# Properties

 Label 5625.2.a.b.1.1 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1875) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 5625.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61803 q^{2} +0.618034 q^{4} -2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q-1.61803 q^{2} +0.618034 q^{4} -2.00000 q^{7} +2.23607 q^{8} +3.00000 q^{11} -1.00000 q^{13} +3.23607 q^{14} -4.85410 q^{16} +4.23607 q^{17} -6.70820 q^{19} -4.85410 q^{22} -5.38197 q^{23} +1.61803 q^{26} -1.23607 q^{28} +3.61803 q^{29} +8.70820 q^{31} +3.38197 q^{32} -6.85410 q^{34} -2.00000 q^{37} +10.8541 q^{38} +9.38197 q^{41} -7.38197 q^{43} +1.85410 q^{44} +8.70820 q^{46} +4.76393 q^{47} -3.00000 q^{49} -0.618034 q^{52} -11.2361 q^{53} -4.47214 q^{56} -5.85410 q^{58} -3.94427 q^{59} +8.70820 q^{61} -14.0902 q^{62} +4.23607 q^{64} -13.1803 q^{67} +2.61803 q^{68} -10.0902 q^{71} +15.7082 q^{73} +3.23607 q^{74} -4.14590 q^{76} -6.00000 q^{77} -9.14590 q^{79} -15.1803 q^{82} -9.00000 q^{83} +11.9443 q^{86} +6.70820 q^{88} +11.1803 q^{89} +2.00000 q^{91} -3.32624 q^{92} -7.70820 q^{94} +3.85410 q^{97} +4.85410 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 4 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 - 4 * q^7 $$2 q - q^{2} - q^{4} - 4 q^{7} + 6 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{22} - 13 q^{23} + q^{26} + 2 q^{28} + 5 q^{29} + 4 q^{31} + 9 q^{32} - 7 q^{34} - 4 q^{37} + 15 q^{38} + 21 q^{41} - 17 q^{43} - 3 q^{44} + 4 q^{46} + 14 q^{47} - 6 q^{49} + q^{52} - 18 q^{53} - 5 q^{58} + 10 q^{59} + 4 q^{61} - 17 q^{62} + 4 q^{64} - 4 q^{67} + 3 q^{68} - 9 q^{71} + 18 q^{73} + 2 q^{74} - 15 q^{76} - 12 q^{77} - 25 q^{79} - 8 q^{82} - 18 q^{83} + 6 q^{86} + 4 q^{91} + 9 q^{92} - 2 q^{94} + q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 4 * q^7 + 6 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^22 - 13 * q^23 + q^26 + 2 * q^28 + 5 * q^29 + 4 * q^31 + 9 * q^32 - 7 * q^34 - 4 * q^37 + 15 * q^38 + 21 * q^41 - 17 * q^43 - 3 * q^44 + 4 * q^46 + 14 * q^47 - 6 * q^49 + q^52 - 18 * q^53 - 5 * q^58 + 10 * q^59 + 4 * q^61 - 17 * q^62 + 4 * q^64 - 4 * q^67 + 3 * q^68 - 9 * q^71 + 18 * q^73 + 2 * q^74 - 15 * q^76 - 12 * q^77 - 25 * q^79 - 8 * q^82 - 18 * q^83 + 6 * q^86 + 4 * q^91 + 9 * q^92 - 2 * q^94 + q^97 + 3 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61803 −1.14412 −0.572061 0.820211i $$-0.693856\pi$$
−0.572061 + 0.820211i $$0.693856\pi$$
$$3$$ 0 0
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 3.23607 0.864876
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 4.23607 1.02740 0.513699 0.857971i $$-0.328275\pi$$
0.513699 + 0.857971i $$0.328275\pi$$
$$18$$ 0 0
$$19$$ −6.70820 −1.53897 −0.769484 0.638666i $$-0.779486\pi$$
−0.769484 + 0.638666i $$0.779486\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.85410 −1.03490
$$23$$ −5.38197 −1.12222 −0.561109 0.827742i $$-0.689625\pi$$
−0.561109 + 0.827742i $$0.689625\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.61803 0.317323
$$27$$ 0 0
$$28$$ −1.23607 −0.233595
$$29$$ 3.61803 0.671852 0.335926 0.941888i $$-0.390951\pi$$
0.335926 + 0.941888i $$0.390951\pi$$
$$30$$ 0 0
$$31$$ 8.70820 1.56404 0.782020 0.623254i $$-0.214190\pi$$
0.782020 + 0.623254i $$0.214190\pi$$
$$32$$ 3.38197 0.597853
$$33$$ 0 0
$$34$$ −6.85410 −1.17547
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 10.8541 1.76077
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.38197 1.46522 0.732608 0.680650i $$-0.238303\pi$$
0.732608 + 0.680650i $$0.238303\pi$$
$$42$$ 0 0
$$43$$ −7.38197 −1.12574 −0.562870 0.826546i $$-0.690303\pi$$
−0.562870 + 0.826546i $$0.690303\pi$$
$$44$$ 1.85410 0.279516
$$45$$ 0 0
$$46$$ 8.70820 1.28395
$$47$$ 4.76393 0.694891 0.347445 0.937700i $$-0.387049\pi$$
0.347445 + 0.937700i $$0.387049\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.618034 −0.0857059
$$53$$ −11.2361 −1.54339 −0.771696 0.635991i $$-0.780591\pi$$
−0.771696 + 0.635991i $$0.780591\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.47214 −0.597614
$$57$$ 0 0
$$58$$ −5.85410 −0.768681
$$59$$ −3.94427 −0.513500 −0.256750 0.966478i $$-0.582652\pi$$
−0.256750 + 0.966478i $$0.582652\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ −14.0902 −1.78945
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.1803 −1.61023 −0.805117 0.593115i $$-0.797898\pi$$
−0.805117 + 0.593115i $$0.797898\pi$$
$$68$$ 2.61803 0.317483
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.0902 −1.19748 −0.598741 0.800942i $$-0.704332\pi$$
