Properties

 Label 5376.2.c.bh.2689.2 Level $5376$ Weight $2$ Character 5376.2689 Analytic conductor $42.928$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5376.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$42.9275761266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 672) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 2689.2 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5376.2689 Dual form 5376.2.c.bh.2689.3

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +5.46410i q^{11} +2.00000i q^{13} +3.46410 q^{15} +7.46410 q^{17} +6.92820i q^{19} +1.00000i q^{21} +5.46410 q^{23} -7.00000 q^{25} +1.00000i q^{27} +8.92820i q^{29} -2.92820 q^{31} +5.46410 q^{33} -3.46410i q^{35} +2.00000i q^{37} +2.00000 q^{39} -4.53590 q^{41} -8.00000i q^{43} -3.46410i q^{45} +2.92820 q^{47} +1.00000 q^{49} -7.46410i q^{51} +2.00000i q^{53} -18.9282 q^{55} +6.92820 q^{57} -14.9282i q^{59} +4.92820i q^{61} +1.00000 q^{63} -6.92820 q^{65} -10.9282i q^{67} -5.46410i q^{69} +2.53590 q^{71} +0.928203 q^{73} +7.00000i q^{75} -5.46410i q^{77} -2.92820 q^{79} +1.00000 q^{81} +4.00000i q^{83} +25.8564i q^{85} +8.92820 q^{87} +3.46410 q^{89} -2.00000i q^{91} +2.92820i q^{93} -24.0000 q^{95} +4.92820 q^{97} -5.46410i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{7} - 4q^{9} + 16q^{17} + 8q^{23} - 28q^{25} + 16q^{31} + 8q^{33} + 8q^{39} - 32q^{41} - 16q^{47} + 4q^{49} - 48q^{55} + 4q^{63} + 24q^{71} - 24q^{73} + 16q^{79} + 4q^{81} + 8q^{87} - 96q^{95} - 8q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$$.

 $$n$$ $$1793$$ $$2815$$ $$4609$$ $$5125$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 3.46410i 1.54919i 0.632456 + 0.774597i $$0.282047\pi$$
−0.632456 + 0.774597i $$0.717953\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 5.46410i 1.64749i 0.566961 + 0.823744i $$0.308119\pi$$
−0.566961 + 0.823744i $$0.691881\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 3.46410 0.894427
$$16$$ 0 0
$$17$$ 7.46410 1.81031 0.905155 0.425081i $$-0.139754\pi$$
0.905155 + 0.425081i $$0.139754\pi$$
$$18$$ 0 0
$$19$$ 6.92820i 1.58944i 0.606977 + 0.794719i $$0.292382\pi$$
−0.606977 + 0.794719i $$0.707618\pi$$
$$20$$ 0 0
$$21$$ 1.00000i 0.218218i
$$22$$ 0 0
$$23$$ 5.46410 1.13934 0.569672 0.821872i $$-0.307070\pi$$
0.569672 + 0.821872i $$0.307070\pi$$
$$24$$ 0 0
$$25$$ −7.00000 −1.40000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 8.92820i 1.65793i 0.559304 + 0.828963i $$0.311069\pi$$
−0.559304 + 0.828963i $$0.688931\pi$$
$$30$$ 0 0
$$31$$ −2.92820 −0.525921 −0.262960 0.964807i $$-0.584699\pi$$
−0.262960 + 0.964807i $$0.584699\pi$$
$$32$$ 0 0
$$33$$ 5.46410 0.951178
$$34$$ 0 0
$$35$$ − 3.46410i − 0.585540i
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −4.53590 −0.708388 −0.354194 0.935172i $$-0.615245\pi$$
−0.354194 + 0.935172i $$0.615245\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ − 3.46410i − 0.516398i
$$46$$ 0 0
$$47$$ 2.92820 0.427122 0.213561 0.976930i $$-0.431494\pi$$
0.213561 + 0.976930i $$0.431494\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ − 7.46410i − 1.04518i
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ −18.9282 −2.55228
$$56$$ 0 0
$$57$$ 6.92820 0.917663
$$58$$ 0 0
$$59$$ − 14.9282i − 1.94349i −0.236040 0.971743i $$-0.575850\pi$$
0.236040 0.971743i $$-0.424150\pi$$
$$60$$ 0 0
$$61$$ 4.92820i 0.630992i 0.948927 + 0.315496i $$0.102171\pi$$
−0.948927 + 0.315496i $$0.897829\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −6.92820 −0.859338
$$66$$ 0 0
$$67$$ − 10.9282i − 1.33509i −0.744568 0.667546i $$-0.767345\pi$$
0.744568 0.667546i $$-0.232655\pi$$
$$68$$ 0 0
$$69$$ − 5.46410i − 0.657801i
$$70$$ 0 0
$$71$$ 2.53590 0.300956 0.150478 0.988613i $$-0.451919\pi$$
0.150478 + 0.988613i $$0.451919\pi$$
$$72$$ 0 0
$$73$$ 0.928203 0.108638 0.0543190 0.998524i $$-0.482701\pi$$
0.0543190 + 0.998524i $$0.482701\pi$$
$$74$$ 0 0
$$75$$ 7.00000i 0.808290i
$$76$$ 0 0
$$77$$ − 5.46410i − 0.622692i
$$78$$ 0 0
$$79$$ −2.92820 −0.329449 −0.164724 0.986340i $$-0.552673\pi$$
−0.164724 + 0.986340i $$0.552673\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 25.8564i 2.80452i
