# Properties

 Label 1521.4.a.u.1.3 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} +O(q^{10})$$ $$q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} -48.1187 q^{10} +25.8785 q^{11} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -154.712 q^{19} -111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} +109.348 q^{28} +20.4522 q^{29} -266.424 q^{31} -256.984 q^{32} +85.5599 q^{34} -128.661 q^{35} -115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +759.390 q^{46} -467.365 q^{47} -216.341 q^{49} +23.9714 q^{50} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +609.204 q^{67} +197.505 q^{68} -541.542 q^{70} +248.038 q^{71} -852.765 q^{73} -488.181 q^{74} -1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} +541.043 q^{80} +1646.51 q^{82} -435.432 q^{83} -232.389 q^{85} +637.281 q^{86} +186.924 q^{88} +259.233 q^{89} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} -1225.17 q^{97} -910.589 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 $$3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 - 4 * q^10 - 16 * q^11 + 176 * q^14 - 110 * q^16 + 146 * q^17 - 94 * q^19 - 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 80 * q^28 + 2 * q^29 - 302 * q^31 + 154 * q^32 - 164 * q^34 - 80 * q^35 - 374 * q^37 - 312 * q^38 - 516 * q^40 + 480 * q^41 - 260 * q^43 + 712 * q^44 + 1104 * q^46 - 24 * q^47 + 447 * q^49 + 814 * q^50 + 678 * q^53 - 1552 * q^55 - 96 * q^56 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 74 * q^67 + 460 * q^68 - 1216 * q^70 - 948 * q^71 + 222 * q^73 - 1724 * q^74 - 2392 * q^76 - 112 * q^77 - 24 * q^79 + 1100 * q^80 + 564 * q^82 - 796 * q^83 + 248 * q^85 + 1800 * q^86 + 1608 * q^88 + 1436 * q^89 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 - 3242 * q^97 - 5070 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.20905 1.48812 0.744062 0.668111i $$-0.232897\pi$$
0.744062 + 0.668111i $$0.232897\pi$$
$$3$$ 0 0
$$4$$ 9.71610 1.21451
$$5$$ −11.4322 −1.02253 −0.511264 0.859424i $$-0.670822\pi$$
−0.511264 + 0.859424i $$0.670822\pi$$
$$6$$ 0 0
$$7$$ 11.2543 0.607675 0.303838 0.952724i $$-0.401732\pi$$
0.303838 + 0.952724i $$0.401732\pi$$
$$8$$ 7.22315 0.319221
$$9$$ 0 0
$$10$$ −48.1187 −1.52165
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 47.3699 0.904296
$$15$$ 0 0
$$16$$ −47.3262 −0.739472
$$17$$ 20.3276 0.290010 0.145005 0.989431i $$-0.453680\pi$$
0.145005 + 0.989431i $$0.453680\pi$$
$$18$$ 0 0
$$19$$ −154.712 −1.86807 −0.934035 0.357181i $$-0.883738\pi$$
−0.934035 + 0.357181i $$0.883738\pi$$
$$20$$ −111.076 −1.24187
$$21$$ 0 0
$$22$$ 108.924 1.05558
$$23$$ 180.418 1.63565 0.817823 0.575471i $$-0.195181\pi$$
0.817823 + 0.575471i $$0.195181\pi$$
$$24$$ 0 0
$$25$$ 5.69520 0.0455616
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 109.348 0.738029
$$29$$ 20.4522 0.130961 0.0654806 0.997854i $$-0.479142\pi$$
0.0654806 + 0.997854i $$0.479142\pi$$
$$30$$ 0 0
$$31$$ −266.424 −1.54359 −0.771794 0.635873i $$-0.780640\pi$$
−0.771794 + 0.635873i $$0.780640\pi$$
$$32$$ −256.984 −1.41965
$$33$$ 0 0
$$34$$ 85.5599 0.431571
$$35$$ −128.661 −0.621364
$$36$$ 0 0
$$37$$ −115.984 −0.515340 −0.257670 0.966233i $$-0.582955\pi$$
−0.257670 + 0.966233i $$0.582955\pi$$
$$38$$ −651.190 −2.77992
$$39$$ 0 0
$$40$$ −82.5765 −0.326412
$$41$$ 391.184 1.49006 0.745032 0.667029i $$-0.232434\pi$$
0.745032 + 0.667029i $$0.232434\pi$$
$$42$$ 0 0
$$43$$ 151.407 0.536963 0.268482 0.963285i $$-0.413478\pi$$
0.268482 + 0.963285i $$0.413478\pi$$
$$44$$ 251.438 0.861494
$$45$$ 0 0
$$46$$ 759.390 2.43404
$$47$$ −467.365 −1.45047 −0.725236 0.688500i $$-0.758269\pi$$
−0.725236 + 0.688500i $$0.758269\pi$$
$$48$$ 0 0
$$49$$ −216.341 −0.630731
$$50$$ 23.9714 0.0678012
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −79.9842 −0.207296 −0.103648 0.994614i $$-0.533051\pi$$
−0.103648 + 0.994614i $$0.533051\pi$$
$$54$$ 0 0
$$55$$ −295.848 −0.725312
$$56$$ 81.2915 0.193983
$$57$$ 0 0
$$58$$ 86.0843 0.194887
$$59$$ −873.710 −1.92792 −0.963960 0.266045i $$-0.914283\pi$$
−0.963960 + 0.266045i $$0.914283\pi$$
$$60$$ 0 0
$$61$$ −187.068 −0.392649 −0.196325 0.980539i $$-0.562901\pi$$
−0.196325 + 0.980539i $$0.562901\pi$$
$$62$$ −1121.39 −2.29705
$$63$$ 0 0
$$64$$ −703.047 −1.37314
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 609.204 1.11084 0.555418 0.831571i $$-0.312558\pi$$
0.555418 + 0.831571i $$0.312558\pi$$
$$68$$ 197.505 0.352221
$$69$$ 0 0
$$70$$ −541.542 −0.924667
$$71$$ 248.038 0.414601 0.207301 0.978277i $$-0.433532\pi$$
0.207301 + 0.978277i $$0.433532\pi$$
$$72$$ 0 0
