# Properties

 Label 2450.2.c.h.99.2 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.h.99.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} -7.00000 q^{19} +3.00000i q^{22} +1.00000 q^{24} -2.00000 q^{26} +5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +8.00000i q^{37} -7.00000i q^{38} -2.00000 q^{39} +9.00000 q^{41} -8.00000i q^{43} -3.00000 q^{44} +6.00000i q^{47} +1.00000i q^{48} +3.00000 q^{51} -2.00000i q^{52} +12.0000i q^{53} -5.00000 q^{54} -7.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -3.00000 q^{66} -7.00000i q^{67} +3.00000i q^{68} +6.00000 q^{71} -2.00000i q^{72} +5.00000i q^{73} -8.00000 q^{74} +7.00000 q^{76} -2.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} -15.0000 q^{89} +4.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 6 q^{11} + 2 q^{16} - 14 q^{19} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{36} - 4 q^{39} + 18 q^{41} - 6 q^{44} + 6 q^{51} - 10 q^{54} + 24 q^{59} + 20 q^{61} - 2 q^{64} - 6 q^{66} + 12 q^{71} - 16 q^{74} + 14 q^{76} - 28 q^{79} + 2 q^{81} + 16 q^{86} - 30 q^{89} - 12 q^{94} - 2 q^{96} + 12 q^{99} + O(q^{100})$$

## 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
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 2.00000i 0.471405i
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000i 0.639602i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ − 7.00000i − 1.13555i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ − 2.00000i − 0.277350i
$$53$$ 12.0000i 1.64833i 0.566352 + 0.824163i $$0.308354\pi$$
−0.566352 + 0.824163i $$0.691646\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 7.00000i − 0.927173i
$$58$$ 6.00000i 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ − 7.00000i − 0.855186i −0.903971 0.427593i $$-0.859362\pi$$
0.903971 0.427593i $$-0.140638\pi$$
$$68$$ 3.00000i 0.363803i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ − 2.00000i − 0.235702i
$$73$$ 5.00000i 0.585206i 0.956234 + 0.292603i $$0.0945214\pi$$
−0.956234 + 0.292603i $$0.905479\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 7.00000 0.802955
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 9.00000i 0.993884i
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 6.00000i 0.643268i
$$88$$ − 3.00000i − 0.319801i
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000i 0.414781i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 3.00000i 0.297044i
$$103$$ 20.0000i 1.97066i 0.170664 + 0.985329i $$0.445409\pi$$
−0.170664 + 0.985329i $$0.554591\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ − 5.00000i − 0.481125i
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ 9.00000i 0.846649i 0.905978 + 0.423324i $$0.139137\pi$$
−0.905978 + 0.423324i $$0.860863\pi$$
$$114$$ 7.00000 0.655610
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 4.00000i 0.369800i
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 10.0000i 0.905357i
$$123$$ 9.00000i 0.811503i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ − 3.00000i − 0.261116i
$$133$$ 0 0
$$134$$ 7.00000 0.604708
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ 21.0000i 1.79415i 0.441877 + 0.897076i $$0.354313\pi$$
−0.441877 + 0.897076i $$0.645687\pi$$
$$138$$ 0 0
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 6.00000i 0.503509i
$$143$$ 6.00000i 0.501745i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ −5.00000 −0.413803
$$147$$ 0 0
$$148$$ − 8.00000i − 0.657596i
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 7.00000i 0.567775i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 20.0000i − 1.59617i −0.602542 0.798087i $$-0.705846\pi$$
0.602542 0.798087i $$-0.294154\pi$$
$$158$$ − 14.0000i − 1.11378i
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 5.00000i − 0.391630i −0.980641 0.195815i $$-0.937265\pi$$
0.980641 0.195815i $$-0.0627352\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −14.0000 −1.07061
$$172$$ 8.00000i 0.609994i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 12.0000i 0.901975i
$$178$$ − 15.0000i − 1.12430i
$$179$$ −3.00000 −0.224231 −0.112115 0.993695i $$-0.535763\pi$$
−0.112115 + 0.993695i $$0.535763\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ − 9.00000i − 0.658145i
$$188$$ − 6.00000i − 0.437595i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 5.00000i − 0.359908i −0.983675 0.179954i $$-0.942405\pi$$
0.983675 0.179954i $$-0.0575949\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 7.00000 0.493742
$$202$$ 0 0
$$203$$ 0 0
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ −20.0000 −1.39347
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 17.0000 1.17033 0.585164 0.810915i $$-0.301030\pi$$
