# Properties

 Label 1521.4.a.m.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-8.00000 q^{4} -5.19615 q^{5} +10.3923 q^{7} +O(q^{10})$$ $$q-8.00000 q^{4} -5.19615 q^{5} +10.3923 q^{7} +51.9615 q^{11} +64.0000 q^{16} -117.000 q^{17} +24.2487 q^{19} +41.5692 q^{20} +18.0000 q^{23} -98.0000 q^{25} -83.1384 q^{28} +99.0000 q^{29} +193.990 q^{31} -54.0000 q^{35} -112.583 q^{37} +36.3731 q^{41} +82.0000 q^{43} -415.692 q^{44} -72.7461 q^{47} -235.000 q^{49} +261.000 q^{53} -270.000 q^{55} +789.815 q^{59} -719.000 q^{61} -512.000 q^{64} -703.213 q^{67} +936.000 q^{68} -467.654 q^{71} -684.160 q^{73} -193.990 q^{76} +540.000 q^{77} -440.000 q^{79} -332.554 q^{80} -1195.12 q^{83} +607.950 q^{85} +1517.28 q^{89} -144.000 q^{92} -126.000 q^{95} +1157.01 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4}+O(q^{10})$$ 2 * q - 16 * q^4 $$2 q - 16 q^{4} + 128 q^{16} - 234 q^{17} + 36 q^{23} - 196 q^{25} + 198 q^{29} - 108 q^{35} + 164 q^{43} - 470 q^{49} + 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} + 1872 q^{68} + 1080 q^{77} - 880 q^{79} - 288 q^{92} - 252 q^{95}+O(q^{100})$$ 2 * q - 16 * q^4 + 128 * q^16 - 234 * q^17 + 36 * q^23 - 196 * q^25 + 198 * q^29 - 108 * q^35 + 164 * q^43 - 470 * q^49 + 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 + 1872 * q^68 + 1080 * q^77 - 880 * q^79 - 288 * q^92 - 252 * q^95

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0
$$4$$ −8.00000 −1.00000
$$5$$ −5.19615 −0.464758 −0.232379 0.972625i $$-0.574651\pi$$
−0.232379 + 0.972625i $$0.574651\pi$$
$$6$$ 0 0
$$7$$ 10.3923 0.561132 0.280566 0.959835i $$-0.409478\pi$$
0.280566 + 0.959835i $$0.409478\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 51.9615 1.42427 0.712136 0.702042i $$-0.247728\pi$$
0.712136 + 0.702042i $$0.247728\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 64.0000 1.00000
$$17$$ −117.000 −1.66922 −0.834608 0.550845i $$-0.814306\pi$$
−0.834608 + 0.550845i $$0.814306\pi$$
$$18$$ 0 0
$$19$$ 24.2487 0.292791 0.146396 0.989226i $$-0.453233\pi$$
0.146396 + 0.989226i $$0.453233\pi$$
$$20$$ 41.5692 0.464758
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 18.0000 0.163185 0.0815926 0.996666i $$-0.473999\pi$$
0.0815926 + 0.996666i $$0.473999\pi$$
$$24$$ 0 0
$$25$$ −98.0000 −0.784000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −83.1384 −0.561132
$$29$$ 99.0000 0.633925 0.316963 0.948438i $$-0.397337\pi$$
0.316963 + 0.948438i $$0.397337\pi$$
$$30$$ 0 0
$$31$$ 193.990 1.12392 0.561961 0.827164i $$-0.310047\pi$$
0.561961 + 0.827164i $$0.310047\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −54.0000 −0.260790
$$36$$ 0 0
$$37$$ −112.583 −0.500232 −0.250116 0.968216i $$-0.580469\pi$$
−0.250116 + 0.968216i $$0.580469\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 36.3731 0.138549 0.0692746 0.997598i $$-0.477932\pi$$
0.0692746 + 0.997598i $$0.477932\pi$$
$$42$$ 0 0
$$43$$ 82.0000 0.290811 0.145406 0.989372i $$-0.453551\pi$$
0.145406 + 0.989372i $$0.453551\pi$$
$$44$$ −415.692 −1.42427
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −72.7461 −0.225768 −0.112884 0.993608i $$-0.536009\pi$$
−0.112884 + 0.993608i $$0.536009\pi$$
$$48$$ 0 0
$$49$$ −235.000 −0.685131
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 261.000 0.676436 0.338218 0.941068i $$-0.390176\pi$$
0.338218 + 0.941068i $$0.390176\pi$$
$$54$$ 0 0
$$55$$ −270.000 −0.661942
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 789.815 1.74280 0.871400 0.490574i $$-0.163213\pi$$
0.871400 + 0.490574i $$0.163213\pi$$
$$60$$ 0 0
$$61$$ −719.000 −1.50916 −0.754578 0.656210i $$-0.772158\pi$$
−0.754578 + 0.656210i $$0.772158\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −703.213 −1.28226 −0.641128 0.767434i $$-0.721533\pi$$
−0.641128 + 0.767434i $$0.721533\pi$$
$$68$$ 936.000 1.66922
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −467.654 −0.781694 −0.390847 0.920456i $$-0.627818\pi$$
−0.390847 + 0.920456i $$0.627818\pi$$
$$72$$ 0 0
$$73$$ −684.160 −1.09692 −0.548458 0.836178i $$-0.684785\pi$$
−0.548458 + 0.836178i $$0.684785\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −193.990 −0.292791
$$77$$ 540.000 0.799204
$$78$$ 0 0
$$79$$ −440.000 −0.626631 −0.313316 0.949649i $$-0.601440\pi$$
−0.313316 + 0.949649i $$0.601440\pi$$
$$80$$ −332.554 −0.464758
