# Properties

 Label 1176.4.a.bd.1.2 Level $1176$ Weight $4$ Character 1176.1 Self dual yes Analytic conductor $69.386$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.3862461668$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-8.69515$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +54.0066 q^{11} -50.2350 q^{13} +0.385773 q^{15} -131.487 q^{17} +91.5094 q^{19} +179.487 q^{23} -124.983 q^{25} +27.0000 q^{27} -69.8961 q^{29} +326.847 q^{31} +162.020 q^{33} +301.698 q^{37} -150.705 q^{39} +296.048 q^{41} -144.302 q^{43} +1.15732 q^{45} -360.085 q^{47} -394.462 q^{51} +1.83513 q^{53} +6.94477 q^{55} +274.528 q^{57} -53.2386 q^{59} -108.121 q^{61} -6.45978 q^{65} +842.007 q^{67} +538.462 q^{69} -241.111 q^{71} -206.983 q^{73} -374.950 q^{75} +559.962 q^{79} +81.0000 q^{81} +986.652 q^{83} -16.9081 q^{85} -209.688 q^{87} -443.366 q^{89} +980.541 q^{93} +11.7673 q^{95} -740.815 q^{97} +486.059 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 4q^{5} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 4q^{5} + 36q^{9} + 14q^{11} + 22q^{13} + 12q^{15} + 96q^{17} - 26q^{19} + 96q^{23} + 110q^{25} + 108q^{27} - 76q^{29} + 238q^{31} + 42q^{33} + 562q^{37} + 66q^{39} + 428q^{41} - 258q^{43} + 36q^{45} - 80q^{47} + 288q^{51} + 1476q^{55} - 78q^{57} + 262q^{59} - 276q^{61} + 2196q^{65} + 150q^{67} + 288q^{69} - 848q^{71} - 218q^{73} + 330q^{75} + 1762q^{79} + 324q^{81} + 3450q^{83} + 1452q^{85} - 228q^{87} - 344q^{89} + 714q^{93} + 2004q^{95} - 622q^{97} + 126q^{99} + O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 0.128591 0.0115015 0.00575077 0.999983i $$-0.498169\pi$$
0.00575077 + 0.999983i $$0.498169\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 54.0066 1.48033 0.740163 0.672427i $$-0.234748\pi$$
0.740163 + 0.672427i $$0.234748\pi$$
$$12$$ 0 0
$$13$$ −50.2350 −1.07175 −0.535873 0.844299i $$-0.680017\pi$$
−0.535873 + 0.844299i $$0.680017\pi$$
$$14$$ 0 0
$$15$$ 0.385773 0.00664042
$$16$$ 0 0
$$17$$ −131.487 −1.87590 −0.937951 0.346767i $$-0.887279\pi$$
−0.937951 + 0.346767i $$0.887279\pi$$
$$18$$ 0 0
$$19$$ 91.5094 1.10493 0.552466 0.833536i $$-0.313687\pi$$
0.552466 + 0.833536i $$0.313687\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 179.487 1.62720 0.813602 0.581423i $$-0.197504\pi$$
0.813602 + 0.581423i $$0.197504\pi$$
$$24$$ 0 0
$$25$$ −124.983 −0.999868
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −69.8961 −0.447565 −0.223782 0.974639i $$-0.571841\pi$$
−0.223782 + 0.974639i $$0.571841\pi$$
$$30$$ 0 0
$$31$$ 326.847 1.89366 0.946829 0.321736i $$-0.104266\pi$$
0.946829 + 0.321736i $$0.104266\pi$$
$$32$$ 0 0
$$33$$ 162.020 0.854667
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 301.698 1.34051 0.670254 0.742132i $$-0.266185\pi$$
0.670254 + 0.742132i $$0.266185\pi$$
$$38$$ 0 0
$$39$$ −150.705 −0.618773
$$40$$ 0 0
$$41$$ 296.048 1.12768 0.563840 0.825884i $$-0.309324\pi$$
0.563840 + 0.825884i $$0.309324\pi$$
$$42$$ 0 0
$$43$$ −144.302 −0.511764 −0.255882 0.966708i $$-0.582366\pi$$
−0.255882 + 0.966708i $$0.582366\pi$$
$$44$$ 0 0
$$45$$ 1.15732 0.00383385
$$46$$ 0 0
$$47$$ −360.085 −1.11753 −0.558764 0.829326i $$-0.688724\pi$$
−0.558764 + 0.829326i $$0.688724\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −394.462 −1.08305
$$52$$ 0 0
$$53$$ 1.83513 0.00475613 0.00237807 0.999997i $$-0.499243\pi$$
0.00237807 + 0.999997i $$0.499243\pi$$
$$54$$ 0 0
$$55$$ 6.94477 0.0170260
$$56$$ 0 0
$$57$$ 274.528 0.637932
$$58$$ 0 0
$$59$$ −53.2386 −0.117476 −0.0587379 0.998273i $$-0.518708\pi$$
−0.0587379 + 0.998273i $$0.518708\pi$$
$$60$$ 0 0
$$61$$ −108.121 −0.226942 −0.113471 0.993541i $$-0.536197\pi$$
−0.113471 + 0.993541i $$0.536197\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.45978 −0.0123267
$$66$$ 0 0
$$67$$ 842.007 1.53534 0.767668 0.640848i $$-0.221417\pi$$
0.767668 + 0.640848i $$0.221417\pi$$
$$68$$ 0 0
$$69$$ 538.462 0.939466
$$70$$ 0 0
$$71$$ −241.111 −0.403023 −0.201511 0.979486i $$-0.564585\pi$$
−0.201511 + 0.979486i $$0.564585\pi$$
$$72$$ 0 0
$$73$$ −206.983 −0.331857 −0.165929 0.986138i $$-0.553062\pi$$
−0.165929 + 0.986138i $$0.553062\pi$$
$$74$$ 0 0
