# Properties

 Label 637.4.a.d.1.3 Level $637$ Weight $4$ Character 637.1 Self dual yes Analytic conductor $37.584$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.5842166737$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5364412.1 Defining polynomial: $$x^{4} - 27x^{2} - 24x + 76$$ x^4 - 27*x^2 - 24*x + 76 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.32361$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.323612 q^{2} -1.75980 q^{3} -7.89528 q^{4} +5.91876 q^{5} -0.569491 q^{6} -5.14391 q^{8} -23.9031 q^{9} +O(q^{10})$$ $$q+0.323612 q^{2} -1.75980 q^{3} -7.89528 q^{4} +5.91876 q^{5} -0.569491 q^{6} -5.14391 q^{8} -23.9031 q^{9} +1.91538 q^{10} -60.8448 q^{11} +13.8941 q^{12} +13.0000 q^{13} -10.4158 q^{15} +61.4976 q^{16} +1.36094 q^{17} -7.73534 q^{18} +4.20658 q^{19} -46.7303 q^{20} -19.6901 q^{22} -10.6695 q^{23} +9.05222 q^{24} -89.9682 q^{25} +4.20696 q^{26} +89.5791 q^{27} +124.031 q^{29} -3.37068 q^{30} -90.9367 q^{31} +61.0526 q^{32} +107.074 q^{33} +0.440417 q^{34} +188.722 q^{36} +101.085 q^{37} +1.36130 q^{38} -22.8773 q^{39} -30.4456 q^{40} -235.305 q^{41} +6.98363 q^{43} +480.386 q^{44} -141.477 q^{45} -3.45278 q^{46} +243.878 q^{47} -108.223 q^{48} -29.1148 q^{50} -2.39498 q^{51} -102.639 q^{52} -282.024 q^{53} +28.9889 q^{54} -360.126 q^{55} -7.40271 q^{57} +40.1380 q^{58} +675.683 q^{59} +82.2357 q^{60} -87.6018 q^{61} -29.4282 q^{62} -472.223 q^{64} +76.9439 q^{65} +34.6506 q^{66} +122.719 q^{67} -10.7450 q^{68} +18.7761 q^{69} +35.3966 q^{71} +122.955 q^{72} +1073.70 q^{73} +32.7124 q^{74} +158.326 q^{75} -33.2121 q^{76} -7.40339 q^{78} +811.612 q^{79} +363.990 q^{80} +487.743 q^{81} -76.1477 q^{82} +1102.66 q^{83} +8.05509 q^{85} +2.25999 q^{86} -218.269 q^{87} +312.980 q^{88} +1093.27 q^{89} -45.7837 q^{90} +84.2387 q^{92} +160.030 q^{93} +78.9220 q^{94} +24.8977 q^{95} -107.440 q^{96} -911.865 q^{97} +1454.38 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 5 * q^3 + 26 * q^4 + 36 * q^5 + 45 * q^6 - 30 * q^8 + 21 * q^9 $$4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9} + 44 q^{10} - 95 q^{11} + 17 q^{12} + 52 q^{13} - 16 q^{15} + 58 q^{16} + 146 q^{17} + 65 q^{18} + 48 q^{19} + 474 q^{20} - 143 q^{22} - 121 q^{23} + 469 q^{24} + 506 q^{25} - 52 q^{26} + 83 q^{27} - 440 q^{29} + 1548 q^{30} + 283 q^{31} - 114 q^{32} - 227 q^{33} - 1234 q^{34} + 755 q^{36} - 209 q^{37} - 440 q^{38} + 65 q^{39} - 754 q^{40} + 93 q^{41} + 526 q^{43} + 217 q^{44} + 768 q^{45} - 841 q^{46} + 783 q^{47} - 1407 q^{48} + 446 q^{50} - 672 q^{51} + 338 q^{52} - 340 q^{53} + 199 q^{54} - 756 q^{55} - 1014 q^{57} + 1916 q^{58} + 922 q^{59} - 396 q^{60} + 141 q^{61} - 1745 q^{62} - 1510 q^{64} + 468 q^{65} - 503 q^{66} - 523 q^{67} + 1710 q^{68} - 1595 q^{69} + 1468 q^{71} - 9 q^{72} + 47 q^{73} - 2249 q^{74} + 1547 q^{75} + 1382 q^{76} + 585 q^{78} + 1025 q^{79} + 2538 q^{80} - 1772 q^{81} + 1561 q^{82} + 1190 q^{83} - 568 q^{85} + 738 q^{86} - 720 q^{87} - 555 q^{88} + 2962 q^{89} + 1960 q^{90} - 599 q^{92} - 763 q^{93} - 317 q^{94} + 2082 q^{95} + 45 q^{96} - 2715 q^{97} + 586 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 5 * q^3 + 26 * q^4 + 36 * q^5 + 45 * q^6 - 30 * q^8 + 21 * q^9 + 44 * q^10 - 95 * q^11 + 17 * q^12 + 52 * q^13 - 16 * q^15 + 58 * q^16 + 146 * q^17 + 65 * q^18 + 48 * q^19 + 474 * q^20 - 143 * q^22 - 121 * q^23 + 469 * q^24 + 506 * q^25 - 52 * q^26 + 83 * q^27 - 440 * q^29 + 1548 * q^30 + 283 * q^31 - 114 * q^32 - 227 * q^33 - 1234 * q^34 + 755 * q^36 - 209 * q^37 - 440 * q^38 + 65 * q^39 - 754 * q^40 + 93 * q^41 + 526 * q^43 + 217 * q^44 + 768 * q^45 - 841 * q^46 + 783 * q^47 - 1407 * q^48 + 446 * q^50 - 672 * q^51 + 338 * q^52 - 340 * q^53 + 199 * q^54 - 756 * q^55 - 1014 * q^57 + 1916 * q^58 + 922 * q^59 - 396 * q^60 + 141 * q^61 - 1745 * q^62 - 1510 * q^64 + 468 * q^65 - 503 * q^66 - 523 * q^67 + 1710 * q^68 - 1595 * q^69 + 1468 * q^71 - 9 * q^72 + 47 * q^73 - 2249 * q^74 + 1547 * q^75 + 1382 * q^76 + 585 * q^78 + 1025 * q^79 + 2538 * q^80 - 1772 * q^81 + 1561 * q^82 + 1190 * q^83 - 568 * q^85 + 738 * q^86 - 720 * q^87 - 555 * q^88 + 2962 * q^89 + 1960 * q^90 - 599 * q^92 - 763 * q^93 - 317 * q^94 + 2082 * q^95 + 45 * q^96 - 2715 * q^97 + 586 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.323612 0.114414 0.0572071 0.998362i $$-0.481780\pi$$
0.0572071 + 0.998362i $$0.481780\pi$$
$$3$$ −1.75980 −0.338673 −0.169336 0.985558i $$-0.554162\pi$$
−0.169336 + 0.985558i $$0.554162\pi$$
$$4$$ −7.89528 −0.986909
$$5$$ 5.91876 0.529390 0.264695 0.964332i $$-0.414729\pi$$
0.264695 + 0.964332i $$0.414729\pi$$
$$6$$ −0.569491 −0.0387490
$$7$$ 0 0
$$8$$ −5.14391 −0.227331
$$9$$ −23.9031 −0.885301
$$10$$ 1.91538 0.0605698
$$11$$ −60.8448 −1.66776 −0.833882 0.551943i $$-0.813886\pi$$
−0.833882 + 0.551943i $$0.813886\pi$$
$$12$$ 13.8941 0.334239
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ −10.4158 −0.179290
$$16$$ 61.4976 0.960900
$$17$$ 1.36094 0.0194163 0.00970813 0.999953i $$-0.496910\pi$$
0.00970813 + 0.999953i $$0.496910\pi$$
$$18$$ −7.73534 −0.101291
$$19$$ 4.20658 0.0507923 0.0253962 0.999677i $$-0.491915\pi$$
0.0253962 + 0.999677i $$0.491915\pi$$
$$20$$ −46.7303 −0.522460
$$21$$ 0 0
