# Properties

 Label 1150.2.a.q.1.3 Level $1150$ Weight $2$ Character 1150.1 Self dual yes Analytic conductor $9.183$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.11903$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.11903 q^{3} +1.00000 q^{4} -3.11903 q^{6} -4.50973 q^{7} -1.00000 q^{8} +6.72833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.11903 q^{3} +1.00000 q^{4} -3.11903 q^{6} -4.50973 q^{7} -1.00000 q^{8} +6.72833 q^{9} +4.33763 q^{11} +3.11903 q^{12} +3.72833 q^{13} +4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} -6.72833 q^{18} +4.50973 q^{19} -14.0660 q^{21} -4.33763 q^{22} +1.00000 q^{23} -3.11903 q^{24} -3.72833 q^{26} +11.6288 q^{27} -4.50973 q^{28} -8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +13.5292 q^{33} +1.11903 q^{34} +6.72833 q^{36} +0.781399 q^{37} -4.50973 q^{38} +11.6288 q^{39} +3.90043 q^{41} +14.0660 q^{42} -8.00000 q^{43} +4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +3.11903 q^{48} +13.3376 q^{49} -3.49027 q^{51} +3.72833 q^{52} +6.00000 q^{53} -11.6288 q^{54} +4.50973 q^{56} +14.0660 q^{57} +8.23805 q^{58} -2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} -30.3429 q^{63} +1.00000 q^{64} -13.5292 q^{66} -2.43720 q^{67} -1.11903 q^{68} +3.11903 q^{69} +7.11903 q^{71} -6.72833 q^{72} +9.45665 q^{73} -0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -11.6288 q^{78} -14.9133 q^{79} +16.0854 q^{81} -3.90043 q^{82} -2.78140 q^{83} -14.0660 q^{84} +8.00000 q^{86} -25.6947 q^{87} -4.33763 q^{88} -7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} +5.39070 q^{93} -11.4567 q^{94} -3.11903 q^{96} +0.642920 q^{97} -13.3376 q^{98} +29.1850 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 3q^{7} - 3q^{8} + 10q^{9} + O(q^{10})$$ $$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 3q^{7} - 3q^{8} + 10q^{9} + 3q^{11} - q^{12} + q^{13} + 3q^{14} + 3q^{16} + 7q^{17} - 10q^{18} + 3q^{19} - 22q^{21} - 3q^{22} + 3q^{23} + q^{24} - q^{26} + 14q^{27} - 3q^{28} - 4q^{29} - 5q^{31} - 3q^{32} + 9q^{33} - 7q^{34} + 10q^{36} + 2q^{37} - 3q^{38} + 14q^{39} + q^{41} + 22q^{42} - 24q^{43} + 3q^{44} - 3q^{46} + 14q^{47} - q^{48} + 30q^{49} - 21q^{51} + q^{52} + 18q^{53} - 14q^{54} + 3q^{56} + 22q^{57} + 4q^{58} + 14q^{59} + q^{61} + 5q^{62} - 8q^{63} + 3q^{64} - 9q^{66} - 8q^{67} + 7q^{68} - q^{69} + 11q^{71} - 10q^{72} + 8q^{73} - 2q^{74} + 3q^{76} + 24q^{77} - 14q^{78} - 4q^{79} + 7q^{81} - q^{82} - 8q^{83} - 22q^{84} + 24q^{86} - 36q^{87} - 3q^{88} + 18q^{89} + q^{91} + 3q^{92} + 16q^{93} - 14q^{94} + q^{96} + 33q^{97} - 30q^{98} + 57q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 3.11903 1.80077 0.900385 0.435093i $$-0.143285\pi$$
0.900385 + 0.435093i $$0.143285\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −3.11903 −1.27334
$$7$$ −4.50973 −1.70452 −0.852258 0.523122i $$-0.824767\pi$$
−0.852258 + 0.523122i $$0.824767\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.72833 2.24278
$$10$$ 0 0
$$11$$ 4.33763 1.30784 0.653922 0.756562i $$-0.273122\pi$$
0.653922 + 0.756562i $$0.273122\pi$$
$$12$$ 3.11903 0.900385
$$13$$ 3.72833 1.03405 0.517026 0.855970i $$-0.327039\pi$$
0.517026 + 0.855970i $$0.327039\pi$$
$$14$$ 4.50973 1.20527
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.11903 −0.271404 −0.135702 0.990750i $$-0.543329\pi$$
−0.135702 + 0.990750i $$0.543329\pi$$
$$18$$ −6.72833 −1.58588
$$19$$ 4.50973 1.03460 0.517301 0.855803i $$-0.326937\pi$$
0.517301 + 0.855803i $$0.326937\pi$$
$$20$$ 0 0
$$21$$ −14.0660 −3.06944
$$22$$ −4.33763 −0.924785
$$23$$ 1.00000 0.208514
$$24$$ −3.11903 −0.636669
$$25$$ 0 0
$$26$$ −3.72833 −0.731185
$$27$$ 11.6288 2.23795
$$28$$ −4.50973 −0.852258
$$29$$ −8.23805 −1.52977 −0.764884 0.644168i $$-0.777204\pi$$
−0.764884 + 0.644168i $$0.777204\pi$$
$$30$$ 0 0
$$31$$ 1.72833 0.310417 0.155208 0.987882i $$-0.450395\pi$$
0.155208 + 0.987882i $$0.450395\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 13.5292 2.35513
$$34$$ 1.11903 0.191911
$$35$$ 0 0
$$36$$ 6.72833 1.12139
$$37$$ 0.781399 0.128461 0.0642306 0.997935i $$-0.479541\pi$$
0.0642306 + 0.997935i $$0.479541\pi$$
$$38$$ −4.50973 −0.731574
$$39$$ 11.6288 1.86209
$$40$$ 0 0
$$41$$ 3.90043 0.609144 0.304572 0.952489i $$-0.401487\pi$$
0.304572 + 0.952489i $$0.401487\pi$$
$$42$$ 14.0660 2.17042
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 4.33763 0.653922
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 11.4567 1.67112 0.835562 0.549396i $$-0.185142\pi$$
0.835562 + 0.549396i $$0.185142\pi$$
$$48$$ 3.11903 0.450193
$$49$$ 13.3376 1.90538
$$50$$ 0 0
$$51$$ −3.49027 −0.488736
$$52$$ 3.72833 0.517026
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −11.6288 −1.58247
$$55$$ 0 0
$$56$$ 4.50973 0.602637
$$57$$ 14.0660 1.86308
$$58$$ 8.23805 1.08171
$$59$$ −2.23805 −0.291370 −0.145685 0.989331i $$-0.546539\pi$$
