# Properties

 Label 2352.3.m.f.1471.4 Level $2352$ Weight $3$ Character 2352.1471 Analytic conductor $64.087$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2352.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.0873581775$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1471.4 Root $$1.39564 - 0.228425i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1471 Dual form 2352.3.m.f.1471.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.73205i q^{3} +3.58258 q^{5} -3.00000 q^{9} +O(q^{10})$$ $$q+1.73205i q^{3} +3.58258 q^{5} -3.00000 q^{9} +0.913701i q^{11} +1.16515 q^{13} +6.20520i q^{15} +26.7477 q^{17} -17.5112i q^{19} +27.1805i q^{23} -12.1652 q^{25} -5.19615i q^{27} -2.00000 q^{29} -45.6054i q^{31} -1.58258 q^{33} +47.4955 q^{37} +2.01810i q^{39} +42.5735 q^{41} +14.6192i q^{43} -10.7477 q^{45} +8.37420i q^{47} +46.3284i q^{51} -41.8258 q^{53} +3.27340i q^{55} +30.3303 q^{57} +27.0296i q^{59} +11.0091 q^{61} +4.17424 q^{65} -71.8722i q^{67} -47.0780 q^{69} -55.6561i q^{71} +95.4955 q^{73} -21.0707i q^{75} +63.7998i q^{79} +9.00000 q^{81} +32.5118i q^{83} +95.8258 q^{85} -3.46410i q^{87} +120.904 q^{89} +78.9909 q^{93} -62.7352i q^{95} -107.495 q^{97} -2.74110i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{5} - 12q^{9} - 32q^{13} + 52q^{17} - 12q^{25} - 8q^{29} + 12q^{33} + 80q^{37} - 68q^{41} + 12q^{45} + 16q^{53} + 48q^{57} + 264q^{61} + 200q^{65} - 60q^{69} + 272q^{73} + 36q^{81} + 200q^{85} + 172q^{89} + 96q^{93} - 320q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ 1.73205i 0.577350i
$$4$$ 0 0
$$5$$ 3.58258 0.716515 0.358258 0.933623i $$-0.383371\pi$$
0.358258 + 0.933623i $$0.383371\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ 0.913701i 0.0830637i 0.999137 + 0.0415318i $$0.0132238\pi$$
−0.999137 + 0.0415318i $$0.986776\pi$$
$$12$$ 0 0
$$13$$ 1.16515 0.0896270 0.0448135 0.998995i $$-0.485731\pi$$
0.0448135 + 0.998995i $$0.485731\pi$$
$$14$$ 0 0
$$15$$ 6.20520i 0.413680i
$$16$$ 0 0
$$17$$ 26.7477 1.57340 0.786698 0.617338i $$-0.211789\pi$$
0.786698 + 0.617338i $$0.211789\pi$$
$$18$$ 0 0
$$19$$ − 17.5112i − 0.921643i −0.887493 0.460821i $$-0.847555\pi$$
0.887493 0.460821i $$-0.152445\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 27.1805i 1.18176i 0.806759 + 0.590881i $$0.201220\pi$$
−0.806759 + 0.590881i $$0.798780\pi$$
$$24$$ 0 0
$$25$$ −12.1652 −0.486606
$$26$$ 0 0
$$27$$ − 5.19615i − 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.0689655 −0.0344828 0.999405i $$-0.510978\pi$$
−0.0344828 + 0.999405i $$0.510978\pi$$
$$30$$ 0 0
$$31$$ − 45.6054i − 1.47114i −0.677447 0.735571i $$-0.736914\pi$$
0.677447 0.735571i $$-0.263086\pi$$
$$32$$ 0 0
$$33$$ −1.58258 −0.0479568
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 47.4955 1.28366 0.641830 0.766847i $$-0.278175\pi$$
0.641830 + 0.766847i $$0.278175\pi$$
$$38$$ 0 0
$$39$$ 2.01810i 0.0517462i
$$40$$ 0 0
$$41$$ 42.5735 1.03838 0.519189 0.854660i $$-0.326234\pi$$
0.519189 + 0.854660i $$0.326234\pi$$
$$42$$ 0 0
$$43$$ 14.6192i 0.339982i 0.985446 + 0.169991i $$0.0543738\pi$$
−0.985446 + 0.169991i $$0.945626\pi$$
$$44$$ 0 0
$$45$$ −10.7477 −0.238838
$$46$$ 0 0
$$47$$ 8.37420i 0.178175i 0.996024 + 0.0890873i $$0.0283950\pi$$
−0.996024 + 0.0890873i $$0.971605\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 46.3284i 0.908400i
$$52$$ 0 0
$$53$$ −41.8258 −0.789165 −0.394583 0.918860i $$-0.629111\pi$$
−0.394583 + 0.918860i $$0.629111\pi$$
$$54$$ 0 0
$$55$$ 3.27340i 0.0595164i
$$56$$ 0 0
$$57$$ 30.3303 0.532111
$$58$$ 0 0
$$59$$ 27.0296i 0.458129i 0.973411 + 0.229065i $$0.0735667\pi$$
−0.973411 + 0.229065i $$0.926433\pi$$
$$60$$ 0 0
$$61$$ 11.0091 0.180477 0.0902385 0.995920i $$-0.471237\pi$$
0.0902385 + 0.995920i $$0.471237\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.17424 0.0642191
$$66$$ 0 0
$$67$$ − 71.8722i − 1.07272i −0.843989 0.536360i $$-0.819799\pi$$
0.843989 0.536360i $$-0.180201\pi$$
$$68$$ 0 0
$$69$$ −47.0780 −0.682290
$$70$$ 0 0
$$71$$ − 55.6561i − 0.783889i −0.919989 0.391945i $$-0.871803\pi$$
0.919989 0.391945i $$-0.128197\pi$$
$$72$$ 0 0
$$73$$ 95.4955 1.30816 0.654078 0.756427i $$-0.273057\pi$$
0.654078 + 0.756427i $$0.273057\pi$$
$$74$$ 0 0
