# Properties

 Label 4560.2.a.br.1.3 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.51414$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.32088 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.32088 q^{7} +1.00000 q^{9} -3.70739 q^{11} +3.32088 q^{13} -1.00000 q^{15} +1.61350 q^{17} +1.00000 q^{19} -1.32088 q^{21} +7.67004 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.70739 q^{29} -9.67004 q^{31} +3.70739 q^{33} +1.32088 q^{35} +5.96265 q^{37} -3.32088 q^{39} +2.29261 q^{41} +8.73566 q^{43} +1.00000 q^{45} -11.6700 q^{47} -5.25526 q^{49} -1.61350 q^{51} +10.4431 q^{53} -3.70739 q^{55} -1.00000 q^{57} -12.6983 q^{59} +9.02827 q^{61} +1.32088 q^{63} +3.32088 q^{65} +13.2835 q^{67} -7.67004 q^{69} +8.69832 q^{71} -12.0565 q^{73} -1.00000 q^{75} -4.89703 q^{77} +14.0565 q^{79} +1.00000 q^{81} -5.02827 q^{83} +1.61350 q^{85} +1.70739 q^{87} -1.70739 q^{89} +4.38650 q^{91} +9.67004 q^{93} +1.00000 q^{95} -0.679116 q^{97} -3.70739 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 + 3 * q^19 + 4 * q^21 - 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^33 - 4 * q^35 - 6 * q^37 - 2 * q^39 + 12 * q^41 + 8 * q^43 + 3 * q^45 - 6 * q^47 + 3 * q^49 - 2 * q^51 + 8 * q^53 - 6 * q^55 - 3 * q^57 + 4 * q^59 + 14 * q^61 - 4 * q^63 + 2 * q^65 + 8 * q^67 + 6 * q^69 - 16 * q^71 - 10 * q^73 - 3 * q^75 + 20 * q^77 + 16 * q^79 + 3 * q^81 - 2 * q^83 + 2 * q^85 + 16 * q^91 + 3 * q^95 - 10 * q^97 - 6 * 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 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.32088 0.499247 0.249624 0.968343i $$-0.419693\pi$$
0.249624 + 0.968343i $$0.419693\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.70739 −1.11782 −0.558910 0.829228i $$-0.688780\pi$$
−0.558910 + 0.829228i $$0.688780\pi$$
$$12$$ 0 0
$$13$$ 3.32088 0.921048 0.460524 0.887647i $$-0.347662\pi$$
0.460524 + 0.887647i $$0.347662\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 1.61350 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.32088 −0.288241
$$22$$ 0 0
$$23$$ 7.67004 1.59931 0.799657 0.600457i $$-0.205015\pi$$
0.799657 + 0.600457i $$0.205015\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −1.70739 −0.317054 −0.158527 0.987355i $$-0.550675\pi$$
−0.158527 + 0.987355i $$0.550675\pi$$
$$30$$ 0 0
$$31$$ −9.67004 −1.73679 −0.868395 0.495872i $$-0.834848\pi$$
−0.868395 + 0.495872i $$0.834848\pi$$
$$32$$ 0 0
$$33$$ 3.70739 0.645374
$$34$$ 0 0
$$35$$ 1.32088 0.223270
$$36$$ 0 0
$$37$$ 5.96265 0.980254 0.490127 0.871651i $$-0.336950\pi$$
0.490127 + 0.871651i $$0.336950\pi$$
$$38$$ 0 0
$$39$$ −3.32088 −0.531767
$$40$$ 0 0
$$41$$ 2.29261 0.358046 0.179023 0.983845i $$-0.442706\pi$$
0.179023 + 0.983845i $$0.442706\pi$$
$$42$$ 0 0
$$43$$ 8.73566 1.33218 0.666088 0.745873i $$-0.267967\pi$$
0.666088 + 0.745873i $$0.267967\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −11.6700 −1.70225 −0.851125 0.524963i $$-0.824079\pi$$
−0.851125 + 0.524963i $$0.824079\pi$$
$$48$$ 0 0
$$49$$ −5.25526 −0.750752
$$50$$ 0 0
$$51$$ −1.61350 −0.225935
$$52$$ 0 0
$$53$$ 10.4431 1.43446 0.717232 0.696835i $$-0.245409\pi$$
0.717232 + 0.696835i $$0.245409\pi$$
$$54$$ 0 0
$$55$$ −3.70739 −0.499904
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −12.6983 −1.65318 −0.826590 0.562805i $$-0.809722\pi$$
−0.826590 + 0.562805i $$0.809722\pi$$
$$60$$ 0 0
$$61$$ 9.02827 1.15595 0.577976 0.816054i $$-0.303843\pi$$
0.577976 + 0.816054i $$0.303843\pi$$
$$62$$ 0 0
$$63$$ 1.32088 0.166416
$$64$$ 0 0
$$65$$ 3.32088 0.411905
$$66$$ 0 0
$$67$$ 13.2835 1.62284 0.811421 0.584462i $$-0.198694\pi$$
0.811421 + 0.584462i $$0.198694\pi$$
$$68$$ 0 0
$$69$$ −7.67004 −0.923365
$$70$$ 0 0
$$71$$ 8.69832 1.03230 0.516150 0.856498i $$-0.327365\pi$$
0.516150 + 0.856498i $$0.327365\pi$$
$$72$$ 0 0
$$73$$ −12.0565 −1.41111 −0.705556 0.708654i $$-0.749303\pi$$
−0.705556 + 0.708654i $$0.749303\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −4.89703 −0.558069
$$78$$ 0 0
$$79$$ 14.0565 1.58149 0.790743 0.612149i $$-0.209695\pi$$
