# Properties

 Label 169.4.b.a.168.1 Level $169$ Weight $4$ Character 169.168 Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 168.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 169.168 Dual form 169.4.b.a.168.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} +O(q^{10})$$ $$q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} -35.0000 q^{10} +26.0000i q^{11} +119.000 q^{12} +65.0000 q^{14} +49.0000i q^{15} +89.0000 q^{16} -77.0000 q^{17} -110.000i q^{18} -126.000i q^{19} +119.000i q^{20} -91.0000i q^{21} +130.000 q^{22} +96.0000 q^{23} -315.000i q^{24} +76.0000 q^{25} +35.0000 q^{27} -221.000i q^{28} -82.0000 q^{29} +245.000 q^{30} +196.000i q^{31} -85.0000i q^{32} -182.000i q^{33} +385.000i q^{34} +91.0000 q^{35} -374.000 q^{36} +131.000i q^{37} -630.000 q^{38} +315.000 q^{40} +336.000i q^{41} -455.000 q^{42} +201.000 q^{43} -442.000i q^{44} -154.000i q^{45} -480.000i q^{46} +105.000i q^{47} -623.000 q^{48} +174.000 q^{49} -380.000i q^{50} +539.000 q^{51} -432.000 q^{53} -175.000i q^{54} +182.000 q^{55} -585.000 q^{56} +882.000i q^{57} +410.000i q^{58} +294.000i q^{59} -833.000i q^{60} -56.0000 q^{61} +980.000 q^{62} +286.000i q^{63} +287.000 q^{64} -910.000 q^{66} +478.000i q^{67} +1309.00 q^{68} -672.000 q^{69} -455.000i q^{70} +9.00000i q^{71} +990.000i q^{72} -98.0000i q^{73} +655.000 q^{74} -532.000 q^{75} +2142.00i q^{76} -338.000 q^{77} +1304.00 q^{79} -623.000i q^{80} -839.000 q^{81} +1680.00 q^{82} -308.000i q^{83} +1547.00i q^{84} +539.000i q^{85} -1005.00i q^{86} +574.000 q^{87} -1170.00 q^{88} +1190.00i q^{89} -770.000 q^{90} -1632.00 q^{92} -1372.00i q^{93} +525.000 q^{94} -882.000 q^{95} +595.000i q^{96} +70.0000i q^{97} -870.000i q^{98} +572.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} - 70 q^{10} + 238 q^{12} + 130 q^{14} + 178 q^{16} - 154 q^{17} + 260 q^{22} + 192 q^{23} + 152 q^{25} + 70 q^{27} - 164 q^{29} + 490 q^{30} + 182 q^{35} - 748 q^{36} - 1260 q^{38} + 630 q^{40} - 910 q^{42} + 402 q^{43} - 1246 q^{48} + 348 q^{49} + 1078 q^{51} - 864 q^{53} + 364 q^{55} - 1170 q^{56} - 112 q^{61} + 1960 q^{62} + 574 q^{64} - 1820 q^{66} + 2618 q^{68} - 1344 q^{69} + 1310 q^{74} - 1064 q^{75} - 676 q^{77} + 2608 q^{79} - 1678 q^{81} + 3360 q^{82} + 1148 q^{87} - 2340 q^{88} - 1540 q^{90} - 3264 q^{92} + 1050 q^{94} - 1764 q^{95}+O(q^{100})$$ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 - 70 * q^10 + 238 * q^12 + 130 * q^14 + 178 * q^16 - 154 * q^17 + 260 * q^22 + 192 * q^23 + 152 * q^25 + 70 * q^27 - 164 * q^29 + 490 * q^30 + 182 * q^35 - 748 * q^36 - 1260 * q^38 + 630 * q^40 - 910 * q^42 + 402 * q^43 - 1246 * q^48 + 348 * q^49 + 1078 * q^51 - 864 * q^53 + 364 * q^55 - 1170 * q^56 - 112 * q^61 + 1960 * q^62 + 574 * q^64 - 1820 * q^66 + 2618 * q^68 - 1344 * q^69 + 1310 * q^74 - 1064 * q^75 - 676 * q^77 + 2608 * q^79 - 1678 * q^81 + 3360 * q^82 + 1148 * q^87 - 2340 * q^88 - 1540 * q^90 - 3264 * q^92 + 1050 * q^94 - 1764 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 5.00000i − 1.76777i −0.467707 0.883883i $$-0.654920\pi$$
0.467707 0.883883i $$-0.345080\pi$$
$$3$$ −7.00000 −1.34715 −0.673575 0.739119i $$-0.735242\pi$$
−0.673575 + 0.739119i $$0.735242\pi$$
$$4$$ −17.0000 −2.12500
$$5$$ − 7.00000i − 0.626099i −0.949737 0.313050i $$-0.898649\pi$$
0.949737 0.313050i $$-0.101351\pi$$
$$6$$ 35.0000i 2.38145i
$$7$$ 13.0000i 0.701934i 0.936388 + 0.350967i $$0.114147\pi$$
−0.936388 + 0.350967i $$0.885853\pi$$
$$8$$ 45.0000i 1.98874i
$$9$$ 22.0000 0.814815
$$10$$ −35.0000 −1.10680
$$11$$ 26.0000i 0.712663i 0.934360 + 0.356332i $$0.115973\pi$$
−0.934360 + 0.356332i $$0.884027\pi$$
$$12$$ 119.000 2.86270
$$13$$ 0 0
$$14$$ 65.0000 1.24086
$$15$$ 49.0000i 0.843450i
$$16$$ 89.0000 1.39062
$$17$$ −77.0000 −1.09854 −0.549272 0.835644i $$-0.685095\pi$$
−0.549272 + 0.835644i $$0.685095\pi$$
$$18$$ − 110.000i − 1.44040i
$$19$$ − 126.000i − 1.52139i −0.649110 0.760694i $$-0.724859\pi$$
0.649110 0.760694i $$-0.275141\pi$$
$$20$$ 119.000i 1.33046i
$$21$$ − 91.0000i − 0.945611i
$$22$$ 130.000 1.25982
$$23$$ 96.0000 0.870321 0.435161 0.900353i $$-0.356692\pi$$
0.435161 + 0.900353i $$0.356692\pi$$
$$24$$ − 315.000i − 2.67913i
$$25$$ 76.0000 0.608000
$$26$$ 0 0
$$27$$ 35.0000 0.249472
$$28$$ − 221.000i − 1.49161i
$$29$$ −82.0000 −0.525070 −0.262535 0.964923i $$-0.584558\pi$$
−0.262535 + 0.964923i $$0.584558\pi$$
$$30$$ 245.000 1.49102
$$31$$ 196.000i 1.13557i 0.823177 + 0.567785i $$0.192199\pi$$
−0.823177 + 0.567785i $$0.807801\pi$$
$$32$$ − 85.0000i − 0.469563i
$$33$$ − 182.000i − 0.960065i
$$34$$ 385.000i 1.94197i
$$35$$ 91.0000 0.439480
$$36$$ −374.000 −1.73148
$$37$$ 131.000i 0.582061i 0.956714 + 0.291031i $$0.0939982\pi$$
−0.956714 + 0.291031i $$0.906002\pi$$
$$38$$ −630.000 −2.68946
$$39$$ 0 0
$$40$$ 315.000 1.24515
$$41$$ 336.000i 1.27986i 0.768432 + 0.639932i $$0.221037\pi$$
−0.768432 + 0.639932i $$0.778963\pi$$
$$42$$ −455.000 −1.67162
$$43$$ 201.000 0.712842 0.356421 0.934325i $$-0.383997\pi$$
0.356421 + 0.934325i $$0.383997\pi$$
$$44$$ − 442.000i − 1.51441i
$$45$$ − 154.000i − 0.510155i
$$46$$ − 480.000i − 1.53852i
$$47$$ 105.000i 0.325869i 0.986637 + 0.162934i $$0.0520959\pi$$
−0.986637 + 0.162934i $$0.947904\pi$$
$$48$$ −623.000 −1.87338
$$49$$ 174.000 0.507289
$$50$$ − 380.000i − 1.07480i
$$51$$ 539.000 1.47990
$$52$$ 0 0
$$53$$ −432.000 −1.11962 −0.559809 0.828622i $$-0.689126\pi$$
