# Properties

 Label 150.4.a.i.1.1 Level $150$ Weight $4$ Character 150.1 Self dual yes Analytic conductor $8.850$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.85028650086$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +16.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +16.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{11} +12.0000 q^{12} -38.0000 q^{13} +32.0000 q^{14} +16.0000 q^{16} +126.000 q^{17} +18.0000 q^{18} +20.0000 q^{19} +48.0000 q^{21} +24.0000 q^{22} -168.000 q^{23} +24.0000 q^{24} -76.0000 q^{26} +27.0000 q^{27} +64.0000 q^{28} +30.0000 q^{29} -88.0000 q^{31} +32.0000 q^{32} +36.0000 q^{33} +252.000 q^{34} +36.0000 q^{36} -254.000 q^{37} +40.0000 q^{38} -114.000 q^{39} +42.0000 q^{41} +96.0000 q^{42} +52.0000 q^{43} +48.0000 q^{44} -336.000 q^{46} +96.0000 q^{47} +48.0000 q^{48} -87.0000 q^{49} +378.000 q^{51} -152.000 q^{52} -198.000 q^{53} +54.0000 q^{54} +128.000 q^{56} +60.0000 q^{57} +60.0000 q^{58} -660.000 q^{59} -538.000 q^{61} -176.000 q^{62} +144.000 q^{63} +64.0000 q^{64} +72.0000 q^{66} -884.000 q^{67} +504.000 q^{68} -504.000 q^{69} +792.000 q^{71} +72.0000 q^{72} -218.000 q^{73} -508.000 q^{74} +80.0000 q^{76} +192.000 q^{77} -228.000 q^{78} -520.000 q^{79} +81.0000 q^{81} +84.0000 q^{82} +492.000 q^{83} +192.000 q^{84} +104.000 q^{86} +90.0000 q^{87} +96.0000 q^{88} +810.000 q^{89} -608.000 q^{91} -672.000 q^{92} -264.000 q^{93} +192.000 q^{94} +96.0000 q^{96} -1154.00 q^{97} -174.000 q^{98} +108.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 6.00000 0.408248
$$7$$ 16.0000 0.863919 0.431959 0.901893i $$-0.357822\pi$$
0.431959 + 0.901893i $$0.357822\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 12.0000 0.328921 0.164461 0.986384i $$-0.447412\pi$$
0.164461 + 0.986384i $$0.447412\pi$$
$$12$$ 12.0000 0.288675
$$13$$ −38.0000 −0.810716 −0.405358 0.914158i $$-0.632853\pi$$
−0.405358 + 0.914158i $$0.632853\pi$$
$$14$$ 32.0000 0.610883
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 126.000 1.79762 0.898808 0.438342i $$-0.144434\pi$$
0.898808 + 0.438342i $$0.144434\pi$$
$$18$$ 18.0000 0.235702
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ 48.0000 0.498784
$$22$$ 24.0000 0.232583
$$23$$ −168.000 −1.52306 −0.761531 0.648129i $$-0.775552\pi$$
−0.761531 + 0.648129i $$0.775552\pi$$
$$24$$ 24.0000 0.204124
$$25$$ 0 0
$$26$$ −76.0000 −0.573263
$$27$$ 27.0000 0.192450
$$28$$ 64.0000 0.431959
$$29$$ 30.0000 0.192099 0.0960493 0.995377i $$-0.469379\pi$$
0.0960493 + 0.995377i $$0.469379\pi$$
$$30$$ 0 0
$$31$$ −88.0000 −0.509847 −0.254924 0.966961i $$-0.582050\pi$$
−0.254924 + 0.966961i $$0.582050\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 36.0000 0.189903
$$34$$ 252.000 1.27111
$$35$$ 0 0
$$36$$ 36.0000 0.166667
$$37$$ −254.000 −1.12858 −0.564288 0.825578i $$-0.690849\pi$$
−0.564288 + 0.825578i $$0.690849\pi$$
$$38$$ 40.0000 0.170759
$$39$$ −114.000 −0.468067
$$40$$ 0 0
$$41$$ 42.0000 0.159983 0.0799914 0.996796i $$-0.474511\pi$$
0.0799914 + 0.996796i $$0.474511\pi$$
$$42$$ 96.0000 0.352693
$$43$$ 52.0000 0.184417 0.0922084 0.995740i $$-0.470607\pi$$
0.0922084 + 0.995740i $$0.470607\pi$$
$$44$$ 48.0000 0.164461
$$45$$ 0 0
$$46$$ −336.000 −1.07697
$$47$$ 96.0000 0.297937 0.148969 0.988842i $$-0.452405\pi$$
0.148969 + 0.988842i $$0.452405\pi$$
$$48$$ 48.0000 0.144338
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ 378.000 1.03785
$$52$$ −152.000 −0.405358
$$53$$ −198.000 −0.513158 −0.256579 0.966523i $$-0.582595\pi$$
−0.256579 + 0.966523i $$0.582595\pi$$
$$54$$ 54.0000 0.136083
$$55$$ 0 0
$$56$$ 128.000 0.305441
$$57$$ 60.0000 0.139424
$$58$$ 60.0000 0.135834
$$59$$ −660.000 −1.45635 −0.728175 0.685391i $$-0.759631\pi$$
−0.728175 + 0.685391i $$0.759631\pi$$
$$60$$ 0 0
$$61$$ −538.000 −1.12924 −0.564622 0.825350i $$-0.690978\pi$$
−0.564622 + 0.825350i $$0.690978\pi$$
$$62$$ −176.000 −0.360516
$$63$$ 144.000 0.287973
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 72.0000 0.134282
$$67$$ −884.000 −1.61191 −0.805954 0.591979i $$-0.798347\pi$$
−0.805954 + 0.591979i $$0.798347\pi$$
$$68$$ 504.000 0.898808
$$69$$ −504.000 −0.879340
$$70$$ 0 0
$$71$$ 792.000 1.32385 0.661923 0.749572i $$-0.269740\pi$$
0.661923 + 0.749572i $$0.269740\pi$$
$$72$$ 72.0000 0.117851
$$73$$ −218.000 −0.349520 −0.174760 0.984611i $$-0.555915\pi$$
−0.174760 + 0.984611i $$0.555915\pi$$
$$74$$ −508.000 −0.798024
$$75$$ 0 0
$$76$$ 80.0000 0.120745
$$77$$ 192.000 0.284161
$$78$$ −228.000 −0.330973
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 84.0000 0.113125
$$83$$ 492.000 0.650651 0.325325 0.945602i $$-0.394526\pi$$
0.325325 + 0.945602i $$0.394526\pi$$
$$84$$ 192.000 0.249392
$$85$$ 0 0
$$86$$ 104.000 0.130402
$$87$$ 90.0000 0.110908
$$88$$ 96.0000 0.116291
$$89$$ 810.000 0.964717 0.482359 0.875974i $$-0.339780\pi$$
0.482359 + 0.875974i $$0.339780\pi$$
$$90$$ 0 0
$$91$$ −608.000 −0.700393
$$92$$ −672.000 −0.761531
$$93$$ −264.000 −0.294360