0.790743 + 0.612149i $$0.209695\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −5.02827 −0.551925 −0.275962 0.961168i $$-0.588997\pi$$
−0.275962 + 0.961168i $$0.588997\pi$$
$$84$$ 0 0
$$85$$ 1.61350 0.175008
$$86$$ 0 0
$$87$$ 1.70739 0.183051
$$88$$ 0 0
$$89$$ −1.70739 −0.180983 −0.0904915 0.995897i $$-0.528844\pi$$
−0.0904915 + 0.995897i $$0.528844\pi$$
$$90$$ 0 0
$$91$$ 4.38650 0.459831
$$92$$ 0 0
$$93$$ 9.67004 1.00274
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −0.679116 −0.0689537 −0.0344769 0.999405i $$-0.510977\pi$$
−0.0344769 + 0.999405i $$0.510977\pi$$
$$98$$ 0 0
$$99$$ −3.70739 −0.372607
$$100$$ 0 0
$$101$$ 12.6418 1.25790 0.628952 0.777445i $$-0.283484\pi$$
0.628952 + 0.777445i $$0.283484\pi$$
$$102$$ 0 0
$$103$$ −6.05655 −0.596769 −0.298385 0.954446i $$-0.596448\pi$$
−0.298385 + 0.954446i $$0.596448\pi$$
$$104$$ 0 0
$$105$$ −1.32088 −0.128905
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 10.6983 1.02471 0.512356 0.858773i $$-0.328773\pi$$
0.512356 + 0.858773i $$0.328773\pi$$
$$110$$ 0 0
$$111$$ −5.96265 −0.565950
$$112$$ 0 0
$$113$$ −16.3118 −1.53449 −0.767243 0.641356i $$-0.778372\pi$$
−0.767243 + 0.641356i $$0.778372\pi$$
$$114$$ 0 0
$$115$$ 7.67004 0.715235
$$116$$ 0 0
$$117$$ 3.32088 0.307016
$$118$$ 0 0
$$119$$ 2.13124 0.195371
$$120$$ 0 0
$$121$$ 2.74474 0.249521
$$122$$ 0 0
$$123$$ −2.29261 −0.206718
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 0 0
$$129$$ −8.73566 −0.769132
$$130$$ 0 0
$$131$$ −4.29261 −0.375047 −0.187524 0.982260i $$-0.560046\pi$$
−0.187524 + 0.982260i $$0.560046\pi$$
$$132$$ 0 0
$$133$$ 1.32088 0.114535
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ 8.58522 0.728189 0.364094 0.931362i $$-0.381379\pi$$
0.364094 + 0.931362i $$0.381379\pi$$
$$140$$ 0 0
$$141$$ 11.6700 0.982795
$$142$$ 0 0
$$143$$ −12.3118 −1.02957
$$144$$ 0 0
$$145$$ −1.70739 −0.141791
$$146$$ 0 0
$$147$$ 5.25526 0.433447
$$148$$ 0 0
$$149$$ 12.0565 0.987711 0.493855 0.869544i $$-0.335587\pi$$
0.493855 + 0.869544i $$0.335587\pi$$
$$150$$ 0 0
$$151$$ 22.9536 1.86794 0.933968 0.357357i $$-0.116322\pi$$
0.933968 + 0.357357i $$0.116322\pi$$
$$152$$ 0 0
$$153$$ 1.61350 0.130443
$$154$$ 0 0
$$155$$ −9.67004 −0.776717
$$156$$ 0 0
$$157$$ −11.8688 −0.947230 −0.473615 0.880732i $$-0.657051\pi$$
−0.473615 + 0.880732i $$0.657051\pi$$
$$158$$ 0 0
$$159$$ −10.4431 −0.828188
$$160$$ 0 0
$$161$$ 10.1312 0.798454
$$162$$ 0 0
$$163$$ 0.0373465 0.00292520 0.00146260 0.999999i $$-0.499534\pi$$
0.00146260 + 0.999999i $$0.499534\pi$$
$$164$$ 0 0
$$165$$ 3.70739 0.288620
$$166$$ 0 0
$$167$$ 11.0283 0.853393 0.426697 0.904395i $$-0.359677\pi$$
0.426697 + 0.904395i $$0.359677\pi$$
$$168$$ 0 0
$$169$$ −1.97173 −0.151671
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −17.8578 −1.35771 −0.678853 0.734274i $$-0.737523\pi$$
−0.678853 + 0.734274i $$0.737523\pi$$
$$174$$ 0 0
$$175$$ 1.32088 0.0998495
$$176$$ 0 0
$$177$$ 12.6983 0.954464
$$178$$ 0 0
$$179$$ 19.9253 1.48929 0.744644 0.667462i $$-0.232619\pi$$
0.744644 + 0.667462i $$0.232619\pi$$
$$180$$ 0 0
$$181$$ −15.4713 −1.14997 −0.574987 0.818162i $$-0.694993\pi$$
−0.574987 + 0.818162i $$0.694993\pi$$
$$182$$ 0 0
$$183$$ −9.02827 −0.667389
$$184$$ 0 0
$$185$$ 5.96265 0.438383
$$186$$ 0 0
$$187$$ −5.98185 −0.437437
$$188$$ 0 0
$$189$$ −1.32088 −0.0960802
$$190$$ 0 0
$$191$$ −18.3492 −1.32770 −0.663849 0.747866i $$-0.731078\pi$$
−0.663849 + 0.747866i $$0.731078\pi$$
$$192$$ 0 0
$$193$$ 12.0192 0.865161 0.432581 0.901595i $$-0.357603\pi$$
0.432581 + 0.901595i $$0.357603\pi$$
$$194$$ 0 0
$$195$$ −3.32088 −0.237813
$$196$$ 0 0
$$197$$ 4.84049 0.344870 0.172435 0.985021i $$-0.444836\pi$$
0.172435 + 0.985021i $$0.444836\pi$$
$$198$$ 0 0
$$199$$ 20.6983 1.46726 0.733632 0.679547i $$-0.237823\pi$$
0.733632 + 0.679547i $$0.237823\pi$$
$$200$$ 0 0
$$201$$ −13.2835 −0.936949
$$202$$ 0 0
$$203$$ −2.25526 −0.158289
$$204$$ 0 0
$$205$$ 2.29261 0.160123
$$206$$ 0 0
$$207$$ 7.67004 0.533105
$$208$$ 0 0
$$209$$ −3.70739 −0.256445
$$210$$ 0 0
$$211$$ −2.05655 −0.141579 −0.0707893 0.997491i $$-0.522552\pi$$