−0.559809 + 0.828622i $$0.689126\pi$$
$$54$$ − 175.000i − 0.441009i
$$55$$ 182.000 0.446198
$$56$$ −585.000 −1.39596
$$57$$ 882.000i 2.04954i
$$58$$ 410.000i 0.928201i
$$59$$ 294.000i 0.648738i 0.945931 + 0.324369i $$0.105152\pi$$
−0.945931 + 0.324369i $$0.894848\pi$$
$$60$$ − 833.000i − 1.79233i
$$61$$ −56.0000 −0.117542 −0.0587710 0.998271i $$-0.518718\pi$$
−0.0587710 + 0.998271i $$0.518718\pi$$
$$62$$ 980.000 2.00742
$$63$$ 286.000i 0.571946i
$$64$$ 287.000 0.560547
$$65$$ 0 0
$$66$$ −910.000 −1.69717
$$67$$ 478.000i 0.871597i 0.900044 + 0.435798i $$0.143534\pi$$
−0.900044 + 0.435798i $$0.856466\pi$$
$$68$$ 1309.00 2.33441
$$69$$ −672.000 −1.17245
$$70$$ − 455.000i − 0.776899i
$$71$$ 9.00000i 0.0150437i 0.999972 + 0.00752186i $$0.00239430\pi$$
−0.999972 + 0.00752186i $$0.997606\pi$$
$$72$$ 990.000i 1.62045i
$$73$$ − 98.0000i − 0.157124i −0.996909 0.0785619i $$-0.974967\pi$$
0.996909 0.0785619i $$-0.0250328\pi$$
$$74$$ 655.000 1.02895
$$75$$ −532.000 −0.819068
$$76$$ 2142.00i 3.23295i
$$77$$ −338.000 −0.500243
$$78$$ 0 0
$$79$$ 1304.00 1.85711 0.928554 0.371198i $$-0.121053\pi$$
0.928554 + 0.371198i $$0.121053\pi$$
$$80$$ − 623.000i − 0.870669i
$$81$$ −839.000 −1.15089
$$82$$ 1680.00 2.26250
$$83$$ − 308.000i − 0.407318i −0.979042 0.203659i $$-0.934717\pi$$
0.979042 0.203659i $$-0.0652834\pi$$
$$84$$ 1547.00i 2.00942i
$$85$$ 539.000i 0.687797i
$$86$$ − 1005.00i − 1.26014i
$$87$$ 574.000 0.707348
$$88$$ −1170.00 −1.41730
$$89$$ 1190.00i 1.41730i 0.705560 + 0.708650i $$0.250696\pi$$
−0.705560 + 0.708650i $$0.749304\pi$$
$$90$$ −770.000 −0.901835
$$91$$ 0 0
$$92$$ −1632.00 −1.84943
$$93$$ − 1372.00i − 1.52978i
$$94$$ 525.000 0.576060
$$95$$ −882.000 −0.952540
$$96$$ 595.000i 0.632572i
$$97$$ 70.0000i 0.0732724i 0.999329 + 0.0366362i $$0.0116643\pi$$
−0.999329 + 0.0366362i $$0.988336\pi$$
$$98$$ − 870.000i − 0.896768i
$$99$$ 572.000i 0.580689i
$$100$$ −1292.00 −1.29200
$$101$$ −420.000 −0.413778 −0.206889 0.978364i $$-0.566334\pi$$
−0.206889 + 0.978364i $$0.566334\pi$$
$$102$$ − 2695.00i − 2.61613i
$$103$$ −588.000 −0.562499 −0.281249 0.959635i $$-0.590749\pi$$
−0.281249 + 0.959635i $$0.590749\pi$$
$$104$$ 0 0
$$105$$ −637.000 −0.592046
$$106$$ 2160.00i 1.97922i
$$107$$ −684.000 −0.617989 −0.308994 0.951064i $$-0.599992\pi$$
−0.308994 + 0.951064i $$0.599992\pi$$
$$108$$ −595.000 −0.530129
$$109$$ 373.000i 0.327770i 0.986479 + 0.163885i $$0.0524026\pi$$
−0.986479 + 0.163885i $$0.947597\pi$$
$$110$$ − 910.000i − 0.788774i
$$111$$ − 917.000i − 0.784124i
$$112$$ 1157.00i 0.976127i
$$113$$ −1734.00 −1.44355 −0.721774 0.692128i $$-0.756673\pi$$
−0.721774 + 0.692128i $$0.756673\pi$$
$$114$$ 4410.00 3.62311
$$115$$ − 672.000i − 0.544907i
$$116$$ 1394.00 1.11577
$$117$$ 0 0
$$118$$ 1470.00 1.14682
$$119$$ − 1001.00i − 0.771105i
$$120$$ −2205.00 −1.67740
$$121$$ 655.000 0.492111
$$122$$ 280.000i 0.207787i
$$123$$ − 2352.00i − 1.72417i
$$124$$ − 3332.00i − 2.41308i
$$125$$ − 1407.00i − 1.00677i
$$126$$ 1430.00 1.01107
$$127$$ −1892.00 −1.32195 −0.660976 0.750407i $$-0.729857\pi$$
−0.660976 + 0.750407i $$0.729857\pi$$
$$128$$ − 2115.00i − 1.46048i
$$129$$ −1407.00 −0.960306
$$130$$ 0 0
$$131$$ 1435.00 0.957073 0.478536 0.878068i $$-0.341167\pi$$
0.478536 + 0.878068i $$0.341167\pi$$
$$132$$ 3094.00i 2.04014i
$$133$$ 1638.00 1.06791
$$134$$ 2390.00 1.54078
$$135$$ − 245.000i − 0.156194i
$$136$$ − 3465.00i − 2.18472i
$$137$$ 1776.00i 1.10755i 0.832667 + 0.553773i $$0.186813\pi$$
−0.832667 + 0.553773i $$0.813187\pi$$
$$138$$ 3360.00i 2.07262i
$$139$$ −1869.00 −1.14048 −0.570239 0.821479i $$-0.693150\pi$$
−0.570239 + 0.821479i $$0.693150\pi$$
$$140$$ −1547.00 −0.933895
$$141$$ − 735.000i − 0.438994i
$$142$$ 45.0000 0.0265938
$$143$$ 0 0
$$144$$ 1958.00 1.13310
$$145$$ 574.000i 0.328746i
$$146$$ −490.000 −0.277758
$$147$$ −1218.00 −0.683394
$$148$$ − 2227.00i − 1.23688i
$$149$$ 2466.00i 1.35586i 0.735128 + 0.677928i $$0.237122\pi$$
−0.735128 + 0.677928i $$0.762878\pi$$
$$150$$ 2660.00i 1.44792i
$$151$$ 3323.00i 1.79087i 0.445189 + 0.895437i $$0.353137\pi$$
−0.445189 + 0.895437i $$0.646863\pi$$
$$152$$ 5670.00 3.02564
$$153$$ −1694.00 −0.895110
$$154$$ 1690.00i 0.884312i
$$155$$ 1372.00 0.710979
$$156$$ 0 0
$$157$$ −2730.00 −1.38776 −0.693878 0.720092i $$-0.744099\pi$$
−0.693878 + 0.720092i $$0.744099\pi$$
$$158$$ − 6520.00i − 3.28293i
$$159$$ 3024.00 1.50829
$$160$$ −595.000 −0.293993
$$161$$ 1248.00i 0.610908i
$$162$$ 4195.00i 2.03451i
$$163$$ 544.000i 0.261407i 0.991421 + 0.130704i $$0.0417236\pi$$
−0.991421 + 0.130704i $$0.958276\pi$$
$$164$$ − 5712.00i − 2.71971i
$$165$$ −1274.00 −0.601096
$$166$$ −1540.00 −0.720043
$$167$$ − 1624.00i − 0.752508i −0.926516 0.376254i $$-0.877212\pi$$
0.926516 0.376254i $$-0.122788\pi$$
$$168$$ 4095.00 1.88057
$$169$$ 0 0
$$170$$ 2695.00 1.21587
$$171$$ − 2772.00i − 1.23965i
$$172$$ −3417.00 −1.51479
$$173$$ 336.000 0.147662 0.0738312 0.997271i $$-0.476477\pi$$
0.0738312 + 0.997271i $$0.476477\pi$$
$$174$$ − 2870.00i − 1.25043i
$$175$$ 988.000i 0.426776i
$$176$$ 2314.00i 0.991047i
$$177$$ − 2058.00i − 0.873948i
$$178$$ 5950.00 2.50546
$$179$$ 3029.00 1.26479 0.632397 0.774645i $$-0.282071\pi$$
0.632397 + 0.774645i $$0.282071\pi$$
$$180$$ 2618.00i 1.08408i
$$181$$ 28.0000 0.0114985 0.00574924 0.999983i $$-0.498170\pi$$
0.00574924 + 0.999983i $$0.498170\pi$$
$$182$$ 0 0
$$183$$ 392.000 0.158347
$$184$$ 4320.00i 1.73084i
