# Properties

 Label 1815.4.a.k.1.2 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +10.3923 q^{6} +27.7128 q^{7} -13.8564 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +10.3923 q^{6} +27.7128 q^{7} -13.8564 q^{8} +9.00000 q^{9} -17.3205 q^{10} +12.0000 q^{12} -58.8897 q^{13} +96.0000 q^{14} -15.0000 q^{15} -80.0000 q^{16} +19.0526 q^{17} +31.1769 q^{18} -45.0333 q^{19} -20.0000 q^{20} +83.1384 q^{21} -75.0000 q^{23} -41.5692 q^{24} +25.0000 q^{25} -204.000 q^{26} +27.0000 q^{27} +110.851 q^{28} -128.172 q^{29} -51.9615 q^{30} -263.000 q^{31} -166.277 q^{32} +66.0000 q^{34} -138.564 q^{35} +36.0000 q^{36} -308.000 q^{37} -156.000 q^{38} -176.669 q^{39} +69.2820 q^{40} +162.813 q^{41} +288.000 q^{42} -38.1051 q^{43} -45.0000 q^{45} -259.808 q^{46} +93.0000 q^{47} -240.000 q^{48} +425.000 q^{49} +86.6025 q^{50} +57.1577 q^{51} -235.559 q^{52} +525.000 q^{53} +93.5307 q^{54} -384.000 q^{56} -135.100 q^{57} -444.000 q^{58} +498.000 q^{59} -60.0000 q^{60} -441.673 q^{61} -911.059 q^{62} +249.415 q^{63} +64.0000 q^{64} +294.449 q^{65} +316.000 q^{67} +76.2102 q^{68} -225.000 q^{69} -480.000 q^{70} -288.000 q^{71} -124.708 q^{72} -928.379 q^{73} -1066.94 q^{74} +75.0000 q^{75} -180.133 q^{76} -612.000 q^{78} -1137.96 q^{79} +400.000 q^{80} +81.0000 q^{81} +564.000 q^{82} +571.577 q^{83} +332.554 q^{84} -95.2628 q^{85} -132.000 q^{86} -384.515 q^{87} -180.000 q^{89} -155.885 q^{90} -1632.00 q^{91} -300.000 q^{92} -789.000 q^{93} +322.161 q^{94} +225.167 q^{95} -498.831 q^{96} -904.000 q^{97} +1472.24 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + 24q^{12} + 192q^{14} - 30q^{15} - 160q^{16} - 40q^{20} - 150q^{23} + 50q^{25} - 408q^{26} + 54q^{27} - 526q^{31} + 132q^{34} + 72q^{36} - 616q^{37} - 312q^{38} + 576q^{42} - 90q^{45} + 186q^{47} - 480q^{48} + 850q^{49} + 1050q^{53} - 768q^{56} - 888q^{58} + 996q^{59} - 120q^{60} + 128q^{64} + 632q^{67} - 450q^{69} - 960q^{70} - 576q^{71} + 150q^{75} - 1224q^{78} + 800q^{80} + 162q^{81} + 1128q^{82} - 264q^{86} - 360q^{89} - 3264q^{91} - 600q^{92} - 1578q^{93} - 1808q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.46410 1.22474 0.612372 0.790569i $$-0.290215\pi$$
0.612372 + 0.790569i $$0.290215\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 4.00000 0.500000
$$5$$ −5.00000 −0.447214
$$6$$ 10.3923 0.707107
$$7$$ 27.7128 1.49635 0.748176 0.663501i $$-0.230930\pi$$
0.748176 + 0.663501i $$0.230930\pi$$
$$8$$ −13.8564 −0.612372
$$9$$ 9.00000 0.333333
$$10$$ −17.3205 −0.547723
$$11$$ 0 0
$$12$$ 12.0000 0.288675
$$13$$ −58.8897 −1.25639 −0.628195 0.778056i $$-0.716206\pi$$
−0.628195 + 0.778056i $$0.716206\pi$$
$$14$$ 96.0000 1.83265
$$15$$ −15.0000 −0.258199
$$16$$ −80.0000 −1.25000
$$17$$ 19.0526 0.271819 0.135910 0.990721i $$-0.456604\pi$$
0.135910 + 0.990721i $$0.456604\pi$$
$$18$$ 31.1769 0.408248
$$19$$ −45.0333 −0.543755 −0.271878 0.962332i $$-0.587645\pi$$
−0.271878 + 0.962332i $$0.587645\pi$$
$$20$$ −20.0000 −0.223607
$$21$$ 83.1384 0.863919
$$22$$ 0 0
$$23$$ −75.0000 −0.679938 −0.339969 0.940437i $$-0.610417\pi$$
−0.339969 + 0.940437i $$0.610417\pi$$
$$24$$ −41.5692 −0.353553
$$25$$ 25.0000 0.200000
$$26$$ −204.000 −1.53876
$$27$$ 27.0000 0.192450
$$28$$ 110.851 0.748176
$$29$$ −128.172 −0.820721 −0.410360 0.911923i $$-0.634597\pi$$
−0.410360 + 0.911923i $$0.634597\pi$$
$$30$$ −51.9615 −0.316228
$$31$$ −263.000 −1.52375 −0.761874 0.647725i $$-0.775721\pi$$
−0.761874 + 0.647725i $$0.775721\pi$$
$$32$$ −166.277 −0.918559
$$33$$ 0 0
$$34$$ 66.0000 0.332909
$$35$$ −138.564 −0.669189
$$36$$ 36.0000 0.166667
$$37$$ −308.000 −1.36851 −0.684255 0.729243i $$-0.739873\pi$$
−0.684255 + 0.729243i $$0.739873\pi$$
$$38$$ −156.000 −0.665962
$$39$$ −176.669 −0.725377
$$40$$ 69.2820 0.273861
$$41$$ 162.813 0.620173 0.310086 0.950708i $$-0.399642\pi$$
0.310086 + 0.950708i $$0.399642\pi$$
$$42$$ 288.000 1.05808
$$43$$ −38.1051 −0.135139 −0.0675695 0.997715i $$-0.521524\pi$$
−0.0675695 + 0.997715i $$0.521524\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ −259.808 −0.832751
$$47$$ 93.0000 0.288626 0.144313 0.989532i $$-0.453903\pi$$
0.144313 + 0.989532i $$0.453903\pi$$
$$48$$ −240.000 −0.721688
$$49$$ 425.000 1.23907
$$50$$ 86.6025 0.244949
$$51$$ 57.1577 0.156935
$$52$$ −235.559 −0.628195
$$53$$ 525.000 1.36065 0.680324 0.732912i $$-0.261839\pi$$
0.680324 + 0.732912i $$0.261839\pi$$
$$54$$ 93.5307 0.235702
$$55$$ 0 0
$$56$$ −384.000 −0.916324
$$57$$ −135.100 −0.313937
$$58$$ −444.000 −1.00517
$$59$$ 498.000 1.09888 0.549441 0.835532i $$-0.314841\pi$$
0.549441 + 0.835532i $$0.314841\pi$$
$$60$$ −60.0000 −0.129099
$$61$$ −441.673 −0.927056 −0.463528 0.886082i $$-0.653417\pi$$
−0.463528 + 0.886082i $$0.653417\pi$$
$$62$$ −911.059 −1.86620
$$63$$ 249.415 0.498784
$$64$$ 64.0000 0.125000
$$65$$ 294.449 0.561875
$$66$$ 0 0
$$67$$ 316.000 0.576202 0.288101 0.957600i $$-0.406976\pi$$
0.288101 + 0.957600i $$0.406976\pi$$
$$68$$ 76.2102 0.135910
