# Properties

 Label 2205.4.a.x.1.1 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.31662$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} +O(q^{10})$$ $$q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} -21.5831 q^{10} -19.7335 q^{11} -71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} -136.332 q^{19} +53.1662 q^{20} +85.1821 q^{22} +100.865 q^{23} +25.0000 q^{25} +307.916 q^{26} +288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +135.251 q^{34} +309.931 q^{37} +588.496 q^{38} -56.8338 q^{40} -181.662 q^{41} -18.2005 q^{43} -209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -107.916 q^{50} -758.496 q^{52} +127.995 q^{53} -98.6675 q^{55} -1244.04 q^{58} +322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -356.662 q^{65} -84.3960 q^{67} -333.166 q^{68} +315.736 q^{71} +1093.32 q^{73} -1337.86 q^{74} -1449.66 q^{76} +1233.19 q^{79} -180.000 q^{80} +784.169 q^{82} -643.325 q^{83} -156.662 q^{85} +78.5647 q^{86} +224.306 q^{88} +1140.32 q^{89} +1072.53 q^{92} +637.414 q^{94} -681.662 q^{95} -1411.99 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 $$2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 - 10 * q^10 - 66 * q^11 - 10 * q^13 - 72 * q^16 + 70 * q^17 - 140 * q^19 + 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 + 450 * q^26 + 258 * q^29 + 20 * q^31 + 360 * q^32 + 370 * q^34 + 328 * q^37 + 580 * q^38 - 180 * q^40 + 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 - 30 * q^47 - 50 * q^50 - 920 * q^52 - 540 * q^53 - 330 * q^55 - 1314 * q^58 + 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 - 600 * q^68 + 1056 * q^71 + 860 * q^73 - 1296 * q^74 - 1440 * q^76 + 158 * q^79 - 360 * q^80 + 1900 * q^82 + 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 910 * q^94 - 700 * q^95 - 1630 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.31662 −1.52616 −0.763079 0.646306i $$-0.776313\pi$$
−0.763079 + 0.646306i $$0.776313\pi$$
$$3$$ 0 0
$$4$$ 10.6332 1.32916
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −11.3668 −0.502344
$$9$$ 0 0
$$10$$ −21.5831 −0.682518
$$11$$ −19.7335 −0.540898 −0.270449 0.962734i $$-0.587172\pi$$
−0.270449 + 0.962734i $$0.587172\pi$$
$$12$$ 0 0
$$13$$ −71.3325 −1.52185 −0.760926 0.648839i $$-0.775255\pi$$
−0.760926 + 0.648839i $$0.775255\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −36.0000 −0.562500
$$17$$ −31.3325 −0.447014 −0.223507 0.974702i $$-0.571751\pi$$
−0.223507 + 0.974702i $$0.571751\pi$$
$$18$$ 0 0
$$19$$ −136.332 −1.64615 −0.823074 0.567934i $$-0.807743\pi$$
−0.823074 + 0.567934i $$0.807743\pi$$
$$20$$ 53.1662 0.594417
$$21$$ 0 0
$$22$$ 85.1821 0.825495
$$23$$ 100.865 0.914431 0.457215 0.889356i $$-0.348847\pi$$
0.457215 + 0.889356i $$0.348847\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 307.916 2.32259
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 288.198 1.84541 0.922707 0.385501i $$-0.125971\pi$$
0.922707 + 0.385501i $$0.125971\pi$$
$$30$$ 0 0
$$31$$ 208.997 1.21087 0.605436 0.795894i $$-0.292999\pi$$
0.605436 + 0.795894i $$0.292999\pi$$
$$32$$ 246.332 1.36081
$$33$$ 0 0
$$34$$ 135.251 0.682214
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 309.931 1.37709 0.688546 0.725192i $$-0.258249\pi$$
0.688546 + 0.725192i $$0.258249\pi$$
$$38$$ 588.496 2.51228
$$39$$ 0 0
$$40$$ −56.8338 −0.224655
$$41$$ −181.662 −0.691973 −0.345987 0.938239i $$-0.612456\pi$$
−0.345987 + 0.938239i $$0.612456\pi$$
$$42$$ 0 0
$$43$$ −18.2005 −0.0645477 −0.0322738 0.999479i $$-0.510275\pi$$
−0.0322738 + 0.999479i $$0.510275\pi$$
$$44$$ −209.831 −0.718937
$$45$$ 0 0
$$46$$ −435.398 −1.39557
$$47$$ −147.665 −0.458280 −0.229140 0.973393i $$-0.573591\pi$$
−0.229140 + 0.973393i $$0.573591\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −107.916 −0.305231
$$51$$ 0 0
$$52$$ −758.496 −2.02278
$$53$$ 127.995 0.331726 0.165863 0.986149i $$-0.446959\pi$$
0.165863 + 0.986149i $$0.446959\pi$$
$$54$$ 0 0
$$55$$ −98.6675 −0.241897
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −1244.04 −2.81639
$$59$$ 322.665 0.711990 0.355995 0.934488i $$-0.384142\pi$$
0.355995 + 0.934488i $$0.384142\pi$$
$$60$$ 0 0
$$61$$ −341.003 −0.715752 −0.357876 0.933769i $$-0.616499\pi$$
−0.357876 + 0.933769i $$0.616499\pi$$
$$62$$ −902.164 −1.84798
$$63$$ 0 0
$$64$$ −775.325 −1.51431
$$65$$ −356.662 −0.680593
$$66$$ 0 0
$$67$$ −84.3960 −0.153890 −0.0769449 0.997035i $$-0.524517\pi$$
−0.0769449 + 0.997035i $$0.524517\pi$$
$$68$$ −333.166 −0.594152
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 315.736 0.527760 0.263880 0.964555i $$-0.414998\pi$$
0.263880 + 0.964555i $$0.414998\pi$$
