ T^{42} $ T^42
$23$	$ T^{42} $ T^42
$29$	$ T^{42} - 2 T^{41} + \cdots + 1 $ T^42 - 2T^41 + 4T^40 - 8T^39 + 16T^38 - 32T^37 + 64T^36 - 128T^35 + 256T^34 - 512T^33 + 1024T^32 - 2048T^31 + 4096T^30 - 7246T^29 + 14492T^28 - 28984T^27 + 27223T^26 - 54446T^25 + 108892T^24 - 46730T^23 + 93460T^22 - 186920T^21 + 40504T^20 - 80535T^19 + 161070T^18 - 17528T^17 + 101749T^16 - 203498T^15 + 256066T^14 + 102768T^13 - 205536T^12 + 454717T^11 + 43575T^10 - 87107T^9 + 166646T^8 + 1592T^7 - 6022T^6 + 12818T^5 + 1239T^4 - 113T^3 + 183T^2 - 22*T + 1
$31$	$ T^{42} $ T^42
$37$	$ T^{42} + 2 T^{41} + \cdots + 1 $ T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 + 128T^35 + 256T^34 + 512T^33 + 1024T^32 + 758T^31 + 1516T^30 + 3032T^29 - 85T^28 - 170T^27 - 340T^26 - 2400T^25 - 4800T^24 - 11793T^23 - 23629T^22 - 47258T^21 + 112529T^20 + 225058T^19 + 450116T^18 - 112074T^17 - 224148T^16 - 448855T^15 + 72843T^14 + 145686T^13 + 202362T^12 + 94049T^11 + 188098T^10 - 1817T^9 + 39065T^8 + 78087T^7 + 2019T^6 + 1243T^5 + 4292T^4 + 27T^3 + 140T^2 + 22*T + 1
$41$	$ T^{42} + 2 T^{41} + \cdots + 1 $ T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 - 87T^35 - 174T^34 - 348T^33 - 696T^32 - 1392T^31 - 2784T^30 - 5568T^29 + 2753T^28 + 5506T^27 + 9507T^26 + 19014T^25 + 38028T^24 + 75669T^23 + 151338T^22 + 27476T^21 + 54952T^20 - 24858T^19 - 49716T^18 - 99948T^17 - 225481T^16 - 450962T^15 + 167744T^14 + 335488T^13 + 11356T^12 + 22669T^11 + 71095T^10 + 65521T^9 + 130999T^8 + 18360T^7 + 36720T^6 + 1157T^5 + 2916T^4 + 586T^3 + 54T^2 - 21*T + 1
$43$	$ T^{42} $ T^42
$47$	$ T^{42} $ T^42
$53$	$ T^{42} - 43 T^{27} + \cdots + 43 $ T^42 - 43T^27 + 6020T^25 - 114036T^23 + 722228T^21 - 2138906T^19 + 3513917T^17 - 3535675T^15 + 2314260T^13 + 301T^12 - 1021293T^11 + 11739T^10 + 309256T^9 + 61490T^8 - 64328T^7 + 76024T^6 + 9030T^5 + 26316T^4 - 817T^3 + 2451T^2 + 43T + 43
$59$	$ T^{42} $ T^42
$61$	$ T^{42} + 301 T^{27} + \cdots + 43 $ T^42 + 301T^27 + 11739T^26 + 61490T^25 + 76024T^24 + 26316T^23 + 2451T^22 + 43T^21 - 43T^13 + 6020T^12 - 114036T^11 + 722228T^10 - 2138906T^9 + 3513917T^8 - 3535675T^7 + 2314260T^6 - 1021293T^5 + 309256T^4 - 64328T^3 + 9030T^2 - 817T + 43
$67$	$ T^{42} $ T^42
$71$	$ T^{42} $ T^42
$73$	$ T^{42} + 2 T^{41} + \cdots + 1 $ T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 - 87T^35 - 174T^34 - 348T^33 - 696T^32 - 1392T^31 - 2784T^30 - 5568T^29 + 2753T^28 + 5506T^27 + 9507T^26 + 19014T^25 + 38028T^24 + 75669T^23 + 151338T^22 + 27476T^21 + 54952T^20 - 24858T^19 - 49716T^18 - 99948T^17 - 225481T^16 - 450962T^15 + 167744T^14 + 335488T^13 + 11356T^12 + 22669T^11 + 71095T^10 + 65521T^9 + 130999T^8 + 18360T^7 + 36720T^6 + 1157T^5 + 2916T^4 + 586T^3 + 54T^2 - 21*T + 1
