# Properties

 Label 3072.2.d.f.1537.4 Level $3072$ Weight $2$ Character 3072.1537 Analytic conductor $24.530$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1537.4 Root $$0.500000 + 0.691860i$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1537 Dual form 3072.2.d.f.1537.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +2.54266i q^{11} +1.95687i q^{13} +3.79793 q^{15} +0.224777 q^{17} +0.224777i q^{19} +2.15894i q^{21} +2.82843 q^{23} -9.42429 q^{25} +1.00000i q^{27} +2.62636i q^{29} +1.84106 q^{31} +2.54266 q^{33} -8.19951i q^{35} +5.18944i q^{37} +1.95687 q^{39} -5.88163 q^{41} -10.9670i q^{43} -3.79793i q^{45} +2.82843 q^{47} -2.33897 q^{49} -0.224777i q^{51} +10.6264i q^{53} -9.65685 q^{55} +0.224777 q^{57} -5.65685i q^{59} -8.46742i q^{61} +2.15894 q^{63} -7.43208 q^{65} +14.7422i q^{67} -2.82843i q^{69} +4.31788 q^{71} -5.97474 q^{73} +9.42429i q^{75} -5.48946i q^{77} -15.0075 q^{79} +1.00000 q^{81} -14.3059i q^{83} +0.853690i q^{85} +2.62636 q^{87} +1.42847 q^{89} -4.22478i q^{91} -1.84106i q^{93} -0.853690 q^{95} -16.3990 q^{97} -2.54266i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} - 8q^{9} + O(q^{10})$$ $$8q - 8q^{7} - 8q^{9} + 8q^{15} - 8q^{25} + 24q^{31} - 16q^{39} + 8q^{49} - 32q^{55} + 8q^{63} - 16q^{65} + 16q^{71} + 16q^{73} + 24q^{79} + 8q^{81} - 24q^{87} + 16q^{89} - 48q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ 3.79793i 1.69849i 0.528001 + 0.849244i $$0.322942\pi$$
−0.528001 + 0.849244i $$0.677058\pi$$
$$6$$ 0 0
$$7$$ −2.15894 −0.816003 −0.408002 0.912981i $$-0.633774\pi$$
−0.408002 + 0.912981i $$0.633774\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.54266i 0.766641i 0.923615 + 0.383321i $$0.125220\pi$$
−0.923615 + 0.383321i $$0.874780\pi$$
$$12$$ 0 0
$$13$$ 1.95687i 0.542739i 0.962475 + 0.271370i $$0.0874766\pi$$
−0.962475 + 0.271370i $$0.912523\pi$$
$$14$$ 0 0
$$15$$ 3.79793 0.980622
$$16$$ 0 0
$$17$$ 0.224777 0.0545165 0.0272583 0.999628i $$-0.491322\pi$$
0.0272583 + 0.999628i $$0.491322\pi$$
$$18$$ 0 0
$$19$$ 0.224777i 0.0515675i 0.999668 + 0.0257837i $$0.00820813\pi$$
−0.999668 + 0.0257837i $$0.991792\pi$$
$$20$$ 0 0
$$21$$ 2.15894i 0.471120i
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ −9.42429 −1.88486
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 2.62636i 0.487703i 0.969813 + 0.243851i $$0.0784109\pi$$
−0.969813 + 0.243851i $$0.921589\pi$$
$$30$$ 0 0
$$31$$ 1.84106 0.330664 0.165332 0.986238i $$-0.447130\pi$$
0.165332 + 0.986238i $$0.447130\pi$$
$$32$$ 0 0
$$33$$ 2.54266 0.442620
$$34$$ 0 0
$$35$$ − 8.19951i − 1.38597i
$$36$$ 0 0
$$37$$ 5.18944i 0.853138i 0.904455 + 0.426569i $$0.140278\pi$$
−0.904455 + 0.426569i $$0.859722\pi$$
$$38$$ 0 0
$$39$$ 1.95687 0.313351
$$40$$ 0 0
$$41$$ −5.88163 −0.918557 −0.459278 0.888292i $$-0.651892\pi$$
−0.459278 + 0.888292i $$0.651892\pi$$
$$42$$ 0 0
$$43$$ − 10.9670i − 1.67244i −0.548391 0.836222i $$-0.684759\pi$$
0.548391 0.836222i $$-0.315241\pi$$
$$44$$ 0 0
$$45$$ − 3.79793i − 0.566162i
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ −2.33897 −0.334139
$$50$$ 0 0
$$51$$ − 0.224777i − 0.0314751i
$$52$$ 0 0
$$53$$ 10.6264i 1.45964i 0.683638 + 0.729821i $$0.260397\pi$$
−0.683638 + 0.729821i $$0.739603\pi$$
$$54$$ 0 0
$$55$$ −9.65685 −1.30213
$$56$$ 0 0
$$57$$ 0.224777 0.0297725
$$58$$ 0 0
$$59$$ − 5.65685i − 0.736460i −0.929735 0.368230i $$-0.879964\pi$$
0.929735 0.368230i $$-0.120036\pi$$
$$60$$ 0 0
$$61$$ − 8.46742i − 1.08414i −0.840333 0.542071i $$-0.817640\pi$$
0.840333 0.542071i $$-0.182360\pi$$
$$62$$ 0 0
$$63$$ 2.15894 0.272001
$$64$$ 0 0
$$65$$ −7.43208 −0.921836
$$66$$ 0 0
$$67$$ 14.7422i 1.80104i 0.434811 + 0.900522i $$0.356815\pi$$
−0.434811 + 0.900522i $$0.643185\pi$$
$$68$$ 0 0
$$69$$ − 2.82843i − 0.340503i
$$70$$ 0 0
$$71$$ 4.31788 0.512438 0.256219 0.966619i $$-0.417523\pi$$
0.256219 + 0.966619i $$0.417523\pi$$
$$72$$ 0 0
$$73$$ −5.97474 −0.699290 −0.349645 0.936882i $$-0.613698\pi$$
−0.349645 + 0.936882i $$0.613698\pi$$
$$74$$ 0 0
$$75$$ 9.42429i 1.08822i
$$76$$ 0 0
$$77$$ − 5.48946i − 0.625582i
$$78$$ 0 0
$$79$$ −15.0075 −1.68848 −0.844239 0.535966i $$-0.819947\pi$$
−0.844239 + 0.535966i $$0.819947\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 14.3059i − 1.57028i −0.619319 0.785140i $$-0.712591\pi$$
0.619319 0.785140i $$-0.287409\pi$$
$$84$$ 0 0
$$85$$ 0.853690i 0.0925956i
$$86$$ 0 0
$$87$$ 2.62636 0.281575
$$88$$ 0 0
