# Properties

 Label 507.4.b.e.337.2 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-17})$$ Defining polynomial: $$x^{4} - 17x^{2} + 289$$ x^4 - 17*x^2 + 289 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.2 Root $$-3.57071 - 2.06155i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.e.337.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +55.4243 q^{10} -40.4962i q^{11} +27.0000 q^{12} +129.556 q^{14} -40.3271i q^{15} -55.0000 q^{16} +43.1414 q^{17} -37.1080i q^{18} -26.9779i q^{19} -120.981i q^{20} -94.2656i q^{21} -166.970 q^{22} +19.0100 q^{23} -12.3693i q^{24} -55.6971 q^{25} -27.0000 q^{27} -282.797i q^{28} -154.111 q^{29} -166.273 q^{30} +308.270i q^{31} +259.756i q^{32} +121.489i q^{33} -177.877i q^{34} -422.384 q^{35} -81.0000 q^{36} -43.5116i q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384i q^{41} -388.667 q^{42} -342.121 q^{43} +364.466i q^{44} +120.981i q^{45} -78.3802i q^{46} -133.468i q^{47} +165.000 q^{48} -644.334 q^{49} +229.645i q^{50} -129.424 q^{51} -438.454 q^{53} +111.324i q^{54} +544.364 q^{55} -129.556 q^{56} +80.9338i q^{57} +635.418i q^{58} -590.553i q^{59} +362.944i q^{60} -541.304 q^{61} +1271.03 q^{62} +282.797i q^{63} +631.000 q^{64} +500.910 q^{66} -230.345i q^{67} -388.273 q^{68} -57.0300 q^{69} +1741.54i q^{70} +449.412i q^{71} +37.1080i q^{72} +389.711i q^{73} -179.403 q^{74} +167.091 q^{75} +242.801i q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330i q^{80} +81.0000 q^{81} +197.243 q^{82} -1300.24i q^{83} +848.391i q^{84} +579.923i q^{85} +1410.60i q^{86} +462.334 q^{87} +166.970 q^{88} +925.045i q^{89} +498.819 q^{90} -171.090 q^{92} -924.811i q^{93} -550.301 q^{94} +362.647 q^{95} -779.267i q^{96} -1560.49i q^{97} +2656.66i q^{98} -364.466i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 $$4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 4.12311i − 1.45774i −0.684653 0.728869i $$-0.740046\pi$$
0.684653 0.728869i $$-0.259954\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −9.00000 −1.12500
$$5$$ 13.4424i 1.20232i 0.799128 + 0.601161i $$0.205295\pi$$
−0.799128 + 0.601161i $$0.794705\pi$$
$$6$$ 12.3693i 0.841625i
$$7$$ 31.4219i 1.69662i 0.529498 + 0.848311i $$0.322380\pi$$
−0.529498 + 0.848311i $$0.677620\pi$$
$$8$$ 4.12311i 0.182217i
$$9$$ 9.00000 0.333333
$$10$$ 55.4243 1.75267
$$11$$ − 40.4962i − 1.11001i −0.831849 0.555003i $$-0.812717\pi$$
0.831849 0.555003i $$-0.187283\pi$$
$$12$$ 27.0000 0.649519
$$13$$ 0 0
$$14$$ 129.556 2.47323
$$15$$ − 40.3271i − 0.694161i
$$16$$ −55.0000 −0.859375
$$17$$ 43.1414 0.615490 0.307745 0.951469i $$-0.400426\pi$$
0.307745 + 0.951469i $$0.400426\pi$$
$$18$$ − 37.1080i − 0.485913i
$$19$$ − 26.9779i − 0.325745i −0.986647 0.162873i $$-0.947924\pi$$
0.986647 0.162873i $$-0.0520760\pi$$
$$20$$ − 120.981i − 1.35261i
$$21$$ − 94.2656i − 0.979545i
$$22$$ −166.970 −1.61810
$$23$$ 19.0100 0.172342 0.0861709 0.996280i $$-0.472537\pi$$
0.0861709 + 0.996280i $$0.472537\pi$$
$$24$$ − 12.3693i − 0.105203i
$$25$$ −55.6971 −0.445577
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 282.797i − 1.90870i
$$29$$ −154.111 −0.986820 −0.493410 0.869797i $$-0.664250\pi$$
−0.493410 + 0.869797i $$0.664250\pi$$
$$30$$ −166.273 −1.01190
$$31$$ 308.270i 1.78603i 0.450025 + 0.893016i $$0.351415\pi$$
−0.450025 + 0.893016i $$0.648585\pi$$
$$32$$ 259.756i 1.43496i
$$33$$ 121.489i 0.640862i
$$34$$ − 177.877i − 0.897223i
$$35$$ −422.384 −2.03988
$$36$$ −81.0000 −0.375000
$$37$$ − 43.5116i − 0.193331i −0.995317 0.0966657i $$-0.969182\pi$$
0.995317 0.0966657i $$-0.0308178\pi$$
$$38$$ −111.233 −0.474851
$$39$$ 0 0
$$40$$ −55.4243 −0.219084
$$41$$ 47.8384i 0.182222i 0.995841 + 0.0911110i $$0.0290418\pi$$
−0.995841 + 0.0911110i $$0.970958\pi$$
$$42$$ −388.667 −1.42792
$$43$$ −342.121 −1.21333 −0.606663 0.794959i $$-0.707492\pi$$
−0.606663 + 0.794959i $$0.707492\pi$$
$$44$$ 364.466i 1.24876i
$$45$$ 120.981i 0.400774i
$$46$$ − 78.3802i − 0.251229i
$$47$$ − 133.468i − 0.414218i −0.978318 0.207109i $$-0.933594\pi$$
0.978318 0.207109i $$-0.0664055\pi$$
$$48$$ 165.000 0.496160
$$49$$ −644.334 −1.87853
$$50$$ 229.645i 0.649535i
$$51$$ −129.424 −0.355353
$$52$$ 0 0
$$53$$ −438.454 −1.13635 −0.568173 0.822909i $$-0.692350\pi$$
−0.568173 + 0.822909i $$0.692350\pi$$
$$54$$ 111.324i 0.280542i
$$55$$ 544.364 1.33458
$$56$$ −129.556 −0.309154
$$57$$ 80.9338i 0.188069i
$$58$$ 635.418i 1.43852i
$$59$$ − 590.553i − 1.30311i −0.758601 0.651555i $$-0.774117\pi$$
0.758601 0.651555i $$-0.225883\pi$$
$$60$$ 362.944i 0.780931i
$$61$$ −541.304 −1.13618 −0.568089 0.822967i $$-0.692317\pi$$
−0.568089 + 0.822967i $$0.692317\pi$$
$$62$$ 1271.03 2.60357
$$63$$ 282.797i 0.565541i
$$64$$ 631.000 1.23242
$$65$$ 0 0
$$66$$ 500.910 0.934208
$$67$$ − 230.345i − 0.420018i −0.977700 0.210009i $$-0.932651\pi$$
0.977700 0.210009i $$-0.0673493\pi$$
$$68$$ −388.273 −0.692426
$$69$$ −57.0300 −0.0995015
$$70$$ 1741.54i 2.97362i
$$71$$ 449.412i 0.751203i 0.926781 + 0.375601i $$0.122564\pi$$
−0.926781 + 0.375601i $$0.877436\pi$$
$$72$$ 37.1080i 0.0607391i
$$73$$ 389.711i 0.624826i 0.949946 + 0.312413i $$0.101137\pi$$
−0.949946 + 0.312413i $$0.898863\pi$$
$$74$$ −179.403 −0.281826
$$75$$ 167.091 0.257254