$$75$$ − 21.0707i − 0.280942i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 63.7998i 0.807593i 0.914849 + 0.403796i $$0.132310\pi$$
−0.914849 + 0.403796i $$0.867690\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ 32.5118i 0.391709i 0.980633 + 0.195854i $$0.0627480\pi$$
−0.980633 + 0.195854i $$0.937252\pi$$
$$84$$ 0 0
$$85$$ 95.8258 1.12736
$$86$$ 0 0
$$87$$ − 3.46410i − 0.0398173i
$$88$$ 0 0
$$89$$ 120.904 1.35847 0.679235 0.733921i $$-0.262312\pi$$
0.679235 + 0.733921i $$0.262312\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 78.9909 0.849365
$$94$$ 0 0
$$95$$ − 62.7352i − 0.660371i
$$96$$ 0 0
$$97$$ −107.495 −1.10820 −0.554100 0.832450i $$-0.686938\pi$$
−0.554100 + 0.832450i $$0.686938\pi$$
$$98$$ 0 0
$$99$$ − 2.74110i − 0.0276879i
$$100$$ 0 0
$$101$$ 104.417 1.03384 0.516918 0.856035i $$-0.327079\pi$$
0.516918 + 0.856035i $$0.327079\pi$$
$$102$$ 0 0
$$103$$ 119.305i 1.15830i 0.815220 + 0.579151i $$0.196616\pi$$
−0.815220 + 0.579151i $$0.803384\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 67.0019i − 0.626186i −0.949722 0.313093i $$-0.898635\pi$$
0.949722 0.313093i $$-0.101365\pi$$
$$108$$ 0 0
$$109$$ 161.303 1.47984 0.739922 0.672693i $$-0.234862\pi$$
0.739922 + 0.672693i $$0.234862\pi$$
$$110$$ 0 0
$$111$$ 82.2645i 0.741122i
$$112$$ 0 0
$$113$$ 32.6788 0.289193 0.144596 0.989491i $$-0.453812\pi$$
0.144596 + 0.989491i $$0.453812\pi$$
$$114$$ 0 0
$$115$$ 97.3762i 0.846750i
$$116$$ 0 0
$$117$$ −3.49545 −0.0298757
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 120.165 0.993100
$$122$$ 0 0
$$123$$ 73.7394i 0.599508i
$$124$$ 0 0
$$125$$ −133.147 −1.06518
$$126$$ 0 0
$$127$$ 56.4902i 0.444805i 0.974955 + 0.222402i $$0.0713899\pi$$
−0.974955 + 0.222402i $$0.928610\pi$$
$$128$$ 0 0
$$129$$ −25.3212 −0.196288
$$130$$ 0 0
$$131$$ 140.471i 1.07230i 0.844123 + 0.536149i $$0.180122\pi$$
−0.844123 + 0.536149i $$0.819878\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ − 18.6156i − 0.137893i
$$136$$ 0 0
$$137$$ −228.156 −1.66537 −0.832686 0.553745i $$-0.813198\pi$$
−0.832686 + 0.553745i $$0.813198\pi$$
$$138$$ 0 0
$$139$$ − 176.256i − 1.26803i −0.773320 0.634015i $$-0.781406\pi$$
0.773320 0.634015i $$-0.218594\pi$$
$$140$$ 0 0
$$141$$ −14.5045 −0.102869
$$142$$ 0 0
$$143$$ 1.06460i 0.00744475i
$$144$$ 0 0
$$145$$ −7.16515 −0.0494148
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 274.835 1.84453 0.922265 0.386559i $$-0.126337\pi$$
0.922265 + 0.386559i $$0.126337\pi$$
$$150$$ 0 0
$$151$$ − 175.875i − 1.16473i −0.812926 0.582367i $$-0.802127\pi$$
0.812926 0.582367i $$-0.197873\pi$$
$$152$$ 0 0
$$153$$ −80.2432 −0.524465
$$154$$ 0 0
$$155$$ − 163.385i − 1.05410i
$$156$$ 0 0
$$157$$ −55.9818 −0.356572 −0.178286 0.983979i $$-0.557055\pi$$
−0.178286 + 0.983979i $$0.557055\pi$$
$$158$$ 0 0
$$159$$ − 72.4443i − 0.455625i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 237.386i 1.45636i 0.685387 + 0.728179i $$0.259633\pi$$
−0.685387 + 0.728179i $$0.740367\pi$$
$$164$$ 0 0
$$165$$ −5.66970 −0.0343618
$$166$$ 0 0
$$167$$ − 237.386i − 1.42147i −0.703457 0.710737i $$-0.748361\pi$$
0.703457 0.710737i $$-0.251639\pi$$
$$168$$ 0 0
$$169$$ −167.642 −0.991967
$$170$$ 0 0
$$171$$ 52.5336i 0.307214i
$$172$$ 0 0
$$173$$ 173.078 1.00045 0.500226 0.865895i $$-0.333250\pi$$
0.500226 + 0.865895i $$0.333250\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −46.8167 −0.264501
$$178$$ 0 0
$$179$$ 136.284i 0.761363i 0.924706 + 0.380681i $$0.124311\pi$$
−0.924706 + 0.380681i $$0.875689\pi$$
$$180$$ 0 0
$$181$$ −80.5045 −0.444776 −0.222388 0.974958i $$-0.571385\pi$$
−0.222388 + 0.974958i $$0.571385\pi$$
$$182$$ 0 0
$$183$$ 19.0683i 0.104198i
$$184$$ 0 0
$$185$$ 170.156 0.919762
$$186$$ 0 0
$$187$$ 24.4394i 0.130692i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 131.183i 0.686823i 0.939185 + 0.343411i $$0.111583\pi$$
−0.939185 + 0.343411i $$0.888417\pi$$
$$192$$ 0 0
$$193$$ −171.495 −0.888577 −0.444289 0.895884i $$-0.646544\pi$$
−0.444289 + 0.895884i $$0.646544\pi$$
$$194$$ 0 0
$$195$$ 7.23000i 0.0370769i
$$196$$ 0 0
$$197$$ 57.1652 0.290178 0.145089 0.989419i $$-0.453653\pi$$
0.145089 + 0.989419i $$0.453653\pi$$
$$198$$ 0 0
$$199$$ − 6.70601i − 0.0336985i −0.999858 0.0168493i $$-0.994636\pi$$
