# Properties

 Label 147.4.a.l.1.1 Level $147$ Weight $4$ Character 147.1 Self dual yes Analytic conductor $8.673$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.57516.1 Defining polynomial: $$x^{3} - x^{2} - 24 x + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.55637$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.55637 q^{2} -3.00000 q^{3} +12.7605 q^{4} +17.8732 q^{5} +13.6691 q^{6} -21.6905 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.55637 q^{2} -3.00000 q^{3} +12.7605 q^{4} +17.8732 q^{5} +13.6691 q^{6} -21.6905 q^{8} +9.00000 q^{9} -81.4369 q^{10} -11.3942 q^{11} -38.2814 q^{12} -13.0987 q^{13} -53.6196 q^{15} -3.25412 q^{16} +53.2674 q^{17} -41.0073 q^{18} -42.4223 q^{19} +228.071 q^{20} +51.9159 q^{22} +152.085 q^{23} +65.0714 q^{24} +194.451 q^{25} +59.6823 q^{26} -27.0000 q^{27} +186.493 q^{29} +244.311 q^{30} -157.874 q^{31} +188.351 q^{32} +34.1825 q^{33} -242.706 q^{34} +114.844 q^{36} +3.74588 q^{37} +193.291 q^{38} +39.2960 q^{39} -387.678 q^{40} -39.3230 q^{41} +429.439 q^{43} -145.395 q^{44} +160.859 q^{45} -692.957 q^{46} +21.1869 q^{47} +9.76236 q^{48} -885.992 q^{50} -159.802 q^{51} -167.145 q^{52} +365.904 q^{53} +123.022 q^{54} -203.650 q^{55} +127.267 q^{57} -849.732 q^{58} -226.578 q^{59} -684.212 q^{60} +651.973 q^{61} +719.331 q^{62} -832.161 q^{64} -234.115 q^{65} -155.748 q^{66} +145.433 q^{67} +679.717 q^{68} -456.256 q^{69} -368.962 q^{71} -195.214 q^{72} +608.906 q^{73} -17.0676 q^{74} -583.354 q^{75} -541.328 q^{76} -179.047 q^{78} +910.237 q^{79} -58.1615 q^{80} +81.0000 q^{81} +179.170 q^{82} -327.929 q^{83} +952.058 q^{85} -1956.68 q^{86} -559.480 q^{87} +247.144 q^{88} -37.6118 q^{89} -732.932 q^{90} +1940.68 q^{92} +473.621 q^{93} -96.5352 q^{94} -758.222 q^{95} -565.052 q^{96} +722.013 q^{97} -102.547 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} + O(q^{10})$$ $$3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} - 55q^{10} + 35q^{11} - 75q^{12} + 62q^{13} - 33q^{15} + 241q^{16} + 48q^{17} + 9q^{18} - 202q^{19} + 439q^{20} - 7q^{22} + 216q^{23} - 117q^{24} + 130q^{25} + 274q^{26} - 81q^{27} + 53q^{29} + 165q^{30} - 95q^{31} + 683q^{32} - 105q^{33} - 24q^{34} + 225q^{36} + 262q^{37} - 398q^{38} - 186q^{39} + 21q^{40} + 244q^{41} + 360q^{43} - 905q^{44} + 99q^{45} - 1056q^{46} - 210q^{47} - 723q^{48} - 1378q^{50} - 144q^{51} + 324q^{52} + 393q^{53} - 27q^{54} - 1031q^{55} + 606q^{57} - 1249q^{58} + 1143q^{59} - 1317q^{60} - 70q^{61} + 1059q^{62} - 399q^{64} - 472q^{65} + 21q^{66} - 628q^{67} + 1944q^{68} - 648q^{69} + 318q^{71} + 351q^{72} + 988q^{73} + 1002q^{74} - 390q^{75} - 2340q^{76} - 822q^{78} + 861q^{79} + 175q^{80} + 243q^{81} + 124q^{82} + 519q^{83} + 1800q^{85} - 3208q^{86} - 159q^{87} - 891q^{88} + 1766q^{89} - 495q^{90} - 672q^{92} + 285q^{93} - 3294q^{94} - 736q^{95} - 2049q^{96} + 19q^{97} + 315q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.55637 −1.61092 −0.805459 0.592651i $$-0.798081\pi$$
−0.805459 + 0.592651i $$0.798081\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 12.7605 1.59506
$$5$$ 17.8732 1.59863 0.799314 0.600914i $$-0.205196\pi$$
0.799314 + 0.600914i $$0.205196\pi$$
$$6$$ 13.6691 0.930064
$$7$$ 0 0
$$8$$ −21.6905 −0.958592
$$9$$ 9.00000 0.333333
$$10$$ −81.4369 −2.57526
$$11$$ −11.3942 −0.312315 −0.156158 0.987732i $$-0.549911\pi$$
−0.156158 + 0.987732i $$0.549911\pi$$
$$12$$ −38.2814 −0.920908
$$13$$ −13.0987 −0.279455 −0.139728 0.990190i $$-0.544623\pi$$
−0.139728 + 0.990190i $$0.544623\pi$$
$$14$$ 0 0
$$15$$ −53.6196 −0.922968
$$16$$ −3.25412 −0.0508456
$$17$$ 53.2674 0.759955 0.379977 0.924996i $$-0.375932\pi$$
0.379977 + 0.924996i $$0.375932\pi$$
$$18$$ −41.0073 −0.536973
$$19$$ −42.4223 −0.512228 −0.256114 0.966647i $$-0.582442\pi$$
−0.256114 + 0.966647i $$0.582442\pi$$
$$20$$ 228.071 2.54991
$$21$$ 0 0
$$22$$ 51.9159 0.503114
$$23$$ 152.085 1.37878 0.689391 0.724389i $$-0.257878\pi$$
0.689391 + 0.724389i $$0.257878\pi$$
$$24$$ 65.0714 0.553443
$$25$$ 194.451 1.55561
$$26$$ 59.6823 0.450180
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 186.493 1.19417 0.597085 0.802178i $$-0.296325\pi$$
0.597085 + 0.802178i $$0.296325\pi$$
$$30$$ 244.311 1.48683
$$31$$ −157.874 −0.914676 −0.457338 0.889293i $$-0.651197\pi$$
−0.457338 + 0.889293i $$0.651197\pi$$
$$32$$ 188.351 1.04050
$$33$$ 34.1825 0.180315
$$34$$ −242.706 −1.22423
$$35$$ 0 0
$$36$$ 114.844 0.531686
$$37$$ 3.74588 0.0166438 0.00832188 0.999965i $$-0.497351\pi$$
0.00832188 + 0.999965i $$0.497351\pi$$
$$38$$ 193.291 0.825158
$$39$$ 39.2960 0.161344
$$40$$ −387.678 −1.53243
$$41$$ −39.3230 −0.149786 −0.0748930 0.997192i $$-0.523862\pi$$
−0.0748930 + 0.997192i $$0.523862\pi$$
$$42$$ 0 0
$$43$$ 429.439 1.52300 0.761498 0.648168i $$-0.224464\pi$$
0.761498 + 0.648168i $$0.224464\pi$$
$$44$$ −145.395 −0.498161
$$45$$ 160.859 0.532876
$$46$$ −692.957 −2.22111
$$47$$ 21.1869 0.0657537 0.0328768 0.999459i $$-0.489533\pi$$
0.0328768 + 0.999459i $$0.489533\pi$$
$$48$$ 9.76236 0.0293557
$$49$$ 0 0
$$50$$ −885.992 −2.50596
$$51$$ −159.802 −0.438760