−0.0707893 + 0.997491i $$0.522552\pi$$
$$212$$ 0 0
$$213$$ −8.69832 −0.595999
$$214$$ 0 0
$$215$$ 8.73566 0.595767
$$216$$ 0 0
$$217$$ −12.7730 −0.867088
$$218$$ 0 0
$$219$$ 12.0565 0.814706
$$220$$ 0 0
$$221$$ 5.35823 0.360434
$$222$$ 0 0
$$223$$ −6.05655 −0.405576 −0.202788 0.979223i $$-0.565000\pi$$
−0.202788 + 0.979223i $$0.565000\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 28.3118 1.87912 0.939560 0.342383i $$-0.111234\pi$$
0.939560 + 0.342383i $$0.111234\pi$$
$$228$$ 0 0
$$229$$ −1.80128 −0.119032 −0.0595161 0.998227i $$-0.518956\pi$$
−0.0595161 + 0.998227i $$0.518956\pi$$
$$230$$ 0 0
$$231$$ 4.89703 0.322201
$$232$$ 0 0
$$233$$ −3.94345 −0.258344 −0.129172 0.991622i $$-0.541232\pi$$
−0.129172 + 0.991622i $$0.541232\pi$$
$$234$$ 0 0
$$235$$ −11.6700 −0.761270
$$236$$ 0 0
$$237$$ −14.0565 −0.913071
$$238$$ 0 0
$$239$$ 16.8031 1.08690 0.543452 0.839440i $$-0.317117\pi$$
0.543452 + 0.839440i $$0.317117\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −5.25526 −0.335747
$$246$$ 0 0
$$247$$ 3.32088 0.211303
$$248$$ 0 0
$$249$$ 5.02827 0.318654
$$250$$ 0 0
$$251$$ −24.9909 −1.57741 −0.788707 0.614770i $$-0.789249\pi$$
−0.788707 + 0.614770i $$0.789249\pi$$
$$252$$ 0 0
$$253$$ −28.4358 −1.78775
$$254$$ 0 0
$$255$$ −1.61350 −0.101041
$$256$$ 0 0
$$257$$ −2.91518 −0.181844 −0.0909219 0.995858i $$-0.528981\pi$$
−0.0909219 + 0.995858i $$0.528981\pi$$
$$258$$ 0 0
$$259$$ 7.87598 0.489389
$$260$$ 0 0
$$261$$ −1.70739 −0.105685
$$262$$ 0 0
$$263$$ 22.3118 1.37581 0.687903 0.725803i $$-0.258532\pi$$
0.687903 + 0.725803i $$0.258532\pi$$
$$264$$ 0 0
$$265$$ 10.4431 0.641512
$$266$$ 0 0
$$267$$ 1.70739 0.104491
$$268$$ 0 0
$$269$$ 15.0656 0.918567 0.459284 0.888290i $$-0.348106\pi$$
0.459284 + 0.888290i $$0.348106\pi$$
$$270$$ 0 0
$$271$$ −5.47133 −0.332359 −0.166180 0.986095i $$-0.553143\pi$$
−0.166180 + 0.986095i $$0.553143\pi$$
$$272$$ 0 0
$$273$$ −4.38650 −0.265483
$$274$$ 0 0
$$275$$ −3.70739 −0.223564
$$276$$ 0 0
$$277$$ −10.7730 −0.647287 −0.323644 0.946179i $$-0.604908\pi$$
−0.323644 + 0.946179i $$0.604908\pi$$
$$278$$ 0 0
$$279$$ −9.67004 −0.578930
$$280$$ 0 0
$$281$$ 17.8205 1.06308 0.531540 0.847033i $$-0.321613\pi$$
0.531540 + 0.847033i $$0.321613\pi$$
$$282$$ 0 0
$$283$$ 30.2070 1.79562 0.897810 0.440384i $$-0.145158\pi$$
0.897810 + 0.440384i $$0.145158\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ 3.02827 0.178753
$$288$$ 0 0
$$289$$ −14.3966 −0.846861
$$290$$ 0 0
$$291$$ 0.679116 0.0398105
$$292$$ 0 0
$$293$$ −18.2553 −1.06648 −0.533242 0.845963i $$-0.679026\pi$$
−0.533242 + 0.845963i $$0.679026\pi$$
$$294$$ 0 0
$$295$$ −12.6983 −0.739325
$$296$$ 0 0
$$297$$ 3.70739 0.215125
$$298$$ 0 0
$$299$$ 25.4713 1.47304
$$300$$ 0 0
$$301$$ 11.5388 0.665085
$$302$$ 0 0
$$303$$ −12.6418 −0.726251
$$304$$ 0 0
$$305$$ 9.02827 0.516957
$$306$$ 0 0
$$307$$ 16.5852 0.946569 0.473284 0.880910i $$-0.343068\pi$$
0.473284 + 0.880910i $$0.343068\pi$$
$$308$$ 0 0
$$309$$ 6.05655 0.344545
$$310$$ 0 0
$$311$$ −11.5196 −0.653217 −0.326608 0.945160i $$-0.605906\pi$$
−0.326608 + 0.945160i $$0.605906\pi$$
$$312$$ 0 0
$$313$$ 32.7549 1.85141 0.925707 0.378241i $$-0.123471\pi$$
0.925707 + 0.378241i $$0.123471\pi$$
$$314$$ 0 0
$$315$$ 1.32088 0.0744234
$$316$$ 0 0
$$317$$ 15.2161 0.854619 0.427310 0.904105i $$-0.359461\pi$$
0.427310 + 0.904105i $$0.359461\pi$$
$$318$$ 0 0
$$319$$ 6.32996 0.354410
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.61350 0.0897773
$$324$$ 0 0
$$325$$ 3.32088 0.184210
$$326$$ 0 0
$$327$$ −10.6983 −0.591618
$$328$$ 0 0
$$329$$ −15.4148 −0.849844
$$330$$ 0 0
$$331$$ 26.9536 1.48150 0.740751 0.671779i $$-0.234470\pi$$
0.740751 + 0.671779i $$0.234470\pi$$
$$332$$ 0 0
$$333$$ 5.96265 0.326751
$$334$$ 0 0
$$335$$ 13.2835 0.725757
$$336$$ 0 0
$$337$$ −17.5652 −0.956839 −0.478419 0.878132i $$-0.658790\pi$$
−0.478419 + 0.878132i $$0.658790\pi$$
$$338$$ 0 0
$$339$$ 16.3118 0.885936
$$340$$ 0 0
$$341$$ 35.8506 1.94142
$$342$$ 0 0
$$343$$ −16.1878 −0.874058
$$344$$ 0 0
$$345$$ −7.67004 −0.412941
$$346$$ 0 0
