# Properties

 Label 1521.4.a.z.1.3 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - 2x^{3} - 13x^{2} + 14x - 2$$ x^4 - 2*x^3 - 13*x^2 + 14*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.829502$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.12311 q^{2} +9.00000 q^{4} -13.4424 q^{5} -31.4219 q^{7} +4.12311 q^{8} +O(q^{10})$$ $$q+4.12311 q^{2} +9.00000 q^{4} -13.4424 q^{5} -31.4219 q^{7} +4.12311 q^{8} -55.4243 q^{10} -40.4962 q^{11} -129.556 q^{14} -55.0000 q^{16} +43.1414 q^{17} -26.9779 q^{19} -120.981 q^{20} -166.970 q^{22} +19.0100 q^{23} +55.6971 q^{25} -282.797 q^{28} +154.111 q^{29} +308.270 q^{31} -259.756 q^{32} +177.877 q^{34} +422.384 q^{35} +43.5116 q^{37} -111.233 q^{38} -55.4243 q^{40} -47.8384 q^{41} +342.121 q^{43} -364.466 q^{44} +78.3802 q^{46} -133.468 q^{47} +644.334 q^{49} +229.645 q^{50} +438.454 q^{53} +544.364 q^{55} -129.556 q^{56} +635.418 q^{58} -590.553 q^{59} -541.304 q^{61} +1271.03 q^{62} -631.000 q^{64} -230.345 q^{67} +388.273 q^{68} +1741.54 q^{70} -449.412 q^{71} -389.711 q^{73} +179.403 q^{74} -242.801 q^{76} +1272.47 q^{77} -897.820 q^{79} +739.330 q^{80} -197.243 q^{82} +1300.24 q^{83} -579.923 q^{85} +1410.60 q^{86} -166.970 q^{88} +925.045 q^{89} +171.090 q^{92} -550.301 q^{94} +362.647 q^{95} -1560.49 q^{97} +2656.66 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 36 q^{4}+O(q^{10})$$ 4 * q + 36 * q^4 $$4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100})$$ 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.12311 1.45774 0.728869 0.684653i $$-0.240046\pi$$
0.728869 + 0.684653i $$0.240046\pi$$
$$3$$ 0 0
$$4$$ 9.00000 1.12500
$$5$$ −13.4424 −1.20232 −0.601161 0.799128i $$-0.705295\pi$$
−0.601161 + 0.799128i $$0.705295\pi$$
$$6$$ 0 0
$$7$$ −31.4219 −1.69662 −0.848311 0.529498i $$-0.822380\pi$$
−0.848311 + 0.529498i $$0.822380\pi$$
$$8$$ 4.12311 0.182217
$$9$$ 0 0
$$10$$ −55.4243 −1.75267
$$11$$ −40.4962 −1.11001 −0.555003 0.831849i $$-0.687283\pi$$
−0.555003 + 0.831849i $$0.687283\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −129.556 −2.47323
$$15$$ 0 0
$$16$$ −55.0000 −0.859375
$$17$$ 43.1414 0.615490 0.307745 0.951469i $$-0.400426\pi$$
0.307745 + 0.951469i $$0.400426\pi$$
$$18$$ 0 0
$$19$$ −26.9779 −0.325745 −0.162873 0.986647i $$-0.552076\pi$$
−0.162873 + 0.986647i $$0.552076\pi$$
$$20$$ −120.981 −1.35261
$$21$$ 0 0
$$22$$ −166.970 −1.61810
$$23$$ 19.0100 0.172342 0.0861709 0.996280i $$-0.472537\pi$$
0.0861709 + 0.996280i $$0.472537\pi$$
$$24$$ 0 0
$$25$$ 55.6971 0.445577
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −282.797 −1.90870
$$29$$ 154.111 0.986820 0.493410 0.869797i $$-0.335750\pi$$
0.493410 + 0.869797i $$0.335750\pi$$
$$30$$ 0 0
$$31$$ 308.270 1.78603 0.893016 0.450025i $$-0.148585\pi$$
0.893016 + 0.450025i $$0.148585\pi$$
$$32$$ −259.756 −1.43496
$$33$$ 0 0
$$34$$ 177.877 0.897223
$$35$$ 422.384 2.03988
$$36$$ 0 0
$$37$$ 43.5116 0.193331 0.0966657 0.995317i $$-0.469182\pi$$
0.0966657 + 0.995317i $$0.469182\pi$$
$$38$$ −111.233 −0.474851
$$39$$ 0 0
$$40$$ −55.4243 −0.219084
$$41$$ −47.8384 −0.182222 −0.0911110 0.995841i $$-0.529042\pi$$
−0.0911110 + 0.995841i $$0.529042\pi$$
$$42$$ 0 0
$$43$$ 342.121 1.21333 0.606663 0.794959i $$-0.292508\pi$$
0.606663 + 0.794959i $$0.292508\pi$$
$$44$$ −364.466 −1.24876
$$45$$ 0 0
$$46$$ 78.3802 0.251229
$$47$$ −133.468 −0.414218 −0.207109 0.978318i $$-0.566406\pi$$
−0.207109 + 0.978318i $$0.566406\pi$$
$$48$$ 0 0
$$49$$ 644.334 1.87853
$$50$$ 229.645 0.649535
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 438.454 1.13635 0.568173 0.822909i $$-0.307650\pi$$
0.568173 + 0.822909i $$0.307650\pi$$
$$54$$ 0 0
$$55$$ 544.364 1.33458
$$56$$ −129.556 −0.309154
$$57$$ 0 0
$$58$$ 635.418 1.43852
$$59$$ −590.553 −1.30311 −0.651555 0.758601i $$-0.725883\pi$$
−0.651555 + 0.758601i $$0.725883\pi$$
$$60$$ 0 0
$$61$$ −541.304 −1.13618 −0.568089 0.822967i $$-0.692317\pi$$
−0.568089 + 0.822967i $$0.692317\pi$$
$$62$$ 1271.03 2.60357
$$63$$ 0 0
$$64$$ −631.000 −1.23242
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −230.345 −0.420018 −0.210009 0.977700i $$-0.567349\pi$$
−0.210009 + 0.977700i $$0.567349\pi$$
$$68$$ 388.273 0.692426
$$69$$ 0 0
$$70$$ 1741.54 2.97362
$$71$$ −449.412 −0.751203 −0.375601 0.926781i $$-0.622564\pi$$
−0.375601 + 0.926781i $$0.622564\pi$$
$$72$$ 0 0
$$73$$ −389.711 −0.624826 −0.312413 0.949946i $$-0.601137\pi$$
−0.312413 + 0.949946i $$0.601137\pi$$
$$74$$ 179.403 0.281826
$$75$$ 0 0
$$76$$ −242.801 −0.366464
$$77$$ 1272.47 1.88326
$$78$$ 0 0