$$76$$ 242.801i 0.366464i
$$77$$ 1272.47 1.88326
$$78$$ 0 0
$$79$$ −897.820 −1.27864 −0.639321 0.768940i $$-0.720784\pi$$
−0.639321 + 0.768940i $$0.720784\pi$$
$$80$$ − 739.330i − 1.03325i
$$81$$ 81.0000 0.111111
$$82$$ 197.243 0.265632
$$83$$ − 1300.24i − 1.71952i −0.510700 0.859759i $$-0.670614\pi$$
0.510700 0.859759i $$-0.329386\pi$$
$$84$$ 848.391i 1.10199i
$$85$$ 579.923i 0.740017i
$$86$$ 1410.60i 1.76871i
$$87$$ 462.334 0.569741
$$88$$ 166.970 0.202262
$$89$$ 925.045i 1.10174i 0.834592 + 0.550869i $$0.185703\pi$$
−0.834592 + 0.550869i $$0.814297\pi$$
$$90$$ 498.819 0.584223
$$91$$ 0 0
$$92$$ −171.090 −0.193884
$$93$$ − 924.811i − 1.03117i
$$94$$ −550.301 −0.603822
$$95$$ 362.647 0.391651
$$96$$ − 779.267i − 0.828475i
$$97$$ − 1560.49i − 1.63344i −0.577031 0.816722i $$-0.695789\pi$$
0.577031 0.816722i $$-0.304211\pi$$
$$98$$ 2656.66i 2.73840i
$$99$$ − 364.466i − 0.370002i
$$100$$ 501.274 0.501274
$$101$$ −958.840 −0.944635 −0.472318 0.881428i $$-0.656582\pi$$
−0.472318 + 0.881428i $$0.656582\pi$$
$$102$$ 533.630i 0.518012i
$$103$$ −635.153 −0.607606 −0.303803 0.952735i $$-0.598257\pi$$
−0.303803 + 0.952735i $$0.598257\pi$$
$$104$$ 0 0
$$105$$ 1267.15 1.17773
$$106$$ 1807.79i 1.65649i
$$107$$ −1448.32 −1.30855 −0.654275 0.756257i $$-0.727026\pi$$
−0.654275 + 0.756257i $$0.727026\pi$$
$$108$$ 243.000 0.216506
$$109$$ − 331.084i − 0.290937i −0.989363 0.145468i $$-0.953531\pi$$
0.989363 0.145468i $$-0.0464689\pi$$
$$110$$ − 2244.47i − 1.94547i
$$111$$ 130.535i 0.111620i
$$112$$ − 1728.20i − 1.45803i
$$113$$ 695.204 0.578755 0.289378 0.957215i $$-0.406552\pi$$
0.289378 + 0.957215i $$0.406552\pi$$
$$114$$ 333.699 0.274156
$$115$$ 255.539i 0.207210i
$$116$$ 1387.00 1.11017
$$117$$ 0 0
$$118$$ −2434.91 −1.89959
$$119$$ 1355.58i 1.04425i
$$120$$ 166.273 0.126488
$$121$$ −308.940 −0.232111
$$122$$ 2231.85i 1.65625i
$$123$$ − 143.515i − 0.105206i
$$124$$ − 2774.43i − 2.00929i
$$125$$ 931.594i 0.666595i
$$126$$ 1166.00 0.824410
$$127$$ −247.154 −0.172688 −0.0863441 0.996265i $$-0.527518\pi$$
−0.0863441 + 0.996265i $$0.527518\pi$$
$$128$$ − 523.634i − 0.361587i
$$129$$ 1026.36 0.700514
$$130$$ 0 0
$$131$$ 472.243 0.314962 0.157481 0.987522i $$-0.449663\pi$$
0.157481 + 0.987522i $$0.449663\pi$$
$$132$$ − 1093.40i − 0.720969i
$$133$$ 847.697 0.552667
$$134$$ −949.739 −0.612275
$$135$$ − 362.944i − 0.231387i
$$136$$ 177.877i 0.112153i
$$137$$ 1830.70i 1.14166i 0.821069 + 0.570829i $$0.193378\pi$$
−0.821069 + 0.570829i $$0.806622\pi$$
$$138$$ 235.141i 0.145047i
$$139$$ −100.000 −0.0610208 −0.0305104 0.999534i $$-0.509713\pi$$
−0.0305104 + 0.999534i $$0.509713\pi$$
$$140$$ 3801.46 2.29487
$$141$$ 400.403i 0.239149i
$$142$$ 1852.97 1.09506
$$143$$ 0 0
$$144$$ −495.000 −0.286458
$$145$$ − 2071.62i − 1.18647i
$$146$$ 1606.82 0.910832
$$147$$ 1933.00 1.08457
$$148$$ 391.604i 0.217498i
$$149$$ 149.557i 0.0822293i 0.999154 + 0.0411147i $$0.0130909\pi$$
−0.999154 + 0.0411147i $$0.986909\pi$$
$$150$$ − 688.936i − 0.375009i
$$151$$ 800.032i 0.431163i 0.976486 + 0.215582i $$0.0691647\pi$$
−0.976486 + 0.215582i $$0.930835\pi$$
$$152$$ 111.233 0.0593564
$$153$$ 388.273 0.205163
$$154$$ − 5246.51i − 2.74530i
$$155$$ −4143.88 −2.14739
$$156$$ 0 0
$$157$$ −2706.16 −1.37564 −0.687818 0.725884i $$-0.741431\pi$$
−0.687818 + 0.725884i $$0.741431\pi$$
$$158$$ 3701.81i 1.86392i
$$159$$ 1315.36 0.656070
$$160$$ −3491.73 −1.72528
$$161$$ 597.330i 0.292399i
$$162$$ − 333.972i − 0.161971i
$$163$$ − 3678.25i − 1.76750i −0.467959 0.883750i $$-0.655010\pi$$
0.467959 0.883750i $$-0.344990\pi$$
$$164$$ − 430.546i − 0.205000i
$$165$$ −1633.09 −0.770522
$$166$$ −5361.03 −2.50661
$$167$$ 3223.11i 1.49348i 0.665114 + 0.746742i $$0.268383\pi$$
−0.665114 + 0.746742i $$0.731617\pi$$
$$168$$ 388.667 0.178490
$$169$$ 0 0
$$170$$ 2391.08 1.07875
$$171$$ − 242.801i − 0.108582i
$$172$$ 3079.09 1.36499
$$173$$ 2689.54 1.18198 0.590988 0.806680i $$-0.298738\pi$$
0.590988 + 0.806680i $$0.298738\pi$$
$$174$$ − 1906.25i − 0.830533i
$$175$$ − 1750.11i − 0.755976i
$$176$$ 2227.29i 0.953911i
$$177$$ 1771.66i 0.752351i
$$178$$ 3814.06 1.60604
$$179$$ 1524.04 0.636381 0.318191 0.948027i $$-0.396925\pi$$
0.318191 + 0.948027i $$0.396925\pi$$
$$180$$ − 1088.83i − 0.450871i
$$181$$ −476.881 −0.195836 −0.0979180 0.995194i $$-0.531218\pi$$
−0.0979180 + 0.995194i $$0.531218\pi$$
$$182$$ 0 0
$$183$$ 1623.91 0.655973
$$184$$ 78.3802i 0.0314036i
$$185$$ 584.899 0.232446
$$186$$ −3813.09 −1.50317
$$187$$ − 1747.06i − 0.683197i
$$188$$ 1201.21i 0.465996i
$$189$$ − 848.391i − 0.326515i
$$190$$ − 1495.23i − 0.570924i
$$191$$ −1369.74 −0.518906 −0.259453 0.965756i $$-0.583542\pi$$
−0.259453 + 0.965756i $$0.583542\pi$$
$$192$$ −1893.00 −0.711539
$$193$$ 2144.72i 0.799898i 0.916537 + 0.399949i $$0.130972\pi$$
−0.916537 + 0.399949i $$0.869028\pi$$
$$194$$ −6434.08 −2.38113
$$195$$ 0 0
$$196$$ 5799.01 2.11334
$$197$$ − 239.739i − 0.0867040i −0.999060 0.0433520i $$-0.986196\pi$$
0.999060 0.0433520i $$-0.0138037\pi$$
$$198$$ −1502.73 −0.539365
$$199$$ 1589.94 0.566371 0.283185 0.959065i $$-0.408609\pi$$
0.283185 + 0.959065i $$0.408609\pi$$
$$200$$ − 229.645i − 0.0811918i
$$201$$ 691.036i 0.242497i