$$94$$ 192.000 0.210673
$$95$$ 0 0
$$96$$ 96.0000 0.102062
$$97$$ −1154.00 −1.20795 −0.603974 0.797004i $$-0.706417\pi$$
−0.603974 + 0.797004i $$0.706417\pi$$
$$98$$ −174.000 −0.179354
$$99$$ 108.000 0.109640
$$100$$ 0 0
$$101$$ −618.000 −0.608845 −0.304422 0.952537i $$-0.598463\pi$$
−0.304422 + 0.952537i $$0.598463\pi$$
$$102$$ 756.000 0.733874
$$103$$ −128.000 −0.122449 −0.0612243 0.998124i $$-0.519501\pi$$
−0.0612243 + 0.998124i $$0.519501\pi$$
$$104$$ −304.000 −0.286631
$$105$$ 0 0
$$106$$ −396.000 −0.362858
$$107$$ 1476.00 1.33355 0.666777 0.745257i $$-0.267673\pi$$
0.666777 + 0.745257i $$0.267673\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 1190.00 1.04570 0.522850 0.852425i $$-0.324869\pi$$
0.522850 + 0.852425i $$0.324869\pi$$
$$110$$ 0 0
$$111$$ −762.000 −0.651584
$$112$$ 256.000 0.215980
$$113$$ 462.000 0.384613 0.192307 0.981335i $$-0.438403\pi$$
0.192307 + 0.981335i $$0.438403\pi$$
$$114$$ 120.000 0.0985880
$$115$$ 0 0
$$116$$ 120.000 0.0960493
$$117$$ −342.000 −0.270239
$$118$$ −1320.00 −1.02980
$$119$$ 2016.00 1.55300
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ −1076.00 −0.798496
$$123$$ 126.000 0.0923662
$$124$$ −352.000 −0.254924
$$125$$ 0 0
$$126$$ 288.000 0.203628
$$127$$ 2536.00 1.77192 0.885959 0.463763i $$-0.153501\pi$$
0.885959 + 0.463763i $$0.153501\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 156.000 0.106473
$$130$$ 0 0
$$131$$ 2292.00 1.52865 0.764324 0.644832i $$-0.223073\pi$$
0.764324 + 0.644832i $$0.223073\pi$$
$$132$$ 144.000 0.0949514
$$133$$ 320.000 0.208628
$$134$$ −1768.00 −1.13979
$$135$$ 0 0
$$136$$ 1008.00 0.635554
$$137$$ 726.000 0.452747 0.226374 0.974041i $$-0.427313\pi$$
0.226374 + 0.974041i $$0.427313\pi$$
$$138$$ −1008.00 −0.621787
$$139$$ 380.000 0.231879 0.115939 0.993256i $$-0.463012\pi$$
0.115939 + 0.993256i $$0.463012\pi$$
$$140$$ 0 0
$$141$$ 288.000 0.172014
$$142$$ 1584.00 0.936101
$$143$$ −456.000 −0.266662
$$144$$ 144.000 0.0833333
$$145$$ 0 0
$$146$$ −436.000 −0.247148
$$147$$ −261.000 −0.146442
$$148$$ −1016.00 −0.564288
$$149$$ 1590.00 0.874214 0.437107 0.899410i $$-0.356003\pi$$
0.437107 + 0.899410i $$0.356003\pi$$
$$150$$ 0 0
$$151$$ 2432.00 1.31068 0.655342 0.755332i $$-0.272524\pi$$
0.655342 + 0.755332i $$0.272524\pi$$
$$152$$ 160.000 0.0853797
$$153$$ 1134.00 0.599206
$$154$$ 384.000 0.200932
$$155$$ 0 0
$$156$$ −456.000 −0.234033
$$157$$ −614.000 −0.312118 −0.156059 0.987748i $$-0.549879\pi$$
−0.156059 + 0.987748i $$0.549879\pi$$
$$158$$ −1040.00 −0.523658
$$159$$ −594.000 −0.296272
$$160$$ 0 0
$$161$$ −2688.00 −1.31580
$$162$$ 162.000 0.0785674
$$163$$ 1852.00 0.889938 0.444969 0.895546i $$-0.353215\pi$$
0.444969 + 0.895546i $$0.353215\pi$$
$$164$$ 168.000 0.0799914
$$165$$ 0 0
$$166$$ 984.000 0.460080
$$167$$ 2136.00 0.989752 0.494876 0.868964i $$-0.335213\pi$$
0.494876 + 0.868964i $$0.335213\pi$$
$$168$$ 384.000 0.176347
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 208.000 0.0922084
$$173$$ −1758.00 −0.772591 −0.386296 0.922375i $$-0.626246\pi$$
−0.386296 + 0.922375i $$0.626246\pi$$
$$174$$ 180.000 0.0784239
$$175$$ 0 0
$$176$$ 192.000 0.0822304
$$177$$ −1980.00 −0.840824
$$178$$ 1620.00 0.682158
$$179$$ −540.000 −0.225483 −0.112742 0.993624i $$-0.535963\pi$$
−0.112742 + 0.993624i $$0.535963\pi$$
$$180$$ 0 0
$$181$$ 1982.00 0.813928 0.406964 0.913444i $$-0.366588\pi$$
0.406964 + 0.913444i $$0.366588\pi$$
$$182$$ −1216.00 −0.495252
$$183$$ −1614.00 −0.651969
$$184$$ −1344.00 −0.538484
$$185$$ 0 0
$$186$$ −528.000 −0.208144
$$187$$ 1512.00 0.591275
$$188$$ 384.000 0.148969
$$189$$ 432.000 0.166261
$$190$$ 0 0
$$191$$ −2688.00 −1.01831 −0.509154 0.860675i $$-0.670042\pi$$
−0.509154 + 0.860675i $$0.670042\pi$$
$$192$$ 192.000 0.0721688
$$193$$ 2302.00 0.858557 0.429279 0.903172i $$-0.358768\pi$$
0.429279 + 0.903172i $$0.358768\pi$$
$$194$$ −2308.00 −0.854148
$$195$$ 0 0
$$196$$ −348.000 −0.126822
$$197$$ −4374.00 −1.58190 −0.790951 0.611880i $$-0.790414\pi$$
−0.790951 + 0.611880i $$0.790414\pi$$
$$198$$ 216.000 0.0775275
$$199$$ −1600.00 −0.569955 −0.284977 0.958534i $$-0.591986\pi$$
−0.284977 + 0.958534i $$0.591986\pi$$
$$200$$ 0 0
$$201$$ −2652.00 −0.930635
$$202$$ −1236.00 −0.430518
$$203$$ 480.000 0.165958
$$204$$ 1512.00 0.518927
$$205$$ 0 0
$$206$$ −256.000 −0.0865843
$$207$$ −1512.00 −0.507687
$$208$$ −608.000 −0.202679
$$209$$ 240.000 0.0794313
$$210$$ 0 0
$$211$$ 3332.00 1.08713 0.543565 0.839367i $$-0.317074\pi$$
0.543565 + 0.839367i $$0.317074\pi$$
$$212$$ −792.000 −0.256579
$$213$$ 2376.00 0.764323
$$214$$ 2952.00 0.942965
$$215$$ 0 0
$$216$$ 216.000 0.0680414
$$217$$ −1408.00 −0.440467
$$218$$ 2380.00 0.739422
$$219$$ −654.000 −0.201796
$$220$$ 0 0
$$221$$ −4788.00 −1.45736
$$222$$ −1524.00 −0.460740
$$223$$ −2648.00 −0.795171 −0.397586 0.917565i $$-0.630152\pi$$
−0.397586 + 0.917565i $$0.630152\pi$$
$$224$$ 512.000 0.152721
$$225$$ 0 0
$$226$$ 924.000 0.271963
$$227$$ −2244.00 −0.656121 −0.328061 0.944657i $$-0.606395\pi$$
