# Properties

 Label 7600.2.a.bi.1.2 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.6863055362$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.36333 q^{3} -0.636672 q^{7} -1.14134 q^{9} +O(q^{10})$$ $$q-1.36333 q^{3} -0.636672 q^{7} -1.14134 q^{9} -3.50466 q^{11} -0.141336 q^{13} +2.14134 q^{17} -1.00000 q^{19} +0.867993 q^{21} -4.91934 q^{23} +5.64600 q^{27} +7.15066 q^{29} +7.78734 q^{31} +4.77801 q^{33} +3.27334 q^{37} +0.192688 q^{39} -4.23132 q^{41} -2.49534 q^{43} +10.2827 q^{47} -6.59465 q^{49} -2.91934 q^{51} +8.14134 q^{53} +1.36333 q^{57} +5.64600 q^{59} -6.49534 q^{61} +0.726656 q^{63} -8.37266 q^{67} +6.70668 q^{69} +8.95798 q^{71} -3.69735 q^{73} +2.23132 q^{77} +4.17997 q^{79} -4.27334 q^{81} +9.00933 q^{83} -9.74870 q^{87} -6.77801 q^{89} +0.0899847 q^{91} -10.6167 q^{93} +14.5653 q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{3} - 4q^{7} + 5q^{9} + O(q^{10})$$ $$3q - 2q^{3} - 4q^{7} + 5q^{9} + 8q^{13} - 2q^{17} - 3q^{19} - 10q^{21} - 2q^{27} - 8q^{29} - 4q^{31} + 8q^{33} + 14q^{37} - 10q^{39} + 2q^{41} - 18q^{43} + 14q^{47} - 3q^{49} + 6q^{51} + 16q^{53} + 2q^{57} - 2q^{59} - 30q^{61} - 2q^{63} - 2q^{67} - 22q^{69} + 8q^{71} + 10q^{73} - 8q^{77} - 17q^{81} + 6q^{83} - 6q^{87} - 14q^{89} - 6q^{91} + 4q^{93} + 10q^{97} + 12q^{99} + O(q^{100})$$

## 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
$$3$$ −1.36333 −0.787118 −0.393559 0.919299i $$-0.628756\pi$$
−0.393559 + 0.919299i $$0.628756\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.636672 −0.240639 −0.120320 0.992735i $$-0.538392\pi$$
−0.120320 + 0.992735i $$0.538392\pi$$
$$8$$ 0 0
$$9$$ −1.14134 −0.380445
$$10$$ 0 0
$$11$$ −3.50466 −1.05670 −0.528348 0.849028i $$-0.677188\pi$$
−0.528348 + 0.849028i $$0.677188\pi$$
$$12$$ 0 0
$$13$$ −0.141336 −0.0391996 −0.0195998 0.999808i $$-0.506239\pi$$
−0.0195998 + 0.999808i $$0.506239\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.14134 0.519350 0.259675 0.965696i $$-0.416385\pi$$
0.259675 + 0.965696i $$0.416385\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.867993 0.189412
$$22$$ 0 0
$$23$$ −4.91934 −1.02575 −0.512877 0.858462i $$-0.671420\pi$$
−0.512877 + 0.858462i $$0.671420\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.64600 1.08657
$$28$$ 0 0
$$29$$ 7.15066 1.32785 0.663923 0.747801i $$-0.268890\pi$$
0.663923 + 0.747801i $$0.268890\pi$$
$$30$$ 0 0
$$31$$ 7.78734 1.39865 0.699323 0.714805i $$-0.253485\pi$$
0.699323 + 0.714805i $$0.253485\pi$$
$$32$$ 0 0
$$33$$ 4.77801 0.831744
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.27334 0.538134 0.269067 0.963121i $$-0.413285\pi$$
0.269067 + 0.963121i $$0.413285\pi$$
$$38$$ 0 0
$$39$$ 0.192688 0.0308547
$$40$$ 0 0
$$41$$ −4.23132 −0.660821 −0.330411 0.943837i $$-0.607187\pi$$
−0.330411 + 0.943837i $$0.607187\pi$$
$$42$$ 0 0
$$43$$ −2.49534 −0.380535 −0.190268 0.981732i $$-0.560936\pi$$
−0.190268 + 0.981732i $$0.560936\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.2827 1.49988 0.749941 0.661505i $$-0.230082\pi$$
0.749941 + 0.661505i $$0.230082\pi$$
$$48$$ 0 0
$$49$$ −6.59465 −0.942093
$$50$$ 0 0
$$51$$ −2.91934 −0.408790
$$52$$ 0 0
$$53$$ 8.14134 1.11830 0.559149 0.829067i $$-0.311128\pi$$
0.559149 + 0.829067i $$0.311128\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.36333 0.180577
$$58$$ 0 0
$$59$$ 5.64600 0.735047 0.367523 0.930014i $$-0.380206\pi$$
0.367523 + 0.930014i $$0.380206\pi$$
$$60$$ 0 0
$$61$$ −6.49534 −0.831643 −0.415821 0.909446i $$-0.636506\pi$$
−0.415821 + 0.909446i $$0.636506\pi$$
$$62$$ 0 0
$$63$$ 0.726656 0.0915501
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.37266 −1.02288 −0.511441 0.859318i $$-0.670888\pi$$
−0.511441 + 0.859318i $$0.670888\pi$$
$$68$$ 0 0
$$69$$ 6.70668 0.807389
$$70$$ 0 0
$$71$$ 8.95798 1.06312 0.531558 0.847022i $$-0.321607\pi$$
0.531558 + 0.847022i $$0.321607\pi$$
$$72$$ 0 0
$$73$$ −3.69735 −0.432742 −0.216371 0.976311i $$-0.569422\pi$$
−0.216371 + 0.976311i $$0.569422\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.23132 0.254283
$$78$$ 0 0
$$79$$ 4.17997 0.470283 0.235142 0.971961i $$-0.424445\pi$$
