# Properties

 Label 4600.2.a.bi.1.1 Level $4600$ Weight $2$ Character 4600.1 Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.49931$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} +O(q^{10})$$ $$q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} -4.10909 q^{11} -0.0122971 q^{13} +0.155378 q^{17} +4.32471 q^{19} -7.31743 q^{21} -1.00000 q^{23} -0.616262 q^{27} -6.79978 q^{29} +2.20713 q^{31} +10.2699 q^{33} -4.60741 q^{37} +0.0307342 q^{39} -7.67826 q^{41} +8.38997 q^{43} +6.38116 q^{47} +1.57186 q^{49} -0.388339 q^{51} -7.80358 q^{53} -10.8088 q^{57} -14.1275 q^{59} +7.05297 q^{61} +9.50523 q^{63} +7.31363 q^{67} +2.49931 q^{69} -5.84061 q^{71} +0.727674 q^{73} -12.0305 q^{77} +4.81003 q^{79} -8.19948 q^{81} +7.75797 q^{83} +16.9948 q^{87} +6.77560 q^{89} -0.0360030 q^{91} -5.51631 q^{93} +14.8705 q^{97} -13.3405 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} - 7q^{11} - 7q^{13} - 7q^{19} - 6q^{21} - 7q^{23} - 11q^{29} - 10q^{31} - 19q^{33} - 19q^{37} - 24q^{39} - 16q^{41} + 6q^{43} + 6q^{47} - 17q^{49} - 7q^{51} - 15q^{53} - 8q^{57} - 11q^{59} + 5q^{61} + 13q^{63} + 9q^{67} - 3q^{69} - 14q^{71} - 10q^{73} - 6q^{77} - 32q^{79} - 5q^{81} + q^{83} + 10q^{87} - 24q^{89} - 7q^{91} - 26q^{93} + 7q^{97} - 61q^{99} + O(q^{100})$$

## 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
$$3$$ −2.49931 −1.44298 −0.721490 0.692425i $$-0.756542\pi$$
−0.721490 + 0.692425i $$0.756542\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.92777 1.10659 0.553297 0.832984i $$-0.313369\pi$$
0.553297 + 0.832984i $$0.313369\pi$$
$$8$$ 0 0
$$9$$ 3.24657 1.08219
$$10$$ 0 0
$$11$$ −4.10909 −1.23894 −0.619468 0.785022i $$-0.712652\pi$$
−0.619468 + 0.785022i $$0.712652\pi$$
$$12$$ 0 0
$$13$$ −0.0122971 −0.00341059 −0.00170530 0.999999i $$-0.500543\pi$$
−0.00170530 + 0.999999i $$0.500543\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.155378 0.0376847 0.0188424 0.999822i $$-0.494002\pi$$
0.0188424 + 0.999822i $$0.494002\pi$$
$$18$$ 0 0
$$19$$ 4.32471 0.992158 0.496079 0.868277i $$-0.334773\pi$$
0.496079 + 0.868277i $$0.334773\pi$$
$$20$$ 0 0
$$21$$ −7.31743 −1.59679
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −0.616262 −0.118600
$$28$$ 0 0
$$29$$ −6.79978 −1.26269 −0.631344 0.775503i $$-0.717496\pi$$
−0.631344 + 0.775503i $$0.717496\pi$$
$$30$$ 0 0
$$31$$ 2.20713 0.396412 0.198206 0.980160i $$-0.436489\pi$$
0.198206 + 0.980160i $$0.436489\pi$$
$$32$$ 0 0
$$33$$ 10.2699 1.78776
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.60741 −0.757453 −0.378726 0.925509i $$-0.623638\pi$$
−0.378726 + 0.925509i $$0.623638\pi$$
$$38$$ 0 0
$$39$$ 0.0307342 0.00492142
$$40$$ 0 0
$$41$$ −7.67826 −1.19914 −0.599571 0.800321i $$-0.704662\pi$$
−0.599571 + 0.800321i $$0.704662\pi$$
$$42$$ 0 0
$$43$$ 8.38997 1.27946 0.639729 0.768600i $$-0.279046\pi$$
0.639729 + 0.768600i $$0.279046\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.38116 0.930787 0.465394 0.885104i $$-0.345913\pi$$
0.465394 + 0.885104i $$0.345913\pi$$
$$48$$ 0 0
$$49$$ 1.57186 0.224551
$$50$$ 0 0
$$51$$ −0.388339 −0.0543783
$$52$$ 0 0
$$53$$ −7.80358 −1.07190 −0.535952 0.844248i $$-0.680047\pi$$
−0.535952 + 0.844248i $$0.680047\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −10.8088 −1.43166
$$58$$ 0 0
$$59$$ −14.1275 −1.83924 −0.919622 0.392805i $$-0.871505\pi$$
−0.919622 + 0.392805i $$0.871505\pi$$
$$60$$ 0 0
$$61$$ 7.05297 0.903040 0.451520 0.892261i $$-0.350882\pi$$
0.451520 + 0.892261i $$0.350882\pi$$
$$62$$ 0 0
$$63$$ 9.50523 1.19755
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.31363 0.893502 0.446751 0.894658i $$-0.352581\pi$$
0.446751 + 0.894658i $$0.352581\pi$$
$$68$$ 0 0
$$69$$ 2.49931 0.300882
$$70$$ 0 0
$$71$$ −5.84061 −0.693153 −0.346576 0.938022i $$-0.612656\pi$$
−0.346576 + 0.938022i $$0.612656\pi$$
$$72$$ 0 0
$$73$$ 0.727674 0.0851678 0.0425839 0.999093i $$-0.486441\pi$$
0.0425839 + 0.999093i $$0.486441\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −12.0305 −1.37100
$$78$$ 0 0
$$79$$ 4.81003 0.541171 0.270586 0.962696i $$-0.412783\pi$$
