# Properties

 Label 3072.2.a.n.1.4 Level $3072$ Weight $2$ Character 3072.1 Self dual yes Analytic conductor $24.530$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.68554$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +3.79793 q^{5} -2.15894 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +3.79793 q^{5} -2.15894 q^{7} +1.00000 q^{9} -2.54266 q^{11} -1.95687 q^{13} -3.79793 q^{15} +0.224777 q^{17} +0.224777 q^{19} +2.15894 q^{21} +2.82843 q^{23} +9.42429 q^{25} -1.00000 q^{27} -2.62636 q^{29} -1.84106 q^{31} +2.54266 q^{33} -8.19951 q^{35} +5.18944 q^{37} +1.95687 q^{39} +5.88163 q^{41} +10.9670 q^{43} +3.79793 q^{45} -2.82843 q^{47} -2.33897 q^{49} -0.224777 q^{51} +10.6264 q^{53} -9.65685 q^{55} -0.224777 q^{57} +5.65685 q^{59} +8.46742 q^{61} -2.15894 q^{63} -7.43208 q^{65} +14.7422 q^{67} -2.82843 q^{69} +4.31788 q^{71} +5.97474 q^{73} -9.42429 q^{75} +5.48946 q^{77} +15.0075 q^{79} +1.00000 q^{81} -14.3059 q^{83} +0.853690 q^{85} +2.62636 q^{87} -1.42847 q^{89} +4.22478 q^{91} +1.84106 q^{93} +0.853690 q^{95} -16.3990 q^{97} -2.54266 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + 8q^{13} - 4q^{15} + 4q^{21} + 4q^{25} - 4q^{27} + 12q^{29} - 12q^{31} + 16q^{37} - 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} - 16q^{55} + 16q^{61} - 4q^{63} - 8q^{65} + 16q^{67} + 8q^{71} - 8q^{73} - 4q^{75} + 24q^{77} - 12q^{79} + 4q^{81} + 24q^{85} - 12q^{87} - 8q^{89} + 16q^{91} + 12q^{93} + 24q^{95} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 3.79793 1.69849 0.849244 0.528001i $$-0.177058\pi$$
0.849244 + 0.528001i $$0.177058\pi$$
$$6$$ 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
$$11$$ −2.54266 −0.766641 −0.383321 0.923615i $$-0.625220\pi$$
−0.383321 + 0.923615i $$0.625220\pi$$
$$12$$ 0 0
$$13$$ −1.95687 −0.542739 −0.271370 0.962475i $$-0.587477\pi$$
−0.271370 + 0.962475i $$0.587477\pi$$
$$14$$ 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
$$19$$ 0.224777 0.0515675 0.0257837 0.999668i $$-0.491792\pi$$
0.0257837 + 0.999668i $$0.491792\pi$$
$$20$$ 0 0
$$21$$ 2.15894 0.471120
$$22$$ 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
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.62636 −0.487703 −0.243851 0.969813i $$-0.578411\pi$$
−0.243851 + 0.969813i $$0.578411\pi$$
$$30$$ 0 0
$$31$$ −1.84106 −0.330664 −0.165332 0.986238i $$-0.552870\pi$$
−0.165332 + 0.986238i $$0.552870\pi$$
$$32$$ 0 0
$$33$$ 2.54266 0.442620
$$34$$ 0 0
$$35$$ −8.19951 −1.38597
$$36$$ 0 0
$$37$$ 5.18944 0.853138 0.426569 0.904455i $$-0.359722\pi$$
0.426569 + 0.904455i $$0.359722\pi$$
$$38$$ 0 0
$$39$$ 1.95687 0.313351
$$40$$ 0 0
$$41$$ 5.88163 0.918557 0.459278 0.888292i $$-0.348108\pi$$
0.459278 + 0.888292i $$0.348108\pi$$
$$42$$ 0 0
$$43$$ 10.9670 1.67244 0.836222 0.548391i $$-0.184759\pi$$
0.836222 + 0.548391i $$0.184759\pi$$
$$44$$ 0 0
$$45$$ 3.79793 0.566162
$$46$$ 0 0
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ −2.33897 −0.334139
$$50$$ 0 0
$$51$$ −0.224777 −0.0314751
$$52$$ 0 0
$$53$$ 10.6264 1.45964 0.729821 0.683638i $$-0.239603\pi$$
0.729821 + 0.683638i $$0.239603\pi$$
$$54$$ 0 0
$$55$$ −9.65685 −1.30213
$$56$$ 0 0
$$57$$ −0.224777 −0.0297725
$$58$$ 0 0
$$59$$ 5.65685 0.736460 0.368230 0.929735i $$-0.379964\pi$$
0.368230 + 0.929735i $$0.379964\pi$$
$$60$$ 0 0
$$61$$ 8.46742 1.08414 0.542071 0.840333i $$-0.317640\pi$$
0.542071 + 0.840333i $$0.317640\pi$$
$$62$$ 0 0
$$63$$ −2.15894 −0.272001
$$64$$ 0 0
$$65$$ −7.43208 −0.921836
$$66$$ 0 0
$$67$$ 14.7422 1.80104 0.900522 0.434811i $$-0.143185\pi$$
0.900522 + 0.434811i $$0.143185\pi$$
$$68$$ 0 0
$$69$$ −2.82843 −0.340503
$$70$$ 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 $$-0.386302\pi$$
0.349645 + 0.936882i $$0.386302\pi$$
$$74$$ 0 0
$$75$$ −9.42429 −1.08822
$$76$$ 0 0
$$77$$ 5.48946 0.625582
$$78$$ 0 0
