# Properties

 Label 735.4.a.o.1.1 Level $735$ Weight $4$ Character 735.1 Self dual yes Analytic conductor $43.366$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.3664038542$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 735.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} -5.00000 q^{5} -11.4853 q^{6} +5.14214 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} -5.00000 q^{5} -11.4853 q^{6} +5.14214 q^{8} +9.00000 q^{9} +19.1421 q^{10} +48.5685 q^{11} +19.9706 q^{12} +43.6569 q^{13} -15.0000 q^{15} -72.9411 q^{16} +67.6569 q^{17} -34.4558 q^{18} +93.2548 q^{19} -33.2843 q^{20} -185.941 q^{22} -104.167 q^{23} +15.4264 q^{24} +25.0000 q^{25} -167.137 q^{26} +27.0000 q^{27} -58.7351 q^{29} +57.4264 q^{30} +9.08831 q^{31} +238.113 q^{32} +145.706 q^{33} -259.019 q^{34} +59.9117 q^{36} -252.676 q^{37} -357.019 q^{38} +130.971 q^{39} -25.7107 q^{40} -276.274 q^{41} -92.6375 q^{43} +323.314 q^{44} -45.0000 q^{45} +398.794 q^{46} +582.794 q^{47} -218.823 q^{48} -95.7107 q^{50} +202.971 q^{51} +290.617 q^{52} +623.019 q^{53} -103.368 q^{54} -242.843 q^{55} +279.765 q^{57} +224.863 q^{58} +524.999 q^{59} -99.8528 q^{60} +352.794 q^{61} -34.7939 q^{62} -328.068 q^{64} -218.284 q^{65} -557.823 q^{66} -736.520 q^{67} +450.382 q^{68} -312.500 q^{69} -492.264 q^{71} +46.2792 q^{72} -1164.75 q^{73} +967.352 q^{74} +75.0000 q^{75} +620.784 q^{76} -501.411 q^{78} -872.195 q^{79} +364.706 q^{80} +81.0000 q^{81} +1057.70 q^{82} +529.588 q^{83} -338.284 q^{85} +354.656 q^{86} -176.205 q^{87} +249.746 q^{88} +385.216 q^{89} +172.279 q^{90} -693.421 q^{92} +27.2649 q^{93} -2231.18 q^{94} -466.274 q^{95} +714.338 q^{96} +463.892 q^{97} +437.117 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 10q^{10} - 16q^{11} + 6q^{12} + 76q^{13} - 30q^{15} - 78q^{16} + 124q^{17} - 18q^{18} + 96q^{19} - 10q^{20} - 304q^{22} - 16q^{23} - 54q^{24} + 50q^{25} - 108q^{26} + 54q^{27} + 188q^{29} + 30q^{30} + 120q^{31} + 414q^{32} - 48q^{33} - 156q^{34} + 18q^{36} - 132q^{37} - 352q^{38} + 228q^{39} + 90q^{40} - 100q^{41} - 536q^{43} + 624q^{44} - 90q^{45} + 560q^{46} + 928q^{47} - 234q^{48} - 50q^{50} + 372q^{51} + 140q^{52} + 884q^{53} - 54q^{54} + 80q^{55} + 288q^{57} + 676q^{58} - 104q^{59} - 30q^{60} + 468q^{61} + 168q^{62} + 34q^{64} - 380q^{65} - 912q^{66} - 1688q^{67} + 188q^{68} - 48q^{69} - 136q^{71} - 162q^{72} - 508q^{73} + 1188q^{74} + 150q^{75} + 608q^{76} - 324q^{78} - 432q^{79} + 390q^{80} + 162q^{81} + 1380q^{82} + 584q^{83} - 620q^{85} - 456q^{86} + 564q^{87} + 1744q^{88} + 1404q^{89} + 90q^{90} - 1104q^{92} + 360q^{93} - 1600q^{94} - 480q^{95} + 1242q^{96} + 1188q^{97} - 144q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.82843 −1.35355 −0.676777 0.736188i $$-0.736624\pi$$
−0.676777 + 0.736188i $$0.736624\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 6.65685 0.832107
$$5$$ −5.00000 −0.447214
$$6$$ −11.4853 −0.781474
$$7$$ 0 0
$$8$$ 5.14214 0.227252
$$9$$ 9.00000 0.333333
$$10$$ 19.1421 0.605327
$$11$$ 48.5685 1.33127 0.665635 0.746278i $$-0.268161\pi$$
0.665635 + 0.746278i $$0.268161\pi$$
$$12$$ 19.9706 0.480417
$$13$$ 43.6569 0.931403 0.465701 0.884942i $$-0.345802\pi$$
0.465701 + 0.884942i $$0.345802\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ −72.9411 −1.13971
$$17$$ 67.6569 0.965247 0.482623 0.875828i $$-0.339684\pi$$
0.482623 + 0.875828i $$0.339684\pi$$
$$18$$ −34.4558 −0.451184
$$19$$ 93.2548 1.12601 0.563003 0.826455i $$-0.309646\pi$$
0.563003 + 0.826455i $$0.309646\pi$$
$$20$$ −33.2843 −0.372129
$$21$$ 0 0
$$22$$ −185.941 −1.80194
$$23$$ −104.167 −0.944357 −0.472179 0.881503i $$-0.656532\pi$$
−0.472179 + 0.881503i $$0.656532\pi$$
$$24$$ 15.4264 0.131204
$$25$$ 25.0000 0.200000
$$26$$ −167.137 −1.26070
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −58.7351 −0.376098 −0.188049 0.982160i $$-0.560216\pi$$
−0.188049 + 0.982160i $$0.560216\pi$$
$$30$$ 57.4264 0.349486
$$31$$ 9.08831 0.0526551 0.0263276 0.999653i $$-0.491619\pi$$
0.0263276 + 0.999653i $$0.491619\pi$$
$$32$$ 238.113 1.31540
$$33$$ 145.706 0.768609
$$34$$ −259.019 −1.30651
$$35$$ 0 0
$$36$$ 59.9117 0.277369
$$37$$ −252.676 −1.12269 −0.561347 0.827580i $$-0.689717\pi$$
−0.561347 + 0.827580i $$0.689717\pi$$
$$38$$ −357.019 −1.52411
$$39$$ 130.971 0.537745
$$40$$ −25.7107 −0.101630
$$41$$ −276.274 −1.05236 −0.526180 0.850373i $$-0.676376\pi$$
−0.526180 + 0.850373i $$0.676376\pi$$
$$42$$ 0 0
$$43$$ −92.6375 −0.328537 −0.164268 0.986416i $$-0.552526\pi$$
−0.164268 + 0.986416i $$0.552526\pi$$
$$44$$ 323.314 1.10776
$$45$$ −45.0000 −0.149071
$$46$$ 398.794 1.27824
$$47$$ 582.794 1.80871 0.904354 0.426784i $$-0.140354\pi$$
0.904354 + 0.426784i $$0.140354\pi$$
$$48$$ −218.823 −0.658009
$$49$$ 0 0
$$50$$ −95.7107 −0.270711
$$51$$ 202.971 0.557286
$$52$$ 290.617 0.775026
$$53$$ 623.019 1.61468 0.807342 0.590083i $$-0.200905\pi$$
0.807342 + 0.590083i $$0.200905\pi$$
$$54$$ −103.368 −0.260491
$$55$$ −242.843 −0.595362
$$56$$ 0 0
