# Properties

 Label 18.6.a.b.1.1 Level $18$ Weight $6$ Character 18.1 Self dual yes Analytic conductor $2.887$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 18.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.88690875663$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 18.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +16.0000 q^{4} +66.0000 q^{5} +176.000 q^{7} -64.0000 q^{8} +O(q^{10})$$ $$q-4.00000 q^{2} +16.0000 q^{4} +66.0000 q^{5} +176.000 q^{7} -64.0000 q^{8} -264.000 q^{10} +60.0000 q^{11} -658.000 q^{13} -704.000 q^{14} +256.000 q^{16} +414.000 q^{17} +956.000 q^{19} +1056.00 q^{20} -240.000 q^{22} -600.000 q^{23} +1231.00 q^{25} +2632.00 q^{26} +2816.00 q^{28} -5574.00 q^{29} -3592.00 q^{31} -1024.00 q^{32} -1656.00 q^{34} +11616.0 q^{35} -8458.00 q^{37} -3824.00 q^{38} -4224.00 q^{40} -19194.0 q^{41} +13316.0 q^{43} +960.000 q^{44} +2400.00 q^{46} +19680.0 q^{47} +14169.0 q^{49} -4924.00 q^{50} -10528.0 q^{52} +31266.0 q^{53} +3960.00 q^{55} -11264.0 q^{56} +22296.0 q^{58} -26340.0 q^{59} -31090.0 q^{61} +14368.0 q^{62} +4096.00 q^{64} -43428.0 q^{65} -16804.0 q^{67} +6624.00 q^{68} -46464.0 q^{70} -6120.00 q^{71} -25558.0 q^{73} +33832.0 q^{74} +15296.0 q^{76} +10560.0 q^{77} +74408.0 q^{79} +16896.0 q^{80} +76776.0 q^{82} +6468.00 q^{83} +27324.0 q^{85} -53264.0 q^{86} -3840.00 q^{88} +32742.0 q^{89} -115808. q^{91} -9600.00 q^{92} -78720.0 q^{94} +63096.0 q^{95} +166082. q^{97} -56676.0 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −0.707107
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ 66.0000 1.18064 0.590322 0.807168i $$-0.299001\pi$$
0.590322 + 0.807168i $$0.299001\pi$$
$$6$$ 0 0
$$7$$ 176.000 1.35759 0.678793 0.734329i $$-0.262503\pi$$
0.678793 + 0.734329i $$0.262503\pi$$
$$8$$ −64.0000 −0.353553
$$9$$ 0 0
$$10$$ −264.000 −0.834841
$$11$$ 60.0000 0.149510 0.0747549 0.997202i $$-0.476183\pi$$
0.0747549 + 0.997202i $$0.476183\pi$$
$$12$$ 0 0
$$13$$ −658.000 −1.07986 −0.539930 0.841710i $$-0.681549\pi$$
−0.539930 + 0.841710i $$0.681549\pi$$
$$14$$ −704.000 −0.959959
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 414.000 0.347439 0.173719 0.984795i $$-0.444421\pi$$
0.173719 + 0.984795i $$0.444421\pi$$
$$18$$ 0 0
$$19$$ 956.000 0.607539 0.303769 0.952746i $$-0.401755\pi$$
0.303769 + 0.952746i $$0.401755\pi$$
$$20$$ 1056.00 0.590322
$$21$$ 0 0
$$22$$ −240.000 −0.105719
$$23$$ −600.000 −0.236500 −0.118250 0.992984i $$-0.537728\pi$$
−0.118250 + 0.992984i $$0.537728\pi$$
$$24$$ 0 0
$$25$$ 1231.00 0.393920
$$26$$ 2632.00 0.763576