$$89$$ 1.42847 0.151417 0.0757086 0.997130i $$-0.475878\pi$$
0.0757086 + 0.997130i $$0.475878\pi$$
$$90$$ 0 0
$$91$$ − 4.22478i − 0.442877i
$$92$$ 0 0
$$93$$ − 1.84106i − 0.190909i
$$94$$ 0 0
$$95$$ −0.853690 −0.0875867
$$96$$ 0 0
$$97$$ −16.3990 −1.66507 −0.832535 0.553973i $$-0.813111\pi$$
−0.832535 + 0.553973i $$0.813111\pi$$
$$98$$ 0 0
$$99$$ − 2.54266i − 0.255547i
$$100$$ 0 0
$$101$$ − 0.115816i − 0.0115241i −0.999983 0.00576206i $$-0.998166\pi$$
0.999983 0.00576206i $$-0.00183413\pi$$
$$102$$ 0 0
$$103$$ −13.3507 −1.31548 −0.657740 0.753245i $$-0.728488\pi$$
−0.657740 + 0.753245i $$0.728488\pi$$
$$104$$ 0 0
$$105$$ −8.19951 −0.800191
$$106$$ 0 0
$$107$$ − 10.2926i − 0.995025i −0.867457 0.497513i $$-0.834247\pi$$
0.867457 0.497513i $$-0.165753\pi$$
$$108$$ 0 0
$$109$$ − 9.95687i − 0.953696i −0.878986 0.476848i $$-0.841779\pi$$
0.878986 0.476848i $$-0.158221\pi$$
$$110$$ 0 0
$$111$$ 5.18944 0.492559
$$112$$ 0 0
$$113$$ −18.8486 −1.77313 −0.886563 0.462608i $$-0.846914\pi$$
−0.886563 + 0.462608i $$0.846914\pi$$
$$114$$ 0 0
$$115$$ 10.7422i 1.00171i
$$116$$ 0 0
$$117$$ − 1.95687i − 0.180913i
$$118$$ 0 0
$$119$$ −0.485281 −0.0444857
$$120$$ 0 0
$$121$$ 4.53488 0.412261
$$122$$ 0 0
$$123$$ 5.88163i 0.530329i
$$124$$ 0 0
$$125$$ − 16.8032i − 1.50292i
$$126$$ 0 0
$$127$$ −3.81580 −0.338597 −0.169299 0.985565i $$-0.554150\pi$$
−0.169299 + 0.985565i $$0.554150\pi$$
$$128$$ 0 0
$$129$$ −10.9670 −0.965586
$$130$$ 0 0
$$131$$ 1.08532i 0.0948250i 0.998875 + 0.0474125i $$0.0150975\pi$$
−0.998875 + 0.0474125i $$0.984902\pi$$
$$132$$ 0 0
$$133$$ − 0.485281i − 0.0420792i
$$134$$ 0 0
$$135$$ −3.79793 −0.326874
$$136$$ 0 0
$$137$$ 5.31010 0.453672 0.226836 0.973933i $$-0.427162\pi$$
0.226836 + 0.973933i $$0.427162\pi$$
$$138$$ 0 0
$$139$$ 12.3990i 1.05167i 0.850586 + 0.525836i $$0.176247\pi$$
−0.850586 + 0.525836i $$0.823753\pi$$
$$140$$ 0 0
$$141$$ − 2.82843i − 0.238197i
$$142$$ 0 0
$$143$$ −4.97567 −0.416086
$$144$$ 0 0
$$145$$ −9.97474 −0.828357
$$146$$ 0 0
$$147$$ 2.33897i 0.192915i
$$148$$ 0 0
$$149$$ − 1.45479i − 0.119181i −0.998223 0.0595904i $$-0.981021\pi$$
0.998223 0.0595904i $$-0.0189795\pi$$
$$150$$ 0 0
$$151$$ −2.03696 −0.165766 −0.0828829 0.996559i $$-0.526413\pi$$
−0.0828829 + 0.996559i $$0.526413\pi$$
$$152$$ 0 0
$$153$$ −0.224777 −0.0181722
$$154$$ 0 0
$$155$$ 6.99222i 0.561628i
$$156$$ 0 0
$$157$$ − 8.61790i − 0.687784i −0.939009 0.343892i $$-0.888255\pi$$
0.939009 0.343892i $$-0.111745\pi$$
$$158$$ 0 0
$$159$$ 10.6264 0.842725
$$160$$ 0 0
$$161$$ −6.10641 −0.481252
$$162$$ 0 0
$$163$$ − 4.86054i − 0.380707i −0.981716 0.190354i $$-0.939037\pi$$
0.981716 0.190354i $$-0.0609634\pi$$
$$164$$ 0 0
$$165$$ 9.65685i 0.751785i
$$166$$ 0 0
$$167$$ 21.7023 1.67937 0.839686 0.543072i $$-0.182739\pi$$
0.839686 + 0.543072i $$0.182739\pi$$
$$168$$ 0 0
$$169$$ 9.17064 0.705434
$$170$$ 0 0
$$171$$ − 0.224777i − 0.0171892i
$$172$$ 0 0
$$173$$ − 12.3695i − 0.940433i −0.882551 0.470217i $$-0.844176\pi$$
0.882551 0.470217i $$-0.155824\pi$$
$$174$$ 0 0
$$175$$ 20.3465 1.53805
$$176$$ 0 0
$$177$$ −5.65685 −0.425195
$$178$$ 0 0
$$179$$ 11.6413i 0.870111i 0.900404 + 0.435055i $$0.143271\pi$$
−0.900404 + 0.435055i $$0.856729\pi$$
$$180$$ 0 0
$$181$$ 9.50732i 0.706673i 0.935496 + 0.353337i $$0.114953\pi$$
−0.935496 + 0.353337i $$0.885047\pi$$
$$182$$ 0 0
$$183$$ −8.46742 −0.625930
$$184$$ 0 0
$$185$$ −19.7091 −1.44904
$$186$$ 0 0
$$187$$ 0.571533i 0.0417946i
$$188$$ 0 0
$$189$$ − 2.15894i − 0.157040i
$$190$$ 0 0
$$191$$ −20.8032 −1.50526 −0.752632 0.658441i $$-0.771216\pi$$
−0.752632 + 0.658441i $$0.771216\pi$$
$$192$$ 0 0
$$193$$ 14.1454 1.01821 0.509103 0.860705i $$-0.329977\pi$$
0.509103 + 0.860705i $$0.329977\pi$$
$$194$$ 0 0
$$195$$ 7.43208i 0.532222i
$$196$$ 0 0
$$197$$ 3.43463i 0.244707i 0.992487 + 0.122354i $$0.0390442\pi$$
−0.992487 + 0.122354i $$0.960956\pi$$
$$198$$ 0 0
$$199$$ −0.306182 −0.0217047 −0.0108523 0.999941i $$-0.503454\pi$$
−0.0108523 + 0.999941i $$0.503454\pi$$
$$200$$ 0 0
$$201$$ 14.7422 1.03983
$$202$$ 0 0
$$203$$ − 5.67016i − 0.397967i
$$204$$ 0 0
$$205$$ − 22.3380i − 1.56016i
$$206$$ 0 0
$$207$$ −2.82843 −0.196589
$$208$$ 0 0
$$209$$ −0.571533 −0.0395337
$$210$$ 0 0
$$211$$ 10.2284i 0.704151i 0.935972 + 0.352076i $$0.114524\pi$$
−0.935972 + 0.352076i $$0.885476\pi$$
$$212$$ 0 0
$$213$$ − 4.31788i − 0.295856i
$$214$$ 0 0
$$215$$ 41.6517 2.84063
$$216$$ 0 0
$$217$$ −3.97474 −0.269823