−0.145685 + 0.989331i $$0.546539\pi$$
$$60$$ 0 0
$$61$$ 3.55623 0.455329 0.227664 0.973740i $$-0.426891\pi$$
0.227664 + 0.973740i $$0.426891\pi$$
$$62$$ −1.72833 −0.219498
$$63$$ −30.3429 −3.82285
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −13.5292 −1.66533
$$67$$ −2.43720 −0.297752 −0.148876 0.988856i $$-0.547565\pi$$
−0.148876 + 0.988856i $$0.547565\pi$$
$$68$$ −1.11903 −0.135702
$$69$$ 3.11903 0.375487
$$70$$ 0 0
$$71$$ 7.11903 0.844873 0.422437 0.906393i $$-0.361175\pi$$
0.422437 + 0.906393i $$0.361175\pi$$
$$72$$ −6.72833 −0.792941
$$73$$ 9.45665 1.10682 0.553409 0.832910i $$-0.313327\pi$$
0.553409 + 0.832910i $$0.313327\pi$$
$$74$$ −0.781399 −0.0908357
$$75$$ 0 0
$$76$$ 4.50973 0.517301
$$77$$ −19.5615 −2.22924
$$78$$ −11.6288 −1.31670
$$79$$ −14.9133 −1.67788 −0.838939 0.544225i $$-0.816824\pi$$
−0.838939 + 0.544225i $$0.816824\pi$$
$$80$$ 0 0
$$81$$ 16.0854 1.78727
$$82$$ −3.90043 −0.430730
$$83$$ −2.78140 −0.305298 −0.152649 0.988280i $$-0.548780\pi$$
−0.152649 + 0.988280i $$0.548780\pi$$
$$84$$ −14.0660 −1.53472
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ −25.6947 −2.75476
$$88$$ −4.33763 −0.462393
$$89$$ −7.69471 −0.815637 −0.407819 0.913063i $$-0.633710\pi$$
−0.407819 + 0.913063i $$0.633710\pi$$
$$90$$ 0 0
$$91$$ −16.8137 −1.76256
$$92$$ 1.00000 0.104257
$$93$$ 5.39070 0.558989
$$94$$ −11.4567 −1.18166
$$95$$ 0 0
$$96$$ −3.11903 −0.318334
$$97$$ 0.642920 0.0652786 0.0326393 0.999467i $$-0.489609\pi$$
0.0326393 + 0.999467i $$0.489609\pi$$
$$98$$ −13.3376 −1.34730
$$99$$ 29.1850 2.93320
$$100$$ 0 0
$$101$$ −8.23805 −0.819717 −0.409858 0.912149i $$-0.634422\pi$$
−0.409858 + 0.912149i $$0.634422\pi$$
$$102$$ 3.49027 0.345589
$$103$$ −12.3376 −1.21566 −0.607831 0.794066i $$-0.707960\pi$$
−0.607831 + 0.794066i $$0.707960\pi$$
$$104$$ −3.72833 −0.365593
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 15.9328 1.54028 0.770139 0.637876i $$-0.220187\pi$$
0.770139 + 0.637876i $$0.220187\pi$$
$$108$$ 11.6288 1.11898
$$109$$ −1.49027 −0.142742 −0.0713712 0.997450i $$-0.522737\pi$$
−0.0713712 + 0.997450i $$0.522737\pi$$
$$110$$ 0 0
$$111$$ 2.43720 0.231329
$$112$$ −4.50973 −0.426129
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −14.0660 −1.31740
$$115$$ 0 0
$$116$$ −8.23805 −0.764884
$$117$$ 25.0854 2.31915
$$118$$ 2.23805 0.206030
$$119$$ 5.04650 0.462612
$$120$$ 0 0
$$121$$ 7.81502 0.710456
$$122$$ −3.55623 −0.321966
$$123$$ 12.1655 1.09693
$$124$$ 1.72833 0.155208
$$125$$ 0 0
$$126$$ 30.3429 2.70316
$$127$$ 0.675256 0.0599193 0.0299597 0.999551i $$-0.490462\pi$$
0.0299597 + 0.999551i $$0.490462\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −24.9522 −2.19692
$$130$$ 0 0
$$131$$ −13.6947 −1.19651 −0.598256 0.801305i $$-0.704139\pi$$
−0.598256 + 0.801305i $$0.704139\pi$$
$$132$$ 13.5292 1.17756
$$133$$ −20.3376 −1.76350
$$134$$ 2.43720 0.210542
$$135$$ 0 0
$$136$$ 1.11903 0.0959557
$$137$$ −7.52918 −0.643261 −0.321631 0.946865i $$-0.604231\pi$$
−0.321631 + 0.946865i $$0.604231\pi$$
$$138$$ −3.11903 −0.265509
$$139$$ 4.67526 0.396550 0.198275 0.980146i $$-0.436466\pi$$
0.198275 + 0.980146i $$0.436466\pi$$
$$140$$ 0 0
$$141$$ 35.7336 3.00931
$$142$$ −7.11903 −0.597415
$$143$$ 16.1721 1.35238
$$144$$ 6.72833 0.560694
$$145$$ 0 0
$$146$$ −9.45665 −0.782638
$$147$$ 41.6004 3.43114
$$148$$ 0.781399 0.0642306
$$149$$ 7.52918 0.616814 0.308407 0.951254i $$-0.400204\pi$$
0.308407 + 0.951254i $$0.400204\pi$$
$$150$$ 0 0
$$151$$ −13.3571 −1.08698 −0.543492 0.839414i $$-0.682898\pi$$
−0.543492 + 0.839414i $$0.682898\pi$$
$$152$$ −4.50973 −0.365787
$$153$$ −7.52918 −0.608698
$$154$$ 19.5615 1.57631
$$155$$ 0 0
$$156$$ 11.6288 0.931045
$$157$$ −16.2381 −1.29594 −0.647969 0.761667i $$-0.724381\pi$$
−0.647969 + 0.761667i $$0.724381\pi$$
$$158$$ 14.9133 1.18644
$$159$$ 18.7142 1.48413
$$160$$ 0 0
$$161$$ −4.50973 −0.355416
$$162$$ −16.0854 −1.26379
$$163$$ 3.29112 0.257781 0.128890 0.991659i $$-0.458858\pi$$
0.128890 + 0.991659i $$0.458858\pi$$
$$164$$ 3.90043 0.304572
$$165$$ 0 0
$$166$$ 2.78140 0.215878
$$167$$ −22.9133 −1.77309 −0.886543 0.462647i $$-0.846900\pi$$
−0.886543 + 0.462647i $$0.846900\pi$$
$$168$$ 14.0660 1.08521
$$169$$ 0.900425 0.0692635
$$170$$ 0 0
$$171$$ 30.3429 2.32038
$$172$$ −8.00000 −0.609994
$$173$$ −0.575681 −0.0437683 −0.0218841 0.999761i $$-0.506966\pi$$
−0.0218841 + 0.999761i $$0.506966\pi$$
$$174$$ 25.6947 1.94791
$$175$$ 0 0
$$176$$ 4.33763 0.326961
$$177$$ −6.98055 −0.524690
$$178$$ 7.69471 0.576743
$$179$$ 5.01945 0.375171 0.187586 0.982248i $$-0.439934\pi$$
0.187586 + 0.982248i $$0.439934\pi$$
$$180$$ 0 0
$$181$$ −11.5292 −0.856957 −0.428479 0.903552i $$-0.640950\pi$$
−0.428479 + 0.903552i $$0.640950\pi$$
$$182$$ 16.8137 1.24632
$$183$$ 11.0920 0.819942
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −5.39070 −0.395265
