# Properties

 Label 2160.4.a.bv.1.2 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 141x^{2} + 200x + 3500$$ x^4 - x^3 - 141*x^2 + 200*x + 3500 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-4.73555$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -11.2995 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -11.2995 q^{7} -61.3933 q^{11} -75.7226 q^{13} +11.1657 q^{17} -71.5271 q^{19} -126.156 q^{23} +25.0000 q^{25} -235.502 q^{29} -110.923 q^{31} -56.4973 q^{35} +434.358 q^{37} -1.15792 q^{41} +77.6254 q^{43} +231.442 q^{47} -215.322 q^{49} +500.296 q^{53} -306.967 q^{55} +334.718 q^{59} +147.171 q^{61} -378.613 q^{65} -84.9722 q^{67} +101.308 q^{71} -50.4773 q^{73} +693.712 q^{77} +818.225 q^{79} -206.044 q^{83} +55.8283 q^{85} -648.508 q^{89} +855.625 q^{91} -357.636 q^{95} +1885.00 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 20 q^{5} - 14 q^{7}+O(q^{10})$$ 4 * q + 20 * q^5 - 14 * q^7 $$4 q + 20 q^{5} - 14 q^{7} + 4 q^{11} + 30 q^{13} - 28 q^{17} - 78 q^{19} - 182 q^{23} + 100 q^{25} + 202 q^{29} + 76 q^{31} - 70 q^{35} + 302 q^{37} + 380 q^{41} - 178 q^{43} - 114 q^{47} + 958 q^{49} - 256 q^{53} + 20 q^{55} + 204 q^{59} + 766 q^{61} + 150 q^{65} - 330 q^{67} + 1060 q^{71} + 1442 q^{73} + 216 q^{77} - 742 q^{79} + 768 q^{83} - 140 q^{85} - 400 q^{89} - 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100})$$ 4 * q + 20 * q^5 - 14 * q^7 + 4 * q^11 + 30 * q^13 - 28 * q^17 - 78 * q^19 - 182 * q^23 + 100 * q^25 + 202 * q^29 + 76 * q^31 - 70 * q^35 + 302 * q^37 + 380 * q^41 - 178 * q^43 - 114 * q^47 + 958 * q^49 - 256 * q^53 + 20 * q^55 + 204 * q^59 + 766 * q^61 + 150 * q^65 - 330 * q^67 + 1060 * q^71 + 1442 * q^73 + 216 * q^77 - 742 * q^79 + 768 * q^83 - 140 * q^85 - 400 * q^89 - 3066 * q^91 - 390 * q^95 + 3338 * 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
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −11.2995 −0.610114 −0.305057 0.952334i $$-0.598675\pi$$
−0.305057 + 0.952334i $$0.598675\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −61.3933 −1.68280 −0.841399 0.540414i $$-0.818268\pi$$
−0.841399 + 0.540414i $$0.818268\pi$$
$$12$$ 0 0
$$13$$ −75.7226 −1.61551 −0.807757 0.589516i $$-0.799319\pi$$
−0.807757 + 0.589516i $$0.799319\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 11.1657 0.159298 0.0796491 0.996823i $$-0.474620\pi$$
0.0796491 + 0.996823i $$0.474620\pi$$
$$18$$ 0 0
$$19$$ −71.5271 −0.863655 −0.431828 0.901956i $$-0.642131\pi$$
−0.431828 + 0.901956i $$0.642131\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −126.156 −1.14371 −0.571855 0.820355i $$-0.693776\pi$$
−0.571855 + 0.820355i $$0.693776\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −235.502 −1.50798 −0.753992 0.656883i $$-0.771874\pi$$
−0.753992 + 0.656883i $$0.771874\pi$$
$$30$$ 0 0
$$31$$ −110.923 −0.642654 −0.321327 0.946968i $$-0.604129\pi$$
−0.321327 + 0.946968i $$0.604129\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −56.4973 −0.272851
$$36$$ 0 0
$$37$$ 434.358 1.92995 0.964973 0.262349i $$-0.0844971\pi$$
0.964973 + 0.262349i $$0.0844971\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.15792 −0.00441066 −0.00220533 0.999998i $$-0.500702\pi$$
−0.00220533 + 0.999998i $$0.500702\pi$$
$$42$$ 0 0
$$43$$ 77.6254 0.275297 0.137648 0.990481i $$-0.456046\pi$$
0.137648 + 0.990481i $$0.456046\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 231.442 0.718282 0.359141 0.933283i $$-0.383070\pi$$
0.359141 + 0.933283i $$0.383070\pi$$
$$48$$ 0 0
$$49$$ −215.322 −0.627761
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 500.296 1.29662 0.648310 0.761376i $$-0.275476\pi$$
0.648310 + 0.761376i $$0.275476\pi$$
$$54$$ 0 0
$$55$$ −306.967 −0.752571
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 334.718 0.738587 0.369293 0.929313i $$-0.379600\pi$$
0.369293 + 0.929313i $$0.379600\pi$$
$$60$$ 0 0
$$61$$ 147.171 0.308906 0.154453 0.988000i $$-0.450638\pi$$
0.154453 + 0.988000i $$0.450638\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −378.613 −0.722480
