# Properties

 Label 1856.4.a.y.1.5 Level $1856$ Weight $4$ Character 1856.1 Self dual yes Analytic conductor $109.508$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.13458092.1 Defining polynomial: $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ x^5 - x^4 - 14*x^3 + 18*x^2 + 20*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$3.03898$$ of defining polynomial Character $$\chi$$ $$=$$ 1856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.46343 q^{3} -2.14270 q^{5} +20.3573 q^{7} +14.7760 q^{9} +O(q^{10})$$ $$q+6.46343 q^{3} -2.14270 q^{5} +20.3573 q^{7} +14.7760 q^{9} -52.0703 q^{11} -7.04574 q^{13} -13.8492 q^{15} +28.7724 q^{17} -76.4208 q^{19} +131.578 q^{21} +59.7251 q^{23} -120.409 q^{25} -79.0092 q^{27} +29.0000 q^{29} -3.25229 q^{31} -336.553 q^{33} -43.6196 q^{35} -150.673 q^{37} -45.5397 q^{39} -92.3254 q^{41} +100.703 q^{43} -31.6605 q^{45} +324.003 q^{47} +71.4197 q^{49} +185.969 q^{51} -374.774 q^{53} +111.571 q^{55} -493.941 q^{57} -489.567 q^{59} -221.508 q^{61} +300.799 q^{63} +15.0969 q^{65} +427.538 q^{67} +386.029 q^{69} -898.999 q^{71} -1087.35 q^{73} -778.254 q^{75} -1060.01 q^{77} -798.018 q^{79} -909.622 q^{81} +436.713 q^{83} -61.6507 q^{85} +187.440 q^{87} +456.763 q^{89} -143.432 q^{91} -21.0209 q^{93} +163.747 q^{95} +803.714 q^{97} -769.390 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10})$$ 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9 $$5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100})$$ 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9 - 12 * q^11 - 14 * q^13 - 74 * q^15 + 66 * q^17 - 214 * q^19 + 164 * q^23 + 207 * q^25 - 362 * q^27 + 145 * q^29 + 420 * q^31 - 576 * q^33 + 52 * q^35 - 378 * q^37 - 374 * q^39 - 1158 * q^41 + 204 * q^43 + 1506 * q^45 + 248 * q^47 - 283 * q^49 - 228 * q^51 + 554 * q^53 + 546 * q^55 + 44 * q^57 - 440 * q^59 - 618 * q^61 + 804 * q^63 - 1656 * q^65 - 1164 * q^67 + 1968 * q^69 - 692 * q^71 - 1950 * q^73 - 3074 * q^75 + 1616 * q^77 + 272 * q^79 + 1801 * q^81 - 512 * q^83 + 1628 * q^85 - 232 * q^87 + 866 * q^89 - 2580 * q^91 + 40 * q^93 + 2244 * q^95 + 1562 * q^97 + 238 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 6.46343 1.24389 0.621944 0.783062i $$-0.286343\pi$$
0.621944 + 0.783062i $$0.286343\pi$$
$$4$$ 0 0
$$5$$ −2.14270 −0.191649 −0.0958246 0.995398i $$-0.530549\pi$$
−0.0958246 + 0.995398i $$0.530549\pi$$
$$6$$ 0 0
$$7$$ 20.3573 1.09919 0.549595 0.835431i $$-0.314782\pi$$
0.549595 + 0.835431i $$0.314782\pi$$
$$8$$ 0 0
$$9$$ 14.7760 0.547258
$$10$$ 0 0
$$11$$ −52.0703 −1.42725 −0.713627 0.700526i $$-0.752949\pi$$
−0.713627 + 0.700526i $$0.752949\pi$$
$$12$$ 0 0
$$13$$ −7.04574 −0.150318 −0.0751591 0.997172i $$-0.523946\pi$$
−0.0751591 + 0.997172i $$0.523946\pi$$
$$14$$ 0 0
$$15$$ −13.8492 −0.238390
$$16$$ 0 0
$$17$$ 28.7724 0.410490 0.205245 0.978711i $$-0.434201\pi$$
0.205245 + 0.978711i $$0.434201\pi$$
$$18$$ 0 0
$$19$$ −76.4208 −0.922744 −0.461372 0.887207i $$-0.652643\pi$$
−0.461372 + 0.887207i $$0.652643\pi$$
$$20$$ 0 0
$$21$$ 131.578 1.36727
$$22$$ 0 0
$$23$$ 59.7251 0.541458 0.270729 0.962656i $$-0.412735\pi$$
0.270729 + 0.962656i $$0.412735\pi$$
$$24$$ 0 0
$$25$$ −120.409 −0.963271
$$26$$ 0 0
$$27$$ −79.0092 −0.563160
$$28$$ 0 0
$$29$$ 29.0000 0.185695
$$30$$ 0 0
$$31$$ −3.25229 −0.0188428 −0.00942142 0.999956i $$-0.502999\pi$$
−0.00942142 + 0.999956i $$0.502999\pi$$
$$32$$ 0 0
$$33$$ −336.553 −1.77534
$$34$$ 0 0
$$35$$ −43.6196 −0.210659
$$36$$ 0 0
$$37$$ −150.673 −0.669471 −0.334736 0.942312i $$-0.608647\pi$$
−0.334736 + 0.942312i $$0.608647\pi$$
$$38$$ 0 0
$$39$$ −45.5397 −0.186979
$$40$$ 0 0
$$41$$ −92.3254 −0.351678 −0.175839 0.984419i $$-0.556264\pi$$
−0.175839 + 0.984419i $$0.556264\pi$$
$$42$$ 0 0
$$43$$ 100.703 0.357142 0.178571 0.983927i $$-0.442852\pi$$
0.178571 + 0.983927i $$0.442852\pi$$
$$44$$ 0 0
$$45$$ −31.6605 −0.104882
$$46$$ 0 0
$$47$$ 324.003 1.00555 0.502774 0.864418i $$-0.332313\pi$$
0.502774 + 0.864418i $$0.332313\pi$$
$$48$$ 0 0
$$49$$ 71.4197 0.208221
$$50$$ 0 0
$$51$$ 185.969 0.510604
$$52$$ 0 0
$$53$$ −374.774 −0.971305 −0.485653 0.874152i $$-0.661418\pi$$
−0.485653 + 0.874152i $$0.661418\pi$$
$$54$$ 0 0
$$55$$ 111.571 0.273532
$$56$$ 0 0
$$57$$ −493.941 −1.14779
$$58$$ 0 0
