# Properties

 Label 1521.4.a.r.1.2 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +O(q^{10})$$ $$q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +1.43845 q^{10} +64.7386 q^{11} -46.5616 q^{14} -50.4233 q^{16} +25.5464 q^{17} +107.970 q^{19} -0.807764 q^{20} +165.831 q^{22} -73.2614 q^{23} -124.685 q^{25} +26.1468 q^{28} -175.909 q^{29} +113.093 q^{31} +64.2547 q^{32} +65.4384 q^{34} -10.2074 q^{35} -114.808 q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} -187.663 q^{46} -31.9479 q^{47} -12.5937 q^{49} -319.386 q^{50} -2.84658 q^{53} +36.3542 q^{55} +439.469 q^{56} -450.600 q^{58} +71.6325 q^{59} -920.695 q^{61} +289.693 q^{62} +567.978 q^{64} +444.280 q^{67} -36.7471 q^{68} -26.1468 q^{70} -541.719 q^{71} -764.004 q^{73} -294.086 q^{74} -155.309 q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153 q^{80} -178.401 q^{82} +603.797 q^{83} +14.3457 q^{85} +1122.73 q^{86} -1565.19 q^{88} -1159.88 q^{89} +105.383 q^{92} -81.8362 q^{94} +60.6307 q^{95} -583.269 q^{97} -32.2595 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.56155 0.905646 0.452823 0.891601i $$-0.350417\pi$$
0.452823 + 0.891601i $$0.350417\pi$$
$$3$$ 0 0
$$4$$ −1.43845 −0.179806
$$5$$ 0.561553 0.0502268 0.0251134 0.999685i $$-0.492005\pi$$
0.0251134 + 0.999685i $$0.492005\pi$$
$$6$$ 0 0
$$7$$ −18.1771 −0.981470 −0.490735 0.871309i $$-0.663272\pi$$
−0.490735 + 0.871309i $$0.663272\pi$$
$$8$$ −24.1771 −1.06849
$$9$$ 0 0
$$10$$ 1.43845 0.0454877
$$11$$ 64.7386 1.77449 0.887247 0.461295i $$-0.152615\pi$$
0.887247 + 0.461295i $$0.152615\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −46.5616 −0.888864
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ 25.5464 0.364465 0.182233 0.983255i $$-0.441668\pi$$
0.182233 + 0.983255i $$0.441668\pi$$
$$18$$ 0 0
$$19$$ 107.970 1.30368 0.651841 0.758356i $$-0.273997\pi$$
0.651841 + 0.758356i $$0.273997\pi$$
$$20$$ −0.807764 −0.00903108
$$21$$ 0 0
$$22$$ 165.831 1.60706
$$23$$ −73.2614 −0.664176 −0.332088 0.943248i $$-0.607753\pi$$
−0.332088 + 0.943248i $$0.607753\pi$$
$$24$$ 0 0
$$25$$ −124.685 −0.997477
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 26.1468 0.176474
$$29$$ −175.909 −1.12640 −0.563198 0.826322i $$-0.690429\pi$$
−0.563198 + 0.826322i $$0.690429\pi$$
$$30$$ 0 0
$$31$$ 113.093 0.655228 0.327614 0.944812i $$-0.393755\pi$$
0.327614 + 0.944812i $$0.393755\pi$$
$$32$$ 64.2547 0.354961
$$33$$ 0 0
$$34$$ 65.4384 0.330077
$$35$$ −10.2074 −0.0492961
$$36$$ 0 0
$$37$$ −114.808 −0.510116 −0.255058 0.966926i $$-0.582095\pi$$
−0.255058 + 0.966926i $$0.582095\pi$$
$$38$$ 276.570 1.18067
$$39$$ 0 0
$$40$$ −13.5767 −0.0536666
$$41$$ −69.6458 −0.265289 −0.132645 0.991164i $$-0.542347\pi$$
−0.132645 + 0.991164i $$0.542347\pi$$
$$42$$ 0 0
$$43$$ 438.302 1.55443 0.777214 0.629236i $$-0.216632\pi$$
0.777214 + 0.629236i $$0.216632\pi$$
$$44$$ −93.1231 −0.319064
$$45$$ 0 0
$$46$$ −187.663 −0.601508
$$47$$ −31.9479 −0.0991506 −0.0495753 0.998770i $$-0.515787\pi$$
−0.0495753 + 0.998770i $$0.515787\pi$$
$$48$$ 0 0
$$49$$ −12.5937 −0.0367164
$$50$$ −319.386 −0.903361
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.84658 −0.00737752 −0.00368876 0.999993i $$-0.501174\pi$$
−0.00368876 + 0.999993i $$0.501174\pi$$
$$54$$ 0 0
$$55$$ 36.3542 0.0891272
$$56$$ 439.469 1.04869
$$57$$ 0 0
$$58$$ −450.600 −1.02012
$$59$$ 71.6325 0.158064 0.0790319 0.996872i $$-0.474817\pi$$
0.0790319 + 0.996872i $$0.474817\pi$$
$$60$$ 0 0
$$61$$ −920.695 −1.93251 −0.966253 0.257593i $$-0.917071\pi$$
−0.966253 + 0.257593i $$0.917071\pi$$
$$62$$ 289.693 0.593404
$$63$$ 0 0
$$64$$ 567.978 1.10933
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 444.280 0.810112 0.405056 0.914292i $$-0.367252\pi$$
0.405056 + 0.914292i $$0.367252\pi$$
$$68$$ −36.7471 −0.0655330
$$69$$ 0 0
$$70$$ −26.1468 −0.0446448
$$71$$ −541.719 −0.905496 −0.452748 0.891639i $$-0.649556\pi$$
−0.452748 + 0.891639i $$0.649556\pi$$
$$72$$ 0 0
$$73$$ −764.004 −1.22493 −0.612465 0.790498i $$-0.709822\pi$$
−0.612465 + 0.790498i $$0.709822\pi$$
$$74$$ −294.086 −0.461984
$$75$$ 0 0
$$76$$ −155.309 −0.234410
$$77$$ −1176.76 −1.74161
$$78$$ 0 0
$$79$$ −421.538 −0.600338 −0.300169 0.953886i $$-0.597043\pi$$
−0.300169 + 0.953886i $$0.597043\pi$$
$$80$$ −28.3153 −0.0395719
$$81$$ 0 0
$$82$$ −178.401 −0.240258
$$83$$ 603.797 0.798498 0.399249 0.916842i $$-0.369271\pi$$
0.399249 + 0.916842i $$0.369271\pi$$
$$84$$ 0 0
$$85$$ 14.3457 0.0183059
$$86$$ 1122.73 1.40776
$$87$$ 0 0
$$88$$ −1565.19 −1.89602
$$89$$ −1159.88 −1.38143 −0.690715 0.723127i $$-0.742704\pi$$
−0.690715 + 0.723127i $$0.742704\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 105.383 0.119423
