# Properties

 Label 384.2.d.c.193.1 Level $384$ Weight $2$ Character 384.193 Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.1 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.2.d.c.193.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -2.82843i q^{5} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -2.82843i q^{5} +2.82843 q^{7} -1.00000 q^{9} +4.00000i q^{11} -5.65685i q^{13} -2.82843 q^{15} -2.00000 q^{17} -4.00000i q^{19} -2.82843i q^{21} -5.65685 q^{23} -3.00000 q^{25} +1.00000i q^{27} -2.82843i q^{29} +8.48528 q^{31} +4.00000 q^{33} -8.00000i q^{35} -5.65685 q^{39} +10.0000 q^{41} +12.0000i q^{43} +2.82843i q^{45} -5.65685 q^{47} +1.00000 q^{49} +2.00000i q^{51} +2.82843i q^{53} +11.3137 q^{55} -4.00000 q^{57} -4.00000i q^{59} +11.3137i q^{61} -2.82843 q^{63} -16.0000 q^{65} -4.00000i q^{67} +5.65685i q^{69} +5.65685 q^{71} -2.00000 q^{73} +3.00000i q^{75} +11.3137i q^{77} +8.48528 q^{79} +1.00000 q^{81} +4.00000i q^{83} +5.65685i q^{85} -2.82843 q^{87} +6.00000 q^{89} -16.0000i q^{91} -8.48528i q^{93} -11.3137 q^{95} +14.0000 q^{97} -4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 8q^{17} - 12q^{25} + 16q^{33} + 40q^{41} + 4q^{49} - 16q^{57} - 64q^{65} - 8q^{73} + 4q^{81} + 24q^{89} + 56q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ − 2.82843i − 1.26491i −0.774597 0.632456i $$-0.782047\pi$$
0.774597 0.632456i $$-0.217953\pi$$
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000i 1.20605i 0.797724 + 0.603023i $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ − 5.65685i − 1.56893i −0.620174 0.784465i $$-0.712938\pi$$
0.620174 0.784465i $$-0.287062\pi$$
$$14$$ 0 0
$$15$$ −2.82843 −0.730297
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ − 2.82843i − 0.617213i
$$22$$ 0 0
$$23$$ −5.65685 −1.17954 −0.589768 0.807573i $$-0.700781\pi$$
−0.589768 + 0.807573i $$0.700781\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ − 2.82843i − 0.525226i −0.964901 0.262613i $$-0.915416\pi$$
0.964901 0.262613i $$-0.0845842\pi$$
$$30$$ 0 0
$$31$$ 8.48528 1.52400 0.762001 0.647576i $$-0.224217\pi$$
0.762001 + 0.647576i $$0.224217\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ − 8.00000i − 1.35225i
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −5.65685 −0.905822
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 0 0
$$45$$ 2.82843i 0.421637i
$$46$$ 0 0
$$47$$ −5.65685 −0.825137 −0.412568 0.910927i $$-0.635368\pi$$
−0.412568 + 0.910927i $$0.635368\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000i 0.280056i
$$52$$ 0 0
$$53$$ 2.82843i 0.388514i 0.980951 + 0.194257i $$0.0622296\pi$$
−0.980951 + 0.194257i $$0.937770\pi$$
$$54$$ 0 0
$$55$$ 11.3137 1.52554
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ 11.3137i 1.44857i 0.689500 + 0.724286i $$0.257830\pi$$
−0.689500 + 0.724286i $$0.742170\pi$$
$$62$$ 0 0
$$63$$ −2.82843 −0.356348
$$64$$ 0 0
$$65$$ −16.0000 −1.98456
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 5.65685i 0.681005i
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 3.00000i 0.346410i
$$76$$ 0 0
$$77$$ 11.3137i 1.28932i
$$78$$ 0 0
$$79$$ 8.48528 0.954669 0.477334 0.878722i $$-0.341603\pi$$
0.477334 + 0.878722i $$0.341603\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 5.65685i 0.613572i
$$86$$ 0 0
$$87$$ −2.82843 −0.303239
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ − 16.0000i − 1.67726i
$$92$$ 0 0
$$93$$ − 8.48528i − 0.879883i
$$94$$ 0 0
$$95$$ −11.3137 −1.16076
