# Properties

 Label 6762.2.a.y.1.1 Level $6762$ Weight $2$ Character 6762.1 Self dual yes Analytic conductor $53.995$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6762.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$53.9948418468$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6762.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -3.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} +3.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -9.00000 q^{37} -3.00000 q^{39} -3.00000 q^{40} -9.00000 q^{41} -3.00000 q^{43} +4.00000 q^{44} -3.00000 q^{45} +1.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -4.00000 q^{51} +3.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -12.0000 q^{55} +3.00000 q^{58} -6.00000 q^{59} +3.00000 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -9.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -9.00000 q^{74} -4.00000 q^{75} -3.00000 q^{78} +8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -4.00000 q^{83} -12.0000 q^{85} -3.00000 q^{86} -3.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -6.00000 q^{93} +7.00000 q^{94} -1.00000 q^{96} +7.00000 q^{97} +4.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −3.00000 −0.948683
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 4.00000 0.800000
$$26$$ 3.00000 0.588348
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 3.00000 0.547723
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −9.00000 −1.47959 −0.739795 0.672832i $$-0.765078\pi$$
−0.739795 + 0.672832i $$0.765078\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ −3.00000 −0.474342
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ −3.00000 −0.457496 −0.228748 0.973486i $$-0.573463\pi$$
−0.228748 + 0.973486i $$0.573463\pi$$
$$44$$ 4.00000 0.603023
$$45$$ −3.00000 −0.447214
$$46$$ 1.00000 0.147442
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ −4.00000 −0.560112
$$52$$ 3.00000 0.416025
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 3.00000 0.387298
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −9.00000 −1.11631
$$66$$ −4.00000 −0.492366
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ −9.00000 −1.04623
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ −3.00000 −0.339683
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ −9.00000 −0.993884
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ −3.00000 −0.323498
$$87$$ −3.00000 −0.321634
$$88$$ 4.00000 0.426401
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 1.00000 0.104257
$$93$$ −6.00000 −0.622171
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 4.00000 0.400000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ −5.00000 −0.492665 −0.246332 0.969185i $$-0.579225\pi$$
−0.246332 + 0.969185i $$0.579225\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ −4.00000 −0.388514
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 3.00000 0.287348 0.143674 0.989625i $$-0.454108\pi$$
0.143674 + 0.989625i $$0.454108\pi$$
$$110$$ −12.0000 −1.14416
$$111$$ 9.00000 0.854242
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 3.00000 0.278543
$$117$$ 3.00000 0.277350
$$118$$ −6.00000 −0.552345
$$119$$ 0 0
$$120$$ 3.00000 0.273861
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 9.00000 0.811503
$$124$$ 6.00000 0.538816
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 3.00000 0.264135
$$130$$ −9.00000 −0.789352
