# Properties

 Label 800.6.c.k Level 800 Weight 6 Character orbit 800.c Analytic conductor 128.307 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$128.307055850$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.6140289600.1 Defining polynomial: $$x^{6} - 2 x^{5} + 32 x^{4} + 116 x^{3} + 256 x^{2} + 2778 x + 7605$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \beta_{1} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( -157 - 6 \beta_{2} + 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -2 \beta_{1} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( -157 - 6 \beta_{2} + 2 \beta_{4} ) q^{9} + ( 131 - 3 \beta_{4} ) q^{11} + ( -53 \beta_{1} - 20 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -211 \beta_{1} + 60 \beta_{3} + 6 \beta_{5} ) q^{17} + ( 1072 + 25 \beta_{2} - \beta_{4} ) q^{19} + ( -260 + 36 \beta_{2} + 4 \beta_{4} ) q^{21} + ( -1009 \beta_{1} - 43 \beta_{3} - 7 \beta_{5} ) q^{23} + ( 2368 \beta_{1} - 182 \beta_{3} + 20 \beta_{5} ) q^{27} + ( -118 + 20 \beta_{2} + 52 \beta_{4} ) q^{29} + ( -1061 + 135 \beta_{2} - 42 \beta_{4} ) q^{31} + ( -526 \beta_{1} + 380 \beta_{3} + 6 \beta_{5} ) q^{33} + ( 2019 \beta_{1} - 200 \beta_{3} + 76 \beta_{5} ) q^{37} + ( 7083 - 55 \beta_{2} - 44 \beta_{4} ) q^{39} + ( 4154 + 34 \beta_{2} - 22 \beta_{4} ) q^{41} + ( -4460 \beta_{1} + 221 \beta_{3} + 14 \beta_{5} ) q^{43} + ( 6093 \beta_{1} + 77 \beta_{3} - 25 \beta_{5} ) q^{47} + ( 1311 + 158 \beta_{2} - 74 \beta_{4} ) q^{49} + ( -23839 - 205 \beta_{2} + 132 \beta_{4} ) q^{51} + ( -3869 \beta_{1} + 1140 \beta_{3} + 114 \beta_{5} ) q^{53} + ( -12082 \beta_{1} + 1580 \beta_{3} - 98 \beta_{5} ) q^{57} + ( 11730 - 35 \beta_{2} + 185 \beta_{4} ) q^{59} + ( -7726 + 992 \beta_{2} - 32 \beta_{4} ) q^{61} + ( -13555 \beta_{1} + 263 \beta_{3} + 91 \beta_{5} ) q^{63} + ( -2404 \beta_{1} + 2345 \beta_{3} - 82 \beta_{5} ) q^{67} + ( 8604 - 1124 \beta_{2} - 100 \beta_{4} ) q^{69} + ( -29517 - 515 \beta_{2} + 56 \beta_{4} ) q^{71} + ( 10295 \beta_{1} + 2340 \beta_{3} - 150 \beta_{5} ) q^{73} + ( 22426 \beta_{1} + 860 \beta_{3} - 106 \beta_{5} ) q^{77} + ( -30922 + 350 \beta_{2} - 164 \beta_{4} ) q^{79} + ( 51337 + 2458 \beta_{2} + 162 \beta_{4} ) q^{81} + ( -4194 \beta_{1} - 2923 \beta_{3} - 376 \beta_{5} ) q^{83} + ( -3068 \beta_{1} - 4094 \beta_{3} - 184 \beta_{5} ) q^{87} + ( -57474 + 300 \beta_{2} - 36 \beta_{4} ) q^{89} + ( 35217 - 1055 \beta_{2} - 106 \beta_{4} ) q^{91} + ( -54764 \beta_{1} + 4720 \beta_{3} - 456 \beta_{5} ) q^{93} + ( -28155 \beta_{1} - 1420 \beta_{3} - 430 \beta_{5} ) q^{97} + ( -117731 - 1800 \beta_{2} + 43 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 934q^{9} + O(q^{10})$$ $$6q - 934q^{9} + 792q^{11} + 6384q^{19} - 1640q^{21} - 852q^{29} - 6552q^{31} + 42696q^{39} + 24900q^{41} + 7698q^{49} - 142888q^{51} + 70080q^{59} - 48276q^{61} + 54072q^{69} - 176184q^{71} - 185904q^{79} + 302782q^{81} - 345372q^{89} + 213624q^{91} - 702872q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 32 