$$79$$ 15.0075 1.68848 0.844239 0.535966i $$-0.180053\pi$$
0.844239 + 0.535966i $$0.180053\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −14.3059 −1.57028 −0.785140 0.619319i $$-0.787409\pi$$
−0.785140 + 0.619319i $$0.787409\pi$$
$$84$$ 0 0
$$85$$ 0.853690 0.0925956
$$86$$ 0 0
$$87$$ 2.62636 0.281575
$$88$$ 0 0
$$89$$ −1.42847 −0.151417 −0.0757086 0.997130i $$-0.524122\pi$$
−0.0757086 + 0.997130i $$0.524122\pi$$
$$90$$ 0 0
$$91$$ 4.22478 0.442877
$$92$$ 0 0
$$93$$ 1.84106 0.190909
$$94$$ 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
$$99$$ −2.54266 −0.255547
$$100$$ 0 0
$$101$$ −0.115816 −0.0115241 −0.00576206 0.999983i $$-0.501834\pi$$
−0.00576206 + 0.999983i $$0.501834\pi$$
$$102$$ 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
$$107$$ 10.2926 0.995025 0.497513 0.867457i $$-0.334247\pi$$
0.497513 + 0.867457i $$0.334247\pi$$
$$108$$ 0 0
$$109$$ 9.95687 0.953696 0.476848 0.878986i $$-0.341779\pi$$
0.476848 + 0.878986i $$0.341779\pi$$
$$110$$ 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
$$115$$ 10.7422 1.00171
$$116$$ 0 0
$$117$$ −1.95687 −0.180913
$$118$$ 0 0
$$119$$ −0.485281 −0.0444857
$$120$$ 0 0
$$121$$ −4.53488 −0.412261
$$122$$ 0 0
$$123$$ −5.88163 −0.530329
$$124$$ 0 0
$$125$$ 16.8032 1.50292
$$126$$ 0 0
$$127$$ 3.81580 0.338597 0.169299 0.985565i $$-0.445850\pi$$
0.169299 + 0.985565i $$0.445850\pi$$
$$128$$ 0 0
$$129$$ −10.9670 −0.965586
$$130$$ 0 0
$$131$$ 1.08532 0.0948250 0.0474125 0.998875i $$-0.484902\pi$$
0.0474125 + 0.998875i $$0.484902\pi$$
$$132$$ 0 0
$$133$$ −0.485281 −0.0420792
$$134$$ 0 0
$$135$$ −3.79793 −0.326874
$$136$$ 0 0
$$137$$ −5.31010 −0.453672 −0.226836 0.973933i $$-0.572838\pi$$
−0.226836 + 0.973933i $$0.572838\pi$$
$$138$$ 0 0
$$139$$ −12.3990 −1.05167 −0.525836 0.850586i $$-0.676247\pi$$
−0.525836 + 0.850586i $$0.676247\pi$$
$$140$$ 0 0
$$141$$ 2.82843 0.238197
$$142$$ 0 0
$$143$$ 4.97567 0.416086
$$144$$ 0 0
$$145$$ −9.97474 −0.828357
$$146$$ 0 0
$$147$$ 2.33897 0.192915
$$148$$ 0 0
$$149$$ −1.45479 −0.119181 −0.0595904 0.998223i $$-0.518979\pi$$
−0.0595904 + 0.998223i $$0.518979\pi$$
$$150$$ 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
$$155$$ −6.99222 −0.561628
$$156$$ 0 0
$$157$$ 8.61790 0.687784 0.343892 0.939009i $$-0.388255\pi$$
0.343892 + 0.939009i $$0.388255\pi$$
$$158$$ 0 0
$$159$$ −10.6264 −0.842725
$$160$$ 0 0
$$161$$ −6.10641 −0.481252
$$162$$ 0 0
$$163$$ −4.86054 −0.380707 −0.190354 0.981716i $$-0.560963\pi$$
−0.190354 + 0.981716i $$0.560963\pi$$
$$164$$ 0 0
$$165$$ 9.65685 0.751785
$$166$$ 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
$$171$$ 0.224777 0.0171892
$$172$$ 0 0
$$173$$ 12.3695 0.940433 0.470217 0.882551i $$-0.344176\pi$$
0.470217 + 0.882551i $$0.344176\pi$$
$$174$$ 0 0
$$175$$ −20.3465 −1.53805
$$176$$ 0 0
$$177$$ −5.65685 −0.425195
$$178$$ 0 0
$$179$$ 11.6413 0.870111 0.435055 0.900404i $$-0.356729\pi$$
0.435055 + 0.900404i $$0.356729\pi$$
$$180$$ 0 0
$$181$$ 9.50732 0.706673 0.353337 0.935496i $$-0.385047\pi$$
0.353337 + 0.935496i $$0.385047\pi$$
$$182$$ 0 0
$$183$$ −8.46742 −0.625930
$$184$$ 0 0
$$185$$ 19.7091 1.44904
$$186$$ 0 0
$$187$$ −0.571533 −0.0417946
$$188$$ 0 0
$$189$$ 2.15894 0.157040
$$190$$ 0 0
$$191$$ 20.8032 1.50526 0.752632 0.658441i $$-0.228784\pi$$
0.752632 + 0.658441i $$0.228784\pi$$
$$192$$ 0 0
$$193$$ 14.1454 1.01821 0.509103 0.860705i $$-0.329977\pi$$
0.509103 + 0.860705i $$0.329977\pi$$
$$194$$ 0 0
$$195$$ 7.43208 0.532222
$$196$$ 0 0
$$197$$ 3.43463 0.244707 0.122354 0.992487i $$-0.460956\pi$$
0.122354 + 0.992487i $$0.460956\pi$$
$$198$$ 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
$$203$$ 5.67016 0.397967
$$204$$ 0 0
$$205$$ 22.3380 1.56016
$$206$$ 0 0
$$207$$ 2.82843 0.196589
$$208$$ 0 0
$$209$$ −0.571533 −0.0395337
$$210$$ 0 0
$$211$$ 10.2284 0.704151 0.352076 0.935972i $$-0.385476\pi$$
