# Properties

 Label 1872.4.a.bk.1.3 Level $1872$ Weight $4$ Character 1872.1 Self dual yes Analytic conductor $110.452$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.451575531$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 1872.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+11.4322 q^{5} +11.2543 q^{7} +O(q^{10})$$ $$q+11.4322 q^{5} +11.2543 q^{7} +25.8785 q^{11} +13.0000 q^{13} +20.3276 q^{17} -154.712 q^{19} -180.418 q^{23} +5.69520 q^{25} +20.4522 q^{29} -266.424 q^{31} +128.661 q^{35} +115.984 q^{37} -391.184 q^{41} -151.407 q^{43} -467.365 q^{47} -216.341 q^{49} -79.9842 q^{53} +295.848 q^{55} -873.710 q^{59} -187.068 q^{61} +148.619 q^{65} +609.204 q^{67} +248.038 q^{71} +852.765 q^{73} +291.244 q^{77} +331.221 q^{79} -435.432 q^{83} +232.389 q^{85} -259.233 q^{89} +146.306 q^{91} -1768.70 q^{95} +1225.17 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{5} - 30 q^{7}+O(q^{10})$$ 3 * q - 4 * q^5 - 30 * q^7 $$3 q - 4 q^{5} - 30 q^{7} - 16 q^{11} + 39 q^{13} + 146 q^{17} - 94 q^{19} - 48 q^{23} + 145 q^{25} + 2 q^{29} - 302 q^{31} + 80 q^{35} + 374 q^{37} - 480 q^{41} + 260 q^{43} - 24 q^{47} + 447 q^{49} + 678 q^{53} + 1552 q^{55} - 1788 q^{59} + 230 q^{61} - 52 q^{65} - 74 q^{67} - 948 q^{71} - 222 q^{73} - 112 q^{77} + 24 q^{79} - 796 q^{83} - 248 q^{85} - 1436 q^{89} - 390 q^{91} - 4032 q^{95} + 3242 q^{97}+O(q^{100})$$ 3 * q - 4 * q^5 - 30 * q^7 - 16 * q^11 + 39 * q^13 + 146 * q^17 - 94 * q^19 - 48 * q^23 + 145 * q^25 + 2 * q^29 - 302 * q^31 + 80 * q^35 + 374 * q^37 - 480 * q^41 + 260 * q^43 - 24 * q^47 + 447 * q^49 + 678 * q^53 + 1552 * q^55 - 1788 * q^59 + 230 * q^61 - 52 * q^65 - 74 * q^67 - 948 * q^71 - 222 * q^73 - 112 * q^77 + 24 * q^79 - 796 * q^83 - 248 * q^85 - 1436 * q^89 - 390 * q^91 - 4032 * q^95 + 3242 * 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
$$5$$ 11.4322 1.02253 0.511264 0.859424i $$-0.329178\pi$$
0.511264 + 0.859424i $$0.329178\pi$$
$$6$$ 0 0
$$7$$ 11.2543 0.607675 0.303838 0.952724i $$-0.401732\pi$$
0.303838 + 0.952724i $$0.401732\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 20.3276 0.290010 0.145005 0.989431i $$-0.453680\pi$$
0.145005 + 0.989431i $$0.453680\pi$$
$$18$$ 0 0
$$19$$ −154.712 −1.86807 −0.934035 0.357181i $$-0.883738\pi$$
−0.934035 + 0.357181i $$0.883738\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −180.418 −1.63565 −0.817823 0.575471i $$-0.804819\pi$$
−0.817823 + 0.575471i $$0.804819\pi$$
$$24$$ 0 0
$$25$$ 5.69520 0.0455616
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 20.4522 0.130961 0.0654806 0.997854i $$-0.479142\pi$$
0.0654806 + 0.997854i $$0.479142\pi$$
$$30$$ 0 0
$$31$$ −266.424 −1.54359 −0.771794 0.635873i $$-0.780640\pi$$
−0.771794 + 0.635873i $$0.780640\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 128.661 0.621364
$$36$$ 0 0
$$37$$ 115.984 0.515340 0.257670 0.966233i $$-0.417045\pi$$
0.257670 + 0.966233i $$0.417045\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −391.184 −1.49006 −0.745032 0.667029i $$-0.767566\pi$$
−0.745032 + 0.667029i $$0.767566\pi$$
$$42$$ 0 0
$$43$$ −151.407 −0.536963 −0.268482 0.963285i $$-0.586522\pi$$
−0.268482 + 0.963285i $$0.586522\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −467.365 −1.45047 −0.725236 0.688500i $$-0.758269\pi$$
−0.725236 + 0.688500i $$0.758269\pi$$
$$48$$ 0 0
$$49$$ −216.341 −0.630731
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −79.9842 −0.207296 −0.103648 0.994614i $$-0.533051\pi$$
−0.103648 + 0.994614i $$0.533051\pi$$
$$54$$ 0 0
$$55$$ 295.848 0.725312
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −873.710 −1.92792 −0.963960 0.266045i $$-0.914283\pi$$
−0.963960 + 0.266045i $$0.914283\pi$$
$$60$$ 0 0
$$61$$ −187.068 −0.392649 −0.196325 0.980539i $$-0.562901\pi$$
−0.196325 + 0.980539i $$0.562901\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 148.619 0.283598
$$66$$ 0 0
$$67$$ 609.204 1.11084 0.555418 0.831571i $$-0.312558\pi$$
0.555418 + 0.831571i $$0.312558\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 248.038 0.414601 0.207301 0.978277i $$-0.433532\pi$$