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1195.12 −1.58049 −0.790247 0.612789i $$-0.790048\pi$$
−0.790247 + 0.612789i $$0.790048\pi$$
$$84$$ 0 0
$$85$$ 607.950 0.775781
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1517.28 1.80709 0.903545 0.428493i $$-0.140955\pi$$
0.903545 + 0.428493i $$0.140955\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −144.000 −0.163185
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −126.000 −0.136077
$$96$$ 0 0
$$97$$ 1157.01 1.21110 0.605549 0.795808i $$-0.292953\pi$$
0.605549 + 0.795808i $$0.292953\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 784.000 0.784000
$$101$$ 1575.00 1.55167 0.775833 0.630938i $$-0.217330\pi$$
0.775833 + 0.630938i $$0.217330\pi$$
$$102$$ 0 0
$$103$$ −794.000 −0.759565 −0.379782 0.925076i $$-0.624001\pi$$
−0.379782 + 0.925076i $$0.624001\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −450.000 −0.406571 −0.203286 0.979119i $$-0.565162\pi$$
−0.203286 + 0.979119i $$0.565162\pi$$
$$108$$ 0 0
$$109$$ 595.825 0.523576 0.261788 0.965125i $$-0.415688\pi$$
0.261788 + 0.965125i $$0.415688\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 665.108 0.561132
$$113$$ 1701.00 1.41608 0.708038 0.706174i $$-0.249580\pi$$
0.708038 + 0.706174i $$0.249580\pi$$
$$114$$ 0 0
$$115$$ −93.5307 −0.0758416
$$116$$ −792.000 −0.633925
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1215.90 −0.936650
$$120$$ 0 0
$$121$$ 1369.00 1.02855
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −1551.92 −1.12392
$$125$$ 1158.74 0.829128
$$126$$ 0 0
$$127$$ 1664.00 1.16265 0.581323 0.813673i $$-0.302535\pi$$
0.581323 + 0.813673i $$0.302535\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1476.00 0.984418 0.492209 0.870477i $$-0.336190\pi$$
0.492209 + 0.870477i $$0.336190\pi$$
$$132$$ 0 0
$$133$$ 252.000 0.164295
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1013.25 −0.631882 −0.315941 0.948779i $$-0.602320\pi$$
−0.315941 + 0.948779i $$0.602320\pi$$
$$138$$ 0 0
$$139$$ 1124.00 0.685874 0.342937 0.939358i $$-0.388578\pi$$
0.342937 + 0.939358i $$0.388578\pi$$
$$140$$ 432.000 0.260790
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −514.419 −0.294622
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 900.666 0.500232
$$149$$ 3268.38 1.79702 0.898510 0.438952i $$-0.144650\pi$$
0.898510 + 0.438952i $$0.144650\pi$$
$$150$$ 0 0
$$151$$ 1638.52 0.883052 0.441526 0.897248i $$-0.354437\pi$$
0.441526 + 0.897248i $$0.354437\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1008.00 −0.522352
$$156$$ 0 0
$$157$$ 1259.00 0.639995 0.319997 0.947418i $$-0.396318\pi$$
0.319997 + 0.947418i $$0.396318\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 187.061 0.0915684
$$162$$ 0 0
$$163$$ 2951.41 1.41824 0.709118 0.705089i $$-0.249093\pi$$
0.709118 + 0.705089i $$0.249093\pi$$
$$164$$ −290.985 −0.138549
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3138.48 −1.45427 −0.727133 0.686496i $$-0.759148\pi$$
−0.727133 + 0.686496i $$0.759148\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −656.000 −0.290811
$$173$$ 4266.00 1.87479 0.937393 0.348273i $$-0.113232\pi$$
0.937393 + 0.348273i $$0.113232\pi$$
$$174$$ 0 0
$$175$$ −1018.45 −0.439927
$$176$$ 3325.54 1.42427
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3006.00 1.25519 0.627595 0.778540i $$-0.284039\pi$$
0.627595 + 0.778540i $$0.284039\pi$$
$$180$$ 0 0
$$181$$ 1873.00 0.769166 0.384583 0.923090i $$-0.374345\pi$$
0.384583 + 0.923090i $$0.374345\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 585.000 0.232487
$$186$$ 0 0
$$187$$ −6079.50 −2.37742
$$188$$ 581.969 0.225768
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2736.00 1.03649 0.518246 0.855232i $$-0.326585\pi$$
0.518246 + 0.855232i $$0.326585\pi$$
$$192$$ 0 0
$$193$$ 2603.27 0.970920 0.485460 0.874259i $$-0.338652\pi$$
0.485460 + 0.874259i $$0.338652\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1880.00 0.685131
$$197$$ −3720.45 −1.34554 −0.672768 0.739853i $$-0.734895\pi$$
−0.672768 + 0.739853i $$0.734895\pi$$
$$198$$ 0 0
$$199$$ −1198.00 −0.426754 −0.213377 0.976970i $$-0.568446\pi$$
−0.213377 + 0.976970i $$0.568446\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1028.84 0.355716
$$204$$ 0 0
$$205$$ −189.000 −0.0643919
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1260.00 0.417014