$$66$$ 0 0
$$67$$ −84.9722 −0.154940 −0.0774702 0.996995i $$-0.524684\pi$$
−0.0774702 + 0.996995i $$0.524684\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 101.308 0.169339 0.0846695 0.996409i $$-0.473017\pi$$
0.0846695 + 0.996409i $$0.473017\pi$$
$$72$$ 0 0
$$73$$ −50.4773 −0.0809304 −0.0404652 0.999181i $$-0.512884\pi$$
−0.0404652 + 0.999181i $$0.512884\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 693.712 1.02670
$$78$$ 0 0
$$79$$ 818.225 1.16528 0.582642 0.812729i $$-0.302019\pi$$
0.582642 + 0.812729i $$0.302019\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −206.044 −0.272486 −0.136243 0.990675i $$-0.543503\pi$$
−0.136243 + 0.990675i $$0.543503\pi$$
$$84$$ 0 0
$$85$$ 55.8283 0.0712403
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −648.508 −0.772379 −0.386189 0.922420i $$-0.626209\pi$$
−0.386189 + 0.922420i $$0.626209\pi$$
$$90$$ 0 0
$$91$$ 855.625 0.985647
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −357.636 −0.386238
$$96$$ 0 0
$$97$$ 1885.00 1.97313 0.986563 0.163381i $$-0.0522399\pi$$
0.986563 + 0.163381i $$0.0522399\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1923.79 1.89529 0.947647 0.319321i $$-0.103455\pi$$
0.947647 + 0.319321i $$0.103455\pi$$
$$102$$ 0 0
$$103$$ −1808.72 −1.73028 −0.865139 0.501531i $$-0.832770\pi$$
−0.865139 + 0.501531i $$0.832770\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1342.31 1.21277 0.606385 0.795171i $$-0.292619\pi$$
0.606385 + 0.795171i $$0.292619\pi$$
$$108$$ 0 0
$$109$$ 1173.14 1.03088 0.515442 0.856924i $$-0.327628\pi$$
0.515442 + 0.856924i $$0.327628\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −284.667 −0.236984 −0.118492 0.992955i $$-0.537806\pi$$
−0.118492 + 0.992955i $$0.537806\pi$$
$$114$$ 0 0
$$115$$ −630.779 −0.511483
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −126.166 −0.0971900
$$120$$ 0 0
$$121$$ 2438.14 1.83181
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2413.23 −1.68614 −0.843068 0.537807i $$-0.819253\pi$$
−0.843068 + 0.537807i $$0.819253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 118.912 0.0793085 0.0396543 0.999213i $$-0.487374\pi$$
0.0396543 + 0.999213i $$0.487374\pi$$
$$132$$ 0 0
$$133$$ 808.218 0.526928
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1097.16 −0.684212 −0.342106 0.939661i $$-0.611140\pi$$
−0.342106 + 0.939661i $$0.611140\pi$$
$$138$$ 0 0
$$139$$ 355.233 0.216766 0.108383 0.994109i $$-0.465433\pi$$
0.108383 + 0.994109i $$0.465433\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4648.86 2.71858
$$144$$ 0 0
$$145$$ −1177.51 −0.674391
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2001.55 −1.10049 −0.550246 0.835002i $$-0.685466\pi$$
−0.550246 + 0.835002i $$0.685466\pi$$
$$150$$ 0 0
$$151$$ −2652.70 −1.42963 −0.714813 0.699316i $$-0.753488\pi$$
−0.714813 + 0.699316i $$0.753488\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −554.613 −0.287404
$$156$$ 0 0
$$157$$ −3087.09 −1.56928 −0.784640 0.619952i $$-0.787152\pi$$
−0.784640 + 0.619952i $$0.787152\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1425.49 0.697793
$$162$$ 0 0
$$163$$ −1139.42 −0.547523 −0.273761 0.961798i $$-0.588268\pi$$
−0.273761 + 0.961798i $$0.588268\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2531.03 −1.17280 −0.586398 0.810023i $$-0.699455\pi$$
−0.586398 + 0.810023i $$0.699455\pi$$
$$168$$ 0 0
$$169$$ 3536.92 1.60988
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2872.89 1.26255 0.631277 0.775557i $$-0.282531\pi$$
0.631277 + 0.775557i $$0.282531\pi$$
$$174$$ 0 0
$$175$$ −282.487 −0.122023
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3827.16 −1.59807 −0.799036 0.601283i $$-0.794657\pi$$
−0.799036 + 0.601283i $$0.794657\pi$$
$$180$$ 0 0
$$181$$ −659.781 −0.270946 −0.135473 0.990781i $$-0.543255\pi$$
−0.135473 + 0.990781i $$0.543255\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2171.79 0.863098
$$186$$ 0 0
$$187$$ −685.497 −0.268067
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4376.11 1.65782 0.828911 0.559380i $$-0.188961\pi$$