0.207301 + 0.978277i $$0.433532\pi$$
$$72$$ 0 0
$$73$$ 852.765 1.36724 0.683621 0.729838i $$-0.260404\pi$$
0.683621 + 0.729838i $$0.260404\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 291.244 0.431044
$$78$$ 0 0
$$79$$ 331.221 0.471712 0.235856 0.971788i $$-0.424211\pi$$
0.235856 + 0.971788i $$0.424211\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −435.432 −0.575842 −0.287921 0.957654i $$-0.592964\pi$$
−0.287921 + 0.957654i $$0.592964\pi$$
$$84$$ 0 0
$$85$$ 232.389 0.296543
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −259.233 −0.308749 −0.154375 0.988012i $$-0.549336\pi$$
−0.154375 + 0.988012i $$0.549336\pi$$
$$90$$ 0 0
$$91$$ 146.306 0.168539
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1768.70 −1.91015
$$96$$ 0 0
$$97$$ 1225.17 1.28245 0.641223 0.767355i $$-0.278428\pi$$
0.641223 + 0.767355i $$0.278428\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −645.416 −0.635855 −0.317927 0.948115i $$-0.602987\pi$$
−0.317927 + 0.948115i $$0.602987\pi$$
$$102$$ 0 0
$$103$$ 511.137 0.488969 0.244484 0.969653i $$-0.421381\pi$$
0.244484 + 0.969653i $$0.421381\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 608.195 0.549499 0.274750 0.961516i $$-0.411405\pi$$
0.274750 + 0.961516i $$0.411405\pi$$
$$108$$ 0 0
$$109$$ −1300.04 −1.14239 −0.571197 0.820813i $$-0.693521\pi$$
−0.571197 + 0.820813i $$0.693521\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −42.1953 −0.0351274 −0.0175637 0.999846i $$-0.505591\pi$$
−0.0175637 + 0.999846i $$0.505591\pi$$
$$114$$ 0 0
$$115$$ −2062.58 −1.67249
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 228.773 0.176232
$$120$$ 0 0
$$121$$ −661.303 −0.496847
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1363.92 −0.975939
$$126$$ 0 0
$$127$$ 311.018 0.217310 0.108655 0.994080i $$-0.465346\pi$$
0.108655 + 0.994080i $$0.465346\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2000.98 1.33456 0.667278 0.744809i $$-0.267459\pi$$
0.667278 + 0.744809i $$0.267459\pi$$
$$132$$ 0 0
$$133$$ −1741.17 −1.13518
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1038.53 −0.647644 −0.323822 0.946118i $$-0.604968\pi$$
−0.323822 + 0.946118i $$0.604968\pi$$
$$138$$ 0 0
$$139$$ 2858.46 1.74426 0.872128 0.489277i $$-0.162739\pi$$
0.872128 + 0.489277i $$0.162739\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 336.421 0.196734
$$144$$ 0 0
$$145$$ 233.814 0.133911
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −743.479 −0.408780 −0.204390 0.978890i $$-0.565521\pi$$
−0.204390 + 0.978890i $$0.565521\pi$$
$$150$$ 0 0
$$151$$ −2277.24 −1.22728 −0.613640 0.789586i $$-0.710295\pi$$
−0.613640 + 0.789586i $$0.710295\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3045.82 −1.57836
$$156$$ 0 0
$$157$$ 3173.51 1.61321 0.806605 0.591091i $$-0.201303\pi$$
0.806605 + 0.591091i $$0.201303\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2030.48 −0.993941
$$162$$ 0 0
$$163$$ 2314.65 1.11225 0.556126 0.831098i $$-0.312287\pi$$
0.556126 + 0.831098i $$0.312287\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2665.65 −1.23517 −0.617587 0.786502i $$-0.711890\pi$$
−0.617587 + 0.786502i $$0.711890\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 165.243 0.0726198 0.0363099 0.999341i $$-0.488440\pi$$
0.0363099 + 0.999341i $$0.488440\pi$$
$$174$$ 0 0
$$175$$ 64.0954 0.0276866
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 712.339 0.297446 0.148723 0.988879i $$-0.452484\pi$$
0.148723 + 0.988879i $$0.452484\pi$$
$$180$$ 0 0
$$181$$ 2206.53 0.906133 0.453066 0.891477i $$-0.350330\pi$$
0.453066 + 0.891477i $$0.350330\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1325.95 0.526949
$$186$$ 0 0
$$187$$ 526.048 0.205714
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1470.64 0.557129 0.278565 0.960417i $$-0.410141\pi$$
0.278565 + 0.960417i $$0.410141\pi$$
$$192$$ 0 0
$$193$$ 369.560 0.137832 0.0689158 0.997622i $$-0.478046\pi$$
0.0689158 + 0.997622i $$0.478046\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4273.41 1.54552 0.772761 0.634697i $$-0.218875\pi$$