$$210$$ 0 0
$$211$$ 2392.00 0.780436 0.390218 0.920722i $$-0.372400\pi$$
0.390218 + 0.920722i $$0.372400\pi$$
$$212$$ −2088.00 −0.676436
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −426.084 −0.135157
$$216$$ 0 0
$$217$$ 2016.00 0.630668
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 2160.00 0.661942
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2036.89 −0.611661 −0.305830 0.952086i $$-0.598934\pi$$
−0.305830 + 0.952086i $$0.598934\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2151.21 0.628990 0.314495 0.949259i $$-0.398165\pi$$
0.314495 + 0.949259i $$0.398165\pi$$
$$228$$ 0 0
$$229$$ −3471.03 −1.00162 −0.500812 0.865556i $$-0.666965\pi$$
−0.500812 + 0.865556i $$0.666965\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1854.00 −0.521286 −0.260643 0.965435i $$-0.583935\pi$$
−0.260643 + 0.965435i $$0.583935\pi$$
$$234$$ 0 0
$$235$$ 378.000 0.104928
$$236$$ −6318.52 −1.74280
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4458.30 1.20662 0.603312 0.797505i $$-0.293847\pi$$
0.603312 + 0.797505i $$0.293847\pi$$
$$240$$ 0 0
$$241$$ −417.424 −0.111571 −0.0557856 0.998443i $$-0.517766\pi$$
−0.0557856 + 0.998443i $$0.517766\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 5752.00 1.50916
$$245$$ 1221.10 0.318420
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4104.00 1.03204 0.516020 0.856576i $$-0.327413\pi$$
0.516020 + 0.856576i $$0.327413\pi$$
$$252$$ 0 0
$$253$$ 935.307 0.232420
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 1989.00 0.482764 0.241382 0.970430i $$-0.422399\pi$$
0.241382 + 0.970430i $$0.422399\pi$$
$$258$$ 0 0
$$259$$ −1170.00 −0.280696
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −738.000 −0.173031 −0.0865153 0.996251i $$-0.527573\pi$$
−0.0865153 + 0.996251i $$0.527573\pi$$
$$264$$ 0 0
$$265$$ −1356.20 −0.314379
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 5625.70 1.28226
$$269$$ 2106.00 0.477342 0.238671 0.971100i $$-0.423288\pi$$
0.238671 + 0.971100i $$0.423288\pi$$
$$270$$ 0 0
$$271$$ −685.892 −0.153745 −0.0768727 0.997041i $$-0.524493\pi$$
−0.0768727 + 0.997041i $$0.524493\pi$$
$$272$$ −7488.00 −1.66922
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −5092.23 −1.11663
$$276$$ 0 0
$$277$$ −3665.00 −0.794977 −0.397488 0.917607i $$-0.630118\pi$$
−0.397488 + 0.917607i $$0.630118\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1719.93 −0.365132 −0.182566 0.983194i $$-0.558440\pi$$
−0.182566 + 0.983194i $$0.558440\pi$$
$$282$$ 0 0
$$283$$ 1826.00 0.383549 0.191775 0.981439i $$-0.438576\pi$$
0.191775 + 0.981439i $$0.438576\pi$$
$$284$$ 3741.23 0.781694
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 378.000 0.0777444
$$288$$ 0 0
$$289$$ 8776.00 1.78628
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 5473.28 1.09692
$$293$$ 504.027 0.100497 0.0502484 0.998737i $$-0.483999\pi$$
0.0502484 + 0.998737i $$0.483999\pi$$
$$294$$ 0 0
$$295$$ −4104.00 −0.809980
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 852.169 0.163183
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 1551.92 0.292791
$$305$$ 3736.03 0.701392
$$306$$ 0 0
$$307$$ −1950.29 −0.362570 −0.181285 0.983431i $$-0.558026\pi$$
−0.181285 + 0.983431i $$0.558026\pi$$
$$308$$ −4320.00 −0.799204
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −3798.00 −0.692491 −0.346246 0.938144i $$-0.612544\pi$$
−0.346246 + 0.938144i $$0.612544\pi$$
$$312$$ 0 0
$$313$$ 1378.00 0.248847 0.124424 0.992229i $$-0.460292\pi$$
0.124424 + 0.992229i $$0.460292\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 3520.00 0.626631
$$317$$ −7103.14 −1.25852 −0.629262 0.777193i $$-0.716643\pi$$
−0.629262 + 0.777193i $$0.716643\pi$$
$$318$$ 0 0
$$319$$ 5144.19 0.902882
$$320$$ 2660.43 0.464758
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2837.10 −0.488732
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −756.000 −0.126686
$$330$$ 0 0
$$331$$ −10073.6 −1.67280 −0.836398 0.548122i $$-0.815343\pi$$
−0.836398 + 0.548122i $$0.815343\pi$$
$$332$$ 9560.92 1.58049
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 3654.00 0.595938
$$336$$ 0 0
$$337$$ −9001.00 −1.45494 −0.727471 0.686138i $$-0.759305\pi$$
−0.727471 + 0.686138i $$0.759305\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −4863.60 −0.775781
$$341$$ 10080.0 1.60077