0.235142 + 0.971961i $$0.424445\pi$$
$$80$$ 0 0
$$81$$ −4.27334 −0.474816
$$82$$ 0 0
$$83$$ 9.00933 0.988902 0.494451 0.869205i $$-0.335369\pi$$
0.494451 + 0.869205i $$0.335369\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −9.74870 −1.04517
$$88$$ 0 0
$$89$$ −6.77801 −0.718467 −0.359234 0.933248i $$-0.616962\pi$$
−0.359234 + 0.933248i $$0.616962\pi$$
$$90$$ 0 0
$$91$$ 0.0899847 0.00943296
$$92$$ 0 0
$$93$$ −10.6167 −1.10090
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.5653 1.47889 0.739443 0.673219i $$-0.235089\pi$$
0.739443 + 0.673219i $$0.235089\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −16.6167 −1.65342 −0.826712 0.562626i $$-0.809791\pi$$
−0.826712 + 0.562626i $$0.809791\pi$$
$$102$$ 0 0
$$103$$ −9.06068 −0.892775 −0.446388 0.894840i $$-0.647290\pi$$
−0.446388 + 0.894840i $$0.647290\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.0899847 −0.00869915 −0.00434958 0.999991i $$-0.501385\pi$$
−0.00434958 + 0.999991i $$0.501385\pi$$
$$108$$ 0 0
$$109$$ −13.5946 −1.30213 −0.651066 0.759021i $$-0.725678\pi$$
−0.651066 + 0.759021i $$0.725678\pi$$
$$110$$ 0 0
$$111$$ −4.46264 −0.423575
$$112$$ 0 0
$$113$$ 11.5233 1.08402 0.542011 0.840371i $$-0.317663\pi$$
0.542011 + 0.840371i $$0.317663\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.161312 0.0149133
$$118$$ 0 0
$$119$$ −1.36333 −0.124976
$$120$$ 0 0
$$121$$ 1.28267 0.116607
$$122$$ 0 0
$$123$$ 5.76868 0.520144
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 3.29200 0.292118 0.146059 0.989276i $$-0.453341\pi$$
0.146059 + 0.989276i $$0.453341\pi$$
$$128$$ 0 0
$$129$$ 3.40196 0.299526
$$130$$ 0 0
$$131$$ −18.0187 −1.57430 −0.787149 0.616763i $$-0.788444\pi$$
−0.787149 + 0.616763i $$0.788444\pi$$
$$132$$ 0 0
$$133$$ 0.636672 0.0552064
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.4240 −1.23233 −0.616163 0.787619i $$-0.711314\pi$$
−0.616163 + 0.787619i $$0.711314\pi$$
$$138$$ 0 0
$$139$$ −15.4720 −1.31232 −0.656158 0.754624i $$-0.727819\pi$$
−0.656158 + 0.754624i $$0.727819\pi$$
$$140$$ 0 0
$$141$$ −14.0187 −1.18058
$$142$$ 0 0
$$143$$ 0.495336 0.0414220
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.99067 0.741538
$$148$$ 0 0
$$149$$ −17.1893 −1.40820 −0.704101 0.710100i $$-0.748650\pi$$
−0.704101 + 0.710100i $$0.748650\pi$$
$$150$$ 0 0
$$151$$ −3.29200 −0.267899 −0.133950 0.990988i $$-0.542766\pi$$
−0.133950 + 0.990988i $$0.542766\pi$$
$$152$$ 0 0
$$153$$ −2.44398 −0.197584
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.1893 1.21224 0.606119 0.795374i $$-0.292726\pi$$
0.606119 + 0.795374i $$0.292726\pi$$
$$158$$ 0 0
$$159$$ −11.0993 −0.880233
$$160$$ 0 0
$$161$$ 3.13201 0.246837
$$162$$ 0 0
$$163$$ 14.0700 1.10205 0.551024 0.834489i $$-0.314237\pi$$
0.551024 + 0.834489i $$0.314237\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.7967 1.14500 0.572500 0.819905i $$-0.305974\pi$$
0.572500 + 0.819905i $$0.305974\pi$$
$$168$$ 0 0
$$169$$ −12.9800 −0.998463
$$170$$ 0 0
$$171$$ 1.14134 0.0872802
$$172$$ 0 0
$$173$$ −17.2920 −1.31469 −0.657343 0.753591i $$-0.728320\pi$$
−0.657343 + 0.753591i $$0.728320\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −7.69735 −0.578568
$$178$$ 0 0
$$179$$ −17.7360 −1.32565 −0.662825 0.748774i $$-0.730643\pi$$
−0.662825 + 0.748774i $$0.730643\pi$$
$$180$$ 0 0
$$181$$ 6.17997 0.459354 0.229677 0.973267i $$-0.426233\pi$$
0.229677 + 0.973267i $$0.426233\pi$$
$$182$$ 0 0
$$183$$ 8.85527 0.654601
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −7.50466 −0.548795
$$188$$ 0 0
$$189$$ −3.59465 −0.261472
$$190$$ 0 0
$$191$$ −14.6367 −1.05907 −0.529536 0.848287i $$-0.677634\pi$$
−0.529536 + 0.848287i $$0.677634\pi$$
$$192$$ 0 0
$$193$$ 20.0187 1.44097 0.720487 0.693468i $$-0.243918\pi$$
0.720487 + 0.693468i $$0.243918\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −9.94865 −0.708812 −0.354406 0.935092i $$-0.615317\pi$$
−0.354406 + 0.935092i $$0.615317\pi$$
$$198$$ 0 0
$$199$$ −9.74870 −0.691067 −0.345534 0.938406i $$-0.612302\pi$$
−0.345534 + 0.938406i $$0.612302\pi$$
$$200$$ 0 0
$$201$$ 11.4147 0.805129
$$202$$ 0 0
$$203$$ −4.55263 −0.319532
