# Properties

 Label 8.6.b.a Level 8 Weight 6 Character orbit 8.b Analytic conductor 1.283 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.28307055850$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.218489.1 Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 8 x + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{5} + ( -32 - 2 \beta_{2} - 6 \beta_{3} ) q^{6} + ( 36 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{7} + ( -58 - 2 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{8} + ( -65 - 40 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{5} + ( -32 - 2 \beta_{2} - 6 \beta_{3} ) q^{6} + ( 36 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{7} + ( -58 - 2 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{8} + ( -65 - 40 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{9} + ( 152 - 8 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{10} + ( -16 - 37 \beta_{1} + 16 \beta_{2} + 5 \beta_{3} ) q^{11} + ( 418 + 26 \beta_{1} + 10 \beta_{2} + 22 \beta_{3} ) q^{12} + ( 6 + 44 \beta_{1} - 6 \beta_{2} - 32 \beta_{3} ) q^{13} + ( -600 - 24 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{14} + ( -92 + 20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{15} + ( -804 + 76 \beta_{1} - 4 \beta_{2} - 28 \beta_{3} ) q^{16} + ( 74 + 40 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{17} + ( 1193 + 41 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{18} + ( 48 + 103 \beta_{1} - 48 \beta_{2} - 7 \beta_{3} ) q^{19} + ( 1044 - 188 \beta_{1} - 28 \beta_{2} - 36 \beta_{3} ) q^{20} + ( -24 - 208 \beta_{1} + 24 \beta_{2} + 160 \beta_{3} ) q^{21} + ( -1376 + 64 \beta_{1} + 86 \beta_{2} + 2 \beta_{3} ) q^{22} + ( 428 - 260 \beta_{1} - 52 \beta_{2} - 52 \beta_{3} ) q^{23} + ( -2164 - 388 \beta_{1} - 52 \beta_{2} - 44 \beta_{3} ) q^{24} + ( 629 + 400 \beta_{1} + 80 \beta_{2} + 80 \beta_{3} ) q^{25} + ( 1480 - 24 \beta_{1} + 28 \beta_{2} + 180 \beta_{3} ) q^{26} + ( 48 + 166 \beta_{1} - 48 \beta_{2} - 70 \beta_{3} ) q^{27} + ( 1480 + 552 \beta_{1} - 88 \beta_{2} - 168 \beta_{3} ) q^{28} + ( -82 + 92 \beta_{1} + 82 \beta_{2} - 256 \beta_{3} ) q^{29} + ( -472 + 104 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{30} + ( -3376 - 240 \beta_{1} - 48 \beta_{2} - 48 \beta_{3} ) q^{31} + ( 1848 + 792 \beta_{1} - 8 \beta_{2} + 200 \beta_{3} ) q^{32} + ( -804 - 360 \beta_{1} - 72 \beta_{2} - 72 \beta_{3} ) q^{33} + ( -1202 - 50 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{34} + ( -224 - 432 \beta_{1} + 224 \beta_{2} - 16 \beta_{3} ) q^{35} + ( -2925 - 1097 \beta_{1} + 183 \beta_{2} + 329 \beta_{3} ) q^{36} + ( 114 + 68 \beta_{1} - 114 \beta_{2} + 160 \beta_{3} ) q^{37} + ( 3872 - 192 \beta_{1} - 274 \beta_{2} - 54 \beta_{3} ) q^{38} + ( 9580 + 1340 \beta_{1} + 268 \beta_{2} + 268 \beta_{3} ) q^{39} + ( 3384 - 1128 \beta_{1} + 184 \beta_{2} - 120 \beta_{3} ) q^{40} + ( -1958 - 1360 \beta_{1} - 272 \beta_{2} - 272 \beta_{3} ) q^{41} + ( -6944 + 96 \beta_{1} - 176 \beta_{2} - 912 \beta_{3} ) q^{42} + ( 96 - 229 \beta_{1} - 96 \beta_{2} + 421 \beta_{3} ) q^{43} + ( -6262 + 1634 \beta_{1} + 274 \beta_{2} + 398 \beta_{3} ) q^{44} + ( 462 + 892 \beta_{1} - 462 \beta_{2} + 32 \beta_{3} ) q^{45} + ( 6904 - 584 \beta_{1} + 208 \beta_{2} - 208 \beta_{3} ) q^{46} + ( -13032 + 1080 \beta_{1} + 216 \beta_{2} + 216 \beta_{3} ) q^{47} + ( 9720 + 2008 \beta_{1} + 312 \beta_{2} - 376 \beta_{3} ) q^{48} + ( 3033 + 960 \beta_{1} + 192 \beta_{2} + 192 \beta_{3} ) q^{49} + ( -11909 - 389 \beta_{1} - 320 \beta_{2} + 320 \beta_{3} ) q^{50} + ( -48 - 418 \beta_{1} + 48 \beta_{2} + 322 \beta_{3} ) q^{51} + ( -10244 - 1396 \beta_{1} - 404 \beta_{2} - 812 \beta_{3} ) q^{52} + ( -262 + 404 \beta_{1} + 262 \beta_{2} - 928 \beta_{3} ) q^{53} + ( 5888 - 192 \beta_{1} - 148 \beta_{2} + 324 \beta_{3} ) q^{54} + ( 19508 - 3100 \beta_{1} - 620 \beta_{2} - 620 \beta_{3} ) q^{55} + ( 9840 - 1744 \beta_{1} - 400 \beta_{2} + 1040 \beta_{3} ) q^{56} + ( -52 + 760 \beta_{1} + 152 \beta_{2} + 152 \beta_{3} ) q^{57} + ( 1960 + 328 \beta_{1} + 1004 \beta_{2} + 1700 \beta_{3} ) q^{58} + ( -256 + 915 \beta_{1} + 256 \beta_{2} - 1427 \beta_{3} ) q^{59} + ( 840 + 424 \beta_{1} - 216 \beta_{2} - 40 \beta_{3} ) q^{60} + ( -1314 - 4644 \beta_{1} + 1314 \beta_{2} + 2016 \beta_{3} ) q^{61} + ( 10144 + 3232 \beta_{1} + 192 \beta_{2} - 192 \beta_{3} ) q^{62} + ( -39428 - 1780 \beta_{1} - 356 \beta_{2} - 356 \beta_{3} ) q^{63} + ( -11536 - 1872 \beta_{1} - 1424 \beta_{2} - 240 \beta_{3} ) q^{64} + ( -4544 + 560 \beta_{1} + 112 \beta_{2} + 112 \beta_{3} ) q^{65} + ( 10956 + 588 \beta_{1} + 288 \beta_{2} - 288 \beta_{3} ) q^{66} + ( 1584 + 4527 \beta_{1} - 1584 \beta_{2} - 1359 \beta_{3} ) q^{67} + ( 2970 + 1106 \beta_{1} - 174 \beta_{2} - 338 \beta_{3} ) q^{68} + ( 312 + 1808 \beta_{1} - 312 \beta_{2} - 1184 \beta_{3} ) q^{69} + ( -16512 + 896 \beta_{1} + 1376 \beta_{2} + 544 \beta_{3} ) q^{70} + ( 52356 + 1140 \beta_{1} + 228 \beta_{2} + 228 \beta_{3} ) q^{71} + ( -20086 + 3474 \beta_{1} + 842 \beta_{2} - 2010 \beta_{3} ) q^{72} + ( 13618 + 6040 \beta_{1} + 1208 \beta_{2} + 1208 \beta_{3} ) q^{73} + ( 3544 - 456 \beta_{1} - 1004 \beta_{2} - 1188 \beta_{3} ) q^{74} + ( -480 - 4069 \beta_{1} + 480 \beta_{2} + 3109 \beta_{3} ) q^{75} + ( 22130 - 4694 \beta_{1} - 742 \beta_{2} - 1018 \beta_{3} ) q^{76} + ( 1672 + 2416 \beta_{1} - 1672 \beta_{2} + 928 \beta_{3} ) q^{77} + ( -47368 - 8776 \beta_{1} - 1072 \beta_{2} + 1072 \beta_{3} ) q^{78} + ( -63000 - 1720 \beta_{1} - 344 \beta_{2} - 344 \beta_{3} ) q^{79} + ( -10192 - 2832 \beta_{1} + 2224 \beta_{2} + 208 \beta_{3} ) q^{80} + ( 3557 - 6440 \beta_{1} - 1288 \beta_{2} - 1288 \beta_{3} ) q^{81} + ( 40310 + 1142 \beta_{1} + 1088 \beta_{2} - 1088 \beta_{3} ) q^{82} + ( -1408 - 5385 \beta_{1} + 1408 \beta_{2} + 2569 \beta_{3} ) q^{83} + ( 54352 + 6416 \beta_{1} + 1936 \beta_{2} + 3952 \beta_{3} ) q^{84} + ( -444 - 856 \beta_{1} + 444 \beta_{2} - 32 \beta_{3} ) q^{85} + ( -6176 - 384 \beta_{1} - 1418 \beta_{2} - 2718 \beta_{3} ) q^{86} + ( 82620 + 9420 \beta_{1} + 1884 \beta_{2} + 1884 \beta_{3} ) q^{87} + ( -37892 + 7084 \beta_{1} - 1732 \beta_{2} + 740 \beta_{3} ) q^{88} + ( -28670 - 12520 \beta_{1} - 2504 \beta_{2} - 2504 \beta_{3} ) q^{89} + ( 34088 - 1848 \beta_{1} - 2836 \beta_{2} - 1116 \beta_{3} ) q^{90} + ( 96 + 5360 \beta_{1} - 96 \beta_{2} - 5168 \beta_{3} ) q^{91} + ( -14760 - 6280 \beta_{1} + 2040 \beta_{2} + 1288 \beta_{3} ) q^{92} + ( 288 + 5440 \beta_{1} - 288 \beta_{2} - 4864 \beta_{3} ) q^{93} + ( -17424 + 13680 \beta_{1} - 864 \beta_{2} + 864 \beta_{3} ) q^{94} + ( -59260 + 9460 \beta_{1} + 1892 \beta_{2} + 1892 \beta_{3} ) q^{95} + ( -30736 - 8784 \beta_{1} + 368 \beta_{2} + 5136 \beta_{3} ) q^{96} + ( -21606 + 5480 \beta_{1} + 1096 \beta_{2} + 1096 \beta_{3} ) q^{97} + ( -30105 - 2457 \beta_{1} - 768 \beta_{2} + 768 \beta_{3} ) q^{98} + ( -3456 - 5091 \beta_{1} + 3456 \beta_{2} - 1821 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + 632q^{10} + 1576q^{12} - 2384q^{14} - 416q^{15} - 3312q^{16} + 200q^{17} + 4754q^{18} + 4624q^{20} - 5636q^{22} + 2336q^{23} - 7792q^{24} + 1556q^{25} + 5608q^{26} + 5152q^{28} - 2128q^{30} - 12928q^{31} + 5408q^{32} - 2352q^{33} - 4772q^{34} - 10164q^{36} + 15980q^{38} + 35104q^{39} + 16032q^{40} - 4568q^{41} - 26144q^{42} - 29112q^{44} + 29200q^{46} - 54720q^{47} + 35616q^{48} + 9828q^{49} - 47498q^{50} - 36560q^{52} + 23288q^{54} + 85472q^{55} + 40768q^{56} - 2032q^{57} + 3784q^{58} + 2592q^{60} + 34496q^{62} - 153440q^{63} - 41920q^{64} - 19520q^{65} + 43224q^{66} + 10344q^{68} - 68928q^{70} + 206688q^{71} - 83272q^{72} + 39976q^{73} + 17464q^{74} + 99944q^{76} - 174064q^{78} - 247872q^{79} - 35520q^{80} + 29684q^{81} + 161132q^{82} + 196672q^{84} - 18500q^{86} + 307872q^{87} - 167216q^{88} - 84632q^{89} + 142280q^{90} - 49056q^{92} - 98784q^{94} - 259744q^{95} - 115648q^{96} - 99576q^{97} - 117042q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 8 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 2 \nu + 4$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{2} + 6 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 7 \nu^{2} + 6 \nu - 20$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 3$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{3} - \beta_{2} + 3 \beta_{1} + 13$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} + \beta_{2} - 27 \beta_{1} + 51$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 2.38600 − 1.51888i 2.38600 + 1.51888i −1.88600 − 2.10784i −1.88600 + 2.10784i
−4.77200 3.03776i 23.6095i 13.5440 + 28.9924i 1.38521i −71.7200 + 112.665i 160.704 23.4400 179.495i −314.408 4.20793 6.61022i
5.2 −4.77200 + 3.03776i 23.6095i 13.5440 28.9924i 1.38521i −71.7200 112.665i 160.704 23.4400 + 179.495i −314.408 4.20793 + 6.61022i
