Newform orbit 7616.2.a.p

• Label: 7616.2.a.p
• Level: $7616$
• Weight: $2$
• Character orbit: 7616.a
• Self dual: yes
• Analytic conductor: $60.814$
• Analytic rank: $0$
• 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
Defining polynomial:	$x^{2} - x - 1$
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

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

1.61803
−0.618034

0

−1.61803

0

−0.618034

0

1.00000

0

−0.381966

0

1.2

0

0.618034

0

1.61803

0

1.00000

0

−2.61803

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.p		2
4.b	odd	2	1		7616.2.a.u		2
8.b	even	2	1		1904.2.a.j		2
8.d	odd	2	1		476.2.a.b	✓	2
24.f	even	2	1		4284.2.a.m		2
56.e	even	2	1		3332.2.a.l		2
136.e	odd	2	1		8092.2.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
476.2.a.b	✓	2	8.d	odd	2	1
1904.2.a.j		2	8.b	even	2	1
3332.2.a.l		2	56.e	even	2	1
4284.2.a.m		2	24.f	even	2	1
7616.2.a.p		2	1.a	even	1	1	trivial
7616.2.a.u		2	4.b	odd	2	1
8092.2.a.m		2	136.e	odd	2	1

Hecke kernels

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

$T_{3}^{2} + T_{3} - 1$

$T_{5}^{2} - T_{5} - 1$

$T_{11}^{2} + 6T_{11} + 4$

$T_{19}^{2} + 8T_{19} - 4$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + T - 1$
$5$	$T^{2} - T - 1$
$7$	$(T - 1)^{2}$
$11$	$T^{2} + 6T + 4$