LMFDB - Newform orbit 180.3.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,3,Mod(19,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("180.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	180.f (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:	$4.90464475849$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
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 $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} - 4 q^{4} + ( - 2 \beta - 3) q^{5} - 4 \beta q^{8} +O(q^{10}) $ q + b * q^2 - 4 * q^4 + (-2b - 3) q^5 - 4b q^8	$ q + \beta q^{2} - 4 q^{4} + ( - 2 \beta - 3) q^{5} - 4 \beta q^{8} + ( - 3 \beta + 8) q^{10} - 12 \beta q^{13} + 16 q^{16} - 8 \beta q^{17} + (8 \beta + 12) q^{20} + (12 \beta - 7) q^{25} + 48 q^{26} - 42 q^{29} + 16 \beta q^{32} + 32 q^{34} - 12 \beta q^{37} + (12 \beta - 32) q^{40} + 18 q^{41} - 49 q^{49} + ( - 7 \beta - 48) q^{50} + 48 \beta q^{52} - 28 \beta q^{53} - 42 \beta q^{58} + 22 q^{61} - 64 q^{64} + (36 \beta - 96) q^{65} + 32 \beta q^{68} + 48 \beta q^{73} + 48 q^{74} + ( - 32 \beta - 48) q^{80} + 18 \beta q^{82} + (24 \beta - 64) q^{85} + 78 q^{89} - 72 \beta q^{97} - 49 \beta q^{98} +O(q^{100}) $ q + b * q^2 - 4 * q^4 + (-2b - 3) q^5 - 4b q^8 + (-3b + 8) q^10 - 12b q^13 + 16 * q^16 - 8b q^17 + (8b + 12) q^20 + (12b - 7) q^25 + 48 * q^26 - 42 * q^29 + 16b q^32 + 32 * q^34 - 12b q^37 + (12b - 32) q^40 + 18 * q^41 - 49 * q^49 + (-7b - 48) q^50 + 48b q^52 - 28b q^53 - 42b q^58 + 22 * q^61 - 64 * q^64 + (36b - 96) q^65 + 32b q^68 + 48b q^73 + 48 * q^74 + (-32b - 48) q^80 + 18b q^82 + (24b - 64) q^85 + 78 * q^89 - 72b q^97 - 49b q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 6 q^{5}+O(q^{10}) $ 2 * q - 8 * q^4 - 6 * q^5	$ 2 q - 8 q^{4} - 6 q^{5} + 16 q^{10} + 32 q^{16} + 24 q^{20} - 14 q^{25} + 96 q^{26} - 84 q^{29} + 64 q^{34} - 64 q^{40} + 36 q^{41} - 98 q^{49} - 96 q^{50} + 44 q^{61} - 128 q^{64} - 192 q^{65} + 96 q^{74} - 96 q^{80} - 128 q^{85} + 156 q^{89}+O(q^{100}) $ 2 * q - 8 * q^4 - 6 * q^5 + 16 * q^10 + 32 * q^16 + 24 * q^20 - 14 * q^25 + 96 * q^26 - 84 * q^29 + 64 * q^34 - 64 * q^40 + 36 * q^41 - 98 * q^49 - 96 * q^50 + 44 * q^61 - 128 * q^64 - 192 * q^65 + 96 * q^74 - 96 * q^80 - 128 * q^85 + 156 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$-1$	$-1$	$1$

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} $

19.1

	−	1.00000i
		1.00000i

−

2.00000i

−4.00000

−3.00000

4.00000i

8.00000i

8.00000

6.00000i

19.2

2.00000i

−4.00000

−3.00000

−

4.00000i

−

8.00000i

8.00000

−

6.00000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	180.3.f.c		2
3.b	odd	2	1		20.3.d.c	✓	2
4.b	odd	2	1	CM	180.3.f.c		2
5.b	even	2	1	inner	180.3.f.c		2
5.c	odd	4	1		900.3.c.a		1
5.c	odd	4	1		900.3.c.d		1
12.b	even	2	1		20.3.d.c	✓	2
15.d	odd	2	1		20.3.d.c	✓	2
15.e	even	4	1		100.3.b.a		1
15.e	even	4	1		100.3.b.b		1
20.d	odd	2	1	inner	180.3.f.c		2
20.e	even	4	1		900.3.c.a		1
20.e	even	4	1		900.3.c.d		1
24.f	even	2	1		320.3.h.d		2
24.h	odd	2	1		320.3.h.d		2
48.i	odd	4	1		1280.3.e.a		2
48.i	odd	4	1		1280.3.e.d		2
48.k	even	4	1		1280.3.e.a		2
48.k	even	4	1		1280.3.e.d		2
60.h	even	2	1		20.3.d.c	✓	2
60.l	odd	4	1		100.3.b.a		1
60.l	odd	4	1		100.3.b.b		1
120.i	odd	2	1		320.3.h.d		2
120.m	even	2	1		320.3.h.d		2
120.q	odd	4	1		1600.3.b.a		1
120.q	odd	4	1		1600.3.b.c		1
120.w	even	4	1		1600.3.b.a		1
120.w	even	4	1		1600.3.b.c		1
240.t	even	4	1		1280.3.e.a		2
240.t	even	4	1		1280.3.e.d		2
240.bm	odd	4	1		1280.3.e.a		2
240.bm	odd	4	1		1280.3.e.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.3.d.c	✓	2	3.b	odd	2	1
20.3.d.c	✓	2	12.b	even	2	1
20.3.d.c	✓	2	15.d	odd	2	1
20.3.d.c	✓	2	60.h	even	2	1
100.3.b.a		1	15.e	even	4	1
100.3.b.a		1	60.l	odd	4	1
100.3.b.b		1	15.e	even	4	1
100.3.b.b		1	60.l	odd	4	1
180.3.f.c		2	1.a	even	1	1	trivial
180.3.f.c		2	4.b	odd	2	1	CM
180.3.f.c		2	5.b	even	2	1	inner
180.3.f.c		2	20.d	odd	2	1	inner
320.3.h.d		2	24.f	even	2	1
320.3.h.d		2	24.h	odd	2	1
320.3.h.d		2	120.i	odd	2	1
320.3.h.d		2	120.m	even	2	1
900.3.c.a		1	5.c	odd	4	1
900.3.c.a		1	20.e	even	4	1
900.3.c.d		1	5.c	odd	4	1
900.3.c.d		1	20.e	even	4	1
1280.3.e.a		2	48.i	odd	4	1
1280.3.e.a		2	48.k	even	4	1
1280.3.e.a		2	240.t	even	4	1
1280.3.e.a		2	240.bm	odd	4	1
1280.3.e.d		2	48.i	odd	4	1
1280.3.e.d		2	48.k	even	4	1
1280.3.e.d		2	240.t	even	4	1
1280.3.e.d		2	240.bm	odd	4	1
1600.3.b.a		1	120.q	odd	4	1
1600.3.b.a		1	120.w	even	4	1
1600.3.b.c		1	120.q	odd	4	1
1600.3.b.c		1	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(180, [\chi])$:

$ T_{7} $

$ T_{13}^{2} + 576 $

$ T_{23} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4 $ T^2 + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 6T + 25 $ T^2 + 6*T + 25
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 576 $ T^2 + 576
$17$	$ T^{2} + 256 $ T^2 + 256
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ (T + 42)^{2} $ (T + 42)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 576 $ T^2 + 576
$41$	$ (T - 18)^{2} $ (T - 18)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + 3136 $ T^2 + 3136
$59$	$ T^{2} $ T^2
$61$	$ (T - 22)^{2} $ (T - 22)^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 9216 $ T^2 + 9216
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T - 78)^{2} $ (T - 78)^2
$97$	$ T^{2} + 20736 $ T^2 + 20736
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Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 4 q^{4} + ( - 2 \beta - 3) q^{5} - 4 \beta q^{8} +O(q^{10}) \) q + b * q^2 - 4 * q^4 + (-2b - 3) q^5 - 4b q^8	\( q + \beta q^{2} - 4 q^{4} + ( - 2 \beta - 3) q^{5} - 4 \beta q^{8} + ( - 3 \beta + 8) q^{10} - 12 \beta q^{13} + 16 q^{16} - 8 \beta q^{17} + (8 \beta + 12) q^{20} + (12 \beta - 7) q^{25} + 48 q^{26} - 42 q^{29} + 16 \beta q^{32} + 32 q^{34} - 12 \beta q^{37} + (12 \beta - 32) q^{40} + 18 q^{41} - 49 q^{49} + ( - 7 \beta - 48) q^{50} + 48 \beta q^{52} - 28 \beta q^{53} - 42 \beta q^{58} + 22 q^{61} - 64 q^{64} + (36 \beta - 96) q^{65} + 32 \beta q^{68} + 48 \beta q^{73} + 48 q^{74} + ( - 32 \beta - 48) q^{80} + 18 \beta q^{82} + (24 \beta - 64) q^{85} + 78 q^{89} - 72 \beta q^{97} - 49 \beta q^{98} +O(q^{100}) \) q + b * q^2 - 4 * q^4 + (-2b - 3) q^5 - 4b q^8 + (-3b + 8) q^10 - 12b q^13 + 16 * q^16 - 8b q^17 + (8b + 12) q^20 + (12b - 7) q^25 + 48 * q^26 - 42 * q^29 + 16b q^32 + 32 * q^34 - 12b q^37 + (12b - 32) q^40 + 18 * q^41 - 49 * q^49 + (-7b - 48) q^50 + 48b q^52 - 28b q^53 - 42b q^58 + 22 * q^61 - 64 * q^64 + (36b - 96) q^65 + 32b q^68 + 48b q^73 + 48 * q^74 + (-32b - 48) q^80 + 18b q^82 + (24b - 64) q^85 + 78 * q^89 - 72b q^97 - 49b q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 6 q^{5}+O(q^{10}) \) 2 * q - 8 * q^4 - 6 * q^5	\( 2 q - 8 q^{4} - 6 q^{5} + 16 q^{10} + 32 q^{16} + 24 q^{20} - 14 q^{25} + 96 q^{26} - 84 q^{29} + 64 q^{34} - 64 q^{40} + 36 q^{41} - 98 q^{49} - 96 q^{50} + 44 q^{61} - 128 q^{64} - 192 q^{65} + 96 q^{74} - 96 q^{80} - 128 q^{85} + 156 q^{89}+O(q^{100}) \) 2 * q - 8 * q^4 - 6 * q^5 + 16 * q^10 + 32 * q^16 + 24 * q^20 - 14 * q^25 + 96 * q^26 - 84 * q^29 + 64 * q^34 - 64 * q^40 + 36 * q^41 - 98 * q^49 - 96 * q^50 + 44 * q^61 - 128 * q^64 - 192 * q^65 + 96 * q^74 - 96 * q^80 - 128 * q^85 + 156 * q^89

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