$$22$$ −19.6901 −0.190816
$$23$$ −10.6695 −0.0967281 −0.0483640 0.998830i $$-0.515401\pi$$
−0.0483640 + 0.998830i $$0.515401\pi$$
$$24$$ 9.05222 0.0769907
$$25$$ −89.9682 −0.719746
$$26$$ 4.20696 0.0317328
$$27$$ 89.5791 0.638500
$$28$$ 0 0
$$29$$ 124.031 0.794207 0.397103 0.917774i $$-0.370015\pi$$
0.397103 + 0.917774i $$0.370015\pi$$
$$30$$ −3.37068 −0.0205133
$$31$$ −90.9367 −0.526862 −0.263431 0.964678i $$-0.584854\pi$$
−0.263431 + 0.964678i $$0.584854\pi$$
$$32$$ 61.0526 0.337271
$$33$$ 107.074 0.564826
$$34$$ 0.440417 0.00222150
$$35$$ 0 0
$$36$$ 188.722 0.873712
$$37$$ 101.085 0.449143 0.224572 0.974458i $$-0.427902\pi$$
0.224572 + 0.974458i $$0.427902\pi$$
$$38$$ 1.36130 0.00581137
$$39$$ −22.8773 −0.0939309
$$40$$ −30.4456 −0.120347
$$41$$ −235.305 −0.896306 −0.448153 0.893957i $$-0.647918\pi$$
−0.448153 + 0.893957i $$0.647918\pi$$
$$42$$ 0 0
$$43$$ 6.98363 0.0247673 0.0123836 0.999923i $$-0.496058\pi$$
0.0123836 + 0.999923i $$0.496058\pi$$
$$44$$ 480.386 1.64593
$$45$$ −141.477 −0.468670
$$46$$ −3.45278 −0.0110671
$$47$$ 243.878 0.756879 0.378440 0.925626i $$-0.376461\pi$$
0.378440 + 0.925626i $$0.376461\pi$$
$$48$$ −108.223 −0.325430
$$49$$ 0 0
$$50$$ −29.1148 −0.0823491
$$51$$ −2.39498 −0.00657576
$$52$$ −102.639 −0.273719
$$53$$ −282.024 −0.730924 −0.365462 0.930826i $$-0.619089\pi$$
−0.365462 + 0.930826i $$0.619089\pi$$
$$54$$ 28.9889 0.0730535
$$55$$ −360.126 −0.882898
$$56$$ 0 0
$$57$$ −7.40271 −0.0172020
$$58$$ 40.1380 0.0908685
$$59$$ 675.683 1.49096 0.745479 0.666529i $$-0.232221\pi$$
0.745479 + 0.666529i $$0.232221\pi$$
$$60$$ 82.2357 0.176943
$$61$$ −87.6018 −0.183873 −0.0919365 0.995765i $$-0.529306\pi$$
−0.0919365 + 0.995765i $$0.529306\pi$$
$$62$$ −29.4282 −0.0602805
$$63$$ 0 0
$$64$$ −472.223 −0.922311
$$65$$ 76.9439 0.146826
$$66$$ 34.6506 0.0646241
$$67$$ 122.719 0.223768 0.111884 0.993721i $$-0.464311\pi$$
0.111884 + 0.993721i $$0.464311\pi$$
$$68$$ −10.7450 −0.0191621
$$69$$ 18.7761 0.0327592
$$70$$ 0 0
$$71$$ 35.3966 0.0591663 0.0295831 0.999562i $$-0.490582\pi$$
0.0295831 + 0.999562i $$0.490582\pi$$
$$72$$ 122.955 0.201256
$$73$$ 1073.70 1.72147 0.860733 0.509057i $$-0.170006\pi$$
0.860733 + 0.509057i $$0.170006\pi$$
$$74$$ 32.7124 0.0513884
$$75$$ 158.326 0.243758
$$76$$ −33.2121 −0.0501274
$$77$$ 0 0
$$78$$ −7.40339 −0.0107470
$$79$$ 811.612 1.15587 0.577934 0.816084i $$-0.303859\pi$$
0.577934 + 0.816084i $$0.303859\pi$$
$$80$$ 363.990 0.508691
$$81$$ 487.743 0.669058
$$82$$ −76.1477 −0.102550
$$83$$ 1102.66 1.45823 0.729116 0.684391i $$-0.239932\pi$$
0.729116 + 0.684391i $$0.239932\pi$$
$$84$$ 0 0
$$85$$ 8.05509 0.0102788
$$86$$ 2.25999 0.00283373
$$87$$ −218.269 −0.268976
$$88$$ 312.980 0.379134
$$89$$ 1093.27 1.30209 0.651044 0.759040i $$-0.274331\pi$$
0.651044 + 0.759040i $$0.274331\pi$$
$$90$$ −45.7837 −0.0536225
$$91$$ 0 0
$$92$$ 84.2387 0.0954618
$$93$$ 160.030 0.178434
$$94$$ 78.9220 0.0865977
$$95$$ 24.8977 0.0268890
$$96$$ −107.440 −0.114225
$$97$$ −911.865 −0.954494 −0.477247 0.878769i $$-0.658365\pi$$
−0.477247 + 0.878769i $$0.658365\pi$$
$$98$$ 0 0
$$99$$ 1454.38 1.47647
$$100$$ 710.324 0.710324
$$101$$ 1708.38 1.68307 0.841535 0.540202i $$-0.181652\pi$$
0.841535 + 0.540202i $$0.181652\pi$$
$$102$$ −0.775044 −0.000752360 0
$$103$$ 833.431 0.797285 0.398643 0.917106i $$-0.369481\pi$$
0.398643 + 0.917106i $$0.369481\pi$$
$$104$$ −66.8708 −0.0630502
$$105$$ 0 0
$$106$$ −91.2664 −0.0836281
$$107$$ 296.677 0.268045 0.134022 0.990978i $$-0.457211\pi$$
0.134022 + 0.990978i $$0.457211\pi$$
$$108$$ −707.251 −0.630142
$$109$$ −2012.35 −1.76833 −0.884166 0.467173i $$-0.845272\pi$$
−0.884166 + 0.467173i $$0.845272\pi$$
$$110$$ −116.541 −0.101016
$$111$$ −177.889 −0.152113
$$112$$ 0 0
$$113$$ −459.400 −0.382448 −0.191224 0.981546i $$-0.561246\pi$$
−0.191224 + 0.981546i $$0.561246\pi$$
$$114$$ −2.39561 −0.00196815
$$115$$ −63.1503 −0.0512069
$$116$$ −979.259 −0.783810
$$117$$ −310.741 −0.245538
$$118$$ 218.659 0.170587
$$119$$ 0 0
$$120$$ 53.5780 0.0407581
$$121$$ 2371.09 1.78143
$$122$$ −28.3490 −0.0210377
$$123$$ 414.089 0.303554
$$124$$ 717.971 0.519965
$$125$$ −1272.35 −0.910417
$$126$$ 0 0
$$127$$ 2573.78 1.79832 0.899158 0.437624i $$-0.144180\pi$$
0.899158 + 0.437624i $$0.144180\pi$$
$$128$$ −641.238 −0.442797
$$129$$ −12.2898 −0.00838801
$$130$$ 24.9000 0.0167990
$$131$$ −599.610 −0.399910 −0.199955 0.979805i $$-0.564080\pi$$
−0.199955 + 0.979805i $$0.564080\pi$$
$$132$$ −845.382 −0.557432
$$133$$ 0 0
$$134$$ 39.7133 0.0256023
$$135$$ 530.197 0.338016
$$136$$ −7.00055 −0.00441391
$$137$$ −1501.25 −0.936207 −0.468104 0.883674i $$-0.655063\pi$$
−0.468104 + 0.883674i $$0.655063\pi$$
$$138$$ 6.07619 0.00374811
$$139$$ −2652.59 −1.61863 −0.809316 0.587374i $$-0.800162\pi$$
−0.809316 + 0.587374i $$0.800162\pi$$
$$140$$ 0 0
$$141$$ −429.176 −0.256334
$$142$$ 11.4548 0.00676946
$$143$$ −790.982 −0.462554
$$144$$ −1469.98 −0.850685
$$145$$ 734.111 0.420445
$$146$$ 347.462 0.196960
$$147$$ 0 0
$$148$$ −798.095 −0.443264
$$149$$ 1316.52 0.723852 0.361926 0.932207i $$-0.382119\pi$$
0.361926 + 0.932207i $$0.382119\pi$$
$$150$$ 51.2361 0.0278894
$$151$$ 2051.26 1.10549 0.552744 0.833351i $$-0.313581\pi$$
0.552744 + 0.833351i $$0.313581\pi$$