0.270586 + 0.962696i $$0.412783\pi$$
$$80$$ 0 0
$$81$$ −8.19948 −0.911054
$$82$$ 0 0
$$83$$ 7.75797 0.851548 0.425774 0.904830i $$-0.360002\pi$$
0.425774 + 0.904830i $$0.360002\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 16.9948 1.82203
$$88$$ 0 0
$$89$$ 6.77560 0.718212 0.359106 0.933297i $$-0.383082\pi$$
0.359106 + 0.933297i $$0.383082\pi$$
$$90$$ 0 0
$$91$$ −0.0360030 −0.00377414
$$92$$ 0 0
$$93$$ −5.51631 −0.572014
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.8705 1.50987 0.754935 0.655799i $$-0.227668\pi$$
0.754935 + 0.655799i $$0.227668\pi$$
$$98$$ 0 0
$$99$$ −13.3405 −1.34077
$$100$$ 0 0
$$101$$ −13.0023 −1.29378 −0.646889 0.762584i $$-0.723930\pi$$
−0.646889 + 0.762584i $$0.723930\pi$$
$$102$$ 0 0
$$103$$ −1.85814 −0.183088 −0.0915442 0.995801i $$-0.529180\pi$$
−0.0915442 + 0.995801i $$0.529180\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.471267 0.0455591 0.0227795 0.999741i $$-0.492748\pi$$
0.0227795 + 0.999741i $$0.492748\pi$$
$$108$$ 0 0
$$109$$ −1.83022 −0.175303 −0.0876515 0.996151i $$-0.527936\pi$$
−0.0876515 + 0.996151i $$0.527936\pi$$
$$110$$ 0 0
$$111$$ 11.5154 1.09299
$$112$$ 0 0
$$113$$ 7.81151 0.734845 0.367422 0.930054i $$-0.380240\pi$$
0.367422 + 0.930054i $$0.380240\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −0.0399233 −0.00369091
$$118$$ 0 0
$$119$$ 0.454912 0.0417017
$$120$$ 0 0
$$121$$ 5.88461 0.534964
$$122$$ 0 0
$$123$$ 19.1904 1.73034
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.62237 −0.676376 −0.338188 0.941079i $$-0.609814\pi$$
−0.338188 + 0.941079i $$0.609814\pi$$
$$128$$ 0 0
$$129$$ −20.9692 −1.84623
$$130$$ 0 0
$$131$$ 19.4575 1.70001 0.850007 0.526772i $$-0.176598\pi$$
0.850007 + 0.526772i $$0.176598\pi$$
$$132$$ 0 0
$$133$$ 12.6618 1.09792
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.3163 1.39399 0.696997 0.717074i $$-0.254519\pi$$
0.696997 + 0.717074i $$0.254519\pi$$
$$138$$ 0 0
$$139$$ −19.9289 −1.69035 −0.845175 0.534490i $$-0.820504\pi$$
−0.845175 + 0.534490i $$0.820504\pi$$
$$140$$ 0 0
$$141$$ −15.9485 −1.34311
$$142$$ 0 0
$$143$$ 0.0505297 0.00422551
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.92857 −0.324023
$$148$$ 0 0
$$149$$ −8.04540 −0.659105 −0.329552 0.944137i $$-0.606898\pi$$
−0.329552 + 0.944137i $$0.606898\pi$$
$$150$$ 0 0
$$151$$ 3.40963 0.277472 0.138736 0.990329i $$-0.455696\pi$$
0.138736 + 0.990329i $$0.455696\pi$$
$$152$$ 0 0
$$153$$ 0.504446 0.0407821
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −21.1868 −1.69089 −0.845446 0.534061i $$-0.820665\pi$$
−0.845446 + 0.534061i $$0.820665\pi$$
$$158$$ 0 0
$$159$$ 19.5036 1.54674
$$160$$ 0 0
$$161$$ −2.92777 −0.230741
$$162$$ 0 0
$$163$$ −20.9487 −1.64083 −0.820415 0.571769i $$-0.806257\pi$$
−0.820415 + 0.571769i $$0.806257\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.9861 0.850131 0.425065 0.905163i $$-0.360251\pi$$
0.425065 + 0.905163i $$0.360251\pi$$
$$168$$ 0 0
$$169$$ −12.9998 −0.999988
$$170$$ 0 0
$$171$$ 14.0405 1.07370
$$172$$ 0 0
$$173$$ −4.31110 −0.327767 −0.163884 0.986480i $$-0.552402\pi$$
−0.163884 + 0.986480i $$0.552402\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 35.3091 2.65399
$$178$$ 0 0
$$179$$ −19.0791 −1.42604 −0.713018 0.701146i $$-0.752672\pi$$
−0.713018 + 0.701146i $$0.752672\pi$$
$$180$$ 0 0
$$181$$ 4.09312 0.304239 0.152120 0.988362i $$-0.451390\pi$$
0.152120 + 0.988362i $$0.451390\pi$$
$$182$$ 0 0
$$183$$ −17.6276 −1.30307
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.638462 −0.0466890
$$188$$ 0 0
$$189$$ −1.80428 −0.131242
$$190$$ 0 0
$$191$$ 6.21032 0.449363 0.224682 0.974432i $$-0.427866\pi$$
0.224682 + 0.974432i $$0.427866\pi$$
$$192$$ 0 0
$$193$$ −13.2674 −0.955005 −0.477503 0.878630i $$-0.658458\pi$$
−0.477503 + 0.878630i $$0.658458\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.8836 −0.989163 −0.494582 0.869131i $$-0.664679\pi$$
−0.494582 + 0.869131i $$0.664679\pi$$
$$198$$ 0 0
$$199$$ −22.8004 −1.61628 −0.808139 0.588992i $$-0.799525\pi$$
−0.808139 + 0.588992i $$0.799525\pi$$
$$200$$ 0 0
$$201$$ −18.2791 −1.28931
$$202$$ 0 0