−0.598741 + 0.800942i $$0.704332\pi$$
$$72$$ 0 0
$$73$$ 15.7082 1.83851 0.919253 0.393667i $$-0.128794\pi$$
0.919253 + 0.393667i $$0.128794\pi$$
$$74$$ 3.23607 0.376185
$$75$$ 0 0
$$76$$ −4.14590 −0.475567
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −9.14590 −1.02899 −0.514497 0.857492i $$-0.672021\pi$$
−0.514497 + 0.857492i $$0.672021\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −15.1803 −1.67639
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 11.9443 1.28798
$$87$$ 0 0
$$88$$ 6.70820 0.715097
$$89$$ 11.1803 1.18511 0.592557 0.805529i $$-0.298119\pi$$
0.592557 + 0.805529i $$0.298119\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ −3.32624 −0.346784
$$93$$ 0 0
$$94$$ −7.70820 −0.795041
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.85410 0.391325 0.195662 0.980671i $$-0.437314\pi$$
0.195662 + 0.980671i $$0.437314\pi$$
$$98$$ 4.85410 0.490338
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.38197 0.933541 0.466770 0.884379i $$-0.345417\pi$$
0.466770 + 0.884379i $$0.345417\pi$$
$$102$$ 0 0
$$103$$ −14.4164 −1.42049 −0.710245 0.703954i $$-0.751416\pi$$
−0.710245 + 0.703954i $$0.751416\pi$$
$$104$$ −2.23607 −0.219265
$$105$$ 0 0
$$106$$ 18.1803 1.76583
$$107$$ 1.14590 0.110778 0.0553891 0.998465i $$-0.482360\pi$$
0.0553891 + 0.998465i $$0.482360\pi$$
$$108$$ 0 0
$$109$$ −4.14590 −0.397105 −0.198553 0.980090i $$-0.563624\pi$$
−0.198553 + 0.980090i $$0.563624\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 9.70820 0.917339
$$113$$ 3.76393 0.354081 0.177040 0.984204i $$-0.443348\pi$$
0.177040 + 0.984204i $$0.443348\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.23607 0.207614
$$117$$ 0 0
$$118$$ 6.38197 0.587508
$$119$$ −8.47214 −0.776639
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −14.0902 −1.27566
$$123$$ 0 0
$$124$$ 5.38197 0.483315
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.6525 1.21146 0.605731 0.795670i $$-0.292881\pi$$
0.605731 + 0.795670i $$0.292881\pi$$
$$128$$ −13.6180 −1.20368
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.1803 1.23894 0.619471 0.785020i $$-0.287347\pi$$
0.619471 + 0.785020i $$0.287347\pi$$
$$132$$ 0 0
$$133$$ 13.4164 1.16335
$$134$$ 21.3262 1.84231
$$135$$ 0 0
$$136$$ 9.47214 0.812229
$$137$$ −0.437694 −0.0373947 −0.0186974 0.999825i $$-0.505952\pi$$
−0.0186974 + 0.999825i $$0.505952\pi$$
$$138$$ 0 0
$$139$$ 13.4164 1.13796 0.568982 0.822350i $$-0.307337\pi$$
0.568982 + 0.822350i $$0.307337\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.3262 1.37007
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −25.4164 −2.10348
$$147$$ 0 0
$$148$$ −1.23607 −0.101604
$$149$$ 13.0902 1.07239 0.536194 0.844095i $$-0.319861\pi$$
0.536194 + 0.844095i $$0.319861\pi$$
$$150$$ 0 0
$$151$$ −6.61803 −0.538568 −0.269284 0.963061i $$-0.586787\pi$$
−0.269284 + 0.963061i $$0.586787\pi$$
$$152$$ −15.0000 −1.21666
$$153$$ 0 0
$$154$$ 9.70820 0.782309
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.85410 −0.227782 −0.113891 0.993493i $$-0.536331\pi$$
−0.113891 + 0.993493i $$0.536331\pi$$
$$158$$ 14.7984 1.17730
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 10.7639 0.848317
$$162$$ 0 0
$$163$$ 18.2705 1.43106 0.715528 0.698584i $$-0.246186\pi$$
0.715528 + 0.698584i $$0.246186\pi$$
$$164$$ 5.79837 0.452777
$$165$$ 0 0
$$166$$ 14.5623 1.13025
$$167$$ −17.7984 −1.37728 −0.688640 0.725104i $$-0.741792\pi$$
−0.688640 + 0.725104i $$0.741792\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.56231 −0.347873
$$173$$ −0.909830 −0.0691731 −0.0345865 0.999402i $$-0.511011\pi$$
−0.0345865 + 0.999402i $$0.511011\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −14.5623 −1.09768
$$177$$ 0 0
$$178$$ −18.0902 −1.35592
$$179$$ 15.6525 1.16992 0.584960 0.811062i $$-0.301110\pi$$
0.584960 + 0.811062i $$0.301110\pi$$
$$180$$ 0 0
$$181$$ −12.4721 −0.927047 −0.463523 0.886085i $$-0.653415\pi$$
−0.463523 + 0.886085i $$0.653415\pi$$
$$182$$ −3.23607 −0.239873
$$183$$ 0 0
$$184$$ −12.0344 −0.887191
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.7082 0.929316
$$188$$ 2.94427 0.214733
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.3262 −1.25368 −0.626841 0.779147i $$-0.715653\pi$$
−0.626841 + 0.779147i $$0.715653\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ −6.23607 −0.447724
$$195$$ 0 0
$$196$$ −1.85410 −0.132436
$$197$$ 0.0901699 0.00642434 0.00321217 0.999995i $$-0.498978\pi$$
0.00321217 + 0.999995i $$0.498978\pi$$
$$198$$ 0 0
