# Properties

 Label 507.4.a.k.1.2 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $1$ 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - 2x^{3} - 13x^{2} + 14x - 2$$ x^4 - 2*x^3 - 13*x^2 + 14*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.170498$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.12311 −1.45774 −0.728869 0.684653i $$-0.759954\pi$$
−0.728869 + 0.684653i $$0.759954\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 9.00000 1.12500
$$5$$ 13.4424 1.20232 0.601161 0.799128i $$-0.294705\pi$$
0.601161 + 0.799128i $$0.294705\pi$$
$$6$$ 12.3693 0.841625
$$7$$ −31.4219 −1.69662 −0.848311 0.529498i $$-0.822380\pi$$
−0.848311 + 0.529498i $$0.822380\pi$$
$$8$$ −4.12311 −0.182217
$$9$$ 9.00000 0.333333
$$10$$ −55.4243 −1.75267
$$11$$ 40.4962 1.11001 0.555003 0.831849i $$-0.312717\pi$$
0.555003 + 0.831849i $$0.312717\pi$$
$$12$$ −27.0000 −0.649519
$$13$$ 0 0
$$14$$ 129.556 2.47323
$$15$$ −40.3271 −0.694161
$$16$$ −55.0000 −0.859375
$$17$$ −43.1414 −0.615490 −0.307745 0.951469i $$-0.599574\pi$$
−0.307745 + 0.951469i $$0.599574\pi$$
$$18$$ −37.1080 −0.485913
$$19$$ −26.9779 −0.325745 −0.162873 0.986647i $$-0.552076\pi$$
−0.162873 + 0.986647i $$0.552076\pi$$
$$20$$ 120.981 1.35261
$$21$$ 94.2656 0.979545
$$22$$ −166.970 −1.61810
$$23$$ −19.0100 −0.172342 −0.0861709 0.996280i $$-0.527463\pi$$
−0.0861709 + 0.996280i $$0.527463\pi$$
$$24$$ 12.3693 0.105203
$$25$$ 55.6971 0.445577
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ −282.797 −1.90870
$$29$$ −154.111 −0.986820 −0.493410 0.869797i $$-0.664250\pi$$
−0.493410 + 0.869797i $$0.664250\pi$$
$$30$$ 166.273 1.01190
$$31$$ 308.270 1.78603 0.893016 0.450025i $$-0.148585\pi$$
0.893016 + 0.450025i $$0.148585\pi$$
$$32$$ 259.756 1.43496
$$33$$ −121.489 −0.640862
$$34$$ 177.877 0.897223
$$35$$ −422.384 −2.03988
$$36$$ 81.0000 0.375000
$$37$$ 43.5116 0.193331 0.0966657 0.995317i $$-0.469182\pi$$
0.0966657 + 0.995317i $$0.469182\pi$$
$$38$$ 111.233 0.474851
$$39$$ 0 0
$$40$$ −55.4243 −0.219084
$$41$$ 47.8384 0.182222 0.0911110 0.995841i $$-0.470958\pi$$
0.0911110 + 0.995841i $$0.470958\pi$$
$$42$$ −388.667 −1.42792
$$43$$ 342.121 1.21333 0.606663 0.794959i $$-0.292508\pi$$
0.606663 + 0.794959i $$0.292508\pi$$
$$44$$ 364.466 1.24876
$$45$$ 120.981 0.400774
$$46$$ 78.3802 0.251229
$$47$$ 133.468 0.414218 0.207109 0.978318i $$-0.433594\pi$$
0.207109 + 0.978318i $$0.433594\pi$$
$$48$$ 165.000 0.496160
$$49$$ 644.334 1.87853
$$50$$ −229.645 −0.649535
$$51$$ 129.424 0.355353
$$52$$ 0 0
$$53$$ −438.454 −1.13635 −0.568173 0.822909i $$-0.692350\pi$$
−0.568173 + 0.822909i $$0.692350\pi$$
$$54$$ 111.324 0.280542
$$55$$ 544.364 1.33458
$$56$$ 129.556 0.309154
$$57$$ 80.9338 0.188069
$$58$$ 635.418 1.43852
$$59$$ 590.553 1.30311 0.651555 0.758601i $$-0.274117\pi$$
0.651555 + 0.758601i $$0.274117\pi$$
$$60$$ −362.944 −0.780931
$$61$$ −541.304 −1.13618 −0.568089 0.822967i $$-0.692317\pi$$
−0.568089 + 0.822967i $$0.692317\pi$$
$$62$$ −1271.03 −2.60357
$$63$$ −282.797 −0.565541
$$64$$ −631.000 −1.23242
$$65$$ 0 0
$$66$$ 500.910 0.934208
$$67$$ −230.345 −0.420018 −0.210009 0.977700i $$-0.567349\pi$$
−0.210009 + 0.977700i $$0.567349\pi$$
$$68$$ −388.273 −0.692426
$$69$$ 57.0300 0.0995015
$$70$$ 1741.54 2.97362
$$71$$ 449.412 0.751203 0.375601 0.926781i $$-0.377436\pi$$
0.375601 + 0.926781i $$0.377436\pi$$
$$72$$ −37.1080 −0.0607391
$$73$$ −389.711 −0.624826 −0.312413 0.949946i $$-0.601137\pi$$
−0.312413 + 0.949946i $$0.601137\pi$$
$$74$$ −179.403 −0.281826
$$75$$ −167.091 −0.257254
$$76$$ −242.801 −0.366464
$$77$$ −1272.47 −1.88326
$$78$$ 0 0
$$79$$ −897.820 −1.27864 −0.639321 0.768940i $$-0.720784\pi$$
−0.639321 + 0.768940i $$0.720784\pi$$
$$80$$ −739.330 −1.03325
$$81$$ 81.0000 0.111111
$$82$$ −197.243 −0.265632
$$83$$ −1300.24 −1.71952 −0.859759 0.510700i $$-0.829386\pi$$
−0.859759 + 0.510700i $$0.829386\pi$$
$$84$$ 848.391 1.10199
$$85$$ −579.923 −0.740017
$$86$$ −1410.60 −1.76871
$$87$$ 462.334 0.569741
$$88$$ −166.970 −0.202262
$$89$$ −925.045 −1.10174 −0.550869 0.834592i $$-0.685703\pi$$
−0.550869 + 0.834592i $$0.685703\pi$$
$$90$$ −498.819 −0.584223
$$91$$ 0 0
$$92$$ −171.090 −0.193884
$$93$$ −924.811 −1.03117
$$94$$ −550.301 −0.603822
$$95$$ −362.647 −0.391651
$$96$$ −779.267 −0.828475
$$97$$ −1560.49 −1.63344 −0.816722 0.577031i $$-0.804211\pi$$
−0.816722 + 0.577031i $$0.804211\pi$$
$$98$$ −2656.66 −2.73840
$$99$$ 364.466 0.370002
$$100$$ 501.274 0.501274
$$101$$ 958.840 0.944635 0.472318 0.881428i $$-0.343418\pi$$
0.472318 + 0.881428i $$0.343418\pi$$
$$102$$ −533.630 −0.518012
$$103$$ 635.153 0.607606 0.303803 0.952735i $$-0.401743\pi$$
0.303803 + 0.952735i $$0.401743\pi$$
$$104$$ 0 0
$$105$$ 1267.15 1.17773
$$106$$ 1807.79 1.65649
$$107$$ −1448.32 −1.30855 −0.654275 0.756257i $$-0.727026\pi$$
