# Properties

 Label 117.4.a.f.1.1 Level $117$ Weight $4$ Character 117.1 Self dual yes Analytic conductor $6.903$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ 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 1.1 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 117.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.20905 q^{2} +9.71610 q^{4} +11.4322 q^{5} -11.2543 q^{7} -7.22315 q^{8} +O(q^{10})$$ $$q-4.20905 q^{2} +9.71610 q^{4} +11.4322 q^{5} -11.2543 q^{7} -7.22315 q^{8} -48.1187 q^{10} -25.8785 q^{11} +13.0000 q^{13} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} +154.712 q^{19} +111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} -54.7176 q^{26} -109.348 q^{28} +20.4522 q^{29} +266.424 q^{31} +256.984 q^{32} -85.5599 q^{34} -128.661 q^{35} +115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} -391.184 q^{41} +151.407 q^{43} -251.438 q^{44} -759.390 q^{46} +467.365 q^{47} -216.341 q^{49} -23.9714 q^{50} +126.309 q^{52} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} -86.0843 q^{58} +873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +148.619 q^{65} -609.204 q^{67} +197.505 q^{68} +541.542 q^{70} -248.038 q^{71} +852.765 q^{73} -488.181 q^{74} +1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} -541.043 q^{80} +1646.51 q^{82} +435.432 q^{83} +232.389 q^{85} -637.281 q^{86} +186.924 q^{88} -259.233 q^{89} -146.306 q^{91} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} +1225.17 q^{97} +910.589 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 $$3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26} + 80 q^{28} + 2 q^{29} + 302 q^{31} - 154 q^{32} + 164 q^{34} - 80 q^{35} + 374 q^{37} - 312 q^{38} - 516 q^{40} - 480 q^{41} - 260 q^{43} - 712 q^{44} - 1104 q^{46} + 24 q^{47} + 447 q^{49} - 814 q^{50} + 130 q^{52} + 678 q^{53} - 1552 q^{55} - 96 q^{56} - 628 q^{58} + 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 52 q^{65} + 74 q^{67} + 460 q^{68} + 1216 q^{70} + 948 q^{71} - 222 q^{73} - 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 24 q^{79} - 1100 q^{80} + 564 q^{82} + 796 q^{83} - 248 q^{85} - 1800 q^{86} + 1608 q^{88} - 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} + 3242 q^{97} + 5070 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 - 4 * q^10 + 16 * q^11 + 39 * q^13 + 176 * q^14 - 110 * q^16 + 146 * q^17 + 94 * q^19 + 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 26 * q^26 + 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 + 130 * q^52 + 678 * q^53 - 1552 * q^55 - 96 * q^56 - 628 * q^58 + 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 52 * q^65 + 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 + 390 * q^91 + 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 $$-0.767103\pi$$
−0.744062 + 0.668111i $$0.767103\pi$$
$$3$$ 0 0
$$4$$ 9.71610 1.21451
$$5$$ 11.4322 1.02253 0.511264 0.859424i $$-0.329178\pi$$
0.511264 + 0.859424i $$0.329178\pi$$
$$6$$ 0 0
$$7$$ −11.2543 −0.607675 −0.303838 0.952724i $$-0.598268\pi$$
−0.303838 + 0.952724i $$0.598268\pi$$
$$8$$ −7.22315 −0.319221
$$9$$ 0 0
$$10$$ −48.1187 −1.52165
$$11$$ −25.8785 −0.709333 −0.354666 0.934993i $$-0.615406\pi$$