$$185$$ 917.000 0.364428
$$186$$ −6860.00 −2.70430
$$187$$ − 2002.00i − 0.782892i
$$188$$ − 1785.00i − 0.692471i
$$189$$ 455.000i 0.175113i
$$190$$ 4410.00i 1.68387i
$$191$$ 422.000 0.159868 0.0799342 0.996800i $$-0.474529\pi$$
0.0799342 + 0.996800i $$0.474529\pi$$
$$192$$ −2009.00 −0.755141
$$193$$ − 492.000i − 0.183497i −0.995782 0.0917485i $$-0.970754\pi$$
0.995782 0.0917485i $$-0.0292456\pi$$
$$194$$ 350.000 0.129529
$$195$$ 0 0
$$196$$ −2958.00 −1.07799
$$197$$ 2991.00i 1.08173i 0.841111 + 0.540863i $$0.181902\pi$$
−0.841111 + 0.540863i $$0.818098\pi$$
$$198$$ 2860.00 1.02652
$$199$$ 70.0000 0.0249355 0.0124678 0.999922i $$-0.496031\pi$$
0.0124678 + 0.999922i $$0.496031\pi$$
$$200$$ 3420.00i 1.20915i
$$201$$ − 3346.00i − 1.17417i
$$202$$ 2100.00i 0.731463i
$$203$$ − 1066.00i − 0.368564i
$$204$$ −9163.00 −3.14480
$$205$$ 2352.00 0.801321
$$206$$ 2940.00i 0.994367i
$$207$$ 2112.00 0.709150
$$208$$ 0 0
$$209$$ 3276.00 1.08424
$$210$$ 3185.00i 1.04660i
$$211$$ 2851.00 0.930194 0.465097 0.885260i $$-0.346019\pi$$
0.465097 + 0.885260i $$0.346019\pi$$
$$212$$ 7344.00 2.37919
$$213$$ − 63.0000i − 0.0202661i
$$214$$ 3420.00i 1.09246i
$$215$$ − 1407.00i − 0.446310i
$$216$$ 1575.00i 0.496135i
$$217$$ −2548.00 −0.797095
$$218$$ 1865.00 0.579421
$$219$$ 686.000i 0.211669i
$$220$$ −3094.00 −0.948170
$$221$$ 0 0
$$222$$ −4585.00 −1.38615
$$223$$ 217.000i 0.0651632i 0.999469 + 0.0325816i $$0.0103729\pi$$
−0.999469 + 0.0325816i $$0.989627\pi$$
$$224$$ 1105.00 0.329602
$$225$$ 1672.00 0.495407
$$226$$ 8670.00i 2.55186i
$$227$$ − 2576.00i − 0.753194i −0.926377 0.376597i $$-0.877094\pi$$
0.926377 0.376597i $$-0.122906\pi$$
$$228$$ − 14994.0i − 4.35527i
$$229$$ − 455.000i − 0.131298i −0.997843 0.0656490i $$-0.979088\pi$$
0.997843 0.0656490i $$-0.0209118\pi$$
$$230$$ −3360.00 −0.963269
$$231$$ 2366.00 0.673902
$$232$$ − 3690.00i − 1.04423i
$$233$$ −3061.00 −0.860656 −0.430328 0.902673i $$-0.641602\pi$$
−0.430328 + 0.902673i $$0.641602\pi$$
$$234$$ 0 0
$$235$$ 735.000 0.204026
$$236$$ − 4998.00i − 1.37857i
$$237$$ −9128.00 −2.50180
$$238$$ −5005.00 −1.36313
$$239$$ − 3477.00i − 0.941039i −0.882389 0.470520i $$-0.844066\pi$$
0.882389 0.470520i $$-0.155934\pi$$
$$240$$ 4361.00i 1.17292i
$$241$$ 1610.00i 0.430329i 0.976578 + 0.215164i $$0.0690287\pi$$
−0.976578 + 0.215164i $$0.930971\pi$$
$$242$$ − 3275.00i − 0.869938i
$$243$$ 4928.00 1.30095
$$244$$ 952.000 0.249777
$$245$$ − 1218.00i − 0.317613i
$$246$$ −11760.0 −3.04793
$$247$$ 0 0
$$248$$ −8820.00 −2.25835
$$249$$ 2156.00i 0.548719i
$$250$$ −7035.00 −1.77973
$$251$$ −1008.00 −0.253484 −0.126742 0.991936i $$-0.540452\pi$$
−0.126742 + 0.991936i $$0.540452\pi$$
$$252$$ − 4862.00i − 1.21539i
$$253$$ 2496.00i 0.620246i
$$254$$ 9460.00i 2.33690i
$$255$$ − 3773.00i − 0.926566i
$$256$$ −8279.00 −2.02124
$$257$$ −6041.00 −1.46625 −0.733127 0.680092i $$-0.761940\pi$$
−0.733127 + 0.680092i $$0.761940\pi$$
$$258$$ 7035.00i 1.69760i
$$259$$ −1703.00 −0.408569
$$260$$ 0 0
$$261$$ −1804.00 −0.427834
$$262$$ − 7175.00i − 1.69188i
$$263$$ −3708.00 −0.869373 −0.434686 0.900582i $$-0.643141\pi$$
−0.434686 + 0.900582i $$0.643141\pi$$
$$264$$ 8190.00 1.90932
$$265$$ 3024.00i 0.700992i
$$266$$ − 8190.00i − 1.88782i
$$267$$ − 8330.00i − 1.90932i
$$268$$ − 8126.00i − 1.85214i
$$269$$ 8344.00 1.89124 0.945618 0.325278i $$-0.105458\pi$$
0.945618 + 0.325278i $$0.105458\pi$$
$$270$$ −1225.00 −0.276115
$$271$$ 1617.00i 0.362457i 0.983441 + 0.181228i $$0.0580073\pi$$
−0.983441 + 0.181228i $$0.941993\pi$$
$$272$$ −6853.00 −1.52766
$$273$$ 0 0
$$274$$ 8880.00 1.95788
$$275$$ 1976.00i 0.433299i
$$276$$ 11424.0 2.49146
$$277$$ 3820.00 0.828598 0.414299 0.910141i $$-0.364027\pi$$
0.414299 + 0.910141i $$0.364027\pi$$
$$278$$ 9345.00i 2.01610i
$$279$$ 4312.00i 0.925278i
$$280$$ 4095.00i 0.874011i
$$281$$ 6214.00i 1.31920i 0.751615 + 0.659602i $$0.229275\pi$$
−0.751615 + 0.659602i $$0.770725\pi$$
$$282$$ −3675.00 −0.776039
$$283$$ 5292.00 1.11158 0.555789 0.831323i $$-0.312416\pi$$
0.555789 + 0.831323i $$0.312416\pi$$
$$284$$ − 153.000i − 0.0319679i
$$285$$ 6174.00 1.28321
$$286$$ 0 0
$$287$$ −4368.00 −0.898379
$$288$$ − 1870.00i − 0.382607i
$$289$$ 1016.00 0.206798
$$290$$ 2870.00 0.581146
$$291$$ − 490.000i − 0.0987090i
$$292$$ 1666.00i 0.333888i
$$293$$ 903.000i 0.180047i 0.995940 + 0.0900236i $$0.0286942\pi$$
−0.995940 + 0.0900236i $$0.971306\pi$$
$$294$$ 6090.00i 1.20808i
$$295$$ 2058.00 0.406174
$$296$$ −5895.00 −1.15757
$$297$$ 910.000i 0.177790i
$$298$$ 12330.0 2.39684
$$299$$ 0 0
$$300$$ 9044.00 1.74052
$$301$$ 2613.00i 0.500368i
$$302$$ 16615.0 3.16585
$$303$$ 2940.00 0.557421
$$304$$ − 11214.0i − 2.11568i
$$305$$ 392.000i 0.0735930i
$$306$$ 8470.00i 1.58235i
$$307$$ − 2114.00i − 0.393004i −0.980503 0.196502i $$-0.937042\pi$$
0.980503 0.196502i $$-0.0629583\pi$$
$$308$$ 5746.00 1.06302
$$309$$ 4116.00 0.757770
$$310$$ − 6860.00i − 1.25684i
$$311$$ −3402.00 −0.620288 −0.310144 0.950690i $$-0.600377\pi$$
−0.310144 + 0.950690i $$0.600377\pi$$
$$312$$ 0 0
$$313$$ −10689.0 −1.93028 −0.965141 0.261732i $$-0.915706\pi$$
−0.965141 + 0.261732i $$0.915706\pi$$
$$314$$ 13650.0i 2.45323i
$$315$$ 2002.00 0.358095
$$316$$ −22168.0 −3.94635
$$317$$ − 7054.00i − 1.24982i −0.780698 0.624909i $$-0.785136\pi$$
0.780698 0.624909i $$-0.214864\pi$$
$$318$$ − 15120.0i − 2.66631i
$$319$$ − 2132.00i − 0.374198i
$$320$$ − 2009.00i − 0.350958i
$$321$$ 4788.00 0.832524