$$203$$ −19.9082 −1.39728
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −3.24657 −0.225652
$$208$$ 0 0
$$209$$ −17.7706 −1.22922
$$210$$ 0 0
$$211$$ −0.698944 −0.0481173 −0.0240587 0.999711i $$-0.507659\pi$$
−0.0240587 + 0.999711i $$0.507659\pi$$
$$212$$ 0 0
$$213$$ 14.5975 1.00021
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6.46197 0.438667
$$218$$ 0 0
$$219$$ −1.81869 −0.122895
$$220$$ 0 0
$$221$$ −0.00191069 −0.000128527 0
$$222$$ 0 0
$$223$$ 27.3719 1.83296 0.916480 0.400082i $$-0.131018\pi$$
0.916480 + 0.400082i $$0.131018\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −10.6070 −0.704011 −0.352005 0.935998i $$-0.614500\pi$$
−0.352005 + 0.935998i $$0.614500\pi$$
$$228$$ 0 0
$$229$$ 3.11433 0.205801 0.102900 0.994692i $$-0.467188\pi$$
0.102900 + 0.994692i $$0.467188\pi$$
$$230$$ 0 0
$$231$$ 30.0680 1.97833
$$232$$ 0 0
$$233$$ −23.5295 −1.54147 −0.770735 0.637156i $$-0.780111\pi$$
−0.770735 + 0.637156i $$0.780111\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −12.0218 −0.780899
$$238$$ 0 0
$$239$$ 24.4214 1.57969 0.789844 0.613308i $$-0.210162\pi$$
0.789844 + 0.613308i $$0.210162\pi$$
$$240$$ 0 0
$$241$$ 27.5525 1.77481 0.887407 0.460986i $$-0.152504\pi$$
0.887407 + 0.460986i $$0.152504\pi$$
$$242$$ 0 0
$$243$$ 22.3419 1.43323
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.0531813 −0.00338385
$$248$$ 0 0
$$249$$ −19.3896 −1.22877
$$250$$ 0 0
$$251$$ −25.3692 −1.60129 −0.800645 0.599139i $$-0.795510\pi$$
−0.800645 + 0.599139i $$0.795510\pi$$
$$252$$ 0 0
$$253$$ 4.10909 0.258336
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −19.8501 −1.23822 −0.619108 0.785306i $$-0.712506\pi$$
−0.619108 + 0.785306i $$0.712506\pi$$
$$258$$ 0 0
$$259$$ −13.4894 −0.838193
$$260$$ 0 0
$$261$$ −22.0760 −1.36647
$$262$$ 0 0
$$263$$ −22.9087 −1.41261 −0.706306 0.707906i $$-0.749640\pi$$
−0.706306 + 0.707906i $$0.749640\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −16.9344 −1.03637
$$268$$ 0 0
$$269$$ 1.03048 0.0628295 0.0314148 0.999506i $$-0.489999\pi$$
0.0314148 + 0.999506i $$0.489999\pi$$
$$270$$ 0 0
$$271$$ −3.79576 −0.230576 −0.115288 0.993332i $$-0.536779\pi$$
−0.115288 + 0.993332i $$0.536779\pi$$
$$272$$ 0 0
$$273$$ 0.0899829 0.00544601
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.48782 0.0893946 0.0446973 0.999001i $$-0.485768\pi$$
0.0446973 + 0.999001i $$0.485768\pi$$
$$278$$ 0 0
$$279$$ 7.16560 0.428993
$$280$$ 0 0
$$281$$ −10.2796 −0.613228 −0.306614 0.951834i $$-0.599196\pi$$
−0.306614 + 0.951834i $$0.599196\pi$$
$$282$$ 0 0
$$283$$ 4.62427 0.274884 0.137442 0.990510i $$-0.456112\pi$$
0.137442 + 0.990510i $$0.456112\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −22.4802 −1.32697
$$288$$ 0 0
$$289$$ −16.9759 −0.998580
$$290$$ 0 0
$$291$$ −37.1661 −2.17871
$$292$$ 0 0
$$293$$ −16.3067 −0.952650 −0.476325 0.879269i $$-0.658031\pi$$
−0.476325 + 0.879269i $$0.658031\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.53228 0.146938
$$298$$ 0 0
$$299$$ 0.0122971 0.000711158 0
$$300$$ 0 0
$$301$$ 24.5639 1.41584
$$302$$ 0 0
$$303$$ 32.4969 1.86690
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −9.28659 −0.530014 −0.265007 0.964246i $$-0.585374\pi$$
−0.265007 + 0.964246i $$0.585374\pi$$
$$308$$ 0 0
$$309$$ 4.64409 0.264193
$$310$$ 0 0
$$311$$ −24.3713 −1.38197 −0.690984 0.722870i $$-0.742822\pi$$
−0.690984 + 0.722870i $$0.742822\pi$$
$$312$$ 0 0
$$313$$ −15.0048 −0.848119 −0.424060 0.905634i $$-0.639395\pi$$
−0.424060 + 0.905634i $$0.639395\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7.84747 0.440758 0.220379 0.975414i $$-0.429271\pi$$
0.220379 + 0.975414i $$0.429271\pi$$
$$318$$ 0 0
$$319$$ 27.9409 1.56439
$$320$$ 0 0
$$321$$ −1.17784 −0.0657408
$$322$$ 0 0
$$323$$ 0.671966 0.0373892
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4.57429 0.252959
$$328$$ 0 0
$$329$$ 18.6826 1.03000
$$330$$ 0 0
$$331$$ 9.28156 0.510160 0.255080 0.966920i $$-0.417898\pi$$
0.255080 + 0.966920i $$0.417898\pi$$
$$332$$ 0 0
$$333$$ −14.9583 −0.819709
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.3760 −1.43679 −0.718397 0.695634i $$-0.755124\pi$$
