# Properties

 Label 2541.2.a.bi.1.2 Level $2541$ Weight $2$ Character 2541.1 Self dual yes Analytic conductor $20.290$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 2541.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} -0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} -0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +0.637062 q^{10} +1.97196 q^{12} -3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} -0.334898 q^{17} +0.167449 q^{18} -8.13941 q^{19} -7.50235 q^{20} -1.00000 q^{21} -1.66510 q^{23} +0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} -1.97196 q^{28} -0.195488 q^{29} -0.637062 q^{30} -9.94392 q^{31} +1.97196 q^{32} -0.0560785 q^{34} +3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} +3.80451 q^{39} -2.53039 q^{40} +6.27882 q^{41} -0.167449 q^{42} -2.33490 q^{43} +3.80451 q^{45} -0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} +1.58647 q^{50} +0.334898 q^{51} +7.50235 q^{52} +7.94392 q^{53} -0.167449 q^{54} -0.665102 q^{56} +8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} -6.00000 q^{61} -1.66510 q^{62} +1.00000 q^{63} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} +0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} -0.665102 q^{72} -4.19549 q^{73} -0.749219 q^{74} -9.47431 q^{75} +16.0506 q^{76} +0.637062 q^{78} -3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} +13.9439 q^{83} +1.97196 q^{84} -1.27412 q^{85} -0.390977 q^{86} +0.195488 q^{87} -9.88784 q^{89} +0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} -2.03273 q^{94} -30.9665 q^{95} -1.97196 q^{96} +0.0560785 q^{97} +0.167449 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 9 q^{10} - 6 q^{12} + 12 q^{16} - 12 q^{19} - 21 q^{20} - 3 q^{21} - 6 q^{23} + 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} + 6 q^{28} - 12 q^{29} + 9 q^{30} - 6 q^{31} - 6 q^{32} - 24 q^{34} + 6 q^{36} - 15 q^{38} - 18 q^{40} - 6 q^{41} - 6 q^{43} + 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} + 39 q^{50} + 21 q^{52} - 3 q^{56} + 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} - 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 30 q^{65} + 12 q^{67} + 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} - 3 q^{72} - 24 q^{73} - 39 q^{74} - 15 q^{75} + 3 q^{76} - 9 q^{78} - 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} + 18 q^{83} - 6 q^{84} + 18 q^{85} - 24 q^{86} + 12 q^{87} + 18 q^{89} - 9 q^{90} - 18 q^{92} + 6 q^{93} - 15 q^{94} - 12 q^{95} + 6 q^{96} + 24 q^{97}+O(q^{100})$$ 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 9 * q^10 - 6 * q^12 + 12 * q^16 - 12 * q^19 - 21 * q^20 - 3 * q^21 - 6 * q^23 + 3 * q^24 + 15 * q^25 + 9 * q^26 - 3 * q^27 + 6 * q^28 - 12 * q^29 + 9 * q^30 - 6 * q^31 - 6 * q^32 - 24 * q^34 + 6 * q^36 - 15 * q^38 - 18 * q^40 - 6 * q^41 - 6 * q^43 + 24 * q^46 - 24 * q^47 - 12 * q^48 + 3 * q^49 + 39 * q^50 + 21 * q^52 - 3 * q^56 + 12 * q^57 - 9 * q^58 - 24 * q^59 + 21 * q^60 - 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 30 * q^65 + 12 * q^67 + 6 * q^68 + 6 * q^69 - 9 * q^70 + 12 * q^71 - 3 * q^72 - 24 * q^73 - 39 * q^74 - 15 * q^75 + 3 * q^76 - 9 * q^78 - 12 * q^79 + 9 * q^80 + 3 * q^81 + 30 * q^82 + 18 * q^83 - 6 * q^84 + 18 * q^85 - 24 * q^86 + 12 * q^87 + 18 * q^89 - 9 * q^90 - 18 * q^92 + 6 * q^93 - 15 * q^94 - 12 * q^95 + 6 * q^96 + 24 * q^97

## 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
$$5$$ 3.80451 1.70143 0.850715 0.525628i $$-0.176170\pi$$
0.850715 + 0.525628i $$0.176170\pi$$
$$6$$ −0.167449 −0.0683608
$$7$$ 1.00000 0.377964
$$8$$ −0.665102 −0.235149
$$9$$ 1.00000 0.333333
$$10$$ 0.637062 0.201457
$$11$$ 0 0
$$12$$ 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
$$15$$ −3.80451 −0.982321
$$16$$ 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 $$-0.883399\pi$$
−0.933654 + 0.358175i $$0.883399\pi$$
$$20$$ −7.50235 −1.67758
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −1.66510 −0.347198 −0.173599 0.984816i $$-0.555540\pi$$
−0.173599 + 0.984816i $$0.555540\pi$$
$$24$$ 0.665102 0.135763
