# Properties

 Label 6048.2.j.a.5615.2 Level 6048 Weight 2 Character 6048.5615 Analytic conductor 48.294 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6048 = 2^{5} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6048.j (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.2935231425$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.56070144.2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 1512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 5615.2 Root $$0.500000 + 1.19293i$$ Character $$\chi$$ = 6048.5615 Dual form 6048.2.j.a.5615.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.38587 q^{5} +1.00000i q^{7} +O(q^{10})$$ $$q-4.38587 q^{5} +1.00000i q^{7} -5.11792i q^{11} -5.21068i q^{13} -2.00000i q^{17} +6.50379 q^{19} -4.63929 q^{23} +14.2358 q^{25} -1.26795 q^{29} -4.21068i q^{31} -4.38587i q^{35} -1.98547i q^{37} +7.37134i q^{41} -3.71753 q^{43} +2.57558 q^{47} -1.00000 q^{49} +10.2358 q^{53} +22.4465i q^{55} +3.50379i q^{59} -4.53590i q^{61} +22.8533i q^{65} -5.21068 q^{67} -1.36071 q^{71} +15.4465 q^{73} +5.11792 q^{77} -3.98241i q^{79} -11.6893i q^{83} +8.77174i q^{85} -14.8645i q^{89} +5.21068 q^{91} -28.5247 q^{95} +4.42136 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} - 16q^{19} - 16q^{23} + 32q^{25} - 24q^{29} - 8q^{43} + 8q^{47} - 8q^{49} - 8q^{67} - 32q^{71} + 8q^{73} + 8q^{91} - 112q^{95} - 32q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$$.

 $$n$$ $$2593$$ $$3781$$ $$3809$$ $$4159$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −4.38587 −1.96142 −0.980710 0.195469i $$-0.937377\pi$$
−0.980710 + 0.195469i $$0.937377\pi$$
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 5.11792i − 1.54311i −0.636162 0.771555i $$-0.719479\pi$$
0.636162 0.771555i $$-0.280521\pi$$
$$12$$ 0 0
$$13$$ − 5.21068i − 1.44518i −0.691276 0.722591i $$-0.742951\pi$$
0.691276 0.722591i $$-0.257049\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ 6.50379 1.49207 0.746035 0.665906i $$-0.231955\pi$$
0.746035 + 0.665906i $$0.231955\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.63929 −0.967359 −0.483680 0.875245i $$-0.660700\pi$$
−0.483680 + 0.875245i $$0.660700\pi$$
$$24$$ 0 0
$$25$$ 14.2358 2.84717
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.26795 −0.235452 −0.117726 0.993046i $$-0.537560\pi$$
−0.117726 + 0.993046i $$0.537560\pi$$
$$30$$ 0 0
$$31$$ − 4.21068i − 0.756260i −0.925752 0.378130i $$-0.876567\pi$$
0.925752 0.378130i $$-0.123433\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 4.38587i − 0.741347i
$$36$$ 0 0
$$37$$ − 1.98547i − 0.326410i −0.986592 0.163205i $$-0.947817\pi$$
0.986592 0.163205i $$-0.0521832\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.37134i 1.15121i 0.817728 + 0.575605i $$0.195233\pi$$
−0.817728 + 0.575605i $$0.804767\pi$$
$$42$$ 0 0
$$43$$ −3.71753 −0.566917 −0.283459 0.958984i $$-0.591482\pi$$
−0.283459 + 0.958984i $$0.591482\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.57558 0.375687 0.187844 0.982199i $$-0.439850\pi$$
0.187844 + 0.982199i $$0.439850\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.2358 1.40600 0.703000 0.711190i $$-0.251843\pi$$
0.703000 + 0.711190i $$0.251843\pi$$
$$54$$ 0 0
$$55$$ 22.4465i 3.02669i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.50379i 0.456154i 0.973643 + 0.228077i $$0.0732438\pi$$
−0.973643 + 0.228077i $$0.926756\pi$$
$$60$$ 0 0
$$61$$ − 4.53590i − 0.580762i −0.956911 0.290381i $$-0.906218\pi$$
0.956911 0.290381i $$-0.0937821\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 22.8533i 2.83461i
$$66$$ 0 0
$$67$$ −5.21068 −0.636586 −0.318293 0.947992i $$-0.603110\pi$$
−0.318293 + 0.947992i $$0.603110\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.36071 −0.161486 −0.0807432 0.996735i $$-0.525729\pi$$
−0.0807432 + 0.996735i $$0.525729\pi$$
$$72$$ 0 0
$$73$$ 15.4465 1.80788 0.903939 0.427662i $$-0.140662\pi$$
0.903939 + 0.427662i $$0.140662\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.11792 0.583241
$$78$$ 0 0
$$79$$ − 3.98241i − 0.448057i −0.974583 0.224028i $$-0.928079\pi$$
