# Properties

 Label 722.2.a.a.1.1 Level $722$ Weight $2$ Character 722.1 Self dual yes Analytic conductor $5.765$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.76519902594$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 722.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} +3.00000 q^{13} +3.00000 q^{14} -6.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -6.00000 q^{18} +2.00000 q^{20} +9.00000 q^{21} +2.00000 q^{22} +5.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -3.00000 q^{26} -9.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} +6.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -6.00000 q^{37} -9.00000 q^{39} -2.00000 q^{40} -12.0000 q^{41} -9.00000 q^{42} -10.0000 q^{43} -2.00000 q^{44} +12.0000 q^{45} -5.00000 q^{46} -8.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} +3.00000 q^{52} +3.00000 q^{53} +9.00000 q^{54} -4.00000 q^{55} +3.00000 q^{56} -3.00000 q^{58} -3.00000 q^{59} -6.00000 q^{60} -6.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} -15.0000 q^{67} -1.00000 q^{68} -15.0000 q^{69} +6.00000 q^{70} -6.00000 q^{72} -11.0000 q^{73} +6.00000 q^{74} +3.00000 q^{75} +6.00000 q^{77} +9.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +12.0000 q^{82} +2.00000 q^{83} +9.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} -9.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -12.0000 q^{90} -9.00000 q^{91} +5.00000 q^{92} -18.0000 q^{93} +8.00000 q^{94} +3.00000 q^{96} -12.0000 q^{97} -2.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 3.00000 1.22474
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.00000 2.00000
$$10$$ −2.00000 −0.632456
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −3.00000 −0.866025
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 3.00000 0.801784
$$15$$ −6.00000 −1.54919
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ −6.00000 −1.41421
$$19$$ 0 0
$$20$$ 2.00000 0.447214
$$21$$ 9.00000 1.96396
$$22$$ 2.00000 0.426401
$$23$$ 5.00000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −1.00000 −0.200000
$$26$$ −3.00000 −0.588348
$$27$$ −9.00000 −1.73205
$$28$$ −3.00000 −0.566947
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 6.00000 1.09545
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.00000 1.04447
$$34$$ 1.00000 0.171499
$$35$$ −6.00000 −1.01419
$$36$$ 6.00000 1.00000
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −9.00000 −1.44115
$$40$$ −2.00000 −0.316228
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ −9.00000 −1.38873
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 12.0000 1.78885
$$46$$ −5.00000 −0.737210
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ 2.00000 0.285714
$$50$$ 1.00000 0.141421
$$51$$ 3.00000 0.420084
$$52$$ 3.00000 0.416025
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 9.00000 1.22474
$$55$$ −4.00000 −0.539360
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ −6.00000 −0.774597
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ −18.0000 −2.26779
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ −6.00000 −0.738549
$$67$$ −15.0000 −1.83254 −0.916271 0.400559i $$-0.868816\pi$$
−0.916271 + 0.400559i $$0.868816\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −15.0000 −1.80579
$$70$$ 6.00000 0.717137
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −6.00000 −0.707107
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 6.00000 0.683763
$$78$$ 9.00000 1.01905
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 9.00000 1.00000
$$82$$ 12.0000 1.32518
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 9.00000 0.981981
$$85$$ −2.00000 −0.216930
$$86$$ 10.0000 1.07833
$$87$$ −9.00000 −0.964901
$$88$$ 2.00000 0.213201
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −12.0000 −1.26491
$$91$$ −9.00000 −0.943456
$$92$$ 5.00000 0.521286
$$93$$ −18.0000 −1.86651
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 3.00000 0.306186
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −12.0000 −1.20605
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 18.0000 1.75662