$$52$$ −167.145 −0.445748
$$53$$ 365.904 0.948317 0.474158 0.880440i $$-0.342752\pi$$
0.474158 + 0.880440i $$0.342752\pi$$
$$54$$ 123.022 0.310021
$$55$$ −203.650 −0.499276
$$56$$ 0 0
$$57$$ 127.267 0.295735
$$58$$ −849.732 −1.92371
$$59$$ −226.578 −0.499964 −0.249982 0.968250i $$-0.580425\pi$$
−0.249982 + 0.968250i $$0.580425\pi$$
$$60$$ −684.212 −1.47219
$$61$$ 651.973 1.36847 0.684235 0.729262i $$-0.260136\pi$$
0.684235 + 0.729262i $$0.260136\pi$$
$$62$$ 719.331 1.47347
$$63$$ 0 0
$$64$$ −832.161 −1.62532
$$65$$ −234.115 −0.446745
$$66$$ −155.748 −0.290473
$$67$$ 145.433 0.265186 0.132593 0.991171i $$-0.457670\pi$$
0.132593 + 0.991171i $$0.457670\pi$$
$$68$$ 679.717 1.21217
$$69$$ −456.256 −0.796041
$$70$$ 0 0
$$71$$ −368.962 −0.616728 −0.308364 0.951268i $$-0.599782\pi$$
−0.308364 + 0.951268i $$0.599782\pi$$
$$72$$ −195.214 −0.319531
$$73$$ 608.906 0.976261 0.488130 0.872771i $$-0.337679\pi$$
0.488130 + 0.872771i $$0.337679\pi$$
$$74$$ −17.0676 −0.0268117
$$75$$ −583.354 −0.898133
$$76$$ −541.328 −0.817034
$$77$$ 0 0
$$78$$ −179.047 −0.259911
$$79$$ 910.237 1.29633 0.648163 0.761502i $$-0.275538\pi$$
0.648163 + 0.761502i $$0.275538\pi$$
$$80$$ −58.1615 −0.0812832
$$81$$ 81.0000 0.111111
$$82$$ 179.170 0.241293
$$83$$ −327.929 −0.433674 −0.216837 0.976208i $$-0.569574\pi$$
−0.216837 + 0.976208i $$0.569574\pi$$
$$84$$ 0 0
$$85$$ 952.058 1.21489
$$86$$ −1956.68 −2.45342
$$87$$ −559.480 −0.689455
$$88$$ 247.144 0.299383
$$89$$ −37.6118 −0.0447960 −0.0223980 0.999749i $$-0.507130\pi$$
−0.0223980 + 0.999749i $$0.507130\pi$$
$$90$$ −732.932 −0.858420
$$91$$ 0 0
$$92$$ 1940.68 2.19924
$$93$$ 473.621 0.528089
$$94$$ −96.5352 −0.105924
$$95$$ −758.222 −0.818863
$$96$$ −565.052 −0.600733
$$97$$ 722.013 0.755766 0.377883 0.925853i $$-0.376652\pi$$
0.377883 + 0.925853i $$0.376652\pi$$
$$98$$ 0 0
$$99$$ −102.547 −0.104105
$$100$$ 2481.29 2.48129
$$101$$ 1518.67 1.49617 0.748087 0.663601i $$-0.230973\pi$$
0.748087 + 0.663601i $$0.230973\pi$$
$$102$$ 728.117 0.706807
$$103$$ −1051.88 −1.00626 −0.503132 0.864210i $$-0.667819\pi$$
−0.503132 + 0.864210i $$0.667819\pi$$
$$104$$ 284.116 0.267883
$$105$$ 0 0
$$106$$ −1667.19 −1.52766
$$107$$ 766.520 0.692545 0.346273 0.938134i $$-0.387447\pi$$
0.346273 + 0.938134i $$0.387447\pi$$
$$108$$ −344.533 −0.306969
$$109$$ −1427.05 −1.25400 −0.627002 0.779018i $$-0.715718\pi$$
−0.627002 + 0.779018i $$0.715718\pi$$
$$110$$ 927.904 0.804292
$$111$$ −11.2376 −0.00960928
$$112$$ 0 0
$$113$$ 362.564 0.301833 0.150917 0.988546i $$-0.451778\pi$$
0.150917 + 0.988546i $$0.451778\pi$$
$$114$$ −579.874 −0.476405
$$115$$ 2718.25 2.20416
$$116$$ 2379.74 1.90477
$$117$$ −117.888 −0.0931517
$$118$$ 1032.37 0.805402
$$119$$ 0 0
$$120$$ 1163.03 0.884750
$$121$$ −1201.17 −0.902459
$$122$$ −2970.63 −2.20449
$$123$$ 117.969 0.0864789
$$124$$ −2014.54 −1.45896
$$125$$ 1241.32 0.888216
$$126$$ 0 0
$$127$$ 974.777 0.681082 0.340541 0.940230i $$-0.389390\pi$$
0.340541 + 0.940230i $$0.389390\pi$$
$$128$$ 2284.83 1.57775
$$129$$ −1288.32 −0.879302
$$130$$ 1066.71 0.719670
$$131$$ −1792.70 −1.19564 −0.597821 0.801629i $$-0.703967\pi$$
−0.597821 + 0.801629i $$0.703967\pi$$
$$132$$ 436.184 0.287613
$$133$$ 0 0
$$134$$ −662.647 −0.427194
$$135$$ −482.577 −0.307656
$$136$$ −1155.39 −0.728486
$$137$$ 1684.42 1.05043 0.525217 0.850969i $$-0.323984\pi$$
0.525217 + 0.850969i $$0.323984\pi$$
$$138$$ 2078.87 1.28236
$$139$$ 315.089 0.192270 0.0961350 0.995368i $$-0.469352\pi$$
0.0961350 + 0.995368i $$0.469352\pi$$
$$140$$ 0 0
$$141$$ −63.5606 −0.0379629
$$142$$ 1681.13 0.993499
$$143$$ 149.248 0.0872781
$$144$$ −29.2871 −0.0169485
$$145$$ 3333.23 1.90903
$$146$$ −2774.40 −1.57268
$$147$$ 0 0
$$148$$ 47.7992 0.0265478
$$149$$ 1893.77 1.04123 0.520617 0.853790i $$-0.325702\pi$$
0.520617 + 0.853790i $$0.325702\pi$$
$$150$$ 2657.98 1.44682
$$151$$ −2011.84 −1.08425 −0.542124 0.840299i $$-0.682380\pi$$
−0.542124 + 0.840299i $$0.682380\pi$$
$$152$$ 920.159 0.491018
$$153$$ 479.406 0.253318
$$154$$ 0 0
$$155$$ −2821.71 −1.46223
$$156$$ 501.436 0.257352
$$157$$ −3828.50 −1.94616 −0.973082 0.230460i $$-0.925977\pi$$
−0.973082 + 0.230460i $$0.925977\pi$$
$$158$$ −4147.38 −2.08827
$$159$$ −1097.71 −0.547511
$$160$$ 3366.43 1.66337
$$161$$ 0 0
$$162$$ −369.066 −0.178991
$$163$$ −3509.26 −1.68630 −0.843148 0.537682i $$-0.819300\pi$$
−0.843148 + 0.537682i $$0.819300\pi$$
$$164$$ −501.780 −0.238917
$$165$$ 610.950 0.288257
$$166$$ 1494.17 0.698613
$$167$$ −343.008 −0.158939 −0.0794694 0.996837i $$-0.525323\pi$$
−0.0794694 + 0.996837i $$0.525323\pi$$
$$168$$ 0 0
$$169$$ −2025.42 −0.921905
$$170$$ −4337.93 −1.95708
$$171$$ −381.800 −0.170743
$$172$$ 5479.84 2.42927
$$173$$ −4187.21 −1.84016 −0.920081 0.391729i $$-0.871877\pi$$
−0.920081 + 0.391729i $$0.871877\pi$$
$$174$$ 2549.20 1.11066
$$175$$ 0 0
$$176$$ 37.0779 0.0158798
$$177$$ 679.733 0.288654
$$178$$ 171.373 0.0721627
$$179$$ −1970.29 −0.822716 −0.411358 0.911474i $$-0.634945\pi$$
−0.411358 + 0.911474i $$0.634945\pi$$
$$180$$ 2052.63 0.849969