$$27$$ 0 0
$$28$$ 2816.00 0.678793
$$29$$ −5574.00 −1.23076 −0.615378 0.788232i $$-0.710997\pi$$
−0.615378 + 0.788232i $$0.710997\pi$$
$$30$$ 0 0
$$31$$ −3592.00 −0.671324 −0.335662 0.941983i $$-0.608960\pi$$
−0.335662 + 0.941983i $$0.608960\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ 0 0
$$34$$ −1656.00 −0.245676
$$35$$ 11616.0 1.60283
$$36$$ 0 0
$$37$$ −8458.00 −1.01570 −0.507848 0.861447i $$-0.669559\pi$$
−0.507848 + 0.861447i $$0.669559\pi$$
$$38$$ −3824.00 −0.429595
$$39$$ 0 0
$$40$$ −4224.00 −0.417421
$$41$$ −19194.0 −1.78322 −0.891612 0.452800i $$-0.850425\pi$$
−0.891612 + 0.452800i $$0.850425\pi$$
$$42$$ 0 0
$$43$$ 13316.0 1.09825 0.549127 0.835739i $$-0.314960\pi$$
0.549127 + 0.835739i $$0.314960\pi$$
$$44$$ 960.000 0.0747549
$$45$$ 0 0
$$46$$ 2400.00 0.167231
$$47$$ 19680.0 1.29951 0.649756 0.760143i $$-0.274871\pi$$
0.649756 + 0.760143i $$0.274871\pi$$
$$48$$ 0 0
$$49$$ 14169.0 0.843042
$$50$$ −4924.00 −0.278544
$$51$$ 0 0
$$52$$ −10528.0 −0.539930
$$53$$ 31266.0 1.52891 0.764456 0.644676i $$-0.223008\pi$$
0.764456 + 0.644676i $$0.223008\pi$$
$$54$$ 0 0
$$55$$ 3960.00 0.176518
$$56$$ −11264.0 −0.479979
$$57$$ 0 0
$$58$$ 22296.0 0.870276
$$59$$ −26340.0 −0.985112 −0.492556 0.870281i $$-0.663937\pi$$
−0.492556 + 0.870281i $$0.663937\pi$$
$$60$$ 0 0
$$61$$ −31090.0 −1.06978 −0.534892 0.844920i $$-0.679648\pi$$
−0.534892 + 0.844920i $$0.679648\pi$$
$$62$$ 14368.0 0.474698
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −43428.0 −1.27493
$$66$$ 0 0
$$67$$ −16804.0 −0.457326 −0.228663 0.973506i $$-0.573435\pi$$
−0.228663 + 0.973506i $$0.573435\pi$$
$$68$$ 6624.00 0.173719
$$69$$ 0 0
$$70$$ −46464.0 −1.13337
$$71$$ −6120.00 −0.144081 −0.0720403 0.997402i $$-0.522951\pi$$
−0.0720403 + 0.997402i $$0.522951\pi$$
$$72$$ 0 0
$$73$$ −25558.0 −0.561332 −0.280666 0.959806i $$-0.590555\pi$$
−0.280666 + 0.959806i $$0.590555\pi$$
$$74$$ 33832.0 0.718205
$$75$$ 0 0
$$76$$ 15296.0 0.303769
$$77$$ 10560.0 0.202972
$$78$$ 0 0
$$79$$ 74408.0 1.34138 0.670690 0.741738i $$-0.265998\pi$$
0.670690 + 0.741738i $$0.265998\pi$$
$$80$$ 16896.0 0.295161
$$81$$ 0 0
$$82$$ 76776.0 1.26093
$$83$$ 6468.00 0.103056 0.0515282 0.998672i $$-0.483591\pi$$
0.0515282 + 0.998672i $$0.483591\pi$$
$$84$$ 0 0
$$85$$ 27324.0 0.410201
$$86$$ −53264.0 −0.776583
$$87$$ 0 0
$$88$$ −3840.00 −0.0528597
$$89$$ 32742.0 0.438157 0.219079 0.975707i $$-0.429695\pi$$
0.219079 + 0.975707i $$0.429695\pi$$
$$90$$ 0 0
$$91$$ −115808. −1.46600
$$92$$ −9600.00 −0.118250
$$93$$ 0 0
$$94$$ −78720.0 −0.918894
