# Properties

 Label 800.6.a.n Level 800 Weight 6 Character orbit 800.a Self dual yes Analytic conductor 128.307 Analytic rank 1 Dimension 3 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: $$3$$ Coefficient field: 3.3.39180.1 Defining polynomial: $$x^{3} - x^{2} - 36 x - 24$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - \beta_{1} ) q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + ( 151 + 12 \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -3 - \beta_{1} ) q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + ( 151 + 12 \beta_{1} + 2 \beta_{2} ) q^{9} + ( -131 - 3 \beta_{2} ) q^{11} + ( 124 - 20 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -368 - 60 \beta_{1} + 6 \beta_{2} ) q^{17} + ( 1047 + 50 \beta_{1} + \beta_{2} ) q^{19} + ( -296 + 72 \beta_{1} - 4 \beta_{2} ) q^{21} + ( -2054 + 43 \beta_{1} - 7 \beta_{2} ) q^{23} + ( -4534 - 182 \beta_{1} - 20 \beta_{2} ) q^{27} + ( 138 - 40 \beta_{1} + 52 \beta_{2} ) q^{29} + ( 1196 - 270 \beta_{1} - 42 \beta_{2} ) q^{31} + ( 678 + 380 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 3762 + 200 \beta_{1} + 76 \beta_{2} ) q^{37} + ( 7138 - 110 \beta_{1} + 44 \beta_{2} ) q^{39} + ( 4120 + 68 \beta_{1} + 22 \beta_{2} ) q^{41} + ( -8713 - 221 \beta_{1} + 14 \beta_{2} ) q^{43} + ( -12288 + 77 \beta_{1} + 25 \beta_{2} ) q^{47} + ( -1153 - 316 \beta_{1} - 74 \beta_{2} ) q^{49} + ( 23634 + 410 \beta_{1} + 132 \beta_{2} ) q^{51} + ( 6712 + 1140 \beta_{1} - 114 \beta_{2} ) q^{53} + ( -22486 - 1580 \beta_{1} - 98 \beta_{2} ) q^{57} + ( 11765 - 70 \beta_{1} - 185 \beta_{2} ) q^{59} + ( -8718 + 1984 \beta_{1} + 32 \beta_{2} ) q^{61} + ( -26938 - 263 \beta_{1} + 91 \beta_{2} ) q^{63} + ( 2381 + 2345 \beta_{1} + 82 \beta_{2} ) q^{67} + ( -9728 + 2248 \beta_{1} - 100 \beta_{2} ) q^{69} + ( 29002 + 1030 \beta_{1} + 56 \beta_{2} ) q^{71} + ( -23080 + 2340 \beta_{1} + 150 \beta_{2} ) q^{73} + ( 45818 - 860 \beta_{1} - 106 \beta_{2} ) q^{77} + ( -31272 + 700 \beta_{1} + 164 \beta_{2} ) q^{79} + ( 48879 + 4916 \beta_{1} - 162 \beta_{2} ) q^{81} + ( -10935 + 2923 \beta_{1} - 376 \beta_{2} ) q^{83} + ( 10046 - 4094 \beta_{1} + 184 \beta_{2} ) q^{87} + ( 57774 - 600 \beta_{1} - 36 \beta_{2} ) q^{89} + ( -36272 + 2110 \beta_{1} - 106 \beta_{2} ) q^{91} + ( 104352 + 4720 \beta_{1} + 456 \beta_{2} ) q^{93} + ( -57300 + 1420 \beta_{1} - 430 \beta_{2} ) q^{97} + ( -115931 - 3600 \beta_{1} - 43 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 10q^{3} + 6q^{7} + 467q^{9} + O(q^{10})$$ $$3q - 10q^{3} + 6q^{7} + 467q^{9} - 396q^{11} + 354q^{13} - 1158q^{17} + 3192q^{19} - 820q^{21} - 6126q^{23} - 13804q^{27} + 426q^{29} + 3276q^{31} + 2408q^{33} + 11562q^{37} + 21348q^{39} + 12450q^{41} - 26346q^{43} - 36762q^{47} - 3849q^{49} + 71444q^{51} + 21162q^{53} - 69136q^{57} + 35040q^{59} - 24138q^{61} - 80986q^{63} + 9570q^{67} - 27036q^{69} + 88092q^{71} - 66750q^{73} + 136488q^{77} - 92952q^{79} + 151391q^{81} - 30258q^{83} + 26228q^{87} + 172686q^{89} - 106812q^{91} + 318232q^{93} - 170910q^{97} - 351436q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 36 x - 24$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$8 \nu^{2} - 16 \nu - 189$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1} + 193$$$$)/8$$

## 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
 6.80681 −0.688934 −5.11788
0 −29.2272 0 0 0 −44.5253 0 611.232 0
1.2 0 0.755735 0 0 0 172.424 0 −242.429 0
1.3 0 18.4715 0 0 0 −121.899 0 98.1968 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.n 3
4.b odd 2 1 800.6.a.o 3
5.b even 2 1 160.6.a.g yes 3
5.c odd 4 2 800.6.c.j 6
20.d odd 2 1 160.6.a.f 3