$$202$$ 3953.40i 1.37703i
$$203$$ − 4842.47i − 1.67426i
$$204$$ 1164.82 0.399773
$$205$$ −643.061 −0.219090
$$206$$ 2618.80i 0.885731i
$$207$$ 171.090 0.0574472
$$208$$ 0 0
$$209$$ −1092.50 −0.361579
$$210$$ − 5224.61i − 1.71682i
$$211$$ −1872.85 −0.611055 −0.305527 0.952183i $$-0.598833\pi$$
−0.305527 + 0.952183i $$0.598833\pi$$
$$212$$ 3946.09 1.27839
$$213$$ − 1348.24i − 0.433707i
$$214$$ 5971.59i 1.90752i
$$215$$ − 4598.92i − 1.45881i
$$216$$ − 111.324i − 0.0350677i
$$217$$ −9686.43 −3.03022
$$218$$ −1365.09 −0.424109
$$219$$ − 1169.13i − 0.360743i
$$220$$ −4899.28 −1.50141
$$221$$ 0 0
$$222$$ 538.209 0.162713
$$223$$ − 56.1283i − 0.0168548i −0.999964 0.00842742i $$-0.997317\pi$$
0.999964 0.00842742i $$-0.00268256\pi$$
$$224$$ −8162.01 −2.43459
$$225$$ −501.274 −0.148526
$$226$$ − 2866.40i − 0.843673i
$$227$$ − 667.390i − 0.195137i −0.995229 0.0975687i $$-0.968893\pi$$
0.995229 0.0975687i $$-0.0311066\pi$$
$$228$$ − 728.404i − 0.211578i
$$229$$ − 723.299i − 0.208720i −0.994540 0.104360i $$-0.966721\pi$$
0.994540 0.104360i $$-0.0332795\pi$$
$$230$$ 1053.62 0.302058
$$231$$ −3817.40 −1.08730
$$232$$ − 635.418i − 0.179816i
$$233$$ 275.451 0.0774482 0.0387241 0.999250i $$-0.487671\pi$$
0.0387241 + 0.999250i $$0.487671\pi$$
$$234$$ 0 0
$$235$$ 1794.12 0.498024
$$236$$ 5314.98i 1.46600i
$$237$$ 2693.46 0.738224
$$238$$ 5589.22 1.52225
$$239$$ − 1529.39i − 0.413925i −0.978349 0.206963i $$-0.933642\pi$$
0.978349 0.206963i $$-0.0663579\pi$$
$$240$$ 2217.99i 0.596544i
$$241$$ − 975.526i − 0.260743i −0.991465 0.130372i $$-0.958383\pi$$
0.991465 0.130372i $$-0.0416170\pi$$
$$242$$ 1273.79i 0.338357i
$$243$$ −243.000 −0.0641500
$$244$$ 4871.74 1.27820
$$245$$ − 8661.38i − 2.25859i
$$246$$ −591.729 −0.153363
$$247$$ 0 0
$$248$$ −1271.03 −0.325446
$$249$$ 3900.72i 0.992765i
$$250$$ 3841.06 0.971720
$$251$$ 1874.14 0.471294 0.235647 0.971839i $$-0.424279\pi$$
0.235647 + 0.971839i $$0.424279\pi$$
$$252$$ − 2545.17i − 0.636233i
$$253$$ − 769.832i − 0.191300i
$$254$$ 1019.04i 0.251734i
$$255$$ − 1739.77i − 0.427249i
$$256$$ 2889.00 0.705322
$$257$$ 1818.19 0.441305 0.220653 0.975352i $$-0.429181\pi$$
0.220653 + 0.975352i $$0.429181\pi$$
$$258$$ − 4231.81i − 1.02117i
$$259$$ 1367.22 0.328010
$$260$$ 0 0
$$261$$ −1387.00 −0.328940
$$262$$ − 1947.11i − 0.459132i
$$263$$ 673.799 0.157978 0.0789890 0.996875i $$-0.474831\pi$$
0.0789890 + 0.996875i $$0.474831\pi$$
$$264$$ −500.910 −0.116776
$$265$$ − 5893.86i − 1.36625i
$$266$$ − 3495.14i − 0.805643i
$$267$$ − 2775.14i − 0.636088i
$$268$$ 2073.11i 0.472520i
$$269$$ −3356.40 −0.760756 −0.380378 0.924831i $$-0.624206\pi$$
−0.380378 + 0.924831i $$0.624206\pi$$
$$270$$ −1496.46 −0.337301
$$271$$ − 8915.55i − 1.99845i −0.0393133 0.999227i $$-0.512517\pi$$
0.0393133 0.999227i $$-0.487483\pi$$
$$272$$ −2372.78 −0.528937
$$273$$ 0 0
$$274$$ 7548.16 1.66424
$$275$$ 2255.52i 0.494593i
$$276$$ 513.270 0.111939
$$277$$ −4017.31 −0.871396 −0.435698 0.900093i $$-0.643498\pi$$
−0.435698 + 0.900093i $$0.643498\pi$$
$$278$$ 412.311i 0.0889523i
$$279$$ 2774.43i 0.595344i
$$280$$ − 1741.54i − 0.371702i
$$281$$ − 1841.12i − 0.390860i −0.980718 0.195430i $$-0.937390\pi$$
0.980718 0.195430i $$-0.0626103\pi$$
$$282$$ 1650.90 0.348617
$$283$$ −4849.40 −1.01861 −0.509305 0.860586i $$-0.670098\pi$$
−0.509305 + 0.860586i $$0.670098\pi$$
$$284$$ − 4044.71i − 0.845103i
$$285$$ −1087.94 −0.226120
$$286$$ 0 0
$$287$$ −1503.17 −0.309162
$$288$$ 2337.80i 0.478320i
$$289$$ −3051.82 −0.621172
$$290$$ −8541.52 −1.72957
$$291$$ 4681.48i 0.943070i
$$292$$ − 3507.40i − 0.702929i
$$293$$ 1413.85i 0.281905i 0.990016 + 0.140953i $$0.0450165\pi$$
−0.990016 + 0.140953i $$0.954983\pi$$
$$294$$ − 7969.97i − 1.58101i
$$295$$ 7938.43 1.56676
$$296$$ 179.403 0.0352283
$$297$$ 1093.40i 0.213621i
$$298$$ 616.639 0.119869
$$299$$ 0 0
$$300$$ −1503.82 −0.289411
$$301$$ − 10750.1i − 2.05856i
$$302$$ 3298.62 0.628523
$$303$$ 2876.52 0.545385
$$304$$ 1483.79i 0.279937i
$$305$$ − 7276.41i − 1.36605i
$$306$$ − 1600.89i − 0.299074i
$$307$$ 4625.64i 0.859932i 0.902845 + 0.429966i $$0.141474\pi$$
−0.902845 + 0.429966i $$0.858526\pi$$
$$308$$ −11452.2 −2.11867
$$309$$ 1905.46 0.350802
$$310$$ 17085.7i 3.13032i
$$311$$ 6060.79 1.10507 0.552534 0.833490i $$-0.313661\pi$$
0.552534 + 0.833490i $$0.313661\pi$$
$$312$$ 0 0
$$313$$ 969.946 0.175158 0.0875792 0.996158i $$-0.472087\pi$$
0.0875792 + 0.996158i $$0.472087\pi$$
$$314$$ 11157.8i 2.00532i
$$315$$ −3801.46 −0.679962
$$316$$ 8080.38 1.43847
$$317$$ 8741.63i 1.54883i 0.632679 + 0.774414i $$0.281955\pi$$
−0.632679 + 0.774414i $$0.718045\pi$$
$$318$$ − 5423.38i − 0.956378i
$$319$$ 6240.92i 1.09537i
$$320$$ 8482.13i 1.48177i
$$321$$ 4344.97 0.755491
$$322$$ 2462.85 0.426241
$$323$$ − 1163.87i − 0.200493i
$$324$$ −729.000 −0.125000
$$325$$ 0 0
$$326$$ −15165.8 −2.57655
$$327$$ 993.252i 0.167972i
$$328$$ −197.243 −0.0332040
$$329$$ 4193.81 0.702772
$$330$$ 6733.41i 1.12322i
$$331$$ 6987.76i 1.16037i 0.814485 + 0.580184i $$0.197019\pi$$
−0.814485 + 0.580184i $$0.802981\pi$$
$$332$$ 11702.2i 1.93446i
$$333$$ − 391.604i − 0.0644438i
$$334$$ 13289.2 2.17711
$$335$$ 3096.39 0.504996
$$336$$ 5184.61i 0.841797i