0.772761 + 0.634697i $$0.218875\pi$$
$$198$$ 0 0
$$199$$ −4154.31 −1.47985 −0.739927 0.672687i $$-0.765140\pi$$
−0.739927 + 0.672687i $$0.765140\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 230.175 0.0795819
$$204$$ 0 0
$$205$$ −4472.09 −1.52363
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4003.71 −1.32508
$$210$$ 0 0
$$211$$ −1231.59 −0.401830 −0.200915 0.979609i $$-0.564392\pi$$
−0.200915 + 0.979609i $$0.564392\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1730.92 −0.549059
$$216$$ 0 0
$$217$$ −2998.42 −0.938000
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 264.259 0.0804343
$$222$$ 0 0
$$223$$ 2187.24 0.656809 0.328404 0.944537i $$-0.393489\pi$$
0.328404 + 0.944537i $$0.393489\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4138.67 1.21010 0.605051 0.796187i $$-0.293153\pi$$
0.605051 + 0.796187i $$0.293153\pi$$
$$228$$ 0 0
$$229$$ −835.354 −0.241056 −0.120528 0.992710i $$-0.538459\pi$$
−0.120528 + 0.992710i $$0.538459\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3685.51 −1.03625 −0.518124 0.855305i $$-0.673370\pi$$
−0.518124 + 0.855305i $$0.673370\pi$$
$$234$$ 0 0
$$235$$ −5343.01 −1.48315
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3026.21 0.819034 0.409517 0.912303i $$-0.365697\pi$$
0.409517 + 0.912303i $$0.365697\pi$$
$$240$$ 0 0
$$241$$ 3265.58 0.872839 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −2473.25 −0.644940
$$246$$ 0 0
$$247$$ −2011.25 −0.518110
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6363.16 −1.60016 −0.800078 0.599897i $$-0.795208\pi$$
−0.800078 + 0.599897i $$0.795208\pi$$
$$252$$ 0 0
$$253$$ −4668.96 −1.16022
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6085.36 1.47702 0.738511 0.674242i $$-0.235529\pi$$
0.738511 + 0.674242i $$0.235529\pi$$
$$258$$ 0 0
$$259$$ 1305.31 0.313159
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 123.227 0.0288916 0.0144458 0.999896i $$-0.495402\pi$$
0.0144458 + 0.999896i $$0.495402\pi$$
$$264$$ 0 0
$$265$$ −914.395 −0.211965
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 1935.79 0.438763 0.219381 0.975639i $$-0.429596\pi$$
0.219381 + 0.975639i $$0.429596\pi$$
$$270$$ 0 0
$$271$$ 4612.69 1.03395 0.516976 0.856000i $$-0.327058\pi$$
0.516976 + 0.856000i $$0.327058\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 147.383 0.0323183
$$276$$ 0 0
$$277$$ −5834.30 −1.26552 −0.632761 0.774347i $$-0.718078\pi$$
−0.632761 + 0.774347i $$0.718078\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4691.91 −0.996071 −0.498036 0.867157i $$-0.665945\pi$$
−0.498036 + 0.867157i $$0.665945\pi$$
$$282$$ 0 0
$$283$$ −3465.60 −0.727945 −0.363973 0.931410i $$-0.618580\pi$$
−0.363973 + 0.931410i $$0.618580\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4402.50 −0.905475
$$288$$ 0 0
$$289$$ −4499.79 −0.915894
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −2677.31 −0.533822 −0.266911 0.963721i $$-0.586003\pi$$
−0.266911 + 0.963721i $$0.586003\pi$$
$$294$$ 0 0
$$295$$ −9988.43 −1.97135
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2345.44 −0.453646
$$300$$ 0 0
$$301$$ −1703.98 −0.326299
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −2138.60 −0.401494
$$306$$ 0 0
$$307$$ −471.915 −0.0877316 −0.0438658 0.999037i $$-0.513967\pi$$
−0.0438658 + 0.999037i $$0.513967\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1518.52 −0.276872 −0.138436 0.990371i $$-0.544207\pi$$
−0.138436 + 0.990371i $$0.544207\pi$$
$$312$$ 0 0
$$313$$ 4049.86 0.731348 0.365674 0.930743i $$-0.380839\pi$$
0.365674 + 0.930743i $$0.380839\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3253.96 −0.576532 −0.288266 0.957550i $$-0.593079\pi$$
−0.288266 + 0.957550i $$0.593079\pi$$
$$318$$ 0 0
$$319$$ 529.272 0.0928951
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3144.92 −0.541759
$$324$$ 0 0
$$325$$ 74.0375 0.0126365
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −5259.86 −0.881415
$$330$$ 0 0
$$331$$ 3422.45 0.568322 0.284161 0.958777i $$-0.408285\pi$$