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 3.00000 0.258199
$$136$$ 4.00000 0.342997
$$137$$ −15.0000 −1.28154 −0.640768 0.767734i $$-0.721384\pi$$
−0.640768 + 0.767734i $$0.721384\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ −7.00000 −0.589506
$$142$$ −6.00000 −0.503509
$$143$$ 12.0000 1.00349
$$144$$ 1.00000 0.0833333
$$145$$ −9.00000 −0.747409
$$146$$ 8.00000 0.662085
$$147$$ 0 0
$$148$$ −9.00000 −0.739795
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −18.0000 −1.44579
$$156$$ −3.00000 −0.240192
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 4.00000 0.317221
$$160$$ −3.00000 −0.237171
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 12.0000 0.934199
$$166$$ −4.00000 −0.310460
$$167$$ 20.0000 1.54765 0.773823 0.633402i $$-0.218342\pi$$
0.773823 + 0.633402i $$0.218342\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ −12.0000 −0.920358
$$171$$ 0 0
$$172$$ −3.00000 −0.228748
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 6.00000 0.450988
$$178$$ 14.0000 1.04934
$$179$$ 19.0000 1.42013 0.710063 0.704138i $$-0.248666\pi$$
0.710063 + 0.704138i $$0.248666\pi$$
$$180$$ −3.00000 −0.223607
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 1.00000 0.0737210
$$185$$ 27.0000 1.98508
$$186$$ −6.00000 −0.439941
$$187$$ 16.0000 1.17004
$$188$$ 7.00000 0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −17.0000 −1.22369 −0.611843 0.790979i $$-0.709572\pi$$
−0.611843 + 0.790979i $$0.709572\pi$$
$$194$$ 7.00000 0.502571
$$195$$ 9.00000 0.644503
$$196$$ 0 0
$$197$$ −23.0000 −1.63868 −0.819341 0.573306i $$-0.805660\pi$$
−0.819341 + 0.573306i $$0.805660\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 4.00000 0.282843
$$201$$ −4.00000 −0.282138
$$202$$ 14.0000 0.985037
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ 27.0000 1.88576
$$206$$ −5.00000 −0.348367
$$207$$ 1.00000 0.0695048
$$208$$ 3.00000 0.208013
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −4.00000 −0.274721
$$213$$ 6.00000 0.411113
$$214$$ 8.00000 0.546869
$$215$$ 9.00000 0.613795
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 3.00000 0.203186
$$219$$ −8.00000 −0.540590
$$220$$ −12.0000 −0.809040
$$221$$ 12.0000 0.807207
$$222$$ 9.00000 0.604040
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 9.00000 0.598671
$$227$$ −11.0000 −0.730096 −0.365048 0.930989i $$-0.618947\pi$$
−0.365048 + 0.930989i $$0.618947\pi$$
$$228$$ 0 0
$$229$$ −12.0000 −0.792982 −0.396491 0.918039i $$-0.629772\pi$$
−0.396491 + 0.918039i $$0.629772\pi$$
$$230$$ −3.00000 −0.197814
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 3.00000 0.196116
$$235$$ −21.0000 −1.36989
$$236$$ −6.00000 −0.390567
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 5.00000 0.322078 0.161039 0.986948i $$-0.448515\pi$$
0.161039 + 0.986948i $$0.448515\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 9.00000 0.573819
$$247$$ 0 0
$$248$$ 6.00000 0.381000
$$249$$ 4.00000 0.253490
$$250$$ 3.00000 0.189737
$$251$$ −19.0000 −1.19927 −0.599635 0.800274i $$-0.704687\pi$$
−0.599635 + 0.800274i $$0.704687\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 7.00000 0.439219
$$255$$ 12.0000 0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 3.00000 0.186772
$$259$$ 0 0
$$260$$ −9.00000 −0.558156
$$261$$ 3.00000 0.185695
$$262$$ 6.00000 0.370681
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ −14.0000 −0.856786
$$268$$ 4.00000 0.244339
