# Properties

 Label 1225.4.a.q.1.1 Level $1225$ Weight $4$ Character 1225.1 Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ 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 $$\chi$$ $$=$$ 1225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +19.7335 q^{11} +53.1662 q^{12} +71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} +8.63325 q^{18} -136.332 q^{19} -85.1821 q^{22} +100.865 q^{23} -56.8338 q^{24} -307.916 q^{26} -145.000 q^{27} -288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +98.6675 q^{33} +135.251 q^{34} -21.2665 q^{36} -309.931 q^{37} +588.496 q^{38} +356.662 q^{39} +181.662 q^{41} +18.2005 q^{43} +209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -180.000 q^{48} -156.662 q^{51} +758.496 q^{52} +127.995 q^{53} +625.911 q^{54} -681.662 q^{57} +1244.04 q^{58} -322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -425.911 q^{66} +84.3960 q^{67} -333.166 q^{68} +504.327 q^{69} -315.736 q^{71} +22.7335 q^{72} -1093.32 q^{73} +1337.86 q^{74} -1449.66 q^{76} -1539.58 q^{78} +1233.19 q^{79} -671.000 q^{81} -784.169 q^{82} -643.325 q^{83} -78.5647 q^{86} -1440.99 q^{87} -224.306 q^{88} -1140.32 q^{89} +1072.53 q^{92} +1044.99 q^{93} +637.414 q^{94} +1231.66 q^{96} +1411.99 q^{97} -39.4670 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 $$2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} + 40 q^{12} + 10 q^{13} - 72 q^{16} + 70 q^{17} + 4 q^{18} - 140 q^{19} + 22 q^{22} + 16 q^{23} - 180 q^{24} - 450 q^{26} - 290 q^{27} - 258 q^{29} + 20 q^{31} + 360 q^{32} + 330 q^{33} + 370 q^{34} - 16 q^{36} - 328 q^{37} + 580 q^{38} + 50 q^{39} - 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} - 30 q^{47} - 360 q^{48} + 350 q^{51} + 920 q^{52} - 540 q^{53} + 290 q^{54} - 700 q^{57} + 1314 q^{58} - 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} + 110 q^{66} - 468 q^{67} - 600 q^{68} + 80 q^{69} - 1056 q^{71} + 72 q^{72} - 860 q^{73} + 1296 q^{74} - 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} - 1900 q^{82} + 40 q^{83} + 148 q^{86} - 1290 q^{87} - 1364 q^{88} + 240 q^{89} + 1296 q^{92} + 100 q^{93} + 910 q^{94} + 1800 q^{96} + 1630 q^{97} - 132 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 + 40 * q^12 + 10 * q^13 - 72 * q^16 + 70 * q^17 + 4 * q^18 - 140 * q^19 + 22 * q^22 + 16 * q^23 - 180 * q^24 - 450 * q^26 - 290 * q^27 - 258 * q^29 + 20 * q^31 + 360 * q^32 + 330 * q^33 + 370 * q^34 - 16 * q^36 - 328 * q^37 + 580 * q^38 + 50 * q^39 - 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 - 30 * q^47 - 360 * q^48 + 350 * q^51 + 920 * q^52 - 540 * q^53 + 290 * q^54 - 700 * q^57 + 1314 * q^58 - 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 + 110 * q^66 - 468 * q^67 - 600 * q^68 + 80 * q^69 - 1056 * q^71 + 72 * q^72 - 860 * q^73 + 1296 * q^74 - 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 - 1900 * q^82 + 40 * q^83 + 148 * q^86 - 1290 * q^87 - 1364 * q^88 + 240 * q^89 + 1296 * q^92 + 100 * q^93 + 910 * q^94 + 1800 * q^96 + 1630 * q^97 - 132 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.31662 −1.52616 −0.763079 0.646306i $$-0.776313\pi$$
−0.763079 + 0.646306i $$0.776313\pi$$
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 10.6332 1.32916
$$5$$ 0 0
$$6$$ −21.5831 −1.46855
$$7$$ 0 0
$$8$$ −11.3668 −0.502344
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ 19.7335 0.540898 0.270449 0.962734i $$-0.412828\pi$$
0.270449 + 0.962734i $$0.412828\pi$$
$$12$$ 53.1662 1.27898
$$13$$ 71.3325 1.52185 0.760926 0.648839i $$-0.224745\pi$$
0.760926 + 0.648839i $$0.224745\pi$$
$$14$$ 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 $$0.571751\pi$$
$$18$$ 8.63325 0.113049
$$19$$ −136.332 −1.64615 −0.823074 0.567934i $$-0.807743\pi$$
−0.823074 + 0.567934i $$0.807743\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −85.1821 −0.825495
$$23$$ 100.865 0.914431 0.457215 0.889356i $$-0.348847\pi$$
0.457215 + 0.889356i $$0.348847\pi$$
$$24$$ −56.8338 −0.483381
$$25$$ 0 0
$$26$$ −307.916 −2.32259
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ −288.198 −1.84541 −0.922707 0.385501i $$-0.874029\pi$$
−0.922707 + 0.385501i $$0.874029\pi$$
$$30$$ 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
$$33$$ 98.6675 0.520479
$$34$$ 135.251 0.682214
$$35$$ 0 0
$$36$$ −21.2665 −0.0984560
