# Properties

 Label 975.4.a.l.1.1 Level $975$ Weight $4$ Character 975.1 Self dual yes Analytic conductor $57.527$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 975.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.5268622556$$ 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 $$\chi$$ $$=$$ 975.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +25.8785 q^{11} -29.1483 q^{12} -13.0000 q^{13} -47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -37.8814 q^{18} +154.712 q^{19} -33.7629 q^{21} -108.924 q^{22} +180.418 q^{23} +21.6695 q^{24} +54.7176 q^{26} -27.0000 q^{27} +109.348 q^{28} -20.4522 q^{29} +266.424 q^{31} +256.984 q^{32} -77.6355 q^{33} -85.5599 q^{34} +87.4449 q^{36} -115.984 q^{37} -651.190 q^{38} +39.0000 q^{39} +391.184 q^{41} +142.110 q^{42} -151.407 q^{43} +251.438 q^{44} -759.390 q^{46} +467.365 q^{47} +141.979 q^{48} -216.341 q^{49} -60.9828 q^{51} -126.309 q^{52} -79.9842 q^{53} +113.644 q^{54} -81.2915 q^{56} -464.136 q^{57} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} +101.289 q^{63} -703.047 q^{64} +326.772 q^{66} +609.204 q^{67} +197.505 q^{68} -541.255 q^{69} +248.038 q^{71} -65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} +1503.20 q^{76} +291.244 q^{77} -164.153 q^{78} -331.221 q^{79} +81.0000 q^{81} -1646.51 q^{82} +435.432 q^{83} -328.044 q^{84} +637.281 q^{86} +61.3566 q^{87} -186.924 q^{88} +259.233 q^{89} -146.306 q^{91} +1752.96 q^{92} -799.273 q^{93} -1967.16 q^{94} -770.951 q^{96} -1225.17 q^{97} +910.589 q^{98} +232.907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 9 * q^3 + 10 * q^4 + 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 $$3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 16 q^{11} - 30 q^{12} - 39 q^{13} - 176 q^{14} - 110 q^{16} + 146 q^{17} - 18 q^{18} + 94 q^{19} + 90 q^{21} + 56 q^{22} + 48 q^{23} - 18 q^{24} + 26 q^{26} - 81 q^{27} - 80 q^{28} - 2 q^{29} + 302 q^{31} - 154 q^{32} + 48 q^{33} + 164 q^{34} + 90 q^{36} - 374 q^{37} - 312 q^{38} + 117 q^{39} + 480 q^{41} + 528 q^{42} + 260 q^{43} + 712 q^{44} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 447 q^{49} - 438 q^{51} - 130 q^{52} + 678 q^{53} + 54 q^{54} + 96 q^{56} - 282 q^{57} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} + 460 q^{68} - 144 q^{69} - 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 78 q^{78} - 24 q^{79} + 243 q^{81} - 564 q^{82} + 796 q^{83} + 240 q^{84} + 1800 q^{86} + 6 q^{87} - 1608 q^{88} + 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 906 q^{93} - 1920 q^{94} + 462 q^{96} - 3242 q^{97} + 5070 q^{98} - 144 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 9 * q^3 + 10 * q^4 + 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 - 16 * q^11 - 30 * q^12 - 39 * q^13 - 176 * q^14 - 110 * q^16 + 146 * q^17 - 18 * q^18 + 94 * q^19 + 90 * q^21 + 56 * q^22 + 48 * q^23 - 18 * q^24 + 26 * q^26 - 81 * q^27 - 80 * q^28 - 2 * q^29 + 302 * q^31 - 154 * q^32 + 48 * q^33 + 164 * q^34 + 90 * q^36 - 374 * q^37 - 312 * q^38 + 117 * q^39 + 480 * q^41 + 528 * q^42 + 260 * q^43 + 712 * q^44 - 1104 * q^46 + 24 * q^47 + 330 * q^48 + 447 * q^49 - 438 * q^51 - 130 * q^52 + 678 * q^53 + 54 * q^54 + 96 * q^56 - 282 * q^57 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 270 * q^63 - 750 * q^64 - 168 * q^66 - 74 * q^67 + 460 * q^68 - 144 * q^69 - 948 * q^71 + 54 * q^72 + 222 * q^73 + 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 78 * q^78 - 24 * q^79 + 243 * q^81 - 564 * q^82 + 796 * q^83 + 240 * q^84 + 1800 * q^86 + 6 * q^87 - 1608 * q^88 + 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 906 * q^93 - 1920 * q^94 + 462 * q^96 - 3242 * q^97 + 5070 * q^98 - 144 * q^99

## 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 $$0.767103\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 9.71610 1.21451
$$5$$ 0 0
$$6$$ 12.6271 0.859169
$$7$$ 11.2543 0.607675 0.303838 0.952724i $$-0.401732\pi$$
0.303838 + 0.952724i $$0.401732\pi$$
$$8$$ −7.22315 −0.319221
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ −29.1483 −0.701199
$$13$$ −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 $$0.453680\pi$$
$$18$$ −37.8814 −0.496041
$$19$$ 154.712 1.86807 0.934035 0.357181i $$-0.116262\pi$$
0.934035 + 0.357181i $$0.116262\pi$$
$$20$$ 0 0
$$21$$ −33.7629 −0.350841