$$199$$ 11.7082 0.829973 0.414986 0.909828i $$-0.363786\pi$$
0.414986 + 0.909828i $$0.363786\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −15.1803 −1.06808
$$203$$ −7.23607 −0.507872
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 23.3262 1.62522
$$207$$ 0 0
$$208$$ 4.85410 0.336571
$$209$$ −20.1246 −1.39205
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ −6.94427 −0.476935
$$213$$ 0 0
$$214$$ −1.85410 −0.126744
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −17.4164 −1.18230
$$218$$ 6.70820 0.454337
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.23607 −0.284949
$$222$$ 0 0
$$223$$ −10.1459 −0.679420 −0.339710 0.940530i $$-0.610329\pi$$
−0.339710 + 0.940530i $$0.610329\pi$$
$$224$$ −6.76393 −0.451934
$$225$$ 0 0
$$226$$ −6.09017 −0.405112
$$227$$ −5.76393 −0.382566 −0.191283 0.981535i $$-0.561265\pi$$
−0.191283 + 0.981535i $$0.561265\pi$$
$$228$$ 0 0
$$229$$ −16.1803 −1.06923 −0.534613 0.845097i $$-0.679543\pi$$
−0.534613 + 0.845097i $$0.679543\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 8.09017 0.531146
$$233$$ −10.1803 −0.666936 −0.333468 0.942761i $$-0.608219\pi$$
−0.333468 + 0.942761i $$0.608219\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2.43769 −0.158680
$$237$$ 0 0
$$238$$ 13.7082 0.888571
$$239$$ −21.3820 −1.38308 −0.691542 0.722336i $$-0.743068\pi$$
−0.691542 + 0.722336i $$0.743068\pi$$
$$240$$ 0 0
$$241$$ 7.32624 0.471924 0.235962 0.971762i $$-0.424176\pi$$
0.235962 + 0.971762i $$0.424176\pi$$
$$242$$ 3.23607 0.208022
$$243$$ 0 0
$$244$$ 5.38197 0.344545
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.70820 0.426833
$$248$$ 19.4721 1.23648
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.9787 1.19793 0.598963 0.800777i $$-0.295580\pi$$
0.598963 + 0.800777i $$0.295580\pi$$
$$252$$ 0 0
$$253$$ −16.1459 −1.01508
$$254$$ −22.0902 −1.38606
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −31.2148 −1.94712 −0.973562 0.228422i $$-0.926644\pi$$
−0.973562 + 0.228422i $$0.926644\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −22.9443 −1.41750
$$263$$ 12.5066 0.771189 0.385594 0.922668i $$-0.373996\pi$$
0.385594 + 0.922668i $$0.373996\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −21.7082 −1.33102
$$267$$ 0 0
$$268$$ −8.14590 −0.497590
$$269$$ 20.5279 1.25161 0.625803 0.779981i $$-0.284771\pi$$
0.625803 + 0.779981i $$0.284771\pi$$
$$270$$ 0 0
$$271$$ −11.4164 −0.693497 −0.346749 0.937958i $$-0.612714\pi$$
−0.346749 + 0.937958i $$0.612714\pi$$
$$272$$ −20.5623 −1.24677
$$273$$ 0 0
$$274$$ 0.708204 0.0427842
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0557 −0.784443 −0.392221 0.919871i $$-0.628293\pi$$
−0.392221 + 0.919871i $$0.628293\pi$$
$$278$$ −21.7082 −1.30197
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.1803 0.845928 0.422964 0.906146i $$-0.360990\pi$$
0.422964 + 0.906146i $$0.360990\pi$$
$$282$$ 0 0
$$283$$ 2.29180 0.136233 0.0681166 0.997677i $$-0.478301\pi$$
0.0681166 + 0.997677i $$0.478301\pi$$
$$284$$ −6.23607 −0.370043
$$285$$ 0 0
$$286$$ 4.85410 0.287029
$$287$$ −18.7639 −1.10760
$$288$$ 0 0
$$289$$ 0.944272 0.0555454
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 9.70820 0.568130
$$293$$ 6.32624 0.369583 0.184791 0.982778i $$-0.440839\pi$$
0.184791 + 0.982778i $$0.440839\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −4.47214 −0.259938
$$297$$ 0 0
$$298$$ −21.1803 −1.22694
$$299$$ 5.38197 0.311247
$$300$$ 0 0
$$301$$ 14.7639 0.850979
$$302$$ 10.7082 0.616188
$$303$$ 0 0
$$304$$ 32.5623 1.86758
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.85410 0.505330 0.252665 0.967554i $$-0.418693\pi$$
0.252665 + 0.967554i $$0.418693\pi$$
$$308$$ −3.70820 −0.211295
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 13.5279 0.767095 0.383547 0.923521i $$-0.374702\pi$$
0.383547 + 0.923521i $$0.374702\pi$$
$$312$$ 0 0
$$313$$ 2.29180 0.129540 0.0647700 0.997900i $$-0.479369\pi$$
0.0647700 + 0.997900i $$0.479369\pi$$
$$314$$ 4.61803 0.260611
$$315$$ 0 0
$$316$$ −5.65248 −0.317977
$$317$$ −20.5623 −1.15489 −0.577447 0.816428i $$-0.695951\pi$$
−0.577447 + 0.816428i $$0.695951\pi$$
$$318$$ 0 0
$$319$$ 10.8541 0.607713
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −17.4164 −0.970578
$$323$$ −28.4164 −1.58113
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −29.5623 −1.63730
$$327$$ 0 0
$$328$$ 20.9787 1.15836
$$329$$ −9.52786 −0.525288
$$330$$ 0 0
$$331$$ −30.6869 −1.68671 −0.843353 0.537360i $$-0.819422\pi$$
−0.843353 + 0.537360i $$0.819422\pi$$
$$332$$ −5.56231 −0.305271
