Properties

 Label 1280.2.d.g.641.1 Level $1280$ Weight $2$ Character 1280.641 Analytic conductor $10.221$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 641.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1280.641 Dual form 1280.2.d.g.641.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -2.00000i q^{13} -2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} +6.00000 q^{23} -1.00000 q^{25} -4.00000i q^{27} -6.00000i q^{29} +4.00000 q^{31} -2.00000i q^{35} +2.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +10.0000i q^{43} +1.00000i q^{45} +6.00000 q^{47} -3.00000 q^{49} +12.0000i q^{51} -6.00000i q^{53} -8.00000 q^{57} -12.0000i q^{59} -2.00000i q^{61} -2.00000 q^{63} -2.00000 q^{65} +2.00000i q^{67} -12.0000i q^{69} -12.0000 q^{71} -2.00000 q^{73} +2.00000i q^{75} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +6.00000i q^{85} -12.0000 q^{87} +6.00000 q^{89} -4.00000i q^{91} -8.00000i q^{93} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} - 4q^{15} - 12q^{17} + 12q^{23} - 2q^{25} + 8q^{31} - 8q^{39} - 12q^{41} + 12q^{47} - 6q^{49} - 16q^{57} - 4q^{63} - 4q^{65} - 24q^{71} - 4q^{73} - 16q^{79} - 22q^{81} - 24q^{87} + 12q^{89} - 8q^{95} + 4q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 0 0
$$5$$ − 1.00000i − 0.447214i
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ − 4.00000i − 0.872872i
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 2.00000i − 0.338062i
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 10.0000i 1.52499i 0.646997 + 0.762493i $$0.276025\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 12.0000i 1.68034i
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ − 2.00000i − 0.256074i −0.991769 0.128037i $$-0.959132\pi$$
0.991769 0.128037i $$-0.0408676\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 0 0
$$69$$ − 12.0000i − 1.44463i
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 2.00000i 0.230940i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 6.00000i 0.650791i
$$86$$ 0 0
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ − 4.00000i − 0.419314i
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000i 0.597022i 0.954406 + 0.298511i $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ − 6.00000i − 0.559503i
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ 20.0000 1.76090
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 4.00000i 0.339276i 0.985506 + 0.169638i $$0.0542598\pi$$
−0.985506 + 0.169638i $$0.945740\pi$$
$$140$$ 0 0
$$141$$ − 12.0000i − 1.01058i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 6.00000i 0.494872i
$$148$$ 0 0
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ − 4.00000i − 0.321288i
$$156$$ 0 0
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ − 10.0000i − 0.783260i −0.920123 0.391630i $$-0.871911\pi$$
0.920123 0.391630i $$-0.128089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ −24.0000 −1.80395
$$178$$ 0 0
$$179$$ − 12.0000i − 0.896922i −0.893802 0.448461i $$-0.851972\pi$$
0.893802 0.448461i $$-0.148028\pi$$
$$180$$ 0 0
$$181$$ − 10.0000i − 0.743294i −0.928374 0.371647i $$-0.878793\pi$$
0.928374 0.371647i $$-0.121207\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 8.00000i − 0.581914i
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 0 0
$$195$$ 4.00000i 0.286446i
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ − 12.0000i − 0.842235i
$$204$$ 0 0
$$205$$ 6.00000i 0.419058i
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ − 16.0000i − 1.10149i −0.834675 0.550743i $$-0.814345\pi$$
