Properties

 Label 3150.2.g.f.2899.1 Level $3150$ Weight $2$ Character 3150.2899 Analytic conductor $25.153$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3150.g (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ 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 2899.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3150.2899 Dual form 3150.2.g.f.2899.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +7.00000 q^{19} +3.00000i q^{22} -2.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +8.00000i q^{37} -7.00000i q^{38} +9.00000 q^{41} -8.00000i q^{43} +3.00000 q^{44} +6.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -12.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -7.00000i q^{67} +3.00000i q^{68} -6.00000 q^{71} -5.00000i q^{73} +8.00000 q^{74} -7.00000 q^{76} -3.00000i q^{77} -14.0000 q^{79} -9.00000i q^{82} -9.00000i q^{83} -8.00000 q^{86} -3.00000i q^{88} -15.0000 q^{89} +2.00000 q^{91} +6.00000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 14 q^{19} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 6 q^{34} + 18 q^{41} + 6 q^{44} - 2 q^{49} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} - 12 q^{71} + 16 q^{74} - 14 q^{76} - 28 q^{79} - 16 q^{86} - 30 q^{89} + 4 q^{91} + 12 q^{94} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000i 0.639602i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ − 7.00000i − 1.13555i
$$39$$ 0 0
$$40$$ 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 $$0.855830\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 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 $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 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 $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ − 5.00000i − 0.585206i −0.956234 0.292603i $$-0.905479\pi$$
0.956234 0.292603i $$-0.0945214\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ − 3.00000i − 0.341882i
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 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
$$87$$ 0 0
$$88$$ − 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
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 20.0000i − 1.97066i −0.170664 0.985329i $$-0.554591\pi$$
0.170664 0.985329i $$-0.445409\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ − 3.00000i − 0.290021i −0.989430 0.145010i $$-0.953678\pi$$
0.989430 0.145010i $$-0.0463216\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 9.00000i − 0.846649i −0.905978 0.423324i $$-0.860863\pi$$
0.905978 0.423324i $$-0.139137\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ − 12.0000i − 1.10469i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 10.0000i 0.905357i
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 7.00000i 0.606977i
$$134$$ −7.00000 −0.604708
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ − 21.0000i − 1.79415i −0.441877 0.897076i $$-0.645687\pi$$
0.441877 0.897076i $$-0.354313\pi$$
$$138$$ 0 0
$$139$$ 7.00000 0.593732 0.296866 0.954919i $$-0.404058\pi$$
0.296866 + 0.954919i $$0.404058\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000i 0.503509i
$$143$$ 6.00000i 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −5.00000 −0.413803
$$147$$ 0 0
$$148$$ − 8.00000i − 0.657596i
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 20.0000i 1.59617i 0.602542 + 0.798087i $$0.294154\pi$$
−0.602542 + 0.798087i $$0.705846\pi$$
$$158$$ 14.0000i 1.11378i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 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
$$171$$ 0 0
$$172$$ 8.00000i 0.609994i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 15.0000i 1.12430i
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ − 6.00000i − 0.437595i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ − 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
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 6.00000i − 0.421117i
$$204$$ 0 0
$$205$$ 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
$$213$$ 0 0
$$214$$ −3.00000 −0.205076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 4.00000i − 0.271538i
$$218$$ 14.0000i 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −9.00000 −0.598671
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ − 3.00000i − 0.194461i
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 14.0000i − 0.890799i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 0 0
$$250$$ 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 $$0.614792\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6.00000i 0.369976i 0.982741 + 0.184988i $$0.0592246\pi$$
−0.982741 + 0.184988i $$0.940775\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 7.00000 0.429198
$$267$$ 0 0
$$268$$ 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 $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ − 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 1.00000i 0.0594438i 0.999558 + 0.0297219i $$0.00946217\pi$$
−0.999558 + 0.0297219i $$0.990538\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 9.00000i 0.531253i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ − 12.0000i − 0.695141i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ − 8.00000i − 0.460348i
$$303$$ 0 0
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.00000i − 0.399511i −0.979846 0.199756i $$-0.935985\pi$$
0.979846 0.199756i $$-0.0640148\pi$$
$$308$$ 3.00000i 0.170941i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 20.0000 1.12867
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 21.0000i − 1.16847i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −5.00000 −0.276924
$$327$$ 0 0
$$328$$ 9.00000i 0.496942i
$$329$$ −6.00000 −0.330791
$$330$$ 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
$$333$$ 0 0
$$334$$ −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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 12.0000 0.649836