$$181$$ −3613.10 −1.48376 −0.741878 0.670535i $$-0.766065\pi$$
−0.741878 + 0.670535i $$0.766065\pi$$
$$182$$ 0 0
$$183$$ −1955.92 −0.790086
$$184$$ −3298.80 −1.32169
$$185$$ 66.9509 0.0266072
$$186$$ −2157.99 −0.850708
$$187$$ −606.936 −0.237345
$$188$$ 270.355 0.104881
$$189$$ 0 0
$$190$$ 3454.74 1.31912
$$191$$ 1907.77 0.722729 0.361365 0.932425i $$-0.382311\pi$$
0.361365 + 0.932425i $$0.382311\pi$$
$$192$$ 2496.48 0.938376
$$193$$ 2399.93 0.895080 0.447540 0.894264i $$-0.352300\pi$$
0.447540 + 0.894264i $$0.352300\pi$$
$$194$$ −3289.75 −1.21748
$$195$$ 702.346 0.257928
$$196$$ 0 0
$$197$$ 1514.32 0.547668 0.273834 0.961777i $$-0.411708\pi$$
0.273834 + 0.961777i $$0.411708\pi$$
$$198$$ 467.243 0.167705
$$199$$ 1367.78 0.487232 0.243616 0.969872i $$-0.421666\pi$$
0.243616 + 0.969872i $$0.421666\pi$$
$$200$$ −4217.74 −1.49120
$$201$$ −436.300 −0.153105
$$202$$ −6919.63 −2.41021
$$203$$ 0 0
$$204$$ −2039.15 −0.699848
$$205$$ −702.828 −0.239452
$$206$$ 4792.76 1.62101
$$207$$ 1368.77 0.459594
$$208$$ 42.6246 0.0142091
$$209$$ 483.366 0.159977
$$210$$ 0 0
$$211$$ 4302.52 1.40378 0.701891 0.712285i $$-0.252339\pi$$
0.701891 + 0.712285i $$0.252339\pi$$
$$212$$ 4669.11 1.51262
$$213$$ 1106.89 0.356068
$$214$$ −3492.55 −1.11563
$$215$$ 7675.45 2.43470
$$216$$ 585.642 0.184481
$$217$$ 0 0
$$218$$ 6502.16 2.02010
$$219$$ −1826.72 −0.563644
$$220$$ −2598.67 −0.796374
$$221$$ −697.731 −0.212373
$$222$$ 51.2028 0.0154798
$$223$$ −1497.19 −0.449592 −0.224796 0.974406i $$-0.572172\pi$$
−0.224796 + 0.974406i $$0.572172\pi$$
$$224$$ 0 0
$$225$$ 1750.06 0.518537
$$226$$ −1651.97 −0.486229
$$227$$ 1603.32 0.468795 0.234397 0.972141i $$-0.424688\pi$$
0.234397 + 0.972141i $$0.424688\pi$$
$$228$$ 1623.98 0.471715
$$229$$ 1010.52 0.291603 0.145802 0.989314i $$-0.453424\pi$$
0.145802 + 0.989314i $$0.453424\pi$$
$$230$$ −12385.4 −3.55072
$$231$$ 0 0
$$232$$ −4045.13 −1.14472
$$233$$ 198.217 0.0557323 0.0278661 0.999612i $$-0.491129\pi$$
0.0278661 + 0.999612i $$0.491129\pi$$
$$234$$ 537.141 0.150060
$$235$$ 378.677 0.105116
$$236$$ −2891.24 −0.797472
$$237$$ −2730.71 −0.748434
$$238$$ 0 0
$$239$$ −1201.19 −0.325098 −0.162549 0.986700i $$-0.551972\pi$$
−0.162549 + 0.986700i $$0.551972\pi$$
$$240$$ 174.485 0.0469289
$$241$$ −2732.69 −0.730407 −0.365204 0.930928i $$-0.619001\pi$$
−0.365204 + 0.930928i $$0.619001\pi$$
$$242$$ 5472.99 1.45379
$$243$$ −243.000 −0.0641500
$$244$$ 8319.49 2.18279
$$245$$ 0 0
$$246$$ −537.510 −0.139311
$$247$$ 555.675 0.143145
$$248$$ 3424.35 0.876801
$$249$$ 983.788 0.250382
$$250$$ −5655.91 −1.43084
$$251$$ 7565.82 1.90259 0.951295 0.308281i $$-0.0997537\pi$$
0.951295 + 0.308281i $$0.0997537\pi$$
$$252$$ 0 0
$$253$$ −1732.88 −0.430615
$$254$$ −4441.44 −1.09717
$$255$$ −2856.18 −0.701414
$$256$$ −3753.22 −0.916313
$$257$$ 5008.68 1.21569 0.607846 0.794055i $$-0.292034\pi$$
0.607846 + 0.794055i $$0.292034\pi$$
$$258$$ 5870.04 1.41648
$$259$$ 0 0
$$260$$ −2987.42 −0.712584
$$261$$ 1678.44 0.398057
$$262$$ 8168.21 1.92608
$$263$$ −6248.81 −1.46509 −0.732544 0.680720i $$-0.761667\pi$$
−0.732544 + 0.680720i $$0.761667\pi$$
$$264$$ −741.433 −0.172849
$$265$$ 6539.88 1.51601
$$266$$ 0 0
$$267$$ 112.835 0.0258630
$$268$$ 1855.80 0.422988
$$269$$ −3588.45 −0.813351 −0.406676 0.913573i $$-0.633312\pi$$
−0.406676 + 0.913573i $$0.633312\pi$$
$$270$$ 2198.80 0.495609
$$271$$ −1983.14 −0.444529 −0.222264 0.974986i $$-0.571345\pi$$
−0.222264 + 0.974986i $$0.571345\pi$$
$$272$$ −173.338 −0.0386404
$$273$$ 0 0
$$274$$ −7674.81 −1.69216
$$275$$ −2215.61 −0.485841
$$276$$ −5822.05 −1.26973
$$277$$ 7363.91 1.59731 0.798654 0.601790i $$-0.205546\pi$$
0.798654 + 0.601790i $$0.205546\pi$$
$$278$$ −1435.66 −0.309731
$$279$$ −1420.86 −0.304892
$$280$$ 0 0
$$281$$ −5312.05 −1.12772 −0.563861 0.825869i $$-0.690685\pi$$
−0.563861 + 0.825869i $$0.690685\pi$$
$$282$$ 289.606 0.0611552
$$283$$ −1091.76 −0.229324 −0.114662 0.993405i $$-0.536578\pi$$
−0.114662 + 0.993405i $$0.536578\pi$$
$$284$$ −4708.13 −0.983718
$$285$$ 2274.67 0.472770
$$286$$ −680.030 −0.140598
$$287$$ 0 0
$$288$$ 1695.16 0.346833
$$289$$ −2075.59 −0.422469
$$290$$ −15187.4 −3.07530
$$291$$ −2166.04 −0.436341
$$292$$ 7769.92 1.55719
$$293$$ −7191.86 −1.43397 −0.716985 0.697089i $$-0.754478\pi$$
−0.716985 + 0.697089i $$0.754478\pi$$
$$294$$ 0 0
$$295$$ −4049.67 −0.799257
$$296$$ −81.2499 −0.0159546
$$297$$ 307.642 0.0601051
$$298$$ −8628.73 −1.67734
$$299$$ −1992.12 −0.385308
$$300$$ −7443.88 −1.43257
$$301$$ 0 0
$$302$$ 9166.69 1.74663
$$303$$ −4556.02 −0.863816
$$304$$ 138.047 0.0260446
$$305$$ 11652.9 2.18767
$$306$$ −2184.35 −0.408075
$$307$$ 541.355 0.100641 0.0503204 0.998733i $$-0.483976\pi$$
0.0503204 + 0.998733i $$0.483976\pi$$
$$308$$ 0 0
$$309$$ 3155.65 0.580966
$$310$$ 12856.7 2.35553
$$311$$ 54.0168 0.00984892 0.00492446 0.999988i $$-0.498432\pi$$
0.00492446 + 0.999988i $$0.498432\pi$$
$$312$$ −852.348 −0.154663
$$313$$ 3772.94 0.681340 0.340670 0.940183i $$-0.389346\pi$$
0.340670 + 0.940183i $$0.389346\pi$$
$$314$$ 17444.1 3.13511
$$315$$ 0 0
$$316$$ 11615.1 2.06772