0.352076 + 0.935972i $$0.385476\pi$$
$$212$$ 0 0
$$213$$ −4.31788 −0.295856
$$214$$ 0 0
$$215$$ 41.6517 2.84063
$$216$$ 0 0
$$217$$ 3.97474 0.269823
$$218$$ 0 0
$$219$$ −5.97474 −0.403735
$$220$$ 0 0
$$221$$ −0.439861 −0.0295883
$$222$$ 0 0
$$223$$ 1.71908 0.115118 0.0575591 0.998342i $$-0.481668\pi$$
0.0575591 + 0.998342i $$0.481668\pi$$
$$224$$ 0 0
$$225$$ 9.42429 0.628286
$$226$$ 0 0
$$227$$ −14.3059 −0.949518 −0.474759 0.880116i $$-0.657465\pi$$
−0.474759 + 0.880116i $$0.657465\pi$$
$$228$$ 0 0
$$229$$ 16.9981 1.12327 0.561634 0.827386i $$-0.310173\pi$$
0.561634 + 0.827386i $$0.310173\pi$$
$$230$$ 0 0
$$231$$ −5.48946 −0.361180
$$232$$ 0 0
$$233$$ 13.3779 0.876418 0.438209 0.898873i $$-0.355613\pi$$
0.438209 + 0.898873i $$0.355613\pi$$
$$234$$ 0 0
$$235$$ −10.7422 −0.700742
$$236$$ 0 0
$$237$$ −15.0075 −0.974844
$$238$$ 0 0
$$239$$ 13.3675 0.864670 0.432335 0.901713i $$-0.357690\pi$$
0.432335 + 0.901713i $$0.357690\pi$$
$$240$$ 0 0
$$241$$ −0.211474 −0.0136222 −0.00681112 0.999977i $$-0.502168\pi$$
−0.00681112 + 0.999977i $$0.502168\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −8.88325 −0.567530
$$246$$ 0 0
$$247$$ −0.439861 −0.0279877
$$248$$ 0 0
$$249$$ 14.3059 0.906601
$$250$$ 0 0
$$251$$ 14.7555 0.931358 0.465679 0.884954i $$-0.345810\pi$$
0.465679 + 0.884954i $$0.345810\pi$$
$$252$$ 0 0
$$253$$ −7.19173 −0.452140
$$254$$ 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
$$259$$ −11.2037 −0.696163
$$260$$ 0 0
$$261$$ −2.62636 −0.162568
$$262$$ 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
$$267$$ 1.42847 0.0874208
$$268$$ 0 0
$$269$$ −20.4694 −1.24804 −0.624021 0.781407i $$-0.714502\pi$$
−0.624021 + 0.781407i $$0.714502\pi$$
$$270$$ 0 0
$$271$$ −14.0370 −0.852685 −0.426342 0.904562i $$-0.640198\pi$$
−0.426342 + 0.904562i $$0.640198\pi$$
$$272$$ 0 0
$$273$$ −4.22478 −0.255695
$$274$$ 0 0
$$275$$ −23.9628 −1.44501
$$276$$ 0 0
$$277$$ 13.4211 0.806394 0.403197 0.915113i $$-0.367899\pi$$
0.403197 + 0.915113i $$0.367899\pi$$
$$278$$ 0 0
$$279$$ −1.84106 −0.110221
$$280$$ 0 0
$$281$$ 3.89359 0.232272 0.116136 0.993233i $$-0.462949\pi$$
0.116136 + 0.993233i $$0.462949\pi$$
$$282$$ 0 0
$$283$$ −17.6569 −1.04959 −0.524796 0.851228i $$-0.675858\pi$$
−0.524796 + 0.851228i $$0.675858\pi$$
$$284$$ 0 0
$$285$$ −0.853690 −0.0505682
$$286$$ 0 0
$$287$$ −12.6981 −0.749545
$$288$$ 0 0
$$289$$ −16.9495 −0.997028
$$290$$ 0 0
$$291$$ 16.3990 0.961328
$$292$$ 0 0
$$293$$ 15.7759 0.921639 0.460819 0.887494i $$-0.347556\pi$$
0.460819 + 0.887494i $$0.347556\pi$$
$$294$$ 0 0
$$295$$ 21.4844 1.25087
$$296$$ 0 0
$$297$$ 2.54266 0.147540
$$298$$ 0 0
$$299$$ −5.53488 −0.320090
$$300$$ 0 0
$$301$$ −23.6770 −1.36472
$$302$$ 0 0
$$303$$ 0.115816 0.00665345
$$304$$ 0 0
$$305$$ 32.1587 1.84140
$$306$$ 0 0
$$307$$ −7.64129 −0.436111 −0.218056 0.975936i $$-0.569971\pi$$
−0.218056 + 0.975936i $$0.569971\pi$$
$$308$$ 0 0
$$309$$ 13.3507 0.759493
$$310$$ 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 $$-0.655544\pi$$
−0.469441 + 0.882964i $$0.655544\pi$$
$$314$$ 0 0
$$315$$ −8.19951 −0.461990
$$316$$ 0 0
$$317$$ 2.56213 0.143903 0.0719517 0.997408i $$-0.477077\pi$$
0.0719517 + 0.997408i $$0.477077\pi$$
$$318$$ 0 0
$$319$$ 6.67794 0.373893
$$320$$ 0 0
$$321$$ −10.2926 −0.574478
$$322$$ 0 0
$$323$$ 0.0505249 0.00281128
$$324$$ 0 0
$$325$$ −18.4422 −1.02299
$$326$$ 0 0
$$327$$ −9.95687 −0.550616
$$328$$ 0 0
$$329$$ 6.10641 0.336657
$$330$$ 0 0
$$331$$ −19.1275 −1.05134 −0.525671 0.850688i $$-0.676186\pi$$
−0.525671 + 0.850688i $$0.676186\pi$$
$$332$$ 0 0
$$333$$ 5.18944 0.284379
$$334$$ 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
$$339$$ 18.8486 1.02371
$$340$$ 0 0
$$341$$ 4.68119 0.253500
$$342$$ 0 0
$$343$$ 20.1623 1.08866
$$344$$ 0 0
$$345$$ −10.7422 −0.578339
$$346$$ 0 0
$$347$$ 29.4068 1.57864 0.789320 0.613982i $$-0.210433\pi$$