−0.654275 + 0.756257i $$0.727026\pi$$
$$108$$ −243.000 −0.216506
$$109$$ −331.084 −0.290937 −0.145468 0.989363i $$-0.546469\pi$$
−0.145468 + 0.989363i $$0.546469\pi$$
$$110$$ −2244.47 −1.94547
$$111$$ −130.535 −0.111620
$$112$$ 1728.20 1.45803
$$113$$ 695.204 0.578755 0.289378 0.957215i $$-0.406552\pi$$
0.289378 + 0.957215i $$0.406552\pi$$
$$114$$ −333.699 −0.274156
$$115$$ −255.539 −0.207210
$$116$$ −1387.00 −1.11017
$$117$$ 0 0
$$118$$ −2434.91 −1.89959
$$119$$ 1355.58 1.04425
$$120$$ 166.273 0.126488
$$121$$ 308.940 0.232111
$$122$$ 2231.85 1.65625
$$123$$ −143.515 −0.105206
$$124$$ 2774.43 2.00929
$$125$$ −931.594 −0.666595
$$126$$ 1166.00 0.824410
$$127$$ 247.154 0.172688 0.0863441 0.996265i $$-0.472482\pi$$
0.0863441 + 0.996265i $$0.472482\pi$$
$$128$$ 523.634 0.361587
$$129$$ −1026.36 −0.700514
$$130$$ 0 0
$$131$$ 472.243 0.314962 0.157481 0.987522i $$-0.449663\pi$$
0.157481 + 0.987522i $$0.449663\pi$$
$$132$$ −1093.40 −0.720969
$$133$$ 847.697 0.552667
$$134$$ 949.739 0.612275
$$135$$ −362.944 −0.231387
$$136$$ 177.877 0.112153
$$137$$ −1830.70 −1.14166 −0.570829 0.821069i $$-0.693378\pi$$
−0.570829 + 0.821069i $$0.693378\pi$$
$$138$$ −235.141 −0.145047
$$139$$ −100.000 −0.0610208 −0.0305104 0.999534i $$-0.509713\pi$$
−0.0305104 + 0.999534i $$0.509713\pi$$
$$140$$ −3801.46 −2.29487
$$141$$ −400.403 −0.239149
$$142$$ −1852.97 −1.09506
$$143$$ 0 0
$$144$$ −495.000 −0.286458
$$145$$ −2071.62 −1.18647
$$146$$ 1606.82 0.910832
$$147$$ −1933.00 −1.08457
$$148$$ 391.604 0.217498
$$149$$ 149.557 0.0822293 0.0411147 0.999154i $$-0.486909\pi$$
0.0411147 + 0.999154i $$0.486909\pi$$
$$150$$ 688.936 0.375009
$$151$$ −800.032 −0.431163 −0.215582 0.976486i $$-0.569165\pi$$
−0.215582 + 0.976486i $$0.569165\pi$$
$$152$$ 111.233 0.0593564
$$153$$ −388.273 −0.205163
$$154$$ 5246.51 2.74530
$$155$$ 4143.88 2.14739
$$156$$ 0 0
$$157$$ −2706.16 −1.37564 −0.687818 0.725884i $$-0.741431\pi$$
−0.687818 + 0.725884i $$0.741431\pi$$
$$158$$ 3701.81 1.86392
$$159$$ 1315.36 0.656070
$$160$$ 3491.73 1.72528
$$161$$ 597.330 0.292399
$$162$$ −333.972 −0.161971
$$163$$ 3678.25 1.76750 0.883750 0.467959i $$-0.155010\pi$$
0.883750 + 0.467959i $$0.155010\pi$$
$$164$$ 430.546 0.205000
$$165$$ −1633.09 −0.770522
$$166$$ 5361.03 2.50661
$$167$$ −3223.11 −1.49348 −0.746742 0.665114i $$-0.768383\pi$$
−0.746742 + 0.665114i $$0.768383\pi$$
$$168$$ −388.667 −0.178490
$$169$$ 0 0
$$170$$ 2391.08 1.07875
$$171$$ −242.801 −0.108582
$$172$$ 3079.09 1.36499
$$173$$ −2689.54 −1.18198 −0.590988 0.806680i $$-0.701262\pi$$
−0.590988 + 0.806680i $$0.701262\pi$$
$$174$$ −1906.25 −0.830533
$$175$$ −1750.11 −0.755976
$$176$$ −2227.29 −0.953911
$$177$$ −1771.66 −0.752351
$$178$$ 3814.06 1.60604
$$179$$ −1524.04 −0.636381 −0.318191 0.948027i $$-0.603075\pi$$
−0.318191 + 0.948027i $$0.603075\pi$$
$$180$$ 1088.83 0.450871
$$181$$ 476.881 0.195836 0.0979180 0.995194i $$-0.468782\pi$$
0.0979180 + 0.995194i $$0.468782\pi$$
$$182$$ 0 0
$$183$$ 1623.91 0.655973
$$184$$ 78.3802 0.0314036
$$185$$ 584.899 0.232446
$$186$$ 3813.09 1.50317
$$187$$ −1747.06 −0.683197
$$188$$ 1201.21 0.465996
$$189$$ 848.391 0.326515
$$190$$ 1495.23 0.570924
$$191$$ −1369.74 −0.518906 −0.259453 0.965756i $$-0.583542\pi$$
−0.259453 + 0.965756i $$0.583542\pi$$
$$192$$ 1893.00 0.711539
$$193$$ −2144.72 −0.799898 −0.399949 0.916537i $$-0.630972\pi$$
−0.399949 + 0.916537i $$0.630972\pi$$
$$194$$ 6434.08 2.38113
$$195$$ 0 0
$$196$$ 5799.01 2.11334
$$197$$ −239.739 −0.0867040 −0.0433520 0.999060i $$-0.513804\pi$$
−0.0433520 + 0.999060i $$0.513804\pi$$
$$198$$ −1502.73 −0.539365
$$199$$ −1589.94 −0.566371 −0.283185 0.959065i $$-0.591391\pi$$
−0.283185 + 0.959065i $$0.591391\pi$$
$$200$$ −229.645 −0.0811918
$$201$$ 691.036 0.242497
$$202$$ −3953.40 −1.37703
$$203$$ 4842.47 1.67426
$$204$$ 1164.82 0.399773
$$205$$ 643.061 0.219090
$$206$$ −2618.80 −0.885731
$$207$$ −171.090 −0.0574472
$$208$$ 0 0
$$209$$ −1092.50 −0.361579
$$210$$ −5224.61 −1.71682
$$211$$ −1872.85 −0.611055 −0.305527 0.952183i $$-0.598833\pi$$
−0.305527 + 0.952183i $$0.598833\pi$$
$$212$$ −3946.09 −1.27839
$$213$$ −1348.24 −0.433707
$$214$$ 5971.59 1.90752
$$215$$ 4598.92 1.45881
$$216$$ 111.324 0.0350677
$$217$$ −9686.43 −3.03022
$$218$$ 1365.09 0.424109
$$219$$ 1169.13 0.360743
$$220$$ 4899.28 1.50141
$$221$$ 0 0
$$222$$ 538.209 0.162713
$$223$$ −56.1283 −0.0168548 −0.00842742 0.999964i $$-0.502683\pi$$
−0.00842742 + 0.999964i $$0.502683\pi$$
$$224$$ −8162.01 −2.43459
$$225$$ 501.274 0.148526
$$226$$ −2866.40 −0.843673
$$227$$ −667.390 −0.195137 −0.0975687 0.995229i $$-0.531107\pi$$
−0.0975687 + 0.995229i $$0.531107\pi$$
$$228$$ 728.404 0.211578
$$229$$ 723.299 0.208720 0.104360 0.994540i $$-0.466721\pi$$
0.104360 + 0.994540i $$0.466721\pi$$
$$230$$ 1053.62 0.302058
$$231$$ 3817.40 1.08730
$$232$$ 635.418 0.179816