x^{4} + 116 x^{3} + 256 x^{2} + 2778 x + 7605$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$7 \nu^{5} - 275 \nu^{4} + 2174 \nu^{3} - 5515 \nu^{2} + 8137 \nu + 106320$$$$)/58125$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{5} + 256 \nu^{3} + 1440 \nu^{2} - 72 \nu + 25205$$$$)/625$$ $$\beta_{3}$$ $$=$$ $$($$$$-173 \nu^{5} + 685 \nu^{4} - 6166 \nu^{3} + 2645 \nu^{2} - 27323 \nu - 212280$$$$)/11625$$ $$\beta_{4}$$ $$=$$ $$($$$$-16 \nu^{5} + 200 \nu^{4} - 112 \nu^{3} - 680 \nu^{2} + 20544 \nu + 12965$$$$)/625$$ $$\beta_{5}$$ $$=$$ $$($$$$1061 \nu^{5} - 1825 \nu^{4} + 32802 \nu^{3} + 89655 \nu^{2} + 693051 \nu + 1899360$$$$)/19375$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} + 10$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{5} - 2 \beta_{4} + 22 \beta_{3} + 8 \beta_{2} + 79 \beta_{1} - 318$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 28 \beta_{2} + 395 \beta_{1} - 1408$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$-115 \beta_{5} + 106 \beta_{4} - 454 \beta_{3} + 204 \beta_{2} - 2959 \beta_{1} - 2030$$$$)/32$$ $$\nu^{5}$$ $$=$$ $$($$$$-211 \beta_{5} + 104 \beta_{4} - 4070 \beta_{3} - 750 \beta_{2} - 39455 \beta_{1} + 46622$$$$)/32$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −2.90341 + 0.978064i 3.05894 − 4.88658i 0.844467 + 4.86464i 0.844467 − 4.86464i 3.05894 + 4.88658i −2.90341 − 0.978064i
0 29.2272i 0 0 0 44.5253i 0 −611.232 0
449.2 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.3 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.4 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.5 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.6 0 29.2272i 0 0 0 44.5253i 0 −611.232 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.c.k 6
4.b odd 2 1 800.6.c.j 6
5.b even 2 1 inner 800.6.c.k 6
5.c odd 4 1 160.6.a.f 3
5.c odd 4 1 800.6.a.o 3
20.d odd 2 1 800.6.c.j 6
20.e even 4 1 160.6.a.g yes 3
20.e even 4 1 800.6.a.n 3
40.i odd 4 1 320.6.a.y 3
40.k even 4 1 320.6.a.x 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 5.c odd 4 1
160.6.a.g yes 3 20.e even 4 1
320.6.a.x 3 40.k even 4 1
320.6.a.y 3 40.i odd 4 1
800.6.a.n 3 20.e even 4 1
800.6.a.o 3 5.c odd 4 1
800.6.c.j 6 4.b odd 2 1
800.6.c.j 6 20.d odd 2 1
800.6.c.k 6 1.a even 1 1 trivial
800.6.c.k 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{6} + 1196 T_{3}^{4} + 292144 T_{3}^{2} + 166464$$ $$T_{11}^{3} - 396 T_{11}^{2} - 155280 T_{11} + 59934400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 262 T^{2} + 15367 T^{4} - 5058036 T^{6} + 907405983 T^{8} - 913537513062 T^{10} + 205891132094649 T^{12}$$
$5$ 1
$7$ $$1 - 54270 T^{2} + 1636355967 T^{4} - 32963917489060 T^{6} + 462230059230960783 T^{8} -$$$$43\!\cdots\!70$$$$T^{10} +$$$$22\!\cdots\!49$$$$T^{12}$$
$11$ $$( 1 - 396 T + 327873 T^{2} - 67617992 T^{3} + 52804274523 T^{4} - 10271220141996 T^{5} + 4177248169415651 T^{6} )^{2}$$
$13$ $$1 - 1565442 T^{2} + 1135881445383 T^{4} - 513409425866197436 T^{6} +$$$$15\!\cdots\!67$$$$T^{8} -$$$$29\!\cdots\!42$$$$T^{10} +$$$$26\!\cdots\!49$$$$T^{12}$$