$$79$$ −897.820 −1.27864 −0.639321 0.768940i $$-0.720784\pi$$
−0.639321 + 0.768940i $$0.720784\pi$$
$$80$$ 739.330 1.03325
$$81$$ 0 0
$$82$$ −197.243 −0.265632
$$83$$ 1300.24 1.71952 0.859759 0.510700i $$-0.170614\pi$$
0.859759 + 0.510700i $$0.170614\pi$$
$$84$$ 0 0
$$85$$ −579.923 −0.740017
$$86$$ 1410.60 1.76871
$$87$$ 0 0
$$88$$ −166.970 −0.202262
$$89$$ 925.045 1.10174 0.550869 0.834592i $$-0.314297\pi$$
0.550869 + 0.834592i $$0.314297\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 171.090 0.193884
$$93$$ 0 0
$$94$$ −550.301 −0.603822
$$95$$ 362.647 0.391651
$$96$$ 0 0
$$97$$ −1560.49 −1.63344 −0.816722 0.577031i $$-0.804211\pi$$
−0.816722 + 0.577031i $$0.804211\pi$$
$$98$$ 2656.66 2.73840
$$99$$ 0 0
$$100$$ 501.274 0.501274
$$101$$ −958.840 −0.944635 −0.472318 0.881428i $$-0.656582\pi$$
−0.472318 + 0.881428i $$0.656582\pi$$
$$102$$ 0 0
$$103$$ 635.153 0.607606 0.303803 0.952735i $$-0.401743\pi$$
0.303803 + 0.952735i $$0.401743\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1807.79 1.65649
$$107$$ 1448.32 1.30855 0.654275 0.756257i $$-0.272974\pi$$
0.654275 + 0.756257i $$0.272974\pi$$
$$108$$ 0 0
$$109$$ −331.084 −0.290937 −0.145468 0.989363i $$-0.546469\pi$$
−0.145468 + 0.989363i $$0.546469\pi$$
$$110$$ 2244.47 1.94547
$$111$$ 0 0
$$112$$ 1728.20 1.45803
$$113$$ −695.204 −0.578755 −0.289378 0.957215i $$-0.593448\pi$$
−0.289378 + 0.957215i $$0.593448\pi$$
$$114$$ 0 0
$$115$$ −255.539 −0.207210
$$116$$ 1387.00 1.11017
$$117$$ 0 0
$$118$$ −2434.91 −1.89959
$$119$$ −1355.58 −1.04425
$$120$$ 0 0
$$121$$ 308.940 0.232111
$$122$$ −2231.85 −1.65625
$$123$$ 0 0
$$124$$ 2774.43 2.00929
$$125$$ 931.594 0.666595
$$126$$ 0 0
$$127$$ 247.154 0.172688 0.0863441 0.996265i $$-0.472482\pi$$
0.0863441 + 0.996265i $$0.472482\pi$$
$$128$$ −523.634 −0.361587
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −472.243 −0.314962 −0.157481 0.987522i $$-0.550337\pi$$
−0.157481 + 0.987522i $$0.550337\pi$$
$$132$$ 0 0
$$133$$ 847.697 0.552667
$$134$$ −949.739 −0.612275
$$135$$ 0 0
$$136$$ 177.877 0.112153
$$137$$ 1830.70 1.14166 0.570829 0.821069i $$-0.306622\pi$$
0.570829 + 0.821069i $$0.306622\pi$$
$$138$$ 0 0
$$139$$ −100.000 −0.0610208 −0.0305104 0.999534i $$-0.509713\pi$$
−0.0305104 + 0.999534i $$0.509713\pi$$
$$140$$ 3801.46 2.29487
$$141$$ 0 0
$$142$$ −1852.97 −1.09506
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2071.62 −1.18647
$$146$$ −1606.82 −0.910832
$$147$$ 0 0
$$148$$ 391.604 0.217498
$$149$$ −149.557 −0.0822293 −0.0411147 0.999154i $$-0.513091\pi$$
−0.0411147 + 0.999154i $$0.513091\pi$$
$$150$$ 0 0
$$151$$ −800.032 −0.431163 −0.215582 0.976486i $$-0.569165\pi$$
−0.215582 + 0.976486i $$0.569165\pi$$
$$152$$ −111.233 −0.0593564
$$153$$ 0 0
$$154$$ 5246.51 2.74530
$$155$$ −4143.88 −2.14739
$$156$$ 0 0
$$157$$ −2706.16 −1.37564 −0.687818 0.725884i $$-0.741431\pi$$
−0.687818 + 0.725884i $$0.741431\pi$$
$$158$$ −3701.81 −1.86392
$$159$$ 0 0
$$160$$ 3491.73 1.72528
$$161$$ −597.330 −0.292399
$$162$$ 0 0
$$163$$ 3678.25 1.76750 0.883750 0.467959i $$-0.155010\pi$$
0.883750 + 0.467959i $$0.155010\pi$$
$$164$$ −430.546 −0.205000
$$165$$ 0 0
$$166$$ 5361.03 2.50661
$$167$$ 3223.11 1.49348 0.746742 0.665114i $$-0.231617\pi$$
0.746742 + 0.665114i $$0.231617\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −2391.08 −1.07875
$$171$$ 0 0
$$172$$ 3079.09 1.36499
$$173$$ 2689.54 1.18198 0.590988 0.806680i $$-0.298738\pi$$
0.590988 + 0.806680i $$0.298738\pi$$
$$174$$ 0 0
$$175$$ −1750.11 −0.755976
$$176$$ 2227.29 0.953911
$$177$$ 0 0
$$178$$ 3814.06 1.60604
$$179$$ 1524.04 0.636381 0.318191 0.948027i $$-0.396925\pi$$
0.318191 + 0.948027i $$0.396925\pi$$
$$180$$ 0 0
$$181$$ 476.881 0.195836 0.0979180 0.995194i $$-0.468782\pi$$
0.0979180 + 0.995194i $$0.468782\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 78.3802 0.0314036
$$185$$ −584.899 −0.232446
$$186$$ 0 0
$$187$$ −1747.06 −0.683197
$$188$$ −1201.21 −0.465996
$$189$$ 0 0
$$190$$ 1495.23 0.570924
$$191$$ 1369.74 0.518906 0.259453 0.965756i $$-0.416458\pi$$
0.259453 + 0.965756i $$0.416458\pi$$
$$192$$ 0 0
$$193$$ −2144.72 −0.799898 −0.399949 0.916537i $$-0.630972\pi$$
−0.399949 + 0.916537i $$0.630972\pi$$
$$194$$ −6434.08 −2.38113
$$195$$ 0 0
$$196$$ 5799.01 2.11334
$$197$$ 239.739 0.0867040 0.0433520 0.999060i $$-0.486196\pi$$
0.0433520 + 0.999060i $$0.486196\pi$$
$$198$$ 0 0
$$199$$ −1589.94 −0.566371 −0.283185 0.959065i $$-0.591391\pi$$
−0.283185 + 0.959065i $$0.591391\pi$$
$$200$$ 229.645 0.0811918
$$201$$ 0 0
$$202$$ −3953.40 −1.37703