−0.718397 + 0.695634i $$0.755124\pi$$
$$338$$ 0 0
$$339$$ −19.5234 −1.06037
$$340$$ 0 0
$$341$$ −9.06928 −0.491129
$$342$$ 0 0
$$343$$ −15.8924 −0.858107
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −18.3316 −0.984093 −0.492047 0.870569i $$-0.663751\pi$$
−0.492047 + 0.870569i $$0.663751\pi$$
$$348$$ 0 0
$$349$$ −32.2759 −1.72769 −0.863845 0.503758i $$-0.831950\pi$$
−0.863845 + 0.503758i $$0.831950\pi$$
$$350$$ 0 0
$$351$$ 0.00757822 0.000404495 0
$$352$$ 0 0
$$353$$ −29.5653 −1.57360 −0.786800 0.617208i $$-0.788264\pi$$
−0.786800 + 0.617208i $$0.788264\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1.13697 −0.0601747
$$358$$ 0 0
$$359$$ 33.4169 1.76368 0.881838 0.471552i $$-0.156306\pi$$
0.881838 + 0.471552i $$0.156306\pi$$
$$360$$ 0 0
$$361$$ −0.296842 −0.0156232
$$362$$ 0 0
$$363$$ −14.7075 −0.771943
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.4764 −1.32986 −0.664928 0.746908i $$-0.731538\pi$$
−0.664928 + 0.746908i $$0.731538\pi$$
$$368$$ 0 0
$$369$$ −24.9280 −1.29770
$$370$$ 0 0
$$371$$ −22.8471 −1.18616
$$372$$ 0 0
$$373$$ −3.28172 −0.169921 −0.0849605 0.996384i $$-0.527076\pi$$
−0.0849605 + 0.996384i $$0.527076\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0.0836174 0.00430651
$$378$$ 0 0
$$379$$ 5.66762 0.291126 0.145563 0.989349i $$-0.453501\pi$$
0.145563 + 0.989349i $$0.453501\pi$$
$$380$$ 0 0
$$381$$ 19.0507 0.975997
$$382$$ 0 0
$$383$$ 34.4343 1.75951 0.879755 0.475428i $$-0.157707\pi$$
0.879755 + 0.475428i $$0.157707\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 27.2387 1.38462
$$388$$ 0 0
$$389$$ −14.3737 −0.728773 −0.364387 0.931248i $$-0.618721\pi$$
−0.364387 + 0.931248i $$0.618721\pi$$
$$390$$ 0 0
$$391$$ −0.155378 −0.00785781
$$392$$ 0 0
$$393$$ −48.6305 −2.45309
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.6149 −0.733500 −0.366750 0.930319i $$-0.619530\pi$$
−0.366750 + 0.930319i $$0.619530\pi$$
$$398$$ 0 0
$$399$$ −31.6458 −1.58427
$$400$$ 0 0
$$401$$ −3.65553 −0.182548 −0.0912742 0.995826i $$-0.529094\pi$$
−0.0912742 + 0.995826i $$0.529094\pi$$
$$402$$ 0 0
$$403$$ −0.0271412 −0.00135200
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 18.9322 0.938436
$$408$$ 0 0
$$409$$ 4.40926 0.218024 0.109012 0.994040i $$-0.465231\pi$$
0.109012 + 0.994040i $$0.465231\pi$$
$$410$$ 0 0
$$411$$ −40.7795 −2.01150
$$412$$ 0 0
$$413$$ −41.3621 −2.03530
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 49.8087 2.43914
$$418$$ 0 0
$$419$$ 32.5906 1.59216 0.796078 0.605195i $$-0.206905\pi$$
0.796078 + 0.605195i $$0.206905\pi$$
$$420$$ 0 0
$$421$$ 4.23954 0.206622 0.103311 0.994649i $$-0.467056\pi$$
0.103311 + 0.994649i $$0.467056\pi$$
$$422$$ 0 0
$$423$$ 20.7169 1.00729
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 20.6495 0.999299
$$428$$ 0 0
$$429$$ −0.126290 −0.00609732
$$430$$ 0 0
$$431$$ 1.81123 0.0872436 0.0436218 0.999048i $$-0.486110\pi$$
0.0436218 + 0.999048i $$0.486110\pi$$
$$432$$ 0 0
$$433$$ −1.76263 −0.0847068 −0.0423534 0.999103i $$-0.513486\pi$$
−0.0423534 + 0.999103i $$0.513486\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.32471 −0.206879
$$438$$ 0 0
$$439$$ −26.7848 −1.27837 −0.639184 0.769054i $$-0.720728\pi$$
−0.639184 + 0.769054i $$0.720728\pi$$
$$440$$ 0 0
$$441$$ 5.10316 0.243008
$$442$$ 0 0
$$443$$ 16.6631 0.791686 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 20.1080 0.951075
$$448$$ 0 0
$$449$$ −34.1236 −1.61039 −0.805196 0.593008i $$-0.797940\pi$$
−0.805196 + 0.593008i $$0.797940\pi$$
$$450$$ 0 0
$$451$$ 31.5507 1.48566
$$452$$ 0 0
$$453$$ −8.52174 −0.400386
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −24.5980 −1.15065 −0.575324 0.817926i $$-0.695124\pi$$
−0.575324 + 0.817926i $$0.695124\pi$$
$$458$$ 0 0
$$459$$ −0.0957537 −0.00446940
$$460$$ 0 0
$$461$$ −10.7297 −0.499733 −0.249866 0.968280i $$-0.580387\pi$$
−0.249866 + 0.968280i $$0.580387\pi$$
$$462$$ 0 0
$$463$$ −22.3124 −1.03695 −0.518473 0.855094i $$-0.673499\pi$$
−0.518473 + 0.855094i $$0.673499\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −20.0218 −0.926497 −0.463248 0.886229i $$-0.653316\pi$$
