# Properties

 Label 1680.4.a.bo.1.1 Level $1680$ Weight $4$ Character 1680.1 Self dual yes Analytic conductor $99.123$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$99.1232088096$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1680.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -48.5685 q^{11} -43.6569 q^{13} +15.0000 q^{15} -67.6569 q^{17} +93.2548 q^{19} +21.0000 q^{21} +104.167 q^{23} +25.0000 q^{25} +27.0000 q^{27} -58.7351 q^{29} +9.08831 q^{31} -145.706 q^{33} +35.0000 q^{35} -252.676 q^{37} -130.971 q^{39} +276.274 q^{41} +92.6375 q^{43} +45.0000 q^{45} +582.794 q^{47} +49.0000 q^{49} -202.971 q^{51} +623.019 q^{53} -242.843 q^{55} +279.765 q^{57} +524.999 q^{59} -352.794 q^{61} +63.0000 q^{63} -218.284 q^{65} +736.520 q^{67} +312.500 q^{69} +492.264 q^{71} +1164.75 q^{73} +75.0000 q^{75} -339.980 q^{77} +872.195 q^{79} +81.0000 q^{81} +529.588 q^{83} -338.284 q^{85} -176.205 q^{87} -385.216 q^{89} -305.598 q^{91} +27.2649 q^{93} +466.274 q^{95} -463.892 q^{97} -437.117 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + 16q^{11} - 76q^{13} + 30q^{15} - 124q^{17} + 96q^{19} + 42q^{21} + 16q^{23} + 50q^{25} + 54q^{27} + 188q^{29} + 120q^{31} + 48q^{33} + 70q^{35} - 132q^{37} - 228q^{39} + 100q^{41} + 536q^{43} + 90q^{45} + 928q^{47} + 98q^{49} - 372q^{51} + 884q^{53} + 80q^{55} + 288q^{57} - 104q^{59} - 468q^{61} + 126q^{63} - 380q^{65} + 1688q^{67} + 48q^{69} + 136q^{71} + 508q^{73} + 150q^{75} + 112q^{77} + 432q^{79} + 162q^{81} + 584q^{83} - 620q^{85} + 564q^{87} - 1404q^{89} - 532q^{91} + 360q^{93} + 480q^{95} - 1188q^{97} + 144q^{99} + O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −48.5685 −1.33127 −0.665635 0.746278i $$-0.731839\pi$$
−0.665635 + 0.746278i $$0.731839\pi$$
$$12$$ 0 0
$$13$$ −43.6569 −0.931403 −0.465701 0.884942i $$-0.654198\pi$$
−0.465701 + 0.884942i $$0.654198\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ −67.6569 −0.965247 −0.482623 0.875828i $$-0.660316\pi$$
−0.482623 + 0.875828i $$0.660316\pi$$
$$18$$ 0 0
$$19$$ 93.2548 1.12601 0.563003 0.826455i $$-0.309646\pi$$
0.563003 + 0.826455i $$0.309646\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 0 0
$$23$$ 104.167 0.944357 0.472179 0.881503i $$-0.343468\pi$$
0.472179 + 0.881503i $$0.343468\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −58.7351 −0.376098 −0.188049 0.982160i $$-0.560216\pi$$
−0.188049 + 0.982160i $$0.560216\pi$$
$$30$$ 0 0
$$31$$ 9.08831 0.0526551 0.0263276 0.999653i $$-0.491619\pi$$
0.0263276 + 0.999653i $$0.491619\pi$$
$$32$$ 0 0
$$33$$ −145.706 −0.768609
$$34$$ 0 0
$$35$$ 35.0000 0.169031
$$36$$ 0 0
$$37$$ −252.676 −1.12269 −0.561347 0.827580i $$-0.689717\pi$$
−0.561347 + 0.827580i $$0.689717\pi$$
$$38$$ 0 0
$$39$$ −130.971 −0.537745
$$40$$ 0 0
$$41$$ 276.274 1.05236 0.526180 0.850373i $$-0.323624\pi$$
0.526180 + 0.850373i $$0.323624\pi$$
$$42$$ 0 0
$$43$$ 92.6375 0.328537 0.164268 0.986416i $$-0.447474\pi$$
0.164268 + 0.986416i $$0.447474\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 582.794 1.80871 0.904354 0.426784i $$-0.140354\pi$$
0.904354 + 0.426784i $$0.140354\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −202.971 −0.557286
$$52$$ 0 0
$$53$$ 623.019 1.61468 0.807342 0.590083i $$-0.200905\pi$$
0.807342 + 0.590083i $$0.200905\pi$$
$$54$$ 0 0
$$55$$ −242.843 −0.595362
$$56$$ 0 0
$$57$$ 279.765 0.650100
$$58$$ 0 0
$$59$$ 524.999 1.15846 0.579229 0.815165i $$-0.303354\pi$$
0.579229 + 0.815165i $$0.303354\pi$$
$$60$$ 0 0
$$61$$ −352.794 −0.740502 −0.370251 0.928932i $$-0.620728\pi$$
−0.370251 + 0.928932i $$0.620728\pi$$
$$62$$ 0 0
$$63$$ 63.0000 0.125988
$$64$$ 0 0
$$65$$ −218.284 −0.416536
$$66$$ 0 0
$$67$$ 736.520 1.34299 0.671494 0.741010i $$-0.265653\pi$$
0.671494 + 0.741010i $$0.265653\pi$$
$$68$$ 0 0
$$69$$ 312.500 0.545225
$$70$$ 0 0
$$71$$ 492.264 0.822831 0.411415 0.911448i $$-0.365035\pi$$
0.411415 + 0.911448i $$0.365035\pi$$
$$72$$ 0 0
$$73$$ 1164.75 1.86745 0.933727 0.357987i $$-0.116537\pi$$
0.933727 + 0.357987i $$0.116537\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ −339.980 −0.503173
$$78$$ 0 0
$$79$$ 872.195 1.24215 0.621074 0.783752i $$-0.286697\pi$$