0.789320 + 0.613982i $$0.210433\pi$$
$$348$$ 0 0
$$349$$ −27.2738 −1.45993 −0.729967 0.683482i $$-0.760465\pi$$
−0.729967 + 0.683482i $$0.760465\pi$$
$$350$$ 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
$$355$$ 16.3990 0.870370
$$356$$ 0 0
$$357$$ 0.485281 0.0256838
$$358$$ 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
$$363$$ 4.53488 0.238019
$$364$$ 0 0
$$365$$ 22.6917 1.18774
$$366$$ 0 0
$$367$$ −27.4474 −1.43274 −0.716371 0.697720i $$-0.754198\pi$$
−0.716371 + 0.697720i $$0.754198\pi$$
$$368$$ 0 0
$$369$$ 5.88163 0.306186
$$370$$ 0 0
$$371$$ −22.9417 −1.19107
$$372$$ 0 0
$$373$$ 17.8518 0.924332 0.462166 0.886794i $$-0.347072\pi$$
0.462166 + 0.886794i $$0.347072\pi$$
$$374$$ 0 0
$$375$$ −16.8032 −0.867712
$$376$$ 0 0
$$377$$ 5.13946 0.264695
$$378$$ 0 0
$$379$$ −16.5018 −0.847642 −0.423821 0.905746i $$-0.639311\pi$$
−0.423821 + 0.905746i $$0.639311\pi$$
$$380$$ 0 0
$$381$$ −3.81580 −0.195489
$$382$$ 0 0
$$383$$ 17.1885 0.878291 0.439145 0.898416i $$-0.355281\pi$$
0.439145 + 0.898416i $$0.355281\pi$$
$$384$$ 0 0
$$385$$ 20.8486 1.06254
$$386$$ 0 0
$$387$$ 10.9670 0.557482
$$388$$ 0 0
$$389$$ 2.66209 0.134973 0.0674866 0.997720i $$-0.478502\pi$$
0.0674866 + 0.997720i $$0.478502\pi$$
$$390$$ 0 0
$$391$$ 0.635767 0.0321521
$$392$$ 0 0
$$393$$ −1.08532 −0.0547472
$$394$$ 0 0
$$395$$ 56.9976 2.86786
$$396$$ 0 0
$$397$$ −11.8959 −0.597037 −0.298519 0.954404i $$-0.596493\pi$$
−0.298519 + 0.954404i $$0.596493\pi$$
$$398$$ 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
$$403$$ 3.60272 0.179464
$$404$$ 0 0
$$405$$ 3.79793 0.188721
$$406$$ 0 0
$$407$$ −13.1950 −0.654051
$$408$$ 0 0
$$409$$ 13.7211 0.678464 0.339232 0.940703i $$-0.389833\pi$$
0.339232 + 0.940703i $$0.389833\pi$$
$$410$$ 0 0
$$411$$ 5.31010 0.261928
$$412$$ 0 0
$$413$$ −12.2128 −0.600953
$$414$$ 0 0
$$415$$ −54.3329 −2.66710
$$416$$ 0 0
$$417$$ 12.3990 0.607183
$$418$$ 0 0
$$419$$ 13.1629 0.643048 0.321524 0.946901i $$-0.395805\pi$$
0.321524 + 0.946901i $$0.395805\pi$$
$$420$$ 0 0
$$421$$ −11.9413 −0.581984 −0.290992 0.956726i $$-0.593985\pi$$
−0.290992 + 0.956726i $$0.593985\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ 2.11837 0.102756
$$426$$ 0 0
$$427$$ −18.2807 −0.884663
$$428$$ 0 0
$$429$$ −4.97567 −0.240227
$$430$$ 0 0
$$431$$ −30.6054 −1.47421 −0.737105 0.675778i $$-0.763808\pi$$
−0.737105 + 0.675778i $$0.763808\pi$$
$$432$$ 0 0
$$433$$ 15.3137 0.735930 0.367965 0.929840i $$-0.380055\pi$$
0.367965 + 0.929840i $$0.380055\pi$$
$$434$$ 0 0
$$435$$ 9.97474 0.478252
$$436$$ 0 0
$$437$$ 0.635767 0.0304128
$$438$$ 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
$$443$$ 3.23617 0.153755 0.0768776 0.997041i $$-0.475505\pi$$
0.0768776 + 0.997041i $$0.475505\pi$$
$$444$$ 0 0
$$445$$ −5.42522 −0.257180
$$446$$ 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
$$451$$ −14.9550 −0.704203
$$452$$ 0 0
$$453$$ 2.03696 0.0957049
$$454$$ 0 0
$$455$$ 16.0454 0.752221
$$456$$ 0 0
$$457$$ −10.9147 −0.510567 −0.255284 0.966866i $$-0.582169\pi$$
−0.255284 + 0.966866i $$0.582169\pi$$
$$458$$ 0 0
$$459$$ −0.224777 −0.0104917
$$460$$ 0 0
$$461$$ −25.2181 −1.17452 −0.587261 0.809398i $$-0.699794\pi$$
−0.587261 + 0.809398i $$0.699794\pi$$
$$462$$ 0 0
$$463$$ 22.4937 1.04537 0.522686 0.852525i $$-0.324930\pi$$
0.522686 + 0.852525i $$0.324930\pi$$
$$464$$ 0 0
$$465$$ 6.99222 0.324256
$$466$$ 0 0
$$467$$ 34.2482 1.58482 0.792408 0.609991i $$-0.208827\pi$$
0.792408 + 0.609991i $$0.208827\pi$$
$$468$$ 0 0
$$469$$ −31.8275 −1.46966
$$470$$ 0 0
$$471$$ −8.61790 −0.397092
$$472$$ 0 0
$$473$$ −27.8852 −1.28216
$$474$$ 0 0
$$475$$ 2.11837 0.0971974
$$476$$ 0 0
$$477$$ 10.6264 0.486548
$$478$$ 0 0
$$479$$ 36.2362 1.65568 0.827838 0.560968i $$-0.189571\pi$$
0.827838 + 0.560968i $$0.189571\pi$$
$$480$$ 0 0
$$481$$ −10.1551 −0.463032
