# Properties

 Label 2450.2.c.w.99.4 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 490) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.4 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.w.99.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} +0.585786 q^{6} -1.00000i q^{8} +2.65685 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} +0.585786 q^{6} -1.00000i q^{8} +2.65685 q^{9} +4.82843 q^{11} +0.585786i q^{12} -0.828427i q^{13} +1.00000 q^{16} +5.41421i q^{17} +2.65685i q^{18} +3.41421 q^{19} +4.82843i q^{22} -6.82843i q^{23} -0.585786 q^{24} +0.828427 q^{26} -3.31371i q^{27} -0.828427 q^{29} -2.82843 q^{31} +1.00000i q^{32} -2.82843i q^{33} -5.41421 q^{34} -2.65685 q^{36} -3.65685i q^{37} +3.41421i q^{38} -0.485281 q^{39} -11.0711 q^{41} -3.17157i q^{43} -4.82843 q^{44} +6.82843 q^{46} +10.8284i q^{47} -0.585786i q^{48} +3.17157 q^{51} +0.828427i q^{52} -10.4853i q^{53} +3.31371 q^{54} -2.00000i q^{57} -0.828427i q^{58} +11.4142 q^{59} +13.3137 q^{61} -2.82843i q^{62} -1.00000 q^{64} +2.82843 q^{66} +9.65685i q^{67} -5.41421i q^{68} -4.00000 q^{69} +12.4853 q^{71} -2.65685i q^{72} +6.58579i q^{73} +3.65685 q^{74} -3.41421 q^{76} -0.485281i q^{78} +1.17157 q^{79} +6.02944 q^{81} -11.0711i q^{82} -6.24264i q^{83} +3.17157 q^{86} +0.485281i q^{87} -4.82843i q^{88} -12.7279 q^{89} +6.82843i q^{92} +1.65685i q^{93} -10.8284 q^{94} +0.585786 q^{96} -16.2426i q^{97} +12.8284 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 8q^{6} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 8q^{6} - 12q^{9} + 8q^{11} + 4q^{16} + 8q^{19} - 8q^{24} - 8q^{26} + 8q^{29} - 16q^{34} + 12q^{36} + 32q^{39} - 16q^{41} - 8q^{44} + 16q^{46} + 24q^{51} - 32q^{54} + 40q^{59} + 8q^{61} - 4q^{64} - 16q^{69} + 16q^{71} - 8q^{74} - 8q^{76} + 16q^{79} + 92q^{81} + 24q^{86} - 32q^{94} + 8q^{96} + 40q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 0.585786i − 0.338204i −0.985599 0.169102i $$-0.945913\pi$$
0.985599 0.169102i $$-0.0540867\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0.585786 0.239146
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ 2.65685 0.885618
$$10$$ 0 0
$$11$$ 4.82843 1.45583 0.727913 0.685670i $$-0.240491\pi$$
0.727913 + 0.685670i $$0.240491\pi$$
$$12$$ 0.585786i 0.169102i
$$13$$ − 0.828427i − 0.229764i −0.993379 0.114882i $$-0.963351\pi$$
0.993379 0.114882i $$-0.0366490\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.41421i 1.31314i 0.754265 + 0.656570i $$0.227993\pi$$
−0.754265 + 0.656570i $$0.772007\pi$$
$$18$$ 2.65685i 0.626227i
$$19$$ 3.41421 0.783274 0.391637 0.920120i $$-0.371909\pi$$
0.391637 + 0.920120i $$0.371909\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.82843i 1.02942i
$$23$$ − 6.82843i − 1.42383i −0.702268 0.711913i $$-0.747829\pi$$
0.702268 0.711913i $$-0.252171\pi$$
$$24$$ −0.585786 −0.119573
$$25$$ 0 0
$$26$$ 0.828427 0.162468
$$27$$ − 3.31371i − 0.637723i
$$28$$ 0 0
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ 0 0
$$31$$ −2.82843 −0.508001 −0.254000 0.967204i $$-0.581746\pi$$
−0.254000 + 0.967204i $$0.581746\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 2.82843i − 0.492366i
$$34$$ −5.41421 −0.928530
$$35$$ 0 0
$$36$$ −2.65685 −0.442809
$$37$$ − 3.65685i − 0.601183i −0.953753 0.300592i $$-0.902816\pi$$
0.953753 0.300592i $$-0.0971841\pi$$
$$38$$ 3.41421i 0.553859i
$$39$$ −0.485281 −0.0777072
$$40$$ 0 0
$$41$$ −11.0711 −1.72901 −0.864505 0.502624i $$-0.832368\pi$$
−0.864505 + 0.502624i $$0.832368\pi$$
$$42$$ 0 0
$$43$$ − 3.17157i − 0.483660i −0.970319 0.241830i $$-0.922252\pi$$
0.970319 0.241830i $$-0.0777477\pi$$
$$44$$ −4.82843 −0.727913
$$45$$ 0 0
$$46$$ 6.82843 1.00680
$$47$$ 10.8284i 1.57949i 0.613436 + 0.789744i $$0.289787\pi$$
−0.613436 + 0.789744i $$0.710213\pi$$
$$48$$ − 0.585786i − 0.0845510i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.17157 0.444109
$$52$$ 0.828427i 0.114882i
$$53$$ − 10.4853i − 1.44026i −0.693837 0.720132i $$-0.744081\pi$$
0.693837 0.720132i $$-0.255919\pi$$
$$54$$ 3.31371 0.450939
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ − 0.828427i − 0.108778i
$$59$$ 11.4142 1.48600 0.743002 0.669289i $$-0.233401\pi$$
0.743002 + 0.669289i $$0.233401\pi$$
$$60$$ 0 0
$$61$$ 13.3137 1.70465 0.852323 0.523016i $$-0.175193\pi$$
0.852323 + 0.523016i $$0.175193\pi$$
$$62$$ − 2.82843i − 0.359211i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.82843 0.348155
$$67$$ 9.65685i 1.17977i 0.807486 + 0.589886i $$0.200827\pi$$
−0.807486 + 0.589886i $$0.799173\pi$$
$$68$$ − 5.41421i − 0.656570i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 12.4853 1.48173 0.740865 0.671654i $$-0.234416\pi$$
0.740865 + 0.671654i $$0.234416\pi$$
$$72$$ − 2.65685i − 0.313113i
$$73$$ 6.58579i 0.770808i 0.922748 + 0.385404i $$0.125938\pi$$
−0.922748 + 0.385404i $$0.874062\pi$$