$$75$$ −374.950 −0.577274
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 559.962 0.797477 0.398738 0.917065i $$-0.369448\pi$$
0.398738 + 0.917065i $$0.369448\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 986.652 1.30481 0.652404 0.757871i $$-0.273760\pi$$
0.652404 + 0.757871i $$0.273760\pi$$
$$84$$ 0 0
$$85$$ −16.9081 −0.0215758
$$86$$ 0 0
$$87$$ −209.688 −0.258402
$$88$$ 0 0
$$89$$ −443.366 −0.528053 −0.264026 0.964515i $$-0.585051\pi$$
−0.264026 + 0.964515i $$0.585051\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 980.541 1.09330
$$94$$ 0 0
$$95$$ 11.7673 0.0127084
$$96$$ 0 0
$$97$$ −740.815 −0.775447 −0.387723 0.921776i $$-0.626738\pi$$
−0.387723 + 0.921776i $$0.626738\pi$$
$$98$$ 0 0
$$99$$ 486.059 0.493442
$$100$$ 0 0
$$101$$ 743.777 0.732758 0.366379 0.930466i $$-0.380597\pi$$
0.366379 + 0.930466i $$0.380597\pi$$
$$102$$ 0 0
$$103$$ 104.651 0.100112 0.0500559 0.998746i $$-0.484060\pi$$
0.0500559 + 0.998746i $$0.484060\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 513.700 0.464124 0.232062 0.972701i $$-0.425453\pi$$
0.232062 + 0.972701i $$0.425453\pi$$
$$108$$ 0 0
$$109$$ 975.038 0.856805 0.428402 0.903588i $$-0.359077\pi$$
0.428402 + 0.903588i $$0.359077\pi$$
$$110$$ 0 0
$$111$$ 905.093 0.773942
$$112$$ 0 0
$$113$$ 1926.07 1.60345 0.801723 0.597696i $$-0.203917\pi$$
0.801723 + 0.597696i $$0.203917\pi$$
$$114$$ 0 0
$$115$$ 23.0805 0.0187153
$$116$$ 0 0
$$117$$ −452.115 −0.357248
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1585.71 1.19137
$$122$$ 0 0
$$123$$ 888.143 0.651066
$$124$$ 0 0
$$125$$ −32.1457 −0.0230016
$$126$$ 0 0
$$127$$ −1125.95 −0.786709 −0.393355 0.919387i $$-0.628686\pi$$
−0.393355 + 0.919387i $$0.628686\pi$$
$$128$$ 0 0
$$129$$ −432.906 −0.295467
$$130$$ 0 0
$$131$$ 1493.24 0.995918 0.497959 0.867201i $$-0.334083\pi$$
0.497959 + 0.867201i $$0.334083\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.47196 0.00221347
$$136$$ 0 0
$$137$$ 1460.77 0.910965 0.455483 0.890245i $$-0.349467\pi$$
0.455483 + 0.890245i $$0.349467\pi$$
$$138$$ 0 0
$$139$$ 2225.85 1.35823 0.679116 0.734031i $$-0.262363\pi$$
0.679116 + 0.734031i $$0.262363\pi$$
$$140$$ 0 0
$$141$$ −1080.26 −0.645206
$$142$$ 0 0
$$143$$ −2713.02 −1.58653
$$144$$ 0 0
$$145$$ −8.98802 −0.00514769
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 394.681 0.217003 0.108502 0.994096i $$-0.465395\pi$$
0.108502 + 0.994096i $$0.465395\pi$$
$$150$$ 0 0
$$151$$ −3124.20 −1.68373 −0.841867 0.539685i $$-0.818543\pi$$
−0.841867 + 0.539685i $$0.818543\pi$$
$$152$$ 0 0
$$153$$ −1183.39 −0.625301
$$154$$ 0 0
$$155$$ 42.0296 0.0217800
$$156$$ 0 0
$$157$$ −3600.71 −1.83037 −0.915183 0.403038i $$-0.867954\pi$$
−0.915183 + 0.403038i $$0.867954\pi$$
$$158$$ 0 0
$$159$$ 5.50540 0.00274596
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1974.02 0.948573 0.474287 0.880370i $$-0.342706\pi$$
0.474287 + 0.880370i $$0.342706\pi$$
$$164$$ 0 0
$$165$$ 20.8343 0.00982999
$$166$$ 0 0
$$167$$ 1067.09 0.494453 0.247227 0.968958i $$-0.420481\pi$$
0.247227 + 0.968958i $$0.420481\pi$$
$$168$$ 0 0
$$169$$ 326.559 0.148638
$$170$$ 0 0
$$171$$ 823.584 0.368310
$$172$$ 0 0
$$173$$ −274.655 −0.120703 −0.0603515 0.998177i $$-0.519222\pi$$
−0.0603515 + 0.998177i $$0.519222\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −159.716 −0.0678247
$$178$$ 0 0
$$179$$ 554.332 0.231468 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$180$$ 0 0
$$181$$ −685.436 −0.281481 −0.140741 0.990047i $$-0.544948\pi$$
−0.140741 + 0.990047i $$0.544948\pi$$
$$182$$ 0 0
$$183$$ −324.363 −0.131025
$$184$$ 0 0
$$185$$ 38.7956 0.0154179
$$186$$ 0 0
$$187$$ −7101.18 −2.77695
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4898.87 1.85586 0.927932 0.372750i $$-0.121585\pi$$
0.927932 + 0.372750i $$0.121585\pi$$
$$192$$ 0 0
$$193$$ 3140.26 1.17120 0.585598 0.810601i $$-0.300860\pi$$
0.585598 + 0.810601i $$0.300860\pi$$
$$194$$ 0 0
$$195$$ −19.3793 −0.00711684
$$196$$ 0 0
$$197$$ −227.412 −0.0822460 −0.0411230 0.999154i $$-0.513094\pi$$
−0.0411230 + 0.999154i $$0.513094\pi$$
$$198$$ 0 0
$$199$$ −1214.50 −0.432631 −0.216316 0.976323i $$-0.569404\pi$$