$$72$$ 0 0
$$73$$ 1093.32 1.75293 0.876466 0.481464i $$-0.159895\pi$$
0.876466 + 0.481464i $$0.159895\pi$$
$$74$$ −1337.86 −2.10166
$$75$$ 0 0
$$76$$ −1449.66 −2.18799
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1233.19 1.75626 0.878128 0.478426i $$-0.158793\pi$$
0.878128 + 0.478426i $$0.158793\pi$$
$$80$$ −180.000 −0.251558
$$81$$ 0 0
$$82$$ 784.169 1.05606
$$83$$ −643.325 −0.850772 −0.425386 0.905012i $$-0.639862\pi$$
−0.425386 + 0.905012i $$0.639862\pi$$
$$84$$ 0 0
$$85$$ −156.662 −0.199911
$$86$$ 78.5647 0.0985099
$$87$$ 0 0
$$88$$ 224.306 0.271717
$$89$$ 1140.32 1.35813 0.679064 0.734079i $$-0.262386\pi$$
0.679064 + 0.734079i $$0.262386\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1072.53 1.21542
$$93$$ 0 0
$$94$$ 637.414 0.699407
$$95$$ −681.662 −0.736180
$$96$$ 0 0
$$97$$ −1411.99 −1.47800 −0.739001 0.673705i $$-0.764702\pi$$
−0.739001 + 0.673705i $$0.764702\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 265.831 0.265831
$$101$$ −545.673 −0.537589 −0.268794 0.963198i $$-0.586625\pi$$
−0.268794 + 0.963198i $$0.586625\pi$$
$$102$$ 0 0
$$103$$ −780.990 −0.747119 −0.373559 0.927606i $$-0.621863\pi$$
−0.373559 + 0.927606i $$0.621863\pi$$
$$104$$ 810.819 0.764493
$$105$$ 0 0
$$106$$ −552.506 −0.506266
$$107$$ 620.660 0.560761 0.280381 0.959889i $$-0.409539\pi$$
0.280381 + 0.959889i $$0.409539\pi$$
$$108$$ 0 0
$$109$$ 4.18794 0.00368011 0.00184005 0.999998i $$-0.499414\pi$$
0.00184005 + 0.999998i $$0.499414\pi$$
$$110$$ 425.911 0.369173
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1413.53 −1.17676 −0.588379 0.808585i $$-0.700234\pi$$
−0.588379 + 0.808585i $$0.700234\pi$$
$$114$$ 0 0
$$115$$ 504.327 0.408946
$$116$$ 3064.48 2.45284
$$117$$ 0 0
$$118$$ −1392.82 −1.08661
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −941.589 −0.707430
$$122$$ 1471.98 1.09235
$$123$$ 0 0
$$124$$ 2222.32 1.60944
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2046.26 −1.42974 −0.714868 0.699259i $$-0.753513\pi$$
−0.714868 + 0.699259i $$0.753513\pi$$
$$128$$ 1376.13 0.950262
$$129$$ 0 0
$$130$$ 1539.58 1.03869
$$131$$ 1140.32 0.760534 0.380267 0.924877i $$-0.375832\pi$$
0.380267 + 0.924877i $$0.375832\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 364.306 0.234860
$$135$$ 0 0
$$136$$ 356.149 0.224555
$$137$$ 490.343 0.305787 0.152893 0.988243i $$-0.451141\pi$$
0.152893 + 0.988243i $$0.451141\pi$$
$$138$$ 0 0
$$139$$ −2800.00 −1.70858 −0.854291 0.519795i $$-0.826008\pi$$
−0.854291 + 0.519795i $$0.826008\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1362.91 −0.805445
$$143$$ 1407.64 0.823166
$$144$$ 0 0
$$145$$ 1440.99 0.825294
$$146$$ −4719.47 −2.67525
$$147$$ 0 0
$$148$$ 3295.58 1.83037
$$149$$ −1166.12 −0.641154 −0.320577 0.947222i $$-0.603877\pi$$
−0.320577 + 0.947222i $$0.603877\pi$$
$$150$$ 0 0
$$151$$ 959.581 0.517150 0.258575 0.965991i $$-0.416747\pi$$
0.258575 + 0.965991i $$0.416747\pi$$
$$152$$ 1549.66 0.826933
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1044.99 0.541519
$$156$$ 0 0
$$157$$ 1020.66 0.518838 0.259419 0.965765i $$-0.416469\pi$$
0.259419 + 0.965765i $$0.416469\pi$$
$$158$$ −5323.20 −2.68032
$$159$$ 0 0
$$160$$ 1231.66 0.608572
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1566.02 −0.752514 −0.376257 0.926515i $$-0.622789\pi$$
−0.376257 + 0.926515i $$0.622789\pi$$
$$164$$ −1931.66 −0.919741
$$165$$ 0 0
$$166$$ 2776.99 1.29841
$$167$$ −1130.30 −0.523746 −0.261873 0.965102i $$-0.584340\pi$$
−0.261873 + 0.965102i $$0.584340\pi$$
$$168$$ 0 0
$$169$$ 2891.32 1.31603
$$170$$ 676.253 0.305096
$$171$$ 0 0
$$172$$ −193.530 −0.0857940
$$173$$ 2543.34 1.11772 0.558862 0.829260i $$-0.311238\pi$$
0.558862 + 0.829260i $$0.311238\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 710.406 0.304255
$$177$$ 0 0
$$178$$ −4922.32 −2.07272
$$179$$ 1210.65 0.505521 0.252760 0.967529i $$-0.418662\pi$$
0.252760 + 0.967529i $$0.418662\pi$$
$$180$$ 0 0
$$181$$ 3031.32 1.24484 0.622421 0.782683i $$-0.286149\pi$$
0.622421 + 0.782683i $$0.286149\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1146.51 −0.459359
$$185$$ 1549.66 0.615854
$$186$$ 0 0
$$187$$ 618.300 0.241789
$$188$$ −1570.16 −0.609125
$$189$$ 0 0
$$190$$ 2942.48 1.12353
$$191$$ 2168.64 0.821555 0.410778 0.911736i $$-0.365257\pi$$
0.410778 + 0.911736i $$0.365257\pi$$
$$192$$ 0 0
$$193$$ 1490.48 0.555892 0.277946 0.960597i $$-0.410346\pi$$
0.277946 + 0.960597i $$0.410346\pi$$
$$194$$ 6095.04 2.25566
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3380.57 −1.22262 −0.611308 0.791393i $$-0.709356\pi$$
−0.611308 + 0.791393i $$0.709356\pi$$
