# Properties

 Label 2300.2.a.n.1.6 Level $2300$ Weight $2$ Character 2300.1 Self dual yes Analytic conductor $18.366$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.3655924649$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.143376304.1 Defining polynomial: $$x^{6} - 12x^{4} + 22x^{2} - 6x - 1$$ x^6 - 12*x^4 + 22*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$1.26443$$ of defining polynomial Character $$\chi$$ $$=$$ 2300.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.40050 q^{3} -4.41307 q^{7} +2.76241 q^{9} +O(q^{10})$$ $$q+2.40050 q^{3} -4.41307 q^{7} +2.76241 q^{9} +2.29289 q^{11} -6.92936 q^{13} -1.51387 q^{17} +2.89920 q^{19} -10.5936 q^{21} -1.00000 q^{23} -0.570328 q^{27} -7.68764 q^{29} +3.85746 q^{31} +5.50408 q^{33} -8.62830 q^{37} -16.6340 q^{39} -6.44324 q^{41} +3.48497 q^{43} +6.19747 q^{47} +12.4752 q^{49} -3.63405 q^{51} -2.17710 q^{53} +6.95953 q^{57} -11.7637 q^{59} -5.11443 q^{61} -12.1907 q^{63} -9.94597 q^{67} -2.40050 q^{69} +3.41407 q^{71} +8.95307 q^{73} -10.1187 q^{77} -1.92694 q^{79} -9.65631 q^{81} -8.04131 q^{83} -18.4542 q^{87} -1.09273 q^{89} +30.5798 q^{91} +9.25985 q^{93} +16.9208 q^{97} +6.33390 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} + 2 q^{11} - 8 q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} + 9 q^{31} + 10 q^{33} - 21 q^{37} - 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49} - 12 q^{51} + q^{53} - 12 q^{57} - 11 q^{59} - 4 q^{61} - 19 q^{63} - 25 q^{67} + 4 q^{69} - 17 q^{71} + 14 q^{73} - 20 q^{77} + 10 q^{79} + 14 q^{81} - 21 q^{83} - 64 q^{87} - 24 q^{89} - 4 q^{91} - 4 q^{97} - 16 q^{99}+O(q^{100})$$ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 + 2 * q^11 - 8 * q^13 - 5 * q^17 + 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 + 9 * q^31 + 10 * q^33 - 21 * q^37 - 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 - 12 * q^51 + q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 - 17 * q^71 + 14 * q^73 - 20 * q^77 + 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 - 4 * q^91 - 4 * q^97 - 16 * q^99

## 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
$$3$$ 2.40050 1.38593 0.692965 0.720971i $$-0.256304\pi$$
0.692965 + 0.720971i $$0.256304\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.41307 −1.66798 −0.833992 0.551777i $$-0.813950\pi$$
−0.833992 + 0.551777i $$0.813950\pi$$
$$8$$ 0 0
$$9$$ 2.76241 0.920804
$$10$$ 0 0
$$11$$ 2.29289 0.691332 0.345666 0.938358i $$-0.387653\pi$$
0.345666 + 0.938358i $$0.387653\pi$$
$$12$$ 0 0
$$13$$ −6.92936 −1.92186 −0.960930 0.276792i $$-0.910729\pi$$
−0.960930 + 0.276792i $$0.910729\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.51387 −0.367168 −0.183584 0.983004i $$-0.558770\pi$$
−0.183584 + 0.983004i $$0.558770\pi$$
$$18$$ 0 0
$$19$$ 2.89920 0.665121 0.332561 0.943082i $$-0.392087\pi$$
0.332561 + 0.943082i $$0.392087\pi$$
$$20$$ 0 0
$$21$$ −10.5936 −2.31171
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −0.570328 −0.109760
$$28$$ 0 0
$$29$$ −7.68764 −1.42756 −0.713779 0.700371i $$-0.753018\pi$$
−0.713779 + 0.700371i $$0.753018\pi$$
$$30$$ 0 0
$$31$$ 3.85746 0.692821 0.346410 0.938083i $$-0.387401\pi$$
0.346410 + 0.938083i $$0.387401\pi$$
$$32$$ 0 0
$$33$$ 5.50408 0.958138
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.62830 −1.41848 −0.709242 0.704965i $$-0.750963\pi$$
−0.709242 + 0.704965i $$0.750963\pi$$
$$38$$ 0 0
$$39$$ −16.6340 −2.66356
$$40$$ 0 0
$$41$$ −6.44324 −1.00626 −0.503132 0.864209i $$-0.667819\pi$$
−0.503132 + 0.864209i $$0.667819\pi$$
$$42$$ 0 0
$$43$$ 3.48497 0.531453 0.265727 0.964048i $$-0.414388\pi$$
0.265727 + 0.964048i $$0.414388\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.19747 0.903994 0.451997 0.892019i $$-0.350712\pi$$
0.451997 + 0.892019i $$0.350712\pi$$
$$48$$ 0 0
$$49$$ 12.4752 1.78217
$$50$$ 0 0
$$51$$ −3.63405 −0.508869
$$52$$ 0 0
$$53$$ −2.17710 −0.299047 −0.149524 0.988758i $$-0.547774\pi$$
−0.149524 + 0.988758i $$0.547774\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.95953 0.921812
$$58$$ 0 0
$$59$$ −11.7637 −1.53150 −0.765750 0.643138i $$-0.777632\pi$$
−0.765750 + 0.643138i $$0.777632\pi$$
$$60$$ 0 0