$$317$$ 1719.24 0.304612 0.152306 0.988333i $$-0.451330\pi$$
0.152306 + 0.988333i $$0.451330\pi$$
$$318$$ 5001.58 0.881996
$$319$$ −2124.93 −0.372957
$$320$$ −14873.4 −2.59827
$$321$$ −2299.56 −0.399841
$$322$$ 0 0
$$323$$ −2259.72 −0.389270
$$324$$ 1033.60 0.177229
$$325$$ −2547.06 −0.434724
$$326$$ 15989.5 2.71649
$$327$$ 4281.15 0.724000
$$328$$ 852.934 0.143584
$$329$$ 0 0
$$330$$ −2783.71 −0.464358
$$331$$ 8408.21 1.39625 0.698123 0.715978i $$-0.254019\pi$$
0.698123 + 0.715978i $$0.254019\pi$$
$$332$$ −4184.53 −0.691735
$$333$$ 33.7129 0.00554792
$$334$$ 1562.87 0.256037
$$335$$ 2599.36 0.423935
$$336$$ 0 0
$$337$$ 2789.46 0.450894 0.225447 0.974255i $$-0.427616\pi$$
0.225447 + 0.974255i $$0.427616\pi$$
$$338$$ 9228.58 1.48511
$$339$$ −1087.69 −0.174263
$$340$$ 12148.7 1.93781
$$341$$ 1798.84 0.285667
$$342$$ 1739.62 0.275053
$$343$$ 0 0
$$344$$ −9314.72 −1.45993
$$345$$ −8154.76 −1.27257
$$346$$ 19078.5 2.96435
$$347$$ −3471.96 −0.537132 −0.268566 0.963261i $$-0.586550\pi$$
−0.268566 + 0.963261i $$0.586550\pi$$
$$348$$ −7139.23 −1.09972
$$349$$ −6626.12 −1.01630 −0.508149 0.861269i $$-0.669670\pi$$
−0.508149 + 0.861269i $$0.669670\pi$$
$$350$$ 0 0
$$351$$ 353.664 0.0537812
$$352$$ −2146.10 −0.324964
$$353$$ 9468.40 1.42763 0.713813 0.700337i $$-0.246967\pi$$
0.713813 + 0.700337i $$0.246967\pi$$
$$354$$ −3097.11 −0.464999
$$355$$ −6594.53 −0.985919
$$356$$ −479.944 −0.0714522
$$357$$ 0 0
$$358$$ 8977.35 1.32533
$$359$$ −6279.55 −0.923182 −0.461591 0.887093i $$-0.652721\pi$$
−0.461591 + 0.887093i $$0.652721\pi$$
$$360$$ −3489.10 −0.510811
$$361$$ −5059.35 −0.737622
$$362$$ 16462.6 2.39021
$$363$$ 3603.52 0.521035
$$364$$ 0 0
$$365$$ 10883.1 1.56068
$$366$$ 8911.89 1.27276
$$367$$ −10827.8 −1.54008 −0.770038 0.637998i $$-0.779763\pi$$
−0.770038 + 0.637998i $$0.779763\pi$$
$$368$$ −494.904 −0.0701050
$$369$$ −353.907 −0.0499286
$$370$$ −305.053 −0.0428620
$$371$$ 0 0
$$372$$ 6043.63 0.842332
$$373$$ 5239.23 0.727284 0.363642 0.931539i $$-0.381533\pi$$
0.363642 + 0.931539i $$0.381533\pi$$
$$374$$ 2765.42 0.382344
$$375$$ −3723.96 −0.512812
$$376$$ −459.553 −0.0630310
$$377$$ −2442.81 −0.333717
$$378$$ 0 0
$$379$$ −11050.4 −1.49768 −0.748839 0.662751i $$-0.769389\pi$$
−0.748839 + 0.662751i $$0.769389\pi$$
$$380$$ −9675.27 −1.30613
$$381$$ −2924.33 −0.393223
$$382$$ −8692.49 −1.16426
$$383$$ −10468.0 −1.39658 −0.698292 0.715813i $$-0.746056\pi$$
−0.698292 + 0.715813i $$0.746056\pi$$
$$384$$ −6854.48 −0.910915
$$385$$ 0 0
$$386$$ −10934.9 −1.44190
$$387$$ 3864.95 0.507665
$$388$$ 9213.22 1.20549
$$389$$ −11614.0 −1.51377 −0.756884 0.653550i $$-0.773279\pi$$
−0.756884 + 0.653550i $$0.773279\pi$$
$$390$$ −3200.14 −0.415501
$$391$$ 8101.19 1.04781
$$392$$ 0 0
$$393$$ 5378.11 0.690304
$$394$$ −6899.78 −0.882248
$$395$$ 16268.9 2.07234
$$396$$ −1308.55 −0.166054
$$397$$ −6707.30 −0.847934 −0.423967 0.905678i $$-0.639363\pi$$
−0.423967 + 0.905678i $$0.639363\pi$$
$$398$$ −6232.10 −0.784892
$$399$$ 0 0
$$400$$ −632.768 −0.0790960
$$401$$ 5526.38 0.688215 0.344107 0.938930i $$-0.388182\pi$$
0.344107 + 0.938930i $$0.388182\pi$$
$$402$$ 1987.94 0.246640
$$403$$ 2067.94 0.255611
$$404$$ 19379.0 2.38649
$$405$$ 1447.73 0.177625
$$406$$ 0 0
$$407$$ −42.6811 −0.00519810
$$408$$ 3466.18 0.420592
$$409$$ −1318.91 −0.159452 −0.0797258 0.996817i $$-0.525404\pi$$
−0.0797258 + 0.996817i $$0.525404\pi$$
$$410$$ 3202.34 0.385738
$$411$$ −5053.25 −0.606468
$$412$$ −13422.5 −1.60505
$$413$$ 0 0
$$414$$ −6236.61 −0.740369
$$415$$ −5861.15 −0.693283
$$416$$ −2467.14 −0.290773
$$417$$ −945.268 −0.111007
$$418$$ −2202.39 −0.257709
$$419$$ 3656.13 0.426286 0.213143 0.977021i $$-0.431630\pi$$
0.213143 + 0.977021i $$0.431630\pi$$
$$420$$ 0 0
$$421$$ −135.389 −0.0156733 −0.00783663 0.999969i $$-0.502495\pi$$
−0.00783663 + 0.999969i $$0.502495\pi$$
$$422$$ −19603.9 −2.26138
$$423$$ 190.682 0.0219179
$$424$$ −7936.63 −0.909049
$$425$$ 10357.9 1.18219
$$426$$ −5043.38 −0.573597
$$427$$ 0 0
$$428$$ 9781.16 1.10465
$$429$$ −447.745 −0.0503900
$$430$$ −34972.1 −3.92211
$$431$$ 8389.16 0.937568 0.468784 0.883313i $$-0.344692\pi$$
0.468784 + 0.883313i $$0.344692\pi$$
$$432$$ 87.8612 0.00978524
$$433$$ −8243.02 −0.914859 −0.457430 0.889246i $$-0.651230\pi$$
−0.457430 + 0.889246i $$0.651230\pi$$
$$434$$ 0 0
$$435$$ −9999.70 −1.10218
$$436$$ −18209.8 −2.00021
$$437$$ −6451.81 −0.706252
$$438$$ 8323.19 0.907985
$$439$$ −18283.2 −1.98772 −0.993859 0.110649i $$-0.964707\pi$$
−0.993859 + 0.110649i $$0.964707\pi$$
$$440$$ 4417.26 0.478602
$$441$$ 0 0
$$442$$ 3179.12 0.342116
$$443$$ 1210.44 0.129818 0.0649092 0.997891i $$-0.479324\pi$$
0.0649092 + 0.997891i $$0.479324\pi$$
$$444$$ −143.398 −0.0153274
$$445$$ −672.243 −0.0716121
$$446$$ 6821.72 0.724256
$$447$$ −5681.32 −0.601157
$$448$$ 0 0
$$449$$ −8301.16 −0.872508 −0.436254 0.899824i $$-0.643695\pi$$
−0.436254 + 0.899824i $$0.643695\pi$$
$$450$$ −7973.93 −0.835321
$$451$$ 448.052 0.0467804
$$452$$ 4626.49 0.481442
$$453$$ 6035.52 0.625990
$$454$$ −7305.34 −0.755190
$$455$$ 0 0
$$456$$ −2760.48 −0.283489
