# Properties

 Label 8550.2.a.cp.1.2 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.993.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.25342$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +4.50684 q^{11} +5.33131 q^{13} +0.0778929 q^{14} +1.00000 q^{16} +7.33131 q^{17} +1.00000 q^{19} +4.50684 q^{22} +3.40920 q^{23} +5.33131 q^{26} +0.0778929 q^{28} +1.33131 q^{29} -2.50684 q^{31} +1.00000 q^{32} +7.33131 q^{34} +5.50684 q^{37} +1.00000 q^{38} -0.506836 q^{43} +4.50684 q^{44} +3.40920 q^{46} -5.66262 q^{47} -6.99393 q^{49} +5.33131 q^{52} -12.9358 q^{53} +0.0778929 q^{56} +1.33131 q^{58} -7.56499 q^{59} -2.15579 q^{61} -2.50684 q^{62} +1.00000 q^{64} +4.58473 q^{67} +7.33131 q^{68} +10.8579 q^{71} +5.09763 q^{73} +5.50684 q^{74} +1.00000 q^{76} +0.351050 q^{77} +17.0137 q^{79} -13.1695 q^{83} -0.506836 q^{86} +4.50684 q^{88} -15.0137 q^{89} +0.415271 q^{91} +3.40920 q^{92} -5.66262 q^{94} -7.67629 q^{97} -6.99393 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 2q^{11} + 6q^{13} - 2q^{14} + 3q^{16} + 12q^{17} + 3q^{19} - 2q^{22} - 2q^{23} + 6q^{26} - 2q^{28} - 6q^{29} + 8q^{31} + 3q^{32} + 12q^{34} + q^{37} + 3q^{38} + 14q^{43} - 2q^{44} - 2q^{46} + 3q^{47} + 9q^{49} + 6q^{52} - 10q^{53} - 2q^{56} - 6q^{58} - 6q^{59} - 2q^{61} + 8q^{62} + 3q^{64} - 4q^{67} + 12q^{68} + 6q^{71} + 12q^{73} + q^{74} + 3q^{76} - 10q^{77} + 20q^{79} - 4q^{83} + 14q^{86} - 2q^{88} - 14q^{89} + 19q^{91} - 2q^{92} + 3q^{94} + 28q^{97} + 9q^{98} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.0778929 0.0294407 0.0147204 0.999892i $$-0.495314\pi$$
0.0147204 + 0.999892i $$0.495314\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.50684 1.35886 0.679431 0.733739i $$-0.262227\pi$$
0.679431 + 0.733739i $$0.262227\pi$$
$$12$$ 0 0
$$13$$ 5.33131 1.47864 0.739320 0.673354i $$-0.235147\pi$$
0.739320 + 0.673354i $$0.235147\pi$$
$$14$$ 0.0778929 0.0208177
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 7.33131 1.77810 0.889052 0.457806i $$-0.151365\pi$$
0.889052 + 0.457806i $$0.151365\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.50684 0.960861
$$23$$ 3.40920 0.710868 0.355434 0.934701i $$-0.384333\pi$$
0.355434 + 0.934701i $$0.384333\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.33131 1.04556
$$27$$ 0 0
$$28$$ 0.0778929 0.0147204
$$29$$ 1.33131 0.247218 0.123609 0.992331i $$-0.460553\pi$$
0.123609 + 0.992331i $$0.460553\pi$$
$$30$$ 0 0
$$31$$ −2.50684 −0.450241 −0.225121 0.974331i $$-0.572278\pi$$
−0.225121 + 0.974331i $$0.572278\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 7.33131 1.25731
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.50684 0.905318 0.452659 0.891684i $$-0.350475\pi$$
0.452659 + 0.891684i $$0.350475\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −0.506836 −0.0772918 −0.0386459 0.999253i $$-0.512304\pi$$
−0.0386459 + 0.999253i $$0.512304\pi$$
$$44$$ 4.50684 0.679431
$$45$$ 0 0
$$46$$ 3.40920 0.502660
$$47$$ −5.66262 −0.825978 −0.412989 0.910736i $$-0.635515\pi$$
−0.412989 + 0.910736i $$0.635515\pi$$
$$48$$ 0 0
$$49$$ −6.99393 −0.999133
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.33131 0.739320
$$53$$ −12.9358 −1.77687 −0.888433 0.459006i $$-0.848206\pi$$
−0.888433 + 0.459006i $$0.848206\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0.0778929 0.0104089
$$57$$ 0 0
$$58$$ 1.33131 0.174810
$$59$$ −7.56499 −0.984878 −0.492439 0.870347i $$-0.663894\pi$$
−0.492439 + 0.870347i $$0.663894\pi$$
$$60$$ 0 0
$$61$$ −2.15579 −0.276020 −0.138010 0.990431i $$-0.544071\pi$$
−0.138010 + 0.990431i $$0.544071\pi$$
$$62$$ −2.50684 −0.318369
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.58473 0.560114 0.280057 0.959983i $$-0.409647\pi$$
0.280057 + 0.959983i $$0.409647\pi$$
$$68$$ 7.33131 0.889052
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.8579 1.28859 0.644297 0.764775i $$-0.277150\pi$$
0.644297 + 0.764775i $$0.277150\pi$$
$$72$$ 0 0
$$73$$ 5.09763 0.596633 0.298316 0.954467i $$-0.403575\pi$$
0.298316 + 0.954467i $$0.403575\pi$$
$$74$$ 5.50684 0.640157
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0.351050 0.0400059
$$78$$ 0 0
$$79$$ 17.0137 1.91419 0.957094 0.289778i $$-0.0935815\pi$$
0.957094 + 0.289778i $$0.0935815\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −13.1695 −1.44554 −0.722768 0.691091i $$-0.757130\pi$$
−0.722768 + 0.691091i $$0.757130\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −0.506836 −0.0546535
$$87$$ 0 0
$$88$$ 4.50684 0.480430