0.999858 0.0168493i $$-0.00536354\pi$$
$$200$$ 0 0
$$201$$ 124.486 0.619335
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 152.523 0.744013
$$206$$ 0 0
$$207$$ − 81.5415i − 0.393920i
$$208$$ 0 0
$$209$$ 16.0000 0.0765550
$$210$$ 0 0
$$211$$ 178.767i 0.847236i 0.905841 + 0.423618i $$0.139240\pi$$
−0.905841 + 0.423618i $$0.860760\pi$$
$$212$$ 0 0
$$213$$ 96.3992 0.452579
$$214$$ 0 0
$$215$$ 52.3744i 0.243602i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 165.403i 0.755265i
$$220$$ 0 0
$$221$$ 31.1652 0.141019
$$222$$ 0 0
$$223$$ 114.506i 0.513480i 0.966480 + 0.256740i $$0.0826484\pi$$
−0.966480 + 0.256740i $$0.917352\pi$$
$$224$$ 0 0
$$225$$ 36.4955 0.162202
$$226$$ 0 0
$$227$$ 307.813i 1.35600i 0.735061 + 0.678001i $$0.237154\pi$$
−0.735061 + 0.678001i $$0.762846\pi$$
$$228$$ 0 0
$$229$$ 356.156 1.55527 0.777633 0.628718i $$-0.216420\pi$$
0.777633 + 0.628718i $$0.216420\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 213.514 0.916368 0.458184 0.888857i $$-0.348500\pi$$
0.458184 + 0.888857i $$0.348500\pi$$
$$234$$ 0 0
$$235$$ 30.0012i 0.127665i
$$236$$ 0 0
$$237$$ −110.505 −0.466264
$$238$$ 0 0
$$239$$ 402.988i 1.68614i 0.537801 + 0.843072i $$0.319255\pi$$
−0.537801 + 0.843072i $$0.680745\pi$$
$$240$$ 0 0
$$241$$ 344.468 1.42933 0.714664 0.699468i $$-0.246579\pi$$
0.714664 + 0.699468i $$0.246579\pi$$
$$242$$ 0 0
$$243$$ 15.5885i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 20.4032i − 0.0826041i
$$248$$ 0 0
$$249$$ −56.3121 −0.226153
$$250$$ 0 0
$$251$$ − 491.601i − 1.95857i −0.202491 0.979284i $$-0.564904\pi$$
0.202491 0.979284i $$-0.435096\pi$$
$$252$$ 0 0
$$253$$ −24.8348 −0.0981615
$$254$$ 0 0
$$255$$ 165.975i 0.650883i
$$256$$ 0 0
$$257$$ −335.372 −1.30495 −0.652475 0.757811i $$-0.726269\pi$$
−0.652475 + 0.757811i $$0.726269\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.0229885
$$262$$ 0 0
$$263$$ 484.601i 1.84259i 0.388865 + 0.921295i $$0.372867\pi$$
−0.388865 + 0.921295i $$0.627133\pi$$
$$264$$ 0 0
$$265$$ −149.844 −0.565449
$$266$$ 0 0
$$267$$ 209.412i 0.784313i
$$268$$ 0 0
$$269$$ 27.9311 0.103833 0.0519165 0.998651i $$-0.483467\pi$$
0.0519165 + 0.998651i $$0.483467\pi$$
$$270$$ 0 0
$$271$$ − 16.5262i − 0.0609823i −0.999535 0.0304912i $$-0.990293\pi$$
0.999535 0.0304912i $$-0.00970714\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 11.1153i − 0.0404193i
$$276$$ 0 0
$$277$$ −359.459 −1.29769 −0.648843 0.760922i $$-0.724747\pi$$
−0.648843 + 0.760922i $$0.724747\pi$$
$$278$$ 0 0
$$279$$ 136.816i 0.490381i
$$280$$ 0 0
$$281$$ 259.147 0.922231 0.461116 0.887340i $$-0.347449\pi$$
0.461116 + 0.887340i $$0.347449\pi$$
$$282$$ 0 0
$$283$$ 177.241i 0.626294i 0.949705 + 0.313147i $$0.101383\pi$$
−0.949705 + 0.313147i $$0.898617\pi$$
$$284$$ 0 0
$$285$$ 108.661 0.381265
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 426.441 1.47557
$$290$$ 0 0
$$291$$ − 186.188i − 0.639820i
$$292$$ 0 0
$$293$$ 296.069 1.01047 0.505237 0.862981i $$-0.331405\pi$$
0.505237 + 0.862981i $$0.331405\pi$$
$$294$$ 0 0
$$295$$ 96.8356i 0.328256i
$$296$$ 0 0
$$297$$ 4.74773 0.0159856
$$298$$ 0 0
$$299$$ 31.6694i 0.105918i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 180.856i 0.596885i
$$304$$ 0 0
$$305$$ 39.4409 0.129314
$$306$$ 0 0
$$307$$ 170.854i 0.556527i 0.960505 + 0.278263i $$0.0897588\pi$$
−0.960505 + 0.278263i $$0.910241\pi$$
$$308$$ 0 0
$$309$$ −206.642 −0.668746
$$310$$ 0 0
$$311$$ − 419.967i − 1.35038i −0.737645 0.675188i $$-0.764062\pi$$
0.737645 0.675188i $$-0.235938\pi$$
$$312$$ 0 0
$$313$$ 238.624 0.762378 0.381189 0.924497i $$-0.375515\pi$$
0.381189 + 0.924497i $$0.375515\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 585.441 1.84682 0.923408 0.383819i $$-0.125391\pi$$
0.923408 + 0.383819i $$0.125391\pi$$
$$318$$ 0 0
$$319$$ − 1.82740i − 0.00572853i
$$320$$ 0 0
$$321$$ 116.051 0.361529
$$322$$ 0 0
$$323$$ − 468.385i − 1.45011i
$$324$$ 0 0
$$325$$ −14.1742 −0.0436131
$$326$$ 0 0
$$327$$ 279.385i 0.854389i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 71.4112i 0.215744i 0.994165 + 0.107872i $$0.0344037\pi$$
−0.994165 + 0.107872i $$0.965596\pi$$
$$332$$ 0 0
$$333$$ −142.486 −0.427887
$$334$$ 0 0
$$335$$ − 257.488i − 0.768620i
$$336$$ 0 0
