# Properties

 Label 1815.4.a.r.1.3 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.47528.1 Defining polynomial: $$x^{3} - x^{2} - 26 x - 22$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-4.06484$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.06484 q^{2} +3.00000 q^{3} +17.6526 q^{4} -5.00000 q^{5} +15.1945 q^{6} -27.4348 q^{7} +48.8887 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+5.06484 q^{2} +3.00000 q^{3} +17.6526 q^{4} -5.00000 q^{5} +15.1945 q^{6} -27.4348 q^{7} +48.8887 q^{8} +9.00000 q^{9} -25.3242 q^{10} +52.9577 q^{12} -22.6949 q^{13} -138.953 q^{14} -15.0000 q^{15} +106.392 q^{16} +41.1755 q^{17} +45.5835 q^{18} +142.128 q^{19} -88.2628 q^{20} -82.3044 q^{21} +176.166 q^{23} +146.666 q^{24} +25.0000 q^{25} -114.946 q^{26} +27.0000 q^{27} -484.295 q^{28} -76.2044 q^{29} -75.9725 q^{30} +197.373 q^{31} +147.751 q^{32} +208.547 q^{34} +137.174 q^{35} +158.873 q^{36} +367.297 q^{37} +719.856 q^{38} -68.0846 q^{39} -244.443 q^{40} +238.279 q^{41} -416.858 q^{42} -30.2905 q^{43} -45.0000 q^{45} +892.254 q^{46} +137.390 q^{47} +319.177 q^{48} +409.668 q^{49} +126.621 q^{50} +123.526 q^{51} -400.623 q^{52} +638.665 q^{53} +136.751 q^{54} -1341.25 q^{56} +426.385 q^{57} -385.963 q^{58} +103.146 q^{59} -264.788 q^{60} -605.596 q^{61} +999.662 q^{62} -246.913 q^{63} -102.804 q^{64} +113.474 q^{65} -704.925 q^{67} +726.852 q^{68} +528.499 q^{69} +694.764 q^{70} -782.162 q^{71} +439.998 q^{72} +243.132 q^{73} +1860.30 q^{74} +75.0000 q^{75} +2508.93 q^{76} -344.837 q^{78} +532.874 q^{79} -531.962 q^{80} +81.0000 q^{81} +1206.84 q^{82} -1204.91 q^{83} -1452.88 q^{84} -205.877 q^{85} -153.416 q^{86} -228.613 q^{87} +1058.49 q^{89} -227.918 q^{90} +622.629 q^{91} +3109.79 q^{92} +592.119 q^{93} +695.857 q^{94} -710.641 q^{95} +443.254 q^{96} -85.1964 q^{97} +2074.90 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} + O(q^{10})$$ $$3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} - 10 q^{10} + 90 q^{12} - 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} + 104 q^{17} + 18 q^{18} + 58 q^{19} - 150 q^{20} - 30 q^{21} + 120 q^{23} - 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} - 676 q^{28} + 220 q^{29} - 30 q^{30} + 248 q^{31} + 258 q^{32} - 80 q^{34} + 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} - 342 q^{39} + 90 q^{40} - 156 q^{41} - 204 q^{42} - 122 q^{43} - 135 q^{45} + 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} + 50 q^{50} + 312 q^{51} - 520 q^{52} + 282 q^{53} + 54 q^{54} - 1644 q^{56} + 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} - 414 q^{61} + 2448 q^{62} - 90 q^{63} - 58 q^{64} + 570 q^{65} - 428 q^{67} + 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} - 162 q^{72} - 618 q^{73} + 1612 q^{74} + 225 q^{75} + 2752 q^{76} - 360 q^{78} + 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} - 2028 q^{84} - 520 q^{85} - 1548 q^{86} + 660 q^{87} + 790 q^{89} - 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} + 424 q^{94} - 290 q^{95} + 774 q^{96} + 2074 q^{97} + 3978 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.06484 1.79069 0.895345 0.445373i $$-0.146929\pi$$
0.895345 + 0.445373i $$0.146929\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 17.6526 2.20657
$$5$$ −5.00000 −0.447214
$$6$$ 15.1945 1.03386
$$7$$ −27.4348 −1.48134 −0.740670 0.671869i $$-0.765492\pi$$
−0.740670 + 0.671869i $$0.765492\pi$$
$$8$$ 48.8887 2.16059
$$9$$ 9.00000 0.333333
$$10$$ −25.3242 −0.800821
$$11$$ 0 0
$$12$$ 52.9577 1.27396
$$13$$ −22.6949 −0.484187 −0.242093 0.970253i $$-0.577834\pi$$
−0.242093 + 0.970253i $$0.577834\pi$$
$$14$$ −138.953 −2.65262
$$15$$ −15.0000 −0.258199
$$16$$ 106.392 1.66238
$$17$$ 41.1755 0.587442 0.293721 0.955891i $$-0.405106\pi$$
0.293721 + 0.955891i $$0.405106\pi$$
$$18$$ 45.5835 0.596897
$$19$$ 142.128 1.71613 0.858064 0.513542i $$-0.171667\pi$$
0.858064 + 0.513542i $$0.171667\pi$$
$$20$$ −88.2628 −0.986808
$$21$$ −82.3044 −0.855252
$$22$$ 0 0
$$23$$ 176.166 1.59710 0.798548 0.601931i $$-0.205602\pi$$
0.798548 + 0.601931i $$0.205602\pi$$
$$24$$ 146.666 1.24742
$$25$$ 25.0000 0.200000
$$26$$ −114.946 −0.867028
$$27$$ 27.0000 0.192450
$$28$$ −484.295 −3.26868
$$29$$ −76.2044 −0.487959 −0.243979 0.969780i $$-0.578453\pi$$
−0.243979 + 0.969780i $$0.578453\pi$$
$$30$$ −75.9725 −0.462354
$$31$$ 197.373 1.14352 0.571762 0.820420i $$-0.306260\pi$$
0.571762 + 0.820420i $$0.306260\pi$$
$$32$$ 147.751 0.816218
$$33$$ 0 0
$$34$$ 208.547 1.05193
$$35$$ 137.174 0.662475
$$36$$ 158.873 0.735523
$$37$$ 367.297 1.63198 0.815989 0.578067i $$-0.196193\pi$$
0.815989 + 0.578067i $$0.196193\pi$$
$$38$$ 719.856 3.07305
$$39$$ −68.0846 −0.279545
$$40$$ −244.443 −0.966247
$$41$$ 238.279 0.907633 0.453817 0.891095i $$-0.350062\pi$$
0.453817 + 0.891095i $$0.350062\pi$$
$$42$$ −416.858 −1.53149
$$43$$ −30.2905 −0.107425 −0.0537123 0.998556i $$-0.517105\pi$$
−0.0537123 + 0.998556i $$0.517105\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 892.254 2.85990
$$47$$ 137.390 0.426391 0.213195 0.977010i $$-0.431613\pi$$
0.213195 + 0.977010i $$0.431613\pi$$