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 21.0000i 1.12734i 0.826000 + 0.563670i $$0.190611\pi$$
−0.826000 + 0.563670i $$0.809389\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3.00000i 0.159901i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 15.0000 0.794998
$$357$$ 0 0
$$358$$ − 3.00000i − 0.158555i
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ − 2.00000i − 0.105118i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 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
$$381$$ 0 0
$$382$$ − 6.00000i − 0.306987i
$$383$$ − 30.0000i − 1.53293i −0.642287 0.766464i $$-0.722014\pi$$
0.642287 0.766464i $$-0.277986\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.00000 −0.254493
$$387$$ 0 0
$$388$$ 10.0000i 0.507673i
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ 14.0000i 0.701757i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ − 24.0000i − 1.18964i
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 20.0000i 0.985329i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 10.0000i − 0.483934i
$$428$$ 3.00000i 0.145010i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ − 11.0000i − 0.528626i −0.964437 0.264313i $$-0.914855\pi$$
0.964437 0.264313i $$-0.0851452\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000i 0.285391i
$$443$$ 21.0000i 0.997740i 0.866677 + 0.498870i $$0.166252\pi$$
−0.866677 + 0.498870i $$0.833748\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −15.0000 −0.707894 −0.353947 0.935266i $$-0.615161\pi$$
−0.353947 + 0.935266i $$0.615161\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ 9.00000i 0.423324i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ 26.0000i 1.21490i
$$459$$ 0 0
$$460$$ 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 $$-0.313345\pi$$
$$468$$ 0 0
$$469$$ 7.00000 0.323230
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000i 0.552345i
$$473$$ 24.0000i 1.10352i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ 0 0
$$478$$ 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
$$486$$ 0 0
$$487$$ − 34.0000i − 1.54069i −0.637629 0.770344i $$-0.720085\pi$$
0.637629 0.770344i $$-0.279915\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 18.0000i 0.810679i
$$494$$ −14.0000 −0.629890
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ − 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
$$511$$ 5.00000 0.221187
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 18.0000i − 0.791639i
$$518$$ 8.00000i 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ 7.00000i 0.306089i 0.988219 + 0.153044i $$0.0489077\pi$$
−0.988219 + 0.153044i $$0.951092\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ 12.0000i 0.522728i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 7.00000i − 0.303488i
$$533$$ − 18.0000i − 0.779667i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.00000 0.302354
$$537$$ 0 0
$$538$$ 6.00000i 0.258678i
$$539$$ 3.00000 0.129219
$$540$$ 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
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ 0 0
$$553$$ − 14.0000i − 0.595341i
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −7.00000 −0.296866
$$557$$ − 36.0000i − 1.52537i −0.646771 0.762684i $$-0.723881\pi$$
0.646771 0.762684i $$-0.276119\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 18.0000i − 0.759284i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.00000 0.0420331
$$567$$ 0 0
$$568$$ − 6.00000i − 0.251754i
$$569$$ −27.0000 −1.13190 −0.565949 0.824440i $$-0.691490\pi$$
−0.565949 + 0.824440i $$0.691490\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ 9.00000 0.375653
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 7.00000i − 0.291414i −0.989328 0.145707i $$-0.953454\pi$$
0.989328 0.145707i $$-0.0465456\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ 0 0
$$583$$ 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
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ − 27.0000i − 1.10876i −0.832265 0.554379i $$-0.812956\pi$$
0.832265 0.554379i $$-0.187044\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 44.0000i 1.78590i 0.450151 + 0.892952i $$0.351370\pi$$
−0.450151 + 0.892952i $$0.648630\pi$$
$$608$$ − 7.00000i − 0.283887i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ − 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
$$616$$ 3.00000 0.120873
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 18.0000i − 0.721734i
$$623$$ − 15.0000i − 0.600962i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ − 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
$$633$$ 0 0
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ − 18.0000i − 0.712627i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ 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 $$0.843616\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.00000i 0.195815i
$$653$$ 36.0000i 1.40879i 0.709809 + 0.704394i $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 9.00000 0.351391
$$657$$ 0 0
$$658$$ 6.00000i 0.233904i
$$659$$ 9.00000 0.350590 0.175295 0.984516i $$-0.443912\pi$$
0.175295 + 0.984516i $$0.443912\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 25.0000i 0.971653i
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 12.0000i 0.464294i
$$669$$ 0 0
$$670$$ 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 $$\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 12.0000i − 0.459504i
$$683$$ 3.00000i 0.114792i 0.998351 + 0.0573959i $$0.0182797\pi$$
−0.998351 + 0.0573959i $$0.981720\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ − 8.00000i − 0.304997i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −19.0000 −0.722794 −0.361397 0.932412i $$-0.617700\pi$$
−0.361397 + 0.932412i $$0.617700\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ 21.0000 0.797149
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 27.0000i − 1.02270i