$$322$$ 6240.00 1.07994
$$323$$ 9702.00i 1.67131i
$$324$$ 14263.0 2.44564
$$325$$ 0 0
$$326$$ 2720.00 0.462107
$$327$$ − 2611.00i − 0.441555i
$$328$$ −15120.0 −2.54531
$$329$$ −1365.00 −0.228738
$$330$$ 6370.00i 1.06260i
$$331$$ 9704.00i 1.61142i 0.592310 + 0.805710i $$0.298216\pi$$
−0.592310 + 0.805710i $$0.701784\pi$$
$$332$$ 5236.00i 0.865551i
$$333$$ 2882.00i 0.474272i
$$334$$ −8120.00 −1.33026
$$335$$ 3346.00 0.545706
$$336$$ − 8099.00i − 1.31499i
$$337$$ 10449.0 1.68900 0.844500 0.535555i $$-0.179897\pi$$
0.844500 + 0.535555i $$0.179897\pi$$
$$338$$ 0 0
$$339$$ 12138.0 1.94468
$$340$$ − 9163.00i − 1.46157i
$$341$$ −5096.00 −0.809278
$$342$$ −13860.0 −2.19141
$$343$$ 6721.00i 1.05802i
$$344$$ 9045.00i 1.41766i
$$345$$ 4704.00i 0.734072i
$$346$$ − 1680.00i − 0.261033i
$$347$$ −621.000 −0.0960721 −0.0480361 0.998846i $$-0.515296\pi$$
−0.0480361 + 0.998846i $$0.515296\pi$$
$$348$$ −9758.00 −1.50311
$$349$$ − 12481.0i − 1.91431i −0.289584 0.957153i $$-0.593517\pi$$
0.289584 0.957153i $$-0.406483\pi$$
$$350$$ 4940.00 0.754440
$$351$$ 0 0
$$352$$ 2210.00 0.334640
$$353$$ − 1400.00i − 0.211089i −0.994415 0.105545i $$-0.966341\pi$$
0.994415 0.105545i $$-0.0336586\pi$$
$$354$$ −10290.0 −1.54494
$$355$$ 63.0000 0.00941885
$$356$$ − 20230.0i − 3.01176i
$$357$$ 7007.00i 1.03879i
$$358$$ − 15145.0i − 2.23586i
$$359$$ 4968.00i 0.730365i 0.930936 + 0.365182i $$0.118993\pi$$
−0.930936 + 0.365182i $$0.881007\pi$$
$$360$$ 6930.00 1.01456
$$361$$ −9017.00 −1.31462
$$362$$ − 140.000i − 0.0203266i
$$363$$ −4585.00 −0.662948
$$364$$ 0 0
$$365$$ −686.000 −0.0983750
$$366$$ − 1960.00i − 0.279920i
$$367$$ 8722.00 1.24056 0.620279 0.784381i $$-0.287019\pi$$
0.620279 + 0.784381i $$0.287019\pi$$
$$368$$ 8544.00 1.21029
$$369$$ 7392.00i 1.04285i
$$370$$ − 4585.00i − 0.644224i
$$371$$ − 5616.00i − 0.785898i
$$372$$ 23324.0i 3.25079i
$$373$$ 10012.0 1.38982 0.694908 0.719098i $$-0.255445\pi$$
0.694908 + 0.719098i $$0.255445\pi$$
$$374$$ −10010.0 −1.38397
$$375$$ 9849.00i 1.35627i
$$376$$ −4725.00 −0.648067
$$377$$ 0 0
$$378$$ 2275.00 0.309559
$$379$$ − 3372.00i − 0.457013i −0.973542 0.228507i $$-0.926616\pi$$
0.973542 0.228507i $$-0.0733843\pi$$
$$380$$ 14994.0 2.02415
$$381$$ 13244.0 1.78087
$$382$$ − 2110.00i − 0.282610i
$$383$$ − 847.000i − 0.113002i −0.998403 0.0565009i $$-0.982006\pi$$
0.998403 0.0565009i $$-0.0179944\pi$$
$$384$$ 14805.0i 1.96749i
$$385$$ 2366.00i 0.313201i
$$386$$ −2460.00 −0.324380
$$387$$ 4422.00 0.580834
$$388$$ − 1190.00i − 0.155704i
$$389$$ −11314.0 −1.47466 −0.737330 0.675533i $$-0.763914\pi$$
−0.737330 + 0.675533i $$0.763914\pi$$
$$390$$ 0 0
$$391$$ −7392.00 −0.956086
$$392$$ 7830.00i 1.00886i
$$393$$ −10045.0 −1.28932
$$394$$ 14955.0 1.91224
$$395$$ − 9128.00i − 1.16273i
$$396$$ − 9724.00i − 1.23396i
$$397$$ − 1862.00i − 0.235393i −0.993050 0.117697i $$-0.962449\pi$$
0.993050 0.117697i $$-0.0375510\pi$$
$$398$$ − 350.000i − 0.0440802i
$$399$$ −11466.0 −1.43864
$$400$$ 6764.00 0.845500
$$401$$ − 6820.00i − 0.849313i −0.905355 0.424657i $$-0.860395\pi$$
0.905355 0.424657i $$-0.139605\pi$$
$$402$$ −16730.0 −2.07566
$$403$$ 0 0
$$404$$ 7140.00 0.879278
$$405$$ 5873.00i 0.720572i
$$406$$ −5330.00 −0.651536
$$407$$ −3406.00 −0.414814
$$408$$ 24255.0i 2.94314i
$$409$$ − 12992.0i − 1.57069i −0.619057 0.785346i $$-0.712485\pi$$
0.619057 0.785346i $$-0.287515\pi$$
$$410$$ − 11760.0i − 1.41655i
$$411$$ − 12432.0i − 1.49203i
$$412$$ 9996.00 1.19531
$$413$$ −3822.00 −0.455371
$$414$$ − 10560.0i − 1.25361i
$$415$$ −2156.00 −0.255021
$$416$$ 0 0
$$417$$ 13083.0 1.53640
$$418$$ − 16380.0i − 1.91668i
$$419$$ −7343.00 −0.856155 −0.428078 0.903742i $$-0.640809\pi$$
−0.428078 + 0.903742i $$0.640809\pi$$
$$420$$ 10829.0 1.25810
$$421$$ − 5059.00i − 0.585655i −0.956165 0.292827i $$-0.905404\pi$$
0.956165 0.292827i $$-0.0945961\pi$$
$$422$$ − 14255.0i − 1.64437i
$$423$$ 2310.00i 0.265523i
$$424$$ − 19440.0i − 2.22663i
$$425$$ −5852.00 −0.667915
$$426$$ −315.000 −0.0358258
$$427$$ − 728.000i − 0.0825068i
$$428$$ 11628.0 1.31323
$$429$$ 0 0
$$430$$ −7035.00 −0.788972
$$431$$ 3243.00i 0.362436i 0.983443 + 0.181218i $$0.0580039\pi$$
−0.983443 + 0.181218i $$0.941996\pi$$
$$432$$ 3115.00 0.346922
$$433$$ −11599.0 −1.28733 −0.643663 0.765309i $$-0.722586\pi$$
−0.643663 + 0.765309i $$0.722586\pi$$
$$434$$ 12740.0i 1.40908i
$$435$$ − 4018.00i − 0.442870i
$$436$$ − 6341.00i − 0.696511i
$$437$$ − 12096.0i − 1.32410i
$$438$$ 3430.00 0.374182
$$439$$ 17374.0 1.88887 0.944437 0.328692i $$-0.106608\pi$$
0.944437 + 0.328692i $$0.106608\pi$$
$$440$$ 8190.00i 0.887370i
$$441$$ 3828.00 0.413346
$$442$$ 0 0
$$443$$ 989.000 0.106070 0.0530348 0.998593i $$-0.483111\pi$$
0.0530348 + 0.998593i $$0.483111\pi$$
$$444$$ 15589.0i 1.66626i
$$445$$ 8330.00 0.887370
$$446$$ 1085.00 0.115193
$$447$$ − 17262.0i − 1.82654i
$$448$$ 3731.00i 0.393467i
$$449$$ 14474.0i 1.52131i 0.649154 + 0.760657i $$0.275123\pi$$
−0.649154 + 0.760657i $$0.724877\pi$$
$$450$$ − 8360.00i − 0.875765i
$$451$$ −8736.00 −0.912111
$$452$$ 29478.0 3.06754
$$453$$ − 23261.0i − 2.41258i
$$454$$ −12880.0 −1.33147
$$455$$ 0 0
$$456$$ −39690.0 −4.07600
$$457$$ − 1594.00i − 0.163160i −0.996667 0.0815801i $$-0.974003\pi$$
0.996667 0.0815801i $$-0.0259966\pi$$
$$458$$ −2275.00 −0.232104
$$459$$ −2695.00 −0.274056
$$460$$ 11424.0i 1.15793i
$$461$$ − 5915.00i − 0.597590i −0.954317 0.298795i $$-0.903415\pi$$