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 3.00000 0.182574
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −15.0000 −0.906183
$$275$$ 16.0000 0.964836
$$276$$ −1.00000 −0.0601929
$$277$$ 20.0000 1.20168 0.600842 0.799368i $$-0.294832\pi$$
0.600842 + 0.799368i $$0.294832\pi$$
$$278$$ −9.00000 −0.539784
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 23.0000 1.37206 0.686032 0.727571i $$-0.259351\pi$$
0.686032 + 0.727571i $$0.259351\pi$$
$$282$$ −7.00000 −0.416844
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −9.00000 −0.528498
$$291$$ −7.00000 −0.410347
$$292$$ 8.00000 0.468165
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 18.0000 1.04800
$$296$$ −9.00000 −0.523114
$$297$$ −4.00000 −0.232104
$$298$$ 16.0000 0.926855
$$299$$ 3.00000 0.173494
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ 15.0000 0.863153
$$303$$ −14.0000 −0.804279
$$304$$ 0 0
$$305$$ 30.0000 1.71780
$$306$$ 4.00000 0.228665
$$307$$ −15.0000 −0.856095 −0.428048 0.903756i $$-0.640798\pi$$
−0.428048 + 0.903756i $$0.640798\pi$$
$$308$$ 0 0
$$309$$ 5.00000 0.284440
$$310$$ −18.0000 −1.02233
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ −3.00000 −0.169842
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 8.00000 0.451466
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 5.00000 0.280828 0.140414 0.990093i $$-0.455157\pi$$
0.140414 + 0.990093i $$0.455157\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 12.0000 0.671871
$$320$$ −3.00000 −0.167705
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 12.0000 0.665640
$$326$$ 16.0000 0.886158
$$327$$ −3.00000 −0.165900
$$328$$ −9.00000 −0.496942
$$329$$ 0 0
$$330$$ 12.0000 0.660578
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −9.00000 −0.493197
$$334$$ 20.0000 1.09435
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ −4.00000 −0.217571
$$339$$ −9.00000 −0.488813
$$340$$ −12.0000 −0.650791
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −3.00000 −0.161749
$$345$$ 3.00000 0.161515
$$346$$ 6.00000 0.322562
$$347$$ 23.0000 1.23470 0.617352 0.786687i $$-0.288205\pi$$
0.617352 + 0.786687i $$0.288205\pi$$
$$348$$ −3.00000 −0.160817
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −3.00000 −0.160128
$$352$$ 4.00000 0.213201
$$353$$ 29.0000 1.54351 0.771757 0.635917i $$-0.219378\pi$$
0.771757 + 0.635917i $$0.219378\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 18.0000 0.955341
$$356$$ 14.0000 0.741999
$$357$$ 0 0
$$358$$ 19.0000 1.00418
$$359$$ −1.00000 −0.0527780 −0.0263890 0.999652i $$-0.508401\pi$$
−0.0263890 + 0.999652i $$0.508401\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −19.0000 −1.00000
$$362$$ 2.00000 0.105118
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ −24.0000 −1.25622
$$366$$ 10.0000 0.522708
$$367$$ −31.0000 −1.61819 −0.809093 0.587680i $$-0.800041\pi$$
−0.809093 + 0.587680i $$0.800041\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ −9.00000 −0.468521
$$370$$ 27.0000 1.40366
$$371$$ 0 0
$$372$$ −6.00000 −0.311086
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 16.0000 0.827340
$$375$$ −3.00000 −0.154919
$$376$$ 7.00000 0.360997
$$377$$ 9.00000 0.463524
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ −12.0000 −0.613973
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −17.0000 −0.865277
$$387$$ −3.00000 −0.152499
$$388$$ 7.00000 0.355371
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 9.00000 0.455733
