Properties

 Label 525.4.d.m.274.4 Level $525$ Weight $4$ Character 525.274 Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 274.4 Root $$2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.m.274.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+3.56155i q^{2} +3.00000i q^{3} -4.68466 q^{4} -10.6847 q^{6} -7.00000i q^{7} +11.8078i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+3.56155i q^{2} +3.00000i q^{3} -4.68466 q^{4} -10.6847 q^{6} -7.00000i q^{7} +11.8078i q^{8} -9.00000 q^{9} -5.19224 q^{11} -14.0540i q^{12} -54.5464i q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619i q^{17} -32.0540i q^{18} -87.4470 q^{19} +21.0000 q^{21} -18.4924i q^{22} -176.477i q^{23} -35.4233 q^{24} +194.270 q^{26} -27.0000i q^{27} +32.7926i q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793i q^{32} -15.5767i q^{33} -57.5616 q^{34} +42.1619 q^{36} +17.3305i q^{37} -311.447i q^{38} +163.639 q^{39} +210.270 q^{41} +74.7926i q^{42} -521.570i q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417i q^{47} -238.594i q^{48} -49.0000 q^{49} -48.4858 q^{51} +255.531i q^{52} +108.978i q^{53} +96.1619 q^{54} +82.6543 q^{56} -262.341i q^{57} -506.348i q^{58} -210.365 q^{59} -674.304 q^{61} -335.855i q^{62} +63.0000i q^{63} +36.1449 q^{64} +55.4773 q^{66} +324.929i q^{67} -75.7131i q^{68} +529.432 q^{69} +793.965 q^{71} -106.270i q^{72} -315.417i q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457i q^{77} +582.810i q^{78} +425.840 q^{79} +81.0000 q^{81} +748.887i q^{82} +283.029i q^{83} -98.3778 q^{84} +1857.60 q^{86} -426.511i q^{87} -61.3087i q^{88} +843.131 q^{89} -381.825 q^{91} +826.736i q^{92} -282.901i q^{93} +375.447 q^{94} +566.378 q^{96} +1537.33i q^{97} -174.516i q^{98} +46.7301 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q + 6 * q^4 - 18 * q^6 - 36 * q^9 $$4 q + 6 q^{4} - 18 q^{6} - 36 q^{9} - 62 q^{11} + 42 q^{14} - 46 q^{16} + 112 q^{19} + 84 q^{21} - 18 q^{24} + 406 q^{26} + 124 q^{29} - 270 q^{31} - 222 q^{34} - 54 q^{36} + 234 q^{39} + 470 q^{41} - 348 q^{44} + 1170 q^{46} - 196 q^{49} + 474 q^{51} + 162 q^{54} + 42 q^{56} + 882 q^{59} - 446 q^{61} + 862 q^{64} + 24 q^{66} + 1524 q^{69} + 1238 q^{71} - 16 q^{74} + 3024 q^{76} + 854 q^{79} + 324 q^{81} + 126 q^{84} + 3398 q^{86} - 932 q^{89} - 546 q^{91} + 1040 q^{94} + 1746 q^{96} + 558 q^{99}+O(q^{100})$$ 4 * q + 6 * q^4 - 18 * q^6 - 36 * q^9 - 62 * q^11 + 42 * q^14 - 46 * q^16 + 112 * q^19 + 84 * q^21 - 18 * q^24 + 406 * q^26 + 124 * q^29 - 270 * q^31 - 222 * q^34 - 54 * q^36 + 234 * q^39 + 470 * q^41 - 348 * q^44 + 1170 * q^46 - 196 * q^49 + 474 * q^51 + 162 * q^54 + 42 * q^56 + 882 * q^59 - 446 * q^61 + 862 * q^64 + 24 * q^66 + 1524 * q^69 + 1238 * q^71 - 16 * q^74 + 3024 * q^76 + 854 * q^79 + 324 * q^81 + 126 * q^84 + 3398 * q^86 - 932 * q^89 - 546 * q^91 + 1040 * q^94 + 1746 * q^96 + 558 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.56155i 1.25920i 0.776920 + 0.629600i $$0.216781\pi$$
−0.776920 + 0.629600i $$0.783219\pi$$
$$3$$ 3.00000i 0.577350i
$$4$$ −4.68466 −0.585582
$$5$$ 0 0
$$6$$ −10.6847 −0.726999
$$7$$ − 7.00000i − 0.377964i
$$8$$ 11.8078i 0.521834i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −5.19224 −0.142320 −0.0711599 0.997465i $$-0.522670\pi$$
−0.0711599 + 0.997465i $$0.522670\pi$$
$$12$$ − 14.0540i − 0.338086i
$$13$$ − 54.5464i − 1.16373i −0.813287 0.581863i $$-0.802324\pi$$
0.813287 0.581863i $$-0.197676\pi$$
$$14$$ 24.9309 0.475933
$$15$$ 0 0
$$16$$ −79.5312 −1.24268
$$17$$ 16.1619i 0.230579i 0.993332 + 0.115289i $$0.0367796\pi$$
−0.993332 + 0.115289i $$0.963220\pi$$
$$18$$ − 32.0540i − 0.419733i
$$19$$ −87.4470 −1.05588 −0.527940 0.849282i $$-0.677035\pi$$
−0.527940 + 0.849282i $$0.677035\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ − 18.4924i − 0.179209i
$$23$$ − 176.477i − 1.59992i −0.600056 0.799958i $$-0.704855\pi$$
0.600056 0.799958i $$-0.295145\pi$$
$$24$$ −35.4233 −0.301281
$$25$$ 0 0
$$26$$ 194.270 1.46536
$$27$$ − 27.0000i − 0.192450i
$$28$$ 32.7926i 0.221329i
$$29$$ −142.170 −0.910358 −0.455179 0.890400i $$-0.650425\pi$$
−0.455179 + 0.890400i $$0.650425\pi$$
$$30$$ 0 0
$$31$$ −94.3002 −0.546349 −0.273174 0.961965i $$-0.588074\pi$$
−0.273174 + 0.961965i $$0.588074\pi$$
$$32$$ − 188.793i − 1.04294i
$$33$$ − 15.5767i − 0.0821684i
$$34$$ −57.5616 −0.290345
$$35$$ 0 0
$$36$$ 42.1619 0.195194
$$37$$ 17.3305i 0.0770031i 0.999259 + 0.0385016i $$0.0122585\pi$$
−0.999259 + 0.0385016i $$0.987742\pi$$
$$38$$ − 311.447i − 1.32956i
$$39$$ 163.639 0.671878
$$40$$ 0 0
$$41$$ 210.270 0.800942 0.400471 0.916309i $$-0.368846\pi$$
0.400471 + 0.916309i $$0.368846\pi$$
$$42$$ 74.7926i 0.274780i
$$43$$ − 521.570i − 1.84974i −0.380287 0.924868i $$-0.624175\pi$$
0.380287 0.924868i $$-0.375825\pi$$
$$44$$ 24.3239 0.0833400
$$45$$ 0 0
$$46$$ 628.533 2.01461
$$47$$ − 105.417i − 0.327162i −0.986530 0.163581i $$-0.947696\pi$$
0.986530 0.163581i $$-0.0523045\pi$$
$$48$$ − 238.594i − 0.717459i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −48.4858 −0.133125
$$52$$ 255.531i 0.681458i