$$22$$ −108.924 −1.05558
$$23$$ 180.418 1.63565 0.817823 0.575471i $$-0.195181\pi$$
0.817823 + 0.575471i $$0.195181\pi$$
$$24$$ 21.6695 0.184302
$$25$$ 0 0
$$26$$ 54.7176 0.412731
$$27$$ −27.0000 −0.192450
$$28$$ 109.348 0.738029
$$29$$ −20.4522 −0.130961 −0.0654806 0.997854i $$-0.520858\pi$$
−0.0654806 + 0.997854i $$0.520858\pi$$
$$30$$ 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
$$33$$ −77.6355 −0.409534
$$34$$ −85.5599 −0.431571
$$35$$ 0 0
$$36$$ 87.4449 0.404837
$$37$$ −115.984 −0.515340 −0.257670 0.966233i $$-0.582955\pi$$
−0.257670 + 0.966233i $$0.582955\pi$$
$$38$$ −651.190 −2.77992
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ 391.184 1.49006 0.745032 0.667029i $$-0.232434\pi$$
0.745032 + 0.667029i $$0.232434\pi$$
$$42$$ 142.110 0.522095
$$43$$ −151.407 −0.536963 −0.268482 0.963285i $$-0.586522\pi$$
−0.268482 + 0.963285i $$0.586522\pi$$
$$44$$ 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 $$0.241731\pi$$
$$48$$ 141.979 0.426934
$$49$$ −216.341 −0.630731
$$50$$ 0 0
$$51$$ −60.9828 −0.167437
$$52$$ −126.309 −0.336845
$$53$$ −79.9842 −0.207296 −0.103648 0.994614i $$-0.533051\pi$$
−0.103648 + 0.994614i $$0.533051\pi$$
$$54$$ 113.644 0.286390
$$55$$ 0 0
$$56$$ −81.2915 −0.193983
$$57$$ −464.136 −1.07853
$$58$$ 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
$$63$$ 101.289 0.202558
$$64$$ −703.047 −1.37314
$$65$$ 0 0
$$66$$ 326.772 0.609437
$$67$$ 609.204 1.11084 0.555418 0.831571i $$-0.312558\pi$$
0.555418 + 0.831571i $$0.312558\pi$$
$$68$$ 197.505 0.352221
$$69$$ −541.255 −0.944340
$$70$$ 0 0
$$71$$ 248.038 0.414601 0.207301 0.978277i $$-0.433532\pi$$
0.207301 + 0.978277i $$0.433532\pi$$
$$72$$ −65.0084 −0.106407
$$73$$ −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
$$78$$ −164.153 −0.238291
$$79$$ −331.221 −0.471712 −0.235856 0.971788i $$-0.575789\pi$$
−0.235856 + 0.971788i $$0.575789\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −1646.51 −2.21740
$$83$$ 435.432 0.575842 0.287921 0.957654i $$-0.407036\pi$$
0.287921 + 0.957654i $$0.407036\pi$$
$$84$$ −328.044 −0.426101
$$85$$ 0 0
$$86$$ 637.281 0.799067
$$87$$ 61.3566 0.0756105
$$88$$ −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
$$91$$ −146.306 −0.168539
$$92$$ 1752.96 1.98651
$$93$$ −799.273 −0.891191
$$94$$ −1967.16 −2.15848
$$95$$ 0 0
$$96$$ −770.951 −0.819634
$$97$$ −1225.17 −1.28245 −0.641223 0.767355i $$-0.721572\pi$$
−0.641223 + 0.767355i $$0.721572\pi$$
$$98$$ 910.589 0.938606
$$99$$ 232.907 0.236444
$$100$$ 0 0
$$101$$ 645.416 0.635855 0.317927 0.948115i $$-0.397013\pi$$
0.317927 + 0.948115i $$0.397013\pi$$
$$102$$ 256.680 0.249167
$$103$$ 511.137 0.488969 0.244484 0.969653i $$-0.421381\pi$$
0.244484 + 0.969653i $$0.421381\pi$$
$$104$$ 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 $$0.588595\pi$$
$$108$$ −262.335 −0.233733
$$109$$ −1300.04 −1.14239 −0.571197 0.820813i $$-0.693521\pi$$
−0.571197 + 0.820813i $$0.693521\pi$$
$$110$$ 0 0
$$111$$ 347.951 0.297532
$$112$$ −532.623 −0.449359
$$113$$ −42.1953 −0.0351274 −0.0175637 0.999846i $$-0.505591\pi$$
−0.0175637 + 0.999846i $$0.505591\pi$$
$$114$$ 1953.57 1.60499
$$115$$ 0 0
$$116$$ −198.716 −0.159054
$$117$$ −117.000 −0.0924500
$$118$$ 3677.49 2.86899
$$119$$ 228.773 0.176232
$$120$$ 0 0
$$121$$ −661.303 −0.496847
$$122$$ 787.378 0.584311
$$123$$ −1173.55 −0.860289
$$124$$ 2588.61 1.87471
$$125$$ 0 0
$$126$$ −426.329 −0.301432
$$127$$ 311.018 0.217310 0.108655 0.994080i $$-0.465346\pi$$
0.108655 + 0.994080i $$0.465346\pi$$
$$128$$ 903.291 0.623753
$$129$$ 454.222 0.310016
$$130$$ 0 0
$$131$$ 2000.98 1.33456 0.667278 0.744809i $$-0.267459\pi$$
0.667278 + 0.744809i $$0.267459\pi$$
$$132$$ −754.314 −0.497384
$$133$$ 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 $$0.604968\pi$$
$$138$$ 2278.17 1.40529
$$139$$ −2858.46 −1.74426 −0.872128 0.489277i $$-0.837261\pi$$
−0.872128 + 0.489277i $$0.837261\pi$$
$$140$$ 0 0
$$141$$ −1402.09 −0.837430
$$142$$ −1044.00 −0.616978
$$143$$ −336.421 −0.196734
$$144$$ −425.936 −0.246491
$$145$$ 0 0
$$146$$ 3589.33 2.03462
$$147$$ 649.022 0.364153
$$148$$ −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 $$-0.289705\pi$$
0.613640 + 0.789586i $$0.289705\pi$$
$$152$$ −1117.51 −0.596328
$$153$$ 182.948 0.0966700
$$154$$ −1225.86 −0.641447