$$347$$ 34.3118 1.84195 0.920977 0.389616i $$-0.127392\pi$$
0.920977 + 0.389616i $$0.127392\pi$$
$$348$$ 0 0
$$349$$ 35.2835 1.88868 0.944342 0.328965i $$-0.106700\pi$$
0.944342 + 0.328965i $$0.106700\pi$$
$$350$$ 0 0
$$351$$ −3.32088 −0.177256
$$352$$ 0 0
$$353$$ −8.71646 −0.463930 −0.231965 0.972724i $$-0.574516\pi$$
−0.231965 + 0.972724i $$0.574516\pi$$
$$354$$ 0 0
$$355$$ 8.69832 0.461659
$$356$$ 0 0
$$357$$ −2.13124 −0.112797
$$358$$ 0 0
$$359$$ −30.4623 −1.60774 −0.803868 0.594808i $$-0.797228\pi$$
−0.803868 + 0.594808i $$0.797228\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −2.74474 −0.144061
$$364$$ 0 0
$$365$$ −12.0565 −0.631069
$$366$$ 0 0
$$367$$ 27.5652 1.43889 0.719446 0.694548i $$-0.244396\pi$$
0.719446 + 0.694548i $$0.244396\pi$$
$$368$$ 0 0
$$369$$ 2.29261 0.119349
$$370$$ 0 0
$$371$$ 13.7941 0.716152
$$372$$ 0 0
$$373$$ 16.0939 0.833310 0.416655 0.909065i $$-0.363202\pi$$
0.416655 + 0.909065i $$0.363202\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −5.67004 −0.292022
$$378$$ 0 0
$$379$$ −9.01013 −0.462819 −0.231410 0.972856i $$-0.574334\pi$$
−0.231410 + 0.972856i $$0.574334\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 23.3401 1.19262 0.596311 0.802753i $$-0.296632\pi$$
0.596311 + 0.802753i $$0.296632\pi$$
$$384$$ 0 0
$$385$$ −4.89703 −0.249576
$$386$$ 0 0
$$387$$ 8.73566 0.444059
$$388$$ 0 0
$$389$$ 20.0565 1.01691 0.508454 0.861089i $$-0.330217\pi$$
0.508454 + 0.861089i $$0.330217\pi$$
$$390$$ 0 0
$$391$$ 12.3756 0.625860
$$392$$ 0 0
$$393$$ 4.29261 0.216534
$$394$$ 0 0
$$395$$ 14.0565 0.707262
$$396$$ 0 0
$$397$$ −21.7375 −1.09097 −0.545487 0.838119i $$-0.683655\pi$$
−0.545487 + 0.838119i $$0.683655\pi$$
$$398$$ 0 0
$$399$$ −1.32088 −0.0661269
$$400$$ 0 0
$$401$$ −7.17872 −0.358488 −0.179244 0.983805i $$-0.557365\pi$$
−0.179244 + 0.983805i $$0.557365\pi$$
$$402$$ 0 0
$$403$$ −32.1131 −1.59967
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −22.1059 −1.09575
$$408$$ 0 0
$$409$$ 16.2443 0.803231 0.401615 0.915808i $$-0.368449\pi$$
0.401615 + 0.915808i $$0.368449\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ −16.7730 −0.825346
$$414$$ 0 0
$$415$$ −5.02827 −0.246828
$$416$$ 0 0
$$417$$ −8.58522 −0.420420
$$418$$ 0 0
$$419$$ −16.9909 −0.830061 −0.415031 0.909807i $$-0.636229\pi$$
−0.415031 + 0.909807i $$0.636229\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ −11.6700 −0.567417
$$424$$ 0 0
$$425$$ 1.61350 0.0782660
$$426$$ 0 0
$$427$$ 11.9253 0.577106
$$428$$ 0 0
$$429$$ 12.3118 0.594420
$$430$$ 0 0
$$431$$ −22.6418 −1.09062 −0.545308 0.838236i $$-0.683587\pi$$
−0.545308 + 0.838236i $$0.683587\pi$$
$$432$$ 0 0
$$433$$ −12.2070 −0.586630 −0.293315 0.956016i $$-0.594759\pi$$
−0.293315 + 0.956016i $$0.594759\pi$$
$$434$$ 0 0
$$435$$ 1.70739 0.0818631
$$436$$ 0 0
$$437$$ 7.67004 0.366908
$$438$$ 0 0
$$439$$ 31.8506 1.52015 0.760073 0.649837i $$-0.225163\pi$$
0.760073 + 0.649837i $$0.225163\pi$$
$$440$$ 0 0
$$441$$ −5.25526 −0.250251
$$442$$ 0 0
$$443$$ −8.95358 −0.425397 −0.212699 0.977118i $$-0.568225\pi$$
−0.212699 + 0.977118i $$0.568225\pi$$
$$444$$ 0 0
$$445$$ −1.70739 −0.0809380
$$446$$ 0 0
$$447$$ −12.0565 −0.570255
$$448$$ 0 0
$$449$$ 5.51960 0.260486 0.130243 0.991482i $$-0.458424\pi$$
0.130243 + 0.991482i $$0.458424\pi$$
$$450$$ 0 0
$$451$$ −8.49960 −0.400231
$$452$$ 0 0
$$453$$ −22.9536 −1.07845
$$454$$ 0 0
$$455$$ 4.38650 0.205643
$$456$$ 0 0
$$457$$ 0.131241 0.00613918 0.00306959 0.999995i $$-0.499023\pi$$
0.00306959 + 0.999995i $$0.499023\pi$$
$$458$$ 0 0
$$459$$ −1.61350 −0.0753115
$$460$$ 0 0
$$461$$ −14.1131 −0.657312 −0.328656 0.944450i $$-0.606596\pi$$
−0.328656 + 0.944450i $$0.606596\pi$$
$$462$$ 0 0
$$463$$ −5.98080 −0.277951 −0.138976 0.990296i $$-0.544381\pi$$
−0.138976 + 0.990296i $$0.544381\pi$$
$$464$$ 0 0
$$465$$ 9.67004 0.448437
$$466$$ 0 0
$$467$$ 6.38650 0.295532 0.147766 0.989022i $$-0.452792\pi$$
0.147766 + 0.989022i $$0.452792\pi$$
$$468$$ 0 0
$$469$$ 17.5460 0.810200
$$470$$ 0 0
$$471$$ 11.8688 0.546884
$$472$$ 0 0
$$473$$ −32.3865 −1.48913