$$61$$ −5.11443 −0.654835 −0.327418 0.944880i $$-0.606178\pi$$
−0.327418 + 0.944880i $$0.606178\pi$$
$$62$$ 0 0
$$63$$ −12.1907 −1.53589
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.94597 −1.21509 −0.607547 0.794284i $$-0.707846\pi$$
−0.607547 + 0.794284i $$0.707846\pi$$
$$68$$ 0 0
$$69$$ −2.40050 −0.288987
$$70$$ 0 0
$$71$$ 3.41407 0.405175 0.202588 0.979264i $$-0.435065\pi$$
0.202588 + 0.979264i $$0.435065\pi$$
$$72$$ 0 0
$$73$$ 8.95307 1.04788 0.523939 0.851756i $$-0.324462\pi$$
0.523939 + 0.851756i $$0.324462\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −10.1187 −1.15313
$$78$$ 0 0
$$79$$ −1.92694 −0.216798 −0.108399 0.994107i $$-0.534572\pi$$
−0.108399 + 0.994107i $$0.534572\pi$$
$$80$$ 0 0
$$81$$ −9.65631 −1.07292
$$82$$ 0 0
$$83$$ −8.04131 −0.882648 −0.441324 0.897348i $$-0.645491\pi$$
−0.441324 + 0.897348i $$0.645491\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −18.4542 −1.97850
$$88$$ 0 0
$$89$$ −1.09273 −0.115829 −0.0579147 0.998322i $$-0.518445\pi$$
−0.0579147 + 0.998322i $$0.518445\pi$$
$$90$$ 0 0
$$91$$ 30.5798 3.20563
$$92$$ 0 0
$$93$$ 9.25985 0.960201
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.9208 1.71805 0.859026 0.511933i $$-0.171070\pi$$
0.859026 + 0.511933i $$0.171070\pi$$
$$98$$ 0 0
$$99$$ 6.33390 0.636581
$$100$$ 0 0
$$101$$ 12.9497 1.28855 0.644274 0.764795i $$-0.277160\pi$$
0.644274 + 0.764795i $$0.277160\pi$$
$$102$$ 0 0
$$103$$ −10.9754 −1.08144 −0.540721 0.841202i $$-0.681848\pi$$
−0.540721 + 0.841202i $$0.681848\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −19.8821 −1.92208 −0.961040 0.276411i $$-0.910855\pi$$
−0.961040 + 0.276411i $$0.910855\pi$$
$$108$$ 0 0
$$109$$ 0.427249 0.0409231 0.0204615 0.999791i $$-0.493486\pi$$
0.0204615 + 0.999791i $$0.493486\pi$$
$$110$$ 0 0
$$111$$ −20.7123 −1.96592
$$112$$ 0 0
$$113$$ 5.38533 0.506609 0.253304 0.967387i $$-0.418483\pi$$
0.253304 + 0.967387i $$0.418483\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −19.1418 −1.76966
$$118$$ 0 0
$$119$$ 6.68082 0.612430
$$120$$ 0 0
$$121$$ −5.74267 −0.522061
$$122$$ 0 0
$$123$$ −15.4670 −1.39461
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.16014 −0.546624 −0.273312 0.961925i $$-0.588119\pi$$
−0.273312 + 0.961925i $$0.588119\pi$$
$$128$$ 0 0
$$129$$ 8.36569 0.736558
$$130$$ 0 0
$$131$$ 14.6325 1.27845 0.639225 0.769020i $$-0.279255\pi$$
0.639225 + 0.769020i $$0.279255\pi$$
$$132$$ 0 0
$$133$$ −12.7944 −1.10941
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −0.972256 −0.0830654 −0.0415327 0.999137i $$-0.513224\pi$$
−0.0415327 + 0.999137i $$0.513224\pi$$
$$138$$ 0 0
$$139$$ 7.74312 0.656763 0.328382 0.944545i $$-0.393497\pi$$
0.328382 + 0.944545i $$0.393497\pi$$
$$140$$ 0 0
$$141$$ 14.8770 1.25287
$$142$$ 0 0
$$143$$ −15.8883 −1.32864
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 29.9467 2.46996
$$148$$ 0 0
$$149$$ −17.6823 −1.44859 −0.724293 0.689492i $$-0.757834\pi$$
−0.724293 + 0.689492i $$0.757834\pi$$
$$150$$ 0 0
$$151$$ −2.01286 −0.163804 −0.0819022 0.996640i $$-0.526100\pi$$
−0.0819022 + 0.996640i $$0.526100\pi$$
$$152$$ 0 0
$$153$$ −4.18194 −0.338090
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.82703 0.465048 0.232524 0.972591i $$-0.425302\pi$$
0.232524 + 0.972591i $$0.425302\pi$$
$$158$$ 0 0
$$159$$ −5.22612 −0.414459
$$160$$ 0 0
$$161$$ 4.41307 0.347799
$$162$$ 0 0
$$163$$ 6.75147 0.528816 0.264408 0.964411i $$-0.414823\pi$$
0.264408 + 0.964411i $$0.414823\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 23.4541 1.81493 0.907466 0.420125i $$-0.138014\pi$$
0.907466 + 0.420125i $$0.138014\pi$$
$$168$$ 0 0
$$169$$ 35.0161 2.69355
$$170$$ 0 0
$$171$$ 8.00878 0.612447
$$172$$ 0 0
$$173$$ 8.28850 0.630163 0.315081 0.949065i $$-0.397968\pi$$
0.315081 + 0.949065i $$0.397968\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −28.2387 −2.12255
$$178$$ 0 0
$$179$$ 1.04002 0.0777345 0.0388673 0.999244i $$-0.487625\pi$$
0.0388673 + 0.999244i $$0.487625\pi$$
$$180$$ 0 0
$$181$$ 2.71850 0.202065 0.101032 0.994883i $$-0.467785\pi$$
0.101032 + 0.994883i $$0.467785\pi$$
$$182$$ 0 0
$$183$$ −12.2772 −0.907557
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.47114 −0.253835
$$188$$ 0 0