$$203$$ −4842.47 −1.67426
$$204$$ 0 0
$$205$$ 643.061 0.219090
$$206$$ 2618.80 0.885731
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1092.50 0.361579
$$210$$ 0 0
$$211$$ −1872.85 −0.611055 −0.305527 0.952183i $$-0.598833\pi$$
−0.305527 + 0.952183i $$0.598833\pi$$
$$212$$ 3946.09 1.27839
$$213$$ 0 0
$$214$$ 5971.59 1.90752
$$215$$ −4598.92 −1.45881
$$216$$ 0 0
$$217$$ −9686.43 −3.03022
$$218$$ −1365.09 −0.424109
$$219$$ 0 0
$$220$$ 4899.28 1.50141
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −56.1283 −0.0168548 −0.00842742 0.999964i $$-0.502683\pi$$
−0.00842742 + 0.999964i $$0.502683\pi$$
$$224$$ 8162.01 2.43459
$$225$$ 0 0
$$226$$ −2866.40 −0.843673
$$227$$ 667.390 0.195137 0.0975687 0.995229i $$-0.468893\pi$$
0.0975687 + 0.995229i $$0.468893\pi$$
$$228$$ 0 0
$$229$$ 723.299 0.208720 0.104360 0.994540i $$-0.466721\pi$$
0.104360 + 0.994540i $$0.466721\pi$$
$$230$$ −1053.62 −0.302058
$$231$$ 0 0
$$232$$ 635.418 0.179816
$$233$$ 275.451 0.0774482 0.0387241 0.999250i $$-0.487671\pi$$
0.0387241 + 0.999250i $$0.487671\pi$$
$$234$$ 0 0
$$235$$ 1794.12 0.498024
$$236$$ −5314.98 −1.46600
$$237$$ 0 0
$$238$$ −5589.22 −1.52225
$$239$$ 1529.39 0.413925 0.206963 0.978349i $$-0.433642\pi$$
0.206963 + 0.978349i $$0.433642\pi$$
$$240$$ 0 0
$$241$$ 975.526 0.260743 0.130372 0.991465i $$-0.458383\pi$$
0.130372 + 0.991465i $$0.458383\pi$$
$$242$$ 1273.79 0.338357
$$243$$ 0 0
$$244$$ −4871.74 −1.27820
$$245$$ −8661.38 −2.25859
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 1271.03 0.325446
$$249$$ 0 0
$$250$$ 3841.06 0.971720
$$251$$ 1874.14 0.471294 0.235647 0.971839i $$-0.424279\pi$$
0.235647 + 0.971839i $$0.424279\pi$$
$$252$$ 0 0
$$253$$ −769.832 −0.191300
$$254$$ 1019.04 0.251734
$$255$$ 0 0
$$256$$ 2889.00 0.705322
$$257$$ 1818.19 0.441305 0.220653 0.975352i $$-0.429181\pi$$
0.220653 + 0.975352i $$0.429181\pi$$
$$258$$ 0 0
$$259$$ −1367.22 −0.328010
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −1947.11 −0.459132
$$263$$ −673.799 −0.157978 −0.0789890 0.996875i $$-0.525169\pi$$
−0.0789890 + 0.996875i $$0.525169\pi$$
$$264$$ 0 0
$$265$$ −5893.86 −1.36625
$$266$$ 3495.14 0.805643
$$267$$ 0 0
$$268$$ −2073.11 −0.472520
$$269$$ 3356.40 0.760756 0.380378 0.924831i $$-0.375794\pi$$
0.380378 + 0.924831i $$0.375794\pi$$
$$270$$ 0 0
$$271$$ 8915.55 1.99845 0.999227 0.0393133i $$-0.0125170\pi$$
0.999227 + 0.0393133i $$0.0125170\pi$$
$$272$$ −2372.78 −0.528937
$$273$$ 0 0
$$274$$ 7548.16 1.66424
$$275$$ −2255.52 −0.494593
$$276$$ 0 0
$$277$$ 4017.31 0.871396 0.435698 0.900093i $$-0.356502\pi$$
0.435698 + 0.900093i $$0.356502\pi$$
$$278$$ −412.311 −0.0889523
$$279$$ 0 0
$$280$$ 1741.54 0.371702
$$281$$ −1841.12 −0.390860 −0.195430 0.980718i $$-0.562610\pi$$
−0.195430 + 0.980718i $$0.562610\pi$$
$$282$$ 0 0
$$283$$ 4849.40 1.01861 0.509305 0.860586i $$-0.329902\pi$$
0.509305 + 0.860586i $$0.329902\pi$$
$$284$$ −4044.71 −0.845103
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1503.17 0.309162
$$288$$ 0 0
$$289$$ −3051.82 −0.621172
$$290$$ −8541.52 −1.72957
$$291$$ 0 0
$$292$$ −3507.40 −0.702929
$$293$$ 1413.85 0.281905 0.140953 0.990016i $$-0.454983\pi$$
0.140953 + 0.990016i $$0.454983\pi$$
$$294$$ 0 0
$$295$$ 7938.43 1.56676
$$296$$ 179.403 0.0352283
$$297$$ 0 0
$$298$$ −616.639 −0.119869
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −10750.1 −2.05856
$$302$$ −3298.62 −0.628523
$$303$$ 0 0
$$304$$ 1483.79 0.279937
$$305$$ 7276.41 1.36605
$$306$$ 0 0
$$307$$ −4625.64 −0.859932 −0.429966 0.902845i $$-0.641474\pi$$
−0.429966 + 0.902845i $$0.641474\pi$$
$$308$$ 11452.2 2.11867
$$309$$ 0 0
$$310$$ −17085.7 −3.13032
$$311$$ 6060.79 1.10507 0.552534 0.833490i $$-0.313661\pi$$
0.552534 + 0.833490i $$0.313661\pi$$
$$312$$ 0 0
$$313$$ 969.946 0.175158 0.0875792 0.996158i $$-0.472087\pi$$
0.0875792 + 0.996158i $$0.472087\pi$$
$$314$$ −11157.8 −2.00532
$$315$$ 0 0
$$316$$ −8080.38 −1.43847
$$317$$ −8741.63 −1.54883 −0.774414 0.632679i $$-0.781955\pi$$
−0.774414 + 0.632679i $$0.781955\pi$$
$$318$$ 0 0
$$319$$ −6240.92 −1.09537
$$320$$ 8482.13 1.48177
$$321$$ 0 0
$$322$$ −2462.85 −0.426241
$$323$$ −1163.87 −0.200493
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 15165.8 2.57655
$$327$$ 0 0
$$328$$ −197.243 −0.0332040
$$329$$ 4193.81 0.702772
$$330$$ 0 0
$$331$$ 6987.76 1.16037 0.580184 0.814485i $$-0.302981\pi$$
0.580184 + 0.814485i $$0.302981\pi$$
$$332$$ 11702.2 1.93446
$$333$$ 0 0
$$334$$ 13289.2 2.17711
$$335$$ 3096.39 0.504996
$$336$$ 0 0
$$337$$ −4156.59 −0.671881 −0.335940 0.941883i $$-0.609054\pi$$