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ − 4.00000i − 0.402015i
$$100$$ 0 0
$$101$$ 14.1421i 1.40720i 0.710599 + 0.703598i $$0.248424\pi$$
−0.710599 + 0.703598i $$0.751576\pi$$
$$102$$ 0 0
$$103$$ −2.82843 −0.278693 −0.139347 0.990244i $$-0.544500\pi$$
−0.139347 + 0.990244i $$0.544500\pi$$
$$104$$ 0 0
$$105$$ −8.00000 −0.780720
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 5.65685i 0.541828i 0.962604 + 0.270914i $$0.0873260\pi$$
−0.962604 + 0.270914i $$0.912674\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 16.0000i 1.49201i
$$116$$ 0 0
$$117$$ 5.65685i 0.522976i
$$118$$ 0 0
$$119$$ −5.65685 −0.518563
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 10.0000i − 0.901670i
$$124$$ 0 0
$$125$$ − 5.65685i − 0.505964i
$$126$$ 0 0
$$127$$ −8.48528 −0.752947 −0.376473 0.926427i $$-0.622863\pi$$
−0.376473 + 0.926427i $$0.622863\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 12.0000i 1.04844i 0.851581 + 0.524222i $$0.175644\pi$$
−0.851581 + 0.524222i $$0.824356\pi$$
$$132$$ 0 0
$$133$$ − 11.3137i − 0.981023i
$$134$$ 0 0
$$135$$ 2.82843 0.243432
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ − 4.00000i − 0.339276i −0.985506 0.169638i $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 0 0
$$141$$ 5.65685i 0.476393i
$$142$$ 0 0
$$143$$ 22.6274 1.89220
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ − 1.00000i − 0.0824786i
$$148$$ 0 0
$$149$$ − 14.1421i − 1.15857i −0.815125 0.579284i $$-0.803332\pi$$
0.815125 0.579284i $$-0.196668\pi$$
$$150$$ 0 0
$$151$$ −8.48528 −0.690522 −0.345261 0.938507i $$-0.612210\pi$$
−0.345261 + 0.938507i $$0.612210\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ − 24.0000i − 1.92773i
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 2.82843 0.224309
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ − 11.3137i − 0.880771i
$$166$$ 0 0
$$167$$ −11.3137 −0.875481 −0.437741 0.899101i $$-0.644221\pi$$
−0.437741 + 0.899101i $$0.644221\pi$$
$$168$$ 0 0
$$169$$ −19.0000 −1.46154
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ − 19.7990i − 1.50529i −0.658427 0.752645i $$-0.728778\pi$$
0.658427 0.752645i $$-0.271222\pi$$
$$174$$ 0 0
$$175$$ −8.48528 −0.641427
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 12.0000i 0.896922i 0.893802 + 0.448461i $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ − 5.65685i − 0.420471i −0.977651 0.210235i $$-0.932577\pi$$
0.977651 0.210235i $$-0.0674230\pi$$
$$182$$ 0 0
$$183$$ 11.3137 0.836333
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 2.82843i 0.205738i
$$190$$ 0 0
$$191$$ 11.3137 0.818631 0.409316 0.912393i $$-0.365768\pi$$
0.409316 + 0.912393i $$0.365768\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 16.0000i 1.14578i
$$196$$ 0 0
$$197$$ − 8.48528i − 0.604551i −0.953221 0.302276i $$-0.902254\pi$$
0.953221 0.302276i $$-0.0977463\pi$$
$$198$$ 0 0
$$199$$ −2.82843 −0.200502 −0.100251 0.994962i $$-0.531965\pi$$
−0.100251 + 0.994962i $$0.531965\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ − 8.00000i − 0.561490i
$$204$$ 0 0
$$205$$ − 28.2843i − 1.97546i
$$206$$ 0 0
$$207$$ 5.65685 0.393179
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 12.0000i 0.826114i 0.910705 + 0.413057i $$0.135539\pi$$
−0.910705 + 0.413057i $$0.864461\pi$$
$$212$$ 0 0
$$213$$ − 5.65685i − 0.387601i
$$214$$ 0 0
$$215$$ 33.9411 2.31477
$$216$$ 0 0
$$217$$ 24.0000 1.62923
$$218$$ 0 0
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 11.3137i 0.761042i
