# Properties

 Label 1911.4.a.k.1.3 Level $1911$ Weight $4$ Character 1911.1 Self dual yes Analytic conductor $112.753$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1911.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$112.752650021$$ 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 $$\chi$$ $$=$$ 1911.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +7.22315 q^{8} +9.00000 q^{9} +48.1187 q^{10} +25.8785 q^{11} -29.1483 q^{12} -13.0000 q^{13} -34.2966 q^{15} -47.3262 q^{16} +20.3276 q^{17} +37.8814 q^{18} -154.712 q^{19} +111.076 q^{20} +108.924 q^{22} -180.418 q^{23} -21.6695 q^{24} +5.69520 q^{25} -54.7176 q^{26} -27.0000 q^{27} -20.4522 q^{29} -144.356 q^{30} -266.424 q^{31} -256.984 q^{32} -77.6355 q^{33} +85.5599 q^{34} +87.4449 q^{36} +115.984 q^{37} -651.190 q^{38} +39.0000 q^{39} +82.5765 q^{40} -391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +102.890 q^{45} -759.390 q^{46} +467.365 q^{47} +141.979 q^{48} +23.9714 q^{50} -60.9828 q^{51} -126.309 q^{52} +79.9842 q^{53} -113.644 q^{54} +295.848 q^{55} +464.136 q^{57} -86.0843 q^{58} +873.710 q^{59} -333.229 q^{60} +187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} -148.619 q^{65} -326.772 q^{66} -609.204 q^{67} +197.505 q^{68} +541.255 q^{69} +248.038 q^{71} +65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} -17.0856 q^{75} -1503.20 q^{76} +164.153 q^{78} -331.221 q^{79} -541.043 q^{80} +81.0000 q^{81} -1646.51 q^{82} +435.432 q^{83} +232.389 q^{85} +637.281 q^{86} +61.3566 q^{87} +186.924 q^{88} -259.233 q^{89} +433.068 q^{90} -1752.96 q^{92} +799.273 q^{93} +1967.16 q^{94} -1768.70 q^{95} +770.951 q^{96} -1225.17 q^{97} +232.907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 6 * q^8 + 27 * q^9 $$3 q + 2 q^{2} - 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{8} + 27 q^{9} + 4 q^{10} - 16 q^{11} - 30 q^{12} - 39 q^{13} + 12 q^{15} - 110 q^{16} + 146 q^{17} + 18 q^{18} - 94 q^{19} + 244 q^{20} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} - 26 q^{26} - 81 q^{27} - 2 q^{29} - 12 q^{30} - 302 q^{31} + 154 q^{32} + 48 q^{33} - 164 q^{34} + 90 q^{36} + 374 q^{37} - 312 q^{38} + 117 q^{39} + 516 q^{40} - 480 q^{41} - 260 q^{43} + 712 q^{44} - 36 q^{45} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 814 q^{50} - 438 q^{51} - 130 q^{52} - 678 q^{53} - 54 q^{54} + 1552 q^{55} + 282 q^{57} - 628 q^{58} + 1788 q^{59} - 732 q^{60} - 230 q^{61} - 1952 q^{62} - 750 q^{64} + 52 q^{65} + 168 q^{66} + 74 q^{67} + 460 q^{68} + 144 q^{69} - 948 q^{71} - 54 q^{72} + 222 q^{73} + 1724 q^{74} - 435 q^{75} - 2392 q^{76} + 78 q^{78} - 24 q^{79} - 1100 q^{80} + 243 q^{81} - 564 q^{82} + 796 q^{83} - 248 q^{85} + 1800 q^{86} + 6 q^{87} + 1608 q^{88} - 1436 q^{89} + 36 q^{90} - 1296 q^{92} + 906 q^{93} + 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} - 144 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 6 * q^8 + 27 * q^9 + 4 * q^10 - 16 * q^11 - 30 * q^12 - 39 * q^13 + 12 * q^15 - 110 * q^16 + 146 * q^17 + 18 * q^18 - 94 * q^19 + 244 * q^20 - 56 * q^22 - 48 * q^23 + 18 * q^24 + 145 * q^25 - 26 * q^26 - 81 * q^27 - 2 * q^29 - 12 * q^30 - 302 * q^31 + 154 * q^32 + 48 * q^33 - 164 * q^34 + 90 * q^36 + 374 * q^37 - 312 * q^38 + 117 * q^39 + 516 * q^40 - 480 * q^41 - 260 * q^43 + 712 * q^44 - 36 * q^45 - 1104 * q^46 + 24 * q^47 + 330 * q^48 + 814 * q^50 - 438 * q^51 - 130 * q^52 - 678 * q^53 - 54 * q^54 + 1552 * q^55 + 282 * q^57 - 628 * q^58 + 1788 * q^59 - 732 * q^60 - 230 * q^61 - 1952 * q^62 - 750 * q^64 + 52 * q^65 + 168 * q^66 + 74 * q^67 + 460 * q^68 + 144 * q^69 - 948 * q^71 - 54 * q^72 + 222 * q^73 + 1724 * q^74 - 435 * q^75 - 2392 * q^76 + 78 * q^78 - 24 * q^79 - 1100 * q^80 + 243 * q^81 - 564 * q^82 + 796 * q^83 - 248 * q^85 + 1800 * q^86 + 6 * q^87 + 1608 * q^88 - 1436 * q^89 + 36 * q^90 - 1296 * q^92 + 906 * q^93 + 1920 * q^94 - 4032 * q^95 - 462 * q^96 - 3242 * q^97 - 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 $$-0.232897\pi$$
