LMFDB - Newform orbit 495.3.h.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(109,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.109"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{-21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 41x^{2} - 40x + 505 $ x^4 - 2x^3 + 41x^2 - 40*x + 505
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - 5 \beta_{2} q^{7} - 3 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2}) q^{10} + ( - \beta_{3} + 4) q^{11} - 4 \beta_{2} q^{13} - 25 q^{14} - 19 q^{16}+ \cdots + 76 \beta_{2} q^{98}+O(q^{100}) $ q + b2 * q^2 + q^4 + (-b1 + 2) * q^5 - 5b2 q^7 - 3b2 q^8 + (b3 + 2b2) q^10 + (-b3 + 4) * q^11 - 4b2 q^13 - 25 * q^14 - 19 * q^16 - 7b2 q^17 + b3 * q^19 + (-b1 + 2) * q^20 + (4b2 + 5b1) * q^22 + 6b1 q^23 + (-4b1 - 17) q^25 - 20 * q^26 - 5b2 q^28 + b3 * q^29 - 3 * q^31 - 7b2 q^32 - 35 * q^34 + (-5b3 - 10b2) * q^35 - b1 * q^37 - 5b1 q^38 + (-3b3 - 6b2) * q^40 + 2b3 q^41 - 10b2 q^43 + (-b3 + 4) * q^44 - 6b3 q^46 - 14b1 q^47 + 76 * q^49 + (4b3 - 17b2) * q^50 - 4b2 q^52 + b1 * q^53 + (-2b3 + 21b2 - 4b1 + 8) q^55 + 75 * q^56 - 5b1 q^58 + 18 * q^59 + 7b3 q^61 - 3b2 q^62 + 41 * q^64 + (-4b3 - 8b2) * q^65 - 6b1 q^67 - 7b2 q^68 + (25b1 - 50) q^70 - 27 * q^71 - 26b2 q^73 + b3 * q^74 + b3 * q^76 + (-20b2 - 25b1) * q^77 - 6b3 q^79 + (19b1 - 38) q^80 - 10b1 q^82 - 32b2 q^83 + (-7b3 - 14b2) * q^85 - 50 * q^86 + (-12b2 - 15b1) * q^88 - 37 * q^89 + 100 * q^91 + 6b1 q^92 + 14b3 q^94 + (2b3 - 21b2) * q^95 - 26b1 q^97 + 76b2 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{5} + 16 q^{11} - 100 q^{14} - 76 q^{16} + 8 q^{20} - 68 q^{25} - 80 q^{26} - 12 q^{31} - 140 q^{34} + 16 q^{44} + 304 q^{49} + 32 q^{55} + 300 q^{56} + 72 q^{59} + 164 q^{64} - 200 q^{70}+ \cdots + 400 q^{91}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^5 + 16 * q^11 - 100 * q^14 - 76 * q^16 + 8 * q^20 - 68 * q^25 - 80 * q^26 - 12 * q^31 - 140 * q^34 + 16 * q^44 + 304 * q^49 + 32 * q^55 + 300 * q^56 + 72 * q^59 + 164 * q^64 - 200 * q^70 - 108 * q^71 - 152 * q^80 - 200 * q^86 - 148 * q^89 + 400 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} - 2x^{3} + 41x^{2} - 40x + 505 $ x^4 - 2*x^3 + 41*x^2 - 40*x + 505 :

$\beta_{1}$	$=$	$ ( -2\nu^{3} + 3\nu^{2} - 125\nu + 62 ) / 89 $ (-2v^3 + 3v^2 - 125*v + 62) / 89
$\beta_{2}$	$=$	$ ( -4\nu^{3} + 6\nu^{2} - 72\nu + 35 ) / 89 $ (-4v^3 + 6v^2 - 72*v + 35) / 89
$\beta_{3}$	$=$	$ \nu^{2} - \nu + 20 $ v^2 - v + 20

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{2} - 2\beta _1 + 1 ) / 2 $ (b2 - 2*b1 + 1) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{3} + \beta_{2} - 2\beta _1 - 39 ) / 2 $ (2b3 + b2 - 2b1 - 39) / 2
$\nu^{3}$	$=$	$ ( 3\beta_{3} - 61\beta_{2} + 33\beta _1 - 59 ) / 2 $ (3b3 - 61b2 + 33*b1 - 59) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$-1$

