# Properties

 Label 1840.4.a.m.1.1 Level $1840$ Weight $4$ Character 1840.1 Self dual yes Analytic conductor $108.564$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$108.563514411$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 68 x^{2} - 111 x + 342$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$8.73081$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +O(q^{10})$$ $$q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +32.1352 q^{11} +40.0811 q^{13} +38.6540 q^{15} +126.165 q^{17} -0.232742 q^{19} +182.155 q^{21} +23.0000 q^{23} +25.0000 q^{25} -44.5711 q^{27} -137.226 q^{29} -112.866 q^{31} -248.431 q^{33} +117.811 q^{35} +45.7057 q^{37} -309.859 q^{39} -135.385 q^{41} -543.528 q^{43} -163.827 q^{45} -26.4344 q^{47} +212.180 q^{49} -975.359 q^{51} +43.6958 q^{53} -160.676 q^{55} +1.79928 q^{57} -202.248 q^{59} +150.279 q^{61} -772.026 q^{63} -200.405 q^{65} +420.722 q^{67} -177.809 q^{69} -667.381 q^{71} +602.960 q^{73} -193.270 q^{75} -757.178 q^{77} +1378.88 q^{79} -540.095 q^{81} +485.178 q^{83} -630.826 q^{85} +1060.86 q^{87} -1127.71 q^{89} -944.400 q^{91} +872.545 q^{93} +1.16371 q^{95} -1486.24 q^{97} +1052.92 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99} + O(q^{100})$$

## 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
$$3$$ −7.73081 −1.48779 −0.743897 0.668294i $$-0.767025\pi$$
−0.743897 + 0.668294i $$0.767025\pi$$
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −23.5622 −1.27224 −0.636121 0.771589i $$-0.719462\pi$$
−0.636121 + 0.771589i $$0.719462\pi$$
$$8$$ 0 0
$$9$$ 32.7654 1.21353
$$10$$ 0 0
$$11$$ 32.1352 0.880830 0.440415 0.897794i $$-0.354831\pi$$
0.440415 + 0.897794i $$0.354831\pi$$
$$12$$ 0 0
$$13$$ 40.0811 0.855115 0.427557 0.903988i $$-0.359374\pi$$
0.427557 + 0.903988i $$0.359374\pi$$
$$14$$ 0 0
$$15$$ 38.6540 0.665362
$$16$$ 0 0
$$17$$ 126.165 1.79997 0.899987 0.435916i $$-0.143576\pi$$
0.899987 + 0.435916i $$0.143576\pi$$
$$18$$ 0 0
$$19$$ −0.232742 −0.00281024 −0.00140512 0.999999i $$-0.500447\pi$$
−0.00140512 + 0.999999i $$0.500447\pi$$
$$20$$ 0 0
$$21$$ 182.155 1.89283
$$22$$ 0 0
$$23$$ 23.0000 0.208514
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −44.5711 −0.317693
$$28$$ 0 0
$$29$$ −137.226 −0.878695 −0.439347 0.898317i $$-0.644790\pi$$
−0.439347 + 0.898317i $$0.644790\pi$$
$$30$$ 0 0
$$31$$ −112.866 −0.653914 −0.326957 0.945039i $$-0.606023\pi$$
−0.326957 + 0.945039i $$0.606023\pi$$
$$32$$ 0 0
$$33$$ −248.431 −1.31049
$$34$$ 0 0
$$35$$ 117.811 0.568964
$$36$$ 0 0
$$37$$ 45.7057 0.203080 0.101540 0.994831i $$-0.467623\pi$$
0.101540 + 0.994831i $$0.467623\pi$$
$$38$$ 0 0
$$39$$ −309.859 −1.27223
$$40$$ 0 0
$$41$$ −135.385 −0.515696 −0.257848 0.966186i $$-0.583013\pi$$
−0.257848 + 0.966186i $$0.583013\pi$$
$$42$$ 0 0
$$43$$ −543.528 −1.92761 −0.963805 0.266608i $$-0.914097\pi$$
−0.963805 + 0.266608i $$0.914097\pi$$
$$44$$ 0 0
$$45$$ −163.827 −0.542708
$$46$$ 0 0
$$47$$ −26.4344 −0.0820394 −0.0410197 0.999158i $$-0.513061\pi$$
−0.0410197 + 0.999158i $$0.513061\pi$$
$$48$$ 0 0
$$49$$ 212.180 0.618599
$$50$$ 0 0
$$51$$ −975.359 −2.67799
$$52$$ 0 0
$$53$$ 43.6958 0.113247 0.0566235 0.998396i $$-0.481967\pi$$
0.0566235 + 0.998396i $$0.481967\pi$$
$$54$$ 0 0
$$55$$ −160.676 −0.393919
$$56$$ 0 0
$$57$$ 1.79928 0.00418106
$$58$$ 0 0
$$59$$ −202.248 −0.446278 −0.223139 0.974787i $$-0.571630\pi$$
−0.223139 + 0.974787i $$0.571630\pi$$
$$60$$ 0 0
$$61$$ 150.279 0.315430 0.157715 0.987485i $$-0.449587\pi$$
0.157715 + 0.987485i $$0.449587\pi$$
$$62$$ 0 0
$$63$$ −772.026 −1.54391
$$64$$ 0 0
$$65$$ −200.405 −0.382419
$$66$$ 0 0
$$67$$ 420.722 0.767155 0.383577 0.923509i $$-0.374692\pi$$
0.383577 + 0.923509i $$0.374692\pi$$
$$68$$ 0 0
$$69$$ −177.809 −0.310227
$$70$$ 0 0
$$71$$ −667.381 −1.11554 −0.557771 0.829995i $$-0.688343\pi$$
−0.557771 + 0.829995i $$0.688343\pi$$
$$72$$ 0 0
$$73$$ 602.960 0.966728 0.483364 0.875420i $$-0.339415\pi$$
0.483364 + 0.875420i $$0.339415\pi$$
$$74$$ 0 0
$$75$$ −193.270 −0.297559
$$76$$ 0 0
$$77$$ −757.178 −1.12063
$$78$$ 0 0
$$79$$ 1378.88 1.96374 0.981872 0.189545i $$-0.0607013\pi$$
0.981872 + 0.189545i $$0.0607013\pi$$
