# Properties

 Label 5775.2.a.bw.1.2 Level $5775$ Weight $2$ Character 5775.1 Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5775.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1136071673$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 Defining polynomial: $$x^{3} - 6x - 1$$ x^3 - 6*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.167449$$ of defining polynomial Character $$\chi$$ $$=$$ 5775.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.167449 q^{2} +1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.167449 q^{2} +1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.97196 q^{12} -3.80451 q^{13} +0.167449 q^{14} +3.83255 q^{16} -0.334898 q^{17} +0.167449 q^{18} +8.13941 q^{19} +1.00000 q^{21} +0.167449 q^{22} +1.66510 q^{23} -0.665102 q^{24} -0.637062 q^{26} +1.00000 q^{27} -1.97196 q^{28} +0.195488 q^{29} -9.94392 q^{31} +1.97196 q^{32} +1.00000 q^{33} -0.0560785 q^{34} -1.97196 q^{36} +4.47431 q^{37} +1.36294 q^{38} -3.80451 q^{39} -6.27882 q^{41} +0.167449 q^{42} -2.33490 q^{43} -1.97196 q^{44} +0.278820 q^{46} +12.1394 q^{47} +3.83255 q^{48} +1.00000 q^{49} -0.334898 q^{51} +7.50235 q^{52} -7.94392 q^{53} +0.167449 q^{54} -0.665102 q^{56} +8.13941 q^{57} +0.0327344 q^{58} +3.74843 q^{59} +6.00000 q^{61} -1.66510 q^{62} +1.00000 q^{63} -7.33490 q^{64} +0.167449 q^{66} +0.139410 q^{67} +0.660406 q^{68} +1.66510 q^{69} +4.66980 q^{71} -0.665102 q^{72} -4.19549 q^{73} +0.749219 q^{74} -16.0506 q^{76} +1.00000 q^{77} -0.637062 q^{78} +3.33020 q^{79} +1.00000 q^{81} -1.05138 q^{82} +13.9439 q^{83} -1.97196 q^{84} -0.390977 q^{86} +0.195488 q^{87} -0.665102 q^{88} -9.88784 q^{89} -3.80451 q^{91} -3.28352 q^{92} -9.94392 q^{93} +2.03273 q^{94} +1.97196 q^{96} -0.0560785 q^{97} +0.167449 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^12 + 12 * q^16 + 12 * q^19 + 3 * q^21 + 6 * q^23 - 3 * q^24 + 9 * q^26 + 3 * q^27 + 6 * q^28 + 12 * q^29 - 6 * q^31 - 6 * q^32 + 3 * q^33 - 24 * q^34 + 6 * q^36 + 15 * q^38 + 6 * q^41 - 6 * q^43 + 6 * q^44 - 24 * q^46 + 24 * q^47 + 12 * q^48 + 3 * q^49 + 21 * q^52 - 3 * q^56 + 12 * q^57 + 9 * q^58 - 24 * q^59 + 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 12 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^71 - 3 * q^72 - 24 * q^73 + 39 * q^74 - 3 * q^76 + 3 * q^77 + 9 * q^78 + 12 * q^79 + 3 * q^81 - 30 * q^82 + 18 * q^83 + 6 * q^84 - 24 * q^86 + 12 * q^87 - 3 * q^88 + 18 * q^89 + 18 * q^92 - 6 * q^93 + 15 * q^94 - 6 * q^96 - 24 * q^97 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.167449 0.118404 0.0592022 0.998246i $$-0.481144\pi$$
0.0592022 + 0.998246i $$0.481144\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.97196 −0.985980
$$5$$ 0 0
$$6$$ 0.167449 0.0683608
$$7$$ 1.00000 0.377964
$$8$$ −0.665102 −0.235149
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −1.97196 −0.569256
$$13$$ −3.80451 −1.05518 −0.527591 0.849499i $$-0.676905\pi$$
−0.527591 + 0.849499i $$0.676905\pi$$
$$14$$ 0.167449 0.0447527
$$15$$ 0 0
$$16$$ 3.83255 0.958138
$$17$$ −0.334898 −0.0812248 −0.0406124 0.999175i $$-0.512931\pi$$
−0.0406124 + 0.999175i $$0.512931\pi$$
$$18$$ 0.167449 0.0394682
$$19$$ 8.13941 1.86731 0.933654 0.358175i $$-0.116601\pi$$
0.933654 + 0.358175i $$0.116601\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0.167449 0.0357003
$$23$$ 1.66510 0.347198 0.173599 0.984816i $$-0.444460\pi$$
0.173599 + 0.984816i $$0.444460\pi$$
$$24$$ −0.665102 −0.135763
$$25$$ 0 0
$$26$$ −0.637062 −0.124938
$$27$$ 1.00000 0.192450
$$28$$ −1.97196 −0.372666
$$29$$ 0.195488 0.0363013 0.0181506 0.999835i $$-0.494222\pi$$
0.0181506 + 0.999835i $$0.494222\pi$$
$$30$$ 0 0
$$31$$ −9.94392 −1.78598 −0.892991 0.450075i $$-0.851397\pi$$
−0.892991 + 0.450075i $$0.851397\pi$$
$$32$$ 1.97196 0.348597
$$33$$ 1.00000 0.174078
$$34$$ −0.0560785 −0.00961738
$$35$$ 0 0
$$36$$ −1.97196 −0.328660
$$37$$ 4.47431 0.735572 0.367786 0.929911i $$-0.380116\pi$$
0.367786 + 0.929911i $$0.380116\pi$$
$$38$$ 1.36294 0.221098
$$39$$ −3.80451 −0.609209
$$40$$ 0 0
$$41$$ −6.27882 −0.980587 −0.490293 0.871557i $$-0.663110\pi$$
−0.490293 + 0.871557i $$0.663110\pi$$
$$42$$ 0.167449 0.0258380
$$43$$ −2.33490 −0.356069 −0.178034 0.984024i $$-0.556974\pi$$
−0.178034 + 0.984024i $$0.556974\pi$$
$$44$$ −1.97196 −0.297284
$$45$$ 0 0
$$46$$ 0.278820 0.0411098
$$47$$ 12.1394 1.77071 0.885357 0.464911i $$-0.153914\pi$$
0.885357 + 0.464911i $$0.153914\pi$$
$$48$$ 3.83255 0.553181
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −0.334898 −0.0468952
$$52$$ 7.50235 1.04039
$$53$$ −7.94392 −1.09118 −0.545591 0.838052i $$-0.683695\pi$$
−0.545591 + 0.838052i $$0.683695\pi$$
$$54$$ 0.167449 0.0227869
$$55$$ 0 0
$$56$$ −0.665102 −0.0888779
$$57$$ 8.13941 1.07809
$$58$$ 0.0327344 0.00429823
$$59$$ 3.74843 0.488004 0.244002 0.969775i $$-0.421540\pi$$
