# Properties

 Label 1014.2.a.m.1.1 Level $1014$ Weight $2$ Character 1014.1 Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.24698$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} +0.356896 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} +0.356896 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.692021 q^{10} +2.93900 q^{11} +1.00000 q^{12} -0.356896 q^{14} -0.692021 q^{15} +1.00000 q^{16} +6.71379 q^{17} -1.00000 q^{18} -7.20775 q^{19} -0.692021 q^{20} +0.356896 q^{21} -2.93900 q^{22} +2.39612 q^{23} -1.00000 q^{24} -4.52111 q^{25} +1.00000 q^{27} +0.356896 q^{28} +7.82908 q^{29} +0.692021 q^{30} -2.76271 q^{31} -1.00000 q^{32} +2.93900 q^{33} -6.71379 q^{34} -0.246980 q^{35} +1.00000 q^{36} +10.0978 q^{37} +7.20775 q^{38} +0.692021 q^{40} +4.89008 q^{41} -0.356896 q^{42} +6.59179 q^{43} +2.93900 q^{44} -0.692021 q^{45} -2.39612 q^{46} +4.98792 q^{47} +1.00000 q^{48} -6.87263 q^{49} +4.52111 q^{50} +6.71379 q^{51} -8.88769 q^{53} -1.00000 q^{54} -2.03385 q^{55} -0.356896 q^{56} -7.20775 q^{57} -7.82908 q^{58} +1.64310 q^{59} -0.692021 q^{60} -6.49396 q^{61} +2.76271 q^{62} +0.356896 q^{63} +1.00000 q^{64} -2.93900 q^{66} +13.5254 q^{67} +6.71379 q^{68} +2.39612 q^{69} +0.246980 q^{70} +6.81163 q^{71} -1.00000 q^{72} +3.18598 q^{73} -10.0978 q^{74} -4.52111 q^{75} -7.20775 q^{76} +1.04892 q^{77} +15.0465 q^{79} -0.692021 q^{80} +1.00000 q^{81} -4.89008 q^{82} -14.8267 q^{83} +0.356896 q^{84} -4.64609 q^{85} -6.59179 q^{86} +7.82908 q^{87} -2.93900 q^{88} +0.396125 q^{89} +0.692021 q^{90} +2.39612 q^{92} -2.76271 q^{93} -4.98792 q^{94} +4.98792 q^{95} -1.00000 q^{96} -0.417895 q^{97} +6.87263 q^{98} +2.93900 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} - q^{11} + 3 q^{12} + 3 q^{14} + 3 q^{15} + 3 q^{16} + 12 q^{17} - 3 q^{18} - 4 q^{19} + 3 q^{20} - 3 q^{21} + q^{22} + 16 q^{23} - 3 q^{24} + 2 q^{25} + 3 q^{27} - 3 q^{28} + 13 q^{29} - 3 q^{30} + 9 q^{31} - 3 q^{32} - q^{33} - 12 q^{34} + 4 q^{35} + 3 q^{36} + 12 q^{37} + 4 q^{38} - 3 q^{40} + 14 q^{41} + 3 q^{42} - 8 q^{43} - q^{44} + 3 q^{45} - 16 q^{46} - 4 q^{47} + 3 q^{48} - 4 q^{49} - 2 q^{50} + 12 q^{51} + 15 q^{53} - 3 q^{54} - 22 q^{55} + 3 q^{56} - 4 q^{57} - 13 q^{58} + 9 q^{59} + 3 q^{60} - 10 q^{61} - 9 q^{62} - 3 q^{63} + 3 q^{64} + q^{66} + 6 q^{67} + 12 q^{68} + 16 q^{69} - 4 q^{70} - 6 q^{71} - 3 q^{72} - 5 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 6 q^{77} - 5 q^{79} + 3 q^{80} + 3 q^{81} - 14 q^{82} + 7 q^{83} - 3 q^{84} + 26 q^{85} + 8 q^{86} + 13 q^{87} + q^{88} + 10 q^{89} - 3 q^{90} + 16 q^{92} + 9 q^{93} + 4 q^{94} - 4 q^{95} - 3 q^{96} - 7 q^{97} + 4 q^{98} - q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 - q^11 + 3 * q^12 + 3 * q^14 + 3 * q^15 + 3 * q^16 + 12 * q^17 - 3 * q^18 - 4 * q^19 + 3 * q^20 - 3 * q^21 + q^22 + 16 * q^23 - 3 * q^24 + 2 * q^25 + 3 * q^27 - 3 * q^28 + 13 * q^29 - 3 * q^30 + 9 * q^31 - 3 * q^32 - q^33 - 12 * q^34 + 4 * q^35 + 3 * q^36 + 12 * q^37 + 4 * q^38 - 3 * q^40 + 14 * q^41 + 3 * q^42 - 8 * q^43 - q^44 + 3 * q^45 - 16 * q^46 - 4 * q^47 + 3 * q^48 - 4 * q^49 - 2 * q^50 + 12 * q^51 + 15 * q^53 - 3 * q^54 - 22 * q^55 + 3 * q^56 - 4 * q^57 - 13 * q^58 + 9 * q^59 + 3 * q^60 - 10 * q^61 - 9 * q^62 - 3 * q^63 + 3 * q^64 + q^66 + 6 * q^67 + 12 * q^68 + 16 * q^69 - 4 * q^70 - 6 * q^71 - 3 * q^72 - 5 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^76 - 6 * q^77 - 5 * q^79 + 3 * q^80 + 3 * q^81 - 14 * q^82 + 7 * q^83 - 3 * q^84 + 26 * q^85 + 8 * q^86 + 13 * q^87 + q^88 + 10 * q^89 - 3 * q^90 + 16 * q^92 + 9 * q^93 + 4 * q^94 - 4 * q^95 - 3 * q^96 - 7 * q^97 + 4 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −0.692021 −0.309481 −0.154741 0.987955i $$-0.549454\pi$$
−0.154741 + 0.987955i $$0.549454\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0.356896 0.134894 0.0674470 0.997723i $$-0.478515\pi$$
0.0674470 + 0.997723i $$0.478515\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0.692021 0.218836
$$11$$ 2.93900 0.886142 0.443071 0.896486i $$-0.353889\pi$$
0.443071 + 0.896486i $$0.353889\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −0.356896 −0.0953844
$$15$$ −0.692021 −0.178679
$$16$$ 1.00000 0.250000
$$17$$ 6.71379 1.62833 0.814167 0.580631i $$-0.197194\pi$$
0.814167 + 0.580631i $$0.197194\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −7.20775 −1.65357 −0.826786 0.562517i $$-0.809833\pi$$
−0.826786 + 0.562517i $$0.809833\pi$$
$$20$$ −0.692021 −0.154741
$$21$$ 0.356896 0.0778811
$$22$$ −2.93900 −0.626597
$$23$$ 2.39612 0.499627 0.249813 0.968294i $$-0.419631\pi$$
0.249813 + 0.968294i $$0.419631\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.52111 −0.904221
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0.356896 0.0674470
$$29$$ 7.82908 1.45382 0.726912 0.686730i $$-0.240955\pi$$
0.726912 + 0.686730i $$0.240955\pi$$
$$30$$ 0.692021 0.126345
$$31$$ −2.76271 −0.496197 −0.248099 0.968735i $$-0.579806\pi$$
−0.248099 + 0.968735i $$0.579806\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.93900 0.511614
$$34$$ −6.71379 −1.15141
$$35$$ −0.246980 −0.0417472
$$36$$ 1.00000 0.166667
$$37$$ 10.0978 1.66007 0.830037 0.557708i $$-0.188319\pi$$
0.830037 + 0.557708i $$0.188319\pi$$
$$38$$ 7.20775 1.16925
$$39$$ 0 0
$$40$$ 0.692021 0.109418
$$41$$ 4.89008 0.763703 0.381851 0.924224i $$-0.375287\pi$$
