# Properties

 Label 2160.4.a.bn.1.2 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.47977.1 Defining polynomial: $$x^{3} - x^{2} - 60x - 44$$ x^3 - x^2 - 60*x - 44 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$8.58548$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -9.46549 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -9.46549 q^{7} -49.0474 q^{11} +55.4438 q^{13} -27.9783 q^{17} -26.3965 q^{19} -9.53451 q^{23} +25.0000 q^{25} +218.379 q^{29} +158.961 q^{31} -47.3274 q^{35} +189.655 q^{37} -246.146 q^{41} -40.2327 q^{43} +213.965 q^{47} -253.405 q^{49} -283.310 q^{53} -245.237 q^{55} +92.0000 q^{59} +370.172 q^{61} +277.219 q^{65} -1087.53 q^{67} -333.681 q^{71} -606.791 q^{73} +464.257 q^{77} +44.3751 q^{79} +107.732 q^{83} -139.892 q^{85} +257.819 q^{89} -524.803 q^{91} -131.982 q^{95} +1312.05 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} - 9 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 - 9 * q^7 $$3 q + 15 q^{5} - 9 q^{7} - 18 q^{11} - 21 q^{13} + 84 q^{17} - 21 q^{19} - 48 q^{23} + 75 q^{25} - 36 q^{29} - 324 q^{31} - 45 q^{35} + 33 q^{37} + 114 q^{41} - 282 q^{43} - 282 q^{47} + 228 q^{49} + 222 q^{53} - 90 q^{55} + 276 q^{59} + 303 q^{61} - 105 q^{65} - 1035 q^{67} - 510 q^{71} + 447 q^{73} - 1578 q^{77} - 777 q^{79} - 78 q^{83} + 420 q^{85} - 324 q^{89} - 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 - 9 * q^7 - 18 * q^11 - 21 * q^13 + 84 * q^17 - 21 * q^19 - 48 * q^23 + 75 * q^25 - 36 * q^29 - 324 * q^31 - 45 * q^35 + 33 * q^37 + 114 * q^41 - 282 * q^43 - 282 * q^47 + 228 * q^49 + 222 * q^53 - 90 * q^55 + 276 * q^59 + 303 * q^61 - 105 * q^65 - 1035 * q^67 - 510 * q^71 + 447 * q^73 - 1578 * q^77 - 777 * q^79 - 78 * q^83 + 420 * q^85 - 324 * q^89 - 1995 * q^91 - 105 * q^95 + 1191 * 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
$$7$$ −9.46549 −0.511088 −0.255544 0.966797i $$-0.582255\pi$$
−0.255544 + 0.966797i $$0.582255\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −49.0474 −1.34439 −0.672197 0.740372i $$-0.734649\pi$$
−0.672197 + 0.740372i $$0.734649\pi$$
$$12$$ 0 0
$$13$$ 55.4438 1.18287 0.591437 0.806351i $$-0.298561\pi$$
0.591437 + 0.806351i $$0.298561\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −27.9783 −0.399162 −0.199581 0.979881i $$-0.563958\pi$$
−0.199581 + 0.979881i $$0.563958\pi$$
$$18$$ 0 0
$$19$$ −26.3965 −0.318724 −0.159362 0.987220i $$-0.550944\pi$$
−0.159362 + 0.987220i $$0.550944\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −9.53451 −0.0864384 −0.0432192 0.999066i $$-0.513761\pi$$
−0.0432192 + 0.999066i $$0.513761\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 218.379 1.39834 0.699171 0.714954i $$-0.253552\pi$$
0.699171 + 0.714954i $$0.253552\pi$$
$$30$$ 0 0
$$31$$ 158.961 0.920974 0.460487 0.887666i $$-0.347675\pi$$
0.460487 + 0.887666i $$0.347675\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −47.3274 −0.228566
$$36$$ 0 0
$$37$$ 189.655 0.842678 0.421339 0.906903i $$-0.361560\pi$$
0.421339 + 0.906903i $$0.361560\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −246.146 −0.937599 −0.468800 0.883304i $$-0.655313\pi$$
−0.468800 + 0.883304i $$0.655313\pi$$
$$42$$ 0 0
$$43$$ −40.2327 −0.142684 −0.0713422 0.997452i $$-0.522728\pi$$
−0.0713422 + 0.997452i $$0.522728\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 213.965 0.664042 0.332021 0.943272i $$-0.392270\pi$$
0.332021 + 0.943272i $$0.392270\pi$$
$$48$$ 0 0
$$49$$ −253.405 −0.738789
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −283.310 −0.734257 −0.367128 0.930170i $$-0.619659\pi$$
−0.367128 + 0.930170i $$0.619659\pi$$
$$54$$ 0 0
$$55$$ −245.237 −0.601232
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 92.0000 0.203006 0.101503 0.994835i $$-0.467635\pi$$
0.101503 + 0.994835i $$0.467635\pi$$
$$60$$ 0 0
$$61$$ 370.172 0.776978 0.388489 0.921453i $$-0.372997\pi$$
0.388489 + 0.921453i $$0.372997\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 277.219 0.528997
$$66$$ 0 0
$$67$$ −1087.53 −1.98302 −0.991510 0.130029i $$-0.958493\pi$$
