Properties

 Label 864.3.g.d.703.7 Level $864$ Weight $3$ Character 864.703 Analytic conductor $23.542$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.g (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 63*x^4 - 74*x^3 + 70*x^2 - 38*x + 13 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 703.7 Root $$0.500000 + 0.564882i$$ of defining polynomial Character $$\chi$$ $$=$$ 864.703 Dual form 864.3.g.d.703.8

$q$-expansion

 $$f(q)$$ $$=$$ $$q+6.42091 q^{5} -13.8102i q^{7} +O(q^{10})$$ $$q+6.42091 q^{5} -13.8102i q^{7} +13.0350i q^{11} +7.16454 q^{13} +31.5918 q^{17} +16.4875i q^{19} -16.9778i q^{23} +16.2281 q^{25} -42.4562 q^{29} -29.6836i q^{31} -88.6741i q^{35} +39.3037 q^{37} +39.8856 q^{41} +16.3291i q^{43} -57.8888i q^{47} -141.722 q^{49} +46.4110 q^{53} +83.6964i q^{55} -14.2179i q^{59} -63.7479 q^{61} +46.0028 q^{65} -32.5634i q^{67} -22.4738i q^{71} +24.9252 q^{73} +180.016 q^{77} +61.9501i q^{79} -44.7901i q^{83} +202.848 q^{85} -1.95333 q^{89} -98.9437i q^{91} +105.865i q^{95} -44.5813 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5}+O(q^{10})$$ 8 * q + 8 * q^5 $$8 q + 8 q^{5} + 8 q^{13} + 24 q^{17} + 24 q^{25} - 128 q^{29} + 24 q^{37} + 160 q^{41} - 144 q^{49} - 48 q^{53} - 136 q^{61} - 280 q^{65} + 72 q^{73} + 520 q^{77} + 96 q^{85} - 168 q^{89} + 104 q^{97}+O(q^{100})$$ 8 * q + 8 * q^5 + 8 * q^13 + 24 * q^17 + 24 * q^25 - 128 * q^29 + 24 * q^37 + 160 * q^41 - 144 * q^49 - 48 * q^53 - 136 * q^61 - 280 * q^65 + 72 * q^73 + 520 * q^77 + 96 * q^85 - 168 * q^89 + 104 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 6.42091 1.28418 0.642091 0.766628i $$-0.278067\pi$$
0.642091 + 0.766628i $$0.278067\pi$$
$$6$$ 0 0
$$7$$ − 13.8102i − 1.97289i −0.164102 0.986443i $$-0.552473\pi$$
0.164102 0.986443i $$-0.447527\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 13.0350i 1.18500i 0.805572 + 0.592498i $$0.201858\pi$$
−0.805572 + 0.592498i $$0.798142\pi$$
$$12$$ 0 0
$$13$$ 7.16454 0.551118 0.275559 0.961284i $$-0.411137\pi$$
0.275559 + 0.961284i $$0.411137\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 31.5918 1.85834 0.929171 0.369651i $$-0.120523\pi$$
0.929171 + 0.369651i $$0.120523\pi$$
$$18$$ 0 0
$$19$$ 16.4875i 0.867763i 0.900970 + 0.433881i $$0.142856\pi$$
−0.900970 + 0.433881i $$0.857144\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 16.9778i − 0.738163i −0.929397 0.369082i $$-0.879672\pi$$
0.929397 0.369082i $$-0.120328\pi$$
$$24$$ 0 0
$$25$$ 16.2281 0.649124
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −42.4562 −1.46401 −0.732004 0.681301i $$-0.761415\pi$$
−0.732004 + 0.681301i $$0.761415\pi$$
$$30$$ 0 0
$$31$$ − 29.6836i − 0.957537i −0.877941 0.478769i $$-0.841083\pi$$
0.877941 0.478769i $$-0.158917\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 88.6741i − 2.53355i
$$36$$ 0 0
$$37$$ 39.3037 1.06226 0.531131 0.847289i $$-0.321767\pi$$
0.531131 + 0.847289i $$0.321767\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 39.8856 0.972819 0.486409 0.873731i $$-0.338306\pi$$
0.486409 + 0.873731i $$0.338306\pi$$
$$42$$ 0 0
$$43$$ 16.3291i 0.379746i 0.981809 + 0.189873i $$0.0608076\pi$$
−0.981809 + 0.189873i $$0.939192\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 57.8888i − 1.23168i −0.787872 0.615839i $$-0.788817\pi$$
0.787872 0.615839i $$-0.211183\pi$$
$$48$$ 0 0
$$49$$ −141.722 −2.89228
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 46.4110 0.875680 0.437840 0.899053i $$-0.355744\pi$$
0.437840 + 0.899053i $$0.355744\pi$$
$$54$$ 0 0
$$55$$ 83.6964i 1.52175i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 14.2179i − 0.240981i −0.992714 0.120491i $$-0.961553\pi$$
0.992714 0.120491i $$-0.0384468\pi$$
$$60$$ 0 0
$$61$$ −63.7479 −1.04505 −0.522524 0.852625i $$-0.675009\pi$$
−0.522524 + 0.852625i $$0.675009\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 46.0028 0.707736
$$66$$ 0 0
$$67$$ − 32.5634i − 0.486022i −0.970024 0.243011i $$-0.921865\pi$$
0.970024 0.243011i $$-0.0781350\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 22.4738i − 0.316532i −0.987397 0.158266i $$-0.949410\pi$$
