# Properties

 Label 1200.2.f.a.49.1 Level $1200$ Weight $2$ Character 1200.49 Analytic conductor $9.582$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.2.f.a.49.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +5.00000i q^{13} +6.00000i q^{17} +5.00000 q^{19} -1.00000 q^{21} +6.00000i q^{23} +1.00000i q^{27} +6.00000 q^{29} +1.00000 q^{31} +6.00000i q^{33} -2.00000i q^{37} +5.00000 q^{39} +1.00000i q^{43} +6.00000i q^{47} +6.00000 q^{49} +6.00000 q^{51} -12.0000i q^{53} -5.00000i q^{57} -6.00000 q^{59} -13.0000 q^{61} +1.00000i q^{63} +11.0000i q^{67} +6.00000 q^{69} +2.00000i q^{73} +6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -6.00000i q^{87} +5.00000 q^{91} -1.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 12q^{11} + 10q^{19} - 2q^{21} + 12q^{29} + 2q^{31} + 10q^{39} + 12q^{49} + 12q^{51} - 12q^{59} - 26q^{61} + 12q^{69} + 16q^{79} + 2q^{81} + 10q^{91} + 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ 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
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ 6.00000i 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 5.00000 0.800641
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 5.00000i − 0.662266i
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 0.125988i
$$64$$ 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
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.00000i 0.683763i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 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
$$87$$ − 6.00000i − 0.643268i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ 0 0
$$93$$ − 1.00000i − 0.103695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.00000i 0.710742i 0.934725 + 0.355371i $$0.115646\pi$$
−0.934725 + 0.355371i $$0.884354\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 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
$$117$$ − 5.00000i − 0.462250i
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ − 5.00000i − 0.433555i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ − 30.0000i − 2.50873i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 6.00000i − 0.494872i
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 23.0000i − 1.83560i −0.397043 0.917800i $$-0.629964\pi$$
0.397043 0.917800i $$-0.370036\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ − 5.00000i − 0.391630i −0.980641 0.195815i $$-0.937265\pi$$
0.980641 0.195815i $$-0.0627352\pi$$
$$164$$ 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
$$171$$ −5.00000 −0.382360
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000i 0.450988i
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 13.0000i 0.960988i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 36.0000i − 2.63258i
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 11.0000i 0.791797i 0.918294 + 0.395899i $$0.129567\pi$$
−0.918294 + 0.395899i $$0.870433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ 11.0000 0.775880
$$202$$ 0 0
$$203$$ − 6.00000i − 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ −30.0000 −2.07514
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 1.00000i − 0.0678844i
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −30.0000 −2.01802
$$222$$ 0 0
$$223$$ 25.0000i 1.67412i 0.547108 + 0.837062i $$0.315729\pi$$
−0.547108 + 0.837062i $$0.684271\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 25.0000i 1.59071i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ − 36.0000i − 2.26330i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 12.0000i − 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ − 5.00000i − 0.302614i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.00000i 0.420589i 0.977638 + 0.210295i $$0.0674423\pi$$
−0.977638 + 0.210295i $$0.932558\pi$$
$$278$$ 0 0
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 17.0000i − 1.01055i −0.862960 0.505273i $$-0.831392\pi$$
0.862960 0.505273i $$-0.168608\pi$$
$$284$$ 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
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 6.00000i − 0.348155i
$$298$$ 0 0
$$299$$ −30.0000 −1.73494
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 12.0000i 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 17.0000i 0.970241i 0.874447 + 0.485121i $$0.161224\pi$$
−0.874447 + 0.485121i $$0.838776\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 11.0000i 0.621757i 0.950450 + 0.310878i $$0.100623\pi$$