0.585164 + 0.810915i $$0.301030\pi$$
$$212$$ − 12.0000i − 0.824163i
$$213$$ 6.00000i 0.411113i
$$214$$ −3.00000 −0.205076
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 0 0
$$218$$ − 14.0000i − 0.948200i
$$219$$ −5.00000 −0.337869
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ − 8.00000i − 0.536925i
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −9.00000 −0.598671
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 7.00000i 0.463586i
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ − 14.0000i − 0.909398i
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 16.0000i 1.02640i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −9.00000 −0.573819
$$247$$ − 14.0000i − 0.890799i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 9.00000 0.570352
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000i 1.87135i 0.352865 + 0.935674i $$0.385208\pi$$
−0.352865 + 0.935674i $$0.614792\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 15.0000i − 0.917985i
$$268$$ 7.00000i 0.427593i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ − 3.00000i − 0.181902i
$$273$$ 0 0
$$274$$ −21.0000 −1.26866
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 7.00000i − 0.419832i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 6.00000i − 0.357295i
$$283$$ − 1.00000i − 0.0594438i −0.999558 0.0297219i $$-0.990538\pi$$
0.999558 0.0297219i $$-0.00946217\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 2.00000i 0.117851i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ − 5.00000i − 0.292603i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 15.0000i 0.870388i
$$298$$ − 12.0000i − 0.695141i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ 0 0
$$309$$ −20.0000 −1.13776
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 20.0000 1.12867
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ − 12.0000i − 0.672927i
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ −3.00000 −0.167444
$$322$$ 0 0
$$323$$ 21.0000i 1.16847i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 5.00000 0.276924
$$327$$ − 14.0000i − 0.774202i
$$328$$ − 9.00000i − 0.496942i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 9.00000i 0.493939i
$$333$$ 16.0000i 0.876795i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 13.0000i − 0.708155i −0.935216 0.354078i $$-0.884795\pi$$
0.935216 0.354078i $$-0.115205\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ −9.00000 −0.488813
$$340$$ 0 0
$$341$$ 12.0000 0.649836
$$342$$ − 14.0000i − 0.757033i
$$343$$ 0 0
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 21.0000i − 1.12734i −0.826000 0.563670i $$-0.809389\pi$$
0.826000 0.563670i $$-0.190611\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ −10.0000 −0.533761
$$352$$ 3.00000i 0.159901i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 15.0000 0.794998
$$357$$ 0 0
$$358$$ − 3.00000i − 0.158555i
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ − 2.00000i − 0.105118i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 4.00000i − 0.207390i
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −17.0000 −0.873231 −0.436616 0.899648i $$-0.643823\pi$$
−0.436616 + 0.899648i $$0.643823\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ − 6.00000i − 0.306987i
$$383$$ − 30.0000i − 1.53293i −0.642287 0.766464i $$-0.722014\pi$$
0.642287 0.766464i $$-0.277986\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 5.00000 0.254493
$$387$$ − 16.0000i − 0.813326i
$$388$$ − 10.0000i − 0.507673i
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 14.0000i 0.701757i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 7.00000i 0.349128i
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000i 1.18964i
$$408$$ − 3.00000i − 0.148522i
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ −21.0000 −1.03585
$$412$$ − 20.0000i − 0.985329i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 7.00000i − 0.342791i
$$418$$ − 21.0000i − 1.02714i
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ 17.0000i 0.827547i
$$423$$ 12.0000i 0.583460i
$$424$$ 12.0000 0.582772
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 0 0
$$428$$ − 3.00000i − 0.145010i
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 5.00000i 0.240563i
$$433$$ 11.0000i 0.528626i 0.964437 + 0.264313i $$0.0851452\pi$$
−0.964437 + 0.264313i $$0.914855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ − 5.00000i − 0.238909i
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000i 0.285391i
$$443$$ − 21.0000i − 0.997740i −0.866677 0.498870i $$-0.833748\pi$$
0.866677 0.498870i $$-0.166252\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ − 12.0000i − 0.567581i