$$189$$ 2.51690 0.183077
$$190$$ 0 0
$$191$$ 8.04858 0.582375 0.291187 0.956666i $$-0.405950\pi$$
0.291187 + 0.956666i $$0.405950\pi$$
$$192$$ 0 0
$$193$$ 16.0276 1.15370 0.576848 0.816852i $$-0.304283\pi$$
0.576848 + 0.816852i $$0.304283\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.06316 −0.289488 −0.144744 0.989469i $$-0.546236\pi$$
−0.144744 + 0.989469i $$0.546236\pi$$
$$198$$ 0 0
$$199$$ 18.7042 1.32590 0.662952 0.748662i $$-0.269303\pi$$
0.662952 + 0.748662i $$0.269303\pi$$
$$200$$ 0 0
$$201$$ −23.8753 −1.68404
$$202$$ 0 0
$$203$$ 33.9261 2.38114
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.76241 −0.192001
$$208$$ 0 0
$$209$$ 6.64753 0.459819
$$210$$ 0 0
$$211$$ 7.87839 0.542371 0.271185 0.962527i $$-0.412584\pi$$
0.271185 + 0.962527i $$0.412584\pi$$
$$212$$ 0 0
$$213$$ 8.19548 0.561545
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −17.0232 −1.15561
$$218$$ 0 0
$$219$$ 21.4919 1.45229
$$220$$ 0 0
$$221$$ 10.4902 0.705645
$$222$$ 0 0
$$223$$ 2.29263 0.153526 0.0767628 0.997049i $$-0.475542\pi$$
0.0767628 + 0.997049i $$0.475542\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.8325 −0.851725 −0.425862 0.904788i $$-0.640029\pi$$
−0.425862 + 0.904788i $$0.640029\pi$$
$$228$$ 0 0
$$229$$ −16.2528 −1.07401 −0.537007 0.843578i $$-0.680445\pi$$
−0.537007 + 0.843578i $$0.680445\pi$$
$$230$$ 0 0
$$231$$ −24.2899 −1.59816
$$232$$ 0 0
$$233$$ −21.5410 −1.41120 −0.705598 0.708613i $$-0.749321\pi$$
−0.705598 + 0.708613i $$0.749321\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.62563 −0.300467
$$238$$ 0 0
$$239$$ −11.0622 −0.715555 −0.357778 0.933807i $$-0.616465\pi$$
−0.357778 + 0.933807i $$0.616465\pi$$
$$240$$ 0 0
$$241$$ 18.2164 1.17342 0.586710 0.809797i $$-0.300423\pi$$
0.586710 + 0.809797i $$0.300423\pi$$
$$242$$ 0 0
$$243$$ −21.4690 −1.37724
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.0896 −1.27827
$$248$$ 0 0
$$249$$ −19.3032 −1.22329
$$250$$ 0 0
$$251$$ −10.0182 −0.632345 −0.316172 0.948702i $$-0.602398\pi$$
−0.316172 + 0.948702i $$0.602398\pi$$
$$252$$ 0 0
$$253$$ −2.29289 −0.144153
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.3021 0.892141 0.446070 0.894998i $$-0.352823\pi$$
0.446070 + 0.894998i $$0.352823\pi$$
$$258$$ 0 0
$$259$$ 38.0773 2.36601
$$260$$ 0 0
$$261$$ −21.2364 −1.31450
$$262$$ 0 0
$$263$$ 9.44361 0.582318 0.291159 0.956675i $$-0.405959\pi$$
0.291159 + 0.956675i $$0.405959\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.62311 −0.160531
$$268$$ 0 0
$$269$$ 23.7630 1.44886 0.724428 0.689350i $$-0.242104\pi$$
0.724428 + 0.689350i $$0.242104\pi$$
$$270$$ 0 0
$$271$$ −17.5939 −1.06875 −0.534375 0.845247i $$-0.679453\pi$$
−0.534375 + 0.845247i $$0.679453\pi$$
$$272$$ 0 0
$$273$$ 73.4068 4.44278
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 29.6582 1.78199 0.890995 0.454014i $$-0.150008\pi$$
0.890995 + 0.454014i $$0.150008\pi$$
$$278$$ 0 0
$$279$$ 10.6559 0.637952
$$280$$ 0 0
$$281$$ −8.25484 −0.492442 −0.246221 0.969214i $$-0.579189\pi$$
−0.246221 + 0.969214i $$0.579189\pi$$
$$282$$ 0 0
$$283$$ −9.21317 −0.547666 −0.273833 0.961777i $$-0.588292\pi$$
−0.273833 + 0.961777i $$0.588292\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 28.4344 1.67843
$$288$$ 0 0
$$289$$ −14.7082 −0.865188
$$290$$ 0 0
$$291$$ 40.6185 2.38110
$$292$$ 0 0
$$293$$ −11.1138 −0.649277 −0.324639 0.945838i $$-0.605243\pi$$
−0.324639 + 0.945838i $$0.605243\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1.30770 −0.0758804
$$298$$ 0 0
$$299$$ 6.92936 0.400735
$$300$$ 0 0
$$301$$ −15.3794 −0.886455
$$302$$ 0 0
$$303$$ 31.0859 1.78584
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.7800 −1.07183 −0.535916 0.844272i $$-0.680033\pi$$
−0.535916 + 0.844272i $$0.680033\pi$$
$$308$$ 0 0
$$309$$ −26.3465 −1.49880
$$310$$ 0 0
$$311$$ 2.51258 0.142475 0.0712377 0.997459i $$-0.477305\pi$$
0.0712377 + 0.997459i $$0.477305\pi$$
$$312$$ 0 0
$$313$$ −19.9278 −1.12638 −0.563192 0.826326i $$-0.690427\pi$$
−0.563192 + 0.826326i $$0.690427\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.8849 1.56617 0.783087 0.621912i $$-0.213644\pi$$
0.783087 + 0.621912i $$0.213644\pi$$
$$318$$ 0 0
$$319$$ −17.6269 −0.986916