0.828911 + 0.559380i $$0.188961\pi$$
$$192$$ 0 0
$$193$$ −3802.02 −1.41801 −0.709004 0.705204i $$-0.750855\pi$$
−0.709004 + 0.705204i $$0.750855\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3340.27 1.20804 0.604021 0.796968i $$-0.293564\pi$$
0.604021 + 0.796968i $$0.293564\pi$$
$$198$$ 0 0
$$199$$ −4274.83 −1.52279 −0.761393 0.648290i $$-0.775484\pi$$
−0.761393 + 0.648290i $$0.775484\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2661.04 0.920042
$$204$$ 0 0
$$205$$ −5.78960 −0.00197251
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4391.29 1.45336
$$210$$ 0 0
$$211$$ 126.348 0.0412233 0.0206117 0.999788i $$-0.493439\pi$$
0.0206117 + 0.999788i $$0.493439\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 388.127 0.123117
$$216$$ 0 0
$$217$$ 1253.36 0.392092
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −845.493 −0.257348
$$222$$ 0 0
$$223$$ −3423.31 −1.02799 −0.513996 0.857793i $$-0.671835\pi$$
−0.513996 + 0.857793i $$0.671835\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2145.09 0.627200 0.313600 0.949555i $$-0.398465\pi$$
0.313600 + 0.949555i $$0.398465\pi$$
$$228$$ 0 0
$$229$$ 3524.09 1.01694 0.508468 0.861081i $$-0.330212\pi$$
0.508468 + 0.861081i $$0.330212\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1479.58 0.416011 0.208005 0.978128i $$-0.433303\pi$$
0.208005 + 0.978128i $$0.433303\pi$$
$$234$$ 0 0
$$235$$ 1157.21 0.321226
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3075.33 0.832328 0.416164 0.909290i $$-0.363374\pi$$
0.416164 + 0.909290i $$0.363374\pi$$
$$240$$ 0 0
$$241$$ 2600.22 0.694998 0.347499 0.937680i $$-0.387031\pi$$
0.347499 + 0.937680i $$0.387031\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1076.61 −0.280743
$$246$$ 0 0
$$247$$ 5416.22 1.39525
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1186.65 −0.298410 −0.149205 0.988806i $$-0.547671\pi$$
−0.149205 + 0.988806i $$0.547671\pi$$
$$252$$ 0 0
$$253$$ 7745.13 1.92463
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3364.26 0.816562 0.408281 0.912856i $$-0.366128\pi$$
0.408281 + 0.912856i $$0.366128\pi$$
$$258$$ 0 0
$$259$$ −4908.01 −1.17749
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1655.19 0.388074 0.194037 0.980994i $$-0.437842\pi$$
0.194037 + 0.980994i $$0.437842\pi$$
$$264$$ 0 0
$$265$$ 2501.48 0.579867
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −2646.65 −0.599886 −0.299943 0.953957i $$-0.596968\pi$$
−0.299943 + 0.953957i $$0.596968\pi$$
$$270$$ 0 0
$$271$$ 3798.31 0.851405 0.425703 0.904863i $$-0.360027\pi$$
0.425703 + 0.904863i $$0.360027\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1534.83 −0.336560
$$276$$ 0 0
$$277$$ −5471.39 −1.18680 −0.593400 0.804908i $$-0.702215\pi$$
−0.593400 + 0.804908i $$0.702215\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3611.68 −0.766743 −0.383371 0.923594i $$-0.625237\pi$$
−0.383371 + 0.923594i $$0.625237\pi$$
$$282$$ 0 0
$$283$$ 1914.83 0.402208 0.201104 0.979570i $$-0.435547\pi$$
0.201104 + 0.979570i $$0.435547\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 13.0839 0.00269100
$$288$$ 0 0
$$289$$ −4788.33 −0.974624
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1706.82 0.340319 0.170159 0.985417i $$-0.445572\pi$$
0.170159 + 0.985417i $$0.445572\pi$$
$$294$$ 0 0
$$295$$ 1673.59 0.330306
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 9552.85 1.84768
$$300$$ 0 0
$$301$$ −877.126 −0.167962
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 735.853 0.138147
$$306$$ 0 0
$$307$$ −5888.56 −1.09472 −0.547358 0.836898i $$-0.684366\pi$$
−0.547358 + 0.836898i $$0.684366\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4902.98 0.893962 0.446981 0.894543i $$-0.352499\pi$$
0.446981 + 0.894543i $$0.352499\pi$$
$$312$$ 0 0
$$313$$ 6209.11 1.12128 0.560638 0.828061i $$-0.310556\pi$$
0.560638 + 0.828061i $$0.310556\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2427.25 0.430056 0.215028 0.976608i $$-0.431016\pi$$
0.215028 + 0.976608i $$0.431016\pi$$
$$318$$ 0 0
$$319$$ 14458.2 2.53763