$$53$$ 108.978i 0.282440i 0.989978 + 0.141220i $$0.0451025\pi$$
−0.989978 + 0.141220i $$0.954898\pi$$
$$54$$ 96.1619 0.242333
$$55$$ 0 0
$$56$$ 82.6543 0.197235
$$57$$ − 262.341i − 0.609612i
$$58$$ − 506.348i − 1.14632i
$$59$$ −210.365 −0.464189 −0.232094 0.972693i $$-0.574558\pi$$
−0.232094 + 0.972693i $$0.574558\pi$$
$$60$$ 0 0
$$61$$ −674.304 −1.41534 −0.707670 0.706543i $$-0.750254\pi$$
−0.707670 + 0.706543i $$0.750254\pi$$
$$62$$ − 335.855i − 0.687962i
$$63$$ 63.0000i 0.125988i
$$64$$ 36.1449 0.0705955
$$65$$ 0 0
$$66$$ 55.4773 0.103466
$$67$$ 324.929i 0.592484i 0.955113 + 0.296242i $$0.0957334\pi$$
−0.955113 + 0.296242i $$0.904267\pi$$
$$68$$ − 75.7131i − 0.135023i
$$69$$ 529.432 0.923712
$$70$$ 0 0
$$71$$ 793.965 1.32713 0.663565 0.748118i $$-0.269042\pi$$
0.663565 + 0.748118i $$0.269042\pi$$
$$72$$ − 106.270i − 0.173945i
$$73$$ − 315.417i − 0.505709i −0.967504 0.252854i $$-0.918631\pi$$
0.967504 0.252854i $$-0.0813693\pi$$
$$74$$ −61.7235 −0.0969623
$$75$$ 0 0
$$76$$ 409.659 0.618304
$$77$$ 36.3457i 0.0537918i
$$78$$ 582.810i 0.846028i
$$79$$ 425.840 0.606465 0.303233 0.952917i $$-0.401934\pi$$
0.303233 + 0.952917i $$0.401934\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 748.887i 1.00855i
$$83$$ 283.029i 0.374295i 0.982332 + 0.187148i $$0.0599243\pi$$
−0.982332 + 0.187148i $$0.940076\pi$$
$$84$$ −98.3778 −0.127785
$$85$$ 0 0
$$86$$ 1857.60 2.32919
$$87$$ − 426.511i − 0.525596i
$$88$$ − 61.3087i − 0.0742674i
$$89$$ 843.131 1.00418 0.502088 0.864817i $$-0.332565\pi$$
0.502088 + 0.864817i $$0.332565\pi$$
$$90$$ 0 0
$$91$$ −381.825 −0.439847
$$92$$ 826.736i 0.936882i
$$93$$ − 282.901i − 0.315435i
$$94$$ 375.447 0.411962
$$95$$ 0 0
$$96$$ 566.378 0.602143
$$97$$ 1537.33i 1.60920i 0.593816 + 0.804601i $$0.297621\pi$$
−0.593816 + 0.804601i $$0.702379\pi$$
$$98$$ − 174.516i − 0.179886i
$$99$$ 46.7301 0.0474399
$$100$$ 0 0
$$101$$ −1589.99 −1.56644 −0.783219 0.621745i $$-0.786424\pi$$
−0.783219 + 0.621745i $$0.786424\pi$$
$$102$$ − 172.685i − 0.167631i
$$103$$ 164.793i 0.157646i 0.996889 + 0.0788228i $$0.0251161\pi$$
−0.996889 + 0.0788228i $$0.974884\pi$$
$$104$$ 644.071 0.607273
$$105$$ 0 0
$$106$$ −388.132 −0.355648
$$107$$ 1184.08i 1.06981i 0.844913 + 0.534904i $$0.179652\pi$$
−0.844913 + 0.534904i $$0.820348\pi$$
$$108$$ 126.486i 0.112695i
$$109$$ −333.247 −0.292837 −0.146419 0.989223i $$-0.546775\pi$$
−0.146419 + 0.989223i $$0.546775\pi$$
$$110$$ 0 0
$$111$$ −51.9915 −0.0444578
$$112$$ 556.719i 0.469687i
$$113$$ − 1881.49i − 1.56634i −0.621810 0.783168i $$-0.713602\pi$$
0.621810 0.783168i $$-0.286398\pi$$
$$114$$ 934.341 0.767623
$$115$$ 0 0
$$116$$ 666.020 0.533090
$$117$$ 490.918i 0.387909i
$$118$$ − 749.224i − 0.584506i
$$119$$ 113.133 0.0871507
$$120$$ 0 0
$$121$$ −1304.04 −0.979745
$$122$$ − 2401.57i − 1.78220i
$$123$$ 630.810i 0.462424i
$$124$$ 441.764 0.319932
$$125$$ 0 0
$$126$$ −224.378 −0.158644
$$127$$ 1638.79i 1.14503i 0.819893 + 0.572516i $$0.194033\pi$$
−0.819893 + 0.572516i $$0.805967\pi$$
$$128$$ − 1381.61i − 0.954048i
$$129$$ 1564.71 1.06795
$$130$$ 0 0
$$131$$ −598.142 −0.398931 −0.199465 0.979905i $$-0.563921\pi$$
−0.199465 + 0.979905i $$0.563921\pi$$
$$132$$ 72.9716i 0.0481164i
$$133$$ 612.129i 0.399085i
$$134$$ −1157.25 −0.746055
$$135$$ 0 0
$$136$$ −190.836 −0.120324
$$137$$ − 1005.25i − 0.626894i −0.949606 0.313447i $$-0.898516\pi$$
0.949606 0.313447i $$-0.101484\pi$$
$$138$$ 1885.60i 1.16314i
$$139$$ 1875.01 1.14414 0.572072 0.820204i $$-0.306140\pi$$
0.572072 + 0.820204i $$0.306140\pi$$
$$140$$ 0 0
$$141$$ 316.250 0.188887
$$142$$ 2827.75i 1.67112i
$$143$$ 283.218i 0.165621i
$$144$$ 715.781 0.414225
$$145$$ 0 0
$$146$$ 1123.37 0.636788
$$147$$ − 147.000i − 0.0824786i
$$148$$ − 81.1875i − 0.0450917i
$$149$$ 1051.80 0.578299 0.289150 0.957284i $$-0.406627\pi$$
0.289150 + 0.957284i $$0.406627\pi$$
$$150$$ 0 0
$$151$$ −750.383 −0.404406 −0.202203 0.979344i $$-0.564810\pi$$
−0.202203 + 0.979344i $$0.564810\pi$$
$$152$$ − 1032.55i − 0.550994i
$$153$$ − 145.457i − 0.0768597i
$$154$$ −129.447 −0.0677346
$$155$$ 0 0
$$156$$ −766.594 −0.393440
$$157$$ − 1453.90i − 0.739067i −0.929217 0.369533i $$-0.879518\pi$$
0.929217 0.369533i $$-0.120482\pi$$
$$158$$ 1516.65i 0.763660i
$$159$$ −326.935 −0.163067
$$160$$ 0 0
$$161$$ −1235.34 −0.604711
$$162$$ 288.486i 0.139911i
$$163$$ − 1300.49i − 0.624921i −0.949931 0.312461i $$-0.898847\pi$$
0.949931 0.312461i $$-0.101153\pi$$
$$164$$ −985.043 −0.469018
$$165$$ 0 0
$$166$$ −1008.02 −0.471312
$$167$$ − 2111.46i − 0.978381i −0.872177 0.489191i $$-0.837292\pi$$
0.872177 0.489191i $$-0.162708\pi$$
$$168$$ 247.963i 0.113874i
$$169$$ −778.310 −0.354260
$$170$$ 0 0
$$171$$ 787.023 0.351960
$$172$$ 2443.38i 1.08317i
$$173$$ − 335.292i − 0.147351i −0.997282 0.0736756i $$-0.976527\pi$$
0.997282 0.0736756i $$-0.0234729\pi$$
$$174$$ 1519.04 0.661829
$$175$$ 0 0
$$176$$ 412.945 0.176857
$$177$$ − 631.094i − 0.267999i
$$178$$ 3002.85i 1.26446i
$$179$$ −2322.23 −0.969672 −0.484836 0.874605i $$-0.661121\pi$$