0.744062 + 0.668111i $$0.232897\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 9.71610 1.21451
$$5$$ 11.4322 1.02253 0.511264 0.859424i $$-0.329178\pi$$
0.511264 + 0.859424i $$0.329178\pi$$
$$6$$ −12.6271 −0.859169
$$7$$ 0 0
$$8$$ 7.22315 0.319221
$$9$$ 9.00000 0.333333
$$10$$ 48.1187 1.52165
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ −29.1483 −0.701199
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ −34.2966 −0.590356
$$16$$ −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 $$-0.883738\pi$$
−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 $$-0.804819\pi$$
−0.817823 + 0.575471i $$0.804819\pi$$
$$24$$ −21.6695 −0.184302
$$25$$ 5.69520 0.0455616
$$26$$ −54.7176 −0.412731
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −20.4522 −0.130961 −0.0654806 0.997854i $$-0.520858\pi$$
−0.0654806 + 0.997854i $$0.520858\pi$$
$$30$$ −144.356 −0.878523
$$31$$ −266.424 −1.54359 −0.771794 0.635873i $$-0.780640\pi$$
−0.771794 + 0.635873i $$0.780640\pi$$
$$32$$ −256.984 −1.41965
$$33$$ −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 $$-0.417045\pi$$
0.257670 + 0.966233i $$0.417045\pi$$
$$38$$ −651.190 −2.77992
$$39$$ 39.0000 0.160128
$$40$$ 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
$$45$$ 102.890 0.340842
$$46$$ −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
$$49$$ 0 0
$$50$$ 23.9714 0.0678012
$$51$$ −60.9828 −0.167437
$$52$$ −126.309 −0.336845
$$53$$ 79.9842 0.207296 0.103648 0.994614i $$-0.466949\pi$$
0.103648 + 0.994614i $$0.466949\pi$$
$$54$$ −113.644 −0.286390
$$55$$ 295.848 0.725312
$$56$$ 0 0
$$57$$ 464.136 1.07853
$$58$$ −86.0843 −0.194887
$$59$$ 873.710 1.92792 0.963960 0.266045i $$-0.0857171\pi$$
0.963960 + 0.266045i $$0.0857171\pi$$
$$60$$ −333.229 −0.716995
$$61$$ 187.068 0.392649 0.196325 0.980539i $$-0.437099\pi$$
0.196325 + 0.980539i $$0.437099\pi$$
$$62$$ −1121.39 −2.29705
$$63$$ 0 0
$$64$$ −703.047 −1.37314
$$65$$ −148.619 −0.283598
$$66$$ −326.772 −0.609437
$$67$$ −609.204 −1.11084 −0.555418 0.831571i $$-0.687442\pi$$
−0.555418 + 0.831571i $$0.687442\pi$$
$$68$$ 197.505 0.352221
$$69$$ 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
$$75$$ −17.0856 −0.0263050
$$76$$ −1503.20 −2.26880
$$77$$ 0 0
$$78$$ 164.153 0.238291
$$79$$ −331.221 −0.471712 −0.235856 0.971788i $$-0.575789\pi$$
−0.235856 + 0.971788i $$0.575789\pi$$
$$80$$ −541.043 −0.756130
$$81$$ 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 $$0.407036\pi$$
$$84$$ 0 0
$$85$$ 232.389 0.296543
$$86$$ 637.281 0.799067
$$87$$ 61.3566 0.0756105
$$88$$ 186.924 0.226434
$$89$$ −259.233 −0.308749 −0.154375 0.988012i $$-0.549336\pi$$
−0.154375 + 0.988012i $$0.549336\pi$$
$$90$$ 433.068 0.507216
$$91$$ 0 0
$$92$$ −1752.96 −1.98651
$$93$$ 799.273 0.891191
$$94$$ 1967.16 2.15848
$$95$$ −1768.70 −1.91015
$$96$$ 770.951 0.819634
$$97$$ −1225.17 −1.28245 −0.641223 0.767355i $$-0.721572\pi$$
−0.641223 + 0.767355i $$0.721572\pi$$
$$98$$ 0 0
$$99$$ 232.907 0.236444
$$100$$ 55.3351 0.0553351
$$101$$ −645.416 −0.635855 −0.317927 0.948115i $$-0.602987\pi$$
−0.317927 + 0.948115i $$0.602987\pi$$
$$102$$ −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 $$-0.411405\pi$$
0.274750 + 0.961516i $$0.411405\pi$$
$$108$$ −262.335 −0.233733
$$109$$ −1300.04 −1.14239 −0.571197 0.820813i $$-0.693521\pi$$
−0.571197 + 0.820813i $$0.693521\pi$$
$$110$$ 1245.24 1.07935
$$111$$ −347.951 −0.297532
$$112$$ 0 0
$$113$$ 42.1953 0.0351274 0.0175637 0.999846i $$-0.494409\pi$$
0.0175637 + 0.999846i $$0.494409\pi$$
$$114$$ 1953.57 1.60499
$$115$$ −2062.58 −1.67249
$$116$$ −198.716 −0.159054
$$117$$ −117.000 −0.0924500