0.834675 0.550743i $$-0.185655\pi$$
$$212$$ 0 0
$$213$$ 24.0000i 1.64445i
$$214$$ 0 0
$$215$$ 10.0000 0.681994
$$216$$ 0 0
$$217$$ 8.00000 0.543075
$$218$$ 0 0
$$219$$ 4.00000i 0.270295i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ − 6.00000i − 0.398234i −0.979976 0.199117i $$-0.936193\pi$$
0.979976 0.199117i $$-0.0638074\pi$$
$$228$$ 0 0
$$229$$ 14.0000i 0.925146i 0.886581 + 0.462573i $$0.153074\pi$$
−0.886581 + 0.462573i $$0.846926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ − 6.00000i − 0.391397i
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 3.00000i 0.191663i
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 12.0000 0.751469
$$256$$ 0 0
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 4.00000i 0.248548i
$$260$$ 0 0
$$261$$ 6.00000i 0.371391i
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ 0 0
$$269$$ − 18.0000i − 1.09748i −0.835993 0.548740i $$-0.815108\pi$$
0.835993 0.548740i $$-0.184892\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 0 0
$$285$$ 8.00000i 0.473879i
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ − 4.00000i − 0.234484i
$$292$$ 0 0
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 12.0000i − 0.693978i
$$300$$ 0 0
$$301$$ 20.0000i 1.15278i
$$302$$ 0 0
$$303$$ 12.0000 0.689382
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 0 0
$$309$$ − 28.0000i − 1.59286i
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ 2.00000i 0.112687i
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 0 0
$$325$$ 2.00000i 0.110940i
$$326$$ 0 0
$$327$$ −4.00000 −0.221201
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ − 8.00000i − 0.439720i −0.975531 0.219860i $$-0.929440\pi$$
0.975531 0.219860i $$-0.0705600\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 2.00000 0.109272
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 12.0000i 0.651751i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ 0 0
$$347$$ 30.0000i 1.61048i 0.592946 + 0.805242i $$0.297965\pi$$
−0.592946 + 0.805242i $$0.702035\pi$$
$$348$$ 0 0
$$349$$ 10.0000i 0.535288i 0.963518 + 0.267644i $$0.0862451\pi$$
−0.963518 + 0.267644i $$0.913755\pi$$
$$350$$ 0 0
$$351$$ −8.00000 −0.427008
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 12.0000i 0.636894i
$$356$$ 0 0
$$357$$ 24.0000i 1.27021i
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 22.0000i − 1.15470i
$$364$$ 0 0
$$365$$ 2.00000i 0.104685i
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ − 12.0000i − 0.623009i
$$372$$ 0 0
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ 0 0
$$375$$ 2.00000 0.103280
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 28.0000i 1.43826i 0.694874 + 0.719132i $$0.255460\pi$$
−0.694874 + 0.719132i $$0.744540\pi$$
$$380$$ 0 0
$$381$$ 4.00000i 0.204926i
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 10.0000i − 0.508329i
$$388$$ 0 0
$$389$$ − 6.00000i − 0.304212i −0.988364 0.152106i $$-0.951394\pi$$
0.988364 0.152106i $$-0.0486055\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.00000i 0.402524i
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 0 0
$$405$$ 11.0000i 0.546594i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 0 0
$$411$$ 36.0000i 1.77575i
$$412$$ 0 0
$$413$$ − 24.0000i − 1.18096i
$$414$$ 0 0
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 36.0000i 1.75872i 0.476162 + 0.879358i $$0.342028\pi$$
−0.476162 + 0.879358i $$0.657972\pi$$
$$420$$ 0 0
$$421$$ 26.0000i 1.26716i 0.773676 + 0.633581i $$0.218416\pi$$