−0.484836 + 0.874605i $$0.661121\pi$$
$$180$$ 0 0
$$181$$ −1525.59 −0.626500 −0.313250 0.949671i $$-0.601418\pi$$
−0.313250 + 0.949671i $$0.601418\pi$$
$$182$$ − 1359.89i − 0.553855i
$$183$$ − 2022.91i − 0.817147i
$$184$$ 2083.80 0.834891
$$185$$ 0 0
$$186$$ 1007.57 0.397195
$$187$$ − 83.9165i − 0.0328160i
$$188$$ 493.841i 0.191580i
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ 293.912 0.111344 0.0556721 0.998449i $$-0.482270\pi$$
0.0556721 + 0.998449i $$0.482270\pi$$
$$192$$ 108.435i 0.0407583i
$$193$$ 3664.91i 1.36687i 0.730012 + 0.683435i $$0.239515\pi$$
−0.730012 + 0.683435i $$0.760485\pi$$
$$194$$ −5475.29 −2.02630
$$195$$ 0 0
$$196$$ 229.548 0.0836546
$$197$$ − 5101.89i − 1.84515i −0.385816 0.922576i $$-0.626080\pi$$
0.385816 0.922576i $$-0.373920\pi$$
$$198$$ 166.432i 0.0597363i
$$199$$ −5025.86 −1.79032 −0.895161 0.445743i $$-0.852939\pi$$
−0.895161 + 0.445743i $$0.852939\pi$$
$$200$$ 0 0
$$201$$ −974.787 −0.342071
$$202$$ − 5662.85i − 1.97246i
$$203$$ 995.193i 0.344083i
$$204$$ 227.139 0.0779556
$$205$$ 0 0
$$206$$ −586.918 −0.198507
$$207$$ 1588.30i 0.533305i
$$208$$ 4338.14i 1.44614i
$$209$$ 454.045 0.150273
$$210$$ 0 0
$$211$$ −3267.98 −1.06624 −0.533122 0.846039i $$-0.678981\pi$$
−0.533122 + 0.846039i $$0.678981\pi$$
$$212$$ − 510.526i − 0.165392i
$$213$$ 2381.89i 0.766219i
$$214$$ −4217.17 −1.34710
$$215$$ 0 0
$$216$$ 318.810 0.100427
$$217$$ 660.101i 0.206500i
$$218$$ − 1186.88i − 0.368741i
$$219$$ 946.250 0.291971
$$220$$ 0 0
$$221$$ 881.575 0.268331
$$222$$ − 185.170i − 0.0559812i
$$223$$ − 5457.65i − 1.63888i −0.573162 0.819442i $$-0.694283\pi$$
0.573162 0.819442i $$-0.305717\pi$$
$$224$$ −1321.55 −0.394195
$$225$$ 0 0
$$226$$ 6701.04 1.97233
$$227$$ 281.023i 0.0821682i 0.999156 + 0.0410841i $$0.0130812\pi$$
−0.999156 + 0.0410841i $$0.986919\pi$$
$$228$$ 1228.98i 0.356978i
$$229$$ −2776.64 −0.801248 −0.400624 0.916243i $$-0.631207\pi$$
−0.400624 + 0.916243i $$0.631207\pi$$
$$230$$ 0 0
$$231$$ −109.037 −0.0310567
$$232$$ − 1678.71i − 0.475056i
$$233$$ − 5781.09i − 1.62546i −0.582642 0.812729i $$-0.697981\pi$$
0.582642 0.812729i $$-0.302019\pi$$
$$234$$ −1748.43 −0.488455
$$235$$ 0 0
$$236$$ 985.486 0.271821
$$237$$ 1277.52i 0.350143i
$$238$$ 402.931i 0.109740i
$$239$$ −1588.17 −0.429833 −0.214916 0.976632i $$-0.568948\pi$$
−0.214916 + 0.976632i $$0.568948\pi$$
$$240$$ 0 0
$$241$$ −4330.01 −1.15735 −0.578673 0.815560i $$-0.696429\pi$$
−0.578673 + 0.815560i $$0.696429\pi$$
$$242$$ − 4644.41i − 1.23369i
$$243$$ 243.000i 0.0641500i
$$244$$ 3158.88 0.828798
$$245$$ 0 0
$$246$$ −2246.66 −0.582284
$$247$$ 4769.92i 1.22876i
$$248$$ − 1113.47i − 0.285104i
$$249$$ −849.088 −0.216099
$$250$$ 0 0
$$251$$ 1400.53 0.352195 0.176097 0.984373i $$-0.443653\pi$$
0.176097 + 0.984373i $$0.443653\pi$$
$$252$$ − 295.133i − 0.0737764i
$$253$$ 916.312i 0.227700i
$$254$$ −5836.64 −1.44182
$$255$$ 0 0
$$256$$ 5209.83 1.27193
$$257$$ 4304.86i 1.04486i 0.852681 + 0.522431i $$0.174975\pi$$
−0.852681 + 0.522431i $$0.825025\pi$$
$$258$$ 5572.80i 1.34476i
$$259$$ 121.313 0.0291045
$$260$$ 0 0
$$261$$ 1279.53 0.303453
$$262$$ − 2130.31i − 0.502333i
$$263$$ 1724.69i 0.404369i 0.979347 + 0.202184i $$0.0648041\pi$$
−0.979347 + 0.202184i $$0.935196\pi$$
$$264$$ 183.926 0.0428783
$$265$$ 0 0
$$266$$ −2180.13 −0.502527
$$267$$ 2529.39i 0.579761i
$$268$$ − 1522.18i − 0.346948i
$$269$$ −8004.82 −1.81436 −0.907180 0.420744i $$-0.861769\pi$$
−0.907180 + 0.420744i $$0.861769\pi$$
$$270$$ 0 0
$$271$$ −1963.65 −0.440160 −0.220080 0.975482i $$-0.570632\pi$$
−0.220080 + 0.975482i $$0.570632\pi$$
$$272$$ − 1285.38i − 0.286535i
$$273$$ − 1145.47i − 0.253946i
$$274$$ 3580.26 0.789384
$$275$$ 0 0
$$276$$ −2480.21 −0.540909
$$277$$ − 3278.33i − 0.711104i −0.934656 0.355552i $$-0.884293\pi$$
0.934656 0.355552i $$-0.115707\pi$$
$$278$$ 6677.93i 1.44070i
$$279$$ 848.702 0.182116
$$280$$ 0 0
$$281$$ 2859.04 0.606961 0.303480 0.952838i $$-0.401851\pi$$
0.303480 + 0.952838i $$0.401851\pi$$
$$282$$ 1126.34i 0.237846i
$$283$$ 5433.66i 1.14134i 0.821181 + 0.570668i $$0.193315\pi$$
−0.821181 + 0.570668i $$0.806685\pi$$
$$284$$ −3719.45 −0.777144
$$285$$ 0 0
$$286$$ −1008.70 −0.208550
$$287$$ − 1471.89i − 0.302728i
$$288$$ 1699.13i 0.347647i
$$289$$ 4651.79 0.946833
$$290$$ 0 0
$$291$$ −4612.00 −0.929073
$$292$$ 1477.62i 0.296134i
$$293$$ − 8583.43i − 1.71143i −0.517447 0.855715i $$-0.673117\pi$$
0.517447 0.855715i $$-0.326883\pi$$
$$294$$ 523.548 0.103857
$$295$$ 0 0
$$296$$ −204.634 −0.0401829
$$297$$ 140.190i 0.0273895i
$$298$$ 3746.03i 0.728194i
$$299$$ −9626.20 −1.86186
$$300$$ 0 0
$$301$$ −3650.99 −0.699135
$$302$$ − 2672.53i − 0.509228i
$$303$$ − 4769.98i − 0.904384i
$$304$$ 6954.77 1.31212
$$305$$ 0 0
$$306$$ 518.054 0.0967816
$$307$$ − 5269.83i − 0.979691i −0.871809 0.489846i $$-0.837053\pi$$
0.871809 0.489846i $$-0.162947\pi$$
$$308$$ − 170.267i − 0.0314995i
$$309$$ −494.378 −0.0910167
$$310$$ 0 0
$$311$$ −4761.43 −0.868154 −0.434077 0.900876i $$-0.642925\pi$$
−0.434077 + 0.900876i $$0.642925\pi$$
$$312$$ 1932.21i 0.350609i
