# Properties

 Label 768.5.g.c.511.1 Level $768$ Weight $5$ Character 768.511 Analytic conductor $79.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 768.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.3881316484$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 511.1 Root $$1.32288 + 2.29129i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.511 Dual form 768.5.g.c.511.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} +O(q^{10})$$ $$q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} -181.283i q^{11} -183.498 q^{13} +206.542i q^{15} +427.992 q^{17} +668.558i q^{19} +239.247 q^{21} +882.463i q^{23} +954.984 q^{25} +140.296i q^{27} -807.247 q^{29} +391.276i q^{31} -941.976 q^{33} -1830.17i q^{35} -466.980 q^{37} +953.484i q^{39} -2159.93 q^{41} +509.182i q^{43} +1073.22 q^{45} -2056.83i q^{47} +281.031 q^{49} -2223.91i q^{51} +753.725 q^{53} +7205.84i q^{55} +3473.93 q^{57} -1309.43i q^{59} -801.913 q^{61} -1243.16i q^{63} +7293.87 q^{65} -505.813i q^{67} +4585.41 q^{69} -2170.94i q^{71} +2297.97 q^{73} -4962.24i q^{75} +8346.85 q^{77} -10705.3i q^{79} +729.000 q^{81} +2977.81i q^{83} -17012.3 q^{85} +4194.58i q^{87} +6493.92 q^{89} -8448.82i q^{91} +2033.13 q^{93} -26574.5i q^{95} -8189.92 q^{97} +4894.65i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{5} - 108 q^{9}+O(q^{10})$$ 4 * q - 32 * q^5 - 108 * q^9 $$4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97}+O(q^{100})$$ 4 * q - 32 * q^5 - 108 * q^9 - 480 * q^13 + 696 * q^17 + 576 * q^21 + 1788 * q^25 - 2848 * q^29 - 720 * q^33 + 672 * q^37 + 504 * q^41 + 864 * q^45 + 5188 * q^49 - 160 * q^53 + 4752 * q^57 + 7968 * q^61 + 11904 * q^65 + 6912 * q^69 + 5128 * q^73 + 14592 * q^77 + 2916 * q^81 - 37824 * q^85 + 15816 * q^89 - 7488 * q^93 - 22600 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 5.19615i − 0.577350i
$$4$$ 0 0
$$5$$ −39.7490 −1.58996 −0.794980 0.606635i $$-0.792519\pi$$
−0.794980 + 0.606635i $$0.792519\pi$$
$$6$$ 0 0
$$7$$ 46.0431i 0.939655i 0.882758 + 0.469828i $$0.155684\pi$$
−0.882758 + 0.469828i $$0.844316\pi$$
$$8$$ 0 0
$$9$$ −27.0000 −0.333333
$$10$$ 0 0
$$11$$ − 181.283i − 1.49821i −0.662451 0.749105i $$-0.730484\pi$$
0.662451 0.749105i $$-0.269516\pi$$
$$12$$ 0 0
$$13$$ −183.498 −1.08579 −0.542894 0.839801i $$-0.682671\pi$$
−0.542894 + 0.839801i $$0.682671\pi$$
$$14$$ 0 0
$$15$$ 206.542i 0.917964i
$$16$$ 0 0
$$17$$ 427.992 1.48094 0.740471 0.672089i $$-0.234603\pi$$
0.740471 + 0.672089i $$0.234603\pi$$
$$18$$ 0 0
$$19$$ 668.558i 1.85196i 0.377571 + 0.925981i $$0.376759\pi$$
−0.377571 + 0.925981i $$0.623241\pi$$
$$20$$ 0 0
$$21$$ 239.247 0.542510
$$22$$ 0 0
$$23$$ 882.463i 1.66817i 0.551635 + 0.834086i $$0.314004\pi$$
−0.551635 + 0.834086i $$0.685996\pi$$
$$24$$ 0 0
$$25$$ 954.984 1.52797
$$26$$ 0 0
$$27$$ 140.296i 0.192450i
$$28$$ 0 0
$$29$$ −807.247 −0.959866 −0.479933 0.877305i $$-0.659339\pi$$
−0.479933 + 0.877305i $$0.659339\pi$$
$$30$$ 0 0
$$31$$ 391.276i 0.407155i 0.979059 + 0.203577i $$0.0652569\pi$$
−0.979059 + 0.203577i $$0.934743\pi$$
$$32$$ 0 0
$$33$$ −941.976 −0.864992
$$34$$ 0 0
$$35$$ − 1830.17i − 1.49402i
$$36$$ 0 0
$$37$$ −466.980 −0.341111 −0.170555 0.985348i $$-0.554556\pi$$
−0.170555 + 0.985348i $$0.554556\pi$$
$$38$$ 0 0
$$39$$ 953.484i 0.626880i
$$40$$ 0 0
$$41$$ −2159.93 −1.28491 −0.642454 0.766325i $$-0.722083\pi$$
−0.642454 + 0.766325i $$0.722083\pi$$
$$42$$ 0 0
$$43$$ 509.182i 0.275382i 0.990475 + 0.137691i $$0.0439682\pi$$
−0.990475 + 0.137691i $$0.956032\pi$$
$$44$$ 0 0
$$45$$ 1073.22 0.529987
$$46$$ 0 0
$$47$$ − 2056.83i − 0.931116i −0.885017 0.465558i $$-0.845854\pi$$
0.885017 0.465558i $$-0.154146\pi$$
$$48$$ 0 0
$$49$$ 281.031 0.117048
$$50$$ 0 0
$$51$$ − 2223.91i − 0.855022i
$$52$$ 0 0
$$53$$ 753.725 0.268325 0.134163 0.990959i $$-0.457166\pi$$
0.134163 + 0.990959i $$0.457166\pi$$
$$54$$ 0 0
$$55$$ 7205.84i 2.38210i
$$56$$ 0 0
$$57$$ 3473.93 1.06923
$$58$$ 0 0
$$59$$ − 1309.43i − 0.376165i −0.982153 0.188083i $$-0.939773\pi$$
0.982153 0.188083i $$-0.0602272\pi$$
$$60$$ 0 0
$$61$$ −801.913 −0.215510 −0.107755 0.994177i $$-0.534366\pi$$
−0.107755 + 0.994177i $$0.534366\pi$$
$$62$$ 0 0
$$63$$ − 1243.16i − 0.313218i
$$64$$ 0 0
$$65$$ 7293.87 1.72636
$$66$$ 0 0
$$67$$ − 505.813i − 0.112678i −0.998412 0.0563392i $$-0.982057\pi$$
0.998412 0.0563392i $$-0.0179428\pi$$
$$68$$ 0 0
$$69$$ 4585.41 0.963119