$$337$$ 4156.59 0.671881 0.335940 0.941883i $$-0.390946\pi$$
0.335940 + 0.941883i $$0.390946\pi$$
$$338$$ 0 0
$$339$$ −2085.61 −0.334144
$$340$$ − 5219.30i − 0.832519i
$$341$$ 12483.8 1.98250
$$342$$ −1001.10 −0.158284
$$343$$ − 9468.49i − 1.49053i
$$344$$ − 1410.60i − 0.221089i
$$345$$ − 766.618i − 0.119633i
$$346$$ − 11089.3i − 1.72301i
$$347$$ −312.513 −0.0483475 −0.0241737 0.999708i $$-0.507695\pi$$
−0.0241737 + 0.999708i $$0.507695\pi$$
$$348$$ −4161.01 −0.640958
$$349$$ 4458.75i 0.683872i 0.939723 + 0.341936i $$0.111083\pi$$
−0.939723 + 0.341936i $$0.888917\pi$$
$$350$$ −7215.88 −1.10201
$$351$$ 0 0
$$352$$ 10519.1 1.59281
$$353$$ − 2249.17i − 0.339126i −0.985519 0.169563i $$-0.945764\pi$$
0.985519 0.169563i $$-0.0542356\pi$$
$$354$$ 7304.74 1.09673
$$355$$ −6041.16 −0.903187
$$356$$ − 8325.41i − 1.23945i
$$357$$ − 4066.75i − 0.602900i
$$358$$ − 6283.78i − 0.927677i
$$359$$ 7842.79i 1.15300i 0.817098 + 0.576499i $$0.195582\pi$$
−0.817098 + 0.576499i $$0.804418\pi$$
$$360$$ −498.819 −0.0730279
$$361$$ 6131.19 0.893890
$$362$$ 1966.23i 0.285478i
$$363$$ 926.820 0.134009
$$364$$ 0 0
$$365$$ −5238.64 −0.751241
$$366$$ − 6695.56i − 0.956237i
$$367$$ 6660.24 0.947307 0.473653 0.880711i $$-0.342935\pi$$
0.473653 + 0.880711i $$0.342935\pi$$
$$368$$ −1045.55 −0.148106
$$369$$ 430.546i 0.0607407i
$$370$$ − 2411.60i − 0.338846i
$$371$$ − 13777.1i − 1.92795i
$$372$$ 8323.30i 1.16006i
$$373$$ −36.9873 −0.00513439 −0.00256720 0.999997i $$-0.500817\pi$$
−0.00256720 + 0.999997i $$0.500817\pi$$
$$374$$ −7203.32 −0.995923
$$375$$ − 2794.78i − 0.384859i
$$376$$ 550.301 0.0754777
$$377$$ 0 0
$$378$$ −3498.00 −0.475973
$$379$$ 12079.9i 1.63721i 0.574360 + 0.818603i $$0.305251\pi$$
−0.574360 + 0.818603i $$0.694749\pi$$
$$380$$ −3263.82 −0.440607
$$381$$ 741.463 0.0997015
$$382$$ 5647.59i 0.756429i
$$383$$ − 10567.4i − 1.40984i −0.709287 0.704919i $$-0.750983\pi$$
0.709287 0.704919i $$-0.249017\pi$$
$$384$$ 1570.90i 0.208763i
$$385$$ 17104.9i 2.26428i
$$386$$ 8842.90 1.16604
$$387$$ −3079.09 −0.404442
$$388$$ 14044.4i 1.83763i
$$389$$ 9757.49 1.27179 0.635893 0.771778i $$-0.280632\pi$$
0.635893 + 0.771778i $$0.280632\pi$$
$$390$$ 0 0
$$391$$ 820.119 0.106075
$$392$$ − 2656.66i − 0.342300i
$$393$$ −1416.73 −0.181844
$$394$$ −988.469 −0.126392
$$395$$ − 12068.8i − 1.53734i
$$396$$ 3280.19i 0.416252i
$$397$$ 14200.5i 1.79522i 0.440786 + 0.897612i $$0.354700\pi$$
−0.440786 + 0.897612i $$0.645300\pi$$
$$398$$ − 6555.48i − 0.825620i
$$399$$ −2543.09 −0.319082
$$400$$ 3063.34 0.382918
$$401$$ 12676.4i 1.57863i 0.613992 + 0.789313i $$0.289563\pi$$
−0.613992 + 0.789313i $$0.710437\pi$$
$$402$$ 2849.22 0.353497
$$403$$ 0 0
$$404$$ 8629.56 1.06271
$$405$$ 1088.83i 0.133591i
$$406$$ −19966.0 −2.44063
$$407$$ −1762.05 −0.214599
$$408$$ − 533.630i − 0.0647515i
$$409$$ 1533.68i 0.185417i 0.995693 + 0.0927083i $$0.0295524\pi$$
−0.995693 + 0.0927083i $$0.970448\pi$$
$$410$$ 2651.41i 0.319375i
$$411$$ − 5492.09i − 0.659136i
$$412$$ 5716.38 0.683557
$$413$$ 18556.3 2.21089
$$414$$ − 705.422i − 0.0837430i
$$415$$ 17478.3 2.06741
$$416$$ 0 0
$$417$$ 300.000 0.0352304
$$418$$ 4504.50i 0.527087i
$$419$$ −2165.89 −0.252532 −0.126266 0.991996i $$-0.540299\pi$$
−0.126266 + 0.991996i $$0.540299\pi$$
$$420$$ −11404.4 −1.32494
$$421$$ 734.575i 0.0850380i 0.999096 + 0.0425190i $$0.0135383\pi$$
−0.999096 + 0.0425190i $$0.986462\pi$$
$$422$$ 7721.98i 0.890758i
$$423$$ − 1201.21i − 0.138073i
$$424$$ − 1807.79i − 0.207062i
$$425$$ −2402.85 −0.274248
$$426$$ −5558.92 −0.632231
$$427$$ − 17008.8i − 1.92767i
$$428$$ 13034.9 1.47212
$$429$$ 0 0
$$430$$ −18961.8 −2.12656
$$431$$ − 13709.3i − 1.53215i −0.642754 0.766073i $$-0.722208\pi$$
0.642754 0.766073i $$-0.277792\pi$$
$$432$$ 1485.00 0.165387
$$433$$ −10049.9 −1.11540 −0.557701 0.830042i $$-0.688316\pi$$
−0.557701 + 0.830042i $$0.688316\pi$$
$$434$$ 39938.2i 4.41727i
$$435$$ 6214.87i 0.685011i
$$436$$ 2979.76i 0.327304i
$$437$$ − 512.850i − 0.0561395i
$$438$$ −4820.46 −0.525869
$$439$$ 8133.47 0.884258 0.442129 0.896951i $$-0.354223\pi$$
0.442129 + 0.896951i $$0.354223\pi$$
$$440$$ 2244.47i 0.243184i
$$441$$ −5799.01 −0.626175
$$442$$ 0 0
$$443$$ −2370.78 −0.254264 −0.127132 0.991886i $$-0.540577\pi$$
−0.127132 + 0.991886i $$0.540577\pi$$
$$444$$ − 1174.81i − 0.125572i
$$445$$ −12434.8 −1.32464
$$446$$ −231.423 −0.0245699
$$447$$ − 448.670i − 0.0474751i
$$448$$ 19827.2i 2.09095i
$$449$$ − 12923.2i − 1.35832i −0.733992 0.679158i $$-0.762345\pi$$
0.733992 0.679158i $$-0.237655\pi$$
$$450$$ 2066.81i 0.216512i
$$451$$ 1937.27 0.202267
$$452$$ −6256.84 −0.651099
$$453$$ − 2400.10i − 0.248932i
$$454$$ −2751.72 −0.284459
$$455$$ 0 0
$$456$$ −333.699 −0.0342694
$$457$$ 8401.26i 0.859944i 0.902842 + 0.429972i $$0.141477\pi$$
−0.902842 + 0.429972i $$0.858523\pi$$
$$458$$ −2982.24 −0.304260
$$459$$ −1164.82 −0.118451
$$460$$ − 2299.85i − 0.233111i
$$461$$ 17627.3i 1.78088i 0.455098 + 0.890441i $$0.349604\pi$$
−0.455098 + 0.890441i $$0.650396\pi$$
$$462$$ 15739.5i 1.58500i
$$463$$ 5461.81i 0.548233i 0.961697 + 0.274116i $$0.0883853\pi$$
−0.961697 + 0.274116i $$0.911615\pi$$
$$464$$ 8476.13 0.848048
$$465$$ 12431.6 1.23979
$$466$$ − 1135.72i − 0.112899i