$$37$$ −309.931 −1.37709 −0.688546 0.725192i $$-0.741751\pi$$
−0.688546 + 0.725192i $$0.741751\pi$$
$$38$$ 588.496 2.51228
$$39$$ 356.662 1.46440
$$40$$ 0 0
$$41$$ 181.662 0.691973 0.345987 0.938239i $$-0.387544\pi$$
0.345987 + 0.938239i $$0.387544\pi$$
$$42$$ 0 0
$$43$$ 18.2005 0.0645477 0.0322738 0.999479i $$-0.489725\pi$$
0.0322738 + 0.999479i $$0.489725\pi$$
$$44$$ 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 $$0.573591\pi$$
$$48$$ −180.000 −0.541266
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −156.662 −0.430140
$$52$$ 758.496 2.02278
$$53$$ 127.995 0.331726 0.165863 0.986149i $$-0.446959\pi$$
0.165863 + 0.986149i $$0.446959\pi$$
$$54$$ 625.911 1.57733
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −681.662 −1.58401
$$58$$ 1244.04 2.81639
$$59$$ −322.665 −0.711990 −0.355995 0.934488i $$-0.615858\pi$$
−0.355995 + 0.934488i $$0.615858\pi$$
$$60$$ 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
$$65$$ 0 0
$$66$$ −425.911 −0.794333
$$67$$ 84.3960 0.153890 0.0769449 0.997035i $$-0.475483\pi$$
0.0769449 + 0.997035i $$0.475483\pi$$
$$68$$ −333.166 −0.594152
$$69$$ 504.327 0.879911
$$70$$ 0 0
$$71$$ −315.736 −0.527760 −0.263880 0.964555i $$-0.585002\pi$$
−0.263880 + 0.964555i $$0.585002\pi$$
$$72$$ 22.7335 0.0372107
$$73$$ −1093.32 −1.75293 −0.876466 0.481464i $$-0.840105\pi$$
−0.876466 + 0.481464i $$0.840105\pi$$
$$74$$ 1337.86 2.10166
$$75$$ 0 0
$$76$$ −1449.66 −2.18799
$$77$$ 0 0
$$78$$ −1539.58 −2.23491
$$79$$ 1233.19 1.75626 0.878128 0.478426i $$-0.158793\pi$$
0.878128 + 0.478426i $$0.158793\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ −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
$$85$$ 0 0
$$86$$ −78.5647 −0.0985099
$$87$$ −1440.99 −1.77575
$$88$$ −224.306 −0.271717
$$89$$ −1140.32 −1.35813 −0.679064 0.734079i $$-0.737614\pi$$
−0.679064 + 0.734079i $$0.737614\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1072.53 1.21542
$$93$$ 1044.99 1.16516
$$94$$ 637.414 0.699407
$$95$$ 0 0
$$96$$ 1231.66 1.30944
$$97$$ 1411.99 1.47800 0.739001 0.673705i $$-0.235298\pi$$
0.739001 + 0.673705i $$0.235298\pi$$
$$98$$ 0 0
$$99$$ −39.4670 −0.0400665
$$100$$ 0 0
$$101$$ 545.673 0.537589 0.268794 0.963198i $$-0.413375\pi$$
0.268794 + 0.963198i $$0.413375\pi$$
$$102$$ 676.253 0.656461
$$103$$ 780.990 0.747119 0.373559 0.927606i $$-0.378137\pi$$
0.373559 + 0.927606i $$0.378137\pi$$
$$104$$ −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 $$0.409539\pi$$
$$108$$ −1541.82 −1.37372
$$109$$ 4.18794 0.00368011 0.00184005 0.999998i $$-0.499414\pi$$
0.00184005 + 0.999998i $$0.499414\pi$$
$$110$$ 0 0
$$111$$ −1549.66 −1.32511
$$112$$ 0 0
$$113$$ −1413.53 −1.17676 −0.588379 0.808585i $$-0.700234\pi$$
−0.588379 + 0.808585i $$0.700234\pi$$
$$114$$ 2942.48 2.41744
$$115$$ 0 0
$$116$$ −3064.48 −2.45284
$$117$$ −142.665 −0.112730
$$118$$ 1392.82 1.08661
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −941.589 −0.707430
$$122$$ 1471.98 1.09235
$$123$$ 908.312 0.665852
$$124$$ 2222.32 1.60944
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2046.26 1.42974 0.714868 0.699259i $$-0.246487\pi$$
0.714868 + 0.699259i $$0.246487\pi$$
$$128$$ 1376.13 0.950262
$$129$$ 91.0025 0.0621110
$$130$$ 0 0
$$131$$ −1140.32 −0.760534 −0.380267 0.924877i $$-0.624168\pi$$
−0.380267 + 0.924877i $$0.624168\pi$$
$$132$$ 1049.16 0.691798
$$133$$ 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 $$0.451141\pi$$
$$138$$ −2176.99 −1.34288
$$139$$ −2800.00 −1.70858 −0.854291 0.519795i $$-0.826008\pi$$
−0.854291 + 0.519795i $$0.826008\pi$$
$$140$$ 0 0
$$141$$ −738.325 −0.440980
$$142$$ 1362.91 0.805445
$$143$$ 1407.64 0.823166
$$144$$ 72.0000 0.0416667
$$145$$ 0 0
$$146$$ 4719.47 2.67525
$$147$$ 0 0
$$148$$ −3295.58 −1.83037
$$149$$ 1166.12 0.641154 0.320577 0.947222i $$-0.396123\pi$$
0.320577 + 0.947222i $$0.396123\pi$$
$$150$$ 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
$$153$$ 62.6650 0.0331122
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3792.48 1.94642
$$157$$ −1020.66 −0.518838 −0.259419 0.965765i $$-0.583531\pi$$
−0.259419 + 0.965765i $$0.583531\pi$$
$$158$$ −5323.20 −2.68032
$$159$$ 639.975 0.319203
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 2896.46 1.40473
$$163$$ 1566.02 0.752514 0.376257 0.926515i $$-0.377211\pi$$