−0.463248 + 0.886229i $$0.653316\pi$$
$$468$$ 0 0
$$469$$ 21.4127 0.988744
$$470$$ 0 0
$$471$$ 52.9525 2.43992
$$472$$ 0 0
$$473$$ −34.4751 −1.58517
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −25.3349 −1.16001
$$478$$ 0 0
$$479$$ −7.61905 −0.348123 −0.174062 0.984735i $$-0.555689\pi$$
−0.174062 + 0.984735i $$0.555689\pi$$
$$480$$ 0 0
$$481$$ 0.0566576 0.00258336
$$482$$ 0 0
$$483$$ 7.31743 0.332954
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 21.5163 0.974997 0.487499 0.873124i $$-0.337909\pi$$
0.487499 + 0.873124i $$0.337909\pi$$
$$488$$ 0 0
$$489$$ 52.3574 2.36768
$$490$$ 0 0
$$491$$ −10.3705 −0.468015 −0.234007 0.972235i $$-0.575184\pi$$
−0.234007 + 0.972235i $$0.575184\pi$$
$$492$$ 0 0
$$493$$ −1.05654 −0.0475841
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −17.1000 −0.767039
$$498$$ 0 0
$$499$$ −16.2850 −0.729018 −0.364509 0.931200i $$-0.618763\pi$$
−0.364509 + 0.931200i $$0.618763\pi$$
$$500$$ 0 0
$$501$$ −27.4577 −1.22672
$$502$$ 0 0
$$503$$ 16.9326 0.754986 0.377493 0.926012i $$-0.376786\pi$$
0.377493 + 0.926012i $$0.376786\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 32.4907 1.44296
$$508$$ 0 0
$$509$$ 39.0075 1.72898 0.864488 0.502654i $$-0.167643\pi$$
0.864488 + 0.502654i $$0.167643\pi$$
$$510$$ 0 0
$$511$$ 2.13046 0.0942462
$$512$$ 0 0
$$513$$ −2.66516 −0.117670
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −26.2207 −1.15319
$$518$$ 0 0
$$519$$ 10.7748 0.472961
$$520$$ 0 0
$$521$$ 33.1380 1.45180 0.725902 0.687798i $$-0.241423\pi$$
0.725902 + 0.687798i $$0.241423\pi$$
$$522$$ 0 0
$$523$$ 12.4056 0.542460 0.271230 0.962515i $$-0.412570\pi$$
0.271230 + 0.962515i $$0.412570\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0.342939 0.0149387
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −45.8660 −1.99041
$$532$$ 0 0
$$533$$ 0.0944201 0.00408979
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 47.6846 2.05774
$$538$$ 0 0
$$539$$ −6.45891 −0.278205
$$540$$ 0 0
$$541$$ −8.53472 −0.366936 −0.183468 0.983026i $$-0.558732\pi$$
−0.183468 + 0.983026i $$0.558732\pi$$
$$542$$ 0 0
$$543$$ −10.2300 −0.439011
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6.45828 0.276136 0.138068 0.990423i $$-0.455911\pi$$
0.138068 + 0.990423i $$0.455911\pi$$
$$548$$ 0 0
$$549$$ 22.8980 0.977262
$$550$$ 0 0
$$551$$ −29.4071 −1.25279
$$552$$ 0 0
$$553$$ 14.0827 0.598857
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −4.08413 −0.173050 −0.0865251 0.996250i $$-0.527576\pi$$
−0.0865251 + 0.996250i $$0.527576\pi$$
$$558$$ 0 0
$$559$$ −0.103172 −0.00436371
$$560$$ 0 0
$$561$$ 1.59572 0.0673713
$$562$$ 0 0
$$563$$ −15.4494 −0.651114 −0.325557 0.945522i $$-0.605552\pi$$
−0.325557 + 0.945522i $$0.605552\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −24.0062 −1.00817
$$568$$ 0 0
$$569$$ 42.2633 1.77177 0.885886 0.463903i $$-0.153552\pi$$
0.885886 + 0.463903i $$0.153552\pi$$
$$570$$ 0 0
$$571$$ −2.74497 −0.114873 −0.0574367 0.998349i $$-0.518293\pi$$
−0.0574367 + 0.998349i $$0.518293\pi$$
$$572$$ 0 0
$$573$$ −15.5215 −0.648422
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.93871 0.372123 0.186062 0.982538i $$-0.440428\pi$$
0.186062 + 0.982538i $$0.440428\pi$$
$$578$$ 0 0
$$579$$ 33.1593 1.37805
$$580$$ 0 0
$$581$$ 22.7136 0.942318
$$582$$ 0 0
$$583$$ 32.0656 1.32802
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.90821 0.243858 0.121929 0.992539i $$-0.461092\pi$$
0.121929 + 0.992539i $$0.461092\pi$$
$$588$$ 0 0
$$589$$ 9.54520 0.393303
$$590$$ 0 0
$$591$$ 34.6994 1.42734
$$592$$ 0 0
$$593$$ 27.8823 1.14499 0.572494 0.819909i $$-0.305976\pi$$
0.572494 + 0.819909i $$0.305976\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 56.9854 2.33226
$$598$$ 0 0
$$599$$ 10.5644 0.431648 0.215824 0.976432i $$-0.430756\pi$$
0.215824 + 0.976432i $$0.430756\pi$$
$$600$$ 0 0
$$601$$ 35.8910 1.46403 0.732013 0.681291i $$-0.238581\pi$$
0.732013 + 0.681291i $$0.238581\pi$$
$$602$$ 0 0
$$603$$ 23.7442 0.966940
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −29.0640 −1.17967 −0.589835 0.807524i $$-0.700807\pi$$