$$218$$ 0 0
$$219$$ 5.97474i 0.403735i
$$220$$ 0 0
$$221$$ 0.439861i 0.0295883i
$$222$$ 0 0
$$223$$ −1.71908 −0.115118 −0.0575591 0.998342i $$-0.518332\pi$$
−0.0575591 + 0.998342i $$0.518332\pi$$
$$224$$ 0 0
$$225$$ 9.42429 0.628286
$$226$$ 0 0
$$227$$ − 14.3059i − 0.949518i −0.880116 0.474759i $$-0.842535\pi$$
0.880116 0.474759i $$-0.157465\pi$$
$$228$$ 0 0
$$229$$ 16.9981i 1.12327i 0.827386 + 0.561634i $$0.189827\pi$$
−0.827386 + 0.561634i $$0.810173\pi$$
$$230$$ 0 0
$$231$$ −5.48946 −0.361180
$$232$$ 0 0
$$233$$ −13.3779 −0.876418 −0.438209 0.898873i $$-0.644387\pi$$
−0.438209 + 0.898873i $$0.644387\pi$$
$$234$$ 0 0
$$235$$ 10.7422i 0.700742i
$$236$$ 0 0
$$237$$ 15.0075i 0.974844i
$$238$$ 0 0
$$239$$ −13.3675 −0.864670 −0.432335 0.901713i $$-0.642310\pi$$
−0.432335 + 0.901713i $$0.642310\pi$$
$$240$$ 0 0
$$241$$ −0.211474 −0.0136222 −0.00681112 0.999977i $$-0.502168\pi$$
−0.00681112 + 0.999977i $$0.502168\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 8.88325i − 0.567530i
$$246$$ 0 0
$$247$$ −0.439861 −0.0279877
$$248$$ 0 0
$$249$$ −14.3059 −0.906601
$$250$$ 0 0
$$251$$ − 14.7555i − 0.931358i −0.884954 0.465679i $$-0.845810\pi$$
0.884954 0.465679i $$-0.154190\pi$$
$$252$$ 0 0
$$253$$ 7.19173i 0.452140i
$$254$$ 0 0
$$255$$ 0.853690 0.0534601
$$256$$ 0 0
$$257$$ −0.742176 −0.0462957 −0.0231478 0.999732i $$-0.507369\pi$$
−0.0231478 + 0.999732i $$0.507369\pi$$
$$258$$ 0 0
$$259$$ − 11.2037i − 0.696163i
$$260$$ 0 0
$$261$$ − 2.62636i − 0.162568i
$$262$$ 0 0
$$263$$ −5.48435 −0.338180 −0.169090 0.985601i $$-0.554083\pi$$
−0.169090 + 0.985601i $$0.554083\pi$$
$$264$$ 0 0
$$265$$ −40.3582 −2.47918
$$266$$ 0 0
$$267$$ − 1.42847i − 0.0874208i
$$268$$ 0 0
$$269$$ 20.4694i 1.24804i 0.781407 + 0.624021i $$0.214502\pi$$
−0.781407 + 0.624021i $$0.785498\pi$$
$$270$$ 0 0
$$271$$ 14.0370 0.852685 0.426342 0.904562i $$-0.359802\pi$$
0.426342 + 0.904562i $$0.359802\pi$$
$$272$$ 0 0
$$273$$ −4.22478 −0.255695
$$274$$ 0 0
$$275$$ − 23.9628i − 1.44501i
$$276$$ 0 0
$$277$$ 13.4211i 0.806394i 0.915113 + 0.403197i $$0.132101\pi$$
−0.915113 + 0.403197i $$0.867899\pi$$
$$278$$ 0 0
$$279$$ −1.84106 −0.110221
$$280$$ 0 0
$$281$$ −3.89359 −0.232272 −0.116136 0.993233i $$-0.537051\pi$$
−0.116136 + 0.993233i $$0.537051\pi$$
$$282$$ 0 0
$$283$$ 17.6569i 1.04959i 0.851228 + 0.524796i $$0.175858\pi$$
−0.851228 + 0.524796i $$0.824142\pi$$
$$284$$ 0 0
$$285$$ 0.853690i 0.0505682i
$$286$$ 0 0
$$287$$ 12.6981 0.749545
$$288$$ 0 0
$$289$$ −16.9495 −0.997028
$$290$$ 0 0
$$291$$ 16.3990i 0.961328i
$$292$$ 0 0
$$293$$ 15.7759i 0.921639i 0.887494 + 0.460819i $$0.152444\pi$$
−0.887494 + 0.460819i $$0.847556\pi$$
$$294$$ 0 0
$$295$$ 21.4844 1.25087
$$296$$ 0 0
$$297$$ −2.54266 −0.147540
$$298$$ 0 0
$$299$$ 5.53488i 0.320090i
$$300$$ 0 0
$$301$$ 23.6770i 1.36472i
$$302$$ 0 0
$$303$$ −0.115816 −0.00665345
$$304$$ 0 0
$$305$$ 32.1587 1.84140
$$306$$ 0 0
$$307$$ − 7.64129i − 0.436111i −0.975936 0.218056i $$-0.930029\pi$$
0.975936 0.218056i $$-0.0699714\pi$$
$$308$$ 0 0
$$309$$ 13.3507i 0.759493i
$$310$$ 0 0
$$311$$ −24.1623 −1.37012 −0.685059 0.728488i $$-0.740224\pi$$
−0.685059 + 0.728488i $$0.740224\pi$$
$$312$$ 0 0
$$313$$ 16.6105 0.938881 0.469441 0.882964i $$-0.344456\pi$$
0.469441 + 0.882964i $$0.344456\pi$$
$$314$$ 0 0
$$315$$ 8.19951i 0.461990i
$$316$$ 0 0
$$317$$ − 2.56213i − 0.143903i −0.997408 0.0719517i $$-0.977077\pi$$
0.997408 0.0719517i $$-0.0229227\pi$$
$$318$$ 0 0
$$319$$ −6.67794 −0.373893
$$320$$ 0 0
$$321$$ −10.2926 −0.574478
$$322$$ 0 0
$$323$$ 0.0505249i 0.00281128i
$$324$$ 0 0
$$325$$ − 18.4422i − 1.02299i
$$326$$ 0 0
$$327$$ −9.95687 −0.550616
$$328$$ 0 0
$$329$$ −6.10641 −0.336657
$$330$$ 0 0
$$331$$ 19.1275i 1.05134i 0.850688 + 0.525671i $$0.176186\pi$$
−0.850688 + 0.525671i $$0.823814\pi$$
$$332$$ 0 0
$$333$$ − 5.18944i − 0.284379i
$$334$$ 0 0
$$335$$ −55.9898 −3.05905
$$336$$ 0 0
$$337$$ 1.12615 0.0613454 0.0306727 0.999529i $$-0.490235\pi$$
0.0306727 + 0.999529i $$0.490235\pi$$
$$338$$ 0 0
$$339$$ 18.8486i 1.02371i
$$340$$ 0 0
$$341$$ 4.68119i 0.253500i
$$342$$ 0 0
$$343$$ 20.1623 1.08866
$$344$$ 0 0
$$345$$ 10.7422 0.578339
$$346$$ 0 0
$$347$$ − 29.4068i − 1.57864i −0.613982 0.789320i $$-0.710433\pi$$
0.613982 0.789320i $$-0.289567\pi$$
$$348$$ 0 0
$$349$$ 27.2738i 1.45993i 0.683482 + 0.729967i $$0.260465\pi$$
−0.683482 + 0.729967i $$0.739535\pi$$
$$350$$ 0 0
$$351$$ −1.95687 −0.104450
$$352$$ 0 0