$$48$$ 319.177 0.959777
$$49$$ 409.668 1.19437
$$50$$ 126.621 0.358138
$$51$$ 123.526 0.339160
$$52$$ −400.623 −1.06839
$$53$$ 638.665 1.65523 0.827617 0.561293i $$-0.189696\pi$$
0.827617 + 0.561293i $$0.189696\pi$$
$$54$$ 136.751 0.344618
$$55$$ 0 0
$$56$$ −1341.25 −3.20057
$$57$$ 426.385 0.990807
$$58$$ −385.963 −0.873783
$$59$$ 103.146 0.227601 0.113800 0.993504i $$-0.463698\pi$$
0.113800 + 0.993504i $$0.463698\pi$$
$$60$$ −264.788 −0.569734
$$61$$ −605.596 −1.27113 −0.635563 0.772049i $$-0.719232\pi$$
−0.635563 + 0.772049i $$0.719232\pi$$
$$62$$ 999.662 2.04770
$$63$$ −246.913 −0.493780
$$64$$ −102.804 −0.200789
$$65$$ 113.474 0.216535
$$66$$ 0 0
$$67$$ −704.925 −1.28538 −0.642688 0.766128i $$-0.722181\pi$$
−0.642688 + 0.766128i $$0.722181\pi$$
$$68$$ 726.852 1.29623
$$69$$ 528.499 0.922084
$$70$$ 694.764 1.18629
$$71$$ −782.162 −1.30740 −0.653701 0.756753i $$-0.726785\pi$$
−0.653701 + 0.756753i $$0.726785\pi$$
$$72$$ 439.998 0.720198
$$73$$ 243.132 0.389814 0.194907 0.980822i $$-0.437560\pi$$
0.194907 + 0.980822i $$0.437560\pi$$
$$74$$ 1860.30 2.92237
$$75$$ 75.0000 0.115470
$$76$$ 2508.93 3.78676
$$77$$ 0 0
$$78$$ −344.837 −0.500579
$$79$$ 532.874 0.758899 0.379449 0.925212i $$-0.376113\pi$$
0.379449 + 0.925212i $$0.376113\pi$$
$$80$$ −531.962 −0.743440
$$81$$ 81.0000 0.111111
$$82$$ 1206.84 1.62529
$$83$$ −1204.91 −1.59344 −0.796722 0.604345i $$-0.793435\pi$$
−0.796722 + 0.604345i $$0.793435\pi$$
$$84$$ −1452.88 −1.88717
$$85$$ −205.877 −0.262712
$$86$$ −153.416 −0.192364
$$87$$ −228.613 −0.281723
$$88$$ 0 0
$$89$$ 1058.49 1.26067 0.630337 0.776322i $$-0.282917\pi$$
0.630337 + 0.776322i $$0.282917\pi$$
$$90$$ −227.918 −0.266940
$$91$$ 622.629 0.717245
$$92$$ 3109.79 3.52411
$$93$$ 592.119 0.660214
$$94$$ 695.857 0.763533
$$95$$ −710.641 −0.767476
$$96$$ 443.254 0.471244
$$97$$ −85.1964 −0.0891792 −0.0445896 0.999005i $$-0.514198\pi$$
−0.0445896 + 0.999005i $$0.514198\pi$$
$$98$$ 2074.90 2.13874
$$99$$ 0 0
$$100$$ 441.314 0.441314
$$101$$ 7.81823 0.00770241 0.00385120 0.999993i $$-0.498774\pi$$
0.00385120 + 0.999993i $$0.498774\pi$$
$$102$$ 625.641 0.607330
$$103$$ 12.4770 0.0119359 0.00596794 0.999982i $$-0.498100\pi$$
0.00596794 + 0.999982i $$0.498100\pi$$
$$104$$ −1109.52 −1.04613
$$105$$ 411.522 0.382480
$$106$$ 3234.73 2.96401
$$107$$ 1376.81 1.24393 0.621967 0.783043i $$-0.286334\pi$$
0.621967 + 0.783043i $$0.286334\pi$$
$$108$$ 476.619 0.424655
$$109$$ −610.189 −0.536197 −0.268099 0.963391i $$-0.586395\pi$$
−0.268099 + 0.963391i $$0.586395\pi$$
$$110$$ 0 0
$$111$$ 1101.89 0.942223
$$112$$ −2918.86 −2.46255
$$113$$ −36.7727 −0.0306132 −0.0153066 0.999883i $$-0.504872\pi$$
−0.0153066 + 0.999883i $$0.504872\pi$$
$$114$$ 2159.57 1.77423
$$115$$ −880.832 −0.714243
$$116$$ −1345.20 −1.07672
$$117$$ −204.254 −0.161396
$$118$$ 522.416 0.407562
$$119$$ −1129.64 −0.870201
$$120$$ −733.330 −0.557863
$$121$$ 0 0
$$122$$ −3067.25 −2.27619
$$123$$ 714.838 0.524022
$$124$$ 3484.14 2.52326
$$125$$ −125.000 −0.0894427
$$126$$ −1250.57 −0.884207
$$127$$ 54.5310 0.0381011 0.0190506 0.999819i $$-0.493936\pi$$
0.0190506 + 0.999819i $$0.493936\pi$$
$$128$$ −1702.69 −1.17577
$$129$$ −90.8715 −0.0620216
$$130$$ 574.729 0.387747
$$131$$ −2377.83 −1.58589 −0.792947 0.609290i $$-0.791454\pi$$
−0.792947 + 0.609290i $$0.791454\pi$$
$$132$$ 0 0
$$133$$ −3899.26 −2.54217
$$134$$ −3570.33 −2.30171
$$135$$ −135.000 −0.0860663
$$136$$ 2013.01 1.26922
$$137$$ −3078.41 −1.91976 −0.959878 0.280417i $$-0.909527\pi$$
−0.959878 + 0.280417i $$0.909527\pi$$
$$138$$ 2676.76 1.65117
$$139$$ 1197.18 0.730530 0.365265 0.930904i $$-0.380978\pi$$
0.365265 + 0.930904i $$0.380978\pi$$
$$140$$ 2421.47 1.46180
$$141$$ 412.169 0.246177
$$142$$ −3961.52 −2.34115
$$143$$ 0 0
$$144$$ 957.532 0.554128
$$145$$ 381.022 0.218222
$$146$$ 1231.42 0.698035
$$147$$ 1229.00 0.689569
$$148$$ 6483.73 3.60108
$$149$$ 749.456 0.412066 0.206033 0.978545i $$-0.433945\pi$$
0.206033 + 0.978545i $$0.433945\pi$$
$$150$$ 379.863 0.206771
$$151$$ 2645.72 1.42586 0.712931 0.701234i $$-0.247367\pi$$
0.712931 + 0.701234i $$0.247367\pi$$
$$152$$ 6948.46 3.70786
$$153$$ 370.579 0.195814
$$154$$ 0 0
$$155$$ −986.865 −0.511399
$$156$$ −1201.87 −0.616836
$$157$$ −3475.74 −1.76684 −0.883422 0.468578i $$-0.844767\pi$$
−0.883422 + 0.468578i $$0.844767\pi$$
$$158$$ 2698.92 1.35895
$$159$$ 1916.00 0.955650
$$160$$ −738.757 −0.365024
$$161$$ −4833.09 −2.36584
$$162$$ 410.252 0.198966
$$163$$ 3518.89 1.69092 0.845462 0.534035i $$-0.179325\pi$$
0.845462 + 0.534035i $$0.179325\pi$$
$$164$$ 4206.24 2.00276
$$165$$ 0 0
$$166$$ −6102.67 −2.85337
$$167$$ −250.304 −0.115983 −0.0579914 0.998317i $$-0.518470\pi$$
−0.0579914 + 0.998317i $$0.518470\pi$$
$$168$$ −4023.75 −1.84785
$$169$$ −1681.94 −0.765563
$$170$$ −1042.73 −0.470436
$$171$$ 1279.15 0.572043
$$172$$ −534.705 −0.237040
$$173$$ −941.985 −0.413976 −0.206988 0.978344i $$-0.566366\pi$$
−0.206988 + 0.978344i $$0.566366\pi$$
$$174$$ −1157.89 −0.504479
$$175$$ −685.870 −0.296268
$$176$$ 0 0
$$177$$ 309.437 0.131405
$$178$$ 5361.09 2.25748