$$106$$ −3.00000 −0.291386
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ −9.00000 −0.866025
$$109$$ −3.00000 −0.287348 −0.143674 0.989625i $$-0.545892\pi$$
−0.143674 + 0.989625i $$0.545892\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 18.0000 1.70848
$$112$$ −3.00000 −0.283473
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ 10.0000 0.932505
$$116$$ 3.00000 0.278543
$$117$$ 18.0000 1.66410
$$118$$ 3.00000 0.276172
$$119$$ 3.00000 0.275010
$$120$$ 6.00000 0.547723
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 36.0000 3.24601
$$124$$ 6.00000 0.538816
$$125$$ −12.0000 −1.07331
$$126$$ 18.0000 1.60357
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 30.0000 2.64135
$$130$$ −6.00000 −0.526235
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 6.00000 0.522233
$$133$$ 0 0
$$134$$ 15.0000 1.29580
$$135$$ −18.0000 −1.54919
$$136$$ 1.00000 0.0857493
$$137$$ 19.0000 1.62328 0.811640 0.584158i $$-0.198575\pi$$
0.811640 + 0.584158i $$0.198575\pi$$
$$138$$ 15.0000 1.27688
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ 6.00000 0.500000
$$145$$ 6.00000 0.498273
$$146$$ 11.0000 0.910366
$$147$$ −6.00000 −0.494872
$$148$$ −6.00000 −0.493197
$$149$$ −8.00000 −0.655386 −0.327693 0.944784i $$-0.606271\pi$$
−0.327693 + 0.944784i $$0.606271\pi$$
$$150$$ −3.00000 −0.244949
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ −6.00000 −0.483494
$$155$$ 12.0000 0.963863
$$156$$ −9.00000 −0.720577
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ −9.00000 −0.713746
$$160$$ −2.00000 −0.158114
$$161$$ −15.0000 −1.18217
$$162$$ −9.00000 −0.707107
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 12.0000 0.934199
$$166$$ −2.00000 −0.155230
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −9.00000 −0.694365
$$169$$ −4.00000 −0.307692
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 9.00000 0.682288
$$175$$ 3.00000 0.226779
$$176$$ −2.00000 −0.150756
$$177$$ 9.00000 0.676481
$$178$$ 6.00000 0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 12.0000 0.894427
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 9.00000 0.667124
$$183$$ 0 0
$$184$$ −5.00000 −0.368605
$$185$$ −12.0000 −0.882258
$$186$$ 18.0000 1.31982
$$187$$ 2.00000 0.146254
$$188$$ −8.00000 −0.583460
$$189$$ 27.0000 1.96396
$$190$$ 0 0
$$191$$ −11.0000 −0.795932 −0.397966 0.917400i $$-0.630284\pi$$
−0.397966 + 0.917400i $$0.630284\pi$$
$$192$$ −3.00000 −0.216506
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 12.0000 0.861550
$$195$$ −18.0000 −1.28901
$$196$$ 2.00000 0.142857
$$197$$ 4.00000 0.284988 0.142494 0.989796i $$-0.454488\pi$$
0.142494 + 0.989796i $$0.454488\pi$$
$$198$$ 12.0000 0.852803
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 45.0000 3.17406
$$202$$ −10.0000 −0.703598
$$203$$ −9.00000 −0.631676
$$204$$ 3.00000 0.210042
$$205$$ −24.0000 −1.67623
$$206$$ 6.00000 0.418040
$$207$$ 30.0000 2.08514
$$208$$ 3.00000 0.208013
$$209$$ 0 0
$$210$$ −18.0000 −1.24212
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 3.00000 0.206041
$$213$$ 0 0
$$214$$ −3.00000 −0.205076
$$215$$ −20.0000 −1.36399
$$216$$ 9.00000 0.612372
$$217$$ −18.0000 −1.22192
$$218$$ 3.00000 0.203186
$$219$$ 33.0000 2.22993
$$220$$ −4.00000 −0.269680
$$221$$ −3.00000 −0.201802
$$222$$ −18.0000 −1.20808
$$223$$ −18.0000 −1.20537 −0.602685 0.797980i $$-0.705902\pi$$
−0.602685 + 0.797980i $$0.705902\pi$$
$$224$$ 3.00000 0.200446
$$225$$ −6.00000 −0.400000
$$226$$ 12.0000 0.798228
$$227$$ −3.00000 −0.199117 −0.0995585 0.995032i $$-0.531743\pi$$
−0.0995585 + 0.995032i $$0.531743\pi$$
$$228$$ 0 0
$$229$$ −12.0000 −0.792982 −0.396491 0.918039i $$-0.629772\pi$$
−0.396491 + 0.918039i $$0.629772\pi$$
$$230$$ −10.0000 −0.659380
$$231$$ −18.0000 −1.18431
$$232$$ −3.00000 −0.196960
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ −18.0000 −1.17670
$$235$$ −16.0000 −1.04372
$$236$$ −3.00000 −0.195283
$$237$$ −36.0000 −2.33845
$$238$$ −3.00000 −0.194461
$$239$$ 1.00000 0.0646846 0.0323423 0.999477i $$-0.489703\pi$$
0.0323423 + 0.999477i $$0.489703\pi$$