$$320$$ 0 0
$$321$$ −47.7271 −2.66387
$$322$$ 0 0
$$323$$ −4.38901 −0.244211
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.02561 0.0567165
$$328$$ 0 0
$$329$$ −27.3499 −1.50785
$$330$$ 0 0
$$331$$ −28.5409 −1.56875 −0.784375 0.620287i $$-0.787016\pi$$
−0.784375 + 0.620287i $$0.787016\pi$$
$$332$$ 0 0
$$333$$ −23.8349 −1.30615
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3.71371 −0.202298 −0.101149 0.994871i $$-0.532252\pi$$
−0.101149 + 0.994871i $$0.532252\pi$$
$$338$$ 0 0
$$339$$ 12.9275 0.702125
$$340$$ 0 0
$$341$$ 8.84472 0.478969
$$342$$ 0 0
$$343$$ −24.1624 −1.30464
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −14.1459 −0.759393 −0.379696 0.925111i $$-0.623972\pi$$
−0.379696 + 0.925111i $$0.623972\pi$$
$$348$$ 0 0
$$349$$ −21.2802 −1.13910 −0.569550 0.821957i $$-0.692883\pi$$
−0.569550 + 0.821957i $$0.692883\pi$$
$$350$$ 0 0
$$351$$ 3.95201 0.210943
$$352$$ 0 0
$$353$$ −24.4641 −1.30209 −0.651046 0.759038i $$-0.725669\pi$$
−0.651046 + 0.759038i $$0.725669\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 16.0373 0.848786
$$358$$ 0 0
$$359$$ −23.0252 −1.21522 −0.607612 0.794234i $$-0.707872\pi$$
−0.607612 + 0.794234i $$0.707872\pi$$
$$360$$ 0 0
$$361$$ −10.5947 −0.557613
$$362$$ 0 0
$$363$$ −13.7853 −0.723540
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −0.478057 −0.0249544 −0.0124772 0.999922i $$-0.503972\pi$$
−0.0124772 + 0.999922i $$0.503972\pi$$
$$368$$ 0 0
$$369$$ −17.7989 −0.926573
$$370$$ 0 0
$$371$$ 9.60767 0.498806
$$372$$ 0 0
$$373$$ −27.3508 −1.41617 −0.708085 0.706127i $$-0.750441\pi$$
−0.708085 + 0.706127i $$0.750441\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 53.2704 2.74357
$$378$$ 0 0
$$379$$ −7.35358 −0.377728 −0.188864 0.982003i $$-0.560481\pi$$
−0.188864 + 0.982003i $$0.560481\pi$$
$$380$$ 0 0
$$381$$ −14.7874 −0.757583
$$382$$ 0 0
$$383$$ 3.34108 0.170721 0.0853606 0.996350i $$-0.472796\pi$$
0.0853606 + 0.996350i $$0.472796\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 9.62693 0.489364
$$388$$ 0 0
$$389$$ 0.366568 0.0185858 0.00929288 0.999957i $$-0.497042\pi$$
0.00929288 + 0.999957i $$0.497042\pi$$
$$390$$ 0 0
$$391$$ 1.51387 0.0765598
$$392$$ 0 0
$$393$$ 35.1254 1.77184
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.3222 0.568245 0.284122 0.958788i $$-0.408298\pi$$
0.284122 + 0.958788i $$0.408298\pi$$
$$398$$ 0 0
$$399$$ −30.7129 −1.53757
$$400$$ 0 0
$$401$$ 37.1673 1.85605 0.928024 0.372520i $$-0.121506\pi$$
0.928024 + 0.372520i $$0.121506\pi$$
$$402$$ 0 0
$$403$$ −26.7298 −1.33150
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −19.7837 −0.980643
$$408$$ 0 0
$$409$$ −13.7745 −0.681107 −0.340553 0.940225i $$-0.610614\pi$$
−0.340553 + 0.940225i $$0.610614\pi$$
$$410$$ 0 0
$$411$$ −2.33390 −0.115123
$$412$$ 0 0
$$413$$ 51.9139 2.55452
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 18.5874 0.910228
$$418$$ 0 0
$$419$$ 7.50726 0.366753 0.183377 0.983043i $$-0.441297\pi$$
0.183377 + 0.983043i $$0.441297\pi$$
$$420$$ 0 0
$$421$$ −7.65685 −0.373172 −0.186586 0.982439i $$-0.559742\pi$$
−0.186586 + 0.982439i $$0.559742\pi$$
$$422$$ 0 0
$$423$$ 17.1200 0.832402
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 22.5703 1.09225
$$428$$ 0 0
$$429$$ −38.1398 −1.84141
$$430$$ 0 0
$$431$$ 6.22764 0.299975 0.149987 0.988688i $$-0.452077\pi$$
0.149987 + 0.988688i $$0.452077\pi$$
$$432$$ 0 0
$$433$$ −0.704087 −0.0338362 −0.0169181 0.999857i $$-0.505385\pi$$
−0.0169181 + 0.999857i $$0.505385\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2.89920 −0.138687
$$438$$ 0 0
$$439$$ −34.8570 −1.66363 −0.831816 0.555052i $$-0.812699\pi$$
−0.831816 + 0.555052i $$0.812699\pi$$
$$440$$ 0 0
$$441$$ 34.4616 1.64103
$$442$$ 0 0
$$443$$ 24.8165 1.17907 0.589533 0.807745i $$-0.299312\pi$$
0.589533 + 0.807745i $$0.299312\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −42.4463 −2.00764
$$448$$ 0 0
$$449$$ 6.77738 0.319844 0.159922 0.987130i $$-0.448876\pi$$
0.159922 + 0.987130i $$0.448876\pi$$
$$450$$ 0 0
$$451$$ −14.7736 −0.695662
$$452$$ 0 0
$$453$$ −4.83188 −0.227021
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.47169 −0.0688429 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$458$$ 0 0
