# Properties

 Label 3072.2.d.i.1537.6 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 1537.6 Root $$0.500000 - 0.0297061i$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1537 Dual form 3072.2.d.i.1537.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -0.473626i q^{5} +4.55765 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -0.473626i q^{5} +4.55765 q^{7} -1.00000 q^{9} -3.49824i q^{11} +0.0840215i q^{13} +0.473626 q^{15} -3.61706 q^{17} +3.61706i q^{19} +4.55765i q^{21} +2.82843 q^{23} +4.77568 q^{25} -1.00000i q^{27} -7.30205i q^{29} +0.557647 q^{31} +3.49824 q^{33} -2.15862i q^{35} +6.20285i q^{37} -0.0840215 q^{39} +9.27391 q^{41} -2.27744i q^{43} +0.473626i q^{45} +2.82843 q^{47} +13.7721 q^{49} -3.61706i q^{51} +0.697947i q^{53} -1.65685 q^{55} -3.61706 q^{57} -5.65685i q^{59} +3.85970i q^{61} -4.55765 q^{63} +0.0397948 q^{65} -5.33962i q^{67} +2.82843i q^{69} -9.11529 q^{71} +0.541560 q^{73} +4.77568i q^{75} -15.9437i q^{77} -10.9937 q^{79} +1.00000 q^{81} -15.0496i q^{83} +1.71313i q^{85} +7.30205 q^{87} +14.6533 q^{89} +0.382941i q^{91} +0.557647i q^{93} +1.71313 q^{95} +4.31724 q^{97} +3.49824i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{7} - 8q^{9} + O(q^{10})$$ $$8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100})$$

## 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
$$5$$ − 0.473626i − 0.211812i −0.994376 0.105906i $$-0.966226\pi$$
0.994376 0.105906i $$-0.0337742\pi$$
$$6$$ 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
$$11$$ − 3.49824i − 1.05476i −0.849630 0.527379i $$-0.823175\pi$$
0.849630 0.527379i $$-0.176825\pi$$
$$12$$ 0 0
$$13$$ 0.0840215i 0.0233034i 0.999932 + 0.0116517i $$0.00370893\pi$$
−0.999932 + 0.0116517i $$0.996291\pi$$
$$14$$ 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
$$19$$ 3.61706i 0.829810i 0.909865 + 0.414905i $$0.136185\pi$$
−0.909865 + 0.414905i $$0.863815\pi$$
$$20$$ 0 0
$$21$$ 4.55765i 0.994560i
$$22$$ 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
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ − 7.30205i − 1.35596i −0.735082 0.677979i $$-0.762856\pi$$
0.735082 0.677979i $$-0.237144\pi$$
$$30$$ 0 0
$$31$$ 0.557647 0.100156 0.0500782 0.998745i $$-0.484053\pi$$
0.0500782 + 0.998745i $$0.484053\pi$$
$$32$$ 0 0
$$33$$ 3.49824 0.608965
$$34$$ 0 0
$$35$$ − 2.15862i − 0.364873i
$$36$$ 0 0
$$37$$ 6.20285i 1.01974i 0.860251 + 0.509871i $$0.170307\pi$$
−0.860251 + 0.509871i $$0.829693\pi$$
$$38$$ 0 0
$$39$$ −0.0840215 −0.0134542
$$40$$ 0 0
$$41$$ 9.27391 1.44834 0.724171 0.689620i $$-0.242223\pi$$
0.724171 + 0.689620i $$0.242223\pi$$
$$42$$ 0 0
$$43$$ − 2.27744i − 0.347307i −0.984807 0.173653i $$-0.944443\pi$$
0.984807 0.173653i $$-0.0555573\pi$$
$$44$$ 0 0
$$45$$ 0.473626i 0.0706040i
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ 13.7721 1.96745
$$50$$ 0 0
$$51$$ − 3.61706i − 0.506490i
$$52$$ 0 0
$$53$$ 0.697947i 0.0958704i 0.998850 + 0.0479352i $$0.0152641\pi$$
−0.998850 + 0.0479352i $$0.984736\pi$$
$$54$$ 0 0
$$55$$ −1.65685 −0.223410
$$56$$ 0 0
$$57$$ −3.61706 −0.479091
$$58$$ 0 0
$$59$$ − 5.65685i − 0.736460i −0.929735 0.368230i $$-0.879964\pi$$
0.929735 0.368230i $$-0.120036\pi$$
$$60$$ 0 0
$$61$$ 3.85970i 0.494184i 0.968992 + 0.247092i $$0.0794750\pi$$
−0.968992 + 0.247092i $$0.920525\pi$$
$$62$$ 0 0
$$63$$ −4.55765 −0.574210
$$64$$ 0 0
$$65$$ 0.0397948 0.00493593
$$66$$ 0 0
$$67$$ − 5.33962i − 0.652338i −0.945311 0.326169i $$-0.894242\pi$$
0.945311 0.326169i $$-0.105758\pi$$
$$68$$ 0 0
$$69$$ 2.82843i 0.340503i
$$70$$ 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 $$-0.489910\pi$$
0.0316924 + 0.999498i $$0.489910\pi$$
$$74$$ 0 0
$$75$$ 4.77568i 0.551448i
$$76$$ 0 0
$$77$$ − 15.9437i − 1.81696i
$$78$$ 0 0
$$79$$ −10.9937 −1.23689 −0.618445 0.785828i $$-0.712237\pi$$
−0.618445 + 0.785828i $$0.712237\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 15.0496i − 1.65191i −0.563738 0.825954i $$-0.690637\pi$$