$$187$$ −4.85392 −0.354954
$$188$$ 11.4567 0.835562
$$189$$ −52.4425 −3.81463
$$190$$ 0 0
$$191$$ −18.7142 −1.35411 −0.677055 0.735933i $$-0.736744\pi$$
−0.677055 + 0.735933i $$0.736744\pi$$
$$192$$ 3.11903 0.225096
$$193$$ −23.4956 −1.69125 −0.845624 0.533780i $$-0.820771\pi$$
−0.845624 + 0.533780i $$0.820771\pi$$
$$194$$ −0.642920 −0.0461590
$$195$$ 0 0
$$196$$ 13.3376 0.952688
$$197$$ 18.1385 1.29231 0.646157 0.763205i $$-0.276375\pi$$
0.646157 + 0.763205i $$0.276375\pi$$
$$198$$ −29.1850 −2.07409
$$199$$ −23.2575 −1.64868 −0.824340 0.566094i $$-0.808454\pi$$
−0.824340 + 0.566094i $$0.808454\pi$$
$$200$$ 0 0
$$201$$ −7.60170 −0.536183
$$202$$ 8.23805 0.579627
$$203$$ 37.1514 2.60751
$$204$$ −3.49027 −0.244368
$$205$$ 0 0
$$206$$ 12.3376 0.859603
$$207$$ 6.72833 0.467651
$$208$$ 3.72833 0.258513
$$209$$ 19.5615 1.35310
$$210$$ 0 0
$$211$$ 4.34420 0.299067 0.149533 0.988757i $$-0.452223\pi$$
0.149533 + 0.988757i $$0.452223\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 22.2044 1.52142
$$214$$ −15.9328 −1.08914
$$215$$ 0 0
$$216$$ −11.6288 −0.791236
$$217$$ −7.79428 −0.529110
$$218$$ 1.49027 0.100934
$$219$$ 29.4956 1.99313
$$220$$ 0 0
$$221$$ −4.17210 −0.280646
$$222$$ −2.43720 −0.163574
$$223$$ −12.4761 −0.835462 −0.417731 0.908571i $$-0.637175\pi$$
−0.417731 + 0.908571i $$0.637175\pi$$
$$224$$ 4.50973 0.301319
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −15.9328 −1.05749 −0.528747 0.848779i $$-0.677338\pi$$
−0.528747 + 0.848779i $$0.677338\pi$$
$$228$$ 14.0660 0.931541
$$229$$ −3.56280 −0.235436 −0.117718 0.993047i $$-0.537558\pi$$
−0.117718 + 0.993047i $$0.537558\pi$$
$$230$$ 0 0
$$231$$ −61.0129 −4.01435
$$232$$ 8.23805 0.540855
$$233$$ 27.4956 1.80129 0.900647 0.434552i $$-0.143093\pi$$
0.900647 + 0.434552i $$0.143093\pi$$
$$234$$ −25.0854 −1.63988
$$235$$ 0 0
$$236$$ −2.23805 −0.145685
$$237$$ −46.5150 −3.02147
$$238$$ −5.04650 −0.327116
$$239$$ 10.0389 0.649363 0.324681 0.945823i $$-0.394743\pi$$
0.324681 + 0.945823i $$0.394743\pi$$
$$240$$ 0 0
$$241$$ −23.6947 −1.52631 −0.763155 0.646215i $$-0.776351\pi$$
−0.763155 + 0.646215i $$0.776351\pi$$
$$242$$ −7.81502 −0.502368
$$243$$ 15.2846 0.980505
$$244$$ 3.55623 0.227664
$$245$$ 0 0
$$246$$ −12.1655 −0.775646
$$247$$ 16.8137 1.06983
$$248$$ −1.72833 −0.109749
$$249$$ −8.67526 −0.549772
$$250$$ 0 0
$$251$$ 12.4425 0.785363 0.392681 0.919675i $$-0.371548\pi$$
0.392681 + 0.919675i $$0.371548\pi$$
$$252$$ −30.3429 −1.91142
$$253$$ 4.33763 0.272704
$$254$$ −0.675256 −0.0423693
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −5.45665 −0.340377 −0.170188 0.985412i $$-0.554438\pi$$
−0.170188 + 0.985412i $$0.554438\pi$$
$$258$$ 24.9522 1.55346
$$259$$ −3.52389 −0.218964
$$260$$ 0 0
$$261$$ −55.4283 −3.43093
$$262$$ 13.6947 0.846062
$$263$$ −0.138479 −0.00853895 −0.00426948 0.999991i $$-0.501359\pi$$
−0.00426948 + 0.999991i $$0.501359\pi$$
$$264$$ −13.5292 −0.832663
$$265$$ 0 0
$$266$$ 20.3376 1.24698
$$267$$ −24.0000 −1.46878
$$268$$ −2.43720 −0.148876
$$269$$ 14.6753 0.894766 0.447383 0.894342i $$-0.352356\pi$$
0.447383 + 0.894342i $$0.352356\pi$$
$$270$$ 0 0
$$271$$ −8.31058 −0.504832 −0.252416 0.967619i $$-0.581225\pi$$
−0.252416 + 0.967619i $$0.581225\pi$$
$$272$$ −1.11903 −0.0678510
$$273$$ −52.4425 −3.17396
$$274$$ 7.52918 0.454854
$$275$$ 0 0
$$276$$ 3.11903 0.187743
$$277$$ −12.9133 −0.775886 −0.387943 0.921683i $$-0.626814\pi$$
−0.387943 + 0.921683i $$0.626814\pi$$
$$278$$ −4.67526 −0.280403
$$279$$ 11.6288 0.696195
$$280$$ 0 0
$$281$$ 2.67526 0.159592 0.0797962 0.996811i $$-0.474573\pi$$
0.0797962 + 0.996811i $$0.474573\pi$$
$$282$$ −35.7336 −2.12791
$$283$$ −0.742495 −0.0441367 −0.0220684 0.999756i $$-0.507025\pi$$
−0.0220684 + 0.999756i $$0.507025\pi$$
$$284$$ 7.11903 0.422437
$$285$$ 0 0
$$286$$ −16.1721 −0.956276
$$287$$ −17.5898 −1.03830
$$288$$ −6.72833 −0.396470
$$289$$ −15.7478 −0.926340
$$290$$ 0 0
$$291$$ 2.00528 0.117552
$$292$$ 9.45665 0.553409
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −41.6004 −2.42619
$$295$$ 0 0
$$296$$ −0.781399 −0.0454179
$$297$$ 50.4412 2.92690
$$298$$ −7.52918 −0.436154
$$299$$ 3.72833 0.215615
$$300$$ 0 0
$$301$$ 36.0778 2.07949
$$302$$ 13.3571 0.768614
$$303$$ −25.6947 −1.47612
$$304$$ 4.50973 0.258651
$$305$$ 0 0
$$306$$ 7.52918 0.430414
$$307$$ −30.5084 −1.74121 −0.870604 0.491984i $$-0.836272\pi$$
−0.870604 + 0.491984i $$0.836272\pi$$
$$308$$ −19.5615 −1.11462
$$309$$ −38.4814 −2.18913
$$310$$ 0 0
$$311$$ 5.56280 0.315437 0.157719 0.987484i $$-0.449586\pi$$
0.157719 + 0.987484i $$0.449586\pi$$
$$312$$ −11.6288 −0.658348
$$313$$ −4.07252 −0.230193 −0.115096 0.993354i $$-0.536718\pi$$
−0.115096 + 0.993354i $$0.536718\pi$$
$$314$$ 16.2381 0.916366
$$315$$ 0 0
$$316$$ −14.9133 −0.838939
$$317$$ 6.16553 0.346291 0.173145 0.984896i $$-0.444607\pi$$
0.173145 + 0.984896i $$0.444607\pi$$