−0.354666 + 0.934993i $$0.615406\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 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 $$-0.116262\pi$$
0.934035 + 0.357181i $$0.116262\pi$$
$$20$$ 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
$$26$$ −54.7176 −0.412731
$$27$$ 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 $$-0.219360\pi$$
0.771794 + 0.635873i $$0.219360\pi$$
$$32$$ 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 $$-0.417045\pi$$
0.257670 + 0.966233i $$0.417045\pi$$
$$38$$ −651.190 −2.77992
$$39$$ 0 0
$$40$$ −82.5765 −0.326412
$$41$$ −391.184 −1.49006 −0.745032 0.667029i $$-0.767566\pi$$
−0.745032 + 0.667029i $$0.767566\pi$$
$$42$$ 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 $$-0.241731\pi$$
0.725236 + 0.688500i $$0.241731\pi$$
$$48$$ 0 0
$$49$$ −216.341 −0.630731
$$50$$ −23.9714 −0.0678012
$$51$$ 0 0
$$52$$ 126.309 0.336845
$$53$$ −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 $$-0.0857171\pi$$
0.963960 + 0.266045i $$0.0857171\pi$$
$$60$$ 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
$$65$$ 148.619 0.283598
$$66$$ 0 0
$$67$$ −609.204 −1.11084 −0.555418 0.831571i $$-0.687442\pi$$
−0.555418 + 0.831571i $$0.687442\pi$$
$$68$$ 197.505 0.352221
$$69$$ 0 0
$$70$$ 541.542 0.924667
$$71$$ −248.038 −0.414601 −0.207301 0.978277i $$-0.566468\pi$$
−0.207301 + 0.978277i $$0.566468\pi$$
$$72$$ 0 0
$$73$$ 852.765 1.36724 0.683621 0.729838i $$-0.260404\pi$$
0.683621 + 0.729838i $$0.260404\pi$$
$$74$$ −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 $$-0.407036\pi$$
0.287921 + 0.957654i $$0.407036\pi$$
$$84$$ 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 $$-0.549336\pi$$
−0.154375 + 0.988012i $$0.549336\pi$$
$$90$$ 0 0
$$91$$ −146.306 −0.168539
$$92$$ 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 $$-0.278428\pi$$
0.641223 + 0.767355i $$0.278428\pi$$
$$98$$ 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 $$0.578619\pi$$
$$104$$ −93.9010 −0.0885360
$$105$$ 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 $$-0.693521\pi$$
−0.571197 + 0.820813i $$0.693521\pi$$
$$110$$ 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
$$130$$ −625.543 −0.422029
$$131$$ −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 $$-0.604968\pi$$
−0.323822 + 0.946118i $$0.604968\pi$$
$$138$$ 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
$$143$$ −336.421 −0.196734
$$144$$ 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 $$-0.565521\pi$$
−0.204390 + 0.978890i $$0.565521\pi$$
$$150$$ 0 0
$$151$$ 2277.24 1.22728 0.613640 0.789586i $$-0.289705\pi$$
0.613640 + 0.789586i $$0.289705\pi$$
$$152$$ −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 $$-0.687713\pi$$
−0.556126 + 0.831098i $$0.687713\pi$$
$$164$$ −3800.78 −1.80970
$$165$$ 0 0
$$166$$ −1832.76 −0.856925
$$167$$ 2665.65 1.23517 0.617587 0.786502i $$-0.288110\pi$$
0.617587 + 0.786502i $$0.288110\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ −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 $$0.350330\pi$$
$$182$$ 615.809 0.250806
$$183$$ 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 $$-0.478046\pi$$
0.0689158 + 0.997622i $$0.478046\pi$$
$$194$$ −5156.80 −1.90844
$$195$$ 0 0