$$337$$ 416.955 1.23725 0.618627 0.785685i $$-0.287689\pi$$
0.618627 + 0.785685i $$0.287689\pi$$
$$338$$ 0 0
$$339$$ 56.6013i 0.166966i
$$340$$ 0 0
$$341$$ 41.6697 0.122199
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −168.661 −0.488871
$$346$$ 0 0
$$347$$ − 546.812i − 1.57583i −0.615785 0.787914i $$-0.711161\pi$$
0.615785 0.787914i $$-0.288839\pi$$
$$348$$ 0 0
$$349$$ 97.6151 0.279700 0.139850 0.990173i $$-0.455338\pi$$
0.139850 + 0.990173i $$0.455338\pi$$
$$350$$ 0 0
$$351$$ − 6.05430i − 0.0172487i
$$352$$ 0 0
$$353$$ −315.858 −0.894783 −0.447391 0.894338i $$-0.647647\pi$$
−0.447391 + 0.894338i $$0.647647\pi$$
$$354$$ 0 0
$$355$$ − 199.392i − 0.561668i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 31.2797i − 0.0871301i −0.999051 0.0435650i $$-0.986128\pi$$
0.999051 0.0435650i $$-0.0138716\pi$$
$$360$$ 0 0
$$361$$ 54.3576 0.150575
$$362$$ 0 0
$$363$$ 208.132i 0.573367i
$$364$$ 0 0
$$365$$ 342.120 0.937314
$$366$$ 0 0
$$367$$ 501.437i 1.36631i 0.730271 + 0.683157i $$0.239394\pi$$
−0.730271 + 0.683157i $$0.760606\pi$$
$$368$$ 0 0
$$369$$ −127.720 −0.346126
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −257.267 −0.689723 −0.344861 0.938654i $$-0.612074\pi$$
−0.344861 + 0.938654i $$0.612074\pi$$
$$374$$ 0 0
$$375$$ − 230.617i − 0.614980i
$$376$$ 0 0
$$377$$ −2.33030 −0.00618117
$$378$$ 0 0
$$379$$ − 169.169i − 0.446356i −0.974778 0.223178i $$-0.928357\pi$$
0.974778 0.223178i $$-0.0716431\pi$$
$$380$$ 0 0
$$381$$ −97.8439 −0.256808
$$382$$ 0 0
$$383$$ 436.858i 1.14062i 0.821429 + 0.570311i $$0.193177\pi$$
−0.821429 + 0.570311i $$0.806823\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 43.8576i − 0.113327i
$$388$$ 0 0
$$389$$ −215.982 −0.555223 −0.277612 0.960693i $$-0.589543\pi$$
−0.277612 + 0.960693i $$0.589543\pi$$
$$390$$ 0 0
$$391$$ 727.017i 1.85938i
$$392$$ 0 0
$$393$$ −243.303 −0.619092
$$394$$ 0 0
$$395$$ 228.568i 0.578652i
$$396$$ 0 0
$$397$$ −256.606 −0.646363 −0.323181 0.946337i $$-0.604752\pi$$
−0.323181 + 0.946337i $$0.604752\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 557.441 1.39013 0.695063 0.718948i $$-0.255376\pi$$
0.695063 + 0.718948i $$0.255376\pi$$
$$402$$ 0 0
$$403$$ − 53.1372i − 0.131854i
$$404$$ 0 0
$$405$$ 32.2432 0.0796128
$$406$$ 0 0
$$407$$ 43.3966i 0.106626i
$$408$$ 0 0
$$409$$ −16.1561 −0.0395014 −0.0197507 0.999805i $$-0.506287\pi$$
−0.0197507 + 0.999805i $$0.506287\pi$$
$$410$$ 0 0
$$411$$ − 395.178i − 0.961503i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 116.476i 0.280665i
$$416$$ 0 0
$$417$$ 305.285 0.732098
$$418$$ 0 0
$$419$$ − 582.652i − 1.39058i −0.718730 0.695289i $$-0.755276\pi$$
0.718730 0.695289i $$-0.244724\pi$$
$$420$$ 0 0
$$421$$ −662.762 −1.57426 −0.787128 0.616789i $$-0.788433\pi$$
−0.787128 + 0.616789i $$0.788433\pi$$
$$422$$ 0 0
$$423$$ − 25.1226i − 0.0593915i
$$424$$ 0 0
$$425$$ −325.390 −0.765624
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.84394 −0.00429823
$$430$$ 0 0
$$431$$ 286.336i 0.664354i 0.943217 + 0.332177i $$0.107783\pi$$
−0.943217 + 0.332177i $$0.892217\pi$$
$$432$$ 0 0
$$433$$ −554.900 −1.28152 −0.640762 0.767739i $$-0.721382\pi$$
−0.640762 + 0.767739i $$0.721382\pi$$
$$434$$ 0 0
$$435$$ − 12.4104i − 0.0285297i
$$436$$ 0 0
$$437$$ 475.964 1.08916
$$438$$ 0 0
$$439$$ 364.843i 0.831078i 0.909575 + 0.415539i $$0.136407\pi$$
−0.909575 + 0.415539i $$0.863593\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 445.241i − 1.00506i −0.864560 0.502529i $$-0.832403\pi$$
0.864560 0.502529i $$-0.167597\pi$$
$$444$$ 0 0
$$445$$ 433.147 0.973364
$$446$$ 0 0
$$447$$ 476.028i 1.06494i
$$448$$ 0 0
$$449$$ 256.955 0.572282 0.286141 0.958188i $$-0.407627\pi$$
0.286141 + 0.958188i $$0.407627\pi$$
$$450$$ 0 0
$$451$$ 38.8994i 0.0862515i
$$452$$ 0 0
$$453$$ 304.624 0.672460
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −643.982 −1.40915 −0.704575 0.709629i $$-0.748862\pi$$
−0.704575 + 0.709629i $$0.748862\pi$$
$$458$$ 0 0
$$459$$ − 138.985i − 0.302800i
$$460$$ 0 0
$$461$$ −30.8856 −0.0669970 −0.0334985 0.999439i $$-0.510665\pi$$
−0.0334985 + 0.999439i $$0.510665\pi$$
$$462$$ 0 0
$$463$$ 435.794i 0.941239i 0.882336 + 0.470619i $$0.155970\pi$$
−0.882336 + 0.470619i $$0.844030\pi$$
$$464$$ 0 0
$$465$$ 282.991 0.608583
