LMFDB - Newform orbit 2848.1.bu.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2848,1,Mod(271,2848)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2848, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 8]))

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

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

chi := DirichletCharacter("2848.271");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2848 = 2^{5} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2848.bu (of order $22$, degree $10$, 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:	$1.42133715598$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 712)
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

$q$-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_{22}^{10} - \zeta_{22}^{2}) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} - \zeta_{22}) q^{9}+O(q^{10}) $ q + (-z^10 - z^2) * q^3 + (-z^9 + z^4 - z) * q^9	$ q + ( - \zeta_{22}^{10} - \zeta_{22}^{2}) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} - \zeta_{22}) q^{9} + (\zeta_{22}^{7} - 1) q^{11} + (\zeta_{22}^{6} + 1) q^{17} + ( - \zeta_{22}^{10} + \zeta_{22}^{3}) q^{19} + \zeta_{22}^{4} q^{25} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} + \cdots + 1) q^{27}+ \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{99}+O(q^{100}) $ q + (-z^10 - z^2) * q^3 + (-z^9 + z^4 - z) * q^9 + (z^7 - 1) * q^11 + (z^6 + 1) * q^17 + (-z^10 + z^3) * q^19 + z^4 * q^25 + (-z^8 - z^6 + z^3 + 1) * q^27 + (z^10 - z^9 + z^6 + z^2) * q^33 - z^5 * q^41 + (-z^6 + z) * q^43 + z^4 * q^49 + (-z^10 - z^8 + z^5 - z^2) * q^51 + (-z^9 - z^5 + z^2 - z) * q^57 + (-z^8 - z^2) * q^59 + (z^7 + z^5) * q^67 + (z^6 - z^3) * q^73 + (-z^6 + z^3) * q^75 + (z^10 + z^8 - z^7 - z^5 + z^2) * q^81 + (z^9 - z^8) * q^83 - z^7 * q^89 + (-z^3 - z) * q^97 + (z^9 - z^8 + z^5 - z^4 + z - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 3 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 3 * q^9	$ 10 q + 2 q^{3} - 3 q^{9} - 9 q^{11} + 9 q^{17} + 2 q^{19} - q^{25} - 7 q^{27} - 4 q^{33} - 2 q^{41} + 2 q^{43} - q^{49} + 4 q^{51} - 4 q^{57} + 2 q^{59} + 2 q^{67} - 2 q^{73} + 2 q^{75} - 5 q^{81} + 2 q^{83} - q^{89} - 2 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 3 * q^9 - 9 * q^11 + 9 * q^17 + 2 * q^19 - q^25 - 7 * q^27 - 4 * q^33 - 2 * q^41 + 2 * q^43 - q^49 + 4 * q^51 - 4 * q^57 + 2 * q^59 + 2 * q^67 - 2 * q^73 + 2 * q^75 - 5 * q^81 + 2 * q^83 - q^89 - 2 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2848\mathbb{Z}\right)^\times$.

$n$	$357$	$1247$	$1249$
$\chi(n)$	$-1$	$-1$	$-\zeta_{22}$

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} $

271.1

0.654861	−	0.755750i
−0.841254	+	0.540641i
−0.841254	−	0.540641i
0.142315	−	0.989821i
0.959493	−	0.281733i
0.654861	+	0.755750i
−0.415415	+	0.909632i
−0.415415	−	0.909632i
0.142315	+	0.989821i
0.959493	+	0.281733i

