LMFDB - Newform orbit 3675.1.cy.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3675,1,Mod(143,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(84)) chi = DirichletCharacter(H, H._module([42, 63, 62])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3675.143"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3675.cy (of order $84$, degree $24$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.83406392143$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$2$ over $\Q(\zeta_{84})$
Coefficient field:	$\Q(\zeta_{168})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{44} - x^{36} - x^{32} + x^{24} - x^{16} - x^{12} + x^{4} + 1 $ x^48 + x^44 - x^36 - x^32 + x^24 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} + \cdots)$

$q$-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_{168}^{59} q^{3} - \zeta_{168}^{2} q^{4} + \zeta_{168}^{29} q^{7} - \zeta_{168}^{34} q^{9} - \zeta_{168}^{61} q^{12} + ( - \zeta_{168}^{67} + \zeta_{168}^{47}) q^{13} + \zeta_{168}^{4} q^{16} + \cdots + (\zeta_{168}^{41} + \zeta_{168}) q^{97} +O(q^{100}) $ q + z^59 * q^3 - z^2 * q^4 + z^29 * q^7 - z^34 * q^9 - z^61 * q^12 + (-z^67 + z^47) * q^13 + z^4 * q^16 + (z^78 - z^62) * q^19 - z^4 * q^21 + z^9 * q^27 - z^31 * q^28 + (-z^68 + z^44) * q^31 + z^36 * q^36 + (-z^77 + z^45) * q^37 + (z^42 - z^22) * q^39 + (-z^75 + z^3) * q^43 + z^63 * q^48 + z^58 * q^49 + (z^69 - z^49) * q^52 + (-z^53 + z^37) * q^57 + (z^80 - 1) * q^61 - z^63 * q^63 - z^6 * q^64 + (z^81 - z^73) * q^67 + (z^67 - z^15) * q^73 + (-z^80 + z^64) * q^76 + (-z^30 + z^26) * q^79 + z^68 * q^81 + z^6 * q^84 + (z^76 + z^12) * q^91 + (z^43 - z^19) * q^93 + (z^41 + z) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{16} + 4 q^{21} + 8 q^{36} - 44 q^{61} - 4 q^{81} + 4 q^{91}+O(q^{100}) $ 48 * q - 4 * q^16 + 4 * q^21 + 8 * q^36 - 44 * q^61 - 4 * q^81 + 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times$.

$n$	$1177$	$1226$	$2551$
$\chi(n)$	$-\zeta_{168}^{42}$	$-1$	$\zeta_{168}^{76}$

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} $

143.1

−0.185912	−	0.982566i
0.185912	+	0.982566i
−0.185912	+	0.982566i
0.185912	−	0.982566i
0.0373912	−	0.999301i
−0.0373912	+	0.999301i
−0.804598	−	0.593820i
0.804598	+	0.593820i
0.884115	−	0.467269i
−0.884115	+	0.467269i
−0.467269	+	0.884115i
0.467269	−	0.884115i
0.0373912	+	0.999301i
−0.0373912	−	0.999301i
−0.652287	+	0.757972i
0.652287	−	0.757972i
−0.916562	−	0.399892i
0.916562	+	0.399892i
−0.804598	+	0.593820i
0.804598	−	0.593820i