$$70$$ 0 0
$$71$$ − 2170.94i − 0.430658i −0.976542 0.215329i $$-0.930918\pi$$
0.976542 0.215329i $$-0.0690823\pi$$
$$72$$ 0 0
$$73$$ 2297.97 0.431219 0.215610 0.976480i $$-0.430826\pi$$
0.215610 + 0.976480i $$0.430826\pi$$
$$74$$ 0 0
$$75$$ − 4962.24i − 0.882177i
$$76$$ 0 0
$$77$$ 8346.85 1.40780
$$78$$ 0 0
$$79$$ − 10705.3i − 1.71532i −0.514219 0.857659i $$-0.671918\pi$$
0.514219 0.857659i $$-0.328082\pi$$
$$80$$ 0 0
$$81$$ 729.000 0.111111
$$82$$ 0 0
$$83$$ 2977.81i 0.432256i 0.976365 + 0.216128i $$0.0693429\pi$$
−0.976365 + 0.216128i $$0.930657\pi$$
$$84$$ 0 0
$$85$$ −17012.3 −2.35464
$$86$$ 0 0
$$87$$ 4194.58i 0.554179i
$$88$$ 0 0
$$89$$ 6493.92 0.819836 0.409918 0.912122i $$-0.365557\pi$$
0.409918 + 0.912122i $$0.365557\pi$$
$$90$$ 0 0
$$91$$ − 8448.82i − 1.02027i
$$92$$ 0 0
$$93$$ 2033.13 0.235071
$$94$$ 0 0
$$95$$ − 26574.5i − 2.94455i
$$96$$ 0 0
$$97$$ −8189.92 −0.870435 −0.435217 0.900325i $$-0.643328\pi$$
−0.435217 + 0.900325i $$0.643328\pi$$
$$98$$ 0 0
$$99$$ 4894.65i 0.499403i
$$100$$ 0 0
$$101$$ 15576.5 1.52696 0.763479 0.645833i $$-0.223490\pi$$
0.763479 + 0.645833i $$0.223490\pi$$
$$102$$ 0 0
$$103$$ − 6628.58i − 0.624807i −0.949950 0.312403i $$-0.898866\pi$$
0.949950 0.312403i $$-0.101134\pi$$
$$104$$ 0 0
$$105$$ −9509.83 −0.862570
$$106$$ 0 0
$$107$$ − 11796.4i − 1.03035i −0.857086 0.515173i $$-0.827728\pi$$
0.857086 0.515173i $$-0.172272\pi$$
$$108$$ 0 0
$$109$$ −13286.1 −1.11826 −0.559130 0.829080i $$-0.688865\pi$$
−0.559130 + 0.829080i $$0.688865\pi$$
$$110$$ 0 0
$$111$$ 2426.50i 0.196940i
$$112$$ 0 0
$$113$$ −11713.6 −0.917350 −0.458675 0.888604i $$-0.651676\pi$$
−0.458675 + 0.888604i $$0.651676\pi$$
$$114$$ 0 0
$$115$$ − 35077.0i − 2.65233i
$$116$$ 0 0
$$117$$ 4954.45 0.361929
$$118$$ 0 0
$$119$$ 19706.1i 1.39157i
$$120$$ 0 0
$$121$$ −18222.7 −1.24463
$$122$$ 0 0
$$123$$ 11223.3i 0.741842i
$$124$$ 0 0
$$125$$ −13116.5 −0.839459
$$126$$ 0 0
$$127$$ − 5640.55i − 0.349715i −0.984594 0.174858i $$-0.944054\pi$$
0.984594 0.174858i $$-0.0559465\pi$$
$$128$$ 0 0
$$129$$ 2645.79 0.158992
$$130$$ 0 0
$$131$$ − 31922.5i − 1.86018i −0.367335 0.930089i $$-0.619730\pi$$
0.367335 0.930089i $$-0.380270\pi$$
$$132$$ 0 0
$$133$$ −30782.5 −1.74021
$$134$$ 0 0
$$135$$ − 5576.63i − 0.305988i
$$136$$ 0 0
$$137$$ 10515.6 0.560263 0.280132 0.959962i $$-0.409622\pi$$
0.280132 + 0.959962i $$0.409622\pi$$
$$138$$ 0 0
$$139$$ − 14985.9i − 0.775626i −0.921738 0.387813i $$-0.873231\pi$$
0.921738 0.387813i $$-0.126769\pi$$
$$140$$ 0 0
$$141$$ −10687.6 −0.537580
$$142$$ 0 0
$$143$$ 33265.2i 1.62674i
$$144$$ 0 0
$$145$$ 32087.3 1.52615
$$146$$ 0 0
$$147$$ − 1460.28i − 0.0675775i
$$148$$ 0 0
$$149$$ 11795.4 0.531298 0.265649 0.964070i $$-0.414414\pi$$
0.265649 + 0.964070i $$0.414414\pi$$
$$150$$ 0 0
$$151$$ 33792.9i 1.48208i 0.671460 + 0.741040i $$0.265667\pi$$
−0.671460 + 0.741040i $$0.734333\pi$$
$$152$$ 0 0
$$153$$ −11555.8 −0.493647
$$154$$ 0 0
$$155$$ − 15552.8i − 0.647360i
$$156$$ 0 0
$$157$$ 28788.9 1.16796 0.583978 0.811770i $$-0.301496\pi$$
0.583978 + 0.811770i $$0.301496\pi$$
$$158$$ 0 0
$$159$$ − 3916.47i − 0.154918i
$$160$$ 0 0
$$161$$ −40631.3 −1.56751
$$162$$ 0 0
$$163$$ 13409.0i 0.504684i 0.967638 + 0.252342i $$0.0812008\pi$$
−0.967638 + 0.252342i $$0.918799\pi$$
$$164$$ 0 0
$$165$$ 37442.6 1.37530
$$166$$ 0 0
$$167$$ − 13206.7i − 0.473546i −0.971565 0.236773i $$-0.923910\pi$$
0.971565 0.236773i $$-0.0760898\pi$$
$$168$$ 0 0
$$169$$ 5110.53 0.178934
$$170$$ 0 0
$$171$$ − 18051.1i − 0.617320i
$$172$$ 0 0
$$173$$ 364.956 0.0121940 0.00609702 0.999981i $$-0.498059\pi$$
0.00609702 + 0.999981i $$0.498059\pi$$
$$174$$ 0 0
$$175$$ 43970.5i 1.43577i
$$176$$ 0 0
$$177$$ −6804.00 −0.217179
$$178$$ 0 0
$$179$$ − 14372.0i − 0.448549i −0.974526 0.224275i $$-0.927999\pi$$
0.974526 0.224275i $$-0.0720013\pi$$
$$180$$ 0 0
$$181$$ 4050.54 0.123639 0.0618195 0.998087i $$-0.480310\pi$$
0.0618195 + 0.998087i $$0.480310\pi$$
$$182$$ 0 0
$$183$$ 4166.86i 0.124425i
$$184$$ 0 0
$$185$$ 18562.0 0.542352
$$186$$ 0 0
$$187$$ − 77587.9i − 2.21876i
$$188$$ 0 0
$$189$$ −6459.67 −0.180837
$$190$$ 0 0
$$191$$ 49880.9i 1.36731i 0.729805 + 0.683656i $$0.239611\pi$$
−0.729805 + 0.683656i $$0.760389\pi$$
$$192$$ 0 0
$$193$$ −48425.7 −1.30005 −0.650026 0.759912i $$-0.725242\pi$$
−0.650026 + 0.759912i $$0.725242\pi$$
$$194$$ 0 0
$$195$$ − 37900.0i − 0.996714i
$$196$$ 0 0