$$80$$ 0 0
$$81$$ −540.095 −0.740871
$$82$$ 0 0
$$83$$ 485.178 0.641629 0.320815 0.947142i $$-0.396043\pi$$
0.320815 + 0.947142i $$0.396043\pi$$
$$84$$ 0 0
$$85$$ −630.826 −0.804973
$$86$$ 0 0
$$87$$ 1060.86 1.30732
$$88$$ 0 0
$$89$$ −1127.71 −1.34311 −0.671556 0.740954i $$-0.734374\pi$$
−0.671556 + 0.740954i $$0.734374\pi$$
$$90$$ 0 0
$$91$$ −944.400 −1.08791
$$92$$ 0 0
$$93$$ 872.545 0.972890
$$94$$ 0 0
$$95$$ 1.16371 0.00125678
$$96$$ 0 0
$$97$$ −1486.24 −1.55572 −0.777858 0.628441i $$-0.783694\pi$$
−0.777858 + 0.628441i $$0.783694\pi$$
$$98$$ 0 0
$$99$$ 1052.92 1.06892
$$100$$ 0 0
$$101$$ 1888.86 1.86087 0.930437 0.366451i $$-0.119427\pi$$
0.930437 + 0.366451i $$0.119427\pi$$
$$102$$ 0 0
$$103$$ 1497.17 1.43224 0.716118 0.697979i $$-0.245917\pi$$
0.716118 + 0.697979i $$0.245917\pi$$
$$104$$ 0 0
$$105$$ −910.776 −0.846501
$$106$$ 0 0
$$107$$ 585.150 0.528678 0.264339 0.964430i $$-0.414846\pi$$
0.264339 + 0.964430i $$0.414846\pi$$
$$108$$ 0 0
$$109$$ −139.166 −0.122290 −0.0611452 0.998129i $$-0.519475\pi$$
−0.0611452 + 0.998129i $$0.519475\pi$$
$$110$$ 0 0
$$111$$ −353.342 −0.302142
$$112$$ 0 0
$$113$$ −1293.18 −1.07656 −0.538282 0.842765i $$-0.680927\pi$$
−0.538282 + 0.842765i $$0.680927\pi$$
$$114$$ 0 0
$$115$$ −115.000 −0.0932505
$$116$$ 0 0
$$117$$ 1313.27 1.03771
$$118$$ 0 0
$$119$$ −2972.74 −2.29000
$$120$$ 0 0
$$121$$ −298.328 −0.224138
$$122$$ 0 0
$$123$$ 1046.63 0.767250
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1298.43 0.907221 0.453610 0.891200i $$-0.350136\pi$$
0.453610 + 0.891200i $$0.350136\pi$$
$$128$$ 0 0
$$129$$ 4201.91 2.86789
$$130$$ 0 0
$$131$$ 525.247 0.350313 0.175157 0.984541i $$-0.443957\pi$$
0.175157 + 0.984541i $$0.443957\pi$$
$$132$$ 0 0
$$133$$ 5.48392 0.00357531
$$134$$ 0 0
$$135$$ 222.855 0.142077
$$136$$ 0 0
$$137$$ −2428.12 −1.51422 −0.757109 0.653288i $$-0.773389\pi$$
−0.757109 + 0.653288i $$0.773389\pi$$
$$138$$ 0 0
$$139$$ −2456.46 −1.49895 −0.749476 0.662031i $$-0.769695\pi$$
−0.749476 + 0.662031i $$0.769695\pi$$
$$140$$ 0 0
$$141$$ 204.359 0.122058
$$142$$ 0 0
$$143$$ 1288.01 0.753211
$$144$$ 0 0
$$145$$ 686.128 0.392964
$$146$$ 0 0
$$147$$ −1640.32 −0.920349
$$148$$ 0 0
$$149$$ −2405.39 −1.32253 −0.661266 0.750151i $$-0.729981\pi$$
−0.661266 + 0.750151i $$0.729981\pi$$
$$150$$ 0 0
$$151$$ 649.276 0.349916 0.174958 0.984576i $$-0.444021\pi$$
0.174958 + 0.984576i $$0.444021\pi$$
$$152$$ 0 0
$$153$$ 4133.85 2.18433
$$154$$ 0 0
$$155$$ 564.330 0.292439
$$156$$ 0 0
$$157$$ 3665.04 1.86307 0.931536 0.363650i $$-0.118470\pi$$
0.931536 + 0.363650i $$0.118470\pi$$
$$158$$ 0 0
$$159$$ −337.804 −0.168488
$$160$$ 0 0
$$161$$ −541.932 −0.265281
$$162$$ 0 0
$$163$$ 1055.27 0.507088 0.253544 0.967324i $$-0.418404\pi$$
0.253544 + 0.967324i $$0.418404\pi$$
$$164$$ 0 0
$$165$$ 1242.16 0.586071
$$166$$ 0 0
$$167$$ 731.805 0.339095 0.169547 0.985522i $$-0.445769\pi$$
0.169547 + 0.985522i $$0.445769\pi$$
$$168$$ 0 0
$$169$$ −590.507 −0.268779
$$170$$ 0 0
$$171$$ −7.62587 −0.00341032
$$172$$ 0 0
$$173$$ −521.773 −0.229304 −0.114652 0.993406i $$-0.536575\pi$$
−0.114652 + 0.993406i $$0.536575\pi$$
$$174$$ 0 0
$$175$$ −589.056 −0.254448
$$176$$ 0 0
$$177$$ 1563.54 0.663970
$$178$$ 0 0
$$179$$ 1183.07 0.494005 0.247002 0.969015i $$-0.420554\pi$$
0.247002 + 0.969015i $$0.420554\pi$$
$$180$$ 0 0
$$181$$ 1723.92 0.707946 0.353973 0.935256i $$-0.384830\pi$$
0.353973 + 0.935256i $$0.384830\pi$$
$$182$$ 0 0
$$183$$ −1161.78 −0.469295
$$184$$ 0 0
$$185$$ −228.529 −0.0908204
$$186$$ 0 0
$$187$$ 4054.35 1.58547
$$188$$ 0 0
$$189$$ 1050.20 0.404182
$$190$$ 0 0
$$191$$ 2798.12 1.06002 0.530012 0.847990i $$-0.322187\pi$$
0.530012 + 0.847990i $$0.322187\pi$$
$$192$$ 0 0
$$193$$ −497.686 −0.185618 −0.0928089 0.995684i $$-0.529585\pi$$
−0.0928089 + 0.995684i $$0.529585\pi$$
$$194$$ 0 0
$$195$$ 1549.30 0.568961
$$196$$ 0 0
$$197$$ 296.489 0.107228 0.0536141 0.998562i $$-0.482926\pi$$
0.0536141 + 0.998562i $$0.482926\pi$$
$$198$$ 0 0
$$199$$ −2347.43 −0.836207 −0.418103 0.908399i $$-0.637305\pi$$
−0.418103 + 0.908399i $$0.637305\pi$$
$$200$$ 0 0
$$201$$ −3252.52 −1.14137
$$202$$ 0 0
$$203$$ 3233.34 1.11791