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 5.61462 0.390243
$$208$$ 0 0
$$209$$ 3.50466 0.242423
$$210$$ 0 0
$$211$$ 20.7580 1.42904 0.714521 0.699614i $$-0.246645\pi$$
0.714521 + 0.699614i $$0.246645\pi$$
$$212$$ 0 0
$$213$$ −12.2127 −0.836798
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.95798 −0.336569
$$218$$ 0 0
$$219$$ 5.04070 0.340619
$$220$$ 0 0
$$221$$ −0.302648 −0.0203583
$$222$$ 0 0
$$223$$ 10.7267 0.718310 0.359155 0.933278i $$-0.383065\pi$$
0.359155 + 0.933278i $$0.383065\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.5526 −0.833147 −0.416574 0.909102i $$-0.636769\pi$$
−0.416574 + 0.909102i $$0.636769\pi$$
$$228$$ 0 0
$$229$$ 25.4720 1.68324 0.841618 0.540074i $$-0.181604\pi$$
0.841618 + 0.540074i $$0.181604\pi$$
$$230$$ 0 0
$$231$$ −3.04202 −0.200150
$$232$$ 0 0
$$233$$ −3.11203 −0.203876 −0.101938 0.994791i $$-0.532504\pi$$
−0.101938 + 0.994791i $$0.532504\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −5.69867 −0.370168
$$238$$ 0 0
$$239$$ 1.54330 0.0998276 0.0499138 0.998754i $$-0.484105\pi$$
0.0499138 + 0.998754i $$0.484105\pi$$
$$240$$ 0 0
$$241$$ 10.2827 0.662365 0.331183 0.943567i $$-0.392552\pi$$
0.331183 + 0.943567i $$0.392552\pi$$
$$242$$ 0 0
$$243$$ −11.1120 −0.712837
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.141336 0.00899300
$$248$$ 0 0
$$249$$ −12.2827 −0.778383
$$250$$ 0 0
$$251$$ 2.51399 0.158682 0.0793409 0.996848i $$-0.474718\pi$$
0.0793409 + 0.996848i $$0.474718\pi$$
$$252$$ 0 0
$$253$$ 17.2406 1.08391
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −24.7967 −1.54677 −0.773387 0.633934i $$-0.781439\pi$$
−0.773387 + 0.633934i $$0.781439\pi$$
$$258$$ 0 0
$$259$$ −2.08405 −0.129496
$$260$$ 0 0
$$261$$ −8.16131 −0.505173
$$262$$ 0 0
$$263$$ 22.5653 1.39144 0.695719 0.718314i $$-0.255086\pi$$
0.695719 + 0.718314i $$0.255086\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 9.24065 0.565519
$$268$$ 0 0
$$269$$ −26.5653 −1.61972 −0.809859 0.586625i $$-0.800456\pi$$
−0.809859 + 0.586625i $$0.800456\pi$$
$$270$$ 0 0
$$271$$ 24.9380 1.51488 0.757438 0.652907i $$-0.226451\pi$$
0.757438 + 0.652907i $$0.226451\pi$$
$$272$$ 0 0
$$273$$ −0.122679 −0.00742485
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.5467 1.11436 0.557181 0.830391i $$-0.311883\pi$$
0.557181 + 0.830391i $$0.311883\pi$$
$$278$$ 0 0
$$279$$ −8.88797 −0.532109
$$280$$ 0 0
$$281$$ 24.7967 1.47925 0.739623 0.673022i $$-0.235004\pi$$
0.739623 + 0.673022i $$0.235004\pi$$
$$282$$ 0 0
$$283$$ −13.5747 −0.806931 −0.403465 0.914995i $$-0.632194\pi$$
−0.403465 + 0.914995i $$0.632194\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.69396 0.159020
$$288$$ 0 0
$$289$$ −12.4147 −0.730275
$$290$$ 0 0
$$291$$ −19.8573 −1.16406
$$292$$ 0 0
$$293$$ −15.6133 −0.912139 −0.456070 0.889944i $$-0.650743\pi$$
−0.456070 + 0.889944i $$0.650743\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −19.7873 −1.14818
$$298$$ 0 0
$$299$$ 0.695281 0.0402091
$$300$$ 0 0
$$301$$ 1.58871 0.0915717
$$302$$ 0 0
$$303$$ 22.6540 1.30144
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −34.0187 −1.94155 −0.970774 0.239997i $$-0.922854\pi$$
−0.970774 + 0.239997i $$0.922854\pi$$
$$308$$ 0 0
$$309$$ 12.3527 0.702719
$$310$$ 0 0
$$311$$ −8.93800 −0.506828 −0.253414 0.967358i $$-0.581553\pi$$
−0.253414 + 0.967358i $$0.581553\pi$$
$$312$$ 0 0
$$313$$ −18.4240 −1.04139 −0.520693 0.853744i $$-0.674326\pi$$
−0.520693 + 0.853744i $$0.674326\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 33.5547 1.88462 0.942310 0.334742i $$-0.108649\pi$$
0.942310 + 0.334742i $$0.108649\pi$$
$$318$$ 0 0
$$319$$ −25.0607 −1.40313
$$320$$ 0 0
$$321$$ 0.122679 0.00684726
$$322$$ 0 0
$$323$$ −2.14134 −0.119147
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 18.5340 1.02493
$$328$$ 0 0
$$329$$ −6.54669 −0.360931
$$330$$ 0 0
$$331$$ −2.25130 −0.123742 −0.0618712 0.998084i $$-0.519707\pi$$
−0.0618712 + 0.998084i $$0.519707\pi$$
$$332$$ 0 0
$$333$$ −3.73599 −0.204731
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −21.3620 −1.16366 −0.581831 0.813309i $$-0.697664\pi$$
−0.581831 + 0.813309i $$0.697664\pi$$