$$313$$ 7602.95i 1.37298i 0.727137 + 0.686492i $$0.240850\pi$$
−0.727137 + 0.686492i $$0.759150\pi$$
$$314$$ 5178.13 0.930632
$$315$$ 0 0
$$316$$ −1994.91 −0.355135
$$317$$ 8064.55i 1.42886i 0.699704 + 0.714432i $$0.253315\pi$$
−0.699704 + 0.714432i $$0.746685\pi$$
$$318$$ − 1164.39i − 0.205333i
$$319$$ 738.182 0.129562
$$320$$ 0 0
$$321$$ −3552.24 −0.617654
$$322$$ − 4399.73i − 0.761452i
$$323$$ − 1413.31i − 0.243464i
$$324$$ −379.457 −0.0650647
$$325$$ 0 0
$$326$$ 4631.76 0.786901
$$327$$ − 999.741i − 0.169070i
$$328$$ 2482.82i 0.417959i
$$329$$ −737.917 −0.123655
$$330$$ 0 0
$$331$$ 6960.79 1.15589 0.577945 0.816076i $$-0.303855\pi$$
0.577945 + 0.816076i $$0.303855\pi$$
$$332$$ − 1325.90i − 0.219181i
$$333$$ − 155.974i − 0.0256677i
$$334$$ 7520.08 1.23198
$$335$$ 0 0
$$336$$ −1670.16 −0.271174
$$337$$ 4731.61i 0.764828i 0.923991 + 0.382414i $$0.124907\pi$$
−0.923991 + 0.382414i $$0.875093\pi$$
$$338$$ − 2771.99i − 0.446084i
$$339$$ 5644.48 0.904325
$$340$$ 0 0
$$341$$ 489.629 0.0777563
$$342$$ 2803.02i 0.443187i
$$343$$ 343.000i 0.0539949i
$$344$$ 6158.58 0.965256
$$345$$ 0 0
$$346$$ 1194.16 0.185544
$$347$$ 9796.67i 1.51560i 0.652487 + 0.757800i $$0.273726\pi$$
−0.652487 + 0.757800i $$0.726274\pi$$
$$348$$ 1998.06i 0.307779i
$$349$$ −12702.4 −1.94827 −0.974134 0.225971i $$-0.927445\pi$$
−0.974134 + 0.225971i $$0.927445\pi$$
$$350$$ 0 0
$$351$$ −1472.75 −0.223959
$$352$$ 980.256i 0.148431i
$$353$$ 9970.21i 1.50329i 0.659569 + 0.751644i $$0.270739\pi$$
−0.659569 + 0.751644i $$0.729261\pi$$
$$354$$ 2247.67 0.337465
$$355$$ 0 0
$$356$$ −3949.78 −0.588028
$$357$$ 339.400i 0.0503165i
$$358$$ − 8270.73i − 1.22101i
$$359$$ −4388.21 −0.645128 −0.322564 0.946548i $$-0.604545\pi$$
−0.322564 + 0.946548i $$0.604545\pi$$
$$360$$ 0 0
$$361$$ 787.970 0.114881
$$362$$ − 5433.49i − 0.788889i
$$363$$ − 3912.12i − 0.565656i
$$364$$ 1788.72 0.257567
$$365$$ 0 0
$$366$$ 7204.71 1.02895
$$367$$ − 9441.30i − 1.34287i −0.741065 0.671433i $$-0.765679\pi$$
0.741065 0.671433i $$-0.234321\pi$$
$$368$$ 14035.5i 1.98818i
$$369$$ −1892.43 −0.266981
$$370$$ 0 0
$$371$$ 762.847 0.106752
$$372$$ 1325.29i 0.184713i
$$373$$ 3219.40i 0.446901i 0.974715 + 0.223451i $$0.0717322\pi$$
−0.974715 + 0.223451i $$0.928268\pi$$
$$374$$ 298.873 0.0413218
$$375$$ 0 0
$$376$$ 1244.73 0.170724
$$377$$ 7754.89i 1.05941i
$$378$$ − 673.133i − 0.0915933i
$$379$$ 14011.4 1.89899 0.949495 0.313783i $$-0.101597\pi$$
0.949495 + 0.313783i $$0.101597\pi$$
$$380$$ 0 0
$$381$$ −4916.37 −0.661085
$$382$$ 1046.78i 0.140204i
$$383$$ − 5322.87i − 0.710147i −0.934838 0.355073i $$-0.884456\pi$$
0.934838 0.355073i $$-0.115544\pi$$
$$384$$ 4144.83 0.550820
$$385$$ 0 0
$$386$$ −13052.8 −1.72116
$$387$$ 4694.13i 0.616579i
$$388$$ − 7201.88i − 0.942320i
$$389$$ 3844.51 0.501091 0.250545 0.968105i $$-0.419390\pi$$
0.250545 + 0.968105i $$0.419390\pi$$
$$390$$ 0 0
$$391$$ 2852.21 0.368907
$$392$$ − 578.580i − 0.0745478i
$$393$$ − 1794.43i − 0.230323i
$$394$$ 18170.7 2.32341
$$395$$ 0 0
$$396$$ −218.915 −0.0277800
$$397$$ − 8046.40i − 1.01722i −0.860996 0.508611i $$-0.830159\pi$$
0.860996 0.508611i $$-0.169841\pi$$
$$398$$ − 17899.9i − 2.25437i
$$399$$ −1836.39 −0.230412
$$400$$ 0 0
$$401$$ 7741.38 0.964055 0.482027 0.876156i $$-0.339901\pi$$
0.482027 + 0.876156i $$0.339901\pi$$
$$402$$ − 3471.76i − 0.430735i
$$403$$ 5143.74i 0.635801i
$$404$$ 7448.58 0.917279
$$405$$ 0 0
$$406$$ −3544.43 −0.433269
$$407$$ − 89.9840i − 0.0109591i
$$408$$ − 572.509i − 0.0694691i
$$409$$ 8966.94 1.08407 0.542037 0.840354i $$-0.317653\pi$$
0.542037 + 0.840354i $$0.317653\pi$$
$$410$$ 0 0
$$411$$ 3015.76 0.361937
$$412$$ − 771.997i − 0.0923145i
$$413$$ 1472.55i 0.175447i
$$414$$ −5656.80 −0.671537
$$415$$ 0 0
$$416$$ −10298.0 −1.21370
$$417$$ 5625.02i 0.660571i
$$418$$ 1617.11i 0.189223i
$$419$$ 12413.6 1.44736 0.723681 0.690135i $$-0.242449\pi$$
0.723681 + 0.690135i $$0.242449\pi$$
$$420$$ 0 0
$$421$$ −1672.14 −0.193575 −0.0967875 0.995305i $$-0.530857\pi$$
−0.0967875 + 0.995305i $$0.530857\pi$$
$$422$$ − 11639.1i − 1.34261i
$$423$$ 948.750i 0.109054i
$$424$$ −1286.79 −0.147387
$$425$$ 0 0
$$426$$ −8483.24 −0.964823
$$427$$ 4720.13i 0.534948i
$$428$$ − 5547.02i − 0.626461i
$$429$$ −849.653 −0.0956216
$$430$$ 0 0
$$431$$ 16021.8 1.79059 0.895296 0.445472i $$-0.146964\pi$$
0.895296 + 0.445472i $$0.146964\pi$$
$$432$$ 2147.34i 0.239153i
$$433$$ − 10882.7i − 1.20782i −0.797051 0.603912i $$-0.793608\pi$$
0.797051 0.603912i $$-0.206392\pi$$
$$434$$ −2350.99 −0.260025
$$435$$ 0 0
$$436$$ 1561.15 0.171480
$$437$$ 15432.4i 1.68932i
$$438$$ 3370.12i 0.367650i
$$439$$ −7738.40 −0.841307 −0.420653 0.907221i $$-0.638199\pi$$
−0.420653 + 0.907221i $$0.638199\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 3139.78i 0.337882i
$$443$$ − 8766.56i − 0.940207i −0.882611 0.470103i $$-0.844217\pi$$
0.882611 0.470103i $$-0.155783\pi$$
$$444$$ 243.562 0.0260337
$$445$$ 0 0
$$446$$ 19437.7 2.06368
$$447$$ 3155.39i 0.333881i
$$448$$ − 253.014i − 0.0266826i
$$449$$ −3099.58 −0.325786 −0.162893 0.986644i $$-0.552083\pi$$
