# Properties

 Label 1856.2.a.y.1.3 Level $1856$ Weight $2$ Character 1856.1 Self dual yes Analytic conductor $14.820$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ 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 232) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 1856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +O(q^{10})$$ $$q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +4.49084 q^{11} -0.103084 q^{13} +4.49084 q^{15} +2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} +4.49084 q^{27} -1.00000 q^{29} +6.76156 q^{31} +12.4017 q^{33} -5.25240 q^{37} -0.284672 q^{39} +5.79383 q^{41} -10.0140 q^{43} +7.52311 q^{45} -11.5371 q^{47} -7.00000 q^{49} +5.52311 q^{51} +7.14931 q^{53} +7.30299 q^{55} -20.0279 q^{57} -1.52311 q^{59} -9.04623 q^{61} -0.167635 q^{65} +15.0462 q^{67} +15.2524 q^{69} +12.0279 q^{71} -1.79383 q^{73} -6.50479 q^{75} +1.98605 q^{79} -1.47689 q^{81} +6.47689 q^{83} +3.25240 q^{85} -2.76156 q^{87} +12.7110 q^{89} +18.6724 q^{93} -11.7938 q^{95} +1.25240 q^{97} +20.7755 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 4 q^{5} + 5 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 4 * q^5 + 5 * q^9 $$3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43} + 10 q^{45} + 2 q^{47} - 21 q^{49} + 4 q^{51} + 26 q^{55} - 12 q^{57} + 8 q^{59} - 2 q^{61} - 2 q^{65} + 20 q^{67} + 28 q^{69} - 12 q^{71} + 2 q^{73} + 16 q^{75} + 30 q^{79} - 17 q^{81} + 32 q^{83} - 8 q^{85} - 2 q^{87} + 10 q^{89} + 22 q^{93} - 28 q^{95} - 14 q^{97} + 32 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 4 * q^5 + 5 * q^9 + 2 * q^11 - 4 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 + 2 * q^27 - 3 * q^29 + 14 * q^31 - 2 * q^33 + 2 * q^37 + 18 * q^39 + 10 * q^41 - 6 * q^43 + 10 * q^45 + 2 * q^47 - 21 * q^49 + 4 * q^51 + 26 * q^55 - 12 * q^57 + 8 * q^59 - 2 * q^61 - 2 * q^65 + 20 * q^67 + 28 * q^69 - 12 * q^71 + 2 * q^73 + 16 * q^75 + 30 * q^79 - 17 * q^81 + 32 * q^83 - 8 * q^85 - 2 * q^87 + 10 * q^89 + 22 * q^93 - 28 * q^95 - 14 * q^97 + 32 * 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
$$3$$ 2.76156 1.59439 0.797193 0.603725i $$-0.206317\pi$$
0.797193 + 0.603725i $$0.206317\pi$$
$$4$$ 0 0
$$5$$ 1.62620 0.727258 0.363629 0.931544i $$-0.381538\pi$$
0.363629 + 0.931544i $$0.381538\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 4.62620 1.54207
$$10$$ 0 0
$$11$$ 4.49084 1.35404 0.677019 0.735965i $$-0.263271\pi$$
0.677019 + 0.735965i $$0.263271\pi$$
$$12$$ 0 0
$$13$$ −0.103084 −0.0285903 −0.0142951 0.999898i $$-0.504550\pi$$
−0.0142951 + 0.999898i $$0.504550\pi$$
$$14$$ 0 0
$$15$$ 4.49084 1.15953
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −7.25240 −1.66381 −0.831907 0.554915i $$-0.812751\pi$$
−0.831907 + 0.554915i $$0.812751\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.52311 1.15165 0.575824 0.817573i $$-0.304681\pi$$
0.575824 + 0.817573i $$0.304681\pi$$
$$24$$ 0 0
$$25$$ −2.35548 −0.471096
$$26$$ 0 0
$$27$$ 4.49084 0.864262
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 6.76156 1.21441 0.607206 0.794545i $$-0.292290\pi$$
0.607206 + 0.794545i $$0.292290\pi$$
$$32$$ 0 0
$$33$$ 12.4017 2.15886
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.25240 −0.863489 −0.431744 0.901996i $$-0.642102\pi$$
−0.431744 + 0.901996i $$0.642102\pi$$
$$38$$ 0 0
$$39$$ −0.284672 −0.0455839
$$40$$ 0 0
$$41$$ 5.79383 0.904845 0.452422 0.891804i $$-0.350560\pi$$
0.452422 + 0.891804i $$0.350560\pi$$
$$42$$ 0 0
$$43$$ −10.0140 −1.52711 −0.763557 0.645741i $$-0.776549\pi$$
−0.763557 + 0.645741i $$0.776549\pi$$
$$44$$ 0 0
$$45$$ 7.52311 1.12148
$$46$$ 0 0
$$47$$ −11.5371 −1.68285 −0.841427 0.540371i $$-0.818284\pi$$
−0.841427 + 0.540371i $$0.818284\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 5.52311 0.773391
$$52$$ 0 0
$$53$$ 7.14931 0.982034 0.491017 0.871150i $$-0.336625\pi$$
0.491017 + 0.871150i $$0.336625\pi$$
$$54$$ 0 0
$$55$$ 7.30299 0.984735
$$56$$ 0 0
$$57$$ −20.0279 −2.65276
$$58$$ 0 0
$$59$$ −1.52311 −0.198293 −0.0991463 0.995073i $$-0.531611\pi$$
−0.0991463 + 0.995073i $$0.531611\pi$$
$$60$$ 0 0
$$61$$ −9.04623 −1.15825 −0.579125 0.815238i $$-0.696606\pi$$