$$342$$ 0 0
$$343$$ −6006.75 −0.945581
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3294.00 −0.509600 −0.254800 0.966994i $$-0.582010\pi$$
−0.254800 + 0.966994i $$0.582010\pi$$
$$348$$ 0 0
$$349$$ 10544.7 1.61732 0.808662 0.588273i $$-0.200192\pi$$
0.808662 + 0.588273i $$0.200192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2478.56 0.373713 0.186856 0.982387i $$-0.440170\pi$$
0.186856 + 0.982387i $$0.440170\pi$$
$$354$$ 0 0
$$355$$ 2430.00 0.363299
$$356$$ −12138.2 −1.80709
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −5414.39 −0.795991 −0.397995 0.917387i $$-0.630294\pi$$
−0.397995 + 0.917387i $$0.630294\pi$$
$$360$$ 0 0
$$361$$ −6271.00 −0.914273
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3555.00 0.509801
$$366$$ 0 0
$$367$$ −9946.00 −1.41465 −0.707326 0.706888i $$-0.750099\pi$$
−0.707326 + 0.706888i $$0.750099\pi$$
$$368$$ 1152.00 0.163185
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2712.39 0.379570
$$372$$ 0 0
$$373$$ −7301.00 −1.01349 −0.506745 0.862096i $$-0.669151\pi$$
−0.506745 + 0.862096i $$0.669151\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −3422.53 −0.463862 −0.231931 0.972732i $$-0.574504\pi$$
−0.231931 + 0.972732i $$0.574504\pi$$
$$380$$ 1008.00 0.136077
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −5778.12 −0.770883 −0.385442 0.922732i $$-0.625951\pi$$
−0.385442 + 0.922732i $$0.625951\pi$$
$$384$$ 0 0
$$385$$ −2805.92 −0.371436
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −9256.08 −1.21110
$$389$$ 9153.00 1.19300 0.596498 0.802614i $$-0.296558\pi$$
0.596498 + 0.802614i $$0.296558\pi$$
$$390$$ 0 0
$$391$$ −2106.00 −0.272391
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2286.31 0.291232
$$396$$ 0 0
$$397$$ 2023.04 0.255751 0.127876 0.991790i $$-0.459184\pi$$
0.127876 + 0.991790i $$0.459184\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −6272.00 −0.784000
$$401$$ 8308.65 1.03470 0.517349 0.855774i $$-0.326919\pi$$
0.517349 + 0.855774i $$0.326919\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −12600.0 −1.55167
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −5850.00 −0.712466
$$408$$ 0 0
$$409$$ 10418.3 1.25954 0.629769 0.776782i $$-0.283150\pi$$
0.629769 + 0.776782i $$0.283150\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 6352.00 0.759565
$$413$$ 8208.00 0.977940
$$414$$ 0 0
$$415$$ 6210.00 0.734547
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4176.00 −0.486900 −0.243450 0.969913i $$-0.578279\pi$$
−0.243450 + 0.969913i $$0.578279\pi$$
$$420$$ 0 0
$$421$$ 14471.3 1.67527 0.837633 0.546233i $$-0.183939\pi$$
0.837633 + 0.546233i $$0.183939\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 11466.0 1.30867
$$426$$ 0 0
$$427$$ −7472.07 −0.846835
$$428$$ 3600.00 0.406571
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6578.33 0.735190 0.367595 0.929986i $$-0.380181\pi$$
0.367595 + 0.929986i $$0.380181\pi$$
$$432$$ 0 0
$$433$$ 6605.00 0.733062 0.366531 0.930406i $$-0.380545\pi$$
0.366531 + 0.930406i $$0.380545\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4766.60 −0.523576
$$437$$ 436.477 0.0477792
$$438$$ 0 0
$$439$$ 8542.00 0.928673 0.464336 0.885659i $$-0.346293\pi$$
0.464336 + 0.885659i $$0.346293\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14328.0 1.53667 0.768334 0.640049i $$-0.221086\pi$$
0.768334 + 0.640049i $$0.221086\pi$$
$$444$$ 0 0
$$445$$ −7884.00 −0.839859
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −5320.86 −0.561132
$$449$$ −3013.77 −0.316767 −0.158384 0.987378i $$-0.550628\pi$$
−0.158384 + 0.987378i $$0.550628\pi$$
$$450$$ 0 0
$$451$$ 1890.00 0.197332
$$452$$ −13608.0 −1.41608
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2887.33 −0.295544 −0.147772 0.989021i $$-0.547210\pi$$
−0.147772 + 0.989021i $$0.547210\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 748.246 0.0758416
$$461$$ 3600.93 0.363801 0.181900 0.983317i $$-0.441775\pi$$
0.181900 + 0.983317i $$0.441775\pi$$
$$462$$ 0 0
$$463$$ −2677.75 −0.268781 −0.134391 0.990928i $$-0.542908\pi$$
−0.134391 + 0.990928i $$0.542908\pi$$
$$464$$ 6336.00 0.633925
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 13878.0 1.37515 0.687577 0.726111i $$-0.258674\pi$$
0.687577 + 0.726111i $$0.258674\pi$$
$$468$$ 0 0
$$469$$ −7308.00 −0.719514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4260.84 0.414194