−0.950450 + 0.310878i $$0.899377\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 24.0000i − 1.34797i −0.738743 0.673987i $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ 0 0
$$319$$ −36.0000 −2.01561
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 30.0000i 1.66924i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 7.00000i − 0.387101i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 17.0000i − 0.926049i −0.886345 0.463025i $$-0.846764\pi$$
0.886345 0.463025i $$-0.153236\pi$$
$$338$$ 0 0
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ − 13.0000i − 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 18.0000i − 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6.00000i − 0.317554i
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ − 25.0000i − 1.31216i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 31.0000i − 1.61819i −0.587680 0.809093i $$-0.699959\pi$$
0.587680 0.809093i $$-0.300041\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ − 1.00000i − 0.0517780i −0.999665 0.0258890i $$-0.991758\pi$$
0.999665 0.0258890i $$-0.00824165\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 30.0000i 1.54508i
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 1.00000i − 0.0508329i
$$388$$ 0 0
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 12.0000i 0.605320i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.0000i 0.652451i 0.945292 + 0.326226i $$0.105777\pi$$
−0.945292 + 0.326226i $$0.894223\pi$$
$$398$$ 0 0
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 0 0
$$403$$ 5.00000i 0.249068i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12.0000i 0.594818i
$$408$$ 0 0
$$409$$ 19.0000 0.939490 0.469745 0.882802i $$-0.344346\pi$$
0.469745 + 0.882802i $$0.344346\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 0 0
$$413$$ 6.00000i 0.295241i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000i 0.195881i
$$418$$ 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
$$423$$ − 6.00000i − 0.291730i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 13.0000i 0.629114i
$$428$$ 0 0
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ 11.0000i 0.528626i 0.964437 + 0.264313i $$0.0851452\pi$$
−0.964437 + 0.264313i $$0.914855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.0000i 1.43509i
$$438$$ 0 0
$$439$$ 23.0000 1.09773 0.548865 0.835911i $$-0.315060\pi$$
0.548865 + 0.835911i $$0.315060\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 18.0000i 0.851371i
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 5.00000i 0.234920i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 36.0000 1.67669 0.838344 0.545142i $$-0.183524\pi$$
0.838344 + 0.545142i $$0.183524\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ 0 0
$$469$$ 11.0000 0.507933
$$470$$ 0 0
$$471$$ −23.0000 −1.05978
$$472$$ 0 0
$$473$$ − 6.00000i − 0.275880i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ − 6.00000i − 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7.00000i − 0.317200i −0.987343 0.158600i $$-0.949302\pi$$
0.987343 0.158600i $$-0.0506981\pi$$
$$488$$ 0 0
$$489$$ −5.00000 −0.226108
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ 36.0000i 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 29.0000 1.29822 0.649109 0.760695i $$-0.275142\pi$$
0.649109 + 0.760695i $$0.275142\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ 5.00000i 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 36.0000i − 1.58328i
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ − 17.0000i − 0.743358i −0.928361 0.371679i $$-0.878782\pi$$
0.928361 0.371679i $$-0.121218\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000i 0.261364i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 0 0
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ −7.00000 −0.300954 −0.150477 0.988614i $$-0.548081\pi$$
−0.150477 + 0.988614i $$0.548081\pi$$
$$542$$ 0 0
$$543$$ − 5.00000i − 0.214571i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 0 0
$$549$$ 13.0000 0.554826
$$550$$ 0 0
$$551$$ 30.0000 1.27804
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ 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
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ −36.0000 −1.51992
$$562$$ 0 0
$$563$$ 30.0000i 1.26435i 0.774826 + 0.632175i $$0.217837\pi$$