$$57$$ 279.765 0.650100
$$58$$ 224.863 0.509068
$$59$$ 524.999 1.15846 0.579229 0.815165i $$-0.303354\pi$$
0.579229 + 0.815165i $$0.303354\pi$$
$$60$$ −99.8528 −0.214849
$$61$$ 352.794 0.740502 0.370251 0.928932i $$-0.379272\pi$$
0.370251 + 0.928932i $$0.379272\pi$$
$$62$$ −34.7939 −0.0712715
$$63$$ 0 0
$$64$$ −328.068 −0.640758
$$65$$ −218.284 −0.416536
$$66$$ −557.823 −1.04035
$$67$$ −736.520 −1.34299 −0.671494 0.741010i $$-0.734347\pi$$
−0.671494 + 0.741010i $$0.734347\pi$$
$$68$$ 450.382 0.803188
$$69$$ −312.500 −0.545225
$$70$$ 0 0
$$71$$ −492.264 −0.822831 −0.411415 0.911448i $$-0.634965\pi$$
−0.411415 + 0.911448i $$0.634965\pi$$
$$72$$ 46.2792 0.0757508
$$73$$ −1164.75 −1.86745 −0.933727 0.357987i $$-0.883463\pi$$
−0.933727 + 0.357987i $$0.883463\pi$$
$$74$$ 967.352 1.51963
$$75$$ 75.0000 0.115470
$$76$$ 620.784 0.936958
$$77$$ 0 0
$$78$$ −501.411 −0.727867
$$79$$ −872.195 −1.24215 −0.621074 0.783752i $$-0.713303\pi$$
−0.621074 + 0.783752i $$0.713303\pi$$
$$80$$ 364.706 0.509692
$$81$$ 81.0000 0.111111
$$82$$ 1057.70 1.42443
$$83$$ 529.588 0.700359 0.350180 0.936683i $$-0.386121\pi$$
0.350180 + 0.936683i $$0.386121\pi$$
$$84$$ 0 0
$$85$$ −338.284 −0.431672
$$86$$ 354.656 0.444692
$$87$$ −176.205 −0.217140
$$88$$ 249.746 0.302534
$$89$$ 385.216 0.458796 0.229398 0.973333i $$-0.426324\pi$$
0.229398 + 0.973333i $$0.426324\pi$$
$$90$$ 172.279 0.201776
$$91$$ 0 0
$$92$$ −693.421 −0.785806
$$93$$ 27.2649 0.0304005
$$94$$ −2231.18 −2.44818
$$95$$ −466.274 −0.503565
$$96$$ 714.338 0.759446
$$97$$ 463.892 0.485579 0.242789 0.970079i $$-0.421938\pi$$
0.242789 + 0.970079i $$0.421938\pi$$
$$98$$ 0 0
$$99$$ 437.117 0.443757
$$100$$ 166.421 0.166421
$$101$$ 432.725 0.426314 0.213157 0.977018i $$-0.431625\pi$$
0.213157 + 0.977018i $$0.431625\pi$$
$$102$$ −777.058 −0.754316
$$103$$ −512.626 −0.490393 −0.245197 0.969473i $$-0.578853\pi$$
−0.245197 + 0.969473i $$0.578853\pi$$
$$104$$ 224.489 0.211663
$$105$$ 0 0
$$106$$ −2385.18 −2.18556
$$107$$ 1963.09 1.77363 0.886817 0.462122i $$-0.152912\pi$$
0.886817 + 0.462122i $$0.152912\pi$$
$$108$$ 179.735 0.160139
$$109$$ 545.176 0.479068 0.239534 0.970888i $$-0.423005\pi$$
0.239534 + 0.970888i $$0.423005\pi$$
$$110$$ 929.706 0.805854
$$111$$ −758.029 −0.648188
$$112$$ 0 0
$$113$$ −231.823 −0.192992 −0.0964961 0.995333i $$-0.530764\pi$$
−0.0964961 + 0.995333i $$0.530764\pi$$
$$114$$ −1071.06 −0.879945
$$115$$ 520.833 0.422329
$$116$$ −390.991 −0.312953
$$117$$ 392.912 0.310468
$$118$$ −2009.92 −1.56804
$$119$$ 0 0
$$120$$ −77.1320 −0.0586763
$$121$$ 1027.90 0.772279
$$122$$ −1350.65 −1.00231
$$123$$ −828.823 −0.607581
$$124$$ 60.4996 0.0438147
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 2372.90 1.65796 0.828979 0.559280i $$-0.188922\pi$$
0.828979 + 0.559280i $$0.188922\pi$$
$$128$$ −648.917 −0.448099
$$129$$ −277.913 −0.189681
$$130$$ 835.685 0.563804
$$131$$ −1200.04 −0.800364 −0.400182 0.916436i $$-0.631053\pi$$
−0.400182 + 0.916436i $$0.631053\pi$$
$$132$$ 969.941 0.639565
$$133$$ 0 0
$$134$$ 2819.71 1.81781
$$135$$ −135.000 −0.0860663
$$136$$ 347.901 0.219355
$$137$$ 2781.25 1.73444 0.867221 0.497924i $$-0.165904\pi$$
0.867221 + 0.497924i $$0.165904\pi$$
$$138$$ 1196.38 0.737991
$$139$$ 1245.60 0.760078 0.380039 0.924971i $$-0.375911\pi$$
0.380039 + 0.924971i $$0.375911\pi$$
$$140$$ 0 0
$$141$$ 1748.38 1.04426
$$142$$ 1884.60 1.11375
$$143$$ 2120.35 1.23995
$$144$$ −656.470 −0.379902
$$145$$ 293.675 0.168196
$$146$$ 4459.17 2.52770
$$147$$ 0 0
$$148$$ −1682.03 −0.934202
$$149$$ 19.4046 0.0106690 0.00533452 0.999986i $$-0.498302\pi$$
0.00533452 + 0.999986i $$0.498302\pi$$
$$150$$ −287.132 −0.156295
$$151$$ −2349.80 −1.26638 −0.633192 0.773995i $$-0.718256\pi$$
−0.633192 + 0.773995i $$0.718256\pi$$
$$152$$ 479.529 0.255888
$$153$$ 608.912 0.321749
$$154$$ 0 0
$$155$$ −45.4416 −0.0235481
$$156$$ 871.852 0.447462
$$157$$ 3898.46 1.98172 0.990862 0.134880i $$-0.0430650\pi$$
0.990862 + 0.134880i $$0.0430650\pi$$
$$158$$ 3339.14 1.68131
$$159$$ 1869.06 0.932239
$$160$$ −1190.56 −0.588264
$$161$$ 0 0
$$162$$ −310.103 −0.150395
$$163$$ 1527.54 0.734024 0.367012 0.930216i $$-0.380381\pi$$
0.367012 + 0.930216i $$0.380381\pi$$
$$164$$ −1839.12 −0.875676
$$165$$ −728.528 −0.343732
$$166$$ −2027.49 −0.947974
$$167$$ 998.518 0.462681 0.231340 0.972873i $$-0.425689\pi$$
0.231340 + 0.972873i $$0.425689\pi$$
$$168$$ 0 0
$$169$$ −291.079 −0.132489
$$170$$ 1295.10 0.584290
$$171$$ 839.294 0.375336
$$172$$ −616.674 −0.273378
$$173$$ −685.253 −0.301149 −0.150575 0.988599i $$-0.548112\pi$$
−0.150575 + 0.988599i $$0.548112\pi$$
$$174$$ 674.589 0.293911
$$175$$ 0 0
$$176$$ −3542.64 −1.51725
$$177$$ 1575.00 0.668836
$$178$$ −1474.77 −0.621005
$$179$$ 1025.58 0.428245 0.214122 0.976807i $$-0.431311\pi$$
0.214122 + 0.976807i $$0.431311\pi$$
$$180$$ −299.558 −0.124043
$$181$$ −2899.40 −1.19067 −0.595333 0.803479i $$-0.702980\pi$$
−0.595333 + 0.803479i $$0.702980\pi$$
$$182$$ 0 0
$$183$$ 1058.38 0.427529
