# Properties

 Label 3330.2.a.bd.1.1 Level $3330$ Weight $2$ Character 3330.1 Self dual yes Analytic conductor $26.590$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 3330.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.46410 q^{11} -5.46410 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +5.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} -5.46410 q^{26} -4.73205 q^{28} -4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} -5.46410 q^{34} +4.73205 q^{35} +1.00000 q^{37} +6.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} +6.92820 q^{43} +5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} +15.3923 q^{49} +1.00000 q^{50} -5.46410 q^{52} +6.00000 q^{53} -5.46410 q^{55} -4.73205 q^{56} -4.92820 q^{58} +10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} +1.00000 q^{64} +5.46410 q^{65} -3.66025 q^{67} -5.46410 q^{68} +4.73205 q^{70} -2.92820 q^{71} -0.928203 q^{73} +1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} +8.73205 q^{79} -1.00000 q^{80} +2.00000 q^{82} +8.73205 q^{83} +5.46410 q^{85} +6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} +4.73205 q^{94} -6.19615 q^{95} -2.00000 q^{97} +15.3923 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 2 * q^25 - 4 * q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 + 2 * q^37 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 + 16 * q^46 + 6 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 4 * q^55 - 6 * q^56 + 4 * q^58 + 10 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 4 * q^68 + 6 * q^70 + 8 * q^71 + 12 * q^73 + 2 * q^74 + 2 * q^76 - 24 * q^77 + 14 * q^79 - 2 * q^80 + 4 * q^82 + 14 * q^83 + 4 * q^85 + 4 * q^88 + 4 * q^89 + 24 * q^91 + 16 * q^92 + 6 * q^94 - 2 * q^95 - 4 * q^97 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.73205 −1.78855 −0.894274 0.447521i $$-0.852307\pi$$
−0.894274 + 0.447521i $$0.852307\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 5.46410 1.64749 0.823744 0.566961i $$-0.191881\pi$$
0.823744 + 0.566961i $$0.191881\pi$$
$$12$$ 0 0
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ −4.73205 −1.26469
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −5.46410 −1.32524 −0.662620 0.748956i $$-0.730555\pi$$
−0.662620 + 0.748956i $$0.730555\pi$$
$$18$$ 0 0
$$19$$ 6.19615 1.42149 0.710747 0.703447i $$-0.248357\pi$$
0.710747 + 0.703447i $$0.248357\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 5.46410 1.16495
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −5.46410 −1.07160
$$27$$ 0 0
$$28$$ −4.73205 −0.894274
$$29$$ −4.92820 −0.915144 −0.457572 0.889172i $$-0.651281\pi$$
−0.457572 + 0.889172i $$0.651281\pi$$
$$30$$ 0 0
$$31$$ 0.732051 0.131480 0.0657401 0.997837i $$-0.479059\pi$$
0.0657401 + 0.997837i $$0.479059\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −5.46410 −0.937086
$$35$$ 4.73205 0.799863
$$36$$ 0 0
$$37$$ 1.00000 0.164399
$$38$$ 6.19615 1.00515
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 6.92820 1.05654 0.528271 0.849076i $$-0.322841\pi$$
0.528271 + 0.849076i $$0.322841\pi$$
$$44$$ 5.46410 0.823744
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 4.73205 0.690241 0.345120 0.938558i $$-0.387838\pi$$
0.345120 + 0.938558i $$0.387838\pi$$
$$48$$ 0 0
$$49$$ 15.3923 2.19890
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −5.46410 −0.757735
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −5.46410 −0.736779
$$56$$ −4.73205 −0.632347
$$57$$ 0 0
$$58$$ −4.92820 −0.647105
$$59$$ 10.1962 1.32743 0.663713 0.747987i $$-0.268980\pi$$
0.663713 + 0.747987i $$0.268980\pi$$
$$60$$ 0 0
$$61$$ −4.92820 −0.630992 −0.315496 0.948927i $$-0.602171\pi$$
−0.315496 + 0.948927i $$0.602171\pi$$
$$62$$ 0.732051 0.0929705
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 5.46410 0.677738
$$66$$ 0 0
$$67$$ −3.66025 −0.447171 −0.223586 0.974684i $$-0.571776\pi$$
−0.223586 + 0.974684i $$0.571776\pi$$
$$68$$ −5.46410 −0.662620
$$69$$ 0 0
$$70$$ 4.73205 0.565588
$$71$$ −2.92820 −0.347514 −0.173757 0.984789i $$-0.555591\pi$$
−0.173757 + 0.984789i $$0.555591\pi$$
$$72$$ 0 0
$$73$$ −0.928203 −0.108638 −0.0543190 0.998524i $$-0.517299\pi$$
−0.0543190 + 0.998524i $$0.517299\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 6.19615 0.710747
$$77$$ −25.8564 −2.94661
$$78$$ 0 0