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −798.648 −0.137579
$$324$$ 0 0
$$325$$ −1893.07 −0.323103
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2615.17 −0.438234
$$330$$ 0 0
$$331$$ −6058.23 −1.00601 −0.503006 0.864283i $$-0.667773\pi$$
−0.503006 + 0.864283i $$0.667773\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −424.861 −0.0692915
$$336$$ 0 0
$$337$$ 117.584 0.0190065 0.00950326 0.999955i $$-0.496975\pi$$
0.00950326 + 0.999955i $$0.496975\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6809.90 1.08146
$$342$$ 0 0
$$343$$ 6308.74 0.993119
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1596.08 0.246922 0.123461 0.992349i $$-0.460601\pi$$
0.123461 + 0.992349i $$0.460601\pi$$
$$348$$ 0 0
$$349$$ 3555.89 0.545394 0.272697 0.962100i $$-0.412084\pi$$
0.272697 + 0.962100i $$0.412084\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2973.17 −0.448289 −0.224145 0.974556i $$-0.571959\pi$$
−0.224145 + 0.974556i $$0.571959\pi$$
$$354$$ 0 0
$$355$$ 506.541 0.0757307
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12834.7 1.88688 0.943442 0.331537i $$-0.107567\pi$$
0.943442 + 0.331537i $$0.107567\pi$$
$$360$$ 0 0
$$361$$ −1742.87 −0.254099
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −252.386 −0.0361932
$$366$$ 0 0
$$367$$ −7503.89 −1.06730 −0.533651 0.845705i $$-0.679181\pi$$
−0.533651 + 0.845705i $$0.679181\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5653.07 −0.791086
$$372$$ 0 0
$$373$$ −10834.1 −1.50394 −0.751970 0.659197i $$-0.770896\pi$$
−0.751970 + 0.659197i $$0.770896\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 17832.8 2.43617
$$378$$ 0 0
$$379$$ −1526.64 −0.206909 −0.103454 0.994634i $$-0.532990\pi$$
−0.103454 + 0.994634i $$0.532990\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −8794.30 −1.17328 −0.586642 0.809846i $$-0.699550\pi$$
−0.586642 + 0.809846i $$0.699550\pi$$
$$384$$ 0 0
$$385$$ 3468.56 0.459154
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 7955.47 1.03691 0.518456 0.855104i $$-0.326507\pi$$
0.518456 + 0.855104i $$0.326507\pi$$
$$390$$ 0 0
$$391$$ −1408.61 −0.182191
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4091.12 0.521131
$$396$$ 0 0
$$397$$ −3359.28 −0.424679 −0.212339 0.977196i $$-0.568108\pi$$
−0.212339 + 0.977196i $$0.568108\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3387.77 −0.421888 −0.210944 0.977498i $$-0.567654\pi$$
−0.210944 + 0.977498i $$0.567654\pi$$
$$402$$ 0 0
$$403$$ 8399.34 1.03822
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −26666.7 −3.24771
$$408$$ 0 0
$$409$$ −7919.60 −0.957454 −0.478727 0.877964i $$-0.658902\pi$$
−0.478727 + 0.877964i $$0.658902\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −3782.14 −0.450622
$$414$$ 0 0
$$415$$ −1030.22 −0.121859
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10671.6 1.24425 0.622127 0.782917i $$-0.286269\pi$$
0.622127 + 0.782917i $$0.286269\pi$$
$$420$$ 0 0
$$421$$ −8445.59 −0.977703 −0.488852 0.872367i $$-0.662584\pi$$
−0.488852 + 0.872367i $$0.662584\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 279.141 0.0318596
$$426$$ 0 0
$$427$$ −1662.95 −0.188468
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 11206.9 1.25248 0.626241 0.779630i $$-0.284593\pi$$
0.626241 + 0.779630i $$0.284593\pi$$
$$432$$ 0 0
$$433$$ −5119.10 −0.568148 −0.284074 0.958802i $$-0.591686\pi$$
−0.284074 + 0.958802i $$0.591686\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 9023.57 0.987771
$$438$$ 0 0
$$439$$ 14018.6 1.52408 0.762039 0.647531i $$-0.224198\pi$$
0.762039 + 0.647531i $$0.224198\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10393.3 1.11468 0.557338 0.830286i $$-0.311823\pi$$
0.557338 + 0.830286i $$0.311823\pi$$
$$444$$ 0 0
$$445$$ −3242.54 −0.345418
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2522.28 −0.265108 −0.132554 0.991176i $$-0.542318\pi$$
−0.132554 + 0.991176i $$0.542318\pi$$
$$450$$ 0 0
$$451$$ 71.0886 0.00742225
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4278.13 0.440795
$$456$$ 0 0
