# Properties

 Label 4851.2.a.bp.1.2 Level $4851$ Weight $2$ Character 4851.1 Self dual yes Analytic conductor $38.735$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.7354300205$$ 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 $$\chi$$ $$=$$ 4851.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +O(q^{10})$$ $$q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +0.637062 q^{10} -1.00000 q^{11} -3.80451 q^{13} +3.83255 q^{16} +0.334898 q^{17} -8.13941 q^{19} -7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} -0.637062 q^{26} -0.195488 q^{29} +9.94392 q^{31} +1.97196 q^{32} +0.0560785 q^{34} -4.47431 q^{37} -1.36294 q^{38} -2.53039 q^{40} -6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} -12.1394 q^{47} +1.58647 q^{50} +7.50235 q^{52} -7.94392 q^{53} -3.80451 q^{55} -0.0327344 q^{58} +3.74843 q^{59} -6.00000 q^{61} +1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} -0.660406 q^{68} -4.66980 q^{71} -4.19549 q^{73} -0.749219 q^{74} +16.0506 q^{76} +3.33020 q^{79} +14.5810 q^{80} -1.05138 q^{82} -13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} -9.88784 q^{89} -3.28352 q^{92} -2.03273 q^{94} -30.9665 q^{95} -0.0560785 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 6 q^{4} - 3 q^{8}+O(q^{10})$$ 3 * q + 6 * q^4 - 3 * q^8 $$3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100})$$ 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 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 $$0.481144\pi$$
$$3$$ 0 0
$$4$$ −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 0
$$7$$ 0 0
$$8$$ −0.665102 −0.235149
$$9$$ 0 0
$$10$$ 0.637062 0.201457
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −3.80451 −1.05518 −0.527591 0.849499i $$-0.676905\pi$$
−0.527591 + 0.849499i $$0.676905\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.83255 0.958138
$$17$$ 0.334898 0.0812248 0.0406124 0.999175i $$-0.487069\pi$$
0.0406124 + 0.999175i $$0.487069\pi$$
$$18$$ 0 0
$$19$$ −8.13941 −1.86731 −0.933654 0.358175i $$-0.883399\pi$$
−0.933654 + 0.358175i $$0.883399\pi$$
$$20$$ −7.50235 −1.67758
$$21$$ 0 0
$$22$$ −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 0
$$25$$ 9.47431 1.89486
$$26$$ −0.637062 −0.124938
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −0.195488 −0.0363013 −0.0181506 0.999835i $$-0.505778\pi$$
−0.0181506 + 0.999835i $$0.505778\pi$$
$$30$$ 0 0
$$31$$ 9.94392 1.78598 0.892991 0.450075i $$-0.148603\pi$$
0.892991 + 0.450075i $$0.148603\pi$$
$$32$$ 1.97196 0.348597
$$33$$ 0 0
$$34$$ 0.0560785 0.00961738
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.47431 −0.735572 −0.367786 0.929911i $$-0.619884\pi$$
−0.367786 + 0.929911i $$0.619884\pi$$
$$38$$ −1.36294 −0.221098
$$39$$ 0 0
$$40$$ −2.53039 −0.400089
$$41$$ −6.27882 −0.980587 −0.490293 0.871557i $$-0.663110\pi$$
−0.490293 + 0.871557i $$0.663110\pi$$
$$42$$ 0 0
$$43$$ 2.33490 0.356069 0.178034 0.984024i $$-0.443026\pi$$
0.178034 + 0.984024i $$0.443026\pi$$
$$44$$ 1.97196 0.297284
$$45$$ 0 0
$$46$$ 0.278820 0.0411098
$$47$$ −12.1394 −1.77071 −0.885357 0.464911i $$-0.846086\pi$$
−0.885357 + 0.464911i $$0.846086\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.58647 0.224360
$$51$$ 0 0
$$52$$ 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 0
$$55$$ −3.80451 −0.513000
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −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 $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 1.66510 0.211468
$$63$$ 0 0
$$64$$ −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
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.66980 −0.554203 −0.277101 0.960841i $$-0.589374\pi$$
−0.277101 + 0.960841i $$0.589374\pi$$
$$72$$ 0 0
$$73$$ −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
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 3.33020 0.374677 0.187339 0.982295i $$-0.440014\pi$$