0.284161 + 0.958777i $$0.408285\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 6964.54 1.13586
$$336$$ 0 0
$$337$$ −9301.67 −1.50354 −0.751772 0.659423i $$-0.770801\pi$$
−0.751772 + 0.659423i $$0.770801\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −6894.66 −1.09492
$$342$$ 0 0
$$343$$ −6294.99 −0.990955
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 216.898 0.0335554 0.0167777 0.999859i $$-0.494659\pi$$
0.0167777 + 0.999859i $$0.494659\pi$$
$$348$$ 0 0
$$349$$ −4809.84 −0.737721 −0.368861 0.929485i $$-0.620252\pi$$
−0.368861 + 0.929485i $$0.620252\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2859.64 0.431170 0.215585 0.976485i $$-0.430834\pi$$
0.215585 + 0.976485i $$0.430834\pi$$
$$354$$ 0 0
$$355$$ 2835.62 0.423941
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3686.04 0.541899 0.270949 0.962594i $$-0.412662\pi$$
0.270949 + 0.962594i $$0.412662\pi$$
$$360$$ 0 0
$$361$$ 17076.8 2.48969
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9748.98 1.39804
$$366$$ 0 0
$$367$$ 3470.59 0.493633 0.246816 0.969062i $$-0.420616\pi$$
0.246816 + 0.969062i $$0.420616\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −900.166 −0.125968
$$372$$ 0 0
$$373$$ −11963.4 −1.66070 −0.830352 0.557240i $$-0.811860\pi$$
−0.830352 + 0.557240i $$0.811860\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 265.879 0.0363221
$$378$$ 0 0
$$379$$ −345.604 −0.0468403 −0.0234202 0.999726i $$-0.507456\pi$$
−0.0234202 + 0.999726i $$0.507456\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3386.40 −0.451793 −0.225897 0.974151i $$-0.572531\pi$$
−0.225897 + 0.974151i $$0.572531\pi$$
$$384$$ 0 0
$$385$$ 3329.56 0.440754
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1629.88 0.212438 0.106219 0.994343i $$-0.466126\pi$$
0.106219 + 0.994343i $$0.466126\pi$$
$$390$$ 0 0
$$391$$ −3667.47 −0.474353
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3786.58 0.482338
$$396$$ 0 0
$$397$$ 7938.94 1.00364 0.501819 0.864973i $$-0.332664\pi$$
0.501819 + 0.864973i $$0.332664\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −214.402 −0.0267001 −0.0133500 0.999911i $$-0.504250\pi$$
−0.0133500 + 0.999911i $$0.504250\pi$$
$$402$$ 0 0
$$403$$ −3463.52 −0.428114
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3001.48 0.365548
$$408$$ 0 0
$$409$$ −4783.73 −0.578338 −0.289169 0.957278i $$-0.593379\pi$$
−0.289169 + 0.957278i $$0.593379\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −9832.99 −1.17155
$$414$$ 0 0
$$415$$ −4977.95 −0.588815
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −9903.67 −1.15472 −0.577358 0.816491i $$-0.695916\pi$$
−0.577358 + 0.816491i $$0.695916\pi$$
$$420$$ 0 0
$$421$$ −12120.6 −1.40314 −0.701572 0.712598i $$-0.747518\pi$$
−0.701572 + 0.712598i $$0.747518\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 115.770 0.0132133
$$426$$ 0 0
$$427$$ −2105.32 −0.238603
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −13672.6 −1.52805 −0.764023 0.645189i $$-0.776779\pi$$
−0.764023 + 0.645189i $$0.776779\pi$$
$$432$$ 0 0
$$433$$ 7113.10 0.789455 0.394727 0.918798i $$-0.370839\pi$$
0.394727 + 0.918798i $$0.370839\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 27912.9 3.05550
$$438$$ 0 0
$$439$$ 6022.04 0.654707 0.327353 0.944902i $$-0.393843\pi$$
0.327353 + 0.944902i $$0.393843\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12994.4 −1.39364 −0.696821 0.717245i $$-0.745403\pi$$
−0.696821 + 0.717245i $$0.745403\pi$$
$$444$$ 0 0
$$445$$ −2963.61 −0.315704
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10984.3 −1.15452 −0.577260 0.816560i $$-0.695878\pi$$
−0.577260 + 0.816560i $$0.695878\pi$$
$$450$$ 0 0
$$451$$ −10123.2 −1.05695
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1672.60 0.172335
$$456$$ 0 0
$$457$$ 9834.10 1.00661 0.503304 0.864109i $$-0.332118\pi$$
0.503304 + 0.864109i $$0.332118\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3401.42 −0.343644 −0.171822 0.985128i $$-0.554965\pi$$
−0.171822 + 0.985128i $$0.554965\pi$$
$$462$$ 0 0
$$463$$ −1739.42 −0.174596 −0.0872979 0.996182i $$-0.527823\pi$$