$$79$$ 8.73205 0.982432 0.491216 0.871038i $$-0.336552\pi$$
0.491216 + 0.871038i $$0.336552\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ 8.73205 0.958467 0.479234 0.877687i $$-0.340915\pi$$
0.479234 + 0.877687i $$0.340915\pi$$
$$84$$ 0 0
$$85$$ 5.46410 0.592665
$$86$$ 6.92820 0.747087
$$87$$ 0 0
$$88$$ 5.46410 0.582475
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 25.8564 2.71049
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 4.73205 0.488074
$$95$$ −6.19615 −0.635712
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 15.3923 1.55486
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 9.46410 0.941713 0.470857 0.882210i $$-0.343945\pi$$
0.470857 + 0.882210i $$0.343945\pi$$
$$102$$ 0 0
$$103$$ −6.53590 −0.644001 −0.322001 0.946739i $$-0.604355\pi$$
−0.322001 + 0.946739i $$0.604355\pi$$
$$104$$ −5.46410 −0.535799
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −3.26795 −0.315925 −0.157962 0.987445i $$-0.550492\pi$$
−0.157962 + 0.987445i $$0.550492\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ −5.46410 −0.520982
$$111$$ 0 0
$$112$$ −4.73205 −0.447137
$$113$$ 10.5359 0.991134 0.495567 0.868570i $$-0.334960\pi$$
0.495567 + 0.868570i $$0.334960\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ −4.92820 −0.457572
$$117$$ 0 0
$$118$$ 10.1962 0.938632
$$119$$ 25.8564 2.37025
$$120$$ 0 0
$$121$$ 18.8564 1.71422
$$122$$ −4.92820 −0.446179
$$123$$ 0 0
$$124$$ 0.732051 0.0657401
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 3.66025 0.324795 0.162398 0.986725i $$-0.448077\pi$$
0.162398 + 0.986725i $$0.448077\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 5.46410 0.479233
$$131$$ 18.5885 1.62408 0.812041 0.583601i $$-0.198357\pi$$
0.812041 + 0.583601i $$0.198357\pi$$
$$132$$ 0 0
$$133$$ −29.3205 −2.54241
$$134$$ −3.66025 −0.316198
$$135$$ 0 0
$$136$$ −5.46410 −0.468543
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ 6.92820 0.587643 0.293821 0.955860i $$-0.405073\pi$$
0.293821 + 0.955860i $$0.405073\pi$$
$$140$$ 4.73205 0.399931
$$141$$ 0 0
$$142$$ −2.92820 −0.245729
$$143$$ −29.8564 −2.49672
$$144$$ 0 0
$$145$$ 4.92820 0.409265
$$146$$ −0.928203 −0.0768186
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ −4.39230 −0.359832 −0.179916 0.983682i $$-0.557583\pi$$
−0.179916 + 0.983682i $$0.557583\pi$$
$$150$$ 0 0
$$151$$ −12.3923 −1.00847 −0.504236 0.863566i $$-0.668226\pi$$
−0.504236 + 0.863566i $$0.668226\pi$$
$$152$$ 6.19615 0.502574
$$153$$ 0 0
$$154$$ −25.8564 −2.08357
$$155$$ −0.732051 −0.0587997
$$156$$ 0 0
$$157$$ 3.07180 0.245156 0.122578 0.992459i $$-0.460884\pi$$
0.122578 + 0.992459i $$0.460884\pi$$
$$158$$ 8.73205 0.694685
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −37.8564 −2.98350
$$162$$ 0 0
$$163$$ −11.3205 −0.886691 −0.443345 0.896351i $$-0.646208\pi$$
−0.443345 + 0.896351i $$0.646208\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 8.73205 0.677739
$$167$$ 1.46410 0.113296 0.0566478 0.998394i $$-0.481959\pi$$
0.0566478 + 0.998394i $$0.481959\pi$$
$$168$$ 0 0
$$169$$ 16.8564 1.29665
$$170$$ 5.46410 0.419077
$$171$$ 0 0
$$172$$ 6.92820 0.528271
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 0 0
$$175$$ −4.73205 −0.357709
$$176$$ 5.46410 0.411872
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 0.339746 0.0253938 0.0126969 0.999919i $$-0.495958\pi$$
0.0126969 + 0.999919i $$0.495958\pi$$
$$180$$ 0 0
$$181$$ 5.46410 0.406143 0.203072 0.979164i $$-0.434908\pi$$
0.203072 + 0.979164i $$0.434908\pi$$
$$182$$ 25.8564 1.91660
$$183$$ 0 0
$$184$$ 8.00000 0.589768
$$185$$ −1.00000 −0.0735215
$$186$$ 0 0
$$187$$ −29.8564 −2.18332
$$188$$ 4.73205 0.345120
$$189$$ 0 0
$$190$$ −6.19615 −0.449516
$$191$$ 8.73205 0.631829 0.315915 0.948788i $$-0.397689\pi$$
0.315915 + 0.948788i $$0.397689\pi$$
$$192$$ 0 0
$$193$$ 15.8564 1.14137 0.570685 0.821169i $$-0.306678\pi$$
0.570685 + 0.821169i $$0.306678\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 15.3923 1.09945
$$197$$ 22.7846 1.62334 0.811668 0.584119i $$-0.198560\pi$$
0.811668 + 0.584119i $$0.198560\pi$$
$$198$$ 0 0
$$199$$ −12.0526 −0.854383 −0.427192 0.904161i $$-0.640497\pi$$
−0.427192 + 0.904161i $$0.640497\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 9.46410 0.665892
$$203$$ 23.3205 1.63678
