# Properties

 Label 1150.4.a.p.1.1 Level $1150$ Weight $4$ Character 1150.1 Self dual yes Analytic conductor $67.852$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.8521965066$$ 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 $$\chi$$ $$=$$ 1150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -7.73081 q^{3} +4.00000 q^{4} -15.4616 q^{6} -23.5622 q^{7} +8.00000 q^{8} +32.7654 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -7.73081 q^{3} +4.00000 q^{4} -15.4616 q^{6} -23.5622 q^{7} +8.00000 q^{8} +32.7654 q^{9} -32.1352 q^{11} -30.9232 q^{12} -40.0811 q^{13} -47.1245 q^{14} +16.0000 q^{16} -126.165 q^{17} +65.5308 q^{18} +0.232742 q^{19} +182.155 q^{21} -64.2704 q^{22} +23.0000 q^{23} -61.8465 q^{24} -80.1621 q^{26} -44.5711 q^{27} -94.2490 q^{28} -137.226 q^{29} +112.866 q^{31} +32.0000 q^{32} +248.431 q^{33} -252.331 q^{34} +131.062 q^{36} -45.7057 q^{37} +0.465483 q^{38} +309.859 q^{39} -135.385 q^{41} +364.310 q^{42} -543.528 q^{43} -128.541 q^{44} +46.0000 q^{46} -26.4344 q^{47} -123.693 q^{48} +212.180 q^{49} +975.359 q^{51} -160.324 q^{52} -43.6958 q^{53} -89.1422 q^{54} -188.498 q^{56} -1.79928 q^{57} -274.451 q^{58} +202.248 q^{59} +150.279 q^{61} +225.732 q^{62} -772.026 q^{63} +64.0000 q^{64} +496.862 q^{66} +420.722 q^{67} -504.661 q^{68} -177.809 q^{69} +667.381 q^{71} +262.123 q^{72} -602.960 q^{73} -91.4115 q^{74} +0.930966 q^{76} +757.178 q^{77} +619.718 q^{78} -1378.88 q^{79} -540.095 q^{81} -270.769 q^{82} +485.178 q^{83} +728.621 q^{84} -1087.06 q^{86} +1060.86 q^{87} -257.082 q^{88} -1127.71 q^{89} +944.400 q^{91} +92.0000 q^{92} -872.545 q^{93} -52.8688 q^{94} -247.386 q^{96} +1486.24 q^{97} +424.359 q^{98} -1052.92 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 8 q^{6} + q^{7} + 32 q^{8} + 32 q^{9} + O(q^{10})$$ $$4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 8 q^{6} + q^{7} + 32 q^{8} + 32 q^{9} - 39 q^{11} + 16 q^{12} + 20 q^{13} + 2 q^{14} + 64 q^{16} + 23 q^{17} + 64 q^{18} + 53 q^{19} + 300 q^{21} - 78 q^{22} + 92 q^{23} + 32 q^{24} + 40 q^{26} - 137 q^{27} + 4 q^{28} + 161 q^{29} + 388 q^{31} + 128 q^{32} - 87 q^{33} + 46 q^{34} + 128 q^{36} - 466 q^{37} + 106 q^{38} + 1047 q^{39} + 484 q^{41} + 600 q^{42} - 894 q^{43} - 156 q^{44} + 184 q^{46} + 265 q^{47} + 64 q^{48} + 1643 q^{49} + 1825 q^{51} + 80 q^{52} - 576 q^{53} - 274 q^{54} + 8 q^{56} - 178 q^{57} + 322 q^{58} - 94 q^{59} + 1153 q^{61} + 776 q^{62} - 60 q^{63} + 256 q^{64} - 174 q^{66} + 1472 q^{67} + 92 q^{68} + 92 q^{69} + 200 q^{71} + 256 q^{72} - 1147 q^{73} - 932 q^{74} + 212 q^{76} + 2176 q^{77} + 2094 q^{78} - 908 q^{79} - 1056 q^{81} + 968 q^{82} + 1048 q^{83} + 1200 q^{84} - 1788 q^{86} + 2167 q^{87} - 312 q^{88} - 1784 q^{89} + 2329 q^{91} + 368 q^{92} - 1483 q^{93} + 530 q^{94} + 128 q^{96} + 2047 q^{97} + 3286 q^{98} - 2665 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ −7.73081 −1.48779 −0.743897 0.668294i $$-0.767025\pi$$
−0.743897 + 0.668294i $$0.767025\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ −15.4616 −1.05203
$$7$$ −23.5622 −1.27224 −0.636121 0.771589i $$-0.719462\pi$$
−0.636121 + 0.771589i $$0.719462\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 32.7654 1.21353
$$10$$ 0 0
$$11$$ −32.1352 −0.880830 −0.440415 0.897794i $$-0.645169\pi$$
−0.440415 + 0.897794i $$0.645169\pi$$
$$12$$ −30.9232 −0.743897
$$13$$ −40.0811 −0.855115 −0.427557 0.903988i $$-0.640626\pi$$
−0.427557 + 0.903988i $$0.640626\pi$$
$$14$$ −47.1245 −0.899611
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −126.165 −1.79997 −0.899987 0.435916i $$-0.856424\pi$$
−0.899987 + 0.435916i $$0.856424\pi$$
$$18$$ 65.5308 0.858097
$$19$$ 0.232742 0.00281024 0.00140512 0.999999i $$-0.499553\pi$$
0.00140512 + 0.999999i $$0.499553\pi$$
$$20$$ 0 0
$$21$$ 182.155 1.89283
$$22$$ −64.2704 −0.622841
$$23$$ 23.0000 0.208514
$$24$$ −61.8465 −0.526015
$$25$$ 0 0
$$26$$ −80.1621 −0.604657
$$27$$ −44.5711 −0.317693
$$28$$ −94.2490 −0.636121
$$29$$ −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 $$-0.393977\pi$$
0.326957 + 0.945039i $$0.393977\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 248.431 1.31049
$$34$$ −252.331 −1.27277
$$35$$ 0 0
$$36$$ 131.062 0.606766
$$37$$ −45.7057 −0.203080 −0.101540 0.994831i $$-0.532377\pi$$
−0.101540 + 0.994831i $$0.532377\pi$$
$$38$$ 0.465483 0.00198714
$$39$$ 309.859 1.27223
$$40$$ 0 0
$$41$$ −135.385 −0.515696 −0.257848 0.966186i $$-0.583013\pi$$
−0.257848 + 0.966186i $$0.583013\pi$$
$$42$$ 364.310 1.33844
$$43$$ −543.528 −1.92761 −0.963805 0.266608i $$-0.914097\pi$$
−0.963805 + 0.266608i $$0.914097\pi$$
$$44$$ −128.541 −0.440415
$$45$$ 0 0
$$46$$ 46.0000 0.147442
$$47$$ −26.4344 −0.0820394 −0.0410197 0.999158i $$-0.513061\pi$$