$$197$$ 5556.74 0.143182 0.0715908 0.997434i $$-0.477192\pi$$
0.0715908 + 0.997434i $$0.477192\pi$$
$$198$$ 0 0
$$199$$ − 60594.7i − 1.53013i −0.643952 0.765066i $$-0.722706\pi$$
0.643952 0.765066i $$-0.277294\pi$$
$$200$$ 0 0
$$201$$ −2628.28 −0.0650549
$$202$$ 0 0
$$203$$ − 37168.2i − 0.901943i
$$204$$ 0 0
$$205$$ 85855.1 2.04295
$$206$$ 0 0
$$207$$ − 23826.5i − 0.556057i
$$208$$ 0 0
$$209$$ 121198. 2.77463
$$210$$ 0 0
$$211$$ − 27539.9i − 0.618583i −0.950967 0.309292i $$-0.899908\pi$$
0.950967 0.309292i $$-0.100092\pi$$
$$212$$ 0 0
$$213$$ −11280.6 −0.248640
$$214$$ 0 0
$$215$$ − 20239.5i − 0.437847i
$$216$$ 0 0
$$217$$ −18015.6 −0.382585
$$218$$ 0 0
$$219$$ − 11940.6i − 0.248965i
$$220$$ 0 0
$$221$$ −78535.7 −1.60799
$$222$$ 0 0
$$223$$ 3021.35i 0.0607564i 0.999538 + 0.0303782i $$0.00967116\pi$$
−0.999538 + 0.0303782i $$0.990329\pi$$
$$224$$ 0 0
$$225$$ −25784.6 −0.509325
$$226$$ 0 0
$$227$$ 7077.28i 0.137346i 0.997639 + 0.0686728i $$0.0218765\pi$$
−0.997639 + 0.0686728i $$0.978124\pi$$
$$228$$ 0 0
$$229$$ −102761. −1.95956 −0.979778 0.200088i $$-0.935877\pi$$
−0.979778 + 0.200088i $$0.935877\pi$$
$$230$$ 0 0
$$231$$ − 43371.5i − 0.812795i
$$232$$ 0 0
$$233$$ −22976.6 −0.423227 −0.211613 0.977353i $$-0.567872\pi$$
−0.211613 + 0.977353i $$0.567872\pi$$
$$234$$ 0 0
$$235$$ 81757.1i 1.48044i
$$236$$ 0 0
$$237$$ −55626.4 −0.990339
$$238$$ 0 0
$$239$$ 35566.7i 0.622656i 0.950303 + 0.311328i $$0.100774\pi$$
−0.950303 + 0.311328i $$0.899226\pi$$
$$240$$ 0 0
$$241$$ −92683.4 −1.59576 −0.797881 0.602816i $$-0.794045\pi$$
−0.797881 + 0.602816i $$0.794045\pi$$
$$242$$ 0 0
$$243$$ − 3788.00i − 0.0641500i
$$244$$ 0 0
$$245$$ −11170.7 −0.186101
$$246$$ 0 0
$$247$$ − 122679.i − 2.01084i
$$248$$ 0 0
$$249$$ 15473.2 0.249563
$$250$$ 0 0
$$251$$ 27590.5i 0.437937i 0.975732 + 0.218968i $$0.0702692\pi$$
−0.975732 + 0.218968i $$0.929731\pi$$
$$252$$ 0 0
$$253$$ 159976. 2.49927
$$254$$ 0 0
$$255$$ 88398.3i 1.35945i
$$256$$ 0 0
$$257$$ 92304.6 1.39752 0.698758 0.715358i $$-0.253736\pi$$
0.698758 + 0.715358i $$0.253736\pi$$
$$258$$ 0 0
$$259$$ − 21501.2i − 0.320526i
$$260$$ 0 0
$$261$$ 21795.7 0.319955
$$262$$ 0 0
$$263$$ − 13884.4i − 0.200732i −0.994951 0.100366i $$-0.967999\pi$$
0.994951 0.100366i $$-0.0320014\pi$$
$$264$$ 0 0
$$265$$ −29959.8 −0.426626
$$266$$ 0 0
$$267$$ − 33743.4i − 0.473333i
$$268$$ 0 0
$$269$$ −52698.1 −0.728266 −0.364133 0.931347i $$-0.618635\pi$$
−0.364133 + 0.931347i $$0.618635\pi$$
$$270$$ 0 0
$$271$$ − 47028.8i − 0.640361i −0.947356 0.320181i $$-0.896256\pi$$
0.947356 0.320181i $$-0.103744\pi$$
$$272$$ 0 0
$$273$$ −43901.4 −0.589051
$$274$$ 0 0
$$275$$ − 173123.i − 2.28923i
$$276$$ 0 0
$$277$$ 108393. 1.41268 0.706340 0.707873i $$-0.250345\pi$$
0.706340 + 0.707873i $$0.250345\pi$$
$$278$$ 0 0
$$279$$ − 10564.4i − 0.135718i
$$280$$ 0 0
$$281$$ 73090.7 0.925656 0.462828 0.886448i $$-0.346835\pi$$
0.462828 + 0.886448i $$0.346835\pi$$
$$282$$ 0 0
$$283$$ 44763.1i 0.558917i 0.960158 + 0.279458i $$0.0901549\pi$$
−0.960158 + 0.279458i $$0.909845\pi$$
$$284$$ 0 0
$$285$$ −138085. −1.70003
$$286$$ 0 0
$$287$$ − 99449.9i − 1.20737i
$$288$$ 0 0
$$289$$ 99656.3 1.19319
$$290$$ 0 0
$$291$$ 42556.1i 0.502546i
$$292$$ 0 0
$$293$$ 42030.2 0.489583 0.244792 0.969576i $$-0.421280\pi$$
0.244792 + 0.969576i $$0.421280\pi$$
$$294$$ 0 0
$$295$$ 52048.6i 0.598088i
$$296$$ 0 0
$$297$$ 25433.4 0.288331
$$298$$ 0 0
$$299$$ − 161930.i − 1.81128i
$$300$$ 0 0
$$301$$ −23444.3 −0.258765
$$302$$ 0 0
$$303$$ − 80937.9i − 0.881590i
$$304$$ 0 0
$$305$$ 31875.3 0.342653
$$306$$ 0 0
$$307$$ − 8820.31i − 0.0935852i −0.998905 0.0467926i $$-0.985100\pi$$
0.998905 0.0467926i $$-0.0149000\pi$$
$$308$$ 0 0
$$309$$ −34443.1 −0.360732
$$310$$ 0 0
$$311$$ − 155059.i − 1.60315i −0.597893 0.801576i $$-0.703995\pi$$
0.597893 0.801576i $$-0.296005\pi$$
$$312$$ 0 0
$$313$$ 179666. 1.83390 0.916951 0.398999i $$-0.130642\pi$$
0.916951 + 0.398999i $$0.130642\pi$$
$$314$$ 0 0
$$315$$ 49414.6i 0.498005i
$$316$$ 0 0
$$317$$ −6188.77 −0.0615865 −0.0307933 0.999526i $$-0.509803\pi$$
−0.0307933 + 0.999526i $$0.509803\pi$$
$$318$$ 0 0
$$319$$ 146341.i 1.43808i
$$320$$ 0 0
$$321$$ −61296.0 −0.594870
$$322$$ 0 0
$$323$$ 286138.i 2.74265i
$$324$$ 0 0
$$325$$ −175238. −1.65906
$$326$$ 0 0
$$327$$ 69036.4i 0.645628i
$$328$$ 0 0
$$329$$ 94703.1 0.874928
$$330$$ 0 0
$$331$$ − 132159.i − 1.20626i −0.797643 0.603130i $$-0.793920\pi$$