0.974583 0.224028i $$-0.0719208\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 11.6893i − 1.28307i −0.767095 0.641534i $$-0.778298\pi$$
0.767095 0.641534i $$-0.221702\pi$$
$$84$$ 0 0
$$85$$ 8.77174i 0.951428i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 14.8645i − 1.57563i −0.615910 0.787817i $$-0.711211\pi$$
0.615910 0.787817i $$-0.288789\pi$$
$$90$$ 0 0
$$91$$ 5.21068 0.546227
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −28.5247 −2.92658
$$96$$ 0 0
$$97$$ 4.42136 0.448921 0.224460 0.974483i $$-0.427938\pi$$
0.224460 + 0.974483i $$0.427938\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ − 1.67478i − 0.165021i −0.996590 0.0825105i $$-0.973706\pi$$
0.996590 0.0825105i $$-0.0262938\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10.2649i 0.992344i 0.868224 + 0.496172i $$0.165262\pi$$
−0.868224 + 0.496172i $$0.834738\pi$$
$$108$$ 0 0
$$109$$ 1.29311i 0.123857i 0.998081 + 0.0619287i $$0.0197251\pi$$
−0.998081 + 0.0619287i $$0.980275\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 8.07937i − 0.760043i −0.924978 0.380022i $$-0.875917\pi$$
0.924978 0.380022i $$-0.124083\pi$$
$$114$$ 0 0
$$115$$ 20.3473 1.89740
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −15.1931 −1.38119
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −40.5072 −3.62307
$$126$$ 0 0
$$127$$ 13.4465i 1.19319i 0.802544 + 0.596593i $$0.203479\pi$$
−0.802544 + 0.596593i $$0.796521\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 1.77480i − 0.155065i −0.996990 0.0775323i $$-0.975296\pi$$
0.996990 0.0775323i $$-0.0247041\pi$$
$$132$$ 0 0
$$133$$ 6.50379i 0.563950i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 14.5756i − 1.24528i −0.782510 0.622638i $$-0.786061\pi$$
0.782510 0.622638i $$-0.213939\pi$$
$$138$$ 0 0
$$139$$ −15.0076 −1.27293 −0.636463 0.771307i $$-0.719603\pi$$
−0.636463 + 0.771307i $$0.719603\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −26.6678 −2.23008
$$144$$ 0 0
$$145$$ 5.56106 0.461821
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 5.33669 0.437198 0.218599 0.975815i $$-0.429851\pi$$
0.218599 + 0.975815i $$0.429851\pi$$
$$150$$ 0 0
$$151$$ 17.5725i 1.43003i 0.699108 + 0.715016i $$0.253581\pi$$
−0.699108 + 0.715016i $$0.746419\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 18.4675i 1.48334i
$$156$$ 0 0
$$157$$ − 13.8259i − 1.10343i −0.834032 0.551715i $$-0.813973\pi$$
0.834032 0.551715i $$-0.186027\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 4.63929i − 0.365627i
$$162$$ 0 0
$$163$$ −8.67478 −0.679461 −0.339731 0.940523i $$-0.610336\pi$$
−0.339731 + 0.940523i $$0.610336\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.8855 −1.22925 −0.614627 0.788818i $$-0.710693\pi$$
−0.614627 + 0.788818i $$0.710693\pi$$
$$168$$ 0 0
$$169$$ −14.1512 −1.08855
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −4.22940 −0.321555 −0.160778 0.986991i $$-0.551400\pi$$
−0.160778 + 0.986991i $$0.551400\pi$$
$$174$$ 0 0
$$175$$ 14.2358i 1.07613i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 6.45041i − 0.482126i −0.970509 0.241063i $$-0.922504\pi$$
0.970509 0.241063i $$-0.0774961\pi$$
$$180$$ 0 0
$$181$$ − 10.5068i − 0.780968i −0.920610 0.390484i $$-0.872308\pi$$
0.920610 0.390484i $$-0.127692\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.70803i 0.640227i
$$186$$ 0 0
$$187$$ −10.2358 −0.748519
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.25986 0.597663 0.298831 0.954306i $$-0.403403\pi$$
0.298831 + 0.954306i $$0.403403\pi$$
$$192$$ 0 0
$$193$$ −8.07937 −0.581566 −0.290783 0.956789i $$-0.593916\pi$$
−0.290783 + 0.956789i $$0.593916\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.04275 −0.501775 −0.250887 0.968016i $$-0.580722\pi$$
−0.250887 + 0.968016i $$0.580722\pi$$
$$198$$ 0 0
$$199$$ − 20.1336i − 1.42723i −0.700537 0.713616i $$-0.747056\pi$$
0.700537 0.713616i $$-0.252944\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 1.26795i − 0.0889926i
$$204$$ 0 0
