LMFDB - Newform orbit 300.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,4,Mod(1,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("300.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	300.a (trivial)

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:	yes
Analytic conductor:	$17.7005730017$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 3 q^{3} + 28 q^{7} + 9 q^{9}+O(q^{10}) $

$ q + 3 q^{3} + 28 q^{7} + 9 q^{9} - 24 q^{11} + 70 q^{13} - 102 q^{17} + 20 q^{19} + 84 q^{21} + 72 q^{23} + 27 q^{27} + 306 q^{29} - 136 q^{31} - 72 q^{33} + 214 q^{37} + 210 q^{39} - 150 q^{41} + 292 q^{43} + 72 q^{47} + 441 q^{49} - 306 q^{51} + 414 q^{53} + 60 q^{57} - 744 q^{59} - 418 q^{61} + 252 q^{63} - 188 q^{67} + 216 q^{69} + 480 q^{71} - 434 q^{73} - 672 q^{77} + 1352 q^{79} + 81 q^{81} + 612 q^{83} + 918 q^{87} - 30 q^{89} + 1960 q^{91} - 408 q^{93} + 286 q^{97} - 216 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $

1.1

3.00000

28.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.4.a.i		1
3.b	odd	2	1		900.4.a.q		1
4.b	odd	2	1		1200.4.a.a		1
5.b	even	2	1		60.4.a.a	✓	1
5.c	odd	4	2		300.4.d.b		2
15.d	odd	2	1		180.4.a.d		1
15.e	even	4	2		900.4.d.h		2
20.d	odd	2	1		240.4.a.i		1
20.e	even	4	2		1200.4.f.n		2
40.e	odd	2	1		960.4.a.r		1
40.f	even	2	1		960.4.a.bc		1
45.h	odd	6	2		1620.4.i.f		2
45.j	even	6	2		1620.4.i.l		2
60.h	even	2	1		720.4.a.bb		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.4.a.a	✓	1	5.b	even	2	1
180.4.a.d		1	15.d	odd	2	1
240.4.a.i		1	20.d	odd	2	1
300.4.a.i		1	1.a	even	1	1	trivial
300.4.d.b		2	5.c	odd	4	2
720.4.a.bb		1	60.h	even	2	1
900.4.a.q		1	3.b	odd	2	1
900.4.d.h		2	15.e	even	4	2
960.4.a.r		1	40.e	odd	2	1
960.4.a.bc		1	40.f	even	2	1
1200.4.a.a		1	4.b	odd	2	1
1200.4.f.n		2	20.e	even	4	2
1620.4.i.f		2	45.h	odd	6	2
1620.4.i.l		2	45.j	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 28 $ T7 - 28 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(300))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 3 $ T - 3
$5$	$ T $ T
$7$	$ T - 28 $ T - 28
$11$	$ T + 24 $ T + 24
$13$	$ T - 70 $ T - 70
$17$	$ T + 102 $ T + 102
$19$	$ T - 20 $ T - 20
$23$	$ T - 72 $ T - 72
$29$	$ T - 306 $ T - 306
$31$	$ T + 136 $ T + 136
$37$	$ T - 214 $ T - 214
$41$	$ T + 150 $ T + 150
$43$	$ T - 292 $ T - 292
$47$	$ T - 72 $ T - 72
$53$	$ T - 414 $ T - 414
$59$	$ T + 744 $ T + 744
$61$	$ T + 418 $ T + 418
$67$	$ T + 188 $ T + 188
$71$	$ T - 480 $ T - 480
$73$	$ T + 434 $ T + 434
$79$	$ T - 1352 $ T - 1352
$83$	$ T - 612 $ T - 612
$89$	$ T + 30 $ T + 30
$97$	$ T - 286 $ T - 286
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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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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