−0.579125 + 0.815238i $$0.696606\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.167635 −0.0207925
$$66$$ 0 0
$$67$$ 15.0462 1.83819 0.919095 0.394037i $$-0.128922\pi$$
0.919095 + 0.394037i $$0.128922\pi$$
$$68$$ 0 0
$$69$$ 15.2524 1.83617
$$70$$ 0 0
$$71$$ 12.0279 1.42745 0.713725 0.700426i $$-0.247007\pi$$
0.713725 + 0.700426i $$0.247007\pi$$
$$72$$ 0 0
$$73$$ −1.79383 −0.209952 −0.104976 0.994475i $$-0.533477\pi$$
−0.104976 + 0.994475i $$0.533477\pi$$
$$74$$ 0 0
$$75$$ −6.50479 −0.751109
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.98605 0.223448 0.111724 0.993739i $$-0.464363\pi$$
0.111724 + 0.993739i $$0.464363\pi$$
$$80$$ 0 0
$$81$$ −1.47689 −0.164098
$$82$$ 0 0
$$83$$ 6.47689 0.710931 0.355465 0.934689i $$-0.384322\pi$$
0.355465 + 0.934689i $$0.384322\pi$$
$$84$$ 0 0
$$85$$ 3.25240 0.352772
$$86$$ 0 0
$$87$$ −2.76156 −0.296070
$$88$$ 0 0
$$89$$ 12.7110 1.34736 0.673680 0.739024i $$-0.264713\pi$$
0.673680 + 0.739024i $$0.264713\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 18.6724 1.93624
$$94$$ 0 0
$$95$$ −11.7938 −1.21002
$$96$$ 0 0
$$97$$ 1.25240 0.127162 0.0635808 0.997977i $$-0.479748\pi$$
0.0635808 + 0.997977i $$0.479748\pi$$
$$98$$ 0 0
$$99$$ 20.7755 2.08802
$$100$$ 0 0
$$101$$ −16.2986 −1.62177 −0.810887 0.585203i $$-0.801015\pi$$
−0.810887 + 0.585203i $$0.801015\pi$$
$$102$$ 0 0
$$103$$ 5.52311 0.544209 0.272104 0.962268i $$-0.412280\pi$$
0.272104 + 0.962268i $$0.412280\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.541436 0.0523426 0.0261713 0.999657i $$-0.491668\pi$$
0.0261713 + 0.999657i $$0.491668\pi$$
$$108$$ 0 0
$$109$$ −15.9248 −1.52532 −0.762661 0.646799i $$-0.776107\pi$$
−0.762661 + 0.646799i $$0.776107\pi$$
$$110$$ 0 0
$$111$$ −14.5048 −1.37673
$$112$$ 0 0
$$113$$ −12.5048 −1.17635 −0.588176 0.808733i $$-0.700154\pi$$
−0.588176 + 0.808733i $$0.700154\pi$$
$$114$$ 0 0
$$115$$ 8.98168 0.837546
$$116$$ 0 0
$$117$$ −0.476886 −0.0440881
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 9.16763 0.833421
$$122$$ 0 0
$$123$$ 16.0000 1.44267
$$124$$ 0 0
$$125$$ −11.9615 −1.06987
$$126$$ 0 0
$$127$$ 0.206167 0.0182944 0.00914720 0.999958i $$-0.497088\pi$$
0.00914720 + 0.999958i $$0.497088\pi$$
$$128$$ 0 0
$$129$$ −27.6541 −2.43481
$$130$$ 0 0
$$131$$ 15.2524 1.33261 0.666304 0.745680i $$-0.267875\pi$$
0.666304 + 0.745680i $$0.267875\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 7.30299 0.628542
$$136$$ 0 0
$$137$$ −5.25240 −0.448742 −0.224371 0.974504i $$-0.572033\pi$$
−0.224371 + 0.974504i $$0.572033\pi$$
$$138$$ 0 0
$$139$$ −12.5693 −1.06612 −0.533059 0.846078i $$-0.678958\pi$$
−0.533059 + 0.846078i $$0.678958\pi$$
$$140$$ 0 0
$$141$$ −31.8603 −2.68312
$$142$$ 0 0
$$143$$ −0.462932 −0.0387123
$$144$$ 0 0
$$145$$ −1.62620 −0.135048
$$146$$ 0 0
$$147$$ −19.3309 −1.59439
$$148$$ 0 0
$$149$$ 0.309251 0.0253348 0.0126674 0.999920i $$-0.495968\pi$$
0.0126674 + 0.999920i $$0.495968\pi$$
$$150$$ 0 0
$$151$$ 11.4586 0.932485 0.466242 0.884657i $$-0.345607\pi$$
0.466242 + 0.884657i $$0.345607\pi$$
$$152$$ 0 0
$$153$$ 9.25240 0.748012
$$154$$ 0 0
$$155$$ 10.9956 0.883190
$$156$$ 0 0
$$157$$ −13.2524 −1.05766 −0.528828 0.848729i $$-0.677368\pi$$
−0.528828 + 0.848729i $$0.677368\pi$$
$$158$$ 0 0
$$159$$ 19.7432 1.56574
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3.17389 −0.248598 −0.124299 0.992245i $$-0.539668\pi$$
−0.124299 + 0.992245i $$0.539668\pi$$
$$164$$ 0 0
$$165$$ 20.1676 1.57005
$$166$$ 0 0
$$167$$ 12.0279 0.930747 0.465374 0.885114i $$-0.345920\pi$$
0.465374 + 0.885114i $$0.345920\pi$$
$$168$$ 0 0
$$169$$ −12.9894 −0.999183
$$170$$ 0 0
$$171$$ −33.5510 −2.56571
$$172$$ 0 0
$$173$$ −12.5048 −0.950722 −0.475361 0.879791i $$-0.657682\pi$$
−0.475361 + 0.879791i $$0.657682\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.20617 −0.316155
$$178$$ 0 0
$$179$$ 8.02791 0.600034 0.300017 0.953934i $$-0.403008\pi$$
0.300017 + 0.953934i $$0.403008\pi$$
$$180$$ 0 0
$$181$$ 11.9248 0.886365 0.443183 0.896431i $$-0.353849\pi$$
0.443183 + 0.896431i $$0.353849\pi$$
$$182$$ 0 0
$$183$$ −24.9817 −1.84670
$$184$$ 0 0
$$185$$ −8.54144 −0.627979
$$186$$ 0 0
$$187$$ 8.98168 0.656805
