# Properties

 Label 3072.2.a.t.1.2 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 1.2 Root $$0.334904$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -0.473626 q^{5} +4.55765 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -0.473626 q^{5} +4.55765 q^{7} +1.00000 q^{9} +3.49824 q^{11} -0.0840215 q^{13} -0.473626 q^{15} -3.61706 q^{17} +3.61706 q^{19} +4.55765 q^{21} +2.82843 q^{23} -4.77568 q^{25} +1.00000 q^{27} +7.30205 q^{29} -0.557647 q^{31} +3.49824 q^{33} -2.15862 q^{35} +6.20285 q^{37} -0.0840215 q^{39} -9.27391 q^{41} +2.27744 q^{43} -0.473626 q^{45} -2.82843 q^{47} +13.7721 q^{49} -3.61706 q^{51} +0.697947 q^{53} -1.65685 q^{55} +3.61706 q^{57} +5.65685 q^{59} -3.85970 q^{61} +4.55765 q^{63} +0.0397948 q^{65} -5.33962 q^{67} +2.82843 q^{69} -9.11529 q^{71} -0.541560 q^{73} -4.77568 q^{75} +15.9437 q^{77} +10.9937 q^{79} +1.00000 q^{81} -15.0496 q^{83} +1.71313 q^{85} +7.30205 q^{87} -14.6533 q^{89} -0.382941 q^{91} -0.557647 q^{93} -1.71313 q^{95} +4.31724 q^{97} +3.49824 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 4q^{5} + 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 4q^{5} + 4q^{7} + 4q^{9} + 8q^{13} + 4q^{15} + 4q^{21} + 4q^{25} + 4q^{27} + 12q^{29} + 12q^{31} + 16q^{37} + 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} + 16q^{55} + 16q^{61} + 4q^{63} - 8q^{65} - 16q^{67} - 8q^{71} - 8q^{73} + 4q^{75} + 24q^{77} + 12q^{79} + 4q^{81} + 24q^{85} + 12q^{87} - 8q^{89} - 16q^{91} + 12q^{93} - 24q^{95} + O(q^{100})$$

## 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
$$5$$ −0.473626 −0.211812 −0.105906 0.994376i $$-0.533774\pi$$
−0.105906 + 0.994376i $$0.533774\pi$$
$$6$$ 0 0
$$7$$ 4.55765 1.72263 0.861314 0.508072i $$-0.169642\pi$$
0.861314 + 0.508072i $$0.169642\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.49824 1.05476 0.527379 0.849630i $$-0.323175\pi$$
0.527379 + 0.849630i $$0.323175\pi$$
$$12$$ 0 0
$$13$$ −0.0840215 −0.0233034 −0.0116517 0.999932i $$-0.503709\pi$$
−0.0116517 + 0.999932i $$0.503709\pi$$
$$14$$ 0 0
$$15$$ −0.473626 −0.122290
$$16$$ 0 0
$$17$$ −3.61706 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$18$$ 0 0
$$19$$ 3.61706 0.829810 0.414905 0.909865i $$-0.363815\pi$$
0.414905 + 0.909865i $$0.363815\pi$$
$$20$$ 0 0
$$21$$ 4.55765 0.994560
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ −4.77568 −0.955136
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 7.30205 1.35596 0.677979 0.735082i $$-0.262856\pi$$
0.677979 + 0.735082i $$0.262856\pi$$
$$30$$ 0 0
$$31$$ −0.557647 −0.100156 −0.0500782 0.998745i $$-0.515947\pi$$
−0.0500782 + 0.998745i $$0.515947\pi$$
$$32$$ 0 0
$$33$$ 3.49824 0.608965
$$34$$ 0 0
$$35$$ −2.15862 −0.364873
$$36$$ 0 0
$$37$$ 6.20285 1.01974 0.509871 0.860251i $$-0.329693\pi$$
0.509871 + 0.860251i $$0.329693\pi$$
$$38$$ 0 0
$$39$$ −0.0840215 −0.0134542
$$40$$ 0 0
$$41$$ −9.27391 −1.44834 −0.724171 0.689620i $$-0.757777\pi$$
−0.724171 + 0.689620i $$0.757777\pi$$
$$42$$ 0 0
$$43$$ 2.27744 0.347307 0.173653 0.984807i $$-0.444443\pi$$
0.173653 + 0.984807i $$0.444443\pi$$
$$44$$ 0 0
$$45$$ −0.473626 −0.0706040
$$46$$ 0 0
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ 13.7721 1.96745
$$50$$ 0 0
$$51$$ −3.61706 −0.506490
$$52$$ 0 0
$$53$$ 0.697947 0.0958704 0.0479352 0.998850i $$-0.484736\pi$$
0.0479352 + 0.998850i $$0.484736\pi$$
$$54$$ 0 0
$$55$$ −1.65685 −0.223410
$$56$$ 0 0
$$57$$ 3.61706 0.479091
$$58$$ 0 0
$$59$$ 5.65685 0.736460 0.368230 0.929735i $$-0.379964\pi$$
0.368230 + 0.929735i $$0.379964\pi$$
$$60$$ 0 0
$$61$$ −3.85970 −0.494184 −0.247092 0.968992i $$-0.579475\pi$$
−0.247092 + 0.968992i $$0.579475\pi$$
$$62$$ 0 0
$$63$$ 4.55765 0.574210
$$64$$ 0 0
$$65$$ 0.0397948 0.00493593
$$66$$ 0 0
$$67$$ −5.33962 −0.652338 −0.326169 0.945311i $$-0.605758\pi$$
−0.326169 + 0.945311i $$0.605758\pi$$
$$68$$ 0 0
$$69$$ 2.82843 0.340503
$$70$$ 0 0
$$71$$ −9.11529 −1.08179 −0.540893 0.841091i $$-0.681914\pi$$
−0.540893 + 0.841091i $$0.681914\pi$$
$$72$$ 0 0
$$73$$ −0.541560 −0.0633848 −0.0316924 0.999498i $$-0.510090\pi$$
−0.0316924 + 0.999498i $$0.510090\pi$$
$$74$$ 0 0
$$75$$ −4.77568 −0.551448
$$76$$ 0 0
$$77$$ 15.9437 1.81696
$$78$$ 0 0
$$79$$ 10.9937 1.23689 0.618445 0.785828i $$-0.287763\pi$$
0.618445 + 0.785828i $$0.287763\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −15.0496 −1.65191 −0.825954 0.563738i $$-0.809363\pi$$