−0.773676 + 0.633581i $$0.781584\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ − 4.00000i − 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 12.0000i 0.575356i
$$436$$ 0 0
$$437$$ − 24.0000i − 1.14808i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ − 6.00000i − 0.285069i −0.989790 0.142534i $$-0.954475\pi$$
0.989790 0.142534i $$-0.0455251\pi$$
$$444$$ 0 0
$$445$$ − 6.00000i − 0.284427i
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 40.0000i − 1.87936i
$$454$$ 0 0
$$455$$ −4.00000 −0.187523
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 0 0
$$459$$ 24.0000i 1.12022i
$$460$$ 0 0
$$461$$ − 30.0000i − 1.39724i −0.715493 0.698620i $$-0.753798\pi$$
0.715493 0.698620i $$-0.246202\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ − 30.0000i − 1.38823i −0.719862 0.694117i $$-0.755795\pi$$
0.719862 0.694117i $$-0.244205\pi$$
$$468$$ 0 0
$$469$$ 4.00000i 0.184703i
$$470$$ 0 0
$$471$$ 44.0000 2.02741
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000i 0.183533i
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ − 24.0000i − 1.09204i
$$484$$ 0 0
$$485$$ − 2.00000i − 0.0908153i
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 36.0000i 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ − 36.0000i − 1.60836i
$$502$$ 0 0
$$503$$ −18.0000 −0.802580 −0.401290 0.915951i $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ − 18.0000i − 0.799408i
$$508$$ 0 0
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 0 0
$$513$$ −16.0000 −0.706417
$$514$$ 0 0
$$515$$ − 14.0000i − 0.616914i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 0 0
$$525$$ 4.00000i 0.174574i
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 12.0000i 0.520756i
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 14.0000i − 0.601907i −0.953639 0.300954i $$-0.902695\pi$$
0.953639 0.300954i $$-0.0973049\pi$$
$$542$$ 0 0
$$543$$ −20.0000 −0.858282
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ 26.0000i 1.11168i 0.831289 + 0.555840i $$0.187603\pi$$
−0.831289 + 0.555840i $$0.812397\pi$$
$$548$$ 0 0
$$549$$ 2.00000i 0.0853579i
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ − 4.00000i − 0.169791i
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ 20.0000 0.845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 6.00000i 0.252422i
$$566$$ 0 0
$$567$$ −22.0000 −0.923913
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ − 8.00000i − 0.334790i −0.985890 0.167395i $$-0.946465\pi$$
0.985890 0.167395i $$-0.0535355\pi$$
$$572$$ 0 0
$$573$$ − 24.0000i − 1.00261i
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 0 0
$$579$$ − 52.0000i − 2.16105i
$$580$$ 0 0
$$581$$ 12.0000i 0.497844i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ 6.00000i 0.247647i 0.992304 + 0.123823i $$0.0395156\pi$$
−0.992304 + 0.123823i $$0.960484\pi$$
$$588$$ 0 0
$$589$$ − 16.0000i − 0.659269i
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 12.0000i 0.491952i
$$596$$ 0 0
$$597$$ − 16.0000i − 0.654836i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ − 2.00000i − 0.0814463i
$$604$$ 0 0
$$605$$ − 11.0000i − 0.447214i
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ − 12.0000i − 0.485468i
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ − 20.0000i − 0.803868i −0.915669 0.401934i $$-0.868338\pi$$
0.915669 0.401934i $$-0.131662\pi$$
$$620$$ 0 0
$$621$$ − 24.0000i − 0.963087i
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 12.0000i − 0.478471i
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 0 0
$$633$$ −32.0000 −1.27189
$$634$$ 0 0
$$635$$ 2.00000i 0.0793676i