−0.991510 + 0.130029i $$0.958493\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −333.681 −0.557755 −0.278877 0.960327i $$-0.589962\pi$$
−0.278877 + 0.960327i $$0.589962\pi$$
$$72$$ 0 0
$$73$$ −606.791 −0.972871 −0.486435 0.873717i $$-0.661703\pi$$
−0.486435 + 0.873717i $$0.661703\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 464.257 0.687104
$$78$$ 0 0
$$79$$ 44.3751 0.0631973 0.0315986 0.999501i $$-0.489940\pi$$
0.0315986 + 0.999501i $$0.489940\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 107.732 0.142471 0.0712355 0.997460i $$-0.477306\pi$$
0.0712355 + 0.997460i $$0.477306\pi$$
$$84$$ 0 0
$$85$$ −139.892 −0.178510
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 257.819 0.307065 0.153532 0.988144i $$-0.450935\pi$$
0.153532 + 0.988144i $$0.450935\pi$$
$$90$$ 0 0
$$91$$ −524.803 −0.604553
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −131.982 −0.142538
$$96$$ 0 0
$$97$$ 1312.05 1.37339 0.686693 0.726947i $$-0.259062\pi$$
0.686693 + 0.726947i $$0.259062\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −63.3293 −0.0623911 −0.0311955 0.999513i $$-0.509931\pi$$
−0.0311955 + 0.999513i $$0.509931\pi$$
$$102$$ 0 0
$$103$$ −1219.85 −1.16695 −0.583474 0.812132i $$-0.698307\pi$$
−0.583474 + 0.812132i $$0.698307\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 42.3947 0.0383033 0.0191517 0.999817i $$-0.493903\pi$$
0.0191517 + 0.999817i $$0.493903\pi$$
$$108$$ 0 0
$$109$$ 266.741 0.234396 0.117198 0.993109i $$-0.462609\pi$$
0.117198 + 0.993109i $$0.462609\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1871.60 −1.55810 −0.779049 0.626964i $$-0.784297\pi$$
−0.779049 + 0.626964i $$0.784297\pi$$
$$114$$ 0 0
$$115$$ −47.6726 −0.0386564
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 264.829 0.204007
$$120$$ 0 0
$$121$$ 1074.64 0.807397
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1232.11 0.860881 0.430441 0.902619i $$-0.358358\pi$$
0.430441 + 0.902619i $$0.358358\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1209.17 −0.806458 −0.403229 0.915099i $$-0.632112\pi$$
−0.403229 + 0.915099i $$0.632112\pi$$
$$132$$ 0 0
$$133$$ 249.855 0.162896
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2629.66 −1.63990 −0.819952 0.572432i $$-0.806000\pi$$
−0.819952 + 0.572432i $$0.806000\pi$$
$$138$$ 0 0
$$139$$ −2215.06 −1.35165 −0.675823 0.737064i $$-0.736212\pi$$
−0.675823 + 0.737064i $$0.736212\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2719.37 −1.59025
$$144$$ 0 0
$$145$$ 1091.89 0.625358
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2446.94 −1.34538 −0.672689 0.739925i $$-0.734861\pi$$
−0.672689 + 0.739925i $$0.734861\pi$$
$$150$$ 0 0
$$151$$ −119.365 −0.0643298 −0.0321649 0.999483i $$-0.510240\pi$$
−0.0321649 + 0.999483i $$0.510240\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 794.804 0.411872
$$156$$ 0 0
$$157$$ 2212.86 1.12487 0.562437 0.826840i $$-0.309864\pi$$
0.562437 + 0.826840i $$0.309864\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 90.2488 0.0441777
$$162$$ 0 0
$$163$$ −3480.54 −1.67250 −0.836249 0.548349i $$-0.815256\pi$$
−0.836249 + 0.548349i $$0.815256\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2477.15 −1.14783 −0.573916 0.818914i $$-0.694576\pi$$
−0.573916 + 0.818914i $$0.694576\pi$$
$$168$$ 0 0
$$169$$ 877.018 0.399189
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −26.7446 −0.0117535 −0.00587676 0.999983i $$-0.501871\pi$$
−0.00587676 + 0.999983i $$0.501871\pi$$
$$174$$ 0 0
$$175$$ −236.637 −0.102218
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 1400.75 0.584901 0.292451 0.956281i $$-0.405529\pi$$
0.292451 + 0.956281i $$0.405529\pi$$
$$180$$ 0 0
$$181$$ 3492.48 1.43422 0.717111 0.696959i $$-0.245464\pi$$
0.717111 + 0.696959i $$0.245464\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 948.275 0.376857
$$186$$ 0 0
$$187$$ 1372.26 0.536631
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2728.16 −1.03352 −0.516762 0.856129i $$-0.672863\pi$$
−0.516762 + 0.856129i $$0.672863\pi$$