0.987397 0.158266i $$-0.0505904\pi$$
$$72$$ 0 0
$$73$$ 24.9252 0.341441 0.170721 0.985319i $$-0.445390\pi$$
0.170721 + 0.985319i $$0.445390\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 180.016 2.33786
$$78$$ 0 0
$$79$$ 61.9501i 0.784178i 0.919927 + 0.392089i $$0.128247\pi$$
−0.919927 + 0.392089i $$0.871753\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 44.7901i − 0.539640i −0.962911 0.269820i $$-0.913036\pi$$
0.962911 0.269820i $$-0.0869642\pi$$
$$84$$ 0 0
$$85$$ 202.848 2.38645
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1.95333 −0.0219475 −0.0109738 0.999940i $$-0.503493\pi$$
−0.0109738 + 0.999940i $$0.503493\pi$$
$$90$$ 0 0
$$91$$ − 98.9437i − 1.08729i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 105.865i 1.11437i
$$96$$ 0 0
$$97$$ −44.5813 −0.459601 −0.229800 0.973238i $$-0.573807\pi$$
−0.229800 + 0.973238i $$0.573807\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 138.393 1.37022 0.685112 0.728438i $$-0.259753\pi$$
0.685112 + 0.728438i $$0.259753\pi$$
$$102$$ 0 0
$$103$$ 19.5698i 0.189998i 0.995477 + 0.0949991i $$0.0302848\pi$$
−0.995477 + 0.0949991i $$0.969715\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 39.6245i 0.370323i 0.982708 + 0.185161i $$0.0592808\pi$$
−0.982708 + 0.185161i $$0.940719\pi$$
$$108$$ 0 0
$$109$$ 45.2012 0.414690 0.207345 0.978268i $$-0.433518\pi$$
0.207345 + 0.978268i $$0.433518\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 111.845 0.989776 0.494888 0.868957i $$-0.335209\pi$$
0.494888 + 0.868957i $$0.335209\pi$$
$$114$$ 0 0
$$115$$ − 109.013i − 0.947936i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 436.289i − 3.66630i
$$120$$ 0 0
$$121$$ −48.9103 −0.404218
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −56.3235 −0.450588
$$126$$ 0 0
$$127$$ − 110.064i − 0.866642i −0.901240 0.433321i $$-0.857342\pi$$
0.901240 0.433321i $$-0.142658\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 14.3672i − 0.109673i −0.998495 0.0548367i $$-0.982536\pi$$
0.998495 0.0548367i $$-0.0174638\pi$$
$$132$$ 0 0
$$133$$ 227.696 1.71200
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −170.048 −1.24123 −0.620613 0.784117i $$-0.713116\pi$$
−0.620613 + 0.784117i $$0.713116\pi$$
$$138$$ 0 0
$$139$$ 213.171i 1.53360i 0.641884 + 0.766802i $$0.278153\pi$$
−0.641884 + 0.766802i $$0.721847\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 93.3895i 0.653073i
$$144$$ 0 0
$$145$$ −272.608 −1.88005
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −84.5143 −0.567210 −0.283605 0.958941i $$-0.591530\pi$$
−0.283605 + 0.958941i $$0.591530\pi$$
$$150$$ 0 0
$$151$$ − 180.407i − 1.19475i −0.801962 0.597376i $$-0.796210\pi$$
0.801962 0.597376i $$-0.203790\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 190.596i − 1.22965i
$$156$$ 0 0
$$157$$ 49.7071 0.316605 0.158303 0.987391i $$-0.449398\pi$$
0.158303 + 0.987391i $$0.449398\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −234.466 −1.45631
$$162$$ 0 0
$$163$$ − 49.3718i − 0.302894i −0.988465 0.151447i $$-0.951607\pi$$
0.988465 0.151447i $$-0.0483933\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 161.617i 0.967766i 0.875133 + 0.483883i $$0.160774\pi$$
−0.875133 + 0.483883i $$0.839226\pi$$
$$168$$ 0 0
$$169$$ −117.669 −0.696269
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −63.6058 −0.367664 −0.183832 0.982958i $$-0.558850\pi$$
−0.183832 + 0.982958i $$0.558850\pi$$
$$174$$ 0 0
$$175$$ − 224.114i − 1.28065i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 203.780i − 1.13844i −0.822186 0.569219i $$-0.807246\pi$$
0.822186 0.569219i $$-0.192754\pi$$
$$180$$ 0 0
$$181$$ 23.1886 0.128114 0.0640570 0.997946i $$-0.479596\pi$$
0.0640570 + 0.997946i $$0.479596\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 252.366 1.36414
$$186$$ 0 0
$$187$$ 411.798i 2.20213i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 247.017i 1.29328i 0.762793 + 0.646642i $$0.223827\pi$$
−0.762793 + 0.646642i $$0.776173\pi$$
$$192$$ 0 0
$$193$$ 126.813 0.657063 0.328531 0.944493i $$-0.393446\pi$$
0.328531 + 0.944493i $$0.393446\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 47.1824 0.239505 0.119752 0.992804i $$-0.461790\pi$$