−0.825954 + 0.563738i $$0.809363\pi$$
$$84$$ 0 0
$$85$$ 1.71313 0.185815
$$86$$ 0 0
$$87$$ 7.30205 0.782862
$$88$$ 0 0
$$89$$ −14.6533 −1.55325 −0.776625 0.629964i $$-0.783070\pi$$
−0.776625 + 0.629964i $$0.783070\pi$$
$$90$$ 0 0
$$91$$ −0.382941 −0.0401431
$$92$$ 0 0
$$93$$ −0.557647 −0.0578253
$$94$$ 0 0
$$95$$ −1.71313 −0.175764
$$96$$ 0 0
$$97$$ 4.31724 0.438349 0.219175 0.975686i $$-0.429664\pi$$
0.219175 + 0.975686i $$0.429664\pi$$
$$98$$ 0 0
$$99$$ 3.49824 0.351586
$$100$$ 0 0
$$101$$ −0.641669 −0.0638484 −0.0319242 0.999490i $$-0.510164\pi$$
−0.0319242 + 0.999490i $$0.510164\pi$$
$$102$$ 0 0
$$103$$ −1.33686 −0.131724 −0.0658622 0.997829i $$-0.520980\pi$$
−0.0658622 + 0.997829i $$0.520980\pi$$
$$104$$ 0 0
$$105$$ −2.15862 −0.210660
$$106$$ 0 0
$$107$$ −8.57373 −0.828854 −0.414427 0.910083i $$-0.636018\pi$$
−0.414427 + 0.910083i $$0.636018\pi$$
$$108$$ 0 0
$$109$$ 8.08402 0.774309 0.387154 0.922015i $$-0.373458\pi$$
0.387154 + 0.922015i $$0.373458\pi$$
$$110$$ 0 0
$$111$$ 6.20285 0.588748
$$112$$ 0 0
$$113$$ 9.55136 0.898516 0.449258 0.893402i $$-0.351688\pi$$
0.449258 + 0.893402i $$0.351688\pi$$
$$114$$ 0 0
$$115$$ −1.33962 −0.124920
$$116$$ 0 0
$$117$$ −0.0840215 −0.00776779
$$118$$ 0 0
$$119$$ −16.4853 −1.51120
$$120$$ 0 0
$$121$$ 1.23765 0.112514
$$122$$ 0 0
$$123$$ −9.27391 −0.836201
$$124$$ 0 0
$$125$$ 4.63001 0.414121
$$126$$ 0 0
$$127$$ 5.09921 0.452481 0.226241 0.974071i $$-0.427356\pi$$
0.226241 + 0.974071i $$0.427356\pi$$
$$128$$ 0 0
$$129$$ 2.27744 0.200518
$$130$$ 0 0
$$131$$ −2.99647 −0.261803 −0.130901 0.991395i $$-0.541787\pi$$
−0.130901 + 0.991395i $$0.541787\pi$$
$$132$$ 0 0
$$133$$ 16.4853 1.42946
$$134$$ 0 0
$$135$$ −0.473626 −0.0407632
$$136$$ 0 0
$$137$$ −3.37941 −0.288723 −0.144361 0.989525i $$-0.546113\pi$$
−0.144361 + 0.989525i $$0.546113\pi$$
$$138$$ 0 0
$$139$$ −8.31724 −0.705459 −0.352729 0.935725i $$-0.614746\pi$$
−0.352729 + 0.935725i $$0.614746\pi$$
$$140$$ 0 0
$$141$$ −2.82843 −0.238197
$$142$$ 0 0
$$143$$ −0.293927 −0.0245794
$$144$$ 0 0
$$145$$ −3.45844 −0.287208
$$146$$ 0 0
$$147$$ 13.7721 1.13591
$$148$$ 0 0
$$149$$ 14.1305 1.15761 0.578807 0.815465i $$-0.303518\pi$$
0.578807 + 0.815465i $$0.303518\pi$$
$$150$$ 0 0
$$151$$ 9.97685 0.811905 0.405952 0.913894i $$-0.366940\pi$$
0.405952 + 0.913894i $$0.366940\pi$$
$$152$$ 0 0
$$153$$ −3.61706 −0.292422
$$154$$ 0 0
$$155$$ 0.264116 0.0212143
$$156$$ 0 0
$$157$$ 22.8562 1.82412 0.912060 0.410056i $$-0.134491\pi$$
0.912060 + 0.410056i $$0.134491\pi$$
$$158$$ 0 0
$$159$$ 0.697947 0.0553508
$$160$$ 0 0
$$161$$ 12.8910 1.01595
$$162$$ 0 0
$$163$$ 10.6135 0.831316 0.415658 0.909521i $$-0.363551\pi$$
0.415658 + 0.909521i $$0.363551\pi$$
$$164$$ 0 0
$$165$$ −1.65685 −0.128986
$$166$$ 0 0
$$167$$ 5.83822 0.451775 0.225888 0.974153i $$-0.427472\pi$$
0.225888 + 0.974153i $$0.427472\pi$$
$$168$$ 0 0
$$169$$ −12.9929 −0.999457
$$170$$ 0 0
$$171$$ 3.61706 0.276603
$$172$$ 0 0
$$173$$ −5.12695 −0.389795 −0.194897 0.980824i $$-0.562437\pi$$
−0.194897 + 0.980824i $$0.562437\pi$$
$$174$$ 0 0
$$175$$ −21.7659 −1.64534
$$176$$ 0 0
$$177$$ 5.65685 0.425195
$$178$$ 0 0
$$179$$ 13.1286 0.981279 0.490640 0.871363i $$-0.336763\pi$$
0.490640 + 0.871363i $$0.336763\pi$$
$$180$$ 0 0
$$181$$ 15.3181 1.13859 0.569294 0.822134i $$-0.307217\pi$$
0.569294 + 0.822134i $$0.307217\pi$$
$$182$$ 0 0
$$183$$ −3.85970 −0.285317
$$184$$ 0 0
$$185$$ −2.93783 −0.215993
$$186$$ 0 0
$$187$$ −12.6533 −0.925303
$$188$$ 0 0
$$189$$ 4.55765 0.331520
$$190$$ 0 0
$$191$$ −8.63001 −0.624446 −0.312223 0.950009i $$-0.601074\pi$$
−0.312223 + 0.950009i $$0.601074\pi$$
$$192$$ 0 0
$$193$$ 11.4514 0.824288 0.412144 0.911119i $$-0.364780\pi$$
0.412144 + 0.911119i $$0.364780\pi$$
$$194$$ 0 0
$$195$$ 0.0397948 0.00284976
$$196$$ 0 0
$$197$$ 10.5925 0.754681 0.377340 0.926075i $$-0.376839\pi$$
0.377340 + 0.926075i $$0.376839\pi$$
$$198$$ 0 0
$$199$$ 3.68000 0.260868 0.130434 0.991457i $$-0.458363\pi$$
0.130434 + 0.991457i $$0.458363\pi$$
$$200$$ 0 0
$$201$$ −5.33962 −0.376627
$$202$$ 0 0
$$203$$ 33.2802 2.33581
$$204$$ 0 0
$$205$$ 4.39236 0.306776
$$206$$ 0 0
$$207$$ 2.82843 0.196589
$$208$$ 0 0
$$209$$ 12.6533 0.875249
$$210$$ 0 0
$$211$$ 14.3102 0.985153 0.492577 0.870269i $$-0.336055\pi$$
0.492577 + 0.870269i $$0.336055\pi$$
$$212$$ 0 0