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 10.4431 0.478155
$$478$$ 0 0
$$479$$ −18.8597 −0.861721 −0.430861 0.902419i $$-0.641790\pi$$
−0.430861 + 0.902419i $$0.641790\pi$$
$$480$$ 0 0
$$481$$ 19.8013 0.902861
$$482$$ 0 0
$$483$$ −10.1312 −0.460987
$$484$$ 0 0
$$485$$ −0.679116 −0.0308370
$$486$$ 0 0
$$487$$ −10.1312 −0.459090 −0.229545 0.973298i $$-0.573724\pi$$
−0.229545 + 0.973298i $$0.573724\pi$$
$$488$$ 0 0
$$489$$ −0.0373465 −0.00168887
$$490$$ 0 0
$$491$$ −1.06562 −0.0480908 −0.0240454 0.999711i $$-0.507655\pi$$
−0.0240454 + 0.999711i $$0.507655\pi$$
$$492$$ 0 0
$$493$$ −2.75486 −0.124073
$$494$$ 0 0
$$495$$ −3.70739 −0.166635
$$496$$ 0 0
$$497$$ 11.4895 0.515373
$$498$$ 0 0
$$499$$ 14.2443 0.637664 0.318832 0.947811i $$-0.396709\pi$$
0.318832 + 0.947811i $$0.396709\pi$$
$$500$$ 0 0
$$501$$ −11.0283 −0.492707
$$502$$ 0 0
$$503$$ −30.9717 −1.38096 −0.690481 0.723351i $$-0.742601\pi$$
−0.690481 + 0.723351i $$0.742601\pi$$
$$504$$ 0 0
$$505$$ 12.6418 0.562551
$$506$$ 0 0
$$507$$ 1.97173 0.0875674
$$508$$ 0 0
$$509$$ −30.4057 −1.34771 −0.673855 0.738864i $$-0.735363\pi$$
−0.673855 + 0.738864i $$0.735363\pi$$
$$510$$ 0 0
$$511$$ −15.9253 −0.704494
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ −6.05655 −0.266883
$$516$$ 0 0
$$517$$ 43.2654 1.90281
$$518$$ 0 0
$$519$$ 17.8578 0.783872
$$520$$ 0 0
$$521$$ −4.34916 −0.190540 −0.0952700 0.995451i $$-0.530371\pi$$
−0.0952700 + 0.995451i $$0.530371\pi$$
$$522$$ 0 0
$$523$$ 8.69832 0.380351 0.190175 0.981750i $$-0.439094\pi$$
0.190175 + 0.981750i $$0.439094\pi$$
$$524$$ 0 0
$$525$$ −1.32088 −0.0576481
$$526$$ 0 0
$$527$$ −15.6026 −0.679659
$$528$$ 0 0
$$529$$ 35.8296 1.55781
$$530$$ 0 0
$$531$$ −12.6983 −0.551060
$$532$$ 0 0
$$533$$ 7.61350 0.329777
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −19.9253 −0.859840
$$538$$ 0 0
$$539$$ 19.4833 0.839206
$$540$$ 0 0
$$541$$ 8.45398 0.363465 0.181733 0.983348i $$-0.441830\pi$$
0.181733 + 0.983348i $$0.441830\pi$$
$$542$$ 0 0
$$543$$ 15.4713 0.663938
$$544$$ 0 0
$$545$$ 10.6983 0.458266
$$546$$ 0 0
$$547$$ −11.5279 −0.492896 −0.246448 0.969156i $$-0.579264\pi$$
−0.246448 + 0.969156i $$0.579264\pi$$
$$548$$ 0 0
$$549$$ 9.02827 0.385317
$$550$$ 0 0
$$551$$ −1.70739 −0.0727372
$$552$$ 0 0
$$553$$ 18.5671 0.789552
$$554$$ 0 0
$$555$$ −5.96265 −0.253101
$$556$$ 0 0
$$557$$ 1.22699 0.0519892 0.0259946 0.999662i $$-0.491725\pi$$
0.0259946 + 0.999662i $$0.491725\pi$$
$$558$$ 0 0
$$559$$ 29.0101 1.22700
$$560$$ 0 0
$$561$$ 5.98185 0.252554
$$562$$ 0 0
$$563$$ −19.9144 −0.839291 −0.419646 0.907688i $$-0.637846\pi$$
−0.419646 + 0.907688i $$0.637846\pi$$
$$564$$ 0 0
$$565$$ −16.3118 −0.686243
$$566$$ 0 0
$$567$$ 1.32088 0.0554719
$$568$$ 0 0
$$569$$ −27.5015 −1.15292 −0.576460 0.817125i $$-0.695567\pi$$
−0.576460 + 0.817125i $$0.695567\pi$$
$$570$$ 0 0
$$571$$ 9.47133 0.396363 0.198181 0.980165i $$-0.436497\pi$$
0.198181 + 0.980165i $$0.436497\pi$$
$$572$$ 0 0
$$573$$ 18.3492 0.766547
$$574$$ 0 0
$$575$$ 7.67004 0.319863
$$576$$ 0 0
$$577$$ 36.6418 1.52542 0.762708 0.646743i $$-0.223869\pi$$
0.762708 + 0.646743i $$0.223869\pi$$
$$578$$ 0 0
$$579$$ −12.0192 −0.499501
$$580$$ 0 0
$$581$$ −6.64177 −0.275547
$$582$$ 0 0
$$583$$ −38.7165 −1.60347
$$584$$ 0 0
$$585$$ 3.32088 0.137302
$$586$$ 0 0
$$587$$ 3.67004 0.151479 0.0757394 0.997128i $$-0.475868\pi$$
0.0757394 + 0.997128i $$0.475868\pi$$
$$588$$ 0 0
$$589$$ −9.67004 −0.398447
$$590$$ 0 0
$$591$$ −4.84049 −0.199111
$$592$$ 0 0
$$593$$ −17.2270 −0.707428 −0.353714 0.935354i $$-0.615081\pi$$
−0.353714 + 0.935354i $$0.615081\pi$$
$$594$$ 0 0
$$595$$ 2.13124 0.0873724
$$596$$ 0 0
$$597$$ −20.6983 −0.847126
$$598$$ 0 0
$$599$$ 8.11310 0.331492 0.165746 0.986168i $$-0.446997\pi$$
0.165746 + 0.986168i $$0.446997\pi$$
$$600$$ 0 0
$$601$$ 7.35823 0.300149 0.150074 0.988675i $$-0.452049\pi$$
0.150074 + 0.988675i $$0.452049\pi$$
$$602$$ 0 0
$$603$$ 13.2835 0.540947
$$604$$ 0 0
$$605$$ 2.74474 0.111589
$$606$$ 0 0
$$607$$ −19.9253 −0.808743 −0.404372 0.914595i $$-0.632510\pi$$
−0.404372 + 0.914595i $$0.632510\pi$$
