# Properties

 Label 2205.4.a.w.1.2 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ Analytic rank $0$ 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: $$0$$ 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 1.2 Root $$3.31662$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.31662 q^{2} -2.63325 q^{4} -5.00000 q^{5} -24.6332 q^{8} +O(q^{10})$$ $$q+2.31662 q^{2} -2.63325 q^{4} -5.00000 q^{5} -24.6332 q^{8} -11.5831 q^{10} -46.2665 q^{11} -61.3325 q^{13} -36.0000 q^{16} -101.332 q^{17} +3.66750 q^{19} +13.1662 q^{20} -107.182 q^{22} -84.8655 q^{23} +25.0000 q^{25} -142.084 q^{26} -30.1980 q^{29} +188.997 q^{31} +113.668 q^{32} -234.749 q^{34} +18.0685 q^{37} +8.49623 q^{38} +123.166 q^{40} -481.662 q^{41} -97.7995 q^{43} +121.831 q^{44} -196.602 q^{46} -117.665 q^{47} +57.9156 q^{50} +161.504 q^{52} -667.995 q^{53} +231.332 q^{55} -69.9574 q^{58} -57.3350 q^{59} +738.997 q^{61} +437.836 q^{62} +551.325 q^{64} +306.662 q^{65} +552.396 q^{67} +266.834 q^{68} +740.264 q^{71} +233.325 q^{73} +41.8580 q^{74} -9.65745 q^{76} -1075.19 q^{79} +180.000 q^{80} -1115.83 q^{82} -683.325 q^{83} +506.662 q^{85} -226.565 q^{86} +1139.69 q^{88} +1380.32 q^{89} +223.472 q^{92} -272.586 q^{94} -18.3375 q^{95} +218.008 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 8 q^{4} - 10 q^{5} - 36 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.31662 0.819051 0.409525 0.912299i $$-0.365694\pi$$
0.409525 + 0.912299i $$0.365694\pi$$
$$3$$ 0 0
$$4$$ −2.63325 −0.329156
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −24.6332 −1.08865
$$9$$ 0 0
$$10$$ −11.5831 −0.366291
$$11$$ −46.2665 −1.26817 −0.634085 0.773263i $$-0.718623\pi$$
−0.634085 + 0.773263i $$0.718623\pi$$
$$12$$ 0 0
$$13$$ −61.3325 −1.30851 −0.654253 0.756276i $$-0.727017\pi$$
−0.654253 + 0.756276i $$0.727017\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −36.0000 −0.562500
$$17$$ −101.332 −1.44569 −0.722845 0.691010i $$-0.757166\pi$$
−0.722845 + 0.691010i $$0.757166\pi$$
$$18$$ 0 0
$$19$$ 3.66750 0.0442833 0.0221417 0.999755i $$-0.492952\pi$$
0.0221417 + 0.999755i $$0.492952\pi$$
$$20$$ 13.1662 0.147203
$$21$$ 0 0
$$22$$ −107.182 −1.03870
$$23$$ −84.8655 −0.769377 −0.384689 0.923046i $$-0.625691\pi$$
−0.384689 + 0.923046i $$0.625691\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −142.084 −1.07173
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −30.1980 −0.193366 −0.0966832 0.995315i $$-0.530823\pi$$
−0.0966832 + 0.995315i $$0.530823\pi$$
$$30$$ 0 0
$$31$$ 188.997 1.09500 0.547499 0.836806i $$-0.315580\pi$$
0.547499 + 0.836806i $$0.315580\pi$$
$$32$$ 113.668 0.627930
$$33$$ 0 0
$$34$$ −234.749 −1.18409
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 18.0685 0.0802823 0.0401411 0.999194i $$-0.487219\pi$$
0.0401411 + 0.999194i $$0.487219\pi$$
$$38$$ 8.49623 0.0362703
$$39$$ 0 0
$$40$$ 123.166 0.486857
$$41$$ −481.662 −1.83471 −0.917354 0.398072i $$-0.869679\pi$$
−0.917354 + 0.398072i $$0.869679\pi$$
$$42$$ 0 0
$$43$$ −97.7995 −0.346844 −0.173422 0.984848i $$-0.555482\pi$$
−0.173422 + 0.984848i $$0.555482\pi$$
$$44$$ 121.831 0.417426
$$45$$ 0 0
$$46$$ −196.602 −0.630159
$$47$$ −117.665 −0.365175 −0.182587 0.983190i $$-0.558447\pi$$
−0.182587 + 0.983190i $$0.558447\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 57.9156 0.163810
$$51$$ 0 0
$$52$$ 161.504 0.430703
$$53$$ −667.995 −1.73125 −0.865624 0.500694i $$-0.833078\pi$$
−0.865624 + 0.500694i $$0.833078\pi$$
$$54$$ 0 0
$$55$$ 231.332 0.567143
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −69.9574 −0.158377
$$59$$ −57.3350 −0.126515 −0.0632575 0.997997i $$-0.520149\pi$$
−0.0632575 + 0.997997i $$0.520149\pi$$
$$60$$ 0 0
$$61$$ 738.997 1.55113 0.775565 0.631268i $$-0.217465\pi$$
0.775565 + 0.631268i $$0.217465\pi$$
$$62$$ 437.836 0.896859
$$63$$ 0 0
$$64$$ 551.325 1.07681
$$65$$ 306.662 0.585182
$$66$$ 0 0
$$67$$ 552.396 1.00725 0.503626 0.863922i $$-0.331999\pi$$
0.503626 + 0.863922i $$0.331999\pi$$
$$68$$ 266.834 0.475858
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 740.264 1.23737 0.618684 0.785640i $$-0.287666\pi$$
0.618684 + 0.785640i $$0.287666\pi$$
$$72$$ 0 0
$$73$$ 233.325 0.374091 0.187045 0.982351i $$-0.440109\pi$$
0.187045 + 0.982351i $$0.440109\pi$$
$$74$$ 41.8580 0.0657553
$$75$$ 0 0
$$76$$ −9.65745 −0.0145761
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1075.19 −1.53124 −0.765619 0.643294i $$-0.777567\pi$$
−0.765619 + 0.643294i $$0.777567\pi$$
$$80$$ 180.000 0.251558
$$81$$ 0 0
$$82$$ −1115.83 −1.50272