$$74$$ 3.65685 0.425101
$$75$$ 0 0
$$76$$ −3.41421 −0.391637
$$77$$ 0 0
$$78$$ − 0.485281i − 0.0549473i
$$79$$ 1.17157 0.131812 0.0659061 0.997826i $$-0.479006\pi$$
0.0659061 + 0.997826i $$0.479006\pi$$
$$80$$ 0 0
$$81$$ 6.02944 0.669937
$$82$$ − 11.0711i − 1.22259i
$$83$$ − 6.24264i − 0.685219i −0.939478 0.342609i $$-0.888689\pi$$
0.939478 0.342609i $$-0.111311\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.17157 0.341999
$$87$$ 0.485281i 0.0520276i
$$88$$ − 4.82843i − 0.514712i
$$89$$ −12.7279 −1.34916 −0.674579 0.738203i $$-0.735675\pi$$
−0.674579 + 0.738203i $$0.735675\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.82843i 0.711913i
$$93$$ 1.65685i 0.171808i
$$94$$ −10.8284 −1.11687
$$95$$ 0 0
$$96$$ 0.585786 0.0597866
$$97$$ − 16.2426i − 1.64919i −0.565723 0.824595i $$-0.691403\pi$$
0.565723 0.824595i $$-0.308597\pi$$
$$98$$ 0 0
$$99$$ 12.8284 1.28931
$$100$$ 0 0
$$101$$ 9.31371 0.926749 0.463374 0.886163i $$-0.346639\pi$$
0.463374 + 0.886163i $$0.346639\pi$$
$$102$$ 3.17157i 0.314033i
$$103$$ 9.17157i 0.903702i 0.892093 + 0.451851i $$0.149236\pi$$
−0.892093 + 0.451851i $$0.850764\pi$$
$$104$$ −0.828427 −0.0812340
$$105$$ 0 0
$$106$$ 10.4853 1.01842
$$107$$ 1.65685i 0.160174i 0.996788 + 0.0800871i $$0.0255198\pi$$
−0.996788 + 0.0800871i $$0.974480\pi$$
$$108$$ 3.31371i 0.318862i
$$109$$ 14.4853 1.38744 0.693719 0.720246i $$-0.255971\pi$$
0.693719 + 0.720246i $$0.255971\pi$$
$$110$$ 0 0
$$111$$ −2.14214 −0.203323
$$112$$ 0 0
$$113$$ 7.31371i 0.688016i 0.938967 + 0.344008i $$0.111785\pi$$
−0.938967 + 0.344008i $$0.888215\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 0.828427 0.0769175
$$117$$ − 2.20101i − 0.203483i
$$118$$ 11.4142i 1.05076i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 13.3137i 1.20537i
$$123$$ 6.48528i 0.584758i
$$124$$ 2.82843 0.254000
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.82843i 0.250982i 0.992095 + 0.125491i $$0.0400507\pi$$
−0.992095 + 0.125491i $$0.959949\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −1.85786 −0.163576
$$130$$ 0 0
$$131$$ −2.24264 −0.195940 −0.0979702 0.995189i $$-0.531235\pi$$
−0.0979702 + 0.995189i $$0.531235\pi$$
$$132$$ 2.82843i 0.246183i
$$133$$ 0 0
$$134$$ −9.65685 −0.834225
$$135$$ 0 0
$$136$$ 5.41421 0.464265
$$137$$ 16.0000i 1.36697i 0.729964 + 0.683486i $$0.239537\pi$$
−0.729964 + 0.683486i $$0.760463\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 0.100505 0.00852473 0.00426236 0.999991i $$-0.498643\pi$$
0.00426236 + 0.999991i $$0.498643\pi$$
$$140$$ 0 0
$$141$$ 6.34315 0.534189
$$142$$ 12.4853i 1.04774i
$$143$$ − 4.00000i − 0.334497i
$$144$$ 2.65685 0.221405
$$145$$ 0 0
$$146$$ −6.58579 −0.545044
$$147$$ 0 0
$$148$$ 3.65685i 0.300592i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 11.3137 0.920697 0.460348 0.887738i $$-0.347725\pi$$
0.460348 + 0.887738i $$0.347725\pi$$
$$152$$ − 3.41421i − 0.276929i
$$153$$ 14.3848i 1.16294i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0.485281 0.0388536
$$157$$ − 10.4853i − 0.836817i −0.908259 0.418408i $$-0.862588\pi$$
0.908259 0.418408i $$-0.137412\pi$$
$$158$$ 1.17157i 0.0932053i
$$159$$ −6.14214 −0.487103
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 6.02944i 0.473717i
$$163$$ 8.14214i 0.637741i 0.947798 + 0.318871i $$0.103304\pi$$
−0.947798 + 0.318871i $$0.896696\pi$$
$$164$$ 11.0711 0.864505
$$165$$ 0 0
$$166$$ 6.24264 0.484523
$$167$$ − 23.7990i − 1.84162i −0.390010 0.920811i $$-0.627529\pi$$
0.390010 0.920811i $$-0.372471\pi$$
$$168$$ 0 0
$$169$$ 12.3137 0.947208
$$170$$ 0 0
$$171$$ 9.07107 0.693682
$$172$$ 3.17157i 0.241830i
$$173$$ 3.17157i 0.241130i 0.992705 + 0.120565i $$0.0384707\pi$$
−0.992705 + 0.120565i $$0.961529\pi$$
$$174$$ −0.485281 −0.0367891
$$175$$ 0 0
$$176$$ 4.82843 0.363956
$$177$$ − 6.68629i − 0.502572i
$$178$$ − 12.7279i − 0.953998i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −14.4853 −1.07668 −0.538341 0.842727i $$-0.680949\pi$$
−0.538341 + 0.842727i $$0.680949\pi$$
$$182$$ 0 0
$$183$$ − 7.79899i − 0.576518i
$$184$$ −6.82843 −0.503398
$$185$$ 0 0
$$186$$ −1.65685 −0.121486
$$187$$ 26.1421i 1.91170i
$$188$$ − 10.8284i − 0.789744i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.1421 −1.31272 −0.656359 0.754448i $$-0.727904\pi$$
−0.656359 + 0.754448i $$0.727904\pi$$
$$192$$ 0.585786i 0.0422755i
$$193$$ − 5.65685i − 0.407189i −0.979055 0.203595i $$-0.934738\pi$$
0.979055 0.203595i $$-0.0652625\pi$$
$$194$$ 16.2426 1.16615
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 13.7990i − 0.983137i −0.870839 0.491569i $$-0.836424\pi$$
0.870839 0.491569i $$-0.163576\pi$$
$$198$$ 12.8284i 0.911677i
$$199$$ −0.485281 −0.0344007 −0.0172003 0.999852i $$-0.505475\pi$$
−0.0172003 + 0.999852i $$0.505475\pi$$
$$200$$ 0 0
$$201$$ 5.65685 0.399004
$$202$$ 9.31371i 0.655310i
$$203$$ 0 0