0.376257 + 0.926515i $$0.377211\pi$$
$$164$$ 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
$$170$$ 0 0
$$171$$ 272.665 0.121937
$$172$$ 193.530 0.0857940
$$173$$ 2543.34 1.11772 0.558862 0.829260i $$-0.311238\pi$$
0.558862 + 0.829260i $$0.311238\pi$$
$$174$$ 6220.21 2.71008
$$175$$ 0 0
$$176$$ −710.406 −0.304255
$$177$$ −1613.32 −0.685113
$$178$$ 4922.32 2.07272
$$179$$ −1210.65 −0.505521 −0.252760 0.967529i $$-0.581338\pi$$
−0.252760 + 0.967529i $$0.581338\pi$$
$$180$$ 0 0
$$181$$ 3031.32 1.24484 0.622421 0.782683i $$-0.286149\pi$$
0.622421 + 0.782683i $$0.286149\pi$$
$$182$$ 0 0
$$183$$ −1705.01 −0.688733
$$184$$ −1146.51 −0.459359
$$185$$ 0 0
$$186$$ −4510.82 −1.77822
$$187$$ −618.300 −0.241789
$$188$$ −1570.16 −0.609125
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2168.64 −0.821555 −0.410778 0.911736i $$-0.634743\pi$$
−0.410778 + 0.911736i $$0.634743\pi$$
$$192$$ −3876.62 −1.45714
$$193$$ −1490.48 −0.555892 −0.277946 0.960597i $$-0.589654\pi$$
−0.277946 + 0.960597i $$0.589654\pi$$
$$194$$ −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 $$0.709356\pi$$
$$198$$ 170.364 0.0611478
$$199$$ −4595.33 −1.63696 −0.818478 0.574538i $$-0.805182\pi$$
−0.818478 + 0.574538i $$0.805182\pi$$
$$200$$ 0 0
$$201$$ 421.980 0.148080
$$202$$ −2355.46 −0.820445
$$203$$ 0 0
$$204$$ −1665.83 −0.571723
$$205$$ 0 0
$$206$$ −3371.24 −1.14022
$$207$$ −201.731 −0.0677356
$$208$$ −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
$$213$$ −1578.68 −0.507837
$$214$$ −2679.16 −0.855810
$$215$$ 0 0
$$216$$ 1648.18 0.519187
$$217$$ 0 0
$$218$$ −18.0778 −0.00561643
$$219$$ −5466.62 −1.68676
$$220$$ 0 0
$$221$$ −2235.03 −0.680290
$$222$$ 6689.29 2.02232
$$223$$ 3792.97 1.13900 0.569498 0.821993i $$-0.307138\pi$$
0.569498 + 0.821993i $$0.307138\pi$$
$$224$$ 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 $$0.306260\pi$$
$$228$$ −7248.29 −2.10539
$$229$$ −354.327 −0.102247 −0.0511236 0.998692i $$-0.516280\pi$$
−0.0511236 + 0.998692i $$0.516280\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3275.87 0.927033
$$233$$ −6492.48 −1.82548 −0.912739 0.408543i $$-0.866037\pi$$
−0.912739 + 0.408543i $$0.866037\pi$$
$$234$$ 615.831 0.172043
$$235$$ 0 0
$$236$$ −3430.98 −0.946346
$$237$$ 6165.93 1.68996
$$238$$ 0 0
$$239$$ −342.688 −0.0927474 −0.0463737 0.998924i $$-0.514766\pi$$
−0.0463737 + 0.998924i $$0.514766\pi$$
$$240$$ 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
$$243$$ 560.000 0.147835
$$244$$ −3625.96 −0.951347
$$245$$ 0 0
$$246$$ −3920.84 −1.01619
$$247$$ −9724.94 −2.50519
$$248$$ −2375.62 −0.608275
$$249$$ −3216.62 −0.818656
$$250$$ 0 0
$$251$$ −3989.29 −1.00319 −0.501597 0.865101i $$-0.667254\pi$$
−0.501597 + 0.865101i $$0.667254\pi$$
$$252$$ 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 $$0.410305\pi$$
$$258$$ −392.824 −0.0947912
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 576.396 0.136697
$$262$$ 4922.32 1.16070
$$263$$ −6360.47 −1.49127 −0.745634 0.666356i $$-0.767853\pi$$
−0.745634 + 0.666356i $$0.767853\pi$$
$$264$$ −1121.53 −0.261460
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −5701.59 −1.30686
$$268$$ 897.404 0.204543
$$269$$ 991.345 0.224697 0.112348 0.993669i $$-0.464163\pi$$
0.112348 + 0.993669i $$0.464163\pi$$
$$270$$ 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
$$275$$ 0 0
$$276$$ 5362.64 1.16954
$$277$$ 3538.63 0.767566 0.383783 0.923423i $$-0.374621\pi$$
0.383783 + 0.923423i $$0.374621\pi$$
$$278$$ 12086.5 2.60756
$$279$$ −417.995 −0.0896943
$$280$$ 0 0
$$281$$ −4663.20 −0.989975 −0.494988 0.868900i $$-0.664827\pi$$
−0.494988 + 0.868900i $$0.664827\pi$$
$$282$$ 3187.07 0.673005
$$283$$ −2104.95 −0.442142 −0.221071 0.975258i $$-0.570955\pi$$
−0.221071 + 0.975258i $$0.570955\pi$$
$$284$$ −3357.30 −0.701476
$$285$$ 0 0
$$286$$ −6076.25 −1.25628
$$287$$ 0 0
$$288$$ −492.665 −0.100801
$$289$$ −3931.27 −0.800178
$$290$$ 0 0
$$291$$ 7059.96 1.42221
$$292$$ −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
$$295$$ 0 0
$$296$$ 3522.91 0.691774
$$297$$ −2861.36 −0.559033
$$298$$ −5033.69 −0.978503
$$299$$ 7194.99 1.39163
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4142.15 −0.789252
$$303$$ 2728.36 0.517295
$$304$$ 4907.97 0.925958
$$305$$ 0 0
$$306$$ −270.501 −0.0505344
$$307$$ −2672.97 −0.496920 −0.248460 0.968642i $$-0.579924\pi$$