$$155$$ 0 0
$$156$$ 378.928 0.194478
$$157$$ −3173.51 −1.61321 −0.806605 0.591091i $$-0.798697\pi$$
−0.806605 + 0.591091i $$0.798697\pi$$
$$158$$ 1394.12 0.701966
$$159$$ 239.953 0.119682
$$160$$ 0 0
$$161$$ 2030.48 0.993941
$$162$$ −340.933 −0.165347
$$163$$ 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 $$-0.288110\pi$$
0.617587 + 0.786502i $$0.288110\pi$$
$$168$$ 243.874 0.111996
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 1392.41 0.622690
$$172$$ −1471.09 −0.652148
$$173$$ 165.243 0.0726198 0.0363099 0.999341i $$-0.488440\pi$$
0.0363099 + 0.999341i $$0.488440\pi$$
$$174$$ −258.253 −0.112518
$$175$$ 0 0
$$176$$ −1224.73 −0.524532
$$177$$ 2621.13 1.11309
$$178$$ −1091.13 −0.459457
$$179$$ 712.339 0.297446 0.148723 0.988879i $$-0.452484\pi$$
0.148723 + 0.988879i $$0.452484\pi$$
$$180$$ 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
$$183$$ 561.204 0.226696
$$184$$ −1303.19 −0.522132
$$185$$ 0 0
$$186$$ 3364.18 1.32620
$$187$$ 526.048 0.205714
$$188$$ 4540.96 1.76162
$$189$$ −303.866 −0.116947
$$190$$ 0 0
$$191$$ 1470.64 0.557129 0.278565 0.960417i $$-0.410141\pi$$
0.278565 + 0.960417i $$0.410141\pi$$
$$192$$ 2109.14 0.792782
$$193$$ −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 $$-0.218875\pi$$
0.772761 + 0.634697i $$0.218875\pi$$
$$198$$ −980.315 −0.351858
$$199$$ 4154.31 1.47985 0.739927 0.672687i $$-0.234860\pi$$
0.739927 + 0.672687i $$0.234860\pi$$
$$200$$ 0 0
$$201$$ −1827.61 −0.641342
$$202$$ −2716.59 −0.946230
$$203$$ −230.175 −0.0795819
$$204$$ −592.515 −0.203355
$$205$$ 0 0
$$206$$ −2151.40 −0.727646
$$207$$ 1623.77 0.545215
$$208$$ 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
$$213$$ −744.114 −0.239370
$$214$$ 2559.92 0.817723
$$215$$ 0 0
$$216$$ 195.025 0.0614341
$$217$$ 2998.42 0.938000
$$218$$ 5471.92 1.70002
$$219$$ 2558.30 0.789377
$$220$$ 0 0
$$221$$ −264.259 −0.0804343
$$222$$ −1464.54 −0.442764
$$223$$ 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 $$-0.706847\pi$$
−0.605051 + 0.796187i $$0.706847\pi$$
$$228$$ −4509.59 −1.30989
$$229$$ −835.354 −0.241056 −0.120528 0.992710i $$-0.538459\pi$$
−0.120528 + 0.992710i $$0.538459\pi$$
$$230$$ 0 0
$$231$$ −873.733 −0.248863
$$232$$ 147.729 0.0418056
$$233$$ −3685.51 −1.03625 −0.518124 0.855305i $$-0.673370\pi$$
−0.518124 + 0.855305i $$0.673370\pi$$
$$234$$ 492.459 0.137577
$$235$$ 0 0
$$236$$ −8489.05 −2.34148
$$237$$ 993.662 0.272343
$$238$$ −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 $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$242$$ 2783.46 0.739370
$$243$$ −243.000 −0.0641500
$$244$$ −1817.57 −0.476877
$$245$$ 0 0
$$246$$ 4939.53 1.28022
$$247$$ −2011.25 −0.518110
$$248$$ −1924.42 −0.492746
$$249$$ −1306.30 −0.332463
$$250$$ 0 0
$$251$$ −6363.16 −1.60016 −0.800078 0.599897i $$-0.795208\pi$$
−0.800078 + 0.599897i $$0.795208\pi$$
$$252$$ 984.131 0.246010
$$253$$ 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 $$0.235529\pi$$
$$258$$ −1911.84 −0.461342
$$259$$ −1305.31 −0.313159
$$260$$ 0 0
$$261$$ −184.070 −0.0436538
$$262$$ −8422.24 −1.98598
$$263$$ −123.227 −0.0288916 −0.0144458 0.999896i $$-0.504598\pi$$
−0.0144458 + 0.999896i $$0.504598\pi$$
$$264$$ 560.773 0.130732
$$265$$ 0 0
$$266$$ −7328.69 −1.68929
$$267$$ −777.700 −0.178256
$$268$$ 5919.08 1.34913
$$269$$ −1935.79 −0.438763 −0.219381 0.975639i $$-0.570404\pi$$
−0.219381 + 0.975639i $$0.570404\pi$$
$$270$$ 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
$$273$$ 438.918 0.0973059
$$274$$ 4371.20 0.963774
$$275$$ 0 0
$$276$$ −5258.89 −1.14691
$$277$$ 5834.30 1.26552 0.632761 0.774347i $$-0.281922\pi$$
0.632761 + 0.774347i $$0.281922\pi$$
$$278$$ 12031.4 2.59567
$$279$$ 2397.82 0.514529
$$280$$ 0 0
$$281$$ 4691.91 0.996071 0.498036 0.867157i $$-0.334055\pi$$
0.498036 + 0.867157i $$0.334055\pi$$
$$282$$ 5901.49 1.24620
$$283$$ −3465.60 −0.727945 −0.363973 0.931410i $$-0.618580\pi$$
−0.363973 + 0.931410i $$0.618580\pi$$
$$284$$ 2409.96 0.503539
$$285$$ 0 0
$$286$$ 1416.01 0.292764
$$287$$ 4402.50 0.905475
$$288$$ 2312.85 0.473216
$$289$$ −4499.79 −0.915894
$$290$$ 0 0
$$291$$ 3675.51 0.740420
$$292$$ −8285.55 −1.66053
$$293$$ −2677.31 −0.533822 −0.266911 0.963721i $$-0.586003\pi$$
−0.266911 + 0.963721i $$0.586003\pi$$
$$294$$ −2731.77 −0.541904
$$295$$ 0 0