$$69$$ −225.000 −0.392563
$$70$$ −480.000 −0.819585
$$71$$ −288.000 −0.481399 −0.240699 0.970600i $$-0.577377\pi$$
−0.240699 + 0.970600i $$0.577377\pi$$
$$72$$ −124.708 −0.204124
$$73$$ −928.379 −1.48847 −0.744237 0.667916i $$-0.767187\pi$$
−0.744237 + 0.667916i $$0.767187\pi$$
$$74$$ −1066.94 −1.67608
$$75$$ 75.0000 0.115470
$$76$$ −180.133 −0.271878
$$77$$ 0 0
$$78$$ −612.000 −0.888402
$$79$$ −1137.96 −1.62064 −0.810318 0.585991i $$-0.800706\pi$$
−0.810318 + 0.585991i $$0.800706\pi$$
$$80$$ 400.000 0.559017
$$81$$ 81.0000 0.111111
$$82$$ 564.000 0.759553
$$83$$ 571.577 0.755888 0.377944 0.925828i $$-0.376631\pi$$
0.377944 + 0.925828i $$0.376631\pi$$
$$84$$ 332.554 0.431959
$$85$$ −95.2628 −0.121561
$$86$$ −132.000 −0.165511
$$87$$ −384.515 −0.473843
$$88$$ 0 0
$$89$$ −180.000 −0.214382 −0.107191 0.994238i $$-0.534186\pi$$
−0.107191 + 0.994238i $$0.534186\pi$$
$$90$$ −155.885 −0.182574
$$91$$ −1632.00 −1.88000
$$92$$ −300.000 −0.339969
$$93$$ −789.000 −0.879736
$$94$$ 322.161 0.353494
$$95$$ 225.167 0.243175
$$96$$ −498.831 −0.530330
$$97$$ −904.000 −0.946261 −0.473130 0.880992i $$-0.656876\pi$$
−0.473130 + 0.880992i $$0.656876\pi$$
$$98$$ 1472.24 1.51754
$$99$$ 0 0
$$100$$ 100.000 0.100000
$$101$$ −1046.16 −1.03066 −0.515330 0.856992i $$-0.672331\pi$$
−0.515330 + 0.856992i $$0.672331\pi$$
$$102$$ 198.000 0.192205
$$103$$ 1352.00 1.29336 0.646682 0.762760i $$-0.276156\pi$$
0.646682 + 0.762760i $$0.276156\pi$$
$$104$$ 816.000 0.769379
$$105$$ −415.692 −0.386356
$$106$$ 1818.65 1.66645
$$107$$ −1047.89 −0.946761 −0.473380 0.880858i $$-0.656966\pi$$
−0.473380 + 0.880858i $$0.656966\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 1870.61 1.64378 0.821892 0.569644i $$-0.192919\pi$$
0.821892 + 0.569644i $$0.192919\pi$$
$$110$$ 0 0
$$111$$ −924.000 −0.790110
$$112$$ −2217.03 −1.87044
$$113$$ −1971.00 −1.64085 −0.820425 0.571754i $$-0.806263\pi$$
−0.820425 + 0.571754i $$0.806263\pi$$
$$114$$ −468.000 −0.384493
$$115$$ 375.000 0.304078
$$116$$ −512.687 −0.410360
$$117$$ −530.008 −0.418797
$$118$$ 1725.12 1.34585
$$119$$ 528.000 0.406737
$$120$$ 207.846 0.158114
$$121$$ 0 0
$$122$$ −1530.00 −1.13541
$$123$$ 488.438 0.358057
$$124$$ −1052.00 −0.761874
$$125$$ −125.000 −0.0894427
$$126$$ 864.000 0.610883
$$127$$ 945.700 0.660766 0.330383 0.943847i $$-0.392822\pi$$
0.330383 + 0.943847i $$0.392822\pi$$
$$128$$ 1551.92 1.07165
$$129$$ −114.315 −0.0780225
$$130$$ 1020.00 0.688153
$$131$$ −980.341 −0.653838 −0.326919 0.945052i $$-0.606010\pi$$
−0.326919 + 0.945052i $$0.606010\pi$$
$$132$$ 0 0
$$133$$ −1248.00 −0.813649
$$134$$ 1094.66 0.705701
$$135$$ −135.000 −0.0860663
$$136$$ −264.000 −0.166455
$$137$$ −2169.00 −1.35263 −0.676315 0.736613i $$-0.736424\pi$$
−0.676315 + 0.736613i $$0.736424\pi$$
$$138$$ −779.423 −0.480789
$$139$$ 1002.86 0.611951 0.305976 0.952039i $$-0.401017\pi$$
0.305976 + 0.952039i $$0.401017\pi$$
$$140$$ −554.256 −0.334594
$$141$$ 279.000 0.166639
$$142$$ −997.661 −0.589591
$$143$$ 0 0
$$144$$ −720.000 −0.416667
$$145$$ 640.859 0.367037
$$146$$ −3216.00 −1.82300
$$147$$ 1275.00 0.715376
$$148$$ −1232.00 −0.684255
$$149$$ 536.936 0.295218 0.147609 0.989046i $$-0.452842\pi$$
0.147609 + 0.989046i $$0.452842\pi$$
$$150$$ 259.808 0.141421
$$151$$ 226.899 0.122283 0.0611416 0.998129i $$-0.480526\pi$$
0.0611416 + 0.998129i $$0.480526\pi$$
$$152$$ 624.000 0.332981
$$153$$ 171.473 0.0906064
$$154$$ 0 0
$$155$$ 1315.00 0.681441
$$156$$ −706.677 −0.362689
$$157$$ −2576.00 −1.30947 −0.654736 0.755857i $$-0.727220\pi$$
−0.654736 + 0.755857i $$0.727220\pi$$
$$158$$ −3942.00 −1.98487
$$159$$ 1575.00 0.785570
$$160$$ 831.384 0.410792
$$161$$ −2078.46 −1.01743
$$162$$ 280.592 0.136083
$$163$$ 1694.00 0.814014 0.407007 0.913425i $$-0.366572\pi$$
0.407007 + 0.913425i $$0.366572\pi$$
$$164$$ 651.251 0.310086
$$165$$ 0 0
$$166$$ 1980.00 0.925770
$$167$$ −874.686 −0.405301 −0.202650 0.979251i $$-0.564955\pi$$
−0.202650 + 0.979251i $$0.564955\pi$$
$$168$$ −1152.00 −0.529040
$$169$$ 1271.00 0.578516
$$170$$ −330.000 −0.148881
$$171$$ −405.300 −0.181252
$$172$$ −152.420 −0.0675695
$$173$$ −2868.28 −1.26053 −0.630263 0.776382i $$-0.717053\pi$$
−0.630263 + 0.776382i $$0.717053\pi$$
$$174$$ −1332.00 −0.580337
$$175$$ 692.820 0.299270
$$176$$ 0 0
$$177$$ 1494.00 0.634440
$$178$$ −623.538 −0.262563
$$179$$ −4416.00 −1.84395 −0.921976 0.387247i $$-0.873426\pi$$
−0.921976 + 0.387247i $$0.873426\pi$$
$$180$$ −180.000 −0.0745356
$$181$$ 4430.00 1.81922 0.909611 0.415460i $$-0.136379\pi$$
0.909611 + 0.415460i $$0.136379\pi$$
$$182$$ −5653.41 −2.30252
$$183$$ −1325.02 −0.535236
$$184$$ 1039.23 0.416375
$$185$$ 1540.00 0.612016
$$186$$ −2733.18 −1.07745
$$187$$ 0 0
$$188$$ 372.000 0.144313
$$189$$ 748.246 0.287973
$$190$$ 780.000 0.297827
$$191$$ −4632.00 −1.75476 −0.877382 0.479793i $$-0.840712\pi$$
−0.877382 + 0.479793i $$0.840712\pi$$
$$192$$ 192.000 0.0721688
$$193$$ 3076.12 1.14728 0.573638 0.819109i $$-0.305532\pi$$