−0.0410197 + 0.999158i $$0.513061\pi$$
$$48$$ −123.693 −0.371949
$$49$$ 212.180 0.618599
$$50$$ 0 0
$$51$$ 975.359 2.67799
$$52$$ −160.324 −0.427557
$$53$$ −43.6958 −0.113247 −0.0566235 0.998396i $$-0.518033\pi$$
−0.0566235 + 0.998396i $$0.518033\pi$$
$$54$$ −89.1422 −0.224643
$$55$$ 0 0
$$56$$ −188.498 −0.449805
$$57$$ −1.79928 −0.00418106
$$58$$ −274.451 −0.621331
$$59$$ 202.248 0.446278 0.223139 0.974787i $$-0.428370\pi$$
0.223139 + 0.974787i $$0.428370\pi$$
$$60$$ 0 0
$$61$$ 150.279 0.315430 0.157715 0.987485i $$-0.449587\pi$$
0.157715 + 0.987485i $$0.449587\pi$$
$$62$$ 225.732 0.462387
$$63$$ −772.026 −1.54391
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 496.862 0.926659
$$67$$ 420.722 0.767155 0.383577 0.923509i $$-0.374692\pi$$
0.383577 + 0.923509i $$0.374692\pi$$
$$68$$ −504.661 −0.899987
$$69$$ −177.809 −0.310227
$$70$$ 0 0
$$71$$ 667.381 1.11554 0.557771 0.829995i $$-0.311657\pi$$
0.557771 + 0.829995i $$0.311657\pi$$
$$72$$ 262.123 0.429049
$$73$$ −602.960 −0.966728 −0.483364 0.875420i $$-0.660585\pi$$
−0.483364 + 0.875420i $$0.660585\pi$$
$$74$$ −91.4115 −0.143600
$$75$$ 0 0
$$76$$ 0.930966 0.00140512
$$77$$ 757.178 1.12063
$$78$$ 619.718 0.899606
$$79$$ −1378.88 −1.96374 −0.981872 0.189545i $$-0.939299\pi$$
−0.981872 + 0.189545i $$0.939299\pi$$
$$80$$ 0 0
$$81$$ −540.095 −0.740871
$$82$$ −270.769 −0.364652
$$83$$ 485.178 0.641629 0.320815 0.947142i $$-0.396043\pi$$
0.320815 + 0.947142i $$0.396043\pi$$
$$84$$ 728.621 0.946417
$$85$$ 0 0
$$86$$ −1087.06 −1.36303
$$87$$ 1060.86 1.30732
$$88$$ −257.082 −0.311420
$$89$$ −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
$$92$$ 92.0000 0.104257
$$93$$ −872.545 −0.972890
$$94$$ −52.8688 −0.0580106
$$95$$ 0 0
$$96$$ −247.386 −0.263007
$$97$$ 1486.24 1.55572 0.777858 0.628441i $$-0.216306\pi$$
0.777858 + 0.628441i $$0.216306\pi$$
$$98$$ 424.359 0.437416
$$99$$ −1052.92 −1.06892
$$100$$ 0 0
$$101$$ 1888.86 1.86087 0.930437 0.366451i $$-0.119427\pi$$
0.930437 + 0.366451i $$0.119427\pi$$
$$102$$ 1950.72 1.89363
$$103$$ 1497.17 1.43224 0.716118 0.697979i $$-0.245917\pi$$
0.716118 + 0.697979i $$0.245917\pi$$
$$104$$ −320.649 −0.302329
$$105$$ 0 0
$$106$$ −87.3917 −0.0800777
$$107$$ 585.150 0.528678 0.264339 0.964430i $$-0.414846\pi$$
0.264339 + 0.964430i $$0.414846\pi$$
$$108$$ −178.284 −0.158846
$$109$$ −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
$$112$$ −376.996 −0.318060
$$113$$ 1293.18 1.07656 0.538282 0.842765i $$-0.319073\pi$$
0.538282 + 0.842765i $$0.319073\pi$$
$$114$$ −3.59856 −0.00295646
$$115$$ 0 0
$$116$$ −548.902 −0.439347
$$117$$ −1313.27 −1.03771
$$118$$ 404.495 0.315566
$$119$$ 2972.74 2.29000
$$120$$ 0 0
$$121$$ −298.328 −0.224138
$$122$$ 300.557 0.223043
$$123$$ 1046.63 0.767250
$$124$$ 451.464 0.326957
$$125$$ 0 0
$$126$$ −1544.05 −1.09171
$$127$$ 1298.43 0.907221 0.453610 0.891200i $$-0.350136\pi$$
0.453610 + 0.891200i $$0.350136\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 4201.91 2.86789
$$130$$ 0 0
$$131$$ −525.247 −0.350313 −0.175157 0.984541i $$-0.556043\pi$$
−0.175157 + 0.984541i $$0.556043\pi$$
$$132$$ 993.725 0.655247
$$133$$ −5.48392 −0.00357531
$$134$$ 841.444 0.542460
$$135$$ 0 0
$$136$$ −1009.32 −0.636387
$$137$$ 2428.12 1.51422 0.757109 0.653288i $$-0.226611\pi$$
0.757109 + 0.653288i $$0.226611\pi$$
$$138$$ −355.617 −0.219363
$$139$$ 2456.46 1.49895 0.749476 0.662031i $$-0.230305\pi$$
0.749476 + 0.662031i $$0.230305\pi$$
$$140$$ 0 0
$$141$$ 204.359 0.122058
$$142$$ 1334.76 0.788808
$$143$$ 1288.01 0.753211
$$144$$ 524.246 0.303383
$$145$$ 0 0
$$146$$ −1205.92 −0.683580
$$147$$ −1640.32 −0.920349
$$148$$ −182.823 −0.101540
$$149$$ −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 $$-0.555979\pi$$
−0.174958 + 0.984576i $$0.555979\pi$$
$$152$$ 1.86193 0.000993570 0
$$153$$ −4133.85 −2.18433
$$154$$ 1514.36 0.792404
$$155$$ 0 0
$$156$$ 1239.44 0.636117
$$157$$ −3665.04 −1.86307 −0.931536 0.363650i $$-0.881530\pi$$
−0.931536 + 0.363650i $$0.881530\pi$$
$$158$$ −2757.75 −1.38858
$$159$$ 337.804 0.168488
$$160$$ 0 0
$$161$$ −541.932 −0.265281
$$162$$ −1080.19 −0.523875
$$163$$ 1055.27 0.507088 0.253544 0.967324i $$-0.418404\pi$$
0.253544 + 0.967324i $$0.418404\pi$$
$$164$$ −541.539 −0.257848
$$165$$ 0 0
$$166$$ 970.356 0.453700
$$167$$ 731.805 0.339095 0.169547 0.985522i $$-0.445769\pi$$
0.169547 + 0.985522i $$0.445769\pi$$
$$168$$ 1457.24 0.669218
$$169$$ −590.507 −0.268779
$$170$$ 0 0
$$171$$ 7.62587 0.00341032
$$172$$ −2174.11 −0.963805
$$173$$ 521.773 0.229304 0.114652 0.993406i $$-0.463425\pi$$
0.114652 + 0.993406i $$0.463425\pi$$
$$174$$ 2121.73 0.924413
$$175$$ 0 0
$$176$$ −514.163 −0.220208
$$177$$ −1563.54 −0.663970