0.381851 + 0.924224i $$0.375287\pi$$
$$42$$ −0.356896 −0.0550702
$$43$$ 6.59179 1.00524 0.502620 0.864508i $$-0.332370\pi$$
0.502620 + 0.864508i $$0.332370\pi$$
$$44$$ 2.93900 0.443071
$$45$$ −0.692021 −0.103160
$$46$$ −2.39612 −0.353289
$$47$$ 4.98792 0.727563 0.363781 0.931484i $$-0.381486\pi$$
0.363781 + 0.931484i $$0.381486\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.87263 −0.981804
$$50$$ 4.52111 0.639381
$$51$$ 6.71379 0.940119
$$52$$ 0 0
$$53$$ −8.88769 −1.22082 −0.610409 0.792086i $$-0.708995\pi$$
−0.610409 + 0.792086i $$0.708995\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −2.03385 −0.274245
$$56$$ −0.356896 −0.0476922
$$57$$ −7.20775 −0.954690
$$58$$ −7.82908 −1.02801
$$59$$ 1.64310 0.213914 0.106957 0.994264i $$-0.465889\pi$$
0.106957 + 0.994264i $$0.465889\pi$$
$$60$$ −0.692021 −0.0893396
$$61$$ −6.49396 −0.831466 −0.415733 0.909487i $$-0.636475\pi$$
−0.415733 + 0.909487i $$0.636475\pi$$
$$62$$ 2.76271 0.350864
$$63$$ 0.356896 0.0449647
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.93900 −0.361766
$$67$$ 13.5254 1.65239 0.826196 0.563382i $$-0.190500\pi$$
0.826196 + 0.563382i $$0.190500\pi$$
$$68$$ 6.71379 0.814167
$$69$$ 2.39612 0.288459
$$70$$ 0.246980 0.0295197
$$71$$ 6.81163 0.808391 0.404196 0.914673i $$-0.367551\pi$$
0.404196 + 0.914673i $$0.367551\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 3.18598 0.372891 0.186445 0.982465i $$-0.440303\pi$$
0.186445 + 0.982465i $$0.440303\pi$$
$$74$$ −10.0978 −1.17385
$$75$$ −4.52111 −0.522052
$$76$$ −7.20775 −0.826786
$$77$$ 1.04892 0.119535
$$78$$ 0 0
$$79$$ 15.0465 1.69287 0.846433 0.532495i $$-0.178745\pi$$
0.846433 + 0.532495i $$0.178745\pi$$
$$80$$ −0.692021 −0.0773704
$$81$$ 1.00000 0.111111
$$82$$ −4.89008 −0.540019
$$83$$ −14.8267 −1.62744 −0.813720 0.581256i $$-0.802561\pi$$
−0.813720 + 0.581256i $$0.802561\pi$$
$$84$$ 0.356896 0.0389405
$$85$$ −4.64609 −0.503939
$$86$$ −6.59179 −0.710811
$$87$$ 7.82908 0.839366
$$88$$ −2.93900 −0.313299
$$89$$ 0.396125 0.0419891 0.0209946 0.999780i $$-0.493317\pi$$
0.0209946 + 0.999780i $$0.493317\pi$$
$$90$$ 0.692021 0.0729455
$$91$$ 0 0
$$92$$ 2.39612 0.249813
$$93$$ −2.76271 −0.286480
$$94$$ −4.98792 −0.514465
$$95$$ 4.98792 0.511750
$$96$$ −1.00000 −0.102062
$$97$$ −0.417895 −0.0424308 −0.0212154 0.999775i $$-0.506754\pi$$
−0.0212154 + 0.999775i $$0.506754\pi$$
$$98$$ 6.87263 0.694240
$$99$$ 2.93900 0.295381
$$100$$ −4.52111 −0.452111
$$101$$ 10.0151 0.996536 0.498268 0.867023i $$-0.333970\pi$$
0.498268 + 0.867023i $$0.333970\pi$$
$$102$$ −6.71379 −0.664764
$$103$$ −9.62565 −0.948443 −0.474222 0.880406i $$-0.657270\pi$$
−0.474222 + 0.880406i $$0.657270\pi$$
$$104$$ 0 0
$$105$$ −0.246980 −0.0241027
$$106$$ 8.88769 0.863249
$$107$$ −6.63102 −0.641045 −0.320523 0.947241i $$-0.603859\pi$$
−0.320523 + 0.947241i $$0.603859\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 12.9879 1.24402 0.622008 0.783011i $$-0.286317\pi$$
0.622008 + 0.783011i $$0.286317\pi$$
$$110$$ 2.03385 0.193920
$$111$$ 10.0978 0.958444
$$112$$ 0.356896 0.0337235
$$113$$ 0.792249 0.0745285 0.0372643 0.999305i $$-0.488136\pi$$
0.0372643 + 0.999305i $$0.488136\pi$$
$$114$$ 7.20775 0.675068
$$115$$ −1.65817 −0.154625
$$116$$ 7.82908 0.726912
$$117$$ 0 0
$$118$$ −1.64310 −0.151260
$$119$$ 2.39612 0.219652
$$120$$ 0.692021 0.0631726
$$121$$ −2.36227 −0.214752
$$122$$ 6.49396 0.587935
$$123$$ 4.89008 0.440924
$$124$$ −2.76271 −0.248099
$$125$$ 6.58881 0.589321
$$126$$ −0.356896 −0.0317948
$$127$$ −18.2174 −1.61654 −0.808268 0.588815i $$-0.799595\pi$$
−0.808268 + 0.588815i $$0.799595\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 6.59179 0.580375
$$130$$ 0 0
$$131$$ 2.73556 0.239007 0.119504 0.992834i $$-0.461870\pi$$
0.119504 + 0.992834i $$0.461870\pi$$
$$132$$ 2.93900 0.255807
$$133$$ −2.57242 −0.223057
$$134$$ −13.5254 −1.16842
$$135$$ −0.692021 −0.0595597
$$136$$ −6.71379 −0.575703
$$137$$ −7.64742 −0.653363 −0.326681 0.945135i $$-0.605930\pi$$
−0.326681 + 0.945135i $$0.605930\pi$$
$$138$$ −2.39612 −0.203972
$$139$$ 3.38404 0.287031 0.143515 0.989648i $$-0.454159\pi$$
0.143515 + 0.989648i $$0.454159\pi$$
$$140$$ −0.246980 −0.0208736
$$141$$ 4.98792 0.420059
$$142$$ −6.81163 −0.571619
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −5.41789 −0.449932
$$146$$ −3.18598 −0.263674
$$147$$ −6.87263 −0.566845
$$148$$ 10.0978 0.830037
$$149$$ 20.8170 1.70540 0.852698 0.522405i $$-0.174965\pi$$
0.852698 + 0.522405i $$0.174965\pi$$
$$150$$ 4.52111 0.369147
$$151$$ 0.895461 0.0728715 0.0364358 0.999336i $$-0.488400\pi$$
0.0364358 + 0.999336i $$0.488400\pi$$
$$152$$ 7.20775 0.584626
$$153$$ 6.71379 0.542778
$$154$$ −1.04892 −0.0845242
$$155$$ 1.91185 0.153564
$$156$$ 0 0
$$157$$ 8.59179 0.685700 0.342850 0.939390i $$-0.388608\pi$$
0.342850 + 0.939390i $$0.388608\pi$$
$$158$$ −15.0465 −1.19704
$$159$$ −8.88769 −0.704840
$$160$$ 0.692021 0.0547091
$$161$$ 0.855167 0.0673966
$$162$$ −1.00000 −0.0785674
$$163$$ −1.72587 −0.135181 −0.0675904 0.997713i $$-0.521531\pi$$
−0.0675904 + 0.997713i $$0.521531\pi$$
$$164$$ 4.89008 0.381851
$$165$$ −2.03385 −0.158335
$$166$$ 14.8267 1.15077
$$167$$ −21.1400 −1.63587 −0.817933 0.575314i $$-0.804880\pi$$
−0.817933 + 0.575314i $$0.804880\pi$$
$$168$$ −0.356896 −0.0275351
$$169$$ 0 0
$$170$$ 4.64609 0.356339
$$171$$ −7.20775 −0.551190
$$172$$ 6.59179 0.502620