$$204$$ 0 0
$$205$$ 676.923 0.230626
$$206$$ 0 0
$$207$$ 753.604 0.253039
$$208$$ 0 0
$$209$$ −7.47920 −0.00247535
$$210$$ 0 0
$$211$$ −2838.61 −0.926153 −0.463076 0.886318i $$-0.653254\pi$$
−0.463076 + 0.886318i $$0.653254\pi$$
$$212$$ 0 0
$$213$$ 5159.39 1.65970
$$214$$ 0 0
$$215$$ 2717.64 0.862053
$$216$$ 0 0
$$217$$ 2659.38 0.831937
$$218$$ 0 0
$$219$$ −4661.37 −1.43829
$$220$$ 0 0
$$221$$ 5056.84 1.53918
$$222$$ 0 0
$$223$$ 2124.68 0.638024 0.319012 0.947751i $$-0.396649\pi$$
0.319012 + 0.947751i $$0.396649\pi$$
$$224$$ 0 0
$$225$$ 819.135 0.242707
$$226$$ 0 0
$$227$$ −1551.97 −0.453780 −0.226890 0.973920i $$-0.572856\pi$$
−0.226890 + 0.973920i $$0.572856\pi$$
$$228$$ 0 0
$$229$$ −158.531 −0.0457469 −0.0228735 0.999738i $$-0.507281\pi$$
−0.0228735 + 0.999738i $$0.507281\pi$$
$$230$$ 0 0
$$231$$ 5853.60 1.66727
$$232$$ 0 0
$$233$$ −728.999 −0.204971 −0.102486 0.994734i $$-0.532680\pi$$
−0.102486 + 0.994734i $$0.532680\pi$$
$$234$$ 0 0
$$235$$ 132.172 0.0366891
$$236$$ 0 0
$$237$$ −10659.8 −2.92165
$$238$$ 0 0
$$239$$ 6201.60 1.67844 0.839222 0.543789i $$-0.183011\pi$$
0.839222 + 0.543789i $$0.183011\pi$$
$$240$$ 0 0
$$241$$ −7223.04 −1.93061 −0.965305 0.261126i $$-0.915906\pi$$
−0.965305 + 0.261126i $$0.915906\pi$$
$$242$$ 0 0
$$243$$ 5378.79 1.41996
$$244$$ 0 0
$$245$$ −1060.90 −0.276646
$$246$$ 0 0
$$247$$ −9.32853 −0.00240308
$$248$$ 0 0
$$249$$ −3750.82 −0.954612
$$250$$ 0 0
$$251$$ −5042.78 −1.26812 −0.634059 0.773285i $$-0.718612\pi$$
−0.634059 + 0.773285i $$0.718612\pi$$
$$252$$ 0 0
$$253$$ 739.110 0.183666
$$254$$ 0 0
$$255$$ 4876.80 1.19763
$$256$$ 0 0
$$257$$ 2981.15 0.723577 0.361788 0.932260i $$-0.382166\pi$$
0.361788 + 0.932260i $$0.382166\pi$$
$$258$$ 0 0
$$259$$ −1076.93 −0.258367
$$260$$ 0 0
$$261$$ −4496.25 −1.06632
$$262$$ 0 0
$$263$$ −7242.86 −1.69815 −0.849076 0.528271i $$-0.822841\pi$$
−0.849076 + 0.528271i $$0.822841\pi$$
$$264$$ 0 0
$$265$$ −218.479 −0.0506456
$$266$$ 0 0
$$267$$ 8718.10 1.99827
$$268$$ 0 0
$$269$$ −2567.58 −0.581962 −0.290981 0.956729i $$-0.593982\pi$$
−0.290981 + 0.956729i $$0.593982\pi$$
$$270$$ 0 0
$$271$$ −8066.27 −1.80808 −0.904042 0.427443i $$-0.859414\pi$$
−0.904042 + 0.427443i $$0.859414\pi$$
$$272$$ 0 0
$$273$$ 7300.98 1.61859
$$274$$ 0 0
$$275$$ 803.380 0.176166
$$276$$ 0 0
$$277$$ 8991.54 1.95036 0.975179 0.221418i $$-0.0710684\pi$$
0.975179 + 0.221418i $$0.0710684\pi$$
$$278$$ 0 0
$$279$$ −3698.10 −0.793546
$$280$$ 0 0
$$281$$ 968.130 0.205529 0.102765 0.994706i $$-0.467231\pi$$
0.102765 + 0.994706i $$0.467231\pi$$
$$282$$ 0 0
$$283$$ 2252.65 0.473167 0.236583 0.971611i $$-0.423972\pi$$
0.236583 + 0.971611i $$0.423972\pi$$
$$284$$ 0 0
$$285$$ −8.99640 −0.00186983
$$286$$ 0 0
$$287$$ 3189.97 0.656090
$$288$$ 0 0
$$289$$ 11004.7 2.23991
$$290$$ 0 0
$$291$$ 11489.8 2.31458
$$292$$ 0 0
$$293$$ 2734.35 0.545196 0.272598 0.962128i $$-0.412117\pi$$
0.272598 + 0.962128i $$0.412117\pi$$
$$294$$ 0 0
$$295$$ 1011.24 0.199582
$$296$$ 0 0
$$297$$ −1432.30 −0.279834
$$298$$ 0 0
$$299$$ 921.865 0.178304
$$300$$ 0 0
$$301$$ 12806.7 2.45239
$$302$$ 0 0
$$303$$ −14602.4 −2.76860
$$304$$ 0 0
$$305$$ −751.394 −0.141065
$$306$$ 0 0
$$307$$ 7977.09 1.48299 0.741493 0.670961i $$-0.234118\pi$$
0.741493 + 0.670961i $$0.234118\pi$$
$$308$$ 0 0
$$309$$ −11574.3 −2.13087
$$310$$ 0 0
$$311$$ −7729.43 −1.40931 −0.704655 0.709550i $$-0.748898\pi$$
−0.704655 + 0.709550i $$0.748898\pi$$
$$312$$ 0 0
$$313$$ 5506.25 0.994350 0.497175 0.867650i $$-0.334371\pi$$
0.497175 + 0.867650i $$0.334371\pi$$
$$314$$ 0 0
$$315$$ 3860.13 0.690456
$$316$$ 0 0
$$317$$ 5231.37 0.926887 0.463443 0.886126i $$-0.346614\pi$$
0.463443 + 0.886126i $$0.346614\pi$$
$$318$$ 0 0
$$319$$ −4409.77 −0.773981
$$320$$ 0 0
$$321$$ −4523.68 −0.786565
$$322$$ 0 0
$$323$$ −29.3639 −0.00505836
$$324$$ 0 0
$$325$$ 1002.03 0.171023
$$326$$ 0 0
$$327$$ 1075.86 0.181943
$$328$$ 0 0
$$329$$ 622.854 0.104374
$$330$$ 0 0
$$331$$ 3551.61 0.589771 0.294885 0.955533i $$-0.404719\pi$$
0.294885 + 0.955533i $$0.404719\pi$$
$$332$$ 0 0
$$333$$ 1497.57 0.246445
$$334$$ 0 0
$$335$$ −2103.61 −0.343082
$$336$$ 0 0
$$337$$ −7002.99 −1.13198 −0.565990 0.824412i $$-0.691506\pi$$