0.954317 0.298795i $$-0.0965847\pi$$
$$462$$ − 11830.0i − 1.19130i
$$463$$ 11072.0i 1.11136i 0.831396 + 0.555680i $$0.187542\pi$$
−0.831396 + 0.555680i $$0.812458\pi$$
$$464$$ −7298.00 −0.730175
$$465$$ −9604.00 −0.957795
$$466$$ 15305.0i 1.52144i
$$467$$ −1260.00 −0.124852 −0.0624260 0.998050i $$-0.519884\pi$$
−0.0624260 + 0.998050i $$0.519884\pi$$
$$468$$ 0 0
$$469$$ −6214.00 −0.611804
$$470$$ − 3675.00i − 0.360670i
$$471$$ 19110.0 1.86952
$$472$$ −13230.0 −1.29017
$$473$$ 5226.00i 0.508016i
$$474$$ 45640.0i 4.42260i
$$475$$ − 9576.00i − 0.925004i
$$476$$ 17017.0i 1.63860i
$$477$$ −9504.00 −0.912281
$$478$$ −17385.0 −1.66354
$$479$$ 12033.0i 1.14781i 0.818921 + 0.573906i $$0.194572\pi$$
−0.818921 + 0.573906i $$0.805428\pi$$
$$480$$ 4165.00 0.396053
$$481$$ 0 0
$$482$$ 8050.00 0.760721
$$483$$ − 8736.00i − 0.822985i
$$484$$ −11135.0 −1.04574
$$485$$ 490.000 0.0458758
$$486$$ − 24640.0i − 2.29978i
$$487$$ − 2280.00i − 0.212149i −0.994358 0.106075i $$-0.966172\pi$$
0.994358 0.106075i $$-0.0338282\pi$$
$$488$$ − 2520.00i − 0.233760i
$$489$$ − 3808.00i − 0.352155i
$$490$$ −6090.00 −0.561466
$$491$$ −16767.0 −1.54111 −0.770554 0.637375i $$-0.780020\pi$$
−0.770554 + 0.637375i $$0.780020\pi$$
$$492$$ 39984.0i 3.66386i
$$493$$ 6314.00 0.576812
$$494$$ 0 0
$$495$$ 4004.00 0.363569
$$496$$ 17444.0i 1.57915i
$$497$$ −117.000 −0.0105597
$$498$$ 10780.0 0.970007
$$499$$ 12840.0i 1.15190i 0.817485 + 0.575949i $$0.195367\pi$$
−0.817485 + 0.575949i $$0.804633\pi$$
$$500$$ 23919.0i 2.13938i
$$501$$ 11368.0i 1.01374i
$$502$$ 5040.00i 0.448100i
$$503$$ −2198.00 −0.194839 −0.0974195 0.995243i $$-0.531059\pi$$
−0.0974195 + 0.995243i $$0.531059\pi$$
$$504$$ −12870.0 −1.13745
$$505$$ 2940.00i 0.259066i
$$506$$ 12480.0 1.09645
$$507$$ 0 0
$$508$$ 32164.0 2.80915
$$509$$ − 17066.0i − 1.48612i −0.669223 0.743062i $$-0.733373\pi$$
0.669223 0.743062i $$-0.266627\pi$$
$$510$$ −18865.0 −1.63795
$$511$$ 1274.00 0.110290
$$512$$ 24475.0i 2.11260i
$$513$$ − 4410.00i − 0.379544i
$$514$$ 30205.0i 2.59200i
$$515$$ 4116.00i 0.352180i
$$516$$ 23919.0 2.04065
$$517$$ −2730.00 −0.232235
$$518$$ 8515.00i 0.722254i
$$519$$ −2352.00 −0.198924
$$520$$ 0 0
$$521$$ 2583.00 0.217204 0.108602 0.994085i $$-0.465363\pi$$
0.108602 + 0.994085i $$0.465363\pi$$
$$522$$ 9020.00i 0.756312i
$$523$$ 18620.0 1.55678 0.778390 0.627781i $$-0.216037\pi$$
0.778390 + 0.627781i $$0.216037\pi$$
$$524$$ −24395.0 −2.03378
$$525$$ − 6916.00i − 0.574931i
$$526$$ 18540.0i 1.53685i
$$527$$ − 15092.0i − 1.24747i
$$528$$ − 16198.0i − 1.33509i
$$529$$ −2951.00 −0.242541
$$530$$ 15120.0 1.23919
$$531$$ 6468.00i 0.528601i
$$532$$ −27846.0 −2.26932
$$533$$ 0 0
$$534$$ −41650.0 −3.37523
$$535$$ 4788.00i 0.386922i
$$536$$ −21510.0 −1.73338
$$537$$ −21203.0 −1.70387
$$538$$ − 41720.0i − 3.34327i
$$539$$ 4524.00i 0.361526i
$$540$$ 4165.00i 0.331913i
$$541$$ 16833.0i 1.33772i 0.743388 + 0.668861i $$0.233218\pi$$
−0.743388 + 0.668861i $$0.766782\pi$$
$$542$$ 8085.00 0.640739
$$543$$ −196.000 −0.0154902
$$544$$ 6545.00i 0.515836i
$$545$$ 2611.00 0.205216
$$546$$ 0 0
$$547$$ −8615.00 −0.673402 −0.336701 0.941612i $$-0.609311\pi$$
−0.336701 + 0.941612i $$0.609311\pi$$
$$548$$ − 30192.0i − 2.35354i
$$549$$ −1232.00 −0.0957750
$$550$$ 9880.00 0.765972
$$551$$ 10332.0i 0.798835i
$$552$$ − 30240.0i − 2.33170i
$$553$$ 16952.0i 1.30357i
$$554$$ − 19100.0i − 1.46477i
$$555$$ −6419.00 −0.490939
$$556$$ 31773.0 2.42352
$$557$$ − 8535.00i − 0.649263i −0.945841 0.324632i $$-0.894760\pi$$
0.945841 0.324632i $$-0.105240\pi$$
$$558$$ 21560.0 1.63568
$$559$$ 0 0
$$560$$ 8099.00 0.611152
$$561$$ 14014.0i 1.05467i
$$562$$ 31070.0 2.33204
$$563$$ 4641.00 0.347415 0.173708 0.984797i $$-0.444425\pi$$
0.173708 + 0.984797i $$0.444425\pi$$
$$564$$ 12495.0i 0.932862i
$$565$$ 12138.0i 0.903804i
$$566$$ − 26460.0i − 1.96501i
$$567$$ − 10907.0i − 0.807850i
$$568$$ −405.000 −0.0299180
$$569$$ 4793.00 0.353134 0.176567 0.984289i $$-0.443501\pi$$
0.176567 + 0.984289i $$0.443501\pi$$
$$570$$ − 30870.0i − 2.26842i
$$571$$ 5563.00 0.407713 0.203857 0.979001i $$-0.434652\pi$$
0.203857 + 0.979001i $$0.434652\pi$$
$$572$$ 0 0
$$573$$ −2954.00 −0.215367
$$574$$ 21840.0i 1.58813i
$$575$$ 7296.00 0.529155
$$576$$ 6314.00 0.456742
$$577$$ 24038.0i 1.73434i 0.498011 + 0.867171i $$0.334064\pi$$
−0.498011 + 0.867171i $$0.665936\pi$$
$$578$$ − 5080.00i − 0.365571i
$$579$$ 3444.00i 0.247198i
$$580$$ − 9758.00i − 0.698584i
$$581$$ 4004.00 0.285910
$$582$$ −2450.00 −0.174494
$$583$$ − 11232.0i − 0.797911i
$$584$$ 4410.00 0.312478
$$585$$ 0 0
$$586$$ 4515.00 0.318281
$$587$$ − 21224.0i − 1.49235i −0.665751 0.746174i $$-0.731889\pi$$
0.665751 0.746174i $$-0.268111\pi$$
$$588$$ 20706.0 1.45221
$$589$$ 24696.0 1.72764
$$590$$ − 10290.0i − 0.718021i
$$591$$ − 20937.0i − 1.45725i
$$592$$ 11659.0i 0.809429i
$$593$$ − 4354.00i − 0.301513i −0.988571 0.150757i $$-0.951829\pi$$
0.988571 0.150757i $$-0.0481710\pi$$
$$594$$ 4550.00 0.314291
$$595$$ −7007.00 −0.482788
$$596$$ − 41922.0i − 2.88119i
$$597$$ −490.000 −0.0335919
$$598$$ 0 0
$$599$$ 7310.00 0.498629 0.249314 0.968423i $$-0.419795\pi$$
0.249314 + 0.968423i $$0.419795\pi$$
$$600$$ − 23940.0i − 1.62891i
$$601$$ −7595.00 −0.515485 −0.257743 0.966214i $$-0.582979\pi$$
−0.257743 + 0.966214i $$0.582979\pi$$
$$602$$ 13065.0 0.884534
$$603$$ 10516.0i 0.710190i
$$604$$ − 56491.0i − 3.80561i
$$605$$ − 4585.00i − 0.308110i