$$93$$ 0 0
$$94$$ −81.8362 −0.0897953
$$95$$ 60.6307 0.0654798
$$96$$ 0 0
$$97$$ −583.269 −0.610536 −0.305268 0.952267i $$-0.598746\pi$$
−0.305268 + 0.952267i $$0.598746\pi$$
$$98$$ −32.2595 −0.0332521
$$99$$ 0 0
$$100$$ 179.352 0.179352
$$101$$ −921.740 −0.908085 −0.454043 0.890980i $$-0.650019\pi$$
−0.454043 + 0.890980i $$0.650019\pi$$
$$102$$ 0 0
$$103$$ −930.712 −0.890347 −0.445174 0.895444i $$-0.646858\pi$$
−0.445174 + 0.895444i $$0.646858\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −7.29168 −0.00668142
$$107$$ −857.383 −0.774638 −0.387319 0.921946i $$-0.626599\pi$$
−0.387319 + 0.921946i $$0.626599\pi$$
$$108$$ 0 0
$$109$$ −671.853 −0.590384 −0.295192 0.955438i $$-0.595384\pi$$
−0.295192 + 0.955438i $$0.595384\pi$$
$$110$$ 93.1231 0.0807176
$$111$$ 0 0
$$112$$ 916.548 0.773265
$$113$$ −641.474 −0.534024 −0.267012 0.963693i $$-0.586036\pi$$
−0.267012 + 0.963693i $$0.586036\pi$$
$$114$$ 0 0
$$115$$ −41.1401 −0.0333594
$$116$$ 253.036 0.202533
$$117$$ 0 0
$$118$$ 183.491 0.143150
$$119$$ −464.359 −0.357712
$$120$$ 0 0
$$121$$ 2860.09 2.14883
$$122$$ −2358.41 −1.75017
$$123$$ 0 0
$$124$$ −162.678 −0.117814
$$125$$ −140.211 −0.100327
$$126$$ 0 0
$$127$$ −553.174 −0.386506 −0.193253 0.981149i $$-0.561904\pi$$
−0.193253 + 0.981149i $$0.561904\pi$$
$$128$$ 940.868 0.649702
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2056.40 −1.37152 −0.685758 0.727830i $$-0.740529\pi$$
−0.685758 + 0.727830i $$0.740529\pi$$
$$132$$ 0 0
$$133$$ −1962.57 −1.27952
$$134$$ 1138.05 0.733674
$$135$$ 0 0
$$136$$ −617.637 −0.389426
$$137$$ −1808.57 −1.12786 −0.563928 0.825824i $$-0.690710\pi$$
−0.563928 + 0.825824i $$0.690710\pi$$
$$138$$ 0 0
$$139$$ 1493.64 0.911428 0.455714 0.890126i $$-0.349384\pi$$
0.455714 + 0.890126i $$0.349384\pi$$
$$140$$ 14.6828 0.00886373
$$141$$ 0 0
$$142$$ −1387.64 −0.820058
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −98.7822 −0.0565753
$$146$$ −1957.04 −1.10935
$$147$$ 0 0
$$148$$ 165.145 0.0917218
$$149$$ −2759.02 −1.51696 −0.758482 0.651694i $$-0.774059\pi$$
−0.758482 + 0.651694i $$0.774059\pi$$
$$150$$ 0 0
$$151$$ 976.355 0.526190 0.263095 0.964770i $$-0.415257\pi$$
0.263095 + 0.964770i $$0.415257\pi$$
$$152$$ −2610.39 −1.39297
$$153$$ 0 0
$$154$$ −3014.33 −1.57728
$$155$$ 63.5076 0.0329100
$$156$$ 0 0
$$157$$ −564.875 −0.287146 −0.143573 0.989640i $$-0.545859\pi$$
−0.143573 + 0.989640i $$0.545859\pi$$
$$158$$ −1079.79 −0.543694
$$159$$ 0 0
$$160$$ 36.0824 0.0178285
$$161$$ 1331.68 0.651869
$$162$$ 0 0
$$163$$ −1508.53 −0.724892 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$164$$ 100.182 0.0477005
$$165$$ 0 0
$$166$$ 1546.66 0.723157
$$167$$ 592.521 0.274555 0.137277 0.990533i $$-0.456165\pi$$
0.137277 + 0.990533i $$0.456165\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 36.7471 0.0165787
$$171$$ 0 0
$$172$$ −630.474 −0.279495
$$173$$ 4495.57 1.97568 0.987838 0.155488i $$-0.0496952\pi$$
0.987838 + 0.155488i $$0.0496952\pi$$
$$174$$ 0 0
$$175$$ 2266.40 0.978994
$$176$$ −3264.34 −1.39806
$$177$$ 0 0
$$178$$ −2971.10 −1.25109
$$179$$ 154.285 0.0644235 0.0322117 0.999481i $$-0.489745\pi$$
0.0322117 + 0.999481i $$0.489745\pi$$
$$180$$ 0 0
$$181$$ 1071.35 0.439959 0.219979 0.975505i $$-0.429401\pi$$
0.219979 + 0.975505i $$0.429401\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1771.25 0.709663
$$185$$ −64.4706 −0.0256215
$$186$$ 0 0
$$187$$ 1653.84 0.646742
$$188$$ 45.9554 0.0178279
$$189$$ 0 0
$$190$$ 155.309 0.0593015
$$191$$ −677.203 −0.256548 −0.128274 0.991739i $$-0.540944\pi$$
−0.128274 + 0.991739i $$0.540944\pi$$
$$192$$ 0 0
$$193$$ −1321.68 −0.492936 −0.246468 0.969151i $$-0.579270\pi$$
−0.246468 + 0.969151i $$0.579270\pi$$
$$194$$ −1494.07 −0.552929
$$195$$ 0 0
$$196$$ 18.1154 0.00660183
$$197$$ 1267.37 0.458356 0.229178 0.973385i $$-0.426396\pi$$
0.229178 + 0.973385i $$0.426396\pi$$
$$198$$ 0 0
$$199$$ 2396.24 0.853593 0.426796 0.904348i $$-0.359642\pi$$
0.426796 + 0.904348i $$0.359642\pi$$
$$200$$ 3014.51 1.06579
$$201$$ 0 0
$$202$$ −2361.09 −0.822403
$$203$$ 3197.51 1.10552
$$204$$ 0 0
$$205$$ −39.1098 −0.0133246
$$206$$ −2384.07 −0.806339
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 6989.81 2.31337
$$210$$ 0 0
$$211$$ −91.5539 −0.0298712 −0.0149356 0.999888i $$-0.504754\pi$$
−0.0149356 + 0.999888i $$0.504754\pi$$
$$212$$ 4.09466 0.00132652
$$213$$ 0 0
$$214$$ −2196.23 −0.701548
$$215$$ 246.130 0.0780740
$$216$$ 0 0
$$217$$ −2055.70 −0.643087
$$218$$ −1720.99 −0.534679
$$219$$ 0 0
$$220$$ −52.2935 −0.0160256