−0.565990 + 0.824412i $$0.691506\pi$$
$$338$$ 0 0
$$339$$ 9997.29 1.60171
$$340$$ 0 0
$$341$$ −3626.97 −0.575987
$$342$$ 0 0
$$343$$ 3082.42 0.485234
$$344$$ 0 0
$$345$$ 889.043 0.138738
$$346$$ 0 0
$$347$$ 10268.2 1.58854 0.794272 0.607562i $$-0.207852\pi$$
0.794272 + 0.607562i $$0.207852\pi$$
$$348$$ 0 0
$$349$$ 4515.58 0.692588 0.346294 0.938126i $$-0.387440\pi$$
0.346294 + 0.938126i $$0.387440\pi$$
$$350$$ 0 0
$$351$$ −1786.46 −0.271664
$$352$$ 0 0
$$353$$ 7838.63 1.18189 0.590947 0.806711i $$-0.298754\pi$$
0.590947 + 0.806711i $$0.298754\pi$$
$$354$$ 0 0
$$355$$ 3336.90 0.498886
$$356$$ 0 0
$$357$$ 22981.7 3.40705
$$358$$ 0 0
$$359$$ 12464.8 1.83250 0.916250 0.400606i $$-0.131201\pi$$
0.916250 + 0.400606i $$0.131201\pi$$
$$360$$ 0 0
$$361$$ −6858.95 −0.999992
$$362$$ 0 0
$$363$$ 2306.32 0.333472
$$364$$ 0 0
$$365$$ −3014.80 −0.432334
$$366$$ 0 0
$$367$$ −3936.14 −0.559850 −0.279925 0.960022i $$-0.590310\pi$$
−0.279925 + 0.960022i $$0.590310\pi$$
$$368$$ 0 0
$$369$$ −4435.93 −0.625814
$$370$$ 0 0
$$371$$ −1029.57 −0.144077
$$372$$ 0 0
$$373$$ 1920.72 0.266625 0.133313 0.991074i $$-0.457439\pi$$
0.133313 + 0.991074i $$0.457439\pi$$
$$374$$ 0 0
$$375$$ 966.351 0.133072
$$376$$ 0 0
$$377$$ −5500.15 −0.751385
$$378$$ 0 0
$$379$$ 13074.5 1.77201 0.886004 0.463678i $$-0.153470\pi$$
0.886004 + 0.463678i $$0.153470\pi$$
$$380$$ 0 0
$$381$$ −10037.9 −1.34976
$$382$$ 0 0
$$383$$ −8838.22 −1.17914 −0.589571 0.807716i $$-0.700703\pi$$
−0.589571 + 0.807716i $$0.700703\pi$$
$$384$$ 0 0
$$385$$ 3785.89 0.501160
$$386$$ 0 0
$$387$$ −17808.9 −2.33922
$$388$$ 0 0
$$389$$ −12687.3 −1.65365 −0.826827 0.562456i $$-0.809856\pi$$
−0.826827 + 0.562456i $$0.809856\pi$$
$$390$$ 0 0
$$391$$ 2901.80 0.375321
$$392$$ 0 0
$$393$$ −4060.58 −0.521194
$$394$$ 0 0
$$395$$ −6894.38 −0.878213
$$396$$ 0 0
$$397$$ 8060.49 1.01900 0.509501 0.860470i $$-0.329830\pi$$
0.509501 + 0.860470i $$0.329830\pi$$
$$398$$ 0 0
$$399$$ −42.3951 −0.00531932
$$400$$ 0 0
$$401$$ 12325.1 1.53488 0.767440 0.641121i $$-0.221530\pi$$
0.767440 + 0.641121i $$0.221530\pi$$
$$402$$ 0 0
$$403$$ −4523.79 −0.559171
$$404$$ 0 0
$$405$$ 2700.47 0.331328
$$406$$ 0 0
$$407$$ 1468.76 0.178879
$$408$$ 0 0
$$409$$ −10806.7 −1.30650 −0.653250 0.757143i $$-0.726595\pi$$
−0.653250 + 0.757143i $$0.726595\pi$$
$$410$$ 0 0
$$411$$ 18771.3 2.25285
$$412$$ 0 0
$$413$$ 4765.41 0.567774
$$414$$ 0 0
$$415$$ −2425.89 −0.286945
$$416$$ 0 0
$$417$$ 18990.4 2.23013
$$418$$ 0 0
$$419$$ 1823.74 0.212639 0.106319 0.994332i $$-0.466093\pi$$
0.106319 + 0.994332i $$0.466093\pi$$
$$420$$ 0 0
$$421$$ 5995.43 0.694060 0.347030 0.937854i $$-0.387190\pi$$
0.347030 + 0.937854i $$0.387190\pi$$
$$422$$ 0 0
$$423$$ −866.133 −0.0995575
$$424$$ 0 0
$$425$$ 3154.13 0.359995
$$426$$ 0 0
$$427$$ −3540.91 −0.401303
$$428$$ 0 0
$$429$$ −9957.39 −1.12062
$$430$$ 0 0
$$431$$ −3887.80 −0.434499 −0.217249 0.976116i $$-0.569708\pi$$
−0.217249 + 0.976116i $$0.569708\pi$$
$$432$$ 0 0
$$433$$ −4422.93 −0.490884 −0.245442 0.969411i $$-0.578933\pi$$
−0.245442 + 0.969411i $$0.578933\pi$$
$$434$$ 0 0
$$435$$ −5304.32 −0.584650
$$436$$ 0 0
$$437$$ −5.35306 −0.000585976 0
$$438$$ 0 0
$$439$$ 1748.08 0.190048 0.0950242 0.995475i $$-0.469707\pi$$
0.0950242 + 0.995475i $$0.469707\pi$$
$$440$$ 0 0
$$441$$ 6952.14 0.750690
$$442$$ 0 0
$$443$$ −3371.31 −0.361571 −0.180785 0.983523i $$-0.557864\pi$$
−0.180785 + 0.983523i $$0.557864\pi$$
$$444$$ 0 0
$$445$$ 5638.55 0.600658
$$446$$ 0 0
$$447$$ 18595.6 1.96766
$$448$$ 0 0
$$449$$ 13314.7 1.39947 0.699734 0.714404i $$-0.253302\pi$$
0.699734 + 0.714404i $$0.253302\pi$$
$$450$$ 0 0
$$451$$ −4350.61 −0.454240
$$452$$ 0 0
$$453$$ −5019.43 −0.520603
$$454$$ 0 0
$$455$$ 4722.00 0.486529
$$456$$ 0 0
$$457$$ 3767.20 0.385607 0.192803 0.981237i $$-0.438242\pi$$
0.192803 + 0.981237i $$0.438242\pi$$
$$458$$ 0 0
$$459$$ −5623.32 −0.571839
$$460$$ 0 0
$$461$$ 9674.06 0.977366 0.488683 0.872461i $$-0.337477\pi$$
0.488683 + 0.872461i $$0.337477\pi$$
$$462$$ 0 0
$$463$$ −2977.73 −0.298892 −0.149446 0.988770i $$-0.547749\pi$$
−0.149446 + 0.988770i $$0.547749\pi$$
$$464$$ 0 0