−0.335940 + 0.941883i $$0.609054\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −5219.30 −0.832519
$$341$$ −12483.8 −1.98250
$$342$$ 0 0
$$343$$ −9468.49 −1.49053
$$344$$ 1410.60 0.221089
$$345$$ 0 0
$$346$$ 11089.3 1.72301
$$347$$ 312.513 0.0483475 0.0241737 0.999708i $$-0.492305\pi$$
0.0241737 + 0.999708i $$0.492305\pi$$
$$348$$ 0 0
$$349$$ −4458.75 −0.683872 −0.341936 0.939723i $$-0.611083\pi$$
−0.341936 + 0.939723i $$0.611083\pi$$
$$350$$ −7215.88 −1.10201
$$351$$ 0 0
$$352$$ 10519.1 1.59281
$$353$$ 2249.17 0.339126 0.169563 0.985519i $$-0.445764\pi$$
0.169563 + 0.985519i $$0.445764\pi$$
$$354$$ 0 0
$$355$$ 6041.16 0.903187
$$356$$ 8325.41 1.23945
$$357$$ 0 0
$$358$$ 6283.78 0.927677
$$359$$ 7842.79 1.15300 0.576499 0.817098i $$-0.304418\pi$$
0.576499 + 0.817098i $$0.304418\pi$$
$$360$$ 0 0
$$361$$ −6131.19 −0.893890
$$362$$ 1966.23 0.285478
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5238.64 0.751241
$$366$$ 0 0
$$367$$ 6660.24 0.947307 0.473653 0.880711i $$-0.342935\pi$$
0.473653 + 0.880711i $$0.342935\pi$$
$$368$$ −1045.55 −0.148106
$$369$$ 0 0
$$370$$ −2411.60 −0.338846
$$371$$ −13777.1 −1.92795
$$372$$ 0 0
$$373$$ −36.9873 −0.00513439 −0.00256720 0.999997i $$-0.500817\pi$$
−0.00256720 + 0.999997i $$0.500817\pi$$
$$374$$ −7203.32 −0.995923
$$375$$ 0 0
$$376$$ −550.301 −0.0754777
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12079.9 1.63721 0.818603 0.574360i $$-0.194749\pi$$
0.818603 + 0.574360i $$0.194749\pi$$
$$380$$ 3263.82 0.440607
$$381$$ 0 0
$$382$$ 5647.59 0.756429
$$383$$ 10567.4 1.40984 0.704919 0.709287i $$-0.250983\pi$$
0.704919 + 0.709287i $$0.250983\pi$$
$$384$$ 0 0
$$385$$ −17104.9 −2.26428
$$386$$ −8842.90 −1.16604
$$387$$ 0 0
$$388$$ −14044.4 −1.83763
$$389$$ 9757.49 1.27179 0.635893 0.771778i $$-0.280632\pi$$
0.635893 + 0.771778i $$0.280632\pi$$
$$390$$ 0 0
$$391$$ 820.119 0.106075
$$392$$ 2656.66 0.342300
$$393$$ 0 0
$$394$$ 988.469 0.126392
$$395$$ 12068.8 1.53734
$$396$$ 0 0
$$397$$ −14200.5 −1.79522 −0.897612 0.440786i $$-0.854700\pi$$
−0.897612 + 0.440786i $$0.854700\pi$$
$$398$$ −6555.48 −0.825620
$$399$$ 0 0
$$400$$ −3063.34 −0.382918
$$401$$ 12676.4 1.57863 0.789313 0.613992i $$-0.210437\pi$$
0.789313 + 0.613992i $$0.210437\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −8629.56 −1.06271
$$405$$ 0 0
$$406$$ −19966.0 −2.44063
$$407$$ −1762.05 −0.214599
$$408$$ 0 0
$$409$$ 1533.68 0.185417 0.0927083 0.995693i $$-0.470448\pi$$
0.0927083 + 0.995693i $$0.470448\pi$$
$$410$$ 2651.41 0.319375
$$411$$ 0 0
$$412$$ 5716.38 0.683557
$$413$$ 18556.3 2.21089
$$414$$ 0 0
$$415$$ −17478.3 −2.06741
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 4504.50 0.527087
$$419$$ 2165.89 0.252532 0.126266 0.991996i $$-0.459701\pi$$
0.126266 + 0.991996i $$0.459701\pi$$
$$420$$ 0 0
$$421$$ 734.575 0.0850380 0.0425190 0.999096i $$-0.486462\pi$$
0.0425190 + 0.999096i $$0.486462\pi$$
$$422$$ −7721.98 −0.890758
$$423$$ 0 0
$$424$$ 1807.79 0.207062
$$425$$ 2402.85 0.274248
$$426$$ 0 0
$$427$$ 17008.8 1.92767
$$428$$ 13034.9 1.47212
$$429$$ 0 0
$$430$$ −18961.8 −2.12656
$$431$$ 13709.3 1.53215 0.766073 0.642754i $$-0.222208\pi$$
0.766073 + 0.642754i $$0.222208\pi$$
$$432$$ 0 0
$$433$$ 10049.9 1.11540 0.557701 0.830042i $$-0.311684\pi$$
0.557701 + 0.830042i $$0.311684\pi$$
$$434$$ −39938.2 −4.41727
$$435$$ 0 0
$$436$$ −2979.76 −0.327304
$$437$$ −512.850 −0.0561395
$$438$$ 0 0
$$439$$ −8133.47 −0.884258 −0.442129 0.896951i $$-0.645777\pi$$
−0.442129 + 0.896951i $$0.645777\pi$$
$$440$$ 2244.47 0.243184
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2370.78 0.254264 0.127132 0.991886i $$-0.459423\pi$$
0.127132 + 0.991886i $$0.459423\pi$$
$$444$$ 0 0
$$445$$ −12434.8 −1.32464
$$446$$ −231.423 −0.0245699
$$447$$ 0 0
$$448$$ 19827.2 2.09095
$$449$$ −12923.2 −1.35832 −0.679158 0.733992i $$-0.737655\pi$$
−0.679158 + 0.733992i $$0.737655\pi$$
$$450$$ 0 0
$$451$$ 1937.27 0.202267
$$452$$ −6256.84 −0.651099
$$453$$ 0 0
$$454$$ 2751.72 0.284459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8401.26 0.859944 0.429972 0.902842i $$-0.358523\pi$$
0.429972 + 0.902842i $$0.358523\pi$$
$$458$$ 2982.24 0.304260
$$459$$ 0 0
$$460$$ −2299.85 −0.233111
$$461$$ −17627.3 −1.78088 −0.890441 0.455098i $$-0.849604\pi$$
−0.890441 + 0.455098i $$0.849604\pi$$
$$462$$ 0 0
$$463$$ −5461.81 −0.548233 −0.274116 0.961697i $$-0.588385\pi$$
−0.274116 + 0.961697i $$0.588385\pi$$
$$464$$ −8476.13 −0.848048
$$465$$ 0 0
$$466$$ 1135.72 0.112899
$$467$$ −8262.19 −0.818691 −0.409345 0.912379i $$-0.634243\pi$$