$$178$$ −2255.42 −0.949723
$$179$$ −1183.07 −0.494005 −0.247002 0.969015i $$-0.579446\pi$$
−0.247002 + 0.969015i $$0.579446\pi$$
$$180$$ 0 0
$$181$$ 1723.92 0.707946 0.353973 0.935256i $$-0.384830\pi$$
0.353973 + 0.935256i $$0.384830\pi$$
$$182$$ 1888.80 0.769270
$$183$$ −1161.78 −0.469295
$$184$$ 184.000 0.0737210
$$185$$ 0 0
$$186$$ −1745.09 −0.687937
$$187$$ 4054.35 1.58547
$$188$$ −105.738 −0.0410197
$$189$$ 1050.20 0.404182
$$190$$ 0 0
$$191$$ −2798.12 −1.06002 −0.530012 0.847990i $$-0.677813\pi$$
−0.530012 + 0.847990i $$0.677813\pi$$
$$192$$ −494.772 −0.185974
$$193$$ 497.686 0.185618 0.0928089 0.995684i $$-0.470415\pi$$
0.0928089 + 0.995684i $$0.470415\pi$$
$$194$$ 2972.47 1.10006
$$195$$ 0 0
$$196$$ 848.718 0.309300
$$197$$ −296.489 −0.107228 −0.0536141 0.998562i $$-0.517074\pi$$
−0.0536141 + 0.998562i $$0.517074\pi$$
$$198$$ −2105.85 −0.755838
$$199$$ 2347.43 0.836207 0.418103 0.908399i $$-0.362695\pi$$
0.418103 + 0.908399i $$0.362695\pi$$
$$200$$ 0 0
$$201$$ −3252.52 −1.14137
$$202$$ 3777.71 1.31584
$$203$$ 3233.34 1.11791
$$204$$ 3901.44 1.33900
$$205$$ 0 0
$$206$$ 2994.34 1.01274
$$207$$ 753.604 0.253039
$$208$$ −641.297 −0.213779
$$209$$ −7.47920 −0.00247535
$$210$$ 0 0
$$211$$ 2838.61 0.926153 0.463076 0.886318i $$-0.346746\pi$$
0.463076 + 0.886318i $$0.346746\pi$$
$$212$$ −174.783 −0.0566235
$$213$$ −5159.39 −1.65970
$$214$$ 1170.30 0.373832
$$215$$ 0 0
$$216$$ −356.569 −0.112321
$$217$$ −2659.38 −0.831937
$$218$$ −278.331 −0.0864723
$$219$$ 4661.37 1.43829
$$220$$ 0 0
$$221$$ 5056.84 1.53918
$$222$$ 706.685 0.213647
$$223$$ 2124.68 0.638024 0.319012 0.947751i $$-0.396649\pi$$
0.319012 + 0.947751i $$0.396649\pi$$
$$224$$ −753.992 −0.224903
$$225$$ 0 0
$$226$$ 2586.35 0.761246
$$227$$ −1551.97 −0.453780 −0.226890 0.973920i $$-0.572856\pi$$
−0.226890 + 0.973920i $$0.572856\pi$$
$$228$$ −7.19712 −0.00209053
$$229$$ −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
$$232$$ −1097.80 −0.310666
$$233$$ 728.999 0.204971 0.102486 0.994734i $$-0.467320\pi$$
0.102486 + 0.994734i $$0.467320\pi$$
$$234$$ −2626.54 −0.733772
$$235$$ 0 0
$$236$$ 808.991 0.223139
$$237$$ 10659.8 2.92165
$$238$$ 5945.47 1.61928
$$239$$ −6201.60 −1.67844 −0.839222 0.543789i $$-0.816989\pi$$
−0.839222 + 0.543789i $$0.816989\pi$$
$$240$$ 0 0
$$241$$ −7223.04 −1.93061 −0.965305 0.261126i $$-0.915906\pi$$
−0.965305 + 0.261126i $$0.915906\pi$$
$$242$$ −596.656 −0.158490
$$243$$ 5378.79 1.41996
$$244$$ 601.115 0.157715
$$245$$ 0 0
$$246$$ 2093.27 0.542527
$$247$$ −9.32853 −0.00240308
$$248$$ 902.928 0.231193
$$249$$ −3750.82 −0.954612
$$250$$ 0 0
$$251$$ 5042.78 1.26812 0.634059 0.773285i $$-0.281388\pi$$
0.634059 + 0.773285i $$0.281388\pi$$
$$252$$ −3088.10 −0.771954
$$253$$ −739.110 −0.183666
$$254$$ 2596.86 0.641502
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −2981.15 −0.723577 −0.361788 0.932260i $$-0.617834\pi$$
−0.361788 + 0.932260i $$0.617834\pi$$
$$258$$ 8403.82 2.02790
$$259$$ 1076.93 0.258367
$$260$$ 0 0
$$261$$ −4496.25 −1.06632
$$262$$ −1050.49 −0.247709
$$263$$ −7242.86 −1.69815 −0.849076 0.528271i $$-0.822841\pi$$
−0.849076 + 0.528271i $$0.822841\pi$$
$$264$$ 1987.45 0.463330
$$265$$ 0 0
$$266$$ −10.9678 −0.00252812
$$267$$ 8718.10 1.99827
$$268$$ 1682.89 0.383577
$$269$$ −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 $$-0.140586\pi$$
0.904042 + 0.427443i $$0.140586\pi$$
$$272$$ −2018.64 −0.449994
$$273$$ −7300.98 −1.61859
$$274$$ 4856.23 1.07071
$$275$$ 0 0
$$276$$ −711.234 −0.155113
$$277$$ −8991.54 −1.95036 −0.975179 0.221418i $$-0.928932\pi$$
−0.975179 + 0.221418i $$0.928932\pi$$
$$278$$ 4912.92 1.05992
$$279$$ 3698.10 0.793546
$$280$$ 0 0
$$281$$ 968.130 0.205529 0.102765 0.994706i $$-0.467231\pi$$
0.102765 + 0.994706i $$0.467231\pi$$
$$282$$ 408.718 0.0863079
$$283$$ 2252.65 0.473167 0.236583 0.971611i $$-0.423972\pi$$
0.236583 + 0.971611i $$0.423972\pi$$
$$284$$ 2669.52 0.557771
$$285$$ 0 0
$$286$$ 2576.03 0.532600
$$287$$ 3189.97 0.656090
$$288$$ 1048.49 0.214524
$$289$$ 11004.7 2.23991
$$290$$ 0 0
$$291$$ −11489.8 −2.31458
$$292$$ −2411.84 −0.483364
$$293$$ −2734.35 −0.545196 −0.272598 0.962128i $$-0.587883\pi$$
−0.272598 + 0.962128i $$0.587883\pi$$
$$294$$ −3280.64 −0.650785
$$295$$ 0 0
$$296$$ −365.646 −0.0717998
$$297$$ 1432.30 0.279834
$$298$$ −4810.78 −0.935172
$$299$$ −921.865 −0.178304
$$300$$ 0 0
$$301$$ 12806.7 2.45239
$$302$$ −1298.55 −0.247428
$$303$$ −14602.4 −2.76860
$$304$$ 3.72387 0.000702560 0
$$305$$ 0 0
$$306$$ −8267.71 −1.54455
$$307$$ 7977.09 1.48299 0.741493 0.670961i $$-0.234118\pi$$
0.741493 + 0.670961i $$0.234118\pi$$
$$308$$ 3028.71 0.560314
$$309$$ −11574.3 −2.13087
$$310$$ 0 0
$$311$$ 7729.43 1.40931 0.704655 0.709550i $$-0.251102\pi$$
