# Properties

 Label 1350.4.a.u.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

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +4.00000 q^{4} +4.00000 q^{7} +8.00000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} +4.00000 q^{4} +4.00000 q^{7} +8.00000 q^{8} -42.0000 q^{11} -20.0000 q^{13} +8.00000 q^{14} +16.0000 q^{16} +93.0000 q^{17} +59.0000 q^{19} -84.0000 q^{22} +9.00000 q^{23} -40.0000 q^{26} +16.0000 q^{28} -120.000 q^{29} +47.0000 q^{31} +32.0000 q^{32} +186.000 q^{34} +262.000 q^{37} +118.000 q^{38} -126.000 q^{41} +178.000 q^{43} -168.000 q^{44} +18.0000 q^{46} +144.000 q^{47} -327.000 q^{49} -80.0000 q^{52} +741.000 q^{53} +32.0000 q^{56} -240.000 q^{58} +444.000 q^{59} +221.000 q^{61} +94.0000 q^{62} +64.0000 q^{64} +538.000 q^{67} +372.000 q^{68} -690.000 q^{71} +1126.00 q^{73} +524.000 q^{74} +236.000 q^{76} -168.000 q^{77} +665.000 q^{79} -252.000 q^{82} +75.0000 q^{83} +356.000 q^{86} -336.000 q^{88} +1086.00 q^{89} -80.0000 q^{91} +36.0000 q^{92} +288.000 q^{94} -1544.00 q^{97} -654.000 q^{98} +O(q^{100})$$

## 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
$$7$$ 4.00000 0.215980 0.107990 0.994152i $$-0.465559\pi$$
0.107990 + 0.994152i $$0.465559\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −42.0000 −1.15123 −0.575613 0.817723i $$-0.695236\pi$$
−0.575613 + 0.817723i $$0.695236\pi$$
$$12$$ 0 0
$$13$$ −20.0000 −0.426692 −0.213346 0.976977i $$-0.568436\pi$$
−0.213346 + 0.976977i $$0.568436\pi$$
$$14$$ 8.00000 0.152721
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 93.0000 1.32681 0.663406 0.748259i $$-0.269110\pi$$
0.663406 + 0.748259i $$0.269110\pi$$
$$18$$ 0 0
$$19$$ 59.0000 0.712396 0.356198 0.934410i $$-0.384073\pi$$
0.356198 + 0.934410i $$0.384073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −84.0000 −0.814039
$$23$$ 9.00000 0.0815926 0.0407963 0.999167i $$-0.487011\pi$$
0.0407963 + 0.999167i $$0.487011\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −40.0000 −0.301717
$$27$$ 0 0
$$28$$ 16.0000 0.107990
$$29$$ −120.000 −0.768395 −0.384197 0.923251i $$-0.625522\pi$$
−0.384197 + 0.923251i $$0.625522\pi$$
$$30$$ 0 0
$$31$$ 47.0000 0.272305 0.136152 0.990688i $$-0.456526\pi$$
0.136152 + 0.990688i $$0.456526\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 0 0
$$34$$ 186.000 0.938198
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 262.000 1.16412 0.582061 0.813145i $$-0.302246\pi$$
0.582061 + 0.813145i $$0.302246\pi$$
$$38$$ 118.000 0.503740
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −126.000 −0.479949 −0.239974 0.970779i $$-0.577139\pi$$
−0.239974 + 0.970779i $$0.577139\pi$$
$$42$$ 0 0
$$43$$ 178.000 0.631273 0.315637 0.948880i $$-0.397782\pi$$
0.315637 + 0.948880i $$0.397782\pi$$
$$44$$ −168.000 −0.575613
$$45$$ 0 0
$$46$$ 18.0000 0.0576947
$$47$$ 144.000 0.446906 0.223453 0.974715i $$-0.428267\pi$$
0.223453 + 0.974715i $$0.428267\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −80.0000 −0.213346
$$53$$ 741.000 1.92046 0.960228 0.279217i $$-0.0900748\pi$$
0.960228 + 0.279217i $$0.0900748\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 32.0000 0.0763604
$$57$$ 0 0
$$58$$ −240.000 −0.543337
$$59$$ 444.000 0.979727 0.489863 0.871799i $$-0.337047\pi$$
0.489863 + 0.871799i $$0.337047\pi$$
$$60$$ 0 0
$$61$$ 221.000 0.463871 0.231936 0.972731i $$-0.425494\pi$$
0.231936 + 0.972731i $$0.425494\pi$$
$$62$$ 94.0000 0.192549
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 538.000 0.981002 0.490501 0.871441i $$-0.336814\pi$$
0.490501 + 0.871441i $$0.336814\pi$$
$$68$$ 372.000 0.663406
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −690.000 −1.15335 −0.576676 0.816973i $$-0.695650\pi$$
−0.576676 + 0.816973i $$0.695650\pi$$
$$72$$ 0 0
$$73$$ 1126.00 1.80532 0.902660 0.430355i $$-0.141612\pi$$
0.902660 + 0.430355i $$0.141612\pi$$
$$74$$ 524.000 0.823159
$$75$$ 0 0
$$76$$ 236.000 0.356198
$$77$$ −168.000 −0.248641
$$78$$ 0 0
$$79$$ 665.000 0.947068 0.473534 0.880776i $$-0.342978\pi$$
0.473534 + 0.880776i $$0.342978\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −252.000 −0.339375
$$83$$ 75.0000 0.0991846 0.0495923 0.998770i $$-0.484208\pi$$
0.0495923 + 0.998770i $$0.484208\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 356.000 0.446378
$$87$$ 0 0
$$88$$ −336.000 −0.407020
$$89$$ 1086.00 1.29344 0.646718 0.762729i $$-0.276141\pi$$
0.646718 + 0.762729i $$0.276141\pi$$
$$90$$ 0 0
$$91$$ −80.0000 −0.0921569
$$92$$ 36.0000 0.0407963
$$93$$ 0 0
$$94$$ 288.000 0.316010