$$240$$ −6.00000 −0.387298
$$241$$ −24.0000 −1.54598 −0.772988 0.634421i $$-0.781239\pi$$
−0.772988 + 0.634421i $$0.781239\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.00000 0.255551
$$246$$ −36.0000 −2.29528
$$247$$ 0 0
$$248$$ −6.00000 −0.381000
$$249$$ −6.00000 −0.380235
$$250$$ 12.0000 0.758947
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ −18.0000 −1.13389
$$253$$ −10.0000 −0.628695
$$254$$ 12.0000 0.752947
$$255$$ 6.00000 0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ −30.0000 −1.86772
$$259$$ 18.0000 1.11847
$$260$$ 6.00000 0.372104
$$261$$ 18.0000 1.11417
$$262$$ −14.0000 −0.864923
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 18.0000 1.10158
$$268$$ −15.0000 −0.916271
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 18.0000 1.09545
$$271$$ −11.0000 −0.668202 −0.334101 0.942537i $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 27.0000 1.63411
$$274$$ −19.0000 −1.14783
$$275$$ 2.00000 0.120605
$$276$$ −15.0000 −0.902894
$$277$$ 30.0000 1.80253 0.901263 0.433273i $$-0.142641\pi$$
0.901263 + 0.433273i $$0.142641\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ 36.0000 2.15526
$$280$$ 6.00000 0.358569
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ −24.0000 −1.42918
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 36.0000 2.12501
$$288$$ −6.00000 −0.353553
$$289$$ −16.0000 −0.941176
$$290$$ −6.00000 −0.352332
$$291$$ 36.0000 2.11036
$$292$$ −11.0000 −0.643726
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 6.00000 0.349927
$$295$$ −6.00000 −0.349334
$$296$$ 6.00000 0.348743
$$297$$ 18.0000 1.04447
$$298$$ 8.00000 0.463428
$$299$$ 15.0000 0.867472
$$300$$ 3.00000 0.173205
$$301$$ 30.0000 1.72917
$$302$$ −18.0000 −1.03578
$$303$$ −30.0000 −1.72345
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 18.0000 1.02398
$$310$$ −12.0000 −0.681554
$$311$$ −11.0000 −0.623753 −0.311876 0.950123i $$-0.600957\pi$$
−0.311876 + 0.950123i $$0.600957\pi$$
$$312$$ 9.00000 0.509525
$$313$$ 21.0000 1.18699 0.593495 0.804838i $$-0.297748\pi$$
0.593495 + 0.804838i $$0.297748\pi$$
$$314$$ 0 0
$$315$$ −36.0000 −2.02837
$$316$$ 12.0000 0.675053
$$317$$ 33.0000 1.85346 0.926732 0.375722i $$-0.122605\pi$$
0.926732 + 0.375722i $$0.122605\pi$$
$$318$$ 9.00000 0.504695
$$319$$ −6.00000 −0.335936
$$320$$ 2.00000 0.111803
$$321$$ −9.00000 −0.502331
$$322$$ 15.0000 0.835917
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ −3.00000 −0.166410
$$326$$ 6.00000 0.332309
$$327$$ 9.00000 0.497701
$$328$$ 12.0000 0.662589
$$329$$ 24.0000 1.32316
$$330$$ −12.0000 −0.660578
$$331$$ −9.00000 −0.494685 −0.247342 0.968928i $$-0.579557\pi$$
−0.247342 + 0.968928i $$0.579557\pi$$
$$332$$ 2.00000 0.109764
$$333$$ −36.0000 −1.97279
$$334$$ −12.0000 −0.656611
$$335$$ −30.0000 −1.63908
$$336$$ 9.00000 0.490990
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 4.00000 0.217571
$$339$$ 36.0000 1.95525
$$340$$ −2.00000 −0.108465
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 10.0000 0.539164
$$345$$ −30.0000 −1.61515
$$346$$ −18.0000 −0.967686
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ −9.00000 −0.482451
$$349$$ 28.0000 1.49881 0.749403 0.662114i $$-0.230341\pi$$
0.749403 + 0.662114i $$0.230341\pi$$
$$350$$ −3.00000 −0.160357
$$351$$ −27.0000 −1.44115
$$352$$ 2.00000 0.106600
$$353$$ −31.0000 −1.64996 −0.824982 0.565159i $$-0.808815\pi$$
−0.824982 + 0.565159i $$0.808815\pi$$
$$354$$ −9.00000 −0.478345
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −9.00000 −0.476331
$$358$$ −12.0000 −0.634220
$$359$$ 19.0000 1.00278 0.501391 0.865221i $$-0.332822\pi$$
0.501391 + 0.865221i $$0.332822\pi$$
$$360$$ −12.0000 −0.632456
$$361$$ 0 0
$$362$$ 18.0000 0.946059
$$363$$ 21.0000 1.10221
$$364$$ −9.00000 −0.471728
$$365$$ −22.0000 −1.15153
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 5.00000 0.260643
$$369$$ −72.0000 −3.74817
$$370$$ 12.0000 0.623850
$$371$$ −9.00000 −0.467257
$$372$$ −18.0000 −0.933257
$$373$$ 21.0000 1.08734 0.543669 0.839299i $$-0.317035\pi$$
0.543669 + 0.839299i $$0.317035\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 36.0000 1.85903
