# Properties

 Label 1350.4.a.e.1.1 Level $1350$ Weight $4$ Character 1350.1 Self dual yes Analytic conductor $79.653$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.6525785077$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1350.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} -8.00000 q^{8} -3.00000 q^{11} -47.0000 q^{13} +28.0000 q^{14} +16.0000 q^{16} -39.0000 q^{17} +32.0000 q^{19} +6.00000 q^{22} -99.0000 q^{23} +94.0000 q^{26} -56.0000 q^{28} -51.0000 q^{29} +83.0000 q^{31} -32.0000 q^{32} +78.0000 q^{34} -314.000 q^{37} -64.0000 q^{38} +108.000 q^{41} -299.000 q^{43} -12.0000 q^{44} +198.000 q^{46} +531.000 q^{47} -147.000 q^{49} -188.000 q^{52} +564.000 q^{53} +112.000 q^{56} +102.000 q^{58} -12.0000 q^{59} +230.000 q^{61} -166.000 q^{62} +64.0000 q^{64} +268.000 q^{67} -156.000 q^{68} -120.000 q^{71} -1106.00 q^{73} +628.000 q^{74} +128.000 q^{76} +42.0000 q^{77} -739.000 q^{79} -216.000 q^{82} +1086.00 q^{83} +598.000 q^{86} +24.0000 q^{88} +120.000 q^{89} +658.000 q^{91} -396.000 q^{92} -1062.00 q^{94} +1642.00 q^{97} +294.000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −14.0000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 −0.0822304 −0.0411152 0.999154i $$-0.513091\pi$$
−0.0411152 + 0.999154i $$0.513091\pi$$
$$12$$ 0 0
$$13$$ −47.0000 −1.00273 −0.501364 0.865237i $$-0.667168\pi$$
−0.501364 + 0.865237i $$0.667168\pi$$
$$14$$ 28.0000 0.534522
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −39.0000 −0.556405 −0.278203 0.960522i $$-0.589739\pi$$
−0.278203 + 0.960522i $$0.589739\pi$$
$$18$$ 0 0
$$19$$ 32.0000 0.386384 0.193192 0.981161i $$-0.438116\pi$$
0.193192 + 0.981161i $$0.438116\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.00000 0.0581456
$$23$$ −99.0000 −0.897519 −0.448759 0.893653i $$-0.648134\pi$$
−0.448759 + 0.893653i $$0.648134\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 94.0000 0.709035
$$27$$ 0 0
$$28$$ −56.0000 −0.377964
$$29$$ −51.0000 −0.326568 −0.163284 0.986579i $$-0.552209\pi$$
−0.163284 + 0.986579i $$0.552209\pi$$
$$30$$ 0 0
$$31$$ 83.0000 0.480879 0.240439 0.970664i $$-0.422708\pi$$
0.240439 + 0.970664i $$0.422708\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ 78.0000 0.393438
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −314.000 −1.39517 −0.697585 0.716502i $$-0.745742\pi$$
−0.697585 + 0.716502i $$0.745742\pi$$
$$38$$ −64.0000 −0.273215
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 108.000 0.411385 0.205692 0.978617i $$-0.434055\pi$$
0.205692 + 0.978617i $$0.434055\pi$$
$$42$$ 0 0
$$43$$ −299.000 −1.06040 −0.530199 0.847874i $$-0.677883\pi$$
−0.530199 + 0.847874i $$0.677883\pi$$
$$44$$ −12.0000 −0.0411152
$$45$$ 0 0
$$46$$ 198.000 0.634641
$$47$$ 531.000 1.64796 0.823982 0.566616i $$-0.191748\pi$$
0.823982 + 0.566616i $$0.191748\pi$$
$$48$$ 0 0
$$49$$ −147.000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −188.000 −0.501364
$$53$$ 564.000 1.46172 0.730862 0.682525i $$-0.239118\pi$$
0.730862 + 0.682525i $$0.239118\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 112.000 0.267261
$$57$$ 0 0
$$58$$ 102.000 0.230918
$$59$$ −12.0000 −0.0264791 −0.0132396 0.999912i $$-0.504214\pi$$
−0.0132396 + 0.999912i $$0.504214\pi$$
$$60$$ 0 0
$$61$$ 230.000 0.482762 0.241381 0.970430i $$-0.422400\pi$$
0.241381 + 0.970430i $$0.422400\pi$$
$$62$$ −166.000 −0.340033
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 268.000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −156.000 −0.278203
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −120.000 −0.200583 −0.100291 0.994958i $$-0.531978\pi$$
−0.100291 + 0.994958i $$0.531978\pi$$
$$72$$ 0 0
$$73$$ −1106.00 −1.77325 −0.886627 0.462486i $$-0.846958\pi$$
−0.886627 + 0.462486i $$0.846958\pi$$
$$74$$ 628.000 0.986534
$$75$$ 0 0
$$76$$ 128.000 0.193192
$$77$$ 42.0000 0.0621603
$$78$$ 0 0
$$79$$ −739.000 −1.05246 −0.526228 0.850344i $$-0.676394\pi$$
−0.526228 + 0.850344i $$0.676394\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −216.000 −0.290893
$$83$$ 1086.00 1.43619 0.718096 0.695944i $$-0.245014\pi$$
0.718096 + 0.695944i $$0.245014\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 598.000 0.749814
$$87$$ 0 0
$$88$$ 24.0000 0.0290728
$$89$$ 120.000 0.142921 0.0714605 0.997443i $$-0.477234\pi$$
0.0714605 + 0.997443i $$0.477234\pi$$
$$90$$ 0 0
$$91$$ 658.000 0.757991
$$92$$ −396.000 −0.448759
$$93$$ 0 0
$$94$$ −1062.00 −1.16529
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1642.00 1.71876 0.859381 0.511336i $$-0.170849\pi$$