$$296$$ 837.767 0.164507
$$297$$ −698.720 −0.136511
$$298$$ −3129.34 −0.608315
$$299$$ −2345.44 −0.453646
$$300$$ 0 0
$$301$$ −1703.98 −0.326299
$$302$$ −9585.02 −1.82634
$$303$$ −1936.25 −0.367111
$$304$$ −7321.93 −1.38139
$$305$$ 0 0
$$306$$ −770.039 −0.143857
$$307$$ −471.915 −0.0877316 −0.0438658 0.999037i $$-0.513967\pi$$
−0.0438658 + 0.999037i $$0.513967\pi$$
$$308$$ 2829.76 0.523508
$$309$$ −1533.41 −0.282306
$$310$$ 0 0
$$311$$ −1518.52 −0.276872 −0.138436 0.990371i $$-0.544207\pi$$
−0.138436 + 0.990371i $$0.544207\pi$$
$$312$$ −281.703 −0.0511163
$$313$$ −4049.86 −0.731348 −0.365674 0.930743i $$-0.619161\pi$$
−0.365674 + 0.930743i $$0.619161\pi$$
$$314$$ 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 $$0.593079\pi$$
$$318$$ −1009.97 −0.178102
$$319$$ −529.272 −0.0928951
$$320$$ 0 0
$$321$$ 1824.59 0.317254
$$322$$ −8546.40 −1.47911
$$323$$ 3144.92 0.541759
$$324$$ 787.004 0.134946
$$325$$ 0 0
$$326$$ −9742.46 −1.65517
$$327$$ 3900.11 0.659561
$$328$$ −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
$$333$$ −1043.85 −0.171780
$$334$$ −11219.8 −1.83809
$$335$$ 0 0
$$336$$ 1597.87 0.259437
$$337$$ 9301.67 1.50354 0.751772 0.659423i $$-0.229199\pi$$
0.751772 + 0.659423i $$0.229199\pi$$
$$338$$ −711.329 −0.114471
$$339$$ 126.586 0.0202808
$$340$$ 0 0
$$341$$ 6894.66 1.09492
$$342$$ −5860.71 −0.926640
$$343$$ −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 $$0.505341\pi$$
$$348$$ 596.147 0.0918299
$$349$$ −4809.84 −0.737721 −0.368861 0.929485i $$-0.620252\pi$$
−0.368861 + 0.929485i $$0.620252\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 6650.35 1.00700
$$353$$ 2859.64 0.431170 0.215585 0.976485i $$-0.430834\pi$$
0.215585 + 0.976485i $$0.430834\pi$$
$$354$$ −11032.5 −1.65641
$$355$$ 0 0
$$356$$ 2518.74 0.374980
$$357$$ −686.319 −0.101747
$$358$$ −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
$$363$$ 1983.91 0.286855
$$364$$ −1421.52 −0.204692
$$365$$ 0 0
$$366$$ −2362.14 −0.337352
$$367$$ 3470.59 0.493633 0.246816 0.969062i $$-0.420616\pi$$
0.246816 + 0.969062i $$0.420616\pi$$
$$368$$ −8538.52 −1.20951
$$369$$ 3520.65 0.496688
$$370$$ 0 0
$$371$$ −900.166 −0.125968
$$372$$ −7765.82 −1.08236
$$373$$ 11963.4 1.66070 0.830352 0.557240i $$-0.188140\pi$$
0.830352 + 0.557240i $$0.188140\pi$$
$$374$$ −2214.16 −0.306127
$$375$$ 0 0
$$376$$ −3375.85 −0.463021
$$377$$ 265.879 0.0363221
$$378$$ 1278.99 0.174032
$$379$$ 345.604 0.0468403 0.0234202 0.999726i $$-0.492544\pi$$
0.0234202 + 0.999726i $$0.492544\pi$$
$$380$$ 0 0
$$381$$ −933.055 −0.125464
$$382$$ −6189.99 −0.829078
$$383$$ 3386.40 0.451793 0.225897 0.974151i $$-0.427469\pi$$
0.225897 + 0.974151i $$0.427469\pi$$
$$384$$ −2709.87 −0.360124
$$385$$ 0 0
$$386$$ 1555.49 0.205110
$$387$$ −1362.67 −0.178988
$$388$$ −11903.9 −1.55755
$$389$$ −1629.88 −0.212438 −0.106219 0.994343i $$-0.533874\pi$$
−0.106219 + 0.994343i $$0.533874\pi$$
$$390$$ 0 0
$$391$$ 3667.47 0.474353
$$392$$ 1562.66 0.201343
$$393$$ −6002.95 −0.770506
$$394$$ −17987.0 −2.29993
$$395$$ 0 0
$$396$$ 2262.94 0.287165
$$397$$ −7938.94 −1.00364 −0.501819 0.864973i $$-0.667336\pi$$
−0.501819 + 0.864973i $$0.667336\pi$$
$$398$$ −17485.7 −2.20221
$$399$$ −5223.52 −0.655396
$$400$$ 0 0
$$401$$ 214.402 0.0267001 0.0133500 0.999911i $$-0.495750\pi$$
0.0133500 + 0.999911i $$0.495750\pi$$
$$402$$ 7692.51 0.954396
$$403$$ −3463.52 −0.428114
$$404$$ 6270.93 0.772253
$$405$$ 0 0
$$406$$ 968.819 0.118428
$$407$$ −3001.48 −0.365548
$$408$$ 440.488 0.0534495
$$409$$ −4783.73 −0.578338 −0.289169 0.957278i $$-0.593379\pi$$
−0.289169 + 0.957278i $$0.593379\pi$$
$$410$$ 0 0
$$411$$ 3115.58 0.373917
$$412$$ 4966.25 0.593859
$$413$$ −9832.99 −1.17155
$$414$$ −6834.51 −0.811347
$$415$$ 0 0
$$416$$ −3340.79 −0.393739
$$417$$ 8575.39 1.00705
$$418$$ −16851.8 −1.97189
$$419$$ −9903.67 −1.15472 −0.577358 0.816491i $$-0.695916\pi$$
−0.577358 + 0.816491i $$0.695916\pi$$
$$420$$ 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
$$423$$ 4206.28 0.483491
$$424$$ 577.738 0.0661732
$$425$$ 0 0
$$426$$ 3132.01 0.356213
$$427$$ −2105.32 −0.238603
$$428$$ −5909.28 −0.667374
$$429$$ 1009.26 0.113584
$$430$$ 0 0
$$431$$ −13672.6 −1.52805 −0.764023 0.645189i $$-0.776779\pi$$
−0.764023 + 0.645189i $$0.776779\pi$$
$$432$$ 1277.81 0.142311