$$376$$ 8.00000 0.412568
$$377$$ 9.00000 0.463524
$$378$$ −27.0000 −1.38873
$$379$$ −3.00000 −0.154100 −0.0770498 0.997027i $$-0.524550\pi$$
−0.0770498 + 0.997027i $$0.524550\pi$$
$$380$$ 0 0
$$381$$ 36.0000 1.84434
$$382$$ 11.0000 0.562809
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 12.0000 0.611577
$$386$$ 6.00000 0.305392
$$387$$ −60.0000 −3.04997
$$388$$ −12.0000 −0.609208
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 18.0000 0.911465
$$391$$ −5.00000 −0.252861
$$392$$ −2.00000 −0.101015
$$393$$ −42.0000 −2.11862
$$394$$ −4.00000 −0.201517
$$395$$ 24.0000 1.20757
$$396$$ −12.0000 −0.603023
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 7.00000 0.350878
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ −45.0000 −2.24440
$$403$$ 18.0000 0.896644
$$404$$ 10.0000 0.497519
$$405$$ 18.0000 0.894427
$$406$$ 9.00000 0.446663
$$407$$ 12.0000 0.594818
$$408$$ −3.00000 −0.148522
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 24.0000 1.18528
$$411$$ −57.0000 −2.81160
$$412$$ −6.00000 −0.295599
$$413$$ 9.00000 0.442861
$$414$$ −30.0000 −1.47442
$$415$$ 4.00000 0.196352
$$416$$ −3.00000 −0.147087
$$417$$ −18.0000 −0.881464
$$418$$ 0 0
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 18.0000 0.878310
$$421$$ 27.0000 1.31590 0.657950 0.753062i $$-0.271424\pi$$
0.657950 + 0.753062i $$0.271424\pi$$
$$422$$ 3.00000 0.146038
$$423$$ −48.0000 −2.33384
$$424$$ −3.00000 −0.145693
$$425$$ 1.00000 0.0485071
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 3.00000 0.145010
$$429$$ 18.0000 0.869048
$$430$$ 20.0000 0.964486
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −9.00000 −0.433013
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 18.0000 0.864028
$$435$$ −18.0000 −0.863034
$$436$$ −3.00000 −0.143674
$$437$$ 0 0
$$438$$ −33.0000 −1.57680
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 12.0000 0.571429
$$442$$ 3.00000 0.142695
$$443$$ −22.0000 −1.04525 −0.522626 0.852562i $$-0.675047\pi$$
−0.522626 + 0.852562i $$0.675047\pi$$
$$444$$ 18.0000 0.854242
$$445$$ −12.0000 −0.568855
$$446$$ 18.0000 0.852325
$$447$$ 24.0000 1.13516
$$448$$ −3.00000 −0.141737
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 6.00000 0.282843
$$451$$ 24.0000 1.13012
$$452$$ −12.0000 −0.564433
$$453$$ −54.0000 −2.53714
$$454$$ 3.00000 0.140797
$$455$$ −18.0000 −0.843853
$$456$$ 0 0
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ 12.0000 0.560723
$$459$$ 9.00000 0.420084
$$460$$ 10.0000 0.466252
$$461$$ 4.00000 0.186299 0.0931493 0.995652i $$-0.470307\pi$$
0.0931493 + 0.995652i $$0.470307\pi$$
$$462$$ 18.0000 0.837436
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 3.00000 0.139272
$$465$$ −36.0000 −1.66946
$$466$$ −14.0000 −0.648537
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 18.0000 0.832050
$$469$$ 45.0000 2.07791
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ 3.00000 0.138086
$$473$$ 20.0000 0.919601
$$474$$ 36.0000 1.65353
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 18.0000 0.824163
$$478$$ −1.00000 −0.0457389
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 6.00000 0.273861
$$481$$ −18.0000 −0.820729
$$482$$ 24.0000 1.09317
$$483$$ 45.0000 2.04757
$$484$$ −7.00000 −0.318182
$$485$$ −24.0000 −1.08978
$$486$$ 0 0
$$487$$ 18.0000 0.815658 0.407829 0.913058i $$-0.366286\pi$$
0.407829 + 0.913058i $$0.366286\pi$$
$$488$$ 0 0
$$489$$ 18.0000 0.813988
$$490$$ −4.00000 −0.180702
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 36.0000 1.62301
$$493$$ −3.00000 −0.135113
$$494$$ 0 0
$$495$$ −24.0000 −1.07872
$$496$$ 6.00000 0.269408
$$497$$ 0 0
$$498$$ 6.00000 0.268866
$$499$$ 18.0000 0.805791 0.402895 0.915246i $$-0.368004\pi$$
0.402895 + 0.915246i $$0.368004\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −36.0000 −1.60836
$$502$$ 20.0000 0.892644
$$503$$ 1.00000 0.0445878 0.0222939 0.999751i $$-0.492903\pi$$
0.0222939 + 0.999751i $$0.492903\pi$$
$$504$$ 18.0000 0.801784
$$505$$ 20.0000 0.889988
$$506$$ 10.0000 0.444554
$$507$$ 12.0000 0.532939