0.797643 0.603130i $$-0.206080\pi$$
$$332$$ 0 0
$$333$$ 12608.5 0.113704
$$334$$ 0 0
$$335$$ 20105.6i 0.179154i
$$336$$ 0 0
$$337$$ 70040.0 0.616718 0.308359 0.951270i $$-0.400220\pi$$
0.308359 + 0.951270i $$0.400220\pi$$
$$338$$ 0 0
$$339$$ 60865.8i 0.529632i
$$340$$ 0 0
$$341$$ 70931.8 0.610004
$$342$$ 0 0
$$343$$ 123489.i 1.04964i
$$344$$ 0 0
$$345$$ −182266. −1.53132
$$346$$ 0 0
$$347$$ 157722.i 1.30988i 0.755680 + 0.654942i $$0.227307\pi$$
−0.755680 + 0.654942i $$0.772693\pi$$
$$348$$ 0 0
$$349$$ 198747. 1.63174 0.815868 0.578238i $$-0.196259\pi$$
0.815868 + 0.578238i $$0.196259\pi$$
$$350$$ 0 0
$$351$$ − 25744.1i − 0.208960i
$$352$$ 0 0
$$353$$ 13376.1 0.107345 0.0536724 0.998559i $$-0.482907\pi$$
0.0536724 + 0.998559i $$0.482907\pi$$
$$354$$ 0 0
$$355$$ 86292.9i 0.684729i
$$356$$ 0 0
$$357$$ 102396. 0.803426
$$358$$ 0 0
$$359$$ 190011.i 1.47432i 0.675721 + 0.737158i $$0.263832\pi$$
−0.675721 + 0.737158i $$0.736168\pi$$
$$360$$ 0 0
$$361$$ −316649. −2.42976
$$362$$ 0 0
$$363$$ 94687.8i 0.718590i
$$364$$ 0 0
$$365$$ −91342.0 −0.685622
$$366$$ 0 0
$$367$$ − 94140.5i − 0.698947i −0.936946 0.349474i $$-0.886360\pi$$
0.936946 0.349474i $$-0.113640\pi$$
$$368$$ 0 0
$$369$$ 58318.1 0.428302
$$370$$ 0 0
$$371$$ 34703.9i 0.252133i
$$372$$ 0 0
$$373$$ 66152.4 0.475475 0.237738 0.971329i $$-0.423594\pi$$
0.237738 + 0.971329i $$0.423594\pi$$
$$374$$ 0 0
$$375$$ 68155.6i 0.484662i
$$376$$ 0 0
$$377$$ 148128. 1.04221
$$378$$ 0 0
$$379$$ 64395.2i 0.448306i 0.974554 + 0.224153i $$0.0719616\pi$$
−0.974554 + 0.224153i $$0.928038\pi$$
$$380$$ 0 0
$$381$$ −29309.2 −0.201908
$$382$$ 0 0
$$383$$ − 30714.5i − 0.209385i −0.994505 0.104693i $$-0.966614\pi$$
0.994505 0.104693i $$-0.0333859\pi$$
$$384$$ 0 0
$$385$$ −331779. −2.23835
$$386$$ 0 0
$$387$$ − 13747.9i − 0.0917941i
$$388$$ 0 0
$$389$$ 199122. 1.31589 0.657945 0.753066i $$-0.271426\pi$$
0.657945 + 0.753066i $$0.271426\pi$$
$$390$$ 0 0
$$391$$ 377687.i 2.47046i
$$392$$ 0 0
$$393$$ −165874. −1.07397
$$394$$ 0 0
$$395$$ 425525.i 2.72729i
$$396$$ 0 0
$$397$$ 68565.8 0.435037 0.217519 0.976056i $$-0.430204\pi$$
0.217519 + 0.976056i $$0.430204\pi$$
$$398$$ 0 0
$$399$$ 159951.i 1.00471i
$$400$$ 0 0
$$401$$ −21797.1 −0.135553 −0.0677765 0.997701i $$-0.521590\pi$$
−0.0677765 + 0.997701i $$0.521590\pi$$
$$402$$ 0 0
$$403$$ − 71798.3i − 0.442084i
$$404$$ 0 0
$$405$$ −28977.0 −0.176662
$$406$$ 0 0
$$407$$ 84655.8i 0.511055i
$$408$$ 0 0
$$409$$ −230211. −1.37620 −0.688098 0.725618i $$-0.741554\pi$$
−0.688098 + 0.725618i $$0.741554\pi$$
$$410$$ 0 0
$$411$$ − 54640.6i − 0.323468i
$$412$$ 0 0
$$413$$ 60290.3 0.353465
$$414$$ 0 0
$$415$$ − 118365.i − 0.687270i
$$416$$ 0 0
$$417$$ −77868.9 −0.447808
$$418$$ 0 0
$$419$$ − 17360.5i − 0.0988859i −0.998777 0.0494430i $$-0.984255\pi$$
0.998777 0.0494430i $$-0.0157446\pi$$
$$420$$ 0 0
$$421$$ −186829. −1.05410 −0.527048 0.849836i $$-0.676701\pi$$
−0.527048 + 0.849836i $$0.676701\pi$$
$$422$$ 0 0
$$423$$ 55534.5i 0.310372i
$$424$$ 0 0
$$425$$ 408726. 2.26284
$$426$$ 0 0
$$427$$ − 36922.6i − 0.202505i
$$428$$ 0 0
$$429$$ 172851. 0.939197
$$430$$ 0 0
$$431$$ − 153753.i − 0.827693i −0.910347 0.413846i $$-0.864185\pi$$
0.910347 0.413846i $$-0.135815\pi$$
$$432$$ 0 0
$$433$$ 9168.87 0.0489035 0.0244517 0.999701i $$-0.492216\pi$$
0.0244517 + 0.999701i $$0.492216\pi$$
$$434$$ 0 0
$$435$$ − 166730.i − 0.881122i
$$436$$ 0 0
$$437$$ −589978. −3.08939
$$438$$ 0 0
$$439$$ − 178760.i − 0.927561i −0.885950 0.463780i $$-0.846493\pi$$
0.885950 0.463780i $$-0.153507\pi$$
$$440$$ 0 0
$$441$$ −7587.85 −0.0390159
$$442$$ 0 0
$$443$$ − 17798.2i − 0.0906920i −0.998971 0.0453460i $$-0.985561\pi$$
0.998971 0.0453460i $$-0.0144390\pi$$
$$444$$ 0 0
$$445$$ −258127. −1.30351
$$446$$ 0 0
$$447$$ − 61290.5i − 0.306745i
$$448$$ 0 0
$$449$$ 242915. 1.20493 0.602464 0.798146i $$-0.294186\pi$$
0.602464 + 0.798146i $$0.294186\pi$$
$$450$$ 0 0
$$451$$ 391559.i 1.92506i
$$452$$ 0 0
$$453$$ 175593. 0.855680
$$454$$ 0 0
$$455$$ 335832.i 1.62218i
$$456$$ 0 0
$$457$$ 190762. 0.913396 0.456698 0.889622i $$-0.349032\pi$$
0.456698 + 0.889622i $$0.349032\pi$$
$$458$$ 0 0
$$459$$ 60045.6i 0.285007i
$$460$$ 0 0
$$461$$ 155121. 0.729909 0.364955 0.931025i $$-0.381084\pi$$
0.364955 + 0.931025i $$0.381084\pi$$
$$462$$ 0 0
$$463$$ − 230925.i − 1.07723i −0.842551 0.538616i $$-0.818947\pi$$