$$457$$ −12293.8 −1.25838 −0.629188 0.777253i $$-0.716612\pi$$
−0.629188 + 0.777253i $$0.716612\pi$$
$$458$$ −4604.31 −0.469749
$$459$$ −1438.22 −0.146253
$$460$$ 34686.2 3.51577
$$461$$ 19434.2 1.96343 0.981717 0.190346i $$-0.0609609\pi$$
0.981717 + 0.190346i $$0.0609609\pi$$
$$462$$ 0 0
$$463$$ −12491.1 −1.25380 −0.626902 0.779098i $$-0.715678\pi$$
−0.626902 + 0.779098i $$0.715678\pi$$
$$464$$ −606.871 −0.0607183
$$465$$ 8465.13 0.844217
$$466$$ −903.149 −0.0897802
$$467$$ 3385.17 0.335433 0.167716 0.985835i $$-0.446361\pi$$
0.167716 + 0.985835i $$0.446361\pi$$
$$468$$ −1504.31 −0.148583
$$469$$ 0 0
$$470$$ −1725.39 −0.169333
$$471$$ 11485.5 1.12362
$$472$$ 4914.57 0.479262
$$473$$ −4893.09 −0.475654
$$474$$ 12442.1 1.20567
$$475$$ −8249.07 −0.796828
$$476$$ 0 0
$$477$$ 3293.14 0.316106
$$478$$ 5473.05 0.523706
$$479$$ 5979.41 0.570368 0.285184 0.958473i $$-0.407945\pi$$
0.285184 + 0.958473i $$0.407945\pi$$
$$480$$ −10099.3 −0.960349
$$481$$ −49.0661 −0.00465119
$$482$$ 12451.1 1.17663
$$483$$ 0 0
$$484$$ −15327.5 −1.43948
$$485$$ 12904.7 1.20819
$$486$$ 1107.20 0.103340
$$487$$ −1114.96 −0.103745 −0.0518725 0.998654i $$-0.516519\pi$$
−0.0518725 + 0.998654i $$0.516519\pi$$
$$488$$ −14141.6 −1.31180
$$489$$ 10527.8 0.973583
$$490$$ 0 0
$$491$$ 1086.23 0.0998387 0.0499194 0.998753i $$-0.484104\pi$$
0.0499194 + 0.998753i $$0.484104\pi$$
$$492$$ 1505.34 0.137939
$$493$$ 9934.01 0.907516
$$494$$ −2531.86 −0.230595
$$495$$ −1832.85 −0.166425
$$496$$ 513.740 0.0465073
$$497$$ 0 0
$$498$$ −4482.50 −0.403344
$$499$$ −2213.50 −0.198577 −0.0992884 0.995059i $$-0.531657\pi$$
−0.0992884 + 0.995059i $$0.531657\pi$$
$$500$$ 15839.8 1.41676
$$501$$ 1029.02 0.0917634
$$502$$ −34472.6 −3.06492
$$503$$ −2643.32 −0.234314 −0.117157 0.993113i $$-0.537378\pi$$
−0.117157 + 0.993113i $$0.537378\pi$$
$$504$$ 0 0
$$505$$ 27143.5 2.39183
$$506$$ 7895.66 0.693685
$$507$$ 6076.27 0.532262
$$508$$ 12438.6 1.08637
$$509$$ −665.169 −0.0579236 −0.0289618 0.999581i $$-0.509220\pi$$
−0.0289618 + 0.999581i $$0.509220\pi$$
$$510$$ 13013.8 1.12992
$$511$$ 0 0
$$512$$ −1177.58 −0.101645
$$513$$ 1145.40 0.0985784
$$514$$ −22821.4 −1.95838
$$515$$ −18800.5 −1.60864
$$516$$ −16439.5 −1.40254
$$517$$ −241.406 −0.0205359
$$518$$ 0 0
$$519$$ 12561.6 1.06242
$$520$$ 5078.07 0.428246
$$521$$ −11762.0 −0.989063 −0.494531 0.869160i $$-0.664660\pi$$
−0.494531 + 0.869160i $$0.664660\pi$$
$$522$$ −7647.59 −0.641237
$$523$$ 10122.6 0.846330 0.423165 0.906053i $$-0.360919\pi$$
0.423165 + 0.906053i $$0.360919\pi$$
$$524$$ −22875.7 −1.90712
$$525$$ 0 0
$$526$$ 28471.9 2.36014
$$527$$ −8409.52 −0.695112
$$528$$ −111.234 −0.00916823
$$529$$ 10963.0 0.901042
$$530$$ −29798.1 −2.44216
$$531$$ −2039.20 −0.166655
$$532$$ 0 0
$$533$$ 515.079 0.0418585
$$534$$ −514.119 −0.0416631
$$535$$ 13700.2 1.10712
$$536$$ −3154.51 −0.254206
$$537$$ 5910.86 0.474995
$$538$$ 16350.3 1.31024
$$539$$ 0 0
$$540$$ −6157.90 −0.490730
$$541$$ 16118.0 1.28090 0.640449 0.768001i $$-0.278748\pi$$
0.640449 + 0.768001i $$0.278748\pi$$
$$542$$ 9035.92 0.716100
$$543$$ 10839.3 0.856647
$$544$$ 10032.9 0.790733
$$545$$ −25505.9 −2.00469
$$546$$ 0 0
$$547$$ −626.100 −0.0489399 −0.0244699 0.999701i $$-0.507790\pi$$
−0.0244699 + 0.999701i $$0.507790\pi$$
$$548$$ 21493.9 1.67550
$$549$$ 5867.76 0.456156
$$550$$ 10095.1 0.782650
$$551$$ −7911.47 −0.611688
$$552$$ 9896.41 0.763078
$$553$$ 0 0
$$554$$ −33552.7 −2.57313
$$555$$ −200.853 −0.0153617
$$556$$ 4020.69 0.306682
$$557$$ 20771.2 1.58008 0.790039 0.613057i $$-0.210060\pi$$
0.790039 + 0.613057i $$0.210060\pi$$
$$558$$ 6473.97 0.491156
$$559$$ −5625.08 −0.425609
$$560$$ 0 0
$$561$$ 1820.81 0.137031
$$562$$ 24203.6 1.81667
$$563$$ −5521.72 −0.413344 −0.206672 0.978410i $$-0.566263\pi$$
−0.206672 + 0.978410i $$0.566263\pi$$
$$564$$ −811.064 −0.0605531
$$565$$ 6480.18 0.482519
$$566$$ 4974.48 0.369422
$$567$$ 0 0
$$568$$ 8002.95 0.591191
$$569$$ 7574.81 0.558089 0.279044 0.960278i $$-0.409982\pi$$
0.279044 + 0.960278i $$0.409982\pi$$
$$570$$ −10364.2 −0.761595
$$571$$ −331.248 −0.0242772 −0.0121386 0.999926i $$-0.503864\pi$$
−0.0121386 + 0.999926i $$0.503864\pi$$
$$572$$ 1904.48 0.139214
$$573$$ −5723.31 −0.417268
$$574$$ 0 0
$$575$$ 29573.2 2.14485
$$576$$ −7489.45 −0.541772
$$577$$ 2038.11 0.147050 0.0735248 0.997293i $$-0.476575\pi$$
0.0735248 + 0.997293i $$0.476575\pi$$
$$578$$ 9457.14 0.680563
$$579$$ −7199.78 −0.516775
$$580$$ 42533.6 3.04502
$$581$$ 0 0
$$582$$ 9869.26 0.702911
$$583$$ −4169.17 −0.296174
$$584$$ −13207.4 −0.935835
$$585$$ −2107.04 −0.148915
$$586$$ 32768.8 2.31001
$$587$$ 5232.90 0.367947 0.183973 0.982931i $$-0.441104\pi$$
0.183973 + 0.982931i $$0.441104\pi$$
$$588$$ 0 0
$$589$$ 6697.36 0.468523
$$590$$ 18451.8 1.28754
$$591$$ −4542.95 −0.316196
$$592$$ −12.1895 −0.000846262 0
$$593$$ −5720.24 −0.396125 −0.198062 0.980189i $$-0.563465\pi$$
−0.198062 + 0.980189i $$0.563465\pi$$
$$594$$ −1401.73 −0.0968244
$$595$$ 0 0
$$596$$ 24165.5 1.66083
$$597$$ −4103.33 −0.281304
$$598$$ 9076.81 0.620700
$$599$$ 18088.4 1.23384 0.616922 0.787024i $$-0.288379\pi$$