0.244002 + 0.969775i $$0.421540\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −1.66510 −0.211468
$$63$$ 1.00000 0.125988
$$64$$ −7.33490 −0.916862
$$65$$ 0 0
$$66$$ 0.167449 0.0206116
$$67$$ 0.139410 0.0170316 0.00851582 0.999964i $$-0.497289\pi$$
0.00851582 + 0.999964i $$0.497289\pi$$
$$68$$ 0.660406 0.0800860
$$69$$ 1.66510 0.200455
$$70$$ 0 0
$$71$$ 4.66980 0.554203 0.277101 0.960841i $$-0.410626\pi$$
0.277101 + 0.960841i $$0.410626\pi$$
$$72$$ −0.665102 −0.0783830
$$73$$ −4.19549 −0.491045 −0.245522 0.969391i $$-0.578959\pi$$
−0.245522 + 0.969391i $$0.578959\pi$$
$$74$$ 0.749219 0.0870950
$$75$$ 0 0
$$76$$ −16.0506 −1.84113
$$77$$ 1.00000 0.113961
$$78$$ −0.637062 −0.0721331
$$79$$ 3.33020 0.374677 0.187339 0.982295i $$-0.440014\pi$$
0.187339 + 0.982295i $$0.440014\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −1.05138 −0.116106
$$83$$ 13.9439 1.53054 0.765272 0.643707i $$-0.222604\pi$$
0.765272 + 0.643707i $$0.222604\pi$$
$$84$$ −1.97196 −0.215159
$$85$$ 0 0
$$86$$ −0.390977 −0.0421601
$$87$$ 0.195488 0.0209586
$$88$$ −0.665102 −0.0709001
$$89$$ −9.88784 −1.04811 −0.524055 0.851685i $$-0.675581\pi$$
−0.524055 + 0.851685i $$0.675581\pi$$
$$90$$ 0 0
$$91$$ −3.80451 −0.398821
$$92$$ −3.28352 −0.342330
$$93$$ −9.94392 −1.03114
$$94$$ 2.03273 0.209661
$$95$$ 0 0
$$96$$ 1.97196 0.201262
$$97$$ −0.0560785 −0.00569391 −0.00284695 0.999996i $$-0.500906\pi$$
−0.00284695 + 0.999996i $$0.500906\pi$$
$$98$$ 0.167449 0.0169149
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −18.8831 −1.87894 −0.939472 0.342626i $$-0.888683\pi$$
−0.939472 + 0.342626i $$0.888683\pi$$
$$102$$ −0.0560785 −0.00555260
$$103$$ 8.27882 0.815736 0.407868 0.913041i $$-0.366272\pi$$
0.407868 + 0.913041i $$0.366272\pi$$
$$104$$ 2.53039 0.248125
$$105$$ 0 0
$$106$$ −1.33020 −0.129201
$$107$$ 8.13941 0.786866 0.393433 0.919353i $$-0.371287\pi$$
0.393433 + 0.919353i $$0.371287\pi$$
$$108$$ −1.97196 −0.189752
$$109$$ 11.5529 1.10657 0.553286 0.832992i $$-0.313374\pi$$
0.553286 + 0.832992i $$0.313374\pi$$
$$110$$ 0 0
$$111$$ 4.47431 0.424683
$$112$$ 3.83255 0.362142
$$113$$ 1.33020 0.125135 0.0625675 0.998041i $$-0.480071\pi$$
0.0625675 + 0.998041i $$0.480071\pi$$
$$114$$ 1.36294 0.127651
$$115$$ 0 0
$$116$$ −0.385496 −0.0357924
$$117$$ −3.80451 −0.351727
$$118$$ 0.627672 0.0577819
$$119$$ −0.334898 −0.0307001
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 1.00470 0.0909608
$$123$$ −6.27882 −0.566142
$$124$$ 19.6090 1.76094
$$125$$ 0 0
$$126$$ 0.167449 0.0149176
$$127$$ −14.6137 −1.29676 −0.648379 0.761318i $$-0.724553\pi$$
−0.648379 + 0.761318i $$0.724553\pi$$
$$128$$ −5.17214 −0.457157
$$129$$ −2.33490 −0.205576
$$130$$ 0 0
$$131$$ 20.5576 1.79613 0.898065 0.439863i $$-0.144973\pi$$
0.898065 + 0.439863i $$0.144973\pi$$
$$132$$ −1.97196 −0.171637
$$133$$ 8.13941 0.705776
$$134$$ 0.0233441 0.00201662
$$135$$ 0 0
$$136$$ 0.222741 0.0190999
$$137$$ 16.2227 1.38600 0.693001 0.720936i $$-0.256288\pi$$
0.693001 + 0.720936i $$0.256288\pi$$
$$138$$ 0.278820 0.0237347
$$139$$ 22.5482 1.91252 0.956259 0.292522i $$-0.0944945\pi$$
0.956259 + 0.292522i $$0.0944945\pi$$
$$140$$ 0 0
$$141$$ 12.1394 1.02232
$$142$$ 0.781954 0.0656201
$$143$$ −3.80451 −0.318149
$$144$$ 3.83255 0.319379
$$145$$ 0 0
$$146$$ −0.702531 −0.0581419
$$147$$ 1.00000 0.0824786
$$148$$ −8.82316 −0.725259
$$149$$ 8.19549 0.671401 0.335700 0.941969i $$-0.391027\pi$$
0.335700 + 0.941969i $$0.391027\pi$$
$$150$$ 0 0
$$151$$ −13.2741 −1.08023 −0.540116 0.841590i $$-0.681620\pi$$
−0.540116 + 0.841590i $$0.681620\pi$$
$$152$$ −5.41353 −0.439096
$$153$$ −0.334898 −0.0270749
$$154$$ 0.167449 0.0134934
$$155$$ 0 0
$$156$$ 7.50235 0.600669
$$157$$ 16.9392 1.35190 0.675949 0.736949i $$-0.263734\pi$$
0.675949 + 0.736949i $$0.263734\pi$$
$$158$$ 0.557640 0.0443634
$$159$$ −7.94392 −0.629994
$$160$$ 0 0
$$161$$ 1.66510 0.131228
$$162$$ 0.167449 0.0131561
$$163$$ 6.79982 0.532603 0.266301 0.963890i $$-0.414198\pi$$
0.266301 + 0.963890i $$0.414198\pi$$
$$164$$ 12.3816 0.966839
$$165$$ 0 0
$$166$$ 2.33490 0.181223
$$167$$ 18.2227 1.41012 0.705059 0.709149i $$-0.250920\pi$$
0.705059 + 0.709149i $$0.250920\pi$$
$$168$$ −0.665102 −0.0513137
$$169$$ 1.47431 0.113408
$$170$$ 0 0
$$171$$ 8.13941 0.622436
$$172$$ 4.60433 0.351077
$$173$$ 1.72118 0.130859 0.0654294 0.997857i $$-0.479158\pi$$
0.0654294 + 0.997857i $$0.479158\pi$$
$$174$$ 0.0327344 0.00248159
$$175$$ 0 0
$$176$$ 3.83255 0.288889
$$177$$ 3.74843 0.281749
$$178$$ −1.65571 −0.124101
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0.725875 0.0539539 0.0269769 0.999636i $$-0.491412\pi$$
0.0269769 + 0.999636i $$0.491412\pi$$
$$182$$ −0.637062 −0.0472222
$$183$$ 6.00000 0.443533
$$184$$ −1.10746 −0.0816432
$$185$$ 0 0
$$186$$ −1.66510 −0.122091
$$187$$ −0.334898 −0.0244902
$$188$$ −23.9384 −1.74589
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 5.27412 0.381622 0.190811 0.981627i $$-0.438888\pi$$