$$338$$ 0 0
$$339$$ −15.7101 −0.853254
$$340$$ 0 0
$$341$$ −27.2920 −1.47794
$$342$$ 0 0
$$343$$ 8.65533 0.467344
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11.5560 −0.620359 −0.310180 0.950678i $$-0.600389\pi$$
−0.310180 + 0.950678i $$0.600389\pi$$
$$348$$ 0 0
$$349$$ −17.1120 −0.915986 −0.457993 0.888956i $$-0.651432\pi$$
−0.457993 + 0.888956i $$0.651432\pi$$
$$350$$ 0 0
$$351$$ −0.797984 −0.0425932
$$352$$ 0 0
$$353$$ 11.6974 0.622587 0.311294 0.950314i $$-0.399238\pi$$
0.311294 + 0.950314i $$0.399238\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.85866 0.0983709
$$358$$ 0 0
$$359$$ 4.47536 0.236200 0.118100 0.993002i $$-0.462320\pi$$
0.118100 + 0.993002i $$0.462320\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −1.74870 −0.0917831
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −18.7453 −0.978497 −0.489249 0.872144i $$-0.662729\pi$$
−0.489249 + 0.872144i $$0.662729\pi$$
$$368$$ 0 0
$$369$$ 4.82936 0.251406
$$370$$ 0 0
$$371$$ −5.18336 −0.269107
$$372$$ 0 0
$$373$$ 1.69735 0.0878855 0.0439428 0.999034i $$-0.486008\pi$$
0.0439428 + 0.999034i $$0.486008\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.01065 −0.0520510
$$378$$ 0 0
$$379$$ −2.63667 −0.135437 −0.0677184 0.997704i $$-0.521572\pi$$
−0.0677184 + 0.997704i $$0.521572\pi$$
$$380$$ 0 0
$$381$$ −4.48808 −0.229931
$$382$$ 0 0
$$383$$ −12.4953 −0.638482 −0.319241 0.947674i $$-0.603428\pi$$
−0.319241 + 0.947674i $$0.603428\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.84802 0.144773
$$388$$ 0 0
$$389$$ −4.51399 −0.228869 −0.114434 0.993431i $$-0.536506\pi$$
−0.114434 + 0.993431i $$0.536506\pi$$
$$390$$ 0 0
$$391$$ −10.5340 −0.532726
$$392$$ 0 0
$$393$$ 24.5653 1.23916
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −35.6774 −1.79060 −0.895298 0.445468i $$-0.853037\pi$$
−0.895298 + 0.445468i $$0.853037\pi$$
$$398$$ 0 0
$$399$$ −0.867993 −0.0434540
$$400$$ 0 0
$$401$$ 15.3434 0.766210 0.383105 0.923705i $$-0.374855\pi$$
0.383105 + 0.923705i $$0.374855\pi$$
$$402$$ 0 0
$$403$$ −1.10063 −0.0548264
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −11.4720 −0.568644
$$408$$ 0 0
$$409$$ 29.3620 1.45186 0.725929 0.687770i $$-0.241410\pi$$
0.725929 + 0.687770i $$0.241410\pi$$
$$410$$ 0 0
$$411$$ 19.6647 0.969986
$$412$$ 0 0
$$413$$ −3.59465 −0.176881
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 21.0934 1.03295
$$418$$ 0 0
$$419$$ −25.1379 −1.22807 −0.614035 0.789279i $$-0.710454\pi$$
−0.614035 + 0.789279i $$0.710454\pi$$
$$420$$ 0 0
$$421$$ 14.5454 0.708898 0.354449 0.935075i $$-0.384668\pi$$
0.354449 + 0.935075i $$0.384668\pi$$
$$422$$ 0 0
$$423$$ −11.7360 −0.570623
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.13540 0.200126
$$428$$ 0 0
$$429$$ −0.675305 −0.0326040
$$430$$ 0 0
$$431$$ −19.4020 −0.934560 −0.467280 0.884109i $$-0.654766\pi$$
−0.467280 + 0.884109i $$0.654766\pi$$
$$432$$ 0 0
$$433$$ 5.50466 0.264537 0.132269 0.991214i $$-0.457774\pi$$
0.132269 + 0.991214i $$0.457774\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.91934 0.235324
$$438$$ 0 0
$$439$$ −12.2500 −0.584660 −0.292330 0.956318i $$-0.594430\pi$$
−0.292330 + 0.956318i $$0.594430\pi$$
$$440$$ 0 0
$$441$$ 7.52671 0.358415
$$442$$ 0 0
$$443$$ −31.6006 −1.50139 −0.750695 0.660649i $$-0.770281\pi$$
−0.750695 + 0.660649i $$0.770281\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 23.4347 1.10842
$$448$$ 0 0
$$449$$ −36.0187 −1.69983 −0.849913 0.526923i $$-0.823345\pi$$
−0.849913 + 0.526923i $$0.823345\pi$$
$$450$$ 0 0
$$451$$ 14.8294 0.698287
$$452$$ 0 0
$$453$$ 4.48808 0.210868
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.1413 1.03573 0.517864 0.855463i $$-0.326727\pi$$
0.517864 + 0.855463i $$0.326727\pi$$
$$458$$ 0 0
$$459$$ 12.0900 0.564312
$$460$$ 0 0
$$461$$ −2.31537 −0.107837 −0.0539187 0.998545i $$-0.517171\pi$$
−0.0539187 + 0.998545i $$0.517171\pi$$
$$462$$ 0 0
$$463$$ −15.8387 −0.736086 −0.368043 0.929809i $$-0.619972\pi$$
−0.368043 + 0.929809i $$0.619972\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −23.1379 −1.07070 −0.535348 0.844631i $$-0.679820\pi$$
−0.535348 + 0.844631i $$0.679820\pi$$
