# Properties

 Label 4400.2.b.k.4049.2 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.k.4049.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} +8.00000 q^{19} +2.00000 q^{21} -3.00000i q^{23} +5.00000i q^{27} -5.00000 q^{31} +1.00000i q^{33} -1.00000i q^{37} -4.00000 q^{39} -10.0000i q^{43} +3.00000 q^{49} -6.00000 q^{51} +6.00000i q^{53} +8.00000i q^{57} +3.00000 q^{59} -4.00000 q^{61} -4.00000i q^{63} +1.00000i q^{67} +3.00000 q^{69} -15.0000 q^{71} +4.00000i q^{73} -2.00000i q^{77} +2.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +9.00000 q^{89} +8.00000 q^{91} -5.00000i q^{93} -7.00000i q^{97} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 2 q^{11} + 16 q^{19} + 4 q^{21} - 10 q^{31} - 8 q^{39} + 6 q^{49} - 12 q^{51} + 6 q^{59} - 8 q^{61} + 6 q^{69} - 30 q^{71} + 4 q^{79} + 2 q^{81} + 18 q^{89} + 16 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 2 * q^11 + 16 * q^19 + 4 * q^21 - 10 * q^31 - 8 * q^39 + 6 * q^49 - 12 * q^51 + 6 * q^59 - 8 * q^61 + 6 * q^69 - 30 * q^71 + 4 * q^79 + 2 * q^81 + 18 * q^89 + 16 * q^91 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ 1.00000i 0.174078i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ − 10.0000i − 1.52499i −0.646997 0.762493i $$-0.723975\pi$$
0.646997 0.762493i $$-0.276025\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 8.00000i 1.05963i
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ − 4.00000i − 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.00000i 0.122169i 0.998133 + 0.0610847i $$0.0194560\pi$$
−0.998133 + 0.0610847i $$0.980544\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ −15.0000 −1.78017 −0.890086 0.455792i $$-0.849356\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.00000i − 0.227921i
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.00000 0.953998 0.476999 0.878904i $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 0 0
$$93$$ − 5.00000i − 0.518476i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.00000i − 0.710742i −0.934725 0.355371i $$-0.884354\pi$$
0.934725 0.355371i $$-0.115646\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 6.00000i − 0.580042i −0.957020 0.290021i $$-0.906338\pi$$
0.957020 0.290021i $$-0.0936623\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ 0 0
$$113$$ 15.0000i 1.41108i 0.708669 + 0.705541i $$0.249296\pi$$
−0.708669 + 0.705541i $$0.750704\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 8.00000i 0.739600i
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 0 0
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ − 16.0000i − 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 9.00000i 0.768922i 0.923141 + 0.384461i $$0.125613\pi$$
−0.923141 + 0.384461i $$0.874387\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.00000i 0.247436i
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 12.0000i 0.970143i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.00000i 0.399043i 0.979893 + 0.199522i $$0.0639388\pi$$
−0.979893 + 0.199522i $$0.936061\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 16.0000 1.22355
$$172$$ 0 0
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.00000i 0.225494i
$$178$$ 0 0
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.00000i 0.438763i
$$188$$ 0 0
$$189$$ 10.0000 0.727393
$$190$$ 0 0
$$191$$ 21.0000 1.51951 0.759753 0.650211i $$-0.225320\pi$$
0.759753 + 0.650211i $$0.225320\pi$$
$$192$$ 0 0
$$193$$ − 20.0000i − 1.43963i −0.694165 0.719816i $$-0.744226\pi$$
0.694165 0.719816i $$-0.255774\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −1.00000 −0.0705346
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ − 15.0000i − 1.02778i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 10.0000i 0.678844i
$$218$$ 0 0
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ 17.0000i 1.13840i 0.822198 + 0.569202i $$0.192748\pi$$
−0.822198 + 0.569202i $$0.807252\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 6.00000i − 0.398234i −0.979976 0.199117i $$-0.936193\pi$$
0.979976 0.199117i $$-0.0638074\pi$$
$$228$$ 0 0
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ 0 0