0.187339 + 0.982295i $$0.440014\pi$$
$$80$$ 14.5810 1.63020
$$81$$ 0 0
$$82$$ −1.05138 −0.116106
$$83$$ −13.9439 −1.53054 −0.765272 0.643707i $$-0.777396\pi$$
−0.765272 + 0.643707i $$0.777396\pi$$
$$84$$ 0 0
$$85$$ 1.27412 0.138198
$$86$$ 0.390977 0.0421601
$$87$$ 0 0
$$88$$ 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
$$91$$ 0 0
$$92$$ −3.28352 −0.342330
$$93$$ 0 0
$$94$$ −2.03273 −0.209661
$$95$$ −30.9665 −3.17709
$$96$$ 0 0
$$97$$ −0.0560785 −0.00569391 −0.00284695 0.999996i $$-0.500906\pi$$
−0.00284695 + 0.999996i $$0.500906\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −18.6830 −1.86830
$$101$$ −18.8831 −1.87894 −0.939472 0.342626i $$-0.888683\pi$$
−0.939472 + 0.342626i $$0.888683\pi$$
$$102$$ 0 0
$$103$$ 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 $$0.371287\pi$$
$$108$$ 0 0
$$109$$ 11.5529 1.10657 0.553286 0.832992i $$-0.313374\pi$$
0.553286 + 0.832992i $$0.313374\pi$$
$$110$$ −0.637062 −0.0607415
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.33020 0.125135 0.0625675 0.998041i $$-0.480071\pi$$
0.0625675 + 0.998041i $$0.480071\pi$$
$$114$$ 0 0
$$115$$ 6.33490 0.590732
$$116$$ 0.385496 0.0357924
$$117$$ 0 0
$$118$$ 0.627672 0.0577819
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −1.00470 −0.0909608
$$123$$ 0 0
$$124$$ −19.6090 −1.76094
$$125$$ 17.0226 1.52254
$$126$$ 0 0
$$127$$ 14.6137 1.29676 0.648379 0.761318i $$-0.275447\pi$$
0.648379 + 0.761318i $$0.275447\pi$$
$$128$$ −5.17214 −0.457157
$$129$$ 0 0
$$130$$ −2.42371 −0.212574
$$131$$ 20.5576 1.79613 0.898065 0.439863i $$-0.144973\pi$$
0.898065 + 0.439863i $$0.144973\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −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 0
$$139$$ −22.5482 −1.91252 −0.956259 0.292522i $$-0.905506\pi$$
−0.956259 + 0.292522i $$0.905506\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −0.781954 −0.0656201
$$143$$ 3.80451 0.318149
$$144$$ 0 0
$$145$$ −0.743738 −0.0617641
$$146$$ −0.702531 −0.0581419
$$147$$ 0 0
$$148$$ 8.82316 0.725259
$$149$$ −8.19549 −0.671401 −0.335700 0.941969i $$-0.608973\pi$$
−0.335700 + 0.941969i $$0.608973\pi$$
$$150$$ 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 0
$$154$$ 0 0
$$155$$ 37.8318 3.03872
$$156$$ 0 0
$$157$$ 16.9392 1.35190 0.675949 0.736949i $$-0.263734\pi$$
0.675949 + 0.736949i $$0.263734\pi$$
$$158$$ 0.557640 0.0443634
$$159$$ 0 0
$$160$$ 7.50235 0.593113
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −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 $$-0.749080\pi$$
−0.705059 + 0.709149i $$0.749080\pi$$
$$168$$ 0 0
$$169$$ 1.47431 0.113408
$$170$$ 0.213351 0.0163633
$$171$$ 0 0
$$172$$ −4.60433 −0.351077
$$173$$ −1.72118 −0.130859 −0.0654294 0.997857i $$-0.520842\pi$$
−0.0654294 + 0.997857i $$0.520842\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.83255 −0.288889
$$177$$ 0 0
$$178$$ −1.65571 −0.124101
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −0.725875 −0.0539539 −0.0269769 0.999636i $$-0.508588\pi$$
−0.0269769 + 0.999636i $$0.508588\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1.10746 −0.0816432
$$185$$ −17.0226 −1.25152
$$186$$ 0 0
$$187$$ −0.334898 −0.0244902
$$188$$ 23.9384 1.74589
$$189$$ 0 0
$$190$$ −5.18531 −0.376182
$$191$$ −5.27412 −0.381622 −0.190811 0.981627i $$-0.561112\pi$$
−0.190811 + 0.981627i $$0.561112\pi$$
$$192$$ 0 0
$$193$$ −19.8318 −1.42752 −0.713761 0.700390i $$-0.753010\pi$$
−0.713761 + 0.700390i $$0.753010\pi$$
$$194$$ −0.00939029 −0.000674184 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −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 $$-0.659498\pi$$
−0.480371 + 0.877065i $$0.659498\pi$$
$$200$$ −6.30138 −0.445575
$$201$$ 0 0
$$202$$ −3.16197 −0.222475
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −23.8878 −1.66840
$$206$$ 1.38628 0.0965868
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ 1.36294 0.0931685