$$482$$ 0 0
$$483$$ 6.10641 0.277851
$$484$$ 0 0
$$485$$ −62.2824 −2.82810
$$486$$ 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
$$491$$ −8.63577 −0.389727 −0.194863 0.980830i $$-0.562426\pi$$
−0.194863 + 0.980830i $$0.562426\pi$$
$$492$$ 0 0
$$493$$ −0.590346 −0.0265879
$$494$$ 0 0
$$495$$ −9.65685 −0.434043
$$496$$ 0 0
$$497$$ −9.32206 −0.418151
$$498$$ 0 0
$$499$$ −27.8275 −1.24573 −0.622865 0.782329i $$-0.714031\pi$$
−0.622865 + 0.782329i $$0.714031\pi$$
$$500$$ 0 0
$$501$$ −21.7023 −0.969586
$$502$$ 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
$$507$$ 9.17064 0.407283
$$508$$ 0 0
$$509$$ 2.45386 0.108765 0.0543826 0.998520i $$-0.482681\pi$$
0.0543826 + 0.998520i $$0.482681\pi$$
$$510$$ 0 0
$$511$$ −12.8991 −0.570623
$$512$$ 0 0
$$513$$ −0.224777 −0.00992417
$$514$$ 0 0
$$515$$ −50.7050 −2.23433
$$516$$ 0 0
$$517$$ 7.19173 0.316292
$$518$$ 0 0
$$519$$ −12.3695 −0.542959
$$520$$ 0 0
$$521$$ −33.5944 −1.47180 −0.735898 0.677092i $$-0.763240\pi$$
−0.735898 + 0.677092i $$0.763240\pi$$
$$522$$ 0 0
$$523$$ 30.8522 1.34907 0.674536 0.738242i $$-0.264344\pi$$
0.674536 + 0.738242i $$0.264344\pi$$
$$524$$ 0 0
$$525$$ 20.3465 0.887994
$$526$$ 0 0
$$527$$ −0.413828 −0.0180266
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 5.65685 0.245487
$$532$$ 0 0
$$533$$ −11.5096 −0.498537
$$534$$ 0 0
$$535$$ 39.0907 1.69004
$$536$$ 0 0
$$537$$ −11.6413 −0.502359
$$538$$ 0 0
$$539$$ 5.94721 0.256164
$$540$$ 0 0
$$541$$ 38.4825 1.65449 0.827245 0.561841i $$-0.189907\pi$$
0.827245 + 0.561841i $$0.189907\pi$$
$$542$$ 0 0
$$543$$ −9.50732 −0.407998
$$544$$ 0 0
$$545$$ 37.8155 1.61984
$$546$$ 0 0
$$547$$ 9.61829 0.411248 0.205624 0.978631i $$-0.434078\pi$$
0.205624 + 0.978631i $$0.434078\pi$$
$$548$$ 0 0
$$549$$ 8.46742 0.361381
$$550$$ 0 0
$$551$$ −0.590346 −0.0251496
$$552$$ 0 0
$$553$$ −32.4004 −1.37780
$$554$$ 0 0
$$555$$ −19.7091 −0.836606
$$556$$ 0 0
$$557$$ 6.07174 0.257268 0.128634 0.991692i $$-0.458941\pi$$
0.128634 + 0.991692i $$0.458941\pi$$
$$558$$ 0 0
$$559$$ −21.4609 −0.907701
$$560$$ 0 0
$$561$$ 0.571533 0.0241301
$$562$$ 0 0
$$563$$ −14.2554 −0.600793 −0.300397 0.953814i $$-0.597119\pi$$
−0.300397 + 0.953814i $$0.597119\pi$$
$$564$$ 0 0
$$565$$ −71.5857 −3.01163
$$566$$ 0 0
$$567$$ −2.15894 −0.0906670
$$568$$ 0 0
$$569$$ −32.5018 −1.36255 −0.681274 0.732029i $$-0.738574\pi$$
−0.681274 + 0.732029i $$0.738574\pi$$
$$570$$ 0 0
$$571$$ −12.9706 −0.542801 −0.271401 0.962466i $$-0.587487\pi$$
−0.271401 + 0.962466i $$0.587487\pi$$
$$572$$ 0 0
$$573$$ −20.8032 −0.869065
$$574$$ 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
$$579$$ −14.1454 −0.587862
$$580$$ 0 0
$$581$$ 30.8857 1.28135
$$582$$ 0 0
$$583$$ −27.0192 −1.11902
$$584$$ 0 0
$$585$$ −7.43208 −0.307279
$$586$$ 0 0
$$587$$ −9.13585 −0.377077 −0.188538 0.982066i $$-0.560375\pi$$
−0.188538 + 0.982066i $$0.560375\pi$$
$$588$$ 0 0
$$589$$ −0.413828 −0.0170515
$$590$$ 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
$$595$$ −1.84307 −0.0755583
$$596$$ 0 0
$$597$$ 0.306182 0.0125312
$$598$$ 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 $$-0.565212\pi$$
−0.203439 + 0.979088i $$0.565212\pi$$
$$602$$ 0 0
$$603$$ 14.7422 0.600348
$$604$$ 0 0
$$605$$ −17.2232 −0.700221
$$606$$ 0 0
$$607$$ −4.51900 −0.183421 −0.0917103 0.995786i $$-0.529233\pi$$
−0.0917103 + 0.995786i $$0.529233\pi$$
$$608$$ 0 0
$$609$$ −5.67016 −0.229766
$$610$$ 0 0
$$611$$ 5.53488 0.223917
$$612$$ 0 0
$$613$$ 11.9316 0.481913 0.240957 0.970536i $$-0.422539\pi$$
0.240957 + 0.970536i $$0.422539\pi$$
$$614$$ 0 0
$$615$$ −22.3380 −0.900757
$$616$$ 0 0
$$617$$ −32.1201 −1.29311 −0.646554 0.762869i $$-0.723790\pi$$
−0.646554 + 0.762869i $$0.723790\pi$$
$$618$$ 0 0
$$619$$ −21.2715 −0.854975 −0.427488 0.904021i $$-0.640601\pi$$
−0.427488 + 0.904021i $$0.640601\pi$$
$$620$$ 0 0