−0.589835 + 0.807524i $$0.700807\pi$$
$$608$$ 0 0
$$609$$ 49.7569 2.01625
$$610$$ 0 0
$$611$$ −0.0784695 −0.00317454
$$612$$ 0 0
$$613$$ 24.4282 0.986644 0.493322 0.869847i $$-0.335782\pi$$
0.493322 + 0.869847i $$0.335782\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29.5614 1.19010 0.595049 0.803689i $$-0.297133\pi$$
0.595049 + 0.803689i $$0.297133\pi$$
$$618$$ 0 0
$$619$$ −30.5880 −1.22944 −0.614718 0.788747i $$-0.710730\pi$$
−0.614718 + 0.788747i $$0.710730\pi$$
$$620$$ 0 0
$$621$$ 0.616262 0.0247298
$$622$$ 0 0
$$623$$ 19.8374 0.794769
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 44.4144 1.77374
$$628$$ 0 0
$$629$$ −0.715890 −0.0285444
$$630$$ 0 0
$$631$$ 10.5713 0.420835 0.210418 0.977612i $$-0.432518\pi$$
0.210418 + 0.977612i $$0.432518\pi$$
$$632$$ 0 0
$$633$$ 1.74688 0.0694323
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −0.0193293 −0.000765853 0
$$638$$ 0 0
$$639$$ −18.9620 −0.750123
$$640$$ 0 0
$$641$$ −20.4738 −0.808666 −0.404333 0.914612i $$-0.632496\pi$$
−0.404333 + 0.914612i $$0.632496\pi$$
$$642$$ 0 0
$$643$$ 18.4989 0.729525 0.364763 0.931101i $$-0.381150\pi$$
0.364763 + 0.931101i $$0.381150\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.6328 0.732531 0.366266 0.930510i $$-0.380636\pi$$
0.366266 + 0.930510i $$0.380636\pi$$
$$648$$ 0 0
$$649$$ 58.0511 2.27871
$$650$$ 0 0
$$651$$ −16.1505 −0.632988
$$652$$ 0 0
$$653$$ 30.5290 1.19469 0.597347 0.801983i $$-0.296222\pi$$
0.597347 + 0.801983i $$0.296222\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.36245 0.0921678
$$658$$ 0 0
$$659$$ −26.6231 −1.03709 −0.518545 0.855051i $$-0.673526\pi$$
−0.518545 + 0.855051i $$0.673526\pi$$
$$660$$ 0 0
$$661$$ −13.0529 −0.507700 −0.253850 0.967244i $$-0.581697\pi$$
−0.253850 + 0.967244i $$0.581697\pi$$
$$662$$ 0 0
$$663$$ 0.00477543 0.000185462 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.79978 0.263289
$$668$$ 0 0
$$669$$ −68.4110 −2.64492
$$670$$ 0 0
$$671$$ −28.9813 −1.11881
$$672$$ 0 0
$$673$$ 28.9120 1.11448 0.557238 0.830353i $$-0.311861\pi$$
0.557238 + 0.830353i $$0.311861\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −12.4824 −0.479738 −0.239869 0.970805i $$-0.577104\pi$$
−0.239869 + 0.970805i $$0.577104\pi$$
$$678$$ 0 0
$$679$$ 43.5375 1.67081
$$680$$ 0 0
$$681$$ 26.5102 1.01587
$$682$$ 0 0
$$683$$ 40.0241 1.53148 0.765740 0.643150i $$-0.222373\pi$$
0.765740 + 0.643150i $$0.222373\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −7.78369 −0.296966
$$688$$ 0 0
$$689$$ 0.0959612 0.00365583
$$690$$ 0 0
$$691$$ −38.9938 −1.48340 −0.741698 0.670734i $$-0.765979\pi$$
−0.741698 + 0.670734i $$0.765979\pi$$
$$692$$ 0 0
$$693$$ −39.0578 −1.48368
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.19303 −0.0451894
$$698$$ 0 0
$$699$$ 58.8077 2.22431
$$700$$ 0 0
$$701$$ 21.9042 0.827309 0.413655 0.910434i $$-0.364252\pi$$
0.413655 + 0.910434i $$0.364252\pi$$
$$702$$ 0 0
$$703$$ −19.9257 −0.751513
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −38.0678 −1.43169
$$708$$ 0 0
$$709$$ 32.1360 1.20689 0.603447 0.797403i $$-0.293793\pi$$
0.603447 + 0.797403i $$0.293793\pi$$
$$710$$ 0 0
$$711$$ 15.6161 0.585650
$$712$$ 0 0
$$713$$ −2.20713 −0.0826576
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −61.0367 −2.27946
$$718$$ 0 0
$$719$$ −24.4018 −0.910033 −0.455017 0.890483i $$-0.650367\pi$$
−0.455017 + 0.890483i $$0.650367\pi$$
$$720$$ 0 0
$$721$$ −5.44023 −0.202605
$$722$$ 0 0
$$723$$ −68.8625 −2.56102
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −34.8573 −1.29279 −0.646393 0.763005i $$-0.723723\pi$$
−0.646393 + 0.763005i $$0.723723\pi$$
$$728$$ 0 0
$$729$$ −31.2409 −1.15707
$$730$$ 0 0
$$731$$ 1.30362 0.0482161
$$732$$ 0 0
$$733$$ 30.7343 1.13520 0.567598 0.823306i $$-0.307873\pi$$
0.567598 + 0.823306i $$0.307873\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −30.0523 −1.10699
$$738$$ 0 0
$$739$$ −38.7573 −1.42571 −0.712855 0.701312i $$-0.752598\pi$$
−0.712855 + 0.701312i $$0.752598\pi$$
$$740$$ 0 0
$$741$$ 0.132917 0.00488282
$$742$$ 0 0