$$196$$ −2101.99 −0.766031
$$197$$ 4273.41 1.54552 0.772761 0.634697i $$-0.218875\pi$$
0.772761 + 0.634697i $$0.218875\pi$$
$$198$$ 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
$$208$$ −615.241 −0.205093
$$209$$ −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
$$221$$ 264.259 0.0804343
$$222$$ 0 0
$$223$$ −2187.24 −0.656809 −0.328404 0.944537i $$-0.606511\pi$$
−0.328404 + 0.944537i $$0.606511\pi$$
$$224$$ −2892.17 −0.862684
$$225$$ 0 0
$$226$$ 177.602 0.0522739
$$227$$ −4138.67 −1.21010 −0.605051 0.796187i $$-0.706847\pi$$
−0.605051 + 0.796187i $$0.706847\pi$$
$$228$$ 0 0
$$229$$ −835.354 −0.241056 −0.120528 0.992710i $$-0.538459\pi$$
−0.120528 + 0.992710i $$0.538459\pi$$
$$230$$ −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 $$-0.634303\pi$$
−0.409517 + 0.912303i $$0.634303\pi$$
$$240$$ 0 0
$$241$$ 3265.58 0.872839 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$242$$ 2783.46 0.739370
$$243$$ 0 0
$$244$$ −1817.57 −0.476877
$$245$$ −2473.25 −0.644940
$$246$$ 0 0
$$247$$ 2011.25 0.518110
$$248$$ −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
$$260$$ 1443.99 0.344433
$$261$$ 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 $$-0.672942\pi$$
−0.516976 + 0.856000i $$0.672942\pi$$
$$272$$ −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 $$-0.665945\pi$$
−0.498036 + 0.867157i $$0.665945\pi$$
$$282$$ 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
$$286$$ 1416.01 0.292764
$$287$$ 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 $$-0.586003\pi$$
−0.266911 + 0.963721i $$0.586003\pi$$
$$294$$ 0 0
$$295$$ 9988.43 1.97135
$$296$$ −837.767 −0.164507
$$297$$ 0 0
$$298$$ 3129.34 0.608315
$$299$$ 2345.44 0.453646
$$300$$ 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 $$-0.486033\pi$$
0.0438658 + 0.999037i $$0.486033\pi$$
$$308$$ 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 $$-0.593079\pi$$
−0.288266 + 0.957550i $$0.593079\pi$$
$$318$$ 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
$$325$$ 74.0375 0.0126365
$$326$$ 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 $$-0.591715\pi$$
−0.284161 + 0.958777i $$0.591715\pi$$
$$332$$ 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 $$0.770801\pi$$
$$338$$ −711.329 −0.114471
$$339$$ 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 $$-0.620252\pi$$
−0.368861 + 0.929485i $$0.620252\pi$$
$$350$$ 269.781 0.0412011
$$351$$ 0 0
$$352$$ −6650.35 −1.00700
$$353$$ 2859.64 0.431170 0.215585 0.976485i $$-0.430834\pi$$
0.215585 + 0.976485i $$0.430834\pi$$
$$354$$ 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 $$-0.587338\pi$$
−0.270949 + 0.962594i $$0.587338\pi$$
$$360$$ 0 0
$$361$$ 17076.8 2.48969
$$362$$ −9287.39 −1.34844
$$363$$ 0 0
$$364$$ −1421.52 −0.204692
$$365$$ 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
$$377$$ 265.879 0.0363221
$$378$$ 0 0
$$379$$ 345.604 0.0468403 0.0234202 0.999726i $$-0.492544\pi$$
0.0234202 + 0.999726i $$0.492544\pi$$
$$380$$ 17184.8 2.31990
$$381$$ 0 0
$$382$$ 6189.99 0.829078
$$383$$ 3386.40 0.451793 0.225897 0.974151i $$-0.427469\pi$$
0.225897 + 0.974151i $$0.427469\pi$$
$$384$$ 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 $$-0.332664\pi$$
0.501819 + 0.864973i $$0.332664\pi$$
