LMFDB - Newform orbit 2368.1.ck.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,1,Mod(127,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2368.127");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2368.ck (of order $18$, degree $6$, not 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:	$1.18178594991$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 148)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.899194740203776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{5} + \zeta_{18}^{2} q^{9}+O(q^{10}) $ q + (z^5 + z^3) * q^5 + z^2 * q^9	$ q + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{5} + \zeta_{18}^{2} q^{9} - \zeta_{18}^{7} q^{13} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{17} + (\zeta_{18}^{8} + \zeta_{18}^{6} - \zeta_{18}) q^{25} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{29} + \zeta_{18}^{7} q^{37} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{41} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{45} + \zeta_{18}^{8} q^{49} - \zeta_{18}^{5} q^{53} + (\zeta_{18}^{5} - 1) q^{61} + (\zeta_{18}^{3} + \zeta_{18}) q^{65} - q^{73} + \zeta_{18}^{4} q^{81} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \cdots - 1) q^{85} + \cdots + ( - \zeta_{18}^{5} - \zeta_{18}) q^{97} +O(q^{100}) $ q + (z^5 + z^3) * q^5 + z^2 * q^9 - z^7 * q^13 + (-z^7 + z^6) * q^17 + (z^8 + z^6 - z) * q^25 + (-z^8 - z^4) * q^29 + z^7 * q^37 + (-z^3 + z^2) * q^41 + (z^7 + z^5) * q^45 + z^8 * q^49 - z^5 * q^53 + (z^5 - 1) * q^61 + (z^3 + z) * q^65 - q^73 + z^4 * q^81 + (z^3 - z^2 + z - 1) * q^85 + (-z + 1) * q^89 + (-z^5 - z) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5}+O(q^{10}) $ 6 * q + 3 * q^5	$ 6 q + 3 q^{5} - 3 q^{17} - 3 q^{25} - 3 q^{41} - 6 q^{61} + 3 q^{65} - 6 q^{73} - 3 q^{85} + 6 q^{89}+O(q^{100}) $ 6 * q + 3 * q^5 - 3 * q^17 - 3 * q^25 - 3 * q^41 - 6 * q^61 + 3 * q^65 - 6 * q^73 - 3 * q^85 + 6 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$.