−0.0872979 + 0.996182i $$0.527823\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −7958.82 −0.788630 −0.394315 0.918975i $$-0.629018\pi$$
−0.394315 + 0.918975i $$0.629018\pi$$
$$468$$ 0 0
$$469$$ 6856.16 0.675028
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −3918.20 −0.380886
$$474$$ 0 0
$$475$$ −881.114 −0.0851122
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 8431.98 0.804315 0.402158 0.915570i $$-0.368260\pi$$
0.402158 + 0.915570i $$0.368260\pi$$
$$480$$ 0 0
$$481$$ 1507.79 0.142930
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 14006.4 1.31133
$$486$$ 0 0
$$487$$ 11684.7 1.08723 0.543617 0.839334i $$-0.317055\pi$$
0.543617 + 0.839334i $$0.317055\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3954.70 −0.363489 −0.181745 0.983346i $$-0.558174\pi$$
−0.181745 + 0.983346i $$0.558174\pi$$
$$492$$ 0 0
$$493$$ 415.744 0.0379801
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2791.49 0.251943
$$498$$ 0 0
$$499$$ 5690.37 0.510493 0.255246 0.966876i $$-0.417843\pi$$
0.255246 + 0.966876i $$0.417843\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 10859.1 0.962595 0.481298 0.876557i $$-0.340166\pi$$
0.481298 + 0.876557i $$0.340166\pi$$
$$504$$ 0 0
$$505$$ −7378.53 −0.650178
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 18558.6 1.61610 0.808049 0.589115i $$-0.200524\pi$$
0.808049 + 0.589115i $$0.200524\pi$$
$$510$$ 0 0
$$511$$ 9597.27 0.830838
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 5843.42 0.499984
$$516$$ 0 0
$$517$$ −12094.7 −1.02887
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −17297.5 −1.45454 −0.727271 0.686350i $$-0.759212\pi$$
−0.727271 + 0.686350i $$0.759212\pi$$
$$522$$ 0 0
$$523$$ 5016.11 0.419386 0.209693 0.977767i $$-0.432753\pi$$
0.209693 + 0.977767i $$0.432753\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5415.77 −0.447656
$$528$$ 0 0
$$529$$ 20383.8 1.67533
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −5085.39 −0.413269
$$534$$ 0 0
$$535$$ 6953.01 0.561878
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −5598.57 −0.447398
$$540$$ 0 0
$$541$$ 17642.3 1.40204 0.701018 0.713144i $$-0.252729\pi$$
0.701018 + 0.713144i $$0.252729\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −14862.3 −1.16813
$$546$$ 0 0
$$547$$ 18414.9 1.43943 0.719713 0.694271i $$-0.244273\pi$$
0.719713 + 0.694271i $$0.244273\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3164.20 −0.244645
$$552$$ 0 0
$$553$$ 3727.66 0.286648
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −8179.15 −0.622193 −0.311096 0.950378i $$-0.600696\pi$$
−0.311096 + 0.950378i $$0.600696\pi$$
$$558$$ 0 0
$$559$$ −1968.30 −0.148927
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1880.07 −0.140738 −0.0703690 0.997521i $$-0.522418\pi$$
−0.0703690 + 0.997521i $$0.522418\pi$$
$$564$$ 0 0
$$565$$ −482.385 −0.0359187
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10118.3 −0.745485 −0.372743 0.927935i $$-0.621583\pi$$
−0.372743 + 0.927935i $$0.621583\pi$$
$$570$$ 0 0
$$571$$ −23428.9 −1.71711 −0.858555 0.512721i $$-0.828638\pi$$
−0.858555 + 0.512721i $$0.828638\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1027.52 −0.0745225
$$576$$ 0 0
$$577$$ 20508.1 1.47966 0.739831 0.672793i $$-0.234906\pi$$
0.739831 + 0.672793i $$0.234906\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4900.49 −0.349925
$$582$$ 0 0
$$583$$ −2069.87 −0.147042
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5968.43 −0.419665 −0.209833 0.977737i $$-0.567292\pi$$
−0.209833 + 0.977737i $$0.567292\pi$$
$$588$$ 0 0
$$589$$ 41219.0 2.88353
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 14659.5 1.01517 0.507584 0.861602i $$-0.330539\pi$$
0.507584 + 0.861602i $$0.330539\pi$$
$$594$$ 0 0
$$595$$ 2615.38 0.180202
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 23635.9 1.61225 0.806125 0.591746i $$-0.201561\pi$$
0.806125 + 0.591746i $$0.201561\pi$$
$$600$$ 0 0
$$601$$ −11527.0 −0.782356 −0.391178 0.920315i $$-0.627932\pi$$