$$204$$ −3.17157 −0.222055
$$205$$ 0 0
$$206$$ −9.17157 −0.639014
$$207$$ − 18.1421i − 1.26097i
$$208$$ − 0.828427i − 0.0574411i
$$209$$ 16.4853 1.14031
$$210$$ 0 0
$$211$$ −26.6274 −1.83311 −0.916553 0.399912i $$-0.869041\pi$$
−0.916553 + 0.399912i $$0.869041\pi$$
$$212$$ 10.4853i 0.720132i
$$213$$ − 7.31371i − 0.501127i
$$214$$ −1.65685 −0.113260
$$215$$ 0 0
$$216$$ −3.31371 −0.225469
$$217$$ 0 0
$$218$$ 14.4853i 0.981067i
$$219$$ 3.85786 0.260690
$$220$$ 0 0
$$221$$ 4.48528 0.301713
$$222$$ − 2.14214i − 0.143771i
$$223$$ − 15.3137i − 1.02548i −0.858543 0.512741i $$-0.828630\pi$$
0.858543 0.512741i $$-0.171370\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7.31371 −0.486501
$$227$$ 9.75736i 0.647619i 0.946122 + 0.323809i $$0.104964\pi$$
−0.946122 + 0.323809i $$0.895036\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ 12.1421 0.802375 0.401187 0.915996i $$-0.368598\pi$$
0.401187 + 0.915996i $$0.368598\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0.828427i 0.0543889i
$$233$$ 0.686292i 0.0449605i 0.999747 + 0.0224802i $$0.00715628\pi$$
−0.999747 + 0.0224802i $$0.992844\pi$$
$$234$$ 2.20101 0.143885
$$235$$ 0 0
$$236$$ −11.4142 −0.743002
$$237$$ − 0.686292i − 0.0445794i
$$238$$ 0 0
$$239$$ 9.65685 0.624650 0.312325 0.949975i $$-0.398892\pi$$
0.312325 + 0.949975i $$0.398892\pi$$
$$240$$ 0 0
$$241$$ 10.5858 0.681890 0.340945 0.940083i $$-0.389253\pi$$
0.340945 + 0.940083i $$0.389253\pi$$
$$242$$ 12.3137i 0.791555i
$$243$$ − 13.4731i − 0.864299i
$$244$$ −13.3137 −0.852323
$$245$$ 0 0
$$246$$ −6.48528 −0.413486
$$247$$ − 2.82843i − 0.179969i
$$248$$ 2.82843i 0.179605i
$$249$$ −3.65685 −0.231744
$$250$$ 0 0
$$251$$ −3.41421 −0.215503 −0.107752 0.994178i $$-0.534365\pi$$
−0.107752 + 0.994178i $$0.534365\pi$$
$$252$$ 0 0
$$253$$ − 32.9706i − 2.07284i
$$254$$ −2.82843 −0.177471
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 9.89949i 0.617514i 0.951141 + 0.308757i $$0.0999129\pi$$
−0.951141 + 0.308757i $$0.900087\pi$$
$$258$$ − 1.85786i − 0.115666i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.20101 −0.136239
$$262$$ − 2.24264i − 0.138551i
$$263$$ 28.0000i 1.72655i 0.504730 + 0.863277i $$0.331592\pi$$
−0.504730 + 0.863277i $$0.668408\pi$$
$$264$$ −2.82843 −0.174078
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 7.45584i 0.456290i
$$268$$ − 9.65685i − 0.589886i
$$269$$ 1.51472 0.0923540 0.0461770 0.998933i $$-0.485296\pi$$
0.0461770 + 0.998933i $$0.485296\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 5.41421i 0.328285i
$$273$$ 0 0
$$274$$ −16.0000 −0.966595
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 20.1421i − 1.21022i −0.796140 0.605112i $$-0.793128\pi$$
0.796140 0.605112i $$-0.206872\pi$$
$$278$$ 0.100505i 0.00602789i
$$279$$ −7.51472 −0.449894
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 6.34315i 0.377729i
$$283$$ − 6.24264i − 0.371086i −0.982636 0.185543i $$-0.940596\pi$$
0.982636 0.185543i $$-0.0594045\pi$$
$$284$$ −12.4853 −0.740865
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 2.65685i 0.156557i
$$289$$ −12.3137 −0.724336
$$290$$ 0 0
$$291$$ −9.51472 −0.557763
$$292$$ − 6.58579i − 0.385404i
$$293$$ 19.6569i 1.14837i 0.818727 + 0.574183i $$0.194680\pi$$
−0.818727 + 0.574183i $$0.805320\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −3.65685 −0.212550
$$297$$ − 16.0000i − 0.928414i
$$298$$ 6.00000i 0.347571i
$$299$$ −5.65685 −0.327144
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 11.3137i 0.651031i
$$303$$ − 5.45584i − 0.313430i
$$304$$ 3.41421 0.195819
$$305$$ 0 0
$$306$$ −14.3848 −0.822323
$$307$$ 29.0711i 1.65917i 0.558378 + 0.829587i $$0.311424\pi$$
−0.558378 + 0.829587i $$0.688576\pi$$
$$308$$ 0 0
$$309$$ 5.37258 0.305636
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0.485281i 0.0274736i
$$313$$ 22.3848i 1.26526i 0.774453 + 0.632631i $$0.218025\pi$$
−0.774453 + 0.632631i $$0.781975\pi$$
$$314$$ 10.4853 0.591719
$$315$$ 0 0
$$316$$ −1.17157 −0.0659061
$$317$$ 6.48528i 0.364250i 0.983275 + 0.182125i $$0.0582975\pi$$
−0.983275 + 0.182125i $$0.941702\pi$$
$$318$$ − 6.14214i − 0.344434i
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 0.970563 0.0541715
$$322$$ 0 0
$$323$$ 18.4853i 1.02855i
$$324$$ −6.02944 −0.334969
$$325$$ 0 0
$$326$$ −8.14214 −0.450951
$$327$$ − 8.48528i − 0.469237i
$$328$$ 11.0711i 0.611297i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5.79899 −0.318741 −0.159371 0.987219i $$-0.550946\pi$$
−0.159371 + 0.987219i $$0.550946\pi$$
$$332$$ 6.24264i 0.342609i
$$333$$ − 9.71573i − 0.532419i
$$334$$ 23.7990 1.30222
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 6.00000i − 0.326841i −0.986557 0.163420i $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ 12.3137i 0.669777i
$$339$$ 4.28427 0.232690
$$340$$ 0 0
$$341$$ −13.6569 −0.739560
$$342$$ 9.07107i 0.490507i