$$25$$ 9.47431 1.89486
$$26$$ −0.637062 −0.124938
$$27$$ −1.00000 −0.192450
$$28$$ −1.97196 −0.372666
$$29$$ −0.195488 −0.0363013 −0.0181506 0.999835i $$-0.505778\pi$$
−0.0181506 + 0.999835i $$0.505778\pi$$
$$30$$ −0.637062 −0.116311
$$31$$ −9.94392 −1.78598 −0.892991 0.450075i $$-0.851397\pi$$
−0.892991 + 0.450075i $$0.851397\pi$$
$$32$$ 1.97196 0.348597
$$33$$ 0 0
$$34$$ −0.0560785 −0.00961738
$$35$$ 3.80451 0.643080
$$36$$ −1.97196 −0.328660
$$37$$ −4.47431 −0.735572 −0.367786 0.929911i $$-0.619884\pi$$
−0.367786 + 0.929911i $$0.619884\pi$$
$$38$$ −1.36294 −0.221098
$$39$$ 3.80451 0.609209
$$40$$ −2.53039 −0.400089
$$41$$ 6.27882 0.980587 0.490293 0.871557i $$-0.336890\pi$$
0.490293 + 0.871557i $$0.336890\pi$$
$$42$$ −0.167449 −0.0258380
$$43$$ −2.33490 −0.356069 −0.178034 0.984024i $$-0.556974\pi$$
−0.178034 + 0.984024i $$0.556974\pi$$
$$44$$ 0 0
$$45$$ 3.80451 0.567143
$$46$$ −0.278820 −0.0411098
$$47$$ −12.1394 −1.77071 −0.885357 0.464911i $$-0.846086\pi$$
−0.885357 + 0.464911i $$0.846086\pi$$
$$48$$ −3.83255 −0.553181
$$49$$ 1.00000 0.142857
$$50$$ 1.58647 0.224360
$$51$$ 0.334898 0.0468952
$$52$$ 7.50235 1.04039
$$53$$ 7.94392 1.09118 0.545591 0.838052i $$-0.316305\pi$$
0.545591 + 0.838052i $$0.316305\pi$$
$$54$$ −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 $$0.421540\pi$$
$$60$$ 7.50235 0.968549
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −1.66510 −0.211468
$$63$$ 1.00000 0.125988
$$64$$ −7.33490 −0.916862
$$65$$ −14.4743 −1.79532
$$66$$ 0 0
$$67$$ −0.139410 −0.0170316 −0.00851582 0.999964i $$-0.502711\pi$$
−0.00851582 + 0.999964i $$0.502711\pi$$
$$68$$ 0.660406 0.0800860
$$69$$ 1.66510 0.200455
$$70$$ 0.637062 0.0761435
$$71$$ 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
$$75$$ −9.47431 −1.09400
$$76$$ 16.0506 1.84113
$$77$$ 0 0
$$78$$ 0.637062 0.0721331
$$79$$ −3.33020 −0.374677 −0.187339 0.982295i $$-0.559986\pi$$
−0.187339 + 0.982295i $$0.559986\pi$$
$$80$$ 14.5810 1.63020
$$81$$ 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
$$85$$ −1.27412 −0.138198
$$86$$ −0.390977 −0.0421601
$$87$$ 0.195488 0.0209586
$$88$$ 0 0
$$89$$ −9.88784 −1.04811 −0.524055 0.851685i $$-0.675581\pi$$
−0.524055 + 0.851685i $$0.675581\pi$$
$$90$$ 0.637062 0.0671523
$$91$$ −3.80451 −0.398821
$$92$$ 3.28352 0.342330
$$93$$ 9.94392 1.03114
$$94$$ −2.03273 −0.209661
$$95$$ −30.9665 −3.17709
$$96$$ −1.97196 −0.201262
$$97$$ 0.0560785 0.00569391 0.00284695 0.999996i $$-0.499094\pi$$
0.00284695 + 0.999996i $$0.499094\pi$$
$$98$$ 0.167449 0.0169149
$$99$$ 0 0
$$100$$ −18.6830 −1.86830
$$101$$ 18.8831 1.87894 0.939472 0.342626i $$-0.111317\pi$$
0.939472 + 0.342626i $$0.111317\pi$$
$$102$$ 0.0560785 0.00555260
$$103$$ −8.27882 −0.815736 −0.407868 0.913041i $$-0.633728\pi$$
−0.407868 + 0.913041i $$0.633728\pi$$
$$104$$ 2.53039 0.248125
$$105$$ −3.80451 −0.371282
$$106$$ 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 $$-0.686626\pi$$
−0.553286 + 0.832992i $$0.686626\pi$$
$$110$$ 0 0
$$111$$ 4.47431 0.424683
$$112$$ 3.83255 0.362142
$$113$$ −1.33020 −0.125135 −0.0625675 0.998041i $$-0.519929\pi$$
−0.0625675 + 0.998041i $$0.519929\pi$$
$$114$$ 1.36294 0.127651
$$115$$ −6.33490 −0.590732
$$116$$ 0.385496 0.0357924
$$117$$ −3.80451 −0.351727
$$118$$ 0.627672 0.0577819
$$119$$ −0.334898 −0.0307001
$$120$$ 2.53039 0.230992
$$121$$ 0 0
$$122$$ −1.00470 −0.0909608
$$123$$ −6.27882 −0.566142
$$124$$ 19.6090 1.76094
$$125$$ 17.0226 1.52254
$$126$$ 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
$$130$$ −2.42371 −0.212574
$$131$$ −20.5576 −1.79613 −0.898065 0.439863i $$-0.855027\pi$$
−0.898065 + 0.439863i $$0.855027\pi$$
$$132$$ 0 0
$$133$$ −8.13941 −0.705776
$$134$$ −0.0233441 −0.00201662
$$135$$ −3.80451 −0.327440
$$136$$ 0.222741 0.0190999
$$137$$ −16.2227 −1.38600 −0.693001 0.720936i $$-0.743712\pi$$
−0.693001 + 0.720936i $$0.743712\pi$$
$$138$$ 0.278820 0.0237347
$$139$$ −22.5482 −1.91252 −0.956259 0.292522i $$-0.905506\pi$$
−0.956259 + 0.292522i $$0.905506\pi$$
$$140$$ −7.50235 −0.634064
$$141$$ 12.1394 1.02232