$$222$$ 0 0
$$223$$ −19.7990 −1.32584 −0.662919 0.748691i $$-0.730683\pi$$
−0.662919 + 0.748691i $$0.730683\pi$$
$$224$$ 0 0
$$225$$ 3.00000 0.200000
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 5.65685i 0.373815i 0.982377 + 0.186908i $$0.0598465\pi$$
−0.982377 + 0.186908i $$0.940153\pi$$
$$230$$ 0 0
$$231$$ 11.3137 0.744387
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 16.0000i 1.04372i
$$236$$ 0 0
$$237$$ − 8.48528i − 0.551178i
$$238$$ 0 0
$$239$$ −22.6274 −1.46365 −0.731823 0.681495i $$-0.761330\pi$$
−0.731823 + 0.681495i $$0.761330\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 2.82843i − 0.180702i
$$246$$ 0 0
$$247$$ −22.6274 −1.43975
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ − 12.0000i − 0.757433i −0.925513 0.378717i $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 0 0
$$253$$ − 22.6274i − 1.42257i
$$254$$ 0 0
$$255$$ 5.65685 0.354246
$$256$$ 0 0
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.82843i 0.175075i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 8.00000 0.491436
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 0 0
$$269$$ 8.48528i 0.517357i 0.965964 + 0.258678i $$0.0832870\pi$$
−0.965964 + 0.258678i $$0.916713\pi$$
$$270$$ 0 0
$$271$$ −19.7990 −1.20270 −0.601351 0.798985i $$-0.705371\pi$$
−0.601351 + 0.798985i $$0.705371\pi$$
$$272$$ 0 0
$$273$$ −16.0000 −0.968364
$$274$$ 0 0
$$275$$ − 12.0000i − 0.723627i
$$276$$ 0 0
$$277$$ 16.9706i 1.01966i 0.860274 + 0.509831i $$0.170292\pi$$
−0.860274 + 0.509831i $$0.829708\pi$$
$$278$$ 0 0
$$279$$ −8.48528 −0.508001
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 0 0
$$285$$ 11.3137i 0.670166i
$$286$$ 0 0
$$287$$ 28.2843 1.66957
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 14.0000i − 0.820695i
$$292$$ 0 0
$$293$$ − 19.7990i − 1.15667i −0.815800 0.578335i $$-0.803703\pi$$
0.815800 0.578335i $$-0.196297\pi$$
$$294$$ 0 0
$$295$$ −11.3137 −0.658710
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 32.0000i 1.85061i
$$300$$ 0 0
$$301$$ 33.9411i 1.95633i
$$302$$ 0 0
$$303$$ 14.1421 0.812444
$$304$$ 0 0
$$305$$ 32.0000 1.83231
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ 2.82843i 0.160904i
$$310$$ 0 0
$$311$$ −11.3137 −0.641542 −0.320771 0.947157i $$-0.603942\pi$$
−0.320771 + 0.947157i $$0.603942\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ 8.00000i 0.450749i
$$316$$ 0 0
$$317$$ 8.48528i 0.476581i 0.971194 + 0.238290i $$0.0765870\pi$$
−0.971194 + 0.238290i $$0.923413\pi$$
$$318$$ 0 0
$$319$$ 11.3137 0.633446
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 16.9706i 0.941357i
$$326$$ 0 0
$$327$$ 5.65685 0.312825
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −11.3137 −0.618134
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ − 2.00000i − 0.108625i
$$340$$ 0 0
$$341$$ 33.9411i 1.83801i
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 36.0000i 1.93258i 0.257454 + 0.966291i $$0.417117\pi$$
−0.257454 + 0.966291i $$0.582883\pi$$
$$348$$ 0 0
$$349$$ 11.3137i 0.605609i 0.953053 + 0.302804i $$0.0979229\pi$$
−0.953053 + 0.302804i $$0.902077\pi$$
$$350$$ 0 0
$$351$$ 5.65685 0.301941
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ − 16.0000i − 0.849192i
$$356$$ 0 0
$$357$$ 5.65685i 0.299392i
$$358$$ 0 0
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 5.65685i 0.296093i
$$366$$ 0 0
$$367$$ 8.48528 0.442928 0.221464 0.975169i $$-0.428916\pi$$