−0.328061 + 0.944657i $$0.606395\pi$$
$$228$$ 240.000 0.0697122
$$229$$ −5650.00 −1.63040 −0.815202 0.579177i $$-0.803374\pi$$
−0.815202 + 0.579177i $$0.803374\pi$$
$$230$$ 0 0
$$231$$ 576.000 0.164061
$$232$$ 240.000 0.0679171
$$233$$ −4698.00 −1.32093 −0.660464 0.750858i $$-0.729640\pi$$
−0.660464 + 0.750858i $$0.729640\pi$$
$$234$$ −684.000 −0.191088
$$235$$ 0 0
$$236$$ −2640.00 −0.728175
$$237$$ −1560.00 −0.427565
$$238$$ 4032.00 1.09813
$$239$$ −1200.00 −0.324776 −0.162388 0.986727i $$-0.551920\pi$$
−0.162388 + 0.986727i $$0.551920\pi$$
$$240$$ 0 0
$$241$$ −718.000 −0.191911 −0.0959553 0.995386i $$-0.530591\pi$$
−0.0959553 + 0.995386i $$0.530591\pi$$
$$242$$ −2374.00 −0.630605
$$243$$ 243.000 0.0641500
$$244$$ −2152.00 −0.564622
$$245$$ 0 0
$$246$$ 252.000 0.0653127
$$247$$ −760.000 −0.195780
$$248$$ −704.000 −0.180258
$$249$$ 1476.00 0.375653
$$250$$ 0 0
$$251$$ 6012.00 1.51185 0.755924 0.654659i $$-0.227188\pi$$
0.755924 + 0.654659i $$0.227188\pi$$
$$252$$ 576.000 0.143986
$$253$$ −2016.00 −0.500968
$$254$$ 5072.00 1.25294
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 2046.00 0.496599 0.248300 0.968683i $$-0.420128\pi$$
0.248300 + 0.968683i $$0.420128\pi$$
$$258$$ 312.000 0.0752879
$$259$$ −4064.00 −0.974999
$$260$$ 0 0
$$261$$ 270.000 0.0640329
$$262$$ 4584.00 1.08092
$$263$$ 6072.00 1.42363 0.711817 0.702365i $$-0.247873\pi$$
0.711817 + 0.702365i $$0.247873\pi$$
$$264$$ 288.000 0.0671408
$$265$$ 0 0
$$266$$ 640.000 0.147522
$$267$$ 2430.00 0.556980
$$268$$ −3536.00 −0.805954
$$269$$ −6930.00 −1.57074 −0.785371 0.619025i $$-0.787528\pi$$
−0.785371 + 0.619025i $$0.787528\pi$$
$$270$$ 0 0
$$271$$ 1352.00 0.303056 0.151528 0.988453i $$-0.451581\pi$$
0.151528 + 0.988453i $$0.451581\pi$$
$$272$$ 2016.00 0.449404
$$273$$ −1824.00 −0.404372
$$274$$ 1452.00 0.320141
$$275$$ 0 0
$$276$$ −2016.00 −0.439670
$$277$$ 1186.00 0.257256 0.128628 0.991693i $$-0.458943\pi$$
0.128628 + 0.991693i $$0.458943\pi$$
$$278$$ 760.000 0.163963
$$279$$ −792.000 −0.169949
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 576.000 0.121632
$$283$$ −2828.00 −0.594018 −0.297009 0.954875i $$-0.595989\pi$$
−0.297009 + 0.954875i $$0.595989\pi$$
$$284$$ 3168.00 0.661923
$$285$$ 0 0
$$286$$ −912.000 −0.188558
$$287$$ 672.000 0.138212
$$288$$ 288.000 0.0589256
$$289$$ 10963.0 2.23143
$$290$$ 0 0
$$291$$ −3462.00 −0.697409
$$292$$ −872.000 −0.174760
$$293$$ −4758.00 −0.948687 −0.474344 0.880340i $$-0.657315\pi$$
−0.474344 + 0.880340i $$0.657315\pi$$
$$294$$ −522.000 −0.103550
$$295$$ 0 0
$$296$$ −2032.00 −0.399012
$$297$$ 324.000 0.0633010
$$298$$ 3180.00 0.618163
$$299$$ 6384.00 1.23477
$$300$$ 0 0
$$301$$ 832.000 0.159321
$$302$$ 4864.00 0.926794
$$303$$ −1854.00 −0.351517
$$304$$ 320.000 0.0603726
$$305$$ 0 0
$$306$$ 2268.00 0.423702
$$307$$ 8476.00 1.57574 0.787868 0.615844i $$-0.211185\pi$$
0.787868 + 0.615844i $$0.211185\pi$$
$$308$$ 768.000 0.142081
$$309$$ −384.000 −0.0706958
$$310$$ 0 0
$$311$$ 4632.00 0.844555 0.422278 0.906467i $$-0.361231\pi$$
0.422278 + 0.906467i $$0.361231\pi$$
$$312$$ −912.000 −0.165487
$$313$$ 4822.00 0.870785 0.435392 0.900241i $$-0.356610\pi$$
0.435392 + 0.900241i $$0.356610\pi$$
$$314$$ −1228.00 −0.220701
$$315$$ 0 0
$$316$$ −2080.00 −0.370282
$$317$$ 3426.00 0.607014 0.303507 0.952829i $$-0.401842\pi$$
0.303507 + 0.952829i $$0.401842\pi$$
$$318$$ −1188.00 −0.209496
$$319$$ 360.000 0.0631854
$$320$$ 0 0
$$321$$ 4428.00 0.769928
$$322$$ −5376.00 −0.930412
$$323$$ 2520.00 0.434107
$$324$$ 324.000 0.0555556
$$325$$ 0 0
$$326$$ 3704.00 0.629281
$$327$$ 3570.00 0.603735
$$328$$ 336.000 0.0565625
$$329$$ 1536.00 0.257393
$$330$$ 0 0
$$331$$ −2788.00 −0.462968 −0.231484 0.972839i $$-0.574358\pi$$
−0.231484 + 0.972839i $$0.574358\pi$$
$$332$$ 1968.00 0.325325
$$333$$ −2286.00 −0.376192
$$334$$ 4272.00 0.699861
$$335$$ 0 0
$$336$$ 768.000 0.124696
$$337$$ −434.000 −0.0701528 −0.0350764 0.999385i $$-0.511167\pi$$
−0.0350764 + 0.999385i $$0.511167\pi$$
$$338$$ −1506.00 −0.242354
$$339$$ 1386.00 0.222057
$$340$$ 0 0
$$341$$ −1056.00 −0.167700
$$342$$ 360.000 0.0569198
$$343$$ −6880.00 −1.08305
$$344$$ 416.000 0.0652012
$$345$$ 0 0
$$346$$ −3516.00 −0.546304
$$347$$ −6684.00 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$348$$ 360.000 0.0554541
$$349$$ 2630.00 0.403383 0.201692 0.979449i $$-0.435356\pi$$
0.201692 + 0.979449i $$0.435356\pi$$
$$350$$ 0 0
$$351$$ −1026.00 −0.156022
$$352$$ 384.000 0.0581456
$$353$$ 7422.00 1.11907 0.559537 0.828805i $$-0.310979\pi$$
0.559537 + 0.828805i $$0.310979\pi$$
$$354$$ −3960.00 −0.594553
$$355$$ 0 0
$$356$$ 3240.00 0.482359
$$357$$ 6048.00 0.896622
$$358$$ −1080.00 −0.159441
$$359$$ −10440.0 −1.53482 −0.767412 0.641154i $$-0.778456\pi$$
−0.767412 + 0.641154i $$0.778456\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 3964.00 0.575534
$$363$$ −3561.00 −0.514887
$$364$$ −2432.00 −0.350196
$$365$$ 0 0
$$366$$ −3228.00 −0.461012