$$192$$ 0 0
$$193$$ 2439.82 0.909959 0.454980 0.890502i $$-0.349647\pi$$
0.454980 + 0.890502i $$0.349647\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 765.906 0.276998 0.138499 0.990363i $$-0.455772\pi$$
0.138499 + 0.990363i $$0.455772\pi$$
$$198$$ 0 0
$$199$$ −754.031 −0.268602 −0.134301 0.990941i $$-0.542879\pi$$
−0.134301 + 0.990941i $$0.542879\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2067.06 −0.714676
$$204$$ 0 0
$$205$$ −1230.73 −0.419307
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1294.68 0.428491
$$210$$ 0 0
$$211$$ 1024.68 0.334320 0.167160 0.985930i $$-0.446540\pi$$
0.167160 + 0.985930i $$0.446540\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −201.163 −0.0638104
$$216$$ 0 0
$$217$$ −1504.64 −0.470699
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1551.23 −0.472157
$$222$$ 0 0
$$223$$ 1938.61 0.582147 0.291074 0.956701i $$-0.405988\pi$$
0.291074 + 0.956701i $$0.405988\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5876.07 −1.71810 −0.859050 0.511892i $$-0.828945\pi$$
−0.859050 + 0.511892i $$0.828945\pi$$
$$228$$ 0 0
$$229$$ 5061.06 1.46046 0.730228 0.683204i $$-0.239414\pi$$
0.730228 + 0.683204i $$0.239414\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1241.58 −0.349093 −0.174547 0.984649i $$-0.555846\pi$$
−0.174547 + 0.984649i $$0.555846\pi$$
$$234$$ 0 0
$$235$$ 1069.82 0.296968
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3141.68 0.850286 0.425143 0.905126i $$-0.360224\pi$$
0.425143 + 0.905126i $$0.360224\pi$$
$$240$$ 0 0
$$241$$ −2682.15 −0.716898 −0.358449 0.933549i $$-0.616694\pi$$
−0.358449 + 0.933549i $$0.616694\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1267.02 −0.330396
$$246$$ 0 0
$$247$$ −1463.52 −0.377010
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4159.18 1.04592 0.522959 0.852358i $$-0.324828\pi$$
0.522959 + 0.852358i $$0.324828\pi$$
$$252$$ 0 0
$$253$$ 467.643 0.116207
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2820.31 0.684537 0.342268 0.939602i $$-0.388805\pi$$
0.342268 + 0.939602i $$0.388805\pi$$
$$258$$ 0 0
$$259$$ −1795.18 −0.430683
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 27.1884 0.00637457 0.00318728 0.999995i $$-0.498985\pi$$
0.00318728 + 0.999995i $$0.498985\pi$$
$$264$$ 0 0
$$265$$ −1416.55 −0.328370
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4955.54 −1.12321 −0.561607 0.827404i $$-0.689817\pi$$
−0.561607 + 0.827404i $$0.689817\pi$$
$$270$$ 0 0
$$271$$ −6619.08 −1.48369 −0.741846 0.670570i $$-0.766050\pi$$
−0.741846 + 0.670570i $$0.766050\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1226.18 −0.268879
$$276$$ 0 0
$$277$$ −2999.18 −0.650554 −0.325277 0.945619i $$-0.605457\pi$$
−0.325277 + 0.945619i $$0.605457\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3387.06 −0.719058 −0.359529 0.933134i $$-0.617063\pi$$
−0.359529 + 0.933134i $$0.617063\pi$$
$$282$$ 0 0
$$283$$ −79.4525 −0.0166889 −0.00834446 0.999965i $$-0.502656\pi$$
−0.00834446 + 0.999965i $$0.502656\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2329.89 0.479196
$$288$$ 0 0
$$289$$ −4130.21 −0.840670
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6045.37 −1.20537 −0.602686 0.797978i $$-0.705903\pi$$
−0.602686 + 0.797978i $$0.705903\pi$$
$$294$$ 0 0
$$295$$ 460.000 0.0907872
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −528.630 −0.102246
$$300$$ 0 0
$$301$$ 380.822 0.0729243
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1850.86 0.347475
$$306$$ 0 0
$$307$$ 1529.87 0.284412 0.142206 0.989837i $$-0.454580\pi$$
0.142206 + 0.989837i $$0.454580\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6583.78 −1.20042 −0.600212 0.799841i $$-0.704917\pi$$
−0.600212 + 0.799841i $$0.704917\pi$$
$$312$$ 0 0
$$313$$ 110.352 0.0199281 0.00996403 0.999950i $$-0.496828\pi$$
0.00996403 + 0.999950i $$0.496828\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4003.48 −0.709331 −0.354666 0.934993i $$-0.615405\pi$$
−0.354666 + 0.934993i $$0.615405\pi$$
$$318$$ 0 0