$$743$$ 43.0371 1.57888 0.789438 0.613830i $$-0.210372\pi$$
0.789438 + 0.613830i $$0.210372\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 25.1868 0.921538
$$748$$ 0 0
$$749$$ 1.37976 0.0504154
$$750$$ 0 0
$$751$$ −35.3761 −1.29089 −0.645447 0.763805i $$-0.723329\pi$$
−0.645447 + 0.763805i $$0.723329\pi$$
$$752$$ 0 0
$$753$$ 63.4056 2.31063
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −51.2605 −1.86309 −0.931547 0.363621i $$-0.881540\pi$$
−0.931547 + 0.363621i $$0.881540\pi$$
$$758$$ 0 0
$$759$$ −10.2699 −0.372774
$$760$$ 0 0
$$761$$ −13.4331 −0.486950 −0.243475 0.969907i $$-0.578287\pi$$
−0.243475 + 0.969907i $$0.578287\pi$$
$$762$$ 0 0
$$763$$ −5.35846 −0.193989
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0.173727 0.00627291
$$768$$ 0 0
$$769$$ −24.7779 −0.893512 −0.446756 0.894656i $$-0.647421\pi$$
−0.446756 + 0.894656i $$0.647421\pi$$
$$770$$ 0 0
$$771$$ 49.6117 1.78672
$$772$$ 0 0
$$773$$ 11.9946 0.431417 0.215708 0.976458i $$-0.430794\pi$$
0.215708 + 0.976458i $$0.430794\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 33.7144 1.20950
$$778$$ 0 0
$$779$$ −33.2063 −1.18974
$$780$$ 0 0
$$781$$ 23.9996 0.858772
$$782$$ 0 0
$$783$$ 4.19045 0.149755
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 10.4110 0.371114 0.185557 0.982634i $$-0.440591\pi$$
0.185557 + 0.982634i $$0.440591\pi$$
$$788$$ 0 0
$$789$$ 57.2561 2.03837
$$790$$ 0 0
$$791$$ 22.8703 0.813175
$$792$$ 0 0
$$793$$ −0.0867308 −0.00307990
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 28.0848 0.994814 0.497407 0.867517i $$-0.334286\pi$$
0.497407 + 0.867517i $$0.334286\pi$$
$$798$$ 0 0
$$799$$ 0.991492 0.0350765
$$800$$ 0 0
$$801$$ 21.9975 0.777243
$$802$$ 0 0
$$803$$ −2.99008 −0.105517
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −2.57550 −0.0906618
$$808$$ 0 0
$$809$$ −11.1238 −0.391092 −0.195546 0.980695i $$-0.562648\pi$$
−0.195546 + 0.980695i $$0.562648\pi$$
$$810$$ 0 0
$$811$$ −28.7572 −1.00980 −0.504900 0.863178i $$-0.668471\pi$$
−0.504900 + 0.863178i $$0.668471\pi$$
$$812$$ 0 0
$$813$$ 9.48679 0.332716
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 36.2842 1.26942
$$818$$ 0 0
$$819$$ −0.116886 −0.00408434
$$820$$ 0 0
$$821$$ 37.0808 1.29413 0.647064 0.762435i $$-0.275996\pi$$
0.647064 + 0.762435i $$0.275996\pi$$
$$822$$ 0 0
$$823$$ −10.9501 −0.381698 −0.190849 0.981619i $$-0.561124\pi$$
−0.190849 + 0.981619i $$0.561124\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 13.9973 0.486732 0.243366 0.969934i $$-0.421748\pi$$
0.243366 + 0.969934i $$0.421748\pi$$
$$828$$ 0 0
$$829$$ 16.0622 0.557864 0.278932 0.960311i $$-0.410020\pi$$
0.278932 + 0.960311i $$0.410020\pi$$
$$830$$ 0 0
$$831$$ −3.71853 −0.128995
$$832$$ 0 0
$$833$$ 0.244233 0.00846216
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.36017 −0.0470144
$$838$$ 0 0
$$839$$ −9.49187 −0.327696 −0.163848 0.986486i $$-0.552391\pi$$
−0.163848 + 0.986486i $$0.552391\pi$$
$$840$$ 0 0
$$841$$ 17.2371 0.594381
$$842$$ 0 0
$$843$$ 25.6919 0.884876
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 17.2288 0.591988
$$848$$ 0 0
$$849$$ −11.5575 −0.396653
$$850$$ 0 0
$$851$$ 4.60741 0.157940
$$852$$ 0 0
$$853$$ −3.45965 −0.118456 −0.0592280 0.998244i $$-0.518864\pi$$
−0.0592280 + 0.998244i $$0.518864\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −0.729973 −0.0249354 −0.0124677 0.999922i $$-0.503969\pi$$
−0.0124677 + 0.999922i $$0.503969\pi$$
$$858$$ 0 0
$$859$$ 0.357241 0.0121889 0.00609446 0.999981i $$-0.498060\pi$$
0.00609446 + 0.999981i $$0.498060\pi$$
$$860$$ 0 0
$$861$$ 56.1851 1.91478
$$862$$ 0 0
$$863$$ −16.8358 −0.573098 −0.286549 0.958066i $$-0.592508\pi$$
−0.286549 + 0.958066i $$0.592508\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 42.4280 1.44093
$$868$$ 0 0
$$869$$ −19.7649 −0.670477
$$870$$ 0 0
$$871$$ −0.0899362 −0.00304737
$$872$$ 0 0
$$873$$ 48.2782 1.63397
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.06147 0.272217 0.136108 0.990694i $$-0.456540\pi$$
0.136108 + 0.990694i $$0.456540\pi$$
$$878$$ 0 0
$$879$$ 40.7557 1.37466
$$880$$ 0 0