$$466$$ 0 0
$$467$$ 22.0084i 0.0471272i 0.999722 + 0.0235636i $$0.00750123\pi$$
−0.999722 + 0.0235636i $$0.992499\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ − 96.9634i − 0.205867i
$$472$$ 0 0
$$473$$ −13.3576 −0.0282401
$$474$$ 0 0
$$475$$ 213.027i 0.448477i
$$476$$ 0 0
$$477$$ 125.477 0.263055
$$478$$ 0 0
$$479$$ − 662.677i − 1.38346i −0.722157 0.691729i $$-0.756849\pi$$
0.722157 0.691729i $$-0.243151\pi$$
$$480$$ 0 0
$$481$$ 55.3394 0.115051
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −385.111 −0.794042
$$486$$ 0 0
$$487$$ − 24.5024i − 0.0503129i −0.999684 0.0251565i $$-0.991992\pi$$
0.999684 0.0251565i $$-0.00800840\pi$$
$$488$$ 0 0
$$489$$ −411.165 −0.840829
$$490$$ 0 0
$$491$$ − 684.136i − 1.39335i −0.717386 0.696676i $$-0.754661\pi$$
0.717386 0.696676i $$-0.245339\pi$$
$$492$$ 0 0
$$493$$ −53.4955 −0.108510
$$494$$ 0 0
$$495$$ − 9.82020i − 0.0198388i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 818.768i 1.64082i 0.571777 + 0.820409i $$0.306254\pi$$
−0.571777 + 0.820409i $$0.693746\pi$$
$$500$$ 0 0
$$501$$ 411.165 0.820689
$$502$$ 0 0
$$503$$ − 693.298i − 1.37833i −0.724606 0.689163i $$-0.757978\pi$$
0.724606 0.689163i $$-0.242022\pi$$
$$504$$ 0 0
$$505$$ 374.083 0.740759
$$506$$ 0 0
$$507$$ − 290.365i − 0.572712i
$$508$$ 0 0
$$509$$ 87.5462 0.171996 0.0859982 0.996295i $$-0.472592\pi$$
0.0859982 + 0.996295i $$0.472592\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −90.9909 −0.177370
$$514$$ 0 0
$$515$$ 427.419i 0.829941i
$$516$$ 0 0
$$517$$ −7.65151 −0.0147998
$$518$$ 0 0
$$519$$ 299.780i 0.577611i
$$520$$ 0 0
$$521$$ −190.573 −0.365784 −0.182892 0.983133i $$-0.558546\pi$$
−0.182892 + 0.983133i $$0.558546\pi$$
$$522$$ 0 0
$$523$$ − 767.379i − 1.46726i −0.679547 0.733632i $$-0.737824\pi$$
0.679547 0.733632i $$-0.262176\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1219.84i − 2.31469i
$$528$$ 0 0
$$529$$ −209.780 −0.396560
$$530$$ 0 0
$$531$$ − 81.0888i − 0.152710i
$$532$$ 0 0
$$533$$ 49.6046 0.0930667
$$534$$ 0 0
$$535$$ − 240.040i − 0.448672i
$$536$$ 0 0
$$537$$ −236.051 −0.439573
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −491.495 −0.908494 −0.454247 0.890876i $$-0.650092\pi$$
−0.454247 + 0.890876i $$0.650092\pi$$
$$542$$ 0 0
$$543$$ − 139.438i − 0.256792i
$$544$$ 0 0
$$545$$ 577.880 1.06033
$$546$$ 0 0
$$547$$ − 1043.66i − 1.90798i −0.299839 0.953990i $$-0.596933\pi$$
0.299839 0.953990i $$-0.403067\pi$$
$$548$$ 0 0
$$549$$ −33.0273 −0.0601590
$$550$$ 0 0
$$551$$ 35.0224i 0.0635616i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 294.719i 0.531025i
$$556$$ 0 0
$$557$$ 149.826 0.268987 0.134493 0.990914i $$-0.457059\pi$$
0.134493 + 0.990914i $$0.457059\pi$$
$$558$$ 0 0
$$559$$ 17.0336i 0.0304715i
$$560$$ 0 0
$$561$$ −42.3303 −0.0754551
$$562$$ 0 0
$$563$$ − 863.309i − 1.53341i −0.642000 0.766704i $$-0.721895\pi$$
0.642000 0.766704i $$-0.278105\pi$$
$$564$$ 0 0
$$565$$ 117.074 0.207211
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −765.056 −1.34456 −0.672281 0.740296i $$-0.734685\pi$$
−0.672281 + 0.740296i $$0.734685\pi$$
$$570$$ 0 0
$$571$$ − 124.850i − 0.218652i −0.994006 0.109326i $$-0.965131\pi$$
0.994006 0.109326i $$-0.0348692\pi$$
$$572$$ 0 0
$$573$$ −227.216 −0.396537
$$574$$ 0 0
$$575$$ − 330.655i − 0.575052i
$$576$$ 0 0
$$577$$ −930.900 −1.61334 −0.806672 0.590999i $$-0.798734\pi$$
−0.806672 + 0.590999i $$0.798734\pi$$
$$578$$ 0 0
$$579$$ − 297.039i − 0.513020i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 38.2162i − 0.0655510i
$$584$$ 0 0
$$585$$ −12.5227 −0.0214064
$$586$$ 0 0
$$587$$ − 333.681i − 0.568452i −0.958757 0.284226i $$-0.908263\pi$$
0.958757 0.284226i $$-0.0917366\pi$$
$$588$$ 0 0
$$589$$ −798.606 −1.35587
$$590$$ 0 0
$$591$$ 99.0129i 0.167535i
$$592$$ 0 0
$$593$$ −244.977 −0.413114 −0.206557 0.978435i $$-0.566226\pi$$
−0.206557 + 0.978435i $$0.566226\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.6151 0.0194559
$$598$$ 0 0
$$599$$ 597.598i 0.997660i 0.866700 + 0.498830i $$0.166237\pi$$
−0.866700 + 0.498830i $$0.833763\pi$$
$$600$$ 0 0
$$601$$ 236.955 0.394267 0.197134 0.980377i $$-0.436837\pi$$
0.197134 + 0.980377i $$0.436837\pi$$
$$602$$ 0 0
$$603$$ 215.617i 0.357573i
$$604$$ 0 0
$$605$$ 430.501 0.711571