$$508$$ −12.0000 −0.532414
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 33.0000 1.45983
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ −12.0000 −0.528783
$$516$$ 30.0000 1.32068
$$517$$ 16.0000 0.703679
$$518$$ −18.0000 −0.790875
$$519$$ −54.0000 −2.37034
$$520$$ −6.00000 −0.263117
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ −9.00000 −0.393543 −0.196771 0.980449i $$-0.563046\pi$$
−0.196771 + 0.980449i $$0.563046\pi$$
$$524$$ 14.0000 0.611593
$$525$$ −9.00000 −0.392792
$$526$$ 8.00000 0.348817
$$527$$ −6.00000 −0.261364
$$528$$ 6.00000 0.261116
$$529$$ 2.00000 0.0869565
$$530$$ −6.00000 −0.260623
$$531$$ −18.0000 −0.781133
$$532$$ 0 0
$$533$$ −36.0000 −1.55933
$$534$$ −18.0000 −0.778936
$$535$$ 6.00000 0.259403
$$536$$ 15.0000 0.647901
$$537$$ −36.0000 −1.55351
$$538$$ −6.00000 −0.258678
$$539$$ −4.00000 −0.172292
$$540$$ −18.0000 −0.774597
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 11.0000 0.472490
$$543$$ 54.0000 2.31736
$$544$$ 1.00000 0.0428746
$$545$$ −6.00000 −0.257012
$$546$$ −27.0000 −1.15549
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ 19.0000 0.811640
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ 0 0
$$552$$ 15.0000 0.638442
$$553$$ −36.0000 −1.53088
$$554$$ −30.0000 −1.27458
$$555$$ 36.0000 1.52811
$$556$$ 6.00000 0.254457
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ −36.0000 −1.52400
$$559$$ −30.0000 −1.26886
$$560$$ −6.00000 −0.253546
$$561$$ −6.00000 −0.253320
$$562$$ 12.0000 0.506189
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 24.0000 1.01058
$$565$$ −24.0000 −1.00969
$$566$$ 14.0000 0.588464
$$567$$ −27.0000 −1.13389
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ 33.0000 1.37859
$$574$$ −36.0000 −1.50261
$$575$$ −5.00000 −0.208514
$$576$$ 6.00000 0.250000
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 18.0000 0.748054
$$580$$ 6.00000 0.249136
$$581$$ −6.00000 −0.248922
$$582$$ −36.0000 −1.49225
$$583$$ −6.00000 −0.248495
$$584$$ 11.0000 0.455183
$$585$$ 36.0000 1.48842
$$586$$ −9.00000 −0.371787
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ 0 0
$$590$$ 6.00000 0.247016
$$591$$ −12.0000 −0.493614
$$592$$ −6.00000 −0.246598
$$593$$ −2.00000 −0.0821302 −0.0410651 0.999156i $$-0.513075\pi$$
−0.0410651 + 0.999156i $$0.513075\pi$$
$$594$$ −18.0000 −0.738549
$$595$$ 6.00000 0.245976
$$596$$ −8.00000 −0.327693
$$597$$ 21.0000 0.859473
$$598$$ −15.0000 −0.613396
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 6.00000 0.244745 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$602$$ −30.0000 −1.22271
$$603$$ −90.0000 −3.66508
$$604$$ 18.0000 0.732410
$$605$$ −14.0000 −0.569181
$$606$$ 30.0000 1.21867
$$607$$ 12.0000 0.487065 0.243532 0.969893i $$-0.421694\pi$$
0.243532 + 0.969893i $$0.421694\pi$$
$$608$$ 0 0
$$609$$ 27.0000 1.09410
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ −6.00000 −0.242536
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 72.0000 2.90332
$$616$$ −6.00000 −0.241747
$$617$$ 10.0000 0.402585 0.201292 0.979531i $$-0.435486\pi$$
0.201292 + 0.979531i $$0.435486\pi$$
$$618$$ −18.0000 −0.724066
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 12.0000 0.481932
$$621$$ −45.0000 −1.80579
$$622$$ 11.0000 0.441060
$$623$$ 18.0000 0.721155
$$624$$ −9.00000 −0.360288
$$625$$ −19.0000 −0.760000
$$626$$ −21.0000 −0.839329
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 36.0000 1.43427
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 9.00000 0.357718
$$634$$ −33.0000 −1.31060
$$635$$ −24.0000 −0.952411
$$636$$ −9.00000 −0.356873
$$637$$ 6.00000 0.237729
$$638$$ 6.00000 0.237542
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 9.00000 0.355202
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ −15.0000 −0.591083
$$645$$ 60.0000 2.36250
$$646$$ 0 0
$$647$$ −23.0000 −0.904223 −0.452112 0.891961i $$-0.649329\pi$$
−0.452112 + 0.891961i $$0.649329\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ 6.00000 0.235521
$$650$$ 3.00000 0.117670
$$651$$ 54.0000 2.11643
$$652$$ −6.00000 −0.234978