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.71096 −0.485588 −0.242794 0.970078i $$-0.578064\pi$$
−0.242794 + 0.970078i $$0.578064\pi$$
$$192$$ 0 0
$$193$$ 12.2986 0.885274 0.442637 0.896701i $$-0.354043\pi$$
0.442637 + 0.896701i $$0.354043\pi$$
$$194$$ 0 0
$$195$$ −0.462932 −0.0331513
$$196$$ 0 0
$$197$$ −4.50479 −0.320953 −0.160477 0.987040i $$-0.551303\pi$$
−0.160477 + 0.987040i $$0.551303\pi$$
$$198$$ 0 0
$$199$$ 19.4586 1.37938 0.689690 0.724104i $$-0.257747\pi$$
0.689690 + 0.724104i $$0.257747\pi$$
$$200$$ 0 0
$$201$$ 41.5510 2.93078
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.42192 0.658055
$$206$$ 0 0
$$207$$ 25.5510 1.77592
$$208$$ 0 0
$$209$$ −32.5693 −2.25287
$$210$$ 0 0
$$211$$ 1.77988 0.122532 0.0612660 0.998121i $$-0.480486\pi$$
0.0612660 + 0.998121i $$0.480486\pi$$
$$212$$ 0 0
$$213$$ 33.2158 2.27591
$$214$$ 0 0
$$215$$ −16.2847 −1.11061
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −4.95377 −0.334745
$$220$$ 0 0
$$221$$ −0.206167 −0.0138683
$$222$$ 0 0
$$223$$ −26.5327 −1.77676 −0.888380 0.459108i $$-0.848169\pi$$
−0.888380 + 0.459108i $$0.848169\pi$$
$$224$$ 0 0
$$225$$ −10.8969 −0.726461
$$226$$ 0 0
$$227$$ −5.39401 −0.358013 −0.179007 0.983848i $$-0.557288\pi$$
−0.179007 + 0.983848i $$0.557288\pi$$
$$228$$ 0 0
$$229$$ −16.2986 −1.07704 −0.538522 0.842612i $$-0.681017\pi$$
−0.538522 + 0.842612i $$0.681017\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.64452 0.566321 0.283161 0.959072i $$-0.408617\pi$$
0.283161 + 0.959072i $$0.408617\pi$$
$$234$$ 0 0
$$235$$ −18.7616 −1.22387
$$236$$ 0 0
$$237$$ 5.48458 0.356262
$$238$$ 0 0
$$239$$ −13.5231 −0.874738 −0.437369 0.899282i $$-0.644090\pi$$
−0.437369 + 0.899282i $$0.644090\pi$$
$$240$$ 0 0
$$241$$ 8.30925 0.535246 0.267623 0.963524i $$-0.413762\pi$$
0.267623 + 0.963524i $$0.413762\pi$$
$$242$$ 0 0
$$243$$ −17.5510 −1.12590
$$244$$ 0 0
$$245$$ −11.3834 −0.727258
$$246$$ 0 0
$$247$$ 0.747604 0.0475689
$$248$$ 0 0
$$249$$ 17.8863 1.13350
$$250$$ 0 0
$$251$$ −19.7432 −1.24618 −0.623091 0.782149i $$-0.714123\pi$$
−0.623091 + 0.782149i $$0.714123\pi$$
$$252$$ 0 0
$$253$$ 24.8034 1.55938
$$254$$ 0 0
$$255$$ 8.98168 0.562454
$$256$$ 0 0
$$257$$ 21.4200 1.33614 0.668072 0.744096i $$-0.267120\pi$$
0.668072 + 0.744096i $$0.267120\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.62620 −0.286354
$$262$$ 0 0
$$263$$ −12.2847 −0.757505 −0.378753 0.925498i $$-0.623647\pi$$
−0.378753 + 0.925498i $$0.623647\pi$$
$$264$$ 0 0
$$265$$ 11.6262 0.714192
$$266$$ 0 0
$$267$$ 35.1020 2.14821
$$268$$ 0 0
$$269$$ 22.5972 1.37778 0.688889 0.724867i $$-0.258099\pi$$
0.688889 + 0.724867i $$0.258099\pi$$
$$270$$ 0 0
$$271$$ 23.7432 1.44230 0.721149 0.692780i $$-0.243614\pi$$
0.721149 + 0.692780i $$0.243614\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −10.5781 −0.637882
$$276$$ 0 0
$$277$$ 1.58767 0.0953936 0.0476968 0.998862i $$-0.484812\pi$$
0.0476968 + 0.998862i $$0.484812\pi$$
$$278$$ 0 0
$$279$$ 31.2803 1.87270
$$280$$ 0 0
$$281$$ −19.8969 −1.18695 −0.593475 0.804852i $$-0.702245\pi$$
−0.593475 + 0.804852i $$0.702245\pi$$
$$282$$ 0 0
$$283$$ 20.9817 1.24723 0.623616 0.781731i $$-0.285663\pi$$
0.623616 + 0.781731i $$0.285663\pi$$
$$284$$ 0 0
$$285$$ −32.5693 −1.92924
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 3.45856 0.202745
$$292$$ 0 0
$$293$$ −4.50479 −0.263173 −0.131586 0.991305i $$-0.542007\pi$$
−0.131586 + 0.991305i $$0.542007\pi$$
$$294$$ 0 0
$$295$$ −2.47689 −0.144210
$$296$$ 0 0
$$297$$ 20.1676 1.17024
$$298$$ 0 0
$$299$$ −0.569343 −0.0329260
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −45.0096 −2.58573
$$304$$ 0 0
$$305$$ −14.7110 −0.842347
$$306$$ 0 0
$$307$$ −9.60162 −0.547993 −0.273997 0.961731i $$-0.588346\pi$$
−0.273997 + 0.961731i $$0.588346\pi$$
$$308$$ 0 0
$$309$$ 15.2524 0.867679
$$310$$ 0 0
$$311$$ 30.2986 1.71808 0.859039 0.511911i $$-0.171062\pi$$
0.859039 + 0.511911i $$0.171062\pi$$
$$312$$ 0 0
$$313$$ 2.37380 0.134175 0.0670876 0.997747i $$-0.478629\pi$$
0.0670876 + 0.997747i $$0.478629\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 11.5510 0.648770 0.324385 0.945925i $$-0.394843\pi$$
0.324385 + 0.945925i $$0.394843\pi$$
$$318$$ 0 0
$$319$$ −4.49084 −0.251439
$$320$$ 0 0