$$457$$ 10879.1 1.11357 0.556785 0.830657i $$-0.312035\pi$$
0.556785 + 0.830657i $$0.312035\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −288.174 −0.0291141 −0.0145570 0.999894i $$-0.504634\pi$$
−0.0145570 + 0.999894i $$0.504634\pi$$
$$462$$ 0 0
$$463$$ 3397.84 0.341060 0.170530 0.985352i $$-0.445452\pi$$
0.170530 + 0.985352i $$0.445452\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 16484.3 1.63341 0.816703 0.577058i $$-0.195799\pi$$
0.816703 + 0.577058i $$0.195799\pi$$
$$468$$ 0 0
$$469$$ 960.140 0.0945313
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −4765.68 −0.463269
$$474$$ 0 0
$$475$$ −1788.18 −0.172731
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 8583.18 0.818738 0.409369 0.912369i $$-0.365749\pi$$
0.409369 + 0.912369i $$0.365749\pi$$
$$480$$ 0 0
$$481$$ −32890.7 −3.11785
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 9425.02 0.882409
$$486$$ 0 0
$$487$$ 6076.49 0.565404 0.282702 0.959208i $$-0.408769\pi$$
0.282702 + 0.959208i $$0.408769\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1652.84 −0.151918 −0.0759589 0.997111i $$-0.524202\pi$$
−0.0759589 + 0.997111i $$0.524202\pi$$
$$492$$ 0 0
$$493$$ −2629.53 −0.240219
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1144.73 −0.103316
$$498$$ 0 0
$$499$$ 18468.0 1.65680 0.828400 0.560137i $$-0.189252\pi$$
0.828400 + 0.560137i $$0.189252\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3044.06 0.269837 0.134918 0.990857i $$-0.456923\pi$$
0.134918 + 0.990857i $$0.456923\pi$$
$$504$$ 0 0
$$505$$ 9618.97 0.847601
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 8238.76 0.717440 0.358720 0.933445i $$-0.383213\pi$$
0.358720 + 0.933445i $$0.383213\pi$$
$$510$$ 0 0
$$511$$ 570.366 0.0493767
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −9043.61 −0.773804
$$516$$ 0 0
$$517$$ −14209.0 −1.20872
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 11609.7 0.976261 0.488131 0.872771i $$-0.337679\pi$$
0.488131 + 0.872771i $$0.337679\pi$$
$$522$$ 0 0
$$523$$ 10413.6 0.870662 0.435331 0.900270i $$-0.356631\pi$$
0.435331 + 0.900270i $$0.356631\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1238.52 −0.102374
$$528$$ 0 0
$$529$$ 3748.31 0.308072
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 87.6808 0.00712547
$$534$$ 0 0
$$535$$ 6711.57 0.542367
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 13219.3 1.05640
$$540$$ 0 0
$$541$$ 15045.2 1.19565 0.597823 0.801628i $$-0.296032\pi$$
0.597823 + 0.801628i $$0.296032\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5865.70 0.461026
$$546$$ 0 0
$$547$$ 8132.94 0.635721 0.317860 0.948138i $$-0.397036\pi$$
0.317860 + 0.948138i $$0.397036\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 16844.8 1.30238
$$552$$ 0 0
$$553$$ −9245.50 −0.710956
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 19419.4 1.47725 0.738623 0.674118i $$-0.235476\pi$$
0.738623 + 0.674118i $$0.235476\pi$$
$$558$$ 0 0
$$559$$ −5878.00 −0.444746
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −5696.48 −0.426426 −0.213213 0.977006i $$-0.568393\pi$$
−0.213213 + 0.977006i $$0.568393\pi$$
$$564$$ 0 0
$$565$$ −1423.33 −0.105982
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18615.2 −1.37151 −0.685754 0.727833i $$-0.740527\pi$$
−0.685754 + 0.727833i $$0.740527\pi$$
$$570$$ 0 0
$$571$$ 11250.1 0.824521 0.412260 0.911066i $$-0.364739\pi$$
0.412260 + 0.911066i $$0.364739\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3153.90 −0.228742
$$576$$ 0 0
$$577$$ 2336.40 0.168571 0.0842856 0.996442i $$-0.473139\pi$$
0.0842856 + 0.996442i $$0.473139\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2328.19 0.166247
$$582$$ 0 0
$$583$$ −30714.8 −2.18195
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −20277.2 −1.42578 −0.712889 0.701277i $$-0.752614\pi$$
−0.712889 + 0.701277i $$0.752614\pi$$
$$588$$ 0 0
$$589$$ 7933.97 0.555031
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −25051.8 −1.73483 −0.867414 0.497587i $$-0.834220\pi$$
−0.867414 + 0.497587i $$0.834220\pi$$