$$398$$ −17485.7 −2.20221
$$399$$ 0 0
$$400$$ −269.532 −0.0336915
$$401$$ −214.402 −0.0267001 −0.0133500 0.999911i $$-0.504250\pi$$
−0.0133500 + 0.999911i $$0.504250\pi$$
$$402$$ 0 0
$$403$$ 3463.52 0.428114
$$404$$ −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 $$-0.593379\pi$$
−0.289169 + 0.957278i $$0.593379\pi$$
$$410$$ 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
$$416$$ 3340.79 0.393739
$$417$$ 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 $$-0.747518\pi$$
−0.701572 + 0.712598i $$0.747518\pi$$
$$422$$ −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 $$-0.223221\pi$$
0.764023 + 0.645189i $$0.223221\pi$$
$$432$$ 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
$$442$$ −1112.28 −0.119696
$$443$$ 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 $$-0.695878\pi$$
−0.577260 + 0.816560i $$0.695878\pi$$
$$450$$ 0 0
$$451$$ 10123.2 1.05695
$$452$$ −409.973 −0.0426627
$$453$$ 0 0
$$454$$ 17419.9 1.80078
$$455$$ −1672.60 −0.172335
$$456$$ 0 0
$$457$$ 9834.10 1.00661 0.503304 0.864109i $$-0.332118\pi$$
0.503304 + 0.864109i $$0.332118\pi$$
$$458$$ 3516.05 0.358721
$$459$$ 0 0
$$460$$ 20040.2 2.03126
$$461$$ −3401.42 −0.343644 −0.171822 0.985128i $$-0.554965\pi$$
−0.171822 + 0.985128i $$0.554965\pi$$
$$462$$ 0 0
$$463$$ 1739.42 0.174596 0.0872979 0.996182i $$-0.472177\pi$$
0.0872979 + 0.996182i $$0.472177\pi$$
$$464$$ −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 $$-0.631740\pi$$
−0.402158 + 0.915570i $$0.631740\pi$$
$$480$$ 0 0
$$481$$ 1507.79 0.142930
$$482$$ −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 $$-0.682945\pi$$
−0.543617 + 0.839334i $$0.682945\pi$$
$$488$$ 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
$$494$$ −8465.47 −0.771011
$$495$$ 0 0
$$496$$ −12608.8 −1.14144
$$497$$ 2791.49 0.251943
$$498$$ 0 0
$$499$$ −5690.37 −0.510493 −0.255246 0.966876i $$-0.582157\pi$$
−0.255246 + 0.966876i $$0.582157\pi$$
$$500$$ −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 $$-0.200524\pi$$
0.808049 + 0.589115i $$0.200524\pi$$
$$510$$ 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
$$520$$ −1073.49 −0.0905305
$$521$$ −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
$$533$$ −5085.39 −0.413269
$$534$$ 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 $$-0.252729\pi$$
0.701018 + 0.713144i $$0.252729\pi$$
$$542$$ 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 $$-0.600696\pi$$
−0.311096 + 0.950378i $$0.600696\pi$$
$$558$$ 0 0
$$559$$ 1968.30 0.148927
$$560$$ 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 $$0.171362\pi$$
$$572$$ −3268.70 −0.238935
$$573$$ 0 0
$$574$$ −18530.3 −1.34746
$$575$$ 1027.52 0.0745225
$$576$$ 0 0
$$577$$ 20508.1 1.47966 0.739831 0.672793i $$-0.234906\pi$$
0.739831 + 0.672793i $$0.234906\pi$$
$$578$$ 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 $$-0.432708\pi$$
0.209833 + 0.977737i $$0.432708\pi$$
$$588$$ 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 $$-0.330539\pi$$
0.507584 + 0.861602i $$0.330539\pi$$
$$594$$ 0 0
$$595$$ −2615.38 −0.180202
$$596$$ −7223.72 −0.496468
$$597$$ 0 0
$$598$$ −9872.07 −0.675082
$$599$$ −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