$$621$$ −2.82843 −0.113501
$$622$$ 0 0
$$623$$ 3.08398 0.123557
$$624$$ 0 0
$$625$$ 16.6958 0.667833
$$626$$ 0 0
$$627$$ 0.571533 0.0228248
$$628$$ 0 0
$$629$$ 1.16647 0.0465101
$$630$$ 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
$$635$$ 14.4921 0.575103
$$636$$ 0 0
$$637$$ 4.57707 0.181350
$$638$$ 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
$$643$$ −23.4807 −0.925990 −0.462995 0.886361i $$-0.653225\pi$$
−0.462995 + 0.886361i $$0.653225\pi$$
$$644$$ 0 0
$$645$$ −41.6517 −1.64004
$$646$$ 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
$$651$$ −3.97474 −0.155782
$$652$$ 0 0
$$653$$ −1.39055 −0.0544165 −0.0272083 0.999630i $$-0.508662\pi$$
−0.0272083 + 0.999630i $$0.508662\pi$$
$$654$$ 0 0
$$655$$ 4.12198 0.161059
$$656$$ 0 0
$$657$$ 5.97474 0.233097
$$658$$ 0 0
$$659$$ −25.5349 −0.994698 −0.497349 0.867551i $$-0.665693\pi$$
−0.497349 + 0.867551i $$0.665693\pi$$
$$660$$ 0 0
$$661$$ −6.44726 −0.250769 −0.125385 0.992108i $$-0.540017\pi$$
−0.125385 + 0.992108i $$0.540017\pi$$
$$662$$ 0 0
$$663$$ 0.439861 0.0170828
$$664$$ 0 0
$$665$$ −1.84307 −0.0714710
$$666$$ 0 0
$$667$$ −7.42847 −0.287631
$$668$$ 0 0
$$669$$ −1.71908 −0.0664635
$$670$$ 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
$$675$$ −9.42429 −0.362741
$$676$$ 0 0
$$677$$ −33.5262 −1.28852 −0.644259 0.764807i $$-0.722834\pi$$
−0.644259 + 0.764807i $$0.722834\pi$$
$$678$$ 0 0
$$679$$ 35.4045 1.35870
$$680$$ 0 0
$$681$$ 14.3059 0.548204
$$682$$ 0 0
$$683$$ 25.2206 0.965040 0.482520 0.875885i $$-0.339722\pi$$
0.482520 + 0.875885i $$0.339722\pi$$
$$684$$ 0 0
$$685$$ −20.1674 −0.770557
$$686$$ 0 0
$$687$$ −16.9981 −0.648519
$$688$$ 0 0
$$689$$ −20.7945 −0.792205
$$690$$ 0 0
$$691$$ 15.3523 0.584028 0.292014 0.956414i $$-0.405675\pi$$
0.292014 + 0.956414i $$0.405675\pi$$
$$692$$ 0 0
$$693$$ 5.48946 0.208527
$$694$$ 0 0
$$695$$ −47.0907 −1.78625
$$696$$ 0 0
$$697$$ 1.32206 0.0500765
$$698$$ 0 0
$$699$$ −13.3779 −0.506000
$$700$$ 0 0
$$701$$ −8.61079 −0.325225 −0.162613 0.986690i $$-0.551992\pi$$
−0.162613 + 0.986690i $$0.551992\pi$$
$$702$$ 0 0
$$703$$ 1.16647 0.0439942
$$704$$ 0 0
$$705$$ 10.7422 0.404574
$$706$$ 0 0
$$707$$ 0.250040 0.00940372
$$708$$ 0 0
$$709$$ −32.3624 −1.21539 −0.607697 0.794169i $$-0.707906\pi$$
−0.607697 + 0.794169i $$0.707906\pi$$
$$710$$ 0 0
$$711$$ 15.0075 0.562826
$$712$$ 0 0
$$713$$ −5.20730 −0.195015
$$714$$ 0 0
$$715$$ 18.8972 0.706717
$$716$$ 0 0
$$717$$ −13.3675 −0.499218
$$718$$ 0 0
$$719$$ −1.46744 −0.0547262 −0.0273631 0.999626i $$-0.508711\pi$$
−0.0273631 + 0.999626i $$0.508711\pi$$
$$720$$ 0 0
$$721$$ 28.8233 1.07344
$$722$$ 0 0
$$723$$ 0.211474 0.00786481
$$724$$ 0 0
$$725$$ −24.7516 −0.919251
$$726$$ 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
$$731$$ 2.46512 0.0911759
$$732$$ 0 0
$$733$$ −17.5624 −0.648683 −0.324342 0.945940i $$-0.605143\pi$$
−0.324342 + 0.945940i $$0.605143\pi$$
$$734$$ 0 0
$$735$$ 8.88325 0.327664
$$736$$ 0 0
$$737$$ −37.4844 −1.38075
$$738$$ 0 0
$$739$$ 20.7266 0.762441 0.381220 0.924484i $$-0.375504\pi$$
0.381220 + 0.924484i $$0.375504\pi$$
$$740$$ 0 0
$$741$$ 0.439861 0.0161587
$$742$$ 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
$$747$$ −14.3059 −0.523426
$$748$$ 0 0
$$749$$ −22.2212 −0.811944
$$750$$ 0 0
$$751$$ 29.7594 1.08594 0.542968 0.839753i $$-0.317301\pi$$
0.542968 + 0.839753i $$0.317301\pi$$
$$752$$ 0 0
$$753$$ −14.7555 −0.537720
$$754$$ 0 0
$$755$$ −7.73625 −0.281551
$$756$$ 0 0
$$757$$ 22.1119 0.803672 0.401836 0.915712i $$-0.368372\pi$$
0.401836 + 0.915712i $$0.368372\pi$$
$$758$$ 0 0
$$759$$ 7.19173 0.261043
$$760$$ 0 0
$$761$$ −4.55957 −0.165284 −0.0826422 0.996579i $$-0.526336\pi$$
−0.0826422 + 0.996579i $$0.526336\pi$$
$$762$$ 0 0
$$763$$ −21.4963 −0.778219
$$764$$ 0 0
$$765$$ 0.853690 0.0308652
$$766$$ 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