$$73$$ −852.765 −1.36724 −0.683621 0.729838i $$-0.739596\pi$$
−0.683621 + 0.729838i $$0.739596\pi$$
$$74$$ −488.181 −0.766890
$$75$$ 0 0
$$76$$ −1503.20 −2.26880
$$77$$ 291.244 0.431044
$$78$$ 0 0
$$79$$ −331.221 −0.471712 −0.235856 0.971788i $$-0.575789\pi$$
−0.235856 + 0.971788i $$0.575789\pi$$
$$80$$ 541.043 0.756130
$$81$$ 0 0
$$82$$ 1646.51 2.21740
$$83$$ −435.432 −0.575842 −0.287921 0.957654i $$-0.592964\pi$$
−0.287921 + 0.957654i $$0.592964\pi$$
$$84$$ 0 0
$$85$$ −232.389 −0.296543
$$86$$ 637.281 0.799067
$$87$$ 0 0
$$88$$ 186.924 0.226434
$$89$$ 259.233 0.308749 0.154375 0.988012i $$-0.450664\pi$$
0.154375 + 0.988012i $$0.450664\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1752.96 1.98651
$$93$$ 0 0
$$94$$ −1967.16 −2.15848
$$95$$ 1768.70 1.91015
$$96$$ 0 0
$$97$$ −1225.17 −1.28245 −0.641223 0.767355i $$-0.721572\pi$$
−0.641223 + 0.767355i $$0.721572\pi$$
$$98$$ −910.589 −0.938606
$$99$$ 0 0
$$100$$ 55.3351 0.0553351
$$101$$ −645.416 −0.635855 −0.317927 0.948115i $$-0.602987\pi$$
−0.317927 + 0.948115i $$0.602987\pi$$
$$102$$ 0 0
$$103$$ −511.137 −0.488969 −0.244484 0.969653i $$-0.578619\pi$$
−0.244484 + 0.969653i $$0.578619\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −336.657 −0.308482
$$107$$ −608.195 −0.549499 −0.274750 0.961516i $$-0.588595\pi$$
−0.274750 + 0.961516i $$0.588595\pi$$
$$108$$ 0 0
$$109$$ 1300.04 1.14239 0.571197 0.820813i $$-0.306479\pi$$
0.571197 + 0.820813i $$0.306479\pi$$
$$110$$ −1245.24 −1.07935
$$111$$ 0 0
$$112$$ −532.623 −0.449359
$$113$$ −42.1953 −0.0351274 −0.0175637 0.999846i $$-0.505591\pi$$
−0.0175637 + 0.999846i $$0.505591\pi$$
$$114$$ 0 0
$$115$$ −2062.58 −1.67249
$$116$$ 198.716 0.159054
$$117$$ 0 0
$$118$$ −3677.49 −2.86899
$$119$$ 228.773 0.176232
$$120$$ 0 0
$$121$$ −661.303 −0.496847
$$122$$ −787.378 −0.584311
$$123$$ 0 0
$$124$$ −2588.61 −1.87471
$$125$$ 1363.92 0.975939
$$126$$ 0 0
$$127$$ −311.018 −0.217310 −0.108655 0.994080i $$-0.534654\pi$$
−0.108655 + 0.994080i $$0.534654\pi$$
$$128$$ −903.291 −0.623753
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2000.98 −1.33456 −0.667278 0.744809i $$-0.732541\pi$$
−0.667278 + 0.744809i $$0.732541\pi$$
$$132$$ 0 0
$$133$$ −1741.17 −1.13518
$$134$$ 2564.17 1.65306
$$135$$ 0 0
$$136$$ 146.829 0.0925773
$$137$$ 1038.53 0.647644 0.323822 0.946118i $$-0.395032\pi$$
0.323822 + 0.946118i $$0.395032\pi$$
$$138$$ 0 0
$$139$$ −2858.46 −1.74426 −0.872128 0.489277i $$-0.837261\pi$$
−0.872128 + 0.489277i $$0.837261\pi$$
$$140$$ −1250.09 −0.754655
$$141$$ 0 0
$$142$$ 1044.00 0.616978
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −233.814 −0.133911
$$146$$ −3589.33 −2.03462
$$147$$ 0 0
$$148$$ −1126.91 −0.625887
$$149$$ 743.479 0.408780 0.204390 0.978890i $$-0.434479\pi$$
0.204390 + 0.978890i $$0.434479\pi$$
$$150$$ 0 0
$$151$$ −2277.24 −1.22728 −0.613640 0.789586i $$-0.710295\pi$$
−0.613640 + 0.789586i $$0.710295\pi$$
$$152$$ −1117.51 −0.596328
$$153$$ 0 0
$$154$$ 1225.86 0.641447
$$155$$ 3045.82 1.57836
$$156$$ 0 0
$$157$$ 3173.51 1.61321 0.806605 0.591091i $$-0.201303\pi$$
0.806605 + 0.591091i $$0.201303\pi$$
$$158$$ −1394.12 −0.701966
$$159$$ 0 0
$$160$$ 2937.89 1.45163
$$161$$ 2030.48 0.993941
$$162$$ 0 0
$$163$$ 2314.65 1.11225 0.556126 0.831098i $$-0.312287\pi$$
0.556126 + 0.831098i $$0.312287\pi$$
$$164$$ 3800.78 1.80970
$$165$$ 0 0
$$166$$ −1832.76 −0.856925
$$167$$ −2665.65 −1.23517 −0.617587 0.786502i $$-0.711890\pi$$
−0.617587 + 0.786502i $$0.711890\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −978.138 −0.441293
$$171$$ 0 0
$$172$$ 1471.09 0.652148
$$173$$ 165.243 0.0726198 0.0363099 0.999341i $$-0.488440\pi$$
0.0363099 + 0.999341i $$0.488440\pi$$
$$174$$ 0 0
$$175$$ 64.0954 0.0276866
$$176$$ −1224.73 −0.524532
$$177$$ 0 0
$$178$$ 1091.13 0.459457
$$179$$ −712.339 −0.297446 −0.148723 0.988879i $$-0.547516\pi$$
−0.148723 + 0.988879i $$0.547516\pi$$
$$180$$ 0 0
$$181$$ 2206.53 0.906133 0.453066 0.891477i $$-0.350330\pi$$
0.453066 + 0.891477i $$0.350330\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1303.19 0.522132
$$185$$ 1325.95 0.526949
$$186$$ 0 0
$$187$$ 526.048 0.205714
$$188$$ −4540.96 −1.76162
$$189$$ 0 0
$$190$$ 7444.54 2.84254
$$191$$ −1470.64 −0.557129 −0.278565 0.960417i $$-0.589859\pi$$
−0.278565 + 0.960417i $$0.589859\pi$$
$$192$$ 0 0
$$193$$ −369.560 −0.137832 −0.0689158 0.997622i $$-0.521954\pi$$
−0.0689158 + 0.997622i $$0.521954\pi$$
$$194$$ −5156.80 −1.90844
$$195$$ 0 0
$$196$$ −2101.99 −0.766031
$$197$$ −4273.41 −1.54552 −0.772761 0.634697i $$-0.781125\pi$$
−0.772761 + 0.634697i $$0.781125\pi$$
$$198$$ 0 0