$$233$$ −275.451 −0.0774482 −0.0387241 0.999250i $$-0.512329\pi$$
−0.0387241 + 0.999250i $$0.512329\pi$$
$$234$$ 0 0
$$235$$ 1794.12 0.498024
$$236$$ 5314.98 1.46600
$$237$$ 2693.46 0.738224
$$238$$ −5589.22 −1.52225
$$239$$ −1529.39 −0.413925 −0.206963 0.978349i $$-0.566358\pi$$
−0.206963 + 0.978349i $$0.566358\pi$$
$$240$$ 2217.99 0.596544
$$241$$ 975.526 0.260743 0.130372 0.991465i $$-0.458383\pi$$
0.130372 + 0.991465i $$0.458383\pi$$
$$242$$ −1273.79 −0.338357
$$243$$ −243.000 −0.0641500
$$244$$ −4871.74 −1.27820
$$245$$ 8661.38 2.25859
$$246$$ 591.729 0.153363
$$247$$ 0 0
$$248$$ −1271.03 −0.325446
$$249$$ 3900.72 0.992765
$$250$$ 3841.06 0.971720
$$251$$ −1874.14 −0.471294 −0.235647 0.971839i $$-0.575721\pi$$
−0.235647 + 0.971839i $$0.575721\pi$$
$$252$$ −2545.17 −0.636233
$$253$$ −769.832 −0.191300
$$254$$ −1019.04 −0.251734
$$255$$ 1739.77 0.427249
$$256$$ 2889.00 0.705322
$$257$$ −1818.19 −0.441305 −0.220653 0.975352i $$-0.570819\pi$$
−0.220653 + 0.975352i $$0.570819\pi$$
$$258$$ 4231.81 1.02117
$$259$$ −1367.22 −0.328010
$$260$$ 0 0
$$261$$ −1387.00 −0.328940
$$262$$ −1947.11 −0.459132
$$263$$ 673.799 0.157978 0.0789890 0.996875i $$-0.474831\pi$$
0.0789890 + 0.996875i $$0.474831\pi$$
$$264$$ 500.910 0.116776
$$265$$ −5893.86 −1.36625
$$266$$ −3495.14 −0.805643
$$267$$ 2775.14 0.636088
$$268$$ −2073.11 −0.472520
$$269$$ −3356.40 −0.760756 −0.380378 0.924831i $$-0.624206\pi$$
−0.380378 + 0.924831i $$0.624206\pi$$
$$270$$ 1496.46 0.337301
$$271$$ 8915.55 1.99845 0.999227 0.0393133i $$-0.0125170\pi$$
0.999227 + 0.0393133i $$0.0125170\pi$$
$$272$$ 2372.78 0.528937
$$273$$ 0 0
$$274$$ 7548.16 1.66424
$$275$$ 2255.52 0.494593
$$276$$ 513.270 0.111939
$$277$$ 4017.31 0.871396 0.435698 0.900093i $$-0.356502\pi$$
0.435698 + 0.900093i $$0.356502\pi$$
$$278$$ 412.311 0.0889523
$$279$$ 2774.43 0.595344
$$280$$ 1741.54 0.371702
$$281$$ 1841.12 0.390860 0.195430 0.980718i $$-0.437390\pi$$
0.195430 + 0.980718i $$0.437390\pi$$
$$282$$ 1650.90 0.348617
$$283$$ 4849.40 1.01861 0.509305 0.860586i $$-0.329902\pi$$
0.509305 + 0.860586i $$0.329902\pi$$
$$284$$ 4044.71 0.845103
$$285$$ 1087.94 0.226120
$$286$$ 0 0
$$287$$ −1503.17 −0.309162
$$288$$ 2337.80 0.478320
$$289$$ −3051.82 −0.621172
$$290$$ 8541.52 1.72957
$$291$$ 4681.48 0.943070
$$292$$ −3507.40 −0.702929
$$293$$ −1413.85 −0.281905 −0.140953 0.990016i $$-0.545017\pi$$
−0.140953 + 0.990016i $$0.545017\pi$$
$$294$$ 7969.97 1.58101
$$295$$ 7938.43 1.56676
$$296$$ −179.403 −0.0352283
$$297$$ −1093.40 −0.213621
$$298$$ −616.639 −0.119869
$$299$$ 0 0
$$300$$ −1503.82 −0.289411
$$301$$ −10750.1 −2.05856
$$302$$ 3298.62 0.628523
$$303$$ −2876.52 −0.545385
$$304$$ 1483.79 0.279937
$$305$$ −7276.41 −1.36605
$$306$$ 1600.89 0.299074
$$307$$ −4625.64 −0.859932 −0.429966 0.902845i $$-0.641474\pi$$
−0.429966 + 0.902845i $$0.641474\pi$$
$$308$$ −11452.2 −2.11867
$$309$$ −1905.46 −0.350802
$$310$$ −17085.7 −3.13032
$$311$$ −6060.79 −1.10507 −0.552534 0.833490i $$-0.686339\pi$$
−0.552534 + 0.833490i $$0.686339\pi$$
$$312$$ 0 0
$$313$$ 969.946 0.175158 0.0875792 0.996158i $$-0.472087\pi$$
0.0875792 + 0.996158i $$0.472087\pi$$
$$314$$ 11157.8 2.00532
$$315$$ −3801.46 −0.679962
$$316$$ −8080.38 −1.43847
$$317$$ 8741.63 1.54883 0.774414 0.632679i $$-0.218045\pi$$
0.774414 + 0.632679i $$0.218045\pi$$
$$318$$ −5423.38 −0.956378
$$319$$ −6240.92 −1.09537
$$320$$ −8482.13 −1.48177
$$321$$ 4344.97 0.755491
$$322$$ −2462.85 −0.426241
$$323$$ 1163.87 0.200493
$$324$$ 729.000 0.125000
$$325$$ 0 0
$$326$$ −15165.8 −2.57655
$$327$$ 993.252 0.167972
$$328$$ −197.243 −0.0332040
$$329$$ −4193.81 −0.702772
$$330$$ 6733.41 1.12322
$$331$$ 6987.76 1.16037 0.580184 0.814485i $$-0.302981\pi$$
0.580184 + 0.814485i $$0.302981\pi$$
$$332$$ −11702.2 −1.93446
$$333$$ 391.604 0.0644438
$$334$$ 13289.2 2.17711
$$335$$ −3096.39 −0.504996
$$336$$ −5184.61 −0.841797
$$337$$ −4156.59 −0.671881 −0.335940 0.941883i $$-0.609054\pi$$
−0.335940 + 0.941883i $$0.609054\pi$$
$$338$$ 0 0
$$339$$ −2085.61 −0.334144
$$340$$ −5219.30 −0.832519
$$341$$ 12483.8 1.98250
$$342$$ 1001.10 0.158284
$$343$$ −9468.49 −1.49053
$$344$$ −1410.60 −0.221089
$$345$$ 766.618 0.119633
$$346$$ 11089.3 1.72301
$$347$$ −312.513 −0.0483475 −0.0241737 0.999708i $$-0.507695\pi$$
−0.0241737 + 0.999708i $$0.507695\pi$$
$$348$$ 4161.01 0.640958
$$349$$ −4458.75 −0.683872 −0.341936 0.939723i $$-0.611083\pi$$
−0.341936 + 0.939723i $$0.611083\pi$$
$$350$$ 7215.88 1.10201
$$351$$ 0 0
$$352$$ 10519.1 1.59281
$$353$$ −2249.17 −0.339126 −0.169563 0.985519i $$-0.554236\pi$$
−0.169563 + 0.985519i $$0.554236\pi$$
$$354$$ 7304.74 1.09673
$$355$$ 6041.16 0.903187
$$356$$ −8325.41 −1.23945
$$357$$ −4066.75 −0.602900
$$358$$ 6283.78 0.927677
$$359$$ −7842.79 −1.15300 −0.576499 0.817098i $$-0.695582\pi$$