0.221464 + 0.975169i $$0.428916\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 8.00000i 0.415339i
$$372$$ 0 0
$$373$$ − 33.9411i − 1.75740i −0.477370 0.878702i $$-0.658410\pi$$
0.477370 0.878702i $$-0.341590\pi$$
$$374$$ 0 0
$$375$$ −5.65685 −0.292119
$$376$$ 0 0
$$377$$ −16.0000 −0.824042
$$378$$ 0 0
$$379$$ 28.0000i 1.43826i 0.694874 + 0.719132i $$0.255460\pi$$
−0.694874 + 0.719132i $$0.744540\pi$$
$$380$$ 0 0
$$381$$ 8.48528i 0.434714i
$$382$$ 0 0
$$383$$ −11.3137 −0.578103 −0.289052 0.957313i $$-0.593340\pi$$
−0.289052 + 0.957313i $$0.593340\pi$$
$$384$$ 0 0
$$385$$ 32.0000 1.63087
$$386$$ 0 0
$$387$$ − 12.0000i − 0.609994i
$$388$$ 0 0
$$389$$ 31.1127i 1.57748i 0.614729 + 0.788738i $$0.289265\pi$$
−0.614729 + 0.788738i $$0.710735\pi$$
$$390$$ 0 0
$$391$$ 11.3137 0.572159
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ − 24.0000i − 1.20757i
$$396$$ 0 0
$$397$$ 33.9411i 1.70346i 0.523984 + 0.851728i $$0.324445\pi$$
−0.523984 + 0.851728i $$0.675555\pi$$
$$398$$ 0 0
$$399$$ −11.3137 −0.566394
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ − 48.0000i − 2.39105i
$$404$$ 0 0
$$405$$ − 2.82843i − 0.140546i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ − 10.0000i − 0.493264i
$$412$$ 0 0
$$413$$ − 11.3137i − 0.556711i
$$414$$ 0 0
$$415$$ 11.3137 0.555368
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ − 28.0000i − 1.36789i −0.729534 0.683945i $$-0.760263\pi$$
0.729534 0.683945i $$-0.239737\pi$$
$$420$$ 0 0
$$421$$ 16.9706i 0.827095i 0.910483 + 0.413547i $$0.135710\pi$$
−0.910483 + 0.413547i $$0.864290\pi$$
$$422$$ 0 0
$$423$$ 5.65685 0.275046
$$424$$ 0 0
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 32.0000i 1.54859i
$$428$$ 0 0
$$429$$ − 22.6274i − 1.09246i
$$430$$ 0 0
$$431$$ 28.2843 1.36241 0.681203 0.732095i $$-0.261457\pi$$
0.681203 + 0.732095i $$0.261457\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 8.00000i 0.383571i
$$436$$ 0 0
$$437$$ 22.6274i 1.08242i
$$438$$ 0 0
$$439$$ 31.1127 1.48493 0.742464 0.669886i $$-0.233657\pi$$
0.742464 + 0.669886i $$0.233657\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 0 0
$$445$$ − 16.9706i − 0.804482i
$$446$$ 0 0
$$447$$ −14.1421 −0.668900
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 40.0000i 1.88353i
$$452$$ 0 0
$$453$$ 8.48528i 0.398673i
$$454$$ 0 0
$$455$$ −45.2548 −2.12158
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 0 0
$$459$$ − 2.00000i − 0.0933520i
$$460$$ 0 0
$$461$$ − 8.48528i − 0.395199i −0.980283 0.197599i $$-0.936685\pi$$
0.980283 0.197599i $$-0.0633145\pi$$
$$462$$ 0 0
$$463$$ −19.7990 −0.920137 −0.460069 0.887883i $$-0.652175\pi$$
−0.460069 + 0.887883i $$0.652175\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 0 0
$$469$$ − 11.3137i − 0.522419i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −48.0000 −2.20704
$$474$$ 0 0
$$475$$ 12.0000i 0.550598i
$$476$$ 0 0
$$477$$ − 2.82843i − 0.129505i
$$478$$ 0 0
$$479$$ −28.2843 −1.29234 −0.646171 0.763193i $$-0.723631\pi$$
−0.646171 + 0.763193i $$0.723631\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 16.0000i 0.728025i
$$484$$ 0 0
$$485$$ − 39.5980i − 1.79805i
$$486$$ 0 0
$$487$$ 14.1421 0.640841 0.320421 0.947275i $$-0.396176\pi$$
0.320421 + 0.947275i $$0.396176\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ − 20.0000i − 0.902587i −0.892375 0.451294i $$-0.850963\pi$$
0.892375 0.451294i $$-0.149037\pi$$
$$492$$ 0 0
$$493$$ 5.65685i 0.254772i
$$494$$ 0 0
$$495$$ −11.3137 −0.508513
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ 11.3137i 0.505459i
