LMFDB - Newform orbit 3800.2.d.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9}+O(q^{10}) $ q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9	\( q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9} + ( - \beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{2}) q^{17} - q^{19} + (\beta_{3} - \beta_1 + 1) q^{21} + (\beta_{5} - 2 \beta_{4}) q^{23} + \beta_{5} q^{27} + ( - \beta_1 - 6) q^{29} - 2 \beta_1 q^{31} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{37} + (\beta_{3} + \beta_1 - 3) q^{39} + (\beta_{3} + 2 \beta_1 + 3) q^{41} + (2 \beta_{4} - 2 \beta_{2}) q^{43} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{47} + ( - \beta_{3} - \beta_1 - 2) q^{49} - \beta_1 q^{51} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{53} + \beta_{4} q^{57} + ( - 2 \beta_{3} - \beta_1 - 6) q^{59} + (\beta_{3} + 3) q^{61} + ( - 2 \beta_{4} - 4 \beta_{2}) q^{63} + (2 \beta_{5} - 5 \beta_{4}) q^{67} + (\beta_{3} + \beta_1 - 7) q^{69} + 4 \beta_1 q^{71} + (\beta_{5} - 4 \beta_{4} + \beta_{2}) q^{73} + ( - \beta_{3} - 1) q^{79} + (\beta_{3} + 2 \beta_1 - 2) q^{81} + (2 \beta_{5} - 2 \beta_{2}) q^{83} + ( - \beta_{5} + 4 \beta_{4}) q^{87} + (\beta_{3} + 2 \beta_1 - 1) q^{89} + (\beta_{3} + \beta_1 + 1) q^{91} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{93} + ( - 3 \beta_{5} + \beta_{4} + \beta_{2}) q^{97}+O(q^{100}) \) q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9 + (-b4 + b2) * q^13 + (b5 - b2) * q^17 - q^19 + (b3 - b1 + 1) * q^21 + (b5 - 2b4) q^23 + b5 * q^27 + (-b1 - 6) * q^29 - 2b1 q^31 + (-b5 - b4 - b2) * q^37 + (b3 + b1 - 3) * q^39 + (b3 + 2b1 + 3) q^41 + (2b4 - 2b2) * q^43 + (-2b5 - 2b4) * q^47 + (-b3 - b1 - 2) * q^49 - b1 * q^51 + (-2b5 + b4 + b2) q^53 + b4 * q^57 + (-2b3 - b1 - 6) q^59 + (b3 + 3) * q^61 + (-2b4 - 4b2) * q^63 + (2b5 - 5b4) * q^67 + (b3 + b1 - 7) * q^69 + 4b1 q^71 + (b5 - 4b4 + b2) q^73 + (-b3 - 1) * q^79 + (b3 + 2b1 - 2) q^81 + (2b5 - 2b2) * q^83 + (-b5 + 4b4) q^87 + (b3 + 2b1 - 1) q^89 + (b3 + b1 + 1) * q^91 + (-2b5 - 4b4) * q^93 + (-3b5 + b4 + b2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) $ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8*x^4 + 16*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu^{4} + 3\nu^{2} - 3 $ v^4 + 3*v^2 - 3
$\beta_{2}$	$=$	$ 2\nu^{3} + 8\nu $ 2v^3 + 8v
$\beta_{3}$	$=$	$ 2\nu^{4} + 10\nu^{2} + 5 $ 2v^4 + 10v^2 + 5
$\beta_{4}$	$=$	$ \nu^{5} + 7\nu^{3} + 11\nu $ v^5 + 7v^3 + 11v
$\beta_{5}$	$=$	$ \nu^{5} + 7\nu^{3} + 13\nu $ v^5 + 7v^3 + 13v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{5} - \beta_{4} ) / 2 $ (b5 - b4) / 2
$\nu^{2}$	$=$	$ ( \beta_{3} - 2\beta _1 - 11 ) / 4 $ (b3 - 2*b1 - 11) / 4
$\nu^{3}$	$=$	$ ( -4\beta_{5} + 4\beta_{4} + \beta_{2} ) / 2 $ (-4b5 + 4b4 + b2) / 2
$\nu^{4}$	$=$	$ ( -3\beta_{3} + 10\beta _1 + 45 ) / 4 $ (-3b3 + 10b1 + 45) / 4
$\nu^{5}$	$=$	$ ( 17\beta_{5} - 15\beta_{4} - 7\beta_{2} ) / 2 $ (17b5 - 15b4 - 7*b2) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