$$118$$ 3677.49 2.86899
$$119$$ 0 0
$$120$$ −247.729 −0.188454
$$121$$ −661.303 −0.496847
$$122$$ 787.378 0.584311
$$123$$ 1173.55 0.860289
$$124$$ −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
$$129$$ −454.222 −0.310016
$$130$$ −625.543 −0.422029
$$131$$ −2000.98 −1.33456 −0.667278 0.744809i $$-0.732541\pi$$
−0.667278 + 0.744809i $$0.732541\pi$$
$$132$$ −754.314 −0.497384
$$133$$ 0 0
$$134$$ −2564.17 −1.65306
$$135$$ −308.669 −0.196785
$$136$$ 146.829 0.0925773
$$137$$ 1038.53 0.647644 0.323822 0.946118i $$-0.395032\pi$$
0.323822 + 0.946118i $$0.395032\pi$$
$$138$$ 2278.17 1.40529
$$139$$ 2858.46 1.74426 0.872128 0.489277i $$-0.162739\pi$$
0.872128 + 0.489277i $$0.162739\pi$$
$$140$$ 0 0
$$141$$ −1402.09 −0.837430
$$142$$ 1044.00 0.616978
$$143$$ −336.421 −0.196734
$$144$$ −425.936 −0.246491
$$145$$ −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 $$0.434479\pi$$
$$150$$ −71.9141 −0.0391451
$$151$$ 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
$$154$$ 0 0
$$155$$ −3045.82 −1.57836
$$156$$ 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
$$160$$ −2937.89 −1.45163
$$161$$ 0 0
$$162$$ 340.933 0.165347
$$163$$ −2314.65 −1.11225 −0.556126 0.831098i $$-0.687713\pi$$
−0.556126 + 0.831098i $$0.687713\pi$$
$$164$$ −3800.78 −1.80970
$$165$$ −887.545 −0.418759
$$166$$ 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
$$171$$ −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 $$0.452484\pi$$
$$180$$ 999.688 0.413957
$$181$$ −2206.53 −0.906133 −0.453066 0.891477i $$-0.649670\pi$$
−0.453066 + 0.891477i $$0.649670\pi$$
$$182$$ 0 0
$$183$$ −561.204 −0.226696
$$184$$ −1303.19 −0.522132
$$185$$ 1325.95 0.526949
$$186$$ 3364.18 1.32620
$$187$$ 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 $$-0.410141\pi$$
0.278565 + 0.960417i $$0.410141\pi$$
$$192$$ 2109.14 0.792782
$$193$$ 369.560 0.137832 0.0689158 0.997622i $$-0.478046\pi$$
0.0689158 + 0.997622i $$0.478046\pi$$
$$194$$ −5156.80 −1.90844
$$195$$ 445.856 0.163735
$$196$$ 0 0
$$197$$ −4273.41 −1.54552 −0.772761 0.634697i $$-0.781125\pi$$
−0.772761 + 0.634697i $$0.781125\pi$$
$$198$$ 980.315 0.351858
$$199$$ −4154.31 −1.47985 −0.739927 0.672687i $$-0.765140\pi$$
−0.739927 + 0.672687i $$0.765140\pi$$
$$200$$ 41.1373 0.0145442
$$201$$ 1827.61 0.641342
$$202$$ −2716.59 −0.946230
$$203$$ 0 0
$$204$$ −592.515 −0.203355
$$205$$ −4472.09 −1.52363
$$206$$ 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
$$215$$ 1730.92 0.549059
$$216$$ −195.025 −0.0614341
$$217$$ 0 0
$$218$$ −5471.92 −1.70002
$$219$$ 2558.30 0.789377
$$220$$ 2874.49 0.880901
$$221$$ −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 $$0.393489\pi$$
$$224$$ 0 0
$$225$$ 51.2568 0.0151872
$$226$$ 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 $$-0.461541\pi$$
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 $$-0.326630\pi$$
0.518124 + 0.855305i $$0.326630\pi$$
$$234$$ −492.459 −0.137577
$$235$$ 5343.01 1.48315
$$236$$ 8489.05 2.34148
$$237$$ 993.662 0.272343
$$238$$ 0 0
$$239$$ 3026.21 0.819034 0.409517 0.912303i $$-0.365697\pi$$
0.409517 + 0.912303i $$0.365697\pi$$
$$240$$ 1623.13 0.436552
$$241$$ −3265.58 −0.872839 −0.436420 0.899743i $$-0.643754\pi$$
−0.436420 + 0.899743i $$0.643754\pi$$
$$242$$ −2783.46 −0.739370
$$243$$ −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
$$250$$ −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
$$255$$ −697.168 −0.171209
$$256$$ 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
$$259$$ 0 0
$$260$$ −1443.99 −0.344433
$$261$$ −184.070 −0.0436538
$$262$$ −8422.24 −1.98598
$$263$$ 123.227 0.0288916 0.0144458 0.999896i $$-0.495402\pi$$
0.0144458 + 0.999896i $$0.495402\pi$$
$$264$$ −560.773 −0.130732
$$265$$ 914.395 0.211965
$$266$$ 0 0
$$267$$ 777.700 0.178256
$$268$$ −5919.08 −1.34913