$$343$$ 0 0
$$344$$ −3.17157 −0.171000
$$345$$ 0 0
$$346$$ −3.17157 −0.170505
$$347$$ 8.82843i 0.473935i 0.971518 + 0.236967i $$0.0761535\pi$$
−0.971518 + 0.236967i $$0.923847\pi$$
$$348$$ − 0.485281i − 0.0260138i
$$349$$ −14.4853 −0.775379 −0.387690 0.921790i $$-0.626727\pi$$
−0.387690 + 0.921790i $$0.626727\pi$$
$$350$$ 0 0
$$351$$ −2.74517 −0.146526
$$352$$ 4.82843i 0.257356i
$$353$$ − 34.3848i − 1.83012i −0.403321 0.915058i $$-0.632144\pi$$
0.403321 0.915058i $$-0.367856\pi$$
$$354$$ 6.68629 0.355372
$$355$$ 0 0
$$356$$ 12.7279 0.674579
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −7.34315 −0.386481
$$362$$ − 14.4853i − 0.761329i
$$363$$ − 7.21320i − 0.378595i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 7.79899 0.407660
$$367$$ − 8.97056i − 0.468260i −0.972205 0.234130i $$-0.924776\pi$$
0.972205 0.234130i $$-0.0752241\pi$$
$$368$$ − 6.82843i − 0.355956i
$$369$$ −29.4142 −1.53124
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 1.65685i − 0.0859039i
$$373$$ − 13.5147i − 0.699766i −0.936793 0.349883i $$-0.886221\pi$$
0.936793 0.349883i $$-0.113779\pi$$
$$374$$ −26.1421 −1.35178
$$375$$ 0 0
$$376$$ 10.8284 0.558433
$$377$$ 0.686292i 0.0353458i
$$378$$ 0 0
$$379$$ −17.5147 −0.899671 −0.449835 0.893112i $$-0.648517\pi$$
−0.449835 + 0.893112i $$0.648517\pi$$
$$380$$ 0 0
$$381$$ 1.65685 0.0848832
$$382$$ − 18.1421i − 0.928232i
$$383$$ 15.5147i 0.792765i 0.918085 + 0.396383i $$0.129735\pi$$
−0.918085 + 0.396383i $$0.870265\pi$$
$$384$$ −0.585786 −0.0298933
$$385$$ 0 0
$$386$$ 5.65685 0.287926
$$387$$ − 8.42641i − 0.428338i
$$388$$ 16.2426i 0.824595i
$$389$$ 0.142136 0.00720656 0.00360328 0.999994i $$-0.498853\pi$$
0.00360328 + 0.999994i $$0.498853\pi$$
$$390$$ 0 0
$$391$$ 36.9706 1.86968
$$392$$ 0 0
$$393$$ 1.31371i 0.0662678i
$$394$$ 13.7990 0.695183
$$395$$ 0 0
$$396$$ −12.8284 −0.644653
$$397$$ − 5.79899i − 0.291043i −0.989355 0.145521i $$-0.953514\pi$$
0.989355 0.145521i $$-0.0464860\pi$$
$$398$$ − 0.485281i − 0.0243250i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 5.65685i 0.282138i
$$403$$ 2.34315i 0.116720i
$$404$$ −9.31371 −0.463374
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 17.6569i − 0.875218i
$$408$$ − 3.17157i − 0.157016i
$$409$$ −13.4142 −0.663290 −0.331645 0.943404i $$-0.607604\pi$$
−0.331645 + 0.943404i $$0.607604\pi$$
$$410$$ 0 0
$$411$$ 9.37258 0.462315
$$412$$ − 9.17157i − 0.451851i
$$413$$ 0 0
$$414$$ 18.1421 0.891637
$$415$$ 0 0
$$416$$ 0.828427 0.0406170
$$417$$ − 0.0588745i − 0.00288310i
$$418$$ 16.4853i 0.806321i
$$419$$ 32.8701 1.60581 0.802904 0.596109i $$-0.203287\pi$$
0.802904 + 0.596109i $$0.203287\pi$$
$$420$$ 0 0
$$421$$ −5.31371 −0.258974 −0.129487 0.991581i $$-0.541333\pi$$
−0.129487 + 0.991581i $$0.541333\pi$$
$$422$$ − 26.6274i − 1.29620i
$$423$$ 28.7696i 1.39882i
$$424$$ −10.4853 −0.509210
$$425$$ 0 0
$$426$$ 7.31371 0.354350
$$427$$ 0 0
$$428$$ − 1.65685i − 0.0800871i
$$429$$ −2.34315 −0.113128
$$430$$ 0 0
$$431$$ −33.6569 −1.62119 −0.810597 0.585605i $$-0.800857\pi$$
−0.810597 + 0.585605i $$0.800857\pi$$
$$432$$ − 3.31371i − 0.159431i
$$433$$ − 13.4142i − 0.644646i −0.946630 0.322323i $$-0.895536\pi$$
0.946630 0.322323i $$-0.104464\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.4853 −0.693719
$$437$$ − 23.3137i − 1.11525i
$$438$$ 3.85786i 0.184336i
$$439$$ 8.97056 0.428142 0.214071 0.976818i $$-0.431328\pi$$
0.214071 + 0.976818i $$0.431328\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.48528i 0.213343i
$$443$$ 36.9706i 1.75652i 0.478179 + 0.878262i $$0.341297\pi$$
−0.478179 + 0.878262i $$0.658703\pi$$
$$444$$ 2.14214 0.101661
$$445$$ 0 0
$$446$$ 15.3137 0.725125
$$447$$ − 3.51472i − 0.166240i
$$448$$ 0 0
$$449$$ −28.6274 −1.35101 −0.675506 0.737355i $$-0.736075\pi$$
−0.675506 + 0.737355i $$0.736075\pi$$
$$450$$ 0 0
$$451$$ −53.4558 −2.51714
$$452$$ − 7.31371i − 0.344008i
$$453$$ − 6.62742i − 0.311383i
$$454$$ −9.75736 −0.457936
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ − 10.3431i − 0.483832i −0.970297 0.241916i $$-0.922224\pi$$
0.970297 0.241916i $$-0.0777758\pi$$
$$458$$ 12.1421i 0.567365i
$$459$$ 17.9411 0.837420
$$460$$ 0 0
$$461$$ 7.17157 0.334013 0.167007 0.985956i $$-0.446590\pi$$
0.167007 + 0.985956i $$0.446590\pi$$
$$462$$ 0 0
$$463$$ 16.9706i 0.788689i 0.918963 + 0.394344i $$0.129028\pi$$
−0.918963 + 0.394344i $$0.870972\pi$$
$$464$$ −0.828427 −0.0384588
$$465$$ 0 0
$$466$$ −0.686292 −0.0317918
$$467$$ 3.89949i 0.180447i 0.995922 + 0.0902236i $$0.0287582\pi$$
−0.995922 + 0.0902236i $$0.971242\pi$$
$$468$$ 2.20101i 0.101742i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.14214 −0.283015
$$472$$ − 11.4142i − 0.525382i
$$473$$ − 15.3137i − 0.704125i
$$474$$ 0.686292 0.0315224
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 27.8579i − 1.27552i
$$478$$ 9.65685i 0.441694i