$$142$$ 0.781954 0.0656201
$$143$$ 0 0
$$144$$ 3.83255 0.319379
$$145$$ −0.743738 −0.0617641
$$146$$ −0.702531 −0.0581419
$$147$$ −1.00000 −0.0824786
$$148$$ 8.82316 0.725259
$$149$$ −8.19549 −0.671401 −0.335700 0.941969i $$-0.608973\pi$$
−0.335700 + 0.941969i $$0.608973\pi$$
$$150$$ −1.58647 −0.129534
$$151$$ 13.2741 1.08023 0.540116 0.841590i $$-0.318380\pi$$
0.540116 + 0.841590i $$0.318380\pi$$
$$152$$ 5.41353 0.439096
$$153$$ −0.334898 −0.0270749
$$154$$ 0 0
$$155$$ −37.8318 −3.03872
$$156$$ −7.50235 −0.600669
$$157$$ −16.9392 −1.35190 −0.675949 0.736949i $$-0.736266\pi$$
−0.675949 + 0.736949i $$0.736266\pi$$
$$158$$ −0.557640 −0.0443634
$$159$$ −7.94392 −0.629994
$$160$$ 7.50235 0.593113
$$161$$ −1.66510 −0.131228
$$162$$ 0.167449 0.0131561
$$163$$ −6.79982 −0.532603 −0.266301 0.963890i $$-0.585802\pi$$
−0.266301 + 0.963890i $$0.585802\pi$$
$$164$$ −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
$$170$$ −0.213351 −0.0163633
$$171$$ −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
$$175$$ 9.47431 0.716190
$$176$$ 0 0
$$177$$ −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 $$0.648028\pi$$
$$180$$ −7.50235 −0.559192
$$181$$ 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
$$185$$ −17.0226 −1.25152
$$186$$ 1.66510 0.122091
$$187$$ 0 0
$$188$$ 23.9384 1.74589
$$189$$ −1.00000 −0.0727393
$$190$$ −5.18531 −0.376182
$$191$$ 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
$$195$$ 14.4743 1.03653
$$196$$ −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 0
$$199$$ 13.5529 0.960743 0.480371 0.877065i $$-0.340502\pi$$
0.480371 + 0.877065i $$0.340502\pi$$
$$200$$ −6.30138 −0.445575
$$201$$ 0.139410 0.00983322
$$202$$ 3.16197 0.222475
$$203$$ −0.195488 −0.0137206
$$204$$ −0.660406 −0.0462377
$$205$$ 23.8878 1.66840
$$206$$ −1.38628 −0.0965868
$$207$$ −1.66510 −0.115733
$$208$$ −14.5810 −1.01101
$$209$$ 0 0
$$210$$ −0.637062 −0.0439615
$$211$$ 4.27882 0.294566 0.147283 0.989094i $$-0.452947\pi$$
0.147283 + 0.989094i $$0.452947\pi$$
$$212$$ −15.6651 −1.07588
$$213$$ −4.66980 −0.319969
$$214$$ 1.36294 0.0931685
$$215$$ −8.88315 −0.605826
$$216$$ 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 $$-0.611730\pi$$
−0.343846 + 0.939026i $$0.611730\pi$$
$$224$$ 1.97196 0.131757
$$225$$ 9.47431 0.631621
$$226$$ −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 $$0.415461\pi$$
$$230$$ −1.06077 −0.0699453
$$231$$ 0 0
$$232$$ 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
$$235$$ −46.1845 −3.01275
$$236$$ −7.39176 −0.481163
$$237$$ 3.33020 0.216320
$$238$$ −0.0560785 −0.00363503
$$239$$ −10.7998 −0.698582 −0.349291 0.937014i $$-0.613578\pi$$
−0.349291 + 0.937014i $$0.613578\pi$$
$$240$$ −14.5810 −0.941198
$$241$$ 12.0833 0.778356 0.389178 0.921163i $$-0.372759\pi$$
0.389178 + 0.921163i $$0.372759\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 11.8318 0.757451
$$245$$ 3.80451 0.243061
$$246$$ −1.05138 −0.0670338
$$247$$ 30.9665 1.97035
$$248$$ 6.61372 0.419972
$$249$$ −13.9439 −0.883660
$$250$$ 2.85041 0.180276
$$251$$ 4.80921 0.303554 0.151777 0.988415i $$-0.451500\pi$$
0.151777 + 0.988415i $$0.451500\pi$$
$$252$$ −1.97196 −0.124222
$$253$$ 0 0
$$254$$ −2.44706 −0.153542
$$255$$ 1.27412 0.0797888
$$256$$ 13.8037 0.862733
$$257$$ −16.7531 −1.04503 −0.522516 0.852630i $$-0.675006\pi$$
−0.522516 + 0.852630i $$0.675006\pi$$
$$258$$ 0.390977 0.0243412
$$259$$ −4.47431 −0.278020
$$260$$ 28.5428 1.77015
$$261$$ −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 0
$$265$$ 30.2227 1.85657
$$266$$ −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 $$0.779139\pi$$
$$270$$ −0.637062 −0.0387704
$$271$$ −4.13941 −0.251451 −0.125726 0.992065i $$-0.540126\pi$$
−0.125726 + 0.992065i $$0.540126\pi$$
$$272$$ −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
$$280$$ −2.53039 −0.151220
$$281$$ −6.74374 −0.402298 −0.201149 0.979561i $$-0.564467\pi$$
−0.201149 + 0.979561i $$0.564467\pi$$
$$282$$ 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
$$285$$ 30.9665 1.83430
$$286$$ 0 0