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ −6.53590 −0.455378
$$207$$ 0 0
$$208$$ −5.46410 −0.378867
$$209$$ 33.8564 2.34190
$$210$$ 0 0
$$211$$ −17.8564 −1.22929 −0.614643 0.788806i $$-0.710700\pi$$
−0.614643 + 0.788806i $$0.710700\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −3.26795 −0.223392
$$215$$ −6.92820 −0.472500
$$216$$ 0 0
$$217$$ −3.46410 −0.235159
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ −5.46410 −0.368390
$$221$$ 29.8564 2.00836
$$222$$ 0 0
$$223$$ 16.0526 1.07496 0.537479 0.843277i $$-0.319377\pi$$
0.537479 + 0.843277i $$0.319377\pi$$
$$224$$ −4.73205 −0.316173
$$225$$ 0 0
$$226$$ 10.5359 0.700838
$$227$$ −24.3923 −1.61897 −0.809487 0.587138i $$-0.800255\pi$$
−0.809487 + 0.587138i $$0.800255\pi$$
$$228$$ 0 0
$$229$$ −11.8564 −0.783493 −0.391747 0.920073i $$-0.628129\pi$$
−0.391747 + 0.920073i $$0.628129\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ −4.92820 −0.323552
$$233$$ −28.9282 −1.89515 −0.947575 0.319534i $$-0.896474\pi$$
−0.947575 + 0.319534i $$0.896474\pi$$
$$234$$ 0 0
$$235$$ −4.73205 −0.308685
$$236$$ 10.1962 0.663713
$$237$$ 0 0
$$238$$ 25.8564 1.67602
$$239$$ 20.7321 1.34104 0.670522 0.741889i $$-0.266070\pi$$
0.670522 + 0.741889i $$0.266070\pi$$
$$240$$ 0 0
$$241$$ 4.92820 0.317453 0.158727 0.987323i $$-0.449261\pi$$
0.158727 + 0.987323i $$0.449261\pi$$
$$242$$ 18.8564 1.21214
$$243$$ 0 0
$$244$$ −4.92820 −0.315496
$$245$$ −15.3923 −0.983378
$$246$$ 0 0
$$247$$ −33.8564 −2.15423
$$248$$ 0.732051 0.0464853
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −19.2679 −1.21618 −0.608091 0.793867i $$-0.708064\pi$$
−0.608091 + 0.793867i $$0.708064\pi$$
$$252$$ 0 0
$$253$$ 43.7128 2.74820
$$254$$ 3.66025 0.229665
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.5359 1.15624 0.578119 0.815953i $$-0.303787\pi$$
0.578119 + 0.815953i $$0.303787\pi$$
$$258$$ 0 0
$$259$$ −4.73205 −0.294035
$$260$$ 5.46410 0.338869
$$261$$ 0 0
$$262$$ 18.5885 1.14840
$$263$$ 8.05256 0.496542 0.248271 0.968691i $$-0.420138\pi$$
0.248271 + 0.968691i $$0.420138\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ −29.3205 −1.79776
$$267$$ 0 0
$$268$$ −3.66025 −0.223586
$$269$$ −20.3923 −1.24334 −0.621670 0.783279i $$-0.713546\pi$$
−0.621670 + 0.783279i $$0.713546\pi$$
$$270$$ 0 0
$$271$$ −24.7846 −1.50556 −0.752779 0.658273i $$-0.771287\pi$$
−0.752779 + 0.658273i $$0.771287\pi$$
$$272$$ −5.46410 −0.331310
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 5.46410 0.329498
$$276$$ 0 0
$$277$$ 22.2487 1.33680 0.668398 0.743804i $$-0.266980\pi$$
0.668398 + 0.743804i $$0.266980\pi$$
$$278$$ 6.92820 0.415526
$$279$$ 0 0
$$280$$ 4.73205 0.282794
$$281$$ 8.92820 0.532612 0.266306 0.963889i $$-0.414197\pi$$
0.266306 + 0.963889i $$0.414197\pi$$
$$282$$ 0 0
$$283$$ 16.3923 0.974421 0.487211 0.873284i $$-0.338014\pi$$
0.487211 + 0.873284i $$0.338014\pi$$
$$284$$ −2.92820 −0.173757
$$285$$ 0 0
$$286$$ −29.8564 −1.76545
$$287$$ −9.46410 −0.558648
$$288$$ 0 0
$$289$$ 12.8564 0.756259
$$290$$ 4.92820 0.289394
$$291$$ 0 0
$$292$$ −0.928203 −0.0543190
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −10.1962 −0.593643
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −4.39230 −0.254439
$$299$$ −43.7128 −2.52798
$$300$$ 0 0
$$301$$ −32.7846 −1.88967
$$302$$ −12.3923 −0.713097
$$303$$ 0 0
$$304$$ 6.19615 0.355374
$$305$$ 4.92820 0.282188
$$306$$ 0 0
$$307$$ 18.5885 1.06090 0.530450 0.847716i $$-0.322023\pi$$
0.530450 + 0.847716i $$0.322023\pi$$
$$308$$ −25.8564 −1.47331
$$309$$ 0 0
$$310$$ −0.732051 −0.0415777
$$311$$ −2.87564 −0.163063 −0.0815314 0.996671i $$-0.525981\pi$$
−0.0815314 + 0.996671i $$0.525981\pi$$
$$312$$ 0 0
$$313$$ −23.8564 −1.34844 −0.674222 0.738529i $$-0.735521\pi$$
−0.674222 + 0.738529i $$0.735521\pi$$
$$314$$ 3.07180 0.173352
$$315$$ 0 0
$$316$$ 8.73205 0.491216
$$317$$ 4.14359 0.232727 0.116364 0.993207i $$-0.462876\pi$$
0.116364 + 0.993207i $$0.462876\pi$$
$$318$$ 0 0
$$319$$ −26.9282 −1.50769
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −37.8564 −2.10966
$$323$$ −33.8564 −1.88382
$$324$$ 0 0
$$325$$ −5.46410 −0.303094
$$326$$ −11.3205 −0.626985
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ −22.3923 −1.23453
$$330$$ 0 0
$$331$$ −33.1244 −1.82068 −0.910340 0.413862i $$-0.864180\pi$$
−0.910340 + 0.413862i $$0.864180\pi$$