$$479$$ −22.8284 −1.04306 −0.521529 0.853234i $$-0.674638\pi$$
−0.521529 + 0.853234i $$0.674638\pi$$
$$480$$ 0 0
$$481$$ −3.02944 −0.138130
$$482$$ 10.5858i 0.482169i
$$483$$ 0 0
$$484$$ −12.3137 −0.559714
$$485$$ 0 0
$$486$$ 13.4731 0.611152
$$487$$ 7.79899i 0.353406i 0.984264 + 0.176703i $$0.0565432\pi$$
−0.984264 + 0.176703i $$0.943457\pi$$
$$488$$ − 13.3137i − 0.602683i
$$489$$ 4.76955 0.215687
$$490$$ 0 0
$$491$$ −24.2843 −1.09593 −0.547967 0.836500i $$-0.684598\pi$$
−0.547967 + 0.836500i $$0.684598\pi$$
$$492$$ − 6.48528i − 0.292379i
$$493$$ − 4.48528i − 0.202007i
$$494$$ 2.82843 0.127257
$$495$$ 0 0
$$496$$ −2.82843 −0.127000
$$497$$ 0 0
$$498$$ − 3.65685i − 0.163868i
$$499$$ −41.6569 −1.86482 −0.932408 0.361406i $$-0.882297\pi$$
−0.932408 + 0.361406i $$0.882297\pi$$
$$500$$ 0 0
$$501$$ −13.9411 −0.622844
$$502$$ − 3.41421i − 0.152384i
$$503$$ − 6.34315i − 0.282827i −0.989951 0.141413i $$-0.954835\pi$$
0.989951 0.141413i $$-0.0451647\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.9706 1.46572
$$507$$ − 7.21320i − 0.320350i
$$508$$ − 2.82843i − 0.125491i
$$509$$ −33.7990 −1.49811 −0.749057 0.662506i $$-0.769493\pi$$
−0.749057 + 0.662506i $$0.769493\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 11.3137i − 0.499512i
$$514$$ −9.89949 −0.436648
$$515$$ 0 0
$$516$$ 1.85786 0.0817879
$$517$$ 52.2843i 2.29946i
$$518$$ 0 0
$$519$$ 1.85786 0.0815512
$$520$$ 0 0
$$521$$ −4.92893 −0.215940 −0.107970 0.994154i $$-0.534435\pi$$
−0.107970 + 0.994154i $$0.534435\pi$$
$$522$$ − 2.20101i − 0.0963356i
$$523$$ − 4.10051i − 0.179303i −0.995973 0.0896513i $$-0.971425\pi$$
0.995973 0.0896513i $$-0.0285753\pi$$
$$524$$ 2.24264 0.0979702
$$525$$ 0 0
$$526$$ −28.0000 −1.22086
$$527$$ − 15.3137i − 0.667076i
$$528$$ − 2.82843i − 0.123091i
$$529$$ −23.6274 −1.02728
$$530$$ 0 0
$$531$$ 30.3259 1.31603
$$532$$ 0 0
$$533$$ 9.17157i 0.397265i
$$534$$ −7.45584 −0.322646
$$535$$ 0 0
$$536$$ 9.65685 0.417113
$$537$$ 2.34315i 0.101114i
$$538$$ 1.51472i 0.0653042i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18.9706 0.815608 0.407804 0.913069i $$-0.366295\pi$$
0.407804 + 0.913069i $$0.366295\pi$$
$$542$$ − 12.0000i − 0.515444i
$$543$$ 8.48528i 0.364138i
$$544$$ −5.41421 −0.232132
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.48528i − 0.277291i −0.990342 0.138645i $$-0.955725\pi$$
0.990342 0.138645i $$-0.0442748\pi$$
$$548$$ − 16.0000i − 0.683486i
$$549$$ 35.3726 1.50967
$$550$$ 0 0
$$551$$ −2.82843 −0.120495
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ 20.1421 0.855757
$$555$$ 0 0
$$556$$ −0.100505 −0.00426236
$$557$$ − 20.8284i − 0.882529i −0.897377 0.441264i $$-0.854530\pi$$
0.897377 0.441264i $$-0.145470\pi$$
$$558$$ − 7.51472i − 0.318123i
$$559$$ −2.62742 −0.111128
$$560$$ 0 0
$$561$$ 15.3137 0.646545
$$562$$ 8.00000i 0.337460i
$$563$$ 39.4142i 1.66111i 0.556936 + 0.830556i $$0.311977\pi$$
−0.556936 + 0.830556i $$0.688023\pi$$
$$564$$ −6.34315 −0.267095
$$565$$ 0 0
$$566$$ 6.24264 0.262398
$$567$$ 0 0
$$568$$ − 12.4853i − 0.523871i
$$569$$ 6.68629 0.280304 0.140152 0.990130i $$-0.455241\pi$$
0.140152 + 0.990130i $$0.455241\pi$$
$$570$$ 0 0
$$571$$ −41.7990 −1.74923 −0.874617 0.484815i $$-0.838887\pi$$
−0.874617 + 0.484815i $$0.838887\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 10.6274i 0.443967i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −2.65685 −0.110702
$$577$$ 25.8995i 1.07821i 0.842239 + 0.539105i $$0.181237\pi$$
−0.842239 + 0.539105i $$0.818763\pi$$
$$578$$ − 12.3137i − 0.512183i
$$579$$ −3.31371 −0.137713
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 9.51472i − 0.394398i
$$583$$ − 50.6274i − 2.09677i
$$584$$ 6.58579 0.272522
$$585$$ 0 0
$$586$$ −19.6569 −0.812017
$$587$$ 2.92893i 0.120890i 0.998172 + 0.0604450i $$0.0192520\pi$$
−0.998172 + 0.0604450i $$0.980748\pi$$
$$588$$ 0 0
$$589$$ −9.65685 −0.397904
$$590$$ 0 0
$$591$$ −8.08326 −0.332501
$$592$$ − 3.65685i − 0.150296i
$$593$$ − 28.7279i − 1.17971i −0.807508 0.589857i $$-0.799184\pi$$
0.807508 0.589857i $$-0.200816\pi$$
$$594$$ 16.0000 0.656488
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 0.284271i 0.0116344i
$$598$$ − 5.65685i − 0.231326i
$$599$$ 5.17157 0.211305 0.105652 0.994403i $$-0.466307\pi$$
0.105652 + 0.994403i $$0.466307\pi$$
$$600$$ 0 0
$$601$$ −9.41421 −0.384014 −0.192007 0.981394i $$-0.561500\pi$$
−0.192007 + 0.981394i $$0.561500\pi$$
$$602$$ 0 0
$$603$$ 25.6569i 1.04483i
$$604$$ −11.3137 −0.460348
$$605$$ 0 0
$$606$$ 5.45584 0.221629
$$607$$ − 40.2843i − 1.63509i −0.575866 0.817544i $$-0.695335\pi$$
0.575866 0.817544i $$-0.304665\pi$$
$$608$$ 3.41421i 0.138465i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.97056 0.362910
$$612$$ − 14.3848i − 0.581470i
$$613$$ − 23.6569i − 0.955491i −0.878498 0.477746i $$-0.841454\pi$$
0.878498 0.477746i $$-0.158546\pi$$