$$321$$ 1.49521 0.0834544
$$322$$ 0 0
$$323$$ −14.5048 −0.807068
$$324$$ 0 0
$$325$$ 0.242812 0.0134688
$$326$$ 0 0
$$327$$ −43.9773 −2.43195
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.3126 −0.676761 −0.338380 0.941009i $$-0.609879\pi$$
−0.338380 + 0.941009i $$0.609879\pi$$
$$332$$ 0 0
$$333$$ −24.2986 −1.33156
$$334$$ 0 0
$$335$$ 24.4681 1.33684
$$336$$ 0 0
$$337$$ 6.95377 0.378796 0.189398 0.981900i $$-0.439346\pi$$
0.189398 + 0.981900i $$0.439346\pi$$
$$338$$ 0 0
$$339$$ −34.5327 −1.87556
$$340$$ 0 0
$$341$$ 30.3651 1.64436
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 24.8034 1.33537
$$346$$ 0 0
$$347$$ −16.4402 −0.882558 −0.441279 0.897370i $$-0.645475\pi$$
−0.441279 + 0.897370i $$0.645475\pi$$
$$348$$ 0 0
$$349$$ −22.9431 −1.22812 −0.614059 0.789260i $$-0.710464\pi$$
−0.614059 + 0.789260i $$0.710464\pi$$
$$350$$ 0 0
$$351$$ −0.462932 −0.0247095
$$352$$ 0 0
$$353$$ 8.09246 0.430718 0.215359 0.976535i $$-0.430908\pi$$
0.215359 + 0.976535i $$0.430908\pi$$
$$354$$ 0 0
$$355$$ 19.5598 1.03812
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −20.5187 −1.08294 −0.541469 0.840721i $$-0.682132\pi$$
−0.541469 + 0.840721i $$0.682132\pi$$
$$360$$ 0 0
$$361$$ 33.5972 1.76828
$$362$$ 0 0
$$363$$ 25.3169 1.32880
$$364$$ 0 0
$$365$$ −2.91713 −0.152689
$$366$$ 0 0
$$367$$ −1.75719 −0.0917245 −0.0458622 0.998948i $$-0.514604\pi$$
−0.0458622 + 0.998948i $$0.514604\pi$$
$$368$$ 0 0
$$369$$ 26.8034 1.39533
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −28.1310 −1.45657 −0.728284 0.685276i $$-0.759682\pi$$
−0.728284 + 0.685276i $$0.759682\pi$$
$$374$$ 0 0
$$375$$ −33.0323 −1.70578
$$376$$ 0 0
$$377$$ 0.103084 0.00530908
$$378$$ 0 0
$$379$$ 24.8034 1.27407 0.637033 0.770837i $$-0.280161\pi$$
0.637033 + 0.770837i $$0.280161\pi$$
$$380$$ 0 0
$$381$$ 0.569343 0.0291683
$$382$$ 0 0
$$383$$ −30.5048 −1.55872 −0.779361 0.626575i $$-0.784456\pi$$
−0.779361 + 0.626575i $$0.784456\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −46.3265 −2.35491
$$388$$ 0 0
$$389$$ 32.8401 1.66506 0.832529 0.553982i $$-0.186892\pi$$
0.832529 + 0.553982i $$0.186892\pi$$
$$390$$ 0 0
$$391$$ 11.0462 0.558632
$$392$$ 0 0
$$393$$ 42.1204 2.12469
$$394$$ 0 0
$$395$$ 3.22971 0.162504
$$396$$ 0 0
$$397$$ 28.3372 1.42220 0.711101 0.703090i $$-0.248197\pi$$
0.711101 + 0.703090i $$0.248197\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.94315 −0.346724 −0.173362 0.984858i $$-0.555463\pi$$
−0.173362 + 0.984858i $$0.555463\pi$$
$$402$$ 0 0
$$403$$ −0.697006 −0.0347204
$$404$$ 0 0
$$405$$ −2.40171 −0.119342
$$406$$ 0 0
$$407$$ −23.5877 −1.16920
$$408$$ 0 0
$$409$$ −39.1387 −1.93528 −0.967642 0.252328i $$-0.918804\pi$$
−0.967642 + 0.252328i $$0.918804\pi$$
$$410$$ 0 0
$$411$$ −14.5048 −0.715469
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 10.5327 0.517030
$$416$$ 0 0
$$417$$ −34.7110 −1.69980
$$418$$ 0 0
$$419$$ 15.6156 0.762871 0.381435 0.924396i $$-0.375430\pi$$
0.381435 + 0.924396i $$0.375430\pi$$
$$420$$ 0 0
$$421$$ 8.50479 0.414498 0.207249 0.978288i $$-0.433549\pi$$
0.207249 + 0.978288i $$0.433549\pi$$
$$422$$ 0 0
$$423$$ −53.3728 −2.59507
$$424$$ 0 0
$$425$$ −4.71096 −0.228515
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.27841 −0.0617224
$$430$$ 0 0
$$431$$ 6.60599 0.318199 0.159100 0.987263i $$-0.449141\pi$$
0.159100 + 0.987263i $$0.449141\pi$$
$$432$$ 0 0
$$433$$ −2.20617 −0.106022 −0.0530108 0.998594i $$-0.516882\pi$$
−0.0530108 + 0.998594i $$0.516882\pi$$
$$434$$ 0 0
$$435$$ −4.49084 −0.215319
$$436$$ 0 0
$$437$$ −40.0558 −1.91613
$$438$$ 0 0
$$439$$ −4.44024 −0.211921 −0.105961 0.994370i $$-0.533792\pi$$
−0.105961 + 0.994370i $$0.533792\pi$$
$$440$$ 0 0
$$441$$ −32.3834 −1.54207
$$442$$ 0 0
$$443$$ −21.7572 −1.03372 −0.516858 0.856071i $$-0.672898\pi$$
−0.516858 + 0.856071i $$0.672898\pi$$
$$444$$ 0 0
$$445$$ 20.6705 0.979877
$$446$$ 0 0
$$447$$ 0.854015 0.0403935
$$448$$ 0 0
$$449$$ −34.5972 −1.63275 −0.816373 0.577526i $$-0.804018\pi$$
−0.816373 + 0.577526i $$0.804018\pi$$
$$450$$ 0 0
$$451$$ 26.0192 1.22519
$$452$$ 0 0
$$453$$ 31.6435 1.48674
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −28.0925 −1.31411 −0.657055 0.753843i $$-0.728198\pi$$