$$287$$ 6.27882 0.370627
$$288$$ 1.97196 0.116199
$$289$$ −16.8878 −0.993403
$$290$$ −0.124538 −0.00731314
$$291$$ −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
$$295$$ 14.2610 0.830305
$$296$$ 2.97587 0.172969
$$297$$ 0 0
$$298$$ −1.37233 −0.0794968
$$299$$ 6.33490 0.366357
$$300$$ 18.6830 1.07866
$$301$$ −2.33490 −0.134581
$$302$$ 2.22274 0.127904
$$303$$ −18.8831 −1.08481
$$304$$ −31.1947 −1.78914
$$305$$ −22.8271 −1.30707
$$306$$ −0.0560785 −0.00320579
$$307$$ 12.5576 0.716702 0.358351 0.933587i $$-0.383339\pi$$
0.358351 + 0.933587i $$0.383339\pi$$
$$308$$ 0 0
$$309$$ 8.27882 0.470966
$$310$$ −6.33490 −0.359798
$$311$$ 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 $$-0.526978\pi$$
−0.0846523 + 0.996411i $$0.526978\pi$$
$$314$$ −2.83646 −0.160071
$$315$$ 3.80451 0.214360
$$316$$ 6.56703 0.369424
$$317$$ 9.93453 0.557979 0.278989 0.960294i $$-0.410001\pi$$
0.278989 + 0.960294i $$0.410001\pi$$
$$318$$ −1.33020 −0.0745941
$$319$$ 0 0
$$320$$ −27.9057 −1.55998
$$321$$ −8.13941 −0.454298
$$322$$ −0.278820 −0.0155380
$$323$$ 2.72588 0.151672
$$324$$ −1.97196 −0.109553
$$325$$ −36.0451 −1.99942
$$326$$ −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
$$335$$ −0.530387 −0.0289781
$$336$$ −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
$$340$$ 2.51252 0.136261
$$341$$ 0 0
$$342$$ −1.36294 −0.0736992
$$343$$ 1.00000 0.0539949
$$344$$ 1.55294 0.0837292
$$345$$ 6.33490 0.341059
$$346$$ 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 $$-0.475646\pi$$
0.0764363 + 0.997074i $$0.475646\pi$$
$$350$$ 1.58647 0.0848001
$$351$$ 3.80451 0.203070
$$352$$ 0 0
$$353$$ 24.6410 1.31151 0.655753 0.754975i $$-0.272351\pi$$
0.655753 + 0.754975i $$0.272351\pi$$
$$354$$ −0.627672 −0.0333604
$$355$$ 17.7663 0.942937
$$356$$ 19.4984 1.03342
$$357$$ 0.334898 0.0177247
$$358$$ −2.00939 −0.106200
$$359$$ 11.8878 0.627416 0.313708 0.949519i $$-0.398429\pi$$
0.313708 + 0.949519i $$0.398429\pi$$
$$360$$ −2.53039 −0.133363
$$361$$ 47.2500 2.48684
$$362$$ 0.121547 0.00638838
$$363$$ 0 0
$$364$$ 7.50235 0.393230
$$365$$ −15.9618 −0.835478
$$366$$ 1.00470 0.0525163
$$367$$ 3.60902 0.188389 0.0941947 0.995554i $$-0.469972\pi$$
0.0941947 + 0.995554i $$0.469972\pi$$
$$368$$ −6.38159 −0.332663
$$369$$ 6.27882 0.326862
$$370$$ −2.85041 −0.148186
$$371$$ 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 0
$$375$$ −17.0226 −0.879041
$$376$$ 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 $$0.295209\pi$$
$$380$$ 61.0647 3.13255
$$381$$ 14.6137 0.748683
$$382$$ 0.883148 0.0451858
$$383$$ −15.2180 −0.777606 −0.388803 0.921321i $$-0.627111\pi$$
−0.388803 + 0.921321i $$0.627111\pi$$
$$384$$ 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 $$0.469851\pi$$
$$390$$ 2.42371 0.122729
$$391$$ 0.557640 0.0282011
$$392$$ −0.665102 −0.0335927
$$393$$ 20.5576 1.03700
$$394$$ −0.447055 −0.0225223
$$395$$ −12.6698 −0.637487
$$396$$ 0 0
$$397$$ 20.8925 1.04857 0.524283 0.851544i $$-0.324333\pi$$
0.524283 + 0.851544i $$0.324333\pi$$
$$398$$ 2.26943 0.113756
$$399$$ 8.13941 0.407480
$$400$$ 36.3108 1.81554
$$401$$ 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
$$405$$ 3.80451 0.189048
$$406$$ −0.0327344 −0.00162458
$$407$$ 0 0
$$408$$ −0.222741 −0.0110273
$$409$$ 23.8972 1.18164 0.590821 0.806803i $$-0.298804\pi$$
0.590821 + 0.806803i $$0.298804\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 16.2227 0.800209
$$412$$ 16.3255 0.804300
$$413$$ 3.74843 0.184448
$$414$$ −0.278820 −0.0137033
$$415$$ 53.0498 2.60411
$$416$$ −7.50235 −0.367833
$$417$$ 22.5482 1.10419
$$418$$ 0 0
$$419$$ −30.2967 −1.48009 −0.740045 0.672557i $$-0.765196\pi$$
−0.740045 + 0.672557i $$0.765196\pi$$
$$420$$ 7.50235 0.366077
$$421$$ −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
$$425$$ −3.17293 −0.153910
$$426$$ −0.781954 −0.0378858
$$427$$ −6.00000 −0.290360
$$428$$ −16.0506 −0.775835
$$429$$ 0 0
$$430$$ −1.48748 −0.0717325
$$431$$ −3.07864 −0.148293 −0.0741463 0.997247i $$-0.523623\pi$$
−0.0741463 + 0.997247i $$0.523623\pi$$
$$432$$ −3.83255 −0.184394