$$318$$ −18.7142 −1.04944
$$319$$ −35.7336 −2.00070
$$320$$ 0 0
$$321$$ 49.6947 2.77369
$$322$$ 4.50973 0.251317
$$323$$ −5.04650 −0.280795
$$324$$ 16.0854 0.893634
$$325$$ 0 0
$$326$$ −3.29112 −0.182279
$$327$$ −4.64820 −0.257046
$$328$$ −3.90043 −0.215365
$$329$$ −51.6664 −2.84846
$$330$$ 0 0
$$331$$ 27.5886 1.51640 0.758202 0.652019i $$-0.226078\pi$$
0.758202 + 0.652019i $$0.226078\pi$$
$$332$$ −2.78140 −0.152649
$$333$$ 5.25751 0.288110
$$334$$ 22.9133 1.25376
$$335$$ 0 0
$$336$$ −14.0660 −0.767361
$$337$$ 17.4230 0.949093 0.474547 0.880230i $$-0.342612\pi$$
0.474547 + 0.880230i $$0.342612\pi$$
$$338$$ −0.900425 −0.0489767
$$339$$ 18.7142 1.01641
$$340$$ 0 0
$$341$$ 7.49684 0.405977
$$342$$ −30.3429 −1.64076
$$343$$ −28.5810 −1.54323
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 0.575681 0.0309488
$$347$$ −4.88097 −0.262024 −0.131012 0.991381i $$-0.541823\pi$$
−0.131012 + 0.991381i $$0.541823\pi$$
$$348$$ −25.6947 −1.37738
$$349$$ 24.0389 1.28677 0.643387 0.765542i $$-0.277529\pi$$
0.643387 + 0.765542i $$0.277529\pi$$
$$350$$ 0 0
$$351$$ 43.3558 2.31416
$$352$$ −4.33763 −0.231196
$$353$$ 14.3442 0.763464 0.381732 0.924273i $$-0.375328\pi$$
0.381732 + 0.924273i $$0.375328\pi$$
$$354$$ 6.98055 0.371012
$$355$$ 0 0
$$356$$ −7.69471 −0.407819
$$357$$ 15.7402 0.833059
$$358$$ −5.01945 −0.265286
$$359$$ 26.7814 1.41347 0.706734 0.707479i $$-0.250168\pi$$
0.706734 + 0.707479i $$0.250168\pi$$
$$360$$ 0 0
$$361$$ 1.33763 0.0704015
$$362$$ 11.5292 0.605960
$$363$$ 24.3752 1.27937
$$364$$ −16.8137 −0.881279
$$365$$ 0 0
$$366$$ −11.0920 −0.579787
$$367$$ 20.4761 1.06884 0.534422 0.845218i $$-0.320529\pi$$
0.534422 + 0.845218i $$0.320529\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 26.2433 1.36617
$$370$$ 0 0
$$371$$ −27.0584 −1.40480
$$372$$ 5.39070 0.279495
$$373$$ 3.89386 0.201616 0.100808 0.994906i $$-0.467857\pi$$
0.100808 + 0.994906i $$0.467857\pi$$
$$374$$ 4.85392 0.250990
$$375$$ 0 0
$$376$$ −11.4567 −0.590832
$$377$$ −30.7142 −1.58186
$$378$$ 52.4425 2.69735
$$379$$ −30.3765 −1.56034 −0.780169 0.625569i $$-0.784867\pi$$
−0.780169 + 0.625569i $$0.784867\pi$$
$$380$$ 0 0
$$381$$ 2.10614 0.107901
$$382$$ 18.7142 0.957500
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −3.11903 −0.159167
$$385$$ 0 0
$$386$$ 23.4956 1.19589
$$387$$ −53.8266 −2.73616
$$388$$ 0.642920 0.0326393
$$389$$ −18.6818 −0.947206 −0.473603 0.880738i $$-0.657047\pi$$
−0.473603 + 0.880738i $$0.657047\pi$$
$$390$$ 0 0
$$391$$ −1.11903 −0.0565916
$$392$$ −13.3376 −0.673652
$$393$$ −42.7142 −2.15464
$$394$$ −18.1385 −0.913803
$$395$$ 0 0
$$396$$ 29.1850 1.46660
$$397$$ 28.5757 1.43417 0.717086 0.696985i $$-0.245475\pi$$
0.717086 + 0.696985i $$0.245475\pi$$
$$398$$ 23.2575 1.16579
$$399$$ −63.4336 −3.17565
$$400$$ 0 0
$$401$$ 12.1061 0.604552 0.302276 0.953220i $$-0.402254\pi$$
0.302276 + 0.953220i $$0.402254\pi$$
$$402$$ 7.60170 0.379138
$$403$$ 6.44377 0.320987
$$404$$ −8.23805 −0.409858
$$405$$ 0 0
$$406$$ −37.1514 −1.84379
$$407$$ 3.38942 0.168007
$$408$$ 3.49027 0.172794
$$409$$ 25.2911 1.25057 0.625283 0.780398i $$-0.284984\pi$$
0.625283 + 0.780398i $$0.284984\pi$$
$$410$$ 0 0
$$411$$ −23.4837 −1.15837
$$412$$ −12.3376 −0.607831
$$413$$ 10.0930 0.496644
$$414$$ −6.72833 −0.330679
$$415$$ 0 0
$$416$$ −3.72833 −0.182796
$$417$$ 14.5822 0.714096
$$418$$ −19.5615 −0.956785
$$419$$ −17.3505 −0.847628 −0.423814 0.905749i $$-0.639309\pi$$
−0.423814 + 0.905749i $$0.639309\pi$$
$$420$$ 0 0
$$421$$ 21.4230 1.04409 0.522047 0.852916i $$-0.325168\pi$$
0.522047 + 0.852916i $$0.325168\pi$$
$$422$$ −4.34420 −0.211472
$$423$$ 77.0841 3.74796
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −22.2044 −1.07581
$$427$$ −16.0376 −0.776115
$$428$$ 15.9328 0.770139
$$429$$ 50.4412 2.43532
$$430$$ 0 0
$$431$$ 22.5822 1.08775 0.543874 0.839167i $$-0.316957\pi$$
0.543874 + 0.839167i $$0.316957\pi$$
$$432$$ 11.6288 0.559489
$$433$$ −1.01417 −0.0487378 −0.0243689 0.999703i $$-0.507758\pi$$
−0.0243689 + 0.999703i $$0.507758\pi$$
$$434$$ 7.79428 0.374138
$$435$$ 0 0
$$436$$ −1.49027 −0.0713712
$$437$$ 4.50973 0.215729
$$438$$ −29.4956 −1.40935
$$439$$ −26.7478 −1.27660 −0.638301 0.769787i $$-0.720362\pi$$
−0.638301 + 0.769787i $$0.720362\pi$$
$$440$$ 0 0
$$441$$ 89.7399 4.27333
$$442$$ 4.17210 0.198446
$$443$$ −10.2044 −0.484827 −0.242414 0.970173i $$-0.577939\pi$$
−0.242414 + 0.970173i $$0.577939\pi$$
$$444$$ 2.43720 0.115665
$$445$$ 0 0
$$446$$ 12.4761 0.590761
$$447$$ 23.4837 1.11074
$$448$$ −4.50973 −0.213065
$$449$$ 38.7867 1.83046 0.915228 0.402936i $$-0.132010\pi$$
0.915228 + 0.402936i $$0.132010\pi$$
$$450$$ 0 0
$$451$$ 16.9186 0.796665
$$452$$ 6.00000 0.282216
$$453$$ −41.6611 −1.95741
$$454$$ 15.9328 0.747762
$$455$$ 0 0
$$456$$ −14.0660 −0.658699
$$457$$ −34.9522 −1.63500 −0.817498 0.575932i $$-0.804639\pi$$