−0.216316 + 0.976323i $$0.569404\pi$$
$$200$$ 0 0
$$201$$ 2526.02 0.886427
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 38.0691 0.0129700
$$206$$ 0 0
$$207$$ 1615.39 0.542401
$$208$$ 0 0
$$209$$ 4942.11 1.63566
$$210$$ 0 0
$$211$$ 5116.07 1.66922 0.834608 0.550844i $$-0.185694\pi$$
0.834608 + 0.550844i $$0.185694\pi$$
$$212$$ 0 0
$$213$$ −723.333 −0.232685
$$214$$ 0 0
$$215$$ −18.5560 −0.00588608
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −620.950 −0.191598
$$220$$ 0 0
$$221$$ 6605.27 2.01049
$$222$$ 0 0
$$223$$ 119.384 0.0358499 0.0179250 0.999839i $$-0.494294\pi$$
0.0179250 + 0.999839i $$0.494294\pi$$
$$224$$ 0 0
$$225$$ −1124.85 −0.333289
$$226$$ 0 0
$$227$$ 1992.00 0.582439 0.291220 0.956656i $$-0.405939\pi$$
0.291220 + 0.956656i $$0.405939\pi$$
$$228$$ 0 0
$$229$$ 703.476 0.203000 0.101500 0.994836i $$-0.467636\pi$$
0.101500 + 0.994836i $$0.467636\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 961.783 0.270423 0.135211 0.990817i $$-0.456829\pi$$
0.135211 + 0.990817i $$0.456829\pi$$
$$234$$ 0 0
$$235$$ −46.3038 −0.0128533
$$236$$ 0 0
$$237$$ 1679.89 0.460423
$$238$$ 0 0
$$239$$ −4464.71 −1.20836 −0.604179 0.796848i $$-0.706499\pi$$
−0.604179 + 0.796848i $$0.706499\pi$$
$$240$$ 0 0
$$241$$ −435.311 −0.116352 −0.0581761 0.998306i $$-0.518528\pi$$
−0.0581761 + 0.998306i $$0.518528\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4596.98 −1.18421
$$248$$ 0 0
$$249$$ 2959.95 0.753331
$$250$$ 0 0
$$251$$ 863.003 0.217021 0.108510 0.994095i $$-0.465392\pi$$
0.108510 + 0.994095i $$0.465392\pi$$
$$252$$ 0 0
$$253$$ 9693.49 2.40879
$$254$$ 0 0
$$255$$ −50.7243 −0.0124568
$$256$$ 0 0
$$257$$ −536.693 −0.130264 −0.0651322 0.997877i $$-0.520747\pi$$
−0.0651322 + 0.997877i $$0.520747\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −629.065 −0.149188
$$262$$ 0 0
$$263$$ 371.908 0.0871971 0.0435986 0.999049i $$-0.486118\pi$$
0.0435986 + 0.999049i $$0.486118\pi$$
$$264$$ 0 0
$$265$$ 0.235982 5.47029e−5 0
$$266$$ 0 0
$$267$$ −1330.10 −0.304871
$$268$$ 0 0
$$269$$ −4505.61 −1.02123 −0.510616 0.859809i $$-0.670583\pi$$
−0.510616 + 0.859809i $$0.670583\pi$$
$$270$$ 0 0
$$271$$ −396.115 −0.0887908 −0.0443954 0.999014i $$-0.514136\pi$$
−0.0443954 + 0.999014i $$0.514136\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6749.93 −1.48013
$$276$$ 0 0
$$277$$ −6508.74 −1.41181 −0.705907 0.708305i $$-0.749460\pi$$
−0.705907 + 0.708305i $$0.749460\pi$$
$$278$$ 0 0
$$279$$ 2941.62 0.631220
$$280$$ 0 0
$$281$$ 2785.80 0.591413 0.295707 0.955279i $$-0.404445\pi$$
0.295707 + 0.955279i $$0.404445\pi$$
$$282$$ 0 0
$$283$$ 0.611846 0.000128518 0 6.42588e−5 1.00000i $$-0.499980\pi$$
6.42588e−5 1.00000i $$0.499980\pi$$
$$284$$ 0 0
$$285$$ 35.3019 0.00733720
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 12375.9 2.51901
$$290$$ 0 0
$$291$$ −2222.44 −0.447704
$$292$$ 0 0
$$293$$ −4145.98 −0.826657 −0.413329 0.910582i $$-0.635634\pi$$
−0.413329 + 0.910582i $$0.635634\pi$$
$$294$$ 0 0
$$295$$ −6.84601 −0.00135115
$$296$$ 0 0
$$297$$ 1458.18 0.284889
$$298$$ 0 0
$$299$$ −9016.55 −1.74395
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2231.33 0.423058
$$304$$ 0 0
$$305$$ −13.9034 −0.00261019
$$306$$ 0 0
$$307$$ 1960.53 0.364473 0.182236 0.983255i $$-0.441666\pi$$
0.182236 + 0.983255i $$0.441666\pi$$
$$308$$ 0 0
$$309$$ 313.952 0.0577996
$$310$$ 0 0
$$311$$ 5207.57 0.949499 0.474749 0.880121i $$-0.342539\pi$$
0.474749 + 0.880121i $$0.342539\pi$$
$$312$$ 0 0
$$313$$ −3991.71 −0.720846 −0.360423 0.932789i $$-0.617368\pi$$
−0.360423 + 0.932789i $$0.617368\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1805.04 0.319814 0.159907 0.987132i $$-0.448881\pi$$
0.159907 + 0.987132i $$0.448881\pi$$
$$318$$ 0 0
$$319$$ −3774.85 −0.662543
$$320$$ 0 0
$$321$$ 1541.10 0.267962
$$322$$ 0 0
$$323$$ −12032.3 −2.07274
$$324$$ 0 0
$$325$$ 6278.55 1.07160
$$326$$ 0 0
$$327$$ 2925.11 0.494676
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 6906.18 1.14682 0.573411 0.819268i $$-0.305620\pi$$
0.573411 + 0.819268i $$0.305620\pi$$
$$332$$ 0 0
$$333$$ 2715.28 0.446836
$$334$$ 0 0
$$335$$ 108.275 0.0176587
$$336$$ 0 0