$$433$$ 16.9392 0.814047 0.407024 0.913418i $$-0.366567\pi$$
0.407024 + 0.913418i $$0.366567\pi$$
$$434$$ −1.66510 −0.0799274
$$435$$ 0.743738 0.0356595
$$436$$ 22.7820 1.09106
$$437$$ 13.5529 0.648325
$$438$$ 0.702531 0.0335682
$$439$$ 11.0786 0.528754 0.264377 0.964419i $$-0.414834\pi$$
0.264377 + 0.964419i $$0.414834\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0.213351 0.0101481
$$443$$ −14.5482 −0.691208 −0.345604 0.938380i $$-0.612326\pi$$
−0.345604 + 0.938380i $$0.612326\pi$$
$$444$$ −8.82316 −0.418729
$$445$$ −37.6184 −1.78328
$$446$$ −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 $$0.275352\pi$$
$$450$$ 1.58647 0.0747867
$$451$$ 0 0
$$452$$ 2.62311 0.123381
$$453$$ −13.2741 −0.623673
$$454$$ 0.0654688 0.00307260
$$455$$ −14.4743 −0.678566
$$456$$ −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
$$460$$ 12.4922 0.582450
$$461$$ −8.26943 −0.385146 −0.192573 0.981283i $$-0.561683\pi$$
−0.192573 + 0.981283i $$0.561683\pi$$
$$462$$ 0 0
$$463$$ 9.73904 0.452612 0.226306 0.974056i $$-0.427335\pi$$
0.226306 + 0.974056i $$0.427335\pi$$
$$464$$ −0.749219 −0.0347816
$$465$$ 37.8318 1.75441
$$466$$ −4.44706 −0.206006
$$467$$ −40.6970 −1.88323 −0.941617 0.336685i $$-0.890694\pi$$
−0.941617 + 0.336685i $$0.890694\pi$$
$$468$$ 7.50235 0.346796
$$469$$ −0.139410 −0.00643735
$$470$$ −7.73356 −0.356723
$$471$$ 16.9392 0.780518
$$472$$ −2.49309 −0.114754
$$473$$ 0 0
$$474$$ 0.557640 0.0256132
$$475$$ −77.1153 −3.53829
$$476$$ 0.660406 0.0302697
$$477$$ 7.94392 0.363727
$$478$$ −1.80842 −0.0827152
$$479$$ −37.8318 −1.72858 −0.864289 0.502996i $$-0.832231\pi$$
−0.864289 + 0.502996i $$0.832231\pi$$
$$480$$ −7.50235 −0.342434
$$481$$ 17.0226 0.776162
$$482$$ 2.02334 0.0921608
$$483$$ 1.66510 0.0757647
$$484$$ 0 0
$$485$$ 0.213351 0.00968778
$$486$$ −0.167449 −0.00759565
$$487$$ 40.5576 1.83784 0.918921 0.394442i $$-0.129062\pi$$
0.918921 + 0.394442i $$0.129062\pi$$
$$488$$ 3.99061 0.180646
$$489$$ 6.79982 0.307498
$$490$$ 0.637062 0.0287795
$$491$$ −31.0786 −1.40256 −0.701280 0.712886i $$-0.747388\pi$$
−0.701280 + 0.712886i $$0.747388\pi$$
$$492$$ 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 $$0.388304\pi$$
$$500$$ −33.5678 −1.50120
$$501$$ −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
$$505$$ 71.8412 3.19689
$$506$$ 0 0
$$507$$ −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 $$0.630614\pi$$
$$510$$ 0.213351 0.00944735
$$511$$ −4.19549 −0.185597
$$512$$ 12.6557 0.559309
$$513$$ 8.13941 0.359364
$$514$$ −2.80530 −0.123736
$$515$$ −31.4969 −1.38792
$$516$$ −4.60433 −0.202694
$$517$$ 0 0
$$518$$ −0.749219 −0.0329188
$$519$$ −1.72118 −0.0755514
$$520$$ 9.62689 0.422167
$$521$$ 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
$$525$$ −9.47431 −0.413493
$$526$$ 2.03273 0.0886314
$$527$$ 3.33020 0.145066
$$528$$ 0 0
$$529$$ −20.2274 −0.879454
$$530$$ 5.06077 0.219826
$$531$$ 3.74843 0.162668
$$532$$ 16.0506 0.695882
$$533$$ −23.8878 −1.03470
$$534$$ 1.65571 0.0716496
$$535$$ 30.9665 1.33880
$$536$$ 0.0927218 0.00400497
$$537$$ 12.0000 0.517838
$$538$$ −4.22274 −0.182055
$$539$$ 0 0
$$540$$ 7.50235 0.322850
$$541$$ 4.26943 0.183557 0.0917786 0.995779i $$-0.470745\pi$$
0.0917786 + 0.995779i $$0.470745\pi$$
$$542$$ −0.693141 −0.0297729
$$543$$ −0.725875 −0.0311503
$$544$$ −0.660406 −0.0283147
$$545$$ −43.9533 −1.88275
$$546$$ 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
$$555$$ 17.0226 0.722567
$$556$$ 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
$$560$$ 14.5810 0.616159
$$561$$ 0 0
$$562$$ −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
$$565$$ −5.06077 −0.212908
$$566$$ −3.91119 −0.164399
$$567$$ 1.00000 0.0419961
$$568$$ −3.10589 −0.130320
$$569$$ 29.2180 1.22488 0.612442 0.790515i $$-0.290187\pi$$
0.612442 + 0.790515i $$0.290187\pi$$
$$570$$ 5.18531 0.217189
$$571$$ −32.4455 −1.35780 −0.678901 0.734230i $$-0.737543\pi$$
−0.678901 + 0.734230i $$0.737543\pi$$
$$572$$ 0 0
$$573$$ −5.27412 −0.220330
$$574$$ 1.05138 0.0438839
$$575$$ −15.7757 −0.657892
$$576$$ −7.33490 −0.305621