$$319$$ −10710.9 −1.87992
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 738.529 0.127222
$$324$$ 0 0
$$325$$ 1386.10 0.236575
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2025.28 −0.339384
$$330$$ 0 0
$$331$$ 3066.52 0.509219 0.254609 0.967044i $$-0.418053\pi$$
0.254609 + 0.967044i $$0.418053\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −5437.63 −0.886834
$$336$$ 0 0
$$337$$ 10714.9 1.73199 0.865993 0.500057i $$-0.166687\pi$$
0.865993 + 0.500057i $$0.166687\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7796.61 −1.23815
$$342$$ 0 0
$$343$$ 5645.26 0.888674
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9008.73 −1.39370 −0.696850 0.717217i $$-0.745416\pi$$
−0.696850 + 0.717217i $$0.745416\pi$$
$$348$$ 0 0
$$349$$ −9529.14 −1.46156 −0.730778 0.682615i $$-0.760843\pi$$
−0.730778 + 0.682615i $$0.760843\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −982.433 −0.148129 −0.0740646 0.997253i $$-0.523597\pi$$
−0.0740646 + 0.997253i $$0.523597\pi$$
$$354$$ 0 0
$$355$$ −1668.40 −0.249436
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −9887.90 −1.45366 −0.726829 0.686818i $$-0.759007\pi$$
−0.726829 + 0.686818i $$0.759007\pi$$
$$360$$ 0 0
$$361$$ −6162.23 −0.898415
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3033.96 −0.435081
$$366$$ 0 0
$$367$$ 3348.54 0.476274 0.238137 0.971232i $$-0.423463\pi$$
0.238137 + 0.971232i $$0.423463\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2681.67 0.375270
$$372$$ 0 0
$$373$$ 9572.04 1.32874 0.664372 0.747402i $$-0.268699\pi$$
0.664372 + 0.747402i $$0.268699\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12107.8 1.65406
$$378$$ 0 0
$$379$$ 9976.30 1.35211 0.676053 0.736853i $$-0.263689\pi$$
0.676053 + 0.736853i $$0.263689\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7619.84 1.01659 0.508297 0.861182i $$-0.330275\pi$$
0.508297 + 0.861182i $$0.330275\pi$$
$$384$$ 0 0
$$385$$ 2321.29 0.307282
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −7802.78 −1.01701 −0.508504 0.861059i $$-0.669801\pi$$
−0.508504 + 0.861059i $$0.669801\pi$$
$$390$$ 0 0
$$391$$ 266.760 0.0345029
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 221.875 0.0282627
$$396$$ 0 0
$$397$$ −8073.51 −1.02065 −0.510325 0.859982i $$-0.670475\pi$$
−0.510325 + 0.859982i $$0.670475\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6306.80 −0.785403 −0.392702 0.919666i $$-0.628459\pi$$
−0.392702 + 0.919666i $$0.628459\pi$$
$$402$$ 0 0
$$403$$ 8813.39 1.08940
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9302.08 −1.13289
$$408$$ 0 0
$$409$$ −15362.3 −1.85726 −0.928629 0.371010i $$-0.879012\pi$$
−0.928629 + 0.371010i $$0.879012\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −870.825 −0.103754
$$414$$ 0 0
$$415$$ 538.659 0.0637150
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −13791.3 −1.60799 −0.803994 0.594637i $$-0.797296\pi$$
−0.803994 + 0.594637i $$0.797296\pi$$
$$420$$ 0 0
$$421$$ −9093.23 −1.05268 −0.526339 0.850275i $$-0.676436\pi$$
−0.526339 + 0.850275i $$0.676436\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −699.459 −0.0798323
$$426$$ 0 0
$$427$$ −3503.86 −0.397104
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −400.347 −0.0447425 −0.0223713 0.999750i $$-0.507122\pi$$
−0.0223713 + 0.999750i $$0.507122\pi$$
$$432$$ 0 0
$$433$$ −10063.9 −1.11695 −0.558474 0.829522i $$-0.688613\pi$$
−0.558474 + 0.829522i $$0.688613\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 251.677 0.0275500
$$438$$ 0 0
$$439$$ 11684.8 1.27035 0.635176 0.772368i $$-0.280928\pi$$
0.635176 + 0.772368i $$0.280928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3698.68 0.396681 0.198341 0.980133i $$-0.436445\pi$$
0.198341 + 0.980133i $$0.436445\pi$$
$$444$$ 0 0
$$445$$ 1289.09 0.137323
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −11318.1 −1.18961 −0.594804 0.803871i $$-0.702770\pi$$
−0.594804 + 0.803871i $$0.702770\pi$$
$$450$$ 0 0
$$451$$ 12072.8 1.26050
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2624.01 −0.270364