$$184$$ −535.638 −0.214608
$$185$$ 1263.38 0.502084
$$186$$ −104.382 −0.0411486
$$187$$ 3285.99 1.28500
$$188$$ 3879.57 1.50504
$$189$$ 0 0
$$190$$ 1785.10 0.681603
$$191$$ 1074.18 0.406939 0.203469 0.979081i $$-0.434778\pi$$
0.203469 + 0.979081i $$0.434778\pi$$
$$192$$ −984.204 −0.369942
$$193$$ 898.999 0.335292 0.167646 0.985847i $$-0.446383\pi$$
0.167646 + 0.985847i $$0.446383\pi$$
$$194$$ −1775.98 −0.657257
$$195$$ −654.853 −0.240487
$$196$$ 0 0
$$197$$ 3063.63 1.10799 0.553996 0.832519i $$-0.313102\pi$$
0.553996 + 0.832519i $$0.313102\pi$$
$$198$$ −1673.47 −0.600648
$$199$$ 949.522 0.338240 0.169120 0.985595i $$-0.445907\pi$$
0.169120 + 0.985595i $$0.445907\pi$$
$$200$$ 128.553 0.0454505
$$201$$ −2209.56 −0.775375
$$202$$ −1656.66 −0.577039
$$203$$ 0 0
$$204$$ 1351.15 0.463721
$$205$$ 1381.37 0.470630
$$206$$ 1962.55 0.663774
$$207$$ −937.499 −0.314786
$$208$$ −3184.38 −1.06152
$$209$$ 4529.25 1.49902
$$210$$ 0 0
$$211$$ 2306.64 0.752587 0.376294 0.926500i $$-0.377198\pi$$
0.376294 + 0.926500i $$0.377198\pi$$
$$212$$ 4147.35 1.34359
$$213$$ −1476.79 −0.475062
$$214$$ −7515.53 −2.40071
$$215$$ 463.188 0.146926
$$216$$ 138.838 0.0437348
$$217$$ 0 0
$$218$$ −2087.17 −0.648444
$$219$$ −3494.26 −1.07817
$$220$$ −1616.57 −0.495405
$$221$$ 2953.69 0.899033
$$222$$ 2902.06 0.877357
$$223$$ 3227.61 0.969222 0.484611 0.874730i $$-0.338961\pi$$
0.484611 + 0.874730i $$0.338961\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 887.519 0.261225
$$227$$ 637.820 0.186492 0.0932458 0.995643i $$-0.470276\pi$$
0.0932458 + 0.995643i $$0.470276\pi$$
$$228$$ 1862.35 0.540953
$$229$$ −544.774 −0.157204 −0.0786019 0.996906i $$-0.525046\pi$$
−0.0786019 + 0.996906i $$0.525046\pi$$
$$230$$ −1993.97 −0.571646
$$231$$ 0 0
$$232$$ −302.024 −0.0854691
$$233$$ −5748.54 −1.61631 −0.808154 0.588972i $$-0.799533\pi$$
−0.808154 + 0.588972i $$0.799533\pi$$
$$234$$ −1504.23 −0.420234
$$235$$ −2913.97 −0.808878
$$236$$ 3494.84 0.963961
$$237$$ −2616.59 −0.717154
$$238$$ 0 0
$$239$$ −2678.10 −0.724820 −0.362410 0.932019i $$-0.618046\pi$$
−0.362410 + 0.932019i $$0.618046\pi$$
$$240$$ 1094.12 0.294271
$$241$$ 2202.16 0.588604 0.294302 0.955713i $$-0.404913\pi$$
0.294302 + 0.955713i $$0.404913\pi$$
$$242$$ −3935.25 −1.04532
$$243$$ 243.000 0.0641500
$$244$$ 2348.50 0.616177
$$245$$ 0 0
$$246$$ 3173.09 0.822393
$$247$$ 4071.21 1.04877
$$248$$ 46.7333 0.0119660
$$249$$ 1588.76 0.404353
$$250$$ 478.553 0.121065
$$251$$ 5716.90 1.43764 0.718820 0.695196i $$-0.244683\pi$$
0.718820 + 0.695196i $$0.244683\pi$$
$$252$$ 0 0
$$253$$ −5059.22 −1.25719
$$254$$ −9084.47 −2.24413
$$255$$ −1014.85 −0.249226
$$256$$ 5108.88 1.24728
$$257$$ −4724.29 −1.14666 −0.573332 0.819323i $$-0.694350\pi$$
−0.573332 + 0.819323i $$0.694350\pi$$
$$258$$ 1063.97 0.256743
$$259$$ 0 0
$$260$$ −1453.09 −0.346602
$$261$$ −528.616 −0.125366
$$262$$ 4594.25 1.08334
$$263$$ 5975.36 1.40097 0.700487 0.713665i $$-0.252966\pi$$
0.700487 + 0.713665i $$0.252966\pi$$
$$264$$ 749.238 0.174668
$$265$$ −3115.10 −0.722109
$$266$$ 0 0
$$267$$ 1155.65 0.264886
$$268$$ −4902.90 −1.11751
$$269$$ −4486.11 −1.01681 −0.508407 0.861117i $$-0.669766\pi$$
−0.508407 + 0.861117i $$0.669766\pi$$
$$270$$ 516.838 0.116495
$$271$$ −3827.68 −0.857989 −0.428994 0.903307i $$-0.641132\pi$$
−0.428994 + 0.903307i $$0.641132\pi$$
$$272$$ −4934.97 −1.10010
$$273$$ 0 0
$$274$$ −10647.8 −2.34766
$$275$$ 1214.21 0.266254
$$276$$ −2080.26 −0.453685
$$277$$ −3420.54 −0.741950 −0.370975 0.928643i $$-0.620976\pi$$
−0.370975 + 0.928643i $$0.620976\pi$$
$$278$$ −4768.71 −1.02881
$$279$$ 81.7948 0.0175517
$$280$$ 0 0
$$281$$ 5235.92 1.11156 0.555781 0.831329i $$-0.312419\pi$$
0.555781 + 0.831329i $$0.312419\pi$$
$$282$$ −6693.55 −1.41346
$$283$$ 6985.88 1.46738 0.733688 0.679486i $$-0.237797\pi$$
0.733688 + 0.679486i $$0.237797\pi$$
$$284$$ −3276.93 −0.684683
$$285$$ −1398.82 −0.290734
$$286$$ −8117.60 −1.67834
$$287$$ 0 0
$$288$$ 2143.01 0.438466
$$289$$ −335.550 −0.0682984
$$290$$ −1124.31 −0.227662
$$291$$ 1391.68 0.280349
$$292$$ −7753.59 −1.55392
$$293$$ −7399.70 −1.47541 −0.737705 0.675123i $$-0.764090\pi$$
−0.737705 + 0.675123i $$0.764090\pi$$
$$294$$ 0 0
$$295$$ −2625.00 −0.518078
$$296$$ −1299.30 −0.255135
$$297$$ 1311.35 0.256203
$$298$$ −74.2892 −0.0144411
$$299$$ −4547.58 −0.879577
$$300$$ 499.264 0.0960834
$$301$$ 0 0
$$302$$ 8996.04 1.71412
$$303$$ 1298.17 0.246133
$$304$$ −6802.11 −1.28332
$$305$$ −1763.97 −0.331163
$$306$$ −2331.17 −0.435504
$$307$$ −2668.64 −0.496116 −0.248058 0.968745i $$-0.579792\pi$$
−0.248058 + 0.968745i $$0.579792\pi$$
$$308$$ 0 0
$$309$$ −1537.88 −0.283129
$$310$$ 173.970 0.0318736
$$311$$ −6189.25 −1.12849 −0.564244 0.825608i $$-0.690832\pi$$
−0.564244 + 0.825608i $$0.690832\pi$$
$$312$$ 673.468 0.122204
$$313$$ 2921.59 0.527598 0.263799 0.964578i $$-0.415024\pi$$
0.263799 + 0.964578i $$0.415024\pi$$
$$314$$ −14925.0 −2.68237
$$315$$ 0 0
$$316$$ −5806.08 −1.03360
$$317$$ 9825.56 1.74088 0.870439 0.492276i $$-0.163835\pi$$