$$269$$ 1935.79 0.438763 0.219381 0.975639i $$-0.429596\pi$$
0.219381 + 0.975639i $$0.429596\pi$$
$$270$$ −1299.20 −0.292841
$$271$$ 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
$$276$$ 5258.89 1.14691
$$277$$ −5834.30 −1.26552 −0.632761 0.774347i $$-0.718078\pi$$
−0.632761 + 0.774347i $$0.718078\pi$$
$$278$$ 12031.4 2.59567
$$279$$ −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
$$285$$ 5306.09 1.10283
$$286$$ −1416.01 −0.292764
$$287$$ 0 0
$$288$$ −2312.85 −0.473216
$$289$$ −4499.79 −0.915894
$$290$$ −984.133 −0.199277
$$291$$ 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 $$0.586003\pi$$
$$294$$ 0 0
$$295$$ 9988.43 1.97135
$$296$$ 837.767 0.164507
$$297$$ −698.720 −0.136511
$$298$$ 3129.34 0.608315
$$299$$ 2345.44 0.453646
$$300$$ −166.005 −0.0319477
$$301$$ 0 0
$$302$$ 9585.02 1.82634
$$303$$ 1936.25 0.367111
$$304$$ 7321.93 1.38139
$$305$$ 2138.60 0.401494
$$306$$ 770.039 0.143857
$$307$$ −471.915 −0.0877316 −0.0438658 0.999037i $$-0.513967\pi$$
−0.0438658 + 0.999037i $$0.513967\pi$$
$$308$$ 0 0
$$309$$ −1533.41 −0.282306
$$310$$ −12820.0 −2.34880
$$311$$ 1518.52 0.276872 0.138436 0.990371i $$-0.455793\pi$$
0.138436 + 0.990371i $$0.455793\pi$$
$$312$$ 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 $$-0.406921\pi$$
0.288266 + 0.957550i $$0.406921\pi$$
$$318$$ −1009.97 −0.178102
$$319$$ −529.272 −0.0928951
$$320$$ −8037.37 −1.40407
$$321$$ −1824.59 −0.317254
$$322$$ 0 0
$$323$$ −3144.92 −0.541759
$$324$$ 787.004 0.134946
$$325$$ −74.0375 −0.0126365
$$326$$ −9742.46 −1.65517
$$327$$ 3900.11 0.659561
$$328$$ −2825.58 −0.475660
$$329$$ 0 0
$$330$$ −3735.72 −0.623165
$$331$$ −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
$$335$$ −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
$$339$$ −126.586 −0.0202808
$$340$$ 2257.92 0.360155
$$341$$ −6894.66 −1.09492
$$342$$ −5860.71 −0.926640
$$343$$ 0 0
$$344$$ 1093.64 0.171410
$$345$$ 6187.74 0.965613
$$346$$ 695.518 0.108067
$$347$$ 216.898 0.0335554 0.0167777 0.999859i $$-0.494659\pi$$
0.0167777 + 0.999859i $$0.494659\pi$$
$$348$$ 596.147 0.0918299
$$349$$ 4809.84 0.737721 0.368861 0.929485i $$-0.379748\pi$$
0.368861 + 0.929485i $$0.379748\pi$$
$$350$$ 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
$$355$$ 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 $$0.412662\pi$$
$$360$$ 743.188 0.108804
$$361$$ 17076.8 2.48969
$$362$$ −9287.39 −1.34844
$$363$$ 1983.91 0.286855
$$364$$ 0 0
$$365$$ −9748.98 −1.39804
$$366$$ −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
$$370$$ 5580.98 0.784166
$$371$$ 0 0
$$372$$ 7765.82 1.08236
$$373$$ −11963.4 −1.66070 −0.830352 0.557240i $$-0.811860\pi$$
−0.830352 + 0.557240i $$0.811860\pi$$
$$374$$ 2214.16 0.306127
$$375$$ 4091.75 0.563459
$$376$$ 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
$$381$$ 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 $$0.533874\pi$$
$$390$$ 1876.63 0.243659
$$391$$ −3667.47 −0.474353
$$392$$ 0 0
$$393$$ 6002.95 0.770506
$$394$$ −17987.0 −2.29993
$$395$$ −3786.58 −0.482338
$$396$$ 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
$$399$$ 0 0
$$400$$ −269.532 −0.0336915
$$401$$ 214.402 0.0267001 0.0133500 0.999911i $$-0.495750\pi$$
0.0133500 + 0.999911i $$0.495750\pi$$
$$402$$ 7692.51 0.954396
$$403$$ 3463.52 0.428114
$$404$$ −6270.93 −0.772253
$$405$$ 926.008 0.113614
$$406$$ 0 0
$$407$$ 3001.48 0.365548
$$408$$ −440.488 −0.0534495
$$409$$ 4783.73 0.578338 0.289169 0.957278i $$-0.406621\pi$$
0.289169 + 0.957278i $$0.406621\pi$$
$$410$$ −18823.2 −2.26735
$$411$$ −3115.58 −0.373917
$$412$$ 4966.25 0.593859
$$413$$ 0 0
$$414$$ −6834.51 −0.811347
$$415$$ 4977.95 0.588815
$$416$$ 3340.79 0.393739
$$417$$ −8575.39 −1.00705
$$418$$ −16851.8 −1.97189
$$419$$ 9903.67 1.15472 0.577358 0.816491i $$-0.304084\pi$$