0.563738 0.825954i $$-0.309363\pi$$
$$84$$ 0 0
$$85$$ 1.71313i 0.185815i
$$86$$ 0 0
$$87$$ 7.30205 0.782862
$$88$$ 0 0
$$89$$ 14.6533 1.55325 0.776625 0.629964i $$-0.216930\pi$$
0.776625 + 0.629964i $$0.216930\pi$$
$$90$$ 0 0
$$91$$ 0.382941i 0.0401431i
$$92$$ 0 0
$$93$$ 0.557647i 0.0578253i
$$94$$ 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
$$99$$ 3.49824i 0.351586i
$$100$$ 0 0
$$101$$ − 0.641669i − 0.0638484i −0.999490 0.0319242i $$-0.989836\pi$$
0.999490 0.0319242i $$-0.0101635\pi$$
$$102$$ 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
$$107$$ 8.57373i 0.828854i 0.910083 + 0.414427i $$0.136018\pi$$
−0.910083 + 0.414427i $$0.863982\pi$$
$$108$$ 0 0
$$109$$ − 8.08402i − 0.774309i −0.922015 0.387154i $$-0.873458\pi$$
0.922015 0.387154i $$-0.126542\pi$$
$$110$$ 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
$$115$$ − 1.33962i − 0.124920i
$$116$$ 0 0
$$117$$ − 0.0840215i − 0.00776779i
$$118$$ 0 0
$$119$$ −16.4853 −1.51120
$$120$$ 0 0
$$121$$ −1.23765 −0.112514
$$122$$ 0 0
$$123$$ 9.27391i 0.836201i
$$124$$ 0 0
$$125$$ − 4.63001i − 0.414121i
$$126$$ 0 0
$$127$$ −5.09921 −0.452481 −0.226241 0.974071i $$-0.572644\pi$$
−0.226241 + 0.974071i $$0.572644\pi$$
$$128$$ 0 0
$$129$$ 2.27744 0.200518
$$130$$ 0 0
$$131$$ − 2.99647i − 0.261803i −0.991395 0.130901i $$-0.958213\pi$$
0.991395 0.130901i $$-0.0417871\pi$$
$$132$$ 0 0
$$133$$ 16.4853i 1.42946i
$$134$$ 0 0
$$135$$ −0.473626 −0.0407632
$$136$$ 0 0
$$137$$ 3.37941 0.288723 0.144361 0.989525i $$-0.453887\pi$$
0.144361 + 0.989525i $$0.453887\pi$$
$$138$$ 0 0
$$139$$ 8.31724i 0.705459i 0.935725 + 0.352729i $$0.114746\pi$$
−0.935725 + 0.352729i $$0.885254\pi$$
$$140$$ 0 0
$$141$$ 2.82843i 0.238197i
$$142$$ 0 0
$$143$$ 0.293927 0.0245794
$$144$$ 0 0
$$145$$ −3.45844 −0.287208
$$146$$ 0 0
$$147$$ 13.7721i 1.13591i
$$148$$ 0 0
$$149$$ 14.1305i 1.15761i 0.815465 + 0.578807i $$0.196482\pi$$
−0.815465 + 0.578807i $$0.803518\pi$$
$$150$$ 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
$$155$$ − 0.264116i − 0.0212143i
$$156$$ 0 0
$$157$$ − 22.8562i − 1.82412i −0.410056 0.912060i $$-0.634491\pi$$
0.410056 0.912060i $$-0.365509\pi$$
$$158$$ 0 0
$$159$$ −0.697947 −0.0553508
$$160$$ 0 0
$$161$$ 12.8910 1.01595
$$162$$ 0 0
$$163$$ 10.6135i 0.831316i 0.909521 + 0.415658i $$0.136449\pi$$
−0.909521 + 0.415658i $$0.863551\pi$$
$$164$$ 0 0
$$165$$ − 1.65685i − 0.128986i
$$166$$ 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
$$171$$ − 3.61706i − 0.276603i
$$172$$ 0 0
$$173$$ 5.12695i 0.389795i 0.980824 + 0.194897i $$0.0624374\pi$$
−0.980824 + 0.194897i $$0.937563\pi$$
$$174$$ 0 0
$$175$$ 21.7659 1.64534
$$176$$ 0 0
$$177$$ 5.65685 0.425195
$$178$$ 0 0
$$179$$ 13.1286i 0.981279i 0.871363 + 0.490640i $$0.163237\pi$$
−0.871363 + 0.490640i $$0.836763\pi$$
$$180$$ 0 0
$$181$$ 15.3181i 1.13859i 0.822134 + 0.569294i $$0.192783\pi$$
−0.822134 + 0.569294i $$0.807217\pi$$
$$182$$ 0 0
$$183$$ −3.85970 −0.285317
$$184$$ 0 0
$$185$$ 2.93783 0.215993
$$186$$ 0 0
$$187$$ 12.6533i 0.925303i
$$188$$ 0 0
$$189$$ − 4.55765i − 0.331520i
$$190$$ 0 0
$$191$$ 8.63001 0.624446 0.312223 0.950009i $$-0.398926\pi$$
0.312223 + 0.950009i $$0.398926\pi$$
$$192$$ 0 0
$$193$$ 11.4514 0.824288 0.412144 0.911119i $$-0.364780\pi$$
0.412144 + 0.911119i $$0.364780\pi$$
$$194$$ 0 0
$$195$$ 0.0397948i 0.00284976i
$$196$$ 0 0
$$197$$ 10.5925i 0.754681i 0.926075 + 0.377340i $$0.123161\pi$$
−0.926075 + 0.377340i $$0.876839\pi$$
$$198$$ 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
$$203$$ − 33.2802i − 2.33581i
$$204$$ 0 0
$$205$$ − 4.39236i − 0.306776i
$$206$$ 0 0
$$207$$ −2.82843 −0.196589
$$208$$ 0 0
$$209$$ 12.6533 0.875249
$$210$$ 0 0
$$211$$ 14.3102i 0.985153i 0.870269 + 0.492577i $$0.163945\pi$$
−0.870269 + 0.492577i $$0.836055\pi$$
$$212$$ 0 0
$$213$$ − 9.11529i − 0.624570i
$$214$$ 0 0
$$215$$ −1.07866 −0.0735637
$$216$$ 0 0
$$217$$ 2.54156 0.172532