$$467$$ −8262.19 −0.818691 −0.409345 0.912379i $$-0.634243\pi$$
−0.409345 + 0.912379i $$0.634243\pi$$
$$468$$ 0 0
$$469$$ 7237.89 0.712611
$$470$$ − 7397.35i − 0.725988i
$$471$$ 8118.47 0.794223
$$472$$ 2434.91 0.237449
$$473$$ 13854.6i 1.34680i
$$474$$ − 11105.4i − 1.07614i
$$475$$ 1502.59i 0.145145i
$$476$$ − 12200.3i − 1.17479i
$$477$$ −3946.09 −0.378782
$$478$$ −6305.84 −0.603395
$$479$$ 1575.87i 0.150320i 0.997171 + 0.0751601i $$0.0239468\pi$$
−0.997171 + 0.0751601i $$0.976053\pi$$
$$480$$ 10475.2 0.996093
$$481$$ 0 0
$$482$$ −4022.20 −0.380095
$$483$$ − 1791.99i − 0.168816i
$$484$$ 2780.46 0.261125
$$485$$ 20976.7 1.96393
$$486$$ 1001.91i 0.0935139i
$$487$$ − 12595.7i − 1.17200i −0.810310 0.586001i $$-0.800701\pi$$
0.810310 0.586001i $$-0.199299\pi$$
$$488$$ − 2231.85i − 0.207031i
$$489$$ 11034.7i 1.02047i
$$490$$ −35711.8 −3.29244
$$491$$ 1071.21 0.0984586 0.0492293 0.998788i $$-0.484323\pi$$
0.0492293 + 0.998788i $$0.484323\pi$$
$$492$$ 1291.64i 0.118357i
$$493$$ −6648.59 −0.607378
$$494$$ 0 0
$$495$$ 4899.28 0.444861
$$496$$ − 16954.9i − 1.53487i
$$497$$ −14121.4 −1.27451
$$498$$ 16083.1 1.44719
$$499$$ − 1422.30i − 0.127597i −0.997963 0.0637985i $$-0.979678\pi$$
0.997963 0.0637985i $$-0.0203215\pi$$
$$500$$ − 8384.35i − 0.749919i
$$501$$ − 9669.33i − 0.862263i
$$502$$ − 7727.28i − 0.687022i
$$503$$ −9349.34 −0.828760 −0.414380 0.910104i $$-0.636002\pi$$
−0.414380 + 0.910104i $$0.636002\pi$$
$$504$$ −1166.00 −0.103051
$$505$$ − 12889.1i − 1.13576i
$$506$$ −3174.10 −0.278866
$$507$$ 0 0
$$508$$ 2224.39 0.194274
$$509$$ 13736.3i 1.19617i 0.801432 + 0.598086i $$0.204072\pi$$
−0.801432 + 0.598086i $$0.795928\pi$$
$$510$$ −7173.25 −0.622817
$$511$$ −12245.5 −1.06009
$$512$$ − 16100.7i − 1.38976i
$$513$$ 728.404i 0.0626897i
$$514$$ − 7496.58i − 0.643307i
$$515$$ − 8537.96i − 0.730538i
$$516$$ −9237.28 −0.788079
$$517$$ −5404.93 −0.459785
$$518$$ − 5637.17i − 0.478153i
$$519$$ −8068.62 −0.682414
$$520$$ 0 0
$$521$$ 11052.3 0.929386 0.464693 0.885472i $$-0.346165\pi$$
0.464693 + 0.885472i $$0.346165\pi$$
$$522$$ 5718.76i 0.479508i
$$523$$ 6477.04 0.541532 0.270766 0.962645i $$-0.412723\pi$$
0.270766 + 0.962645i $$0.412723\pi$$
$$524$$ −4250.19 −0.354332
$$525$$ 5250.33i 0.436463i
$$526$$ − 2778.14i − 0.230290i
$$527$$ 13299.2i 1.09929i
$$528$$ − 6681.87i − 0.550741i
$$529$$ −11805.6 −0.970298
$$530$$ −24301.0 −1.99164
$$531$$ − 5314.98i − 0.434370i
$$532$$ −7629.27 −0.621750
$$533$$ 0 0
$$534$$ −11442.2 −0.927250
$$535$$ − 19468.9i − 1.57330i
$$536$$ 949.739 0.0765344
$$537$$ −4572.12 −0.367415
$$538$$ 13838.8i 1.10898i
$$539$$ 26093.1i 2.08517i
$$540$$ 3266.49i 0.260310i
$$541$$ − 18341.5i − 1.45761i −0.684723 0.728803i $$-0.740077\pi$$
0.684723 0.728803i $$-0.259923\pi$$
$$542$$ −36759.7 −2.91322
$$543$$ 1430.64 0.113066
$$544$$ 11206.2i 0.883204i
$$545$$ 4450.55 0.349799
$$546$$ 0 0
$$547$$ −18943.1 −1.48071 −0.740356 0.672215i $$-0.765343\pi$$
−0.740356 + 0.672215i $$0.765343\pi$$
$$548$$ − 16476.3i − 1.28436i
$$549$$ −4871.74 −0.378726
$$550$$ 9299.75 0.720987
$$551$$ 4157.61i 0.321452i
$$552$$ − 235.141i − 0.0181309i
$$553$$ − 28211.2i − 2.16937i
$$554$$ 16563.8i 1.27027i
$$555$$ −1754.70 −0.134203
$$556$$ 900.000 0.0686484
$$557$$ − 415.532i − 0.0316098i −0.999875 0.0158049i $$-0.994969\pi$$
0.999875 0.0158049i $$-0.00503106\pi$$
$$558$$ 11439.3 0.867856
$$559$$ 0 0
$$560$$ 23231.1 1.75303
$$561$$ 5241.19i 0.394444i
$$562$$ −7591.12 −0.569772
$$563$$ −18291.8 −1.36929 −0.684643 0.728879i $$-0.740042\pi$$
−0.684643 + 0.728879i $$0.740042\pi$$
$$564$$ − 3603.63i − 0.269043i
$$565$$ 9345.19i 0.695850i
$$566$$ 19994.6i 1.48487i
$$567$$ 2545.17i 0.188514i
$$568$$ −1852.97 −0.136882
$$569$$ −4347.47 −0.320308 −0.160154 0.987092i $$-0.551199\pi$$
−0.160154 + 0.987092i $$0.551199\pi$$
$$570$$ 4485.70i 0.329623i
$$571$$ 16756.0 1.22805 0.614024 0.789288i $$-0.289550\pi$$
0.614024 + 0.789288i $$0.289550\pi$$
$$572$$ 0 0
$$573$$ 4109.23 0.299591
$$574$$ 6197.74i 0.450677i
$$575$$ −1058.80 −0.0767915
$$576$$ 5679.00 0.410807
$$577$$ 19974.7i 1.44117i 0.693364 + 0.720587i $$0.256128\pi$$
−0.693364 + 0.720587i $$0.743872\pi$$
$$578$$ 12583.0i 0.905506i
$$579$$ − 6434.16i − 0.461821i
$$580$$ 18644.6i 1.33478i
$$581$$ 40856.0 2.91737
$$582$$ 19302.2 1.37475
$$583$$ 17755.7i 1.26135i
$$584$$ −1606.82 −0.113854
$$585$$ 0 0
$$586$$ 5829.47 0.410944
$$587$$ − 15748.7i − 1.10735i −0.832732 0.553677i $$-0.813224\pi$$
0.832732 0.553677i $$-0.186776\pi$$
$$588$$ −17397.0 −1.22014
$$589$$ 8316.50 0.581792
$$590$$ − 32731.0i − 2.28392i
$$591$$ 719.217i 0.0500586i
$$592$$ 2393.14i 0.166144i
$$593$$ − 13318.4i − 0.922297i −0.887323 0.461148i $$-0.847438\pi$$
0.887323 0.461148i $$-0.152562\pi$$
$$594$$ 4508.19 0.311403
$$595$$ −18222.3 −1.25553
$$596$$ − 1346.01i − 0.0925080i
$$597$$ −4769.82 −0.326994
$$598$$ 0 0
$$599$$ 2970.80 0.202644 0.101322 0.994854i $$-0.467693\pi$$
0.101322 + 0.994854i $$0.467693\pi$$
$$600$$ 688.936i 0.0468761i
$$601$$ 10632.6 0.721654 0.360827 0.932633i $$-0.382495\pi$$
0.360827 + 0.932633i $$0.382495\pi$$
$$602$$ −44323.8 −3.00083
$$603$$ − 2073.11i − 0.140006i
$$604$$ − 7200.29i − 0.485059i
$$605$$ − 4152.88i − 0.279072i
$$606$$ − 11860.2i − 0.795029i