$$213$$ −9.11529 −0.624570
$$214$$ 0 0
$$215$$ −1.07866 −0.0735637
$$216$$ 0 0
$$217$$ −2.54156 −0.172532
$$218$$ 0 0
$$219$$ −0.541560 −0.0365952
$$220$$ 0 0
$$221$$ 0.303911 0.0204433
$$222$$ 0 0
$$223$$ −4.86156 −0.325554 −0.162777 0.986663i $$-0.552045\pi$$
−0.162777 + 0.986663i $$0.552045\pi$$
$$224$$ 0 0
$$225$$ −4.77568 −0.318379
$$226$$ 0 0
$$227$$ −15.0496 −0.998877 −0.499438 0.866349i $$-0.666460\pi$$
−0.499438 + 0.866349i $$0.666460\pi$$
$$228$$ 0 0
$$229$$ −28.5264 −1.88507 −0.942537 0.334101i $$-0.891567\pi$$
−0.942537 + 0.334101i $$0.891567\pi$$
$$230$$ 0 0
$$231$$ 15.9437 1.04902
$$232$$ 0 0
$$233$$ 13.5702 0.889014 0.444507 0.895775i $$-0.353379\pi$$
0.444507 + 0.895775i $$0.353379\pi$$
$$234$$ 0 0
$$235$$ 1.33962 0.0873869
$$236$$ 0 0
$$237$$ 10.9937 0.714118
$$238$$ 0 0
$$239$$ −29.3629 −1.89933 −0.949665 0.313267i $$-0.898576\pi$$
−0.949665 + 0.313267i $$0.898576\pi$$
$$240$$ 0 0
$$241$$ −24.0063 −1.54638 −0.773190 0.634175i $$-0.781340\pi$$
−0.773190 + 0.634175i $$0.781340\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −6.52284 −0.416729
$$246$$ 0 0
$$247$$ −0.303911 −0.0193374
$$248$$ 0 0
$$249$$ −15.0496 −0.953729
$$250$$ 0 0
$$251$$ 22.2837 1.40654 0.703268 0.710925i $$-0.251724\pi$$
0.703268 + 0.710925i $$0.251724\pi$$
$$252$$ 0 0
$$253$$ 9.89450 0.622062
$$254$$ 0 0
$$255$$ 1.71313 0.107281
$$256$$ 0 0
$$257$$ 8.66038 0.540220 0.270110 0.962829i $$-0.412940\pi$$
0.270110 + 0.962829i $$0.412940\pi$$
$$258$$ 0 0
$$259$$ 28.2704 1.75664
$$260$$ 0 0
$$261$$ 7.30205 0.451986
$$262$$ 0 0
$$263$$ −13.3208 −0.821394 −0.410697 0.911772i $$-0.634715\pi$$
−0.410697 + 0.911772i $$0.634715\pi$$
$$264$$ 0 0
$$265$$ −0.330566 −0.0203065
$$266$$ 0 0
$$267$$ −14.6533 −0.896769
$$268$$ 0 0
$$269$$ −16.5058 −1.00638 −0.503188 0.864177i $$-0.667840\pi$$
−0.503188 + 0.864177i $$0.667840\pi$$
$$270$$ 0 0
$$271$$ 21.9769 1.33500 0.667499 0.744610i $$-0.267365\pi$$
0.667499 + 0.744610i $$0.267365\pi$$
$$272$$ 0 0
$$273$$ −0.382941 −0.0231766
$$274$$ 0 0
$$275$$ −16.7064 −1.00744
$$276$$ 0 0
$$277$$ 15.4862 0.930475 0.465237 0.885186i $$-0.345969\pi$$
0.465237 + 0.885186i $$0.345969\pi$$
$$278$$ 0 0
$$279$$ −0.557647 −0.0333855
$$280$$ 0 0
$$281$$ 22.8910 1.36556 0.682780 0.730624i $$-0.260771\pi$$
0.682780 + 0.730624i $$0.260771\pi$$
$$282$$ 0 0
$$283$$ 6.34315 0.377061 0.188530 0.982067i $$-0.439628\pi$$
0.188530 + 0.982067i $$0.439628\pi$$
$$284$$ 0 0
$$285$$ −1.71313 −0.101477
$$286$$ 0 0
$$287$$ −42.2672 −2.49496
$$288$$ 0 0
$$289$$ −3.91688 −0.230405
$$290$$ 0 0
$$291$$ 4.31724 0.253081
$$292$$ 0 0
$$293$$ 30.5783 1.78641 0.893203 0.449654i $$-0.148453\pi$$
0.893203 + 0.449654i $$0.148453\pi$$
$$294$$ 0 0
$$295$$ −2.67923 −0.155991
$$296$$ 0 0
$$297$$ 3.49824 0.202988
$$298$$ 0 0
$$299$$ −0.237649 −0.0137436
$$300$$ 0 0
$$301$$ 10.3798 0.598281
$$302$$ 0 0
$$303$$ −0.641669 −0.0368629
$$304$$ 0 0
$$305$$ 1.82805 0.104674
$$306$$ 0 0
$$307$$ −17.1286 −0.977582 −0.488791 0.872401i $$-0.662562\pi$$
−0.488791 + 0.872401i $$0.662562\pi$$
$$308$$ 0 0
$$309$$ −1.33686 −0.0760511
$$310$$ 0 0
$$311$$ −26.8651 −1.52338 −0.761689 0.647943i $$-0.775630\pi$$
−0.761689 + 0.647943i $$0.775630\pi$$
$$312$$ 0 0
$$313$$ −19.6890 −1.11289 −0.556445 0.830885i $$-0.687835\pi$$
−0.556445 + 0.830885i $$0.687835\pi$$
$$314$$ 0 0
$$315$$ −2.15862 −0.121624
$$316$$ 0 0
$$317$$ −30.1860 −1.69541 −0.847706 0.530466i $$-0.822017\pi$$
−0.847706 + 0.530466i $$0.822017\pi$$
$$318$$ 0 0
$$319$$ 25.5443 1.43021
$$320$$ 0 0
$$321$$ −8.57373 −0.478539
$$322$$ 0 0
$$323$$ −13.0831 −0.727964
$$324$$ 0 0
$$325$$ 0.401260 0.0222579
$$326$$ 0 0
$$327$$ 8.08402 0.447047
$$328$$ 0 0
$$329$$ −12.8910 −0.710702
$$330$$ 0 0
$$331$$ −20.7784 −1.14209 −0.571043 0.820920i $$-0.693461\pi$$
−0.571043 + 0.820920i $$0.693461\pi$$
$$332$$ 0 0
$$333$$ 6.20285 0.339914
$$334$$ 0 0
$$335$$ 2.52898 0.138173
$$336$$ 0 0
$$337$$ 23.0098 1.25342 0.626712 0.779251i $$-0.284400\pi$$
0.626712 + 0.779251i $$0.284400\pi$$
$$338$$ 0 0
$$339$$ 9.55136 0.518759
$$340$$ 0 0
$$341$$ −1.95078 −0.105641
$$342$$ 0 0
$$343$$ 30.8651 1.66656
$$344$$ 0 0
$$345$$ −1.33962 −0.0721225
$$346$$ 0 0
$$347$$ −15.4186 −0.827716 −0.413858 0.910341i $$-0.635819\pi$$
−0.413858 + 0.910341i $$0.635819\pi$$
$$348$$ 0 0
$$349$$ −28.3638 −1.51828 −0.759140 0.650927i $$-0.774380\pi$$