$$218$$ 0 0
$$219$$ 0.541560i 0.0365952i
$$220$$ 0 0
$$221$$ − 0.303911i − 0.0204433i
$$222$$ 0 0
$$223$$ 4.86156 0.325554 0.162777 0.986663i $$-0.447955\pi$$
0.162777 + 0.986663i $$0.447955\pi$$
$$224$$ 0 0
$$225$$ −4.77568 −0.318379
$$226$$ 0 0
$$227$$ − 15.0496i − 0.998877i −0.866349 0.499438i $$-0.833540\pi$$
0.866349 0.499438i $$-0.166460\pi$$
$$228$$ 0 0
$$229$$ − 28.5264i − 1.88507i −0.334101 0.942537i $$-0.608433\pi$$
0.334101 0.942537i $$-0.391567\pi$$
$$230$$ 0 0
$$231$$ 15.9437 1.04902
$$232$$ 0 0
$$233$$ −13.5702 −0.889014 −0.444507 0.895775i $$-0.646621\pi$$
−0.444507 + 0.895775i $$0.646621\pi$$
$$234$$ 0 0
$$235$$ − 1.33962i − 0.0873869i
$$236$$ 0 0
$$237$$ − 10.9937i − 0.714118i
$$238$$ 0 0
$$239$$ 29.3629 1.89933 0.949665 0.313267i $$-0.101424\pi$$
0.949665 + 0.313267i $$0.101424\pi$$
$$240$$ 0 0
$$241$$ −24.0063 −1.54638 −0.773190 0.634175i $$-0.781340\pi$$
−0.773190 + 0.634175i $$0.781340\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ − 6.52284i − 0.416729i
$$246$$ 0 0
$$247$$ −0.303911 −0.0193374
$$248$$ 0 0
$$249$$ 15.0496 0.953729
$$250$$ 0 0
$$251$$ − 22.2837i − 1.40654i −0.710925 0.703268i $$-0.751724\pi$$
0.710925 0.703268i $$-0.248276\pi$$
$$252$$ 0 0
$$253$$ − 9.89450i − 0.622062i
$$254$$ 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
$$259$$ 28.2704i 1.75664i
$$260$$ 0 0
$$261$$ 7.30205i 0.451986i
$$262$$ 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
$$267$$ 14.6533i 0.896769i
$$268$$ 0 0
$$269$$ 16.5058i 1.00638i 0.864177 + 0.503188i $$0.167840\pi$$
−0.864177 + 0.503188i $$0.832160\pi$$
$$270$$ 0 0
$$271$$ −21.9769 −1.33500 −0.667499 0.744610i $$-0.732635\pi$$
−0.667499 + 0.744610i $$0.732635\pi$$
$$272$$ 0 0
$$273$$ −0.382941 −0.0231766
$$274$$ 0 0
$$275$$ − 16.7064i − 1.00744i
$$276$$ 0 0
$$277$$ 15.4862i 0.930475i 0.885186 + 0.465237i $$0.154031\pi$$
−0.885186 + 0.465237i $$0.845969\pi$$
$$278$$ 0 0
$$279$$ −0.557647 −0.0333855
$$280$$ 0 0
$$281$$ −22.8910 −1.36556 −0.682780 0.730624i $$-0.739229\pi$$
−0.682780 + 0.730624i $$0.739229\pi$$
$$282$$ 0 0
$$283$$ − 6.34315i − 0.377061i −0.982067 0.188530i $$-0.939628\pi$$
0.982067 0.188530i $$-0.0603724\pi$$
$$284$$ 0 0
$$285$$ 1.71313i 0.101477i
$$286$$ 0 0
$$287$$ 42.2672 2.49496
$$288$$ 0 0
$$289$$ −3.91688 −0.230405
$$290$$ 0 0
$$291$$ 4.31724i 0.253081i
$$292$$ 0 0
$$293$$ 30.5783i 1.78641i 0.449654 + 0.893203i $$0.351547\pi$$
−0.449654 + 0.893203i $$0.648453\pi$$
$$294$$ 0 0
$$295$$ −2.67923 −0.155991
$$296$$ 0 0
$$297$$ −3.49824 −0.202988
$$298$$ 0 0
$$299$$ 0.237649i 0.0137436i
$$300$$ 0 0
$$301$$ − 10.3798i − 0.598281i
$$302$$ 0 0
$$303$$ 0.641669 0.0368629
$$304$$ 0 0
$$305$$ 1.82805 0.104674
$$306$$ 0 0
$$307$$ − 17.1286i − 0.977582i −0.872401 0.488791i $$-0.837438\pi$$
0.872401 0.488791i $$-0.162562\pi$$
$$308$$ 0 0
$$309$$ − 1.33686i − 0.0760511i
$$310$$ 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 $$-0.312165\pi$$
0.556445 + 0.830885i $$0.312165\pi$$
$$314$$ 0 0
$$315$$ 2.15862i 0.121624i
$$316$$ 0 0
$$317$$ 30.1860i 1.69541i 0.530466 + 0.847706i $$0.322017\pi$$
−0.530466 + 0.847706i $$0.677983\pi$$
$$318$$ 0 0
$$319$$ −25.5443 −1.43021
$$320$$ 0 0
$$321$$ −8.57373 −0.478539
$$322$$ 0 0
$$323$$ − 13.0831i − 0.727964i
$$324$$ 0 0
$$325$$ 0.401260i 0.0222579i
$$326$$ 0 0
$$327$$ 8.08402 0.447047
$$328$$ 0 0
$$329$$ 12.8910 0.710702
$$330$$ 0 0
$$331$$ 20.7784i 1.14209i 0.820920 + 0.571043i $$0.193461\pi$$
−0.820920 + 0.571043i $$0.806539\pi$$
$$332$$ 0 0
$$333$$ − 6.20285i − 0.339914i
$$334$$ 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
$$339$$ 9.55136i 0.518759i
$$340$$ 0 0
$$341$$ − 1.95078i − 0.105641i
$$342$$ 0 0
$$343$$ 30.8651 1.66656
$$344$$ 0 0
$$345$$ 1.33962 0.0721225
$$346$$ 0 0
$$347$$ 15.4186i 0.827716i 0.910341 + 0.413858i $$0.135819\pi$$
−0.910341 + 0.413858i $$0.864181\pi$$
$$348$$ 0 0
$$349$$ 28.3638i 1.51828i 0.650927 + 0.759140i $$0.274380\pi$$