$$205$$ − 32.3297i − 2.25801i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 33.2859i − 2.30243i
$$210$$ 0 0
$$211$$ −7.54347 −0.519314 −0.259657 0.965701i $$-0.583610\pi$$
−0.259657 + 0.965701i $$0.583610\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 16.3046 1.11196
$$216$$ 0 0
$$217$$ 4.21068 0.285839
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −10.4214 −0.701016
$$222$$ 0 0
$$223$$ − 5.48926i − 0.367588i −0.982965 0.183794i $$-0.941162\pi$$
0.982965 0.183794i $$-0.0588380\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0.127416i 0.00845689i 0.999991 + 0.00422845i $$0.00134596\pi$$
−0.999991 + 0.00422845i $$0.998654\pi$$
$$228$$ 0 0
$$229$$ − 6.53590i − 0.431904i −0.976404 0.215952i $$-0.930714\pi$$
0.976404 0.215952i $$-0.0692855\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 10.5359i − 0.690230i −0.938560 0.345115i $$-0.887840\pi$$
0.938560 0.345115i $$-0.112160\pi$$
$$234$$ 0 0
$$235$$ −11.2962 −0.736881
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.3862 1.57741 0.788706 0.614771i $$-0.210752\pi$$
0.788706 + 0.614771i $$0.210752\pi$$
$$240$$ 0 0
$$241$$ −10.8603 −0.699573 −0.349787 0.936829i $$-0.613746\pi$$
−0.349787 + 0.936829i $$0.613746\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.38587 0.280203
$$246$$ 0 0
$$247$$ − 33.8891i − 2.15631i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 3.73511i − 0.235758i −0.993028 0.117879i $$-0.962390\pi$$
0.993028 0.117879i $$-0.0376095\pi$$
$$252$$ 0 0
$$253$$ 23.7435i 1.49274i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 7.52781i − 0.469572i −0.972047 0.234786i $$-0.924561\pi$$
0.972047 0.234786i $$-0.0754389\pi$$
$$258$$ 0 0
$$259$$ 1.98547 0.123371
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −16.7187 −1.03092 −0.515458 0.856915i $$-0.672378\pi$$
−0.515458 + 0.856915i $$0.672378\pi$$
$$264$$ 0 0
$$265$$ −44.8930 −2.75776
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 4.11486 0.250887 0.125444 0.992101i $$-0.459965\pi$$
0.125444 + 0.992101i $$0.459965\pi$$
$$270$$ 0 0
$$271$$ 4.31293i 0.261992i 0.991383 + 0.130996i $$0.0418175\pi$$
−0.991383 + 0.130996i $$0.958182\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 72.8579i − 4.39349i
$$276$$ 0 0
$$277$$ 27.7564i 1.66772i 0.551976 + 0.833860i $$0.313874\pi$$
−0.551976 + 0.833860i $$0.686126\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 21.8961i 1.30621i 0.757267 + 0.653106i $$0.226534\pi$$
−0.757267 + 0.653106i $$0.773466\pi$$
$$282$$ 0 0
$$283$$ −15.3786 −0.914164 −0.457082 0.889425i $$-0.651105\pi$$
−0.457082 + 0.889425i $$0.651105\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.37134 −0.435117
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0.408482 0.0238638 0.0119319 0.999929i $$-0.496202\pi$$
0.0119319 + 0.999929i $$0.496202\pi$$
$$294$$ 0 0
$$295$$ − 15.3671i − 0.894710i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 24.1739i 1.39801i
$$300$$ 0 0
$$301$$ − 3.71753i − 0.214275i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 19.8939i 1.13912i
$$306$$ 0 0
$$307$$ −9.99694 −0.570555 −0.285278 0.958445i $$-0.592086\pi$$
−0.285278 + 0.958445i $$0.592086\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −19.9603 −1.13185 −0.565923 0.824458i $$-0.691480\pi$$
−0.565923 + 0.824458i $$0.691480\pi$$
$$312$$ 0 0
$$313$$ 23.4465 1.32528 0.662638 0.748940i $$-0.269437\pi$$
0.662638 + 0.748940i $$0.269437\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −20.7740 −1.16678 −0.583391 0.812191i $$-0.698275\pi$$
−0.583391 + 0.812191i $$0.698275\pi$$
$$318$$ 0 0
$$319$$ 6.48926i 0.363329i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 13.0076i − 0.723761i
$$324$$ 0 0
$$325$$ − 74.1784i − 4.11468i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 2.57558i 0.141997i
$$330$$ 0 0
$$331$$ −19.0366 −1.04635 −0.523174 0.852226i $$-0.675252\pi$$
−0.523174 + 0.852226i $$0.675252\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 22.8533 1.24861
$$336$$ 0 0
$$337$$ 2.88546 0.157181 0.0785905 0.996907i $$-0.474958\pi$$