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1544.00 −1.61618 −0.808090 0.589059i $$-0.799499\pi$$
−0.808090 + 0.589059i $$0.799499\pi$$
$$98$$ −654.000 −0.674122
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 132.000 0.130044 0.0650222 0.997884i $$-0.479288\pi$$
0.0650222 + 0.997884i $$0.479288\pi$$
$$102$$ 0 0
$$103$$ 892.000 0.853314 0.426657 0.904413i $$-0.359691\pi$$
0.426657 + 0.904413i $$0.359691\pi$$
$$104$$ −160.000 −0.150859
$$105$$ 0 0
$$106$$ 1482.00 1.35797
$$107$$ −1140.00 −1.02998 −0.514990 0.857196i $$-0.672205\pi$$
−0.514990 + 0.857196i $$0.672205\pi$$
$$108$$ 0 0
$$109$$ −1735.00 −1.52461 −0.762307 0.647216i $$-0.775933\pi$$
−0.762307 + 0.647216i $$0.775933\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 64.0000 0.0539949
$$113$$ −1434.00 −1.19380 −0.596900 0.802316i $$-0.703601\pi$$
−0.596900 + 0.802316i $$0.703601\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −480.000 −0.384197
$$117$$ 0 0
$$118$$ 888.000 0.692771
$$119$$ 372.000 0.286565
$$120$$ 0 0
$$121$$ 433.000 0.325319
$$122$$ 442.000 0.328007
$$123$$ 0 0
$$124$$ 188.000 0.136152
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −686.000 −0.479312 −0.239656 0.970858i $$-0.577035\pi$$
−0.239656 + 0.970858i $$0.577035\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 114.000 0.0760323 0.0380161 0.999277i $$-0.487896\pi$$
0.0380161 + 0.999277i $$0.487896\pi$$
$$132$$ 0 0
$$133$$ 236.000 0.153863
$$134$$ 1076.00 0.693673
$$135$$ 0 0
$$136$$ 744.000 0.469099
$$137$$ 159.000 0.0991554 0.0495777 0.998770i $$-0.484212\pi$$
0.0495777 + 0.998770i $$0.484212\pi$$
$$138$$ 0 0
$$139$$ 2276.00 1.38883 0.694417 0.719573i $$-0.255663\pi$$
0.694417 + 0.719573i $$0.255663\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1380.00 −0.815542
$$143$$ 840.000 0.491219
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2252.00 1.27655
$$147$$ 0 0
$$148$$ 1048.00 0.582061
$$149$$ 1398.00 0.768648 0.384324 0.923198i $$-0.374434\pi$$
0.384324 + 0.923198i $$0.374434\pi$$
$$150$$ 0 0
$$151$$ 2624.00 1.41416 0.707080 0.707134i $$-0.250012\pi$$
0.707080 + 0.707134i $$0.250012\pi$$
$$152$$ 472.000 0.251870
$$153$$ 0 0
$$154$$ −336.000 −0.175816
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 394.000 0.200284 0.100142 0.994973i $$-0.468070\pi$$
0.100142 + 0.994973i $$0.468070\pi$$
$$158$$ 1330.00 0.669678
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 36.0000 0.0176223
$$162$$ 0 0
$$163$$ 3346.00 1.60785 0.803923 0.594733i $$-0.202742\pi$$
0.803923 + 0.594733i $$0.202742\pi$$
$$164$$ −504.000 −0.239974
$$165$$ 0 0
$$166$$ 150.000 0.0701341
$$167$$ −1491.00 −0.690881 −0.345440 0.938441i $$-0.612270\pi$$
−0.345440 + 0.938441i $$0.612270\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 712.000 0.315637
$$173$$ 2403.00 1.05605 0.528025 0.849229i $$-0.322933\pi$$
0.528025 + 0.849229i $$0.322933\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −672.000 −0.287806
$$177$$ 0 0
$$178$$ 2172.00 0.914597
$$179$$ 2640.00 1.10236 0.551181 0.834386i $$-0.314177\pi$$
0.551181 + 0.834386i $$0.314177\pi$$
$$180$$ 0 0
$$181$$ 1073.00 0.440638 0.220319 0.975428i $$-0.429290\pi$$
0.220319 + 0.975428i $$0.429290\pi$$
$$182$$ −160.000 −0.0651648
$$183$$ 0 0
$$184$$ 72.0000 0.0288473
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3906.00 −1.52746
$$188$$ 576.000 0.223453
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1470.00 −0.556887 −0.278444 0.960453i $$-0.589819\pi$$
−0.278444 + 0.960453i $$0.589819\pi$$
$$192$$ 0 0
$$193$$ 4720.00 1.76038 0.880189 0.474623i $$-0.157416\pi$$
0.880189 + 0.474623i $$0.157416\pi$$
$$194$$ −3088.00 −1.14281
$$195$$ 0 0
$$196$$ −1308.00 −0.476676
$$197$$ −765.000 −0.276670 −0.138335 0.990385i $$-0.544175\pi$$
−0.138335 + 0.990385i $$0.544175\pi$$
$$198$$ 0 0
$$199$$ 668.000 0.237956 0.118978 0.992897i $$-0.462038\pi$$
0.118978 + 0.992897i $$0.462038\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 264.000 0.0919553
$$203$$ −480.000 −0.165958
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 1784.00 0.603384
$$207$$ 0 0
$$208$$ −320.000 −0.106673
$$209$$ −2478.00 −0.820128
$$210$$ 0 0
$$211$$ 4601.00 1.50117 0.750583 0.660777i $$-0.229773\pi$$
0.750583 + 0.660777i $$0.229773\pi$$
$$212$$ 2964.00 0.960228
$$213$$ 0 0
$$214$$ −2280.00 −0.728307
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 188.000 0.0588123
$$218$$ −3470.00 −1.07806
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1860.00 −0.566141
$$222$$ 0 0
$$223$$ 2158.00 0.648029 0.324014 0.946052i $$-0.394967\pi$$
0.324014 + 0.946052i $$0.394967\pi$$