−0.759140 + 0.650927i $$0.774380\pi$$
$$350$$ 0 0
$$351$$ −0.0840215 −0.00448474
$$352$$ 0 0
$$353$$ −12.2117 −0.649965 −0.324983 0.945720i $$-0.605358\pi$$
−0.324983 + 0.945720i $$0.605358\pi$$
$$354$$ 0 0
$$355$$ 4.31724 0.229135
$$356$$ 0 0
$$357$$ −16.4853 −0.872494
$$358$$ 0 0
$$359$$ −33.4780 −1.76690 −0.883452 0.468522i $$-0.844786\pi$$
−0.883452 + 0.468522i $$0.844786\pi$$
$$360$$ 0 0
$$361$$ −5.91688 −0.311415
$$362$$ 0 0
$$363$$ 1.23765 0.0649597
$$364$$ 0 0
$$365$$ 0.256497 0.0134256
$$366$$ 0 0
$$367$$ 0.702379 0.0366639 0.0183319 0.999832i $$-0.494164\pi$$
0.0183319 + 0.999832i $$0.494164\pi$$
$$368$$ 0 0
$$369$$ −9.27391 −0.482781
$$370$$ 0 0
$$371$$ 3.18100 0.165149
$$372$$ 0 0
$$373$$ −26.8132 −1.38834 −0.694168 0.719813i $$-0.744227\pi$$
−0.694168 + 0.719813i $$0.744227\pi$$
$$374$$ 0 0
$$375$$ 4.63001 0.239093
$$376$$ 0 0
$$377$$ −0.613530 −0.0315984
$$378$$ 0 0
$$379$$ −2.51509 −0.129192 −0.0645958 0.997912i $$-0.520576\pi$$
−0.0645958 + 0.997912i $$0.520576\pi$$
$$380$$ 0 0
$$381$$ 5.09921 0.261240
$$382$$ 0 0
$$383$$ 25.4880 1.30238 0.651188 0.758916i $$-0.274271\pi$$
0.651188 + 0.758916i $$0.274271\pi$$
$$384$$ 0 0
$$385$$ −7.55136 −0.384853
$$386$$ 0 0
$$387$$ 2.27744 0.115769
$$388$$ 0 0
$$389$$ −16.5532 −0.839281 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$390$$ 0 0
$$391$$ −10.2306 −0.517383
$$392$$ 0 0
$$393$$ −2.99647 −0.151152
$$394$$ 0 0
$$395$$ −5.20690 −0.261988
$$396$$ 0 0
$$397$$ −12.7936 −0.642094 −0.321047 0.947063i $$-0.604035\pi$$
−0.321047 + 0.947063i $$0.604035\pi$$
$$398$$ 0 0
$$399$$ 16.4853 0.825296
$$400$$ 0 0
$$401$$ 18.0853 0.903137 0.451568 0.892237i $$-0.350865\pi$$
0.451568 + 0.892237i $$0.350865\pi$$
$$402$$ 0 0
$$403$$ 0.0468544 0.00233398
$$404$$ 0 0
$$405$$ −0.473626 −0.0235347
$$406$$ 0 0
$$407$$ 21.6990 1.07558
$$408$$ 0 0
$$409$$ 25.2271 1.24740 0.623699 0.781665i $$-0.285629\pi$$
0.623699 + 0.781665i $$0.285629\pi$$
$$410$$ 0 0
$$411$$ −3.37941 −0.166694
$$412$$ 0 0
$$413$$ 25.7819 1.26865
$$414$$ 0 0
$$415$$ 7.12787 0.349894
$$416$$ 0 0
$$417$$ −8.31724 −0.407297
$$418$$ 0 0
$$419$$ −10.2571 −0.501090 −0.250545 0.968105i $$-0.580610\pi$$
−0.250545 + 0.968105i $$0.580610\pi$$
$$420$$ 0 0
$$421$$ 3.38775 0.165109 0.0825543 0.996587i $$-0.473692\pi$$
0.0825543 + 0.996587i $$0.473692\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ 17.2739 0.837908
$$426$$ 0 0
$$427$$ −17.5912 −0.851296
$$428$$ 0 0
$$429$$ −0.293927 −0.0141909
$$430$$ 0 0
$$431$$ 4.42454 0.213123 0.106561 0.994306i $$-0.466016\pi$$
0.106561 + 0.994306i $$0.466016\pi$$
$$432$$ 0 0
$$433$$ −7.31371 −0.351474 −0.175737 0.984437i $$-0.556231\pi$$
−0.175737 + 0.984437i $$0.556231\pi$$
$$434$$ 0 0
$$435$$ −3.45844 −0.165820
$$436$$ 0 0
$$437$$ 10.2306 0.489395
$$438$$ 0 0
$$439$$ −29.6533 −1.41527 −0.707637 0.706576i $$-0.750239\pi$$
−0.707637 + 0.706576i $$0.750239\pi$$
$$440$$ 0 0
$$441$$ 13.7721 0.655817
$$442$$ 0 0
$$443$$ 14.5743 0.692446 0.346223 0.938152i $$-0.387464\pi$$
0.346223 + 0.938152i $$0.387464\pi$$
$$444$$ 0 0
$$445$$ 6.94019 0.328997
$$446$$ 0 0
$$447$$ 14.1305 0.668349
$$448$$ 0 0
$$449$$ −6.48844 −0.306208 −0.153104 0.988210i $$-0.548927\pi$$
−0.153104 + 0.988210i $$0.548927\pi$$
$$450$$ 0 0
$$451$$ −32.4423 −1.52765
$$452$$ 0 0
$$453$$ 9.97685 0.468753
$$454$$ 0 0
$$455$$ 0.181370 0.00850278
$$456$$ 0 0
$$457$$ −9.00353 −0.421167 −0.210584 0.977576i $$-0.567536\pi$$
−0.210584 + 0.977576i $$0.567536\pi$$
$$458$$ 0 0
$$459$$ −3.61706 −0.168830
$$460$$ 0 0
$$461$$ 20.6783 0.963085 0.481542 0.876423i $$-0.340077\pi$$
0.481542 + 0.876423i $$0.340077\pi$$
$$462$$ 0 0
$$463$$ 18.6435 0.866437 0.433219 0.901289i $$-0.357378\pi$$
0.433219 + 0.901289i $$0.357378\pi$$
$$464$$ 0 0
$$465$$ 0.264116 0.0122481
$$466$$ 0 0
$$467$$ −33.2535 −1.53879 −0.769395 0.638773i $$-0.779442\pi$$
−0.769395 + 0.638773i $$0.779442\pi$$
$$468$$ 0 0
$$469$$ −24.3361 −1.12374
$$470$$ 0 0
$$471$$ 22.8562 1.05316
$$472$$ 0 0
$$473$$ 7.96703 0.366325
$$474$$ 0 0
$$475$$ −17.2739 −0.792582
$$476$$ 0 0
$$477$$ 0.697947 0.0319568
$$478$$ 0 0
$$479$$ −1.08864 −0.0497412 −0.0248706 0.999691i $$-0.507917\pi$$
−0.0248706 + 0.999691i $$0.507917\pi$$
$$480$$ 0 0
$$481$$ −0.521173 −0.0237634
$$482$$ 0 0
$$483$$ 12.8910 0.586560
$$484$$ 0 0
$$485$$ −2.04476 −0.0928476
$$486$$ 0 0
$$487$$ 35.3298 1.60095 0.800473 0.599369i $$-0.204582\pi$$