$$614$$ −29.0711 −1.17321
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10.6863i 0.430214i 0.976590 + 0.215107i $$0.0690100\pi$$
−0.976590 + 0.215107i $$0.930990\pi$$
$$618$$ 5.37258i 0.216117i
$$619$$ 14.9289 0.600044 0.300022 0.953932i $$-0.403006\pi$$
0.300022 + 0.953932i $$0.403006\pi$$
$$620$$ 0 0
$$621$$ −22.6274 −0.908007
$$622$$ − 4.00000i − 0.160385i
$$623$$ 0 0
$$624$$ −0.485281 −0.0194268
$$625$$ 0 0
$$626$$ −22.3848 −0.894676
$$627$$ − 9.65685i − 0.385658i
$$628$$ 10.4853i 0.418408i
$$629$$ 19.7990 0.789437
$$630$$ 0 0
$$631$$ 4.48528 0.178556 0.0892781 0.996007i $$-0.471544\pi$$
0.0892781 + 0.996007i $$0.471544\pi$$
$$632$$ − 1.17157i − 0.0466027i
$$633$$ 15.5980i 0.619964i
$$634$$ −6.48528 −0.257563
$$635$$ 0 0
$$636$$ 6.14214 0.243552
$$637$$ 0 0
$$638$$ − 4.00000i − 0.158362i
$$639$$ 33.1716 1.31225
$$640$$ 0 0
$$641$$ 20.6274 0.814734 0.407367 0.913265i $$-0.366447\pi$$
0.407367 + 0.913265i $$0.366447\pi$$
$$642$$ 0.970563i 0.0383051i
$$643$$ − 47.2132i − 1.86191i −0.365138 0.930953i $$-0.618978\pi$$
0.365138 0.930953i $$-0.381022\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −18.4853 −0.727294
$$647$$ 39.1127i 1.53768i 0.639442 + 0.768839i $$0.279165\pi$$
−0.639442 + 0.768839i $$0.720835\pi$$
$$648$$ − 6.02944i − 0.236859i
$$649$$ 55.1127 2.16336
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 8.14214i − 0.318871i
$$653$$ 15.6569i 0.612700i 0.951919 + 0.306350i $$0.0991078\pi$$
−0.951919 + 0.306350i $$0.900892\pi$$
$$654$$ 8.48528 0.331801
$$655$$ 0 0
$$656$$ −11.0711 −0.432253
$$657$$ 17.4975i 0.682642i
$$658$$ 0 0
$$659$$ 32.8284 1.27881 0.639407 0.768868i $$-0.279180\pi$$
0.639407 + 0.768868i $$0.279180\pi$$
$$660$$ 0 0
$$661$$ −18.2843 −0.711176 −0.355588 0.934643i $$-0.615719\pi$$
−0.355588 + 0.934643i $$0.615719\pi$$
$$662$$ − 5.79899i − 0.225384i
$$663$$ − 2.62742i − 0.102040i
$$664$$ −6.24264 −0.242261
$$665$$ 0 0
$$666$$ 9.71573 0.376477
$$667$$ 5.65685i 0.219034i
$$668$$ 23.7990i 0.920811i
$$669$$ −8.97056 −0.346822
$$670$$ 0 0
$$671$$ 64.2843 2.48167
$$672$$ 0 0
$$673$$ − 48.0000i − 1.85026i −0.379646 0.925132i $$-0.623954\pi$$
0.379646 0.925132i $$-0.376046\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ −12.3137 −0.473604
$$677$$ − 11.4558i − 0.440284i −0.975468 0.220142i $$-0.929348\pi$$
0.975468 0.220142i $$-0.0706521\pi$$
$$678$$ 4.28427i 0.164536i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 5.71573 0.219027
$$682$$ − 13.6569i − 0.522948i
$$683$$ 22.3431i 0.854937i 0.904030 + 0.427468i $$0.140594\pi$$
−0.904030 + 0.427468i $$0.859406\pi$$
$$684$$ −9.07107 −0.346841
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 7.11270i − 0.271366i
$$688$$ − 3.17157i − 0.120915i
$$689$$ −8.68629 −0.330921
$$690$$ 0 0
$$691$$ −10.2426 −0.389648 −0.194824 0.980838i $$-0.562414\pi$$
−0.194824 + 0.980838i $$0.562414\pi$$
$$692$$ − 3.17157i − 0.120565i
$$693$$ 0 0
$$694$$ −8.82843 −0.335123
$$695$$ 0 0
$$696$$ 0.485281 0.0183945
$$697$$ − 59.9411i − 2.27043i
$$698$$ − 14.4853i − 0.548276i
$$699$$ 0.402020 0.0152058
$$700$$ 0 0
$$701$$ −14.4853 −0.547102 −0.273551 0.961858i $$-0.588198\pi$$
−0.273551 + 0.961858i $$0.588198\pi$$
$$702$$ − 2.74517i − 0.103610i
$$703$$ − 12.4853i − 0.470891i
$$704$$ −4.82843 −0.181978
$$705$$ 0 0
$$706$$ 34.3848 1.29409
$$707$$ 0 0
$$708$$ 6.68629i 0.251286i
$$709$$ 17.1127 0.642681 0.321340 0.946964i $$-0.395867\pi$$
0.321340 + 0.946964i $$0.395867\pi$$
$$710$$ 0 0
$$711$$ 3.11270 0.116735
$$712$$ 12.7279i 0.476999i
$$713$$ 19.3137i 0.723304i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ − 5.65685i − 0.211259i
$$718$$ 28.2843i 1.05556i
$$719$$ −9.45584 −0.352643 −0.176322 0.984333i $$-0.556420\pi$$
−0.176322 + 0.984333i $$0.556420\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 7.34315i − 0.273284i
$$723$$ − 6.20101i − 0.230618i
$$724$$ 14.4853 0.538341
$$725$$ 0 0
$$726$$ 7.21320 0.267707
$$727$$ − 20.4853i − 0.759757i −0.925036 0.379879i $$-0.875966\pi$$
0.925036 0.379879i $$-0.124034\pi$$
$$728$$ 0 0
$$729$$ 10.1960 0.377628
$$730$$ 0 0
$$731$$ 17.1716 0.635114
$$732$$ 7.79899i 0.288259i
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ 8.97056 0.331110
$$735$$ 0 0
$$736$$ 6.82843 0.251699
$$737$$ 46.6274i 1.71754i
$$738$$ − 29.4142i − 1.08275i
$$739$$ −8.82843 −0.324759 −0.162379 0.986728i $$-0.551917\pi$$
−0.162379 + 0.986728i $$0.551917\pi$$
$$740$$ 0 0
$$741$$ −1.65685 −0.0608661
$$742$$ 0 0
$$743$$ 12.2010i 0.447612i 0.974634 + 0.223806i $$0.0718481\pi$$
−0.974634 + 0.223806i $$0.928152\pi$$
$$744$$ 1.65685 0.0607432
$$745$$ 0 0
$$746$$ 13.5147 0.494809
$$747$$ − 16.5858i − 0.606842i
$$748$$ − 26.1421i − 0.955851i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.6863 −0.608891 −0.304446 0.952530i $$-0.598471\pi$$
−0.304446 + 0.952530i $$0.598471\pi$$