$$152$$ −21.6382 −0.0115467
$$153$$ −32.5307 −0.0171892
$$154$$ 0 0
$$155$$ −538.233 −0.278916
$$156$$ 180.623 0.0927013
$$157$$ −1313.72 −0.667809 −0.333904 0.942607i $$-0.608366\pi$$
−0.333904 + 0.942607i $$0.608366\pi$$
$$158$$ 262.648 0.132248
$$159$$ 496.305 0.247544
$$160$$ 361.356 0.178548
$$161$$ 0 0
$$162$$ 157.840 0.0765498
$$163$$ −1850.44 −0.889190 −0.444595 0.895732i $$-0.646652\pi$$
−0.444595 + 0.895732i $$0.646652\pi$$
$$164$$ 1857.80 0.884572
$$165$$ 633.748 0.299013
$$166$$ 356.836 0.166842
$$167$$ 1090.13 0.505130 0.252565 0.967580i $$-0.418726\pi$$
0.252565 + 0.967580i $$0.418726\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 2.60672 0.00117604
$$171$$ −100.550 −0.0449665
$$172$$ −55.1377 −0.0244431
$$173$$ −2306.38 −1.01359 −0.506795 0.862066i $$-0.669170\pi$$
−0.506795 + 0.862066i $$0.669170\pi$$
$$174$$ −70.6346 −0.0307747
$$175$$ 0 0
$$176$$ −3741.81 −1.60255
$$177$$ −1189.06 −0.504947
$$178$$ 353.794 0.148977
$$179$$ −363.408 −0.151745 −0.0758726 0.997118i $$-0.524174\pi$$
−0.0758726 + 0.997118i $$0.524174\pi$$
$$180$$ 1117.00 0.462535
$$181$$ 3863.47 1.58657 0.793287 0.608848i $$-0.208368\pi$$
0.793287 + 0.608848i $$0.208368\pi$$
$$182$$ 0 0
$$183$$ 154.161 0.0622728
$$184$$ 54.8829 0.0219893
$$185$$ 598.299 0.237772
$$186$$ 51.7877 0.0204154
$$187$$ −82.8061 −0.0323817
$$188$$ −1925.49 −0.746971
$$189$$ 0 0
$$190$$ 8.05721 0.00307648
$$191$$ −1304.09 −0.494035 −0.247017 0.969011i $$-0.579450\pi$$
−0.247017 + 0.969011i $$0.579450\pi$$
$$192$$ 831.016 0.312362
$$193$$ −96.9311 −0.0361516 −0.0180758 0.999837i $$-0.505754\pi$$
−0.0180758 + 0.999837i $$0.505754\pi$$
$$194$$ −295.091 −0.109208
$$195$$ −135.406 −0.0497261
$$196$$ 0 0
$$197$$ −3135.09 −1.13384 −0.566919 0.823774i $$-0.691865\pi$$
−0.566919 + 0.823774i $$0.691865\pi$$
$$198$$ 470.655 0.168929
$$199$$ −3431.34 −1.22232 −0.611159 0.791508i $$-0.709296\pi$$
−0.611159 + 0.791508i $$0.709296\pi$$
$$200$$ 462.788 0.163620
$$201$$ −215.960 −0.0757842
$$202$$ 552.853 0.192567
$$203$$ 0 0
$$204$$ 18.9090 0.00648968
$$205$$ −1392.72 −0.474496
$$206$$ 269.708 0.0912207
$$207$$ 255.034 0.0856334
$$208$$ 799.468 0.266506
$$209$$ −255.948 −0.0847096
$$210$$ 0 0
$$211$$ 4599.00 1.50051 0.750256 0.661147i $$-0.229930\pi$$
0.750256 + 0.661147i $$0.229930\pi$$
$$212$$ 2226.66 0.721356
$$213$$ −62.2908 −0.0200380
$$214$$ 96.0082 0.0306681
$$215$$ 41.3345 0.0131116
$$216$$ −460.786 −0.145151
$$217$$ 0 0
$$218$$ −651.221 −0.202322
$$219$$ −1889.49 −0.583014
$$220$$ 2843.29 0.871340
$$221$$ 17.6922 0.00538510
$$222$$ −57.5671 −0.0174038
$$223$$ 4769.56 1.43226 0.716129 0.697968i $$-0.245912\pi$$
0.716129 + 0.697968i $$0.245912\pi$$
$$224$$ 0 0
$$225$$ 2150.52 0.637192
$$226$$ −148.667 −0.0437575
$$227$$ −5426.06 −1.58652 −0.793260 0.608883i $$-0.791618\pi$$
−0.793260 + 0.608883i $$0.791618\pi$$
$$228$$ 58.4464 0.0169768
$$229$$ 3946.45 1.13881 0.569407 0.822056i $$-0.307173\pi$$
0.569407 + 0.822056i $$0.307173\pi$$
$$230$$ −20.4362 −0.00585880
$$231$$ 0 0
$$232$$ −638.004 −0.180547
$$233$$ −4611.04 −1.29648 −0.648238 0.761437i $$-0.724494\pi$$
−0.648238 + 0.761437i $$0.724494\pi$$
$$234$$ −100.559 −0.0280931
$$235$$ 1443.46 0.400685
$$236$$ −5334.71 −1.47144
$$237$$ −1428.27 −0.391461
$$238$$ 0 0
$$239$$ −1663.03 −0.450094 −0.225047 0.974348i $$-0.572253\pi$$
−0.225047 + 0.974348i $$0.572253\pi$$
$$240$$ −640.547 −0.172280
$$241$$ −2964.29 −0.792311 −0.396156 0.918183i $$-0.629656\pi$$
−0.396156 + 0.918183i $$0.629656\pi$$
$$242$$ 767.313 0.203821
$$243$$ −3276.96 −0.865092
$$244$$ 691.640 0.181466
$$245$$ 0 0
$$246$$ 134.004 0.0347309
$$247$$ 54.6855 0.0140873
$$248$$ 467.770 0.119772
$$249$$ −1940.46 −0.493863
$$250$$ −411.747 −0.104165
$$251$$ 3483.07 0.875893 0.437947 0.899001i $$-0.355706\pi$$
0.437947 + 0.899001i $$0.355706\pi$$
$$252$$ 0 0
$$253$$ 649.184 0.161320
$$254$$ 832.907 0.205753
$$255$$ −14.1753 −0.00348114
$$256$$ 3570.27 0.871649
$$257$$ −4983.36 −1.20955 −0.604774 0.796397i $$-0.706736\pi$$
−0.604774 + 0.796397i $$0.706736\pi$$
$$258$$ −3.97712 −0.000959707 0
$$259$$ 0 0
$$260$$ −607.494 −0.144904
$$261$$ −2964.73 −0.703112
$$262$$ −194.041 −0.0457553
$$263$$ 7413.00 1.73804 0.869021 0.494776i $$-0.164750\pi$$
0.869021 + 0.494776i $$0.164750\pi$$
$$264$$ −550.780 −0.128402
$$265$$ −1669.23 −0.386944
$$266$$ 0 0
$$267$$ −1923.92 −0.440982
$$268$$ −968.898 −0.220839
$$269$$ 7815.61 1.77147 0.885736 0.464189i $$-0.153654\pi$$
0.885736 + 0.464189i $$0.153654\pi$$
$$270$$ 171.578 0.0386738
$$271$$ 940.210 0.210752 0.105376 0.994432i $$-0.466395\pi$$
0.105376 + 0.994432i $$0.466395\pi$$
$$272$$ 83.6945 0.0186571
$$273$$ 0 0
$$274$$ −485.823 −0.107115
$$275$$ 5474.10 1.20037
$$276$$ −148.243 −0.0323303
$$277$$ 1086.00 0.235565 0.117783 0.993039i $$-0.462421\pi$$
0.117783 + 0.993039i $$0.462421\pi$$
$$278$$ −858.411 −0.185194
$$279$$ 2173.67 0.466431
$$280$$ 0 0
$$281$$ 81.9702 0.0174019 0.00870095 0.999962i $$-0.497230\pi$$
0.00870095 + 0.999962i $$0.497230\pi$$
$$282$$ −138.887 −0.0293283
$$283$$ −8459.84 −1.77698 −0.888490 0.458896i $$-0.848245\pi$$
−0.888490 + 0.458896i $$0.848245\pi$$
$$284$$ −279.466 −0.0583918
$$285$$ −43.8149 −0.00910656
$$286$$ −255.972 −0.0529228