$$353$$ 25.5908 1.36206 0.681029 0.732256i $$-0.261533\pi$$
0.681029 + 0.732256i $$0.261533\pi$$
$$354$$ 0 0
$$355$$ 16.3990i 0.870370i
$$356$$ 0 0
$$357$$ 0.485281i 0.0256838i
$$358$$ 0 0
$$359$$ −3.77296 −0.199129 −0.0995645 0.995031i $$-0.531745\pi$$
−0.0995645 + 0.995031i $$0.531745\pi$$
$$360$$ 0 0
$$361$$ 18.9495 0.997341
$$362$$ 0 0
$$363$$ − 4.53488i − 0.238019i
$$364$$ 0 0
$$365$$ − 22.6917i − 1.18774i
$$366$$ 0 0
$$367$$ 27.4474 1.43274 0.716371 0.697720i $$-0.245802\pi$$
0.716371 + 0.697720i $$0.245802\pi$$
$$368$$ 0 0
$$369$$ 5.88163 0.306186
$$370$$ 0 0
$$371$$ − 22.9417i − 1.19107i
$$372$$ 0 0
$$373$$ 17.8518i 0.924332i 0.886794 + 0.462166i $$0.152928\pi$$
−0.886794 + 0.462166i $$0.847072\pi$$
$$374$$ 0 0
$$375$$ −16.8032 −0.867712
$$376$$ 0 0
$$377$$ −5.13946 −0.264695
$$378$$ 0 0
$$379$$ 16.5018i 0.847642i 0.905746 + 0.423821i $$0.139311\pi$$
−0.905746 + 0.423821i $$0.860689\pi$$
$$380$$ 0 0
$$381$$ 3.81580i 0.195489i
$$382$$ 0 0
$$383$$ −17.1885 −0.878291 −0.439145 0.898416i $$-0.644719\pi$$
−0.439145 + 0.898416i $$0.644719\pi$$
$$384$$ 0 0
$$385$$ 20.8486 1.06254
$$386$$ 0 0
$$387$$ 10.9670i 0.557482i
$$388$$ 0 0
$$389$$ 2.66209i 0.134973i 0.997720 + 0.0674866i $$0.0214980\pi$$
−0.997720 + 0.0674866i $$0.978502\pi$$
$$390$$ 0 0
$$391$$ 0.635767 0.0321521
$$392$$ 0 0
$$393$$ 1.08532 0.0547472
$$394$$ 0 0
$$395$$ − 56.9976i − 2.86786i
$$396$$ 0 0
$$397$$ 11.8959i 0.597037i 0.954404 + 0.298519i $$0.0964925\pi$$
−0.954404 + 0.298519i $$0.903507\pi$$
$$398$$ 0 0
$$399$$ −0.485281 −0.0242945
$$400$$ 0 0
$$401$$ −1.12389 −0.0561242 −0.0280621 0.999606i $$-0.508934\pi$$
−0.0280621 + 0.999606i $$0.508934\pi$$
$$402$$ 0 0
$$403$$ 3.60272i 0.179464i
$$404$$ 0 0
$$405$$ 3.79793i 0.188721i
$$406$$ 0 0
$$407$$ −13.1950 −0.654051
$$408$$ 0 0
$$409$$ −13.7211 −0.678464 −0.339232 0.940703i $$-0.610167\pi$$
−0.339232 + 0.940703i $$0.610167\pi$$
$$410$$ 0 0
$$411$$ − 5.31010i − 0.261928i
$$412$$ 0 0
$$413$$ 12.2128i 0.600953i
$$414$$ 0 0
$$415$$ 54.3329 2.66710
$$416$$ 0 0
$$417$$ 12.3990 0.607183
$$418$$ 0 0
$$419$$ 13.1629i 0.643048i 0.946901 + 0.321524i $$0.104195\pi$$
−0.946901 + 0.321524i $$0.895805\pi$$
$$420$$ 0 0
$$421$$ − 11.9413i − 0.581984i −0.956726 0.290992i $$-0.906015\pi$$
0.956726 0.290992i $$-0.0939852\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ −2.11837 −0.102756
$$426$$ 0 0
$$427$$ 18.2807i 0.884663i
$$428$$ 0 0
$$429$$ 4.97567i 0.240227i
$$430$$ 0 0
$$431$$ 30.6054 1.47421 0.737105 0.675778i $$-0.236192\pi$$
0.737105 + 0.675778i $$0.236192\pi$$
$$432$$ 0 0
$$433$$ 15.3137 0.735930 0.367965 0.929840i $$-0.380055\pi$$
0.367965 + 0.929840i $$0.380055\pi$$
$$434$$ 0 0
$$435$$ 9.97474i 0.478252i
$$436$$ 0 0
$$437$$ 0.635767i 0.0304128i
$$438$$ 0 0
$$439$$ −33.3676 −1.59255 −0.796274 0.604936i $$-0.793199\pi$$
−0.796274 + 0.604936i $$0.793199\pi$$
$$440$$ 0 0
$$441$$ 2.33897 0.111380
$$442$$ 0 0
$$443$$ − 3.23617i − 0.153755i −0.997041 0.0768776i $$-0.975505\pi$$
0.997041 0.0768776i $$-0.0244951\pi$$
$$444$$ 0 0
$$445$$ 5.42522i 0.257180i
$$446$$ 0 0
$$447$$ −1.45479 −0.0688091
$$448$$ 0 0
$$449$$ −27.4165 −1.29387 −0.646933 0.762547i $$-0.723948\pi$$
−0.646933 + 0.762547i $$0.723948\pi$$
$$450$$ 0 0
$$451$$ − 14.9550i − 0.704203i
$$452$$ 0 0
$$453$$ 2.03696i 0.0957049i
$$454$$ 0 0
$$455$$ 16.0454 0.752221
$$456$$ 0 0
$$457$$ 10.9147 0.510567 0.255284 0.966866i $$-0.417831\pi$$
0.255284 + 0.966866i $$0.417831\pi$$
$$458$$ 0 0
$$459$$ 0.224777i 0.0104917i
$$460$$ 0 0
$$461$$ 25.2181i 1.17452i 0.809398 + 0.587261i $$0.199794\pi$$
−0.809398 + 0.587261i $$0.800206\pi$$
$$462$$ 0 0
$$463$$ −22.4937 −1.04537 −0.522686 0.852525i $$-0.675070\pi$$
−0.522686 + 0.852525i $$0.675070\pi$$
$$464$$ 0 0
$$465$$ 6.99222 0.324256
$$466$$ 0 0
$$467$$ 34.2482i 1.58482i 0.609991 + 0.792408i $$0.291173\pi$$
−0.609991 + 0.792408i $$0.708827\pi$$
$$468$$ 0 0
$$469$$ − 31.8275i − 1.46966i
$$470$$ 0 0
$$471$$ −8.61790 −0.397092
$$472$$ 0 0
$$473$$ 27.8852 1.28216
$$474$$ 0 0
$$475$$ − 2.11837i − 0.0971974i
$$476$$ 0 0
$$477$$ − 10.6264i − 0.486548i
$$478$$ 0 0
$$479$$ −36.2362 −1.65568 −0.827838 0.560968i $$-0.810429\pi$$
−0.827838 + 0.560968i $$0.810429\pi$$
$$480$$ 0 0
$$481$$ −10.1551 −0.463032
$$482$$ 0 0
$$483$$ 6.10641i 0.277851i
$$484$$ 0 0
$$485$$ − 62.2824i − 2.82810i
$$486$$ 0 0
$$487$$ −16.8200 −0.762186 −0.381093 0.924537i $$-0.624452\pi$$
−0.381093 + 0.924537i $$0.624452\pi$$