$$198$$ 0 0
$$199$$ −4595.33 −1.63696 −0.818478 0.574538i $$-0.805182\pi$$
−0.818478 + 0.574538i $$0.805182\pi$$
$$200$$ −284.169 −0.100469
$$201$$ 0 0
$$202$$ 2355.46 0.820445
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −908.312 −0.309460
$$206$$ 3371.24 1.14022
$$207$$ 0 0
$$208$$ 2567.97 0.856042
$$209$$ 2690.32 0.890398
$$210$$ 0 0
$$211$$ −4988.66 −1.62765 −0.813824 0.581112i $$-0.802618\pi$$
−0.813824 + 0.581112i $$0.802618\pi$$
$$212$$ 1361.00 0.440915
$$213$$ 0 0
$$214$$ −2679.16 −0.855810
$$215$$ −91.0025 −0.0288666
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0778 −0.00561643
$$219$$ 0 0
$$220$$ −1049.16 −0.321519
$$221$$ 2235.03 0.680290
$$222$$ 0 0
$$223$$ −3792.97 −1.13900 −0.569498 0.821993i $$-0.692862\pi$$
−0.569498 + 0.821993i $$0.692862\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 6101.68 1.79592
$$227$$ 3910.96 1.14352 0.571762 0.820420i $$-0.306260\pi$$
0.571762 + 0.820420i $$0.306260\pi$$
$$228$$ 0 0
$$229$$ −354.327 −0.102247 −0.0511236 0.998692i $$-0.516280\pi$$
−0.0511236 + 0.998692i $$0.516280\pi$$
$$230$$ −2176.99 −0.624116
$$231$$ 0 0
$$232$$ −3275.87 −0.927033
$$233$$ −6492.48 −1.82548 −0.912739 0.408543i $$-0.866037\pi$$
−0.912739 + 0.408543i $$0.866037\pi$$
$$234$$ 0 0
$$235$$ −738.325 −0.204949
$$236$$ 3430.98 0.946346
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 342.688 0.0927474 0.0463737 0.998924i $$-0.485234\pi$$
0.0463737 + 0.998924i $$0.485234\pi$$
$$240$$ 0 0
$$241$$ −2313.67 −0.618408 −0.309204 0.950996i $$-0.600063\pi$$
−0.309204 + 0.950996i $$0.600063\pi$$
$$242$$ 4064.49 1.07965
$$243$$ 0 0
$$244$$ −3625.96 −0.951347
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9724.94 2.50519
$$248$$ −2375.62 −0.608275
$$249$$ 0 0
$$250$$ −539.578 −0.136504
$$251$$ 3989.29 1.00319 0.501597 0.865101i $$-0.332746\pi$$
0.501597 + 0.865101i $$0.332746\pi$$
$$252$$ 0 0
$$253$$ −1990.43 −0.494614
$$254$$ 8832.95 2.18200
$$255$$ 0 0
$$256$$ 262.376 0.0640566
$$257$$ 2291.32 0.556142 0.278071 0.960560i $$-0.410305\pi$$
0.278071 + 0.960560i $$0.410305\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −3792.48 −0.904614
$$261$$ 0 0
$$262$$ −4922.32 −1.16070
$$263$$ −6360.47 −1.49127 −0.745634 0.666356i $$-0.767853\pi$$
−0.745634 + 0.666356i $$0.767853\pi$$
$$264$$ 0 0
$$265$$ 639.975 0.148352
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −897.404 −0.204543
$$269$$ −991.345 −0.224697 −0.112348 0.993669i $$-0.535837\pi$$
−0.112348 + 0.993669i $$0.535837\pi$$
$$270$$ 0 0
$$271$$ −730.977 −0.163851 −0.0819257 0.996638i $$-0.526107\pi$$
−0.0819257 + 0.996638i $$0.526107\pi$$
$$272$$ 1127.97 0.251446
$$273$$ 0 0
$$274$$ −2116.62 −0.466679
$$275$$ −493.338 −0.108180
$$276$$ 0 0
$$277$$ −3538.63 −0.767566 −0.383783 0.923423i $$-0.625379\pi$$
−0.383783 + 0.923423i $$0.625379\pi$$
$$278$$ 12086.5 2.60756
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4663.20 0.989975 0.494988 0.868900i $$-0.335173\pi$$
0.494988 + 0.868900i $$0.335173\pi$$
$$282$$ 0 0
$$283$$ 2104.95 0.442142 0.221071 0.975258i $$-0.429045\pi$$
0.221071 + 0.975258i $$0.429045\pi$$
$$284$$ 3357.30 0.701476
$$285$$ 0 0
$$286$$ −6076.25 −1.25628
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3931.27 −0.800178
$$290$$ −6220.21 −1.25953
$$291$$ 0 0
$$292$$ 11625.6 2.32992
$$293$$ −6594.66 −1.31489 −0.657447 0.753501i $$-0.728364\pi$$
−0.657447 + 0.753501i $$0.728364\pi$$
$$294$$ 0 0
$$295$$ 1613.32 0.318412
$$296$$ −3522.91 −0.691774
$$297$$ 0 0
$$298$$ 5033.69 0.978503
$$299$$ −7194.99 −1.39163
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4142.15 −0.789252
$$303$$ 0 0
$$304$$ 4907.97 0.925958
$$305$$ −1705.01 −0.320094
$$306$$ 0 0
$$307$$ 2672.97 0.496920 0.248460 0.968642i $$-0.420076\pi$$
0.248460 + 0.968642i $$0.420076\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4510.82 −0.826443
$$311$$ −855.698 −0.156020 −0.0780099 0.996953i $$-0.524857\pi$$
−0.0780099 + 0.996953i $$0.524857\pi$$
$$312$$ 0 0
$$313$$ 3349.99 0.604960 0.302480 0.953156i $$-0.402185\pi$$
0.302480 + 0.953156i $$0.402185\pi$$
$$314$$ −4405.81 −0.791828
$$315$$ 0 0
$$316$$ 13112.8 2.33434
$$317$$ −7633.05 −1.35241 −0.676206 0.736712i $$-0.736377\pi$$
−0.676206 + 0.736712i $$0.736377\pi$$
$$318$$ 0 0
$$319$$ −5687.16 −0.998180
$$320$$ −3876.62 −0.677218
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4271.64 0.735852
$$324$$ 0 0
$$325$$ −1783.31 −0.304370
$$326$$ 6759.90 1.14845
$$327$$ 0 0
$$328$$ 2064.91 0.347609
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −3321.70 −0.551593 −0.275796 0.961216i $$-0.588942\pi$$