0.573638 + 0.819109i $$0.305532\pi$$
$$194$$ −3131.55 −1.15893
$$195$$ 883.346 0.324399
$$196$$ 1700.00 0.619534
$$197$$ 1406.43 0.508648 0.254324 0.967119i $$-0.418147\pi$$
0.254324 + 0.967119i $$0.418147\pi$$
$$198$$ 0 0
$$199$$ 4753.00 1.69312 0.846561 0.532292i $$-0.178669\pi$$
0.846561 + 0.532292i $$0.178669\pi$$
$$200$$ −346.410 −0.122474
$$201$$ 948.000 0.332670
$$202$$ −3624.00 −1.26230
$$203$$ −3552.00 −1.22809
$$204$$ 228.631 0.0784674
$$205$$ −814.064 −0.277350
$$206$$ 4683.47 1.58404
$$207$$ −675.000 −0.226646
$$208$$ 4711.18 1.57049
$$209$$ 0 0
$$210$$ −1440.00 −0.473188
$$211$$ −3670.22 −1.19748 −0.598739 0.800944i $$-0.704332\pi$$
−0.598739 + 0.800944i $$0.704332\pi$$
$$212$$ 2100.00 0.680324
$$213$$ −864.000 −0.277936
$$214$$ −3630.00 −1.15954
$$215$$ 190.526 0.0604360
$$216$$ −374.123 −0.117851
$$217$$ −7288.47 −2.28006
$$218$$ 6480.00 2.01322
$$219$$ −2785.14 −0.859371
$$220$$ 0 0
$$221$$ −1122.00 −0.341511
$$222$$ −3200.83 −0.967683
$$223$$ −1606.00 −0.482268 −0.241134 0.970492i $$-0.577519\pi$$
−0.241134 + 0.970492i $$0.577519\pi$$
$$224$$ −4608.00 −1.37449
$$225$$ 225.000 0.0666667
$$226$$ −6827.74 −2.00962
$$227$$ 4581.27 1.33951 0.669757 0.742580i $$-0.266398\pi$$
0.669757 + 0.742580i $$0.266398\pi$$
$$228$$ −540.400 −0.156969
$$229$$ 4021.00 1.16033 0.580164 0.814500i $$-0.302988\pi$$
0.580164 + 0.814500i $$0.302988\pi$$
$$230$$ 1299.04 0.372418
$$231$$ 0 0
$$232$$ 1776.00 0.502587
$$233$$ 1498.22 0.421253 0.210626 0.977567i $$-0.432450\pi$$
0.210626 + 0.977567i $$0.432450\pi$$
$$234$$ −1836.00 −0.512919
$$235$$ −465.000 −0.129078
$$236$$ 1992.00 0.549441
$$237$$ −3413.87 −0.935674
$$238$$ 1829.05 0.498149
$$239$$ −543.864 −0.147195 −0.0735976 0.997288i $$-0.523448\pi$$
−0.0735976 + 0.997288i $$0.523448\pi$$
$$240$$ 1200.00 0.322749
$$241$$ −2270.72 −0.606929 −0.303464 0.952843i $$-0.598143\pi$$
−0.303464 + 0.952843i $$0.598143\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ −1766.69 −0.463528
$$245$$ −2125.00 −0.554128
$$246$$ 1692.00 0.438528
$$247$$ 2652.00 0.683169
$$248$$ 3644.23 0.933101
$$249$$ 1714.73 0.436412
$$250$$ −433.013 −0.109545
$$251$$ −4746.00 −1.19349 −0.596743 0.802433i $$-0.703539\pi$$
−0.596743 + 0.802433i $$0.703539\pi$$
$$252$$ 997.661 0.249392
$$253$$ 0 0
$$254$$ 3276.00 0.809270
$$255$$ −285.788 −0.0701834
$$256$$ 4864.00 1.18750
$$257$$ −2691.00 −0.653152 −0.326576 0.945171i $$-0.605895\pi$$
−0.326576 + 0.945171i $$0.605895\pi$$
$$258$$ −396.000 −0.0955577
$$259$$ −8535.55 −2.04777
$$260$$ 1177.79 0.280937
$$261$$ −1153.55 −0.273574
$$262$$ −3396.00 −0.800785
$$263$$ 3909.24 0.916555 0.458278 0.888809i $$-0.348467\pi$$
0.458278 + 0.888809i $$0.348467\pi$$
$$264$$ 0 0
$$265$$ −2625.00 −0.608500
$$266$$ −4323.20 −0.996513
$$267$$ −540.000 −0.123773
$$268$$ 1264.00 0.288101
$$269$$ −7848.00 −1.77881 −0.889407 0.457116i $$-0.848882\pi$$
−0.889407 + 0.457116i $$0.848882\pi$$
$$270$$ −467.654 −0.105409
$$271$$ 4823.76 1.08126 0.540632 0.841259i $$-0.318185\pi$$
0.540632 + 0.841259i $$0.318185\pi$$
$$272$$ −1524.20 −0.339774
$$273$$ −4896.00 −1.08542
$$274$$ −7513.64 −1.65663
$$275$$ 0 0
$$276$$ −900.000 −0.196281
$$277$$ 4842.81 1.05046 0.525228 0.850961i $$-0.323980\pi$$
0.525228 + 0.850961i $$0.323980\pi$$
$$278$$ 3474.00 0.749484
$$279$$ −2367.00 −0.507916
$$280$$ 1920.00 0.409793
$$281$$ 7676.45 1.62967 0.814837 0.579690i $$-0.196826\pi$$
0.814837 + 0.579690i $$0.196826\pi$$
$$282$$ 966.484 0.204090
$$283$$ −7430.50 −1.56077 −0.780384 0.625301i $$-0.784976\pi$$
−0.780384 + 0.625301i $$0.784976\pi$$
$$284$$ −1152.00 −0.240699
$$285$$ 675.500 0.140397
$$286$$ 0 0
$$287$$ 4512.00 0.927996
$$288$$ −1496.49 −0.306186
$$289$$ −4550.00 −0.926114
$$290$$ 2220.00 0.449527
$$291$$ −2712.00 −0.546324
$$292$$ −3713.52 −0.744237
$$293$$ 4460.03 0.889276 0.444638 0.895710i $$-0.353332\pi$$
0.444638 + 0.895710i $$0.353332\pi$$
$$294$$ 4416.73 0.876153
$$295$$ −2490.00 −0.491435
$$296$$ 4267.77 0.838038
$$297$$ 0 0
$$298$$ 1860.00 0.361567
$$299$$ 4416.73 0.854268
$$300$$ 300.000 0.0577350
$$301$$ −1056.00 −0.202215
$$302$$ 786.000 0.149766
$$303$$ −3138.48 −0.595052
$$304$$ 3602.67 0.679694
$$305$$ 2208.36 0.414592
$$306$$ 594.000 0.110970
$$307$$ −827.920 −0.153915 −0.0769575 0.997034i $$-0.524521\pi$$
−0.0769575 + 0.997034i $$0.524521\pi$$
$$308$$ 0 0
$$309$$ 4056.00 0.746724
$$310$$ 4555.29 0.834591
$$311$$ 6300.00 1.14868 0.574341 0.818616i $$-0.305258\pi$$
0.574341 + 0.818616i $$0.305258\pi$$
$$312$$ 2448.00 0.444201
$$313$$ −2318.00 −0.418598 −0.209299 0.977852i $$-0.567118\pi$$
−0.209299 + 0.977852i $$0.567118\pi$$
$$314$$ −8923.53 −1.60377
$$315$$ −1247.08 −0.223063
$$316$$ −4551.83 −0.810318
$$317$$ 1767.00 0.313074 0.156537 0.987672i $$-0.449967\pi$$
0.156537 + 0.987672i $$0.449967\pi$$
$$318$$ 5455.96 0.962123
$$319$$ 0 0
$$320$$ −320.000 −0.0559017
$$321$$ −3143.67 −0.546613
$$322$$ −7200.00 −1.24609
$$323$$ −858.000 −0.147803
$$324$$ 324.000 0.0555556