$$488$$ 0 0
$$489$$ −4.86054 −0.219801
$$490$$ 0 0
$$491$$ 8.63577i 0.389727i 0.980830 + 0.194863i $$0.0624263\pi$$
−0.980830 + 0.194863i $$0.937574\pi$$
$$492$$ 0 0
$$493$$ 0.590346i 0.0265879i
$$494$$ 0 0
$$495$$ 9.65685 0.434043
$$496$$ 0 0
$$497$$ −9.32206 −0.418151
$$498$$ 0 0
$$499$$ − 27.8275i − 1.24573i −0.782329 0.622865i $$-0.785969\pi$$
0.782329 0.622865i $$-0.214031\pi$$
$$500$$ 0 0
$$501$$ − 21.7023i − 0.969586i
$$502$$ 0 0
$$503$$ 25.7308 1.14728 0.573639 0.819108i $$-0.305531\pi$$
0.573639 + 0.819108i $$0.305531\pi$$
$$504$$ 0 0
$$505$$ 0.439861 0.0195736
$$506$$ 0 0
$$507$$ − 9.17064i − 0.407283i
$$508$$ 0 0
$$509$$ − 2.45386i − 0.108765i −0.998520 0.0543826i $$-0.982681\pi$$
0.998520 0.0543826i $$-0.0173191\pi$$
$$510$$ 0 0
$$511$$ 12.8991 0.570623
$$512$$ 0 0
$$513$$ −0.224777 −0.00992417
$$514$$ 0 0
$$515$$ − 50.7050i − 2.23433i
$$516$$ 0 0
$$517$$ 7.19173i 0.316292i
$$518$$ 0 0
$$519$$ −12.3695 −0.542959
$$520$$ 0 0
$$521$$ 33.5944 1.47180 0.735898 0.677092i $$-0.236760\pi$$
0.735898 + 0.677092i $$0.236760\pi$$
$$522$$ 0 0
$$523$$ − 30.8522i − 1.34907i −0.738242 0.674536i $$-0.764344\pi$$
0.738242 0.674536i $$-0.235656\pi$$
$$524$$ 0 0
$$525$$ − 20.3465i − 0.887994i
$$526$$ 0 0
$$527$$ 0.413828 0.0180266
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 5.65685i 0.245487i
$$532$$ 0 0
$$533$$ − 11.5096i − 0.498537i
$$534$$ 0 0
$$535$$ 39.0907 1.69004
$$536$$ 0 0
$$537$$ 11.6413 0.502359
$$538$$ 0 0
$$539$$ − 5.94721i − 0.256164i
$$540$$ 0 0
$$541$$ − 38.4825i − 1.65449i −0.561841 0.827245i $$-0.689907\pi$$
0.561841 0.827245i $$-0.310093\pi$$
$$542$$ 0 0
$$543$$ 9.50732 0.407998
$$544$$ 0 0
$$545$$ 37.8155 1.61984
$$546$$ 0 0
$$547$$ 9.61829i 0.411248i 0.978631 + 0.205624i $$0.0659224\pi$$
−0.978631 + 0.205624i $$0.934078\pi$$
$$548$$ 0 0
$$549$$ 8.46742i 0.361381i
$$550$$ 0 0
$$551$$ −0.590346 −0.0251496
$$552$$ 0 0
$$553$$ 32.4004 1.37780
$$554$$ 0 0
$$555$$ 19.7091i 0.836606i
$$556$$ 0 0
$$557$$ − 6.07174i − 0.257268i −0.991692 0.128634i $$-0.958941\pi$$
0.991692 0.128634i $$-0.0410592\pi$$
$$558$$ 0 0
$$559$$ 21.4609 0.907701
$$560$$ 0 0
$$561$$ 0.571533 0.0241301
$$562$$ 0 0
$$563$$ − 14.2554i − 0.600793i −0.953814 0.300397i $$-0.902881\pi$$
0.953814 0.300397i $$-0.0971191\pi$$
$$564$$ 0 0
$$565$$ − 71.5857i − 3.01163i
$$566$$ 0 0
$$567$$ −2.15894 −0.0906670
$$568$$ 0 0
$$569$$ 32.5018 1.36255 0.681274 0.732029i $$-0.261426\pi$$
0.681274 + 0.732029i $$0.261426\pi$$
$$570$$ 0 0
$$571$$ 12.9706i 0.542801i 0.962466 + 0.271401i $$0.0874868\pi$$
−0.962466 + 0.271401i $$0.912513\pi$$
$$572$$ 0 0
$$573$$ 20.8032i 0.869065i
$$574$$ 0 0
$$575$$ −26.6559 −1.11163
$$576$$ 0 0
$$577$$ 11.7536 0.489308 0.244654 0.969611i $$-0.421326\pi$$
0.244654 + 0.969611i $$0.421326\pi$$
$$578$$ 0 0
$$579$$ − 14.1454i − 0.587862i
$$580$$ 0 0
$$581$$ 30.8857i 1.28135i
$$582$$ 0 0
$$583$$ −27.0192 −1.11902
$$584$$ 0 0
$$585$$ 7.43208 0.307279
$$586$$ 0 0
$$587$$ 9.13585i 0.377077i 0.982066 + 0.188538i $$0.0603750\pi$$
−0.982066 + 0.188538i $$0.939625\pi$$
$$588$$ 0 0
$$589$$ 0.413828i 0.0170515i
$$590$$ 0 0
$$591$$ 3.43463 0.141282
$$592$$ 0 0
$$593$$ −5.49270 −0.225558 −0.112779 0.993620i $$-0.535975\pi$$
−0.112779 + 0.993620i $$0.535975\pi$$
$$594$$ 0 0
$$595$$ − 1.84307i − 0.0755583i
$$596$$ 0 0
$$597$$ 0.306182i 0.0125312i
$$598$$ 0 0
$$599$$ 36.4348 1.48868 0.744342 0.667799i $$-0.232763\pi$$
0.744342 + 0.667799i $$0.232763\pi$$
$$600$$ 0 0
$$601$$ 9.97474 0.406878 0.203439 0.979088i $$-0.434788\pi$$
0.203439 + 0.979088i $$0.434788\pi$$
$$602$$ 0 0
$$603$$ − 14.7422i − 0.600348i
$$604$$ 0 0
$$605$$ 17.2232i 0.700221i
$$606$$ 0 0
$$607$$ 4.51900 0.183421 0.0917103 0.995786i $$-0.470767\pi$$
0.0917103 + 0.995786i $$0.470767\pi$$
$$608$$ 0 0
$$609$$ −5.67016 −0.229766
$$610$$ 0 0
$$611$$ 5.53488i 0.223917i
$$612$$ 0 0
$$613$$ 11.9316i 0.481913i 0.970536 + 0.240957i $$0.0774612\pi$$
−0.970536 + 0.240957i $$0.922539\pi$$
$$614$$ 0 0
$$615$$ −22.3380 −0.900757
$$616$$ 0 0
$$617$$ 32.1201 1.29311 0.646554 0.762869i $$-0.276210\pi$$
0.646554 + 0.762869i $$0.276210\pi$$
$$618$$ 0 0
$$619$$ 21.2715i 0.854975i 0.904021 + 0.427488i $$0.140601\pi$$
−0.904021 + 0.427488i $$0.859399\pi$$
$$620$$ 0 0
$$621$$ 2.82843i 0.113501i
$$622$$ 0 0
$$623$$ −3.08398 −0.123557
$$624$$ 0 0
$$625$$ 16.6958 0.667833
$$626$$ 0 0
$$627$$ 0.571533i 0.0228248i
$$628$$ 0 0
$$629$$ 1.16647i 0.0465101i