$$215$$ 8.88315 0.605826
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 1.93453 0.131023
$$219$$ 0 0
$$220$$ 7.50235 0.505808
$$221$$ −1.27412 −0.0857069
$$222$$ 0 0
$$223$$ 10.2694 0.687692 0.343846 0.939026i $$-0.388270\pi$$
0.343846 + 0.939026i $$0.388270\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0.222741 0.0148165
$$227$$ −0.390977 −0.0259500 −0.0129750 0.999916i $$-0.504130\pi$$
−0.0129750 + 0.999916i $$0.504130\pi$$
$$228$$ 0 0
$$229$$ −7.94392 −0.524949 −0.262475 0.964939i $$-0.584539\pi$$
−0.262475 + 0.964939i $$0.584539\pi$$
$$230$$ 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 0
$$235$$ −46.1845 −3.01275
$$236$$ −7.39176 −0.481163
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.7998 −0.698582 −0.349291 0.937014i $$-0.613578\pi$$
−0.349291 + 0.937014i $$0.613578\pi$$
$$240$$ 0 0
$$241$$ 12.0833 0.778356 0.389178 0.921163i $$-0.372759\pi$$
0.389178 + 0.921163i $$0.372759\pi$$
$$242$$ 0.167449 0.0107640
$$243$$ 0 0
$$244$$ 11.8318 0.757451
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 30.9665 1.97035
$$248$$ −6.61372 −0.419972
$$249$$ 0 0
$$250$$ 2.85041 0.180276
$$251$$ 4.80921 0.303554 0.151777 0.988415i $$-0.451500\pi$$
0.151777 + 0.988415i $$0.451500\pi$$
$$252$$ 0 0
$$253$$ −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 $$-0.675006\pi$$
−0.522516 + 0.852630i $$0.675006\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 28.5428 1.77015
$$261$$ 0 0
$$262$$ 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
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 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 $$-0.540126\pi$$
−0.125726 + 0.992065i $$0.540126\pi$$
$$272$$ 1.28352 0.0778245
$$273$$ 0 0
$$274$$ 2.71648 0.164109
$$275$$ −9.47431 −0.571322
$$276$$ 0 0
$$277$$ −18.2788 −1.09827 −0.549134 0.835734i $$-0.685042\pi$$
−0.549134 + 0.835734i $$0.685042\pi$$
$$278$$ −3.77569 −0.226451
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.74374 −0.402298 −0.201149 0.979561i $$-0.564467\pi$$
−0.201149 + 0.979561i $$0.564467\pi$$
$$282$$ 0 0
$$283$$ −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
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.8878 −0.993403
$$290$$ −0.124538 −0.00731314
$$291$$ 0 0
$$292$$ 8.27334 0.484161
$$293$$ −14.1667 −0.827625 −0.413813 0.910362i $$-0.635803\pi$$
−0.413813 + 0.910362i $$0.635803\pi$$
$$294$$ 0 0
$$295$$ 14.2610 0.830305
$$296$$ 2.97587 0.172969
$$297$$ 0 0
$$298$$ −1.37233 −0.0794968
$$299$$ −6.33490 −0.366357
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −2.22274 −0.127904
$$303$$ 0 0
$$304$$ −31.1947 −1.78914
$$305$$ −22.8271 −1.30707
$$306$$ 0 0
$$307$$ 12.5576 0.716702 0.358351 0.933587i $$-0.383339\pi$$
0.358351 + 0.933587i $$0.383339\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 6.33490 0.359798
$$311$$ 9.33959 0.529600 0.264800 0.964303i $$-0.414694\pi$$
0.264800 + 0.964303i $$0.414694\pi$$
$$312$$ 0 0
$$313$$ 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 $$0.589999\pi$$
$$318$$ 0 0
$$319$$ 0.195488 0.0109453
$$320$$ −27.9057 −1.55998
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.72588 −0.151672
$$324$$ 0 0
$$325$$ −36.0451 −1.99942
$$326$$ −1.13862 −0.0630625
$$327$$ 0 0
$$328$$ 4.17605 0.230584
$$329$$ 0 0
$$330$$ 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
$$333$$ 0 0
$$334$$ −3.05138 −0.166964
$$335$$ −0.530387 −0.0289781
$$336$$ 0 0
$$337$$ 13.2835 0.723599 0.361800 0.932256i $$-0.382162\pi$$
0.361800 + 0.932256i $$0.382162\pi$$
$$338$$ 0.246872 0.0134281
$$339$$ 0 0
$$340$$ −2.51252 −0.136261
$$341$$ −9.94392 −0.538494
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −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 0
$$349$$ 2.85589 0.152873 0.0764363 0.997074i $$-0.475646\pi$$
0.0764363 + 0.997074i $$0.475646\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.97196 −0.105106