$$333$$ 0 0
$$334$$ 28.7984 1.57578
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −33.1803 −1.80745 −0.903724 0.428115i $$-0.859178\pi$$
−0.903724 + 0.428115i $$0.859178\pi$$
$$338$$ 19.4164 1.05611
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 26.1246 1.41473
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −16.5066 −0.889975
$$345$$ 0 0
$$346$$ 1.47214 0.0791425
$$347$$ −12.2705 −0.658715 −0.329358 0.944205i $$-0.606832\pi$$
−0.329358 + 0.944205i $$0.606832\pi$$
$$348$$ 0 0
$$349$$ 7.23607 0.387338 0.193669 0.981067i $$-0.437961\pi$$
0.193669 + 0.981067i $$0.437961\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 10.1459 0.540778
$$353$$ 12.3820 0.659026 0.329513 0.944151i $$-0.393116\pi$$
0.329513 + 0.944151i $$0.393116\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.90983 0.366220
$$357$$ 0 0
$$358$$ −25.3262 −1.33853
$$359$$ −23.9443 −1.26373 −0.631865 0.775078i $$-0.717710\pi$$
−0.631865 + 0.775078i $$0.717710\pi$$
$$360$$ 0 0
$$361$$ 26.0000 1.36842
$$362$$ 20.1803 1.06066
$$363$$ 0 0
$$364$$ 1.23607 0.0647876
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −12.5279 −0.653949 −0.326975 0.945033i $$-0.606029\pi$$
−0.326975 + 0.945033i $$0.606029\pi$$
$$368$$ 26.1246 1.36184
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 22.4721 1.16670
$$372$$ 0 0
$$373$$ 17.4164 0.901787 0.450894 0.892578i $$-0.351105\pi$$
0.450894 + 0.892578i $$0.351105\pi$$
$$374$$ −20.5623 −1.06325
$$375$$ 0 0
$$376$$ 10.6525 0.549359
$$377$$ −3.61803 −0.186338
$$378$$ 0 0
$$379$$ −13.6180 −0.699511 −0.349756 0.936841i $$-0.613735\pi$$
−0.349756 + 0.936841i $$0.613735\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 28.0344 1.43437
$$383$$ −5.05573 −0.258336 −0.129168 0.991623i $$-0.541231\pi$$
−0.129168 + 0.991623i $$0.541231\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 17.7984 0.905913
$$387$$ 0 0
$$388$$ 2.38197 0.120926
$$389$$ −0.652476 −0.0330818 −0.0165409 0.999863i $$-0.505265\pi$$
−0.0165409 + 0.999863i $$0.505265\pi$$
$$390$$ 0 0
$$391$$ −22.7984 −1.15296
$$392$$ −6.70820 −0.338815
$$393$$ 0 0
$$394$$ −0.145898 −0.00735024
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.52786 −0.126870 −0.0634349 0.997986i $$-0.520206\pi$$
−0.0634349 + 0.997986i $$0.520206\pi$$
$$398$$ −18.9443 −0.949591
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −36.2705 −1.81126 −0.905631 0.424066i $$-0.860603\pi$$
−0.905631 + 0.424066i $$0.860603\pi$$
$$402$$ 0 0
$$403$$ −8.70820 −0.433787
$$404$$ 5.79837 0.288480
$$405$$ 0 0
$$406$$ 11.7082 0.581068
$$407$$ −6.00000 −0.297409
$$408$$ 0 0
$$409$$ −5.12461 −0.253396 −0.126698 0.991941i $$-0.540438\pi$$
−0.126698 + 0.991941i $$0.540438\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.90983 −0.438956
$$413$$ 7.88854 0.388170
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3.38197 −0.165815
$$417$$ 0 0
$$418$$ 32.5623 1.59267
$$419$$ −0.326238 −0.0159378 −0.00796888 0.999968i $$-0.502537\pi$$
−0.00796888 + 0.999968i $$0.502537\pi$$
$$420$$ 0 0
$$421$$ −30.3607 −1.47969 −0.739844 0.672778i $$-0.765101\pi$$
−0.739844 + 0.672778i $$0.765101\pi$$
$$422$$ 4.85410 0.236294
$$423$$ 0 0
$$424$$ −25.1246 −1.22016
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −17.4164 −0.842839
$$428$$ 0.708204 0.0342323
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −29.7639 −1.43368 −0.716839 0.697239i $$-0.754412\pi$$
−0.716839 + 0.697239i $$0.754412\pi$$
$$432$$ 0 0
$$433$$ 3.47214 0.166860 0.0834301 0.996514i $$-0.473412\pi$$
0.0834301 + 0.996514i $$0.473412\pi$$
$$434$$ 28.1803 1.35270
$$435$$ 0 0
$$436$$ −2.56231 −0.122712
$$437$$ 36.1033 1.72706
$$438$$ 0 0
$$439$$ −32.0344 −1.52892 −0.764460 0.644671i $$-0.776994\pi$$
−0.764460 + 0.644671i $$0.776994\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.85410 0.326016
$$443$$ 19.4164 0.922501 0.461251 0.887270i $$-0.347401\pi$$
0.461251 + 0.887270i $$0.347401\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16.4164 0.777339
$$447$$ 0 0
$$448$$ −8.47214 −0.400271
$$449$$ −16.5066 −0.778994 −0.389497 0.921028i $$-0.627351\pi$$
−0.389497 + 0.921028i $$0.627351\pi$$
$$450$$ 0 0
$$451$$ 28.1459 1.32534
$$452$$ 2.32624 0.109417
$$453$$ 0 0
$$454$$ 9.32624 0.437702
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9.88854 −0.462567 −0.231283 0.972886i $$-0.574292\pi$$
−0.231283 + 0.972886i $$0.574292\pi$$
$$458$$ 26.1803 1.22333
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 19.1803 0.893317 0.446659 0.894704i $$-0.352614\pi$$