−0.576499 + 0.817098i $$0.695582\pi$$
$$360$$ −498.819 −0.0730279
$$361$$ −6131.19 −0.893890
$$362$$ −1966.23 −0.285478
$$363$$ −926.820 −0.134009
$$364$$ 0 0
$$365$$ −5238.64 −0.751241
$$366$$ −6695.56 −0.956237
$$367$$ 6660.24 0.947307 0.473653 0.880711i $$-0.342935\pi$$
0.473653 + 0.880711i $$0.342935\pi$$
$$368$$ 1045.55 0.148106
$$369$$ 430.546 0.0607407
$$370$$ −2411.60 −0.338846
$$371$$ 13777.1 1.92795
$$372$$ −8323.30 −1.16006
$$373$$ −36.9873 −0.00513439 −0.00256720 0.999997i $$-0.500817\pi$$
−0.00256720 + 0.999997i $$0.500817\pi$$
$$374$$ 7203.32 0.995923
$$375$$ 2794.78 0.384859
$$376$$ −550.301 −0.0754777
$$377$$ 0 0
$$378$$ −3498.00 −0.475973
$$379$$ 12079.9 1.63721 0.818603 0.574360i $$-0.194749\pi$$
0.818603 + 0.574360i $$0.194749\pi$$
$$380$$ −3263.82 −0.440607
$$381$$ −741.463 −0.0997015
$$382$$ 5647.59 0.756429
$$383$$ −10567.4 −1.40984 −0.704919 0.709287i $$-0.749017\pi$$
−0.704919 + 0.709287i $$0.749017\pi$$
$$384$$ −1570.90 −0.208763
$$385$$ −17104.9 −2.26428
$$386$$ 8842.90 1.16604
$$387$$ 3079.09 0.404442
$$388$$ −14044.4 −1.83763
$$389$$ −9757.49 −1.27179 −0.635893 0.771778i $$-0.719368\pi$$
−0.635893 + 0.771778i $$0.719368\pi$$
$$390$$ 0 0
$$391$$ 820.119 0.106075
$$392$$ −2656.66 −0.342300
$$393$$ −1416.73 −0.181844
$$394$$ 988.469 0.126392
$$395$$ −12068.8 −1.53734
$$396$$ 3280.19 0.416252
$$397$$ −14200.5 −1.79522 −0.897612 0.440786i $$-0.854700\pi$$
−0.897612 + 0.440786i $$0.854700\pi$$
$$398$$ 6555.48 0.825620
$$399$$ −2543.09 −0.319082
$$400$$ −3063.34 −0.382918
$$401$$ −12676.4 −1.57863 −0.789313 0.613992i $$-0.789563\pi$$
−0.789313 + 0.613992i $$0.789563\pi$$
$$402$$ −2849.22 −0.353497
$$403$$ 0 0
$$404$$ 8629.56 1.06271
$$405$$ 1088.83 0.133591
$$406$$ −19966.0 −2.44063
$$407$$ 1762.05 0.214599
$$408$$ −533.630 −0.0647515
$$409$$ 1533.68 0.185417 0.0927083 0.995693i $$-0.470448\pi$$
0.0927083 + 0.995693i $$0.470448\pi$$
$$410$$ −2651.41 −0.319375
$$411$$ 5492.09 0.659136
$$412$$ 5716.38 0.683557
$$413$$ −18556.3 −2.21089
$$414$$ 705.422 0.0837430
$$415$$ −17478.3 −2.06741
$$416$$ 0 0
$$417$$ 300.000 0.0352304
$$418$$ 4504.50 0.527087
$$419$$ −2165.89 −0.252532 −0.126266 0.991996i $$-0.540299\pi$$
−0.126266 + 0.991996i $$0.540299\pi$$
$$420$$ 11404.4 1.32494
$$421$$ 734.575 0.0850380 0.0425190 0.999096i $$-0.486462\pi$$
0.0425190 + 0.999096i $$0.486462\pi$$
$$422$$ 7721.98 0.890758
$$423$$ 1201.21 0.138073
$$424$$ 1807.79 0.207062
$$425$$ −2402.85 −0.274248
$$426$$ 5558.92 0.632231
$$427$$ 17008.8 1.92767
$$428$$ −13034.9 −1.47212
$$429$$ 0 0
$$430$$ −18961.8 −2.12656
$$431$$ −13709.3 −1.53215 −0.766073 0.642754i $$-0.777792\pi$$
−0.766073 + 0.642754i $$0.777792\pi$$
$$432$$ 1485.00 0.165387
$$433$$ 10049.9 1.11540 0.557701 0.830042i $$-0.311684\pi$$
0.557701 + 0.830042i $$0.311684\pi$$
$$434$$ 39938.2 4.41727
$$435$$ 6214.87 0.685011
$$436$$ −2979.76 −0.327304
$$437$$ 512.850 0.0561395
$$438$$ −4820.46 −0.525869
$$439$$ −8133.47 −0.884258 −0.442129 0.896951i $$-0.645777\pi$$
−0.442129 + 0.896951i $$0.645777\pi$$
$$440$$ −2244.47 −0.243184
$$441$$ 5799.01 0.626175
$$442$$ 0 0
$$443$$ −2370.78 −0.254264 −0.127132 0.991886i $$-0.540577\pi$$
−0.127132 + 0.991886i $$0.540577\pi$$
$$444$$ −1174.81 −0.125572
$$445$$ −12434.8 −1.32464
$$446$$ 231.423 0.0245699
$$447$$ −448.670 −0.0474751
$$448$$ 19827.2 2.09095
$$449$$ 12923.2 1.35832 0.679158 0.733992i $$-0.262345\pi$$
0.679158 + 0.733992i $$0.262345\pi$$
$$450$$ −2066.81 −0.216512
$$451$$ 1937.27 0.202267
$$452$$ 6256.84 0.651099
$$453$$ 2400.10 0.248932
$$454$$ 2751.72 0.284459
$$455$$ 0 0
$$456$$ −333.699 −0.0342694
$$457$$ 8401.26 0.859944 0.429972 0.902842i $$-0.358523\pi$$
0.429972 + 0.902842i $$0.358523\pi$$
$$458$$ −2982.24 −0.304260
$$459$$ 1164.82 0.118451
$$460$$ −2299.85 −0.233111
$$461$$ 17627.3 1.78088 0.890441 0.455098i $$-0.150396\pi$$
0.890441 + 0.455098i $$0.150396\pi$$
$$462$$ −15739.5 −1.58500
$$463$$ −5461.81 −0.548233 −0.274116 0.961697i $$-0.588385\pi$$
−0.274116 + 0.961697i $$0.588385\pi$$
$$464$$ 8476.13 0.848048
$$465$$ −12431.6 −1.23979
$$466$$ 1135.72 0.112899
$$467$$ 8262.19 0.818691 0.409345 0.912379i $$-0.365757\pi$$
0.409345 + 0.912379i $$0.365757\pi$$
$$468$$ 0 0
$$469$$ 7237.89 0.712611
$$470$$ −7397.35 −0.725988
$$471$$ 8118.47 0.794223
$$472$$ −2434.91 −0.237449
$$473$$ 13854.6 1.34680
$$474$$ −11105.4 −1.07614
$$475$$ −1502.59 −0.145145
$$476$$ 12200.3 1.17479
$$477$$ −3946.09 −0.378782
$$478$$ 6305.84 0.603395
$$479$$ −1575.87 −0.150320 −0.0751601 0.997171i $$-0.523947\pi$$
−0.0751601 + 0.997171i $$0.523947\pi$$
$$480$$ −10475.2 −0.996093
$$481$$ 0 0
$$482$$ −4022.20 −0.380095
$$483$$ −1791.99 −0.168816
$$484$$ 2780.46 0.261125
$$485$$ −20976.7 −1.96393
$$486$$ 1001.91 0.0935139
$$487$$ −12595.7 −1.17200 −0.586001 0.810310i $$-0.699299\pi$$
−0.586001 + 0.810310i $$0.699299\pi$$