$$448$$ 0 0
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 27.0000 1.27138
$$452$$ − 9.00000i − 0.423324i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −7.00000 −0.327805
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ 26.0000i 1.21490i
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 24.0000i − 1.10352i
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 24.0000i 1.09888i
$$478$$ 12.0000i 0.548867i
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 25.0000i 1.13872i
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ − 34.0000i − 1.54069i −0.637629 0.770344i $$-0.720085\pi$$
0.637629 0.770344i $$-0.279915\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 5.00000 0.226108
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ − 9.00000i − 0.405751i
$$493$$ − 18.0000i − 0.810679i
$$494$$ 14.0000 0.629890
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 9.00000i 0.403300i
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ − 15.0000i − 0.669483i
$$503$$ − 6.00000i − 0.267527i −0.991013 0.133763i $$-0.957294\pi$$
0.991013 0.133763i $$-0.0427062\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ 42.0000 1.86162 0.930809 0.365507i $$-0.119104\pi$$
0.930809 + 0.365507i $$0.119104\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 35.0000i − 1.54529i
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 12.0000i 0.525226i
$$523$$ − 7.00000i − 0.306089i −0.988219 0.153044i $$-0.951092\pi$$
0.988219 0.153044i $$-0.0489077\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ − 12.0000i − 0.522728i
$$528$$ 3.00000i 0.130558i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ 18.0000i 0.779667i
$$534$$ 15.0000 0.649113
$$535$$ 0 0
$$536$$ −7.00000 −0.302354
$$537$$ − 3.00000i − 0.129460i
$$538$$ − 6.00000i − 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ − 2.00000i − 0.0859074i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.0000i 1.49649i 0.663421 + 0.748246i $$0.269104\pi$$
−0.663421 + 0.748246i $$0.730896\pi$$
$$548$$ − 21.0000i − 0.897076i
$$549$$ 20.0000 0.853579
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 7.00000 0.296866
$$557$$ 36.0000i 1.52537i 0.646771 + 0.762684i $$0.276119\pi$$
−0.646771 + 0.762684i $$0.723881\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 9.00000 0.379980
$$562$$ − 18.0000i − 0.759284i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ 1.00000 0.0420331
$$567$$ 0 0
$$568$$ − 6.00000i − 0.251754i
$$569$$ 27.0000 1.13190 0.565949 0.824440i $$-0.308510\pi$$
0.565949 + 0.824440i $$0.308510\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 6.00000i − 0.250873i
$$573$$ − 6.00000i − 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ 7.00000i 0.291414i 0.989328 + 0.145707i $$0.0465456\pi$$
−0.989328 + 0.145707i $$0.953454\pi$$
$$578$$ 8.00000i 0.332756i
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 10.0000i − 0.414513i
$$583$$ 36.0000i 1.49097i
$$584$$ 5.00000 0.206901
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 39.0000i − 1.60970i −0.593477 0.804851i $$-0.702245\pi$$
0.593477 0.804851i $$-0.297755\pi$$
$$588$$ 0 0
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 8.00000i 0.328798i
$$593$$ − 27.0000i − 1.10876i −0.832265 0.554379i $$-0.812956\pi$$
0.832265 0.554379i $$-0.187044\pi$$
$$594$$ −15.0000 −0.615457
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ 14.0000i 0.572982i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 7.00000 0.285536 0.142768 0.989756i $$-0.454400\pi$$
0.142768 + 0.989756i $$0.454400\pi$$
$$602$$ 0 0
$$603$$ − 14.0000i − 0.570124i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 44.0000i − 1.78590i −0.450151 0.892952i $$-0.648630\pi$$
0.450151 0.892952i $$-0.351370\pi$$
$$608$$ − 7.00000i − 0.283887i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 6.00000i 0.242536i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ − 20.0000i − 0.804518i
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000i 0.721734i
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ − 21.0000i − 0.838659i
$$628$$ 20.0000i 0.798087i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 14.0000i 0.556890i
$$633$$ 17.0000i 0.675689i
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 0 0
$$638$$ 18.0000i 0.712627i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ − 3.00000i − 0.118401i
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −21.0000 −0.826234
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.00000i 0.195815i
$$653$$ − 36.0000i − 1.40879i −0.709809 0.704394i $$-0.751219\pi$$
0.709809 0.704394i $$-0.248781\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ 9.00000 0.351391