$$752$$ 10.8284i 0.394872i
$$753$$ 2.00000i 0.0728841i
$$754$$ −0.686292 −0.0249933
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 7.65685i 0.278293i 0.990272 + 0.139147i $$0.0444359\pi$$
−0.990272 + 0.139147i $$0.955564\pi$$
$$758$$ − 17.5147i − 0.636163i
$$759$$ −19.3137 −0.701043
$$760$$ 0 0
$$761$$ 14.3848 0.521448 0.260724 0.965413i $$-0.416039\pi$$
0.260724 + 0.965413i $$0.416039\pi$$
$$762$$ 1.65685i 0.0600215i
$$763$$ 0 0
$$764$$ 18.1421 0.656359
$$765$$ 0 0
$$766$$ −15.5147 −0.560570
$$767$$ − 9.45584i − 0.341431i
$$768$$ − 0.585786i − 0.0211377i
$$769$$ −11.5563 −0.416733 −0.208366 0.978051i $$-0.566815\pi$$
−0.208366 + 0.978051i $$0.566815\pi$$
$$770$$ 0 0
$$771$$ 5.79899 0.208846
$$772$$ 5.65685i 0.203595i
$$773$$ 2.00000i 0.0719350i 0.999353 + 0.0359675i $$0.0114513\pi$$
−0.999353 + 0.0359675i $$0.988549\pi$$
$$774$$ 8.42641 0.302881
$$775$$ 0 0
$$776$$ −16.2426 −0.583077
$$777$$ 0 0
$$778$$ 0.142136i 0.00509581i
$$779$$ −37.7990 −1.35429
$$780$$ 0 0
$$781$$ 60.2843 2.15714
$$782$$ 36.9706i 1.32206i
$$783$$ 2.74517i 0.0981042i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −1.31371 −0.0468584
$$787$$ − 26.7279i − 0.952748i −0.879243 0.476374i $$-0.841951\pi$$
0.879243 0.476374i $$-0.158049\pi$$
$$788$$ 13.7990i 0.491569i
$$789$$ 16.4020 0.583927
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 12.8284i − 0.455838i
$$793$$ − 11.0294i − 0.391667i
$$794$$ 5.79899 0.205798
$$795$$ 0 0
$$796$$ 0.485281 0.0172003
$$797$$ − 2.20101i − 0.0779638i −0.999240 0.0389819i $$-0.987589\pi$$
0.999240 0.0389819i $$-0.0124115\pi$$
$$798$$ 0 0
$$799$$ −58.6274 −2.07409
$$800$$ 0 0
$$801$$ −33.8162 −1.19484
$$802$$ − 6.00000i − 0.211867i
$$803$$ 31.7990i 1.12216i
$$804$$ −5.65685 −0.199502
$$805$$ 0 0
$$806$$ −2.34315 −0.0825338
$$807$$ − 0.887302i − 0.0312345i
$$808$$ − 9.31371i − 0.327655i
$$809$$ −36.9706 −1.29982 −0.649908 0.760013i $$-0.725193\pi$$
−0.649908 + 0.760013i $$0.725193\pi$$
$$810$$ 0 0
$$811$$ −35.4142 −1.24356 −0.621781 0.783191i $$-0.713590\pi$$
−0.621781 + 0.783191i $$0.713590\pi$$
$$812$$ 0 0
$$813$$ 7.02944i 0.246533i
$$814$$ 17.6569 0.618872
$$815$$ 0 0
$$816$$ 3.17157 0.111027
$$817$$ − 10.8284i − 0.378839i
$$818$$ − 13.4142i − 0.469017i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −5.31371 −0.185450 −0.0927249 0.995692i $$-0.529558\pi$$
−0.0927249 + 0.995692i $$0.529558\pi$$
$$822$$ 9.37258i 0.326906i
$$823$$ − 36.2843i − 1.26479i −0.774646 0.632395i $$-0.782072\pi$$
0.774646 0.632395i $$-0.217928\pi$$
$$824$$ 9.17157 0.319507
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 50.6274i 1.76049i 0.474522 + 0.880244i $$0.342621\pi$$
−0.474522 + 0.880244i $$0.657379\pi$$
$$828$$ 18.1421i 0.630483i
$$829$$ −38.9706 −1.35350 −0.676752 0.736211i $$-0.736613\pi$$
−0.676752 + 0.736211i $$0.736613\pi$$
$$830$$ 0 0
$$831$$ −11.7990 −0.409302
$$832$$ 0.828427i 0.0287205i
$$833$$ 0 0
$$834$$ 0.0588745 0.00203866
$$835$$ 0 0
$$836$$ −16.4853 −0.570155
$$837$$ 9.37258i 0.323964i
$$838$$ 32.8701i 1.13548i
$$839$$ −13.8579 −0.478427 −0.239213 0.970967i $$-0.576890\pi$$
−0.239213 + 0.970967i $$0.576890\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ − 5.31371i − 0.183122i
$$843$$ − 4.68629i − 0.161404i
$$844$$ 26.6274 0.916553
$$845$$ 0 0
$$846$$ −28.7696 −0.989118
$$847$$ 0 0
$$848$$ − 10.4853i − 0.360066i
$$849$$ −3.65685 −0.125503
$$850$$ 0 0
$$851$$ −24.9706 −0.855980
$$852$$ 7.31371i 0.250564i
$$853$$ 48.8284i 1.67185i 0.548841 + 0.835927i $$0.315069\pi$$
−0.548841 + 0.835927i $$0.684931\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 1.65685 0.0566301
$$857$$ − 19.0711i − 0.651455i −0.945464 0.325728i $$-0.894391\pi$$
0.945464 0.325728i $$-0.105609\pi$$
$$858$$ − 2.34315i − 0.0799937i
$$859$$ 35.2132 1.20146 0.600729 0.799452i $$-0.294877\pi$$
0.600729 + 0.799452i $$0.294877\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 33.6569i − 1.14636i
$$863$$ 28.9706i 0.986169i 0.869981 + 0.493085i $$0.164131\pi$$
−0.869981 + 0.493085i $$0.835869\pi$$
$$864$$ 3.31371 0.112735
$$865$$ 0 0
$$866$$ 13.4142 0.455834
$$867$$ 7.21320i 0.244973i
$$868$$ 0 0
$$869$$ 5.65685 0.191896
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ − 14.4853i − 0.490534i
$$873$$ − 43.1543i − 1.46055i
$$874$$ 23.3137 0.788598
$$875$$ 0 0
$$876$$ −3.85786 −0.130345
$$877$$ − 26.2843i − 0.887557i −0.896137 0.443778i $$-0.853638\pi$$
0.896137 0.443778i $$-0.146362\pi$$
$$878$$ 8.97056i 0.302742i
$$879$$ 11.5147 0.388382
$$880$$ 0 0
$$881$$ −34.3848 −1.15845 −0.579226 0.815167i $$-0.696645\pi$$
−0.579226 + 0.815167i $$0.696645\pi$$
$$882$$ 0 0
$$883$$ − 30.3431i − 1.02113i −0.859840 0.510564i $$-0.829437\pi$$
0.859840 0.510564i $$-0.170563\pi$$
$$884$$ −4.48528 −0.150856
$$885$$ 0 0
$$886$$ −36.9706 −1.24205
$$887$$ 7.11270i 0.238821i 0.992845 + 0.119411i $$0.0381005\pi$$