0.119752 + 0.992804i $$0.461790\pi$$
$$198$$ 0 0
$$199$$ 298.835i 1.50168i 0.660482 + 0.750842i $$0.270352\pi$$
−0.660482 + 0.750842i $$0.729648\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 586.329i 2.88832i
$$204$$ 0 0
$$205$$ 256.102 1.24928
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −214.914 −1.02830
$$210$$ 0 0
$$211$$ 270.034i 1.27978i 0.768466 + 0.639891i $$0.221020\pi$$
−0.768466 + 0.639891i $$0.778980\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 104.848i 0.487663i
$$216$$ 0 0
$$217$$ −409.937 −1.88911
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 226.341 1.02417
$$222$$ 0 0
$$223$$ 171.763i 0.770237i 0.922867 + 0.385118i $$0.125839\pi$$
−0.922867 + 0.385118i $$0.874161\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 146.057i 0.643424i 0.946838 + 0.321712i $$0.104258\pi$$
−0.946838 + 0.321712i $$0.895742\pi$$
$$228$$ 0 0
$$229$$ −325.016 −1.41928 −0.709641 0.704563i $$-0.751143\pi$$
−0.709641 + 0.704563i $$0.751143\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 405.093 1.73860 0.869299 0.494286i $$-0.164570\pi$$
0.869299 + 0.494286i $$0.164570\pi$$
$$234$$ 0 0
$$235$$ − 371.699i − 1.58170i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 85.3071i 0.356933i 0.983946 + 0.178467i $$0.0571137\pi$$
−0.983946 + 0.178467i $$0.942886\pi$$
$$240$$ 0 0
$$241$$ −357.900 −1.48506 −0.742532 0.669811i $$-0.766375\pi$$
−0.742532 + 0.669811i $$0.766375\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −909.983 −3.71422
$$246$$ 0 0
$$247$$ 118.125i 0.478240i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 88.8097i 0.353823i 0.984227 + 0.176912i $$0.0566107\pi$$
−0.984227 + 0.176912i $$0.943389\pi$$
$$252$$ 0 0
$$253$$ 221.304 0.874721
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 237.843 0.925460 0.462730 0.886499i $$-0.346870\pi$$
0.462730 + 0.886499i $$0.346870\pi$$
$$258$$ 0 0
$$259$$ − 542.793i − 2.09572i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 309.702i 1.17757i 0.808289 + 0.588786i $$0.200394\pi$$
−0.808289 + 0.588786i $$0.799606\pi$$
$$264$$ 0 0
$$265$$ 298.001 1.12453
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −490.940 −1.82505 −0.912527 0.409016i $$-0.865872\pi$$
−0.912527 + 0.409016i $$0.865872\pi$$
$$270$$ 0 0
$$271$$ − 0.493459i − 0.00182088i −1.00000 0.000910442i $$-0.999710\pi$$
1.00000 0.000910442i $$-0.000289803\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 211.533i 0.769210i
$$276$$ 0 0
$$277$$ −10.8497 −0.0391686 −0.0195843 0.999808i $$-0.506234\pi$$
−0.0195843 + 0.999808i $$0.506234\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −351.417 −1.25059 −0.625297 0.780387i $$-0.715022\pi$$
−0.625297 + 0.780387i $$0.715022\pi$$
$$282$$ 0 0
$$283$$ − 62.5357i − 0.220974i −0.993878 0.110487i $$-0.964759\pi$$
0.993878 0.110487i $$-0.0352411\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 550.828i − 1.91926i
$$288$$ 0 0
$$289$$ 709.042 2.45343
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 74.3035 0.253596 0.126798 0.991929i $$-0.459530\pi$$
0.126798 + 0.991929i $$0.459530\pi$$
$$294$$ 0 0
$$295$$ − 91.2919i − 0.309464i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 121.638i − 0.406815i
$$300$$ 0 0
$$301$$ 225.508 0.749195
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −409.320 −1.34203
$$306$$ 0 0
$$307$$ − 460.555i − 1.50018i −0.661336 0.750089i $$-0.730010\pi$$
0.661336 0.750089i $$-0.269990\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 433.687i 1.39449i 0.716832 + 0.697246i $$0.245591\pi$$
−0.716832 + 0.697246i $$0.754409\pi$$
$$312$$ 0 0
$$313$$ −434.686 −1.38877 −0.694387 0.719602i $$-0.744324\pi$$
−0.694387 + 0.719602i $$0.744324\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −212.120 −0.669149 −0.334574 0.942369i $$-0.608593\pi$$
−0.334574 + 0.942369i $$0.608593\pi$$
$$318$$ 0 0
$$319$$ − 553.415i − 1.73484i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 520.870i 1.61260i
$$324$$ 0 0
$$325$$ 116.267 0.357744
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −799.457 −2.42996