$$86$$ 0 0
$$87$$ 8.92820 0.957204
$$88$$ 0 0
$$89$$ 3.46410 0.367194 0.183597 0.983002i $$-0.441226\pi$$
0.183597 + 0.983002i $$0.441226\pi$$
$$90$$ 0 0
$$91$$ − 2.00000i − 0.209657i
$$92$$ 0 0
$$93$$ 2.92820i 0.303641i
$$94$$ 0 0
$$95$$ −24.0000 −2.46235
$$96$$ 0 0
$$97$$ 4.92820 0.500383 0.250192 0.968196i $$-0.419506\pi$$
0.250192 + 0.968196i $$0.419506\pi$$
$$98$$ 0 0
$$99$$ − 5.46410i − 0.549163i
$$100$$ 0 0
$$101$$ 8.53590i 0.849354i 0.905345 + 0.424677i $$0.139612\pi$$
−0.905345 + 0.424677i $$0.860388\pi$$
$$102$$ 0 0
$$103$$ 2.92820 0.288524 0.144262 0.989539i $$-0.453919\pi$$
0.144262 + 0.989539i $$0.453919\pi$$
$$104$$ 0 0
$$105$$ −3.46410 −0.338062
$$106$$ 0 0
$$107$$ − 8.39230i − 0.811315i −0.914025 0.405657i $$-0.867043\pi$$
0.914025 0.405657i $$-0.132957\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 7.85641 0.739069 0.369534 0.929217i $$-0.379517\pi$$
0.369534 + 0.929217i $$0.379517\pi$$
$$114$$ 0 0
$$115$$ 18.9282i 1.76506i
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −7.46410 −0.684233
$$120$$ 0 0
$$121$$ −18.8564 −1.71422
$$122$$ 0 0
$$123$$ 4.53590i 0.408988i
$$124$$ 0 0
$$125$$ − 6.92820i − 0.619677i
$$126$$ 0 0
$$127$$ 10.9282 0.969721 0.484861 0.874591i $$-0.338870\pi$$
0.484861 + 0.874591i $$0.338870\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ − 6.92820i − 0.600751i
$$134$$ 0 0
$$135$$ −3.46410 −0.298142
$$136$$ 0 0
$$137$$ −4.92820 −0.421045 −0.210522 0.977589i $$-0.567516\pi$$
−0.210522 + 0.977589i $$0.567516\pi$$
$$138$$ 0 0
$$139$$ 9.85641i 0.836009i 0.908445 + 0.418005i $$0.137270\pi$$
−0.908445 + 0.418005i $$0.862730\pi$$
$$140$$ 0 0
$$141$$ − 2.92820i − 0.246599i
$$142$$ 0 0
$$143$$ −10.9282 −0.913862
$$144$$ 0 0
$$145$$ −30.9282 −2.56845
$$146$$ 0 0
$$147$$ − 1.00000i − 0.0824786i
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ −5.85641 −0.476588 −0.238294 0.971193i $$-0.576588\pi$$
−0.238294 + 0.971193i $$0.576588\pi$$
$$152$$ 0 0
$$153$$ −7.46410 −0.603437
$$154$$ 0 0
$$155$$ − 10.1436i − 0.814753i
$$156$$ 0 0
$$157$$ 12.9282i 1.03178i 0.856654 + 0.515891i $$0.172539\pi$$
−0.856654 + 0.515891i $$0.827461\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ −5.46410 −0.430632
$$162$$ 0 0
$$163$$ 2.92820i 0.229355i 0.993403 + 0.114677i $$0.0365834\pi$$
−0.993403 + 0.114677i $$0.963417\pi$$
$$164$$ 0 0
$$165$$ 18.9282i 1.47356i
$$166$$ 0 0
$$167$$ 2.92820 0.226591 0.113296 0.993561i $$-0.463859\pi$$
0.113296 + 0.993561i $$0.463859\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ − 6.92820i − 0.529813i
$$172$$ 0 0
$$173$$ 1.60770i 0.122231i 0.998131 + 0.0611154i $$0.0194658\pi$$
−0.998131 + 0.0611154i $$0.980534\pi$$
$$174$$ 0 0
$$175$$ 7.00000 0.529150
$$176$$ 0 0
$$177$$ −14.9282 −1.12207
$$178$$ 0 0
$$179$$ − 2.53590i − 0.189542i −0.995499 0.0947710i $$-0.969788\pi$$
0.995499 0.0947710i $$-0.0302119\pi$$
$$180$$ 0 0
$$181$$ 6.00000i 0.445976i 0.974821 + 0.222988i $$0.0715812\pi$$
−0.974821 + 0.222988i $$0.928419\pi$$
$$182$$ 0 0
$$183$$ 4.92820 0.364303
$$184$$ 0 0
$$185$$ −6.92820 −0.509372
$$186$$ 0 0
$$187$$ 40.7846i 2.98247i
$$188$$ 0 0
$$189$$ − 1.00000i − 0.0727393i
$$190$$ 0 0
$$191$$ −2.53590 −0.183491 −0.0917456 0.995782i $$-0.529245\pi$$
−0.0917456 + 0.995782i $$0.529245\pi$$
$$192$$ 0 0
$$193$$ −11.8564 −0.853443 −0.426721 0.904383i $$-0.640332\pi$$
−0.426721 + 0.904383i $$0.640332\pi$$
$$194$$ 0 0
$$195$$ 6.92820i 0.496139i
$$196$$ 0 0
$$197$$ − 19.8564i − 1.41471i −0.706858 0.707355i $$-0.749888\pi$$
0.706858 0.707355i $$-0.250112\pi$$
$$198$$ 0 0
$$199$$ −21.8564 −1.54936 −0.774680 0.632354i $$-0.782089\pi$$
−0.774680 + 0.632354i $$0.782089\pi$$
$$200$$ 0 0
$$201$$ −10.9282 −0.770816
$$202$$ 0 0
$$203$$ − 8.92820i − 0.626637i
$$204$$ 0 0
$$205$$ − 15.7128i − 1.09743i
$$206$$ 0 0
$$207$$ −5.46410 −0.379781
$$208$$ 0 0
$$209$$ −37.8564 −2.61858
$$210$$ 0 0
$$211$$ 16.0000i 1.10149i 0.834675 + 0.550743i $$0.185655\pi$$
−0.834675 + 0.550743i $$0.814345\pi$$
$$212$$ 0 0
$$213$$ − 2.53590i − 0.173757i
$$214$$ 0 0
$$215$$ 27.7128 1.89000
$$216$$ 0 0
$$217$$ 2.92820 0.198779