0.859381 + 0.511336i $$0.170849\pi$$
$$98$$ 294.000 0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −33.0000 −0.0325111 −0.0162556 0.999868i $$-0.505175\pi$$
−0.0162556 + 0.999868i $$0.505175\pi$$
$$102$$ 0 0
$$103$$ 1198.00 1.14604 0.573022 0.819540i $$-0.305771\pi$$
0.573022 + 0.819540i $$0.305771\pi$$
$$104$$ 376.000 0.354518
$$105$$ 0 0
$$106$$ −1128.00 −1.03359
$$107$$ −1542.00 −1.39318 −0.696592 0.717467i $$-0.745301\pi$$
−0.696592 + 0.717467i $$0.745301\pi$$
$$108$$ 0 0
$$109$$ −556.000 −0.488579 −0.244290 0.969702i $$-0.578555\pi$$
−0.244290 + 0.969702i $$0.578555\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −224.000 −0.188982
$$113$$ 1605.00 1.33616 0.668078 0.744091i $$-0.267117\pi$$
0.668078 + 0.744091i $$0.267117\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −204.000 −0.163284
$$117$$ 0 0
$$118$$ 24.0000 0.0187236
$$119$$ 546.000 0.420603
$$120$$ 0 0
$$121$$ −1322.00 −0.993238
$$122$$ −460.000 −0.341364
$$123$$ 0 0
$$124$$ 332.000 0.240439
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1334.00 −0.932074 −0.466037 0.884765i $$-0.654319\pi$$
−0.466037 + 0.884765i $$0.654319\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2883.00 1.92282 0.961408 0.275127i $$-0.0887199\pi$$
0.961408 + 0.275127i $$0.0887199\pi$$
$$132$$ 0 0
$$133$$ −448.000 −0.292079
$$134$$ −536.000 −0.345547
$$135$$ 0 0
$$136$$ 312.000 0.196719
$$137$$ 282.000 0.175860 0.0879302 0.996127i $$-0.471975\pi$$
0.0879302 + 0.996127i $$0.471975\pi$$
$$138$$ 0 0
$$139$$ −2494.00 −1.52186 −0.760929 0.648835i $$-0.775257\pi$$
−0.760929 + 0.648835i $$0.775257\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 240.000 0.141833
$$143$$ 141.000 0.0824546
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2212.00 1.25388
$$147$$ 0 0
$$148$$ −1256.00 −0.697585
$$149$$ −2595.00 −1.42678 −0.713392 0.700766i $$-0.752842\pi$$
−0.713392 + 0.700766i $$0.752842\pi$$
$$150$$ 0 0
$$151$$ 1229.00 0.662348 0.331174 0.943570i $$-0.392555\pi$$
0.331174 + 0.943570i $$0.392555\pi$$
$$152$$ −256.000 −0.136608
$$153$$ 0 0
$$154$$ −84.0000 −0.0439540
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1591.00 0.808762 0.404381 0.914591i $$-0.367487\pi$$
0.404381 + 0.914591i $$0.367487\pi$$
$$158$$ 1478.00 0.744199
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1386.00 0.678460
$$162$$ 0 0
$$163$$ 457.000 0.219601 0.109801 0.993954i $$-0.464979\pi$$
0.109801 + 0.993954i $$0.464979\pi$$
$$164$$ 432.000 0.205692
$$165$$ 0 0
$$166$$ −2172.00 −1.01554
$$167$$ −1164.00 −0.539359 −0.269680 0.962950i $$-0.586918\pi$$
−0.269680 + 0.962950i $$0.586918\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.00546199
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1196.00 −0.530199
$$173$$ 3942.00 1.73240 0.866199 0.499700i $$-0.166556\pi$$
0.866199 + 0.499700i $$0.166556\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −48.0000 −0.0205576
$$177$$ 0 0
$$178$$ −240.000 −0.101060
$$179$$ 1212.00 0.506085 0.253042 0.967455i $$-0.418569\pi$$
0.253042 + 0.967455i $$0.418569\pi$$
$$180$$ 0 0
$$181$$ 2288.00 0.939590 0.469795 0.882776i $$-0.344328\pi$$
0.469795 + 0.882776i $$0.344328\pi$$
$$182$$ −1316.00 −0.535980
$$183$$ 0 0
$$184$$ 792.000 0.317321
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 117.000 0.0457534
$$188$$ 2124.00 0.823982
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1938.00 0.734182 0.367091 0.930185i $$-0.380354\pi$$
0.367091 + 0.930185i $$0.380354\pi$$
$$192$$ 0 0
$$193$$ 1498.00 0.558696 0.279348 0.960190i $$-0.409882\pi$$
0.279348 + 0.960190i $$0.409882\pi$$
$$194$$ −3284.00 −1.21535
$$195$$ 0 0
$$196$$ −588.000 −0.214286
$$197$$ −2124.00 −0.768166 −0.384083 0.923299i $$-0.625482\pi$$
−0.384083 + 0.923299i $$0.625482\pi$$
$$198$$ 0 0
$$199$$ −385.000 −0.137145 −0.0685727 0.997646i $$-0.521845\pi$$
−0.0685727 + 0.997646i $$0.521845\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 66.0000 0.0229888
$$203$$ 714.000 0.246862
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2396.00 −0.810375
$$207$$ 0 0
$$208$$ −752.000 −0.250682
$$209$$ −96.0000 −0.0317725
$$210$$ 0 0
$$211$$ 3170.00 1.03427 0.517137 0.855903i $$-0.326998\pi$$
0.517137 + 0.855903i $$0.326998\pi$$
$$212$$ 2256.00 0.730862
$$213$$ 0 0
$$214$$ 3084.00 0.985130
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1162.00 −0.363510
$$218$$ 1112.00 0.345478
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1833.00 0.557923
$$222$$ 0 0
$$223$$ −1388.00 −0.416804 −0.208402 0.978043i $$-0.566826\pi$$
−0.208402 + 0.978043i $$0.566826\pi$$