0.621074 + 0.783752i $$0.286697\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 529.588 0.700359 0.350180 0.936683i $$-0.386121\pi$$
0.350180 + 0.936683i $$0.386121\pi$$
$$84$$ 0 0
$$85$$ −338.284 −0.431672
$$86$$ 0 0
$$87$$ −176.205 −0.217140
$$88$$ 0 0
$$89$$ −385.216 −0.458796 −0.229398 0.973333i $$-0.573676\pi$$
−0.229398 + 0.973333i $$0.573676\pi$$
$$90$$ 0 0
$$91$$ −305.598 −0.352037
$$92$$ 0 0
$$93$$ 27.2649 0.0304005
$$94$$ 0 0
$$95$$ 466.274 0.503565
$$96$$ 0 0
$$97$$ −463.892 −0.485579 −0.242789 0.970079i $$-0.578062\pi$$
−0.242789 + 0.970079i $$0.578062\pi$$
$$98$$ 0 0
$$99$$ −437.117 −0.443757
$$100$$ 0 0
$$101$$ −432.725 −0.426314 −0.213157 0.977018i $$-0.568375\pi$$
−0.213157 + 0.977018i $$0.568375\pi$$
$$102$$ 0 0
$$103$$ −512.626 −0.490393 −0.245197 0.969473i $$-0.578853\pi$$
−0.245197 + 0.969473i $$0.578853\pi$$
$$104$$ 0 0
$$105$$ 105.000 0.0975900
$$106$$ 0 0
$$107$$ −1963.09 −1.77363 −0.886817 0.462122i $$-0.847088\pi$$
−0.886817 + 0.462122i $$0.847088\pi$$
$$108$$ 0 0
$$109$$ 545.176 0.479068 0.239534 0.970888i $$-0.423005\pi$$
0.239534 + 0.970888i $$0.423005\pi$$
$$110$$ 0 0
$$111$$ −758.029 −0.648188
$$112$$ 0 0
$$113$$ −231.823 −0.192992 −0.0964961 0.995333i $$-0.530764\pi$$
−0.0964961 + 0.995333i $$0.530764\pi$$
$$114$$ 0 0
$$115$$ 520.833 0.422329
$$116$$ 0 0
$$117$$ −392.912 −0.310468
$$118$$ 0 0
$$119$$ −473.598 −0.364829
$$120$$ 0 0
$$121$$ 1027.90 0.772279
$$122$$ 0 0
$$123$$ 828.823 0.607581
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2372.90 −1.65796 −0.828979 0.559280i $$-0.811078\pi$$
−0.828979 + 0.559280i $$0.811078\pi$$
$$128$$ 0 0
$$129$$ 277.913 0.189681
$$130$$ 0 0
$$131$$ −1200.04 −0.800364 −0.400182 0.916436i $$-0.631053\pi$$
−0.400182 + 0.916436i $$0.631053\pi$$
$$132$$ 0 0
$$133$$ 652.784 0.425591
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 2781.25 1.73444 0.867221 0.497924i $$-0.165904\pi$$
0.867221 + 0.497924i $$0.165904\pi$$
$$138$$ 0 0
$$139$$ 1245.60 0.760078 0.380039 0.924971i $$-0.375911\pi$$
0.380039 + 0.924971i $$0.375911\pi$$
$$140$$ 0 0
$$141$$ 1748.38 1.04426
$$142$$ 0 0
$$143$$ 2120.35 1.23995
$$144$$ 0 0
$$145$$ −293.675 −0.168196
$$146$$ 0 0
$$147$$ 147.000 0.0824786
$$148$$ 0 0
$$149$$ 19.4046 0.0106690 0.00533452 0.999986i $$-0.498302\pi$$
0.00533452 + 0.999986i $$0.498302\pi$$
$$150$$ 0 0
$$151$$ 2349.80 1.26638 0.633192 0.773995i $$-0.281744\pi$$
0.633192 + 0.773995i $$0.281744\pi$$
$$152$$ 0 0
$$153$$ −608.912 −0.321749
$$154$$ 0 0
$$155$$ 45.4416 0.0235481
$$156$$ 0 0
$$157$$ −3898.46 −1.98172 −0.990862 0.134880i $$-0.956935\pi$$
−0.990862 + 0.134880i $$0.956935\pi$$
$$158$$ 0 0
$$159$$ 1869.06 0.932239
$$160$$ 0 0
$$161$$ 729.166 0.356934
$$162$$ 0 0
$$163$$ −1527.54 −0.734024 −0.367012 0.930216i $$-0.619619\pi$$
−0.367012 + 0.930216i $$0.619619\pi$$
$$164$$ 0 0
$$165$$ −728.528 −0.343732
$$166$$ 0 0
$$167$$ 998.518 0.462681 0.231340 0.972873i $$-0.425689\pi$$
0.231340 + 0.972873i $$0.425689\pi$$
$$168$$ 0 0
$$169$$ −291.079 −0.132489
$$170$$ 0 0
$$171$$ 839.294 0.375336
$$172$$ 0 0
$$173$$ 685.253 0.301149 0.150575 0.988599i $$-0.451888\pi$$
0.150575 + 0.988599i $$0.451888\pi$$
$$174$$ 0 0
$$175$$ 175.000 0.0755929
$$176$$ 0 0
$$177$$ 1575.00 0.668836
$$178$$ 0 0
$$179$$ −1025.58 −0.428245 −0.214122 0.976807i $$-0.568689\pi$$
−0.214122 + 0.976807i $$0.568689\pi$$
$$180$$ 0 0
$$181$$ 2899.40 1.19067 0.595333 0.803479i $$-0.297020\pi$$
0.595333 + 0.803479i $$0.297020\pi$$
$$182$$ 0 0
$$183$$ −1058.38 −0.427529
$$184$$ 0 0
$$185$$ −1263.38 −0.502084
$$186$$ 0 0
$$187$$ 3285.99 1.28500
$$188$$ 0 0
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −1074.18 −0.406939 −0.203469 0.979081i $$-0.565222\pi$$
−0.203469 + 0.979081i $$0.565222\pi$$
$$192$$ 0 0
$$193$$ 898.999 0.335292 0.167646 0.985847i $$-0.446383\pi$$
0.167646 + 0.985847i $$0.446383\pi$$
$$194$$ 0 0
$$195$$ −654.853 −0.240487
$$196$$ 0 0
$$197$$ 3063.63 1.10799 0.553996 0.832519i $$-0.313102\pi$$
0.553996 + 0.832519i $$0.313102\pi$$
$$198$$ 0 0
$$199$$ 949.522 0.338240 0.169120 0.985595i $$-0.445907\pi$$
0.169120 + 0.985595i $$0.445907\pi$$
$$200$$ 0 0
$$201$$ 2209.56 0.775375
$$202$$ 0 0
