LMFDB - Newform orbit 200.3.g.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,3,Mod(51,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("200.51");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	200.g (of order $2$, degree $1$, 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:	$5.44960528721$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} - 4 q^{4} - 3 \beta q^{7} - 4 \beta q^{8} - 9 q^{9} +O(q^{10}) $ q + b * q^2 - 4 * q^4 - 3b q^7 - 4b q^8 - 9 * q^9	$ q + \beta q^{2} - 4 q^{4} - 3 \beta q^{7} - 4 \beta q^{8} - 9 q^{9} - 18 q^{11} - 3 \beta q^{13} + 12 q^{14} + 16 q^{16} - 9 \beta q^{18} + 2 q^{19} - 18 \beta q^{22} - 13 \beta q^{23} + 12 q^{26} + 12 \beta q^{28} + 16 \beta q^{32} + 36 q^{36} + 27 \beta q^{37} + 2 \beta q^{38} - 78 q^{41} + 72 q^{44} + 52 q^{46} - 43 \beta q^{47} + 13 q^{49} + 12 \beta q^{52} + 37 \beta q^{53} - 48 q^{56} - 78 q^{59} + 27 \beta q^{63} - 64 q^{64} + 36 \beta q^{72} - 108 q^{74} - 8 q^{76} + 54 \beta q^{77} + 81 q^{81} - 78 \beta q^{82} + 72 \beta q^{88} - 18 q^{89} - 36 q^{91} + 52 \beta q^{92} + 172 q^{94} + 13 \beta q^{98} + 162 q^{99} +O(q^{100}) $ q + b * q^2 - 4 * q^4 - 3b q^7 - 4b q^8 - 9 * q^9 - 18 * q^11 - 3b q^13 + 12 * q^14 + 16 * q^16 - 9b q^18 + 2 * q^19 - 18b q^22 - 13b q^23 + 12 * q^26 + 12b q^28 + 16b q^32 + 36 * q^36 + 27b q^37 + 2b q^38 - 78 * q^41 + 72 * q^44 + 52 * q^46 - 43b q^47 + 13 * q^49 + 12b q^52 + 37b q^53 - 48 * q^56 - 78 * q^59 + 27b q^63 - 64 * q^64 + 36b q^72 - 108 * q^74 - 8 * q^76 + 54b q^77 + 81 * q^81 - 78b q^82 + 72b q^88 - 18 * q^89 - 36 * q^91 + 52b q^92 + 172 * q^94 + 13b q^98 + 162 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 - 18 * q^9	$ 2 q - 8 q^{4} - 18 q^{9} - 36 q^{11} + 24 q^{14} + 32 q^{16} + 4 q^{19} + 24 q^{26} + 72 q^{36} - 156 q^{41} + 144 q^{44} + 104 q^{46} + 26 q^{49} - 96 q^{56} - 156 q^{59} - 128 q^{64} - 216 q^{74} - 16 q^{76} + 162 q^{81} - 36 q^{89} - 72 q^{91} + 344 q^{94} + 324 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 18 * q^9 - 36 * q^11 + 24 * q^14 + 32 * q^16 + 4 * q^19 + 24 * q^26 + 72 * q^36 - 156 * q^41 + 144 * q^44 + 104 * q^46 + 26 * q^49 - 96 * q^56 - 156 * q^59 - 128 * q^64 - 216 * q^74 - 16 * q^76 + 162 * q^81 - 36 * q^89 - 72 * q^91 + 344 * q^94 + 324 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$-1$	$1$