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-\zeta_{18}^{7}$	$-1$	$1$

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} $

127.1

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

0.326352

−

1.85083i

0.766044

−

0.642788i

255.1

−0.266044

0.223238i

−0.939693

0.342020i

895.1

0.326352

1.85083i

0.766044

0.642788i

959.1

1.43969

0.524005i

0.173648

−

0.984808i

2047.1

1.43969

−

0.524005i

0.173648

0.984808i

2303.1

−0.266044

−

0.223238i

−0.939693

−

0.342020i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
37.f	even	9	1	inner
148.p	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2368.1.ck.a		6
4.b	odd	2	1	CM	2368.1.ck.a		6
8.b	even	2	1		148.1.p.a	✓	6
8.d	odd	2	1		148.1.p.a	✓	6
24.f	even	2	1		1332.1.cz.a		6
24.h	odd	2	1		1332.1.cz.a		6
37.f	even	9	1	inner	2368.1.ck.a		6
40.e	odd	2	1		3700.1.ce.a		6
40.f	even	2	1		3700.1.ce.a		6
40.i	odd	4	2		3700.1.cb.a		12
40.k	even	4	2		3700.1.cb.a		12
148.p	odd	18	1	inner	2368.1.ck.a		6
296.bb	odd	18	1		148.1.p.a	✓	6
296.bc	even	18	1		148.1.p.a	✓	6
888.ca	odd	18	1		1332.1.cz.a		6
888.cf	even	18	1		1332.1.cz.a		6
1480.de	odd	18	1		3700.1.ce.a		6
1480.dp	even	18	1		3700.1.ce.a		6
1480.dw	even	36	2		3700.1.cb.a		12
1480.ej	odd	36	2		3700.1.cb.a		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
148.1.p.a	✓	6	8.b	even	2	1
148.1.p.a	✓	6	8.d	odd	2	1
148.1.p.a	✓	6	296.bb	odd	18	1
148.1.p.a	✓	6	296.bc	even	18	1
1332.1.cz.a		6	24.f	even	2	1
1332.1.cz.a		6	24.h	odd	2	1
1332.1.cz.a		6	888.ca	odd	18	1
1332.1.cz.a		6	888.cf	even	18	1
2368.1.ck.a		6	1.a	even	1	1	trivial
2368.1.ck.a		6	4.b	odd	2	1	CM
2368.1.ck.a		6	37.f	even	9	1	inner
2368.1.ck.a		6	148.p	odd	18	1	inner
3700.1.cb.a		12	40.i	odd	4	2
3700.1.cb.a		12	40.k	even	4	2
3700.1.cb.a		12	1480.dw	even	36	2
3700.1.cb.a		12	1480.ej	odd	36	2
3700.1.ce.a		6	40.e	odd	2	1
3700.1.ce.a		6	40.f	even	2	1
3700.1.ce.a		6	1480.de	odd	18	1
3700.1.ce.a		6	1480.dp	even	18	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2368, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 + 6T^4 - 8T^3 + 3T^2 + 3*T + 1
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$17$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$19$	$ T^{6} $ T^6
$23$	$ T^{6} $ T^6
$29$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$31$	$ T^{6} $ T^6
$37$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$41$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$43$	$ T^{6} $ T^6
$47$	$ T^{6} $ T^6
$53$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$59$	$ T^{6} $ T^6
$61$	$ T^{6} + 6 T^{5} + \cdots + 1 $ T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1
$67$	$ T^{6} $ T^6
$71$	$ T^{6} $ T^6
$73$	$ (T + 1)^{6} $ (T + 1)^6
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$97$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
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Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2368.ck (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{5} + \zeta_{18}^{2} q^{9}+O(q^{10}) \) q + (z^5 + z^3) * q^5 + z^2 * q^9	\( q + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{5} + \zeta_{18}^{2} q^{9} - \zeta_{18}^{7} q^{13} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{17} + (\zeta_{18}^{8} + \zeta_{18}^{6} - \zeta_{18}) q^{25} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{29} + \zeta_{18}^{7} q^{37} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{41} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{45} + \zeta_{18}^{8} q^{49} - \zeta_{18}^{5} q^{53} + (\zeta_{18}^{5} - 1) q^{61} + (\zeta_{18}^{3} + \zeta_{18}) q^{65} - q^{73} + \zeta_{18}^{4} q^{81} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \cdots - 1) q^{85} + \cdots + ( - \zeta_{18}^{5} - \zeta_{18}) q^{97} +O(q^{100}) \) q + (z^5 + z^3) * q^5 + z^2 * q^9 - z^7 * q^13 + (-z^7 + z^6) * q^17 + (z^8 + z^6 - z) * q^25 + (-z^8 - z^4) * q^29 + z^7 * q^37 + (-z^3 + z^2) * q^41 + (z^7 + z^5) * q^45 + z^8 * q^49 - z^5 * q^53 + (z^5 - 1) * q^61 + (z^3 + z) * q^65 - q^73 + z^4 * q^81 + (z^3 - z^2 + z - 1) * q^85 + (-z + 1) * q^89 + (-z^5 - z) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5}+O(q^{10}) \) 6 * q + 3 * q^5	\( 6 q + 3 q^{5} - 3 q^{17} - 3 q^{25} - 3 q^{41} - 6 q^{61} + 3 q^{65} - 6 q^{73} - 3 q^{85} + 6 q^{89}+O(q^{100}) \) 6 * q + 3 * q^5 - 3 * q^17 - 3 * q^25 - 3 * q^41 - 6 * q^61 + 3 * q^65 - 6 * q^73 - 3 * q^85 + 6 * q^89

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