−0.650927 + 0.759140i $$0.725620\pi$$
$$350$$ 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
$$355$$ 4.31724i 0.229135i
$$356$$ 0 0
$$357$$ − 16.4853i − 0.872494i
$$358$$ 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
$$363$$ − 1.23765i − 0.0649597i
$$364$$ 0 0
$$365$$ − 0.256497i − 0.0134256i
$$366$$ 0 0
$$367$$ −0.702379 −0.0366639 −0.0183319 0.999832i $$-0.505836\pi$$
−0.0183319 + 0.999832i $$0.505836\pi$$
$$368$$ 0 0
$$369$$ −9.27391 −0.482781
$$370$$ 0 0
$$371$$ 3.18100i 0.165149i
$$372$$ 0 0
$$373$$ − 26.8132i − 1.38834i −0.719813 0.694168i $$-0.755773\pi$$
0.719813 0.694168i $$-0.244227\pi$$
$$374$$ 0 0
$$375$$ 4.63001 0.239093
$$376$$ 0 0
$$377$$ 0.613530 0.0315984
$$378$$ 0 0
$$379$$ 2.51509i 0.129192i 0.997912 + 0.0645958i $$0.0205758\pi$$
−0.997912 + 0.0645958i $$0.979424\pi$$
$$380$$ 0 0
$$381$$ − 5.09921i − 0.261240i
$$382$$ 0 0
$$383$$ −25.4880 −1.30238 −0.651188 0.758916i $$-0.725729\pi$$
−0.651188 + 0.758916i $$0.725729\pi$$
$$384$$ 0 0
$$385$$ −7.55136 −0.384853
$$386$$ 0 0
$$387$$ 2.27744i 0.115769i
$$388$$ 0 0
$$389$$ − 16.5532i − 0.839281i −0.907690 0.419641i $$-0.862156\pi$$
0.907690 0.419641i $$-0.137844\pi$$
$$390$$ 0 0
$$391$$ −10.2306 −0.517383
$$392$$ 0 0
$$393$$ 2.99647 0.151152
$$394$$ 0 0
$$395$$ 5.20690i 0.261988i
$$396$$ 0 0
$$397$$ 12.7936i 0.642094i 0.947063 + 0.321047i $$0.104035\pi$$
−0.947063 + 0.321047i $$0.895965\pi$$
$$398$$ 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
$$403$$ 0.0468544i 0.00233398i
$$404$$ 0 0
$$405$$ − 0.473626i − 0.0235347i
$$406$$ 0 0
$$407$$ 21.6990 1.07558
$$408$$ 0 0
$$409$$ −25.2271 −1.24740 −0.623699 0.781665i $$-0.714371\pi$$
−0.623699 + 0.781665i $$0.714371\pi$$
$$410$$ 0 0
$$411$$ 3.37941i 0.166694i
$$412$$ 0 0
$$413$$ − 25.7819i − 1.26865i
$$414$$ 0 0
$$415$$ −7.12787 −0.349894
$$416$$ 0 0
$$417$$ −8.31724 −0.407297
$$418$$ 0 0
$$419$$ − 10.2571i − 0.501090i −0.968105 0.250545i $$-0.919390\pi$$
0.968105 0.250545i $$-0.0806098\pi$$
$$420$$ 0 0
$$421$$ 3.38775i 0.165109i 0.996587 + 0.0825543i $$0.0263078\pi$$
−0.996587 + 0.0825543i $$0.973692\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ −17.2739 −0.837908
$$426$$ 0 0
$$427$$ 17.5912i 0.851296i
$$428$$ 0 0
$$429$$ 0.293927i 0.0141909i
$$430$$ 0 0
$$431$$ −4.42454 −0.213123 −0.106561 0.994306i $$-0.533984\pi$$
−0.106561 + 0.994306i $$0.533984\pi$$
$$432$$ 0 0
$$433$$ −7.31371 −0.351474 −0.175737 0.984437i $$-0.556231\pi$$
−0.175737 + 0.984437i $$0.556231\pi$$
$$434$$ 0 0
$$435$$ − 3.45844i − 0.165820i
$$436$$ 0 0
$$437$$ 10.2306i 0.489395i
$$438$$ 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
$$443$$ − 14.5743i − 0.692446i −0.938152 0.346223i $$-0.887464\pi$$
0.938152 0.346223i $$-0.112536\pi$$
$$444$$ 0 0
$$445$$ − 6.94019i − 0.328997i
$$446$$ 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
$$451$$ − 32.4423i − 1.52765i
$$452$$ 0 0
$$453$$ 9.97685i 0.468753i
$$454$$ 0 0
$$455$$ 0.181370 0.00850278
$$456$$ 0 0
$$457$$ 9.00353 0.421167 0.210584 0.977576i $$-0.432464\pi$$
0.210584 + 0.977576i $$0.432464\pi$$
$$458$$ 0 0
$$459$$ 3.61706i 0.168830i
$$460$$ 0 0
$$461$$ − 20.6783i − 0.963085i −0.876423 0.481542i $$-0.840077\pi$$
0.876423 0.481542i $$-0.159923\pi$$
$$462$$ 0 0
$$463$$ −18.6435 −0.866437 −0.433219 0.901289i $$-0.642622\pi$$
−0.433219 + 0.901289i $$0.642622\pi$$
$$464$$ 0 0
$$465$$ 0.264116 0.0122481
$$466$$ 0 0
$$467$$ − 33.2535i − 1.53879i −0.638773 0.769395i $$-0.720558\pi$$
0.638773 0.769395i $$-0.279442\pi$$
$$468$$ 0 0
$$469$$ − 24.3361i − 1.12374i
$$470$$ 0 0
$$471$$ 22.8562 1.05316
$$472$$ 0 0
$$473$$ −7.96703 −0.366325
$$474$$ 0 0
$$475$$ 17.2739i 0.792582i
$$476$$ 0 0
$$477$$ − 0.697947i − 0.0319568i
$$478$$ 0 0
$$479$$ 1.08864 0.0497412 0.0248706 0.999691i $$-0.492083\pi$$
0.0248706 + 0.999691i $$0.492083\pi$$
$$480$$ 0 0
$$481$$ −0.521173 −0.0237634
$$482$$ 0 0
$$483$$ 12.8910i 0.586560i
$$484$$ 0 0
$$485$$ − 2.04476i − 0.0928476i