$$179$$ 336.037 0.140316 0.0701582 0.997536i $$-0.477650\pi$$
0.0701582 + 0.997536i $$0.477650\pi$$
$$180$$ −794.365 −0.328936
$$181$$ 1107.45 0.454784 0.227392 0.973803i $$-0.426980\pi$$
0.227392 + 0.973803i $$0.426980\pi$$
$$182$$ 3153.52 1.28436
$$183$$ −1816.79 −0.733885
$$184$$ 8612.53 3.45068
$$185$$ −1836.48 −0.729843
$$186$$ 2998.98 1.18224
$$187$$ 0 0
$$188$$ 2425.28 0.940861
$$189$$ −740.740 −0.285084
$$190$$ −3599.28 −1.37431
$$191$$ −4243.01 −1.60740 −0.803700 0.595035i $$-0.797138\pi$$
−0.803700 + 0.595035i $$0.797138\pi$$
$$192$$ −308.411 −0.115925
$$193$$ 3324.23 1.23981 0.619905 0.784677i $$-0.287171\pi$$
0.619905 + 0.784677i $$0.287171\pi$$
$$194$$ −431.506 −0.159692
$$195$$ 340.423 0.125016
$$196$$ 7231.69 2.63546
$$197$$ −2677.14 −0.968213 −0.484107 0.875009i $$-0.660855\pi$$
−0.484107 + 0.875009i $$0.660855\pi$$
$$198$$ 0 0
$$199$$ 2779.90 0.990261 0.495131 0.868819i $$-0.335120\pi$$
0.495131 + 0.868819i $$0.335120\pi$$
$$200$$ 1222.22 0.432119
$$201$$ −2114.77 −0.742113
$$202$$ 39.5981 0.0137926
$$203$$ 2090.65 0.722833
$$204$$ 2180.56 0.748380
$$205$$ −1191.40 −0.405906
$$206$$ 63.1940 0.0213735
$$207$$ 1585.50 0.532365
$$208$$ −2414.56 −0.804903
$$209$$ 0 0
$$210$$ 2084.29 0.684904
$$211$$ −3056.15 −0.997129 −0.498564 0.866853i $$-0.666139\pi$$
−0.498564 + 0.866853i $$0.666139\pi$$
$$212$$ 11274.1 3.65239
$$213$$ −2346.49 −0.754829
$$214$$ 6973.30 2.22750
$$215$$ 151.453 0.0480417
$$216$$ 1319.99 0.415806
$$217$$ −5414.89 −1.69395
$$218$$ −3090.51 −0.960163
$$219$$ 729.395 0.225059
$$220$$ 0 0
$$221$$ −934.472 −0.284432
$$222$$ 5580.89 1.68723
$$223$$ 571.375 0.171579 0.0857895 0.996313i $$-0.472659\pi$$
0.0857895 + 0.996313i $$0.472659\pi$$
$$224$$ −4053.53 −1.20910
$$225$$ 225.000 0.0666667
$$226$$ −186.248 −0.0548187
$$227$$ 1094.40 0.319989 0.159995 0.987118i $$-0.448852\pi$$
0.159995 + 0.987118i $$0.448852\pi$$
$$228$$ 7526.78 2.18629
$$229$$ −645.240 −0.186195 −0.0930975 0.995657i $$-0.529677\pi$$
−0.0930975 + 0.995657i $$0.529677\pi$$
$$230$$ −4461.27 −1.27899
$$231$$ 0 0
$$232$$ −3725.53 −1.05428
$$233$$ 2337.39 0.657201 0.328600 0.944469i $$-0.393423\pi$$
0.328600 + 0.944469i $$0.393423\pi$$
$$234$$ −1034.51 −0.289009
$$235$$ −686.949 −0.190688
$$236$$ 1820.79 0.502217
$$237$$ 1598.62 0.438151
$$238$$ −5721.44 −1.55826
$$239$$ 3656.91 0.989733 0.494866 0.868969i $$-0.335217\pi$$
0.494866 + 0.868969i $$0.335217\pi$$
$$240$$ −1595.89 −0.429225
$$241$$ 389.034 0.103983 0.0519914 0.998648i $$-0.483443\pi$$
0.0519914 + 0.998648i $$0.483443\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ −10690.3 −2.80483
$$245$$ −2048.34 −0.534138
$$246$$ 3620.53 0.938361
$$247$$ −3225.58 −0.830927
$$248$$ 9649.30 2.47069
$$249$$ −3614.73 −0.919976
$$250$$ −633.104 −0.160164
$$251$$ 2299.55 0.578271 0.289135 0.957288i $$-0.406632\pi$$
0.289135 + 0.957288i $$0.406632\pi$$
$$252$$ −4358.65 −1.08956
$$253$$ 0 0
$$254$$ 276.191 0.0682273
$$255$$ −617.632 −0.151677
$$256$$ −7801.44 −1.90465
$$257$$ 4921.61 1.19456 0.597279 0.802033i $$-0.296248\pi$$
0.597279 + 0.802033i $$0.296248\pi$$
$$258$$ −460.249 −0.111062
$$259$$ −10076.7 −2.41752
$$260$$ 2003.11 0.477799
$$261$$ −685.840 −0.162653
$$262$$ −12043.3 −2.83985
$$263$$ 2575.61 0.603875 0.301938 0.953328i $$-0.402367\pi$$
0.301938 + 0.953328i $$0.402367\pi$$
$$264$$ 0 0
$$265$$ −3193.33 −0.740243
$$266$$ −19749.1 −4.55224
$$267$$ 3175.48 0.727850
$$268$$ −12443.7 −2.83627
$$269$$ −4794.97 −1.08682 −0.543410 0.839468i $$-0.682867\pi$$
−0.543410 + 0.839468i $$0.682867\pi$$
$$270$$ −683.753 −0.154118
$$271$$ −2729.47 −0.611821 −0.305910 0.952060i $$-0.598961\pi$$
−0.305910 + 0.952060i $$0.598961\pi$$
$$272$$ 4380.76 0.976553
$$273$$ 1867.89 0.414102
$$274$$ −15591.7 −3.43769
$$275$$ 0 0
$$276$$ 9329.36 2.03464
$$277$$ 3761.45 0.815898 0.407949 0.913005i $$-0.366244\pi$$
0.407949 + 0.913005i $$0.366244\pi$$
$$278$$ 6063.53 1.30815
$$279$$ 1776.36 0.381174
$$280$$ 6706.25 1.43134
$$281$$ 6434.87 1.36609 0.683046 0.730375i $$-0.260655\pi$$
0.683046 + 0.730375i $$0.260655\pi$$
$$282$$ 2087.57 0.440826
$$283$$ −3335.65 −0.700649 −0.350325 0.936628i $$-0.613929\pi$$
−0.350325 + 0.936628i $$0.613929\pi$$
$$284$$ −13807.2 −2.88487
$$285$$ −2131.92 −0.443103
$$286$$ 0 0
$$287$$ −6537.14 −1.34451
$$288$$ 1329.76 0.272073
$$289$$ −3217.58 −0.654912
$$290$$ 1929.81 0.390768
$$291$$ −255.589 −0.0514876
$$292$$ 4291.89 0.860151
$$293$$ 2878.93 0.574023 0.287012 0.957927i $$-0.407338\pi$$
0.287012 + 0.957927i $$0.407338\pi$$
$$294$$ 6224.71 1.23480
$$295$$ −515.729 −0.101786
$$296$$ 17956.6 3.52604
$$297$$ 0 0
$$298$$ 3795.87 0.737883
$$299$$ −3998.07 −0.773293
$$300$$ 1323.94 0.254793
$$301$$ 831.014 0.159132
$$302$$ 13400.1 2.55328
$$303$$ 23.4547 0.00444699
$$304$$ 15121.4 2.85286
$$305$$ 3027.98 0.568465
$$306$$ 1876.92 0.350642
$$307$$ 8154.28 1.51593 0.757963 0.652297i $$-0.226195\pi$$
0.757963 + 0.652297i $$0.226195\pi$$
$$308$$ 0 0
$$309$$ 37.4310 0.00689119
$$310$$ −4998.31 −0.915757
$$311$$ 3818.24 0.696182 0.348091 0.937461i $$-0.386830\pi$$
0.348091 + 0.937461i $$0.386830\pi$$