0.870439 + 0.492276i $$0.163835\pi$$
$$318$$ −7155.55 −1.26183
$$319$$ −2852.68 −0.500687
$$320$$ 1640.34 0.286556
$$321$$ 5889.26 1.02401
$$322$$ 0 0
$$323$$ 6309.33 1.08687
$$324$$ 539.205 0.0924563
$$325$$ 1091.42 0.186281
$$326$$ −5848.07 −0.993541
$$327$$ 1635.53 0.276590
$$328$$ −1420.64 −0.239151
$$329$$ 0 0
$$330$$ 2789.12 0.465260
$$331$$ −9258.17 −1.53739 −0.768693 0.639618i $$-0.779093\pi$$
−0.768693 + 0.639618i $$0.779093\pi$$
$$332$$ 3525.39 0.582774
$$333$$ −2274.09 −0.374232
$$334$$ −3822.75 −0.626263
$$335$$ 3682.60 0.600603
$$336$$ 0 0
$$337$$ −3693.98 −0.597103 −0.298552 0.954394i $$-0.596503\pi$$
−0.298552 + 0.954394i $$0.596503\pi$$
$$338$$ 1114.38 0.179331
$$339$$ −695.470 −0.111424
$$340$$ −2251.91 −0.359197
$$341$$ 441.406 0.0700982
$$342$$ −3213.17 −0.508037
$$343$$ 0 0
$$344$$ −476.355 −0.0746608
$$345$$ 1562.50 0.243832
$$346$$ 2623.44 0.407622
$$347$$ 3832.83 0.592960 0.296480 0.955039i $$-0.404187\pi$$
0.296480 + 0.955039i $$0.404187\pi$$
$$348$$ −1172.97 −0.180684
$$349$$ −8325.22 −1.27690 −0.638451 0.769662i $$-0.720425\pi$$
−0.638451 + 0.769662i $$0.720425\pi$$
$$350$$ 0 0
$$351$$ 1178.74 0.179248
$$352$$ 11564.8 1.75115
$$353$$ 8991.52 1.35572 0.677862 0.735189i $$-0.262907\pi$$
0.677862 + 0.735189i $$0.262907\pi$$
$$354$$ −6029.76 −0.905306
$$355$$ 2461.32 0.367981
$$356$$ 2564.33 0.381767
$$357$$ 0 0
$$358$$ −3926.38 −0.579652
$$359$$ −12893.8 −1.89557 −0.947783 0.318917i $$-0.896681\pi$$
−0.947783 + 0.318917i $$0.896681\pi$$
$$360$$ −231.396 −0.0338768
$$361$$ 1837.46 0.267891
$$362$$ 11100.1 1.61163
$$363$$ 3083.71 0.445875
$$364$$ 0 0
$$365$$ 5823.77 0.835150
$$366$$ −4051.94 −0.578684
$$367$$ 7480.17 1.06393 0.531964 0.846767i $$-0.321454\pi$$
0.531964 + 0.846767i $$0.321454\pi$$
$$368$$ 7598.02 1.07629
$$369$$ −2486.47 −0.350787
$$370$$ −4836.76 −0.679598
$$371$$ 0 0
$$372$$ 181.499 0.0252964
$$373$$ −3523.32 −0.489090 −0.244545 0.969638i $$-0.578639\pi$$
−0.244545 + 0.969638i $$0.578639\pi$$
$$374$$ −12580.2 −1.73932
$$375$$ −375.000 −0.0516398
$$376$$ 2996.81 0.411033
$$377$$ −2564.19 −0.350298
$$378$$ 0 0
$$379$$ 13515.4 1.83177 0.915886 0.401438i $$-0.131490\pi$$
0.915886 + 0.401438i $$0.131490\pi$$
$$380$$ −3103.92 −0.419020
$$381$$ 7118.69 0.957222
$$382$$ −4112.44 −0.550813
$$383$$ 657.182 0.0876774 0.0438387 0.999039i $$-0.486041\pi$$
0.0438387 + 0.999039i $$0.486041\pi$$
$$384$$ −1946.75 −0.258710
$$385$$ 0 0
$$386$$ −3441.75 −0.453836
$$387$$ −833.738 −0.109512
$$388$$ 3088.06 0.404053
$$389$$ −9741.87 −1.26975 −0.634875 0.772615i $$-0.718948\pi$$
−0.634875 + 0.772615i $$0.718948\pi$$
$$390$$ 2507.06 0.325512
$$391$$ −7047.58 −0.911538
$$392$$ 0 0
$$393$$ −3600.11 −0.462091
$$394$$ −11728.9 −1.49973
$$395$$ 4360.98 0.555505
$$396$$ 2909.82 0.369253
$$397$$ −4407.42 −0.557184 −0.278592 0.960410i $$-0.589868\pi$$
−0.278592 + 0.960410i $$0.589868\pi$$
$$398$$ −3635.18 −0.457827
$$399$$ 0 0
$$400$$ −1823.53 −0.227941
$$401$$ −11569.5 −1.44078 −0.720391 0.693568i $$-0.756037\pi$$
−0.720391 + 0.693568i $$0.756037\pi$$
$$402$$ 8459.14 1.04951
$$403$$ 396.767 0.0490431
$$404$$ 2880.59 0.354739
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −12272.1 −1.49461
$$408$$ 1043.70 0.126645
$$409$$ 3083.03 0.372729 0.186364 0.982481i $$-0.440330\pi$$
0.186364 + 0.982481i $$0.440330\pi$$
$$410$$ −5288.48 −0.637023
$$411$$ 8343.76 1.00138
$$412$$ −3412.47 −0.408060
$$413$$ 0 0
$$414$$ 3589.15 0.426079
$$415$$ −2647.94 −0.313210
$$416$$ 10395.3 1.22517
$$417$$ 3736.81 0.438831
$$418$$ −17339.9 −2.02900
$$419$$ 5415.21 0.631385 0.315692 0.948862i $$-0.397763\pi$$
0.315692 + 0.948862i $$0.397763\pi$$
$$420$$ 0 0
$$421$$ 4188.34 0.484863 0.242432 0.970168i $$-0.422055\pi$$
0.242432 + 0.970168i $$0.422055\pi$$
$$422$$ −8830.82 −1.01867
$$423$$ 5245.15 0.602902
$$424$$ 3203.65 0.366941
$$425$$ 1691.42 0.193049
$$426$$ 5653.79 0.643021
$$427$$ 0 0
$$428$$ 13068.0 1.47585
$$429$$ 6361.05 0.715884
$$430$$ −1773.28 −0.198872
$$431$$ −9108.41 −1.01795 −0.508975 0.860781i $$-0.669976\pi$$
−0.508975 + 0.860781i $$0.669976\pi$$
$$432$$ −1969.41 −0.219336
$$433$$ 16847.0 1.86979 0.934893 0.354930i $$-0.115495\pi$$
0.934893 + 0.354930i $$0.115495\pi$$
$$434$$ 0 0
$$435$$ 881.026 0.0971080
$$436$$ 3629.16 0.398635
$$437$$ −9714.03 −1.06335
$$438$$ 13377.5 1.45937
$$439$$ −8434.14 −0.916946 −0.458473 0.888708i $$-0.651603\pi$$
−0.458473 + 0.888708i $$0.651603\pi$$
$$440$$ −1248.73 −0.135297
$$441$$ 0 0
$$442$$ −11308.0 −1.21689
$$443$$ −4298.49 −0.461010 −0.230505 0.973071i $$-0.574038\pi$$
−0.230505 + 0.973071i $$0.574038\pi$$
$$444$$ −5046.09 −0.539362
$$445$$ −1926.08 −0.205180
$$446$$ −12356.7 −1.31189
$$447$$ 58.2139 0.00615978
$$448$$ 0 0
$$449$$ 10545.0 1.10835 0.554173 0.832402i $$-0.313035\pi$$
0.554173 + 0.832402i $$0.313035\pi$$
$$450$$ −861.396 −0.0902369
$$451$$ −13418.2 −1.40098
$$452$$ −1543.21 −0.160590
$$453$$ −7049.40 −0.731147
$$454$$ −2441.85 −0.252426