−0.409345 + 0.912379i $$0.634243\pi$$
$$468$$ 0 0
$$469$$ 7237.89 0.712611
$$470$$ 7397.35 0.725988
$$471$$ 0 0
$$472$$ −2434.91 −0.237449
$$473$$ −13854.6 −1.34680
$$474$$ 0 0
$$475$$ −1502.59 −0.145145
$$476$$ −12200.3 −1.17479
$$477$$ 0 0
$$478$$ 6305.84 0.603395
$$479$$ 1575.87 0.150320 0.0751601 0.997171i $$-0.476053\pi$$
0.0751601 + 0.997171i $$0.476053\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 4022.20 0.380095
$$483$$ 0 0
$$484$$ 2780.46 0.261125
$$485$$ 20976.7 1.96393
$$486$$ 0 0
$$487$$ −12595.7 −1.17200 −0.586001 0.810310i $$-0.699299\pi$$
−0.586001 + 0.810310i $$0.699299\pi$$
$$488$$ −2231.85 −0.207031
$$489$$ 0 0
$$490$$ −35711.8 −3.29244
$$491$$ 1071.21 0.0984586 0.0492293 0.998788i $$-0.484323\pi$$
0.0492293 + 0.998788i $$0.484323\pi$$
$$492$$ 0 0
$$493$$ 6648.59 0.607378
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −16954.9 −1.53487
$$497$$ 14121.4 1.27451
$$498$$ 0 0
$$499$$ −1422.30 −0.127597 −0.0637985 0.997963i $$-0.520322\pi$$
−0.0637985 + 0.997963i $$0.520322\pi$$
$$500$$ 8384.35 0.749919
$$501$$ 0 0
$$502$$ 7727.28 0.687022
$$503$$ 9349.34 0.828760 0.414380 0.910104i $$-0.363998\pi$$
0.414380 + 0.910104i $$0.363998\pi$$
$$504$$ 0 0
$$505$$ 12889.1 1.13576
$$506$$ −3174.10 −0.278866
$$507$$ 0 0
$$508$$ 2224.39 0.194274
$$509$$ −13736.3 −1.19617 −0.598086 0.801432i $$-0.704072\pi$$
−0.598086 + 0.801432i $$0.704072\pi$$
$$510$$ 0 0
$$511$$ 12245.5 1.06009
$$512$$ 16100.7 1.38976
$$513$$ 0 0
$$514$$ 7496.58 0.643307
$$515$$ −8537.96 −0.730538
$$516$$ 0 0
$$517$$ 5404.93 0.459785
$$518$$ −5637.17 −0.478153
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11052.3 −0.929386 −0.464693 0.885472i $$-0.653835\pi$$
−0.464693 + 0.885472i $$0.653835\pi$$
$$522$$ 0 0
$$523$$ 6477.04 0.541532 0.270766 0.962645i $$-0.412723\pi$$
0.270766 + 0.962645i $$0.412723\pi$$
$$524$$ −4250.19 −0.354332
$$525$$ 0 0
$$526$$ −2778.14 −0.230290
$$527$$ 13299.2 1.09929
$$528$$ 0 0
$$529$$ −11805.6 −0.970298
$$530$$ −24301.0 −1.99164
$$531$$ 0 0
$$532$$ 7629.27 0.621750
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −19468.9 −1.57330
$$536$$ −949.739 −0.0765344
$$537$$ 0 0
$$538$$ 13838.8 1.10898
$$539$$ −26093.1 −2.08517
$$540$$ 0 0
$$541$$ 18341.5 1.45761 0.728803 0.684723i $$-0.240077\pi$$
0.728803 + 0.684723i $$0.240077\pi$$
$$542$$ 36759.7 2.91322
$$543$$ 0 0
$$544$$ −11206.2 −0.883204
$$545$$ 4450.55 0.349799
$$546$$ 0 0
$$547$$ −18943.1 −1.48071 −0.740356 0.672215i $$-0.765343\pi$$
−0.740356 + 0.672215i $$0.765343\pi$$
$$548$$ 16476.3 1.28436
$$549$$ 0 0
$$550$$ −9299.75 −0.720987
$$551$$ −4157.61 −0.321452
$$552$$ 0 0
$$553$$ 28211.2 2.16937
$$554$$ 16563.8 1.27027
$$555$$ 0 0
$$556$$ −900.000 −0.0686484
$$557$$ −415.532 −0.0316098 −0.0158049 0.999875i $$-0.505031\pi$$
−0.0158049 + 0.999875i $$0.505031\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −23231.1 −1.75303
$$561$$ 0 0
$$562$$ −7591.12 −0.569772
$$563$$ −18291.8 −1.36929 −0.684643 0.728879i $$-0.740042\pi$$
−0.684643 + 0.728879i $$0.740042\pi$$
$$564$$ 0 0
$$565$$ 9345.19 0.695850
$$566$$ 19994.6 1.48487
$$567$$ 0 0
$$568$$ −1852.97 −0.136882
$$569$$ −4347.47 −0.320308 −0.160154 0.987092i $$-0.551199\pi$$
−0.160154 + 0.987092i $$0.551199\pi$$
$$570$$ 0 0
$$571$$ −16756.0 −1.22805 −0.614024 0.789288i $$-0.710450\pi$$
−0.614024 + 0.789288i $$0.710450\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 6197.74 0.450677
$$575$$ 1058.80 0.0767915
$$576$$ 0 0
$$577$$ 19974.7 1.44117 0.720587 0.693364i $$-0.243872\pi$$
0.720587 + 0.693364i $$0.243872\pi$$
$$578$$ −12583.0 −0.905506
$$579$$ 0 0
$$580$$ −18644.6 −1.33478
$$581$$ −40856.0 −2.91737
$$582$$ 0 0
$$583$$ −17755.7 −1.26135
$$584$$ −1606.82 −0.113854
$$585$$ 0 0
$$586$$ 5829.47 0.410944
$$587$$ 15748.7 1.10735 0.553677 0.832732i $$-0.313224\pi$$
0.553677 + 0.832732i $$0.313224\pi$$
$$588$$ 0 0
$$589$$ −8316.50 −0.581792
$$590$$ 32731.0 2.28392
$$591$$ 0 0
$$592$$ −2393.14 −0.166144
$$593$$ −13318.4 −0.922297 −0.461148 0.887323i $$-0.652562\pi$$
−0.461148 + 0.887323i $$0.652562\pi$$
$$594$$ 0 0
$$595$$ 18222.3 1.25553
$$596$$ −1346.01 −0.0925080
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2970.80 −0.202644 −0.101322 0.994854i $$-0.532307\pi$$
−0.101322 + 0.994854i $$0.532307\pi$$
$$600$$ 0 0
$$601$$ 10632.6 0.721654 0.360827 0.932633i $$-0.382495\pi$$
0.360827 + 0.932633i $$0.382495\pi$$
$$602$$ −44323.8 −3.00083
$$603$$ 0 0
$$604$$ −7200.29 −0.485059
$$605$$ −4152.88 −0.279072
$$606$$ 0 0
$$607$$ 11587.9 0.774856 0.387428 0.921900i $$-0.373364\pi$$