0.842551 0.538616i $$-0.181053\pi$$
$$464$$ 0 0
$$465$$ −80814.9 −0.373754
$$466$$ 0 0
$$467$$ 154112.i 0.706647i 0.935501 + 0.353323i $$0.114948\pi$$
−0.935501 + 0.353323i $$0.885052\pi$$
$$468$$ 0 0
$$469$$ 23289.2 0.105879
$$470$$ 0 0
$$471$$ − 149592.i − 0.674319i
$$472$$ 0 0
$$473$$ 92306.3 0.412581
$$474$$ 0 0
$$475$$ 638462.i 2.82975i
$$476$$ 0 0
$$477$$ −20350.6 −0.0894417
$$478$$ 0 0
$$479$$ − 405668.i − 1.76807i −0.467420 0.884035i $$-0.654816\pi$$
0.467420 0.884035i $$-0.345184\pi$$
$$480$$ 0 0
$$481$$ 85690.0 0.370373
$$482$$ 0 0
$$483$$ 211127.i 0.905000i
$$484$$ 0 0
$$485$$ 325541. 1.38396
$$486$$ 0 0
$$487$$ − 426114.i − 1.79667i −0.439314 0.898334i $$-0.644778\pi$$
0.439314 0.898334i $$-0.355222\pi$$
$$488$$ 0 0
$$489$$ 69675.0 0.291379
$$490$$ 0 0
$$491$$ − 273125.i − 1.13292i −0.824090 0.566459i $$-0.808313\pi$$
0.824090 0.566459i $$-0.191687\pi$$
$$492$$ 0 0
$$493$$ −345495. −1.42151
$$494$$ 0 0
$$495$$ − 194558.i − 0.794032i
$$496$$ 0 0
$$497$$ 99957.1 0.404670
$$498$$ 0 0
$$499$$ 103548.i 0.415855i 0.978144 + 0.207928i $$0.0666719\pi$$
−0.978144 + 0.207928i $$0.933328\pi$$
$$500$$ 0 0
$$501$$ −68624.2 −0.273402
$$502$$ 0 0
$$503$$ − 217063.i − 0.857925i −0.903322 0.428963i $$-0.858879\pi$$
0.903322 0.428963i $$-0.141121\pi$$
$$504$$ 0 0
$$505$$ −619151. −2.42780
$$506$$ 0 0
$$507$$ − 26555.1i − 0.103307i
$$508$$ 0 0
$$509$$ −115098. −0.444256 −0.222128 0.975018i $$-0.571300\pi$$
−0.222128 + 0.975018i $$0.571300\pi$$
$$510$$ 0 0
$$511$$ 105806.i 0.405198i
$$512$$ 0 0
$$513$$ −93796.1 −0.356410
$$514$$ 0 0
$$515$$ 263479.i 0.993418i
$$516$$ 0 0
$$517$$ −372870. −1.39501
$$518$$ 0 0
$$519$$ − 1896.37i − 0.00704024i
$$520$$ 0 0
$$521$$ −288543. −1.06300 −0.531502 0.847057i $$-0.678372\pi$$
−0.531502 + 0.847057i $$0.678372\pi$$
$$522$$ 0 0
$$523$$ − 176826.i − 0.646461i −0.946320 0.323231i $$-0.895231\pi$$
0.946320 0.323231i $$-0.104769\pi$$
$$524$$ 0 0
$$525$$ 228477. 0.828942
$$526$$ 0 0
$$527$$ 167463.i 0.602973i
$$528$$ 0 0
$$529$$ −498900. −1.78280
$$530$$ 0 0
$$531$$ 35354.6i 0.125388i
$$532$$ 0 0
$$533$$ 396343. 1.39514
$$534$$ 0 0
$$535$$ 468896.i 1.63821i
$$536$$ 0 0
$$537$$ −74678.9 −0.258970
$$538$$ 0 0
$$539$$ − 50946.4i − 0.175362i
$$540$$ 0 0
$$541$$ −425220. −1.45284 −0.726422 0.687249i $$-0.758818\pi$$
−0.726422 + 0.687249i $$0.758818\pi$$
$$542$$ 0 0
$$543$$ − 21047.2i − 0.0713830i
$$544$$ 0 0
$$545$$ 528108. 1.77799
$$546$$ 0 0
$$547$$ 495505.i 1.65605i 0.560692 + 0.828024i $$0.310535\pi$$
−0.560692 + 0.828024i $$0.689465\pi$$
$$548$$ 0 0
$$549$$ 21651.7 0.0718367
$$550$$ 0 0
$$551$$ − 539691.i − 1.77763i
$$552$$ 0 0
$$553$$ 492905. 1.61181
$$554$$ 0 0
$$555$$ − 96451.0i − 0.313127i
$$556$$ 0 0
$$557$$ 257895. 0.831251 0.415626 0.909536i $$-0.363563\pi$$
0.415626 + 0.909536i $$0.363563\pi$$
$$558$$ 0 0
$$559$$ − 93433.9i − 0.299007i
$$560$$ 0 0
$$561$$ −403158. −1.28100
$$562$$ 0 0
$$563$$ − 417799.i − 1.31811i −0.752096 0.659054i $$-0.770957\pi$$
0.752096 0.659054i $$-0.229043\pi$$
$$564$$ 0 0
$$565$$ 465606. 1.45855
$$566$$ 0 0
$$567$$ 33565.4i 0.104406i
$$568$$ 0 0
$$569$$ 100935. 0.311756 0.155878 0.987776i $$-0.450179\pi$$
0.155878 + 0.987776i $$0.450179\pi$$
$$570$$ 0 0
$$571$$ 453320.i 1.39038i 0.718828 + 0.695188i $$0.244679\pi$$
−0.718828 + 0.695188i $$0.755321\pi$$
$$572$$ 0 0
$$573$$ 259189. 0.789418
$$574$$ 0 0
$$575$$ 842738.i 2.54892i
$$576$$ 0 0
$$577$$ 12119.6 0.0364031 0.0182015 0.999834i $$-0.494206\pi$$
0.0182015 + 0.999834i $$0.494206\pi$$
$$578$$ 0 0
$$579$$ 251627.i 0.750586i
$$580$$ 0 0
$$581$$ −137108. −0.406172
$$582$$ 0 0
$$583$$ − 136638.i − 0.402008i
$$584$$ 0 0
$$585$$ −196934. −0.575453
$$586$$ 0 0
$$587$$ 327468.i 0.950370i 0.879886 + 0.475185i $$0.157619\pi$$
−0.879886 + 0.475185i $$0.842381\pi$$
$$588$$ 0 0
$$589$$ −261591. −0.754035
$$590$$ 0 0
$$591$$ − 28873.7i − 0.0826660i
$$592$$ 0 0
$$593$$ 610182. 1.73520 0.867601 0.497260i $$-0.165661\pi$$
0.867601 + 0.497260i $$0.165661\pi$$
$$594$$ 0 0
$$595$$ − 783298.i − 2.21255i
$$596$$ 0 0
$$597$$ −314859. −0.883422
$$598$$ 0 0
$$599$$ 499941.i 1.39337i 0.717379 + 0.696683i $$0.245342\pi$$
−0.717379 + 0.696683i $$0.754658\pi$$
$$600$$ 0 0
$$601$$ 486965. 1.34818 0.674091 0.738649i $$-0.264536\pi$$
0.674091 + 0.738649i $$0.264536\pi$$
$$602$$ 0 0
$$603$$ 13657.0i 0.0375595i
$$604$$ 0 0
$$605$$ 724334. 1.97892
$$606$$ 0 0