$$455$$ 0 0
$$456$$ 1438.59 0.147737
$$457$$ 11952.4 1.22344 0.611719 0.791075i $$-0.290478\pi$$
0.611719 + 0.791075i $$0.290478\pi$$
$$458$$ 2085.63 0.212784
$$459$$ 1826.74 0.185762
$$460$$ 3467.11 0.351423
$$461$$ 17200.9 1.73780 0.868900 0.494988i $$-0.164827\pi$$
0.868900 + 0.494988i $$0.164827\pi$$
$$462$$ 0 0
$$463$$ −10368.7 −1.04076 −0.520381 0.853934i $$-0.674210\pi$$
−0.520381 + 0.853934i $$0.674210\pi$$
$$464$$ 4284.20 0.428640
$$465$$ −136.325 −0.0135955
$$466$$ 22007.9 2.18776
$$467$$ 16879.5 1.67257 0.836284 0.548296i $$-0.184723\pi$$
0.836284 + 0.548296i $$0.184723\pi$$
$$468$$ 2615.56 0.258342
$$469$$ 0 0
$$470$$ 11155.9 1.09486
$$471$$ 11695.4 1.14415
$$472$$ 2699.62 0.263263
$$473$$ −4499.27 −0.437371
$$474$$ 10017.4 0.970706
$$475$$ 2331.37 0.225201
$$476$$ 0 0
$$477$$ 5607.17 0.538228
$$478$$ 10252.9 0.981082
$$479$$ −7329.12 −0.699115 −0.349558 0.936915i $$-0.613668\pi$$
−0.349558 + 0.936915i $$0.613668\pi$$
$$480$$ −3571.69 −0.339635
$$481$$ −11031.0 −1.04568
$$482$$ −8430.80 −0.796706
$$483$$ 0 0
$$484$$ 6842.60 0.642619
$$485$$ −2319.46 −0.217157
$$486$$ −930.308 −0.0868305
$$487$$ −17209.5 −1.60131 −0.800655 0.599125i $$-0.795515\pi$$
−0.800655 + 0.599125i $$0.795515\pi$$
$$488$$ 1814.11 0.168281
$$489$$ 4582.61 0.423789
$$490$$ 0 0
$$491$$ 11392.5 1.04712 0.523560 0.851989i $$-0.324604\pi$$
0.523560 + 0.851989i $$0.324604\pi$$
$$492$$ −5517.35 −0.505572
$$493$$ −3973.83 −0.363027
$$494$$ −15586.3 −1.41956
$$495$$ −2185.58 −0.198454
$$496$$ −662.912 −0.0600113
$$497$$ 0 0
$$498$$ −6082.47 −0.547313
$$499$$ 19079.4 1.71164 0.855822 0.517271i $$-0.173052\pi$$
0.855822 + 0.517271i $$0.173052\pi$$
$$500$$ −832.107 −0.0744259
$$501$$ 2995.55 0.267129
$$502$$ −21886.7 −1.94592
$$503$$ 13499.4 1.19663 0.598317 0.801259i $$-0.295836\pi$$
0.598317 + 0.801259i $$0.295836\pi$$
$$504$$ 0 0
$$505$$ −2163.62 −0.190654
$$506$$ 19368.8 1.70168
$$507$$ −873.237 −0.0764928
$$508$$ 15796.0 1.37960
$$509$$ 4328.85 0.376960 0.188480 0.982077i $$-0.439644\pi$$
0.188480 + 0.982077i $$0.439644\pi$$
$$510$$ 3885.29 0.337340
$$511$$ 0 0
$$512$$ −14367.6 −1.24017
$$513$$ 2517.88 0.216700
$$514$$ 18086.6 1.55207
$$515$$ 2563.13 0.219311
$$516$$ −1850.02 −0.157835
$$517$$ 28305.5 2.40788
$$518$$ 0 0
$$519$$ −2055.76 −0.173869
$$520$$ −1122.45 −0.0946588
$$521$$ −19395.7 −1.63098 −0.815490 0.578771i $$-0.803532\pi$$
−0.815490 + 0.578771i $$0.803532\pi$$
$$522$$ 2023.77 0.169689
$$523$$ −20413.8 −1.70675 −0.853377 0.521294i $$-0.825450\pi$$
−0.853377 + 0.521294i $$0.825450\pi$$
$$524$$ −7988.47 −0.665989
$$525$$ 0 0
$$526$$ −22876.2 −1.89629
$$527$$ 614.887 0.0508252
$$528$$ −10627.9 −0.875987
$$529$$ −1316.34 −0.108189
$$530$$ 11925.9 0.977413
$$531$$ 4724.99 0.386153
$$532$$ 0 0
$$533$$ −12061.3 −0.980171
$$534$$ −4424.32 −0.358537
$$535$$ −9815.43 −0.793193
$$536$$ −3787.28 −0.305197
$$537$$ 3076.75 0.247247
$$538$$ 17174.8 1.37631
$$539$$ 0 0
$$540$$ −898.675 −0.0716163
$$541$$ −4919.18 −0.390928 −0.195464 0.980711i $$-0.562621\pi$$
−0.195464 + 0.980711i $$0.562621\pi$$
$$542$$ 14654.0 1.16133
$$543$$ −8698.19 −0.687431
$$544$$ 16110.0 1.26969
$$545$$ −2725.88 −0.214246
$$546$$ 0 0
$$547$$ 15334.2 1.19862 0.599308 0.800518i $$-0.295442\pi$$
0.599308 + 0.800518i $$0.295442\pi$$
$$548$$ 18514.4 1.44324
$$549$$ 3175.15 0.246834
$$550$$ −4648.53 −0.360389
$$551$$ −5477.33 −0.423488
$$552$$ −1606.92 −0.123904
$$553$$ 0 0
$$554$$ 13095.3 1.00427
$$555$$ 3790.14 0.289879
$$556$$ 8291.81 0.632466
$$557$$ −8613.78 −0.655256 −0.327628 0.944807i $$-0.606249\pi$$
−0.327628 + 0.944807i $$0.606249\pi$$
$$558$$ −313.145 −0.0237572
$$559$$ −4044.26 −0.306000
$$560$$ 0 0
$$561$$ 9857.98 0.741897
$$562$$ −20045.3 −1.50456
$$563$$ 2320.81 0.173731 0.0868654 0.996220i $$-0.472315\pi$$
0.0868654 + 0.996220i $$0.472315\pi$$
$$564$$ 11638.7 0.868934
$$565$$ 1159.12 0.0863087
$$566$$ −26744.9 −1.98617
$$567$$ 0 0
$$568$$ −2531.29 −0.186990
$$569$$ −1736.04 −0.127906 −0.0639529 0.997953i $$-0.520371\pi$$
−0.0639529 + 0.997953i $$0.520371\pi$$
$$570$$ 5355.29 0.393524
$$571$$ 23897.8 1.75148 0.875738 0.482786i $$-0.160375\pi$$
0.875738 + 0.482786i $$0.160375\pi$$
$$572$$ 14114.9 1.03177
$$573$$ 3222.55 0.234946
$$574$$ 0 0
$$575$$ −2604.16 −0.188871
$$576$$ −2952.61 −0.213586
$$577$$ −8029.26 −0.579311 −0.289655 0.957131i $$-0.593541\pi$$
−0.289655 + 0.957131i $$0.593541\pi$$
$$578$$ 1284.63 0.0924455
$$579$$ 2697.00 0.193581
$$580$$ 1954.95 0.139957
$$581$$ 0 0
$$582$$ −5327.93 −0.379467
$$583$$ 30259.1 2.14958
$$584$$ −5989.32 −0.424383
$$585$$ −1964.56 −0.138845
$$586$$ 28329.2 1.99705
$$587$$ 8015.14 0.563578 0.281789 0.959476i $$-0.409072\pi$$
0.281789 + 0.959476i $$0.409072\pi$$
$$588$$ 0 0
$$589$$ 847.529 0.0592900
$$590$$ 10049.6 0.701247
$$591$$ 9190.88 0.639700
$$592$$ 18430.5 1.27954
$$593$$ −12820.3 −0.887801 −0.443901 0.896076i $$-0.646406\pi$$
−0.443901 + 0.896076i $$0.646406\pi$$
$$594$$ −5020.41 −0.346784
$$595$$ 0 0
$$596$$ 129.174 0.00887779