0.190811 + 0.981627i $$0.438888\pi$$
$$192$$ −7.33490 −0.529351
$$193$$ 19.8318 1.42752 0.713761 0.700390i $$-0.246990\pi$$
0.713761 + 0.700390i $$0.246990\pi$$
$$194$$ −0.00939029 −0.000674184 0
$$195$$ 0 0
$$196$$ −1.97196 −0.140854
$$197$$ −2.66980 −0.190215 −0.0951076 0.995467i $$-0.530320\pi$$
−0.0951076 + 0.995467i $$0.530320\pi$$
$$198$$ 0.167449 0.0119001
$$199$$ 13.5529 0.960743 0.480371 0.877065i $$-0.340502\pi$$
0.480371 + 0.877065i $$0.340502\pi$$
$$200$$ 0 0
$$201$$ 0.139410 0.00983322
$$202$$ −3.16197 −0.222475
$$203$$ 0.195488 0.0137206
$$204$$ 0.660406 0.0462377
$$205$$ 0 0
$$206$$ 1.38628 0.0965868
$$207$$ 1.66510 0.115733
$$208$$ −14.5810 −1.01101
$$209$$ 8.13941 0.563015
$$210$$ 0 0
$$211$$ −4.27882 −0.294566 −0.147283 0.989094i $$-0.547053\pi$$
−0.147283 + 0.989094i $$0.547053\pi$$
$$212$$ 15.6651 1.07588
$$213$$ 4.66980 0.319969
$$214$$ 1.36294 0.0931685
$$215$$ 0 0
$$216$$ −0.665102 −0.0452544
$$217$$ −9.94392 −0.675037
$$218$$ 1.93453 0.131023
$$219$$ −4.19549 −0.283505
$$220$$ 0 0
$$221$$ 1.27412 0.0857069
$$222$$ 0.749219 0.0502843
$$223$$ 10.2694 0.687692 0.343846 0.939026i $$-0.388270\pi$$
0.343846 + 0.939026i $$0.388270\pi$$
$$224$$ 1.97196 0.131757
$$225$$ 0 0
$$226$$ 0.222741 0.0148165
$$227$$ 0.390977 0.0259500 0.0129750 0.999916i $$-0.495870\pi$$
0.0129750 + 0.999916i $$0.495870\pi$$
$$228$$ −16.0506 −1.06298
$$229$$ 7.94392 0.524949 0.262475 0.964939i $$-0.415461\pi$$
0.262475 + 0.964939i $$0.415461\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ −0.130020 −0.00853621
$$233$$ −26.5576 −1.73985 −0.869924 0.493185i $$-0.835833\pi$$
−0.869924 + 0.493185i $$0.835833\pi$$
$$234$$ −0.637062 −0.0416461
$$235$$ 0 0
$$236$$ −7.39176 −0.481163
$$237$$ 3.33020 0.216320
$$238$$ −0.0560785 −0.00363503
$$239$$ 10.7998 0.698582 0.349291 0.937014i $$-0.386422\pi$$
0.349291 + 0.937014i $$0.386422\pi$$
$$240$$ 0 0
$$241$$ −12.0833 −0.778356 −0.389178 0.921163i $$-0.627241\pi$$
−0.389178 + 0.921163i $$0.627241\pi$$
$$242$$ 0.167449 0.0107640
$$243$$ 1.00000 0.0641500
$$244$$ −11.8318 −0.757451
$$245$$ 0 0
$$246$$ −1.05138 −0.0670338
$$247$$ −30.9665 −1.97035
$$248$$ 6.61372 0.419972
$$249$$ 13.9439 0.883660
$$250$$ 0 0
$$251$$ 4.80921 0.303554 0.151777 0.988415i $$-0.451500\pi$$
0.151777 + 0.988415i $$0.451500\pi$$
$$252$$ −1.97196 −0.124222
$$253$$ 1.66510 0.104684
$$254$$ −2.44706 −0.153542
$$255$$ 0 0
$$256$$ 13.8037 0.862733
$$257$$ 16.7531 1.04503 0.522516 0.852630i $$-0.324994\pi$$
0.522516 + 0.852630i $$0.324994\pi$$
$$258$$ −0.390977 −0.0243412
$$259$$ 4.47431 0.278020
$$260$$ 0 0
$$261$$ 0.195488 0.0121004
$$262$$ 3.44236 0.212670
$$263$$ 12.1394 0.748548 0.374274 0.927318i $$-0.377892\pi$$
0.374274 + 0.927318i $$0.377892\pi$$
$$264$$ −0.665102 −0.0409342
$$265$$ 0 0
$$266$$ 1.36294 0.0835671
$$267$$ −9.88784 −0.605126
$$268$$ −0.274911 −0.0167929
$$269$$ −25.2180 −1.53757 −0.768786 0.639506i $$-0.779139\pi$$
−0.768786 + 0.639506i $$0.779139\pi$$
$$270$$ 0 0
$$271$$ 4.13941 0.251451 0.125726 0.992065i $$-0.459874\pi$$
0.125726 + 0.992065i $$0.459874\pi$$
$$272$$ −1.28352 −0.0778245
$$273$$ −3.80451 −0.230260
$$274$$ 2.71648 0.164109
$$275$$ 0 0
$$276$$ −3.28352 −0.197644
$$277$$ 18.2788 1.09827 0.549134 0.835734i $$-0.314958\pi$$
0.549134 + 0.835734i $$0.314958\pi$$
$$278$$ 3.77569 0.226451
$$279$$ −9.94392 −0.595327
$$280$$ 0 0
$$281$$ 6.74374 0.402298 0.201149 0.979561i $$-0.435533\pi$$
0.201149 + 0.979561i $$0.435533\pi$$
$$282$$ 2.03273 0.121048
$$283$$ −23.3575 −1.38846 −0.694228 0.719755i $$-0.744254\pi$$
−0.694228 + 0.719755i $$0.744254\pi$$
$$284$$ −9.20866 −0.546433
$$285$$ 0 0
$$286$$ −0.637062 −0.0376703
$$287$$ −6.27882 −0.370627
$$288$$ 1.97196 0.116199
$$289$$ −16.8878 −0.993403
$$290$$ 0 0
$$291$$ −0.0560785 −0.00328738
$$292$$ 8.27334 0.484161
$$293$$ 14.1667 0.827625 0.413813 0.910362i $$-0.364197\pi$$
0.413813 + 0.910362i $$0.364197\pi$$
$$294$$ 0.167449 0.00976584
$$295$$ 0 0
$$296$$ −2.97587 −0.172969
$$297$$ 1.00000 0.0580259
$$298$$ 1.37233 0.0794968
$$299$$ −6.33490 −0.366357
$$300$$ 0 0
$$301$$ −2.33490 −0.134581
$$302$$ −2.22274 −0.127904
$$303$$ −18.8831 −1.08481
$$304$$ 31.1947 1.78914
$$305$$ 0 0
$$306$$ −0.0560785 −0.00320579
$$307$$ 12.5576 0.716702 0.358351 0.933587i $$-0.383339\pi$$
0.358351 + 0.933587i $$0.383339\pi$$
$$308$$ −1.97196 −0.112363
$$309$$ 8.27882 0.470966
$$310$$ 0 0
$$311$$ 9.33959 0.529600 0.264800 0.964303i $$-0.414694\pi$$
0.264800 + 0.964303i $$0.414694\pi$$
$$312$$ 2.53039 0.143255
$$313$$ 2.99530 0.169305 0.0846523 0.996411i $$-0.473022\pi$$
0.0846523 + 0.996411i $$0.473022\pi$$
$$314$$ 2.83646 0.160071
$$315$$ 0 0
$$316$$ −6.56703 −0.369424
$$317$$ −9.93453 −0.557979 −0.278989 0.960294i $$-0.589999\pi$$
−0.278989 + 0.960294i $$0.589999\pi$$
$$318$$ −1.33020 −0.0745941
$$319$$ 0.195488 0.0109453
$$320$$ 0 0
$$321$$ 8.13941 0.454298