−0.162893 + 0.986644i $$0.552083\pi$$
$$450$$ 0 0
$$451$$ −1091.77 −0.113990
$$452$$ 8814.15i 0.917219i
$$453$$ − 2251.15i − 0.233484i
$$454$$ −1000.88 −0.103466
$$455$$ 0 0
$$456$$ 3097.66 0.318117
$$457$$ − 6122.94i − 0.626737i −0.949632 0.313369i $$-0.898542\pi$$
0.949632 0.313369i $$-0.101458\pi$$
$$458$$ − 9889.16i − 1.00893i
$$459$$ 436.372 0.0443749
$$460$$ 0 0
$$461$$ −10412.2 −1.05194 −0.525970 0.850503i $$-0.676297\pi$$
−0.525970 + 0.850503i $$0.676297\pi$$
$$462$$ − 388.341i − 0.0391066i
$$463$$ 11278.5i 1.13209i 0.824376 + 0.566043i $$0.191526\pi$$
−0.824376 + 0.566043i $$0.808474\pi$$
$$464$$ 11307.0 1.13128
$$465$$ 0 0
$$466$$ 20589.6 2.04677
$$467$$ 14923.2i 1.47872i 0.673311 + 0.739359i $$0.264872\pi$$
−0.673311 + 0.739359i $$0.735128\pi$$
$$468$$ − 2299.78i − 0.227153i
$$469$$ 2274.50 0.223938
$$470$$ 0 0
$$471$$ 4361.69 0.426701
$$472$$ − 2483.93i − 0.242230i
$$473$$ 2708.11i 0.263254i
$$474$$ −4549.95 −0.440899
$$475$$ 0 0
$$476$$ −529.992 −0.0510339
$$477$$ − 980.804i − 0.0941466i
$$478$$ − 5656.34i − 0.541245i
$$479$$ 4674.21 0.445867 0.222933 0.974834i $$-0.428437\pi$$
0.222933 + 0.974834i $$0.428437\pi$$
$$480$$ 0 0
$$481$$ 945.316 0.0896106
$$482$$ − 15421.6i − 1.45733i
$$483$$ − 3706.02i − 0.349130i
$$484$$ 6108.99 0.573721
$$485$$ 0 0
$$486$$ −865.457 −0.0807777
$$487$$ 17081.7i 1.58941i 0.606994 + 0.794706i $$0.292375\pi$$
−0.606994 + 0.794706i $$0.707625\pi$$
$$488$$ − 7962.02i − 0.738573i
$$489$$ 3901.47 0.360799
$$490$$ 0 0
$$491$$ 18203.9 1.67318 0.836588 0.547832i $$-0.184547\pi$$
0.836588 + 0.547832i $$0.184547\pi$$
$$492$$ − 2955.13i − 0.270787i
$$493$$ − 2297.75i − 0.209909i
$$494$$ −16988.3 −1.54725
$$495$$ 0 0
$$496$$ 7499.81 0.678934
$$497$$ − 5557.75i − 0.501608i
$$498$$ − 3024.07i − 0.272112i
$$499$$ −7109.47 −0.637803 −0.318901 0.947788i $$-0.603314\pi$$
−0.318901 + 0.947788i $$0.603314\pi$$
$$500$$ 0 0
$$501$$ 6334.38 0.564869
$$502$$ 4988.07i 0.443483i
$$503$$ 15402.0i 1.36529i 0.730748 + 0.682647i $$0.239171\pi$$
−0.730748 + 0.682647i $$0.760829\pi$$
$$504$$ −743.889 −0.0657450
$$505$$ 0 0
$$506$$ −3263.49 −0.286719
$$507$$ − 2334.93i − 0.204532i
$$508$$ − 7677.17i − 0.670511i
$$509$$ −6404.72 −0.557730 −0.278865 0.960330i $$-0.589958\pi$$
−0.278865 + 0.960330i $$0.589958\pi$$
$$510$$ 0 0
$$511$$ −2207.92 −0.191140
$$512$$ 7502.22i 0.647567i
$$513$$ 2361.07i 0.203204i
$$514$$ −15332.0 −1.31569
$$515$$ 0 0
$$516$$ −7330.13 −0.625370
$$517$$ 547.348i 0.0465616i
$$518$$ 432.064i 0.0366483i
$$519$$ 1005.88 0.0850732
$$520$$ 0 0
$$521$$ −8916.72 −0.749806 −0.374903 0.927064i $$-0.622324\pi$$
−0.374903 + 0.927064i $$0.622324\pi$$
$$522$$ 4557.13i 0.382107i
$$523$$ 6929.40i 0.579353i 0.957125 + 0.289677i $$0.0935478\pi$$
−0.957125 + 0.289677i $$0.906452\pi$$
$$524$$ 2802.09 0.233607
$$525$$ 0 0
$$526$$ −6142.58 −0.509181
$$527$$ − 1524.07i − 0.125977i
$$528$$ 1238.83i 0.102109i
$$529$$ −18977.2 −1.55973
$$530$$ 0 0
$$531$$ 1893.28 0.154730
$$532$$ − 2867.61i − 0.233697i
$$533$$ − 11469.5i − 0.932078i
$$534$$ −9008.56 −0.730035
$$535$$ 0 0
$$536$$ −3836.69 −0.309178
$$537$$ − 6966.68i − 0.559840i
$$538$$ − 28509.6i − 2.28464i
$$539$$ 254.420 0.0203314
$$540$$ 0 0
$$541$$ −6929.23 −0.550667 −0.275334 0.961349i $$-0.588788\pi$$
−0.275334 + 0.961349i $$0.588788\pi$$
$$542$$ − 6993.65i − 0.554249i
$$543$$ − 4576.78i − 0.361710i
$$544$$ 3051.25 0.240480
$$545$$ 0 0
$$546$$ 4079.67 0.319769
$$547$$ − 8509.95i − 0.665190i −0.943070 0.332595i $$-0.892076\pi$$
0.943070 0.332595i $$-0.107924\pi$$
$$548$$ 4709.26i 0.367098i
$$549$$ 6068.74 0.471780
$$550$$ 0 0
$$551$$ 12432.4 0.961228
$$552$$ 6251.41i 0.482024i
$$553$$ − 2980.88i − 0.229222i
$$554$$ 11676.0 0.895422
$$555$$ 0 0
$$556$$ −8783.76 −0.669990
$$557$$ − 4043.94i − 0.307625i −0.988100 0.153813i $$-0.950845\pi$$
0.988100 0.153813i $$-0.0491552\pi$$
$$558$$ 3022.70i 0.229321i
$$559$$ −28449.8 −2.15259
$$560$$ 0 0
$$561$$ 251.750 0.0189463
$$562$$ 10182.6i 0.764285i
$$563$$ − 15878.9i − 1.18866i −0.804221 0.594331i $$-0.797417\pi$$
0.804221 0.594331i $$-0.202583\pi$$
$$564$$ −1481.52 −0.110609
$$565$$ 0 0
$$566$$ −19352.3 −1.43717
$$567$$ − 567.000i − 0.0419961i
$$568$$ 9374.95i 0.692543i
$$569$$ −11611.6 −0.855510 −0.427755 0.903895i $$-0.640695\pi$$
−0.427755 + 0.903895i $$0.640695\pi$$
$$570$$ 0 0
$$571$$ 17395.7 1.27493 0.637466 0.770478i $$-0.279982\pi$$
0.637466 + 0.770478i $$0.279982\pi$$
$$572$$ − 1326.78i − 0.0969850i
$$573$$ 881.736i 0.0642846i
$$574$$ 5242.21 0.381195
$$575$$ 0 0
$$576$$ −325.304 −0.0235318
$$577$$ − 11474.0i − 0.827848i −0.910311 0.413924i $$-0.864158\pi$$
0.910311 0.413924i $$-0.135842\pi$$
$$578$$ 16567.6i 1.19225i
$$579$$ −10994.7 −0.789162
$$580$$ 0 0
$$581$$ 1981.20 0.141470
$$582$$ − 16425.9i − 1.16989i
$$583$$ − 565.841i − 0.0401968i
$$584$$ 3724.37 0.263896
$$585$$ 0 0
$$586$$ 30570.3 2.15503
$$587$$ 11870.4i 0.834659i 0.908755 + 0.417330i $$0.137034\pi$$
−0.908755 + 0.417330i $$0.862966\pi$$
$$588$$ 688.645i 0.0482980i
$$589$$ 8246.26 0.576878
$$590$$ 0 0
