# Properties

 Label 2088.4.a.f.1.1 Level $2088$ Weight $4$ Character 2088.1 Self dual yes Analytic conductor $123.196$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2088 = 2^{3} \cdot 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2088.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$123.195988092$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198$$ x^5 - x^4 - 34*x^3 + 74*x^2 + 94*x - 198 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 232) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.69859$$ of defining polynomial Character $$\chi$$ $$=$$ 2088.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-16.3850 q^{5} +5.74609 q^{7} +O(q^{10})$$ $$q-16.3850 q^{5} +5.74609 q^{7} -23.3693 q^{11} -18.0318 q^{13} +24.1314 q^{17} -12.0191 q^{19} +144.669 q^{23} +143.469 q^{25} -29.0000 q^{29} +6.27664 q^{31} -94.1498 q^{35} +28.6658 q^{37} +436.340 q^{41} +495.080 q^{43} -351.351 q^{47} -309.982 q^{49} +58.0933 q^{53} +382.906 q^{55} -485.535 q^{59} +607.926 q^{61} +295.451 q^{65} -296.974 q^{67} +330.178 q^{71} -662.978 q^{73} -134.282 q^{77} +145.394 q^{79} -1077.01 q^{83} -395.393 q^{85} +851.923 q^{89} -103.612 q^{91} +196.933 q^{95} -227.768 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 10 q^{5} + 32 q^{7}+O(q^{10})$$ 5 * q - 10 * q^5 + 32 * q^7 $$5 q - 10 q^{5} + 32 q^{7} - 36 q^{11} + 26 q^{13} - 82 q^{17} + 156 q^{19} - 336 q^{23} + 151 q^{25} - 145 q^{29} + 432 q^{31} - 600 q^{35} - 18 q^{37} - 82 q^{41} + 340 q^{43} - 680 q^{47} - 115 q^{49} + 102 q^{53} + 736 q^{55} - 924 q^{59} - 618 q^{61} + 704 q^{65} + 44 q^{67} - 1032 q^{71} - 1078 q^{73} + 888 q^{77} + 200 q^{79} - 452 q^{83} - 1700 q^{85} + 1790 q^{89} - 1128 q^{91} - 1024 q^{95} - 2518 q^{97}+O(q^{100})$$ 5 * q - 10 * q^5 + 32 * q^7 - 36 * q^11 + 26 * q^13 - 82 * q^17 + 156 * q^19 - 336 * q^23 + 151 * q^25 - 145 * q^29 + 432 * q^31 - 600 * q^35 - 18 * q^37 - 82 * q^41 + 340 * q^43 - 680 * q^47 - 115 * q^49 + 102 * q^53 + 736 * q^55 - 924 * q^59 - 618 * q^61 + 704 * q^65 + 44 * q^67 - 1032 * q^71 - 1078 * q^73 + 888 * q^77 + 200 * q^79 - 452 * q^83 - 1700 * q^85 + 1790 * q^89 - 1128 * q^91 - 1024 * q^95 - 2518 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −16.3850 −1.46552 −0.732760 0.680487i $$-0.761768\pi$$
−0.732760 + 0.680487i $$0.761768\pi$$
$$6$$ 0 0
$$7$$ 5.74609 0.310260 0.155130 0.987894i $$-0.450420\pi$$
0.155130 + 0.987894i $$0.450420\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −23.3693 −0.640554 −0.320277 0.947324i $$-0.603776\pi$$
−0.320277 + 0.947324i $$0.603776\pi$$
$$12$$ 0 0
$$13$$ −18.0318 −0.384701 −0.192351 0.981326i $$-0.561611\pi$$
−0.192351 + 0.981326i $$0.561611\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 24.1314 0.344278 0.172139 0.985073i $$-0.444932\pi$$
0.172139 + 0.985073i $$0.444932\pi$$
$$18$$ 0 0
$$19$$ −12.0191 −0.145125 −0.0725624 0.997364i $$-0.523118\pi$$
−0.0725624 + 0.997364i $$0.523118\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 144.669 1.31155 0.655776 0.754956i $$-0.272342\pi$$
0.655776 + 0.754956i $$0.272342\pi$$
$$24$$ 0 0
$$25$$ 143.469 1.14775
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −29.0000 −0.185695
$$30$$ 0 0
$$31$$ 6.27664 0.0363651 0.0181826 0.999835i $$-0.494212\pi$$
0.0181826 + 0.999835i $$0.494212\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −94.1498 −0.454692
$$36$$ 0 0
$$37$$ 28.6658 0.127368 0.0636842 0.997970i $$-0.479715\pi$$
0.0636842 + 0.997970i $$0.479715\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 436.340 1.66207 0.831035 0.556220i $$-0.187749\pi$$
0.831035 + 0.556220i $$0.187749\pi$$
$$42$$ 0 0
$$43$$ 495.080 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −351.351 −1.09042 −0.545211 0.838299i $$-0.683550\pi$$
−0.545211 + 0.838299i $$0.683550\pi$$
$$48$$ 0 0
$$49$$ −309.982 −0.903739
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 58.0933 0.150561 0.0752804 0.997162i $$-0.476015\pi$$
0.0752804 + 0.997162i $$0.476015\pi$$
$$54$$ 0 0
$$55$$ 382.906 0.938746
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −485.535 −1.07138 −0.535689 0.844415i $$-0.679948\pi$$