−0.248460 + 0.968642i $$0.579924\pi$$
$$308$$ 0 0
$$309$$ 3904.95 0.718915
$$310$$ 0 0
$$311$$ 855.698 0.156020 0.0780099 0.996953i $$-0.475143\pi$$
0.0780099 + 0.996953i $$0.475143\pi$$
$$312$$ −4054.09 −0.735634
$$313$$ −3349.99 −0.604960 −0.302480 0.953156i $$-0.597815\pi$$
−0.302480 + 0.953156i $$0.597815\pi$$
$$314$$ 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 $$0.736377\pi$$
$$318$$ −2762.53 −0.487154
$$319$$ −5687.16 −0.998180
$$320$$ 0 0
$$321$$ 3103.30 0.539593
$$322$$ 0 0
$$323$$ 4271.64 0.735852
$$324$$ −7134.91 −1.22341
$$325$$ 0 0
$$326$$ −6759.90 −1.14845
$$327$$ 20.9397 0.00354119
$$328$$ −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
$$333$$ 619.863 0.102007
$$334$$ 4879.10 0.799319
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2233.98 0.361107 0.180553 0.983565i $$-0.442211\pi$$
0.180553 + 0.983565i $$0.442211\pi$$
$$338$$ −12480.8 −2.00847
$$339$$ −7067.65 −1.13234
$$340$$ 0 0
$$341$$ 4124.25 0.654958
$$342$$ −1176.99 −0.186095
$$343$$ 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 $$0.562664\pi$$
$$348$$ −15322.4 −2.36025
$$349$$ −1291.00 −0.198011 −0.0990054 0.995087i $$-0.531566\pi$$
−0.0990054 + 0.995087i $$0.531566\pi$$
$$350$$ 0 0
$$351$$ −10343.2 −1.57288
$$352$$ 4861.00 0.736058
$$353$$ −7768.64 −1.17134 −0.585670 0.810550i $$-0.699169\pi$$
−0.585670 + 0.810550i $$0.699169\pi$$
$$354$$ 6964.12 1.04559
$$355$$ 0 0
$$356$$ −12125.3 −1.80516
$$357$$ 0 0
$$358$$ 5225.92 0.771504
$$359$$ −2284.14 −0.335800 −0.167900 0.985804i $$-0.553699\pi$$
−0.167900 + 0.985804i $$0.553699\pi$$
$$360$$ 0 0
$$361$$ 11727.5 1.70980
$$362$$ −13085.1 −1.89982
$$363$$ −4707.94 −0.680725
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 7359.90 1.05112
$$367$$ 10707.0 1.52289 0.761446 0.648229i $$-0.224490\pi$$
0.761446 + 0.648229i $$0.224490\pi$$
$$368$$ −3631.16 −0.514367
$$369$$ −363.325 −0.0512573
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 11111.6 1.54868
$$373$$ 830.429 0.115276 0.0576381 0.998338i $$-0.481643\pi$$
0.0576381 + 0.998338i $$0.481643\pi$$
$$374$$ 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 $$0.384145\pi$$
$$380$$ 0 0
$$381$$ 10231.3 1.37576
$$382$$ 9361.19 1.25382
$$383$$ −11243.9 −1.50010 −0.750048 0.661383i $$-0.769970\pi$$
−0.750048 + 0.661383i $$0.769970\pi$$
$$384$$ 6880.63 0.914390
$$385$$ 0 0
$$386$$ 6433.84 0.848378
$$387$$ −36.4010 −0.00478131
$$388$$ 15014.1 1.96449
$$389$$ −8506.85 −1.10878 −0.554389 0.832258i $$-0.687048\pi$$
−0.554389 + 0.832258i $$0.687048\pi$$
$$390$$ 0 0
$$391$$ −3160.37 −0.408764
$$392$$ 0 0
$$393$$ −5701.59 −0.731824
$$394$$ 14592.6 1.86590
$$395$$ 0 0
$$396$$ −419.662 −0.0532546
$$397$$ 3123.24 0.394838 0.197419 0.980319i $$-0.436744\pi$$
0.197419 + 0.980319i $$0.436744\pi$$
$$398$$ 19836.3 2.49825
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −11255.3 −1.40166 −0.700828 0.713330i $$-0.747186\pi$$
−0.700828 + 0.713330i $$0.747186\pi$$
$$402$$ −1821.53 −0.225994
$$403$$ 14908.3 1.84277
$$404$$ 5802.27 0.714539
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6116.03 −0.744866
$$408$$ 1780.74 0.216078
$$409$$ 7919.92 0.957494 0.478747 0.877953i $$-0.341091\pi$$
0.478747 + 0.877953i $$0.341091\pi$$
$$410$$ 0 0
$$411$$ 2451.71 0.294243
$$412$$ 8304.46 0.993037
$$413$$ 0 0
$$414$$ 870.797 0.103375
$$415$$ 0 0
$$416$$ 17571.5 2.07095
$$417$$ −14000.0 −1.64408
$$418$$ 11613.1 1.35889
$$419$$ 5257.28 0.612972 0.306486 0.951875i $$-0.400847\pi$$
0.306486 + 0.951875i $$0.400847\pi$$
$$420$$ 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
$$423$$ 295.330 0.0339467
$$424$$ −1454.89 −0.166640
$$425$$ 0 0
$$426$$ 6814.57 0.775040
$$427$$ 0 0
$$428$$ 6599.63 0.745339
$$429$$ 7038.20 0.792092
$$430$$ 0 0
$$431$$ −15291.2 −1.70893 −0.854467 0.519506i $$-0.826116\pi$$
−0.854467 + 0.519506i $$0.826116\pi$$
$$432$$ 5220.00 0.581360
$$433$$ 187.260 0.0207832 0.0103916 0.999946i $$-0.496692\pi$$
0.0103916 + 0.999946i $$0.496692\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 44.5314 0.00489144
$$437$$ −13751.2 −1.50529
$$438$$ 23597.4 2.57426
$$439$$ −3587.92 −0.390073 −0.195037 0.980796i $$-0.562483\pi$$
−0.195037 + 0.980796i $$0.562483\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9647.76 1.03823
$$443$$ −4915.65 −0.527200 −0.263600 0.964632i $$-0.584910\pi$$