0.800473 + 0.599369i $$0.204582\pi$$
$$488$$ 0 0
$$489$$ 10.6135 0.479960
$$490$$ 0 0
$$491$$ 18.2306 0.822735 0.411367 0.911470i $$-0.365051\pi$$
0.411367 + 0.911470i $$0.365051\pi$$
$$492$$ 0 0
$$493$$ −26.4120 −1.18953
$$494$$ 0 0
$$495$$ −1.65685 −0.0744701
$$496$$ 0 0
$$497$$ −41.5443 −1.86352
$$498$$ 0 0
$$499$$ 20.3361 0.910368 0.455184 0.890397i $$-0.349573\pi$$
0.455184 + 0.890397i $$0.349573\pi$$
$$500$$ 0 0
$$501$$ 5.83822 0.260833
$$502$$ 0 0
$$503$$ −30.2969 −1.35087 −0.675435 0.737420i $$-0.736044\pi$$
−0.675435 + 0.737420i $$0.736044\pi$$
$$504$$ 0 0
$$505$$ 0.303911 0.0135239
$$506$$ 0 0
$$507$$ −12.9929 −0.577037
$$508$$ 0 0
$$509$$ −14.9660 −0.663355 −0.331677 0.943393i $$-0.607615\pi$$
−0.331677 + 0.943393i $$0.607615\pi$$
$$510$$ 0 0
$$511$$ −2.46824 −0.109188
$$512$$ 0 0
$$513$$ 3.61706 0.159697
$$514$$ 0 0
$$515$$ 0.633169 0.0279008
$$516$$ 0 0
$$517$$ −9.89450 −0.435160
$$518$$ 0 0
$$519$$ −5.12695 −0.225048
$$520$$ 0 0
$$521$$ 24.9049 1.09110 0.545551 0.838078i $$-0.316320\pi$$
0.545551 + 0.838078i $$0.316320\pi$$
$$522$$ 0 0
$$523$$ 18.2445 0.797775 0.398888 0.917000i $$-0.369396\pi$$
0.398888 + 0.917000i $$0.369396\pi$$
$$524$$ 0 0
$$525$$ −21.7659 −0.949940
$$526$$ 0 0
$$527$$ 2.01704 0.0878638
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 5.65685 0.245487
$$532$$ 0 0
$$533$$ 0.779208 0.0337513
$$534$$ 0 0
$$535$$ 4.06074 0.175561
$$536$$ 0 0
$$537$$ 13.1286 0.566542
$$538$$ 0 0
$$539$$ 48.1782 2.07518
$$540$$ 0 0
$$541$$ −25.8471 −1.11125 −0.555627 0.831432i $$-0.687522\pi$$
−0.555627 + 0.831432i $$0.687522\pi$$
$$542$$ 0 0
$$543$$ 15.3181 0.657364
$$544$$ 0 0
$$545$$ −3.82880 −0.164008
$$546$$ 0 0
$$547$$ −19.4249 −0.830549 −0.415275 0.909696i $$-0.636315\pi$$
−0.415275 + 0.909696i $$0.636315\pi$$
$$548$$ 0 0
$$549$$ −3.85970 −0.164728
$$550$$ 0 0
$$551$$ 26.4120 1.12519
$$552$$ 0 0
$$553$$ 50.1055 2.13070
$$554$$ 0 0
$$555$$ −2.93783 −0.124704
$$556$$ 0 0
$$557$$ −38.9652 −1.65101 −0.825504 0.564397i $$-0.809109\pi$$
−0.825504 + 0.564397i $$0.809109\pi$$
$$558$$ 0 0
$$559$$ −0.191354 −0.00809342
$$560$$ 0 0
$$561$$ −12.6533 −0.534224
$$562$$ 0 0
$$563$$ −28.1327 −1.18565 −0.592826 0.805330i $$-0.701988\pi$$
−0.592826 + 0.805330i $$0.701988\pi$$
$$564$$ 0 0
$$565$$ −4.52377 −0.190316
$$566$$ 0 0
$$567$$ 4.55765 0.191403
$$568$$ 0 0
$$569$$ −13.4849 −0.565317 −0.282658 0.959221i $$-0.591216\pi$$
−0.282658 + 0.959221i $$0.591216\pi$$
$$570$$ 0 0
$$571$$ −20.9706 −0.877591 −0.438795 0.898587i $$-0.644595\pi$$
−0.438795 + 0.898587i $$0.644595\pi$$
$$572$$ 0 0
$$573$$ −8.63001 −0.360524
$$574$$ 0 0
$$575$$ −13.5077 −0.563308
$$576$$ 0 0
$$577$$ −11.6176 −0.483648 −0.241824 0.970320i $$-0.577746\pi$$
−0.241824 + 0.970320i $$0.577746\pi$$
$$578$$ 0 0
$$579$$ 11.4514 0.475903
$$580$$ 0 0
$$581$$ −68.5907 −2.84562
$$582$$ 0 0
$$583$$ 2.44158 0.101120
$$584$$ 0 0
$$585$$ 0.0397948 0.00164531
$$586$$ 0 0
$$587$$ 24.0796 0.993871 0.496936 0.867787i $$-0.334459\pi$$
0.496936 + 0.867787i $$0.334459\pi$$
$$588$$ 0 0
$$589$$ −2.01704 −0.0831108
$$590$$ 0 0
$$591$$ 10.5925 0.435715
$$592$$ 0 0
$$593$$ −41.5372 −1.70573 −0.852865 0.522132i $$-0.825137\pi$$
−0.852865 + 0.522132i $$0.825137\pi$$
$$594$$ 0 0
$$595$$ 7.80785 0.320091
$$596$$ 0 0
$$597$$ 3.68000 0.150612
$$598$$ 0 0
$$599$$ −6.43160 −0.262788 −0.131394 0.991330i $$-0.541945\pi$$
−0.131394 + 0.991330i $$0.541945\pi$$
$$600$$ 0 0
$$601$$ −3.45844 −0.141073 −0.0705364 0.997509i $$-0.522471\pi$$
−0.0705364 + 0.997509i $$0.522471\pi$$
$$602$$ 0 0
$$603$$ −5.33962 −0.217446
$$604$$ 0 0
$$605$$ −0.586182 −0.0238317
$$606$$ 0 0
$$607$$ −30.1019 −1.22180 −0.610900 0.791708i $$-0.709192\pi$$
−0.610900 + 0.791708i $$0.709192\pi$$
$$608$$ 0 0
$$609$$ 33.2802 1.34858
$$610$$ 0 0
$$611$$ 0.237649 0.00961424
$$612$$ 0 0
$$613$$ 3.54246 0.143079 0.0715393 0.997438i $$-0.477209\pi$$
0.0715393 + 0.997438i $$0.477209\pi$$
$$614$$ 0 0
$$615$$ 4.39236 0.177117
$$616$$ 0 0
$$617$$ −22.9098 −0.922315 −0.461157 0.887318i $$-0.652566\pi$$
−0.461157 + 0.887318i $$0.652566\pi$$
$$618$$ 0 0
$$619$$ 40.4612 1.62627 0.813136 0.582074i $$-0.197758\pi$$
0.813136 + 0.582074i $$0.197758\pi$$
$$620$$ 0 0
$$621$$ 2.82843 0.113501
$$622$$ 0 0
$$623$$ −66.7847 −2.67567
$$624$$ 0 0
$$625$$ 21.6855 0.867420
$$626$$ 0 0
$$627$$ 12.6533 0.505325
$$628$$ 0 0