$$594$$ 0 0
$$595$$ −630.830 −0.0434647
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −26572.3 −1.81254 −0.906271 0.422697i $$-0.861084\pi$$
−0.906271 + 0.422697i $$0.861084\pi$$
$$600$$ 0 0
$$601$$ 4224.74 0.286740 0.143370 0.989669i $$-0.454206\pi$$
0.143370 + 0.989669i $$0.454206\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 12190.7 0.819211
$$606$$ 0 0
$$607$$ 17428.2 1.16539 0.582693 0.812692i $$-0.301999\pi$$
0.582693 + 0.812692i $$0.301999\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −17525.4 −1.16039
$$612$$ 0 0
$$613$$ −7100.27 −0.467826 −0.233913 0.972258i $$-0.575153\pi$$
−0.233913 + 0.972258i $$0.575153\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −26045.2 −1.69941 −0.849707 0.527255i $$-0.823221\pi$$
−0.849707 + 0.527255i $$0.823221\pi$$
$$618$$ 0 0
$$619$$ −8865.32 −0.575650 −0.287825 0.957683i $$-0.592932\pi$$
−0.287825 + 0.957683i $$0.592932\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 7327.79 0.471239
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4849.89 0.307437
$$630$$ 0 0
$$631$$ −177.883 −0.0112225 −0.00561126 0.999984i $$-0.501786\pi$$
−0.00561126 + 0.999984i $$0.501786\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −12066.1 −0.754063
$$636$$ 0 0
$$637$$ 16304.8 1.01416
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −27742.7 −1.70947 −0.854736 0.519064i $$-0.826281\pi$$
−0.854736 + 0.519064i $$0.826281\pi$$
$$642$$ 0 0
$$643$$ −9182.34 −0.563166 −0.281583 0.959537i $$-0.590860\pi$$
−0.281583 + 0.959537i $$0.590860\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16429.9 −0.998341 −0.499171 0.866504i $$-0.666362\pi$$
−0.499171 + 0.866504i $$0.666362\pi$$
$$648$$ 0 0
$$649$$ −20549.5 −1.24289
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −10965.8 −0.657158 −0.328579 0.944477i $$-0.606570\pi$$
−0.328579 + 0.944477i $$0.606570\pi$$
$$654$$ 0 0
$$655$$ 594.561 0.0354678
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 23252.7 1.37450 0.687250 0.726421i $$-0.258818\pi$$
0.687250 + 0.726421i $$0.258818\pi$$
$$660$$ 0 0
$$661$$ −2215.22 −0.130351 −0.0651755 0.997874i $$-0.520761\pi$$
−0.0651755 + 0.997874i $$0.520761\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4041.09 0.235649
$$666$$ 0 0
$$667$$ 29709.9 1.72470
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9035.30 −0.519827
$$672$$ 0 0
$$673$$ −20009.1 −1.14605 −0.573026 0.819537i $$-0.694230\pi$$
−0.573026 + 0.819537i $$0.694230\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18842.2 1.06967 0.534834 0.844957i $$-0.320374\pi$$
0.534834 + 0.844957i $$0.320374\pi$$
$$678$$ 0 0
$$679$$ −21299.5 −1.20383
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 20258.1 1.13493 0.567463 0.823399i $$-0.307925\pi$$
0.567463 + 0.823399i $$0.307925\pi$$
$$684$$ 0 0
$$685$$ −5485.82 −0.305989
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −37883.7 −2.09471
$$690$$ 0 0
$$691$$ −22738.2 −1.25181 −0.625905 0.779899i $$-0.715270\pi$$
−0.625905 + 0.779899i $$0.715270\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1776.17 0.0969407
$$696$$ 0 0
$$697$$ −12.9289 −0.000702610 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8306.55 −0.447552 −0.223776 0.974641i $$-0.571838\pi$$
−0.223776 + 0.974641i $$0.571838\pi$$
$$702$$ 0 0
$$703$$ −31068.4 −1.66681
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −21737.8 −1.15634
$$708$$ 0 0
$$709$$ 24716.6 1.30924 0.654620 0.755958i $$-0.272829\pi$$
0.654620 + 0.755958i $$0.272829\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 13993.5 0.735009
$$714$$ 0 0
$$715$$ 23244.3 1.21579
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 9888.23 0.512891 0.256446 0.966559i $$-0.417449\pi$$
0.256446 + 0.966559i $$0.417449\pi$$
$$720$$ 0 0
$$721$$ 20437.6 1.05567
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5887.54 −0.301597
$$726$$ 0 0
$$727$$ −6335.29 −0.323195 −0.161598 0.986857i $$-0.551665\pi$$
−0.161598 + 0.986857i $$0.551665\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 866.739 0.0438543