$$468$$ 0 0
$$469$$ 5.33063 0.246146
$$470$$ 0 0
$$471$$ −20.7080 −0.954174
$$472$$ 0 0
$$473$$ 8.74531 0.402110
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.29200 −0.425451
$$478$$ 0 0
$$479$$ 10.1214 0.462457 0.231228 0.972900i $$-0.425726\pi$$
0.231228 + 0.972900i $$0.425726\pi$$
$$480$$ 0 0
$$481$$ −0.462642 −0.0210946
$$482$$ 0 0
$$483$$ −4.26995 −0.194290
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −20.3854 −0.923750 −0.461875 0.886945i $$-0.652823\pi$$
−0.461875 + 0.886945i $$0.652823\pi$$
$$488$$ 0 0
$$489$$ −19.1820 −0.867442
$$490$$ 0 0
$$491$$ 7.78734 0.351438 0.175719 0.984440i $$-0.443775\pi$$
0.175719 + 0.984440i $$0.443775\pi$$
$$492$$ 0 0
$$493$$ 15.3120 0.689617
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −5.70329 −0.255828
$$498$$ 0 0
$$499$$ −4.31537 −0.193182 −0.0965912 0.995324i $$-0.530794\pi$$
−0.0965912 + 0.995324i $$0.530794\pi$$
$$500$$ 0 0
$$501$$ −20.1727 −0.901250
$$502$$ 0 0
$$503$$ −18.5526 −0.827221 −0.413610 0.910454i $$-0.635732\pi$$
−0.413610 + 0.910454i $$0.635732\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 17.6960 0.785908
$$508$$ 0 0
$$509$$ −7.73599 −0.342892 −0.171446 0.985194i $$-0.554844\pi$$
−0.171446 + 0.985194i $$0.554844\pi$$
$$510$$ 0 0
$$511$$ 2.35400 0.104135
$$512$$ 0 0
$$513$$ −5.64600 −0.249277
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −36.0373 −1.58492
$$518$$ 0 0
$$519$$ 23.5747 1.03481
$$520$$ 0 0
$$521$$ 15.2080 0.666273 0.333136 0.942879i $$-0.391893\pi$$
0.333136 + 0.942879i $$0.391893\pi$$
$$522$$ 0 0
$$523$$ 18.2113 0.796327 0.398163 0.917315i $$-0.369648\pi$$
0.398163 + 0.917315i $$0.369648\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.6753 0.726388
$$528$$ 0 0
$$529$$ 1.19995 0.0521715
$$530$$ 0 0
$$531$$ −6.44398 −0.279645
$$532$$ 0 0
$$533$$ 0.598038 0.0259039
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 24.1800 1.04344
$$538$$ 0 0
$$539$$ 23.1120 0.995506
$$540$$ 0 0
$$541$$ 16.5140 0.709992 0.354996 0.934868i $$-0.384482\pi$$
0.354996 + 0.934868i $$0.384482\pi$$
$$542$$ 0 0
$$543$$ −8.42533 −0.361565
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.2827 −0.696197 −0.348098 0.937458i $$-0.613172\pi$$
−0.348098 + 0.937458i $$0.613172\pi$$
$$548$$ 0 0
$$549$$ 7.41336 0.316395
$$550$$ 0 0
$$551$$ −7.15066 −0.304629
$$552$$ 0 0
$$553$$ −2.66127 −0.113169
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 37.4533 1.58695 0.793474 0.608604i $$-0.208270\pi$$
0.793474 + 0.608604i $$0.208270\pi$$
$$558$$ 0 0
$$559$$ 0.352681 0.0149168
$$560$$ 0 0
$$561$$ 10.2313 0.431967
$$562$$ 0 0
$$563$$ 29.1307 1.22771 0.613856 0.789418i $$-0.289618\pi$$
0.613856 + 0.789418i $$0.289618\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.72072 0.114259
$$568$$ 0 0
$$569$$ 14.8480 0.622461 0.311231 0.950334i $$-0.399259\pi$$
0.311231 + 0.950334i $$0.399259\pi$$
$$570$$ 0 0
$$571$$ −41.9087 −1.75382 −0.876912 0.480651i $$-0.840401\pi$$
−0.876912 + 0.480651i $$0.840401\pi$$
$$572$$ 0 0
$$573$$ 19.9546 0.833615
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 16.4427 0.684517 0.342259 0.939606i $$-0.388808\pi$$
0.342259 + 0.939606i $$0.388808\pi$$
$$578$$ 0 0
$$579$$ −27.2920 −1.13422
$$580$$ 0 0
$$581$$ −5.73599 −0.237969
$$582$$ 0 0
$$583$$ −28.5327 −1.18170
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 42.5327 1.75551 0.877755 0.479109i $$-0.159040\pi$$
0.877755 + 0.479109i $$0.159040\pi$$
$$588$$ 0 0
$$589$$ −7.78734 −0.320872
$$590$$ 0 0
$$591$$ 13.5633 0.557919
$$592$$ 0 0
$$593$$ 3.92273 0.161087 0.0805437 0.996751i $$-0.474334\pi$$
0.0805437 + 0.996751i $$0.474334\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13.2907 0.543951
$$598$$ 0 0
$$599$$ 10.7594 0.439615 0.219808 0.975543i $$-0.429457\pi$$
0.219808 + 0.975543i $$0.429457\pi$$
$$600$$ 0 0
$$601$$ 25.2220 1.02883 0.514413 0.857542i $$-0.328010\pi$$
0.514413 + 0.857542i $$0.328010\pi$$
$$602$$ 0 0
$$603$$ 9.55602 0.389151
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −39.2920 −1.59481 −0.797407 0.603442i $$-0.793795\pi$$
−0.797407 + 0.603442i $$0.793795\pi$$
$$608$$ 0 0