$$173$$ 9.35450 0.711210 0.355605 0.934636i $$-0.384275\pi$$
0.355605 + 0.934636i $$0.384275\pi$$
$$174$$ −7.82908 −0.593521
$$175$$ −1.61356 −0.121974
$$176$$ 2.93900 0.221536
$$177$$ 1.64310 0.123503
$$178$$ −0.396125 −0.0296908
$$179$$ 3.17523 0.237328 0.118664 0.992934i $$-0.462139\pi$$
0.118664 + 0.992934i $$0.462139\pi$$
$$180$$ −0.692021 −0.0515802
$$181$$ −19.7995 −1.47169 −0.735844 0.677151i $$-0.763214\pi$$
−0.735844 + 0.677151i $$0.763214\pi$$
$$182$$ 0 0
$$183$$ −6.49396 −0.480047
$$184$$ −2.39612 −0.176645
$$185$$ −6.98792 −0.513762
$$186$$ 2.76271 0.202572
$$187$$ 19.7318 1.44294
$$188$$ 4.98792 0.363781
$$189$$ 0.356896 0.0259604
$$190$$ −4.98792 −0.361862
$$191$$ 15.2620 1.10432 0.552161 0.833737i $$-0.313803\pi$$
0.552161 + 0.833737i $$0.313803\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −4.76809 −0.343214 −0.171607 0.985165i $$-0.554896\pi$$
−0.171607 + 0.985165i $$0.554896\pi$$
$$194$$ 0.417895 0.0300031
$$195$$ 0 0
$$196$$ −6.87263 −0.490902
$$197$$ −12.2349 −0.871700 −0.435850 0.900019i $$-0.643552\pi$$
−0.435850 + 0.900019i $$0.643552\pi$$
$$198$$ −2.93900 −0.208866
$$199$$ −11.8485 −0.839915 −0.419958 0.907544i $$-0.637955\pi$$
−0.419958 + 0.907544i $$0.637955\pi$$
$$200$$ 4.52111 0.319690
$$201$$ 13.5254 0.954009
$$202$$ −10.0151 −0.704658
$$203$$ 2.79417 0.196112
$$204$$ 6.71379 0.470059
$$205$$ −3.38404 −0.236352
$$206$$ 9.62565 0.670651
$$207$$ 2.39612 0.166542
$$208$$ 0 0
$$209$$ −21.1836 −1.46530
$$210$$ 0.246980 0.0170432
$$211$$ −17.2620 −1.18837 −0.594184 0.804329i $$-0.702525\pi$$
−0.594184 + 0.804329i $$0.702525\pi$$
$$212$$ −8.88769 −0.610409
$$213$$ 6.81163 0.466725
$$214$$ 6.63102 0.453287
$$215$$ −4.56166 −0.311103
$$216$$ −1.00000 −0.0680414
$$217$$ −0.985999 −0.0669340
$$218$$ −12.9879 −0.879653
$$219$$ 3.18598 0.215289
$$220$$ −2.03385 −0.137122
$$221$$ 0 0
$$222$$ −10.0978 −0.677722
$$223$$ −6.76809 −0.453225 −0.226612 0.973985i $$-0.572765\pi$$
−0.226612 + 0.973985i $$0.572765\pi$$
$$224$$ −0.356896 −0.0238461
$$225$$ −4.52111 −0.301407
$$226$$ −0.792249 −0.0526996
$$227$$ −23.6799 −1.57169 −0.785846 0.618422i $$-0.787772\pi$$
−0.785846 + 0.618422i $$0.787772\pi$$
$$228$$ −7.20775 −0.477345
$$229$$ 8.29829 0.548366 0.274183 0.961677i $$-0.411593\pi$$
0.274183 + 0.961677i $$0.411593\pi$$
$$230$$ 1.65817 0.109336
$$231$$ 1.04892 0.0690137
$$232$$ −7.82908 −0.514005
$$233$$ −23.9651 −1.57000 −0.785002 0.619493i $$-0.787338\pi$$
−0.785002 + 0.619493i $$0.787338\pi$$
$$234$$ 0 0
$$235$$ −3.45175 −0.225167
$$236$$ 1.64310 0.106957
$$237$$ 15.0465 0.977377
$$238$$ −2.39612 −0.155318
$$239$$ −12.6160 −0.816058 −0.408029 0.912969i $$-0.633784\pi$$
−0.408029 + 0.912969i $$0.633784\pi$$
$$240$$ −0.692021 −0.0446698
$$241$$ −26.3937 −1.70017 −0.850085 0.526646i $$-0.823449\pi$$
−0.850085 + 0.526646i $$0.823449\pi$$
$$242$$ 2.36227 0.151853
$$243$$ 1.00000 0.0641500
$$244$$ −6.49396 −0.415733
$$245$$ 4.75600 0.303850
$$246$$ −4.89008 −0.311780
$$247$$ 0 0
$$248$$ 2.76271 0.175432
$$249$$ −14.8267 −0.939603
$$250$$ −6.58881 −0.416713
$$251$$ −30.0344 −1.89576 −0.947879 0.318632i $$-0.896777\pi$$
−0.947879 + 0.318632i $$0.896777\pi$$
$$252$$ 0.356896 0.0224823
$$253$$ 7.04221 0.442740
$$254$$ 18.2174 1.14306
$$255$$ −4.64609 −0.290949
$$256$$ 1.00000 0.0625000
$$257$$ 11.2620 0.702507 0.351254 0.936280i $$-0.385756\pi$$
0.351254 + 0.936280i $$0.385756\pi$$
$$258$$ −6.59179 −0.410387
$$259$$ 3.60388 0.223934
$$260$$ 0 0
$$261$$ 7.82908 0.484608
$$262$$ −2.73556 −0.169004
$$263$$ −5.54958 −0.342202 −0.171101 0.985254i $$-0.554732\pi$$
−0.171101 + 0.985254i $$0.554732\pi$$
$$264$$ −2.93900 −0.180883
$$265$$ 6.15047 0.377821
$$266$$ 2.57242 0.157725
$$267$$ 0.396125 0.0242424
$$268$$ 13.5254 0.826196
$$269$$ 16.6872 1.01744 0.508719 0.860932i $$-0.330119\pi$$
0.508719 + 0.860932i $$0.330119\pi$$
$$270$$ 0.692021 0.0421151
$$271$$ 6.61356 0.401745 0.200873 0.979617i $$-0.435622\pi$$
0.200873 + 0.979617i $$0.435622\pi$$
$$272$$ 6.71379 0.407083
$$273$$ 0 0
$$274$$ 7.64742 0.461997
$$275$$ −13.2875 −0.801269
$$276$$ 2.39612 0.144230
$$277$$ −21.7995 −1.30981 −0.654904 0.755712i $$-0.727291\pi$$
−0.654904 + 0.755712i $$0.727291\pi$$
$$278$$ −3.38404 −0.202961
$$279$$ −2.76271 −0.165399
$$280$$ 0.246980 0.0147599
$$281$$ −20.5918 −1.22840 −0.614202 0.789149i $$-0.710522\pi$$
−0.614202 + 0.789149i $$0.710522\pi$$
$$282$$ −4.98792 −0.297026
$$283$$ −13.0121 −0.773488 −0.386744 0.922187i $$-0.626400\pi$$
−0.386744 + 0.922187i $$0.626400\pi$$
$$284$$ 6.81163 0.404196
$$285$$ 4.98792 0.295459
$$286$$ 0 0
$$287$$ 1.74525 0.103019
$$288$$ −1.00000 −0.0589256
$$289$$ 28.0750 1.65147
$$290$$ 5.41789 0.318150
$$291$$ −0.417895 −0.0244974
$$292$$ 3.18598 0.186445
$$293$$ 14.9390 0.872746 0.436373 0.899766i $$-0.356263\pi$$
0.436373 + 0.899766i $$0.356263\pi$$
$$294$$ 6.87263 0.400820
$$295$$ −1.13706 −0.0662024
$$296$$ −10.0978 −0.586925
$$297$$ 2.93900 0.170538
$$298$$ −20.8170 −1.20590
$$299$$ 0 0
$$300$$ −4.52111 −0.261026
$$301$$ 2.35258 0.135601
$$302$$ −0.895461 −0.0515280
$$303$$ 10.0151 0.575350
$$304$$ −7.20775 −0.413393
$$305$$ 4.49396 0.257323
$$306$$ −6.71379 −0.383802
$$307$$ −26.0301 −1.48562 −0.742809 0.669503i $$-0.766507\pi$$
−0.742809 + 0.669503i $$0.766507\pi$$
$$308$$ 1.04892 0.0597676
$$309$$ −9.62565 −0.547584
$$310$$ −1.91185 −0.108586
