# Properties

 Label 800.6.a.h 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: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 7 \beta q^{7} -203 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 7 \beta q^{7} -203 q^{9} + 114 \beta q^{11} + 146 q^{13} + 702 q^{17} -432 \beta q^{19} + 280 q^{21} -647 \beta q^{23} -446 \beta q^{27} -4010 q^{29} -722 \beta q^{31} + 4560 q^{33} + 14778 q^{37} + 146 \beta q^{39} -4350 q^{41} + 1965 \beta q^{43} + 951 \beta q^{47} -14847 q^{49} + 702 \beta q^{51} + 18154 q^{53} -17280 q^{57} + 3116 \beta q^{59} -42130 q^{61} -1421 \beta q^{63} + 2559 \beta q^{67} -25880 q^{69} -7186 \beta q^{71} -26266 q^{73} + 31920 q^{77} + 1372 \beta q^{79} + 31489 q^{81} -15615 \beta q^{83} -4010 \beta q^{87} + 30570 q^{89} + 1022 \beta q^{91} -28880 q^{93} -66882 q^{97} -23142 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 406q^{9} + O(q^{10})$$ $$2q - 406q^{9} + 292q^{13} + 1404q^{17} + 560q^{21} - 8020q^{29} + 9120q^{33} + 29556q^{37} - 8700q^{41} - 29694q^{49} + 36308q^{53} - 34560q^{57} - 84260q^{61} - 51760q^{69} - 52532q^{73} + 63840q^{77} + 62978q^{81} + 61140q^{89} - 57760q^{93} - 133764q^{97} + O(q^{100})$$

## 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
 −3.16228 3.16228
0 −6.32456 0 0 0 −44.2719 0 −203.000 0
1.2 0 6.32456 0 0 0 44.2719 0 −203.000 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.h 2
4.b odd 2 1 inner 800.6.a.h 2
5.b even 2 1 160.6.a.d 2
5.c odd 4 2 800.6.c.i 4
20.d odd 2 1 160.6.a.d 2
20.e even 4 2 800.6.c.i 4
40.e odd 2 1 320.6.a.s 2
40.f even 2 1 320.6.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.d 2 5.b even 2 1
160.6.a.d 2 20.d odd 2 1
320.6.a.s 2 40.e odd 2 1
320.6.a.s 2 40.f even 2 1
800.6.a.h 2 1.a even 1 1 trivial
800.6.a.h 2 4.b odd 2 1 inner
800.6.c.i 4 5.c odd 4 2
800.6.c.i 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3}^{2} - 40$$ $$T_{11}^{2} - 519840$$ $$T_{13} - 146$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 446 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 31654 T^{2} + 282475249 T^{4}$$
$11$ $$1 - 197738 T^{2} + 25937424601 T^{4}$$
$13$ $$( 1 - 146 T + 371293 T^{2} )^{2}$$
$17$ $$( 1 - 702 T + 1419857 T^{2} )^{2}$$
$19$ $$1 - 2512762 T^{2} + 6131066257801 T^{4}$$
$23$ $$1 - 3871674 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 4010 T + 20511149 T^{2} )^{2}$$
$31$ $$1 + 36406942 T^{2} + 819628286980801 T^{4}$$
$37$ $$( 1 - 14778 T + 69343957 T^{2} )^{2}$$
$41$ $$( 1 + 4350 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 139567886 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 422513974 T^{2} + 52599132235830049 T^{4}$$
$53$ $$( 1 - 18154 T + 418195493 T^{2} )^{2}$$
$59$ $$1 + 1041470358 T^{2} + 511116753300641401 T^{4}$$
$61$ $$( 1 + 42130 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 2438310974 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$1 + 1542914862 T^{2} + 3255243551009881201 T^{4}$$
$73$ $$( 1 + 26266 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 + 6078817438 T^{2} + 9468276082626847201 T^{4}$$
$83$ $$1 - 1875047714 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 30570 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 + 66882 T + 8587340257 T^{2} )^{2}$$