0.704655 + 0.709550i $$0.251102\pi$$
$$312$$ 2478.87 0.449803
$$313$$ −5506.25 −0.994350 −0.497175 0.867650i $$-0.665629\pi$$
−0.497175 + 0.867650i $$0.665629\pi$$
$$314$$ −7330.08 −1.31739
$$315$$ 0 0
$$316$$ −5515.51 −0.981872
$$317$$ −5231.37 −0.926887 −0.463443 0.886126i $$-0.653386\pi$$
−0.463443 + 0.886126i $$0.653386\pi$$
$$318$$ 675.608 0.119139
$$319$$ 4409.77 0.773981
$$320$$ 0 0
$$321$$ −4523.68 −0.786565
$$322$$ −1083.86 −0.187582
$$323$$ −29.3639 −0.00505836
$$324$$ −2160.38 −0.370435
$$325$$ 0 0
$$326$$ 2110.55 0.358566
$$327$$ 1075.86 0.181943
$$328$$ −1083.08 −0.182326
$$329$$ 622.854 0.104374
$$330$$ 0 0
$$331$$ −3551.61 −0.589771 −0.294885 0.955533i $$-0.595281\pi$$
−0.294885 + 0.955533i $$0.595281\pi$$
$$332$$ 1940.71 0.320815
$$333$$ −1497.57 −0.246445
$$334$$ 1463.61 0.239776
$$335$$ 0 0
$$336$$ 2914.48 0.473209
$$337$$ 7002.99 1.13198 0.565990 0.824412i $$-0.308494\pi$$
0.565990 + 0.824412i $$0.308494\pi$$
$$338$$ −1181.01 −0.190055
$$339$$ −9997.29 −1.60171
$$340$$ 0 0
$$341$$ −3626.97 −0.575987
$$342$$ 15.2517 0.00241146
$$343$$ 3082.42 0.485234
$$344$$ −4348.22 −0.681513
$$345$$ 0 0
$$346$$ 1043.55 0.162143
$$347$$ 10268.2 1.58854 0.794272 0.607562i $$-0.207852\pi$$
0.794272 + 0.607562i $$0.207852\pi$$
$$348$$ 4243.46 0.653659
$$349$$ 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
$$352$$ −1028.33 −0.155710
$$353$$ −7838.63 −1.18189 −0.590947 0.806711i $$-0.701246\pi$$
−0.590947 + 0.806711i $$0.701246\pi$$
$$354$$ −3127.08 −0.469498
$$355$$ 0 0
$$356$$ −4510.84 −0.671556
$$357$$ −22981.7 −3.40705
$$358$$ −2366.14 −0.349314
$$359$$ −12464.8 −1.83250 −0.916250 0.400606i $$-0.868799\pi$$
−0.916250 + 0.400606i $$0.868799\pi$$
$$360$$ 0 0
$$361$$ −6858.95 −0.999992
$$362$$ 3447.85 0.500594
$$363$$ 2306.32 0.333472
$$364$$ 3777.60 0.543956
$$365$$ 0 0
$$366$$ −2323.55 −0.331842
$$367$$ −3936.14 −0.559850 −0.279925 0.960022i $$-0.590310\pi$$
−0.279925 + 0.960022i $$0.590310\pi$$
$$368$$ 368.000 0.0521286
$$369$$ −4435.93 −0.625814
$$370$$ 0 0
$$371$$ 1029.57 0.144077
$$372$$ −3490.18 −0.486445
$$373$$ −1920.72 −0.266625 −0.133313 0.991074i $$-0.542561\pi$$
−0.133313 + 0.991074i $$0.542561\pi$$
$$374$$ 8108.69 1.12110
$$375$$ 0 0
$$376$$ −211.475 −0.0290053
$$377$$ 5500.15 0.751385
$$378$$ 2100.39 0.285800
$$379$$ −13074.5 −1.77201 −0.886004 0.463678i $$-0.846530\pi$$
−0.886004 + 0.463678i $$0.846530\pi$$
$$380$$ 0 0
$$381$$ −10037.9 −1.34976
$$382$$ −5596.23 −0.749550
$$383$$ −8838.22 −1.17914 −0.589571 0.807716i $$-0.700703\pi$$
−0.589571 + 0.807716i $$0.700703\pi$$
$$384$$ −989.543 −0.131504
$$385$$ 0 0
$$386$$ 995.372 0.131252
$$387$$ −17808.9 −2.33922
$$388$$ 5944.94 0.777858
$$389$$ −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
$$392$$ 1697.44 0.218708
$$393$$ 4060.58 0.521194
$$394$$ −592.977 −0.0758218
$$395$$ 0 0
$$396$$ −4211.69 −0.534458
$$397$$ −8060.49 −1.01900 −0.509501 0.860470i $$-0.670170\pi$$
−0.509501 + 0.860470i $$0.670170\pi$$
$$398$$ 4694.87 0.591287
$$399$$ 42.3951 0.00531932
$$400$$ 0 0
$$401$$ 12325.1 1.53488 0.767440 0.641121i $$-0.221530\pi$$
0.767440 + 0.641121i $$0.221530\pi$$
$$402$$ −6505.04 −0.807070
$$403$$ −4523.79 −0.559171
$$404$$ 7555.43 0.930437
$$405$$ 0 0
$$406$$ 6466.69 0.790483
$$407$$ 1468.76 0.178879
$$408$$ 7802.87 0.946813
$$409$$ −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
$$412$$ 5988.67 0.716118
$$413$$ −4765.41 −0.567774
$$414$$ 1507.21 0.178926
$$415$$ 0 0
$$416$$ −1282.59 −0.151164
$$417$$ −18990.4 −2.23013
$$418$$ −14.9584 −0.00175033
$$419$$ −1823.74 −0.212639 −0.106319 0.994332i $$-0.533907\pi$$
−0.106319 + 0.994332i $$0.533907\pi$$
$$420$$ 0 0
$$421$$ 5995.43 0.694060 0.347030 0.937854i $$-0.387190\pi$$
0.347030 + 0.937854i $$0.387190\pi$$
$$422$$ 5677.23 0.654889
$$423$$ −866.133 −0.0995575
$$424$$ −349.567 −0.0400388
$$425$$ 0 0
$$426$$ −10318.8 −1.17358
$$427$$ −3540.91 −0.401303
$$428$$ 2340.60 0.264339
$$429$$ −9957.39 −1.12062
$$430$$ 0 0
$$431$$ 3887.80 0.434499 0.217249 0.976116i $$-0.430292\pi$$
0.217249 + 0.976116i $$0.430292\pi$$
$$432$$ −713.137 −0.0794232
$$433$$ 4422.93 0.490884 0.245442 0.969411i $$-0.421067\pi$$
0.245442 + 0.969411i $$0.421067\pi$$
$$434$$ −5318.75 −0.588268
$$435$$ 0 0
$$436$$ −556.662 −0.0611452
$$437$$ 5.35306 0.000585976 0
$$438$$ 9322.74 1.01703
$$439$$ −1748.08 −0.190048 −0.0950242 0.995475i $$-0.530293\pi$$
−0.0950242 + 0.995475i $$0.530293\pi$$
$$440$$ 0 0
$$441$$ 6952.14 0.750690
$$442$$ 10113.7 1.08837
$$443$$ −3371.31 −0.361571 −0.180785 0.983523i $$-0.557864\pi$$
−0.180785 + 0.983523i $$0.557864\pi$$
$$444$$ 1413.37 0.151071
$$445$$ 0 0
$$446$$ 4249.37 0.451151
$$447$$ 18595.6 1.96766
$$448$$ −1507.98 −0.159030