$$89$$ −15.0137 −1.59145 −0.795723 0.605661i $$-0.792909\pi$$
−0.795723 + 0.605661i $$0.792909\pi$$
$$90$$ 0 0
$$91$$ 0.415271 0.0435322
$$92$$ 3.40920 0.355434
$$93$$ 0 0
$$94$$ −5.66262 −0.584055
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.67629 −0.779410 −0.389705 0.920940i $$-0.627423\pi$$
−0.389705 + 0.920940i $$0.627423\pi$$
$$98$$ −6.99393 −0.706494
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.15579 0.413516 0.206758 0.978392i $$-0.433709\pi$$
0.206758 + 0.978392i $$0.433709\pi$$
$$102$$ 0 0
$$103$$ −2.35105 −0.231656 −0.115828 0.993269i $$-0.536952\pi$$
−0.115828 + 0.993269i $$0.536952\pi$$
$$104$$ 5.33131 0.522778
$$105$$ 0 0
$$106$$ −12.9358 −1.25643
$$107$$ −14.0334 −1.35666 −0.678331 0.734757i $$-0.737296\pi$$
−0.678331 + 0.734757i $$0.737296\pi$$
$$108$$ 0 0
$$109$$ −0.0778929 −0.00746078 −0.00373039 0.999993i $$-0.501187\pi$$
−0.00373039 + 0.999993i $$0.501187\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0.0778929 0.00736018
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.33131 0.123609
$$117$$ 0 0
$$118$$ −7.56499 −0.696414
$$119$$ 0.571057 0.0523487
$$120$$ 0 0
$$121$$ 9.31157 0.846506
$$122$$ −2.15579 −0.195176
$$123$$ 0 0
$$124$$ −2.50684 −0.225121
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −17.8321 −1.58234 −0.791171 0.611596i $$-0.790528\pi$$
−0.791171 + 0.611596i $$0.790528\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.49316 0.130458 0.0652292 0.997870i $$-0.479222\pi$$
0.0652292 + 0.997870i $$0.479222\pi$$
$$132$$ 0 0
$$133$$ 0.0778929 0.00675417
$$134$$ 4.58473 0.396060
$$135$$ 0 0
$$136$$ 7.33131 0.628655
$$137$$ −8.42894 −0.720133 −0.360067 0.932927i $$-0.617246\pi$$
−0.360067 + 0.932927i $$0.617246\pi$$
$$138$$ 0 0
$$139$$ 8.81841 0.747968 0.373984 0.927435i $$-0.377992\pi$$
0.373984 + 0.927435i $$0.377992\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 10.8579 0.911174
$$143$$ 24.0273 2.00927
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 5.09763 0.421883
$$147$$ 0 0
$$148$$ 5.50684 0.452659
$$149$$ −17.6763 −1.44810 −0.724049 0.689748i $$-0.757721\pi$$
−0.724049 + 0.689748i $$0.757721\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 0.351050 0.0282884
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −0.506836 −0.0404499 −0.0202250 0.999795i $$-0.506438\pi$$
−0.0202250 + 0.999795i $$0.506438\pi$$
$$158$$ 17.0137 1.35354
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0.265553 0.0209285
$$162$$ 0 0
$$163$$ −0.830542 −0.0650531 −0.0325265 0.999471i $$-0.510355\pi$$
−0.0325265 + 0.999471i $$0.510355\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −13.1695 −1.02215
$$167$$ 16.5068 1.27734 0.638669 0.769482i $$-0.279485\pi$$
0.638669 + 0.769482i $$0.279485\pi$$
$$168$$ 0 0
$$169$$ 15.4229 1.18638
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −0.506836 −0.0386459
$$173$$ 18.8321 1.43178 0.715888 0.698215i $$-0.246022\pi$$
0.715888 + 0.698215i $$0.246022\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.50684 0.339716
$$177$$ 0 0
$$178$$ −15.0137 −1.12532
$$179$$ 7.15579 0.534849 0.267424 0.963579i $$-0.413827\pi$$
0.267424 + 0.963579i $$0.413827\pi$$
$$180$$ 0 0
$$181$$ 12.5205 0.930642 0.465321 0.885142i $$-0.345939\pi$$
0.465321 + 0.885142i $$0.345939\pi$$
$$182$$ 0.415271 0.0307819
$$183$$ 0 0
$$184$$ 3.40920 0.251330
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 33.0410 2.41620
$$188$$ −5.66262 −0.412989
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.90237 0.354723 0.177361 0.984146i $$-0.443244\pi$$
0.177361 + 0.984146i $$0.443244\pi$$
$$192$$ 0 0
$$193$$ −18.1558 −1.30688 −0.653441 0.756977i $$-0.726675\pi$$
−0.653441 + 0.756977i $$0.726675\pi$$
$$194$$ −7.67629 −0.551126
$$195$$ 0 0
$$196$$ −6.99393 −0.499567
$$197$$ −2.98633 −0.212767 −0.106384 0.994325i $$-0.533927\pi$$
−0.106384 + 0.994325i $$0.533927\pi$$
$$198$$ 0 0
$$199$$ −3.06422 −0.217217 −0.108608 0.994085i $$-0.534639\pi$$
−0.108608 + 0.994085i $$0.534639\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 4.15579 0.292400
$$203$$ 0.103700 0.00727829
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.35105 −0.163805
$$207$$ 0 0
$$208$$ 5.33131 0.369660
$$209$$ 4.50684 0.311744
$$210$$ 0 0
$$211$$ 19.2534 1.32546 0.662730 0.748858i $$-0.269398\pi$$
0.662730 + 0.748858i $$0.269398\pi$$
$$212$$ −12.9358 −0.888433
$$213$$ 0 0
$$214$$ −14.0334 −0.959304
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.195265 −0.0132554
$$218$$ −0.0778929 −0.00527557
