# Properties

 Label 150.4.c.d.49.1 Level $150$ Weight $4$ Character 150.49 Analytic conductor $8.850$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 150.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.85028650086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 150.49 Dual form 150.4.c.d.49.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} -16.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} -16.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} +12.0000 q^{11} -12.0000i q^{12} -38.0000i q^{13} -32.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} +18.0000i q^{18} -20.0000 q^{19} +48.0000 q^{21} -24.0000i q^{22} -168.000i q^{23} -24.0000 q^{24} -76.0000 q^{26} -27.0000i q^{27} +64.0000i q^{28} -30.0000 q^{29} -88.0000 q^{31} -32.0000i q^{32} +36.0000i q^{33} -252.000 q^{34} +36.0000 q^{36} +254.000i q^{37} +40.0000i q^{38} +114.000 q^{39} +42.0000 q^{41} -96.0000i q^{42} +52.0000i q^{43} -48.0000 q^{44} -336.000 q^{46} -96.0000i q^{47} +48.0000i q^{48} +87.0000 q^{49} +378.000 q^{51} +152.000i q^{52} -198.000i q^{53} -54.0000 q^{54} +128.000 q^{56} -60.0000i q^{57} +60.0000i q^{58} +660.000 q^{59} -538.000 q^{61} +176.000i q^{62} +144.000i q^{63} -64.0000 q^{64} +72.0000 q^{66} +884.000i q^{67} +504.000i q^{68} +504.000 q^{69} +792.000 q^{71} -72.0000i q^{72} -218.000i q^{73} +508.000 q^{74} +80.0000 q^{76} -192.000i q^{77} -228.000i q^{78} +520.000 q^{79} +81.0000 q^{81} -84.0000i q^{82} +492.000i q^{83} -192.000 q^{84} +104.000 q^{86} -90.0000i q^{87} +96.0000i q^{88} -810.000 q^{89} -608.000 q^{91} +672.000i q^{92} -264.000i q^{93} -192.000 q^{94} +96.0000 q^{96} +1154.00i q^{97} -174.000i q^{98} -108.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} + 24q^{11} - 64q^{14} + 32q^{16} - 40q^{19} + 96q^{21} - 48q^{24} - 152q^{26} - 60q^{29} - 176q^{31} - 504q^{34} + 72q^{36} + 228q^{39} + 84q^{41} - 96q^{44} - 672q^{46} + 174q^{49} + 756q^{51} - 108q^{54} + 256q^{56} + 1320q^{59} - 1076q^{61} - 128q^{64} + 144q^{66} + 1008q^{69} + 1584q^{71} + 1016q^{74} + 160q^{76} + 1040q^{79} + 162q^{81} - 384q^{84} + 208q^{86} - 1620q^{89} - 1216q^{91} - 384q^{94} + 192q^{96} - 216q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i
$$3$$ 3.00000i 0.577350i
$$4$$ −4.00000 −0.500000
$$5$$ 0 0
$$6$$ 6.00000 0.408248
$$7$$ − 16.0000i − 0.863919i −0.901893 0.431959i $$-0.857822\pi$$
0.901893 0.431959i $$-0.142178\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 12.0000 0.328921 0.164461 0.986384i $$-0.447412\pi$$
0.164461 + 0.986384i $$0.447412\pi$$
$$12$$ − 12.0000i − 0.288675i
$$13$$ − 38.0000i − 0.810716i −0.914158 0.405358i $$-0.867147\pi$$
0.914158 0.405358i $$-0.132853\pi$$
$$14$$ −32.0000 −0.610883
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ − 126.000i − 1.79762i −0.438342 0.898808i $$-0.644434\pi$$
0.438342 0.898808i $$-0.355566\pi$$
$$18$$ 18.0000i 0.235702i
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 0 0
$$21$$ 48.0000 0.498784
$$22$$ − 24.0000i − 0.232583i
$$23$$ − 168.000i − 1.52306i −0.648129 0.761531i $$-0.724448\pi$$
0.648129 0.761531i $$-0.275552\pi$$
$$24$$ −24.0000 −0.204124
$$25$$ 0 0
$$26$$ −76.0000 −0.573263
$$27$$ − 27.0000i − 0.192450i
$$28$$ 64.0000i 0.431959i
$$29$$ −30.0000 −0.192099 −0.0960493 0.995377i $$-0.530621\pi$$
−0.0960493 + 0.995377i $$0.530621\pi$$
$$30$$ 0 0
$$31$$ −88.0000 −0.509847 −0.254924 0.966961i $$-0.582050\pi$$
−0.254924 + 0.966961i $$0.582050\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ 36.0000i 0.189903i
$$34$$ −252.000 −1.27111
$$35$$ 0 0
$$36$$ 36.0000 0.166667
$$37$$ 254.000i 1.12858i 0.825578 + 0.564288i $$0.190849\pi$$
−0.825578 + 0.564288i $$0.809151\pi$$
$$38$$ 40.0000i 0.170759i
$$39$$ 114.000 0.468067
$$40$$ 0 0
$$41$$ 42.0000 0.159983 0.0799914 0.996796i $$-0.474511\pi$$
0.0799914 + 0.996796i $$0.474511\pi$$
$$42$$ − 96.0000i − 0.352693i
$$43$$ 52.0000i 0.184417i 0.995740 + 0.0922084i $$0.0293926\pi$$
−0.995740 + 0.0922084i $$0.970607\pi$$
$$44$$ −48.0000 −0.164461
$$45$$ 0 0
$$46$$ −336.000 −1.07697
$$47$$ − 96.0000i − 0.297937i −0.988842 0.148969i $$-0.952405\pi$$
0.988842 0.148969i $$-0.0475953\pi$$
$$48$$ 48.0000i 0.144338i
$$49$$ 87.0000 0.253644
$$50$$ 0 0
$$51$$ 378.000 1.03785
$$52$$ 152.000i 0.405358i
$$53$$ − 198.000i − 0.513158i −0.966523 0.256579i $$-0.917405\pi$$
0.966523 0.256579i $$-0.0825954\pi$$
$$54$$ −54.0000 −0.136083
$$55$$ 0 0
$$56$$ 128.000 0.305441
$$57$$ − 60.0000i − 0.139424i
$$58$$ 60.0000i 0.135834i
$$59$$ 660.000 1.45635 0.728175 0.685391i $$-0.240369\pi$$
0.728175 + 0.685391i $$0.240369\pi$$
$$60$$ 0 0
$$61$$ −538.000 −1.12924 −0.564622 0.825350i $$-0.690978\pi$$
−0.564622 + 0.825350i $$0.690978\pi$$
$$62$$ 176.000i 0.360516i
$$63$$ 144.000i 0.287973i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 72.0000 0.134282
$$67$$ 884.000i 1.61191i 0.591979 + 0.805954i $$0.298347\pi$$
−0.591979 + 0.805954i $$0.701653\pi$$
$$68$$ 504.000i 0.898808i
$$69$$ 504.000 0.879340
$$70$$ 0 0
$$71$$ 792.000 1.32385 0.661923 0.749572i $$-0.269740\pi$$
0.661923 + 0.749572i $$0.269740\pi$$
$$72$$ − 72.0000i − 0.117851i
$$73$$ − 218.000i − 0.349520i −0.984611 0.174760i $$-0.944085\pi$$
0.984611 0.174760i $$-0.0559150\pi$$