$$456$$ 0 0
$$457$$ −2436.98 −0.249446 −0.124723 0.992192i $$-0.539804\pi$$
−0.124723 + 0.992192i $$0.539804\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 8575.54 0.866384 0.433192 0.901302i $$-0.357387\pi$$
0.433192 + 0.901302i $$0.357387\pi$$
$$462$$ 0 0
$$463$$ 4711.30 0.472900 0.236450 0.971644i $$-0.424016\pi$$
0.236450 + 0.971644i $$0.424016\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −11075.4 −1.09745 −0.548726 0.836002i $$-0.684887\pi$$
−0.548726 + 0.836002i $$0.684887\pi$$
$$468$$ 0 0
$$469$$ 10294.0 1.01350
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1973.31 0.191824
$$474$$ 0 0
$$475$$ −659.911 −0.0637449
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 17455.1 1.66502 0.832510 0.554010i $$-0.186903\pi$$
0.832510 + 0.554010i $$0.186903\pi$$
$$480$$ 0 0
$$481$$ 10515.2 0.996781
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6560.25 0.614197
$$486$$ 0 0
$$487$$ −8285.77 −0.770974 −0.385487 0.922713i $$-0.625966\pi$$
−0.385487 + 0.922713i $$0.625966\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7055.06 0.648453 0.324226 0.945980i $$-0.394896\pi$$
0.324226 + 0.945980i $$0.394896\pi$$
$$492$$ 0 0
$$493$$ −6109.88 −0.558165
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3158.45 0.285062
$$498$$ 0 0
$$499$$ −2778.49 −0.249263 −0.124632 0.992203i $$-0.539775\pi$$
−0.124632 + 0.992203i $$0.539775\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 5445.52 0.482711 0.241356 0.970437i $$-0.422408\pi$$
0.241356 + 0.970437i $$0.422408\pi$$
$$504$$ 0 0
$$505$$ −316.646 −0.0279021
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −16567.4 −1.44270 −0.721352 0.692569i $$-0.756479\pi$$
−0.721352 + 0.692569i $$0.756479\pi$$
$$510$$ 0 0
$$511$$ 5743.58 0.497223
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6099.26 −0.521875
$$516$$ 0 0
$$517$$ −10494.4 −0.892734
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3805.03 0.319964 0.159982 0.987120i $$-0.448856\pi$$
0.159982 + 0.987120i $$0.448856\pi$$
$$522$$ 0 0
$$523$$ 7489.80 0.626207 0.313104 0.949719i $$-0.398631\pi$$
0.313104 + 0.949719i $$0.398631\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4447.46 −0.367617
$$528$$ 0 0
$$529$$ −12076.1 −0.992528
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −13647.3 −1.10906
$$534$$ 0 0
$$535$$ 211.974 0.0171298
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12428.8 0.993224
$$540$$ 0 0
$$541$$ −2063.73 −0.164005 −0.0820026 0.996632i $$-0.526132\pi$$
−0.0820026 + 0.996632i $$0.526132\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1333.71 0.104825
$$546$$ 0 0
$$547$$ 12954.6 1.01261 0.506307 0.862353i $$-0.331010\pi$$
0.506307 + 0.862353i $$0.331010\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5764.43 −0.445686
$$552$$ 0 0
$$553$$ −420.032 −0.0322994
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14340.8 1.09091 0.545457 0.838139i $$-0.316356\pi$$
0.545457 + 0.838139i $$0.316356\pi$$
$$558$$ 0 0
$$559$$ −2230.65 −0.168777
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −12827.3 −0.960222 −0.480111 0.877208i $$-0.659404\pi$$
−0.480111 + 0.877208i $$0.659404\pi$$
$$564$$ 0 0
$$565$$ −9357.98 −0.696802
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −15102.9 −1.11274 −0.556370 0.830935i $$-0.687806\pi$$
−0.556370 + 0.830935i $$0.687806\pi$$
$$570$$ 0 0
$$571$$ −22553.6 −1.65295 −0.826477 0.562970i $$-0.809658\pi$$
−0.826477 + 0.562970i $$0.809658\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −238.363 −0.0172877
$$576$$ 0 0
$$577$$ 5007.38 0.361283 0.180641 0.983549i $$-0.442183\pi$$
0.180641 + 0.983549i $$0.442183\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1019.73 −0.0728153
$$582$$ 0 0
$$583$$ 13895.6 0.987131
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −24975.4 −1.75612 −0.878062 0.478547i $$-0.841163\pi$$
−0.878062 + 0.478547i $$0.841163\pi$$
$$588$$ 0 0
$$589$$ −4196.00 −0.293537
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 19247.2 1.33287 0.666433 0.745565i $$-0.267820\pi$$