$$224$$ 448.000 0.133631
$$225$$ 0 0
$$226$$ −3210.00 −0.944805
$$227$$ −4644.00 −1.35786 −0.678928 0.734205i $$-0.737555\pi$$
−0.678928 + 0.734205i $$0.737555\pi$$
$$228$$ 0 0
$$229$$ 4736.00 1.36665 0.683327 0.730113i $$-0.260532\pi$$
0.683327 + 0.730113i $$0.260532\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 408.000 0.115459
$$233$$ 2814.00 0.791207 0.395604 0.918421i $$-0.370535\pi$$
0.395604 + 0.918421i $$0.370535\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −48.0000 −0.0132396
$$237$$ 0 0
$$238$$ −1092.00 −0.297411
$$239$$ 2202.00 0.595965 0.297982 0.954571i $$-0.403686\pi$$
0.297982 + 0.954571i $$0.403686\pi$$
$$240$$ 0 0
$$241$$ 3485.00 0.931488 0.465744 0.884920i $$-0.345787\pi$$
0.465744 + 0.884920i $$0.345787\pi$$
$$242$$ 2644.00 0.702325
$$243$$ 0 0
$$244$$ 920.000 0.241381
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1504.00 −0.387438
$$248$$ −664.000 −0.170016
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6345.00 1.59559 0.797795 0.602929i $$-0.206000\pi$$
0.797795 + 0.602929i $$0.206000\pi$$
$$252$$ 0 0
$$253$$ 297.000 0.0738033
$$254$$ 2668.00 0.659076
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 525.000 0.127426 0.0637132 0.997968i $$-0.479706\pi$$
0.0637132 + 0.997968i $$0.479706\pi$$
$$258$$ 0 0
$$259$$ 4396.00 1.05465
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −5766.00 −1.35964
$$263$$ 5196.00 1.21825 0.609124 0.793075i $$-0.291521\pi$$
0.609124 + 0.793075i $$0.291521\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 896.000 0.206531
$$267$$ 0 0
$$268$$ 1072.00 0.244339
$$269$$ 7479.00 1.69518 0.847589 0.530654i $$-0.178054\pi$$
0.847589 + 0.530654i $$0.178054\pi$$
$$270$$ 0 0
$$271$$ −856.000 −0.191876 −0.0959378 0.995387i $$-0.530585\pi$$
−0.0959378 + 0.995387i $$0.530585\pi$$
$$272$$ −624.000 −0.139101
$$273$$ 0 0
$$274$$ −564.000 −0.124352
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7054.00 1.53009 0.765043 0.643979i $$-0.222718\pi$$
0.765043 + 0.643979i $$0.222718\pi$$
$$278$$ 4988.00 1.07612
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1014.00 −0.215268 −0.107634 0.994191i $$-0.534327\pi$$
−0.107634 + 0.994191i $$0.534327\pi$$
$$282$$ 0 0
$$283$$ −992.000 −0.208368 −0.104184 0.994558i $$-0.533223\pi$$
−0.104184 + 0.994558i $$0.533223\pi$$
$$284$$ −480.000 −0.100291
$$285$$ 0 0
$$286$$ −282.000 −0.0583042
$$287$$ −1512.00 −0.310977
$$288$$ 0 0
$$289$$ −3392.00 −0.690413
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4424.00 −0.886627
$$293$$ −4950.00 −0.986970 −0.493485 0.869754i $$-0.664277\pi$$
−0.493485 + 0.869754i $$0.664277\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2512.00 0.493267
$$297$$ 0 0
$$298$$ 5190.00 1.00889
$$299$$ 4653.00 0.899966
$$300$$ 0 0
$$301$$ 4186.00 0.801585
$$302$$ −2458.00 −0.468351
$$303$$ 0 0
$$304$$ 512.000 0.0965961
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4777.00 0.888071 0.444035 0.896009i $$-0.353546\pi$$
0.444035 + 0.896009i $$0.353546\pi$$
$$308$$ 168.000 0.0310802
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7692.00 1.40249 0.701243 0.712922i $$-0.252629\pi$$
0.701243 + 0.712922i $$0.252629\pi$$
$$312$$ 0 0
$$313$$ 2932.00 0.529477 0.264739 0.964320i $$-0.414714\pi$$
0.264739 + 0.964320i $$0.414714\pi$$
$$314$$ −3182.00 −0.571881
$$315$$ 0 0
$$316$$ −2956.00 −0.526228
$$317$$ 8352.00 1.47980 0.739898 0.672720i $$-0.234874\pi$$
0.739898 + 0.672720i $$0.234874\pi$$
$$318$$ 0 0
$$319$$ 153.000 0.0268538
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −2772.00 −0.479744
$$323$$ −1248.00 −0.214986
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −914.000 −0.155282
$$327$$ 0 0
$$328$$ −864.000 −0.145446
$$329$$ −7434.00 −1.24574
$$330$$ 0 0
$$331$$ −3070.00 −0.509796 −0.254898 0.966968i $$-0.582042\pi$$
−0.254898 + 0.966968i $$0.582042\pi$$
$$332$$ 4344.00 0.718096
$$333$$ 0 0
$$334$$ 2328.00 0.381385
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1672.00 0.270266 0.135133 0.990827i $$-0.456854\pi$$
0.135133 + 0.990827i $$0.456854\pi$$
$$338$$ −24.0000 −0.00386221
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −249.000 −0.0395428
$$342$$ 0 0
$$343$$ 6860.00 1.07990
$$344$$ 2392.00 0.374907
$$345$$ 0 0
$$346$$ −7884.00 −1.22499
$$347$$ −5076.00 −0.785285 −0.392643 0.919691i $$-0.628439\pi$$
−0.392643 + 0.919691i $$0.628439\pi$$
$$348$$ 0 0
$$349$$ 8594.00 1.31813 0.659063 0.752087i $$-0.270953\pi$$
0.659063 + 0.752087i $$0.270953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 96.0000 0.0145364