$$74$$ 508.000 0.798024
$$75$$ 0 0
$$76$$ 80.0000 0.120745
$$77$$ − 192.000i − 0.284161i
$$78$$ − 228.000i − 0.330973i
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 84.0000i − 0.113125i
$$83$$ 492.000i 0.650651i 0.945602 + 0.325325i $$0.105474\pi$$
−0.945602 + 0.325325i $$0.894526\pi$$
$$84$$ −192.000 −0.249392
$$85$$ 0 0
$$86$$ 104.000 0.130402
$$87$$ − 90.0000i − 0.110908i
$$88$$ 96.0000i 0.116291i
$$89$$ −810.000 −0.964717 −0.482359 0.875974i $$-0.660220\pi$$
−0.482359 + 0.875974i $$0.660220\pi$$
$$90$$ 0 0
$$91$$ −608.000 −0.700393
$$92$$ 672.000i 0.761531i
$$93$$ − 264.000i − 0.294360i
$$94$$ −192.000 −0.210673
$$95$$ 0 0
$$96$$ 96.0000 0.102062
$$97$$ 1154.00i 1.20795i 0.797004 + 0.603974i $$0.206417\pi$$
−0.797004 + 0.603974i $$0.793583\pi$$
$$98$$ − 174.000i − 0.179354i
$$99$$ −108.000 −0.109640
$$100$$ 0 0
$$101$$ −618.000 −0.608845 −0.304422 0.952537i $$-0.598463\pi$$
−0.304422 + 0.952537i $$0.598463\pi$$
$$102$$ − 756.000i − 0.733874i
$$103$$ − 128.000i − 0.122449i −0.998124 0.0612243i $$-0.980499\pi$$
0.998124 0.0612243i $$-0.0195005\pi$$
$$104$$ 304.000 0.286631
$$105$$ 0 0
$$106$$ −396.000 −0.362858
$$107$$ − 1476.00i − 1.33355i −0.745257 0.666777i $$-0.767673\pi$$
0.745257 0.666777i $$-0.232327\pi$$
$$108$$ 108.000i 0.0962250i
$$109$$ −1190.00 −1.04570 −0.522850 0.852425i $$-0.675131\pi$$
−0.522850 + 0.852425i $$0.675131\pi$$
$$110$$ 0 0
$$111$$ −762.000 −0.651584
$$112$$ − 256.000i − 0.215980i
$$113$$ 462.000i 0.384613i 0.981335 + 0.192307i $$0.0615968\pi$$
−0.981335 + 0.192307i $$0.938403\pi$$
$$114$$ −120.000 −0.0985880
$$115$$ 0 0
$$116$$ 120.000 0.0960493
$$117$$ 342.000i 0.270239i
$$118$$ − 1320.00i − 1.02980i
$$119$$ −2016.00 −1.55300
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ 1076.00i 0.798496i
$$123$$ 126.000i 0.0923662i
$$124$$ 352.000 0.254924
$$125$$ 0 0
$$126$$ 288.000 0.203628
$$127$$ − 2536.00i − 1.77192i −0.463763 0.885959i $$-0.653501\pi$$
0.463763 0.885959i $$-0.346499\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ −156.000 −0.106473
$$130$$ 0 0
$$131$$ 2292.00 1.52865 0.764324 0.644832i $$-0.223073\pi$$
0.764324 + 0.644832i $$0.223073\pi$$
$$132$$ − 144.000i − 0.0949514i
$$133$$ 320.000i 0.208628i
$$134$$ 1768.00 1.13979
$$135$$ 0 0
$$136$$ 1008.00 0.635554
$$137$$ − 726.000i − 0.452747i −0.974041 0.226374i $$-0.927313\pi$$
0.974041 0.226374i $$-0.0726870\pi$$
$$138$$ − 1008.00i − 0.621787i
$$139$$ −380.000 −0.231879 −0.115939 0.993256i $$-0.536988\pi$$
−0.115939 + 0.993256i $$0.536988\pi$$
$$140$$ 0 0
$$141$$ 288.000 0.172014
$$142$$ − 1584.00i − 0.936101i
$$143$$ − 456.000i − 0.266662i
$$144$$ −144.000 −0.0833333
$$145$$ 0 0
$$146$$ −436.000 −0.247148
$$147$$ 261.000i 0.146442i
$$148$$ − 1016.00i − 0.564288i
$$149$$ −1590.00 −0.874214 −0.437107 0.899410i $$-0.643997\pi$$
−0.437107 + 0.899410i $$0.643997\pi$$
$$150$$ 0 0
$$151$$ 2432.00 1.31068 0.655342 0.755332i $$-0.272524\pi$$
0.655342 + 0.755332i $$0.272524\pi$$
$$152$$ − 160.000i − 0.0853797i
$$153$$ 1134.00i 0.599206i
$$154$$ −384.000 −0.200932
$$155$$ 0 0
$$156$$ −456.000 −0.234033
$$157$$ 614.000i 0.312118i 0.987748 + 0.156059i $$0.0498790\pi$$
−0.987748 + 0.156059i $$0.950121\pi$$
$$158$$ − 1040.00i − 0.523658i
$$159$$ 594.000 0.296272
$$160$$ 0 0
$$161$$ −2688.00 −1.31580
$$162$$ − 162.000i − 0.0785674i
$$163$$ 1852.00i 0.889938i 0.895546 + 0.444969i $$0.146785\pi$$
−0.895546 + 0.444969i $$0.853215\pi$$
$$164$$ −168.000 −0.0799914
$$165$$ 0 0
$$166$$ 984.000 0.460080
$$167$$ − 2136.00i − 0.989752i −0.868964 0.494876i $$-0.835213\pi$$
0.868964 0.494876i $$-0.164787\pi$$
$$168$$ 384.000i 0.176347i
$$169$$ 753.000 0.342740
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ − 208.000i − 0.0922084i
$$173$$ − 1758.00i − 0.772591i −0.922375 0.386296i $$-0.873754\pi$$
0.922375 0.386296i $$-0.126246\pi$$
$$174$$ −180.000 −0.0784239
$$175$$ 0 0
$$176$$ 192.000 0.0822304
$$177$$ 1980.00i 0.840824i
$$178$$ 1620.00i 0.682158i
$$179$$ 540.000 0.225483 0.112742 0.993624i $$-0.464037\pi$$
0.112742 + 0.993624i $$0.464037\pi$$
$$180$$ 0 0
$$181$$ 1982.00 0.813928 0.406964 0.913444i $$-0.366588\pi$$
0.406964 + 0.913444i $$0.366588\pi$$
$$182$$ 1216.00i 0.495252i
$$183$$ − 1614.00i − 0.651969i
$$184$$ 1344.00 0.538484
$$185$$ 0 0
$$186$$ −528.000 −0.208144
$$187$$ − 1512.00i − 0.591275i
$$188$$ 384.000i 0.148969i
$$189$$ −432.000 −0.166261
$$190$$ 0 0
$$191$$ −2688.00 −1.01831 −0.509154 0.860675i $$-0.670042\pi$$
−0.509154 + 0.860675i $$0.670042\pi$$
$$192$$ − 192.000i − 0.0721688i
$$193$$ 2302.00i 0.858557i 0.903172 + 0.429279i $$0.141232\pi$$
−0.903172 + 0.429279i $$0.858768\pi$$
$$194$$ 2308.00 0.854148
$$195$$ 0 0
$$196$$ −348.000 −0.126822
$$197$$ 4374.00i 1.58190i 0.611880 + 0.790951i $$0.290414\pi$$
−0.611880 + 0.790951i $$0.709586\pi$$
$$198$$ 216.000i 0.0775275i
$$199$$ 1600.00 0.569955 0.284977 0.958534i $$-0.408014\pi$$
0.284977 + 0.958534i $$0.408014\pi$$
$$200$$ 0 0
$$201$$ −2652.00 −0.930635
$$202$$ 1236.00i 0.430518i
$$203$$ 480.000i 0.165958i
$$204$$ −1512.00 −0.518927
$$205$$ 0 0
$$206$$ −256.000 −0.0865843
$$207$$ 1512.00i 0.507687i
$$208$$ − 608.000i − 0.202679i
$$209$$ −240.000 −0.0794313
$$210$$ 0 0
$$211$$ 3332.00 1.08713 0.543565 0.839367i $$-0.317074\pi$$