$$330$$ 0 0
$$331$$ 578.614i 1.74808i 0.485854 + 0.874040i $$0.338509\pi$$
−0.485854 + 0.874040i $$0.661491\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 209.087i − 0.624140i
$$336$$ 0 0
$$337$$ −214.736 −0.637198 −0.318599 0.947890i $$-0.603212\pi$$
−0.318599 + 0.947890i $$0.603212\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 386.925 1.13468
$$342$$ 0 0
$$343$$ 1280.51i 3.73326i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 280.616i − 0.808692i −0.914606 0.404346i $$-0.867499\pi$$
0.914606 0.404346i $$-0.132501\pi$$
$$348$$ 0 0
$$349$$ 232.447 0.666036 0.333018 0.942920i $$-0.391933\pi$$
0.333018 + 0.942920i $$0.391933\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −97.0279 −0.274866 −0.137433 0.990511i $$-0.543885\pi$$
−0.137433 + 0.990511i $$0.543885\pi$$
$$354$$ 0 0
$$355$$ − 144.302i − 0.406485i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 37.7190i − 0.105067i −0.998619 0.0525334i $$-0.983270\pi$$
0.998619 0.0525334i $$-0.0167296\pi$$
$$360$$ 0 0
$$361$$ 89.1625 0.246988
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 160.043 0.438473
$$366$$ 0 0
$$367$$ − 303.506i − 0.826991i −0.910506 0.413496i $$-0.864308\pi$$
0.910506 0.413496i $$-0.135692\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 640.946i − 1.72762i
$$372$$ 0 0
$$373$$ −576.215 −1.54481 −0.772406 0.635129i $$-0.780947\pi$$
−0.772406 + 0.635129i $$0.780947\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −304.179 −0.806841
$$378$$ 0 0
$$379$$ 36.0808i 0.0952001i 0.998866 + 0.0476000i $$0.0151573\pi$$
−0.998866 + 0.0476000i $$0.984843\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 521.053i 1.36045i 0.733003 + 0.680225i $$0.238118\pi$$
−0.733003 + 0.680225i $$0.761882\pi$$
$$384$$ 0 0
$$385$$ 1155.86 3.00224
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 472.918 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$390$$ 0 0
$$391$$ − 536.358i − 1.37176i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 397.776i 1.00703i
$$396$$ 0 0
$$397$$ −209.784 −0.528423 −0.264212 0.964465i $$-0.585112\pi$$
−0.264212 + 0.964465i $$0.585112\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 664.907 1.65812 0.829062 0.559157i $$-0.188875\pi$$
0.829062 + 0.559157i $$0.188875\pi$$
$$402$$ 0 0
$$403$$ − 212.670i − 0.527716i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 512.323i 1.25878i
$$408$$ 0 0
$$409$$ −573.373 −1.40189 −0.700945 0.713215i $$-0.747238\pi$$
−0.700945 + 0.713215i $$0.747238\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −196.352 −0.475429
$$414$$ 0 0
$$415$$ − 287.594i − 0.692996i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 672.621i 1.60530i 0.596450 + 0.802651i $$0.296578\pi$$
−0.596450 + 0.802651i $$0.703422\pi$$
$$420$$ 0 0
$$421$$ −659.810 −1.56724 −0.783622 0.621238i $$-0.786630\pi$$
−0.783622 + 0.621238i $$0.786630\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 512.675 1.20629
$$426$$ 0 0
$$427$$ 880.372i 2.06176i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 360.903i 0.837361i 0.908134 + 0.418681i $$0.137507\pi$$
−0.908134 + 0.418681i $$0.862493\pi$$
$$432$$ 0 0
$$433$$ −319.790 −0.738546 −0.369273 0.929321i $$-0.620393\pi$$
−0.369273 + 0.929321i $$0.620393\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 279.921 0.640551
$$438$$ 0 0
$$439$$ 80.6635i 0.183744i 0.995771 + 0.0918719i $$0.0292850\pi$$
−0.995771 + 0.0918719i $$0.970715\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 551.362i − 1.24461i −0.782775 0.622305i $$-0.786197\pi$$
0.782775 0.622305i $$-0.213803\pi$$
$$444$$ 0 0
$$445$$ −12.5422 −0.0281847
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −43.0735 −0.0959321 −0.0479661 0.998849i $$-0.515274\pi$$
−0.0479661 + 0.998849i $$0.515274\pi$$
$$450$$ 0 0
$$451$$ 519.907i 1.15279i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 635.309i − 1.39628i
$$456$$ 0 0
$$457$$ 205.936 0.450626 0.225313 0.974286i $$-0.427659\pi$$
0.225313 + 0.974286i $$0.427659\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 650.729 1.41156 0.705780 0.708431i $$-0.250597\pi$$