−0.535689 + 0.844415i $$0.679948\pi$$
$$60$$ 0 0
$$61$$ 607.926 1.27602 0.638008 0.770030i $$-0.279759\pi$$
0.638008 + 0.770030i $$0.279759\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 295.451 0.563788
$$66$$ 0 0
$$67$$ −296.974 −0.541510 −0.270755 0.962648i $$-0.587273\pi$$
−0.270755 + 0.962648i $$0.587273\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 330.178 0.551901 0.275950 0.961172i $$-0.411007\pi$$
0.275950 + 0.961172i $$0.411007\pi$$
$$72$$ 0 0
$$73$$ −662.978 −1.06296 −0.531478 0.847072i $$-0.678363\pi$$
−0.531478 + 0.847072i $$0.678363\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −134.282 −0.198738
$$78$$ 0 0
$$79$$ 145.394 0.207065 0.103532 0.994626i $$-0.466985\pi$$
0.103532 + 0.994626i $$0.466985\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1077.01 −1.42431 −0.712154 0.702023i $$-0.752280\pi$$
−0.712154 + 0.702023i $$0.752280\pi$$
$$84$$ 0 0
$$85$$ −395.393 −0.504546
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 851.923 1.01465 0.507324 0.861755i $$-0.330635\pi$$
0.507324 + 0.861755i $$0.330635\pi$$
$$90$$ 0 0
$$91$$ −103.612 −0.119357
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 196.933 0.212684
$$96$$ 0 0
$$97$$ −227.768 −0.238416 −0.119208 0.992869i $$-0.538036\pi$$
−0.119208 + 0.992869i $$0.538036\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 77.3495 0.0762036 0.0381018 0.999274i $$-0.487869\pi$$
0.0381018 + 0.999274i $$0.487869\pi$$
$$102$$ 0 0
$$103$$ −751.327 −0.718742 −0.359371 0.933195i $$-0.617009\pi$$
−0.359371 + 0.933195i $$0.617009\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 761.498 0.688007 0.344004 0.938968i $$-0.388217\pi$$
0.344004 + 0.938968i $$0.388217\pi$$
$$108$$ 0 0
$$109$$ −661.001 −0.580848 −0.290424 0.956898i $$-0.593796\pi$$
−0.290424 + 0.956898i $$0.593796\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2266.45 1.88681 0.943404 0.331646i $$-0.107604\pi$$
0.943404 + 0.331646i $$0.107604\pi$$
$$114$$ 0 0
$$115$$ −2370.41 −1.92211
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 138.661 0.106815
$$120$$ 0 0
$$121$$ −784.877 −0.589690
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −302.612 −0.216532
$$126$$ 0 0
$$127$$ 420.522 0.293821 0.146911 0.989150i $$-0.453067\pi$$
0.146911 + 0.989150i $$0.453067\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 208.443 0.139021 0.0695103 0.997581i $$-0.477856\pi$$
0.0695103 + 0.997581i $$0.477856\pi$$
$$132$$ 0 0
$$133$$ −69.0629 −0.0450264
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 188.285 0.117418 0.0587089 0.998275i $$-0.481302\pi$$
0.0587089 + 0.998275i $$0.481302\pi$$
$$138$$ 0 0
$$139$$ −1460.15 −0.890996 −0.445498 0.895283i $$-0.646973\pi$$
−0.445498 + 0.895283i $$0.646973\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 421.390 0.246422
$$144$$ 0 0
$$145$$ 475.166 0.272140
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −665.203 −0.365742 −0.182871 0.983137i $$-0.558539\pi$$
−0.182871 + 0.983137i $$0.558539\pi$$
$$150$$ 0 0
$$151$$ 3248.63 1.75079 0.875396 0.483406i $$-0.160601\pi$$
0.875396 + 0.483406i $$0.160601\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −102.843 −0.0532938
$$156$$ 0 0
$$157$$ 1907.56 0.969681 0.484840 0.874603i $$-0.338878\pi$$
0.484840 + 0.874603i $$0.338878\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 831.284 0.406921
$$162$$ 0 0
$$163$$ −2961.93 −1.42329 −0.711645 0.702539i $$-0.752050\pi$$
−0.711645 + 0.702539i $$0.752050\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2952.52 −1.36810 −0.684051 0.729434i $$-0.739783\pi$$
−0.684051 + 0.729434i $$0.739783\pi$$
$$168$$ 0 0
$$169$$ −1871.85 −0.852005
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3500.71 −1.53846 −0.769232 0.638969i $$-0.779361\pi$$
−0.769232 + 0.638969i $$0.779361\pi$$
$$174$$ 0 0
$$175$$ 824.385 0.356101
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −4562.48 −1.90511 −0.952557 0.304359i $$-0.901558\pi$$
−0.952557 + 0.304359i $$0.901558\pi$$
$$180$$ 0 0
$$181$$ 326.513 0.134086 0.0670430 0.997750i $$-0.478644\pi$$
