LMFDB - Newform orbit 855.3.g.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,3,Mod(379,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, 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("855.379"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$855 = 3^{2} \cdot 5 \cdot 19$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	855.g (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,9] 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:	$23.2970626028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 5$ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 95)
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 $\beta = \frac{1}{2}(1 + \sqrt{-19})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 4 q^{4} + ( - \beta + 5) q^{5} + ( - 6 \beta + 3) q^{7} + 3 q^{11} + 16 q^{16} + ( - 14 \beta + 7) q^{17} - 19 q^{19} + (4 \beta - 20) q^{20} + (16 \beta - 8) q^{23} + ( - 9 \beta + 20) q^{25} + (24 \beta - 12) q^{28}+ \cdots + (19 \beta - 95) q^{95}+O(q^{100})$ q - 4 * q^4 + (-b + 5) * q^5 + (-6b + 3) q^7 + 3 * q^11 + 16 * q^16 + (-14b + 7) q^17 - 19 * q^19 + (4b - 20) q^20 + (16b - 8) q^23 + (-9b + 20) q^25 + (24b - 12) q^28 + (-27b - 15) q^35 + (-6b + 3) q^43 - 12 * q^44 + (26b - 13) q^47 - 122 * q^49 + (-3b + 15) q^55 - 103 * q^61 - 64 * q^64 + (56b - 28) q^68 + (-66b + 33) q^73 + 76 * q^76 + (-18b + 9) q^77 + (-16b + 80) q^80 + (-64b + 32) q^83 + (-63b - 35) q^85 + (-64b + 32) q^92 + (19b - 95) q^95
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 9 q^{5} + 6 q^{11} + 32 q^{16} - 38 q^{19} - 36 q^{20} + 31 q^{25} - 57 q^{35} - 24 q^{44} - 244 q^{49} + 27 q^{55} - 206 q^{61} - 128 q^{64} + 152 q^{76} + 144 q^{80} - 133 q^{85} - 171 q^{95}+O(q^{100})$ 2 * q - 8 * q^4 + 9 * q^5 + 6 * q^11 + 32 * q^16 - 38 * q^19 - 36 * q^20 + 31 * q^25 - 57 * q^35 - 24 * q^44 - 244 * q^49 + 27 * q^55 - 206 * q^61 - 128 * q^64 + 152 * q^76 + 144 * q^80 - 133 * q^85 - 171 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$ .

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$1$	$-1$

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}

379.1

0.500000	+	2.17945i
0.500000	−	2.17945i

−4.00000

4.50000

−

2.17945i

−

13.0767i

379.2

−4.00000

4.50000

2.17945i

13.0767i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by $\Q(\sqrt{-19})$
5.b	even	2	1	inner
95.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	855.3.g.a		2
3.b	odd	2	1		95.3.d.a	✓	2
5.b	even	2	1	inner	855.3.g.a		2
15.d	odd	2	1		95.3.d.a	✓	2
15.e	even	4	2		475.3.c.c		2
19.b	odd	2	1	CM	855.3.g.a		2
57.d	even	2	1		95.3.d.a	✓	2
95.d	odd	2	1	inner	855.3.g.a		2
285.b	even	2	1		95.3.d.a	✓	2
285.j	odd	4	2		475.3.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.3.d.a	✓	2	3.b	odd	2	1
95.3.d.a	✓	2	15.d	odd	2	1
95.3.d.a	✓	2	57.d	even	2	1
95.3.d.a	✓	2	285.b	even	2	1
475.3.c.c		2	15.e	even	4	2
475.3.c.c		2	285.j	odd	4	2
855.3.g.a		2	1.a	even	1	1	trivial
855.3.g.a		2	5.b	even	2	1	inner
855.3.g.a		2	19.b	odd	2	1	CM
855.3.g.a		2	95.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(855, [\chi])$ :

T_{2}

T2

T_{11} - 3

T11 - 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - 9T + 25$ T^2 - 9*T + 25
$7$	$T^{2} + 171$ T^2 + 171
$11$	$(T - 3)^{2}$ (T - 3)^2
$13$	$T^{2}$ T^2
$17$	$T^{2} + 931$ T^2 + 931
$19$	$(T + 19)^{2}$ (T + 19)^2
$23$	$T^{2} + 1216$ T^2 + 1216
$29$	$T^{2}$ T^2
$31$	$T^{2}$ T^2
$37$	$T^{2}$ T^2
$41$	$T^{2}$ T^2
$43$	$T^{2} + 171$ T^2 + 171
$47$	$T^{2} + 3211$ T^2 + 3211
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$(T + 103)^{2}$ (T + 103)^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2} + 20691$ T^2 + 20691
$79$	$T^{2}$ T^2
$83$	$T^{2} + 19456$ T^2 + 19456
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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