$$459$$ 0.863404 0.0403003
$$460$$ 0 0
$$461$$ 25.8037 1.20180 0.600899 0.799325i $$-0.294809\pi$$
0.600899 + 0.799325i $$0.294809\pi$$
$$462$$ 0 0
$$463$$ 15.3469 0.713231 0.356616 0.934251i $$-0.383931\pi$$
0.356616 + 0.934251i $$0.383931\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10.2933 −0.476315 −0.238157 0.971227i $$-0.576543\pi$$
−0.238157 + 0.971227i $$0.576543\pi$$
$$468$$ 0 0
$$469$$ 43.8922 2.02676
$$470$$ 0 0
$$471$$ 13.9878 0.644524
$$472$$ 0 0
$$473$$ 7.99065 0.367410
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.01404 −0.275364
$$478$$ 0 0
$$479$$ 4.50453 0.205817 0.102909 0.994691i $$-0.467185\pi$$
0.102909 + 0.994691i $$0.467185\pi$$
$$480$$ 0 0
$$481$$ 59.7886 2.72613
$$482$$ 0 0
$$483$$ 10.5936 0.482025
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 27.2669 1.23558 0.617790 0.786343i $$-0.288028\pi$$
0.617790 + 0.786343i $$0.288028\pi$$
$$488$$ 0 0
$$489$$ 16.2069 0.732902
$$490$$ 0 0
$$491$$ 24.4866 1.10507 0.552533 0.833491i $$-0.313661\pi$$
0.552533 + 0.833491i $$0.313661\pi$$
$$492$$ 0 0
$$493$$ 11.6381 0.524154
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −15.0665 −0.675826
$$498$$ 0 0
$$499$$ −24.1892 −1.08286 −0.541429 0.840746i $$-0.682117\pi$$
−0.541429 + 0.840746i $$0.682117\pi$$
$$500$$ 0 0
$$501$$ 56.3016 2.51537
$$502$$ 0 0
$$503$$ −30.6230 −1.36541 −0.682706 0.730693i $$-0.739197\pi$$
−0.682706 + 0.730693i $$0.739197\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 84.0562 3.73307
$$508$$ 0 0
$$509$$ −2.56826 −0.113836 −0.0569181 0.998379i $$-0.518127\pi$$
−0.0569181 + 0.998379i $$0.518127\pi$$
$$510$$ 0 0
$$511$$ −39.5105 −1.74784
$$512$$ 0 0
$$513$$ −1.65349 −0.0730035
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.2101 0.624960
$$518$$ 0 0
$$519$$ 19.8966 0.873362
$$520$$ 0 0
$$521$$ −44.1355 −1.93361 −0.966807 0.255509i $$-0.917757\pi$$
−0.966807 + 0.255509i $$0.917757\pi$$
$$522$$ 0 0
$$523$$ −42.2884 −1.84914 −0.924572 0.381008i $$-0.875577\pi$$
−0.924572 + 0.381008i $$0.875577\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.83970 −0.254381
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −32.4961 −1.41021
$$532$$ 0 0
$$533$$ 44.6475 1.93390
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 2.49656 0.107735
$$538$$ 0 0
$$539$$ 28.6042 1.23207
$$540$$ 0 0
$$541$$ 10.2776 0.441871 0.220935 0.975288i $$-0.429089\pi$$
0.220935 + 0.975288i $$0.429089\pi$$
$$542$$ 0 0
$$543$$ 6.52577 0.280048
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3.87305 0.165600 0.0827998 0.996566i $$-0.473614\pi$$
0.0827998 + 0.996566i $$0.473614\pi$$
$$548$$ 0 0
$$549$$ −14.1282 −0.602975
$$550$$ 0 0
$$551$$ −22.2880 −0.949500
$$552$$ 0 0
$$553$$ 8.50373 0.361615
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.62364 0.407767 0.203883 0.978995i $$-0.434644\pi$$
0.203883 + 0.978995i $$0.434644\pi$$
$$558$$ 0 0
$$559$$ −24.1486 −1.02138
$$560$$ 0 0
$$561$$ −8.33248 −0.351797
$$562$$ 0 0
$$563$$ −3.35865 −0.141550 −0.0707751 0.997492i $$-0.522547\pi$$
−0.0707751 + 0.997492i $$0.522547\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 42.6140 1.78962
$$568$$ 0 0
$$569$$ −4.26939 −0.178982 −0.0894910 0.995988i $$-0.528524\pi$$
−0.0894910 + 0.995988i $$0.528524\pi$$
$$570$$ 0 0
$$571$$ 16.4567 0.688689 0.344345 0.938843i $$-0.388101\pi$$
0.344345 + 0.938843i $$0.388101\pi$$
$$572$$ 0 0
$$573$$ 19.3206 0.807131
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.98780 −0.0827530 −0.0413765 0.999144i $$-0.513174\pi$$
−0.0413765 + 0.999144i $$0.513174\pi$$
$$578$$ 0 0
$$579$$ 38.4744 1.59894
$$580$$ 0 0
$$581$$ 35.4869 1.47224
$$582$$ 0 0
$$583$$ −4.99184 −0.206741
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 28.9720 1.19580 0.597900 0.801570i $$-0.296002\pi$$
0.597900 + 0.801570i $$0.296002\pi$$
$$588$$ 0 0
$$589$$ 11.1835 0.460810
$$590$$ 0 0
$$591$$ −9.75363 −0.401211
$$592$$ 0 0
$$593$$ 19.4799 0.799944 0.399972 0.916527i $$-0.369020\pi$$
0.399972 + 0.916527i $$0.369020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 44.8994 1.83761
$$598$$ 0 0
$$599$$ 37.8442 1.54627 0.773136 0.634240i $$-0.218687\pi$$
0.773136 + 0.634240i $$0.218687\pi$$