0.616922 + 0.787024i $$0.288379\pi$$
$$600$$ 12653.2 0.860943
$$601$$ −1821.43 −0.123623 −0.0618117 0.998088i $$-0.519688\pi$$
−0.0618117 + 0.998088i $$0.519688\pi$$
$$602$$ 0 0
$$603$$ 1308.90 0.0883955
$$604$$ −25672.0 −1.72944
$$605$$ −21468.8 −1.44270
$$606$$ 20758.9 1.39154
$$607$$ 2372.20 0.158624 0.0793120 0.996850i $$-0.474728\pi$$
0.0793120 + 0.996850i $$0.474728\pi$$
$$608$$ −7990.26 −0.532974
$$609$$ 0 0
$$610$$ −53094.7 −3.52416
$$611$$ −277.520 −0.0183752
$$612$$ 6117.45 0.404058
$$613$$ 9725.08 0.640770 0.320385 0.947287i $$-0.396188\pi$$
0.320385 + 0.947287i $$0.396188\pi$$
$$614$$ −2466.61 −0.162124
$$615$$ 2108.48 0.138248
$$616$$ 0 0
$$617$$ −5329.51 −0.347744 −0.173872 0.984768i $$-0.555628\pi$$
−0.173872 + 0.984768i $$0.555628\pi$$
$$618$$ −14378.3 −0.935890
$$619$$ −15976.6 −1.03740 −0.518702 0.854955i $$-0.673585\pi$$
−0.518702 + 0.854955i $$0.673585\pi$$
$$620$$ −36006.3 −2.33234
$$621$$ −4106.31 −0.265347
$$622$$ −246.120 −0.0158658
$$623$$ 0 0
$$624$$ −127.874 −0.00820361
$$625$$ −2120.06 −0.135684
$$626$$ −17190.9 −1.09758
$$627$$ −1450.10 −0.0923625
$$628$$ −48853.5 −3.10425
$$629$$ 199.533 0.0126485
$$630$$ 0 0
$$631$$ −4199.98 −0.264974 −0.132487 0.991185i $$-0.542296\pi$$
−0.132487 + 0.991185i $$0.542296\pi$$
$$632$$ −19743.5 −1.24265
$$633$$ −12907.6 −0.810474
$$634$$ −7833.47 −0.490705
$$635$$ 17422.4 1.08880
$$636$$ −14007.3 −0.873312
$$637$$ 0 0
$$638$$ 9681.97 0.600804
$$639$$ −3320.66 −0.205576
$$640$$ 40837.2 2.52224
$$641$$ 2648.51 0.163198 0.0815988 0.996665i $$-0.473997\pi$$
0.0815988 + 0.996665i $$0.473997\pi$$
$$642$$ 10477.6 0.644112
$$643$$ 13.4305 0.000823715 0 0.000411857 1.00000i $$-0.499869\pi$$
0.000411857 1.00000i $$0.499869\pi$$
$$644$$ 0 0
$$645$$ −23026.3 −1.40568
$$646$$ 10296.1 0.627083
$$647$$ 11624.1 0.706324 0.353162 0.935562i $$-0.385106\pi$$
0.353162 + 0.935562i $$0.385106\pi$$
$$648$$ −1756.93 −0.106510
$$649$$ 2581.66 0.156146
$$650$$ 11605.3 0.700305
$$651$$ 0 0
$$652$$ −44779.8 −2.68974
$$653$$ −28516.6 −1.70894 −0.854471 0.519499i $$-0.826119\pi$$
−0.854471 + 0.519499i $$0.826119\pi$$
$$654$$ −19506.5 −1.16630
$$655$$ −32041.3 −1.91139
$$656$$ 127.962 0.00761595
$$657$$ 5480.15 0.325420
$$658$$ 0 0
$$659$$ 18048.6 1.06688 0.533440 0.845838i $$-0.320899\pi$$
0.533440 + 0.845838i $$0.320899\pi$$
$$660$$ 7796.01 0.459787
$$661$$ 17841.4 1.04985 0.524926 0.851148i $$-0.324093\pi$$
0.524926 + 0.851148i $$0.324093\pi$$
$$662$$ −38310.9 −2.24924
$$663$$ 2093.19 0.122614
$$664$$ 7112.94 0.415716
$$665$$ 0 0
$$666$$ −153.608 −0.00893725
$$667$$ 28362.9 1.64650
$$668$$ −4376.95 −0.253517
$$669$$ 4491.56 0.259572
$$670$$ −11843.6 −0.682924
$$671$$ −7428.68 −0.427394
$$672$$ 0 0
$$673$$ −6826.13 −0.390978 −0.195489 0.980706i $$-0.562629\pi$$
−0.195489 + 0.980706i $$0.562629\pi$$
$$674$$ −12709.8 −0.726354
$$675$$ −5250.19 −0.299378
$$676$$ −25845.4 −1.47049
$$677$$ −21286.9 −1.20845 −0.604225 0.796814i $$-0.706517\pi$$
−0.604225 + 0.796814i $$0.706517\pi$$
$$678$$ 4955.92 0.280724
$$679$$ 0 0
$$680$$ −20650.6 −1.16458
$$681$$ −4809.97 −0.270659
$$682$$ −8196.16 −0.460187
$$683$$ −20696.8 −1.15951 −0.579753 0.814793i $$-0.696851\pi$$
−0.579753 + 0.814793i $$0.696851\pi$$
$$684$$ −4871.95 −0.272345
$$685$$ 30105.9 1.67925
$$686$$ 0 0
$$687$$ −3031.56 −0.168357
$$688$$ −1397.44 −0.0774376
$$689$$ −4792.86 −0.265012
$$690$$ 37156.1 2.05001
$$691$$ 31341.9 1.72548 0.862738 0.505652i $$-0.168748\pi$$
0.862738 + 0.505652i $$0.168748\pi$$
$$692$$ −53430.8 −2.93517
$$693$$ 0 0
$$694$$ 15819.5 0.865276
$$695$$ 5631.66 0.307368
$$696$$ 12135.4 0.660906
$$697$$ −2094.63 −0.113831
$$698$$ 30191.0 1.63717
$$699$$ −594.651 −0.0321770
$$700$$ 0 0
$$701$$ −9213.32 −0.496408 −0.248204 0.968708i $$-0.579840\pi$$
−0.248204 + 0.968708i $$0.579840\pi$$
$$702$$ −1611.42 −0.0866371
$$703$$ −158.909 −0.00852541
$$704$$ 9481.77 0.507610
$$705$$ −1136.03 −0.0606886
$$706$$ −43141.5 −2.29979
$$707$$ 0 0
$$708$$ 8673.71 0.460421
$$709$$ 14516.5 0.768942 0.384471 0.923137i $$-0.374384\pi$$
0.384471 + 0.923137i $$0.374384\pi$$
$$710$$ 30047.1 1.58824
$$711$$ 8192.14 0.432108
$$712$$ 815.817 0.0429411
$$713$$ −24010.3 −1.26114
$$714$$ 0 0
$$715$$ 2667.54 0.139525
$$716$$ −25141.8 −1.31228
$$717$$ 3603.56 0.187695
$$718$$ 28611.9 1.48717
$$719$$ 25882.4 1.34249 0.671246 0.741235i $$-0.265760\pi$$
0.671246 + 0.741235i $$0.265760\pi$$
$$720$$ −523.454 −0.0270944
$$721$$ 0 0
$$722$$ 23052.3 1.18825
$$723$$ 8198.08 0.421701
$$724$$ −46104.9 −2.36668
$$725$$ 36263.9 1.85767
$$726$$ −16419.0 −0.839345
$$727$$ 32181.2 1.64172 0.820862 0.571127i $$-0.193494\pi$$
0.820862 + 0.571127i $$0.193494\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −49587.4 −2.51412
$$731$$ 22875.1 1.15741
$$732$$ −24958.5 −1.26023
$$733$$ 20836.1 1.04993 0.524966 0.851123i $$-0.324078\pi$$
0.524966 + 0.851123i $$0.324078\pi$$
$$734$$ 49335.5 2.48094
$$735$$ 0 0
$$736$$ 28645.4 1.43462
$$737$$ −1657.09 −0.0828217
$$738$$ 1612.53 0.0804310
$$739$$ 26434.9 1.31586 0.657931 0.753078i $$-0.271432\pi$$