−0.391178 + 0.920315i $$0.627932\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −7560.15 −0.508039
$$606$$ 0 0
$$607$$ 5098.56 0.340930 0.170465 0.985364i $$-0.445473\pi$$
0.170465 + 0.985364i $$0.445473\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6075.74 −0.402288
$$612$$ 0 0
$$613$$ 1516.39 0.0999128 0.0499564 0.998751i $$-0.484092\pi$$
0.0499564 + 0.998751i $$0.484092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18539.3 −1.20966 −0.604832 0.796353i $$-0.706760\pi$$
−0.604832 + 0.796353i $$0.706760\pi$$
$$618$$ 0 0
$$619$$ −25684.9 −1.66779 −0.833897 0.551920i $$-0.813895\pi$$
−0.833897 + 0.551920i $$0.813895\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2917.49 −0.187619
$$624$$ 0 0
$$625$$ −16304.5 −1.04349
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2357.67 0.149454
$$630$$ 0 0
$$631$$ 22410.9 1.41389 0.706945 0.707269i $$-0.250073\pi$$
0.706945 + 0.707269i $$0.250073\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 3555.62 0.222206
$$636$$ 0 0
$$637$$ −2812.43 −0.174933
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6827.81 −0.420721 −0.210361 0.977624i $$-0.567464\pi$$
−0.210361 + 0.977624i $$0.567464\pi$$
$$642$$ 0 0
$$643$$ 23264.3 1.42684 0.713418 0.700738i $$-0.247146\pi$$
0.713418 + 0.700738i $$0.247146\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 14745.9 0.896014 0.448007 0.894030i $$-0.352134\pi$$
0.448007 + 0.894030i $$0.352134\pi$$
$$648$$ 0 0
$$649$$ −22610.3 −1.36754
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −10909.0 −0.653755 −0.326878 0.945067i $$-0.605997\pi$$
−0.326878 + 0.945067i $$0.605997\pi$$
$$654$$ 0 0
$$655$$ 22875.7 1.36462
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ 0 0
$$661$$ 2224.23 0.130881 0.0654406 0.997856i $$-0.479155\pi$$
0.0654406 + 0.997856i $$0.479155\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −19905.4 −1.16075
$$666$$ 0 0
$$667$$ −3689.95 −0.214206
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4841.04 −0.278519
$$672$$ 0 0
$$673$$ −24152.5 −1.38337 −0.691687 0.722197i $$-0.743132\pi$$
−0.691687 + 0.722197i $$0.743132\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15310.7 0.869187 0.434593 0.900627i $$-0.356892\pi$$
0.434593 + 0.900627i $$0.356892\pi$$
$$678$$ 0 0
$$679$$ 13788.4 0.779310
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 11399.6 0.638646 0.319323 0.947646i $$-0.396545\pi$$
0.319323 + 0.947646i $$0.396545\pi$$
$$684$$ 0 0
$$685$$ −11872.6 −0.662233
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1039.79 −0.0574935
$$690$$ 0 0
$$691$$ 3323.23 0.182955 0.0914773 0.995807i $$-0.470841\pi$$
0.0914773 + 0.995807i $$0.470841\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 32678.5 1.78355
$$696$$ 0 0
$$697$$ −7951.82 −0.432133
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 12670.4 0.682673 0.341336 0.939941i $$-0.389120\pi$$
0.341336 + 0.939941i $$0.389120\pi$$
$$702$$ 0 0
$$703$$ −17944.0 −0.962692
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −7263.71 −0.386393
$$708$$ 0 0
$$709$$ 13075.2 0.692594 0.346297 0.938125i $$-0.387439\pi$$
0.346297 + 0.938125i $$0.387439\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 48067.8 2.52476
$$714$$ 0 0
$$715$$ 3846.03 0.201165
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −2988.41 −0.155005 −0.0775026 0.996992i $$-0.524695\pi$$
−0.0775026 + 0.996992i $$0.524695\pi$$
$$720$$ 0 0
$$721$$ 5752.48 0.297134
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 116.479 0.00596680
$$726$$ 0 0
$$727$$ 5507.46 0.280963 0.140482 0.990083i $$-0.455135\pi$$
0.140482 + 0.990083i $$0.455135\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3077.75 −0.155725
$$732$$ 0 0
$$733$$ −36585.2 −1.84353 −0.921764 0.387751i $$-0.873252\pi$$
−0.921764 + 0.387751i $$0.873252\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15765.3 0.787953
$$738$$ 0 0
$$739$$ −6425.89 −0.319865 −0.159933 0.987128i $$-0.551128\pi$$
−0.159933 + 0.987128i $$0.551128\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20411.0 1.00782 0.503908 0.863757i $$-0.331895\pi$$