$$325$$ −1472.24 −0.251278
$$326$$ 5868.19 0.996960
$$327$$ 5611.84 0.949039
$$328$$ −2256.00 −0.379777
$$329$$ 2577.29 0.431887
$$330$$ 0 0
$$331$$ −1309.00 −0.217369 −0.108685 0.994076i $$-0.534664\pi$$
−0.108685 + 0.994076i $$0.534664\pi$$
$$332$$ 2286.31 0.377944
$$333$$ −2772.00 −0.456170
$$334$$ −3030.00 −0.496390
$$335$$ −1580.00 −0.257685
$$336$$ −6651.08 −1.07990
$$337$$ 7278.08 1.17645 0.588223 0.808699i $$-0.299828\pi$$
0.588223 + 0.808699i $$0.299828\pi$$
$$338$$ 4402.87 0.708535
$$339$$ −5913.00 −0.947345
$$340$$ −381.051 −0.0607806
$$341$$ 0 0
$$342$$ −1404.00 −0.221987
$$343$$ 2272.45 0.357728
$$344$$ 528.000 0.0827554
$$345$$ 1125.00 0.175559
$$346$$ −9936.00 −1.54382
$$347$$ 1536.33 0.237679 0.118839 0.992914i $$-0.462083\pi$$
0.118839 + 0.992914i $$0.462083\pi$$
$$348$$ −1538.06 −0.236922
$$349$$ 10806.3 1.65744 0.828719 0.559664i $$-0.189070\pi$$
0.828719 + 0.559664i $$0.189070\pi$$
$$350$$ 2400.00 0.366530
$$351$$ −1590.02 −0.241792
$$352$$ 0 0
$$353$$ −8319.00 −1.25432 −0.627161 0.778890i $$-0.715783\pi$$
−0.627161 + 0.778890i $$0.715783\pi$$
$$354$$ 5175.37 0.777027
$$355$$ 1440.00 0.215288
$$356$$ −720.000 −0.107191
$$357$$ 1584.00 0.234830
$$358$$ −15297.5 −2.25837
$$359$$ −10142.9 −1.49115 −0.745573 0.666424i $$-0.767824\pi$$
−0.745573 + 0.666424i $$0.767824\pi$$
$$360$$ 623.538 0.0912871
$$361$$ −4831.00 −0.704330
$$362$$ 15346.0 2.22808
$$363$$ 0 0
$$364$$ −6528.00 −0.940000
$$365$$ 4641.90 0.665666
$$366$$ −4590.00 −0.655528
$$367$$ 1136.00 0.161577 0.0807884 0.996731i $$-0.474256\pi$$
0.0807884 + 0.996731i $$0.474256\pi$$
$$368$$ 6000.00 0.849923
$$369$$ 1465.31 0.206724
$$370$$ 5334.72 0.749564
$$371$$ 14549.2 2.03601
$$372$$ −3156.00 −0.439868
$$373$$ 9976.61 1.38490 0.692452 0.721464i $$-0.256530\pi$$
0.692452 + 0.721464i $$0.256530\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ −1288.65 −0.176747
$$377$$ 7548.00 1.03115
$$378$$ 2592.00 0.352693
$$379$$ 12929.0 1.75229 0.876145 0.482047i $$-0.160107\pi$$
0.876145 + 0.482047i $$0.160107\pi$$
$$380$$ 900.666 0.121587
$$381$$ 2837.10 0.381493
$$382$$ −16045.7 −2.14914
$$383$$ 360.000 0.0480291 0.0240145 0.999712i $$-0.492355\pi$$
0.0240145 + 0.999712i $$0.492355\pi$$
$$384$$ 4655.75 0.618718
$$385$$ 0 0
$$386$$ 10656.0 1.40512
$$387$$ −342.946 −0.0450463
$$388$$ −3616.00 −0.473130
$$389$$ −3918.00 −0.510670 −0.255335 0.966853i $$-0.582186\pi$$
−0.255335 + 0.966853i $$0.582186\pi$$
$$390$$ 3060.00 0.397305
$$391$$ −1428.94 −0.184820
$$392$$ −5888.97 −0.758771
$$393$$ −2941.02 −0.377494
$$394$$ 4872.00 0.622964
$$395$$ 5689.79 0.724770
$$396$$ 0 0
$$397$$ 5546.00 0.701123 0.350561 0.936540i $$-0.385991\pi$$
0.350561 + 0.936540i $$0.385991\pi$$
$$398$$ 16464.9 2.07364
$$399$$ −3744.00 −0.469761
$$400$$ −2000.00 −0.250000
$$401$$ −672.000 −0.0836860 −0.0418430 0.999124i $$-0.513323\pi$$
−0.0418430 + 0.999124i $$0.513323\pi$$
$$402$$ 3283.97 0.407436
$$403$$ 15488.0 1.91442
$$404$$ −4184.63 −0.515330
$$405$$ −405.000 −0.0496904
$$406$$ −12304.5 −1.50409
$$407$$ 0 0
$$408$$ −792.000 −0.0961026
$$409$$ −226.899 −0.0274313 −0.0137157 0.999906i $$-0.504366\pi$$
−0.0137157 + 0.999906i $$0.504366\pi$$
$$410$$ −2820.00 −0.339683
$$411$$ −6507.00 −0.780941
$$412$$ 5408.00 0.646682
$$413$$ 13801.0 1.64431
$$414$$ −2338.27 −0.277584
$$415$$ −2857.88 −0.338043
$$416$$ 9792.00 1.15407
$$417$$ 3008.57 0.353310
$$418$$ 0 0
$$419$$ 4470.00 0.521178 0.260589 0.965450i $$-0.416083\pi$$
0.260589 + 0.965450i $$0.416083\pi$$
$$420$$ −1662.77 −0.193178
$$421$$ 5539.00 0.641222 0.320611 0.947211i $$-0.396112\pi$$
0.320611 + 0.947211i $$0.396112\pi$$
$$422$$ −12714.0 −1.46661
$$423$$ 837.000 0.0962088
$$424$$ −7274.61 −0.833223
$$425$$ 476.314 0.0543638
$$426$$ −2992.98 −0.340400
$$427$$ −12240.0 −1.38720
$$428$$ −4191.56 −0.473380
$$429$$ 0 0
$$430$$ 660.000 0.0740187
$$431$$ −9980.08 −1.11537 −0.557684 0.830054i $$-0.688310\pi$$
−0.557684 + 0.830054i $$0.688310\pi$$
$$432$$ −2160.00 −0.240563
$$433$$ 13526.0 1.50120 0.750598 0.660759i $$-0.229765\pi$$
0.750598 + 0.660759i $$0.229765\pi$$
$$434$$ −25248.0 −2.79249
$$435$$ 1922.58 0.211909
$$436$$ 7482.46 0.821892
$$437$$ 3377.50 0.369720
$$438$$ −9648.00 −1.05251
$$439$$ −11869.7 −1.29046 −0.645230 0.763988i $$-0.723238\pi$$
−0.645230 + 0.763988i $$0.723238\pi$$
$$440$$ 0 0
$$441$$ 3825.00 0.413022
$$442$$ −3886.72 −0.418264
$$443$$ −12108.0 −1.29857 −0.649287 0.760543i $$-0.724933\pi$$
−0.649287 + 0.760543i $$0.724933\pi$$
$$444$$ −3696.00 −0.395055
$$445$$ 900.000 0.0958744
$$446$$ −5563.35 −0.590655
$$447$$ 1610.81 0.170444
$$448$$ 1773.62 0.187044
$$449$$ −5586.00 −0.587126 −0.293563 0.955940i $$-0.594841\pi$$
−0.293563 + 0.955940i $$0.594841\pi$$
$$450$$ 779.423 0.0816497
$$451$$ 0 0
$$452$$ −7884.00 −0.820425
$$453$$ 680.696 0.0706002
$$454$$ 15870.0 1.64056
$$455$$ 8160.00 0.840762
$$456$$ 1872.00 0.192247
$$457$$ 17767.4 1.81865 0.909325 0.416087i $$-0.136599\pi$$