$$311$$ −4.81163 −0.272842 −0.136421 0.990651i $$-0.543560\pi$$
−0.136421 + 0.990651i $$0.543560\pi$$
$$312$$ 0 0
$$313$$ −26.0411 −1.47193 −0.735966 0.677018i $$-0.763272\pi$$
−0.735966 + 0.677018i $$0.763272\pi$$
$$314$$ −8.59179 −0.484863
$$315$$ −0.246980 −0.0139157
$$316$$ 15.0465 0.846433
$$317$$ 11.5211 0.647090 0.323545 0.946213i $$-0.395125\pi$$
0.323545 + 0.946213i $$0.395125\pi$$
$$318$$ 8.88769 0.498397
$$319$$ 23.0097 1.28830
$$320$$ −0.692021 −0.0386852
$$321$$ −6.63102 −0.370108
$$322$$ −0.855167 −0.0476566
$$323$$ −48.3913 −2.69257
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 1.72587 0.0955873
$$327$$ 12.9879 0.718234
$$328$$ −4.89008 −0.270010
$$329$$ 1.78017 0.0981438
$$330$$ 2.03385 0.111960
$$331$$ −3.43834 −0.188988 −0.0944940 0.995525i $$-0.530123\pi$$
−0.0944940 + 0.995525i $$0.530123\pi$$
$$332$$ −14.8267 −0.813720
$$333$$ 10.0978 0.553358
$$334$$ 21.1400 1.15673
$$335$$ −9.35988 −0.511385
$$336$$ 0.356896 0.0194703
$$337$$ 8.20105 0.446739 0.223370 0.974734i $$-0.428294\pi$$
0.223370 + 0.974734i $$0.428294\pi$$
$$338$$ 0 0
$$339$$ 0.792249 0.0430291
$$340$$ −4.64609 −0.251970
$$341$$ −8.11960 −0.439701
$$342$$ 7.20775 0.389751
$$343$$ −4.95108 −0.267333
$$344$$ −6.59179 −0.355406
$$345$$ −1.65817 −0.0892729
$$346$$ −9.35450 −0.502901
$$347$$ 7.86294 0.422105 0.211052 0.977475i $$-0.432311\pi$$
0.211052 + 0.977475i $$0.432311\pi$$
$$348$$ 7.82908 0.419683
$$349$$ 18.7245 1.00230 0.501151 0.865360i $$-0.332910\pi$$
0.501151 + 0.865360i $$0.332910\pi$$
$$350$$ 1.61356 0.0862486
$$351$$ 0 0
$$352$$ −2.93900 −0.156649
$$353$$ 31.5448 1.67896 0.839480 0.543391i $$-0.182860\pi$$
0.839480 + 0.543391i $$0.182860\pi$$
$$354$$ −1.64310 −0.0873300
$$355$$ −4.71379 −0.250182
$$356$$ 0.396125 0.0209946
$$357$$ 2.39612 0.126816
$$358$$ −3.17523 −0.167816
$$359$$ −2.39612 −0.126463 −0.0632313 0.997999i $$-0.520141\pi$$
−0.0632313 + 0.997999i $$0.520141\pi$$
$$360$$ 0.692021 0.0364727
$$361$$ 32.9517 1.73430
$$362$$ 19.7995 1.04064
$$363$$ −2.36227 −0.123987
$$364$$ 0 0
$$365$$ −2.20477 −0.115403
$$366$$ 6.49396 0.339445
$$367$$ −0.00431187 −0.000225078 0 −0.000112539 1.00000i $$-0.500036\pi$$
−0.000112539 1.00000i $$0.500036\pi$$
$$368$$ 2.39612 0.124907
$$369$$ 4.89008 0.254568
$$370$$ 6.98792 0.363285
$$371$$ −3.17198 −0.164681
$$372$$ −2.76271 −0.143240
$$373$$ −32.3129 −1.67310 −0.836549 0.547892i $$-0.815430\pi$$
−0.836549 + 0.547892i $$0.815430\pi$$
$$374$$ −19.7318 −1.02031
$$375$$ 6.58881 0.340245
$$376$$ −4.98792 −0.257232
$$377$$ 0 0
$$378$$ −0.356896 −0.0183567
$$379$$ −19.7560 −1.01480 −0.507399 0.861711i $$-0.669393\pi$$
−0.507399 + 0.861711i $$0.669393\pi$$
$$380$$ 4.98792 0.255875
$$381$$ −18.2174 −0.933308
$$382$$ −15.2620 −0.780874
$$383$$ −28.8116 −1.47221 −0.736103 0.676870i $$-0.763336\pi$$
−0.736103 + 0.676870i $$0.763336\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −0.725873 −0.0369939
$$386$$ 4.76809 0.242689
$$387$$ 6.59179 0.335080
$$388$$ −0.417895 −0.0212154
$$389$$ 34.7821 1.76352 0.881761 0.471697i $$-0.156358\pi$$
0.881761 + 0.471697i $$0.156358\pi$$
$$390$$ 0 0
$$391$$ 16.0871 0.813559
$$392$$ 6.87263 0.347120
$$393$$ 2.73556 0.137991
$$394$$ 12.2349 0.616385
$$395$$ −10.4125 −0.523911
$$396$$ 2.93900 0.147690
$$397$$ 5.15346 0.258645 0.129322 0.991603i $$-0.458720\pi$$
0.129322 + 0.991603i $$0.458720\pi$$
$$398$$ 11.8485 0.593910
$$399$$ −2.57242 −0.128782
$$400$$ −4.52111 −0.226055
$$401$$ −13.3250 −0.665417 −0.332708 0.943030i $$-0.607962\pi$$
−0.332708 + 0.943030i $$0.607962\pi$$
$$402$$ −13.5254 −0.674587
$$403$$ 0 0
$$404$$ 10.0151 0.498268
$$405$$ −0.692021 −0.0343868
$$406$$ −2.79417 −0.138672
$$407$$ 29.6775 1.47106
$$408$$ −6.71379 −0.332382
$$409$$ 24.0237 1.18789 0.593947 0.804504i $$-0.297569\pi$$
0.593947 + 0.804504i $$0.297569\pi$$
$$410$$ 3.38404 0.167126
$$411$$ −7.64742 −0.377219
$$412$$ −9.62565 −0.474222
$$413$$ 0.586417 0.0288557
$$414$$ −2.39612 −0.117763
$$415$$ 10.2604 0.503663
$$416$$ 0 0
$$417$$ 3.38404 0.165717
$$418$$ 21.1836 1.03612
$$419$$ 13.8049 0.674415 0.337207 0.941430i $$-0.390518\pi$$
0.337207 + 0.941430i $$0.390518\pi$$
$$420$$ −0.246980 −0.0120514
$$421$$ −7.72587 −0.376536 −0.188268 0.982118i $$-0.560287\pi$$
−0.188268 + 0.982118i $$0.560287\pi$$
$$422$$ 17.2620 0.840303
$$423$$ 4.98792 0.242521
$$424$$ 8.88769 0.431624
$$425$$ −30.3538 −1.47237
$$426$$ −6.81163 −0.330024
$$427$$ −2.31767 −0.112160
$$428$$ −6.63102 −0.320523
$$429$$ 0 0
$$430$$ 4.56166 0.219983
$$431$$ −0.640120 −0.0308335 −0.0154168 0.999881i $$-0.504907\pi$$
−0.0154168 + 0.999881i $$0.504907\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 21.2760 1.02246 0.511231 0.859443i $$-0.329190\pi$$
0.511231 + 0.859443i $$0.329190\pi$$
$$434$$ 0.985999 0.0473295
$$435$$ −5.41789 −0.259768
$$436$$ 12.9879 0.622008
$$437$$ −17.2707 −0.826168
$$438$$ −3.18598 −0.152232
$$439$$ 12.7181 0.607002 0.303501 0.952831i $$-0.401844\pi$$
0.303501 + 0.952831i $$0.401844\pi$$
$$440$$ 2.03385 0.0969601
$$441$$ −6.87263 −0.327268
$$442$$ 0 0
$$443$$ 22.5972 1.07362 0.536812 0.843702i $$-0.319628\pi$$
0.536812 + 0.843702i $$0.319628\pi$$
$$444$$ 10.0978 0.479222
$$445$$ −0.274127 −0.0129949
$$446$$ 6.76809 0.320478
$$447$$ 20.8170 0.984610
$$448$$ 0.356896 0.0168617
$$449$$ −11.6474 −0.549676 −0.274838 0.961491i $$-0.588624\pi$$
−0.274838 + 0.961491i $$0.588624\pi$$