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ −23.0000 −1.15872
$$395$$ −24.0000 −1.20757
$$396$$ 4.00000 0.201008
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 5.00000 0.250627
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 18.0000 0.896644
$$404$$ 14.0000 0.696526
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ −36.0000 −1.78445
$$408$$ −4.00000 −0.198030
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 27.0000 1.33343
$$411$$ 15.0000 0.739895
$$412$$ −5.00000 −0.246332
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 12.0000 0.589057
$$416$$ 3.00000 0.147087
$$417$$ 9.00000 0.440732
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −21.0000 −1.02348 −0.511739 0.859141i $$-0.670998\pi$$
−0.511739 + 0.859141i $$0.670998\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 7.00000 0.340352
$$424$$ −4.00000 −0.194257
$$425$$ 16.0000 0.776114
$$426$$ 6.00000 0.290701
$$427$$ 0 0
$$428$$ 8.00000 0.386695
$$429$$ −12.0000 −0.579365
$$430$$ 9.00000 0.434019
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −23.0000 −1.10531 −0.552655 0.833410i $$-0.686385\pi$$
−0.552655 + 0.833410i $$0.686385\pi$$
$$434$$ 0 0
$$435$$ 9.00000 0.431517
$$436$$ 3.00000 0.143674
$$437$$ 0 0
$$438$$ −8.00000 −0.382255
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ −12.0000 −0.572078
$$441$$ 0 0
$$442$$ 12.0000 0.570782
$$443$$ 25.0000 1.18779 0.593893 0.804544i $$-0.297590\pi$$
0.593893 + 0.804544i $$0.297590\pi$$
$$444$$ 9.00000 0.427121
$$445$$ −42.0000 −1.99099
$$446$$ 14.0000 0.662919
$$447$$ −16.0000 −0.756774
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 4.00000 0.188562
$$451$$ −36.0000 −1.69517
$$452$$ 9.00000 0.423324
$$453$$ −15.0000 −0.704761
$$454$$ −11.0000 −0.516256
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 24.0000 1.12267 0.561336 0.827588i $$-0.310287\pi$$
0.561336 + 0.827588i $$0.310287\pi$$
$$458$$ −12.0000 −0.560723
$$459$$ −4.00000 −0.186704
$$460$$ −3.00000 −0.139876
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 0 0
$$463$$ 13.0000 0.604161 0.302081 0.953282i $$-0.402319\pi$$
0.302081 + 0.953282i $$0.402319\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 18.0000 0.834730
$$466$$ −10.0000 −0.463241
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 3.00000 0.138675
$$469$$ 0 0
$$470$$ −21.0000 −0.968658
$$471$$ −8.00000 −0.368621
$$472$$ −6.00000 −0.276172
$$473$$ −12.0000 −0.551761
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −4.00000 −0.183147
$$478$$ −12.0000 −0.548867
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 3.00000 0.136931
$$481$$ −27.0000 −1.23109
$$482$$ 5.00000 0.227744
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −21.0000 −0.953561
$$486$$ −1.00000 −0.0453609
$$487$$ 13.0000 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 9.00000 0.405751
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 6.00000 0.269408
$$497$$ 0 0
$$498$$ 4.00000 0.179244
$$499$$ 34.0000 1.52205 0.761025 0.648723i $$-0.224697\pi$$
0.761025 + 0.648723i $$0.224697\pi$$
$$500$$ 3.00000 0.134164
$$501$$ −20.0000 −0.893534
$$502$$ −19.0000 −0.848012
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ −42.0000 −1.86898
$$506$$ 4.00000 0.177822
$$507$$ 4.00000 0.177646
$$508$$ 7.00000 0.310575
$$509$$ −44.0000 −1.95027 −0.975133 0.221621i $$-0.928865\pi$$
−0.975133 + 0.221621i $$0.928865\pi$$
$$510$$ 12.0000 0.531369
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 26.0000 1.14681
