LMFDB - Newform orbit 4000.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (of order $2$, degree $1$, 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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
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 a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9}+O(q^{10}) $ q - b3 * q^3 + (-b7 + b3) * q^7 + (-b6 - b1 - 1) * q^9	$ q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9} + (\beta_{6} + 3) q^{21} + ( - 2 \beta_{7} - \beta_{5} + \cdots + \beta_{2}) q^{23}+ \cdots + ( - 2 \beta_{6} - 4 \beta_{4} - 3 \beta_1) q^{89}+O(q^{100}) $ q - b3 * q^3 + (-b7 + b3) * q^7 + (-b6 - b1 - 1) * q^9 + (b6 + 3) * q^21 + (-2b7 - b5 - 2b3 + b2) * q^23 + (-b7 + 2b5 + 2b3 - 2b2) q^27 + (-2b6 + b4 - 3b1 - 3) * q^29 + (-3b4 - b1 - 3) q^41 + (4b5 + 2b3 - b2) * q^43 + (-2b5 - b3 - 2b2) * q^47 + (-b4 + 3b1 + 2) q^49 + (2b6 - 3b1 + 6) * q^61 + (-2b7 - b5 - 4b3 + 2b2) q^63 + (-5b7 + 5b5 + 4b3 - 4b2) * q^67 + (-2b6 - b4 - 6b1 - 9) * q^69 + (-b6 + 2b4 + 2b1 + 2) * q^81 + (-2b7 + 5b5 + 2b3 + 5b2) * q^83 + (-b7 + 2b5 + 10b3 - 3b2) q^87 + (-2b6 - 4b4 - 3b1) q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 24 q^{21} - 12 q^{29} - 20 q^{41} + 4 q^{49} + 60 q^{61} - 48 q^{69} + 8 q^{81} + 12 q^{89}+O(q^{100}) $ 8 * q - 4 * q^9 + 24 * q^21 - 12 * q^29 - 20 * q^41 + 4 * q^49 + 60 * q^61 - 48 * q^69 + 8 * q^81 + 12 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ -\zeta_{20}^{6} + \zeta_{20}^{4} $ -v^6 + v^4
$\beta_{2}$	$=$	$ \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} $ v^6 - v^5 + v^4
$\beta_{3}$	$=$	$ \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} $ v^6 + v^5 + v^4
$\beta_{4}$	$=$	$ -2\zeta_{20}^{7} + 2\zeta_{20}^{3} $ -2v^7 + 2v^3
$\beta_{5}$	$=$	$ \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + 2\zeta_{20}^{2} - 1 $ v^7 + v^6 - v^5 - v^4 + v^3 + 2*v^2 - 1
$\beta_{6}$	$=$	$ -2\zeta_{20}^{7} + 2\zeta_{20}^{5} - 2\zeta_{20}^{3} + 4\zeta_{20} $ -2v^7 + 2v^5 - 2v^3 + 4v
$\beta_{7}$	$=$	$ -\zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{3} + 2\zeta_{20}^{2} - 1 $ -v^7 + v^6 + v^5 - v^4 - v^3 + 2*v^2 - 1

Display $\zeta_{20}^j$ in terms of $\beta_i$

$\zeta_{20}$	$=$	$ ( -\beta_{7} + \beta_{6} + \beta_{5} ) / 4 $ (-b7 + b6 + b5) / 4
$\zeta_{20}^{2}$	$=$	$ ( \beta_{7} + \beta_{5} + 2\beta _1 + 2 ) / 4 $ (b7 + b5 + 2*b1 + 2) / 4
$\zeta_{20}^{3}$	$=$	$ ( -\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 4 $ (-b7 + b5 + b4 + b3 - b2) / 4
$\zeta_{20}^{4}$	$=$	$ ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 $ (b3 + b2 + 2*b1) / 4
$\zeta_{20}^{5}$	$=$	$ ( \beta_{3} - \beta_{2} ) / 2 $ (b3 - b2) / 2
$\zeta_{20}^{6}$	$=$	$ ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 $ (b3 + b2 - 2*b1) / 4
$\zeta_{20}^{7}$	$=$	$ ( -\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 $ (-b7 + b5 - b4 + b3 - b2) / 4

Display $\beta_i$ in terms of $\zeta_{20}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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} $

1249.1

0.951057	+	0.309017i
0.587785	−	0.809017i
−0.951057	−	0.309017i
−0.587785	+	0.809017i
−0.587785	−	0.809017i
−0.951057	+	0.309017i
0.587785	+	0.809017i
0.951057	−	0.309017i