$$629$$ −22.4361 −0.894584
$$630$$ 0 0
$$631$$ −11.1851 −0.445270 −0.222635 0.974902i $$-0.571466\pi$$
−0.222635 + 0.974902i $$0.571466\pi$$
$$632$$ 0 0
$$633$$ 14.3102 0.568779
$$634$$ 0 0
$$635$$ −2.41512 −0.0958409
$$636$$ 0 0
$$637$$ −1.15716 −0.0458482
$$638$$ 0 0
$$639$$ −9.11529 −0.360595
$$640$$ 0 0
$$641$$ −6.69312 −0.264362 −0.132181 0.991226i $$-0.542198\pi$$
−0.132181 + 0.991226i $$0.542198\pi$$
$$642$$ 0 0
$$643$$ 25.3724 1.00059 0.500294 0.865856i $$-0.333225\pi$$
0.500294 + 0.865856i $$0.333225\pi$$
$$644$$ 0 0
$$645$$ −1.07866 −0.0424720
$$646$$ 0 0
$$647$$ 6.72999 0.264583 0.132292 0.991211i $$-0.457766\pi$$
0.132292 + 0.991211i $$0.457766\pi$$
$$648$$ 0 0
$$649$$ 19.7890 0.776786
$$650$$ 0 0
$$651$$ −2.54156 −0.0996116
$$652$$ 0 0
$$653$$ 37.0144 1.44849 0.724243 0.689545i $$-0.242190\pi$$
0.724243 + 0.689545i $$0.242190\pi$$
$$654$$ 0 0
$$655$$ 1.41921 0.0554529
$$656$$ 0 0
$$657$$ −0.541560 −0.0211283
$$658$$ 0 0
$$659$$ 19.7624 0.769832 0.384916 0.922952i $$-0.374230\pi$$
0.384916 + 0.922952i $$0.374230\pi$$
$$660$$ 0 0
$$661$$ −16.8632 −0.655904 −0.327952 0.944694i $$-0.606358\pi$$
−0.327952 + 0.944694i $$0.606358\pi$$
$$662$$ 0 0
$$663$$ 0.303911 0.0118029
$$664$$ 0 0
$$665$$ −7.80785 −0.302776
$$666$$ 0 0
$$667$$ 20.6533 0.799700
$$668$$ 0 0
$$669$$ −4.86156 −0.187959
$$670$$ 0 0
$$671$$ −13.5021 −0.521244
$$672$$ 0 0
$$673$$ −37.3066 −1.43807 −0.719033 0.694976i $$-0.755415\pi$$
−0.719033 + 0.694976i $$0.755415\pi$$
$$674$$ 0 0
$$675$$ −4.77568 −0.183816
$$676$$ 0 0
$$677$$ 0.632805 0.0243207 0.0121603 0.999926i $$-0.496129\pi$$
0.0121603 + 0.999926i $$0.496129\pi$$
$$678$$ 0 0
$$679$$ 19.6764 0.755113
$$680$$ 0 0
$$681$$ −15.0496 −0.576702
$$682$$ 0 0
$$683$$ 6.04606 0.231346 0.115673 0.993287i $$-0.463098\pi$$
0.115673 + 0.993287i $$0.463098\pi$$
$$684$$ 0 0
$$685$$ 1.60058 0.0611549
$$686$$ 0 0
$$687$$ −28.5264 −1.08835
$$688$$ 0 0
$$689$$ −0.0586426 −0.00223410
$$690$$ 0 0
$$691$$ 28.3955 1.08021 0.540107 0.841596i $$-0.318384\pi$$
0.540107 + 0.841596i $$0.318384\pi$$
$$692$$ 0 0
$$693$$ 15.9437 0.605652
$$694$$ 0 0
$$695$$ 3.93926 0.149425
$$696$$ 0 0
$$697$$ 33.5443 1.27058
$$698$$ 0 0
$$699$$ 13.5702 0.513272
$$700$$ 0 0
$$701$$ 14.7738 0.558000 0.279000 0.960291i $$-0.409997\pi$$
0.279000 + 0.960291i $$0.409997\pi$$
$$702$$ 0 0
$$703$$ 22.4361 0.846192
$$704$$ 0 0
$$705$$ 1.33962 0.0504529
$$706$$ 0 0
$$707$$ −2.92450 −0.109987
$$708$$ 0 0
$$709$$ 22.7569 0.854655 0.427327 0.904097i $$-0.359455\pi$$
0.427327 + 0.904097i $$0.359455\pi$$
$$710$$ 0 0
$$711$$ 10.9937 0.412296
$$712$$ 0 0
$$713$$ −1.57726 −0.0590690
$$714$$ 0 0
$$715$$ 0.139211 0.00520621
$$716$$ 0 0
$$717$$ −29.3629 −1.09658
$$718$$ 0 0
$$719$$ 30.9957 1.15594 0.577972 0.816057i $$-0.303844\pi$$
0.577972 + 0.816057i $$0.303844\pi$$
$$720$$ 0 0
$$721$$ −6.09292 −0.226912
$$722$$ 0 0
$$723$$ −24.0063 −0.892803
$$724$$ 0 0
$$725$$ −34.8723 −1.29512
$$726$$ 0 0
$$727$$ 41.1117 1.52475 0.762375 0.647135i $$-0.224033\pi$$
0.762375 + 0.647135i $$0.224033\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.23765 −0.304680
$$732$$ 0 0
$$733$$ −0.206562 −0.00762954 −0.00381477 0.999993i $$-0.501214\pi$$
−0.00381477 + 0.999993i $$0.501214\pi$$
$$734$$ 0 0
$$735$$ −6.52284 −0.240599
$$736$$ 0 0
$$737$$ −18.6792 −0.688058
$$738$$ 0 0
$$739$$ 2.13215 0.0784325 0.0392162 0.999231i $$-0.487514\pi$$
0.0392162 + 0.999231i $$0.487514\pi$$
$$740$$ 0 0
$$741$$ −0.303911 −0.0111644
$$742$$ 0 0
$$743$$ −40.5175 −1.48644 −0.743221 0.669046i $$-0.766703\pi$$
−0.743221 + 0.669046i $$0.766703\pi$$
$$744$$ 0 0
$$745$$ −6.69256 −0.245196
$$746$$ 0 0
$$747$$ −15.0496 −0.550636
$$748$$ 0 0
$$749$$ −39.0761 −1.42781
$$750$$ 0 0
$$751$$ 12.5843 0.459208 0.229604 0.973284i $$-0.426257\pi$$
0.229604 + 0.973284i $$0.426257\pi$$
$$752$$ 0 0
$$753$$ 22.2837 0.812064
$$754$$ 0 0
$$755$$ −4.72529 −0.171971
$$756$$ 0 0
$$757$$ 10.6052 0.385452 0.192726 0.981253i $$-0.438267\pi$$
0.192726 + 0.981253i $$0.438267\pi$$
$$758$$ 0 0
$$759$$ 9.89450 0.359148
$$760$$ 0 0
$$761$$ 42.8182 1.55216 0.776079 0.630635i $$-0.217206\pi$$
0.776079 + 0.630635i $$0.217206\pi$$
$$762$$ 0 0
$$763$$ 36.8441 1.33385
$$764$$ 0 0
$$765$$ 1.71313 0.0619384
$$766$$ 0 0
$$767$$ −0.475298 −0.0171620
$$768$$ 0 0
$$769$$ 12.7455 0.459614 0.229807 0.973236i $$-0.426190\pi$$