$$474$$ 0 0
$$475$$ −2376.37 −0.229548
$$476$$ 9727.20 0.936650
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1101.58 0.105079 0.0525393 0.998619i $$-0.483269\pi$$
0.0525393 + 0.998619i $$0.483269\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −10952.0 −1.02855
$$485$$ −6012.00 −0.562868
$$486$$ 0 0
$$487$$ 17123.1 1.59326 0.796632 0.604464i $$-0.206613\pi$$
0.796632 + 0.604464i $$0.206613\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 450.000 0.0413609 0.0206805 0.999786i $$-0.493417\pi$$
0.0206805 + 0.999786i $$0.493417\pi$$
$$492$$ 0 0
$$493$$ −11583.0 −1.05816
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 12415.3 1.12392
$$497$$ −4860.00 −0.438633
$$498$$ 0 0
$$499$$ 13219.0 1.18590 0.592950 0.805239i $$-0.297963\pi$$
0.592950 + 0.805239i $$0.297963\pi$$
$$500$$ −9269.94 −0.829128
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −5346.00 −0.473889 −0.236945 0.971523i $$-0.576146\pi$$
−0.236945 + 0.971523i $$0.576146\pi$$
$$504$$ 0 0
$$505$$ −8183.94 −0.721150
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −13312.0 −1.16265
$$509$$ 5866.46 0.510857 0.255428 0.966828i $$-0.417783\pi$$
0.255428 + 0.966828i $$0.417783\pi$$
$$510$$ 0 0
$$511$$ −7110.00 −0.615514
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4125.75 0.353014
$$516$$ 0 0
$$517$$ −3780.00 −0.321556
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 9657.00 0.812055 0.406028 0.913861i $$-0.366914\pi$$
0.406028 + 0.913861i $$0.366914\pi$$
$$522$$ 0 0
$$523$$ 21626.0 1.80811 0.904053 0.427421i $$-0.140578\pi$$
0.904053 + 0.427421i $$0.140578\pi$$
$$524$$ −11808.0 −0.984418
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −22696.8 −1.87607
$$528$$ 0 0
$$529$$ −11843.0 −0.973371
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2016.00 −0.164295
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 2338.27 0.188957
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12211.0 −0.975813
$$540$$ 0 0
$$541$$ 5371.09 0.426841 0.213421 0.976960i $$-0.431540\pi$$
0.213421 + 0.976960i $$0.431540\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −3096.00 −0.243336
$$546$$ 0 0
$$547$$ 16946.0 1.32460 0.662302 0.749237i $$-0.269579\pi$$
0.662302 + 0.749237i $$0.269579\pi$$
$$548$$ 8106.00 0.631882
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2400.62 0.185608
$$552$$ 0 0
$$553$$ −4572.61 −0.351623
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −8992.00 −0.685874
$$557$$ 3860.74 0.293689 0.146845 0.989160i $$-0.453088\pi$$
0.146845 + 0.989160i $$0.453088\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −3456.00 −0.260790
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 21672.0 1.62232 0.811160 0.584825i $$-0.198837\pi$$
0.811160 + 0.584825i $$0.198837\pi$$
$$564$$ 0 0
$$565$$ −8838.66 −0.658133
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1386.00 −0.102116 −0.0510581 0.998696i $$-0.516259\pi$$
−0.0510581 + 0.998696i $$0.516259\pi$$
$$570$$ 0 0
$$571$$ −1162.00 −0.0851632 −0.0425816 0.999093i $$-0.513558\pi$$
−0.0425816 + 0.999093i $$0.513558\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1764.00 −0.127937
$$576$$ 0 0
$$577$$ −8045.38 −0.580474 −0.290237 0.956955i $$-0.593734\pi$$
−0.290237 + 0.956955i $$0.593734\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 4115.35 0.294622
$$581$$ −12420.0 −0.886865
$$582$$ 0 0
$$583$$ 13562.0 0.963429
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27622.7 1.94227 0.971135 0.238530i $$-0.0766654\pi$$
0.971135 + 0.238530i $$0.0766654\pi$$
$$588$$ 0 0
$$589$$ 4704.00 0.329075
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −7205.33 −0.500232
$$593$$ −275.396 −0.0190711 −0.00953555 0.999955i $$-0.503035\pi$$
−0.00953555 + 0.999955i $$0.503035\pi$$
$$594$$ 0 0
$$595$$ 6318.00 0.435316
$$596$$ −26147.0 −1.79702
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −22356.0 −1.52494 −0.762472 0.647021i $$-0.776014\pi$$
−0.762472 + 0.647021i $$0.776014\pi$$
$$600$$ 0 0
$$601$$ −18083.0 −1.22732 −0.613661 0.789569i $$-0.710304\pi$$
−0.613661 + 0.789569i $$0.710304\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −13108.2 −0.883052
$$605$$ −7113.53 −0.478027
$$606$$ 0 0
$$607$$ 5480.00 0.366435 0.183218 0.983072i $$-0.441349\pi$$