$$322$$ 0.278820 0.0155380
$$323$$ −2.72588 −0.151672
$$324$$ −1.97196 −0.109553
$$325$$ 0 0
$$326$$ 1.13862 0.0630625
$$327$$ 11.5529 0.638879
$$328$$ 4.17605 0.230584
$$329$$ 12.1394 0.669267
$$330$$ 0 0
$$331$$ 22.5482 1.23936 0.619682 0.784853i $$-0.287262\pi$$
0.619682 + 0.784853i $$0.287262\pi$$
$$332$$ −27.4969 −1.50909
$$333$$ 4.47431 0.245191
$$334$$ 3.05138 0.166964
$$335$$ 0 0
$$336$$ 3.83255 0.209083
$$337$$ −13.2835 −0.723599 −0.361800 0.932256i $$-0.617838\pi$$
−0.361800 + 0.932256i $$0.617838\pi$$
$$338$$ 0.246872 0.0134281
$$339$$ 1.33020 0.0722467
$$340$$ 0 0
$$341$$ −9.94392 −0.538494
$$342$$ 1.36294 0.0736992
$$343$$ 1.00000 0.0539949
$$344$$ 1.55294 0.0837292
$$345$$ 0 0
$$346$$ 0.288210 0.0154943
$$347$$ 27.7757 1.49108 0.745538 0.666463i $$-0.232192\pi$$
0.745538 + 0.666463i $$0.232192\pi$$
$$348$$ −0.385496 −0.0206647
$$349$$ −2.85589 −0.152873 −0.0764363 0.997074i $$-0.524354\pi$$
−0.0764363 + 0.997074i $$0.524354\pi$$
$$350$$ 0 0
$$351$$ −3.80451 −0.203070
$$352$$ 1.97196 0.105106
$$353$$ −24.6410 −1.31151 −0.655753 0.754975i $$-0.727649\pi$$
−0.655753 + 0.754975i $$0.727649\pi$$
$$354$$ 0.627672 0.0333604
$$355$$ 0 0
$$356$$ 19.4984 1.03342
$$357$$ −0.334898 −0.0177247
$$358$$ −2.00939 −0.106200
$$359$$ −11.8878 −0.627416 −0.313708 0.949519i $$-0.601571\pi$$
−0.313708 + 0.949519i $$0.601571\pi$$
$$360$$ 0 0
$$361$$ 47.2500 2.48684
$$362$$ 0.121547 0.00638838
$$363$$ 1.00000 0.0524864
$$364$$ 7.50235 0.393230
$$365$$ 0 0
$$366$$ 1.00470 0.0525163
$$367$$ −3.60902 −0.188389 −0.0941947 0.995554i $$-0.530028\pi$$
−0.0941947 + 0.995554i $$0.530028\pi$$
$$368$$ 6.38159 0.332663
$$369$$ −6.27882 −0.326862
$$370$$ 0 0
$$371$$ −7.94392 −0.412428
$$372$$ 19.6090 1.01668
$$373$$ −12.3349 −0.638677 −0.319338 0.947641i $$-0.603461\pi$$
−0.319338 + 0.947641i $$0.603461\pi$$
$$374$$ −0.0560785 −0.00289975
$$375$$ 0 0
$$376$$ −8.07394 −0.416382
$$377$$ −0.743738 −0.0383045
$$378$$ 0.167449 0.00861266
$$379$$ 23.3575 1.19979 0.599896 0.800078i $$-0.295209\pi$$
0.599896 + 0.800078i $$0.295209\pi$$
$$380$$ 0 0
$$381$$ −14.6137 −0.748683
$$382$$ 0.883148 0.0451858
$$383$$ 15.2180 0.777606 0.388803 0.921321i $$-0.372889\pi$$
0.388803 + 0.921321i $$0.372889\pi$$
$$384$$ −5.17214 −0.263940
$$385$$ 0 0
$$386$$ 3.32081 0.169025
$$387$$ −2.33490 −0.118690
$$388$$ 0.110585 0.00561408
$$389$$ 3.73057 0.189147 0.0945737 0.995518i $$-0.469851\pi$$
0.0945737 + 0.995518i $$0.469851\pi$$
$$390$$ 0 0
$$391$$ −0.557640 −0.0282011
$$392$$ −0.665102 −0.0335927
$$393$$ 20.5576 1.03700
$$394$$ −0.447055 −0.0225223
$$395$$ 0 0
$$396$$ −1.97196 −0.0990948
$$397$$ −20.8925 −1.04857 −0.524283 0.851544i $$-0.675667\pi$$
−0.524283 + 0.851544i $$0.675667\pi$$
$$398$$ 2.26943 0.113756
$$399$$ 8.13941 0.407480
$$400$$ 0 0
$$401$$ 23.2741 1.16225 0.581127 0.813813i $$-0.302612\pi$$
0.581127 + 0.813813i $$0.302612\pi$$
$$402$$ 0.0233441 0.00116430
$$403$$ 37.8318 1.88453
$$404$$ 37.2368 1.85260
$$405$$ 0 0
$$406$$ 0.0327344 0.00162458
$$407$$ 4.47431 0.221783
$$408$$ 0.222741 0.0110273
$$409$$ −23.8972 −1.18164 −0.590821 0.806803i $$-0.701196\pi$$
−0.590821 + 0.806803i $$0.701196\pi$$
$$410$$ 0 0
$$411$$ 16.2227 0.800209
$$412$$ −16.3255 −0.804300
$$413$$ 3.74843 0.184448
$$414$$ 0.278820 0.0137033
$$415$$ 0 0
$$416$$ −7.50235 −0.367833
$$417$$ 22.5482 1.10419
$$418$$ 1.36294 0.0666635
$$419$$ −30.2967 −1.48009 −0.740045 0.672557i $$-0.765196\pi$$
−0.740045 + 0.672557i $$0.765196\pi$$
$$420$$ 0 0
$$421$$ −15.5257 −0.756676 −0.378338 0.925668i $$-0.623504\pi$$
−0.378338 + 0.925668i $$0.623504\pi$$
$$422$$ −0.716485 −0.0348779
$$423$$ 12.1394 0.590238
$$424$$ 5.28352 0.256590
$$425$$ 0 0
$$426$$ 0.781954 0.0378858
$$427$$ 6.00000 0.290360
$$428$$ −16.0506 −0.775835
$$429$$ −3.80451 −0.183684
$$430$$ 0 0
$$431$$ 3.07864 0.148293 0.0741463 0.997247i $$-0.476377\pi$$
0.0741463 + 0.997247i $$0.476377\pi$$
$$432$$ 3.83255 0.184394
$$433$$ −16.9392 −0.814047 −0.407024 0.913418i $$-0.633433\pi$$
−0.407024 + 0.913418i $$0.633433\pi$$
$$434$$ −1.66510 −0.0799274
$$435$$ 0 0
$$436$$ −22.7820 −1.09106
$$437$$ 13.5529 0.648325
$$438$$ −0.702531 −0.0335682
$$439$$ −11.0786 −0.528754 −0.264377 0.964419i $$-0.585166\pi$$
−0.264377 + 0.964419i $$0.585166\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0.213351 0.0101481
$$443$$ 14.5482 0.691208 0.345604 0.938380i $$-0.387674\pi$$
0.345604 + 0.938380i $$0.387674\pi$$
$$444$$ −8.82316 −0.418729
$$445$$ 0 0
$$446$$ 1.71961 0.0814258
$$447$$ 8.19549 0.387633
$$448$$ −7.33490 −0.346541
$$449$$ 27.4875 1.29721 0.648607 0.761123i $$-0.275352\pi$$
0.648607 + 0.761123i $$0.275352\pi$$
$$450$$ 0 0
$$451$$ −6.27882 −0.295658
$$452$$ −2.62311 −0.123381
$$453$$ −13.2741 −0.623673
$$454$$ 0.0654688 0.00307260
$$455$$ 0 0
$$456$$ −5.41353 −0.253512
$$457$$ 7.94392 0.371601 0.185800 0.982587i $$-0.440512\pi$$