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 39.0855 2.62918
$$222$$ 0 0
$$223$$ 14.5068 0.971450 0.485725 0.874112i $$-0.338556\pi$$
0.485725 + 0.874112i $$0.338556\pi$$
$$224$$ 0.0778929 0.00520444
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −21.5984 −1.43354 −0.716768 0.697312i $$-0.754379\pi$$
−0.716768 + 0.697312i $$0.754379\pi$$
$$228$$ 0 0
$$229$$ −19.0137 −1.25646 −0.628229 0.778028i $$-0.716220\pi$$
−0.628229 + 0.778028i $$0.716220\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.33131 0.0874048
$$233$$ −6.01367 −0.393969 −0.196984 0.980407i $$-0.563115\pi$$
−0.196984 + 0.980407i $$0.563115\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −7.56499 −0.492439
$$237$$ 0 0
$$238$$ 0.571057 0.0370161
$$239$$ −15.4092 −0.996739 −0.498369 0.866965i $$-0.666068\pi$$
−0.498369 + 0.866965i $$0.666068\pi$$
$$240$$ 0 0
$$241$$ −4.81841 −0.310381 −0.155190 0.987885i $$-0.549599\pi$$
−0.155190 + 0.987885i $$0.549599\pi$$
$$242$$ 9.31157 0.598570
$$243$$ 0 0
$$244$$ −2.15579 −0.138010
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.33131 0.339223
$$248$$ −2.50684 −0.159184
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.52051 −0.0959736 −0.0479868 0.998848i $$-0.515281\pi$$
−0.0479868 + 0.998848i $$0.515281\pi$$
$$252$$ 0 0
$$253$$ 15.3647 0.965972
$$254$$ −17.8321 −1.11888
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 16.5068 1.02967 0.514834 0.857290i $$-0.327854\pi$$
0.514834 + 0.857290i $$0.327854\pi$$
$$258$$ 0 0
$$259$$ 0.428943 0.0266532
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1.49316 0.0922480
$$263$$ −18.0273 −1.11161 −0.555807 0.831311i $$-0.687591\pi$$
−0.555807 + 0.831311i $$0.687591\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.0778929 0.00477592
$$267$$ 0 0
$$268$$ 4.58473 0.280057
$$269$$ −20.2089 −1.23216 −0.616080 0.787683i $$-0.711280\pi$$
−0.616080 + 0.787683i $$0.711280\pi$$
$$270$$ 0 0
$$271$$ 6.08396 0.369574 0.184787 0.982779i $$-0.440840\pi$$
0.184787 + 0.982779i $$0.440840\pi$$
$$272$$ 7.33131 0.444526
$$273$$ 0 0
$$274$$ −8.42894 −0.509211
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 31.3647 1.88452 0.942262 0.334877i $$-0.108695\pi$$
0.942262 + 0.334877i $$0.108695\pi$$
$$278$$ 8.81841 0.528893
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −11.3252 −0.675607 −0.337804 0.941217i $$-0.609684\pi$$
−0.337804 + 0.941217i $$0.609684\pi$$
$$282$$ 0 0
$$283$$ −26.1437 −1.55408 −0.777039 0.629452i $$-0.783279\pi$$
−0.777039 + 0.629452i $$0.783279\pi$$
$$284$$ 10.8579 0.644297
$$285$$ 0 0
$$286$$ 24.0273 1.42077
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 36.7481 2.16165
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 5.09763 0.298316
$$293$$ 1.33131 0.0777760 0.0388880 0.999244i $$-0.487618\pi$$
0.0388880 + 0.999244i $$0.487618\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 5.50684 0.320078
$$297$$ 0 0
$$298$$ −17.6763 −1.02396
$$299$$ 18.1755 1.05112
$$300$$ 0 0
$$301$$ −0.0394789 −0.00227553
$$302$$ 20.0000 1.15087
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.84421 0.162328 0.0811639 0.996701i $$-0.474136\pi$$
0.0811639 + 0.996701i $$0.474136\pi$$
$$308$$ 0.351050 0.0200030
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.3895 0.929361 0.464681 0.885478i $$-0.346169\pi$$
0.464681 + 0.885478i $$0.346169\pi$$
$$312$$ 0 0
$$313$$ 21.0471 1.18965 0.594826 0.803855i $$-0.297221\pi$$
0.594826 + 0.803855i $$0.297221\pi$$
$$314$$ −0.506836 −0.0286024
$$315$$ 0 0
$$316$$ 17.0137 0.957094
$$317$$ −30.8902 −1.73497 −0.867484 0.497465i $$-0.834264\pi$$
−0.867484 + 0.497465i $$0.834264\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0.265553 0.0147987
$$323$$ 7.33131 0.407925
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −0.830542 −0.0459995
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −0.441078 −0.0243174
$$330$$ 0 0
$$331$$ 28.3845 1.56015 0.780076 0.625685i $$-0.215181\pi$$
0.780076 + 0.625685i $$0.215181\pi$$
$$332$$ −13.1695 −0.722768
$$333$$ 0 0
$$334$$ 16.5068 0.903214
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 17.8716 0.973526 0.486763 0.873534i $$-0.338178\pi$$
0.486763 + 0.873534i $$0.338178\pi$$
$$338$$ 15.4229 0.838894
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −11.2979 −0.611816
$$342$$ 0 0
$$343$$ −1.09003 −0.0588560
$$344$$ −0.506836 −0.0273268
$$345$$ 0 0
$$346$$ 18.8321 1.01242
$$347$$ 33.0410 1.77373 0.886867 0.462024i $$-0.152877\pi$$
0.886867 + 0.462024i $$0.152877\pi$$
