LMFDB - Newform orbit 3200.2.c.z

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(2049,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, 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("3200.2049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - 2 \beta_{3} q^{9}+O(q^{10}) $ q + (b2 + b1) * q^3 + (-2b2 + 2b1) * q^7 - 2b3 q^9	\( q + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - 2 \beta_{3} q^{9} + ( - 3 \beta_{3} - 1) q^{11} - 4 \beta_{2} q^{13} + (2 \beta_{2} - 3 \beta_1) q^{17} + ( - \beta_{3} + 3) q^{19} + 2 q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} - \beta_1) q^{27} + 8 q^{29} + (2 \beta_{3} - 2) q^{31} + ( - 4 \beta_{2} - 7 \beta_1) q^{33} + (4 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{3} + 8) q^{39} + (4 \beta_{3} + 5) q^{41} + 10 \beta_1 q^{43} + (4 \beta_{2} + 4 \beta_1) q^{47} + (8 \beta_{3} - 5) q^{49} + (\beta_{3} - 1) q^{51} + (4 \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{2} + \beta_1) q^{57} + ( - 4 \beta_{3} + 2) q^{59} - 6 q^{61} + ( - 4 \beta_{2} + 8 \beta_1) q^{63} + ( - 3 \beta_{2} + 7 \beta_1) q^{67} + (4 \beta_{3} + 6) q^{69} + (4 \beta_{3} + 4) q^{71} + ( - 2 \beta_{2} - 3 \beta_1) q^{73} + ( - 4 \beta_{2} + 10 \beta_1) q^{77} + (2 \beta_{3} + 10) q^{79} + ( - 6 \beta_{3} - 1) q^{81} + (3 \beta_{2} - 3 \beta_1) q^{83} + (8 \beta_{2} + 8 \beta_1) q^{87} + (2 \beta_{3} - 9) q^{89} + (8 \beta_{3} - 16) q^{91} + 2 \beta_1 q^{93} + ( - 8 \beta_{2} - 6 \beta_1) q^{97} + (2 \beta_{3} + 12) q^{99}+O(q^{100}) \) q + (b2 + b1) * q^3 + (-2b2 + 2b1) * q^7 - 2b3 q^9 + (-3b3 - 1) q^11 - 4b2 q^13 + (2b2 - 3b1) * q^17 + (-b3 + 3) * q^19 + 2 * q^21 + (-2b2 - 2b1) * q^23 + (b2 - b1) * q^27 + 8 * q^29 + (2b3 - 2) q^31 + (-4b2 - 7b1) * q^33 + (4b2 + 2b1) * q^37 + (4b3 + 8) q^39 + (4b3 + 5) q^41 + 10b1 q^43 + (4b2 + 4b1) * q^47 + (8b3 - 5) q^49 + (b3 - 1) * q^51 + (4b2 + 2b1) * q^53 + (2b2 + b1) q^57 + (-4b3 + 2) q^59 - 6 * q^61 + (-4b2 + 8b1) * q^63 + (-3b2 + 7b1) * q^67 + (4b3 + 6) q^69 + (4b3 + 4) q^71 + (-2b2 - 3b1) * q^73 + (-4b2 + 10b1) * q^77 + (2b3 + 10) q^79 + (-6b3 - 1) q^81 + (3b2 - 3b1) * q^83 + (8b2 + 8b1) * q^87 + (2b3 - 9) q^89 + (8b3 - 16) q^91 + 2b1 q^93 + (-8b2 - 6b1) * q^97 + (2b3 + 12) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 12 q^{19} + 8 q^{21} + 32 q^{29} - 8 q^{31} + 32 q^{39} + 20 q^{41} - 20 q^{49} - 4 q^{51} + 8 q^{59} - 24 q^{61} + 24 q^{69} + 16 q^{71} + 40 q^{79} - 4 q^{81} - 36 q^{89} - 64 q^{91} + 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^11 + 12 * q^19 + 8 * q^21 + 32 * q^29 - 8 * q^31 + 32 * q^39 + 20 * q^41 - 20 * q^49 - 4 * q^51 + 8 * q^59 - 24 * q^61 + 24 * q^69 + 16 * q^71 + 40 * q^79 - 4 * q^81 - 36 * q^89 - 64 * q^91 + 48 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{8}^{2} $ v^2
$\beta_{2}$	$=$	$ \zeta_{8}^{3} + \zeta_{8} $ v^3 + v
$\beta_{3}$	$=$	$ -\zeta_{8}^{3} + \zeta_{8} $ -v^3 + v