$$312$$ −3328.57 −0.603984
$$313$$ 2527.23 0.456381 0.228191 0.973616i $$-0.426719\pi$$
0.228191 + 0.973616i $$0.426719\pi$$
$$314$$ −17604.1 −3.16387
$$315$$ 1234.57 0.220825
$$316$$ 9406.59 1.67456
$$317$$ −11084.7 −1.96398 −0.981989 0.188937i $$-0.939496\pi$$
−0.981989 + 0.188937i $$0.939496\pi$$
$$318$$ 9704.20 1.71127
$$319$$ 0 0
$$320$$ 514.019 0.0897954
$$321$$ 4130.42 0.718186
$$322$$ −24478.8 −4.23649
$$323$$ 5852.19 1.00813
$$324$$ 1429.86 0.245174
$$325$$ −567.372 −0.0968373
$$326$$ 17822.6 3.02792
$$327$$ −1830.57 −0.309574
$$328$$ 11649.1 1.96103
$$329$$ −3769.26 −0.631629
$$330$$ 0 0
$$331$$ −9417.70 −1.56388 −0.781939 0.623355i $$-0.785769\pi$$
−0.781939 + 0.623355i $$0.785769\pi$$
$$332$$ −21269.7 −3.51605
$$333$$ 3305.67 0.543993
$$334$$ −1267.75 −0.207689
$$335$$ 3524.62 0.574838
$$336$$ −8756.57 −1.42176
$$337$$ 11263.2 1.82061 0.910305 0.413938i $$-0.135847\pi$$
0.910305 + 0.413938i $$0.135847\pi$$
$$338$$ −8518.76 −1.37089
$$339$$ −110.318 −0.0176745
$$340$$ −3634.26 −0.579693
$$341$$ 0 0
$$342$$ 6478.70 1.02435
$$343$$ −1829.03 −0.287925
$$344$$ −1480.86 −0.232101
$$345$$ −2642.49 −0.412369
$$346$$ −4771.00 −0.741302
$$347$$ 4213.25 0.651814 0.325907 0.945402i $$-0.394330\pi$$
0.325907 + 0.945402i $$0.394330\pi$$
$$348$$ −4035.61 −0.621642
$$349$$ −4987.07 −0.764905 −0.382452 0.923975i $$-0.624920\pi$$
−0.382452 + 0.923975i $$0.624920\pi$$
$$350$$ −3473.82 −0.530524
$$351$$ −612.762 −0.0931817
$$352$$ 0 0
$$353$$ −7633.47 −1.15096 −0.575480 0.817816i $$-0.695185\pi$$
−0.575480 + 0.817816i $$0.695185\pi$$
$$354$$ 1567.25 0.235306
$$355$$ 3910.81 0.584688
$$356$$ 18685.1 2.78177
$$357$$ −3388.92 −0.502411
$$358$$ 1701.97 0.251263
$$359$$ 3114.76 0.457913 0.228957 0.973437i $$-0.426469\pi$$
0.228957 + 0.973437i $$0.426469\pi$$
$$360$$ −2199.99 −0.322082
$$361$$ 13341.4 1.94510
$$362$$ 5609.04 0.814378
$$363$$ 0 0
$$364$$ 10991.0 1.58265
$$365$$ −1215.66 −0.174330
$$366$$ −9201.74 −1.31416
$$367$$ 5931.48 0.843653 0.421827 0.906677i $$-0.361389\pi$$
0.421827 + 0.906677i $$0.361389\pi$$
$$368$$ 18742.8 2.65499
$$369$$ 2144.51 0.302544
$$370$$ −9301.49 −1.30692
$$371$$ −17521.7 −2.45196
$$372$$ 10452.4 1.45681
$$373$$ −13918.9 −1.93215 −0.966073 0.258267i $$-0.916848\pi$$
−0.966073 + 0.258267i $$0.916848\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 6716.80 0.921257
$$377$$ 1729.45 0.236263
$$378$$ −3751.72 −0.510497
$$379$$ −12267.3 −1.66261 −0.831303 0.555820i $$-0.812404\pi$$
−0.831303 + 0.555820i $$0.812404\pi$$
$$380$$ −12544.6 −1.69349
$$381$$ 163.593 0.0219977
$$382$$ −21490.1 −2.87835
$$383$$ −6935.04 −0.925233 −0.462616 0.886559i $$-0.653089\pi$$
−0.462616 + 0.886559i $$0.653089\pi$$
$$384$$ −5108.08 −0.678830
$$385$$ 0 0
$$386$$ 16836.7 2.22012
$$387$$ −272.615 −0.0358082
$$388$$ −1503.93 −0.196780
$$389$$ 2775.18 0.361715 0.180858 0.983509i $$-0.442113\pi$$
0.180858 + 0.983509i $$0.442113\pi$$
$$390$$ 1724.19 0.223866
$$391$$ 7253.73 0.938202
$$392$$ 20028.1 2.58054
$$393$$ −7133.50 −0.915617
$$394$$ −13559.3 −1.73377
$$395$$ −2664.37 −0.339390
$$396$$ 0 0
$$397$$ 10539.5 1.33240 0.666198 0.745775i $$-0.267921\pi$$
0.666198 + 0.745775i $$0.267921\pi$$
$$398$$ 14079.7 1.77325
$$399$$ −11697.8 −1.46772
$$400$$ 2659.81 0.332477
$$401$$ −12295.3 −1.53117 −0.765585 0.643334i $$-0.777550\pi$$
−0.765585 + 0.643334i $$0.777550\pi$$
$$402$$ −10711.0 −1.32889
$$403$$ −4479.35 −0.553679
$$404$$ 138.012 0.0169959
$$405$$ −405.000 −0.0496904
$$406$$ 10588.8 1.29437
$$407$$ 0 0
$$408$$ 6039.04 0.732787
$$409$$ 11545.7 1.39584 0.697918 0.716178i $$-0.254110\pi$$
0.697918 + 0.716178i $$0.254110\pi$$
$$410$$ −6034.22 −0.726852
$$411$$ −9235.24 −1.10837
$$412$$ 220.251 0.0263374
$$413$$ −2829.78 −0.337154
$$414$$ 8030.28 0.953301
$$415$$ 6024.54 0.712610
$$416$$ −3353.20 −0.395202
$$417$$ 3591.55 0.421772
$$418$$ 0 0
$$419$$ 1851.51 0.215876 0.107938 0.994158i $$-0.465575\pi$$
0.107938 + 0.994158i $$0.465575\pi$$
$$420$$ 7264.42 0.843970
$$421$$ −1303.60 −0.150911 −0.0754557 0.997149i $$-0.524041\pi$$
−0.0754557 + 0.997149i $$0.524041\pi$$
$$422$$ −15478.9 −1.78555
$$423$$ 1236.51 0.142130
$$424$$ 31223.5 3.57629
$$425$$ 1029.39 0.117488
$$426$$ −11884.6 −1.35166
$$427$$ 16614.4 1.88297
$$428$$ 24304.2 2.74483
$$429$$ 0 0
$$430$$ 767.082 0.0860279
$$431$$ −8228.85 −0.919652 −0.459826 0.888009i $$-0.652088\pi$$
−0.459826 + 0.888009i $$0.652088\pi$$
$$432$$ 2872.60 0.319926
$$433$$ 5830.21 0.647072 0.323536 0.946216i $$-0.395128\pi$$
0.323536 + 0.946216i $$0.395128\pi$$
$$434$$ −27425.5 −3.03333
$$435$$ 1143.07 0.125990
$$436$$ −10771.4 −1.18316
$$437$$ 25038.2 2.74082
$$438$$ 3694.26 0.403011
$$439$$ −14261.0 −1.55044 −0.775218 0.631694i $$-0.782360\pi$$
−0.775218 + 0.631694i $$0.782360\pi$$
$$440$$ 0 0
$$441$$ 3687.01 0.398123
$$442$$ −4732.95 −0.509329
$$443$$ 12025.7 1.28975 0.644875 0.764288i $$-0.276910\pi$$
0.644875 + 0.764288i $$0.276910\pi$$
$$444$$ 19451.2 2.07908
$$445$$ −5292.46 −0.563790
$$446$$ 2893.92 0.307245
$$447$$ 2248.37 0.237907
$$448$$ 2820.40 0.297436