$$218$$ 0 0
$$219$$ − 0.928203i − 0.0627222i
$$220$$ 0 0
$$221$$ 14.9282i 1.00418i
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ 7.00000 0.466667
$$226$$ 0 0
$$227$$ 9.07180i 0.602116i 0.953606 + 0.301058i $$0.0973398\pi$$
−0.953606 + 0.301058i $$0.902660\pi$$
$$228$$ 0 0
$$229$$ 0.143594i 0.00948893i 0.999989 + 0.00474446i $$0.00151022\pi$$
−0.999989 + 0.00474446i $$0.998490\pi$$
$$230$$ 0 0
$$231$$ −5.46410 −0.359511
$$232$$ 0 0
$$233$$ 0.928203 0.0608086 0.0304043 0.999538i $$-0.490321\pi$$
0.0304043 + 0.999538i $$0.490321\pi$$
$$234$$ 0 0
$$235$$ 10.1436i 0.661695i
$$236$$ 0 0
$$237$$ 2.92820i 0.190207i
$$238$$ 0 0
$$239$$ −21.4641 −1.38840 −0.694199 0.719783i $$-0.744241\pi$$
−0.694199 + 0.719783i $$0.744241\pi$$
$$240$$ 0 0
$$241$$ −16.9282 −1.09044 −0.545221 0.838293i $$-0.683554\pi$$
−0.545221 + 0.838293i $$0.683554\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 3.46410i 0.221313i
$$246$$ 0 0
$$247$$ −13.8564 −0.881662
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ − 22.9282i − 1.44722i −0.690212 0.723608i $$-0.742483\pi$$
0.690212 0.723608i $$-0.257517\pi$$
$$252$$ 0 0
$$253$$ 29.8564i 1.87706i
$$254$$ 0 0
$$255$$ 25.8564 1.61919
$$256$$ 0 0
$$257$$ 4.53590 0.282942 0.141471 0.989942i $$-0.454817\pi$$
0.141471 + 0.989942i $$0.454817\pi$$
$$258$$ 0 0
$$259$$ − 2.00000i − 0.124274i
$$260$$ 0 0
$$261$$ − 8.92820i − 0.552642i
$$262$$ 0 0
$$263$$ −8.39230 −0.517492 −0.258746 0.965945i $$-0.583309\pi$$
−0.258746 + 0.965945i $$0.583309\pi$$
$$264$$ 0 0
$$265$$ −6.92820 −0.425596
$$266$$ 0 0
$$267$$ − 3.46410i − 0.212000i
$$268$$ 0 0
$$269$$ − 3.46410i − 0.211210i −0.994408 0.105605i $$-0.966322\pi$$
0.994408 0.105605i $$-0.0336779\pi$$
$$270$$ 0 0
$$271$$ 10.9282 0.663841 0.331921 0.943307i $$-0.392303\pi$$
0.331921 + 0.943307i $$0.392303\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ − 38.2487i − 2.30648i
$$276$$ 0 0
$$277$$ − 27.8564i − 1.67373i −0.547410 0.836865i $$-0.684386\pi$$
0.547410 0.836865i $$-0.315614\pi$$
$$278$$ 0 0
$$279$$ 2.92820 0.175307
$$280$$ 0 0
$$281$$ 0.928203 0.0553720 0.0276860 0.999617i $$-0.491186\pi$$
0.0276860 + 0.999617i $$0.491186\pi$$
$$282$$ 0 0
$$283$$ − 1.07180i − 0.0637117i −0.999492 0.0318559i $$-0.989858\pi$$
0.999492 0.0318559i $$-0.0101417\pi$$
$$284$$ 0 0
$$285$$ 24.0000i 1.42164i
$$286$$ 0 0
$$287$$ 4.53590 0.267746
$$288$$ 0 0
$$289$$ 38.7128 2.27722
$$290$$ 0 0
$$291$$ − 4.92820i − 0.288896i
$$292$$ 0 0
$$293$$ − 23.4641i − 1.37079i −0.728173 0.685394i $$-0.759630\pi$$
0.728173 0.685394i $$-0.240370\pi$$
$$294$$ 0 0
$$295$$ 51.7128 3.01084
$$296$$ 0 0
$$297$$ −5.46410 −0.317059
$$298$$ 0 0
$$299$$ 10.9282i 0.631994i
$$300$$ 0 0
$$301$$ 8.00000i 0.461112i
$$302$$ 0 0
$$303$$ 8.53590 0.490375
$$304$$ 0 0
$$305$$ −17.0718 −0.977528
$$306$$ 0 0
$$307$$ − 9.07180i − 0.517755i −0.965910 0.258877i $$-0.916647\pi$$
0.965910 0.258877i $$-0.0833526\pi$$
$$308$$ 0 0
$$309$$ − 2.92820i − 0.166580i
$$310$$ 0 0
$$311$$ 5.07180 0.287595 0.143798 0.989607i $$-0.454069\pi$$
0.143798 + 0.989607i $$0.454069\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 3.46410i 0.195180i
$$316$$ 0 0
$$317$$ 27.8564i 1.56457i 0.622920 + 0.782286i $$0.285946\pi$$
−0.622920 + 0.782286i $$0.714054\pi$$
$$318$$ 0 0
$$319$$ −48.7846 −2.73141
$$320$$ 0 0
$$321$$ −8.39230 −0.468413
$$322$$ 0 0
$$323$$ 51.7128i 2.87738i
$$324$$ 0 0
$$325$$ − 14.0000i − 0.776580i
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ −2.92820 −0.161437
$$330$$ 0 0
$$331$$ 24.0000i 1.31916i 0.751635 + 0.659580i $$0.229266\pi$$
−0.751635 + 0.659580i $$0.770734\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 37.8564 2.06832
$$336$$ 0 0
$$337$$ −19.8564 −1.08165 −0.540824 0.841136i $$-0.681887\pi$$
−0.540824 + 0.841136i $$0.681887\pi$$
$$338$$ 0 0
$$339$$ − 7.85641i − 0.426701i
$$340$$ 0 0
$$341$$ − 16.0000i − 0.866449i
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 18.9282 1.01906
$$346$$ 0 0
$$347$$ − 11.3205i − 0.607717i −0.952717 0.303858i $$-0.901725\pi$$
0.952717 0.303858i $$-0.0982750\pi$$
$$348$$ 0 0