$$224$$ 128.000 0.0381802
$$225$$ 0 0
$$226$$ −2868.00 −0.844144
$$227$$ 3123.00 0.913131 0.456566 0.889690i $$-0.349079\pi$$
0.456566 + 0.889690i $$0.349079\pi$$
$$228$$ 0 0
$$229$$ 2027.00 0.584925 0.292463 0.956277i $$-0.405525\pi$$
0.292463 + 0.956277i $$0.405525\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −960.000 −0.271668
$$233$$ −438.000 −0.123152 −0.0615758 0.998102i $$-0.519613\pi$$
−0.0615758 + 0.998102i $$0.519613\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1776.00 0.489863
$$237$$ 0 0
$$238$$ 744.000 0.202632
$$239$$ −6414.00 −1.73593 −0.867965 0.496626i $$-0.834572\pi$$
−0.867965 + 0.496626i $$0.834572\pi$$
$$240$$ 0 0
$$241$$ 3431.00 0.917055 0.458527 0.888680i $$-0.348377\pi$$
0.458527 + 0.888680i $$0.348377\pi$$
$$242$$ 866.000 0.230035
$$243$$ 0 0
$$244$$ 884.000 0.231936
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1180.00 −0.303974
$$248$$ 376.000 0.0962743
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −7308.00 −1.83776 −0.918878 0.394541i $$-0.870904\pi$$
−0.918878 + 0.394541i $$0.870904\pi$$
$$252$$ 0 0
$$253$$ −378.000 −0.0939314
$$254$$ −1372.00 −0.338925
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −3729.00 −0.905092 −0.452546 0.891741i $$-0.649484\pi$$
−0.452546 + 0.891741i $$0.649484\pi$$
$$258$$ 0 0
$$259$$ 1048.00 0.251427
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 228.000 0.0537629
$$263$$ −1956.00 −0.458601 −0.229301 0.973356i $$-0.573644\pi$$
−0.229301 + 0.973356i $$0.573644\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 472.000 0.108798
$$267$$ 0 0
$$268$$ 2152.00 0.490501
$$269$$ −990.000 −0.224392 −0.112196 0.993686i $$-0.535788\pi$$
−0.112196 + 0.993686i $$0.535788\pi$$
$$270$$ 0 0
$$271$$ 8495.00 1.90419 0.952093 0.305808i $$-0.0989266\pi$$
0.952093 + 0.305808i $$0.0989266\pi$$
$$272$$ 1488.00 0.331703
$$273$$ 0 0
$$274$$ 318.000 0.0701134
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1366.00 0.296300 0.148150 0.988965i $$-0.452668\pi$$
0.148150 + 0.988965i $$0.452668\pi$$
$$278$$ 4552.00 0.982053
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5520.00 −1.17187 −0.585935 0.810358i $$-0.699273\pi$$
−0.585935 + 0.810358i $$0.699273\pi$$
$$282$$ 0 0
$$283$$ −5438.00 −1.14225 −0.571123 0.820865i $$-0.693492\pi$$
−0.571123 + 0.820865i $$0.693492\pi$$
$$284$$ −2760.00 −0.576676
$$285$$ 0 0
$$286$$ 1680.00 0.347344
$$287$$ −504.000 −0.103659
$$288$$ 0 0
$$289$$ 3736.00 0.760432
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4504.00 0.902660
$$293$$ −8253.00 −1.64555 −0.822774 0.568369i $$-0.807575\pi$$
−0.822774 + 0.568369i $$0.807575\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2096.00 0.411579
$$297$$ 0 0
$$298$$ 2796.00 0.543517
$$299$$ −180.000 −0.0348149
$$300$$ 0 0
$$301$$ 712.000 0.136342
$$302$$ 5248.00 0.999962
$$303$$ 0 0
$$304$$ 944.000 0.178099
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −9290.00 −1.72706 −0.863531 0.504295i $$-0.831752\pi$$
−0.863531 + 0.504295i $$0.831752\pi$$
$$308$$ −672.000 −0.124321
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8112.00 1.47907 0.739533 0.673121i $$-0.235047\pi$$
0.739533 + 0.673121i $$0.235047\pi$$
$$312$$ 0 0
$$313$$ 7900.00 1.42663 0.713314 0.700845i $$-0.247193\pi$$
0.713314 + 0.700845i $$0.247193\pi$$
$$314$$ 788.000 0.141622
$$315$$ 0 0
$$316$$ 2660.00 0.473534
$$317$$ −4419.00 −0.782952 −0.391476 0.920188i $$-0.628035\pi$$
−0.391476 + 0.920188i $$0.628035\pi$$
$$318$$ 0 0
$$319$$ 5040.00 0.884595
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 72.0000 0.0124609
$$323$$ 5487.00 0.945216
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 6692.00 1.13692
$$327$$ 0 0
$$328$$ −1008.00 −0.169687
$$329$$ 576.000 0.0965225
$$330$$ 0 0
$$331$$ −8200.00 −1.36167 −0.680835 0.732437i $$-0.738383\pi$$
−0.680835 + 0.732437i $$0.738383\pi$$
$$332$$ 300.000 0.0495923
$$333$$ 0 0
$$334$$ −2982.00 −0.488526
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9556.00 1.54465 0.772327 0.635225i $$-0.219093\pi$$
0.772327 + 0.635225i $$0.219093\pi$$
$$338$$ −3594.00 −0.578366
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1974.00 −0.313484
$$342$$ 0 0
$$343$$ −2680.00 −0.421885
$$344$$ 1424.00 0.223189
$$345$$ 0 0
$$346$$ 4806.00 0.746740
$$347$$ −10116.0 −1.56500 −0.782500 0.622650i $$-0.786056\pi$$
−0.782500 + 0.622650i $$0.786056\pi$$
$$348$$ 0 0
$$349$$ −6751.00 −1.03545 −0.517726 0.855546i $$-0.673221\pi$$
−0.517726 + 0.855546i $$0.673221\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1344.00 −0.203510
