LMFDB - Newform orbit 1200.4.f.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 3 i q^{3} + 24 i q^{7} - 9 q^{9}+O(q^{10}) $ q + 3i q^3 + 24i q^7 - 9 * q^9	$ q + 3 i q^{3} + 24 i q^{7} - 9 q^{9} + 28 q^{11} + 74 i q^{13} + 82 i q^{17} + 92 q^{19} - 72 q^{21} + 8 i q^{23} - 27 i q^{27} + 138 q^{29} - 80 q^{31} + 84 i q^{33} + 30 i q^{37} - 222 q^{39} + 282 q^{41} + 4 i q^{43} - 240 i q^{47} - 233 q^{49} - 246 q^{51} + 130 i q^{53} + 276 i q^{57} + 596 q^{59} - 218 q^{61} - 216 i q^{63} + 436 i q^{67} - 24 q^{69} - 856 q^{71} + 998 i q^{73} + 672 i q^{77} - 32 q^{79} + 81 q^{81} - 1508 i q^{83} + 414 i q^{87} + 246 q^{89} - 1776 q^{91} - 240 i q^{93} + 866 i q^{97} - 252 q^{99} +O(q^{100}) $ q + 3i q^3 + 24i q^7 - 9 * q^9 + 28 * q^11 + 74i q^13 + 82i q^17 + 92 * q^19 - 72 * q^21 + 8i q^23 - 27i q^27 + 138 * q^29 - 80 * q^31 + 84i q^33 + 30i q^37 - 222 * q^39 + 282 * q^41 + 4i q^43 - 240i q^47 - 233 * q^49 - 246 * q^51 + 130i q^53 + 276i q^57 + 596 * q^59 - 218 * q^61 - 216i q^63 + 436i q^67 - 24 * q^69 - 856 * q^71 + 998i q^73 + 672i q^77 - 32 * q^79 + 81 * q^81 - 1508i q^83 + 414i q^87 + 246 * q^89 - 1776 * q^91 - 240i q^93 + 866i q^97 - 252 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 56 q^{11} + 184 q^{19} - 144 q^{21} + 276 q^{29} - 160 q^{31} - 444 q^{39} + 564 q^{41} - 466 q^{49} - 492 q^{51} + 1192 q^{59} - 436 q^{61} - 48 q^{69} - 1712 q^{71} - 64 q^{79} + 162 q^{81} + 492 q^{89} - 3552 q^{91} - 504 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 56 * q^11 + 184 * q^19 - 144 * q^21 + 276 * q^29 - 160 * q^31 - 444 * q^39 + 564 * q^41 - 466 * q^49 - 492 * q^51 + 1192 * q^59 - 436 * q^61 - 48 * q^69 - 1712 * q^71 - 64 * q^79 + 162 * q^81 + 492 * q^89 - 3552 * q^91 - 504 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

49.1

	−	1.00000i
		1.00000i

−

3.00000i

−

24.0000i

−9.00000

49.2

3.00000i

24.0000i

−9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1200.4.f.p		2
4.b	odd	2	1		600.4.f.b		2
5.b	even	2	1	inner	1200.4.f.p		2
5.c	odd	4	1		48.4.a.b		1
5.c	odd	4	1		1200.4.a.u		1
12.b	even	2	1		1800.4.f.q		2
15.e	even	4	1		144.4.a.b		1
20.d	odd	2	1		600.4.f.b		2
20.e	even	4	1		24.4.a.a	✓	1
20.e	even	4	1		600.4.a.h		1
35.f	even	4	1		2352.4.a.w		1
40.i	odd	4	1		192.4.a.g		1
40.k	even	4	1		192.4.a.a		1
60.h	even	2	1		1800.4.f.q		2
60.l	odd	4	1		72.4.a.b		1
60.l	odd	4	1		1800.4.a.bg		1
80.i	odd	4	1		768.4.d.b		2
80.j	even	4	1		768.4.d.o		2
80.s	even	4	1		768.4.d.o		2
80.t	odd	4	1		768.4.d.b		2
120.q	odd	4	1		576.4.a.u		1
120.w	even	4	1		576.4.a.v		1
140.j	odd	4	1		1176.4.a.a		1
180.v	odd	12	2		648.4.i.k		2
180.x	even	12	2		648.4.i.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.4.a.a	✓	1	20.e	even	4	1
48.4.a.b		1	5.c	odd	4	1
72.4.a.b		1	60.l	odd	4	1
144.4.a.b		1	15.e	even	4	1
192.4.a.a		1	40.k	even	4	1
192.4.a.g		1	40.i	odd	4	1
576.4.a.u		1	120.q	odd	4	1
576.4.a.v		1	120.w	even	4	1
600.4.a.h		1	20.e	even	4	1
600.4.f.b		2	4.b	odd	2	1
600.4.f.b		2	20.d	odd	2	1
648.4.i.b		2	180.x	even	12	2
648.4.i.k		2	180.v	odd	12	2
768.4.d.b		2	80.i	odd	4	1
768.4.d.b		2	80.t	odd	4	1
768.4.d.o		2	80.j	even	4	1
768.4.d.o		2	80.s	even	4	1
1176.4.a.a		1	140.j	odd	4	1
1200.4.a.u		1	5.c	odd	4	1
1200.4.f.p		2	1.a	even	1	1	trivial
1200.4.f.p		2	5.b	even	2	1	inner
1800.4.a.bg		1	60.l	odd	4	1
1800.4.f.q		2	12.b	even	2	1
1800.4.f.q		2	60.h	even	2	1
2352.4.a.w		1	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1200, [\chi])$:

$ T_{7}^{2} + 576 $

$ T_{11} - 28 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 9 $ T^2 + 9
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 576 $ T^2 + 576
$11$	$ (T - 28)^{2} $ (T - 28)^2
$13$	$ T^{2} + 5476 $ T^2 + 5476
$17$	$ T^{2} + 6724 $ T^2 + 6724
$19$	$ (T - 92)^{2} $ (T - 92)^2
$23$	$ T^{2} + 64 $ T^2 + 64
$29$	$ (T - 138)^{2} $ (T - 138)^2
$31$	$ (T + 80)^{2} $ (T + 80)^2
$37$	$ T^{2} + 900 $ T^2 + 900
$41$	$ (T - 282)^{2} $ (T - 282)^2
$43$	$ T^{2} + 16 $ T^2 + 16
$47$	$ T^{2} + 57600 $ T^2 + 57600
$53$	$ T^{2} + 16900 $ T^2 + 16900
$59$	$ (T - 596)^{2} $ (T - 596)^2
$61$	$ (T + 218)^{2} $ (T + 218)^2
$67$	$ T^{2} + 190096 $ T^2 + 190096
$71$	$ (T + 856)^{2} $ (T + 856)^2
$73$	$ T^{2} + 996004 $ T^2 + 996004
$79$	$ (T + 32)^{2} $ (T + 32)^2
$83$	$ T^{2} + 2274064 $ T^2 + 2274064
$89$	$ (T - 246)^{2} $ (T - 246)^2
$97$	$ T^{2} + 749956 $ T^2 + 749956
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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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 3 i q^{3} + 24 i q^{7} - 9 q^{9}+O(q^{10}) \) q + 3i q^3 + 24i q^7 - 9 * q^9	\( q + 3 i q^{3} + 24 i q^{7} - 9 q^{9} + 28 q^{11} + 74 i q^{13} + 82 i q^{17} + 92 q^{19} - 72 q^{21} + 8 i q^{23} - 27 i q^{27} + 138 q^{29} - 80 q^{31} + 84 i q^{33} + 30 i q^{37} - 222 q^{39} + 282 q^{41} + 4 i q^{43} - 240 i q^{47} - 233 q^{49} - 246 q^{51} + 130 i q^{53} + 276 i q^{57} + 596 q^{59} - 218 q^{61} - 216 i q^{63} + 436 i q^{67} - 24 q^{69} - 856 q^{71} + 998 i q^{73} + 672 i q^{77} - 32 q^{79} + 81 q^{81} - 1508 i q^{83} + 414 i q^{87} + 246 q^{89} - 1776 q^{91} - 240 i q^{93} + 866 i q^{97} - 252 q^{99} +O(q^{100}) \) q + 3i q^3 + 24i q^7 - 9 * q^9 + 28 * q^11 + 74i q^13 + 82i q^17 + 92 * q^19 - 72 * q^21 + 8i q^23 - 27i q^27 + 138 * q^29 - 80 * q^31 + 84i q^33 + 30i q^37 - 222 * q^39 + 282 * q^41 + 4i q^43 - 240i q^47 - 233 * q^49 - 246 * q^51 + 130i q^53 + 276i q^57 + 596 * q^59 - 218 * q^61 - 216i q^63 + 436i q^67 - 24 * q^69 - 856 * q^71 + 998i q^73 + 672i q^77 - 32 * q^79 + 81 * q^81 - 1508i q^83 + 414i q^87 + 246 * q^89 - 1776 * q^91 - 240i q^93 + 866i q^97 - 252 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 56 q^{11} + 184 q^{19} - 144 q^{21} + 276 q^{29} - 160 q^{31} - 444 q^{39} + 564 q^{41} - 466 q^{49} - 492 q^{51} + 1192 q^{59} - 436 q^{61} - 48 q^{69} - 1712 q^{71} - 64 q^{79} + 162 q^{81} + 492 q^{89} - 3552 q^{91} - 504 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 56 * q^11 + 184 * q^19 - 144 * q^21 + 276 * q^29 - 160 * q^31 - 444 * q^39 + 564 * q^41 - 466 * q^49 - 492 * q^51 + 1192 * q^59 - 436 * q^61 - 48 * q^69 - 1712 * q^71 - 64 * q^79 + 162 * q^81 + 492 * q^89 - 3552 * q^91 - 504 * q^99

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