0.446659 + 0.894704i $$0.352614\pi$$
$$462$$ 0 0
$$463$$ −33.6869 −1.56556 −0.782782 0.622296i $$-0.786200\pi$$
−0.782782 + 0.622296i $$0.786200\pi$$
$$464$$ −17.5623 −0.815310
$$465$$ 0 0
$$466$$ 16.4721 0.763057
$$467$$ 10.4164 0.482014 0.241007 0.970523i $$-0.422522\pi$$
0.241007 + 0.970523i $$0.422522\pi$$
$$468$$ 0 0
$$469$$ 26.3607 1.21722
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −8.81966 −0.405958
$$473$$ −22.1459 −1.01827
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −5.23607 −0.239995
$$477$$ 0 0
$$478$$ 34.5967 1.58242
$$479$$ −4.79837 −0.219243 −0.109622 0.993973i $$-0.534964\pi$$
−0.109622 + 0.993973i $$0.534964\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −11.8541 −0.539940
$$483$$ 0 0
$$484$$ −1.23607 −0.0561849
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.6180 0.753035 0.376518 0.926410i $$-0.377121\pi$$
0.376518 + 0.926410i $$0.377121\pi$$
$$488$$ 19.4721 0.881462
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −22.3262 −1.00757 −0.503785 0.863829i $$-0.668059\pi$$
−0.503785 + 0.863829i $$0.668059\pi$$
$$492$$ 0 0
$$493$$ 15.3262 0.690259
$$494$$ −10.8541 −0.488349
$$495$$ 0 0
$$496$$ −42.2705 −1.89800
$$497$$ 20.1803 0.905212
$$498$$ 0 0
$$499$$ −15.0000 −0.671492 −0.335746 0.941953i $$-0.608988\pi$$
−0.335746 + 0.941953i $$0.608988\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −30.7082 −1.37057
$$503$$ 3.96556 0.176815 0.0884077 0.996084i $$-0.471822\pi$$
0.0884077 + 0.996084i $$0.471822\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 26.1246 1.16138
$$507$$ 0 0
$$508$$ 8.43769 0.374362
$$509$$ −32.8885 −1.45776 −0.728880 0.684642i $$-0.759959\pi$$
−0.728880 + 0.684642i $$0.759959\pi$$
$$510$$ 0 0
$$511$$ −31.4164 −1.38978
$$512$$ 5.29180 0.233867
$$513$$ 0 0
$$514$$ 50.5066 2.22775
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.2918 0.628552
$$518$$ −6.47214 −0.284369
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.0902 −1.75638 −0.878191 0.478310i $$-0.841250\pi$$
−0.878191 + 0.478310i $$0.841250\pi$$
$$522$$ 0 0
$$523$$ 1.56231 0.0683149 0.0341574 0.999416i $$-0.489125\pi$$
0.0341574 + 0.999416i $$0.489125\pi$$
$$524$$ 8.76393 0.382854
$$525$$ 0 0
$$526$$ −20.2361 −0.882334
$$527$$ 36.8885 1.60689
$$528$$ 0 0
$$529$$ 5.96556 0.259372
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 8.29180 0.359495
$$533$$ −9.38197 −0.406378
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −29.4721 −1.27300
$$537$$ 0 0
$$538$$ −33.2148 −1.43199
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −26.2918 −1.13037 −0.565186 0.824963i $$-0.691196\pi$$
−0.565186 + 0.824963i $$0.691196\pi$$
$$542$$ 18.4721 0.793446
$$543$$ 0 0
$$544$$ 14.3262 0.614232
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 24.7082 1.05645 0.528223 0.849106i $$-0.322858\pi$$
0.528223 + 0.849106i $$0.322858\pi$$
$$548$$ −0.270510 −0.0115556
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.2705 −1.03396
$$552$$ 0 0
$$553$$ 18.2918 0.777846
$$554$$ 21.1246 0.897499
$$555$$ 0 0
$$556$$ 8.29180 0.351650
$$557$$ 37.6525 1.59539 0.797693 0.603063i $$-0.206053\pi$$
0.797693 + 0.603063i $$0.206053\pi$$
$$558$$ 0 0
$$559$$ 7.38197 0.312224
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −22.9443 −0.967846
$$563$$ −9.00000 −0.379305 −0.189652 0.981851i $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −3.70820 −0.155867
$$567$$ 0 0
$$568$$ −22.5623 −0.946693
$$569$$ 10.8541 0.455028 0.227514 0.973775i $$-0.426940\pi$$
0.227514 + 0.973775i $$0.426940\pi$$
$$570$$ 0 0
$$571$$ −38.1246 −1.59547 −0.797733 0.603011i $$-0.793967\pi$$
−0.797733 + 0.603011i $$0.793967\pi$$
$$572$$ −1.85410 −0.0775239
$$573$$ 0 0
$$574$$ 30.3607 1.26723
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 3.72949 0.155261 0.0776304 0.996982i $$-0.475265\pi$$
0.0776304 + 0.996982i $$0.475265\pi$$
$$578$$ −1.52786 −0.0635508
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ −33.7082 −1.39605
$$584$$ 35.1246 1.45347
$$585$$ 0 0
$$586$$ −10.2361 −0.422848
$$587$$ −39.3050 −1.62229 −0.811144 0.584846i $$-0.801155\pi$$
−0.811144 + 0.584846i $$0.801155\pi$$
$$588$$ 0 0
$$589$$ −58.4164 −2.40701
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 9.70820 0.399005
$$593$$ −17.6180 −0.723486 −0.361743 0.932278i $$-0.617818\pi$$
−0.361743 + 0.932278i $$0.617818\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 8.09017 0.331386
$$597$$ 0 0
$$598$$ −8.70820 −0.356105
$$599$$ −39.2705 −1.60455 −0.802275 0.596955i $$-0.796377\pi$$