$$337$$ −6081.36 −0.983006 −0.491503 0.870876i $$-0.663552\pi$$
−0.491503 + 0.870876i $$0.663552\pi$$
$$338$$ 0 0
$$339$$ 5778.21 0.925750
$$340$$ 0 0
$$341$$ 17651.9 2.80323
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 69.2414 0.0108053
$$346$$ 0 0
$$347$$ −7787.02 −1.20469 −0.602347 0.798234i $$-0.705768\pi$$
−0.602347 + 0.798234i $$0.705768\pi$$
$$348$$ 0 0
$$349$$ −1928.25 −0.295751 −0.147875 0.989006i $$-0.547243\pi$$
−0.147875 + 0.989006i $$0.547243\pi$$
$$350$$ 0 0
$$351$$ −1356.35 −0.206258
$$352$$ 0 0
$$353$$ −3404.27 −0.513289 −0.256645 0.966506i $$-0.582617\pi$$
−0.256645 + 0.966506i $$0.582617\pi$$
$$354$$ 0 0
$$355$$ −31.0047 −0.00463538
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4417.05 0.649368 0.324684 0.945823i $$-0.394742\pi$$
0.324684 + 0.945823i $$0.394742\pi$$
$$360$$ 0 0
$$361$$ 1514.97 0.220873
$$362$$ 0 0
$$363$$ 4757.13 0.687837
$$364$$ 0 0
$$365$$ −26.6162 −0.00381687
$$366$$ 0 0
$$367$$ −5076.65 −0.722068 −0.361034 0.932553i $$-0.617576\pi$$
−0.361034 + 0.932553i $$0.617576\pi$$
$$368$$ 0 0
$$369$$ 2664.43 0.375893
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6876.64 −0.954582 −0.477291 0.878745i $$-0.658381\pi$$
−0.477291 + 0.878745i $$0.658381\pi$$
$$374$$ 0 0
$$375$$ −96.4370 −0.0132800
$$376$$ 0 0
$$377$$ 3511.23 0.479676
$$378$$ 0 0
$$379$$ −9285.61 −1.25850 −0.629248 0.777205i $$-0.716637\pi$$
−0.629248 + 0.777205i $$0.716637\pi$$
$$380$$ 0 0
$$381$$ −3377.85 −0.454207
$$382$$ 0 0
$$383$$ −7681.65 −1.02484 −0.512420 0.858735i $$-0.671251\pi$$
−0.512420 + 0.858735i $$0.671251\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1298.72 −0.170588
$$388$$ 0 0
$$389$$ 8627.72 1.12453 0.562266 0.826956i $$-0.309930\pi$$
0.562266 + 0.826956i $$0.309930\pi$$
$$390$$ 0 0
$$391$$ −23600.3 −3.05247
$$392$$ 0 0
$$393$$ 4479.73 0.574993
$$394$$ 0 0
$$395$$ 72.0062 0.00917221
$$396$$ 0 0
$$397$$ 5735.61 0.725094 0.362547 0.931966i $$-0.381907\pi$$
0.362547 + 0.931966i $$0.381907\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7217.79 −0.898850 −0.449425 0.893318i $$-0.648371\pi$$
−0.449425 + 0.893318i $$0.648371\pi$$
$$402$$ 0 0
$$403$$ −16419.2 −2.02952
$$404$$ 0 0
$$405$$ 10.4159 0.00127795
$$406$$ 0 0
$$407$$ 16293.7 1.98439
$$408$$ 0 0
$$409$$ −10802.9 −1.30604 −0.653019 0.757341i $$-0.726498\pi$$
−0.653019 + 0.757341i $$0.726498\pi$$
$$410$$ 0 0
$$411$$ 4382.32 0.525946
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 126.875 0.0150073
$$416$$ 0 0
$$417$$ 6677.55 0.784175
$$418$$ 0 0
$$419$$ 13257.1 1.54571 0.772856 0.634582i $$-0.218828\pi$$
0.772856 + 0.634582i $$0.218828\pi$$
$$420$$ 0 0
$$421$$ −6252.11 −0.723774 −0.361887 0.932222i $$-0.617867\pi$$
−0.361887 + 0.932222i $$0.617867\pi$$
$$422$$ 0 0
$$423$$ −3240.77 −0.372510
$$424$$ 0 0
$$425$$ 16433.7 1.87565
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −8139.07 −0.915986
$$430$$ 0 0
$$431$$ 4949.08 0.553106 0.276553 0.960999i $$-0.410808\pi$$
0.276553 + 0.960999i $$0.410808\pi$$
$$432$$ 0 0
$$433$$ −16602.8 −1.84267 −0.921337 0.388764i $$-0.872902\pi$$
−0.921337 + 0.388764i $$0.872902\pi$$
$$434$$ 0 0
$$435$$ −26.9641 −0.00297202
$$436$$ 0 0
$$437$$ 16424.8 1.79795
$$438$$ 0 0
$$439$$ −4708.38 −0.511887 −0.255944 0.966692i $$-0.582386\pi$$
−0.255944 + 0.966692i $$0.582386\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1700.44 −0.182370 −0.0911852 0.995834i $$-0.529066\pi$$
−0.0911852 + 0.995834i $$0.529066\pi$$
$$444$$ 0 0
$$445$$ −57.0129 −0.00607342
$$446$$ 0 0
$$447$$ 1184.04 0.125287
$$448$$ 0 0
$$449$$ 10050.2 1.05635 0.528173 0.849137i $$-0.322877\pi$$
0.528173 + 0.849137i $$0.322877\pi$$
$$450$$ 0 0
$$451$$ 15988.5 1.66933
$$452$$ 0 0
$$453$$ −9372.60 −0.972104
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10991.9 −1.12512 −0.562560 0.826757i $$-0.690183\pi$$
−0.562560 + 0.826757i $$0.690183\pi$$
$$458$$ 0 0
$$459$$ −3550.16 −0.361018
$$460$$ 0 0
$$461$$ −548.440 −0.0554086 −0.0277043 0.999616i $$-0.508820\pi$$
−0.0277043 + 0.999616i $$0.508820\pi$$
$$462$$ 0 0
$$463$$ 4028.04 0.404317 0.202159 0.979353i $$-0.435204\pi$$
0.202159 + 0.979353i $$0.435204\pi$$
$$464$$ 0 0
