Properties

 Label 7360.2.a.bn Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$58.7698958877$$ 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 230) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} - q^{5} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} + ( 2 - 3 \beta ) q^{11} + ( -1 + 5 \beta ) q^{13} -\beta q^{15} + ( -2 + 5 \beta ) q^{17} + ( -3 + 3 \beta ) q^{19} + q^{21} - q^{23} + q^{25} + ( 1 - 4 \beta ) q^{27} + ( 6 + 2 \beta ) q^{29} + ( -1 - 5 \beta ) q^{31} + ( -3 - \beta ) q^{33} + ( 1 - \beta ) q^{35} -4 \beta q^{37} + ( 5 + 4 \beta ) q^{39} + ( -8 + 7 \beta ) q^{41} + ( 2 - \beta ) q^{45} + ( -6 + 6 \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 5 + 3 \beta ) q^{51} + ( 6 - 4 \beta ) q^{53} + ( -2 + 3 \beta ) q^{55} + 3 q^{57} + ( -8 + 6 \beta ) q^{59} + ( -2 + 7 \beta ) q^{61} + ( 3 - 2 \beta ) q^{63} + ( 1 - 5 \beta ) q^{65} + ( 8 + 4 \beta ) q^{67} -\beta q^{69} + ( -4 + 5 \beta ) q^{71} + 2 \beta q^{73} + \beta q^{75} + ( -5 + 2 \beta ) q^{77} + ( -8 + 4 \beta ) q^{79} + ( 2 - 6 \beta ) q^{81} + ( 6 - 8 \beta ) q^{83} + ( 2 - 5 \beta ) q^{85} + ( 2 + 8 \beta ) q^{87} + ( -4 - 4 \beta ) q^{89} + ( 6 - \beta ) q^{91} + ( -5 - 6 \beta ) q^{93} + ( 3 - 3 \beta ) q^{95} + ( 14 - \beta ) q^{97} + ( -7 + 5 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{5} - q^{7} - 3q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{5} - q^{7} - 3q^{9} + q^{11} + 3q^{13} - q^{15} + q^{17} - 3q^{19} + 2q^{21} - 2q^{23} + 2q^{25} - 2q^{27} + 14q^{29} - 7q^{31} - 7q^{33} + q^{35} - 4q^{37} + 14q^{39} - 9q^{41} + 3q^{45} - 6q^{47} - 11q^{49} + 13q^{51} + 8q^{53} - q^{55} + 6q^{57} - 10q^{59} + 3q^{61} + 4q^{63} - 3q^{65} + 20q^{67} - q^{69} - 3q^{71} + 2q^{73} + q^{75} - 8q^{77} - 12q^{79} - 2q^{81} + 4q^{83} - q^{85} + 12q^{87} - 12q^{89} + 11q^{91} - 16q^{93} + 3q^{95} + 27q^{97} - 9q^{99} + O(q^{100})$$

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
 −0.618034 1.61803
0 −0.618034 0 −1.00000 0 −1.61803 0 −2.61803 0
1.2 0 1.61803 0 −1.00000 0 0.618034 0 −0.381966 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$1$$

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 7360.2.a.bn 2
4.b odd 2 1 7360.2.a.bh 2
8.b even 2 1 1840.2.a.l 2
8.d odd 2 1 230.2.a.c 2
24.f even 2 1 2070.2.a.u 2
40.e odd 2 1 1150.2.a.j 2
40.f even 2 1 9200.2.a.bu 2
40.k even 4 2 1150.2.b.i 4
184.h even 2 1 5290.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 8.d odd 2 1
1150.2.a.j 2 40.e odd 2 1
1150.2.b.i 4 40.k even 4 2
1840.2.a.l 2 8.b even 2 1
2070.2.a.u 2 24.f even 2 1
5290.2.a.o 2 184.h even 2 1
7360.2.a.bh 2 4.b odd 2 1
7360.2.a.bn 2 1.a even 1 1 trivial
9200.2.a.bu 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{7}^{2} + T_{7} - 1$$ $$T_{11}^{2} - T_{11} - 11$$ $$T_{13}^{2} - 3 T_{13} - 29$$ $$T_{17}^{2} - T_{17} - 31$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 - T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-1 + T + T^{2}$$
$11$ $$-11 - T + T^{2}$$
$13$ $$-29 - 3 T + T^{2}$$
$17$ $$-31 - T + T^{2}$$
$19$ $$-9 + 3 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$44 - 14 T + T^{2}$$
$31$ $$-19 + 7 T + T^{2}$$
$37$ $$-16 + 4 T + T^{2}$$
$41$ $$-41 + 9 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$-36 + 6 T + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-20 + 10 T + T^{2}$$
$61$ $$-59 - 3 T + T^{2}$$
$67$ $$80 - 20 T + T^{2}$$
$71$ $$-29 + 3 T + T^{2}$$
$73$ $$-4 - 2 T + T^{2}$$
$79$ $$16 + 12 T + T^{2}$$
$83$ $$-76 - 4 T + T^{2}$$
$89$ $$16 + 12 T + T^{2}$$
$97$ $$181 - 27 T + T^{2}$$