$$59$$ −489.567 −1.08027 −0.540137 0.841577i $$-0.681628\pi$$
−0.540137 + 0.841577i $$0.681628\pi$$
$$60$$ 0 0
$$61$$ −221.508 −0.464937 −0.232468 0.972604i $$-0.574680\pi$$
−0.232468 + 0.972604i $$0.574680\pi$$
$$62$$ 0 0
$$63$$ 300.799 0.601541
$$64$$ 0 0
$$65$$ 15.0969 0.0288083
$$66$$ 0 0
$$67$$ 427.538 0.779584 0.389792 0.920903i $$-0.372547\pi$$
0.389792 + 0.920903i $$0.372547\pi$$
$$68$$ 0 0
$$69$$ 386.029 0.673514
$$70$$ 0 0
$$71$$ −898.999 −1.50270 −0.751349 0.659905i $$-0.770596\pi$$
−0.751349 + 0.659905i $$0.770596\pi$$
$$72$$ 0 0
$$73$$ −1087.35 −1.74335 −0.871673 0.490087i $$-0.836965\pi$$
−0.871673 + 0.490087i $$0.836965\pi$$
$$74$$ 0 0
$$75$$ −778.254 −1.19820
$$76$$ 0 0
$$77$$ −1060.01 −1.56882
$$78$$ 0 0
$$79$$ −798.018 −1.13651 −0.568253 0.822854i $$-0.692381\pi$$
−0.568253 + 0.822854i $$0.692381\pi$$
$$80$$ 0 0
$$81$$ −909.622 −1.24777
$$82$$ 0 0
$$83$$ 436.713 0.577536 0.288768 0.957399i $$-0.406754\pi$$
0.288768 + 0.957399i $$0.406754\pi$$
$$84$$ 0 0
$$85$$ −61.6507 −0.0786701
$$86$$ 0 0
$$87$$ 187.440 0.230984
$$88$$ 0 0
$$89$$ 456.763 0.544009 0.272004 0.962296i $$-0.412313\pi$$
0.272004 + 0.962296i $$0.412313\pi$$
$$90$$ 0 0
$$91$$ −143.432 −0.165228
$$92$$ 0 0
$$93$$ −21.0209 −0.0234384
$$94$$ 0 0
$$95$$ 163.747 0.176843
$$96$$ 0 0
$$97$$ 803.714 0.841287 0.420643 0.907226i $$-0.361804\pi$$
0.420643 + 0.907226i $$0.361804\pi$$
$$98$$ 0 0
$$99$$ −769.390 −0.781077
$$100$$ 0 0
$$101$$ 738.800 0.727855 0.363928 0.931427i $$-0.381436\pi$$
0.363928 + 0.931427i $$0.381436\pi$$
$$102$$ 0 0
$$103$$ 2031.60 1.94349 0.971743 0.236042i $$-0.0758501\pi$$
0.971743 + 0.236042i $$0.0758501\pi$$
$$104$$ 0 0
$$105$$ −281.933 −0.262036
$$106$$ 0 0
$$107$$ −1594.33 −1.44047 −0.720233 0.693732i $$-0.755965\pi$$
−0.720233 + 0.693732i $$0.755965\pi$$
$$108$$ 0 0
$$109$$ 229.668 0.201818 0.100909 0.994896i $$-0.467825\pi$$
0.100909 + 0.994896i $$0.467825\pi$$
$$110$$ 0 0
$$111$$ −973.863 −0.832748
$$112$$ 0 0
$$113$$ −1584.73 −1.31928 −0.659640 0.751582i $$-0.729291\pi$$
−0.659640 + 0.751582i $$0.729291\pi$$
$$114$$ 0 0
$$115$$ −127.973 −0.103770
$$116$$ 0 0
$$117$$ −104.108 −0.0822629
$$118$$ 0 0
$$119$$ 585.728 0.451207
$$120$$ 0 0
$$121$$ 1380.32 1.03705
$$122$$ 0 0
$$123$$ −596.739 −0.437448
$$124$$ 0 0
$$125$$ 525.838 0.376259
$$126$$ 0 0
$$127$$ 621.184 0.434025 0.217013 0.976169i $$-0.430369\pi$$
0.217013 + 0.976169i $$0.430369\pi$$
$$128$$ 0 0
$$129$$ 650.890 0.444245
$$130$$ 0 0
$$131$$ 1504.34 1.00332 0.501659 0.865066i $$-0.332723\pi$$
0.501659 + 0.865066i $$0.332723\pi$$
$$132$$ 0 0
$$133$$ −1555.72 −1.01427
$$134$$ 0 0
$$135$$ 169.293 0.107929
$$136$$ 0 0
$$137$$ −29.3812 −0.0183227 −0.00916135 0.999958i $$-0.502916\pi$$
−0.00916135 + 0.999958i $$0.502916\pi$$
$$138$$ 0 0
$$139$$ −1262.60 −0.770450 −0.385225 0.922823i $$-0.625876\pi$$
−0.385225 + 0.922823i $$0.625876\pi$$
$$140$$ 0 0
$$141$$ 2094.17 1.25079
$$142$$ 0 0
$$143$$ 366.874 0.214542
$$144$$ 0 0
$$145$$ −62.1384 −0.0355883
$$146$$ 0 0
$$147$$ 461.616 0.259003
$$148$$ 0 0
$$149$$ −826.526 −0.454441 −0.227220 0.973843i $$-0.572964\pi$$
−0.227220 + 0.973843i $$0.572964\pi$$
$$150$$ 0 0
$$151$$ −2632.03 −1.41849 −0.709243 0.704964i $$-0.750963\pi$$
−0.709243 + 0.704964i $$0.750963\pi$$
$$152$$ 0 0
$$153$$ 425.140 0.224644
$$154$$ 0 0
$$155$$ 6.96868 0.00361121
$$156$$ 0 0
$$157$$ −234.540 −0.119225 −0.0596124 0.998222i $$-0.518986\pi$$
−0.0596124 + 0.998222i $$0.518986\pi$$
$$158$$ 0 0
$$159$$ −2422.33 −1.20820
$$160$$ 0 0
$$161$$ 1215.84 0.595166
$$162$$ 0 0
$$163$$ 1650.78 0.793246 0.396623 0.917982i $$-0.370182\pi$$
0.396623 + 0.917982i $$0.370182\pi$$
$$164$$ 0 0
$$165$$ 721.133 0.340243
$$166$$ 0 0
$$167$$ −3684.28 −1.70717 −0.853587 0.520950i $$-0.825578\pi$$
−0.853587 + 0.520950i $$0.825578\pi$$
$$168$$ 0 0
$$169$$ −2147.36 −0.977404
$$170$$ 0 0
$$171$$ −1129.19 −0.504979
$$172$$ 0 0
$$173$$ 3073.58 1.35075 0.675375 0.737474i $$-0.263982\pi$$
0.675375 + 0.737474i $$0.263982\pi$$
$$174$$ 0 0
$$175$$ −2451.20 −1.05882
$$176$$ 0 0
$$177$$ −3164.28 −1.34374
$$178$$ 0 0
$$179$$ −2980.70 −1.24463 −0.622313 0.782769i $$-0.713807\pi$$
−0.622313 + 0.782769i $$0.713807\pi$$
$$180$$ 0 0
$$181$$ −3145.07 −1.29155 −0.645777 0.763526i $$-0.723467\pi$$
−0.645777 + 0.763526i $$0.723467\pi$$
$$182$$ 0 0