−0.263600 + 0.964632i $$0.584910\pi$$
$$444$$ −16477.9 −1.76128
$$445$$ 0 0
$$446$$ −16372.8 −1.73829
$$447$$ 5830.58 0.616951
$$448$$ 0 0
$$449$$ 7091.12 0.745324 0.372662 0.927967i $$-0.378445\pi$$
0.372662 + 0.927967i $$0.378445\pi$$
$$450$$ 0 0
$$451$$ 3584.84 0.374287
$$452$$ −15030.4 −1.56410
$$453$$ 4797.91 0.497628
$$454$$ −16882.2 −1.74520
$$455$$ 0 0
$$456$$ 7748.29 0.795717
$$457$$ −5051.81 −0.517098 −0.258549 0.965998i $$-0.583244\pi$$
−0.258549 + 0.965998i $$0.583244\pi$$
$$458$$ 1529.50 0.156045
$$459$$ 4543.21 0.462002
$$460$$ 0 0
$$461$$ 16681.3 1.68531 0.842653 0.538456i $$-0.180992\pi$$
0.842653 + 0.538456i $$0.180992\pi$$
$$462$$ 0 0
$$463$$ −15569.6 −1.56280 −0.781402 0.624027i $$-0.785495\pi$$
−0.781402 + 0.624027i $$0.785495\pi$$
$$464$$ 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 $$0.447269\pi$$
$$468$$ −1516.99 −0.149835
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −5103.30 −0.499252
$$472$$ 3667.65 0.357664
$$473$$ 359.160 0.0349137
$$474$$ −26616.0 −2.57914
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −255.990 −0.0245723
$$478$$ 1479.25 0.141547
$$479$$ 14607.5 1.39339 0.696695 0.717367i $$-0.254653\pi$$
0.696695 + 0.717367i $$0.254653\pi$$
$$480$$ 0 0
$$481$$ −22108.2 −2.09573
$$482$$ 9987.23 0.943789
$$483$$ 0 0
$$484$$ −10012.2 −0.940285
$$485$$ 0 0
$$486$$ −2417.31 −0.225620
$$487$$ 1879.49 0.174882 0.0874412 0.996170i $$-0.472131\pi$$
0.0874412 + 0.996170i $$0.472131\pi$$
$$488$$ 3876.09 0.359554
$$489$$ 7830.08 0.724107
$$490$$ 0 0
$$491$$ 3221.13 0.296064 0.148032 0.988983i $$-0.452706\pi$$
0.148032 + 0.988983i $$0.452706\pi$$
$$492$$ 9658.31 0.885021
$$493$$ 9029.96 0.824927
$$494$$ 41978.9 3.82332
$$495$$ 0 0
$$496$$ −7523.91 −0.681116
$$497$$ 0 0
$$498$$ 13885.0 1.24940
$$499$$ 9713.81 0.871443 0.435721 0.900082i $$-0.356493\pi$$
0.435721 + 0.900082i $$0.356493\pi$$
$$500$$ 0 0
$$501$$ −5651.52 −0.503975
$$502$$ 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
$$505$$ 0 0
$$506$$ −8591.94 −0.754858
$$507$$ 14456.6 1.26635
$$508$$ 21758.4 1.90034
$$509$$ 18974.5 1.65232 0.826158 0.563439i $$-0.190522\pi$$
0.826158 + 0.563439i $$0.190522\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −12141.6 −1.04802
$$513$$ 19768.2 1.70134
$$514$$ −9890.77 −0.848761
$$515$$ 0 0
$$516$$ 967.652 0.0825553
$$517$$ −2913.95 −0.247883
$$518$$ 0 0
$$519$$ 12716.7 1.07553
$$520$$ 0 0
$$521$$ −17523.6 −1.47355 −0.736777 0.676136i $$-0.763653\pi$$
−0.736777 + 0.676136i $$0.763653\pi$$
$$522$$ −2488.09 −0.208622
$$523$$ 15218.6 1.27239 0.636197 0.771527i $$-0.280507\pi$$
0.636197 + 0.771527i $$0.280507\pi$$
$$524$$ −12125.3 −1.01087
$$525$$ 0 0
$$526$$ 27455.8 2.27591
$$527$$ −6548.41 −0.541278
$$528$$ −3552.03 −0.292769
$$529$$ −1993.15 −0.163816
$$530$$ 0 0
$$531$$ 645.330 0.0527400
$$532$$ 0 0
$$533$$ 12958.4 1.05308
$$534$$ 24611.6 1.99447
$$535$$ 0 0
$$536$$ −959.308 −0.0773056
$$537$$ −6053.25 −0.486438
$$538$$ −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
$$543$$ 15156.6 1.19785
$$544$$ −7718.21 −0.608301
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8690.70 0.679319 0.339660 0.940548i $$-0.389688\pi$$
0.339660 + 0.940548i $$0.389688\pi$$
$$548$$ 5213.93 0.406438
$$549$$ 682.005 0.0530187
$$550$$ 0 0
$$551$$ 39290.8 3.03783
$$552$$ −5732.56 −0.442018
$$553$$ 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 $$0.409480\pi$$
$$558$$ 1804.33 0.136888
$$559$$ 1298.29 0.0982320
$$560$$ 0 0
$$561$$ −3091.50 −0.232662
$$562$$ 20129.3 1.51086
$$563$$ 12875.4 0.963825 0.481913 0.876219i $$-0.339942\pi$$
0.481913 + 0.876219i $$0.339942\pi$$
$$564$$ −7850.79 −0.586131
$$565$$ 0 0
$$566$$ 9086.28 0.674779
$$567$$ 0 0
$$568$$ 3588.89 0.265117
$$569$$ −12064.6 −0.888882 −0.444441 0.895808i $$-0.646598\pi$$
−0.444441 + 0.895808i $$0.646598\pi$$
$$570$$ 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
$$573$$ −10843.2 −0.790542
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1550.65 0.112171
$$577$$ −9846.08 −0.710394 −0.355197 0.934791i $$-0.615586\pi$$
−0.355197 + 0.934791i $$0.615586\pi$$
$$578$$ 16969.8 1.22120
$$579$$ −7452.40 −0.534907
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −30475.2 −2.17051
$$583$$ 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
$$590$$ 0 0
