# Properties

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

# 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: $$\Q(\sqrt{70})$$ Defining polynomial: $$x^{2} - 70$$ 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 $$\beta = 2\sqrt{70}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 + \beta ) q^{3} + ( 52 - \beta ) q^{7} + ( 53 + 8 \beta ) q^{9} +O(q^{10})$$ $$q + ( 4 + \beta ) q^{3} + ( 52 - \beta ) q^{7} + ( 53 + 8 \beta ) q^{9} + ( -160 + 10 \beta ) q^{11} + ( 50 - 40 \beta ) q^{13} + ( -290 - 40 \beta ) q^{17} + ( -360 + 40 \beta ) q^{19} + ( -72 + 48 \beta ) q^{21} + ( 844 + 97 \beta ) q^{23} + ( 1480 - 158 \beta ) q^{27} + ( 54 - 80 \beta ) q^{29} + ( -4920 - 130 \beta ) q^{31} + ( 2160 - 120 \beta ) q^{33} + ( -3270 + 560 \beta ) q^{37} + ( -11000 - 110 \beta ) q^{39} + ( -5310 - 808 \beta ) q^{41} + ( 12836 - 579 \beta ) q^{43} + ( 14148 + 383 \beta ) q^{47} + ( -13823 - 104 \beta ) q^{49} + ( -12360 - 450 \beta ) q^{51} + ( -15670 - 1080 \beta ) q^{53} + ( 9760 - 200 \beta ) q^{57} + ( -15400 + 20 \beta ) q^{59} + ( 12270 - 1184 \beta ) q^{61} + ( 516 + 363 \beta ) q^{63} + ( 17292 + 495 \beta ) q^{67} + ( 30536 + 1232 \beta ) q^{69} + ( 6200 + 2990 \beta ) q^{71} + ( 3590 - 3720 \beta ) q^{73} + ( -11120 + 680 \beta ) q^{77} + ( -35920 + 220 \beta ) q^{79} + ( -51199 - 1096 \beta ) q^{81} + ( -15964 + 2817 \beta ) q^{83} + ( -22184 - 266 \beta ) q^{87} + ( -20374 - 3280 \beta ) q^{89} + ( 13800 - 2130 \beta ) q^{91} + ( -56080 - 5440 \beta ) q^{93} + ( 95070 + 4840 \beta ) q^{97} + ( 13920 - 750 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{3} + 104q^{7} + 106q^{9} + O(q^{10})$$ $$2q + 8q^{3} + 104q^{7} + 106q^{9} - 320q^{11} + 100q^{13} - 580q^{17} - 720q^{19} - 144q^{21} + 1688q^{23} + 2960q^{27} + 108q^{29} - 9840q^{31} + 4320q^{33} - 6540q^{37} - 22000q^{39} - 10620q^{41} + 25672q^{43} + 28296q^{47} - 27646q^{49} - 24720q^{51} - 31340q^{53} + 19520q^{57} - 30800q^{59} + 24540q^{61} + 1032q^{63} + 34584q^{67} + 61072q^{69} + 12400q^{71} + 7180q^{73} - 22240q^{77} - 71840q^{79} - 102398q^{81} - 31928q^{83} - 44368q^{87} - 40748q^{89} + 27600q^{91} - 112160q^{93} + 190140q^{97} + 27840q^{99} + O(q^{100})$$

## 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
 −8.36660 8.36660
0 −12.7332 0 0 0 68.7332 0 −80.8656 0
1.2 0 20.7332 0 0 0 35.2668 0 186.866 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 800.6.a.l 2
4.b odd 2 1 800.6.a.g 2
5.b even 2 1 160.6.a.a 2
5.c odd 4 2 800.6.c.f 4
20.d odd 2 1 160.6.a.e yes 2
20.e even 4 2 800.6.c.g 4
40.e odd 2 1 320.6.a.r 2
40.f even 2 1 320.6.a.v 2

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

 $$T_{3}^{2} - 8 T_{3} - 264$$ $$T_{11}^{2} + 320 T_{11} - 2400$$ $$T_{13}^{2} - 100 T_{13} - 445500$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 8 T + 222 T^{2} - 1944 T^{3} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 104 T + 36038 T^{2} - 1747928 T^{3} + 282475249 T^{4}$$
$11$ $$1 + 320 T + 319702 T^{2} + 51536320 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 100 T + 297086 T^{2} - 37129300 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 580 T + 2475814 T^{2} + 823517060 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 720 T + 4633798 T^{2} + 1782791280 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 - 1688 T + 10950502 T^{2} - 10864546984 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 108 T + 39233214 T^{2} - 2215204092 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 9840 T + 76732702 T^{2} + 281710845840 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 + 6540 T + 61572814 T^{2} + 453509478780 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 25672 T + 364912302 T^{2} - 3774000748696 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 - 28296 T + 617782998 T^{2} - 6489546318072 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 + 31340 T + 755347886 T^{2} + 13106246750620 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 + 30800 T + 1666896598 T^{2} + 22019668409200 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 34584 T + 2930656478 T^{2} - 46692726700488 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 - 12400 T + 1143670702 T^{2} - 22372443952400 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 - 7180 T + 284279286 T^{2} - 14884654037740 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 + 71840 T + 7430807198 T^{2} + 221055731704160 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 31928 T + 5910993662 T^{2} + 125765689649704 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 - 190140 T + 19653817414 T^{2} - 1632796876465980 T^{3} + 73742412689492826049 T^{4}$$