$$771$$ 0.742176 0.0267288
$$772$$ 0 0
$$773$$ 26.4611 0.951739 0.475869 0.879516i $$-0.342133\pi$$
0.475869 + 0.879516i $$0.342133\pi$$
$$774$$ 0 0
$$775$$ −17.3507 −0.623255
$$776$$ 0 0
$$777$$ 11.2037 0.401930
$$778$$ 0 0
$$779$$ 1.32206 0.0473676
$$780$$ 0 0
$$781$$ −10.9789 −0.392856
$$782$$ 0 0
$$783$$ 2.62636 0.0938584
$$784$$ 0 0
$$785$$ 32.7302 1.16819
$$786$$ 0 0
$$787$$ 18.9164 0.674298 0.337149 0.941451i $$-0.390537\pi$$
0.337149 + 0.941451i $$0.390537\pi$$
$$788$$ 0 0
$$789$$ 5.48435 0.195248
$$790$$ 0 0
$$791$$ 40.6930 1.44688
$$792$$ 0 0
$$793$$ −16.5697 −0.588406
$$794$$ 0 0
$$795$$ −40.3582 −1.43136
$$796$$ 0 0
$$797$$ −47.8065 −1.69339 −0.846697 0.532075i $$-0.821412\pi$$
−0.846697 + 0.532075i $$0.821412\pi$$
$$798$$ 0 0
$$799$$ −0.635767 −0.0224918
$$800$$ 0 0
$$801$$ −1.42847 −0.0504724
$$802$$ 0 0
$$803$$ −15.1917 −0.536105
$$804$$ 0 0
$$805$$ −23.1917 −0.817401
$$806$$ 0 0
$$807$$ 20.4694 0.720558
$$808$$ 0 0
$$809$$ 29.9862 1.05426 0.527129 0.849785i $$-0.323268\pi$$
0.527129 + 0.849785i $$0.323268\pi$$
$$810$$ 0 0
$$811$$ 11.3899 0.399954 0.199977 0.979801i $$-0.435913\pi$$
0.199977 + 0.979801i $$0.435913\pi$$
$$812$$ 0 0
$$813$$ 14.0370 0.492298
$$814$$ 0 0
$$815$$ −18.4600 −0.646626
$$816$$ 0 0
$$817$$ 2.46512 0.0862438
$$818$$ 0 0
$$819$$ 4.22478 0.147626
$$820$$ 0 0
$$821$$ 19.6929 0.687286 0.343643 0.939100i $$-0.388339\pi$$
0.343643 + 0.939100i $$0.388339\pi$$
$$822$$ 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
$$827$$ 10.7927 0.375299 0.187649 0.982236i $$-0.439913\pi$$
0.187649 + 0.982236i $$0.439913\pi$$
$$828$$ 0 0
$$829$$ −45.9421 −1.59563 −0.797817 0.602900i $$-0.794012\pi$$
−0.797817 + 0.602900i $$0.794012\pi$$
$$830$$ 0 0
$$831$$ −13.4211 −0.465572
$$832$$ 0 0
$$833$$ −0.525748 −0.0182161
$$834$$ 0 0
$$835$$ 82.4238 2.85239
$$836$$ 0 0
$$837$$ 1.84106 0.0636363
$$838$$ 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
$$843$$ −3.89359 −0.134102
$$844$$ 0 0
$$845$$ −34.8295 −1.19817
$$846$$ 0 0
$$847$$ 9.79053 0.336407
$$848$$ 0 0
$$849$$ 17.6569 0.605982
$$850$$ 0 0
$$851$$ 14.6779 0.503153
$$852$$ 0 0
$$853$$ 55.0728 1.88566 0.942829 0.333278i $$-0.108155\pi$$
0.942829 + 0.333278i $$0.108155\pi$$
$$854$$ 0 0
$$855$$ 0.853690 0.0291956
$$856$$ 0 0
$$857$$ −1.79079 −0.0611723 −0.0305861 0.999532i $$-0.509737\pi$$
−0.0305861 + 0.999532i $$0.509737\pi$$
$$858$$ 0 0
$$859$$ 25.5468 0.871647 0.435823 0.900032i $$-0.356457\pi$$
0.435823 + 0.900032i $$0.356457\pi$$
$$860$$ 0 0
$$861$$ 12.6981 0.432750
$$862$$ 0 0
$$863$$ −42.1150 −1.43361 −0.716806 0.697273i $$-0.754397\pi$$
−0.716806 + 0.697273i $$0.754397\pi$$
$$864$$ 0 0
$$865$$ 46.9784 1.59731
$$866$$ 0 0
$$867$$ 16.9495 0.575634
$$868$$ 0 0
$$869$$ −38.1590 −1.29446
$$870$$ 0 0
$$871$$ −28.8486 −0.977497
$$872$$ 0 0
$$873$$ −16.3990 −0.555023
$$874$$ 0 0
$$875$$ −36.2771 −1.22639
$$876$$ 0 0
$$877$$ −1.01044 −0.0341203 −0.0170601 0.999854i $$-0.505431\pi$$
−0.0170601 + 0.999854i $$0.505431\pi$$
$$878$$ 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
$$883$$ 1.82389 0.0613787 0.0306894 0.999529i $$-0.490230\pi$$
0.0306894 + 0.999529i $$0.490230\pi$$
$$884$$ 0 0
$$885$$ −21.4844 −0.722189
$$886$$ 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
$$891$$ −2.54266 −0.0851823
$$892$$ 0 0
$$893$$ −0.635767 −0.0212751
$$894$$ 0 0
$$895$$ 44.2128 1.47787
$$896$$ 0 0
$$897$$ 5.53488 0.184804
$$898$$ 0 0
$$899$$ 4.83528 0.161266
$$900$$ 0 0
$$901$$ 2.38857 0.0795747
$$902$$ 0 0
$$903$$ 23.6770 0.787922
$$904$$ 0 0
$$905$$ 36.1082 1.20028
$$906$$ 0 0
$$907$$ 4.06248 0.134893 0.0674463 0.997723i $$-0.478515\pi$$
0.0674463 + 0.997723i $$0.478515\pi$$
$$908$$ 0 0
$$909$$ −0.115816 −0.00384137
$$910$$ 0 0
$$911$$ 42.3784 1.40406 0.702029 0.712149i $$-0.252278\pi$$
0.702029 + 0.712149i $$0.252278\pi$$
