LMFDB - Newform orbit 225.4.b.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$13.2754297513$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 9)
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 $\beta = 10i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 8 q^{4} + 2 \beta q^{7}+O(q^{10}) $ q + 8 * q^4 + 2b q^7	$ q + 8 q^{4} + 2 \beta q^{7} + 7 \beta q^{13} + 64 q^{16} - 56 q^{19} + 16 \beta q^{28} + 308 q^{31} + 11 \beta q^{37} + 52 \beta q^{43} - 57 q^{49} + 56 \beta q^{52} + 182 q^{61} + 512 q^{64} - 88 \beta q^{67} - 119 \beta q^{73} - 448 q^{76} - 884 q^{79} - 1400 q^{91} - 133 \beta q^{97} +O(q^{100}) $ q + 8 * q^4 + 2b q^7 + 7b q^13 + 64 * q^16 - 56 * q^19 + 16b q^28 + 308 * q^31 + 11b q^37 + 52b q^43 - 57 * q^49 + 56b q^52 + 182 * q^61 + 512 * q^64 - 88b q^67 - 119b q^73 - 448 * q^76 - 884 * q^79 - 1400 * q^91 - 133b q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4}+O(q^{10}) $ 2 * q + 16 * q^4	$ 2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91}+O(q^{100}) $ 2 * q + 16 * q^4 + 128 * q^16 - 112 * q^19 + 616 * q^31 - 114 * q^49 + 364 * q^61 + 1024 * q^64 - 896 * q^76 - 1768 * q^79 - 2800 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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

8.00000

−

20.0000i

199.2

8.00000

20.0000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.4.b.g		2
3.b	odd	2	1	CM	225.4.b.g		2
5.b	even	2	1	inner	225.4.b.g		2
5.c	odd	4	1		9.4.a.a	✓	1
5.c	odd	4	1		225.4.a.d		1
15.d	odd	2	1	inner	225.4.b.g		2
15.e	even	4	1		9.4.a.a	✓	1
15.e	even	4	1		225.4.a.d		1
20.e	even	4	1		144.4.a.d		1
35.f	even	4	1		441.4.a.f		1
35.k	even	12	2		441.4.e.j		2
35.l	odd	12	2		441.4.e.i		2
40.i	odd	4	1		576.4.a.m		1
40.k	even	4	1		576.4.a.l		1
45.k	odd	12	2		81.4.c.b		2
45.l	even	12	2		81.4.c.b		2
55.e	even	4	1		1089.4.a.g		1
60.l	odd	4	1		144.4.a.d		1
65.h	odd	4	1		1521.4.a.g		1
105.k	odd	4	1		441.4.a.f		1
105.w	odd	12	2		441.4.e.j		2
105.x	even	12	2		441.4.e.i		2
120.q	odd	4	1		576.4.a.l		1
120.w	even	4	1		576.4.a.m		1
165.l	odd	4	1		1089.4.a.g		1
195.s	even	4	1		1521.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.a.a	✓	1	5.c	odd	4	1
9.4.a.a	✓	1	15.e	even	4	1
81.4.c.b		2	45.k	odd	12	2
81.4.c.b		2	45.l	even	12	2
144.4.a.d		1	20.e	even	4	1
144.4.a.d		1	60.l	odd	4	1
225.4.a.d		1	5.c	odd	4	1
225.4.a.d		1	15.e	even	4	1
225.4.b.g		2	1.a	even	1	1	trivial
225.4.b.g		2	3.b	odd	2	1	CM
225.4.b.g		2	5.b	even	2	1	inner
225.4.b.g		2	15.d	odd	2	1	inner
441.4.a.f		1	35.f	even	4	1
441.4.a.f		1	105.k	odd	4	1
441.4.e.i		2	35.l	odd	12	2
441.4.e.i		2	105.x	even	12	2
441.4.e.j		2	35.k	even	12	2
441.4.e.j		2	105.w	odd	12	2
576.4.a.l		1	40.k	even	4	1
576.4.a.l		1	120.q	odd	4	1
576.4.a.m		1	40.i	odd	4	1
576.4.a.m		1	120.w	even	4	1
1089.4.a.g		1	55.e	even	4	1
1089.4.a.g		1	165.l	odd	4	1
1521.4.a.g		1	65.h	odd	4	1
1521.4.a.g		1	195.s	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(225, [\chi])$:

$ T_{2} $

$ T_{7}^{2} + 400 $

$ T_{11} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 400 $ T^2 + 400
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 4900 $ T^2 + 4900
$17$	$ T^{2} $ T^2
$19$	$ (T + 56)^{2} $ (T + 56)^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ (T - 308)^{2} $ (T - 308)^2
$37$	$ T^{2} + 12100 $ T^2 + 12100
$41$	$ T^{2} $ T^2
$43$	$ T^{2} + 270400 $ T^2 + 270400
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ (T - 182)^{2} $ (T - 182)^2
$67$	$ T^{2} + 774400 $ T^2 + 774400
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 1416100 $ T^2 + 1416100
$79$	$ (T + 884)^{2} $ (T + 884)^2
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} + 1768900 $ T^2 + 1768900
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Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 8 q^{4} + 2 \beta q^{7}+O(q^{10}) \) q + 8 * q^4 + 2b q^7	\( q + 8 q^{4} + 2 \beta q^{7} + 7 \beta q^{13} + 64 q^{16} - 56 q^{19} + 16 \beta q^{28} + 308 q^{31} + 11 \beta q^{37} + 52 \beta q^{43} - 57 q^{49} + 56 \beta q^{52} + 182 q^{61} + 512 q^{64} - 88 \beta q^{67} - 119 \beta q^{73} - 448 q^{76} - 884 q^{79} - 1400 q^{91} - 133 \beta q^{97} +O(q^{100}) \) q + 8 * q^4 + 2b q^7 + 7b q^13 + 64 * q^16 - 56 * q^19 + 16b q^28 + 308 * q^31 + 11b q^37 + 52b q^43 - 57 * q^49 + 56b q^52 + 182 * q^61 + 512 * q^64 - 88b q^67 - 119b q^73 - 448 * q^76 - 884 * q^79 - 1400 * q^91 - 133b q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4}+O(q^{10}) \) 2 * q + 16 * q^4	\( 2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91}+O(q^{100}) \) 2 * q + 16 * q^4 + 128 * q^16 - 112 * q^19 + 616 * q^31 - 114 * q^49 + 364 * q^61 + 1024 * q^64 - 896 * q^76 - 1768 * q^79 - 2800 * q^91

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