−0.999301

0.0373912i

0.930874

−

0.365341i

0.757972

−

0.652287i

0.997204

−

0.0747301i

143.2

0.999301

−

0.0373912i

0.930874

−

0.365341i

−0.757972

0.652287i

0.997204

−

0.0747301i

257.1

−0.999301

−

0.0373912i

0.930874

0.365341i

0.757972

0.652287i

0.997204

0.0747301i

257.2

0.999301

0.0373912i

0.930874

0.365341i

−0.757972

−

0.652287i

0.997204

0.0747301i

332.1

−0.804598

−

0.593820i

0.997204

0.0747301i

0.884115

−

0.467269i

0.294755

0.955573i

332.2

0.804598

0.593820i

0.997204

0.0747301i

−0.884115

0.467269i

0.294755

0.955573i

593.1

−0.982566

0.185912i

−0.294755

−

0.955573i

−0.916562

0.399892i

0.930874

−

0.365341i

593.2

0.982566

−

0.185912i

−0.294755

−

0.955573i

0.916562

−

0.399892i

0.930874

−

0.365341i

782.1

−0.916562

0.399892i

−0.563320

0.826239i

0.0373912

−

0.999301i

0.680173

−

0.733052i

782.2

0.916562

−

0.399892i

−0.563320

0.826239i

−0.0373912

0.999301i

0.680173

−

0.733052i

857.1

−0.399892

0.916562i

0.563320

0.826239i

−0.999301

0.0373912i

−0.680173

−

0.733052i

857.2

0.399892

−

0.916562i

0.563320

0.826239i

0.999301

−

0.0373912i

−0.680173

−

0.733052i

1118.1

−0.804598

0.593820i

0.997204

−

0.0747301i

0.884115

0.467269i

0.294755

−

0.955573i

1118.2

0.804598

−

0.593820i

0.997204

−

0.0747301i

−0.884115

−

0.467269i

0.294755

−

0.955573i

1193.1

−0.884115

0.467269i

0.149042

0.988831i

−0.982566

−

0.185912i

0.563320

−

0.826239i

1193.2

0.884115

−

0.467269i

0.149042

0.988831i

0.982566

0.185912i

0.563320

−

0.826239i

1307.1

−0.652287

0.757972i

−0.680173

−

0.733052i

−0.804598

0.593820i

−0.149042

−

0.988831i

1307.2

0.652287

−

0.757972i

−0.680173

−

0.733052i

0.804598

−

0.593820i

−0.149042

−

0.988831i

1382.1

−0.982566

−

0.185912i

−0.294755

0.955573i

−0.916562

−

0.399892i

0.930874

0.365341i

1382.2

0.982566

0.185912i

−0.294755

0.955573i

0.916562

0.399892i

0.930874

0.365341i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner
49.h	odd	42	1	inner
147.o	even	42	1	inner
245.u	odd	42	1	inner
245.x	even	84	2	inner
735.bp	even	42	1	inner
735.bs	odd	84	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3675.1.cy.a	✓	48
3.b	odd	2	1	CM	3675.1.cy.a	✓	48
5.b	even	2	1	inner	3675.1.cy.a	✓	48
5.c	odd	4	2	inner	3675.1.cy.a	✓	48
15.d	odd	2	1	inner	3675.1.cy.a	✓	48
15.e	even	4	2	inner	3675.1.cy.a	✓	48
49.h	odd	42	1	inner	3675.1.cy.a	✓	48
147.o	even	42	1	inner	3675.1.cy.a	✓	48
245.u	odd	42	1	inner	3675.1.cy.a	✓	48
245.x	even	84	2	inner	3675.1.cy.a	✓	48
735.bp	even	42	1	inner	3675.1.cy.a	✓	48
735.bs	odd	84	2	inner	3675.1.cy.a	✓	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3675.1.cy.a	✓	48	1.a	even	1	1	trivial
3675.1.cy.a	✓	48	3.b	odd	2	1	CM
3675.1.cy.a	✓	48	5.b	even	2	1	inner
3675.1.cy.a	✓	48	5.c	odd	4	2	inner
3675.1.cy.a	✓	48	15.d	odd	2	1	inner
3675.1.cy.a	✓	48	15.e	even	4	2	inner
3675.1.cy.a	✓	48	49.h	odd	42	1	inner
3675.1.cy.a	✓	48	147.o	even	42	1	inner
3675.1.cy.a	✓	48	245.u	odd	42	1	inner
3675.1.cy.a	✓	48	245.x	even	84	2	inner
3675.1.cy.a	✓	48	735.bp	even	42	1	inner
3675.1.cy.a	✓	48	735.bs	odd	84	2	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3675, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{48} $ T^48