−0.802275 + 0.596955i $$0.796377\pi$$
$$600$$ 0 0
$$601$$ −24.7082 −1.00787 −0.503934 0.863742i $$-0.668115\pi$$
−0.503934 + 0.863742i $$0.668115\pi$$
$$602$$ −23.8885 −0.973624
$$603$$ 0 0
$$604$$ −4.09017 −0.166427
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.8541 −0.927619 −0.463810 0.885935i $$-0.653518\pi$$
−0.463810 + 0.885935i $$0.653518\pi$$
$$608$$ −22.6869 −0.920076
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.76393 −0.192728
$$612$$ 0 0
$$613$$ −5.87539 −0.237305 −0.118652 0.992936i $$-0.537857\pi$$
−0.118652 + 0.992936i $$0.537857\pi$$
$$614$$ −14.3262 −0.578160
$$615$$ 0 0
$$616$$ −13.4164 −0.540562
$$617$$ −25.2361 −1.01597 −0.507983 0.861367i $$-0.669609\pi$$
−0.507983 + 0.861367i $$0.669609\pi$$
$$618$$ 0 0
$$619$$ 34.2705 1.37745 0.688724 0.725024i $$-0.258171\pi$$
0.688724 + 0.725024i $$0.258171\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −21.8885 −0.877651
$$623$$ −22.3607 −0.895862
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −3.70820 −0.148210
$$627$$ 0 0
$$628$$ −1.76393 −0.0703886
$$629$$ −8.47214 −0.337806
$$630$$ 0 0
$$631$$ −10.7639 −0.428505 −0.214253 0.976778i $$-0.568732\pi$$
−0.214253 + 0.976778i $$0.568732\pi$$
$$632$$ −20.4508 −0.813491
$$633$$ 0 0
$$634$$ 33.2705 1.32134
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ −17.5623 −0.695298
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23.3262 0.921331 0.460666 0.887574i $$-0.347611\pi$$
0.460666 + 0.887574i $$0.347611\pi$$
$$642$$ 0 0
$$643$$ 5.90983 0.233061 0.116530 0.993187i $$-0.462823\pi$$
0.116530 + 0.993187i $$0.462823\pi$$
$$644$$ 6.65248 0.262144
$$645$$ 0 0
$$646$$ 45.9787 1.80901
$$647$$ 19.0344 0.748321 0.374161 0.927364i $$-0.377931\pi$$
0.374161 + 0.927364i $$0.377931\pi$$
$$648$$ 0 0
$$649$$ −11.8328 −0.464479
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 11.2918 0.442221
$$653$$ −29.6525 −1.16039 −0.580196 0.814477i $$-0.697024\pi$$
−0.580196 + 0.814477i $$0.697024\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −45.5410 −1.77808
$$657$$ 0 0
$$658$$ 15.4164 0.600994
$$659$$ 2.23607 0.0871048 0.0435524 0.999051i $$-0.486132\pi$$
0.0435524 + 0.999051i $$0.486132\pi$$
$$660$$ 0 0
$$661$$ 4.88854 0.190142 0.0950712 0.995470i $$-0.469692\pi$$
0.0950712 + 0.995470i $$0.469692\pi$$
$$662$$ 49.6525 1.92980
$$663$$ 0 0
$$664$$ −20.1246 −0.780986
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −19.4721 −0.753964
$$668$$ −11.0000 −0.425603
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 26.1246 1.00853
$$672$$ 0 0
$$673$$ −46.7771 −1.80312 −0.901562 0.432650i $$-0.857579\pi$$
−0.901562 + 0.432650i $$0.857579\pi$$
$$674$$ 53.6869 2.06794
$$675$$ 0 0
$$676$$ −7.41641 −0.285246
$$677$$ 44.8885 1.72521 0.862603 0.505881i $$-0.168832\pi$$
0.862603 + 0.505881i $$0.168832\pi$$
$$678$$ 0 0
$$679$$ −7.70820 −0.295814
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −42.2705 −1.61862
$$683$$ 0.596748 0.0228339 0.0114170 0.999935i $$-0.496366\pi$$
0.0114170 + 0.999935i $$0.496366\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −32.3607 −1.23554
$$687$$ 0 0
$$688$$ 35.8328 1.36611
$$689$$ 11.2361 0.428060
$$690$$ 0 0
$$691$$ 15.0902 0.574057 0.287029 0.957922i $$-0.407333\pi$$
0.287029 + 0.957922i $$0.407333\pi$$
$$692$$ −0.562306 −0.0213757
$$693$$ 0 0
$$694$$ 19.8541 0.753651
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 39.7426 1.50536
$$698$$ −11.7082 −0.443162
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 48.6525 1.83758 0.918789 0.394748i $$-0.129168\pi$$
0.918789 + 0.394748i $$0.129168\pi$$
$$702$$ 0 0
$$703$$ 13.4164 0.506009
$$704$$ 12.7082 0.478958
$$705$$ 0 0
$$706$$ −20.0344 −0.754006
$$707$$ −18.7639 −0.705690
$$708$$ 0 0
$$709$$ −5.20163 −0.195351 −0.0976756 0.995218i $$-0.531141\pi$$
−0.0976756 + 0.995218i $$0.531141\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 25.0000 0.936915
$$713$$ −46.8673 −1.75519
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9.67376 0.361525
$$717$$ 0 0
$$718$$ 38.7426 1.44586
$$719$$ 35.1246 1.30993 0.654963 0.755661i $$-0.272684\pi$$
0.654963 + 0.755661i $$0.272684\pi$$
$$720$$ 0 0
$$721$$ 28.8328 1.07379
$$722$$ −42.0689 −1.56564
$$723$$ 0 0
$$724$$ −7.70820 −0.286473
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −14.4377 −0.535464 −0.267732 0.963493i $$-0.586274\pi$$
−0.267732 + 0.963493i $$0.586274\pi$$
$$728$$ 4.47214 0.165748
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −31.2705 −1.15658
$$732$$ 0 0