$$353$$ 24.6410 1.31151 0.655753 0.754975i $$-0.272351\pi$$
0.655753 + 0.754975i $$0.272351\pi$$
$$354$$ 0 0
$$355$$ −17.7663 −0.942937
$$356$$ 19.4984 1.03342
$$357$$ 0 0
$$358$$ 2.00939 0.106200
$$359$$ 11.8878 0.627416 0.313708 0.949519i $$-0.398429\pi$$
0.313708 + 0.949519i $$0.398429\pi$$
$$360$$ 0 0
$$361$$ 47.2500 2.48684
$$362$$ −0.121547 −0.00638838
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.9618 −0.835478
$$366$$ 0 0
$$367$$ −3.60902 −0.188389 −0.0941947 0.995554i $$-0.530028\pi$$
−0.0941947 + 0.995554i $$0.530028\pi$$
$$368$$ 6.38159 0.332663
$$369$$ 0 0
$$370$$ −2.85041 −0.148186
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.3349 0.638677 0.319338 0.947641i $$-0.396539\pi$$
0.319338 + 0.947641i $$0.396539\pi$$
$$374$$ −0.0560785 −0.00289975
$$375$$ 0 0
$$376$$ 8.07394 0.416382
$$377$$ 0.743738 0.0383045
$$378$$ 0 0
$$379$$ 23.3575 1.19979 0.599896 0.800078i $$-0.295209\pi$$
0.599896 + 0.800078i $$0.295209\pi$$
$$380$$ 61.0647 3.13255
$$381$$ 0 0
$$382$$ −0.883148 −0.0451858
$$383$$ −15.2180 −0.777606 −0.388803 0.921321i $$-0.627111\pi$$
−0.388803 + 0.921321i $$0.627111\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.32081 −0.169025
$$387$$ 0 0
$$388$$ 0.110585 0.00561408
$$389$$ −3.73057 −0.189147 −0.0945737 0.995518i $$-0.530149\pi$$
−0.0945737 + 0.995518i $$0.530149\pi$$
$$390$$ 0 0
$$391$$ 0.557640 0.0282011
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −0.447055 −0.0225223
$$395$$ 12.6698 0.637487
$$396$$ 0 0
$$397$$ −20.8925 −1.04857 −0.524283 0.851544i $$-0.675667\pi$$
−0.524283 + 0.851544i $$0.675667\pi$$
$$398$$ −2.26943 −0.113756
$$399$$ 0 0
$$400$$ 36.3108 1.81554
$$401$$ −23.2741 −1.16225 −0.581127 0.813813i $$-0.697388\pi$$
−0.581127 + 0.813813i $$0.697388\pi$$
$$402$$ 0 0
$$403$$ −37.8318 −1.88453
$$404$$ 37.2368 1.85260
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.47431 0.221783
$$408$$ 0 0
$$409$$ 23.8972 1.18164 0.590821 0.806803i $$-0.298804\pi$$
0.590821 + 0.806803i $$0.298804\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 0 0
$$412$$ −16.3255 −0.804300
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −53.0498 −2.60411
$$416$$ −7.50235 −0.367833
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 5.28352 0.256590
$$425$$ 3.17293 0.153910
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −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 $$0.523623\pi$$
$$432$$ 0 0
$$433$$ −16.9392 −0.814047 −0.407024 0.913418i $$-0.633433\pi$$
−0.407024 + 0.913418i $$0.633433\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −22.7820 −1.09106
$$437$$ −13.5529 −0.648325
$$438$$ 0 0
$$439$$ 11.0786 0.528754 0.264377 0.964419i $$-0.414834\pi$$
0.264377 + 0.964419i $$0.414834\pi$$
$$440$$ 2.53039 0.120631
$$441$$ 0 0
$$442$$ −0.213351 −0.0101481
$$443$$ 14.5482 0.691208 0.345604 0.938380i $$-0.387674\pi$$
0.345604 + 0.938380i $$0.387674\pi$$
$$444$$ 0 0
$$445$$ −37.6184 −1.78328
$$446$$ 1.71961 0.0814258
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −27.4875 −1.29721 −0.648607 0.761123i $$-0.724648\pi$$
−0.648607 + 0.761123i $$0.724648\pi$$
$$450$$ 0 0
$$451$$ 6.27882 0.295658
$$452$$ −2.62311 −0.123381
$$453$$ 0 0
$$454$$ −0.0654688 −0.00307260
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −7.94392 −0.371601 −0.185800 0.982587i $$-0.559488\pi$$
−0.185800 + 0.982587i $$0.559488\pi$$
$$458$$ −1.33020 −0.0621563
$$459$$ 0 0
$$460$$ −12.4922 −0.582450
$$461$$ 8.26943 0.385146 0.192573 0.981283i $$-0.438317\pi$$
0.192573 + 0.981283i $$0.438317\pi$$
$$462$$ 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
$$465$$ 0 0
$$466$$ −4.44706 −0.206006
$$467$$ −40.6970 −1.88323 −0.941617 0.336685i $$-0.890694\pi$$
−0.941617 + 0.336685i $$0.890694\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −7.73356 −0.356723
$$471$$ 0 0