0.666433 + 0.745565i $$0.267820\pi$$
$$594$$ 0 0
$$595$$ 1324.14 0.0912346
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6000.84 0.409328 0.204664 0.978832i $$-0.434390\pi$$
0.204664 + 0.978832i $$0.434390\pi$$
$$600$$ 0 0
$$601$$ −8576.08 −0.582073 −0.291036 0.956712i $$-0.594000\pi$$
−0.291036 + 0.956712i $$0.594000\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 5373.22 0.361079
$$606$$ 0 0
$$607$$ 2960.22 0.197943 0.0989715 0.995090i $$-0.468445\pi$$
0.0989715 + 0.995090i $$0.468445\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 11863.0 0.785477
$$612$$ 0 0
$$613$$ 6483.88 0.427212 0.213606 0.976920i $$-0.431479\pi$$
0.213606 + 0.976920i $$0.431479\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1169.84 0.0763304 0.0381652 0.999271i $$-0.487849\pi$$
0.0381652 + 0.999271i $$0.487849\pi$$
$$618$$ 0 0
$$619$$ −2930.97 −0.190316 −0.0951580 0.995462i $$-0.530336\pi$$
−0.0951580 + 0.995462i $$0.530336\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2440.38 −0.156937
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −5306.23 −0.336365
$$630$$ 0 0
$$631$$ 29359.9 1.85230 0.926150 0.377156i $$-0.123098\pi$$
0.926150 + 0.377156i $$0.123098\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 6160.54 0.384998
$$636$$ 0 0
$$637$$ −14049.7 −0.873894
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14661.8 0.903443 0.451721 0.892159i $$-0.350810\pi$$
0.451721 + 0.892159i $$0.350810\pi$$
$$642$$ 0 0
$$643$$ −11141.3 −0.683312 −0.341656 0.939825i $$-0.610988\pi$$
−0.341656 + 0.939825i $$0.610988\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17044.5 1.03569 0.517844 0.855475i $$-0.326735\pi$$
0.517844 + 0.855475i $$0.326735\pi$$
$$648$$ 0 0
$$649$$ −4512.36 −0.272921
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −19960.2 −1.19618 −0.598089 0.801430i $$-0.704073\pi$$
−0.598089 + 0.801430i $$0.704073\pi$$
$$654$$ 0 0
$$655$$ −6045.87 −0.360659
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13422.1 0.793399 0.396700 0.917948i $$-0.370155\pi$$
0.396700 + 0.917948i $$0.370155\pi$$
$$660$$ 0 0
$$661$$ 13349.5 0.785531 0.392765 0.919639i $$-0.371518\pi$$
0.392765 + 0.919639i $$0.371518\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1249.28 0.0728494
$$666$$ 0 0
$$667$$ −2082.14 −0.120871
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −18156.0 −1.04456
$$672$$ 0 0
$$673$$ 6927.58 0.396788 0.198394 0.980122i $$-0.436427\pi$$
0.198394 + 0.980122i $$0.436427\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6607.86 0.375126 0.187563 0.982253i $$-0.439941\pi$$
0.187563 + 0.982253i $$0.439941\pi$$
$$678$$ 0 0
$$679$$ −12419.2 −0.701922
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 15422.7 0.864031 0.432015 0.901866i $$-0.357803\pi$$
0.432015 + 0.901866i $$0.357803\pi$$
$$684$$ 0 0
$$685$$ −13148.3 −0.733388
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −15707.8 −0.868533
$$690$$ 0 0
$$691$$ 16254.6 0.894870 0.447435 0.894317i $$-0.352338\pi$$
0.447435 + 0.894317i $$0.352338\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −11075.3 −0.604475
$$696$$ 0 0
$$697$$ 6886.76 0.374254
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −10389.8 −0.559796 −0.279898 0.960030i $$-0.590301\pi$$
−0.279898 + 0.960030i $$0.590301\pi$$
$$702$$ 0 0
$$703$$ −5006.22 −0.268582
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 599.442 0.0318873
$$708$$ 0 0
$$709$$ 7530.22 0.398877 0.199438 0.979910i $$-0.436088\pi$$
0.199438 + 0.979910i $$0.436088\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1515.61 −0.0796075
$$714$$ 0 0
$$715$$ −13596.9 −0.711181
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 33888.4 1.75775 0.878875 0.477051i $$-0.158294\pi$$
0.878875 + 0.477051i $$0.158294\pi$$
$$720$$ 0 0
$$721$$ 11546.5 0.596413
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5459.47 0.279669
$$726$$ 0 0
$$727$$ 13863.3 0.707239 0.353619 0.935389i $$-0.384951\pi$$
0.353619 + 0.935389i $$0.384951\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1125.64 0.0569541