$$653$$ −10.0000 −0.391330 −0.195665 0.980671i $$-0.562687\pi$$
−0.195665 + 0.980671i $$0.562687\pi$$
$$654$$ −9.00000 −0.351928
$$655$$ 28.0000 1.09405
$$656$$ −12.0000 −0.468521
$$657$$ −66.0000 −2.57491
$$658$$ −24.0000 −0.935617
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 12.0000 0.467099
$$661$$ −15.0000 −0.583432 −0.291716 0.956505i $$-0.594226\pi$$
−0.291716 + 0.956505i $$0.594226\pi$$
$$662$$ 9.00000 0.349795
$$663$$ 9.00000 0.349531
$$664$$ −2.00000 −0.0776151
$$665$$ 0 0
$$666$$ 36.0000 1.39497
$$667$$ 15.0000 0.580802
$$668$$ 12.0000 0.464294
$$669$$ 54.0000 2.08776
$$670$$ 30.0000 1.15900
$$671$$ 0 0
$$672$$ −9.00000 −0.347183
$$673$$ 48.0000 1.85026 0.925132 0.379646i $$-0.123954\pi$$
0.925132 + 0.379646i $$0.123954\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 9.00000 0.346410
$$676$$ −4.00000 −0.153846
$$677$$ −3.00000 −0.115299 −0.0576497 0.998337i $$-0.518361\pi$$
−0.0576497 + 0.998337i $$0.518361\pi$$
$$678$$ −36.0000 −1.38257
$$679$$ 36.0000 1.38155
$$680$$ 2.00000 0.0766965
$$681$$ 9.00000 0.344881
$$682$$ 12.0000 0.459504
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 38.0000 1.45191
$$686$$ −15.0000 −0.572703
$$687$$ 36.0000 1.37349
$$688$$ −10.0000 −0.381246
$$689$$ 9.00000 0.342873
$$690$$ 30.0000 1.14208
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 36.0000 1.36753
$$694$$ −16.0000 −0.607352
$$695$$ 12.0000 0.455186
$$696$$ 9.00000 0.341144
$$697$$ 12.0000 0.454532
$$698$$ −28.0000 −1.05982
$$699$$ −42.0000 −1.58859
$$700$$ 3.00000 0.113389
$$701$$ −40.0000 −1.51078 −0.755390 0.655276i $$-0.772552\pi$$
−0.755390 + 0.655276i $$0.772552\pi$$
$$702$$ 27.0000 1.01905
$$703$$ 0 0
$$704$$ −2.00000 −0.0753778
$$705$$ 48.0000 1.80778
$$706$$ 31.0000 1.16670
$$707$$ −30.0000 −1.12827
$$708$$ 9.00000 0.338241
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ 72.0000 2.70021
$$712$$ 6.00000 0.224860
$$713$$ 30.0000 1.12351
$$714$$ 9.00000 0.336817
$$715$$ −12.0000 −0.448775
$$716$$ 12.0000 0.448461
$$717$$ −3.00000 −0.112037
$$718$$ −19.0000 −0.709074
$$719$$ −43.0000 −1.60363 −0.801815 0.597573i $$-0.796132\pi$$
−0.801815 + 0.597573i $$0.796132\pi$$
$$720$$ 12.0000 0.447214
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 72.0000 2.67771
$$724$$ −18.0000 −0.668965
$$725$$ −3.00000 −0.111417
$$726$$ −21.0000 −0.779383
$$727$$ −35.0000 −1.29808 −0.649039 0.760755i $$-0.724829\pi$$
−0.649039 + 0.760755i $$0.724829\pi$$
$$728$$ 9.00000 0.333562
$$729$$ −27.0000 −1.00000
$$730$$ 22.0000 0.814257
$$731$$ 10.0000 0.369863
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ −12.0000 −0.442627
$$736$$ −5.00000 −0.184302
$$737$$ 30.0000 1.10506
$$738$$ 72.0000 2.65036
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 9.00000 0.330400
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 18.0000 0.659912
$$745$$ −16.0000 −0.586195
$$746$$ −21.0000 −0.768865
$$747$$ 12.0000 0.439057
$$748$$ 2.00000 0.0731272
$$749$$ −9.00000 −0.328853
$$750$$ −36.0000 −1.31453
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 60.0000 2.18652
$$754$$ −9.00000 −0.327761
$$755$$ 36.0000 1.31017
$$756$$ 27.0000 0.981981
$$757$$ 12.0000 0.436147 0.218074 0.975932i $$-0.430023\pi$$
0.218074 + 0.975932i $$0.430023\pi$$
$$758$$ 3.00000 0.108965
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ −36.0000 −1.30414
$$763$$ 9.00000 0.325822
$$764$$ −11.0000 −0.397966
$$765$$ −12.0000 −0.433861
$$766$$ 18.0000 0.650366
$$767$$ −9.00000 −0.324971
$$768$$ −3.00000 −0.108253
$$769$$ −15.0000 −0.540914 −0.270457 0.962732i $$-0.587175\pi$$
−0.270457 + 0.962732i $$0.587175\pi$$
$$770$$ −12.0000 −0.432450
$$771$$ −54.0000 −1.94476
$$772$$ −6.00000 −0.215945
$$773$$ −15.0000 −0.539513 −0.269756 0.962929i $$-0.586943\pi$$
−0.269756 + 0.962929i $$0.586943\pi$$
$$774$$ 60.0000 2.15666
$$775$$ −6.00000 −0.215526
$$776$$ 12.0000 0.430775
$$777$$ −54.0000 −1.93724
$$778$$ −26.0000 −0.932145
$$779$$ 0 0
$$780$$ −18.0000 −0.644503
$$781$$ 0 0
$$782$$ 5.00000 0.178800
$$783$$ −27.0000 −0.964901
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 42.0000 1.49809