$$609$$ 6.20672 0.251509
$$610$$ 0 0
$$611$$ −1.45331 −0.0587947
$$612$$ 0 0
$$613$$ 9.80599 0.396060 0.198030 0.980196i $$-0.436546\pi$$
0.198030 + 0.980196i $$0.436546\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −35.0093 −1.40942 −0.704711 0.709494i $$-0.748923\pi$$
−0.704711 + 0.709494i $$0.748923\pi$$
$$618$$ 0 0
$$619$$ −13.4206 −0.539420 −0.269710 0.962942i $$-0.586928\pi$$
−0.269710 + 0.962942i $$0.586928\pi$$
$$620$$ 0 0
$$621$$ −27.7746 −1.11456
$$622$$ 0 0
$$623$$ 4.31537 0.172891
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4.77801 −0.190815
$$628$$ 0 0
$$629$$ 7.00933 0.279480
$$630$$ 0 0
$$631$$ −40.5254 −1.61329 −0.806645 0.591036i $$-0.798719\pi$$
−0.806645 + 0.591036i $$0.798719\pi$$
$$632$$ 0 0
$$633$$ −28.3000 −1.12482
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.932062 0.0369296
$$638$$ 0 0
$$639$$ −10.2241 −0.404458
$$640$$ 0 0
$$641$$ −26.0700 −1.02970 −0.514852 0.857279i $$-0.672153\pi$$
−0.514852 + 0.857279i $$0.672153\pi$$
$$642$$ 0 0
$$643$$ −30.1400 −1.18861 −0.594303 0.804241i $$-0.702572\pi$$
−0.594303 + 0.804241i $$0.702572\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 20.1086 0.790552 0.395276 0.918562i $$-0.370649\pi$$
0.395276 + 0.918562i $$0.370649\pi$$
$$648$$ 0 0
$$649$$ −19.7873 −0.776721
$$650$$ 0 0
$$651$$ 6.75935 0.264920
$$652$$ 0 0
$$653$$ −28.0373 −1.09718 −0.548592 0.836090i $$-0.684836\pi$$
−0.548592 + 0.836090i $$0.684836\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 4.21992 0.164635
$$658$$ 0 0
$$659$$ −4.90069 −0.190904 −0.0954518 0.995434i $$-0.530430\pi$$
−0.0954518 + 0.995434i $$0.530430\pi$$
$$660$$ 0 0
$$661$$ −8.03863 −0.312667 −0.156333 0.987704i $$-0.549967\pi$$
−0.156333 + 0.987704i $$0.549967\pi$$
$$662$$ 0 0
$$663$$ 0.412609 0.0160244
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −35.1766 −1.36204
$$668$$ 0 0
$$669$$ −14.6240 −0.565395
$$670$$ 0 0
$$671$$ 22.7640 0.878793
$$672$$ 0 0
$$673$$ −4.82936 −0.186158 −0.0930791 0.995659i $$-0.529671\pi$$
−0.0930791 + 0.995659i $$0.529671\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 12.8094 0.492305 0.246152 0.969231i $$-0.420834\pi$$
0.246152 + 0.969231i $$0.420834\pi$$
$$678$$ 0 0
$$679$$ −9.27334 −0.355878
$$680$$ 0 0
$$681$$ 17.1133 0.655785
$$682$$ 0 0
$$683$$ −37.1307 −1.42077 −0.710383 0.703815i $$-0.751478\pi$$
−0.710383 + 0.703815i $$0.751478\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −34.7267 −1.32490
$$688$$ 0 0
$$689$$ −1.15066 −0.0438368
$$690$$ 0 0
$$691$$ 18.1986 0.692308 0.346154 0.938178i $$-0.387487\pi$$
0.346154 + 0.938178i $$0.387487\pi$$
$$692$$ 0 0
$$693$$ −2.54669 −0.0967406
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.06068 −0.343198
$$698$$ 0 0
$$699$$ 4.24272 0.160474
$$700$$ 0 0
$$701$$ −26.2827 −0.992683 −0.496341 0.868127i $$-0.665324\pi$$
−0.496341 + 0.868127i $$0.665324\pi$$
$$702$$ 0 0
$$703$$ −3.27334 −0.123456
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10.5794 0.397879
$$708$$ 0 0
$$709$$ −14.9253 −0.560531 −0.280265 0.959923i $$-0.590422\pi$$
−0.280265 + 0.959923i $$0.590422\pi$$
$$710$$ 0 0
$$711$$ −4.77075 −0.178917
$$712$$ 0 0
$$713$$ −38.3086 −1.43467
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −2.10402 −0.0785761
$$718$$ 0 0
$$719$$ −32.3327 −1.20581 −0.602903 0.797814i $$-0.705989\pi$$
−0.602903 + 0.797814i $$0.705989\pi$$
$$720$$ 0 0
$$721$$ 5.76868 0.214837
$$722$$ 0 0
$$723$$ −14.0187 −0.521359
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 42.0246 1.55861 0.779303 0.626647i $$-0.215573\pi$$
0.779303 + 0.626647i $$0.215573\pi$$
$$728$$ 0 0
$$729$$ 27.9694 1.03590
$$730$$ 0 0
$$731$$ −5.34335 −0.197631
$$732$$ 0 0
$$733$$ 26.5840 0.981903 0.490951 0.871187i $$-0.336649\pi$$
0.490951 + 0.871187i $$0.336649\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 29.3434 1.08088
$$738$$ 0 0
$$739$$ −8.14728 −0.299702 −0.149851 0.988709i $$-0.547879\pi$$
−0.149851 + 0.988709i $$0.547879\pi$$
$$740$$ 0 0
$$741$$ −0.192688 −0.00707855
$$742$$ 0 0
$$743$$ −35.8247 −1.31428 −0.657139 0.753769i $$-0.728234\pi$$
−0.657139 + 0.753769i $$0.728234\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −10.2827 −0.376223