$$203$$ −411.145 −0.142151
$$204$$ 0 0
$$205$$ 1381.37 0.470630
$$206$$ 0 0
$$207$$ 937.499 0.314786
$$208$$ 0 0
$$209$$ −4529.25 −1.49902
$$210$$ 0 0
$$211$$ −2306.64 −0.752587 −0.376294 0.926500i $$-0.622802\pi$$
−0.376294 + 0.926500i $$0.622802\pi$$
$$212$$ 0 0
$$213$$ 1476.79 0.475062
$$214$$ 0 0
$$215$$ 463.188 0.146926
$$216$$ 0 0
$$217$$ 63.6182 0.0199018
$$218$$ 0 0
$$219$$ 3494.26 1.07817
$$220$$ 0 0
$$221$$ 2953.69 0.899033
$$222$$ 0 0
$$223$$ 3227.61 0.969222 0.484611 0.874730i $$-0.338961\pi$$
0.484611 + 0.874730i $$0.338961\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 637.820 0.186492 0.0932458 0.995643i $$-0.470276\pi$$
0.0932458 + 0.995643i $$0.470276\pi$$
$$228$$ 0 0
$$229$$ 544.774 0.157204 0.0786019 0.996906i $$-0.474954\pi$$
0.0786019 + 0.996906i $$0.474954\pi$$
$$230$$ 0 0
$$231$$ −1019.94 −0.290507
$$232$$ 0 0
$$233$$ −5748.54 −1.61631 −0.808154 0.588972i $$-0.799533\pi$$
−0.808154 + 0.588972i $$0.799533\pi$$
$$234$$ 0 0
$$235$$ 2913.97 0.808878
$$236$$ 0 0
$$237$$ 2616.59 0.717154
$$238$$ 0 0
$$239$$ 2678.10 0.724820 0.362410 0.932019i $$-0.381954\pi$$
0.362410 + 0.932019i $$0.381954\pi$$
$$240$$ 0 0
$$241$$ −2202.16 −0.588604 −0.294302 0.955713i $$-0.595087\pi$$
−0.294302 + 0.955713i $$0.595087\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 245.000 0.0638877
$$246$$ 0 0
$$247$$ −4071.21 −1.04877
$$248$$ 0 0
$$249$$ 1588.76 0.404353
$$250$$ 0 0
$$251$$ 5716.90 1.43764 0.718820 0.695196i $$-0.244683\pi$$
0.718820 + 0.695196i $$0.244683\pi$$
$$252$$ 0 0
$$253$$ −5059.22 −1.25719
$$254$$ 0 0
$$255$$ −1014.85 −0.249226
$$256$$ 0 0
$$257$$ 4724.29 1.14666 0.573332 0.819323i $$-0.305650\pi$$
0.573332 + 0.819323i $$0.305650\pi$$
$$258$$ 0 0
$$259$$ −1768.73 −0.424339
$$260$$ 0 0
$$261$$ −528.616 −0.125366
$$262$$ 0 0
$$263$$ −5975.36 −1.40097 −0.700487 0.713665i $$-0.747034\pi$$
−0.700487 + 0.713665i $$0.747034\pi$$
$$264$$ 0 0
$$265$$ 3115.10 0.722109
$$266$$ 0 0
$$267$$ −1155.65 −0.264886
$$268$$ 0 0
$$269$$ 4486.11 1.01681 0.508407 0.861117i $$-0.330234\pi$$
0.508407 + 0.861117i $$0.330234\pi$$
$$270$$ 0 0
$$271$$ −3827.68 −0.857989 −0.428994 0.903307i $$-0.641132\pi$$
−0.428994 + 0.903307i $$0.641132\pi$$
$$272$$ 0 0
$$273$$ −916.794 −0.203249
$$274$$ 0 0
$$275$$ −1214.21 −0.266254
$$276$$ 0 0
$$277$$ −3420.54 −0.741950 −0.370975 0.928643i $$-0.620976\pi$$
−0.370975 + 0.928643i $$0.620976\pi$$
$$278$$ 0 0
$$279$$ 81.7948 0.0175517
$$280$$ 0 0
$$281$$ 5235.92 1.11156 0.555781 0.831329i $$-0.312419\pi$$
0.555781 + 0.831329i $$0.312419\pi$$
$$282$$ 0 0
$$283$$ 6985.88 1.46738 0.733688 0.679486i $$-0.237797\pi$$
0.733688 + 0.679486i $$0.237797\pi$$
$$284$$ 0 0
$$285$$ 1398.82 0.290734
$$286$$ 0 0
$$287$$ 1933.92 0.397755
$$288$$ 0 0
$$289$$ −335.550 −0.0682984
$$290$$ 0 0
$$291$$ −1391.68 −0.280349
$$292$$ 0 0
$$293$$ 7399.70 1.47541 0.737705 0.675123i $$-0.235910\pi$$
0.737705 + 0.675123i $$0.235910\pi$$
$$294$$ 0 0
$$295$$ 2625.00 0.518078
$$296$$ 0 0
$$297$$ −1311.35 −0.256203
$$298$$ 0 0
$$299$$ −4547.58 −0.879577
$$300$$ 0 0
$$301$$ 648.463 0.124175
$$302$$ 0 0
$$303$$ −1298.17 −0.246133
$$304$$ 0 0
$$305$$ −1763.97 −0.331163
$$306$$ 0 0
$$307$$ −2668.64 −0.496116 −0.248058 0.968745i $$-0.579792\pi$$
−0.248058 + 0.968745i $$0.579792\pi$$
$$308$$ 0 0
$$309$$ −1537.88 −0.283129
$$310$$ 0 0
$$311$$ −6189.25 −1.12849 −0.564244 0.825608i $$-0.690832\pi$$
−0.564244 + 0.825608i $$0.690832\pi$$
$$312$$ 0 0
$$313$$ −2921.59 −0.527598 −0.263799 0.964578i $$-0.584976\pi$$
−0.263799 + 0.964578i $$0.584976\pi$$
$$314$$ 0 0
$$315$$ 315.000 0.0563436
$$316$$ 0 0
$$317$$ 9825.56 1.74088 0.870439 0.492276i $$-0.163835\pi$$
0.870439 + 0.492276i $$0.163835\pi$$
$$318$$ 0 0
$$319$$ 2852.68 0.500687
$$320$$ 0 0
$$321$$ −5889.26 −1.02401
$$322$$ 0 0
$$323$$ −6309.33 −1.08687
$$324$$ 0 0
$$325$$ −1091.42 −0.186281
$$326$$ 0 0
$$327$$ 1635.53 0.276590
$$328$$ 0 0
$$329$$ 4079.56 0.683627
$$330$$ 0 0
$$331$$ 9258.17 1.53739 0.768693 0.639618i $$-0.220907\pi$$
0.768693 + 0.639618i $$0.220907\pi$$
$$332$$ 0 0
$$333$$ −2274.09 −0.374232
$$334$$ 0 0
$$335$$ 3682.60 0.600603
$$336$$ 0 0
$$337$$ −3693.98 −0.597103 −0.298552 0.954394i $$-0.596503\pi$$