$$608$$ 0 0
$$609$$ 2.25526 0.0913879
$$610$$ 0 0
$$611$$ −38.7549 −1.56785
$$612$$ 0 0
$$613$$ 39.3219 1.58820 0.794099 0.607788i $$-0.207943\pi$$
0.794099 + 0.607788i $$0.207943\pi$$
$$614$$ 0 0
$$615$$ −2.29261 −0.0924470
$$616$$ 0 0
$$617$$ 0.0674757 0.00271647 0.00135823 0.999999i $$-0.499568\pi$$
0.00135823 + 0.999999i $$0.499568\pi$$
$$618$$ 0 0
$$619$$ −38.8680 −1.56224 −0.781118 0.624384i $$-0.785350\pi$$
−0.781118 + 0.624384i $$0.785350\pi$$
$$620$$ 0 0
$$621$$ −7.67004 −0.307788
$$622$$ 0 0
$$623$$ −2.25526 −0.0903552
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 3.70739 0.148059
$$628$$ 0 0
$$629$$ 9.62071 0.383603
$$630$$ 0 0
$$631$$ −29.2835 −1.16576 −0.582880 0.812559i $$-0.698074\pi$$
−0.582880 + 0.812559i $$0.698074\pi$$
$$632$$ 0 0
$$633$$ 2.05655 0.0817404
$$634$$ 0 0
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ −17.4521 −0.691478
$$638$$ 0 0
$$639$$ 8.69832 0.344100
$$640$$ 0 0
$$641$$ 10.4057 0.411001 0.205500 0.978657i $$-0.434118\pi$$
0.205500 + 0.978657i $$0.434118\pi$$
$$642$$ 0 0
$$643$$ 11.3774 0.448682 0.224341 0.974511i $$-0.427977\pi$$
0.224341 + 0.974511i $$0.427977\pi$$
$$644$$ 0 0
$$645$$ −8.73566 −0.343966
$$646$$ 0 0
$$647$$ −45.5388 −1.79032 −0.895158 0.445750i $$-0.852937\pi$$
−0.895158 + 0.445750i $$0.852937\pi$$
$$648$$ 0 0
$$649$$ 47.0776 1.84796
$$650$$ 0 0
$$651$$ 12.7730 0.500614
$$652$$ 0 0
$$653$$ −2.27341 −0.0889654 −0.0444827 0.999010i $$-0.514164\pi$$
−0.0444827 + 0.999010i $$0.514164\pi$$
$$654$$ 0 0
$$655$$ −4.29261 −0.167726
$$656$$ 0 0
$$657$$ −12.0565 −0.470371
$$658$$ 0 0
$$659$$ 8.77301 0.341748 0.170874 0.985293i $$-0.445341\pi$$
0.170874 + 0.985293i $$0.445341\pi$$
$$660$$ 0 0
$$661$$ −44.4924 −1.73055 −0.865277 0.501295i $$-0.832857\pi$$
−0.865277 + 0.501295i $$0.832857\pi$$
$$662$$ 0 0
$$663$$ −5.35823 −0.208096
$$664$$ 0 0
$$665$$ 1.32088 0.0512217
$$666$$ 0 0
$$667$$ −13.0957 −0.507069
$$668$$ 0 0
$$669$$ 6.05655 0.234160
$$670$$ 0 0
$$671$$ −33.4713 −1.29215
$$672$$ 0 0
$$673$$ −5.96265 −0.229843 −0.114922 0.993375i $$-0.536662\pi$$
−0.114922 + 0.993375i $$0.536662\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 6.32996 0.243280 0.121640 0.992574i $$-0.461185\pi$$
0.121640 + 0.992574i $$0.461185\pi$$
$$678$$ 0 0
$$679$$ −0.897033 −0.0344250
$$680$$ 0 0
$$681$$ −28.3118 −1.08491
$$682$$ 0 0
$$683$$ −39.8506 −1.52484 −0.762421 0.647082i $$-0.775989\pi$$
−0.762421 + 0.647082i $$0.775989\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ 1.80128 0.0687233
$$688$$ 0 0
$$689$$ 34.6802 1.32121
$$690$$ 0 0
$$691$$ −8.07469 −0.307176 −0.153588 0.988135i $$-0.549083\pi$$
−0.153588 + 0.988135i $$0.549083\pi$$
$$692$$ 0 0
$$693$$ −4.89703 −0.186023
$$694$$ 0 0
$$695$$ 8.58522 0.325656
$$696$$ 0 0
$$697$$ 3.69912 0.140114
$$698$$ 0 0
$$699$$ 3.94345 0.149155
$$700$$ 0 0
$$701$$ −2.77301 −0.104735 −0.0523676 0.998628i $$-0.516677\pi$$
−0.0523676 + 0.998628i $$0.516677\pi$$
$$702$$ 0 0
$$703$$ 5.96265 0.224886
$$704$$ 0 0
$$705$$ 11.6700 0.439519
$$706$$ 0 0
$$707$$ 16.6983 0.628005
$$708$$ 0 0
$$709$$ 10.1987 0.383021 0.191510 0.981491i $$-0.438661\pi$$
0.191510 + 0.981491i $$0.438661\pi$$
$$710$$ 0 0
$$711$$ 14.0565 0.527162
$$712$$ 0 0
$$713$$ −74.1696 −2.77767
$$714$$ 0 0
$$715$$ −12.3118 −0.460436
$$716$$ 0 0
$$717$$ −16.8031 −0.627525
$$718$$ 0 0
$$719$$ −34.2745 −1.27822 −0.639111 0.769115i $$-0.720698\pi$$
−0.639111 + 0.769115i $$0.720698\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ −14.0000 −0.520666
$$724$$ 0 0
$$725$$ −1.70739 −0.0634108
$$726$$ 0 0
$$727$$ −22.0939 −0.819417 −0.409709 0.912216i $$-0.634370\pi$$
−0.409709 + 0.912216i $$0.634370\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 14.0950 0.521321
$$732$$ 0 0
$$733$$ −13.3401 −0.492727 −0.246364 0.969177i $$-0.579236\pi$$
−0.246364 + 0.969177i $$0.579236\pi$$
$$734$$ 0 0
$$735$$ 5.25526 0.193843
$$736$$ 0 0
$$737$$ −49.2472 −1.81405
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ −3.32088 −0.121996
$$742$$ 0 0
$$743$$ −32.8861 −1.20647 −0.603237 0.797562i $$-0.706123\pi$$