$$787$$ 9.00000 0.320815 0.160408 0.987051i $$-0.448719\pi$$
0.160408 + 0.987051i $$0.448719\pi$$
$$788$$ 4.00000 0.142494
$$789$$ 24.0000 0.854423
$$790$$ −24.0000 −0.853882
$$791$$ 36.0000 1.28001
$$792$$ 12.0000 0.426401
$$793$$ 0 0
$$794$$ −2.00000 −0.0709773
$$795$$ −18.0000 −0.638394
$$796$$ −7.00000 −0.248108
$$797$$ −33.0000 −1.16892 −0.584460 0.811423i $$-0.698694\pi$$
−0.584460 + 0.811423i $$0.698694\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 1.00000 0.0353553
$$801$$ −36.0000 −1.27200
$$802$$ 36.0000 1.27120
$$803$$ 22.0000 0.776363
$$804$$ 45.0000 1.58703
$$805$$ −30.0000 −1.05736
$$806$$ −18.0000 −0.634023
$$807$$ −18.0000 −0.633630
$$808$$ −10.0000 −0.351799
$$809$$ −11.0000 −0.386739 −0.193370 0.981126i $$-0.561942\pi$$
−0.193370 + 0.981126i $$0.561942\pi$$
$$810$$ −18.0000 −0.632456
$$811$$ −33.0000 −1.15879 −0.579393 0.815048i $$-0.696710\pi$$
−0.579393 + 0.815048i $$0.696710\pi$$
$$812$$ −9.00000 −0.315838
$$813$$ 33.0000 1.15736
$$814$$ −12.0000 −0.420600
$$815$$ −12.0000 −0.420342
$$816$$ 3.00000 0.105021
$$817$$ 0 0
$$818$$ 6.00000 0.209785
$$819$$ −54.0000 −1.88691
$$820$$ −24.0000 −0.838116
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ 57.0000 1.98810
$$823$$ 3.00000 0.104573 0.0522867 0.998632i $$-0.483349\pi$$
0.0522867 + 0.998632i $$0.483349\pi$$
$$824$$ 6.00000 0.209020
$$825$$ −6.00000 −0.208893
$$826$$ −9.00000 −0.313150
$$827$$ −15.0000 −0.521601 −0.260801 0.965393i $$-0.583986\pi$$
−0.260801 + 0.965393i $$0.583986\pi$$
$$828$$ 30.0000 1.04257
$$829$$ −51.0000 −1.77130 −0.885652 0.464350i $$-0.846288\pi$$
−0.885652 + 0.464350i $$0.846288\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ −90.0000 −3.12207
$$832$$ 3.00000 0.104006
$$833$$ −2.00000 −0.0692959
$$834$$ 18.0000 0.623289
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ −54.0000 −1.86651
$$838$$ 14.0000 0.483622
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ −18.0000 −0.621059
$$841$$ −20.0000 −0.689655
$$842$$ −27.0000 −0.930481
$$843$$ 36.0000 1.23991
$$844$$ −3.00000 −0.103264
$$845$$ −8.00000 −0.275208
$$846$$ 48.0000 1.65027
$$847$$ 21.0000 0.721569
$$848$$ 3.00000 0.103020
$$849$$ 42.0000 1.44144
$$850$$ −1.00000 −0.0342997
$$851$$ −30.0000 −1.02839
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −3.00000 −0.102538
$$857$$ −24.0000 −0.819824 −0.409912 0.912125i $$-0.634441\pi$$
−0.409912 + 0.912125i $$0.634441\pi$$
$$858$$ −18.0000 −0.614510
$$859$$ 54.0000 1.84246 0.921228 0.389023i $$-0.127187\pi$$
0.921228 + 0.389023i $$0.127187\pi$$
$$860$$ −20.0000 −0.681994
$$861$$ −108.000 −3.68063
$$862$$ 24.0000 0.817443
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 9.00000 0.306186
$$865$$ 36.0000 1.22404
$$866$$ 30.0000 1.01944
$$867$$ 48.0000 1.63017
$$868$$ −18.0000 −0.610960
$$869$$ −24.0000 −0.814144
$$870$$ 18.0000 0.610257
$$871$$ −45.0000 −1.52477
$$872$$ 3.00000 0.101593
$$873$$ −72.0000 −2.43683
$$874$$ 0 0
$$875$$ 36.0000 1.21702
$$876$$ 33.0000 1.11497
$$877$$ −27.0000 −0.911725 −0.455863 0.890050i $$-0.650669\pi$$
−0.455863 + 0.890050i $$0.650669\pi$$
$$878$$ 0 0
$$879$$ −27.0000 −0.910687
$$880$$ −4.00000 −0.134840
$$881$$ 34.0000 1.14549 0.572745 0.819734i $$-0.305879\pi$$
0.572745 + 0.819734i $$0.305879\pi$$
$$882$$ −12.0000 −0.404061
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 18.0000 0.605063
$$886$$ 22.0000 0.739104
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −18.0000 −0.604040
$$889$$ 36.0000 1.20740
$$890$$ 12.0000 0.402241
$$891$$ −18.0000 −0.603023
$$892$$ −18.0000 −0.602685
$$893$$ 0 0
$$894$$ −24.0000 −0.802680
$$895$$ 24.0000 0.802232
$$896$$ 3.00000 0.100223
$$897$$ −45.0000 −1.50251
$$898$$ 6.00000 0.200223
$$899$$ 18.0000 0.600334
$$900$$ −6.00000 −0.200000
$$901$$ −3.00000 −0.0999445
$$902$$ −24.0000 −0.799113
$$903$$ −90.0000 −2.99501
$$904$$ 12.0000 0.399114
$$905$$ −36.0000 −1.19668
$$906$$ 54.0000 1.79403
$$907$$ 15.0000 0.498067 0.249033 0.968495i $$-0.419887\pi$$
0.249033 + 0.968495i $$0.419887\pi$$
$$908$$ −3.00000 −0.0995585
$$909$$ 60.0000 1.99007
$$910$$ 18.0000 0.596694