$$332$$ 8.73205 0.479234
$$333$$ 0 0
$$334$$ 1.46410 0.0801121
$$335$$ 3.66025 0.199981
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 16.8564 0.916868
$$339$$ 0 0
$$340$$ 5.46410 0.296333
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ −39.7128 −2.14429
$$344$$ 6.92820 0.373544
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ −30.9282 −1.66031 −0.830156 0.557530i $$-0.811749\pi$$
−0.830156 + 0.557530i $$0.811749\pi$$
$$348$$ 0 0
$$349$$ −15.3205 −0.820088 −0.410044 0.912066i $$-0.634487\pi$$
−0.410044 + 0.912066i $$0.634487\pi$$
$$350$$ −4.73205 −0.252939
$$351$$ 0 0
$$352$$ 5.46410 0.291238
$$353$$ 11.8564 0.631053 0.315526 0.948917i $$-0.397819\pi$$
0.315526 + 0.948917i $$0.397819\pi$$
$$354$$ 0 0
$$355$$ 2.92820 0.155413
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 0.339746 0.0179561
$$359$$ 12.3923 0.654041 0.327020 0.945017i $$-0.393955\pi$$
0.327020 + 0.945017i $$0.393955\pi$$
$$360$$ 0 0
$$361$$ 19.3923 1.02065
$$362$$ 5.46410 0.287187
$$363$$ 0 0
$$364$$ 25.8564 1.35524
$$365$$ 0.928203 0.0485844
$$366$$ 0 0
$$367$$ −4.33975 −0.226533 −0.113266 0.993565i $$-0.536131\pi$$
−0.113266 + 0.993565i $$0.536131\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 0 0
$$370$$ −1.00000 −0.0519875
$$371$$ −28.3923 −1.47406
$$372$$ 0 0
$$373$$ −10.7846 −0.558406 −0.279203 0.960232i $$-0.590070\pi$$
−0.279203 + 0.960232i $$0.590070\pi$$
$$374$$ −29.8564 −1.54384
$$375$$ 0 0
$$376$$ 4.73205 0.244037
$$377$$ 26.9282 1.38687
$$378$$ 0 0
$$379$$ 32.3923 1.66388 0.831940 0.554865i $$-0.187230\pi$$
0.831940 + 0.554865i $$0.187230\pi$$
$$380$$ −6.19615 −0.317856
$$381$$ 0 0
$$382$$ 8.73205 0.446771
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 25.8564 1.31776
$$386$$ 15.8564 0.807070
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 11.8564 0.601144 0.300572 0.953759i $$-0.402822\pi$$
0.300572 + 0.953759i $$0.402822\pi$$
$$390$$ 0 0
$$391$$ −43.7128 −2.21065
$$392$$ 15.3923 0.777429
$$393$$ 0 0
$$394$$ 22.7846 1.14787
$$395$$ −8.73205 −0.439357
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −12.0526 −0.604140
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 32.9282 1.64436 0.822178 0.569230i $$-0.192759\pi$$
0.822178 + 0.569230i $$0.192759\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 9.46410 0.470857
$$405$$ 0 0
$$406$$ 23.3205 1.15738
$$407$$ 5.46410 0.270845
$$408$$ 0 0
$$409$$ 3.07180 0.151891 0.0759453 0.997112i $$-0.475803\pi$$
0.0759453 + 0.997112i $$0.475803\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 0 0
$$412$$ −6.53590 −0.322001
$$413$$ −48.2487 −2.37416
$$414$$ 0 0
$$415$$ −8.73205 −0.428640
$$416$$ −5.46410 −0.267900
$$417$$ 0 0
$$418$$ 33.8564 1.65597
$$419$$ −14.2487 −0.696095 −0.348048 0.937477i $$-0.613155\pi$$
−0.348048 + 0.937477i $$0.613155\pi$$
$$420$$ 0 0
$$421$$ −0.143594 −0.00699832 −0.00349916 0.999994i $$-0.501114\pi$$
−0.00349916 + 0.999994i $$0.501114\pi$$
$$422$$ −17.8564 −0.869236
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −5.46410 −0.265048
$$426$$ 0 0
$$427$$ 23.3205 1.12856
$$428$$ −3.26795 −0.157962
$$429$$ 0 0
$$430$$ −6.92820 −0.334108
$$431$$ 2.19615 0.105785 0.0528925 0.998600i $$-0.483156\pi$$
0.0528925 + 0.998600i $$0.483156\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ −3.46410 −0.166282
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 49.5692 2.37122
$$438$$ 0 0
$$439$$ 13.5167 0.645115 0.322558 0.946550i $$-0.395457\pi$$
0.322558 + 0.946550i $$0.395457\pi$$
$$440$$ −5.46410 −0.260491
$$441$$ 0 0
$$442$$ 29.8564 1.42012
$$443$$ 14.8756 0.706763 0.353382 0.935479i $$-0.385032\pi$$
0.353382 + 0.935479i $$0.385032\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 16.0526 0.760111
$$447$$ 0 0
$$448$$ −4.73205 −0.223568
$$449$$ 21.7128 1.02469 0.512345 0.858779i $$-0.328777\pi$$
0.512345 + 0.858779i $$0.328777\pi$$
$$450$$ 0 0
$$451$$ 10.9282 0.514589
$$452$$ 10.5359 0.495567
$$453$$ 0 0
$$454$$ −24.3923 −1.14479
$$455$$ −25.8564 −1.21217
$$456$$ 0 0
$$457$$ −31.8564 −1.49018 −0.745090 0.666964i $$-0.767593\pi$$
−0.745090 + 0.666964i $$0.767593\pi$$
$$458$$ −11.8564 −0.554013
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −14.7846 −0.688588 −0.344294 0.938862i $$-0.611882\pi$$
−0.344294 + 0.938862i $$0.611882\pi$$
$$462$$ 0 0