$$488$$ 2231.85 0.207031
$$489$$ −11034.7 −1.02047
$$490$$ −35711.8 −3.29244
$$491$$ −1071.21 −0.0984586 −0.0492293 0.998788i $$-0.515677\pi$$
−0.0492293 + 0.998788i $$0.515677\pi$$
$$492$$ −1291.64 −0.118357
$$493$$ 6648.59 0.607378
$$494$$ 0 0
$$495$$ 4899.28 0.444861
$$496$$ −16954.9 −1.53487
$$497$$ −14121.4 −1.27451
$$498$$ −16083.1 −1.44719
$$499$$ −1422.30 −0.127597 −0.0637985 0.997963i $$-0.520322\pi$$
−0.0637985 + 0.997963i $$0.520322\pi$$
$$500$$ −8384.35 −0.749919
$$501$$ 9669.33 0.862263
$$502$$ 7727.28 0.687022
$$503$$ −9349.34 −0.828760 −0.414380 0.910104i $$-0.636002\pi$$
−0.414380 + 0.910104i $$0.636002\pi$$
$$504$$ 1166.00 0.103051
$$505$$ 12889.1 1.13576
$$506$$ 3174.10 0.278866
$$507$$ 0 0
$$508$$ 2224.39 0.194274
$$509$$ 13736.3 1.19617 0.598086 0.801432i $$-0.295928\pi$$
0.598086 + 0.801432i $$0.295928\pi$$
$$510$$ −7173.25 −0.622817
$$511$$ 12245.5 1.06009
$$512$$ −16100.7 −1.38976
$$513$$ 728.404 0.0626897
$$514$$ 7496.58 0.643307
$$515$$ 8537.96 0.730538
$$516$$ −9237.28 −0.788079
$$517$$ 5404.93 0.459785
$$518$$ 5637.17 0.478153
$$519$$ 8068.62 0.682414
$$520$$ 0 0
$$521$$ 11052.3 0.929386 0.464693 0.885472i $$-0.346165\pi$$
0.464693 + 0.885472i $$0.346165\pi$$
$$522$$ 5718.76 0.479508
$$523$$ 6477.04 0.541532 0.270766 0.962645i $$-0.412723\pi$$
0.270766 + 0.962645i $$0.412723\pi$$
$$524$$ 4250.19 0.354332
$$525$$ 5250.33 0.436463
$$526$$ −2778.14 −0.230290
$$527$$ −13299.2 −1.09929
$$528$$ 6681.87 0.550741
$$529$$ −11805.6 −0.970298
$$530$$ 24301.0 1.99164
$$531$$ 5314.98 0.434370
$$532$$ 7629.27 0.621750
$$533$$ 0 0
$$534$$ −11442.2 −0.927250
$$535$$ −19468.9 −1.57330
$$536$$ 949.739 0.0765344
$$537$$ 4572.12 0.367415
$$538$$ 13838.8 1.10898
$$539$$ 26093.1 2.08517
$$540$$ −3266.49 −0.260310
$$541$$ 18341.5 1.45761 0.728803 0.684723i $$-0.240077\pi$$
0.728803 + 0.684723i $$0.240077\pi$$
$$542$$ −36759.7 −2.91322
$$543$$ −1430.64 −0.113066
$$544$$ −11206.2 −0.883204
$$545$$ −4450.55 −0.349799
$$546$$ 0 0
$$547$$ −18943.1 −1.48071 −0.740356 0.672215i $$-0.765343\pi$$
−0.740356 + 0.672215i $$0.765343\pi$$
$$548$$ −16476.3 −1.28436
$$549$$ −4871.74 −0.378726
$$550$$ −9299.75 −0.720987
$$551$$ 4157.61 0.321452
$$552$$ −235.141 −0.0181309
$$553$$ 28211.2 2.16937
$$554$$ −16563.8 −1.27027
$$555$$ −1754.70 −0.134203
$$556$$ −900.000 −0.0686484
$$557$$ 415.532 0.0316098 0.0158049 0.999875i $$-0.494969\pi$$
0.0158049 + 0.999875i $$0.494969\pi$$
$$558$$ −11439.3 −0.867856
$$559$$ 0 0
$$560$$ 23231.1 1.75303
$$561$$ 5241.19 0.394444
$$562$$ −7591.12 −0.569772
$$563$$ 18291.8 1.36929 0.684643 0.728879i $$-0.259958\pi$$
0.684643 + 0.728879i $$0.259958\pi$$
$$564$$ −3603.63 −0.269043
$$565$$ 9345.19 0.695850
$$566$$ −19994.6 −1.48487
$$567$$ −2545.17 −0.188514
$$568$$ −1852.97 −0.136882
$$569$$ 4347.47 0.320308 0.160154 0.987092i $$-0.448801\pi$$
0.160154 + 0.987092i $$0.448801\pi$$
$$570$$ −4485.70 −0.329623
$$571$$ −16756.0 −1.22805 −0.614024 0.789288i $$-0.710450\pi$$
−0.614024 + 0.789288i $$0.710450\pi$$
$$572$$ 0 0
$$573$$ 4109.23 0.299591
$$574$$ 6197.74 0.450677
$$575$$ −1058.80 −0.0767915
$$576$$ −5679.00 −0.410807
$$577$$ 19974.7 1.44117 0.720587 0.693364i $$-0.243872\pi$$
0.720587 + 0.693364i $$0.243872\pi$$
$$578$$ 12583.0 0.905506
$$579$$ 6434.16 0.461821
$$580$$ −18644.6 −1.33478
$$581$$ 40856.0 2.91737
$$582$$ −19302.2 −1.37475
$$583$$ −17755.7 −1.26135
$$584$$ 1606.82 0.113854
$$585$$ 0 0
$$586$$ 5829.47 0.410944
$$587$$ −15748.7 −1.10735 −0.553677 0.832732i $$-0.686776\pi$$
−0.553677 + 0.832732i $$0.686776\pi$$
$$588$$ −17397.0 −1.22014
$$589$$ −8316.50 −0.581792
$$590$$ −32731.0 −2.28392
$$591$$ 719.217 0.0500586
$$592$$ −2393.14 −0.166144
$$593$$ 13318.4 0.922297 0.461148 0.887323i $$-0.347438\pi$$
0.461148 + 0.887323i $$0.347438\pi$$
$$594$$ 4508.19 0.311403
$$595$$ 18222.3 1.25553
$$596$$ 1346.01 0.0925080
$$597$$ 4769.82 0.326994
$$598$$ 0 0
$$599$$ 2970.80 0.202644 0.101322 0.994854i $$-0.467693\pi$$
0.101322 + 0.994854i $$0.467693\pi$$
$$600$$ 688.936 0.0468761
$$601$$ 10632.6 0.721654 0.360827 0.932633i $$-0.382495\pi$$
0.360827 + 0.932633i $$0.382495\pi$$
$$602$$ 44323.8 3.00083
$$603$$ −2073.11 −0.140006
$$604$$ −7200.29 −0.485059
$$605$$ 4152.88 0.279072
$$606$$ 11860.2 0.795029
$$607$$ 11587.9 0.774856 0.387428 0.921900i $$-0.373364\pi$$
0.387428 + 0.921900i $$0.373364\pi$$
$$608$$ −7007.67 −0.467432
$$609$$ −14527.4 −0.966634
$$610$$ 30001.4 1.99135
$$611$$ 0 0
$$612$$ −3494.46 −0.230809
$$613$$ 20792.3 1.36998 0.684988 0.728555i $$-0.259808\pi$$
0.684988 + 0.728555i $$0.259808\pi$$
$$614$$ 19072.0 1.25356
$$615$$ −1929.18 −0.126491
$$616$$ 5246.51 0.343162
$$617$$ −1562.78 −0.101969 −0.0509846 0.998699i $$-0.516236\pi$$
−0.0509846 + 0.998699i $$0.516236\pi$$
$$618$$ 7856.41 0.511377
$$619$$ 758.406 0.0492454 0.0246227 0.999697i $$-0.492162\pi$$