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ −9.00000 −0.350590 −0.175295 0.984516i $$-0.556088\pi$$
−0.175295 + 0.984516i $$0.556088\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ − 25.0000i − 0.971653i
$$663$$ 6.00000i 0.233021i
$$664$$ −9.00000 −0.349268
$$665$$ 0 0
$$666$$ −16.0000 −0.619987
$$667$$ 0 0
$$668$$ 12.0000i 0.464294i
$$669$$ −14.0000 −0.541271
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ 13.0000 0.500741
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ − 9.00000i − 0.345643i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 12.0000i 0.459504i
$$683$$ − 3.00000i − 0.114792i −0.998351 0.0573959i $$-0.981720\pi$$
0.998351 0.0573959i $$-0.0182797\pi$$
$$684$$ 14.0000 0.535303
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 26.0000i 0.991962i
$$688$$ − 8.00000i − 0.304997i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 19.0000 0.722794 0.361397 0.932412i $$-0.382300\pi$$
0.361397 + 0.932412i $$0.382300\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ 21.0000 0.797149
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ − 27.0000i − 1.02270i
$$698$$ 8.00000i 0.302804i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ − 10.0000i − 0.377426i
$$703$$ − 56.0000i − 2.11208i
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ − 12.0000i − 0.450988i
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 0 0
$$711$$ −28.0000 −1.05008
$$712$$ 15.0000i 0.562149i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3.00000 0.112115
$$717$$ 12.0000i 0.448148i
$$718$$ 6.00000i 0.223918i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 30.0000i 1.11648i
$$723$$ 25.0000i 0.929760i
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 34.0000i 1.26099i 0.776193 + 0.630495i $$0.217148\pi$$
−0.776193 + 0.630495i $$0.782852\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ − 10.0000i − 0.369611i
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 21.0000i − 0.773545i
$$738$$ 18.0000i 0.662589i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 14.0000 0.514303
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ − 18.0000i − 0.658586i
$$748$$ 9.00000i 0.329073i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −46.0000 −1.67856 −0.839282 0.543696i $$-0.817024\pi$$
−0.839282 + 0.543696i $$0.817024\pi$$
$$752$$ 6.00000i 0.218797i
$$753$$ − 15.0000i − 0.546630i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 34.0000i − 1.23575i −0.786276 0.617876i $$-0.787994\pi$$
0.786276 0.617876i $$-0.212006\pi$$
$$758$$ − 17.0000i − 0.617468i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9.00000 0.326250 0.163125 0.986605i $$-0.447843\pi$$
0.163125 + 0.986605i $$0.447843\pi$$
$$762$$ − 2.00000i − 0.0724524i
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 30.0000 1.08394
$$767$$ 24.0000i 0.866590i
$$768$$ 1.00000i 0.0360844i
$$769$$ 23.0000 0.829401 0.414701 0.909958i $$-0.363886\pi$$
0.414701 + 0.909958i $$0.363886\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 5.00000i 0.179954i
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 16.0000 0.575108
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 24.0000i 0.860442i
$$779$$ −63.0000 −2.25721
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ 30.0000i 1.07211i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 6.00000i − 0.213201i
$$793$$ 20.0000i 0.710221i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ − 27.0000i − 0.953403i
$$803$$ 15.0000i 0.529339i
$$804$$ −7.00000 −0.246871
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ − 6.00000i − 0.211210i
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 0 0
$$813$$ − 2.00000i − 0.0701431i
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 3.00000 0.105021
$$817$$ 56.0000i 1.95919i
$$818$$ − 25.0000i − 0.874105i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ − 21.0000i − 0.732459i
$$823$$ − 26.0000i − 0.906303i −0.891434 0.453152i $$-0.850300\pi$$
0.891434 0.453152i $$-0.149700\pi$$
$$824$$ 20.0000 0.696733
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.00000i 0.312961i 0.987681 + 0.156480i $$0.0500148\pi$$
−0.987681 + 0.156480i $$0.949985\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 0 0
$$834$$ 7.00000 0.242390
$$835$$ 0 0
$$836$$ 21.0000 0.726300
$$837$$ 20.0000i 0.691301i
$$838$$ − 3.00000i − 0.103633i
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 20.0000i 0.689246i
$$843$$ − 18.0000i − 0.619953i
$$844$$ −17.0000 −0.585164
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 0 0
$$848$$ 12.0000i 0.412082i
$$849$$ 1.00000 0.0343199
$$850$$ 0 0
$$851$$ 0 0