0.657931 + 0.753078i $$0.271432\pi$$
$$740$$ 854.325 0.0424400
$$741$$ −1667.03 −0.0826447
$$742$$ 0 0
$$743$$ 9954.69 0.491524 0.245762 0.969330i $$-0.420962\pi$$
0.245762 + 0.969330i $$0.420962\pi$$
$$744$$ −10273.1 −0.506221
$$745$$ 33847.8 1.66455
$$746$$ −23871.8 −1.17160
$$747$$ −2951.36 −0.144558
$$748$$ −7744.79 −0.378580
$$749$$ 0 0
$$750$$ 16967.7 0.826098
$$751$$ −33204.5 −1.61338 −0.806692 0.590973i $$-0.798744\pi$$
−0.806692 + 0.590973i $$0.798744\pi$$
$$752$$ −68.9446 −0.00334329
$$753$$ −22697.5 −1.09846
$$754$$ 11130.4 0.537591
$$755$$ −35958.1 −1.73331
$$756$$ 0 0
$$757$$ 1964.06 0.0942998 0.0471499 0.998888i $$-0.484986\pi$$
0.0471499 + 0.998888i $$0.484986\pi$$
$$758$$ 50349.6 2.41264
$$759$$ 5198.65 0.248615
$$760$$ 16446.2 0.784955
$$761$$ −38553.8 −1.83650 −0.918248 0.396005i $$-0.870396\pi$$
−0.918248 + 0.396005i $$0.870396\pi$$
$$762$$ 13324.3 0.633450
$$763$$ 0 0
$$764$$ 24344.0 1.15280
$$765$$ 8568.53 0.404962
$$766$$ 47696.2 2.24978
$$767$$ 2967.87 0.139718
$$768$$ 11259.7 0.529034
$$769$$ −19715.0 −0.924501 −0.462251 0.886749i $$-0.652958\pi$$
−0.462251 + 0.886749i $$0.652958\pi$$
$$770$$ 0 0
$$771$$ −15026.0 −0.701880
$$772$$ 30624.2 1.42771
$$773$$ −14700.7 −0.684019 −0.342010 0.939696i $$-0.611108\pi$$
−0.342010 + 0.939696i $$0.611108\pi$$
$$774$$ −17610.1 −0.817807
$$775$$ −30698.8 −1.42288
$$776$$ −15660.8 −0.724471
$$777$$ 0 0
$$778$$ 52917.8 2.43856
$$779$$ 1668.17 0.0767246
$$780$$ 8962.26 0.411411
$$781$$ 4204.01 0.192614
$$782$$ −36912.0 −1.68794
$$783$$ −5035.32 −0.229818
$$784$$ 0 0
$$785$$ −68427.6 −3.11119
$$786$$ −24504.6 −1.11202
$$787$$ −23918.6 −1.08336 −0.541681 0.840584i $$-0.682212\pi$$
−0.541681 + 0.840584i $$0.682212\pi$$
$$788$$ 19323.4 0.873562
$$789$$ 18746.4 0.845869
$$790$$ −74126.9 −3.33837
$$791$$ 0 0
$$792$$ 2224.30 0.0997942
$$793$$ −8539.98 −0.382426
$$794$$ 30560.9 1.36595
$$795$$ −19619.6 −0.875266
$$796$$ 17453.5 0.777164
$$797$$ 38252.7 1.70010 0.850051 0.526700i $$-0.176571\pi$$
0.850051 + 0.526700i $$0.176571\pi$$
$$798$$ 0 0
$$799$$ 1128.57 0.0499698
$$800$$ 36625.1 1.61861
$$801$$ −338.506 −0.0149320
$$802$$ −25180.2 −1.10866
$$803$$ −6937.96 −0.304901
$$804$$ −5567.39 −0.244212
$$805$$ 0 0
$$806$$ −9422.27 −0.411769
$$807$$ 10765.3 0.469589
$$808$$ −32940.7 −1.43422
$$809$$ −31435.6 −1.36615 −0.683075 0.730348i $$-0.739358\pi$$
−0.683075 + 0.730348i $$0.739358\pi$$
$$810$$ −6596.39 −0.286140
$$811$$ 11467.0 0.496501 0.248250 0.968696i $$-0.420144\pi$$
0.248250 + 0.968696i $$0.420144\pi$$
$$812$$ 0 0
$$813$$ 5949.43 0.256649
$$814$$ 194.471 0.00837371
$$815$$ −62721.7 −2.69576
$$816$$ 520.015 0.0223090
$$817$$ −18217.8 −0.780121
$$818$$ 6009.42 0.256864
$$819$$ 0 0
$$820$$ −8968.42 −0.381940
$$821$$ −5030.57 −0.213847 −0.106923 0.994267i $$-0.534100\pi$$
−0.106923 + 0.994267i $$0.534100\pi$$
$$822$$ 23024.4 0.976970
$$823$$ −13985.2 −0.592336 −0.296168 0.955136i $$-0.595709\pi$$
−0.296168 + 0.955136i $$0.595709\pi$$
$$824$$ 22815.8 0.964596
$$825$$ 6646.83 0.280500
$$826$$ 0 0
$$827$$ −13939.5 −0.586125 −0.293063 0.956093i $$-0.594674\pi$$
−0.293063 + 0.956093i $$0.594674\pi$$
$$828$$ 17466.1 0.733080
$$829$$ −20104.4 −0.842286 −0.421143 0.906994i $$-0.638371\pi$$
−0.421143 + 0.906994i $$0.638371\pi$$
$$830$$ 26705.5 1.11682
$$831$$ −22091.7 −0.922207
$$832$$ 10900.2 0.454203
$$833$$ 0 0
$$834$$ 4306.99 0.178823
$$835$$ −6130.66 −0.254084
$$836$$ 6167.98 0.255172
$$837$$ 4262.59 0.176030
$$838$$ −16658.7 −0.686711
$$839$$ 15949.5 0.656302 0.328151 0.944625i $$-0.393575\pi$$
0.328151 + 0.944625i $$0.393575\pi$$
$$840$$ 0 0
$$841$$ 10390.8 0.426043
$$842$$ 616.881 0.0252484
$$843$$ 15936.1 0.651091
$$844$$ 54902.2 2.23911
$$845$$ −36200.8 −1.47378
$$846$$ −868.817 −0.0353080
$$847$$ 0 0
$$848$$ −1190.70 −0.0482177
$$849$$ 3275.29 0.132400
$$850$$ −47194.5 −1.90442
$$851$$ 569.694 0.0229481
$$852$$ 14124.4 0.567950
$$853$$ 11802.0 0.473730 0.236865 0.971543i $$-0.423880\pi$$
0.236865 + 0.971543i $$0.423880\pi$$
$$854$$ 0 0
$$855$$ −6824.00 −0.272954
$$856$$ −16626.2 −0.663868
$$857$$ 9595.28 0.382460 0.191230 0.981545i $$-0.438752\pi$$
0.191230 + 0.981545i $$0.438752\pi$$
$$858$$ 2040.09 0.0811742
$$859$$ 21840.9 0.867521 0.433760 0.901028i $$-0.357186\pi$$
0.433760 + 0.901028i $$0.357186\pi$$
$$860$$ 97942.3 3.88350
$$861$$ 0 0
$$862$$ −38224.1 −1.51035
$$863$$ −26531.6 −1.04652 −0.523260 0.852173i $$-0.675284\pi$$
−0.523260 + 0.852173i $$0.675284\pi$$
$$864$$ −5085.47 −0.200244
$$865$$ −74838.9 −2.94173
$$866$$ 37558.2 1.47376
$$867$$ 6226.77 0.243912
$$868$$ 0 0
$$869$$ −10371.4 −0.404862
$$870$$ 45562.3 1.77552
$$871$$ −1904.98 −0.0741077
$$872$$ 30953.3 1.20208
$$873$$ 6498.11 0.251922
$$874$$ 29396.8 1.13771
$$875$$ 0 0
$$876$$ −23309.8 −0.899046
$$877$$ −6832.46 −0.263074 −0.131537 0.991311i $$-0.541991\pi$$
−0.131537 + 0.991311i $$0.541991\pi$$
$$878$$ 83304.9 3.20205
$$879$$ 21575.6 0.827903
$$880$$ 662.701 0.0253860
$$881$$ −3994.77 −0.152766 −0.0763832 0.997079i $$-0.524337\pi$$
−0.0763832 + 0.997079i $$0.524337\pi$$
