LMFDB - Newform orbit 150.4.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,4,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	150.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:	$8.85028650086$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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} - 3 q^{3} + 4 q^{4} - 6 q^{6} - 2 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $

$ q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} - 2 q^{7} + 8 q^{8} + 9 q^{9} + 70 q^{11} - 12 q^{12} + 54 q^{13} - 4 q^{14} + 16 q^{16} - 22 q^{17} + 18 q^{18} + 24 q^{19} + 6 q^{21} + 140 q^{22} - 100 q^{23} - 24 q^{24} + 108 q^{26} - 27 q^{27} - 8 q^{28} + 216 q^{29} + 208 q^{31} + 32 q^{32} - 210 q^{33} - 44 q^{34} + 36 q^{36} - 254 q^{37} + 48 q^{38} - 162 q^{39} - 206 q^{41} + 12 q^{42} + 292 q^{43} + 280 q^{44} - 200 q^{46} - 320 q^{47} - 48 q^{48} - 339 q^{49} + 66 q^{51} + 216 q^{52} - 402 q^{53} - 54 q^{54} - 16 q^{56} - 72 q^{57} + 432 q^{58} - 370 q^{59} - 550 q^{61} + 416 q^{62} - 18 q^{63} + 64 q^{64} - 420 q^{66} + 728 q^{67} - 88 q^{68} + 300 q^{69} - 540 q^{71} + 72 q^{72} + 604 q^{73} - 508 q^{74} + 96 q^{76} - 140 q^{77} - 324 q^{78} + 792 q^{79} + 81 q^{81} - 412 q^{82} + 404 q^{83} + 24 q^{84} + 584 q^{86} - 648 q^{87} + 560 q^{88} - 938 q^{89} - 108 q^{91} - 400 q^{92} - 624 q^{93} - 640 q^{94} - 96 q^{96} + 56 q^{97} - 678 q^{98} + 630 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

−3.00000

4.00000

−6.00000

−2.00000

8.00000

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	150.4.a.f		1
3.b	odd	2	1		450.4.a.e		1
4.b	odd	2	1		1200.4.a.bc		1
5.b	even	2	1		150.4.a.d		1
5.c	odd	4	2		30.4.c.a	✓	2
15.d	odd	2	1		450.4.a.p		1
15.e	even	4	2		90.4.c.a		2
20.d	odd	2	1		1200.4.a.h		1
20.e	even	4	2		240.4.f.d		2
40.i	odd	4	2		960.4.f.c		2
40.k	even	4	2		960.4.f.d		2
60.l	odd	4	2		720.4.f.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.4.c.a	✓	2	5.c	odd	4	2
90.4.c.a		2	15.e	even	4	2
150.4.a.d		1	5.b	even	2	1
150.4.a.f		1	1.a	even	1	1	trivial
240.4.f.d		2	20.e	even	4	2
450.4.a.e		1	3.b	odd	2	1
450.4.a.p		1	15.d	odd	2	1
720.4.f.c		2	60.l	odd	4	2
960.4.f.c		2	40.i	odd	4	2
960.4.f.d		2	40.k	even	4	2
1200.4.a.h		1	20.d	odd	2	1
1200.4.a.bc		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 2 $ T7 + 2 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(150))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 3 $ T + 3
$5$	$ T $ T
$7$	$ T + 2 $ T + 2
$11$	$ T - 70 $ T - 70
$13$	$ T - 54 $ T - 54
$17$	$ T + 22 $ T + 22
$19$	$ T - 24 $ T - 24
$23$	$ T + 100 $ T + 100
$29$	$ T - 216 $ T - 216
$31$	$ T - 208 $ T - 208
$37$	$ T + 254 $ T + 254
$41$	$ T + 206 $ T + 206
$43$	$ T - 292 $ T - 292
$47$	$ T + 320 $ T + 320
$53$	$ T + 402 $ T + 402
$59$	$ T + 370 $ T + 370
$61$	$ T + 550 $ T + 550
$67$	$ T - 728 $ T - 728
$71$	$ T + 540 $ T + 540
$73$	$ T - 604 $ T - 604
$79$	$ T - 792 $ T - 792
$83$	$ T - 404 $ T - 404
$89$	$ T + 938 $ T + 938
$97$	$ T - 56 $ T - 56
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Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

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