$$183$$ −1431.70 −0.578329
$$184$$ 0 0
$$185$$ 322.847 0.128304
$$186$$ 0 0
$$187$$ −1498.19 −0.585874
$$188$$ 0 0
$$189$$ −1608.41 −0.619021
$$190$$ 0 0
$$191$$ −2125.56 −0.805236 −0.402618 0.915368i $$-0.631900\pi$$
−0.402618 + 0.915368i $$0.631900\pi$$
$$192$$ 0 0
$$193$$ 3085.29 1.15069 0.575347 0.817909i $$-0.304867\pi$$
0.575347 + 0.817909i $$0.304867\pi$$
$$194$$ 0 0
$$195$$ 97.5779 0.0358344
$$196$$ 0 0
$$197$$ 2484.45 0.898526 0.449263 0.893400i $$-0.351687\pi$$
0.449263 + 0.893400i $$0.351687\pi$$
$$198$$ 0 0
$$199$$ 5489.57 1.95550 0.977752 0.209764i $$-0.0672696\pi$$
0.977752 + 0.209764i $$0.0672696\pi$$
$$200$$ 0 0
$$201$$ 2763.37 0.969715
$$202$$ 0 0
$$203$$ 590.362 0.204115
$$204$$ 0 0
$$205$$ 197.826 0.0673988
$$206$$ 0 0
$$207$$ 882.496 0.296318
$$208$$ 0 0
$$209$$ 3979.26 1.31699
$$210$$ 0 0
$$211$$ 1884.64 0.614902 0.307451 0.951564i $$-0.400524\pi$$
0.307451 + 0.951564i $$0.400524\pi$$
$$212$$ 0 0
$$213$$ −5810.62 −1.86919
$$214$$ 0 0
$$215$$ −215.777 −0.0684460
$$216$$ 0 0
$$217$$ −66.2078 −0.0207119
$$218$$ 0 0
$$219$$ −7027.99 −2.16853
$$220$$ 0 0
$$221$$ −202.723 −0.0617041
$$222$$ 0 0
$$223$$ −5208.50 −1.56407 −0.782033 0.623237i $$-0.785817\pi$$
−0.782033 + 0.623237i $$0.785817\pi$$
$$224$$ 0 0
$$225$$ −1779.16 −0.527158
$$226$$ 0 0
$$227$$ 5243.29 1.53308 0.766540 0.642196i $$-0.221977\pi$$
0.766540 + 0.642196i $$0.221977\pi$$
$$228$$ 0 0
$$229$$ 846.348 0.244228 0.122114 0.992516i $$-0.461033\pi$$
0.122114 + 0.992516i $$0.461033\pi$$
$$230$$ 0 0
$$231$$ −6851.31 −1.95144
$$232$$ 0 0
$$233$$ 823.620 0.231576 0.115788 0.993274i $$-0.463061\pi$$
0.115788 + 0.993274i $$0.463061\pi$$
$$234$$ 0 0
$$235$$ −694.243 −0.192712
$$236$$ 0 0
$$237$$ −5157.93 −1.41369
$$238$$ 0 0
$$239$$ −2471.50 −0.668903 −0.334452 0.942413i $$-0.608551\pi$$
−0.334452 + 0.942413i $$0.608551\pi$$
$$240$$ 0 0
$$241$$ −1148.79 −0.307055 −0.153527 0.988144i $$-0.549063\pi$$
−0.153527 + 0.988144i $$0.549063\pi$$
$$242$$ 0 0
$$243$$ −3746.03 −0.988922
$$244$$ 0 0
$$245$$ −153.031 −0.0399053
$$246$$ 0 0
$$247$$ 538.441 0.138705
$$248$$ 0 0
$$249$$ 2822.67 0.718391
$$250$$ 0 0
$$251$$ 1686.20 0.424032 0.212016 0.977266i $$-0.431997\pi$$
0.212016 + 0.977266i $$0.431997\pi$$
$$252$$ 0 0
$$253$$ −3109.91 −0.772799
$$254$$ 0 0
$$255$$ −398.475 −0.0978568
$$256$$ 0 0
$$257$$ 3593.97 0.872318 0.436159 0.899870i $$-0.356338\pi$$
0.436159 + 0.899870i $$0.356338\pi$$
$$258$$ 0 0
$$259$$ −3067.29 −0.735877
$$260$$ 0 0
$$261$$ 428.503 0.101623
$$262$$ 0 0
$$263$$ −2321.87 −0.544382 −0.272191 0.962243i $$-0.587748\pi$$
−0.272191 + 0.962243i $$0.587748\pi$$
$$264$$ 0 0
$$265$$ 803.029 0.186150
$$266$$ 0 0
$$267$$ 2952.26 0.676686
$$268$$ 0 0
$$269$$ −2365.41 −0.536140 −0.268070 0.963399i $$-0.586386\pi$$
−0.268070 + 0.963399i $$0.586386\pi$$
$$270$$ 0 0
$$271$$ 1732.51 0.388348 0.194174 0.980967i $$-0.437797\pi$$
0.194174 + 0.980967i $$0.437797\pi$$
$$272$$ 0 0
$$273$$ −927.065 −0.205526
$$274$$ 0 0
$$275$$ 6269.73 1.37483
$$276$$ 0 0
$$277$$ 688.616 0.149368 0.0746839 0.997207i $$-0.476205\pi$$
0.0746839 + 0.997207i $$0.476205\pi$$
$$278$$ 0 0
$$279$$ −48.0557 −0.0103119
$$280$$ 0 0
$$281$$ 2224.44 0.472238 0.236119 0.971724i $$-0.424125\pi$$
0.236119 + 0.971724i $$0.424125\pi$$
$$282$$ 0 0
$$283$$ −5037.52 −1.05813 −0.529063 0.848582i $$-0.677457\pi$$
−0.529063 + 0.848582i $$0.677457\pi$$
$$284$$ 0 0
$$285$$ 1058.37 0.219973
$$286$$ 0 0
$$287$$ −1879.50 −0.386561
$$288$$ 0 0
$$289$$ −4085.15 −0.831498
$$290$$ 0 0
$$291$$ 5194.75 1.04647
$$292$$ 0 0
$$293$$ 6761.01 1.34806 0.674032 0.738702i $$-0.264561\pi$$
0.674032 + 0.738702i $$0.264561\pi$$
$$294$$ 0 0
$$295$$ 1049.00 0.207034
$$296$$ 0 0
$$297$$ 4114.03 0.803773
$$298$$ 0 0
$$299$$ −420.807 −0.0813910
$$300$$ 0 0
$$301$$ 2050.05 0.392568
$$302$$ 0 0
$$303$$ 4775.19 0.905371
$$304$$ 0 0
$$305$$ 474.625 0.0891047
$$306$$ 0 0
$$307$$ −8860.99 −1.64731 −0.823653 0.567093i $$-0.808068\pi$$
−0.823653 + 0.567093i $$0.808068\pi$$
$$308$$ 0 0
$$309$$ 13131.1 2.41748
$$310$$ 0 0
$$311$$ 4127.77 0.752619 0.376309 0.926494i $$-0.377193\pi$$
0.376309 + 0.926494i $$0.377193\pi$$
$$312$$ 0 0
$$313$$ −5384.53 −0.972369 −0.486184 0.873856i $$-0.661612\pi$$
−0.486184 + 0.873856i $$0.661612\pi$$
$$314$$ 0 0
$$315$$ −644.522 −0.115285