$$450$$ 4.52111 0.213127
$$451$$ 14.3720 0.676749
$$452$$ 0.792249 0.0372643
$$453$$ 0.895461 0.0420724
$$454$$ 23.6799 1.11135
$$455$$ 0 0
$$456$$ 7.20775 0.337534
$$457$$ 21.1890 0.991178 0.495589 0.868557i $$-0.334952\pi$$
0.495589 + 0.868557i $$0.334952\pi$$
$$458$$ −8.29829 −0.387754
$$459$$ 6.71379 0.313373
$$460$$ −1.65817 −0.0773126
$$461$$ −24.0694 −1.12102 −0.560511 0.828147i $$-0.689395\pi$$
−0.560511 + 0.828147i $$0.689395\pi$$
$$462$$ −1.04892 −0.0488001
$$463$$ 18.1715 0.844502 0.422251 0.906479i $$-0.361240\pi$$
0.422251 + 0.906479i $$0.361240\pi$$
$$464$$ 7.82908 0.363456
$$465$$ 1.91185 0.0886601
$$466$$ 23.9651 1.11016
$$467$$ −2.93123 −0.135641 −0.0678206 0.997698i $$-0.521605\pi$$
−0.0678206 + 0.997698i $$0.521605\pi$$
$$468$$ 0 0
$$469$$ 4.82717 0.222898
$$470$$ 3.45175 0.159217
$$471$$ 8.59179 0.395889
$$472$$ −1.64310 −0.0756300
$$473$$ 19.3733 0.890785
$$474$$ −15.0465 −0.691110
$$475$$ 32.5870 1.49519
$$476$$ 2.39612 0.109826
$$477$$ −8.88769 −0.406939
$$478$$ 12.6160 0.577040
$$479$$ 30.7090 1.40313 0.701565 0.712605i $$-0.252485\pi$$
0.701565 + 0.712605i $$0.252485\pi$$
$$480$$ 0.692021 0.0315863
$$481$$ 0 0
$$482$$ 26.3937 1.20220
$$483$$ 0.855167 0.0389114
$$484$$ −2.36227 −0.107376
$$485$$ 0.289192 0.0131315
$$486$$ −1.00000 −0.0453609
$$487$$ −24.1497 −1.09433 −0.547164 0.837025i $$-0.684293\pi$$
−0.547164 + 0.837025i $$0.684293\pi$$
$$488$$ 6.49396 0.293968
$$489$$ −1.72587 −0.0780467
$$490$$ −4.75600 −0.214854
$$491$$ 14.5972 0.658761 0.329381 0.944197i $$-0.393160\pi$$
0.329381 + 0.944197i $$0.393160\pi$$
$$492$$ 4.89008 0.220462
$$493$$ 52.5628 2.36731
$$494$$ 0 0
$$495$$ −2.03385 −0.0914148
$$496$$ −2.76271 −0.124049
$$497$$ 2.43104 0.109047
$$498$$ 14.8267 0.664400
$$499$$ −6.85517 −0.306879 −0.153440 0.988158i $$-0.549035\pi$$
−0.153440 + 0.988158i $$0.549035\pi$$
$$500$$ 6.58881 0.294661
$$501$$ −21.1400 −0.944468
$$502$$ 30.0344 1.34050
$$503$$ −26.7332 −1.19197 −0.595987 0.802994i $$-0.703239\pi$$
−0.595987 + 0.802994i $$0.703239\pi$$
$$504$$ −0.356896 −0.0158974
$$505$$ −6.93064 −0.308409
$$506$$ −7.04221 −0.313065
$$507$$ 0 0
$$508$$ −18.2174 −0.808268
$$509$$ 20.6595 0.915716 0.457858 0.889025i $$-0.348617\pi$$
0.457858 + 0.889025i $$0.348617\pi$$
$$510$$ 4.64609 0.205732
$$511$$ 1.13706 0.0503007
$$512$$ −1.00000 −0.0441942
$$513$$ −7.20775 −0.318230
$$514$$ −11.2620 −0.496748
$$515$$ 6.66115 0.293525
$$516$$ 6.59179 0.290188
$$517$$ 14.6595 0.644724
$$518$$ −3.60388 −0.158345
$$519$$ 9.35450 0.410617
$$520$$ 0 0
$$521$$ −15.0965 −0.661390 −0.330695 0.943738i $$-0.607283\pi$$
−0.330695 + 0.943738i $$0.607283\pi$$
$$522$$ −7.82908 −0.342670
$$523$$ 0.0349168 0.00152680 0.000763402 1.00000i $$-0.499757\pi$$
0.000763402 1.00000i $$0.499757\pi$$
$$524$$ 2.73556 0.119504
$$525$$ −1.61356 −0.0704217
$$526$$ 5.54958 0.241973
$$527$$ −18.5483 −0.807975
$$528$$ 2.93900 0.127904
$$529$$ −17.2586 −0.750373
$$530$$ −6.15047 −0.267159
$$531$$ 1.64310 0.0713046
$$532$$ −2.57242 −0.111528
$$533$$ 0 0
$$534$$ −0.396125 −0.0171420
$$535$$ 4.58881 0.198392
$$536$$ −13.5254 −0.584209
$$537$$ 3.17523 0.137021
$$538$$ −16.6872 −0.719438
$$539$$ −20.1987 −0.870018
$$540$$ −0.692021 −0.0297799
$$541$$ −13.0858 −0.562600 −0.281300 0.959620i $$-0.590766\pi$$
−0.281300 + 0.959620i $$0.590766\pi$$
$$542$$ −6.61356 −0.284077
$$543$$ −19.7995 −0.849680
$$544$$ −6.71379 −0.287851
$$545$$ −8.98792 −0.385000
$$546$$ 0 0
$$547$$ −5.97584 −0.255508 −0.127754 0.991806i $$-0.540777\pi$$
−0.127754 + 0.991806i $$0.540777\pi$$
$$548$$ −7.64742 −0.326681
$$549$$ −6.49396 −0.277155
$$550$$ 13.2875 0.566582
$$551$$ −56.4301 −2.40400
$$552$$ −2.39612 −0.101986
$$553$$ 5.37004 0.228357
$$554$$ 21.7995 0.926174
$$555$$ −6.98792 −0.296621
$$556$$ 3.38404 0.143515
$$557$$ 10.4397 0.442343 0.221171 0.975235i $$-0.429012\pi$$
0.221171 + 0.975235i $$0.429012\pi$$
$$558$$ 2.76271 0.116955
$$559$$ 0 0
$$560$$ −0.246980 −0.0104368
$$561$$ 19.7318 0.833079
$$562$$ 20.5918 0.868612
$$563$$ −5.66056 −0.238564 −0.119282 0.992860i $$-0.538059\pi$$
−0.119282 + 0.992860i $$0.538059\pi$$
$$564$$ 4.98792 0.210029
$$565$$ −0.548253 −0.0230652
$$566$$ 13.0121 0.546939
$$567$$ 0.356896 0.0149882
$$568$$ −6.81163 −0.285809
$$569$$ 20.6568 0.865980 0.432990 0.901399i $$-0.357459\pi$$
0.432990 + 0.901399i $$0.357459\pi$$
$$570$$ −4.98792 −0.208921
$$571$$ 44.3672 1.85671 0.928354 0.371697i $$-0.121224\pi$$
0.928354 + 0.371697i $$0.121224\pi$$
$$572$$ 0 0
$$573$$ 15.2620 0.637581
$$574$$ −1.74525 −0.0728454
$$575$$ −10.8331 −0.451773
$$576$$ 1.00000 0.0416667
$$577$$ −29.4426 −1.22571 −0.612857 0.790194i $$-0.709980\pi$$
−0.612857 + 0.790194i $$0.709980\pi$$
$$578$$ −28.0750 −1.16777
$$579$$ −4.76809 −0.198155
$$580$$ −5.41789 −0.224966
$$581$$ −5.29159 −0.219532
$$582$$ 0.417895 0.0173223
$$583$$ −26.1209 −1.08182
$$584$$ −3.18598 −0.131837
$$585$$ 0 0
$$586$$ −14.9390 −0.617124
$$587$$ 19.5636 0.807475 0.403738 0.914875i $$-0.367711\pi$$
0.403738 + 0.914875i $$0.367711\pi$$
$$588$$ −6.87263 −0.283422
$$589$$ 19.9129 0.820498
$$590$$ 1.13706 0.0468122
$$591$$ −12.2349 −0.503276
$$592$$ 10.0978 0.415018
$$593$$ 10.8793 0.446761 0.223380 0.974731i $$-0.428291\pi$$
0.223380 + 0.974731i $$0.428291\pi$$
$$594$$ −2.93900 −0.120589
$$595$$ −1.65817 −0.0679783
$$596$$ 20.8170 0.852698
$$597$$ −11.8485 −0.484925