0.229807 + 0.973236i $$0.426190\pi$$
$$770$$ 0 0
$$771$$ 8.66038 0.311896
$$772$$ 0 0
$$773$$ −32.3522 −1.16363 −0.581814 0.813322i $$-0.697657\pi$$
−0.581814 + 0.813322i $$0.697657\pi$$
$$774$$ 0 0
$$775$$ 2.66314 0.0956630
$$776$$ 0 0
$$777$$ 28.2704 1.01419
$$778$$ 0 0
$$779$$ −33.5443 −1.20185
$$780$$ 0 0
$$781$$ −31.8874 −1.14102
$$782$$ 0 0
$$783$$ 7.30205 0.260954
$$784$$ 0 0
$$785$$ −10.8253 −0.386370
$$786$$ 0 0
$$787$$ 7.36056 0.262376 0.131188 0.991358i $$-0.458121\pi$$
0.131188 + 0.991358i $$0.458121\pi$$
$$788$$ 0 0
$$789$$ −13.3208 −0.474232
$$790$$ 0 0
$$791$$ 43.5317 1.54781
$$792$$ 0 0
$$793$$ 0.324298 0.0115162
$$794$$ 0 0
$$795$$ −0.330566 −0.0117240
$$796$$ 0 0
$$797$$ −24.0627 −0.852344 −0.426172 0.904642i $$-0.640138\pi$$
−0.426172 + 0.904642i $$0.640138\pi$$
$$798$$ 0 0
$$799$$ 10.2306 0.361932
$$800$$ 0 0
$$801$$ −14.6533 −0.517750
$$802$$ 0 0
$$803$$ −1.89450 −0.0668556
$$804$$ 0 0
$$805$$ −6.10550 −0.215190
$$806$$ 0 0
$$807$$ −16.5058 −0.581032
$$808$$ 0 0
$$809$$ −7.83586 −0.275494 −0.137747 0.990467i $$-0.543986\pi$$
−0.137747 + 0.990467i $$0.543986\pi$$
$$810$$ 0 0
$$811$$ −45.7351 −1.60598 −0.802988 0.595995i $$-0.796758\pi$$
−0.802988 + 0.595995i $$0.796758\pi$$
$$812$$ 0 0
$$813$$ 21.9769 0.770762
$$814$$ 0 0
$$815$$ −5.02684 −0.176083
$$816$$ 0 0
$$817$$ 8.23765 0.288199
$$818$$ 0 0
$$819$$ −0.382941 −0.0133810
$$820$$ 0 0
$$821$$ −27.3709 −0.955250 −0.477625 0.878564i $$-0.658502\pi$$
−0.477625 + 0.878564i $$0.658502\pi$$
$$822$$ 0 0
$$823$$ −28.8560 −1.00586 −0.502929 0.864328i $$-0.667744\pi$$
−0.502929 + 0.864328i $$0.667744\pi$$
$$824$$ 0 0
$$825$$ −16.7064 −0.581644
$$826$$ 0 0
$$827$$ −14.4227 −0.501528 −0.250764 0.968048i $$-0.580682\pi$$
−0.250764 + 0.968048i $$0.580682\pi$$
$$828$$ 0 0
$$829$$ −21.7497 −0.755400 −0.377700 0.925928i $$-0.623285\pi$$
−0.377700 + 0.925928i $$0.623285\pi$$
$$830$$ 0 0
$$831$$ 15.4862 0.537210
$$832$$ 0 0
$$833$$ −49.8147 −1.72598
$$834$$ 0 0
$$835$$ −2.76513 −0.0956914
$$836$$ 0 0
$$837$$ −0.557647 −0.0192751
$$838$$ 0 0
$$839$$ −44.4557 −1.53478 −0.767390 0.641181i $$-0.778445\pi$$
−0.767390 + 0.641181i $$0.778445\pi$$
$$840$$ 0 0
$$841$$ 24.3200 0.838620
$$842$$ 0 0
$$843$$ 22.8910 0.788407
$$844$$ 0 0
$$845$$ 6.15379 0.211697
$$846$$ 0 0
$$847$$ 5.64077 0.193819
$$848$$ 0 0
$$849$$ 6.34315 0.217696
$$850$$ 0 0
$$851$$ 17.5443 0.601411
$$852$$ 0 0
$$853$$ 16.5648 0.567169 0.283585 0.958947i $$-0.408476\pi$$
0.283585 + 0.958947i $$0.408476\pi$$
$$854$$ 0 0
$$855$$ −1.71313 −0.0585879
$$856$$ 0 0
$$857$$ −19.0888 −0.652062 −0.326031 0.945359i $$-0.605711\pi$$
−0.326031 + 0.945359i $$0.605711\pi$$
$$858$$ 0 0
$$859$$ −53.9272 −1.83997 −0.919987 0.391949i $$-0.871801\pi$$
−0.919987 + 0.391949i $$0.871801\pi$$
$$860$$ 0 0
$$861$$ −42.2672 −1.44046
$$862$$ 0 0
$$863$$ 3.64533 0.124089 0.0620443 0.998073i $$-0.480238\pi$$
0.0620443 + 0.998073i $$0.480238\pi$$
$$864$$ 0 0
$$865$$ 2.42826 0.0825632
$$866$$ 0 0
$$867$$ −3.91688 −0.133024
$$868$$ 0 0
$$869$$ 38.4586 1.30462
$$870$$ 0 0
$$871$$ 0.448643 0.0152017
$$872$$ 0 0
$$873$$ 4.31724 0.146116
$$874$$ 0 0
$$875$$ 21.1020 0.713377
$$876$$ 0 0
$$877$$ 56.6481 1.91287 0.956435 0.291945i $$-0.0943023\pi$$
0.956435 + 0.291945i $$0.0943023\pi$$
$$878$$ 0 0
$$879$$ 30.5783 1.03138
$$880$$ 0 0
$$881$$ 20.0118 0.674214 0.337107 0.941466i $$-0.390552\pi$$
0.337107 + 0.941466i $$0.390552\pi$$
$$882$$ 0 0
$$883$$ −15.0292 −0.505773 −0.252887 0.967496i $$-0.581380\pi$$
−0.252887 + 0.967496i $$0.581380\pi$$
$$884$$ 0 0
$$885$$ −2.67923 −0.0900614
$$886$$ 0 0
$$887$$ −26.1180 −0.876958 −0.438479 0.898742i $$-0.644483\pi$$
−0.438479 + 0.898742i $$0.644483\pi$$
$$888$$ 0 0
$$889$$ 23.2404 0.779458
$$890$$ 0 0
$$891$$ 3.49824 0.117195
$$892$$ 0 0
$$893$$ −10.2306 −0.342354
$$894$$ 0 0
$$895$$ −6.21805 −0.207847
$$896$$ 0 0
$$897$$ −0.237649 −0.00793486
$$898$$ 0 0
$$899$$ −4.07197 −0.135808
$$900$$ 0 0
$$901$$ −2.52452 −0.0841038
$$902$$ 0 0
$$903$$ 10.3798 0.345418
$$904$$ 0 0
$$905$$ −7.25507 −0.241167
$$906$$ 0 0
$$907$$ −51.2480 −1.70166 −0.850831 0.525439i $$-0.823901\pi$$
−0.850831 + 0.525439i $$0.823901\pi$$
$$908$$ 0 0
$$909$$ −0.641669 −0.0212828
$$910$$ 0 0
$$911$$ 21.0535 0.697533 0.348767 0.937210i $$-0.386601\pi$$
0.348767 + 0.937210i $$0.386601\pi$$
$$912$$ 0 0
$$913$$ −52.6470 −1.74236
$$914$$ 0 0