$$591$$ 15305.7 1.06530
$$592$$ − 1378.32i − 0.0956899i
$$593$$ 5760.65i 0.398923i 0.979906 + 0.199462i $$0.0639193\pi$$
−0.979906 + 0.199462i $$0.936081\pi$$
$$594$$ −499.295 −0.0344888
$$595$$ 0 0
$$596$$ −4927.31 −0.338642
$$597$$ − 15077.6i − 1.03364i
$$598$$ − 34284.2i − 2.34446i
$$599$$ 21696.5 1.47996 0.739978 0.672631i $$-0.234836\pi$$
0.739978 + 0.672631i $$0.234836\pi$$
$$600$$ 0 0
$$601$$ −12403.0 −0.841812 −0.420906 0.907104i $$-0.638288\pi$$
−0.420906 + 0.907104i $$0.638288\pi$$
$$602$$ − 13003.2i − 0.880350i
$$603$$ − 2924.36i − 0.197495i
$$604$$ 3515.29 0.236813
$$605$$ 0 0
$$606$$ 16988.5 1.13880
$$607$$ 17066.5i 1.14120i 0.821228 + 0.570600i $$0.193289\pi$$
−0.821228 + 0.570600i $$0.806711\pi$$
$$608$$ 16509.3i 1.10122i
$$609$$ −2985.58 −0.198656
$$610$$ 0 0
$$611$$ −5750.10 −0.380727
$$612$$ 681.418i 0.0450077i
$$613$$ − 2707.50i − 0.178393i −0.996014 0.0891965i $$-0.971570\pi$$
0.996014 0.0891965i $$-0.0284299\pi$$
$$614$$ 18768.8 1.23363
$$615$$ 0 0
$$616$$ −429.161 −0.0280704
$$617$$ − 23226.3i − 1.51549i −0.652553 0.757743i $$-0.726302\pi$$
0.652553 0.757743i $$-0.273698\pi$$
$$618$$ − 1760.75i − 0.114608i
$$619$$ 2298.43 0.149243 0.0746216 0.997212i $$-0.476225\pi$$
0.0746216 + 0.997212i $$0.476225\pi$$
$$620$$ 0 0
$$621$$ −4764.89 −0.307904
$$622$$ − 16958.1i − 1.09318i
$$623$$ − 5901.91i − 0.379543i
$$624$$ −13014.4 −0.834926
$$625$$ 0 0
$$626$$ −27078.3 −1.72886
$$627$$ 1362.14i 0.0867599i
$$628$$ 6811.01i 0.432785i
$$629$$ −280.094 −0.0177553
$$630$$ 0 0
$$631$$ −663.913 −0.0418858 −0.0209429 0.999781i $$-0.506667\pi$$
−0.0209429 + 0.999781i $$0.506667\pi$$
$$632$$ 5028.22i 0.316474i
$$633$$ − 9803.95i − 0.615596i
$$634$$ −28722.3 −1.79923
$$635$$ 0 0
$$636$$ 1531.58 0.0954890
$$637$$ 2672.77i 0.166247i
$$638$$ 2629.08i 0.163144i
$$639$$ −7145.68 −0.442377
$$640$$ 0 0
$$641$$ 15215.6 0.937566 0.468783 0.883313i $$-0.344693\pi$$
0.468783 + 0.883313i $$0.344693\pi$$
$$642$$ − 12651.5i − 0.777749i
$$643$$ − 12904.0i − 0.791420i −0.918375 0.395710i $$-0.870498\pi$$
0.918375 0.395710i $$-0.129502\pi$$
$$644$$ 5787.15 0.354108
$$645$$ 0 0
$$646$$ 5033.58 0.306569
$$647$$ − 9425.54i − 0.572730i −0.958121 0.286365i $$-0.907553\pi$$
0.958121 0.286365i $$-0.0924470\pi$$
$$648$$ 956.429i 0.0579816i
$$649$$ 1092.26 0.0660632
$$650$$ 0 0
$$651$$ −1980.30 −0.119223
$$652$$ 6092.35i 0.365943i
$$653$$ 29894.7i 1.79153i 0.444528 + 0.895765i $$0.353372\pi$$
−0.444528 + 0.895765i $$0.646628\pi$$
$$654$$ 3560.63 0.212892
$$655$$ 0 0
$$656$$ −16723.0 −0.995312
$$657$$ 2838.75i 0.168570i
$$658$$ − 2628.13i − 0.155707i
$$659$$ 11593.6 0.685313 0.342656 0.939461i $$-0.388673\pi$$
0.342656 + 0.939461i $$0.388673\pi$$
$$660$$ 0 0
$$661$$ −17149.2 −1.00911 −0.504557 0.863378i $$-0.668344\pi$$
−0.504557 + 0.863378i $$0.668344\pi$$
$$662$$ 24791.2i 1.45550i
$$663$$ 2644.72i 0.154921i
$$664$$ −3341.94 −0.195320
$$665$$ 0 0
$$666$$ 555.511 0.0323208
$$667$$ 25089.9i 1.45650i
$$668$$ 9891.47i 0.572923i
$$669$$ 16372.9 0.946210
$$670$$ 0 0
$$671$$ 3501.15 0.201431
$$672$$ − 3964.64i − 0.227589i
$$673$$ 16475.0i 0.943633i 0.881697 + 0.471817i $$0.156402\pi$$
−0.881697 + 0.471817i $$0.843598\pi$$
$$674$$ −16851.9 −0.963071
$$675$$ 0 0
$$676$$ 3646.11 0.207448
$$677$$ − 4559.89i − 0.258864i −0.991588 0.129432i $$-0.958685\pi$$
0.991588 0.129432i $$-0.0413154\pi$$
$$678$$ 20103.1i 1.13872i
$$679$$ 10761.3 0.608221
$$680$$ 0 0
$$681$$ −843.070 −0.0474398
$$682$$ 1743.84i 0.0979106i
$$683$$ 27895.9i 1.56282i 0.624017 + 0.781411i $$0.285500\pi$$
−0.624017 + 0.781411i $$0.714500\pi$$
$$684$$ −3686.93 −0.206101
$$685$$ 0 0
$$686$$ −1221.61 −0.0679904
$$687$$ − 8329.93i − 0.462601i
$$688$$ 41481.1i 2.29862i
$$689$$ 5944.37 0.328683
$$690$$ 0 0
$$691$$ −28178.5 −1.55132 −0.775659 0.631152i $$-0.782582\pi$$
−0.775659 + 0.631152i $$0.782582\pi$$
$$692$$ 1570.73i 0.0862862i
$$693$$ − 327.111i − 0.0179306i
$$694$$ −34891.4 −1.90844
$$695$$ 0 0
$$696$$ 5036.14 0.274274
$$697$$ 3398.37i 0.184680i
$$698$$ − 45240.4i − 2.45326i
$$699$$ 17343.3 0.938458
$$700$$ 0 0
$$701$$ 3912.96 0.210828 0.105414 0.994428i $$-0.466383\pi$$
0.105414 + 0.994428i $$0.466383\pi$$
$$702$$ − 5245.29i − 0.282009i
$$703$$ − 1515.50i − 0.0813060i
$$704$$ −187.673 −0.0100471
$$705$$ 0 0
$$706$$ −35509.4 −1.89294
$$707$$ 11130.0i 0.592058i
$$708$$ 2956.46i 0.156936i
$$709$$ 7782.72 0.412251 0.206126 0.978526i $$-0.433914\pi$$
0.206126 + 0.978526i $$0.433914\pi$$
$$710$$ 0 0
$$711$$ −3832.56 −0.202155
$$712$$ 9955.49i 0.524014i
$$713$$ 16641.8i 0.874112i
$$714$$ −1208.79 −0.0633584
$$715$$ 0 0
$$716$$ 10878.8 0.567823
$$717$$ − 4764.50i − 0.248164i
$$718$$ − 15628.9i − 0.812345i
$$719$$ 27868.8 1.44552 0.722762 0.691097i $$-0.242872\pi$$
0.722762 + 0.691097i $$0.242872\pi$$
$$720$$ 0 0
$$721$$ 1153.55 0.0595844
$$722$$ 2806.40i 0.144658i
$$723$$ − 12990.0i − 0.668194i
$$724$$ 7146.89 0.366868
$$725$$ 0 0
$$726$$ 13933.2 0.712274
$$727$$ − 34202.4i − 1.74484i −0.488759 0.872419i $$-0.662550\pi$$
0.488759 0.872419i $$-0.337450\pi$$