$$83$$ −683.325 −0.903671 −0.451835 0.892101i $$-0.649231\pi$$
−0.451835 + 0.892101i $$0.649231\pi$$
$$84$$ 0 0
$$85$$ 506.662 0.646532
$$86$$ −226.565 −0.284083
$$87$$ 0 0
$$88$$ 1139.69 1.38059
$$89$$ 1380.32 1.64397 0.821985 0.569509i $$-0.192867\pi$$
0.821985 + 0.569509i $$0.192867\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 223.472 0.253245
$$93$$ 0 0
$$94$$ −272.586 −0.299096
$$95$$ −18.3375 −0.0198041
$$96$$ 0 0
$$97$$ 218.008 0.228199 0.114100 0.993469i $$-0.463602\pi$$
0.114100 + 0.993469i $$0.463602\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −65.8312 −0.0658312
$$101$$ 1474.33 1.45249 0.726243 0.687438i $$-0.241265\pi$$
0.726243 + 0.687438i $$0.241265\pi$$
$$102$$ 0 0
$$103$$ −810.990 −0.775818 −0.387909 0.921698i $$-0.626802\pi$$
−0.387909 + 0.921698i $$0.626802\pi$$
$$104$$ 1510.82 1.42450
$$105$$ 0 0
$$106$$ −1547.49 −1.41798
$$107$$ −440.660 −0.398133 −0.199066 0.979986i $$-0.563791\pi$$
−0.199066 + 0.979986i $$0.563791\pi$$
$$108$$ 0 0
$$109$$ −1906.19 −1.67504 −0.837522 0.546404i $$-0.815996\pi$$
−0.837522 + 0.546404i $$0.815996\pi$$
$$110$$ 535.911 0.464519
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −962.470 −0.801252 −0.400626 0.916242i $$-0.631207\pi$$
−0.400626 + 0.916242i $$0.631207\pi$$
$$114$$ 0 0
$$115$$ 424.327 0.344076
$$116$$ 79.5188 0.0636478
$$117$$ 0 0
$$118$$ −132.824 −0.103622
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 809.589 0.608256
$$122$$ 1711.98 1.27045
$$123$$ 0 0
$$124$$ −497.678 −0.360426
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1621.74 −1.13312 −0.566558 0.824022i $$-0.691725\pi$$
−0.566558 + 0.824022i $$0.691725\pi$$
$$128$$ 367.873 0.254029
$$129$$ 0 0
$$130$$ 710.422 0.479293
$$131$$ 1380.32 0.920602 0.460301 0.887763i $$-0.347741\pi$$
0.460301 + 0.887763i $$0.347741\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1279.69 0.824991
$$135$$ 0 0
$$136$$ 2496.15 1.57385
$$137$$ 1949.66 1.21584 0.607921 0.793997i $$-0.292004\pi$$
0.607921 + 0.793997i $$0.292004\pi$$
$$138$$ 0 0
$$139$$ 2800.00 1.70858 0.854291 0.519795i $$-0.173992\pi$$
0.854291 + 0.519795i $$0.173992\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1714.91 1.01347
$$143$$ 2837.64 1.65941
$$144$$ 0 0
$$145$$ 150.990 0.0864761
$$146$$ 540.526 0.306399
$$147$$ 0 0
$$148$$ −47.5789 −0.0264254
$$149$$ 1434.12 0.788506 0.394253 0.919002i $$-0.371003\pi$$
0.394253 + 0.919002i $$0.371003\pi$$
$$150$$ 0 0
$$151$$ −1985.58 −1.07009 −0.535047 0.844822i $$-0.679706\pi$$
−0.535047 + 0.844822i $$0.679706\pi$$
$$152$$ −90.3425 −0.0482089
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −944.987 −0.489698
$$156$$ 0 0
$$157$$ 40.6600 0.0206689 0.0103345 0.999947i $$-0.496710\pi$$
0.0103345 + 0.999947i $$0.496710\pi$$
$$158$$ −2490.80 −1.25416
$$159$$ 0 0
$$160$$ −568.338 −0.280819
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3953.98 −1.90000 −0.950000 0.312250i $$-0.898917\pi$$
−0.950000 + 0.312250i $$0.898917\pi$$
$$164$$ 1268.34 0.603906
$$165$$ 0 0
$$166$$ −1583.01 −0.740152
$$167$$ −3380.30 −1.56632 −0.783161 0.621819i $$-0.786394\pi$$
−0.783161 + 0.621819i $$0.786394\pi$$
$$168$$ 0 0
$$169$$ 1564.68 0.712187
$$170$$ 1173.75 0.529543
$$171$$ 0 0
$$172$$ 257.530 0.114166
$$173$$ −3206.66 −1.40924 −0.704619 0.709586i $$-0.748882\pi$$
−0.704619 + 0.709586i $$0.748882\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1665.59 0.713346
$$177$$ 0 0
$$178$$ 3197.68 1.34649
$$179$$ −1442.65 −0.602395 −0.301198 0.953562i $$-0.597386\pi$$
−0.301198 + 0.953562i $$0.597386\pi$$
$$180$$ 0 0
$$181$$ −908.680 −0.373158 −0.186579 0.982440i $$-0.559740\pi$$
−0.186579 + 0.982440i $$0.559740\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2090.51 0.837580
$$185$$ −90.3425 −0.0359033
$$186$$ 0 0
$$187$$ 4688.30 1.83338
$$188$$ 309.841 0.120199
$$189$$ 0 0
$$190$$ −42.4812 −0.0162206
$$191$$ −2474.64 −0.937479 −0.468739 0.883336i $$-0.655292\pi$$
−0.468739 + 0.883336i $$0.655292\pi$$
$$192$$ 0 0
$$193$$ 3533.52 1.31787 0.658934 0.752201i $$-0.271008\pi$$
0.658934 + 0.752201i $$0.271008\pi$$
$$194$$ 505.042 0.186907
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1952.57 0.706165 0.353083 0.935592i $$-0.385133\pi$$
0.353083 + 0.935592i $$0.385133\pi$$
$$198$$ 0 0
$$199$$ 4064.67 1.44792 0.723962 0.689840i $$-0.242319\pi$$
0.723962 + 0.689840i $$0.242319\pi$$
$$200$$ −615.831 −0.217729
$$201$$ 0 0
$$202$$ 3415.46 1.18966
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2408.31 0.820507
$$206$$ −1878.76 −0.635434
$$207$$ 0 0
$$208$$ 2207.97 0.736034