0.705780 + 0.708431i $$0.250597\pi$$
$$462$$ 0 0
$$463$$ − 333.871i − 0.721103i −0.932739 0.360552i $$-0.882588\pi$$
0.932739 0.360552i $$-0.117412\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 359.357i 0.769502i 0.923020 + 0.384751i $$0.125713\pi$$
−0.923020 + 0.384751i $$0.874287\pi$$
$$468$$ 0 0
$$469$$ −449.708 −0.958865
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −212.849 −0.449998
$$474$$ 0 0
$$475$$ 267.561i 0.563286i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 635.502i − 1.32673i −0.748297 0.663363i $$-0.769128\pi$$
0.748297 0.663363i $$-0.230872\pi$$
$$480$$ 0 0
$$481$$ 281.593 0.585432
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −286.252 −0.590211
$$486$$ 0 0
$$487$$ 233.959i 0.480409i 0.970722 + 0.240205i $$0.0772146\pi$$
−0.970722 + 0.240205i $$0.922785\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 494.173i − 1.00646i −0.864152 0.503231i $$-0.832145\pi$$
0.864152 0.503231i $$-0.167855\pi$$
$$492$$ 0 0
$$493$$ −1341.27 −2.72063
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −310.368 −0.624482
$$498$$ 0 0
$$499$$ 468.260i 0.938396i 0.883093 + 0.469198i $$0.155457\pi$$
−0.883093 + 0.469198i $$0.844543\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 59.2449i − 0.117783i −0.998264 0.0588916i $$-0.981243\pi$$
0.998264 0.0588916i $$-0.0187566\pi$$
$$504$$ 0 0
$$505$$ 888.607 1.75962
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −161.243 −0.316784 −0.158392 0.987376i $$-0.550631\pi$$
−0.158392 + 0.987376i $$0.550631\pi$$
$$510$$ 0 0
$$511$$ − 344.222i − 0.673625i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 125.656i 0.243992i
$$516$$ 0 0
$$517$$ 754.579 1.45953
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 350.872 0.673458 0.336729 0.941602i $$-0.390679\pi$$
0.336729 + 0.941602i $$0.390679\pi$$
$$522$$ 0 0
$$523$$ − 729.342i − 1.39454i −0.716811 0.697268i $$-0.754399\pi$$
0.716811 0.697268i $$-0.245601\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 937.760i − 1.77943i
$$528$$ 0 0
$$529$$ 240.756 0.455115
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 285.762 0.536138
$$534$$ 0 0
$$535$$ 254.426i 0.475562i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 1847.34i − 3.42734i
$$540$$ 0 0
$$541$$ −365.728 −0.676021 −0.338011 0.941142i $$-0.609754\pi$$
−0.338011 + 0.941142i $$0.609754\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 290.233 0.532537
$$546$$ 0 0
$$547$$ − 468.636i − 0.856738i −0.903604 0.428369i $$-0.859088\pi$$
0.903604 0.428369i $$-0.140912\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 699.997i − 1.27041i
$$552$$ 0 0
$$553$$ 855.543 1.54709
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 288.006 0.517066 0.258533 0.966002i $$-0.416761\pi$$
0.258533 + 0.966002i $$0.416761\pi$$
$$558$$ 0 0
$$559$$ 116.990i 0.209285i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 158.644i − 0.281783i −0.990025 0.140892i $$-0.955003\pi$$
0.990025 0.140892i $$-0.0449969\pi$$
$$564$$ 0 0
$$565$$ 718.145 1.27105
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 438.309 0.770315 0.385157 0.922851i $$-0.374147\pi$$
0.385157 + 0.922851i $$0.374147\pi$$
$$570$$ 0 0
$$571$$ 683.282i 1.19664i 0.801257 + 0.598321i $$0.204165\pi$$
−0.801257 + 0.598321i $$0.795835\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 275.517i − 0.479160i
$$576$$ 0 0
$$577$$ −166.370 −0.288337 −0.144168 0.989553i $$-0.546051\pi$$
−0.144168 + 0.989553i $$0.546051\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −618.561 −1.06465
$$582$$ 0 0
$$583$$ 604.966i 1.03768i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1095.17i 1.86570i 0.360265 + 0.932850i $$0.382686\pi$$
−0.360265 + 0.932850i $$0.617314\pi$$
$$588$$ 0 0
$$589$$ 489.409 0.830915
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 131.905 0.222437 0.111218 0.993796i $$-0.464525\pi$$
0.111218 + 0.993796i $$0.464525\pi$$
$$594$$ 0 0
$$595$$ − 2801.38i − 4.70819i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 228.590i 0.381619i 0.981627 + 0.190809i $$0.0611112\pi$$
−0.981627 + 0.190809i $$0.938889\pi$$
$$600$$ 0 0