$$515$$ 15.0000 0.660979
$$516$$ 3.00000 0.132068
$$517$$ 28.0000 1.23144
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ −9.00000 −0.394676
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 3.00000 0.131306
$$523$$ −42.0000 −1.83653 −0.918266 0.395964i $$-0.870410\pi$$
−0.918266 + 0.395964i $$0.870410\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 21.0000 0.915644
$$527$$ 24.0000 1.04546
$$528$$ −4.00000 −0.174078
$$529$$ 1.00000 0.0434783
$$530$$ 12.0000 0.521247
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ −27.0000 −1.16950
$$534$$ −14.0000 −0.605839
$$535$$ −24.0000 −1.03761
$$536$$ 4.00000 0.172774
$$537$$ −19.0000 −0.819911
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 3.00000 0.129099
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −2.00000 −0.0858282
$$544$$ 4.00000 0.171499
$$545$$ −9.00000 −0.385518
$$546$$ 0 0
$$547$$ 22.0000 0.940652 0.470326 0.882493i $$-0.344136\pi$$
0.470326 + 0.882493i $$0.344136\pi$$
$$548$$ −15.0000 −0.640768
$$549$$ −10.0000 −0.426790
$$550$$ 16.0000 0.682242
$$551$$ 0 0
$$552$$ −1.00000 −0.0425628
$$553$$ 0 0
$$554$$ 20.0000 0.849719
$$555$$ −27.0000 −1.14609
$$556$$ −9.00000 −0.381685
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 23.0000 0.970196
$$563$$ −15.0000 −0.632175 −0.316087 0.948730i $$-0.602369\pi$$
−0.316087 + 0.948730i $$0.602369\pi$$
$$564$$ −7.00000 −0.294753
$$565$$ −27.0000 −1.13590
$$566$$ 6.00000 0.252199
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −13.0000 −0.544988 −0.272494 0.962157i $$-0.587849\pi$$
−0.272494 + 0.962157i $$0.587849\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 17.0000 0.706496
$$580$$ −9.00000 −0.373705
$$581$$ 0 0
$$582$$ −7.00000 −0.290159
$$583$$ −16.0000 −0.662652
$$584$$ 8.00000 0.331042
$$585$$ −9.00000 −0.372104
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 18.0000 0.741048
$$591$$ 23.0000 0.946094
$$592$$ −9.00000 −0.369898
$$593$$ −29.0000 −1.19089 −0.595444 0.803397i $$-0.703024\pi$$
−0.595444 + 0.803397i $$0.703024\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 16.0000 0.655386
$$597$$ −5.00000 −0.204636
$$598$$ 3.00000 0.122679
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 15.0000 0.610341
$$605$$ −15.0000 −0.609837
$$606$$ −14.0000 −0.568711
$$607$$ −30.0000 −1.21766 −0.608831 0.793300i $$-0.708361\pi$$
−0.608831 + 0.793300i $$0.708361\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 30.0000 1.21466
$$611$$ 21.0000 0.849569
$$612$$ 4.00000 0.161690
$$613$$ 25.0000 1.00974 0.504870 0.863195i $$-0.331540\pi$$
0.504870 + 0.863195i $$0.331540\pi$$
$$614$$ −15.0000 −0.605351
$$615$$ −27.0000 −1.08875
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 5.00000 0.201129
$$619$$ −2.00000 −0.0803868 −0.0401934 0.999192i $$-0.512797\pi$$
−0.0401934 + 0.999192i $$0.512797\pi$$
$$620$$ −18.0000 −0.722897
$$621$$ −1.00000 −0.0401286
$$622$$ 28.0000 1.12270
$$623$$ 0 0
$$624$$ −3.00000 −0.120096
$$625$$ −29.0000 −1.16000
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 12.0000 0.476957
$$634$$ 5.00000 0.198575
$$635$$ −21.0000 −0.833360
$$636$$ 4.00000 0.158610
$$637$$ 0 0
$$638$$ 12.0000 0.475085
$$639$$ −6.00000 −0.237356
$$640$$ −3.00000 −0.118585
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ −8.00000 −0.315735
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ −9.00000 −0.354375