$$316$$ 0 0
$$317$$ 5379.49 0.953129 0.476565 0.879139i $$-0.341882\pi$$
0.476565 + 0.879139i $$0.341882\pi$$
$$318$$ 0 0
$$319$$ −1510.04 −0.265034
$$320$$ 0 0
$$321$$ −10304.9 −1.79178
$$322$$ 0 0
$$323$$ −2198.81 −0.378778
$$324$$ 0 0
$$325$$ 848.369 0.144797
$$326$$ 0 0
$$327$$ 1484.44 0.251039
$$328$$ 0 0
$$329$$ 6595.83 1.10529
$$330$$ 0 0
$$331$$ −5825.09 −0.967298 −0.483649 0.875262i $$-0.660689\pi$$
−0.483649 + 0.875262i $$0.660689\pi$$
$$332$$ 0 0
$$333$$ −2226.34 −0.366374
$$334$$ 0 0
$$335$$ −916.087 −0.149407
$$336$$ 0 0
$$337$$ −3627.26 −0.586318 −0.293159 0.956064i $$-0.594707\pi$$
−0.293159 + 0.956064i $$0.594707\pi$$
$$338$$ 0 0
$$339$$ −10242.8 −1.64104
$$340$$ 0 0
$$341$$ 169.348 0.0268935
$$342$$ 0 0
$$343$$ −5528.64 −0.870317
$$344$$ 0 0
$$345$$ −827.145 −0.129078
$$346$$ 0 0
$$347$$ −1172.68 −0.181421 −0.0907104 0.995877i $$-0.528914\pi$$
−0.0907104 + 0.995877i $$0.528914\pi$$
$$348$$ 0 0
$$349$$ 173.078 0.0265463 0.0132731 0.999912i $$-0.495775\pi$$
0.0132731 + 0.999912i $$0.495775\pi$$
$$350$$ 0 0
$$351$$ 556.678 0.0846532
$$352$$ 0 0
$$353$$ 13008.4 1.96137 0.980687 0.195582i $$-0.0626596\pi$$
0.980687 + 0.195582i $$0.0626596\pi$$
$$354$$ 0 0
$$355$$ 1926.29 0.287991
$$356$$ 0 0
$$357$$ 3785.82 0.561251
$$358$$ 0 0
$$359$$ −4487.40 −0.659710 −0.329855 0.944032i $$-0.607000\pi$$
−0.329855 + 0.944032i $$0.607000\pi$$
$$360$$ 0 0
$$361$$ −1018.85 −0.148543
$$362$$ 0 0
$$363$$ 8921.60 1.28998
$$364$$ 0 0
$$365$$ 2329.86 0.334111
$$366$$ 0 0
$$367$$ 1674.11 0.238114 0.119057 0.992887i $$-0.462013\pi$$
0.119057 + 0.992887i $$0.462013\pi$$
$$368$$ 0 0
$$369$$ −1364.20 −0.192459
$$370$$ 0 0
$$371$$ −7629.39 −1.06765
$$372$$ 0 0
$$373$$ 6800.89 0.944066 0.472033 0.881581i $$-0.343520\pi$$
0.472033 + 0.881581i $$0.343520\pi$$
$$374$$ 0 0
$$375$$ 3398.72 0.468024
$$376$$ 0 0
$$377$$ −204.326 −0.0279134
$$378$$ 0 0
$$379$$ −10216.7 −1.38468 −0.692341 0.721571i $$-0.743421\pi$$
−0.692341 + 0.721571i $$0.743421\pi$$
$$380$$ 0 0
$$381$$ 4014.98 0.539879
$$382$$ 0 0
$$383$$ −7184.47 −0.958509 −0.479255 0.877676i $$-0.659093\pi$$
−0.479255 + 0.877676i $$0.659093\pi$$
$$384$$ 0 0
$$385$$ 2271.29 0.300664
$$386$$ 0 0
$$387$$ 1487.99 0.195449
$$388$$ 0 0
$$389$$ 3848.67 0.501633 0.250817 0.968035i $$-0.419301\pi$$
0.250817 + 0.968035i $$0.419301\pi$$
$$390$$ 0 0
$$391$$ 1718.43 0.222263
$$392$$ 0 0
$$393$$ 9723.18 1.24801
$$394$$ 0 0
$$395$$ 1709.91 0.217810
$$396$$ 0 0
$$397$$ 1622.51 0.205117 0.102559 0.994727i $$-0.467297\pi$$
0.102559 + 0.994727i $$0.467297\pi$$
$$398$$ 0 0
$$399$$ −10055.3 −1.26164
$$400$$ 0 0
$$401$$ 13842.9 1.72389 0.861947 0.506999i $$-0.169245\pi$$
0.861947 + 0.506999i $$0.169245\pi$$
$$402$$ 0 0
$$403$$ 22.9148 0.00283242
$$404$$ 0 0
$$405$$ 1949.05 0.239133
$$406$$ 0 0
$$407$$ 7845.58 0.955506
$$408$$ 0 0
$$409$$ 3357.24 0.405880 0.202940 0.979191i $$-0.434950\pi$$
0.202940 + 0.979191i $$0.434950\pi$$
$$410$$ 0 0
$$411$$ −189.904 −0.0227914
$$412$$ 0 0
$$413$$ −9966.26 −1.18743
$$414$$ 0 0
$$415$$ −935.746 −0.110684
$$416$$ 0 0
$$417$$ −8160.74 −0.958353
$$418$$ 0 0
$$419$$ −1652.11 −0.192627 −0.0963134 0.995351i $$-0.530705\pi$$
−0.0963134 + 0.995351i $$0.530705\pi$$
$$420$$ 0 0
$$421$$ 6405.13 0.741490 0.370745 0.928735i $$-0.379102\pi$$
0.370745 + 0.928735i $$0.379102\pi$$
$$422$$ 0 0
$$423$$ 4787.46 0.550294
$$424$$ 0 0
$$425$$ −3464.45 −0.395413
$$426$$ 0 0
$$427$$ −4509.30 −0.511054
$$428$$ 0 0
$$429$$ 2371.27 0.266867
$$430$$ 0 0
$$431$$ 252.536 0.0282233 0.0141116 0.999900i $$-0.495508\pi$$
0.0141116 + 0.999900i $$0.495508\pi$$
$$432$$ 0 0
$$433$$ −12779.1 −1.41831 −0.709153 0.705055i $$-0.750922\pi$$
−0.709153 + 0.705055i $$0.750922\pi$$
$$434$$ 0 0
$$435$$ −401.627 −0.0442679
$$436$$ 0 0
$$437$$ −4564.24 −0.499628
$$438$$ 0 0
$$439$$ −3760.84 −0.408873 −0.204437 0.978880i $$-0.565536\pi$$
−0.204437 + 0.978880i $$0.565536\pi$$
$$440$$ 0 0
$$441$$ 1055.29 0.113950
$$442$$ 0 0
$$443$$ −3713.77 −0.398299 −0.199149 0.979969i $$-0.563818\pi$$
−0.199149 + 0.979969i $$0.563818\pi$$
$$444$$ 0 0
$$445$$ −978.707 −0.104259
$$446$$ 0 0
$$447$$ −5342.20 −0.565274
$$448$$ 0 0
$$449$$ −12043.9 −1.26589 −0.632947 0.774195i $$-0.718155\pi$$
−0.632947 + 0.774195i $$0.718155\pi$$
$$450$$ 0 0
$$451$$ 4807.41 0.501934