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} $

109.1

−0.618034	−	4.58258i
−0.618034	+	4.58258i
1.61803	−	4.58258i
1.61803	+	4.58258i

−2.23607

1.00000

2.00000

−

4.58258i

11.1803

6.70820

−4.47214

10.2470i

109.2

−2.23607

1.00000

2.00000

4.58258i

11.1803

6.70820

−4.47214

−

10.2470i

109.3

2.23607

1.00000

2.00000

−

4.58258i

−11.1803

−6.70820

4.47214

−

10.2470i

109.4

2.23607

1.00000

2.00000

4.58258i

−11.1803

−6.70820

4.47214

10.2470i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
11.b	odd	2	1	inner
55.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	495.3.h.d		4
3.b	odd	2	1		55.3.d.d	✓	4
5.b	even	2	1	inner	495.3.h.d		4
11.b	odd	2	1	inner	495.3.h.d		4
12.b	even	2	1		880.3.i.d		4
15.d	odd	2	1		55.3.d.d	✓	4
15.e	even	4	2		275.3.c.e		4
33.d	even	2	1		55.3.d.d	✓	4
55.d	odd	2	1	inner	495.3.h.d		4
60.h	even	2	1		880.3.i.d		4
132.d	odd	2	1		880.3.i.d		4
165.d	even	2	1		55.3.d.d	✓	4
165.l	odd	4	2		275.3.c.e		4
660.g	odd	2	1		880.3.i.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.3.d.d	✓	4	3.b	odd	2	1
55.3.d.d	✓	4	15.d	odd	2	1
55.3.d.d	✓	4	33.d	even	2	1
55.3.d.d	✓	4	165.d	even	2	1
275.3.c.e		4	15.e	even	4	2
275.3.c.e		4	165.l	odd	4	2
495.3.h.d		4	1.a	even	1	1	trivial
495.3.h.d		4	5.b	even	2	1	inner
495.3.h.d		4	11.b	odd	2	1	inner
495.3.h.d		4	55.d	odd	2	1	inner
880.3.i.d		4	12.b	even	2	1
880.3.i.d		4	60.h	even	2	1
880.3.i.d		4	132.d	odd	2	1
880.3.i.d		4	660.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(495, [\chi])$:

$ T_{2}^{2} - 5 $

T2^2 - 5

$ T_{7}^{2} - 125 $

T7^2 - 125

$ T_{89} + 37 $

T89 + 37

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 5)^{2} $ (T^2 - 5)^2
$3$	$ T^{4} $ T^4
$5$	$ (T^{2} - 4 T + 25)^{2} $ (T^2 - 4*T + 25)^2
$7$	$ (T^{2} - 125)^{2} $ (T^2 - 125)^2
$11$	$ (T^{2} - 8 T + 121)^{2} $ (T^2 - 8*T + 121)^2
$13$	$ (T^{2} - 80)^{2} $ (T^2 - 80)^2
$17$	$ (T^{2} - 245)^{2} $ (T^2 - 245)^2
$19$	$ (T^{2} + 105)^{2} $ (T^2 + 105)^2
$23$	$ (T^{2} + 756)^{2} $ (T^2 + 756)^2
$29$	$ (T^{2} + 105)^{2} $ (T^2 + 105)^2
$31$	$ (T + 3)^{4} $ (T + 3)^4
$37$	$ (T^{2} + 21)^{2} $ (T^2 + 21)^2
$41$	$ (T^{2} + 420)^{2} $ (T^2 + 420)^2
$43$	$ (T^{2} - 500)^{2} $ (T^2 - 500)^2
$47$	$ (T^{2} + 4116)^{2} $ (T^2 + 4116)^2
$53$	$ (T^{2} + 21)^{2} $ (T^2 + 21)^2
$59$	$ (T - 18)^{4} $ (T - 18)^4
$61$	$ (T^{2} + 5145)^{2} $ (T^2 + 5145)^2
$67$	$ (T^{2} + 756)^{2} $ (T^2 + 756)^2
$71$	$ (T + 27)^{4} $ (T + 27)^4
$73$	$ (T^{2} - 3380)^{2} $ (T^2 - 3380)^2
$79$	$ (T^{2} + 3780)^{2} $ (T^2 + 3780)^2
$83$	$ (T^{2} - 5120)^{2} $ (T^2 - 5120)^2
$89$	$ (T + 37)^{4} $ (T + 37)^4
$97$	$ (T^{2} + 14196)^{2} $ (T^2 + 14196)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - 5 \beta_{2} q^{7} - 3 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2}) q^{10} + ( - \beta_{3} + 4) q^{11} - 4 \beta_{2} q^{13} - 25 q^{14} - 19 q^{16}+ \cdots + 76 \beta_{2} q^{98}+O(q^{100}) \) q + b2 * q^2 + q^4 + (-b1 + 2) * q^5 - 5b2 q^7 - 3b2 q^8 + (b3 + 2b2) q^10 + (-b3 + 4) * q^11 - 4b2 q^13 - 25 * q^14 - 19 * q^16 - 7b2 q^17 + b3 * q^19 + (-b1 + 2) * q^20 + (4b2 + 5b1) * q^22 + 6b1 q^23 + (-4b1 - 17) q^25 - 20 * q^26 - 5b2 q^28 + b3 * q^29 - 3 * q^31 - 7b2 q^32 - 35 * q^34 + (-5b3 - 10b2) * q^35 - b1 * q^37 - 5b1 q^38 + (-3b3 - 6b2) * q^40 + 2b3 q^41 - 10b2 q^43 + (-b3 + 4) * q^44 - 6b3 q^46 - 14b1 q^47 + 76 * q^49 + (4b3 - 17b2) * q^50 - 4b2 q^52 + b1 * q^53 + (-2b3 + 21b2 - 4b1 + 8) q^55 + 75 * q^56 - 5b1 q^58 + 18 * q^59 + 7b3 q^61 - 3b2 q^62 + 41 * q^64 + (-4b3 - 8b2) * q^65 - 6b1 q^67 - 7b2 q^68 + (25b1 - 50) q^70 - 27 * q^71 - 26b2 q^73 + b3 * q^74 + b3 * q^76 + (-20b2 - 25b1) * q^77 - 6b3 q^79 + (19b1 - 38) q^80 - 10b1 q^82 - 32b2 q^83 + (-7b3 - 14b2) * q^85 - 50 * q^86 + (-12b2 - 15b1) * q^88 - 37 * q^89 + 100 * q^91 + 6b1 q^92 + 14b3 q^94 + (2b3 - 21b2) * q^95 - 26b1 q^97 + 76b2 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{5} + 16 q^{11} - 100 q^{14} - 76 q^{16} + 8 q^{20} - 68 q^{25} - 80 q^{26} - 12 q^{31} - 140 q^{34} + 16 q^{44} + 304 q^{49} + 32 q^{55} + 300 q^{56} + 72 q^{59} + 164 q^{64} - 200 q^{70}+ \cdots + 400 q^{91}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^5 + 16 * q^11 - 100 * q^14 - 76 * q^16 + 8 * q^20 - 68 * q^25 - 80 * q^26 - 12 * q^31 - 140 * q^34 + 16 * q^44 + 304 * q^49 + 32 * q^55 + 300 * q^56 + 72 * q^59 + 164 * q^64 - 200 * q^70 - 108 * q^71 - 152 * q^80 - 200 * q^86 - 148 * q^89 + 400 * q^91