−0.992845 + 0.119411i $$0.961900\pi$$
$$888$$ 2.14214i 0.0718854i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 29.1127 0.975312
$$892$$ 15.3137i 0.512741i
$$893$$ 36.9706i 1.23717i
$$894$$ 3.51472 0.117550
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.31371i 0.110642i
$$898$$ − 28.6274i − 0.955309i
$$899$$ 2.34315 0.0781483
$$900$$ 0 0
$$901$$ 56.7696 1.89127
$$902$$ − 53.4558i − 1.77988i
$$903$$ 0 0
$$904$$ 7.31371 0.243250
$$905$$ 0 0
$$906$$ 6.62742 0.220181
$$907$$ 56.2843i 1.86889i 0.356109 + 0.934444i $$0.384103\pi$$
−0.356109 + 0.934444i $$0.615897\pi$$
$$908$$ − 9.75736i − 0.323809i
$$909$$ 24.7452 0.820745
$$910$$ 0 0
$$911$$ −20.2843 −0.672048 −0.336024 0.941853i $$-0.609082\pi$$
−0.336024 + 0.941853i $$0.609082\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ − 30.1421i − 0.997559i
$$914$$ 10.3431 0.342121
$$915$$ 0 0
$$916$$ −12.1421 −0.401187
$$917$$ 0 0
$$918$$ 17.9411i 0.592145i
$$919$$ −32.4853 −1.07159 −0.535795 0.844348i $$-0.679988\pi$$
−0.535795 + 0.844348i $$0.679988\pi$$
$$920$$ 0 0
$$921$$ 17.0294 0.561139
$$922$$ 7.17157i 0.236183i
$$923$$ − 10.3431i − 0.340449i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.9706 −0.557687
$$927$$ 24.3675i 0.800335i
$$928$$ − 0.828427i − 0.0271945i
$$929$$ 25.2132 0.827218 0.413609 0.910455i $$-0.364268\pi$$
0.413609 + 0.910455i $$0.364268\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 0.686292i − 0.0224802i
$$933$$ 2.34315i 0.0767111i
$$934$$ −3.89949 −0.127595
$$935$$ 0 0
$$936$$ −2.20101 −0.0719423
$$937$$ 11.7574i 0.384096i 0.981386 + 0.192048i $$0.0615130\pi$$
−0.981386 + 0.192048i $$0.938487\pi$$
$$938$$ 0 0
$$939$$ 13.1127 0.427917
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ − 6.14214i − 0.200122i
$$943$$ 75.5980i 2.46181i
$$944$$ 11.4142 0.371501
$$945$$ 0 0
$$946$$ 15.3137 0.497892
$$947$$ − 0.828427i − 0.0269203i −0.999909 0.0134601i $$-0.995715\pi$$
0.999909 0.0134601i $$-0.00428462\pi$$
$$948$$ 0.686292i 0.0222897i
$$949$$ 5.45584 0.177104
$$950$$ 0 0
$$951$$ 3.79899 0.123191
$$952$$ 0 0
$$953$$ 11.6569i 0.377603i 0.982015 + 0.188801i $$0.0604602\pi$$
−0.982015 + 0.188801i $$0.939540\pi$$
$$954$$ 27.8579 0.901932
$$955$$ 0 0
$$956$$ −9.65685 −0.312325
$$957$$ 2.34315i 0.0757431i
$$958$$ − 22.8284i − 0.737553i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23.0000 −0.741935
$$962$$ − 3.02944i − 0.0976730i
$$963$$ 4.40202i 0.141853i
$$964$$ −10.5858 −0.340945
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 13.4558i − 0.432711i −0.976315 0.216355i $$-0.930583\pi$$
0.976315 0.216355i $$-0.0694170\pi$$
$$968$$ − 12.3137i − 0.395778i
$$969$$ 10.8284 0.347859
$$970$$ 0 0
$$971$$ −37.3553 −1.19879 −0.599395 0.800453i $$-0.704592\pi$$
−0.599395 + 0.800453i $$0.704592\pi$$
$$972$$ 13.4731i 0.432150i
$$973$$ 0 0
$$974$$ −7.79899 −0.249896
$$975$$ 0 0
$$976$$ 13.3137 0.426161
$$977$$ 35.3137i 1.12979i 0.825164 + 0.564893i $$0.191082\pi$$
−0.825164 + 0.564893i $$0.808918\pi$$
$$978$$ 4.76955i 0.152513i
$$979$$ −61.4558 −1.96414
$$980$$ 0 0
$$981$$ 38.4853 1.22874
$$982$$ − 24.2843i − 0.774942i
$$983$$ 51.7990i 1.65213i 0.563574 + 0.826066i $$0.309426\pi$$
−0.563574 + 0.826066i $$0.690574\pi$$
$$984$$ 6.48528 0.206743
$$985$$ 0 0
$$986$$ 4.48528 0.142840
$$987$$ 0 0
$$988$$ 2.82843i 0.0899843i
$$989$$ −21.6569 −0.688648
$$990$$ 0 0
$$991$$ −28.7696 −0.913895 −0.456947 0.889494i $$-0.651057\pi$$
−0.456947 + 0.889494i $$0.651057\pi$$
$$992$$ − 2.82843i − 0.0898027i
$$993$$ 3.39697i 0.107800i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 3.65685 0.115872
$$997$$ 38.2843i 1.21248i 0.795284 + 0.606238i $$0.207322\pi$$
−0.795284 + 0.606238i $$0.792678\pi$$
$$998$$ − 41.6569i − 1.31862i
$$999$$ −12.1177 −0.383389
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 2450.2.c.w.99.4 4
5.2 odd 4 490.2.a.l.1.2 2
5.3 odd 4 2450.2.a.bs.1.1 2
5.4 even 2 inner 2450.2.c.w.99.1 4
7.6 odd 2 2450.2.c.t.99.3 4
15.2 even 4 4410.2.a.by.1.1 2
20.7 even 4 3920.2.a.ca.1.1 2
35.2 odd 12 490.2.e.j.361.1 4
35.12 even 12 490.2.e.i.361.2 4
35.13 even 4 2450.2.a.bn.1.2 2
35.17 even 12 490.2.e.i.471.2 4
35.27 even 4 490.2.a.m.1.1 yes 2
35.32 odd 12 490.2.e.j.471.1 4
35.34 odd 2 2450.2.c.t.99.2 4
105.62 odd 4 4410.2.a.bt.1.1 2
140.27 odd 4 3920.2.a.bm.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.2 2 5.2 odd 4
490.2.a.m.1.1 yes 2 35.27 even 4
490.2.e.i.361.2 4 35.12 even 12
490.2.e.i.471.2 4 35.17 even 12
490.2.e.j.361.1 4 35.2 odd 12
490.2.e.j.471.1 4 35.32 odd 12
2450.2.a.bn.1.2 2 35.13 even 4
2450.2.a.bs.1.1 2 5.3 odd 4
2450.2.c.t.99.2 4 35.34 odd 2
2450.2.c.t.99.3 4 7.6 odd 2
2450.2.c.w.99.1 4 5.4 even 2 inner
2450.2.c.w.99.4 4 1.1 even 1 trivial
3920.2.a.bm.1.2 2 140.27 odd 4
3920.2.a.ca.1.1 2 20.7 even 4
4410.2.a.bt.1.1 2 105.62 odd 4
4410.2.a.by.1.1 2 15.2 even 4