$$465$$ −4362.73 −0.435089
$$466$$ 0 0
$$467$$ 10701.0 1.06035 0.530176 0.847888i $$-0.322126\pi$$
0.530176 + 0.847888i $$0.322126\pi$$
$$468$$ 0 0
$$469$$ −9913.16 −0.976006
$$470$$ 0 0
$$471$$ −28333.7 −2.77187
$$472$$ 0 0
$$473$$ −17466.4 −1.69790
$$474$$ 0 0
$$475$$ −5.81854 −0.000562048 0
$$476$$ 0 0
$$477$$ 1431.71 0.137429
$$478$$ 0 0
$$479$$ −2337.15 −0.222937 −0.111469 0.993768i $$-0.535555\pi$$
−0.111469 + 0.993768i $$0.535555\pi$$
$$480$$ 0 0
$$481$$ 1831.94 0.173657
$$482$$ 0 0
$$483$$ 4189.57 0.394683
$$484$$ 0 0
$$485$$ 7431.18 0.695737
$$486$$ 0 0
$$487$$ 7183.85 0.668442 0.334221 0.942495i $$-0.391527\pi$$
0.334221 + 0.942495i $$0.391527\pi$$
$$488$$ 0 0
$$489$$ −8158.12 −0.754443
$$490$$ 0 0
$$491$$ 12084.2 1.11070 0.555350 0.831617i $$-0.312584\pi$$
0.555350 + 0.831617i $$0.312584\pi$$
$$492$$ 0 0
$$493$$ −17313.1 −1.58163
$$494$$ 0 0
$$495$$ −5264.61 −0.478034
$$496$$ 0 0
$$497$$ 15725.0 1.41924
$$498$$ 0 0
$$499$$ −7145.78 −0.641060 −0.320530 0.947238i $$-0.603861\pi$$
−0.320530 + 0.947238i $$0.603861\pi$$
$$500$$ 0 0
$$501$$ −5657.45 −0.504503
$$502$$ 0 0
$$503$$ −20436.2 −1.81154 −0.905770 0.423771i $$-0.860706\pi$$
−0.905770 + 0.423771i $$0.860706\pi$$
$$504$$ 0 0
$$505$$ −9444.29 −0.832208
$$506$$ 0 0
$$507$$ 4565.10 0.399888
$$508$$ 0 0
$$509$$ 19721.7 1.71738 0.858690 0.512495i $$-0.171279\pi$$
0.858690 + 0.512495i $$0.171279\pi$$
$$510$$ 0 0
$$511$$ −14207.1 −1.22991
$$512$$ 0 0
$$513$$ 10.3735 0.000892794 0
$$514$$ 0 0
$$515$$ −7485.84 −0.640516
$$516$$ 0 0
$$517$$ −849.475 −0.0722628
$$518$$ 0 0
$$519$$ 4033.73 0.341158
$$520$$ 0 0
$$521$$ 3483.23 0.292904 0.146452 0.989218i $$-0.453215\pi$$
0.146452 + 0.989218i $$0.453215\pi$$
$$522$$ 0 0
$$523$$ −15689.0 −1.31172 −0.655862 0.754881i $$-0.727695\pi$$
−0.655862 + 0.754881i $$0.727695\pi$$
$$524$$ 0 0
$$525$$ 4553.88 0.378567
$$526$$ 0 0
$$527$$ −14239.8 −1.17703
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ 0 0
$$531$$ −6626.72 −0.541573
$$532$$ 0 0
$$533$$ −5426.36 −0.440979
$$534$$ 0 0
$$535$$ −2925.75 −0.236432
$$536$$ 0 0
$$537$$ −9146.09 −0.734977
$$538$$ 0 0
$$539$$ 6818.43 0.544881
$$540$$ 0 0
$$541$$ 7620.73 0.605621 0.302810 0.953051i $$-0.402075\pi$$
0.302810 + 0.953051i $$0.402075\pi$$
$$542$$ 0 0
$$543$$ −13327.3 −1.05328
$$544$$ 0 0
$$545$$ 695.828 0.0546899
$$546$$ 0 0
$$547$$ −5925.62 −0.463183 −0.231592 0.972813i $$-0.574393\pi$$
−0.231592 + 0.972813i $$0.574393\pi$$
$$548$$ 0 0
$$549$$ 4923.94 0.382784
$$550$$ 0 0
$$551$$ 31.9381 0.00246934
$$552$$ 0 0
$$553$$ −32489.4 −2.49836
$$554$$ 0 0
$$555$$ 1766.71 0.135122
$$556$$ 0 0
$$557$$ −5456.16 −0.415054 −0.207527 0.978229i $$-0.566541\pi$$
−0.207527 + 0.978229i $$0.566541\pi$$
$$558$$ 0 0
$$559$$ −21785.2 −1.64833
$$560$$ 0 0
$$561$$ −31343.4 −2.35886
$$562$$ 0 0
$$563$$ 9194.29 0.688265 0.344132 0.938921i $$-0.388173\pi$$
0.344132 + 0.938921i $$0.388173\pi$$
$$564$$ 0 0
$$565$$ 6465.88 0.481454
$$566$$ 0 0
$$567$$ 12725.8 0.942567
$$568$$ 0 0
$$569$$ 338.831 0.0249640 0.0124820 0.999922i $$-0.496027\pi$$
0.0124820 + 0.999922i $$0.496027\pi$$
$$570$$ 0 0
$$571$$ 1725.34 0.126451 0.0632254 0.997999i $$-0.479861\pi$$
0.0632254 + 0.997999i $$0.479861\pi$$
$$572$$ 0 0
$$573$$ −21631.7 −1.57710
$$574$$ 0 0
$$575$$ 575.000 0.0417029
$$576$$ 0 0
$$577$$ 23300.0 1.68109 0.840547 0.541738i $$-0.182234\pi$$
0.840547 + 0.541738i $$0.182234\pi$$
$$578$$ 0 0
$$579$$ 3847.52 0.276161
$$580$$ 0 0
$$581$$ −11431.9 −0.816307
$$582$$ 0 0
$$583$$ 1404.18 0.0997513
$$584$$ 0 0
$$585$$ −6566.36 −0.464078
$$586$$ 0 0
$$587$$ −15653.3 −1.10065 −0.550325 0.834950i $$-0.685496\pi$$
−0.550325 + 0.834950i $$0.685496\pi$$
$$588$$ 0 0
$$589$$ 26.2686 0.00183766
$$590$$ 0 0
$$591$$ −2292.10 −0.159533
$$592$$ 0 0
$$593$$ −2658.08 −0.184072 −0.0920358 0.995756i $$-0.529337\pi$$
−0.0920358 + 0.995756i $$0.529337\pi$$
$$594$$ 0 0
$$595$$ 14863.7 1.02412
$$596$$ 0 0
$$597$$ 18147.6 1.24410
$$598$$ 0 0
$$599$$ −19417.6 −1.32451 −0.662256 0.749278i $$-0.730401\pi$$
−0.662256 + 0.749278i $$0.730401\pi$$
$$600$$ 0 0
$$601$$ −18469.0 −1.25352 −0.626760 0.779213i $$-0.715619\pi$$
−0.626760 + 0.779213i $$0.715619\pi$$