$$353$$ 12711.0 1.91654 0.958269 0.285866i $$-0.0922813\pi$$
0.958269 + 0.285866i $$0.0922813\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 480.000 0.0714605
$$357$$ 0 0
$$358$$ −2424.00 −0.357856
$$359$$ 1464.00 0.215228 0.107614 0.994193i $$-0.465679\pi$$
0.107614 + 0.994193i $$0.465679\pi$$
$$360$$ 0 0
$$361$$ −5835.00 −0.850707
$$362$$ −4576.00 −0.664390
$$363$$ 0 0
$$364$$ 2632.00 0.378995
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7630.00 1.08524 0.542620 0.839979i $$-0.317433\pi$$
0.542620 + 0.839979i $$0.317433\pi$$
$$368$$ −1584.00 −0.224380
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7896.00 −1.10496
$$372$$ 0 0
$$373$$ 3883.00 0.539019 0.269510 0.962998i $$-0.413138\pi$$
0.269510 + 0.962998i $$0.413138\pi$$
$$374$$ −234.000 −0.0323525
$$375$$ 0 0
$$376$$ −4248.00 −0.582643
$$377$$ 2397.00 0.327458
$$378$$ 0 0
$$379$$ −13768.0 −1.86600 −0.933001 0.359874i $$-0.882820\pi$$
−0.933001 + 0.359874i $$0.882820\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −3876.00 −0.519145
$$383$$ −14139.0 −1.88634 −0.943171 0.332307i $$-0.892173\pi$$
−0.943171 + 0.332307i $$0.892173\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2996.00 −0.395058
$$387$$ 0 0
$$388$$ 6568.00 0.859381
$$389$$ −567.000 −0.0739024 −0.0369512 0.999317i $$-0.511765\pi$$
−0.0369512 + 0.999317i $$0.511765\pi$$
$$390$$ 0 0
$$391$$ 3861.00 0.499384
$$392$$ 1176.00 0.151523
$$393$$ 0 0
$$394$$ 4248.00 0.543176
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6685.00 0.845115 0.422557 0.906336i $$-0.361133\pi$$
0.422557 + 0.906336i $$0.361133\pi$$
$$398$$ 770.000 0.0969764
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4572.00 −0.569364 −0.284682 0.958622i $$-0.591888\pi$$
−0.284682 + 0.958622i $$0.591888\pi$$
$$402$$ 0 0
$$403$$ −3901.00 −0.482190
$$404$$ −132.000 −0.0162556
$$405$$ 0 0
$$406$$ −1428.00 −0.174558
$$407$$ 942.000 0.114725
$$408$$ 0 0
$$409$$ −25.0000 −0.00302242 −0.00151121 0.999999i $$-0.500481\pi$$
−0.00151121 + 0.999999i $$0.500481\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4792.00 0.573022
$$413$$ 168.000 0.0200163
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1504.00 0.177259
$$417$$ 0 0
$$418$$ 192.000 0.0224666
$$419$$ −12453.0 −1.45195 −0.725977 0.687719i $$-0.758612\pi$$
−0.725977 + 0.687719i $$0.758612\pi$$
$$420$$ 0 0
$$421$$ 5048.00 0.584381 0.292191 0.956360i $$-0.405616\pi$$
0.292191 + 0.956360i $$0.405616\pi$$
$$422$$ −6340.00 −0.731342
$$423$$ 0 0
$$424$$ −4512.00 −0.516797
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3220.00 −0.364934
$$428$$ −6168.00 −0.696592
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5400.00 −0.603501 −0.301750 0.953387i $$-0.597571\pi$$
−0.301750 + 0.953387i $$0.597571\pi$$
$$432$$ 0 0
$$433$$ 6298.00 0.698990 0.349495 0.936938i $$-0.386353\pi$$
0.349495 + 0.936938i $$0.386353\pi$$
$$434$$ 2324.00 0.257040
$$435$$ 0 0
$$436$$ −2224.00 −0.244290
$$437$$ −3168.00 −0.346787
$$438$$ 0 0
$$439$$ −6208.00 −0.674924 −0.337462 0.941339i $$-0.609568\pi$$
−0.337462 + 0.941339i $$0.609568\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3666.00 −0.394511
$$443$$ −3360.00 −0.360358 −0.180179 0.983634i $$-0.557668\pi$$
−0.180179 + 0.983634i $$0.557668\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2776.00 0.294725
$$447$$ 0 0
$$448$$ −896.000 −0.0944911
$$449$$ −14394.0 −1.51291 −0.756453 0.654048i $$-0.773069\pi$$
−0.756453 + 0.654048i $$0.773069\pi$$
$$450$$ 0 0
$$451$$ −324.000 −0.0338283
$$452$$ 6420.00 0.668078
$$453$$ 0 0
$$454$$ 9288.00 0.960149
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 916.000 0.0937608 0.0468804 0.998901i $$-0.485072\pi$$
0.0468804 + 0.998901i $$0.485072\pi$$
$$458$$ −9472.00 −0.966370
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8550.00 −0.863803 −0.431902 0.901921i $$-0.642157\pi$$
−0.431902 + 0.901921i $$0.642157\pi$$
$$462$$ 0 0
$$463$$ −3734.00 −0.374803 −0.187401 0.982283i $$-0.560007\pi$$
−0.187401 + 0.982283i $$0.560007\pi$$
$$464$$ −816.000 −0.0816419
$$465$$ 0 0
$$466$$ −5628.00 −0.559468
$$467$$ 9840.00 0.975034 0.487517 0.873113i $$-0.337903\pi$$
0.487517 + 0.873113i $$0.337903\pi$$
$$468$$ 0 0
$$469$$ −3752.00 −0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 96.0000 0.00936178
$$473$$ 897.000 0.0871968
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2184.00 0.210301
$$477$$ 0 0
$$478$$ −4404.00 −0.421411
$$479$$ 17280.0 1.64832 0.824158 0.566360i $$-0.191649\pi$$
0.824158 + 0.566360i $$0.191649\pi$$
$$480$$ 0 0
$$481$$ 14758.0 1.39897
$$482$$ −6970.00 −0.658661