Introduction

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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	495.3.h.d

Level	$495$
Weight	$3$
Character orbit	495.h
Analytic conductor	$13.488$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(13.4877730858\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{5}, \sqrt{-21})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - 2x^{3} + 41x^{2} - 40x + 505 \) x^4 - 2x^3 + 41x^2 - 40*x + 505
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -2\nu^{3} + 3\nu^{2} - 125\nu + 62 ) / 89 \) (-2v^3 + 3v^2 - 125*v + 62) / 89
\(\beta_{2}\)	\(=\)	\( ( -4\nu^{3} + 6\nu^{2} - 72\nu + 35 ) / 89 \) (-4v^3 + 6v^2 - 72*v + 35) / 89
\(\beta_{3}\)	\(=\)	\( \nu^{2} - \nu + 20 \) v^2 - v + 20

\(\nu\)	\(=\)	\( ( \beta_{2} - 2\beta _1 + 1 ) / 2 \) (b2 - 2*b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{3} + \beta_{2} - 2\beta _1 - 39 ) / 2 \) (2b3 + b2 - 2b1 - 39) / 2
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{3} - 61\beta_{2} + 33\beta _1 - 59 ) / 2 \) (3b3 - 61b2 + 33*b1 - 59) / 2

$p$	$F_p(T)$
$2$	\( (T^{2} - 5)^{2} \) (T^2 - 5)^2
$3$	\( T^{4} \) T^4
$5$	\( (T^{2} - 4 T + 25)^{2} \) (T^2 - 4*T + 25)^2
$7$	\( (T^{2} - 125)^{2} \) (T^2 - 125)^2
$11$	\( (T^{2} - 8 T + 121)^{2} \) (T^2 - 8*T + 121)^2
$13$	\( (T^{2} - 80)^{2} \) (T^2 - 80)^2
$17$	\( (T^{2} - 245)^{2} \) (T^2 - 245)^2
$19$	\( (T^{2} + 105)^{2} \) (T^2 + 105)^2
$23$	\( (T^{2} + 756)^{2} \) (T^2 + 756)^2
$29$	\( (T^{2} + 105)^{2} \) (T^2 + 105)^2
$31$	\( (T + 3)^{4} \) (T + 3)^4
$37$	\( (T^{2} + 21)^{2} \) (T^2 + 21)^2
$41$	\( (T^{2} + 420)^{2} \) (T^2 + 420)^2
$43$	\( (T^{2} - 500)^{2} \) (T^2 - 500)^2
$47$	\( (T^{2} + 4116)^{2} \) (T^2 + 4116)^2
$53$	\( (T^{2} + 21)^{2} \) (T^2 + 21)^2
$59$	\( (T - 18)^{4} \) (T - 18)^4
$61$	\( (T^{2} + 5145)^{2} \) (T^2 + 5145)^2
$67$	\( (T^{2} + 756)^{2} \) (T^2 + 756)^2
$71$	\( (T + 27)^{4} \) (T + 27)^4
$73$	\( (T^{2} - 3380)^{2} \) (T^2 - 3380)^2
$79$	\( (T^{2} + 3780)^{2} \) (T^2 + 3780)^2
$83$	\( (T^{2} - 5120)^{2} \) (T^2 - 5120)^2
$89$	\( (T + 37)^{4} \) (T + 37)^4
$97$	\( (T^{2} + 14196)^{2} \) (T^2 + 14196)^2
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\(n\)	\(46\)	\(56\)	\(397\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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