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −24.0000 −0.942082
$$650$$ 12.0000 0.470679
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ 31.0000 1.21312 0.606562 0.795036i $$-0.292548\pi$$
0.606562 + 0.795036i $$0.292548\pi$$
$$654$$ −3.00000 −0.117309
$$655$$ −18.0000 −0.703318
$$656$$ −9.00000 −0.351391
$$657$$ 8.00000 0.312110
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 12.0000 0.467099
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ 8.00000 0.310929
$$663$$ −12.0000 −0.466041
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −9.00000 −0.348743
$$667$$ 3.00000 0.116160
$$668$$ 20.0000 0.773823
$$669$$ −14.0000 −0.541271
$$670$$ −12.0000 −0.463600
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ −4.00000 −0.153960
$$676$$ −4.00000 −0.153846
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ −9.00000 −0.345643
$$679$$ 0 0
$$680$$ −12.0000 −0.460179
$$681$$ 11.0000 0.421521
$$682$$ 24.0000 0.919007
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 45.0000 1.71936
$$686$$ 0 0
$$687$$ 12.0000 0.457829
$$688$$ −3.00000 −0.114374
$$689$$ −12.0000 −0.457164
$$690$$ 3.00000 0.114208
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 23.0000 0.873068
$$695$$ 27.0000 1.02417
$$696$$ −3.00000 −0.113715
$$697$$ −36.0000 −1.36360
$$698$$ 26.0000 0.984115
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ −3.00000 −0.113228
$$703$$ 0 0
$$704$$ 4.00000 0.150756
$$705$$ 21.0000 0.790906
$$706$$ 29.0000 1.09143
$$707$$ 0 0
$$708$$ 6.00000 0.225494
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 18.0000 0.675528
$$711$$ 8.00000 0.300023
$$712$$ 14.0000 0.524672
$$713$$ 6.00000 0.224702
$$714$$ 0 0
$$715$$ −36.0000 −1.34632
$$716$$ 19.0000 0.710063
$$717$$ 12.0000 0.448148
$$718$$ −1.00000 −0.0373197
$$719$$ 7.00000 0.261056 0.130528 0.991445i $$-0.458333\pi$$
0.130528 + 0.991445i $$0.458333\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 0 0
$$722$$ −19.0000 −0.707107
$$723$$ −5.00000 −0.185952
$$724$$ 2.00000 0.0743294
$$725$$ 12.0000 0.445669
$$726$$ −5.00000 −0.185567
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −24.0000 −0.888280
$$731$$ −12.0000 −0.443836
$$732$$ 10.0000 0.369611
$$733$$ 16.0000 0.590973 0.295487 0.955347i $$-0.404518\pi$$
0.295487 + 0.955347i $$0.404518\pi$$
$$734$$ −31.0000 −1.14423
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 16.0000 0.589368
$$738$$ −9.00000 −0.331295
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 27.0000 0.992540
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ −48.0000 −1.75858
$$746$$ 22.0000 0.805477
$$747$$ −4.00000 −0.146352
$$748$$ 16.0000 0.585018
$$749$$ 0 0
$$750$$ −3.00000 −0.109545
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 19.0000 0.692398
$$754$$ 9.00000 0.327761
$$755$$ −45.0000 −1.63772
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −25.0000 −0.908041
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ −7.00000 −0.253583
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ −12.0000 −0.433861
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ −1.00000 −0.0360844
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ −26.0000 −0.936367
$$772$$ −17.0000 −0.611843
$$773$$ −19.0000 −0.683383 −0.341691 0.939812i $$-0.611000\pi$$
−0.341691 + 0.939812i $$0.611000\pi$$
$$774$$ −3.00000 −0.107833
$$775$$ 24.0000 0.862105
$$776$$ 7.00000 0.251285
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 0 0
$$780$$ 9.00000 0.322252
