# Properties

 Label 5400.2.f.e.649.1 Level $5400$ Weight $2$ Character 5400.649 Analytic conductor $43.119$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5400 = 2^{3} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5400.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$43.1192170915$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5400.649 Dual form 5400.2.f.e.649.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{7} +O(q^{10})$$ $$q-3.00000i q^{7} -4.00000 q^{11} -1.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} -4.00000 q^{31} -9.00000i q^{37} +8.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000i q^{53} +4.00000 q^{59} -5.00000 q^{61} +11.0000i q^{67} -8.00000 q^{71} -1.00000i q^{73} +12.0000i q^{77} +5.00000 q^{79} +8.00000i q^{83} +12.0000 q^{89} -3.00000 q^{91} +5.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q - 8 q^{11} + 2 q^{19} - 8 q^{31} - 4 q^{49} + 8 q^{59} - 10 q^{61} - 16 q^{71} + 10 q^{79} + 24 q^{89} - 6 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$2701$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 9.00000i − 1.47959i −0.672832 0.739795i $$-0.734922\pi$$
0.672832 0.739795i $$-0.265078\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 8.00000i − 1.09888i −0.835532 0.549442i $$-0.814840\pi$$
0.835532 0.549442i $$-0.185160\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −5.00000 −0.640184 −0.320092 0.947386i $$-0.603714\pi$$
−0.320092 + 0.947386i $$0.603714\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.0000i 1.34386i 0.740613 + 0.671932i $$0.234535\pi$$
−0.740613 + 0.671932i $$0.765465\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ − 1.00000i − 0.117041i −0.998286 0.0585206i $$-0.981362\pi$$
0.998286 0.0585206i $$-0.0186383\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 12.0000i 1.36753i
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000i 0.507673i 0.967247 + 0.253837i $$0.0816925\pi$$
−0.967247 + 0.253837i $$0.918307\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 1.00000i − 0.0985329i −0.998786 0.0492665i $$-0.984312\pi$$
0.998786 0.0492665i $$-0.0156884\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ − 3.00000i − 0.260133i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ 15.0000i 1.17489i 0.809264 + 0.587445i $$0.199866\pi$$
−0.809264 + 0.587445i $$0.800134\pi$$
$$164$$ 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
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 21.0000 1.56092 0.780459 0.625207i $$-0.214986\pi$$
0.780459 + 0.625207i $$0.214986\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 16.0000i − 1.17004i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ − 23.0000i − 1.65558i −0.561041 0.827788i $$-0.689599\pi$$
0.561041 0.827788i $$-0.310401\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000i 0.814613i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ − 4.00000i − 0.267860i −0.990991 0.133930i $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 24.0000i 1.59294i 0.604681 + 0.796468i $$0.293301\pi$$
−0.604681 + 0.796468i $$0.706699\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 16.0000i 1.04819i 0.851658 + 0.524097i $$0.175597\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −15.0000 −0.966235 −0.483117 0.875556i $$-0.660496\pi$$
−0.483117 + 0.875556i $$0.660496\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 1.00000i − 0.0636285i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.0000 1.00991 0.504956 0.863145i $$-0.331509\pi$$
0.504956 + 0.863145i $$0.331509\pi$$
$$252$$ 0 0
$$253$$ − 16.0000i − 1.00591i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ 0 0
$$259$$ −27.0000 −1.67770
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ −25.0000 −1.51864 −0.759321 0.650716i $$-0.774469\pi$$
−0.759321 + 0.650716i $$0.774469\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −32.0000 −1.90896 −0.954480 0.298275i $$-0.903589\pi$$
−0.954480 + 0.298275i $$0.903589\pi$$
$$282$$ 0 0
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 12.0000i 0.701047i 0.936554 + 0.350524i $$0.113996\pi$$
−0.936554 + 0.350524i $$0.886004\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ − 3.00000i − 0.169570i −0.996399 0.0847850i $$-0.972980\pi$$