$$601$$ 945.987 1.57402 0.787010 0.616940i $$-0.211628\pi$$
0.787010 + 0.616940i $$0.211628\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −314.049 −0.519089
$$606$$ 0 0
$$607$$ − 539.641i − 0.889029i −0.895772 0.444514i $$-0.853376\pi$$
0.895772 0.444514i $$-0.146624\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 414.747i − 0.678800i
$$612$$ 0 0
$$613$$ −305.962 −0.499123 −0.249561 0.968359i $$-0.580286\pi$$
−0.249561 + 0.968359i $$0.580286\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 409.095 0.663038 0.331519 0.943449i $$-0.392439\pi$$
0.331519 + 0.943449i $$0.392439\pi$$
$$618$$ 0 0
$$619$$ 453.938i 0.733341i 0.930351 + 0.366670i $$0.119502\pi$$
−0.930351 + 0.366670i $$0.880498\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 26.9759i 0.0433000i
$$624$$ 0 0
$$625$$ −767.351 −1.22776
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1241.68 1.97405
$$630$$ 0 0
$$631$$ 393.383i 0.623428i 0.950176 + 0.311714i $$0.100903\pi$$
−0.950176 + 0.311714i $$0.899097\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 706.709i − 1.11293i
$$636$$ 0 0
$$637$$ −1015.37 −1.59399
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 982.724 1.53311 0.766555 0.642179i $$-0.221969\pi$$
0.766555 + 0.642179i $$0.221969\pi$$
$$642$$ 0 0
$$643$$ − 1078.21i − 1.67684i −0.545025 0.838420i $$-0.683480\pi$$
0.545025 0.838420i $$-0.316520\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 38.4217i 0.0593844i 0.999559 + 0.0296922i $$0.00945271\pi$$
−0.999559 + 0.0296922i $$0.990547\pi$$
$$648$$ 0 0
$$649$$ 185.330 0.285562
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −317.953 −0.486911 −0.243456 0.969912i $$-0.578281\pi$$
−0.243456 + 0.969912i $$0.578281\pi$$
$$654$$ 0 0
$$655$$ − 92.2507i − 0.140841i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 792.921i 1.20322i 0.798790 + 0.601609i $$0.205474\pi$$
−0.798790 + 0.601609i $$0.794526\pi$$
$$660$$ 0 0
$$661$$ −1029.55 −1.55757 −0.778786 0.627290i $$-0.784164\pi$$
−0.778786 + 0.627290i $$0.784164\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1462.01 2.19852
$$666$$ 0 0
$$667$$ 720.811i 1.08068i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 830.952i − 1.23838i
$$672$$ 0 0
$$673$$ −6.46119 −0.00960059 −0.00480029 0.999988i $$-0.501528\pi$$
−0.00480029 + 0.999988i $$0.501528\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 21.6650 0.0320015 0.0160008 0.999872i $$-0.494907\pi$$
0.0160008 + 0.999872i $$0.494907\pi$$
$$678$$ 0 0
$$679$$ 615.676i 0.906740i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 503.008i 0.736469i 0.929733 + 0.368234i $$0.120038\pi$$
−0.929733 + 0.368234i $$0.879962\pi$$
$$684$$ 0 0
$$685$$ −1091.86 −1.59396
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 332.514 0.482603
$$690$$ 0 0
$$691$$ 514.362i 0.744374i 0.928158 + 0.372187i $$0.121392\pi$$
−0.928158 + 0.372187i $$0.878608\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1368.75i 1.96943i
$$696$$ 0 0
$$697$$ 1260.06 1.80783
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 457.404 0.652502 0.326251 0.945283i $$-0.394214\pi$$
0.326251 + 0.945283i $$0.394214\pi$$
$$702$$ 0 0
$$703$$ 648.020i 0.921792i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 1911.23i − 2.70330i
$$708$$ 0 0
$$709$$ 390.207 0.550363 0.275182 0.961392i $$-0.411262\pi$$
0.275182 + 0.961392i $$0.411262\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −503.962 −0.706819
$$714$$ 0 0
$$715$$ 599.646i 0.838665i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 979.508i 1.36232i 0.732134 + 0.681160i $$0.238524\pi$$
−0.732134 + 0.681160i $$0.761476\pi$$
$$720$$ 0 0
$$721$$ 270.263 0.374845
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −688.984 −0.950323
$$726$$ 0 0
$$727$$ 553.802i 0.761764i 0.924624 + 0.380882i $$0.124380\pi$$
−0.924624 + 0.380882i $$0.875620\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 515.865i 0.705697i
$$732$$ 0 0
$$733$$ 250.392 0.341599 0.170800 0.985306i $$-0.445365\pi$$
0.170800 + 0.985306i $$0.445365\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 424.463 0.575934
$$738$$ 0 0
