Properties

 Label 160.6.a.b Level 160 Weight 6 Character orbit 160.a Self dual yes Analytic conductor 25.661 Analytic rank 0 Dimension 2 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$25.6614111701$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 6 - 12 \beta ) q^{3} -25 q^{5} + ( 62 - 124 \beta ) q^{7} -63 q^{9} +O(q^{10})$$ $$q + ( 6 - 12 \beta ) q^{3} -25 q^{5} + ( 62 - 124 \beta ) q^{7} -63 q^{9} + ( 116 - 232 \beta ) q^{11} + 154 q^{13} + ( -150 + 300 \beta ) q^{15} + 178 q^{17} + ( 432 - 864 \beta ) q^{19} + 1860 q^{21} + ( 1178 - 2356 \beta ) q^{23} + 625 q^{25} + ( -1836 + 3672 \beta ) q^{27} + 4110 q^{29} + ( 1412 - 2824 \beta ) q^{31} + 3480 q^{33} + ( -1550 + 3100 \beta ) q^{35} + 7442 q^{37} + ( 924 - 1848 \beta ) q^{39} + 7270 q^{41} + ( -8010 + 16020 \beta ) q^{43} + 1575 q^{45} + ( -3314 + 6628 \beta ) q^{47} + 2413 q^{49} + ( 1068 - 2136 \beta ) q^{51} + 32226 q^{53} + ( -2900 + 5800 \beta ) q^{55} + 12960 q^{57} + ( 15224 - 30448 \beta ) q^{59} + 26770 q^{61} + ( -3906 + 7812 \beta ) q^{63} -3850 q^{65} + ( 22274 - 44548 \beta ) q^{67} + 35340 q^{69} + ( 24196 - 48392 \beta ) q^{71} -18534 q^{73} + ( 3750 - 7500 \beta ) q^{75} + 35960 q^{77} + ( -38792 + 77584 \beta ) q^{79} -39771 q^{81} + ( -35170 + 70340 \beta ) q^{83} -4450 q^{85} + ( 24660 - 49320 \beta ) q^{87} -107590 q^{89} + ( 9548 - 19096 \beta ) q^{91} + 42360 q^{93} + ( -10800 + 21600 \beta ) q^{95} -108838 q^{97} + ( -7308 + 14616 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 50q^{5} - 126q^{9} + O(q^{10})$$ $$2q - 50q^{5} - 126q^{9} + 308q^{13} + 356q^{17} + 3720q^{21} + 1250q^{25} + 8220q^{29} + 6960q^{33} + 14884q^{37} + 14540q^{41} + 3150q^{45} + 4826q^{49} + 64452q^{53} + 25920q^{57} + 53540q^{61} - 7700q^{65} + 70680q^{69} - 37068q^{73} + 71920q^{77} - 79542q^{81} - 8900q^{85} - 215180q^{89} + 84720q^{93} - 217676q^{97} + O(q^{100})$$

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
 1.61803 −0.618034
0 −13.4164 0 −25.0000 0 −138.636 0 −63.0000 0
1.2 0 13.4164 0 −25.0000 0 138.636 0 −63.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.a.b 2
4.b odd 2 1 inner 160.6.a.b 2
5.b even 2 1 800.6.a.i 2
5.c odd 4 2 800.6.c.h 4
8.b even 2 1 320.6.a.t 2
8.d odd 2 1 320.6.a.t 2
20.d odd 2 1 800.6.a.i 2
20.e even 4 2 800.6.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 1.a even 1 1 trivial
160.6.a.b 2 4.b odd 2 1 inner
320.6.a.t 2 8.b even 2 1
320.6.a.t 2 8.d odd 2 1
800.6.a.i 2 5.b even 2 1
800.6.a.i 2 20.d odd 2 1
800.6.c.h 4 5.c odd 4 2
800.6.c.h 4 20.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 180$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(160))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 306 T^{2} + 59049 T^{4}$$
$5$ $$( 1 + 25 T )^{2}$$
$7$ $$1 + 14394 T^{2} + 282475249 T^{4}$$
$11$ $$1 + 254822 T^{2} + 25937424601 T^{4}$$
$13$ $$( 1 - 154 T + 371293 T^{2} )^{2}$$
$17$ $$( 1 - 178 T + 1419857 T^{2} )^{2}$$
$19$ $$1 + 4019078 T^{2} + 6131066257801 T^{4}$$
$23$ $$1 + 5934266 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 4110 T + 20511149 T^{2} )^{2}$$
$31$ $$1 + 47289582 T^{2} + 819628286980801 T^{4}$$
$37$ $$( 1 - 7442 T + 69343957 T^{2} )^{2}$$
$41$ $$( 1 - 7270 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 26783614 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 403777034 T^{2} + 52599132235830049 T^{4}$$
$53$ $$( 1 - 32226 T + 418195493 T^{2} )^{2}$$
$59$ $$1 + 270997718 T^{2} + 511116753300641401 T^{4}$$
$61$ $$( 1 - 26770 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 219594834 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$1 + 681226622 T^{2} + 3255243551009881201 T^{4}$$
$73$ $$( 1 + 18534 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 - 1369983522 T^{2} + 9468276082626847201 T^{4}$$
$83$ $$1 + 1693436786 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 107590 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 + 108838 T + 8587340257 T^{2} )^{2}$$