$$602$$ 0 0
$$603$$ 13785.1 0.930968
$$604$$ 0 0
$$605$$ 1491.64 0.100238
$$606$$ 0 0
$$607$$ −3968.56 −0.265369 −0.132684 0.991158i $$-0.542360\pi$$
−0.132684 + 0.991158i $$0.542360\pi$$
$$608$$ 0 0
$$609$$ −24996.4 −1.66322
$$610$$ 0 0
$$611$$ −1059.52 −0.0701531
$$612$$ 0 0
$$613$$ −11478.7 −0.756311 −0.378156 0.925742i $$-0.623442\pi$$
−0.378156 + 0.925742i $$0.623442\pi$$
$$614$$ 0 0
$$615$$ −5233.16 −0.343124
$$616$$ 0 0
$$617$$ −15691.1 −1.02382 −0.511911 0.859039i $$-0.671062\pi$$
−0.511911 + 0.859039i $$0.671062\pi$$
$$618$$ 0 0
$$619$$ 18249.4 1.18499 0.592494 0.805575i $$-0.298144\pi$$
0.592494 + 0.805575i $$0.298144\pi$$
$$620$$ 0 0
$$621$$ −1025.14 −0.0662436
$$622$$ 0 0
$$623$$ 26571.4 1.70876
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 57.8203 0.00368281
$$628$$ 0 0
$$629$$ 5766.48 0.365540
$$630$$ 0 0
$$631$$ 23528.8 1.48442 0.742208 0.670169i $$-0.233778\pi$$
0.742208 + 0.670169i $$0.233778\pi$$
$$632$$ 0 0
$$633$$ 21944.8 1.37793
$$634$$ 0 0
$$635$$ −6492.15 −0.405722
$$636$$ 0 0
$$637$$ 8504.38 0.528973
$$638$$ 0 0
$$639$$ −21867.0 −1.35375
$$640$$ 0 0
$$641$$ 3748.79 0.230996 0.115498 0.993308i $$-0.463154\pi$$
0.115498 + 0.993308i $$0.463154\pi$$
$$642$$ 0 0
$$643$$ −15924.3 −0.976660 −0.488330 0.872659i $$-0.662394\pi$$
−0.488330 + 0.872659i $$0.662394\pi$$
$$644$$ 0 0
$$645$$ −21009.5 −1.28256
$$646$$ 0 0
$$647$$ 2854.11 0.173426 0.0867129 0.996233i $$-0.472364\pi$$
0.0867129 + 0.996233i $$0.472364\pi$$
$$648$$ 0 0
$$649$$ −6499.27 −0.393095
$$650$$ 0 0
$$651$$ −20559.1 −1.23775
$$652$$ 0 0
$$653$$ 7925.97 0.474988 0.237494 0.971389i $$-0.423674\pi$$
0.237494 + 0.971389i $$0.423674\pi$$
$$654$$ 0 0
$$655$$ −2626.24 −0.156665
$$656$$ 0 0
$$657$$ 19756.2 1.17316
$$658$$ 0 0
$$659$$ 5799.43 0.342813 0.171406 0.985200i $$-0.445169\pi$$
0.171406 + 0.985200i $$0.445169\pi$$
$$660$$ 0 0
$$661$$ 22324.8 1.31367 0.656833 0.754036i $$-0.271896\pi$$
0.656833 + 0.754036i $$0.271896\pi$$
$$662$$ 0 0
$$663$$ −39093.4 −2.28999
$$664$$ 0 0
$$665$$ −27.4196 −0.00159893
$$666$$ 0 0
$$667$$ −3156.19 −0.183221
$$668$$ 0 0
$$669$$ −16425.5 −0.949249
$$670$$ 0 0
$$671$$ 4829.24 0.277840
$$672$$ 0 0
$$673$$ −23253.5 −1.33188 −0.665940 0.746005i $$-0.731969\pi$$
−0.665940 + 0.746005i $$0.731969\pi$$
$$674$$ 0 0
$$675$$ −1114.28 −0.0635386
$$676$$ 0 0
$$677$$ −19003.7 −1.07884 −0.539419 0.842038i $$-0.681356\pi$$
−0.539419 + 0.842038i $$0.681356\pi$$
$$678$$ 0 0
$$679$$ 35019.1 1.97925
$$680$$ 0 0
$$681$$ 11998.0 0.675131
$$682$$ 0 0
$$683$$ 11222.5 0.628722 0.314361 0.949304i $$-0.398210\pi$$
0.314361 + 0.949304i $$0.398210\pi$$
$$684$$ 0 0
$$685$$ 12140.6 0.677179
$$686$$ 0 0
$$687$$ 1225.58 0.0680620
$$688$$ 0 0
$$689$$ 1751.38 0.0968391
$$690$$ 0 0
$$691$$ −4297.68 −0.236601 −0.118301 0.992978i $$-0.537745\pi$$
−0.118301 + 0.992978i $$0.537745\pi$$
$$692$$ 0 0
$$693$$ −24809.2 −1.35992
$$694$$ 0 0
$$695$$ 12282.3 0.670352
$$696$$ 0 0
$$697$$ −17080.8 −0.928240
$$698$$ 0 0
$$699$$ 5635.75 0.304955
$$700$$ 0 0
$$701$$ 25769.6 1.38845 0.694225 0.719758i $$-0.255747\pi$$
0.694225 + 0.719758i $$0.255747\pi$$
$$702$$ 0 0
$$703$$ −10.6376 −0.000570705 0
$$704$$ 0 0
$$705$$ −1021.80 −0.0545859
$$706$$ 0 0
$$707$$ −44505.7 −2.36748
$$708$$ 0 0
$$709$$ −4900.01 −0.259554 −0.129777 0.991543i $$-0.541426\pi$$
−0.129777 + 0.991543i $$0.541426\pi$$
$$710$$ 0 0
$$711$$ 45179.4 2.38307
$$712$$ 0 0
$$713$$ −2595.92 −0.136350
$$714$$ 0 0
$$715$$ −6440.07 −0.336846
$$716$$ 0 0
$$717$$ −47943.4 −2.49718
$$718$$ 0 0
$$719$$ −7631.85 −0.395855 −0.197928 0.980217i $$-0.563421\pi$$
−0.197928 + 0.980217i $$0.563421\pi$$
$$720$$ 0 0
$$721$$ −35276.6 −1.82215
$$722$$ 0 0
$$723$$ 55839.9 2.87235
$$724$$ 0 0
$$725$$ −3430.64 −0.175739
$$726$$ 0 0
$$727$$ 36985.6 1.88682 0.943410 0.331628i $$-0.107598\pi$$
0.943410 + 0.331628i $$0.107598\pi$$
$$728$$ 0 0
$$729$$ −26999.8 −1.37173
$$730$$ 0 0
$$731$$ −68574.3 −3.46965
$$732$$ 0 0
$$733$$ −22227.3 −1.12003 −0.560015 0.828482i $$-0.689205\pi$$
−0.560015 + 0.828482i $$0.689205\pi$$
$$734$$ 0 0
$$735$$ 8201.60 0.411592
$$736$$ 0 0
$$737$$ 13520.0 0.675733
$$738$$ 0 0