$$463$$ 18.9282 0.879668 0.439834 0.898079i $$-0.355037\pi$$
0.439834 + 0.898079i $$0.355037\pi$$
$$464$$ −4.92820 −0.228786
$$465$$ 0 0
$$466$$ −28.9282 −1.34007
$$467$$ 25.1769 1.16505 0.582524 0.812813i $$-0.302065\pi$$
0.582524 + 0.812813i $$0.302065\pi$$
$$468$$ 0 0
$$469$$ 17.3205 0.799787
$$470$$ −4.73205 −0.218273
$$471$$ 0 0
$$472$$ 10.1962 0.469316
$$473$$ 37.8564 1.74064
$$474$$ 0 0
$$475$$ 6.19615 0.284299
$$476$$ 25.8564 1.18513
$$477$$ 0 0
$$478$$ 20.7321 0.948262
$$479$$ −4.05256 −0.185166 −0.0925831 0.995705i $$-0.529512\pi$$
−0.0925831 + 0.995705i $$0.529512\pi$$
$$480$$ 0 0
$$481$$ −5.46410 −0.249142
$$482$$ 4.92820 0.224474
$$483$$ 0 0
$$484$$ 18.8564 0.857109
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 4.39230 0.199034 0.0995172 0.995036i $$-0.468270\pi$$
0.0995172 + 0.995036i $$0.468270\pi$$
$$488$$ −4.92820 −0.223089
$$489$$ 0 0
$$490$$ −15.3923 −0.695353
$$491$$ 14.9282 0.673700 0.336850 0.941558i $$-0.390638\pi$$
0.336850 + 0.941558i $$0.390638\pi$$
$$492$$ 0 0
$$493$$ 26.9282 1.21279
$$494$$ −33.8564 −1.52327
$$495$$ 0 0
$$496$$ 0.732051 0.0328701
$$497$$ 13.8564 0.621545
$$498$$ 0 0
$$499$$ 9.41154 0.421319 0.210659 0.977560i $$-0.432439\pi$$
0.210659 + 0.977560i $$0.432439\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −19.2679 −0.859971
$$503$$ 18.9282 0.843967 0.421983 0.906604i $$-0.361334\pi$$
0.421983 + 0.906604i $$0.361334\pi$$
$$504$$ 0 0
$$505$$ −9.46410 −0.421147
$$506$$ 43.7128 1.94327
$$507$$ 0 0
$$508$$ 3.66025 0.162398
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 4.39230 0.194304
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.5359 0.817583
$$515$$ 6.53590 0.288006
$$516$$ 0 0
$$517$$ 25.8564 1.13716
$$518$$ −4.73205 −0.207914
$$519$$ 0 0
$$520$$ 5.46410 0.239617
$$521$$ −26.5359 −1.16256 −0.581279 0.813704i $$-0.697448\pi$$
−0.581279 + 0.813704i $$0.697448\pi$$
$$522$$ 0 0
$$523$$ 36.7846 1.60848 0.804239 0.594306i $$-0.202573\pi$$
0.804239 + 0.594306i $$0.202573\pi$$
$$524$$ 18.5885 0.812041
$$525$$ 0 0
$$526$$ 8.05256 0.351108
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ −29.3205 −1.27121
$$533$$ −10.9282 −0.473353
$$534$$ 0 0
$$535$$ 3.26795 0.141286
$$536$$ −3.66025 −0.158099
$$537$$ 0 0
$$538$$ −20.3923 −0.879175
$$539$$ 84.1051 3.62266
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ −24.7846 −1.06459
$$543$$ 0 0
$$544$$ −5.46410 −0.234271
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 0 0
$$550$$ 5.46410 0.232990
$$551$$ −30.5359 −1.30087
$$552$$ 0 0
$$553$$ −41.3205 −1.75713
$$554$$ 22.2487 0.945257
$$555$$ 0 0
$$556$$ 6.92820 0.293821
$$557$$ −44.1051 −1.86879 −0.934397 0.356234i $$-0.884061\pi$$
−0.934397 + 0.356234i $$0.884061\pi$$
$$558$$ 0 0
$$559$$ −37.8564 −1.60116
$$560$$ 4.73205 0.199966
$$561$$ 0 0
$$562$$ 8.92820 0.376614
$$563$$ −30.9282 −1.30347 −0.651734 0.758447i $$-0.725958\pi$$
−0.651734 + 0.758447i $$0.725958\pi$$
$$564$$ 0 0
$$565$$ −10.5359 −0.443249
$$566$$ 16.3923 0.689020
$$567$$ 0 0
$$568$$ −2.92820 −0.122865
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −29.8564 −1.24836
$$573$$ 0 0
$$574$$ −9.46410 −0.395024
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ −24.3923 −1.01546 −0.507732 0.861515i $$-0.669516\pi$$
−0.507732 + 0.861515i $$0.669516\pi$$
$$578$$ 12.8564 0.534756
$$579$$ 0 0
$$580$$ 4.92820 0.204633
$$581$$ −41.3205 −1.71426
$$582$$ 0 0
$$583$$ 32.7846 1.35780
$$584$$ −0.928203 −0.0384093
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −44.7846 −1.84846 −0.924229 0.381838i $$-0.875291\pi$$
−0.924229 + 0.381838i $$0.875291\pi$$
$$588$$ 0 0
$$589$$ 4.53590 0.186898
$$590$$ −10.1962 −0.419769
$$591$$ 0 0
$$592$$ 1.00000 0.0410997
$$593$$ 34.7846 1.42843 0.714216 0.699925i $$-0.246783\pi$$
0.714216 + 0.699925i $$0.246783\pi$$
$$594$$ 0 0
$$595$$ −25.8564 −1.06001
$$596$$ −4.39230 −0.179916
$$597$$ 0 0
$$598$$ −43.7128 −1.78755
$$599$$ −9.46410 −0.386693 −0.193346 0.981131i $$-0.561934\pi$$
−0.193346 + 0.981131i $$0.561934\pi$$
$$600$$ 0 0
$$601$$ 27.6077 1.12614 0.563071 0.826409i $$-0.309620\pi$$
0.563071 + 0.826409i $$0.309620\pi$$
$$602$$ −32.7846 −1.33620
$$603$$ 0 0
$$604$$ −12.3923 −0.504236
$$605$$ −18.8564 −0.766622
$$606$$ 0 0
$$607$$ 0.784610 0.0318463 0.0159232 0.999873i $$-0.494931\pi$$
0.0159232 + 0.999873i $$0.494931\pi$$