$$449$$ −7073.28 −0.743449 −0.371725 0.928343i $$-0.621233\pi$$
−0.371725 + 0.928343i $$0.621233\pi$$
$$450$$ 1139.59 0.119379
$$451$$ 0 0
$$452$$ −649.133 −0.0675501
$$453$$ 7937.15 0.823222
$$454$$ 5542.93 0.573001
$$455$$ −3113.15 −0.320762
$$456$$ 20845.4 2.14073
$$457$$ −2732.31 −0.279677 −0.139838 0.990174i $$-0.544658\pi$$
−0.139838 + 0.990174i $$0.544658\pi$$
$$458$$ −3268.04 −0.333418
$$459$$ 1111.74 0.113053
$$460$$ −15548.9 −1.57603
$$461$$ 332.708 0.0336134 0.0168067 0.999859i $$-0.494650\pi$$
0.0168067 + 0.999859i $$0.494650\pi$$
$$462$$ 0 0
$$463$$ 8248.39 0.827937 0.413969 0.910291i $$-0.364142\pi$$
0.413969 + 0.910291i $$0.364142\pi$$
$$464$$ −8107.58 −0.811174
$$465$$ −2960.59 −0.295256
$$466$$ 11838.5 1.17684
$$467$$ 7359.35 0.729229 0.364615 0.931158i $$-0.381201\pi$$
0.364615 + 0.931158i $$0.381201\pi$$
$$468$$ −3605.60 −0.356131
$$469$$ 19339.5 1.90408
$$470$$ −3479.28 −0.341462
$$471$$ −10427.2 −1.02009
$$472$$ 5042.66 0.491752
$$473$$ 0 0
$$474$$ 8096.76 0.784592
$$475$$ 3553.21 0.343226
$$476$$ −19941.0 −1.92016
$$477$$ 5747.99 0.551745
$$478$$ 18521.7 1.77230
$$479$$ 10327.6 0.985140 0.492570 0.870273i $$-0.336058\pi$$
0.492570 + 0.870273i $$0.336058\pi$$
$$480$$ −2216.27 −0.210747
$$481$$ −8335.75 −0.790182
$$482$$ 1970.39 0.186201
$$483$$ −14499.3 −1.36592
$$484$$ 0 0
$$485$$ 425.982 0.0398822
$$486$$ 1230.76 0.114873
$$487$$ 9622.57 0.895360 0.447680 0.894194i $$-0.352250\pi$$
0.447680 + 0.894194i $$0.352250\pi$$
$$488$$ −29606.8 −2.74639
$$489$$ 10556.7 0.976256
$$490$$ −10374.5 −0.956475
$$491$$ 8993.59 0.826629 0.413315 0.910588i $$-0.364371\pi$$
0.413315 + 0.910588i $$0.364371\pi$$
$$492$$ 12618.7 1.15629
$$493$$ −3137.75 −0.286648
$$494$$ −16337.0 −1.48793
$$495$$ 0 0
$$496$$ 20999.0 1.90097
$$497$$ 21458.5 1.93671
$$498$$ −18308.0 −1.64739
$$499$$ 2623.70 0.235377 0.117688 0.993051i $$-0.462452\pi$$
0.117688 + 0.993051i $$0.462452\pi$$
$$500$$ −2206.57 −0.197362
$$501$$ −750.912 −0.0669627
$$502$$ 11646.8 1.03550
$$503$$ −13234.5 −1.17316 −0.586579 0.809892i $$-0.699526\pi$$
−0.586579 + 0.809892i $$0.699526\pi$$
$$504$$ −12071.3 −1.06686
$$505$$ −39.0912 −0.00344462
$$506$$ 0 0
$$507$$ −5045.83 −0.441998
$$508$$ 962.612 0.0840728
$$509$$ −12810.3 −1.11553 −0.557765 0.829999i $$-0.688341\pi$$
−0.557765 + 0.829999i $$0.688341\pi$$
$$510$$ −3128.20 −0.271606
$$511$$ −6670.26 −0.577446
$$512$$ −25891.5 −2.23487
$$513$$ 3837.46 0.330269
$$514$$ 24927.1 2.13908
$$515$$ −62.3850 −0.00533789
$$516$$ −1604.12 −0.136855
$$517$$ 0 0
$$518$$ −51036.9 −4.32902
$$519$$ −2825.95 −0.239009
$$520$$ 5547.61 0.467844
$$521$$ 1151.47 0.0968268 0.0484134 0.998827i $$-0.484584\pi$$
0.0484134 + 0.998827i $$0.484584\pi$$
$$522$$ −3473.67 −0.291261
$$523$$ 17319.1 1.44801 0.724005 0.689794i $$-0.242299\pi$$
0.724005 + 0.689794i $$0.242299\pi$$
$$524$$ −41974.8 −3.49939
$$525$$ −2057.61 −0.171050
$$526$$ 13045.1 1.08135
$$527$$ 8126.92 0.671754
$$528$$ 0 0
$$529$$ 18867.6 1.55072
$$530$$ −16173.7 −1.32555
$$531$$ 928.312 0.0758669
$$532$$ −68831.9 −5.60948
$$533$$ −5407.72 −0.439464
$$534$$ 16083.3 1.30335
$$535$$ −6884.03 −0.556304
$$536$$ −34462.8 −2.77718
$$537$$ 1008.11 0.0810117
$$538$$ −24285.7 −1.94616
$$539$$ 0 0
$$540$$ −2383.10 −0.189911
$$541$$ 7190.73 0.571449 0.285724 0.958312i $$-0.407766\pi$$
0.285724 + 0.958312i $$0.407766\pi$$
$$542$$ −13824.3 −1.09558
$$543$$ 3322.34 0.262570
$$544$$ 6083.73 0.479481
$$545$$ 3050.94 0.239795
$$546$$ 9460.55 0.741527
$$547$$ −18670.5 −1.45940 −0.729701 0.683766i $$-0.760341\pi$$
−0.729701 + 0.683766i $$0.760341\pi$$
$$548$$ −54341.9 −4.23608
$$549$$ −5450.37 −0.423709
$$550$$ 0 0
$$551$$ −10830.8 −0.837400
$$552$$ 25837.6 1.99225
$$553$$ −14619.3 −1.12419
$$554$$ 19051.1 1.46102
$$555$$ −5509.45 −0.421375
$$556$$ 21133.3 1.61197
$$557$$ 5510.44 0.419183 0.209591 0.977789i $$-0.432787\pi$$
0.209591 + 0.977789i $$0.432787\pi$$
$$558$$ 8996.95 0.682565
$$559$$ 687.439 0.0520136
$$560$$ 14594.3 1.10129
$$561$$ 0 0
$$562$$ 32591.5 2.44625
$$563$$ −3576.57 −0.267734 −0.133867 0.990999i $$-0.542740\pi$$
−0.133867 + 0.990999i $$0.542740\pi$$
$$564$$ 7275.84 0.543206
$$565$$ 183.864 0.0136906
$$566$$ −16894.5 −1.25465
$$567$$ −2222.22 −0.164593
$$568$$ −38238.8 −2.82476
$$569$$ −12285.2 −0.905138 −0.452569 0.891729i $$-0.649492\pi$$
−0.452569 + 0.891729i $$0.649492\pi$$
$$570$$ −10797.8 −0.793459
$$571$$ −13889.5 −1.01797 −0.508983 0.860777i $$-0.669978\pi$$
−0.508983 + 0.860777i $$0.669978\pi$$
$$572$$ 0 0
$$573$$ −12729.0 −0.928032
$$574$$ −33109.5 −2.40761
$$575$$ 4404.16 0.319419
$$576$$ −925.234 −0.0669296
$$577$$ 11579.4 0.835457 0.417728 0.908572i $$-0.362826\pi$$
0.417728 + 0.908572i $$0.362826\pi$$
$$578$$ −16296.5 −1.17274
$$579$$ 9972.69 0.715805
$$580$$ 6726.02 0.481522
$$581$$ 33056.4 2.36043
$$582$$ −1294.52 −0.0921984
$$583$$ 0 0
$$584$$ 11886.4 0.842229
$$585$$ 1021.27 0.0721783
$$586$$ 14581.3 1.02790
$$587$$ 26468.0 1.86107 0.930537 0.366199i $$-0.119341\pi$$
0.930537 + 0.366199i $$0.119341\pi$$
$$588$$ 21695.1 1.52158
$$589$$ 28052.3 1.96243
$$590$$ −2612.08 −0.182267
$$591$$ −8031.41 −0.558998