$$483$$ 0 0
$$484$$ −5288.00 −0.496619
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4588.00 0.426904 0.213452 0.976954i $$-0.431529\pi$$
0.213452 + 0.976954i $$0.431529\pi$$
$$488$$ −1840.00 −0.170682
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −636.000 −0.0584568 −0.0292284 0.999573i $$-0.509305\pi$$
−0.0292284 + 0.999573i $$0.509305\pi$$
$$492$$ 0 0
$$493$$ 1989.00 0.181704
$$494$$ 3008.00 0.273960
$$495$$ 0 0
$$496$$ 1328.00 0.120220
$$497$$ 1680.00 0.151626
$$498$$ 0 0
$$499$$ −11716.0 −1.05106 −0.525531 0.850774i $$-0.676133\pi$$
−0.525531 + 0.850774i $$0.676133\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −12690.0 −1.12825
$$503$$ 4653.00 0.412459 0.206230 0.978504i $$-0.433881\pi$$
0.206230 + 0.978504i $$0.433881\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −594.000 −0.0521868
$$507$$ 0 0
$$508$$ −5336.00 −0.466037
$$509$$ −16479.0 −1.43501 −0.717504 0.696555i $$-0.754715\pi$$
−0.717504 + 0.696555i $$0.754715\pi$$
$$510$$ 0 0
$$511$$ 15484.0 1.34045
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ −1050.00 −0.0901041
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1593.00 −0.135513
$$518$$ −8792.00 −0.745750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −3120.00 −0.262360 −0.131180 0.991359i $$-0.541877\pi$$
−0.131180 + 0.991359i $$0.541877\pi$$
$$522$$ 0 0
$$523$$ −17645.0 −1.47526 −0.737631 0.675204i $$-0.764056\pi$$
−0.737631 + 0.675204i $$0.764056\pi$$
$$524$$ 11532.0 0.961408
$$525$$ 0 0
$$526$$ −10392.0 −0.861431
$$527$$ −3237.00 −0.267563
$$528$$ 0 0
$$529$$ −2366.00 −0.194460
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −1792.00 −0.146040
$$533$$ −5076.00 −0.412507
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −2144.00 −0.172774
$$537$$ 0 0
$$538$$ −14958.0 −1.19867
$$539$$ 441.000 0.0352416
$$540$$ 0 0
$$541$$ −2182.00 −0.173404 −0.0867019 0.996234i $$-0.527633\pi$$
−0.0867019 + 0.996234i $$0.527633\pi$$
$$542$$ 1712.00 0.135677
$$543$$ 0 0
$$544$$ 1248.00 0.0983595
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4033.00 0.315244 0.157622 0.987499i $$-0.449617\pi$$
0.157622 + 0.987499i $$0.449617\pi$$
$$548$$ 1128.00 0.0879302
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1632.00 −0.126181
$$552$$ 0 0
$$553$$ 10346.0 0.795582
$$554$$ −14108.0 −1.08193
$$555$$ 0 0
$$556$$ −9976.00 −0.760929
$$557$$ −960.000 −0.0730278 −0.0365139 0.999333i $$-0.511625\pi$$
−0.0365139 + 0.999333i $$0.511625\pi$$
$$558$$ 0 0
$$559$$ 14053.0 1.06329
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2028.00 0.152217
$$563$$ −23754.0 −1.77817 −0.889087 0.457739i $$-0.848660\pi$$
−0.889087 + 0.457739i $$0.848660\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1984.00 0.147339
$$567$$ 0 0
$$568$$ 960.000 0.0709167
$$569$$ 22536.0 1.66038 0.830192 0.557478i $$-0.188231\pi$$
0.830192 + 0.557478i $$0.188231\pi$$
$$570$$ 0 0
$$571$$ 17726.0 1.29914 0.649571 0.760301i $$-0.274949\pi$$
0.649571 + 0.760301i $$0.274949\pi$$
$$572$$ 564.000 0.0412273
$$573$$ 0 0
$$574$$ 3024.00 0.219894
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −17168.0 −1.23867 −0.619336 0.785126i $$-0.712598\pi$$
−0.619336 + 0.785126i $$0.712598\pi$$
$$578$$ 6784.00 0.488196
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −15204.0 −1.08566
$$582$$ 0 0
$$583$$ −1692.00 −0.120198
$$584$$ 8848.00 0.626940
$$585$$ 0 0
$$586$$ 9900.00 0.697893
$$587$$ 7542.00 0.530309 0.265155 0.964206i $$-0.414577\pi$$
0.265155 + 0.964206i $$0.414577\pi$$
$$588$$ 0 0
$$589$$ 2656.00 0.185804
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5024.00 −0.348792
$$593$$ −15543.0 −1.07635 −0.538174 0.842834i $$-0.680886\pi$$
−0.538174 + 0.842834i $$0.680886\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10380.0 −0.713392
$$597$$ 0 0
$$598$$ −9306.00 −0.636372
$$599$$ −16026.0 −1.09316 −0.546581 0.837406i $$-0.684071\pi$$
−0.546581 + 0.837406i $$0.684071\pi$$
$$600$$ 0 0
$$601$$ 10469.0 0.710548 0.355274 0.934762i $$-0.384388\pi$$
0.355274 + 0.934762i $$0.384388\pi$$
$$602$$ −8372.00 −0.566806
$$603$$ 0 0
$$604$$ 4916.00 0.331174
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8074.00 0.539891 0.269945 0.962876i $$-0.412994\pi$$
0.269945 + 0.962876i $$0.412994\pi$$
$$608$$ −1024.00 −0.0683038
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24957.0 −1.65246
$$612$$ 0 0
$$613$$ −26855.0 −1.76943 −0.884717 0.466128i $$-0.845649\pi$$
−0.884717 + 0.466128i $$0.845649\pi$$
$$614$$ −9554.00 −0.627961
$$615$$ 0 0
$$616$$ −336.000 −0.0219770
$$617$$ 24447.0 1.59514 0.797568 0.603229i $$-0.206119\pi$$