$$739$$ 17899.1 0.890974 0.445487 0.895288i $$-0.353030\pi$$
0.445487 + 0.895288i $$0.353030\pi$$
$$740$$ 0 0
$$741$$ 72.1171 0.00357529
$$742$$ 0 0
$$743$$ 29843.4 1.47355 0.736776 0.676137i $$-0.236347\pi$$
0.736776 + 0.676137i $$0.236347\pi$$
$$744$$ 0 0
$$745$$ 12027.0 0.591455
$$746$$ 0 0
$$747$$ 15897.0 0.778638
$$748$$ 0 0
$$749$$ −13787.4 −0.672606
$$750$$ 0 0
$$751$$ 9399.65 0.456722 0.228361 0.973577i $$-0.426663\pi$$
0.228361 + 0.973577i $$0.426663\pi$$
$$752$$ 0 0
$$753$$ 38984.8 1.88670
$$754$$ 0 0
$$755$$ −3246.38 −0.156487
$$756$$ 0 0
$$757$$ 29871.6 1.43422 0.717109 0.696961i $$-0.245465\pi$$
0.717109 + 0.696961i $$0.245465\pi$$
$$758$$ 0 0
$$759$$ −5713.92 −0.273257
$$760$$ 0 0
$$761$$ −273.955 −0.0130497 −0.00652487 0.999979i $$-0.502077\pi$$
−0.00652487 + 0.999979i $$0.502077\pi$$
$$762$$ 0 0
$$763$$ 3279.05 0.155583
$$764$$ 0 0
$$765$$ −20669.3 −0.976861
$$766$$ 0 0
$$767$$ −8106.31 −0.381619
$$768$$ 0 0
$$769$$ −13273.8 −0.622450 −0.311225 0.950336i $$-0.600739\pi$$
−0.311225 + 0.950336i $$0.600739\pi$$
$$770$$ 0 0
$$771$$ −23046.7 −1.07653
$$772$$ 0 0
$$773$$ 34161.1 1.58951 0.794753 0.606933i $$-0.207600\pi$$
0.794753 + 0.606933i $$0.207600\pi$$
$$774$$ 0 0
$$775$$ −2821.65 −0.130783
$$776$$ 0 0
$$777$$ 8325.54 0.384398
$$778$$ 0 0
$$779$$ 31.5096 0.00144923
$$780$$ 0 0
$$781$$ −21446.4 −0.982603
$$782$$ 0 0
$$783$$ 6116.29 0.279155
$$784$$ 0 0
$$785$$ −18325.2 −0.833191
$$786$$ 0 0
$$787$$ 38489.6 1.74334 0.871669 0.490095i $$-0.163038\pi$$
0.871669 + 0.490095i $$0.163038\pi$$
$$788$$ 0 0
$$789$$ 55993.2 2.52650
$$790$$ 0 0
$$791$$ 30470.1 1.36965
$$792$$ 0 0
$$793$$ 6023.33 0.269729
$$794$$ 0 0
$$795$$ 1689.02 0.0753502
$$796$$ 0 0
$$797$$ 17176.8 0.763405 0.381702 0.924285i $$-0.375338\pi$$
0.381702 + 0.924285i $$0.375338\pi$$
$$798$$ 0 0
$$799$$ −3335.10 −0.147669
$$800$$ 0 0
$$801$$ −36949.8 −1.62991
$$802$$ 0 0
$$803$$ 19376.2 0.851523
$$804$$ 0 0
$$805$$ 2709.66 0.118637
$$806$$ 0 0
$$807$$ 19849.4 0.865841
$$808$$ 0 0
$$809$$ 8226.61 0.357518 0.178759 0.983893i $$-0.442792\pi$$
0.178759 + 0.983893i $$0.442792\pi$$
$$810$$ 0 0
$$811$$ −27867.9 −1.20662 −0.603312 0.797505i $$-0.706153\pi$$
−0.603312 + 0.797505i $$0.706153\pi$$
$$812$$ 0 0
$$813$$ 62358.8 2.69006
$$814$$ 0 0
$$815$$ −5276.37 −0.226777
$$816$$ 0 0
$$817$$ 126.502 0.00541705
$$818$$ 0 0
$$819$$ −30943.6 −1.32022
$$820$$ 0 0
$$821$$ −17637.5 −0.749761 −0.374880 0.927073i $$-0.622316\pi$$
−0.374880 + 0.927073i $$0.622316\pi$$
$$822$$ 0 0
$$823$$ −28048.6 −1.18799 −0.593993 0.804470i $$-0.702449\pi$$
−0.593993 + 0.804470i $$0.702449\pi$$
$$824$$ 0 0
$$825$$ −6210.78 −0.262099
$$826$$ 0 0
$$827$$ 32592.9 1.37046 0.685228 0.728329i $$-0.259703\pi$$
0.685228 + 0.728329i $$0.259703\pi$$
$$828$$ 0 0
$$829$$ 18815.9 0.788303 0.394152 0.919045i $$-0.371038\pi$$
0.394152 + 0.919045i $$0.371038\pi$$
$$830$$ 0 0
$$831$$ −69511.9 −2.90173
$$832$$ 0 0
$$833$$ 26769.7 1.11346
$$834$$ 0 0
$$835$$ −3659.03 −0.151648
$$836$$ 0 0
$$837$$ 5030.56 0.207744
$$838$$ 0 0
$$839$$ −7612.72 −0.313254 −0.156627 0.987658i $$-0.550062\pi$$
−0.156627 + 0.987658i $$0.550062\pi$$
$$840$$ 0 0
$$841$$ −5558.14 −0.227896
$$842$$ 0 0
$$843$$ −7484.43 −0.305786
$$844$$ 0 0
$$845$$ 2952.54 0.120202
$$846$$ 0 0
$$847$$ 7029.28 0.285158
$$848$$ 0 0
$$849$$ −17414.8 −0.703975
$$850$$ 0 0
$$851$$ 1051.23 0.0423452
$$852$$ 0 0
$$853$$ −31421.4 −1.26125 −0.630627 0.776086i $$-0.717202\pi$$
−0.630627 + 0.776086i $$0.717202\pi$$
$$854$$ 0 0
$$855$$ 38.1293 0.00152514
$$856$$ 0 0
$$857$$ 20909.5 0.833435 0.416718 0.909036i $$-0.363180\pi$$
0.416718 + 0.909036i $$0.363180\pi$$
$$858$$ 0 0
$$859$$ −23304.5 −0.925655 −0.462828 0.886448i $$-0.653165\pi$$
−0.462828 + 0.886448i $$0.653165\pi$$
$$860$$ 0 0
$$861$$ −24661.0 −0.976127
$$862$$ 0 0
$$863$$ 19146.3 0.755211 0.377606 0.925966i $$-0.376747\pi$$
0.377606 + 0.925966i $$0.376747\pi$$
$$864$$ 0 0
$$865$$ 2608.86 0.102548
$$866$$ 0 0
$$867$$ −85075.0 −3.33252
$$868$$ 0 0
$$869$$ 44310.5 1.72972
$$870$$ 0 0
$$871$$ 16863.0 0.656005
$$872$$ 0 0
$$873$$ −48697.1 −1.88791
$$874$$ 0 0
$$875$$ 2945.28 0.113793
$$876$$ 0 0