$$367$$ −10424.0 −1.48264 −0.741319 0.671153i $$-0.765800\pi$$
−0.741319 + 0.671153i $$0.765800\pi$$
$$368$$ −2688.00 −0.380765
$$369$$ 378.000 0.0533276
$$370$$ 0 0
$$371$$ −3168.00 −0.443327
$$372$$ −1056.00 −0.147180
$$373$$ −3278.00 −0.455036 −0.227518 0.973774i $$-0.573061\pi$$
−0.227518 + 0.973774i $$0.573061\pi$$
$$374$$ 3024.00 0.418094
$$375$$ 0 0
$$376$$ 768.000 0.105337
$$377$$ −1140.00 −0.155737
$$378$$ 864.000 0.117564
$$379$$ 6140.00 0.832165 0.416083 0.909327i $$-0.363403\pi$$
0.416083 + 0.909327i $$0.363403\pi$$
$$380$$ 0 0
$$381$$ 7608.00 1.02302
$$382$$ −5376.00 −0.720053
$$383$$ 3072.00 0.409848 0.204924 0.978778i $$-0.434305\pi$$
0.204924 + 0.978778i $$0.434305\pi$$
$$384$$ 384.000 0.0510310
$$385$$ 0 0
$$386$$ 4604.00 0.607092
$$387$$ 468.000 0.0614723
$$388$$ −4616.00 −0.603974
$$389$$ 6150.00 0.801587 0.400794 0.916168i $$-0.368734\pi$$
0.400794 + 0.916168i $$0.368734\pi$$
$$390$$ 0 0
$$391$$ −21168.0 −2.73788
$$392$$ −696.000 −0.0896768
$$393$$ 6876.00 0.882566
$$394$$ −8748.00 −1.11857
$$395$$ 0 0
$$396$$ 432.000 0.0548202
$$397$$ 106.000 0.0134005 0.00670024 0.999978i $$-0.497867\pi$$
0.00670024 + 0.999978i $$0.497867\pi$$
$$398$$ −3200.00 −0.403019
$$399$$ 960.000 0.120451
$$400$$ 0 0
$$401$$ −1758.00 −0.218929 −0.109464 0.993991i $$-0.534914\pi$$
−0.109464 + 0.993991i $$0.534914\pi$$
$$402$$ −5304.00 −0.658058
$$403$$ 3344.00 0.413341
$$404$$ −2472.00 −0.304422
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ −3048.00 −0.371213
$$408$$ 3024.00 0.366937
$$409$$ −3670.00 −0.443691 −0.221846 0.975082i $$-0.571208\pi$$
−0.221846 + 0.975082i $$0.571208\pi$$
$$410$$ 0 0
$$411$$ 2178.00 0.261394
$$412$$ −512.000 −0.0612243
$$413$$ −10560.0 −1.25817
$$414$$ −3024.00 −0.358989
$$415$$ 0 0
$$416$$ −1216.00 −0.143316
$$417$$ 1140.00 0.133875
$$418$$ 480.000 0.0561664
$$419$$ −9660.00 −1.12631 −0.563153 0.826353i $$-0.690412\pi$$
−0.563153 + 0.826353i $$0.690412\pi$$
$$420$$ 0 0
$$421$$ 8462.00 0.979602 0.489801 0.871834i $$-0.337069\pi$$
0.489801 + 0.871834i $$0.337069\pi$$
$$422$$ 6664.00 0.768717
$$423$$ 864.000 0.0993123
$$424$$ −1584.00 −0.181429
$$425$$ 0 0
$$426$$ 4752.00 0.540458
$$427$$ −8608.00 −0.975575
$$428$$ 5904.00 0.666777
$$429$$ −1368.00 −0.153957
$$430$$ 0 0
$$431$$ 9792.00 1.09435 0.547174 0.837019i $$-0.315704\pi$$
0.547174 + 0.837019i $$0.315704\pi$$
$$432$$ 432.000 0.0481125
$$433$$ 7342.00 0.814859 0.407430 0.913237i $$-0.366425\pi$$
0.407430 + 0.913237i $$0.366425\pi$$
$$434$$ −2816.00 −0.311457
$$435$$ 0 0
$$436$$ 4760.00 0.522850
$$437$$ −3360.00 −0.367805
$$438$$ −1308.00 −0.142691
$$439$$ 10640.0 1.15676 0.578382 0.815766i $$-0.303684\pi$$
0.578382 + 0.815766i $$0.303684\pi$$
$$440$$ 0 0
$$441$$ −783.000 −0.0845481
$$442$$ −9576.00 −1.03051
$$443$$ 17412.0 1.86742 0.933712 0.358024i $$-0.116549\pi$$
0.933712 + 0.358024i $$0.116549\pi$$
$$444$$ −3048.00 −0.325792
$$445$$ 0 0
$$446$$ −5296.00 −0.562271
$$447$$ 4770.00 0.504728
$$448$$ 1024.00 0.107990
$$449$$ −1710.00 −0.179732 −0.0898662 0.995954i $$-0.528644\pi$$
−0.0898662 + 0.995954i $$0.528644\pi$$
$$450$$ 0 0
$$451$$ 504.000 0.0526218
$$452$$ 1848.00 0.192307
$$453$$ 7296.00 0.756724
$$454$$ −4488.00 −0.463948
$$455$$ 0 0
$$456$$ 480.000 0.0492940
$$457$$ 646.000 0.0661239 0.0330619 0.999453i $$-0.489474\pi$$
0.0330619 + 0.999453i $$0.489474\pi$$
$$458$$ −11300.0 −1.15287
$$459$$ 3402.00 0.345952
$$460$$ 0 0
$$461$$ −6018.00 −0.607996 −0.303998 0.952673i $$-0.598322\pi$$
−0.303998 + 0.952673i $$0.598322\pi$$
$$462$$ 1152.00 0.116008
$$463$$ 6712.00 0.673722 0.336861 0.941554i $$-0.390635\pi$$
0.336861 + 0.941554i $$0.390635\pi$$
$$464$$ 480.000 0.0480247
$$465$$ 0 0
$$466$$ −9396.00 −0.934037
$$467$$ −5364.00 −0.531512 −0.265756 0.964040i $$-0.585622\pi$$
−0.265756 + 0.964040i $$0.585622\pi$$
$$468$$ −1368.00 −0.135119
$$469$$ −14144.0 −1.39256
$$470$$ 0 0
$$471$$ −1842.00 −0.180201
$$472$$ −5280.00 −0.514898
$$473$$ 624.000 0.0606587
$$474$$ −3120.00 −0.302334
$$475$$ 0 0
$$476$$ 8064.00 0.776498
$$477$$ −1782.00 −0.171053
$$478$$ −2400.00 −0.229652
$$479$$ 9840.00 0.938624 0.469312 0.883032i $$-0.344502\pi$$
0.469312 + 0.883032i $$0.344502\pi$$
$$480$$ 0 0
$$481$$ 9652.00 0.914955
$$482$$ −1436.00 −0.135701
$$483$$ −8064.00 −0.759678
$$484$$ −4748.00 −0.445905
$$485$$ 0 0
$$486$$ 486.000 0.0453609
$$487$$ −1424.00 −0.132500 −0.0662501 0.997803i $$-0.521104\pi$$
−0.0662501 + 0.997803i $$0.521104\pi$$
$$488$$ −4304.00 −0.399248
$$489$$ 5556.00 0.513806
$$490$$ 0 0
$$491$$ −4548.00 −0.418021 −0.209011 0.977913i $$-0.567024\pi$$
−0.209011 + 0.977913i $$0.567024\pi$$
$$492$$ 504.000 0.0461831
$$493$$ 3780.00 0.345320
$$494$$ −1520.00 −0.138437
$$495$$ 0 0
$$496$$ −1408.00 −0.127462
$$497$$ 12672.0 1.14370
$$498$$ 2952.00 0.265627
$$499$$ 6500.00 0.583126 0.291563 0.956552i $$-0.405825\pi$$
0.291563 + 0.956552i $$0.405825\pi$$
$$500$$ 0 0
$$501$$ 6408.00 0.571434
$$502$$ 12024.0 1.06904