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.00000i 0.129914i
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 0 0
$$243$$ 16.0000i 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 32.0000i 2.03611i
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 9.00000 0.568075 0.284037 0.958813i $$-0.408326\pi$$
0.284037 + 0.958813i $$0.408326\pi$$
$$252$$ 0 0
$$253$$ − 3.00000i − 0.188608i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 18.0000i 1.10993i 0.831875 + 0.554964i $$0.187268\pi$$
−0.831875 + 0.554964i $$0.812732\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 9.00000i 0.550791i
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 8.00000i 0.484182i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000i 0.290129i
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ 0 0
$$303$$ 18.0000i 1.03407i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 1.00000i 0.0565233i 0.999601 + 0.0282617i $$0.00899717\pi$$
−0.999601 + 0.0282617i $$0.991003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 33.0000i 1.85346i 0.375722 + 0.926732i $$0.377395\pi$$
−0.375722 + 0.926732i $$0.622605\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ 48.0000i 2.67079i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.00000i − 0.110600i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 0 0
$$339$$ −15.0000 −0.814688
$$340$$ 0 0
$$341$$ −5.00000 −0.270765
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ 21.0000i 1.11772i 0.829263 + 0.558859i $$0.188761\pi$$
−0.829263 + 0.558859i $$0.811239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 12.0000i 0.635107i
$$358$$ 0 0
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 1.00000i 0.0524864i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.0000i 0.991792i 0.868382 + 0.495896i $$0.165160\pi$$
−0.868382 + 0.495896i $$0.834840\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 29.0000 1.48963 0.744815 0.667271i $$-0.232538\pi$$
0.744815 + 0.667271i $$0.232538\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ − 27.0000i − 1.37964i −0.723983 0.689818i $$-0.757691\pi$$
0.723983 0.689818i $$-0.242309\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 20.0000i − 1.01666i
$$388$$ 0 0
$$389$$ 27.0000 1.36895 0.684477 0.729034i $$-0.260031\pi$$
0.684477 + 0.729034i $$0.260031\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 6.00000i 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ 0 0
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ − 20.0000i − 0.996271i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1.00000i − 0.0495682i
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 0 0
$$413$$ − 6.00000i − 0.295241i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.0000i 0.685583i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ − 29.0000i − 1.39365i −0.717241 0.696826i $$-0.754595\pi$$
0.717241 0.696826i $$-0.245405\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 6.00000 0.285714
$$442$$ 0 0
$$443$$ − 21.0000i − 0.997740i −0.866677 0.498870i $$-0.833748\pi$$
0.866677 0.498870i $$-0.166252\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 10.0000i 0.469841i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 28.0000i − 1.30978i −0.755722 0.654892i $$-0.772714\pi$$
0.755722 0.654892i $$-0.227286\pi$$
$$458$$ 0 0
$$459$$ −30.0000 −1.40028
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 23.0000i 1.06890i 0.845200 + 0.534450i $$0.179481\pi$$
−0.845200 + 0.534450i $$0.820519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 3.00000i − 0.138823i −0.997588 0.0694117i $$-0.977888\pi$$
0.997588 0.0694117i $$-0.0221122\pi$$
$$468$$ 0 0
$$469$$ 2.00000 0.0923514
$$470$$ 0 0
$$471$$ −5.00000 −0.230388
$$472$$ 0 0
$$473$$ − 10.0000i − 0.459800i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ − 6.00000i − 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 29.0000i − 1.31412i −0.753840 0.657058i $$-0.771801\pi$$
0.753840 0.657058i $$-0.228199\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 30.0000i 1.34568i
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ − 30.0000i − 1.33763i −0.743427 0.668817i $$-0.766801\pi$$