$$606$$ 0 0
$$607$$ − 579.236i − 0.954261i −0.878833 0.477130i $$-0.841677\pi$$
0.878833 0.477130i $$-0.158323\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9.75721i 0.0159693i
$$612$$ 0 0
$$613$$ 10.0000 0.0163132 0.00815661 0.999967i $$-0.497404\pi$$
0.00815661 + 0.999967i $$0.497404\pi$$
$$614$$ 0 0
$$615$$ 264.177i 0.429556i
$$616$$ 0 0
$$617$$ −1160.32 −1.88059 −0.940294 0.340364i $$-0.889450\pi$$
−0.940294 + 0.340364i $$0.889450\pi$$
$$618$$ 0 0
$$619$$ − 516.279i − 0.834053i −0.908894 0.417027i $$-0.863072\pi$$
0.908894 0.417027i $$-0.136928\pi$$
$$620$$ 0 0
$$621$$ 141.234 0.227430
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −172.880 −0.276608
$$626$$ 0 0
$$627$$ 27.7128i 0.0441991i
$$628$$ 0 0
$$629$$ 1270.40 2.01971
$$630$$ 0 0
$$631$$ 1212.67i 1.92183i 0.276845 + 0.960915i $$0.410711\pi$$
−0.276845 + 0.960915i $$0.589289\pi$$
$$632$$ 0 0
$$633$$ −309.633 −0.489152
$$634$$ 0 0
$$635$$ 202.381i 0.318709i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 166.968i 0.261296i
$$640$$ 0 0
$$641$$ −40.1197 −0.0625892 −0.0312946 0.999510i $$-0.509963\pi$$
−0.0312946 + 0.999510i $$0.509963\pi$$
$$642$$ 0 0
$$643$$ 198.852i 0.309256i 0.987973 + 0.154628i $$0.0494179\pi$$
−0.987973 + 0.154628i $$0.950582\pi$$
$$644$$ 0 0
$$645$$ −90.7152 −0.140644
$$646$$ 0 0
$$647$$ − 211.484i − 0.326869i −0.986554 0.163435i $$-0.947743\pi$$
0.986554 0.163435i $$-0.0522573\pi$$
$$648$$ 0 0
$$649$$ −24.6970 −0.0380539
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −994.000 −1.52221 −0.761103 0.648632i $$-0.775342\pi$$
−0.761103 + 0.648632i $$0.775342\pi$$
$$654$$ 0 0
$$655$$ 503.248i 0.768318i
$$656$$ 0 0
$$657$$ −286.486 −0.436052
$$658$$ 0 0
$$659$$ − 900.356i − 1.36625i −0.730303 0.683123i $$-0.760621\pi$$
0.730303 0.683123i $$-0.239379\pi$$
$$660$$ 0 0
$$661$$ −683.945 −1.03471 −0.517357 0.855770i $$-0.673084\pi$$
−0.517357 + 0.855770i $$0.673084\pi$$
$$662$$ 0 0
$$663$$ 53.9796i 0.0814172i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 54.3610i − 0.0815008i
$$668$$ 0 0
$$669$$ −198.330 −0.296458
$$670$$ 0 0
$$671$$ 10.0590i 0.0149911i
$$672$$ 0 0
$$673$$ 122.211 0.181591 0.0907954 0.995870i $$-0.471059\pi$$
0.0907954 + 0.995870i $$0.471059\pi$$
$$674$$ 0 0
$$675$$ 63.2120i 0.0936474i
$$676$$ 0 0
$$677$$ −191.931 −0.283502 −0.141751 0.989902i $$-0.545273\pi$$
−0.141751 + 0.989902i $$0.545273\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −533.147 −0.782888
$$682$$ 0 0
$$683$$ − 303.340i − 0.444129i −0.975032 0.222065i $$-0.928720\pi$$
0.975032 0.222065i $$-0.0712796\pi$$
$$684$$ 0 0
$$685$$ −817.386 −1.19326
$$686$$ 0 0
$$687$$ 616.880i 0.897934i
$$688$$ 0 0
$$689$$ −48.7333 −0.0707305
$$690$$ 0 0
$$691$$ − 329.725i − 0.477170i −0.971122 0.238585i $$-0.923316\pi$$
0.971122 0.238585i $$-0.0766836\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 631.451i − 0.908563i
$$696$$ 0 0
$$697$$ 1138.74 1.63378
$$698$$ 0 0
$$699$$ 369.816i 0.529065i
$$700$$ 0 0
$$701$$ 712.918 1.01700 0.508501 0.861061i $$-0.330200\pi$$
0.508501 + 0.861061i $$0.330200\pi$$
$$702$$ 0 0
$$703$$ − 831.703i − 1.18308i
$$704$$ 0 0
$$705$$ −51.9636 −0.0737073
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −975.248 −1.37553 −0.687763 0.725935i $$-0.741407\pi$$
−0.687763 + 0.725935i $$0.741407\pi$$
$$710$$ 0 0
$$711$$ − 191.399i − 0.269198i
$$712$$ 0 0
$$713$$ 1239.58 1.73854
$$714$$ 0 0
$$715$$ 3.81401i 0.00533428i
$$716$$ 0 0
$$717$$ −697.996 −0.973495
$$718$$ 0 0
$$719$$ 30.2864i 0.0421230i 0.999778 + 0.0210615i $$0.00670457\pi$$
−0.999778 + 0.0210615i $$0.993295\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 596.636i 0.825223i
$$724$$ 0 0
$$725$$ 24.3303 0.0335590
$$726$$ 0 0
$$727$$ 1292.40i 1.77771i 0.458186 + 0.888856i $$0.348499\pi$$
−0.458186 + 0.888856i $$0.651501\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −0.0370370
$$730$$ 0 0
$$731$$ 391.031i 0.534926i
$$732$$ 0 0
$$733$$ −1053.37 −1.43706 −0.718532 0.695494i $$-0.755186\pi$$
−0.718532 + 0.695494i $$0.755186\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 65.6697 0.0891041
$$738$$ 0 0
$$739$$ 167.341i 0.226443i 0.993570 + 0.113222i $$0.0361170\pi$$
−0.993570 + 0.113222i $$0.963883\pi$$
$$740$$ 0 0