$$502$$ 0 0
$$503$$ 5.65685 0.252227 0.126113 0.992016i $$-0.459750\pi$$
0.126113 + 0.992016i $$0.459750\pi$$
$$504$$ 0 0
$$505$$ 40.0000 1.77998
$$506$$ 0 0
$$507$$ 19.0000i 0.843820i
$$508$$ 0 0
$$509$$ − 2.82843i − 0.125368i −0.998033 0.0626839i $$-0.980034\pi$$
0.998033 0.0626839i $$-0.0199660\pi$$
$$510$$ 0 0
$$511$$ −5.65685 −0.250244
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 8.00000i 0.352522i
$$516$$ 0 0
$$517$$ − 22.6274i − 0.995153i
$$518$$ 0 0
$$519$$ −19.7990 −0.869079
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 0 0
$$525$$ 8.48528i 0.370328i
$$526$$ 0 0
$$527$$ −16.9706 −0.739249
$$528$$ 0 0
$$529$$ 9.00000 0.391304
$$530$$ 0 0
$$531$$ 4.00000i 0.173585i
$$532$$ 0 0
$$533$$ − 56.5685i − 2.45026i
$$534$$ 0 0
$$535$$ 33.9411 1.46740
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 4.00000i 0.172292i
$$540$$ 0 0
$$541$$ − 16.9706i − 0.729621i −0.931082 0.364811i $$-0.881134\pi$$
0.931082 0.364811i $$-0.118866\pi$$
$$542$$ 0 0
$$543$$ −5.65685 −0.242759
$$544$$ 0 0
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 0 0
$$549$$ − 11.3137i − 0.482857i
$$550$$ 0 0
$$551$$ −11.3137 −0.481980
$$552$$ 0 0
$$553$$ 24.0000 1.02058
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 8.48528i − 0.359533i −0.983709 0.179766i $$-0.942466\pi$$
0.983709 0.179766i $$-0.0575342\pi$$
$$558$$ 0 0
$$559$$ 67.8823 2.87111
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ − 5.65685i − 0.237986i
$$566$$ 0 0
$$567$$ 2.82843 0.118783
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ − 4.00000i − 0.167395i −0.996491 0.0836974i $$-0.973327\pi$$
0.996491 0.0836974i $$-0.0266729\pi$$
$$572$$ 0 0
$$573$$ − 11.3137i − 0.472637i
$$574$$ 0 0
$$575$$ 16.9706 0.707721
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 0 0
$$579$$ − 6.00000i − 0.249351i
$$580$$ 0 0
$$581$$ 11.3137i 0.469372i
$$582$$ 0 0
$$583$$ −11.3137 −0.468566
$$584$$ 0 0
$$585$$ 16.0000 0.661519
$$586$$ 0 0
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ − 33.9411i − 1.39852i
$$590$$ 0 0
$$591$$ −8.48528 −0.349038
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 16.0000i 0.655936i
$$596$$ 0 0
$$597$$ 2.82843i 0.115760i
$$598$$ 0 0
$$599$$ −39.5980 −1.61793 −0.808965 0.587857i $$-0.799972\pi$$
−0.808965 + 0.587857i $$0.799972\pi$$
$$600$$ 0 0
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 14.1421i 0.574960i
$$606$$ 0 0
$$607$$ 14.1421 0.574012 0.287006 0.957929i $$-0.407340\pi$$
0.287006 + 0.957929i $$0.407340\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ 32.0000i 1.29458i
$$612$$ 0 0
$$613$$ − 11.3137i − 0.456956i −0.973549 0.228478i $$-0.926625\pi$$
0.973549 0.228478i $$-0.0733750\pi$$
$$614$$ 0 0
$$615$$ −28.2843 −1.14053
$$616$$ 0 0
$$617$$ 38.0000 1.52982 0.764911 0.644136i $$-0.222783\pi$$
0.764911 + 0.644136i $$0.222783\pi$$
$$618$$ 0 0
$$619$$ − 36.0000i − 1.44696i −0.690344 0.723481i $$-0.742541\pi$$
0.690344 0.723481i $$-0.257459\pi$$
$$620$$ 0 0
$$621$$ − 5.65685i − 0.227002i
$$622$$ 0 0
$$623$$ 16.9706 0.679911
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.48528 −0.337794 −0.168897 0.985634i $$-0.554020\pi$$
−0.168897 + 0.985634i $$0.554020\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ 0 0
$$635$$ 24.0000i 0.952411i
$$636$$ 0 0
$$637$$ − 5.65685i − 0.224133i
$$638$$ 0 0
$$639$$ −5.65685 −0.223782
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ 0 0
$$645$$ − 33.9411i − 1.33643i
$$646$$ 0 0