$$592$$ 39077.6 2.71297
$$593$$ −1059.52 −0.0733716 −0.0366858 0.999327i $$-0.511680\pi$$
−0.0366858 + 0.999327i $$0.511680\pi$$
$$594$$ 0 0
$$595$$ 5648.20 0.389166
$$596$$ 13229.8 0.909253
$$597$$ 8339.70 0.571728
$$598$$ −20249.6 −1.38473
$$599$$ −17858.7 −1.21817 −0.609086 0.793104i $$-0.708464\pi$$
−0.609086 + 0.793104i $$0.708464\pi$$
$$600$$ 3666.65 0.249484
$$601$$ 9650.91 0.655023 0.327511 0.944847i $$-0.393790\pi$$
0.327511 + 0.944847i $$0.393790\pi$$
$$602$$ 4208.95 0.284957
$$603$$ −6344.32 −0.428459
$$604$$ 46703.7 3.14627
$$605$$ 0 0
$$606$$ 118.794 0.00796318
$$607$$ −22754.9 −1.52157 −0.760786 0.649002i $$-0.775187\pi$$
−0.760786 + 0.649002i $$0.775187\pi$$
$$608$$ 20999.6 1.40074
$$609$$ 6271.96 0.417328
$$610$$ 15336.2 1.01794
$$611$$ −3118.04 −0.206453
$$612$$ 6541.67 0.432077
$$613$$ −13074.5 −0.861459 −0.430729 0.902481i $$-0.641744\pi$$
−0.430729 + 0.902481i $$0.641744\pi$$
$$614$$ 41300.1 2.71455
$$615$$ −3574.19 −0.234350
$$616$$ 0 0
$$617$$ −13393.0 −0.873878 −0.436939 0.899491i $$-0.643937\pi$$
−0.436939 + 0.899491i $$0.643937\pi$$
$$618$$ 189.582 0.0123400
$$619$$ −15965.3 −1.03667 −0.518336 0.855177i $$-0.673448\pi$$
−0.518336 + 0.855177i $$0.673448\pi$$
$$620$$ −17420.7 −1.12844
$$621$$ 4756.49 0.307361
$$622$$ 19338.8 1.24665
$$623$$ −29039.5 −1.86749
$$624$$ −7243.69 −0.464711
$$625$$ 625.000 0.0400000
$$626$$ 12800.0 0.817237
$$627$$ 0 0
$$628$$ −61355.8 −3.89867
$$629$$ 15123.6 0.958693
$$630$$ 6252.87 0.395429
$$631$$ 17698.3 1.11657 0.558287 0.829648i $$-0.311459\pi$$
0.558287 + 0.829648i $$0.311459\pi$$
$$632$$ 26051.5 1.63967
$$633$$ −9168.45 −0.575693
$$634$$ −56142.4 −3.51688
$$635$$ −272.655 −0.0170393
$$636$$ 33822.2 2.10871
$$637$$ −9297.37 −0.578297
$$638$$ 0 0
$$639$$ −7039.46 −0.435801
$$640$$ 8513.47 0.525820
$$641$$ 18264.8 1.12546 0.562728 0.826642i $$-0.309752\pi$$
0.562728 + 0.826642i $$0.309752\pi$$
$$642$$ 20919.9 1.28605
$$643$$ −15730.5 −0.964778 −0.482389 0.875957i $$-0.660231\pi$$
−0.482389 + 0.875957i $$0.660231\pi$$
$$644$$ −85316.4 −5.22040
$$645$$ 454.358 0.0277369
$$646$$ 29640.4 1.80524
$$647$$ 21176.6 1.28676 0.643382 0.765545i $$-0.277531\pi$$
0.643382 + 0.765545i $$0.277531\pi$$
$$648$$ 3959.98 0.240066
$$649$$ 0 0
$$650$$ −2873.65 −0.173406
$$651$$ −16244.7 −0.978001
$$652$$ 62117.4 3.73114
$$653$$ −28293.6 −1.69558 −0.847791 0.530331i $$-0.822068\pi$$
−0.847791 + 0.530331i $$0.822068\pi$$
$$654$$ −9271.52 −0.554350
$$655$$ 11889.2 0.709234
$$656$$ 25351.1 1.50883
$$657$$ 2188.18 0.129938
$$658$$ −19090.7 −1.13105
$$659$$ 3894.19 0.230191 0.115096 0.993354i $$-0.463283\pi$$
0.115096 + 0.993354i $$0.463283\pi$$
$$660$$ 0 0
$$661$$ −6063.63 −0.356805 −0.178402 0.983958i $$-0.557093\pi$$
−0.178402 + 0.983958i $$0.557093\pi$$
$$662$$ −47699.1 −2.80042
$$663$$ −2803.42 −0.164217
$$664$$ −58906.4 −3.44279
$$665$$ 19496.3 1.13689
$$666$$ 16742.7 0.974123
$$667$$ −13424.7 −0.779317
$$668$$ −4418.51 −0.255924
$$669$$ 1714.13 0.0990612
$$670$$ 17851.6 1.02936
$$671$$ 0 0
$$672$$ −12160.6 −0.698072
$$673$$ −17297.1 −0.990719 −0.495360 0.868688i $$-0.664964\pi$$
−0.495360 + 0.868688i $$0.664964\pi$$
$$674$$ 57046.3 3.26015
$$675$$ 675.000 0.0384900
$$676$$ −29690.6 −1.68927
$$677$$ 4640.36 0.263432 0.131716 0.991287i $$-0.457951\pi$$
0.131716 + 0.991287i $$0.457951\pi$$
$$678$$ −558.744 −0.0316496
$$679$$ 2337.35 0.132105
$$680$$ −10065.1 −0.567614
$$681$$ 3283.19 0.184746
$$682$$ 0 0
$$683$$ −14694.9 −0.823256 −0.411628 0.911352i $$-0.635040\pi$$
−0.411628 + 0.911352i $$0.635040\pi$$
$$684$$ 22580.3 1.26225
$$685$$ 15392.1 0.858541
$$686$$ −9263.73 −0.515584
$$687$$ −1935.72 −0.107500
$$688$$ −3222.68 −0.178581
$$689$$ −14494.4 −0.801442
$$690$$ −13383.8 −0.738424
$$691$$ −9905.09 −0.545307 −0.272654 0.962112i $$-0.587901\pi$$
−0.272654 + 0.962112i $$0.587901\pi$$
$$692$$ −16628.4 −0.913466
$$693$$ 0 0
$$694$$ 21339.4 1.16720
$$695$$ −5985.91 −0.326703
$$696$$ −11176.6 −0.608689
$$697$$ 9811.25 0.533182
$$698$$ −25258.7 −1.36971
$$699$$ 7012.18 0.379435
$$700$$ −12107.4 −0.653736
$$701$$ 947.946 0.0510748 0.0255374 0.999674i $$-0.491870\pi$$
0.0255374 + 0.999674i $$0.491870\pi$$
$$702$$ −3103.54 −0.166860
$$703$$ 52203.2 2.80069
$$704$$ 0 0
$$705$$ −2060.85 −0.110094
$$706$$ −38662.3 −2.06101
$$707$$ −214.492 −0.0114099
$$708$$ 5462.36 0.289955
$$709$$ −3310.76 −0.175371 −0.0876855 0.996148i $$-0.527947\pi$$
−0.0876855 + 0.996148i $$0.527947\pi$$
$$710$$ 19807.6 1.04699
$$711$$ 4795.87 0.252966
$$712$$ 51748.3 2.72380
$$713$$ 34770.5 1.82632
$$714$$ −17164.3 −0.899662
$$715$$ 0 0
$$716$$ 5931.92 0.309618
$$717$$ 10970.7 0.571423
$$718$$ 15775.8 0.819981
$$719$$ 3061.15 0.158778 0.0793892 0.996844i $$-0.474703\pi$$
0.0793892 + 0.996844i $$0.474703\pi$$
$$720$$ −4787.66 −0.247813
$$721$$ −342.304 −0.0176811
$$722$$ 67572.1 3.48307
$$723$$ 1167.10 0.0600345
$$724$$ 19549.3 1.00351
$$725$$ −1905.11 −0.0975918
$$726$$ 0 0
$$727$$ −7405.09 −0.377771 −0.188886 0.981999i $$-0.560488\pi$$
−0.188886 + 0.981999i $$0.560488\pi$$
$$728$$ 30439.5 1.54967
$$729$$ 729.000 0.0370370