0.797568 + 0.603229i $$0.206119\pi$$
$$618$$ 0 0
$$619$$ 1850.00 0.120126 0.0600628 0.998195i $$-0.480870\pi$$
0.0600628 + 0.998195i $$0.480870\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −15384.0 −0.991708
$$623$$ −1680.00 −0.108038
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −5864.00 −0.374397
$$627$$ 0 0
$$628$$ 6364.00 0.404381
$$629$$ 12246.0 0.776280
$$630$$ 0 0
$$631$$ 21728.0 1.37081 0.685403 0.728164i $$-0.259626\pi$$
0.685403 + 0.728164i $$0.259626\pi$$
$$632$$ 5912.00 0.372099
$$633$$ 0 0
$$634$$ −16704.0 −1.04637
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6909.00 0.429740
$$638$$ −306.000 −0.0189885
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −23862.0 −1.47035 −0.735173 0.677879i $$-0.762899\pi$$
−0.735173 + 0.677879i $$0.762899\pi$$
$$642$$ 0 0
$$643$$ −10523.0 −0.645391 −0.322696 0.946503i $$-0.604589\pi$$
−0.322696 + 0.946503i $$0.604589\pi$$
$$644$$ 5544.00 0.339230
$$645$$ 0 0
$$646$$ 2496.00 0.152018
$$647$$ 5484.00 0.333228 0.166614 0.986022i $$-0.446717\pi$$
0.166614 + 0.986022i $$0.446717\pi$$
$$648$$ 0 0
$$649$$ 36.0000 0.00217739
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1828.00 0.109801
$$653$$ −26784.0 −1.60511 −0.802557 0.596576i $$-0.796527\pi$$
−0.802557 + 0.596576i $$0.796527\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 1728.00 0.102846
$$657$$ 0 0
$$658$$ 14868.0 0.880874
$$659$$ 12120.0 0.716431 0.358216 0.933639i $$-0.383385\pi$$
0.358216 + 0.933639i $$0.383385\pi$$
$$660$$ 0 0
$$661$$ −18226.0 −1.07248 −0.536240 0.844066i $$-0.680156\pi$$
−0.536240 + 0.844066i $$0.680156\pi$$
$$662$$ 6140.00 0.360480
$$663$$ 0 0
$$664$$ −8688.00 −0.507771
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5049.00 0.293101
$$668$$ −4656.00 −0.269680
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −690.000 −0.0396977
$$672$$ 0 0
$$673$$ 11062.0 0.633594 0.316797 0.948493i $$-0.397393\pi$$
0.316797 + 0.948493i $$0.397393\pi$$
$$674$$ −3344.00 −0.191107
$$675$$ 0 0
$$676$$ 48.0000 0.00273100
$$677$$ −9348.00 −0.530684 −0.265342 0.964154i $$-0.585485\pi$$
−0.265342 + 0.964154i $$0.585485\pi$$
$$678$$ 0 0
$$679$$ −22988.0 −1.29926
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 498.000 0.0279610
$$683$$ 19248.0 1.07834 0.539169 0.842198i $$-0.318739\pi$$
0.539169 + 0.842198i $$0.318739\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13720.0 −0.763604
$$687$$ 0 0
$$688$$ −4784.00 −0.265099
$$689$$ −26508.0 −1.46571
$$690$$ 0 0
$$691$$ −17710.0 −0.974993 −0.487496 0.873125i $$-0.662090\pi$$
−0.487496 + 0.873125i $$0.662090\pi$$
$$692$$ 15768.0 0.866199
$$693$$ 0 0
$$694$$ 10152.0 0.555280
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4212.00 −0.228897
$$698$$ −17188.0 −0.932056
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19437.0 1.04725 0.523627 0.851947i $$-0.324578\pi$$
0.523627 + 0.851947i $$0.324578\pi$$
$$702$$ 0 0
$$703$$ −10048.0 −0.539072
$$704$$ −192.000 −0.0102788
$$705$$ 0 0
$$706$$ −25422.0 −1.35520
$$707$$ 462.000 0.0245761
$$708$$ 0 0
$$709$$ −19516.0 −1.03376 −0.516882 0.856057i $$-0.672907\pi$$
−0.516882 + 0.856057i $$0.672907\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −960.000 −0.0505302
$$713$$ −8217.00 −0.431598
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4848.00 0.253042
$$717$$ 0 0
$$718$$ −2928.00 −0.152189
$$719$$ −17358.0 −0.900340 −0.450170 0.892943i $$-0.648637\pi$$
−0.450170 + 0.892943i $$0.648637\pi$$
$$720$$ 0 0
$$721$$ −16772.0 −0.866327
$$722$$ 11670.0 0.601541
$$723$$ 0 0
$$724$$ 9152.00 0.469795
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −24428.0 −1.24620 −0.623098 0.782144i $$-0.714126\pi$$
−0.623098 + 0.782144i $$0.714126\pi$$
$$728$$ −5264.00 −0.267990
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 11661.0 0.590010
$$732$$ 0 0
$$733$$ 21418.0 1.07925 0.539626 0.841905i $$-0.318566\pi$$
0.539626 + 0.841905i $$0.318566\pi$$
$$734$$ −15260.0 −0.767380
$$735$$ 0 0
$$736$$ 3168.00 0.158660
$$737$$ −804.000 −0.0401842
$$738$$ 0 0
$$739$$ −664.000 −0.0330523 −0.0165261 0.999863i $$-0.505261\pi$$
−0.0165261 + 0.999863i $$0.505261\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 15792.0 0.781324
$$743$$ −34209.0 −1.68911 −0.844553 0.535471i $$-0.820134\pi$$
−0.844553 + 0.535471i $$0.820134\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −7766.00 −0.381144
$$747$$ 0 0
$$748$$ 468.000 0.0228767
$$749$$ 21588.0 1.05315
$$750$$ 0 0
$$751$$ 6857.00 0.333176 0.166588 0.986027i $$-0.446725\pi$$
0.166588 + 0.986027i $$0.446725\pi$$