$$349$$ − 30.7846i − 1.64786i −0.566690 0.823931i $$-0.691776\pi$$
0.566690 0.823931i $$-0.308224\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 15.4641 0.823071 0.411536 0.911394i $$-0.364993\pi$$
0.411536 + 0.911394i $$0.364993\pi$$
$$354$$ 0 0
$$355$$ 8.78461i 0.466239i
$$356$$ 0 0
$$357$$ 7.46410i 0.395042i
$$358$$ 0 0
$$359$$ −27.3205 −1.44192 −0.720961 0.692976i $$-0.756299\pi$$
−0.720961 + 0.692976i $$0.756299\pi$$
$$360$$ 0 0
$$361$$ −29.0000 −1.52632
$$362$$ 0 0
$$363$$ 18.8564i 0.989705i
$$364$$ 0 0
$$365$$ 3.21539i 0.168301i
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 4.53590 0.236129
$$370$$ 0 0
$$371$$ − 2.00000i − 0.103835i
$$372$$ 0 0
$$373$$ − 30.0000i − 1.55334i −0.629907 0.776671i $$-0.716907\pi$$
0.629907 0.776671i $$-0.283093\pi$$
$$374$$ 0 0
$$375$$ −6.92820 −0.357771
$$376$$ 0 0
$$377$$ −17.8564 −0.919652
$$378$$ 0 0
$$379$$ − 8.00000i − 0.410932i −0.978664 0.205466i $$-0.934129\pi$$
0.978664 0.205466i $$-0.0658711\pi$$
$$380$$ 0 0
$$381$$ − 10.9282i − 0.559869i
$$382$$ 0 0
$$383$$ 19.7128 1.00728 0.503639 0.863914i $$-0.331994\pi$$
0.503639 + 0.863914i $$0.331994\pi$$
$$384$$ 0 0
$$385$$ 18.9282 0.964671
$$386$$ 0 0
$$387$$ 8.00000i 0.406663i
$$388$$ 0 0
$$389$$ 26.7846i 1.35803i 0.734123 + 0.679017i $$0.237594\pi$$
−0.734123 + 0.679017i $$0.762406\pi$$
$$390$$ 0 0
$$391$$ 40.7846 2.06257
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ − 10.1436i − 0.510380i
$$396$$ 0 0
$$397$$ − 30.7846i − 1.54504i −0.634993 0.772518i $$-0.718997\pi$$
0.634993 0.772518i $$-0.281003\pi$$
$$398$$ 0 0
$$399$$ −6.92820 −0.346844
$$400$$ 0 0
$$401$$ −8.92820 −0.445853 −0.222927 0.974835i $$-0.571561\pi$$
−0.222927 + 0.974835i $$0.571561\pi$$
$$402$$ 0 0
$$403$$ − 5.85641i − 0.291728i
$$404$$ 0 0
$$405$$ 3.46410i 0.172133i
$$406$$ 0 0
$$407$$ −10.9282 −0.541691
$$408$$ 0 0
$$409$$ 8.92820 0.441471 0.220736 0.975334i $$-0.429154\pi$$
0.220736 + 0.975334i $$0.429154\pi$$
$$410$$ 0 0
$$411$$ 4.92820i 0.243090i
$$412$$ 0 0
$$413$$ 14.9282i 0.734569i
$$414$$ 0 0
$$415$$ −13.8564 −0.680184
$$416$$ 0 0
$$417$$ 9.85641 0.482670
$$418$$ 0 0
$$419$$ − 30.9282i − 1.51094i −0.655182 0.755471i $$-0.727408\pi$$
0.655182 0.755471i $$-0.272592\pi$$
$$420$$ 0 0
$$421$$ 31.8564i 1.55259i 0.630372 + 0.776293i $$0.282902\pi$$
−0.630372 + 0.776293i $$0.717098\pi$$
$$422$$ 0 0
$$423$$ −2.92820 −0.142374
$$424$$ 0 0
$$425$$ −52.2487 −2.53443
$$426$$ 0 0
$$427$$ − 4.92820i − 0.238492i
$$428$$ 0 0
$$429$$ 10.9282i 0.527619i
$$430$$ 0 0
$$431$$ −38.2487 −1.84238 −0.921188 0.389118i $$-0.872780\pi$$
−0.921188 + 0.389118i $$0.872780\pi$$
$$432$$ 0 0
$$433$$ −0.143594 −0.00690067 −0.00345033 0.999994i $$-0.501098\pi$$
−0.00345033 + 0.999994i $$0.501098\pi$$
$$434$$ 0 0
$$435$$ 30.9282i 1.48289i
$$436$$ 0 0
$$437$$ 37.8564i 1.81092i
$$438$$ 0 0
$$439$$ −13.8564 −0.661330 −0.330665 0.943748i $$-0.607273\pi$$
−0.330665 + 0.943748i $$0.607273\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ − 7.60770i − 0.361453i −0.983533 0.180726i $$-0.942155\pi$$
0.983533 0.180726i $$-0.0578448\pi$$
$$444$$ 0 0
$$445$$ 12.0000i 0.568855i
$$446$$ 0 0
$$447$$ 18.0000 0.851371
$$448$$ 0 0
$$449$$ −19.8564 −0.937082 −0.468541 0.883442i $$-0.655220\pi$$
−0.468541 + 0.883442i $$0.655220\pi$$
$$450$$ 0 0
$$451$$ − 24.7846i − 1.16706i
$$452$$ 0 0
$$453$$ 5.85641i 0.275158i
$$454$$ 0 0
$$455$$ 6.92820 0.324799
$$456$$ 0 0
$$457$$ 17.7128 0.828570 0.414285 0.910147i $$-0.364032\pi$$
0.414285 + 0.910147i $$0.364032\pi$$
$$458$$ 0 0
$$459$$ 7.46410i 0.348394i
$$460$$ 0 0
$$461$$ 35.1769i 1.63835i 0.573542 + 0.819176i $$0.305569\pi$$
−0.573542 + 0.819176i $$0.694431\pi$$
$$462$$ 0 0
$$463$$ 8.78461 0.408255 0.204128 0.978944i $$-0.434564\pi$$
0.204128 + 0.978944i $$0.434564\pi$$
$$464$$ 0 0
$$465$$ −10.1436 −0.470398
$$466$$ 0 0
$$467$$ 20.7846i 0.961797i 0.876776 + 0.480899i $$0.159689\pi$$
−0.876776 + 0.480899i $$0.840311\pi$$
$$468$$ 0 0
$$469$$ 10.9282i 0.504618i
$$470$$ 0 0
$$471$$ 12.9282 0.595700
$$472$$ 0 0
$$473$$ 43.7128 2.00992
$$474$$ 0 0
$$475$$ − 48.4974i − 2.22521i
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ 18.9282 0.864852 0.432426 0.901670i $$-0.357658\pi$$