$$912$$ 0 0
$$913$$ 36.3751 1.20384
$$914$$ 0 0
$$915$$ −32.1587 −1.06313
$$916$$ 0 0
$$917$$ −2.34315 −0.0773775
$$918$$ 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
$$923$$ −8.44955 −0.278120
$$924$$ 0 0
$$925$$ 48.9068 1.60804
$$926$$ 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
$$931$$ −0.525748 −0.0172307
$$932$$ 0 0
$$933$$ 24.1623 0.791038
$$934$$ 0 0
$$935$$ −2.17064 −0.0709876
$$936$$ 0 0
$$937$$ 54.7669 1.78916 0.894579 0.446910i $$-0.147476\pi$$
0.894579 + 0.446910i $$0.147476\pi$$
$$938$$ 0 0
$$939$$ 16.6105 0.542063
$$940$$ 0 0
$$941$$ 8.89558 0.289988 0.144994 0.989433i $$-0.453684\pi$$
0.144994 + 0.989433i $$0.453684\pi$$
$$942$$ 0 0
$$943$$ 16.6358 0.541735
$$944$$ 0 0
$$945$$ 8.19951 0.266730
$$946$$ 0 0
$$947$$ −15.8541 −0.515189 −0.257595 0.966253i $$-0.582930\pi$$
−0.257595 + 0.966253i $$0.582930\pi$$
$$948$$ 0 0
$$949$$ −11.6918 −0.379532
$$950$$ 0 0
$$951$$ −2.56213 −0.0830826
$$952$$ 0 0
$$953$$ −30.2807 −0.980887 −0.490443 0.871473i $$-0.663165\pi$$
−0.490443 + 0.871473i $$0.663165\pi$$
$$954$$ 0 0
$$955$$ 79.0090 2.55667
$$956$$ 0 0
$$957$$ −6.67794 −0.215867
$$958$$ 0 0
$$959$$ 11.4642 0.370198
$$960$$ 0 0
$$961$$ −27.6105 −0.890661
$$962$$ 0 0
$$963$$ 10.2926 0.331675
$$964$$ 0 0
$$965$$ 53.7232 1.72941
$$966$$ 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
$$971$$ −38.7050 −1.24210 −0.621051 0.783771i $$-0.713294\pi$$
−0.621051 + 0.783771i $$0.713294\pi$$
$$972$$ 0 0
$$973$$ 26.7688 0.858168
$$974$$ 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
$$979$$ 3.63211 0.116083
$$980$$ 0 0
$$981$$ 9.95687 0.317899
$$982$$ 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
$$987$$ −6.10641 −0.194369
$$988$$ 0 0
$$989$$ 31.0192 0.986354
$$990$$ 0 0
$$991$$ 42.3446 1.34512 0.672561 0.740042i $$-0.265194\pi$$
0.672561 + 0.740042i $$0.265194\pi$$
$$992$$ 0 0
$$993$$ 19.1275 0.606993
$$994$$ 0 0
$$995$$ −1.16286 −0.0368651
$$996$$ 0 0
$$997$$ 47.6132 1.50793 0.753963 0.656917i $$-0.228140\pi$$
0.753963 + 0.656917i $$0.228140\pi$$
$$998$$ 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.a.n.1.4 4
3.2 odd 2 9216.2.a.x.1.1 4
4.3 odd 2 3072.2.a.t.1.4 4
8.3 odd 2 3072.2.a.i.1.1 4
8.5 even 2 3072.2.a.o.1.1 4
12.11 even 2 9216.2.a.y.1.1 4
16.3 odd 4 3072.2.d.f.1537.5 8
16.5 even 4 3072.2.d.i.1537.8 8
16.11 odd 4 3072.2.d.f.1537.4 8
16.13 even 4 3072.2.d.i.1537.1 8
24.5 odd 2 9216.2.a.bn.1.4 4
24.11 even 2 9216.2.a.bo.1.4 4
32.3 odd 8 48.2.j.a.13.3 8
32.5 even 8 384.2.j.a.97.4 8
32.11 odd 8 48.2.j.a.37.3 yes 8
32.13 even 8 384.2.j.a.289.4 8
32.19 odd 8 384.2.j.b.289.2 8
32.21 even 8 192.2.j.a.49.1 8
32.27 odd 8 384.2.j.b.97.2 8
32.29 even 8 192.2.j.a.145.1 8
96.5 odd 8 1152.2.k.f.865.1 8
96.11 even 8 144.2.k.b.37.2 8
96.29 odd 8 576.2.k.b.145.4 8
96.35 even 8 144.2.k.b.109.2 8
96.53 odd 8 576.2.k.b.433.4 8
96.59 even 8 1152.2.k.c.865.1 8
96.77 odd 8 1152.2.k.f.289.1 8
96.83 even 8 1152.2.k.c.289.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.3 8 32.3 odd 8
48.2.j.a.37.3 yes 8 32.11 odd 8
144.2.k.b.37.2 8 96.11 even 8
144.2.k.b.109.2 8 96.35 even 8
192.2.j.a.49.1 8 32.21 even 8
192.2.j.a.145.1 8 32.29 even 8
384.2.j.a.97.4 8 32.5 even 8
384.2.j.a.289.4 8 32.13 even 8
384.2.j.b.97.2 8 32.27 odd 8
384.2.j.b.289.2 8 32.19 odd 8
576.2.k.b.145.4 8 96.29 odd 8
576.2.k.b.433.4 8 96.53 odd 8
1152.2.k.c.289.1 8 96.83 even 8
1152.2.k.c.865.1 8 96.59 even 8
1152.2.k.f.289.1 8 96.77 odd 8
1152.2.k.f.865.1 8 96.5 odd 8
3072.2.a.i.1.1 4 8.3 odd 2
3072.2.a.n.1.4 4 1.1 even 1 trivial
3072.2.a.o.1.1 4 8.5 even 2
3072.2.a.t.1.4 4 4.3 odd 2
3072.2.d.f.1537.4 8 16.11 odd 4
3072.2.d.f.1537.5 8 16.3 odd 4
3072.2.d.i.1537.1 8 16.13 even 4
3072.2.d.i.1537.8 8 16.5 even 4
9216.2.a.x.1.1 4 3.2 odd 2
9216.2.a.y.1.1 4 12.11 even 2
9216.2.a.bn.1.4 4 24.5 odd 2
9216.2.a.bo.1.4 4 24.11 even 2