0.0785905 + 0.996907i $$0.474958\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −21.5499 −1.16699
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 30.7033i 1.64824i 0.566415 + 0.824120i $$0.308330\pi$$
−0.566415 + 0.824120i $$0.691670\pi$$
$$348$$ 0 0
$$349$$ 26.5007i 1.41855i 0.704931 + 0.709276i $$0.250978\pi$$
−0.704931 + 0.709276i $$0.749022\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.29955i 0.335291i 0.985847 + 0.167645i $$0.0536164\pi$$
−0.985847 + 0.167645i $$0.946384\pi$$
$$354$$ 0 0
$$355$$ 5.96789 0.316743
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.60770 −0.0848509 −0.0424255 0.999100i $$-0.513508\pi$$
−0.0424255 + 0.999100i $$0.513508\pi$$
$$360$$ 0 0
$$361$$ 23.2992 1.22628
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −67.7464 −3.54601
$$366$$ 0 0
$$367$$ − 17.1763i − 0.896597i −0.893884 0.448298i $$-0.852030\pi$$
0.893884 0.448298i $$-0.147970\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 10.2358i 0.531418i
$$372$$ 0 0
$$373$$ − 2.27941i − 0.118024i −0.998257 0.0590118i $$-0.981205\pi$$
0.998257 0.0590118i $$-0.0187950\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.60688i 0.340271i
$$378$$ 0 0
$$379$$ −20.1389 −1.03446 −0.517232 0.855845i $$-0.673038\pi$$
−0.517232 + 0.855845i $$0.673038\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −22.3068 −1.13982 −0.569912 0.821705i $$-0.693023\pi$$
−0.569912 + 0.821705i $$0.693023\pi$$
$$384$$ 0 0
$$385$$ −22.4465 −1.14398
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.57864 −0.0800404 −0.0400202 0.999199i $$-0.512742\pi$$
−0.0400202 + 0.999199i $$0.512742\pi$$
$$390$$ 0 0
$$391$$ 9.27858i 0.469238i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 17.4663i 0.878827i
$$396$$ 0 0
$$397$$ − 31.4289i − 1.57737i −0.614796 0.788686i $$-0.710762\pi$$
0.614796 0.788686i $$-0.289238\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 26.5404i − 1.32536i −0.748901 0.662682i $$-0.769418\pi$$
0.748901 0.662682i $$-0.230582\pi$$
$$402$$ 0 0
$$403$$ −21.9405 −1.09293
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.1615 −0.503687
$$408$$ 0 0
$$409$$ 32.0037 1.58248 0.791240 0.611506i $$-0.209436\pi$$
0.791240 + 0.611506i $$0.209436\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −3.50379 −0.172410
$$414$$ 0 0
$$415$$ 51.2678i 2.51663i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 20.6572i − 1.00917i −0.863362 0.504585i $$-0.831646\pi$$
0.863362 0.504585i $$-0.168354\pi$$
$$420$$ 0 0
$$421$$ 32.9930i 1.60798i 0.594641 + 0.803991i $$0.297294\pi$$
−0.594641 + 0.803991i $$0.702706\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 28.4717i − 1.38108i
$$426$$ 0 0
$$427$$ 4.53590 0.219508
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.48282 0.312267 0.156133 0.987736i $$-0.450097\pi$$
0.156133 + 0.987736i $$0.450097\pi$$
$$432$$ 0 0
$$433$$ −18.4038 −0.884429 −0.442214 0.896909i $$-0.645807\pi$$
−0.442214 + 0.896909i $$0.645807\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −30.1730 −1.44337
$$438$$ 0 0
$$439$$ − 4.31293i − 0.205845i −0.994689 0.102923i $$-0.967181\pi$$
0.994689 0.102923i $$-0.0328194\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 5.54540i − 0.263470i −0.991285 0.131735i $$-0.957945\pi$$
0.991285 0.131735i $$-0.0420547\pi$$
$$444$$ 0 0
$$445$$ 65.1937i 3.09048i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.2007i 1.42526i 0.701541 + 0.712629i $$0.252496\pi$$
−0.701541 + 0.712629i $$0.747504\pi$$
$$450$$ 0 0
$$451$$ 37.7259 1.77644
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −22.8533 −1.07138
$$456$$ 0 0
$$457$$ −24.3205 −1.13767 −0.568833 0.822453i $$-0.692605\pi$$
−0.568833 + 0.822453i $$0.692605\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 37.3141 1.73789 0.868945 0.494909i $$-0.164799\pi$$
0.868945 + 0.494909i $$0.164799\pi$$
$$462$$ 0 0
$$463$$ 35.2853i 1.63985i 0.572472 + 0.819924i $$0.305985\pi$$
−0.572472 + 0.819924i $$0.694015\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 20.7824i 0.961693i 0.876805 + 0.480847i $$0.159671\pi$$
−0.876805 + 0.480847i $$0.840329\pi$$