$$503$$ −12168.0 −1.07862 −0.539308 0.842108i $$-0.681314\pi$$
−0.539308 + 0.842108i $$0.681314\pi$$
$$504$$ 1152.00 0.101814
$$505$$ 0 0
$$506$$ −4032.00 −0.354238
$$507$$ −2259.00 −0.197881
$$508$$ 10144.0 0.885959
$$509$$ −21090.0 −1.83654 −0.918269 0.395957i $$-0.870413\pi$$
−0.918269 + 0.395957i $$0.870413\pi$$
$$510$$ 0 0
$$511$$ −3488.00 −0.301957
$$512$$ 512.000 0.0441942
$$513$$ 540.000 0.0464748
$$514$$ 4092.00 0.351149
$$515$$ 0 0
$$516$$ 624.000 0.0532366
$$517$$ 1152.00 0.0979979
$$518$$ −8128.00 −0.689428
$$519$$ −5274.00 −0.446056
$$520$$ 0 0
$$521$$ −5238.00 −0.440462 −0.220231 0.975448i $$-0.570681\pi$$
−0.220231 + 0.975448i $$0.570681\pi$$
$$522$$ 540.000 0.0452781
$$523$$ −8588.00 −0.718025 −0.359012 0.933333i $$-0.616886\pi$$
−0.359012 + 0.933333i $$0.616886\pi$$
$$524$$ 9168.00 0.764324
$$525$$ 0 0
$$526$$ 12144.0 1.00666
$$527$$ −11088.0 −0.916510
$$528$$ 576.000 0.0474757
$$529$$ 16057.0 1.31972
$$530$$ 0 0
$$531$$ −5940.00 −0.485450
$$532$$ 1280.00 0.104314
$$533$$ −1596.00 −0.129701
$$534$$ 4860.00 0.393844
$$535$$ 0 0
$$536$$ −7072.00 −0.569895
$$537$$ −1620.00 −0.130183
$$538$$ −13860.0 −1.11068
$$539$$ −1044.00 −0.0834291
$$540$$ 0 0
$$541$$ 3062.00 0.243338 0.121669 0.992571i $$-0.461175\pi$$
0.121669 + 0.992571i $$0.461175\pi$$
$$542$$ 2704.00 0.214293
$$543$$ 5946.00 0.469921
$$544$$ 4032.00 0.317777
$$545$$ 0 0
$$546$$ −3648.00 −0.285934
$$547$$ 8476.00 0.662537 0.331268 0.943537i $$-0.392523\pi$$
0.331268 + 0.943537i $$0.392523\pi$$
$$548$$ 2904.00 0.226374
$$549$$ −4842.00 −0.376414
$$550$$ 0 0
$$551$$ 600.000 0.0463899
$$552$$ −4032.00 −0.310894
$$553$$ −8320.00 −0.639787
$$554$$ 2372.00 0.181907
$$555$$ 0 0
$$556$$ 1520.00 0.115939
$$557$$ 12546.0 0.954383 0.477191 0.878799i $$-0.341655\pi$$
0.477191 + 0.878799i $$0.341655\pi$$
$$558$$ −1584.00 −0.120172
$$559$$ −1976.00 −0.149510
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ 4884.00 0.366582
$$563$$ 12.0000 0.000898294 0 0.000449147 1.00000i $$-0.499857\pi$$
0.000449147 1.00000i $$0.499857\pi$$
$$564$$ 1152.00 0.0860070
$$565$$ 0 0
$$566$$ −5656.00 −0.420034
$$567$$ 1296.00 0.0959910
$$568$$ 6336.00 0.468050
$$569$$ 19290.0 1.42123 0.710614 0.703582i $$-0.248417\pi$$
0.710614 + 0.703582i $$0.248417\pi$$
$$570$$ 0 0
$$571$$ −12148.0 −0.890329 −0.445165 0.895449i $$-0.646855\pi$$
−0.445165 + 0.895449i $$0.646855\pi$$
$$572$$ −1824.00 −0.133331
$$573$$ −8064.00 −0.587920
$$574$$ 1344.00 0.0977308
$$575$$ 0 0
$$576$$ 576.000 0.0416667
$$577$$ 10366.0 0.747907 0.373953 0.927447i $$-0.378002\pi$$
0.373953 + 0.927447i $$0.378002\pi$$
$$578$$ 21926.0 1.57786
$$579$$ 6906.00 0.495688
$$580$$ 0 0
$$581$$ 7872.00 0.562109
$$582$$ −6924.00 −0.493143
$$583$$ −2376.00 −0.168789
$$584$$ −1744.00 −0.123574
$$585$$ 0 0
$$586$$ −9516.00 −0.670823
$$587$$ −7644.00 −0.537482 −0.268741 0.963213i $$-0.586607\pi$$
−0.268741 + 0.963213i $$0.586607\pi$$
$$588$$ −1044.00 −0.0732208
$$589$$ −1760.00 −0.123123
$$590$$ 0 0
$$591$$ −13122.0 −0.913311
$$592$$ −4064.00 −0.282144
$$593$$ −8658.00 −0.599564 −0.299782 0.954008i $$-0.596914\pi$$
−0.299782 + 0.954008i $$0.596914\pi$$
$$594$$ 648.000 0.0447605
$$595$$ 0 0
$$596$$ 6360.00 0.437107
$$597$$ −4800.00 −0.329064
$$598$$ 12768.0 0.873114
$$599$$ 25800.0 1.75987 0.879933 0.475098i $$-0.157587\pi$$
0.879933 + 0.475098i $$0.157587\pi$$
$$600$$ 0 0
$$601$$ 16202.0 1.09966 0.549828 0.835278i $$-0.314693\pi$$
0.549828 + 0.835278i $$0.314693\pi$$
$$602$$ 1664.00 0.112657
$$603$$ −7956.00 −0.537302
$$604$$ 9728.00 0.655342
$$605$$ 0 0
$$606$$ −3708.00 −0.248560
$$607$$ 24136.0 1.61392 0.806960 0.590605i $$-0.201111\pi$$
0.806960 + 0.590605i $$0.201111\pi$$
$$608$$ 640.000 0.0426898
$$609$$ 1440.00 0.0958157
$$610$$ 0 0
$$611$$ −3648.00 −0.241542
$$612$$ 4536.00 0.299603
$$613$$ 4642.00 0.305854 0.152927 0.988237i $$-0.451130\pi$$
0.152927 + 0.988237i $$0.451130\pi$$
$$614$$ 16952.0 1.11421
$$615$$ 0 0
$$616$$ 1536.00 0.100466
$$617$$ 6726.00 0.438863 0.219432 0.975628i $$-0.429580\pi$$
0.219432 + 0.975628i $$0.429580\pi$$
$$618$$ −768.000 −0.0499895
$$619$$ −21220.0 −1.37787 −0.688937 0.724821i $$-0.741922\pi$$
−0.688937 + 0.724821i $$0.741922\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ 9264.00 0.597191
$$623$$ 12960.0 0.833437
$$624$$ −1824.00 −0.117017
$$625$$ 0 0
$$626$$ 9644.00 0.615738
$$627$$ 720.000 0.0458597
$$628$$ −2456.00 −0.156059
$$629$$ −32004.0 −2.02875
$$630$$ 0 0
$$631$$ 29792.0 1.87956 0.939779 0.341783i $$-0.111031\pi$$
0.939779 + 0.341783i $$0.111031\pi$$
$$632$$ −4160.00 −0.261829
$$633$$ 9996.00 0.627655
$$634$$ 6852.00 0.429223
$$635$$ 0 0
$$636$$ −2376.00 −0.148136
$$637$$ 3306.00 0.205633
$$638$$ 720.000 0.0446788
$$639$$ 7128.00 0.441282
$$640$$ 0 0
$$641$$ −10158.0 −0.625923 −0.312962 0.949766i $$-0.601321\pi$$
−0.312962 + 0.949766i $$0.601321\pi$$
$$642$$ 8856.00 0.544421
$$643$$ −29828.0 −1.82940 −0.914698 0.404138i $$-0.867571\pi$$
−0.914698 + 0.404138i $$0.867571\pi$$