−0.657055 + 0.753843i $$0.728198\pi$$
$$458$$ 0 0
$$459$$ 8.98168 0.419229
$$460$$ 0 0
$$461$$ 8.91713 0.415312 0.207656 0.978202i $$-0.433417\pi$$
0.207656 + 0.978202i $$0.433417\pi$$
$$462$$ 0 0
$$463$$ −3.55976 −0.165436 −0.0827180 0.996573i $$-0.526360\pi$$
−0.0827180 + 0.996573i $$0.526360\pi$$
$$464$$ 0 0
$$465$$ 30.3651 1.40815
$$466$$ 0 0
$$467$$ −11.0968 −0.513500 −0.256750 0.966478i $$-0.582652\pi$$
−0.256750 + 0.966478i $$0.582652\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −36.5972 −1.68631
$$472$$ 0 0
$$473$$ −44.9711 −2.06777
$$474$$ 0 0
$$475$$ 17.0829 0.783816
$$476$$ 0 0
$$477$$ 33.0741 1.51436
$$478$$ 0 0
$$479$$ 5.77988 0.264089 0.132045 0.991244i $$-0.457846\pi$$
0.132045 + 0.991244i $$0.457846\pi$$
$$480$$ 0 0
$$481$$ 0.541436 0.0246874
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.03664 0.0924792
$$486$$ 0 0
$$487$$ −11.0462 −0.500552 −0.250276 0.968174i $$-0.580521\pi$$
−0.250276 + 0.968174i $$0.580521\pi$$
$$488$$ 0 0
$$489$$ −8.76488 −0.396362
$$490$$ 0 0
$$491$$ −9.03228 −0.407621 −0.203810 0.979010i $$-0.565333\pi$$
−0.203810 + 0.979010i $$0.565333\pi$$
$$492$$ 0 0
$$493$$ −2.00000 −0.0900755
$$494$$ 0 0
$$495$$ 33.7851 1.51853
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −39.6156 −1.77344 −0.886718 0.462310i $$-0.847021\pi$$
−0.886718 + 0.462310i $$0.847021\pi$$
$$500$$ 0 0
$$501$$ 33.2158 1.48397
$$502$$ 0 0
$$503$$ 26.7895 1.19448 0.597242 0.802061i $$-0.296263\pi$$
0.597242 + 0.802061i $$0.296263\pi$$
$$504$$ 0 0
$$505$$ −26.5048 −1.17945
$$506$$ 0 0
$$507$$ −35.8709 −1.59308
$$508$$ 0 0
$$509$$ 17.1127 0.758506 0.379253 0.925293i $$-0.376181\pi$$
0.379253 + 0.925293i $$0.376181\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −32.5693 −1.43797
$$514$$ 0 0
$$515$$ 8.98168 0.395780
$$516$$ 0 0
$$517$$ −51.8111 −2.27865
$$518$$ 0 0
$$519$$ −34.5327 −1.51582
$$520$$ 0 0
$$521$$ 19.5896 0.858234 0.429117 0.903249i $$-0.358825\pi$$
0.429117 + 0.903249i $$0.358825\pi$$
$$522$$ 0 0
$$523$$ 15.0462 0.657926 0.328963 0.944343i $$-0.393301\pi$$
0.328963 + 0.944343i $$0.393301\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 13.5231 0.589076
$$528$$ 0 0
$$529$$ 7.50479 0.326295
$$530$$ 0 0
$$531$$ −7.04623 −0.305780
$$532$$ 0 0
$$533$$ −0.597250 −0.0258698
$$534$$ 0 0
$$535$$ 0.880483 0.0380666
$$536$$ 0 0
$$537$$ 22.1695 0.956686
$$538$$ 0 0
$$539$$ −31.4359 −1.35404
$$540$$ 0 0
$$541$$ −31.2158 −1.34207 −0.671035 0.741426i $$-0.734150\pi$$
−0.671035 + 0.741426i $$0.734150\pi$$
$$542$$ 0 0
$$543$$ 32.9311 1.41321
$$544$$ 0 0
$$545$$ −25.8969 −1.10930
$$546$$ 0 0
$$547$$ 10.5048 0.449152 0.224576 0.974457i $$-0.427900\pi$$
0.224576 + 0.974457i $$0.427900\pi$$
$$548$$ 0 0
$$549$$ −41.8496 −1.78610
$$550$$ 0 0
$$551$$ 7.25240 0.308962
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −23.5877 −1.00124
$$556$$ 0 0
$$557$$ −27.6801 −1.17284 −0.586422 0.810006i $$-0.699464\pi$$
−0.586422 + 0.810006i $$0.699464\pi$$
$$558$$ 0 0
$$559$$ 1.03228 0.0436606
$$560$$ 0 0
$$561$$ 24.8034 1.04720
$$562$$ 0 0
$$563$$ 13.6508 0.575312 0.287656 0.957734i $$-0.407124\pi$$
0.287656 + 0.957734i $$0.407124\pi$$
$$564$$ 0 0
$$565$$ −20.3353 −0.855511
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 27.1387 1.13771 0.568856 0.822437i $$-0.307386\pi$$
0.568856 + 0.822437i $$0.307386\pi$$
$$570$$ 0 0
$$571$$ 21.9634 0.919138 0.459569 0.888142i $$-0.348004\pi$$
0.459569 + 0.888142i $$0.348004\pi$$
$$572$$ 0 0
$$573$$ −18.5327 −0.774215
$$574$$ 0 0
$$575$$ −13.0096 −0.542537
$$576$$ 0 0
$$577$$ −10.5414 −0.438846 −0.219423 0.975630i $$-0.570417\pi$$
−0.219423 + 0.975630i $$0.570417\pi$$
$$578$$ 0 0
$$579$$ 33.9634 1.41147
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 32.1064 1.32971
$$584$$ 0 0
$$585$$ −0.775511 −0.0320634
$$586$$ 0 0
$$587$$ 14.9450 0.616848 0.308424 0.951249i $$-0.400199\pi$$
0.308424 + 0.951249i $$0.400199\pi$$
$$588$$ 0 0
$$589$$ −49.0375 −2.02055
$$590$$ 0 0
$$591$$ −12.4402 −0.511723
$$592$$ 0 0
$$593$$ 24.8015 1.01848 0.509238 0.860626i $$-0.329927\pi$$
0.509238 + 0.860626i $$0.329927\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 53.7359 2.19927
$$598$$ 0 0