$$468$$ 0 0
$$469$$ − 5.21068i − 0.240607i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 19.0260i 0.874816i
$$474$$ 0 0
$$475$$ 92.5868 4.24818
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 17.1640 0.784245 0.392123 0.919913i $$-0.371741\pi$$
0.392123 + 0.919913i $$0.371741\pi$$
$$480$$ 0 0
$$481$$ −10.3457 −0.471722
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −19.3915 −0.880522
$$486$$ 0 0
$$487$$ − 38.7961i − 1.75802i −0.476805 0.879009i $$-0.658205\pi$$
0.476805 0.879009i $$-0.341795\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 31.1973i 1.40791i 0.710243 + 0.703957i $$0.248585\pi$$
−0.710243 + 0.703957i $$0.751415\pi$$
$$492$$ 0 0
$$493$$ 2.53590i 0.114211i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 1.36071i − 0.0610361i
$$498$$ 0 0
$$499$$ 8.78320 0.393190 0.196595 0.980485i $$-0.437012\pi$$
0.196595 + 0.980485i $$0.437012\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 11.2679 0.502413 0.251207 0.967934i $$-0.419173\pi$$
0.251207 + 0.967934i $$0.419173\pi$$
$$504$$ 0 0
$$505$$ 8.77174 0.390337
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 7.99388 0.354322 0.177161 0.984182i $$-0.443309\pi$$
0.177161 + 0.984182i $$0.443309\pi$$
$$510$$ 0 0
$$511$$ 15.4465i 0.683314i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 7.34536i 0.323675i
$$516$$ 0 0
$$517$$ − 13.1816i − 0.579727i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 15.1797i − 0.665035i −0.943097 0.332517i $$-0.892102\pi$$
0.943097 0.332517i $$-0.107898\pi$$
$$522$$ 0 0
$$523$$ 5.58846 0.244366 0.122183 0.992508i $$-0.461010\pi$$
0.122183 + 0.992508i $$0.461010\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.42136 −0.366840
$$528$$ 0 0
$$529$$ −1.47698 −0.0642163
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 38.4097 1.66371
$$534$$ 0 0
$$535$$ − 45.0204i − 1.94640i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 5.11792i 0.220444i
$$540$$ 0 0
$$541$$ 10.1229i 0.435219i 0.976036 + 0.217610i $$0.0698260\pi$$
−0.976036 + 0.217610i $$0.930174\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 5.67140i − 0.242936i
$$546$$ 0 0
$$547$$ −7.03662 −0.300864 −0.150432 0.988620i $$-0.548067\pi$$
−0.150432 + 0.988620i $$0.548067\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.24647 −0.351311
$$552$$ 0 0
$$553$$ 3.98241 0.169349
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −11.0427 −0.467896 −0.233948 0.972249i $$-0.575165\pi$$
−0.233948 + 0.972249i $$0.575165\pi$$
$$558$$ 0 0
$$559$$ 19.3708i 0.819299i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 19.1115i 0.805453i 0.915320 + 0.402726i $$0.131937\pi$$
−0.915320 + 0.402726i $$0.868063\pi$$
$$564$$ 0 0
$$565$$ 35.4351i 1.49076i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 12.6443i − 0.530077i −0.964238 0.265039i $$-0.914615\pi$$
0.964238 0.265039i $$-0.0853847\pi$$
$$570$$ 0 0
$$571$$ −19.4351 −0.813332 −0.406666 0.913577i $$-0.633309\pi$$
−0.406666 + 0.913577i $$0.633309\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −66.0442 −2.75423
$$576$$ 0 0
$$577$$ −42.8137 −1.78236 −0.891178 0.453654i $$-0.850120\pi$$
−0.891178 + 0.453654i $$0.850120\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 11.6893 0.484954
$$582$$ 0 0
$$583$$ − 52.3862i − 2.16961i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 28.9221i 1.19374i 0.802337 + 0.596871i $$0.203590\pi$$
−0.802337 + 0.596871i $$0.796410\pi$$
$$588$$ 0 0
$$589$$ − 27.3854i − 1.12839i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 41.9140i − 1.72120i −0.509280 0.860601i $$-0.670088\pi$$
0.509280 0.860601i $$-0.329912\pi$$
$$594$$ 0 0
$$595$$ −8.77174 −0.359606
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 20.5829 0.840993 0.420496 0.907294i $$-0.361856\pi$$
0.420496 + 0.907294i $$0.361856\pi$$
$$600$$ 0 0
$$601$$ 5.44652 0.222168 0.111084 0.993811i $$-0.464568\pi$$
0.111084 + 0.993811i $$0.464568\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 66.6349 2.70909
$$606$$ 0 0
$$607$$ − 3.40600i − 0.138245i −0.997608 0.0691226i $$-0.977980\pi$$