$$728$$ − 4508.50i − 0.229527i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 8429.58 0.426510
$$732$$ 9476.65i 0.478507i
$$733$$ 12544.2i 0.632101i 0.948742 + 0.316051i $$0.102357\pi$$
−0.948742 + 0.316051i $$0.897643\pi$$
$$734$$ 33625.7 1.69094
$$735$$ 0 0
$$736$$ −33317.6 −1.66862
$$737$$ − 1687.11i − 0.0843221i
$$738$$ − 6739.99i − 0.336182i
$$739$$ −4563.19 −0.227144 −0.113572 0.993530i $$-0.536229\pi$$
−0.113572 + 0.993530i $$0.536229\pi$$
$$740$$ 0 0
$$741$$ −14309.7 −0.709422
$$742$$ 2716.92i 0.134422i
$$743$$ − 10369.7i − 0.512017i −0.966674 0.256009i $$-0.917592\pi$$
0.966674 0.256009i $$-0.0824076\pi$$
$$744$$ 3340.42 0.164605
$$745$$ 0 0
$$746$$ −11466.1 −0.562738
$$747$$ − 2547.26i − 0.124765i
$$748$$ 393.120i 0.0192164i
$$749$$ 8288.57 0.404349
$$750$$ 0 0
$$751$$ −36808.0 −1.78847 −0.894237 0.447595i $$-0.852281\pi$$
−0.894237 + 0.447595i $$0.852281\pi$$
$$752$$ 8383.92i 0.406556i
$$753$$ 4201.60i 0.203340i
$$754$$ −27619.4 −1.33401
$$755$$ 0 0
$$756$$ 885.400 0.0425948
$$757$$ − 12516.6i − 0.600955i −0.953789 0.300477i $$-0.902854\pi$$
0.953789 0.300477i $$-0.0971460\pi$$
$$758$$ 49902.3i 2.39121i
$$759$$ −2748.93 −0.131462
$$760$$ 0 0
$$761$$ 11745.1 0.559473 0.279736 0.960077i $$-0.409753\pi$$
0.279736 + 0.960077i $$0.409753\pi$$
$$762$$ − 17509.9i − 0.832438i
$$763$$ 2332.73i 0.110682i
$$764$$ −1376.88 −0.0652011
$$765$$ 0 0
$$766$$ 18957.7 0.894216
$$767$$ 11474.6i 0.540189i
$$768$$ 15629.5i 0.734350i
$$769$$ −36497.1 −1.71147 −0.855735 0.517414i $$-0.826895\pi$$
−0.855735 + 0.517414i $$0.826895\pi$$
$$770$$ 0 0
$$771$$ −12914.6 −0.603252
$$772$$ − 17168.8i − 0.800414i
$$773$$ 29858.1i 1.38929i 0.719353 + 0.694644i $$0.244438\pi$$
−0.719353 + 0.694644i $$0.755562\pi$$
$$774$$ −16718.4 −0.776396
$$775$$ 0 0
$$776$$ −18152.5 −0.839737
$$777$$ 363.940i 0.0168035i
$$778$$ 13692.4i 0.630973i
$$779$$ −18387.5 −0.845699
$$780$$ 0 0
$$781$$ −4122.45 −0.188877
$$782$$ 10158.3i 0.464527i
$$783$$ 3838.60i 0.175199i
$$784$$ 3897.03 0.177525
$$785$$ 0 0
$$786$$ 6390.94 0.290022
$$787$$ 3168.00i 0.143491i 0.997423 + 0.0717453i $$0.0228569\pi$$
−0.997423 + 0.0717453i $$0.977143\pi$$
$$788$$ 23900.6i 1.08049i
$$789$$ −5174.07 −0.233463
$$790$$ 0 0
$$791$$ −13170.5 −0.592019
$$792$$ 551.778i 0.0247558i
$$793$$ 36780.8i 1.64707i
$$794$$ 28657.7 1.28089
$$795$$ 0 0
$$796$$ 23544.5 1.04838
$$797$$ 29317.0i 1.30296i 0.758665 + 0.651481i $$0.225852\pi$$
−0.758665 + 0.651481i $$0.774148\pi$$
$$798$$ − 6540.39i − 0.290134i
$$799$$ 1703.74 0.0754366
$$800$$ 0 0
$$801$$ −7588.18 −0.334725
$$802$$ 27571.3i 1.21394i
$$803$$ 1637.72i 0.0719724i
$$804$$ 4566.54 0.200310
$$805$$ 0 0
$$806$$ −18319.7 −0.800600
$$807$$ − 24014.5i − 1.04752i
$$808$$ − 18774.3i − 0.817422i
$$809$$ −16657.3 −0.723904 −0.361952 0.932197i $$-0.617890\pi$$
−0.361952 + 0.932197i $$0.617890\pi$$
$$810$$ 0 0
$$811$$ 5144.55 0.222749 0.111375 0.993779i $$-0.464475\pi$$
0.111375 + 0.993779i $$0.464475\pi$$
$$812$$ − 4662.14i − 0.201489i
$$813$$ − 5890.95i − 0.254126i
$$814$$ 320.483 0.0137997
$$815$$ 0 0
$$816$$ 3856.13 0.165431
$$817$$ 45609.7i 1.95310i
$$818$$ 31936.2i 1.36507i
$$819$$ 3436.42 0.146616
$$820$$ 0 0
$$821$$ −5217.18 −0.221779 −0.110890 0.993833i $$-0.535370\pi$$
−0.110890 + 0.993833i $$0.535370\pi$$
$$822$$ 10740.8i 0.455751i
$$823$$ − 42326.5i − 1.79272i −0.443327 0.896360i $$-0.646202\pi$$
0.443327 0.896360i $$-0.353798\pi$$
$$824$$ −1945.83 −0.0822649
$$825$$ 0 0
$$826$$ −5244.57 −0.220922
$$827$$ − 31675.8i − 1.33189i −0.746000 0.665946i $$-0.768028\pi$$
0.746000 0.665946i $$-0.231972\pi$$
$$828$$ − 7440.62i − 0.312294i
$$829$$ 3471.22 0.145429 0.0727144 0.997353i $$-0.476834\pi$$
0.0727144 + 0.997353i $$0.476834\pi$$
$$830$$ 0 0
$$831$$ 9835.00 0.410556
$$832$$ − 1971.57i − 0.0821539i
$$833$$ − 791.934i − 0.0329399i
$$834$$ −20033.8 −0.831791
$$835$$ 0 0
$$836$$ −2127.05 −0.0879970
$$837$$ 2546.11i 0.105145i
$$838$$ 44211.7i 1.82252i
$$839$$ 20964.9 0.862682 0.431341 0.902189i $$-0.358041\pi$$
0.431341 + 0.902189i $$0.358041\pi$$
$$840$$ 0 0
$$841$$ −4176.57 −0.171248
$$842$$ − 5955.41i − 0.243749i
$$843$$ 8577.12i 0.350429i
$$844$$ 15309.4 0.624373
$$845$$ 0 0
$$846$$ −3379.02 −0.137321
$$847$$ 9128.28i 0.370309i
$$848$$ − 8667.17i − 0.350981i
$$849$$ −16301.0 −0.658950
$$850$$ 0 0
$$851$$ 3058.44 0.123199
$$852$$ − 11158.4i − 0.448685i
$$853$$ − 1084.77i − 0.0435426i −0.999763 0.0217713i $$-0.993069\pi$$
0.999763 0.0217713i $$-0.00693057\pi$$
$$854$$ −16811.0 −0.673607
$$855$$ 0 0
$$856$$ −13981.4 −0.558263
$$857$$ − 12661.6i − 0.504679i −0.967639 0.252340i $$-0.918800\pi$$
0.967639 0.252340i $$-0.0812001\pi$$
$$858$$ − 3026.09i − 0.120407i
$$859$$ 39678.7 1.57604 0.788021 0.615648i $$-0.211106\pi$$
0.788021 + 0.615648i $$0.211106\pi$$
$$860$$ 0 0
$$861$$ 4415.67 0.174780
$$862$$ 57062.6i 2.25471i
$$863$$ 41614.0i 1.64143i 0.571334 + 0.820717i $$0.306426\pi$$
−0.571334 + 0.820717i $$0.693574\pi$$
$$864$$ −5097.40 −0.200714
$$865$$ 0 0
$$866$$ 38759.2 1.52089
$$867$$ 13955.4i 0.546654i
$$868$$ − 3092.35i − 0.120923i