−0.275796 + 0.961216i $$0.588942\pi$$
$$332$$ −6840.63 −1.13081
$$333$$ 0 0
$$334$$ 4879.10 0.799319
$$335$$ −421.980 −0.0688216
$$336$$ 0 0
$$337$$ −2233.98 −0.361107 −0.180553 0.983565i $$-0.557789\pi$$
−0.180553 + 0.983565i $$0.557789\pi$$
$$338$$ −12480.8 −2.00847
$$339$$ 0 0
$$340$$ −1665.83 −0.265713
$$341$$ −4124.25 −0.654958
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 206.881 0.0324252
$$345$$ 0 0
$$346$$ −10978.6 −1.70582
$$347$$ −2528.61 −0.391190 −0.195595 0.980685i $$-0.562664\pi$$
−0.195595 + 0.980685i $$0.562664\pi$$
$$348$$ 0 0
$$349$$ −1291.00 −0.198011 −0.0990054 0.995087i $$-0.531566\pi$$
−0.0990054 + 0.995087i $$0.531566\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4861.00 −0.736058
$$353$$ −7768.64 −1.17134 −0.585670 0.810550i $$-0.699169\pi$$
−0.585670 + 0.810550i $$0.699169\pi$$
$$354$$ 0 0
$$355$$ 1578.68 0.236022
$$356$$ 12125.3 1.80516
$$357$$ 0 0
$$358$$ −5225.92 −0.771504
$$359$$ 2284.14 0.335800 0.167900 0.985804i $$-0.446301\pi$$
0.167900 + 0.985804i $$0.446301\pi$$
$$360$$ 0 0
$$361$$ 11727.5 1.70980
$$362$$ −13085.1 −1.89982
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5466.62 0.783935
$$366$$ 0 0
$$367$$ −10707.0 −1.52289 −0.761446 0.648229i $$-0.775510\pi$$
−0.761446 + 0.648229i $$0.775510\pi$$
$$368$$ −3631.16 −0.514367
$$369$$ 0 0
$$370$$ −6689.29 −0.939891
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −830.429 −0.115276 −0.0576381 0.998338i $$-0.518357\pi$$
−0.0576381 + 0.998338i $$0.518357\pi$$
$$374$$ −2668.97 −0.369008
$$375$$ 0 0
$$376$$ 1678.47 0.230214
$$377$$ −20557.9 −2.80845
$$378$$ 0 0
$$379$$ 5253.17 0.711972 0.355986 0.934491i $$-0.384145\pi$$
0.355986 + 0.934491i $$0.384145\pi$$
$$380$$ −7248.29 −0.978498
$$381$$ 0 0
$$382$$ −9361.19 −1.25382
$$383$$ −11243.9 −1.50010 −0.750048 0.661383i $$-0.769970\pi$$
−0.750048 + 0.661383i $$0.769970\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6433.84 −0.848378
$$387$$ 0 0
$$388$$ −15014.1 −1.96449
$$389$$ 8506.85 1.10878 0.554389 0.832258i $$-0.312952\pi$$
0.554389 + 0.832258i $$0.312952\pi$$
$$390$$ 0 0
$$391$$ −3160.37 −0.408764
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 14592.6 1.86590
$$395$$ 6165.93 0.785421
$$396$$ 0 0
$$397$$ −3123.24 −0.394838 −0.197419 0.980319i $$-0.563256\pi$$
−0.197419 + 0.980319i $$0.563256\pi$$
$$398$$ 19836.3 2.49825
$$399$$ 0 0
$$400$$ −900.000 −0.112500
$$401$$ 11255.3 1.40166 0.700828 0.713330i $$-0.252814\pi$$
0.700828 + 0.713330i $$0.252814\pi$$
$$402$$ 0 0
$$403$$ −14908.3 −1.84277
$$404$$ −5802.27 −0.714539
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6116.03 −0.744866
$$408$$ 0 0
$$409$$ 7919.92 0.957494 0.478747 0.877953i $$-0.341091\pi$$
0.478747 + 0.877953i $$0.341091\pi$$
$$410$$ 3920.84 0.472285
$$411$$ 0 0
$$412$$ −8304.46 −0.993037
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3216.62 −0.380477
$$416$$ −17571.5 −2.07095
$$417$$ 0 0
$$418$$ −11613.1 −1.35889
$$419$$ −5257.28 −0.612972 −0.306486 0.951875i $$-0.599153\pi$$
−0.306486 + 0.951875i $$0.599153\pi$$
$$420$$ 0 0
$$421$$ 1457.36 0.168711 0.0843556 0.996436i $$-0.473117\pi$$
0.0843556 + 0.996436i $$0.473117\pi$$
$$422$$ 21534.2 2.48405
$$423$$ 0 0
$$424$$ −1454.89 −0.166640
$$425$$ −783.312 −0.0894029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 6599.63 0.745339
$$429$$ 0 0
$$430$$ 392.824 0.0440550
$$431$$ 15291.2 1.70893 0.854467 0.519506i $$-0.173884\pi$$
0.854467 + 0.519506i $$0.173884\pi$$
$$432$$ 0 0
$$433$$ −187.260 −0.0207832 −0.0103916 0.999946i $$-0.503308\pi$$
−0.0103916 + 0.999946i $$0.503308\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 44.5314 0.00489144
$$437$$ −13751.2 −1.50529
$$438$$ 0 0
$$439$$ −3587.92 −0.390073 −0.195037 0.980796i $$-0.562483\pi$$
−0.195037 + 0.980796i $$0.562483\pi$$
$$440$$ 1121.53 0.121515
$$441$$ 0 0
$$442$$ −9647.76 −1.03823
$$443$$ −4915.65 −0.527200 −0.263600 0.964632i $$-0.584910\pi$$
−0.263600 + 0.964632i $$0.584910\pi$$
$$444$$ 0 0
$$445$$ 5701.59 0.607373
$$446$$ 16372.8 1.73829
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −7091.12 −0.745324 −0.372662 0.927967i $$-0.621555\pi$$
−0.372662 + 0.927967i $$0.621555\pi$$
$$450$$ 0 0
$$451$$ 3584.84 0.374287
$$452$$ −15030.4 −1.56410
$$453$$ 0 0
$$454$$ −16882.2 −1.74520
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5051.81 0.517098 0.258549 0.965998i $$-0.416756\pi$$
0.258549 + 0.965998i $$0.416756\pi$$
$$458$$ 1529.50 0.156045
$$459$$ 0 0
$$460$$ 5362.64 0.543553
$$461$$ −16681.3 −1.68531 −0.842653 0.538456i $$-0.819008\pi$$
−0.842653 + 0.538456i $$0.819008\pi$$
$$462$$ 0 0