$$472$$ −2.49309 −0.114754
$$473$$ −2.33490 −0.107359
$$474$$ 0 0
$$475$$ −77.1153 −3.53829
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −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
$$483$$ 0 0
$$484$$ −1.97196 −0.0896346
$$485$$ −0.213351 −0.00968778
$$486$$ 0 0
$$487$$ 40.5576 1.83784 0.918921 0.394442i $$-0.129062\pi$$
0.918921 + 0.394442i $$0.129062\pi$$
$$488$$ 3.99061 0.180646
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −31.0786 −1.40256 −0.701280 0.712886i $$-0.747388\pi$$
−0.701280 + 0.712886i $$0.747388\pi$$
$$492$$ 0 0
$$493$$ −0.0654688 −0.00294856
$$494$$ 5.18531 0.233298
$$495$$ 0 0
$$496$$ 38.1106 1.71122
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 15.3575 0.687494 0.343747 0.939062i $$-0.388304\pi$$
0.343747 + 0.939062i $$0.388304\pi$$
$$500$$ −33.5678 −1.50120
$$501$$ 0 0
$$502$$ 0.805298 0.0359422
$$503$$ 14.8925 0.664025 0.332013 0.943275i $$-0.392272\pi$$
0.332013 + 0.943275i $$0.392272\pi$$
$$504$$ 0 0
$$505$$ −71.8412 −3.19689
$$506$$ −0.278820 −0.0123951
$$507$$ 0 0
$$508$$ −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
$$511$$ 0 0
$$512$$ 12.6557 0.559309
$$513$$ 0 0
$$514$$ −2.80530 −0.123736
$$515$$ 31.4969 1.38792
$$516$$ 0 0
$$517$$ 12.1394 0.533891
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 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 0
$$523$$ −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
$$528$$ 0 0
$$529$$ −20.2274 −0.879454
$$530$$ −5.06077 −0.219826
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 23.8878 1.03470
$$534$$ 0 0
$$535$$ 30.9665 1.33880
$$536$$ 0.0927218 0.00400497
$$537$$ 0 0
$$538$$ −4.22274 −0.182055
$$539$$ 0 0
$$540$$ 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 0
$$544$$ 0.660406 0.0283147
$$545$$ 43.9533 1.88275
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −31.9906 −1.36657
$$549$$ 0 0
$$550$$ −1.58647 −0.0676471
$$551$$ 1.59116 0.0677857
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −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 $$0.414867\pi$$
$$558$$ 0 0
$$559$$ −8.88315 −0.375717
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.12923 −0.0476338
$$563$$ −32.7710 −1.38113 −0.690566 0.723269i $$-0.742639\pi$$
−0.690566 + 0.723269i $$0.742639\pi$$
$$564$$ 0 0
$$565$$ 5.06077 0.212908
$$566$$ −3.91119 −0.164399
$$567$$ 0 0
$$568$$ 3.10589 0.130320
$$569$$ 29.2180 1.22488 0.612442 0.790515i $$-0.290187\pi$$
0.612442 + 0.790515i $$0.290187\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 15.7757 0.657892
$$576$$ 0 0
$$577$$ 14.8831 0.619594 0.309797 0.950803i $$-0.399739\pi$$
0.309797 + 0.950803i $$0.399739\pi$$
$$578$$ −2.82786 −0.117623
$$579$$ 0 0
$$580$$ 1.46662 0.0608982
$$581$$ 0 0
$$582$$ 0 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 $$-0.529804\pi$$
−0.0934945 + 0.995620i $$0.529804\pi$$
$$588$$ 0 0
$$589$$ −80.9377 −3.33498
$$590$$ 2.38799 0.0983118
$$591$$ 0 0
$$592$$ −17.1480 −0.704779
$$593$$ 8.28821 0.340356 0.170178 0.985413i $$-0.445566\pi$$
0.170178 + 0.985413i $$0.445566\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 16.1612 0.661988
$$597$$ 0 0
$$598$$ −1.06077 −0.0433783
$$599$$ −1.99061 −0.0813341 −0.0406671 0.999173i $$-0.512948\pi$$
−0.0406671 + 0.999173i $$0.512948\pi$$
$$600$$ 0 0
$$601$$ 8.47431 0.345674 0.172837 0.984950i $$-0.444707\pi$$
0.172837 + 0.984950i $$0.444707\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 26.1761 1.06509
$$605$$ 3.80451 0.154675
$$606$$ 0 0
$$607$$ 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 0
$$610$$ −3.82237 −0.154763
$$611$$ 46.1845 1.86843
$$612$$ 0 0
$$613$$ 3.55294 0.143502 0.0717510 0.997423i $$-0.477141\pi$$
0.0717510 + 0.997423i $$0.477141\pi$$
$$614$$ 2.10277 0.0848608
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.6698 −0.912652 −0.456326 0.889813i $$-0.650835\pi$$
−0.456326 + 0.889813i $$0.650835\pi$$