$$591$$ −16902.8 −1.17646
$$592$$ 11157.5 0.774615
$$593$$ −7387.25 −0.511565 −0.255782 0.966734i $$-0.582333\pi$$
−0.255782 + 0.966734i $$0.582333\pi$$
$$594$$ 12351.4 0.853172
$$595$$ 0 0
$$596$$ 12399.6 0.852194
$$597$$ −22976.6 −1.57516
$$598$$ −31058.1 −2.12384
$$599$$ 1252.73 0.0854510 0.0427255 0.999087i $$-0.486396\pi$$
0.0427255 + 0.999087i $$0.486396\pi$$
$$600$$ 0 0
$$601$$ 1800.81 0.122224 0.0611120 0.998131i $$-0.480535\pi$$
0.0611120 + 0.998131i $$0.480535\pi$$
$$602$$ 0 0
$$603$$ −168.792 −0.0113992
$$604$$ 10203.5 0.687373
$$605$$ 0 0
$$606$$ −11777.3 −0.789473
$$607$$ 2497.06 0.166973 0.0834863 0.996509i $$-0.473395\pi$$
0.0834863 + 0.996509i $$0.473395\pi$$
$$608$$ −33583.1 −2.24009
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10533.3 −0.697434
$$612$$ 666.332 0.0440113
$$613$$ 19750.8 1.30135 0.650674 0.759357i $$-0.274487\pi$$
0.650674 + 0.759357i $$0.274487\pi$$
$$614$$ 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 $$0.684612\pi$$
$$618$$ −16856.2 −1.09718
$$619$$ −26547.4 −1.72379 −0.861897 0.507084i $$-0.830723\pi$$
−0.861897 + 0.507084i $$0.830723\pi$$
$$620$$ 0 0
$$621$$ −14625.5 −0.945090
$$622$$ −3693.73 −0.238111
$$623$$ 0 0
$$624$$ −12839.8 −0.823727
$$625$$ 0 0
$$626$$ 14460.6 0.923264
$$627$$ −13451.6 −0.856786
$$628$$ −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
$$633$$ −24943.3 −1.56620
$$634$$ 32949.0 2.06399
$$635$$ 0 0
$$636$$ 6805.01 0.424271
$$637$$ 0 0
$$638$$ 24549.3 1.52338
$$639$$ 631.472 0.0390933
$$640$$ 0 0
$$641$$ 2452.41 0.151114 0.0755572 0.997141i $$-0.475926\pi$$
0.0755572 + 0.997141i $$0.475926\pi$$
$$642$$ −13395.8 −0.823504
$$643$$ 7074.97 0.433919 0.216959 0.976181i $$-0.430386\pi$$
0.216959 + 0.976181i $$0.430386\pi$$
$$644$$ 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 $$0.467630\pi$$
$$648$$ 7627.09 0.462377
$$649$$ −6367.31 −0.385114
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16651.8 1.00021
$$653$$ 23061.6 1.38204 0.691019 0.722837i $$-0.257162\pi$$
0.691019 + 0.722837i $$0.257162\pi$$
$$654$$ −90.3888 −0.00540441
$$655$$ 0 0
$$656$$ −6539.85 −0.389235
$$657$$ 2186.65 0.129847
$$658$$ 0 0
$$659$$ 1742.64 0.103010 0.0515049 0.998673i $$-0.483598\pi$$
0.0515049 + 0.998673i $$0.483598\pi$$
$$660$$ 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
$$663$$ −11175.1 −0.654609
$$664$$ 7312.51 0.427380
$$665$$ 0 0
$$666$$ −2675.72 −0.155679
$$667$$ −29069.2 −1.68750
$$668$$ −12018.8 −0.696141
$$669$$ 18964.8 1.09600
$$670$$ 0 0
$$671$$ −6729.17 −0.387149
$$672$$ 0 0
$$673$$ 10680.8 0.611760 0.305880 0.952070i $$-0.401049\pi$$
0.305880 + 0.952070i $$0.401049\pi$$
$$674$$ −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 $$0.816878\pi$$
$$678$$ 30508.4 1.72812
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 19554.8 1.10036
$$682$$ −17802.8 −0.999569
$$683$$ −10250.4 −0.574263 −0.287132 0.957891i $$-0.592702\pi$$
−0.287132 + 0.957891i $$0.592702\pi$$
$$684$$ 2899.31 0.162073
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −1771.64 −0.0983875
$$688$$ −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
$$695$$ 0 0
$$696$$ 16379.4 0.892038
$$697$$ −5691.94 −0.309322
$$698$$ 5572.77 0.302196
$$699$$ −32462.4 −1.75657
$$700$$ 0 0
$$701$$ −22086.2 −1.18999 −0.594996 0.803729i $$-0.702846\pi$$
−0.594996 + 0.803729i $$0.702846\pi$$
$$702$$ 44647.8 2.40046
$$703$$ 42253.7 2.26690
$$704$$ −15299.9 −0.819085
$$705$$ 0 0
$$706$$ 33534.3 1.78765
$$707$$ 0 0
$$708$$ −17154.9 −0.910622
$$709$$ −27878.9 −1.47675 −0.738373 0.674392i $$-0.764406\pi$$
−0.738373 + 0.674392i $$0.764406\pi$$
$$710$$ 0 0
$$711$$ −2466.37 −0.130093
$$712$$ 12961.7 0.682248
$$713$$ 21080.6 1.10726
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12873.1 −0.671916
$$717$$ −1713.44 −0.0892462
$$718$$ 9859.76 0.512483
$$719$$ −25863.3 −1.34150 −0.670750 0.741684i $$-0.734027\pi$$
−0.670750 + 0.741684i $$0.734027\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −50623.4 −2.60943
$$723$$ −11568.3 −0.595064
$$724$$ 32232.8 1.65459
$$725$$ 0 0
$$726$$ 20322.4 1.03889
$$727$$ 29157.0 1.48744 0.743722 0.668489i $$-0.233059\pi$$
0.743722 + 0.668489i $$0.233059\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ −570.267 −0.0288538
$$732$$ −18129.8 −0.915434
$$733$$ 11006.1 0.554595 0.277297 0.960784i $$-0.410561\pi$$