$$630$$ 0 0
$$631$$ −36.4685 −1.45179 −0.725894 0.687807i $$-0.758574\pi$$
−0.725894 + 0.687807i $$0.758574\pi$$
$$632$$ 0 0
$$633$$ 10.2284 0.406542
$$634$$ 0 0
$$635$$ − 14.4921i − 0.575103i
$$636$$ 0 0
$$637$$ − 4.57707i − 0.181350i
$$638$$ 0 0
$$639$$ −4.31788 −0.170813
$$640$$ 0 0
$$641$$ 14.0036 0.553109 0.276555 0.960998i $$-0.410807\pi$$
0.276555 + 0.960998i $$0.410807\pi$$
$$642$$ 0 0
$$643$$ − 23.4807i − 0.925990i −0.886361 0.462995i $$-0.846775\pi$$
0.886361 0.462995i $$-0.153225\pi$$
$$644$$ 0 0
$$645$$ − 41.6517i − 1.64004i
$$646$$ 0 0
$$647$$ 12.1908 0.479270 0.239635 0.970863i $$-0.422972\pi$$
0.239635 + 0.970863i $$0.422972\pi$$
$$648$$ 0 0
$$649$$ 14.3835 0.564600
$$650$$ 0 0
$$651$$ 3.97474i 0.155782i
$$652$$ 0 0
$$653$$ 1.39055i 0.0544165i 0.999630 + 0.0272083i $$0.00866173\pi$$
−0.999630 + 0.0272083i $$0.991338\pi$$
$$654$$ 0 0
$$655$$ −4.12198 −0.161059
$$656$$ 0 0
$$657$$ 5.97474 0.233097
$$658$$ 0 0
$$659$$ − 25.5349i − 0.994698i −0.867551 0.497349i $$-0.834307\pi$$
0.867551 0.497349i $$-0.165693\pi$$
$$660$$ 0 0
$$661$$ − 6.44726i − 0.250769i −0.992108 0.125385i $$-0.959983\pi$$
0.992108 0.125385i $$-0.0400165\pi$$
$$662$$ 0 0
$$663$$ 0.439861 0.0170828
$$664$$ 0 0
$$665$$ 1.84307 0.0714710
$$666$$ 0 0
$$667$$ 7.42847i 0.287631i
$$668$$ 0 0
$$669$$ 1.71908i 0.0664635i
$$670$$ 0 0
$$671$$ 21.5298 0.831148
$$672$$ 0 0
$$673$$ −10.8569 −0.418504 −0.209252 0.977862i $$-0.567103\pi$$
−0.209252 + 0.977862i $$0.567103\pi$$
$$674$$ 0 0
$$675$$ − 9.42429i − 0.362741i
$$676$$ 0 0
$$677$$ − 33.5262i − 1.28852i −0.764807 0.644259i $$-0.777166\pi$$
0.764807 0.644259i $$-0.222834\pi$$
$$678$$ 0 0
$$679$$ 35.4045 1.35870
$$680$$ 0 0
$$681$$ −14.3059 −0.548204
$$682$$ 0 0
$$683$$ − 25.2206i − 0.965040i −0.875885 0.482520i $$-0.839722\pi$$
0.875885 0.482520i $$-0.160278\pi$$
$$684$$ 0 0
$$685$$ 20.1674i 0.770557i
$$686$$ 0 0
$$687$$ 16.9981 0.648519
$$688$$ 0 0
$$689$$ −20.7945 −0.792205
$$690$$ 0 0
$$691$$ 15.3523i 0.584028i 0.956414 + 0.292014i $$0.0943254\pi$$
−0.956414 + 0.292014i $$0.905675\pi$$
$$692$$ 0 0
$$693$$ 5.48946i 0.208527i
$$694$$ 0 0
$$695$$ −47.0907 −1.78625
$$696$$ 0 0
$$697$$ −1.32206 −0.0500765
$$698$$ 0 0
$$699$$ 13.3779i 0.506000i
$$700$$ 0 0
$$701$$ 8.61079i 0.325225i 0.986690 + 0.162613i $$0.0519921\pi$$
−0.986690 + 0.162613i $$0.948008\pi$$
$$702$$ 0 0
$$703$$ −1.16647 −0.0439942
$$704$$ 0 0
$$705$$ 10.7422 0.404574
$$706$$ 0 0
$$707$$ 0.250040i 0.00940372i
$$708$$ 0 0
$$709$$ − 32.3624i − 1.21539i −0.794169 0.607697i $$-0.792094\pi$$
0.794169 0.607697i $$-0.207906\pi$$
$$710$$ 0 0
$$711$$ 15.0075 0.562826
$$712$$ 0 0
$$713$$ 5.20730 0.195015
$$714$$ 0 0
$$715$$ − 18.8972i − 0.706717i
$$716$$ 0 0
$$717$$ 13.3675i 0.499218i
$$718$$ 0 0
$$719$$ 1.46744 0.0547262 0.0273631 0.999626i $$-0.491289\pi$$
0.0273631 + 0.999626i $$0.491289\pi$$
$$720$$ 0 0
$$721$$ 28.8233 1.07344
$$722$$ 0 0
$$723$$ 0.211474i 0.00786481i
$$724$$ 0 0
$$725$$ − 24.7516i − 0.919251i
$$726$$ 0 0
$$727$$ 15.3928 0.570889 0.285445 0.958395i $$-0.407859\pi$$
0.285445 + 0.958395i $$0.407859\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 2.46512i − 0.0911759i
$$732$$ 0 0
$$733$$ 17.5624i 0.648683i 0.945940 + 0.324342i $$0.105143\pi$$
−0.945940 + 0.324342i $$0.894857\pi$$
$$734$$ 0 0
$$735$$ −8.88325 −0.327664
$$736$$ 0 0
$$737$$ −37.4844 −1.38075
$$738$$ 0 0
$$739$$ 20.7266i 0.762441i 0.924484 + 0.381220i $$0.124496\pi$$
−0.924484 + 0.381220i $$0.875504\pi$$
$$740$$ 0 0
$$741$$ 0.439861i 0.0161587i
$$742$$ 0 0
$$743$$ 31.7821 1.16597 0.582986 0.812482i $$-0.301884\pi$$
0.582986 + 0.812482i $$0.301884\pi$$
$$744$$ 0 0
$$745$$ 5.52518 0.202427
$$746$$ 0 0
$$747$$ 14.3059i 0.523426i
$$748$$ 0 0
$$749$$ 22.2212i 0.811944i
$$750$$ 0 0
$$751$$ −29.7594 −1.08594 −0.542968 0.839753i $$-0.682699\pi$$
−0.542968 + 0.839753i $$0.682699\pi$$
$$752$$ 0 0
$$753$$ −14.7555 −0.537720
$$754$$ 0 0
$$755$$ − 7.73625i − 0.281551i
$$756$$ 0 0
$$757$$ 22.1119i 0.803672i 0.915712 + 0.401836i $$0.131628\pi$$
−0.915712 + 0.401836i $$0.868372\pi$$
$$758$$ 0 0
$$759$$ 7.19173 0.261043
$$760$$ 0 0
$$761$$ 4.55957 0.165284 0.0826422 0.996579i $$-0.473664\pi$$
0.0826422 + 0.996579i $$0.473664\pi$$
$$762$$ 0 0
$$763$$ 21.4963i 0.778219i
$$764$$ 0 0
$$765$$ − 0.853690i − 0.0308652i
$$766$$ 0 0
$$767$$ 11.0698 0.399706
$$768$$ 0 0
$$769$$ 36.5794 1.31909 0.659543 0.751667i $$-0.270750\pi$$
0.659543 + 0.751667i $$0.270750\pi$$