0.996399 0.0847850i $$-0.0270203\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.0000i 1.34797i 0.738743 + 0.673987i $$0.235420\pi$$
−0.738743 + 0.673987i $$0.764580\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.00000i 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 36.0000 1.98474
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 3.00000i − 0.163420i −0.996656 0.0817102i $$-0.973962\pi$$
0.996656 0.0817102i $$-0.0260382\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 0 0
$$349$$ 33.0000 1.76645 0.883225 0.468950i $$-0.155368\pi$$
0.883225 + 0.468950i $$0.155368\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8.00000i 0.425797i 0.977074 + 0.212899i $$0.0682904\pi$$
−0.977074 + 0.212899i $$0.931710\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 23.0000i − 1.20059i −0.799779 0.600295i $$-0.795050\pi$$
0.799779 0.600295i $$-0.204950\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 0 0
$$373$$ 1.00000i 0.0517780i 0.999665 + 0.0258890i $$0.00824165\pi$$
−0.999665 + 0.0258890i $$0.991758\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −11.0000 −0.565032 −0.282516 0.959263i $$-0.591169\pi$$
−0.282516 + 0.959263i $$0.591169\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 36.0000i 1.78445i
$$408$$ 0 0
$$409$$ 39.0000 1.92843 0.964213 0.265129i $$-0.0854146\pi$$
0.964213 + 0.265129i $$0.0854146\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.0000i 0.725901i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 28.0000 1.34871 0.674356 0.738406i $$-0.264421\pi$$
0.674356 + 0.738406i $$0.264421\pi$$
$$432$$ 0 0
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.00000i 0.191346i
$$438$$ 0 0
$$439$$ −36.0000 −1.71819 −0.859093 0.511819i $$-0.828972\pi$$
−0.859093 + 0.511819i $$0.828972\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 8.00000i 0.380091i 0.981775 + 0.190046i $$0.0608636\pi$$
−0.981775 + 0.190046i $$0.939136\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000i 1.77757i 0.458329 + 0.888783i $$0.348448\pi$$
−0.458329 + 0.888783i $$0.651552\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ 19.0000i 0.883005i 0.897260 + 0.441502i $$0.145554\pi$$
−0.897260 + 0.441502i $$0.854446\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 0 0
$$469$$ 33.0000 1.52380
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 32.0000i − 1.47136i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 0 0
$$481$$ −9.00000 −0.410365
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 11.0000i − 0.498458i −0.968445 0.249229i $$-0.919823\pi$$
0.968445 0.249229i $$-0.0801771\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 36.0000i 1.60516i 0.596544 + 0.802580i $$0.296540\pi$$
−0.596544 + 0.802580i $$0.703460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −12.0000 −0.531891 −0.265945 0.963988i $$-0.585684\pi$$
−0.265945 + 0.963988i $$0.585684\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 48.0000i − 2.11104i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ 0 0
$$523$$ 29.0000i 1.26808i 0.773300 + 0.634041i $$0.218605\pi$$
−0.773300 + 0.634041i $$0.781395\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 8.00000 0.344584
$$540$$ 0 0
$$541$$ 9.00000 0.386940 0.193470 0.981106i $$-0.438026\pi$$
0.193470 + 0.981106i $$0.438026\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 7.00000i − 0.299298i −0.988739 0.149649i $$-0.952186\pi$$
0.988739 0.149649i $$-0.0478144\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 15.0000i − 0.637865i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 28.0000i − 1.18640i −0.805056 0.593199i $$-0.797865\pi$$
0.805056 0.593199i $$-0.202135\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 32.0000i − 1.34864i −0.738440 0.674320i $$-0.764437\pi$$
0.738440 0.674320i $$-0.235563\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.0000 1.17382 0.586911 0.809652i $$-0.300344\pi$$
0.586911 + 0.809652i $$0.300344\pi$$
$$570$$ 0 0
$$571$$ −33.0000 −1.38101 −0.690504 0.723329i $$-0.742611\pi$$
−0.690504 + 0.723329i $$0.742611\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 13.0000i − 0.541197i −0.962692 0.270599i $$-0.912778\pi$$
0.962692 0.270599i $$-0.0872216\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 32.0000i 1.32530i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 40.0000i 1.64260i 0.570494 + 0.821302i $$0.306752\pi$$