$$287$$ 0 0
$$288$$ −1459.35 −0.298586
$$289$$ −4911.15 −0.999623
$$290$$ 237.567 0.0481049
$$291$$ 1604.70 0.323261
$$292$$ −8477.15 −1.69893
$$293$$ 3439.10 0.685714 0.342857 0.939388i $$-0.388605\pi$$
0.342857 + 0.939388i $$0.388605\pi$$
$$294$$ 0 0
$$295$$ 3999.21 0.789298
$$296$$ −519.973 −0.102104
$$297$$ −5450.42 −1.06487
$$298$$ 426.043 0.0828189
$$299$$ −138.704 −0.0268275
$$300$$ −1250.02 −0.240567
$$301$$ 0 0
$$302$$ 663.811 0.126484
$$303$$ −3006.40 −0.570010
$$304$$ 258.694 0.0488063
$$305$$ −518.494 −0.0973406
$$306$$ −10.5273 −0.00196669
$$307$$ 1076.49 0.200126 0.100063 0.994981i $$-0.468096\pi$$
0.100063 + 0.994981i $$0.468096\pi$$
$$308$$ 0 0
$$309$$ −1466.67 −0.270019
$$310$$ −174.179 −0.0319119
$$311$$ 7001.34 1.27656 0.638279 0.769805i $$-0.279647\pi$$
0.638279 + 0.769805i $$0.279647\pi$$
$$312$$ 117.679 0.0213534
$$313$$ 7431.60 1.34204 0.671020 0.741439i $$-0.265856\pi$$
0.671020 + 0.741439i $$0.265856\pi$$
$$314$$ −425.135 −0.0764068
$$315$$ 0 0
$$316$$ −6407.90 −1.14074
$$317$$ −7181.40 −1.27239 −0.636195 0.771528i $$-0.719493\pi$$
−0.636195 + 0.771528i $$0.719493\pi$$
$$318$$ 160.610 0.0283226
$$319$$ −7546.64 −1.32455
$$320$$ −2794.98 −0.488263
$$321$$ −522.090 −0.0907795
$$322$$ 0 0
$$323$$ 5.72490 0.000986198 0
$$324$$ −3850.87 −0.660300
$$325$$ −1169.59 −0.199622
$$326$$ −598.826 −0.101736
$$327$$ 3541.32 0.598886
$$328$$ 1210.39 0.203758
$$329$$ 0 0
$$330$$ 205.089 0.0342114
$$331$$ −3742.93 −0.621540 −0.310770 0.950485i $$-0.600587\pi$$
−0.310770 + 0.950485i $$0.600587\pi$$
$$332$$ −8705.84 −1.43914
$$333$$ −2416.25 −0.397627
$$334$$ 352.779 0.0577940
$$335$$ 726.343 0.118461
$$336$$ 0 0
$$337$$ 10859.5 1.75535 0.877674 0.479258i $$-0.159094\pi$$
0.877674 + 0.479258i $$0.159094\pi$$
$$338$$ 54.6905 0.00880109
$$339$$ 808.449 0.129525
$$340$$ −63.5971 −0.0101442
$$341$$ 5533.03 0.878681
$$342$$ −32.5393 −0.00514481
$$343$$ 0 0
$$344$$ −35.9231 −0.00563037
$$345$$ 111.132 0.0173424
$$346$$ −746.374 −0.115969
$$347$$ 267.989 0.0414594 0.0207297 0.999785i $$-0.493401\pi$$
0.0207297 + 0.999785i $$0.493401\pi$$
$$348$$ 1723.30 0.265455
$$349$$ 5082.15 0.779488 0.389744 0.920923i $$-0.372563\pi$$
0.389744 + 0.920923i $$0.372563\pi$$
$$350$$ 0 0
$$351$$ 1164.53 0.177088
$$352$$ −3714.73 −0.562489
$$353$$ −9356.72 −1.41079 −0.705394 0.708815i $$-0.749230\pi$$
−0.705394 + 0.708815i $$0.749230\pi$$
$$354$$ −384.796 −0.0577731
$$355$$ 209.504 0.0313221
$$356$$ −8631.63 −1.28504
$$357$$ 0 0
$$358$$ −117.603 −0.0173618
$$359$$ −499.977 −0.0735036 −0.0367518 0.999324i $$-0.511701\pi$$
−0.0367518 + 0.999324i $$0.511701\pi$$
$$360$$ 727.744 0.106543
$$361$$ −6841.30 −0.997420
$$362$$ 1250.27 0.181527
$$363$$ −4172.63 −0.603323
$$364$$ 0 0
$$365$$ 6354.97 0.911327
$$366$$ 49.8884 0.00712489
$$367$$ 2754.73 0.391814 0.195907 0.980622i $$-0.437235\pi$$
0.195907 + 0.980622i $$0.437235\pi$$
$$368$$ −656.149 −0.0929460
$$369$$ 5624.53 0.793500
$$370$$ 193.617 0.0272045
$$371$$ 0 0
$$372$$ −1263.48 −0.176098
$$373$$ −9414.45 −1.30687 −0.653434 0.756984i $$-0.726672\pi$$
−0.653434 + 0.756984i $$0.726672\pi$$
$$374$$ −26.7971 −0.00370493
$$375$$ 2239.07 0.308333
$$376$$ −1254.49 −0.172062
$$377$$ 1612.40 0.220273
$$378$$ 0 0
$$379$$ −11295.5 −1.53090 −0.765451 0.643494i $$-0.777484\pi$$
−0.765451 + 0.643494i $$0.777484\pi$$
$$380$$ −196.574 −0.0265370
$$381$$ −4529.33 −0.609041
$$382$$ −422.019 −0.0565246
$$383$$ −265.629 −0.0354386 −0.0177193 0.999843i $$-0.505641\pi$$
−0.0177193 + 0.999843i $$0.505641\pi$$
$$384$$ 1128.45 0.149963
$$385$$ 0 0
$$386$$ −31.3681 −0.00413625
$$387$$ −166.931 −0.0219265
$$388$$ 7199.43 0.941999
$$389$$ 11601.0 1.51207 0.756037 0.654529i $$-0.227133\pi$$
0.756037 + 0.654529i $$0.227133\pi$$
$$390$$ −43.8189 −0.00568937
$$391$$ −14.5206 −0.00187810
$$392$$ 0 0
$$393$$ 1055.19 0.135438
$$394$$ −1014.55 −0.129727
$$395$$ 4803.74 0.611905
$$396$$ −11482.7 −1.45714
$$397$$ −6664.32 −0.842501 −0.421250 0.906944i $$-0.638409\pi$$
−0.421250 + 0.906944i $$0.638409\pi$$
$$398$$ −1110.42 −0.139851
$$399$$ 0 0
$$400$$ −5532.83 −0.691603
$$401$$ 7874.09 0.980582 0.490291 0.871559i $$-0.336891\pi$$
0.490291 + 0.871559i $$0.336891\pi$$
$$402$$ −69.8872 −0.00867079
$$403$$ −1182.18 −0.146125
$$404$$ −13488.1 −1.66104
$$405$$ 2886.84 0.354193
$$406$$ 0 0
$$407$$ −6150.51 −0.749065
$$408$$ 12.3195 0.00149487
$$409$$ 2311.74 0.279483 0.139741 0.990188i $$-0.455373\pi$$
0.139741 + 0.990188i $$0.455373\pi$$
$$410$$ −450.700 −0.0542890
$$411$$ 2641.89 0.317068
$$412$$ −6580.16 −0.786848
$$413$$ 0 0
$$414$$ 82.5323 0.00979768
$$415$$ 6526.41 0.771973
$$416$$ 793.684 0.0935422
$$417$$ 4668.02 0.548186
$$418$$ −82.8280 −0.00969198
$$419$$ 3945.48 0.460023 0.230011 0.973188i $$-0.426124\pi$$
0.230011 + 0.973188i $$0.426124\pi$$
$$420$$ 0 0
$$421$$ −699.284 −0.0809526 −0.0404763 0.999180i $$-0.512888\pi$$
−0.0404763 + 0.999180i $$0.512888\pi$$
$$422$$ 1488.29 0.171680
$$423$$ −5829.46 −0.670066
$$424$$ 1450.70 0.166161
$$425$$ −122.441 −0.0139748
$$426$$ −20.1581 −0.00229263
$$427$$ 0 0
$$428$$ −2342.34 −0.264536
$$429$$ 1391.97 0.156655
$$430$$ 13.3763 0.00150015
$$431$$ 14778.0 1.65158 0.825789 0.563980i $$-0.190730\pi$$