$$433$$ −7113.10 −0.789455 −0.394727 0.918798i $$-0.629161\pi$$
−0.394727 + 0.918798i $$0.629161\pi$$
$$434$$ −12620.5 −1.39586
$$435$$ 0 0
$$436$$ −12631.3 −1.38745
$$437$$ 27912.9 3.05550
$$438$$ −10768.0 −1.17469
$$439$$ −6022.04 −0.654707 −0.327353 0.944902i $$-0.606157\pi$$
−0.327353 + 0.944902i $$0.606157\pi$$
$$440$$ 0 0
$$441$$ −1947.07 −0.210244
$$442$$ 1112.28 0.119696
$$443$$ 12994.4 1.39364 0.696821 0.717245i $$-0.254597\pi$$
0.696821 + 0.717245i $$0.254597\pi$$
$$444$$ 3380.72 0.361356
$$445$$ 0 0
$$446$$ −9206.20 −0.977413
$$447$$ −2230.44 −0.236009
$$448$$ −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
$$453$$ −6831.72 −0.708570
$$454$$ 17419.9 1.80078
$$455$$ 0 0
$$456$$ 3352.52 0.344290
$$457$$ −9834.10 −1.00661 −0.503304 0.864109i $$-0.667882\pi$$
−0.503304 + 0.864109i $$0.667882\pi$$
$$458$$ 3516.05 0.358721
$$459$$ −548.845 −0.0558124
$$460$$ 0 0
$$461$$ 3401.42 0.343644 0.171822 0.985128i $$-0.445035\pi$$
0.171822 + 0.985128i $$0.445035\pi$$
$$462$$ 3677.59 0.370339
$$463$$ −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 $$0.370982\pi$$
$$468$$ −1136.78 −0.112282
$$469$$ 6856.16 0.675028
$$470$$ 0 0
$$471$$ 9520.54 0.931387
$$472$$ 6310.94 0.615433
$$473$$ −3918.20 −0.380886
$$474$$ −4182.37 −0.405280
$$475$$ 0 0
$$476$$ 2222.78 0.214036
$$477$$ −719.858 −0.0690986
$$478$$ −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
$$481$$ 1507.79 0.142930
$$482$$ −13745.0 −1.29889
$$483$$ −6091.45 −0.573852
$$484$$ −6425.29 −0.603427
$$485$$ 0 0
$$486$$ 1022.80 0.0954632
$$487$$ 11684.7 1.08723 0.543617 0.839334i $$-0.317055\pi$$
0.543617 + 0.839334i $$0.317055\pi$$
$$488$$ 1351.22 0.125342
$$489$$ −6943.94 −0.642159
$$490$$ 0 0
$$491$$ −3954.70 −0.363489 −0.181745 0.983346i $$-0.558174\pi$$
−0.181745 + 0.983346i $$0.558174\pi$$
$$492$$ −11402.3 −1.04483
$$493$$ −415.744 −0.0379801
$$494$$ 8465.47 0.771011
$$495$$ 0 0
$$496$$ −12608.8 −1.14144
$$497$$ 2791.49 0.251943
$$498$$ 5498.27 0.494746
$$499$$ −5690.37 −0.510493 −0.255246 0.966876i $$-0.582157\pi$$
−0.255246 + 0.966876i $$0.582157\pi$$
$$500$$ 0 0
$$501$$ −7996.95 −0.713128
$$502$$ 26782.8 2.38123
$$503$$ −10859.1 −0.962595 −0.481298 0.876557i $$-0.659834\pi$$
−0.481298 + 0.876557i $$0.659834\pi$$
$$504$$ −731.623 −0.0646609
$$505$$ 0 0
$$506$$ −19651.9 −1.72655
$$507$$ −507.000 −0.0444116
$$508$$ 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
$$513$$ −4177.22 −0.359510
$$514$$ −25613.6 −2.19799
$$515$$ 0 0
$$516$$ 4413.27 0.376518
$$517$$ 12094.7 1.02887
$$518$$ 5494.13 0.466020
$$519$$ −495.730 −0.0419271
$$520$$ 0 0
$$521$$ 17297.5 1.45454 0.727271 0.686350i $$-0.240788\pi$$
0.727271 + 0.686350i $$0.240788\pi$$
$$522$$ 774.759 0.0649622
$$523$$ 5016.11 0.419386 0.209693 0.977767i $$-0.432753\pi$$
0.209693 + 0.977767i $$0.432753\pi$$
$$524$$ 19441.8 1.62084
$$525$$ 0 0
$$526$$ 518.667 0.0429942
$$527$$ 5415.77 0.447656
$$528$$ 3674.19 0.302839
$$529$$ 20383.8 1.67533
$$530$$ 0 0
$$531$$ −7863.39 −0.642640
$$532$$ 16917.4 1.37869
$$533$$ −5085.39 −0.413269
$$534$$ 3273.38 0.265268
$$535$$ 0 0
$$536$$ −4400.37 −0.354603
$$537$$ −2137.02 −0.171730
$$538$$ 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
$$543$$ −6619.59 −0.523156
$$544$$ 5223.86 0.411712
$$545$$ 0 0
$$546$$ −1847.43 −0.144803
$$547$$ 18414.9 1.43943 0.719713 0.694271i $$-0.244273\pi$$
0.719713 + 0.694271i $$0.244273\pi$$
$$548$$ −10090.4 −0.786571
$$549$$ −1683.61 −0.130883
$$550$$ 0 0
$$551$$ −3164.20 −0.244645
$$552$$ 3909.57 0.301453
$$553$$ −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 $$0.600696\pi$$
$$558$$ −10092.5 −0.765683
$$559$$ 1968.30 0.148927
$$560$$ 0 0
$$561$$ −1578.14 −0.118769
$$562$$ −19748.5 −1.48228
$$563$$ 1880.07 0.140738 0.0703690 0.997521i $$-0.477582\pi$$
0.0703690 + 0.997521i $$0.477582\pi$$
$$564$$ −13622.9 −1.01707
$$565$$ 0 0
$$566$$ 14586.9 1.08327
$$567$$ 911.598 0.0675194
$$568$$ −1791.62 −0.132350
$$569$$ 10118.3 0.745485 0.372743 0.927935i $$-0.378417\pi$$
0.372743 + 0.927935i $$0.378417\pi$$
$$570$$ 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
$$573$$ −4411.92 −0.321659
$$574$$ −18530.3 −1.34746
$$575$$ 0 0
$$576$$ −6327.42 −0.457713
$$577$$ −20508.1 −1.47966 −0.739831 0.672793i $$-0.765094\pi$$
−0.739831 + 0.672793i $$0.765094\pi$$
$$578$$ 18939.8 1.36296