$$465$$ 126.089 0.0125747
$$466$$ 0 0
$$467$$ 2070.69 0.205182 0.102591 0.994724i $$-0.467287\pi$$
0.102591 + 0.994724i $$0.467287\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −10802.1 −1.05676
$$472$$ 0 0
$$473$$ −7793.26 −0.757579
$$474$$ 0 0
$$475$$ −11437.2 −1.10479
$$476$$ 0 0
$$477$$ 16.5162 0.00158538
$$478$$ 0 0
$$479$$ −178.785 −0.0170541 −0.00852704 0.999964i $$-0.502714\pi$$
−0.00852704 + 0.999964i $$0.502714\pi$$
$$480$$ 0 0
$$481$$ −15155.8 −1.43668
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −95.2622 −0.00891883
$$486$$ 0 0
$$487$$ −16597.0 −1.54431 −0.772156 0.635433i $$-0.780822\pi$$
−0.772156 + 0.635433i $$0.780822\pi$$
$$488$$ 0 0
$$489$$ 5922.07 0.547659
$$490$$ 0 0
$$491$$ −4547.22 −0.417949 −0.208975 0.977921i $$-0.567013\pi$$
−0.208975 + 0.977921i $$0.567013\pi$$
$$492$$ 0 0
$$493$$ 9190.45 0.839588
$$494$$ 0 0
$$495$$ 62.5029 0.00567535
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1025.97 0.0920414 0.0460207 0.998940i $$-0.485346\pi$$
0.0460207 + 0.998940i $$0.485346\pi$$
$$500$$ 0 0
$$501$$ 3201.26 0.285473
$$502$$ 0 0
$$503$$ 6745.26 0.597925 0.298962 0.954265i $$-0.403359\pi$$
0.298962 + 0.954265i $$0.403359\pi$$
$$504$$ 0 0
$$505$$ 95.6431 0.00842785
$$506$$ 0 0
$$507$$ 979.676 0.0858164
$$508$$ 0 0
$$509$$ −6001.02 −0.522575 −0.261287 0.965261i $$-0.584147\pi$$
−0.261287 + 0.965261i $$0.584147\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 2470.75 0.212644
$$514$$ 0 0
$$515$$ 13.4571 0.00115144
$$516$$ 0 0
$$517$$ −19447.0 −1.65431
$$518$$ 0 0
$$519$$ −823.964 −0.0696879
$$520$$ 0 0
$$521$$ 4695.21 0.394820 0.197410 0.980321i $$-0.436747\pi$$
0.197410 + 0.980321i $$0.436747\pi$$
$$522$$ 0 0
$$523$$ 771.767 0.0645258 0.0322629 0.999479i $$-0.489729\pi$$
0.0322629 + 0.999479i $$0.489729\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −42976.2 −3.55232
$$528$$ 0 0
$$529$$ 20048.7 1.64779
$$530$$ 0 0
$$531$$ −479.147 −0.0391586
$$532$$ 0 0
$$533$$ −14872.0 −1.20859
$$534$$ 0 0
$$535$$ 66.0573 0.00533814
$$536$$ 0 0
$$537$$ 1663.00 0.133638
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 3385.66 0.269059 0.134529 0.990910i $$-0.457048\pi$$
0.134529 + 0.990910i $$0.457048\pi$$
$$542$$ 0 0
$$543$$ −2056.31 −0.162513
$$544$$ 0 0
$$545$$ 125.381 0.00985457
$$546$$ 0 0
$$547$$ 4988.75 0.389952 0.194976 0.980808i $$-0.437537\pi$$
0.194976 + 0.980808i $$0.437537\pi$$
$$548$$ 0 0
$$549$$ −973.090 −0.0756475
$$550$$ 0 0
$$551$$ −6396.15 −0.494529
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 116.387 0.00890153
$$556$$ 0 0
$$557$$ −20207.3 −1.53718 −0.768591 0.639740i $$-0.779042\pi$$
−0.768591 + 0.639740i $$0.779042\pi$$
$$558$$ 0 0
$$559$$ 7249.02 0.548481
$$560$$ 0 0
$$561$$ −21303.5 −1.60327
$$562$$ 0 0
$$563$$ −11690.4 −0.875118 −0.437559 0.899190i $$-0.644157\pi$$
−0.437559 + 0.899190i $$0.644157\pi$$
$$564$$ 0 0
$$565$$ 247.675 0.0184421
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −17928.9 −1.32095 −0.660473 0.750849i $$-0.729644\pi$$
−0.660473 + 0.750849i $$0.729644\pi$$
$$570$$ 0 0
$$571$$ −15673.7 −1.14873 −0.574364 0.818600i $$-0.694751\pi$$
−0.574364 + 0.818600i $$0.694751\pi$$
$$572$$ 0 0
$$573$$ 14696.6 1.07148
$$574$$ 0 0
$$575$$ −22432.9 −1.62699
$$576$$ 0 0
$$577$$ −13653.0 −0.985064 −0.492532 0.870294i $$-0.663929\pi$$
−0.492532 + 0.870294i $$0.663929\pi$$
$$578$$ 0 0
$$579$$ 9420.78 0.676191
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 99.1093 0.00704063
$$584$$ 0 0
$$585$$ −58.1380 −0.00410891
$$586$$ 0 0
$$587$$ −18021.5 −1.26717 −0.633583 0.773675i $$-0.718417\pi$$
−0.633583 + 0.773675i $$0.718417\pi$$
$$588$$ 0 0
$$589$$ 29909.6 2.09236
$$590$$ 0 0
$$591$$ −682.237 −0.0474847
$$592$$ 0 0
$$593$$ 21137.3 1.46375 0.731877 0.681437i $$-0.238645\pi$$
0.731877 + 0.681437i $$0.238645\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −3643.50 −0.249780
$$598$$ 0 0
$$599$$ 9369.59 0.639117 0.319559 0.947567i $$-0.396465\pi$$
0.319559 + 0.947567i $$0.396465\pi$$
$$600$$ 0 0
$$601$$ −22750.0 −1.54408 −0.772040 0.635573i $$-0.780764\pi$$
−0.772040 + 0.635573i $$0.780764\pi$$
$$602$$ 0 0
$$603$$ 7578.06 0.511779
$$604$$ 0 0
