# Properties

 Label 1728.3.g.j.703.2 Level $1728$ Weight $3$ Character 1728.703 Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ 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: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 703.2 Root $$0.500000 - 0.564882i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.703 Dual form 1728.3.g.j.703.1

## $q$-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 $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$-0.721933\pi$$
−0.642091 + 0.766628i $$0.721933\pi$$
$$6$$ 0 0
$$7$$ 13.8102i 1.97289i 0.164102 + 0.986443i $$0.447527\pi$$
−0.164102 + 0.986443i $$0.552473\pi$$
$$8$$ 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 $$-0.588863\pi$$
−0.275559 + 0.961284i $$0.588863\pi$$
$$14$$ 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 $$0.120328\pi$$
−0.929397 + 0.369082i $$0.879672\pi$$
$$24$$ 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 $$-0.238585\pi$$
0.732004 + 0.681301i $$0.238585\pi$$
$$30$$ 0 0
$$31$$ 29.6836i 0.957537i 0.877941 + 0.478769i $$0.158917\pi$$
−0.877941 + 0.478769i $$0.841083\pi$$
$$32$$ 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 $$-0.678233\pi$$
−0.531131 + 0.847289i $$0.678233\pi$$
$$38$$ 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 $$0.211183\pi$$
−0.787872 + 0.615839i $$0.788817\pi$$
$$48$$ 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 $$-0.644256\pi$$
−0.437840 + 0.899053i $$0.644256\pi$$
$$54$$ 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 $$-0.324991\pi$$
0.522524 + 0.852625i $$0.324991\pi$$
$$62$$ 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 $$0.0505904\pi$$
−0.987397 + 0.158266i $$0.949410\pi$$
$$72$$ 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 $$-0.871753\pi$$
0.919927 0.392089i $$-0.128247\pi$$
$$80$$ 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 $$-0.740247\pi$$
−0.685112 + 0.728438i $$0.740247\pi$$
$$102$$ 0 0
$$103$$ − 19.5698i − 0.189998i −0.995477 0.0949991i $$-0.969715\pi$$
0.995477 0.0949991i $$-0.0302848\pi$$
$$104$$ 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 $$-0.566482\pi$$
−0.207345 + 0.978268i $$0.566482\pi$$
$$110$$ 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 $$0.142658\pi$$
−0.901240 + 0.433321i $$0.857342\pi$$
$$128$$ 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 $$-0.408470\pi$$
0.283605 + 0.958941i $$0.408470\pi$$
$$150$$ 0 0
$$151$$ 180.407i 1.19475i 0.801962 + 0.597376i $$0.203790\pi$$
−0.801962 + 0.597376i $$0.796210\pi$$
$$152$$ 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 $$-0.550602\pi$$
−0.158303 + 0.987391i $$0.550602\pi$$
$$158$$ 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 $$-0.839226\pi$$
0.875133 0.483883i $$-0.160774\pi$$
$$168$$ 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 $$-0.441150\pi$$
0.183832 + 0.982958i $$0.441150\pi$$
$$174$$ 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 $$-0.520404\pi$$
−0.0640570 + 0.997946i $$0.520404\pi$$
$$182$$ 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 $$-0.776173\pi$$
0.762793 0.646642i $$-0.223827\pi$$
$$192$$ 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 $$-0.538210\pi$$
−0.119752 + 0.992804i $$0.538210\pi$$
$$198$$ 0 0
$$199$$ − 298.835i − 1.50168i −0.660482 0.750842i $$-0.729648\pi$$
0.660482 0.750842i $$-0.270352\pi$$
$$200$$ 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 $$-0.874161\pi$$
0.922867 0.385118i $$-0.125839\pi$$
$$224$$ 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 $$-0.248857\pi$$
0.709641 + 0.704563i $$0.248857\pi$$
$$230$$ 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 $$-0.942886\pi$$
0.983946 0.178467i $$-0.0571137\pi$$
$$240$$ 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 $$-0.799606\pi$$
0.808289 0.588786i $$-0.200394\pi$$
$$264$$ 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 $$-0.134128\pi$$
0.912527 + 0.409016i $$0.134128\pi$$
$$270$$ 0 0
$$271$$ 0.493459i 0.00182088i 1.00000 0.000910442i $$0.000289803\pi$$
−1.00000 0.000910442i $$0.999710\pi$$
$$272$$ 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 $$-0.493766\pi$$
0.0195843 + 0.999808i $$0.493766\pi$$
$$278$$ 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 $$-0.540470\pi$$
−0.126798 + 0.991929i $$0.540470\pi$$
$$294$$ 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 $$-0.754409\pi$$
0.716832 0.697246i $$-0.245591\pi$$
$$312$$ 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 $$-0.391407\pi$$
0.334574 + 0.942369i $$0.391407\pi$$
$$318$$ 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 $$-0.608067\pi$$
−0.333018 + 0.942920i $$0.608067\pi$$
$$350$$ 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 $$0.0167296\pi$$
−0.998619 + 0.0525334i $$0.983270\pi$$
$$360$$ 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 $$0.135692\pi$$
−0.910506 + 0.413496i $$0.864308\pi$$
$$368$$ 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 $$-0.219053\pi$$
0.772406 + 0.635129i $$0.219053\pi$$
$$374$$ 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 $$-0.761882\pi$$
0.733003 0.680225i $$-0.238118\pi$$
$$384$$ 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 $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$390$$ 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 $$-0.414888\pi$$
0.264212 + 0.964465i $$0.414888\pi$$
$$398$$ 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 $$-0.213370\pi$$
0.783622 + 0.621238i $$0.213370\pi$$