0.825789 + 0.563980i $$0.190730\pi$$
$$432$$ 5508.89 0.613534
$$433$$ −4217.17 −0.468046 −0.234023 0.972231i $$-0.575189\pi$$
−0.234023 + 0.972231i $$0.575189\pi$$
$$434$$ 0 0
$$435$$ −1291.88 −0.142393
$$436$$ 15888.1 1.74518
$$437$$ −44.8821 −0.00491305
$$438$$ −611.462 −0.0667050
$$439$$ −6372.34 −0.692791 −0.346395 0.938089i $$-0.612594\pi$$
−0.346395 + 0.938089i $$0.612594\pi$$
$$440$$ 1852.45 0.200710
$$441$$ 0 0
$$442$$ 5.72542 0.000616132 0
$$443$$ 7824.17 0.839137 0.419568 0.907724i $$-0.362181\pi$$
0.419568 + 0.907724i $$0.362181\pi$$
$$444$$ 1404.48 0.150121
$$445$$ 6470.78 0.689313
$$446$$ 1543.49 0.163871
$$447$$ −2316.81 −0.245149
$$448$$ 0 0
$$449$$ 3132.31 0.329226 0.164613 0.986358i $$-0.447362\pi$$
0.164613 + 0.986358i $$0.447362\pi$$
$$450$$ 695.935 0.0729038
$$451$$ 14317.1 1.49483
$$452$$ 3627.09 0.377442
$$453$$ −3609.79 −0.374399
$$454$$ −1755.94 −0.181520
$$455$$ 0 0
$$456$$ 38.0788 0.00391054
$$457$$ 2253.14 0.230629 0.115314 0.993329i $$-0.463212\pi$$
0.115314 + 0.993329i $$0.463212\pi$$
$$458$$ 1277.12 0.130297
$$459$$ 121.912 0.0123973
$$460$$ 498.589 0.0505366
$$461$$ 9164.98 0.925934 0.462967 0.886375i $$-0.346785\pi$$
0.462967 + 0.886375i $$0.346785\pi$$
$$462$$ 0 0
$$463$$ 7086.00 0.711263 0.355631 0.934626i $$-0.384266\pi$$
0.355631 + 0.934626i $$0.384266\pi$$
$$464$$ 7627.61 0.763153
$$465$$ 947.180 0.0944611
$$466$$ −1492.19 −0.148335
$$467$$ −11174.0 −1.10721 −0.553607 0.832778i $$-0.686749\pi$$
−0.553607 + 0.832778i $$0.686749\pi$$
$$468$$ 2453.38 0.242324
$$469$$ 0 0
$$470$$ 467.121 0.0458440
$$471$$ 2311.87 0.226169
$$472$$ −3475.65 −0.338940
$$473$$ −424.918 −0.0413060
$$474$$ −462.206 −0.0447887
$$475$$ −378.458 −0.0365576
$$476$$ 0 0
$$477$$ 6741.25 0.647088
$$478$$ −538.176 −0.0514971
$$479$$ −8240.34 −0.786035 −0.393018 0.919531i $$-0.628569\pi$$
−0.393018 + 0.919531i $$0.628569\pi$$
$$480$$ −635.913 −0.0604694
$$481$$ 1314.11 0.124570
$$482$$ −959.282 −0.0906517
$$483$$ 0 0
$$484$$ −18720.4 −1.75811
$$485$$ −5397.12 −0.505300
$$486$$ −1060.47 −0.0989788
$$487$$ −9967.60 −0.927464 −0.463732 0.885975i $$-0.653490\pi$$
−0.463732 + 0.885975i $$0.653490\pi$$
$$488$$ 450.615 0.0418000
$$489$$ 3256.40 0.301144
$$490$$ 0 0
$$491$$ 11339.8 1.04228 0.521138 0.853473i $$-0.325508\pi$$
0.521138 + 0.853473i $$0.325508\pi$$
$$492$$ −3269.35 −0.299581
$$493$$ 168.799 0.0154205
$$494$$ 17.6969 0.00161178
$$495$$ 8608.14 0.781630
$$496$$ −5592.39 −0.506261
$$497$$ 0 0
$$498$$ −627.958 −0.0565050
$$499$$ 8011.01 0.718682 0.359341 0.933206i $$-0.383002\pi$$
0.359341 + 0.933206i $$0.383002\pi$$
$$500$$ 10045.5 0.898499
$$501$$ −1918.40 −0.171074
$$502$$ 1127.16 0.100215
$$503$$ 17925.8 1.58901 0.794503 0.607260i $$-0.207731\pi$$
0.794503 + 0.607260i $$0.207731\pi$$
$$504$$ 0 0
$$505$$ 10111.5 0.891001
$$506$$ 210.084 0.0184572
$$507$$ −297.405 −0.0260517
$$508$$ −20320.7 −1.77477
$$509$$ 13744.2 1.19686 0.598429 0.801176i $$-0.295792\pi$$
0.598429 + 0.801176i $$0.295792\pi$$
$$510$$ −4.58730 −0.000398292 0
$$511$$ 0 0
$$512$$ 6285.29 0.542526
$$513$$ 376.821 0.0324309
$$514$$ −1612.68 −0.138389
$$515$$ 4932.88 0.422075
$$516$$ 97.0310 0.00827820
$$517$$ −14838.7 −1.26230
$$518$$ 0 0
$$519$$ 4058.77 0.343276
$$520$$ −395.792 −0.0333782
$$521$$ −2637.79 −0.221811 −0.110906 0.993831i $$-0.535375\pi$$
−0.110906 + 0.993831i $$0.535375\pi$$
$$522$$ −959.423 −0.0804460
$$523$$ 16059.6 1.34271 0.671353 0.741138i $$-0.265713\pi$$
0.671353 + 0.741138i $$0.265713\pi$$
$$524$$ 4734.08 0.394674
$$525$$ 0 0
$$526$$ 2398.94 0.198857
$$527$$ −123.759 −0.0102297
$$528$$ 6584.81 0.542741
$$529$$ −12053.2 −0.990644
$$530$$ −540.184 −0.0442719
$$531$$ −16150.9 −1.31995
$$532$$ 0 0
$$533$$ −3058.97 −0.248590
$$534$$ −622.605 −0.0504546
$$535$$ 1755.96 0.141900
$$536$$ −631.253 −0.0508694
$$537$$ 639.524 0.0513920
$$538$$ 2529.23 0.202682
$$539$$ 0 0
$$540$$ −4186.05 −0.333591
$$541$$ 10315.9 0.819804 0.409902 0.912130i $$-0.365563\pi$$
0.409902 + 0.912130i $$0.365563\pi$$
$$542$$ 304.263 0.0241130
$$543$$ −6798.92 −0.537329
$$544$$ 83.0890 0.00654855
$$545$$ −11910.6 −0.936138
$$546$$ 0 0
$$547$$ −5327.06 −0.416396 −0.208198 0.978087i $$-0.566760\pi$$
−0.208198 + 0.978087i $$0.566760\pi$$
$$548$$ 11852.8 0.923952
$$549$$ 2093.96 0.162783
$$550$$ 1771.49 0.137339
$$551$$ 521.746 0.0403396
$$552$$ −96.5827 −0.00744716
$$553$$ 0 0
$$554$$ 351.444 0.0269520
$$555$$ −1052.88 −0.0805270
$$556$$ 20942.9 1.59744
$$557$$ 11816.4 0.898884 0.449442 0.893310i $$-0.351623\pi$$
0.449442 + 0.893310i $$0.351623\pi$$
$$558$$ 703.427 0.0533664
$$559$$ 90.7872 0.00686921
$$560$$ 0 0
$$561$$ 145.722 0.0109668
$$562$$ 26.5266 0.00199102
$$563$$ 20627.2 1.54411 0.772054 0.635557i $$-0.219230\pi$$
0.772054 + 0.635557i $$0.219230\pi$$
$$564$$ 3388.46 0.252979
$$565$$ −2719.08 −0.202465
$$566$$ −2737.71 −0.203312
$$567$$ 0 0
$$568$$ −182.077 −0.0134503
$$569$$ −13529.3 −0.996798 −0.498399 0.866948i $$-0.666078\pi$$
−0.498399 + 0.866948i $$0.666078\pi$$
$$570$$ −14.1790 −0.00104192
$$571$$ −13230.8 −0.969690 −0.484845 0.874600i $$-0.661124\pi$$
−0.484845 + 0.874600i $$0.661124\pi$$
$$572$$ 6245.02 0.456499
$$573$$ 2294.93 0.167316
$$574$$ 0 0
$$575$$ 959.917 0.0696196
$$576$$ 11287.6 0.816523