$$348$$ 0 0
$$349$$ −19.7158 −1.05536 −0.527681 0.849443i $$-0.676938\pi$$
−0.527681 + 0.849443i $$0.676938\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.50684 0.240215
$$353$$ −5.90997 −0.314556 −0.157278 0.987554i $$-0.550272\pi$$
−0.157278 + 0.987554i $$0.550272\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −15.0137 −0.795723
$$357$$ 0 0
$$358$$ 7.15579 0.378195
$$359$$ 7.00760 0.369847 0.184924 0.982753i $$-0.440796\pi$$
0.184924 + 0.982753i $$0.440796\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 12.5205 0.658063
$$363$$ 0 0
$$364$$ 0.415271 0.0217661
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −17.7158 −0.924756 −0.462378 0.886683i $$-0.653004\pi$$
−0.462378 + 0.886683i $$0.653004\pi$$
$$368$$ 3.40920 0.177717
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.00760 −0.0523122
$$372$$ 0 0
$$373$$ −15.4487 −0.799902 −0.399951 0.916536i $$-0.630973\pi$$
−0.399951 + 0.916536i $$0.630973\pi$$
$$374$$ 33.0410 1.70851
$$375$$ 0 0
$$376$$ −5.66262 −0.292027
$$377$$ 7.09763 0.365547
$$378$$ 0 0
$$379$$ 34.4563 1.76990 0.884950 0.465685i $$-0.154192\pi$$
0.884950 + 0.465685i $$0.154192\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 4.90237 0.250827
$$383$$ −12.0273 −0.614569 −0.307284 0.951618i $$-0.599420\pi$$
−0.307284 + 0.951618i $$0.599420\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.1558 −0.924105
$$387$$ 0 0
$$388$$ −7.67629 −0.389705
$$389$$ 15.3647 0.779022 0.389511 0.921022i $$-0.372644\pi$$
0.389511 + 0.921022i $$0.372644\pi$$
$$390$$ 0 0
$$391$$ 24.9939 1.26400
$$392$$ −6.99393 −0.353247
$$393$$ 0 0
$$394$$ −2.98633 −0.150449
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −1.32524 −0.0665121 −0.0332560 0.999447i $$-0.510588\pi$$
−0.0332560 + 0.999447i $$0.510588\pi$$
$$398$$ −3.06422 −0.153596
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.46736 0.322964 0.161482 0.986876i $$-0.448373\pi$$
0.161482 + 0.986876i $$0.448373\pi$$
$$402$$ 0 0
$$403$$ −13.3647 −0.665744
$$404$$ 4.15579 0.206758
$$405$$ 0 0
$$406$$ 0.103700 0.00514653
$$407$$ 24.8184 1.23020
$$408$$ 0 0
$$409$$ −1.36472 −0.0674812 −0.0337406 0.999431i $$-0.510742\pi$$
−0.0337406 + 0.999431i $$0.510742\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.35105 −0.115828
$$413$$ −0.589259 −0.0289955
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.33131 0.261389
$$417$$ 0 0
$$418$$ 4.50684 0.220437
$$419$$ −16.6231 −0.812094 −0.406047 0.913852i $$-0.633093\pi$$
−0.406047 + 0.913852i $$0.633093\pi$$
$$420$$ 0 0
$$421$$ −31.1128 −1.51635 −0.758174 0.652053i $$-0.773908\pi$$
−0.758174 + 0.652053i $$0.773908\pi$$
$$422$$ 19.2534 0.937242
$$423$$ 0 0
$$424$$ −12.9358 −0.628217
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −0.167920 −0.00812623
$$428$$ −14.0334 −0.678331
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 10.1831 0.490504 0.245252 0.969459i $$-0.421129\pi$$
0.245252 + 0.969459i $$0.421129\pi$$
$$432$$ 0 0
$$433$$ −22.1953 −1.06664 −0.533318 0.845915i $$-0.679055\pi$$
−0.533318 + 0.845915i $$0.679055\pi$$
$$434$$ −0.195265 −0.00937300
$$435$$ 0 0
$$436$$ −0.0778929 −0.00373039
$$437$$ 3.40920 0.163084
$$438$$ 0 0
$$439$$ −9.32524 −0.445070 −0.222535 0.974925i $$-0.571433\pi$$
−0.222535 + 0.974925i $$0.571433\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 39.0855 1.85911
$$443$$ −13.9879 −0.664584 −0.332292 0.943177i $$-0.607822\pi$$
−0.332292 + 0.943177i $$0.607822\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 14.5068 0.686919
$$447$$ 0 0
$$448$$ 0.0778929 0.00368009
$$449$$ −13.4932 −0.636782 −0.318391 0.947960i $$-0.603142\pi$$
−0.318391 + 0.947960i $$0.603142\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −21.5984 −1.01366
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9.68236 −0.452922 −0.226461 0.974020i $$-0.572716\pi$$
−0.226461 + 0.974020i $$0.572716\pi$$
$$458$$ −19.0137 −0.888451
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8.66262 −0.403459 −0.201729 0.979441i $$-0.564656\pi$$
−0.201729 + 0.979441i $$0.564656\pi$$
$$462$$ 0 0
$$463$$ −28.0015 −1.30134 −0.650671 0.759360i $$-0.725512\pi$$
−0.650671 + 0.759360i $$0.725512\pi$$
$$464$$ 1.33131 0.0618046
$$465$$ 0 0
$$466$$ −6.01367 −0.278578
$$467$$ 16.9742 0.785472 0.392736 0.919651i $$-0.371529\pi$$
0.392736 + 0.919651i $$0.371529\pi$$
$$468$$ 0 0
$$469$$ 0.357118 0.0164902
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −7.56499 −0.348207
$$473$$ −2.28423 −0.105029