5.3 3.77200 4.21569i 3.25452i −3.54400 31.8031i 73.9600i 13.7200 + 12.2760i −112.704 −147.440 105.021i 232.408 311.792 + 278.977i
5.4 3.77200 + 4.21569i 3.25452i −3.54400 + 31.8031i 73.9600i 13.7200 12.2760i −112.704 −147.440 + 105.021i 232.408 311.792 278.977i
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.6.b.a 4
3.b odd 2 1 72.6.d.b 4
4.b odd 2 1 32.6.b.a 4
5.b even 2 1 200.6.d.a 4
5.c odd 4 2 200.6.f.a 8
8.b even 2 1 inner 8.6.b.a 4
8.d odd 2 1 32.6.b.a 4
12.b even 2 1 288.6.d.b 4
16.e even 4 2 256.6.a.k 4
16.f odd 4 2 256.6.a.n 4
20.d odd 2 1 800.6.d.a 4
20.e even 4 2 800.6.f.a 8
24.f even 2 1 288.6.d.b 4
24.h odd 2 1 72.6.d.b 4
40.e odd 2 1 800.6.d.a 4
40.f even 2 1 200.6.d.a 4
40.i odd 4 2 200.6.f.a 8
40.k even 4 2 800.6.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a 4 1.a even 1 1 trivial
8.6.b.a 4 8.b even 2 1 inner
32.6.b.a 4 4.b odd 2 1
32.6.b.a 4 8.d odd 2 1
72.6.d.b 4 3.b odd 2 1
72.6.d.b 4 24.h odd 2 1
200.6.d.a 4 5.b even 2 1
200.6.d.a 4 40.f even 2 1
200.6.f.a 8 5.c odd 4 2
200.6.f.a 8 40.i odd 4 2
256.6.a.k 4 16.e even 4 2
256.6.a.n 4 16.f odd 4 2
288.6.d.b 4 12.b even 2 1
288.6.d.b 4 24.f even 2 1
800.6.d.a 4 20.d odd 2 1
800.6.d.a 4 40.e odd 2 1
800.6.f.a 8 20.e even 4 2
800.6.f.a 8 40.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 8 T^{2} + 64 T^{3} + 1024 T^{4}$$
$3$ $$1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8}$$
$5$ $$1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8}$$
$7$ $$( 1 - 48 T + 15502 T^{2} - 806736 T^{3} + 282475249 T^{4} )^{2}$$
$11$ $$1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$( 1 - 1168 T + 10055470 T^{2} - 7517648624 T^{3} + 41426511213649 T^{4} )^{2}$$
$29$ $$1 - 31255380 T^{2} + 976386653995702 T^{4} -$$$$13\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$( 1 + 6464 T + 65013054 T^{2} + 185058832064 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 241262580 T^{2} + 24149916431784598 T^{4} -$$$$11\!\cdots\!20$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 466346868 T^{2} + 96250708269010006 T^{4} -$$$$10\!\cdots\!32$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$( 1 + 27360 T + 591338206 T^{2} + 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2}$$
$53$ $$1 - 1039152180 T^{2} + 595616955270391126 T^{4} -$$$$18\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$1 - 1537424180 T^{2} + 1399694789142612374 T^{4} -$$$$78\!\cdots\!80$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$1 + 741098540 T^{2} + 1478044222094100534 T^{4} +$$$$52\!\cdots\!40$$$$T^{6} +$$$$50\!\cdots\!01$$$$T^{8}$$
$67$ $$1 - 1366835860 T^{2} + 3274116308996825526 T^{4} -$$$$24\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 103344 T + 6217736974 T^{2} - 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 + 123936 T + 9855929374 T^{2} + 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 10047855188 T^{2} + 55381071937674414326 T^{4} -$$$$15\!\cdots\!12$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2}$$