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} $

51.1

	−	1.00000i
		1.00000i

−

2.00000i

−4.00000

6.00000i

8.00000i

−9.00000

51.2

2.00000i

−4.00000

−

6.00000i

−

8.00000i

−9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by $\Q(\sqrt{-10}) $
5.b	even	2	1	inner
8.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	200.3.g.c		2
4.b	odd	2	1		800.3.g.c		2
5.b	even	2	1	inner	200.3.g.c		2
5.c	odd	4	1		40.3.e.a	✓	1
5.c	odd	4	1		40.3.e.b	yes	1
8.b	even	2	1		800.3.g.c		2
8.d	odd	2	1	inner	200.3.g.c		2
15.e	even	4	1		360.3.p.a		1
15.e	even	4	1		360.3.p.b		1
20.d	odd	2	1		800.3.g.c		2
20.e	even	4	1		160.3.e.a		1
20.e	even	4	1		160.3.e.b		1
40.e	odd	2	1	CM	200.3.g.c		2
40.f	even	2	1		800.3.g.c		2
40.i	odd	4	1		160.3.e.a		1
40.i	odd	4	1		160.3.e.b		1
40.k	even	4	1		40.3.e.a	✓	1
40.k	even	4	1		40.3.e.b	yes	1
60.l	odd	4	1		1440.3.p.a		1
60.l	odd	4	1		1440.3.p.b		1
80.i	odd	4	1		1280.3.h.b		2
80.i	odd	4	1		1280.3.h.c		2
80.j	even	4	1		1280.3.h.b		2
80.j	even	4	1		1280.3.h.c		2
80.s	even	4	1		1280.3.h.b		2
80.s	even	4	1		1280.3.h.c		2
80.t	odd	4	1		1280.3.h.b		2
80.t	odd	4	1		1280.3.h.c		2
120.q	odd	4	1		360.3.p.a		1
120.q	odd	4	1		360.3.p.b		1
120.w	even	4	1		1440.3.p.a		1
120.w	even	4	1		1440.3.p.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.3.e.a	✓	1	5.c	odd	4	1
40.3.e.a	✓	1	40.k	even	4	1
40.3.e.b	yes	1	5.c	odd	4	1
40.3.e.b	yes	1	40.k	even	4	1
160.3.e.a		1	20.e	even	4	1
160.3.e.a		1	40.i	odd	4	1
160.3.e.b		1	20.e	even	4	1
160.3.e.b		1	40.i	odd	4	1
200.3.g.c		2	1.a	even	1	1	trivial
200.3.g.c		2	5.b	even	2	1	inner
200.3.g.c		2	8.d	odd	2	1	inner
200.3.g.c		2	40.e	odd	2	1	CM
360.3.p.a		1	15.e	even	4	1
360.3.p.a		1	120.q	odd	4	1
360.3.p.b		1	15.e	even	4	1
360.3.p.b		1	120.q	odd	4	1
800.3.g.c		2	4.b	odd	2	1
800.3.g.c		2	8.b	even	2	1
800.3.g.c		2	20.d	odd	2	1
800.3.g.c		2	40.f	even	2	1
1280.3.h.b		2	80.i	odd	4	1
1280.3.h.b		2	80.j	even	4	1
1280.3.h.b		2	80.s	even	4	1
1280.3.h.b		2	80.t	odd	4	1
1280.3.h.c		2	80.i	odd	4	1
1280.3.h.c		2	80.j	even	4	1
1280.3.h.c		2	80.s	even	4	1
1280.3.h.c		2	80.t	odd	4	1
1440.3.p.a		1	60.l	odd	4	1
1440.3.p.a		1	120.w	even	4	1
1440.3.p.b		1	60.l	odd	4	1
1440.3.p.b		1	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} $ T3 acting on $S_{3}^{\mathrm{new}}(200, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4 $ T^2 + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 36 $ T^2 + 36
$11$	$ (T + 18)^{2} $ (T + 18)^2
$13$	$ T^{2} + 36 $ T^2 + 36
$17$	$ T^{2} $ T^2
$19$	$ (T - 2)^{2} $ (T - 2)^2
$23$	$ T^{2} + 676 $ T^2 + 676
$29$	$ T^{2} $ T^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 2916 $ T^2 + 2916
$41$	$ (T + 78)^{2} $ (T + 78)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} + 7396 $ T^2 + 7396
$53$	$ T^{2} + 5476 $ T^2 + 5476
$59$	$ (T + 78)^{2} $ (T + 78)^2
$61$	$ T^{2} $ T^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T + 18)^{2} $ (T + 18)^2
$97$	$ T^{2} $ T^2
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Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	200.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 4 q^{4} - 3 \beta q^{7} - 4 \beta q^{8} - 9 q^{9} +O(q^{10}) \) q + b * q^2 - 4 * q^4 - 3b q^7 - 4b q^8 - 9 * q^9	\( q + \beta q^{2} - 4 q^{4} - 3 \beta q^{7} - 4 \beta q^{8} - 9 q^{9} - 18 q^{11} - 3 \beta q^{13} + 12 q^{14} + 16 q^{16} - 9 \beta q^{18} + 2 q^{19} - 18 \beta q^{22} - 13 \beta q^{23} + 12 q^{26} + 12 \beta q^{28} + 16 \beta q^{32} + 36 q^{36} + 27 \beta q^{37} + 2 \beta q^{38} - 78 q^{41} + 72 q^{44} + 52 q^{46} - 43 \beta q^{47} + 13 q^{49} + 12 \beta q^{52} + 37 \beta q^{53} - 48 q^{56} - 78 q^{59} + 27 \beta q^{63} - 64 q^{64} + 36 \beta q^{72} - 108 q^{74} - 8 q^{76} + 54 \beta q^{77} + 81 q^{81} - 78 \beta q^{82} + 72 \beta q^{88} - 18 q^{89} - 36 q^{91} + 52 \beta q^{92} + 172 q^{94} + 13 \beta q^{98} + 162 q^{99} +O(q^{100}) \) q + b * q^2 - 4 * q^4 - 3b q^7 - 4b q^8 - 9 * q^9 - 18 * q^11 - 3b q^13 + 12 * q^14 + 16 * q^16 - 9b q^18 + 2 * q^19 - 18b q^22 - 13b q^23 + 12 * q^26 + 12b q^28 + 16b q^32 + 36 * q^36 + 27b q^37 + 2b q^38 - 78 * q^41 + 72 * q^44 + 52 * q^46 - 43b q^47 + 13 * q^49 + 12b q^52 + 37b q^53 - 48 * q^56 - 78 * q^59 + 27b q^63 - 64 * q^64 + 36b q^72 - 108 * q^74 - 8 * q^76 + 54b q^77 + 81 * q^81 - 78b q^82 + 72b q^88 - 18 * q^89 - 36 * q^91 + 52b q^92 + 172 * q^94 + 13b q^98 + 162 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 - 18 * q^9	\( 2 q - 8 q^{4} - 18 q^{9} - 36 q^{11} + 24 q^{14} + 32 q^{16} + 4 q^{19} + 24 q^{26} + 72 q^{36} - 156 q^{41} + 144 q^{44} + 104 q^{46} + 26 q^{49} - 96 q^{56} - 156 q^{59} - 128 q^{64} - 216 q^{74} - 16 q^{76} + 162 q^{81} - 36 q^{89} - 72 q^{91} + 344 q^{94} + 324 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 18 * q^9 - 36 * q^11 + 24 * q^14 + 32 * q^16 + 4 * q^19 + 24 * q^26 + 72 * q^36 - 156 * q^41 + 144 * q^44 + 104 * q^46 + 26 * q^49 - 96 * q^56 - 156 * q^59 - 128 * q^64 - 216 * q^74 - 16 * q^76 + 162 * q^81 - 36 * q^89 - 72 * q^91 + 344 * q^94 + 324 * q^99

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