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ 0 0
$$645$$ − 20.0000i − 0.787499i
$$646$$ 0 0
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ − 16.0000i − 0.627089i
$$652$$ 0 0
$$653$$ − 42.0000i − 1.64359i −0.569785 0.821794i $$-0.692974\pi$$
0.569785 0.821794i $$-0.307026\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 36.0000i 1.40236i 0.712984 + 0.701180i $$0.247343\pi$$
−0.712984 + 0.701180i $$0.752657\pi$$
$$660$$ 0 0
$$661$$ − 22.0000i − 0.855701i −0.903850 0.427850i $$-0.859271\pi$$
0.903850 0.427850i $$-0.140729\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ − 36.0000i − 1.39393i
$$668$$ 0 0
$$669$$ − 20.0000i − 0.773245i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ 0 0
$$675$$ 4.00000i 0.153960i
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 42.0000i 1.60709i 0.595247 + 0.803543i $$0.297054\pi$$
−0.595247 + 0.803543i $$0.702946\pi$$
$$684$$ 0 0
$$685$$ 18.0000i 0.687745i
$$686$$ 0 0
$$687$$ 28.0000 1.06827
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 8.00000i 0.304334i 0.988355 + 0.152167i $$0.0486252\pi$$
−0.988355 + 0.152167i $$0.951375\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ − 12.0000i − 0.453882i
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i 0.824033 + 0.566542i $$0.191719\pi$$
−0.824033 + 0.566542i $$0.808281\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ − 34.0000i − 1.27690i −0.769665 0.638448i $$-0.779577\pi$$
0.769665 0.638448i $$-0.220423\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 48.0000i − 1.79259i
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 0 0
$$723$$ − 28.0000i − 1.04133i
$$724$$ 0 0
$$725$$ 6.00000i 0.222834i
$$726$$ 0 0
$$727$$ −46.0000 −1.70605 −0.853023 0.521874i $$-0.825233\pi$$
−0.853023 + 0.521874i $$0.825233\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ − 60.0000i − 2.21918i
$$732$$ 0 0
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.0000i 0.735712i 0.929883 + 0.367856i $$0.119908\pi$$
−0.929883 + 0.367856i $$0.880092\pi$$
$$740$$ 0 0
$$741$$ 16.0000i 0.587775i
$$742$$ 0 0
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 20.0000i − 0.727875i
$$756$$ 0 0
$$757$$ − 22.0000i − 0.799604i −0.916602 0.399802i $$-0.869079\pi$$
0.916602 0.399802i $$-0.130921\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ − 4.00000i − 0.144810i
$$764$$ 0 0
$$765$$ − 6.00000i − 0.216930i
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 12.0000i 0.432169i
$$772$$ 0 0
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 8.00000 0.286998
$$778$$ 0 0
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ 0 0
$$785$$ 22.0000 0.785214
$$786$$ 0 0
$$787$$ 26.0000i 0.926800i 0.886149 + 0.463400i $$0.153371\pi$$
−0.886149 + 0.463400i $$0.846629\pi$$
$$788$$ 0 0
$$789$$ 36.0000i 1.28163i
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 12.0000i 0.425596i
$$796$$ 0 0
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ − 12.0000i − 0.422944i
$$806$$ 0 0
$$807$$ −36.0000 −1.26726
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 16.0000i 0.561836i 0.959732 + 0.280918i $$0.0906389\pi$$
−0.959732 + 0.280918i $$0.909361\pi$$
$$812$$ 0 0
$$813$$ 40.0000i 1.40286i
$$814$$ 0 0
$$815$$ −10.0000 −0.350285
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 0 0
$$819$$ 4.00000i 0.139771i
$$820$$ 0 0
$$821$$ − 54.0000i − 1.88461i −0.334751 0.942306i $$-0.608652\pi$$
0.334751 0.942306i $$-0.391348\pi$$
$$822$$ 0 0
$$823$$ 38.0000 1.32460 0.662298 0.749240i $$-0.269581\pi$$