$$449$$ 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
$$452$$ 5172.70 0.538282
$$453$$ 5019.43 0.520603
$$454$$ −3103.94 −0.320871
$$455$$ 0 0
$$456$$ −14.3942 −0.00147823
$$457$$ −3767.20 −0.385607 −0.192803 0.981237i $$-0.561758\pi$$
−0.192803 + 0.981237i $$0.561758\pi$$
$$458$$ −317.063 −0.0323480
$$459$$ 5623.32 0.571839
$$460$$ 0 0
$$461$$ 9674.06 0.977366 0.488683 0.872461i $$-0.337477\pi$$
0.488683 + 0.872461i $$0.337477\pi$$
$$462$$ −11707.2 −1.17893
$$463$$ −2977.73 −0.298892 −0.149446 0.988770i $$-0.547749\pi$$
−0.149446 + 0.988770i $$0.547749\pi$$
$$464$$ −2195.61 −0.219674
$$465$$ 0 0
$$466$$ 1458.00 0.144937
$$467$$ 10701.0 1.06035 0.530176 0.847888i $$-0.322126\pi$$
0.530176 + 0.847888i $$0.322126\pi$$
$$468$$ −5253.09 −0.518855
$$469$$ −9913.16 −0.976006
$$470$$ 0 0
$$471$$ 28333.7 2.77187
$$472$$ 1617.98 0.157783
$$473$$ 17466.4 1.69790
$$474$$ 21319.7 2.06592
$$475$$ 0 0
$$476$$ 11890.9 1.14500
$$477$$ −1431.71 −0.137429
$$478$$ −12403.2 −1.18684
$$479$$ 2337.15 0.222937 0.111469 0.993768i $$-0.464445\pi$$
0.111469 + 0.993768i $$0.464445\pi$$
$$480$$ 0 0
$$481$$ 1831.94 0.173657
$$482$$ −14446.1 −1.36515
$$483$$ 4189.57 0.394683
$$484$$ −1193.31 −0.112069
$$485$$ 0 0
$$486$$ 10757.6 1.00406
$$487$$ 7183.85 0.668442 0.334221 0.942495i $$-0.391527\pi$$
0.334221 + 0.942495i $$0.391527\pi$$
$$488$$ 1202.23 0.111521
$$489$$ −8158.12 −0.754443
$$490$$ 0 0
$$491$$ −12084.2 −1.11070 −0.555350 0.831617i $$-0.687416\pi$$
−0.555350 + 0.831617i $$0.687416\pi$$
$$492$$ 4186.53 0.383625
$$493$$ 17313.1 1.58163
$$494$$ −18.6571 −0.00169923
$$495$$ 0 0
$$496$$ 1805.86 0.163478
$$497$$ −15725.0 −1.41924
$$498$$ −7501.64 −0.675013
$$499$$ 7145.78 0.641060 0.320530 0.947238i $$-0.396139\pi$$
0.320530 + 0.947238i $$0.396139\pi$$
$$500$$ 0 0
$$501$$ −5657.45 −0.504503
$$502$$ 10085.6 0.896695
$$503$$ −20436.2 −1.81154 −0.905770 0.423771i $$-0.860706\pi$$
−0.905770 + 0.423771i $$0.860706\pi$$
$$504$$ −6176.21 −0.545854
$$505$$ 0 0
$$506$$ −1478.22 −0.129871
$$507$$ 4565.10 0.399888
$$508$$ 5193.72 0.453610
$$509$$ 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
$$512$$ 512.000 0.0441942
$$513$$ −10.3735 −0.000892794 0
$$514$$ −5962.30 −0.511646
$$515$$ 0 0
$$516$$ 16807.6 1.43394
$$517$$ 849.475 0.0722628
$$518$$ 2153.86 0.182693
$$519$$ −4033.73 −0.341158
$$520$$ 0 0
$$521$$ 3483.23 0.292904 0.146452 0.989218i $$-0.453215\pi$$
0.146452 + 0.989218i $$0.453215\pi$$
$$522$$ −8992.50 −0.754006
$$523$$ −15689.0 −1.31172 −0.655862 0.754881i $$-0.727695\pi$$
−0.655862 + 0.754881i $$0.727695\pi$$
$$524$$ −2100.99 −0.175157
$$525$$ 0 0
$$526$$ −14485.7 −1.20077
$$527$$ −14239.8 −1.17703
$$528$$ 3974.90 0.327624
$$529$$ 529.000 0.0434783
$$530$$ 0 0
$$531$$ 6626.72 0.541573
$$532$$ −21.9357 −0.00178765
$$533$$ 5426.36 0.440979
$$534$$ 17436.2 1.41299
$$535$$ 0 0
$$536$$ 3365.78 0.271230
$$537$$ 9146.09 0.734977
$$538$$ −5135.15 −0.411510
$$539$$ −6818.43 −0.544881
$$540$$ 0 0
$$541$$ 7620.73 0.605621 0.302810 0.953051i $$-0.402075\pi$$
0.302810 + 0.953051i $$0.402075\pi$$
$$542$$ 16132.5 1.27851
$$543$$ −13327.3 −1.05328
$$544$$ −4037.29 −0.318194
$$545$$ 0 0
$$546$$ −14602.0 −1.14452
$$547$$ −5925.62 −0.463183 −0.231592 0.972813i $$-0.574393\pi$$
−0.231592 + 0.972813i $$0.574393\pi$$
$$548$$ 9712.47 0.757109
$$549$$ 4923.94 0.382784
$$550$$ 0 0
$$551$$ −31.9381 −0.00246934
$$552$$ −1422.47 −0.109682
$$553$$ 32489.4 2.49836
$$554$$ −17983.1 −1.37911
$$555$$ 0 0
$$556$$ 9825.85 0.749476
$$557$$ 5456.16 0.415054 0.207527 0.978229i $$-0.433459\pi$$
0.207527 + 0.978229i $$0.433459\pi$$
$$558$$ 7396.20 0.561122
$$559$$ 21785.2 1.64833
$$560$$ 0 0
$$561$$ −31343.4 −2.35886
$$562$$ 1936.26 0.145331
$$563$$ 9194.29 0.688265 0.344132 0.938921i $$-0.388173\pi$$
0.344132 + 0.938921i $$0.388173\pi$$
$$564$$ 817.437 0.0610289
$$565$$ 0 0
$$566$$ 4505.30 0.334580
$$567$$ 12725.8 0.942567
$$568$$ 5339.05 0.394404
$$569$$ 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 $$-0.520139\pi$$
−0.0632254 + 0.997999i $$0.520139\pi$$
$$572$$ 5152.06 0.376605
$$573$$ 21631.7 1.57710
$$574$$ 6379.93 0.463926
$$575$$ 0 0
$$576$$ 2096.98 0.151692
$$577$$ −23300.0 −1.68109 −0.840547 0.541738i $$-0.817766\pi$$
−0.840547 + 0.541738i $$0.817766\pi$$
$$578$$ 22009.3 1.58385
$$579$$ −3847.52 −0.276161
$$580$$ 0 0
$$581$$ −11431.9 −0.816307
$$582$$ −22979.6 −1.63666
$$583$$ 1404.18 0.0997513
$$584$$ −4823.68 −0.341790
$$585$$ 0 0
$$586$$ −5468.70 −0.385512
$$587$$ −15653.3 −1.10065 −0.550325 0.834950i $$-0.685496\pi$$
−0.550325 + 0.834950i $$0.685496\pi$$
$$588$$ −6561.28 −0.460174
$$589$$ 26.2686 0.00183766
$$590$$ 0 0
$$591$$ 2292.10 0.159533
$$592$$ −731.292 −0.0507701
