LMFDB - Newform orbit 416.3.x.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,3,Mod(303,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 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("416.303"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	416.x (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-4] 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:	$11.3351789974$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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 primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \zeta_{6} + 1) q^{3} - 2 q^{5} - 5 \zeta_{6} q^{7} + 8 \zeta_{6} q^{9} + ( - 7 \zeta_{6} - 7) q^{11} - 13 q^{13} + (2 \zeta_{6} - 2) q^{15} + 25 \zeta_{6} q^{17} + (7 \zeta_{6} - 14) q^{19} - 5 q^{21} + \cdots + ( - 112 \zeta_{6} + 56) q^{99} +O(q^{100})$ q + (-z + 1) * q^3 - 2 * q^5 - 5z q^7 + 8z q^9 + (-7z - 7) q^11 - 13 * q^13 + (2z - 2) q^15 + 25z q^17 + (7z - 14) q^19 - 5 * q^21 + (-5z - 5) q^23 - 21 * q^25 + 17 * q^27 + (-13z - 13) q^29 - 10 * q^31 + (7z - 14) q^33 + 10z q^35 + (61z - 61) q^37 + (13z - 13) q^39 + (z + 1) * q^41 + 41z q^43 - 16z q^45 - 82 * q^47 + (-24z + 24) q^49 + 25 * q^51 + (-48z + 24) q^53 + (14z + 14) q^55 + (14z - 7) q^57 + (31z - 62) q^59 + (-35z + 70) q^61 + (-40z + 40) q^63 + 26 * q^65 + (-7z - 7) q^67 + (5z - 10) q^69 - 29z q^71 + (-48z + 24) q^73 + (21z - 21) q^75 + (70z - 35) q^77 + (144z - 72) q^79 + (55z - 55) q^81 - 50z q^85 + (13z - 26) q^87 + (-23z - 23) q^89 + 65z q^91 + (10z - 10) q^93 + (-14z + 28) q^95 + (-z + 2) * q^97 + (-112z + 56) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{3} - 4 q^{5} - 5 q^{7} + 8 q^{9} - 21 q^{11} - 26 q^{13} - 2 q^{15} + 25 q^{17} - 21 q^{19} - 10 q^{21} - 15 q^{23} - 42 q^{25} + 34 q^{27} - 39 q^{29} - 20 q^{31} - 21 q^{33} + 10 q^{35} - 61 q^{37}+ \cdots + 3 q^{97}+O(q^{100})$ 2 * q + q^3 - 4 * q^5 - 5 * q^7 + 8 * q^9 - 21 * q^11 - 26 * q^13 - 2 * q^15 + 25 * q^17 - 21 * q^19 - 10 * q^21 - 15 * q^23 - 42 * q^25 + 34 * q^27 - 39 * q^29 - 20 * q^31 - 21 * q^33 + 10 * q^35 - 61 * q^37 - 13 * q^39 + 3 * q^41 + 41 * q^43 - 16 * q^45 - 164 * q^47 + 24 * q^49 + 50 * q^51 + 42 * q^55 - 93 * q^59 + 105 * q^61 + 40 * q^63 + 52 * q^65 - 21 * q^67 - 15 * q^69 - 29 * q^71 - 21 * q^75 - 55 * q^81 - 50 * q^85 - 39 * q^87 - 69 * q^89 + 65 * q^91 - 10 * q^93 + 42 * q^95 + 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$ .

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$\zeta_{6}$

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}

303.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

−2.00000

−2.50000

−

4.33013i

4.00000

6.92820i

335.1

0.500000

0.866025i

−2.00000

−2.50000

4.33013i

4.00000

−

6.92820i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.3.x.a		2
4.b	odd	2	1		104.3.p.b	yes	2
8.b	even	2	1		104.3.p.a	✓	2
8.d	odd	2	1		416.3.x.b		2
13.e	even	6	1		416.3.x.b		2
52.i	odd	6	1		104.3.p.a	✓	2
104.p	odd	6	1	inner	416.3.x.a		2
104.s	even	6	1		104.3.p.b	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.3.p.a	✓	2	8.b	even	2	1
104.3.p.a	✓	2	52.i	odd	6	1
104.3.p.b	yes	2	4.b	odd	2	1
104.3.p.b	yes	2	104.s	even	6	1
416.3.x.a		2	1.a	even	1	1	trivial
416.3.x.a		2	104.p	odd	6	1	inner
416.3.x.b		2	8.d	odd	2	1
416.3.x.b		2	13.e	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(416, [\chi])$ :

T_{3}^{2} - T_{3} + 1

T3^2 - T3 + 1

T_{5} + 2

T5 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - T + 1$ T^2 - T + 1
$5$	$(T + 2)^{2}$ (T + 2)^2
$7$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$11$	$T^{2} + 21T + 147$ T^2 + 21*T + 147
$13$	$(T + 13)^{2}$ (T + 13)^2
$17$	$T^{2} - 25T + 625$ T^2 - 25*T + 625
$19$	$T^{2} + 21T + 147$ T^2 + 21*T + 147
$23$	$T^{2} + 15T + 75$ T^2 + 15*T + 75
$29$	$T^{2} + 39T + 507$ T^2 + 39*T + 507
$31$	$(T + 10)^{2}$ (T + 10)^2
$37$	$T^{2} + 61T + 3721$ T^2 + 61*T + 3721
$41$	$T^{2} - 3T + 3$ T^2 - 3*T + 3
$43$	$T^{2} - 41T + 1681$ T^2 - 41*T + 1681
$47$	$(T + 82)^{2}$ (T + 82)^2
$53$	$T^{2} + 1728$ T^2 + 1728
$59$	$T^{2} + 93T + 2883$ T^2 + 93*T + 2883
$61$	$T^{2} - 105T + 3675$ T^2 - 105*T + 3675
$67$	$T^{2} + 21T + 147$ T^2 + 21*T + 147
$71$	$T^{2} + 29T + 841$ T^2 + 29*T + 841
$73$	$T^{2} + 1728$ T^2 + 1728
$79$	$T^{2} + 15552$ T^2 + 15552
$83$	$T^{2}$ T^2
$89$	$T^{2} + 69T + 1587$ T^2 + 69*T + 1587
$97$	$T^{2} - 3T + 3$ T^2 - 3*T + 3
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