0.185800 + 0.982587i $$0.440512\pi$$
$$458$$ 1.33020 0.0621563
$$459$$ −0.334898 −0.0156317
$$460$$ 0 0
$$461$$ 8.26943 0.385146 0.192573 0.981283i $$-0.438317\pi$$
0.192573 + 0.981283i $$0.438317\pi$$
$$462$$ 0.167449 0.00779044
$$463$$ −9.73904 −0.452612 −0.226306 0.974056i $$-0.572665\pi$$
−0.226306 + 0.974056i $$0.572665\pi$$
$$464$$ 0.749219 0.0347816
$$465$$ 0 0
$$466$$ −4.44706 −0.206006
$$467$$ 40.6970 1.88323 0.941617 0.336685i $$-0.109306\pi$$
0.941617 + 0.336685i $$0.109306\pi$$
$$468$$ 7.50235 0.346796
$$469$$ 0.139410 0.00643735
$$470$$ 0 0
$$471$$ 16.9392 0.780518
$$472$$ −2.49309 −0.114754
$$473$$ −2.33490 −0.107359
$$474$$ 0.557640 0.0256132
$$475$$ 0 0
$$476$$ 0.660406 0.0302697
$$477$$ −7.94392 −0.363727
$$478$$ 1.80842 0.0827152
$$479$$ 37.8318 1.72858 0.864289 0.502996i $$-0.167769\pi$$
0.864289 + 0.502996i $$0.167769\pi$$
$$480$$ 0 0
$$481$$ −17.0226 −0.776162
$$482$$ −2.02334 −0.0921608
$$483$$ 1.66510 0.0757647
$$484$$ −1.97196 −0.0896346
$$485$$ 0 0
$$486$$ 0.167449 0.00759565
$$487$$ −40.5576 −1.83784 −0.918921 0.394442i $$-0.870938\pi$$
−0.918921 + 0.394442i $$0.870938\pi$$
$$488$$ −3.99061 −0.180646
$$489$$ 6.79982 0.307498
$$490$$ 0 0
$$491$$ 31.0786 1.40256 0.701280 0.712886i $$-0.252612\pi$$
0.701280 + 0.712886i $$0.252612\pi$$
$$492$$ 12.3816 0.558205
$$493$$ −0.0654688 −0.00294856
$$494$$ −5.18531 −0.233298
$$495$$ 0 0
$$496$$ −38.1106 −1.71122
$$497$$ 4.66980 0.209469
$$498$$ 2.33490 0.104629
$$499$$ 15.3575 0.687494 0.343747 0.939062i $$-0.388304\pi$$
0.343747 + 0.939062i $$0.388304\pi$$
$$500$$ 0 0
$$501$$ 18.2227 0.814132
$$502$$ 0.805298 0.0359422
$$503$$ −14.8925 −0.664025 −0.332013 0.943275i $$-0.607728\pi$$
−0.332013 + 0.943275i $$0.607728\pi$$
$$504$$ −0.665102 −0.0296260
$$505$$ 0 0
$$506$$ 0.278820 0.0123951
$$507$$ 1.47431 0.0654763
$$508$$ 28.8177 1.27858
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −4.19549 −0.185597
$$512$$ 12.6557 0.559309
$$513$$ 8.13941 0.359364
$$514$$ 2.80530 0.123736
$$515$$ 0 0
$$516$$ 4.60433 0.202694
$$517$$ 12.1394 0.533891
$$518$$ 0.749219 0.0329188
$$519$$ 1.72118 0.0755514
$$520$$ 0 0
$$521$$ 27.6924 1.21322 0.606612 0.794998i $$-0.292528\pi$$
0.606612 + 0.794998i $$0.292528\pi$$
$$522$$ 0.0327344 0.00143274
$$523$$ −18.4088 −0.804962 −0.402481 0.915428i $$-0.631852\pi$$
−0.402481 + 0.915428i $$0.631852\pi$$
$$524$$ −40.5389 −1.77095
$$525$$ 0 0
$$526$$ 2.03273 0.0886314
$$527$$ 3.33020 0.145066
$$528$$ 3.83255 0.166790
$$529$$ −20.2274 −0.879454
$$530$$ 0 0
$$531$$ 3.74843 0.162668
$$532$$ −16.0506 −0.695882
$$533$$ 23.8878 1.03470
$$534$$ −1.65571 −0.0716496
$$535$$ 0 0
$$536$$ −0.0927218 −0.00400497
$$537$$ −12.0000 −0.517838
$$538$$ −4.22274 −0.182055
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −4.26943 −0.183557 −0.0917786 0.995779i $$-0.529255\pi$$
−0.0917786 + 0.995779i $$0.529255\pi$$
$$542$$ 0.693141 0.0297729
$$543$$ 0.725875 0.0311503
$$544$$ −0.660406 −0.0283147
$$545$$ 0 0
$$546$$ −0.637062 −0.0272638
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −31.9906 −1.36657
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 1.59116 0.0677857
$$552$$ −1.10746 −0.0471367
$$553$$ 3.33020 0.141615
$$554$$ 3.06077 0.130040
$$555$$ 0 0
$$556$$ −44.4643 −1.88570
$$557$$ 12.4743 0.528553 0.264277 0.964447i $$-0.414867\pi$$
0.264277 + 0.964447i $$0.414867\pi$$
$$558$$ −1.66510 −0.0704894
$$559$$ 8.88315 0.375717
$$560$$ 0 0
$$561$$ −0.334898 −0.0141394
$$562$$ 1.12923 0.0476338
$$563$$ 32.7710 1.38113 0.690566 0.723269i $$-0.257361\pi$$
0.690566 + 0.723269i $$0.257361\pi$$
$$564$$ −23.9384 −1.00799
$$565$$ 0 0
$$566$$ −3.91119 −0.164399
$$567$$ 1.00000 0.0419961
$$568$$ −3.10589 −0.130320
$$569$$ −29.2180 −1.22488 −0.612442 0.790515i $$-0.709813\pi$$
−0.612442 + 0.790515i $$0.709813\pi$$
$$570$$ 0 0
$$571$$ 32.4455 1.35780 0.678901 0.734230i $$-0.262457\pi$$
0.678901 + 0.734230i $$0.262457\pi$$
$$572$$ 7.50235 0.313689
$$573$$ 5.27412 0.220330
$$574$$ −1.05138 −0.0438839
$$575$$ 0 0
$$576$$ −7.33490 −0.305621
$$577$$ 14.8831 0.619594 0.309797 0.950803i $$-0.399739\pi$$
0.309797 + 0.950803i $$0.399739\pi$$
$$578$$ −2.82786 −0.117623
$$579$$ 19.8318 0.824180
$$580$$ 0 0
$$581$$ 13.9439 0.578491
$$582$$ −0.00939029 −0.000389240 0
$$583$$ −7.94392 −0.329004
$$584$$ 2.79043 0.115469
$$585$$ 0 0
$$586$$ 2.37220 0.0979945
$$587$$ 4.53039 0.186989 0.0934945 0.995620i $$-0.470196\pi$$
0.0934945 + 0.995620i $$0.470196\pi$$
$$588$$ −1.97196 −0.0813223
$$589$$ −80.9377 −3.33498
$$590$$ 0 0
$$591$$ −2.66980 −0.109821
$$592$$ 17.1480 0.704779
$$593$$ −8.28821 −0.340356 −0.170178 0.985413i $$-0.554434\pi$$
−0.170178 + 0.985413i $$0.554434\pi$$
$$594$$ 0.167449 0.00687052
$$595$$ 0 0
$$596$$ −16.1612 −0.661988
$$597$$ 13.5529 0.554685
$$598$$ −1.06077 −0.0433783
$$599$$ 1.99061 0.0813341 0.0406671 0.999173i $$-0.487052\pi$$
0.0406671 + 0.999173i $$0.487052\pi$$