$$353$$ −4062.00 −0.612460 −0.306230 0.951958i $$-0.599068\pi$$
−0.306230 + 0.951958i $$0.599068\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 4344.00 0.646718
$$357$$ 0 0
$$358$$ 5280.00 0.779488
$$359$$ 8778.00 1.29049 0.645244 0.763977i $$-0.276756\pi$$
0.645244 + 0.763977i $$0.276756\pi$$
$$360$$ 0 0
$$361$$ −3378.00 −0.492492
$$362$$ 2146.00 0.311578
$$363$$ 0 0
$$364$$ −320.000 −0.0460785
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −956.000 −0.135975 −0.0679875 0.997686i $$-0.521658\pi$$
−0.0679875 + 0.997686i $$0.521658\pi$$
$$368$$ 144.000 0.0203981
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2964.00 0.414780
$$372$$ 0 0
$$373$$ −2300.00 −0.319275 −0.159637 0.987176i $$-0.551032\pi$$
−0.159637 + 0.987176i $$0.551032\pi$$
$$374$$ −7812.00 −1.08008
$$375$$ 0 0
$$376$$ 1152.00 0.158005
$$377$$ 2400.00 0.327868
$$378$$ 0 0
$$379$$ 29.0000 0.00393042 0.00196521 0.999998i $$-0.499374\pi$$
0.00196521 + 0.999998i $$0.499374\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −2940.00 −0.393779
$$383$$ −8127.00 −1.08426 −0.542128 0.840296i $$-0.682381\pi$$
−0.542128 + 0.840296i $$0.682381\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 9440.00 1.24478
$$387$$ 0 0
$$388$$ −6176.00 −0.808090
$$389$$ −7938.00 −1.03463 −0.517317 0.855794i $$-0.673069\pi$$
−0.517317 + 0.855794i $$0.673069\pi$$
$$390$$ 0 0
$$391$$ 837.000 0.108258
$$392$$ −2616.00 −0.337061
$$393$$ 0 0
$$394$$ −1530.00 −0.195635
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −272.000 −0.0343861 −0.0171931 0.999852i $$-0.505473\pi$$
−0.0171931 + 0.999852i $$0.505473\pi$$
$$398$$ 1336.00 0.168260
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4554.00 0.567122 0.283561 0.958954i $$-0.408484\pi$$
0.283561 + 0.958954i $$0.408484\pi$$
$$402$$ 0 0
$$403$$ −940.000 −0.116190
$$404$$ 528.000 0.0650222
$$405$$ 0 0
$$406$$ −960.000 −0.117350
$$407$$ −11004.0 −1.34017
$$408$$ 0 0
$$409$$ 1001.00 0.121018 0.0605089 0.998168i $$-0.480728\pi$$
0.0605089 + 0.998168i $$0.480728\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 3568.00 0.426657
$$413$$ 1776.00 0.211601
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −640.000 −0.0754293
$$417$$ 0 0
$$418$$ −4956.00 −0.579918
$$419$$ −1794.00 −0.209171 −0.104585 0.994516i $$-0.533352\pi$$
−0.104585 + 0.994516i $$0.533352\pi$$
$$420$$ 0 0
$$421$$ −16129.0 −1.86717 −0.933586 0.358354i $$-0.883338\pi$$
−0.933586 + 0.358354i $$0.883338\pi$$
$$422$$ 9202.00 1.06148
$$423$$ 0 0
$$424$$ 5928.00 0.678984
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 884.000 0.100187
$$428$$ −4560.00 −0.514990
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −13356.0 −1.49266 −0.746329 0.665577i $$-0.768186\pi$$
−0.746329 + 0.665577i $$0.768186\pi$$
$$432$$ 0 0
$$433$$ 11500.0 1.27634 0.638169 0.769896i $$-0.279692\pi$$
0.638169 + 0.769896i $$0.279692\pi$$
$$434$$ 376.000 0.0415866
$$435$$ 0 0
$$436$$ −6940.00 −0.762307
$$437$$ 531.000 0.0581263
$$438$$ 0 0
$$439$$ −11149.0 −1.21210 −0.606051 0.795426i $$-0.707247\pi$$
−0.606051 + 0.795426i $$0.707247\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3720.00 −0.400322
$$443$$ −3849.00 −0.412803 −0.206401 0.978467i $$-0.566175\pi$$
−0.206401 + 0.978467i $$0.566175\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 4316.00 0.458225
$$447$$ 0 0
$$448$$ 256.000 0.0269975
$$449$$ 18048.0 1.89697 0.948483 0.316828i $$-0.102618\pi$$
0.948483 + 0.316828i $$0.102618\pi$$
$$450$$ 0 0
$$451$$ 5292.00 0.552529
$$452$$ −5736.00 −0.596900
$$453$$ 0 0
$$454$$ 6246.00 0.645681
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4264.00 0.436458 0.218229 0.975898i $$-0.429972\pi$$
0.218229 + 0.975898i $$0.429972\pi$$
$$458$$ 4054.00 0.413605
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −10242.0 −1.03475 −0.517373 0.855760i $$-0.673090\pi$$
−0.517373 + 0.855760i $$0.673090\pi$$
$$462$$ 0 0
$$463$$ −3302.00 −0.331441 −0.165720 0.986173i $$-0.552995\pi$$
−0.165720 + 0.986173i $$0.552995\pi$$
$$464$$ −1920.00 −0.192099
$$465$$ 0 0
$$466$$ −876.000 −0.0870814
$$467$$ 1923.00 0.190548 0.0952739 0.995451i $$-0.469627\pi$$
0.0952739 + 0.995451i $$0.469627\pi$$
$$468$$ 0 0
$$469$$ 2152.00 0.211877
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3552.00 0.346386
$$473$$ −7476.00 −0.726738
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1488.00 0.143282
$$477$$ 0 0
$$478$$ −12828.0 −1.22749
$$479$$ 15246.0 1.45430 0.727148 0.686481i $$-0.240846\pi$$
0.727148 + 0.686481i $$0.240846\pi$$
$$480$$ 0 0
$$481$$ −5240.00 −0.496722
$$482$$ 6862.00 0.648455