$$600$$ 0 0
$$601$$ −20.2347 −0.825389 −0.412695 0.910869i $$-0.635412\pi$$
−0.412695 + 0.910869i $$0.635412\pi$$
$$602$$ 0 0
$$603$$ −27.4749 −1.11886
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −24.4075 −0.990670 −0.495335 0.868702i $$-0.664955\pi$$
−0.495335 + 0.868702i $$0.664955\pi$$
$$608$$ 0 0
$$609$$ 81.4396 3.30010
$$610$$ 0 0
$$611$$ −42.9445 −1.73735
$$612$$ 0 0
$$613$$ −10.6964 −0.432025 −0.216012 0.976391i $$-0.569305\pi$$
−0.216012 + 0.976391i $$0.569305\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.4738 1.10605 0.553026 0.833164i $$-0.313473\pi$$
0.553026 + 0.833164i $$0.313473\pi$$
$$618$$ 0 0
$$619$$ −0.389330 −0.0156485 −0.00782425 0.999969i $$-0.502491\pi$$
−0.00782425 + 0.999969i $$0.502491\pi$$
$$620$$ 0 0
$$621$$ 0.570328 0.0228865
$$622$$ 0 0
$$623$$ 4.82230 0.193201
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 15.9574 0.637278
$$628$$ 0 0
$$629$$ 13.0621 0.520822
$$630$$ 0 0
$$631$$ −11.9330 −0.475047 −0.237523 0.971382i $$-0.576336\pi$$
−0.237523 + 0.971382i $$0.576336\pi$$
$$632$$ 0 0
$$633$$ 18.9121 0.751689
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −86.4451 −3.42508
$$638$$ 0 0
$$639$$ 9.43106 0.373087
$$640$$ 0 0
$$641$$ 22.8525 0.902621 0.451311 0.892367i $$-0.350957\pi$$
0.451311 + 0.892367i $$0.350957\pi$$
$$642$$ 0 0
$$643$$ −34.9279 −1.37742 −0.688711 0.725036i $$-0.741823\pi$$
−0.688711 + 0.725036i $$0.741823\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −26.3738 −1.03686 −0.518430 0.855120i $$-0.673483\pi$$
−0.518430 + 0.855120i $$0.673483\pi$$
$$648$$ 0 0
$$649$$ −26.9728 −1.05877
$$650$$ 0 0
$$651$$ −40.8643 −1.60160
$$652$$ 0 0
$$653$$ 2.17022 0.0849274 0.0424637 0.999098i $$-0.486479\pi$$
0.0424637 + 0.999098i $$0.486479\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 24.7321 0.964891
$$658$$ 0 0
$$659$$ 19.7820 0.770596 0.385298 0.922792i $$-0.374099\pi$$
0.385298 + 0.922792i $$0.374099\pi$$
$$660$$ 0 0
$$661$$ 45.7505 1.77949 0.889743 0.456461i $$-0.150883\pi$$
0.889743 + 0.456461i $$0.150883\pi$$
$$662$$ 0 0
$$663$$ 25.1817 0.977976
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.68764 0.297666
$$668$$ 0 0
$$669$$ 5.50346 0.212776
$$670$$ 0 0
$$671$$ −11.7268 −0.452708
$$672$$ 0 0
$$673$$ −6.99940 −0.269807 −0.134904 0.990859i $$-0.543072\pi$$
−0.134904 + 0.990859i $$0.543072\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −26.7178 −1.02685 −0.513425 0.858134i $$-0.671624\pi$$
−0.513425 + 0.858134i $$0.671624\pi$$
$$678$$ 0 0
$$679$$ −74.6728 −2.86568
$$680$$ 0 0
$$681$$ −30.8045 −1.18043
$$682$$ 0 0
$$683$$ 19.8947 0.761248 0.380624 0.924730i $$-0.375709\pi$$
0.380624 + 0.924730i $$0.375709\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −39.0148 −1.48851
$$688$$ 0 0
$$689$$ 15.0859 0.574727
$$690$$ 0 0
$$691$$ 1.24438 0.0473385 0.0236693 0.999720i $$-0.492465\pi$$
0.0236693 + 0.999720i $$0.492465\pi$$
$$692$$ 0 0
$$693$$ −27.9520 −1.06181
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 9.75424 0.369468
$$698$$ 0 0
$$699$$ −51.7091 −1.95582
$$700$$ 0 0
$$701$$ −29.8454 −1.12724 −0.563622 0.826033i $$-0.690592\pi$$
−0.563622 + 0.826033i $$0.690592\pi$$
$$702$$ 0 0
$$703$$ −25.0151 −0.943464
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −57.1481 −2.14928
$$708$$ 0 0
$$709$$ 25.2255 0.947362 0.473681 0.880697i $$-0.342925\pi$$
0.473681 + 0.880697i $$0.342925\pi$$
$$710$$ 0 0
$$711$$ −5.32301 −0.199628
$$712$$ 0 0
$$713$$ −3.85746 −0.144463
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −26.5549 −0.991710
$$718$$ 0 0
$$719$$ −23.7787 −0.886797 −0.443398 0.896325i $$-0.646227\pi$$
−0.443398 + 0.896325i $$0.646227\pi$$
$$720$$ 0 0
$$721$$ 48.4353 1.80383
$$722$$ 0 0
$$723$$ 43.7285 1.62628
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −36.5849 −1.35686 −0.678429 0.734666i $$-0.737339\pi$$
−0.678429 + 0.734666i $$0.737339\pi$$
$$728$$ 0 0
$$729$$ −22.5675 −0.835833
$$730$$ 0 0
$$731$$ −5.27580 −0.195133
$$732$$ 0 0
$$733$$ 21.7593 0.803698 0.401849 0.915706i $$-0.368368\pi$$
0.401849 + 0.915706i $$0.368368\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −22.8050 −0.840032
$$738$$ 0 0
$$739$$ 2.72303 0.100168 0.0500842 0.998745i $$-0.484051\pi$$