0.662298 + 0.749240i $$0.269581\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.0000i 1.04320i 0.853189 + 0.521601i $$0.174665\pi$$
−0.853189 + 0.521601i $$0.825335\pi$$
$$828$$ 0 0
$$829$$ − 2.00000i − 0.0694629i −0.999397 0.0347314i $$-0.988942\pi$$
0.999397 0.0347314i $$-0.0110576\pi$$
$$830$$ 0 0
$$831$$ 52.0000 1.80386
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ − 18.0000i − 0.622916i
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 0 0
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 12.0000i 0.413302i
$$844$$ 0 0
$$845$$ − 9.00000i − 0.309609i
$$846$$ 0 0
$$847$$ 22.0000 0.755929
$$848$$ 0 0
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ 12.0000i 0.411355i
$$852$$ 0 0
$$853$$ 50.0000i 1.71197i 0.517003 + 0.855984i $$0.327048\pi$$
−0.517003 + 0.855984i $$0.672952\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 4.00000i 0.136478i 0.997669 + 0.0682391i $$0.0217381\pi$$
−0.997669 + 0.0682391i $$0.978262\pi$$
$$860$$ 0 0
$$861$$ 24.0000i 0.817918i
$$862$$ 0 0
$$863$$ −6.00000 −0.204242 −0.102121 0.994772i $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ − 38.0000i − 1.29055i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 2.00000i 0.0676123i
$$876$$ 0 0
$$877$$ − 26.0000i − 0.877958i −0.898497 0.438979i $$-0.855340\pi$$
0.898497 0.438979i $$-0.144660\pi$$
$$878$$ 0 0
$$879$$ −60.0000 −2.02375
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ 14.0000i 0.471138i 0.971858 + 0.235569i $$0.0756953\pi$$
−0.971858 + 0.235569i $$0.924305\pi$$
$$884$$ 0 0
$$885$$ 24.0000i 0.806751i
$$886$$ 0 0
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 24.0000i − 0.803129i
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ −24.0000 −0.801337
$$898$$ 0 0
$$899$$ − 24.0000i − 0.800445i
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ 40.0000 1.33112
$$904$$ 0 0
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ 46.0000i 1.52740i 0.645568 + 0.763702i $$0.276621\pi$$
−0.645568 + 0.763702i $$0.723379\pi$$
$$908$$ 0 0
$$909$$ − 6.00000i − 0.199007i
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 4.00000i 0.132236i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 0 0
$$923$$ 24.0000i 0.789970i
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 0 0
$$927$$ −14.0000 −0.459820
$$928$$ 0 0
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ 12.0000i 0.393284i
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ − 44.0000i − 1.43589i
$$940$$ 0 0
$$941$$ 18.0000i 0.586783i 0.955992 + 0.293392i $$0.0947840\pi$$
−0.955992 + 0.293392i $$0.905216\pi$$
$$942$$ 0 0
$$943$$ −36.0000 −1.17232
$$944$$ 0 0
$$945$$ −8.00000 −0.260240
$$946$$ 0 0
$$947$$ 18.0000i 0.584921i 0.956278 + 0.292461i $$0.0944741\pi$$
−0.956278 + 0.292461i $$0.905526\pi$$
$$948$$ 0 0
$$949$$ 4.00000i 0.129845i
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ − 12.0000i − 0.388311i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 6.00000i − 0.193347i
$$964$$ 0 0
$$965$$ − 26.0000i − 0.836970i
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 48.0000 1.54198
$$970$$ 0 0
$$971$$ 24.0000i 0.770197i 0.922876 + 0.385098i $$0.125832\pi$$
−0.922876 + 0.385098i $$0.874168\pi$$
$$972$$ 0 0
$$973$$ 8.00000i 0.256468i
$$974$$ 0 0
$$975$$ 4.00000 0.128103
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ 0 0
$$983$$ −18.0000 −0.574111 −0.287055 0.957914i $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ 0 0
$$989$$ 60.0000i 1.90789i
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 0 0
$$993$$ −16.0000 −0.507745
$$994$$ 0 0