$$199$$ 4154.31 1.47985 0.739927 0.672687i $$-0.234860\pi$$
0.739927 + 0.672687i $$0.234860\pi$$
$$200$$ 41.1373 0.0145442
$$201$$ 0 0
$$202$$ −2716.59 −0.946230
$$203$$ 230.175 0.0795819
$$204$$ 0 0
$$205$$ −4472.09 −1.52363
$$206$$ −2151.40 −0.727646
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4003.71 −1.32508
$$210$$ 0 0
$$211$$ 1231.59 0.401830 0.200915 0.979609i $$-0.435608\pi$$
0.200915 + 0.979609i $$0.435608\pi$$
$$212$$ −777.134 −0.251763
$$213$$ 0 0
$$214$$ −2559.92 −0.817723
$$215$$ −1730.92 −0.549059
$$216$$ 0 0
$$217$$ −2998.42 −0.938000
$$218$$ 5471.92 1.70002
$$219$$ 0 0
$$220$$ −2874.49 −0.880901
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 2187.24 0.656809 0.328404 0.944537i $$-0.393489\pi$$
0.328404 + 0.944537i $$0.393489\pi$$
$$224$$ −2892.17 −0.862684
$$225$$ 0 0
$$226$$ −177.602 −0.0522739
$$227$$ 4138.67 1.21010 0.605051 0.796187i $$-0.293153\pi$$
0.605051 + 0.796187i $$0.293153\pi$$
$$228$$ 0 0
$$229$$ 835.354 0.241056 0.120528 0.992710i $$-0.461541\pi$$
0.120528 + 0.992710i $$0.461541\pi$$
$$230$$ −8681.50 −2.48887
$$231$$ 0 0
$$232$$ 147.729 0.0418056
$$233$$ −3685.51 −1.03625 −0.518124 0.855305i $$-0.673370\pi$$
−0.518124 + 0.855305i $$0.673370\pi$$
$$234$$ 0 0
$$235$$ 5343.01 1.48315
$$236$$ −8489.05 −2.34148
$$237$$ 0 0
$$238$$ 962.917 0.262255
$$239$$ 3026.21 0.819034 0.409517 0.912303i $$-0.365697\pi$$
0.409517 + 0.912303i $$0.365697\pi$$
$$240$$ 0 0
$$241$$ −3265.58 −0.872839 −0.436420 0.899743i $$-0.643754\pi$$
−0.436420 + 0.899743i $$0.643754\pi$$
$$242$$ −2783.46 −0.739370
$$243$$ 0 0
$$244$$ −1817.57 −0.476877
$$245$$ 2473.25 0.644940
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −1924.42 −0.492746
$$249$$ 0 0
$$250$$ 5740.79 1.45232
$$251$$ 6363.16 1.60016 0.800078 0.599897i $$-0.204792\pi$$
0.800078 + 0.599897i $$0.204792\pi$$
$$252$$ 0 0
$$253$$ 4668.96 1.16022
$$254$$ −1309.09 −0.323385
$$255$$ 0 0
$$256$$ 1822.38 0.444917
$$257$$ 6085.36 1.47702 0.738511 0.674242i $$-0.235529\pi$$
0.738511 + 0.674242i $$0.235529\pi$$
$$258$$ 0 0
$$259$$ −1305.31 −0.313159
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8422.24 −1.98598
$$263$$ −123.227 −0.0288916 −0.0144458 0.999896i $$-0.504598\pi$$
−0.0144458 + 0.999896i $$0.504598\pi$$
$$264$$ 0 0
$$265$$ 914.395 0.211965
$$266$$ −7328.69 −1.68929
$$267$$ 0 0
$$268$$ 5919.08 1.34913
$$269$$ 1935.79 0.438763 0.219381 0.975639i $$-0.429596\pi$$
0.219381 + 0.975639i $$0.429596\pi$$
$$270$$ 0 0
$$271$$ 4612.69 1.03395 0.516976 0.856000i $$-0.327058\pi$$
0.516976 + 0.856000i $$0.327058\pi$$
$$272$$ −962.028 −0.214454
$$273$$ 0 0
$$274$$ 4371.20 0.963774
$$275$$ 147.383 0.0323183
$$276$$ 0 0
$$277$$ −5834.30 −1.26552 −0.632761 0.774347i $$-0.718078\pi$$
−0.632761 + 0.774347i $$0.718078\pi$$
$$278$$ −12031.4 −2.59567
$$279$$ 0 0
$$280$$ −929.341 −0.198353
$$281$$ 4691.91 0.996071 0.498036 0.867157i $$-0.334055\pi$$
0.498036 + 0.867157i $$0.334055\pi$$
$$282$$ 0 0
$$283$$ 3465.60 0.727945 0.363973 0.931410i $$-0.381420\pi$$
0.363973 + 0.931410i $$0.381420\pi$$
$$284$$ 2409.96 0.503539
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4402.50 0.905475
$$288$$ 0 0
$$289$$ −4499.79 −0.915894
$$290$$ −984.133 −0.199277
$$291$$ 0 0
$$292$$ −8285.55 −1.66053
$$293$$ 2677.31 0.533822 0.266911 0.963721i $$-0.413997\pi$$
0.266911 + 0.963721i $$0.413997\pi$$
$$294$$ 0 0
$$295$$ 9988.43 1.97135
$$296$$ −837.767 −0.164507
$$297$$ 0 0
$$298$$ 3129.34 0.608315
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1703.98 0.326299
$$302$$ −9585.02 −1.82634
$$303$$ 0 0
$$304$$ 7321.93 1.38139
$$305$$ 2138.60 0.401494
$$306$$ 0 0
$$307$$ −471.915 −0.0877316 −0.0438658 0.999037i $$-0.513967\pi$$
−0.0438658 + 0.999037i $$0.513967\pi$$
$$308$$ 2829.76 0.523508
$$309$$ 0 0
$$310$$ 12820.0 2.34880
$$311$$ 1518.52 0.276872 0.138436 0.990371i $$-0.455793\pi$$
0.138436 + 0.990371i $$0.455793\pi$$
$$312$$ 0 0
$$313$$ 4049.86 0.731348 0.365674 0.930743i $$-0.380839\pi$$
0.365674 + 0.930743i $$0.380839\pi$$
$$314$$ 13357.5 2.40066
$$315$$ 0 0
$$316$$ −3218.17 −0.572900
$$317$$ 3253.96 0.576532 0.288266 0.957550i $$-0.406921\pi$$
0.288266 + 0.957550i $$0.406921\pi$$
$$318$$ 0 0
$$319$$ 529.272 0.0928951
$$320$$ 8037.37 1.40407
$$321$$ 0 0
$$322$$ 8546.40 1.47911
$$323$$ −3144.92 −0.541759
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9742.46 1.65517
$$327$$ 0 0
$$328$$ 2825.58 0.475660
$$329$$ −5259.86 −0.881415
$$330$$ 0 0
$$331$$ 3422.45 0.568322 0.284161 0.958777i $$-0.408285\pi$$
0.284161 + 0.958777i $$0.408285\pi$$
$$332$$ −4230.71 −0.699368
$$333$$ 0 0