$$452$$ 0 0
$$453$$ −17011.9 −1.76444
$$454$$ 0 0
$$455$$ 307.333 0.0316659
$$456$$ 0 0
$$457$$ −12995.7 −1.33023 −0.665113 0.746743i $$-0.731617\pi$$
−0.665113 + 0.746743i $$0.731617\pi$$
$$458$$ 0 0
$$459$$ −2273.28 −0.231172
$$460$$ 0 0
$$461$$ −7874.30 −0.795538 −0.397769 0.917486i $$-0.630215\pi$$
−0.397769 + 0.917486i $$0.630215\pi$$
$$462$$ 0 0
$$463$$ −3466.08 −0.347910 −0.173955 0.984754i $$-0.555655\pi$$
−0.173955 + 0.984754i $$0.555655\pi$$
$$464$$ 0 0
$$465$$ 45.0416 0.00449195
$$466$$ 0 0
$$467$$ −14835.9 −1.47007 −0.735034 0.678030i $$-0.762834\pi$$
−0.735034 + 0.678030i $$0.762834\pi$$
$$468$$ 0 0
$$469$$ 8703.53 0.856912
$$470$$ 0 0
$$471$$ −1515.93 −0.148302
$$472$$ 0 0
$$473$$ −5243.66 −0.509733
$$474$$ 0 0
$$475$$ 9201.74 0.888853
$$476$$ 0 0
$$477$$ −5537.65 −0.531555
$$478$$ 0 0
$$479$$ 8119.86 0.774542 0.387271 0.921966i $$-0.373418\pi$$
0.387271 + 0.921966i $$0.373418\pi$$
$$480$$ 0 0
$$481$$ 1061.60 0.100634
$$482$$ 0 0
$$483$$ 7858.51 0.740320
$$484$$ 0 0
$$485$$ −1722.12 −0.161232
$$486$$ 0 0
$$487$$ 9698.92 0.902464 0.451232 0.892407i $$-0.350985\pi$$
0.451232 + 0.892407i $$0.350985\pi$$
$$488$$ 0 0
$$489$$ 10669.7 0.986709
$$490$$ 0 0
$$491$$ 6844.25 0.629076 0.314538 0.949245i $$-0.398150\pi$$
0.314538 + 0.949245i $$0.398150\pi$$
$$492$$ 0 0
$$493$$ 834.400 0.0762261
$$494$$ 0 0
$$495$$ 1648.57 0.149693
$$496$$ 0 0
$$497$$ −18301.2 −1.65175
$$498$$ 0 0
$$499$$ 18603.4 1.66895 0.834473 0.551049i $$-0.185772\pi$$
0.834473 + 0.551049i $$0.185772\pi$$
$$500$$ 0 0
$$501$$ −23813.1 −2.12353
$$502$$ 0 0
$$503$$ 10624.8 0.941822 0.470911 0.882181i $$-0.343925\pi$$
0.470911 + 0.882181i $$0.343925\pi$$
$$504$$ 0 0
$$505$$ −1583.03 −0.139493
$$506$$ 0 0
$$507$$ −13879.3 −1.21578
$$508$$ 0 0
$$509$$ 7931.22 0.690658 0.345329 0.938482i $$-0.387767\pi$$
0.345329 + 0.938482i $$0.387767\pi$$
$$510$$ 0 0
$$511$$ −22135.4 −1.91627
$$512$$ 0 0
$$513$$ 6037.95 0.519653
$$514$$ 0 0
$$515$$ −4353.10 −0.372467
$$516$$ 0 0
$$517$$ −16871.0 −1.43517
$$518$$ 0 0
$$519$$ 19865.9 1.68018
$$520$$ 0 0
$$521$$ 13771.1 1.15801 0.579005 0.815324i $$-0.303441\pi$$
0.579005 + 0.815324i $$0.303441\pi$$
$$522$$ 0 0
$$523$$ −20397.1 −1.70536 −0.852680 0.522433i $$-0.825024\pi$$
−0.852680 + 0.522433i $$0.825024\pi$$
$$524$$ 0 0
$$525$$ −15843.2 −1.31705
$$526$$ 0 0
$$527$$ −93.5761 −0.00773480
$$528$$ 0 0
$$529$$ −8599.91 −0.706823
$$530$$ 0 0
$$531$$ −7233.83 −0.591189
$$532$$ 0 0
$$533$$ 650.501 0.0528636
$$534$$ 0 0
$$535$$ 3416.18 0.276064
$$536$$ 0 0
$$537$$ −19265.6 −1.54817
$$538$$ 0 0
$$539$$ −3718.85 −0.297184
$$540$$ 0 0
$$541$$ −1883.72 −0.149699 −0.0748496 0.997195i $$-0.523848\pi$$
−0.0748496 + 0.997195i $$0.523848\pi$$
$$542$$ 0 0
$$543$$ −20328.0 −1.60655
$$544$$ 0 0
$$545$$ −492.110 −0.0386783
$$546$$ 0 0
$$547$$ 9079.06 0.709676 0.354838 0.934928i $$-0.384536\pi$$
0.354838 + 0.934928i $$0.384536\pi$$
$$548$$ 0 0
$$549$$ −3272.99 −0.254440
$$550$$ 0 0
$$551$$ −2216.20 −0.171349
$$552$$ 0 0
$$553$$ −16245.5 −1.24924
$$554$$ 0 0
$$555$$ 2086.70 0.159595
$$556$$ 0 0
$$557$$ 5982.38 0.455084 0.227542 0.973768i $$-0.426931\pi$$
0.227542 + 0.973768i $$0.426931\pi$$
$$558$$ 0 0
$$559$$ −709.530 −0.0536850
$$560$$ 0 0
$$561$$ −9683.44 −0.728762
$$562$$ 0 0
$$563$$ 7837.88 0.586727 0.293363 0.956001i $$-0.405225\pi$$
0.293363 + 0.956001i $$0.405225\pi$$
$$564$$ 0 0
$$565$$ 3395.60 0.252839
$$566$$ 0 0
$$567$$ −18517.4 −1.37153
$$568$$ 0 0
$$569$$ 10019.2 0.738182 0.369091 0.929393i $$-0.379669\pi$$
0.369091 + 0.929393i $$0.379669\pi$$
$$570$$ 0 0
$$571$$ −8832.38 −0.647327 −0.323663 0.946172i $$-0.604915\pi$$
−0.323663 + 0.946172i $$0.604915\pi$$
$$572$$ 0 0
$$573$$ −13738.4 −1.00162
$$574$$ 0 0
$$575$$ −7191.43 −0.521571
$$576$$ 0 0
$$577$$ −17583.3 −1.26863 −0.634317 0.773073i $$-0.718719\pi$$
−0.634317 + 0.773073i $$0.718719\pi$$
$$578$$ 0 0
$$579$$ 19941.6 1.43133
$$580$$ 0 0
$$581$$ 8890.30 0.634823
$$582$$ 0 0
$$583$$ 19514.6 1.38630
$$584$$ 0 0
$$585$$ 223.072 0.0157656
$$586$$ 0 0
$$587$$ 10120.7 0.711630 0.355815 0.934556i $$-0.384203\pi$$
0.355815 + 0.934556i $$0.384203\pi$$
$$588$$ 0 0
$$589$$ 248.543 0.0173871
$$590$$ 0 0
$$591$$ 16058.1 1.11767
$$592$$ 0 0
$$593$$ 3597.06 0.249095 0.124548 0.992214i $$-0.460252\pi$$