$$882$$ 0 0
$$883$$ 13727.0 0.523161 0.261580 0.965182i $$-0.415756\pi$$
0.261580 + 0.965182i $$0.415756\pi$$
$$884$$ −8903.38 −0.338748
$$885$$ 12149.0 0.461451
$$886$$ −5515.19 −0.209127
$$887$$ 44119.4 1.67011 0.835054 0.550169i $$-0.185437\pi$$
0.835054 + 0.550169i $$0.185437\pi$$
$$888$$ 243.750 0.00921138
$$889$$ 0 0
$$890$$ 3062.99 0.115361
$$891$$ −922.926 −0.0347017
$$892$$ −19104.8 −0.717125
$$893$$ −898.796 −0.0336809
$$894$$ 25886.2 0.968415
$$895$$ −35215.3 −1.31522
$$896$$ 0 0
$$897$$ 5976.35 0.222458
$$898$$ 37823.1 1.40554
$$899$$ −29442.4 −1.09228
$$900$$ 22331.6 0.827098
$$901$$ 19490.7 0.720678
$$902$$ −2041.49 −0.0753594
$$903$$ 0 0
$$904$$ −7864.18 −0.289335
$$905$$ −64577.7 −2.37197
$$906$$ −27500.1 −1.00842
$$907$$ 36905.8 1.35109 0.675545 0.737319i $$-0.263908\pi$$
0.675545 + 0.737319i $$0.263908\pi$$
$$908$$ 20459.2 0.747755
$$909$$ 13668.1 0.498725
$$910$$ 0 0
$$911$$ −3169.56 −0.115271 −0.0576356 0.998338i $$-0.518356\pi$$
−0.0576356 + 0.998338i $$0.518356\pi$$
$$912$$ −414.141 −0.0150368
$$913$$ 3736.48 0.135443
$$914$$ 56014.9 2.02714
$$915$$ −34958.6 −1.26305
$$916$$ 12894.7 0.465124
$$917$$ 0 0
$$918$$ 6553.05 0.235602
$$919$$ 8727.00 0.313250 0.156625 0.987658i $$-0.449939\pi$$
0.156625 + 0.987658i $$0.449939\pi$$
$$920$$ −58960.2 −2.11289
$$921$$ −1624.06 −0.0581050
$$922$$ −88549.5 −3.16293
$$923$$ 4832.91 0.172348
$$924$$ 0 0
$$925$$ 728.392 0.0258912
$$926$$ 56914.1 2.01978
$$927$$ −9466.95 −0.335421
$$928$$ 35126.1 1.24253
$$929$$ 19405.1 0.685317 0.342659 0.939460i $$-0.388673\pi$$
0.342659 + 0.939460i $$0.388673\pi$$
$$930$$ −38570.2 −1.35997
$$931$$ 0 0
$$932$$ 2529.34 0.0888963
$$933$$ −162.050 −0.00568628
$$934$$ −15424.1 −0.540355
$$935$$ −10847.9 −0.379427
$$936$$ 2557.05 0.0892945
$$937$$ −615.692 −0.0214662 −0.0107331 0.999942i $$-0.503417\pi$$
−0.0107331 + 0.999942i $$0.503417\pi$$
$$938$$ 0 0
$$939$$ −11318.8 −0.393372
$$940$$ 4832.10 0.167666
$$941$$ −29602.0 −1.02550 −0.512751 0.858537i $$-0.671374\pi$$
−0.512751 + 0.858537i $$0.671374\pi$$
$$942$$ −52332.2 −1.81006
$$943$$ −5980.46 −0.206522
$$944$$ 737.310 0.0254210
$$945$$ 0 0
$$946$$ 22294.7 0.766240
$$947$$ 13537.4 0.464527 0.232264 0.972653i $$-0.425387\pi$$
0.232264 + 0.972653i $$0.425387\pi$$
$$948$$ −34845.2 −1.19380
$$949$$ −7975.85 −0.272821
$$950$$ 37585.8 1.28363
$$951$$ −5157.71 −0.175868
$$952$$ 0 0
$$953$$ 33468.5 1.13762 0.568810 0.822469i $$-0.307404\pi$$
0.568810 + 0.822469i $$0.307404\pi$$
$$954$$ −15004.7 −0.509220
$$955$$ 34097.9 1.15538
$$956$$ −15327.7 −0.518550
$$957$$ 6374.80 0.215327
$$958$$ −27244.4 −0.918817
$$959$$ 0 0
$$960$$ 44620.2 1.50011
$$961$$ −4866.88 −0.163368
$$962$$ 223.563 0.00749268
$$963$$ 6898.68 0.230848
$$964$$ −34870.4 −1.16504
$$965$$ 42894.4 1.43090
$$966$$ 0 0
$$967$$ −55733.5 −1.85343 −0.926715 0.375764i $$-0.877380\pi$$
−0.926715 + 0.375764i $$0.877380\pi$$
$$968$$ 26054.0 0.865090
$$969$$ 6779.17 0.224745
$$970$$ −58798.4 −1.94629
$$971$$ −19491.3 −0.644187 −0.322094 0.946708i $$-0.604387\pi$$
−0.322094 + 0.946708i $$0.604387\pi$$
$$972$$ −3100.79 −0.102323
$$973$$ 0 0
$$974$$ 5080.18 0.167125
$$975$$ 7641.17 0.250988
$$976$$ −2121.60 −0.0695807
$$977$$ −6241.56 −0.204386 −0.102193 0.994765i $$-0.532586\pi$$
−0.102193 + 0.994765i $$0.532586\pi$$
$$978$$ −47968.4 −1.56836
$$979$$ 428.554 0.0139905
$$980$$ 0 0
$$981$$ −12843.4 −0.418001
$$982$$ −4949.25 −0.160832
$$983$$ 59694.6 1.93689 0.968444 0.249231i $$-0.0801778\pi$$
0.968444 + 0.249231i $$0.0801778\pi$$
$$984$$ −2558.80 −0.0828980
$$985$$ 27065.7 0.875517
$$986$$ −45263.0 −1.46193
$$987$$ 0 0
$$988$$ 7090.68 0.228324
$$989$$ 65311.4 2.09988
$$990$$ 8351.14 0.268097
$$991$$ 15561.6 0.498821 0.249411 0.968398i $$-0.419763\pi$$
0.249411 + 0.968398i $$0.419763\pi$$
$$992$$ −29735.6 −0.951720
$$993$$ −25224.6 −0.806123
$$994$$ 0 0
$$995$$ 24446.6 0.778903
$$996$$ 12553.6 0.399374
$$997$$ 19884.9 0.631657 0.315829 0.948816i $$-0.397718\pi$$
0.315829 + 0.948816i $$0.397718\pi$$
$$998$$ 10085.5 0.319891
$$999$$ −101.139 −0.00320309
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 147.4.a.l.1.1 3
3.2 odd 2 441.4.a.s.1.3 3
4.3 odd 2 2352.4.a.ci.1.3 3
7.2 even 3 21.4.e.b.4.3 6
7.3 odd 6 147.4.e.n.79.3 6
7.4 even 3 21.4.e.b.16.3 yes 6
7.5 odd 6 147.4.e.n.67.3 6
7.6 odd 2 147.4.a.m.1.1 3
21.2 odd 6 63.4.e.c.46.1 6
21.5 even 6 441.4.e.w.361.1 6
21.11 odd 6 63.4.e.c.37.1 6
21.17 even 6 441.4.e.w.226.1 6
21.20 even 2 441.4.a.t.1.3 3
28.11 odd 6 336.4.q.k.289.1 6
28.23 odd 6 336.4.q.k.193.1 6
28.27 even 2 2352.4.a.cg.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.3 6 7.2 even 3
21.4.e.b.16.3 yes 6 7.4 even 3
63.4.e.c.37.1 6 21.11 odd 6
63.4.e.c.46.1 6 21.2 odd 6
147.4.a.l.1.1 3 1.1 even 1 trivial
147.4.a.m.1.1 3 7.6 odd 2
147.4.e.n.67.3 6 7.5 odd 6
147.4.e.n.79.3 6 7.3 odd 6
336.4.q.k.193.1 6 28.23 odd 6
336.4.q.k.289.1 6 28.11 odd 6
441.4.a.s.1.3 3 3.2 odd 2
441.4.a.t.1.3 3 21.20 even 2
441.4.e.w.226.1 6 21.17 even 6
441.4.e.w.361.1 6 21.5 even 6
2352.4.a.cg.1.1 3 28.27 even 2
2352.4.a.ci.1.3 3 4.3 odd 2