0.0246227 + 0.999697i $$0.492162\pi$$
$$620$$ 37294.9 2.41581
$$621$$ 513.270 0.0331672
$$622$$ 24989.3 1.61090
$$623$$ 29066.7 1.86923
$$624$$ 0 0
$$625$$ −19485.0 −1.24704
$$626$$ −3999.19 −0.255335
$$627$$ 3277.51 0.208758
$$628$$ −24355.4 −1.54759
$$629$$ −1877.15 −0.118994
$$630$$ 15673.8 0.991206
$$631$$ −14265.2 −0.899981 −0.449990 0.893033i $$-0.648573\pi$$
−0.449990 + 0.893033i $$0.648573\pi$$
$$632$$ 3701.81 0.232990
$$633$$ 5618.56 0.352793
$$634$$ −36042.7 −2.25779
$$635$$ 3322.34 0.207627
$$636$$ 11838.3 0.738078
$$637$$ 0 0
$$638$$ 25732.0 1.59677
$$639$$ 4044.71 0.250401
$$640$$ 7038.88 0.434744
$$641$$ 3985.64 0.245590 0.122795 0.992432i $$-0.460814\pi$$
0.122795 + 0.992432i $$0.460814\pi$$
$$642$$ −17914.8 −1.10131
$$643$$ −8156.55 −0.500254 −0.250127 0.968213i $$-0.580472\pi$$
−0.250127 + 0.968213i $$0.580472\pi$$
$$644$$ 5375.97 0.328949
$$645$$ −13796.8 −0.842243
$$646$$ −4798.74 −0.292266
$$647$$ 11279.2 0.685368 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$648$$ −333.972 −0.0202464
$$649$$ 23915.2 1.44646
$$650$$ 0 0
$$651$$ 29059.3 1.74950
$$652$$ 33104.2 1.98844
$$653$$ 6565.75 0.393473 0.196736 0.980456i $$-0.436966\pi$$
0.196736 + 0.980456i $$0.436966\pi$$
$$654$$ −4095.28 −0.244860
$$655$$ 6348.06 0.378686
$$656$$ −2631.11 −0.156597
$$657$$ −3507.40 −0.208275
$$658$$ 17291.5 1.02446
$$659$$ 4799.35 0.283696 0.141848 0.989888i $$-0.454696\pi$$
0.141848 + 0.989888i $$0.454696\pi$$
$$660$$ −14697.8 −0.866837
$$661$$ 15593.6 0.917581 0.458790 0.888545i $$-0.348283\pi$$
0.458790 + 0.888545i $$0.348283\pi$$
$$662$$ −28811.3 −1.69151
$$663$$ 0 0
$$664$$ 5361.03 0.313326
$$665$$ 11395.1 0.664483
$$666$$ −1614.63 −0.0939422
$$667$$ 2929.66 0.170070
$$668$$ −29008.0 −1.68017
$$669$$ 168.385 0.00973115
$$670$$ 12766.7 0.736152
$$671$$ −21920.8 −1.26116
$$672$$ 24486.0 1.40561
$$673$$ −2205.54 −0.126326 −0.0631630 0.998003i $$-0.520119\pi$$
−0.0631630 + 0.998003i $$0.520119\pi$$
$$674$$ 17138.1 0.979426
$$675$$ −1503.82 −0.0857514
$$676$$ 0 0
$$677$$ −15046.4 −0.854182 −0.427091 0.904209i $$-0.640462\pi$$
−0.427091 + 0.904209i $$0.640462\pi$$
$$678$$ 8599.20 0.487095
$$679$$ 49033.6 2.77134
$$680$$ 2391.08 0.134844
$$681$$ 2002.17 0.112663
$$682$$ −51471.9 −2.88997
$$683$$ 30632.5 1.71614 0.858068 0.513537i $$-0.171665\pi$$
0.858068 + 0.513537i $$0.171665\pi$$
$$684$$ −2185.21 −0.122155
$$685$$ −24608.9 −1.37264
$$686$$ 39039.6 2.17280
$$687$$ −2169.90 −0.120505
$$688$$ −18816.7 −1.04270
$$689$$ 0 0
$$690$$ −3160.85 −0.174393
$$691$$ 2175.72 0.119780 0.0598901 0.998205i $$-0.480925\pi$$
0.0598901 + 0.998205i $$0.480925\pi$$
$$692$$ −24205.9 −1.32972
$$693$$ −11452.2 −0.627753
$$694$$ 1288.52 0.0704780
$$695$$ −1344.24 −0.0733666
$$696$$ −1906.25 −0.103817
$$697$$ −2063.82 −0.112156
$$698$$ 18383.9 0.996906
$$699$$ 826.354 0.0447147
$$700$$ −15751.0 −0.850473
$$701$$ −32718.2 −1.76284 −0.881419 0.472335i $$-0.843411\pi$$
−0.881419 + 0.472335i $$0.843411\pi$$
$$702$$ 0 0
$$703$$ −1173.85 −0.0629768
$$704$$ −25553.1 −1.36799
$$705$$ −5382.36 −0.287534
$$706$$ 9273.58 0.494356
$$707$$ −30128.6 −1.60269
$$708$$ −15944.9 −0.846395
$$709$$ −25219.7 −1.33589 −0.667945 0.744211i $$-0.732826\pi$$
−0.667945 + 0.744211i $$0.732826\pi$$
$$710$$ −24908.3 −1.31661
$$711$$ −8080.38 −0.426214
$$712$$ 3814.06 0.200756
$$713$$ −5860.22 −0.307808
$$714$$ 16767.7 0.878871
$$715$$ 0 0
$$716$$ −13716.4 −0.715929
$$717$$ 4588.18 0.238980
$$718$$ 32336.6 1.68077
$$719$$ −35466.2 −1.83959 −0.919796 0.392398i $$-0.871646\pi$$
−0.919796 + 0.392398i $$0.871646\pi$$
$$720$$ −6653.97 −0.344415
$$721$$ −19957.7 −1.03088
$$722$$ 25279.5 1.30306
$$723$$ −2926.58 −0.150540
$$724$$ 4291.93 0.220315
$$725$$ −8583.57 −0.439704
$$726$$ 3821.38 0.195351
$$727$$ −14262.2 −0.727588 −0.363794 0.931479i $$-0.618519\pi$$
−0.363794 + 0.931479i $$0.618519\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 21599.5 1.09511
$$731$$ −14759.6 −0.746790
$$732$$ 14615.2 0.737970
$$733$$ 16022.5 0.807371 0.403685 0.914898i $$-0.367729\pi$$
0.403685 + 0.914898i $$0.367729\pi$$
$$734$$ −27460.9 −1.38093
$$735$$ −25984.1 −1.30400
$$736$$ −4937.96 −0.247304
$$737$$ −9328.11 −0.466222
$$738$$ −1775.19 −0.0885440
$$739$$ 3796.21 0.188966 0.0944830 0.995526i $$-0.469880\pi$$
0.0944830 + 0.995526i $$0.469880\pi$$
$$740$$ 5264.09 0.261502
$$741$$ 0 0
$$742$$ −56804.3 −2.81044
$$743$$ −30329.3 −1.49754 −0.748772 0.662827i $$-0.769356\pi$$
−0.748772 + 0.662827i $$0.769356\pi$$
$$744$$ 3813.09 0.187896
$$745$$ 2010.40 0.0988661
$$746$$ 152.502 0.00748460
$$747$$ −11702.2 −0.573173
$$748$$ −15723.6 −0.768597
$$749$$ 45509.1 2.22011
$$750$$ −11523.2 −0.561023
$$751$$ 21551.9 1.04719 0.523595 0.851967i $$-0.324591\pi$$
0.523595 + 0.851967i $$0.324591\pi$$
$$752$$ −7340.72 −0.355969