0.909325 + 0.416087i $$0.136599\pi$$
$$458$$ 13929.2 1.42111
$$459$$ 514.419 0.0523116
$$460$$ 1500.00 0.152039
$$461$$ 17594.2 1.77753 0.888766 0.458361i $$-0.151563\pi$$
0.888766 + 0.458361i $$0.151563\pi$$
$$462$$ 0 0
$$463$$ −622.000 −0.0624337 −0.0312168 0.999513i $$-0.509938\pi$$
−0.0312168 + 0.999513i $$0.509938\pi$$
$$464$$ 10253.7 1.02590
$$465$$ 3945.00 0.393430
$$466$$ 5190.00 0.515927
$$467$$ −14019.0 −1.38913 −0.694563 0.719432i $$-0.744402\pi$$
−0.694563 + 0.719432i $$0.744402\pi$$
$$468$$ −2120.03 −0.209398
$$469$$ 8757.25 0.862201
$$470$$ −1610.81 −0.158087
$$471$$ −7728.00 −0.756024
$$472$$ −6900.49 −0.672925
$$473$$ 0 0
$$474$$ −11826.0 −1.14596
$$475$$ −1125.83 −0.108751
$$476$$ 2112.00 0.203368
$$477$$ 4725.00 0.453549
$$478$$ −1884.00 −0.180276
$$479$$ 13735.2 1.31018 0.655089 0.755551i $$-0.272631\pi$$
0.655089 + 0.755551i $$0.272631\pi$$
$$480$$ 2494.15 0.237171
$$481$$ 18138.0 1.71938
$$482$$ −7866.00 −0.743333
$$483$$ −6235.38 −0.587411
$$484$$ 0 0
$$485$$ 4520.00 0.423181
$$486$$ 841.777 0.0785674
$$487$$ 6982.00 0.649660 0.324830 0.945772i $$-0.394693\pi$$
0.324830 + 0.945772i $$0.394693\pi$$
$$488$$ 6120.00 0.567704
$$489$$ 5082.00 0.469971
$$490$$ −7361.22 −0.678665
$$491$$ −6609.51 −0.607501 −0.303750 0.952752i $$-0.598239\pi$$
−0.303750 + 0.952752i $$0.598239\pi$$
$$492$$ 1953.75 0.179028
$$493$$ −2442.00 −0.223088
$$494$$ 9186.80 0.836708
$$495$$ 0 0
$$496$$ 21040.0 1.90469
$$497$$ −7981.29 −0.720342
$$498$$ 5940.00 0.534494
$$499$$ −6104.00 −0.547600 −0.273800 0.961787i $$-0.588281\pi$$
−0.273800 + 0.961787i $$0.588281\pi$$
$$500$$ −500.000 −0.0447214
$$501$$ −2624.06 −0.234000
$$502$$ −16440.6 −1.46172
$$503$$ 12496.7 1.10776 0.553879 0.832597i $$-0.313147\pi$$
0.553879 + 0.832597i $$0.313147\pi$$
$$504$$ −3456.00 −0.305441
$$505$$ 5230.79 0.460925
$$506$$ 0 0
$$507$$ 3813.00 0.334006
$$508$$ 3782.80 0.330383
$$509$$ 5904.00 0.514126 0.257063 0.966395i $$-0.417245\pi$$
0.257063 + 0.966395i $$0.417245\pi$$
$$510$$ −990.000 −0.0859567
$$511$$ −25728.0 −2.22728
$$512$$ 4434.05 0.382733
$$513$$ −1215.90 −0.104646
$$514$$ −9321.90 −0.799944
$$515$$ −6760.00 −0.578410
$$516$$ −457.261 −0.0390113
$$517$$ 0 0
$$518$$ −29568.0 −2.50800
$$519$$ −8604.83 −0.727765
$$520$$ −4080.00 −0.344077
$$521$$ −22878.0 −1.92381 −0.961903 0.273389i $$-0.911855\pi$$
−0.961903 + 0.273389i $$0.911855\pi$$
$$522$$ −3996.00 −0.335058
$$523$$ 6858.92 0.573460 0.286730 0.958011i $$-0.407432\pi$$
0.286730 + 0.958011i $$0.407432\pi$$
$$524$$ −3921.36 −0.326919
$$525$$ 2078.46 0.172784
$$526$$ 13542.0 1.12255
$$527$$ −5010.82 −0.414184
$$528$$ 0 0
$$529$$ −6542.00 −0.537684
$$530$$ −9093.27 −0.745257
$$531$$ 4482.00 0.366294
$$532$$ −4992.00 −0.406825
$$533$$ −9588.00 −0.779179
$$534$$ −1870.61 −0.151591
$$535$$ 5239.45 0.423404
$$536$$ −4378.62 −0.352850
$$537$$ −13248.0 −1.06461
$$538$$ −27186.3 −2.17859
$$539$$ 0 0
$$540$$ −540.000 −0.0430331
$$541$$ −4240.06 −0.336958 −0.168479 0.985705i $$-0.553886\pi$$
−0.168479 + 0.985705i $$0.553886\pi$$
$$542$$ 16710.0 1.32427
$$543$$ 13290.0 1.05033
$$544$$ −3168.00 −0.249682
$$545$$ −9353.07 −0.735122
$$546$$ −16960.2 −1.32936
$$547$$ 11261.8 0.880292 0.440146 0.897926i $$-0.354927\pi$$
0.440146 + 0.897926i $$0.354927\pi$$
$$548$$ −8676.00 −0.676315
$$549$$ −3975.06 −0.309019
$$550$$ 0 0
$$551$$ 5772.00 0.446271
$$552$$ 3117.69 0.240394
$$553$$ −31536.0 −2.42504
$$554$$ 16776.0 1.28654
$$555$$ 4620.00 0.353348
$$556$$ 4011.43 0.305976
$$557$$ 14138.7 1.07554 0.537771 0.843091i $$-0.319266\pi$$
0.537771 + 0.843091i $$0.319266\pi$$
$$558$$ −8199.53 −0.622068
$$559$$ 2244.00 0.169787
$$560$$ 11085.1 0.836486
$$561$$ 0 0
$$562$$ 26592.0 1.99594
$$563$$ −6460.55 −0.483623 −0.241811 0.970323i $$-0.577742\pi$$
−0.241811 + 0.970323i $$0.577742\pi$$
$$564$$ 1116.00 0.0833193
$$565$$ 9855.00 0.733811
$$566$$ −25740.0 −1.91154
$$567$$ 2244.74 0.166261
$$568$$ 3990.65 0.294795
$$569$$ −12037.8 −0.886905 −0.443452 0.896298i $$-0.646246\pi$$
−0.443452 + 0.896298i $$0.646246\pi$$
$$570$$ 2340.00 0.171951
$$571$$ −17013.9 −1.24695 −0.623477 0.781841i $$-0.714281\pi$$
−0.623477 + 0.781841i $$0.714281\pi$$
$$572$$ 0 0
$$573$$ −13896.0 −1.01311
$$574$$ 15630.0 1.13656
$$575$$ −1875.00 −0.135988
$$576$$ 576.000 0.0416667
$$577$$ −9788.00 −0.706204 −0.353102 0.935585i $$-0.614873\pi$$
−0.353102 + 0.935585i $$0.614873\pi$$
$$578$$ −15761.7 −1.13425
$$579$$ 9228.37 0.662380
$$580$$ 2563.44 0.183519
$$581$$ 15840.0 1.13107
$$582$$ −9394.64 −0.669107
$$583$$ 0 0
$$584$$ 12864.0 0.911500
$$585$$ 2650.04 0.187292
$$586$$ 15450.0 1.08914
$$587$$ −27633.0 −1.94299 −0.971496 0.237057i $$-0.923817\pi$$
−0.971496 + 0.237057i $$0.923817\pi$$
$$588$$ 5100.00 0.357688
$$589$$ 11843.8 0.828546
$$590$$ −8625.61 −0.601883
$$591$$ 4219.28 0.293668
$$592$$ 24640.0 1.71064
$$593$$ 1960.68 0.135777 0.0678883 0.997693i $$-0.478374\pi$$
0.0678883 + 0.997693i $$0.478374\pi$$
$$594$$ 0 0
$$595$$ −2640.00 −0.181898