$$334$$ −11219.8 −1.83809
$$335$$ −6964.54 −1.13586
$$336$$ 0 0
$$337$$ −9301.67 −1.50354 −0.751772 0.659423i $$-0.770801\pi$$
−0.751772 + 0.659423i $$0.770801\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −2257.92 −0.360155
$$341$$ −6894.66 −1.09492
$$342$$ 0 0
$$343$$ −6294.99 −0.990955
$$344$$ 1093.64 0.171410
$$345$$ 0 0
$$346$$ 695.518 0.108067
$$347$$ −216.898 −0.0335554 −0.0167777 0.999859i $$-0.505341\pi$$
−0.0167777 + 0.999859i $$0.505341\pi$$
$$348$$ 0 0
$$349$$ 4809.84 0.737721 0.368861 0.929485i $$-0.379748\pi$$
0.368861 + 0.929485i $$0.379748\pi$$
$$350$$ 269.781 0.0412011
$$351$$ 0 0
$$352$$ −6650.35 −1.00700
$$353$$ −2859.64 −0.431170 −0.215585 0.976485i $$-0.569166\pi$$
−0.215585 + 0.976485i $$0.569166\pi$$
$$354$$ 0 0
$$355$$ −2835.62 −0.423941
$$356$$ 2518.74 0.374980
$$357$$ 0 0
$$358$$ −2998.27 −0.442636
$$359$$ 3686.04 0.541899 0.270949 0.962594i $$-0.412662\pi$$
0.270949 + 0.962594i $$0.412662\pi$$
$$360$$ 0 0
$$361$$ 17076.8 2.48969
$$362$$ 9287.39 1.34844
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9748.98 1.39804
$$366$$ 0 0
$$367$$ −3470.59 −0.493633 −0.246816 0.969062i $$-0.579384\pi$$
−0.246816 + 0.969062i $$0.579384\pi$$
$$368$$ −8538.52 −1.20951
$$369$$ 0 0
$$370$$ 5580.98 0.784166
$$371$$ −900.166 −0.125968
$$372$$ 0 0
$$373$$ −11963.4 −1.66070 −0.830352 0.557240i $$-0.811860\pi$$
−0.830352 + 0.557240i $$0.811860\pi$$
$$374$$ 2214.16 0.306127
$$375$$ 0 0
$$376$$ −3375.85 −0.463021
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −345.604 −0.0468403 −0.0234202 0.999726i $$-0.507456\pi$$
−0.0234202 + 0.999726i $$0.507456\pi$$
$$380$$ 17184.8 2.31990
$$381$$ 0 0
$$382$$ −6189.99 −0.829078
$$383$$ −3386.40 −0.451793 −0.225897 0.974151i $$-0.572531\pi$$
−0.225897 + 0.974151i $$0.572531\pi$$
$$384$$ 0 0
$$385$$ −3329.56 −0.440754
$$386$$ −1555.49 −0.205110
$$387$$ 0 0
$$388$$ −11903.9 −1.55755
$$389$$ 1629.88 0.212438 0.106219 0.994343i $$-0.466126\pi$$
0.106219 + 0.994343i $$0.466126\pi$$
$$390$$ 0 0
$$391$$ 3667.47 0.474353
$$392$$ −1562.66 −0.201343
$$393$$ 0 0
$$394$$ −17987.0 −2.29993
$$395$$ 3786.58 0.482338
$$396$$ 0 0
$$397$$ −7938.94 −1.00364 −0.501819 0.864973i $$-0.667336\pi$$
−0.501819 + 0.864973i $$0.667336\pi$$
$$398$$ 17485.7 2.20221
$$399$$ 0 0
$$400$$ −269.532 −0.0336915
$$401$$ 214.402 0.0267001 0.0133500 0.999911i $$-0.495750\pi$$
0.0133500 + 0.999911i $$0.495750\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −6270.93 −0.772253
$$405$$ 0 0
$$406$$ 968.819 0.118428
$$407$$ −3001.48 −0.365548
$$408$$ 0 0
$$409$$ 4783.73 0.578338 0.289169 0.957278i $$-0.406621\pi$$
0.289169 + 0.957278i $$0.406621\pi$$
$$410$$ −18823.2 −2.26735
$$411$$ 0 0
$$412$$ −4966.25 −0.593859
$$413$$ −9832.99 −1.17155
$$414$$ 0 0
$$415$$ 4977.95 0.588815
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −16851.8 −1.97189
$$419$$ 9903.67 1.15472 0.577358 0.816491i $$-0.304084\pi$$
0.577358 + 0.816491i $$0.304084\pi$$
$$420$$ 0 0
$$421$$ 12120.6 1.40314 0.701572 0.712598i $$-0.252482\pi$$
0.701572 + 0.712598i $$0.252482\pi$$
$$422$$ 5183.82 0.597973
$$423$$ 0 0
$$424$$ −577.738 −0.0661732
$$425$$ 115.770 0.0132133
$$426$$ 0 0
$$427$$ −2105.32 −0.238603
$$428$$ −5909.28 −0.667374
$$429$$ 0 0
$$430$$ −7285.53 −0.817068
$$431$$ −13672.6 −1.52805 −0.764023 0.645189i $$-0.776779\pi$$
−0.764023 + 0.645189i $$0.776779\pi$$
$$432$$ 0 0
$$433$$ 7113.10 0.789455 0.394727 0.918798i $$-0.370839\pi$$
0.394727 + 0.918798i $$0.370839\pi$$
$$434$$ −12620.5 −1.39586
$$435$$ 0 0
$$436$$ 12631.3 1.38745
$$437$$ −27912.9 −3.05550
$$438$$ 0 0
$$439$$ −6022.04 −0.654707 −0.327353 0.944902i $$-0.606157\pi$$
−0.327353 + 0.944902i $$0.606157\pi$$
$$440$$ −2136.96 −0.231535
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12994.4 1.39364 0.696821 0.717245i $$-0.254597\pi$$
0.696821 + 0.717245i $$0.254597\pi$$
$$444$$ 0 0
$$445$$ −2963.61 −0.315704
$$446$$ 9206.20 0.977413
$$447$$ 0 0
$$448$$ −7912.30 −0.834422
$$449$$ 10984.3 1.15452 0.577260 0.816560i $$-0.304122\pi$$
0.577260 + 0.816560i $$0.304122\pi$$
$$450$$ 0 0
$$451$$ 10123.2 1.05695
$$452$$ −409.973 −0.0426627
$$453$$ 0 0
$$454$$ 17419.9 1.80078
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9834.10 −1.00661 −0.503304 0.864109i $$-0.667882\pi$$
−0.503304 + 0.864109i $$0.667882\pi$$
$$458$$ 3516.05 0.358721
$$459$$ 0 0
$$460$$ −20040.2 −2.03126
$$461$$ 3401.42 0.343644 0.171822 0.985128i $$-0.445035\pi$$
0.171822 + 0.985128i $$0.445035\pi$$
$$462$$ 0 0
$$463$$ −1739.42 −0.174596 −0.0872979 0.996182i $$-0.527823\pi$$
−0.0872979 + 0.996182i $$0.527823\pi$$
