Newform orbit 7616.2.a.t

• Label: 7616.2.a.t
• Level: $7616$
• Weight: $2$
• Character orbit: 7616.a
• Self dual: yes
• Analytic conductor: $60.814$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(60.8140661794\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^3 + (b - 1) * q^5 - q^7 + b * q^9 - 4 * q^11 + 2*b * q^13 + 3 * q^15 - q^17 + (-2*b - 4) * q^19 - b * q^21 + 4 * q^23 + (-b - 1) * q^25 + (-2*b + 3) * q^27 - 4*b * q^29 + (b + 5) * q^31 - 4*b * q^33 + (-b + 1) * q^35 + (-6*b + 4) * q^37 + (2*b + 6) * q^39 - 5*b * q^41 + (b + 2) * q^43 + 3 * q^45 + (-2*b + 2) * q^47 + q^49 - b * q^51 + (b - 3) * q^53 + (-4*b + 4) * q^55 + (-6*b - 6) * q^57 + 8 * q^59 + (-5*b - 4) * q^61 - b * q^63 + 6 * q^65 + (b - 5) * q^67 + 4*b * q^69 + (2*b - 8) * q^71 + (5*b - 4) * q^73 + (-2*b - 3) * q^75 + 4 * q^77 + (-2*b + 2) * q^79 + (-2*b - 6) * q^81 + (-4*b + 6) * q^83 + (-b + 1) * q^85 + (-4*b - 12) * q^87 + (6*b - 6) * q^89 - 2*b * q^91 + (6*b + 3) * q^93 + (-4*b - 2) * q^95 + (-b - 11) * q^97 - 4*b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^3 - q^5 - 2 * q^7 + q^9 - 8 * q^11 + 2 * q^13 + 6 * q^15 - 2 * q^17 - 10 * q^19 - q^21 + 8 * q^23 - 3 * q^25 + 4 * q^27 - 4 * q^29 + 11 * q^31 - 4 * q^33 + q^35 + 2 * q^37 + 14 * q^39 - 5 * q^41 + 5 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 - q^51 - 5 * q^53 + 4 * q^55 - 18 * q^57 + 16 * q^59 - 13 * q^61 - q^63 + 12 * q^65 - 9 * q^67 + 4 * q^69 - 14 * q^71 - 3 * q^73 - 8 * q^75 + 8 * q^77 + 2 * q^79 - 14 * q^81 + 8 * q^83 + q^85 - 28 * q^87 - 6 * q^89 - 2 * q^91 + 12 * q^93 - 8 * q^95 - 23 * q^97 - 4 * q^99

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

−1.30278
2.30278

0

−1.30278

0

−2.30278

0

−1.00000

0

−1.30278

0

1.2

0

2.30278

0

1.30278

0

−1.00000

0

2.30278

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(7\)	\( +1 \)
\(17\)	\( +1 \)

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	7616.2.a.t		2
4.b	odd	2	1		7616.2.a.o		2
8.b	even	2	1		476.2.a.c	✓	2
8.d	odd	2	1		1904.2.a.k		2
24.h	odd	2	1		4284.2.a.l		2
56.h	odd	2	1		3332.2.a.k		2
136.h	even	2	1		8092.2.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
476.2.a.c	✓	2	8.b	even	2	1
1904.2.a.k		2	8.d	odd	2	1
3332.2.a.k		2	56.h	odd	2	1
4284.2.a.l		2	24.h	odd	2	1
7616.2.a.o		2	4.b	odd	2	1
7616.2.a.t		2	1.a	even	1	1	trivial
8092.2.a.l		2	136.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7616))\):

\( T_{3}^{2} - T_{3} - 3 \)

\( T_{5}^{2} + T_{5} - 3 \)

\( T_{11} + 4 \)

\( T_{19}^{2} + 10T_{19} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T - 3 \)
$5$	\( T^{2} + T - 3 \)
$7$	\( (T + 1)^{2} \)
$11$	\( (T + 4)^{2} \)