$$483$$ 0 0
$$484$$ 1732.00 0.162660
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8206.00 0.763551 0.381776 0.924255i $$-0.375313\pi$$
0.381776 + 0.924255i $$0.375313\pi$$
$$488$$ 1768.00 0.164003
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −16806.0 −1.54469 −0.772346 0.635202i $$-0.780917\pi$$
−0.772346 + 0.635202i $$0.780917\pi$$
$$492$$ 0 0
$$493$$ −11160.0 −1.01952
$$494$$ −2360.00 −0.214942
$$495$$ 0 0
$$496$$ 752.000 0.0680762
$$497$$ −2760.00 −0.249100
$$498$$ 0 0
$$499$$ −5425.00 −0.486686 −0.243343 0.969940i $$-0.578244\pi$$
−0.243343 + 0.969940i $$0.578244\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −14616.0 −1.29949
$$503$$ 19665.0 1.74318 0.871589 0.490236i $$-0.163090\pi$$
0.871589 + 0.490236i $$0.163090\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −756.000 −0.0664196
$$507$$ 0 0
$$508$$ −2744.00 −0.239656
$$509$$ −14724.0 −1.28218 −0.641090 0.767466i $$-0.721518\pi$$
−0.641090 + 0.767466i $$0.721518\pi$$
$$510$$ 0 0
$$511$$ 4504.00 0.389912
$$512$$ 512.000 0.0441942
$$513$$ 0 0
$$514$$ −7458.00 −0.639997
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −6048.00 −0.514489
$$518$$ 2096.00 0.177786
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2058.00 0.173057 0.0865284 0.996249i $$-0.472423\pi$$
0.0865284 + 0.996249i $$0.472423\pi$$
$$522$$ 0 0
$$523$$ −11912.0 −0.995938 −0.497969 0.867195i $$-0.665921\pi$$
−0.497969 + 0.867195i $$0.665921\pi$$
$$524$$ 456.000 0.0380161
$$525$$ 0 0
$$526$$ −3912.00 −0.324280
$$527$$ 4371.00 0.361297
$$528$$ 0 0
$$529$$ −12086.0 −0.993343
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 944.000 0.0769316
$$533$$ 2520.00 0.204790
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4304.00 0.346837
$$537$$ 0 0
$$538$$ −1980.00 −0.158669
$$539$$ 13734.0 1.09752
$$540$$ 0 0
$$541$$ −5170.00 −0.410861 −0.205430 0.978672i $$-0.565859\pi$$
−0.205430 + 0.978672i $$0.565859\pi$$
$$542$$ 16990.0 1.34646
$$543$$ 0 0
$$544$$ 2976.00 0.234550
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4186.00 0.327204 0.163602 0.986526i $$-0.447689\pi$$
0.163602 + 0.986526i $$0.447689\pi$$
$$548$$ 636.000 0.0495777
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7080.00 −0.547401
$$552$$ 0 0
$$553$$ 2660.00 0.204547
$$554$$ 2732.00 0.209515
$$555$$ 0 0
$$556$$ 9104.00 0.694417
$$557$$ −13026.0 −0.990896 −0.495448 0.868637i $$-0.664996\pi$$
−0.495448 + 0.868637i $$0.664996\pi$$
$$558$$ 0 0
$$559$$ −3560.00 −0.269359
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −11040.0 −0.828638
$$563$$ 10668.0 0.798584 0.399292 0.916824i $$-0.369256\pi$$
0.399292 + 0.916824i $$0.369256\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −10876.0 −0.807690
$$567$$ 0 0
$$568$$ −5520.00 −0.407771
$$569$$ −15372.0 −1.13256 −0.566281 0.824212i $$-0.691618\pi$$
−0.566281 + 0.824212i $$0.691618\pi$$
$$570$$ 0 0
$$571$$ −14989.0 −1.09855 −0.549273 0.835643i $$-0.685095\pi$$
−0.549273 + 0.835643i $$0.685095\pi$$
$$572$$ 3360.00 0.245610
$$573$$ 0 0
$$574$$ −1008.00 −0.0732981
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1066.00 0.0769119 0.0384559 0.999260i $$-0.487756\pi$$
0.0384559 + 0.999260i $$0.487756\pi$$
$$578$$ 7472.00 0.537706
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 300.000 0.0214219
$$582$$ 0 0
$$583$$ −31122.0 −2.21088
$$584$$ 9008.00 0.638277
$$585$$ 0 0
$$586$$ −16506.0 −1.16358
$$587$$ 621.000 0.0436651 0.0218325 0.999762i $$-0.493050\pi$$
0.0218325 + 0.999762i $$0.493050\pi$$
$$588$$ 0 0
$$589$$ 2773.00 0.193989
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 4192.00 0.291031
$$593$$ 20187.0 1.39794 0.698972 0.715149i $$-0.253641\pi$$
0.698972 + 0.715149i $$0.253641\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 5592.00 0.384324
$$597$$ 0 0
$$598$$ −360.000 −0.0246179
$$599$$ −18228.0 −1.24337 −0.621683 0.783269i $$-0.713551\pi$$
−0.621683 + 0.783269i $$0.713551\pi$$
$$600$$ 0 0
$$601$$ −11743.0 −0.797017 −0.398508 0.917165i $$-0.630472\pi$$
−0.398508 + 0.917165i $$0.630472\pi$$
$$602$$ 1424.00 0.0964085
$$603$$ 0 0
$$604$$ 10496.0 0.707080
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 24418.0 1.63278 0.816389 0.577503i $$-0.195973\pi$$
0.816389 + 0.577503i $$0.195973\pi$$
$$608$$ 1888.00 0.125935
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2880.00 −0.190691
$$612$$ 0 0
$$613$$ −2672.00 −0.176054 −0.0880270 0.996118i $$-0.528056\pi$$
−0.0880270 + 0.996118i $$0.528056\pi$$
$$614$$ −18580.0 −1.22122
$$615$$ 0 0
$$616$$ −1344.00 −0.0879080
$$617$$ 8601.00 0.561205 0.280602 0.959824i $$-0.409466\pi$$