$$644$$ −10752.0 −0.657901
$$645$$ 0 0
$$646$$ 5040.00 0.306960
$$647$$ −1944.00 −0.118124 −0.0590622 0.998254i $$-0.518811\pi$$
−0.0590622 + 0.998254i $$0.518811\pi$$
$$648$$ 648.000 0.0392837
$$649$$ −7920.00 −0.479025
$$650$$ 0 0
$$651$$ −4224.00 −0.254304
$$652$$ 7408.00 0.444969
$$653$$ −26718.0 −1.60116 −0.800579 0.599227i $$-0.795475\pi$$
−0.800579 + 0.599227i $$0.795475\pi$$
$$654$$ 7140.00 0.426905
$$655$$ 0 0
$$656$$ 672.000 0.0399957
$$657$$ −1962.00 −0.116507
$$658$$ 3072.00 0.182005
$$659$$ 4260.00 0.251815 0.125907 0.992042i $$-0.459816\pi$$
0.125907 + 0.992042i $$0.459816\pi$$
$$660$$ 0 0
$$661$$ 22862.0 1.34528 0.672639 0.739971i $$-0.265161\pi$$
0.672639 + 0.739971i $$0.265161\pi$$
$$662$$ −5576.00 −0.327368
$$663$$ −14364.0 −0.841405
$$664$$ 3936.00 0.230040
$$665$$ 0 0
$$666$$ −4572.00 −0.266008
$$667$$ −5040.00 −0.292578
$$668$$ 8544.00 0.494876
$$669$$ −7944.00 −0.459092
$$670$$ 0 0
$$671$$ −6456.00 −0.371432
$$672$$ 1536.00 0.0881733
$$673$$ 32542.0 1.86390 0.931948 0.362592i $$-0.118108\pi$$
0.931948 + 0.362592i $$0.118108\pi$$
$$674$$ −868.000 −0.0496055
$$675$$ 0 0
$$676$$ −3012.00 −0.171370
$$677$$ −14214.0 −0.806925 −0.403463 0.914996i $$-0.632193\pi$$
−0.403463 + 0.914996i $$0.632193\pi$$
$$678$$ 2772.00 0.157018
$$679$$ −18464.0 −1.04357
$$680$$ 0 0
$$681$$ −6732.00 −0.378812
$$682$$ −2112.00 −0.118582
$$683$$ 7092.00 0.397317 0.198659 0.980069i $$-0.436341\pi$$
0.198659 + 0.980069i $$0.436341\pi$$
$$684$$ 720.000 0.0402484
$$685$$ 0 0
$$686$$ −13760.0 −0.765830
$$687$$ −16950.0 −0.941314
$$688$$ 832.000 0.0461042
$$689$$ 7524.00 0.416026
$$690$$ 0 0
$$691$$ −13228.0 −0.728244 −0.364122 0.931351i $$-0.618631\pi$$
−0.364122 + 0.931351i $$0.618631\pi$$
$$692$$ −7032.00 −0.386296
$$693$$ 1728.00 0.0947205
$$694$$ −13368.0 −0.731185
$$695$$ 0 0
$$696$$ 720.000 0.0392120
$$697$$ 5292.00 0.287588
$$698$$ 5260.00 0.285235
$$699$$ −14094.0 −0.762638
$$700$$ 0 0
$$701$$ 28062.0 1.51196 0.755982 0.654592i $$-0.227160\pi$$
0.755982 + 0.654592i $$0.227160\pi$$
$$702$$ −2052.00 −0.110324
$$703$$ −5080.00 −0.272540
$$704$$ 768.000 0.0411152
$$705$$ 0 0
$$706$$ 14844.0 0.791305
$$707$$ −9888.00 −0.525992
$$708$$ −7920.00 −0.420412
$$709$$ −27250.0 −1.44343 −0.721717 0.692188i $$-0.756647\pi$$
−0.721717 + 0.692188i $$0.756647\pi$$
$$710$$ 0 0
$$711$$ −4680.00 −0.246855
$$712$$ 6480.00 0.341079
$$713$$ 14784.0 0.776529
$$714$$ 12096.0 0.634008
$$715$$ 0 0
$$716$$ −2160.00 −0.112742
$$717$$ −3600.00 −0.187510
$$718$$ −20880.0 −1.08529
$$719$$ −14400.0 −0.746912 −0.373456 0.927648i $$-0.621827\pi$$
−0.373456 + 0.927648i $$0.621827\pi$$
$$720$$ 0 0
$$721$$ −2048.00 −0.105786
$$722$$ −12918.0 −0.665870
$$723$$ −2154.00 −0.110800
$$724$$ 7928.00 0.406964
$$725$$ 0 0
$$726$$ −7122.00 −0.364080
$$727$$ −17984.0 −0.917455 −0.458727 0.888577i $$-0.651695\pi$$
−0.458727 + 0.888577i $$0.651695\pi$$
$$728$$ −4864.00 −0.247626
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 6552.00 0.331511
$$732$$ −6456.00 −0.325984
$$733$$ −16598.0 −0.836373 −0.418186 0.908361i $$-0.637334\pi$$
−0.418186 + 0.908361i $$0.637334\pi$$
$$734$$ −20848.0 −1.04838
$$735$$ 0 0
$$736$$ −5376.00 −0.269242
$$737$$ −10608.0 −0.530191
$$738$$ 756.000 0.0377083
$$739$$ 1460.00 0.0726752 0.0363376 0.999340i $$-0.488431\pi$$
0.0363376 + 0.999340i $$0.488431\pi$$
$$740$$ 0 0
$$741$$ −2280.00 −0.113034
$$742$$ −6336.00 −0.313480
$$743$$ 30072.0 1.48484 0.742419 0.669936i $$-0.233678\pi$$
0.742419 + 0.669936i $$0.233678\pi$$
$$744$$ −2112.00 −0.104072
$$745$$ 0 0
$$746$$ −6556.00 −0.321759
$$747$$ 4428.00 0.216884
$$748$$ 6048.00 0.295637
$$749$$ 23616.0 1.15208
$$750$$ 0 0
$$751$$ −18088.0 −0.878882 −0.439441 0.898271i $$-0.644823\pi$$
−0.439441 + 0.898271i $$0.644823\pi$$
$$752$$ 1536.00 0.0744843
$$753$$ 18036.0 0.872866
$$754$$ −2280.00 −0.110123
$$755$$ 0 0
$$756$$ 1728.00 0.0831306
$$757$$ −24734.0 −1.18755 −0.593773 0.804633i $$-0.702362\pi$$
−0.593773 + 0.804633i $$0.702362\pi$$
$$758$$ 12280.0 0.588430
$$759$$ −6048.00 −0.289234
$$760$$ 0 0
$$761$$ −22278.0 −1.06120 −0.530602 0.847621i $$-0.678034\pi$$
−0.530602 + 0.847621i $$0.678034\pi$$
$$762$$ 15216.0 0.723383
$$763$$ 19040.0 0.903400
$$764$$ −10752.0 −0.509154
$$765$$ 0 0
$$766$$ 6144.00 0.289806
$$767$$ 25080.0 1.18069
$$768$$ 768.000 0.0360844
$$769$$ 16130.0 0.756388 0.378194 0.925726i $$-0.376545\pi$$
0.378194 + 0.925726i $$0.376545\pi$$
$$770$$ 0 0
$$771$$ 6138.00 0.286712
$$772$$ 9208.00 0.429279
$$773$$ −29718.0 −1.38277 −0.691386 0.722486i $$-0.742999\pi$$
−0.691386 + 0.722486i $$0.742999\pi$$
$$774$$ 936.000 0.0434675
$$775$$ 0 0
$$776$$ −9232.00 −0.427074
$$777$$ −12192.0 −0.562916
$$778$$ 12300.0 0.566808
$$779$$ 840.000 0.0386343
$$780$$ 0 0
$$781$$ 9504.00 0.435442
$$782$$ −42336.0 −1.93597
$$783$$ 810.000 0.0369694
$$784$$ −1392.00 −0.0634111
$$785$$ 0 0
$$786$$ 13752.0 0.624068
$$787$$ −9524.00 −0.431377 −0.215689 0.976462i $$-0.569200\pi$$