0.997608 0.0691226i $$-0.0220200\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 13.4205i − 0.542937i
$$612$$ 0 0
$$613$$ 23.0992i 0.932968i 0.884530 + 0.466484i $$0.154479\pi$$
−0.884530 + 0.466484i $$0.845521\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 25.4457i 1.02440i 0.858865 + 0.512202i $$0.171170\pi$$
−0.858865 + 0.512202i $$0.828830\pi$$
$$618$$ 0 0
$$619$$ 18.7167 0.752287 0.376144 0.926561i $$-0.377250\pi$$
0.376144 + 0.926561i $$0.377250\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 14.8645 0.595533
$$624$$ 0 0
$$625$$ 106.480 4.25920
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.97095 −0.158332
$$630$$ 0 0
$$631$$ 31.5549i 1.25618i 0.778140 + 0.628091i $$0.216163\pi$$
−0.778140 + 0.628091i $$0.783837\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 58.9746i − 2.34034i
$$636$$ 0 0
$$637$$ 5.21068i 0.206455i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.53508i 0.376613i 0.982110 + 0.188306i $$0.0602998\pi$$
−0.982110 + 0.188306i $$0.939700\pi$$
$$642$$ 0 0
$$643$$ 37.8243 1.49164 0.745822 0.666145i $$-0.232057\pi$$
0.745822 + 0.666145i $$0.232057\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −19.4663 −0.765301 −0.382650 0.923893i $$-0.624989\pi$$
−0.382650 + 0.923893i $$0.624989\pi$$
$$648$$ 0 0
$$649$$ 17.9321 0.703896
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −9.58928 −0.375257 −0.187629 0.982240i $$-0.560080\pi$$
−0.187629 + 0.982240i $$0.560080\pi$$
$$654$$ 0 0
$$655$$ 7.78402i 0.304147i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 13.0615i − 0.508803i −0.967099 0.254402i $$-0.918122\pi$$
0.967099 0.254402i $$-0.0818785\pi$$
$$660$$ 0 0
$$661$$ 39.8807i 1.55118i 0.631236 + 0.775591i $$0.282548\pi$$
−0.631236 + 0.775591i $$0.717452\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 28.5247i − 1.10614i
$$666$$ 0 0
$$667$$ 5.88239 0.227767
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −23.2144 −0.896180
$$672$$ 0 0
$$673$$ −12.8992 −0.497226 −0.248613 0.968603i $$-0.579975\pi$$
−0.248613 + 0.968603i $$0.579975\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30.1713 1.15958 0.579789 0.814767i $$-0.303135\pi$$
0.579789 + 0.814767i $$0.303135\pi$$
$$678$$ 0 0
$$679$$ 4.42136i 0.169676i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 6.61719i − 0.253200i −0.991954 0.126600i $$-0.959594\pi$$
0.991954 0.126600i $$-0.0404064\pi$$
$$684$$ 0 0
$$685$$ 63.9266i 2.44251i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 53.3357i − 2.03193i
$$690$$ 0 0
$$691$$ −40.6701 −1.54716 −0.773581 0.633697i $$-0.781537\pi$$
−0.773581 + 0.633697i $$0.781537\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 65.8212 2.49674
$$696$$ 0 0
$$697$$ 14.7427 0.558419
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 18.4320 0.696167 0.348083 0.937464i $$-0.386833\pi$$
0.348083 + 0.937464i $$0.386833\pi$$
$$702$$ 0 0
$$703$$ − 12.9131i − 0.487027i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 2.00000i − 0.0752177i
$$708$$ 0 0
$$709$$ − 21.1221i − 0.793258i −0.917979 0.396629i $$-0.870180\pi$$
0.917979 0.396629i $$-0.129820\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 19.5346i 0.731575i
$$714$$ 0 0
$$715$$ 116.962 4.37411
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −24.1319 −0.899969 −0.449985 0.893036i $$-0.648571\pi$$
−0.449985 + 0.893036i $$0.648571\pi$$
$$720$$ 0 0
$$721$$ 1.67478 0.0623721
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −18.0503 −0.670372
$$726$$ 0 0
$$727$$ − 18.3923i − 0.682133i −0.940039 0.341066i $$-0.889212\pi$$
0.940039 0.341066i $$-0.110788\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7.43505i 0.274995i
$$732$$ 0 0
$$733$$ 34.5602i 1.27651i 0.769824 + 0.638256i $$0.220344\pi$$
−0.769824 + 0.638256i $$0.779656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 26.6678i 0.982322i
$$738$$ 0 0
$$739$$ −45.0366 −1.65670 −0.828350 0.560212i $$-0.810720\pi$$
−0.828350 + 0.560212i $$0.810720\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −7.83850 −0.287567 −0.143783 0.989609i $$-0.545927\pi$$