$$596$$ 2147.74 0.147609
$$597$$ 14259.0 0.977524
$$598$$ 15300.0 1.04626
$$599$$ 20496.0 1.39807 0.699035 0.715088i $$-0.253613\pi$$
0.699035 + 0.715088i $$0.253613\pi$$
$$600$$ −1039.23 −0.0707107
$$601$$ −10399.2 −0.705813 −0.352906 0.935659i $$-0.614807\pi$$
−0.352906 + 0.935659i $$0.614807\pi$$
$$602$$ −3658.09 −0.247662
$$603$$ 2844.00 0.192067
$$604$$ 907.595 0.0611416
$$605$$ 0 0
$$606$$ −10872.0 −0.728787
$$607$$ −1953.75 −0.130643 −0.0653216 0.997864i $$-0.520807\pi$$
−0.0653216 + 0.997864i $$0.520807\pi$$
$$608$$ 7488.00 0.499471
$$609$$ −10656.0 −0.709036
$$610$$ 7650.00 0.507770
$$611$$ −5476.74 −0.362627
$$612$$ 685.892 0.0453032
$$613$$ −5203.08 −0.342823 −0.171411 0.985200i $$-0.554833\pi$$
−0.171411 + 0.985200i $$0.554833\pi$$
$$614$$ −2868.00 −0.188507
$$615$$ −2442.19 −0.160128
$$616$$ 0 0
$$617$$ 23046.0 1.50372 0.751861 0.659321i $$-0.229156\pi$$
0.751861 + 0.659321i $$0.229156\pi$$
$$618$$ 14050.4 0.914547
$$619$$ −17584.0 −1.14178 −0.570889 0.821027i $$-0.693401\pi$$
−0.570889 + 0.821027i $$0.693401\pi$$
$$620$$ 5260.00 0.340720
$$621$$ −2025.00 −0.130854
$$622$$ 21823.8 1.40684
$$623$$ −4988.31 −0.320790
$$624$$ 14133.5 0.906721
$$625$$ 625.000 0.0400000
$$626$$ −8029.79 −0.512675
$$627$$ 0 0
$$628$$ −10304.0 −0.654736
$$629$$ −5868.19 −0.371987
$$630$$ −4320.00 −0.273195
$$631$$ 6925.00 0.436894 0.218447 0.975849i $$-0.429901\pi$$
0.218447 + 0.975849i $$0.429901\pi$$
$$632$$ 15768.0 0.992433
$$633$$ −11010.6 −0.691365
$$634$$ 6121.07 0.383436
$$635$$ −4728.50 −0.295504
$$636$$ 6300.00 0.392785
$$637$$ −25028.1 −1.55675
$$638$$ 0 0
$$639$$ −2592.00 −0.160466
$$640$$ −7759.59 −0.479257
$$641$$ 5388.00 0.332002 0.166001 0.986126i $$-0.446915\pi$$
0.166001 + 0.986126i $$0.446915\pi$$
$$642$$ −10890.0 −0.669461
$$643$$ 17882.0 1.09673 0.548365 0.836239i $$-0.315251\pi$$
0.548365 + 0.836239i $$0.315251\pi$$
$$644$$ −8313.84 −0.508713
$$645$$ 571.577 0.0348927
$$646$$ −2972.20 −0.181021
$$647$$ 4989.00 0.303150 0.151575 0.988446i $$-0.451566\pi$$
0.151575 + 0.988446i $$0.451566\pi$$
$$648$$ −1122.37 −0.0680414
$$649$$ 0 0
$$650$$ −5100.00 −0.307751
$$651$$ −21865.4 −1.31639
$$652$$ 6776.00 0.407007
$$653$$ 9258.00 0.554814 0.277407 0.960752i $$-0.410525\pi$$
0.277407 + 0.960752i $$0.410525\pi$$
$$654$$ 19440.0 1.16233
$$655$$ 4901.70 0.292405
$$656$$ −13025.0 −0.775216
$$657$$ −8355.41 −0.496158
$$658$$ 8928.00 0.528951
$$659$$ −595.825 −0.0352201 −0.0176101 0.999845i $$-0.505606\pi$$
−0.0176101 + 0.999845i $$0.505606\pi$$
$$660$$ 0 0
$$661$$ 11410.0 0.671403 0.335702 0.941968i $$-0.391027\pi$$
0.335702 + 0.941968i $$0.391027\pi$$
$$662$$ −4534.51 −0.266222
$$663$$ −3366.00 −0.197171
$$664$$ −7920.00 −0.462885
$$665$$ 6240.00 0.363875
$$666$$ −9602.49 −0.558692
$$667$$ 9612.88 0.558039
$$668$$ −3498.74 −0.202650
$$669$$ −4818.00 −0.278437
$$670$$ −5473.28 −0.315599
$$671$$ 0 0
$$672$$ −13824.0 −0.793560
$$673$$ 24858.4 1.42380 0.711902 0.702278i $$-0.247834\pi$$
0.711902 + 0.702278i $$0.247834\pi$$
$$674$$ 25212.0 1.44085
$$675$$ 675.000 0.0384900
$$676$$ 5084.00 0.289258
$$677$$ 17805.5 1.01081 0.505406 0.862881i $$-0.331343\pi$$
0.505406 + 0.862881i $$0.331343\pi$$
$$678$$ −20483.2 −1.16026
$$679$$ −25052.4 −1.41594
$$680$$ 1320.00 0.0744407
$$681$$ 13743.8 0.773369
$$682$$ 0 0
$$683$$ −25164.0 −1.40977 −0.704886 0.709321i $$-0.749002\pi$$
−0.704886 + 0.709321i $$0.749002\pi$$
$$684$$ −1621.20 −0.0906259
$$685$$ 10845.0 0.604914
$$686$$ 7872.00 0.438126
$$687$$ 12063.0 0.669916
$$688$$ 3048.41 0.168924
$$689$$ −30917.1 −1.70950
$$690$$ 3897.11 0.215015
$$691$$ −2707.00 −0.149029 −0.0745146 0.997220i $$-0.523741\pi$$
−0.0745146 + 0.997220i $$0.523741\pi$$
$$692$$ −11473.1 −0.630263
$$693$$ 0 0
$$694$$ 5322.00 0.291096
$$695$$ −5014.29 −0.273673
$$696$$ 5328.00 0.290169
$$697$$ 3102.00 0.168575
$$698$$ 37434.0 2.02994
$$699$$ 4494.67 0.243210
$$700$$ 2771.28 0.149635
$$701$$ 3775.87 0.203442 0.101721 0.994813i $$-0.467565\pi$$
0.101721 + 0.994813i $$0.467565\pi$$
$$702$$ −5508.00 −0.296134
$$703$$ 13870.3 0.744135
$$704$$ 0 0
$$705$$ −1395.00 −0.0745230
$$706$$ −28817.9 −1.53622
$$707$$ −28992.0 −1.54223
$$708$$ 5976.00 0.317220
$$709$$ 8119.00 0.430064 0.215032 0.976607i $$-0.431014\pi$$
0.215032 + 0.976607i $$0.431014\pi$$
$$710$$ 4988.31 0.263673
$$711$$ −10241.6 −0.540212
$$712$$ 2494.15 0.131281
$$713$$ 19725.0 1.03605
$$714$$ 5487.14 0.287606
$$715$$ 0 0
$$716$$ −17664.0 −0.921976
$$717$$ −1631.59 −0.0849831
$$718$$ −35136.0 −1.82627
$$719$$ 27042.0 1.40264 0.701319 0.712848i $$-0.252595\pi$$
0.701319 + 0.712848i $$0.252595\pi$$
$$720$$ 3600.00 0.186339
$$721$$ 37467.7 1.93533
$$722$$ −16735.1 −0.862625
$$723$$ −6812.16 −0.350411
$$724$$ 17720.0 0.909611
$$725$$ −3204.29 −0.164144
$$726$$ 0 0
$$727$$ −12290.0 −0.626975 −0.313488 0.949592i $$-0.601497\pi$$
−0.313488 + 0.949592i $$0.601497\pi$$
$$728$$ 22613.7 1.15126
$$729$$ 729.000 0.0370370
$$730$$ 16080.0 0.815271