$$739$$ 403.959i 0.546630i 0.961925 + 0.273315i $$0.0881201\pi$$
−0.961925 + 0.273315i $$0.911880\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 142.091i 0.191240i 0.995418 + 0.0956201i $$0.0304834\pi$$
−0.995418 + 0.0956201i $$0.969517\pi$$
$$744$$ 0 0
$$745$$ −542.659 −0.728401
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 547.223 0.730605
$$750$$ 0 0
$$751$$ − 667.242i − 0.888471i −0.895910 0.444235i $$-0.853475\pi$$
0.895910 0.444235i $$-0.146525\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 1158.38i − 1.53428i
$$756$$ 0 0
$$757$$ 970.866 1.28252 0.641259 0.767325i $$-0.278413\pi$$
0.641259 + 0.767325i $$0.278413\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −295.323 −0.388072 −0.194036 0.980994i $$-0.562158\pi$$
−0.194036 + 0.980994i $$0.562158\pi$$
$$762$$ 0 0
$$763$$ − 624.238i − 0.818136i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 101.865i − 0.132809i
$$768$$ 0 0
$$769$$ −552.240 −0.718128 −0.359064 0.933313i $$-0.616904\pi$$
−0.359064 + 0.933313i $$0.616904\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −797.819 −1.03211 −0.516053 0.856556i $$-0.672599\pi$$
−0.516053 + 0.856556i $$0.672599\pi$$
$$774$$ 0 0
$$775$$ − 481.709i − 0.621561i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 657.613i 0.844176i
$$780$$ 0 0
$$781$$ 292.945 0.375090
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 319.165 0.406579
$$786$$ 0 0
$$787$$ 1140.72i 1.44946i 0.689033 + 0.724730i $$0.258035\pi$$
−0.689033 + 0.724730i $$0.741965\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 1544.60i − 1.95272i
$$792$$ 0 0
$$793$$ −456.724 −0.575945
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1453.75 1.82403 0.912016 0.410154i $$-0.134525\pi$$
0.912016 + 0.410154i $$0.134525\pi$$
$$798$$ 0 0
$$799$$ − 1828.81i − 2.28888i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 324.899i 0.404607i
$$804$$ 0 0
$$805$$ −1505.49 −1.87017
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −981.664 −1.21343 −0.606714 0.794920i $$-0.707513\pi$$
−0.606714 + 0.794920i $$0.707513\pi$$
$$810$$ 0 0
$$811$$ 1090.81i 1.34502i 0.740087 + 0.672511i $$0.234784\pi$$
−0.740087 + 0.672511i $$0.765216\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 317.012i − 0.388972i
$$816$$ 0 0
$$817$$ −269.225 −0.329529
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 282.456 0.344039 0.172019 0.985094i $$-0.444971\pi$$
0.172019 + 0.985094i $$0.444971\pi$$
$$822$$ 0 0
$$823$$ 497.382i 0.604352i 0.953252 + 0.302176i $$0.0977131\pi$$
−0.953252 + 0.302176i $$0.902287\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1159.09i 1.40156i 0.713379 + 0.700778i $$0.247164\pi$$
−0.713379 + 0.700778i $$0.752836\pi$$
$$828$$ 0 0
$$829$$ −806.908 −0.973351 −0.486675 0.873583i $$-0.661791\pi$$
−0.486675 + 0.873583i $$0.661791\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4477.25 −5.37485
$$834$$ 0 0
$$835$$ 1037.73i 1.24279i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1045.71i − 1.24638i −0.782071 0.623189i $$-0.785837\pi$$
0.782071 0.623189i $$-0.214163\pi$$
$$840$$ 0 0
$$841$$ 961.530 1.14332
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −755.545 −0.894136
$$846$$ 0 0
$$847$$ 675.462i 0.797476i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 667.289i − 0.784123i
$$852$$ 0 0
$$853$$ 1126.10 1.32017 0.660083 0.751193i $$-0.270521\pi$$
0.660083 + 0.751193i $$0.270521\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −112.853 −0.131684 −0.0658419 0.997830i $$-0.520973\pi$$
−0.0658419 + 0.997830i $$0.520973\pi$$
$$858$$ 0 0
$$859$$ 679.360i 0.790873i 0.918493 + 0.395437i $$0.129407\pi$$
−0.918493 + 0.395437i $$0.870593\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 560.002i 0.648901i 0.945903 + 0.324451i $$0.105179\pi$$
−0.945903 + 0.324451i $$0.894821\pi$$
$$864$$ 0 0
$$865$$ −408.408 −0.472147
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −807.517 −0.929249
$$870$$ 0 0
$$871$$ − 233.302i − 0.267855i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 777.840i 0.888960i
$$876$$ 0 0
$$877$$ −775.431 −0.884186 −0.442093 0.896969i $$-0.645764\pi$$