−0.817498 + 0.575932i $$0.804639\pi$$
$$458$$ 3.56280 0.166479
$$459$$ −13.0129 −0.607389
$$460$$ 0 0
$$461$$ 16.3700 0.762425 0.381213 0.924487i $$-0.375507\pi$$
0.381213 + 0.924487i $$0.375507\pi$$
$$462$$ 61.0129 2.83858
$$463$$ −29.2186 −1.35790 −0.678952 0.734183i $$-0.737565\pi$$
−0.678952 + 0.734183i $$0.737565\pi$$
$$464$$ −8.23805 −0.382442
$$465$$ 0 0
$$466$$ −27.4956 −1.27371
$$467$$ −24.2770 −1.12340 −0.561702 0.827340i $$-0.689853\pi$$
−0.561702 + 0.827340i $$0.689853\pi$$
$$468$$ 25.0854 1.15957
$$469$$ 10.9911 0.507523
$$470$$ 0 0
$$471$$ −50.6469 −2.33369
$$472$$ 2.23805 0.103015
$$473$$ −34.7010 −1.59555
$$474$$ 46.5150 2.13651
$$475$$ 0 0
$$476$$ 5.04650 0.231306
$$477$$ 40.3700 1.84841
$$478$$ −10.0389 −0.459169
$$479$$ 24.6080 1.12437 0.562185 0.827012i $$-0.309961\pi$$
0.562185 + 0.827012i $$0.309961\pi$$
$$480$$ 0 0
$$481$$ 2.91331 0.132835
$$482$$ 23.6947 1.07926
$$483$$ −14.0660 −0.640023
$$484$$ 7.81502 0.355228
$$485$$ 0 0
$$486$$ −15.2846 −0.693322
$$487$$ 30.2381 1.37022 0.685108 0.728441i $$-0.259755\pi$$
0.685108 + 0.728441i $$0.259755\pi$$
$$488$$ −3.55623 −0.160983
$$489$$ 10.2651 0.464204
$$490$$ 0 0
$$491$$ 12.3311 0.556493 0.278246 0.960510i $$-0.410247\pi$$
0.278246 + 0.960510i $$0.410247\pi$$
$$492$$ 12.1655 0.548464
$$493$$ 9.21860 0.415185
$$494$$ −16.8137 −0.756486
$$495$$ 0 0
$$496$$ 1.72833 0.0776042
$$497$$ −32.1049 −1.44010
$$498$$ 8.67526 0.388748
$$499$$ −26.9133 −1.20481 −0.602403 0.798192i $$-0.705790\pi$$
−0.602403 + 0.798192i $$0.705790\pi$$
$$500$$ 0 0
$$501$$ −71.4672 −3.19292
$$502$$ −12.4425 −0.555335
$$503$$ −20.5097 −0.914483 −0.457242 0.889342i $$-0.651163\pi$$
−0.457242 + 0.889342i $$0.651163\pi$$
$$504$$ 30.3429 1.35158
$$505$$ 0 0
$$506$$ −4.33763 −0.192831
$$507$$ 2.80845 0.124728
$$508$$ 0.675256 0.0299597
$$509$$ −36.7142 −1.62733 −0.813663 0.581336i $$-0.802530\pi$$
−0.813663 + 0.581336i $$0.802530\pi$$
$$510$$ 0 0
$$511$$ −42.6469 −1.88659
$$512$$ −1.00000 −0.0441942
$$513$$ 52.4425 2.31539
$$514$$ 5.45665 0.240683
$$515$$ 0 0
$$516$$ −24.9522 −1.09846
$$517$$ 49.6947 2.18557
$$518$$ 3.52389 0.154831
$$519$$ −1.79557 −0.0788166
$$520$$ 0 0
$$521$$ −4.91331 −0.215256 −0.107628 0.994191i $$-0.534326\pi$$
−0.107628 + 0.994191i $$0.534326\pi$$
$$522$$ 55.4283 2.42603
$$523$$ 0.344196 0.0150506 0.00752531 0.999972i $$-0.497605\pi$$
0.00752531 + 0.999972i $$0.497605\pi$$
$$524$$ −13.6947 −0.598256
$$525$$ 0 0
$$526$$ 0.138479 0.00603795
$$527$$ −1.93404 −0.0842483
$$528$$ 13.5292 0.588782
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −15.0584 −0.653477
$$532$$ −20.3376 −0.881748
$$533$$ 14.5421 0.629887
$$534$$ 24.0000 1.03858
$$535$$ 0 0
$$536$$ 2.43720 0.105271
$$537$$ 15.6558 0.675598
$$538$$ −14.6753 −0.632695
$$539$$ 57.8537 2.49193
$$540$$ 0 0
$$541$$ −6.13191 −0.263631 −0.131816 0.991274i $$-0.542081\pi$$
−0.131816 + 0.991274i $$0.542081\pi$$
$$542$$ 8.31058 0.356970
$$543$$ −35.9598 −1.54318
$$544$$ 1.11903 0.0479779
$$545$$ 0 0
$$546$$ 52.4425 2.24433
$$547$$ 9.18498 0.392721 0.196361 0.980532i $$-0.437088\pi$$
0.196361 + 0.980532i $$0.437088\pi$$
$$548$$ −7.52918 −0.321631
$$549$$ 23.9275 1.02120
$$550$$ 0 0
$$551$$ −37.1514 −1.58270
$$552$$ −3.11903 −0.132755
$$553$$ 67.2549 2.85997
$$554$$ 12.9133 0.548634
$$555$$ 0 0
$$556$$ 4.67526 0.198275
$$557$$ 4.30529 0.182421 0.0912105 0.995832i $$-0.470926\pi$$
0.0912105 + 0.995832i $$0.470926\pi$$
$$558$$ −11.6288 −0.492284
$$559$$ −29.8266 −1.26153
$$560$$ 0 0
$$561$$ −15.1395 −0.639191
$$562$$ −2.67526 −0.112849
$$563$$ −11.1256 −0.468888 −0.234444 0.972130i $$-0.575327\pi$$
−0.234444 + 0.972130i $$0.575327\pi$$
$$564$$ 35.7336 1.50466
$$565$$ 0 0
$$566$$ 0.742495 0.0312094
$$567$$ −72.5408 −3.04643
$$568$$ −7.11903 −0.298708
$$569$$ −16.0389 −0.672386 −0.336193 0.941793i $$-0.609139\pi$$
−0.336193 + 0.941793i $$0.609139\pi$$
$$570$$ 0 0
$$571$$ 17.9004 0.749109 0.374555 0.927205i $$-0.377796\pi$$
0.374555 + 0.927205i $$0.377796\pi$$
$$572$$ 16.1721 0.676189
$$573$$ −58.3700 −2.43844
$$574$$ 17.5898 0.734186
$$575$$ 0 0
$$576$$ 6.72833 0.280347
$$577$$ 9.12559 0.379903 0.189952 0.981793i $$-0.439167\pi$$
0.189952 + 0.981793i $$0.439167\pi$$
$$578$$ 15.7478 0.655021
$$579$$ −73.2833 −3.04555
$$580$$ 0 0
$$581$$ 12.5433 0.520386
$$582$$ −2.00528 −0.0831217
$$583$$ 26.0258 1.07788
$$584$$ −9.45665 −0.391319
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 33.6340 1.38823 0.694113 0.719866i $$-0.255797\pi$$
0.694113 + 0.719866i $$0.255797\pi$$
$$588$$ 41.6004 1.71557
$$589$$ 7.79428 0.321158
$$590$$ 0 0
$$591$$ 56.5744 2.32716
$$592$$ 0.781399 0.0321153
$$593$$ 17.4567 0.716859 0.358429 0.933557i $$-0.383312\pi$$
0.358429 + 0.933557i $$0.383312\pi$$
$$594$$ −50.4412 −2.06963
$$595$$ 0 0
$$596$$ 7.52918 0.308407
$$597$$ −72.5408 −2.96890
$$598$$ −3.72833 −0.152463
