LMFDB - Newform orbit 350.4.a.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,4,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.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:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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 + 2 q^{2} - 8 q^{3} + 4 q^{4} - 16 q^{6} + 7 q^{7} + 8 q^{8} + 37 q^{9}+O(q^{10}) $

$ q + 2 q^{2} - 8 q^{3} + 4 q^{4} - 16 q^{6} + 7 q^{7} + 8 q^{8} + 37 q^{9} - 28 q^{11} - 32 q^{12} - 18 q^{13} + 14 q^{14} + 16 q^{16} - 74 q^{17} + 74 q^{18} + 80 q^{19} - 56 q^{21} - 56 q^{22} + 112 q^{23} - 64 q^{24} - 36 q^{26} - 80 q^{27} + 28 q^{28} + 190 q^{29} + 72 q^{31} + 32 q^{32} + 224 q^{33} - 148 q^{34} + 148 q^{36} + 346 q^{37} + 160 q^{38} + 144 q^{39} + 162 q^{41} - 112 q^{42} + 412 q^{43} - 112 q^{44} + 224 q^{46} - 24 q^{47} - 128 q^{48} + 49 q^{49} + 592 q^{51} - 72 q^{52} - 318 q^{53} - 160 q^{54} + 56 q^{56} - 640 q^{57} + 380 q^{58} - 200 q^{59} - 198 q^{61} + 144 q^{62} + 259 q^{63} + 64 q^{64} + 448 q^{66} + 716 q^{67} - 296 q^{68} - 896 q^{69} + 392 q^{71} + 296 q^{72} - 538 q^{73} + 692 q^{74} + 320 q^{76} - 196 q^{77} + 288 q^{78} + 240 q^{79} - 359 q^{81} + 324 q^{82} + 1072 q^{83} - 224 q^{84} + 824 q^{86} - 1520 q^{87} - 224 q^{88} + 810 q^{89} - 126 q^{91} + 448 q^{92} - 576 q^{93} - 48 q^{94} - 256 q^{96} - 1354 q^{97} + 98 q^{98} - 1036 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

2.00000

−8.00000

4.00000

−16.0000

7.00000

8.00000

37.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$5$	$1$
$7$	$-1$

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	350.4.a.l		1
5.b	even	2	1		14.4.a.a	✓	1
5.c	odd	4	2		350.4.c.b		2
7.b	odd	2	1		2450.4.a.bo		1
15.d	odd	2	1		126.4.a.h		1
20.d	odd	2	1		112.4.a.a		1
35.c	odd	2	1		98.4.a.a		1
35.i	odd	6	2		98.4.c.f		2
35.j	even	6	2		98.4.c.d		2
40.e	odd	2	1		448.4.a.o		1
40.f	even	2	1		448.4.a.b		1
55.d	odd	2	1		1694.4.a.g		1
60.h	even	2	1		1008.4.a.s		1
65.d	even	2	1		2366.4.a.h		1
105.g	even	2	1		882.4.a.i		1
105.o	odd	6	2		882.4.g.b		2
105.p	even	6	2		882.4.g.k		2
140.c	even	2	1		784.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.a.a	✓	1	5.b	even	2	1
98.4.a.a		1	35.c	odd	2	1
98.4.c.d		2	35.j	even	6	2
98.4.c.f		2	35.i	odd	6	2
112.4.a.a		1	20.d	odd	2	1
126.4.a.h		1	15.d	odd	2	1
350.4.a.l		1	1.a	even	1	1	trivial
350.4.c.b		2	5.c	odd	4	2
448.4.a.b		1	40.f	even	2	1
448.4.a.o		1	40.e	odd	2	1
784.4.a.s		1	140.c	even	2	1
882.4.a.i		1	105.g	even	2	1
882.4.g.b		2	105.o	odd	6	2
882.4.g.k		2	105.p	even	6	2
1008.4.a.s		1	60.h	even	2	1
1694.4.a.g		1	55.d	odd	2	1
2366.4.a.h		1	65.d	even	2	1
2450.4.a.bo		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(350))$:

$ T_{3} + 8 $

$ T_{11} + 28 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 8 $ T + 8
$5$	$ T $ T
$7$	$ T - 7 $ T - 7
$11$	$ T + 28 $ T + 28
$13$	$ T + 18 $ T + 18
$17$	$ T + 74 $ T + 74
$19$	$ T - 80 $ T - 80
$23$	$ T - 112 $ T - 112
$29$	$ T - 190 $ T - 190
$31$	$ T - 72 $ T - 72
$37$	$ T - 346 $ T - 346
$41$	$ T - 162 $ T - 162
$43$	$ T - 412 $ T - 412
$47$	$ T + 24 $ T + 24
$53$	$ T + 318 $ T + 318
$59$	$ T + 200 $ T + 200
$61$	$ T + 198 $ T + 198
$67$	$ T - 716 $ T - 716
$71$	$ T - 392 $ T - 392
$73$	$ T + 538 $ T + 538
$79$	$ T - 240 $ T - 240
$83$	$ T - 1072 $ T - 1072
$89$	$ T - 810 $ T - 810
$97$	$ T + 1354 $ T + 1354
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

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