0.0670430 + 0.997750i $$0.478644\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −469.690 −0.186661
$$186$$ 0 0
$$187$$ −563.933 −0.220529
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2623.16 −0.993744 −0.496872 0.867824i $$-0.665518\pi$$
−0.496872 + 0.867824i $$0.665518\pi$$
$$192$$ 0 0
$$193$$ 894.480 0.333607 0.166803 0.985990i $$-0.446656\pi$$
0.166803 + 0.985990i $$0.446656\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2182.51 −0.789326 −0.394663 0.918826i $$-0.629139\pi$$
−0.394663 + 0.918826i $$0.629139\pi$$
$$198$$ 0 0
$$199$$ −635.874 −0.226512 −0.113256 0.993566i $$-0.536128\pi$$
−0.113256 + 0.993566i $$0.536128\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −166.637 −0.0576138
$$204$$ 0 0
$$205$$ −7149.44 −2.43580
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 280.878 0.0929604
$$210$$ 0 0
$$211$$ −745.334 −0.243180 −0.121590 0.992580i $$-0.538799\pi$$
−0.121590 + 0.992580i $$0.538799\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8111.90 −2.57315
$$216$$ 0 0
$$217$$ 36.0661 0.0112826
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −435.132 −0.132444
$$222$$ 0 0
$$223$$ 159.305 0.0478380 0.0239190 0.999714i $$-0.492386\pi$$
0.0239190 + 0.999714i $$0.492386\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3699.17 1.08160 0.540798 0.841153i $$-0.318122\pi$$
0.540798 + 0.841153i $$0.318122\pi$$
$$228$$ 0 0
$$229$$ −6713.39 −1.93726 −0.968632 0.248499i $$-0.920063\pi$$
−0.968632 + 0.248499i $$0.920063\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1250.99 −0.351740 −0.175870 0.984413i $$-0.556274\pi$$
−0.175870 + 0.984413i $$0.556274\pi$$
$$234$$ 0 0
$$235$$ 5756.89 1.59803
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3073.10 0.831726 0.415863 0.909427i $$-0.363480\pi$$
0.415863 + 0.909427i $$0.363480\pi$$
$$240$$ 0 0
$$241$$ 4745.39 1.26837 0.634185 0.773181i $$-0.281336\pi$$
0.634185 + 0.773181i $$0.281336\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 5079.07 1.32445
$$246$$ 0 0
$$247$$ 216.726 0.0558298
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1919.33 −0.482658 −0.241329 0.970443i $$-0.577583\pi$$
−0.241329 + 0.970443i $$0.577583\pi$$
$$252$$ 0 0
$$253$$ −3380.82 −0.840120
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4680.45 −1.13602 −0.568012 0.823020i $$-0.692287\pi$$
−0.568012 + 0.823020i $$0.692287\pi$$
$$258$$ 0 0
$$259$$ 164.716 0.0395173
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4404.71 1.03272 0.516361 0.856371i $$-0.327286\pi$$
0.516361 + 0.856371i $$0.327286\pi$$
$$264$$ 0 0
$$265$$ −951.859 −0.220650
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −8244.86 −1.86877 −0.934383 0.356271i $$-0.884048\pi$$
−0.934383 + 0.356271i $$0.884048\pi$$
$$270$$ 0 0
$$271$$ −1050.02 −0.235366 −0.117683 0.993051i $$-0.537547\pi$$
−0.117683 + 0.993051i $$0.537547\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3352.76 −0.735197
$$276$$ 0 0
$$277$$ 3630.09 0.787405 0.393703 0.919238i $$-0.371194\pi$$
0.393703 + 0.919238i $$0.371194\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6718.58 −1.42632 −0.713162 0.700999i $$-0.752738\pi$$
−0.713162 + 0.700999i $$0.752738\pi$$
$$282$$ 0 0
$$283$$ −1096.86 −0.230394 −0.115197 0.993343i $$-0.536750\pi$$
−0.115197 + 0.993343i $$0.536750\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2507.25 0.515673
$$288$$ 0 0
$$289$$ −4330.68 −0.881473
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1991.65 0.397111 0.198556 0.980090i $$-0.436375\pi$$
0.198556 + 0.980090i $$0.436375\pi$$
$$294$$ 0 0
$$295$$ 7955.50 1.57013
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2608.65 −0.504556
$$300$$ 0 0
$$301$$ 2844.78 0.544751
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −9960.88 −1.87003
$$306$$ 0 0
$$307$$ −3847.15 −0.715207 −0.357604 0.933873i $$-0.616406\pi$$
−0.357604 + 0.933873i $$0.616406\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8281.96 −1.51005 −0.755027 0.655693i $$-0.772376\pi$$
−0.755027 + 0.655693i $$0.772376\pi$$
$$312$$ 0 0
$$313$$ −8930.42 −1.61271 −0.806354 0.591434i $$-0.798562\pi$$