−0.298552 + 0.954394i $$0.596503\pi$$
$$338$$ 0 0
$$339$$ −695.470 −0.111424
$$340$$ 0 0
$$341$$ −441.406 −0.0700982
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ 1562.50 0.243832
$$346$$ 0 0
$$347$$ −3832.83 −0.592960 −0.296480 0.955039i $$-0.595813\pi$$
−0.296480 + 0.955039i $$0.595813\pi$$
$$348$$ 0 0
$$349$$ 8325.22 1.27690 0.638451 0.769662i $$-0.279575\pi$$
0.638451 + 0.769662i $$0.279575\pi$$
$$350$$ 0 0
$$351$$ −1178.74 −0.179248
$$352$$ 0 0
$$353$$ −8991.52 −1.35572 −0.677862 0.735189i $$-0.737093\pi$$
−0.677862 + 0.735189i $$0.737093\pi$$
$$354$$ 0 0
$$355$$ 2461.32 0.367981
$$356$$ 0 0
$$357$$ −1420.79 −0.210634
$$358$$ 0 0
$$359$$ 12893.8 1.89557 0.947783 0.318917i $$-0.103319\pi$$
0.947783 + 0.318917i $$0.103319\pi$$
$$360$$ 0 0
$$361$$ 1837.46 0.267891
$$362$$ 0 0
$$363$$ 3083.71 0.445875
$$364$$ 0 0
$$365$$ 5823.77 0.835150
$$366$$ 0 0
$$367$$ 7480.17 1.06393 0.531964 0.846767i $$-0.321454\pi$$
0.531964 + 0.846767i $$0.321454\pi$$
$$368$$ 0 0
$$369$$ 2486.47 0.350787
$$370$$ 0 0
$$371$$ 4361.14 0.610293
$$372$$ 0 0
$$373$$ −3523.32 −0.489090 −0.244545 0.969638i $$-0.578639\pi$$
−0.244545 + 0.969638i $$0.578639\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 2564.19 0.350298
$$378$$ 0 0
$$379$$ −13515.4 −1.83177 −0.915886 0.401438i $$-0.868510\pi$$
−0.915886 + 0.401438i $$0.868510\pi$$
$$380$$ 0 0
$$381$$ −7118.69 −0.957222
$$382$$ 0 0
$$383$$ 657.182 0.0876774 0.0438387 0.999039i $$-0.486041\pi$$
0.0438387 + 0.999039i $$0.486041\pi$$
$$384$$ 0 0
$$385$$ −1699.90 −0.225026
$$386$$ 0 0
$$387$$ 833.738 0.109512
$$388$$ 0 0
$$389$$ −9741.87 −1.26975 −0.634875 0.772615i $$-0.718948\pi$$
−0.634875 + 0.772615i $$0.718948\pi$$
$$390$$ 0 0
$$391$$ −7047.58 −0.911538
$$392$$ 0 0
$$393$$ −3600.11 −0.462091
$$394$$ 0 0
$$395$$ 4360.98 0.555505
$$396$$ 0 0
$$397$$ 4407.42 0.557184 0.278592 0.960410i $$-0.410132\pi$$
0.278592 + 0.960410i $$0.410132\pi$$
$$398$$ 0 0
$$399$$ 1958.35 0.245715
$$400$$ 0 0
$$401$$ −11569.5 −1.44078 −0.720391 0.693568i $$-0.756037\pi$$
−0.720391 + 0.693568i $$0.756037\pi$$
$$402$$ 0 0
$$403$$ −396.767 −0.0490431
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 12272.1 1.49461
$$408$$ 0 0
$$409$$ −3083.03 −0.372729 −0.186364 0.982481i $$-0.559670\pi$$
−0.186364 + 0.982481i $$0.559670\pi$$
$$410$$ 0 0
$$411$$ 8343.76 1.00138
$$412$$ 0 0
$$413$$ 3674.99 0.437856
$$414$$ 0 0
$$415$$ 2647.94 0.313210
$$416$$ 0 0
$$417$$ 3736.81 0.438831
$$418$$ 0 0
$$419$$ 5415.21 0.631385 0.315692 0.948862i $$-0.397763\pi$$
0.315692 + 0.948862i $$0.397763\pi$$
$$420$$ 0 0
$$421$$ 4188.34 0.484863 0.242432 0.970168i $$-0.422055\pi$$
0.242432 + 0.970168i $$0.422055\pi$$
$$422$$ 0 0
$$423$$ 5245.15 0.602902
$$424$$ 0 0
$$425$$ −1691.42 −0.193049
$$426$$ 0 0
$$427$$ −2469.56 −0.279884
$$428$$ 0 0
$$429$$ 6361.05 0.715884
$$430$$ 0 0
$$431$$ 9108.41 1.01795 0.508975 0.860781i $$-0.330024\pi$$
0.508975 + 0.860781i $$0.330024\pi$$
$$432$$ 0 0
$$433$$ −16847.0 −1.86979 −0.934893 0.354930i $$-0.884505\pi$$
−0.934893 + 0.354930i $$0.884505\pi$$
$$434$$ 0 0
$$435$$ −881.026 −0.0971080
$$436$$ 0 0
$$437$$ 9714.03 1.06335
$$438$$ 0 0
$$439$$ −8434.14 −0.916946 −0.458473 0.888708i $$-0.651603\pi$$
−0.458473 + 0.888708i $$0.651603\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ 4298.49 0.461010 0.230505 0.973071i $$-0.425962\pi$$
0.230505 + 0.973071i $$0.425962\pi$$
$$444$$ 0 0
$$445$$ −1926.08 −0.205180
$$446$$ 0 0
$$447$$ 58.2139 0.00615978
$$448$$ 0 0
$$449$$ 10545.0 1.10835 0.554173 0.832402i $$-0.313035\pi$$
0.554173 + 0.832402i $$0.313035\pi$$
$$450$$ 0 0
$$451$$ −13418.2 −1.40098
$$452$$ 0 0
$$453$$ 7049.40 0.731147
$$454$$ 0 0
$$455$$ −1527.99 −0.157436
$$456$$ 0 0
$$457$$ 11952.4 1.22344 0.611719 0.791075i $$-0.290478\pi$$
0.611719 + 0.791075i $$0.290478\pi$$
$$458$$ 0 0
$$459$$ −1826.74 −0.185762
$$460$$ 0 0
$$461$$ −17200.9 −1.73780 −0.868900 0.494988i $$-0.835173\pi$$
−0.868900 + 0.494988i $$0.835173\pi$$
$$462$$ 0 0
$$463$$ 10368.7 1.04076 0.520381 0.853934i $$-0.325790\pi$$
0.520381 + 0.853934i $$0.325790\pi$$
$$464$$ 0 0
$$465$$ 136.325 0.0135955
$$466$$ 0 0
$$467$$ 16879.5 1.67257 0.836284 0.548296i $$-0.184723\pi$$