$$608$$ 6.19615 0.251287
$$609$$ 0 0
$$610$$ 4.92820 0.199537
$$611$$ −25.8564 −1.04604
$$612$$ 0 0
$$613$$ 16.9282 0.683724 0.341862 0.939750i $$-0.388943\pi$$
0.341862 + 0.939750i $$0.388943\pi$$
$$614$$ 18.5885 0.750169
$$615$$ 0 0
$$616$$ −25.8564 −1.04178
$$617$$ 0.928203 0.0373681 0.0186840 0.999825i $$-0.494052\pi$$
0.0186840 + 0.999825i $$0.494052\pi$$
$$618$$ 0 0
$$619$$ −17.8564 −0.717710 −0.358855 0.933393i $$-0.616833\pi$$
−0.358855 + 0.933393i $$0.616833\pi$$
$$620$$ −0.732051 −0.0293999
$$621$$ 0 0
$$622$$ −2.87564 −0.115303
$$623$$ −9.46410 −0.379171
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −23.8564 −0.953494
$$627$$ 0 0
$$628$$ 3.07180 0.122578
$$629$$ −5.46410 −0.217868
$$630$$ 0 0
$$631$$ 30.9808 1.23332 0.616662 0.787228i $$-0.288484\pi$$
0.616662 + 0.787228i $$0.288484\pi$$
$$632$$ 8.73205 0.347342
$$633$$ 0 0
$$634$$ 4.14359 0.164563
$$635$$ −3.66025 −0.145253
$$636$$ 0 0
$$637$$ −84.1051 −3.33237
$$638$$ −26.9282 −1.06610
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 6.53590 0.258152 0.129076 0.991635i $$-0.458799\pi$$
0.129076 + 0.991635i $$0.458799\pi$$
$$642$$ 0 0
$$643$$ 34.5359 1.36196 0.680981 0.732301i $$-0.261553\pi$$
0.680981 + 0.732301i $$0.261553\pi$$
$$644$$ −37.8564 −1.49175
$$645$$ 0 0
$$646$$ −33.8564 −1.33206
$$647$$ 35.7128 1.40402 0.702008 0.712169i $$-0.252287\pi$$
0.702008 + 0.712169i $$0.252287\pi$$
$$648$$ 0 0
$$649$$ 55.7128 2.18692
$$650$$ −5.46410 −0.214320
$$651$$ 0 0
$$652$$ −11.3205 −0.443345
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −18.5885 −0.726311
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ −22.3923 −0.872943
$$659$$ 6.92820 0.269884 0.134942 0.990853i $$-0.456915\pi$$
0.134942 + 0.990853i $$0.456915\pi$$
$$660$$ 0 0
$$661$$ −31.0718 −1.20855 −0.604276 0.796775i $$-0.706538\pi$$
−0.604276 + 0.796775i $$0.706538\pi$$
$$662$$ −33.1244 −1.28741
$$663$$ 0 0
$$664$$ 8.73205 0.338869
$$665$$ 29.3205 1.13700
$$666$$ 0 0
$$667$$ −39.4256 −1.52657
$$668$$ 1.46410 0.0566478
$$669$$ 0 0
$$670$$ 3.66025 0.141408
$$671$$ −26.9282 −1.03955
$$672$$ 0 0
$$673$$ −32.9282 −1.26929 −0.634644 0.772804i $$-0.718853\pi$$
−0.634644 + 0.772804i $$0.718853\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ 16.8564 0.648323
$$677$$ 4.14359 0.159251 0.0796256 0.996825i $$-0.474628\pi$$
0.0796256 + 0.996825i $$0.474628\pi$$
$$678$$ 0 0
$$679$$ 9.46410 0.363199
$$680$$ 5.46410 0.209539
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ 4.78461 0.183078 0.0915390 0.995801i $$-0.470821\pi$$
0.0915390 + 0.995801i $$0.470821\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ −39.7128 −1.51624
$$687$$ 0 0
$$688$$ 6.92820 0.264135
$$689$$ −32.7846 −1.24899
$$690$$ 0 0
$$691$$ −0.392305 −0.0149240 −0.00746199 0.999972i $$-0.502375\pi$$
−0.00746199 + 0.999972i $$0.502375\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 0 0
$$694$$ −30.9282 −1.17402
$$695$$ −6.92820 −0.262802
$$696$$ 0 0
$$697$$ −10.9282 −0.413935
$$698$$ −15.3205 −0.579890
$$699$$ 0 0
$$700$$ −4.73205 −0.178855
$$701$$ 11.8564 0.447810 0.223905 0.974611i $$-0.428119\pi$$
0.223905 + 0.974611i $$0.428119\pi$$
$$702$$ 0 0
$$703$$ 6.19615 0.233692
$$704$$ 5.46410 0.205936
$$705$$ 0 0
$$706$$ 11.8564 0.446222
$$707$$ −44.7846 −1.68430
$$708$$ 0 0
$$709$$ 43.8564 1.64706 0.823531 0.567271i $$-0.192001\pi$$
0.823531 + 0.567271i $$0.192001\pi$$
$$710$$ 2.92820 0.109894
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ 5.85641 0.219324
$$714$$ 0 0
$$715$$ 29.8564 1.11657
$$716$$ 0.339746 0.0126969
$$717$$ 0 0
$$718$$ 12.3923 0.462477
$$719$$ 12.3923 0.462155 0.231077 0.972935i $$-0.425775\pi$$
0.231077 + 0.972935i $$0.425775\pi$$
$$720$$ 0 0
$$721$$ 30.9282 1.15183
$$722$$ 19.3923 0.721707
$$723$$ 0 0
$$724$$ 5.46410 0.203072
$$725$$ −4.92820 −0.183029
$$726$$ 0 0
$$727$$ 8.78461 0.325803 0.162902 0.986642i $$-0.447915\pi$$
0.162902 + 0.986642i $$0.447915\pi$$
$$728$$ 25.8564 0.958302
$$729$$ 0 0
$$730$$ 0.928203 0.0343543
$$731$$ −37.8564 −1.40017
$$732$$ 0 0
$$733$$ 27.8564 1.02890 0.514450 0.857520i $$-0.327996\pi$$
0.514450 + 0.857520i $$0.327996\pi$$
$$734$$ −4.33975 −0.160183
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −20.0000 −0.736709
$$738$$ 0 0
$$739$$ 6.92820 0.254858 0.127429 0.991848i $$-0.459327\pi$$