$$611$$ 6075.74 0.402288
$$612$$ 0 0
$$613$$ 1516.39 0.0999128 0.0499564 0.998751i $$-0.484092\pi$$
0.0499564 + 0.998751i $$0.484092\pi$$
$$614$$ −1986.31 −0.130556
$$615$$ 0 0
$$616$$ −2103.70 −0.137598
$$617$$ −18539.3 −1.20966 −0.604832 0.796353i $$-0.706760\pi$$
−0.604832 + 0.796353i $$0.706760\pi$$
$$618$$ 0 0
$$619$$ 25684.9 1.66779 0.833897 0.551920i $$-0.186105\pi$$
0.833897 + 0.551920i $$0.186105\pi$$
$$620$$ 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 $$-0.749927\pi$$
−0.706945 + 0.707269i $$0.749927\pi$$
$$632$$ 2392.46 0.150580
$$633$$ 0 0
$$634$$ 13696.1 0.857950
$$635$$ −3555.62 −0.222206
$$636$$ 0 0
$$637$$ −2812.43 −0.174933
$$638$$ 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 $$-0.752854\pi$$
−0.713418 + 0.700738i $$0.752854\pi$$
$$644$$ −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
$$650$$ −311.628 −0.0188047
$$651$$ 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 $$-0.479155\pi$$
0.0654406 + 0.997856i $$0.479155\pi$$
$$662$$ 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
$$676$$ 1642.02 0.0934240
$$677$$ 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 $$-0.603455\pi$$
−0.319323 + 0.947646i $$0.603455\pi$$
$$684$$ 0 0
$$685$$ −11872.6 −0.662233
$$686$$ −26495.9 −1.47466
$$687$$ 0 0
$$688$$ −7165.54 −0.397069
$$689$$ −1039.79 −0.0574935
$$690$$ 0 0
$$691$$ −3323.23 −0.182955 −0.0914773 0.995807i $$-0.529159\pi$$
−0.0914773 + 0.995807i $$0.529159\pi$$
$$692$$ 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 $$-0.387439\pi$$
0.346297 + 0.938125i $$0.387439\pi$$
$$710$$ 11935.3 0.630877
$$711$$ 0 0
$$712$$ 1872.48 0.0985592
$$713$$ 48067.8 2.52476
$$714$$ 0 0
$$715$$ −3846.03 −0.201165
$$716$$ −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 $$0.544865\pi$$
$$728$$ 1056.79 0.0538011
$$729$$ 0 0
$$730$$ −41033.9 −2.08046
$$731$$ 3077.75 0.155725
$$732$$ 0 0
$$733$$ −36585.2 −1.84353 −0.921764 0.387751i $$-0.873252\pi$$
−0.921764 + 0.387751i $$0.873252\pi$$
$$734$$ 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 $$-0.448872\pi$$
0.159933 + 0.987128i $$0.448872\pi$$
$$740$$ 12883.0 0.639986
$$741$$ 0 0
$$742$$ −3788.84 −0.187457
$$743$$ −20411.0 −1.00782 −0.503908 0.863757i $$-0.668105\pi$$
−0.503908 + 0.863757i $$0.668105\pi$$
$$744$$ 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
$$754$$ −1119.10 −0.0540518
$$755$$ 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 $$-0.324672\pi$$
0.523378 + 0.852101i $$0.324672\pi$$
$$762$$ 0 0
$$763$$ 14631.0 0.694204
$$764$$ −14288.9 −0.676641
$$765$$ 0 0
$$766$$ −14253.5 −0.672324
$$767$$ 11358.2 0.534709
$$768$$ 0 0
$$769$$ 22987.4 1.07795 0.538977 0.842320i $$-0.318811\pi$$
0.538977 + 0.842320i $$0.318811\pi$$
$$770$$ −14014.3 −0.655897
$$771$$ 0 0
$$772$$ 3590.68 0.167398
$$773$$ −31970.9 −1.48760 −0.743799 0.668404i $$-0.766978\pi$$
−0.743799 + 0.668404i $$0.766978\pi$$
$$774$$ 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 $$-0.455978\pi$$
0.137857 + 0.990452i $$0.455978\pi$$
$$788$$ 41520.9 1.87706
$$789$$ 0 0
$$790$$ 15937.9 0.717779
$$791$$ 474.878 0.0213460
$$792$$ 0 0
$$793$$ −2431.88 −0.108901
$$794$$ −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
$$806$$ −14578.1 −0.637087