0.124548 + 0.992214i $$0.460252\pi$$
$$594$$ 0 0
$$595$$ −1255.04 −0.0864734
$$596$$ 0 0
$$597$$ 35481.5 2.43243
$$598$$ 0 0
$$599$$ −7987.90 −0.544870 −0.272435 0.962174i $$-0.587829\pi$$
−0.272435 + 0.962174i $$0.587829\pi$$
$$600$$ 0 0
$$601$$ 2646.60 0.179629 0.0898147 0.995958i $$-0.471373\pi$$
0.0898147 + 0.995958i $$0.471373\pi$$
$$602$$ 0 0
$$603$$ 6317.29 0.426634
$$604$$ 0 0
$$605$$ −2957.61 −0.198751
$$606$$ 0 0
$$607$$ −15181.8 −1.01517 −0.507587 0.861600i $$-0.669463\pi$$
−0.507587 + 0.861600i $$0.669463\pi$$
$$608$$ 0 0
$$609$$ 3815.76 0.253896
$$610$$ 0 0
$$611$$ −2282.84 −0.151152
$$612$$ 0 0
$$613$$ 9721.86 0.640558 0.320279 0.947323i $$-0.396223\pi$$
0.320279 + 0.947323i $$0.396223\pi$$
$$614$$ 0 0
$$615$$ 1278.63 0.0838366
$$616$$ 0 0
$$617$$ −14150.2 −0.923282 −0.461641 0.887067i $$-0.652739\pi$$
−0.461641 + 0.887067i $$0.652739\pi$$
$$618$$ 0 0
$$619$$ 10804.4 0.701560 0.350780 0.936458i $$-0.385917\pi$$
0.350780 + 0.936458i $$0.385917\pi$$
$$620$$ 0 0
$$621$$ −4718.83 −0.304928
$$622$$ 0 0
$$623$$ 9298.46 0.597970
$$624$$ 0 0
$$625$$ 13924.4 0.891161
$$626$$ 0 0
$$627$$ 25719.7 1.63819
$$628$$ 0 0
$$629$$ −4335.22 −0.274811
$$630$$ 0 0
$$631$$ 20692.1 1.30545 0.652725 0.757595i $$-0.273626\pi$$
0.652725 + 0.757595i $$0.273626\pi$$
$$632$$ 0 0
$$633$$ 12181.3 0.764869
$$634$$ 0 0
$$635$$ −1331.01 −0.0831805
$$636$$ 0 0
$$637$$ −503.204 −0.0312993
$$638$$ 0 0
$$639$$ −13283.6 −0.822364
$$640$$ 0 0
$$641$$ −1995.59 −0.122966 −0.0614828 0.998108i $$-0.519583\pi$$
−0.0614828 + 0.998108i $$0.519583\pi$$
$$642$$ 0 0
$$643$$ −10911.2 −0.669203 −0.334602 0.942360i $$-0.608602\pi$$
−0.334602 + 0.942360i $$0.608602\pi$$
$$644$$ 0 0
$$645$$ −1394.66 −0.0851392
$$646$$ 0 0
$$647$$ 20046.3 1.21808 0.609042 0.793138i $$-0.291554\pi$$
0.609042 + 0.793138i $$0.291554\pi$$
$$648$$ 0 0
$$649$$ 25491.9 1.54183
$$650$$ 0 0
$$651$$ −427.930 −0.0257633
$$652$$ 0 0
$$653$$ −20205.6 −1.21088 −0.605441 0.795890i $$-0.707003\pi$$
−0.605441 + 0.795890i $$0.707003\pi$$
$$654$$ 0 0
$$655$$ −3223.35 −0.192285
$$656$$ 0 0
$$657$$ −16066.6 −0.954061
$$658$$ 0 0
$$659$$ 9268.73 0.547888 0.273944 0.961746i $$-0.411672\pi$$
0.273944 + 0.961746i $$0.411672\pi$$
$$660$$ 0 0
$$661$$ 18209.9 1.07153 0.535765 0.844367i $$-0.320023\pi$$
0.535765 + 0.844367i $$0.320023\pi$$
$$662$$ 0 0
$$663$$ −1310.29 −0.0767531
$$664$$ 0 0
$$665$$ 3333.45 0.194384
$$666$$ 0 0
$$667$$ 1732.03 0.100546
$$668$$ 0 0
$$669$$ −33664.8 −1.94552
$$670$$ 0 0
$$671$$ 11534.0 0.663583
$$672$$ 0 0
$$673$$ −63.9154 −0.00366086 −0.00183043 0.999998i $$-0.500583\pi$$
−0.00183043 + 0.999998i $$0.500583\pi$$
$$674$$ 0 0
$$675$$ 9513.40 0.542476
$$676$$ 0 0
$$677$$ −11436.2 −0.649231 −0.324616 0.945846i $$-0.605235\pi$$
−0.324616 + 0.945846i $$0.605235\pi$$
$$678$$ 0 0
$$679$$ 16361.5 0.924735
$$680$$ 0 0
$$681$$ 33889.6 1.90698
$$682$$ 0 0
$$683$$ 22719.5 1.27282 0.636410 0.771351i $$-0.280419\pi$$
0.636410 + 0.771351i $$0.280419\pi$$
$$684$$ 0 0
$$685$$ 62.9553 0.00351153
$$686$$ 0 0
$$687$$ 5470.31 0.303793
$$688$$ 0 0
$$689$$ 2640.56 0.146005
$$690$$ 0 0
$$691$$ −10495.0 −0.577783 −0.288891 0.957362i $$-0.593287\pi$$
−0.288891 + 0.957362i $$0.593287\pi$$
$$692$$ 0 0
$$693$$ −15662.7 −0.858552
$$694$$ 0 0
$$695$$ 2705.38 0.147656
$$696$$ 0 0
$$697$$ −2656.42 −0.144360
$$698$$ 0 0
$$699$$ 5323.41 0.288054
$$700$$ 0 0
$$701$$ 6530.49 0.351859 0.175930 0.984403i $$-0.443707\pi$$
0.175930 + 0.984403i $$0.443707\pi$$
$$702$$ 0 0
$$703$$ 11514.5 0.617751
$$704$$ 0 0
$$705$$ −4487.19 −0.239713
$$706$$ 0 0
$$707$$ 15040.0 0.800052
$$708$$ 0 0
$$709$$ 16597.7 0.879183 0.439591 0.898198i $$-0.355123\pi$$
0.439591 + 0.898198i $$0.355123\pi$$
$$710$$ 0 0
$$711$$ −11791.5 −0.621962
$$712$$ 0 0
$$713$$ −194.243 −0.0102026
$$714$$ 0 0
$$715$$ −786.102 −0.0411168
$$716$$ 0 0
$$717$$ −15974.4 −0.832041
$$718$$ 0 0
$$719$$ 6525.25 0.338457 0.169229 0.985577i $$-0.445872\pi$$
0.169229 + 0.985577i $$0.445872\pi$$
$$720$$ 0 0
$$721$$ 41357.8 2.13626
$$722$$ 0 0
$$723$$ −7425.14 −0.381942
$$724$$ 0 0
$$725$$ −3491.86 −0.178875
$$726$$ 0 0
$$727$$ 1066.84 0.0544249 0.0272125 0.999630i $$-0.491337\pi$$
0.0272125 + 0.999630i $$0.491337\pi$$
$$728$$ 0 0
$$729$$ 347.561 0.0176579
$$730$$ 0 0
$$731$$ 2897.48 0.146603
$$732$$ 0 0