0.543565 + 0.839367i $$0.317074\pi$$
$$212$$ 792.000i 0.256579i
$$213$$ 2376.00i 0.764323i
$$214$$ −2952.00 −0.942965
$$215$$ 0 0
$$216$$ 216.000 0.0680414
$$217$$ 1408.00i 0.440467i
$$218$$ 2380.00i 0.739422i
$$219$$ 654.000 0.201796
$$220$$ 0 0
$$221$$ −4788.00 −1.45736
$$222$$ 1524.00i 0.460740i
$$223$$ − 2648.00i − 0.795171i −0.917565 0.397586i $$-0.869848\pi$$
0.917565 0.397586i $$-0.130152\pi$$
$$224$$ −512.000 −0.152721
$$225$$ 0 0
$$226$$ 924.000 0.271963
$$227$$ 2244.00i 0.656121i 0.944657 + 0.328061i $$0.106395\pi$$
−0.944657 + 0.328061i $$0.893605\pi$$
$$228$$ 240.000i 0.0697122i
$$229$$ 5650.00 1.63040 0.815202 0.579177i $$-0.196626\pi$$
0.815202 + 0.579177i $$0.196626\pi$$
$$230$$ 0 0
$$231$$ 576.000 0.164061
$$232$$ − 240.000i − 0.0679171i
$$233$$ − 4698.00i − 1.32093i −0.750858 0.660464i $$-0.770360\pi$$
0.750858 0.660464i $$-0.229640\pi$$
$$234$$ 684.000 0.191088
$$235$$ 0 0
$$236$$ −2640.00 −0.728175
$$237$$ 1560.00i 0.427565i
$$238$$ 4032.00i 1.09813i
$$239$$ 1200.00 0.324776 0.162388 0.986727i $$-0.448080\pi$$
0.162388 + 0.986727i $$0.448080\pi$$
$$240$$ 0 0
$$241$$ −718.000 −0.191911 −0.0959553 0.995386i $$-0.530591\pi$$
−0.0959553 + 0.995386i $$0.530591\pi$$
$$242$$ 2374.00i 0.630605i
$$243$$ 243.000i 0.0641500i
$$244$$ 2152.00 0.564622
$$245$$ 0 0
$$246$$ 252.000 0.0653127
$$247$$ 760.000i 0.195780i
$$248$$ − 704.000i − 0.180258i
$$249$$ −1476.00 −0.375653
$$250$$ 0 0
$$251$$ 6012.00 1.51185 0.755924 0.654659i $$-0.227188\pi$$
0.755924 + 0.654659i $$0.227188\pi$$
$$252$$ − 576.000i − 0.143986i
$$253$$ − 2016.00i − 0.500968i
$$254$$ −5072.00 −1.25294
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ − 2046.00i − 0.496599i −0.968683 0.248300i $$-0.920128\pi$$
0.968683 0.248300i $$-0.0798717\pi$$
$$258$$ 312.000i 0.0752879i
$$259$$ 4064.00 0.974999
$$260$$ 0 0
$$261$$ 270.000 0.0640329
$$262$$ − 4584.00i − 1.08092i
$$263$$ 6072.00i 1.42363i 0.702365 + 0.711817i $$0.252127\pi$$
−0.702365 + 0.711817i $$0.747873\pi$$
$$264$$ −288.000 −0.0671408
$$265$$ 0 0
$$266$$ 640.000 0.147522
$$267$$ − 2430.00i − 0.556980i
$$268$$ − 3536.00i − 0.805954i
$$269$$ 6930.00 1.57074 0.785371 0.619025i $$-0.212472\pi$$
0.785371 + 0.619025i $$0.212472\pi$$
$$270$$ 0 0
$$271$$ 1352.00 0.303056 0.151528 0.988453i $$-0.451581\pi$$
0.151528 + 0.988453i $$0.451581\pi$$
$$272$$ − 2016.00i − 0.449404i
$$273$$ − 1824.00i − 0.404372i
$$274$$ −1452.00 −0.320141
$$275$$ 0 0
$$276$$ −2016.00 −0.439670
$$277$$ − 1186.00i − 0.257256i −0.991693 0.128628i $$-0.958943\pi$$
0.991693 0.128628i $$-0.0410573\pi$$
$$278$$ 760.000i 0.163963i
$$279$$ 792.000 0.169949
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ − 576.000i − 0.121632i
$$283$$ − 2828.00i − 0.594018i −0.954875 0.297009i $$-0.904011\pi$$
0.954875 0.297009i $$-0.0959892\pi$$
$$284$$ −3168.00 −0.661923
$$285$$ 0 0
$$286$$ −912.000 −0.188558
$$287$$ − 672.000i − 0.138212i
$$288$$ 288.000i 0.0589256i
$$289$$ −10963.0 −2.23143
$$290$$ 0 0
$$291$$ −3462.00 −0.697409
$$292$$ 872.000i 0.174760i
$$293$$ − 4758.00i − 0.948687i −0.880340 0.474344i $$-0.842685\pi$$
0.880340 0.474344i $$-0.157315\pi$$
$$294$$ 522.000 0.103550
$$295$$ 0 0
$$296$$ −2032.00 −0.399012
$$297$$ − 324.000i − 0.0633010i
$$298$$ 3180.00i 0.618163i
$$299$$ −6384.00 −1.23477
$$300$$ 0 0
$$301$$ 832.000 0.159321
$$302$$ − 4864.00i − 0.926794i
$$303$$ − 1854.00i − 0.351517i
$$304$$ −320.000 −0.0603726
$$305$$ 0 0
$$306$$ 2268.00 0.423702
$$307$$ − 8476.00i − 1.57574i −0.615844 0.787868i $$-0.711185\pi$$
0.615844 0.787868i $$-0.288815\pi$$
$$308$$ 768.000i 0.142081i
$$309$$ 384.000 0.0706958
$$310$$ 0 0
$$311$$ 4632.00 0.844555 0.422278 0.906467i $$-0.361231\pi$$
0.422278 + 0.906467i $$0.361231\pi$$
$$312$$ 912.000i 0.165487i
$$313$$ 4822.00i 0.870785i 0.900241 + 0.435392i $$0.143390\pi$$
−0.900241 + 0.435392i $$0.856610\pi$$
$$314$$ 1228.00 0.220701
$$315$$ 0 0
$$316$$ −2080.00 −0.370282
$$317$$ − 3426.00i − 0.607014i −0.952829 0.303507i $$-0.901842\pi$$
0.952829 0.303507i $$-0.0981575\pi$$
$$318$$ − 1188.00i − 0.209496i
$$319$$ −360.000 −0.0631854
$$320$$ 0 0
$$321$$ 4428.00 0.769928
$$322$$ 5376.00i 0.930412i
$$323$$ 2520.00i 0.434107i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ 3704.00 0.629281
$$327$$ − 3570.00i − 0.603735i
$$328$$ 336.000i 0.0565625i
$$329$$ −1536.00 −0.257393
$$330$$ 0 0
$$331$$ −2788.00 −0.462968 −0.231484 0.972839i $$-0.574358\pi$$
−0.231484 + 0.972839i $$0.574358\pi$$
$$332$$ − 1968.00i − 0.325325i
$$333$$ − 2286.00i − 0.376192i
$$334$$ −4272.00 −0.699861
$$335$$ 0 0
$$336$$ 768.000 0.124696
$$337$$ 434.000i 0.0701528i 0.999385 + 0.0350764i $$0.0111675\pi$$
−0.999385 + 0.0350764i $$0.988833\pi$$
$$338$$ − 1506.00i − 0.242354i
$$339$$ −1386.00 −0.222057
$$340$$ 0 0
$$341$$ −1056.00 −0.167700
$$342$$ − 360.000i − 0.0569198i
$$343$$ − 6880.00i − 1.08305i
$$344$$ −416.000 −0.0652012
$$345$$ 0 0
$$346$$ −3516.00 −0.546304
$$347$$ 6684.00i 1.03405i 0.855970 + 0.517026i $$0.172961\pi$$
−0.855970 + 0.517026i $$0.827039\pi$$
$$348$$ 360.000i 0.0554541i
$$349$$ −2630.00 −0.403383 −0.201692 0.979449i $$-0.564644\pi$$
−0.201692 + 0.979449i $$0.564644\pi$$
$$350$$ 0 0
$$351$$ −1026.00 −0.156022
$$352$$ − 384.000i − 0.0581456i