−0.442093 + 0.896969i $$0.645764\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1024.25 −1.16260 −0.581298 0.813691i $$-0.697455\pi$$
−0.581298 + 0.813691i $$0.697455\pi$$
$$882$$ 0 0
$$883$$ − 1301.28i − 1.47370i −0.676057 0.736850i $$-0.736313\pi$$
0.676057 0.736850i $$-0.263687\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1284.02i − 1.44760i −0.690010 0.723800i $$-0.742394\pi$$
0.690010 0.723800i $$-0.257606\pi$$
$$888$$ 0 0
$$889$$ −1520.00 −1.70979
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 954.442 1.06880
$$894$$ 0 0
$$895$$ − 1308.46i − 1.46196i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1260.26i 1.40184i
$$900$$ 0 0
$$901$$ 1466.21 1.62731
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 148.892 0.164522
$$906$$ 0 0
$$907$$ − 1042.84i − 1.14976i −0.818236 0.574882i $$-0.805048\pi$$
0.818236 0.574882i $$-0.194952\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 857.109i 0.940844i 0.882442 + 0.470422i $$0.155898\pi$$
−0.882442 + 0.470422i $$0.844102\pi$$
$$912$$ 0 0
$$913$$ 583.838 0.639472
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −198.414 −0.216373
$$918$$ 0 0
$$919$$ 451.721i 0.491536i 0.969329 + 0.245768i $$0.0790401\pi$$
−0.969329 + 0.245768i $$0.920960\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 161.014i − 0.174447i
$$924$$ 0 0
$$925$$ 637.825 0.689541
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1826.21 1.96578 0.982890 0.184191i $$-0.0589666\pi$$
0.982890 + 0.184191i $$0.0589666\pi$$
$$930$$ 0 0
$$931$$ − 2336.64i − 2.50981i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2644.12i 2.82794i
$$936$$ 0 0
$$937$$ −264.729 −0.282529 −0.141264 0.989972i $$-0.545117\pi$$
−0.141264 + 0.989972i $$0.545117\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1401.16 −1.48901 −0.744505 0.667616i $$-0.767315\pi$$
−0.744505 + 0.667616i $$0.767315\pi$$
$$942$$ 0 0
$$943$$ − 677.167i − 0.718099i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 389.952i − 0.411776i −0.978576 0.205888i $$-0.933992\pi$$
0.978576 0.205888i $$-0.0660082\pi$$
$$948$$ 0 0
$$949$$ 178.578 0.188175
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1291.61 −1.35531 −0.677656 0.735379i $$-0.737004\pi$$
−0.677656 + 0.735379i $$0.737004\pi$$
$$954$$ 0 0
$$955$$ 1586.08i 1.66081i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 2348.40i 2.44880i
$$960$$ 0 0
$$961$$ 79.8811 0.0831229
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 814.256 0.843789
$$966$$ 0 0
$$967$$ − 1802.57i − 1.86408i −0.362349 0.932042i $$-0.618025\pi$$
0.362349 0.932042i $$-0.381975\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 473.260i − 0.487395i −0.969851 0.243697i $$-0.921640\pi$$
0.969851 0.243697i $$-0.0783603\pi$$
$$972$$ 0 0
$$973$$ 2943.93 3.02563
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 656.147 0.671594 0.335797 0.941934i $$-0.390994\pi$$
0.335797 + 0.941934i $$0.390994\pi$$
$$978$$ 0 0
$$979$$ − 25.4616i − 0.0260078i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 267.000i − 0.271617i −0.990735 0.135809i $$-0.956637\pi$$
0.990735 0.135809i $$-0.0433632\pi$$
$$984$$ 0 0
$$985$$ 302.954 0.307568
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 277.231 0.280314
$$990$$ 0 0
$$991$$ − 549.499i − 0.554489i −0.960799 0.277245i $$-0.910579\pi$$
0.960799 0.277245i $$-0.0894213\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1918.79i 1.92844i
$$996$$ 0 0
$$997$$ 546.254 0.547898 0.273949 0.961744i $$-0.411670\pi$$
0.273949 + 0.961744i $$0.411670\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 864.3.g.d.703.7 yes 8
3.2 odd 2 864.3.g.b.703.1 8
4.3 odd 2 inner 864.3.g.d.703.8 yes 8
8.3 odd 2 1728.3.g.j.703.2 8
8.5 even 2 1728.3.g.j.703.1 8
12.11 even 2 864.3.g.b.703.2 yes 8
24.5 odd 2 1728.3.g.m.703.7 8
24.11 even 2 1728.3.g.m.703.8 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.1 8 3.2 odd 2
864.3.g.b.703.2 yes 8 12.11 even 2
864.3.g.d.703.7 yes 8 1.1 even 1 trivial
864.3.g.d.703.8 yes 8 4.3 odd 2 inner
1728.3.g.j.703.1 8 8.5 even 2
1728.3.g.j.703.2 8 8.3 odd 2
1728.3.g.m.703.7 8 24.5 odd 2
1728.3.g.m.703.8 8 24.11 even 2