$$598$$ 0 0
$$599$$ 16.0543 0.655961 0.327980 0.944685i $$-0.393632\pi$$
0.327980 + 0.944685i $$0.393632\pi$$
$$600$$ 4.52111 0.184573
$$601$$ 12.8955 0.526017 0.263008 0.964794i $$-0.415285\pi$$
0.263008 + 0.964794i $$0.415285\pi$$
$$602$$ −2.35258 −0.0958842
$$603$$ 13.5254 0.550798
$$604$$ 0.895461 0.0364358
$$605$$ 1.63474 0.0664618
$$606$$ −10.0151 −0.406834
$$607$$ 44.4741 1.80515 0.902574 0.430534i $$-0.141675\pi$$
0.902574 + 0.430534i $$0.141675\pi$$
$$608$$ 7.20775 0.292313
$$609$$ 2.79417 0.113225
$$610$$ −4.49396 −0.181955
$$611$$ 0 0
$$612$$ 6.71379 0.271389
$$613$$ −42.0253 −1.69739 −0.848694 0.528884i $$-0.822610\pi$$
−0.848694 + 0.528884i $$0.822610\pi$$
$$614$$ 26.0301 1.05049
$$615$$ −3.38404 −0.136458
$$616$$ −1.04892 −0.0422621
$$617$$ 16.6655 0.670926 0.335463 0.942053i $$-0.391107\pi$$
0.335463 + 0.942053i $$0.391107\pi$$
$$618$$ 9.62565 0.387200
$$619$$ −39.7512 −1.59774 −0.798868 0.601506i $$-0.794568\pi$$
−0.798868 + 0.601506i $$0.794568\pi$$
$$620$$ 1.91185 0.0767819
$$621$$ 2.39612 0.0961532
$$622$$ 4.81163 0.192929
$$623$$ 0.141375 0.00566408
$$624$$ 0 0
$$625$$ 18.0459 0.721837
$$626$$ 26.0411 1.04081
$$627$$ −21.1836 −0.845991
$$628$$ 8.59179 0.342850
$$629$$ 67.7948 2.70315
$$630$$ 0.246980 0.00983990
$$631$$ −14.5767 −0.580290 −0.290145 0.956983i $$-0.593704\pi$$
−0.290145 + 0.956983i $$0.593704\pi$$
$$632$$ −15.0465 −0.598519
$$633$$ −17.2620 −0.686105
$$634$$ −11.5211 −0.457562
$$635$$ 12.6069 0.500288
$$636$$ −8.88769 −0.352420
$$637$$ 0 0
$$638$$ −23.0097 −0.910962
$$639$$ 6.81163 0.269464
$$640$$ 0.692021 0.0273546
$$641$$ 38.4349 1.51809 0.759043 0.651040i $$-0.225667\pi$$
0.759043 + 0.651040i $$0.225667\pi$$
$$642$$ 6.63102 0.261706
$$643$$ 1.04221 0.0411009 0.0205504 0.999789i $$-0.493458\pi$$
0.0205504 + 0.999789i $$0.493458\pi$$
$$644$$ 0.855167 0.0336983
$$645$$ −4.56166 −0.179615
$$646$$ 48.3913 1.90393
$$647$$ −28.1608 −1.10711 −0.553557 0.832811i $$-0.686730\pi$$
−0.553557 + 0.832811i $$0.686730\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 4.82908 0.189558
$$650$$ 0 0
$$651$$ −0.985999 −0.0386444
$$652$$ −1.72587 −0.0675904
$$653$$ 10.6203 0.415603 0.207802 0.978171i $$-0.433369\pi$$
0.207802 + 0.978171i $$0.433369\pi$$
$$654$$ −12.9879 −0.507868
$$655$$ −1.89307 −0.0739683
$$656$$ 4.89008 0.190926
$$657$$ 3.18598 0.124297
$$658$$ −1.78017 −0.0693982
$$659$$ −40.3629 −1.57231 −0.786157 0.618027i $$-0.787932\pi$$
−0.786157 + 0.618027i $$0.787932\pi$$
$$660$$ −2.03385 −0.0791676
$$661$$ 31.9168 1.24142 0.620709 0.784041i $$-0.286845\pi$$
0.620709 + 0.784041i $$0.286845\pi$$
$$662$$ 3.43834 0.133635
$$663$$ 0 0
$$664$$ 14.8267 0.575387
$$665$$ 1.78017 0.0690319
$$666$$ −10.0978 −0.391283
$$667$$ 18.7595 0.726369
$$668$$ −21.1400 −0.817933
$$669$$ −6.76809 −0.261669
$$670$$ 9.35988 0.361604
$$671$$ −19.0858 −0.736797
$$672$$ −0.356896 −0.0137676
$$673$$ 3.82802 0.147559 0.0737797 0.997275i $$-0.476494\pi$$
0.0737797 + 0.997275i $$0.476494\pi$$
$$674$$ −8.20105 −0.315892
$$675$$ −4.52111 −0.174017
$$676$$ 0 0
$$677$$ 1.78927 0.0687670 0.0343835 0.999409i $$-0.489053\pi$$
0.0343835 + 0.999409i $$0.489053\pi$$
$$678$$ −0.792249 −0.0304261
$$679$$ −0.149145 −0.00572366
$$680$$ 4.64609 0.178169
$$681$$ −23.6799 −0.907417
$$682$$ 8.11960 0.310916
$$683$$ −0.759725 −0.0290701 −0.0145350 0.999894i $$-0.504627\pi$$
−0.0145350 + 0.999894i $$0.504627\pi$$
$$684$$ −7.20775 −0.275595
$$685$$ 5.29218 0.202204
$$686$$ 4.95108 0.189033
$$687$$ 8.29829 0.316600
$$688$$ 6.59179 0.251310
$$689$$ 0 0
$$690$$ 1.65817 0.0631254
$$691$$ −4.65950 −0.177256 −0.0886278 0.996065i $$-0.528248\pi$$
−0.0886278 + 0.996065i $$0.528248\pi$$
$$692$$ 9.35450 0.355605
$$693$$ 1.04892 0.0398451
$$694$$ −7.86294 −0.298473
$$695$$ −2.34183 −0.0888307
$$696$$ −7.82908 −0.296761
$$697$$ 32.8310 1.24356
$$698$$ −18.7245 −0.708734
$$699$$ −23.9651 −0.906443
$$700$$ −1.61356 −0.0609870
$$701$$ −24.3284 −0.918872 −0.459436 0.888211i $$-0.651948\pi$$
−0.459436 + 0.888211i $$0.651948\pi$$
$$702$$ 0 0
$$703$$ −72.7827 −2.74505
$$704$$ 2.93900 0.110768
$$705$$ −3.45175 −0.130000
$$706$$ −31.5448 −1.18720
$$707$$ 3.57434 0.134427
$$708$$ 1.64310 0.0617516
$$709$$ 29.3927 1.10386 0.551932 0.833889i $$-0.313891\pi$$
0.551932 + 0.833889i $$0.313891\pi$$
$$710$$ 4.71379 0.176905
$$711$$ 15.0465 0.564289
$$712$$ −0.396125 −0.0148454
$$713$$ −6.61979 −0.247913
$$714$$ −2.39612 −0.0896727
$$715$$ 0 0
$$716$$ 3.17523 0.118664
$$717$$ −12.6160 −0.471152
$$718$$ 2.39612 0.0894226
$$719$$ −51.4878 −1.92017 −0.960086 0.279704i $$-0.909764\pi$$
−0.960086 + 0.279704i $$0.909764\pi$$
$$720$$ −0.692021 −0.0257901
$$721$$ −3.43535 −0.127939
$$722$$ −32.9517 −1.22633
$$723$$ −26.3937 −0.981593
$$724$$ −19.7995 −0.735844
$$725$$ −35.3961 −1.31458
$$726$$ 2.36227 0.0876722
$$727$$ −3.67324 −0.136233 −0.0681164 0.997677i $$-0.521699\pi$$
−0.0681164 + 0.997677i $$0.521699\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.20477 0.0816021
$$731$$ 44.2559 1.63686
$$732$$ −6.49396 −0.240024
$$733$$ −18.3612 −0.678187 −0.339093 0.940753i $$-0.610120\pi$$
−0.339093 + 0.940753i $$0.610120\pi$$
$$734$$ 0.00431187 0.000159154 0
$$735$$ 4.75600 0.175428
$$736$$ −2.39612 −0.0883223
$$737$$ 39.7512 1.46425
$$738$$ −4.89008 −0.180006
$$739$$ −1.68233 −0.0618856 −0.0309428 0.999521i $$-0.509851\pi$$