$$579$$ 1108.68 0.0795771
$$580$$ 0 0
$$581$$ 4900.49 0.349925
$$582$$ −15470.4 −1.10184
$$583$$ −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 $$0.432708\pi$$
$$588$$ 6305.97 0.442268
$$589$$ 41219.0 2.88353
$$590$$ 0 0
$$591$$ −12820.2 −0.892308
$$592$$ 5489.06 0.381080
$$593$$ 14659.5 1.01517 0.507584 0.861602i $$-0.330539\pi$$
0.507584 + 0.861602i $$0.330539\pi$$
$$594$$ 2940.95 0.203146
$$595$$ 0 0
$$596$$ 7223.72 0.496468
$$597$$ −12462.9 −0.854394
$$598$$ 9872.07 0.675082
$$599$$ 23635.9 1.61225 0.806125 0.591746i $$-0.201561\pi$$
0.806125 + 0.591746i $$0.201561\pi$$
$$600$$ 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
$$603$$ 5482.83 0.370279
$$604$$ 22125.9 1.49055
$$605$$ 0 0
$$606$$ 8149.77 0.546306
$$607$$ 5098.56 0.340930 0.170465 0.985364i $$-0.445473\pi$$
0.170465 + 0.985364i $$0.445473\pi$$
$$608$$ 39758.4 2.65200
$$609$$ 690.525 0.0459466
$$610$$ 0 0
$$611$$ −6075.74 −0.402288
$$612$$ 1777.55 0.117407
$$613$$ −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 $$-0.706760\pi$$
−0.604832 + 0.796353i $$0.706760\pi$$
$$618$$ 6454.20 0.420107
$$619$$ 25684.9 1.66779 0.833897 0.551920i $$-0.186105\pi$$
0.833897 + 0.551920i $$0.186105\pi$$
$$620$$ 0 0
$$621$$ −4871.30 −0.314780
$$622$$ 6391.51 0.412020
$$623$$ 2917.49 0.187619
$$624$$ −1845.72 −0.118410
$$625$$ 0 0
$$626$$ 17046.1 1.08834
$$627$$ −12011.1 −0.765038
$$628$$ −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
$$633$$ −3694.77 −0.231997
$$634$$ 13696.1 0.857950
$$635$$ 0 0
$$636$$ 2331.40 0.145356
$$637$$ 2812.43 0.174933
$$638$$ 2227.73 0.138239
$$639$$ 2232.34 0.138200
$$640$$ 0 0
$$641$$ 6827.81 0.420721 0.210361 0.977624i $$-0.432536\pi$$
0.210361 + 0.977624i $$0.432536\pi$$
$$642$$ −7679.77 −0.472113
$$643$$ 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 $$0.647866\pi$$
$$648$$ −585.075 −0.0354690
$$649$$ −22610.3 −1.36754
$$650$$ 0 0
$$651$$ −8995.26 −0.541554
$$652$$ 22489.3 1.35084
$$653$$ −10909.0 −0.653755 −0.326878 0.945067i $$-0.605997\pi$$
−0.326878 + 0.945067i $$0.605997\pi$$
$$654$$ −16415.8 −0.981509
$$655$$ 0 0
$$656$$ −18513.2 −1.10186
$$657$$ −7674.89 −0.455747
$$658$$ −22139.0 −1.31166
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ 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
$$663$$ 792.776 0.0464387
$$664$$ −3145.19 −0.183821
$$665$$ 0 0
$$666$$ 4393.63 0.255630
$$667$$ −3689.95 −0.214206
$$668$$ 25899.7 1.50013
$$669$$ −6561.72 −0.379209
$$670$$ 0 0
$$671$$ −4841.04 −0.278519
$$672$$ −8676.51 −0.498071
$$673$$ 24152.5 1.38337 0.691687 0.722197i $$-0.256868\pi$$
0.691687 + 0.722197i $$0.256868\pi$$
$$674$$ −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 $$0.356892\pi$$
$$678$$ −532.806 −0.0301804
$$679$$ −13788.4 −0.779310
$$680$$ 0 0
$$681$$ 12416.0 0.698652
$$682$$ −29020.0 −1.62937
$$683$$ −11399.6 −0.638646 −0.319323 0.947646i $$-0.603455\pi$$
−0.319323 + 0.947646i $$0.603455\pi$$
$$684$$ 13528.8 0.756265
$$685$$ 0 0
$$686$$ 26495.9 1.47466
$$687$$ 2506.06 0.139174
$$688$$ 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
$$693$$ 2621.20 0.143681
$$694$$ 912.936 0.0499345
$$695$$ 0 0
$$696$$ −443.188 −0.0241365
$$697$$ 7951.82 0.432133
$$698$$ 20244.8 1.09782
$$699$$ 11056.5 0.598279
$$700$$ 0 0
$$701$$ −12670.4 −0.682673 −0.341336 0.939941i $$-0.610880\pi$$
−0.341336 + 0.939941i $$0.610880\pi$$
$$702$$ −1477.38 −0.0794302
$$703$$ −17944.0 −0.962692
$$704$$ −18193.8 −0.974012
$$705$$ 0 0
$$706$$ −12036.4 −0.641635
$$707$$ 7263.71 0.386393
$$708$$ 25467.2 1.35186
$$709$$ 13075.2 0.692594 0.346297 0.938125i $$-0.387439\pi$$
0.346297 + 0.938125i $$0.387439\pi$$
$$710$$ 0 0
$$711$$ −2980.99 −0.157237
$$712$$ −1872.48 −0.0985592
$$713$$ 48067.8 2.52476
$$714$$ 2888.75 0.151413
$$715$$ 0 0
$$716$$ 6921.16 0.361251
$$717$$ −9078.62 −0.472869
$$718$$ −15514.7 −0.806412
$$719$$ −2988.41 −0.155005 −0.0775026 0.996992i $$-0.524695\pi$$
−0.0775026 + 0.996992i $$0.524695\pi$$
$$720$$ 0 0
$$721$$ 5752.48 0.297134
$$722$$ −71877.0 −3.70496
$$723$$ −9796.73 −0.503934
$$724$$ 21438.9 1.10051
$$725$$ 0 0
$$726$$ −8350.37 −0.426875
$$727$$ 5507.46 0.280963 0.140482 0.990083i $$-0.455135\pi$$
0.140482 + 0.990083i $$0.455135\pi$$
$$728$$ 1056.79 0.0538011
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −3077.75 −0.155725