$$607$$ − 41693.4i − 0.113159i −0.998398 0.0565796i $$-0.981981\pi$$
0.998398 0.0565796i $$-0.0180195\pi$$
$$608$$ 0 0
$$609$$ −193131. −0.520737
$$610$$ 0 0
$$611$$ 377425.i 1.01099i
$$612$$ 0 0
$$613$$ 542042. 1.44249 0.721244 0.692681i $$-0.243571\pi$$
0.721244 + 0.692681i $$0.243571\pi$$
$$614$$ 0 0
$$615$$ − 446116.i − 1.17950i
$$616$$ 0 0
$$617$$ 146814. 0.385653 0.192827 0.981233i $$-0.438235\pi$$
0.192827 + 0.981233i $$0.438235\pi$$
$$618$$ 0 0
$$619$$ 72720.6i 0.189791i 0.995487 + 0.0948956i $$0.0302517\pi$$
−0.995487 + 0.0948956i $$0.969748\pi$$
$$620$$ 0 0
$$621$$ −123806. −0.321040
$$622$$ 0 0
$$623$$ 299000.i 0.770363i
$$624$$ 0 0
$$625$$ −75495.2 −0.193268
$$626$$ 0 0
$$627$$ − 629766.i − 1.60193i
$$628$$ 0 0
$$629$$ −199864. −0.505165
$$630$$ 0 0
$$631$$ − 568570.i − 1.42799i −0.700151 0.713995i $$-0.746884\pi$$
0.700151 0.713995i $$-0.253116\pi$$
$$632$$ 0 0
$$633$$ −143102. −0.357139
$$634$$ 0 0
$$635$$ 224206.i 0.556033i
$$636$$ 0 0
$$637$$ −51568.7 −0.127089
$$638$$ 0 0
$$639$$ 58615.5i 0.143553i
$$640$$ 0 0
$$641$$ 292494. 0.711870 0.355935 0.934511i $$-0.384162\pi$$
0.355935 + 0.934511i $$0.384162\pi$$
$$642$$ 0 0
$$643$$ − 103161.i − 0.249514i −0.992187 0.124757i $$-0.960185\pi$$
0.992187 0.124757i $$-0.0398151\pi$$
$$644$$ 0 0
$$645$$ −105167. −0.252791
$$646$$ 0 0
$$647$$ − 372569.i − 0.890017i −0.895526 0.445009i $$-0.853201\pi$$
0.895526 0.445009i $$-0.146799\pi$$
$$648$$ 0 0
$$649$$ −237378. −0.563574
$$650$$ 0 0
$$651$$ 93611.6i 0.220886i
$$652$$ 0 0
$$653$$ −85619.0 −0.200791 −0.100395 0.994948i $$-0.532011\pi$$
−0.100395 + 0.994948i $$0.532011\pi$$
$$654$$ 0 0
$$655$$ 1.26889e6i 2.95761i
$$656$$ 0 0
$$657$$ −62045.1 −0.143740
$$658$$ 0 0
$$659$$ − 258812.i − 0.595954i −0.954573 0.297977i $$-0.903688\pi$$
0.954573 0.297977i $$-0.0963119\pi$$
$$660$$ 0 0
$$661$$ 327660. 0.749928 0.374964 0.927039i $$-0.377655\pi$$
0.374964 + 0.927039i $$0.377655\pi$$
$$662$$ 0 0
$$663$$ 408084.i 0.928372i
$$664$$ 0 0
$$665$$ 1.22357e6 2.76686
$$666$$ 0 0
$$667$$ − 712366.i − 1.60122i
$$668$$ 0 0
$$669$$ 15699.4 0.0350777
$$670$$ 0 0
$$671$$ 145374.i 0.322880i
$$672$$ 0 0
$$673$$ 137121. 0.302742 0.151371 0.988477i $$-0.451631\pi$$
0.151371 + 0.988477i $$0.451631\pi$$
$$674$$ 0 0
$$675$$ 133981.i 0.294059i
$$676$$ 0 0
$$677$$ −573451. −1.25118 −0.625589 0.780153i $$-0.715141\pi$$
−0.625589 + 0.780153i $$0.715141\pi$$
$$678$$ 0 0
$$679$$ − 377089.i − 0.817909i
$$680$$ 0 0
$$681$$ 36774.6 0.0792965
$$682$$ 0 0
$$683$$ − 175872.i − 0.377012i −0.982072 0.188506i $$-0.939636\pi$$
0.982072 0.188506i $$-0.0603644\pi$$
$$684$$ 0 0
$$685$$ −417984. −0.890797
$$686$$ 0 0
$$687$$ 533962.i 1.13135i
$$688$$ 0 0
$$689$$ −138307. −0.291344
$$690$$ 0 0
$$691$$ 7216.66i 0.0151140i 0.999971 + 0.00755702i $$0.00240550\pi$$
−0.999971 + 0.00755702i $$0.997595\pi$$
$$692$$ 0 0
$$693$$ −225365. −0.469267
$$694$$ 0 0
$$695$$ 595673.i 1.23321i
$$696$$ 0 0
$$697$$ −924433. −1.90287
$$698$$ 0 0
$$699$$ 119390.i 0.244350i
$$700$$ 0 0
$$701$$ −502543. −1.02267 −0.511337 0.859380i $$-0.670850\pi$$
−0.511337 + 0.859380i $$0.670850\pi$$
$$702$$ 0 0
$$703$$ − 312203.i − 0.631723i
$$704$$ 0 0
$$705$$ 424823. 0.854731
$$706$$ 0 0
$$707$$ 717191.i 1.43481i
$$708$$ 0 0
$$709$$ 95591.1 0.190163 0.0950813 0.995470i $$-0.469689\pi$$
0.0950813 + 0.995470i $$0.469689\pi$$
$$710$$ 0 0
$$711$$ 289043.i 0.571772i
$$712$$ 0 0
$$713$$ −345286. −0.679204
$$714$$ 0 0
$$715$$ − 1.32226e6i − 2.58645i
$$716$$ 0 0
$$717$$ 184810. 0.359491
$$718$$ 0 0
$$719$$ − 475789.i − 0.920357i −0.887826 0.460178i $$-0.847785\pi$$
0.887826 0.460178i $$-0.152215\pi$$
$$720$$ 0 0
$$721$$ 305200. 0.587103
$$722$$ 0 0
$$723$$ 481597.i 0.921313i
$$724$$ 0 0
$$725$$ −770908. −1.46665
$$726$$ 0 0
$$727$$ 193262.i 0.365661i 0.983144 + 0.182830i $$0.0585259\pi$$
−0.983144 + 0.182830i $$0.941474\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −0.0370370
$$730$$ 0 0
$$731$$ 217926.i 0.407825i
$$732$$ 0 0
$$733$$ 650799. 1.21126 0.605632 0.795745i $$-0.292920\pi$$
0.605632 + 0.795745i $$0.292920\pi$$
$$734$$ 0 0
$$735$$ 58044.8i 0.107446i
$$736$$ 0 0
$$737$$ −91695.6 −0.168816
$$738$$ 0 0
$$739$$ − 99472.6i − 0.182144i −0.995844 0.0910719i $$-0.970971\pi$$
0.995844 0.0910719i $$-0.0290293\pi$$
$$740$$ 0 0
$$741$$ −637459. −1.16096
$$742$$ 0 0
$$743$$ − 715681.i − 1.29641i −0.761466 0.648205i $$-0.775520\pi$$