0.0500842 + 0.998745i $$0.484051\pi$$
$$740$$ 0 0
$$741$$ −48.2251 −1.77159
$$742$$ 0 0
$$743$$ 32.0161 1.17456 0.587278 0.809385i $$-0.300200\pi$$
0.587278 + 0.809385i $$0.300200\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −22.2134 −0.812746
$$748$$ 0 0
$$749$$ 87.7413 3.20600
$$750$$ 0 0
$$751$$ −36.0949 −1.31712 −0.658561 0.752527i $$-0.728835\pi$$
−0.658561 + 0.752527i $$0.728835\pi$$
$$752$$ 0 0
$$753$$ −24.0488 −0.876386
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 46.8188 1.70166 0.850830 0.525442i $$-0.176100\pi$$
0.850830 + 0.525442i $$0.176100\pi$$
$$758$$ 0 0
$$759$$ −5.50408 −0.199786
$$760$$ 0 0
$$761$$ −28.8182 −1.04466 −0.522330 0.852744i $$-0.674937\pi$$
−0.522330 + 0.852744i $$0.674937\pi$$
$$762$$ 0 0
$$763$$ −1.88548 −0.0682590
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 81.5148 2.94333
$$768$$ 0 0
$$769$$ 51.6243 1.86162 0.930811 0.365501i $$-0.119102\pi$$
0.930811 + 0.365501i $$0.119102\pi$$
$$770$$ 0 0
$$771$$ 34.3322 1.23645
$$772$$ 0 0
$$773$$ −25.9610 −0.933753 −0.466877 0.884322i $$-0.654621\pi$$
−0.466877 + 0.884322i $$0.654621\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 91.4046 3.27912
$$778$$ 0 0
$$779$$ −18.6802 −0.669288
$$780$$ 0 0
$$781$$ 7.82807 0.280110
$$782$$ 0 0
$$783$$ 4.38448 0.156688
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −53.2258 −1.89730 −0.948648 0.316333i $$-0.897548\pi$$
−0.948648 + 0.316333i $$0.897548\pi$$
$$788$$ 0 0
$$789$$ 22.6694 0.807053
$$790$$ 0 0
$$791$$ −23.7658 −0.845015
$$792$$ 0 0
$$793$$ 35.4397 1.25850
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −31.1874 −1.10471 −0.552357 0.833608i $$-0.686271\pi$$
−0.552357 + 0.833608i $$0.686271\pi$$
$$798$$ 0 0
$$799$$ −9.38218 −0.331918
$$800$$ 0 0
$$801$$ −3.01858 −0.106656
$$802$$ 0 0
$$803$$ 20.5284 0.724431
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 57.0432 2.00802
$$808$$ 0 0
$$809$$ −1.60251 −0.0563414 −0.0281707 0.999603i $$-0.508968\pi$$
−0.0281707 + 0.999603i $$0.508968\pi$$
$$810$$ 0 0
$$811$$ −36.9545 −1.29765 −0.648824 0.760938i $$-0.724739\pi$$
−0.648824 + 0.760938i $$0.724739\pi$$
$$812$$ 0 0
$$813$$ −42.2341 −1.48121
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10.1036 0.353481
$$818$$ 0 0
$$819$$ 84.4739 2.95176
$$820$$ 0 0
$$821$$ −16.5583 −0.577890 −0.288945 0.957346i $$-0.593304\pi$$
−0.288945 + 0.957346i $$0.593304\pi$$
$$822$$ 0 0
$$823$$ 7.63201 0.266035 0.133018 0.991114i $$-0.457533\pi$$
0.133018 + 0.991114i $$0.457533\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 19.1969 0.667543 0.333772 0.942654i $$-0.391679\pi$$
0.333772 + 0.942654i $$0.391679\pi$$
$$828$$ 0 0
$$829$$ −6.32413 −0.219646 −0.109823 0.993951i $$-0.535028\pi$$
−0.109823 + 0.993951i $$0.535028\pi$$
$$830$$ 0 0
$$831$$ 71.1946 2.46971
$$832$$ 0 0
$$833$$ −18.8858 −0.654355
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −2.20002 −0.0760438
$$838$$ 0 0
$$839$$ 23.1158 0.798046 0.399023 0.916941i $$-0.369349\pi$$
0.399023 + 0.916941i $$0.369349\pi$$
$$840$$ 0 0
$$841$$ 30.0997 1.03792
$$842$$ 0 0
$$843$$ −19.8158 −0.682491
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.3428 0.870789
$$848$$ 0 0
$$849$$ −22.1163 −0.759028
$$850$$ 0 0
$$851$$ 8.62830 0.295774
$$852$$ 0 0
$$853$$ 37.7049 1.29099 0.645496 0.763764i $$-0.276651\pi$$
0.645496 + 0.763764i $$0.276651\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −19.7133 −0.673395 −0.336697 0.941613i $$-0.609310\pi$$
−0.336697 + 0.941613i $$0.609310\pi$$
$$858$$ 0 0
$$859$$ 2.00609 0.0684471 0.0342235 0.999414i $$-0.489104\pi$$
0.0342235 + 0.999414i $$0.489104\pi$$
$$860$$ 0 0
$$861$$ 68.2570 2.32619
$$862$$ 0 0
$$863$$ −16.8127 −0.572312 −0.286156 0.958183i $$-0.592377\pi$$
−0.286156 + 0.958183i $$0.592377\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −35.3071 −1.19909
$$868$$ 0 0
$$869$$ −4.41826 −0.149879
$$870$$ 0 0
$$871$$ 68.9192 2.33524
$$872$$ 0 0
$$873$$ 46.7424 1.58199
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14.8922 −0.502874 −0.251437 0.967874i $$-0.580903\pi$$
−0.251437 + 0.967874i $$0.580903\pi$$
$$878$$ 0 0
$$879$$ −26.6788 −0.899853
$$880$$ 0 0