0.387428 + 0.921900i $$0.373364\pi$$
$$608$$ 7007.67 0.467432
$$609$$ 0 0
$$610$$ 30001.4 1.99135
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20792.3 1.36998 0.684988 0.728555i $$-0.259808\pi$$
0.684988 + 0.728555i $$0.259808\pi$$
$$614$$ −19072.0 −1.25356
$$615$$ 0 0
$$616$$ 5246.51 0.343162
$$617$$ 1562.78 0.101969 0.0509846 0.998699i $$-0.483764\pi$$
0.0509846 + 0.998699i $$0.483764\pi$$
$$618$$ 0 0
$$619$$ 758.406 0.0492454 0.0246227 0.999697i $$-0.492162\pi$$
0.0246227 + 0.999697i $$0.492162\pi$$
$$620$$ −37294.9 −2.41581
$$621$$ 0 0
$$622$$ 24989.3 1.61090
$$623$$ −29066.7 −1.86923
$$624$$ 0 0
$$625$$ −19485.0 −1.24704
$$626$$ 3999.19 0.255335
$$627$$ 0 0
$$628$$ −24355.4 −1.54759
$$629$$ 1877.15 0.118994
$$630$$ 0 0
$$631$$ −14265.2 −0.899981 −0.449990 0.893033i $$-0.648573\pi$$
−0.449990 + 0.893033i $$0.648573\pi$$
$$632$$ −3701.81 −0.232990
$$633$$ 0 0
$$634$$ −36042.7 −2.25779
$$635$$ −3322.34 −0.207627
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −25732.0 −1.59677
$$639$$ 0 0
$$640$$ 7038.88 0.434744
$$641$$ −3985.64 −0.245590 −0.122795 0.992432i $$-0.539186\pi$$
−0.122795 + 0.992432i $$0.539186\pi$$
$$642$$ 0 0
$$643$$ −8156.55 −0.500254 −0.250127 0.968213i $$-0.580472\pi$$
−0.250127 + 0.968213i $$0.580472\pi$$
$$644$$ −5375.97 −0.328949
$$645$$ 0 0
$$646$$ −4798.74 −0.292266
$$647$$ −11279.2 −0.685368 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$648$$ 0 0
$$649$$ 23915.2 1.44646
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 33104.2 1.98844
$$653$$ −6565.75 −0.393473 −0.196736 0.980456i $$-0.563034\pi$$
−0.196736 + 0.980456i $$0.563034\pi$$
$$654$$ 0 0
$$655$$ 6348.06 0.378686
$$656$$ 2631.11 0.156597
$$657$$ 0 0
$$658$$ 17291.5 1.02446
$$659$$ −4799.35 −0.283696 −0.141848 0.989888i $$-0.545304\pi$$
−0.141848 + 0.989888i $$0.545304\pi$$
$$660$$ 0 0
$$661$$ 15593.6 0.917581 0.458790 0.888545i $$-0.348283\pi$$
0.458790 + 0.888545i $$0.348283\pi$$
$$662$$ 28811.3 1.69151
$$663$$ 0 0
$$664$$ 5361.03 0.313326
$$665$$ −11395.1 −0.664483
$$666$$ 0 0
$$667$$ 2929.66 0.170070
$$668$$ 29008.0 1.68017
$$669$$ 0 0
$$670$$ 12766.7 0.736152
$$671$$ 21920.8 1.26116
$$672$$ 0 0
$$673$$ −2205.54 −0.126326 −0.0631630 0.998003i $$-0.520119\pi$$
−0.0631630 + 0.998003i $$0.520119\pi$$
$$674$$ −17138.1 −0.979426
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15046.4 0.854182 0.427091 0.904209i $$-0.359538\pi$$
0.427091 + 0.904209i $$0.359538\pi$$
$$678$$ 0 0
$$679$$ 49033.6 2.77134
$$680$$ −2391.08 −0.134844
$$681$$ 0 0
$$682$$ −51471.9 −2.88997
$$683$$ −30632.5 −1.71614 −0.858068 0.513537i $$-0.828335\pi$$
−0.858068 + 0.513537i $$0.828335\pi$$
$$684$$ 0 0
$$685$$ −24608.9 −1.37264
$$686$$ −39039.6 −2.17280
$$687$$ 0 0
$$688$$ −18816.7 −1.04270
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2175.72 0.119780 0.0598901 0.998205i $$-0.480925\pi$$
0.0598901 + 0.998205i $$0.480925\pi$$
$$692$$ 24205.9 1.32972
$$693$$ 0 0
$$694$$ 1288.52 0.0704780
$$695$$ 1344.24 0.0733666
$$696$$ 0 0
$$697$$ −2063.82 −0.112156
$$698$$ −18383.9 −0.996906
$$699$$ 0 0
$$700$$ −15751.0 −0.850473
$$701$$ 32718.2 1.76284 0.881419 0.472335i $$-0.156589\pi$$
0.881419 + 0.472335i $$0.156589\pi$$
$$702$$ 0 0
$$703$$ −1173.85 −0.0629768
$$704$$ 25553.1 1.36799
$$705$$ 0 0
$$706$$ 9273.58 0.494356
$$707$$ 30128.6 1.60269
$$708$$ 0 0
$$709$$ −25219.7 −1.33589 −0.667945 0.744211i $$-0.732826\pi$$
−0.667945 + 0.744211i $$0.732826\pi$$
$$710$$ 24908.3 1.31661
$$711$$ 0 0
$$712$$ 3814.06 0.200756
$$713$$ 5860.22 0.307808
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 13716.4 0.715929
$$717$$ 0 0
$$718$$ 32336.6 1.68077
$$719$$ 35466.2 1.83959 0.919796 0.392398i $$-0.128354\pi$$
0.919796 + 0.392398i $$0.128354\pi$$
$$720$$ 0 0
$$721$$ −19957.7 −1.03088
$$722$$ −25279.5 −1.30306
$$723$$ 0 0
$$724$$ 4291.93 0.220315
$$725$$ 8583.57 0.439704
$$726$$ 0 0
$$727$$ −14262.2 −0.727588 −0.363794 0.931479i $$-0.618519\pi$$
−0.363794 + 0.931479i $$0.618519\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 21599.5 1.09511
$$731$$ 14759.6 0.746790
$$732$$ 0 0
$$733$$ 16022.5 0.807371 0.403685 0.914898i $$-0.367729\pi$$
0.403685 + 0.914898i $$0.367729\pi$$
$$734$$ 27460.9 1.38093
$$735$$ 0 0
$$736$$ −4937.96 −0.247304
$$737$$ 9328.11 0.466222
$$738$$ 0 0
$$739$$ 3796.21 0.188966 0.0944830 0.995526i $$-0.469880\pi$$
0.0944830 + 0.995526i $$0.469880\pi$$
$$740$$ −5264.09 −0.261502
$$741$$ 0 0
$$742$$ −56804.3 −2.81044
$$743$$ 30329.3 1.49754 0.748772 0.662827i $$-0.230644\pi$$
0.748772 + 0.662827i $$0.230644\pi$$