$$353$$ 7422.00i 1.11907i 0.828805 + 0.559537i $$0.189021\pi$$
−0.828805 + 0.559537i $$0.810979\pi$$
$$354$$ 3960.00 0.594553
$$355$$ 0 0
$$356$$ 3240.00 0.482359
$$357$$ − 6048.00i − 0.896622i
$$358$$ − 1080.00i − 0.159441i
$$359$$ 10440.0 1.53482 0.767412 0.641154i $$-0.221544\pi$$
0.767412 + 0.641154i $$0.221544\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ − 3964.00i − 0.575534i
$$363$$ − 3561.00i − 0.514887i
$$364$$ 2432.00 0.350196
$$365$$ 0 0
$$366$$ −3228.00 −0.461012
$$367$$ 10424.0i 1.48264i 0.671153 + 0.741319i $$0.265800\pi$$
−0.671153 + 0.741319i $$0.734200\pi$$
$$368$$ − 2688.00i − 0.380765i
$$369$$ −378.000 −0.0533276
$$370$$ 0 0
$$371$$ −3168.00 −0.443327
$$372$$ 1056.00i 0.147180i
$$373$$ − 3278.00i − 0.455036i −0.973774 0.227518i $$-0.926939\pi$$
0.973774 0.227518i $$-0.0730610\pi$$
$$374$$ −3024.00 −0.418094
$$375$$ 0 0
$$376$$ 768.000 0.105337
$$377$$ 1140.00i 0.155737i
$$378$$ 864.000i 0.117564i
$$379$$ −6140.00 −0.832165 −0.416083 0.909327i $$-0.636597\pi$$
−0.416083 + 0.909327i $$0.636597\pi$$
$$380$$ 0 0
$$381$$ 7608.00 1.02302
$$382$$ 5376.00i 0.720053i
$$383$$ 3072.00i 0.409848i 0.978778 + 0.204924i $$0.0656948\pi$$
−0.978778 + 0.204924i $$0.934305\pi$$
$$384$$ −384.000 −0.0510310
$$385$$ 0 0
$$386$$ 4604.00 0.607092
$$387$$ − 468.000i − 0.0614723i
$$388$$ − 4616.00i − 0.603974i
$$389$$ −6150.00 −0.801587 −0.400794 0.916168i $$-0.631266\pi$$
−0.400794 + 0.916168i $$0.631266\pi$$
$$390$$ 0 0
$$391$$ −21168.0 −2.73788
$$392$$ 696.000i 0.0896768i
$$393$$ 6876.00i 0.882566i
$$394$$ 8748.00 1.11857
$$395$$ 0 0
$$396$$ 432.000 0.0548202
$$397$$ − 106.000i − 0.0134005i −0.999978 0.00670024i $$-0.997867\pi$$
0.999978 0.00670024i $$-0.00213277\pi$$
$$398$$ − 3200.00i − 0.403019i
$$399$$ −960.000 −0.120451
$$400$$ 0 0
$$401$$ −1758.00 −0.218929 −0.109464 0.993991i $$-0.534914\pi$$
−0.109464 + 0.993991i $$0.534914\pi$$
$$402$$ 5304.00i 0.658058i
$$403$$ 3344.00i 0.413341i
$$404$$ 2472.00 0.304422
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ 3048.00i 0.371213i
$$408$$ 3024.00i 0.366937i
$$409$$ 3670.00 0.443691 0.221846 0.975082i $$-0.428792\pi$$
0.221846 + 0.975082i $$0.428792\pi$$
$$410$$ 0 0
$$411$$ 2178.00 0.261394
$$412$$ 512.000i 0.0612243i
$$413$$ − 10560.0i − 1.25817i
$$414$$ 3024.00 0.358989
$$415$$ 0 0
$$416$$ −1216.00 −0.143316
$$417$$ − 1140.00i − 0.133875i
$$418$$ 480.000i 0.0561664i
$$419$$ 9660.00 1.12631 0.563153 0.826353i $$-0.309588\pi$$
0.563153 + 0.826353i $$0.309588\pi$$
$$420$$ 0 0
$$421$$ 8462.00 0.979602 0.489801 0.871834i $$-0.337069\pi$$
0.489801 + 0.871834i $$0.337069\pi$$
$$422$$ − 6664.00i − 0.768717i
$$423$$ 864.000i 0.0993123i
$$424$$ 1584.00 0.181429
$$425$$ 0 0
$$426$$ 4752.00 0.540458
$$427$$ 8608.00i 0.975575i
$$428$$ 5904.00i 0.666777i
$$429$$ 1368.00 0.153957
$$430$$ 0 0
$$431$$ 9792.00 1.09435 0.547174 0.837019i $$-0.315704\pi$$
0.547174 + 0.837019i $$0.315704\pi$$
$$432$$ − 432.000i − 0.0481125i
$$433$$ 7342.00i 0.814859i 0.913237 + 0.407430i $$0.133575\pi$$
−0.913237 + 0.407430i $$0.866425\pi$$
$$434$$ 2816.00 0.311457
$$435$$ 0 0
$$436$$ 4760.00 0.522850
$$437$$ 3360.00i 0.367805i
$$438$$ − 1308.00i − 0.142691i
$$439$$ −10640.0 −1.15676 −0.578382 0.815766i $$-0.696316\pi$$
−0.578382 + 0.815766i $$0.696316\pi$$
$$440$$ 0 0
$$441$$ −783.000 −0.0845481
$$442$$ 9576.00i 1.03051i
$$443$$ 17412.0i 1.86742i 0.358024 + 0.933712i $$0.383451\pi$$
−0.358024 + 0.933712i $$0.616549\pi$$
$$444$$ 3048.00 0.325792
$$445$$ 0 0
$$446$$ −5296.00 −0.562271
$$447$$ − 4770.00i − 0.504728i
$$448$$ 1024.00i 0.107990i
$$449$$ 1710.00 0.179732 0.0898662 0.995954i $$-0.471356\pi$$
0.0898662 + 0.995954i $$0.471356\pi$$
$$450$$ 0 0
$$451$$ 504.000 0.0526218
$$452$$ − 1848.00i − 0.192307i
$$453$$ 7296.00i 0.756724i
$$454$$ 4488.00 0.463948
$$455$$ 0 0
$$456$$ 480.000 0.0492940
$$457$$ − 646.000i − 0.0661239i −0.999453 0.0330619i $$-0.989474\pi$$
0.999453 0.0330619i $$-0.0105259\pi$$
$$458$$ − 11300.0i − 1.15287i
$$459$$ −3402.00 −0.345952
$$460$$ 0 0
$$461$$ −6018.00 −0.607996 −0.303998 0.952673i $$-0.598322\pi$$
−0.303998 + 0.952673i $$0.598322\pi$$
$$462$$ − 1152.00i − 0.116008i
$$463$$ 6712.00i 0.673722i 0.941554 + 0.336861i $$0.109365\pi$$
−0.941554 + 0.336861i $$0.890635\pi$$
$$464$$ −480.000 −0.0480247
$$465$$ 0 0
$$466$$ −9396.00 −0.934037
$$467$$ 5364.00i 0.531512i 0.964040 + 0.265756i $$0.0856216\pi$$
−0.964040 + 0.265756i $$0.914378\pi$$
$$468$$ − 1368.00i − 0.135119i
$$469$$ 14144.0 1.39256
$$470$$ 0 0
$$471$$ −1842.00 −0.180201
$$472$$ 5280.00i 0.514898i
$$473$$ 624.000i 0.0606587i
$$474$$ 3120.00 0.302334
$$475$$ 0 0
$$476$$ 8064.00 0.776498
$$477$$ 1782.00i 0.171053i
$$478$$ − 2400.00i − 0.229652i
$$479$$ −9840.00 −0.938624 −0.469312 0.883032i $$-0.655498\pi$$
−0.469312 + 0.883032i $$0.655498\pi$$
$$480$$ 0 0
$$481$$ 9652.00 0.914955
$$482$$ 1436.00i 0.135701i
$$483$$ − 8064.00i − 0.759678i
$$484$$ 4748.00 0.445905
$$485$$ 0 0
$$486$$ 486.000 0.0453609
$$487$$ 1424.00i 0.132500i 0.997803 + 0.0662501i $$0.0211035\pi$$
−0.997803 + 0.0662501i $$0.978896\pi$$
$$488$$ − 4304.00i − 0.399248i
$$489$$ −5556.00 −0.513806
$$490$$ 0 0
$$491$$ −4548.00 −0.418021 −0.209011 0.977913i $$-0.567024\pi$$