$$748$$ 0 0
$$749$$ 0.0572907 0.00209336
$$750$$ 0 0
$$751$$ 14.8994 0.543686 0.271843 0.962342i $$-0.412367\pi$$
0.271843 + 0.962342i $$0.412367\pi$$
$$752$$ 0 0
$$753$$ −3.42740 −0.124901
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −47.7920 −1.73703 −0.868514 0.495664i $$-0.834925\pi$$
−0.868514 + 0.495664i $$0.834925\pi$$
$$758$$ 0 0
$$759$$ −23.5047 −0.853165
$$760$$ 0 0
$$761$$ −38.9053 −1.41032 −0.705158 0.709050i $$-0.749124\pi$$
−0.705158 + 0.709050i $$0.749124\pi$$
$$762$$ 0 0
$$763$$ 8.65533 0.313344
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −0.797984 −0.0288135
$$768$$ 0 0
$$769$$ 8.74663 0.315412 0.157706 0.987486i $$-0.449590\pi$$
0.157706 + 0.987486i $$0.449590\pi$$
$$770$$ 0 0
$$771$$ 33.8060 1.21749
$$772$$ 0 0
$$773$$ −23.4707 −0.844181 −0.422090 0.906554i $$-0.638703\pi$$
−0.422090 + 0.906554i $$0.638703\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2.84124 0.101929
$$778$$ 0 0
$$779$$ 4.23132 0.151603
$$780$$ 0 0
$$781$$ −31.3947 −1.12339
$$782$$ 0 0
$$783$$ 40.3727 1.44280
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.26063 −0.0449364 −0.0224682 0.999748i $$-0.507152\pi$$
−0.0224682 + 0.999748i $$0.507152\pi$$
$$788$$ 0 0
$$789$$ −30.7640 −1.09523
$$790$$ 0 0
$$791$$ −7.33657 −0.260859
$$792$$ 0 0
$$793$$ 0.918026 0.0326000
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −38.6481 −1.36898 −0.684492 0.729020i $$-0.739976\pi$$
−0.684492 + 0.729020i $$0.739976\pi$$
$$798$$ 0 0
$$799$$ 22.0187 0.778964
$$800$$ 0 0
$$801$$ 7.73599 0.273338
$$802$$ 0 0
$$803$$ 12.9580 0.457277
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 36.2173 1.27491
$$808$$ 0 0
$$809$$ 51.6506 1.81594 0.907970 0.419036i $$-0.137632\pi$$
0.907970 + 0.419036i $$0.137632\pi$$
$$810$$ 0 0
$$811$$ −8.19269 −0.287684 −0.143842 0.989601i $$-0.545946\pi$$
−0.143842 + 0.989601i $$0.545946\pi$$
$$812$$ 0 0
$$813$$ −33.9987 −1.19239
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.49534 0.0873007
$$818$$ 0 0
$$819$$ −0.102703 −0.00358873
$$820$$ 0 0
$$821$$ −49.9600 −1.74362 −0.871809 0.489846i $$-0.837053\pi$$
−0.871809 + 0.489846i $$0.837053\pi$$
$$822$$ 0 0
$$823$$ 16.8421 0.587078 0.293539 0.955947i $$-0.405167\pi$$
0.293539 + 0.955947i $$0.405167\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 41.7487 1.45174 0.725872 0.687829i $$-0.241436\pi$$
0.725872 + 0.687829i $$0.241436\pi$$
$$828$$ 0 0
$$829$$ −5.98002 −0.207695 −0.103847 0.994593i $$-0.533115\pi$$
−0.103847 + 0.994593i $$0.533115\pi$$
$$830$$ 0 0
$$831$$ −25.2852 −0.877135
$$832$$ 0 0
$$833$$ −14.1214 −0.489276
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 43.9673 1.51973
$$838$$ 0 0
$$839$$ −32.0373 −1.10605 −0.553025 0.833164i $$-0.686527\pi$$
−0.553025 + 0.833164i $$0.686527\pi$$
$$840$$ 0 0
$$841$$ 22.1320 0.763173
$$842$$ 0 0
$$843$$ −33.8060 −1.16434
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −0.816641 −0.0280601
$$848$$ 0 0
$$849$$ 18.5067 0.635150
$$850$$ 0 0
$$851$$ −16.1027 −0.551994
$$852$$ 0 0
$$853$$ −40.8480 −1.39861 −0.699305 0.714824i $$-0.746507\pi$$
−0.699305 + 0.714824i $$0.746507\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.8667 0.986067 0.493033 0.870010i $$-0.335888\pi$$
0.493033 + 0.870010i $$0.335888\pi$$
$$858$$ 0 0
$$859$$ 11.5047 0.392534 0.196267 0.980550i $$-0.437118\pi$$
0.196267 + 0.980550i $$0.437118\pi$$
$$860$$ 0 0
$$861$$ −3.67276 −0.125167
$$862$$ 0 0
$$863$$ 31.6074 1.07593 0.537964 0.842968i $$-0.319194\pi$$
0.537964 + 0.842968i $$0.319194\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 16.9253 0.574813
$$868$$ 0 0
$$869$$ −14.6494 −0.496947
$$870$$ 0 0
$$871$$ 1.18336 0.0400966
$$872$$ 0 0
$$873$$ −16.6240 −0.562636
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 12.0641 0.407375 0.203687 0.979036i $$-0.434707\pi$$
0.203687 + 0.979036i $$0.434707\pi$$
$$878$$ 0 0
$$879$$ 21.2861 0.717961
$$880$$ 0 0
$$881$$ −22.8480 −0.769769 −0.384885 0.922965i $$-0.625759\pi$$
−0.384885 + 0.922965i $$0.625759\pi$$
$$882$$ 0 0
$$883$$ 34.1773 1.15016 0.575079 0.818098i $$-0.304971\pi$$