$$463$$ 15569.6 1.56280 0.781402 0.624027i $$-0.214505\pi$$
0.781402 + 0.624027i $$0.214505\pi$$
$$464$$ −10375.1 −1.03805
$$465$$ 0 0
$$466$$ 28025.6 2.78597
$$467$$ 3328.35 0.329802 0.164901 0.986310i $$-0.447269\pi$$
0.164901 + 0.986310i $$0.447269\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 3187.07 0.312784
$$471$$ 0 0
$$472$$ −3667.65 −0.357664
$$473$$ 359.160 0.0349137
$$474$$ 0 0
$$475$$ −3408.31 −0.329230
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −1479.25 −0.141547
$$479$$ −14607.5 −1.39339 −0.696695 0.717367i $$-0.745347\pi$$
−0.696695 + 0.717367i $$0.745347\pi$$
$$480$$ 0 0
$$481$$ −22108.2 −2.09573
$$482$$ 9987.23 0.943789
$$483$$ 0 0
$$484$$ −10012.2 −0.940285
$$485$$ −7059.96 −0.660982
$$486$$ 0 0
$$487$$ −1879.49 −0.174882 −0.0874412 0.996170i $$-0.527869\pi$$
−0.0874412 + 0.996170i $$0.527869\pi$$
$$488$$ 3876.09 0.359554
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3221.13 −0.296064 −0.148032 0.988983i $$-0.547294\pi$$
−0.148032 + 0.988983i $$0.547294\pi$$
$$492$$ 0 0
$$493$$ −9029.96 −0.824927
$$494$$ −41978.9 −3.82332
$$495$$ 0 0
$$496$$ −7523.91 −0.681116
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9713.81 0.871443 0.435721 0.900082i $$-0.356493\pi$$
0.435721 + 0.900082i $$0.356493\pi$$
$$500$$ 1329.16 0.118883
$$501$$ 0 0
$$502$$ −17220.3 −1.53103
$$503$$ 2078.32 0.184230 0.0921152 0.995748i $$-0.470637\pi$$
0.0921152 + 0.995748i $$0.470637\pi$$
$$504$$ 0 0
$$505$$ −2728.36 −0.240417
$$506$$ 8591.94 0.754858
$$507$$ 0 0
$$508$$ −21758.4 −1.90034
$$509$$ −18974.5 −1.65232 −0.826158 0.563439i $$-0.809478\pi$$
−0.826158 + 0.563439i $$0.809478\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −12141.6 −1.04802
$$513$$ 0 0
$$514$$ −9890.77 −0.848761
$$515$$ −3904.95 −0.334122
$$516$$ 0 0
$$517$$ 2913.95 0.247883
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4054.09 0.341892
$$521$$ 17523.6 1.47355 0.736777 0.676136i $$-0.236347\pi$$
0.736777 + 0.676136i $$0.236347\pi$$
$$522$$ 0 0
$$523$$ −15218.6 −1.27239 −0.636197 0.771527i $$-0.719493\pi$$
−0.636197 + 0.771527i $$0.719493\pi$$
$$524$$ 12125.3 1.01087
$$525$$ 0 0
$$526$$ 27455.8 2.27591
$$527$$ −6548.41 −0.541278
$$528$$ 0 0
$$529$$ −1993.15 −0.163816
$$530$$ −2762.53 −0.226409
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12958.4 1.05308
$$534$$ 0 0
$$535$$ 3103.30 0.250780
$$536$$ 959.308 0.0773056
$$537$$ 0 0
$$538$$ 4279.26 0.342922
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −15559.3 −1.23650 −0.618249 0.785983i $$-0.712158\pi$$
−0.618249 + 0.785983i $$0.712158\pi$$
$$542$$ 3155.36 0.250063
$$543$$ 0 0
$$544$$ −7718.21 −0.608301
$$545$$ 20.9397 0.00164579
$$546$$ 0 0
$$547$$ −8690.70 −0.679319 −0.339660 0.940548i $$-0.610312\pi$$
−0.339660 + 0.940548i $$0.610312\pi$$
$$548$$ 5213.93 0.406438
$$549$$ 0 0
$$550$$ 2129.55 0.165099
$$551$$ −39290.8 −3.03783
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 15274.9 1.17143
$$555$$ 0 0
$$556$$ −29773.1 −2.27097
$$557$$ 7376.26 0.561117 0.280559 0.959837i $$-0.409480\pi$$
0.280559 + 0.959837i $$0.409480\pi$$
$$558$$ 0 0
$$559$$ 1298.29 0.0982320
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −20129.3 −1.51086
$$563$$ 12875.4 0.963825 0.481913 0.876219i $$-0.339942\pi$$
0.481913 + 0.876219i $$0.339942\pi$$
$$564$$ 0 0
$$565$$ −7067.65 −0.526263
$$566$$ −9086.28 −0.674779
$$567$$ 0 0
$$568$$ −3588.89 −0.265117
$$569$$ 12064.6 0.888882 0.444441 0.895808i $$-0.353402\pi$$
0.444441 + 0.895808i $$0.353402\pi$$
$$570$$ 0 0
$$571$$ 23745.6 1.74032 0.870158 0.492772i $$-0.164016\pi$$
0.870158 + 0.492772i $$0.164016\pi$$
$$572$$ 14967.8 1.09412
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2521.64 0.182886
$$576$$ 0 0
$$577$$ 9846.08 0.710394 0.355197 0.934791i $$-0.384414\pi$$
0.355197 + 0.934791i $$0.384414\pi$$
$$578$$ 16969.8 1.22120
$$579$$ 0 0
$$580$$ 15322.4 1.09695
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2525.79 −0.179430
$$584$$ −12427.6 −0.880575
$$585$$ 0 0
$$586$$ 28466.7 2.00674
$$587$$ 10074.7 0.708392 0.354196 0.935171i $$-0.384755\pi$$
0.354196 + 0.935171i $$0.384755\pi$$
$$588$$ 0 0
$$589$$ −28493.1 −1.99328
$$590$$ −6964.12 −0.485946
$$591$$ 0 0
$$592$$ −11157.5 −0.774615
$$593$$ −7387.25 −0.511565 −0.255782 0.966734i $$-0.582333\pi$$
−0.255782 + 0.966734i $$0.582333\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12399.6 −0.852194
$$597$$ 0 0
$$598$$ 31058.1 2.12384
$$599$$ −1252.73 −0.0854510 −0.0427255 0.999087i $$-0.513604\pi$$
−0.0427255 + 0.999087i $$0.513604\pi$$
$$600$$ 0 0
$$601$$ 1800.81 0.122224 0.0611120 0.998131i $$-0.480535\pi$$