$$995$$ − 8.00000i − 0.253617i
$$996$$ 0 0
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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 1280.2.d.g.641.1 2
4.3 odd 2 1280.2.d.c.641.2 2
8.3 odd 2 1280.2.d.c.641.1 2
8.5 even 2 inner 1280.2.d.g.641.2 2
16.3 odd 4 20.2.a.a.1.1 1
16.5 even 4 320.2.a.a.1.1 1
16.11 odd 4 320.2.a.f.1.1 1
16.13 even 4 80.2.a.b.1.1 1
48.5 odd 4 2880.2.a.f.1.1 1
48.11 even 4 2880.2.a.m.1.1 1
48.29 odd 4 720.2.a.h.1.1 1
48.35 even 4 180.2.a.a.1.1 1
80.3 even 4 100.2.c.a.49.1 2
80.13 odd 4 400.2.c.b.49.2 2
80.19 odd 4 100.2.a.a.1.1 1
80.27 even 4 1600.2.c.d.449.1 2
80.29 even 4 400.2.a.c.1.1 1
80.37 odd 4 1600.2.c.e.449.2 2
80.43 even 4 1600.2.c.d.449.2 2
80.53 odd 4 1600.2.c.e.449.1 2
80.59 odd 4 1600.2.a.c.1.1 1
80.67 even 4 100.2.c.a.49.2 2
80.69 even 4 1600.2.a.w.1.1 1
80.77 odd 4 400.2.c.b.49.1 2
112.3 even 12 980.2.i.c.961.1 2
112.13 odd 4 3920.2.a.h.1.1 1
112.19 even 12 980.2.i.c.361.1 2
112.51 odd 12 980.2.i.i.361.1 2
112.67 odd 12 980.2.i.i.961.1 2
112.83 even 4 980.2.a.h.1.1 1
144.67 odd 12 1620.2.i.h.541.1 2
144.83 even 12 1620.2.i.b.1081.1 2
144.115 odd 12 1620.2.i.h.1081.1 2
144.131 even 12 1620.2.i.b.541.1 2
176.109 odd 4 9680.2.a.ba.1.1 1
176.131 even 4 2420.2.a.a.1.1 1
208.51 odd 4 3380.2.a.c.1.1 1
208.83 even 4 3380.2.f.b.3041.2 2
208.99 even 4 3380.2.f.b.3041.1 2
240.29 odd 4 3600.2.a.be.1.1 1
240.77 even 4 3600.2.f.j.2449.1 2
240.83 odd 4 900.2.d.c.649.1 2
240.173 even 4 3600.2.f.j.2449.2 2
240.179 even 4 900.2.a.b.1.1 1
240.227 odd 4 900.2.d.c.649.2 2
272.67 odd 4 5780.2.a.f.1.1 1
272.115 odd 4 5780.2.c.a.5201.1 2
272.259 odd 4 5780.2.c.a.5201.2 2
304.227 even 4 7220.2.a.f.1.1 1
336.83 odd 4 8820.2.a.g.1.1 1
560.83 odd 4 4900.2.e.f.2549.2 2
560.307 odd 4 4900.2.e.f.2549.1 2
560.419 even 4 4900.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
20.2.a.a.1.1 1 16.3 odd 4
80.2.a.b.1.1 1 16.13 even 4
100.2.a.a.1.1 1 80.19 odd 4
100.2.c.a.49.1 2 80.3 even 4
100.2.c.a.49.2 2 80.67 even 4
180.2.a.a.1.1 1 48.35 even 4
320.2.a.a.1.1 1 16.5 even 4
320.2.a.f.1.1 1 16.11 odd 4
400.2.a.c.1.1 1 80.29 even 4
400.2.c.b.49.1 2 80.77 odd 4
400.2.c.b.49.2 2 80.13 odd 4
720.2.a.h.1.1 1 48.29 odd 4
900.2.a.b.1.1 1 240.179 even 4
900.2.d.c.649.1 2 240.83 odd 4
900.2.d.c.649.2 2 240.227 odd 4
980.2.a.h.1.1 1 112.83 even 4
980.2.i.c.361.1 2 112.19 even 12
980.2.i.c.961.1 2 112.3 even 12
980.2.i.i.361.1 2 112.51 odd 12
980.2.i.i.961.1 2 112.67 odd 12
1280.2.d.c.641.1 2 8.3 odd 2
1280.2.d.c.641.2 2 4.3 odd 2
1280.2.d.g.641.1 2 1.1 even 1 trivial
1280.2.d.g.641.2 2 8.5 even 2 inner
1600.2.a.c.1.1 1 80.59 odd 4
1600.2.a.w.1.1 1 80.69 even 4
1600.2.c.d.449.1 2 80.27 even 4
1600.2.c.d.449.2 2 80.43 even 4
1600.2.c.e.449.1 2 80.53 odd 4
1600.2.c.e.449.2 2 80.37 odd 4
1620.2.i.b.541.1 2 144.131 even 12
1620.2.i.b.1081.1 2 144.83 even 12
1620.2.i.h.541.1 2 144.67 odd 12
1620.2.i.h.1081.1 2 144.115 odd 12
2420.2.a.a.1.1 1 176.131 even 4
2880.2.a.f.1.1 1 48.5 odd 4
2880.2.a.m.1.1 1 48.11 even 4
3380.2.a.c.1.1 1 208.51 odd 4
3380.2.f.b.3041.1 2 208.99 even 4
3380.2.f.b.3041.2 2 208.83 even 4
3600.2.a.be.1.1 1 240.29 odd 4
3600.2.f.j.2449.1 2 240.77 even 4
3600.2.f.j.2449.2 2 240.173 even 4
3920.2.a.h.1.1 1 112.13 odd 4
4900.2.a.e.1.1 1 560.419 even 4
4900.2.e.f.2549.1 2 560.307 odd 4
4900.2.e.f.2549.2 2 560.83 odd 4
5780.2.a.f.1.1 1 272.67 odd 4
5780.2.c.a.5201.1 2 272.115 odd 4
5780.2.c.a.5201.2 2 272.259 odd 4
7220.2.a.f.1.1 1 304.227 even 4
8820.2.a.g.1.1 1 336.83 odd 4
9680.2.a.ba.1.1 1 176.109 odd 4