$$599$$ −11.5951 −0.473764 −0.236882 0.971538i $$-0.576126\pi$$
−0.236882 + 0.971538i $$0.576126\pi$$
$$600$$ 0 0
$$601$$ −31.6611 −1.29148 −0.645741 0.763556i $$-0.723452\pi$$
−0.645741 + 0.763556i $$0.723452\pi$$
$$602$$ −36.0778 −1.47042
$$603$$ −16.3983 −0.667790
$$604$$ −13.3571 −0.543492
$$605$$ 0 0
$$606$$ 25.6947 1.04378
$$607$$ 36.0778 1.46435 0.732177 0.681115i $$-0.238505\pi$$
0.732177 + 0.681115i $$0.238505\pi$$
$$608$$ −4.50973 −0.182894
$$609$$ 115.876 4.69554
$$610$$ 0 0
$$611$$ 42.7142 1.72803
$$612$$ −7.52918 −0.304349
$$613$$ 32.0389 1.29404 0.647020 0.762473i $$-0.276015\pi$$
0.647020 + 0.762473i $$0.276015\pi$$
$$614$$ 30.5084 1.23122
$$615$$ 0 0
$$616$$ 19.5615 0.788156
$$617$$ 13.3960 0.539302 0.269651 0.962958i $$-0.413092\pi$$
0.269651 + 0.962958i $$0.413092\pi$$
$$618$$ 38.4814 1.54795
$$619$$ 37.1309 1.49242 0.746208 0.665713i $$-0.231872\pi$$
0.746208 + 0.665713i $$0.231872\pi$$
$$620$$ 0 0
$$621$$ 11.6288 0.466646
$$622$$ −5.56280 −0.223048
$$623$$ 34.7010 1.39027
$$624$$ 11.6288 0.465523
$$625$$ 0 0
$$626$$ 4.07252 0.162771
$$627$$ 61.0129 2.43662
$$628$$ −16.2381 −0.647969
$$629$$ −0.874406 −0.0348648
$$630$$ 0 0
$$631$$ 11.1125 0.442380 0.221190 0.975231i $$-0.429006\pi$$
0.221190 + 0.975231i $$0.429006\pi$$
$$632$$ 14.9133 0.593220
$$633$$ 13.5497 0.538551
$$634$$ −6.16553 −0.244864
$$635$$ 0 0
$$636$$ 18.7142 0.742065
$$637$$ 49.7270 1.97026
$$638$$ 35.7336 1.41471
$$639$$ 47.8991 1.89486
$$640$$ 0 0
$$641$$ 12.3831 0.489103 0.244552 0.969636i $$-0.421359\pi$$
0.244552 + 0.969636i $$0.421359\pi$$
$$642$$ −49.6947 −1.96129
$$643$$ 37.4956 1.47868 0.739340 0.673332i $$-0.235138\pi$$
0.739340 + 0.673332i $$0.235138\pi$$
$$644$$ −4.50973 −0.177708
$$645$$ 0 0
$$646$$ 5.04650 0.198552
$$647$$ −14.5691 −0.572771 −0.286385 0.958114i $$-0.592454\pi$$
−0.286385 + 0.958114i $$0.592454\pi$$
$$648$$ −16.0854 −0.631894
$$649$$ −9.70784 −0.381066
$$650$$ 0 0
$$651$$ −24.3106 −0.952807
$$652$$ 3.29112 0.128890
$$653$$ 4.41672 0.172840 0.0864198 0.996259i $$-0.472457\pi$$
0.0864198 + 0.996259i $$0.472457\pi$$
$$654$$ 4.64820 0.181759
$$655$$ 0 0
$$656$$ 3.90043 0.152286
$$657$$ 63.6275 2.48234
$$658$$ 51.6664 2.01416
$$659$$ −31.8655 −1.24130 −0.620652 0.784086i $$-0.713132\pi$$
−0.620652 + 0.784086i $$0.713132\pi$$
$$660$$ 0 0
$$661$$ 33.1190 1.28818 0.644090 0.764949i $$-0.277236\pi$$
0.644090 + 0.764949i $$0.277236\pi$$
$$662$$ −27.5886 −1.07226
$$663$$ −13.0129 −0.505379
$$664$$ 2.78140 0.107939
$$665$$ 0 0
$$666$$ −5.25751 −0.203724
$$667$$ −8.23805 −0.318979
$$668$$ −22.9133 −0.886543
$$669$$ −38.9133 −1.50448
$$670$$ 0 0
$$671$$ 15.4256 0.595499
$$672$$ 14.0660 0.542606
$$673$$ −19.3505 −0.745907 −0.372954 0.927850i $$-0.621655\pi$$
−0.372954 + 0.927850i $$0.621655\pi$$
$$674$$ −17.4230 −0.671110
$$675$$ 0 0
$$676$$ 0.900425 0.0346317
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −18.7142 −0.718713
$$679$$ −2.89939 −0.111268
$$680$$ 0 0
$$681$$ −49.6947 −1.90431
$$682$$ −7.49684 −0.287069
$$683$$ −38.3495 −1.46740 −0.733701 0.679472i $$-0.762209\pi$$
−0.733701 + 0.679472i $$0.762209\pi$$
$$684$$ 30.3429 1.16019
$$685$$ 0 0
$$686$$ 28.5810 1.09123
$$687$$ −11.1125 −0.423967
$$688$$ −8.00000 −0.304997
$$689$$ 22.3700 0.852228
$$690$$ 0 0
$$691$$ −21.0195 −0.799618 −0.399809 0.916599i $$-0.630923\pi$$
−0.399809 + 0.916599i $$0.630923\pi$$
$$692$$ −0.575681 −0.0218841
$$693$$ −131.616 −4.99969
$$694$$ 4.88097 0.185279
$$695$$ 0 0
$$696$$ 25.6947 0.973955
$$697$$ −4.36468 −0.165324
$$698$$ −24.0389 −0.909886
$$699$$ 85.7594 3.24372
$$700$$ 0 0
$$701$$ 19.3169 0.729589 0.364794 0.931088i $$-0.381139\pi$$
0.364794 + 0.931088i $$0.381139\pi$$
$$702$$ −43.3558 −1.63636
$$703$$ 3.52389 0.132906
$$704$$ 4.33763 0.163481
$$705$$ 0 0
$$706$$ −14.3442 −0.539851
$$707$$ 37.1514 1.39722
$$708$$ −6.98055 −0.262345
$$709$$ 12.2315 0.459363 0.229682 0.973266i $$-0.426232\pi$$
0.229682 + 0.973266i $$0.426232\pi$$
$$710$$ 0 0
$$711$$ −100.342 −3.76311
$$712$$ 7.69471 0.288371
$$713$$ 1.72833 0.0647264
$$714$$ −15.7402 −0.589061
$$715$$ 0 0
$$716$$ 5.01945 0.187586
$$717$$ 31.3116 1.16935
$$718$$ −26.7814 −0.999473
$$719$$ −40.6416 −1.51568 −0.757839 0.652442i $$-0.773745\pi$$
−0.757839 + 0.652442i $$0.773745\pi$$
$$720$$ 0 0
$$721$$ 55.6393 2.07212
$$722$$ −1.33763 −0.0497814
$$723$$ −73.9044 −2.74854
$$724$$ −11.5292 −0.428479
$$725$$ 0 0
$$726$$ −24.3752 −0.904650
$$727$$ 23.4501 0.869716 0.434858 0.900499i $$-0.356799\pi$$
0.434858 + 0.900499i $$0.356799\pi$$
$$728$$ 16.8137 0.623158
$$729$$ −0.583281 −0.0216030
$$730$$ 0 0
$$731$$ 8.95221 0.331110
$$732$$ 11.0920 0.409971
$$733$$ 12.5150 0.462252 0.231126 0.972924i $$-0.425759\pi$$
0.231126 + 0.972924i $$0.425759\pi$$
$$734$$ −20.4761 −0.755787
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −10.5717 −0.389413