−0.570494 + 0.821302i $$0.693248\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 5.00000i 0.202944i 0.994838 + 0.101472i $$0.0323552\pi$$
−0.994838 + 0.101472i $$0.967645\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ − 35.0000i − 1.41364i −0.707395 0.706818i $$-0.750130\pi$$
0.707395 0.706818i $$-0.249870\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 36.0000i − 1.44931i −0.689114 0.724653i $$-0.742000\pi$$
0.689114 0.724653i $$-0.258000\pi$$
$$618$$ 0 0
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 36.0000i − 1.44231i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 0 0
$$643$$ 24.0000i 0.946468i 0.880937 + 0.473234i $$0.156913\pi$$
−0.880937 + 0.473234i $$0.843087\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 32.0000i − 1.25805i −0.777385 0.629025i $$-0.783454\pi$$
0.777385 0.629025i $$-0.216546\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.0000i 0.939193i 0.882881 + 0.469596i $$0.155601\pi$$
−0.882881 + 0.469596i $$0.844399\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ − 19.0000i − 0.732396i −0.930537 0.366198i $$-0.880659\pi$$
0.930537 0.366198i $$-0.119341\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 44.0000i 1.69106i 0.533930 + 0.845529i $$0.320715\pi$$
−0.533930 + 0.845529i $$0.679285\pi$$
$$678$$ 0 0
$$679$$ 15.0000 0.575647
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 16.0000i 0.612223i 0.951996 + 0.306111i $$0.0990280\pi$$
−0.951996 + 0.306111i $$0.900972\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ − 9.00000i − 0.339441i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 7.00000 0.262891 0.131445 0.991323i $$-0.458038\pi$$
0.131445 + 0.991323i $$0.458038\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ −3.00000 −0.111726
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.0000i 0.445055i 0.974926 + 0.222528i $$0.0714308\pi$$
−0.974926 + 0.222528i $$0.928569\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −32.0000 −1.18356
$$732$$ 0 0
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 44.0000i − 1.62076i
$$738$$ 0 0
$$739$$ −32.0000 −1.17714 −0.588570 0.808447i $$-0.700309\pi$$
−0.588570 + 0.808447i $$0.700309\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ 3.00000 0.109472 0.0547358 0.998501i $$-0.482568\pi$$
0.0547358 + 0.998501i $$0.482568\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 31.0000i − 1.12671i −0.826214 0.563357i $$-0.809510\pi$$
0.826214 0.563357i $$-0.190490\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4.00000 0.145000 0.0724999 0.997368i $$-0.476902\pi$$
0.0724999 + 0.997368i $$0.476902\pi$$
$$762$$ 0 0
$$763$$ − 42.0000i − 1.52050i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 4.00000i − 0.144432i
$$768$$ 0 0
$$769$$ −47.0000 −1.69486 −0.847432 0.530904i $$-0.821852\pi$$
−0.847432 + 0.530904i $$0.821852\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 36.0000 1.28001
$$792$$ 0 0
$$793$$ 5.00000i 0.177555i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 16.0000i − 0.566749i −0.959009 0.283375i $$-0.908546\pi$$
0.959009 0.283375i $$-0.0914540\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 4.00000i 0.141157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −40.0000 −1.40633 −0.703163 0.711029i $$-0.748229\pi$$
−0.703163 + 0.711029i $$0.748229\pi$$
$$810$$ 0 0
$$811$$ 40.0000 1.40459 0.702295 0.711886i $$-0.252159\pi$$
0.702295 + 0.711886i $$0.252159\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000i 0.279885i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 0 0
$$823$$ 25.0000i 0.871445i 0.900081 + 0.435723i $$0.143507\pi$$
−0.900081 + 0.435723i $$0.856493\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 8.00000i − 0.277184i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 15.0000i − 0.515406i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ 0 0
$$853$$ 1.00000i 0.0342393i 0.999853 + 0.0171197i $$0.00544963\pi$$
−0.999853 + 0.0171197i $$0.994550\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 24.0000i − 0.819824i −0.912125 0.409912i $$-0.865559\pi$$
0.912125 0.409912i $$-0.134441\pi$$
$$858$$ 0 0
$$859$$ 27.0000 0.921228 0.460614 0.887601i $$-0.347629\pi$$