0.183218 + 0.983072i $$0.441349\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 17737.9 1.16872 0.584362 0.811493i $$-0.301345\pi$$
0.584362 + 0.811493i $$0.301345\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9867.49 0.643842 0.321921 0.946767i $$-0.395672\pi$$
0.321921 + 0.946767i $$0.395672\pi$$
$$618$$ 0 0
$$619$$ −4115.35 −0.267221 −0.133611 0.991034i $$-0.542657\pi$$
−0.133611 + 0.991034i $$0.542657\pi$$
$$620$$ 8064.00 0.522352
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 15768.0 1.01402
$$624$$ 0 0
$$625$$ 6229.00 0.398656
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −10072.0 −0.639995
$$629$$ 13172.2 0.834995
$$630$$ 0 0
$$631$$ −12664.8 −0.799011 −0.399506 0.916731i $$-0.630818\pi$$
−0.399506 + 0.916731i $$0.630818\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8646.40 −0.540349
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −3789.00 −0.233473 −0.116737 0.993163i $$-0.537243\pi$$
−0.116737 + 0.993163i $$0.537243\pi$$
$$642$$ 0 0
$$643$$ −16911.7 −1.03722 −0.518611 0.855010i $$-0.673551\pi$$
−0.518611 + 0.855010i $$0.673551\pi$$
$$644$$ −1496.49 −0.0915684
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −27792.0 −1.68874 −0.844371 0.535759i $$-0.820026\pi$$
−0.844371 + 0.535759i $$0.820026\pi$$
$$648$$ 0 0
$$649$$ 41040.0 2.48222
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −23611.3 −1.41824
$$653$$ 594.000 0.0355973 0.0177986 0.999842i $$-0.494334\pi$$
0.0177986 + 0.999842i $$0.494334\pi$$
$$654$$ 0 0
$$655$$ −7669.52 −0.457516
$$656$$ 2327.88 0.138549
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −17748.0 −1.04911 −0.524555 0.851376i $$-0.675768\pi$$
−0.524555 + 0.851376i $$0.675768\pi$$
$$660$$ 0 0
$$661$$ −15791.1 −0.929203 −0.464601 0.885520i $$-0.653802\pi$$
−0.464601 + 0.885520i $$0.653802\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1309.43 −0.0763572
$$666$$ 0 0
$$667$$ 1782.00 0.103447
$$668$$ 25107.8 1.45427
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −37360.3 −2.14945
$$672$$ 0 0
$$673$$ −20933.0 −1.19897 −0.599486 0.800385i $$-0.704628\pi$$
−0.599486 + 0.800385i $$0.704628\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3402.00 −0.193131 −0.0965653 0.995327i $$-0.530786\pi$$
−0.0965653 + 0.995327i $$0.530786\pi$$
$$678$$ 0 0
$$679$$ 12024.0 0.679586
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −24983.1 −1.39964 −0.699818 0.714321i $$-0.746736\pi$$
−0.699818 + 0.714321i $$0.746736\pi$$
$$684$$ 0 0
$$685$$ 5265.00 0.293672
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 5248.00 0.290811
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 13866.8 0.763412 0.381706 0.924284i $$-0.375337\pi$$
0.381706 + 0.924284i $$0.375337\pi$$
$$692$$ −34128.0 −1.87479
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5840.48 −0.318765
$$696$$ 0 0
$$697$$ −4255.65 −0.231269
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 8147.57 0.439927
$$701$$ −21906.0 −1.18028 −0.590141 0.807300i $$-0.700928\pi$$
−0.590141 + 0.807300i $$0.700928\pi$$
$$702$$ 0 0
$$703$$ −2730.00 −0.146464
$$704$$ −26604.3 −1.42427
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16367.9 0.870690
$$708$$ 0 0
$$709$$ −13057.9 −0.691680 −0.345840 0.938294i $$-0.612406\pi$$
−0.345840 + 0.938294i $$0.612406\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3491.81 0.183407
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24048.0 −1.25519
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −14220.0 −0.737575 −0.368788 0.929514i $$-0.620227\pi$$
−0.368788 + 0.929514i $$0.620227\pi$$
$$720$$ 0 0
$$721$$ −8251.49 −0.426216
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −14984.0 −0.769166
$$725$$ −9702.00 −0.496998
$$726$$ 0 0
$$727$$ 5282.00 0.269462 0.134731 0.990882i $$-0.456983\pi$$
0.134731 + 0.990882i $$0.456983\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −9594.00 −0.485427
$$732$$ 0 0
$$733$$ 11419.4 0.575424 0.287712 0.957717i $$-0.407105\pi$$
0.287712 + 0.957717i $$0.407105\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36540.0 −1.82628
$$738$$ 0 0
$$739$$ 20535.2 1.02219 0.511096 0.859524i $$-0.329240\pi$$
0.511096 + 0.859524i $$0.329240\pi$$
$$740$$ −4680.00 −0.232487
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20826.2 1.02832 0.514158 0.857696i $$-0.328105\pi$$
0.514158 + 0.857696i $$0.328105\pi$$