−0.774826 + 0.632175i $$0.782163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −5.00000 −0.209243 −0.104622 0.994512i $$-0.533363\pi$$
−0.104622 + 0.994512i $$0.533363\pi$$
$$572$$ 0 0
$$573$$ − 6.00000i − 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 37.0000i 1.54033i 0.637845 + 0.770165i $$0.279826\pi$$
−0.637845 + 0.770165i $$0.720174\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ 72.0000i 2.98194i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ 5.00000 0.206021
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 24.0000i 0.985562i 0.870153 + 0.492781i $$0.164020\pi$$
−0.870153 + 0.492781i $$0.835980\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.00000i 0.286491i
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 0 0
$$603$$ − 11.0000i − 0.447955i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 0 0
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 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
$$619$$ −31.0000 −1.24600 −0.622998 0.782224i $$-0.714085\pi$$
−0.622998 + 0.782224i $$0.714085\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 30.0000i 1.19808i
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 13.0000 0.517522 0.258761 0.965941i $$-0.416686\pi$$
0.258761 + 0.965941i $$0.416686\pi$$
$$632$$ 0 0
$$633$$ 5.00000i 0.198732i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 30.0000i 1.18864i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ −1.00000 −0.0391931
$$652$$ 0 0
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 0 0
$$663$$ 30.0000i 1.16510i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 36.0000i 1.39393i
$$668$$ 0 0
$$669$$ 25.0000 0.966556
$$670$$ 0 0
$$671$$ 78.0000 3.01116
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 0 0
$$679$$ 7.00000 0.268635
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 19.0000i − 0.724895i
$$688$$ 0 0
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ −29.0000 −1.08912 −0.544559 0.838723i $$-0.683303\pi$$
−0.544559 + 0.838723i $$0.683303\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 6.00000i 0.224702i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 0 0
$$723$$ 7.00000i 0.260333i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 23.0000i 0.853023i 0.904482 + 0.426511i $$0.140258\pi$$
−0.904482 + 0.426511i $$0.859742\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −6.00000 −0.221918
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 66.0000i − 2.43114i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 25.0000 0.918398
$$742$$ 0 0
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.0000i 0.690567i 0.938498 + 0.345283i $$0.112217\pi$$
−0.938498 + 0.345283i $$0.887783\pi$$
$$758$$ 0 0
$$759$$ −36.0000 −1.30672
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ − 7.00000i − 0.253417i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 30.0000i − 1.08324i
$$768$$ 0 0
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 0 0
$$773$$ 48.0000i 1.72644i 0.504828 + 0.863220i $$0.331556\pi$$
−0.504828 + 0.863220i $$0.668444\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2.00000i 0.0717496i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 7.00000i − 0.249523i −0.992187 0.124762i $$-0.960183\pi$$
0.992187 0.124762i $$-0.0398166\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ − 65.0000i − 2.30822i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 12.0000i − 0.423471i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000i 0.211210i
$$808$$ 0 0
$$809$$ −36.0000 −1.26569 −0.632846 0.774277i $$-0.718114\pi$$
−0.632846 + 0.774277i $$0.718114\pi$$
$$810$$ 0 0
$$811$$ −5.00000 −0.175574 −0.0877869 0.996139i $$-0.527979\pi$$
−0.0877869 + 0.996139i $$0.527979\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000i 0.174928i
$$818$$ 0 0
$$819$$ −5.00000 −0.174714
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ 13.0000i 0.453152i 0.973994 + 0.226576i $$0.0727531\pi$$
−0.973994 + 0.226576i $$0.927247\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 7.00000 0.242827
$$832$$ 0 0
$$833$$ 36.0000i 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.00000i 0.0345651i
$$838$$ 0 0
$$839$$ 54.0000 1.86429 0.932144 0.362089i $$-0.117936\pi$$
0.932144 + 0.362089i $$0.117936\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ − 6.00000i − 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 25.0000i − 0.859010i