$$752$$ 8496.00 0.411991
$$753$$ 0 0
$$754$$ −4794.00 −0.231548
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23719.0 1.13881 0.569407 0.822056i $$-0.307173\pi$$
0.569407 + 0.822056i $$0.307173\pi$$
$$758$$ 27536.0 1.31946
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 14418.0 0.686796 0.343398 0.939190i $$-0.388422\pi$$
0.343398 + 0.939190i $$0.388422\pi$$
$$762$$ 0 0
$$763$$ 7784.00 0.369331
$$764$$ 7752.00 0.367091
$$765$$ 0 0
$$766$$ 28278.0 1.33385
$$767$$ 564.000 0.0265513
$$768$$ 0 0
$$769$$ −4849.00 −0.227385 −0.113693 0.993516i $$-0.536268\pi$$
−0.113693 + 0.993516i $$0.536268\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5992.00 0.279348
$$773$$ −36258.0 −1.68708 −0.843538 0.537070i $$-0.819531\pi$$
−0.843538 + 0.537070i $$0.819531\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −13136.0 −0.607674
$$777$$ 0 0
$$778$$ 1134.00 0.0522569
$$779$$ 3456.00 0.158953
$$780$$ 0 0
$$781$$ 360.000 0.0164940
$$782$$ −7722.00 −0.353118
$$783$$ 0 0
$$784$$ −2352.00 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18877.0 0.855009 0.427505 0.904013i $$-0.359393\pi$$
0.427505 + 0.904013i $$0.359393\pi$$
$$788$$ −8496.00 −0.384083
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −22470.0 −1.01004
$$792$$ 0 0
$$793$$ −10810.0 −0.484079
$$794$$ −13370.0 −0.597586
$$795$$ 0 0
$$796$$ −1540.00 −0.0685727
$$797$$ −16200.0 −0.719992 −0.359996 0.932954i $$-0.617222\pi$$
−0.359996 + 0.932954i $$0.617222\pi$$
$$798$$ 0 0
$$799$$ −20709.0 −0.916936
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 9144.00 0.402601
$$803$$ 3318.00 0.145815
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7802.00 0.340960
$$807$$ 0 0
$$808$$ 264.000 0.0114944
$$809$$ 26760.0 1.16296 0.581478 0.813562i $$-0.302475\pi$$
0.581478 + 0.813562i $$0.302475\pi$$
$$810$$ 0 0
$$811$$ −10510.0 −0.455063 −0.227531 0.973771i $$-0.573065\pi$$
−0.227531 + 0.973771i $$0.573065\pi$$
$$812$$ 2856.00 0.123431
$$813$$ 0 0
$$814$$ −1884.00 −0.0811231
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −9568.00 −0.409721
$$818$$ 50.0000 0.00213717
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 28230.0 1.20004 0.600021 0.799985i $$-0.295159\pi$$
0.600021 + 0.799985i $$0.295159\pi$$
$$822$$ 0 0
$$823$$ 39868.0 1.68859 0.844296 0.535877i $$-0.180019\pi$$
0.844296 + 0.535877i $$0.180019\pi$$
$$824$$ −9584.00 −0.405187
$$825$$ 0 0
$$826$$ −336.000 −0.0141537
$$827$$ 32394.0 1.36209 0.681046 0.732241i $$-0.261525\pi$$
0.681046 + 0.732241i $$0.261525\pi$$
$$828$$ 0 0
$$829$$ 34820.0 1.45880 0.729402 0.684085i $$-0.239798\pi$$
0.729402 + 0.684085i $$0.239798\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −3008.00 −0.125341
$$833$$ 5733.00 0.238459
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −384.000 −0.0158863
$$837$$ 0 0
$$838$$ 24906.0 1.02669
$$839$$ 1146.00 0.0471565 0.0235783 0.999722i $$-0.492494\pi$$
0.0235783 + 0.999722i $$0.492494\pi$$
$$840$$ 0 0
$$841$$ −21788.0 −0.893354
$$842$$ −10096.0 −0.413220
$$843$$ 0 0
$$844$$ 12680.0 0.517137
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 18508.0 0.750817
$$848$$ 9024.00 0.365431
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 31086.0 1.25219
$$852$$ 0 0
$$853$$ 19393.0 0.778433 0.389217 0.921146i $$-0.372746\pi$$
0.389217 + 0.921146i $$0.372746\pi$$
$$854$$ 6440.00 0.258047
$$855$$ 0 0
$$856$$ 12336.0 0.492565
$$857$$ −8430.00 −0.336013 −0.168007 0.985786i $$-0.553733\pi$$
−0.168007 + 0.985786i $$0.553733\pi$$
$$858$$ 0 0
$$859$$ 15470.0 0.614470 0.307235 0.951634i $$-0.400596\pi$$
0.307235 + 0.951634i $$0.400596\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10800.0 0.426740
$$863$$ 5871.00 0.231577 0.115789 0.993274i $$-0.463060\pi$$
0.115789 + 0.993274i $$0.463060\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −12596.0 −0.494260
$$867$$ 0 0
$$868$$ −4648.00 −0.181755
$$869$$ 2217.00 0.0865438
$$870$$ 0 0
$$871$$ −12596.0 −0.490011
$$872$$ 4448.00 0.172739
$$873$$ 0 0
$$874$$ 6336.00 0.245216
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11299.0 0.435051 0.217526 0.976055i $$-0.430201\pi$$
0.217526 + 0.976055i $$0.430201\pi$$
$$878$$ 12416.0 0.477243
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −29682.0 −1.13509 −0.567544 0.823343i $$-0.692106\pi$$
−0.567544 + 0.823343i $$0.692106\pi$$
$$882$$ 0 0
$$883$$ −40316.0 −1.53651 −0.768257 0.640142i $$-0.778876\pi$$
−0.768257 + 0.640142i $$0.778876\pi$$
$$884$$ 7332.00 0.278961
$$885$$ 0 0
$$886$$ 6720.00 0.254811
$$887$$ −21945.0 −0.830711 −0.415356 0.909659i $$-0.636343\pi$$