$$869$$ −2211.06 −0.0863120
$$870$$ 0 0
$$871$$ 17723.7 0.689489
$$872$$ − 3934.90i − 0.152813i
$$873$$ − 13836.0i − 0.536400i
$$874$$ −54963.3 −2.12719
$$875$$ 0 0
$$876$$ −4432.86 −0.170973
$$877$$ − 37061.0i − 1.42698i −0.700665 0.713490i $$-0.747113\pi$$
0.700665 0.713490i $$-0.252887\pi$$
$$878$$ − 27560.7i − 1.05937i
$$879$$ 25750.3 0.988095
$$880$$ 0 0
$$881$$ −25468.7 −0.973962 −0.486981 0.873412i $$-0.661902\pi$$
−0.486981 + 0.873412i $$0.661902\pi$$
$$882$$ 1570.64i 0.0599619i
$$883$$ − 34428.3i − 1.31212i −0.754707 0.656062i $$-0.772221\pi$$
0.754707 0.656062i $$-0.227779\pi$$
$$884$$ −4129.88 −0.157130
$$885$$ 0 0
$$886$$ 31222.6 1.18391
$$887$$ 41295.4i 1.56321i 0.623777 + 0.781603i $$0.285597\pi$$
−0.623777 + 0.781603i $$0.714403\pi$$
$$888$$ − 613.903i − 0.0231996i
$$889$$ 11471.5 0.432782
$$890$$ 0 0
$$891$$ −420.571 −0.0158133
$$892$$ 25567.2i 0.959702i
$$893$$ 9218.37i 0.345443i
$$894$$ −11238.1 −0.420423
$$895$$ 0 0
$$896$$ −9671.26 −0.360596
$$897$$ − 28878.6i − 1.07495i
$$898$$ − 11039.3i − 0.410230i
$$899$$ 13406.7 0.497373
$$900$$ 0 0
$$901$$ −1761.30 −0.0651247
$$902$$ − 3888.40i − 0.143536i
$$903$$ − 10953.0i − 0.403646i
$$904$$ 22216.2 0.817368
$$905$$ 0 0
$$906$$ 8017.59 0.294003
$$907$$ − 53733.8i − 1.96715i −0.180509 0.983573i $$-0.557775\pi$$
0.180509 0.983573i $$-0.442225\pi$$
$$908$$ − 1316.50i − 0.0481162i
$$909$$ 14309.9 0.522146
$$910$$ 0 0
$$911$$ 24296.8 0.883634 0.441817 0.897105i $$-0.354334\pi$$
0.441817 + 0.897105i $$0.354334\pi$$
$$912$$ 20864.3i 0.757550i
$$913$$ − 1469.55i − 0.0532696i
$$914$$ 21807.2 0.789187
$$915$$ 0 0
$$916$$ 13007.6 0.469197
$$917$$ 4186.99i 0.150782i
$$918$$ 1554.16i 0.0558769i
$$919$$ −4280.46 −0.153645 −0.0768223 0.997045i $$-0.524477\pi$$
−0.0768223 + 0.997045i $$0.524477\pi$$
$$920$$ 0 0
$$921$$ 15809.5 0.565625
$$922$$ − 37083.6i − 1.32460i
$$923$$ − 43307.9i − 1.54442i
$$924$$ 510.801 0.0181863
$$925$$ 0 0
$$926$$ −40168.9 −1.42552
$$927$$ − 1483.13i − 0.0525485i
$$928$$ 26840.7i 0.949450i
$$929$$ 31884.5 1.12604 0.563022 0.826442i $$-0.309639\pi$$
0.563022 + 0.826442i $$0.309639\pi$$
$$930$$ 0 0
$$931$$ 4284.90 0.150840
$$932$$ 27082.4i 0.951839i
$$933$$ − 14284.3i − 0.501229i
$$934$$ −53149.6 −1.86200
$$935$$ 0 0
$$936$$ −5796.64 −0.202424
$$937$$ − 44523.1i − 1.55230i −0.630548 0.776151i $$-0.717170\pi$$
0.630548 0.776151i $$-0.282830\pi$$
$$938$$ 8100.76i 0.281982i
$$939$$ −22808.8 −0.792693
$$940$$ 0 0
$$941$$ 46374.9 1.60657 0.803283 0.595598i $$-0.203085\pi$$
0.803283 + 0.595598i $$0.203085\pi$$
$$942$$ 15534.4i 0.537301i
$$943$$ − 37107.9i − 1.28144i
$$944$$ 16730.6 0.576836
$$945$$ 0 0
$$946$$ −9645.09 −0.331489
$$947$$ − 20348.2i − 0.698234i −0.937079 0.349117i $$-0.886482\pi$$
0.937079 0.349117i $$-0.113518\pi$$
$$948$$ − 5984.74i − 0.205037i
$$949$$ −17204.8 −0.588507
$$950$$ 0 0
$$951$$ −24193.6 −0.824956
$$952$$ 1335.85i 0.0454782i
$$953$$ 45012.9i 1.53002i 0.644018 + 0.765010i $$0.277266\pi$$
−0.644018 + 0.765010i $$0.722734\pi$$
$$954$$ 3493.18 0.118549
$$955$$ 0 0
$$956$$ 7440.02 0.251702
$$957$$ 2214.55i 0.0748027i
$$958$$ 16647.4i 0.561435i
$$959$$ −7036.76 −0.236944
$$960$$ 0 0
$$961$$ −20898.5 −0.701503
$$962$$ 3366.79i 0.112838i
$$963$$ − 10656.7i − 0.356603i
$$964$$ 20284.6 0.677721
$$965$$ 0 0
$$966$$ 13199.2 0.439624
$$967$$ − 40305.8i − 1.34038i −0.742190 0.670190i $$-0.766213\pi$$
0.742190 0.670190i $$-0.233787\pi$$
$$968$$ − 15397.8i − 0.511265i
$$969$$ 4239.93 0.140564
$$970$$ 0 0
$$971$$ 33991.8 1.12343 0.561713 0.827332i $$-0.310142\pi$$
0.561713 + 0.827332i $$0.310142\pi$$
$$972$$ − 1138.37i − 0.0375651i
$$973$$ − 13125.0i − 0.432445i
$$974$$ −60837.2 −2.00139
$$975$$ 0 0
$$976$$ 53628.2 1.75881
$$977$$ − 18219.0i − 0.596600i −0.954472 0.298300i $$-0.903580\pi$$
0.954472 0.298300i $$-0.0964196\pi$$
$$978$$ 13895.3i 0.454317i
$$979$$ −4377.73 −0.142914
$$980$$ 0 0
$$981$$ 2999.22 0.0976125
$$982$$ 64834.1i 2.10686i
$$983$$ − 7676.89i − 0.249089i −0.992214 0.124545i $$-0.960253\pi$$
0.992214 0.124545i $$-0.0397470\pi$$
$$984$$ −7448.45 −0.241309
$$985$$ 0 0
$$986$$ 8183.55 0.264318
$$987$$ − 2213.75i − 0.0713925i
$$988$$ − 22345.4i − 0.719537i
$$989$$ −92045.3 −2.95942
$$990$$ 0 0
$$991$$ 50585.6 1.62150 0.810748 0.585395i $$-0.199061\pi$$
0.810748 + 0.585395i $$0.199061\pi$$
$$992$$ 17803.2i 0.569810i
$$993$$ 20882.4i 0.667354i
$$994$$ 19794.2 0.631625
$$995$$ 0 0
$$996$$ 3977.69 0.126544
$$997$$ − 53060.5i − 1.68550i −0.538305 0.842750i $$-0.680935\pi$$
0.538305 0.842750i $$-0.319065\pi$$
$$998$$ − 25320.8i − 0.803121i
$$999$$ 467.923 0.0148193
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 525.4.d.m.274.4 4
5.2 odd 4 525.4.a.j.1.1 2
5.3 odd 4 525.4.a.m.1.2 yes 2
5.4 even 2 inner 525.4.d.m.274.1 4
15.2 even 4 1575.4.a.x.1.2 2
15.8 even 4 1575.4.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 5.2 odd 4
525.4.a.m.1.2 yes 2 5.3 odd 4
525.4.d.m.274.1 4 5.4 even 2 inner
525.4.d.m.274.4 4 1.1 even 1 trivial
1575.4.a.o.1.1 2 15.8 even 4
1575.4.a.x.1.2 2 15.2 even 4