$$730$$ −6157.11 −0.312171
$$731$$ −1247.23 −0.0631057
$$732$$ −32071.0 −1.61937
$$733$$ −29791.4 −1.50119 −0.750593 0.660765i $$-0.770232\pi$$
−0.750593 + 0.660765i $$0.770232\pi$$
$$734$$ 30042.0 1.51072
$$735$$ −6145.02 −0.308384
$$736$$ 26028.8 1.30358
$$737$$ 0 0
$$738$$ 10861.6 0.541763
$$739$$ 7150.93 0.355955 0.177978 0.984035i $$-0.443045\pi$$
0.177978 + 0.984035i $$0.443045\pi$$
$$740$$ −32418.6 −1.61045
$$741$$ −9676.75 −0.479736
$$742$$ −88744.3 −4.39071
$$743$$ 1940.69 0.0958239 0.0479120 0.998852i $$-0.484743\pi$$
0.0479120 + 0.998852i $$0.484743\pi$$
$$744$$ 28947.9 1.42645
$$745$$ −3747.28 −0.184282
$$746$$ −70496.7 −3.45988
$$747$$ −10844.2 −0.531148
$$748$$ 0 0
$$749$$ −37772.4 −1.84269
$$750$$ −1899.31 −0.0924708
$$751$$ −29491.2 −1.43295 −0.716476 0.697611i $$-0.754246\pi$$
−0.716476 + 0.697611i $$0.754246\pi$$
$$752$$ 14617.2 0.708824
$$753$$ 6898.64 0.333865
$$754$$ 8759.38 0.423074
$$755$$ −13228.6 −0.637665
$$756$$ −13076.0 −0.629058
$$757$$ 3542.54 0.170087 0.0850436 0.996377i $$-0.472897\pi$$
0.0850436 + 0.996377i $$0.472897\pi$$
$$758$$ −62131.7 −2.97721
$$759$$ 0 0
$$760$$ −34742.3 −1.65820
$$761$$ −8552.86 −0.407412 −0.203706 0.979032i $$-0.565299\pi$$
−0.203706 + 0.979032i $$0.565299\pi$$
$$762$$ 828.572 0.0393911
$$763$$ 16740.4 0.794290
$$764$$ −74900.0 −3.54684
$$765$$ −1852.90 −0.0875707
$$766$$ −35124.9 −1.65680
$$767$$ −2340.88 −0.110201
$$768$$ −23404.3 −1.09965
$$769$$ 3128.73 0.146716 0.0733582 0.997306i $$-0.476628\pi$$
0.0733582 + 0.997306i $$0.476628\pi$$
$$770$$ 0 0
$$771$$ 14764.8 0.689679
$$772$$ 58681.2 2.73573
$$773$$ −5364.51 −0.249609 −0.124805 0.992181i $$-0.539830\pi$$
−0.124805 + 0.992181i $$0.539830\pi$$
$$774$$ −1380.75 −0.0641214
$$775$$ 4934.32 0.228705
$$776$$ −4165.14 −0.192680
$$777$$ −30230.1 −1.39575
$$778$$ 14055.8 0.647719
$$779$$ 33866.2 1.55762
$$780$$ 6009.34 0.275858
$$781$$ 0 0
$$782$$ 36738.9 1.68003
$$783$$ −2057.52 −0.0939077
$$784$$ 43585.6 1.98550
$$785$$ 17378.7 0.790157
$$786$$ −36130.0 −1.63959
$$787$$ −12160.9 −0.550813 −0.275406 0.961328i $$-0.588812\pi$$
−0.275406 + 0.961328i $$0.588812\pi$$
$$788$$ −47258.3 −2.13643
$$789$$ 7726.84 0.348648
$$790$$ −13494.6 −0.607742
$$791$$ 1008.85 0.0453485
$$792$$ 0 0
$$793$$ 13743.9 0.615462
$$794$$ 53380.7 2.38591
$$795$$ −9579.98 −0.427380
$$796$$ 49072.4 2.18508
$$797$$ −581.179 −0.0258299 −0.0129149 0.999917i $$-0.504111\pi$$
−0.0129149 + 0.999917i $$0.504111\pi$$
$$798$$ −59247.3 −2.62824
$$799$$ 5657.09 0.250480
$$800$$ 3693.78 0.163244
$$801$$ 9526.43 0.420225
$$802$$ −62273.8 −2.74185
$$803$$ 0 0
$$804$$ −37331.2 −1.63752
$$805$$ 24165.4 1.05804
$$806$$ −22687.2 −0.991467
$$807$$ −14384.9 −0.627475
$$808$$ 382.223 0.0166418
$$809$$ −4763.37 −0.207010 −0.103505 0.994629i $$-0.533006\pi$$
−0.103505 + 0.994629i $$0.533006\pi$$
$$810$$ −2051.26 −0.0889801
$$811$$ 18055.4 0.781762 0.390881 0.920441i $$-0.372170\pi$$
0.390881 + 0.920441i $$0.372170\pi$$
$$812$$ 36905.4 1.59498
$$813$$ −8188.40 −0.353235
$$814$$ 0 0
$$815$$ −17594.4 −0.756205
$$816$$ 13142.3 0.563813
$$817$$ −4305.14 −0.184354
$$818$$ 58476.9 2.49951
$$819$$ 5603.66 0.239082
$$820$$ −21031.2 −0.895660
$$821$$ −1128.04 −0.0479522 −0.0239761 0.999713i $$-0.507633\pi$$
−0.0239761 + 0.999713i $$0.507633\pi$$
$$822$$ −46775.0 −1.98475
$$823$$ −32124.2 −1.36061 −0.680304 0.732930i $$-0.738152\pi$$
−0.680304 + 0.732930i $$0.738152\pi$$
$$824$$ 609.984 0.0257886
$$825$$ 0 0
$$826$$ −14332.4 −0.603738
$$827$$ −11914.2 −0.500964 −0.250482 0.968121i $$-0.580589\pi$$
−0.250482 + 0.968121i $$0.580589\pi$$
$$828$$ 27988.1 1.17470
$$829$$ −37721.6 −1.58037 −0.790185 0.612868i $$-0.790016\pi$$
−0.790185 + 0.612868i $$0.790016\pi$$
$$830$$ 30513.3 1.27606
$$831$$ 11284.4 0.471059
$$832$$ 2333.12 0.0972192
$$833$$ 16868.3 0.701622
$$834$$ 18190.6 0.755262
$$835$$ 1251.52 0.0518690
$$836$$ 0 0
$$837$$ 5329.07 0.220071
$$838$$ 9377.57 0.386567
$$839$$ −16550.5 −0.681034 −0.340517 0.940238i $$-0.610602\pi$$
−0.340517 + 0.940238i $$0.610602\pi$$
$$840$$ 20118.8 0.826384
$$841$$ −18581.9 −0.761896
$$842$$ −6602.53 −0.270236
$$843$$ 19304.6 0.788714
$$844$$ −53948.9 −2.20023
$$845$$ 8409.71 0.342370
$$846$$ 6262.71 0.254511
$$847$$ 0 0
$$848$$ 67949.2 2.75163
$$849$$ −10006.9 −0.404520
$$850$$ 5213.67 0.210385
$$851$$ 64705.3 2.60643
$$852$$ −41421.5 −1.66558
$$853$$ −1045.52 −0.0419672 −0.0209836 0.999780i $$-0.506680\pi$$
−0.0209836 + 0.999780i $$0.506680\pi$$
$$854$$ 84149.3 3.37181
$$855$$ −6395.77 −0.255825
$$856$$ 67310.2 2.68764
$$857$$ 14016.5 0.558688 0.279344 0.960191i $$-0.409883\pi$$
0.279344 + 0.960191i $$0.409883\pi$$
$$858$$ 0 0
$$859$$ −20476.3 −0.813319 −0.406660 0.913580i $$-0.633306\pi$$
−0.406660 + 0.913580i $$0.633306\pi$$
$$860$$ 2673.53 0.106008
$$861$$ −19611.4 −0.776255
$$862$$ −41677.8 −1.64681
$$863$$ 24083.0 0.949936 0.474968 0.880003i $$-0.342460\pi$$
0.474968 + 0.880003i $$0.342460\pi$$
$$864$$ 3989.29 0.157081
$$865$$ 4709.92 0.185135
$$866$$ 29529.1 1.15871
$$867$$ −9652.75 −0.378114
$$868$$ −95586.6 −3.73781
$$869$$ 0 0
