LMFDB - Newform orbit 160.4.c.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, 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:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \beta_{3} - \beta_1) q^{3} - 5 \beta_{2} q^{5} + ( - 5 \beta_{3} - 6 \beta_1) q^{7} + ( - 14 \beta_{2} - 27) q^{9}+O(q^{10}) $ q + (2b3 - b1) q^3 - 5b2 q^5 + (-5b3 - 6b1) * q^7 + (-14b2 - 27) q^9	$ q + (2 \beta_{3} - \beta_1) q^{3} - 5 \beta_{2} q^{5} + ( - 5 \beta_{3} - 6 \beta_1) q^{7} + ( - 14 \beta_{2} - 27) q^{9} + ( - 15 \beta_{3} - 20 \beta_1) q^{15} + (86 \beta_{2} + 118) q^{21} + (\beta_{3} - 64 \beta_1) q^{23} + 125 q^{25} + ( - 42 \beta_{3} - 56 \beta_1) q^{27} - 306 q^{29} + (80 \beta_{3} - 35 \beta_1) q^{35} - 206 \beta_{2} q^{41} + ( - 104 \beta_{3} - 15 \beta_1) q^{43} + (135 \beta_{2} + 350) q^{45} + ( - 37 \beta_{3} + 190 \beta_1) q^{47} + ( - 198 \beta_{2} - 343) q^{49} - 18 \beta_{2} q^{61} + (359 \beta_{3} + 64 \beta_1) q^{63} + ( - 76 \beta_{3} + 321 \beta_1) q^{67} + (374 \beta_{2} - 154) q^{69} + (250 \beta_{3} - 125 \beta_1) q^{75} + (378 \beta_{2} + 251) q^{81} + (100 \beta_{3} + 377 \beta_1) q^{83} + ( - 612 \beta_{3} + 306 \beta_1) q^{87} + 1386 q^{89}+O(q^{100}) $ q + (2b3 - b1) q^3 - 5b2 q^5 + (-5b3 - 6b1) * q^7 + (-14b2 - 27) q^9 + (-15b3 - 20b1) * q^15 + (86b2 + 118) q^21 + (b3 - 64b1) q^23 + 125 * q^25 + (-42b3 - 56b1) * q^27 - 306 * q^29 + (80b3 - 35b1) * q^35 - 206b2 q^41 + (-104b3 - 15b1) * q^43 + (135b2 + 350) q^45 + (-37b3 + 190b1) * q^47 + (-198b2 - 343) q^49 - 18b2 q^61 + (359b3 + 64b1) * q^63 + (-76b3 + 321b1) * q^67 + (374b2 - 154) q^69 + (250b3 - 125b1) * q^75 + (378b2 + 251) q^81 + (100b3 + 377b1) * q^83 + (-612b3 + 306b1) * q^87 + 1386 * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) $ 4 * q - 108 * q^9 + 472 * q^21 + 500 * q^25 - 1224 * q^29 + 1400 * q^45 - 1372 * q^49 - 616 * q^69 + 1004 * q^81 + 5544 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1 :

$\beta_{1}$	$=$	$ 2\nu $ 2*v
$\beta_{2}$	$=$	$ 2\nu^{2} + 3 $ 2*v^2 + 3
$\beta_{3}$	$=$	$ 4\nu^{3} + 10\nu $ 4v^3 + 10v

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

$\nu$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{2} - 3 ) / 2 $ (b2 - 3) / 2
$\nu^{3}$	$=$	$ ( \beta_{3} - 5\beta_1 ) / 4 $ (b3 - 5*b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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} $