$$605$$ 203.908 0.0137026
$$606$$ 0 0
$$607$$ −5973.65 −0.399445 −0.199722 0.979853i $$-0.564004\pi$$
−0.199722 + 0.979853i $$0.564004\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18088.9 1.19771
$$612$$ 0 0
$$613$$ −23347.4 −1.53832 −0.769162 0.639054i $$-0.779326\pi$$
−0.769162 + 0.639054i $$0.779326\pi$$
$$614$$ 0 0
$$615$$ 114.207 0.00748826
$$616$$ 0 0
$$617$$ 28199.1 1.83996 0.919979 0.391968i $$-0.128206\pi$$
0.919979 + 0.391968i $$0.128206\pi$$
$$618$$ 0 0
$$619$$ −2975.56 −0.193211 −0.0966057 0.995323i $$-0.530799\pi$$
−0.0966057 + 0.995323i $$0.530799\pi$$
$$620$$ 0 0
$$621$$ 4846.16 0.313155
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15618.8 0.999603
$$626$$ 0 0
$$627$$ 14826.3 0.944348
$$628$$ 0 0
$$629$$ −39669.4 −2.51466
$$630$$ 0 0
$$631$$ −2631.33 −0.166009 −0.0830044 0.996549i $$-0.526452\pi$$
−0.0830044 + 0.996549i $$0.526452\pi$$
$$632$$ 0 0
$$633$$ 15348.2 0.963723
$$634$$ 0 0
$$635$$ −144.787 −0.00904837
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2170.00 −0.134341
$$640$$ 0 0
$$641$$ 14494.2 0.893117 0.446559 0.894754i $$-0.352649\pi$$
0.446559 + 0.894754i $$0.352649\pi$$
$$642$$ 0 0
$$643$$ −15176.0 −0.930767 −0.465383 0.885109i $$-0.654084\pi$$
−0.465383 + 0.885109i $$0.654084\pi$$
$$644$$ 0 0
$$645$$ −55.6679 −0.00339833
$$646$$ 0 0
$$647$$ −7569.18 −0.459931 −0.229965 0.973199i $$-0.573861\pi$$
−0.229965 + 0.973199i $$0.573861\pi$$
$$648$$ 0 0
$$649$$ −2875.23 −0.173903
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13933.5 −0.835007 −0.417504 0.908675i $$-0.637095\pi$$
−0.417504 + 0.908675i $$0.637095\pi$$
$$654$$ 0 0
$$655$$ 192.018 0.0114546
$$656$$ 0 0
$$657$$ −1862.85 −0.110619
$$658$$ 0 0
$$659$$ −17015.5 −1.00581 −0.502904 0.864342i $$-0.667735\pi$$
−0.502904 + 0.864342i $$0.667735\pi$$
$$660$$ 0 0
$$661$$ −16535.4 −0.972999 −0.486500 0.873681i $$-0.661726\pi$$
−0.486500 + 0.873681i $$0.661726\pi$$
$$662$$ 0 0
$$663$$ 19815.8 1.16076
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12545.5 −0.728279
$$668$$ 0 0
$$669$$ 358.152 0.0206980
$$670$$ 0 0
$$671$$ −5839.25 −0.335949
$$672$$ 0 0
$$673$$ 4571.05 0.261814 0.130907 0.991395i $$-0.458211\pi$$
0.130907 + 0.991395i $$0.458211\pi$$
$$674$$ 0 0
$$675$$ −3374.55 −0.192425
$$676$$ 0 0
$$677$$ 26284.5 1.49217 0.746083 0.665853i $$-0.231932\pi$$
0.746083 + 0.665853i $$0.231932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 5976.00 0.336271
$$682$$ 0 0
$$683$$ 6797.22 0.380803 0.190402 0.981706i $$-0.439021\pi$$
0.190402 + 0.981706i $$0.439021\pi$$
$$684$$ 0 0
$$685$$ 187.842 0.0104775
$$686$$ 0 0
$$687$$ 2110.43 0.117202
$$688$$ 0 0
$$689$$ −92.1880 −0.00509737
$$690$$ 0 0
$$691$$ 27758.7 1.52821 0.764103 0.645094i $$-0.223182\pi$$
0.764103 + 0.645094i $$0.223182\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 286.225 0.0156218
$$696$$ 0 0
$$697$$ −38926.5 −2.11542
$$698$$ 0 0
$$699$$ 2885.35 0.156129
$$700$$ 0 0
$$701$$ −21638.3 −1.16586 −0.582929 0.812523i $$-0.698093\pi$$
−0.582929 + 0.812523i $$0.698093\pi$$
$$702$$ 0 0
$$703$$ 27608.2 1.48117
$$704$$ 0 0
$$705$$ −138.911 −0.00742086
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16288.7 −0.862815 −0.431408 0.902157i $$-0.641983\pi$$
−0.431408 + 0.902157i $$0.641983\pi$$
$$710$$ 0 0
$$711$$ 5039.66 0.265826
$$712$$ 0 0
$$713$$ 58664.8 3.08137
$$714$$ 0 0
$$715$$ −348.871 −0.0182476
$$716$$ 0 0
$$717$$ −13394.1 −0.697646
$$718$$ 0 0
$$719$$ −3391.74 −0.175926 −0.0879628 0.996124i $$-0.528036\pi$$
−0.0879628 + 0.996124i $$0.528036\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −1305.93 −0.0671760
$$724$$ 0 0
$$725$$ 8735.86 0.447506
$$726$$ 0 0
$$727$$ 23158.7 1.18144 0.590722 0.806875i $$-0.298843\pi$$
0.590722 + 0.806875i $$0.298843\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 18973.9 0.960020
$$732$$ 0 0
$$733$$ −27629.2 −1.39223 −0.696116 0.717929i $$-0.745090\pi$$
−0.696116 + 0.717929i $$0.745090\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 45473.9 2.27280
$$738$$ 0 0
$$739$$ −9461.65 −0.470978 −0.235489 0.971877i $$-0.575669\pi$$
−0.235489 + 0.971877i $$0.575669\pi$$
$$740$$ 0 0
$$741$$ −13790.9 −0.683701