$$464$$ −967.925 −0.0968422
$$465$$ 0 0
$$466$$ −15512.5 −1.54207
$$467$$ 7958.82 0.788630 0.394315 0.918975i $$-0.370982\pi$$
0.394315 + 0.918975i $$0.370982\pi$$
$$468$$ 0 0
$$469$$ 6856.16 0.675028
$$470$$ 22489.0 2.20711
$$471$$ 0 0
$$472$$ −6310.94 −0.615433
$$473$$ 3918.20 0.380886
$$474$$ 0 0
$$475$$ −881.114 −0.0851122
$$476$$ 2222.78 0.214036
$$477$$ 0 0
$$478$$ 12737.5 1.21882
$$479$$ 8431.98 0.804315 0.402158 0.915570i $$-0.368260\pi$$
0.402158 + 0.915570i $$0.368260\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −13745.0 −1.29889
$$483$$ 0 0
$$484$$ −6425.29 −0.603427
$$485$$ 14006.4 1.31133
$$486$$ 0 0
$$487$$ 11684.7 1.08723 0.543617 0.839334i $$-0.317055\pi$$
0.543617 + 0.839334i $$0.317055\pi$$
$$488$$ −1351.22 −0.125342
$$489$$ 0 0
$$490$$ 10410.0 0.959750
$$491$$ 3954.70 0.363489 0.181745 0.983346i $$-0.441826\pi$$
0.181745 + 0.983346i $$0.441826\pi$$
$$492$$ 0 0
$$493$$ 415.744 0.0379801
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 12608.8 1.14144
$$497$$ 2791.49 0.251943
$$498$$ 0 0
$$499$$ 5690.37 0.510493 0.255246 0.966876i $$-0.417843\pi$$
0.255246 + 0.966876i $$0.417843\pi$$
$$500$$ 13251.9 1.18529
$$501$$ 0 0
$$502$$ 26782.8 2.38123
$$503$$ −10859.1 −0.962595 −0.481298 0.876557i $$-0.659834\pi$$
−0.481298 + 0.876557i $$0.659834\pi$$
$$504$$ 0 0
$$505$$ 7378.53 0.650178
$$506$$ 19651.9 1.72655
$$507$$ 0 0
$$508$$ −3021.88 −0.263926
$$509$$ −18558.6 −1.61610 −0.808049 0.589115i $$-0.799476\pi$$
−0.808049 + 0.589115i $$0.799476\pi$$
$$510$$ 0 0
$$511$$ −9597.27 −0.830838
$$512$$ 14896.8 1.28584
$$513$$ 0 0
$$514$$ 25613.6 2.19799
$$515$$ 5843.42 0.499984
$$516$$ 0 0
$$517$$ −12094.7 −1.02887
$$518$$ −5494.13 −0.466020
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −17297.5 −1.45454 −0.727271 0.686350i $$-0.759212\pi$$
−0.727271 + 0.686350i $$0.759212\pi$$
$$522$$ 0 0
$$523$$ −5016.11 −0.419386 −0.209693 0.977767i $$-0.567247\pi$$
−0.209693 + 0.977767i $$0.567247\pi$$
$$524$$ −19441.8 −1.62084
$$525$$ 0 0
$$526$$ −518.667 −0.0429942
$$527$$ −5415.77 −0.447656
$$528$$ 0 0
$$529$$ 20383.8 1.67533
$$530$$ 3848.73 0.315431
$$531$$ 0 0
$$532$$ −16917.4 −1.37869
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 6953.01 0.561878
$$536$$ 4400.37 0.354603
$$537$$ 0 0
$$538$$ 8147.83 0.652933
$$539$$ −5598.57 −0.447398
$$540$$ 0 0
$$541$$ −17642.3 −1.40204 −0.701018 0.713144i $$-0.747271\pi$$
−0.701018 + 0.713144i $$0.747271\pi$$
$$542$$ 19415.0 1.53865
$$543$$ 0 0
$$544$$ −5223.86 −0.411712
$$545$$ −14862.3 −1.16813
$$546$$ 0 0
$$547$$ −18414.9 −1.43943 −0.719713 0.694271i $$-0.755727\pi$$
−0.719713 + 0.694271i $$0.755727\pi$$
$$548$$ 10090.4 0.786571
$$549$$ 0 0
$$550$$ 620.343 0.0480937
$$551$$ −3164.20 −0.244645
$$552$$ 0 0
$$553$$ −3727.66 −0.286648
$$554$$ −24556.9 −1.88325
$$555$$ 0 0
$$556$$ −27773.1 −2.11842
$$557$$ 8179.15 0.622193 0.311096 0.950378i $$-0.399304\pi$$
0.311096 + 0.950378i $$0.399304\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 6089.05 0.459481
$$561$$ 0 0
$$562$$ 19748.5 1.48228
$$563$$ 1880.07 0.140738 0.0703690 0.997521i $$-0.477582\pi$$
0.0703690 + 0.997521i $$0.477582\pi$$
$$564$$ 0 0
$$565$$ 482.385 0.0359187
$$566$$ 14586.9 1.08327
$$567$$ 0 0
$$568$$ 1791.62 0.132350
$$569$$ −10118.3 −0.745485 −0.372743 0.927935i $$-0.621583\pi$$
−0.372743 + 0.927935i $$0.621583\pi$$
$$570$$ 0 0
$$571$$ 23428.9 1.71711 0.858555 0.512721i $$-0.171362\pi$$
0.858555 + 0.512721i $$0.171362\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 18530.3 1.34746
$$575$$ 1027.52 0.0745225
$$576$$ 0 0
$$577$$ −20508.1 −1.47966 −0.739831 0.672793i $$-0.765094\pi$$
−0.739831 + 0.672793i $$0.765094\pi$$
$$578$$ −18939.8 −1.36296
$$579$$ 0 0
$$580$$ −2271.76 −0.162637
$$581$$ −4900.49 −0.349925
$$582$$ 0 0
$$583$$ −2069.87 −0.147042
$$584$$ −6159.65 −0.436452
$$585$$ 0 0
$$586$$ 11268.9 0.794394
$$587$$ −5968.43 −0.419665 −0.209833 0.977737i $$-0.567292\pi$$
−0.209833 + 0.977737i $$0.567292\pi$$
$$588$$ 0 0
$$589$$ 41219.0 2.88353
$$590$$ 42041.8 2.93361
$$591$$ 0 0
$$592$$ 5489.06 0.381080
$$593$$ −14659.5 −1.01517 −0.507584 0.861602i $$-0.669461\pi$$
−0.507584 + 0.861602i $$0.669461\pi$$
$$594$$ 0 0
$$595$$ −2615.38 −0.180202
$$596$$ 7223.72 0.496468
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −23635.9 −1.61225 −0.806125 0.591746i $$-0.798439\pi$$
−0.806125 + 0.591746i $$0.798439\pi$$
$$600$$ 0 0
$$601$$ −11527.0 −0.782356 −0.391178 0.920315i $$-0.627932\pi$$
−0.391178 + 0.920315i $$0.627932\pi$$
$$602$$ 7172.15 0.485573
$$603$$ 0 0
$$604$$ −22125.9 −1.49055
$$605$$ 7560.15 0.508039