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
$$423$$ 4206.28 0.483491
$$424$$ 577.738 0.0661732
$$425$$ 115.770 0.0132133
$$426$$ −3132.01 −0.356213
$$427$$ 0 0
$$428$$ 5909.28 0.667374
$$429$$ 1009.26 0.113584
$$430$$ 7285.53 0.817068
$$431$$ −13672.6 −1.52805 −0.764023 0.645189i $$-0.776779\pi$$
−0.764023 + 0.645189i $$0.776779\pi$$
$$432$$ 1277.81 0.142311
$$433$$ −7113.10 −0.789455 −0.394727 0.918798i $$-0.629161\pi$$
−0.394727 + 0.918798i $$0.629161\pi$$
$$434$$ 0 0
$$435$$ 701.441 0.0773138
$$436$$ −12631.3 −1.38745
$$437$$ 27912.9 3.05550
$$438$$ 10768.0 1.17469
$$439$$ 6022.04 0.654707 0.327353 0.944902i $$-0.393843\pi$$
0.327353 + 0.944902i $$0.393843\pi$$
$$440$$ 2136.96 0.231535
$$441$$ 0 0
$$442$$ −1112.28 −0.119696
$$443$$ −12994.4 −1.39364 −0.696821 0.717245i $$-0.745403\pi$$
−0.696821 + 0.717245i $$0.745403\pi$$
$$444$$ −3380.72 −0.361356
$$445$$ −2963.61 −0.315704
$$446$$ 9206.20 0.977413
$$447$$ −2230.44 −0.236009
$$448$$ 0 0
$$449$$ 10984.3 1.15452 0.577260 0.816560i $$-0.304122\pi$$
0.577260 + 0.816560i $$0.304122\pi$$
$$450$$ 215.742 0.0226004
$$451$$ −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 $$-0.332118\pi$$
0.503304 + 0.864109i $$0.332118\pi$$
$$458$$ 3516.05 0.358721
$$459$$ −548.845 −0.0558124
$$460$$ −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
$$465$$ 9137.45 0.911267
$$466$$ 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
$$469$$ 0 0
$$470$$ 22489.0 2.20711
$$471$$ 9520.54 0.931387
$$472$$ 6310.94 0.615433
$$473$$ 3918.20 0.380886
$$474$$ 4182.37 0.405280
$$475$$ −881.114 −0.0851122
$$476$$ 0 0
$$477$$ 719.858 0.0690986
$$478$$ 12737.5 1.21882
$$479$$ −8431.98 −0.804315 −0.402158 0.915570i $$-0.631740\pi$$
−0.402158 + 0.915570i $$0.631740\pi$$
$$480$$ 8813.66 0.838097
$$481$$ −1507.79 −0.142930
$$482$$ −13745.0 −1.29889
$$483$$ 0 0
$$484$$ −6425.29 −0.603427
$$485$$ −14006.4 −1.31133
$$486$$ −1022.80 −0.0954632
$$487$$ −11684.7 −1.08723 −0.543617 0.839334i $$-0.682945\pi$$
−0.543617 + 0.839334i $$0.682945\pi$$
$$488$$ 1351.22 0.125342
$$489$$ 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
$$495$$ 2662.63 0.241771
$$496$$ 12608.8 1.14144
$$497$$ 0 0
$$498$$ −5498.27 −0.494746
$$499$$ −5690.37 −0.510493 −0.255246 0.966876i $$-0.582157\pi$$
−0.255246 + 0.966876i $$0.582157\pi$$
$$500$$ −13251.9 −1.18529
$$501$$ −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 $$0.659834\pi$$
$$504$$ 0 0
$$505$$ −7378.53 −0.650178
$$506$$ −19651.9 −1.72655
$$507$$ −507.000 −0.0444116
$$508$$ −3021.88 −0.263926
$$509$$ 18558.6 1.61610 0.808049 0.589115i $$-0.200524\pi$$
0.808049 + 0.589115i $$0.200524\pi$$
$$510$$ −2934.41 −0.254780
$$511$$ 0 0
$$512$$ 14896.8 1.28584
$$513$$ 4177.22 0.359510
$$514$$ 25613.6 2.19799
$$515$$ 5843.42 0.499984
$$516$$ −4413.27 −0.376518
$$517$$ 12094.7 1.02887
$$518$$ 0 0
$$519$$ −495.730 −0.0419271
$$520$$ −1073.49 −0.0905305
$$521$$ −17297.5 −1.45454 −0.727271 0.686350i $$-0.759212\pi$$
−0.727271 + 0.686350i $$0.759212\pi$$
$$522$$ −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
$$530$$ 3848.73 0.315431
$$531$$ 7863.39 0.642640
$$532$$ 0 0
$$533$$ 5085.39 0.413269
$$534$$ 3273.38 0.265268
$$535$$ 6953.01 0.561878
$$536$$ −4400.37 −0.354603
$$537$$ −2137.02 −0.171730
$$538$$ 8147.83 0.652933
$$539$$ 0 0
$$540$$ −2999.06 −0.238998
$$541$$ 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
$$545$$ −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
$$549$$ 1683.61 0.130883
$$550$$ 620.343 0.0480937
$$551$$ 3164.20 0.244645
$$552$$ 3909.57 0.301453
$$553$$ 0 0
$$554$$ −24556.9 −1.88325
$$555$$ −3977.84 −0.304234
$$556$$ 27773.1 2.11842
$$557$$ 8179.15 0.622193 0.311096 0.950378i $$-0.399304\pi$$
0.311096 + 0.950378i $$0.399304\pi$$
$$558$$ −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