$$733$$ 6226.75 0.313765 0.156883 0.987617i $$-0.449856\pi$$
0.156883 + 0.987617i $$0.449856\pi$$
$$734$$ 0 0
$$735$$ −989.106 −0.0496377
$$736$$ 0 0
$$737$$ −22262.1 −1.11266
$$738$$ 0 0
$$739$$ −31226.3 −1.55437 −0.777183 0.629274i $$-0.783352\pi$$
−0.777183 + 0.629274i $$0.783352\pi$$
$$740$$ 0 0
$$741$$ 3480.18 0.172534
$$742$$ 0 0
$$743$$ −30790.4 −1.52031 −0.760156 0.649741i $$-0.774877\pi$$
−0.760156 + 0.649741i $$0.774877\pi$$
$$744$$ 0 0
$$745$$ 1771.00 0.0870932
$$746$$ 0 0
$$747$$ 6452.86 0.316061
$$748$$ 0 0
$$749$$ −32456.3 −1.58335
$$750$$ 0 0
$$751$$ 11908.0 0.578598 0.289299 0.957239i $$-0.406578\pi$$
0.289299 + 0.957239i $$0.406578\pi$$
$$752$$ 0 0
$$753$$ 10898.6 0.527448
$$754$$ 0 0
$$755$$ 5639.65 0.271852
$$756$$ 0 0
$$757$$ 7197.74 0.345583 0.172792 0.984958i $$-0.444721\pi$$
0.172792 + 0.984958i $$0.444721\pi$$
$$758$$ 0 0
$$759$$ −20100.7 −0.961275
$$760$$ 0 0
$$761$$ 9185.20 0.437534 0.218767 0.975777i $$-0.429797\pi$$
0.218767 + 0.975777i $$0.429797\pi$$
$$762$$ 0 0
$$763$$ 4675.42 0.221837
$$764$$ 0 0
$$765$$ −910.949 −0.0430528
$$766$$ 0 0
$$767$$ 3449.36 0.162385
$$768$$ 0 0
$$769$$ 32791.1 1.53768 0.768840 0.639441i $$-0.220834\pi$$
0.768840 + 0.639441i $$0.220834\pi$$
$$770$$ 0 0
$$771$$ 23229.4 1.08507
$$772$$ 0 0
$$773$$ 28237.1 1.31386 0.656932 0.753950i $$-0.271854\pi$$
0.656932 + 0.753950i $$0.271854\pi$$
$$774$$ 0 0
$$775$$ 391.604 0.0181508
$$776$$ 0 0
$$777$$ −19825.2 −0.915349
$$778$$ 0 0
$$779$$ 7055.58 0.324509
$$780$$ 0 0
$$781$$ 46811.2 2.14473
$$782$$ 0 0
$$783$$ −2291.27 −0.104576
$$784$$ 0 0
$$785$$ 502.548 0.0228493
$$786$$ 0 0
$$787$$ −12082.3 −0.547254 −0.273627 0.961836i $$-0.588223\pi$$
−0.273627 + 0.961836i $$0.588223\pi$$
$$788$$ 0 0
$$789$$ −15007.2 −0.677151
$$790$$ 0 0
$$791$$ −32260.8 −1.45014
$$792$$ 0 0
$$793$$ 1560.68 0.0698884
$$794$$ 0 0
$$795$$ 5190.33 0.231550
$$796$$ 0 0
$$797$$ 1248.33 0.0554807 0.0277404 0.999615i $$-0.491169\pi$$
0.0277404 + 0.999615i $$0.491169\pi$$
$$798$$ 0 0
$$799$$ 9322.35 0.412767
$$800$$ 0 0
$$801$$ 6749.12 0.297713
$$802$$ 0 0
$$803$$ 56618.5 2.48820
$$804$$ 0 0
$$805$$ −2605.19 −0.114063
$$806$$ 0 0
$$807$$ −15288.7 −0.666898
$$808$$ 0 0
$$809$$ −23512.6 −1.02183 −0.510914 0.859632i $$-0.670693\pi$$
−0.510914 + 0.859632i $$0.670693\pi$$
$$810$$ 0 0
$$811$$ −23248.4 −1.00661 −0.503306 0.864108i $$-0.667883\pi$$
−0.503306 + 0.864108i $$0.667883\pi$$
$$812$$ 0 0
$$813$$ 11197.9 0.483062
$$814$$ 0 0
$$815$$ −3537.13 −0.152025
$$816$$ 0 0
$$817$$ −7695.84 −0.329551
$$818$$ 0 0
$$819$$ −2119.35 −0.0904226
$$820$$ 0 0
$$821$$ −26898.0 −1.14342 −0.571708 0.820457i $$-0.693719\pi$$
−0.571708 + 0.820457i $$0.693719\pi$$
$$822$$ 0 0
$$823$$ 12422.7 0.526158 0.263079 0.964774i $$-0.415262\pi$$
0.263079 + 0.964774i $$0.415262\pi$$
$$824$$ 0 0
$$825$$ 40524.0 1.71014
$$826$$ 0 0
$$827$$ 4915.14 0.206670 0.103335 0.994647i $$-0.467049\pi$$
0.103335 + 0.994647i $$0.467049\pi$$
$$828$$ 0 0
$$829$$ 45226.4 1.89478 0.947392 0.320074i $$-0.103708\pi$$
0.947392 + 0.320074i $$0.103708\pi$$
$$830$$ 0 0
$$831$$ 4450.82 0.185797
$$832$$ 0 0
$$833$$ 2054.92 0.0854725
$$834$$ 0 0
$$835$$ 7894.31 0.327178
$$836$$ 0 0
$$837$$ 256.961 0.0106115
$$838$$ 0 0
$$839$$ 3982.56 0.163877 0.0819387 0.996637i $$-0.473889\pi$$
0.0819387 + 0.996637i $$0.473889\pi$$
$$840$$ 0 0
$$841$$ 841.000 0.0344828
$$842$$ 0 0
$$843$$ 14377.5 0.587411
$$844$$ 0 0
$$845$$ 4601.15 0.187319
$$846$$ 0 0
$$847$$ 28099.6 1.13992
$$848$$ 0 0
$$849$$ −32559.7 −1.31619
$$850$$ 0 0
$$851$$ −8998.94 −0.362491
$$852$$ 0 0
$$853$$ 37142.6 1.49090 0.745450 0.666562i $$-0.232235\pi$$
0.745450 + 0.666562i $$0.232235\pi$$
$$854$$ 0 0
$$855$$ 2419.52 0.0967789
$$856$$ 0 0
$$857$$ 18690.5 0.744990 0.372495 0.928034i $$-0.378502\pi$$
0.372495 + 0.928034i $$0.378502\pi$$
$$858$$ 0 0
$$859$$ 22852.1 0.907687 0.453843 0.891081i $$-0.350053\pi$$
0.453843 + 0.891081i $$0.350053\pi$$
$$860$$ 0 0
$$861$$ −12148.0 −0.480839
$$862$$ 0 0
$$863$$ 41976.4 1.65573 0.827865 0.560928i $$-0.189555\pi$$
0.827865 + 0.560928i $$0.189555\pi$$
$$864$$ 0 0
$$865$$ −6585.77 −0.258870
$$866$$ 0 0
$$867$$ −26404.1 −1.03429
$$868$$ 0 0
$$869$$ 41553.0 1.62208
$$870$$ 0 0
$$871$$ −3012.32 −0.117186
$$872$$ 0 0
$$873$$ 11875.7 0.460401
$$874$$ 0 0