$$770$$ 0 0
$$771$$ 0.742176i 0.0267288i
$$772$$ 0 0
$$773$$ 26.4611i 0.951739i 0.879516 + 0.475869i $$0.157867\pi$$
−0.879516 + 0.475869i $$0.842133\pi$$
$$774$$ 0 0
$$775$$ −17.3507 −0.623255
$$776$$ 0 0
$$777$$ −11.2037 −0.401930
$$778$$ 0 0
$$779$$ − 1.32206i − 0.0473676i
$$780$$ 0 0
$$781$$ 10.9789i 0.392856i
$$782$$ 0 0
$$783$$ −2.62636 −0.0938584
$$784$$ 0 0
$$785$$ 32.7302 1.16819
$$786$$ 0 0
$$787$$ 18.9164i 0.674298i 0.941451 + 0.337149i $$0.109463\pi$$
−0.941451 + 0.337149i $$0.890537\pi$$
$$788$$ 0 0
$$789$$ 5.48435i 0.195248i
$$790$$ 0 0
$$791$$ 40.6930 1.44688
$$792$$ 0 0
$$793$$ 16.5697 0.588406
$$794$$ 0 0
$$795$$ 40.3582i 1.43136i
$$796$$ 0 0
$$797$$ 47.8065i 1.69339i 0.532075 + 0.846697i $$0.321412\pi$$
−0.532075 + 0.846697i $$0.678588\pi$$
$$798$$ 0 0
$$799$$ 0.635767 0.0224918
$$800$$ 0 0
$$801$$ −1.42847 −0.0504724
$$802$$ 0 0
$$803$$ − 15.1917i − 0.536105i
$$804$$ 0 0
$$805$$ − 23.1917i − 0.817401i
$$806$$ 0 0
$$807$$ 20.4694 0.720558
$$808$$ 0 0
$$809$$ −29.9862 −1.05426 −0.527129 0.849785i $$-0.676732\pi$$
−0.527129 + 0.849785i $$0.676732\pi$$
$$810$$ 0 0
$$811$$ − 11.3899i − 0.399954i −0.979801 0.199977i $$-0.935913\pi$$
0.979801 0.199977i $$-0.0640867\pi$$
$$812$$ 0 0
$$813$$ − 14.0370i − 0.492298i
$$814$$ 0 0
$$815$$ 18.4600 0.646626
$$816$$ 0 0
$$817$$ 2.46512 0.0862438
$$818$$ 0 0
$$819$$ 4.22478i 0.147626i
$$820$$ 0 0
$$821$$ 19.6929i 0.687286i 0.939100 + 0.343643i $$0.111661\pi$$
−0.939100 + 0.343643i $$0.888339\pi$$
$$822$$ 0 0
$$823$$ −22.4666 −0.783137 −0.391568 0.920149i $$-0.628067\pi$$
−0.391568 + 0.920149i $$0.628067\pi$$
$$824$$ 0 0
$$825$$ −23.9628 −0.834277
$$826$$ 0 0
$$827$$ − 10.7927i − 0.375299i −0.982236 0.187649i $$-0.939913\pi$$
0.982236 0.187649i $$-0.0600869\pi$$
$$828$$ 0 0
$$829$$ 45.9421i 1.59563i 0.602900 + 0.797817i $$0.294012\pi$$
−0.602900 + 0.797817i $$0.705988\pi$$
$$830$$ 0 0
$$831$$ 13.4211 0.465572
$$832$$ 0 0
$$833$$ −0.525748 −0.0182161
$$834$$ 0 0
$$835$$ 82.4238i 2.85239i
$$836$$ 0 0
$$837$$ 1.84106i 0.0636363i
$$838$$ 0 0
$$839$$ −22.9142 −0.791085 −0.395542 0.918448i $$-0.629443\pi$$
−0.395542 + 0.918448i $$0.629443\pi$$
$$840$$ 0 0
$$841$$ 22.1022 0.762146
$$842$$ 0 0
$$843$$ 3.89359i 0.134102i
$$844$$ 0 0
$$845$$ 34.8295i 1.19817i
$$846$$ 0 0
$$847$$ −9.79053 −0.336407
$$848$$ 0 0
$$849$$ 17.6569 0.605982
$$850$$ 0 0
$$851$$ 14.6779i 0.503153i
$$852$$ 0 0
$$853$$ 55.0728i 1.88566i 0.333278 + 0.942829i $$0.391845\pi$$
−0.333278 + 0.942829i $$0.608155\pi$$
$$854$$ 0 0
$$855$$ 0.853690 0.0291956
$$856$$ 0 0
$$857$$ 1.79079 0.0611723 0.0305861 0.999532i $$-0.490263\pi$$
0.0305861 + 0.999532i $$0.490263\pi$$
$$858$$ 0 0
$$859$$ − 25.5468i − 0.871647i −0.900032 0.435823i $$-0.856457\pi$$
0.900032 0.435823i $$-0.143543\pi$$
$$860$$ 0 0
$$861$$ − 12.6981i − 0.432750i
$$862$$ 0 0
$$863$$ 42.1150 1.43361 0.716806 0.697273i $$-0.245603\pi$$
0.716806 + 0.697273i $$0.245603\pi$$
$$864$$ 0 0
$$865$$ 46.9784 1.59731
$$866$$ 0 0
$$867$$ 16.9495i 0.575634i
$$868$$ 0 0
$$869$$ − 38.1590i − 1.29446i
$$870$$ 0 0
$$871$$ −28.8486 −0.977497
$$872$$ 0 0
$$873$$ 16.3990 0.555023
$$874$$ 0 0
$$875$$ 36.2771i 1.22639i
$$876$$ 0 0
$$877$$ 1.01044i 0.0341203i 0.999854 + 0.0170601i $$0.00543067\pi$$
−0.999854 + 0.0170601i $$0.994569\pi$$
$$878$$ 0 0
$$879$$ 15.7759 0.532108
$$880$$ 0 0
$$881$$ 44.3972 1.49578 0.747889 0.663823i $$-0.231067\pi$$
0.747889 + 0.663823i $$0.231067\pi$$
$$882$$ 0 0
$$883$$ 1.82389i 0.0613787i 0.999529 + 0.0306894i $$0.00977026\pi$$
−0.999529 + 0.0306894i $$0.990230\pi$$
$$884$$ 0 0
$$885$$ − 21.4844i − 0.722189i
$$886$$ 0 0
$$887$$ −4.38532 −0.147245 −0.0736223 0.997286i $$-0.523456\pi$$
−0.0736223 + 0.997286i $$0.523456\pi$$
$$888$$ 0 0
$$889$$ 8.23808 0.276296
$$890$$ 0 0
$$891$$ 2.54266i 0.0851823i
$$892$$ 0 0
$$893$$ 0.635767i 0.0212751i
$$894$$ 0 0
$$895$$ −44.2128 −1.47787
$$896$$ 0 0
$$897$$ 5.53488 0.184804
$$898$$ 0 0
$$899$$ 4.83528i 0.161266i
$$900$$ 0 0
$$901$$ 2.38857i 0.0795747i
$$902$$ 0 0
$$903$$ 23.6770 0.787922
$$904$$ 0 0
$$905$$ −36.1082 −1.20028
$$906$$ 0 0
$$907$$ − 4.06248i − 0.134893i −0.997723 0.0674463i $$-0.978515\pi$$
0.997723 0.0674463i $$-0.0214851\pi$$
$$908$$ 0 0
$$909$$ 0.115816i 0.00384137i
$$910$$ 0 0
$$911$$ −42.3784 −1.40406 −0.702029 0.712149i $$-0.747722\pi$$
−0.702029 + 0.712149i $$0.747722\pi$$
$$912$$ 0 0
$$913$$ 36.3751 1.20384
$$914$$ 0 0