$$486$$ 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
$$491$$ − 18.2306i − 0.822735i −0.911470 0.411367i $$-0.865051\pi$$
0.911470 0.411367i $$-0.134949\pi$$
$$492$$ 0 0
$$493$$ 26.4120i 1.18953i
$$494$$ 0 0
$$495$$ 1.65685 0.0744701
$$496$$ 0 0
$$497$$ −41.5443 −1.86352
$$498$$ 0 0
$$499$$ 20.3361i 0.910368i 0.890397 + 0.455184i $$0.150427\pi$$
−0.890397 + 0.455184i $$0.849573\pi$$
$$500$$ 0 0
$$501$$ 5.83822i 0.260833i
$$502$$ 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
$$507$$ 12.9929i 0.577037i
$$508$$ 0 0
$$509$$ 14.9660i 0.663355i 0.943393 + 0.331677i $$0.107615\pi$$
−0.943393 + 0.331677i $$0.892385\pi$$
$$510$$ 0 0
$$511$$ 2.46824 0.109188
$$512$$ 0 0
$$513$$ 3.61706 0.159697
$$514$$ 0 0
$$515$$ 0.633169i 0.0279008i
$$516$$ 0 0
$$517$$ − 9.89450i − 0.435160i
$$518$$ 0 0
$$519$$ −5.12695 −0.225048
$$520$$ 0 0
$$521$$ −24.9049 −1.09110 −0.545551 0.838078i $$-0.683680\pi$$
−0.545551 + 0.838078i $$0.683680\pi$$
$$522$$ 0 0
$$523$$ − 18.2445i − 0.797775i −0.917000 0.398888i $$-0.869396\pi$$
0.917000 0.398888i $$-0.130604\pi$$
$$524$$ 0 0
$$525$$ 21.7659i 0.949940i
$$526$$ 0 0
$$527$$ −2.01704 −0.0878638
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 5.65685i 0.245487i
$$532$$ 0 0
$$533$$ 0.779208i 0.0337513i
$$534$$ 0 0
$$535$$ 4.06074 0.175561
$$536$$ 0 0
$$537$$ −13.1286 −0.566542
$$538$$ 0 0
$$539$$ − 48.1782i − 2.07518i
$$540$$ 0 0
$$541$$ 25.8471i 1.11125i 0.831432 + 0.555627i $$0.187522\pi$$
−0.831432 + 0.555627i $$0.812478\pi$$
$$542$$ 0 0
$$543$$ −15.3181 −0.657364
$$544$$ 0 0
$$545$$ −3.82880 −0.164008
$$546$$ 0 0
$$547$$ − 19.4249i − 0.830549i −0.909696 0.415275i $$-0.863685\pi$$
0.909696 0.415275i $$-0.136315\pi$$
$$548$$ 0 0
$$549$$ − 3.85970i − 0.164728i
$$550$$ 0 0
$$551$$ 26.4120 1.12519
$$552$$ 0 0
$$553$$ −50.1055 −2.13070
$$554$$ 0 0
$$555$$ 2.93783i 0.124704i
$$556$$ 0 0
$$557$$ 38.9652i 1.65101i 0.564397 + 0.825504i $$0.309109\pi$$
−0.564397 + 0.825504i $$0.690891\pi$$
$$558$$ 0 0
$$559$$ 0.191354 0.00809342
$$560$$ 0 0
$$561$$ −12.6533 −0.534224
$$562$$ 0 0
$$563$$ − 28.1327i − 1.18565i −0.805330 0.592826i $$-0.798012\pi$$
0.805330 0.592826i $$-0.201988\pi$$
$$564$$ 0 0
$$565$$ − 4.52377i − 0.190316i
$$566$$ 0 0
$$567$$ 4.55765 0.191403
$$568$$ 0 0
$$569$$ 13.4849 0.565317 0.282658 0.959221i $$-0.408784\pi$$
0.282658 + 0.959221i $$0.408784\pi$$
$$570$$ 0 0
$$571$$ 20.9706i 0.877591i 0.898587 + 0.438795i $$0.144595\pi$$
−0.898587 + 0.438795i $$0.855405\pi$$
$$572$$ 0 0
$$573$$ 8.63001i 0.360524i
$$574$$ 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
$$579$$ 11.4514i 0.475903i
$$580$$ 0 0
$$581$$ − 68.5907i − 2.84562i
$$582$$ 0 0
$$583$$ 2.44158 0.101120
$$584$$ 0 0
$$585$$ −0.0397948 −0.00164531
$$586$$ 0 0
$$587$$ − 24.0796i − 0.993871i −0.867787 0.496936i $$-0.834459\pi$$
0.867787 0.496936i $$-0.165541\pi$$
$$588$$ 0 0
$$589$$ 2.01704i 0.0831108i
$$590$$ 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
$$595$$ 7.80785i 0.320091i
$$596$$ 0 0
$$597$$ 3.68000i 0.150612i
$$598$$ 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 $$-0.477529\pi$$
0.0705364 + 0.997509i $$0.477529\pi$$
$$602$$ 0 0
$$603$$ 5.33962i 0.217446i
$$604$$ 0 0
$$605$$ 0.586182i 0.0238317i
$$606$$ 0 0
$$607$$ 30.1019 1.22180 0.610900 0.791708i $$-0.290808\pi$$
0.610900 + 0.791708i $$0.290808\pi$$
$$608$$ 0 0
$$609$$ 33.2802 1.34858
$$610$$ 0 0
$$611$$ 0.237649i 0.00961424i
$$612$$ 0 0
$$613$$ 3.54246i 0.143079i 0.997438 + 0.0715393i $$0.0227911\pi$$
−0.997438 + 0.0715393i $$0.977209\pi$$
$$614$$ 0 0
$$615$$ 4.39236 0.177117
$$616$$ 0 0
$$617$$ 22.9098 0.922315 0.461157 0.887318i $$-0.347434\pi$$
0.461157 + 0.887318i $$0.347434\pi$$
$$618$$ 0 0
$$619$$ − 40.4612i − 1.62627i −0.582074 0.813136i $$-0.697758\pi$$
0.582074 0.813136i $$-0.302242\pi$$
$$620$$ 0 0
$$621$$ − 2.82843i − 0.113501i
$$622$$ 0 0
$$623$$ 66.7847 2.67567
$$624$$ 0 0
$$625$$ 21.6855 0.867420