−0.806354 + 0.591434i $$0.798562\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2716.45 0.481297 0.240649 0.970612i $$-0.422640\pi$$
0.240649 + 0.970612i $$0.422640\pi$$
$$318$$ 0 0
$$319$$ 677.709 0.118948
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −290.038 −0.0499633
$$324$$ 0 0
$$325$$ −2587.00 −0.441541
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2018.89 −0.338314
$$330$$ 0 0
$$331$$ 6445.84 1.07038 0.535189 0.844732i $$-0.320240\pi$$
0.535189 + 0.844732i $$0.320240\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4865.92 0.793593
$$336$$ 0 0
$$337$$ 6898.90 1.11515 0.557577 0.830125i $$-0.311731\pi$$
0.557577 + 0.830125i $$0.311731\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −146.681 −0.0232938
$$342$$ 0 0
$$343$$ −3752.10 −0.590653
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4463.85 −0.690583 −0.345291 0.938496i $$-0.612220\pi$$
−0.345291 + 0.938496i $$0.612220\pi$$
$$348$$ 0 0
$$349$$ −9789.78 −1.50153 −0.750766 0.660568i $$-0.770315\pi$$
−0.750766 + 0.660568i $$0.770315\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 3601.80 0.543072 0.271536 0.962428i $$-0.412468\pi$$
0.271536 + 0.962428i $$0.412468\pi$$
$$354$$ 0 0
$$355$$ −5409.98 −0.808822
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11426.7 1.67988 0.839940 0.542679i $$-0.182590\pi$$
0.839940 + 0.542679i $$0.182590\pi$$
$$360$$ 0 0
$$361$$ −6714.54 −0.978939
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10862.9 1.55778
$$366$$ 0 0
$$367$$ 6545.41 0.930975 0.465487 0.885055i $$-0.345879\pi$$
0.465487 + 0.885055i $$0.345879\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 333.809 0.0467129
$$372$$ 0 0
$$373$$ 12488.0 1.73352 0.866761 0.498724i $$-0.166198\pi$$
0.866761 + 0.498724i $$0.166198\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 522.922 0.0714373
$$378$$ 0 0
$$379$$ 10431.7 1.41383 0.706915 0.707298i $$-0.250086\pi$$
0.706915 + 0.707298i $$0.250086\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 9500.35 1.26748 0.633741 0.773545i $$-0.281519\pi$$
0.633741 + 0.773545i $$0.281519\pi$$
$$384$$ 0 0
$$385$$ 2200.21 0.291255
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −12956.1 −1.68869 −0.844345 0.535800i $$-0.820010\pi$$
−0.844345 + 0.535800i $$0.820010\pi$$
$$390$$ 0 0
$$391$$ 3491.08 0.451538
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −2382.28 −0.303457
$$396$$ 0 0
$$397$$ −167.727 −0.0212040 −0.0106020 0.999944i $$-0.503375\pi$$
−0.0106020 + 0.999944i $$0.503375\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −13225.9 −1.64705 −0.823526 0.567279i $$-0.807996\pi$$
−0.823526 + 0.567279i $$0.807996\pi$$
$$402$$ 0 0
$$403$$ −113.179 −0.0139897
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −669.899 −0.0815864
$$408$$ 0 0
$$409$$ −674.413 −0.0815344 −0.0407672 0.999169i $$-0.512980\pi$$
−0.0407672 + 0.999169i $$0.512980\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2789.93 −0.332405
$$414$$ 0 0
$$415$$ 17646.9 2.08735
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12355.6 1.44059 0.720296 0.693667i $$-0.244006\pi$$
0.720296 + 0.693667i $$0.244006\pi$$
$$420$$ 0 0
$$421$$ −8788.18 −1.01736 −0.508682 0.860955i $$-0.669867\pi$$
−0.508682 + 0.860955i $$0.669867\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3462.10 0.395145
$$426$$ 0 0
$$427$$ 3493.20 0.395896
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −15171.7 −1.69559 −0.847793 0.530328i $$-0.822069\pi$$
−0.847793 + 0.530328i $$0.822069\pi$$
$$432$$ 0 0
$$433$$ 3884.76 0.431154 0.215577 0.976487i $$-0.430837\pi$$
0.215577 + 0.976487i $$0.430837\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1738.80 −0.190339
$$438$$ 0 0
$$439$$ 14998.9 1.63066 0.815329 0.578998i $$-0.196556\pi$$
0.815329 + 0.578998i $$0.196556\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4945.82 −0.530435 −0.265218 0.964189i $$-0.585444\pi$$
−0.265218 + 0.964189i $$0.585444\pi$$
$$444$$ 0 0
$$445$$ −13958.8 −1.48699
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −13041.0 −1.37070 −0.685351 0.728213i $$-0.740351\pi$$