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1235.42 −0.370985 −0.185493 0.982646i $$-0.559388\pi$$
−0.185493 + 0.982646i $$0.559388\pi$$
$$224$$ −1167.96 −0.348383
$$225$$ 0 0
$$226$$ −1643.17 −0.483637
$$227$$ 3301.66 0.965370 0.482685 0.875794i $$-0.339662\pi$$
0.482685 + 0.875794i $$0.339662\pi$$
$$228$$ 0 0
$$229$$ −211.283 −0.0609694 −0.0304847 0.999535i $$-0.509705\pi$$
−0.0304847 + 0.999535i $$0.509705\pi$$
$$230$$ −105.383 −0.0302118
$$231$$ 0 0
$$232$$ 4252.97 1.20354
$$233$$ 256.724 0.0721827 0.0360913 0.999348i $$-0.488509\pi$$
0.0360913 + 0.999348i $$0.488509\pi$$
$$234$$ 0 0
$$235$$ −17.9404 −0.00498002
$$236$$ −103.040 −0.0284208
$$237$$ 0 0
$$238$$ −1189.48 −0.323960
$$239$$ −3549.62 −0.960694 −0.480347 0.877078i $$-0.659489\pi$$
−0.480347 + 0.877078i $$0.659489\pi$$
$$240$$ 0 0
$$241$$ 5030.10 1.34447 0.672235 0.740338i $$-0.265335\pi$$
0.672235 + 0.740338i $$0.265335\pi$$
$$242$$ 7326.27 1.94608
$$243$$ 0 0
$$244$$ 1324.37 0.347476
$$245$$ −7.07204 −0.00184415
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2734.25 −0.700102
$$249$$ 0 0
$$250$$ −359.158 −0.0908606
$$251$$ 718.784 0.180754 0.0903770 0.995908i $$-0.471193\pi$$
0.0903770 + 0.995908i $$0.471193\pi$$
$$252$$ 0 0
$$253$$ −4742.84 −1.17858
$$254$$ −1416.98 −0.350038
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ −1280.79 −0.310871 −0.155435 0.987846i $$-0.549678\pi$$
−0.155435 + 0.987846i $$0.549678\pi$$
$$258$$ 0 0
$$259$$ 2086.87 0.500663
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −5267.58 −1.24211
$$263$$ −5225.55 −1.22517 −0.612587 0.790403i $$-0.709871\pi$$
−0.612587 + 0.790403i $$0.709871\pi$$
$$264$$ 0 0
$$265$$ −1.59851 −0.000370549 0
$$266$$ −5027.24 −1.15880
$$267$$ 0 0
$$268$$ −639.074 −0.145663
$$269$$ −6443.80 −1.46054 −0.730270 0.683158i $$-0.760606\pi$$
−0.730270 + 0.683158i $$0.760606\pi$$
$$270$$ 0 0
$$271$$ 3929.93 0.880909 0.440455 0.897775i $$-0.354817\pi$$
0.440455 + 0.897775i $$0.354817\pi$$
$$272$$ −1288.13 −0.287149
$$273$$ 0 0
$$274$$ −4632.74 −1.02144
$$275$$ −8071.91 −1.77002
$$276$$ 0 0
$$277$$ −5884.40 −1.27639 −0.638194 0.769876i $$-0.720318\pi$$
−0.638194 + 0.769876i $$0.720318\pi$$
$$278$$ 3826.03 0.825431
$$279$$ 0 0
$$280$$ 246.785 0.0526722
$$281$$ 3529.79 0.749358 0.374679 0.927155i $$-0.377753\pi$$
0.374679 + 0.927155i $$0.377753\pi$$
$$282$$ 0 0
$$283$$ −2611.00 −0.548438 −0.274219 0.961667i $$-0.588419\pi$$
−0.274219 + 0.961667i $$0.588419\pi$$
$$284$$ 779.234 0.162813
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1265.96 0.260373
$$288$$ 0 0
$$289$$ −4260.38 −0.867165
$$290$$ −253.036 −0.0512372
$$291$$ 0 0
$$292$$ 1098.98 0.220250
$$293$$ −5491.03 −1.09484 −0.547422 0.836857i $$-0.684391\pi$$
−0.547422 + 0.836857i $$0.684391\pi$$
$$294$$ 0 0
$$295$$ 40.2255 0.00793904
$$296$$ 2775.72 0.545052
$$297$$ 0 0
$$298$$ −7067.37 −1.37383
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −7967.05 −1.52563
$$302$$ 2500.99 0.476542
$$303$$ 0 0
$$304$$ −5444.19 −1.02712
$$305$$ −517.019 −0.0970637
$$306$$ 0 0
$$307$$ 7307.59 1.35852 0.679261 0.733897i $$-0.262300\pi$$
0.679261 + 0.733897i $$0.262300\pi$$
$$308$$ 1692.71 0.313152
$$309$$ 0 0
$$310$$ 162.678 0.0298048
$$311$$ −7904.92 −1.44131 −0.720654 0.693295i $$-0.756158\pi$$
−0.720654 + 0.693295i $$0.756158\pi$$
$$312$$ 0 0
$$313$$ 10002.4 1.80629 0.903145 0.429336i $$-0.141252\pi$$
0.903145 + 0.429336i $$0.141252\pi$$
$$314$$ −1446.96 −0.260053
$$315$$ 0 0
$$316$$ 606.360 0.107944
$$317$$ −6230.81 −1.10397 −0.551983 0.833856i $$-0.686129\pi$$
−0.551983 + 0.833856i $$0.686129\pi$$
$$318$$ 0 0
$$319$$ −11388.1 −1.99878
$$320$$ 318.950 0.0557182
$$321$$ 0 0
$$322$$ 3411.16 0.590362
$$323$$ 2758.24 0.475147
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −3864.19 −0.656495
$$327$$ 0 0
$$328$$ 1683.83 0.283458
$$329$$ 580.719 0.0973134
$$330$$ 0 0
$$331$$ 4634.51 0.769594 0.384797 0.923001i $$-0.374271\pi$$
0.384797 + 0.923001i $$0.374271\pi$$
$$332$$ −868.531 −0.143575
$$333$$ 0 0
$$334$$ 1517.77 0.248649
$$335$$ 249.487 0.0406893
$$336$$ 0 0
$$337$$ 3029.82 0.489747 0.244874 0.969555i $$-0.421254\pi$$
0.244874 + 0.969555i $$0.421254\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −20.6355 −0.00329151
$$341$$ 7321.47 1.16270
$$342$$ 0 0
$$343$$ 6463.66 1.01751
$$344$$ −10596.9 −1.66089
$$345$$ 0 0
$$346$$ 11515.6 1.78926
$$347$$ −2841.60 −0.439611 −0.219805 0.975544i $$-0.570542\pi$$
−0.219805 + 0.975544i $$0.570542\pi$$
$$348$$ 0 0
$$349$$ −7565.68 −1.16040 −0.580202 0.814472i $$-0.697027\pi$$
−0.580202 + 0.814472i $$0.697027\pi$$
$$350$$ 5805.51 0.886622
$$351$$ 0 0
$$352$$ 4159.76 0.629875