$$577$$ −21279.3 −1.53530 −0.767650 0.640870i $$-0.778574\pi$$
−0.767650 + 0.640870i $$0.778574\pi$$
$$578$$ −1589.31 −0.114371
$$579$$ 170.579 0.0122435
$$580$$ −5796.00 −0.414941
$$581$$ 0 0
$$582$$ 519.299 0.0369857
$$583$$ 17159.7 1.21901
$$584$$ −5523.01 −0.391342
$$585$$ −1839.20 −0.129986
$$586$$ 1112.93 0.0784555
$$587$$ 13737.6 0.965945 0.482972 0.875636i $$-0.339557\pi$$
0.482972 + 0.875636i $$0.339557\pi$$
$$588$$ 0 0
$$589$$ −382.532 −0.0267606
$$590$$ 1294.19 0.0903070
$$591$$ 5517.12 0.384000
$$592$$ 6216.49 0.431582
$$593$$ 22012.2 1.52434 0.762169 0.647379i $$-0.224135\pi$$
0.762169 + 0.647379i $$0.224135\pi$$
$$594$$ −1763.82 −0.121836
$$595$$ 0 0
$$596$$ −10394.3 −0.714376
$$597$$ 6038.46 0.413966
$$598$$ −44.8862 −0.00306945
$$599$$ 3333.73 0.227400 0.113700 0.993515i $$-0.463730\pi$$
0.113700 + 0.993515i $$0.463730\pi$$
$$600$$ −814.412 −0.0554137
$$601$$ −17140.5 −1.16336 −0.581678 0.813419i $$-0.697604\pi$$
−0.581678 + 0.813419i $$0.697604\pi$$
$$602$$ 0 0
$$603$$ −2933.36 −0.198102
$$604$$ −16195.2 −1.09102
$$605$$ 14033.9 0.943074
$$606$$ −972.907 −0.0652173
$$607$$ 2365.32 0.158164 0.0790818 0.996868i $$-0.474801\pi$$
0.0790818 + 0.996868i $$0.474801\pi$$
$$608$$ 256.822 0.0171308
$$609$$ 0 0
$$610$$ −167.791 −0.0111372
$$611$$ 3170.42 0.209921
$$612$$ 256.839 0.0169642
$$613$$ −165.695 −0.0109174 −0.00545869 0.999985i $$-0.501738\pi$$
−0.00545869 + 0.999985i $$0.501738\pi$$
$$614$$ 348.367 0.0228973
$$615$$ 2450.90 0.160699
$$616$$ 0 0
$$617$$ 5993.71 0.391082 0.195541 0.980696i $$-0.437354\pi$$
0.195541 + 0.980696i $$0.437354\pi$$
$$618$$ −474.632 −0.0308940
$$619$$ 7157.52 0.464758 0.232379 0.972625i $$-0.425349\pi$$
0.232379 + 0.972625i $$0.425349\pi$$
$$620$$ 4249.50 0.275264
$$621$$ −955.764 −0.0617609
$$622$$ 2265.72 0.146056
$$623$$ 0 0
$$624$$ −1406.90 −0.0902582
$$625$$ 3715.31 0.237780
$$626$$ 2404.96 0.153549
$$627$$ 450.416 0.0286888
$$628$$ 10372.2 0.659067
$$629$$ 137.571 0.00872069
$$630$$ 0 0
$$631$$ 25765.6 1.62554 0.812769 0.582586i $$-0.197959\pi$$
0.812769 + 0.582586i $$0.197959\pi$$
$$632$$ −4174.86 −0.262764
$$633$$ −8093.30 −0.508183
$$634$$ −2323.99 −0.145580
$$635$$ 15233.6 0.952011
$$636$$ −3918.46 −0.244304
$$637$$ 0 0
$$638$$ −2442.19 −0.151547
$$639$$ −846.090 −0.0523800
$$640$$ −3795.34 −0.234412
$$641$$ −20628.0 −1.27107 −0.635536 0.772072i $$-0.719221\pi$$
−0.635536 + 0.772072i $$0.719221\pi$$
$$642$$ −168.955 −0.0103865
$$643$$ −25301.9 −1.55180 −0.775901 0.630854i $$-0.782704\pi$$
−0.775901 + 0.630854i $$0.782704\pi$$
$$644$$ 0 0
$$645$$ −72.7402 −0.00444053
$$646$$ 1.85265 0.000112835 0
$$647$$ 14852.8 0.902508 0.451254 0.892396i $$-0.350977\pi$$
0.451254 + 0.892396i $$0.350977\pi$$
$$648$$ −2508.91 −0.152097
$$649$$ −41111.8 −2.48656
$$650$$ −378.493 −0.0228395
$$651$$ 0 0
$$652$$ 14609.8 0.877550
$$653$$ 25682.0 1.53907 0.769537 0.638602i $$-0.220487\pi$$
0.769537 + 0.638602i $$0.220487\pi$$
$$654$$ 1146.02 0.0685210
$$655$$ −3548.95 −0.211708
$$656$$ −14470.7 −0.861260
$$657$$ −25664.8 −1.52402
$$658$$ 0 0
$$659$$ −7290.57 −0.430956 −0.215478 0.976509i $$-0.569131\pi$$
−0.215478 + 0.976509i $$0.569131\pi$$
$$660$$ −5003.61 −0.295099
$$661$$ −16930.0 −0.996222 −0.498111 0.867113i $$-0.665973\pi$$
−0.498111 + 0.867113i $$0.665973\pi$$
$$662$$ −1211.26 −0.0711131
$$663$$ −31.1347 −0.00182379
$$664$$ −5672.00 −0.331501
$$665$$ 0 0
$$666$$ −781.929 −0.0454942
$$667$$ −1323.35 −0.0768221
$$668$$ −8606.87 −0.498517
$$669$$ −8393.45 −0.485067
$$670$$ 235.053 0.0135536
$$671$$ 5330.11 0.306657
$$672$$ 0 0
$$673$$ 16730.2 0.958249 0.479125 0.877747i $$-0.340954\pi$$
0.479125 + 0.877747i $$0.340954\pi$$
$$674$$ 3514.25 0.200837
$$675$$ −8059.27 −0.459558
$$676$$ −1334.30 −0.0759161
$$677$$ 5093.48 0.289156 0.144578 0.989493i $$-0.453818\pi$$
0.144578 + 0.989493i $$0.453818\pi$$
$$678$$ 261.624 0.0148195
$$679$$ 0 0
$$680$$ −41.4346 −0.00233668
$$681$$ 9548.75 0.537311
$$682$$ 1790.56 0.100534
$$683$$ 25619.5 1.43529 0.717645 0.696409i $$-0.245220\pi$$
0.717645 + 0.696409i $$0.245220\pi$$
$$684$$ 793.872 0.0443779
$$685$$ −8885.54 −0.495619
$$686$$ 0 0
$$687$$ −6944.94 −0.385685
$$688$$ 429.476 0.0237989
$$689$$ −3666.31 −0.202722
$$690$$ 35.9635 0.00198422
$$691$$ 31693.2 1.74482 0.872408 0.488778i $$-0.162557\pi$$
0.872408 + 0.488778i $$0.162557\pi$$
$$692$$ 18209.5 1.00032
$$693$$ 0 0
$$694$$ 86.7245 0.00474354
$$695$$ −15700.1 −0.856888
$$696$$ 1122.76 0.0611465
$$697$$ −320.237 −0.0174029
$$698$$ 1644.65 0.0891845
$$699$$ 8114.48 0.439081
$$700$$ 0 0
$$701$$ 2661.14 0.143381 0.0716905 0.997427i $$-0.477161\pi$$
0.0716905 + 0.997427i $$0.477161\pi$$
$$702$$ 376.855 0.0202614
$$703$$ 425.223 0.0228130
$$704$$ 28732.3 1.53820
$$705$$ −2540.19 −0.135701
$$706$$ −3027.95 −0.161414
$$707$$ 0 0
$$708$$ 9387.99 0.498337
$$709$$ −16745.3 −0.887000 −0.443500 0.896274i $$-0.646263\pi$$
−0.443500 + 0.896274i $$0.646263\pi$$
$$710$$ 67.7981 0.00358369
$$711$$ −19400.1 −1.02329
$$712$$ −5623.65 −0.296005
$$713$$ 970.250 0.0509623
$$714$$ 0 0
$$715$$ −4681.64 −0.244872
$$716$$ 2869.21 0.149759
$$717$$ 2926.59 0.152434
$$718$$ −161.799 −0.00840986
$$719$$ −7621.40 −0.395313 −0.197657 0.980271i $$-0.563333\pi$$