0.460614 + 0.887601i $$0.347629\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 44.0000i − 1.49778i −0.662696 0.748889i $$-0.730588\pi$$
0.662696 0.748889i $$-0.269412\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −20.0000 −0.678454
$$870$$ 0 0
$$871$$ 11.0000 0.372721
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 33.0000i − 1.11433i −0.830402 0.557165i $$-0.811889\pi$$
0.830402 0.557165i $$-0.188111\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 28.0000 0.943344 0.471672 0.881774i $$-0.343651\pi$$
0.471672 + 0.881774i $$0.343651\pi$$
$$882$$ 0 0
$$883$$ 7.00000i 0.235569i 0.993039 + 0.117784i $$0.0375792\pi$$
−0.993039 + 0.117784i $$0.962421\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 56.0000i − 1.88030i −0.340766 0.940148i $$-0.610687\pi$$
0.340766 0.940148i $$-0.389313\pi$$
$$888$$ 0 0
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 12.0000i 0.401565i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 32.0000 1.06607
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 5.00000i − 0.166022i −0.996549 0.0830111i $$-0.973546\pi$$
0.996549 0.0830111i $$-0.0264537\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 0 0
$$913$$ − 32.0000i − 1.05905i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 48.0000i − 1.58510i
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 8.00000i 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −40.0000 −1.31236 −0.656179 0.754606i $$-0.727828\pi$$
−0.656179 + 0.754606i $$0.727828\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 51.0000i 1.66610i 0.553200 + 0.833049i $$0.313407\pi$$
−0.553200 + 0.833049i $$0.686593\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −12.0000 −0.391189 −0.195594 0.980685i $$-0.562664\pi$$
−0.195594 + 0.980685i $$0.562664\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ −1.00000 −0.0324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 36.0000i − 1.16615i −0.812417 0.583077i $$-0.801849\pi$$
0.812417 0.583077i $$-0.198151\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 59.0000i 1.89731i 0.316310 + 0.948656i $$0.397556\pi$$
−0.316310 + 0.948656i $$0.602444\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 27.0000i 0.865580i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 24.0000i 0.767828i 0.923369 + 0.383914i $$0.125424\pi$$
−0.923369 + 0.383914i $$0.874576\pi$$
$$978$$ 0 0
$$979$$ −48.0000 −1.53409
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 12.0000i 0.382741i 0.981518 + 0.191370i $$0.0612931\pi$$
−0.981518 + 0.191370i $$0.938707\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 25.0000 0.794151 0.397076 0.917786i $$-0.370025\pi$$
0.397076 + 0.917786i $$0.370025\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 26.0000i − 0.823428i −0.911313 0.411714i $$-0.864930\pi$$
0.911313 0.411714i $$-0.135070\pi$$
$$998$$ 0 0
$$999$$ 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 5400.2.f.e.649.1 2
3.2 odd 2 5400.2.f.v.649.1 2
5.2 odd 4 5400.2.a.bn.1.1 1
5.3 odd 4 216.2.a.a.1.1 1
5.4 even 2 inner 5400.2.f.e.649.2 2
15.2 even 4 5400.2.a.bp.1.1 1
15.8 even 4 216.2.a.d.1.1 yes 1
15.14 odd 2 5400.2.f.v.649.2 2
20.3 even 4 432.2.a.a.1.1 1
40.3 even 4 1728.2.a.bb.1.1 1
40.13 odd 4 1728.2.a.ba.1.1 1
45.13 odd 12 648.2.i.h.217.1 2
45.23 even 12 648.2.i.a.217.1 2
45.38 even 12 648.2.i.a.433.1 2
45.43 odd 12 648.2.i.h.433.1 2
60.23 odd 4 432.2.a.h.1.1 1
120.53 even 4 1728.2.a.a.1.1 1
120.83 odd 4 1728.2.a.b.1.1 1
180.23 odd 12 1296.2.i.a.865.1 2
180.43 even 12 1296.2.i.q.433.1 2
180.83 odd 12 1296.2.i.a.433.1 2
180.103 even 12 1296.2.i.q.865.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.a.a.1.1 1 5.3 odd 4
216.2.a.d.1.1 yes 1 15.8 even 4
432.2.a.a.1.1 1 20.3 even 4
432.2.a.h.1.1 1 60.23 odd 4
648.2.i.a.217.1 2 45.23 even 12
648.2.i.a.433.1 2 45.38 even 12
648.2.i.h.217.1 2 45.13 odd 12
648.2.i.h.433.1 2 45.43 odd 12
1296.2.i.a.433.1 2 180.83 odd 12
1296.2.i.a.865.1 2 180.23 odd 12
1296.2.i.q.433.1 2 180.43 even 12
1296.2.i.q.865.1 2 180.103 even 12
1728.2.a.a.1.1 1 120.53 even 4
1728.2.a.b.1.1 1 120.83 odd 4
1728.2.a.ba.1.1 1 40.13 odd 4
1728.2.a.bb.1.1 1 40.3 even 4
5400.2.a.bn.1.1 1 5.2 odd 4
5400.2.a.bp.1.1 1 15.2 even 4
5400.2.f.e.649.1 2 1.1 even 1 trivial
5400.2.f.e.649.2 2 5.4 even 2 inner
5400.2.f.v.649.1 2 3.2 odd 2
5400.2.f.v.649.2 2 15.14 odd 2