$$753$$ 5622.42 0.272101
$$754$$ 0 0
$$755$$ −10754.3 −0.518397
$$756$$ 7635.52 0.367329
$$757$$ −20417.6 −0.980306 −0.490153 0.871637i $$-0.663059\pi$$
−0.490153 + 0.871637i $$0.663059\pi$$
$$758$$ −49806.5 −2.38662
$$759$$ 2309.50 0.110447
$$760$$ 1495.23 0.0713655
$$761$$ −31375.4 −1.49456 −0.747278 0.664512i $$-0.768640\pi$$
−0.747278 + 0.664512i $$0.768640\pi$$
$$762$$ 3057.13 0.145339
$$763$$ 10403.3 0.493609
$$764$$ −12327.7 −0.583770
$$765$$ −5219.30 −0.246672
$$766$$ 43570.5 2.05518
$$767$$ 0 0
$$768$$ −8667.00 −0.407218
$$769$$ 12452.7 0.583946 0.291973 0.956427i $$-0.405688\pi$$
0.291973 + 0.956427i $$0.405688\pi$$
$$770$$ 70525.5 3.30073
$$771$$ 5454.56 0.254788
$$772$$ −19302.5 −0.899885
$$773$$ −37449.8 −1.74253 −0.871264 0.490815i $$-0.836699\pi$$
−0.871264 + 0.490815i $$0.836699\pi$$
$$774$$ −12695.4 −0.589571
$$775$$ 17169.8 0.795815
$$776$$ 6434.08 0.297642
$$777$$ 4101.65 0.189377
$$778$$ 40231.2 1.85393
$$779$$ −1290.58 −0.0593580
$$780$$ 0 0
$$781$$ 18199.5 0.833839
$$782$$ −3381.44 −0.154629
$$783$$ 4161.01 0.189914
$$784$$ −35438.4 −1.61436
$$785$$ −36377.1 −1.65396
$$786$$ 5841.32 0.265080
$$787$$ −26460.2 −1.19848 −0.599240 0.800570i $$-0.704530\pi$$
−0.599240 + 0.800570i $$0.704530\pi$$
$$788$$ −2157.65 −0.0975420
$$789$$ −2021.40 −0.0912086
$$790$$ 49761.0 2.24104
$$791$$ −21844.6 −0.981928
$$792$$ −1502.73 −0.0674207
$$793$$ 0 0
$$794$$ 58550.3 2.61697
$$795$$ 17681.6 0.788807
$$796$$ −14309.4 −0.637167
$$797$$ 4749.47 0.211085 0.105543 0.994415i $$-0.466342\pi$$
0.105543 + 0.994415i $$0.466342\pi$$
$$798$$ 10485.4 0.465138
$$799$$ −5757.99 −0.254947
$$800$$ 14467.6 0.639386
$$801$$ −8325.41 −0.367246
$$802$$ 52266.1 2.30122
$$803$$ −15781.8 −0.693560
$$804$$ 6219.33 0.272809
$$805$$ 8029.53 0.351557
$$806$$ 0 0
$$807$$ 10069.2 0.439223
$$808$$ −3953.40 −0.172129
$$809$$ −4464.04 −0.194002 −0.0970009 0.995284i $$-0.530925\pi$$
−0.0970009 + 0.995284i $$0.530925\pi$$
$$810$$ −4489.37 −0.194741
$$811$$ −20774.6 −0.899499 −0.449749 0.893155i $$-0.648487\pi$$
−0.449749 + 0.893155i $$0.648487\pi$$
$$812$$ 43582.2 1.88354
$$813$$ −26746.6 −1.15381
$$814$$ −7265.13 −0.312829
$$815$$ 49444.3 2.12510
$$816$$ −7118.34 −0.305382
$$817$$ −9229.73 −0.395235
$$818$$ −6323.51 −0.270289
$$819$$ 0 0
$$820$$ 5787.55 0.246476
$$821$$ 35769.2 1.52053 0.760264 0.649614i $$-0.225070\pi$$
0.760264 + 0.649614i $$0.225070\pi$$
$$822$$ −22644.5 −0.960848
$$823$$ 2945.44 0.124753 0.0623764 0.998053i $$-0.480132\pi$$
0.0623764 + 0.998053i $$0.480132\pi$$
$$824$$ −2618.80 −0.110716
$$825$$ −6766.56 −0.285553
$$826$$ 76509.6 3.22289
$$827$$ 17878.6 0.751753 0.375877 0.926670i $$-0.377342\pi$$
0.375877 + 0.926670i $$0.377342\pi$$
$$828$$ −1539.81 −0.0646281
$$829$$ 13423.8 0.562398 0.281199 0.959649i $$-0.409268\pi$$
0.281199 + 0.959649i $$0.409268\pi$$
$$830$$ 72065.0 3.01375
$$831$$ −12051.9 −0.503101
$$832$$ 0 0
$$833$$ −27797.5 −1.15621
$$834$$ −1236.93 −0.0513566
$$835$$ −43326.2 −1.79565
$$836$$ −9832.53 −0.406776
$$837$$ −8323.30 −0.343722
$$838$$ 8930.21 0.368125
$$839$$ 31542.7 1.29794 0.648971 0.760813i $$-0.275200\pi$$
0.648971 + 0.760813i $$0.275200\pi$$
$$840$$ −5224.61 −0.214602
$$841$$ −638.669 −0.0261867
$$842$$ −3028.73 −0.123963
$$843$$ −5523.35 −0.225663
$$844$$ −16855.7 −0.687437
$$845$$ 0 0
$$846$$ −4952.71 −0.201274
$$847$$ −9707.47 −0.393805
$$848$$ 24115.0 0.976547
$$849$$ −14548.2 −0.588095
$$850$$ 9907.22 0.399782
$$851$$ −827.155 −0.0333191
$$852$$ −12134.1 −0.487920
$$853$$ −21810.6 −0.875476 −0.437738 0.899103i $$-0.644220\pi$$
−0.437738 + 0.899103i $$0.644220\pi$$
$$854$$ −70129.1 −2.81003
$$855$$ −3263.82 −0.130550
$$856$$ 5971.59 0.238440
$$857$$ 33234.6 1.32470 0.662352 0.749193i $$-0.269558\pi$$
0.662352 + 0.749193i $$0.269558\pi$$
$$858$$ 0 0
$$859$$ 42697.7 1.69596 0.847978 0.530032i $$-0.177820\pi$$
0.847978 + 0.530032i $$0.177820\pi$$
$$860$$ 41390.3 1.64116
$$861$$ 4509.52 0.178495
$$862$$ 56525.0 2.23347
$$863$$ 27419.2 1.08153 0.540765 0.841174i $$-0.318135\pi$$
0.540765 + 0.841174i $$0.318135\pi$$
$$864$$ −7013.40 −0.276158
$$865$$ −36153.8 −1.42112
$$866$$ −41437.0 −1.62596
$$867$$ 9155.45 0.358634
$$868$$ −87177.9 −3.40900
$$869$$ −36358.3 −1.41930
$$870$$ −25624.5 −0.998567
$$871$$ 0 0
$$872$$ 1365.09 0.0530137
$$873$$ −14044.4 −0.544482
$$874$$ −2114.54 −0.0818367
$$875$$ 29272.4 1.13096
$$876$$ 10522.2 0.405836
$$877$$ 42604.2 1.64041 0.820206 0.572068i $$-0.193859\pi$$
0.820206 + 0.572068i $$0.193859\pi$$
$$878$$ 33535.2 1.28902
$$879$$ 4241.56 0.162758
$$880$$ −29940.0 −1.14691
$$881$$ −4699.40 −0.179712 −0.0898562 0.995955i $$-0.528641\pi$$
−0.0898562 + 0.995955i $$0.528641\pi$$
$$882$$ −23909.9 −0.912799
$$883$$ 22233.5 0.847358 0.423679 0.905812i $$-0.360738\pi$$
0.423679 + 0.905812i $$0.360738\pi$$
$$884$$ 0 0
$$885$$ −23815.3 −0.904568