$$647$$ 39.5980 1.55676 0.778379 0.627795i $$-0.216042\pi$$
0.778379 + 0.627795i $$0.216042\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ − 24.0000i − 0.940634i
$$652$$ 0 0
$$653$$ − 42.4264i − 1.66027i −0.557560 0.830137i $$-0.688262\pi$$
0.557560 0.830137i $$-0.311738\pi$$
$$654$$ 0 0
$$655$$ 33.9411 1.32619
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ − 4.00000i − 0.155818i −0.996960 0.0779089i $$-0.975176\pi$$
0.996960 0.0779089i $$-0.0248243\pi$$
$$660$$ 0 0
$$661$$ 45.2548i 1.76021i 0.474780 + 0.880105i $$0.342528\pi$$
−0.474780 + 0.880105i $$0.657472\pi$$
$$662$$ 0 0
$$663$$ 11.3137 0.439388
$$664$$ 0 0
$$665$$ −32.0000 −1.24091
$$666$$ 0 0
$$667$$ 16.0000i 0.619522i
$$668$$ 0 0
$$669$$ 19.7990i 0.765473i
$$670$$ 0 0
$$671$$ −45.2548 −1.74704
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 0 0
$$675$$ − 3.00000i − 0.115470i
$$676$$ 0 0
$$677$$ 19.7990i 0.760937i 0.924794 + 0.380468i $$0.124237\pi$$
−0.924794 + 0.380468i $$0.875763\pi$$
$$678$$ 0 0
$$679$$ 39.5980 1.51963
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ 0 0
$$685$$ − 28.2843i − 1.08069i
$$686$$ 0 0
$$687$$ 5.65685 0.215822
$$688$$ 0 0
$$689$$ 16.0000 0.609551
$$690$$ 0 0
$$691$$ 28.0000i 1.06517i 0.846376 + 0.532585i $$0.178779\pi$$
−0.846376 + 0.532585i $$0.821221\pi$$
$$692$$ 0 0
$$693$$ − 11.3137i − 0.429772i
$$694$$ 0 0
$$695$$ −11.3137 −0.429153
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ 0 0
$$699$$ − 6.00000i − 0.226941i
$$700$$ 0 0
$$701$$ 19.7990i 0.747798i 0.927470 + 0.373899i $$0.121979\pi$$
−0.927470 + 0.373899i $$0.878021\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 16.0000 0.602595
$$706$$ 0 0
$$707$$ 40.0000i 1.50435i
$$708$$ 0 0
$$709$$ − 16.9706i − 0.637343i −0.947865 0.318671i $$-0.896763\pi$$
0.947865 0.318671i $$-0.103237\pi$$
$$710$$ 0 0
$$711$$ −8.48528 −0.318223
$$712$$ 0 0
$$713$$ −48.0000 −1.79761
$$714$$ 0 0
$$715$$ − 64.0000i − 2.39346i
$$716$$ 0 0
$$717$$ 22.6274i 0.845036i
$$718$$ 0 0
$$719$$ −16.9706 −0.632895 −0.316448 0.948610i $$-0.602490\pi$$
−0.316448 + 0.948610i $$0.602490\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ − 6.00000i − 0.223142i
$$724$$ 0 0
$$725$$ 8.48528i 0.315135i
$$726$$ 0 0
$$727$$ 25.4558 0.944105 0.472052 0.881570i $$-0.343513\pi$$
0.472052 + 0.881570i $$0.343513\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 24.0000i − 0.887672i
$$732$$ 0 0
$$733$$ 16.9706i 0.626822i 0.949618 + 0.313411i $$0.101472\pi$$
−0.949618 + 0.313411i $$0.898528\pi$$
$$734$$ 0 0
$$735$$ −2.82843 −0.104328
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ − 36.0000i − 1.32428i −0.749380 0.662141i $$-0.769648\pi$$
0.749380 0.662141i $$-0.230352\pi$$
$$740$$ 0 0
$$741$$ 22.6274i 0.831239i
$$742$$ 0 0
$$743$$ 22.6274 0.830119 0.415060 0.909794i $$-0.363761\pi$$
0.415060 + 0.909794i $$0.363761\pi$$
$$744$$ 0 0
$$745$$ −40.0000 −1.46549
$$746$$ 0 0
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 33.9411i 1.24018i
$$750$$ 0 0
$$751$$ −48.0833 −1.75458 −0.877292 0.479958i $$-0.840652\pi$$
−0.877292 + 0.479958i $$0.840652\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 24.0000i 0.873449i
$$756$$ 0 0
$$757$$ 28.2843i 1.02801i 0.857787 + 0.514005i $$0.171839\pi$$
−0.857787 + 0.514005i $$0.828161\pi$$
$$758$$ 0 0
$$759$$ −22.6274 −0.821323
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 16.0000i 0.579239i
$$764$$ 0 0
$$765$$ − 5.65685i − 0.204524i
$$766$$ 0 0
$$767$$ −22.6274 −0.817029