−0.215689 + 0.976462i $$0.569200\pi$$
$$788$$ −17496.0 −0.790951
$$789$$ 18216.0 0.821935
$$790$$ 0 0
$$791$$ 7392.00 0.332275
$$792$$ 864.000 0.0387638
$$793$$ 20444.0 0.915495
$$794$$ 212.000 0.00947556
$$795$$ 0 0
$$796$$ −6400.00 −0.284977
$$797$$ 33906.0 1.50692 0.753458 0.657496i $$-0.228384\pi$$
0.753458 + 0.657496i $$0.228384\pi$$
$$798$$ 1920.00 0.0851720
$$799$$ 12096.0 0.535577
$$800$$ 0 0
$$801$$ 7290.00 0.321572
$$802$$ −3516.00 −0.154806
$$803$$ −2616.00 −0.114965
$$804$$ −10608.0 −0.465318
$$805$$ 0 0
$$806$$ 6688.00 0.292276
$$807$$ −20790.0 −0.906868
$$808$$ −4944.00 −0.215259
$$809$$ −630.000 −0.0273790 −0.0136895 0.999906i $$-0.504358\pi$$
−0.0136895 + 0.999906i $$0.504358\pi$$
$$810$$ 0 0
$$811$$ −20788.0 −0.900081 −0.450040 0.893008i $$-0.648590\pi$$
−0.450040 + 0.893008i $$0.648590\pi$$
$$812$$ 1920.00 0.0829788
$$813$$ 4056.00 0.174969
$$814$$ −6096.00 −0.262487
$$815$$ 0 0
$$816$$ 6048.00 0.259464
$$817$$ 1040.00 0.0445349
$$818$$ −7340.00 −0.313737
$$819$$ −5472.00 −0.233464
$$820$$ 0 0
$$821$$ −43098.0 −1.83207 −0.916036 0.401097i $$-0.868629\pi$$
−0.916036 + 0.401097i $$0.868629\pi$$
$$822$$ 4356.00 0.184833
$$823$$ 14272.0 0.604484 0.302242 0.953231i $$-0.402265\pi$$
0.302242 + 0.953231i $$0.402265\pi$$
$$824$$ −1024.00 −0.0432921
$$825$$ 0 0
$$826$$ −21120.0 −0.889660
$$827$$ −13644.0 −0.573698 −0.286849 0.957976i $$-0.592608\pi$$
−0.286849 + 0.957976i $$0.592608\pi$$
$$828$$ −6048.00 −0.253844
$$829$$ −2410.00 −0.100968 −0.0504842 0.998725i $$-0.516076\pi$$
−0.0504842 + 0.998725i $$0.516076\pi$$
$$830$$ 0 0
$$831$$ 3558.00 0.148527
$$832$$ −2432.00 −0.101339
$$833$$ −10962.0 −0.455955
$$834$$ 2280.00 0.0946642
$$835$$ 0 0
$$836$$ 960.000 0.0397157
$$837$$ −2376.00 −0.0981202
$$838$$ −19320.0 −0.796418
$$839$$ 23160.0 0.953006 0.476503 0.879173i $$-0.341904\pi$$
0.476503 + 0.879173i $$0.341904\pi$$
$$840$$ 0 0
$$841$$ −23489.0 −0.963098
$$842$$ 16924.0 0.692684
$$843$$ 7326.00 0.299313
$$844$$ 13328.0 0.543565
$$845$$ 0 0
$$846$$ 1728.00 0.0702244
$$847$$ −18992.0 −0.770452
$$848$$ −3168.00 −0.128290
$$849$$ −8484.00 −0.342957
$$850$$ 0 0
$$851$$ 42672.0 1.71889
$$852$$ 9504.00 0.382162
$$853$$ −32078.0 −1.28761 −0.643804 0.765190i $$-0.722645\pi$$
−0.643804 + 0.765190i $$0.722645\pi$$
$$854$$ −17216.0 −0.689835
$$855$$ 0 0
$$856$$ 11808.0 0.471483
$$857$$ 14406.0 0.574212 0.287106 0.957899i $$-0.407307\pi$$
0.287106 + 0.957899i $$0.407307\pi$$
$$858$$ −2736.00 −0.108864
$$859$$ 30620.0 1.21623 0.608115 0.793849i $$-0.291926\pi$$
0.608115 + 0.793849i $$0.291926\pi$$
$$860$$ 0 0
$$861$$ 2016.00 0.0797969
$$862$$ 19584.0 0.773821
$$863$$ −17568.0 −0.692957 −0.346478 0.938058i $$-0.612623\pi$$
−0.346478 + 0.938058i $$0.612623\pi$$
$$864$$ 864.000 0.0340207
$$865$$ 0 0
$$866$$ 14684.0 0.576192
$$867$$ 32889.0 1.28831
$$868$$ −5632.00 −0.220233
$$869$$ −6240.00 −0.243587
$$870$$ 0 0
$$871$$ 33592.0 1.30680
$$872$$ 9520.00 0.369711
$$873$$ −10386.0 −0.402649
$$874$$ −6720.00 −0.260077
$$875$$ 0 0
$$876$$ −2616.00 −0.100898
$$877$$ 21706.0 0.835758 0.417879 0.908503i $$-0.362774\pi$$
0.417879 + 0.908503i $$0.362774\pi$$
$$878$$ 21280.0 0.817956
$$879$$ −14274.0 −0.547725
$$880$$ 0 0
$$881$$ −14958.0 −0.572018 −0.286009 0.958227i $$-0.592329\pi$$
−0.286009 + 0.958227i $$0.592329\pi$$
$$882$$ −1566.00 −0.0597845
$$883$$ 32812.0 1.25052 0.625261 0.780415i $$-0.284992\pi$$
0.625261 + 0.780415i $$0.284992\pi$$
$$884$$ −19152.0 −0.728678
$$885$$ 0 0
$$886$$ 34824.0 1.32047
$$887$$ 38856.0 1.47086 0.735432 0.677598i $$-0.236979\pi$$
0.735432 + 0.677598i $$0.236979\pi$$
$$888$$ −6096.00 −0.230370
$$889$$ 40576.0 1.53079
$$890$$ 0 0
$$891$$ 972.000 0.0365468
$$892$$ −10592.0 −0.397586
$$893$$ 1920.00 0.0719489
$$894$$ 9540.00 0.356896
$$895$$ 0 0
$$896$$ 2048.00 0.0763604
$$897$$ 19152.0 0.712895
$$898$$ −3420.00 −0.127090
$$899$$ −2640.00 −0.0979410
$$900$$ 0 0
$$901$$ −24948.0 −0.922462
$$902$$ 1008.00 0.0372092
$$903$$ 2496.00 0.0919841
$$904$$ 3696.00 0.135981
$$905$$ 0 0
$$906$$ 14592.0 0.535085
$$907$$ 28276.0 1.03516 0.517579 0.855635i $$-0.326833\pi$$
0.517579 + 0.855635i $$0.326833\pi$$
$$908$$ −8976.00 −0.328061
$$909$$ −5562.00 −0.202948
$$910$$ 0 0
$$911$$ 8112.00 0.295019 0.147510 0.989061i $$-0.452874\pi$$
0.147510 + 0.989061i $$0.452874\pi$$
$$912$$ 960.000 0.0348561
$$913$$ 5904.00 0.214013
$$914$$ 1292.00 0.0467566
$$915$$ 0 0
$$916$$ −22600.0 −0.815202
$$917$$ 36672.0 1.32063
$$918$$ 6804.00 0.244625
$$919$$ −26080.0 −0.936126 −0.468063 0.883695i $$-0.655048\pi$$
−0.468063 + 0.883695i $$0.655048\pi$$
$$920$$ 0 0
$$921$$ 25428.0 0.909751
$$922$$ −12036.0 −0.429918
$$923$$ −30096.0 −1.07326
$$924$$ 2304.00 0.0820303
$$925$$ 0 0
$$926$$ 13424.0 0.476393
$$927$$ −1152.00 −0.0408162
$$928$$ 960.000 0.0339586
$$929$$ 49170.0 1.73651 0.868254 0.496120i $$-0.165243\pi$$
0.868254 + 0.496120i $$0.165243\pi$$
$$930$$ 0 0
$$931$$ −1740.00 −0.0612526
$$932$$ −18792.0 −0.660464
$$933$$ 13896.0 0.487604