$$742$$ 0 0
$$743$$ −14316.9 −0.706913 −0.353457 0.935451i $$-0.614994\pi$$
−0.353457 + 0.935451i $$0.614994\pi$$
$$744$$ 0 0
$$745$$ 50.7524 0.00249587
$$746$$ 0 0
$$747$$ 8879.86 0.434936
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −7486.07 −0.363742 −0.181871 0.983322i $$-0.558215\pi$$
−0.181871 + 0.983322i $$0.558215\pi$$
$$752$$ 0 0
$$753$$ 2589.01 0.125297
$$754$$ 0 0
$$755$$ −401.744 −0.0193655
$$756$$ 0 0
$$757$$ −17416.8 −0.836227 −0.418114 0.908395i $$-0.637309\pi$$
−0.418114 + 0.908395i $$0.637309\pi$$
$$758$$ 0 0
$$759$$ 29080.5 1.39072
$$760$$ 0 0
$$761$$ −33231.0 −1.58295 −0.791474 0.611203i $$-0.790686\pi$$
−0.791474 + 0.611203i $$0.790686\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −152.173 −0.00719192
$$766$$ 0 0
$$767$$ 2674.44 0.125904
$$768$$ 0 0
$$769$$ 13714.2 0.643103 0.321552 0.946892i $$-0.395796\pi$$
0.321552 + 0.946892i $$0.395796\pi$$
$$770$$ 0 0
$$771$$ −1610.08 −0.0752082
$$772$$ 0 0
$$773$$ 15209.3 0.707685 0.353842 0.935305i $$-0.384875\pi$$
0.353842 + 0.935305i $$0.384875\pi$$
$$774$$ 0 0
$$775$$ −40850.5 −1.89341
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 27091.1 1.24601
$$780$$ 0 0
$$781$$ −13021.6 −0.596605
$$782$$ 0 0
$$783$$ −1887.20 −0.0861339
$$784$$ 0 0
$$785$$ −463.019 −0.0210520
$$786$$ 0 0
$$787$$ −13221.5 −0.598850 −0.299425 0.954120i $$-0.596795\pi$$
−0.299425 + 0.954120i $$0.596795\pi$$
$$788$$ 0 0
$$789$$ 1115.72 0.0503433
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 5431.47 0.243224
$$794$$ 0 0
$$795$$ 0.707946 3.15827e−5 0
$$796$$ 0 0
$$797$$ 13373.9 0.594387 0.297194 0.954817i $$-0.403949\pi$$
0.297194 + 0.954817i $$0.403949\pi$$
$$798$$ 0 0
$$799$$ 47346.6 2.09637
$$800$$ 0 0
$$801$$ −3990.29 −0.176018
$$802$$ 0 0
$$803$$ −11178.5 −0.491257
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −13516.8 −0.589609
$$808$$ 0 0
$$809$$ −11229.3 −0.488011 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$810$$ 0 0
$$811$$ −4301.42 −0.186243 −0.0931215 0.995655i $$-0.529685\pi$$
−0.0931215 + 0.995655i $$0.529685\pi$$
$$812$$ 0 0
$$813$$ −1188.35 −0.0512634
$$814$$ 0 0
$$815$$ 253.842 0.0109101
$$816$$ 0 0
$$817$$ −13205.0 −0.565464
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −40910.6 −1.73909 −0.869543 0.493857i $$-0.835587\pi$$
−0.869543 + 0.493857i $$0.835587\pi$$
$$822$$ 0 0
$$823$$ 28043.2 1.18776 0.593879 0.804554i $$-0.297596\pi$$
0.593879 + 0.804554i $$0.297596\pi$$
$$824$$ 0 0
$$825$$ −20249.8 −0.854554
$$826$$ 0 0
$$827$$ 17598.9 0.739993 0.369996 0.929033i $$-0.379359\pi$$
0.369996 + 0.929033i $$0.379359\pi$$
$$828$$ 0 0
$$829$$ −14565.5 −0.610231 −0.305116 0.952315i $$-0.598695\pi$$
−0.305116 + 0.952315i $$0.598695\pi$$
$$830$$ 0 0
$$831$$ −19526.2 −0.815111
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 137.218 0.00568697
$$836$$ 0 0
$$837$$ 8824.86 0.364435
$$838$$ 0 0
$$839$$ 711.056 0.0292591 0.0146295 0.999893i $$-0.495343\pi$$
0.0146295 + 0.999893i $$0.495343\pi$$
$$840$$ 0 0
$$841$$ −19503.5 −0.799686
$$842$$ 0 0
$$843$$ 8357.41 0.341453
$$844$$ 0 0
$$845$$ 41.9925 0.00170957
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 1.83554 7.41996e−5 0
$$850$$ 0 0
$$851$$ 54150.9 2.18128
$$852$$ 0 0
$$853$$ −7459.48 −0.299423 −0.149712 0.988730i $$-0.547834\pi$$
−0.149712 + 0.988730i $$0.547834\pi$$
$$854$$ 0 0
$$855$$ 105.906 0.00423614
$$856$$ 0 0
$$857$$ 3560.92 0.141936 0.0709678 0.997479i $$-0.477391\pi$$
0.0709678 + 0.997479i $$0.477391\pi$$
$$858$$ 0 0
$$859$$ 13519.1 0.536981 0.268491 0.963282i $$-0.413475\pi$$
0.268491 + 0.963282i $$0.413475\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 20395.9 0.804502 0.402251 0.915530i $$-0.368228\pi$$
0.402251 + 0.915530i $$0.368228\pi$$
$$864$$ 0 0
$$865$$ −35.3182 −0.00138827
$$866$$ 0 0
$$867$$ 37127.7 1.45435
$$868$$ 0 0
$$869$$ 30241.6 1.18053
$$870$$ 0 0
$$871$$ −42298.3 −1.64549
$$872$$ 0 0
$$873$$ −6667.33 −0.258482
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 7197.26 0.277120 0.138560 0.990354i $$-0.455753\pi$$
0.138560 + 0.990354i $$0.455753\pi$$
$$878$$ 0 0
$$879$$ −12437.9 −0.477271
$$880$$ 0 0