0.432426 + 0.901670i $$0.357658\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 5.46410i 0.248625i
$$484$$ 0 0
$$485$$ 17.0718i 0.775190i
$$486$$ 0 0
$$487$$ 27.7128 1.25579 0.627894 0.778299i $$-0.283917\pi$$
0.627894 + 0.778299i $$0.283917\pi$$
$$488$$ 0 0
$$489$$ 2.92820 0.132418
$$490$$ 0 0
$$491$$ − 13.4641i − 0.607626i −0.952732 0.303813i $$-0.901740\pi$$
0.952732 0.303813i $$-0.0982599\pi$$
$$492$$ 0 0
$$493$$ 66.6410i 3.00136i
$$494$$ 0 0
$$495$$ 18.9282 0.850759
$$496$$ 0 0
$$497$$ −2.53590 −0.113751
$$498$$ 0 0
$$499$$ − 5.85641i − 0.262169i −0.991371 0.131084i $$-0.958154\pi$$
0.991371 0.131084i $$-0.0418459\pi$$
$$500$$ 0 0
$$501$$ − 2.92820i − 0.130822i
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −29.5692 −1.31581
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ 8.24871i 0.365618i 0.983148 + 0.182809i $$0.0585189\pi$$
−0.983148 + 0.182809i $$0.941481\pi$$
$$510$$ 0 0
$$511$$ −0.928203 −0.0410613
$$512$$ 0 0
$$513$$ −6.92820 −0.305888
$$514$$ 0 0
$$515$$ 10.1436i 0.446980i
$$516$$ 0 0
$$517$$ 16.0000i 0.703679i
$$518$$ 0 0
$$519$$ 1.60770 0.0705700
$$520$$ 0 0
$$521$$ −9.60770 −0.420921 −0.210460 0.977602i $$-0.567496\pi$$
−0.210460 + 0.977602i $$0.567496\pi$$
$$522$$ 0 0
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 0 0
$$525$$ − 7.00000i − 0.305505i
$$526$$ 0 0
$$527$$ −21.8564 −0.952080
$$528$$ 0 0
$$529$$ 6.85641 0.298105
$$530$$ 0 0
$$531$$ 14.9282i 0.647829i
$$532$$ 0 0
$$533$$ − 9.07180i − 0.392943i
$$534$$ 0 0
$$535$$ 29.0718 1.25688
$$536$$ 0 0
$$537$$ −2.53590 −0.109432
$$538$$ 0 0
$$539$$ 5.46410i 0.235356i
$$540$$ 0 0
$$541$$ 19.8564i 0.853694i 0.904324 + 0.426847i $$0.140376\pi$$
−0.904324 + 0.426847i $$0.859624\pi$$
$$542$$ 0 0
$$543$$ 6.00000 0.257485
$$544$$ 0 0
$$545$$ 6.92820 0.296772
$$546$$ 0 0
$$547$$ 2.92820i 0.125201i 0.998039 + 0.0626005i $$0.0199394\pi$$
−0.998039 + 0.0626005i $$0.980061\pi$$
$$548$$ 0 0
$$549$$ − 4.92820i − 0.210331i
$$550$$ 0 0
$$551$$ −61.8564 −2.63517
$$552$$ 0 0
$$553$$ 2.92820 0.124520
$$554$$ 0 0
$$555$$ 6.92820i 0.294086i
$$556$$ 0 0
$$557$$ − 31.8564i − 1.34980i −0.737910 0.674900i $$-0.764187\pi$$
0.737910 0.674900i $$-0.235813\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 40.7846 1.72193
$$562$$ 0 0
$$563$$ − 46.9282i − 1.97779i −0.148623 0.988894i $$-0.547484\pi$$
0.148623 0.988894i $$-0.452516\pi$$
$$564$$ 0 0
$$565$$ 27.2154i 1.14496i
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 11.0718 0.464154 0.232077 0.972697i $$-0.425448\pi$$
0.232077 + 0.972697i $$0.425448\pi$$
$$570$$ 0 0
$$571$$ − 24.7846i − 1.03720i −0.855016 0.518602i $$-0.826453\pi$$
0.855016 0.518602i $$-0.173547\pi$$
$$572$$ 0 0
$$573$$ 2.53590i 0.105939i
$$574$$ 0 0
$$575$$ −38.2487 −1.59508
$$576$$ 0 0
$$577$$ −3.85641 −0.160544 −0.0802722 0.996773i $$-0.525579\pi$$
−0.0802722 + 0.996773i $$0.525579\pi$$
$$578$$ 0 0
$$579$$ 11.8564i 0.492735i
$$580$$ 0 0
$$581$$ − 4.00000i − 0.165948i
$$582$$ 0 0
$$583$$ −10.9282 −0.452600
$$584$$ 0 0
$$585$$ 6.92820 0.286446
$$586$$ 0 0
$$587$$ 20.7846i 0.857873i 0.903335 + 0.428936i $$0.141112\pi$$
−0.903335 + 0.428936i $$0.858888\pi$$
$$588$$ 0 0
$$589$$ − 20.2872i − 0.835919i
$$590$$ 0 0
$$591$$ −19.8564 −0.816783
$$592$$ 0 0
$$593$$ 15.4641 0.635035 0.317517 0.948252i $$-0.397151\pi$$
0.317517 + 0.948252i $$0.397151\pi$$
$$594$$ 0 0
$$595$$ − 25.8564i − 1.06001i
$$596$$ 0 0
$$597$$ 21.8564i 0.894523i
$$598$$ 0 0
$$599$$ −7.60770 −0.310842 −0.155421 0.987848i $$-0.549673\pi$$
−0.155421 + 0.987848i $$0.549673\pi$$
$$600$$ 0 0
$$601$$ 39.5692 1.61406 0.807031 0.590509i $$-0.201073\pi$$
0.807031 + 0.590509i $$0.201073\pi$$
$$602$$ 0 0
$$603$$ 10.9282i 0.445031i
$$604$$ 0 0
$$605$$ − 65.3205i − 2.65566i
$$606$$ 0 0
$$607$$ 13.8564 0.562414 0.281207 0.959647i $$-0.409265\pi$$
0.281207 + 0.959647i $$0.409265\pi$$
$$608$$ 0 0
$$609$$ −8.92820 −0.361789
$$610$$ 0 0
$$611$$ 5.85641i 0.236925i
$$612$$ 0 0
$$613$$ − 0.143594i − 0.00579969i −0.999996 0.00289984i $$-0.999077\pi$$
0.999996 0.00289984i $$-0.000923050\pi$$
$$614$$ 0 0