$$607$$ 11587.9 0.774856 0.387428 0.921900i $$-0.373364\pi$$
0.387428 + 0.921900i $$0.373364\pi$$
$$608$$ 7007.67 0.467432
$$609$$ 14527.4i 0.966634i
$$610$$ −30001.4 −1.99135
$$611$$ 0 0
$$612$$ −3494.46 −0.230809
$$613$$ 20792.3i 1.36998i 0.728555 + 0.684988i $$0.240192\pi$$
−0.728555 + 0.684988i $$0.759808\pi$$
$$614$$ 19072.0 1.25356
$$615$$ 1929.18 0.126491
$$616$$ 5246.51i 0.343162i
$$617$$ − 1562.78i − 0.101969i −0.998699 0.0509846i $$-0.983764\pi$$
0.998699 0.0509846i $$-0.0162359\pi$$
$$618$$ − 7856.41i − 0.511377i
$$619$$ − 758.406i − 0.0492454i −0.999697 0.0246227i $$-0.992162\pi$$
0.999697 0.0246227i $$-0.00783844\pi$$
$$620$$ 37294.9 2.41581
$$621$$ −513.270 −0.0331672
$$622$$ − 24989.3i − 1.61090i
$$623$$ −29066.7 −1.86923
$$624$$ 0 0
$$625$$ −19485.0 −1.24704
$$626$$ − 3999.19i − 0.255335i
$$627$$ 3277.51 0.208758
$$628$$ 24355.4 1.54759
$$629$$ − 1877.15i − 0.118994i
$$630$$ 15673.8i 0.991206i
$$631$$ 14265.2i 0.899981i 0.893033 + 0.449990i $$0.148573\pi$$
−0.893033 + 0.449990i $$0.851427\pi$$
$$632$$ − 3701.81i − 0.232990i
$$633$$ 5618.56 0.352793
$$634$$ 36042.7 2.25779
$$635$$ − 3322.34i − 0.207627i
$$636$$ −11838.3 −0.738078
$$637$$ 0 0
$$638$$ 25732.0 1.59677
$$639$$ 4044.71i 0.250401i
$$640$$ 7038.88 0.434744
$$641$$ −3985.64 −0.245590 −0.122795 0.992432i $$-0.539186\pi$$
−0.122795 + 0.992432i $$0.539186\pi$$
$$642$$ − 17914.8i − 1.10131i
$$643$$ − 8156.55i − 0.500254i −0.968213 0.250127i $$-0.919528\pi$$
0.968213 0.250127i $$-0.0804723\pi$$
$$644$$ − 5375.97i − 0.328949i
$$645$$ 13796.8i 0.842243i
$$646$$ −4798.74 −0.292266
$$647$$ −11279.2 −0.685368 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$648$$ 333.972i 0.0202464i
$$649$$ −23915.2 −1.44646
$$650$$ 0 0
$$651$$ 29059.3 1.74950
$$652$$ 33104.2i 1.98844i
$$653$$ 6565.75 0.393473 0.196736 0.980456i $$-0.436966\pi$$
0.196736 + 0.980456i $$0.436966\pi$$
$$654$$ 4095.28 0.244860
$$655$$ 6348.06i 0.378686i
$$656$$ − 2631.11i − 0.156597i
$$657$$ 3507.40i 0.208275i
$$658$$ − 17291.5i − 1.02446i
$$659$$ 4799.35 0.283696 0.141848 0.989888i $$-0.454696\pi$$
0.141848 + 0.989888i $$0.454696\pi$$
$$660$$ 14697.8 0.866837
$$661$$ − 15593.6i − 0.917581i −0.888545 0.458790i $$-0.848283\pi$$
0.888545 0.458790i $$-0.151717\pi$$
$$662$$ 28811.3 1.69151
$$663$$ 0 0
$$664$$ 5361.03 0.313326
$$665$$ 11395.1i 0.664483i
$$666$$ −1614.63 −0.0939422
$$667$$ −2929.66 −0.170070
$$668$$ − 29008.0i − 1.68017i
$$669$$ 168.385i 0.00973115i
$$670$$ − 12766.7i − 0.736152i
$$671$$ 21920.8i 1.26116i
$$672$$ 24486.0 1.40561
$$673$$ 2205.54 0.126326 0.0631630 0.998003i $$-0.479881\pi$$
0.0631630 + 0.998003i $$0.479881\pi$$
$$674$$ − 17138.1i − 0.979426i
$$675$$ 1503.82 0.0857514
$$676$$ 0 0
$$677$$ −15046.4 −0.854182 −0.427091 0.904209i $$-0.640462\pi$$
−0.427091 + 0.904209i $$0.640462\pi$$
$$678$$ 8599.20i 0.487095i
$$679$$ 49033.6 2.77134
$$680$$ −2391.08 −0.134844
$$681$$ 2002.17i 0.112663i
$$682$$ − 51471.9i − 2.88997i
$$683$$ − 30632.5i − 1.71614i −0.513537 0.858068i $$-0.671665\pi$$
0.513537 0.858068i $$-0.328335\pi$$
$$684$$ 2185.21i 0.122155i
$$685$$ −24608.9 −1.37264
$$686$$ −39039.6 −2.17280
$$687$$ 2169.90i 0.120505i
$$688$$ 18816.7 1.04270
$$689$$ 0 0
$$690$$ −3160.85 −0.174393
$$691$$ 2175.72i 0.119780i 0.998205 + 0.0598901i $$0.0190750\pi$$
−0.998205 + 0.0598901i $$0.980925\pi$$
$$692$$ −24205.9 −1.32972
$$693$$ 11452.2 0.627753
$$694$$ 1288.52i 0.0704780i
$$695$$ − 1344.24i − 0.0733666i
$$696$$ 1906.25i 0.103817i
$$697$$ 2063.82i 0.112156i
$$698$$ 18383.9 0.996906
$$699$$ −826.354 −0.0447147
$$700$$ 15751.0i 0.850473i
$$701$$ 32718.2 1.76284 0.881419 0.472335i $$-0.156589\pi$$
0.881419 + 0.472335i $$0.156589\pi$$
$$702$$ 0 0
$$703$$ −1173.85 −0.0629768
$$704$$ − 25553.1i − 1.36799i
$$705$$ −5382.36 −0.287534
$$706$$ −9273.58 −0.494356
$$707$$ − 30128.6i − 1.60269i
$$708$$ − 15944.9i − 0.846395i
$$709$$ 25219.7i 1.33589i 0.744211 + 0.667945i $$0.232826\pi$$
−0.744211 + 0.667945i $$0.767174\pi$$
$$710$$ 24908.3i 1.31661i
$$711$$ −8080.38 −0.426214
$$712$$ −3814.06 −0.200756
$$713$$ 5860.22i 0.307808i
$$714$$ −16767.7 −0.878871
$$715$$ 0 0
$$716$$ −13716.4 −0.715929
$$717$$ 4588.18i 0.238980i
$$718$$ 32336.6 1.68077
$$719$$ 35466.2 1.83959 0.919796 0.392398i $$-0.128354\pi$$
0.919796 + 0.392398i $$0.128354\pi$$
$$720$$ − 6653.97i − 0.344415i
$$721$$ − 19957.7i − 1.03088i
$$722$$ − 25279.5i − 1.30306i
$$723$$ 2926.58i 0.150540i
$$724$$ 4291.93 0.220315
$$725$$ 8583.57 0.439704
$$726$$ − 3821.38i − 0.195351i
$$727$$ 14262.2 0.727588 0.363794 0.931479i $$-0.381481\pi$$
0.363794 + 0.931479i $$0.381481\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 21599.5i 1.09511i
$$731$$ −14759.6 −0.746790
$$732$$ −14615.2 −0.737970
$$733$$ 16022.5i 0.807371i 0.914898 + 0.403685i $$0.132271\pi$$
−0.914898 + 0.403685i $$0.867729\pi$$
$$734$$ − 27460.9i − 1.38093i
$$735$$ 25984.1i 1.30400i
$$736$$ 4937.96i 0.247304i
$$737$$ −9328.11 −0.466222
$$738$$ 1775.19 0.0885440
$$739$$ − 3796.21i − 0.188966i −0.995526 0.0944830i $$-0.969880\pi$$
0.995526 0.0944830i $$-0.0301198\pi$$
$$740$$ −5264.09 −0.261502
$$741$$ 0 0
$$742$$ −56804.3 −2.81044
$$743$$ − 30329.3i − 1.49754i −0.662827 0.748772i $$-0.730644\pi$$