$$781$$ −24.0000 −0.858788
$$782$$ 4.00000 0.143040
$$783$$ −3.00000 −0.107211
$$784$$ 0 0
$$785$$ −24.0000 −0.856597
$$786$$ −6.00000 −0.214013
$$787$$ 40.0000 1.42585 0.712923 0.701242i $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ −23.0000 −0.819341
$$789$$ −21.0000 −0.747620
$$790$$ −24.0000 −0.853882
$$791$$ 0 0
$$792$$ 4.00000 0.142134
$$793$$ −30.0000 −1.06533
$$794$$ −34.0000 −1.20661
$$795$$ −12.0000 −0.425596
$$796$$ 5.00000 0.177220
$$797$$ 49.0000 1.73567 0.867835 0.496853i $$-0.165511\pi$$
0.867835 + 0.496853i $$0.165511\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ 4.00000 0.141421
$$801$$ 14.0000 0.494666
$$802$$ 38.0000 1.34183
$$803$$ 32.0000 1.12926
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 18.0000 0.634023
$$807$$ −6.00000 −0.211210
$$808$$ 14.0000 0.492518
$$809$$ −16.0000 −0.562530 −0.281265 0.959630i $$-0.590754\pi$$
−0.281265 + 0.959630i $$0.590754\pi$$
$$810$$ −3.00000 −0.105409
$$811$$ −33.0000 −1.15879 −0.579393 0.815048i $$-0.696710\pi$$
−0.579393 + 0.815048i $$0.696710\pi$$
$$812$$ 0 0
$$813$$ 20.0000 0.701431
$$814$$ −36.0000 −1.26180
$$815$$ −48.0000 −1.68137
$$816$$ −4.00000 −0.140028
$$817$$ 0 0
$$818$$ −16.0000 −0.559427
$$819$$ 0 0
$$820$$ 27.0000 0.942881
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 15.0000 0.523185
$$823$$ 7.00000 0.244005 0.122002 0.992530i $$-0.461068\pi$$
0.122002 + 0.992530i $$0.461068\pi$$
$$824$$ −5.00000 −0.174183
$$825$$ −16.0000 −0.557048
$$826$$ 0 0
$$827$$ −30.0000 −1.04320 −0.521601 0.853189i $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 12.0000 0.416526
$$831$$ −20.0000 −0.693792
$$832$$ 3.00000 0.104006
$$833$$ 0 0
$$834$$ 9.00000 0.311645
$$835$$ −60.0000 −2.07639
$$836$$ 0 0
$$837$$ −6.00000 −0.207390
$$838$$ 4.00000 0.138178
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −21.0000 −0.723708
$$843$$ −23.0000 −0.792162
$$844$$ −12.0000 −0.413057
$$845$$ 12.0000 0.412813
$$846$$ 7.00000 0.240665
$$847$$ 0 0
$$848$$ −4.00000 −0.137361
$$849$$ −6.00000 −0.205919
$$850$$ 16.0000 0.548795
$$851$$ −9.00000 −0.308516
$$852$$ 6.00000 0.205557
$$853$$ −7.00000 −0.239675 −0.119838 0.992793i $$-0.538237\pi$$
−0.119838 + 0.992793i $$0.538237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ −33.0000 −1.12726 −0.563629 0.826028i $$-0.690595\pi$$
−0.563629 + 0.826028i $$0.690595\pi$$
$$858$$ −12.0000 −0.409673
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 9.00000 0.306897
$$861$$ 0 0
$$862$$ 21.0000 0.715263
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −18.0000 −0.612018
$$866$$ −23.0000 −0.781572
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 9.00000 0.305129
$$871$$ 12.0000 0.406604
$$872$$ 3.00000 0.101593
$$873$$ 7.00000 0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −8.00000 −0.270295
$$877$$ −36.0000 −1.21563 −0.607817 0.794077i $$-0.707955\pi$$
−0.607817 + 0.794077i $$0.707955\pi$$
$$878$$ 10.0000 0.337484
$$879$$ 6.00000 0.202375
$$880$$ −12.0000 −0.404520
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 54.0000 1.81724 0.908622 0.417619i $$-0.137135\pi$$
0.908622 + 0.417619i $$0.137135\pi$$
$$884$$ 12.0000 0.403604
$$885$$ −18.0000 −0.605063
$$886$$ 25.0000 0.839891
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 9.00000 0.302020
$$889$$ 0 0
$$890$$ −42.0000 −1.40784
$$891$$ 4.00000 0.134005
$$892$$ 14.0000 0.468755
$$893$$ 0 0
$$894$$ −16.0000 −0.535120