$$934$$ −10728.0 −0.375836
$$935$$ 0 0
$$936$$ −2736.00 −0.0955438
$$937$$ −48314.0 −1.68447 −0.842236 0.539110i $$-0.818761\pi$$
−0.842236 + 0.539110i $$0.818761\pi$$
$$938$$ −28288.0 −0.984687
$$939$$ 14466.0 0.502748
$$940$$ 0 0
$$941$$ 34782.0 1.20495 0.602477 0.798137i $$-0.294181\pi$$
0.602477 + 0.798137i $$0.294181\pi$$
$$942$$ −3684.00 −0.127422
$$943$$ −7056.00 −0.243664
$$944$$ −10560.0 −0.364088
$$945$$ 0 0
$$946$$ 1248.00 0.0428922
$$947$$ 25116.0 0.861838 0.430919 0.902391i $$-0.358190\pi$$
0.430919 + 0.902391i $$0.358190\pi$$
$$948$$ −6240.00 −0.213782
$$949$$ 8284.00 0.283361
$$950$$ 0 0
$$951$$ 10278.0 0.350460
$$952$$ 16128.0 0.549067
$$953$$ 15462.0 0.525565 0.262782 0.964855i $$-0.415360\pi$$
0.262782 + 0.964855i $$0.415360\pi$$
$$954$$ −3564.00 −0.120953
$$955$$ 0 0
$$956$$ −4800.00 −0.162388
$$957$$ 1080.00 0.0364801
$$958$$ 19680.0 0.663708
$$959$$ 11616.0 0.391137
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ 19304.0 0.646971
$$963$$ 13284.0 0.444518
$$964$$ −2872.00 −0.0959553
$$965$$ 0 0
$$966$$ −16128.0 −0.537174
$$967$$ 736.000 0.0244759 0.0122379 0.999925i $$-0.496104\pi$$
0.0122379 + 0.999925i $$0.496104\pi$$
$$968$$ −9496.00 −0.315303
$$969$$ 7560.00 0.250632
$$970$$ 0 0
$$971$$ −29268.0 −0.967307 −0.483653 0.875260i $$-0.660690\pi$$
−0.483653 + 0.875260i $$0.660690\pi$$
$$972$$ 972.000 0.0320750
$$973$$ 6080.00 0.200325
$$974$$ −2848.00 −0.0936918
$$975$$ 0 0
$$976$$ −8608.00 −0.282311
$$977$$ −16674.0 −0.546007 −0.273003 0.962013i $$-0.588017\pi$$
−0.273003 + 0.962013i $$0.588017\pi$$
$$978$$ 11112.0 0.363316
$$979$$ 9720.00 0.317316
$$980$$ 0 0
$$981$$ 10710.0 0.348567
$$982$$ −9096.00 −0.295586
$$983$$ 31272.0 1.01467 0.507336 0.861749i $$-0.330630\pi$$
0.507336 + 0.861749i $$0.330630\pi$$
$$984$$ 1008.00 0.0326564
$$985$$ 0 0
$$986$$ 7560.00 0.244178
$$987$$ 4608.00 0.148606
$$988$$ −3040.00 −0.0978900
$$989$$ −8736.00 −0.280878
$$990$$ 0 0
$$991$$ −15928.0 −0.510565 −0.255282 0.966867i $$-0.582168\pi$$
−0.255282 + 0.966867i $$0.582168\pi$$
$$992$$ −2816.00 −0.0901291
$$993$$ −8364.00 −0.267295
$$994$$ 25344.0 0.808715
$$995$$ 0 0
$$996$$ 5904.00 0.187827
$$997$$ −42014.0 −1.33460 −0.667300 0.744789i $$-0.732550\pi$$
−0.667300 + 0.744789i $$0.732550\pi$$
$$998$$ 13000.0 0.412332
$$999$$ −6858.00 −0.217195
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 150.4.a.i.1.1 1
3.2 odd 2 450.4.a.h.1.1 1
4.3 odd 2 1200.4.a.b.1.1 1
5.2 odd 4 150.4.c.d.49.2 2
5.3 odd 4 150.4.c.d.49.1 2
5.4 even 2 6.4.a.a.1.1 1
15.2 even 4 450.4.c.e.199.1 2
15.8 even 4 450.4.c.e.199.2 2
15.14 odd 2 18.4.a.a.1.1 1
20.3 even 4 1200.4.f.j.49.1 2
20.7 even 4 1200.4.f.j.49.2 2
20.19 odd 2 48.4.a.c.1.1 1
35.4 even 6 294.4.e.h.79.1 2
35.9 even 6 294.4.e.h.67.1 2
35.19 odd 6 294.4.e.g.67.1 2
35.24 odd 6 294.4.e.g.79.1 2
35.34 odd 2 294.4.a.e.1.1 1
40.19 odd 2 192.4.a.c.1.1 1
40.29 even 2 192.4.a.i.1.1 1
45.4 even 6 162.4.c.f.55.1 2
45.14 odd 6 162.4.c.c.55.1 2
45.29 odd 6 162.4.c.c.109.1 2
45.34 even 6 162.4.c.f.109.1 2
55.54 odd 2 726.4.a.f.1.1 1
60.59 even 2 144.4.a.c.1.1 1
65.34 odd 4 1014.4.b.d.337.1 2
65.44 odd 4 1014.4.b.d.337.2 2
65.64 even 2 1014.4.a.g.1.1 1
80.19 odd 4 768.4.d.c.385.2 2
80.29 even 4 768.4.d.n.385.1 2
80.59 odd 4 768.4.d.c.385.1 2
80.69 even 4 768.4.d.n.385.2 2
85.84 even 2 1734.4.a.d.1.1 1
95.94 odd 2 2166.4.a.i.1.1 1
105.44 odd 6 882.4.g.i.361.1 2
105.59 even 6 882.4.g.f.667.1 2
105.74 odd 6 882.4.g.i.667.1 2
105.89 even 6 882.4.g.f.361.1 2
105.104 even 2 882.4.a.n.1.1 1
120.29 odd 2 576.4.a.q.1.1 1
120.59 even 2 576.4.a.r.1.1 1
140.139 even 2 2352.4.a.e.1.1 1
165.164 even 2 2178.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 5.4 even 2
18.4.a.a.1.1 1 15.14 odd 2
48.4.a.c.1.1 1 20.19 odd 2
144.4.a.c.1.1 1 60.59 even 2
150.4.a.i.1.1 1 1.1 even 1 trivial
150.4.c.d.49.1 2 5.3 odd 4
150.4.c.d.49.2 2 5.2 odd 4
162.4.c.c.55.1 2 45.14 odd 6
162.4.c.c.109.1 2 45.29 odd 6
162.4.c.f.55.1 2 45.4 even 6
162.4.c.f.109.1 2 45.34 even 6
192.4.a.c.1.1 1 40.19 odd 2
192.4.a.i.1.1 1 40.29 even 2
294.4.a.e.1.1 1 35.34 odd 2
294.4.e.g.67.1 2 35.19 odd 6
294.4.e.g.79.1 2 35.24 odd 6
294.4.e.h.67.1 2 35.9 even 6
294.4.e.h.79.1 2 35.4 even 6
450.4.a.h.1.1 1 3.2 odd 2
450.4.c.e.199.1 2 15.2 even 4
450.4.c.e.199.2 2 15.8 even 4
576.4.a.q.1.1 1 120.29 odd 2
576.4.a.r.1.1 1 120.59 even 2
726.4.a.f.1.1 1 55.54 odd 2
768.4.d.c.385.1 2 80.59 odd 4
768.4.d.c.385.2 2 80.19 odd 4
768.4.d.n.385.1 2 80.29 even 4
768.4.d.n.385.2 2 80.69 even 4
882.4.a.n.1.1 1 105.104 even 2
882.4.g.f.361.1 2 105.89 even 6
882.4.g.f.667.1 2 105.59 even 6
882.4.g.i.361.1 2 105.44 odd 6
882.4.g.i.667.1 2 105.74 odd 6
1014.4.a.g.1.1 1 65.64 even 2
1014.4.b.d.337.1 2 65.34 odd 4
1014.4.b.d.337.2 2 65.44 odd 4
1200.4.a.b.1.1 1 4.3 odd 2
1200.4.f.j.49.1 2 20.3 even 4
1200.4.f.j.49.2 2 20.7 even 4
1734.4.a.d.1.1 1 85.84 even 2
2166.4.a.i.1.1 1 95.94 odd 2
2178.4.a.e.1.1 1 165.164 even 2
2352.4.a.e.1.1 1 140.139 even 2