−0.209011 + 0.977913i $$0.567024\pi$$
$$492$$ − 504.000i − 0.0461831i
$$493$$ 3780.00i 0.345320i
$$494$$ 1520.00 0.138437
$$495$$ 0 0
$$496$$ −1408.00 −0.127462
$$497$$ − 12672.0i − 1.14370i
$$498$$ 2952.00i 0.265627i
$$499$$ −6500.00 −0.583126 −0.291563 0.956552i $$-0.594175\pi$$
−0.291563 + 0.956552i $$0.594175\pi$$
$$500$$ 0 0
$$501$$ 6408.00 0.571434
$$502$$ − 12024.0i − 1.06904i
$$503$$ − 12168.0i − 1.07862i −0.842108 0.539308i $$-0.818686\pi$$
0.842108 0.539308i $$-0.181314\pi$$
$$504$$ −1152.00 −0.101814
$$505$$ 0 0
$$506$$ −4032.00 −0.354238
$$507$$ 2259.00i 0.197881i
$$508$$ 10144.0i 0.885959i
$$509$$ 21090.0 1.83654 0.918269 0.395957i $$-0.129587\pi$$
0.918269 + 0.395957i $$0.129587\pi$$
$$510$$ 0 0
$$511$$ −3488.00 −0.301957
$$512$$ − 512.000i − 0.0441942i
$$513$$ 540.000i 0.0464748i
$$514$$ −4092.00 −0.351149
$$515$$ 0 0
$$516$$ 624.000 0.0532366
$$517$$ − 1152.00i − 0.0979979i
$$518$$ − 8128.00i − 0.689428i
$$519$$ 5274.00 0.446056
$$520$$ 0 0
$$521$$ −5238.00 −0.440462 −0.220231 0.975448i $$-0.570681\pi$$
−0.220231 + 0.975448i $$0.570681\pi$$
$$522$$ − 540.000i − 0.0452781i
$$523$$ − 8588.00i − 0.718025i −0.933333 0.359012i $$-0.883114\pi$$
0.933333 0.359012i $$-0.116886\pi$$
$$524$$ −9168.00 −0.764324
$$525$$ 0 0
$$526$$ 12144.0 1.00666
$$527$$ 11088.0i 0.916510i
$$528$$ 576.000i 0.0474757i
$$529$$ −16057.0 −1.31972
$$530$$ 0 0
$$531$$ −5940.00 −0.485450
$$532$$ − 1280.00i − 0.104314i
$$533$$ − 1596.00i − 0.129701i
$$534$$ −4860.00 −0.393844
$$535$$ 0 0
$$536$$ −7072.00 −0.569895
$$537$$ 1620.00i 0.130183i
$$538$$ − 13860.0i − 1.11068i
$$539$$ 1044.00 0.0834291
$$540$$ 0 0
$$541$$ 3062.00 0.243338 0.121669 0.992571i $$-0.461175\pi$$
0.121669 + 0.992571i $$0.461175\pi$$
$$542$$ − 2704.00i − 0.214293i
$$543$$ 5946.00i 0.469921i
$$544$$ −4032.00 −0.317777
$$545$$ 0 0
$$546$$ −3648.00 −0.285934
$$547$$ − 8476.00i − 0.662537i −0.943537 0.331268i $$-0.892523\pi$$
0.943537 0.331268i $$-0.107477\pi$$
$$548$$ 2904.00i 0.226374i
$$549$$ 4842.00 0.376414
$$550$$ 0 0
$$551$$ 600.000 0.0463899
$$552$$ 4032.00i 0.310894i
$$553$$ − 8320.00i − 0.639787i
$$554$$ −2372.00 −0.181907
$$555$$ 0 0
$$556$$ 1520.00 0.115939
$$557$$ − 12546.0i − 0.954383i −0.878799 0.477191i $$-0.841655\pi$$
0.878799 0.477191i $$-0.158345\pi$$
$$558$$ − 1584.00i − 0.120172i
$$559$$ 1976.00 0.149510
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ − 4884.00i − 0.366582i
$$563$$ 12.0000i 0 0.000898294i 1.00000 0.000449147i $$0.000142968\pi$$
−1.00000 0.000449147i $$0.999857\pi$$
$$564$$ −1152.00 −0.0860070
$$565$$ 0 0
$$566$$ −5656.00 −0.420034
$$567$$ − 1296.00i − 0.0959910i
$$568$$ 6336.00i 0.468050i
$$569$$ −19290.0 −1.42123 −0.710614 0.703582i $$-0.751583\pi$$
−0.710614 + 0.703582i $$0.751583\pi$$
$$570$$ 0 0
$$571$$ −12148.0 −0.890329 −0.445165 0.895449i $$-0.646855\pi$$
−0.445165 + 0.895449i $$0.646855\pi$$
$$572$$ 1824.00i 0.133331i
$$573$$ − 8064.00i − 0.587920i
$$574$$ −1344.00 −0.0977308
$$575$$ 0 0
$$576$$ 576.000 0.0416667
$$577$$ − 10366.0i − 0.747907i −0.927447 0.373953i $$-0.878002\pi$$
0.927447 0.373953i $$-0.121998\pi$$
$$578$$ 21926.0i 1.57786i
$$579$$ −6906.00 −0.495688
$$580$$ 0 0
$$581$$ 7872.00 0.562109
$$582$$ 6924.00i 0.493143i
$$583$$ − 2376.00i − 0.168789i
$$584$$ 1744.00 0.123574
$$585$$ 0 0
$$586$$ −9516.00 −0.670823
$$587$$ 7644.00i 0.537482i 0.963213 + 0.268741i $$0.0866075\pi$$
−0.963213 + 0.268741i $$0.913393\pi$$
$$588$$ − 1044.00i − 0.0732208i
$$589$$ 1760.00 0.123123
$$590$$ 0 0
$$591$$ −13122.0 −0.913311
$$592$$ 4064.00i 0.282144i
$$593$$ − 8658.00i − 0.599564i −0.954008 0.299782i $$-0.903086\pi$$
0.954008 0.299782i $$-0.0969139\pi$$
$$594$$ −648.000 −0.0447605
$$595$$ 0 0
$$596$$ 6360.00 0.437107
$$597$$ 4800.00i 0.329064i
$$598$$ 12768.0i 0.873114i
$$599$$ −25800.0 −1.75987 −0.879933 0.475098i $$-0.842413\pi$$
−0.879933 + 0.475098i $$0.842413\pi$$
$$600$$ 0 0
$$601$$ 16202.0 1.09966 0.549828 0.835278i $$-0.314693\pi$$
0.549828 + 0.835278i $$0.314693\pi$$
$$602$$ − 1664.00i − 0.112657i
$$603$$ − 7956.00i − 0.537302i
$$604$$ −9728.00 −0.655342
$$605$$ 0 0
$$606$$ −3708.00 −0.248560
$$607$$ − 24136.0i − 1.61392i −0.590605 0.806960i $$-0.701111\pi$$
0.590605 0.806960i $$-0.298889\pi$$
$$608$$ 640.000i 0.0426898i
$$609$$ −1440.00 −0.0958157
$$610$$ 0 0
$$611$$ −3648.00 −0.241542
$$612$$ − 4536.00i − 0.299603i
$$613$$ 4642.00i 0.305854i 0.988237 + 0.152927i $$0.0488700\pi$$
−0.988237 + 0.152927i $$0.951130\pi$$
$$614$$ −16952.0 −1.11421
$$615$$ 0 0
$$616$$ 1536.00 0.100466
$$617$$ − 6726.00i − 0.438863i −0.975628 0.219432i $$-0.929580\pi$$
0.975628 0.219432i $$-0.0704203\pi$$
$$618$$ − 768.000i − 0.0499895i
$$619$$ 21220.0 1.37787 0.688937 0.724821i $$-0.258078\pi$$
0.688937 + 0.724821i $$0.258078\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ − 9264.00i − 0.597191i
$$623$$ 12960.0i 0.833437i
$$624$$ 1824.00 0.117017
$$625$$ 0 0
$$626$$ 9644.00 0.615738
$$627$$ − 720.000i − 0.0458597i
$$628$$ − 2456.00i − 0.156059i
$$629$$ 32004.0 2.02875
$$630$$ 0 0
$$631$$ 29792.0 1.87956 0.939779 0.341783i $$-0.111031\pi$$
0.939779 + 0.341783i $$0.111031\pi$$
$$632$$ 4160.00i 0.261829i
$$633$$ 9996.00i 0.627655i
$$634$$ −6852.00 −0.429223
$$635$$ 0 0