−0.685351 + 0.728213i $$0.740351\pi$$
$$450$$ 0 0
$$451$$ −10196.9 −1.06465
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1697.69 0.174921
$$456$$ 0 0
$$457$$ 3833.64 0.392408 0.196204 0.980563i $$-0.437139\pi$$
0.196204 + 0.980563i $$0.437139\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2806.70 0.283560 0.141780 0.989898i $$-0.454717\pi$$
0.141780 + 0.989898i $$0.454717\pi$$
$$462$$ 0 0
$$463$$ −4240.08 −0.425600 −0.212800 0.977096i $$-0.568258\pi$$
−0.212800 + 0.977096i $$0.568258\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −7538.50 −0.746981 −0.373491 0.927634i $$-0.621839\pi$$
−0.373491 + 0.927634i $$0.621839\pi$$
$$468$$ 0 0
$$469$$ −1706.44 −0.168009
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −11569.7 −1.12468
$$474$$ 0 0
$$475$$ −1724.37 −0.166567
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 14810.8 1.41279 0.706393 0.707820i $$-0.250321\pi$$
0.706393 + 0.707820i $$0.250321\pi$$
$$480$$ 0 0
$$481$$ −516.896 −0.0489988
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3731.98 0.349403
$$486$$ 0 0
$$487$$ 2951.06 0.274590 0.137295 0.990530i $$-0.456159\pi$$
0.137295 + 0.990530i $$0.456159\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −19549.3 −1.79683 −0.898417 0.439143i $$-0.855282\pi$$
−0.898417 + 0.439143i $$0.855282\pi$$
$$492$$ 0 0
$$493$$ −699.810 −0.0639308
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1897.23 0.171233
$$498$$ 0 0
$$499$$ −5713.30 −0.512550 −0.256275 0.966604i $$-0.582495\pi$$
−0.256275 + 0.966604i $$0.582495\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −7379.92 −0.654184 −0.327092 0.944993i $$-0.606069\pi$$
−0.327092 + 0.944993i $$0.606069\pi$$
$$504$$ 0 0
$$505$$ −1267.37 −0.111678
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −4642.94 −0.404312 −0.202156 0.979353i $$-0.564795\pi$$
−0.202156 + 0.979353i $$0.564795\pi$$
$$510$$ 0 0
$$511$$ −3809.53 −0.329792
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 12310.5 1.05333
$$516$$ 0 0
$$517$$ 8210.81 0.698474
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 11184.4 0.940494 0.470247 0.882535i $$-0.344165\pi$$
0.470247 + 0.882535i $$0.344165\pi$$
$$522$$ 0 0
$$523$$ 10752.9 0.899025 0.449512 0.893274i $$-0.351598\pi$$
0.449512 + 0.893274i $$0.351598\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 151.464 0.0125197
$$528$$ 0 0
$$529$$ 8762.26 0.720166
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7867.99 −0.639401
$$534$$ 0 0
$$535$$ −12477.2 −1.00829
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 7244.06 0.578894
$$540$$ 0 0
$$541$$ −20175.9 −1.60338 −0.801691 0.597738i $$-0.796066\pi$$
−0.801691 + 0.597738i $$0.796066\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10830.5 0.851245
$$546$$ 0 0
$$547$$ −9678.22 −0.756510 −0.378255 0.925702i $$-0.623476\pi$$
−0.378255 + 0.925702i $$0.623476\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 348.554 0.0269490
$$552$$ 0 0
$$553$$ 835.447 0.0642438
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −19870.3 −1.51155 −0.755775 0.654832i $$-0.772740\pi$$
−0.755775 + 0.654832i $$0.772740\pi$$
$$558$$ 0 0
$$559$$ −8927.18 −0.675455
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −9424.02 −0.705462 −0.352731 0.935725i $$-0.614747\pi$$
−0.352731 + 0.935725i $$0.614747\pi$$
$$564$$ 0 0
$$565$$ −37135.8 −2.76516
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 14765.7 1.08789 0.543945 0.839121i $$-0.316930\pi$$
0.543945 + 0.839121i $$0.316930\pi$$
$$570$$ 0 0
$$571$$ −10061.1 −0.737380 −0.368690 0.929552i $$-0.620194\pi$$
−0.368690 + 0.929552i $$0.620194\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 20755.6 1.50533
$$576$$ 0 0
$$577$$ −26307.2 −1.89806 −0.949032 0.315179i $$-0.897935\pi$$
−0.949032 + 0.315179i $$0.897935\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6188.62 −0.441905
$$582$$ 0 0
$$583$$ −1357.60 −0.0964424
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −2053.70 −0.144404 −0.0722021 0.997390i $$-0.523003\pi$$
−0.0722021 + 0.997390i $$0.523003\pi$$
$$588$$ 0 0
$$589$$ −75.4397 −0.00527748