0.662827 0.748772i $$-0.269356\pi$$
$$744$$ 3813.09 0.187896
$$745$$ −2010.40 −0.0988661
$$746$$ 152.502i 0.00748460i
$$747$$ − 11702.2i − 0.573173i
$$748$$ 15723.6i 0.768597i
$$749$$ − 45509.1i − 2.22011i
$$750$$ −11523.2 −0.561023
$$751$$ −21551.9 −1.04719 −0.523595 0.851967i $$-0.675409\pi$$
−0.523595 + 0.851967i $$0.675409\pi$$
$$752$$ 7340.72i 0.355969i
$$753$$ −5622.42 −0.272101
$$754$$ 0 0
$$755$$ −10754.3 −0.518397
$$756$$ 7635.52i 0.367329i
$$757$$ −20417.6 −0.980306 −0.490153 0.871637i $$-0.663059\pi$$
−0.490153 + 0.871637i $$0.663059\pi$$
$$758$$ 49806.5 2.38662
$$759$$ 2309.50i 0.110447i
$$760$$ 1495.23i 0.0713655i
$$761$$ 31375.4i 1.49456i 0.664512 + 0.747278i $$0.268640\pi$$
−0.664512 + 0.747278i $$0.731360\pi$$
$$762$$ − 3057.13i − 0.145339i
$$763$$ 10403.3 0.493609
$$764$$ 12327.7 0.583770
$$765$$ 5219.30i 0.246672i
$$766$$ −43570.5 −2.05518
$$767$$ 0 0
$$768$$ −8667.00 −0.407218
$$769$$ 12452.7i 0.583946i 0.956427 + 0.291973i $$0.0943118\pi$$
−0.956427 + 0.291973i $$0.905688\pi$$
$$770$$ 70525.5 3.30073
$$771$$ −5454.56 −0.254788
$$772$$ − 19302.5i − 0.899885i
$$773$$ − 37449.8i − 1.74253i −0.490815 0.871264i $$-0.663301\pi$$
0.490815 0.871264i $$-0.336699\pi$$
$$774$$ 12695.4i 0.589571i
$$775$$ − 17169.8i − 0.795815i
$$776$$ 6434.08 0.297642
$$777$$ −4101.65 −0.189377
$$778$$ − 40231.2i − 1.85393i
$$779$$ 1290.58 0.0593580
$$780$$ 0 0
$$781$$ 18199.5 0.833839
$$782$$ − 3381.44i − 0.154629i
$$783$$ 4161.01 0.189914
$$784$$ 35438.4 1.61436
$$785$$ − 36377.1i − 1.65396i
$$786$$ 5841.32i 0.265080i
$$787$$ 26460.2i 1.19848i 0.800570 + 0.599240i $$0.204530\pi$$
−0.800570 + 0.599240i $$0.795470\pi$$
$$788$$ 2157.65i 0.0975420i
$$789$$ −2021.40 −0.0912086
$$790$$ −49761.0 −2.24104
$$791$$ 21844.6i 0.981928i
$$792$$ 1502.73 0.0674207
$$793$$ 0 0
$$794$$ 58550.3 2.61697
$$795$$ 17681.6i 0.788807i
$$796$$ −14309.4 −0.637167
$$797$$ −4749.47 −0.211085 −0.105543 0.994415i $$-0.533658\pi$$
−0.105543 + 0.994415i $$0.533658\pi$$
$$798$$ 10485.4i 0.465138i
$$799$$ − 5757.99i − 0.254947i
$$800$$ − 14467.6i − 0.639386i
$$801$$ 8325.41i 0.367246i
$$802$$ 52266.1 2.30122
$$803$$ 15781.8 0.693560
$$804$$ − 6219.33i − 0.272809i
$$805$$ −8029.53 −0.351557
$$806$$ 0 0
$$807$$ 10069.2 0.439223
$$808$$ − 3953.40i − 0.172129i
$$809$$ −4464.04 −0.194002 −0.0970009 0.995284i $$-0.530925\pi$$
−0.0970009 + 0.995284i $$0.530925\pi$$
$$810$$ 4489.37 0.194741
$$811$$ − 20774.6i − 0.899499i −0.893155 0.449749i $$-0.851513\pi$$
0.893155 0.449749i $$-0.148487\pi$$
$$812$$ 43582.2i 1.88354i
$$813$$ 26746.6i 1.15381i
$$814$$ 7265.13i 0.312829i
$$815$$ 49444.3 2.12510
$$816$$ 7118.34 0.305382
$$817$$ 9229.73i 0.395235i
$$818$$ 6323.51 0.270289
$$819$$ 0 0
$$820$$ 5787.55 0.246476
$$821$$ 35769.2i 1.52053i 0.649614 + 0.760264i $$0.274930\pi$$
−0.649614 + 0.760264i $$0.725070\pi$$
$$822$$ −22644.5 −0.960848
$$823$$ −2945.44 −0.124753 −0.0623764 0.998053i $$-0.519868\pi$$
−0.0623764 + 0.998053i $$0.519868\pi$$
$$824$$ − 2618.80i − 0.110716i
$$825$$ − 6766.56i − 0.285553i
$$826$$ − 76509.6i − 3.22289i
$$827$$ − 17878.6i − 0.751753i −0.926670 0.375877i $$-0.877342\pi$$
0.926670 0.375877i $$-0.122658\pi$$
$$828$$ −1539.81 −0.0646281
$$829$$ −13423.8 −0.562398 −0.281199 0.959649i $$-0.590732\pi$$
−0.281199 + 0.959649i $$0.590732\pi$$
$$830$$ − 72065.0i − 3.01375i
$$831$$ 12051.9 0.503101
$$832$$ 0 0
$$833$$ −27797.5 −1.15621
$$834$$ − 1236.93i − 0.0513566i
$$835$$ −43326.2 −1.79565
$$836$$ 9832.53 0.406776
$$837$$ − 8323.30i − 0.343722i
$$838$$ 8930.21i 0.368125i
$$839$$ − 31542.7i − 1.29794i −0.760813 0.648971i $$-0.775200\pi$$
0.760813 0.648971i $$-0.224800\pi$$
$$840$$ 5224.61i 0.214602i
$$841$$ −638.669 −0.0261867
$$842$$ 3028.73 0.123963
$$843$$ 5523.35i 0.225663i
$$844$$ 16855.7 0.687437
$$845$$ 0 0
$$846$$ −4952.71 −0.201274
$$847$$ − 9707.47i − 0.393805i
$$848$$ 24115.0 0.976547
$$849$$ 14548.2 0.588095
$$850$$ 9907.22i 0.399782i
$$851$$ − 827.155i − 0.0333191i
$$852$$ 12134.1i 0.487920i
$$853$$ 21810.6i 0.875476i 0.899103 + 0.437738i $$0.144220\pi$$
−0.899103 + 0.437738i $$0.855780\pi$$
$$854$$ −70129.1 −2.81003
$$855$$ 3263.82 0.130550
$$856$$ − 5971.59i − 0.238440i
$$857$$ −33234.6 −1.32470 −0.662352 0.749193i $$-0.730442\pi$$
−0.662352 + 0.749193i $$0.730442\pi$$
$$858$$ 0 0
$$859$$ 42697.7 1.69596 0.847978 0.530032i $$-0.177820\pi$$
0.847978 + 0.530032i $$0.177820\pi$$
$$860$$ 41390.3i 1.64116i
$$861$$ 4509.52 0.178495
$$862$$ −56525.0 −2.23347
$$863$$ 27419.2i 1.08153i 0.841174 + 0.540765i $$0.181865\pi$$
−0.841174 + 0.540765i $$0.818135\pi$$
$$864$$ − 7013.40i − 0.276158i
$$865$$ 36153.8i 1.42112i
$$866$$ 41437.0i 1.62596i
$$867$$ 9155.45 0.358634
$$868$$ 87177.9 3.40900
$$869$$ 36358.3i 1.41930i
$$870$$ 25624.5 0.998567
$$871$$ 0 0
$$872$$ 1365.09 0.0530137
$$873$$ − 14044.4i − 0.544482i
$$874$$ −2114.54 −0.0818367
$$875$$ −29272.4 −1.13096
$$876$$ 10522.2i 0.405836i
$$877$$ 42604.2i 1.64041i 0.572068 + 0.820206i $$0.306141\pi$$
−0.572068 + 0.820206i $$0.693859\pi$$
$$878$$ − 33535.2i − 1.28902i
$$879$$ − 4241.56i − 0.162758i
$$880$$ −29940.0 −1.14691
$$881$$ 4699.40 0.179712 0.0898562 0.995955i $$-0.471359\pi$$