$$600$$ 0 0
$$601$$ −8.47431 −0.345674 −0.172837 0.984950i $$-0.555293\pi$$
−0.172837 + 0.984950i $$0.555293\pi$$
$$602$$ −0.390977 −0.0159350
$$603$$ 0.139410 0.00567721
$$604$$ 26.1761 1.06509
$$605$$ 0 0
$$606$$ −3.16197 −0.128446
$$607$$ 22.9665 0.932181 0.466090 0.884737i $$-0.345662\pi$$
0.466090 + 0.884737i $$0.345662\pi$$
$$608$$ 16.0506 0.650938
$$609$$ 0.195488 0.00792159
$$610$$ 0 0
$$611$$ −46.1845 −1.86843
$$612$$ 0.660406 0.0266953
$$613$$ −3.55294 −0.143502 −0.0717510 0.997423i $$-0.522859\pi$$
−0.0717510 + 0.997423i $$0.522859\pi$$
$$614$$ 2.10277 0.0848608
$$615$$ 0 0
$$616$$ −0.665102 −0.0267977
$$617$$ −22.6698 −0.912652 −0.456326 0.889813i $$-0.650835\pi$$
−0.456326 + 0.889813i $$0.650835\pi$$
$$618$$ 1.38628 0.0557644
$$619$$ 8.71648 0.350345 0.175173 0.984538i $$-0.443952\pi$$
0.175173 + 0.984538i $$0.443952\pi$$
$$620$$ 0 0
$$621$$ 1.66510 0.0668182
$$622$$ 1.56391 0.0627070
$$623$$ −9.88784 −0.396148
$$624$$ −14.5810 −0.583707
$$625$$ 0 0
$$626$$ 0.501561 0.0200464
$$627$$ 8.13941 0.325057
$$628$$ −33.4035 −1.33294
$$629$$ −1.49844 −0.0597467
$$630$$ 0 0
$$631$$ −11.8878 −0.473248 −0.236624 0.971601i $$-0.576041\pi$$
−0.236624 + 0.971601i $$0.576041\pi$$
$$632$$ −2.21492 −0.0881049
$$633$$ −4.27882 −0.170068
$$634$$ −1.66353 −0.0660672
$$635$$ 0 0
$$636$$ 15.6651 0.621162
$$637$$ −3.80451 −0.150740
$$638$$ 0.0327344 0.00129597
$$639$$ 4.66980 0.184734
$$640$$ 0 0
$$641$$ −4.89254 −0.193244 −0.0966218 0.995321i $$-0.530804\pi$$
−0.0966218 + 0.995321i $$0.530804\pi$$
$$642$$ 1.36294 0.0537909
$$643$$ −22.7804 −0.898371 −0.449185 0.893439i $$-0.648286\pi$$
−0.449185 + 0.893439i $$0.648286\pi$$
$$644$$ −3.28352 −0.129389
$$645$$ 0 0
$$646$$ −0.456446 −0.0179586
$$647$$ −31.2453 −1.22838 −0.614190 0.789158i $$-0.710517\pi$$
−0.614190 + 0.789158i $$0.710517\pi$$
$$648$$ −0.665102 −0.0261277
$$649$$ 3.74843 0.147139
$$650$$ 0 0
$$651$$ −9.94392 −0.389733
$$652$$ −13.4090 −0.525136
$$653$$ −28.2882 −1.10700 −0.553502 0.832848i $$-0.686709\pi$$
−0.553502 + 0.832848i $$0.686709\pi$$
$$654$$ 1.93453 0.0756462
$$655$$ 0 0
$$656$$ −24.0639 −0.939537
$$657$$ −4.19549 −0.163682
$$658$$ 2.03273 0.0792442
$$659$$ 30.2967 1.18019 0.590096 0.807333i $$-0.299090\pi$$
0.590096 + 0.807333i $$0.299090\pi$$
$$660$$ 0 0
$$661$$ −35.7196 −1.38933 −0.694666 0.719333i $$-0.744448\pi$$
−0.694666 + 0.719333i $$0.744448\pi$$
$$662$$ 3.77569 0.146746
$$663$$ 1.27412 0.0494829
$$664$$ −9.27412 −0.359906
$$665$$ 0 0
$$666$$ 0.749219 0.0290317
$$667$$ 0.325508 0.0126037
$$668$$ −35.9345 −1.39035
$$669$$ 10.2694 0.397039
$$670$$ 0 0
$$671$$ 6.00000 0.231627
$$672$$ 1.97196 0.0760700
$$673$$ −34.9377 −1.34675 −0.673374 0.739302i $$-0.735156\pi$$
−0.673374 + 0.739302i $$0.735156\pi$$
$$674$$ −2.22431 −0.0856774
$$675$$ 0 0
$$676$$ −2.90728 −0.111818
$$677$$ −20.0094 −0.769023 −0.384512 0.923120i $$-0.625630\pi$$
−0.384512 + 0.923120i $$0.625630\pi$$
$$678$$ 0.222741 0.00855433
$$679$$ −0.0560785 −0.00215209
$$680$$ 0 0
$$681$$ 0.390977 0.0149823
$$682$$ −1.66510 −0.0637600
$$683$$ −25.0498 −0.958504 −0.479252 0.877677i $$-0.659092\pi$$
−0.479252 + 0.877677i $$0.659092\pi$$
$$684$$ −16.0506 −0.613710
$$685$$ 0 0
$$686$$ 0.167449 0.00639324
$$687$$ 7.94392 0.303080
$$688$$ −8.94862 −0.341163
$$689$$ 30.2227 1.15139
$$690$$ 0 0
$$691$$ −38.8271 −1.47705 −0.738526 0.674225i $$-0.764478\pi$$
−0.738526 + 0.674225i $$0.764478\pi$$
$$692$$ −3.39410 −0.129024
$$693$$ 1.00000 0.0379869
$$694$$ 4.65102 0.176550
$$695$$ 0 0
$$696$$ −0.130020 −0.00492838
$$697$$ 2.10277 0.0796480
$$698$$ −0.478217 −0.0181008
$$699$$ −26.5576 −1.00450
$$700$$ 0 0
$$701$$ −41.7757 −1.57785 −0.788923 0.614492i $$-0.789361\pi$$
−0.788923 + 0.614492i $$0.789361\pi$$
$$702$$ −0.637062 −0.0240444
$$703$$ 36.4182 1.37354
$$704$$ −7.33490 −0.276444
$$705$$ 0 0
$$706$$ −4.12611 −0.155288
$$707$$ −18.8831 −0.710174
$$708$$ −7.39176 −0.277799
$$709$$ −13.4041 −0.503403 −0.251702 0.967805i $$-0.580990\pi$$
−0.251702 + 0.967805i $$0.580990\pi$$
$$710$$ 0 0
$$711$$ 3.33020 0.124892
$$712$$ 6.57642 0.246462
$$713$$ −16.5576 −0.620088
$$714$$ −0.0560785 −0.00209868
$$715$$ 0 0
$$716$$ 23.6635 0.884348
$$717$$ 10.7998 0.403327
$$718$$ −1.99061 −0.0742889
$$719$$ 5.47900 0.204332 0.102166 0.994767i $$-0.467423\pi$$
0.102166 + 0.994767i $$0.467423\pi$$
$$720$$ 0 0
$$721$$ 8.27882 0.308319
$$722$$ 7.91197 0.294453
$$723$$ −12.0833 −0.449384
$$724$$ −1.43140 −0.0531975
$$725$$ 0 0
$$726$$ 0.167449 0.00621462
$$727$$ 32.8831 1.21957 0.609784 0.792567i $$-0.291256\pi$$
0.609784 + 0.792567i $$0.291256\pi$$
$$728$$ 2.53039 0.0937824
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0.781954 0.0289216
$$732$$ −11.8318 −0.437315
$$733$$ 35.8972 1.32589 0.662947 0.748666i $$-0.269305\pi$$
0.662947 + 0.748666i $$0.269305\pi$$
$$734$$ −0.604328 −0.0223062
$$735$$ 0 0