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0.571057 0.0261743
$$477$$ 0 0
$$478$$ −15.4092 −0.704801
$$479$$ −10.0532 −0.459340 −0.229670 0.973269i $$-0.573765\pi$$
−0.229670 + 0.973269i $$0.573765\pi$$
$$480$$ 0 0
$$481$$ 29.3587 1.33864
$$482$$ −4.81841 −0.219472
$$483$$ 0 0
$$484$$ 9.31157 0.423253
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −19.2089 −0.870440 −0.435220 0.900324i $$-0.643329\pi$$
−0.435220 + 0.900324i $$0.643329\pi$$
$$488$$ −2.15579 −0.0975878
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −5.32524 −0.240325 −0.120162 0.992754i $$-0.538342\pi$$
−0.120162 + 0.992754i $$0.538342\pi$$
$$492$$ 0 0
$$493$$ 9.76025 0.439580
$$494$$ 5.33131 0.239867
$$495$$ 0 0
$$496$$ −2.50684 −0.112560
$$497$$ 0.845752 0.0379372
$$498$$ 0 0
$$499$$ −11.9605 −0.535426 −0.267713 0.963499i $$-0.586268\pi$$
−0.267713 + 0.963499i $$0.586268\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −1.52051 −0.0678636
$$503$$ −19.3313 −0.861941 −0.430970 0.902366i $$-0.641829\pi$$
−0.430970 + 0.902366i $$0.641829\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 15.3647 0.683045
$$507$$ 0 0
$$508$$ −17.8321 −0.791171
$$509$$ 36.1573 1.60265 0.801323 0.598232i $$-0.204130\pi$$
0.801323 + 0.598232i $$0.204130\pi$$
$$510$$ 0 0
$$511$$ 0.397069 0.0175653
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 16.5068 0.728085
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −25.5205 −1.12239
$$518$$ 0.428943 0.0188467
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 31.5205 1.38094 0.690469 0.723362i $$-0.257404\pi$$
0.690469 + 0.723362i $$0.257404\pi$$
$$522$$ 0 0
$$523$$ −5.59840 −0.244801 −0.122400 0.992481i $$-0.539059\pi$$
−0.122400 + 0.992481i $$0.539059\pi$$
$$524$$ 1.49316 0.0652292
$$525$$ 0 0
$$526$$ −18.0273 −0.786030
$$527$$ −18.3784 −0.800575
$$528$$ 0 0
$$529$$ −11.3773 −0.494667
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.0778929 0.00337708
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.58473 0.198030
$$537$$ 0 0
$$538$$ −20.2089 −0.871269
$$539$$ −31.5205 −1.35768
$$540$$ 0 0
$$541$$ 15.1968 0.653362 0.326681 0.945135i $$-0.394070\pi$$
0.326681 + 0.945135i $$0.394070\pi$$
$$542$$ 6.08396 0.261328
$$543$$ 0 0
$$544$$ 7.33131 0.314327
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 26.6505 1.13949 0.569746 0.821821i $$-0.307041\pi$$
0.569746 + 0.821821i $$0.307041\pi$$
$$548$$ −8.42894 −0.360067
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.33131 0.0567158
$$552$$ 0 0
$$553$$ 1.32524 0.0563551
$$554$$ 31.3647 1.33256
$$555$$ 0 0
$$556$$ 8.81841 0.373984
$$557$$ 12.8458 0.544292 0.272146 0.962256i $$-0.412267\pi$$
0.272146 + 0.962256i $$0.412267\pi$$
$$558$$ 0 0
$$559$$ −2.70210 −0.114287
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −11.3252 −0.477727
$$563$$ 10.1968 0.429744 0.214872 0.976642i $$-0.431067\pi$$
0.214872 + 0.976642i $$0.431067\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −26.1437 −1.09890
$$567$$ 0 0
$$568$$ 10.8579 0.455587
$$569$$ 24.3784 1.02200 0.510998 0.859582i $$-0.329276\pi$$
0.510998 + 0.859582i $$0.329276\pi$$
$$570$$ 0 0
$$571$$ 8.32371 0.348336 0.174168 0.984716i $$-0.444276\pi$$
0.174168 + 0.984716i $$0.444276\pi$$
$$572$$ 24.0273 1.00463
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 7.29290 0.303607 0.151804 0.988411i $$-0.451492\pi$$
0.151804 + 0.988411i $$0.451492\pi$$
$$578$$ 36.7481 1.52852
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.02581 −0.0425576
$$582$$ 0 0
$$583$$ −58.2994 −2.41452
$$584$$ 5.09763 0.210942
$$585$$ 0 0
$$586$$ 1.33131 0.0549959
$$587$$ 5.53264 0.228357 0.114178 0.993460i $$-0.463576\pi$$
0.114178 + 0.993460i $$0.463576\pi$$
$$588$$ 0 0
$$589$$ −2.50684 −0.103292
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 5.50684 0.226330
$$593$$ −26.3252 −1.08105 −0.540524 0.841329i $$-0.681774\pi$$
−0.540524 + 0.841329i $$0.681774\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −17.6763 −0.724049
$$597$$ 0 0
$$598$$ 18.1755 0.743252
$$599$$ −22.2994 −0.911130 −0.455565 0.890202i $$-0.650563\pi$$
−0.455565 + 0.890202i $$0.650563\pi$$
$$600$$ 0 0
$$601$$ 32.4947 1.32549 0.662743 0.748847i $$-0.269392\pi$$
0.662743 + 0.748847i $$0.269392\pi$$
$$602$$ −0.0394789 −0.00160904
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.35105 −0.338959 −0.169479 0.985534i $$-0.554209\pi$$
−0.169479 + 0.985534i $$0.554209\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −30.1892 −1.22132
$$612$$ 0 0