−0.0309428 + 0.999521i $$0.509851\pi$$
$$740$$ −6.98792 −0.256881
$$741$$ 0 0
$$742$$ 3.17198 0.116447
$$743$$ −48.2935 −1.77172 −0.885858 0.463956i $$-0.846430\pi$$
−0.885858 + 0.463956i $$0.846430\pi$$
$$744$$ 2.76271 0.101286
$$745$$ −14.4058 −0.527788
$$746$$ 32.3129 1.18306
$$747$$ −14.8267 −0.542480
$$748$$ 19.7318 0.721468
$$749$$ −2.36658 −0.0864731
$$750$$ −6.58881 −0.240589
$$751$$ 11.7011 0.426980 0.213490 0.976945i $$-0.431517\pi$$
0.213490 + 0.976945i $$0.431517\pi$$
$$752$$ 4.98792 0.181891
$$753$$ −30.0344 −1.09452
$$754$$ 0 0
$$755$$ −0.619678 −0.0225524
$$756$$ 0.356896 0.0129802
$$757$$ 25.1836 0.915313 0.457657 0.889129i $$-0.348689\pi$$
0.457657 + 0.889129i $$0.348689\pi$$
$$758$$ 19.7560 0.717570
$$759$$ 7.04221 0.255616
$$760$$ −4.98792 −0.180931
$$761$$ −22.4155 −0.812561 −0.406281 0.913748i $$-0.633174\pi$$
−0.406281 + 0.913748i $$0.633174\pi$$
$$762$$ 18.2174 0.659948
$$763$$ 4.63533 0.167810
$$764$$ 15.2620 0.552161
$$765$$ −4.64609 −0.167980
$$766$$ 28.8116 1.04101
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 0.132751 0.00478714 0.00239357 0.999997i $$-0.499238\pi$$
0.00239357 + 0.999997i $$0.499238\pi$$
$$770$$ 0.725873 0.0261587
$$771$$ 11.2620 0.405593
$$772$$ −4.76809 −0.171607
$$773$$ 48.0694 1.72893 0.864467 0.502689i $$-0.167656\pi$$
0.864467 + 0.502689i $$0.167656\pi$$
$$774$$ −6.59179 −0.236937
$$775$$ 12.4905 0.448672
$$776$$ 0.417895 0.0150015
$$777$$ 3.60388 0.129288
$$778$$ −34.7821 −1.24700
$$779$$ −35.2465 −1.26284
$$780$$ 0 0
$$781$$ 20.0194 0.716350
$$782$$ −16.0871 −0.575273
$$783$$ 7.82908 0.279789
$$784$$ −6.87263 −0.245451
$$785$$ −5.94571 −0.212211
$$786$$ −2.73556 −0.0975743
$$787$$ −6.20908 −0.221330 −0.110665 0.993858i $$-0.535298\pi$$
−0.110665 + 0.993858i $$0.535298\pi$$
$$788$$ −12.2349 −0.435850
$$789$$ −5.54958 −0.197570
$$790$$ 10.4125 0.370461
$$791$$ 0.282750 0.0100534
$$792$$ −2.93900 −0.104433
$$793$$ 0 0
$$794$$ −5.15346 −0.182889
$$795$$ 6.15047 0.218135
$$796$$ −11.8485 −0.419958
$$797$$ 0.327830 0.0116123 0.00580616 0.999983i $$-0.498152\pi$$
0.00580616 + 0.999983i $$0.498152\pi$$
$$798$$ 2.57242 0.0910626
$$799$$ 33.4878 1.18471
$$800$$ 4.52111 0.159845
$$801$$ 0.396125 0.0139964
$$802$$ 13.3250 0.470521
$$803$$ 9.36360 0.330434
$$804$$ 13.5254 0.477005
$$805$$ −0.591794 −0.0208580
$$806$$ 0 0
$$807$$ 16.6872 0.587419
$$808$$ −10.0151 −0.352329
$$809$$ −37.4383 −1.31626 −0.658131 0.752904i $$-0.728653\pi$$
−0.658131 + 0.752904i $$0.728653\pi$$
$$810$$ 0.692021 0.0243152
$$811$$ −17.1448 −0.602037 −0.301018 0.953618i $$-0.597327\pi$$
−0.301018 + 0.953618i $$0.597327\pi$$
$$812$$ 2.79417 0.0980561
$$813$$ 6.61356 0.231948
$$814$$ −29.6775 −1.04020
$$815$$ 1.19434 0.0418360
$$816$$ 6.71379 0.235030
$$817$$ −47.5120 −1.66223
$$818$$ −24.0237 −0.839969
$$819$$ 0 0
$$820$$ −3.38404 −0.118176
$$821$$ 17.7885 0.620824 0.310412 0.950602i $$-0.399533\pi$$
0.310412 + 0.950602i $$0.399533\pi$$
$$822$$ 7.64742 0.266734
$$823$$ 12.2301 0.426315 0.213157 0.977018i $$-0.431625\pi$$
0.213157 + 0.977018i $$0.431625\pi$$
$$824$$ 9.62565 0.335325
$$825$$ −13.2875 −0.462613
$$826$$ −0.586417 −0.0204041
$$827$$ −20.5623 −0.715020 −0.357510 0.933909i $$-0.616374\pi$$
−0.357510 + 0.933909i $$0.616374\pi$$
$$828$$ 2.39612 0.0832711
$$829$$ −25.4470 −0.883809 −0.441905 0.897062i $$-0.645697\pi$$
−0.441905 + 0.897062i $$0.645697\pi$$
$$830$$ −10.2604 −0.356143
$$831$$ −21.7995 −0.756218
$$832$$ 0 0
$$833$$ −46.1414 −1.59870
$$834$$ −3.38404 −0.117180
$$835$$ 14.6294 0.506270
$$836$$ −21.1836 −0.732650
$$837$$ −2.76271 −0.0954932
$$838$$ −13.8049 −0.476883
$$839$$ −5.76676 −0.199091 −0.0995453 0.995033i $$-0.531739\pi$$
−0.0995453 + 0.995033i $$0.531739\pi$$
$$840$$ 0.246980 0.00852161
$$841$$ 32.2946 1.11361
$$842$$ 7.72587 0.266251
$$843$$ −20.5918 −0.709219
$$844$$ −17.2620 −0.594184
$$845$$ 0 0
$$846$$ −4.98792 −0.171488
$$847$$ −0.843085 −0.0289688
$$848$$ −8.88769 −0.305205
$$849$$ −13.0121 −0.446573
$$850$$ 30.3538 1.04113
$$851$$ 24.1957 0.829417
$$852$$ 6.81163 0.233362
$$853$$ 21.1728 0.724944 0.362472 0.931995i $$-0.381933\pi$$
0.362472 + 0.931995i $$0.381933\pi$$
$$854$$ 2.31767 0.0793089
$$855$$ 4.98792 0.170583
$$856$$ 6.63102 0.226644
$$857$$ 12.0086 0.410207 0.205103 0.978740i $$-0.434247\pi$$
0.205103 + 0.978740i $$0.434247\pi$$
$$858$$ 0 0
$$859$$ −1.66296 −0.0567393 −0.0283697 0.999598i $$-0.509032\pi$$
−0.0283697 + 0.999598i $$0.509032\pi$$
$$860$$ −4.56166 −0.155551
$$861$$ 1.74525 0.0594780
$$862$$ 0.640120 0.0218026
$$863$$ 23.8323 0.811262 0.405631 0.914037i $$-0.367052\pi$$
0.405631 + 0.914037i $$0.367052\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −6.47352 −0.220106
$$866$$ −21.2760 −0.722989
$$867$$ 28.0750 0.953477
$$868$$ −0.985999 −0.0334670
$$869$$ 44.2218 1.50012
$$870$$ 5.41789 0.183684
$$871$$ 0 0
$$872$$ −12.9879 −0.439826
$$873$$ −0.417895 −0.0141436
$$874$$ 17.2707 0.584189
$$875$$ 2.35152 0.0794959
$$876$$ 3.18598 0.107644
$$877$$ 42.3177 1.42897 0.714483 0.699653i $$-0.246662\pi$$
0.714483 + 0.699653i $$0.246662\pi$$
$$878$$ −12.7181 −0.429215
$$879$$ 14.9390 0.503880
$$880$$ −2.03385 −0.0685611
$$881$$ 22.2741 0.750434 0.375217 0.926937i $$-0.377568\pi$$
0.375217 + 0.926937i $$0.377568\pi$$
$$882$$ 6.87263 0.231413
$$883$$ −8.54229 −0.287471 −0.143735 0.989616i $$-0.545911\pi$$