$$606$$ 0 0
$$607$$ −5098.56 −0.340930 −0.170465 0.985364i $$-0.554527\pi$$
−0.170465 + 0.985364i $$0.554527\pi$$
$$608$$ 39758.4 2.65200
$$609$$ 0 0
$$610$$ 9001.47 0.597473
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1516.39 −0.0999128 −0.0499564 0.998751i $$-0.515908\pi$$
−0.0499564 + 0.998751i $$0.515908\pi$$
$$614$$ −1986.31 −0.130556
$$615$$ 0 0
$$616$$ 2103.70 0.137598
$$617$$ 18539.3 1.20966 0.604832 0.796353i $$-0.293240\pi$$
0.604832 + 0.796353i $$0.293240\pi$$
$$618$$ 0 0
$$619$$ −25684.9 −1.66779 −0.833897 0.551920i $$-0.813895\pi$$
−0.833897 + 0.551920i $$0.813895\pi$$
$$620$$ 29593.5 1.91694
$$621$$ 0 0
$$622$$ 6391.51 0.412020
$$623$$ 2917.49 0.187619
$$624$$ 0 0
$$625$$ −16304.5 −1.04349
$$626$$ 17046.1 1.08834
$$627$$ 0 0
$$628$$ 30834.2 1.95926
$$629$$ −2357.67 −0.149454
$$630$$ 0 0
$$631$$ 22410.9 1.41389 0.706945 0.707269i $$-0.250073\pi$$
0.706945 + 0.707269i $$0.250073\pi$$
$$632$$ −2392.46 −0.150580
$$633$$ 0 0
$$634$$ 13696.1 0.857950
$$635$$ 3555.62 0.222206
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 2227.73 0.138239
$$639$$ 0 0
$$640$$ 10326.6 0.637804
$$641$$ −6827.81 −0.420721 −0.210361 0.977624i $$-0.567464\pi$$
−0.210361 + 0.977624i $$0.567464\pi$$
$$642$$ 0 0
$$643$$ 23264.3 1.42684 0.713418 0.700738i $$-0.247146\pi$$
0.713418 + 0.700738i $$0.247146\pi$$
$$644$$ 19728.4 1.20715
$$645$$ 0 0
$$646$$ −13237.1 −0.806204
$$647$$ −14745.9 −0.896014 −0.448007 0.894030i $$-0.647866\pi$$
−0.448007 + 0.894030i $$0.647866\pi$$
$$648$$ 0 0
$$649$$ −22610.3 −1.36754
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 22489.3 1.35084
$$653$$ −10909.0 −0.653755 −0.326878 0.945067i $$-0.605997\pi$$
−0.326878 + 0.945067i $$0.605997\pi$$
$$654$$ 0 0
$$655$$ 22875.7 1.36462
$$656$$ −18513.2 −1.10186
$$657$$ 0 0
$$658$$ −22139.0 −1.31166
$$659$$ 4182.99 0.247263 0.123631 0.992328i $$-0.460546\pi$$
0.123631 + 0.992328i $$0.460546\pi$$
$$660$$ 0 0
$$661$$ −2224.23 −0.130881 −0.0654406 0.997856i $$-0.520845\pi$$
−0.0654406 + 0.997856i $$0.520845\pi$$
$$662$$ 14405.2 0.845734
$$663$$ 0 0
$$664$$ −3145.19 −0.183821
$$665$$ 19905.4 1.16075
$$666$$ 0 0
$$667$$ 3689.95 0.214206
$$668$$ −25899.7 −1.50013
$$669$$ 0 0
$$670$$ −29314.1 −1.69030
$$671$$ −4841.04 −0.278519
$$672$$ 0 0
$$673$$ −24152.5 −1.38337 −0.691687 0.722197i $$-0.743132\pi$$
−0.691687 + 0.722197i $$0.743132\pi$$
$$674$$ −39151.2 −2.23746
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15310.7 0.869187 0.434593 0.900627i $$-0.356892\pi$$
0.434593 + 0.900627i $$0.356892\pi$$
$$678$$ 0 0
$$679$$ −13788.4 −0.779310
$$680$$ −1678.58 −0.0946628
$$681$$ 0 0
$$682$$ −29020.0 −1.62937
$$683$$ 11399.6 0.638646 0.319323 0.947646i $$-0.396545\pi$$
0.319323 + 0.947646i $$0.396545\pi$$
$$684$$ 0 0
$$685$$ −11872.6 −0.662233
$$686$$ −26495.9 −1.47466
$$687$$ 0 0
$$688$$ −7165.54 −0.397069
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 3323.23 0.182955 0.0914773 0.995807i $$-0.470841\pi$$
0.0914773 + 0.995807i $$0.470841\pi$$
$$692$$ 1605.52 0.0881976
$$693$$ 0 0
$$694$$ −912.936 −0.0499345
$$695$$ 32678.5 1.78355
$$696$$ 0 0
$$697$$ 7951.82 0.432133
$$698$$ 20244.8 1.09782
$$699$$ 0 0
$$700$$ 622.758 0.0336257
$$701$$ 12670.4 0.682673 0.341336 0.939941i $$-0.389120\pi$$
0.341336 + 0.939941i $$0.389120\pi$$
$$702$$ 0 0
$$703$$ 17944.0 0.962692
$$704$$ −18193.8 −0.974012
$$705$$ 0 0
$$706$$ −12036.4 −0.641635
$$707$$ −7263.71 −0.386393
$$708$$ 0 0
$$709$$ −13075.2 −0.692594 −0.346297 0.938125i $$-0.612561\pi$$
−0.346297 + 0.938125i $$0.612561\pi$$
$$710$$ −11935.3 −0.630877
$$711$$ 0 0
$$712$$ 1872.48 0.0985592
$$713$$ −48067.8 −2.52476
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6921.16 −0.361251
$$717$$ 0 0
$$718$$ 15514.7 0.806412
$$719$$ 2988.41 0.155005 0.0775026 0.996992i $$-0.475305\pi$$
0.0775026 + 0.996992i $$0.475305\pi$$
$$720$$ 0 0
$$721$$ −5752.48 −0.297134
$$722$$ 71877.0 3.70496
$$723$$ 0 0
$$724$$ 21438.9 1.10051
$$725$$ 116.479 0.00596680
$$726$$ 0 0
$$727$$ −5507.46 −0.280963 −0.140482 0.990083i $$-0.544865\pi$$
−0.140482 + 0.990083i $$0.544865\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 41033.9 2.08046
$$731$$ 3077.75 0.155725
$$732$$ 0 0
$$733$$ 36585.2 1.84353 0.921764 0.387751i $$-0.126748\pi$$
0.921764 + 0.387751i $$0.126748\pi$$
$$734$$ −14607.9 −0.734587
$$735$$ 0 0
$$736$$ −46364.6 −2.32204
$$737$$ 15765.3 0.787953
$$738$$ 0 0
$$739$$ −6425.89 −0.319865 −0.159933 0.987128i $$-0.551128\pi$$
−0.159933 + 0.987128i $$0.551128\pi$$
$$740$$ 12883.0 0.639986
$$741$$ 0 0
$$742$$ −3788.84 −0.187457