$$626$$ 0 0
$$627$$ 12.6533i 0.505325i
$$628$$ 0 0
$$629$$ − 22.4361i − 0.894584i
$$630$$ 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
$$635$$ 2.41512i 0.0958409i
$$636$$ 0 0
$$637$$ 1.15716i 0.0458482i
$$638$$ 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
$$643$$ 25.3724i 1.00059i 0.865856 + 0.500294i $$0.166775\pi$$
−0.865856 + 0.500294i $$0.833225\pi$$
$$644$$ 0 0
$$645$$ − 1.07866i − 0.0424720i
$$646$$ 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
$$651$$ 2.54156i 0.0996116i
$$652$$ 0 0
$$653$$ − 37.0144i − 1.44849i −0.689545 0.724243i $$-0.742190\pi$$
0.689545 0.724243i $$-0.257810\pi$$
$$654$$ 0 0
$$655$$ −1.41921 −0.0554529
$$656$$ 0 0
$$657$$ −0.541560 −0.0211283
$$658$$ 0 0
$$659$$ 19.7624i 0.769832i 0.922952 + 0.384916i $$0.125770\pi$$
−0.922952 + 0.384916i $$0.874230\pi$$
$$660$$ 0 0
$$661$$ − 16.8632i − 0.655904i −0.944694 0.327952i $$-0.893642\pi$$
0.944694 0.327952i $$-0.106358\pi$$
$$662$$ 0 0
$$663$$ 0.303911 0.0118029
$$664$$ 0 0
$$665$$ 7.80785 0.302776
$$666$$ 0 0
$$667$$ − 20.6533i − 0.799700i
$$668$$ 0 0
$$669$$ 4.86156i 0.187959i
$$670$$ 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
$$675$$ − 4.77568i − 0.183816i
$$676$$ 0 0
$$677$$ 0.632805i 0.0243207i 0.999926 + 0.0121603i $$0.00387085\pi$$
−0.999926 + 0.0121603i $$0.996129\pi$$
$$678$$ 0 0
$$679$$ 19.6764 0.755113
$$680$$ 0 0
$$681$$ 15.0496 0.576702
$$682$$ 0 0
$$683$$ − 6.04606i − 0.231346i −0.993287 0.115673i $$-0.963098\pi$$
0.993287 0.115673i $$-0.0369025\pi$$
$$684$$ 0 0
$$685$$ − 1.60058i − 0.0611549i
$$686$$ 0 0
$$687$$ 28.5264 1.08835
$$688$$ 0 0
$$689$$ −0.0586426 −0.00223410
$$690$$ 0 0
$$691$$ 28.3955i 1.08021i 0.841596 + 0.540107i $$0.181616\pi$$
−0.841596 + 0.540107i $$0.818384\pi$$
$$692$$ 0 0
$$693$$ 15.9437i 0.605652i
$$694$$ 0 0
$$695$$ 3.93926 0.149425
$$696$$ 0 0
$$697$$ −33.5443 −1.27058
$$698$$ 0 0
$$699$$ − 13.5702i − 0.513272i
$$700$$ 0 0
$$701$$ − 14.7738i − 0.558000i −0.960291 0.279000i $$-0.909997\pi$$
0.960291 0.279000i $$-0.0900029\pi$$
$$702$$ 0 0
$$703$$ −22.4361 −0.846192
$$704$$ 0 0
$$705$$ 1.33962 0.0504529
$$706$$ 0 0
$$707$$ − 2.92450i − 0.109987i
$$708$$ 0 0
$$709$$ 22.7569i 0.854655i 0.904097 + 0.427327i $$0.140545\pi$$
−0.904097 + 0.427327i $$0.859455\pi$$
$$710$$ 0 0
$$711$$ 10.9937 0.412296
$$712$$ 0 0
$$713$$ 1.57726 0.0590690
$$714$$ 0 0
$$715$$ − 0.139211i − 0.00520621i
$$716$$ 0 0
$$717$$ 29.3629i 1.09658i
$$718$$ 0 0
$$719$$ −30.9957 −1.15594 −0.577972 0.816057i $$-0.696156\pi$$
−0.577972 + 0.816057i $$0.696156\pi$$
$$720$$ 0 0
$$721$$ −6.09292 −0.226912
$$722$$ 0 0
$$723$$ − 24.0063i − 0.892803i
$$724$$ 0 0
$$725$$ − 34.8723i − 1.29512i
$$726$$ 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
$$731$$ 8.23765i 0.304680i
$$732$$ 0 0
$$733$$ 0.206562i 0.00762954i 0.999993 + 0.00381477i $$0.00121428\pi$$
−0.999993 + 0.00381477i $$0.998786\pi$$
$$734$$ 0 0
$$735$$ 6.52284 0.240599
$$736$$ 0 0
$$737$$ −18.6792 −0.688058
$$738$$ 0 0
$$739$$ 2.13215i 0.0784325i 0.999231 + 0.0392162i $$0.0124861\pi$$
−0.999231 + 0.0392162i $$0.987514\pi$$
$$740$$ 0 0
$$741$$ − 0.303911i − 0.0111644i
$$742$$ 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
$$747$$ 15.0496i 0.550636i
$$748$$ 0 0
$$749$$ 39.0761i 1.42781i
$$750$$ 0 0
$$751$$ −12.5843 −0.459208 −0.229604 0.973284i $$-0.573743\pi$$
−0.229604 + 0.973284i $$0.573743\pi$$
$$752$$ 0 0
$$753$$ 22.2837 0.812064
$$754$$ 0 0
$$755$$ − 4.72529i − 0.171971i
$$756$$ 0 0
$$757$$ 10.6052i 0.385452i 0.981253 + 0.192726i $$0.0617329\pi$$
−0.981253 + 0.192726i $$0.938267\pi$$
$$758$$ 0 0
$$759$$ 9.89450 0.359148
$$760$$ 0 0
$$761$$ −42.8182 −1.55216 −0.776079 0.630635i $$-0.782794\pi$$
−0.776079 + 0.630635i $$0.782794\pi$$
$$762$$ 0 0
$$763$$ − 36.8441i − 1.33385i
$$764$$ 0 0
$$765$$ − 1.71313i − 0.0619384i
$$766$$ 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
$$771$$ 8.66038i 0.311896i
$$772$$ 0 0