$$565$$ 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 $$-0.378417\pi$$
0.372743 + 0.927935i $$0.378417\pi$$
$$570$$ 22333.6 1.64114
$$571$$ 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
$$574$$ 0 0
$$575$$ −1027.52 −0.0745225
$$576$$ −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
$$580$$ −2271.76 −0.162637
$$581$$ 0 0
$$582$$ 15470.4 1.10184
$$583$$ 2069.87 0.147042
$$584$$ −6159.65 −0.436452
$$585$$ −1337.57 −0.0945327
$$586$$ −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
$$591$$ 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 $$0.201561\pi$$
$$600$$ −123.412 −0.00839711
$$601$$ 11527.0 0.782356 0.391178 0.920315i $$-0.372068\pi$$
0.391178 + 0.920315i $$0.372068\pi$$
$$602$$ 0 0
$$603$$ −5482.83 −0.370279
$$604$$ 22125.9 1.49055
$$605$$ −7560.15 −0.508039
$$606$$ 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
$$609$$ 0 0
$$610$$ 9001.47 0.597473
$$611$$ −6075.74 −0.402288
$$612$$ 1777.55 0.117407
$$613$$ 1516.39 0.0999128 0.0499564 0.998751i $$-0.484092\pi$$
0.0499564 + 0.998751i $$0.484092\pi$$
$$614$$ −1986.31 −0.130556
$$615$$ 13416.3 0.879668
$$616$$ 0 0
$$617$$ 18539.3 1.20966 0.604832 0.796353i $$-0.293240\pi$$
0.604832 + 0.796353i $$0.293240\pi$$
$$618$$ −6454.20 −0.420107
$$619$$ −25684.9 −1.66779 −0.833897 0.551920i $$-0.813895\pi$$
−0.833897 + 0.551920i $$0.813895\pi$$
$$620$$ −29593.5 −1.91694
$$621$$ 4871.30 0.314780
$$622$$ 6391.51 0.412020
$$623$$ 0 0
$$624$$ −1845.72 −0.118410
$$625$$ −16304.5 −1.04349
$$626$$ −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
$$635$$ −3555.62 −0.222206
$$636$$ −2331.40 −0.145356
$$637$$ 0 0
$$638$$ −2227.73 −0.138239
$$639$$ 2232.34 0.138200
$$640$$ −10326.6 −0.637804
$$641$$ 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 $$0.247146\pi$$
$$644$$ 0 0
$$645$$ −5192.76 −0.316999
$$646$$ −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
$$650$$ −311.628 −0.0188047
$$651$$ 0 0
$$652$$ −22489.3 −1.35084
$$653$$ 10909.0 0.653755 0.326878 0.945067i $$-0.394003\pi$$
0.326878 + 0.945067i $$0.394003\pi$$
$$654$$ 16415.8 0.981509
$$655$$ −22875.7 −1.36462
$$656$$ 18513.2 1.10186
$$657$$ −7674.89 −0.455747
$$658$$ 0 0
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ −8623.47 −0.508588
$$661$$ −2224.23 −0.130881 −0.0654406 0.997856i $$-0.520845\pi$$
−0.0654406 + 0.997856i $$0.520845\pi$$
$$662$$ −14405.2 −0.845734
$$663$$ 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
$$670$$ −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
$$675$$ −153.770 −0.00876833
$$676$$ 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
$$679$$ 0 0
$$680$$ 1678.58 0.0946628
$$681$$ 12416.0 0.698652
$$682$$ −29020.0 −1.62937
$$683$$ 11399.6 0.638646 0.319323 0.947646i $$-0.396545\pi$$
0.319323 + 0.947646i $$0.396545\pi$$
$$684$$ −13528.8 −0.756265
$$685$$ 11872.6 0.662233
$$686$$ 0 0
$$687$$ −2506.06 −0.139174
$$688$$ −7165.54 −0.397069
$$689$$ −1039.79 −0.0574935
$$690$$ 26044.5 1.43695
$$691$$ 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
$$696$$ 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
$$705$$ −16029.0 −0.856295
$$706$$ 12036.4 0.641635
$$707$$ 0 0
$$708$$ −25467.2 −1.35186
$$709$$ 13075.2 0.692594 0.346297 0.938125i $$-0.387439\pi$$
0.346297 + 0.938125i $$0.387439\pi$$
$$710$$ 11935.3 0.630877
$$711$$ −2980.99 −0.157237
$$712$$ −1872.48 −0.0985592
$$713$$ 48067.8 2.52476
$$714$$ 0 0
$$715$$ −3846.03 −0.201165
$$716$$ 6921.16 0.361251
$$717$$ −9078.62 −0.472869
$$718$$ 15514.7 0.806412
$$719$$ 2988.41 0.155005 0.0775026 0.996992i $$-0.475305\pi$$
0.0775026 + 0.996992i $$0.475305\pi$$
$$720$$ −4869.38 −0.252043
$$721$$ 0 0
$$722$$ 71877.0 3.70496
$$723$$ 9796.73 0.503934
$$724$$ −21438.9 −1.10051
$$725$$ −116.479 −0.00596680
$$726$$ 8350.37 0.426875
$$727$$ 5507.46 0.280963 0.140482 0.990083i $$-0.455135\pi$$