$$738$$ −26.2433 −0.966031
$$739$$ −21.3505 −0.785391 −0.392696 0.919668i $$-0.628457\pi$$
−0.392696 + 0.919668i $$0.628457\pi$$
$$740$$ 0 0
$$741$$ 52.4425 1.92652
$$742$$ 27.0584 0.993343
$$743$$ −24.9858 −0.916641 −0.458321 0.888787i $$-0.651549\pi$$
−0.458321 + 0.888787i $$0.651549\pi$$
$$744$$ −5.39070 −0.197633
$$745$$ 0 0
$$746$$ −3.89386 −0.142564
$$747$$ −18.7142 −0.684715
$$748$$ −4.85392 −0.177477
$$749$$ −71.8524 −2.62543
$$750$$ 0 0
$$751$$ −33.6275 −1.22708 −0.613542 0.789662i $$-0.710256\pi$$
−0.613542 + 0.789662i $$0.710256\pi$$
$$752$$ 11.4567 0.417781
$$753$$ 38.8085 1.41426
$$754$$ 30.7142 1.11854
$$755$$ 0 0
$$756$$ −52.4425 −1.90731
$$757$$ 37.1230 1.34926 0.674630 0.738156i $$-0.264303\pi$$
0.674630 + 0.738156i $$0.264303\pi$$
$$758$$ 30.3765 1.10333
$$759$$ 13.5292 0.491078
$$760$$ 0 0
$$761$$ −3.87337 −0.140410 −0.0702048 0.997533i $$-0.522365\pi$$
−0.0702048 + 0.997533i $$0.522365\pi$$
$$762$$ −2.10614 −0.0762975
$$763$$ 6.72073 0.243307
$$764$$ −18.7142 −0.677055
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8.34420 −0.301291
$$768$$ 3.11903 0.112548
$$769$$ 23.1645 0.835333 0.417667 0.908600i $$-0.362848\pi$$
0.417667 + 0.908600i $$0.362848\pi$$
$$770$$ 0 0
$$771$$ −17.0195 −0.612941
$$772$$ −23.4956 −0.845624
$$773$$ 8.78140 0.315845 0.157922 0.987452i $$-0.449520\pi$$
0.157922 + 0.987452i $$0.449520\pi$$
$$774$$ 53.8266 1.93476
$$775$$ 0 0
$$776$$ −0.642920 −0.0230795
$$777$$ −10.9911 −0.394304
$$778$$ 18.6818 0.669776
$$779$$ 17.5898 0.630222
$$780$$ 0 0
$$781$$ 30.8797 1.10496
$$782$$ 1.11903 0.0400163
$$783$$ −95.7983 −3.42355
$$784$$ 13.3376 0.476344
$$785$$ 0 0
$$786$$ 42.7142 1.52356
$$787$$ −49.6275 −1.76903 −0.884514 0.466513i $$-0.845510\pi$$
−0.884514 + 0.466513i $$0.845510\pi$$
$$788$$ 18.1385 0.646157
$$789$$ −0.431918 −0.0153767
$$790$$ 0 0
$$791$$ −27.0584 −0.962084
$$792$$ −29.1850 −1.03704
$$793$$ 13.2588 0.470833
$$794$$ −28.5757 −1.01411
$$795$$ 0 0
$$796$$ −23.2575 −0.824340
$$797$$ −18.3311 −0.649319 −0.324660 0.945831i $$-0.605250\pi$$
−0.324660 + 0.945831i $$0.605250\pi$$
$$798$$ 63.4336 2.24553
$$799$$ −12.8203 −0.453550
$$800$$ 0 0
$$801$$ −51.7725 −1.82929
$$802$$ −12.1061 −0.427483
$$803$$ 41.0195 1.44755
$$804$$ −7.60170 −0.268091
$$805$$ 0 0
$$806$$ −6.44377 −0.226972
$$807$$ 45.7725 1.61127
$$808$$ 8.23805 0.289814
$$809$$ −1.93933 −0.0681832 −0.0340916 0.999419i $$-0.510854\pi$$
−0.0340916 + 0.999419i $$0.510854\pi$$
$$810$$ 0 0
$$811$$ −5.41775 −0.190243 −0.0951215 0.995466i $$-0.530324\pi$$
−0.0951215 + 0.995466i $$0.530324\pi$$
$$812$$ 37.1514 1.30376
$$813$$ −25.9209 −0.909086
$$814$$ −3.38942 −0.118799
$$815$$ 0 0
$$816$$ −3.49027 −0.122184
$$817$$ −36.0778 −1.26220
$$818$$ −25.2911 −0.884283
$$819$$ −113.128 −3.95302
$$820$$ 0 0
$$821$$ 37.8655 1.32152 0.660758 0.750599i $$-0.270235\pi$$
0.660758 + 0.750599i $$0.270235\pi$$
$$822$$ 23.4837 0.819088
$$823$$ −43.7336 −1.52446 −0.762229 0.647308i $$-0.775895\pi$$
−0.762229 + 0.647308i $$0.775895\pi$$
$$824$$ 12.3376 0.429802
$$825$$ 0 0
$$826$$ −10.0930 −0.351181
$$827$$ −28.1991 −0.980581 −0.490290 0.871559i $$-0.663109\pi$$
−0.490290 + 0.871559i $$0.663109\pi$$
$$828$$ 6.72833 0.233826
$$829$$ 1.12559 0.0390935 0.0195468 0.999809i $$-0.493778\pi$$
0.0195468 + 0.999809i $$0.493778\pi$$
$$830$$ 0 0
$$831$$ −40.2770 −1.39719
$$832$$ 3.72833 0.129256
$$833$$ −14.9252 −0.517126
$$834$$ −14.5822 −0.504942
$$835$$ 0 0
$$836$$ 19.5615 0.676549
$$837$$ 20.0983 0.694699
$$838$$ 17.3505 0.599364
$$839$$ 39.9328 1.37863 0.689316 0.724461i $$-0.257911\pi$$
0.689316 + 0.724461i $$0.257911\pi$$
$$840$$ 0 0
$$841$$ 38.8655 1.34019
$$842$$ −21.4230 −0.738287
$$843$$ 8.34420 0.287389
$$844$$ 4.34420 0.149533
$$845$$ 0 0
$$846$$ −77.0841 −2.65021
$$847$$ −35.2436 −1.21098
$$848$$ 6.00000 0.206041
$$849$$ −2.31586 −0.0794801
$$850$$ 0 0
$$851$$ 0.781399 0.0267860
$$852$$ 22.2044 0.760711
$$853$$ −47.9921 −1.64322 −0.821610 0.570050i $$-0.806924\pi$$
−0.821610 + 0.570050i $$0.806924\pi$$
$$854$$ 16.0376 0.548796
$$855$$ 0 0
$$856$$ −15.9328 −0.544571
$$857$$ 43.4283 1.48348 0.741742 0.670686i $$-0.234000\pi$$
0.741742 + 0.670686i $$0.234000\pi$$
$$858$$ −50.4412 −1.72203
$$859$$ 32.5433 1.11036 0.555182 0.831729i $$-0.312648\pi$$
0.555182 + 0.831729i $$0.312648\pi$$
$$860$$ 0 0
$$861$$ −54.8632 −1.86973
$$862$$ −22.5822 −0.769154
$$863$$ −46.1036 −1.56938 −0.784692 0.619886i $$-0.787179\pi$$
−0.784692 + 0.619886i $$0.787179\pi$$
$$864$$ −11.6288 −0.395618
$$865$$ 0 0
$$866$$ 1.01417 0.0344628
$$867$$ −49.1177 −1.66813
$$868$$ −7.79428 −0.264555
$$869$$ −64.6884 −2.19440
$$870$$ 0 0
$$871$$ −9.08669 −0.307891
$$872$$ 1.49027 0.0504670
$$873$$ 4.32578 0.146405
$$874$$ −4.50973 −0.152544
$$875$$ 0 0
$$876$$ 29.4956 0.996563
$$877$$ 24.0996 0.813785 0.406892 0.913476i $$-0.366612\pi$$