$$593$$ 2658.08 0.184072 0.0920358 0.995756i $$-0.470663\pi$$
0.0920358 + 0.995756i $$0.470663\pi$$
$$594$$ 2864.60 0.197872
$$595$$ 0 0
$$596$$ −9621.57 −0.661266
$$597$$ −18147.6 −1.24410
$$598$$ −1843.73 −0.126080
$$599$$ 19417.6 1.32451 0.662256 0.749278i $$-0.269599\pi$$
0.662256 + 0.749278i $$0.269599\pi$$
$$600$$ 0 0
$$601$$ −18469.0 −1.25352 −0.626760 0.779213i $$-0.715619\pi$$
−0.626760 + 0.779213i $$0.715619\pi$$
$$602$$ 25613.5 1.73410
$$603$$ 13785.1 0.930968
$$604$$ −2597.10 −0.174958
$$605$$ 0 0
$$606$$ −29204.8 −1.95770
$$607$$ −3968.56 −0.265369 −0.132684 0.991158i $$-0.542360\pi$$
−0.132684 + 0.991158i $$0.542360\pi$$
$$608$$ 7.44773 0.000496785 0
$$609$$ −24996.4 −1.66322
$$610$$ 0 0
$$611$$ 1059.52 0.0701531
$$612$$ −16535.4 −1.09216
$$613$$ 11478.7 0.756311 0.378156 0.925742i $$-0.376558\pi$$
0.378156 + 0.925742i $$0.376558\pi$$
$$614$$ 15954.2 1.04863
$$615$$ 0 0
$$616$$ 6057.42 0.396202
$$617$$ 15691.1 1.02382 0.511911 0.859039i $$-0.328938\pi$$
0.511911 + 0.859039i $$0.328938\pi$$
$$618$$ −23148.6 −1.50676
$$619$$ −18249.4 −1.18499 −0.592494 0.805575i $$-0.701856\pi$$
−0.592494 + 0.805575i $$0.701856\pi$$
$$620$$ 0 0
$$621$$ −1025.14 −0.0662436
$$622$$ 15458.9 0.996533
$$623$$ 26571.4 1.70876
$$624$$ 4957.75 0.318059
$$625$$ 0 0
$$626$$ −11012.5 −0.703111
$$627$$ 57.8203 0.00368281
$$628$$ −14660.2 −0.931536
$$629$$ 5766.48 0.365540
$$630$$ 0 0
$$631$$ −23528.8 −1.48442 −0.742208 0.670169i $$-0.766222\pi$$
−0.742208 + 0.670169i $$0.766222\pi$$
$$632$$ −11031.0 −0.694288
$$633$$ −21944.8 −1.37793
$$634$$ −10462.7 −0.655408
$$635$$ 0 0
$$636$$ 1351.22 0.0842441
$$637$$ −8504.38 −0.528973
$$638$$ 8819.55 0.547287
$$639$$ 21867.0 1.35375
$$640$$ 0 0
$$641$$ 3748.79 0.230996 0.115498 0.993308i $$-0.463154\pi$$
0.115498 + 0.993308i $$0.463154\pi$$
$$642$$ −9047.36 −0.556185
$$643$$ −15924.3 −0.976660 −0.488330 0.872659i $$-0.662394\pi$$
−0.488330 + 0.872659i $$0.662394\pi$$
$$644$$ −2167.73 −0.132640
$$645$$ 0 0
$$646$$ −58.7278 −0.00357680
$$647$$ 2854.11 0.173426 0.0867129 0.996233i $$-0.472364\pi$$
0.0867129 + 0.996233i $$0.472364\pi$$
$$648$$ −4320.76 −0.261937
$$649$$ −6499.27 −0.393095
$$650$$ 0 0
$$651$$ 20559.1 1.23775
$$652$$ 4221.09 0.253544
$$653$$ −7925.97 −0.474988 −0.237494 0.971389i $$-0.576326\pi$$
−0.237494 + 0.971389i $$0.576326\pi$$
$$654$$ 2151.72 0.128653
$$655$$ 0 0
$$656$$ −2166.15 −0.128924
$$657$$ −19756.2 −1.17316
$$658$$ 1245.71 0.0738035
$$659$$ −5799.43 −0.342813 −0.171406 0.985200i $$-0.554831\pi$$
−0.171406 + 0.985200i $$0.554831\pi$$
$$660$$ 0 0
$$661$$ 22324.8 1.31367 0.656833 0.754036i $$-0.271896\pi$$
0.656833 + 0.754036i $$0.271896\pi$$
$$662$$ −7103.22 −0.417031
$$663$$ −39093.4 −2.28999
$$664$$ 3881.42 0.226850
$$665$$ 0 0
$$666$$ −2995.13 −0.174263
$$667$$ −3156.19 −0.183221
$$668$$ 2927.22 0.169547
$$669$$ −16425.5 −0.949249
$$670$$ 0 0
$$671$$ −4829.24 −0.277840
$$672$$ 5828.97 0.334609
$$673$$ 23253.5 1.33188 0.665940 0.746005i $$-0.268031\pi$$
0.665940 + 0.746005i $$0.268031\pi$$
$$674$$ 14006.0 0.800430
$$675$$ 0 0
$$676$$ −2362.03 −0.134389
$$677$$ 19003.7 1.07884 0.539419 0.842038i $$-0.318644\pi$$
0.539419 + 0.842038i $$0.318644\pi$$
$$678$$ −19994.6 −1.13258
$$679$$ −35019.1 −1.97925
$$680$$ 0 0
$$681$$ 11998.0 0.675131
$$682$$ −7253.94 −0.407284
$$683$$ 11222.5 0.628722 0.314361 0.949304i $$-0.398210\pi$$
0.314361 + 0.949304i $$0.398210\pi$$
$$684$$ 30.5035 0.00170516
$$685$$ 0 0
$$686$$ 6164.85 0.343112
$$687$$ 1225.58 0.0680620
$$688$$ −8696.45 −0.481902
$$689$$ 1751.38 0.0968391
$$690$$ 0 0
$$691$$ 4297.68 0.236601 0.118301 0.992978i $$-0.462255\pi$$
0.118301 + 0.992978i $$0.462255\pi$$
$$692$$ 2087.09 0.114652
$$693$$ 24809.2 1.35992
$$694$$ 20536.4 1.12327
$$695$$ 0 0
$$696$$ 8486.92 0.462206
$$697$$ 17080.8 0.928240
$$698$$ 9031.15 0.489734
$$699$$ −5635.75 −0.304955
$$700$$ 0 0
$$701$$ 25769.6 1.38845 0.694225 0.719758i $$-0.255747\pi$$
0.694225 + 0.719758i $$0.255747\pi$$
$$702$$ 3572.91 0.192095
$$703$$ −10.6376 −0.000570705 0
$$704$$ −2056.65 −0.110104
$$705$$ 0 0
$$706$$ −15677.3 −0.835725
$$707$$ −44505.7 −2.36748
$$708$$ −6254.15 −0.331985
$$709$$ −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
$$712$$ −9021.67 −0.474862
$$713$$ 2595.92 0.136350
$$714$$ −45963.3 −2.40915
$$715$$ 0 0
$$716$$ −4732.28 −0.247002
$$717$$ 47943.4 2.49718
$$718$$ −24929.6 −1.29577
$$719$$ 7631.85 0.395855 0.197928 0.980217i $$-0.436579\pi$$
0.197928 + 0.980217i $$0.436579\pi$$
$$720$$ 0 0
$$721$$ −35276.6 −1.82215
$$722$$ −13717.9 −0.707101
$$723$$ 55839.9 2.87235
$$724$$ 6895.70 0.353973
$$725$$ 0 0
$$726$$ 4612.64 0.235800
$$727$$ 36985.6 1.88682 0.943410 0.331628i $$-0.107598\pi$$
0.943410 + 0.331628i $$0.107598\pi$$