$3$	$ T^{48} + T^{44} + \cdots + 1 $ T^48 + T^44 - T^36 - T^32 + T^24 - T^16 - T^12 + T^4 + 1
$5$	$ T^{48} $ T^48
$7$	$ T^{48} + T^{44} + \cdots + 1 $ T^48 + T^44 - T^36 - T^32 + T^24 - T^16 - T^12 + T^4 + 1
$11$	$ T^{48} $ T^48
$13$	$ T^{48} - 2 T^{44} + \cdots + 1 $ T^48 - 2T^44 + 199T^40 - 1579T^36 + 37161T^32 - 526455T^28 + 3716797T^24 - 12984411T^20 + 44664576T^16 + 16364680T^12 + 4001280T^8 - 3607*T^4 + 1
$17$	$ T^{48} $ T^48
$19$	$ (T^{24} + 11 T^{22} + \cdots + 1)^{2} $ (T^24 + 11T^22 + 77T^20 + 328T^18 + 1018T^16 + 2128T^14 + 3270T^12 + 3283T^10 + 2308T^8 + 804T^6 + 196T^4 + 16*T^2 + 1)^2
$23$	$ T^{48} $ T^48
$29$	$ T^{48} $ T^48
$31$	$ (T^{12} - 7 T^{10} + \cdots + 49)^{4} $ (T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49)^4
$37$	$ T^{48} + 2 T^{44} + \cdots + 1 $ T^48 + 2T^44 - 63T^40 + 1238T^36 + 43328T^32 + 224126T^28 + 807871T^24 + 1387008T^20 + 734709T^16 - 238582T^12 + 15806T^8 + 4*T^4 + 1
$41$	$ T^{48} $ T^48
$43$	$ (T^{24} - 28 T^{20} + \cdots + 2401)^{2} $ (T^24 - 28T^20 + 294T^16 - 1274T^12 + 6517T^8 + 4802*T^4 + 2401)^2
$47$	$ T^{48} $ T^48
$53$	$ T^{48} $ T^48
$59$	$ T^{48} $ T^48
$61$	$ (T^{12} + 11 T^{11} + \cdots + 1)^{4} $ (T^12 + 11T^11 + 55T^10 + 166T^9 + 338T^8 + 490T^7 + 519T^6 + 406T^5 + 235T^4 + 100T^3 + 31T^2 + 6*T + 1)^4
$67$	$ T^{48} - 33 T^{44} + \cdots + 1 $ T^48 - 33T^44 + 749T^40 - 8912T^36 + 76326T^32 - 313824T^28 + 921950T^24 - 1283121T^20 + 1263580T^16 - 159804T^12 + 17304T^8 - 136*T^4 + 1
$71$	$ T^{48} $ T^48
$73$	$ T^{48} - 27 T^{44} + \cdots + 1 $ T^48 - 27T^44 + 462T^40 - 5041T^36 + 49720T^32 - 126287T^28 - 1515030T^24 - 409136T^20 + 49520855T^16 + 26975647T^12 + 4025861T^8 + 1471*T^4 + 1
$79$	$ (T^{24} - 13 T^{22} + \cdots + 1)^{2} $ (T^24 - 13T^22 + 105T^20 - 540T^18 + 2050T^16 - 5544T^14 + 11222T^12 - 15477T^10 + 14832T^8 - 6812T^6 + 2156T^4 - 48*T^2 + 1)^2
$83$	$ T^{48} $ T^48
$89$	$ T^{48} $ T^48
$97$	$ (T^{24} + 41 T^{20} + \cdots + 1)^{2} $ (T^24 + 41T^20 + 596T^16 + 3618T^12 + 8016T^8 + 2008*T^4 + 1)^2
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.cy (of order \(84\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{168}^{59} q^{3} - \zeta_{168}^{2} q^{4} + \zeta_{168}^{29} q^{7} - \zeta_{168}^{34} q^{9} - \zeta_{168}^{61} q^{12} + ( - \zeta_{168}^{67} + \zeta_{168}^{47}) q^{13} + \zeta_{168}^{4} q^{16} + \cdots + (\zeta_{168}^{41} + \zeta_{168}) q^{97} +O(q^{100}) \) q + z^59 * q^3 - z^2 * q^4 + z^29 * q^7 - z^34 * q^9 - z^61 * q^12 + (-z^67 + z^47) * q^13 + z^4 * q^16 + (z^78 - z^62) * q^19 - z^4 * q^21 + z^9 * q^27 - z^31 * q^28 + (-z^68 + z^44) * q^31 + z^36 * q^36 + (-z^77 + z^45) * q^37 + (z^42 - z^22) * q^39 + (-z^75 + z^3) * q^43 + z^63 * q^48 + z^58 * q^49 + (z^69 - z^49) * q^52 + (-z^53 + z^37) * q^57 + (z^80 - 1) * q^61 - z^63 * q^63 - z^6 * q^64 + (z^81 - z^73) * q^67 + (z^67 - z^15) * q^73 + (-z^80 + z^64) * q^76 + (-z^30 + z^26) * q^79 + z^68 * q^81 + z^6 * q^84 + (z^76 + z^12) * q^91 + (z^43 - z^19) * q^93 + (z^41 + z) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{16} + 4 q^{21} + 8 q^{36} - 44 q^{61} - 4 q^{81} + 4 q^{91}+O(q^{100}) \) 48 * q - 4 * q^16 + 4 * q^21 + 8 * q^36 - 44 * q^61 - 4 * q^81 + 4 * q^91