$$733$$ −10.1459 −0.374747 −0.187374 0.982289i $$-0.559998\pi$$
−0.187374 + 0.982289i $$0.559998\pi$$
$$734$$ 20.2705 0.748198
$$735$$ 0 0
$$736$$ −18.2016 −0.670921
$$737$$ −39.5410 −1.45651
$$738$$ 0 0
$$739$$ −1.70820 −0.0628373 −0.0314186 0.999506i $$-0.510003\pi$$
−0.0314186 + 0.999506i $$0.510003\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −36.3607 −1.33484
$$743$$ 16.5279 0.606349 0.303174 0.952935i $$-0.401954\pi$$
0.303174 + 0.952935i $$0.401954\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −28.1803 −1.03176
$$747$$ 0 0
$$748$$ 7.85410 0.287174
$$749$$ −2.29180 −0.0837404
$$750$$ 0 0
$$751$$ 15.2918 0.558006 0.279003 0.960290i $$-0.409996\pi$$
0.279003 + 0.960290i $$0.409996\pi$$
$$752$$ −23.1246 −0.843268
$$753$$ 0 0
$$754$$ 5.85410 0.213194
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32.2705 1.17289 0.586446 0.809988i $$-0.300527\pi$$
0.586446 + 0.809988i $$0.300527\pi$$
$$758$$ 22.0344 0.800327
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.18034 −0.115287 −0.0576436 0.998337i $$-0.518359\pi$$
−0.0576436 + 0.998337i $$0.518359\pi$$
$$762$$ 0 0
$$763$$ 8.29180 0.300183
$$764$$ −10.7082 −0.387409
$$765$$ 0 0
$$766$$ 8.18034 0.295568
$$767$$ 3.94427 0.142419
$$768$$ 0 0
$$769$$ −36.3050 −1.30919 −0.654595 0.755980i $$-0.727161\pi$$
−0.654595 + 0.755980i $$0.727161\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.79837 −0.244679
$$773$$ 41.7771 1.50262 0.751309 0.659951i $$-0.229423\pi$$
0.751309 + 0.659951i $$0.229423\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 8.61803 0.309369
$$777$$ 0 0
$$778$$ 1.05573 0.0378497
$$779$$ −62.9361 −2.25492
$$780$$ 0 0
$$781$$ −30.2705 −1.08316
$$782$$ 36.8885 1.31913
$$783$$ 0 0
$$784$$ 14.5623 0.520082
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17.1459 0.611185 0.305593 0.952162i $$-0.401145\pi$$
0.305593 + 0.952162i $$0.401145\pi$$
$$788$$ 0.0557281 0.00198523
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7.52786 −0.267660
$$792$$ 0 0
$$793$$ −8.70820 −0.309237
$$794$$ 4.09017 0.145155
$$795$$ 0 0
$$796$$ 7.23607 0.256476
$$797$$ 30.0132 1.06312 0.531560 0.847020i $$-0.321606\pi$$
0.531560 + 0.847020i $$0.321606\pi$$
$$798$$ 0 0
$$799$$ 20.1803 0.713929
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 58.6869 2.07231
$$803$$ 47.1246 1.66299
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 14.0902 0.496305
$$807$$ 0 0
$$808$$ 20.9787 0.738029
$$809$$ 39.9230 1.40362 0.701809 0.712365i $$-0.252376\pi$$
0.701809 + 0.712365i $$0.252376\pi$$
$$810$$ 0 0
$$811$$ −33.7771 −1.18607 −0.593037 0.805175i $$-0.702071\pi$$
−0.593037 + 0.805175i $$0.702071\pi$$
$$812$$ −4.47214 −0.156941
$$813$$ 0 0
$$814$$ 9.70820 0.340272
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 49.5197 1.73248
$$818$$ 8.29180 0.289916
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −5.94427 −0.207457 −0.103728 0.994606i $$-0.533077\pi$$
−0.103728 + 0.994606i $$0.533077\pi$$
$$822$$ 0 0
$$823$$ −28.5623 −0.995619 −0.497810 0.867286i $$-0.665862\pi$$
−0.497810 + 0.867286i $$0.665862\pi$$
$$824$$ −32.2361 −1.12300
$$825$$ 0 0
$$826$$ −12.7639 −0.444114
$$827$$ −48.9787 −1.70316 −0.851578 0.524227i $$-0.824354\pi$$
−0.851578 + 0.524227i $$0.824354\pi$$
$$828$$ 0 0
$$829$$ −50.1246 −1.74090 −0.870450 0.492257i $$-0.836172\pi$$
−0.870450 + 0.492257i $$0.836172\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.23607 −0.146859
$$833$$ −12.7082 −0.440313
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.4377 −0.430167
$$837$$ 0 0
$$838$$ 0.527864 0.0182348
$$839$$ 3.21478 0.110987 0.0554933 0.998459i $$-0.482327\pi$$
0.0554933 + 0.998459i $$0.482327\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ 49.1246 1.69295
$$843$$ 0 0
$$844$$ −1.85410 −0.0638208
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.00000 0.137442
$$848$$ 54.5410 1.87295
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 10.7639 0.368983
$$852$$ 0 0
$$853$$ −20.3951 −0.698316 −0.349158 0.937064i $$-0.613532\pi$$
−0.349158 + 0.937064i $$0.613532\pi$$
$$854$$ 28.1803 0.964311
$$855$$ 0 0
$$856$$ 2.56231 0.0875778
$$857$$ −9.05573 −0.309338 −0.154669 0.987966i $$-0.549431\pi$$
−0.154669 + 0.987966i $$0.549431\pi$$
$$858$$ 0 0
$$859$$ 15.1246 0.516045 0.258023 0.966139i $$-0.416929\pi$$
0.258023 + 0.966139i $$0.416929\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 48.1591 1.64030
$$863$$ −13.0689 −0.444870 −0.222435 0.974948i $$-0.571401\pi$$
−0.222435 + 0.974948i $$0.571401\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −5.61803 −0.190909