$$732$$ 5452.71 0.275325
$$733$$ 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
$$738$$ −14818.6 −0.739133
$$739$$ 6425.89 0.319865 0.159933 0.987128i $$-0.448872\pi$$
0.159933 + 0.987128i $$0.448872\pi$$
$$740$$ 0 0
$$741$$ 6033.76 0.299131
$$742$$ 3788.84 0.187457
$$743$$ −20411.0 −1.00782 −0.503908 0.863757i $$-0.668105\pi$$
−0.503908 + 0.863757i $$0.668105\pi$$
$$744$$ 5773.27 0.284487
$$745$$ 0 0
$$746$$ −50354.6 −2.47133
$$747$$ 3918.89 0.191947
$$748$$ 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
$$753$$ 19089.5 0.923850
$$754$$ −1119.10 −0.0540518
$$755$$ 0 0
$$756$$ −2952.39 −0.142034
$$757$$ −9295.39 −0.446297 −0.223148 0.974785i $$-0.571633\pi$$
−0.223148 + 0.974785i $$0.571633\pi$$
$$758$$ −1454.66 −0.0697042
$$759$$ −14006.9 −0.669851
$$760$$ 0 0
$$761$$ −21974.7 −1.04676 −0.523378 0.852101i $$-0.675328\pi$$
−0.523378 + 0.852101i $$0.675328\pi$$
$$762$$ 3927.27 0.186706
$$763$$ −14631.0 −0.694204
$$764$$ 14288.9 0.676641
$$765$$ 0 0
$$766$$ −14253.5 −0.672324
$$767$$ 11358.2 0.534709
$$768$$ −5467.13 −0.256873
$$769$$ 22987.4 1.07795 0.538977 0.842320i $$-0.318811\pi$$
0.538977 + 0.842320i $$0.318811\pi$$
$$770$$ 0 0
$$771$$ −18256.1 −0.852759
$$772$$ −3590.68 −0.167398
$$773$$ −31970.9 −1.48760 −0.743799 0.668404i $$-0.766978\pi$$
−0.743799 + 0.668404i $$0.766978\pi$$
$$774$$ 5735.53 0.266356
$$775$$ 0 0
$$776$$ 8849.59 0.409384
$$777$$ 3915.94 0.180803
$$778$$ 6860.25 0.316133
$$779$$ 60520.8 2.78354
$$780$$ 0 0
$$781$$ 6418.85 0.294090
$$782$$ −15436.6 −0.705896
$$783$$ 552.209 0.0252035
$$784$$ 10238.6 0.466408
$$785$$ 0 0
$$786$$ 25266.7 1.14661
$$787$$ −6087.26 −0.275715 −0.137857 0.990452i $$-0.544022\pi$$
−0.137857 + 0.990452i $$0.544022\pi$$
$$788$$ 41520.9 1.87706
$$789$$ 369.680 0.0166805
$$790$$ 0 0
$$791$$ −474.878 −0.0213460
$$792$$ −1682.32 −0.0754780
$$793$$ 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 $$0.671423\pi$$
$$798$$ 21986.1 0.975311
$$799$$ 9500.41 0.420651
$$800$$ 0 0
$$801$$ 2333.10 0.102916
$$802$$ −902.429 −0.0397330
$$803$$ −22068.3 −0.969829
$$804$$ −17757.3 −0.778918
$$805$$ 0 0
$$806$$ 14578.1 0.637087
$$807$$ 5807.37 0.253320
$$808$$ −4661.94 −0.202978
$$809$$ −32377.8 −1.40710 −0.703550 0.710646i $$-0.748403\pi$$
−0.703550 + 0.710646i $$0.748403\pi$$
$$810$$ 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
$$813$$ 13838.1 0.596952
$$814$$ 12633.4 0.543980
$$815$$ 0 0
$$816$$ 2886.08 0.123815
$$817$$ −23424.5 −1.00308
$$818$$ 20135.0 0.860638
$$819$$ −1316.75 −0.0561796
$$820$$ 0 0
$$821$$ 35355.3 1.50294 0.751468 0.659770i $$-0.229346\pi$$
0.751468 + 0.659770i $$0.229346\pi$$
$$822$$ −13113.6 −0.556435
$$823$$ 12663.3 0.536347 0.268173 0.963371i $$-0.413580\pi$$
0.268173 + 0.963371i $$0.413580\pi$$
$$824$$ −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 $$0.388697\pi$$
$$828$$ 15776.7 0.662170
$$829$$ 13638.9 0.571411 0.285705 0.958318i $$-0.407772\pi$$
0.285705 + 0.958318i $$0.407772\pi$$
$$830$$ 0 0
$$831$$ −17502.9 −0.730649
$$832$$ 9139.61 0.380840
$$833$$ −4397.69 −0.182918
$$834$$ −36094.2 −1.49861
$$835$$ 0 0
$$836$$ 38900.5 1.60933
$$837$$ −7193.46 −0.297064
$$838$$ 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
$$843$$ −14075.7 −0.575082
$$844$$ 11966.3 0.488028
$$845$$ 0 0
$$846$$ −17704.5 −0.719494
$$847$$ −7442.50 −0.301921
$$848$$ 3785.35 0.153289
$$849$$ 10396.8 0.420279
$$850$$ 0 0
$$851$$ −20925.6 −0.842914
$$852$$ −7229.89 −0.290718
$$853$$ −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 $$0.406629\pi$$
$$858$$ −4248.03 −0.169027
$$859$$ 29639.8 1.17730 0.588648 0.808389i $$-0.299660\pi$$
0.588648 + 0.808389i $$0.299660\pi$$
$$860$$ 0 0
$$861$$ −13207.5 −0.522776
$$862$$ 57548.8 2.27392
$$863$$ −21528.8 −0.849186 −0.424593 0.905384i $$-0.639583\pi$$
−0.424593 + 0.905384i $$0.639583\pi$$
$$864$$ −6938.56 −0.273211
$$865$$ 0 0
$$866$$ 29939.4 1.17481
$$867$$ 13499.4 0.528792
$$868$$ 29132.9 1.13921
$$869$$ −8571.50 −0.334601
$$870$$ 0 0
$$871$$ −7919.65 −0.308091
$$872$$ 9390.36 0.364676
$$873$$ −11026.5 −0.427482
$$874$$ −117487. −4.54696
$$875$$ 0 0
$$876$$ 24856.7 0.958708
$$877$$ −14865.3 −0.572366 −0.286183 0.958175i $$-0.592387\pi$$
−0.286183 + 0.958175i $$0.592387\pi$$
$$878$$ 25347.1 0.974285
$$879$$ 8031.92 0.308202
$$880$$ 0 0