0.743427 0.668817i $$-0.233199\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 3.00000i − 0.133235i
$$508$$ 0 0
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 0 0
$$513$$ 40.0000i 1.76604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −27.0000 −1.18289 −0.591446 0.806345i $$-0.701443\pi$$
−0.591446 + 0.806345i $$0.701443\pi$$
$$522$$ 0 0
$$523$$ 8.00000i 0.349816i 0.984585 + 0.174908i $$0.0559627\pi$$
−0.984585 + 0.174908i $$0.944037\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 30.0000i − 1.30682i
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 9.00000i − 0.388379i
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ 0 0
$$543$$ − 13.0000i − 0.557883i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 4.00000i − 0.170097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ 0 0
$$561$$ −6.00000 −0.253320
$$562$$ 0 0
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ 21.0000i 0.877288i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.0000i 0.707719i 0.935299 + 0.353860i $$0.115131\pi$$
−0.935299 + 0.353860i $$0.884869\pi$$
$$578$$ 0 0
$$579$$ 20.0000 0.831172
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 6.00000i 0.248495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −40.0000 −1.64817
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 36.0000i 1.47834i 0.673517 + 0.739171i $$0.264783\pi$$
−0.673517 + 0.739171i $$0.735217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 2.00000i 0.0814463i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 14.0000i − 0.568242i −0.958788 0.284121i $$-0.908298\pi$$
0.958788 0.284121i $$-0.0917018\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 32.0000i − 1.29247i −0.763139 0.646234i $$-0.776343\pi$$
0.763139 0.646234i $$-0.223657\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 0 0
$$621$$ 15.0000 0.601929
$$622$$ 0 0
$$623$$ − 18.0000i − 0.721155i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 8.00000i 0.319489i
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 43.0000 1.71180 0.855901 0.517139i $$-0.173003\pi$$
0.855901 + 0.517139i $$0.173003\pi$$
$$632$$ 0 0
$$633$$ − 20.0000i − 0.794929i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 12.0000i 0.475457i
$$638$$ 0 0
$$639$$ −30.0000 −1.18678
$$640$$ 0 0
$$641$$ 39.0000 1.54041 0.770204 0.637798i $$-0.220155\pi$$
0.770204 + 0.637798i $$0.220155\pi$$
$$642$$ 0 0
$$643$$ − 13.0000i − 0.512670i −0.966588 0.256335i $$-0.917485\pi$$
0.966588 0.256335i $$-0.0825150\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 3.00000i − 0.117942i −0.998260 0.0589711i $$-0.981218\pi$$
0.998260 0.0589711i $$-0.0187820\pi$$
$$648$$ 0 0
$$649$$ 3.00000 0.117760
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ 0 0
$$653$$ − 3.00000i − 0.117399i −0.998276 0.0586995i $$-0.981305\pi$$
0.998276 0.0586995i $$-0.0186954\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.00000i 0.312110i
$$658$$ 0 0
$$659$$ −42.0000 −1.63609 −0.818044 0.575156i $$-0.804941\pi$$
−0.818044 + 0.575156i $$0.804941\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ 0 0
$$663$$ − 24.0000i − 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −17.0000 −0.657258
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ 6.00000 0.229920
$$682$$ 0 0
$$683$$ 48.0000i 1.83667i 0.395805 + 0.918334i $$0.370466\pi$$
−0.395805 + 0.918334i $$0.629534\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 13.0000i 0.495981i
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 1.00000 0.0380418 0.0190209 0.999819i $$-0.493945\pi$$
0.0190209 + 0.999819i $$0.493945\pi$$
$$692$$ 0 0
$$693$$ − 4.00000i − 0.151947i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 36.0000i − 1.35392i
$$708$$ 0 0
$$709$$ 37.0000 1.38956 0.694782 0.719220i $$-0.255501\pi$$
0.694782 + 0.719220i $$0.255501\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ 15.0000i 0.561754i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000i 0.224074i
$$718$$ 0 0
$$719$$ 45.0000 1.67822 0.839108 0.543964i $$-0.183077\pi$$
0.839108 + 0.543964i $$0.183077\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 8.00000i 0.297523i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 17.0000i − 0.630495i −0.949009 0.315248i $$-0.897912\pi$$