$$915$$ − 32.1587i − 1.06313i
$$916$$ 0 0
$$917$$ − 2.34315i − 0.0773775i
$$918$$ 0 0
$$919$$ −44.8603 −1.47980 −0.739902 0.672715i $$-0.765128\pi$$
−0.739902 + 0.672715i $$0.765128\pi$$
$$920$$ 0 0
$$921$$ −7.64129 −0.251789
$$922$$ 0 0
$$923$$ 8.44955i 0.278120i
$$924$$ 0 0
$$925$$ − 48.9068i − 1.60804i
$$926$$ 0 0
$$927$$ 13.3507 0.438494
$$928$$ 0 0
$$929$$ −2.96695 −0.0973426 −0.0486713 0.998815i $$-0.515499\pi$$
−0.0486713 + 0.998815i $$0.515499\pi$$
$$930$$ 0 0
$$931$$ − 0.525748i − 0.0172307i
$$932$$ 0 0
$$933$$ 24.1623i 0.791038i
$$934$$ 0 0
$$935$$ −2.17064 −0.0709876
$$936$$ 0 0
$$937$$ −54.7669 −1.78916 −0.894579 0.446910i $$-0.852524\pi$$
−0.894579 + 0.446910i $$0.852524\pi$$
$$938$$ 0 0
$$939$$ − 16.6105i − 0.542063i
$$940$$ 0 0
$$941$$ − 8.89558i − 0.289988i −0.989433 0.144994i $$-0.953684\pi$$
0.989433 0.144994i $$-0.0463162\pi$$
$$942$$ 0 0
$$943$$ −16.6358 −0.541735
$$944$$ 0 0
$$945$$ 8.19951 0.266730
$$946$$ 0 0
$$947$$ − 15.8541i − 0.515189i −0.966253 0.257595i $$-0.917070\pi$$
0.966253 0.257595i $$-0.0829299\pi$$
$$948$$ 0 0
$$949$$ − 11.6918i − 0.379532i
$$950$$ 0 0
$$951$$ −2.56213 −0.0830826
$$952$$ 0 0
$$953$$ 30.2807 0.980887 0.490443 0.871473i $$-0.336835\pi$$
0.490443 + 0.871473i $$0.336835\pi$$
$$954$$ 0 0
$$955$$ − 79.0090i − 2.55667i
$$956$$ 0 0
$$957$$ 6.67794i 0.215867i
$$958$$ 0 0
$$959$$ −11.4642 −0.370198
$$960$$ 0 0
$$961$$ −27.6105 −0.890661
$$962$$ 0 0
$$963$$ 10.2926i 0.331675i
$$964$$ 0 0
$$965$$ 53.7232i 1.72941i
$$966$$ 0 0
$$967$$ 10.5273 0.338537 0.169268 0.985570i $$-0.445860\pi$$
0.169268 + 0.985570i $$0.445860\pi$$
$$968$$ 0 0
$$969$$ 0.0505249 0.00162309
$$970$$ 0 0
$$971$$ 38.7050i 1.24210i 0.783771 + 0.621051i $$0.213294\pi$$
−0.783771 + 0.621051i $$0.786706\pi$$
$$972$$ 0 0
$$973$$ − 26.7688i − 0.858168i
$$974$$ 0 0
$$975$$ −18.4422 −0.590622
$$976$$ 0 0
$$977$$ 16.9009 0.540706 0.270353 0.962761i $$-0.412860\pi$$
0.270353 + 0.962761i $$0.412860\pi$$
$$978$$ 0 0
$$979$$ 3.63211i 0.116083i
$$980$$ 0 0
$$981$$ 9.95687i 0.317899i
$$982$$ 0 0
$$983$$ 10.0798 0.321496 0.160748 0.986995i $$-0.448609\pi$$
0.160748 + 0.986995i $$0.448609\pi$$
$$984$$ 0 0
$$985$$ −13.0445 −0.415632
$$986$$ 0 0
$$987$$ 6.10641i 0.194369i
$$988$$ 0 0
$$989$$ − 31.0192i − 0.986354i
$$990$$ 0 0
$$991$$ −42.3446 −1.34512 −0.672561 0.740042i $$-0.734806\pi$$
−0.672561 + 0.740042i $$0.734806\pi$$
$$992$$ 0 0
$$993$$ 19.1275 0.606993
$$994$$ 0 0
$$995$$ − 1.16286i − 0.0368651i
$$996$$ 0 0
$$997$$ 47.6132i 1.50793i 0.656917 + 0.753963i $$0.271860\pi$$
−0.656917 + 0.753963i $$0.728140\pi$$
$$998$$ 0 0
$$999$$ −5.18944 −0.164186
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 3072.2.d.f.1537.4 8
4.3 odd 2 3072.2.d.i.1537.8 8
8.3 odd 2 3072.2.d.i.1537.1 8
8.5 even 2 inner 3072.2.d.f.1537.5 8
16.3 odd 4 3072.2.a.n.1.4 4
16.5 even 4 3072.2.a.i.1.1 4
16.11 odd 4 3072.2.a.o.1.1 4
16.13 even 4 3072.2.a.t.1.4 4
32.3 odd 8 192.2.j.a.49.1 8
32.5 even 8 48.2.j.a.13.3 8
32.11 odd 8 384.2.j.a.289.4 8
32.13 even 8 384.2.j.b.97.2 8
32.19 odd 8 384.2.j.a.97.4 8
32.21 even 8 384.2.j.b.289.2 8
32.27 odd 8 192.2.j.a.145.1 8
32.29 even 8 48.2.j.a.37.3 yes 8
48.5 odd 4 9216.2.a.bo.1.4 4
48.11 even 4 9216.2.a.bn.1.4 4
48.29 odd 4 9216.2.a.y.1.1 4
48.35 even 4 9216.2.a.x.1.1 4
96.5 odd 8 144.2.k.b.109.2 8
96.11 even 8 1152.2.k.f.289.1 8
96.29 odd 8 144.2.k.b.37.2 8
96.35 even 8 576.2.k.b.433.4 8
96.53 odd 8 1152.2.k.c.289.1 8
96.59 even 8 576.2.k.b.145.4 8
96.77 odd 8 1152.2.k.c.865.1 8
96.83 even 8 1152.2.k.f.865.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.3 8 32.5 even 8
48.2.j.a.37.3 yes 8 32.29 even 8
144.2.k.b.37.2 8 96.29 odd 8
144.2.k.b.109.2 8 96.5 odd 8
192.2.j.a.49.1 8 32.3 odd 8
192.2.j.a.145.1 8 32.27 odd 8
384.2.j.a.97.4 8 32.19 odd 8
384.2.j.a.289.4 8 32.11 odd 8
384.2.j.b.97.2 8 32.13 even 8
384.2.j.b.289.2 8 32.21 even 8
576.2.k.b.145.4 8 96.59 even 8
576.2.k.b.433.4 8 96.35 even 8
1152.2.k.c.289.1 8 96.53 odd 8
1152.2.k.c.865.1 8 96.77 odd 8
1152.2.k.f.289.1 8 96.11 even 8
1152.2.k.f.865.1 8 96.83 even 8
3072.2.a.i.1.1 4 16.5 even 4
3072.2.a.n.1.4 4 16.3 odd 4
3072.2.a.o.1.1 4 16.11 odd 4
3072.2.a.t.1.4 4 16.13 even 4
3072.2.d.f.1537.4 8 1.1 even 1 trivial
3072.2.d.f.1537.5 8 8.5 even 2 inner
3072.2.d.i.1537.1 8 8.3 odd 2
3072.2.d.i.1537.8 8 4.3 odd 2
9216.2.a.x.1.1 4 48.35 even 4
9216.2.a.y.1.1 4 48.29 odd 4
9216.2.a.bn.1.4 4 48.11 even 4
9216.2.a.bo.1.4 4 48.5 odd 4