0.140482 + 0.990083i $$0.455135\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −41033.9 −2.08046
$$731$$ 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 $$0.448872\pi$$
$$740$$ 12883.0 0.639986
$$741$$ −6033.76 −0.299131
$$742$$ 0 0
$$743$$ 20411.0 1.00782 0.503908 0.863757i $$-0.331895\pi$$
0.503908 + 0.863757i $$0.331895\pi$$
$$744$$ 5773.27 0.284487
$$745$$ 8499.60 0.417988
$$746$$ −50354.6 −2.47133
$$747$$ 3918.89 0.191947
$$748$$ 5111.13 0.249842
$$749$$ 0 0
$$750$$ 17222.4 0.838496
$$751$$ −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
$$755$$ 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
$$759$$ 14006.9 0.669851
$$760$$ −12775.6 −0.609761
$$761$$ 21974.7 1.04676 0.523378 0.852101i $$-0.324672\pi$$
0.523378 + 0.852101i $$0.324672\pi$$
$$762$$ 3927.27 0.186706
$$763$$ 0 0
$$764$$ 14288.9 0.676641
$$765$$ 2091.50 0.0988476
$$766$$ 14253.5 0.672324
$$767$$ −11358.2 −0.534709
$$768$$ −5467.13 −0.256873
$$769$$ −22987.4 −1.07795 −0.538977 0.842320i $$-0.681189\pi$$
−0.538977 + 0.842320i $$0.681189\pi$$
$$770$$ 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
$$775$$ −1517.34 −0.0703283
$$776$$ −8849.59 −0.409384
$$777$$ 0 0
$$778$$ −6860.25 −0.316133
$$779$$ 60520.8 2.78354
$$780$$ 4331.98 0.198859
$$781$$ 6418.85 0.294090
$$782$$ −15436.6 −0.705896
$$783$$ 552.209 0.0252035
$$784$$ 0 0
$$785$$ −36280.2 −1.64955
$$786$$ 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
$$790$$ −15937.9 −0.717779
$$791$$ 0 0
$$792$$ 1682.32 0.0754780
$$793$$ −2431.88 −0.108901
$$794$$ −33415.4 −1.49354
$$795$$ −2743.19 −0.122378
$$796$$ −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
$$801$$ −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 $$0.748403\pi$$
$$810$$ 3897.61 0.169072
$$811$$ −26352.8 −1.14103 −0.570513 0.821288i $$-0.693256\pi$$
−0.570513 + 0.821288i $$0.693256\pi$$
$$812$$ 0 0
$$813$$ −13838.1 −0.596952
$$814$$ 12633.4 0.543980
$$815$$ −26461.5 −1.13731
$$816$$ 2886.08 0.123815
$$817$$ −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 $$0.229346\pi$$
$$822$$ −13113.6 −0.556435
$$823$$ −12663.3 −0.536347 −0.268173 0.963371i $$-0.586420\pi$$
−0.268173 + 0.963371i $$0.586420\pi$$
$$824$$ 3692.02 0.156089
$$825$$ −442.149 −0.0186590
$$826$$ 0 0
$$827$$ −16295.2 −0.685176 −0.342588 0.939486i $$-0.611303\pi$$
−0.342588 + 0.939486i $$0.611303\pi$$
$$828$$ −15776.7 −0.662170
$$829$$ −13638.9 −0.571411 −0.285705 0.958318i $$-0.592228\pi$$
−0.285705 + 0.958318i $$0.592228\pi$$
$$830$$ 20952.4 0.876229
$$831$$ 17502.9 0.730649
$$832$$ 9139.61 0.380840
$$833$$ 0 0
$$834$$ −36094.2 −1.49861
$$835$$ 30474.2 1.26300
$$836$$ −38900.5 −1.60933
$$837$$ 7193.46 0.297064
$$838$$ 41685.0 1.71836
$$839$$ 1890.31 0.0777838 0.0388919 0.999243i $$-0.487617\pi$$
0.0388919 + 0.999243i $$0.487617\pi$$
$$840$$ 0 0
$$841$$ −23970.7 −0.982849
$$842$$ −51016.4 −2.08805
$$843$$ −14075.7 −0.575082
$$844$$ 11966.3 0.488028
$$845$$ 1932.04 0.0786559
$$846$$ 17704.5 0.719494
$$847$$ 0 0
$$848$$ −3785.35 −0.153289
$$849$$ 10396.8 0.420279
$$850$$ 487.280 0.0196630
$$851$$ −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 $$0.510352\pi$$
$$854$$ 0 0
$$855$$ −15918.3 −0.636718
$$856$$ 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 $$-0.700340\pi$$
−0.588648 + 0.808389i $$0.700340\pi$$
$$860$$ 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 $$0.360417\pi$$
$$864$$ 6938.56 0.273211
$$865$$ 1889.10 0.0742557
$$866$$ −29939.4 −1.17481
$$867$$ 13499.4 0.528792
$$868$$ 0 0
$$869$$ −8571.50 −0.334601
$$870$$ 2952.40 0.115053
$$871$$ 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 $$-0.407613\pi$$
0.286183 + 0.958175i $$0.407613\pi$$
$$878$$ 25347.1 0.974285
$$879$$ 8031.92 0.308202
$$880$$ −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