0.949009 0.315248i $$-0.102088\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 60.0000 2.21918
$$732$$ 0 0
$$733$$ 4.00000i 0.147743i 0.997268 + 0.0738717i $$0.0235355\pi$$
−0.997268 + 0.0738717i $$0.976464\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.00000i 0.0368355i
$$738$$ 0 0
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 0 0
$$741$$ −32.0000 −1.17555
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −35.0000 −1.27717 −0.638584 0.769552i $$-0.720480\pi$$
−0.638584 + 0.769552i $$0.720480\pi$$
$$752$$ 0 0
$$753$$ 9.00000i 0.327978i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 22.0000i − 0.799604i −0.916602 0.399802i $$-0.869079\pi$$
0.916602 0.399802i $$-0.130921\pi$$
$$758$$ 0 0
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 0 0
$$763$$ 4.00000i 0.144810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000i 0.433295i
$$768$$ 0 0
$$769$$ −44.0000 −1.58668 −0.793340 0.608778i $$-0.791660\pi$$
−0.793340 + 0.608778i $$0.791660\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 2.00000i − 0.0717496i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −15.0000 −0.536742
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ 0 0
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 30.0000 1.06668
$$792$$ 0 0
$$793$$ − 16.0000i − 0.568177i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 9.00000i 0.318796i 0.987214 + 0.159398i $$0.0509554\pi$$
−0.987214 + 0.159398i $$0.949045\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 4.00000i 0.141157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000i 0.211210i
$$808$$ 0 0
$$809$$ 24.0000 0.843795 0.421898 0.906644i $$-0.361364\pi$$
0.421898 + 0.906644i $$0.361364\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 0 0
$$813$$ − 20.0000i − 0.701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 80.0000i − 2.79885i
$$818$$ 0 0
$$819$$ 16.0000 0.559085
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ − 43.0000i − 1.49889i −0.662069 0.749443i $$-0.730321\pi$$
0.662069 0.749443i $$-0.269679\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ 19.0000 0.659897 0.329949 0.943999i $$-0.392969\pi$$
0.329949 + 0.943999i $$0.392969\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 18.0000i 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 25.0000i − 0.864126i
$$838$$ 0 0
$$839$$ −39.0000 −1.34643 −0.673215 0.739447i $$-0.735087\pi$$
−0.673215 + 0.739447i $$0.735087\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ − 18.0000i − 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ − 38.0000i − 1.30110i −0.759465 0.650548i $$-0.774539\pi$$
0.759465 0.650548i $$-0.225461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24.0000i 0.819824i 0.912125 + 0.409912i $$0.134441\pi$$
−0.912125 + 0.409912i $$0.865559\pi$$
$$858$$ 0 0
$$859$$ −25.0000 −0.852989 −0.426494 0.904490i $$-0.640252\pi$$
−0.426494 + 0.904490i $$0.640252\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 48.0000i 1.63394i 0.576681 + 0.816970i $$0.304348\pi$$
−0.576681 + 0.816970i $$0.695652\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 52.0000i − 1.75592i −0.478738 0.877958i $$-0.658906\pi$$
0.478738 0.877958i $$-0.341094\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 0 0
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 30.0000i − 1.00730i −0.863907 0.503651i $$-0.831990\pi$$
0.863907 0.503651i $$-0.168010\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 12.0000i 0.400668i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 20.0000i − 0.665558i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 0 0
$$909$$ 36.0000 1.19404
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 6.00000i 0.198571i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 12.0000i − 0.396275i
$$918$$ 0 0
$$919$$ −34.0000 −1.12156 −0.560778 0.827966i $$-0.689498\pi$$
−0.560778 + 0.827966i $$0.689498\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ 0 0
$$923$$ − 60.0000i − 1.97492i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 16.0000i 0.525509i