0.280602 + 0.959824i $$0.409466\pi$$
$$618$$ 0 0
$$619$$ 21308.0 1.38359 0.691794 0.722095i $$-0.256821\pi$$
0.691794 + 0.722095i $$0.256821\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16224.0 1.04586
$$623$$ 4344.00 0.279356
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 15800.0 1.00878
$$627$$ 0 0
$$628$$ 1576.00 0.100142
$$629$$ 24366.0 1.54457
$$630$$ 0 0
$$631$$ −19015.0 −1.19964 −0.599822 0.800134i $$-0.704762\pi$$
−0.599822 + 0.800134i $$0.704762\pi$$
$$632$$ 5320.00 0.334839
$$633$$ 0 0
$$634$$ −8838.00 −0.553631
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6540.00 0.406788
$$638$$ 10080.0 0.625503
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −4416.00 −0.272108 −0.136054 0.990701i $$-0.543442\pi$$
−0.136054 + 0.990701i $$0.543442\pi$$
$$642$$ 0 0
$$643$$ −7580.00 −0.464893 −0.232446 0.972609i $$-0.574673\pi$$
−0.232446 + 0.972609i $$0.574673\pi$$
$$644$$ 144.000 0.00881117
$$645$$ 0 0
$$646$$ 10974.0 0.668369
$$647$$ 14901.0 0.905439 0.452719 0.891653i $$-0.350454\pi$$
0.452719 + 0.891653i $$0.350454\pi$$
$$648$$ 0 0
$$649$$ −18648.0 −1.12789
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 13384.0 0.803923
$$653$$ −12915.0 −0.773971 −0.386985 0.922086i $$-0.626484\pi$$
−0.386985 + 0.922086i $$0.626484\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2016.00 −0.119987
$$657$$ 0 0
$$658$$ 1152.00 0.0682517
$$659$$ 28128.0 1.66269 0.831344 0.555758i $$-0.187572\pi$$
0.831344 + 0.555758i $$0.187572\pi$$
$$660$$ 0 0
$$661$$ −8362.00 −0.492049 −0.246024 0.969264i $$-0.579124\pi$$
−0.246024 + 0.969264i $$0.579124\pi$$
$$662$$ −16400.0 −0.962846
$$663$$ 0 0
$$664$$ 600.000 0.0350670
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1080.00 −0.0626953
$$668$$ −5964.00 −0.345440
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9282.00 −0.534020
$$672$$ 0 0
$$673$$ −29708.0 −1.70157 −0.850787 0.525511i $$-0.823874\pi$$
−0.850787 + 0.525511i $$0.823874\pi$$
$$674$$ 19112.0 1.09224
$$675$$ 0 0
$$676$$ −7188.00 −0.408967
$$677$$ −6762.00 −0.383877 −0.191939 0.981407i $$-0.561477\pi$$
−0.191939 + 0.981407i $$0.561477\pi$$
$$678$$ 0 0
$$679$$ −6176.00 −0.349062
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −3948.00 −0.221667
$$683$$ 19155.0 1.07313 0.536563 0.843860i $$-0.319722\pi$$
0.536563 + 0.843860i $$0.319722\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −5360.00 −0.298317
$$687$$ 0 0
$$688$$ 2848.00 0.157818
$$689$$ −14820.0 −0.819444
$$690$$ 0 0
$$691$$ −22975.0 −1.26485 −0.632424 0.774622i $$-0.717940\pi$$
−0.632424 + 0.774622i $$0.717940\pi$$
$$692$$ 9612.00 0.528025
$$693$$ 0 0
$$694$$ −20232.0 −1.10662
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −11718.0 −0.636802
$$698$$ −13502.0 −0.732175
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6450.00 −0.347522 −0.173761 0.984788i $$-0.555592\pi$$
−0.173761 + 0.984788i $$0.555592\pi$$
$$702$$ 0 0
$$703$$ 15458.0 0.829317
$$704$$ −2688.00 −0.143903
$$705$$ 0 0
$$706$$ −8124.00 −0.433075
$$707$$ 528.000 0.0280870
$$708$$ 0 0
$$709$$ 34538.0 1.82948 0.914740 0.404042i $$-0.132395\pi$$
0.914740 + 0.404042i $$0.132395\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 8688.00 0.457299
$$713$$ 423.000 0.0222181
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10560.0 0.551181
$$717$$ 0 0
$$718$$ 17556.0 0.912513
$$719$$ 27114.0 1.40637 0.703186 0.711006i $$-0.251760\pi$$
0.703186 + 0.711006i $$0.251760\pi$$
$$720$$ 0 0
$$721$$ 3568.00 0.184299
$$722$$ −6756.00 −0.348244
$$723$$ 0 0
$$724$$ 4292.00 0.220319
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −236.000 −0.0120396 −0.00601978 0.999982i $$-0.501916\pi$$
−0.00601978 + 0.999982i $$0.501916\pi$$
$$728$$ −640.000 −0.0325824
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 16554.0 0.837581
$$732$$ 0 0
$$733$$ −27128.0 −1.36698 −0.683489 0.729960i $$-0.739538\pi$$
−0.683489 + 0.729960i $$0.739538\pi$$
$$734$$ −1912.00 −0.0961488
$$735$$ 0 0
$$736$$ 288.000 0.0144237
$$737$$ −22596.0 −1.12935
$$738$$ 0 0
$$739$$ 5249.00 0.261282 0.130641 0.991430i $$-0.458296\pi$$
0.130641 + 0.991430i $$0.458296\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 5928.00 0.293293
$$743$$ −13896.0 −0.686130 −0.343065 0.939312i $$-0.611465\pi$$
−0.343065 + 0.939312i $$0.611465\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −4600.00 −0.225761
$$747$$ 0 0
$$748$$ −15624.0 −0.763730
$$749$$ −4560.00 −0.222455
$$750$$ 0 0
$$751$$ 27665.0 1.34422 0.672111 0.740451i $$-0.265388\pi$$
0.672111 + 0.740451i $$0.265388\pi$$