$$886$$ 9774.98 0.370651
$$887$$ −8852.46 −0.335103 −0.167552 0.985863i $$-0.553586\pi$$
−0.167552 + 0.985863i $$0.553586\pi$$
$$888$$ 538.209 0.0203391
$$889$$ −7766.05 −0.292986
$$890$$ 51270.0 1.93098
$$891$$ 3280.19 0.123334
$$892$$ −505.155 −0.0189617
$$893$$ −3600.68 −0.134930
$$894$$ 1849.92 0.0692063
$$895$$ −20486.7 −0.765135
$$896$$ −16453.6 −0.613477
$$897$$ 0 0
$$898$$ −53283.7 −1.98007
$$899$$ −47508.0 −1.76249
$$900$$ 4511.47 0.167091
$$901$$ 18915.5 0.699410
$$902$$ −7987.58 −0.294853
$$903$$ 32250.3 1.18851
$$904$$ −2866.40 −0.105459
$$905$$ 6410.41 0.235458
$$906$$ −9895.85 −0.362878
$$907$$ 37172.4 1.36085 0.680424 0.732818i $$-0.261795\pi$$
0.680424 + 0.732818i $$0.261795\pi$$
$$908$$ −6006.51 −0.219530
$$909$$ 8629.56 0.314878
$$910$$ 0 0
$$911$$ 38035.1 1.38327 0.691635 0.722247i $$-0.256891\pi$$
0.691635 + 0.722247i $$0.256891\pi$$
$$912$$ −4451.36 −0.161622
$$913$$ −52654.8 −1.90867
$$914$$ −34639.3 −1.25357
$$915$$ 21829.2 0.788691
$$916$$ 6509.70 0.234810
$$917$$ −14838.8 −0.534372
$$918$$ −4802.67 −0.172671
$$919$$ −8352.27 −0.299800 −0.149900 0.988701i $$-0.547895\pi$$
−0.149900 + 0.988701i $$0.547895\pi$$
$$920$$ 1053.62 0.0377573
$$921$$ 13876.9 0.496482
$$922$$ −72679.4 −2.59606
$$923$$ 0 0
$$924$$ 34356.6 1.22321
$$925$$ 2423.47 0.0861440
$$926$$ 22519.6 0.799179
$$927$$ 5716.38 0.202535
$$928$$ −40031.3 −1.41605
$$929$$ 20232.1 0.714524 0.357262 0.934004i $$-0.383710\pi$$
0.357262 + 0.934004i $$0.383710\pi$$
$$930$$ 51257.0 1.80729
$$931$$ −17382.8 −0.611921
$$932$$ −2479.06 −0.0871292
$$933$$ 18182.4 0.638011
$$934$$ −34065.9 −1.19344
$$935$$ −23484.7 −0.821423
$$936$$ 0 0
$$937$$ −27766.9 −0.968095 −0.484048 0.875042i $$-0.660834\pi$$
−0.484048 + 0.875042i $$0.660834\pi$$
$$938$$ −29842.6 −1.03880
$$939$$ −2909.84 −0.101128
$$940$$ 16147.1 0.560277
$$941$$ −400.765 −0.0138837 −0.00694185 0.999976i $$-0.502210\pi$$
−0.00694185 + 0.999976i $$0.502210\pi$$
$$942$$ −33473.3 −1.15777
$$943$$ −909.408 −0.0314045
$$944$$ −32480.4 −1.11986
$$945$$ 11404.4 0.392576
$$946$$ −57124.0 −1.96328
$$947$$ 21804.4 0.748201 0.374101 0.927388i $$-0.377951\pi$$
0.374101 + 0.927388i $$0.377951\pi$$
$$948$$ 24241.1 0.830502
$$949$$ 0 0
$$950$$ 6195.35 0.211583
$$951$$ −26224.9 −0.894217
$$952$$ −5589.22 −0.190281
$$953$$ 30480.4 1.03605 0.518026 0.855365i $$-0.326667\pi$$
0.518026 + 0.855365i $$0.326667\pi$$
$$954$$ 16270.1 0.552165
$$955$$ −18412.6 −0.623892
$$956$$ −13764.5 −0.465666
$$957$$ 18722.8 0.632415
$$958$$ 6497.48 0.219127
$$959$$ 57524.0 1.93696
$$960$$ 25446.4 0.855499
$$961$$ 65239.6 2.18991
$$962$$ 0 0
$$963$$ −13034.9 −0.436183
$$964$$ 8779.73 0.293336
$$965$$ −28830.1 −0.961734
$$966$$ 7388.56 0.246090
$$967$$ 23864.1 0.793608 0.396804 0.917903i $$-0.370119\pi$$
0.396804 + 0.917903i $$0.370119\pi$$
$$968$$ −1273.79 −0.0422947
$$969$$ −3491.60 −0.115755
$$970$$ 86489.2 2.86289
$$971$$ −14010.3 −0.463040 −0.231520 0.972830i $$-0.574370\pi$$
−0.231520 + 0.972830i $$0.574370\pi$$
$$972$$ −2187.00 −0.0721688
$$973$$ 3142.19 0.103529
$$974$$ 51933.3 1.70847
$$975$$ 0 0
$$976$$ 29771.7 0.976404
$$977$$ −25448.2 −0.833326 −0.416663 0.909061i $$-0.636800\pi$$
−0.416663 + 0.909061i $$0.636800\pi$$
$$978$$ 45497.4 1.48757
$$979$$ −37460.8 −1.22293
$$980$$ 77952.4 2.54092
$$981$$ −2979.76 −0.0969789
$$982$$ 4416.72 0.143527
$$983$$ 52479.9 1.70280 0.851399 0.524519i $$-0.175755\pi$$
0.851399 + 0.524519i $$0.175755\pi$$
$$984$$ 591.729 0.0191703
$$985$$ −3222.66 −0.104246
$$986$$ −27412.8 −0.885398
$$987$$ 12581.4 0.405746
$$988$$ 0 0
$$989$$ −6503.73 −0.209107
$$990$$ −20200.2 −0.648491
$$991$$ 38048.5 1.21963 0.609814 0.792545i $$-0.291244\pi$$
0.609814 + 0.792545i $$0.291244\pi$$
$$992$$ 80075.0 2.56289
$$993$$ −20963.3 −0.669939
$$994$$ 58223.9 1.85790
$$995$$ −21372.5 −0.680960
$$996$$ 35106.5 1.11686
$$997$$ −31236.6 −0.992251 −0.496125 0.868251i $$-0.665244\pi$$
−0.496125 + 0.868251i $$0.665244\pi$$
$$998$$ 5864.30 0.186003
$$999$$ −1174.81 −0.0372066
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.a.k.1.2 4
3.2 odd 2 1521.4.a.z.1.3 4
13.2 odd 12 39.4.j.b.4.1 4
13.5 odd 4 507.4.b.e.337.3 4
13.7 odd 12 39.4.j.b.10.1 yes 4
13.8 odd 4 507.4.b.e.337.2 4
13.12 even 2 inner 507.4.a.k.1.3 4
39.2 even 12 117.4.q.d.82.2 4
39.20 even 12 117.4.q.d.10.2 4
39.38 odd 2 1521.4.a.z.1.2 4
52.7 even 12 624.4.bv.c.49.2 4
52.15 even 12 624.4.bv.c.433.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.1 4 13.2 odd 12
39.4.j.b.10.1 yes 4 13.7 odd 12
117.4.q.d.10.2 4 39.20 even 12
117.4.q.d.82.2 4 39.2 even 12
507.4.a.k.1.2 4 1.1 even 1 trivial
507.4.a.k.1.3 4 13.12 even 2 inner
507.4.b.e.337.2 4 13.8 odd 4
507.4.b.e.337.3 4 13.5 odd 4
624.4.bv.c.49.2 4 52.7 even 12
624.4.bv.c.433.1 4 52.15 even 12
1521.4.a.z.1.2 4 39.38 odd 2
1521.4.a.z.1.3 4 3.2 odd 2