0.836284 + 0.548296i $$0.184723\pi$$
$$468$$ 0 0
$$469$$ 5155.64 0.507602
$$470$$ 0 0
$$471$$ −11695.4 −1.14415
$$472$$ 0 0
$$473$$ −4499.27 −0.437371
$$474$$ 0 0
$$475$$ 2331.37 0.225201
$$476$$ 0 0
$$477$$ 5607.17 0.538228
$$478$$ 0 0
$$479$$ −7329.12 −0.699115 −0.349558 0.936915i $$-0.613668\pi$$
−0.349558 + 0.936915i $$0.613668\pi$$
$$480$$ 0 0
$$481$$ 11031.0 1.04568
$$482$$ 0 0
$$483$$ 2187.50 0.206076
$$484$$ 0 0
$$485$$ −2319.46 −0.217157
$$486$$ 0 0
$$487$$ 17209.5 1.60131 0.800655 0.599125i $$-0.204485\pi$$
0.800655 + 0.599125i $$0.204485\pi$$
$$488$$ 0 0
$$489$$ −4582.61 −0.423789
$$490$$ 0 0
$$491$$ −11392.5 −1.04712 −0.523560 0.851989i $$-0.675396\pi$$
−0.523560 + 0.851989i $$0.675396\pi$$
$$492$$ 0 0
$$493$$ 3973.83 0.363027
$$494$$ 0 0
$$495$$ −2185.58 −0.198454
$$496$$ 0 0
$$497$$ 3445.85 0.311001
$$498$$ 0 0
$$499$$ −19079.4 −1.71164 −0.855822 0.517271i $$-0.826948\pi$$
−0.855822 + 0.517271i $$0.826948\pi$$
$$500$$ 0 0
$$501$$ 2995.55 0.267129
$$502$$ 0 0
$$503$$ 13499.4 1.19663 0.598317 0.801259i $$-0.295836\pi$$
0.598317 + 0.801259i $$0.295836\pi$$
$$504$$ 0 0
$$505$$ −2163.62 −0.190654
$$506$$ 0 0
$$507$$ −873.237 −0.0764928
$$508$$ 0 0
$$509$$ −4328.85 −0.376960 −0.188480 0.982077i $$-0.560356\pi$$
−0.188480 + 0.982077i $$0.560356\pi$$
$$510$$ 0 0
$$511$$ 8153.27 0.705831
$$512$$ 0 0
$$513$$ 2517.88 0.216700
$$514$$ 0 0
$$515$$ −2563.13 −0.219311
$$516$$ 0 0
$$517$$ −28305.5 −2.40788
$$518$$ 0 0
$$519$$ 2055.76 0.173869
$$520$$ 0 0
$$521$$ 19395.7 1.63098 0.815490 0.578771i $$-0.196468\pi$$
0.815490 + 0.578771i $$0.196468\pi$$
$$522$$ 0 0
$$523$$ −20413.8 −1.70675 −0.853377 0.521294i $$-0.825450\pi$$
−0.853377 + 0.521294i $$0.825450\pi$$
$$524$$ 0 0
$$525$$ 525.000 0.0436436
$$526$$ 0 0
$$527$$ −614.887 −0.0508252
$$528$$ 0 0
$$529$$ −1316.34 −0.108189
$$530$$ 0 0
$$531$$ 4724.99 0.386153
$$532$$ 0 0
$$533$$ −12061.3 −0.980171
$$534$$ 0 0
$$535$$ −9815.43 −0.793193
$$536$$ 0 0
$$537$$ −3076.75 −0.247247
$$538$$ 0 0
$$539$$ −2379.86 −0.190181
$$540$$ 0 0
$$541$$ −4919.18 −0.390928 −0.195464 0.980711i $$-0.562621\pi$$
−0.195464 + 0.980711i $$0.562621\pi$$
$$542$$ 0 0
$$543$$ 8698.19 0.687431
$$544$$ 0 0
$$545$$ 2725.88 0.214246
$$546$$ 0 0
$$547$$ −15334.2 −1.19862 −0.599308 0.800518i $$-0.704558\pi$$
−0.599308 + 0.800518i $$0.704558\pi$$
$$548$$ 0 0
$$549$$ −3175.15 −0.246834
$$550$$ 0 0
$$551$$ −5477.33 −0.423488
$$552$$ 0 0
$$553$$ 6105.37 0.469487
$$554$$ 0 0
$$555$$ −3790.14 −0.289879
$$556$$ 0 0
$$557$$ −8613.78 −0.655256 −0.327628 0.944807i $$-0.606249\pi$$
−0.327628 + 0.944807i $$0.606249\pi$$
$$558$$ 0 0
$$559$$ −4044.26 −0.306000
$$560$$ 0 0
$$561$$ 9857.98 0.741897
$$562$$ 0 0
$$563$$ 2320.81 0.173731 0.0868654 0.996220i $$-0.472315\pi$$
0.0868654 + 0.996220i $$0.472315\pi$$
$$564$$ 0 0
$$565$$ −1159.12 −0.0863087
$$566$$ 0 0
$$567$$ 567.000 0.0419961
$$568$$ 0 0
$$569$$ −1736.04 −0.127906 −0.0639529 0.997953i $$-0.520371\pi$$
−0.0639529 + 0.997953i $$0.520371\pi$$
$$570$$ 0 0
$$571$$ −23897.8 −1.75148 −0.875738 0.482786i $$-0.839625\pi$$
−0.875738 + 0.482786i $$0.839625\pi$$
$$572$$ 0 0
$$573$$ −3222.55 −0.234946
$$574$$ 0 0
$$575$$ 2604.16 0.188871
$$576$$ 0 0
$$577$$ 8029.26 0.579311 0.289655 0.957131i $$-0.406459\pi$$
0.289655 + 0.957131i $$0.406459\pi$$
$$578$$ 0 0
$$579$$ 2697.00 0.193581
$$580$$ 0 0
$$581$$ 3707.12 0.264711
$$582$$ 0 0
$$583$$ −30259.1 −2.14958
$$584$$ 0 0
$$585$$ −1964.56 −0.138845
$$586$$ 0 0
$$587$$ 8015.14 0.563578 0.281789 0.959476i $$-0.409072\pi$$
0.281789 + 0.959476i $$0.409072\pi$$
$$588$$ 0 0
$$589$$ 847.529 0.0592900
$$590$$ 0 0
$$591$$ 9190.88 0.639700
$$592$$ 0 0
$$593$$ 12820.3 0.887801 0.443901 0.896076i $$-0.353594\pi$$
0.443901 + 0.896076i $$0.353594\pi$$
$$594$$ 0 0
$$595$$ −2367.99 −0.163157
$$596$$ 0 0
$$597$$ 2848.57 0.195283
$$598$$ 0 0
$$599$$ 15330.5 1.04572 0.522860 0.852418i $$-0.324865\pi$$
0.522860 + 0.852418i $$0.324865\pi$$
$$600$$ 0 0
$$601$$ 107.658 0.00730694 0.00365347 0.999993i $$-0.498837\pi$$
0.00365347 + 0.999993i $$0.498837\pi$$
$$602$$ 0 0
$$603$$ 6628.68 0.447663
$$604$$ 0 0
$$605$$ 5139.52 0.345374
$$606$$ 0 0