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ −4.00000 −0.132381
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ −42.0000 −1.38696
$$918$$ −9.00000 −0.297044
$$919$$ 15.0000 0.494804 0.247402 0.968913i $$-0.420423\pi$$
0.247402 + 0.968913i $$0.420423\pi$$
$$920$$ −10.0000 −0.329690
$$921$$ 36.0000 1.18624
$$922$$ −4.00000 −0.131733
$$923$$ 0 0
$$924$$ −18.0000 −0.592157
$$925$$ 6.00000 0.197279
$$926$$ −32.0000 −1.05159
$$927$$ −36.0000 −1.18240
$$928$$ −3.00000 −0.0984798
$$929$$ 41.0000 1.34517 0.672583 0.740022i $$-0.265185\pi$$
0.672583 + 0.740022i $$0.265185\pi$$
$$930$$ 36.0000 1.18049
$$931$$ 0 0
$$932$$ 14.0000 0.458585
$$933$$ 33.0000 1.08037
$$934$$ −8.00000 −0.261768
$$935$$ 4.00000 0.130814
$$936$$ −18.0000 −0.588348
$$937$$ −47.0000 −1.53542 −0.767712 0.640796i $$-0.778605\pi$$
−0.767712 + 0.640796i $$0.778605\pi$$
$$938$$ −45.0000 −1.46930
$$939$$ −63.0000 −2.05593
$$940$$ −16.0000 −0.521862
$$941$$ 27.0000 0.880175 0.440087 0.897955i $$-0.354947\pi$$
0.440087 + 0.897955i $$0.354947\pi$$
$$942$$ 0 0
$$943$$ −60.0000 −1.95387
$$944$$ −3.00000 −0.0976417
$$945$$ 54.0000 1.75662
$$946$$ −20.0000 −0.650256
$$947$$ −46.0000 −1.49480 −0.747400 0.664375i $$-0.768698\pi$$
−0.747400 + 0.664375i $$0.768698\pi$$
$$948$$ −36.0000 −1.16923
$$949$$ −33.0000 −1.07123
$$950$$ 0 0
$$951$$ −99.0000 −3.21029
$$952$$ −3.00000 −0.0972306
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ −22.0000 −0.711903
$$956$$ 1.00000 0.0323423
$$957$$ 18.0000 0.581857
$$958$$ 40.0000 1.29234
$$959$$ −57.0000 −1.84063
$$960$$ −6.00000 −0.193649
$$961$$ 5.00000 0.161290
$$962$$ 18.0000 0.580343
$$963$$ 18.0000 0.580042
$$964$$ −24.0000 −0.772988
$$965$$ −12.0000 −0.386294
$$966$$ −45.0000 −1.44785
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ −18.0000 −0.577054
$$974$$ −18.0000 −0.576757
$$975$$ 9.00000 0.288231
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ −18.0000 −0.575577
$$979$$ 12.0000 0.383522
$$980$$ 4.00000 0.127775
$$981$$ −18.0000 −0.574696
$$982$$ −8.00000 −0.255290
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ −36.0000 −1.14764
$$985$$ 8.00000 0.254901
$$986$$ 3.00000 0.0955395
$$987$$ −72.0000 −2.29179
$$988$$ 0 0
$$989$$ −50.0000 −1.58991
$$990$$ 24.0000 0.762770
$$991$$ −6.00000 −0.190596 −0.0952981 0.995449i $$-0.530380\pi$$
−0.0952981 + 0.995449i $$0.530380\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ 27.0000 0.856819
$$994$$ 0 0
$$995$$ −14.0000 −0.443830
$$996$$ −6.00000 −0.190117
$$997$$ 6.00000 0.190022 0.0950110 0.995476i $$-0.469711\pi$$
0.0950110 + 0.995476i $$0.469711\pi$$
$$998$$ −18.0000 −0.569780
$$999$$ 54.0000 1.70848
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 722.2.a.a.1.1 1
3.2 odd 2 6498.2.a.m.1.1 1
4.3 odd 2 5776.2.a.q.1.1 1
19.2 odd 18 722.2.e.g.99.1 6
19.3 odd 18 722.2.e.g.389.1 6
19.4 even 9 722.2.e.h.415.1 6
19.5 even 9 722.2.e.h.595.1 6
19.6 even 9 722.2.e.h.245.1 6
19.7 even 3 722.2.c.g.429.1 2
19.8 odd 6 722.2.c.a.653.1 2
19.9 even 9 722.2.e.h.423.1 6
19.10 odd 18 722.2.e.g.423.1 6
19.11 even 3 722.2.c.g.653.1 2
19.12 odd 6 722.2.c.a.429.1 2
19.13 odd 18 722.2.e.g.245.1 6
19.14 odd 18 722.2.e.g.595.1 6
19.15 odd 18 722.2.e.g.415.1 6
19.16 even 9 722.2.e.h.389.1 6
19.17 even 9 722.2.e.h.99.1 6
19.18 odd 2 722.2.a.f.1.1 yes 1
57.56 even 2 6498.2.a.a.1.1 1
76.75 even 2 5776.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.a.1.1 1 1.1 even 1 trivial
722.2.a.f.1.1 yes 1 19.18 odd 2
722.2.c.a.429.1 2 19.12 odd 6
722.2.c.a.653.1 2 19.8 odd 6
722.2.c.g.429.1 2 19.7 even 3
722.2.c.g.653.1 2 19.11 even 3
722.2.e.g.99.1 6 19.2 odd 18
722.2.e.g.245.1 6 19.13 odd 18
722.2.e.g.389.1 6 19.3 odd 18
722.2.e.g.415.1 6 19.15 odd 18
722.2.e.g.423.1 6 19.10 odd 18
722.2.e.g.595.1 6 19.14 odd 18
722.2.e.h.99.1 6 19.17 even 9
722.2.e.h.245.1 6 19.6 even 9
722.2.e.h.389.1 6 19.16 even 9
722.2.e.h.415.1 6 19.4 even 9
722.2.e.h.423.1 6 19.9 even 9
722.2.e.h.595.1 6 19.5 even 9
5776.2.a.a.1.1 1 76.75 even 2
5776.2.a.q.1.1 1 4.3 odd 2
6498.2.a.a.1.1 1 57.56 even 2
6498.2.a.m.1.1 1 3.2 odd 2