0.503908 + 0.863757i $$0.331895\pi$$
$$744$$ 0 0
$$745$$ −8499.60 −0.417988
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6844.81 0.333917
$$750$$ 0 0
$$751$$ 24259.5 1.17875 0.589375 0.807860i $$-0.299374\pi$$
0.589375 + 0.807860i $$0.299374\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −26033.9 −1.25493
$$756$$ 0 0
$$757$$ 9295.39 0.446297 0.223148 0.974785i $$-0.428367\pi$$
0.223148 + 0.974785i $$0.428367\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 21974.7 1.04676 0.523378 0.852101i $$-0.324672\pi$$
0.523378 + 0.852101i $$0.324672\pi$$
$$762$$ 0 0
$$763$$ −14631.0 −0.694204
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −11358.2 −0.534709
$$768$$ 0 0
$$769$$ 22987.4 1.07795 0.538977 0.842320i $$-0.318811\pi$$
0.538977 + 0.842320i $$0.318811\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −31970.9 −1.48760 −0.743799 0.668404i $$-0.766978\pi$$
−0.743799 + 0.668404i $$0.766978\pi$$
$$774$$ 0 0
$$775$$ −1517.34 −0.0703283
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 60520.8 2.78354
$$780$$ 0 0
$$781$$ 6418.85 0.294090
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 36280.2 1.64955
$$786$$ 0 0
$$787$$ −6087.26 −0.275715 −0.137857 0.990452i $$-0.544022\pi$$
−0.137857 + 0.990452i $$0.544022\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −474.878 −0.0213460
$$792$$ 0 0
$$793$$ −2431.88 −0.108901
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −23080.0 −1.02577 −0.512883 0.858458i $$-0.671423\pi$$
−0.512883 + 0.858458i $$0.671423\pi$$
$$798$$ 0 0
$$799$$ −9500.41 −0.420651
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 22068.3 0.969829
$$804$$ 0 0
$$805$$ −23212.9 −1.01633
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 32377.8 1.40710 0.703550 0.710646i $$-0.251597\pi$$
0.703550 + 0.710646i $$0.251597\pi$$
$$810$$ 0 0
$$811$$ −26352.8 −1.14103 −0.570513 0.821288i $$-0.693256\pi$$
−0.570513 + 0.821288i $$0.693256\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 26461.5 1.13731
$$816$$ 0 0
$$817$$ 23424.5 1.00308
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −35355.3 −1.50294 −0.751468 0.659770i $$-0.770654\pi$$
−0.751468 + 0.659770i $$0.770654\pi$$
$$822$$ 0 0
$$823$$ 12663.3 0.536347 0.268173 0.963371i $$-0.413580\pi$$
0.268173 + 0.963371i $$0.413580\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16295.2 −0.685176 −0.342588 0.939486i $$-0.611303\pi$$
−0.342588 + 0.939486i $$0.611303\pi$$
$$828$$ 0 0
$$829$$ 13638.9 0.571411 0.285705 0.958318i $$-0.407772\pi$$
0.285705 + 0.958318i $$0.407772\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4397.69 −0.182918
$$834$$ 0 0
$$835$$ −30474.2 −1.26300
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −1890.31 −0.0777838 −0.0388919 0.999243i $$-0.512383\pi$$
−0.0388919 + 0.999243i $$0.512383\pi$$
$$840$$ 0 0
$$841$$ −23970.7 −0.982849
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1932.04 0.0786559
$$846$$ 0 0
$$847$$ −7442.50 −0.301921
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −20925.6 −0.842914
$$852$$ 0 0
$$853$$ 1620.21 0.0650351 0.0325175 0.999471i $$-0.489648\pi$$
0.0325175 + 0.999471i $$0.489648\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 14508.4 0.578292 0.289146 0.957285i $$-0.406629\pi$$
0.289146 + 0.957285i $$0.406629\pi$$
$$858$$ 0 0
$$859$$ −29639.8 −1.17730 −0.588648 0.808389i $$-0.700340\pi$$
−0.588648 + 0.808389i $$0.700340\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 21528.8 0.849186 0.424593 0.905384i $$-0.360417\pi$$
0.424593 + 0.905384i $$0.360417\pi$$
$$864$$ 0 0
$$865$$ 1889.10 0.0742557
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 8571.50 0.334601
$$870$$ 0 0
$$871$$ 7919.65 0.308091
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −15349.9 −0.593054
$$876$$ 0 0
$$877$$ 14865.3 0.572366 0.286183 0.958175i $$-0.407613\pi$$
0.286183 + 0.958175i $$0.407613\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 21336.0 0.815921 0.407961 0.913000i $$-0.366240\pi$$
0.407961 + 0.913000i $$0.366240\pi$$
$$882$$ 0 0