$$353$$ −2339.44 −0.352736 −0.176368 0.984324i $$-0.556435\pi$$
−0.176368 + 0.984324i $$0.556435\pi$$
$$354$$ 0 0
$$355$$ −304.204 −0.0454802
$$356$$ 1668.43 0.248389
$$357$$ 0 0
$$358$$ 395.209 0.0583449
$$359$$ −2531.68 −0.372192 −0.186096 0.982532i $$-0.559583\pi$$
−0.186096 + 0.982532i $$0.559583\pi$$
$$360$$ 0 0
$$361$$ 4798.45 0.699585
$$362$$ 2744.31 0.398447
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −429.028 −0.0615243
$$366$$ 0 0
$$367$$ 6577.81 0.935583 0.467792 0.883839i $$-0.345050\pi$$
0.467792 + 0.883839i $$0.345050\pi$$
$$368$$ 3694.08 0.523280
$$369$$ 0 0
$$370$$ −165.145 −0.0232040
$$371$$ 51.7426 0.00724081
$$372$$ 0 0
$$373$$ 2902.72 0.402942 0.201471 0.979495i $$-0.435428\pi$$
0.201471 + 0.979495i $$0.435428\pi$$
$$374$$ 4236.40 0.585719
$$375$$ 0 0
$$376$$ 772.407 0.105941
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1865.73 0.252866 0.126433 0.991975i $$-0.459647\pi$$
0.126433 + 0.991975i $$0.459647\pi$$
$$380$$ −87.2140 −0.0117736
$$381$$ 0 0
$$382$$ −1734.69 −0.232342
$$383$$ 10836.0 1.44567 0.722837 0.691019i $$-0.242838\pi$$
0.722837 + 0.691019i $$0.242838\pi$$
$$384$$ 0 0
$$385$$ −660.813 −0.0874757
$$386$$ −3385.55 −0.446425
$$387$$ 0 0
$$388$$ 839.001 0.109778
$$389$$ 9520.34 1.24088 0.620438 0.784256i $$-0.286955\pi$$
0.620438 + 0.784256i $$0.286955\pi$$
$$390$$ 0 0
$$391$$ −1871.56 −0.242069
$$392$$ 304.480 0.0392310
$$393$$ 0 0
$$394$$ 3246.43 0.415108
$$395$$ −236.716 −0.0301531
$$396$$ 0 0
$$397$$ 10108.8 1.27796 0.638978 0.769225i $$-0.279358\pi$$
0.638978 + 0.769225i $$0.279358\pi$$
$$398$$ 6138.10 0.773053
$$399$$ 0 0
$$400$$ 6287.01 0.785876
$$401$$ 2084.38 0.259573 0.129787 0.991542i $$-0.458571\pi$$
0.129787 + 0.991542i $$0.458571\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 1325.88 0.163279
$$405$$ 0 0
$$406$$ 8190.60 1.00121
$$407$$ −7432.50 −0.905197
$$408$$ 0 0
$$409$$ 9716.53 1.17470 0.587349 0.809334i $$-0.300172\pi$$
0.587349 + 0.809334i $$0.300172\pi$$
$$410$$ −100.182 −0.0120674
$$411$$ 0 0
$$412$$ 1338.78 0.160090
$$413$$ −1302.07 −0.155135
$$414$$ 0 0
$$415$$ 339.064 0.0401060
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 17904.8 2.09510
$$419$$ −13381.9 −1.56026 −0.780129 0.625619i $$-0.784847\pi$$
−0.780129 + 0.625619i $$0.784847\pi$$
$$420$$ 0 0
$$421$$ 9463.37 1.09553 0.547763 0.836633i $$-0.315479\pi$$
0.547763 + 0.836633i $$0.315479\pi$$
$$422$$ −234.520 −0.0270527
$$423$$ 0 0
$$424$$ 68.8221 0.00788278
$$425$$ −3185.24 −0.363546
$$426$$ 0 0
$$427$$ 16735.5 1.89670
$$428$$ 1233.30 0.139285
$$429$$ 0 0
$$430$$ 630.474 0.0707074
$$431$$ 4852.28 0.542288 0.271144 0.962539i $$-0.412598\pi$$
0.271144 + 0.962539i $$0.412598\pi$$
$$432$$ 0 0
$$433$$ −8208.00 −0.910973 −0.455486 0.890243i $$-0.650535\pi$$
−0.455486 + 0.890243i $$0.650535\pi$$
$$434$$ −5265.78 −0.582409
$$435$$ 0 0
$$436$$ 966.425 0.106155
$$437$$ −7910.01 −0.865874
$$438$$ 0 0
$$439$$ −2993.80 −0.325481 −0.162741 0.986669i $$-0.552033\pi$$
−0.162741 + 0.986669i $$0.552033\pi$$
$$440$$ −878.938 −0.0952311
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9743.67 −1.04500 −0.522501 0.852639i $$-0.675001\pi$$
−0.522501 + 0.852639i $$0.675001\pi$$
$$444$$ 0 0
$$445$$ −651.335 −0.0693848
$$446$$ −3164.59 −0.335981
$$447$$ 0 0
$$448$$ −10324.2 −1.08878
$$449$$ −561.459 −0.0590131 −0.0295065 0.999565i $$-0.509394\pi$$
−0.0295065 + 0.999565i $$0.509394\pi$$
$$450$$ 0 0
$$451$$ −4508.78 −0.470754
$$452$$ 922.726 0.0960207
$$453$$ 0 0
$$454$$ 8457.38 0.874283
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13758.4 −1.40830 −0.704148 0.710054i $$-0.748671\pi$$
−0.704148 + 0.710054i $$0.748671\pi$$
$$458$$ −541.213 −0.0552166
$$459$$ 0 0
$$460$$ 59.1779 0.00599823
$$461$$ 12009.2 1.21329 0.606644 0.794974i $$-0.292515\pi$$
0.606644 + 0.794974i $$0.292515\pi$$
$$462$$ 0 0
$$463$$ −13635.7 −1.36870 −0.684348 0.729156i $$-0.739913\pi$$
−0.684348 + 0.729156i $$0.739913\pi$$
$$464$$ 8869.91 0.887447
$$465$$ 0 0
$$466$$ 657.613 0.0653719
$$467$$ −8821.95 −0.874157 −0.437079 0.899423i $$-0.643987\pi$$
−0.437079 + 0.899423i $$0.643987\pi$$
$$468$$ 0 0
$$469$$ −8075.72 −0.795100
$$470$$ −45.9554 −0.00451013
$$471$$ 0 0
$$472$$ −1731.87 −0.168889
$$473$$ 28375.1 2.75832
$$474$$ 0 0
$$475$$ −13462.2 −1.30039
$$476$$ 667.956 0.0643187
$$477$$ 0 0
$$478$$ −9092.54 −0.870049
$$479$$ −14620.0 −1.39459 −0.697293 0.716786i $$-0.745612\pi$$
−0.697293 + 0.716786i $$0.745612\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 12884.9 1.21761
$$483$$ 0 0
$$484$$ −4114.09 −0.386372
$$485$$ −327.536 −0.0306653
$$486$$ 0 0
$$487$$ 9798.86 0.911763 0.455882 0.890040i $$-0.349324\pi$$