$$773$$ − 32.3522i − 1.16363i −0.813322 0.581814i $$-0.802343\pi$$
0.813322 0.581814i $$-0.197657\pi$$
$$774$$ 0 0
$$775$$ 2.66314 0.0956630
$$776$$ 0 0
$$777$$ −28.2704 −1.01419
$$778$$ 0 0
$$779$$ 33.5443i 1.20185i
$$780$$ 0 0
$$781$$ 31.8874i 1.14102i
$$782$$ 0 0
$$783$$ −7.30205 −0.260954
$$784$$ 0 0
$$785$$ −10.8253 −0.386370
$$786$$ 0 0
$$787$$ 7.36056i 0.262376i 0.991358 + 0.131188i $$0.0418791\pi$$
−0.991358 + 0.131188i $$0.958121\pi$$
$$788$$ 0 0
$$789$$ − 13.3208i − 0.474232i
$$790$$ 0 0
$$791$$ 43.5317 1.54781
$$792$$ 0 0
$$793$$ −0.324298 −0.0115162
$$794$$ 0 0
$$795$$ 0.330566i 0.0117240i
$$796$$ 0 0
$$797$$ 24.0627i 0.852344i 0.904642 + 0.426172i $$0.140138\pi$$
−0.904642 + 0.426172i $$0.859862\pi$$
$$798$$ 0 0
$$799$$ −10.2306 −0.361932
$$800$$ 0 0
$$801$$ −14.6533 −0.517750
$$802$$ 0 0
$$803$$ − 1.89450i − 0.0668556i
$$804$$ 0 0
$$805$$ − 6.10550i − 0.215190i
$$806$$ 0 0
$$807$$ −16.5058 −0.581032
$$808$$ 0 0
$$809$$ 7.83586 0.275494 0.137747 0.990467i $$-0.456014\pi$$
0.137747 + 0.990467i $$0.456014\pi$$
$$810$$ 0 0
$$811$$ 45.7351i 1.60598i 0.595995 + 0.802988i $$0.296758\pi$$
−0.595995 + 0.802988i $$0.703242\pi$$
$$812$$ 0 0
$$813$$ − 21.9769i − 0.770762i
$$814$$ 0 0
$$815$$ 5.02684 0.176083
$$816$$ 0 0
$$817$$ 8.23765 0.288199
$$818$$ 0 0
$$819$$ − 0.382941i − 0.0133810i
$$820$$ 0 0
$$821$$ − 27.3709i − 0.955250i −0.878564 0.477625i $$-0.841498\pi$$
0.878564 0.477625i $$-0.158502\pi$$
$$822$$ 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
$$827$$ 14.4227i 0.501528i 0.968048 + 0.250764i $$0.0806818\pi$$
−0.968048 + 0.250764i $$0.919318\pi$$
$$828$$ 0 0
$$829$$ 21.7497i 0.755400i 0.925928 + 0.377700i $$0.123285\pi$$
−0.925928 + 0.377700i $$0.876715\pi$$
$$830$$ 0 0
$$831$$ −15.4862 −0.537210
$$832$$ 0 0
$$833$$ −49.8147 −1.72598
$$834$$ 0 0
$$835$$ − 2.76513i − 0.0956914i
$$836$$ 0 0
$$837$$ − 0.557647i − 0.0192751i
$$838$$ 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
$$843$$ − 22.8910i − 0.788407i
$$844$$ 0 0
$$845$$ − 6.15379i − 0.211697i
$$846$$ 0 0
$$847$$ −5.64077 −0.193819
$$848$$ 0 0
$$849$$ 6.34315 0.217696
$$850$$ 0 0
$$851$$ 17.5443i 0.601411i
$$852$$ 0 0
$$853$$ 16.5648i 0.567169i 0.958947 + 0.283585i $$0.0915237\pi$$
−0.958947 + 0.283585i $$0.908476\pi$$
$$854$$ 0 0
$$855$$ −1.71313 −0.0585879
$$856$$ 0 0
$$857$$ 19.0888 0.652062 0.326031 0.945359i $$-0.394289\pi$$
0.326031 + 0.945359i $$0.394289\pi$$
$$858$$ 0 0
$$859$$ 53.9272i 1.83997i 0.391949 + 0.919987i $$0.371801\pi$$
−0.391949 + 0.919987i $$0.628199\pi$$
$$860$$ 0 0
$$861$$ 42.2672i 1.44046i
$$862$$ 0 0
$$863$$ −3.64533 −0.124089 −0.0620443 0.998073i $$-0.519762\pi$$
−0.0620443 + 0.998073i $$0.519762\pi$$
$$864$$ 0 0
$$865$$ 2.42826 0.0825632
$$866$$ 0 0
$$867$$ − 3.91688i − 0.133024i
$$868$$ 0 0
$$869$$ 38.4586i 1.30462i
$$870$$ 0 0
$$871$$ 0.448643 0.0152017
$$872$$ 0 0
$$873$$ −4.31724 −0.146116
$$874$$ 0 0
$$875$$ − 21.1020i − 0.713377i
$$876$$ 0 0
$$877$$ − 56.6481i − 1.91287i −0.291945 0.956435i $$-0.594302\pi$$
0.291945 0.956435i $$-0.405698\pi$$
$$878$$ 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
$$883$$ − 15.0292i − 0.505773i −0.967496 0.252887i $$-0.918620\pi$$
0.967496 0.252887i $$-0.0813799\pi$$
$$884$$ 0 0
$$885$$ − 2.67923i − 0.0900614i
$$886$$ 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
$$891$$ − 3.49824i − 0.117195i
$$892$$ 0 0
$$893$$ 10.2306i 0.342354i
$$894$$ 0 0
$$895$$ 6.21805 0.207847
$$896$$ 0 0
$$897$$ −0.237649 −0.00793486
$$898$$ 0 0
$$899$$ − 4.07197i − 0.135808i
$$900$$ 0 0
$$901$$ − 2.52452i − 0.0841038i
$$902$$ 0 0
$$903$$ 10.3798 0.345418
$$904$$ 0 0
$$905$$ 7.25507 0.241167
$$906$$ 0 0
$$907$$ 51.2480i 1.70166i 0.525439 + 0.850831i $$0.323901\pi$$
−0.525439 + 0.850831i $$0.676099\pi$$
$$908$$ 0 0
$$909$$ 0.641669i 0.0212828i
$$910$$ 0 0
$$911$$ −21.0535 −0.697533 −0.348767 0.937210i $$-0.613399\pi$$
−0.348767 + 0.937210i $$0.613399\pi$$
$$912$$ 0 0