$$422$$ 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 $$-0.862493\pi$$
0.908134 0.418681i $$-0.137507\pi$$
$$432$$ 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 $$-0.970715\pi$$
0.995771 0.0918719i $$-0.0292850\pi$$
$$440$$ 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 $$-0.749403\pi$$
−0.705780 + 0.708431i $$0.749403\pi$$
$$462$$ 0 0
$$463$$ 333.871i 0.721103i 0.932739 + 0.360552i $$0.117412\pi$$
−0.932739 + 0.360552i $$0.882588\pi$$
$$464$$ 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 $$0.230872\pi$$
−0.748297 + 0.663363i $$0.769128\pi$$
$$480$$ 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 $$-0.922785\pi$$
0.970722 0.240205i $$-0.0772146\pi$$
$$488$$ 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 $$0.0187566\pi$$
−0.998264 + 0.0588916i $$0.981243\pi$$
$$504$$ 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 $$-0.449369\pi$$
0.158392 + 0.987376i $$0.449369\pi$$
$$510$$ 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 $$-0.390246\pi$$
0.338011 + 0.941142i $$0.390246\pi$$
$$542$$ 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 $$-0.583239\pi$$
−0.258533 + 0.966002i $$0.583239\pi$$
$$558$$ 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 $$-0.938889\pi$$
0.981627 0.190809i $$-0.0611112\pi$$
$$600$$ 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 $$0.146624\pi$$
−0.895772 + 0.444514i $$0.853376\pi$$
$$608$$ 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 $$-0.419714\pi$$
0.249561 + 0.968359i $$0.419714\pi$$
$$614$$ 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 $$-0.899097\pi$$
0.950176 0.311714i $$-0.100903\pi$$
$$632$$ 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 $$-0.990547\pi$$
0.999559 0.0296922i $$-0.00945271\pi$$
$$648$$ 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 $$-0.421719\pi$$
0.243456 + 0.969912i $$0.421719\pi$$
$$654$$ 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 $$-0.215836\pi$$
0.778786 + 0.627290i $$0.215836\pi$$
$$662$$ 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 $$-0.505093\pi$$
−0.0160008 + 0.999872i $$0.505093\pi$$
$$678$$ 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 $$-0.605786\pi$$
−0.326251 + 0.945283i $$0.605786\pi$$
$$702$$ 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 $$-0.588738\pi$$
−0.275182 + 0.961392i $$0.588738\pi$$
$$710$$ 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 $$-0.761476\pi$$
0.732134 0.681160i $$-0.238524\pi$$
$$720$$ 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 $$-0.875620\pi$$
0.924624 0.380882i $$-0.124380\pi$$
$$728$$ 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 $$-0.554635\pi$$
−0.170800 + 0.985306i $$0.554635\pi$$
$$734$$ 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 $$-0.969517\pi$$
0.995418 0.0956201i $$-0.0304834\pi$$
$$744$$ 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 $$0.146525\pi$$
−0.895910 + 0.444235i $$0.853475\pi$$
$$752$$ 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 $$-0.721587\pi$$
−0.641259 + 0.767325i $$0.721587\pi$$
$$758$$ 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 $$-0.327401\pi$$
0.516053 + 0.856556i $$0.327401\pi$$
$$774$$ 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 $$-0.865475\pi$$
−0.912016 + 0.410154i $$0.865475\pi$$
$$798$$ 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 $$-0.555029\pi$$
−0.172019 + 0.985094i $$0.555029\pi$$
$$822$$ 0 0
$$823$$ − 497.382i − 0.604352i −0.953252 0.302176i $$-0.902287\pi$$
0.953252 0.302176i $$-0.0977131\pi$$
$$824$$ 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 $$-0.338209\pi$$
0.486675 + 0.873583i $$0.338209\pi$$
$$830$$ 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 $$0.214163\pi$$
−0.782071 + 0.623189i $$0.785837\pi$$
$$840$$ 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 $$-0.729479\pi$$
−0.660083 + 0.751193i $$0.729479\pi$$
$$854$$ 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 $$-0.894821\pi$$
0.945903 0.324451i $$-0.105179\pi$$
$$864$$ 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 $$-0.354236\pi$$
0.442093 + 0.896969i $$0.354236\pi$$
$$878$$ 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 $$0.257606\pi$$
−0.690010 + 0.723800i $$0.742394\pi$$
$$888$$ 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 $$-0.844102\pi$$
0.882442 0.470422i $$-0.155898\pi$$
$$912$$ 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 $$-0.920960\pi$$
0.969329 0.245768i $$-0.0790401\pi$$
$$920$$ 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 $$-0.232685\pi$$
0.744505 + 0.667616i $$0.232685\pi$$
$$942$$ 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 $$0.381975\pi$$
−0.362349 + 0.932042i $$0.618025\pi$$
$$968$$ 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 $$0.0433632\pi$$
−0.990735 + 0.135809i $$0.956637\pi$$
$$984$$ 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 $$0.0894213\pi$$
−0.960799 + 0.277245i $$0.910579\pi$$
$$992$$ 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 $$-0.588330\pi$$
−0.273949 + 0.961744i $$0.588330\pi$$
$$998$$ 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 1728.3.g.j.703.2 8
3.2 odd 2 1728.3.g.m.703.8 8
4.3 odd 2 inner 1728.3.g.j.703.1 8
8.3 odd 2 864.3.g.d.703.7 yes 8
8.5 even 2 864.3.g.d.703.8 yes 8
12.11 even 2 1728.3.g.m.703.7 8
24.5 odd 2 864.3.g.b.703.2 yes 8
24.11 even 2 864.3.g.b.703.1 8

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