0.455882 + 0.890040i $$0.349324\pi$$
$$488$$ 22259.7 2.06486
$$489$$ 0 0
$$490$$ −18.1154 −0.00167014
$$491$$ 10836.1 0.995977 0.497989 0.867184i $$-0.334072\pi$$
0.497989 + 0.867184i $$0.334072\pi$$
$$492$$ 0 0
$$493$$ −4493.84 −0.410532
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −5702.51 −0.516230
$$497$$ 9846.86 0.888717
$$498$$ 0 0
$$499$$ −2589.96 −0.232349 −0.116175 0.993229i $$-0.537063\pi$$
−0.116175 + 0.993229i $$0.537063\pi$$
$$500$$ 201.686 0.0180394
$$501$$ 0 0
$$502$$ 1841.20 0.163699
$$503$$ 17067.5 1.51292 0.756462 0.654038i $$-0.226926\pi$$
0.756462 + 0.654038i $$0.226926\pi$$
$$504$$ 0 0
$$505$$ −517.606 −0.0456102
$$506$$ −12149.0 −1.06737
$$507$$ 0 0
$$508$$ 795.712 0.0694961
$$509$$ −1012.89 −0.0882038 −0.0441019 0.999027i $$-0.514043\pi$$
−0.0441019 + 0.999027i $$0.514043\pi$$
$$510$$ 0 0
$$511$$ 13887.4 1.20223
$$512$$ −12992.6 −1.12148
$$513$$ 0 0
$$514$$ −3280.82 −0.281539
$$515$$ −522.644 −0.0447193
$$516$$ 0 0
$$517$$ −2068.26 −0.175942
$$518$$ 5345.63 0.453424
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 14367.7 1.20818 0.604089 0.796917i $$-0.293537\pi$$
0.604089 + 0.796917i $$0.293537\pi$$
$$522$$ 0 0
$$523$$ −16219.9 −1.35611 −0.678057 0.735010i $$-0.737178\pi$$
−0.678057 + 0.735010i $$0.737178\pi$$
$$524$$ 2958.02 0.246607
$$525$$ 0 0
$$526$$ −13385.5 −1.10957
$$527$$ 2889.11 0.238808
$$528$$ 0 0
$$529$$ −6799.77 −0.558870
$$530$$ −4.09466 −0.000335586 0
$$531$$ 0 0
$$532$$ 2823.06 0.230066
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −481.466 −0.0389076
$$536$$ −10741.4 −0.865593
$$537$$ 0 0
$$538$$ −16506.1 −1.32273
$$539$$ −815.301 −0.0651530
$$540$$ 0 0
$$541$$ −17592.2 −1.39806 −0.699029 0.715094i $$-0.746384\pi$$
−0.699029 + 0.715094i $$0.746384\pi$$
$$542$$ 10066.7 0.797792
$$543$$ 0 0
$$544$$ 1641.48 0.129371
$$545$$ −377.281 −0.0296531
$$546$$ 0 0
$$547$$ 10504.6 0.821103 0.410552 0.911837i $$-0.365336\pi$$
0.410552 + 0.911837i $$0.365336\pi$$
$$548$$ 2601.53 0.202795
$$549$$ 0 0
$$550$$ −20676.6 −1.60301
$$551$$ −18992.8 −1.46846
$$552$$ 0 0
$$553$$ 7662.33 0.589214
$$554$$ −15073.2 −1.15596
$$555$$ 0 0
$$556$$ −2148.52 −0.163880
$$557$$ −507.558 −0.0386102 −0.0193051 0.999814i $$-0.506145\pi$$
−0.0193051 + 0.999814i $$0.506145\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 514.690 0.0388386
$$561$$ 0 0
$$562$$ 9041.75 0.678653
$$563$$ 3443.14 0.257746 0.128873 0.991661i $$-0.458864\pi$$
0.128873 + 0.991661i $$0.458864\pi$$
$$564$$ 0 0
$$565$$ −360.221 −0.0268223
$$566$$ −6688.21 −0.496690
$$567$$ 0 0
$$568$$ 13097.2 0.967509
$$569$$ −23972.2 −1.76620 −0.883098 0.469189i $$-0.844546\pi$$
−0.883098 + 0.469189i $$0.844546\pi$$
$$570$$ 0 0
$$571$$ −7458.32 −0.546622 −0.273311 0.961926i $$-0.588119\pi$$
−0.273311 + 0.961926i $$0.588119\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 3242.82 0.235806
$$575$$ 9134.57 0.662501
$$576$$ 0 0
$$577$$ −5669.57 −0.409059 −0.204530 0.978860i $$-0.565566\pi$$
−0.204530 + 0.978860i $$0.565566\pi$$
$$578$$ −10913.2 −0.785344
$$579$$ 0 0
$$580$$ 142.093 0.0101726
$$581$$ −10975.3 −0.783702
$$582$$ 0 0
$$583$$ −184.284 −0.0130914
$$584$$ 18471.4 1.30882
$$585$$ 0 0
$$586$$ −14065.6 −0.991541
$$587$$ 1017.39 0.0715371 0.0357685 0.999360i $$-0.488612\pi$$
0.0357685 + 0.999360i $$0.488612\pi$$
$$588$$ 0 0
$$589$$ 12210.6 0.854208
$$590$$ 103.040 0.00718996
$$591$$ 0 0
$$592$$ 5788.99 0.401902
$$593$$ −10198.2 −0.706221 −0.353111 0.935582i $$-0.614876\pi$$
−0.353111 + 0.935582i $$0.614876\pi$$
$$594$$ 0 0
$$595$$ −260.762 −0.0179667
$$596$$ 3968.70 0.272759
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12516.3 −0.853763 −0.426881 0.904308i $$-0.640388\pi$$
−0.426881 + 0.904308i $$0.640388\pi$$
$$600$$ 0 0
$$601$$ 9627.46 0.653431 0.326716 0.945123i $$-0.394058\pi$$
0.326716 + 0.945123i $$0.394058\pi$$
$$602$$ −20408.0 −1.38168
$$603$$ 0 0
$$604$$ −1404.44 −0.0946120
$$605$$ 1606.09 0.107929
$$606$$ 0 0
$$607$$ 6667.20 0.445821 0.222910 0.974839i $$-0.428444\pi$$
0.222910 + 0.974839i $$0.428444\pi$$
$$608$$ 6937.56 0.462755
$$609$$ 0 0
$$610$$ −1324.37 −0.0879053
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 23085.4 1.52106 0.760530 0.649302i $$-0.224939\pi$$
0.760530 + 0.649302i $$0.224939\pi$$
$$614$$ 18718.8 1.23034
$$615$$ 0 0
$$616$$ 28450.6 1.86089
$$617$$ 3049.24 0.198959 0.0994796 0.995040i $$-0.468282\pi$$
0.0994796 + 0.995040i $$0.468282\pi$$
$$618$$ 0 0
$$619$$ −7296.58 −0.473787 −0.236894 0.971536i $$-0.576129\pi$$
−0.236894 + 0.971536i $$0.576129\pi$$
$$620$$ −91.3523 −0.00591741
$$621$$ 0 0
$$622$$ −20248.9 −1.30531