$$875$$ 10704.6 0.413581
$$876$$ 0 0
$$877$$ −44394.0 −1.70932 −0.854662 0.519184i $$-0.826236\pi$$
−0.854662 + 0.519184i $$0.826236\pi$$
$$878$$ 0 0
$$879$$ 43699.4 1.67684
$$880$$ 0 0
$$881$$ −6337.13 −0.242342 −0.121171 0.992632i $$-0.538665\pi$$
−0.121171 + 0.992632i $$0.538665\pi$$
$$882$$ 0 0
$$883$$ 2834.24 0.108018 0.0540090 0.998540i $$-0.482800\pi$$
0.0540090 + 0.998540i $$0.482800\pi$$
$$884$$ 0 0
$$885$$ 6780.12 0.257527
$$886$$ 0 0
$$887$$ 76.3532 0.00289029 0.00144515 0.999999i $$-0.499540\pi$$
0.00144515 + 0.999999i $$0.499540\pi$$
$$888$$ 0 0
$$889$$ 12645.6 0.477077
$$890$$ 0 0
$$891$$ 47364.3 1.78088
$$892$$ 0 0
$$893$$ −24760.6 −0.927863
$$894$$ 0 0
$$895$$ 6386.75 0.238531
$$896$$ 0 0
$$897$$ −2719.86 −0.101241
$$898$$ 0 0
$$899$$ −94.3163 −0.00349903
$$900$$ 0 0
$$901$$ −10783.2 −0.398711
$$902$$ 0 0
$$903$$ 13250.4 0.488310
$$904$$ 0 0
$$905$$ 6738.95 0.247525
$$906$$ 0 0
$$907$$ −18209.7 −0.666641 −0.333320 0.942814i $$-0.608169\pi$$
−0.333320 + 0.942814i $$0.608169\pi$$
$$908$$ 0 0
$$909$$ 10916.5 0.398325
$$910$$ 0 0
$$911$$ 33348.8 1.21284 0.606419 0.795145i $$-0.292605\pi$$
0.606419 + 0.795145i $$0.292605\pi$$
$$912$$ 0 0
$$913$$ −22739.8 −0.824291
$$914$$ 0 0
$$915$$ 3067.71 0.110836
$$916$$ 0 0
$$917$$ 30624.2 1.10284
$$918$$ 0 0
$$919$$ 20190.6 0.724730 0.362365 0.932036i $$-0.381969\pi$$
0.362365 + 0.932036i $$0.381969\pi$$
$$920$$ 0 0
$$921$$ −57272.4 −2.04907
$$922$$ 0 0
$$923$$ 6334.11 0.225883
$$924$$ 0 0
$$925$$ 18142.3 0.644882
$$926$$ 0 0
$$927$$ 30018.8 1.06359
$$928$$ 0 0
$$929$$ 9089.58 0.321011 0.160506 0.987035i $$-0.448688\pi$$
0.160506 + 0.987035i $$0.448688\pi$$
$$930$$ 0 0
$$931$$ −5457.95 −0.192134
$$932$$ 0 0
$$933$$ 26679.6 0.936173
$$934$$ 0 0
$$935$$ 3210.17 0.112282
$$936$$ 0 0
$$937$$ 39646.5 1.38228 0.691140 0.722721i $$-0.257109\pi$$
0.691140 + 0.722721i $$0.257109\pi$$
$$938$$ 0 0
$$939$$ −34802.5 −1.20952
$$940$$ 0 0
$$941$$ 51013.4 1.76726 0.883629 0.468188i $$-0.155093\pi$$
0.883629 + 0.468188i $$0.155093\pi$$
$$942$$ 0 0
$$943$$ −5514.14 −0.190419
$$944$$ 0 0
$$945$$ 3446.35 0.118635
$$946$$ 0 0
$$947$$ −43867.1 −1.50527 −0.752634 0.658439i $$-0.771217\pi$$
−0.752634 + 0.658439i $$0.771217\pi$$
$$948$$ 0 0
$$949$$ 7661.16 0.262057
$$950$$ 0 0
$$951$$ 34769.9 1.18559
$$952$$ 0 0
$$953$$ −4167.10 −0.141643 −0.0708214 0.997489i $$-0.522562\pi$$
−0.0708214 + 0.997489i $$0.522562\pi$$
$$954$$ 0 0
$$955$$ 4554.44 0.154323
$$956$$ 0 0
$$957$$ −9760.04 −0.329673
$$958$$ 0 0
$$959$$ −598.123 −0.0201401
$$960$$ 0 0
$$961$$ −29780.4 −0.999645
$$962$$ 0 0
$$963$$ −23557.8 −0.788307
$$964$$ 0 0
$$965$$ −6610.85 −0.220529
$$966$$ 0 0
$$967$$ −21935.0 −0.729454 −0.364727 0.931115i $$-0.618838\pi$$
−0.364727 + 0.931115i $$0.618838\pi$$
$$968$$ 0 0
$$969$$ −14211.9 −0.471157
$$970$$ 0 0
$$971$$ 24169.6 0.798805 0.399403 0.916776i $$-0.369218\pi$$
0.399403 + 0.916776i $$0.369218\pi$$
$$972$$ 0 0
$$973$$ −25703.2 −0.846871
$$974$$ 0 0
$$975$$ 5483.38 0.180111
$$976$$ 0 0
$$977$$ 16106.2 0.527415 0.263707 0.964603i $$-0.415055\pi$$
0.263707 + 0.964603i $$0.415055\pi$$
$$978$$ 0 0
$$979$$ −23783.8 −0.776439
$$980$$ 0 0
$$981$$ 3393.57 0.110447
$$982$$ 0 0
$$983$$ −22527.2 −0.730933 −0.365466 0.930825i $$-0.619090\pi$$
−0.365466 + 0.930825i $$0.619090\pi$$
$$984$$ 0 0
$$985$$ −5323.43 −0.172202
$$986$$ 0 0
$$987$$ 42631.7 1.37486
$$988$$ 0 0
$$989$$ 6014.52 0.193378
$$990$$ 0 0
$$991$$ 8337.94 0.267269 0.133634 0.991031i $$-0.457335\pi$$
0.133634 + 0.991031i $$0.457335\pi$$
$$992$$ 0 0
$$993$$ −37650.1 −1.20321
$$994$$ 0 0
$$995$$ −11762.5 −0.374771
$$996$$ 0 0
$$997$$ 18827.2 0.598058 0.299029 0.954244i $$-0.403337\pi$$
0.299029 + 0.954244i $$0.403337\pi$$
$$998$$ 0 0
$$999$$ 11904.5 0.377020
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 1856.4.a.y.1.5 5
4.3 odd 2 1856.4.a.bb.1.1 5
8.3 odd 2 464.4.a.l.1.5 5
8.5 even 2 29.4.a.b.1.1 5
24.5 odd 2 261.4.a.f.1.5 5
40.29 even 2 725.4.a.c.1.5 5
56.13 odd 2 1421.4.a.e.1.1 5
232.173 even 2 841.4.a.b.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.1 5 8.5 even 2
261.4.a.f.1.5 5 24.5 odd 2
464.4.a.l.1.5 5 8.3 odd 2
725.4.a.c.1.5 5 40.29 even 2
841.4.a.b.1.5 5 232.173 even 2
1421.4.a.e.1.1 5 56.13 odd 2
1856.4.a.y.1.5 5 1.1 even 1 trivial
1856.4.a.bb.1.1 5 4.3 odd 2