$$768$$ 0 0
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ 0 0
$$771$$ 30.0000i 1.08042i
$$772$$ 0 0
$$773$$ 2.82843i 0.101731i 0.998706 + 0.0508657i $$0.0161981\pi$$
−0.998706 + 0.0508657i $$0.983802\pi$$
$$774$$ 0 0
$$775$$ −25.4558 −0.914401
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 40.0000i − 1.43315i
$$780$$ 0 0
$$781$$ 22.6274i 0.809673i
$$782$$ 0 0
$$783$$ 2.82843 0.101080
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 20.0000i − 0.712923i −0.934310 0.356462i $$-0.883983\pi$$
0.934310 0.356462i $$-0.116017\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.65685 0.201135
$$792$$ 0 0
$$793$$ 64.0000 2.27271
$$794$$ 0 0
$$795$$ − 8.00000i − 0.283731i
$$796$$ 0 0
$$797$$ − 19.7990i − 0.701316i −0.936504 0.350658i $$-0.885958\pi$$
0.936504 0.350658i $$-0.114042\pi$$
$$798$$ 0 0
$$799$$ 11.3137 0.400250
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ − 8.00000i − 0.282314i
$$804$$ 0 0
$$805$$ 45.2548i 1.59502i
$$806$$ 0 0
$$807$$ 8.48528 0.298696
$$808$$ 0 0
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ − 20.0000i − 0.702295i −0.936320 0.351147i $$-0.885792\pi$$
0.936320 0.351147i $$-0.114208\pi$$
$$812$$ 0 0
$$813$$ 19.7990i 0.694381i
$$814$$ 0 0
$$815$$ −11.3137 −0.396302
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ 0 0
$$819$$ 16.0000i 0.559085i
$$820$$ 0 0
$$821$$ − 31.1127i − 1.08584i −0.839784 0.542920i $$-0.817319\pi$$
0.839784 0.542920i $$-0.182681\pi$$
$$822$$ 0 0
$$823$$ −19.7990 −0.690149 −0.345075 0.938575i $$-0.612146\pi$$
−0.345075 + 0.938575i $$0.612146\pi$$
$$824$$ 0 0
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 0 0
$$829$$ 50.9117i 1.76824i 0.467264 + 0.884118i $$0.345240\pi$$
−0.467264 + 0.884118i $$0.654760\pi$$
$$830$$ 0 0
$$831$$ 16.9706 0.588702
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 32.0000i 1.10741i
$$836$$ 0 0
$$837$$ 8.48528i 0.293294i
$$838$$ 0 0
$$839$$ −5.65685 −0.195296 −0.0976481 0.995221i $$-0.531132\pi$$
−0.0976481 + 0.995221i $$0.531132\pi$$
$$840$$ 0 0
$$841$$ 21.0000 0.724138
$$842$$ 0 0
$$843$$ 10.0000i 0.344418i
$$844$$ 0 0
$$845$$ 53.7401i 1.84872i
$$846$$ 0 0
$$847$$ −14.1421 −0.485930
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 22.6274i − 0.774748i −0.921923 0.387374i $$-0.873382\pi$$
0.921923 0.387374i $$-0.126618\pi$$
$$854$$ 0 0
$$855$$ 11.3137 0.386921
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 44.0000i 1.50126i 0.660722 + 0.750630i $$0.270250\pi$$
−0.660722 + 0.750630i $$0.729750\pi$$
$$860$$ 0 0
$$861$$ − 28.2843i − 0.963925i
$$862$$ 0 0
$$863$$ 56.5685 1.92562 0.962808 0.270187i $$-0.0870856\pi$$
0.962808 + 0.270187i $$0.0870856\pi$$
$$864$$ 0 0
$$865$$ −56.0000 −1.90406
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 33.9411i 1.15137i
$$870$$ 0 0
$$871$$ −22.6274 −0.766701
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ − 16.0000i − 0.540899i
$$876$$ 0 0
$$877$$ − 11.3137i − 0.382037i −0.981586 0.191018i $$-0.938821\pi$$
0.981586 0.191018i $$-0.0611790\pi$$
$$878$$ 0 0
$$879$$ −19.7990 −0.667803
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ 12.0000i 0.403832i 0.979403 + 0.201916i $$0.0647168\pi$$
−0.979403 + 0.201916i $$0.935283\pi$$
$$884$$ 0 0
$$885$$ 11.3137i 0.380306i
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 4.00000i 0.134005i
$$892$$ 0 0
$$893$$ 22.6274i 0.757198i
$$894$$ 0 0
$$895$$ 33.9411 1.13453
$$896$$ 0 0
$$897$$ 32.0000 1.06845
$$898$$ 0 0
$$899$$ − 24.0000i − 0.800445i