$$599$$ 13.4446 0.549332 0.274666 0.961540i $$-0.411433\pi$$
0.274666 + 0.961540i $$0.411433\pi$$
$$600$$ 0 0
$$601$$ 25.2524 1.03007 0.515033 0.857170i $$-0.327780\pi$$
0.515033 + 0.857170i $$0.327780\pi$$
$$602$$ 0 0
$$603$$ 69.6068 2.83461
$$604$$ 0 0
$$605$$ 14.9084 0.606112
$$606$$ 0 0
$$607$$ 2.78946 0.113221 0.0566104 0.998396i $$-0.481971\pi$$
0.0566104 + 0.998396i $$0.481971\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.18928 0.0481133
$$612$$ 0 0
$$613$$ 43.1772 1.74391 0.871956 0.489585i $$-0.162852\pi$$
0.871956 + 0.489585i $$0.162852\pi$$
$$614$$ 0 0
$$615$$ 26.0192 1.04919
$$616$$ 0 0
$$617$$ 10.7476 0.432682 0.216341 0.976318i $$-0.430588\pi$$
0.216341 + 0.976318i $$0.430588\pi$$
$$618$$ 0 0
$$619$$ 30.3771 1.22096 0.610480 0.792032i $$-0.290977\pi$$
0.610480 + 0.792032i $$0.290977\pi$$
$$620$$ 0 0
$$621$$ 24.8034 0.995327
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −7.67432 −0.306973
$$626$$ 0 0
$$627$$ −89.9421 −3.59194
$$628$$ 0 0
$$629$$ −10.5048 −0.418853
$$630$$ 0 0
$$631$$ 13.5231 0.538347 0.269173 0.963092i $$-0.413250\pi$$
0.269173 + 0.963092i $$0.413250\pi$$
$$632$$ 0 0
$$633$$ 4.91524 0.195363
$$634$$ 0 0
$$635$$ 0.335269 0.0133047
$$636$$ 0 0
$$637$$ 0.721586 0.0285903
$$638$$ 0 0
$$639$$ 55.6435 2.20122
$$640$$ 0 0
$$641$$ −16.6339 −0.656999 −0.328500 0.944504i $$-0.606543\pi$$
−0.328500 + 0.944504i $$0.606543\pi$$
$$642$$ 0 0
$$643$$ 24.9538 0.984081 0.492040 0.870572i $$-0.336251\pi$$
0.492040 + 0.870572i $$0.336251\pi$$
$$644$$ 0 0
$$645$$ −44.9711 −1.77073
$$646$$ 0 0
$$647$$ −10.0646 −0.395678 −0.197839 0.980234i $$-0.563392\pi$$
−0.197839 + 0.980234i $$0.563392\pi$$
$$648$$ 0 0
$$649$$ −6.84006 −0.268496
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −42.2620 −1.65384 −0.826920 0.562320i $$-0.809909\pi$$
−0.826920 + 0.562320i $$0.809909\pi$$
$$654$$ 0 0
$$655$$ 24.8034 0.969150
$$656$$ 0 0
$$657$$ −8.29862 −0.323760
$$658$$ 0 0
$$659$$ −43.3867 −1.69011 −0.845053 0.534682i $$-0.820431\pi$$
−0.845053 + 0.534682i $$0.820431\pi$$
$$660$$ 0 0
$$661$$ 0.0924575 0.00359618 0.00179809 0.999998i $$-0.499428\pi$$
0.00179809 + 0.999998i $$0.499428\pi$$
$$662$$ 0 0
$$663$$ −0.569343 −0.0221115
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5.52311 −0.213856
$$668$$ 0 0
$$669$$ −73.2716 −2.83284
$$670$$ 0 0
$$671$$ −40.6252 −1.56832
$$672$$ 0 0
$$673$$ 1.86027 0.0717082 0.0358541 0.999357i $$-0.488585\pi$$
0.0358541 + 0.999357i $$0.488585\pi$$
$$674$$ 0 0
$$675$$ −10.5781 −0.407151
$$676$$ 0 0
$$677$$ 11.1387 0.428094 0.214047 0.976823i $$-0.431335\pi$$
0.214047 + 0.976823i $$0.431335\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −14.8959 −0.570811
$$682$$ 0 0
$$683$$ −8.54144 −0.326829 −0.163414 0.986558i $$-0.552251\pi$$
−0.163414 + 0.986558i $$0.552251\pi$$
$$684$$ 0 0
$$685$$ −8.54144 −0.326352
$$686$$ 0 0
$$687$$ −45.0096 −1.71722
$$688$$ 0 0
$$689$$ −0.736978 −0.0280766
$$690$$ 0 0
$$691$$ −24.5972 −0.935723 −0.467862 0.883802i $$-0.654975\pi$$
−0.467862 + 0.883802i $$0.654975\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −20.4402 −0.775343
$$696$$ 0 0
$$697$$ 11.5877 0.438914
$$698$$ 0 0
$$699$$ 23.8723 0.902935
$$700$$ 0 0
$$701$$ −3.56165 −0.134522 −0.0672608 0.997735i $$-0.521426\pi$$
−0.0672608 + 0.997735i $$0.521426\pi$$
$$702$$ 0 0
$$703$$ 38.0925 1.43668
$$704$$ 0 0
$$705$$ −51.8111 −1.95132
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 24.3651 0.915049 0.457525 0.889197i $$-0.348736\pi$$
0.457525 + 0.889197i $$0.348736\pi$$
$$710$$ 0 0
$$711$$ 9.18785 0.344571
$$712$$ 0 0
$$713$$ 37.3449 1.39858
$$714$$ 0 0
$$715$$ −0.752820 −0.0281539
$$716$$ 0 0
$$717$$ −37.3449 −1.39467
$$718$$ 0 0
$$719$$ −7.89881 −0.294576 −0.147288 0.989094i $$-0.547054\pi$$
−0.147288 + 0.989094i $$0.547054\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 22.9465 0.853388
$$724$$ 0 0
$$725$$ 2.35548 0.0874803
$$726$$ 0 0
$$727$$ −33.8130 −1.25405 −0.627027 0.778997i $$-0.715729\pi$$
−0.627027 + 0.778997i $$0.715729\pi$$
$$728$$ 0 0
$$729$$ −44.0375 −1.63102
$$730$$ 0 0
$$731$$ −20.0279 −0.740759
$$732$$ 0 0
$$733$$ −45.7205 −1.68873 −0.844363 0.535771i $$-0.820021\pi$$
−0.844363 + 0.535771i $$0.820021\pi$$