0.0611120 + 0.998131i $$0.480535\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 10203.5 0.687373
$$605$$ −4707.94 −0.316372
$$606$$ 0 0
$$607$$ −2497.06 −0.166973 −0.0834863 0.996509i $$-0.526605\pi$$
−0.0834863 + 0.996509i $$0.526605\pi$$
$$608$$ −33583.1 −2.24009
$$609$$ 0 0
$$610$$ 7359.90 0.488514
$$611$$ 10533.3 0.697434
$$612$$ 0 0
$$613$$ −19750.8 −1.30135 −0.650674 0.759357i $$-0.725513\pi$$
−0.650674 + 0.759357i $$0.725513\pi$$
$$614$$ −11538.2 −0.758378
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −16797.4 −1.09601 −0.548004 0.836476i $$-0.684612\pi$$
−0.548004 + 0.836476i $$0.684612\pi$$
$$618$$ 0 0
$$619$$ −26547.4 −1.72379 −0.861897 0.507084i $$-0.830723\pi$$
−0.861897 + 0.507084i $$0.830723\pi$$
$$620$$ 11111.6 0.719763
$$621$$ 0 0
$$622$$ 3693.73 0.238111
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −14460.6 −0.923264
$$627$$ 0 0
$$628$$ 10852.9 0.689616
$$629$$ −9710.93 −0.615580
$$630$$ 0 0
$$631$$ 5394.86 0.340358 0.170179 0.985413i $$-0.445565\pi$$
0.170179 + 0.985413i $$0.445565\pi$$
$$632$$ −14017.3 −0.882245
$$633$$ 0 0
$$634$$ 32949.0 2.06399
$$635$$ −10231.3 −0.639398
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 24549.3 1.52338
$$639$$ 0 0
$$640$$ 6880.63 0.424970
$$641$$ −2452.41 −0.151114 −0.0755572 0.997141i $$-0.524074\pi$$
−0.0755572 + 0.997141i $$0.524074\pi$$
$$642$$ 0 0
$$643$$ −7074.97 −0.433919 −0.216959 0.976181i $$-0.569614\pi$$
−0.216959 + 0.976181i $$0.569614\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −18439.1 −1.12303
$$647$$ 3341.37 0.203034 0.101517 0.994834i $$-0.467630\pi$$
0.101517 + 0.994834i $$0.467630\pi$$
$$648$$ 0 0
$$649$$ −6367.31 −0.385114
$$650$$ 7697.89 0.464517
$$651$$ 0 0
$$652$$ −16651.8 −1.00021
$$653$$ 23061.6 1.38204 0.691019 0.722837i $$-0.257162\pi$$
0.691019 + 0.722837i $$0.257162\pi$$
$$654$$ 0 0
$$655$$ 5701.59 0.340121
$$656$$ 6539.85 0.389235
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1742.64 −0.103010 −0.0515049 0.998673i $$-0.516402\pi$$
−0.0515049 + 0.998673i $$0.516402\pi$$
$$660$$ 0 0
$$661$$ 12576.5 0.740046 0.370023 0.929023i $$-0.379350\pi$$
0.370023 + 0.929023i $$0.379350\pi$$
$$662$$ 14338.5 0.841817
$$663$$ 0 0
$$664$$ 7312.51 0.427380
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 29069.2 1.68750
$$668$$ −12018.8 −0.696141
$$669$$ 0 0
$$670$$ 1821.53 0.105033
$$671$$ 6729.17 0.387149
$$672$$ 0 0
$$673$$ −10680.8 −0.611760 −0.305880 0.952070i $$-0.598951\pi$$
−0.305880 + 0.952070i $$0.598951\pi$$
$$674$$ 9643.27 0.551105
$$675$$ 0 0
$$676$$ 30744.2 1.74921
$$677$$ −29559.1 −1.67806 −0.839032 0.544082i $$-0.816878\pi$$
−0.839032 + 0.544082i $$0.816878\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1780.74 0.100424
$$681$$ 0 0
$$682$$ 17802.8 0.999569
$$683$$ −10250.4 −0.574263 −0.287132 0.957891i $$-0.592702\pi$$
−0.287132 + 0.957891i $$0.592702\pi$$
$$684$$ 0 0
$$685$$ 2451.71 0.136752
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 655.218 0.0363081
$$689$$ −9130.20 −0.504837
$$690$$ 0 0
$$691$$ 8874.04 0.488544 0.244272 0.969707i $$-0.421451\pi$$
0.244272 + 0.969707i $$0.421451\pi$$
$$692$$ 27043.9 1.48563
$$693$$ 0 0
$$694$$ 10915.1 0.597018
$$695$$ −14000.0 −0.764101
$$696$$ 0 0
$$697$$ 5691.94 0.309322
$$698$$ 5572.77 0.302196
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22086.2 1.18999 0.594996 0.803729i $$-0.297154\pi$$
0.594996 + 0.803729i $$0.297154\pi$$
$$702$$ 0 0
$$703$$ −42253.7 −2.26690
$$704$$ 15299.9 0.819085
$$705$$ 0 0
$$706$$ 33534.3 1.78765
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −27878.9 −1.47675 −0.738373 0.674392i $$-0.764406\pi$$
−0.738373 + 0.674392i $$0.764406\pi$$
$$710$$ −6814.57 −0.360206
$$711$$ 0 0
$$712$$ −12961.7 −0.682248
$$713$$ 21080.6 1.10726
$$714$$ 0 0
$$715$$ 7038.20 0.368131
$$716$$ 12873.1 0.671916
$$717$$ 0 0
$$718$$ −9859.76 −0.512483
$$719$$ 25863.3 1.34150 0.670750 0.741684i $$-0.265973\pi$$
0.670750 + 0.741684i $$0.265973\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −50623.4 −2.60943
$$723$$ 0 0
$$724$$ 32232.8 1.65459
$$725$$ 7204.95 0.369083
$$726$$ 0 0
$$727$$ −29157.0 −1.48744 −0.743722 0.668489i $$-0.766941\pi$$
−0.743722 + 0.668489i $$0.766941\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −23597.4 −1.19641
$$731$$ 570.267 0.0288538
$$732$$ 0 0
$$733$$ −11006.1 −0.554595 −0.277297 0.960784i $$-0.589439\pi$$
−0.277297 + 0.960784i $$0.589439\pi$$
$$734$$ 46218.1 2.32417
$$735$$ 0 0
$$736$$ 24846.4 1.24436
$$737$$ 1665.43 0.0832386
$$738$$ 0 0
$$739$$ −37214.4 −1.85244 −0.926221 0.376982i $$-0.876962\pi$$