0.127429 + 0.991848i $$0.459327\pi$$
$$740$$ −1.00000 −0.0367607
$$741$$ 0 0
$$742$$ −28.3923 −1.04231
$$743$$ −17.1244 −0.628232 −0.314116 0.949385i $$-0.601708\pi$$
−0.314116 + 0.949385i $$0.601708\pi$$
$$744$$ 0 0
$$745$$ 4.39230 0.160922
$$746$$ −10.7846 −0.394853
$$747$$ 0 0
$$748$$ −29.8564 −1.09166
$$749$$ 15.4641 0.565046
$$750$$ 0 0
$$751$$ −20.3923 −0.744126 −0.372063 0.928208i $$-0.621349\pi$$
−0.372063 + 0.928208i $$0.621349\pi$$
$$752$$ 4.73205 0.172560
$$753$$ 0 0
$$754$$ 26.9282 0.980667
$$755$$ 12.3923 0.451002
$$756$$ 0 0
$$757$$ 1.71281 0.0622532 0.0311266 0.999515i $$-0.490090\pi$$
0.0311266 + 0.999515i $$0.490090\pi$$
$$758$$ 32.3923 1.17654
$$759$$ 0 0
$$760$$ −6.19615 −0.224758
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ −9.46410 −0.342623
$$764$$ 8.73205 0.315915
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ −55.7128 −2.01167
$$768$$ 0 0
$$769$$ 7.07180 0.255016 0.127508 0.991838i $$-0.459302\pi$$
0.127508 + 0.991838i $$0.459302\pi$$
$$770$$ 25.8564 0.931800
$$771$$ 0 0
$$772$$ 15.8564 0.570685
$$773$$ −2.78461 −0.100155 −0.0500777 0.998745i $$-0.515947\pi$$
−0.0500777 + 0.998745i $$0.515947\pi$$
$$774$$ 0 0
$$775$$ 0.732051 0.0262960
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 11.8564 0.425073
$$779$$ 12.3923 0.444000
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ −43.7128 −1.56317
$$783$$ 0 0
$$784$$ 15.3923 0.549725
$$785$$ −3.07180 −0.109637
$$786$$ 0 0
$$787$$ 22.1962 0.791207 0.395604 0.918421i $$-0.370535\pi$$
0.395604 + 0.918421i $$0.370535\pi$$
$$788$$ 22.7846 0.811668
$$789$$ 0 0
$$790$$ −8.73205 −0.310672
$$791$$ −49.8564 −1.77269
$$792$$ 0 0
$$793$$ 26.9282 0.956249
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ −12.0526 −0.427192
$$797$$ −10.5359 −0.373201 −0.186600 0.982436i $$-0.559747\pi$$
−0.186600 + 0.982436i $$0.559747\pi$$
$$798$$ 0 0
$$799$$ −25.8564 −0.914734
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 32.9282 1.16274
$$803$$ −5.07180 −0.178980
$$804$$ 0 0
$$805$$ 37.8564 1.33426
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 9.46410 0.332946
$$809$$ −43.5692 −1.53181 −0.765906 0.642952i $$-0.777709\pi$$
−0.765906 + 0.642952i $$0.777709\pi$$
$$810$$ 0 0
$$811$$ −41.8564 −1.46978 −0.734889 0.678188i $$-0.762766\pi$$
−0.734889 + 0.678188i $$0.762766\pi$$
$$812$$ 23.3205 0.818389
$$813$$ 0 0
$$814$$ 5.46410 0.191517
$$815$$ 11.3205 0.396540
$$816$$ 0 0
$$817$$ 42.9282 1.50187
$$818$$ 3.07180 0.107403
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ 48.4449 1.68868 0.844341 0.535806i $$-0.179992\pi$$
0.844341 + 0.535806i $$0.179992\pi$$
$$824$$ −6.53590 −0.227689
$$825$$ 0 0
$$826$$ −48.2487 −1.67879
$$827$$ 16.3923 0.570016 0.285008 0.958525i $$-0.408004\pi$$
0.285008 + 0.958525i $$0.408004\pi$$
$$828$$ 0 0
$$829$$ −51.5692 −1.79107 −0.895537 0.444988i $$-0.853208\pi$$
−0.895537 + 0.444988i $$0.853208\pi$$
$$830$$ −8.73205 −0.303094
$$831$$ 0 0
$$832$$ −5.46410 −0.189434
$$833$$ −84.1051 −2.91407
$$834$$ 0 0
$$835$$ −1.46410 −0.0506673
$$836$$ 33.8564 1.17095
$$837$$ 0 0
$$838$$ −14.2487 −0.492214
$$839$$ 8.78461 0.303278 0.151639 0.988436i $$-0.451545\pi$$
0.151639 + 0.988436i $$0.451545\pi$$
$$840$$ 0 0
$$841$$ −4.71281 −0.162511
$$842$$ −0.143594 −0.00494856
$$843$$ 0 0
$$844$$ −17.8564 −0.614643
$$845$$ −16.8564 −0.579878
$$846$$ 0 0
$$847$$ −89.2295 −3.06596
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ −5.46410 −0.187417
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ 23.3205 0.798011
$$855$$ 0 0
$$856$$ −3.26795 −0.111696
$$857$$ 31.8564 1.08819 0.544097 0.839022i $$-0.316872\pi$$
0.544097 + 0.839022i $$0.316872\pi$$
$$858$$ 0 0
$$859$$ 44.4449 1.51644 0.758220 0.651999i $$-0.226069\pi$$
0.758220 + 0.651999i $$0.226069\pi$$
$$860$$ −6.92820 −0.236250
$$861$$ 0 0
$$862$$ 2.19615 0.0748012
$$863$$ 20.4449 0.695951 0.347976 0.937504i $$-0.386869\pi$$
0.347976 + 0.937504i $$0.386869\pi$$
$$864$$ 0 0
$$865$$ 10.0000 0.340010
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ −3.46410 −0.117579
$$869$$ 47.7128 1.61855
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ 49.5692 1.67670
$$875$$ 4.73205 0.159973
$$876$$ 0 0
$$877$$ −9.21539 −0.311182 −0.155591 0.987822i $$-0.549728\pi$$
−0.155591 + 0.987822i $$0.549728\pi$$