$$597$$ 2848.57 0.195283
$$598$$ 17410.1 1.19055
$$599$$ −15330.5 −1.04572 −0.522860 0.852418i $$-0.675135\pi$$
−0.522860 + 0.852418i $$0.675135\pi$$
$$600$$ 385.660 0.0262409
$$601$$ −107.658 −0.00730694 −0.00365347 0.999993i $$-0.501163\pi$$
−0.00365347 + 0.999993i $$0.501163\pi$$
$$602$$ 0 0
$$603$$ −6628.68 −0.447663
$$604$$ −15642.3 −1.05377
$$605$$ −5139.52 −0.345374
$$606$$ −4969.97 −0.333154
$$607$$ 8213.06 0.549189 0.274595 0.961560i $$-0.411456\pi$$
0.274595 + 0.961560i $$0.411456\pi$$
$$608$$ 22205.2 1.48115
$$609$$ 0 0
$$610$$ 6753.23 0.448246
$$611$$ 25443.0 1.68463
$$612$$ 4053.44 0.267729
$$613$$ 1242.60 0.0818732 0.0409366 0.999162i $$-0.486966\pi$$
0.0409366 + 0.999162i $$0.486966\pi$$
$$614$$ 10216.7 0.671519
$$615$$ 4144.11 0.271718
$$616$$ 0 0
$$617$$ −13170.6 −0.859367 −0.429683 0.902980i $$-0.641375\pi$$
−0.429683 + 0.902980i $$0.641375\pi$$
$$618$$ 5887.65 0.383230
$$619$$ −19774.1 −1.28399 −0.641993 0.766711i $$-0.721892\pi$$
−0.641993 + 0.766711i $$0.721892\pi$$
$$620$$ −302.498 −0.0195945
$$621$$ −2812.50 −0.181742
$$622$$ 23695.1 1.52747
$$623$$ 0 0
$$624$$ −9553.14 −0.612871
$$625$$ 625.000 0.0400000
$$626$$ −11185.1 −0.714132
$$627$$ 13587.8 0.865459
$$628$$ 25951.5 1.64901
$$629$$ −17095.3 −1.08368
$$630$$ 0 0
$$631$$ −14308.8 −0.902735 −0.451367 0.892338i $$-0.649064\pi$$
−0.451367 + 0.892338i $$0.649064\pi$$
$$632$$ −4484.95 −0.282281
$$633$$ 6919.93 0.434507
$$634$$ −37616.4 −2.35637
$$635$$ −11864.5 −0.741461
$$636$$ 12442.0 0.775722
$$637$$ 0 0
$$638$$ 10921.3 0.677707
$$639$$ −4430.38 −0.274277
$$640$$ 3244.58 0.200396
$$641$$ 11537.5 0.710925 0.355463 0.934691i $$-0.384323\pi$$
0.355463 + 0.934691i $$0.384323\pi$$
$$642$$ −22546.6 −1.38605
$$643$$ −19603.0 −1.20228 −0.601139 0.799144i $$-0.705286\pi$$
−0.601139 + 0.799144i $$0.705286\pi$$
$$644$$ 0 0
$$645$$ 1389.56 0.0848279
$$646$$ −24154.8 −1.47114
$$647$$ −21650.0 −1.31553 −0.657765 0.753223i $$-0.728498\pi$$
−0.657765 + 0.753223i $$0.728498\pi$$
$$648$$ 416.513 0.0252503
$$649$$ 25498.4 1.54222
$$650$$ −4178.43 −0.252141
$$651$$ 0 0
$$652$$ 10168.6 0.610787
$$653$$ −2927.33 −0.175429 −0.0877145 0.996146i $$-0.527956\pi$$
−0.0877145 + 0.996146i $$0.527956\pi$$
$$654$$ −6261.50 −0.374379
$$655$$ 6000.18 0.357934
$$656$$ 20151.7 1.19938
$$657$$ −10482.8 −0.622484
$$658$$ 0 0
$$659$$ −4778.76 −0.282480 −0.141240 0.989975i $$-0.545109\pi$$
−0.141240 + 0.989975i $$0.545109\pi$$
$$660$$ −4849.71 −0.286022
$$661$$ 31510.3 1.85417 0.927086 0.374849i $$-0.122305\pi$$
0.927086 + 0.374849i $$0.122305\pi$$
$$662$$ 35444.2 2.08093
$$663$$ 8861.06 0.519057
$$664$$ 2723.21 0.159158
$$665$$ 0 0
$$666$$ 8706.17 0.506542
$$667$$ 6118.23 0.355170
$$668$$ 6646.99 0.385000
$$669$$ 9682.82 0.559581
$$670$$ −14098.6 −0.812948
$$671$$ 17134.7 0.985808
$$672$$ 0 0
$$673$$ −8992.15 −0.515040 −0.257520 0.966273i $$-0.582905\pi$$
−0.257520 + 0.966273i $$0.582905\pi$$
$$674$$ 14142.1 0.808211
$$675$$ 675.000 0.0384900
$$676$$ −1937.67 −0.110245
$$677$$ −19340.8 −1.09797 −0.548985 0.835832i $$-0.684986\pi$$
−0.548985 + 0.835832i $$0.684986\pi$$
$$678$$ 2662.56 0.150818
$$679$$ 0 0
$$680$$ −1739.50 −0.0980984
$$681$$ 1913.46 0.107671
$$682$$ −1689.89 −0.0948816
$$683$$ −4255.14 −0.238387 −0.119194 0.992871i $$-0.538031\pi$$
−0.119194 + 0.992871i $$0.538031\pi$$
$$684$$ 5587.05 0.312319
$$685$$ −13906.3 −0.775666
$$686$$ 0 0
$$687$$ −1634.32 −0.0907616
$$688$$ 6757.08 0.374435
$$689$$ 27199.1 1.50392
$$690$$ −5981.91 −0.330040
$$691$$ 17505.5 0.963733 0.481867 0.876245i $$-0.339959\pi$$
0.481867 + 0.876245i $$0.339959\pi$$
$$692$$ −4561.63 −0.250588
$$693$$ 0 0
$$694$$ −14673.7 −0.802603
$$695$$ −6228.02 −0.339917
$$696$$ −906.071 −0.0493456
$$697$$ −18691.8 −1.01579
$$698$$ 31872.5 1.72836
$$699$$ −17245.6 −0.933175
$$700$$ 0 0
$$701$$ −3240.77 −0.174611 −0.0873054 0.996182i $$-0.527826\pi$$
−0.0873054 + 0.996182i $$0.527826\pi$$
$$702$$ −4512.70 −0.242622
$$703$$ −23563.3 −1.26416
$$704$$ −15933.8 −0.853022
$$705$$ −8741.91 −0.467006
$$706$$ −34423.4 −1.83504
$$707$$ 0 0
$$708$$ 10484.5 0.556543
$$709$$ 19949.3 1.05672 0.528358 0.849022i $$-0.322808\pi$$
0.528358 + 0.849022i $$0.322808\pi$$
$$710$$ −9422.99 −0.498082
$$711$$ −7849.76 −0.414049
$$712$$ 1980.83 0.104262
$$713$$ −946.698 −0.0497253
$$714$$ 0 0
$$715$$ −10601.7 −0.554522
$$716$$ 6827.17 0.356345
$$717$$ −8034.30 −0.418475
$$718$$ 49362.9 2.56575
$$719$$ 11259.4 0.584011 0.292006 0.956417i $$-0.405677\pi$$
0.292006 + 0.956417i $$0.405677\pi$$
$$720$$ 3282.35 0.169897
$$721$$ 0 0
$$722$$ −7034.60 −0.362605
$$723$$ 6606.47 0.339830
$$724$$ −19300.9 −0.990761
$$725$$ −1468.38 −0.0752195
$$726$$ −11805.8 −0.603516
$$727$$ 12228.5 0.623840 0.311920 0.950108i $$-0.399028\pi$$
0.311920 + 0.950108i $$0.399028\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −22295.9 −1.13042
$$731$$ −6267.56 −0.317119
$$732$$ 7045.49 0.355750
$$733$$ 26635.1 1.34214 0.671072 0.741392i $$-0.265834\pi$$