$$741$$ 35.3394 0.0476915
$$742$$ 0 0
$$743$$ 618.001i 0.831765i 0.909418 + 0.415883i $$0.136527\pi$$
−0.909418 + 0.415883i $$0.863473\pi$$
$$744$$ 0 0
$$745$$ 984.617 1.32163
$$746$$ 0 0
$$747$$ − 97.5355i − 0.130570i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 1215.11i − 1.61798i −0.587820 0.808992i $$-0.700014\pi$$
0.587820 0.808992i $$-0.299986\pi$$
$$752$$ 0 0
$$753$$ 851.477 1.13078
$$754$$ 0 0
$$755$$ − 630.085i − 0.834550i
$$756$$ 0 0
$$757$$ 1011.80 1.33659 0.668296 0.743896i $$-0.267024\pi$$
0.668296 + 0.743896i $$0.267024\pi$$
$$758$$ 0 0
$$759$$ − 43.0152i − 0.0566735i
$$760$$ 0 0
$$761$$ −209.042 −0.274693 −0.137347 0.990523i $$-0.543857\pi$$
−0.137347 + 0.990523i $$0.543857\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −287.477 −0.375787
$$766$$ 0 0
$$767$$ 31.4936i 0.0410607i
$$768$$ 0 0
$$769$$ −7.00909 −0.00911455 −0.00455728 0.999990i $$-0.501451\pi$$
−0.00455728 + 0.999990i $$0.501451\pi$$
$$770$$ 0 0
$$771$$ − 580.881i − 0.753413i
$$772$$ 0 0
$$773$$ −4.59167 −0.00594006 −0.00297003 0.999996i $$-0.500945\pi$$
−0.00297003 + 0.999996i $$0.500945\pi$$
$$774$$ 0 0
$$775$$ 554.797i 0.715867i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 745.513i − 0.957013i
$$780$$ 0 0
$$781$$ 50.8530 0.0651127
$$782$$ 0 0
$$783$$ 10.3923i 0.0132724i
$$784$$ 0 0
$$785$$ −200.559 −0.255489
$$786$$ 0 0
$$787$$ 732.642i 0.930930i 0.885066 + 0.465465i $$0.154113\pi$$
−0.885066 + 0.465465i $$0.845887\pi$$
$$788$$ 0 0
$$789$$ −839.354 −1.06382
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 12.8273 0.0161756
$$794$$ 0 0
$$795$$ − 259.537i − 0.326462i
$$796$$ 0 0
$$797$$ −78.8856 −0.0989782 −0.0494891 0.998775i $$-0.515759\pi$$
−0.0494891 + 0.998775i $$0.515759\pi$$
$$798$$ 0 0
$$799$$ 223.991i 0.280339i
$$800$$ 0 0
$$801$$ −362.711 −0.452823
$$802$$ 0 0
$$803$$ 87.2542i 0.108660i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 48.3780i 0.0599480i
$$808$$ 0 0
$$809$$ −1022.83 −1.26431 −0.632155 0.774842i $$-0.717830\pi$$
−0.632155 + 0.774842i $$0.717830\pi$$
$$810$$ 0 0
$$811$$ − 73.3182i − 0.0904047i −0.998978 0.0452024i $$-0.985607\pi$$
0.998978 0.0452024i $$-0.0143933\pi$$
$$812$$ 0 0
$$813$$ 28.6242 0.0352082
$$814$$ 0 0
$$815$$ 850.454i 1.04350i
$$816$$ 0 0
$$817$$ 256.000 0.313341
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 336.395 0.409739 0.204869 0.978789i $$-0.434323\pi$$
0.204869 + 0.978789i $$0.434323\pi$$
$$822$$ 0 0
$$823$$ − 333.937i − 0.405755i −0.979204 0.202878i $$-0.934971\pi$$
0.979204 0.202878i $$-0.0650294\pi$$
$$824$$ 0 0
$$825$$ 19.2523 0.0233361
$$826$$ 0 0
$$827$$ − 1652.01i − 1.99759i −0.0490993 0.998794i $$-0.515635\pi$$
0.0490993 0.998794i $$-0.484365\pi$$
$$828$$ 0 0
$$829$$ −1011.07 −1.21963 −0.609816 0.792543i $$-0.708756\pi$$
−0.609816 + 0.792543i $$0.708756\pi$$
$$830$$ 0 0
$$831$$ − 622.601i − 0.749219i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ − 850.454i − 1.01851i
$$836$$ 0 0
$$837$$ −236.973 −0.283122
$$838$$ 0 0
$$839$$ − 337.718i − 0.402524i −0.979537 0.201262i $$-0.935496\pi$$
0.979537 0.201262i $$-0.0645042\pi$$
$$840$$ 0 0
$$841$$ −837.000 −0.995244
$$842$$ 0 0
$$843$$ 448.856i 0.532450i
$$844$$ 0 0
$$845$$ −600.592 −0.710759
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −306.991 −0.361591
$$850$$ 0 0
$$851$$ 1290.95i 1.51698i
$$852$$ 0 0
$$853$$ 1285.51 1.50704 0.753521 0.657424i $$-0.228354\pi$$
0.753521 + 0.657424i $$0.228354\pi$$
$$854$$ 0 0
$$855$$ 188.206i 0.220124i
$$856$$ 0 0
$$857$$ −1169.04 −1.36411 −0.682055 0.731301i $$-0.738913\pi$$
−0.682055 + 0.731301i $$0.738913\pi$$
$$858$$ 0 0
$$859$$ − 1283.62i − 1.49432i −0.664642 0.747162i $$-0.731416\pi$$
0.664642 0.747162i $$-0.268584\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 1019.52i − 1.18137i −0.806904 0.590683i $$-0.798858\pi$$
0.806904 0.590683i $$-0.201142\pi$$
$$864$$ 0 0
$$865$$ 620.065 0.716838
$$866$$ 0 0
$$867$$ 738.617i 0.851923i
$$868$$ 0 0
$$869$$ −58.2939 −0.0670816
$$870$$ 0 0
$$871$$ − 83.7420i − 0.0961447i
$$872$$ 0 0
$$873$$ 322.486 0.369400
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 301.194 0.343437 0.171718 0.985146i $$-0.445068\pi$$
0.171718 + 0.985146i $$0.445068\pi$$
$$878$$ 0 0
$$879$$ 512.806i 0.583398i