−0.603237 + 0.797562i $$0.706123\pi$$
$$744$$ 0 0
$$745$$ 12.0565 0.441718
$$746$$ 0 0
$$747$$ −5.02827 −0.183975
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.4996 −0.602079 −0.301039 0.953612i $$-0.597334\pi$$
−0.301039 + 0.953612i $$0.597334\pi$$
$$752$$ 0 0
$$753$$ 24.9909 0.910720
$$754$$ 0 0
$$755$$ 22.9536 0.835366
$$756$$ 0 0
$$757$$ −42.5489 −1.54647 −0.773234 0.634121i $$-0.781362\pi$$
−0.773234 + 0.634121i $$0.781362\pi$$
$$758$$ 0 0
$$759$$ 28.4358 1.03216
$$760$$ 0 0
$$761$$ 5.34009 0.193578 0.0967890 0.995305i $$-0.469143\pi$$
0.0967890 + 0.995305i $$0.469143\pi$$
$$762$$ 0 0
$$763$$ 14.1312 0.511585
$$764$$ 0 0
$$765$$ 1.61350 0.0583360
$$766$$ 0 0
$$767$$ −42.1696 −1.52266
$$768$$ 0 0
$$769$$ −7.28354 −0.262651 −0.131326 0.991339i $$-0.541923\pi$$
−0.131326 + 0.991339i $$0.541923\pi$$
$$770$$ 0 0
$$771$$ 2.91518 0.104988
$$772$$ 0 0
$$773$$ −18.1806 −0.653910 −0.326955 0.945040i $$-0.606023\pi$$
−0.326955 + 0.945040i $$0.606023\pi$$
$$774$$ 0 0
$$775$$ −9.67004 −0.347358
$$776$$ 0 0
$$777$$ −7.87598 −0.282549
$$778$$ 0 0
$$779$$ 2.29261 0.0821413
$$780$$ 0 0
$$781$$ −32.2480 −1.15393
$$782$$ 0 0
$$783$$ 1.70739 0.0610171
$$784$$ 0 0
$$785$$ −11.8688 −0.423614
$$786$$ 0 0
$$787$$ −20.8114 −0.741847 −0.370923 0.928663i $$-0.620959\pi$$
−0.370923 + 0.928663i $$0.620959\pi$$
$$788$$ 0 0
$$789$$ −22.3118 −0.794322
$$790$$ 0 0
$$791$$ −21.5460 −0.766088
$$792$$ 0 0
$$793$$ 29.9819 1.06469
$$794$$ 0 0
$$795$$ −10.4431 −0.370377
$$796$$ 0 0
$$797$$ 40.9354 1.45001 0.725004 0.688745i $$-0.241838\pi$$
0.725004 + 0.688745i $$0.241838\pi$$
$$798$$ 0 0
$$799$$ −18.8296 −0.666142
$$800$$ 0 0
$$801$$ −1.70739 −0.0603276
$$802$$ 0 0
$$803$$ 44.6983 1.57737
$$804$$ 0 0
$$805$$ 10.1312 0.357079
$$806$$ 0 0
$$807$$ −15.0656 −0.530335
$$808$$ 0 0
$$809$$ −1.48947 −0.0523670 −0.0261835 0.999657i $$-0.508335\pi$$
−0.0261835 + 0.999657i $$0.508335\pi$$
$$810$$ 0 0
$$811$$ −21.5460 −0.756583 −0.378292 0.925687i $$-0.623488\pi$$
−0.378292 + 0.925687i $$0.623488\pi$$
$$812$$ 0 0
$$813$$ 5.47133 0.191888
$$814$$ 0 0
$$815$$ 0.0373465 0.00130819
$$816$$ 0 0
$$817$$ 8.73566 0.305622
$$818$$ 0 0
$$819$$ 4.38650 0.153277
$$820$$ 0 0
$$821$$ −35.9819 −1.25578 −0.627888 0.778304i $$-0.716080\pi$$
−0.627888 + 0.778304i $$0.716080\pi$$
$$822$$ 0 0
$$823$$ −15.8880 −0.553819 −0.276910 0.960896i $$-0.589310\pi$$
−0.276910 + 0.960896i $$0.589310\pi$$
$$824$$ 0 0
$$825$$ 3.70739 0.129075
$$826$$ 0 0
$$827$$ −26.8789 −0.934671 −0.467335 0.884080i $$-0.654786\pi$$
−0.467335 + 0.884080i $$0.654786\pi$$
$$828$$ 0 0
$$829$$ 17.9253 0.622572 0.311286 0.950316i $$-0.399240\pi$$
0.311286 + 0.950316i $$0.399240\pi$$
$$830$$ 0 0
$$831$$ 10.7730 0.373712
$$832$$ 0 0
$$833$$ −8.47934 −0.293792
$$834$$ 0 0
$$835$$ 11.0283 0.381649
$$836$$ 0 0
$$837$$ 9.67004 0.334246
$$838$$ 0 0
$$839$$ −7.30168 −0.252082 −0.126041 0.992025i $$-0.540227\pi$$
−0.126041 + 0.992025i $$0.540227\pi$$
$$840$$ 0 0
$$841$$ −26.0848 −0.899477
$$842$$ 0 0
$$843$$ −17.8205 −0.613770
$$844$$ 0 0
$$845$$ −1.97173 −0.0678294
$$846$$ 0 0
$$847$$ 3.62548 0.124573
$$848$$ 0 0
$$849$$ −30.2070 −1.03670
$$850$$ 0 0
$$851$$ 45.7338 1.56773
$$852$$ 0 0
$$853$$ −37.4148 −1.28106 −0.640529 0.767934i $$-0.721285\pi$$
−0.640529 + 0.767934i $$0.721285\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −10.1806 −0.347762 −0.173881 0.984767i $$-0.555631\pi$$
−0.173881 + 0.984767i $$0.555631\pi$$
$$858$$ 0 0
$$859$$ 40.7367 1.38992 0.694959 0.719049i $$-0.255422\pi$$
0.694959 + 0.719049i $$0.255422\pi$$
$$860$$ 0 0
$$861$$ −3.02827 −0.103203
$$862$$ 0 0
$$863$$ 8.11310 0.276173 0.138086 0.990420i $$-0.455905\pi$$
0.138086 + 0.990420i $$0.455905\pi$$
$$864$$ 0 0
$$865$$ −17.8578 −0.607184
$$866$$ 0 0
$$867$$ 14.3966 0.488935
$$868$$ 0 0
$$869$$ −52.1131 −1.76782
$$870$$ 0 0
$$871$$ 44.1131 1.49472
$$872$$ 0 0
$$873$$ −0.679116 −0.0229846
$$874$$ 0 0
$$875$$ 1.32088 0.0446540
$$876$$ 0 0
$$877$$ 29.8880 1.00924 0.504622 0.863340i $$-0.331632\pi$$
0.504622 + 0.863340i $$0.331632\pi$$
$$878$$ 0 0