$$636$$ −2376.00 −0.148136
$$637$$ − 3306.00i − 0.205633i
$$638$$ 720.000i 0.0446788i
$$639$$ −7128.00 −0.441282
$$640$$ 0 0
$$641$$ −10158.0 −0.625923 −0.312962 0.949766i $$-0.601321\pi$$
−0.312962 + 0.949766i $$0.601321\pi$$
$$642$$ − 8856.00i − 0.544421i
$$643$$ − 29828.0i − 1.82940i −0.404138 0.914698i $$-0.632429\pi$$
0.404138 0.914698i $$-0.367571\pi$$
$$644$$ 10752.0 0.657901
$$645$$ 0 0
$$646$$ 5040.00 0.306960
$$647$$ 1944.00i 0.118124i 0.998254 + 0.0590622i $$0.0188110\pi$$
−0.998254 + 0.0590622i $$0.981189\pi$$
$$648$$ 648.000i 0.0392837i
$$649$$ 7920.00 0.479025
$$650$$ 0 0
$$651$$ −4224.00 −0.254304
$$652$$ − 7408.00i − 0.444969i
$$653$$ − 26718.0i − 1.60116i −0.599227 0.800579i $$-0.704525\pi$$
0.599227 0.800579i $$-0.295475\pi$$
$$654$$ −7140.00 −0.426905
$$655$$ 0 0
$$656$$ 672.000 0.0399957
$$657$$ 1962.00i 0.116507i
$$658$$ 3072.00i 0.182005i
$$659$$ −4260.00 −0.251815 −0.125907 0.992042i $$-0.540184\pi$$
−0.125907 + 0.992042i $$0.540184\pi$$
$$660$$ 0 0
$$661$$ 22862.0 1.34528 0.672639 0.739971i $$-0.265161\pi$$
0.672639 + 0.739971i $$0.265161\pi$$
$$662$$ 5576.00i 0.327368i
$$663$$ − 14364.0i − 0.841405i
$$664$$ −3936.00 −0.230040
$$665$$ 0 0
$$666$$ −4572.00 −0.266008
$$667$$ 5040.00i 0.292578i
$$668$$ 8544.00i 0.494876i
$$669$$ 7944.00 0.459092
$$670$$ 0 0
$$671$$ −6456.00 −0.371432
$$672$$ − 1536.00i − 0.0881733i
$$673$$ 32542.0i 1.86390i 0.362592 + 0.931948i $$0.381892\pi$$
−0.362592 + 0.931948i $$0.618108\pi$$
$$674$$ 868.000 0.0496055
$$675$$ 0 0
$$676$$ −3012.00 −0.171370
$$677$$ 14214.0i 0.806925i 0.914996 + 0.403463i $$0.132193\pi$$
−0.914996 + 0.403463i $$0.867807\pi$$
$$678$$ 2772.00i 0.157018i
$$679$$ 18464.0 1.04357
$$680$$ 0 0
$$681$$ −6732.00 −0.378812
$$682$$ 2112.00i 0.118582i
$$683$$ 7092.00i 0.397317i 0.980069 + 0.198659i $$0.0636585\pi$$
−0.980069 + 0.198659i $$0.936341\pi$$
$$684$$ −720.000 −0.0402484
$$685$$ 0 0
$$686$$ −13760.0 −0.765830
$$687$$ 16950.0i 0.941314i
$$688$$ 832.000i 0.0461042i
$$689$$ −7524.00 −0.416026
$$690$$ 0 0
$$691$$ −13228.0 −0.728244 −0.364122 0.931351i $$-0.618631\pi$$
−0.364122 + 0.931351i $$0.618631\pi$$
$$692$$ 7032.00i 0.386296i
$$693$$ 1728.00i 0.0947205i
$$694$$ 13368.0 0.731185
$$695$$ 0 0
$$696$$ 720.000 0.0392120
$$697$$ − 5292.00i − 0.287588i
$$698$$ 5260.00i 0.285235i
$$699$$ 14094.0 0.762638
$$700$$ 0 0
$$701$$ 28062.0 1.51196 0.755982 0.654592i $$-0.227160\pi$$
0.755982 + 0.654592i $$0.227160\pi$$
$$702$$ 2052.00i 0.110324i
$$703$$ − 5080.00i − 0.272540i
$$704$$ −768.000 −0.0411152
$$705$$ 0 0
$$706$$ 14844.0 0.791305
$$707$$ 9888.00i 0.525992i
$$708$$ − 7920.00i − 0.420412i
$$709$$ 27250.0 1.44343 0.721717 0.692188i $$-0.243353\pi$$
0.721717 + 0.692188i $$0.243353\pi$$
$$710$$ 0 0
$$711$$ −4680.00 −0.246855
$$712$$ − 6480.00i − 0.341079i
$$713$$ 14784.0i 0.776529i
$$714$$ −12096.0 −0.634008
$$715$$ 0 0
$$716$$ −2160.00 −0.112742
$$717$$ 3600.00i 0.187510i
$$718$$ − 20880.0i − 1.08529i
$$719$$ 14400.0 0.746912 0.373456 0.927648i $$-0.378173\pi$$
0.373456 + 0.927648i $$0.378173\pi$$
$$720$$ 0 0
$$721$$ −2048.00 −0.105786
$$722$$ 12918.0i 0.665870i
$$723$$ − 2154.00i − 0.110800i
$$724$$ −7928.00 −0.406964
$$725$$ 0 0
$$726$$ −7122.00 −0.364080
$$727$$ 17984.0i 0.917455i 0.888577 + 0.458727i $$0.151695\pi$$
−0.888577 + 0.458727i $$0.848305\pi$$
$$728$$ − 4864.00i − 0.247626i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 6552.00 0.331511
$$732$$ 6456.00i 0.325984i
$$733$$ − 16598.0i − 0.836373i −0.908361 0.418186i $$-0.862666\pi$$
0.908361 0.418186i $$-0.137334\pi$$
$$734$$ 20848.0 1.04838
$$735$$ 0 0
$$736$$ −5376.00 −0.269242
$$737$$ 10608.0i 0.530191i
$$738$$ 756.000i 0.0377083i
$$739$$ −1460.00 −0.0726752 −0.0363376 0.999340i $$-0.511569\pi$$
−0.0363376 + 0.999340i $$0.511569\pi$$
$$740$$ 0 0
$$741$$ −2280.00 −0.113034
$$742$$ 6336.00i 0.313480i
$$743$$ 30072.0i 1.48484i 0.669936 + 0.742419i $$0.266322\pi$$
−0.669936 + 0.742419i $$0.733678\pi$$
$$744$$ 2112.00 0.104072
$$745$$ 0 0
$$746$$ −6556.00 −0.321759
$$747$$ − 4428.00i − 0.216884i
$$748$$ 6048.00i 0.295637i
$$749$$ −23616.0 −1.15208
$$750$$ 0 0
$$751$$ −18088.0 −0.878882 −0.439441 0.898271i $$-0.644823\pi$$
−0.439441 + 0.898271i $$0.644823\pi$$
$$752$$ − 1536.00i − 0.0744843i
$$753$$ 18036.0i 0.872866i
$$754$$ 2280.00 0.110123
$$755$$ 0 0
$$756$$ 1728.00 0.0831306
$$757$$ 24734.0i 1.18755i 0.804633 + 0.593773i $$0.202362\pi$$
−0.804633 + 0.593773i $$0.797638\pi$$
$$758$$ 12280.0i 0.588430i
$$759$$ 6048.00 0.289234
$$760$$ 0 0
$$761$$ −22278.0 −1.06120 −0.530602 0.847621i $$-0.678034\pi$$
−0.530602 + 0.847621i $$0.678034\pi$$
$$762$$ − 15216.0i − 0.723383i
$$763$$ 19040.0i 0.903400i
$$764$$ 10752.0 0.509154
$$765$$ 0 0
$$766$$ 6144.00 0.289806
$$767$$ − 25080.0i − 1.18069i
$$768$$ 768.000i 0.0360844i
$$769$$ −16130.0 −0.756388 −0.378194 0.925726i $$-0.623455\pi$$
−0.378194 + 0.925726i $$0.623455\pi$$
$$770$$ 0 0
$$771$$ 6138.00 0.286712
$$772$$ − 9208.00i − 0.429279i
$$773$$ − 29718.0i − 1.38277i −0.722486 0.691386i $$-0.757001\pi$$
0.722486 0.691386i $$-0.242999\pi$$
$$774$$ −936.000 −0.0434675
$$775$$ 0 0
$$776$$ −9232.00 −0.427074
$$777$$ 12192.0i 0.562916i
$$778$$ 12300.0i 0.566808i
$$779$$ −840.000 −0.0386343
$$780$$ 0 0