$$736$$ 3.28352 0.121032
$$737$$ 0.139410 0.00513523
$$738$$ −1.05138 −0.0387020
$$739$$ 29.6651 1.09125 0.545624 0.838030i $$-0.316293\pi$$
0.545624 + 0.838030i $$0.316293\pi$$
$$740$$ 0 0
$$741$$ −30.9665 −1.13758
$$742$$ −1.33020 −0.0488333
$$743$$ 18.7998 0.689698 0.344849 0.938658i $$-0.387930\pi$$
0.344849 + 0.938658i $$0.387930\pi$$
$$744$$ 6.61372 0.242471
$$745$$ 0 0
$$746$$ −2.06547 −0.0756222
$$747$$ 13.9439 0.510181
$$748$$ 0.660406 0.0241469
$$749$$ 8.13941 0.297408
$$750$$ 0 0
$$751$$ −9.59116 −0.349986 −0.174993 0.984570i $$-0.555990\pi$$
−0.174993 + 0.984570i $$0.555990\pi$$
$$752$$ 46.5249 1.69659
$$753$$ 4.80921 0.175257
$$754$$ −0.124538 −0.00453542
$$755$$ 0 0
$$756$$ −1.97196 −0.0717195
$$757$$ −2.74374 −0.0997229 −0.0498614 0.998756i $$-0.515878\pi$$
−0.0498614 + 0.998756i $$0.515878\pi$$
$$758$$ 3.91119 0.142061
$$759$$ 1.66510 0.0604394
$$760$$ 0 0
$$761$$ 6.39098 0.231673 0.115836 0.993268i $$-0.463045\pi$$
0.115836 + 0.993268i $$0.463045\pi$$
$$762$$ −2.44706 −0.0886475
$$763$$ 11.5529 0.418245
$$764$$ −10.4004 −0.376272
$$765$$ 0 0
$$766$$ 2.54825 0.0920720
$$767$$ −14.2610 −0.514933
$$768$$ 13.8037 0.498099
$$769$$ 17.7951 0.641708 0.320854 0.947129i $$-0.396030\pi$$
0.320854 + 0.947129i $$0.396030\pi$$
$$770$$ 0 0
$$771$$ 16.7531 0.603349
$$772$$ −39.1075 −1.40751
$$773$$ 31.5257 1.13390 0.566950 0.823752i $$-0.308123\pi$$
0.566950 + 0.823752i $$0.308123\pi$$
$$774$$ −0.390977 −0.0140534
$$775$$ 0 0
$$776$$ 0.0372979 0.00133892
$$777$$ 4.47431 0.160515
$$778$$ 0.624681 0.0223959
$$779$$ −51.1059 −1.83106
$$780$$ 0 0
$$781$$ 4.66980 0.167098
$$782$$ −0.0933763 −0.00333913
$$783$$ 0.195488 0.00698619
$$784$$ 3.83255 0.136877
$$785$$ 0 0
$$786$$ 3.44236 0.122785
$$787$$ −31.8606 −1.13571 −0.567854 0.823130i $$-0.692226\pi$$
−0.567854 + 0.823130i $$0.692226\pi$$
$$788$$ 5.26473 0.187548
$$789$$ 12.1394 0.432174
$$790$$ 0 0
$$791$$ 1.33020 0.0472966
$$792$$ −0.665102 −0.0236334
$$793$$ −22.8271 −0.810613
$$794$$ −3.49844 −0.124155
$$795$$ 0 0
$$796$$ −26.7259 −0.947274
$$797$$ −43.8590 −1.55357 −0.776783 0.629768i $$-0.783150\pi$$
−0.776783 + 0.629768i $$0.783150\pi$$
$$798$$ 1.36294 0.0482475
$$799$$ −4.06547 −0.143826
$$800$$ 0 0
$$801$$ −9.88784 −0.349370
$$802$$ 3.89723 0.137616
$$803$$ −4.19549 −0.148056
$$804$$ −0.274911 −0.00969536
$$805$$ 0 0
$$806$$ 6.33490 0.223137
$$807$$ −25.2180 −0.887717
$$808$$ 12.5592 0.441832
$$809$$ 6.74374 0.237097 0.118549 0.992948i $$-0.462176\pi$$
0.118549 + 0.992948i $$0.462176\pi$$
$$810$$ 0 0
$$811$$ 3.97275 0.139502 0.0697510 0.997564i $$-0.477780\pi$$
0.0697510 + 0.997564i $$0.477780\pi$$
$$812$$ −0.385496 −0.0135282
$$813$$ 4.13941 0.145175
$$814$$ 0.749219 0.0262601
$$815$$ 0 0
$$816$$ −1.28352 −0.0449320
$$817$$ −19.0047 −0.664890
$$818$$ −4.00157 −0.139912
$$819$$ −3.80451 −0.132940
$$820$$ 0 0
$$821$$ 49.5896 1.73069 0.865344 0.501178i $$-0.167100\pi$$
0.865344 + 0.501178i $$0.167100\pi$$
$$822$$ 2.71648 0.0947483
$$823$$ −23.1908 −0.808380 −0.404190 0.914675i $$-0.632447\pi$$
−0.404190 + 0.914675i $$0.632447\pi$$
$$824$$ −5.50626 −0.191820
$$825$$ 0 0
$$826$$ 0.627672 0.0218395
$$827$$ −27.7484 −0.964908 −0.482454 0.875921i $$-0.660254\pi$$
−0.482454 + 0.875921i $$0.660254\pi$$
$$828$$ −3.28352 −0.114110
$$829$$ 0.269430 0.00935768 0.00467884 0.999989i $$-0.498511\pi$$
0.00467884 + 0.999989i $$0.498511\pi$$
$$830$$ 0 0
$$831$$ 18.2788 0.634085
$$832$$ 27.9057 0.967456
$$833$$ −0.334898 −0.0116035
$$834$$ 3.77569 0.130741
$$835$$ 0 0
$$836$$ −16.0506 −0.555122
$$837$$ −9.94392 −0.343712
$$838$$ −5.07316 −0.175249
$$839$$ −27.3575 −0.944484 −0.472242 0.881469i $$-0.656555\pi$$
−0.472242 + 0.881469i $$0.656555\pi$$
$$840$$ 0 0
$$841$$ −28.9618 −0.998682
$$842$$ −2.59976 −0.0895938
$$843$$ 6.74374 0.232267
$$844$$ 8.43767 0.290436
$$845$$ 0 0
$$846$$ 2.03273 0.0698868
$$847$$ 1.00000 0.0343604
$$848$$ −30.4455 −1.04550
$$849$$ −23.3575 −0.801626
$$850$$ 0 0
$$851$$ 7.45018 0.255389
$$852$$ −9.20866 −0.315483
$$853$$ 28.5482 0.977473 0.488737 0.872431i $$-0.337458\pi$$
0.488737 + 0.872431i $$0.337458\pi$$
$$854$$ 1.00470 0.0343800
$$855$$ 0 0
$$856$$ −5.41353 −0.185031
$$857$$ 39.8972 1.36286 0.681432 0.731882i $$-0.261358\pi$$
0.681432 + 0.731882i $$0.261358\pi$$
$$858$$ −0.637062 −0.0217490
$$859$$ 20.5576 0.701418 0.350709 0.936485i $$-0.385941\pi$$
0.350709 + 0.936485i $$0.385941\pi$$
$$860$$ 0 0
$$861$$ −6.27882 −0.213982
$$862$$ 0.515515 0.0175585
$$863$$ 9.66510 0.329004 0.164502 0.986377i $$-0.447398\pi$$
0.164502 + 0.986377i $$0.447398\pi$$
$$864$$ 1.97196 0.0670875
$$865$$ 0 0
$$866$$ −2.83646 −0.0963868
$$867$$ −16.8878 −0.573541
$$868$$ 19.6090 0.665574
$$869$$ 3.33020 0.112969
$$870$$ 0 0
$$871$$ −0.530387 −0.0179715
$$872$$ −7.68388 −0.260209
$$873$$ −0.0560785 −0.00189797
$$874$$ 2.26943 0.0767646
$$875$$ 0 0
$$876$$ 8.27334 0.279530