0.575079 + 0.818098i $$0.304971\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −26.6167 −0.893701 −0.446851 0.894609i $$-0.647454\pi$$
−0.446851 + 0.894609i $$0.647454\pi$$
$$888$$ 0 0
$$889$$ −2.09592 −0.0702950
$$890$$ 0 0
$$891$$ 14.9766 0.501736
$$892$$ 0 0
$$893$$ −10.2827 −0.344097
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −0.947896 −0.0316493
$$898$$ 0 0
$$899$$ 55.6846 1.85719
$$900$$ 0 0
$$901$$ 17.4333 0.580789
$$902$$ 0 0
$$903$$ −2.16593 −0.0720777
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −13.1434 −0.436420 −0.218210 0.975902i $$-0.570022\pi$$
−0.218210 + 0.975902i $$0.570022\pi$$
$$908$$ 0 0
$$909$$ 18.9652 0.629037
$$910$$ 0 0
$$911$$ 32.9253 1.09086 0.545432 0.838155i $$-0.316366\pi$$
0.545432 + 0.838155i $$0.316366\pi$$
$$912$$ 0 0
$$913$$ −31.5747 −1.04497
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 11.4720 0.378838
$$918$$ 0 0
$$919$$ 24.1273 0.795886 0.397943 0.917410i $$-0.369724\pi$$
0.397943 + 0.917410i $$0.369724\pi$$
$$920$$ 0 0
$$921$$ 46.3786 1.52823
$$922$$ 0 0
$$923$$ −1.26609 −0.0416737
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 10.3413 0.339652
$$928$$ 0 0
$$929$$ −15.9359 −0.522841 −0.261420 0.965225i $$-0.584191\pi$$
−0.261420 + 0.965225i $$0.584191\pi$$
$$930$$ 0 0
$$931$$ 6.59465 0.216131
$$932$$ 0 0
$$933$$ 12.1854 0.398933
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.5547 0.769498 0.384749 0.923021i $$-0.374288\pi$$
0.384749 + 0.923021i $$0.374288\pi$$
$$938$$ 0 0
$$939$$ 25.1180 0.819694
$$940$$ 0 0
$$941$$ −25.6974 −0.837710 −0.418855 0.908053i $$-0.637568\pi$$
−0.418855 + 0.908053i $$0.637568\pi$$
$$942$$ 0 0
$$943$$ 20.8153 0.677840
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 16.3013 0.529722 0.264861 0.964287i $$-0.414674\pi$$
0.264861 + 0.964287i $$0.414674\pi$$
$$948$$ 0 0
$$949$$ 0.522569 0.0169633
$$950$$ 0 0
$$951$$ −45.7461 −1.48342
$$952$$ 0 0
$$953$$ 10.2754 0.332853 0.166427 0.986054i $$-0.446777\pi$$
0.166427 + 0.986054i $$0.446777\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 34.1659 1.10443
$$958$$ 0 0
$$959$$ 9.18336 0.296546
$$960$$ 0 0
$$961$$ 29.6426 0.956213
$$962$$ 0 0
$$963$$ 0.102703 0.00330955
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 53.2920 1.71376 0.856878 0.515520i $$-0.172401\pi$$
0.856878 + 0.515520i $$0.172401\pi$$
$$968$$ 0 0
$$969$$ 2.91934 0.0937828
$$970$$ 0 0
$$971$$ −2.54669 −0.0817271 −0.0408635 0.999165i $$-0.513011\pi$$
−0.0408635 + 0.999165i $$0.513011\pi$$
$$972$$ 0 0
$$973$$ 9.85057 0.315795
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −49.2966 −1.57714 −0.788569 0.614946i $$-0.789178\pi$$
−0.788569 + 0.614946i $$0.789178\pi$$
$$978$$ 0 0
$$979$$ 23.7546 0.759202
$$980$$ 0 0
$$981$$ 15.5161 0.495390
$$982$$ 0 0
$$983$$ 36.7453 1.17199 0.585997 0.810313i $$-0.300703\pi$$
0.585997 + 0.810313i $$0.300703\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 8.92528 0.284095
$$988$$ 0 0
$$989$$ 12.2754 0.390335
$$990$$ 0 0
$$991$$ 35.5674 1.12984 0.564918 0.825147i $$-0.308908\pi$$
0.564918 + 0.825147i $$0.308908\pi$$
$$992$$ 0 0
$$993$$ 3.06926 0.0973999
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −48.4299 −1.53379 −0.766896 0.641771i $$-0.778200\pi$$
−0.766896 + 0.641771i $$0.778200\pi$$
$$998$$ 0 0
$$999$$ 18.4813 0.584722
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 7600.2.a.bi.1.2 3
4.3 odd 2 950.2.a.n.1.2 3
5.2 odd 4 1520.2.d.j.609.4 6
5.3 odd 4 1520.2.d.j.609.3 6
5.4 even 2 7600.2.a.cd.1.2 3
12.11 even 2 8550.2.a.ck.1.2 3
20.3 even 4 190.2.b.b.39.2 6
20.7 even 4 190.2.b.b.39.5 yes 6
20.19 odd 2 950.2.a.i.1.2 3
60.23 odd 4 1710.2.d.d.1369.5 6
60.47 odd 4 1710.2.d.d.1369.2 6
60.59 even 2 8550.2.a.cl.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.2 6 20.3 even 4
190.2.b.b.39.5 yes 6 20.7 even 4
950.2.a.i.1.2 3 20.19 odd 2
950.2.a.n.1.2 3 4.3 odd 2
1520.2.d.j.609.3 6 5.3 odd 4
1520.2.d.j.609.4 6 5.2 odd 4
1710.2.d.d.1369.2 6 60.47 odd 4
1710.2.d.d.1369.5 6 60.23 odd 4
7600.2.a.bi.1.2 3 1.1 even 1 trivial
7600.2.a.cd.1.2 3 5.4 even 2
8550.2.a.ck.1.2 3 12.11 even 2
8550.2.a.cl.1.2 3 60.59 even 2