$$623$$ 21083.3 1.35583
$$624$$ 0 0
$$625$$ 15506.8 0.992438
$$626$$ 25621.7 1.63586
$$627$$ 0 0
$$628$$ 812.543 0.0516305
$$629$$ −2932.92 −0.185920
$$630$$ 0 0
$$631$$ 23829.5 1.50339 0.751694 0.659512i $$-0.229237\pi$$
0.751694 + 0.659512i $$0.229237\pi$$
$$632$$ 10191.6 0.641453
$$633$$ 0 0
$$634$$ −15960.5 −0.999801
$$635$$ −310.637 −0.0194130
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −29171.3 −1.81019
$$639$$ 0 0
$$640$$ 528.347 0.0326324
$$641$$ −13405.3 −0.826016 −0.413008 0.910727i $$-0.635522\pi$$
−0.413008 + 0.910727i $$0.635522\pi$$
$$642$$ 0 0
$$643$$ −5251.51 −0.322083 −0.161042 0.986948i $$-0.551485\pi$$
−0.161042 + 0.986948i $$0.551485\pi$$
$$644$$ −1915.55 −0.117210
$$645$$ 0 0
$$646$$ 7065.37 0.430315
$$647$$ −21611.4 −1.31319 −0.656595 0.754244i $$-0.728004\pi$$
−0.656595 + 0.754244i $$0.728004\pi$$
$$648$$ 0 0
$$649$$ 4637.39 0.280483
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 2169.94 0.130340
$$653$$ 21595.8 1.29420 0.647099 0.762406i $$-0.275982\pi$$
0.647099 + 0.762406i $$0.275982\pi$$
$$654$$ 0 0
$$655$$ −1154.78 −0.0688869
$$656$$ 3511.77 0.209012
$$657$$ 0 0
$$658$$ 1487.54 0.0881314
$$659$$ 16642.6 0.983768 0.491884 0.870661i $$-0.336308\pi$$
0.491884 + 0.870661i $$0.336308\pi$$
$$660$$ 0 0
$$661$$ −26981.1 −1.58766 −0.793831 0.608139i $$-0.791916\pi$$
−0.793831 + 0.608139i $$0.791916\pi$$
$$662$$ 11871.5 0.696980
$$663$$ 0 0
$$664$$ −14598.1 −0.853185
$$665$$ −1102.09 −0.0642664
$$666$$ 0 0
$$667$$ 12887.3 0.748126
$$668$$ −852.310 −0.0493665
$$669$$ 0 0
$$670$$ 639.074 0.0368501
$$671$$ −59604.5 −3.42922
$$672$$ 0 0
$$673$$ 11149.2 0.638591 0.319296 0.947655i $$-0.396554\pi$$
0.319296 + 0.947655i $$0.396554\pi$$
$$674$$ 7761.04 0.443537
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3314.33 −0.188154 −0.0940769 0.995565i $$-0.529990\pi$$
−0.0940769 + 0.995565i $$0.529990\pi$$
$$678$$ 0 0
$$679$$ 10602.1 0.599223
$$680$$ −346.836 −0.0195596
$$681$$ 0 0
$$682$$ 18754.3 1.05299
$$683$$ 24505.2 1.37287 0.686433 0.727193i $$-0.259176\pi$$
0.686433 + 0.727193i $$0.259176\pi$$
$$684$$ 0 0
$$685$$ −1015.61 −0.0566486
$$686$$ 16557.0 0.921500
$$687$$ 0 0
$$688$$ −22100.6 −1.22468
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 21752.8 1.19756 0.598782 0.800912i $$-0.295652\pi$$
0.598782 + 0.800912i $$0.295652\pi$$
$$692$$ −6466.64 −0.355238
$$693$$ 0 0
$$694$$ −7278.90 −0.398132
$$695$$ 838.755 0.0457781
$$696$$ 0 0
$$697$$ −1779.20 −0.0966887
$$698$$ −19379.9 −1.05092
$$699$$ 0 0
$$700$$ −3260.10 −0.176029
$$701$$ −34250.9 −1.84542 −0.922709 0.385496i $$-0.874030\pi$$
−0.922709 + 0.385496i $$0.874030\pi$$
$$702$$ 0 0
$$703$$ −12395.8 −0.665028
$$704$$ 36770.1 1.96850
$$705$$ 0 0
$$706$$ −5992.59 −0.319454
$$707$$ 16754.6 0.891259
$$708$$ 0 0
$$709$$ 5527.11 0.292771 0.146386 0.989228i $$-0.453236\pi$$
0.146386 + 0.989228i $$0.453236\pi$$
$$710$$ −779.234 −0.0411889
$$711$$ 0 0
$$712$$ 28042.6 1.47604
$$713$$ −8285.33 −0.435187
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −221.931 −0.0115837
$$717$$ 0 0
$$718$$ −6485.02 −0.337074
$$719$$ 3777.78 0.195949 0.0979745 0.995189i $$-0.468764\pi$$
0.0979745 + 0.995189i $$0.468764\pi$$
$$720$$ 0 0
$$721$$ 16917.6 0.873849
$$722$$ 12291.5 0.633576
$$723$$ 0 0
$$724$$ −1541.08 −0.0791072
$$725$$ 21933.2 1.12355
$$726$$ 0 0
$$727$$ 19076.8 0.973204 0.486602 0.873624i $$-0.338236\pi$$
0.486602 + 0.873624i $$0.338236\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −1098.98 −0.0557192
$$731$$ 11197.0 0.566535
$$732$$ 0 0
$$733$$ −7997.30 −0.402984 −0.201492 0.979490i $$-0.564579\pi$$
−0.201492 + 0.979490i $$0.564579\pi$$
$$734$$ 16849.4 0.847307
$$735$$ 0 0
$$736$$ −4707.39 −0.235756
$$737$$ 28762.1 1.43754
$$738$$ 0 0
$$739$$ −28983.6 −1.44273 −0.721367 0.692553i $$-0.756486\pi$$
−0.721367 + 0.692553i $$0.756486\pi$$
$$740$$ 92.7376 0.00460689
$$741$$ 0 0
$$742$$ 132.541 0.00655761
$$743$$ −19145.4 −0.945324 −0.472662 0.881244i $$-0.656707\pi$$
−0.472662 + 0.881244i $$0.656707\pi$$
$$744$$ 0 0
$$745$$ −1549.33 −0.0761923
$$746$$ 7435.47 0.364922
$$747$$ 0 0
$$748$$ −2378.96 −0.116288
$$749$$ 15584.7 0.760284
$$750$$ 0 0
$$751$$ −25516.9 −1.23985 −0.619923 0.784663i $$-0.712836\pi$$
−0.619923 + 0.784663i $$0.712836\pi$$
$$752$$ 1610.92 0.0781172
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 548.275 0.0264288
$$756$$ 0 0
$$757$$ −17230.6 −0.827289 −0.413645 0.910438i $$-0.635744\pi$$
−0.413645 + 0.910438i $$0.635744\pi$$
$$758$$ 4779.17 0.229007
$$759$$ 0 0
$$760$$ −1465.87 −0.0699642
$$761$$ −2343.06 −0.111611 −0.0558053 0.998442i $$-0.517773\pi$$