$$731$$ −726.000 −0.0367334
$$732$$ −5300.08 −0.267618
$$733$$ −1326.75 −0.0668549 −0.0334275 0.999441i $$-0.510642\pi$$
−0.0334275 + 0.999441i $$0.510642\pi$$
$$734$$ 3935.22 0.197890
$$735$$ −6375.00 −0.319926
$$736$$ 12470.8 0.624563
$$737$$ 0 0
$$738$$ 5076.00 0.253184
$$739$$ 4830.69 0.240460 0.120230 0.992746i $$-0.461637\pi$$
0.120230 + 0.992746i $$0.461637\pi$$
$$740$$ 6160.00 0.306008
$$741$$ 7956.00 0.394428
$$742$$ 50400.0 2.49359
$$743$$ −743.050 −0.0366889 −0.0183445 0.999832i $$-0.505840\pi$$
−0.0183445 + 0.999832i $$0.505840\pi$$
$$744$$ 10932.7 0.538726
$$745$$ −2684.68 −0.132026
$$746$$ 34560.0 1.69615
$$747$$ 5144.19 0.251963
$$748$$ 0 0
$$749$$ −29040.0 −1.41669
$$750$$ −1299.04 −0.0632456
$$751$$ −10835.0 −0.526464 −0.263232 0.964733i $$-0.584789\pi$$
−0.263232 + 0.964733i $$0.584789\pi$$
$$752$$ −7440.00 −0.360783
$$753$$ −14238.0 −0.689059
$$754$$ 26147.0 1.26289
$$755$$ −1134.49 −0.0546867
$$756$$ 2992.98 0.143986
$$757$$ −21050.0 −1.01067 −0.505334 0.862924i $$-0.668631\pi$$
−0.505334 + 0.862924i $$0.668631\pi$$
$$758$$ 44787.4 2.14611
$$759$$ 0 0
$$760$$ −3120.00 −0.148914
$$761$$ −26621.6 −1.26811 −0.634056 0.773287i $$-0.718611\pi$$
−0.634056 + 0.773287i $$0.718611\pi$$
$$762$$ 9828.00 0.467232
$$763$$ 51840.0 2.45968
$$764$$ −18528.0 −0.877382
$$765$$ −857.365 −0.0405204
$$766$$ 1247.08 0.0588234
$$767$$ −29327.1 −1.38063
$$768$$ 14592.0 0.685603
$$769$$ −20654.7 −0.968567 −0.484283 0.874911i $$-0.660920\pi$$
−0.484283 + 0.874911i $$0.660920\pi$$
$$770$$ 0 0
$$771$$ −8073.00 −0.377097
$$772$$ 12304.5 0.573638
$$773$$ 15435.0 0.718187 0.359093 0.933302i $$-0.383086\pi$$
0.359093 + 0.933302i $$0.383086\pi$$
$$774$$ −1188.00 −0.0551703
$$775$$ −6575.00 −0.304750
$$776$$ 12526.2 0.579464
$$777$$ −25606.6 −1.18228
$$778$$ −13572.4 −0.625440
$$779$$ −7332.00 −0.337222
$$780$$ 3533.38 0.162199
$$781$$ 0 0
$$782$$ −4950.00 −0.226358
$$783$$ −3460.64 −0.157948
$$784$$ −34000.0 −1.54883
$$785$$ 12880.0 0.585614
$$786$$ −10188.0 −0.462333
$$787$$ −10149.8 −0.459723 −0.229861 0.973223i $$-0.573827\pi$$
−0.229861 + 0.973223i $$0.573827\pi$$
$$788$$ 5625.70 0.254324
$$789$$ 11727.7 0.529173
$$790$$ 19710.0 0.887659
$$791$$ −54622.0 −2.45529
$$792$$ 0 0
$$793$$ 26010.0 1.16474
$$794$$ 19211.9 0.858697
$$795$$ −7875.00 −0.351318
$$796$$ 19012.0 0.846561
$$797$$ −954.000 −0.0423995 −0.0211998 0.999775i $$-0.506749\pi$$
−0.0211998 + 0.999775i $$0.506749\pi$$
$$798$$ −12969.6 −0.575337
$$799$$ 1771.89 0.0784542
$$800$$ −4156.92 −0.183712
$$801$$ −1620.00 −0.0714605
$$802$$ −2327.88 −0.102494
$$803$$ 0 0
$$804$$ 3792.00 0.166335
$$805$$ 10392.3 0.455007
$$806$$ 53652.0 2.34468
$$807$$ −23544.0 −1.02700
$$808$$ 14496.0 0.631148
$$809$$ 33421.7 1.45246 0.726232 0.687450i $$-0.241270\pi$$
0.726232 + 0.687450i $$0.241270\pi$$
$$810$$ −1402.96 −0.0608581
$$811$$ 3386.16 0.146614 0.0733071 0.997309i $$-0.476645\pi$$
0.0733071 + 0.997309i $$0.476645\pi$$
$$812$$ −14208.0 −0.614043
$$813$$ 14471.3 0.624268
$$814$$ 0 0
$$815$$ −8470.00 −0.364038
$$816$$ −4572.61 −0.196169
$$817$$ 1716.00 0.0734825
$$818$$ −786.000 −0.0335964
$$819$$ −14688.0 −0.626667
$$820$$ −3256.26 −0.138675
$$821$$ −35462.0 −1.50747 −0.753735 0.657179i $$-0.771750\pi$$
−0.753735 + 0.657179i $$0.771750\pi$$
$$822$$ −22540.9 −0.956453
$$823$$ −5668.00 −0.240066 −0.120033 0.992770i $$-0.538300\pi$$
−0.120033 + 0.992770i $$0.538300\pi$$
$$824$$ −18733.9 −0.792021
$$825$$ 0 0
$$826$$ 47808.0 2.01387
$$827$$ 24397.7 1.02586 0.512932 0.858429i $$-0.328559\pi$$
0.512932 + 0.858429i $$0.328559\pi$$
$$828$$ −2700.00 −0.113323
$$829$$ −27889.0 −1.16843 −0.584213 0.811600i $$-0.698597\pi$$
−0.584213 + 0.811600i $$0.698597\pi$$
$$830$$ −9900.00 −0.414017
$$831$$ 14528.4 0.606481
$$832$$ −3768.94 −0.157049
$$833$$ 8097.34 0.336802
$$834$$ 10422.0 0.432715
$$835$$ 4373.43 0.181256
$$836$$ 0 0
$$837$$ −7101.00 −0.293245
$$838$$ 15484.5 0.638311
$$839$$ −38220.0 −1.57271 −0.786353 0.617777i $$-0.788033\pi$$
−0.786353 + 0.617777i $$0.788033\pi$$
$$840$$ 5760.00 0.236594
$$841$$ −7961.00 −0.326418
$$842$$ 19187.7 0.785333
$$843$$ 23029.3 0.940893
$$844$$ −14680.9 −0.598739
$$845$$ −6355.00 −0.258720
$$846$$ 2899.45 0.117831
$$847$$ 0 0
$$848$$ −42000.0 −1.70081
$$849$$ −22291.5 −0.901110
$$850$$ 1650.00 0.0665818
$$851$$ 23100.0 0.930503
$$852$$ −3456.00 −0.138968
$$853$$ 45459.4 1.82474 0.912368 0.409370i $$-0.134252\pi$$
0.912368 + 0.409370i $$0.134252\pi$$
$$854$$ −42400.6 −1.69897
$$855$$ 2026.50 0.0810583
$$856$$ 14520.0 0.579770
$$857$$ 24728.5 0.985658 0.492829 0.870126i $$-0.335963\pi$$
0.492829 + 0.870126i $$0.335963\pi$$
$$858$$ 0 0
$$859$$ −18556.0 −0.737046 −0.368523 0.929619i $$-0.620136\pi$$
−0.368523 + 0.929619i $$0.620136\pi$$
$$860$$ 762.102 0.0302180
$$861$$ 13536.0 0.535779
$$862$$ −34572.0 −1.36604
$$863$$ 41088.0 1.62069 0.810343 0.585956i $$-0.199281\pi$$
0.810343 + 0.585956i $$0.199281\pi$$
$$864$$ −4489.48 −0.176777