−0.197657 + 0.980271i $$0.563333\pi$$
$$720$$ −8700.49 −0.450344
$$721$$ 0 0
$$722$$ −2213.93 −0.114119
$$723$$ 5216.55 0.268334
$$724$$ −30503.2 −1.56580
$$725$$ −11158.9 −0.571627
$$726$$ −1350.31 −0.0690287
$$727$$ 33363.8 1.70206 0.851028 0.525120i $$-0.175979\pi$$
0.851028 + 0.525120i $$0.175979\pi$$
$$728$$ 0 0
$$729$$ −7402.29 −0.376075
$$730$$ 2056.55 0.104269
$$731$$ 9.50431 0.000480888 0
$$732$$ −1217.14 −0.0614576
$$733$$ 10628.8 0.535583 0.267791 0.963477i $$-0.413706\pi$$
0.267791 + 0.963477i $$0.413706\pi$$
$$734$$ 891.465 0.0448291
$$735$$ 0 0
$$736$$ −651.401 −0.0326236
$$737$$ −7466.79 −0.373192
$$738$$ 1820.17 0.0907877
$$739$$ 13321.7 0.663120 0.331560 0.943434i $$-0.392425\pi$$
0.331560 + 0.943434i $$0.392425\pi$$
$$740$$ −4723.74 −0.234660
$$741$$ −96.2352 −0.00477097
$$742$$ 0 0
$$743$$ −1456.36 −0.0719092 −0.0359546 0.999353i $$-0.511447\pi$$
−0.0359546 + 0.999353i $$0.511447\pi$$
$$744$$ −823.179 −0.0405635
$$745$$ 7792.20 0.383200
$$746$$ −3046.63 −0.149524
$$747$$ −26357.1 −1.29097
$$748$$ 653.777 0.0319578
$$749$$ 0 0
$$750$$ 724.590 0.0352777
$$751$$ −3929.23 −0.190918 −0.0954591 0.995433i $$-0.530432\pi$$
−0.0954591 + 0.995433i $$0.530432\pi$$
$$752$$ 14997.9 0.727285
$$753$$ −6129.48 −0.296641
$$754$$ 521.794 0.0252024
$$755$$ 12140.9 0.585235
$$756$$ 0 0
$$757$$ −27434.8 −1.31722 −0.658610 0.752484i $$-0.728855\pi$$
−0.658610 + 0.752484i $$0.728855\pi$$
$$758$$ −3655.37 −0.175157
$$759$$ −1142.43 −0.0546345
$$760$$ −128.072 −0.00611269
$$761$$ 19994.8 0.952445 0.476222 0.879325i $$-0.342006\pi$$
0.476222 + 0.879325i $$0.342006\pi$$
$$762$$ −1465.75 −0.0696829
$$763$$ 0 0
$$764$$ 10296.1 0.487567
$$765$$ −192.542 −0.00909982
$$766$$ −85.9607 −0.00405468
$$767$$ 8783.88 0.413517
$$768$$ −6282.95 −0.295204
$$769$$ 7948.30 0.372722 0.186361 0.982481i $$-0.440331\pi$$
0.186361 + 0.982481i $$0.440331\pi$$
$$770$$ 0 0
$$771$$ 8769.70 0.409641
$$772$$ 765.297 0.0356783
$$773$$ 21350.2 0.993418 0.496709 0.867917i $$-0.334542\pi$$
0.496709 + 0.867917i $$0.334542\pi$$
$$774$$ −54.0208 −0.00250870
$$775$$ 8181.42 0.379207
$$776$$ 4690.55 0.216986
$$777$$ 0 0
$$778$$ 3754.24 0.173003
$$779$$ −989.830 −0.0455255
$$780$$ 1069.06 0.0490752
$$781$$ −2153.70 −0.0986754
$$782$$ −4.69903 −0.000214881 0
$$783$$ 11110.6 0.507101
$$784$$ 0 0
$$785$$ −7775.58 −0.353532
$$786$$ 341.473 0.0154961
$$787$$ 20880.3 0.945744 0.472872 0.881131i $$-0.343217\pi$$
0.472872 + 0.881131i $$0.343217\pi$$
$$788$$ 24752.4 1.11900
$$789$$ −13045.4 −0.588627
$$790$$ 1554.55 0.0700106
$$791$$ 0 0
$$792$$ −7481.20 −0.335647
$$793$$ −1138.82 −0.0509972
$$794$$ −2156.66 −0.0963941
$$795$$ 2937.51 0.131047
$$796$$ 27091.4 1.20632
$$797$$ 40432.4 1.79697 0.898487 0.439001i $$-0.144668\pi$$
0.898487 + 0.439001i $$0.144668\pi$$
$$798$$ 0 0
$$799$$ 331.904 0.0146958
$$800$$ −5492.80 −0.242750
$$801$$ −26132.4 −1.15274
$$802$$ 2548.15 0.112192
$$803$$ −65329.0 −2.87100
$$804$$ 1705.06 0.0747921
$$805$$ 0 0
$$806$$ −382.567 −0.0167188
$$807$$ −13753.9 −0.599950
$$808$$ −8787.74 −0.382614
$$809$$ 19268.3 0.837376 0.418688 0.908130i $$-0.362490\pi$$
0.418688 + 0.908130i $$0.362490\pi$$
$$810$$ 934.216 0.0405247
$$811$$ 37223.2 1.61169 0.805846 0.592125i $$-0.201711\pi$$
0.805846 + 0.592125i $$0.201711\pi$$
$$812$$ 0 0
$$813$$ −1654.58 −0.0713758
$$814$$ −1990.38 −0.0857037
$$815$$ −10952.3 −0.470729
$$816$$ −147.285 −0.00631864
$$817$$ 29.3772 0.00125799
$$818$$ 748.109 0.0319768
$$819$$ 0 0
$$820$$ 10995.9 0.468284
$$821$$ −883.967 −0.0375769 −0.0187885 0.999823i $$-0.505981\pi$$
−0.0187885 + 0.999823i $$0.505981\pi$$
$$822$$ 854.949 0.0362771
$$823$$ 41410.5 1.75392 0.876961 0.480561i $$-0.159567\pi$$
0.876961 + 0.480561i $$0.159567\pi$$
$$824$$ −4287.09 −0.181247
$$825$$ −9633.29 −0.406531
$$826$$ 0 0
$$827$$ −37881.9 −1.59285 −0.796423 0.604739i $$-0.793277\pi$$
−0.796423 + 0.604739i $$0.793277\pi$$
$$828$$ −2013.57 −0.0845124
$$829$$ 4896.97 0.205161 0.102581 0.994725i $$-0.467290\pi$$
0.102581 + 0.994725i $$0.467290\pi$$
$$830$$ 2112.03 0.0883247
$$831$$ −1911.14 −0.0797795
$$832$$ −6138.90 −0.255803
$$833$$ 0 0
$$834$$ 1510.63 0.0627203
$$835$$ 6452.21 0.267411
$$836$$ 2020.78 0.0836007
$$837$$ −8146.03 −0.336401
$$838$$ 1276.81 0.0526331
$$839$$ −93.0019 −0.00382692 −0.00191346 0.999998i $$-0.500609\pi$$
−0.00191346 + 0.999998i $$0.500609\pi$$
$$840$$ 0 0
$$841$$ −9005.30 −0.369236
$$842$$ −226.297 −0.00926212
$$843$$ −144.251 −0.00589355
$$844$$ −36310.4 −1.48087
$$845$$ 1000.27 0.0407223
$$846$$ −1886.48 −0.0766650
$$847$$ 0 0
$$848$$ −17343.8 −0.702345
$$849$$ 14887.6 0.601815
$$850$$ −39.6235 −0.00159891
$$851$$ −1078.53 −0.0434448
$$852$$ 491.803 0.0197757
$$853$$ 8325.23 0.334174 0.167087 0.985942i $$-0.446564\pi$$
0.167087 + 0.985942i $$0.446564\pi$$
$$854$$ 0 0
$$855$$ −595.133 −0.0238048
$$856$$ −1526.08 −0.0609348
$$857$$ 3891.72 0.155121 0.0775606 0.996988i $$-0.475287\pi$$
0.0775606 + 0.996988i $$0.475287\pi$$
$$858$$ 450.458 0.0179235
$$859$$ −35346.9 −1.40398 −0.701991 0.712186i $$-0.747705\pi$$
−0.701991 + 0.712186i $$0.747705\pi$$
$$860$$ −326.347 −0.0129399
$$861$$ 0 0
$$862$$ 4782.33 0.188964
$$863$$ −21075.1 −0.831290 −0.415645 0.909527i $$-0.636444\pi$$