$$883$$ −37538.2 −1.43065 −0.715323 0.698794i $$-0.753720\pi$$
−0.715323 + 0.698794i $$0.753720\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 34575.0 1.30881 0.654406 0.756144i $$-0.272919\pi$$
0.654406 + 0.756144i $$0.272919\pi$$
$$888$$ 0 0
$$889$$ 3500.29 0.132054
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 72306.9 2.70958
$$894$$ 0 0
$$895$$ 8143.61 0.304146
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −5448.96 −0.202150
$$900$$ 0 0
$$901$$ −1625.89 −0.0601178
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25225.5 0.926545
$$906$$ 0 0
$$907$$ 10424.8 0.381641 0.190820 0.981625i $$-0.438885\pi$$
0.190820 + 0.981625i $$0.438885\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 10961.8 0.398661 0.199331 0.979932i $$-0.436123\pi$$
0.199331 + 0.979932i $$0.436123\pi$$
$$912$$ 0 0
$$913$$ −11268.3 −0.408464
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 22519.7 0.810976
$$918$$ 0 0
$$919$$ 10779.2 0.386914 0.193457 0.981109i $$-0.438030\pi$$
0.193457 + 0.981109i $$0.438030\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 3224.49 0.114990
$$924$$ 0 0
$$925$$ 660.549 0.0234797
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −5429.07 −0.191735 −0.0958675 0.995394i $$-0.530563\pi$$
−0.0958675 + 0.995394i $$0.530563\pi$$
$$930$$ 0 0
$$931$$ 33470.5 1.17825
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 6013.88 0.210348
$$936$$ 0 0
$$937$$ −21300.1 −0.742631 −0.371315 0.928507i $$-0.621093\pi$$
−0.371315 + 0.928507i $$0.621093\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −26851.2 −0.930207 −0.465103 0.885256i $$-0.653983\pi$$
−0.465103 + 0.885256i $$0.653983\pi$$
$$942$$ 0 0
$$943$$ 70576.7 2.43722
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8021.68 −0.275258 −0.137629 0.990484i $$-0.543948\pi$$
−0.137629 + 0.990484i $$0.543948\pi$$
$$948$$ 0 0
$$949$$ 11085.9 0.379204
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −35715.0 −1.21398 −0.606990 0.794709i $$-0.707623\pi$$
−0.606990 + 0.794709i $$0.707623\pi$$
$$954$$ 0 0
$$955$$ 16812.6 0.569680
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −11687.9 −0.393557
$$960$$ 0 0
$$961$$ 41190.9 1.38266
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 4224.88 0.140936
$$966$$ 0 0
$$967$$ −53338.8 −1.77380 −0.886898 0.461965i $$-0.847145\pi$$
−0.886898 + 0.461965i $$0.847145\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 23112.9 0.763882 0.381941 0.924187i $$-0.375256\pi$$
0.381941 + 0.924187i $$0.375256\pi$$
$$972$$ 0 0
$$973$$ 32170.0 1.05994
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52874.6 1.73143 0.865715 0.500538i $$-0.166864\pi$$
0.865715 + 0.500538i $$0.166864\pi$$
$$978$$ 0 0
$$979$$ −6708.57 −0.219006
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 45173.1 1.46572 0.732858 0.680381i $$-0.238186\pi$$
0.732858 + 0.680381i $$0.238186\pi$$
$$984$$ 0 0
$$985$$ 48854.5 1.58034
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 27316.7 0.878281
$$990$$ 0 0
$$991$$ −60485.6 −1.93884 −0.969418 0.245414i $$-0.921076\pi$$
−0.969418 + 0.245414i $$0.921076\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −47492.8 −1.51319
$$996$$ 0 0
$$997$$ −18108.1 −0.575214 −0.287607 0.957749i $$-0.592860\pi$$
−0.287607 + 0.957749i $$0.592860\pi$$
$$998$$ 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 1872.4.a.bk.1.3 3
3.2 odd 2 624.4.a.t.1.1 3
4.3 odd 2 117.4.a.f.1.1 3
12.11 even 2 39.4.a.c.1.3 3
24.5 odd 2 2496.4.a.bp.1.3 3
24.11 even 2 2496.4.a.bl.1.3 3
52.51 odd 2 1521.4.a.u.1.3 3
60.59 even 2 975.4.a.l.1.1 3
84.83 odd 2 1911.4.a.k.1.3 3
156.47 odd 4 507.4.b.g.337.6 6
156.83 odd 4 507.4.b.g.337.1 6
156.155 even 2 507.4.a.h.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 12.11 even 2
117.4.a.f.1.1 3 4.3 odd 2
507.4.a.h.1.1 3 156.155 even 2
507.4.b.g.337.1 6 156.83 odd 4
507.4.b.g.337.6 6 156.47 odd 4
624.4.a.t.1.1 3 3.2 odd 2
975.4.a.l.1.1 3 60.59 even 2
1521.4.a.u.1.3 3 52.51 odd 2
1872.4.a.bk.1.3 3 1.1 even 1 trivial
1911.4.a.k.1.3 3 84.83 odd 2
2496.4.a.bl.1.3 3 24.11 even 2
2496.4.a.bp.1.3 3 24.5 odd 2