$$900$$ 0 0
$$901$$ − 5.65685i − 0.188457i
$$902$$ 0 0
$$903$$ 33.9411 1.12949
$$904$$ 0 0
$$905$$ −16.0000 −0.531858
$$906$$ 0 0
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ 0 0
$$909$$ − 14.1421i − 0.469065i
$$910$$ 0 0
$$911$$ 22.6274 0.749680 0.374840 0.927090i $$-0.377698\pi$$
0.374840 + 0.927090i $$0.377698\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ − 32.0000i − 1.05789i
$$916$$ 0 0
$$917$$ 33.9411i 1.12083i
$$918$$ 0 0
$$919$$ 2.82843 0.0933012 0.0466506 0.998911i $$-0.485145\pi$$
0.0466506 + 0.998911i $$0.485145\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ − 32.0000i − 1.05329i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.82843 0.0928977
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ − 4.00000i − 0.131095i
$$932$$ 0 0
$$933$$ 11.3137i 0.370394i
$$934$$ 0 0
$$935$$ −22.6274 −0.739996
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ 14.0000i 0.456873i
$$940$$ 0 0
$$941$$ 14.1421i 0.461020i 0.973070 + 0.230510i $$0.0740395\pi$$
−0.973070 + 0.230510i $$0.925960\pi$$
$$942$$ 0 0
$$943$$ −56.5685 −1.84213
$$944$$ 0 0
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ 11.3137i 0.367259i
$$950$$ 0 0
$$951$$ 8.48528 0.275154
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ − 32.0000i − 1.03550i
$$956$$ 0 0
$$957$$ − 11.3137i − 0.365720i
$$958$$ 0 0
$$959$$ 28.2843 0.913347
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ − 16.9706i − 0.546302i
$$966$$ 0 0
$$967$$ −14.1421 −0.454780 −0.227390 0.973804i $$-0.573019\pi$$
−0.227390 + 0.973804i $$0.573019\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ − 12.0000i − 0.385098i −0.981287 0.192549i $$-0.938325\pi$$
0.981287 0.192549i $$-0.0616755\pi$$
$$972$$ 0 0
$$973$$ − 11.3137i − 0.362701i
$$974$$ 0 0
$$975$$ 16.9706 0.543493
$$976$$ 0 0
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 24.0000i 0.767043i
$$980$$ 0 0
$$981$$ − 5.65685i − 0.180609i
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ −24.0000 −0.764704
$$986$$ 0 0
$$987$$ 16.0000i 0.509286i
$$988$$ 0 0
$$989$$ − 67.8823i − 2.15853i
$$990$$ 0 0
$$991$$ −25.4558 −0.808632 −0.404316 0.914619i $$-0.632490\pi$$
−0.404316 + 0.914619i $$0.632490\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 8.00000i 0.253617i
$$996$$ 0 0
$$997$$ − 33.9411i − 1.07493i −0.843287 0.537463i $$-0.819383\pi$$
0.843287 0.537463i $$-0.180617\pi$$
$$998$$ 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 384.2.d.c.193.1 4
3.2 odd 2 1152.2.d.h.577.4 4
4.3 odd 2 inner 384.2.d.c.193.3 yes 4
8.3 odd 2 inner 384.2.d.c.193.2 yes 4
8.5 even 2 inner 384.2.d.c.193.4 yes 4
12.11 even 2 1152.2.d.h.577.3 4
16.3 odd 4 768.2.a.i.1.1 2
16.5 even 4 768.2.a.i.1.2 2
16.11 odd 4 768.2.a.l.1.2 2
16.13 even 4 768.2.a.l.1.1 2
24.5 odd 2 1152.2.d.h.577.2 4
24.11 even 2 1152.2.d.h.577.1 4
48.5 odd 4 2304.2.a.x.1.1 2
48.11 even 4 2304.2.a.r.1.1 2
48.29 odd 4 2304.2.a.r.1.2 2
48.35 even 4 2304.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.c.193.1 4 1.1 even 1 trivial
384.2.d.c.193.2 yes 4 8.3 odd 2 inner
384.2.d.c.193.3 yes 4 4.3 odd 2 inner
384.2.d.c.193.4 yes 4 8.5 even 2 inner
768.2.a.i.1.1 2 16.3 odd 4
768.2.a.i.1.2 2 16.5 even 4
768.2.a.l.1.1 2 16.13 even 4
768.2.a.l.1.2 2 16.11 odd 4
1152.2.d.h.577.1 4 24.11 even 2
1152.2.d.h.577.2 4 24.5 odd 2
1152.2.d.h.577.3 4 12.11 even 2
1152.2.d.h.577.4 4 3.2 odd 2
2304.2.a.r.1.1 2 48.11 even 4
2304.2.a.r.1.2 2 48.29 odd 4
2304.2.a.x.1.1 2 48.5 odd 4
2304.2.a.x.1.2 2 48.35 even 4