$$732$$ 0 0
$$733$$ 24497.7 1.23444 0.617219 0.786792i $$-0.288259\pi$$
0.617219 + 0.786792i $$0.288259\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 53340.3 2.66596
$$738$$ 0 0
$$739$$ −19029.1 −0.947221 −0.473611 0.880734i $$-0.657050\pi$$
−0.473611 + 0.880734i $$0.657050\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 5639.67 0.278465 0.139233 0.990260i $$-0.455536\pi$$
0.139233 + 0.990260i $$0.455536\pi$$
$$744$$ 0 0
$$745$$ −12234.7 −0.601672
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −401.287 −0.0195764
$$750$$ 0 0
$$751$$ 14566.0 0.707753 0.353876 0.935292i $$-0.384863\pi$$
0.353876 + 0.935292i $$0.384863\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −596.825 −0.0287691
$$756$$ 0 0
$$757$$ −2958.83 −0.142061 −0.0710307 0.997474i $$-0.522629\pi$$
−0.0710307 + 0.997474i $$0.522629\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −281.640 −0.0134158 −0.00670791 0.999978i $$-0.502135\pi$$
−0.00670791 + 0.999978i $$0.502135\pi$$
$$762$$ 0 0
$$763$$ −2524.84 −0.119797
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 5100.83 0.240131
$$768$$ 0 0
$$769$$ 29799.2 1.39738 0.698690 0.715425i $$-0.253767\pi$$
0.698690 + 0.715425i $$0.253767\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −8377.18 −0.389788 −0.194894 0.980824i $$-0.562436\pi$$
−0.194894 + 0.980824i $$0.562436\pi$$
$$774$$ 0 0
$$775$$ 3974.02 0.184195
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6497.39 0.298836
$$780$$ 0 0
$$781$$ 16366.2 0.749843
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 11064.3 0.503059
$$786$$ 0 0
$$787$$ 17368.8 0.786697 0.393348 0.919389i $$-0.371317\pi$$
0.393348 + 0.919389i $$0.371317\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 17715.6 0.796325
$$792$$ 0 0
$$793$$ 20523.7 0.919066
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6003.86 0.266835 0.133418 0.991060i $$-0.457405\pi$$
0.133418 + 0.991060i $$0.457405\pi$$
$$798$$ 0 0
$$799$$ −5986.38 −0.265060
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 29761.5 1.30792
$$804$$ 0 0
$$805$$ 451.244 0.0197568
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 42392.2 1.84231 0.921157 0.389191i $$-0.127246\pi$$
0.921157 + 0.389191i $$0.127246\pi$$
$$810$$ 0 0
$$811$$ 7148.82 0.309530 0.154765 0.987951i $$-0.450538\pi$$
0.154765 + 0.987951i $$0.450538\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −17402.7 −0.747964
$$816$$ 0 0
$$817$$ 1062.00 0.0454770
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25700.4 −1.09251 −0.546254 0.837619i $$-0.683947\pi$$
−0.546254 + 0.837619i $$0.683947\pi$$
$$822$$ 0 0
$$823$$ 19064.7 0.807476 0.403738 0.914875i $$-0.367711\pi$$
0.403738 + 0.914875i $$0.367711\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −41947.6 −1.76380 −0.881900 0.471437i $$-0.843736\pi$$
−0.881900 + 0.471437i $$0.843736\pi$$
$$828$$ 0 0
$$829$$ −31403.8 −1.31568 −0.657841 0.753157i $$-0.728530\pi$$
−0.657841 + 0.753157i $$0.728530\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 7089.84 0.294896
$$834$$ 0 0
$$835$$ −12385.8 −0.513326
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 25495.3 1.04910 0.524550 0.851380i $$-0.324234\pi$$
0.524550 + 0.851380i $$0.324234\pi$$
$$840$$ 0 0
$$841$$ 23300.3 0.955362
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4385.09 0.178523
$$846$$ 0 0
$$847$$ −10172.0 −0.412651
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1808.27 −0.0728398
$$852$$ 0 0
$$853$$ −13537.1 −0.543377 −0.271689 0.962385i $$-0.587582\pi$$
−0.271689 + 0.962385i $$0.587582\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15337.3 0.611332 0.305666 0.952139i $$-0.401121\pi$$
0.305666 + 0.952139i $$0.401121\pi$$
$$858$$ 0 0
$$859$$ −19720.1 −0.783286 −0.391643 0.920117i $$-0.628093\pi$$
−0.391643 + 0.920117i $$0.628093\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 21464.2 0.846641 0.423321 0.905980i $$-0.360864\pi$$
0.423321 + 0.905980i $$0.360864\pi$$
$$864$$ 0 0
$$865$$ −133.723 −0.00525633