$$781$$ 9504.00 0.435442
$$782$$ 42336.0i 1.93597i
$$783$$ 810.000i 0.0369694i
$$784$$ 1392.00 0.0634111
$$785$$ 0 0
$$786$$ 13752.0 0.624068
$$787$$ 9524.00i 0.431377i 0.976462 + 0.215689i $$0.0691996\pi$$
−0.976462 + 0.215689i $$0.930800\pi$$
$$788$$ − 17496.0i − 0.790951i
$$789$$ −18216.0 −0.821935
$$790$$ 0 0
$$791$$ 7392.00 0.332275
$$792$$ − 864.000i − 0.0387638i
$$793$$ 20444.0i 0.915495i
$$794$$ −212.000 −0.00947556
$$795$$ 0 0
$$796$$ −6400.00 −0.284977
$$797$$ − 33906.0i − 1.50692i −0.657496 0.753458i $$-0.728384\pi$$
0.657496 0.753458i $$-0.271616\pi$$
$$798$$ 1920.00i 0.0851720i
$$799$$ −12096.0 −0.535577
$$800$$ 0 0
$$801$$ 7290.00 0.321572
$$802$$ 3516.00i 0.154806i
$$803$$ − 2616.00i − 0.114965i
$$804$$ 10608.0 0.465318
$$805$$ 0 0
$$806$$ 6688.00 0.292276
$$807$$ 20790.0i 0.906868i
$$808$$ − 4944.00i − 0.215259i
$$809$$ 630.000 0.0273790 0.0136895 0.999906i $$-0.495642\pi$$
0.0136895 + 0.999906i $$0.495642\pi$$
$$810$$ 0 0
$$811$$ −20788.0 −0.900081 −0.450040 0.893008i $$-0.648590\pi$$
−0.450040 + 0.893008i $$0.648590\pi$$
$$812$$ − 1920.00i − 0.0829788i
$$813$$ 4056.00i 0.174969i
$$814$$ 6096.00 0.262487
$$815$$ 0 0
$$816$$ 6048.00 0.259464
$$817$$ − 1040.00i − 0.0445349i
$$818$$ − 7340.00i − 0.313737i
$$819$$ 5472.00 0.233464
$$820$$ 0 0
$$821$$ −43098.0 −1.83207 −0.916036 0.401097i $$-0.868629\pi$$
−0.916036 + 0.401097i $$0.868629\pi$$
$$822$$ − 4356.00i − 0.184833i
$$823$$ 14272.0i 0.604484i 0.953231 + 0.302242i $$0.0977351\pi$$
−0.953231 + 0.302242i $$0.902265\pi$$
$$824$$ 1024.00 0.0432921
$$825$$ 0 0
$$826$$ −21120.0 −0.889660
$$827$$ 13644.0i 0.573698i 0.957976 + 0.286849i $$0.0926078\pi$$
−0.957976 + 0.286849i $$0.907392\pi$$
$$828$$ − 6048.00i − 0.253844i
$$829$$ 2410.00 0.100968 0.0504842 0.998725i $$-0.483924\pi$$
0.0504842 + 0.998725i $$0.483924\pi$$
$$830$$ 0 0
$$831$$ 3558.00 0.148527
$$832$$ 2432.00i 0.101339i
$$833$$ − 10962.0i − 0.455955i
$$834$$ −2280.00 −0.0946642
$$835$$ 0 0
$$836$$ 960.000 0.0397157
$$837$$ 2376.00i 0.0981202i
$$838$$ − 19320.0i − 0.796418i
$$839$$ −23160.0 −0.953006 −0.476503 0.879173i $$-0.658096\pi$$
−0.476503 + 0.879173i $$0.658096\pi$$
$$840$$ 0 0
$$841$$ −23489.0 −0.963098
$$842$$ − 16924.0i − 0.692684i
$$843$$ 7326.00i 0.299313i
$$844$$ −13328.0 −0.543565
$$845$$ 0 0
$$846$$ 1728.00 0.0702244
$$847$$ 18992.0i 0.770452i
$$848$$ − 3168.00i − 0.128290i
$$849$$ 8484.00 0.342957
$$850$$ 0 0
$$851$$ 42672.0 1.71889
$$852$$ − 9504.00i − 0.382162i
$$853$$ − 32078.0i − 1.28761i −0.765190 0.643804i $$-0.777355\pi$$
0.765190 0.643804i $$-0.222645\pi$$
$$854$$ 17216.0 0.689835
$$855$$ 0 0
$$856$$ 11808.0 0.471483
$$857$$ − 14406.0i − 0.574212i −0.957899 0.287106i $$-0.907307\pi$$
0.957899 0.287106i $$-0.0926932\pi$$
$$858$$ − 2736.00i − 0.108864i
$$859$$ −30620.0 −1.21623 −0.608115 0.793849i $$-0.708074\pi$$
−0.608115 + 0.793849i $$0.708074\pi$$
$$860$$ 0 0
$$861$$ 2016.00 0.0797969
$$862$$ − 19584.0i − 0.773821i
$$863$$ − 17568.0i − 0.692957i −0.938058 0.346478i $$-0.887377\pi$$
0.938058 0.346478i $$-0.112623\pi$$
$$864$$ −864.000 −0.0340207
$$865$$ 0 0
$$866$$ 14684.0 0.576192
$$867$$ − 32889.0i − 1.28831i
$$868$$ − 5632.00i − 0.220233i
$$869$$ 6240.00 0.243587
$$870$$ 0 0
$$871$$ 33592.0 1.30680
$$872$$ − 9520.00i − 0.369711i
$$873$$ − 10386.0i − 0.402649i
$$874$$ 6720.00 0.260077
$$875$$ 0 0
$$876$$ −2616.00 −0.100898
$$877$$ − 21706.0i − 0.835758i −0.908503 0.417879i $$-0.862774\pi$$
0.908503 0.417879i $$-0.137226\pi$$
$$878$$ 21280.0i 0.817956i
$$879$$ 14274.0 0.547725
$$880$$ 0 0
$$881$$ −14958.0 −0.572018 −0.286009 0.958227i $$-0.592329\pi$$
−0.286009 + 0.958227i $$0.592329\pi$$
$$882$$ 1566.00i 0.0597845i
$$883$$ 32812.0i 1.25052i 0.780415 + 0.625261i $$0.215008\pi$$
−0.780415 + 0.625261i $$0.784992\pi$$
$$884$$ 19152.0 0.728678
$$885$$ 0 0
$$886$$ 34824.0 1.32047
$$887$$ − 38856.0i − 1.47086i −0.677598 0.735432i $$-0.736979\pi$$
0.677598 0.735432i $$-0.263021\pi$$
$$888$$ − 6096.00i − 0.230370i
$$889$$ −40576.0 −1.53079
$$890$$ 0 0
$$891$$ 972.000 0.0365468
$$892$$ 10592.0i 0.397586i
$$893$$ 1920.00i 0.0719489i
$$894$$ −9540.00 −0.356896
$$895$$ 0 0
$$896$$ 2048.00 0.0763604
$$897$$ − 19152.0i − 0.712895i
$$898$$ − 3420.00i − 0.127090i
$$899$$ 2640.00 0.0979410
$$900$$ 0 0
$$901$$ −24948.0 −0.922462
$$902$$ − 1008.00i − 0.0372092i
$$903$$ 2496.00i 0.0919841i
$$904$$ −3696.00 −0.135981
$$905$$ 0 0
$$906$$ 14592.0 0.535085
$$907$$ − 28276.0i − 1.03516i −0.855635 0.517579i $$-0.826833\pi$$
0.855635 0.517579i $$-0.173167\pi$$
$$908$$ − 8976.00i − 0.328061i
$$909$$ 5562.00 0.202948
$$910$$ 0 0
$$911$$ 8112.00 0.295019 0.147510 0.989061i $$-0.452874\pi$$
0.147510 + 0.989061i $$0.452874\pi$$
$$912$$ − 960.000i − 0.0348561i
$$913$$ 5904.00i 0.214013i
$$914$$ −1292.00 −0.0467566
$$915$$ 0 0
$$916$$ −22600.0 −0.815202
$$917$$ − 36672.0i − 1.32063i
$$918$$ 6804.00i 0.244625i
$$919$$ 26080.0 0.936126 0.468063 0.883695i $$-0.344952\pi$$
0.468063 + 0.883695i $$0.344952\pi$$
$$920$$ 0 0
$$921$$ 25428.0 0.909751
$$922$$ 12036.0i 0.429918i
$$923$$ − 30096.0i − 1.07326i
$$924$$ −2304.00 −0.0820303
$$925$$ 0 0
$$926$$ 13424.0 0.476393
$$927$$ 1152.00i 0.0408162i
$$928$$ 960.000i 0.0339586i