0.406892 + 0.913476i $$0.366612\pi$$
$$878$$ 26.7478 0.902694
$$879$$ 18.7142 0.631213
$$880$$ 0 0
$$881$$ 2.34420 0.0789780 0.0394890 0.999220i $$-0.487427\pi$$
0.0394890 + 0.999220i $$0.487427\pi$$
$$882$$ −89.7399 −3.02170
$$883$$ 41.0505 1.38146 0.690730 0.723113i $$-0.257289\pi$$
0.690730 + 0.723113i $$0.257289\pi$$
$$884$$ −4.17210 −0.140323
$$885$$ 0 0
$$886$$ 10.2044 0.342825
$$887$$ −54.7788 −1.83929 −0.919647 0.392747i $$-0.871525\pi$$
−0.919647 + 0.392747i $$0.871525\pi$$
$$888$$ −2.43720 −0.0817872
$$889$$ −3.04522 −0.102133
$$890$$ 0 0
$$891$$ 69.7725 2.33747
$$892$$ −12.4761 −0.417731
$$893$$ 51.6664 1.72895
$$894$$ −23.4837 −0.785413
$$895$$ 0 0
$$896$$ 4.50973 0.150659
$$897$$ 11.6288 0.388273
$$898$$ −38.7867 −1.29433
$$899$$ −14.2381 −0.474866
$$900$$ 0 0
$$901$$ −6.71416 −0.223681
$$902$$ −16.9186 −0.563328
$$903$$ 112.528 3.74469
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 41.6611 1.38410
$$907$$ 10.1061 0.335569 0.167784 0.985824i $$-0.446339\pi$$
0.167784 + 0.985824i $$0.446339\pi$$
$$908$$ −15.9328 −0.528747
$$909$$ −55.4283 −1.83844
$$910$$ 0 0
$$911$$ −25.4178 −0.842128 −0.421064 0.907031i $$-0.638343\pi$$
−0.421064 + 0.907031i $$0.638343\pi$$
$$912$$ 14.0660 0.465770
$$913$$ −12.0647 −0.399282
$$914$$ 34.9522 1.15612
$$915$$ 0 0
$$916$$ −3.56280 −0.117718
$$917$$ 61.7594 2.03947
$$918$$ 13.0129 0.429489
$$919$$ 23.6017 0.778548 0.389274 0.921122i $$-0.372726\pi$$
0.389274 + 0.921122i $$0.372726\pi$$
$$920$$ 0 0
$$921$$ −95.1566 −3.13552
$$922$$ −16.3700 −0.539116
$$923$$ 26.5421 0.873643
$$924$$ −61.0129 −2.00718
$$925$$ 0 0
$$926$$ 29.2186 0.960183
$$927$$ −83.0116 −2.72646
$$928$$ 8.23805 0.270427
$$929$$ −7.08669 −0.232507 −0.116253 0.993220i $$-0.537088\pi$$
−0.116253 + 0.993220i $$0.537088\pi$$
$$930$$ 0 0
$$931$$ 60.1490 1.97131
$$932$$ 27.4956 0.900647
$$933$$ 17.3505 0.568030
$$934$$ 24.2770 0.794366
$$935$$ 0 0
$$936$$ −25.0854 −0.819942
$$937$$ −27.3169 −0.892404 −0.446202 0.894932i $$-0.647224\pi$$
−0.446202 + 0.894932i $$0.647224\pi$$
$$938$$ −10.9911 −0.358873
$$939$$ −12.7023 −0.414524
$$940$$ 0 0
$$941$$ 55.8979 1.82222 0.911109 0.412165i $$-0.135227\pi$$
0.911109 + 0.412165i $$0.135227\pi$$
$$942$$ 50.6469 1.65017
$$943$$ 3.90043 0.127015
$$944$$ −2.23805 −0.0728424
$$945$$ 0 0
$$946$$ 34.7010 1.12823
$$947$$ −37.5939 −1.22164 −0.610818 0.791771i $$-0.709159\pi$$
−0.610818 + 0.791771i $$0.709159\pi$$
$$948$$ −46.5150 −1.51074
$$949$$ 35.2575 1.14451
$$950$$ 0 0
$$951$$ 19.2305 0.623590
$$952$$ −5.04650 −0.163558
$$953$$ 29.3828 0.951804 0.475902 0.879498i $$-0.342122\pi$$
0.475902 + 0.879498i $$0.342122\pi$$
$$954$$ −40.3700 −1.30703
$$955$$ 0 0
$$956$$ 10.0389 0.324681
$$957$$ −111.454 −3.60280
$$958$$ −24.6080 −0.795049
$$959$$ 33.9545 1.09645
$$960$$ 0 0
$$961$$ −28.0129 −0.903641
$$962$$ −2.91331 −0.0939289
$$963$$ 107.201 3.45450
$$964$$ −23.6947 −0.763155
$$965$$ 0 0
$$966$$ 14.0660 0.452565
$$967$$ 49.2292 1.58310 0.791552 0.611102i $$-0.209274\pi$$
0.791552 + 0.611102i $$0.209274\pi$$
$$968$$ −7.81502 −0.251184
$$969$$ −15.7402 −0.505647
$$970$$ 0 0
$$971$$ 9.62347 0.308832 0.154416 0.988006i $$-0.450650\pi$$
0.154416 + 0.988006i $$0.450650\pi$$
$$972$$ 15.2846 0.490252
$$973$$ −21.0841 −0.675926
$$974$$ −30.2381 −0.968890
$$975$$ 0 0
$$976$$ 3.55623 0.113832
$$977$$ −18.9858 −0.607411 −0.303705 0.952766i $$-0.598224\pi$$
−0.303705 + 0.952766i $$0.598224\pi$$
$$978$$ −10.2651 −0.328242
$$979$$ −33.3768 −1.06673
$$980$$ 0 0
$$981$$ −10.0271 −0.320139
$$982$$ −12.3311 −0.393500
$$983$$ −4.33763 −0.138349 −0.0691744 0.997605i $$-0.522037\pi$$
−0.0691744 + 0.997605i $$0.522037\pi$$
$$984$$ −12.1655 −0.387823
$$985$$ 0 0
$$986$$ −9.21860 −0.293580
$$987$$ −161.149 −5.12942
$$988$$ 16.8137 0.534916
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 1.96766 0.0625049 0.0312524 0.999512i $$-0.490050\pi$$
0.0312524 + 0.999512i $$0.490050\pi$$
$$992$$ −1.72833 −0.0548744
$$993$$ 86.0495 2.73070
$$994$$ 32.1049 1.01830
$$995$$ 0 0
$$996$$ −8.67526 −0.274886
$$997$$ −3.96110 −0.125449 −0.0627246 0.998031i $$-0.519979\pi$$
−0.0627246 + 0.998031i $$0.519979\pi$$
$$998$$ 26.9133 0.851926
$$999$$ 9.08669 0.287490
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 1150.2.a.q.1.3 3
4.3 odd 2 9200.2.a.cf.1.1 3
5.2 odd 4 1150.2.b.j.599.1 6
5.3 odd 4 1150.2.b.j.599.6 6
5.4 even 2 230.2.a.d.1.1 3
15.14 odd 2 2070.2.a.z.1.3 3
20.19 odd 2 1840.2.a.r.1.3 3
40.19 odd 2 7360.2.a.ce.1.1 3
40.29 even 2 7360.2.a.bz.1.3 3
115.114 odd 2 5290.2.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 5.4 even 2
1150.2.a.q.1.3 3 1.1 even 1 trivial
1150.2.b.j.599.1 6 5.2 odd 4
1150.2.b.j.599.6 6 5.3 odd 4
1840.2.a.r.1.3 3 20.19 odd 2
2070.2.a.z.1.3 3 15.14 odd 2
5290.2.a.r.1.1 3 115.114 odd 2
7360.2.a.bz.1.3 3 40.29 even 2
7360.2.a.ce.1.1 3 40.19 odd 2
9200.2.a.cf.1.1 3 4.3 odd 2