$$744$$ 0 0
$$745$$ −16983.0 −0.835180
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 48636.0 2.37742
$$749$$ −4676.54 −0.228140
$$750$$ 0 0
$$751$$ 4834.00 0.234880 0.117440 0.993080i $$-0.462531\pi$$
0.117440 + 0.993080i $$0.462531\pi$$
$$752$$ −4655.75 −0.225768
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8514.00 −0.410406
$$756$$ 0 0
$$757$$ 9046.00 0.434323 0.217161 0.976136i $$-0.430320\pi$$
0.217161 + 0.976136i $$0.430320\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12034.3 0.573249 0.286625 0.958043i $$-0.407467\pi$$
0.286625 + 0.958043i $$0.407467\pi$$
$$762$$ 0 0
$$763$$ 6192.00 0.293795
$$764$$ −21888.0 −1.03649
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 37543.9 1.76056 0.880279 0.474457i $$-0.157355\pi$$
0.880279 + 0.474457i $$0.157355\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −20826.2 −0.970920
$$773$$ −15713.2 −0.731130 −0.365565 0.930786i $$-0.619124\pi$$
−0.365565 + 0.930786i $$0.619124\pi$$
$$774$$ 0 0
$$775$$ −19011.0 −0.881155
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 882.000 0.0405660
$$780$$ 0 0
$$781$$ −24300.0 −1.11334
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −15040.0 −0.685131
$$785$$ −6541.96 −0.297443
$$786$$ 0 0
$$787$$ −3755.09 −0.170082 −0.0850409 0.996377i $$-0.527102\pi$$
−0.0850409 + 0.996377i $$0.527102\pi$$
$$788$$ 29763.6 1.34554
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 17677.3 0.794605
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 9584.00 0.426754
$$797$$ −7830.00 −0.347996 −0.173998 0.984746i $$-0.555669\pi$$
−0.173998 + 0.984746i $$0.555669\pi$$
$$798$$ 0 0
$$799$$ 8511.30 0.376856
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −35550.0 −1.56231
$$804$$ 0 0
$$805$$ −972.000 −0.0425571
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6165.00 0.267923 0.133962 0.990987i $$-0.457230\pi$$
0.133962 + 0.990987i $$0.457230\pi$$
$$810$$ 0 0
$$811$$ −29839.8 −1.29201 −0.646003 0.763335i $$-0.723560\pi$$
−0.646003 + 0.763335i $$0.723560\pi$$
$$812$$ −8230.71 −0.355716
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −15336.0 −0.659137
$$816$$ 0 0
$$817$$ 1988.39 0.0851470
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 1512.00 0.0643919
$$821$$ 29763.6 1.26523 0.632616 0.774466i $$-0.281981\pi$$
0.632616 + 0.774466i $$0.281981\pi$$
$$822$$ 0 0
$$823$$ 8920.00 0.377803 0.188901 0.981996i $$-0.439507\pi$$
0.188901 + 0.981996i $$0.439507\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −18041.0 −0.758583 −0.379292 0.925277i $$-0.623832\pi$$
−0.379292 + 0.925277i $$0.623832\pi$$
$$828$$ 0 0
$$829$$ 21023.0 0.880771 0.440385 0.897809i $$-0.354842\pi$$
0.440385 + 0.897809i $$0.354842\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 27495.0 1.14363
$$834$$ 0 0
$$835$$ 16308.0 0.675882
$$836$$ −10080.0 −0.417014
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 22073.3 0.908288 0.454144 0.890928i $$-0.349945\pi$$
0.454144 + 0.890928i $$0.349945\pi$$
$$840$$ 0 0
$$841$$ −14588.0 −0.598139
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −19136.0 −0.780436
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14227.1 0.577152
$$848$$ 16704.0 0.676436
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2026.50 −0.0816304
$$852$$ 0 0
$$853$$ 26609.5 1.06810 0.534051 0.845452i $$-0.320669\pi$$
0.534051 + 0.845452i $$0.320669\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12771.0 −0.509042 −0.254521 0.967067i $$-0.581918\pi$$
−0.254521 + 0.967067i $$0.581918\pi$$
$$858$$ 0 0
$$859$$ 17134.0 0.680564 0.340282 0.940323i $$-0.389477\pi$$
0.340282 + 0.940323i $$0.389477\pi$$
$$860$$ 3408.68 0.135157
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 7929.33 0.312766 0.156383 0.987696i $$-0.450017\pi$$
0.156383 + 0.987696i $$0.450017\pi$$
$$864$$ 0 0
$$865$$ −22166.8 −0.871322
$$866$$ 0 0
$$867$$ 0 0
$$868$$ −16128.0 −0.630668
$$869$$ −22863.1 −0.892493
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 12042.0 0.465250
$$876$$ 0 0
$$877$$ −9864.03 −0.379800 −0.189900 0.981803i $$-0.560816\pi$$
−0.189900 + 0.981803i $$0.560816\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ −17280.0 −0.661942
$$881$$ 29169.0 1.11547 0.557735 0.830019i $$-0.311671\pi$$
0.557735 + 0.830019i $$0.311671\pi$$
$$882$$ 0 0