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −12767.4 −0.884140 −0.442070 0.896981i $$-0.645756\pi$$
−0.442070 + 0.896981i $$0.645756\pi$$
$$594$$ 0 0
$$595$$ −2271.96 −0.156540
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 22284.1 1.52004 0.760021 0.649899i $$-0.225189\pi$$
0.760021 + 0.649899i $$0.225189\pi$$
$$600$$ 0 0
$$601$$ 1145.05 0.0777166 0.0388583 0.999245i $$-0.487628\pi$$
0.0388583 + 0.999245i $$0.487628\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 12860.2 0.864203
$$606$$ 0 0
$$607$$ 5726.01 0.382886 0.191443 0.981504i $$-0.438683\pi$$
0.191443 + 0.981504i $$0.438683\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6335.49 0.419487
$$612$$ 0 0
$$613$$ 17973.2 1.18423 0.592113 0.805855i $$-0.298294\pi$$
0.592113 + 0.805855i $$0.298294\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −28779.6 −1.87783 −0.938916 0.344146i $$-0.888168\pi$$
−0.938916 + 0.344146i $$0.888168\pi$$
$$618$$ 0 0
$$619$$ −20981.6 −1.36239 −0.681197 0.732100i $$-0.738540\pi$$
−0.681197 + 0.732100i $$0.738540\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4895.22 0.314804
$$624$$ 0 0
$$625$$ −12975.3 −0.830419
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 691.746 0.0438501
$$630$$ 0 0
$$631$$ 618.760 0.0390372 0.0195186 0.999809i $$-0.493787\pi$$
0.0195186 + 0.999809i $$0.493787\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6890.26 −0.430601
$$636$$ 0 0
$$637$$ 5589.54 0.347670
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21528.5 1.32656 0.663280 0.748371i $$-0.269164\pi$$
0.663280 + 0.748371i $$0.269164\pi$$
$$642$$ 0 0
$$643$$ 8142.79 0.499410 0.249705 0.968322i $$-0.419666\pi$$
0.249705 + 0.968322i $$0.419666\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12779.6 −0.776538 −0.388269 0.921546i $$-0.626927\pi$$
−0.388269 + 0.921546i $$0.626927\pi$$
$$648$$ 0 0
$$649$$ 11346.6 0.686276
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19463.5 1.16641 0.583205 0.812325i $$-0.301799\pi$$
0.583205 + 0.812325i $$0.301799\pi$$
$$654$$ 0 0
$$655$$ −3415.33 −0.203738
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 23009.3 1.36011 0.680056 0.733160i $$-0.261955\pi$$
0.680056 + 0.733160i $$0.261955\pi$$
$$660$$ 0 0
$$661$$ 4135.64 0.243355 0.121678 0.992570i $$-0.461173\pi$$
0.121678 + 0.992570i $$0.461173\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1131.60 0.0659871
$$666$$ 0 0
$$667$$ −4195.42 −0.243549
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −14206.8 −0.817357
$$672$$ 0 0
$$673$$ 19479.7 1.11573 0.557865 0.829931i $$-0.311621\pi$$
0.557865 + 0.829931i $$0.311621\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −5334.44 −0.302835 −0.151417 0.988470i $$-0.548384\pi$$
−0.151417 + 0.988470i $$0.548384\pi$$
$$678$$ 0 0
$$679$$ −1308.78 −0.0739708
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −8454.59 −0.473654 −0.236827 0.971552i $$-0.576107\pi$$
−0.236827 + 0.971552i $$0.576107\pi$$
$$684$$ 0 0
$$685$$ −3085.05 −0.172078
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1047.53 −0.0579210
$$690$$ 0 0
$$691$$ 10176.2 0.560231 0.280115 0.959966i $$-0.409627\pi$$
0.280115 + 0.959966i $$0.409627\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 23924.6 1.30577
$$696$$ 0 0
$$697$$ 10529.5 0.572214
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20411.6 1.09976 0.549882 0.835243i $$-0.314673\pi$$
0.549882 + 0.835243i $$0.314673\pi$$
$$702$$ 0 0
$$703$$ −344.538 −0.0184843
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 444.457 0.0236429
$$708$$ 0 0
$$709$$ 8169.75 0.432752 0.216376 0.976310i $$-0.430576\pi$$
0.216376 + 0.976310i $$0.430576\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 908.039 0.0476947
$$714$$ 0 0
$$715$$ −6904.48 −0.361137
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −2017.01 −0.104620 −0.0523101 0.998631i $$-0.516658\pi$$
−0.0523101 + 0.998631i $$0.516658\pi$$
$$720$$ 0 0
$$721$$ −4317.19 −0.222997
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4160.60 −0.213132
$$726$$ 0 0
$$727$$ 9836.46 0.501807 0.250904 0.968012i $$-0.419272\pi$$