$$95$$ 63096.0 0.717287
$$96$$ 0 0
$$97$$ 166082. 1.79223 0.896114 0.443824i $$-0.146378\pi$$
0.896114 + 0.443824i $$0.146378\pi$$
$$98$$ −56676.0 −0.596120
$$99$$ 0 0
$$100$$ 19696.0 0.196960
$$101$$ 22002.0 0.214614 0.107307 0.994226i $$-0.465777\pi$$
0.107307 + 0.994226i $$0.465777\pi$$
$$102$$ 0 0
$$103$$ −79264.0 −0.736178 −0.368089 0.929791i $$-0.619988\pi$$
−0.368089 + 0.929791i $$0.619988\pi$$
$$104$$ 42112.0 0.381788
$$105$$ 0 0
$$106$$ −125064. −1.08110
$$107$$ −227988. −1.92510 −0.962548 0.271110i $$-0.912609\pi$$
−0.962548 + 0.271110i $$0.912609\pi$$
$$108$$ 0 0
$$109$$ −8530.00 −0.0687674 −0.0343837 0.999409i $$-0.510947\pi$$
−0.0343837 + 0.999409i $$0.510947\pi$$
$$110$$ −15840.0 −0.124817
$$111$$ 0 0
$$112$$ 45056.0 0.339397
$$113$$ 195438. 1.43984 0.719918 0.694059i $$-0.244179\pi$$
0.719918 + 0.694059i $$0.244179\pi$$
$$114$$ 0 0
$$115$$ −39600.0 −0.279223
$$116$$ −89184.0 −0.615378
$$117$$ 0 0
$$118$$ 105360. 0.696580
$$119$$ 72864.0 0.471678
$$120$$ 0 0
$$121$$ −157451. −0.977647
$$122$$ 124360. 0.756452
$$123$$ 0 0
$$124$$ −57472.0 −0.335662
$$125$$ −125004. −0.715565
$$126$$ 0 0
$$127$$ 173000. 0.951780 0.475890 0.879505i $$-0.342126\pi$$
0.475890 + 0.879505i $$0.342126\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ 0 0
$$130$$ 173712. 0.901512
$$131$$ −151260. −0.770098 −0.385049 0.922896i $$-0.625815\pi$$
−0.385049 + 0.922896i $$0.625815\pi$$
$$132$$ 0 0
$$133$$ 168256. 0.824786
$$134$$ 67216.0 0.323378
$$135$$ 0 0
$$136$$ −26496.0 −0.122838
$$137$$ 128454. 0.584718 0.292359 0.956309i $$-0.405560\pi$$
0.292359 + 0.956309i $$0.405560\pi$$
$$138$$ 0 0
$$139$$ 154196. 0.676918 0.338459 0.940981i $$-0.390094\pi$$
0.338459 + 0.940981i $$0.390094\pi$$
$$140$$ 185856. 0.801413
$$141$$ 0 0
$$142$$ 24480.0 0.101880
$$143$$ −39480.0 −0.161450
$$144$$ 0 0
$$145$$ −367884. −1.45308
$$146$$ 102232. 0.396922
$$147$$ 0 0
$$148$$ −135328. −0.507848
$$149$$ −29454.0 −0.108687 −0.0543436 0.998522i $$-0.517307\pi$$
−0.0543436 + 0.998522i $$0.517307\pi$$
$$150$$ 0 0
$$151$$ −203872. −0.727638 −0.363819 0.931470i $$-0.618527\pi$$
−0.363819 + 0.931470i $$0.618527\pi$$
$$152$$ −61184.0 −0.214797
$$153$$ 0 0
$$154$$ −42240.0 −0.143523
$$155$$ −237072. −0.792594
$$156$$ 0 0
$$157$$ 136142. 0.440801 0.220401 0.975409i $$-0.429263\pi$$
0.220401 + 0.975409i $$0.429263\pi$$
$$158$$ −297632. −0.948499
$$159$$ 0 0
$$160$$ −67584.0 −0.208710
$$161$$ −105600. −0.321070
$$162$$ 0 0
$$163$$ −171124. −0.504478 −0.252239 0.967665i $$-0.581167\pi$$
−0.252239 + 0.967665i $$0.581167\pi$$
$$164$$ −307104. −0.891612