$$209$$ −169.683 −0.0561588
$$210$$ 0 0
$$211$$ −4325.34 −1.41123 −0.705613 0.708598i $$-0.749328\pi$$
−0.705613 + 0.708598i $$0.749328\pi$$
$$212$$ 1759.00 0.569851
$$213$$ 0 0
$$214$$ −1020.84 −0.326091
$$215$$ 488.997 0.155113
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −4415.92 −1.37194
$$219$$ 0 0
$$220$$ −609.156 −0.186679
$$221$$ 6214.97 1.89169
$$222$$ 0 0
$$223$$ −982.970 −0.295177 −0.147589 0.989049i $$-0.547151\pi$$
−0.147589 + 0.989049i $$0.547151\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2229.68 −0.656266
$$227$$ 1660.96 0.485648 0.242824 0.970070i $$-0.421926\pi$$
0.242824 + 0.970070i $$0.421926\pi$$
$$228$$ 0 0
$$229$$ −574.327 −0.165732 −0.0828660 0.996561i $$-0.526407\pi$$
−0.0828660 + 0.996561i $$0.526407\pi$$
$$230$$ 983.008 0.281816
$$231$$ 0 0
$$232$$ 743.875 0.210508
$$233$$ 2316.48 0.651320 0.325660 0.945487i $$-0.394414\pi$$
0.325660 + 0.945487i $$0.394414\pi$$
$$234$$ 0 0
$$235$$ 588.325 0.163311
$$236$$ 150.977 0.0416432
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3659.31 0.990382 0.495191 0.868784i $$-0.335098\pi$$
0.495191 + 0.868784i $$0.335098\pi$$
$$240$$ 0 0
$$241$$ 2446.33 0.653868 0.326934 0.945047i $$-0.393985\pi$$
0.326934 + 0.945047i $$0.393985\pi$$
$$242$$ 1875.51 0.498193
$$243$$ 0 0
$$244$$ −1945.96 −0.510564
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −224.937 −0.0579450
$$248$$ −4655.62 −1.19207
$$249$$ 0 0
$$250$$ −289.578 −0.0732581
$$251$$ 2909.29 0.731605 0.365802 0.930693i $$-0.380795\pi$$
0.365802 + 0.930693i $$0.380795\pi$$
$$252$$ 0 0
$$253$$ 3926.43 0.975702
$$254$$ −3756.95 −0.928080
$$255$$ 0 0
$$256$$ −3558.38 −0.868744
$$257$$ −168.680 −0.0409415 −0.0204708 0.999790i $$-0.506517\pi$$
−0.0204708 + 0.999790i $$0.506517\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −807.519 −0.192616
$$261$$ 0 0
$$262$$ 3197.68 0.754020
$$263$$ 3244.47 0.760695 0.380347 0.924844i $$-0.375804\pi$$
0.380347 + 0.924844i $$0.375804\pi$$
$$264$$ 0 0
$$265$$ 3339.97 0.774238
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1454.60 −0.331543
$$269$$ 2848.65 0.645671 0.322836 0.946455i $$-0.395364\pi$$
0.322836 + 0.946455i $$0.395364\pi$$
$$270$$ 0 0
$$271$$ −2850.98 −0.639057 −0.319529 0.947577i $$-0.603525\pi$$
−0.319529 + 0.947577i $$0.603525\pi$$
$$272$$ 3647.97 0.813201
$$273$$ 0 0
$$274$$ 4516.62 0.995837
$$275$$ −1156.66 −0.253634
$$276$$ 0 0
$$277$$ 2298.63 0.498597 0.249298 0.968427i $$-0.419800\pi$$
0.249298 + 0.968427i $$0.419800\pi$$
$$278$$ 6486.55 1.39942
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6109.20 −1.29695 −0.648477 0.761234i $$-0.724594\pi$$
−0.648477 + 0.761234i $$0.724594\pi$$
$$282$$ 0 0
$$283$$ 5854.95 1.22983 0.614913 0.788595i $$-0.289191\pi$$
0.614913 + 0.788595i $$0.289191\pi$$
$$284$$ −1949.30 −0.407288
$$285$$ 0 0
$$286$$ 6573.75 1.35914
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5355.27 1.09002
$$290$$ 349.787 0.0708283
$$291$$ 0 0
$$292$$ −614.403 −0.123134
$$293$$ 5135.34 1.02392 0.511962 0.859008i $$-0.328919\pi$$
0.511962 + 0.859008i $$0.328919\pi$$
$$294$$ 0 0
$$295$$ 286.675 0.0565792
$$296$$ −445.086 −0.0873990
$$297$$ 0 0
$$298$$ 3322.31 0.645827
$$299$$ 5205.01 1.00673
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4599.85 −0.876462
$$303$$ 0 0
$$304$$ −132.030 −0.0249094
$$305$$ −3694.99 −0.693686
$$306$$ 0 0
$$307$$ 2102.97 0.390954 0.195477 0.980708i $$-0.437375\pi$$
0.195477 + 0.980708i $$0.437375\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2189.18 −0.401088
$$311$$ 5764.30 1.05101 0.525504 0.850791i $$-0.323877\pi$$
0.525504 + 0.850791i $$0.323877\pi$$
$$312$$ 0 0
$$313$$ −1360.01 −0.245599 −0.122799 0.992432i $$-0.539187\pi$$
−0.122799 + 0.992432i $$0.539187\pi$$
$$314$$ 94.1939 0.0169289
$$315$$ 0 0
$$316$$ 2831.23 0.504017
$$317$$ −5138.95 −0.910511 −0.455256 0.890361i $$-0.650452\pi$$
−0.455256 + 0.890361i $$0.650452\pi$$
$$318$$ 0 0
$$319$$ 1397.16 0.245222
$$320$$ −2756.62 −0.481563
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −371.637 −0.0640200
$$324$$ 0 0
$$325$$ −1533.31 −0.261701
$$326$$ −9159.90 −1.55620
$$327$$ 0 0
$$328$$ 11864.9 1.99735
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1825.70 0.303171 0.151585 0.988444i $$-0.451562\pi$$
0.151585 + 0.988444i $$0.451562\pi$$
$$332$$ 1799.37 0.297449
$$333$$ 0 0
$$334$$ −7830.90 −1.28290
$$335$$ −2761.98 −0.450457
$$336$$ 0 0
$$337$$ 153.985 0.0248905 0.0124452 0.999923i $$-0.496038\pi$$
0.0124452 + 0.999923i $$0.496038\pi$$
$$338$$ 3624.76 0.583317
$$339$$ 0 0
$$340$$ −1334.17 −0.212810