−0.143735 + 0.989616i $$0.545911\pi$$
$$884$$ 0 0
$$885$$ −1.13706 −0.0382220
$$886$$ −22.5972 −0.759167
$$887$$ −18.9142 −0.635078 −0.317539 0.948245i $$-0.602856\pi$$
−0.317539 + 0.948245i $$0.602856\pi$$
$$888$$ −10.0978 −0.338861
$$889$$ −6.50173 −0.218061
$$890$$ 0.274127 0.00918875
$$891$$ 2.93900 0.0984602
$$892$$ −6.76809 −0.226612
$$893$$ −35.9517 −1.20308
$$894$$ −20.8170 −0.696225
$$895$$ −2.19733 −0.0734485
$$896$$ −0.356896 −0.0119231
$$897$$ 0 0
$$898$$ 11.6474 0.388679
$$899$$ −21.6295 −0.721384
$$900$$ −4.52111 −0.150704
$$901$$ −59.6701 −1.98790
$$902$$ −14.3720 −0.478534
$$903$$ 2.35258 0.0782891
$$904$$ −0.792249 −0.0263498
$$905$$ 13.7017 0.455460
$$906$$ −0.895461 −0.0297497
$$907$$ 13.9517 0.463258 0.231629 0.972804i $$-0.425595\pi$$
0.231629 + 0.972804i $$0.425595\pi$$
$$908$$ −23.6799 −0.785846
$$909$$ 10.0151 0.332179
$$910$$ 0 0
$$911$$ 45.0422 1.49232 0.746158 0.665769i $$-0.231897\pi$$
0.746158 + 0.665769i $$0.231897\pi$$
$$912$$ −7.20775 −0.238672
$$913$$ −43.5757 −1.44214
$$914$$ −21.1890 −0.700869
$$915$$ 4.49396 0.148566
$$916$$ 8.29829 0.274183
$$917$$ 0.976311 0.0322406
$$918$$ −6.71379 −0.221588
$$919$$ 39.9976 1.31940 0.659700 0.751529i $$-0.270683\pi$$
0.659700 + 0.751529i $$0.270683\pi$$
$$920$$ 1.65817 0.0546682
$$921$$ −26.0301 −0.857722
$$922$$ 24.0694 0.792682
$$923$$ 0 0
$$924$$ 1.04892 0.0345068
$$925$$ −45.6534 −1.50107
$$926$$ −18.1715 −0.597153
$$927$$ −9.62565 −0.316148
$$928$$ −7.82908 −0.257002
$$929$$ −34.6848 −1.13797 −0.568986 0.822347i $$-0.692664\pi$$
−0.568986 + 0.822347i $$0.692664\pi$$
$$930$$ −1.91185 −0.0626922
$$931$$ 49.5362 1.62348
$$932$$ −23.9651 −0.785002
$$933$$ −4.81163 −0.157526
$$934$$ 2.93123 0.0959128
$$935$$ −13.6549 −0.446562
$$936$$ 0 0
$$937$$ −19.1260 −0.624821 −0.312410 0.949947i $$-0.601136\pi$$
−0.312410 + 0.949947i $$0.601136\pi$$
$$938$$ −4.82717 −0.157613
$$939$$ −26.0411 −0.849821
$$940$$ −3.45175 −0.112584
$$941$$ 22.5972 0.736647 0.368323 0.929698i $$-0.379932\pi$$
0.368323 + 0.929698i $$0.379932\pi$$
$$942$$ −8.59179 −0.279936
$$943$$ 11.7172 0.381566
$$944$$ 1.64310 0.0534785
$$945$$ −0.246980 −0.00803425
$$946$$ −19.3733 −0.629880
$$947$$ 27.4359 0.891548 0.445774 0.895145i $$-0.352928\pi$$
0.445774 + 0.895145i $$0.352928\pi$$
$$948$$ 15.0465 0.488688
$$949$$ 0 0
$$950$$ −32.5870 −1.05726
$$951$$ 11.5211 0.373597
$$952$$ −2.39612 −0.0776588
$$953$$ −1.84787 −0.0598584 −0.0299292 0.999552i $$-0.509528\pi$$
−0.0299292 + 0.999552i $$0.509528\pi$$
$$954$$ 8.88769 0.287750
$$955$$ −10.5617 −0.341767
$$956$$ −12.6160 −0.408029
$$957$$ 23.0097 0.743798
$$958$$ −30.7090 −0.992163
$$959$$ −2.72933 −0.0881347
$$960$$ −0.692021 −0.0223349
$$961$$ −23.3674 −0.753788
$$962$$ 0 0
$$963$$ −6.63102 −0.213682
$$964$$ −26.3937 −0.850085
$$965$$ 3.29962 0.106218
$$966$$ −0.855167 −0.0275145
$$967$$ 8.88471 0.285713 0.142856 0.989743i $$-0.454371\pi$$
0.142856 + 0.989743i $$0.454371\pi$$
$$968$$ 2.36227 0.0759263
$$969$$ −48.3913 −1.55455
$$970$$ −0.289192 −0.00928540
$$971$$ 35.0863 1.12597 0.562987 0.826466i $$-0.309652\pi$$
0.562987 + 0.826466i $$0.309652\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 1.20775 0.0387187
$$974$$ 24.1497 0.773807
$$975$$ 0 0
$$976$$ −6.49396 −0.207867
$$977$$ 8.33704 0.266726 0.133363 0.991067i $$-0.457422\pi$$
0.133363 + 0.991067i $$0.457422\pi$$
$$978$$ 1.72587 0.0551873
$$979$$ 1.16421 0.0372083
$$980$$ 4.75600 0.151925
$$981$$ 12.9879 0.414672
$$982$$ −14.5972 −0.465814
$$983$$ 55.6051 1.77353 0.886763 0.462224i $$-0.152949\pi$$
0.886763 + 0.462224i $$0.152949\pi$$
$$984$$ −4.89008 −0.155890
$$985$$ 8.46681 0.269775
$$986$$ −52.5628 −1.67394
$$987$$ 1.78017 0.0566634
$$988$$ 0 0
$$989$$ 15.7948 0.502244
$$990$$ 2.03385 0.0646401
$$991$$ −43.5967 −1.38489 −0.692447 0.721468i $$-0.743468\pi$$
−0.692447 + 0.721468i $$0.743468\pi$$
$$992$$ 2.76271 0.0877161
$$993$$ −3.43834 −0.109112
$$994$$ −2.43104 −0.0771079
$$995$$ 8.19939 0.259938
$$996$$ −14.8267 −0.469802
$$997$$ 22.4590 0.711285 0.355643 0.934622i $$-0.384262\pi$$
0.355643 + 0.934622i $$0.384262\pi$$
$$998$$ 6.85517 0.216997
$$999$$ 10.0978 0.319481
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 1014.2.a.m.1.1 3
3.2 odd 2 3042.2.a.be.1.3 3
4.3 odd 2 8112.2.a.ce.1.1 3
13.2 odd 12 1014.2.i.g.823.3 12
13.3 even 3 1014.2.e.m.529.1 6
13.4 even 6 1014.2.e.k.991.3 6
13.5 odd 4 1014.2.b.g.337.6 6
13.6 odd 12 1014.2.i.g.361.6 12
13.7 odd 12 1014.2.i.g.361.1 12
13.8 odd 4 1014.2.b.g.337.1 6
13.9 even 3 1014.2.e.m.991.1 6
13.10 even 6 1014.2.e.k.529.3 6
13.11 odd 12 1014.2.i.g.823.4 12
13.12 even 2 1014.2.a.o.1.3 yes 3
39.5 even 4 3042.2.b.r.1351.1 6
39.8 even 4 3042.2.b.r.1351.6 6
39.38 odd 2 3042.2.a.bd.1.1 3
52.51 odd 2 8112.2.a.bz.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1014.2.a.m.1.1 3 1.1 even 1 trivial
1014.2.a.o.1.3 yes 3 13.12 even 2
1014.2.b.g.337.1 6 13.8 odd 4
1014.2.b.g.337.6 6 13.5 odd 4
1014.2.e.k.529.3 6 13.10 even 6
1014.2.e.k.991.3 6 13.4 even 6
1014.2.e.m.529.1 6 13.3 even 3
1014.2.e.m.991.1 6 13.9 even 3
1014.2.i.g.361.1 12 13.7 odd 12
1014.2.i.g.361.6 12 13.6 odd 12
1014.2.i.g.823.3 12 13.2 odd 12
1014.2.i.g.823.4 12 13.11 odd 12
3042.2.a.bd.1.1 3 39.38 odd 2
3042.2.a.be.1.3 3 3.2 odd 2
3042.2.b.r.1351.1 6 39.5 even 4
3042.2.b.r.1351.6 6 39.8 even 4
8112.2.a.bz.1.3 3 52.51 odd 2
8112.2.a.ce.1.1 3 4.3 odd 2