−0.926221 + 0.376982i $$0.876962\pi$$
$$740$$ 16477.9 0.818567
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −11214.5 −0.553730 −0.276865 0.960909i $$-0.589296\pi$$
−0.276865 + 0.960909i $$0.589296\pi$$
$$744$$ 0 0
$$745$$ −5830.58 −0.286733
$$746$$ 3584.65 0.175930
$$747$$ 0 0
$$748$$ 6574.54 0.321375
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 6965.26 0.338437 0.169218 0.985579i $$-0.445876\pi$$
0.169218 + 0.985579i $$0.445876\pi$$
$$752$$ 5315.94 0.257782
$$753$$ 0 0
$$754$$ 88740.7 4.28613
$$755$$ 4797.91 0.231276
$$756$$ 0 0
$$757$$ −19352.8 −0.929180 −0.464590 0.885526i $$-0.653798\pi$$
−0.464590 + 0.885526i $$0.653798\pi$$
$$758$$ −22676.0 −1.08658
$$759$$ 0 0
$$760$$ 7748.29 0.369816
$$761$$ 32383.6 1.54258 0.771291 0.636483i $$-0.219612\pi$$
0.771291 + 0.636483i $$0.219612\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 23059.7 1.09198
$$765$$ 0 0
$$766$$ 48535.7 2.28938
$$767$$ −23016.5 −1.08354
$$768$$ 0 0
$$769$$ 25353.9 1.18893 0.594463 0.804123i $$-0.297365\pi$$
0.594463 + 0.804123i $$0.297365\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 15848.6 0.738867
$$773$$ 26117.0 1.21522 0.607610 0.794236i $$-0.292128\pi$$
0.607610 + 0.794236i $$0.292128\pi$$
$$774$$ 0 0
$$775$$ 5224.94 0.242175
$$776$$ 16049.8 0.742465
$$777$$ 0 0
$$778$$ −36720.9 −1.69217
$$779$$ 24766.5 1.13909
$$780$$ 0 0
$$781$$ −6230.58 −0.285464
$$782$$ 13642.1 0.623838
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5103.30 0.232031
$$786$$ 0 0
$$787$$ 2273.38 0.102970 0.0514849 0.998674i $$-0.483605\pi$$
0.0514849 + 0.998674i $$0.483605\pi$$
$$788$$ −35946.4 −1.62505
$$789$$ 0 0
$$790$$ −26616.0 −1.19868
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 24324.6 1.08927
$$794$$ 13481.8 0.602585
$$795$$ 0 0
$$796$$ −48863.3 −2.17577
$$797$$ 2937.42 0.130551 0.0652753 0.997867i $$-0.479207\pi$$
0.0652753 + 0.997867i $$0.479207\pi$$
$$798$$ 0 0
$$799$$ 4626.71 0.204858
$$800$$ 6158.31 0.272162
$$801$$ 0 0
$$802$$ −48585.0 −2.13915
$$803$$ −21575.1 −0.948157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 64353.6 2.81236
$$807$$ 0 0
$$808$$ 6202.52 0.270054
$$809$$ 4317.51 0.187634 0.0938169 0.995589i $$-0.470093\pi$$
0.0938169 + 0.995589i $$0.470093\pi$$
$$810$$ 0 0
$$811$$ −1286.12 −0.0556863 −0.0278432 0.999612i $$-0.508864\pi$$
−0.0278432 + 0.999612i $$0.508864\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 26400.6 1.13678
$$815$$ −7830.08 −0.336534
$$816$$ 0 0
$$817$$ 2481.32 0.106255
$$818$$ −34187.3 −1.46129
$$819$$ 0 0
$$820$$ −9658.31 −0.411321
$$821$$ −26350.2 −1.12013 −0.560066 0.828448i $$-0.689224\pi$$
−0.560066 + 0.828448i $$0.689224\pi$$
$$822$$ 0 0
$$823$$ 9820.05 0.415924 0.207962 0.978137i $$-0.433317\pi$$
0.207962 + 0.978137i $$0.433317\pi$$
$$824$$ 8877.32 0.375311
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30370.7 1.27702 0.638509 0.769615i $$-0.279552\pi$$
0.638509 + 0.769615i $$0.279552\pi$$
$$828$$ 0 0
$$829$$ 30817.7 1.29112 0.645562 0.763708i $$-0.276623\pi$$
0.645562 + 0.763708i $$0.276623\pi$$
$$830$$ 13885.0 0.580668
$$831$$ 0 0
$$832$$ 55305.9 2.30455
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5651.52 −0.234226
$$836$$ 28606.8 1.18348
$$837$$ 0 0
$$838$$ 22693.7 0.935491
$$839$$ −24746.0 −1.01827 −0.509134 0.860688i $$-0.670034\pi$$
−0.509134 + 0.860688i $$0.670034\pi$$
$$840$$ 0 0
$$841$$ 58669.1 2.40556
$$842$$ −6290.88 −0.257480
$$843$$ 0 0
$$844$$ −53045.7 −2.16340
$$845$$ 14456.6 0.588548
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −4607.82 −0.186596
$$849$$ 0 0
$$850$$ 3381.27 0.136443
$$851$$ 31261.4 1.25926
$$852$$ 0 0
$$853$$ 11812.1 0.474135 0.237067 0.971493i $$-0.423814\pi$$
0.237067 + 0.971493i $$0.423814\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −7054.89 −0.281695
$$857$$ −23440.6 −0.934322 −0.467161 0.884172i $$-0.654723\pi$$
−0.467161 + 0.884172i $$0.654723\pi$$
$$858$$ 0 0
$$859$$ −8945.58 −0.355319 −0.177660 0.984092i $$-0.556853\pi$$
−0.177660 + 0.984092i $$0.556853\pi$$
$$860$$ −967.652 −0.0383682
$$861$$ 0 0
$$862$$ −66006.3 −2.60810
$$863$$ −19313.5 −0.761806 −0.380903 0.924615i $$-0.624387\pi$$
−0.380903 + 0.924615i $$0.624387\pi$$
$$864$$ 0 0
$$865$$ 12716.7 0.499862
$$866$$ 808.330 0.0317184
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −24335.1 −0.949955
$$870$$ 0 0
$$871$$ 6020.18 0.234197
$$872$$ −47.6033 −0.00184868
$$873$$ 0 0
$$874$$ 59359.0 2.29731
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12154.8 −0.468001 −0.234001 0.972236i $$-0.575182\pi$$
−0.234001 + 0.972236i $$0.575182\pi$$
$$878$$ 15487.7 0.595313
$$879$$ 0 0