$$341$$ −8744.25 −1.38864
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2409.12 0.377590
$$345$$ 0 0
$$346$$ −7428.63 −1.15424
$$347$$ −4359.39 −0.674421 −0.337211 0.941429i $$-0.609483\pi$$
−0.337211 + 0.941429i $$0.609483\pi$$
$$348$$ 0 0
$$349$$ 1689.00 0.259054 0.129527 0.991576i $$-0.458654\pi$$
0.129527 + 0.991576i $$0.458654\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5259.00 −0.796322
$$353$$ 3921.36 0.591254 0.295627 0.955303i $$-0.404471\pi$$
0.295627 + 0.955303i $$0.404471\pi$$
$$354$$ 0 0
$$355$$ −3701.32 −0.553368
$$356$$ −3634.72 −0.541123
$$357$$ 0 0
$$358$$ −3342.08 −0.493392
$$359$$ 2867.86 0.421616 0.210808 0.977528i $$-0.432391\pi$$
0.210808 + 0.977528i $$0.432391\pi$$
$$360$$ 0 0
$$361$$ −6845.55 −0.998039
$$362$$ −2105.07 −0.305636
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1166.62 −0.167298
$$366$$ 0 0
$$367$$ 11503.0 1.63611 0.818054 0.575142i $$-0.195053\pi$$
0.818054 + 0.575142i $$0.195053\pi$$
$$368$$ 3055.16 0.432775
$$369$$ 0 0
$$370$$ −209.290 −0.0294066
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5086.43 0.706073 0.353037 0.935610i $$-0.385149\pi$$
0.353037 + 0.935610i $$0.385149\pi$$
$$374$$ 10861.0 1.50163
$$375$$ 0 0
$$376$$ 2898.47 0.397546
$$377$$ 1852.12 0.253021
$$378$$ 0 0
$$379$$ 954.827 0.129409 0.0647047 0.997904i $$-0.479389\pi$$
0.0647047 + 0.997904i $$0.479389\pi$$
$$380$$ 48.2873 0.00651864
$$381$$ 0 0
$$382$$ −5732.81 −0.767843
$$383$$ −3083.91 −0.411437 −0.205719 0.978611i $$-0.565953\pi$$
−0.205719 + 0.978611i $$0.565953\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8185.84 1.07940
$$387$$ 0 0
$$388$$ −574.068 −0.0751131
$$389$$ 6331.15 0.825198 0.412599 0.910913i $$-0.364621\pi$$
0.412599 + 0.910913i $$0.364621\pi$$
$$390$$ 0 0
$$391$$ 8599.63 1.11228
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4523.36 0.578385
$$395$$ 5375.93 0.684791
$$396$$ 0 0
$$397$$ −12133.2 −1.53388 −0.766939 0.641720i $$-0.778221\pi$$
−0.766939 + 0.641720i $$0.778221\pi$$
$$398$$ 9416.32 1.18592
$$399$$ 0 0
$$400$$ −900.000 −0.112500
$$401$$ 270.669 0.0337072 0.0168536 0.999858i $$-0.494635\pi$$
0.0168536 + 0.999858i $$0.494635\pi$$
$$402$$ 0 0
$$403$$ −11591.7 −1.43281
$$404$$ −3882.27 −0.478095
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −835.967 −0.101812
$$408$$ 0 0
$$409$$ 4019.92 0.485996 0.242998 0.970027i $$-0.421869\pi$$
0.242998 + 0.970027i $$0.421869\pi$$
$$410$$ 5579.16 0.672036
$$411$$ 0 0
$$412$$ 2135.54 0.255365
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3416.62 0.404134
$$416$$ −6971.51 −0.821650
$$417$$ 0 0
$$418$$ −393.091 −0.0459969
$$419$$ −2437.28 −0.284175 −0.142087 0.989854i $$-0.545381\pi$$
−0.142087 + 0.989854i $$0.545381\pi$$
$$420$$ 0 0
$$421$$ −4751.36 −0.550041 −0.275020 0.961438i $$-0.588685\pi$$
−0.275020 + 0.961438i $$0.588685\pi$$
$$422$$ −10020.2 −1.15586
$$423$$ 0 0
$$424$$ 16454.9 1.88472
$$425$$ −2533.31 −0.289138
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1160.37 0.131048
$$429$$ 0 0
$$430$$ 1132.82 0.127046
$$431$$ −7925.19 −0.885714 −0.442857 0.896592i $$-0.646035\pi$$
−0.442857 + 0.896592i $$0.646035\pi$$
$$432$$ 0 0
$$433$$ −11487.3 −1.27492 −0.637462 0.770481i $$-0.720016\pi$$
−0.637462 + 0.770481i $$0.720016\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 5019.47 0.551351
$$437$$ −311.245 −0.0340706
$$438$$ 0 0
$$439$$ −9147.92 −0.994548 −0.497274 0.867594i $$-0.665666\pi$$
−0.497274 + 0.867594i $$0.665666\pi$$
$$440$$ −5698.47 −0.617418
$$441$$ 0 0
$$442$$ 14397.8 1.54939
$$443$$ −1864.35 −0.199950 −0.0999752 0.994990i $$-0.531876\pi$$
−0.0999752 + 0.994990i $$0.531876\pi$$
$$444$$ 0 0
$$445$$ −6901.59 −0.735206
$$446$$ −2277.17 −0.241765
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4490.88 −0.472022 −0.236011 0.971750i $$-0.575840\pi$$
−0.236011 + 0.971750i $$0.575840\pi$$
$$450$$ 0 0
$$451$$ 22284.8 2.32672
$$452$$ 2534.42 0.263737
$$453$$ 0 0
$$454$$ 3847.83 0.397770
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −14343.8 −1.46822 −0.734109 0.679032i $$-0.762400\pi$$
−0.734109 + 0.679032i $$0.762400\pi$$
$$458$$ −1330.50 −0.135743
$$459$$ 0 0
$$460$$ −1117.36 −0.113255
$$461$$ 14558.7 1.47086 0.735429 0.677602i $$-0.236981\pi$$
0.735429 + 0.677602i $$0.236981\pi$$
$$462$$ 0 0
$$463$$ −1809.56 −0.181636 −0.0908178 0.995868i $$-0.528948\pi$$
−0.0908178 + 0.995868i $$0.528948\pi$$
$$464$$ 1087.13 0.108769
$$465$$ 0 0
$$466$$ 5366.41 0.533464
$$467$$ −5981.65 −0.592715 −0.296357 0.955077i $$-0.595772\pi$$
