LMFDB - Newform orbit 399.2.cb.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(5,399)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 15, 16]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("399.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.cb (of order $18$ , degree $6$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{3} + 1$ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$ -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 $\zeta_{18}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (2 \zeta_{18}^{4} - \zeta_{18}) q^{3} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{4} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{7} - 3 \zeta_{18}^{2} q^{9} + ( - 4 \zeta_{18}^{3} + 2) q^{12} + ( - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + \cdots - 3) q^{13} + \cdots + ( - 22 \zeta_{18}^{4} + 11 \zeta_{18}) q^{97} +O(q^{100})$ q + (2z^4 - z) q^3 + (2z^5 - 2z^2) * q^4 + (-z^5 - 2z^2) q^7 - 3z^2 q^9 + (-4z^3 + 2) q^12 + (-4z^4 + 4z^3 + z - 3) * q^13 + (-4z^4 + 4z) * q^16 + (-5z^3 + 2) q^19 + (-z^3 + 5) * q^21 - 5z^2 q^25 + (-3z^3 + 6) q^27 + (2z^4 + 4z) * q^28 + (z^5 - 5z^4 + 5z^2 - z) * q^31 + 6z q^36 + (-3z^5 - 4z^4 - 4z^2 - 3z) * q^37 + (-2z^5 - 2z^4 + 7z^2 - 5z) * q^39 + (7z^5 + 6z^3 - 6z^2 - 7) q^43 + (4z^5 + 4z^2) * q^48 + (8z^4 - 5z) * q^49 + (-6z^5 + 8z^3 - 2z^2 - 2) q^52 + (-z^4 + 8z) q^57 + (5z^4 - 5z^3 - 9z - 4) q^61 + (9z^4 - 3z) * q^63 + (8z^3 - 8) q^64 + (9z^4 + 9z^3 - 7z - 2) q^67 + (-9z^5 - 9z^3 + 8z^2 + 1) q^73 + (-5z^3 + 10) q^75 + (4z^5 + 6z^2) * q^76 + (3z^5 - 7z^3 + 7z^2 - 3) q^79 + 9z^4 q^81 + (10z^5 - 8z^2) * q^84 + (-9z^5 + 5z^3 + 10z^2 - 11) q^91 + (-7z^5 + 4z^3 + 11z^2 - 11) q^93 + (-22z^4 + 11z) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{13} - 3 q^{19} + 27 q^{21} + 27 q^{27} - 24 q^{43} + 12 q^{52} - 39 q^{61} - 24 q^{64} + 15 q^{67} - 21 q^{73} + 45 q^{75} - 39 q^{79} - 51 q^{91} - 54 q^{93}+O(q^{100})$ 6 * q - 6 * q^13 - 3 * q^19 + 27 * q^21 + 27 * q^27 - 24 * q^43 + 12 * q^52 - 39 * q^61 - 24 * q^64 + 15 * q^67 - 21 * q^73 + 45 * q^75 - 39 * q^79 - 51 * q^91 - 54 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$ .

$n$	$115$	$134$	$211$
$\chi(n)$	$\zeta_{18}^{3}$	$-1$	$\zeta_{18}^{2}$

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}

5.1

0.939693	−	0.342020i
0.939693	+	0.342020i
−0.766044	+	0.642788i
−0.173648	+	0.984808i
−0.766044	−	0.642788i
−0.173648	−	0.984808i

−0.592396

−

1.62760i

−1.87939

−

0.684040i

−1.35844

2.27038i

−2.29813

1.92836i

80.1

−0.592396

1.62760i

−1.87939

0.684040i

−1.35844

−

2.27038i

−2.29813

−

1.92836i

101.1

−1.11334

−

1.32683i

1.53209

1.28558i

−1.28699

2.31164i

−0.520945

2.95442i

131.1

1.70574

0.300767i

0.347296

1.96962i

2.64543

0.0412527i

2.81908

1.02606i

320.1

−1.11334

1.32683i

1.53209

−

1.28558i

−1.28699

−

2.31164i

−0.520945

−

2.95442i

332.1

1.70574

−

0.300767i

0.347296

−

1.96962i

2.64543

−

0.0412527i

2.81908

−

1.02606i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
133.z	odd	18	1	inner
399.cb	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.2.cb.a	✓	6
3.b	odd	2	1	CM	399.2.cb.a	✓	6
7.d	odd	6	1		399.2.ci.a	yes	6
19.e	even	9	1		399.2.ci.a	yes	6
21.g	even	6	1		399.2.ci.a	yes	6
57.l	odd	18	1		399.2.ci.a	yes	6
133.z	odd	18	1	inner	399.2.cb.a	✓	6
399.cb	even	18	1	inner	399.2.cb.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.cb.a	✓	6	1.a	even	1	1	trivial
399.2.cb.a	✓	6	3.b	odd	2	1	CM
399.2.cb.a	✓	6	133.z	odd	18	1	inner
399.2.cb.a	✓	6	399.cb	even	18	1	inner
399.2.ci.a	yes	6	7.d	odd	6	1
399.2.ci.a	yes	6	19.e	even	9	1
399.2.ci.a	yes	6	21.g	even	6	1
399.2.ci.a	yes	6	57.l	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}$ T2 acting on $S_{2}^{\mathrm{new}}(399, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 9T^{3} + 27$ T^6 - 9*T^3 + 27
$5$	$T^{6}$ T^6
$7$	$T^{6} - 37T^{3} + 343$ T^6 - 37*T^3 + 343
$11$	$T^{6}$ T^6
$13$	$T^{6} + 6 T^{5} + \cdots + 867$ T^6 + 6T^5 + 51T^4 + 183T^3 + 1674T^2 + 2295*T + 867
$17$	$T^{6}$ T^6
$19$	$(T^{2} + T + 19)^{3}$ (T^2 + T + 19)^3
$23$	$T^{6}$ T^6
$29$	$T^{6}$ T^6
$31$	$T^{6} - 93 T^{4} + \cdots + 35643$ T^6 - 93T^4 + 8649T^2 - 30411*T + 35643
$37$	$T^{6} + 111 T^{4} + \cdots + 187489$ T^6 + 111T^4 - 866T^3 + 12321T^2 - 48063T + 187489
$41$	$T^{6}$ T^6
$43$	$T^{6} + 24 T^{5} + \cdots + 201601$ T^6 + 24T^5 + 321T^4 + 2647T^3 + 23376T^2 + 111801*T + 201601
$47$	$T^{6}$ T^6
$53$	$T^{6}$ T^6
$59$	$T^{6}$ T^6
$61$	$T^{6} + 39 T^{5} + \cdots + 96123$ T^6 + 39T^5 + 690T^4 + 7674T^3 + 47988T^2 + 90216*T + 96123
$67$	$T^{6} - 15 T^{5} + \cdots + 16129$ T^6 - 15T^5 + 276T^4 - 3142T^3 + 36210T^2 + 46482*T + 16129
$71$	$T^{6}$ T^6
$73$	$T^{6} + 21 T^{5} + \cdots + 711507$ T^6 + 21T^5 + 366T^4 + 3138T^3 + 52344T^2 + 350640*T + 711507
$79$	$T^{6} + 39 T^{5} + \cdots + 253009$ T^6 + 39T^5 + 744T^4 + 9746T^3 + 81822T^2 + 214278*T + 253009
$83$	$T^{6}$ T^6
$89$	$T^{6}$ T^6
$97$	$T^{6} + 11979 T^{3} + 47832147$ T^6 + 11979*T^3 + 47832147
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