0.250904 + 0.968012i $$0.419272\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 11947.0 0.604480
$$732$$ 0 0
$$733$$ 14843.6 0.747969 0.373984 0.927435i $$-0.377991\pi$$
0.373984 + 0.927435i $$0.377991\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6940.06 0.346866
$$738$$ 0 0
$$739$$ 2834.15 0.141077 0.0705385 0.997509i $$-0.477528\pi$$
0.0705385 + 0.997509i $$0.477528\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 22076.2 1.09004 0.545019 0.838424i $$-0.316523\pi$$
0.545019 + 0.838424i $$0.316523\pi$$
$$744$$ 0 0
$$745$$ 10899.4 0.536002
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 4375.63 0.213461
$$750$$ 0 0
$$751$$ 24518.8 1.19135 0.595675 0.803225i $$-0.296885\pi$$
0.595675 + 0.803225i $$0.296885\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −53228.8 −2.56582
$$756$$ 0 0
$$757$$ −19834.9 −0.952326 −0.476163 0.879357i $$-0.657973\pi$$
−0.476163 + 0.879357i $$0.657973\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −17719.2 −0.844048 −0.422024 0.906585i $$-0.638680\pi$$
−0.422024 + 0.906585i $$0.638680\pi$$
$$762$$ 0 0
$$763$$ −3798.17 −0.180214
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8755.07 0.412161
$$768$$ 0 0
$$769$$ −17318.8 −0.812134 −0.406067 0.913843i $$-0.633100\pi$$
−0.406067 + 0.913843i $$0.633100\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −6232.89 −0.290015 −0.145007 0.989431i $$-0.546321\pi$$
−0.145007 + 0.989431i $$0.546321\pi$$
$$774$$ 0 0
$$775$$ 900.502 0.0417381
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5244.42 −0.241208
$$780$$ 0 0
$$781$$ −7716.02 −0.353522
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −31255.4 −1.42109
$$786$$ 0 0
$$787$$ −18625.8 −0.843634 −0.421817 0.906681i $$-0.638607\pi$$
−0.421817 + 0.906681i $$0.638607\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 13023.2 0.585400
$$792$$ 0 0
$$793$$ −10962.0 −0.490885
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 39096.7 1.73761 0.868805 0.495155i $$-0.164889\pi$$
0.868805 + 0.495155i $$0.164889\pi$$
$$798$$ 0 0
$$799$$ −8478.59 −0.375408
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 15493.3 0.680881
$$804$$ 0 0
$$805$$ −13620.6 −0.596352
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 13821.7 0.600672 0.300336 0.953833i $$-0.402901\pi$$
0.300336 + 0.953833i $$0.402901\pi$$
$$810$$ 0 0
$$811$$ −7106.62 −0.307703 −0.153852 0.988094i $$-0.549168\pi$$
−0.153852 + 0.988094i $$0.549168\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 48531.3 2.08586
$$816$$ 0 0
$$817$$ −5950.42 −0.254809
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22099.3 0.939427 0.469714 0.882819i $$-0.344357\pi$$
0.469714 + 0.882819i $$0.344357\pi$$
$$822$$ 0 0
$$823$$ −11652.0 −0.493517 −0.246758 0.969077i $$-0.579365\pi$$
−0.246758 + 0.969077i $$0.579365\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14728.5 0.619298 0.309649 0.950851i $$-0.399788\pi$$
0.309649 + 0.950851i $$0.399788\pi$$
$$828$$ 0 0
$$829$$ −11640.5 −0.487684 −0.243842 0.969815i $$-0.578408\pi$$
−0.243842 + 0.969815i $$0.578408\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −7480.31 −0.311137
$$834$$ 0 0
$$835$$ 48377.1 2.00498
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −17270.0 −0.710638 −0.355319 0.934745i $$-0.615628\pi$$
−0.355319 + 0.934745i $$0.615628\pi$$
$$840$$ 0 0
$$841$$ 841.000 0.0344828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 30670.4 1.24863
$$846$$ 0 0
$$847$$ −4509.98 −0.182957
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4147.07 0.167050
$$852$$ 0 0
$$853$$ 12968.3 0.520547 0.260274 0.965535i $$-0.416187\pi$$
0.260274 + 0.965535i $$0.416187\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 9597.62 0.382554 0.191277 0.981536i $$-0.438737\pi$$
0.191277 + 0.981536i $$0.438737\pi$$
$$858$$ 0 0
$$859$$ 18055.5 0.717164 0.358582 0.933498i $$-0.383260\pi$$
0.358582 + 0.933498i $$0.383260\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1068.84 0.0421594 0.0210797 0.999778i $$-0.493290\pi$$
0.0210797 + 0.999778i $$0.493290\pi$$
$$864$$ 0 0
$$865$$ 57359.3 2.25465