$$752$$ 2304.00 0.111726
$$753$$ 0 0
$$754$$ 4800.00 0.231838
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8122.00 0.389959 0.194980 0.980807i $$-0.437536\pi$$
0.194980 + 0.980807i $$0.437536\pi$$
$$758$$ 58.0000 0.00277923
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10584.0 0.504165 0.252083 0.967706i $$-0.418885\pi$$
0.252083 + 0.967706i $$0.418885\pi$$
$$762$$ 0 0
$$763$$ −6940.00 −0.329286
$$764$$ −5880.00 −0.278444
$$765$$ 0 0
$$766$$ −16254.0 −0.766685
$$767$$ −8880.00 −0.418042
$$768$$ 0 0
$$769$$ −18619.0 −0.873106 −0.436553 0.899679i $$-0.643801\pi$$
−0.436553 + 0.899679i $$0.643801\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18880.0 0.880189
$$773$$ −22251.0 −1.03533 −0.517667 0.855582i $$-0.673199\pi$$
−0.517667 + 0.855582i $$0.673199\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −12352.0 −0.571406
$$777$$ 0 0
$$778$$ −15876.0 −0.731597
$$779$$ −7434.00 −0.341914
$$780$$ 0 0
$$781$$ 28980.0 1.32777
$$782$$ 1674.00 0.0765500
$$783$$ 0 0
$$784$$ −5232.00 −0.238338
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −24854.0 −1.12573 −0.562865 0.826549i $$-0.690301\pi$$
−0.562865 + 0.826549i $$0.690301\pi$$
$$788$$ −3060.00 −0.138335
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5736.00 −0.257837
$$792$$ 0 0
$$793$$ −4420.00 −0.197930
$$794$$ −544.000 −0.0243147
$$795$$ 0 0
$$796$$ 2672.00 0.118978
$$797$$ −3681.00 −0.163598 −0.0817991 0.996649i $$-0.526067\pi$$
−0.0817991 + 0.996649i $$0.526067\pi$$
$$798$$ 0 0
$$799$$ 13392.0 0.592960
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 9108.00 0.401016
$$803$$ −47292.0 −2.07833
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1880.00 −0.0821590
$$807$$ 0 0
$$808$$ 1056.00 0.0459777
$$809$$ −5142.00 −0.223465 −0.111732 0.993738i $$-0.535640\pi$$
−0.111732 + 0.993738i $$0.535640\pi$$
$$810$$ 0 0
$$811$$ −18484.0 −0.800322 −0.400161 0.916445i $$-0.631046\pi$$
−0.400161 + 0.916445i $$0.631046\pi$$
$$812$$ −1920.00 −0.0829788
$$813$$ 0 0
$$814$$ −22008.0 −0.947641
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10502.0 0.449717
$$818$$ 2002.00 0.0855725
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25014.0 1.06333 0.531665 0.846954i $$-0.321566\pi$$
0.531665 + 0.846954i $$0.321566\pi$$
$$822$$ 0 0
$$823$$ 32146.0 1.36153 0.680765 0.732502i $$-0.261648\pi$$
0.680765 + 0.732502i $$0.261648\pi$$
$$824$$ 7136.00 0.301692
$$825$$ 0 0
$$826$$ 3552.00 0.149625
$$827$$ 10977.0 0.461557 0.230779 0.973006i $$-0.425873\pi$$
0.230779 + 0.973006i $$0.425873\pi$$
$$828$$ 0 0
$$829$$ 36602.0 1.53346 0.766731 0.641969i $$-0.221882\pi$$
0.766731 + 0.641969i $$0.221882\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1280.00 −0.0533366
$$833$$ −30411.0 −1.26492
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −9912.00 −0.410064
$$837$$ 0 0
$$838$$ −3588.00 −0.147906
$$839$$ 11076.0 0.455764 0.227882 0.973689i $$-0.426820\pi$$
0.227882 + 0.973689i $$0.426820\pi$$
$$840$$ 0 0
$$841$$ −9989.00 −0.409570
$$842$$ −32258.0 −1.32029
$$843$$ 0 0
$$844$$ 18404.0 0.750583
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1732.00 0.0702624
$$848$$ 11856.0 0.480114
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2358.00 0.0949838
$$852$$ 0 0
$$853$$ −36848.0 −1.47908 −0.739538 0.673115i $$-0.764956\pi$$
−0.739538 + 0.673115i $$0.764956\pi$$
$$854$$ 1768.00 0.0708428
$$855$$ 0 0
$$856$$ −9120.00 −0.364153
$$857$$ 26961.0 1.07464 0.537322 0.843377i $$-0.319436\pi$$
0.537322 + 0.843377i $$0.319436\pi$$
$$858$$ 0 0
$$859$$ −415.000 −0.0164838 −0.00824192 0.999966i $$-0.502624\pi$$
−0.00824192 + 0.999966i $$0.502624\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −26712.0 −1.05547
$$863$$ 45501.0 1.79475 0.897377 0.441265i $$-0.145470\pi$$
0.897377 + 0.441265i $$0.145470\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 23000.0 0.902508
$$867$$ 0 0
$$868$$ 752.000 0.0294062
$$869$$ −27930.0 −1.09029
$$870$$ 0 0
$$871$$ −10760.0 −0.418586
$$872$$ −13880.0 −0.539032
$$873$$ 0 0
$$874$$ 1062.00 0.0411015
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −35042.0 −1.34924 −0.674620 0.738165i $$-0.735693\pi$$
−0.674620 + 0.738165i $$0.735693\pi$$
$$878$$ −22298.0 −0.857085
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1080.00 0.0413009 0.0206505 0.999787i $$-0.493426\pi$$
0.0206505 + 0.999787i $$0.493426\pi$$
$$882$$ 0 0
$$883$$ 20164.0 0.768485 0.384243 0.923232i $$-0.374463\pi$$
0.384243 + 0.923232i $$0.374463\pi$$
$$884$$ −7440.00 −0.283070
$$885$$ 0 0
$$886$$ −7698.00 −0.291895