0.761466 0.648205i $$-0.224480\pi$$
$$744$$ 0 0
$$745$$ −468854. −0.844744
$$746$$ 0 0
$$747$$ − 80401.0i − 0.144085i
$$748$$ 0 0
$$749$$ 543144. 0.968170
$$750$$ 0 0
$$751$$ 603532.i 1.07009i 0.844824 + 0.535045i $$0.179705\pi$$
−0.844824 + 0.535045i $$0.820295\pi$$
$$752$$ 0 0
$$753$$ 143364. 0.252843
$$754$$ 0 0
$$755$$ − 1.34324e6i − 2.35645i
$$756$$ 0 0
$$757$$ 784192. 1.36846 0.684228 0.729268i $$-0.260139\pi$$
0.684228 + 0.729268i $$0.260139\pi$$
$$758$$ 0 0
$$759$$ − 831259.i − 1.44296i
$$760$$ 0 0
$$761$$ 568076. 0.980927 0.490464 0.871462i $$-0.336827\pi$$
0.490464 + 0.871462i $$0.336827\pi$$
$$762$$ 0 0
$$763$$ − 611731.i − 1.05078i
$$764$$ 0 0
$$765$$ 459331. 0.784880
$$766$$ 0 0
$$767$$ 240278.i 0.408435i
$$768$$ 0 0
$$769$$ −531757. −0.899209 −0.449605 0.893228i $$-0.648435\pi$$
−0.449605 + 0.893228i $$0.648435\pi$$
$$770$$ 0 0
$$771$$ − 479629.i − 0.806856i
$$772$$ 0 0
$$773$$ −260161. −0.435394 −0.217697 0.976016i $$-0.569855\pi$$
−0.217697 + 0.976016i $$0.569855\pi$$
$$774$$ 0 0
$$775$$ 373662.i 0.622122i
$$776$$ 0 0
$$777$$ −111724. −0.185056
$$778$$ 0 0
$$779$$ − 1.44404e6i − 2.37960i
$$780$$ 0 0
$$781$$ −393556. −0.645216
$$782$$ 0 0
$$783$$ − 113254.i − 0.184726i
$$784$$ 0 0
$$785$$ −1.14433e6 −1.85700
$$786$$ 0 0
$$787$$ − 724342.i − 1.16948i −0.811219 0.584742i $$-0.801196\pi$$
0.811219 0.584742i $$-0.198804\pi$$
$$788$$ 0 0
$$789$$ −72145.7 −0.115893
$$790$$ 0 0
$$791$$ − 539332.i − 0.861993i
$$792$$ 0 0
$$793$$ 147150. 0.233998
$$794$$ 0 0
$$795$$ 155676.i 0.246313i
$$796$$ 0 0
$$797$$ −1.04249e6 −1.64117 −0.820586 0.571524i $$-0.806353\pi$$
−0.820586 + 0.571524i $$0.806353\pi$$
$$798$$ 0 0
$$799$$ − 880309.i − 1.37893i
$$800$$ 0 0
$$801$$ −175336. −0.273279
$$802$$ 0 0
$$803$$ − 416584.i − 0.646057i
$$804$$ 0 0
$$805$$ 1.61506e6 2.49227
$$806$$ 0 0
$$807$$ 273827.i 0.420465i
$$808$$ 0 0
$$809$$ 647222. 0.988909 0.494455 0.869204i $$-0.335368\pi$$
0.494455 + 0.869204i $$0.335368\pi$$
$$810$$ 0 0
$$811$$ 1.24087e6i 1.88662i 0.331916 + 0.943309i $$0.392305\pi$$
−0.331916 + 0.943309i $$0.607695\pi$$
$$812$$ 0 0
$$813$$ −244369. −0.369713
$$814$$ 0 0
$$815$$ − 532993.i − 0.802428i
$$816$$ 0 0
$$817$$ −340418. −0.509997
$$818$$ 0 0
$$819$$ 228118.i 0.340089i
$$820$$ 0 0
$$821$$ 819276. 1.21547 0.607734 0.794141i $$-0.292079\pi$$
0.607734 + 0.794141i $$0.292079\pi$$
$$822$$ 0 0
$$823$$ 515929.i 0.761710i 0.924635 + 0.380855i $$0.124370\pi$$
−0.924635 + 0.380855i $$0.875630\pi$$
$$824$$ 0 0
$$825$$ −899573. −1.32169
$$826$$ 0 0
$$827$$ − 140674.i − 0.205685i −0.994698 0.102842i $$-0.967206\pi$$
0.994698 0.102842i $$-0.0327938\pi$$
$$828$$ 0 0
$$829$$ −137443. −0.199993 −0.0999964 0.994988i $$-0.531883\pi$$
−0.0999964 + 0.994988i $$0.531883\pi$$
$$830$$ 0 0
$$831$$ − 563229.i − 0.815611i
$$832$$ 0 0
$$833$$ 120279. 0.173341
$$834$$ 0 0
$$835$$ 524955.i 0.752920i
$$836$$ 0 0
$$837$$ −54894.5 −0.0783570
$$838$$ 0 0
$$839$$ − 591422.i − 0.840182i −0.907482 0.420091i $$-0.861998\pi$$
0.907482 0.420091i $$-0.138002\pi$$
$$840$$ 0 0
$$841$$ −55633.2 −0.0786579
$$842$$ 0 0
$$843$$ − 379790.i − 0.534428i
$$844$$ 0 0
$$845$$ −203138. −0.284498
$$846$$ 0 0
$$847$$ − 839029.i − 1.16953i
$$848$$ 0 0
$$849$$ 232596. 0.322691
$$850$$ 0 0
$$851$$ − 412093.i − 0.569031i
$$852$$ 0 0
$$853$$ −169773. −0.233331 −0.116665 0.993171i $$-0.537220\pi$$
−0.116665 + 0.993171i $$0.537220\pi$$
$$854$$ 0 0
$$855$$ 717512.i 0.981515i
$$856$$ 0 0
$$857$$ −1.05083e6 −1.43077 −0.715384 0.698732i $$-0.753748\pi$$
−0.715384 + 0.698732i $$0.753748\pi$$
$$858$$ 0 0
$$859$$ 1.05896e6i 1.43514i 0.696487 + 0.717569i $$0.254745\pi$$
−0.696487 + 0.717569i $$0.745255\pi$$
$$860$$ 0 0
$$861$$ −516757. −0.697075
$$862$$ 0 0
$$863$$ − 1.22011e6i − 1.63824i −0.573621 0.819121i $$-0.694462\pi$$
0.573621 0.819121i $$-0.305538\pi$$
$$864$$ 0 0
$$865$$ −14506.6 −0.0193881
$$866$$ 0 0
$$867$$ − 517829.i − 0.688887i
$$868$$ 0 0
$$869$$ −1.94069e6 −2.56991
$$870$$ 0 0
$$871$$ 92815.8i 0.122345i
$$872$$ 0 0
$$873$$ 221128. 0.290145
$$874$$ 0 0
$$875$$ − 603927.i − 0.788802i
$$876$$ 0 0
$$877$$ 707060. 0.919299 0.459650 0.888100i $$-0.347975\pi$$
0.459650 + 0.888100i $$0.347975\pi$$
$$878$$ 0 0
$$879$$ − 218395.i − 0.282661i
$$880$$ 0 0
$$881$$ 430965. 0.555252 0.277626 0.960689i $$-0.410452\pi$$
0.277626 + 0.960689i $$0.410452\pi$$
$$882$$ 0 0
$$883$$ − 1.08382e6i − 1.39007i −0.718974 0.695037i $$-0.755388\pi$$