$$734$$ 0 0
$$735$$ −31.4359 −1.15953
$$736$$ 0 0
$$737$$ 67.5702 2.48898
$$738$$ 0 0
$$739$$ −8.28467 −0.304757 −0.152378 0.988322i $$-0.548693\pi$$
−0.152378 + 0.988322i $$0.548693\pi$$
$$740$$ 0 0
$$741$$ 2.06455 0.0758432
$$742$$ 0 0
$$743$$ 1.70138 0.0624174 0.0312087 0.999513i $$-0.490064\pi$$
0.0312087 + 0.999513i $$0.490064\pi$$
$$744$$ 0 0
$$745$$ 0.502904 0.0184250
$$746$$ 0 0
$$747$$ 29.9634 1.09630
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.4277 −0.380513 −0.190257 0.981734i $$-0.560932\pi$$
−0.190257 + 0.981734i $$0.560932\pi$$
$$752$$ 0 0
$$753$$ −54.5221 −1.98689
$$754$$ 0 0
$$755$$ 18.6339 0.678157
$$756$$ 0 0
$$757$$ 45.9267 1.66923 0.834617 0.550830i $$-0.185689\pi$$
0.834617 + 0.550830i $$0.185689\pi$$
$$758$$ 0 0
$$759$$ 68.4961 2.48625
$$760$$ 0 0
$$761$$ 1.58767 0.0575528 0.0287764 0.999586i $$-0.490839\pi$$
0.0287764 + 0.999586i $$0.490839\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 15.0462 0.543998
$$766$$ 0 0
$$767$$ 0.157008 0.00566924
$$768$$ 0 0
$$769$$ −8.37569 −0.302035 −0.151018 0.988531i $$-0.548255\pi$$
−0.151018 + 0.988531i $$0.548255\pi$$
$$770$$ 0 0
$$771$$ 59.1526 2.13033
$$772$$ 0 0
$$773$$ −10.2062 −0.367090 −0.183545 0.983011i $$-0.558757\pi$$
−0.183545 + 0.983011i $$0.558757\pi$$
$$774$$ 0 0
$$775$$ −15.9267 −0.572104
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −42.0192 −1.50549
$$780$$ 0 0
$$781$$ 54.0154 1.93282
$$782$$ 0 0
$$783$$ −4.49084 −0.160489
$$784$$ 0 0
$$785$$ −21.5510 −0.769189
$$786$$ 0 0
$$787$$ 26.6618 0.950391 0.475195 0.879880i $$-0.342377\pi$$
0.475195 + 0.879880i $$0.342377\pi$$
$$788$$ 0 0
$$789$$ −33.9248 −1.20776
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0.932519 0.0331147
$$794$$ 0 0
$$795$$ 32.1064 1.13870
$$796$$ 0 0
$$797$$ −25.0462 −0.887183 −0.443591 0.896229i $$-0.646296\pi$$
−0.443591 + 0.896229i $$0.646296\pi$$
$$798$$ 0 0
$$799$$ −23.0741 −0.816304
$$800$$ 0 0
$$801$$ 58.8034 2.07772
$$802$$ 0 0
$$803$$ −8.05581 −0.284283
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 62.4036 2.19671
$$808$$ 0 0
$$809$$ 32.1695 1.13102 0.565510 0.824741i $$-0.308679\pi$$
0.565510 + 0.824741i $$0.308679\pi$$
$$810$$ 0 0
$$811$$ 26.5048 0.930709 0.465355 0.885124i $$-0.345927\pi$$
0.465355 + 0.885124i $$0.345927\pi$$
$$812$$ 0 0
$$813$$ 65.5683 2.29958
$$814$$ 0 0
$$815$$ −5.16138 −0.180795
$$816$$ 0 0
$$817$$ 72.6252 2.54083
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −27.2051 −0.949465 −0.474733 0.880130i $$-0.657455\pi$$
−0.474733 + 0.880130i $$0.657455\pi$$
$$822$$ 0 0
$$823$$ −8.61850 −0.300422 −0.150211 0.988654i $$-0.547995\pi$$
−0.150211 + 0.988654i $$0.547995\pi$$
$$824$$ 0 0
$$825$$ −29.2120 −1.01703
$$826$$ 0 0
$$827$$ −37.8636 −1.31665 −0.658323 0.752735i $$-0.728734\pi$$
−0.658323 + 0.752735i $$0.728734\pi$$
$$828$$ 0 0
$$829$$ 51.9634 1.80476 0.902381 0.430939i $$-0.141818\pi$$
0.902381 + 0.430939i $$0.141818\pi$$
$$830$$ 0 0
$$831$$ 4.38443 0.152094
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ 19.5598 0.676893
$$836$$ 0 0
$$837$$ 30.3651 1.04957
$$838$$ 0 0
$$839$$ −32.0785 −1.10747 −0.553736 0.832692i $$-0.686799\pi$$
−0.553736 + 0.832692i $$0.686799\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ −54.9465 −1.89246
$$844$$ 0 0
$$845$$ −21.1233 −0.726663
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 57.9421 1.98857
$$850$$ 0 0
$$851$$ −29.0096 −0.994436
$$852$$ 0 0
$$853$$ 40.5048 1.38686 0.693429 0.720525i $$-0.256099\pi$$
0.693429 + 0.720525i $$0.256099\pi$$
$$854$$ 0 0
$$855$$ −54.5606 −1.86593
$$856$$ 0 0
$$857$$ 31.3834 1.07204 0.536018 0.844207i $$-0.319928\pi$$
0.536018 + 0.844207i $$0.319928\pi$$
$$858$$ 0 0
$$859$$ 32.5187 1.10953 0.554763 0.832009i $$-0.312809\pi$$
0.554763 + 0.832009i $$0.312809\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46.4036 1.57960 0.789798 0.613367i $$-0.210185\pi$$
0.789798 + 0.613367i $$0.210185\pi$$
$$864$$ 0 0
$$865$$ −20.3353 −0.691420
$$866$$ 0 0
$$867$$ −35.9002 −1.21924
$$868$$ 0 0
$$869$$ 8.91902 0.302557
$$870$$ 0 0
$$871$$ −1.55102 −0.0525543
$$872$$ 0 0
$$873$$ 5.79383 0.196092
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 47.1493 1.59212 0.796060 0.605218i $$-0.206914\pi$$