−0.0558053 + 0.998442i $$0.517773\pi$$
$$762$$ 0 0
$$763$$ 12212.3 0.579444
$$764$$ 974.121 0.0461289
$$765$$ 0 0
$$766$$ 27756.9 1.30927
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 7100.18 0.332950 0.166475 0.986046i $$-0.446761\pi$$
0.166475 + 0.986046i $$0.446761\pi$$
$$770$$ −1692.71 −0.0792219
$$771$$ 0 0
$$772$$ 1901.17 0.0886328
$$773$$ 12270.4 0.570940 0.285470 0.958388i $$-0.407850\pi$$
0.285470 + 0.958388i $$0.407850\pi$$
$$774$$ 0 0
$$775$$ −14100.9 −0.653575
$$776$$ 14101.7 0.652349
$$777$$ 0 0
$$778$$ 24386.9 1.12379
$$779$$ −7519.64 −0.345852
$$780$$ 0 0
$$781$$ −35070.1 −1.60680
$$782$$ −4794.11 −0.219229
$$783$$ 0 0
$$784$$ 635.017 0.0289275
$$785$$ −317.207 −0.0144224
$$786$$ 0 0
$$787$$ −3425.04 −0.155133 −0.0775663 0.996987i $$-0.524715\pi$$
−0.0775663 + 0.996987i $$0.524715\pi$$
$$788$$ −1823.04 −0.0824151
$$789$$ 0 0
$$790$$ −606.360 −0.0273080
$$791$$ 11660.1 0.524129
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 25894.3 1.15737
$$795$$ 0 0
$$796$$ −3446.87 −0.153481
$$797$$ 11781.1 0.523600 0.261800 0.965122i $$-0.415684\pi$$
0.261800 + 0.965122i $$0.415684\pi$$
$$798$$ 0 0
$$799$$ −816.154 −0.0361370
$$800$$ −8011.58 −0.354065
$$801$$ 0 0
$$802$$ 5339.24 0.235081
$$803$$ −49460.6 −2.17363
$$804$$ 0 0
$$805$$ 747.807 0.0327413
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 22285.0 0.970276
$$809$$ −18910.1 −0.821810 −0.410905 0.911678i $$-0.634787\pi$$
−0.410905 + 0.911678i $$0.634787\pi$$
$$810$$ 0 0
$$811$$ −12803.3 −0.554359 −0.277180 0.960818i $$-0.589400\pi$$
−0.277180 + 0.960818i $$0.589400\pi$$
$$812$$ −4599.45 −0.198780
$$813$$ 0 0
$$814$$ −19038.7 −0.819788
$$815$$ −847.121 −0.0364090
$$816$$ 0 0
$$817$$ 47323.3 2.02648
$$818$$ 24889.4 1.06386
$$819$$ 0 0
$$820$$ 56.2574 0.00239585
$$821$$ 19335.1 0.821923 0.410962 0.911653i $$-0.365193\pi$$
0.410962 + 0.911653i $$0.365193\pi$$
$$822$$ 0 0
$$823$$ −2125.90 −0.0900417 −0.0450209 0.998986i $$-0.514335\pi$$
−0.0450209 + 0.998986i $$0.514335\pi$$
$$824$$ 22501.9 0.951324
$$825$$ 0 0
$$826$$ −3335.32 −0.140497
$$827$$ −6989.24 −0.293881 −0.146941 0.989145i $$-0.546943\pi$$
−0.146941 + 0.989145i $$0.546943\pi$$
$$828$$ 0 0
$$829$$ −32649.7 −1.36788 −0.683938 0.729540i $$-0.739734\pi$$
−0.683938 + 0.729540i $$0.739734\pi$$
$$830$$ 868.531 0.0363219
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −321.724 −0.0133819
$$834$$ 0 0
$$835$$ 332.732 0.0137900
$$836$$ −10054.5 −0.415958
$$837$$ 0 0
$$838$$ −34278.4 −1.41304
$$839$$ −4038.23 −0.166168 −0.0830841 0.996543i $$-0.526477\pi$$
−0.0830841 + 0.996543i $$0.526477\pi$$
$$840$$ 0 0
$$841$$ 6555.00 0.268769
$$842$$ 24240.9 0.992159
$$843$$ 0 0
$$844$$ 131.695 0.00537102
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −51988.1 −2.10901
$$848$$ 143.534 0.00581248
$$849$$ 0 0
$$850$$ −8159.17 −0.329244
$$851$$ 8410.97 0.338807
$$852$$ 0 0
$$853$$ −8114.12 −0.325700 −0.162850 0.986651i $$-0.552069\pi$$
−0.162850 + 0.986651i $$0.552069\pi$$
$$854$$ 42869.0 1.71774
$$855$$ 0 0
$$856$$ 20729.0 0.827690
$$857$$ 22298.1 0.888786 0.444393 0.895832i $$-0.353419\pi$$
0.444393 + 0.895832i $$0.353419\pi$$
$$858$$ 0 0
$$859$$ 33550.5 1.33263 0.666315 0.745670i $$-0.267870\pi$$
0.666315 + 0.745670i $$0.267870\pi$$
$$860$$ −354.045 −0.0140382
$$861$$ 0 0
$$862$$ 12429.4 0.491121
$$863$$ −14120.5 −0.556972 −0.278486 0.960440i $$-0.589833\pi$$
−0.278486 + 0.960440i $$0.589833\pi$$
$$864$$ 0 0
$$865$$ 2524.50 0.0992319
$$866$$ −21025.2 −0.825018
$$867$$ 0 0
$$868$$ 2957.01 0.115631
$$869$$ −27289.8 −1.06530
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 16243.4 0.630817
$$873$$ 0 0
$$874$$ −20261.9 −0.784175
$$875$$ 2548.63 0.0984679
$$876$$ 0 0
$$877$$ 1941.69 0.0747619 0.0373809 0.999301i $$-0.488099\pi$$
0.0373809 + 0.999301i $$0.488099\pi$$
$$878$$ −7668.78 −0.294771
$$879$$ 0 0
$$880$$ −1833.10 −0.0702201
$$881$$ 790.231 0.0302197 0.0151099 0.999886i $$-0.495190\pi$$
0.0151099 + 0.999886i $$0.495190\pi$$
$$882$$ 0 0
$$883$$ −36638.6 −1.39636 −0.698180 0.715922i $$-0.746007\pi$$
−0.698180 + 0.715922i $$0.746007\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −24958.9 −0.946401
$$887$$ 40686.3 1.54015 0.770075 0.637954i $$-0.220219\pi$$
0.770075 + 0.637954i $$0.220219\pi$$
$$888$$ 0 0
$$889$$ 10055.1 0.379344
$$890$$ −1668.43 −0.0628381
$$891$$ 0 0
$$892$$ 1777.08 0.0667053
$$893$$ −3449.40 −0.129261
$$894$$ 0 0
$$895$$ 86.6392 0.00323579
$$896$$ −17102.2 −0.637663
$$897$$ 0 0
$$898$$ −1438.21 −0.0534449
$$899$$ −19894.0 −0.738046
$$900$$ 0 0
$$901$$ −72.7200 −0.00268885
$$902$$ −11549.5 −0.426336
$$903$$ 0 0
$$904$$ 15509.0 0.570598