−0.296357 + 0.955077i $$0.595772\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1362.93 0.133760
$$471$$ 0 0
$$472$$ 1412.35 0.137730
$$473$$ 4524.84 0.439857
$$474$$ 0 0
$$475$$ 91.6876 0.00885666
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 8477.25 0.811173
$$479$$ −11527.5 −1.09959 −0.549796 0.835299i $$-0.685295\pi$$
−0.549796 + 0.835299i $$0.685295\pi$$
$$480$$ 0 0
$$481$$ −1108.19 −0.105050
$$482$$ 5667.23 0.535551
$$483$$ 0 0
$$484$$ −2131.85 −0.200211
$$485$$ −1090.04 −0.102054
$$486$$ 0 0
$$487$$ 15791.5 1.46936 0.734682 0.678411i $$-0.237331\pi$$
0.734682 + 0.678411i $$0.237331\pi$$
$$488$$ −18203.9 −1.68863
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13064.9 −1.20083 −0.600417 0.799687i $$-0.704999\pi$$
−0.600417 + 0.799687i $$0.704999\pi$$
$$492$$ 0 0
$$493$$ 3060.04 0.279548
$$494$$ −521.095 −0.0474599
$$495$$ 0 0
$$496$$ −6803.91 −0.615937
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −20135.8 −1.80642 −0.903209 0.429201i $$-0.858795\pi$$
−0.903209 + 0.429201i $$0.858795\pi$$
$$500$$ 329.156 0.0294406
$$501$$ 0 0
$$502$$ 6739.73 0.599221
$$503$$ −751.675 −0.0666313 −0.0333156 0.999445i $$-0.510607\pi$$
−0.0333156 + 0.999445i $$0.510607\pi$$
$$504$$ 0 0
$$505$$ −7371.64 −0.649571
$$506$$ 9096.06 0.799149
$$507$$ 0 0
$$508$$ 4270.44 0.372972
$$509$$ −12334.5 −1.07410 −0.537049 0.843551i $$-0.680461\pi$$
−0.537049 + 0.843551i $$0.680461\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11186.4 −0.965574
$$513$$ 0 0
$$514$$ −390.768 −0.0335332
$$515$$ 4054.95 0.346956
$$516$$ 0 0
$$517$$ 5443.95 0.463104
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −7554.09 −0.637056
$$521$$ −1736.43 −0.146016 −0.0730082 0.997331i $$-0.523260\pi$$
−0.0730082 + 0.997331i $$0.523260\pi$$
$$522$$ 0 0
$$523$$ 1421.42 0.118842 0.0594210 0.998233i $$-0.481075\pi$$
0.0594210 + 0.998233i $$0.481075\pi$$
$$524$$ −3634.72 −0.303022
$$525$$ 0 0
$$526$$ 7516.23 0.623048
$$527$$ −19151.6 −1.58303
$$528$$ 0 0
$$529$$ −4964.85 −0.408059
$$530$$ 7737.47 0.634140
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 29541.6 2.40073
$$534$$ 0 0
$$535$$ 2203.30 0.178050
$$536$$ −13607.3 −1.09654
$$537$$ 0 0
$$538$$ 6599.26 0.528837
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5773.27 0.458802 0.229401 0.973332i $$-0.426323\pi$$
0.229401 + 0.973332i $$0.426323\pi$$
$$542$$ −6604.64 −0.523420
$$543$$ 0 0
$$544$$ −11518.2 −0.907793
$$545$$ 9530.94 0.749102
$$546$$ 0 0
$$547$$ −3941.30 −0.308076 −0.154038 0.988065i $$-0.549228\pi$$
−0.154038 + 0.988065i $$0.549228\pi$$
$$548$$ −5133.93 −0.400202
$$549$$ 0 0
$$550$$ −2679.55 −0.207739
$$551$$ −110.751 −0.00856291
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 5325.06 0.408376
$$555$$ 0 0
$$556$$ −7373.10 −0.562390
$$557$$ 6951.74 0.528823 0.264412 0.964410i $$-0.414822\pi$$
0.264412 + 0.964410i $$0.414822\pi$$
$$558$$ 0 0
$$559$$ 5998.29 0.453847
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −14152.7 −1.06227
$$563$$ −24284.6 −1.81789 −0.908946 0.416913i $$-0.863112\pi$$
−0.908946 + 0.416913i $$0.863112\pi$$
$$564$$ 0 0
$$565$$ 4812.35 0.358331
$$566$$ 13563.7 1.00729
$$567$$ 0 0
$$568$$ −18235.1 −1.34706
$$569$$ 21563.4 1.58873 0.794363 0.607443i $$-0.207805\pi$$
0.794363 + 0.607443i $$0.207805\pi$$
$$570$$ 0 0
$$571$$ −3689.56 −0.270409 −0.135204 0.990818i $$-0.543169\pi$$
−0.135204 + 0.990818i $$0.543169\pi$$
$$572$$ −7472.21 −0.546204
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2121.64 −0.153875
$$576$$ 0 0
$$577$$ −22183.9 −1.60057 −0.800285 0.599620i $$-0.795318\pi$$
−0.800285 + 0.599620i $$0.795318\pi$$
$$578$$ 12406.2 0.892783
$$579$$ 0 0
$$580$$ −397.594 −0.0284641
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 30905.8 2.19552
$$584$$ −5747.55 −0.407252
$$585$$ 0 0
$$586$$ 11896.7 0.838646
$$587$$ −10605.3 −0.745705 −0.372852 0.927891i $$-0.621620\pi$$
−0.372852 + 0.927891i $$0.621620\pi$$
$$588$$ 0 0
$$589$$ 693.149 0.0484902
$$590$$ 664.119 0.0463412
$$591$$ 0 0
$$592$$ −650.466 −0.0451588
$$593$$ −6277.25 −0.434698 −0.217349 0.976094i $$-0.569741\pi$$
−0.217349 + 0.976094i $$0.569741\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −3776.39 −0.259542
$$597$$ 0 0
$$598$$ 12058.1 0.824567
$$599$$ 9970.73 0.680122 0.340061 0.940403i $$-0.389552\pi$$
0.340061 + 0.940403i $$0.389552\pi$$
$$600$$ 0 0
$$601$$ −24619.2 −1.67094 −0.835472 0.549533i $$-0.814806\pi$$
−0.835472 + 0.549533i $$0.814806\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 5228.53 0.352228
$$605$$ −4047.94 −0.272020
$$606$$ 0 0