$$867$$ 0 0
$$868$$ −10.7639 −0.365352
$$869$$ −27.4377 −0.930760
$$870$$ 0 0
$$871$$ 13.1803 0.446599
$$872$$ −9.27051 −0.313939
$$873$$ 0 0
$$874$$ −58.4164 −1.97596
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 43.1246 1.45621 0.728107 0.685463i $$-0.240400\pi$$
0.728107 + 0.685463i $$0.240400\pi$$
$$878$$ 51.8328 1.74927
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −15.0902 −0.508401 −0.254200 0.967152i $$-0.581812\pi$$
−0.254200 + 0.967152i $$0.581812\pi$$
$$882$$ 0 0
$$883$$ 29.2016 0.982713 0.491356 0.870959i $$-0.336501\pi$$
0.491356 + 0.870959i $$0.336501\pi$$
$$884$$ −2.61803 −0.0880540
$$885$$ 0 0
$$886$$ −31.4164 −1.05545
$$887$$ −42.9230 −1.44121 −0.720606 0.693344i $$-0.756136\pi$$
−0.720606 + 0.693344i $$0.756136\pi$$
$$888$$ 0 0
$$889$$ −27.3050 −0.915779
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −6.27051 −0.209952
$$893$$ −31.9574 −1.06941
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 27.2361 0.909893
$$897$$ 0 0
$$898$$ 26.7082 0.891264
$$899$$ 31.5066 1.05080
$$900$$ 0 0
$$901$$ −47.5967 −1.58568
$$902$$ −45.5410 −1.51635
$$903$$ 0 0
$$904$$ 8.41641 0.279926
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ −3.56231 −0.118219
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −14.8885 −0.493279 −0.246640 0.969107i $$-0.579326\pi$$
−0.246640 + 0.969107i $$0.579326\pi$$
$$912$$ 0 0
$$913$$ −27.0000 −0.893570
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −28.3607 −0.936552
$$918$$ 0 0
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −31.0344 −1.02206
$$923$$ 10.0902 0.332122
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 54.5066 1.79120
$$927$$ 0 0
$$928$$ 12.2361 0.401669
$$929$$ 29.5967 0.971038 0.485519 0.874226i $$-0.338631\pi$$
0.485519 + 0.874226i $$0.338631\pi$$
$$930$$ 0 0
$$931$$ 20.1246 0.659558
$$932$$ −6.29180 −0.206095
$$933$$ 0 0
$$934$$ −16.8541 −0.551483
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.4164 −0.340289 −0.170145 0.985419i $$-0.554423\pi$$
−0.170145 + 0.985419i $$0.554423\pi$$
$$938$$ −42.6525 −1.39265
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −57.9787 −1.89005 −0.945026 0.326995i $$-0.893964\pi$$
−0.945026 + 0.326995i $$0.893964\pi$$
$$942$$ 0 0
$$943$$ −50.4934 −1.64429
$$944$$ 19.1459 0.623146
$$945$$ 0 0
$$946$$ 35.8328 1.16503
$$947$$ 23.8328 0.774462 0.387231 0.921983i $$-0.373432\pi$$
0.387231 + 0.921983i $$0.373432\pi$$
$$948$$ 0 0
$$949$$ −15.7082 −0.509910
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −18.9443 −0.613987
$$953$$ 42.0557 1.36232 0.681159 0.732135i $$-0.261476\pi$$
0.681159 + 0.732135i $$0.261476\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −13.2148 −0.427397
$$957$$ 0 0
$$958$$ 7.76393 0.250841
$$959$$ 0.875388 0.0282678
$$960$$ 0 0
$$961$$ 44.8328 1.44622
$$962$$ −3.23607 −0.104335
$$963$$ 0 0
$$964$$ 4.52786 0.145833
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −35.4164 −1.13891 −0.569457 0.822021i $$-0.692847\pi$$
−0.569457 + 0.822021i $$0.692847\pi$$
$$968$$ −4.47214 −0.143740
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −29.8885 −0.959169 −0.479585 0.877496i $$-0.659213\pi$$
−0.479585 + 0.877496i $$0.659213\pi$$
$$972$$ 0 0
$$973$$ −26.8328 −0.860221
$$974$$ −26.8885 −0.861565
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ 37.6525 1.20461 0.602305 0.798266i $$-0.294249\pi$$
0.602305 + 0.798266i $$0.294249\pi$$
$$978$$ 0 0
$$979$$ 33.5410 1.07198
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 36.1246 1.15278
$$983$$ 24.6180 0.785193 0.392597 0.919711i $$-0.371577\pi$$
0.392597 + 0.919711i $$0.371577\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −24.7984 −0.789741
$$987$$ 0 0
$$988$$ 4.14590 0.131899
$$989$$ 39.7295 1.26332
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 29.4508 0.935065
$$993$$ 0 0
$$994$$ −32.6525 −1.03567
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −2.93112 −0.0928294 −0.0464147 0.998922i $$-0.514780\pi$$
−0.0464147 + 0.998922i $$0.514780\pi$$
$$998$$ 24.2705 0.768270
$$999$$ 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 5625.2.a.b.1.1 2
3.2 odd 2 1875.2.a.c.1.2 yes 2
5.4 even 2 5625.2.a.g.1.2 2
15.2 even 4 1875.2.b.a.1249.4 4
15.8 even 4 1875.2.b.a.1249.1 4
15.14 odd 2 1875.2.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.1 2 15.14 odd 2
1875.2.a.c.1.2 yes 2 3.2 odd 2
1875.2.b.a.1249.1 4 15.8 even 4
1875.2.b.a.1249.4 4 15.2 even 4
5625.2.a.b.1.1 2 1.1 even 1 trivial
5625.2.a.g.1.2 2 5.4 even 2