$$905$$ 601.618 0.0220977
$$906$$ 0 0
$$907$$ −10464.4 −0.383093 −0.191547 0.981484i $$-0.561350\pi$$
−0.191547 + 0.981484i $$0.561350\pi$$
$$908$$ −4749.26 −0.173579
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 35611.5 1.29513 0.647563 0.762011i $$-0.275788\pi$$
0.647563 + 0.762011i $$0.275788\pi$$
$$912$$ 0 0
$$913$$ 39089.0 1.41693
$$914$$ −35242.9 −1.27542
$$915$$ 0 0
$$916$$ 303.920 0.0109627
$$917$$ 37379.4 1.34610
$$918$$ 0 0
$$919$$ 1077.25 0.0386674 0.0193337 0.999813i $$-0.493846\pi$$
0.0193337 + 0.999813i $$0.493846\pi$$
$$920$$ 994.648 0.0356441
$$921$$ 0 0
$$922$$ 30762.3 1.09881
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 14314.8 0.508829
$$926$$ −34928.6 −1.23955
$$927$$ 0 0
$$928$$ −11303.0 −0.399826
$$929$$ 55733.8 1.96832 0.984159 0.177290i $$-0.0567330\pi$$
0.984159 + 0.177290i $$0.0567330\pi$$
$$930$$ 0 0
$$931$$ −1359.74 −0.0478665
$$932$$ −369.284 −0.0129789
$$933$$ 0 0
$$934$$ −22597.9 −0.791677
$$935$$ 928.718 0.0324838
$$936$$ 0 0
$$937$$ −3198.60 −0.111519 −0.0557596 0.998444i $$-0.517758\pi$$
−0.0557596 + 0.998444i $$0.517758\pi$$
$$938$$ −20686.4 −0.720079
$$939$$ 0 0
$$940$$ 25.8064 0.000895437 0
$$941$$ 8823.35 0.305667 0.152834 0.988252i $$-0.451160\pi$$
0.152834 + 0.988252i $$0.451160\pi$$
$$942$$ 0 0
$$943$$ 5102.35 0.176199
$$944$$ −3611.95 −0.124533
$$945$$ 0 0
$$946$$ 72684.3 2.49806
$$947$$ 28290.4 0.970766 0.485383 0.874301i $$-0.338680\pi$$
0.485383 + 0.874301i $$0.338680\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −34484.0 −1.17769
$$951$$ 0 0
$$952$$ 11226.8 0.382210
$$953$$ 12399.0 0.421452 0.210726 0.977545i $$-0.432417\pi$$
0.210726 + 0.977545i $$0.432417\pi$$
$$954$$ 0 0
$$955$$ −380.285 −0.0128856
$$956$$ 5105.94 0.172739
$$957$$ 0 0
$$958$$ −37450.0 −1.26300
$$959$$ 32874.5 1.10696
$$960$$ 0 0
$$961$$ −17001.0 −0.570676
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −7235.53 −0.241744
$$965$$ −742.193 −0.0247586
$$966$$ 0 0
$$967$$ 26667.1 0.886820 0.443410 0.896319i $$-0.353769\pi$$
0.443410 + 0.896319i $$0.353769\pi$$
$$968$$ −69148.6 −2.29599
$$969$$ 0 0
$$970$$ −839.001 −0.0277719
$$971$$ −49420.7 −1.63335 −0.816676 0.577096i $$-0.804186\pi$$
−0.816676 + 0.577096i $$0.804186\pi$$
$$972$$ 0 0
$$973$$ −27149.9 −0.894539
$$974$$ 25100.3 0.825735
$$975$$ 0 0
$$976$$ 46424.5 1.52255
$$977$$ 778.759 0.0255012 0.0127506 0.999919i $$-0.495941\pi$$
0.0127506 + 0.999919i $$0.495941\pi$$
$$978$$ 0 0
$$979$$ −75089.2 −2.45134
$$980$$ 10.1728 0.000331589 0
$$981$$ 0 0
$$982$$ 27757.2 0.902003
$$983$$ 5997.90 0.194612 0.0973059 0.995255i $$-0.468977\pi$$
0.0973059 + 0.995255i $$0.468977\pi$$
$$984$$ 0 0
$$985$$ 711.693 0.0230218
$$986$$ −11511.2 −0.371797
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −32110.6 −1.03241
$$990$$ 0 0
$$991$$ 8974.94 0.287688 0.143844 0.989600i $$-0.454054\pi$$
0.143844 + 0.989600i $$0.454054\pi$$
$$992$$ 7266.75 0.232580
$$993$$ 0 0
$$994$$ 25223.3 0.804863
$$995$$ 1345.62 0.0428732
$$996$$ 0 0
$$997$$ 28530.2 0.906280 0.453140 0.891439i $$-0.350304\pi$$
0.453140 + 0.891439i $$0.350304\pi$$
$$998$$ −6634.31 −0.210426
$$999$$ 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 1521.4.a.r.1.2 2
3.2 odd 2 169.4.a.g.1.1 2
13.12 even 2 117.4.a.d.1.1 2
39.2 even 12 169.4.e.f.147.1 8
39.5 even 4 169.4.b.f.168.4 4
39.8 even 4 169.4.b.f.168.1 4
39.11 even 12 169.4.e.f.147.4 8
39.17 odd 6 169.4.c.g.146.1 4
39.20 even 12 169.4.e.f.23.1 8
39.23 odd 6 169.4.c.g.22.1 4
39.29 odd 6 169.4.c.j.22.2 4
39.32 even 12 169.4.e.f.23.4 8
39.35 odd 6 169.4.c.j.146.2 4
39.38 odd 2 13.4.a.b.1.2 2
52.51 odd 2 1872.4.a.bb.1.1 2
156.155 even 2 208.4.a.h.1.2 2
195.38 even 4 325.4.b.e.274.1 4
195.77 even 4 325.4.b.e.274.4 4
195.194 odd 2 325.4.a.f.1.1 2
273.272 even 2 637.4.a.b.1.2 2
312.77 odd 2 832.4.a.s.1.2 2
312.155 even 2 832.4.a.z.1.1 2
429.428 even 2 1573.4.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.2 2 39.38 odd 2
117.4.a.d.1.1 2 13.12 even 2
169.4.a.g.1.1 2 3.2 odd 2
169.4.b.f.168.1 4 39.8 even 4
169.4.b.f.168.4 4 39.5 even 4
169.4.c.g.22.1 4 39.23 odd 6
169.4.c.g.146.1 4 39.17 odd 6
169.4.c.j.22.2 4 39.29 odd 6
169.4.c.j.146.2 4 39.35 odd 6
169.4.e.f.23.1 8 39.20 even 12
169.4.e.f.23.4 8 39.32 even 12
169.4.e.f.147.1 8 39.2 even 12
169.4.e.f.147.4 8 39.11 even 12
208.4.a.h.1.2 2 156.155 even 2
325.4.a.f.1.1 2 195.194 odd 2
325.4.b.e.274.1 4 195.38 even 4
325.4.b.e.274.4 4 195.77 even 4
637.4.a.b.1.2 2 273.272 even 2
832.4.a.s.1.2 2 312.77 odd 2
832.4.a.z.1.1 2 312.155 even 2
1521.4.a.r.1.2 2 1.1 even 1 trivial
1573.4.a.b.1.1 2 429.428 even 2
1872.4.a.bb.1.1 2 52.51 odd 2