$$607$$ 11252.9 0.752460 0.376230 0.926526i $$-0.377220\pi$$
0.376230 + 0.926526i $$0.377220\pi$$
$$608$$ 416.876 0.0278068
$$609$$ 0 0
$$610$$ −8559.90 −0.568164
$$611$$ 7216.69 0.477833
$$612$$ 0 0
$$613$$ −15293.2 −1.00765 −0.503824 0.863807i $$-0.668074\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$614$$ 4871.79 0.320211
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 17589.4 1.14769 0.573843 0.818966i $$-0.305452\pi$$
0.573843 + 0.818966i $$0.305452\pi$$
$$618$$ 0 0
$$619$$ −23467.4 −1.52380 −0.761900 0.647694i $$-0.775733\pi$$
−0.761900 + 0.647694i $$0.775733\pi$$
$$620$$ 2488.39 0.161187
$$621$$ 0 0
$$622$$ 13353.7 0.860829
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −3150.64 −0.201158
$$627$$ 0 0
$$628$$ −107.068 −0.00680330
$$629$$ −1830.93 −0.116063
$$630$$ 0 0
$$631$$ −6040.86 −0.381114 −0.190557 0.981676i $$-0.561029\pi$$
−0.190557 + 0.981676i $$0.561029\pi$$
$$632$$ 26485.3 1.66698
$$633$$ 0 0
$$634$$ −11905.0 −0.745755
$$635$$ 8108.68 0.506745
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 3236.68 0.200849
$$639$$ 0 0
$$640$$ −1839.37 −0.113605
$$641$$ −25111.6 −1.54735 −0.773673 0.633586i $$-0.781582\pi$$
−0.773673 + 0.633586i $$0.781582\pi$$
$$642$$ 0 0
$$643$$ 3095.03 0.189823 0.0949113 0.995486i $$-0.469743\pi$$
0.0949113 + 0.995486i $$0.469743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −860.944 −0.0524356
$$647$$ −9178.63 −0.557727 −0.278863 0.960331i $$-0.589958\pi$$
−0.278863 + 0.960331i $$0.589958\pi$$
$$648$$ 0 0
$$649$$ 2652.69 0.160443
$$650$$ −3552.11 −0.214346
$$651$$ 0 0
$$652$$ 10411.8 0.625397
$$653$$ 14438.4 0.865265 0.432632 0.901570i $$-0.357585\pi$$
0.432632 + 0.901570i $$0.357585\pi$$
$$654$$ 0 0
$$655$$ −6901.59 −0.411706
$$656$$ 17339.8 1.03202
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2900.64 0.171461 0.0857305 0.996318i $$-0.472678\pi$$
0.0857305 + 0.996318i $$0.472678\pi$$
$$660$$ 0 0
$$661$$ 9976.52 0.587053 0.293526 0.955951i $$-0.405171\pi$$
0.293526 + 0.955951i $$0.405171\pi$$
$$662$$ 4229.46 0.248312
$$663$$ 0 0
$$664$$ 16832.5 0.983777
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2562.77 0.148772
$$668$$ 8901.19 0.515565
$$669$$ 0 0
$$670$$ −6398.47 −0.368947
$$671$$ −34190.8 −1.96710
$$672$$ 0 0
$$673$$ 20760.8 1.18911 0.594554 0.804055i $$-0.297329\pi$$
0.594554 + 0.804055i $$0.297329\pi$$
$$674$$ 356.725 0.0203866
$$675$$ 0 0
$$676$$ −4120.18 −0.234421
$$677$$ −3209.13 −0.182181 −0.0910907 0.995843i $$-0.529035\pi$$
−0.0910907 + 0.995843i $$0.529035\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −12480.7 −0.703845
$$681$$ 0 0
$$682$$ −20257.2 −1.13737
$$683$$ −4333.57 −0.242781 −0.121391 0.992605i $$-0.538735\pi$$
−0.121391 + 0.992605i $$0.538735\pi$$
$$684$$ 0 0
$$685$$ −9748.29 −0.543741
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 3520.78 0.195100
$$689$$ 40969.8 2.26535
$$690$$ 0 0
$$691$$ −14446.0 −0.795297 −0.397649 0.917538i $$-0.630174\pi$$
−0.397649 + 0.917538i $$0.630174\pi$$
$$692$$ 8443.94 0.463859
$$693$$ 0 0
$$694$$ −10099.1 −0.552385
$$695$$ −14000.0 −0.764101
$$696$$ 0 0
$$697$$ 48808.1 2.65242
$$698$$ 3912.77 0.212179
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 859.801 0.0463256 0.0231628 0.999732i $$-0.492626\pi$$
0.0231628 + 0.999732i $$0.492626\pi$$
$$702$$ 0 0
$$703$$ 66.2663 0.00355517
$$704$$ −25507.9 −1.36557
$$705$$ 0 0
$$706$$ 9084.31 0.484267
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −7979.13 −0.422655 −0.211327 0.977415i $$-0.567779\pi$$
−0.211327 + 0.977415i $$0.567779\pi$$
$$710$$ −8574.57 −0.453236
$$711$$ 0 0
$$712$$ −34001.7 −1.78970
$$713$$ −16039.4 −0.842467
$$714$$ 0 0
$$715$$ −14188.2 −0.742110
$$716$$ 3798.86 0.198282
$$717$$ 0 0
$$718$$ 6643.76 0.345325
$$719$$ 33703.3 1.74815 0.874076 0.485790i $$-0.161468\pi$$
0.874076 + 0.485790i $$0.161468\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −15858.6 −0.817444
$$723$$ 0 0
$$724$$ 2392.78 0.122827
$$725$$ −754.950 −0.0386733
$$726$$ 0 0
$$727$$ −30277.0 −1.54458 −0.772290 0.635270i $$-0.780889\pi$$
−0.772290 + 0.635270i $$0.780889\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2702.63 −0.137026
$$731$$ 9910.27 0.501429
$$732$$ 0 0
$$733$$ 19363.9 0.975749 0.487874 0.872914i $$-0.337772\pi$$
0.487874 + 0.872914i $$0.337772\pi$$
$$734$$ 26648.1 1.34005
$$735$$ 0 0
$$736$$ −9646.45 −0.483115
$$737$$ −25557.4 −1.27737
$$738$$ 0 0
$$739$$ 24952.4 1.24207 0.621035 0.783783i $$-0.286713\pi$$
0.621035 + 0.783783i $$0.286713\pi$$
$$740$$ 237.894 0.0118178