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	3675.1.cy.a

Level	$3675$
Weight	$1$
Character orbit	3675.cy
Analytic conductor	$1.834$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{42}$
CM discriminant	-3
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.83406392143\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(2\) over \(\Q(\zeta_{84})\)
Coefficient field:	\(\Q(\zeta_{168})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{48} + x^{44} - x^{36} - x^{32} + x^{24} - x^{16} - x^{12} + x^{4} + 1 \) x^48 + x^44 - x^36 - x^32 + x^24 - x^16 - x^12 + x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{42}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{42} + \cdots)\)

\(n\)	\(1177\)	\(1226\)	\(2551\)
\(\chi(n)\)	\(-\zeta_{168}^{42}\)	\(-1\)	\(\zeta_{168}^{76}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner
49.h	odd	42	1	inner
147.o	even	42	1	inner
245.u	odd	42	1	inner
245.x	even	84	2	inner
735.bp	even	42	1	inner
735.bs	odd	84	2	inner

$p$	$F_p(T)$
$2$	\( T^{48} \) T^48
$3$	\( T^{48} + T^{44} + \cdots + 1 \) T^48 + T^44 - T^36 - T^32 + T^24 - T^16 - T^12 + T^4 + 1
$5$	\( T^{48} \) T^48
$7$	\( T^{48} + T^{44} + \cdots + 1 \) T^48 + T^44 - T^36 - T^32 + T^24 - T^16 - T^12 + T^4 + 1
$11$	\( T^{48} \) T^48
$13$	\( T^{48} - 2 T^{44} + \cdots + 1 \) T^48 - 2T^44 + 199T^40 - 1579T^36 + 37161T^32 - 526455T^28 + 3716797T^24 - 12984411T^20 + 44664576T^16 + 16364680T^12 + 4001280T^8 - 3607*T^4 + 1
$17$	\( T^{48} \) T^48
$19$	\( (T^{24} + 11 T^{22} + \cdots + 1)^{2} \) (T^24 + 11T^22 + 77T^20 + 328T^18 + 1018T^16 + 2128T^14 + 3270T^12 + 3283T^10 + 2308T^8 + 804T^6 + 196T^4 + 16*T^2 + 1)^2
$23$	\( T^{48} \) T^48
$29$	\( T^{48} \) T^48
$31$	\( (T^{12} - 7 T^{10} + \cdots + 49)^{4} \) (T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49)^4
$37$	\( T^{48} + 2 T^{44} + \cdots + 1 \) T^48 + 2T^44 - 63T^40 + 1238T^36 + 43328T^32 + 224126T^28 + 807871T^24 + 1387008T^20 + 734709T^16 - 238582T^12 + 15806T^8 + 4*T^4 + 1
$41$	\( T^{48} \) T^48
$43$	\( (T^{24} - 28 T^{20} + \cdots + 2401)^{2} \) (T^24 - 28T^20 + 294T^16 - 1274T^12 + 6517T^8 + 4802*T^4 + 2401)^2
$47$	\( T^{48} \) T^48
$53$	\( T^{48} \) T^48
$59$	\( T^{48} \) T^48
$61$	\( (T^{12} + 11 T^{11} + \cdots + 1)^{4} \) (T^12 + 11T^11 + 55T^10 + 166T^9 + 338T^8 + 490T^7 + 519T^6 + 406T^5 + 235T^4 + 100T^3 + 31T^2 + 6*T + 1)^4
$67$	\( T^{48} - 33 T^{44} + \cdots + 1 \) T^48 - 33T^44 + 749T^40 - 8912T^36 + 76326T^32 - 313824T^28 + 921950T^24 - 1283121T^20 + 1263580T^16 - 159804T^12 + 17304T^8 - 136*T^4 + 1
$71$	\( T^{48} \) T^48
$73$	\( T^{48} - 27 T^{44} + \cdots + 1 \) T^48 - 27T^44 + 462T^40 - 5041T^36 + 49720T^32 - 126287T^28 - 1515030T^24 - 409136T^20 + 49520855T^16 + 26975647T^12 + 4025861T^8 + 1471*T^4 + 1
$79$	\( (T^{24} - 13 T^{22} + \cdots + 1)^{2} \) (T^24 - 13T^22 + 105T^20 - 540T^18 + 2050T^16 - 5544T^14 + 11222T^12 - 15477T^10 + 14832T^8 - 6812T^6 + 2156T^4 - 48*T^2 + 1)^2
$83$	\( T^{48} \) T^48
$89$	\( T^{48} \) T^48
$97$	\( (T^{24} + 41 T^{20} + \cdots + 1)^{2} \) (T^24 + 41T^20 + 596T^16 + 3618T^12 + 8016T^8 + 2008*T^4 + 1)^2
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