$$929$$ −49170.0 −1.73651 −0.868254 0.496120i $$-0.834757\pi$$
−0.868254 + 0.496120i $$0.834757\pi$$
$$930$$ 0 0
$$931$$ −1740.00 −0.0612526
$$932$$ 18792.0i 0.660464i
$$933$$ 13896.0i 0.487604i
$$934$$ 10728.0 0.375836
$$935$$ 0 0
$$936$$ −2736.00 −0.0955438
$$937$$ 48314.0i 1.68447i 0.539110 + 0.842236i $$0.318761\pi$$
−0.539110 + 0.842236i $$0.681239\pi$$
$$938$$ − 28288.0i − 0.984687i
$$939$$ −14466.0 −0.502748
$$940$$ 0 0
$$941$$ 34782.0 1.20495 0.602477 0.798137i $$-0.294181\pi$$
0.602477 + 0.798137i $$0.294181\pi$$
$$942$$ 3684.00i 0.127422i
$$943$$ − 7056.00i − 0.243664i
$$944$$ 10560.0 0.364088
$$945$$ 0 0
$$946$$ 1248.00 0.0428922
$$947$$ − 25116.0i − 0.861838i −0.902391 0.430919i $$-0.858190\pi$$
0.902391 0.430919i $$-0.141810\pi$$
$$948$$ − 6240.00i − 0.213782i
$$949$$ −8284.00 −0.283361
$$950$$ 0 0
$$951$$ 10278.0 0.350460
$$952$$ − 16128.0i − 0.549067i
$$953$$ 15462.0i 0.525565i 0.964855 + 0.262782i $$0.0846401\pi$$
−0.964855 + 0.262782i $$0.915360\pi$$
$$954$$ 3564.00 0.120953
$$955$$ 0 0
$$956$$ −4800.00 −0.162388
$$957$$ − 1080.00i − 0.0364801i
$$958$$ 19680.0i 0.663708i
$$959$$ −11616.0 −0.391137
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ − 19304.0i − 0.646971i
$$963$$ 13284.0i 0.444518i
$$964$$ 2872.00 0.0959553
$$965$$ 0 0
$$966$$ −16128.0 −0.537174
$$967$$ − 736.000i − 0.0244759i −0.999925 0.0122379i $$-0.996104\pi$$
0.999925 0.0122379i $$-0.00389555\pi$$
$$968$$ − 9496.00i − 0.315303i
$$969$$ −7560.00 −0.250632
$$970$$ 0 0
$$971$$ −29268.0 −0.967307 −0.483653 0.875260i $$-0.660690\pi$$
−0.483653 + 0.875260i $$0.660690\pi$$
$$972$$ − 972.000i − 0.0320750i
$$973$$ 6080.00i 0.200325i
$$974$$ 2848.00 0.0936918
$$975$$ 0 0
$$976$$ −8608.00 −0.282311
$$977$$ 16674.0i 0.546007i 0.962013 + 0.273003i $$0.0880170\pi$$
−0.962013 + 0.273003i $$0.911983\pi$$
$$978$$ 11112.0i 0.363316i
$$979$$ −9720.00 −0.317316
$$980$$ 0 0
$$981$$ 10710.0 0.348567
$$982$$ 9096.00i 0.295586i
$$983$$ 31272.0i 1.01467i 0.861749 + 0.507336i $$0.169370\pi$$
−0.861749 + 0.507336i $$0.830630\pi$$
$$984$$ −1008.00 −0.0326564
$$985$$ 0 0
$$986$$ 7560.00 0.244178
$$987$$ − 4608.00i − 0.148606i
$$988$$ − 3040.00i − 0.0978900i
$$989$$ 8736.00 0.280878
$$990$$ 0 0
$$991$$ −15928.0 −0.510565 −0.255282 0.966867i $$-0.582168\pi$$
−0.255282 + 0.966867i $$0.582168\pi$$
$$992$$ 2816.00i 0.0901291i
$$993$$ − 8364.00i − 0.267295i
$$994$$ −25344.0 −0.808715
$$995$$ 0 0
$$996$$ 5904.00 0.187827
$$997$$ 42014.0i 1.33460i 0.744789 + 0.667300i $$0.232550\pi$$
−0.744789 + 0.667300i $$0.767450\pi$$
$$998$$ 13000.0i 0.412332i
$$999$$ 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.c.d.49.1 2
3.2 odd 2 450.4.c.e.199.2 2
4.3 odd 2 1200.4.f.j.49.1 2
5.2 odd 4 150.4.a.i.1.1 1
5.3 odd 4 6.4.a.a.1.1 1
5.4 even 2 inner 150.4.c.d.49.2 2
15.2 even 4 450.4.a.h.1.1 1
15.8 even 4 18.4.a.a.1.1 1
15.14 odd 2 450.4.c.e.199.1 2
20.3 even 4 48.4.a.c.1.1 1
20.7 even 4 1200.4.a.b.1.1 1
20.19 odd 2 1200.4.f.j.49.2 2
35.3 even 12 294.4.e.g.79.1 2
35.13 even 4 294.4.a.e.1.1 1
35.18 odd 12 294.4.e.h.79.1 2
35.23 odd 12 294.4.e.h.67.1 2
35.33 even 12 294.4.e.g.67.1 2
40.3 even 4 192.4.a.c.1.1 1
40.13 odd 4 192.4.a.i.1.1 1
45.13 odd 12 162.4.c.f.55.1 2
45.23 even 12 162.4.c.c.55.1 2
45.38 even 12 162.4.c.c.109.1 2
45.43 odd 12 162.4.c.f.109.1 2
55.43 even 4 726.4.a.f.1.1 1
60.23 odd 4 144.4.a.c.1.1 1
65.8 even 4 1014.4.b.d.337.1 2
65.18 even 4 1014.4.b.d.337.2 2
65.38 odd 4 1014.4.a.g.1.1 1
80.3 even 4 768.4.d.c.385.2 2
80.13 odd 4 768.4.d.n.385.1 2
80.43 even 4 768.4.d.c.385.1 2
80.53 odd 4 768.4.d.n.385.2 2
85.33 odd 4 1734.4.a.d.1.1 1
95.18 even 4 2166.4.a.i.1.1 1
105.23 even 12 882.4.g.i.361.1 2
105.38 odd 12 882.4.g.f.667.1 2
105.53 even 12 882.4.g.i.667.1 2
105.68 odd 12 882.4.g.f.361.1 2
105.83 odd 4 882.4.a.n.1.1 1
120.53 even 4 576.4.a.q.1.1 1
120.83 odd 4 576.4.a.r.1.1 1
140.83 odd 4 2352.4.a.e.1.1 1
165.98 odd 4 2178.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 5.3 odd 4
18.4.a.a.1.1 1 15.8 even 4
48.4.a.c.1.1 1 20.3 even 4
144.4.a.c.1.1 1 60.23 odd 4
150.4.a.i.1.1 1 5.2 odd 4
150.4.c.d.49.1 2 1.1 even 1 trivial
150.4.c.d.49.2 2 5.4 even 2 inner
162.4.c.c.55.1 2 45.23 even 12
162.4.c.c.109.1 2 45.38 even 12
162.4.c.f.55.1 2 45.13 odd 12
162.4.c.f.109.1 2 45.43 odd 12
192.4.a.c.1.1 1 40.3 even 4
192.4.a.i.1.1 1 40.13 odd 4
294.4.a.e.1.1 1 35.13 even 4
294.4.e.g.67.1 2 35.33 even 12
294.4.e.g.79.1 2 35.3 even 12
294.4.e.h.67.1 2 35.23 odd 12
294.4.e.h.79.1 2 35.18 odd 12
450.4.a.h.1.1 1 15.2 even 4
450.4.c.e.199.1 2 15.14 odd 2
450.4.c.e.199.2 2 3.2 odd 2
576.4.a.q.1.1 1 120.53 even 4
576.4.a.r.1.1 1 120.83 odd 4
726.4.a.f.1.1 1 55.43 even 4
768.4.d.c.385.1 2 80.43 even 4
768.4.d.c.385.2 2 80.3 even 4
768.4.d.n.385.1 2 80.13 odd 4
768.4.d.n.385.2 2 80.53 odd 4
882.4.a.n.1.1 1 105.83 odd 4
882.4.g.f.361.1 2 105.68 odd 12
882.4.g.f.667.1 2 105.38 odd 12
882.4.g.i.361.1 2 105.23 even 12
882.4.g.i.667.1 2 105.53 even 12
1014.4.a.g.1.1 1 65.38 odd 4
1014.4.b.d.337.1 2 65.8 even 4
1014.4.b.d.337.2 2 65.18 even 4
1200.4.a.b.1.1 1 20.7 even 4
1200.4.f.j.49.1 2 4.3 odd 2
1200.4.f.j.49.2 2 20.19 odd 2
1734.4.a.d.1.1 1 85.33 odd 4
2166.4.a.i.1.1 1 95.18 even 4
2178.4.a.e.1.1 1 165.98 odd 4
2352.4.a.e.1.1 1 140.83 odd 4