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8154.54 0.402640 0.201320 0.979526i $$-0.435477\pi$$
0.201320 + 0.979526i $$0.435477\pi$$
$$744$$ 0 0
$$745$$ −7170.58 −0.352631
$$746$$ 11783.3 0.578310
$$747$$ 0 0
$$748$$ −12345.5 −0.603469
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4311.26 −0.209481 −0.104740 0.994500i $$-0.533401\pi$$
−0.104740 + 0.994500i $$0.533401\pi$$
$$752$$ 4235.94 0.205411
$$753$$ 0 0
$$754$$ 4290.66 0.207237
$$755$$ 9927.91 0.478561
$$756$$ 0 0
$$757$$ 3624.79 0.174036 0.0870179 0.996207i $$-0.472266\pi$$
0.0870179 + 0.996207i $$0.472266\pi$$
$$758$$ 2211.98 0.105993
$$759$$ 0 0
$$760$$ 451.713 0.0215597
$$761$$ −20576.4 −0.980150 −0.490075 0.871680i $$-0.663031\pi$$
−0.490075 + 0.871680i $$0.663031\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 6516.34 0.308577
$$765$$ 0 0
$$766$$ −7144.26 −0.336988
$$767$$ 3516.50 0.165546
$$768$$ 0 0
$$769$$ −3066.14 −0.143781 −0.0718907 0.997413i $$-0.522903\pi$$
−0.0718907 + 0.997413i $$0.522903\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −9304.64 −0.433784
$$773$$ 19387.0 0.902074 0.451037 0.892505i $$-0.351054\pi$$
0.451037 + 0.892505i $$0.351054\pi$$
$$774$$ 0 0
$$775$$ 4724.94 0.219000
$$776$$ −5370.23 −0.248428
$$777$$ 0 0
$$778$$ 14666.9 0.675879
$$779$$ −1766.50 −0.0812470
$$780$$ 0 0
$$781$$ −34249.4 −1.56919
$$782$$ 19922.1 0.911015
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −203.300 −0.00924342
$$786$$ 0 0
$$787$$ 43363.4 1.96409 0.982044 0.188651i $$-0.0604115\pi$$
0.982044 + 0.188651i $$0.0604115\pi$$
$$788$$ −5141.59 −0.232439
$$789$$ 0 0
$$790$$ 12454.0 0.560878
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −45324.6 −2.02966
$$794$$ −28108.2 −1.25632
$$795$$ 0 0
$$796$$ −10703.3 −0.476593
$$797$$ −17132.6 −0.761439 −0.380720 0.924691i $$-0.624324\pi$$
−0.380720 + 0.924691i $$0.624324\pi$$
$$798$$ 0 0
$$799$$ 11923.3 0.527929
$$800$$ 2841.69 0.125586
$$801$$ 0 0
$$802$$ 627.039 0.0276079
$$803$$ −10795.1 −0.474411
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −26853.6 −1.17355
$$807$$ 0 0
$$808$$ −36317.5 −1.58124
$$809$$ 1080.49 0.0469566 0.0234783 0.999724i $$-0.492526\pi$$
0.0234783 + 0.999724i $$0.492526\pi$$
$$810$$ 0 0
$$811$$ 19593.9 0.848378 0.424189 0.905574i $$-0.360559\pi$$
0.424189 + 0.905574i $$0.360559\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −1936.62 −0.0833889
$$815$$ 19769.9 0.849706
$$816$$ 0 0
$$817$$ −358.680 −0.0153594
$$818$$ 9312.66 0.398056
$$819$$ 0 0
$$820$$ −6341.69 −0.270075
$$821$$ −5123.80 −0.217810 −0.108905 0.994052i $$-0.534734\pi$$
−0.108905 + 0.994052i $$0.534734\pi$$
$$822$$ 0 0
$$823$$ −13184.1 −0.558405 −0.279202 0.960232i $$-0.590070\pi$$
−0.279202 + 0.960232i $$0.590070\pi$$
$$824$$ 19977.3 0.844591
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −24658.7 −1.03684 −0.518421 0.855126i $$-0.673480\pi$$
−0.518421 + 0.855126i $$0.673480\pi$$
$$828$$ 0 0
$$829$$ −28562.3 −1.19664 −0.598318 0.801259i $$-0.704164\pi$$
−0.598318 + 0.801259i $$0.704164\pi$$
$$830$$ 7915.04 0.331006
$$831$$ 0 0
$$832$$ −33814.1 −1.40901
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 16901.5 0.700481
$$836$$ 446.817 0.0184850
$$837$$ 0 0
$$838$$ −5646.27 −0.232753
$$839$$ −31106.0 −1.27997 −0.639987 0.768386i $$-0.721060\pi$$
−0.639987 + 0.768386i $$0.721060\pi$$
$$840$$ 0 0
$$841$$ −23477.1 −0.962609
$$842$$ −11007.1 −0.450511
$$843$$ 0 0
$$844$$ 11389.7 0.464514
$$845$$ −7823.38 −0.318500
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 24047.8 0.973827
$$849$$ 0 0
$$850$$ −5868.73 −0.236819
$$851$$ −1533.39 −0.0617674
$$852$$ 0 0
$$853$$ −20567.9 −0.825596 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 10854.9 0.433426
$$857$$ 6459.44 0.257468 0.128734 0.991679i $$-0.458909\pi$$
0.128734 + 0.991679i $$0.458909\pi$$
$$858$$ 0 0
$$859$$ 48214.4 1.91508 0.957541 0.288298i $$-0.0930895\pi$$
0.957541 + 0.288298i $$0.0930895\pi$$
$$860$$ −1287.65 −0.0510565
$$861$$ 0 0
$$862$$ −18359.7 −0.725445
$$863$$ 31709.5 1.25076 0.625378 0.780322i $$-0.284945\pi$$
0.625378 + 0.780322i $$0.284945\pi$$
$$864$$ 0 0
$$865$$ 16033.3 0.630230
$$866$$ −26611.7 −1.04423
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 49745.1 1.94187
$$870$$ 0 0
$$871$$ −33879.8 −1.31800
$$872$$ 46955.6 1.82353
$$873$$ 0 0
$$874$$ −721.037 −0.0279055
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 25654.8 0.987799 0.493900 0.869519i $$-0.335571\pi$$
0.493900 + 0.869519i $$0.335571\pi$$
$$878$$ −21192.3 −0.814585
$$879$$ 0 0
$$880$$ −8327.97 −0.319018
$$881$$ −11470.4 −0.438647 −0.219323 0.975652i $$-0.570385\pi$$