−0.143783 + 0.989609i $$0.545927\pi$$
$$744$$ 0 0
$$745$$ −23.4060 −0.857529
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −10.2649 −0.375071
$$750$$ 0 0
$$751$$ − 38.4328i − 1.40243i −0.712948 0.701217i $$-0.752641\pi$$
0.712948 0.701217i $$-0.247359\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 77.0708i − 2.80489i
$$756$$ 0 0
$$757$$ 2.24423i 0.0815680i 0.999168 + 0.0407840i $$0.0129855\pi$$
−0.999168 + 0.0407840i $$0.987014\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 28.3946i 1.02930i 0.857400 + 0.514651i $$0.172079\pi$$
−0.857400 + 0.514651i $$0.827921\pi$$
$$762$$ 0 0
$$763$$ −1.29311 −0.0468137
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.2571 0.659226
$$768$$ 0 0
$$769$$ 27.2968 0.984348 0.492174 0.870497i $$-0.336202\pi$$
0.492174 + 0.870497i $$0.336202\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −31.6583 −1.13867 −0.569335 0.822105i $$-0.692799\pi$$
−0.569335 + 0.822105i $$0.692799\pi$$
$$774$$ 0 0
$$775$$ − 59.9425i − 2.15320i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 47.9416i 1.71769i
$$780$$ 0 0
$$781$$ 6.96400i 0.249191i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 60.6388i 2.16429i
$$786$$ 0 0
$$787$$ 24.1375 0.860408 0.430204 0.902732i $$-0.358442\pi$$
0.430204 + 0.902732i $$0.358442\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8.07937 0.287269
$$792$$ 0 0
$$793$$ −23.6351 −0.839307
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23.1867 0.821313 0.410657 0.911790i $$-0.365299\pi$$
0.410657 + 0.911790i $$0.365299\pi$$
$$798$$ 0 0
$$799$$ − 5.15117i − 0.182235i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 79.0540i − 2.78976i
$$804$$ 0 0
$$805$$ 20.3473i 0.717149i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 24.2823i − 0.853719i −0.904318 0.426860i $$-0.859620\pi$$
0.904318 0.426860i $$-0.140380\pi$$
$$810$$ 0 0
$$811$$ −29.1610 −1.02398 −0.511990 0.858991i $$-0.671092\pi$$
−0.511990 + 0.858991i $$0.671092\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 38.0464 1.33271
$$816$$ 0 0
$$817$$ −24.1780 −0.845881
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.6701 −1.48920 −0.744598 0.667513i $$-0.767359\pi$$
−0.744598 + 0.667513i $$0.767359\pi$$
$$822$$ 0 0
$$823$$ − 18.4487i − 0.643083i −0.946896 0.321541i $$-0.895799\pi$$
0.946896 0.321541i $$-0.104201\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 37.8393i 1.31580i 0.753104 + 0.657901i $$0.228556\pi$$
−0.753104 + 0.657901i $$0.771444\pi$$
$$828$$ 0 0
$$829$$ − 46.5463i − 1.61662i −0.588756 0.808310i $$-0.700382\pi$$
0.588756 0.808310i $$-0.299618\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 2.00000i 0.0692959i
$$834$$ 0 0
$$835$$ 69.6715 2.41108
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 12.9388 0.446698 0.223349 0.974739i $$-0.428301\pi$$
0.223349 + 0.974739i $$0.428301\pi$$
$$840$$ 0 0
$$841$$ −27.3923 −0.944562
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 62.0651 2.13511
$$846$$ 0 0
$$847$$ − 15.1931i − 0.522041i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.21119i 0.315756i
$$852$$ 0 0
$$853$$ − 49.9509i − 1.71029i −0.518391 0.855144i $$-0.673469\pi$$
0.518391 0.855144i $$-0.326531\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.4208i 1.10747i 0.832691 + 0.553737i $$0.186799\pi$$
−0.832691 + 0.553737i $$0.813201\pi$$
$$858$$ 0 0
$$859$$ 18.3311 0.625450 0.312725 0.949844i $$-0.398758\pi$$
0.312725 + 0.949844i $$0.398758\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −3.77704 −0.128572 −0.0642859 0.997932i $$-0.520477\pi$$
−0.0642859 + 0.997932i $$0.520477\pi$$
$$864$$ 0 0
$$865$$ 18.5496 0.630705
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −20.3817 −0.691401
$$870$$ 0 0
$$871$$ 27.1512i 0.919982i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 40.5072i − 1.36939i
$$876$$ 0 0
$$877$$ 9.61218i 0.324580i 0.986743 + 0.162290i $$0.0518880\pi$$
−0.986743 + 0.162290i $$0.948112\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22.4722i 0.757107i 0.925579 + 0.378554i $$0.123578\pi$$