$$613$$ 38.2994 1.54690 0.773450 0.633858i $$-0.218529\pi$$
0.773450 + 0.633858i $$0.218529\pi$$
$$614$$ 2.84421 0.114783
$$615$$ 0 0
$$616$$ 0.351050 0.0141442
$$617$$ −14.3526 −0.577813 −0.288907 0.957357i $$-0.593292\pi$$
−0.288907 + 0.957357i $$0.593292\pi$$
$$618$$ 0 0
$$619$$ −8.62314 −0.346593 −0.173297 0.984870i $$-0.555442\pi$$
−0.173297 + 0.984870i $$0.555442\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16.3895 0.657158
$$623$$ −1.16946 −0.0468533
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 21.0471 0.841211
$$627$$ 0 0
$$628$$ −0.506836 −0.0202250
$$629$$ 40.3723 1.60975
$$630$$ 0 0
$$631$$ −10.3374 −0.411525 −0.205762 0.978602i $$-0.565967\pi$$
−0.205762 + 0.978602i $$0.565967\pi$$
$$632$$ 17.0137 0.676768
$$633$$ 0 0
$$634$$ −30.8902 −1.22681
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −37.2868 −1.47736
$$638$$ 6.00000 0.237542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5.88369 −0.232392 −0.116196 0.993226i $$-0.537070\pi$$
−0.116196 + 0.993226i $$0.537070\pi$$
$$642$$ 0 0
$$643$$ 25.5084 1.00595 0.502976 0.864300i $$-0.332238\pi$$
0.502976 + 0.864300i $$0.332238\pi$$
$$644$$ 0.265553 0.0104642
$$645$$ 0 0
$$646$$ 7.33131 0.288447
$$647$$ −5.09157 −0.200170 −0.100085 0.994979i $$-0.531911\pi$$
−0.100085 + 0.994979i $$0.531911\pi$$
$$648$$ 0 0
$$649$$ −34.0942 −1.33831
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −0.830542 −0.0325265
$$653$$ −11.1816 −0.437570 −0.218785 0.975773i $$-0.570209\pi$$
−0.218785 + 0.975773i $$0.570209\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ −0.441078 −0.0171950
$$659$$ 13.7542 0.535787 0.267894 0.963449i $$-0.413672\pi$$
0.267894 + 0.963449i $$0.413672\pi$$
$$660$$ 0 0
$$661$$ −12.6171 −0.490747 −0.245374 0.969429i $$-0.578911\pi$$
−0.245374 + 0.969429i $$0.578911\pi$$
$$662$$ 28.3845 1.10319
$$663$$ 0 0
$$664$$ −13.1695 −0.511074
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.53871 0.175740
$$668$$ 16.5068 0.638669
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9.71577 −0.375073
$$672$$ 0 0
$$673$$ −2.35105 −0.0906263 −0.0453132 0.998973i $$-0.514429\pi$$
−0.0453132 + 0.998973i $$0.514429\pi$$
$$674$$ 17.8716 0.688387
$$675$$ 0 0
$$676$$ 15.4229 0.593188
$$677$$ −23.7663 −0.913414 −0.456707 0.889617i $$-0.650971\pi$$
−0.456707 + 0.889617i $$0.650971\pi$$
$$678$$ 0 0
$$679$$ −0.597928 −0.0229464
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −11.2979 −0.432619
$$683$$ 28.1695 1.07787 0.538937 0.842346i $$-0.318826\pi$$
0.538937 + 0.842346i $$0.318826\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.09003 −0.0416174
$$687$$ 0 0
$$688$$ −0.506836 −0.0193229
$$689$$ −68.9647 −2.62734
$$690$$ 0 0
$$691$$ 2.32371 0.0883979 0.0441990 0.999023i $$-0.485926\pi$$
0.0441990 + 0.999023i $$0.485926\pi$$
$$692$$ 18.8321 0.715888
$$693$$ 0 0
$$694$$ 33.0410 1.25422
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −19.7158 −0.746253
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −23.7036 −0.895274 −0.447637 0.894215i $$-0.647734\pi$$
−0.447637 + 0.894215i $$0.647734\pi$$
$$702$$ 0 0
$$703$$ 5.50684 0.207694
$$704$$ 4.50684 0.169858
$$705$$ 0 0
$$706$$ −5.90997 −0.222425
$$707$$ 0.323706 0.0121742
$$708$$ 0 0
$$709$$ 9.05315 0.339998 0.169999 0.985444i $$-0.445624\pi$$
0.169999 + 0.985444i $$0.445624\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −15.0137 −0.562661
$$713$$ −8.54631 −0.320062
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 7.15579 0.267424
$$717$$ 0 0
$$718$$ 7.00760 0.261521
$$719$$ −35.0734 −1.30802 −0.654008 0.756488i $$-0.726914\pi$$
−0.654008 + 0.756488i $$0.726914\pi$$
$$720$$ 0 0
$$721$$ −0.183130 −0.00682012
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 12.5205 0.465321
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −30.1386 −1.11778 −0.558890 0.829242i $$-0.688773\pi$$
−0.558890 + 0.829242i $$0.688773\pi$$
$$728$$ 0.415271 0.0153910
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.71577 −0.137433
$$732$$ 0 0
$$733$$ −47.8321 −1.76672 −0.883359 0.468697i $$-0.844724\pi$$
−0.883359 + 0.468697i $$0.844724\pi$$
$$734$$ −17.7158 −0.653901
$$735$$ 0 0
$$736$$ 3.40920 0.125665
$$737$$ 20.6626 0.761117
$$738$$ 0 0
$$739$$ −22.4674 −0.826475 −0.413238 0.910623i $$-0.635602\pi$$
−0.413238 + 0.910623i $$0.635602\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −1.00760 −0.0369903
$$743$$ −11.7926 −0.432629 −0.216314 0.976324i $$-0.569404\pi$$