−0.219323 + 0.975652i $$0.570385\pi$$
$$882$$ 0 0
$$883$$ 39124.0 1.49108 0.745542 0.666459i $$-0.232191\pi$$
0.745542 + 0.666459i $$0.232191\pi$$
$$884$$ −16365.6 −0.622663
$$885$$ 0 0
$$886$$ −4319.01 −0.163770
$$887$$ 15585.8 0.589987 0.294994 0.955499i $$-0.404682\pi$$
0.294994 + 0.955499i $$0.404682\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −15988.4 −0.602171
$$891$$ 0 0
$$892$$ 2588.40 0.0971594
$$893$$ −431.537 −0.0161711
$$894$$ 0 0
$$895$$ 7213.25 0.269399
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −10403.7 −0.386610
$$899$$ −5707.34 −0.211736
$$900$$ 0 0
$$901$$ 67689.6 2.50285
$$902$$ 51625.6 1.90570
$$903$$ 0 0
$$904$$ 23708.8 0.872280
$$905$$ 4543.40 0.166881
$$906$$ 0 0
$$907$$ 27596.1 1.01027 0.505134 0.863041i $$-0.331443\pi$$
0.505134 + 0.863041i $$0.331443\pi$$
$$908$$ −4373.73 −0.159854
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 14396.2 0.523565 0.261782 0.965127i $$-0.415690\pi$$
0.261782 + 0.965127i $$0.415690\pi$$
$$912$$ 0 0
$$913$$ 31615.1 1.14601
$$914$$ −33229.2 −1.20254
$$915$$ 0 0
$$916$$ 1512.35 0.0545517
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 10279.6 0.368978 0.184489 0.982835i $$-0.440937\pi$$
0.184489 + 0.982835i $$0.440937\pi$$
$$920$$ −10452.6 −0.374577
$$921$$ 0 0
$$922$$ 33727.0 1.20471
$$923$$ −45402.2 −1.61910
$$924$$ 0 0
$$925$$ 451.713 0.0160565
$$926$$ −4192.06 −0.148769
$$927$$ 0 0
$$928$$ −3432.53 −0.121421
$$929$$ −6499.87 −0.229552 −0.114776 0.993391i $$-0.536615\pi$$
−0.114776 + 0.993391i $$0.536615\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6099.86 −0.214386
$$933$$ 0 0
$$934$$ −13857.2 −0.485463
$$935$$ −23441.5 −0.819913
$$936$$ 0 0
$$937$$ −10269.8 −0.358056 −0.179028 0.983844i $$-0.557295\pi$$
−0.179028 + 0.983844i $$0.557295\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −1549.21 −0.0537548
$$941$$ −34396.2 −1.19159 −0.595794 0.803137i $$-0.703163\pi$$
−0.595794 + 0.803137i $$0.703163\pi$$
$$942$$ 0 0
$$943$$ 40876.5 1.41158
$$944$$ 2064.06 0.0711647
$$945$$ 0 0
$$946$$ 10482.4 0.360265
$$947$$ −27192.1 −0.933078 −0.466539 0.884501i $$-0.654499\pi$$
−0.466539 + 0.884501i $$0.654499\pi$$
$$948$$ 0 0
$$949$$ −14310.4 −0.489500
$$950$$ 212.406 0.00725406
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −49965.2 −1.69836 −0.849178 0.528107i $$-0.822902\pi$$
−0.849178 + 0.528107i $$0.822902\pi$$
$$954$$ 0 0
$$955$$ 12373.2 0.419253
$$956$$ −9635.88 −0.325990
$$957$$ 0 0
$$958$$ −26704.9 −0.900622
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5929.05 0.199022
$$962$$ −2567.25 −0.0860411
$$963$$ 0 0
$$964$$ −6441.80 −0.215225
$$965$$ −17667.6 −0.589368
$$966$$ 0 0
$$967$$ −16755.5 −0.557208 −0.278604 0.960406i $$-0.589872\pi$$
−0.278604 + 0.960406i $$0.589872\pi$$
$$968$$ −19942.8 −0.662176
$$969$$ 0 0
$$970$$ −2525.21 −0.0835872
$$971$$ 37617.4 1.24325 0.621626 0.783314i $$-0.286472\pi$$
0.621626 + 0.783314i $$0.286472\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 36583.0 1.20348
$$975$$ 0 0
$$976$$ −26603.9 −0.872511
$$977$$ 27690.9 0.906767 0.453384 0.891315i $$-0.350217\pi$$
0.453384 + 0.891315i $$0.350217\pi$$
$$978$$ 0 0
$$979$$ −63862.5 −2.08483
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −30266.4 −0.983544
$$983$$ 22754.2 0.738299 0.369149 0.929370i $$-0.379649\pi$$
0.369149 + 0.929370i $$0.379649\pi$$
$$984$$ 0 0
$$985$$ −9762.83 −0.315807
$$986$$ 7088.96 0.228964
$$987$$ 0 0
$$988$$ 592.316 0.0190729
$$989$$ 8299.80 0.266854
$$990$$ 0 0
$$991$$ −55470.9 −1.77809 −0.889047 0.457816i $$-0.848632\pi$$
−0.889047 + 0.457816i $$0.848632\pi$$
$$992$$ 21482.9 0.687583
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −20323.4 −0.647531
$$996$$ 0 0
$$997$$ −15181.9 −0.482264 −0.241132 0.970492i $$-0.577519\pi$$
−0.241132 + 0.970492i $$0.577519\pi$$
$$998$$ −46647.1 −1.47955
$$999$$ 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.w.1.2 2
3.2 odd 2 245.4.a.j.1.1 yes 2
7.6 odd 2 2205.4.a.x.1.2 2
15.14 odd 2 1225.4.a.p.1.2 2
21.2 odd 6 245.4.e.j.116.2 4
21.5 even 6 245.4.e.k.116.2 4
21.11 odd 6 245.4.e.j.226.2 4
21.17 even 6 245.4.e.k.226.2 4
21.20 even 2 245.4.a.i.1.1 2
105.104 even 2 1225.4.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.1 2 21.20 even 2
245.4.a.j.1.1 yes 2 3.2 odd 2
245.4.e.j.116.2 4 21.2 odd 6
245.4.e.j.226.2 4 21.11 odd 6
245.4.e.k.116.2 4 21.5 even 6
245.4.e.k.226.2 4 21.17 even 6
1225.4.a.p.1.2 2 15.14 odd 2
1225.4.a.q.1.2 2 105.104 even 2
2205.4.a.w.1.2 2 1.1 even 1 trivial
2205.4.a.x.1.2 2 7.6 odd 2