$$885$$ −29965.3 −1.13816
$$886$$ −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
$$889$$ 0 0
$$890$$ −12474.0 −0.469807
$$891$$ 2096.16 0.0788148
$$892$$ 21251.4 0.797703
$$893$$ −72306.9 −2.70958
$$894$$ −9388.02 −0.351211
$$895$$ 8143.61 0.304146
$$896$$ 0 0
$$897$$ −7036.32 −0.261913
$$898$$ 46233.3 1.71807
$$899$$ 5448.96 0.202150
$$900$$ 498.016 0.0184450
$$901$$ 1625.89 0.0601178
$$902$$ −42609.2 −1.57287
$$903$$ 0 0
$$904$$ 304.783 0.0112134
$$905$$ −25225.5 −0.926545
$$906$$ −28755.0 −1.05444
$$907$$ −10424.8 −0.381641 −0.190820 0.981625i $$-0.561115\pi$$
−0.190820 + 0.981625i $$0.561115\pi$$
$$908$$ −40211.7 −1.46968
$$909$$ −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
$$915$$ −6415.80 −0.231803
$$916$$ 8116.38 0.292765
$$917$$ 0 0
$$918$$ −2310.12 −0.0830558
$$919$$ −10779.2 −0.386914 −0.193457 0.981109i $$-0.561970\pi$$
−0.193457 + 0.981109i $$0.561970\pi$$
$$920$$ −14898.3 −0.533895
$$921$$ 1415.74 0.0506519
$$922$$ −14316.7 −0.511384
$$923$$ −3224.49 −0.114990
$$924$$ 0 0
$$925$$ 660.549 0.0234797
$$926$$ 7321.32 0.259820
$$927$$ 4600.23 0.162990
$$928$$ 5255.88 0.185919
$$929$$ −5429.07 −0.191735 −0.0958675 0.995394i $$-0.530563\pi$$
−0.0958675 + 0.995394i $$0.530563\pi$$
$$930$$ 38460.0 1.35608
$$931$$ 0 0
$$932$$ 35808.8 1.25854
$$933$$ −4555.55 −0.159852
$$934$$ 33499.1 1.17358
$$935$$ 6013.88 0.210348
$$936$$ −845.109 −0.0295120
$$937$$ 21300.1 0.742631 0.371315 0.928507i $$-0.378907\pi$$
0.371315 + 0.928507i $$0.378907\pi$$
$$938$$ 0 0
$$939$$ 12149.6 0.422244
$$940$$ 51913.2 1.80130
$$941$$ −26851.2 −0.930207 −0.465103 0.885256i $$-0.653983\pi$$
−0.465103 + 0.885256i $$0.653983\pi$$
$$942$$ 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 $$-0.543948\pi$$
−0.137629 + 0.990484i $$0.543948\pi$$
$$948$$ 9654.52 0.330764
$$949$$ 11085.9 0.379204
$$950$$ −3708.65 −0.126658
$$951$$ −9761.88 −0.332861
$$952$$ 0 0
$$953$$ 35715.0 1.21398 0.606990 0.794709i $$-0.292377\pi$$
0.606990 + 0.794709i $$0.292377\pi$$
$$954$$ 3029.92 0.102827
$$955$$ 16812.6 0.569680
$$956$$ 29402.9 0.994726
$$957$$ 1587.82 0.0536330
$$958$$ −35490.6 −1.19692
$$959$$ 0 0
$$960$$ 24112.1 0.810641
$$961$$ 41190.9 1.38266
$$962$$ −6346.35 −0.212697
$$963$$ 5473.76 0.183166
$$964$$ −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
$$969$$ 9434.77 0.312785
$$970$$ −58953.6 −1.95143
$$971$$ −23112.9 −0.763882 −0.381941 0.924187i $$-0.624744\pi$$
−0.381941 + 0.924187i $$0.624744\pi$$
$$972$$ −2361.01 −0.0779110
$$973$$ 0 0
$$974$$ −49181.3 −1.61794
$$975$$ 222.113 0.00729569
$$976$$ −8853.22 −0.290353
$$977$$ −52874.6 −1.73143 −0.865715 0.500538i $$-0.833136\pi$$
−0.865715 + 0.500538i $$0.833136\pi$$
$$978$$ 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
$$985$$ −48854.5 −1.58034
$$986$$ −1749.89 −0.0565190
$$987$$ 0 0
$$988$$ 19541.6 0.629251
$$989$$ −27316.7 −0.878281
$$990$$ 11207.2 0.359785
$$991$$ 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
$$994$$ 0 0
$$995$$ −47492.8 −1.51319
$$996$$ −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 1911.4.a.k.1.3 3
7.6 odd 2 39.4.a.c.1.3 3
21.20 even 2 117.4.a.f.1.1 3
28.27 even 2 624.4.a.t.1.1 3
35.34 odd 2 975.4.a.l.1.1 3
56.13 odd 2 2496.4.a.bl.1.3 3
56.27 even 2 2496.4.a.bp.1.3 3
84.83 odd 2 1872.4.a.bk.1.3 3
91.34 even 4 507.4.b.g.337.6 6
91.83 even 4 507.4.b.g.337.1 6
91.90 odd 2 507.4.a.h.1.1 3
273.272 even 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 7.6 odd 2
117.4.a.f.1.1 3 21.20 even 2
507.4.a.h.1.1 3 91.90 odd 2
507.4.b.g.337.1 6 91.83 even 4
507.4.b.g.337.6 6 91.34 even 4
624.4.a.t.1.1 3 28.27 even 2
975.4.a.l.1.1 3 35.34 odd 2
1521.4.a.u.1.3 3 273.272 even 2
1872.4.a.bk.1.3 3 84.83 odd 2
1911.4.a.k.1.3 3 1.1 even 1 trivial
2496.4.a.bl.1.3 3 56.13 odd 2
2496.4.a.bp.1.3 3 56.27 even 2