$$744$$ 0 0
$$745$$ 2010.40 0.0988661
$$746$$ −152.502 −0.00748460
$$747$$ 0 0
$$748$$ −15723.6 −0.768597
$$749$$ −45509.1 −2.22011
$$750$$ 0 0
$$751$$ 21551.9 1.04719 0.523595 0.851967i $$-0.324591\pi$$
0.523595 + 0.851967i $$0.324591\pi$$
$$752$$ 7340.72 0.355969
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 10754.3 0.518397
$$756$$ 0 0
$$757$$ −20417.6 −0.980306 −0.490153 0.871637i $$-0.663059\pi$$
−0.490153 + 0.871637i $$0.663059\pi$$
$$758$$ 49806.5 2.38662
$$759$$ 0 0
$$760$$ 1495.23 0.0713655
$$761$$ 31375.4 1.49456 0.747278 0.664512i $$-0.231360\pi$$
0.747278 + 0.664512i $$0.231360\pi$$
$$762$$ 0 0
$$763$$ 10403.3 0.493609
$$764$$ 12327.7 0.583770
$$765$$ 0 0
$$766$$ 43570.5 2.05518
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 12452.7 0.583946 0.291973 0.956427i $$-0.405688\pi$$
0.291973 + 0.956427i $$0.405688\pi$$
$$770$$ −70525.5 −3.30073
$$771$$ 0 0
$$772$$ −19302.5 −0.899885
$$773$$ 37449.8 1.74253 0.871264 0.490815i $$-0.163301\pi$$
0.871264 + 0.490815i $$0.163301\pi$$
$$774$$ 0 0
$$775$$ 17169.8 0.795815
$$776$$ −6434.08 −0.297642
$$777$$ 0 0
$$778$$ 40231.2 1.85393
$$779$$ 1290.58 0.0593580
$$780$$ 0 0
$$781$$ 18199.5 0.833839
$$782$$ 3381.44 0.154629
$$783$$ 0 0
$$784$$ −35438.4 −1.61436
$$785$$ 36377.1 1.65396
$$786$$ 0 0
$$787$$ −26460.2 −1.19848 −0.599240 0.800570i $$-0.704530\pi$$
−0.599240 + 0.800570i $$0.704530\pi$$
$$788$$ 2157.65 0.0975420
$$789$$ 0 0
$$790$$ 49761.0 2.24104
$$791$$ 21844.6 0.981928
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −58550.3 −2.61697
$$795$$ 0 0
$$796$$ −14309.4 −0.637167
$$797$$ −4749.47 −0.211085 −0.105543 0.994415i $$-0.533658\pi$$
−0.105543 + 0.994415i $$0.533658\pi$$
$$798$$ 0 0
$$799$$ −5757.99 −0.254947
$$800$$ −14467.6 −0.639386
$$801$$ 0 0
$$802$$ 52266.1 2.30122
$$803$$ 15781.8 0.693560
$$804$$ 0 0
$$805$$ 8029.53 0.351557
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −3953.40 −0.172129
$$809$$ 4464.04 0.194002 0.0970009 0.995284i $$-0.469075\pi$$
0.0970009 + 0.995284i $$0.469075\pi$$
$$810$$ 0 0
$$811$$ −20774.6 −0.899499 −0.449749 0.893155i $$-0.648487\pi$$
−0.449749 + 0.893155i $$0.648487\pi$$
$$812$$ −43582.2 −1.88354
$$813$$ 0 0
$$814$$ −7265.13 −0.312829
$$815$$ −49444.3 −2.12510
$$816$$ 0 0
$$817$$ −9229.73 −0.395235
$$818$$ 6323.51 0.270289
$$819$$ 0 0
$$820$$ 5787.55 0.246476
$$821$$ −35769.2 −1.52053 −0.760264 0.649614i $$-0.774930\pi$$
−0.760264 + 0.649614i $$0.774930\pi$$
$$822$$ 0 0
$$823$$ 2945.44 0.124753 0.0623764 0.998053i $$-0.480132\pi$$
0.0623764 + 0.998053i $$0.480132\pi$$
$$824$$ 2618.80 0.110716
$$825$$ 0 0
$$826$$ 76509.6 3.22289
$$827$$ −17878.6 −0.751753 −0.375877 0.926670i $$-0.622658\pi$$
−0.375877 + 0.926670i $$0.622658\pi$$
$$828$$ 0 0
$$829$$ 13423.8 0.562398 0.281199 0.959649i $$-0.409268\pi$$
0.281199 + 0.959649i $$0.409268\pi$$
$$830$$ −72065.0 −3.01375
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 27797.5 1.15621
$$834$$ 0 0
$$835$$ −43326.2 −1.79565
$$836$$ 9832.53 0.406776
$$837$$ 0 0
$$838$$ 8930.21 0.368125
$$839$$ −31542.7 −1.29794 −0.648971 0.760813i $$-0.724800\pi$$
−0.648971 + 0.760813i $$0.724800\pi$$
$$840$$ 0 0
$$841$$ −638.669 −0.0261867
$$842$$ 3028.73 0.123963
$$843$$ 0 0
$$844$$ −16855.7 −0.687437
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −9707.47 −0.393805
$$848$$ −24115.0 −0.976547
$$849$$ 0 0
$$850$$ 9907.22 0.399782
$$851$$ 827.155 0.0333191
$$852$$ 0 0
$$853$$ −21810.6 −0.875476 −0.437738 0.899103i $$-0.644220\pi$$
−0.437738 + 0.899103i $$0.644220\pi$$
$$854$$ 70129.1 2.81003
$$855$$ 0 0
$$856$$ 5971.59 0.238440
$$857$$ −33234.6 −1.32470 −0.662352 0.749193i $$-0.730442\pi$$
−0.662352 + 0.749193i $$0.730442\pi$$
$$858$$ 0 0
$$859$$ 42697.7 1.69596 0.847978 0.530032i $$-0.177820\pi$$
0.847978 + 0.530032i $$0.177820\pi$$
$$860$$ −41390.3 −1.64116
$$861$$ 0 0
$$862$$ 56525.0 2.23347
$$863$$ −27419.2 −1.08153 −0.540765 0.841174i $$-0.681865\pi$$
−0.540765 + 0.841174i $$0.681865\pi$$
$$864$$ 0 0
$$865$$ −36153.8 −1.42112
$$866$$ 41437.0 1.62596
$$867$$ 0 0
$$868$$ −87177.9 −3.40900
$$869$$ 36358.3 1.41930
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −1365.09 −0.0530137
$$873$$ 0 0
$$874$$ −2114.54 −0.0818367
$$875$$ −29272.4 −1.13096
$$876$$ 0 0
$$877$$ 42604.2 1.64041 0.820206 0.572068i $$-0.193859\pi$$
0.820206 + 0.572068i $$0.193859\pi$$
$$878$$ −33535.2 −1.28902
$$879$$ 0 0
$$880$$ −29940.0 −1.14691
$$881$$ 4699.40 0.179712 0.0898562 0.995955i $$-0.471359\pi$$
0.0898562 + 0.995955i $$0.471359\pi$$
$$882$$ 0 0