−0.216314 + 0.976324i $$0.569404\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −15.4487 −0.565616
$$747$$ 0 0
$$748$$ 33.0410 1.20810
$$749$$ −1.09310 −0.0399411
$$750$$ 0 0
$$751$$ −2.03948 −0.0744216 −0.0372108 0.999307i $$-0.511847\pi$$
−0.0372108 + 0.999307i $$0.511847\pi$$
$$752$$ −5.66262 −0.206495
$$753$$ 0 0
$$754$$ 7.09763 0.258481
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −43.3526 −1.57568 −0.787838 0.615882i $$-0.788800\pi$$
−0.787838 + 0.615882i $$0.788800\pi$$
$$758$$ 34.4563 1.25151
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.62921 −0.240309 −0.120154 0.992755i $$-0.538339\pi$$
−0.120154 + 0.992755i $$0.538339\pi$$
$$762$$ 0 0
$$763$$ −0.00606730 −0.000219651 0
$$764$$ 4.90237 0.177361
$$765$$ 0 0
$$766$$ −12.0273 −0.434566
$$767$$ −40.3313 −1.45628
$$768$$ 0 0
$$769$$ −19.6429 −0.708340 −0.354170 0.935181i $$-0.615237\pi$$
−0.354170 + 0.935181i $$0.615237\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −18.1558 −0.653441
$$773$$ 14.4107 0.518318 0.259159 0.965835i $$-0.416555\pi$$
0.259159 + 0.965835i $$0.416555\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −7.67629 −0.275563
$$777$$ 0 0
$$778$$ 15.3647 0.550852
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 48.9347 1.75102
$$782$$ 24.9939 0.893781
$$783$$ 0 0
$$784$$ −6.99393 −0.249783
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −24.3176 −0.866830 −0.433415 0.901194i $$-0.642692\pi$$
−0.433415 + 0.901194i $$0.642692\pi$$
$$788$$ −2.98633 −0.106384
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −0.467357 −0.0166173
$$792$$ 0 0
$$793$$ −11.4932 −0.408134
$$794$$ −1.32524 −0.0470311
$$795$$ 0 0
$$796$$ −3.06422 −0.108608
$$797$$ −7.30397 −0.258720 −0.129360 0.991598i $$-0.541292\pi$$
−0.129360 + 0.991598i $$0.541292\pi$$
$$798$$ 0 0
$$799$$ −41.5144 −1.46868
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 6.46736 0.228370
$$803$$ 22.9742 0.810742
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −13.3647 −0.470752
$$807$$ 0 0
$$808$$ 4.15579 0.146200
$$809$$ 0.584729 0.0205580 0.0102790 0.999947i $$-0.496728\pi$$
0.0102790 + 0.999947i $$0.496728\pi$$
$$810$$ 0 0
$$811$$ 1.90997 0.0670682 0.0335341 0.999438i $$-0.489324\pi$$
0.0335341 + 0.999438i $$0.489324\pi$$
$$812$$ 0.103700 0.00363914
$$813$$ 0 0
$$814$$ 24.8184 0.869885
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −0.506836 −0.0177319
$$818$$ −1.36472 −0.0477164
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 38.2226 1.33398 0.666989 0.745067i $$-0.267583\pi$$
0.666989 + 0.745067i $$0.267583\pi$$
$$822$$ 0 0
$$823$$ 45.7481 1.59468 0.797340 0.603531i $$-0.206240\pi$$
0.797340 + 0.603531i $$0.206240\pi$$
$$824$$ −2.35105 −0.0819027
$$825$$ 0 0
$$826$$ −0.589259 −0.0205029
$$827$$ −22.6687 −0.788268 −0.394134 0.919053i $$-0.628955\pi$$
−0.394134 + 0.919053i $$0.628955\pi$$
$$828$$ 0 0
$$829$$ −4.63028 −0.160816 −0.0804081 0.996762i $$-0.525622\pi$$
−0.0804081 + 0.996762i $$0.525622\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.33131 0.184830
$$833$$ −51.2747 −1.77656
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 4.50684 0.155872
$$837$$ 0 0
$$838$$ −16.6231 −0.574237
$$839$$ 50.4826 1.74285 0.871426 0.490527i $$-0.163196\pi$$
0.871426 + 0.490527i $$0.163196\pi$$
$$840$$ 0 0
$$841$$ −27.2276 −0.938883
$$842$$ −31.1128 −1.07222
$$843$$ 0 0
$$844$$ 19.2534 0.662730
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.725305 0.0249218
$$848$$ −12.9358 −0.444216
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 18.7739 0.643562
$$852$$ 0 0
$$853$$ −36.2105 −1.23982 −0.619912 0.784672i $$-0.712832\pi$$
−0.619912 + 0.784672i $$0.712832\pi$$
$$854$$ −0.167920 −0.00574611
$$855$$ 0 0
$$856$$ −14.0334 −0.479652
$$857$$ 25.1968 0.860706 0.430353 0.902661i $$-0.358389\pi$$
0.430353 + 0.902661i $$0.358389\pi$$
$$858$$ 0 0
$$859$$ 18.8579 0.643423 0.321711 0.946838i $$-0.395742\pi$$
0.321711 + 0.946838i $$0.395742\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10.1831 0.346839
$$863$$ 15.0137 0.511071 0.255536 0.966800i $$-0.417748\pi$$
0.255536 + 0.966800i $$0.417748\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −22.1953 −0.754226
$$867$$ 0 0
$$868$$ −0.195265 −0.00662771
$$869$$ 76.6778 2.60112
$$870$$ 0 0
$$871$$ 24.4426 0.828206
$$872$$ −0.0778929 −0.00263779
$$873$$ 0 0
$$874$$ 3.40920 0.115318
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −35.1128 −1.18568 −0.592838 0.805322i $$-0.701993\pi$$
−0.592838 + 0.805322i $$0.701993\pi$$
$$878$$ −9.32524 −0.314712