$$877$$ 31389.0 1.20859 0.604293 0.796762i $$-0.293456\pi$$
0.604293 + 0.796762i $$0.293456\pi$$
$$878$$ 0 0
$$879$$ −21138.7 −0.811139
$$880$$ 0 0
$$881$$ 44496.3 1.70161 0.850806 0.525480i $$-0.176114\pi$$
0.850806 + 0.525480i $$0.176114\pi$$
$$882$$ 0 0
$$883$$ −8906.65 −0.339448 −0.169724 0.985492i $$-0.554288\pi$$
−0.169724 + 0.985492i $$0.554288\pi$$
$$884$$ 0 0
$$885$$ −7817.69 −0.296936
$$886$$ 0 0
$$887$$ 36584.7 1.38489 0.692444 0.721472i $$-0.256534\pi$$
0.692444 + 0.721472i $$0.256534\pi$$
$$888$$ 0 0
$$889$$ −30593.9 −1.15420
$$890$$ 0 0
$$891$$ −17356.1 −0.652581
$$892$$ 0 0
$$893$$ 6.15238 0.000230551 0
$$894$$ 0 0
$$895$$ −5915.35 −0.220926
$$896$$ 0 0
$$897$$ −7126.76 −0.265279
$$898$$ 0 0
$$899$$ 15488.1 0.574591
$$900$$ 0 0
$$901$$ 5512.90 0.203842
$$902$$ 0 0
$$903$$ −99006.4 −3.64865
$$904$$ 0 0
$$905$$ −8619.62 −0.316603
$$906$$ 0 0
$$907$$ 21346.1 0.781463 0.390731 0.920505i $$-0.372222\pi$$
0.390731 + 0.920505i $$0.372222\pi$$
$$908$$ 0 0
$$909$$ 61889.1 2.25823
$$910$$ 0 0
$$911$$ 10208.0 0.371246 0.185623 0.982621i $$-0.440570\pi$$
0.185623 + 0.982621i $$0.440570\pi$$
$$912$$ 0 0
$$913$$ 15591.3 0.565166
$$914$$ 0 0
$$915$$ 5808.88 0.209875
$$916$$ 0 0
$$917$$ −12376.0 −0.445683
$$918$$ 0 0
$$919$$ −1758.00 −0.0631025 −0.0315513 0.999502i $$-0.510045\pi$$
−0.0315513 + 0.999502i $$0.510045\pi$$
$$920$$ 0 0
$$921$$ −61669.3 −2.20638
$$922$$ 0 0
$$923$$ −26749.3 −0.953917
$$924$$ 0 0
$$925$$ 1142.64 0.0406161
$$926$$ 0 0
$$927$$ 49055.3 1.73807
$$928$$ 0 0
$$929$$ −845.008 −0.0298426 −0.0149213 0.999889i $$-0.504750\pi$$
−0.0149213 + 0.999889i $$0.504750\pi$$
$$930$$ 0 0
$$931$$ −49.3830 −0.00173841
$$932$$ 0 0
$$933$$ 59754.7 2.09676
$$934$$ 0 0
$$935$$ −20271.7 −0.709045
$$936$$ 0 0
$$937$$ 3851.61 0.134287 0.0671433 0.997743i $$-0.478612\pi$$
0.0671433 + 0.997743i $$0.478612\pi$$
$$938$$ 0 0
$$939$$ −42567.7 −1.47939
$$940$$ 0 0
$$941$$ −4379.36 −0.151714 −0.0758571 0.997119i $$-0.524169\pi$$
−0.0758571 + 0.997119i $$0.524169\pi$$
$$942$$ 0 0
$$943$$ −3113.85 −0.107530
$$944$$ 0 0
$$945$$ −5250.98 −0.180756
$$946$$ 0 0
$$947$$ −29746.9 −1.02074 −0.510371 0.859954i $$-0.670492\pi$$
−0.510371 + 0.859954i $$0.670492\pi$$
$$948$$ 0 0
$$949$$ 24167.3 0.826663
$$950$$ 0 0
$$951$$ −40442.7 −1.37902
$$952$$ 0 0
$$953$$ 4306.61 0.146385 0.0731924 0.997318i $$-0.476681\pi$$
0.0731924 + 0.997318i $$0.476681\pi$$
$$954$$ 0 0
$$955$$ −13990.6 −0.474057
$$956$$ 0 0
$$957$$ 34091.1 1.15152
$$958$$ 0 0
$$959$$ 57211.9 1.92645
$$960$$ 0 0
$$961$$ −17052.3 −0.572397
$$962$$ 0 0
$$963$$ 19172.7 0.641568
$$964$$ 0 0
$$965$$ 2488.43 0.0830108
$$966$$ 0 0
$$967$$ 12233.4 0.406824 0.203412 0.979093i $$-0.434797\pi$$
0.203412 + 0.979093i $$0.434797\pi$$
$$968$$ 0 0
$$969$$ 227.007 0.00752581
$$970$$ 0 0
$$971$$ −48207.7 −1.59326 −0.796631 0.604465i $$-0.793387\pi$$
−0.796631 + 0.604465i $$0.793387\pi$$
$$972$$ 0 0
$$973$$ 57879.8 1.90703
$$974$$ 0 0
$$975$$ −7746.48 −0.254447
$$976$$ 0 0
$$977$$ 28662.9 0.938595 0.469298 0.883040i $$-0.344507\pi$$
0.469298 + 0.883040i $$0.344507\pi$$
$$978$$ 0 0
$$979$$ −36239.2 −1.18305
$$980$$ 0 0
$$981$$ −4559.81 −0.148403
$$982$$ 0 0
$$983$$ 20944.1 0.679567 0.339783 0.940504i $$-0.389646\pi$$
0.339783 + 0.940504i $$0.389646\pi$$
$$984$$ 0 0
$$985$$ −1482.44 −0.0479539
$$986$$ 0 0
$$987$$ −4815.16 −0.155287
$$988$$ 0 0
$$989$$ −12501.1 −0.401934
$$990$$ 0 0
$$991$$ 36440.2 1.16808 0.584038 0.811727i $$-0.301472\pi$$
0.584038 + 0.811727i $$0.301472\pi$$
$$992$$ 0 0
$$993$$ −27456.8 −0.877458
$$994$$ 0 0
$$995$$ 11737.2 0.373963
$$996$$ 0 0
$$997$$ −11945.0 −0.379439 −0.189720 0.981838i $$-0.560758\pi$$
−0.189720 + 0.981838i $$0.560758\pi$$
$$998$$ 0 0
$$999$$ −2037.15 −0.0645172
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 1840.4.a.m.1.1 4
4.3 odd 2 230.4.a.h.1.4 4
12.11 even 2 2070.4.a.bj.1.3 4
20.3 even 4 1150.4.b.n.599.8 8
20.7 even 4 1150.4.b.n.599.1 8
20.19 odd 2 1150.4.a.p.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 4.3 odd 2
1150.4.a.p.1.1 4 20.19 odd 2
1150.4.b.n.599.1 8 20.7 even 4
1150.4.b.n.599.8 8 20.3 even 4
1840.4.a.m.1.1 4 1.1 even 1 trivial
2070.4.a.bj.1.3 4 12.11 even 2