−0.415356 + 0.909659i $$0.636343\pi$$
$$888$$ 0 0
$$889$$ 18676.0 0.704581
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −5552.00 −0.208402
$$893$$ 16992.0 0.636748
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1792.00 0.0668153
$$897$$ 0 0
$$898$$ 28788.0 1.06979
$$899$$ −4233.00 −0.157039
$$900$$ 0 0
$$901$$ −21996.0 −0.813311
$$902$$ 648.000 0.0239202
$$903$$ 0 0
$$904$$ −12840.0 −0.472403
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24911.0 −0.911969 −0.455985 0.889988i $$-0.650713\pi$$
−0.455985 + 0.889988i $$0.650713\pi$$
$$908$$ −18576.0 −0.678928
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −33264.0 −1.20975 −0.604877 0.796319i $$-0.706778\pi$$
−0.604877 + 0.796319i $$0.706778\pi$$
$$912$$ 0 0
$$913$$ −3258.00 −0.118099
$$914$$ −1832.00 −0.0662989
$$915$$ 0 0
$$916$$ 18944.0 0.683327
$$917$$ −40362.0 −1.45351
$$918$$ 0 0
$$919$$ −23191.0 −0.832427 −0.416214 0.909267i $$-0.636643\pi$$
−0.416214 + 0.909267i $$0.636643\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 17100.0 0.610801
$$923$$ 5640.00 0.201130
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 7468.00 0.265026
$$927$$ 0 0
$$928$$ 1632.00 0.0577296
$$929$$ −2160.00 −0.0762834 −0.0381417 0.999272i $$-0.512144\pi$$
−0.0381417 + 0.999272i $$0.512144\pi$$
$$930$$ 0 0
$$931$$ −4704.00 −0.165593
$$932$$ 11256.0 0.395604
$$933$$ 0 0
$$934$$ −19680.0 −0.689453
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2066.00 −0.0720312 −0.0360156 0.999351i $$-0.511467\pi$$
−0.0360156 + 0.999351i $$0.511467\pi$$
$$938$$ 7504.00 0.261209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 22233.0 0.770218 0.385109 0.922871i $$-0.374164\pi$$
0.385109 + 0.922871i $$0.374164\pi$$
$$942$$ 0 0
$$943$$ −10692.0 −0.369225
$$944$$ −192.000 −0.00661978
$$945$$ 0 0
$$946$$ −1794.00 −0.0616575
$$947$$ −17754.0 −0.609216 −0.304608 0.952478i $$-0.598525\pi$$
−0.304608 + 0.952478i $$0.598525\pi$$
$$948$$ 0 0
$$949$$ 51982.0 1.77809
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4368.00 −0.148706
$$953$$ 33891.0 1.15198 0.575990 0.817457i $$-0.304617\pi$$
0.575990 + 0.817457i $$0.304617\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 8808.00 0.297982
$$957$$ 0 0
$$958$$ −34560.0 −1.16554
$$959$$ −3948.00 −0.132938
$$960$$ 0 0
$$961$$ −22902.0 −0.768756
$$962$$ −29516.0 −0.989225
$$963$$ 0 0
$$964$$ 13940.0 0.465744
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −51074.0 −1.69848 −0.849239 0.528008i $$-0.822939\pi$$
−0.849239 + 0.528008i $$0.822939\pi$$
$$968$$ 10576.0 0.351163
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20967.0 −0.692959 −0.346479 0.938058i $$-0.612623\pi$$
−0.346479 + 0.938058i $$0.612623\pi$$
$$972$$ 0 0
$$973$$ 34916.0 1.15042
$$974$$ −9176.00 −0.301867
$$975$$ 0 0
$$976$$ 3680.00 0.120691
$$977$$ 31749.0 1.03965 0.519826 0.854272i $$-0.325997\pi$$
0.519826 + 0.854272i $$0.325997\pi$$
$$978$$ 0 0
$$979$$ −360.000 −0.0117525
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1272.00 0.0413352
$$983$$ 47325.0 1.53554 0.767769 0.640727i $$-0.221367\pi$$
0.767769 + 0.640727i $$0.221367\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −3978.00 −0.128484
$$987$$ 0 0
$$988$$ −6016.00 −0.193719
$$989$$ 29601.0 0.951726
$$990$$ 0 0
$$991$$ 2363.00 0.0757449 0.0378724 0.999283i $$-0.487942\pi$$
0.0378724 + 0.999283i $$0.487942\pi$$
$$992$$ −2656.00 −0.0850081
$$993$$ 0 0
$$994$$ −3360.00 −0.107216
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −45569.0 −1.44753 −0.723764 0.690048i $$-0.757589\pi$$
−0.723764 + 0.690048i $$0.757589\pi$$
$$998$$ 23432.0 0.743213
$$999$$ 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 1350.4.a.e.1.1 1
3.2 odd 2 1350.4.a.r.1.1 1
5.2 odd 4 1350.4.c.j.649.1 2
5.3 odd 4 1350.4.c.j.649.2 2
5.4 even 2 270.4.a.j.1.1 yes 1
15.2 even 4 1350.4.c.k.649.2 2
15.8 even 4 1350.4.c.k.649.1 2
15.14 odd 2 270.4.a.f.1.1 1
20.19 odd 2 2160.4.a.b.1.1 1
45.4 even 6 810.4.e.f.541.1 2
45.14 odd 6 810.4.e.n.541.1 2
45.29 odd 6 810.4.e.n.271.1 2
45.34 even 6 810.4.e.f.271.1 2
60.59 even 2 2160.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.f.1.1 1 15.14 odd 2
270.4.a.j.1.1 yes 1 5.4 even 2
810.4.e.f.271.1 2 45.34 even 6
810.4.e.f.541.1 2 45.4 even 6
810.4.e.n.271.1 2 45.29 odd 6
810.4.e.n.541.1 2 45.14 odd 6
1350.4.a.e.1.1 1 1.1 even 1 trivial
1350.4.a.r.1.1 1 3.2 odd 2
1350.4.c.j.649.1 2 5.2 odd 4
1350.4.c.j.649.2 2 5.3 odd 4
1350.4.c.k.649.1 2 15.8 even 4
1350.4.c.k.649.2 2 15.2 even 4
2160.4.a.b.1.1 1 20.19 odd 2
2160.4.a.l.1.1 1 60.59 even 2