$$618$$ 0 0
$$619$$ −8.71648 −0.350345 −0.175173 0.984538i $$-0.556048\pi$$
−0.175173 + 0.984538i $$0.556048\pi$$
$$620$$ −74.6028 −2.99612
$$621$$ 0 0
$$622$$ 1.56391 0.0627070
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 17.3910 0.695639
$$626$$ 0.501561 0.0200464
$$627$$ 0 0
$$628$$ −33.4035 −1.33294
$$629$$ −1.49844 −0.0597467
$$630$$ 0 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
$$633$$ 0 0
$$634$$ −1.66353 −0.0660672
$$635$$ 55.5981 2.20634
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0.0327344 0.00129597
$$639$$ 0 0
$$640$$ −19.6775 −0.777821
$$641$$ 4.89254 0.193244 0.0966218 0.995321i $$-0.469196\pi$$
0.0966218 + 0.995321i $$0.469196\pi$$
$$642$$ 0 0
$$643$$ −22.7804 −0.898371 −0.449185 0.893439i $$-0.648286\pi$$
−0.449185 + 0.893439i $$0.648286\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −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 0
$$649$$ −3.74843 −0.147139
$$650$$ −6.03573 −0.236741
$$651$$ 0 0
$$652$$ 13.4090 0.525136
$$653$$ −28.2882 −1.10700 −0.553502 0.832848i $$-0.686709\pi$$
−0.553502 + 0.832848i $$0.686709\pi$$
$$654$$ 0 0
$$655$$ 78.2118 3.05599
$$656$$ −24.0639 −0.939537
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −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 $$-0.255552\pi$$
0.694666 + 0.719333i $$0.255552\pi$$
$$662$$ 3.77569 0.146746
$$663$$ 0 0
$$664$$ 9.27412 0.359906
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.325508 −0.0126037
$$668$$ 35.9345 1.39035
$$669$$ 0 0
$$670$$ −0.0888128 −0.00343114
$$671$$ 6.00000 0.231627
$$672$$ 0 0
$$673$$ 34.9377 1.34675 0.673374 0.739302i $$-0.264844\pi$$
0.673374 + 0.739302i $$0.264844\pi$$
$$674$$ 2.22431 0.0856774
$$675$$ 0 0
$$676$$ −2.90728 −0.111818
$$677$$ 20.0094 0.769023 0.384512 0.923120i $$-0.374370\pi$$
0.384512 + 0.923120i $$0.374370\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −0.847422 −0.0324972
$$681$$ 0 0
$$682$$ −1.66510 −0.0637600
$$683$$ −25.0498 −0.958504 −0.479252 0.877677i $$-0.659092\pi$$
−0.479252 + 0.877677i $$0.659092\pi$$
$$684$$ 0 0
$$685$$ 61.7196 2.35818
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 8.94862 0.341163
$$689$$ 30.2227 1.15139
$$690$$ 0 0
$$691$$ 38.8271 1.47705 0.738526 0.674225i $$-0.235522\pi$$
0.738526 + 0.674225i $$0.235522\pi$$
$$692$$ 3.39410 0.129024
$$693$$ 0 0
$$694$$ 4.65102 0.176550
$$695$$ −85.7851 −3.25401
$$696$$ 0 0
$$697$$ −2.10277 −0.0796480
$$698$$ 0.478217 0.0181008
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 41.7757 1.57785 0.788923 0.614492i $$-0.210639\pi$$
0.788923 + 0.614492i $$0.210639\pi$$
$$702$$ 0 0
$$703$$ 36.4182 1.37354
$$704$$ 7.33490 0.276444
$$705$$ 0 0
$$706$$ 4.12611 0.155288
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −13.4041 −0.503403 −0.251702 0.967805i $$-0.580990\pi$$
−0.251702 + 0.967805i $$0.580990\pi$$
$$710$$ −2.97495 −0.111648
$$711$$ 0 0
$$712$$ 6.57642 0.246462
$$713$$ 16.5576 0.620088
$$714$$ 0 0
$$715$$ 14.4743 0.541308
$$716$$ −23.6635 −0.884348
$$717$$ 0 0
$$718$$ 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
$$721$$ 0 0
$$722$$ 7.91197 0.294453
$$723$$ 0 0
$$724$$ 1.43140 0.0531975
$$725$$ −1.85212 −0.0687859
$$726$$ 0 0
$$727$$ 32.8831 1.21957 0.609784 0.792567i $$-0.291256\pi$$
0.609784 + 0.792567i $$0.291256\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2.67279 −0.0989243
$$731$$ 0.781954 0.0289216
$$732$$ 0 0
$$733$$ 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
$$738$$ 0 0
$$739$$ 29.6651 1.09125 0.545624 0.838030i $$-0.316293\pi$$
0.545624 + 0.838030i $$0.316293\pi$$
$$740$$ 33.5678 1.23398
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 18.7998 0.689698 0.344849 0.938658i $$-0.387930\pi$$
0.344849 + 0.938658i $$0.387930\pi$$
$$744$$ 0 0
$$745$$ −31.1798 −1.14234
$$746$$ 2.06547 0.0756222
$$747$$ 0 0
$$748$$ 0.660406 0.0241469
$$749$$ 0 0