$$732$$ 0 0
$$733$$ −13170.9 −0.663680 −0.331840 0.943336i $$-0.607669\pi$$
−0.331840 + 0.943336i $$0.607669\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5216.73 0.260734
$$738$$ 0 0
$$739$$ 31361.7 1.56111 0.780554 0.625088i $$-0.214937\pi$$
0.780554 + 0.625088i $$0.214937\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20002.1 0.987624 0.493812 0.869569i $$-0.335603\pi$$
0.493812 + 0.869569i $$0.335603\pi$$
$$744$$ 0 0
$$745$$ −10007.8 −0.492155
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −15167.4 −0.739927
$$750$$ 0 0
$$751$$ 25698.3 1.24866 0.624331 0.781160i $$-0.285372\pi$$
0.624331 + 0.781160i $$0.285372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −13263.5 −0.639348
$$756$$ 0 0
$$757$$ −33958.5 −1.63044 −0.815219 0.579153i $$-0.803384\pi$$
−0.815219 + 0.579153i $$0.803384\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −13216.4 −0.629559 −0.314780 0.949165i $$-0.601931\pi$$
−0.314780 + 0.949165i $$0.601931\pi$$
$$762$$ 0 0
$$763$$ −13255.9 −0.628957
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −25345.8 −1.19320
$$768$$ 0 0
$$769$$ 3107.27 0.145710 0.0728551 0.997343i $$-0.476789\pi$$
0.0728551 + 0.997343i $$0.476789\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 20568.2 0.957034 0.478517 0.878078i $$-0.341175\pi$$
0.478517 + 0.878078i $$0.341175\pi$$
$$774$$ 0 0
$$775$$ −2773.06 −0.128531
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 82.8228 0.00380929
$$780$$ 0 0
$$781$$ −6219.65 −0.284963
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −15435.5 −0.701803
$$786$$ 0 0
$$787$$ −19353.9 −0.876609 −0.438304 0.898827i $$-0.644421\pi$$
−0.438304 + 0.898827i $$0.644421\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3216.58 0.144587
$$792$$ 0 0
$$793$$ −11144.1 −0.499042
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 29243.9 1.29971 0.649856 0.760057i $$-0.274829\pi$$
0.649856 + 0.760057i $$0.274829\pi$$
$$798$$ 0 0
$$799$$ 2584.20 0.114421
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 3098.97 0.136190
$$804$$ 0 0
$$805$$ 7127.47 0.312063
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31460.5 −1.36723 −0.683617 0.729841i $$-0.739594\pi$$
−0.683617 + 0.729841i $$0.739594\pi$$
$$810$$ 0 0
$$811$$ −22991.8 −0.995503 −0.497751 0.867320i $$-0.665841\pi$$
−0.497751 + 0.867320i $$0.665841\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −5697.10 −0.244860
$$816$$ 0 0
$$817$$ −5552.32 −0.237762
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 32144.1 1.36643 0.683214 0.730218i $$-0.260581\pi$$
0.683214 + 0.730218i $$0.260581\pi$$
$$822$$ 0 0
$$823$$ −908.840 −0.0384935 −0.0192468 0.999815i $$-0.506127\pi$$
−0.0192468 + 0.999815i $$0.506127\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23098.9 0.971255 0.485628 0.874166i $$-0.338591\pi$$
0.485628 + 0.874166i $$0.338591\pi$$
$$828$$ 0 0
$$829$$ 42491.6 1.78021 0.890104 0.455757i $$-0.150631\pi$$
0.890104 + 0.455757i $$0.150631\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2404.21 −0.100001
$$834$$ 0 0
$$835$$ −12655.2 −0.524491
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −12239.5 −0.503642 −0.251821 0.967774i $$-0.581029\pi$$
−0.251821 + 0.967774i $$0.581029\pi$$
$$840$$ 0 0
$$841$$ 31072.0 1.27402
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 17684.6 0.719962
$$846$$ 0 0
$$847$$ −27549.7 −1.11761
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −54796.8 −2.20730
$$852$$ 0 0
$$853$$ 9020.23 0.362071 0.181036 0.983477i $$-0.442055\pi$$
0.181036 + 0.983477i $$0.442055\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 39913.4 1.59092 0.795459 0.606008i $$-0.207230\pi$$
0.795459 + 0.606008i $$0.207230\pi$$
$$858$$ 0 0
$$859$$ −26038.5 −1.03425 −0.517126 0.855910i $$-0.672998\pi$$
−0.517126 + 0.855910i $$0.672998\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −6044.84 −0.238434 −0.119217 0.992868i $$-0.538038\pi$$
−0.119217 + 0.992868i $$0.538038\pi$$
$$864$$ 0 0
$$865$$ 14364.5 0.564631
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −50233.5 −1.96094