0.796060 + 0.605218i $$0.206914\pi$$
$$878$$ 0 0
$$879$$ −12.4402 −0.419599
$$880$$ 0 0
$$881$$ −2.95377 −0.0995151 −0.0497575 0.998761i $$-0.515845\pi$$
−0.0497575 + 0.998761i $$0.515845\pi$$
$$882$$ 0 0
$$883$$ −14.6339 −0.492470 −0.246235 0.969210i $$-0.579193\pi$$
−0.246235 + 0.969210i $$0.579193\pi$$
$$884$$ 0 0
$$885$$ −6.84006 −0.229926
$$886$$ 0 0
$$887$$ 33.3955 1.12131 0.560655 0.828050i $$-0.310549\pi$$
0.560655 + 0.828050i $$0.310549\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −6.63246 −0.222196
$$892$$ 0 0
$$893$$ 83.6714 2.79996
$$894$$ 0 0
$$895$$ 13.0550 0.436379
$$896$$ 0 0
$$897$$ −1.57227 −0.0524967
$$898$$ 0 0
$$899$$ −6.76156 −0.225511
$$900$$ 0 0
$$901$$ 14.2986 0.476356
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.3921 0.644616
$$906$$ 0 0
$$907$$ 7.25240 0.240812 0.120406 0.992725i $$-0.461580\pi$$
0.120406 + 0.992725i $$0.461580\pi$$
$$908$$ 0 0
$$909$$ −75.4007 −2.50088
$$910$$ 0 0
$$911$$ 14.6604 0.485719 0.242860 0.970061i $$-0.421915\pi$$
0.242860 + 0.970061i $$0.421915\pi$$
$$912$$ 0 0
$$913$$ 29.0867 0.962628
$$914$$ 0 0
$$915$$ −40.6252 −1.34303
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 42.1204 1.38942 0.694711 0.719289i $$-0.255532\pi$$
0.694711 + 0.719289i $$0.255532\pi$$
$$920$$ 0 0
$$921$$ −26.5154 −0.873713
$$922$$ 0 0
$$923$$ −1.23988 −0.0408112
$$924$$ 0 0
$$925$$ 12.3719 0.406786
$$926$$ 0 0
$$927$$ 25.5510 0.839206
$$928$$ 0 0
$$929$$ −41.5144 −1.36204 −0.681021 0.732264i $$-0.738464\pi$$
−0.681021 + 0.732264i $$0.738464\pi$$
$$930$$ 0 0
$$931$$ 50.7668 1.66381
$$932$$ 0 0
$$933$$ 83.6714 2.73928
$$934$$ 0 0
$$935$$ 14.6060 0.477667
$$936$$ 0 0
$$937$$ −43.4219 −1.41853 −0.709266 0.704941i $$-0.750974\pi$$
−0.709266 + 0.704941i $$0.750974\pi$$
$$938$$ 0 0
$$939$$ 6.55539 0.213927
$$940$$ 0 0
$$941$$ 13.0848 0.426551 0.213276 0.976992i $$-0.431587\pi$$
0.213276 + 0.976992i $$0.431587\pi$$
$$942$$ 0 0
$$943$$ 32.0000 1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 49.9128 1.62195 0.810973 0.585083i $$-0.198938\pi$$
0.810973 + 0.585083i $$0.198938\pi$$
$$948$$ 0 0
$$949$$ 0.184915 0.00600259
$$950$$ 0 0
$$951$$ 31.8988 1.03439
$$952$$ 0 0
$$953$$ 39.6175 1.28334 0.641668 0.766983i $$-0.278243\pi$$
0.641668 + 0.766983i $$0.278243\pi$$
$$954$$ 0 0
$$955$$ −10.9133 −0.353148
$$956$$ 0 0
$$957$$ −12.4017 −0.400890
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 14.7187 0.474795
$$962$$ 0 0
$$963$$ 2.50479 0.0807158
$$964$$ 0 0
$$965$$ 20.0000 0.643823
$$966$$ 0 0
$$967$$ 37.6574 1.21098 0.605491 0.795852i $$-0.292977\pi$$
0.605491 + 0.795852i $$0.292977\pi$$
$$968$$ 0 0
$$969$$ −40.0558 −1.28678
$$970$$ 0 0
$$971$$ −9.15994 −0.293956 −0.146978 0.989140i $$-0.546955\pi$$
−0.146978 + 0.989140i $$0.546955\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0.670538 0.0214744
$$976$$ 0 0
$$977$$ 45.4758 1.45490 0.727451 0.686160i $$-0.240705\pi$$
0.727451 + 0.686160i $$0.240705\pi$$
$$978$$ 0 0
$$979$$ 57.0829 1.82438
$$980$$ 0 0
$$981$$ −73.6714 −2.35215
$$982$$ 0 0
$$983$$ −27.9494 −0.891447 −0.445724 0.895171i $$-0.647054\pi$$
−0.445724 + 0.895171i $$0.647054\pi$$
$$984$$ 0 0
$$985$$ −7.32568 −0.233416
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −55.3082 −1.75870
$$990$$ 0 0
$$991$$ −13.1108 −0.416478 −0.208239 0.978078i $$-0.566773\pi$$
−0.208239 + 0.978078i $$0.566773\pi$$
$$992$$ 0 0
$$993$$ −34.0019 −1.07902
$$994$$ 0 0
$$995$$ 31.6435 1.00317
$$996$$ 0 0
$$997$$ −28.8401 −0.913374 −0.456687 0.889627i $$-0.650964\pi$$
−0.456687 + 0.889627i $$0.650964\pi$$
$$998$$ 0 0
$$999$$ −23.5877 −0.746281
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 1856.2.a.y.1.3 3
4.3 odd 2 1856.2.a.x.1.1 3
8.3 odd 2 232.2.a.d.1.3 3
8.5 even 2 464.2.a.j.1.1 3
24.5 odd 2 4176.2.a.bu.1.3 3
24.11 even 2 2088.2.a.s.1.3 3
40.19 odd 2 5800.2.a.p.1.1 3
232.115 odd 2 6728.2.a.j.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.a.d.1.3 3 8.3 odd 2
464.2.a.j.1.1 3 8.5 even 2
1856.2.a.x.1.1 3 4.3 odd 2
1856.2.a.y.1.3 3 1.1 even 1 trivial
2088.2.a.s.1.3 3 24.11 even 2
4176.2.a.bu.1.3 3 24.5 odd 2
5800.2.a.p.1.1 3 40.19 odd 2
6728.2.a.j.1.1 3 232.115 odd 2