0.277297 + 0.960784i $$0.410561\pi$$
$$734$$ −46218.1 −2.32417
$$735$$ 0 0
$$736$$ 24846.4 1.24436
$$737$$ 1665.43 0.0832386
$$738$$ 1568.34 0.0782267
$$739$$ −37214.4 −1.85244 −0.926221 0.376982i $$-0.876962\pi$$
−0.926221 + 0.376982i $$0.876962\pi$$
$$740$$ 0 0
$$741$$ −48624.7 −2.41062
$$742$$ 0 0
$$743$$ −11214.5 −0.553730 −0.276865 0.960909i $$-0.589296\pi$$
−0.276865 + 0.960909i $$0.589296\pi$$
$$744$$ −11878.1 −0.585313
$$745$$ 0 0
$$746$$ −3584.65 −0.175930
$$747$$ 1286.65 0.0630202
$$748$$ −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
$$753$$ −19946.4 −0.965324
$$754$$ 88740.7 4.28613
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19352.8 0.929180 0.464590 0.885526i $$-0.346202\pi$$
0.464590 + 0.885526i $$0.346202\pi$$
$$758$$ −22676.0 −1.08658
$$759$$ 9952.15 0.475942
$$760$$ 0 0
$$761$$ −32383.6 −1.54258 −0.771291 0.636483i $$-0.780388\pi$$
−0.771291 + 0.636483i $$0.780388\pi$$
$$762$$ −44164.8 −2.09963
$$763$$ 0 0
$$764$$ −23059.7 −1.09198
$$765$$ 0 0
$$766$$ 48535.7 2.28938
$$767$$ −23016.5 −1.08354
$$768$$ 1311.88 0.0616385
$$769$$ 25353.9 1.18893 0.594463 0.804123i $$-0.297365\pi$$
0.594463 + 0.804123i $$0.297365\pi$$
$$770$$ 0 0
$$771$$ 11456.6 0.535148
$$772$$ −15848.6 −0.738867
$$773$$ 26117.0 1.21522 0.607610 0.794236i $$-0.292128\pi$$
0.607610 + 0.794236i $$0.292128\pi$$
$$774$$ 157.129 0.00729703
$$775$$ 0 0
$$776$$ −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
$$783$$ 41788.7 1.90729
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 24611.6 1.11688
$$787$$ −2273.38 −0.102970 −0.0514849 0.998674i $$-0.516395\pi$$
−0.0514849 + 0.998674i $$0.516395\pi$$
$$788$$ −35946.4 −1.62505
$$789$$ −31802.4 −1.43497
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 448.612 0.0201272
$$793$$ −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
$$800$$ 0 0
$$801$$ 2280.63 0.100602
$$802$$ 48585.0 2.13915
$$803$$ −21575.1 −0.948157
$$804$$ 4487.02 0.196822
$$805$$ 0 0
$$806$$ −64353.6 −2.81236
$$807$$ 4956.73 0.216214
$$808$$ −6202.52 −0.270054
$$809$$ −4317.51 −0.187634 −0.0938169 0.995589i $$-0.529907\pi$$
−0.0938169 + 0.995589i $$0.529907\pi$$
$$810$$ 0 0
$$811$$ −1286.12 −0.0556863 −0.0278432 0.999612i $$-0.508864\pi$$
−0.0278432 + 0.999612i $$0.508864\pi$$
$$812$$ 0 0
$$813$$ −3654.89 −0.157666
$$814$$ 26400.6 1.13678
$$815$$ 0 0
$$816$$ 5639.85 0.241954
$$817$$ −2481.32 −0.106255
$$818$$ −34187.3 −1.46129
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26350.2 1.12013 0.560066 0.828448i $$-0.310776\pi$$
0.560066 + 0.828448i $$0.310776\pi$$
$$822$$ −10583.1 −0.449062
$$823$$ −9820.05 −0.415924 −0.207962 0.978137i $$-0.566683\pi$$
−0.207962 + 0.978137i $$0.566683\pi$$
$$824$$ −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 $$0.279552\pi$$
$$828$$ −2145.06 −0.0900312
$$829$$ 30817.7 1.29112 0.645562 0.763708i $$-0.276623\pi$$
0.645562 + 0.763708i $$0.276623\pi$$
$$830$$ 0 0
$$831$$ 17693.1 0.738590
$$832$$ −55305.9 −2.30455
$$833$$ 0 0
$$834$$ 60432.7 2.50913
$$835$$ 0 0
$$836$$ −28606.8 −1.18348
$$837$$ −30304.6 −1.25147
$$838$$ −22693.7 −0.935491
$$839$$ 24746.0 1.01827 0.509134 0.860688i $$-0.329966\pi$$
0.509134 + 0.860688i $$0.329966\pi$$
$$840$$ 0 0
$$841$$ 58669.1 2.40556
$$842$$ −6290.88 −0.257480
$$843$$ −23316.0 −0.952604
$$844$$ −53045.7 −2.16340
$$845$$ 0 0
$$846$$ −1274.83 −0.0518079
$$847$$ 0 0
$$848$$ −4607.82 −0.186596
$$849$$ −10524.7 −0.425452
$$850$$ 0 0
$$851$$ −31261.4 −1.25926
$$852$$ −16786.5 −0.674995
$$853$$ −11812.1 −0.474135 −0.237067 0.971493i $$-0.576186\pi$$
−0.237067 + 0.971493i $$0.576186\pi$$
$$854$$ 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 $$0.654723\pi$$
$$858$$ −30381.3 −1.20886
$$859$$ −8945.58 −0.355319 −0.177660 0.984092i $$-0.556853\pi$$
−0.177660 + 0.984092i $$0.556853\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 66006.3 2.60810
$$863$$ −19313.5 −0.761806 −0.380903 0.924615i $$-0.624387\pi$$
−0.380903 + 0.924615i $$0.624387\pi$$
$$864$$ −35718.2 −1.40643
$$865$$ 0 0
$$866$$ −808.330 −0.0317184
$$867$$ −19656.4 −0.769972
$$868$$ 0 0
$$869$$ 24335.1 0.949955
$$870$$ 0 0
$$871$$ 6020.18 0.234197
$$872$$ −47.6033 −0.00184868
$$873$$ −2823.98 −0.109482
$$874$$ 59359.0 2.29731
$$875$$ 0 0
$$876$$ −58128.0 −2.24197
$$877$$ 12154.8 0.468001 0.234001 0.972236i $$-0.424818\pi$$