0.797176

1.74557i

−1.75667

2.02730i

655.1

−1.25667

0.368991i

0.601808

−

0.386758i

687.1

−1.25667

−

0.368991i

0.601808

0.386758i

751.1

1.10181

1.27155i

−0.260554

1.81219i

879.1

0.118239

0.822373i

0.297176

−

0.0872586i

1135.1

0.797176

−

1.74557i

−1.75667

−

2.02730i

1647.1

0.239446

−

0.153882i

−0.381761

0.835939i

1871.1

0.239446

0.153882i

−0.381761

−

0.835939i

2063.1

1.10181

−

1.27155i

−0.260554

−

1.81219i

2767.1

0.118239

−

0.822373i

0.297176

0.0872586i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
89.e	even	11	1	inner
712.s	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2848.1.bu.a		10
4.b	odd	2	1		712.1.s.a	✓	10
8.b	even	2	1		712.1.s.a	✓	10
8.d	odd	2	1	CM	2848.1.bu.a		10
89.e	even	11	1	inner	2848.1.bu.a		10
356.l	odd	22	1		712.1.s.a	✓	10
712.s	odd	22	1	inner	2848.1.bu.a		10
712.x	even	22	1		712.1.s.a	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
712.1.s.a	✓	10	4.b	odd	2	1
712.1.s.a	✓	10	8.b	even	2	1
712.1.s.a	✓	10	356.l	odd	22	1
712.1.s.a	✓	10	712.x	even	22	1
2848.1.bu.a		10	1.a	even	1	1	trivial
2848.1.bu.a		10	8.d	odd	2	1	CM
2848.1.bu.a		10	89.e	even	11	1	inner
2848.1.bu.a		10	712.s	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2848, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$5$	$ T^{10} $ T^10
$7$	$ T^{10} $ T^10
$11$	$ T^{10} + 9 T^{9} + \cdots + 1 $ T^10 + 9T^9 + 37T^8 + 91T^7 + 148T^6 + 166T^5 + 130T^4 + 70T^3 + 25T^2 + 5*T + 1
$13$	$ T^{10} $ T^10
$17$	$ T^{10} - 9 T^{9} + \cdots + 1 $ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$19$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$23$	$ T^{10} $ T^10
$29$	$ T^{10} $ T^10
$31$	$ T^{10} $ T^10
$37$	$ T^{10} $ T^10
$41$	$ T^{10} + 2 T^{9} + \cdots + 1024 $ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 64T^4 + 128T^3 + 256T^2 + 512*T + 1024
$43$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$47$	$ T^{10} $ T^10
$53$	$ T^{10} $ T^10
$59$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$61$	$ T^{10} $ T^10
$67$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$71$	$ T^{10} $ T^10
$73$	$ T^{10} + 2 T^{9} + \cdots + 1 $ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$79$	$ T^{10} $ T^10
$83$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$89$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$97$	$ T^{10} + 2 T^{9} + \cdots + 1 $ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
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Level:	\( N \)	\(=\)	\( 2848 = 2^{5} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2848.bu (of order \(22\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{22}^{10} - \zeta_{22}^{2}) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} - \zeta_{22}) q^{9}+O(q^{10}) \) q + (-z^10 - z^2) * q^3 + (-z^9 + z^4 - z) * q^9	\( q + ( - \zeta_{22}^{10} - \zeta_{22}^{2}) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} - \zeta_{22}) q^{9} + (\zeta_{22}^{7} - 1) q^{11} + (\zeta_{22}^{6} + 1) q^{17} + ( - \zeta_{22}^{10} + \zeta_{22}^{3}) q^{19} + \zeta_{22}^{4} q^{25} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} + \cdots + 1) q^{27}+ \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{99}+O(q^{100}) \) q + (-z^10 - z^2) * q^3 + (-z^9 + z^4 - z) * q^9 + (z^7 - 1) * q^11 + (z^6 + 1) * q^17 + (-z^10 + z^3) * q^19 + z^4 * q^25 + (-z^8 - z^6 + z^3 + 1) * q^27 + (z^10 - z^9 + z^6 + z^2) * q^33 - z^5 * q^41 + (-z^6 + z) * q^43 + z^4 * q^49 + (-z^10 - z^8 + z^5 - z^2) * q^51 + (-z^9 - z^5 + z^2 - z) * q^57 + (-z^8 - z^2) * q^59 + (z^7 + z^5) * q^67 + (z^6 - z^3) * q^73 + (-z^6 + z^3) * q^75 + (z^10 + z^8 - z^7 - z^5 + z^2) * q^81 + (z^9 - z^8) * q^83 - z^7 * q^89 + (-z^3 - z) * q^97 + (z^9 - z^8 + z^5 - z^4 + z - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 3 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 3 * q^9	\( 10 q + 2 q^{3} - 3 q^{9} - 9 q^{11} + 9 q^{17} + 2 q^{19} - q^{25} - 7 q^{27} - 4 q^{33} - 2 q^{41} + 2 q^{43} - q^{49} + 4 q^{51} - 4 q^{57} + 2 q^{59} + 2 q^{67} - 2 q^{73} + 2 q^{75} - 5 q^{81} + 2 q^{83} - q^{89} - 2 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 3 * q^9 - 9 * q^11 + 9 * q^17 + 2 * q^19 - q^25 - 7 * q^27 - 4 * q^33 - 2 * q^41 + 2 * q^43 - q^49 + 4 * q^51 - 4 * q^57 + 2 * q^59 + 2 * q^67 - 2 * q^73 + 2 * q^75 - 5 * q^81 + 2 * q^83 - q^89 - 2 * q^97 - 5 * q^99

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