$79$	$ T^{42} $ T^42
$83$	$ T^{42} $ T^42
$89$	$ T^{42} + 2 T^{41} + \cdots + 1 $ T^42 + 2T^41 + 4T^40 + 8T^39 + 16T^38 + 32T^37 + 64T^36 + 128T^35 + 256T^34 + 512T^33 + 1024T^32 + 758T^31 + 1516T^30 + 3032T^29 - 85T^28 - 170T^27 - 340T^26 - 2400T^25 - 4800T^24 - 11793T^23 - 23629T^22 - 47258T^21 + 112529T^20 + 225058T^19 + 450116T^18 - 112074T^17 - 224148T^16 - 448855T^15 + 72843T^14 + 145686T^13 + 202362T^12 + 94049T^11 + 188098T^10 - 1817T^9 + 39065T^8 + 78087T^7 + 2019T^6 + 1243T^5 + 4292T^4 + 27T^3 + 140T^2 + 22*T + 1
$97$	$ T^{42} + 43 T^{34} + \cdots + 43 $ T^42 + 43T^34 - 5676T^29 + 602T^26 + 20468T^24 + 172602T^21 - 16555T^19 + 3010T^18 + 1016262T^16 + 5031T^14 - 349977T^13 + 560462T^11 + 3913T^10 - 688T^9 + 409790T^8 + 39947T^6 + 23779T^5 + 43T^4 - 8815T^3 + 301T^2 + 215T + 43
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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:	\( N \)	\(=\)	\( 2768 = 2^{4} \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2768.w (of order \(86\), degree \(42\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{86}^{23} + \zeta_{86}^{2}) q^{5} - \zeta_{86}^{14} q^{9} +O(q^{10}) \) q + (z^23 + z^2) * q^5 - z^14 * q^9	\( q + (\zeta_{86}^{23} + \zeta_{86}^{2}) q^{5} - \zeta_{86}^{14} q^{9} + (\zeta_{86}^{18} + \zeta_{86}^{8}) q^{13} + ( - \zeta_{86}^{42} - \zeta_{86}^{7}) q^{17} + (\zeta_{86}^{25} + \cdots - \zeta_{86}^{3}) q^{25} + \cdots + ( - \zeta_{86}^{6} - \zeta_{86}) q^{97} +O(q^{100}) \) q + (z^23 + z^2) * q^5 - z^14 * q^9 + (z^18 + z^8) * q^13 + (-z^42 - z^7) * q^17 + (z^25 + z^4 - z^3) * q^25 + (z^35 + z^11) * q^29 + (z^24 - z^17) * q^37 + (-z^33 - z^29) * q^41 + (-z^37 - z^16) * q^45 + z^19 * q^49 + (z^40 - z^34) * q^53 + (z^32 + z^5) * q^61 + (z^41 + z^31 + z^20 + z^10) * q^65 + (z^38 - z^21) * q^73 + z^28 * q^81 + (-z^30 + z^22 - z^9 + z) * q^85 + (z^30 + z^12) * q^89 + (-z^6 - z) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + q^{9}+O(q^{10}) \) 42 * q + q^9	\( 42 q + q^{9} - 2 q^{13} - q^{25} + 2 q^{29} - 2 q^{37} - 2 q^{41} + q^{49} - 2 q^{73} - q^{81} - 2 q^{89}+O(q^{100}) \) 42 * q + q^9 - 2 * q^13 - q^25 + 2 * q^29 - 2 * q^37 - 2 * q^41 + q^49 - 2 * q^73 - q^81 - 2 * q^89

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