$$895$$ −57.0000 −1.90530
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ 30.0000 1.00111
$$899$$ 18.0000 0.600334
$$900$$ 4.00000 0.133333
$$901$$ −16.0000 −0.533037
$$902$$ −36.0000 −1.19867
$$903$$ 0 0
$$904$$ 9.00000 0.299336
$$905$$ −6.00000 −0.199447
$$906$$ −15.0000 −0.498342
$$907$$ −53.0000 −1.75984 −0.879918 0.475125i $$-0.842403\pi$$
−0.879918 + 0.475125i $$0.842403\pi$$
$$908$$ −11.0000 −0.365048
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ −49.0000 −1.62344 −0.811721 0.584045i $$-0.801469\pi$$
−0.811721 + 0.584045i $$0.801469\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 24.0000 0.793849
$$915$$ −30.0000 −0.991769
$$916$$ −12.0000 −0.396491
$$917$$ 0 0
$$918$$ −4.00000 −0.132020
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −3.00000 −0.0989071
$$921$$ 15.0000 0.494267
$$922$$ 22.0000 0.724531
$$923$$ −18.0000 −0.592477
$$924$$ 0 0
$$925$$ −36.0000 −1.18367
$$926$$ 13.0000 0.427207
$$927$$ −5.00000 −0.164222
$$928$$ 3.00000 0.0984798
$$929$$ −9.00000 −0.295280 −0.147640 0.989041i $$-0.547168\pi$$
−0.147640 + 0.989041i $$0.547168\pi$$
$$930$$ 18.0000 0.590243
$$931$$ 0 0
$$932$$ −10.0000 −0.327561
$$933$$ −28.0000 −0.916679
$$934$$ −13.0000 −0.425373
$$935$$ −48.0000 −1.56977
$$936$$ 3.00000 0.0980581
$$937$$ −25.0000 −0.816714 −0.408357 0.912822i $$-0.633898\pi$$
−0.408357 + 0.912822i $$0.633898\pi$$
$$938$$ 0 0
$$939$$ 34.0000 1.10955
$$940$$ −21.0000 −0.684944
$$941$$ −3.00000 −0.0977972 −0.0488986 0.998804i $$-0.515571\pi$$
−0.0488986 + 0.998804i $$0.515571\pi$$
$$942$$ −8.00000 −0.260654
$$943$$ −9.00000 −0.293080
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −31.0000 −1.00736 −0.503682 0.863889i $$-0.668022\pi$$
−0.503682 + 0.863889i $$0.668022\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ −5.00000 −0.162136
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ −4.00000 −0.129505
$$955$$ 36.0000 1.16493
$$956$$ −12.0000 −0.388108
$$957$$ −12.0000 −0.387905
$$958$$ −30.0000 −0.969256
$$959$$ 0 0
$$960$$ 3.00000 0.0968246
$$961$$ 5.00000 0.161290
$$962$$ −27.0000 −0.870515
$$963$$ 8.00000 0.257796
$$964$$ 5.00000 0.161039
$$965$$ 51.0000 1.64175
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −21.0000 −0.674269
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 13.0000 0.416547
$$975$$ −12.0000 −0.384308
$$976$$ −10.0000 −0.320092
$$977$$ 13.0000 0.415907 0.207953 0.978139i $$-0.433320\pi$$
0.207953 + 0.978139i $$0.433320\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 3.00000 0.0957826
$$982$$ −12.0000 −0.382935
$$983$$ 54.0000 1.72233 0.861166 0.508323i $$-0.169735\pi$$
0.861166 + 0.508323i $$0.169735\pi$$
$$984$$ 9.00000 0.286910
$$985$$ 69.0000 2.19852
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ −12.0000 −0.381385
$$991$$ −36.0000 −1.14358 −0.571789 0.820401i $$-0.693750\pi$$
−0.571789 + 0.820401i $$0.693750\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ −15.0000 −0.475532
$$996$$ 4.00000 0.126745
$$997$$ 46.0000 1.45683 0.728417 0.685134i $$-0.240256\pi$$
0.728417 + 0.685134i $$0.240256\pi$$
$$998$$ 34.0000 1.07625
$$999$$ 9.00000 0.284747
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 6762.2.a.y.1.1 1
7.6 odd 2 966.2.a.k.1.1 1
21.20 even 2 2898.2.a.a.1.1 1
28.27 even 2 7728.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.k.1.1 1 7.6 odd 2
2898.2.a.a.1.1 1 21.20 even 2
6762.2.a.y.1.1 1 1.1 even 1 trivial
7728.2.a.j.1.1 1 28.27 even 2