$$848$$ 0 0
$$849$$ −17.0000 −0.583438
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 23.0000i 0.787505i 0.919216 + 0.393753i $$0.128823\pi$$
−0.919216 + 0.393753i $$0.871177\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 12.0000i 0.409912i 0.978771 + 0.204956i $$0.0657052\pi$$
−0.978771 + 0.204956i $$0.934295\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −55.0000 −1.86360
$$872$$ 0 0
$$873$$ − 7.00000i − 0.236914i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 49.0000i 1.65461i 0.561751 + 0.827306i $$0.310128\pi$$
−0.561751 + 0.827306i $$0.689872\pi$$
$$878$$ 0 0
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ − 23.0000i − 0.774012i −0.922077 0.387006i $$-0.873509\pi$$
0.922077 0.387006i $$-0.126491\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 6.00000i − 0.201460i −0.994914 0.100730i $$-0.967882\pi$$
0.994914 0.100730i $$-0.0321179\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 0 0
$$893$$ 30.0000i 1.00391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 30.0000i 1.00167i
$$898$$ 0 0
$$899$$ 6.00000 0.200111
$$900$$ 0 0
$$901$$ 72.0000 2.39867
$$902$$ 0 0
$$903$$ − 1.00000i − 0.0332779i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 0 0
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 0 0
$$913$$ − 36.0000i − 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ 53.0000 1.74831 0.874154 0.485648i $$-0.161416\pi$$
0.874154 + 0.485648i $$0.161416\pi$$
$$920$$ 0 0
$$921$$ 17.0000 0.560169
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4.00000i − 0.131377i
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 30.0000 0.983210
$$932$$ 0 0
$$933$$ − 6.00000i − 0.196431i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 47.0000i − 1.53542i −0.640796 0.767712i $$-0.721395\pi$$
0.640796 0.767712i $$-0.278605\pi$$
$$938$$ 0 0
$$939$$ 11.0000 0.358971
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 42.0000i 1.36482i 0.730971 + 0.682408i $$0.239067\pi$$
−0.730971 + 0.682408i $$0.760933\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ − 24.0000i − 0.777436i −0.921357 0.388718i $$-0.872918\pi$$
0.921357 0.388718i $$-0.127082\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 36.0000i 1.16371i
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 0 0
$$969$$ 30.0000 0.963739
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 0 0
$$973$$ 4.00000i 0.128234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −7.00000 −0.223493
$$982$$ 0 0
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.00000i − 0.190982i
$$988$$ 0 0
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 37.0000 1.17534 0.587672 0.809099i $$-0.300045\pi$$
0.587672 + 0.809099i $$0.300045\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 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
$$999$$ 2.00000 0.0632772
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 1200.2.f.a.49.1 2
3.2 odd 2 3600.2.f.v.2449.1 2
4.3 odd 2 300.2.d.a.49.2 2
5.2 odd 4 1200.2.a.f.1.1 1
5.3 odd 4 1200.2.a.n.1.1 1
5.4 even 2 inner 1200.2.f.a.49.2 2
8.3 odd 2 4800.2.f.b.3649.1 2
8.5 even 2 4800.2.f.bi.3649.2 2
12.11 even 2 900.2.d.a.649.2 2
15.2 even 4 3600.2.a.z.1.1 1
15.8 even 4 3600.2.a.s.1.1 1
15.14 odd 2 3600.2.f.v.2449.2 2
20.3 even 4 300.2.a.b.1.1 1
20.7 even 4 300.2.a.c.1.1 yes 1
20.19 odd 2 300.2.d.a.49.1 2
40.3 even 4 4800.2.a.ce.1.1 1
40.13 odd 4 4800.2.a.p.1.1 1
40.19 odd 2 4800.2.f.b.3649.2 2
40.27 even 4 4800.2.a.o.1.1 1
40.29 even 2 4800.2.f.bi.3649.1 2
40.37 odd 4 4800.2.a.cf.1.1 1
60.23 odd 4 900.2.a.e.1.1 1
60.47 odd 4 900.2.a.c.1.1 1
60.59 even 2 900.2.d.a.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.a.b.1.1 1 20.3 even 4
300.2.a.c.1.1 yes 1 20.7 even 4
300.2.d.a.49.1 2 20.19 odd 2
300.2.d.a.49.2 2 4.3 odd 2
900.2.a.c.1.1 1 60.47 odd 4
900.2.a.e.1.1 1 60.23 odd 4
900.2.d.a.649.1 2 60.59 even 2
900.2.d.a.649.2 2 12.11 even 2
1200.2.a.f.1.1 1 5.2 odd 4
1200.2.a.n.1.1 1 5.3 odd 4
1200.2.f.a.49.1 2 1.1 even 1 trivial
1200.2.f.a.49.2 2 5.4 even 2 inner
3600.2.a.s.1.1 1 15.8 even 4
3600.2.a.z.1.1 1 15.2 even 4
3600.2.f.v.2449.1 2 3.2 odd 2
3600.2.f.v.2449.2 2 15.14 odd 2
4800.2.a.o.1.1 1 40.27 even 4
4800.2.a.p.1.1 1 40.13 odd 4
4800.2.a.ce.1.1 1 40.3 even 4
4800.2.a.cf.1.1 1 40.37 odd 4
4800.2.f.b.3649.1 2 8.3 odd 2
4800.2.f.b.3649.2 2 40.19 odd 2
4800.2.f.bi.3649.1 2 40.29 even 2
4800.2.f.bi.3649.2 2 8.5 even 2