0.671072 + 0.741392i $$0.265834\pi$$
$$734$$ −28637.3 −1.44008
$$735$$ 0 0
$$736$$ −24803.4 −1.24221
$$737$$ −35771.7 −1.78788
$$738$$ 9519.26 0.474809
$$739$$ −6074.00 −0.302349 −0.151174 0.988507i $$-0.548306\pi$$
−0.151174 + 0.988507i $$0.548306\pi$$
$$740$$ 8410.14 0.417788
$$741$$ 12213.6 0.605505
$$742$$ 0 0
$$743$$ −4016.87 −0.198337 −0.0991686 0.995071i $$-0.531618\pi$$
−0.0991686 + 0.995071i $$0.531618\pi$$
$$744$$ 140.200 0.00690858
$$745$$ −97.0231 −0.00477134
$$746$$ 13488.8 0.662010
$$747$$ 4766.29 0.233453
$$748$$ 21874.4 1.06926
$$749$$ 0 0
$$750$$ 1435.66 0.0698972
$$751$$ −23913.2 −1.16192 −0.580962 0.813931i $$-0.697324\pi$$
−0.580962 + 0.813931i $$0.697324\pi$$
$$752$$ −42509.6 −2.06139
$$753$$ 17150.7 0.830022
$$754$$ 9816.81 0.474147
$$755$$ 11749.0 0.566344
$$756$$ 0 0
$$757$$ 31044.9 1.49055 0.745275 0.666758i $$-0.232318\pi$$
0.745275 + 0.666758i $$0.232318\pi$$
$$758$$ −51742.9 −2.47940
$$759$$ −15177.6 −0.725842
$$760$$ −2397.65 −0.114436
$$761$$ −14011.8 −0.667446 −0.333723 0.942671i $$-0.608305\pi$$
−0.333723 + 0.942671i $$0.608305\pi$$
$$762$$ −27253.4 −1.29565
$$763$$ 0 0
$$764$$ 7150.69 0.338616
$$765$$ −3044.56 −0.143891
$$766$$ −2515.97 −0.118676
$$767$$ 22919.8 1.07899
$$768$$ 15326.6 0.720120
$$769$$ −3342.49 −0.156740 −0.0783701 0.996924i $$-0.524972\pi$$
−0.0783701 + 0.996924i $$0.524972\pi$$
$$770$$ 0 0
$$771$$ −14172.9 −0.662027
$$772$$ 5984.51 0.278999
$$773$$ 21074.6 0.980594 0.490297 0.871555i $$-0.336888\pi$$
0.490297 + 0.871555i $$0.336888\pi$$
$$774$$ 3191.90 0.148231
$$775$$ 227.208 0.0105310
$$776$$ 2385.40 0.110349
$$777$$ 0 0
$$778$$ 37296.0 1.71867
$$779$$ −25763.9 −1.18496
$$780$$ −4359.26 −0.200111
$$781$$ −23908.5 −1.09541
$$782$$ 26981.1 1.23382
$$783$$ −1585.85 −0.0723800
$$784$$ 0 0
$$785$$ −19492.3 −0.886254
$$786$$ 13782.8 0.625464
$$787$$ −21394.8 −0.969048 −0.484524 0.874778i $$-0.661007\pi$$
−0.484524 + 0.874778i $$0.661007\pi$$
$$788$$ 20394.1 0.921968
$$789$$ 17926.1 0.808853
$$790$$ −16695.7 −0.751906
$$791$$ 0 0
$$792$$ 2247.71 0.100845
$$793$$ 15401.9 0.689706
$$794$$ 16873.5 0.754179
$$795$$ −9345.29 −0.416910
$$796$$ 6320.83 0.281452
$$797$$ −20645.0 −0.917547 −0.458773 0.888553i $$-0.651711\pi$$
−0.458773 + 0.888553i $$0.651711\pi$$
$$798$$ 0 0
$$799$$ 39430.0 1.74585
$$800$$ 5952.82 0.263080
$$801$$ 3466.95 0.152932
$$802$$ 44293.0 1.95017
$$803$$ −56570.4 −2.48608
$$804$$ −14708.7 −0.645194
$$805$$ 0 0
$$806$$ −1518.99 −0.0663825
$$807$$ −13458.3 −0.587058
$$808$$ 2225.13 0.0968810
$$809$$ −15939.0 −0.692688 −0.346344 0.938108i $$-0.612577\pi$$
−0.346344 + 0.938108i $$0.612577\pi$$
$$810$$ 1550.51 0.0672586
$$811$$ −22829.2 −0.988460 −0.494230 0.869331i $$-0.664550\pi$$
−0.494230 + 0.869331i $$0.664550\pi$$
$$812$$ 0 0
$$813$$ −11483.0 −0.495360
$$814$$ 46982.9 2.02303
$$815$$ −7637.69 −0.328266
$$816$$ −14804.9 −0.635141
$$817$$ −8638.90 −0.369935
$$818$$ −11803.2 −0.504508
$$819$$ 0 0
$$820$$ 9195.58 0.391614
$$821$$ −5700.22 −0.242313 −0.121157 0.992633i $$-0.538660\pi$$
−0.121157 + 0.992633i $$0.538660\pi$$
$$822$$ −31943.5 −1.35542
$$823$$ −32438.0 −1.37390 −0.686948 0.726707i $$-0.741050\pi$$
−0.686948 + 0.726707i $$0.741050\pi$$
$$824$$ −2635.99 −0.111443
$$825$$ 3642.64 0.153722
$$826$$ 0 0
$$827$$ −12762.6 −0.536638 −0.268319 0.963330i $$-0.586468\pi$$
−0.268319 + 0.963330i $$0.586468\pi$$
$$828$$ −6240.79 −0.261935
$$829$$ 30766.7 1.28899 0.644494 0.764609i $$-0.277068\pi$$
0.644494 + 0.764609i $$0.277068\pi$$
$$830$$ 10137.4 0.423947
$$831$$ −10261.6 −0.428365
$$832$$ −14322.4 −0.596804
$$833$$ 0 0
$$834$$ −14306.1 −0.593981
$$835$$ −4992.59 −0.206917
$$836$$ 30150.6 1.24734
$$837$$ 245.384 0.0101335
$$838$$ −20731.7 −0.854613
$$839$$ 9779.71 0.402423 0.201212 0.979548i $$-0.435512\pi$$
0.201212 + 0.979548i $$0.435512\pi$$
$$840$$ 0 0
$$841$$ −20939.2 −0.858551
$$842$$ −16034.8 −0.656288
$$843$$ 15707.8 0.641760
$$844$$ 15355.0 0.626233
$$845$$ 1455.40 0.0592510
$$846$$ −20080.7 −0.816061
$$847$$ 0 0
$$848$$ −45443.7 −1.84026
$$849$$ 20957.6 0.847190
$$850$$ −6475.48 −0.261303
$$851$$ 26320.4 1.06023
$$852$$ −9830.79 −0.395302
$$853$$ −24201.5 −0.971445 −0.485723 0.874113i $$-0.661444\pi$$
−0.485723 + 0.874113i $$0.661444\pi$$
$$854$$ 0 0
$$855$$ −4196.47 −0.167855
$$856$$ 10094.5 0.403062
$$857$$ −21036.7 −0.838507 −0.419254 0.907869i $$-0.637708\pi$$
−0.419254 + 0.907869i $$0.637708\pi$$
$$858$$ −24352.8 −0.968988
$$859$$ 6179.19 0.245438 0.122719 0.992441i $$-0.460839\pi$$
0.122719 + 0.992441i $$0.460839\pi$$
$$860$$ 3083.37 0.122258
$$861$$ 0 0
$$862$$ 34870.9 1.37785
$$863$$ 50256.2 1.98232 0.991160 0.132671i $$-0.0423555\pi$$
0.991160 + 0.132671i $$0.0423555\pi$$
$$864$$ 6429.04 0.253149
$$865$$ 3426.27 0.134678
$$866$$ −64497.7 −2.53085
$$867$$ −1006.65 −0.0394321
$$868$$ 0 0
$$869$$ −42361.2 −1.65363
$$870$$ −3372.94 −0.131441
$$871$$ −32154.1 −1.25086
$$872$$ 2803.37 0.108869
$$873$$ 4175.03 0.161860
$$874$$ 37189.5 1.43930
$$875$$ 0 0