$$615$$ −15.7128 −0.633602
$$616$$ 0 0
$$617$$ 22.7846 0.917274 0.458637 0.888624i $$-0.348338\pi$$
0.458637 + 0.888624i $$0.348338\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 5.46410i 0.219267i
$$622$$ 0 0
$$623$$ −3.46410 −0.138786
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 37.8564i 1.51184i
$$628$$ 0 0
$$629$$ 14.9282i 0.595226i
$$630$$ 0 0
$$631$$ 13.0718 0.520380 0.260190 0.965557i $$-0.416215\pi$$
0.260190 + 0.965557i $$0.416215\pi$$
$$632$$ 0 0
$$633$$ 16.0000 0.635943
$$634$$ 0 0
$$635$$ 37.8564i 1.50229i
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ 0 0
$$639$$ −2.53590 −0.100319
$$640$$ 0 0
$$641$$ 34.7846 1.37391 0.686955 0.726700i $$-0.258947\pi$$
0.686955 + 0.726700i $$0.258947\pi$$
$$642$$ 0 0
$$643$$ − 28.7846i − 1.13515i −0.823320 0.567577i $$-0.807881\pi$$
0.823320 0.567577i $$-0.192119\pi$$
$$644$$ 0 0
$$645$$ − 27.7128i − 1.09119i
$$646$$ 0 0
$$647$$ −18.9282 −0.744144 −0.372072 0.928204i $$-0.621353\pi$$
−0.372072 + 0.928204i $$0.621353\pi$$
$$648$$ 0 0
$$649$$ 81.5692 3.20187
$$650$$ 0 0
$$651$$ − 2.92820i − 0.114765i
$$652$$ 0 0
$$653$$ − 7.07180i − 0.276741i −0.990381 0.138370i $$-0.955814\pi$$
0.990381 0.138370i $$-0.0441864\pi$$
$$654$$ 0 0
$$655$$ 41.5692 1.62424
$$656$$ 0 0
$$657$$ −0.928203 −0.0362127
$$658$$ 0 0
$$659$$ − 21.4641i − 0.836123i −0.908419 0.418061i $$-0.862710\pi$$
0.908419 0.418061i $$-0.137290\pi$$
$$660$$ 0 0
$$661$$ 14.7846i 0.575055i 0.957772 + 0.287527i $$0.0928332\pi$$
−0.957772 + 0.287527i $$0.907167\pi$$
$$662$$ 0 0
$$663$$ 14.9282 0.579763
$$664$$ 0 0
$$665$$ 24.0000 0.930680
$$666$$ 0 0
$$667$$ 48.7846i 1.88895i
$$668$$ 0 0
$$669$$ − 24.0000i − 0.927894i
$$670$$ 0 0
$$671$$ −26.9282 −1.03955
$$672$$ 0 0
$$673$$ 4.14359 0.159724 0.0798619 0.996806i $$-0.474552\pi$$
0.0798619 + 0.996806i $$0.474552\pi$$
$$674$$ 0 0
$$675$$ − 7.00000i − 0.269430i
$$676$$ 0 0
$$677$$ 22.3923i 0.860606i 0.902684 + 0.430303i $$0.141593\pi$$
−0.902684 + 0.430303i $$0.858407\pi$$
$$678$$ 0 0
$$679$$ −4.92820 −0.189127
$$680$$ 0 0
$$681$$ 9.07180 0.347632
$$682$$ 0 0
$$683$$ 30.2487i 1.15743i 0.815528 + 0.578717i $$0.196447\pi$$
−0.815528 + 0.578717i $$0.803553\pi$$
$$684$$ 0 0
$$685$$ − 17.0718i − 0.652280i
$$686$$ 0 0
$$687$$ 0.143594 0.00547844
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 14.1436i 0.538048i 0.963134 + 0.269024i $$0.0867010\pi$$
−0.963134 + 0.269024i $$0.913299\pi$$
$$692$$ 0 0
$$693$$ 5.46410i 0.207564i
$$694$$ 0 0
$$695$$ −34.1436 −1.29514
$$696$$ 0 0
$$697$$ −33.8564 −1.28240
$$698$$ 0 0
$$699$$ − 0.928203i − 0.0351079i
$$700$$ 0 0
$$701$$ − 40.6410i − 1.53499i −0.641055 0.767495i $$-0.721503\pi$$
0.641055 0.767495i $$-0.278497\pi$$
$$702$$ 0 0
$$703$$ −13.8564 −0.522604
$$704$$ 0 0
$$705$$ 10.1436 0.382030
$$706$$ 0 0
$$707$$ − 8.53590i − 0.321025i
$$708$$ 0 0
$$709$$ 37.7128i 1.41633i 0.706045 + 0.708167i $$0.250478\pi$$
−0.706045 + 0.708167i $$0.749522\pi$$
$$710$$ 0 0
$$711$$ 2.92820 0.109816
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ − 37.8564i − 1.41575i
$$716$$ 0 0
$$717$$ 21.4641i 0.801592i
$$718$$ 0 0
$$719$$ −13.8564 −0.516757 −0.258378 0.966044i $$-0.583188\pi$$
−0.258378 + 0.966044i $$0.583188\pi$$
$$720$$ 0 0
$$721$$ −2.92820 −0.109052
$$722$$ 0 0
$$723$$ 16.9282i 0.629567i
$$724$$ 0 0
$$725$$ − 62.4974i − 2.32110i
$$726$$ 0 0
$$727$$ 34.9282 1.29542 0.647708 0.761889i $$-0.275728\pi$$
0.647708 + 0.761889i $$0.275728\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 59.7128i − 2.20856i
$$732$$ 0 0
$$733$$ − 49.7128i − 1.83618i −0.396367 0.918092i $$-0.629729\pi$$
0.396367 0.918092i $$-0.370271\pi$$
$$734$$ 0 0
$$735$$ 3.46410 0.127775
$$736$$ 0 0
$$737$$ 59.7128 2.19955
$$738$$ 0 0
$$739$$ 50.9282i 1.87342i 0.350101 + 0.936712i $$0.386147\pi$$
−0.350101 + 0.936712i $$0.613853\pi$$
$$740$$ 0 0
$$741$$ 13.8564i 0.509028i
$$742$$ 0 0
$$743$$ 32.3923 1.18836 0.594179 0.804333i $$-0.297477\pi$$
0.594179 + 0.804333i $$0.297477\pi$$
$$744$$ 0 0
$$745$$ −62.3538 −2.28447
$$746$$ 0 0
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 8.39230i 0.306648i
$$750$$ 0 0