$$728$$ 7555.20 0.384635
$$729$$ −26999.8 −1.37173
$$730$$ 0 0
$$731$$ 68574.3 3.46965
$$732$$ −4647.10 −0.234647
$$733$$ 22227.3 1.12003 0.560015 0.828482i $$-0.310795\pi$$
0.560015 + 0.828482i $$0.310795\pi$$
$$734$$ −7872.29 −0.395874
$$735$$ 0 0
$$736$$ 736.000 0.0368605
$$737$$ −13520.0 −0.675733
$$738$$ −8871.86 −0.442517
$$739$$ −17899.1 −0.890974 −0.445487 0.895288i $$-0.646970\pi$$
−0.445487 + 0.895288i $$0.646970\pi$$
$$740$$ 0 0
$$741$$ 72.1171 0.00357529
$$742$$ 2059.14 0.101878
$$743$$ 29843.4 1.47355 0.736776 0.676137i $$-0.236347\pi$$
0.736776 + 0.676137i $$0.236347\pi$$
$$744$$ −6980.36 −0.343968
$$745$$ 0 0
$$746$$ −3841.44 −0.188532
$$747$$ 15897.0 0.778638
$$748$$ 16217.4 0.792736
$$749$$ −13787.4 −0.672606
$$750$$ 0 0
$$751$$ −9399.65 −0.456722 −0.228361 0.973577i $$-0.573337\pi$$
−0.228361 + 0.973577i $$0.573337\pi$$
$$752$$ −422.950 −0.0205098
$$753$$ −38984.8 −1.88670
$$754$$ 11000.3 0.531309
$$755$$ 0 0
$$756$$ 4200.78 0.202091
$$757$$ −29871.6 −1.43422 −0.717109 0.696961i $$-0.754535\pi$$
−0.717109 + 0.696961i $$0.754535\pi$$
$$758$$ −26149.0 −1.25300
$$759$$ 5713.92 0.273257
$$760$$ 0 0
$$761$$ −273.955 −0.0130497 −0.00652487 0.999979i $$-0.502077\pi$$
−0.00652487 + 0.999979i $$0.502077\pi$$
$$762$$ −20075.8 −0.954423
$$763$$ 3279.05 0.155583
$$764$$ −11192.5 −0.530012
$$765$$ 0 0
$$766$$ −17676.4 −0.833780
$$767$$ −8106.31 −0.381619
$$768$$ −1979.09 −0.0929872
$$769$$ −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
$$772$$ 1990.74 0.0928089
$$773$$ −34161.1 −1.58951 −0.794753 0.606933i $$-0.792400\pi$$
−0.794753 + 0.606933i $$0.792400\pi$$
$$774$$ −35617.8 −1.65408
$$775$$ 0 0
$$776$$ 11889.9 0.550028
$$777$$ −8325.54 −0.384398
$$778$$ −25374.6 −1.16931
$$779$$ −31.5096 −0.00144923
$$780$$ 0 0
$$781$$ −21446.4 −0.982603
$$782$$ −5803.60 −0.265392
$$783$$ 6116.29 0.279155
$$784$$ 3394.87 0.154650
$$785$$ 0 0
$$786$$ 8121.17 0.368540
$$787$$ 38489.6 1.74334 0.871669 0.490095i $$-0.163038\pi$$
0.871669 + 0.490095i $$0.163038\pi$$
$$788$$ −1185.95 −0.0536141
$$789$$ 55993.2 2.52650
$$790$$ 0 0
$$791$$ −30470.1 −1.36965
$$792$$ −8423.38 −0.377919
$$793$$ −6023.33 −0.269729
$$794$$ −16121.0 −0.720544
$$795$$ 0 0
$$796$$ 9389.73 0.418103
$$797$$ −17176.8 −0.763405 −0.381702 0.924285i $$-0.624662\pi$$
−0.381702 + 0.924285i $$0.624662\pi$$
$$798$$ 84.7902 0.00376133
$$799$$ 3335.10 0.147669
$$800$$ 0 0
$$801$$ −36949.8 −1.62991
$$802$$ 24650.2 1.08532
$$803$$ 19376.2 0.851523
$$804$$ −13010.1 −0.570684
$$805$$ 0 0
$$806$$ −9047.58 −0.395394
$$807$$ 19849.4 0.865841
$$808$$ 15110.9 0.657918
$$809$$ 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 $$-0.293847\pi$$
0.603312 + 0.797505i $$0.293847\pi$$
$$812$$ 12933.4 0.558956
$$813$$ −62358.8 −2.69006
$$814$$ 2937.53 0.126487
$$815$$ 0 0
$$816$$ 15605.7 0.669498
$$817$$ −126.502 −0.00541705
$$818$$ −21613.5 −0.923835
$$819$$ 30943.6 1.32022
$$820$$ 0 0
$$821$$ −17637.5 −0.749761 −0.374880 0.927073i $$-0.622316\pi$$
−0.374880 + 0.927073i $$0.622316\pi$$
$$822$$ −37542.6 −1.59300
$$823$$ −28048.6 −1.18799 −0.593993 0.804470i $$-0.702449\pi$$
−0.593993 + 0.804470i $$0.702449\pi$$
$$824$$ 11977.3 0.506372
$$825$$ 0 0
$$826$$ −9530.82 −0.401477
$$827$$ 32592.9 1.37046 0.685228 0.728329i $$-0.259703\pi$$
0.685228 + 0.728329i $$0.259703\pi$$
$$828$$ 3014.42 0.126520
$$829$$ 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
$$832$$ −2565.19 −0.106889
$$833$$ −26769.7 −1.11346
$$834$$ −37980.9 −1.57694
$$835$$ 0 0
$$836$$ −29.9168 −0.00123767
$$837$$ −5030.56 −0.207744
$$838$$ −3647.48 −0.150358
$$839$$ 7612.72 0.313254 0.156627 0.987658i $$-0.449938\pi$$
0.156627 + 0.987658i $$0.449938\pi$$
$$840$$ 0 0
$$841$$ −5558.14 −0.227896
$$842$$ 11990.9 0.490775
$$843$$ −7484.43 −0.305786
$$844$$ 11354.5 0.463076
$$845$$ 0 0
$$846$$ −1732.27 −0.0703978
$$847$$ 7029.28 0.285158
$$848$$ −699.134 −0.0283117
$$849$$ −17414.8 −0.703975
$$850$$ 0 0
$$851$$ −1051.23 −0.0423452
$$852$$ −20637.6 −0.829849
$$853$$ 31421.4 1.26125 0.630627 0.776086i $$-0.282798\pi$$
0.630627 + 0.776086i $$0.282798\pi$$
$$854$$ −7081.81 −0.283764
$$855$$ 0 0
$$856$$ 4681.20 0.186916
$$857$$ −20909.5 −0.833435 −0.416718 0.909036i $$-0.636820\pi$$
−0.416718 + 0.909036i $$0.636820\pi$$
$$858$$ −19914.8 −0.792400
$$859$$ 23304.5 0.925655 0.462828 0.886448i $$-0.346835\pi$$
0.462828 + 0.886448i $$0.346835\pi$$
$$860$$ 0 0
$$861$$ −24661.0 −0.976127
$$862$$ 7775.61 0.307237
$$863$$ 19146.3 0.755211 0.377606 0.925966i $$-0.376747\pi$$
0.377606 + 0.925966i $$0.376747\pi$$
$$864$$ −1426.27 −0.0561607
$$865$$ 0 0
$$866$$ 8845.87 0.347107
$$867$$ −85075.0 −3.33252
$$868$$ −10637.5 −0.415968
$$869$$ 44310.5 1.72972
$$870$$ 0 0
$$871$$ −16863.0 −0.656005