20.e even 4 2 800.6.c.k 6
40.e odd 2 1 320.6.a.y 3
40.f even 2 1 320.6.a.x 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 20.d odd 2 1
160.6.a.g yes 3 5.b even 2 1
320.6.a.x 3 40.f even 2 1
320.6.a.y 3 40.e odd 2 1
800.6.a.n 3 1.a even 1 1 trivial
800.6.a.o 3 4.b odd 2 1
800.6.c.j 6 5.c odd 4 2
800.6.c.k 6 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3}^{3} + 10 T_{3}^{2} - 548 T_{3} + 408$$ $$T_{11}^{3} + 396 T_{11}^{2} - 155280 T_{11} - 59934400$$ $$T_{13}^{3} - 354 T_{13}^{2} - 268500 T_{13} - 28863000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 10 T + 181 T^{2} + 5268 T^{3} + 43983 T^{4} + 590490 T^{5} + 14348907 T^{6}$$
$5$ 1
$7$ $$1 - 6 T + 27153 T^{2} - 1137532 T^{3} + 456360471 T^{4} - 1694851494 T^{5} + 4747561509943 T^{6}$$
$11$ $$1 + 396 T + 327873 T^{2} + 67617992 T^{3} + 52804274523 T^{4} + 10271220141996 T^{5} + 4177248169415651 T^{6}$$
$13$ $$1 - 354 T + 845379 T^{2} - 291738444 T^{3} + 313883305047 T^{4} - 48801906114546 T^{5} + 51185893014090757 T^{6}$$
$17$ $$1 + 1158 T + 1914111 T^{2} + 544550612 T^{3} + 2717763902127 T^{4} + 2334520936719942 T^{5} + 2862423051509815793 T^{6}$$
$19$ $$1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6}$$
$23$ $$1 + 6126 T + 29722593 T^{2} + 84141222540 T^{3} + 191304803397399 T^{4} + 253778807694813774 T^{5} +$$$$26\!\cdots\!07$$$$T^{6}$$
$29$ $$1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} +$$$$86\!\cdots\!49$$$$T^{6}$$
$31$ $$1 - 3276 T + 2292813 T^{2} + 41639420248 T^{3} + 65641289591763 T^{4} - 2685102268149104076 T^{5} +$$$$23\!\cdots\!51$$$$T^{6}$$
$37$ $$1 - 11562 T + 90623691 T^{2} - 525372493468 T^{3} + 6284205331885287 T^{4} - 55596852513895170138 T^{5} +$$$$33\!\cdots\!93$$$$T^{6}$$
$41$ $$1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} -$$$$16\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6}$$
$43$ $$1 + 26346 T + 640605069 T^{2} + 8098987233524 T^{3} + 94174353771597567 T^{4} +$$$$56\!\cdots\!54$$$$T^{5} +$$$$31\!\cdots\!07$$$$T^{6}$$
$47$ $$1 + 36762 T + 1119958377 T^{2} + 18490559326820 T^{3} + 256856861812773639 T^{4} +$$$$19\!\cdots\!38$$$$T^{5} +$$$$12\!\cdots\!43$$$$T^{6}$$
$53$ $$1 - 21162 T + 395789499 T^{2} + 881498066468 T^{3} + 165517384658528007 T^{4} -$$$$37\!\cdots\!38$$$$T^{5} +$$$$73\!\cdots\!57$$$$T^{6}$$
$59$ $$1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} -$$$$17\!\cdots\!40$$$$T^{5} +$$$$36\!\cdots\!99$$$$T^{6}$$
$61$ $$1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} +$$$$17\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6}$$
$67$ $$1 - 9570 T + 659335509 T^{2} - 83080838420484 T^{3} + 890185424637524463 T^{4} -$$$$17\!\cdots\!30$$$$T^{5} +$$$$24\!\cdots\!43$$$$T^{6}$$
$71$ $$1 - 88092 T + 7289446053 T^{2} - 329541325840584 T^{3} + 13151832521353701603 T^{4} -$$$$28\!\cdots\!92$$$$T^{5} +$$$$58\!\cdots\!51$$$$T^{6}$$
$73$ $$1 + 66750 T + 3875077479 T^{2} + 111360074258500 T^{3} + 8033313042388954047 T^{4} +$$$$28\!\cdots\!50$$$$T^{5} +$$$$89\!\cdots\!57$$$$T^{6}$$
$79$ $$1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 34353638858281213923 T^{4} +$$$$88\!\cdots\!52$$$$T^{5} +$$$$29\!\cdots\!99$$$$T^{6}$$
$83$ $$1 + 30258 T + 4293702405 T^{2} + 426012342532708 T^{3} + 16913068282241846415 T^{4} +$$$$46\!\cdots\!42$$$$T^{5} +$$$$61\!\cdots\!07$$$$T^{6}$$
$89$ $$1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} +$$$$14\!\cdots\!59$$$$T^{4} -$$$$53\!\cdots\!86$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6}$$
$97$ $$1 + 170910 T + 30283966671 T^{2} + 2852314667192740 T^{3} +$$$$26\!\cdots\!47$$$$T^{4} +$$$$12\!\cdots\!90$$$$T^{5} +$$$$63\!\cdots\!93$$$$T^{6}$$