$$880$$ 0 0
$$881$$ 1530.67 1.73743 0.868715 0.495313i $$-0.164947\pi$$
0.868715 + 0.495313i $$0.164947\pi$$
$$882$$ 0 0
$$883$$ − 319.301i − 0.361609i −0.983519 0.180805i $$-0.942130\pi$$
0.983519 0.180805i $$-0.0578702\pi$$
$$884$$ 0 0
$$885$$ −167.724 −0.189519
$$886$$ 0 0
$$887$$ 1315.17i 1.48271i 0.671110 + 0.741357i $$0.265818\pi$$
−0.671110 + 0.741357i $$0.734182\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 8.22330i 0.00922930i
$$892$$ 0 0
$$893$$ 146.642 0.164213
$$894$$ 0 0
$$895$$ 488.248i 0.545528i
$$896$$ 0 0
$$897$$ −54.8530 −0.0611517
$$898$$ 0 0
$$899$$ 91.2108i 0.101458i
$$900$$ 0 0
$$901$$ −1118.74 −1.24167
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −288.414 −0.318689
$$906$$ 0 0
$$907$$ − 1585.93i − 1.74854i −0.485441 0.874270i $$-0.661341\pi$$
0.485441 0.874270i $$-0.338659\pi$$
$$908$$ 0 0
$$909$$ −313.252 −0.344612
$$910$$ 0 0
$$911$$ 350.279i 0.384499i 0.981346 + 0.192250i $$0.0615783\pi$$
−0.981346 + 0.192250i $$0.938422\pi$$
$$912$$ 0 0
$$913$$ −29.7061 −0.0325368
$$914$$ 0 0
$$915$$ 68.3136i 0.0746597i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 431.552i 0.469588i 0.972045 + 0.234794i $$0.0754416\pi$$
−0.972045 + 0.234794i $$0.924558\pi$$
$$920$$ 0 0
$$921$$ −295.927 −0.321311
$$922$$ 0 0
$$923$$ − 64.8478i − 0.0702577i
$$924$$ 0 0
$$925$$ −577.789 −0.624637
$$926$$ 0 0
$$927$$ − 357.915i − 0.386101i
$$928$$ 0 0
$$929$$ −620.802 −0.668248 −0.334124 0.942529i $$-0.608440\pi$$
−0.334124 + 0.942529i $$0.608440\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 727.405 0.779640
$$934$$ 0 0
$$935$$ 87.5560i 0.0936428i
$$936$$ 0 0
$$937$$ 298.936 0.319036 0.159518 0.987195i $$-0.449006\pi$$
0.159518 + 0.987195i $$0.449006\pi$$
$$938$$ 0 0
$$939$$ 413.309i 0.440159i
$$940$$ 0 0
$$941$$ −786.951 −0.836292 −0.418146 0.908380i $$-0.637320\pi$$
−0.418146 + 0.908380i $$0.637320\pi$$
$$942$$ 0 0
$$943$$ 1157.17i 1.22711i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 629.775i − 0.665021i −0.943100 0.332511i $$-0.892104\pi$$
0.943100 0.332511i $$-0.107896\pi$$
$$948$$ 0 0
$$949$$ 111.267 0.117246
$$950$$ 0 0
$$951$$ 1014.01i 1.06626i
$$952$$ 0 0
$$953$$ 465.855 0.488830 0.244415 0.969671i $$-0.421404\pi$$
0.244415 + 0.969671i $$0.421404\pi$$
$$954$$ 0 0
$$955$$ 469.974i 0.492119i
$$956$$ 0 0
$$957$$ 3.16515 0.00330737
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −1118.85 −1.16426
$$962$$ 0 0
$$963$$ 201.006i 0.208729i
$$964$$ 0 0
$$965$$ −614.395 −0.636679
$$966$$ 0 0
$$967$$ 362.538i 0.374910i 0.982273 + 0.187455i $$0.0600239\pi$$
−0.982273 + 0.187455i $$0.939976\pi$$
$$968$$ 0 0
$$969$$ 811.267 0.837220
$$970$$ 0 0
$$971$$ − 1530.12i − 1.57582i −0.615792 0.787908i $$-0.711164\pi$$
0.615792 0.787908i $$-0.288836\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ − 24.5505i − 0.0251800i
$$976$$ 0 0
$$977$$ −841.441 −0.861250 −0.430625 0.902531i $$-0.641707\pi$$
−0.430625 + 0.902531i $$0.641707\pi$$
$$978$$ 0 0
$$979$$ 110.470i 0.112839i
$$980$$ 0 0
$$981$$ −483.909 −0.493281
$$982$$ 0 0
$$983$$ − 1611.63i − 1.63951i −0.572717 0.819753i $$-0.694111\pi$$
0.572717 0.819753i $$-0.305889\pi$$
$$984$$ 0 0
$$985$$ 204.798 0.207917
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −397.358 −0.401777
$$990$$ 0 0
$$991$$ 368.180i 0.371523i 0.982595 + 0.185762i $$0.0594753\pi$$
−0.982595 + 0.185762i $$0.940525\pi$$
$$992$$ 0 0
$$993$$ −123.688 −0.124560
$$994$$ 0 0
$$995$$ − 24.0248i − 0.0241455i
$$996$$ 0 0
$$997$$ −924.642 −0.927425 −0.463712 0.885986i $$-0.653483\pi$$
−0.463712 + 0.885986i $$0.653483\pi$$
$$998$$ 0 0
$$999$$ − 246.794i − 0.247041i
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 2352.3.m.f.1471.4 4
4.3 odd 2 inner 2352.3.m.f.1471.2 4
7.6 odd 2 336.3.m.c.127.1 4
21.20 even 2 1008.3.m.b.127.4 4
28.27 even 2 336.3.m.c.127.3 yes 4
56.13 odd 2 1344.3.m.a.127.4 4
56.27 even 2 1344.3.m.a.127.2 4
84.83 odd 2 1008.3.m.b.127.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.m.c.127.1 4 7.6 odd 2
336.3.m.c.127.3 yes 4 28.27 even 2
1008.3.m.b.127.3 4 84.83 odd 2
1008.3.m.b.127.4 4 21.20 even 2
1344.3.m.a.127.2 4 56.27 even 2
1344.3.m.a.127.4 4 56.13 odd 2
2352.3.m.f.1471.2 4 4.3 odd 2 inner
2352.3.m.f.1471.4 4 1.1 even 1 trivial