$$743$$ 20411.0 1.00782 0.503908 0.863757i $$-0.331895\pi$$
0.503908 + 0.863757i $$0.331895\pi$$
$$744$$ 0 0
$$745$$ −8499.60 −0.417988
$$746$$ −50354.6 −2.47133
$$747$$ 0 0
$$748$$ 5111.13 0.249842
$$749$$ −6844.81 −0.333917
$$750$$ 0 0
$$751$$ −24259.5 −1.17875 −0.589375 0.807860i $$-0.700626\pi$$
−0.589375 + 0.807860i $$0.700626\pi$$
$$752$$ 22118.6 1.07258
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 26033.9 1.25493
$$756$$ 0 0
$$757$$ 9295.39 0.446297 0.223148 0.974785i $$-0.428367\pi$$
0.223148 + 0.974785i $$0.428367\pi$$
$$758$$ −1454.66 −0.0697042
$$759$$ 0 0
$$760$$ 12775.6 0.609761
$$761$$ −21974.7 −1.04676 −0.523378 0.852101i $$-0.675328\pi$$
−0.523378 + 0.852101i $$0.675328\pi$$
$$762$$ 0 0
$$763$$ 14631.0 0.694204
$$764$$ −14288.9 −0.676641
$$765$$ 0 0
$$766$$ −14253.5 −0.672324
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −22987.4 −1.07795 −0.538977 0.842320i $$-0.681189\pi$$
−0.538977 + 0.842320i $$0.681189\pi$$
$$770$$ −14014.3 −0.655897
$$771$$ 0 0
$$772$$ −3590.68 −0.167398
$$773$$ 31970.9 1.48760 0.743799 0.668404i $$-0.233022\pi$$
0.743799 + 0.668404i $$0.233022\pi$$
$$774$$ 0 0
$$775$$ −1517.34 −0.0703283
$$776$$ −8849.59 −0.409384
$$777$$ 0 0
$$778$$ 6860.25 0.316133
$$779$$ −60520.8 −2.78354
$$780$$ 0 0
$$781$$ 6418.85 0.294090
$$782$$ 15436.6 0.705896
$$783$$ 0 0
$$784$$ 10238.6 0.466408
$$785$$ −36280.2 −1.64955
$$786$$ 0 0
$$787$$ −6087.26 −0.275715 −0.137857 0.990452i $$-0.544022\pi$$
−0.137857 + 0.990452i $$0.544022\pi$$
$$788$$ −41520.9 −1.87706
$$789$$ 0 0
$$790$$ 15937.9 0.717779
$$791$$ −474.878 −0.0213460
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −33415.4 −1.49354
$$795$$ 0 0
$$796$$ 40363.6 1.79730
$$797$$ −23080.0 −1.02577 −0.512883 0.858458i $$-0.671423\pi$$
−0.512883 + 0.858458i $$0.671423\pi$$
$$798$$ 0 0
$$799$$ −9500.41 −0.420651
$$800$$ −1463.57 −0.0646813
$$801$$ 0 0
$$802$$ 902.429 0.0397330
$$803$$ −22068.3 −0.969829
$$804$$ 0 0
$$805$$ −23212.9 −1.01633
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −4661.94 −0.202978
$$809$$ 32377.8 1.40710 0.703550 0.710646i $$-0.251597\pi$$
0.703550 + 0.710646i $$0.251597\pi$$
$$810$$ 0 0
$$811$$ −26352.8 −1.14103 −0.570513 0.821288i $$-0.693256\pi$$
−0.570513 + 0.821288i $$0.693256\pi$$
$$812$$ 2236.40 0.0966532
$$813$$ 0 0
$$814$$ −12633.4 −0.543980
$$815$$ −26461.5 −1.13731
$$816$$ 0 0
$$817$$ −23424.5 −1.00308
$$818$$ 20135.0 0.860638
$$819$$ 0 0
$$820$$ −43451.3 −1.85047
$$821$$ 35355.3 1.50294 0.751468 0.659770i $$-0.229346\pi$$
0.751468 + 0.659770i $$0.229346\pi$$
$$822$$ 0 0
$$823$$ −12663.3 −0.536347 −0.268173 0.963371i $$-0.586420\pi$$
−0.268173 + 0.963371i $$0.586420\pi$$
$$824$$ −3692.02 −0.156089
$$825$$ 0 0
$$826$$ −41387.6 −1.74341
$$827$$ −16295.2 −0.685176 −0.342588 0.939486i $$-0.611303\pi$$
−0.342588 + 0.939486i $$0.611303\pi$$
$$828$$ 0 0
$$829$$ 13638.9 0.571411 0.285705 0.958318i $$-0.407772\pi$$
0.285705 + 0.958318i $$0.407772\pi$$
$$830$$ 20952.4 0.876229
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4397.69 −0.182918
$$834$$ 0 0
$$835$$ 30474.2 1.26300
$$836$$ −38900.5 −1.60933
$$837$$ 0 0
$$838$$ 41685.0 1.71836
$$839$$ −1890.31 −0.0777838 −0.0388919 0.999243i $$-0.512383\pi$$
−0.0388919 + 0.999243i $$0.512383\pi$$
$$840$$ 0 0
$$841$$ −23970.7 −0.982849
$$842$$ 51016.4 2.08805
$$843$$ 0 0
$$844$$ 11966.3 0.488028
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −7442.50 −0.301921
$$848$$ 3785.35 0.153289
$$849$$ 0 0
$$850$$ 487.280 0.0196630
$$851$$ −20925.6 −0.842914
$$852$$ 0 0
$$853$$ −1620.21 −0.0650351 −0.0325175 0.999471i $$-0.510352\pi$$
−0.0325175 + 0.999471i $$0.510352\pi$$
$$854$$ −8861.39 −0.355071
$$855$$ 0 0
$$856$$ −4393.08 −0.175412
$$857$$ 14508.4 0.578292 0.289146 0.957285i $$-0.406629\pi$$
0.289146 + 0.957285i $$0.406629\pi$$
$$858$$ 0 0
$$859$$ 29639.8 1.17730 0.588648 0.808389i $$-0.299660\pi$$
0.588648 + 0.808389i $$0.299660\pi$$
$$860$$ −16817.8 −0.666839
$$861$$ 0 0
$$862$$ −57548.8 −2.27392
$$863$$ 21528.8 0.849186 0.424593 0.905384i $$-0.360417\pi$$
0.424593 + 0.905384i $$0.360417\pi$$
$$864$$ 0 0
$$865$$ −1889.10 −0.0742557
$$866$$ 29939.4 1.17481
$$867$$ 0 0
$$868$$ −29132.9 −1.13921
$$869$$ −8571.50 −0.334601
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 9390.36 0.364676
$$873$$ 0 0
$$874$$ −117487. −4.54696
$$875$$ 15349.9 0.593054
$$876$$ 0 0
$$877$$ −14865.3 −0.572366 −0.286183 0.958175i $$-0.592387\pi$$
−0.286183 + 0.958175i $$0.592387\pi$$
$$878$$ −25347.1 −0.974285
$$879$$ 0 0
$$880$$ 14001.4 0.536348
$$881$$ 21336.0 0.815921 0.407961 0.913000i $$-0.366240\pi$$