$$879$$ 18.2553 0.615735
$$880$$ 0 0
$$881$$ 34.8114 1.17283 0.586413 0.810012i $$-0.300540\pi$$
0.586413 + 0.810012i $$0.300540\pi$$
$$882$$ 0 0
$$883$$ −27.8496 −0.937212 −0.468606 0.883407i $$-0.655244\pi$$
−0.468606 + 0.883407i $$0.655244\pi$$
$$884$$ 0 0
$$885$$ 12.6983 0.426849
$$886$$ 0 0
$$887$$ 50.7933 1.70547 0.852735 0.522343i $$-0.174942\pi$$
0.852735 + 0.522343i $$0.174942\pi$$
$$888$$ 0 0
$$889$$ 5.28354 0.177204
$$890$$ 0 0
$$891$$ −3.70739 −0.124202
$$892$$ 0 0
$$893$$ −11.6700 −0.390523
$$894$$ 0 0
$$895$$ 19.9253 0.666030
$$896$$ 0 0
$$897$$ −25.4713 −0.850463
$$898$$ 0 0
$$899$$ 16.5105 0.550657
$$900$$ 0 0
$$901$$ 16.8498 0.561349
$$902$$ 0 0
$$903$$ −11.5388 −0.383987
$$904$$ 0 0
$$905$$ −15.4713 −0.514284
$$906$$ 0 0
$$907$$ 32.3009 1.07253 0.536267 0.844049i $$-0.319834\pi$$
0.536267 + 0.844049i $$0.319834\pi$$
$$908$$ 0 0
$$909$$ 12.6418 0.419301
$$910$$ 0 0
$$911$$ 52.7367 1.74725 0.873623 0.486604i $$-0.161764\pi$$
0.873623 + 0.486604i $$0.161764\pi$$
$$912$$ 0 0
$$913$$ 18.6418 0.616953
$$914$$ 0 0
$$915$$ −9.02827 −0.298466
$$916$$ 0 0
$$917$$ −5.67004 −0.187241
$$918$$ 0 0
$$919$$ −52.5105 −1.73216 −0.866081 0.499903i $$-0.833369\pi$$
−0.866081 + 0.499903i $$0.833369\pi$$
$$920$$ 0 0
$$921$$ −16.5852 −0.546502
$$922$$ 0 0
$$923$$ 28.8861 0.950798
$$924$$ 0 0
$$925$$ 5.96265 0.196051
$$926$$ 0 0
$$927$$ −6.05655 −0.198923
$$928$$ 0 0
$$929$$ −35.8688 −1.17682 −0.588408 0.808564i $$-0.700245\pi$$
−0.588408 + 0.808564i $$0.700245\pi$$
$$930$$ 0 0
$$931$$ −5.25526 −0.172234
$$932$$ 0 0
$$933$$ 11.5196 0.377135
$$934$$ 0 0
$$935$$ −5.98185 −0.195628
$$936$$ 0 0
$$937$$ 10.6983 0.349499 0.174749 0.984613i $$-0.444088\pi$$
0.174749 + 0.984613i $$0.444088\pi$$
$$938$$ 0 0
$$939$$ −32.7549 −1.06891
$$940$$ 0 0
$$941$$ −22.3673 −0.729153 −0.364577 0.931173i $$-0.618786\pi$$
−0.364577 + 0.931173i $$0.618786\pi$$
$$942$$ 0 0
$$943$$ 17.5844 0.572628
$$944$$ 0 0
$$945$$ −1.32088 −0.0429684
$$946$$ 0 0
$$947$$ −34.8223 −1.13157 −0.565787 0.824551i $$-0.691428\pi$$
−0.565787 + 0.824551i $$0.691428\pi$$
$$948$$ 0 0
$$949$$ −40.0384 −1.29970
$$950$$ 0 0
$$951$$ −15.2161 −0.493415
$$952$$ 0 0
$$953$$ −29.4823 −0.955024 −0.477512 0.878625i $$-0.658461\pi$$
−0.477512 + 0.878625i $$0.658461\pi$$
$$954$$ 0 0
$$955$$ −18.3492 −0.593765
$$956$$ 0 0
$$957$$ −6.32996 −0.204618
$$958$$ 0 0
$$959$$ 2.64177 0.0853072
$$960$$ 0 0
$$961$$ 62.5097 2.01644
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 12.0192 0.386912
$$966$$ 0 0
$$967$$ −18.8669 −0.606719 −0.303359 0.952876i $$-0.598108\pi$$
−0.303359 + 0.952876i $$0.598108\pi$$
$$968$$ 0 0
$$969$$ −1.61350 −0.0518329
$$970$$ 0 0
$$971$$ −31.2270 −1.00212 −0.501061 0.865412i $$-0.667057\pi$$
−0.501061 + 0.865412i $$0.667057\pi$$
$$972$$ 0 0
$$973$$ 11.3401 0.363546
$$974$$ 0 0
$$975$$ −3.32088 −0.106353
$$976$$ 0 0
$$977$$ 6.44305 0.206132 0.103066 0.994675i $$-0.467135\pi$$
0.103066 + 0.994675i $$0.467135\pi$$
$$978$$ 0 0
$$979$$ 6.32996 0.202306
$$980$$ 0 0
$$981$$ 10.6983 0.341571
$$982$$ 0 0
$$983$$ −4.57429 −0.145897 −0.0729486 0.997336i $$-0.523241\pi$$
−0.0729486 + 0.997336i $$0.523241\pi$$
$$984$$ 0 0
$$985$$ 4.84049 0.154231
$$986$$ 0 0
$$987$$ 15.4148 0.490658
$$988$$ 0 0
$$989$$ 67.0029 2.13057
$$990$$ 0 0
$$991$$ 38.8296 1.23346 0.616731 0.787174i $$-0.288457\pi$$
0.616731 + 0.787174i $$0.288457\pi$$
$$992$$ 0 0
$$993$$ −26.9536 −0.855346
$$994$$ 0 0
$$995$$ 20.6983 0.656181
$$996$$ 0 0
$$997$$ 54.9245 1.73948 0.869738 0.493513i $$-0.164288\pi$$
0.869738 + 0.493513i $$0.164288\pi$$
$$998$$ 0 0
$$999$$ −5.96265 −0.188650
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 4560.2.a.br.1.3 3
4.3 odd 2 1140.2.a.g.1.1 3
12.11 even 2 3420.2.a.m.1.1 3
20.3 even 4 5700.2.f.q.3649.6 6
20.7 even 4 5700.2.f.q.3649.1 6
20.19 odd 2 5700.2.a.w.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.g.1.1 3 4.3 odd 2
3420.2.a.m.1.1 3 12.11 even 2
4560.2.a.br.1.3 3 1.1 even 1 trivial
5700.2.a.w.1.3 3 20.19 odd 2
5700.2.f.q.3649.1 6 20.7 even 4
5700.2.f.q.3649.6 6 20.3 even 4