$$877$$ 25.7212 0.868543 0.434271 0.900782i $$-0.357006\pi$$
0.434271 + 0.900782i $$0.357006\pi$$
$$878$$ −1.85511 −0.0626069
$$879$$ 14.1667 0.477830
$$880$$ 0 0
$$881$$ −14.6316 −0.492950 −0.246475 0.969149i $$-0.579272\pi$$
−0.246475 + 0.969149i $$0.579272\pi$$
$$882$$ 0.167449 0.00563831
$$883$$ −18.6877 −0.628890 −0.314445 0.949276i $$-0.601818\pi$$
−0.314445 + 0.949276i $$0.601818\pi$$
$$884$$ −2.51252 −0.0845053
$$885$$ 0 0
$$886$$ 2.43609 0.0818421
$$887$$ −38.6137 −1.29652 −0.648261 0.761418i $$-0.724503\pi$$
−0.648261 + 0.761418i $$0.724503\pi$$
$$888$$ −2.97587 −0.0998636
$$889$$ −14.6137 −0.490128
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −20.2509 −0.678051
$$893$$ 98.8076 3.30647
$$894$$ 1.37233 0.0458975
$$895$$ 0 0
$$896$$ −5.17214 −0.172789
$$897$$ −6.33490 −0.211516
$$898$$ 4.60276 0.153596
$$899$$ −1.94392 −0.0648334
$$900$$ 0 0
$$901$$ 2.66041 0.0886310
$$902$$ −1.05138 −0.0350072
$$903$$ −2.33490 −0.0777006
$$904$$ −0.884720 −0.0294254
$$905$$ 0 0
$$906$$ −2.22274 −0.0738456
$$907$$ −49.0965 −1.63022 −0.815111 0.579304i $$-0.803324\pi$$
−0.815111 + 0.579304i $$0.803324\pi$$
$$908$$ −0.770991 −0.0255862
$$909$$ −18.8831 −0.626314
$$910$$ 0 0
$$911$$ 29.3753 0.973248 0.486624 0.873612i $$-0.338228\pi$$
0.486624 + 0.873612i $$0.338228\pi$$
$$912$$ 31.1947 1.03296
$$913$$ 13.9439 0.461476
$$914$$ 1.33020 0.0439992
$$915$$ 0 0
$$916$$ −15.6651 −0.517590
$$917$$ 20.5576 0.678873
$$918$$ −0.0560785 −0.00185087
$$919$$ −27.0047 −0.890803 −0.445401 0.895331i $$-0.646939\pi$$
−0.445401 + 0.895331i $$0.646939\pi$$
$$920$$ 0 0
$$921$$ 12.5576 0.413788
$$922$$ 1.38471 0.0456030
$$923$$ −17.7663 −0.584785
$$924$$ −1.97196 −0.0648727
$$925$$ 0 0
$$926$$ −1.63079 −0.0535912
$$927$$ 8.27882 0.271912
$$928$$ 0.385496 0.0126545
$$929$$ 57.8496 1.89798 0.948992 0.315299i $$-0.102105\pi$$
0.948992 + 0.315299i $$0.102105\pi$$
$$930$$ 0 0
$$931$$ 8.13941 0.266758
$$932$$ 52.3706 1.71546
$$933$$ 9.33959 0.305765
$$934$$ 6.81469 0.222983
$$935$$ 0 0
$$936$$ 2.53039 0.0827083
$$937$$ −25.2180 −0.823838 −0.411919 0.911221i $$-0.635141\pi$$
−0.411919 + 0.911221i $$0.635141\pi$$
$$938$$ 0.0233441 0.000762211 0
$$939$$ 2.99530 0.0977481
$$940$$ 0 0
$$941$$ −34.2788 −1.11746 −0.558729 0.829350i $$-0.688711\pi$$
−0.558729 + 0.829350i $$0.688711\pi$$
$$942$$ 2.83646 0.0924169
$$943$$ −10.4549 −0.340458
$$944$$ 14.3661 0.467575
$$945$$ 0 0
$$946$$ −0.390977 −0.0127118
$$947$$ 18.1573 0.590032 0.295016 0.955492i $$-0.404675\pi$$
0.295016 + 0.955492i $$0.404675\pi$$
$$948$$ −6.56703 −0.213287
$$949$$ 15.9618 0.518141
$$950$$ 0 0
$$951$$ −9.93453 −0.322149
$$952$$ 0.222741 0.00721909
$$953$$ 19.8045 0.641531 0.320766 0.947159i $$-0.396060\pi$$
0.320766 + 0.947159i $$0.396060\pi$$
$$954$$ −1.33020 −0.0430669
$$955$$ 0 0
$$956$$ −21.2968 −0.688788
$$957$$ 0.195488 0.00631924
$$958$$ 6.33490 0.204671
$$959$$ 16.2227 0.523860
$$960$$ 0 0
$$961$$ 67.8816 2.18973
$$962$$ −2.85041 −0.0919010
$$963$$ 8.13941 0.262289
$$964$$ 23.8279 0.767444
$$965$$ 0 0
$$966$$ 0.278820 0.00897088
$$967$$ 0.278820 0.00896624 0.00448312 0.999990i $$-0.498573\pi$$
0.00448312 + 0.999990i $$0.498573\pi$$
$$968$$ −0.665102 −0.0213772
$$969$$ −2.72588 −0.0875677
$$970$$ 0 0
$$971$$ 25.3481 0.813458 0.406729 0.913549i $$-0.366669\pi$$
0.406729 + 0.913549i $$0.366669\pi$$
$$972$$ −1.97196 −0.0632507
$$973$$ 22.5482 0.722864
$$974$$ −6.79134 −0.217609
$$975$$ 0 0
$$976$$ 22.9953 0.736062
$$977$$ −6.71648 −0.214879 −0.107440 0.994212i $$-0.534265\pi$$
−0.107440 + 0.994212i $$0.534265\pi$$
$$978$$ 1.13862 0.0364092
$$979$$ −9.88784 −0.316017
$$980$$ 0 0
$$981$$ 11.5529 0.368857
$$982$$ 5.20409 0.166069
$$983$$ −9.99061 −0.318651 −0.159325 0.987226i $$-0.550932\pi$$
−0.159325 + 0.987226i $$0.550932\pi$$
$$984$$ 4.17605 0.133128
$$985$$ 0 0
$$986$$ −0.0109627 −0.000349123 0
$$987$$ 12.1394 0.386402
$$988$$ 61.0647 1.94273
$$989$$ −3.88784 −0.123626
$$990$$ 0 0
$$991$$ −26.7064 −0.848358 −0.424179 0.905578i $$-0.639437\pi$$
−0.424179 + 0.905578i $$0.639437\pi$$
$$992$$ −19.6090 −0.622587
$$993$$ 22.5482 0.715547
$$994$$ 0.781954 0.0248021
$$995$$ 0 0
$$996$$ −27.4969 −0.871272
$$997$$ −54.3333 −1.72075 −0.860377 0.509658i $$-0.829772\pi$$
−0.860377 + 0.509658i $$0.829772\pi$$
$$998$$ 2.57159 0.0814024
$$999$$ 4.47431 0.141561
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 5775.2.a.bw.1.2 3
5.4 even 2 231.2.a.d.1.2 3
15.14 odd 2 693.2.a.m.1.2 3
20.19 odd 2 3696.2.a.bp.1.3 3
35.34 odd 2 1617.2.a.s.1.2 3
55.54 odd 2 2541.2.a.bi.1.2 3
105.104 even 2 4851.2.a.bp.1.2 3
165.164 even 2 7623.2.a.cb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 5.4 even 2
693.2.a.m.1.2 3 15.14 odd 2
1617.2.a.s.1.2 3 35.34 odd 2
2541.2.a.bi.1.2 3 55.54 odd 2
3696.2.a.bp.1.3 3 20.19 odd 2
4851.2.a.bp.1.2 3 105.104 even 2
5775.2.a.bw.1.2 3 1.1 even 1 trivial
7623.2.a.cb.1.2 3 165.164 even 2