−0.925579 + 0.378554i $$0.876422\pi$$
$$882$$ 0 0
$$883$$ 9.77704 0.329023 0.164512 0.986375i $$-0.447395\pi$$
0.164512 + 0.986375i $$0.447395\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −59.1043 −1.98453 −0.992265 0.124141i $$-0.960383\pi$$
−0.992265 + 0.124141i $$0.960383\pi$$
$$888$$ 0 0
$$889$$ −13.4465 −0.450982
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 16.7510 0.560552
$$894$$ 0 0
$$895$$ 28.2906i 0.945652i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 5.33893i 0.178063i
$$900$$ 0 0
$$901$$ − 20.4717i − 0.682010i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 46.0816i 1.53181i
$$906$$ 0 0
$$907$$ 31.9861 1.06208 0.531040 0.847346i $$-0.321801\pi$$
0.531040 + 0.847346i $$0.321801\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 26.4052 0.874843 0.437421 0.899257i $$-0.355892\pi$$
0.437421 + 0.899257i $$0.355892\pi$$
$$912$$ 0 0
$$913$$ −59.8249 −1.97992
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.77480 0.0586089
$$918$$ 0 0
$$919$$ 46.4656i 1.53276i 0.642389 + 0.766379i $$0.277943\pi$$
−0.642389 + 0.766379i $$0.722057\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 7.09021i 0.233377i
$$924$$ 0 0
$$925$$ − 28.2649i − 0.929344i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 16.7656i 0.550062i 0.961435 + 0.275031i $$0.0886881\pi$$
−0.961435 + 0.275031i $$0.911312\pi$$
$$930$$ 0 0
$$931$$ −6.50379 −0.213153
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 44.8930 1.46816
$$936$$ 0 0
$$937$$ −32.3923 −1.05821 −0.529105 0.848556i $$-0.677472\pi$$
−0.529105 + 0.848556i $$0.677472\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 40.1216 1.30793 0.653964 0.756526i $$-0.273105\pi$$
0.653964 + 0.756526i $$0.273105\pi$$
$$942$$ 0 0
$$943$$ − 34.1978i − 1.11363i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 38.9905i − 1.26702i −0.773734 0.633511i $$-0.781613\pi$$
0.773734 0.633511i $$-0.218387\pi$$
$$948$$ 0 0
$$949$$ − 80.4868i − 2.61271i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 1.39764i − 0.0452739i −0.999744 0.0226370i $$-0.992794\pi$$
0.999744 0.0226370i $$-0.00720619\pi$$
$$954$$ 0 0
$$955$$ −36.2267 −1.17227
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 14.5756 0.470670
$$960$$ 0 0
$$961$$ 13.2702 0.428071
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 35.4351 1.14069
$$966$$ 0 0
$$967$$ 21.9709i 0.706538i 0.935522 + 0.353269i $$0.114930\pi$$
−0.935522 + 0.353269i $$0.885070\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 42.7098i − 1.37062i −0.728251 0.685311i $$-0.759666\pi$$
0.728251 0.685311i $$-0.240334\pi$$
$$972$$ 0 0
$$973$$ − 15.0076i − 0.481121i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 60.3885i − 1.93200i −0.258546 0.965999i $$-0.583243\pi$$
0.258546 0.965999i $$-0.416757\pi$$
$$978$$ 0 0
$$979$$ −76.0753 −2.43138
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −19.6251 −0.625943 −0.312971 0.949763i $$-0.601324\pi$$
−0.312971 + 0.949763i $$0.601324\pi$$
$$984$$ 0 0
$$985$$ 30.8886 0.984191
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 17.2467 0.548413
$$990$$ 0 0
$$991$$ 7.84660i 0.249256i 0.992204 + 0.124628i $$0.0397737\pi$$
−0.992204 + 0.124628i $$0.960226\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 88.3032i 2.79940i
$$996$$ 0 0
$$997$$ − 19.9128i − 0.630646i −0.948984 0.315323i $$-0.897887\pi$$
0.948984 0.315323i $$-0.102113\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.j.a.5615.2 8
3.2 odd 2 6048.2.j.b.5615.8 8
4.3 odd 2 1512.2.j.a.323.7 yes 8
8.3 odd 2 6048.2.j.b.5615.7 8
8.5 even 2 1512.2.j.b.323.4 yes 8
12.11 even 2 1512.2.j.b.323.2 yes 8
24.5 odd 2 1512.2.j.a.323.5 8
24.11 even 2 inner 6048.2.j.a.5615.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.a.323.5 8 24.5 odd 2
1512.2.j.a.323.7 yes 8 4.3 odd 2
1512.2.j.b.323.2 yes 8 12.11 even 2
1512.2.j.b.323.4 yes 8 8.5 even 2
6048.2.j.a.5615.1 8 24.11 even 2 inner
6048.2.j.a.5615.2 8 1.1 even 1 trivial
6048.2.j.b.5615.7 8 8.3 odd 2
6048.2.j.b.5615.8 8 3.2 odd 2