$$915$$ 1.82805 0.0604336
$$916$$ 0 0
$$917$$ −13.6569 −0.450989
$$918$$ 0 0
$$919$$ 17.8839 0.589937 0.294968 0.955507i $$-0.404691\pi$$
0.294968 + 0.955507i $$0.404691\pi$$
$$920$$ 0 0
$$921$$ −17.1286 −0.564407
$$922$$ 0 0
$$923$$ 0.765881 0.0252093
$$924$$ 0 0
$$925$$ −29.6228 −0.973992
$$926$$ 0 0
$$927$$ −1.33686 −0.0439081
$$928$$ 0 0
$$929$$ 10.2774 0.337192 0.168596 0.985685i $$-0.446077\pi$$
0.168596 + 0.985685i $$0.446077\pi$$
$$930$$ 0 0
$$931$$ 49.8147 1.63261
$$932$$ 0 0
$$933$$ −26.8651 −0.879523
$$934$$ 0 0
$$935$$ 5.99294 0.195990
$$936$$ 0 0
$$937$$ −13.5780 −0.443574 −0.221787 0.975095i $$-0.571189\pi$$
−0.221787 + 0.975095i $$0.571189\pi$$
$$938$$ 0 0
$$939$$ −19.6890 −0.642527
$$940$$ 0 0
$$941$$ −5.59890 −0.182519 −0.0912595 0.995827i $$-0.529089\pi$$
−0.0912595 + 0.995827i $$0.529089\pi$$
$$942$$ 0 0
$$943$$ −26.2306 −0.854186
$$944$$ 0 0
$$945$$ −2.15862 −0.0702199
$$946$$ 0 0
$$947$$ −46.9106 −1.52439 −0.762194 0.647348i $$-0.775878\pi$$
−0.762194 + 0.647348i $$0.775878\pi$$
$$948$$ 0 0
$$949$$ 0.0455027 0.00147708
$$950$$ 0 0
$$951$$ −30.1860 −0.978847
$$952$$ 0 0
$$953$$ 5.59115 0.181115 0.0905576 0.995891i $$-0.471135\pi$$
0.0905576 + 0.995891i $$0.471135\pi$$
$$954$$ 0 0
$$955$$ 4.08740 0.132265
$$956$$ 0 0
$$957$$ 25.5443 0.825730
$$958$$ 0 0
$$959$$ −15.4022 −0.497362
$$960$$ 0 0
$$961$$ −30.6890 −0.989969
$$962$$ 0 0
$$963$$ −8.57373 −0.276285
$$964$$ 0 0
$$965$$ −5.42367 −0.174594
$$966$$ 0 0
$$967$$ −30.7561 −0.989048 −0.494524 0.869164i $$-0.664658\pi$$
−0.494524 + 0.869164i $$0.664658\pi$$
$$968$$ 0 0
$$969$$ −13.0831 −0.420290
$$970$$ 0 0
$$971$$ −11.3668 −0.364779 −0.182389 0.983226i $$-0.558383\pi$$
−0.182389 + 0.983226i $$0.558383\pi$$
$$972$$ 0 0
$$973$$ −37.9070 −1.21524
$$974$$ 0 0
$$975$$ 0.401260 0.0128506
$$976$$ 0 0
$$977$$ −22.8323 −0.730471 −0.365235 0.930915i $$-0.619012\pi$$
−0.365235 + 0.930915i $$0.619012\pi$$
$$978$$ 0 0
$$979$$ −51.2608 −1.63830
$$980$$ 0 0
$$981$$ 8.08402 0.258103
$$982$$ 0 0
$$983$$ −46.3557 −1.47852 −0.739258 0.673422i $$-0.764824\pi$$
−0.739258 + 0.673422i $$0.764824\pi$$
$$984$$ 0 0
$$985$$ −5.01686 −0.159850
$$986$$ 0 0
$$987$$ −12.8910 −0.410324
$$988$$ 0 0
$$989$$ 6.44158 0.204830
$$990$$ 0 0
$$991$$ 3.43683 0.109175 0.0545873 0.998509i $$-0.482616\pi$$
0.0545873 + 0.998509i $$0.482616\pi$$
$$992$$ 0 0
$$993$$ −20.7784 −0.659383
$$994$$ 0 0
$$995$$ −1.74294 −0.0552550
$$996$$ 0 0
$$997$$ −31.0320 −0.982794 −0.491397 0.870936i $$-0.663514\pi$$
−0.491397 + 0.870936i $$0.663514\pi$$
$$998$$ 0 0
$$999$$ 6.20285 0.196249
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.t.1.2 4
3.2 odd 2 9216.2.a.y.1.3 4
4.3 odd 2 3072.2.a.n.1.2 4
8.3 odd 2 3072.2.a.o.1.3 4
8.5 even 2 3072.2.a.i.1.3 4
12.11 even 2 9216.2.a.x.1.3 4
16.3 odd 4 3072.2.d.i.1537.3 8
16.5 even 4 3072.2.d.f.1537.2 8
16.11 odd 4 3072.2.d.i.1537.6 8
16.13 even 4 3072.2.d.f.1537.7 8
24.5 odd 2 9216.2.a.bo.1.2 4
24.11 even 2 9216.2.a.bn.1.2 4
32.3 odd 8 384.2.j.a.289.2 8
32.5 even 8 48.2.j.a.37.4 yes 8
32.11 odd 8 384.2.j.a.97.2 8
32.13 even 8 48.2.j.a.13.4 8
32.19 odd 8 192.2.j.a.145.3 8
32.21 even 8 384.2.j.b.97.4 8
32.27 odd 8 192.2.j.a.49.3 8
32.29 even 8 384.2.j.b.289.4 8
96.5 odd 8 144.2.k.b.37.1 8
96.11 even 8 1152.2.k.f.865.2 8
96.29 odd 8 1152.2.k.c.289.2 8
96.35 even 8 1152.2.k.f.289.2 8
96.53 odd 8 1152.2.k.c.865.2 8
96.59 even 8 576.2.k.b.433.3 8
96.77 odd 8 144.2.k.b.109.1 8
96.83 even 8 576.2.k.b.145.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.4 8 32.13 even 8
48.2.j.a.37.4 yes 8 32.5 even 8
144.2.k.b.37.1 8 96.5 odd 8
144.2.k.b.109.1 8 96.77 odd 8
192.2.j.a.49.3 8 32.27 odd 8
192.2.j.a.145.3 8 32.19 odd 8
384.2.j.a.97.2 8 32.11 odd 8
384.2.j.a.289.2 8 32.3 odd 8
384.2.j.b.97.4 8 32.21 even 8
384.2.j.b.289.4 8 32.29 even 8
576.2.k.b.145.3 8 96.83 even 8
576.2.k.b.433.3 8 96.59 even 8
1152.2.k.c.289.2 8 96.29 odd 8
1152.2.k.c.865.2 8 96.53 odd 8
1152.2.k.f.289.2 8 96.35 even 8
1152.2.k.f.865.2 8 96.11 even 8
3072.2.a.i.1.3 4 8.5 even 2
3072.2.a.n.1.2 4 4.3 odd 2
3072.2.a.o.1.3 4 8.3 odd 2
3072.2.a.t.1.2 4 1.1 even 1 trivial
3072.2.d.f.1537.2 8 16.5 even 4
3072.2.d.f.1537.7 8 16.13 even 4
3072.2.d.i.1537.3 8 16.3 odd 4
3072.2.d.i.1537.6 8 16.11 odd 4
9216.2.a.x.1.3 4 12.11 even 2
9216.2.a.y.1.3 4 3.2 odd 2
9216.2.a.bn.1.2 4 24.11 even 2
9216.2.a.bo.1.2 4 24.5 odd 2