$$606$$ − 14700.0i − 0.985391i
$$607$$ −826.000 −0.0552328 −0.0276164 0.999619i $$-0.508792\pi$$
−0.0276164 + 0.999619i $$0.508792\pi$$
$$608$$ −10710.0 −0.714388
$$609$$ 7462.00i 0.496511i
$$610$$ 1960.00 0.130095
$$611$$ 0 0
$$612$$ 28798.0 1.90211
$$613$$ 14590.0i 0.961312i 0.876909 + 0.480656i $$0.159602\pi$$
−0.876909 + 0.480656i $$0.840398\pi$$
$$614$$ −10570.0 −0.694740
$$615$$ −16464.0 −1.07950
$$616$$ − 15210.0i − 0.994851i
$$617$$ 4888.00i 0.318936i 0.987203 + 0.159468i $$0.0509779\pi$$
−0.987203 + 0.159468i $$0.949022\pi$$
$$618$$ − 20580.0i − 1.33956i
$$619$$ 11004.0i 0.714520i 0.934005 + 0.357260i $$0.116289\pi$$
−0.934005 + 0.357260i $$0.883711\pi$$
$$620$$ −23324.0 −1.51083
$$621$$ 3360.00 0.217121
$$622$$ 17010.0i 1.09653i
$$623$$ −15470.0 −0.994851
$$624$$ 0 0
$$625$$ −349.000 −0.0223360
$$626$$ 53445.0i 3.41229i
$$627$$ −22932.0 −1.46063
$$628$$ 46410.0 2.94898
$$629$$ − 10087.0i − 0.639420i
$$630$$ − 10010.0i − 0.633028i
$$631$$ 4975.00i 0.313869i 0.987609 + 0.156935i $$0.0501612\pi$$
−0.987609 + 0.156935i $$0.949839\pi$$
$$632$$ 58680.0i 3.69330i
$$633$$ −19957.0 −1.25311
$$634$$ −35270.0 −2.20939
$$635$$ 13244.0i 0.827673i
$$636$$ −51408.0 −3.20513
$$637$$ 0 0
$$638$$ −10660.0 −0.661494
$$639$$ 198.000i 0.0122578i
$$640$$ −14805.0 −0.914405
$$641$$ −3950.00 −0.243394 −0.121697 0.992567i $$-0.538834\pi$$
−0.121697 + 0.992567i $$0.538834\pi$$
$$642$$ − 23940.0i − 1.47171i
$$643$$ − 3682.00i − 0.225823i −0.993605 0.112911i $$-0.963982\pi$$
0.993605 0.112911i $$-0.0360176\pi$$
$$644$$ − 21216.0i − 1.29818i
$$645$$ 9849.00i 0.601247i
$$646$$ 48510.0 2.95449
$$647$$ −10402.0 −0.632063 −0.316032 0.948749i $$-0.602351\pi$$
−0.316032 + 0.948749i $$0.602351\pi$$
$$648$$ − 37755.0i − 2.28882i
$$649$$ −7644.00 −0.462332
$$650$$ 0 0
$$651$$ 17836.0 1.07381
$$652$$ − 9248.00i − 0.555490i
$$653$$ −31680.0 −1.89852 −0.949260 0.314491i $$-0.898166\pi$$
−0.949260 + 0.314491i $$0.898166\pi$$
$$654$$ −13055.0 −0.780567
$$655$$ − 10045.0i − 0.599222i
$$656$$ 29904.0i 1.77981i
$$657$$ − 2156.00i − 0.128027i
$$658$$ 6825.00i 0.404356i
$$659$$ 21940.0 1.29691 0.648453 0.761255i $$-0.275416\pi$$
0.648453 + 0.761255i $$0.275416\pi$$
$$660$$ 21658.0 1.27733
$$661$$ 31374.0i 1.84615i 0.384616 + 0.923077i $$0.374334\pi$$
−0.384616 + 0.923077i $$0.625666\pi$$
$$662$$ 48520.0 2.84862
$$663$$ 0 0
$$664$$ 13860.0 0.810049
$$665$$ − 11466.0i − 0.668620i
$$666$$ 14410.0 0.838403
$$667$$ −7872.00 −0.456979
$$668$$ 27608.0i 1.59908i
$$669$$ − 1519.00i − 0.0877847i
$$670$$ − 16730.0i − 0.964681i
$$671$$ − 1456.00i − 0.0837679i
$$672$$ −7735.00 −0.444024
$$673$$ −18013.0 −1.03172 −0.515862 0.856672i $$-0.672528\pi$$
−0.515862 + 0.856672i $$0.672528\pi$$
$$674$$ − 52245.0i − 2.98576i
$$675$$ 2660.00 0.151679
$$676$$ 0 0
$$677$$ −10640.0 −0.604030 −0.302015 0.953303i $$-0.597659\pi$$
−0.302015 + 0.953303i $$0.597659\pi$$
$$678$$ − 60690.0i − 3.43774i
$$679$$ −910.000 −0.0514324
$$680$$ −24255.0 −1.36785
$$681$$ 18032.0i 1.01467i
$$682$$ 25480.0i 1.43062i
$$683$$ 9336.00i 0.523034i 0.965199 + 0.261517i $$0.0842227\pi$$
−0.965199 + 0.261517i $$0.915777\pi$$
$$684$$ 47124.0i 2.63426i
$$685$$ 12432.0 0.693434
$$686$$ 33605.0 1.87033
$$687$$ 3185.00i 0.176878i
$$688$$ 17889.0 0.991296
$$689$$ 0 0
$$690$$ 23520.0 1.29767
$$691$$ 4200.00i 0.231224i 0.993294 + 0.115612i $$0.0368829\pi$$
−0.993294 + 0.115612i $$0.963117\pi$$
$$692$$ −5712.00 −0.313783
$$693$$ −7436.00 −0.407605
$$694$$ 3105.00i 0.169833i
$$695$$ 13083.0i 0.714052i
$$696$$ 25830.0i 1.40673i
$$697$$ − 25872.0i − 1.40599i
$$698$$ −62405.0 −3.38405
$$699$$ 21427.0 1.15943
$$700$$ − 16796.0i − 0.906899i
$$701$$ −9872.00 −0.531898 −0.265949 0.963987i $$-0.585685\pi$$
−0.265949 + 0.963987i $$0.585685\pi$$
$$702$$ 0 0
$$703$$ 16506.0 0.885541
$$704$$ 7462.00i 0.399481i
$$705$$ −5145.00 −0.274854
$$706$$ −7000.00 −0.373156
$$707$$ − 5460.00i − 0.290445i
$$708$$ 34986.0i 1.85714i
$$709$$ − 28450.0i − 1.50700i −0.657449 0.753499i $$-0.728364\pi$$
0.657449 0.753499i $$-0.271636\pi$$
$$710$$ − 315.000i − 0.0166503i
$$711$$ 28688.0 1.51320
$$712$$ −53550.0 −2.81864
$$713$$ 18816.0i 0.988310i
$$714$$ 35035.0 1.83635
$$715$$ 0 0
$$716$$ −51493.0 −2.68769
$$717$$ 24339.0i 1.26772i
$$718$$ 24840.0 1.29111
$$719$$ −32718.0 −1.69705 −0.848523 0.529159i $$-0.822507\pi$$
−0.848523 + 0.529159i $$0.822507\pi$$
$$720$$ − 13706.0i − 0.709434i
$$721$$ − 7644.00i − 0.394837i
$$722$$ 45085.0i 2.32395i
$$723$$ − 11270.0i − 0.579718i
$$724$$ −476.000 −0.0244343
$$725$$ −6232.00 −0.319242
$$726$$ 22925.0i 1.17194i
$$727$$ 22834.0 1.16488 0.582439 0.812874i $$-0.302099\pi$$
0.582439 + 0.812874i $$0.302099\pi$$
$$728$$ 0 0
$$729$$ −11843.0 −0.601687
$$730$$ 3430.00i 0.173904i
$$731$$ −15477.0 −0.783088
$$732$$ −6664.00 −0.336487
$$733$$ 7875.00i 0.396821i 0.980119 + 0.198410i $$0.0635779\pi$$
−0.980119 + 0.198410i $$0.936422\pi$$
$$734$$ − 43610.0i − 2.19302i
$$735$$ 8526.00i 0.427872i
$$736$$ − 8160.00i − 0.408671i
$$737$$ −12428.0 −0.621155
$$738$$ 36960.0 1.84352
$$739$$ 2140.00i 0.106524i 0.998581 + 0.0532620i $$0.0169618\pi$$
−0.998581 + 0.0532620i $$0.983038\pi$$
$$740$$ −15589.0 −0.774410
$$741$$ 0 0
$$742$$ −28080.0 −1.38928
$$743$$ 31971.0i 1.57860i 0.614006 + 0.789302i $$0.289557\pi$$
−0.614006 + 0.789302i $$0.710443\pi$$
$$744$$ 61740.0 3.04234
$$745$$ 17262.0 0.848900
$$746$$ − 50060.0i − 2.45687i
$$747$$ − 6776.00i − 0.331889i