−

2.90211i

2.34458i

−5.42226

1249.2

−

2.17557i

2.45965i

−1.73311

1249.3

−

0.902113i

−

0.891491i

2.18619

1249.4

−

0.175571i

3.69572i

2.96917

1249.5

0.175571i

−

3.69572i

2.96917

1249.6

0.902113i

0.891491i

2.18619

1249.7

2.17557i

−

2.45965i

−1.73311

1249.8

2.90211i

−

2.34458i

−5.42226

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.2.c.b		8
4.b	odd	2	1	inner	4000.2.c.b		8
5.b	even	2	1	inner	4000.2.c.b		8
5.c	odd	4	1		4000.2.a.b	✓	4
5.c	odd	4	1		4000.2.a.i	yes	4
20.d	odd	2	1	CM	4000.2.c.b		8
20.e	even	4	1		4000.2.a.b	✓	4
20.e	even	4	1		4000.2.a.i	yes	4
40.i	odd	4	1		8000.2.a.z		4
40.i	odd	4	1		8000.2.a.bs		4
40.k	even	4	1		8000.2.a.z		4
40.k	even	4	1		8000.2.a.bs		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.b	✓	4	5.c	odd	4	1
4000.2.a.b	✓	4	20.e	even	4	1
4000.2.a.i	yes	4	5.c	odd	4	1
4000.2.a.i	yes	4	20.e	even	4	1
4000.2.c.b		8	1.a	even	1	1	trivial
4000.2.c.b		8	4.b	odd	2	1	inner
4000.2.c.b		8	5.b	even	2	1	inner
4000.2.c.b		8	20.d	odd	2	1	CM
8000.2.a.z		4	40.i	odd	4	1
8000.2.a.z		4	40.k	even	4	1
8000.2.a.bs		4	40.i	odd	4	1
8000.2.a.bs		4	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4000, [\chi])$:

$ T_{3}^{8} + 14T_{3}^{6} + 51T_{3}^{4} + 34T_{3}^{2} + 1 $

$ T_{7}^{8} + 26T_{7}^{6} + 211T_{7}^{4} + 606T_{7}^{2} + 361 $

$ T_{11} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 14 T^{6} + \cdots + 1 $ T^8 + 14T^6 + 51T^4 + 34*T^2 + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 26 T^{6} + \cdots + 361 $ T^8 + 26T^6 + 211T^4 + 606*T^2 + 361
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ T^{8} + 154 T^{6} + \cdots + 361 $ T^8 + 154T^6 + 6451T^4 + 47454*T^2 + 361
$29$	$ (T^{4} + 6 T^{3} + \cdots + 281)^{2} $ (T^4 + 6T^3 - 109T^2 - 654*T + 281)^2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ (T^{4} + 10 T^{3} + \cdots + 4705)^{2} $ (T^4 + 10T^3 - 145T^2 - 1030*T + 4705)^2
$43$	$ T^{8} + 254 T^{6} + \cdots + 160801 $ T^8 + 254T^6 + 16771T^4 + 146514*T^2 + 160801
$47$	$ T^{8} + 166 T^{6} + \cdots + 1343281 $ T^8 + 166T^6 + 9211T^4 + 193146*T^2 + 1343281
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} - 30 T^{3} + \cdots - 3895)^{2} $ (T^4 - 30T^3 + 235T^2 + 90*T - 3895)^2
$67$	$ (T^{4} + 268 T^{2} + 13456)^{2} $ (T^4 + 268*T^2 + 13456)^2
$71$	$ T^{8} $ T^8
$73$	$ T^{8} $ T^8
$79$	$ T^{8} $ T^8
$83$	$ T^{8} + \cdots + 3142611481 $ T^8 + 994T^6 + 360571T^4 + 56202054*T^2 + 3142611481
$89$	$ (T^{4} - 6 T^{3} + \cdots + 24881)^{2} $ (T^4 - 6T^3 - 409T^2 + 2454*T + 24881)^2
$97$	$ T^{8} $ T^8
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Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9}+O(q^{10}) \) q - b3 * q^3 + (-b7 + b3) * q^7 + (-b6 - b1 - 1) * q^9	\( q - \beta_{3} q^{3} + ( - \beta_{7} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_1 - 1) q^{9} + (\beta_{6} + 3) q^{21} + ( - 2 \beta_{7} - \beta_{5} + \cdots + \beta_{2}) q^{23}+ \cdots + ( - 2 \beta_{6} - 4 \beta_{4} - 3 \beta_1) q^{89}+O(q^{100}) \) q - b3 * q^3 + (-b7 + b3) * q^7 + (-b6 - b1 - 1) * q^9 + (b6 + 3) * q^21 + (-2b7 - b5 - 2b3 + b2) * q^23 + (-b7 + 2b5 + 2b3 - 2b2) q^27 + (-2b6 + b4 - 3b1 - 3) * q^29 + (-3b4 - b1 - 3) q^41 + (4b5 + 2b3 - b2) * q^43 + (-2b5 - b3 - 2b2) * q^47 + (-b4 + 3b1 + 2) q^49 + (2b6 - 3b1 + 6) * q^61 + (-2b7 - b5 - 4b3 + 2b2) q^63 + (-5b7 + 5b5 + 4b3 - 4b2) * q^67 + (-2b6 - b4 - 6b1 - 9) * q^69 + (-b6 + 2b4 + 2b1 + 2) * q^81 + (-2b7 + 5b5 + 2b3 + 5b2) * q^83 + (-b7 + 2b5 + 10b3 - 3b2) q^87 + (-2b6 - 4b4 - 3b1) q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 24 q^{21} - 12 q^{29} - 20 q^{41} + 4 q^{49} + 60 q^{61} - 48 q^{69} + 8 q^{81} + 12 q^{89}+O(q^{100}) \) 8 * q - 4 * q^9 + 24 * q^21 - 12 * q^29 - 20 * q^41 + 4 * q^49 + 60 * q^61 - 48 * q^69 + 8 * q^81 + 12 * q^89

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