LMFDB - Newform orbit 1575.4.a.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
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 + q^{2} - 7 q^{4} - 7 q^{7} - 15 q^{8}+O(q^{10}) $

$ q + q^{2} - 7 q^{4} - 7 q^{7} - 15 q^{8} - 12 q^{11} + 78 q^{13} - 7 q^{14} + 41 q^{16} - 94 q^{17} + 40 q^{19} - 12 q^{22} + 32 q^{23} + 78 q^{26} + 49 q^{28} + 50 q^{29} - 248 q^{31} + 161 q^{32} - 94 q^{34} + 434 q^{37} + 40 q^{38} - 402 q^{41} + 68 q^{43} + 84 q^{44} + 32 q^{46} + 536 q^{47} + 49 q^{49} - 546 q^{52} + 22 q^{53} + 105 q^{56} + 50 q^{58} + 560 q^{59} - 278 q^{61} - 248 q^{62} - 167 q^{64} + 164 q^{67} + 658 q^{68} - 672 q^{71} - 82 q^{73} + 434 q^{74} - 280 q^{76} + 84 q^{77} - 1000 q^{79} - 402 q^{82} - 448 q^{83} + 68 q^{86} + 180 q^{88} + 870 q^{89} - 546 q^{91} - 224 q^{92} + 536 q^{94} - 1026 q^{97} + 49 q^{98}+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

1.00000

−7.00000

−15.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$5$	$1$
$7$	$1$

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	1575.4.a.g		1
3.b	odd	2	1		175.4.a.a		1
5.b	even	2	1		315.4.a.c		1
15.d	odd	2	1		35.4.a.a	✓	1
15.e	even	4	2		175.4.b.a		2
21.c	even	2	1		1225.4.a.e		1
35.c	odd	2	1		2205.4.a.i		1
60.h	even	2	1		560.4.a.p		1
105.g	even	2	1		245.4.a.d		1
105.o	odd	6	2		245.4.e.e		2
105.p	even	6	2		245.4.e.b		2
120.i	odd	2	1		2240.4.a.bk		1
120.m	even	2	1		2240.4.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.a.a	✓	1	15.d	odd	2	1
175.4.a.a		1	3.b	odd	2	1
175.4.b.a		2	15.e	even	4	2
245.4.a.d		1	105.g	even	2	1
245.4.e.b		2	105.p	even	6	2
245.4.e.e		2	105.o	odd	6	2
315.4.a.c		1	5.b	even	2	1
560.4.a.p		1	60.h	even	2	1
1225.4.a.e		1	21.c	even	2	1
1575.4.a.g		1	1.a	even	1	1	trivial
2205.4.a.i		1	35.c	odd	2	1
2240.4.a.b		1	120.m	even	2	1
2240.4.a.bk		1	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1575))$:

$ T_{2} - 1 $

$ T_{11} + 12 $

$ T_{13} - 78 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 7 $ T + 7
$11$	$ T + 12 $ T + 12
$13$	$ T - 78 $ T - 78
$17$	$ T + 94 $ T + 94
$19$	$ T - 40 $ T - 40
$23$	$ T - 32 $ T - 32
$29$	$ T - 50 $ T - 50
$31$	$ T + 248 $ T + 248
$37$	$ T - 434 $ T - 434
$41$	$ T + 402 $ T + 402
$43$	$ T - 68 $ T - 68
$47$	$ T - 536 $ T - 536
$53$	$ T - 22 $ T - 22
$59$	$ T - 560 $ T - 560
$61$	$ T + 278 $ T + 278
$67$	$ T - 164 $ T - 164
$71$	$ T + 672 $ T + 672
$73$	$ T + 82 $ T + 82
$79$	$ T + 1000 $ T + 1000
$83$	$ T + 448 $ T + 448
$89$	$ T - 870 $ T - 870
$97$	$ T + 1026 $ T + 1026
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Overview	Random
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

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