$$751$$ 27.7128 1.01125 0.505627 0.862752i $$-0.331261\pi$$
0.505627 + 0.862752i $$0.331261\pi$$
$$752$$ 0 0
$$753$$ −22.9282 −0.835550
$$754$$ 0 0
$$755$$ − 20.2872i − 0.738326i
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 0 0
$$759$$ 29.8564 1.08372
$$760$$ 0 0
$$761$$ −20.5359 −0.744426 −0.372213 0.928147i $$-0.621401\pi$$
−0.372213 + 0.928147i $$0.621401\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ 0 0
$$765$$ − 25.8564i − 0.934840i
$$766$$ 0 0
$$767$$ 29.8564 1.07805
$$768$$ 0 0
$$769$$ 31.8564 1.14877 0.574386 0.818585i $$-0.305241\pi$$
0.574386 + 0.818585i $$0.305241\pi$$
$$770$$ 0 0
$$771$$ − 4.53590i − 0.163356i
$$772$$ 0 0
$$773$$ − 29.3205i − 1.05459i −0.849684 0.527293i $$-0.823207\pi$$
0.849684 0.527293i $$-0.176793\pi$$
$$774$$ 0 0
$$775$$ 20.4974 0.736289
$$776$$ 0 0
$$777$$ −2.00000 −0.0717496
$$778$$ 0 0
$$779$$ − 31.4256i − 1.12594i
$$780$$ 0 0
$$781$$ 13.8564i 0.495821i
$$782$$ 0 0
$$783$$ −8.92820 −0.319068
$$784$$ 0 0
$$785$$ −44.7846 −1.59843
$$786$$ 0 0
$$787$$ 31.7128i 1.13044i 0.824940 + 0.565220i $$0.191209\pi$$
−0.824940 + 0.565220i $$0.808791\pi$$
$$788$$ 0 0
$$789$$ 8.39230i 0.298774i
$$790$$ 0 0
$$791$$ −7.85641 −0.279342
$$792$$ 0 0
$$793$$ −9.85641 −0.350011
$$794$$ 0 0
$$795$$ 6.92820i 0.245718i
$$796$$ 0 0
$$797$$ − 14.3923i − 0.509802i −0.966967 0.254901i $$-0.917957\pi$$
0.966967 0.254901i $$-0.0820428\pi$$
$$798$$ 0 0
$$799$$ 21.8564 0.773224
$$800$$ 0 0
$$801$$ −3.46410 −0.122398
$$802$$ 0 0
$$803$$ 5.07180i 0.178980i
$$804$$ 0 0
$$805$$ − 18.9282i − 0.667132i
$$806$$ 0 0
$$807$$ −3.46410 −0.121942
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 45.5692i 1.60015i 0.599899 + 0.800076i $$0.295207\pi$$
−0.599899 + 0.800076i $$0.704793\pi$$
$$812$$ 0 0
$$813$$ − 10.9282i − 0.383269i
$$814$$ 0 0
$$815$$ −10.1436 −0.355315
$$816$$ 0 0
$$817$$ 55.4256 1.93910
$$818$$ 0 0
$$819$$ 2.00000i 0.0698857i
$$820$$ 0 0
$$821$$ 39.8564i 1.39100i 0.718527 + 0.695499i $$0.244817\pi$$
−0.718527 + 0.695499i $$0.755183\pi$$
$$822$$ 0 0
$$823$$ −34.9282 −1.21752 −0.608760 0.793354i $$-0.708333\pi$$
−0.608760 + 0.793354i $$0.708333\pi$$
$$824$$ 0 0
$$825$$ −38.2487 −1.33165
$$826$$ 0 0
$$827$$ 25.1769i 0.875487i 0.899100 + 0.437744i $$0.144222\pi$$
−0.899100 + 0.437744i $$0.855778\pi$$
$$828$$ 0 0
$$829$$ − 33.7128i − 1.17089i −0.810711 0.585447i $$-0.800919\pi$$
0.810711 0.585447i $$-0.199081\pi$$
$$830$$ 0 0
$$831$$ −27.8564 −0.966328
$$832$$ 0 0
$$833$$ 7.46410 0.258616
$$834$$ 0 0
$$835$$ 10.1436i 0.351034i
$$836$$ 0 0
$$837$$ − 2.92820i − 0.101214i
$$838$$ 0 0
$$839$$ 2.92820 0.101093 0.0505464 0.998722i $$-0.483904\pi$$
0.0505464 + 0.998722i $$0.483904\pi$$
$$840$$ 0 0
$$841$$ −50.7128 −1.74872
$$842$$ 0 0
$$843$$ − 0.928203i − 0.0319690i
$$844$$ 0 0
$$845$$ 31.1769i 1.07252i
$$846$$ 0 0
$$847$$ 18.8564 0.647914
$$848$$ 0 0
$$849$$ −1.07180 −0.0367840
$$850$$ 0 0
$$851$$ 10.9282i 0.374614i
$$852$$ 0 0
$$853$$ − 36.9282i − 1.26440i −0.774806 0.632199i $$-0.782153\pi$$
0.774806 0.632199i $$-0.217847\pi$$
$$854$$ 0 0
$$855$$ 24.0000 0.820783
$$856$$ 0 0
$$857$$ 7.17691 0.245159 0.122579 0.992459i $$-0.460883\pi$$
0.122579 + 0.992459i $$0.460883\pi$$
$$858$$ 0 0
$$859$$ − 25.0718i − 0.855439i −0.903912 0.427719i $$-0.859317\pi$$
0.903912 0.427719i $$-0.140683\pi$$
$$860$$ 0 0
$$861$$ − 4.53590i − 0.154583i
$$862$$ 0 0
$$863$$ 2.53590 0.0863230 0.0431615 0.999068i $$-0.486257\pi$$
0.0431615 + 0.999068i $$0.486257\pi$$
$$864$$ 0 0
$$865$$ −5.56922 −0.189359
$$866$$ 0 0
$$867$$ − 38.7128i − 1.31476i
$$868$$ 0 0
$$869$$ − 16.0000i − 0.542763i
$$870$$ 0 0
$$871$$ 21.8564 0.740576
$$872$$ 0 0
$$873$$ −4.92820 −0.166794
$$874$$ 0 0
$$875$$ 6.92820i 0.234216i
$$876$$ 0 0
$$877$$ 35.8564i 1.21078i 0.795927 + 0.605392i $$0.206984\pi$$
−0.795927 + 0.605392i $$0.793016\pi$$
$$878$$ 0 0
$$879$$ −23.4641 −0.791425
$$880$$ 0 0
$$881$$ 21.3205 0.718306 0.359153 0.933279i $$-0.383066\pi$$
0.359153 + 0.933279i $$0.383066\pi$$
$$882$$ 0 0
$$883$$ − 49.5692i − 1.66814i −0.551661 0.834069i $$-0.686006\pi$$
0.551661 0.834069i $$-0.313994\pi$$
$$884$$ 0 0
$$885$$ − 51.7128i − 1.73831i