$$852$$ − 6.00000i − 0.205557i
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ − 15.0000i − 0.512390i −0.966625 0.256195i $$-0.917531\pi$$
0.966625 0.256195i $$-0.0824690\pi$$
$$858$$ − 6.00000i − 0.204837i
$$859$$ −31.0000 −1.05771 −0.528853 0.848713i $$-0.677378\pi$$
−0.528853 + 0.848713i $$0.677378\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 36.0000i − 1.22616i
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −11.0000 −0.373795
$$867$$ 8.00000i 0.271694i
$$868$$ 0 0
$$869$$ −42.0000 −1.42475
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 14.0000i 0.474100i
$$873$$ 20.0000i 0.676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 5.00000 0.168934
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ − 47.0000i − 1.58168i −0.612026 0.790838i $$-0.709645\pi$$
0.612026 0.790838i $$-0.290355\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ 21.0000 0.705509
$$887$$ − 6.00000i − 0.201460i −0.994914 0.100730i $$-0.967882\pi$$
0.994914 0.100730i $$-0.0321179\pi$$
$$888$$ 8.00000i 0.268462i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ − 14.0000i − 0.468755i
$$893$$ − 42.0000i − 1.40548i
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 15.0000i 0.500556i
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 27.0000i 0.899002i
$$903$$ 0 0
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ − 7.00000i − 0.231793i
$$913$$ − 27.0000i − 0.893570i
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 15.0000i 0.495074i
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ − 18.0000i − 0.592798i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 40.0000i 1.31377i
$$928$$ 6.00000i 0.196960i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 18.0000i 0.589294i
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ − 29.0000i − 0.947389i −0.880689 0.473694i $$-0.842920\pi$$
0.880689 0.473694i $$-0.157080\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 20.0000i 0.651635i
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 14.0000i 0.454699i
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ − 57.0000i − 1.84641i −0.384307 0.923206i $$-0.625559\pi$$
0.384307 0.923206i $$-0.374441\pi$$
$$954$$ −24.0000 −0.777029
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 18.0000i 0.581857i
$$958$$ 18.0000i 0.581554i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 16.0000i − 0.515861i
$$963$$ 6.00000i 0.193347i
$$964$$ −25.0000 −0.805196
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 34.0000i − 1.09337i −0.837340 0.546683i $$-0.815890\pi$$
0.837340 0.546683i $$-0.184110\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ −21.0000 −0.674617
$$970$$ 0 0
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ − 16.0000i − 0.513200i
$$973$$ 0 0
$$974$$ 34.0000 1.08943
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ − 3.00000i − 0.0959785i −0.998848 0.0479893i $$-0.984719\pi$$
0.998848 0.0479893i $$-0.0152813\pi$$
$$978$$ 5.00000i 0.159882i
$$979$$ −45.0000 −1.43821
$$980$$ 0 0
$$981$$ −28.0000 −0.893971
$$982$$ − 12.0000i − 0.382935i
$$983$$ 12.0000i 0.382741i 0.981518 + 0.191370i $$0.0612931\pi$$
−0.981518 + 0.191370i $$0.938707\pi$$
$$984$$ 9.00000 0.286910
$$985$$ 0 0
$$986$$ 18.0000 0.573237
$$987$$ 0 0
$$988$$ 14.0000i 0.445399i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ − 25.0000i − 0.793351i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −9.00000 −0.285176
$$997$$ − 62.0000i − 1.96356i −0.190022 0.981780i $$-0.560856\pi$$
0.190022 0.981780i $$-0.439144\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ −40.0000 −1.26554
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.h.99.2 2
5.2 odd 4 2450.2.a.m.1.1 1
5.3 odd 4 2450.2.a.x.1.1 1
5.4 even 2 inner 2450.2.c.h.99.1 2
7.6 odd 2 350.2.c.c.99.2 2
21.20 even 2 3150.2.g.f.2899.1 2
28.27 even 2 2800.2.g.i.449.2 2
35.13 even 4 350.2.a.e.1.1 yes 1
35.27 even 4 350.2.a.a.1.1 1
35.34 odd 2 350.2.c.c.99.1 2
105.62 odd 4 3150.2.a.x.1.1 1
105.83 odd 4 3150.2.a.m.1.1 1
105.104 even 2 3150.2.g.f.2899.2 2
140.27 odd 4 2800.2.a.x.1.1 1
140.83 odd 4 2800.2.a.h.1.1 1
140.139 even 2 2800.2.g.i.449.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 35.27 even 4
350.2.a.e.1.1 yes 1 35.13 even 4
350.2.c.c.99.1 2 35.34 odd 2
350.2.c.c.99.2 2 7.6 odd 2
2450.2.a.m.1.1 1 5.2 odd 4
2450.2.a.x.1.1 1 5.3 odd 4
2450.2.c.h.99.1 2 5.4 even 2 inner
2450.2.c.h.99.2 2 1.1 even 1 trivial
2800.2.a.h.1.1 1 140.83 odd 4
2800.2.a.x.1.1 1 140.27 odd 4
2800.2.g.i.449.1 2 140.139 even 2
2800.2.g.i.449.2 2 28.27 even 2
3150.2.a.m.1.1 1 105.83 odd 4
3150.2.a.x.1.1 1 105.62 odd 4
3150.2.g.f.2899.1 2 21.20 even 2
3150.2.g.f.2899.2 2 105.104 even 2