# Properties

 Label 800.6.a.i Level 800 Weight 6 Character orbit 800.a Self dual yes Analytic conductor 128.307 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$128.307055850$$ Analytic rank: $$1$$ 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: no (minimal twist has level 160) 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.

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

## 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 0 0 −138.636 0 −63.0000 0
1.2 0 13.4164 0 0 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 800.6.a.i 2
4.b odd 2 1 inner 800.6.a.i 2
5.b even 2 1 160.6.a.b 2
5.c odd 4 2 800.6.c.h 4
20.d odd 2 1 160.6.a.b 2
20.e even 4 2 800.6.c.h 4
40.e odd 2 1 320.6.a.t 2
40.f even 2 1 320.6.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 5.b even 2 1
160.6.a.b 2 20.d odd 2 1
320.6.a.t 2 40.e odd 2 1
320.6.a.t 2 40.f even 2 1
800.6.a.i 2 1.a even 1 1 trivial
800.6.a.i 2 4.b odd 2 1 inner
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 intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3}^{2} - 180$$ $$T_{11}^{2} - 67280$$ $$T_{13} + 154$$

## Hecke characteristic polynomials

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