−0.415645 + 0.909527i $$0.636444\pi$$
$$864$$ 5469.04 0.215348
$$865$$ −13650.9 −0.536585
$$866$$ −1364.73 −0.0535512
$$867$$ 8642.61 0.338545
$$868$$ 0 0
$$869$$ −49382.4 −1.92771
$$870$$ −418.070 −0.0162918
$$871$$ 1595.34 0.0620621
$$872$$ 10351.3 0.401996
$$873$$ 21796.4 0.845014
$$874$$ −14.5244 −0.000562122 0
$$875$$ 0 0
$$876$$ 14918.1 0.575382
$$877$$ −47944.7 −1.84604 −0.923021 0.384751i $$-0.874287\pi$$
−0.923021 + 0.384751i $$0.874287\pi$$
$$878$$ −2062.17 −0.0792651
$$879$$ −6052.11 −0.232233
$$880$$ −22146.9 −0.848376
$$881$$ −37133.6 −1.42005 −0.710024 0.704178i $$-0.751316\pi$$
−0.710024 + 0.704178i $$0.751316\pi$$
$$882$$ 0 0
$$883$$ 1686.40 0.0642715 0.0321358 0.999484i $$-0.489769\pi$$
0.0321358 + 0.999484i $$0.489769\pi$$
$$884$$ −139.685 −0.00531461
$$885$$ −7037.79 −0.267314
$$886$$ 2532.00 0.0960092
$$887$$ −11389.6 −0.431146 −0.215573 0.976488i $$-0.569162\pi$$
−0.215573 + 0.976488i $$0.569162\pi$$
$$888$$ 915.046 0.0345799
$$889$$ 0 0
$$890$$ 2094.02 0.0788672
$$891$$ −29676.7 −1.11583
$$892$$ −37657.0 −1.41351
$$893$$ 1025.89 0.0384437
$$894$$ −749.749 −0.0280485
$$895$$ −2150.93 −0.0803325
$$896$$ 0 0
$$897$$ 244.090 0.00908576
$$898$$ 1013.65 0.0376682
$$899$$ −11279.0 −0.418437
$$900$$ −16979.0 −0.628850
$$901$$ −383.818 −0.0141918
$$902$$ 4633.19 0.171029
$$903$$ 0 0
$$904$$ 2363.11 0.0869423
$$905$$ 22867.0 0.839917
$$906$$ −1168.17 −0.0428366
$$907$$ −29663.9 −1.08597 −0.542985 0.839743i $$-0.682706\pi$$
−0.542985 + 0.839743i $$0.682706\pi$$
$$908$$ 42840.2 1.56575
$$909$$ −40835.6 −1.49002
$$910$$ 0 0
$$911$$ 16210.3 0.589541 0.294770 0.955568i $$-0.404757\pi$$
0.294770 + 0.955568i $$0.404757\pi$$
$$912$$ −455.249 −0.0165294
$$913$$ −67091.4 −2.43198
$$914$$ 729.143 0.0263872
$$915$$ 912.444 0.0329666
$$916$$ −31158.3 −1.12391
$$917$$ 0 0
$$918$$ 39.4521 0.00141843
$$919$$ −13025.5 −0.467542 −0.233771 0.972292i $$-0.575107\pi$$
−0.233771 + 0.972292i $$0.575107\pi$$
$$920$$ 324.839 0.0116409
$$921$$ −1894.41 −0.0677773
$$922$$ 2965.90 0.105940
$$923$$ 460.156 0.0164098
$$924$$ 0 0
$$925$$ −9094.46 −0.323269
$$926$$ 2293.12 0.0813785
$$927$$ −19921.6 −0.705837
$$928$$ 7572.42 0.267863
$$929$$ −46834.2 −1.65401 −0.827007 0.562192i $$-0.809958\pi$$
−0.827007 + 0.562192i $$0.809958\pi$$
$$930$$ 306.519 0.0108077
$$931$$ 0 0
$$932$$ 36405.4 1.27951
$$933$$ −12320.9 −0.432335
$$934$$ −3616.03 −0.126681
$$935$$ −490.110 −0.0171426
$$936$$ 1598.42 0.0558184
$$937$$ −24171.1 −0.842727 −0.421363 0.906892i $$-0.638448\pi$$
−0.421363 + 0.906892i $$0.638448\pi$$
$$938$$ 0 0
$$939$$ −13078.1 −0.454513
$$940$$ −11396.5 −0.395439
$$941$$ 32410.4 1.12279 0.561397 0.827547i $$-0.310264\pi$$
0.561397 + 0.827547i $$0.310264\pi$$
$$942$$ 748.150 0.0258769
$$943$$ 2510.59 0.0866979
$$944$$ 41552.9 1.43266
$$945$$ 0 0
$$946$$ −137.509 −0.00472599
$$947$$ −36285.0 −1.24509 −0.622547 0.782582i $$-0.713902\pi$$
−0.622547 + 0.782582i $$0.713902\pi$$
$$948$$ 11276.6 0.386336
$$949$$ 13958.1 0.477449
$$950$$ −122.474 −0.00418271
$$951$$ 12637.8 0.430924
$$952$$ 0 0
$$953$$ 31851.1 1.08264 0.541321 0.840816i $$-0.317924\pi$$
0.541321 + 0.840816i $$0.317924\pi$$
$$954$$ 2181.55 0.0740360
$$955$$ −7718.60 −0.261537
$$956$$ 13130.1 0.444202
$$957$$ 13280.5 0.448588
$$958$$ −2666.68 −0.0899336
$$959$$ 0 0
$$960$$ 4918.59 0.165361
$$961$$ −21521.5 −0.722416
$$962$$ 425.261 0.0142526
$$963$$ −7091.50 −0.237300
$$964$$ 23403.9 0.781939
$$965$$ −573.712 −0.0191383
$$966$$ 0 0
$$967$$ 455.696 0.0151543 0.00757714 0.999971i $$-0.497588\pi$$
0.00757714 + 0.999971i $$0.497588\pi$$
$$968$$ −12196.7 −0.404975
$$969$$ −10.0746 −0.000333998 0
$$970$$ −1746.57 −0.0578135
$$971$$ −10641.2 −0.351690 −0.175845 0.984418i $$-0.556266\pi$$
−0.175845 + 0.984418i $$0.556266\pi$$
$$972$$ 25872.5 0.853767
$$973$$ 0 0
$$974$$ −3225.64 −0.106115
$$975$$ 2058.23 0.0676064
$$976$$ −5387.30 −0.176684
$$977$$ 19412.9 0.635694 0.317847 0.948142i $$-0.397040\pi$$
0.317847 + 0.948142i $$0.397040\pi$$
$$978$$ 1053.81 0.0344552
$$979$$ −66519.5 −2.17158
$$980$$ 0 0
$$981$$ 48101.4 1.56551
$$982$$ 3669.69 0.119251
$$983$$ −44913.0 −1.45728 −0.728639 0.684898i $$-0.759847\pi$$
−0.728639 + 0.684898i $$0.759847\pi$$
$$984$$ −2130.04 −0.0690072
$$985$$ −18555.9 −0.600243
$$986$$ 54.6254 0.00176433
$$987$$ 0 0
$$988$$ −431.757 −0.0139029
$$989$$ −74.5119 −0.00239569
$$990$$ 2785.70 0.0894296
$$991$$ 36377.2 1.16605 0.583027 0.812453i $$-0.301868\pi$$
0.583027 + 0.812453i $$0.301868\pi$$
$$992$$ −5551.93 −0.177695
$$993$$ 6586.79 0.210499
$$994$$ 0 0
$$995$$ −20309.3 −0.647083
$$996$$ 15320.5 0.487398
$$997$$ −6366.39 −0.202232 −0.101116 0.994875i $$-0.532241\pi$$
−0.101116 + 0.994875i $$0.532241\pi$$
$$998$$ 2592.46 0.0822274
$$999$$ 9055.12 0.286778
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 637.4.a.d.1.3 4
7.6 odd 2 91.4.a.b.1.3 4
21.20 even 2 819.4.a.h.1.2 4
28.27 even 2 1456.4.a.s.1.2 4
35.34 odd 2 2275.4.a.h.1.2 4
91.90 odd 2 1183.4.a.e.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
91.4.a.b.1.3 4 7.6 odd 2
637.4.a.d.1.3 4 1.1 even 1 trivial
819.4.a.h.1.2 4 21.20 even 2
1183.4.a.e.1.2 4 91.90 odd 2
1456.4.a.s.1.2 4 28.27 even 2
2275.4.a.h.1.2 4 35.34 odd 2