$$878$$ 13.5167 0.456165
$$879$$ 0 0
$$880$$ −5.46410 −0.184195
$$881$$ −12.6795 −0.427183 −0.213591 0.976923i $$-0.568516\pi$$
−0.213591 + 0.976923i $$0.568516\pi$$
$$882$$ 0 0
$$883$$ 14.2487 0.479507 0.239754 0.970834i $$-0.422933\pi$$
0.239754 + 0.970834i $$0.422933\pi$$
$$884$$ 29.8564 1.00418
$$885$$ 0 0
$$886$$ 14.8756 0.499757
$$887$$ −9.80385 −0.329181 −0.164590 0.986362i $$-0.552630\pi$$
−0.164590 + 0.986362i $$0.552630\pi$$
$$888$$ 0 0
$$889$$ −17.3205 −0.580911
$$890$$ −2.00000 −0.0670402
$$891$$ 0 0
$$892$$ 16.0526 0.537479
$$893$$ 29.3205 0.981173
$$894$$ 0 0
$$895$$ −0.339746 −0.0113565
$$896$$ −4.73205 −0.158087
$$897$$ 0 0
$$898$$ 21.7128 0.724566
$$899$$ −3.60770 −0.120323
$$900$$ 0 0
$$901$$ −32.7846 −1.09221
$$902$$ 10.9282 0.363869
$$903$$ 0 0
$$904$$ 10.5359 0.350419
$$905$$ −5.46410 −0.181633
$$906$$ 0 0
$$907$$ 54.2487 1.80130 0.900649 0.434546i $$-0.143091\pi$$
0.900649 + 0.434546i $$0.143091\pi$$
$$908$$ −24.3923 −0.809487
$$909$$ 0 0
$$910$$ −25.8564 −0.857132
$$911$$ −37.9090 −1.25598 −0.627990 0.778221i $$-0.716122\pi$$
−0.627990 + 0.778221i $$0.716122\pi$$
$$912$$ 0 0
$$913$$ 47.7128 1.57906
$$914$$ −31.8564 −1.05372
$$915$$ 0 0
$$916$$ −11.8564 −0.391747
$$917$$ −87.9615 −2.90475
$$918$$ 0 0
$$919$$ −16.7321 −0.551939 −0.275970 0.961166i $$-0.588999\pi$$
−0.275970 + 0.961166i $$0.588999\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 0 0
$$922$$ −14.7846 −0.486905
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 18.9282 0.622019
$$927$$ 0 0
$$928$$ −4.92820 −0.161776
$$929$$ −11.8564 −0.388996 −0.194498 0.980903i $$-0.562308\pi$$
−0.194498 + 0.980903i $$0.562308\pi$$
$$930$$ 0 0
$$931$$ 95.3731 3.12573
$$932$$ −28.9282 −0.947575
$$933$$ 0 0
$$934$$ 25.1769 0.823814
$$935$$ 29.8564 0.976409
$$936$$ 0 0
$$937$$ −23.8564 −0.779355 −0.389677 0.920951i $$-0.627413\pi$$
−0.389677 + 0.920951i $$0.627413\pi$$
$$938$$ 17.3205 0.565535
$$939$$ 0 0
$$940$$ −4.73205 −0.154342
$$941$$ −7.60770 −0.248004 −0.124002 0.992282i $$-0.539573\pi$$
−0.124002 + 0.992282i $$0.539573\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 10.1962 0.331856
$$945$$ 0 0
$$946$$ 37.8564 1.23082
$$947$$ −17.1769 −0.558175 −0.279087 0.960266i $$-0.590032\pi$$
−0.279087 + 0.960266i $$0.590032\pi$$
$$948$$ 0 0
$$949$$ 5.07180 0.164637
$$950$$ 6.19615 0.201030
$$951$$ 0 0
$$952$$ 25.8564 0.838011
$$953$$ −15.8564 −0.513639 −0.256820 0.966459i $$-0.582675\pi$$
−0.256820 + 0.966459i $$0.582675\pi$$
$$954$$ 0 0
$$955$$ −8.73205 −0.282563
$$956$$ 20.7321 0.670522
$$957$$ 0 0
$$958$$ −4.05256 −0.130932
$$959$$ 9.46410 0.305612
$$960$$ 0 0
$$961$$ −30.4641 −0.982713
$$962$$ −5.46410 −0.176170
$$963$$ 0 0
$$964$$ 4.92820 0.158727
$$965$$ −15.8564 −0.510436
$$966$$ 0 0
$$967$$ −44.3923 −1.42756 −0.713780 0.700370i $$-0.753018\pi$$
−0.713780 + 0.700370i $$0.753018\pi$$
$$968$$ 18.8564 0.606068
$$969$$ 0 0
$$970$$ 2.00000 0.0642161
$$971$$ −1.07180 −0.0343956 −0.0171978 0.999852i $$-0.505474\pi$$
−0.0171978 + 0.999852i $$0.505474\pi$$
$$972$$ 0 0
$$973$$ −32.7846 −1.05103
$$974$$ 4.39230 0.140739
$$975$$ 0 0
$$976$$ −4.92820 −0.157748
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ 0 0
$$979$$ 10.9282 0.349267
$$980$$ −15.3923 −0.491689
$$981$$ 0 0
$$982$$ 14.9282 0.476378
$$983$$ −20.4449 −0.652090 −0.326045 0.945354i $$-0.605716\pi$$
−0.326045 + 0.945354i $$0.605716\pi$$
$$984$$ 0 0
$$985$$ −22.7846 −0.725978
$$986$$ 26.9282 0.857569
$$987$$ 0 0
$$988$$ −33.8564 −1.07712
$$989$$ 55.4256 1.76243
$$990$$ 0 0
$$991$$ 44.0526 1.39938 0.699688 0.714449i $$-0.253322\pi$$
0.699688 + 0.714449i $$0.253322\pi$$
$$992$$ 0.732051 0.0232426
$$993$$ 0 0
$$994$$ 13.8564 0.439499
$$995$$ 12.0526 0.382092
$$996$$ 0 0
$$997$$ −31.8564 −1.00890 −0.504451 0.863440i $$-0.668305\pi$$
−0.504451 + 0.863440i $$0.668305\pi$$
$$998$$ 9.41154 0.297917
$$999$$ 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 3330.2.a.bd.1.1 2
3.2 odd 2 370.2.a.e.1.2 2
12.11 even 2 2960.2.a.q.1.1 2
15.2 even 4 1850.2.b.l.149.1 4
15.8 even 4 1850.2.b.l.149.4 4
15.14 odd 2 1850.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.2 2 3.2 odd 2
1850.2.a.x.1.1 2 15.14 odd 2
1850.2.b.l.149.1 4 15.2 even 4
1850.2.b.l.149.4 4 15.8 even 4
2960.2.a.q.1.1 2 12.11 even 2
3330.2.a.bd.1.1 2 1.1 even 1 trivial