0.234001 + 0.972236i $$0.424818\pi$$
$$878$$ 15487.7 0.595313
$$879$$ −32973.3 −1.26526
$$880$$ 0 0
$$881$$ 29390.4 1.12394 0.561968 0.827159i $$-0.310044\pi$$
0.561968 + 0.827159i $$0.310044\pi$$
$$882$$ 0 0
$$883$$ −4180.02 −0.159308 −0.0796540 0.996823i $$-0.525382\pi$$
−0.0796540 + 0.996823i $$0.525382\pi$$
$$884$$ −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 $$0.364446\pi$$
$$888$$ 17614.6 0.665660
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −13241.2 −0.497863
$$892$$ 40331.6 1.51390
$$893$$ 20131.5 0.754397
$$894$$ −25168.4 −0.941565
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 35974.9 1.33909
$$898$$ −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
$$905$$ 0 0
$$906$$ −20710.8 −0.759458
$$907$$ 8356.11 0.305910 0.152955 0.988233i $$-0.451121\pi$$
0.152955 + 0.988233i $$0.451121\pi$$
$$908$$ 41586.3 1.51992
$$909$$ −1091.35 −0.0398214
$$910$$ 0 0
$$911$$ −4419.80 −0.160740 −0.0803701 0.996765i $$-0.525610\pi$$
−0.0803701 + 0.996765i $$0.525610\pi$$
$$912$$ 24539.8 0.891004
$$913$$ −12695.1 −0.460181
$$914$$ 21806.8 0.789173
$$915$$ 0 0
$$916$$ −3767.65 −0.135903
$$917$$ 0 0
$$918$$ −19611.3 −0.705088
$$919$$ −39257.6 −1.40913 −0.704563 0.709641i $$-0.748857\pi$$
−0.704563 + 0.709641i $$0.748857\pi$$
$$920$$ 0 0
$$921$$ −13364.8 −0.478162
$$922$$ −72007.0 −2.57204
$$923$$ −22522.2 −0.803173
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 67207.9 2.38509
$$927$$ −1561.98 −0.0553421
$$928$$ −70992.5 −2.51125
$$929$$ 13399.9 0.473235 0.236618 0.971603i $$-0.423961\pi$$
0.236618 + 0.971603i $$0.423961\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −69036.1 −2.42635
$$933$$ 4278.49 0.150130
$$934$$ −14367.2 −0.503330
$$935$$ 0 0
$$936$$ 1621.64 0.0566291
$$937$$ 27539.8 0.960176 0.480088 0.877220i $$-0.340605\pi$$
0.480088 + 0.877220i $$0.340605\pi$$
$$938$$ 0 0
$$939$$ −16749.9 −0.582123
$$940$$ 0 0
$$941$$ −14363.8 −0.497605 −0.248802 0.968554i $$-0.580037\pi$$
−0.248802 + 0.968554i $$0.580037\pi$$
$$942$$ 22029.0 0.761937
$$943$$ 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 $$0.465130\pi$$
$$948$$ 65563.8 2.24622
$$949$$ −77989.6 −2.66770
$$950$$ 0 0
$$951$$ −38165.3 −1.30136
$$952$$ 0 0
$$953$$ −958.776 −0.0325895 −0.0162948 0.999867i $$-0.505187\pi$$
−0.0162948 + 0.999867i $$0.505187\pi$$
$$954$$ 1105.01 0.0375012
$$955$$ 0 0
$$956$$ −3643.88 −0.123276
$$957$$ −28435.8 −0.960500
$$958$$ −63055.1 −2.12653
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 13888.9 0.466213
$$962$$ 95432.7 3.19842
$$963$$ −1241.32 −0.0415379
$$964$$ −24601.8 −0.821961
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40104.5 1.33369 0.666843 0.745198i $$-0.267645\pi$$
0.666843 + 0.745198i $$0.267645\pi$$
$$968$$ 10702.8 0.355373
$$969$$ 21358.2 0.708074
$$970$$ 0 0
$$971$$ −12397.4 −0.409732 −0.204866 0.978790i $$-0.565676\pi$$
−0.204866 + 0.978790i $$0.565676\pi$$
$$972$$ 5954.62 0.196496
$$973$$ 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 $$0.763525\pi$$
$$978$$ −33799.5 −1.10510
$$979$$ −22502.5 −0.734608
$$980$$ 0 0
$$981$$ −8.37588 −0.000272601 0
$$982$$ −13904.4 −0.451841
$$983$$ −7895.76 −0.256191 −0.128095 0.991762i $$-0.540886\pi$$
−0.128095 + 0.991762i $$0.540886\pi$$
$$984$$ −10324.6 −0.334487
$$985$$ 0 0
$$986$$ −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
$$993$$ −16608.5 −0.530770
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −34203.2 −1.08812
$$997$$ −6028.06 −0.191485 −0.0957425 0.995406i $$-0.530523\pi$$
−0.0957425 + 0.995406i $$0.530523\pi$$
$$998$$ −41930.9 −1.32996
$$999$$ 44940.1 1.42326
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 1225.4.a.q.1.1 2
5.4 even 2 245.4.a.i.1.2 2
7.6 odd 2 1225.4.a.p.1.1 2
15.14 odd 2 2205.4.a.x.1.1 2
35.4 even 6 245.4.e.k.226.1 4
35.9 even 6 245.4.e.k.116.1 4
35.19 odd 6 245.4.e.j.116.1 4
35.24 odd 6 245.4.e.j.226.1 4
35.34 odd 2 245.4.a.j.1.2 yes 2
105.104 even 2 2205.4.a.w.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.2 2 5.4 even 2
245.4.a.j.1.2 yes 2 35.34 odd 2
245.4.e.j.116.1 4 35.19 odd 6
245.4.e.j.226.1 4 35.24 odd 6
245.4.e.k.116.1 4 35.9 even 6
245.4.e.k.226.1 4 35.4 even 6
1225.4.a.p.1.1 2 7.6 odd 2
1225.4.a.q.1.1 2 1.1 even 1 trivial
2205.4.a.w.1.1 2 105.104 even 2
2205.4.a.x.1.1 2 15.14 odd 2