$$750$$ 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
$$753$$ 0 0
$$754$$ 0.124538 0.00453542
$$755$$ −50.5016 −1.83794
$$756$$ 0 0
$$757$$ 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 $$-0.463045\pi$$
0.115836 + 0.993268i $$0.463045\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 10.4004 0.376272
$$765$$ 0 0
$$766$$ −2.54825 −0.0920720
$$767$$ −14.2610 −0.514933
$$768$$ 0 0
$$769$$ −17.7951 −0.641708 −0.320854 0.947129i $$-0.603970\pi$$
−0.320854 + 0.947129i $$0.603970\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 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 0
$$775$$ 94.2118 3.38419
$$776$$ 0.0372979 0.00133892
$$777$$ 0 0
$$778$$ −0.624681 −0.0223959
$$779$$ 51.1059 1.83106
$$780$$ 0 0
$$781$$ 4.66980 0.167098
$$782$$ 0.0933763 0.00333913
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 64.4455 2.30016
$$786$$ 0 0
$$787$$ −31.8606 −1.13571 −0.567854 0.823130i $$-0.692226\pi$$
−0.567854 + 0.823130i $$0.692226\pi$$
$$788$$ 5.26473 0.187548
$$789$$ 0 0
$$790$$ 2.12155 0.0754813
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 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 $$-0.216850\pi$$
0.776783 + 0.629768i $$0.216850\pi$$
$$798$$ 0 0
$$799$$ −4.06547 −0.143826
$$800$$ 18.6830 0.660543
$$801$$ 0 0
$$802$$ −3.89723 −0.137616
$$803$$ 4.19549 0.148056
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −6.33490 −0.223137
$$807$$ 0 0
$$808$$ 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 0
$$811$$ −3.97275 −0.139502 −0.0697510 0.997564i $$-0.522220\pi$$
−0.0697510 + 0.997564i $$0.522220\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0.749219 0.0262601
$$815$$ −25.8700 −0.906186
$$816$$ 0 0
$$817$$ −19.0047 −0.664890
$$818$$ 4.00157 0.139912
$$819$$ 0 0
$$820$$ 47.1059 1.64501
$$821$$ −49.5896 −1.73069 −0.865344 0.501178i $$-0.832900\pi$$
−0.865344 + 0.501178i $$0.832900\pi$$
$$822$$ 0 0
$$823$$ 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 0
$$827$$ −27.7484 −0.964908 −0.482454 0.875921i $$-0.660254\pi$$
−0.482454 + 0.875921i $$0.660254\pi$$
$$828$$ 0 0
$$829$$ −0.269430 −0.00935768 −0.00467884 0.999989i $$-0.501489\pi$$
−0.00467884 + 0.999989i $$0.501489\pi$$
$$830$$ −8.88315 −0.308339
$$831$$ 0 0
$$832$$ 27.9057 0.967456
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −69.3286 −2.39922
$$836$$ −16.0506 −0.555122
$$837$$ 0 0
$$838$$ −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
$$843$$ 0 0
$$844$$ 8.43767 0.290436
$$845$$ 5.60902 0.192956
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −30.4455 −1.04550
$$849$$ 0 0
$$850$$ 0.531305 0.0182236
$$851$$ −7.45018 −0.255389
$$852$$ 0 0
$$853$$ 28.5482 0.977473 0.488737 0.872431i $$-0.337458\pi$$
0.488737 + 0.872431i $$0.337458\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −5.41353 −0.185031
$$857$$ −39.8972 −1.36286 −0.681432 0.731882i $$-0.738642\pi$$
−0.681432 + 0.731882i $$0.738642\pi$$
$$858$$ 0 0
$$859$$ −20.5576 −0.701418 −0.350709 0.936485i $$-0.614059\pi$$
−0.350709 + 0.936485i $$0.614059\pi$$
$$860$$ −17.5172 −0.597332
$$861$$ 0 0
$$862$$ −0.515515 −0.0175585
$$863$$ 9.66510 0.329004 0.164502 0.986377i $$-0.447398\pi$$
0.164502 + 0.986377i $$0.447398\pi$$
$$864$$ 0 0
$$865$$ −6.54825 −0.222647
$$866$$ −2.83646 −0.0963868
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.33020 −0.112969
$$870$$ 0 0
$$871$$ 0.530387 0.0179715
$$872$$ −7.68388 −0.260209
$$873$$ 0 0
$$874$$ −2.26943 −0.0767646
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −25.7212 −0.868543 −0.434271 0.900782i $$-0.642994\pi$$
−0.434271 + 0.900782i $$0.642994\pi$$
$$878$$ 1.85511 0.0626069
$$879$$ 0 0
$$880$$ −14.5810 −0.491525
$$881$$ −14.6316 −0.492950 −0.246475 0.969149i $$-0.579272\pi$$
−0.246475 + 0.969149i $$0.579272\pi$$
$$882$$ 0 0
$$883$$ 18.6877 0.628890 0.314445 0.949276i $$-0.398182\pi$$
0.314445 + 0.949276i $$0.398182\pi$$