$$876$$ −23260.8 −0.897156
$$877$$ −9175.95 −0.353306 −0.176653 0.984273i $$-0.556527\pi$$
−0.176653 + 0.984273i $$0.556527\pi$$
$$878$$ 32289.5 1.24114
$$879$$ −22199.1 −0.851828
$$880$$ 17713.2 0.678537
$$881$$ 26172.7 1.00089 0.500444 0.865769i $$-0.333170\pi$$
0.500444 + 0.865769i $$0.333170\pi$$
$$882$$ 0 0
$$883$$ 18615.5 0.709471 0.354736 0.934967i $$-0.384571\pi$$
0.354736 + 0.934967i $$0.384571\pi$$
$$884$$ 19662.3 0.748092
$$885$$ −7874.99 −0.299113
$$886$$ 16456.4 0.624001
$$887$$ 12837.8 0.485964 0.242982 0.970031i $$-0.421874\pi$$
0.242982 + 0.970031i $$0.421874\pi$$
$$888$$ −3897.89 −0.147302
$$889$$ 0 0
$$890$$ 7373.86 0.277722
$$891$$ 3934.05 0.147919
$$892$$ 21485.7 0.806496
$$893$$ 54348.4 2.03662
$$894$$ −222.868 −0.00833759
$$895$$ −5127.92 −0.191517
$$896$$ 0 0
$$897$$ −13642.7 −0.507824
$$898$$ −40370.6 −1.50020
$$899$$ −533.803 −0.0198035
$$900$$ 1497.79 0.0554738
$$901$$ 42151.5 1.55857
$$902$$ 51370.7 1.89630
$$903$$ 0 0
$$904$$ −1192.07 −0.0438579
$$905$$ 14497.0 0.532482
$$906$$ 26988.1 0.989647
$$907$$ −26766.1 −0.979885 −0.489942 0.871755i $$-0.662982\pi$$
−0.489942 + 0.871755i $$0.662982\pi$$
$$908$$ 4245.87 0.155181
$$909$$ 3894.52 0.142105
$$910$$ 0 0
$$911$$ 5022.67 0.182666 0.0913328 0.995820i $$-0.470887\pi$$
0.0913328 + 0.995820i $$0.470887\pi$$
$$912$$ −20406.3 −0.740923
$$913$$ 25721.3 0.932367
$$914$$ −45759.0 −1.65599
$$915$$ −5291.91 −0.191197
$$916$$ −3626.48 −0.130810
$$917$$ 0 0
$$918$$ −6993.52 −0.251439
$$919$$ 9541.79 0.342497 0.171248 0.985228i $$-0.445220\pi$$
0.171248 + 0.985228i $$0.445220\pi$$
$$920$$ 2678.19 0.0959754
$$921$$ −8005.93 −0.286432
$$922$$ −65852.4 −2.35220
$$923$$ −21490.7 −0.766387
$$924$$ 0 0
$$925$$ −6316.90 −0.224539
$$926$$ 39695.7 1.40873
$$927$$ −4613.63 −0.163464
$$928$$ −13985.6 −0.494718
$$929$$ 25479.6 0.899846 0.449923 0.893067i $$-0.351451\pi$$
0.449923 + 0.893067i $$0.351451\pi$$
$$930$$ 521.909 0.0184022
$$931$$ 0 0
$$932$$ −38267.2 −1.34494
$$933$$ −18567.7 −0.651533
$$934$$ −64621.9 −2.26391
$$935$$ −16430.0 −0.574671
$$936$$ 2020.41 0.0705545
$$937$$ 33608.3 1.17176 0.585878 0.810399i $$-0.300750\pi$$
0.585878 + 0.810399i $$0.300750\pi$$
$$938$$ 0 0
$$939$$ 8764.77 0.304609
$$940$$ −19397.9 −0.673073
$$941$$ 19173.6 0.664232 0.332116 0.943239i $$-0.392238\pi$$
0.332116 + 0.943239i $$0.392238\pi$$
$$942$$ −44774.9 −1.54867
$$943$$ 28778.5 0.993804
$$944$$ −38294.0 −1.32030
$$945$$ 0 0
$$946$$ 17225.1 0.592005
$$947$$ −979.315 −0.0336045 −0.0168023 0.999859i $$-0.505349\pi$$
−0.0168023 + 0.999859i $$0.505349\pi$$
$$948$$ −17418.2 −0.596749
$$949$$ −50849.5 −1.73935
$$950$$ −8925.48 −0.304822
$$951$$ 29476.7 1.00510
$$952$$ 0 0
$$953$$ 3048.61 0.103624 0.0518122 0.998657i $$-0.483500\pi$$
0.0518122 + 0.998657i $$0.483500\pi$$
$$954$$ −21466.7 −0.728521
$$955$$ −5370.92 −0.181989
$$956$$ −17827.7 −0.603127
$$957$$ −8558.03 −0.289072
$$958$$ 28059.0 0.946290
$$959$$ 0 0
$$960$$ 4921.02 0.165443
$$961$$ −29708.4 −0.997227
$$962$$ 42231.6 1.41538
$$963$$ 17667.8 0.591211
$$964$$ 14659.4 0.489781
$$965$$ −4495.00 −0.149947
$$966$$ 0 0
$$967$$ −14467.9 −0.481133 −0.240567 0.970633i $$-0.577333\pi$$
−0.240567 + 0.970633i $$0.577333\pi$$
$$968$$ 5285.62 0.175502
$$969$$ 18928.0 0.627507
$$970$$ 8879.89 0.293934
$$971$$ −12952.8 −0.428090 −0.214045 0.976824i $$-0.568664\pi$$
−0.214045 + 0.976824i $$0.568664\pi$$
$$972$$ 1617.62 0.0533797
$$973$$ 0 0
$$974$$ 65885.4 2.16746
$$975$$ 3274.26 0.107549
$$976$$ −25733.2 −0.843954
$$977$$ 47244.2 1.54706 0.773529 0.633760i $$-0.218489\pi$$
0.773529 + 0.633760i $$0.218489\pi$$
$$978$$ −17544.2 −0.573621
$$979$$ 18709.4 0.610781
$$980$$ 0 0
$$981$$ 4906.58 0.159689
$$982$$ −43615.3 −1.41733
$$983$$ 1536.84 0.0498651 0.0249326 0.999689i $$-0.492063\pi$$
0.0249326 + 0.999689i $$0.492063\pi$$
$$984$$ −4261.92 −0.138074
$$985$$ −15318.1 −0.495509
$$986$$ 15213.5 0.491376
$$987$$ 0 0
$$988$$ 27101.5 0.872685
$$989$$ 9649.73 0.310256
$$990$$ 8367.35 0.268618
$$991$$ 3785.22 0.121334 0.0606668 0.998158i $$-0.480677\pi$$
0.0606668 + 0.998158i $$0.480677\pi$$
$$992$$ 2164.04 0.0692625
$$993$$ −27774.5 −0.887610
$$994$$ 0 0
$$995$$ −4747.61 −0.151266
$$996$$ 10576.2 0.336465
$$997$$ 25894.9 0.822566 0.411283 0.911508i $$-0.365081\pi$$
0.411283 + 0.911508i $$0.365081\pi$$
$$998$$ −73044.0 −2.31680
$$999$$ −6822.26 −0.216063
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 735.4.a.o.1.1 2
3.2 odd 2 2205.4.a.bb.1.2 2
7.6 odd 2 105.4.a.e.1.1 2
21.20 even 2 315.4.a.k.1.2 2
28.27 even 2 1680.4.a.bo.1.1 2
35.13 even 4 525.4.d.l.274.4 4
35.27 even 4 525.4.d.l.274.1 4
35.34 odd 2 525.4.a.l.1.2 2
105.104 even 2 1575.4.a.q.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 7.6 odd 2
315.4.a.k.1.2 2 21.20 even 2
525.4.a.l.1.2 2 35.34 odd 2
525.4.d.l.274.1 4 35.27 even 4
525.4.d.l.274.4 4 35.13 even 4
735.4.a.o.1.1 2 1.1 even 1 trivial
1575.4.a.q.1.1 2 105.104 even 2
1680.4.a.bo.1.1 2 28.27 even 2
2205.4.a.bb.1.2 2 3.2 odd 2