$$881$$ −21336.0 −0.815921 −0.407961 0.913000i $$-0.633760\pi$$
−0.407961 + 0.913000i $$0.633760\pi$$
$$882$$ 8195.30 0.312869
$$883$$ −37538.2 −1.43065 −0.715323 0.698794i $$-0.753720\pi$$
−0.715323 + 0.698794i $$0.753720\pi$$
$$884$$ −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 $$0.727081\pi$$
$$888$$ −2513.30 −0.0949784
$$889$$ 3500.29 0.132054
$$890$$ 0 0
$$891$$ 2096.16 0.0788148
$$892$$ 21251.4 0.797703
$$893$$ 72306.9 2.70958
$$894$$ 9388.02 0.351211
$$895$$ 0 0
$$896$$ 10165.9 0.379039
$$897$$ 7036.32 0.261913
$$898$$ −46233.3 −1.71807
$$899$$ −5448.96 −0.202150
$$900$$ 0 0
$$901$$ −1625.89 −0.0601178
$$902$$ −42609.2 −1.57287
$$903$$ 5111.95 0.188389
$$904$$ 304.783 0.0112134
$$905$$ 0 0
$$906$$ 28755.0 1.05444
$$907$$ 10424.8 0.381641 0.190820 0.981625i $$-0.438885\pi$$
0.190820 + 0.981625i $$0.438885\pi$$
$$908$$ −40211.7 −1.46968
$$909$$ 5808.75 0.211952
$$910$$ 0 0
$$911$$ 10961.8 0.398661 0.199331 0.979932i $$-0.436123\pi$$
0.199331 + 0.979932i $$0.436123\pi$$
$$912$$ 21965.8 0.797543
$$913$$ 11268.3 0.408464
$$914$$ 41392.2 1.49796
$$915$$ 0 0
$$916$$ −8116.38 −0.292765
$$917$$ 22519.7 0.810976
$$918$$ 2310.12 0.0830558
$$919$$ −10779.2 −0.386914 −0.193457 0.981109i $$-0.561970\pi$$
−0.193457 + 0.981109i $$0.561970\pi$$
$$920$$ 0 0
$$921$$ 1415.74 0.0506519
$$922$$ −14316.7 −0.511384
$$923$$ −3224.49 −0.114990
$$924$$ −8489.28 −0.302248
$$925$$ 0 0
$$926$$ 7321.32 0.259820
$$927$$ 4600.23 0.162990
$$928$$ −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
$$933$$ 4555.55 0.159852
$$934$$ −33499.1 −1.17358
$$935$$ 0 0
$$936$$ 845.109 0.0295120
$$937$$ 21300.1 0.742631 0.371315 0.928507i $$-0.378907\pi$$
0.371315 + 0.928507i $$0.378907\pi$$
$$938$$ −28857.9 −1.00453
$$939$$ 12149.6 0.422244
$$940$$ 0 0
$$941$$ 26851.2 0.930207 0.465103 0.885256i $$-0.346017\pi$$
0.465103 + 0.885256i $$0.346017\pi$$
$$942$$ −40072.4 −1.38602
$$943$$ 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 $$0.456052\pi$$
$$948$$ 9654.52 0.330764
$$949$$ 11085.9 0.379204
$$950$$ 0 0
$$951$$ 9761.88 0.332861
$$952$$ −1652.46 −0.0562569
$$953$$ −35715.0 −1.21398 −0.606990 0.794709i $$-0.707623\pi$$
−0.606990 + 0.794709i $$0.707623\pi$$
$$954$$ 3029.92 0.102827
$$955$$ 0 0
$$956$$ 29402.9 0.994726
$$957$$ 1587.82 0.0536330
$$958$$ −35490.6 −1.19692
$$959$$ −11687.9 −0.393557
$$960$$ 0 0
$$961$$ 41190.9 1.38266
$$962$$ −6346.35 −0.212697
$$963$$ −5473.76 −0.183166
$$964$$ 31728.7 1.06007
$$965$$ 0 0
$$966$$ 25639.2 0.853963
$$967$$ −53338.8 −1.77380 −0.886898 0.461965i $$-0.847145\pi$$
−0.886898 + 0.461965i $$0.847145\pi$$
$$968$$ 4776.69 0.158604
$$969$$ −9434.77 −0.312785
$$970$$ 0 0
$$971$$ 23112.9 0.763882 0.381941 0.924187i $$-0.375256\pi$$
0.381941 + 0.924187i $$0.375256\pi$$
$$972$$ −2361.01 −0.0779110
$$973$$ −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 $$0.166864\pi$$
$$978$$ 29227.4 0.955612
$$979$$ 6708.57 0.219006
$$980$$ 0 0
$$981$$ −11700.3 −0.380798
$$982$$ 16645.5 0.540917
$$983$$ −45173.1 −1.46572 −0.732858 0.680381i $$-0.761814\pi$$
−0.732858 + 0.680381i $$0.761814\pi$$
$$984$$ 8476.73 0.274622
$$985$$ 0 0
$$986$$ 1749.89 0.0565190
$$987$$ −15779.6 −0.508885
$$988$$ −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
$$993$$ 10267.3 0.328121
$$994$$ −11749.5 −0.374922
$$995$$ 0 0
$$996$$ −12692.1 −0.403780
$$997$$ 18108.1 0.575214 0.287607 0.957749i $$-0.407140\pi$$
0.287607 + 0.957749i $$0.407140\pi$$
$$998$$ 23951.1 0.759677
$$999$$ 3131.56 0.0991773
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 975.4.a.l.1.1 3
5.4 even 2 39.4.a.c.1.3 3
15.14 odd 2 117.4.a.f.1.1 3
20.19 odd 2 624.4.a.t.1.1 3
35.34 odd 2 1911.4.a.k.1.3 3
40.19 odd 2 2496.4.a.bp.1.3 3
40.29 even 2 2496.4.a.bl.1.3 3
60.59 even 2 1872.4.a.bk.1.3 3
65.34 odd 4 507.4.b.g.337.6 6
65.44 odd 4 507.4.b.g.337.1 6
65.64 even 2 507.4.a.h.1.1 3
195.194 odd 2 1521.4.a.u.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 5.4 even 2
117.4.a.f.1.1 3 15.14 odd 2
507.4.a.h.1.1 3 65.64 even 2
507.4.b.g.337.1 6 65.44 odd 4
507.4.b.g.337.6 6 65.34 odd 4
624.4.a.t.1.1 3 20.19 odd 2
975.4.a.l.1.1 3 1.1 even 1 trivial
1521.4.a.u.1.3 3 195.194 odd 2
1872.4.a.bk.1.3 3 60.59 even 2
1911.4.a.k.1.3 3 35.34 odd 2
2496.4.a.bl.1.3 3 40.29 even 2
2496.4.a.bp.1.3 3 40.19 odd 2