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 24.0000 0.786568
$$932$$ 0 0
$$933$$ − 12.0000i − 0.392862i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 32.0000i 1.04539i 0.852518 + 0.522697i $$0.175074\pi$$
−0.852518 + 0.522697i $$0.824926\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 27.0000i − 0.877382i −0.898638 0.438691i $$-0.855442\pi$$
0.898638 0.438691i $$-0.144558\pi$$
$$948$$ 0 0
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ −33.0000 −1.07010
$$952$$ 0 0
$$953$$ − 42.0000i − 1.36051i −0.732974 0.680257i $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 0 0
$$969$$ −48.0000 −1.54198
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ − 28.0000i − 0.897639i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 45.0000i 1.43968i 0.694141 + 0.719839i $$0.255784\pi$$
−0.694141 + 0.719839i $$0.744216\pi$$
$$978$$ 0 0
$$979$$ 9.00000 0.287641
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 45.0000i 1.43528i 0.696416 + 0.717639i $$0.254777\pi$$
−0.696416 + 0.717639i $$0.745223\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 7.00000i 0.222138i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 0 0
$$999$$ 5.00000 0.158193
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 4400.2.b.k.4049.2 2
4.3 odd 2 1100.2.b.c.749.1 2
5.2 odd 4 4400.2.a.v.1.1 1
5.3 odd 4 176.2.a.a.1.1 1
5.4 even 2 inner 4400.2.b.k.4049.1 2
12.11 even 2 9900.2.c.g.5149.2 2
15.8 even 4 1584.2.a.p.1.1 1
20.3 even 4 44.2.a.a.1.1 1
20.7 even 4 1100.2.a.b.1.1 1
20.19 odd 2 1100.2.b.c.749.2 2
35.13 even 4 8624.2.a.w.1.1 1
40.3 even 4 704.2.a.f.1.1 1
40.13 odd 4 704.2.a.i.1.1 1
55.43 even 4 1936.2.a.c.1.1 1
60.23 odd 4 396.2.a.c.1.1 1
60.47 odd 4 9900.2.a.h.1.1 1
60.59 even 2 9900.2.c.g.5149.1 2
80.3 even 4 2816.2.c.e.1409.2 2
80.13 odd 4 2816.2.c.k.1409.1 2
80.43 even 4 2816.2.c.e.1409.1 2
80.53 odd 4 2816.2.c.k.1409.2 2
120.53 even 4 6336.2.a.i.1.1 1
120.83 odd 4 6336.2.a.j.1.1 1
140.3 odd 12 2156.2.i.c.177.1 2
140.23 even 12 2156.2.i.b.1145.1 2
140.83 odd 4 2156.2.a.a.1.1 1
140.103 odd 12 2156.2.i.c.1145.1 2
140.123 even 12 2156.2.i.b.177.1 2
180.23 odd 12 3564.2.i.a.1189.1 2
180.43 even 12 3564.2.i.j.2377.1 2
180.83 odd 12 3564.2.i.a.2377.1 2
180.103 even 12 3564.2.i.j.1189.1 2
220.3 even 20 484.2.e.a.9.1 4
220.43 odd 4 484.2.a.a.1.1 1
220.63 odd 20 484.2.e.b.9.1 4
220.83 odd 20 484.2.e.b.245.1 4
220.103 even 20 484.2.e.a.269.1 4
220.123 odd 20 484.2.e.b.81.1 4
220.163 even 20 484.2.e.a.81.1 4
220.183 odd 20 484.2.e.b.269.1 4
220.203 even 20 484.2.e.a.245.1 4
260.103 even 4 7436.2.a.d.1.1 1
440.43 odd 4 7744.2.a.m.1.1 1
440.373 even 4 7744.2.a.bc.1.1 1
660.263 even 4 4356.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
44.2.a.a.1.1 1 20.3 even 4
176.2.a.a.1.1 1 5.3 odd 4
396.2.a.c.1.1 1 60.23 odd 4
484.2.a.a.1.1 1 220.43 odd 4
484.2.e.a.9.1 4 220.3 even 20
484.2.e.a.81.1 4 220.163 even 20
484.2.e.a.245.1 4 220.203 even 20
484.2.e.a.269.1 4 220.103 even 20
484.2.e.b.9.1 4 220.63 odd 20
484.2.e.b.81.1 4 220.123 odd 20
484.2.e.b.245.1 4 220.83 odd 20
484.2.e.b.269.1 4 220.183 odd 20
704.2.a.f.1.1 1 40.3 even 4
704.2.a.i.1.1 1 40.13 odd 4
1100.2.a.b.1.1 1 20.7 even 4
1100.2.b.c.749.1 2 4.3 odd 2
1100.2.b.c.749.2 2 20.19 odd 2
1584.2.a.p.1.1 1 15.8 even 4
1936.2.a.c.1.1 1 55.43 even 4
2156.2.a.a.1.1 1 140.83 odd 4
2156.2.i.b.177.1 2 140.123 even 12
2156.2.i.b.1145.1 2 140.23 even 12
2156.2.i.c.177.1 2 140.3 odd 12
2156.2.i.c.1145.1 2 140.103 odd 12
2816.2.c.e.1409.1 2 80.43 even 4
2816.2.c.e.1409.2 2 80.3 even 4
2816.2.c.k.1409.1 2 80.13 odd 4
2816.2.c.k.1409.2 2 80.53 odd 4
3564.2.i.a.1189.1 2 180.23 odd 12
3564.2.i.a.2377.1 2 180.83 odd 12
3564.2.i.j.1189.1 2 180.103 even 12
3564.2.i.j.2377.1 2 180.43 even 12
4356.2.a.j.1.1 1 660.263 even 4
4400.2.a.v.1.1 1 5.2 odd 4
4400.2.b.k.4049.1 2 5.4 even 2 inner
4400.2.b.k.4049.2 2 1.1 even 1 trivial
6336.2.a.i.1.1 1 120.53 even 4
6336.2.a.j.1.1 1 120.83 odd 4
7436.2.a.d.1.1 1 260.103 even 4
7744.2.a.m.1.1 1 440.43 odd 4
7744.2.a.bc.1.1 1 440.373 even 4
8624.2.a.w.1.1 1 35.13 even 4
9900.2.a.h.1.1 1 60.47 odd 4
9900.2.c.g.5149.1 2 60.59 even 2
9900.2.c.g.5149.2 2 12.11 even 2