$$881$$ −37.7548 −1.27199 −0.635996 0.771693i $$-0.719410\pi$$
−0.635996 + 0.771693i $$0.719410\pi$$
$$882$$ 0 0
$$883$$ 35.2103 1.18492 0.592460 0.805600i $$-0.298157\pi$$
0.592460 + 0.805600i $$0.298157\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.4225 −0.719297 −0.359649 0.933088i $$-0.617103\pi$$
−0.359649 + 0.933088i $$0.617103\pi$$
$$888$$ 0 0
$$889$$ −22.3166 −0.748474
$$890$$ 0 0
$$891$$ 33.6924 1.12874
$$892$$ 0 0
$$893$$ 27.5967 0.923488
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −0.0307342 −0.00102619
$$898$$ 0 0
$$899$$ −15.0080 −0.500545
$$900$$ 0 0
$$901$$ −1.21251 −0.0403944
$$902$$ 0 0
$$903$$ −61.3930 −2.04303
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.7517 1.28673 0.643364 0.765560i $$-0.277538\pi$$
0.643364 + 0.765560i $$0.277538\pi$$
$$908$$ 0 0
$$909$$ −42.2129 −1.40012
$$910$$ 0 0
$$911$$ −47.0069 −1.55741 −0.778704 0.627391i $$-0.784123\pi$$
−0.778704 + 0.627391i $$0.784123\pi$$
$$912$$ 0 0
$$913$$ −31.8782 −1.05501
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 56.9673 1.88123
$$918$$ 0 0
$$919$$ 12.2360 0.403629 0.201814 0.979424i $$-0.435316\pi$$
0.201814 + 0.979424i $$0.435316\pi$$
$$920$$ 0 0
$$921$$ 23.2101 0.764799
$$922$$ 0 0
$$923$$ 0.0718223 0.00236406
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −6.03260 −0.198137
$$928$$ 0 0
$$929$$ −13.0143 −0.426984 −0.213492 0.976945i $$-0.568484\pi$$
−0.213492 + 0.976945i $$0.568484\pi$$
$$930$$ 0 0
$$931$$ 6.79785 0.222790
$$932$$ 0 0
$$933$$ 60.9114 1.99415
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −38.6856 −1.26380 −0.631902 0.775048i $$-0.717726\pi$$
−0.631902 + 0.775048i $$0.717726\pi$$
$$938$$ 0 0
$$939$$ 37.5016 1.22382
$$940$$ 0 0
$$941$$ −43.8241 −1.42862 −0.714312 0.699828i $$-0.753260\pi$$
−0.714312 + 0.699828i $$0.753260\pi$$
$$942$$ 0 0
$$943$$ 7.67826 0.250039
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −43.3854 −1.40984 −0.704918 0.709289i $$-0.749016\pi$$
−0.704918 + 0.709289i $$0.749016\pi$$
$$948$$ 0 0
$$949$$ −0.00894825 −0.000290473 0
$$950$$ 0 0
$$951$$ −19.6133 −0.636005
$$952$$ 0 0
$$953$$ 2.80245 0.0907802 0.0453901 0.998969i $$-0.485547\pi$$
0.0453901 + 0.998969i $$0.485547\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −69.8331 −2.25738
$$958$$ 0 0
$$959$$ 47.7704 1.54259
$$960$$ 0 0
$$961$$ −26.1286 −0.842858
$$962$$ 0 0
$$963$$ 1.53000 0.0493036
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 30.0015 0.964783 0.482391 0.875956i $$-0.339768\pi$$
0.482391 + 0.875956i $$0.339768\pi$$
$$968$$ 0 0
$$969$$ −1.67945 −0.0539518
$$970$$ 0 0
$$971$$ 53.6632 1.72213 0.861067 0.508492i $$-0.169797\pi$$
0.861067 + 0.508492i $$0.169797\pi$$
$$972$$ 0 0
$$973$$ −58.3474 −1.87053
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 46.4590 1.48636 0.743178 0.669094i $$-0.233318\pi$$
0.743178 + 0.669094i $$0.233318\pi$$
$$978$$ 0 0
$$979$$ −27.8415 −0.889819
$$980$$ 0 0
$$981$$ −5.94193 −0.189711
$$982$$ 0 0
$$983$$ −50.8346 −1.62137 −0.810686 0.585481i $$-0.800906\pi$$
−0.810686 + 0.585481i $$0.800906\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −46.6936 −1.48627
$$988$$ 0 0
$$989$$ −8.38997 −0.266786
$$990$$ 0 0
$$991$$ 4.30720 0.136823 0.0684114 0.997657i $$-0.478207\pi$$
0.0684114 + 0.997657i $$0.478207\pi$$
$$992$$ 0 0
$$993$$ −23.1975 −0.736151
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −13.2358 −0.419181 −0.209590 0.977789i $$-0.567213\pi$$
−0.209590 + 0.977789i $$0.567213\pi$$
$$998$$ 0 0
$$999$$ 2.83937 0.0898337
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 4600.2.a.bi.1.1 7
4.3 odd 2 9200.2.a.cz.1.7 7
5.2 odd 4 920.2.e.b.369.13 yes 14
5.3 odd 4 920.2.e.b.369.2 14
5.4 even 2 4600.2.a.bh.1.7 7
20.3 even 4 1840.2.e.g.369.13 14
20.7 even 4 1840.2.e.g.369.2 14
20.19 odd 2 9200.2.a.dc.1.1 7

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.b.369.2 14 5.3 odd 4
920.2.e.b.369.13 yes 14 5.2 odd 4
1840.2.e.g.369.2 14 20.7 even 4
1840.2.e.g.369.13 14 20.3 even 4
4600.2.a.bh.1.7 7 5.4 even 2
4600.2.a.bi.1.1 7 1.1 even 1 trivial
9200.2.a.cz.1.7 7 4.3 odd 2
9200.2.a.dc.1.1 7 20.19 odd 2