$$872$$ −1113.32 −0.0432362
$$873$$ 48697.1 1.88791
$$874$$ 10.7061 0.000414347 0
$$875$$ 0 0
$$876$$ 18645.5 0.719146
$$877$$ −31389.0 −1.20859 −0.604293 0.796762i $$-0.706544\pi$$
−0.604293 + 0.796762i $$0.706544\pi$$
$$878$$ −3496.16 −0.134384
$$879$$ 21138.7 0.811139
$$880$$ 0 0
$$881$$ 44496.3 1.70161 0.850806 0.525480i $$-0.176114\pi$$
0.850806 + 0.525480i $$0.176114\pi$$
$$882$$ 13904.3 0.530818
$$883$$ −8906.65 −0.339448 −0.169724 0.985492i $$-0.554288\pi$$
−0.169724 + 0.985492i $$0.554288\pi$$
$$884$$ 20227.4 0.769592
$$885$$ 0 0
$$886$$ −6742.62 −0.255669
$$887$$ 36584.7 1.38489 0.692444 0.721472i $$-0.256534\pi$$
0.692444 + 0.721472i $$0.256534\pi$$
$$888$$ 2826.74 0.106823
$$889$$ −30593.9 −1.15420
$$890$$ 0 0
$$891$$ 17356.1 0.652581
$$892$$ 8498.74 0.319012
$$893$$ −6.15238 −0.000230551 0
$$894$$ 37191.2 1.39134
$$895$$ 0 0
$$896$$ −3015.97 −0.112451
$$897$$ 7126.76 0.265279
$$898$$ 26629.5 0.989573
$$899$$ −15488.1 −0.574591
$$900$$ 0 0
$$901$$ 5512.90 0.203842
$$902$$ 8701.23 0.321197
$$903$$ −99006.4 −3.64865
$$904$$ 10345.4 0.380623
$$905$$ 0 0
$$906$$ 10038.9 0.368122
$$907$$ 21346.1 0.781463 0.390731 0.920505i $$-0.372222\pi$$
0.390731 + 0.920505i $$0.372222\pi$$
$$908$$ −6207.89 −0.226890
$$909$$ 61889.1 2.25823
$$910$$ 0 0
$$911$$ −10208.0 −0.371246 −0.185623 0.982621i $$-0.559430\pi$$
−0.185623 + 0.982621i $$0.559430\pi$$
$$912$$ −28.7885 −0.00104527
$$913$$ −15591.3 −0.565166
$$914$$ −7534.40 −0.272665
$$915$$ 0 0
$$916$$ −634.125 −0.0228735
$$917$$ 12376.0 0.445683
$$918$$ 11246.6 0.404351
$$919$$ 1758.00 0.0631025 0.0315513 0.999502i $$-0.489955\pi$$
0.0315513 + 0.999502i $$0.489955\pi$$
$$920$$ 0 0
$$921$$ −61669.3 −2.20638
$$922$$ 19348.1 0.691102
$$923$$ −26749.3 −0.953917
$$924$$ −23414.4 −0.833633
$$925$$ 0 0
$$926$$ −5955.46 −0.211348
$$927$$ 49055.3 1.73807
$$928$$ −4391.22 −0.155333
$$929$$ −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
$$932$$ 2915.99 0.102486
$$933$$ −59754.7 −2.09676
$$934$$ 21402.0 0.749782
$$935$$ 0 0
$$936$$ −10506.2 −0.366886
$$937$$ −3851.61 −0.134287 −0.0671433 0.997743i $$-0.521388\pi$$
−0.0671433 + 0.997743i $$0.521388\pi$$
$$938$$ −19826.3 −0.690141
$$939$$ 42567.7 1.47939
$$940$$ 0 0
$$941$$ −4379.36 −0.151714 −0.0758571 0.997119i $$-0.524169\pi$$
−0.0758571 + 0.997119i $$0.524169\pi$$
$$942$$ 56667.5 1.96001
$$943$$ −3113.85 −0.107530
$$944$$ 3235.96 0.111570
$$945$$ 0 0
$$946$$ 34932.8 1.20059
$$947$$ −29746.9 −1.02074 −0.510371 0.859954i $$-0.670492\pi$$
−0.510371 + 0.859954i $$0.670492\pi$$
$$948$$ 42639.3 1.46082
$$949$$ 24167.3 0.826663
$$950$$ 0 0
$$951$$ 40442.7 1.37902
$$952$$ 23781.9 0.809638
$$953$$ −4306.61 −0.146385 −0.0731924 0.997318i $$-0.523319\pi$$
−0.0731924 + 0.997318i $$0.523319\pi$$
$$954$$ −2863.42 −0.0971769
$$955$$ 0 0
$$956$$ −24806.4 −0.839222
$$957$$ −34091.1 −1.15152
$$958$$ 4674.29 0.157640
$$959$$ −57211.9 −1.92645
$$960$$ 0 0
$$961$$ −17052.3 −0.572397
$$962$$ 3663.87 0.122794
$$963$$ 19172.7 0.641568
$$964$$ −28892.2 −0.965305
$$965$$ 0 0
$$966$$ 8379.14 0.279083
$$967$$ 12233.4 0.406824 0.203412 0.979093i $$-0.434797\pi$$
0.203412 + 0.979093i $$0.434797\pi$$
$$968$$ −2386.63 −0.0792449
$$969$$ 227.007 0.00752581
$$970$$ 0 0
$$971$$ 48207.7 1.59326 0.796631 0.604465i $$-0.206613\pi$$
0.796631 + 0.604465i $$0.206613\pi$$
$$972$$ 21515.2 0.709978
$$973$$ −57879.8 −1.90703
$$974$$ 14367.7 0.472660
$$975$$ 0 0
$$976$$ 2404.46 0.0788575
$$977$$ −28662.9 −0.938595 −0.469298 0.883040i $$-0.655493\pi$$
−0.469298 + 0.883040i $$0.655493\pi$$
$$978$$ −16316.2 −0.533472
$$979$$ 36239.2 1.18305
$$980$$ 0 0
$$981$$ −4559.81 −0.148403
$$982$$ −24168.4 −0.785383
$$983$$ 20944.1 0.679567 0.339783 0.940504i $$-0.389646\pi$$
0.339783 + 0.940504i $$0.389646\pi$$
$$984$$ 8373.06 0.271264
$$985$$ 0 0
$$986$$ 34626.2 1.11838
$$987$$ −4815.16 −0.155287
$$988$$ −37.3141 −0.00120154
$$989$$ −12501.1 −0.401934
$$990$$ 0 0
$$991$$ −36440.2 −1.16808 −0.584038 0.811727i $$-0.698528\pi$$
−0.584038 + 0.811727i $$0.698528\pi$$
$$992$$ 3611.71 0.115597
$$993$$ 27456.8 0.877458
$$994$$ −31450.0 −1.00355
$$995$$ 0 0
$$996$$ −15003.3 −0.477306
$$997$$ 11945.0 0.379439 0.189720 0.981838i $$-0.439242\pi$$
0.189720 + 0.981838i $$0.439242\pi$$
$$998$$ 14291.6 0.453298
$$999$$ 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 1150.4.a.p.1.1 4
5.2 odd 4 1150.4.b.n.599.8 8
5.3 odd 4 1150.4.b.n.599.1 8
5.4 even 2 230.4.a.h.1.4 4
15.14 odd 2 2070.4.a.bj.1.3 4
20.19 odd 2 1840.4.a.m.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 5.4 even 2
1150.4.a.p.1.1 4 1.1 even 1 trivial
1150.4.b.n.599.1 8 5.3 odd 4
1150.4.b.n.599.8 8 5.2 odd 4
1840.4.a.m.1.1 4 20.19 odd 2
2070.4.a.bj.1.3 4 15.14 odd 2