$$881$$ −16.5218 −0.556632 −0.278316 0.960490i $$-0.589776\pi$$
−0.278316 + 0.960490i $$0.589776\pi$$
$$882$$ 0 0
$$883$$ −38.7254 −1.30321 −0.651607 0.758557i $$-0.725905\pi$$
−0.651607 + 0.758557i $$0.725905\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −1.16364 −0.0390711 −0.0195355 0.999809i $$-0.506219\pi$$
−0.0195355 + 0.999809i $$0.506219\pi$$
$$888$$ 0 0
$$889$$ 27.1851 0.911760
$$890$$ 0 0
$$891$$ −22.1408 −0.741746
$$892$$ 0 0
$$893$$ 17.9677 0.601266
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 16.6340 0.555392
$$898$$ 0 0
$$899$$ −29.6548 −0.989042
$$900$$ 0 0
$$901$$ 3.29584 0.109800
$$902$$ 0 0
$$903$$ −36.9183 −1.22857
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −30.2796 −1.00542 −0.502709 0.864456i $$-0.667663\pi$$
−0.502709 + 0.864456i $$0.667663\pi$$
$$908$$ 0 0
$$909$$ 35.7725 1.18650
$$910$$ 0 0
$$911$$ −28.2296 −0.935290 −0.467645 0.883916i $$-0.654897\pi$$
−0.467645 + 0.883916i $$0.654897\pi$$
$$912$$ 0 0
$$913$$ −18.4378 −0.610203
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −64.5744 −2.13243
$$918$$ 0 0
$$919$$ 3.28150 0.108247 0.0541233 0.998534i $$-0.482764\pi$$
0.0541233 + 0.998534i $$0.482764\pi$$
$$920$$ 0 0
$$921$$ −45.0814 −1.48548
$$922$$ 0 0
$$923$$ −23.6573 −0.778690
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −30.3187 −0.995796
$$928$$ 0 0
$$929$$ 7.64745 0.250905 0.125452 0.992100i $$-0.459962\pi$$
0.125452 + 0.992100i $$0.459962\pi$$
$$930$$ 0 0
$$931$$ 36.1680 1.18536
$$932$$ 0 0
$$933$$ 6.03146 0.197461
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.95226 0.161783 0.0808917 0.996723i $$-0.474223\pi$$
0.0808917 + 0.996723i $$0.474223\pi$$
$$938$$ 0 0
$$939$$ −47.8366 −1.56109
$$940$$ 0 0
$$941$$ −8.62677 −0.281225 −0.140612 0.990065i $$-0.544907\pi$$
−0.140612 + 0.990065i $$0.544907\pi$$
$$942$$ 0 0
$$943$$ 6.44324 0.209821
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 46.6807 1.51692 0.758460 0.651720i $$-0.225952\pi$$
0.758460 + 0.651720i $$0.225952\pi$$
$$948$$ 0 0
$$949$$ −62.0391 −2.01387
$$950$$ 0 0
$$951$$ 66.9378 2.17061
$$952$$ 0 0
$$953$$ 12.2343 0.396307 0.198154 0.980171i $$-0.436505\pi$$
0.198154 + 0.980171i $$0.436505\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −42.3134 −1.36780
$$958$$ 0 0
$$959$$ 4.29063 0.138552
$$960$$ 0 0
$$961$$ −16.1200 −0.520000
$$962$$ 0 0
$$963$$ −54.9227 −1.76986
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 54.8318 1.76327 0.881635 0.471932i $$-0.156443\pi$$
0.881635 + 0.471932i $$0.156443\pi$$
$$968$$ 0 0
$$969$$ −10.5358 −0.338460
$$970$$ 0 0
$$971$$ 40.3804 1.29587 0.647935 0.761696i $$-0.275633\pi$$
0.647935 + 0.761696i $$0.275633\pi$$
$$972$$ 0 0
$$973$$ −34.1709 −1.09547
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 10.6700 0.341363 0.170681 0.985326i $$-0.445403\pi$$
0.170681 + 0.985326i $$0.445403\pi$$
$$978$$ 0 0
$$979$$ −2.50551 −0.0800765
$$980$$ 0 0
$$981$$ 1.18024 0.0376821
$$982$$ 0 0
$$983$$ −6.27812 −0.200241 −0.100120 0.994975i $$-0.531923\pi$$
−0.100120 + 0.994975i $$0.531923\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −65.6535 −2.08977
$$988$$ 0 0
$$989$$ −3.48497 −0.110816
$$990$$ 0 0
$$991$$ 61.2864 1.94683 0.973413 0.229057i $$-0.0735643\pi$$
0.973413 + 0.229057i $$0.0735643\pi$$
$$992$$ 0 0
$$993$$ −68.5125 −2.17418
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −39.9799 −1.26618 −0.633088 0.774080i $$-0.718213\pi$$
−0.633088 + 0.774080i $$0.718213\pi$$
$$998$$ 0 0
$$999$$ 4.92096 0.155692
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 2300.2.a.n.1.6 6
4.3 odd 2 9200.2.a.cy.1.1 6
5.2 odd 4 460.2.c.a.369.3 12
5.3 odd 4 460.2.c.a.369.10 yes 12
5.4 even 2 2300.2.a.o.1.1 6
15.2 even 4 4140.2.f.b.829.8 12
15.8 even 4 4140.2.f.b.829.7 12
20.3 even 4 1840.2.e.f.369.3 12
20.7 even 4 1840.2.e.f.369.10 12
20.19 odd 2 9200.2.a.cx.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.c.a.369.3 12 5.2 odd 4
460.2.c.a.369.10 yes 12 5.3 odd 4
1840.2.e.f.369.3 12 20.3 even 4
1840.2.e.f.369.10 12 20.7 even 4
2300.2.a.n.1.6 6 1.1 even 1 trivial
2300.2.a.o.1.1 6 5.4 even 2
4140.2.f.b.829.7 12 15.8 even 4
4140.2.f.b.829.8 12 15.2 even 4
9200.2.a.cx.1.6 6 20.19 odd 2
9200.2.a.cy.1.1 6 4.3 odd 2