$$577$$ −14.8831 −0.619594 −0.309797 0.950803i $$-0.600261\pi$$
−0.309797 + 0.950803i $$0.600261\pi$$
$$578$$ −2.82786 −0.117623
$$579$$ −19.8318 −0.824180
$$580$$ 1.46662 0.0608982
$$581$$ 13.9439 0.578491
$$582$$ −0.00939029 −0.000389240 0
$$583$$ 0 0
$$584$$ 2.79043 0.115469
$$585$$ −14.4743 −0.598439
$$586$$ 2.37220 0.0979945
$$587$$ −4.53039 −0.186989 −0.0934945 0.995620i $$-0.529804\pi$$
−0.0934945 + 0.995620i $$0.529804\pi$$
$$588$$ 1.97196 0.0813223
$$589$$ 80.9377 3.33498
$$590$$ 2.38799 0.0983118
$$591$$ 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 0
$$595$$ −1.27412 −0.0522340
$$596$$ 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 $$0.487052\pi$$
$$600$$ 6.30138 0.257253
$$601$$ 8.47431 0.345674 0.172837 0.984950i $$-0.444707\pi$$
0.172837 + 0.984950i $$0.444707\pi$$
$$602$$ −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
$$610$$ −3.82237 −0.154763
$$611$$ 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
$$615$$ −23.8878 −0.963251
$$616$$ 0 0
$$617$$ 22.6698 0.912652 0.456326 0.889813i $$-0.349165\pi$$
0.456326 + 0.889813i $$0.349165\pi$$
$$618$$ 1.38628 0.0557644
$$619$$ 8.71648 0.350345 0.175173 0.984538i $$-0.443952\pi$$
0.175173 + 0.984538i $$0.443952\pi$$
$$620$$ 74.6028 2.99612
$$621$$ 1.66510 0.0668182
$$622$$ 1.56391 0.0627070
$$623$$ −9.88784 −0.396148
$$624$$ 14.5810 0.583707
$$625$$ 17.3910 0.695639
$$626$$ −0.501561 −0.0200464
$$627$$ 0 0
$$628$$ 33.4035 1.33294
$$629$$ 1.49844 0.0597467
$$630$$ 0.637062 0.0253812
$$631$$ −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
$$635$$ −55.5981 −2.20634
$$636$$ 15.6651 0.621162
$$637$$ −3.80451 −0.150740
$$638$$ 0 0
$$639$$ 4.66980 0.184734
$$640$$ −19.6775 −0.777821
$$641$$ −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 $$-0.351714\pi$$
0.449185 + 0.893439i $$0.351714\pi$$
$$644$$ 3.28352 0.129389
$$645$$ 8.88315 0.349774
$$646$$ 0.456446 0.0179586
$$647$$ 31.2453 1.22838 0.614190 0.789158i $$-0.289483\pi$$
0.614190 + 0.789158i $$0.289483\pi$$
$$648$$ −0.665102 −0.0261277
$$649$$ 0 0
$$650$$ −6.03573 −0.236741
$$651$$ 9.94392 0.389733
$$652$$ 13.4090 0.525136
$$653$$ 28.2882 1.10700 0.553502 0.832848i $$-0.313291\pi$$
0.553502 + 0.832848i $$0.313291\pi$$
$$654$$ 1.93453 0.0756462
$$655$$ −78.2118 −3.05599
$$656$$ 24.0639 0.939537
$$657$$ −4.19549 −0.163682
$$658$$ −2.03273 −0.0792442
$$659$$ −30.2967 −1.18019 −0.590096 0.807333i $$-0.700910\pi$$
−0.590096 + 0.807333i $$0.700910\pi$$
$$660$$ 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
$$665$$ −30.9665 −1.20083
$$666$$ −0.749219 −0.0290317
$$667$$ 0.325508 0.0126037
$$668$$ −35.9345 −1.39035
$$669$$ 10.2694 0.397039
$$670$$ −0.0888128 −0.00343114
$$671$$ 0 0
$$672$$ −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
$$675$$ −9.47431 −0.364666
$$676$$ −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
$$680$$ 0.847422 0.0324972
$$681$$ −0.390977 −0.0149823
$$682$$ 0 0
$$683$$ 25.0498 0.958504 0.479252 0.877677i $$-0.340908\pi$$
0.479252 + 0.877677i $$0.340908\pi$$
$$684$$ 16.0506 0.613710
$$685$$ −61.7196 −2.35818
$$686$$ 0.167449 0.00639324
$$687$$ −7.94392 −0.303080
$$688$$ −8.94862 −0.341163
$$689$$ −30.2227 −1.15139
$$690$$ 1.06077 0.0403830
$$691$$ −38.8271 −1.47705 −0.738526 0.674225i $$-0.764478\pi$$
−0.738526 + 0.674225i $$0.764478\pi$$
$$692$$ −3.39410 −0.129024
$$693$$ 0 0
$$694$$ 4.65102 0.176550
$$695$$ −85.7851 −3.25401
$$696$$ −0.130020 −0.00492838
$$697$$ −2.10277 −0.0796480
$$698$$ 0.478217 0.0181008
$$699$$ 26.5576 1.00450
$$700$$ −18.6830 −0.706150
$$701$$ 41.7757 1.57785 0.788923 0.614492i $$-0.210639\pi$$
0.788923 + 0.614492i $$0.210639\pi$$
$$702$$ 0.637062 0.0240444
$$703$$ 36.4182 1.37354
$$704$$ 0 0
$$705$$ 46.1845 1.73941
$$706$$ 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 $$0.580990\pi$$
$$710$$ 2.97495 0.111648
$$711$$ −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 $$0.467423\pi$$
$$720$$ 14.5810 0.543401
$$721$$ −8.27882 −0.308319
$$722$$ 7.91197 0.294453
$$723$$ −12.0833 −0.449384