$$165$$ 0 0
$$166$$ −25872.0 −0.0728718
$$167$$ 676200. 1.87622 0.938110 0.346336i $$-0.112574\pi$$
0.938110 + 0.346336i $$0.112574\pi$$
$$168$$ 0 0
$$169$$ 61671.0 0.166098
$$170$$ −109296. −0.290056
$$171$$ 0 0
$$172$$ 213056. 0.549127
$$173$$ −133158. −0.338261 −0.169131 0.985594i $$-0.554096\pi$$
−0.169131 + 0.985594i $$0.554096\pi$$
$$174$$ 0 0
$$175$$ 216656. 0.534781
$$176$$ 15360.0 0.0373774
$$177$$ 0 0
$$178$$ −130968. −0.309824
$$179$$ 693396. 1.61752 0.808758 0.588141i $$-0.200140\pi$$
0.808758 + 0.588141i $$0.200140\pi$$
$$180$$ 0 0
$$181$$ 377174. 0.855747 0.427873 0.903839i $$-0.359263\pi$$
0.427873 + 0.903839i $$0.359263\pi$$
$$182$$ 463232. 1.03662
$$183$$ 0 0
$$184$$ 38400.0 0.0836155
$$185$$ −558228. −1.19917
$$186$$ 0 0
$$187$$ 24840.0 0.0519455
$$188$$ 314880. 0.649756
$$189$$ 0 0
$$190$$ −252384. −0.507198
$$191$$ 265344. 0.526291 0.263145 0.964756i $$-0.415240\pi$$
0.263145 + 0.964756i $$0.415240\pi$$
$$192$$ 0 0
$$193$$ 295298. 0.570647 0.285323 0.958431i $$-0.407899\pi$$
0.285323 + 0.958431i $$0.407899\pi$$
$$194$$ −664328. −1.26730
$$195$$ 0 0
$$196$$ 226704. 0.421521
$$197$$ −201294. −0.369543 −0.184772 0.982781i $$-0.559155\pi$$
−0.184772 + 0.982781i $$0.559155\pi$$
$$198$$ 0 0
$$199$$ 652448. 1.16792 0.583960 0.811782i $$-0.301502\pi$$
0.583960 + 0.811782i $$0.301502\pi$$
$$200$$ −78784.0 −0.139272
$$201$$ 0 0
$$202$$ −88008.0 −0.151755
$$203$$ −981024. −1.67086
$$204$$ 0 0
$$205$$ −1.26680e6 −2.10535
$$206$$ 317056. 0.520557
$$207$$ 0 0
$$208$$ −168448. −0.269965
$$209$$ 57360.0 0.0908330
$$210$$ 0 0
$$211$$ −1.14706e6 −1.77370 −0.886850 0.462058i $$-0.847111\pi$$
−0.886850 + 0.462058i $$0.847111\pi$$
$$212$$ 500256. 0.764456
$$213$$ 0 0
$$214$$ 911952. 1.36125
$$215$$ 878856. 1.29665
$$216$$ 0 0
$$217$$ −632192. −0.911380
$$218$$ 34120.0 0.0486259
$$219$$ 0 0
$$220$$ 63360.0 0.0882589
$$221$$ −272412. −0.375185
$$222$$ 0 0
$$223$$ 701960. 0.945258 0.472629 0.881262i $$-0.343305\pi$$
0.472629 + 0.881262i $$0.343305\pi$$
$$224$$ −180224. −0.239990
$$225$$ 0 0
$$226$$ −781752. −1.01812
$$227$$ −1.23611e6 −1.59218 −0.796089 0.605179i $$-0.793101\pi$$
−0.796089 + 0.605179i $$0.793101\pi$$
$$228$$ 0 0
$$229$$ 105830. 0.133358 0.0666792 0.997774i $$-0.478760\pi$$
0.0666792 + 0.997774i $$0.478760\pi$$
$$230$$ 158400. 0.197440
$$231$$ 0 0
$$232$$ 356736. 0.435138
$$233$$ 438678. 0.529366 0.264683 0.964335i $$-0.414733\pi$$
0.264683 + 0.964335i $$0.414733\pi$$
$$234$$ 0 0
$$235$$ 1.29888e6 1.53426
$$236$$ −421440. −0.492556
$$237$$ 0 0
$$238$$ −291456. −0.333527