$$884$$ 2.51252 0.0845053
$$885$$ 0 0
$$886$$ 2.43609 0.0818421
$$887$$ 38.6137 1.29652 0.648261 0.761418i $$-0.275497\pi$$
0.648261 + 0.761418i $$0.275497\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −6.29917 −0.211149
$$891$$ 0 0
$$892$$ −20.2509 −0.678051
$$893$$ 98.8076 3.30647
$$894$$ 0 0
$$895$$ 45.6541 1.52605
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −4.60276 −0.153596
$$899$$ −1.94392 −0.0648334
$$900$$ 0 0
$$901$$ −2.66041 −0.0886310
$$902$$ 1.05138 0.0350072
$$903$$ 0 0
$$904$$ −0.884720 −0.0294254
$$905$$ −2.76160 −0.0917987
$$906$$ 0 0
$$907$$ 49.0965 1.63022 0.815111 0.579304i $$-0.196676\pi$$
0.815111 + 0.579304i $$0.196676\pi$$
$$908$$ 0.770991 0.0255862
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −29.3753 −0.973248 −0.486624 0.873612i $$-0.661772\pi$$
−0.486624 + 0.873612i $$0.661772\pi$$
$$912$$ 0 0
$$913$$ 13.9439 0.461476
$$914$$ −1.33020 −0.0439992
$$915$$ 0 0
$$916$$ 15.6651 0.517590
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −27.0047 −0.890803 −0.445401 0.895331i $$-0.646939\pi$$
−0.445401 + 0.895331i $$0.646939\pi$$
$$920$$ −4.21335 −0.138910
$$921$$ 0 0
$$922$$ 1.38471 0.0456030
$$923$$ 17.7663 0.584785
$$924$$ 0 0
$$925$$ −42.3910 −1.39381
$$926$$ 1.63079 0.0535912
$$927$$ 0 0
$$928$$ −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
$$931$$ 0 0
$$932$$ 52.3706 1.71546
$$933$$ 0 0
$$934$$ −6.81469 −0.222983
$$935$$ −1.27412 −0.0416683
$$936$$ 0 0
$$937$$ −25.2180 −0.823838 −0.411919 0.911221i $$-0.635141\pi$$
−0.411919 + 0.911221i $$0.635141\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 91.0741 2.97051
$$941$$ −34.2788 −1.11746 −0.558729 0.829350i $$-0.688711\pi$$
−0.558729 + 0.829350i $$0.688711\pi$$
$$942$$ 0 0
$$943$$ −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 $$0.404675\pi$$
$$948$$ 0 0
$$949$$ 15.9618 0.518141
$$950$$ −12.9129 −0.418950
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 19.8045 0.641531 0.320766 0.947159i $$-0.396060\pi$$
0.320766 + 0.947159i $$0.396060\pi$$
$$954$$ 0 0
$$955$$ −20.0655 −0.649303
$$956$$ 21.2968 0.688788
$$957$$ 0 0
$$958$$ 6.33490 0.204671
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 67.8816 2.18973
$$962$$ 2.85041 0.0919010
$$963$$ 0 0
$$964$$ −23.8279 −0.767444
$$965$$ −75.4502 −2.42883
$$966$$ 0 0
$$967$$ −0.278820 −0.00896624 −0.00448312 0.999990i $$-0.501427\pi$$
−0.00448312 + 0.999990i $$0.501427\pi$$
$$968$$ −0.665102 −0.0213772
$$969$$ 0 0
$$970$$ −0.0357255 −0.00114708
$$971$$ 25.3481 0.813458 0.406729 0.913549i $$-0.366669\pi$$
0.406729 + 0.913549i $$0.366669\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 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 $$0.534265\pi$$
$$978$$ 0 0
$$979$$ 9.88784 0.316017
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −5.20409 −0.166069
$$983$$ 9.99061 0.318651 0.159325 0.987226i $$-0.449068\pi$$
0.159325 + 0.987226i $$0.449068\pi$$
$$984$$ 0 0
$$985$$ −10.1573 −0.323638
$$986$$ −0.0109627 −0.000349123 0
$$987$$ 0 0
$$988$$ −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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −51.5623 −1.63464
$$996$$ 0 0
$$997$$ −54.3333 −1.72075 −0.860377 0.509658i $$-0.829772\pi$$
−0.860377 + 0.509658i $$0.829772\pi$$
$$998$$ 2.57159 0.0814024
$$999$$ 0 0
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 4851.2.a.bp.1.2 3
3.2 odd 2 1617.2.a.s.1.2 3
7.6 odd 2 693.2.a.m.1.2 3
21.20 even 2 231.2.a.d.1.2 3
77.76 even 2 7623.2.a.cb.1.2 3
84.83 odd 2 3696.2.a.bp.1.3 3
105.104 even 2 5775.2.a.bw.1.2 3
231.230 odd 2 2541.2.a.bi.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 21.20 even 2
693.2.a.m.1.2 3 7.6 odd 2
1617.2.a.s.1.2 3 3.2 odd 2
2541.2.a.bi.1.2 3 231.230 odd 2
3696.2.a.bp.1.3 3 84.83 odd 2
4851.2.a.bp.1.2 3 1.1 even 1 trivial
5775.2.a.bw.1.2 3 105.104 even 2
7623.2.a.cb.1.2 3 77.76 even 2