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 41.3389 1.39274 0.696372 0.717681i $$-0.254797\pi$$
0.696372 + 0.717681i $$0.254797\pi$$
$$882$$ 0 0
$$883$$ 12.6353 0.425211 0.212605 0.977138i $$-0.431805\pi$$
0.212605 + 0.977138i $$0.431805\pi$$
$$884$$ 39.0855 1.31459
$$885$$ 0 0
$$886$$ −13.9879 −0.469932
$$887$$ 4.15579 0.139538 0.0697688 0.997563i $$-0.477774\pi$$
0.0697688 + 0.997563i $$0.477774\pi$$
$$888$$ 0 0
$$889$$ −1.38899 −0.0465853
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.5068 0.485725
$$893$$ −5.66262 −0.189492
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0.0778929 0.00260222
$$897$$ 0 0
$$898$$ −13.4932 −0.450273
$$899$$ −3.33738 −0.111308
$$900$$ 0 0
$$901$$ −94.8362 −3.15945
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 41.7542 1.38643 0.693213 0.720733i $$-0.256195\pi$$
0.693213 + 0.720733i $$0.256195\pi$$
$$908$$ −21.5984 −0.716768
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9.12998 0.302490 0.151245 0.988496i $$-0.451672\pi$$
0.151245 + 0.988496i $$0.451672\pi$$
$$912$$ 0 0
$$913$$ −59.3526 −1.96428
$$914$$ −9.68236 −0.320264
$$915$$ 0 0
$$916$$ −19.0137 −0.628229
$$917$$ 0.116307 0.00384079
$$918$$ 0 0
$$919$$ 19.0197 0.627403 0.313702 0.949522i $$-0.398431\pi$$
0.313702 + 0.949522i $$0.398431\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −8.66262 −0.285288
$$923$$ 57.8868 1.90537
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −28.0015 −0.920188
$$927$$ 0 0
$$928$$ 1.33131 0.0437024
$$929$$ 7.77239 0.255004 0.127502 0.991838i $$-0.459304\pi$$
0.127502 + 0.991838i $$0.459304\pi$$
$$930$$ 0 0
$$931$$ −6.99393 −0.229217
$$932$$ −6.01367 −0.196984
$$933$$ 0 0
$$934$$ 16.9742 0.555413
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 45.9818 1.50216 0.751080 0.660211i $$-0.229533\pi$$
0.751080 + 0.660211i $$0.229533\pi$$
$$938$$ 0.357118 0.0116603
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 40.6242 1.32431 0.662156 0.749366i $$-0.269642\pi$$
0.662156 + 0.749366i $$0.269642\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −7.56499 −0.246219
$$945$$ 0 0
$$946$$ −2.28423 −0.0742666
$$947$$ −2.28423 −0.0742274 −0.0371137 0.999311i $$-0.511816\pi$$
−0.0371137 + 0.999311i $$0.511816\pi$$
$$948$$ 0 0
$$949$$ 27.1771 0.882205
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0.571057 0.0185081
$$953$$ −57.5084 −1.86288 −0.931439 0.363896i $$-0.881446\pi$$
−0.931439 + 0.363896i $$0.881446\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −15.4092 −0.498369
$$957$$ 0 0
$$958$$ −10.0532 −0.324803
$$959$$ −0.656555 −0.0212013
$$960$$ 0 0
$$961$$ −24.7158 −0.797283
$$962$$ 29.3587 0.946561
$$963$$ 0 0
$$964$$ −4.81841 −0.155190
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.70210 0.151209 0.0756047 0.997138i $$-0.475911\pi$$
0.0756047 + 0.997138i $$0.475911\pi$$
$$968$$ 9.31157 0.299285
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.9863 −0.480934 −0.240467 0.970657i $$-0.577301\pi$$
−0.240467 + 0.970657i $$0.577301\pi$$
$$972$$ 0 0
$$973$$ 0.686891 0.0220207
$$974$$ −19.2089 −0.615494
$$975$$ 0 0
$$976$$ −2.15579 −0.0690050
$$977$$ 8.89737 0.284652 0.142326 0.989820i $$-0.454542\pi$$
0.142326 + 0.989820i $$0.454542\pi$$
$$978$$ 0 0
$$979$$ −67.6642 −2.16256
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −5.32524 −0.169935
$$983$$ −1.60947 −0.0513341 −0.0256671 0.999671i $$-0.508171\pi$$
−0.0256671 + 0.999671i $$0.508171\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 9.76025 0.310830
$$987$$ 0 0
$$988$$ 5.33131 0.169612
$$989$$ −1.72791 −0.0549443
$$990$$ 0 0
$$991$$ 16.8974 0.536763 0.268381 0.963313i $$-0.413511\pi$$
0.268381 + 0.963313i $$0.413511\pi$$
$$992$$ −2.50684 −0.0795921
$$993$$ 0 0
$$994$$ 0.845752 0.0268256
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −24.1558 −0.765021 −0.382511 0.923951i $$-0.624940\pi$$
−0.382511 + 0.923951i $$0.624940\pi$$
$$998$$ −11.9605 −0.378604
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.cp.1.2 3
3.2 odd 2 950.2.a.j.1.1 3
5.4 even 2 8550.2.a.ci.1.2 3
12.11 even 2 7600.2.a.bz.1.3 3
15.2 even 4 950.2.b.h.799.3 6
15.8 even 4 950.2.b.h.799.4 6
15.14 odd 2 950.2.a.l.1.3 yes 3
60.59 even 2 7600.2.a.bk.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.1 3 3.2 odd 2
950.2.a.l.1.3 yes 3 15.14 odd 2
950.2.b.h.799.3 6 15.2 even 4
950.2.b.h.799.4 6 15.8 even 4
7600.2.a.bk.1.1 3 60.59 even 2
7600.2.a.bz.1.3 3 12.11 even 2
8550.2.a.ci.1.2 3 5.4 even 2
8550.2.a.cp.1.2 3 1.1 even 1 trivial