$$883$$ 22233.5 0.847358 0.423679 0.905812i $$-0.360738\pi$$
0.423679 + 0.905812i $$0.360738\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 9774.98 0.370651
$$887$$ 8852.46 0.335103 0.167552 0.985863i $$-0.446414\pi$$
0.167552 + 0.985863i $$0.446414\pi$$
$$888$$ 0 0
$$889$$ −7766.05 −0.292986
$$890$$ −51270.0 −1.93098
$$891$$ 0 0
$$892$$ −505.155 −0.0189617
$$893$$ 3600.68 0.134930
$$894$$ 0 0
$$895$$ −20486.7 −0.765135
$$896$$ 16453.6 0.613477
$$897$$ 0 0
$$898$$ −53283.7 −1.98007
$$899$$ 47508.0 1.76249
$$900$$ 0 0
$$901$$ 18915.5 0.699410
$$902$$ 7987.58 0.294853
$$903$$ 0 0
$$904$$ −2866.40 −0.105459
$$905$$ −6410.41 −0.235458
$$906$$ 0 0
$$907$$ 37172.4 1.36085 0.680424 0.732818i $$-0.261795\pi$$
0.680424 + 0.732818i $$0.261795\pi$$
$$908$$ 6006.51 0.219530
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38035.1 −1.38327 −0.691635 0.722247i $$-0.743109\pi$$
−0.691635 + 0.722247i $$0.743109\pi$$
$$912$$ 0 0
$$913$$ −52654.8 −1.90867
$$914$$ 34639.3 1.25357
$$915$$ 0 0
$$916$$ 6509.70 0.234810
$$917$$ 14838.8 0.534372
$$918$$ 0 0
$$919$$ −8352.27 −0.299800 −0.149900 0.988701i $$-0.547895\pi$$
−0.149900 + 0.988701i $$0.547895\pi$$
$$920$$ −1053.62 −0.0377573
$$921$$ 0 0
$$922$$ −72679.4 −2.59606
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2423.47 0.0861440
$$926$$ −22519.6 −0.799179
$$927$$ 0 0
$$928$$ −40031.3 −1.41605
$$929$$ −20232.1 −0.714524 −0.357262 0.934004i $$-0.616290\pi$$
−0.357262 + 0.934004i $$0.616290\pi$$
$$930$$ 0 0
$$931$$ −17382.8 −0.611921
$$932$$ 2479.06 0.0871292
$$933$$ 0 0
$$934$$ −34065.9 −1.19344
$$935$$ 23484.7 0.821423
$$936$$ 0 0
$$937$$ −27766.9 −0.968095 −0.484048 0.875042i $$-0.660834\pi$$
−0.484048 + 0.875042i $$0.660834\pi$$
$$938$$ 29842.6 1.03880
$$939$$ 0 0
$$940$$ 16147.1 0.560277
$$941$$ 400.765 0.0138837 0.00694185 0.999976i $$-0.497790\pi$$
0.00694185 + 0.999976i $$0.497790\pi$$
$$942$$ 0 0
$$943$$ −909.408 −0.0314045
$$944$$ 32480.4 1.11986
$$945$$ 0 0
$$946$$ −57124.0 −1.96328
$$947$$ −21804.4 −0.748201 −0.374101 0.927388i $$-0.622049\pi$$
−0.374101 + 0.927388i $$0.622049\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −6195.35 −0.211583
$$951$$ 0 0
$$952$$ −5589.22 −0.190281
$$953$$ −30480.4 −1.03605 −0.518026 0.855365i $$-0.673333\pi$$
−0.518026 + 0.855365i $$0.673333\pi$$
$$954$$ 0 0
$$955$$ −18412.6 −0.623892
$$956$$ 13764.5 0.465666
$$957$$ 0 0
$$958$$ 6497.48 0.219127
$$959$$ −57524.0 −1.93696
$$960$$ 0 0
$$961$$ 65239.6 2.18991
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 8779.73 0.293336
$$965$$ 28830.1 0.961734
$$966$$ 0 0
$$967$$ 23864.1 0.793608 0.396804 0.917903i $$-0.370119\pi$$
0.396804 + 0.917903i $$0.370119\pi$$
$$968$$ 1273.79 0.0422947
$$969$$ 0 0
$$970$$ 86489.2 2.86289
$$971$$ 14010.3 0.463040 0.231520 0.972830i $$-0.425630\pi$$
0.231520 + 0.972830i $$0.425630\pi$$
$$972$$ 0 0
$$973$$ 3142.19 0.103529
$$974$$ −51933.3 −1.70847
$$975$$ 0 0
$$976$$ 29771.7 0.976404
$$977$$ 25448.2 0.833326 0.416663 0.909061i $$-0.363200\pi$$
0.416663 + 0.909061i $$0.363200\pi$$
$$978$$ 0 0
$$979$$ −37460.8 −1.22293
$$980$$ −77952.4 −2.54092
$$981$$ 0 0
$$982$$ 4416.72 0.143527
$$983$$ −52479.9 −1.70280 −0.851399 0.524519i $$-0.824245\pi$$
−0.851399 + 0.524519i $$0.824245\pi$$
$$984$$ 0 0
$$985$$ −3222.66 −0.104246
$$986$$ 27412.8 0.885398
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 6503.73 0.209107
$$990$$ 0 0
$$991$$ 38048.5 1.21963 0.609814 0.792545i $$-0.291244\pi$$
0.609814 + 0.792545i $$0.291244\pi$$
$$992$$ −80075.0 −2.56289
$$993$$ 0 0
$$994$$ 58223.9 1.85790
$$995$$ 21372.5 0.680960
$$996$$ 0 0
$$997$$ −31236.6 −0.992251 −0.496125 0.868251i $$-0.665244\pi$$
−0.496125 + 0.868251i $$0.665244\pi$$
$$998$$ −5864.30 −0.186003
$$999$$ 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.z.1.3 4
3.2 odd 2 507.4.a.k.1.2 4
13.2 odd 12 117.4.q.d.82.2 4
13.7 odd 12 117.4.q.d.10.2 4
13.12 even 2 inner 1521.4.a.z.1.2 4
39.2 even 12 39.4.j.b.4.1 4
39.5 even 4 507.4.b.e.337.3 4
39.8 even 4 507.4.b.e.337.2 4
39.20 even 12 39.4.j.b.10.1 yes 4
39.38 odd 2 507.4.a.k.1.3 4
156.59 odd 12 624.4.bv.c.49.2 4
156.119 odd 12 624.4.bv.c.433.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.1 4 39.2 even 12
39.4.j.b.10.1 yes 4 39.20 even 12
117.4.q.d.10.2 4 13.7 odd 12
117.4.q.d.82.2 4 13.2 odd 12
507.4.a.k.1.2 4 3.2 odd 2
507.4.a.k.1.3 4 39.38 odd 2
507.4.b.e.337.2 4 39.8 even 4
507.4.b.e.337.3 4 39.5 even 4
624.4.bv.c.49.2 4 156.59 odd 12
624.4.bv.c.433.1 4 156.119 odd 12
1521.4.a.z.1.2 4 13.12 even 2 inner
1521.4.a.z.1.3 4 1.1 even 1 trivial