Newform orbit 1900.3.e.a

• Label: 1900.3.e.a
• Level: $1900$
• Weight: $3$
• Character orbit: 1900.e
• Self dual: yes
• Analytic conductor: $51.771$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(51.7712502285\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{57}) \)
Defining polynomial:	\( x^{2} - x - 14 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 3 \beta - 1) q^{7} + 9 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{7} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)	\(951\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1101.1

4.27492
−3.27492

0

0

0

0

0

−13.8248

0

9.00000

0

1101.2

0

0

0

0

0

8.82475

0

9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1900.3.e.a		2
5.b	even	2	1		76.3.c.b	✓	2
5.c	odd	4	2		1900.3.g.a		4
15.d	odd	2	1		684.3.h.a		2
19.b	odd	2	1	CM	1900.3.e.a		2
20.d	odd	2	1		304.3.e.e		2
40.e	odd	2	1		1216.3.e.e		2
40.f	even	2	1		1216.3.e.f		2
60.h	even	2	1		2736.3.o.c		2
95.d	odd	2	1		76.3.c.b	✓	2
95.g	even	4	2		1900.3.g.a		4
285.b	even	2	1		684.3.h.a		2
380.d	even	2	1		304.3.e.e		2
760.b	odd	2	1		1216.3.e.f		2
760.p	even	2	1		1216.3.e.e		2
1140.p	odd	2	1		2736.3.o.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.3.c.b	✓	2	5.b	even	2	1
76.3.c.b	✓	2	95.d	odd	2	1
304.3.e.e		2	20.d	odd	2	1
304.3.e.e		2	380.d	even	2	1
684.3.h.a		2	15.d	odd	2	1
684.3.h.a		2	285.b	even	2	1
1216.3.e.e		2	40.e	odd	2	1
1216.3.e.e		2	760.p	even	2	1
1216.3.e.f		2	40.f	even	2	1
1216.3.e.f		2	760.b	odd	2	1
1900.3.e.a		2	1.a	even	1	1	trivial
1900.3.e.a		2	19.b	odd	2	1	CM
1900.3.g.a		4	5.c	odd	4	2
1900.3.g.a		4	95.g	even	4	2
2736.3.o.c		2	60.h	even	2	1
2736.3.o.c		2	1140.p	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3} \)

\( T_{7}^{2} + 5T_{7} - 122 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 5T - 122 \)
$11$	\( T^{2} + 3T - 354 \)