Newform orbit 1682.4.a.c

• Label: 1682.4.a.c
• Level: $1682$
• Weight: $4$
• Character orbit: 1682.a
• Self dual: yes
• Analytic conductor: $99.241$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(99.2412126297\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (6 \beta - 5) q^{5} + (2 \beta + 2) q^{6} + ( - 8 \beta - 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−2.44949
2.44949

2.00000

−1.44949

4.00000

−19.6969

−2.89898

11.5959

8.00000

−24.8990

−39.3939

1.2

2.00000

3.44949

4.00000

9.69694

6.89898

−27.5959

8.00000

−15.1010

19.3939

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(29\)	\(1\)

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	1682.4.a.c		2
29.b	even	2	1		58.4.a.c	✓	2
87.d	odd	2	1		522.4.a.j		2
116.d	odd	2	1		464.4.a.e		2
145.d	even	2	1		1450.4.a.g		2
232.b	odd	2	1		1856.4.a.i		2
232.g	even	2	1		1856.4.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.4.a.c	✓	2	29.b	even	2	1
464.4.a.e		2	116.d	odd	2	1
522.4.a.j		2	87.d	odd	2	1
1450.4.a.g		2	145.d	even	2	1
1682.4.a.c		2	1.a	even	1	1	trivial
1856.4.a.i		2	232.b	odd	2	1
1856.4.a.l		2	232.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 2T - 5 \)
$5$	\( T^{2} + 10T - 191 \)
$7$	\( T^{2} + 16T - 320 \)
$11$	\( T^{2} - 90T + 1971 \)