$$887$$ −20067.0 −0.759621 −0.379811 0.925064i $$-0.624011\pi$$
−0.379811 + 0.925064i $$0.624011\pi$$
$$888$$ 0 0
$$889$$ −2744.00 −0.103522
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8632.00 0.324014
$$893$$ 8496.00 0.318374
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 512.000 0.0190901
$$897$$ 0 0
$$898$$ 36096.0 1.34136
$$899$$ −5640.00 −0.209238
$$900$$ 0 0
$$901$$ 68913.0 2.54809
$$902$$ 10584.0 0.390697
$$903$$ 0 0
$$904$$ −11472.0 −0.422072
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26524.0 0.971020 0.485510 0.874231i $$-0.338634\pi$$
0.485510 + 0.874231i $$0.338634\pi$$
$$908$$ 12492.0 0.456566
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 35568.0 1.29355 0.646773 0.762683i $$-0.276118\pi$$
0.646773 + 0.762683i $$0.276118\pi$$
$$912$$ 0 0
$$913$$ −3150.00 −0.114184
$$914$$ 8528.00 0.308623
$$915$$ 0 0
$$916$$ 8108.00 0.292463
$$917$$ 456.000 0.0164214
$$918$$ 0 0
$$919$$ −23704.0 −0.850841 −0.425420 0.904996i $$-0.639874\pi$$
−0.425420 + 0.904996i $$0.639874\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −20484.0 −0.731675
$$923$$ 13800.0 0.492126
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −6604.00 −0.234364
$$927$$ 0 0
$$928$$ −3840.00 −0.135834
$$929$$ −40590.0 −1.43349 −0.716746 0.697334i $$-0.754369\pi$$
−0.716746 + 0.697334i $$0.754369\pi$$
$$930$$ 0 0
$$931$$ −19293.0 −0.679165
$$932$$ −1752.00 −0.0615758
$$933$$ 0 0
$$934$$ 3846.00 0.134738
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12964.0 0.451991 0.225995 0.974128i $$-0.427437\pi$$
0.225995 + 0.974128i $$0.427437\pi$$
$$938$$ 4304.00 0.149819
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 29922.0 1.03659 0.518294 0.855203i $$-0.326567\pi$$
0.518294 + 0.855203i $$0.326567\pi$$
$$942$$ 0 0
$$943$$ −1134.00 −0.0391603
$$944$$ 7104.00 0.244932
$$945$$ 0 0
$$946$$ −14952.0 −0.513881
$$947$$ −5241.00 −0.179841 −0.0899206 0.995949i $$-0.528661\pi$$
−0.0899206 + 0.995949i $$0.528661\pi$$
$$948$$ 0 0
$$949$$ −22520.0 −0.770316
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2976.00 0.101316
$$953$$ −26214.0 −0.891033 −0.445517 0.895274i $$-0.646980\pi$$
−0.445517 + 0.895274i $$0.646980\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −25656.0 −0.867965
$$957$$ 0 0
$$958$$ 30492.0 1.02834
$$959$$ 636.000 0.0214155
$$960$$ 0 0
$$961$$ −27582.0 −0.925850
$$962$$ −10480.0 −0.351236
$$963$$ 0 0
$$964$$ 13724.0 0.458527
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −18278.0 −0.607840 −0.303920 0.952698i $$-0.598295\pi$$
−0.303920 + 0.952698i $$0.598295\pi$$
$$968$$ 3464.00 0.115018
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −24942.0 −0.824333 −0.412166 0.911109i $$-0.635228\pi$$
−0.412166 + 0.911109i $$0.635228\pi$$
$$972$$ 0 0
$$973$$ 9104.00 0.299960
$$974$$ 16412.0 0.539912
$$975$$ 0 0
$$976$$ 3536.00 0.115968
$$977$$ 11226.0 0.367607 0.183803 0.982963i $$-0.441159\pi$$
0.183803 + 0.982963i $$0.441159\pi$$
$$978$$ 0 0
$$979$$ −45612.0 −1.48904
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −33612.0 −1.09226
$$983$$ 23073.0 0.748641 0.374321 0.927299i $$-0.377876\pi$$
0.374321 + 0.927299i $$0.377876\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −22320.0 −0.720906
$$987$$ 0 0
$$988$$ −4720.00 −0.151987
$$989$$ 1602.00 0.0515072
$$990$$ 0 0
$$991$$ 22037.0 0.706386 0.353193 0.935551i $$-0.385096\pi$$
0.353193 + 0.935551i $$0.385096\pi$$
$$992$$ 1504.00 0.0481371
$$993$$ 0 0
$$994$$ −5520.00 −0.176141
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19082.0 −0.606151 −0.303076 0.952966i $$-0.598014\pi$$
−0.303076 + 0.952966i $$0.598014\pi$$
$$998$$ −10850.0 −0.344139
$$999$$ 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.u.1.1 1
3.2 odd 2 1350.4.a.g.1.1 1
5.2 odd 4 1350.4.c.d.649.2 2
5.3 odd 4 1350.4.c.d.649.1 2
5.4 even 2 270.4.a.e.1.1 1
15.2 even 4 1350.4.c.q.649.1 2
15.8 even 4 1350.4.c.q.649.2 2
15.14 odd 2 270.4.a.i.1.1 yes 1
20.19 odd 2 2160.4.a.o.1.1 1
45.4 even 6 810.4.e.q.541.1 2
45.14 odd 6 810.4.e.h.541.1 2
45.29 odd 6 810.4.e.h.271.1 2
45.34 even 6 810.4.e.q.271.1 2
60.59 even 2 2160.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.e.1.1 1 5.4 even 2
270.4.a.i.1.1 yes 1 15.14 odd 2
810.4.e.h.271.1 2 45.29 odd 6
810.4.e.h.541.1 2 45.14 odd 6
810.4.e.q.271.1 2 45.34 even 6
810.4.e.q.541.1 2 45.4 even 6
1350.4.a.g.1.1 1 3.2 odd 2
1350.4.a.u.1.1 1 1.1 even 1 trivial
1350.4.c.d.649.1 2 5.3 odd 4
1350.4.c.d.649.2 2 5.2 odd 4
1350.4.c.q.649.1 2 15.2 even 4
1350.4.c.q.649.2 2 15.8 even 4
2160.4.a.e.1.1 1 60.59 even 2
2160.4.a.o.1.1 1 20.19 odd 2