$$870$$ 0 0
$$871$$ 6434.32 0.250308
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1412.43 −0.0545702
$$876$$ 0 0
$$877$$ 28918.4 1.11346 0.556731 0.830693i $$-0.312055\pi$$
0.556731 + 0.830693i $$0.312055\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25949.7 0.992359 0.496179 0.868220i $$-0.334736\pi$$
0.496179 + 0.868220i $$0.334736\pi$$
$$882$$ 0 0
$$883$$ −39784.1 −1.51624 −0.758121 0.652114i $$-0.773882\pi$$
−0.758121 + 0.652114i $$0.773882\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −16227.6 −0.614284 −0.307142 0.951664i $$-0.599373\pi$$
−0.307142 + 0.951664i $$0.599373\pi$$
$$888$$ 0 0
$$889$$ 27268.2 1.02873
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −16554.4 −0.620348
$$894$$ 0 0
$$895$$ −19135.8 −0.714680
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 26122.4 0.969112
$$900$$ 0 0
$$901$$ 5586.13 0.206549
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −3298.91 −0.121171
$$906$$ 0 0
$$907$$ 2107.66 0.0771596 0.0385798 0.999256i $$-0.487717\pi$$
0.0385798 + 0.999256i $$0.487717\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3471.16 −0.126240 −0.0631199 0.998006i $$-0.520105\pi$$
−0.0631199 + 0.998006i $$0.520105\pi$$
$$912$$ 0 0
$$913$$ 12649.8 0.458538
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1343.65 −0.0483872
$$918$$ 0 0
$$919$$ 7620.88 0.273547 0.136773 0.990602i $$-0.456327\pi$$
0.136773 + 0.990602i $$0.456327\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −7671.32 −0.273569
$$924$$ 0 0
$$925$$ 10858.9 0.385989
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −23396.4 −0.826276 −0.413138 0.910669i $$-0.635567\pi$$
−0.413138 + 0.910669i $$0.635567\pi$$
$$930$$ 0 0
$$931$$ 15401.4 0.542169
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −3427.48 −0.119883
$$936$$ 0 0
$$937$$ 23669.1 0.825226 0.412613 0.910906i $$-0.364616\pi$$
0.412613 + 0.910906i $$0.364616\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −15682.0 −0.543272 −0.271636 0.962400i $$-0.587565\pi$$
−0.271636 + 0.962400i $$0.587565\pi$$
$$942$$ 0 0
$$943$$ 146.079 0.00504451
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −28683.3 −0.984247 −0.492123 0.870526i $$-0.663779\pi$$
−0.492123 + 0.870526i $$0.663779\pi$$
$$948$$ 0 0
$$949$$ 3822.27 0.130744
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 20850.6 0.708728 0.354364 0.935108i $$-0.384697\pi$$
0.354364 + 0.935108i $$0.384697\pi$$
$$954$$ 0 0
$$955$$ 21880.5 0.741401
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12397.4 0.417447
$$960$$ 0 0
$$961$$ −17487.2 −0.586996
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −19010.1 −0.634152
$$966$$ 0 0
$$967$$ 5002.87 0.166372 0.0831858 0.996534i $$-0.473491\pi$$
0.0831858 + 0.996534i $$0.473491\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 39413.3 1.30261 0.651304 0.758817i $$-0.274222\pi$$
0.651304 + 0.758817i $$0.274222\pi$$
$$972$$ 0 0
$$973$$ −4013.94 −0.132252
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −11631.3 −0.380879 −0.190439 0.981699i $$-0.560991\pi$$
−0.190439 + 0.981699i $$0.560991\pi$$
$$978$$ 0 0
$$979$$ 39814.1 1.29976
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −38605.2 −1.25261 −0.626304 0.779579i $$-0.715433\pi$$
−0.626304 + 0.779579i $$0.715433\pi$$
$$984$$ 0 0
$$985$$ 16701.3 0.540253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9792.90 −0.314860
$$990$$ 0 0
$$991$$ −382.345 −0.0122559 −0.00612795 0.999981i $$-0.501951\pi$$
−0.00612795 + 0.999981i $$0.501951\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −21374.1 −0.681011
$$996$$ 0 0
$$997$$ 36072.1 1.14585 0.572926 0.819607i $$-0.305808\pi$$
0.572926 + 0.819607i $$0.305808\pi$$
$$998$$ 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 2160.4.a.bv.1.2 4
3.2 odd 2 2160.4.a.bu.1.2 4
4.3 odd 2 1080.4.a.p.1.3 yes 4
12.11 even 2 1080.4.a.o.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.3 4 12.11 even 2
1080.4.a.p.1.3 yes 4 4.3 odd 2
2160.4.a.bu.1.2 4 3.2 odd 2
2160.4.a.bv.1.2 4 1.1 even 1 trivial