$$724$$ −1.43140 −0.0531975
$$725$$ −1.85212 −0.0687859
$$726$$ 0 0
$$727$$ −32.8831 −1.21957 −0.609784 0.792567i $$-0.708744\pi$$
−0.609784 + 0.792567i $$0.708744\pi$$
$$728$$ 2.53039 0.0937824
$$729$$ 1.00000 0.0370370
$$730$$ −2.67279 −0.0989243
$$731$$ 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
$$735$$ −3.80451 −0.140332
$$736$$ −3.28352 −0.121032
$$737$$ 0 0
$$738$$ 1.05138 0.0387020
$$739$$ −29.6651 −1.09125 −0.545624 0.838030i $$-0.683707\pi$$
−0.545624 + 0.838030i $$0.683707\pi$$
$$740$$ 33.5678 1.23398
$$741$$ −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
$$745$$ −31.1798 −1.14234
$$746$$ −2.06547 −0.0756222
$$747$$ 13.9439 0.510181
$$748$$ 0 0
$$749$$ 8.13941 0.297408
$$750$$ −2.85041 −0.104082
$$751$$ −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
$$755$$ 50.5016 1.83794
$$756$$ 1.97196 0.0717195
$$757$$ 2.74374 0.0997229 0.0498614 0.998756i $$-0.484122\pi$$
0.0498614 + 0.998756i $$0.484122\pi$$
$$758$$ 3.91119 0.142061
$$759$$ 0 0
$$760$$ 20.5959 0.747090
$$761$$ −6.39098 −0.231673 −0.115836 0.993268i $$-0.536955\pi$$
−0.115836 + 0.993268i $$0.536955\pi$$
$$762$$ 2.44706 0.0886475
$$763$$ −11.5529 −0.418245
$$764$$ −10.4004 −0.376272
$$765$$ −1.27412 −0.0460661
$$766$$ −2.54825 −0.0920720
$$767$$ −14.2610 −0.514933
$$768$$ −13.8037 −0.498099
$$769$$ −17.7951 −0.641708 −0.320854 0.947129i $$-0.603970\pi$$
−0.320854 + 0.947129i $$0.603970\pi$$
$$770$$ 0 0
$$771$$ 16.7531 0.603349
$$772$$ −39.1075 −1.40751
$$773$$ −31.5257 −1.13390 −0.566950 0.823752i $$-0.691877\pi$$
−0.566950 + 0.823752i $$0.691877\pi$$
$$774$$ −0.390977 −0.0140534
$$775$$ −94.2118 −3.38419
$$776$$ −0.0372979 −0.00133892
$$777$$ 4.47431 0.160515
$$778$$ 0.624681 0.0223959
$$779$$ −51.1059 −1.83106
$$780$$ −28.5428 −1.02200
$$781$$ 0 0
$$782$$ 0.0933763 0.00333913
$$783$$ 0.195488 0.00698619
$$784$$ 3.83255 0.136877
$$785$$ −64.4455 −2.30016
$$786$$ 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
$$790$$ −2.12155 −0.0754813
$$791$$ −1.33020 −0.0472966
$$792$$ 0 0
$$793$$ 22.8271 0.810613
$$794$$ 3.49844 0.124155
$$795$$ −30.2227 −1.07189
$$796$$ −26.7259 −0.947274
$$797$$ 43.8590 1.55357 0.776783 0.629768i $$-0.216850\pi$$
0.776783 + 0.629768i $$0.216850\pi$$
$$798$$ 1.36294 0.0482475
$$799$$ 4.06547 0.143826
$$800$$ 18.6830 0.660543
$$801$$ −9.88784 −0.349370
$$802$$ 3.89723 0.137616
$$803$$ 0 0
$$804$$ −0.274911 −0.00969536
$$805$$ −6.33490 −0.223276
$$806$$ 6.33490 0.223137
$$807$$ 25.2180 0.887717
$$808$$ −12.5592 −0.441832
$$809$$ −6.74374 −0.237097 −0.118549 0.992948i $$-0.537824\pi$$
−0.118549 + 0.992948i $$0.537824\pi$$
$$810$$ 0.637062 0.0223841
$$811$$ −3.97275 −0.139502 −0.0697510 0.997564i $$-0.522220\pi$$
−0.0697510 + 0.997564i $$0.522220\pi$$
$$812$$ 0.385496 0.0135282
$$813$$ 4.13941 0.145175
$$814$$ 0 0
$$815$$ −25.8700 −0.906186
$$816$$ 1.28352 0.0449320
$$817$$ 19.0047 0.664890
$$818$$ 4.00157 0.139912
$$819$$ −3.80451 −0.132940
$$820$$ −47.1059 −1.64501
$$821$$ −49.5896 −1.73069 −0.865344 0.501178i $$-0.832900\pi$$
−0.865344 + 0.501178i $$0.832900\pi$$
$$822$$ 2.71648 0.0947483
$$823$$ 23.1908 0.808380 0.404190 0.914675i $$-0.367553\pi$$
0.404190 + 0.914675i $$0.367553\pi$$
$$824$$ 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 $$0.498511\pi$$
$$830$$ 8.88315 0.308339
$$831$$ −18.2788 −0.634085
$$832$$ 27.9057 0.967456
$$833$$ −0.334898 −0.0116035
$$834$$ 3.77569 0.130741
$$835$$ 69.3286 2.39922
$$836$$ 0 0
$$837$$ 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 $$0.656555\pi$$
$$840$$ 2.53039 0.0873066
$$841$$ −28.9618 −0.998682
$$842$$ −2.59976 −0.0895938
$$843$$ 6.74374 0.232267
$$844$$ −8.43767 −0.290436
$$845$$ 5.60902 0.192956
$$846$$ −2.03273 −0.0698868
$$847$$ 0 0
$$848$$ 30.4455 1.04550
$$849$$ 23.3575 0.801626
$$850$$ −0.531305 −0.0182236
$$851$$ 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
$$855$$ −30.9665 −1.05903
$$856$$ −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 0
$$859$$ 20.5576 0.701418 0.350709 0.936485i $$-0.385941\pi$$
0.350709 + 0.936485i $$0.385941\pi$$