$$607$$ 8213.06 0.549189 0.274595 0.961560i $$-0.411456\pi$$
0.274595 + 0.961560i $$0.411456\pi$$
$$608$$ 0 0
$$609$$ −1233.44 −0.0820712
$$610$$ 0 0
$$611$$ −25443.0 −1.68463
$$612$$ 0 0
$$613$$ 1242.60 0.0818732 0.0409366 0.999162i $$-0.486966\pi$$
0.0409366 + 0.999162i $$0.486966\pi$$
$$614$$ 0 0
$$615$$ 4144.11 0.271718
$$616$$ 0 0
$$617$$ −13170.6 −0.859367 −0.429683 0.902980i $$-0.641375\pi$$
−0.429683 + 0.902980i $$0.641375\pi$$
$$618$$ 0 0
$$619$$ −19774.1 −1.28399 −0.641993 0.766711i $$-0.721892\pi$$
−0.641993 + 0.766711i $$0.721892\pi$$
$$620$$ 0 0
$$621$$ 2812.50 0.181742
$$622$$ 0 0
$$623$$ −2696.51 −0.173409
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −13587.8 −0.865459
$$628$$ 0 0
$$629$$ 17095.3 1.08368
$$630$$ 0 0
$$631$$ 14308.8 0.902735 0.451367 0.892338i $$-0.350936\pi$$
0.451367 + 0.892338i $$0.350936\pi$$
$$632$$ 0 0
$$633$$ −6919.93 −0.434507
$$634$$ 0 0
$$635$$ −11864.5 −0.741461
$$636$$ 0 0
$$637$$ −2139.19 −0.133058
$$638$$ 0 0
$$639$$ 4430.38 0.274277
$$640$$ 0 0
$$641$$ 11537.5 0.710925 0.355463 0.934691i $$-0.384323\pi$$
0.355463 + 0.934691i $$0.384323\pi$$
$$642$$ 0 0
$$643$$ −19603.0 −1.20228 −0.601139 0.799144i $$-0.705286\pi$$
−0.601139 + 0.799144i $$0.705286\pi$$
$$644$$ 0 0
$$645$$ 1389.56 0.0848279
$$646$$ 0 0
$$647$$ −21650.0 −1.31553 −0.657765 0.753223i $$-0.728498\pi$$
−0.657765 + 0.753223i $$0.728498\pi$$
$$648$$ 0 0
$$649$$ −25498.4 −1.54222
$$650$$ 0 0
$$651$$ 190.855 0.0114903
$$652$$ 0 0
$$653$$ −2927.33 −0.175429 −0.0877145 0.996146i $$-0.527956\pi$$
−0.0877145 + 0.996146i $$0.527956\pi$$
$$654$$ 0 0
$$655$$ −6000.18 −0.357934
$$656$$ 0 0
$$657$$ 10482.8 0.622484
$$658$$ 0 0
$$659$$ 4778.76 0.282480 0.141240 0.989975i $$-0.454891\pi$$
0.141240 + 0.989975i $$0.454891\pi$$
$$660$$ 0 0
$$661$$ −31510.3 −1.85417 −0.927086 0.374849i $$-0.877695\pi$$
−0.927086 + 0.374849i $$0.877695\pi$$
$$662$$ 0 0
$$663$$ 8861.06 0.519057
$$664$$ 0 0
$$665$$ 3263.92 0.190330
$$666$$ 0 0
$$667$$ −6118.23 −0.355170
$$668$$ 0 0
$$669$$ 9682.82 0.559581
$$670$$ 0 0
$$671$$ 17134.7 0.985808
$$672$$ 0 0
$$673$$ −8992.15 −0.515040 −0.257520 0.966273i $$-0.582905\pi$$
−0.257520 + 0.966273i $$0.582905\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ 19340.8 1.09797 0.548985 0.835832i $$-0.315014\pi$$
0.548985 + 0.835832i $$0.315014\pi$$
$$678$$ 0 0
$$679$$ −3247.25 −0.183531
$$680$$ 0 0
$$681$$ 1913.46 0.107671
$$682$$ 0 0
$$683$$ 4255.14 0.238387 0.119194 0.992871i $$-0.461969\pi$$
0.119194 + 0.992871i $$0.461969\pi$$
$$684$$ 0 0
$$685$$ 13906.3 0.775666
$$686$$ 0 0
$$687$$ 1634.32 0.0907616
$$688$$ 0 0
$$689$$ −27199.1 −1.50392
$$690$$ 0 0
$$691$$ 17505.5 0.963733 0.481867 0.876245i $$-0.339959\pi$$
0.481867 + 0.876245i $$0.339959\pi$$
$$692$$ 0 0
$$693$$ −3059.82 −0.167724
$$694$$ 0 0
$$695$$ 6228.02 0.339917
$$696$$ 0 0
$$697$$ −18691.8 −1.01579
$$698$$ 0 0
$$699$$ −17245.6 −0.933175
$$700$$ 0 0
$$701$$ −3240.77 −0.174611 −0.0873054 0.996182i $$-0.527826\pi$$
−0.0873054 + 0.996182i $$0.527826\pi$$
$$702$$ 0 0
$$703$$ −23563.3 −1.26416
$$704$$ 0 0
$$705$$ 8741.91 0.467006
$$706$$ 0 0
$$707$$ −3029.07 −0.161132
$$708$$ 0 0
$$709$$ 19949.3 1.05672 0.528358 0.849022i $$-0.322808\pi$$
0.528358 + 0.849022i $$0.322808\pi$$
$$710$$ 0 0
$$711$$ 7849.76 0.414049
$$712$$ 0 0
$$713$$ 946.698 0.0497253
$$714$$ 0 0
$$715$$ 10601.7 0.554522
$$716$$ 0 0
$$717$$ 8034.30 0.418475
$$718$$ 0 0
$$719$$ 11259.4 0.584011 0.292006 0.956417i $$-0.405677\pi$$
0.292006 + 0.956417i $$0.405677\pi$$
$$720$$ 0 0
$$721$$ −3588.38 −0.185351
$$722$$ 0 0
$$723$$ −6606.47 −0.339830
$$724$$ 0 0
$$725$$ −1468.38 −0.0752195
$$726$$ 0 0
$$727$$ 12228.5 0.623840 0.311920 0.950108i $$-0.399028\pi$$
0.311920 + 0.950108i $$0.399028\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −6267.56 −0.317119
$$732$$ 0 0
$$733$$ −26635.1 −1.34214 −0.671072 0.741392i $$-0.734166\pi$$
−0.671072 + 0.741392i $$0.734166\pi$$
$$734$$ 0 0
$$735$$ 735.000 0.0368856
$$736$$ 0 0
$$737$$ −35771.7 −1.78788
$$738$$ 0 0
$$739$$ 6074.00 0.302349 0.151174 0.988507i $$-0.451694\pi$$
0.151174 + 0.988507i $$0.451694\pi$$
$$740$$ 0 0
$$741$$ −12213.6 −0.605505
$$742$$ 0 0