0.0898562 + 0.995955i $$0.471359\pi$$
$$882$$ 23909.9i 0.912799i
$$883$$ −22233.5 −0.847358 −0.423679 0.905812i $$-0.639262\pi$$
−0.423679 + 0.905812i $$0.639262\pi$$
$$884$$ 0 0
$$885$$ −23815.3 −0.904568
$$886$$ 9774.98i 0.370651i
$$887$$ −8852.46 −0.335103 −0.167552 0.985863i $$-0.553586\pi$$
−0.167552 + 0.985863i $$0.553586\pi$$
$$888$$ −538.209 −0.0203391
$$889$$ − 7766.05i − 0.292986i
$$890$$ 51270.0i 1.93098i
$$891$$ − 3280.19i − 0.123334i
$$892$$ 505.155i 0.0189617i
$$893$$ −3600.68 −0.134930
$$894$$ −1849.92 −0.0692063
$$895$$ 20486.7i 0.765135i
$$896$$ 16453.6 0.613477
$$897$$ 0 0
$$898$$ −53283.7 −1.98007
$$899$$ − 47508.0i − 1.76249i
$$900$$ 4511.47 0.167091
$$901$$ −18915.5 −0.699410
$$902$$ − 7987.58i − 0.294853i
$$903$$ 32250.3i 1.18851i
$$904$$ 2866.40i 0.105459i
$$905$$ − 6410.41i − 0.235458i
$$906$$ −9895.85 −0.362878
$$907$$ −37172.4 −1.36085 −0.680424 0.732818i $$-0.738205\pi$$
−0.680424 + 0.732818i $$0.738205\pi$$
$$908$$ 6006.51i 0.219530i
$$909$$ −8629.56 −0.314878
$$910$$ 0 0
$$911$$ 38035.1 1.38327 0.691635 0.722247i $$-0.256891\pi$$
0.691635 + 0.722247i $$0.256891\pi$$
$$912$$ − 4451.36i − 0.161622i
$$913$$ −52654.8 −1.90867
$$914$$ 34639.3 1.25357
$$915$$ 21829.2i 0.788691i
$$916$$ 6509.70i 0.234810i
$$917$$ 14838.8i 0.534372i
$$918$$ 4802.67i 0.172671i
$$919$$ −8352.27 −0.299800 −0.149900 0.988701i $$-0.547895\pi$$
−0.149900 + 0.988701i $$0.547895\pi$$
$$920$$ −1053.62 −0.0377573
$$921$$ − 13876.9i − 0.496482i
$$922$$ 72679.4 2.59606
$$923$$ 0 0
$$924$$ 34356.6 1.22321
$$925$$ 2423.47i 0.0861440i
$$926$$ 22519.6 0.799179
$$927$$ −5716.38 −0.202535
$$928$$ − 40031.3i − 1.41605i
$$929$$ 20232.1i 0.714524i 0.934004 + 0.357262i $$0.116290\pi$$
−0.934004 + 0.357262i $$0.883710\pi$$
$$930$$ − 51257.0i − 1.80729i
$$931$$ 17382.8i 0.611921i
$$932$$ −2479.06 −0.0871292
$$933$$ −18182.4 −0.638011
$$934$$ 34065.9i 1.19344i
$$935$$ 23484.7 0.821423
$$936$$ 0 0
$$937$$ −27766.9 −0.968095 −0.484048 0.875042i $$-0.660834\pi$$
−0.484048 + 0.875042i $$0.660834\pi$$
$$938$$ − 29842.6i − 1.03880i
$$939$$ −2909.84 −0.101128
$$940$$ −16147.1 −0.560277
$$941$$ − 400.765i − 0.0138837i −0.999976 0.00694185i $$-0.997790\pi$$
0.999976 0.00694185i $$-0.00220968\pi$$
$$942$$ − 33473.3i − 1.15777i
$$943$$ 909.408i 0.0314045i
$$944$$ 32480.4i 1.11986i
$$945$$ 11404.4 0.392576
$$946$$ 57124.0 1.96328
$$947$$ − 21804.4i − 0.748201i −0.927388 0.374101i $$-0.877951\pi$$
0.927388 0.374101i $$-0.122049\pi$$
$$948$$ −24241.1 −0.830502
$$949$$ 0 0
$$950$$ 6195.35 0.211583
$$951$$ − 26224.9i − 0.894217i
$$952$$ −5589.22 −0.190281
$$953$$ −30480.4 −1.03605 −0.518026 0.855365i $$-0.673333\pi$$
−0.518026 + 0.855365i $$0.673333\pi$$
$$954$$ 16270.1i 0.552165i
$$955$$ − 18412.6i − 0.623892i
$$956$$ 13764.5i 0.465666i
$$957$$ − 18722.8i − 0.632415i
$$958$$ 6497.48 0.219127
$$959$$ −57524.0 −1.93696
$$960$$ − 25446.4i − 0.855499i
$$961$$ −65239.6 −2.18991
$$962$$ 0 0
$$963$$ −13034.9 −0.436183
$$964$$ 8779.73i 0.293336i
$$965$$ −28830.1 −0.961734
$$966$$ −7388.56 −0.246090
$$967$$ 23864.1i 0.793608i 0.917903 + 0.396804i $$0.129881\pi$$
−0.917903 + 0.396804i $$0.870119\pi$$
$$968$$ − 1273.79i − 0.0422947i
$$969$$ 3491.60i 0.115755i
$$970$$ − 86489.2i − 2.86289i
$$971$$ −14010.3 −0.463040 −0.231520 0.972830i $$-0.574370\pi$$
−0.231520 + 0.972830i $$0.574370\pi$$
$$972$$ 2187.00 0.0721688
$$973$$ − 3142.19i − 0.103529i
$$974$$ −51933.3 −1.70847
$$975$$ 0 0
$$976$$ 29771.7 0.976404
$$977$$ − 25448.2i − 0.833326i −0.909061 0.416663i $$-0.863200\pi$$
0.909061 0.416663i $$-0.136800\pi$$
$$978$$ 45497.4 1.48757
$$979$$ 37460.8 1.22293
$$980$$ 77952.4i 2.54092i
$$981$$ − 2979.76i − 0.0969789i
$$982$$ − 4416.72i − 0.143527i
$$983$$ − 52479.9i − 1.70280i −0.524519 0.851399i $$-0.675755\pi$$
0.524519 0.851399i $$-0.324245\pi$$
$$984$$ 591.729 0.0191703
$$985$$ 3222.66 0.104246
$$986$$ 27412.8i 0.885398i
$$987$$ −12581.4 −0.405746
$$988$$ 0 0
$$989$$ −6503.73 −0.209107
$$990$$ − 20200.2i − 0.648491i
$$991$$ 38048.5 1.21963 0.609814 0.792545i $$-0.291244\pi$$
0.609814 + 0.792545i $$0.291244\pi$$
$$992$$ −80075.0 −2.56289
$$993$$ − 20963.3i − 0.669939i
$$994$$ 58223.9i 1.85790i
$$995$$ 21372.5i 0.680960i
$$996$$ − 35106.5i − 1.11686i
$$997$$ −31236.6 −0.992251 −0.496125 0.868251i $$-0.665244\pi$$
−0.496125 + 0.868251i $$0.665244\pi$$
$$998$$ −5864.30 −0.186003
$$999$$ 1174.81i 0.0372066i
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 507.4.b.e.337.2 4
13.5 odd 4 507.4.a.k.1.2 4
13.8 odd 4 507.4.a.k.1.3 4
13.9 even 3 39.4.j.b.10.1 yes 4
13.10 even 6 39.4.j.b.4.1 4
13.12 even 2 inner 507.4.b.e.337.3 4
39.5 even 4 1521.4.a.z.1.3 4
39.8 even 4 1521.4.a.z.1.2 4
39.23 odd 6 117.4.q.d.82.2 4
39.35 odd 6 117.4.q.d.10.2 4
52.23 odd 6 624.4.bv.c.433.1 4
52.35 odd 6 624.4.bv.c.49.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.1 4 13.10 even 6
39.4.j.b.10.1 yes 4 13.9 even 3
117.4.q.d.10.2 4 39.35 odd 6
117.4.q.d.82.2 4 39.23 odd 6
507.4.a.k.1.2 4 13.5 odd 4
507.4.a.k.1.3 4 13.8 odd 4
507.4.b.e.337.2 4 1.1 even 1 trivial
507.4.b.e.337.3 4 13.12 even 2 inner
624.4.bv.c.49.2 4 52.35 odd 6
624.4.bv.c.433.1 4 52.23 odd 6
1521.4.a.z.1.2 4 39.8 even 4
1521.4.a.z.1.3 4 39.5 even 4