3649.1

		0.254102i
	−	1.86081i
	−	2.11491i
		2.11491i
		1.86081i
	−	0.254102i

−

2.68133i

3.18953i

−4.18953

3649.2

−

2.32340i

−

1.39821i

−2.39821

3649.3

−

0.642074i

−

3.58774i

2.58774

3649.4

0.642074i

3.58774i

2.58774

3649.5

2.32340i

1.39821i

−2.39821

3649.6

2.68133i

−

3.18953i

−4.18953

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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	3800.2.d.l		6
5.b	even	2	1	inner	3800.2.d.l		6
5.c	odd	4	1		760.2.a.j	✓	3
5.c	odd	4	1		3800.2.a.x		3
15.e	even	4	1		6840.2.a.bg		3
20.e	even	4	1		1520.2.a.s		3
20.e	even	4	1		7600.2.a.bq		3
40.i	odd	4	1		6080.2.a.bv		3
40.k	even	4	1		6080.2.a.bq		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.a.j	✓	3	5.c	odd	4	1
1520.2.a.s		3	20.e	even	4	1
3800.2.a.x		3	5.c	odd	4	1
3800.2.d.l		6	1.a	even	1	1	trivial
3800.2.d.l		6	5.b	even	2	1	inner
6080.2.a.bq		3	40.k	even	4	1
6080.2.a.bv		3	40.i	odd	4	1
6840.2.a.bg		3	15.e	even	4	1
7600.2.a.bq		3	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3800, [\chi])$:

$ T_{3}^{6} + 13T_{3}^{4} + 44T_{3}^{2} + 16 $

$ T_{7}^{6} + 25T_{7}^{4} + 176T_{7}^{2} + 256 $

$ T_{11} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 13 T^{4} + \cdots + 16 $ T^6 + 13T^4 + 44T^2 + 16
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 25 T^{4} + \cdots + 256 $ T^6 + 25T^4 + 176T^2 + 256
$11$	$ T^{6} $ T^6
$13$	$ T^{6} + 21 T^{4} + \cdots + 16 $ T^6 + 21T^4 + 44T^2 + 16
$17$	$ T^{6} + 33 T^{4} + \cdots + 16 $ T^6 + 33T^4 + 56T^2 + 16
$19$	$ (T + 1)^{6} $ (T + 1)^6
$23$	$ T^{6} + 65 T^{4} + \cdots + 4096 $ T^6 + 65T^4 + 1152T^2 + 4096
$29$	$ (T^{3} + 17 T^{2} + \cdots + 124)^{2} $ (T^3 + 17T^2 + 84T + 124)^2
$31$	$ (T^{3} - 2 T^{2} + \cdots + 128)^{2} $ (T^3 - 2T^2 - 48T + 128)^2
$37$	$ T^{6} + 64 T^{4} + \cdots + 64 $ T^6 + 64T^4 + 128T^2 + 64
$41$	$ (T^{3} - 6 T^{2} - 52 T + 56)^{2} $ (T^3 - 6T^2 - 52T + 56)^2
$43$	$ T^{6} + 84 T^{4} + \cdots + 1024 $ T^6 + 84T^4 + 704T^2 + 1024
$47$	$ T^{6} + 176 T^{4} + \cdots + 16384 $ T^6 + 176T^4 + 5376T^2 + 16384
$53$	$ T^{6} + 109 T^{4} + \cdots + 2704 $ T^6 + 109T^4 + 1244T^2 + 2704
$59$	$ (T^{3} + 15 T^{2} + \cdots - 784)^{2} $ (T^3 + 15T^2 - 28T - 784)^2
$61$	$ (T^{3} - 8 T^{2} - 4 T + 64)^{2} $ (T^3 - 8T^2 - 4T + 64)^2
$67$	$ T^{6} + 365 T^{4} + \cdots + 1106704 $ T^6 + 365T^4 + 37996T^2 + 1106704
$71$	$ (T^{3} + 4 T^{2} + \cdots - 1024)^{2} $ (T^3 + 4T^2 - 192T - 1024)^2
$73$	$ T^{6} + 209 T^{4} + \cdots + 85264 $ T^6 + 209T^4 + 8248T^2 + 85264
$79$	$ (T^{3} + 2 T^{2} - 24 T - 32)^{2} $ (T^3 + 2T^2 - 24T - 32)^2
$83$	$ T^{6} + 132 T^{4} + \cdots + 1024 $ T^6 + 132T^4 + 896T^2 + 1024
$89$	$ (T^{3} + 6 T^{2} + \cdots - 184)^{2} $ (T^3 + 6T^2 - 52T - 184)^2
$97$	$ T^{6} + 224 T^{4} + \cdots + 87616 $ T^6 + 224T^4 + 8448T^2 + 87616
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Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9}+O(q^{10}) \) q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9	\( q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9} + ( - \beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{2}) q^{17} - q^{19} + (\beta_{3} - \beta_1 + 1) q^{21} + (\beta_{5} - 2 \beta_{4}) q^{23} + \beta_{5} q^{27} + ( - \beta_1 - 6) q^{29} - 2 \beta_1 q^{31} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{37} + (\beta_{3} + \beta_1 - 3) q^{39} + (\beta_{3} + 2 \beta_1 + 3) q^{41} + (2 \beta_{4} - 2 \beta_{2}) q^{43} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{47} + ( - \beta_{3} - \beta_1 - 2) q^{49} - \beta_1 q^{51} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{53} + \beta_{4} q^{57} + ( - 2 \beta_{3} - \beta_1 - 6) q^{59} + (\beta_{3} + 3) q^{61} + ( - 2 \beta_{4} - 4 \beta_{2}) q^{63} + (2 \beta_{5} - 5 \beta_{4}) q^{67} + (\beta_{3} + \beta_1 - 7) q^{69} + 4 \beta_1 q^{71} + (\beta_{5} - 4 \beta_{4} + \beta_{2}) q^{73} + ( - \beta_{3} - 1) q^{79} + (\beta_{3} + 2 \beta_1 - 2) q^{81} + (2 \beta_{5} - 2 \beta_{2}) q^{83} + ( - \beta_{5} + 4 \beta_{4}) q^{87} + (\beta_{3} + 2 \beta_1 - 1) q^{89} + (\beta_{3} + \beta_1 + 1) q^{91} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{93} + ( - 3 \beta_{5} + \beta_{4} + \beta_{2}) q^{97}+O(q^{100}) \) q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9 + (-b4 + b2) * q^13 + (b5 - b2) * q^17 - q^19 + (b3 - b1 + 1) * q^21 + (b5 - 2b4) q^23 + b5 * q^27 + (-b1 - 6) * q^29 - 2b1 q^31 + (-b5 - b4 - b2) * q^37 + (b3 + b1 - 3) * q^39 + (b3 + 2b1 + 3) q^41 + (2b4 - 2b2) * q^43 + (-2b5 - 2b4) * q^47 + (-b3 - b1 - 2) * q^49 - b1 * q^51 + (-2b5 + b4 + b2) q^53 + b4 * q^57 + (-2b3 - b1 - 6) q^59 + (b3 + 3) * q^61 + (-2b4 - 4b2) * q^63 + (2b5 - 5b4) * q^67 + (b3 + b1 - 7) * q^69 + 4b1 q^71 + (b5 - 4b4 + b2) q^73 + (-b3 - 1) * q^79 + (b3 + 2b1 - 2) q^81 + (2b5 - 2b2) * q^83 + (-b5 + 4b4) q^87 + (b3 + 2b1 - 1) q^89 + (b3 + b1 + 1) * q^91 + (-2b5 - 4b4) * q^93 + (-3b5 + b4 + b2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

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