$$748$$ 34034.0i 1.66364i
$$749$$ − 8892.00i − 0.433787i
$$750$$ 49245.0 2.39756
$$751$$ 7432.00 0.361115 0.180558 0.983564i $$-0.442210\pi$$
0.180558 + 0.983564i $$0.442210\pi$$
$$752$$ 9345.00i 0.453161i
$$753$$ 7056.00 0.341481
$$754$$ 0 0
$$755$$ 23261.0 1.12126
$$756$$ − 7735.00i − 0.372115i
$$757$$ 20176.0 0.968704 0.484352 0.874873i $$-0.339055\pi$$
0.484352 + 0.874873i $$0.339055\pi$$
$$758$$ −16860.0 −0.807893
$$759$$ − 17472.0i − 0.835564i
$$760$$ − 39690.0i − 1.89435i
$$761$$ 9478.00i 0.451481i 0.974187 + 0.225741i $$0.0724802\pi$$
−0.974187 + 0.225741i $$0.927520\pi$$
$$762$$ − 66220.0i − 3.14816i
$$763$$ −4849.00 −0.230073
$$764$$ −7174.00 −0.339720
$$765$$ 11858.0i 0.560427i
$$766$$ −4235.00 −0.199761
$$767$$ 0 0
$$768$$ 57953.0 2.72292
$$769$$ − 12096.0i − 0.567221i −0.958940 0.283610i $$-0.908468\pi$$
0.958940 0.283610i $$-0.0915323\pi$$
$$770$$ 11830.0 0.553667
$$771$$ 42287.0 1.97526
$$772$$ 8364.00i 0.389931i
$$773$$ 17941.0i 0.834790i 0.908725 + 0.417395i $$0.137057\pi$$
−0.908725 + 0.417395i $$0.862943\pi$$
$$774$$ − 22110.0i − 1.02678i
$$775$$ 14896.0i 0.690426i
$$776$$ −3150.00 −0.145720
$$777$$ 11921.0 0.550403
$$778$$ 56570.0i 2.60685i
$$779$$ 42336.0 1.94717
$$780$$ 0 0
$$781$$ −234.000 −0.0107211
$$782$$ 36960.0i 1.69014i
$$783$$ −2870.00 −0.130990
$$784$$ 15486.0 0.705448
$$785$$ 19110.0i 0.868873i
$$786$$ 50225.0i 2.27922i
$$787$$ − 6664.00i − 0.301837i −0.988546 0.150919i $$-0.951777\pi$$
0.988546 0.150919i $$-0.0482232\pi$$
$$788$$ − 50847.0i − 2.29867i
$$789$$ 25956.0 1.17118
$$790$$ −45640.0 −2.05544
$$791$$ − 22542.0i − 1.01328i
$$792$$ −25740.0 −1.15484
$$793$$ 0 0
$$794$$ −9310.00 −0.416120
$$795$$ − 21168.0i − 0.944342i
$$796$$ −1190.00 −0.0529880
$$797$$ 1442.00 0.0640882 0.0320441 0.999486i $$-0.489798\pi$$
0.0320441 + 0.999486i $$0.489798\pi$$
$$798$$ 57330.0i 2.54318i
$$799$$ − 8085.00i − 0.357981i
$$800$$ − 6460.00i − 0.285494i
$$801$$ 26180.0i 1.15484i
$$802$$ −34100.0 −1.50139
$$803$$ 2548.00 0.111976
$$804$$ 56882.0i 2.49512i
$$805$$ 8736.00 0.382489
$$806$$ 0 0
$$807$$ −58408.0 −2.54778
$$808$$ − 18900.0i − 0.822896i
$$809$$ 30207.0 1.31276 0.656379 0.754431i $$-0.272087\pi$$
0.656379 + 0.754431i $$0.272087\pi$$
$$810$$ 29365.0 1.27380
$$811$$ 21140.0i 0.915322i 0.889127 + 0.457661i $$0.151313\pi$$
−0.889127 + 0.457661i $$0.848687\pi$$
$$812$$ 18122.0i 0.783199i
$$813$$ − 11319.0i − 0.488284i
$$814$$ 17030.0i 0.733294i
$$815$$ 3808.00 0.163667
$$816$$ 47971.0 2.05799
$$817$$ − 25326.0i − 1.08451i
$$818$$ −64960.0 −2.77662
$$819$$ 0 0
$$820$$ −39984.0 −1.70281
$$821$$ 569.000i 0.0241879i 0.999927 + 0.0120939i $$0.00384971\pi$$
−0.999927 + 0.0120939i $$0.996150\pi$$
$$822$$ −62160.0 −2.63757
$$823$$ 8538.00 0.361623 0.180812 0.983518i $$-0.442128\pi$$
0.180812 + 0.983518i $$0.442128\pi$$
$$824$$ − 26460.0i − 1.11866i
$$825$$ − 13832.0i − 0.583719i
$$826$$ 19110.0i 0.804990i
$$827$$ 32702.0i 1.37504i 0.726164 + 0.687521i $$0.241301\pi$$
−0.726164 + 0.687521i $$0.758699\pi$$
$$828$$ −35904.0 −1.50694
$$829$$ 21154.0 0.886259 0.443130 0.896458i $$-0.353868\pi$$
0.443130 + 0.896458i $$0.353868\pi$$
$$830$$ 10780.0i 0.450818i
$$831$$ −26740.0 −1.11625
$$832$$ 0 0
$$833$$ −13398.0 −0.557279
$$834$$ − 65415.0i − 2.71599i
$$835$$ −11368.0 −0.471145
$$836$$ −55692.0 −2.30400
$$837$$ 6860.00i 0.283293i
$$838$$ 36715.0i 1.51348i
$$839$$ 2184.00i 0.0898690i 0.998990 + 0.0449345i $$0.0143079\pi$$
−0.998990 + 0.0449345i $$0.985692\pi$$
$$840$$ − 28665.0i − 1.17742i
$$841$$ −17665.0 −0.724302
$$842$$ −25295.0 −1.03530
$$843$$ − 43498.0i − 1.77717i
$$844$$ −48467.0 −1.97666
$$845$$ 0 0
$$846$$ 11550.0 0.469382
$$847$$ 8515.00i 0.345430i
$$848$$ −38448.0 −1.55697
$$849$$ −37044.0 −1.49746
$$850$$ 29260.0i 1.18072i
$$851$$ 12576.0i 0.506580i
$$852$$ 1071.00i 0.0430656i
$$853$$ − 36687.0i − 1.47261i −0.676648 0.736307i $$-0.736568\pi$$
0.676648 0.736307i $$-0.263432\pi$$
$$854$$ −3640.00 −0.145853
$$855$$ −19404.0 −0.776144
$$856$$ − 30780.0i − 1.22902i
$$857$$ −36806.0 −1.46706 −0.733529 0.679658i $$-0.762128\pi$$
−0.733529 + 0.679658i $$0.762128\pi$$
$$858$$ 0 0
$$859$$ 4900.00 0.194628 0.0973142 0.995254i $$-0.468975\pi$$
0.0973142 + 0.995254i $$0.468975\pi$$
$$860$$ 23919.0i 0.948408i
$$861$$ 30576.0 1.21025
$$862$$ 16215.0 0.640702
$$863$$ − 13697.0i − 0.540268i −0.962823 0.270134i $$-0.912932\pi$$
0.962823 0.270134i $$-0.0870680\pi$$
$$864$$ − 2975.00i − 0.117143i
$$865$$ − 2352.00i − 0.0924513i
$$866$$ 57995.0i 2.27569i
$$867$$ −7112.00 −0.278588
$$868$$ 43316.0 1.69383
$$869$$ 33904.0i 1.32349i
$$870$$ −20090.0 −0.782891
$$871$$ 0 0
$$872$$ −16785.0 −0.651848
$$873$$ 1540.00i 0.0597034i
$$874$$ −60480.0 −2.34069
$$875$$ 18291.0 0.706684
$$876$$ − 11662.0i − 0.449797i
$$877$$ 6239.00i 0.240224i 0.992760 + 0.120112i $$0.0383253\pi$$
−0.992760 + 0.120112i $$0.961675\pi$$
$$878$$ − 86870.0i − 3.33909i
$$879$$ − 6321.00i − 0.242551i
$$880$$ 16198.0 0.620494
$$881$$ −133.000 −0.00508613 −0.00254307 0.999997i $$-0.500809\pi$$
−0.00254307 + 0.999997i $$0.500809\pi$$
$$882$$ − 19140.0i − 0.730700i
$$883$$ 26003.0 0.991020 0.495510 0.868602i $$-0.334981\pi$$
0.495510 + 0.868602i $$0.334981\pi$$
$$884$$ 0 0
$$885$$ −14406.0 −0.547178
$$886$$ − 4945.00i − 0.187506i
$$887$$ −31248.0 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$888$$ 41265.0 1.55942
$$889$$ − 24596.0i − 0.927923i
$$890$$ − 41650.0i − 1.56866i