$$880$$ 3552.03 0.136067
$$881$$ −29390.4 −1.12394 −0.561968 0.827159i $$-0.689956\pi$$
−0.561968 + 0.827159i $$0.689956\pi$$
$$882$$ 0 0
$$883$$ 4180.02 0.159308 0.0796540 0.996823i $$-0.474618\pi$$
0.0796540 + 0.996823i $$0.474618\pi$$
$$884$$ 23765.6 0.904211
$$885$$ 0 0
$$886$$ 21219.0 0.804590
$$887$$ 21825.8 0.826198 0.413099 0.910686i $$-0.364446\pi$$
0.413099 + 0.910686i $$0.364446\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −24611.6 −0.926947
$$891$$ 0 0
$$892$$ −40331.6 −1.51390
$$893$$ 20131.5 0.754397
$$894$$ 0 0
$$895$$ 6053.25 0.226076
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 30609.7 1.13748
$$899$$ 60232.7 2.23456
$$900$$ 0 0
$$901$$ −4010.40 −0.148286
$$902$$ −15474.4 −0.571221
$$903$$ 0 0
$$904$$ 16067.2 0.591138
$$905$$ 15156.6 0.556710
$$906$$ 0 0
$$907$$ −8356.11 −0.305910 −0.152955 0.988233i $$-0.548879\pi$$
−0.152955 + 0.988233i $$0.548879\pi$$
$$908$$ 41586.3 1.51992
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 4419.80 0.160740 0.0803701 0.996765i $$-0.474390\pi$$
0.0803701 + 0.996765i $$0.474390\pi$$
$$912$$ 0 0
$$913$$ 12695.1 0.460181
$$914$$ −21806.8 −0.789173
$$915$$ 0 0
$$916$$ −3767.65 −0.135903
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −39257.6 −1.40913 −0.704563 0.709641i $$-0.748857\pi$$
−0.704563 + 0.709641i $$0.748857\pi$$
$$920$$ −5732.56 −0.205432
$$921$$ 0 0
$$922$$ 72007.0 2.57204
$$923$$ −22522.2 −0.803173
$$924$$ 0 0
$$925$$ 7748.29 0.275419
$$926$$ −67207.9 −2.38509
$$927$$ 0 0
$$928$$ 70992.5 2.51125
$$929$$ −13399.9 −0.473235 −0.236618 0.971603i $$-0.576039\pi$$
−0.236618 + 0.971603i $$0.576039\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −69036.1 −2.42635
$$933$$ 0 0
$$934$$ −14367.2 −0.503330
$$935$$ 3091.50 0.108131
$$936$$ 0 0
$$937$$ −27539.8 −0.960176 −0.480088 0.877220i $$-0.659395\pi$$
−0.480088 + 0.877220i $$0.659395\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −7850.79 −0.272409
$$941$$ 14363.8 0.497605 0.248802 0.968554i $$-0.419963\pi$$
0.248802 + 0.968554i $$0.419963\pi$$
$$942$$ 0 0
$$943$$ −18323.5 −0.632762
$$944$$ −11615.9 −0.400494
$$945$$ 0 0
$$946$$ −1550.36 −0.0532838
$$947$$ 6372.12 0.218655 0.109327 0.994006i $$-0.465130\pi$$
0.109327 + 0.994006i $$0.465130\pi$$
$$948$$ 0 0
$$949$$ −77989.6 −2.66770
$$950$$ 14712.4 0.502456
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −958.776 −0.0325895 −0.0162948 0.999867i $$-0.505187\pi$$
−0.0162948 + 0.999867i $$0.505187\pi$$
$$954$$ 0 0
$$955$$ 10843.2 0.367411
$$956$$ 3643.88 0.123276
$$957$$ 0 0
$$958$$ 63055.1 2.12653
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 13888.9 0.466213
$$962$$ 95432.7 3.19842
$$963$$ 0 0
$$964$$ −24601.8 −0.821961
$$965$$ 7452.40 0.248602
$$966$$ 0 0
$$967$$ −40104.5 −1.33369 −0.666843 0.745198i $$-0.732355\pi$$
−0.666843 + 0.745198i $$0.732355\pi$$
$$968$$ 10702.8 0.355373
$$969$$ 0 0
$$970$$ 30475.2 1.00876
$$971$$ 12397.4 0.409732 0.204866 0.978790i $$-0.434324\pi$$
0.204866 + 0.978790i $$0.434324\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 8113.05 0.266898
$$975$$ 0 0
$$976$$ 12276.1 0.402611
$$977$$ −44982.9 −1.47301 −0.736506 0.676432i $$-0.763525\pi$$
−0.736506 + 0.676432i $$0.763525\pi$$
$$978$$ 0 0
$$979$$ −22502.5 −0.734608
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 13904.4 0.451841
$$983$$ −7895.76 −0.256191 −0.128095 0.991762i $$-0.540886\pi$$
−0.128095 + 0.991762i $$0.540886\pi$$
$$984$$ 0 0
$$985$$ −16902.8 −0.546771
$$986$$ 38979.0 1.25897
$$987$$ 0 0
$$988$$ 103408. 3.32979
$$989$$ −1835.80 −0.0590244
$$990$$ 0 0
$$991$$ 54534.9 1.74809 0.874046 0.485844i $$-0.161488\pi$$
0.874046 + 0.485844i $$0.161488\pi$$
$$992$$ 51482.9 1.64776
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −22976.6 −0.732069
$$996$$ 0 0
$$997$$ 6028.06 0.191485 0.0957425 0.995406i $$-0.469477\pi$$
0.0957425 + 0.995406i $$0.469477\pi$$
$$998$$ −41930.9 −1.32996
$$999$$ 0 0
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 2205.4.a.x.1.1 2
3.2 odd 2 245.4.a.i.1.2 2
7.6 odd 2 2205.4.a.w.1.1 2
15.14 odd 2 1225.4.a.q.1.1 2
21.2 odd 6 245.4.e.k.116.1 4
21.5 even 6 245.4.e.j.116.1 4
21.11 odd 6 245.4.e.k.226.1 4
21.17 even 6 245.4.e.j.226.1 4
21.20 even 2 245.4.a.j.1.2 yes 2
105.104 even 2 1225.4.a.p.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.2 2 3.2 odd 2
245.4.a.j.1.2 yes 2 21.20 even 2
245.4.e.j.116.1 4 21.5 even 6
245.4.e.j.226.1 4 21.17 even 6
245.4.e.k.116.1 4 21.2 odd 6
245.4.e.k.226.1 4 21.11 odd 6
1225.4.a.p.1.1 2 105.104 even 2
1225.4.a.q.1.1 2 15.14 odd 2
2205.4.a.w.1.1 2 7.6 odd 2
2205.4.a.x.1.1 2 1.1 even 1 trivial