$$870$$ 5789.44 0.225610
$$871$$ 15998.2 0.622362
$$872$$ −29831.3 −1.15850
$$873$$ −766.768 −0.0297264
$$874$$ 126814. 4.90796
$$875$$ 3429.35 0.132495
$$876$$ 12875.7 0.496608
$$877$$ −30432.5 −1.17176 −0.585879 0.810399i $$-0.699250\pi$$
−0.585879 + 0.810399i $$0.699250\pi$$
$$878$$ −72229.7 −2.77635
$$879$$ 8636.79 0.331413
$$880$$ 0 0
$$881$$ 24559.2 0.939183 0.469592 0.882884i $$-0.344401\pi$$
0.469592 + 0.882884i $$0.344401\pi$$
$$882$$ 18674.1 0.712914
$$883$$ 6013.88 0.229200 0.114600 0.993412i $$-0.463441\pi$$
0.114600 + 0.993412i $$0.463441\pi$$
$$884$$ −16495.8 −0.627618
$$885$$ −1547.19 −0.0587662
$$886$$ 60908.3 2.30954
$$887$$ −13395.5 −0.507075 −0.253538 0.967325i $$-0.581594\pi$$
−0.253538 + 0.967325i $$0.581594\pi$$
$$888$$ 53869.9 2.03576
$$889$$ −1496.05 −0.0564407
$$890$$ −26805.4 −1.00957
$$891$$ 0 0
$$892$$ 10086.2 0.378601
$$893$$ 19527.0 0.731741
$$894$$ 11387.6 0.426017
$$895$$ −1680.19 −0.0627514
$$896$$ 46713.1 1.74171
$$897$$ −11994.2 −0.446461
$$898$$ −35825.0 −1.33129
$$899$$ −15040.7 −0.557992
$$900$$ 3971.83 0.147105
$$901$$ 26297.3 0.972354
$$902$$ 0 0
$$903$$ 2493.04 0.0918751
$$904$$ −1797.77 −0.0661426
$$905$$ −5537.24 −0.203386
$$906$$ 40200.3 1.47414
$$907$$ −32078.9 −1.17438 −0.587189 0.809450i $$-0.699766\pi$$
−0.587189 + 0.809450i $$0.699766\pi$$
$$908$$ 19318.9 0.706079
$$909$$ 70.3641 0.00256747
$$910$$ −15767.6 −0.574385
$$911$$ −15992.0 −0.581600 −0.290800 0.956784i $$-0.593921\pi$$
−0.290800 + 0.956784i $$0.593921\pi$$
$$912$$ 45364.1 1.64710
$$913$$ 0 0
$$914$$ −13838.7 −0.500814
$$915$$ 9083.94 0.328203
$$916$$ −11390.1 −0.410852
$$917$$ 65235.4 2.34925
$$918$$ 5630.77 0.202443
$$919$$ −33876.1 −1.21596 −0.607980 0.793952i $$-0.708020\pi$$
−0.607980 + 0.793952i $$0.708020\pi$$
$$920$$ −43062.7 −1.54319
$$921$$ 24462.8 0.875220
$$922$$ 1685.11 0.0601912
$$923$$ 17751.1 0.633026
$$924$$ 0 0
$$925$$ 9182.42 0.326396
$$926$$ 41776.7 1.48258
$$927$$ 112.293 0.00397863
$$928$$ −11259.3 −0.398281
$$929$$ 21163.4 0.747416 0.373708 0.927546i $$-0.378086\pi$$
0.373708 + 0.927546i $$0.378086\pi$$
$$930$$ −14994.9 −0.528713
$$931$$ 58225.4 2.04969
$$932$$ 41261.0 1.45016
$$933$$ 11454.7 0.401941
$$934$$ 37273.9 1.30582
$$935$$ 0 0
$$936$$ −9985.70 −0.348710
$$937$$ 49771.9 1.73530 0.867650 0.497175i $$-0.165629\pi$$
0.867650 + 0.497175i $$0.165629\pi$$
$$938$$ 97951.2 3.40962
$$939$$ 7581.68 0.263492
$$940$$ −12126.4 −0.420766
$$941$$ −32194.6 −1.11532 −0.557659 0.830070i $$-0.688300\pi$$
−0.557659 + 0.830070i $$0.688300\pi$$
$$942$$ −52812.2 −1.82666
$$943$$ 41976.8 1.44958
$$944$$ 10973.9 0.378359
$$945$$ 3703.70 0.127493
$$946$$ 0 0
$$947$$ −30091.9 −1.03258 −0.516291 0.856413i $$-0.672688\pi$$
−0.516291 + 0.856413i $$0.672688\pi$$
$$948$$ 28219.8 0.966810
$$949$$ −5517.84 −0.188742
$$950$$ 17996.4 0.614611
$$951$$ −33254.2 −1.13390
$$952$$ −55226.6 −1.88015
$$953$$ −5710.17 −0.194093 −0.0970465 0.995280i $$-0.530940\pi$$
−0.0970465 + 0.995280i $$0.530940\pi$$
$$954$$ 29112.6 0.988004
$$955$$ 21215.0 0.718851
$$956$$ 64553.9 2.18392
$$957$$ 0 0
$$958$$ 52307.8 1.76408
$$959$$ 84455.7 2.84381
$$960$$ 1542.06 0.0518434
$$961$$ 9165.07 0.307646
$$962$$ −42219.2 −1.41497
$$963$$ 12391.3 0.414645
$$964$$ 6867.44 0.229445
$$965$$ −16621.2 −0.554460
$$966$$ −73436.4 −2.44594
$$967$$ −29638.4 −0.985634 −0.492817 0.870133i $$-0.664033\pi$$
−0.492817 + 0.870133i $$0.664033\pi$$
$$968$$ 0 0
$$969$$ 17556.6 0.582042
$$970$$ 2157.53 0.0714166
$$971$$ 21416.3 0.707808 0.353904 0.935282i $$-0.384854\pi$$
0.353904 + 0.935282i $$0.384854\pi$$
$$972$$ 4289.57 0.141552
$$973$$ −32844.5 −1.08216
$$974$$ 48736.7 1.60331
$$975$$ −1702.12 −0.0559090
$$976$$ −64430.9 −2.11310
$$977$$ 1038.84 0.0340177 0.0170088 0.999855i $$-0.494586\pi$$
0.0170088 + 0.999855i $$0.494586\pi$$
$$978$$ 53467.8 1.74817
$$979$$ 0 0
$$980$$ −36158.5 −1.17861
$$981$$ −5491.70 −0.178732
$$982$$ 45551.1 1.48024
$$983$$ −44173.6 −1.43329 −0.716643 0.697440i $$-0.754322\pi$$
−0.716643 + 0.697440i $$0.754322\pi$$
$$984$$ 34947.4 1.13220
$$985$$ 13385.7 0.432998
$$986$$ −15892.2 −0.513297
$$987$$ −11307.8 −0.364671
$$988$$ −56939.8 −1.83350
$$989$$ −5336.17 −0.171567
$$990$$ 0 0
$$991$$ −23940.9 −0.767414 −0.383707 0.923455i $$-0.625353\pi$$
−0.383707 + 0.923455i $$0.625353\pi$$
$$992$$ 29162.1 0.933365
$$993$$ −28253.1 −0.902905
$$994$$ 108684. 3.46804
$$995$$ −13899.5 −0.442858
$$996$$ −63809.2 −2.02999
$$997$$ −13557.9 −0.430674 −0.215337 0.976540i $$-0.569085\pi$$
−0.215337 + 0.976540i $$0.569085\pi$$
$$998$$ 13288.6 0.421487
$$999$$ 9917.01 0.314074
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 1815.4.a.r.1.3 3
11.10 odd 2 165.4.a.e.1.1 3
33.32 even 2 495.4.a.k.1.3 3
55.32 even 4 825.4.c.k.199.1 6
55.43 even 4 825.4.c.k.199.6 6
55.54 odd 2 825.4.a.r.1.3 3
165.164 even 2 2475.4.a.t.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.e.1.1 3 11.10 odd 2
495.4.a.k.1.3 3 33.32 even 2
825.4.a.r.1.3 3 55.54 odd 2
825.4.c.k.199.1 6 55.32 even 4
825.4.c.k.199.6 6 55.43 even 4
1815.4.a.r.1.3 3 1.1 even 1 trivial
2475.4.a.t.1.1 3 165.164 even 2