$$881$$ 11588.3 0.443156 0.221578 0.975143i $$-0.428879\pi$$
0.221578 + 0.975143i $$0.428879\pi$$
$$882$$ 0 0
$$883$$ 21959.0 0.836898 0.418449 0.908240i $$-0.362574\pi$$
0.418449 + 0.908240i $$0.362574\pi$$
$$884$$ 0 0
$$885$$ −20.5380 −0.000780088 0
$$886$$ 0 0
$$887$$ 12518.9 0.473894 0.236947 0.971523i $$-0.423853\pi$$
0.236947 + 0.971523i $$0.423853\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4374.53 0.164481
$$892$$ 0 0
$$893$$ −32951.2 −1.23479
$$894$$ 0 0
$$895$$ 71.2822 0.00266224
$$896$$ 0 0
$$897$$ −27049.6 −1.00687
$$898$$ 0 0
$$899$$ −22845.3 −0.847535
$$900$$ 0 0
$$901$$ −241.297 −0.00892204
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −88.1410 −0.00323747
$$906$$ 0 0
$$907$$ 28270.0 1.03494 0.517470 0.855701i $$-0.326874\pi$$
0.517470 + 0.855701i $$0.326874\pi$$
$$908$$ 0 0
$$909$$ 6693.99 0.244253
$$910$$ 0 0
$$911$$ −7908.58 −0.287621 −0.143811 0.989605i $$-0.545936\pi$$
−0.143811 + 0.989605i $$0.545936\pi$$
$$912$$ 0 0
$$913$$ 53285.7 1.93154
$$914$$ 0 0
$$915$$ −41.7102 −0.00150699
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 31450.6 1.12890 0.564451 0.825467i $$-0.309088\pi$$
0.564451 + 0.825467i $$0.309088\pi$$
$$920$$ 0 0
$$921$$ 5881.58 0.210428
$$922$$ 0 0
$$923$$ 12112.2 0.431938
$$924$$ 0 0
$$925$$ −37707.2 −1.34033
$$926$$ 0 0
$$927$$ 941.855 0.0333706
$$928$$ 0 0
$$929$$ −3469.19 −0.122519 −0.0612596 0.998122i $$-0.519512\pi$$
−0.0612596 + 0.998122i $$0.519512\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 15622.7 0.548193
$$934$$ 0 0
$$935$$ −913.148 −0.0319392
$$936$$ 0 0
$$937$$ 50935.7 1.77588 0.887939 0.459962i $$-0.152137\pi$$
0.887939 + 0.459962i $$0.152137\pi$$
$$938$$ 0 0
$$939$$ −11975.1 −0.416181
$$940$$ 0 0
$$941$$ 47894.9 1.65922 0.829612 0.558341i $$-0.188562\pi$$
0.829612 + 0.558341i $$0.188562\pi$$
$$942$$ 0 0
$$943$$ 53136.7 1.83496
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 10538.9 0.361634 0.180817 0.983517i $$-0.442126\pi$$
0.180817 + 0.983517i $$0.442126\pi$$
$$948$$ 0 0
$$949$$ 10397.8 0.355667
$$950$$ 0 0
$$951$$ 5415.11 0.184645
$$952$$ 0 0
$$953$$ 31123.5 1.05791 0.528955 0.848650i $$-0.322584\pi$$
0.528955 + 0.848650i $$0.322584\pi$$
$$954$$ 0 0
$$955$$ 629.951 0.0213453
$$956$$ 0 0
$$957$$ −11324.6 −0.382519
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 77037.9 2.58594
$$962$$ 0 0
$$963$$ 4623.30 0.154708
$$964$$ 0 0
$$965$$ 403.810 0.0134706
$$966$$ 0 0
$$967$$ −1626.14 −0.0540776 −0.0270388 0.999634i $$-0.508608\pi$$
−0.0270388 + 0.999634i $$0.508608\pi$$
$$968$$ 0 0
$$969$$ −36096.9 −1.19670
$$970$$ 0 0
$$971$$ 35500.1 1.17328 0.586638 0.809849i $$-0.300451\pi$$
0.586638 + 0.809849i $$0.300451\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 18835.6 0.618691
$$976$$ 0 0
$$977$$ 26570.2 0.870068 0.435034 0.900414i $$-0.356736\pi$$
0.435034 + 0.900414i $$0.356736\pi$$
$$978$$ 0 0
$$979$$ −23944.7 −0.781691
$$980$$ 0 0
$$981$$ 8775.34 0.285602
$$982$$ 0 0
$$983$$ −53851.6 −1.74730 −0.873652 0.486552i $$-0.838254\pi$$
−0.873652 + 0.486552i $$0.838254\pi$$
$$984$$ 0 0
$$985$$ −29.2432 −0.000945955 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −25900.4 −0.832745
$$990$$ 0 0
$$991$$ 2829.75 0.0907063 0.0453531 0.998971i $$-0.485559\pi$$
0.0453531 + 0.998971i $$0.485559\pi$$
$$992$$ 0 0
$$993$$ 20718.6 0.662118
$$994$$ 0 0
$$995$$ −156.174 −0.00497593
$$996$$ 0 0
$$997$$ 12904.3 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$998$$ 0 0
$$999$$ 8145.83 0.257981
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 1176.4.a.bd.1.2 4
4.3 odd 2 2352.4.a.cm.1.2 4
7.2 even 3 168.4.q.f.25.3 8
7.4 even 3 168.4.q.f.121.3 yes 8
7.6 odd 2 1176.4.a.ba.1.3 4
21.2 odd 6 504.4.s.j.361.2 8
21.11 odd 6 504.4.s.j.289.2 8
28.11 odd 6 336.4.q.m.289.3 8
28.23 odd 6 336.4.q.m.193.3 8
28.27 even 2 2352.4.a.cp.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.3 8 7.2 even 3
168.4.q.f.121.3 yes 8 7.4 even 3
336.4.q.m.193.3 8 28.23 odd 6
336.4.q.m.289.3 8 28.11 odd 6
504.4.s.j.289.2 8 21.11 odd 6
504.4.s.j.361.2 8 21.2 odd 6
1176.4.a.ba.1.3 4 7.6 odd 2
1176.4.a.bd.1.2 4 1.1 even 1 trivial
2352.4.a.cm.1.2 4 4.3 odd 2
2352.4.a.cp.1.3 4 28.27 even 2