LMFDB - Newform orbit 495.3.h.b

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 = [2,0,0,2,-10] 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:	yes
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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 $\beta = \sqrt{5}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{2} + q^{4} - 5 q^{5} - 2 \beta q^{7} + 3 \beta q^{8} + 5 \beta q^{10} + 11 q^{11} - 10 \beta q^{13} + 10 q^{14} - 19 q^{16} - 14 \beta q^{17} - 5 q^{20} - 11 \beta q^{22} + 25 q^{25} + 50 q^{26} + \cdots + 29 \beta q^{98} +O(q^{100})$ q - b * q^2 + q^4 - 5 * q^5 - 2b q^7 + 3b q^8 + 5b q^10 + 11 * q^11 - 10b q^13 + 10 * q^14 - 19 * q^16 - 14b q^17 - 5 * q^20 - 11b q^22 + 25 * q^25 + 50 * q^26 - 2b q^28 + 18 * q^31 + 7b q^32 + 70 * q^34 + 10b q^35 - 15b q^40 + 38b q^43 + 11 * q^44 - 29 * q^49 - 25b q^50 - 10b q^52 - 55 * q^55 - 30 * q^56 + 102 * q^59 - 18b q^62 + 41 * q^64 + 50b q^65 - 14b q^68 - 50 * q^70 + 78 * q^71 - 2b q^73 - 22b q^77 + 95 * q^80 + 74b q^83 + 70b q^85 - 190 * q^86 + 33b q^88 - 2 * q^89 + 100 * q^91 + 29b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{4} - 10 q^{5} + 22 q^{11} + 20 q^{14} - 38 q^{16} - 10 q^{20} + 50 q^{25} + 100 q^{26} + 36 q^{31} + 140 q^{34} + 22 q^{44} - 58 q^{49} - 110 q^{55} - 60 q^{56} + 204 q^{59} + 82 q^{64} - 100 q^{70}+ \cdots + 200 q^{91}+O(q^{100})$ 2 * q + 2 * q^4 - 10 * q^5 + 22 * q^11 + 20 * q^14 - 38 * q^16 - 10 * q^20 + 50 * q^25 + 100 * q^26 + 36 * q^31 + 140 * q^34 + 22 * q^44 - 58 * q^49 - 110 * q^55 - 60 * q^56 + 204 * q^59 + 82 * q^64 - 100 * q^70 + 156 * q^71 + 190 * q^80 - 380 * q^86 - 4 * q^89 + 200 * q^91

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

1.61803
−0.618034

−2.23607

1.00000

−5.00000

−4.47214

6.70820

11.1803

109.2

2.23607

1.00000

−5.00000

4.47214

−6.70820

−11.1803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
55.d	odd	2	1	CM by $\Q(\sqrt{-55})$
5.b	even	2	1	inner
11.b	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.b		2
3.b	odd	2	1		55.3.d.b	✓	2
5.b	even	2	1	inner	495.3.h.b		2
11.b	odd	2	1	inner	495.3.h.b		2
12.b	even	2	1		880.3.i.c		2
15.d	odd	2	1		55.3.d.b	✓	2
15.e	even	4	2		275.3.c.c		2
33.d	even	2	1		55.3.d.b	✓	2
55.d	odd	2	1	CM	495.3.h.b		2
60.h	even	2	1		880.3.i.c		2
132.d	odd	2	1		880.3.i.c		2
165.d	even	2	1		55.3.d.b	✓	2
165.l	odd	4	2		275.3.c.c		2
660.g	odd	2	1		880.3.i.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.3.d.b	✓	2	3.b	odd	2	1
55.3.d.b	✓	2	15.d	odd	2	1
55.3.d.b	✓	2	33.d	even	2	1
55.3.d.b	✓	2	165.d	even	2	1
275.3.c.c		2	15.e	even	4	2
275.3.c.c		2	165.l	odd	4	2
495.3.h.b		2	1.a	even	1	1	trivial
495.3.h.b		2	5.b	even	2	1	inner
495.3.h.b		2	11.b	odd	2	1	inner
495.3.h.b		2	55.d	odd	2	1	CM
880.3.i.c		2	12.b	even	2	1
880.3.i.c		2	60.h	even	2	1
880.3.i.c		2	132.d	odd	2	1
880.3.i.c		2	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} - 20

T7^2 - 20

T_{89} + 2

T89 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 5$ T^2 - 5
$3$	$T^{2}$ T^2
$5$	$(T + 5)^{2}$ (T + 5)^2
$7$	$T^{2} - 20$ T^2 - 20
$11$	$(T - 11)^{2}$ (T - 11)^2
$13$	$T^{2} - 500$ T^2 - 500
$17$	$T^{2} - 980$ T^2 - 980
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$(T - 18)^{2}$ (T - 18)^2
$37$	$T^{2}$ T^2
$41$	$T^{2}$ T^2
$43$	$T^{2} - 7220$ T^2 - 7220
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$(T - 102)^{2}$ (T - 102)^2
$61$	$T^{2}$ T^2
$67$	$T^{2}$ T^2
$71$	$(T - 78)^{2}$ (T - 78)^2
$73$	$T^{2} - 20$ T^2 - 20
$79$	$T^{2}$ T^2
$83$	$T^{2} - 27380$ T^2 - 27380
$89$	$(T + 2)^{2}$ (T + 2)^2
$97$	$T^{2}$ T^2
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