$$891$$ − 21814.0i − 0.820198i
$$892$$ − 3689.00i − 0.138472i
$$893$$ 13230.0 0.495773
$$894$$ −86310.0 −3.22890
$$895$$ − 21203.0i − 0.791886i
$$896$$ 27495.0 1.02516
$$897$$ 0 0
$$898$$ 72370.0 2.68933
$$899$$ − 16072.0i − 0.596253i
$$900$$ −28424.0 −1.05274
$$901$$ 33264.0 1.22995
$$902$$ 43680.0i 1.61240i
$$903$$ − 18291.0i − 0.674071i
$$904$$ − 78030.0i − 2.87084i
$$905$$ − 196.000i − 0.00719918i
$$906$$ −116305. −4.26487
$$907$$ 38253.0 1.40041 0.700204 0.713943i $$-0.253092\pi$$
0.700204 + 0.713943i $$0.253092\pi$$
$$908$$ 43792.0i 1.60054i
$$909$$ −9240.00 −0.337152
$$910$$ 0 0
$$911$$ 36374.0 1.32286 0.661429 0.750007i $$-0.269950\pi$$
0.661429 + 0.750007i $$0.269950\pi$$
$$912$$ 78498.0i 2.85014i
$$913$$ 8008.00 0.290281
$$914$$ −7970.00 −0.288429
$$915$$ − 2744.00i − 0.0991408i
$$916$$ 7735.00i 0.279008i
$$917$$ 18655.0i 0.671802i
$$918$$ 13475.0i 0.484468i
$$919$$ −27648.0 −0.992408 −0.496204 0.868206i $$-0.665273\pi$$
−0.496204 + 0.868206i $$0.665273\pi$$
$$920$$ 30240.0 1.08368
$$921$$ 14798.0i 0.529436i
$$922$$ −29575.0 −1.05640
$$923$$ 0 0
$$924$$ −40222.0 −1.43204
$$925$$ 9956.00i 0.353893i
$$926$$ 55360.0 1.96462
$$927$$ −12936.0 −0.458332
$$928$$ 6970.00i 0.246553i
$$929$$ 756.000i 0.0266992i 0.999911 + 0.0133496i $$0.00424944\pi$$
−0.999911 + 0.0133496i $$0.995751\pi$$
$$930$$ 48020.0i 1.69316i
$$931$$ − 21924.0i − 0.771783i
$$932$$ 52037.0 1.82889
$$933$$ 23814.0 0.835622
$$934$$ 6300.00i 0.220709i
$$935$$ −14014.0 −0.490168
$$936$$ 0 0
$$937$$ 20846.0 0.726797 0.363399 0.931634i $$-0.381616\pi$$
0.363399 + 0.931634i $$0.381616\pi$$
$$938$$ 31070.0i 1.08153i
$$939$$ 74823.0 2.60038
$$940$$ −12495.0 −0.433555
$$941$$ − 41321.0i − 1.43148i −0.698365 0.715742i $$-0.746089\pi$$
0.698365 0.715742i $$-0.253911\pi$$
$$942$$ − 95550.0i − 3.30487i
$$943$$ 32256.0i 1.11389i
$$944$$ 26166.0i 0.902151i
$$945$$ 3185.00 0.109638
$$946$$ 26130.0 0.898055
$$947$$ − 54966.0i − 1.88612i −0.332624 0.943060i $$-0.607934\pi$$
0.332624 0.943060i $$-0.392066\pi$$
$$948$$ 155176. 5.31633
$$949$$ 0 0
$$950$$ −47880.0 −1.63519
$$951$$ 49378.0i 1.68369i
$$952$$ 45045.0 1.53353
$$953$$ 44553.0 1.51439 0.757195 0.653189i $$-0.226569\pi$$
0.757195 + 0.653189i $$0.226569\pi$$
$$954$$ 47520.0i 1.61270i
$$955$$ − 2954.00i − 0.100093i
$$956$$ 59109.0i 1.99971i
$$957$$ 14924.0i 0.504101i
$$958$$ 60165.0 2.02906
$$959$$ −23088.0 −0.777425
$$960$$ 14063.0i 0.472793i
$$961$$ −8625.00 −0.289517
$$962$$ 0 0
$$963$$ −15048.0 −0.503546
$$964$$ − 27370.0i − 0.914448i
$$965$$ −3444.00 −0.114887
$$966$$ −43680.0 −1.45485
$$967$$ − 27907.0i − 0.928054i −0.885821 0.464027i $$-0.846404\pi$$
0.885821 0.464027i $$-0.153596\pi$$
$$968$$ 29475.0i 0.978680i
$$969$$ − 67914.0i − 2.25151i
$$970$$ − 2450.00i − 0.0810977i
$$971$$ −16443.0 −0.543441 −0.271720 0.962376i $$-0.587593\pi$$
−0.271720 + 0.962376i $$0.587593\pi$$
$$972$$ −83776.0 −2.76452
$$973$$ − 24297.0i − 0.800541i
$$974$$ −11400.0 −0.375030
$$975$$ 0 0
$$976$$ −4984.00 −0.163457
$$977$$ − 45414.0i − 1.48713i −0.668666 0.743563i $$-0.733134\pi$$
0.668666 0.743563i $$-0.266866\pi$$
$$978$$ −19040.0 −0.622528
$$979$$ −30940.0 −1.01006
$$980$$ 20706.0i 0.674927i
$$981$$ 8206.00i 0.267072i
$$982$$ 83835.0i 2.72432i
$$983$$ 8981.00i 0.291403i 0.989329 + 0.145702i $$0.0465440\pi$$
−0.989329 + 0.145702i $$0.953456\pi$$
$$984$$ 105840. 3.42892
$$985$$ 20937.0 0.677267
$$986$$ − 31570.0i − 1.01967i
$$987$$ 9555.00 0.308145
$$988$$ 0 0
$$989$$ 19296.0 0.620402
$$990$$ − 20020.0i − 0.642704i
$$991$$ −17414.0 −0.558198 −0.279099 0.960262i $$-0.590036\pi$$
−0.279099 + 0.960262i $$0.590036\pi$$
$$992$$ 16660.0 0.533221
$$993$$ − 67928.0i − 2.17083i
$$994$$ 585.000i 0.0186671i
$$995$$ − 490.000i − 0.0156121i
$$996$$ − 36652.0i − 1.16603i
$$997$$ −23702.0 −0.752909 −0.376454 0.926435i $$-0.622857\pi$$
−0.376454 + 0.926435i $$0.622857\pi$$
$$998$$ 64200.0 2.03629
$$999$$ 4585.00i 0.145208i
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 169.4.b.a.168.1 2
13.2 odd 12 169.4.c.e.22.1 2
13.3 even 3 169.4.e.e.147.2 4
13.4 even 6 169.4.e.e.23.2 4
13.5 odd 4 13.4.a.a.1.1 1
13.6 odd 12 169.4.c.e.146.1 2
13.7 odd 12 169.4.c.a.146.1 2
13.8 odd 4 169.4.a.e.1.1 1
13.9 even 3 169.4.e.e.23.1 4
13.10 even 6 169.4.e.e.147.1 4
13.11 odd 12 169.4.c.a.22.1 2
13.12 even 2 inner 169.4.b.a.168.2 2
39.5 even 4 117.4.a.b.1.1 1
39.8 even 4 1521.4.a.a.1.1 1
52.31 even 4 208.4.a.g.1.1 1
65.18 even 4 325.4.b.b.274.2 2
65.44 odd 4 325.4.a.d.1.1 1
65.57 even 4 325.4.b.b.274.1 2
91.83 even 4 637.4.a.a.1.1 1
104.5 odd 4 832.4.a.r.1.1 1
104.83 even 4 832.4.a.a.1.1 1
143.109 even 4 1573.4.a.a.1.1 1
156.83 odd 4 1872.4.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.a.1.1 1 13.5 odd 4
117.4.a.b.1.1 1 39.5 even 4
169.4.a.e.1.1 1 13.8 odd 4
169.4.b.a.168.1 2 1.1 even 1 trivial
169.4.b.a.168.2 2 13.12 even 2 inner
169.4.c.a.22.1 2 13.11 odd 12
169.4.c.a.146.1 2 13.7 odd 12
169.4.c.e.22.1 2 13.2 odd 12
169.4.c.e.146.1 2 13.6 odd 12
169.4.e.e.23.1 4 13.9 even 3
169.4.e.e.23.2 4 13.4 even 6
169.4.e.e.147.1 4 13.10 even 6
169.4.e.e.147.2 4 13.3 even 3
208.4.a.g.1.1 1 52.31 even 4
325.4.a.d.1.1 1 65.44 odd 4
325.4.b.b.274.1 2 65.57 even 4
325.4.b.b.274.2 2 65.18 even 4
637.4.a.a.1.1 1 91.83 even 4
832.4.a.a.1.1 1 104.83 even 4
832.4.a.r.1.1 1 104.5 odd 4
1521.4.a.a.1.1 1 39.8 even 4
1573.4.a.a.1.1 1 143.109 even 4
1872.4.a.k.1.1 1 156.83 odd 4