$$743$$ 4016.87 0.198337 0.0991686 0.995071i $$-0.468382\pi$$
0.0991686 + 0.995071i $$0.468382\pi$$
$$744$$ 0 0
$$745$$ 97.0231 0.00477134
$$746$$ 0 0
$$747$$ 4766.29 0.233453
$$748$$ 0 0
$$749$$ −13741.6 −0.670370
$$750$$ 0 0
$$751$$ 23913.2 1.16192 0.580962 0.813931i $$-0.302676\pi$$
0.580962 + 0.813931i $$0.302676\pi$$
$$752$$ 0 0
$$753$$ 17150.7 0.830022
$$754$$ 0 0
$$755$$ 11749.0 0.566344
$$756$$ 0 0
$$757$$ 31044.9 1.49055 0.745275 0.666758i $$-0.232318\pi$$
0.745275 + 0.666758i $$0.232318\pi$$
$$758$$ 0 0
$$759$$ −15177.6 −0.725842
$$760$$ 0 0
$$761$$ 14011.8 0.667446 0.333723 0.942671i $$-0.391695\pi$$
0.333723 + 0.942671i $$0.391695\pi$$
$$762$$ 0 0
$$763$$ 3816.23 0.181071
$$764$$ 0 0
$$765$$ −3044.56 −0.143891
$$766$$ 0 0
$$767$$ −22919.8 −1.07899
$$768$$ 0 0
$$769$$ 3342.49 0.156740 0.0783701 0.996924i $$-0.475028\pi$$
0.0783701 + 0.996924i $$0.475028\pi$$
$$770$$ 0 0
$$771$$ 14172.9 0.662027
$$772$$ 0 0
$$773$$ −21074.6 −0.980594 −0.490297 0.871555i $$-0.663112\pi$$
−0.490297 + 0.871555i $$0.663112\pi$$
$$774$$ 0 0
$$775$$ 227.208 0.0105310
$$776$$ 0 0
$$777$$ −5306.20 −0.244992
$$778$$ 0 0
$$779$$ 25763.9 1.18496
$$780$$ 0 0
$$781$$ −23908.5 −1.09541
$$782$$ 0 0
$$783$$ −1585.85 −0.0723800
$$784$$ 0 0
$$785$$ −19492.3 −0.886254
$$786$$ 0 0
$$787$$ −21394.8 −0.969048 −0.484524 0.874778i $$-0.661007\pi$$
−0.484524 + 0.874778i $$0.661007\pi$$
$$788$$ 0 0
$$789$$ −17926.1 −0.808853
$$790$$ 0 0
$$791$$ −1622.76 −0.0729442
$$792$$ 0 0
$$793$$ 15401.9 0.689706
$$794$$ 0 0
$$795$$ 9345.29 0.416910
$$796$$ 0 0
$$797$$ 20645.0 0.917547 0.458773 0.888553i $$-0.348289\pi$$
0.458773 + 0.888553i $$0.348289\pi$$
$$798$$ 0 0
$$799$$ −39430.0 −1.74585
$$800$$ 0 0
$$801$$ −3466.95 −0.152932
$$802$$ 0 0
$$803$$ −56570.4 −2.48608
$$804$$ 0 0
$$805$$ 3645.83 0.159626
$$806$$ 0 0
$$807$$ 13458.3 0.587058
$$808$$ 0 0
$$809$$ −15939.0 −0.692688 −0.346344 0.938108i $$-0.612577\pi$$
−0.346344 + 0.938108i $$0.612577\pi$$
$$810$$ 0 0
$$811$$ −22829.2 −0.988460 −0.494230 0.869331i $$-0.664550\pi$$
−0.494230 + 0.869331i $$0.664550\pi$$
$$812$$ 0 0
$$813$$ −11483.0 −0.495360
$$814$$ 0 0
$$815$$ −7637.69 −0.328266
$$816$$ 0 0
$$817$$ 8638.90 0.369935
$$818$$ 0 0
$$819$$ −2750.38 −0.117346
$$820$$ 0 0
$$821$$ −5700.22 −0.242313 −0.121157 0.992633i $$-0.538660\pi$$
−0.121157 + 0.992633i $$0.538660\pi$$
$$822$$ 0 0
$$823$$ 32438.0 1.37390 0.686948 0.726707i $$-0.258950\pi$$
0.686948 + 0.726707i $$0.258950\pi$$
$$824$$ 0 0
$$825$$ −3642.64 −0.153722
$$826$$ 0 0
$$827$$ 12762.6 0.536638 0.268319 0.963330i $$-0.413532\pi$$
0.268319 + 0.963330i $$0.413532\pi$$
$$828$$ 0 0
$$829$$ −30766.7 −1.28899 −0.644494 0.764609i $$-0.722932\pi$$
−0.644494 + 0.764609i $$0.722932\pi$$
$$830$$ 0 0
$$831$$ −10261.6 −0.428365
$$832$$ 0 0
$$833$$ −3315.19 −0.137892
$$834$$ 0 0
$$835$$ 4992.59 0.206917
$$836$$ 0 0
$$837$$ 245.384 0.0101335
$$838$$ 0 0
$$839$$ 9779.71 0.402423 0.201212 0.979548i $$-0.435512\pi$$
0.201212 + 0.979548i $$0.435512\pi$$
$$840$$ 0 0
$$841$$ −20939.2 −0.858551
$$842$$ 0 0
$$843$$ 15707.8 0.641760
$$844$$ 0 0
$$845$$ −1455.40 −0.0592510
$$846$$ 0 0
$$847$$ 7195.32 0.291894
$$848$$ 0 0
$$849$$ 20957.6 0.847190
$$850$$ 0 0
$$851$$ −26320.4 −1.06023
$$852$$ 0 0
$$853$$ 24201.5 0.971445 0.485723 0.874113i $$-0.338556\pi$$
0.485723 + 0.874113i $$0.338556\pi$$
$$854$$ 0 0
$$855$$ 4196.47 0.167855
$$856$$ 0 0
$$857$$ 21036.7 0.838507 0.419254 0.907869i $$-0.362292\pi$$
0.419254 + 0.907869i $$0.362292\pi$$
$$858$$ 0 0
$$859$$ 6179.19 0.245438 0.122719 0.992441i $$-0.460839\pi$$
0.122719 + 0.992441i $$0.460839\pi$$
$$860$$ 0 0
$$861$$ 5801.76 0.229644
$$862$$ 0 0
$$863$$ −50256.2 −1.98232 −0.991160 0.132671i $$-0.957644\pi$$
−0.991160 + 0.132671i $$0.957644\pi$$
$$864$$ 0 0
$$865$$ 3426.27 0.134678
$$866$$ 0 0
$$867$$ −1006.65 −0.0394321
$$868$$ 0 0
$$869$$ −42361.2 −1.65363
$$870$$ 0 0
$$871$$ −32154.1 −1.25086
$$872$$ 0 0
$$873$$ −4175.03 −0.161860
$$874$$ 0 0
$$875$$ 875.000 0.0338062
$$876$$ 0 0
$$877$$ −9175.95 −0.353306 −0.176653 0.984273i $$-0.556527\pi$$
−0.176653 + 0.984273i $$0.556527\pi$$
$$878$$ 0 0
$$879$$ 22199.1 0.851828
$$880$$ 0 0
$$881$$ −26172.7 −1.00089 −0.500444 0.865769i $$-0.666830\pi$$