129.1

	−	0.618034i
		1.61803i
	−	1.61803i
		0.618034i

−

9.23607i

−11.1803

33.5967i

−58.3050

129.2

−

4.76393i

11.1803

−

15.5967i

4.30495

129.3

4.76393i

11.1803

15.5967i

4.30495

129.4

9.23607i

−11.1803

−

33.5967i

−58.3050

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	160.4.c.c	✓	4
3.b	odd	2	1		1440.4.f.g		4
4.b	odd	2	1	inner	160.4.c.c	✓	4
5.b	even	2	1	inner	160.4.c.c	✓	4
5.c	odd	4	1		800.4.a.l		2
5.c	odd	4	1		800.4.a.t		2
8.b	even	2	1		320.4.c.f		4
8.d	odd	2	1		320.4.c.f		4
12.b	even	2	1		1440.4.f.g		4
15.d	odd	2	1		1440.4.f.g		4
20.d	odd	2	1	CM	160.4.c.c	✓	4
20.e	even	4	1		800.4.a.l		2
20.e	even	4	1		800.4.a.t		2
40.e	odd	2	1		320.4.c.f		4
40.f	even	2	1		320.4.c.f		4
40.i	odd	4	1		1600.4.a.cb		2
40.i	odd	4	1		1600.4.a.cp		2
40.k	even	4	1		1600.4.a.cb		2
40.k	even	4	1		1600.4.a.cp		2
60.h	even	2	1		1440.4.f.g		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.4.c.c	✓	4	1.a	even	1	1	trivial
160.4.c.c	✓	4	4.b	odd	2	1	inner
160.4.c.c	✓	4	5.b	even	2	1	inner
160.4.c.c	✓	4	20.d	odd	2	1	CM
320.4.c.f		4	8.b	even	2	1
320.4.c.f		4	8.d	odd	2	1
320.4.c.f		4	40.e	odd	2	1
320.4.c.f		4	40.f	even	2	1
800.4.a.l		2	5.c	odd	4	1
800.4.a.l		2	20.e	even	4	1
800.4.a.t		2	5.c	odd	4	1
800.4.a.t		2	20.e	even	4	1
1440.4.f.g		4	3.b	odd	2	1
1440.4.f.g		4	12.b	even	2	1
1440.4.f.g		4	15.d	odd	2	1
1440.4.f.g		4	60.h	even	2	1
1600.4.a.cb		2	40.i	odd	4	1
1600.4.a.cb		2	40.k	even	4	1
1600.4.a.cp		2	40.i	odd	4	1
1600.4.a.cp		2	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 108T_{3}^{2} + 1936 $ T3^4 + 108*T3^2 + 1936 acting on $S_{4}^{\mathrm{new}}(160, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 108T^{2} + 1936 $ T^4 + 108*T^2 + 1936
$5$	$ (T^{2} - 125)^{2} $ (T^2 - 125)^2
$7$	$ T^{4} + 1372 T^{2} + 274576 $ T^4 + 1372*T^2 + 274576
$11$	$ T^{4} $ T^4
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} + 48668 T^{2} + 235806736 $ T^4 + 48668*T^2 + 235806736
$29$	$ (T + 306)^{4} $ (T + 306)^4
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ (T^{2} - 212180)^{2} $ (T^2 - 212180)^2
$43$	$ T^{4} + 318028 T^{2} + 302899216 $ T^4 + 318028*T^2 + 302899216
$47$	$ T^{4} + 415292 T^{2} + 699285136 $ T^4 + 415292*T^2 + 699285136
$53$	$ T^{4} $ T^4
$59$	$ T^{4} $ T^4
$61$	$ (T^{2} - 1620)^{2} $ (T^2 - 1620)^2
$67$	$ T^{4} + 1203052 T^{2} + 1628176 $ T^4 + 1203052*T^2 + 1628176
$71$	$ T^{4} $ T^4
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} + \cdots + 1280781104656 $ T^4 + 2287148*T^2 + 1280781104656
$89$	$ (T - 1386)^{4} $ (T - 1386)^4
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \beta_{3} - \beta_1) q^{3} - 5 \beta_{2} q^{5} + ( - 5 \beta_{3} - 6 \beta_1) q^{7} + ( - 14 \beta_{2} - 27) q^{9}+O(q^{10}) \) q + (2b3 - b1) q^3 - 5b2 q^5 + (-5b3 - 6b1) * q^7 + (-14b2 - 27) q^9	\( q + (2 \beta_{3} - \beta_1) q^{3} - 5 \beta_{2} q^{5} + ( - 5 \beta_{3} - 6 \beta_1) q^{7} + ( - 14 \beta_{2} - 27) q^{9} + ( - 15 \beta_{3} - 20 \beta_1) q^{15} + (86 \beta_{2} + 118) q^{21} + (\beta_{3} - 64 \beta_1) q^{23} + 125 q^{25} + ( - 42 \beta_{3} - 56 \beta_1) q^{27} - 306 q^{29} + (80 \beta_{3} - 35 \beta_1) q^{35} - 206 \beta_{2} q^{41} + ( - 104 \beta_{3} - 15 \beta_1) q^{43} + (135 \beta_{2} + 350) q^{45} + ( - 37 \beta_{3} + 190 \beta_1) q^{47} + ( - 198 \beta_{2} - 343) q^{49} - 18 \beta_{2} q^{61} + (359 \beta_{3} + 64 \beta_1) q^{63} + ( - 76 \beta_{3} + 321 \beta_1) q^{67} + (374 \beta_{2} - 154) q^{69} + (250 \beta_{3} - 125 \beta_1) q^{75} + (378 \beta_{2} + 251) q^{81} + (100 \beta_{3} + 377 \beta_1) q^{83} + ( - 612 \beta_{3} + 306 \beta_1) q^{87} + 1386 q^{89}+O(q^{100}) \) q + (2b3 - b1) q^3 - 5b2 q^5 + (-5b3 - 6b1) * q^7 + (-14b2 - 27) q^9 + (-15b3 - 20b1) * q^15 + (86b2 + 118) q^21 + (b3 - 64b1) q^23 + 125 * q^25 + (-42b3 - 56b1) * q^27 - 306 * q^29 + (80b3 - 35b1) * q^35 - 206b2 q^41 + (-104b3 - 15b1) * q^43 + (135b2 + 350) q^45 + (-37b3 + 190b1) * q^47 + (-198b2 - 343) q^49 - 18b2 q^61 + (359b3 + 64b1) * q^63 + (-76b3 + 321b1) * q^67 + (374b2 - 154) q^69 + (250b3 - 125b1) * q^75 + (378b2 + 251) q^81 + (100b3 + 377b1) * q^83 + (-612b3 + 306b1) * q^87 + 1386 * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) \) 4 * q - 108 * q^9 + 472 * q^21 + 500 * q^25 - 1224 * q^29 + 1400 * q^45 - 1372 * q^49 - 616 * q^69 + 1004 * q^81 + 5544 * q^89

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