Display $\zeta_{8}^j$ in terms of $\beta_i$

$\zeta_{8}$	$=$	$ ( \beta_{3} + \beta_{2} ) / 2 $ (b3 + b2) / 2
$\zeta_{8}^{2}$	$=$	$ \beta_1 $ b1
$\zeta_{8}^{3}$	$=$	$ ( -\beta_{3} + \beta_{2} ) / 2 $ (-b3 + b2) / 2

Display $\beta_i$ in terms of $\zeta_{8}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

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} $

2049.1

0.707107	−	0.707107i
−0.707107	−	0.707107i
−0.707107	+	0.707107i
0.707107	+	0.707107i

−

2.41421i

0.828427i

−2.82843

2049.2

−

0.414214i

4.82843i

2.82843

2049.3

0.414214i

−

4.82843i

2.82843

2049.4

2.41421i

−

0.828427i

−2.82843

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	3200.2.c.z		4
4.b	odd	2	1		3200.2.c.bb		4
5.b	even	2	1	inner	3200.2.c.z		4
5.c	odd	4	1		3200.2.a.bg	yes	2
5.c	odd	4	1		3200.2.a.bi	yes	2
8.b	even	2	1		3200.2.c.ba		4
8.d	odd	2	1		3200.2.c.y		4
20.d	odd	2	1		3200.2.c.bb		4
20.e	even	4	1		3200.2.a.bh	yes	2
20.e	even	4	1		3200.2.a.bj	yes	2
40.e	odd	2	1		3200.2.c.y		4
40.f	even	2	1		3200.2.c.ba		4
40.i	odd	4	1		3200.2.a.bd	yes	2
40.i	odd	4	1		3200.2.a.bn	yes	2
40.k	even	4	1		3200.2.a.bc	✓	2
40.k	even	4	1		3200.2.a.bm	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3200.2.a.bc	✓	2	40.k	even	4	1
3200.2.a.bd	yes	2	40.i	odd	4	1
3200.2.a.bg	yes	2	5.c	odd	4	1
3200.2.a.bh	yes	2	20.e	even	4	1
3200.2.a.bi	yes	2	5.c	odd	4	1
3200.2.a.bj	yes	2	20.e	even	4	1
3200.2.a.bm	yes	2	40.k	even	4	1
3200.2.a.bn	yes	2	40.i	odd	4	1
3200.2.c.y		4	8.d	odd	2	1
3200.2.c.y		4	40.e	odd	2	1
3200.2.c.z		4	1.a	even	1	1	trivial
3200.2.c.z		4	5.b	even	2	1	inner
3200.2.c.ba		4	8.b	even	2	1
3200.2.c.ba		4	40.f	even	2	1
3200.2.c.bb		4	4.b	odd	2	1
3200.2.c.bb		4	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3200, [\chi])$:

$ T_{3}^{4} + 6T_{3}^{2} + 1 $

$ T_{7}^{4} + 24T_{7}^{2} + 16 $

$ T_{11}^{2} + 2T_{11} - 17 $

$ T_{29} - 8 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 6T^{2} + 1 $ T^4 + 6*T^2 + 1
$5$	$ T^{4} $ T^4
$7$	$ T^{4} + 24T^{2} + 16 $ T^4 + 24*T^2 + 16
$11$	$ (T^{2} + 2 T - 17)^{2} $ (T^2 + 2*T - 17)^2
$13$	$ (T^{2} + 32)^{2} $ (T^2 + 32)^2
$17$	$ T^{4} + 34T^{2} + 1 $ T^4 + 34*T^2 + 1
$19$	$ (T^{2} - 6 T + 7)^{2} $ (T^2 - 6*T + 7)^2
$23$	$ T^{4} + 24T^{2} + 16 $ T^4 + 24*T^2 + 16
$29$	$ (T - 8)^{4} $ (T - 8)^4
$31$	$ (T^{2} + 4 T - 4)^{2} $ (T^2 + 4*T - 4)^2
$37$	$ T^{4} + 72T^{2} + 784 $ T^4 + 72*T^2 + 784
$41$	$ (T^{2} - 10 T - 7)^{2} $ (T^2 - 10*T - 7)^2
$43$	$ (T^{2} + 100)^{2} $ (T^2 + 100)^2
$47$	$ T^{4} + 96T^{2} + 256 $ T^4 + 96*T^2 + 256
$53$	$ T^{4} + 72T^{2} + 784 $ T^4 + 72*T^2 + 784
$59$	$ (T^{2} - 4 T - 28)^{2} $ (T^2 - 4*T - 28)^2
$61$	$ (T + 6)^{4} $ (T + 6)^4
$67$	$ T^{4} + 134T^{2} + 961 $ T^4 + 134*T^2 + 961
$71$	$ (T^{2} - 8 T - 16)^{2} $ (T^2 - 8*T - 16)^2
$73$	$ T^{4} + 34T^{2} + 1 $ T^4 + 34*T^2 + 1
$79$	$ (T^{2} - 20 T + 92)^{2} $ (T^2 - 20*T + 92)^2
$83$	$ T^{4} + 54T^{2} + 81 $ T^4 + 54*T^2 + 81
$89$	$ (T^{2} + 18 T + 73)^{2} $ (T^2 + 18*T + 73)^2
$97$	$ T^{4} + 328T^{2} + 8464 $ T^4 + 328*T^2 + 8464
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Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - 2 \beta_{3} q^{9}+O(q^{10}) \) q + (b2 + b1) * q^3 + (-2b2 + 2b1) * q^7 - 2b3 q^9	\( q + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - 2 \beta_{3} q^{9} + ( - 3 \beta_{3} - 1) q^{11} - 4 \beta_{2} q^{13} + (2 \beta_{2} - 3 \beta_1) q^{17} + ( - \beta_{3} + 3) q^{19} + 2 q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} - \beta_1) q^{27} + 8 q^{29} + (2 \beta_{3} - 2) q^{31} + ( - 4 \beta_{2} - 7 \beta_1) q^{33} + (4 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{3} + 8) q^{39} + (4 \beta_{3} + 5) q^{41} + 10 \beta_1 q^{43} + (4 \beta_{2} + 4 \beta_1) q^{47} + (8 \beta_{3} - 5) q^{49} + (\beta_{3} - 1) q^{51} + (4 \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{2} + \beta_1) q^{57} + ( - 4 \beta_{3} + 2) q^{59} - 6 q^{61} + ( - 4 \beta_{2} + 8 \beta_1) q^{63} + ( - 3 \beta_{2} + 7 \beta_1) q^{67} + (4 \beta_{3} + 6) q^{69} + (4 \beta_{3} + 4) q^{71} + ( - 2 \beta_{2} - 3 \beta_1) q^{73} + ( - 4 \beta_{2} + 10 \beta_1) q^{77} + (2 \beta_{3} + 10) q^{79} + ( - 6 \beta_{3} - 1) q^{81} + (3 \beta_{2} - 3 \beta_1) q^{83} + (8 \beta_{2} + 8 \beta_1) q^{87} + (2 \beta_{3} - 9) q^{89} + (8 \beta_{3} - 16) q^{91} + 2 \beta_1 q^{93} + ( - 8 \beta_{2} - 6 \beta_1) q^{97} + (2 \beta_{3} + 12) q^{99}+O(q^{100}) \) q + (b2 + b1) * q^3 + (-2b2 + 2b1) * q^7 - 2b3 q^9 + (-3b3 - 1) q^11 - 4b2 q^13 + (2b2 - 3b1) * q^17 + (-b3 + 3) * q^19 + 2 * q^21 + (-2b2 - 2b1) * q^23 + (b2 - b1) * q^27 + 8 * q^29 + (2b3 - 2) q^31 + (-4b2 - 7b1) * q^33 + (4b2 + 2b1) * q^37 + (4b3 + 8) q^39 + (4b3 + 5) q^41 + 10b1 q^43 + (4b2 + 4b1) * q^47 + (8b3 - 5) q^49 + (b3 - 1) * q^51 + (4b2 + 2b1) * q^53 + (2b2 + b1) q^57 + (-4b3 + 2) q^59 - 6 * q^61 + (-4b2 + 8b1) * q^63 + (-3b2 + 7b1) * q^67 + (4b3 + 6) q^69 + (4b3 + 4) q^71 + (-2b2 - 3b1) * q^73 + (-4b2 + 10b1) * q^77 + (2b3 + 10) q^79 + (-6b3 - 1) q^81 + (3b2 - 3b1) * q^83 + (8b2 + 8b1) * q^87 + (2b3 - 9) q^89 + (8b3 - 16) q^91 + 2b1 q^93 + (-8b2 - 6b1) * q^97 + (2b3 + 12) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 12 q^{19} + 8 q^{21} + 32 q^{29} - 8 q^{31} + 32 q^{39} + 20 q^{41} - 20 q^{49} - 4 q^{51} + 8 q^{59} - 24 q^{61} + 24 q^{69} + 16 q^{71} + 40 q^{79} - 4 q^{81} - 36 q^{89} - 64 q^{91} + 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^11 + 12 * q^19 + 8 * q^21 + 32 * q^29 - 8 * q^31 + 32 * q^39 + 20 * q^41 - 20 * q^49 - 4 * q^51 + 8 * q^59 - 24 * q^61 + 24 * q^69 + 16 * q^71 + 40 * q^79 - 4 * q^81 - 36 * q^89 - 64 * q^91 + 48 * q^99

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