−0.500444 + 0.865769i $$0.666830\pi$$
$$882$$ 0 0
$$883$$ −18615.5 −0.709471 −0.354736 0.934967i $$-0.615429\pi$$
−0.354736 + 0.934967i $$0.615429\pi$$
$$884$$ 0 0
$$885$$ 7874.99 0.299113
$$886$$ 0 0
$$887$$ 12837.8 0.485964 0.242982 0.970031i $$-0.421874\pi$$
0.242982 + 0.970031i $$0.421874\pi$$
$$888$$ 0 0
$$889$$ −16610.3 −0.626649
$$890$$ 0 0
$$891$$ −3934.05 −0.147919
$$892$$ 0 0
$$893$$ 54348.4 2.03662
$$894$$ 0 0
$$895$$ −5127.92 −0.191517
$$896$$ 0 0
$$897$$ −13642.7 −0.507824
$$898$$ 0 0
$$899$$ −533.803 −0.0198035
$$900$$ 0 0
$$901$$ −42151.5 −1.55857
$$902$$ 0 0
$$903$$ 1945.39 0.0716926
$$904$$ 0 0
$$905$$ 14497.0 0.532482
$$906$$ 0 0
$$907$$ 26766.1 0.979885 0.489942 0.871755i $$-0.337018\pi$$
0.489942 + 0.871755i $$0.337018\pi$$
$$908$$ 0 0
$$909$$ −3894.52 −0.142105
$$910$$ 0 0
$$911$$ −5022.67 −0.182666 −0.0913328 0.995820i $$-0.529113\pi$$
−0.0913328 + 0.995820i $$0.529113\pi$$
$$912$$ 0 0
$$913$$ −25721.3 −0.932367
$$914$$ 0 0
$$915$$ −5291.91 −0.191197
$$916$$ 0 0
$$917$$ −8400.26 −0.302509
$$918$$ 0 0
$$919$$ −9541.79 −0.342497 −0.171248 0.985228i $$-0.554780\pi$$
−0.171248 + 0.985228i $$0.554780\pi$$
$$920$$ 0 0
$$921$$ −8005.93 −0.286432
$$922$$ 0 0
$$923$$ −21490.7 −0.766387
$$924$$ 0 0
$$925$$ −6316.90 −0.224539
$$926$$ 0 0
$$927$$ −4613.63 −0.163464
$$928$$ 0 0
$$929$$ −25479.6 −0.899846 −0.449923 0.893067i $$-0.648549\pi$$
−0.449923 + 0.893067i $$0.648549\pi$$
$$930$$ 0 0
$$931$$ 4569.49 0.160858
$$932$$ 0 0
$$933$$ −18567.7 −0.651533
$$934$$ 0 0
$$935$$ 16430.0 0.574671
$$936$$ 0 0
$$937$$ −33608.3 −1.17176 −0.585878 0.810399i $$-0.699250\pi$$
−0.585878 + 0.810399i $$0.699250\pi$$
$$938$$ 0 0
$$939$$ −8764.77 −0.304609
$$940$$ 0 0
$$941$$ −19173.6 −0.664232 −0.332116 0.943239i $$-0.607762\pi$$
−0.332116 + 0.943239i $$0.607762\pi$$
$$942$$ 0 0
$$943$$ 28778.5 0.993804
$$944$$ 0 0
$$945$$ 945.000 0.0325300
$$946$$ 0 0
$$947$$ 979.315 0.0336045 0.0168023 0.999859i $$-0.494651\pi$$
0.0168023 + 0.999859i $$0.494651\pi$$
$$948$$ 0 0
$$949$$ −50849.5 −1.73935
$$950$$ 0 0
$$951$$ 29476.7 1.00510
$$952$$ 0 0
$$953$$ 3048.61 0.103624 0.0518122 0.998657i $$-0.483500\pi$$
0.0518122 + 0.998657i $$0.483500\pi$$
$$954$$ 0 0
$$955$$ −5370.92 −0.181989
$$956$$ 0 0
$$957$$ 8558.03 0.289072
$$958$$ 0 0
$$959$$ 19468.8 0.655557
$$960$$ 0 0
$$961$$ −29708.4 −0.997227
$$962$$ 0 0
$$963$$ −17667.8 −0.591211
$$964$$ 0 0
$$965$$ 4495.00 0.149947
$$966$$ 0 0
$$967$$ 14467.9 0.481133 0.240567 0.970633i $$-0.422667\pi$$
0.240567 + 0.970633i $$0.422667\pi$$
$$968$$ 0 0
$$969$$ −18928.0 −0.627507
$$970$$ 0 0
$$971$$ −12952.8 −0.428090 −0.214045 0.976824i $$-0.568664\pi$$
−0.214045 + 0.976824i $$0.568664\pi$$
$$972$$ 0 0
$$973$$ 8719.23 0.287282
$$974$$ 0 0
$$975$$ −3274.26 −0.107549
$$976$$ 0 0
$$977$$ 47244.2 1.54706 0.773529 0.633760i $$-0.218489\pi$$
0.773529 + 0.633760i $$0.218489\pi$$
$$978$$ 0 0
$$979$$ 18709.4 0.610781
$$980$$ 0 0
$$981$$ 4906.58 0.159689
$$982$$ 0 0
$$983$$ 1536.84 0.0498651 0.0249326 0.999689i $$-0.492063\pi$$
0.0249326 + 0.999689i $$0.492063\pi$$
$$984$$ 0 0
$$985$$ 15318.1 0.495509
$$986$$ 0 0
$$987$$ 12238.7 0.394692
$$988$$ 0 0
$$989$$ 9649.73 0.310256
$$990$$ 0 0
$$991$$ −3785.22 −0.121334 −0.0606668 0.998158i $$-0.519323\pi$$
−0.0606668 + 0.998158i $$0.519323\pi$$
$$992$$ 0 0
$$993$$ 27774.5 0.887610
$$994$$ 0 0
$$995$$ 4747.61 0.151266
$$996$$ 0 0
$$997$$ −25894.9 −0.822566 −0.411283 0.911508i $$-0.634919\pi$$
−0.411283 + 0.911508i $$0.634919\pi$$
$$998$$ 0 0
$$999$$ −6822.26 −0.216063
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 1680.4.a.bo.1.1 2
4.3 odd 2 105.4.a.e.1.1 2
12.11 even 2 315.4.a.k.1.2 2
20.3 even 4 525.4.d.l.274.4 4
20.7 even 4 525.4.d.l.274.1 4
20.19 odd 2 525.4.a.l.1.2 2
28.27 even 2 735.4.a.o.1.1 2
60.59 even 2 1575.4.a.q.1.1 2
84.83 odd 2 2205.4.a.bb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 4.3 odd 2
315.4.a.k.1.2 2 12.11 even 2
525.4.a.l.1.2 2 20.19 odd 2
525.4.d.l.274.1 4 20.7 even 4
525.4.d.l.274.4 4 20.3 even 4
735.4.a.o.1.1 2 28.27 even 2
1575.4.a.q.1.1 2 60.59 even 2
1680.4.a.bo.1.1 2 1.1 even 1 trivial
2205.4.a.bb.1.2 2 84.83 odd 2