LMFDB - Newform orbit 900.3.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(199,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, 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("900.199");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.f (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:	$24.5232237924$
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 36)
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} + 4 \beta q^{8} +O(q^{10}) $ q - b * q^2 - 4 * q^4 + 4b q^8	$ q - \beta q^{2} - 4 q^{4} + 4 \beta q^{8} - 5 \beta q^{13} + 16 q^{16} - 8 \beta q^{17} - 20 q^{26} - 40 q^{29} - 16 \beta q^{32} - 32 q^{34} + 35 \beta q^{37} - 80 q^{41} - 49 q^{49} + 20 \beta q^{52} - 28 \beta q^{53} + 40 \beta q^{58} - 22 q^{61} - 64 q^{64} + 32 \beta q^{68} + 55 \beta q^{73} + 140 q^{74} + 80 \beta q^{82} - 160 q^{89} + 65 \beta q^{97} + 49 \beta q^{98} +O(q^{100}) $ q - b * q^2 - 4 * q^4 + 4b q^8 - 5b q^13 + 16 * q^16 - 8b q^17 - 20 * q^26 - 40 * q^29 - 16b q^32 - 32 * q^34 + 35b q^37 - 80 * q^41 - 49 * q^49 + 20b q^52 - 28b q^53 + 40b q^58 - 22 * q^61 - 64 * q^64 + 32b q^68 + 55b q^73 + 140 * q^74 + 80b q^82 - 160 * q^89 + 65b q^97 + 49b q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 32 q^{16} - 40 q^{26} - 80 q^{29} - 64 q^{34} - 160 q^{41} - 98 q^{49} - 44 q^{61} - 128 q^{64} + 280 q^{74} - 320 q^{89}+O(q^{100}) $ 2 * q - 8 * q^4 + 32 * q^16 - 40 * q^26 - 80 * q^29 - 64 * q^34 - 160 * q^41 - 98 * q^49 - 44 * q^61 - 128 * q^64 + 280 * q^74 - 320 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$-1$

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} $

199.1

		1.00000i
	−	1.00000i

−

2.00000i

−4.00000

8.00000i

199.2

2.00000i

−4.00000

−

8.00000i

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	900.3.f.a		2
3.b	odd	2	1		900.3.f.b		2
4.b	odd	2	1	CM	900.3.f.a		2
5.b	even	2	1	inner	900.3.f.a		2
5.c	odd	4	1		36.3.d.b	yes	1
5.c	odd	4	1		900.3.c.b		1
12.b	even	2	1		900.3.f.b		2
15.d	odd	2	1		900.3.f.b		2
15.e	even	4	1		36.3.d.a	✓	1
15.e	even	4	1		900.3.c.c		1
20.d	odd	2	1	inner	900.3.f.a		2
20.e	even	4	1		36.3.d.b	yes	1
20.e	even	4	1		900.3.c.b		1
40.i	odd	4	1		576.3.g.c		1
40.k	even	4	1		576.3.g.c		1
45.k	odd	12	2		324.3.f.e		2
45.l	even	12	2		324.3.f.f		2
60.h	even	2	1		900.3.f.b		2
60.l	odd	4	1		36.3.d.a	✓	1
60.l	odd	4	1		900.3.c.c		1
80.i	odd	4	1		2304.3.b.e		2
80.j	even	4	1		2304.3.b.e		2
80.s	even	4	1		2304.3.b.e		2
80.t	odd	4	1		2304.3.b.e		2
120.q	odd	4	1		576.3.g.a		1
120.w	even	4	1		576.3.g.a		1
180.v	odd	12	2		324.3.f.f		2
180.x	even	12	2		324.3.f.e		2
240.z	odd	4	1		2304.3.b.d		2
240.bb	even	4	1		2304.3.b.d		2
240.bd	odd	4	1		2304.3.b.d		2
240.bf	even	4	1		2304.3.b.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.3.d.a	✓	1	15.e	even	4	1
36.3.d.a	✓	1	60.l	odd	4	1
36.3.d.b	yes	1	5.c	odd	4	1
36.3.d.b	yes	1	20.e	even	4	1
324.3.f.e		2	45.k	odd	12	2
324.3.f.e		2	180.x	even	12	2
324.3.f.f		2	45.l	even	12	2
324.3.f.f		2	180.v	odd	12	2
576.3.g.a		1	120.q	odd	4	1
576.3.g.a		1	120.w	even	4	1
576.3.g.c		1	40.i	odd	4	1
576.3.g.c		1	40.k	even	4	1
900.3.c.b		1	5.c	odd	4	1
900.3.c.b		1	20.e	even	4	1
900.3.c.c		1	15.e	even	4	1
900.3.c.c		1	60.l	odd	4	1
900.3.f.a		2	1.a	even	1	1	trivial
900.3.f.a		2	4.b	odd	2	1	CM
900.3.f.a		2	5.b	even	2	1	inner
900.3.f.a		2	20.d	odd	2	1	inner
900.3.f.b		2	3.b	odd	2	1
900.3.f.b		2	12.b	even	2	1
900.3.f.b		2	15.d	odd	2	1
900.3.f.b		2	60.h	even	2	1
2304.3.b.d		2	240.z	odd	4	1
2304.3.b.d		2	240.bb	even	4	1
2304.3.b.d		2	240.bd	odd	4	1
2304.3.b.d		2	240.bf	even	4	1
2304.3.b.e		2	80.i	odd	4	1
2304.3.b.e		2	80.j	even	4	1
2304.3.b.e		2	80.s	even	4	1
2304.3.b.e		2	80.t	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(900, [\chi])$:

$ T_{7} $

$ T_{29} + 40 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4 $ T^2 + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 100 $ T^2 + 100
$17$	$ T^{2} + 256 $ T^2 + 256
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ (T + 40)^{2} $ (T + 40)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 4900 $ T^2 + 4900
$41$	$ (T + 80)^{2} $ (T + 80)^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} + 12100 $ T^2 + 12100
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T + 160)^{2} $ (T + 160)^2
$97$	$ T^{2} + 16900 $ T^2 + 16900
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta q^{2} - 4 q^{4} + 4 \beta q^{8} +O(q^{10}) \) q - b * q^2 - 4 * q^4 + 4b q^8	\( q - \beta q^{2} - 4 q^{4} + 4 \beta q^{8} - 5 \beta q^{13} + 16 q^{16} - 8 \beta q^{17} - 20 q^{26} - 40 q^{29} - 16 \beta q^{32} - 32 q^{34} + 35 \beta q^{37} - 80 q^{41} - 49 q^{49} + 20 \beta q^{52} - 28 \beta q^{53} + 40 \beta q^{58} - 22 q^{61} - 64 q^{64} + 32 \beta q^{68} + 55 \beta q^{73} + 140 q^{74} + 80 \beta q^{82} - 160 q^{89} + 65 \beta q^{97} + 49 \beta q^{98} +O(q^{100}) \) q - b * q^2 - 4 * q^4 + 4b q^8 - 5b q^13 + 16 * q^16 - 8b q^17 - 20 * q^26 - 40 * q^29 - 16b q^32 - 32 * q^34 + 35b q^37 - 80 * q^41 - 49 * q^49 + 20b q^52 - 28b q^53 + 40b q^58 - 22 * q^61 - 64 * q^64 + 32b q^68 + 55b q^73 + 140 * q^74 + 80b q^82 - 160 * q^89 + 65b q^97 + 49b q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 32 q^{16} - 40 q^{26} - 80 q^{29} - 64 q^{34} - 160 q^{41} - 98 q^{49} - 44 q^{61} - 128 q^{64} + 280 q^{74} - 320 q^{89}+O(q^{100}) \) 2 * q - 8 * q^4 + 32 * q^16 - 40 * q^26 - 80 * q^29 - 64 * q^34 - 160 * q^41 - 98 * q^49 - 44 * q^61 - 128 * q^64 + 280 * q^74 - 320 * q^89

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