$$883$$ −928.000 −0.0353677 −0.0176839 0.999844i $$-0.505629\pi$$
−0.0176839 + 0.999844i $$0.505629\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 14400.0 0.545101 0.272551 0.962141i $$-0.412133\pi$$
0.272551 + 0.962141i $$0.412133\pi$$
$$888$$ 0 0
$$889$$ 17292.8 0.652398
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 16295.1 0.611661
$$893$$ −1764.00 −0.0661030
$$894$$ 0 0
$$895$$ −15619.6 −0.583360
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 19205.0 0.712483
$$900$$ 0 0
$$901$$ −30537.0 −1.12912
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9732.39 −0.357476
$$906$$ 0 0
$$907$$ −19684.0 −0.720614 −0.360307 0.932834i $$-0.617328\pi$$
−0.360307 + 0.932834i $$0.617328\pi$$
$$908$$ −17209.7 −0.628990
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24480.0 −0.890295 −0.445147 0.895457i $$-0.646849\pi$$
−0.445147 + 0.895457i $$0.646849\pi$$
$$912$$ 0 0
$$913$$ −62100.0 −2.25105
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 27768.2 1.00162
$$917$$ 15339.0 0.552388
$$918$$ 0 0
$$919$$ 38608.0 1.38581 0.692906 0.721028i $$-0.256330\pi$$
0.692906 + 0.721028i $$0.256330\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 11033.2 0.392182
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 10210.4 0.360596 0.180298 0.983612i $$-0.442294\pi$$
0.180298 + 0.983612i $$0.442294\pi$$
$$930$$ 0 0
$$931$$ −5698.45 −0.200600
$$932$$ 14832.0 0.521286
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 31590.0 1.10492
$$936$$ 0 0
$$937$$ 28495.0 0.993480 0.496740 0.867899i $$-0.334530\pi$$
0.496740 + 0.867899i $$0.334530\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −3024.00 −0.104928
$$941$$ −10724.9 −0.371541 −0.185771 0.982593i $$-0.559478\pi$$
−0.185771 + 0.982593i $$0.559478\pi$$
$$942$$ 0 0
$$943$$ 654.715 0.0226092
$$944$$ 50548.2 1.74280
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 33962.1 1.16538 0.582692 0.812693i $$-0.301999\pi$$
0.582692 + 0.812693i $$0.301999\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 23814.0 0.809456 0.404728 0.914437i $$-0.367366\pi$$
0.404728 + 0.914437i $$0.367366\pi$$
$$954$$ 0 0
$$955$$ −14216.7 −0.481718
$$956$$ −35666.4 −1.20662
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −10530.0 −0.354569
$$960$$ 0 0
$$961$$ 7841.00 0.263200
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 3339.39 0.111571
$$965$$ −13527.0 −0.451243
$$966$$ 0 0
$$967$$ 51549.3 1.71429 0.857143 0.515079i $$-0.172238\pi$$
0.857143 + 0.515079i $$0.172238\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12312.0 0.406911 0.203456 0.979084i $$-0.434783\pi$$
0.203456 + 0.979084i $$0.434783\pi$$
$$972$$ 0 0
$$973$$ 11681.0 0.384865
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −46016.0 −1.50916
$$977$$ −21538.1 −0.705285 −0.352642 0.935758i $$-0.614717\pi$$
−0.352642 + 0.935758i $$0.614717\pi$$
$$978$$ 0 0
$$979$$ 78840.0 2.57379
$$980$$ −9768.77 −0.318420
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −32611.1 −1.05812 −0.529060 0.848585i $$-0.677455\pi$$
−0.529060 + 0.848585i $$0.677455\pi$$
$$984$$ 0 0
$$985$$ 19332.0 0.625349
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1476.00 0.0474561
$$990$$ 0 0
$$991$$ −22330.0 −0.715778 −0.357889 0.933764i $$-0.616503\pi$$
−0.357889 + 0.933764i $$0.616503\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 6224.99 0.198337
$$996$$ 0 0
$$997$$ −24931.0 −0.791949 −0.395974 0.918262i $$-0.629593\pi$$
−0.395974 + 0.918262i $$0.629593\pi$$
$$998$$ 0 0
$$999$$ 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 1521.4.a.m.1.1 2
3.2 odd 2 507.4.a.g.1.2 2
13.2 odd 12 117.4.q.b.82.1 2
13.7 odd 12 117.4.q.b.10.1 2
13.12 even 2 inner 1521.4.a.m.1.2 2
39.2 even 12 39.4.j.a.4.1 2
39.5 even 4 507.4.b.a.337.1 2
39.8 even 4 507.4.b.a.337.2 2
39.20 even 12 39.4.j.a.10.1 yes 2
39.38 odd 2 507.4.a.g.1.1 2
156.59 odd 12 624.4.bv.a.49.1 2
156.119 odd 12 624.4.bv.a.433.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.a.4.1 2 39.2 even 12
39.4.j.a.10.1 yes 2 39.20 even 12
117.4.q.b.10.1 2 13.7 odd 12
117.4.q.b.82.1 2 13.2 odd 12
507.4.a.g.1.1 2 39.38 odd 2
507.4.a.g.1.2 2 3.2 odd 2
507.4.b.a.337.1 2 39.5 even 4
507.4.b.a.337.2 2 39.8 even 4
624.4.bv.a.49.1 2 156.59 odd 12
624.4.bv.a.433.1 2 156.119 odd 12
1521.4.a.m.1.1 2 1.1 even 1 trivial
1521.4.a.m.1.2 2 13.12 even 2 inner