$$698$$ 8.00000i 0.302804i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 56.0000i 2.11208i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 15.0000i − 0.562149i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ 0 0
$$718$$ 6.00000i 0.223918i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 20.0000 0.744839
$$722$$ − 30.0000i − 1.11648i
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 34.0000i − 1.26099i −0.776193 0.630495i $$-0.782852\pi$$
0.776193 0.630495i $$-0.217148\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 40.0000i 1.47743i 0.674016 + 0.738717i $$0.264568\pi$$
−0.674016 + 0.738717i $$0.735432\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0000i 0.773545i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 12.0000i − 0.440534i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ − 9.00000i − 0.329073i
$$749$$ 3.00000 0.109618
$$750$$ 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
$$753$$ 0 0
$$754$$ 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 $$0.447843\pi$$
$$762$$ 0 0
$$763$$ − 14.0000i − 0.506834i
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ −30.0000 −1.08394
$$767$$ − 24.0000i − 0.866590i
$$768$$ 0 0
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.00000i 0.179954i
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 0 0
$$775$$ 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
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ 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
$$801$$ 0 0
$$802$$ − 27.0000i − 0.953403i
$$803$$ 15.0000i 0.529339i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 6.00000i 0.210559i
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 56.0000i − 1.95919i
$$818$$ − 25.0000i − 0.874105i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 0 0
$$823$$ − 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
$$826$$ 12.0000 0.417533
$$827$$ − 9.00000i − 0.312961i −0.987681 0.156480i $$-0.949985\pi$$
0.987681 0.156480i $$-0.0500148\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000i 0.0693375i
$$833$$ 3.00000i 0.103944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 21.0000 0.726300
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ −17.0000 −0.585164
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ − 12.0000i − 0.412082i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ − 15.0000i − 0.512390i −0.966625 0.256195i $$-0.917531\pi$$
0.966625 0.256195i $$-0.0824690\pi$$
$$858$$ 0 0
$$859$$ 31.0000 1.05771 0.528853 0.848713i $$-0.322622\pi$$
0.528853 + 0.848713i $$0.322622\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 36.0000i − 1.22616i
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −11.0000 −0.373795
$$867$$ 0 0
$$868$$ 4.00000i 0.135769i
$$869$$ 42.0000 1.42475
$$870$$ 0 0
$$871$$ −14.0000 −0.474372
$$872$$ − 14.0000i − 0.474100i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ 0 0
$$880$$ 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 $$-0.0321179\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000i 0.468755i
$$893$$ 42.0000i 1.40548i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 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
$$906$$ 0 0
$$907$$ − 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 $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 27.0000i 0.893570i
$$914$$ 17.0000 0.562310
$$915$$ 0 0
$$916$$ 26.0000 0.859064
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 18.0000i 0.592798i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 0 0
$$928$$ 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
$$931$$ −7.00000 −0.229416
$$932$$ − 6.00000i − 0.196537i
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 29.0000i 0.947389i 0.880689 + 0.473694i $$0.157080\pi$$
−0.880689 + 0.473694i $$0.842920\pi$$
$$938$$ − 7.00000i − 0.228558i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 3.00000i 0.0972306i
$$953$$ 57.0000i 1.84641i 0.384307 + 0.923206i $$0.374441\pi$$
−0.384307 + 0.923206i $$0.625559\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ − 18.0000i − 0.581554i
$$959$$ 21.0000 0.678125
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 16.0000i − 0.515861i
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ 0 0
$$973$$ 7.00000i 0.224410i
$$974$$ −34.0000 −1.08943
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 3.00000i 0.0959785i 0.998848 + 0.0479893i $$0.0152813\pi$$
−0.998848 + 0.0479893i $$0.984719\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 12.0000i − 0.382935i
$$983$$ 12.0000i 0.382741i 0.981518 + 0.191370i $$0.0612931\pi$$
−0.981518 + 0.191370i $$0.938707\pi$$
$$984$$ 0 0
$$985$$ 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
$$993$$ 0 0
$$994$$ −6.00000 −0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 62.0000i 1.96356i 0.190022 + 0.981780i $$0.439144\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ − 28.0000i − 0.886325i
$$999$$ 0 0
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 3150.2.g.f.2899.1 2
3.2 odd 2 350.2.c.c.99.2 2
5.2 odd 4 3150.2.a.x.1.1 1
5.3 odd 4 3150.2.a.m.1.1 1
5.4 even 2 inner 3150.2.g.f.2899.2 2
12.11 even 2 2800.2.g.i.449.2 2
15.2 even 4 350.2.a.a.1.1 1
15.8 even 4 350.2.a.e.1.1 yes 1
15.14 odd 2 350.2.c.c.99.1 2
21.20 even 2 2450.2.c.h.99.2 2
60.23 odd 4 2800.2.a.h.1.1 1
60.47 odd 4 2800.2.a.x.1.1 1
60.59 even 2 2800.2.g.i.449.1 2
105.62 odd 4 2450.2.a.m.1.1 1
105.83 odd 4 2450.2.a.x.1.1 1
105.104 even 2 2450.2.c.h.99.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 15.2 even 4
350.2.a.e.1.1 yes 1 15.8 even 4
350.2.c.c.99.1 2 15.14 odd 2
350.2.c.c.99.2 2 3.2 odd 2
2450.2.a.m.1.1 1 105.62 odd 4
2450.2.a.x.1.1 1 105.83 odd 4
2450.2.c.h.99.1 2 105.104 even 2
2450.2.c.h.99.2 2 21.20 even 2
2800.2.a.h.1.1 1 60.23 odd 4
2800.2.a.x.1.1 1 60.47 odd 4
2800.2.g.i.449.1 2 60.59 even 2
2800.2.g.i.449.2 2 12.11 even 2
3150.2.a.m.1.1 1 5.3 odd 4
3150.2.a.x.1.1 1 5.2 odd 4
3150.2.g.f.2899.1 2 1.1 even 1 trivial
3150.2.g.f.2899.2 2 5.4 even 2 inner