0.407961 + 0.913000i $$0.366240\pi$$
$$882$$ 0 0
$$883$$ 37538.2 1.43065 0.715323 0.698794i $$-0.246280\pi$$
0.715323 + 0.698794i $$0.246280\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 54694.1 2.07391
$$887$$ −34575.0 −1.30881 −0.654406 0.756144i $$-0.727081\pi$$
−0.654406 + 0.756144i $$0.727081\pi$$
$$888$$ 0 0
$$889$$ −3500.29 −0.132054
$$890$$ −12474.0 −0.469807
$$891$$ 0 0
$$892$$ 21251.4 0.797703
$$893$$ 72306.9 2.70958
$$894$$ 0 0
$$895$$ 8143.61 0.304146
$$896$$ −10165.9 −0.379039
$$897$$ 0 0
$$898$$ 46233.3 1.71807
$$899$$ −5448.96 −0.202150
$$900$$ 0 0
$$901$$ −1625.89 −0.0601178
$$902$$ 42609.2 1.57287
$$903$$ 0 0
$$904$$ −304.783 −0.0112134
$$905$$ −25225.5 −0.926545
$$906$$ 0 0
$$907$$ −10424.8 −0.381641 −0.190820 0.981625i $$-0.561115\pi$$
−0.190820 + 0.981625i $$0.561115\pi$$
$$908$$ 40211.7 1.46968
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10961.8 −0.398661 −0.199331 0.979932i $$-0.563877\pi$$
−0.199331 + 0.979932i $$0.563877\pi$$
$$912$$ 0 0
$$913$$ −11268.3 −0.408464
$$914$$ −41392.2 −1.49796
$$915$$ 0 0
$$916$$ 8116.38 0.292765
$$917$$ −22519.7 −0.810976
$$918$$ 0 0
$$919$$ −10779.2 −0.386914 −0.193457 0.981109i $$-0.561970\pi$$
−0.193457 + 0.981109i $$0.561970\pi$$
$$920$$ −14898.3 −0.533895
$$921$$ 0 0
$$922$$ 14316.7 0.511384
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −660.549 −0.0234797
$$926$$ −7321.32 −0.259820
$$927$$ 0 0
$$928$$ −5255.88 −0.185919
$$929$$ 5429.07 0.191735 0.0958675 0.995394i $$-0.469437\pi$$
0.0958675 + 0.995394i $$0.469437\pi$$
$$930$$ 0 0
$$931$$ 33470.5 1.17825
$$932$$ −35808.8 −1.25854
$$933$$ 0 0
$$934$$ 33499.1 1.17358
$$935$$ −6013.88 −0.210348
$$936$$ 0 0
$$937$$ −21300.1 −0.742631 −0.371315 0.928507i $$-0.621093\pi$$
−0.371315 + 0.928507i $$0.621093\pi$$
$$938$$ 28857.9 1.00453
$$939$$ 0 0
$$940$$ 51913.2 1.80130
$$941$$ 26851.2 0.930207 0.465103 0.885256i $$-0.346017\pi$$
0.465103 + 0.885256i $$0.346017\pi$$
$$942$$ 0 0
$$943$$ 70576.7 2.43722
$$944$$ 41349.4 1.42564
$$945$$ 0 0
$$946$$ 16491.9 0.566805
$$947$$ −8021.68 −0.275258 −0.137629 0.990484i $$-0.543948\pi$$
−0.137629 + 0.990484i $$0.543948\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −3708.65 −0.126658
$$951$$ 0 0
$$952$$ 1652.46 0.0562569
$$953$$ −35715.0 −1.21398 −0.606990 0.794709i $$-0.707623\pi$$
−0.606990 + 0.794709i $$0.707623\pi$$
$$954$$ 0 0
$$955$$ 16812.6 0.569680
$$956$$ 29402.9 0.994726
$$957$$ 0 0
$$958$$ 35490.6 1.19692
$$959$$ 11687.9 0.393557
$$960$$ 0 0
$$961$$ 41190.9 1.38266
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −31728.7 −1.06007
$$965$$ 4224.88 0.140936
$$966$$ 0 0
$$967$$ −53338.8 −1.77380 −0.886898 0.461965i $$-0.847145\pi$$
−0.886898 + 0.461965i $$0.847145\pi$$
$$968$$ −4776.69 −0.158604
$$969$$ 0 0
$$970$$ 58953.6 1.95143
$$971$$ −23112.9 −0.763882 −0.381941 0.924187i $$-0.624744\pi$$
−0.381941 + 0.924187i $$0.624744\pi$$
$$972$$ 0 0
$$973$$ −32170.0 −1.05994
$$974$$ 49181.3 1.61794
$$975$$ 0 0
$$976$$ 8853.22 0.290353
$$977$$ −52874.6 −1.73143 −0.865715 0.500538i $$-0.833136\pi$$
−0.865715 + 0.500538i $$0.833136\pi$$
$$978$$ 0 0
$$979$$ 6708.57 0.219006
$$980$$ 24030.4 0.783287
$$981$$ 0 0
$$982$$ 16645.5 0.540917
$$983$$ 45173.1 1.46572 0.732858 0.680381i $$-0.238186\pi$$
0.732858 + 0.680381i $$0.238186\pi$$
$$984$$ 0 0
$$985$$ 48854.5 1.58034
$$986$$ 1749.89 0.0565190
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 27316.7 0.878281
$$990$$ 0 0
$$991$$ 60485.6 1.93884 0.969418 0.245414i $$-0.0789239\pi$$
0.969418 + 0.245414i $$0.0789239\pi$$
$$992$$ 68466.7 2.19135
$$993$$ 0 0
$$994$$ 11749.5 0.374922
$$995$$ −47492.8 −1.51319
$$996$$ 0 0
$$997$$ −18108.1 −0.575214 −0.287607 0.957749i $$-0.592860\pi$$
−0.287607 + 0.957749i $$0.592860\pi$$
$$998$$ 23951.1 0.759677
$$999$$ 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.u.1.3 3
3.2 odd 2 507.4.a.h.1.1 3
13.12 even 2 117.4.a.f.1.1 3
39.5 even 4 507.4.b.g.337.6 6
39.8 even 4 507.4.b.g.337.1 6
39.38 odd 2 39.4.a.c.1.3 3
52.51 odd 2 1872.4.a.bk.1.3 3
156.155 even 2 624.4.a.t.1.1 3
195.194 odd 2 975.4.a.l.1.1 3
273.272 even 2 1911.4.a.k.1.3 3
312.77 odd 2 2496.4.a.bl.1.3 3
312.155 even 2 2496.4.a.bp.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 39.38 odd 2
117.4.a.f.1.1 3 13.12 even 2
507.4.a.h.1.1 3 3.2 odd 2
507.4.b.g.337.1 6 39.8 even 4
507.4.b.g.337.6 6 39.5 even 4
624.4.a.t.1.1 3 156.155 even 2
975.4.a.l.1.1 3 195.194 odd 2
1521.4.a.u.1.3 3 1.1 even 1 trivial
1872.4.a.bk.1.3 3 52.51 odd 2
1911.4.a.k.1.3 3 273.272 even 2
2496.4.a.bl.1.3 3 312.77 odd 2
2496.4.a.bp.1.3 3 312.155 even 2