$$886$$ 0 0
$$887$$ −13.0718 −0.438908 −0.219454 0.975623i $$-0.570428\pi$$
−0.219454 + 0.975623i $$0.570428\pi$$
$$888$$ 0 0
$$889$$ −10.9282 −0.366520
$$890$$ 0 0
$$891$$ 5.46410i 0.183054i
$$892$$ 0 0
$$893$$ 20.2872i 0.678885i
$$894$$ 0 0
$$895$$ 8.78461 0.293637
$$896$$ 0 0
$$897$$ 10.9282 0.364882
$$898$$ 0 0
$$899$$ − 26.1436i − 0.871938i
$$900$$ 0 0
$$901$$ 14.9282i 0.497331i
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −20.7846 −0.690904
$$906$$ 0 0
$$907$$ 11.7128i 0.388918i 0.980911 + 0.194459i $$0.0622951\pi$$
−0.980911 + 0.194459i $$0.937705\pi$$
$$908$$ 0 0
$$909$$ − 8.53590i − 0.283118i
$$910$$ 0 0
$$911$$ 33.1769 1.09920 0.549600 0.835428i $$-0.314780\pi$$
0.549600 + 0.835428i $$0.314780\pi$$
$$912$$ 0 0
$$913$$ −21.8564 −0.723341
$$914$$ 0 0
$$915$$ 17.0718i 0.564376i
$$916$$ 0 0
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −9.07180 −0.298926
$$922$$ 0 0
$$923$$ 5.07180i 0.166940i
$$924$$ 0 0
$$925$$ − 14.0000i − 0.460317i
$$926$$ 0 0
$$927$$ −2.92820 −0.0961748
$$928$$ 0 0
$$929$$ 34.3923 1.12837 0.564187 0.825647i $$-0.309190\pi$$
0.564187 + 0.825647i $$0.309190\pi$$
$$930$$ 0 0
$$931$$ 6.92820i 0.227063i
$$932$$ 0 0
$$933$$ − 5.07180i − 0.166043i
$$934$$ 0 0
$$935$$ −141.282 −4.62042
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ 0 0
$$939$$ − 14.0000i − 0.456873i
$$940$$ 0 0
$$941$$ − 43.4641i − 1.41689i −0.705766 0.708445i $$-0.749397\pi$$
0.705766 0.708445i $$-0.250603\pi$$
$$942$$ 0 0
$$943$$ −24.7846 −0.807098
$$944$$ 0 0
$$945$$ 3.46410 0.112687
$$946$$ 0 0
$$947$$ 22.2487i 0.722986i 0.932375 + 0.361493i $$0.117733\pi$$
−0.932375 + 0.361493i $$0.882267\pi$$
$$948$$ 0 0
$$949$$ 1.85641i 0.0602615i
$$950$$ 0 0
$$951$$ 27.8564 0.903306
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ − 8.78461i − 0.284263i
$$956$$ 0 0
$$957$$ 48.7846i 1.57698i
$$958$$ 0 0
$$959$$ 4.92820 0.159140
$$960$$ 0 0
$$961$$ −22.4256 −0.723407
$$962$$ 0 0
$$963$$ 8.39230i 0.270438i
$$964$$ 0 0
$$965$$ − 41.0718i − 1.32215i
$$966$$ 0 0
$$967$$ 21.0718 0.677623 0.338812 0.940854i $$-0.389975\pi$$
0.338812 + 0.940854i $$0.389975\pi$$
$$968$$ 0 0
$$969$$ 51.7128 1.66125
$$970$$ 0 0
$$971$$ − 25.8564i − 0.829772i −0.909873 0.414886i $$-0.863822\pi$$
0.909873 0.414886i $$-0.136178\pi$$
$$972$$ 0 0
$$973$$ − 9.85641i − 0.315982i
$$974$$ 0 0
$$975$$ −14.0000 −0.448359
$$976$$ 0 0
$$977$$ −58.4974 −1.87150 −0.935749 0.352666i $$-0.885275\pi$$
−0.935749 + 0.352666i $$0.885275\pi$$
$$978$$ 0 0
$$979$$ 18.9282i 0.604948i
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ 0 0
$$983$$ 51.7128 1.64938 0.824691 0.565583i $$-0.191349\pi$$
0.824691 + 0.565583i $$0.191349\pi$$
$$984$$ 0 0
$$985$$ 68.7846 2.19166
$$986$$ 0 0
$$987$$ 2.92820i 0.0932057i
$$988$$ 0 0
$$989$$ − 43.7128i − 1.38999i
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ 24.0000 0.761617
$$994$$ 0 0
$$995$$ − 75.7128i − 2.40026i
$$996$$ 0 0
$$997$$ − 44.9282i − 1.42289i −0.702742 0.711445i $$-0.748041\pi$$
0.702742 0.711445i $$-0.251959\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 5376.2.c.bh.2689.2 4
4.3 odd 2 5376.2.c.bn.2689.4 4
8.3 odd 2 5376.2.c.bn.2689.1 4
8.5 even 2 inner 5376.2.c.bh.2689.3 4
16.3 odd 4 1344.2.a.u.1.2 2
16.5 even 4 672.2.a.i.1.1 2
16.11 odd 4 672.2.a.j.1.1 yes 2
16.13 even 4 1344.2.a.v.1.2 2
48.5 odd 4 2016.2.a.t.1.2 2
48.11 even 4 2016.2.a.s.1.2 2
48.29 odd 4 4032.2.a.bs.1.1 2
48.35 even 4 4032.2.a.br.1.1 2
112.13 odd 4 9408.2.a.do.1.1 2
112.27 even 4 4704.2.a.bm.1.2 2
112.69 odd 4 4704.2.a.bn.1.2 2
112.83 even 4 9408.2.a.dx.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.1 2 16.5 even 4
672.2.a.j.1.1 yes 2 16.11 odd 4
1344.2.a.u.1.2 2 16.3 odd 4
1344.2.a.v.1.2 2 16.13 even 4
2016.2.a.s.1.2 2 48.11 even 4
2016.2.a.t.1.2 2 48.5 odd 4
4032.2.a.br.1.1 2 48.35 even 4
4032.2.a.bs.1.1 2 48.29 odd 4
4704.2.a.bm.1.2 2 112.27 even 4
4704.2.a.bn.1.2 2 112.69 odd 4
5376.2.c.bh.2689.2 4 1.1 even 1 trivial
5376.2.c.bh.2689.3 4 8.5 even 2 inner
5376.2.c.bn.2689.1 4 8.3 odd 2
5376.2.c.bn.2689.4 4 4.3 odd 2
9408.2.a.do.1.1 2 112.13 odd 4
9408.2.a.dx.1.1 2 112.83 even 4