$17$ $$1 - 2487258 T^{2} + 7838169507183 T^{4} - 10425763805475099564 T^{6} +$$$$15\!\cdots\!67$$$$T^{8} -$$$$10\!\cdots\!58$$$$T^{10} +$$$$81\!\cdots\!49$$$$T^{12}$$
$19$ $$( 1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6} )^{2}$$
$23$ $$1 - 21917310 T^{2} + 235143882878367 T^{4} -$$$$17\!\cdots\!80$$$$T^{6} +$$$$97\!\cdots\!83$$$$T^{8} -$$$$37\!\cdots\!10$$$$T^{10} +$$$$71\!\cdots\!49$$$$T^{12}$$
$29$ $$( 1 + 426 T - 939693 T^{2} - 141773503364 T^{3} - 19274183137257 T^{4} + 179221281385885626 T^{5} +$$$$86\!\cdots\!49$$$$T^{6} )^{2}$$
$31$ $$( 1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} +$$$$23\!\cdots\!51$$$$T^{6} )^{2}$$
$37$ $$1 - 47567538 T^{2} + 8632350495280023 T^{4} -$$$$24\!\cdots\!84$$$$T^{6} +$$$$41\!\cdots\!27$$$$T^{8} -$$$$10\!\cdots\!38$$$$T^{10} +$$$$11\!\cdots\!49$$$$T^{12}$$
$41$ $$( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} -$$$$16\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6} )^{2}$$
$43$ $$1 - 587098422 T^{2} + 171971726662843287 T^{4} -$$$$31\!\cdots\!16$$$$T^{6} +$$$$37\!\cdots\!63$$$$T^{8} -$$$$27\!\cdots\!22$$$$T^{10} +$$$$10\!\cdots\!49$$$$T^{12}$$
$47$ $$1 - 888472110 T^{2} + 408520605892907727 T^{4} -$$$$11\!\cdots\!80$$$$T^{6} +$$$$21\!\cdots\!23$$$$T^{8} -$$$$24\!\cdots\!10$$$$T^{10} +$$$$14\!\cdots\!49$$$$T^{12}$$
$53$ $$1 - 343748754 T^{2} + 524992621000918647 T^{4} -$$$$11\!\cdots\!64$$$$T^{6} +$$$$91\!\cdots\!03$$$$T^{8} -$$$$10\!\cdots\!54$$$$T^{10} +$$$$53\!\cdots\!49$$$$T^{12}$$
$59$ $$( 1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} -$$$$17\!\cdots\!40$$$$T^{5} +$$$$36\!\cdots\!99$$$$T^{6} )^{2}$$
$61$ $$( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} +$$$$17\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6} )^{2}$$
$67$ $$1 - 1227086118 T^{2} + 624926915335274247 T^{4} +$$$$11\!\cdots\!36$$$$T^{6} +$$$$11\!\cdots\!03$$$$T^{8} -$$$$40\!\cdots\!18$$$$T^{10} +$$$$60\!\cdots\!49$$$$T^{12}$$
$71$ $$( 1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} +$$$$28\!\cdots\!92$$$$T^{5} +$$$$58\!\cdots\!51$$$$T^{6} )^{2}$$
$73$ $$1 - 3294592458 T^{2} + 16216281639521153535 T^{4} -$$$$29\!\cdots\!40$$$$T^{6} +$$$$69\!\cdots\!15$$$$T^{8} -$$$$60\!\cdots\!58$$$$T^{10} +$$$$79\!\cdots\!49$$$$T^{12}$$
$79$ $$( 1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 34353638858281213923 T^{4} +$$$$88\!\cdots\!52$$$$T^{5} +$$$$29\!\cdots\!99$$$$T^{6} )^{2}$$
$83$ $$1 - 7671858246 T^{2} + 26481453986477119527 T^{4} -$$$$57\!\cdots\!28$$$$T^{6} +$$$$41\!\cdots\!23$$$$T^{8} -$$$$18\!\cdots\!46$$$$T^{10} +$$$$37\!\cdots\!49$$$$T^{12}$$
$89$ $$( 1 + 172686 T + 26445328791 T^{2} + 2103593815517412 T^{3} +$$$$14\!\cdots\!59$$$$T^{4} +$$$$53\!\cdots\!86$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6} )^{2}$$
$97$ $$1 - 31357705242 T^{2} +$$$$46\!\cdots\!35$$$$T^{4} -$$$$45\!\cdots\!60$$$$T^{6} +$$$$34\!\cdots\!15$$$$T^{8} -$$$$17\!\cdots\!42$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12}$$