$$913$$ −52.6470 −1.74236
$$914$$ 0 0
$$915$$ 1.82805i 0.0604336i
$$916$$ 0 0
$$917$$ − 13.6569i − 0.450989i
$$918$$ 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
$$923$$ − 0.765881i − 0.0252093i
$$924$$ 0 0
$$925$$ 29.6228i 0.973992i
$$926$$ 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
$$931$$ 49.8147i 1.63261i
$$932$$ 0 0
$$933$$ − 26.8651i − 0.879523i
$$934$$ 0 0
$$935$$ 5.99294 0.195990
$$936$$ 0 0
$$937$$ 13.5780 0.443574 0.221787 0.975095i $$-0.428811\pi$$
0.221787 + 0.975095i $$0.428811\pi$$
$$938$$ 0 0
$$939$$ 19.6890i 0.642527i
$$940$$ 0 0
$$941$$ 5.59890i 0.182519i 0.995827 + 0.0912595i $$0.0290893\pi$$
−0.995827 + 0.0912595i $$0.970911\pi$$
$$942$$ 0 0
$$943$$ 26.2306 0.854186
$$944$$ 0 0
$$945$$ −2.15862 −0.0702199
$$946$$ 0 0
$$947$$ − 46.9106i − 1.52439i −0.647348 0.762194i $$-0.724122\pi$$
0.647348 0.762194i $$-0.275878\pi$$
$$948$$ 0 0
$$949$$ 0.0455027i 0.00147708i
$$950$$ 0 0
$$951$$ −30.1860 −0.978847
$$952$$ 0 0
$$953$$ −5.59115 −0.181115 −0.0905576 0.995891i $$-0.528865\pi$$
−0.0905576 + 0.995891i $$0.528865\pi$$
$$954$$ 0 0
$$955$$ − 4.08740i − 0.132265i
$$956$$ 0 0
$$957$$ − 25.5443i − 0.825730i
$$958$$ 0 0
$$959$$ 15.4022 0.497362
$$960$$ 0 0
$$961$$ −30.6890 −0.989969
$$962$$ 0 0
$$963$$ − 8.57373i − 0.276285i
$$964$$ 0 0
$$965$$ − 5.42367i − 0.174594i
$$966$$ 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
$$971$$ 11.3668i 0.364779i 0.983226 + 0.182389i $$0.0583832\pi$$
−0.983226 + 0.182389i $$0.941617\pi$$
$$972$$ 0 0
$$973$$ 37.9070i 1.21524i
$$974$$ 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
$$979$$ − 51.2608i − 1.63830i
$$980$$ 0 0
$$981$$ 8.08402i 0.258103i
$$982$$ 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
$$987$$ 12.8910i 0.410324i
$$988$$ 0 0
$$989$$ − 6.44158i − 0.204830i
$$990$$ 0 0
$$991$$ −3.43683 −0.109175 −0.0545873 0.998509i $$-0.517384\pi$$
−0.0545873 + 0.998509i $$0.517384\pi$$
$$992$$ 0 0
$$993$$ −20.7784 −0.659383
$$994$$ 0 0
$$995$$ − 1.74294i − 0.0552550i
$$996$$ 0 0
$$997$$ − 31.0320i − 0.982794i −0.870936 0.491397i $$-0.836486\pi$$
0.870936 0.491397i $$-0.163514\pi$$
$$998$$ 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.d.i.1537.6 8
4.3 odd 2 3072.2.d.f.1537.2 8
8.3 odd 2 3072.2.d.f.1537.7 8
8.5 even 2 inner 3072.2.d.i.1537.3 8
16.3 odd 4 3072.2.a.t.1.2 4
16.5 even 4 3072.2.a.o.1.3 4
16.11 odd 4 3072.2.a.i.1.3 4
16.13 even 4 3072.2.a.n.1.2 4
32.3 odd 8 384.2.j.b.97.4 8
32.5 even 8 384.2.j.a.289.2 8
32.11 odd 8 48.2.j.a.13.4 8
32.13 even 8 192.2.j.a.49.3 8
32.19 odd 8 48.2.j.a.37.4 yes 8
32.21 even 8 192.2.j.a.145.3 8
32.27 odd 8 384.2.j.b.289.4 8
32.29 even 8 384.2.j.a.97.2 8
48.5 odd 4 9216.2.a.bn.1.2 4
48.11 even 4 9216.2.a.bo.1.2 4
48.29 odd 4 9216.2.a.x.1.3 4
48.35 even 4 9216.2.a.y.1.3 4
96.5 odd 8 1152.2.k.f.289.2 8
96.11 even 8 144.2.k.b.109.1 8
96.29 odd 8 1152.2.k.f.865.2 8
96.35 even 8 1152.2.k.c.865.2 8
96.53 odd 8 576.2.k.b.145.3 8
96.59 even 8 1152.2.k.c.289.2 8
96.77 odd 8 576.2.k.b.433.3 8
96.83 even 8 144.2.k.b.37.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.4 8 32.11 odd 8
48.2.j.a.37.4 yes 8 32.19 odd 8
144.2.k.b.37.1 8 96.83 even 8
144.2.k.b.109.1 8 96.11 even 8
192.2.j.a.49.3 8 32.13 even 8
192.2.j.a.145.3 8 32.21 even 8
384.2.j.a.97.2 8 32.29 even 8
384.2.j.a.289.2 8 32.5 even 8
384.2.j.b.97.4 8 32.3 odd 8
384.2.j.b.289.4 8 32.27 odd 8
576.2.k.b.145.3 8 96.53 odd 8
576.2.k.b.433.3 8 96.77 odd 8
1152.2.k.c.289.2 8 96.59 even 8
1152.2.k.c.865.2 8 96.35 even 8
1152.2.k.f.289.2 8 96.5 odd 8
1152.2.k.f.865.2 8 96.29 odd 8
3072.2.a.i.1.3 4 16.11 odd 4
3072.2.a.n.1.2 4 16.13 even 4
3072.2.a.o.1.3 4 16.5 even 4
3072.2.a.t.1.2 4 16.3 odd 4
3072.2.d.f.1537.2 8 4.3 odd 2
3072.2.d.f.1537.7 8 8.3 odd 2
3072.2.d.i.1537.3 8 8.5 even 2 inner
3072.2.d.i.1537.6 8 1.1 even 1 trivial
9216.2.a.x.1.3 4 48.29 odd 4
9216.2.a.y.1.3 4 48.35 even 4
9216.2.a.bn.1.2 4 48.5 odd 4
9216.2.a.bo.1.2 4 48.11 even 4