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3397.75 −0.132636
$$870$$ 0 0
$$871$$ 5354.97 0.208319
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1738.84 −0.0671810
$$876$$ 0 0
$$877$$ 50968.4 1.96246 0.981231 0.192837i $$-0.0617688\pi$$
0.981231 + 0.192837i $$0.0617688\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11100.3 0.424495 0.212247 0.977216i $$-0.431922\pi$$
0.212247 + 0.977216i $$0.431922\pi$$
$$882$$ 0 0
$$883$$ 24959.0 0.951232 0.475616 0.879653i $$-0.342225\pi$$
0.475616 + 0.879653i $$0.342225\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 22444.1 0.849604 0.424802 0.905286i $$-0.360344\pi$$
0.424802 + 0.905286i $$0.360344\pi$$
$$888$$ 0 0
$$889$$ 2416.36 0.0911609
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 4222.93 0.158247
$$894$$ 0 0
$$895$$ 74756.3 2.79199
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −182.023 −0.00675283
$$900$$ 0 0
$$901$$ 1401.87 0.0518347
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −5349.93 −0.196506
$$906$$ 0 0
$$907$$ 18929.6 0.692994 0.346497 0.938051i $$-0.387371\pi$$
0.346497 + 0.938051i $$0.387371\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 32210.8 1.17145 0.585726 0.810509i $$-0.300810\pi$$
0.585726 + 0.810509i $$0.300810\pi$$
$$912$$ 0 0
$$913$$ 25169.0 0.912347
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1197.73 0.0431325
$$918$$ 0 0
$$919$$ −13885.5 −0.498411 −0.249206 0.968451i $$-0.580169\pi$$
−0.249206 + 0.968451i $$0.580169\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −5953.70 −0.212317
$$924$$ 0 0
$$925$$ 4112.65 0.146187
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −46561.6 −1.64439 −0.822194 0.569207i $$-0.807250\pi$$
−0.822194 + 0.569207i $$0.807250\pi$$
$$930$$ 0 0
$$931$$ 3725.71 0.131155
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 9240.05 0.323189
$$936$$ 0 0
$$937$$ −3046.54 −0.106218 −0.0531088 0.998589i $$-0.516913\pi$$
−0.0531088 + 0.998589i $$0.516913\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24897.4 −0.862521 −0.431261 0.902227i $$-0.641931\pi$$
−0.431261 + 0.902227i $$0.641931\pi$$
$$942$$ 0 0
$$943$$ 63125.1 2.17989
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2787.52 −0.0956519 −0.0478259 0.998856i $$-0.515229\pi$$
−0.0478259 + 0.998856i $$0.515229\pi$$
$$948$$ 0 0
$$949$$ 11954.7 0.408920
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −45400.9 −1.54321 −0.771605 0.636102i $$-0.780546\pi$$
−0.771605 + 0.636102i $$0.780546\pi$$
$$954$$ 0 0
$$955$$ 42980.5 1.45635
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1081.90 0.0364300
$$960$$ 0 0
$$961$$ −29751.6 −0.998678
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −14656.1 −0.488907
$$966$$ 0 0
$$967$$ −43256.9 −1.43852 −0.719259 0.694742i $$-0.755519\pi$$
−0.719259 + 0.694742i $$0.755519\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −35734.1 −1.18101 −0.590505 0.807034i $$-0.701072\pi$$
−0.590505 + 0.807034i $$0.701072\pi$$
$$972$$ 0 0
$$973$$ −8390.16 −0.276440
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 15675.2 0.513299 0.256650 0.966505i $$-0.417381\pi$$
0.256650 + 0.966505i $$0.417381\pi$$
$$978$$ 0 0
$$979$$ −19908.8 −0.649937
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −17324.1 −0.562109 −0.281055 0.959692i $$-0.590684\pi$$
−0.281055 + 0.959692i $$0.590684\pi$$
$$984$$ 0 0
$$985$$ 35760.4 1.15677
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 71623.0 2.30281
$$990$$ 0 0
$$991$$ 22853.9 0.732572 0.366286 0.930502i $$-0.380629\pi$$
0.366286 + 0.930502i $$0.380629\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 10418.8 0.331958
$$996$$ 0 0
$$997$$ −20703.7 −0.657667 −0.328834 0.944388i $$-0.606656\pi$$
−0.328834 + 0.944388i $$0.606656\pi$$
$$998$$ 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 2088.4.a.f.1.1 5
3.2 odd 2 232.4.a.d.1.4 5
12.11 even 2 464.4.a.m.1.2 5
24.5 odd 2 1856.4.a.z.1.2 5
24.11 even 2 1856.4.a.ba.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.d.1.4 5 3.2 odd 2
464.4.a.m.1.2 5 12.11 even 2
1856.4.a.z.1.2 5 24.5 odd 2
1856.4.a.ba.1.4 5 24.11 even 2
2088.4.a.f.1.1 5 1.1 even 1 trivial