$$807$$ 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 $$-0.306744\pi$$
0.570513 + 0.821288i $$0.306744\pi$$
$$812$$ −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 $$-0.770654\pi$$
−0.751468 + 0.659770i $$0.770654\pi$$
$$822$$ 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 $$-0.388697\pi$$
0.342588 + 0.939486i $$0.388697\pi$$
$$828$$ 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
$$832$$ −9139.61 −0.380840
$$833$$ −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 $$-0.487617\pi$$
0.0388919 + 0.999243i $$0.487617\pi$$
$$840$$ 0 0
$$841$$ −23970.7 −0.982849
$$842$$ 51016.4 2.08805
$$843$$ 0 0
$$844$$ 11966.3 0.488028
$$845$$ 1932.04 0.0786559
$$846$$ 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 $$-0.489648\pi$$
0.0325175 + 0.999471i $$0.489648\pi$$
$$854$$ −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 $$-0.639583\pi$$
−0.424593 + 0.905384i $$0.639583\pi$$
$$864$$ 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
$$871$$ −7919.65 −0.308091
$$872$$ 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 $$-0.407613\pi$$
0.286183 + 0.958175i $$0.407613\pi$$
$$878$$ 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 $$0.246280\pi$$
$$884$$ 2567.57 0.0976884
$$885$$ 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
$$910$$ 7040.05 0.256456
$$911$$ −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
$$923$$ −3224.49 −0.114990
$$924$$ 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 $$-0.530563\pi$$
−0.0958675 + 0.995394i $$0.530563\pi$$
$$930$$ 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 $$-0.653983\pi$$
−0.465103 + 0.885256i $$0.653983\pi$$
$$942$$ 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 $$-0.456052\pi$$
0.137629 + 0.990484i $$0.456052\pi$$
$$948$$ 0 0
$$949$$ 11085.9 0.379204
$$950$$ −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
$$962$$ −6346.35 −0.212697
$$963$$ 0 0
$$964$$ 31728.7 1.06007
$$965$$ 4224.88 0.140936
$$966$$ 0 0
$$967$$ 53338.8 1.77380 0.886898 0.461965i $$-0.152855\pi$$
0.886898 + 0.461965i $$0.152855\pi$$
$$968$$ 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 $$-0.166864\pi$$
0.865715 + 0.500538i $$0.166864\pi$$
$$978$$ 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 $$-0.761814\pi$$
−0.732858 + 0.680381i $$0.761814\pi$$
$$984$$ 0 0
$$985$$ 48854.5 1.58034
$$986$$ −1749.89 −0.0565190
$$987$$ 0 0
$$988$$ 19541.6 0.629251
$$989$$ 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 117.4.a.f.1.1 3
3.2 odd 2 39.4.a.c.1.3 3
4.3 odd 2 1872.4.a.bk.1.3 3
12.11 even 2 624.4.a.t.1.1 3
13.12 even 2 1521.4.a.u.1.3 3
15.14 odd 2 975.4.a.l.1.1 3
21.20 even 2 1911.4.a.k.1.3 3
24.5 odd 2 2496.4.a.bl.1.3 3
24.11 even 2 2496.4.a.bp.1.3 3
39.5 even 4 507.4.b.g.337.1 6
39.8 even 4 507.4.b.g.337.6 6
39.38 odd 2 507.4.a.h.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 3.2 odd 2
117.4.a.f.1.1 3 1.1 even 1 trivial
507.4.a.h.1.1 3 39.38 odd 2
507.4.b.g.337.1 6 39.5 even 4
507.4.b.g.337.6 6 39.8 even 4
624.4.a.t.1.1 3 12.11 even 2
975.4.a.l.1.1 3 15.14 odd 2
1521.4.a.u.1.3 3 13.12 even 2
1872.4.a.bk.1.3 3 4.3 odd 2
1911.4.a.k.1.3 3 21.20 even 2
2496.4.a.bl.1.3 3 24.5 odd 2
2496.4.a.bp.1.3 3 24.11 even 2