$$865$$ 14341.4 0.563724
$$866$$ 46855.4 1.83858
$$867$$ −13650.0 −0.534692
$$868$$ −29153.9 −1.14003
$$869$$ 0 0
$$870$$ 6660.00 0.259535
$$871$$ −18609.2 −0.723935
$$872$$ −25920.0 −1.00661
$$873$$ −8136.00 −0.315420
$$874$$ 11700.0 0.452813
$$875$$ −3464.10 −0.133838
$$876$$ −11140.6 −0.429685
$$877$$ 7520.56 0.289568 0.144784 0.989463i $$-0.453751\pi$$
0.144784 + 0.989463i $$0.453751\pi$$
$$878$$ −41118.0 −1.58048
$$879$$ 13380.1 0.513424
$$880$$ 0 0
$$881$$ 19320.0 0.738828 0.369414 0.929265i $$-0.379558\pi$$
0.369414 + 0.929265i $$0.379558\pi$$
$$882$$ 13250.2 0.505847
$$883$$ 5818.00 0.221734 0.110867 0.993835i $$-0.464637\pi$$
0.110867 + 0.993835i $$0.464637\pi$$
$$884$$ −4488.00 −0.170755
$$885$$ −7470.00 −0.283730
$$886$$ −41943.3 −1.59042
$$887$$ −30667.7 −1.16090 −0.580451 0.814295i $$-0.697124\pi$$
−0.580451 + 0.814295i $$0.697124\pi$$
$$888$$ 12803.3 0.483842
$$889$$ 26208.0 0.988738
$$890$$ 3117.69 0.117422
$$891$$ 0 0
$$892$$ −6424.00 −0.241134
$$893$$ −4188.10 −0.156942
$$894$$ 5580.00 0.208751
$$895$$ 22080.0 0.824640
$$896$$ 43008.0 1.60357
$$897$$ 13250.2 0.493212
$$898$$ −19350.5 −0.719080
$$899$$ 33709.2 1.25057
$$900$$ 900.000 0.0333333
$$901$$ 10002.6 0.369850
$$902$$ 0 0
$$903$$ −3168.00 −0.116749
$$904$$ 27311.0 1.00481
$$905$$ −22150.0 −0.813581
$$906$$ 2358.00 0.0864672
$$907$$ −31474.0 −1.15223 −0.576117 0.817367i $$-0.695433\pi$$
−0.576117 + 0.817367i $$0.695433\pi$$
$$908$$ 18325.1 0.669757
$$909$$ −9415.43 −0.343553
$$910$$ 28267.1 1.02972
$$911$$ 16560.0 0.602258 0.301129 0.953583i $$-0.402636\pi$$
0.301129 + 0.953583i $$0.402636\pi$$
$$912$$ 10808.0 0.392422
$$913$$ 0 0
$$914$$ 61548.0 2.22738
$$915$$ 6625.09 0.239365
$$916$$ 16084.0 0.580164
$$917$$ −27168.0 −0.978371
$$918$$ 1782.00 0.0640684
$$919$$ −13617.4 −0.488788 −0.244394 0.969676i $$-0.578589\pi$$
−0.244394 + 0.969676i $$0.578589\pi$$
$$920$$ −5196.15 −0.186209
$$921$$ −2483.76 −0.0888629
$$922$$ 60948.0 2.17702
$$923$$ 16960.2 0.604825
$$924$$ 0 0
$$925$$ −7700.00 −0.273702
$$926$$ −2154.67 −0.0764653
$$927$$ 12168.0 0.431121
$$928$$ 21312.0 0.753880
$$929$$ −2634.00 −0.0930234 −0.0465117 0.998918i $$-0.514810\pi$$
−0.0465117 + 0.998918i $$0.514810\pi$$
$$930$$ 13665.9 0.481851
$$931$$ −19139.2 −0.673749
$$932$$ 5992.90 0.210626
$$933$$ 18900.0 0.663192
$$934$$ −48563.2 −1.70133
$$935$$ 0 0
$$936$$ 7344.00 0.256460
$$937$$ −48455.9 −1.68942 −0.844709 0.535227i $$-0.820226\pi$$
−0.844709 + 0.535227i $$0.820226\pi$$
$$938$$ 30336.0 1.05598
$$939$$ −6954.00 −0.241678
$$940$$ −1860.00 −0.0645388
$$941$$ −33861.6 −1.17307 −0.586534 0.809925i $$-0.699508\pi$$
−0.586534 + 0.809925i $$0.699508\pi$$
$$942$$ −26770.6 −0.925937
$$943$$ −12211.0 −0.421679
$$944$$ −39840.0 −1.37360
$$945$$ −3741.23 −0.128785
$$946$$ 0 0
$$947$$ −4557.00 −0.156370 −0.0781851 0.996939i $$-0.524913\pi$$
−0.0781851 + 0.996939i $$0.524913\pi$$
$$948$$ −13655.5 −0.467837
$$949$$ 54672.0 1.87010
$$950$$ −3900.00 −0.133192
$$951$$ 5301.00 0.180754
$$952$$ −7316.18 −0.249074
$$953$$ −36601.7 −1.24412 −0.622059 0.782970i $$-0.713704\pi$$
−0.622059 + 0.782970i $$0.713704\pi$$
$$954$$ 16367.9 0.555482
$$955$$ 23160.0 0.784754
$$956$$ −2175.46 −0.0735976
$$957$$ 0 0
$$958$$ 47580.0 1.60463
$$959$$ −60109.1 −2.02401
$$960$$ −960.000 −0.0322749
$$961$$ 39378.0 1.32181
$$962$$ 62832.0 2.10581
$$963$$ −9431.02 −0.315587
$$964$$ −9082.87 −0.303464
$$965$$ −15380.6 −0.513077
$$966$$ −21600.0 −0.719429
$$967$$ −12242.1 −0.407115 −0.203558 0.979063i $$-0.565250\pi$$
−0.203558 + 0.979063i $$0.565250\pi$$
$$968$$ 0 0
$$969$$ −2574.00 −0.0853342
$$970$$ 15657.7 0.518288
$$971$$ 31398.0 1.03770 0.518852 0.854864i $$-0.326360\pi$$
0.518852 + 0.854864i $$0.326360\pi$$
$$972$$ 972.000 0.0320750
$$973$$ 27792.0 0.915694
$$974$$ 24186.4 0.795668
$$975$$ −4416.73 −0.145075
$$976$$ 35333.8 1.15882
$$977$$ −54783.0 −1.79392 −0.896962 0.442108i $$-0.854231\pi$$
−0.896962 + 0.442108i $$0.854231\pi$$
$$978$$ 17604.6 0.575595
$$979$$ 0 0
$$980$$ −8500.00 −0.277064
$$981$$ 16835.5 0.547928
$$982$$ −22896.0 −0.744033
$$983$$ −14325.0 −0.464798 −0.232399 0.972621i $$-0.574658\pi$$
−0.232399 + 0.972621i $$0.574658\pi$$
$$984$$ −6768.00 −0.219264
$$985$$ −7032.13 −0.227474
$$986$$ −8459.34 −0.273225
$$987$$ 7731.87 0.249350
$$988$$ 10608.0 0.341584
$$989$$ 2857.88 0.0918862
$$990$$ 0 0
$$991$$ 17017.0 0.545472 0.272736 0.962089i $$-0.412071\pi$$
0.272736 + 0.962089i $$0.412071\pi$$
$$992$$ 43730.8 1.39965
$$993$$ −3927.00 −0.125498
$$994$$ −27648.0 −0.882235
$$995$$ −23765.0 −0.757187
$$996$$ 6858.92 0.218206
$$997$$ −4964.06 −0.157686 −0.0788432 0.996887i $$-0.525123\pi$$
−0.0788432 + 0.996887i $$0.525123\pi$$
$$998$$ −21144.9 −0.670671
$$999$$ −8316.00 −0.263370
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 1815.4.a.k.1.2 yes 2
11.10 odd 2 inner 1815.4.a.k.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.k.1.1 2 11.10 odd 2 inner
1815.4.a.k.1.2 yes 2 1.1 even 1 trivial