0.718974 0.695037i $$-0.244612\pi$$
$$884$$ 0 0
$$885$$ 270452. 0.345306
$$886$$ 0 0
$$887$$ − 620229.i − 0.788324i −0.919041 0.394162i $$-0.871035\pi$$
0.919041 0.394162i $$-0.128965\pi$$
$$888$$ 0 0
$$889$$ 259709. 0.328612
$$890$$ 0 0
$$891$$ − 132156.i − 0.166468i
$$892$$ 0 0
$$893$$ 1.37511e6 1.72439
$$894$$ 0 0
$$895$$ 571272.i 0.713176i
$$896$$ 0 0
$$897$$ −841414. −1.04574
$$898$$ 0 0
$$899$$ − 315856.i − 0.390814i
$$900$$ 0 0
$$901$$ 322589. 0.397374
$$902$$ 0 0
$$903$$ 121820.i 0.149398i
$$904$$ 0 0
$$905$$ −161005. −0.196581
$$906$$ 0 0
$$907$$ − 155091.i − 0.188526i −0.995547 0.0942629i $$-0.969951\pi$$
0.995547 0.0942629i $$-0.0300494\pi$$
$$908$$ 0 0
$$909$$ −420566. −0.508986
$$910$$ 0 0
$$911$$ − 670134.i − 0.807467i −0.914877 0.403734i $$-0.867712\pi$$
0.914877 0.403734i $$-0.132288\pi$$
$$912$$ 0 0
$$913$$ 539828. 0.647611
$$914$$ 0 0
$$915$$ − 165629.i − 0.197831i
$$916$$ 0 0
$$917$$ 1.46981e6 1.74793
$$918$$ 0 0
$$919$$ − 631844.i − 0.748134i −0.927402 0.374067i $$-0.877963\pi$$
0.927402 0.374067i $$-0.122037\pi$$
$$920$$ 0 0
$$921$$ −45831.7 −0.0540314
$$922$$ 0 0
$$923$$ 398364.i 0.467602i
$$924$$ 0 0
$$925$$ −445959. −0.521208
$$926$$ 0 0
$$927$$ 178972.i 0.208269i
$$928$$ 0 0
$$929$$ 1.16204e6 1.34645 0.673225 0.739437i $$-0.264908\pi$$
0.673225 + 0.739437i $$0.264908\pi$$
$$930$$ 0 0
$$931$$ 187886.i 0.216768i
$$932$$ 0 0
$$933$$ −805708. −0.925580
$$934$$ 0 0
$$935$$ 3.08404e6i 3.52774i
$$936$$ 0 0
$$937$$ 1.18305e6 1.34748 0.673741 0.738967i $$-0.264686\pi$$
0.673741 + 0.738967i $$0.264686\pi$$
$$938$$ 0 0
$$939$$ − 933570.i − 1.05880i
$$940$$ 0 0
$$941$$ 159241. 0.179836 0.0899179 0.995949i $$-0.471340\pi$$
0.0899179 + 0.995949i $$0.471340\pi$$
$$942$$ 0 0
$$943$$ − 1.90606e6i − 2.14345i
$$944$$ 0 0
$$945$$ 256766. 0.287523
$$946$$ 0 0
$$947$$ − 940316.i − 1.04851i −0.851560 0.524257i $$-0.824343\pi$$
0.851560 0.524257i $$-0.175657\pi$$
$$948$$ 0 0
$$949$$ −421673. −0.468213
$$950$$ 0 0
$$951$$ 32157.8i 0.0355570i
$$952$$ 0 0
$$953$$ −577950. −0.636362 −0.318181 0.948030i $$-0.603072\pi$$
−0.318181 + 0.948030i $$0.603072\pi$$
$$954$$ 0 0
$$955$$ − 1.98272e6i − 2.17397i
$$956$$ 0 0
$$957$$ 760408. 0.830276
$$958$$ 0 0
$$959$$ 484170.i 0.526454i
$$960$$ 0 0
$$961$$ 770424. 0.834225
$$962$$ 0 0
$$963$$ 318504.i 0.343449i
$$964$$ 0 0
$$965$$ 1.92487e6 2.06703
$$966$$ 0 0
$$967$$ 785047.i 0.839542i 0.907630 + 0.419771i $$0.137890\pi$$
−0.907630 + 0.419771i $$0.862110\pi$$
$$968$$ 0 0
$$969$$ 1.48681e6 1.58347
$$970$$ 0 0
$$971$$ − 1.05236e6i − 1.11616i −0.829786 0.558082i $$-0.811538\pi$$
0.829786 0.558082i $$-0.188462\pi$$
$$972$$ 0 0
$$973$$ 689996. 0.728821
$$974$$ 0 0
$$975$$ 910562.i 0.957856i
$$976$$ 0 0
$$977$$ 20197.7 0.0211598 0.0105799 0.999944i $$-0.496632\pi$$
0.0105799 + 0.999944i $$0.496632\pi$$
$$978$$ 0 0
$$979$$ − 1.17724e6i − 1.22829i
$$980$$ 0 0
$$981$$ 358723. 0.372754
$$982$$ 0 0
$$983$$ 1.43076e6i 1.48068i 0.672234 + 0.740338i $$0.265335\pi$$
−0.672234 + 0.740338i $$0.734665\pi$$
$$984$$ 0 0
$$985$$ −220875. −0.227653
$$986$$ 0 0
$$987$$ − 492092.i − 0.505140i
$$988$$ 0 0
$$989$$ −449334. −0.459385
$$990$$ 0 0
$$991$$ 787866.i 0.802241i 0.916025 + 0.401120i $$0.131379\pi$$
−0.916025 + 0.401120i $$0.868621\pi$$
$$992$$ 0 0
$$993$$ −686718. −0.696434
$$994$$ 0 0
$$995$$ 2.40858e6i 2.43285i
$$996$$ 0 0
$$997$$ −453709. −0.456444 −0.228222 0.973609i $$-0.573291\pi$$
−0.228222 + 0.973609i $$0.573291\pi$$
$$998$$ 0 0
$$999$$ − 65515.5i − 0.0656468i
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 768.5.g.c.511.1 4
4.3 odd 2 inner 768.5.g.c.511.3 4
8.3 odd 2 768.5.g.g.511.2 4
8.5 even 2 768.5.g.g.511.4 4
16.3 odd 4 384.5.b.c.319.4 yes 8
16.5 even 4 384.5.b.c.319.1 8
16.11 odd 4 384.5.b.c.319.5 yes 8
16.13 even 4 384.5.b.c.319.8 yes 8
48.5 odd 4 1152.5.b.k.703.7 8
48.11 even 4 1152.5.b.k.703.8 8
48.29 odd 4 1152.5.b.k.703.1 8
48.35 even 4 1152.5.b.k.703.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.1 8 16.5 even 4
384.5.b.c.319.4 yes 8 16.3 odd 4
384.5.b.c.319.5 yes 8 16.11 odd 4
384.5.b.c.319.8 yes 8 16.13 even 4
768.5.g.c.511.1 4 1.1 even 1 trivial
768.5.g.c.511.3 4 4.3 odd 2 inner
768.5.g.g.511.2 4 8.3 odd 2
768.5.g.g.511.4 4 8.5 even 2
1152.5.b.k.703.1 8 48.29 odd 4
1152.5.b.k.703.2 8 48.35 even 4
1152.5.b.k.703.7 8 48.5 odd 4
1152.5.b.k.703.8 8 48.11 even 4