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2176.48 −0.0849621
$$870$$ 0 0
$$871$$ −60296.6 −2.34566
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1183.19 −0.0457131
$$876$$ 0 0
$$877$$ 3671.90 0.141381 0.0706905 0.997498i $$-0.477480\pi$$
0.0706905 + 0.997498i $$0.477480\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 48518.9 1.85544 0.927720 0.373276i $$-0.121766\pi$$
0.927720 + 0.373276i $$0.121766\pi$$
$$882$$ 0 0
$$883$$ −13413.5 −0.511210 −0.255605 0.966781i $$-0.582275\pi$$
−0.255605 + 0.966781i $$0.582275\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −50035.0 −1.89404 −0.947018 0.321180i $$-0.895921\pi$$
−0.947018 + 0.321180i $$0.895921\pi$$
$$888$$ 0 0
$$889$$ −11662.5 −0.439986
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −5647.91 −0.211646
$$894$$ 0 0
$$895$$ 7003.77 0.261576
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 34713.7 1.28784
$$900$$ 0 0
$$901$$ 7926.54 0.293087
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 17462.4 0.641403
$$906$$ 0 0
$$907$$ −35905.1 −1.31446 −0.657228 0.753692i $$-0.728271\pi$$
−0.657228 + 0.753692i $$0.728271\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −36938.6 −1.34339 −0.671697 0.740826i $$-0.734434\pi$$
−0.671697 + 0.740826i $$0.734434\pi$$
$$912$$ 0 0
$$913$$ −5283.96 −0.191537
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 11445.4 0.412171
$$918$$ 0 0
$$919$$ −35867.0 −1.28742 −0.643712 0.765268i $$-0.722606\pi$$
−0.643712 + 0.765268i $$0.722606\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −18500.5 −0.659753
$$924$$ 0 0
$$925$$ 4741.37 0.168536
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 11391.2 0.402297 0.201148 0.979561i $$-0.435533\pi$$
0.201148 + 0.979561i $$0.435533\pi$$
$$930$$ 0 0
$$931$$ 6688.98 0.235470
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 6861.32 0.239988
$$936$$ 0 0
$$937$$ 30928.7 1.07833 0.539166 0.842199i $$-0.318740\pi$$
0.539166 + 0.842199i $$0.318740\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −36684.3 −1.27086 −0.635428 0.772160i $$-0.719176\pi$$
−0.635428 + 0.772160i $$0.719176\pi$$
$$942$$ 0 0
$$943$$ 2346.88 0.0810446
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 39211.7 1.34552 0.672761 0.739860i $$-0.265108\pi$$
0.672761 + 0.739860i $$0.265108\pi$$
$$948$$ 0 0
$$949$$ −33642.8 −1.15078
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −30476.1 −1.03591 −0.517953 0.855409i $$-0.673306\pi$$
−0.517953 + 0.855409i $$0.673306\pi$$
$$954$$ 0 0
$$955$$ −13640.8 −0.462206
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24891.0 0.838136
$$960$$ 0 0
$$961$$ −4522.48 −0.151807
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 12199.1 0.406946
$$966$$ 0 0
$$967$$ −48866.8 −1.62508 −0.812538 0.582908i $$-0.801915\pi$$
−0.812538 + 0.582908i $$0.801915\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 54060.7 1.78670 0.893352 0.449358i $$-0.148347\pi$$
0.893352 + 0.449358i $$0.148347\pi$$
$$972$$ 0 0
$$973$$ 20966.6 0.690811
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27233.7 0.891795 0.445898 0.895084i $$-0.352885\pi$$
0.445898 + 0.895084i $$0.352885\pi$$
$$978$$ 0 0
$$979$$ −12645.3 −0.412816
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 949.700 0.0308146 0.0154073 0.999881i $$-0.495096\pi$$
0.0154073 + 0.999881i $$0.495096\pi$$
$$984$$ 0 0
$$985$$ 3829.53 0.123877
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 383.599 0.0123334
$$990$$ 0 0
$$991$$ −37146.4 −1.19071 −0.595356 0.803462i $$-0.702989\pi$$
−0.595356 + 0.803462i $$0.702989\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −3770.15 −0.120123
$$996$$ 0 0
$$997$$ 45360.4 1.44090 0.720451 0.693506i $$-0.243935\pi$$
0.720451 + 0.693506i $$0.243935\pi$$
$$998$$ 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.bn.1.2 3
3.2 odd 2 2160.4.a.bf.1.2 3
4.3 odd 2 1080.4.a.n.1.2 yes 3
12.11 even 2 1080.4.a.h.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.2 3 12.11 even 2
1080.4.a.n.1.2 yes 3 4.3 odd 2
2160.4.a.bf.1.2 3 3.2 odd 2
2160.4.a.bn.1.2 3 1.1 even 1 trivial