$$860$$ 17.5172 0.597332
$$861$$ −6.27882 −0.213982
$$862$$ −0.515515 −0.0175585
$$863$$ −9.66510 −0.329004 −0.164502 0.986377i $$-0.552602\pi$$
−0.164502 + 0.986377i $$0.552602\pi$$
$$864$$ −1.97196 −0.0670875
$$865$$ 6.54825 0.222647
$$866$$ 2.83646 0.0963868
$$867$$ 16.8878 0.573541
$$868$$ 19.6090 0.665574
$$869$$ 0 0
$$870$$ 0.124538 0.00422224
$$871$$ 0.530387 0.0179715
$$872$$ 7.68388 0.260209
$$873$$ 0.0560785 0.00189797
$$874$$ 2.26943 0.0767646
$$875$$ 17.0226 0.575467
$$876$$ −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 $$-0.398182\pi$$
0.314445 + 0.949276i $$0.398182\pi$$
$$884$$ −2.51252 −0.0845053
$$885$$ −14.2610 −0.479377
$$886$$ −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
$$890$$ −6.29917 −0.211149
$$891$$ 0 0
$$892$$ 20.2509 0.678051
$$893$$ 98.8076 3.30647
$$894$$ 1.37233 0.0458975
$$895$$ −45.6541 −1.52605
$$896$$ −5.17214 −0.172789
$$897$$ −6.33490 −0.211516
$$898$$ 4.60276 0.153596
$$899$$ 1.94392 0.0648334
$$900$$ −18.6830 −0.622765
$$901$$ −2.66041 −0.0886310
$$902$$ 0 0
$$903$$ 2.33490 0.0777006
$$904$$ 0.884720 0.0294254
$$905$$ 2.76160 0.0917987
$$906$$ −2.22274 −0.0738456
$$907$$ 49.0965 1.63022 0.815111 0.579304i $$-0.196676\pi$$
0.815111 + 0.579304i $$0.196676\pi$$
$$908$$ −0.770991 −0.0255862
$$909$$ 18.8831 0.626314
$$910$$ −2.42371 −0.0803452
$$911$$ 29.3753 0.973248 0.486624 0.873612i $$-0.338228\pi$$
0.486624 + 0.873612i $$0.338228\pi$$
$$912$$ 31.1947 1.03296
$$913$$ 0 0
$$914$$ 1.33020 0.0439992
$$915$$ 22.8271 0.754640
$$916$$ −15.6651 −0.517590
$$917$$ −20.5576 −0.678873
$$918$$ 0.0560785 0.00185087
$$919$$ 27.0047 0.890803 0.445401 0.895331i $$-0.353061\pi$$
0.445401 + 0.895331i $$0.353061\pi$$
$$920$$ 4.21335 0.138910
$$921$$ −12.5576 −0.413788
$$922$$ −1.38471 −0.0456030
$$923$$ −17.7663 −0.584785
$$924$$ 0 0
$$925$$ −42.3910 −1.39381
$$926$$ 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 $$0.102105\pi$$
$$930$$ 6.33490 0.207730
$$931$$ −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
$$940$$ 91.0741 2.97051
$$941$$ 34.2788 1.11746 0.558729 0.829350i $$-0.311289\pi$$
0.558729 + 0.829350i $$0.311289\pi$$
$$942$$ 2.83646 0.0924169
$$943$$ −10.4549 −0.340458
$$944$$ 14.3661 0.467575
$$945$$ −3.80451 −0.123761
$$946$$ 0 0
$$947$$ −18.1573 −0.590032 −0.295016 0.955492i $$-0.595325\pi$$
−0.295016 + 0.955492i $$0.595325\pi$$
$$948$$ −6.56703 −0.213287
$$949$$ 15.9618 0.518141
$$950$$ −12.9129 −0.418950
$$951$$ −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
$$955$$ 20.0655 0.649303
$$956$$ 21.2968 0.688788
$$957$$ 0 0
$$958$$ −6.33490 −0.204671
$$959$$ −16.2227 −0.523860
$$960$$ 27.9057 0.900653
$$961$$ 67.8816 2.18973
$$962$$ 2.85041 0.0919010
$$963$$ 8.13941 0.262289
$$964$$ −23.8279 −0.767444
$$965$$ 75.4502 2.42883
$$966$$ 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 0
$$969$$ −2.72588 −0.0875677
$$970$$ 0.0357255 0.00114708
$$971$$ 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
$$975$$ 36.0451 1.15437
$$976$$ −22.9953 −0.736062
$$977$$ 6.71648 0.214879 0.107440 0.994212i $$-0.465735\pi$$
0.107440 + 0.994212i $$0.465735\pi$$
$$978$$ 1.13862 0.0364092
$$979$$ 0 0
$$980$$ −7.50235 −0.239654
$$981$$ −11.5529 −0.368857
$$982$$ −5.20409 −0.166069
$$983$$ 9.99061 0.318651 0.159325 0.987226i $$-0.449068\pi$$
0.159325 + 0.987226i $$0.449068\pi$$
$$984$$ 4.17605 0.133128
$$985$$ −10.1573 −0.323638
$$986$$ 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
$$995$$ 51.5623 1.63464
$$996$$ 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 2541.2.a.bi.1.2 3
3.2 odd 2 7623.2.a.cb.1.2 3
11.10 odd 2 231.2.a.d.1.2 3
33.32 even 2 693.2.a.m.1.2 3
44.43 even 2 3696.2.a.bp.1.3 3
55.54 odd 2 5775.2.a.bw.1.2 3
77.76 even 2 1617.2.a.s.1.2 3
231.230 odd 2 4851.2.a.bp.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 11.10 odd 2
693.2.a.m.1.2 3 33.32 even 2
1617.2.a.s.1.2 3 77.76 even 2
2541.2.a.bi.1.2 3 1.1 even 1 trivial
3696.2.a.bp.1.3 3 44.43 even 2
4851.2.a.bp.1.2 3 231.230 odd 2
5775.2.a.bw.1.2 3 55.54 odd 2
7623.2.a.cb.1.2 3 3.2 odd 2