$$239$$ −28464.0 −0.0322330 −0.0161165 0.999870i $$-0.505130\pi$$
−0.0161165 + 0.999870i $$0.505130\pi$$
$$240$$ 0 0
$$241$$ 892562. 0.989910 0.494955 0.868919i $$-0.335185\pi$$
0.494955 + 0.868919i $$0.335185\pi$$
$$242$$ 629804. 0.691301
$$243$$ 0 0
$$244$$ −497440. −0.534892
$$245$$ 935154. 0.995332
$$246$$ 0 0
$$247$$ −629048. −0.656057
$$248$$ 229888. 0.237349
$$249$$ 0 0
$$250$$ 500016. 0.505981
$$251$$ 110124. 0.110331 0.0551655 0.998477i $$-0.482431\pi$$
0.0551655 + 0.998477i $$0.482431\pi$$
$$252$$ 0 0
$$253$$ −36000.0 −0.0353591
$$254$$ −692000. −0.673010
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −140802. −0.132977 −0.0664884 0.997787i $$-0.521180\pi$$
−0.0664884 + 0.997787i $$0.521180\pi$$
$$258$$ 0 0
$$259$$ −1.48861e6 −1.37889
$$260$$ −694848. −0.637465
$$261$$ 0 0
$$262$$ 605040. 0.544541
$$263$$ 938760. 0.836884 0.418442 0.908244i $$-0.362576\pi$$
0.418442 + 0.908244i $$0.362576\pi$$
$$264$$ 0 0
$$265$$ 2.06356e6 1.80510
$$266$$ −673024. −0.583212
$$267$$ 0 0
$$268$$ −268864. −0.228663
$$269$$ 1.11451e6 0.939078 0.469539 0.882912i $$-0.344420\pi$$
0.469539 + 0.882912i $$0.344420\pi$$
$$270$$ 0 0
$$271$$ 567704. 0.469568 0.234784 0.972048i $$-0.424562\pi$$
0.234784 + 0.972048i $$0.424562\pi$$
$$272$$ 105984. 0.0868596
$$273$$ 0 0
$$274$$ −513816. −0.413458
$$275$$ 73860.0 0.0588949
$$276$$ 0 0
$$277$$ −1.21326e6 −0.950066 −0.475033 0.879968i $$-0.657564\pi$$
−0.475033 + 0.879968i $$0.657564\pi$$
$$278$$ −616784. −0.478653
$$279$$ 0 0
$$280$$ −743424. −0.566685
$$281$$ −687738. −0.519586 −0.259793 0.965664i $$-0.583654\pi$$
−0.259793 + 0.965664i $$0.583654\pi$$
$$282$$ 0 0
$$283$$ −830908. −0.616718 −0.308359 0.951270i $$-0.599780\pi$$
−0.308359 + 0.951270i $$0.599780\pi$$
$$284$$ −97920.0 −0.0720403
$$285$$ 0 0
$$286$$ 157920. 0.114162
$$287$$ −3.37814e6 −2.42088
$$288$$ 0 0
$$289$$ −1.24846e6 −0.879286
$$290$$ 1.47154e6 1.02749
$$291$$ 0 0
$$292$$ −408928. −0.280666
$$293$$ 1.31263e6 0.893248 0.446624 0.894722i $$-0.352626\pi$$
0.446624 + 0.894722i $$0.352626\pi$$
$$294$$ 0 0
$$295$$ −1.73844e6 −1.16307
$$296$$ 541312. 0.359102
$$297$$ 0 0
$$298$$ 117816. 0.0768535
$$299$$ 394800. 0.255387
$$300$$ 0 0
$$301$$ 2.34362e6 1.49097
$$302$$ 815488. 0.514518
$$303$$ 0 0
$$304$$ 244736. 0.151885
$$305$$ −2.05194e6 −1.26303
$$306$$ 0 0
$$307$$ 1.69022e6 1.02352 0.511761 0.859128i $$-0.328993\pi$$
0.511761 + 0.859128i $$0.328993\pi$$
$$308$$ 168960. 0.101486
$$309$$ 0 0
$$310$$ 948288. 0.560449
$$311$$ 1.50204e6 0.880604 0.440302 0.897850i $$-0.354871\pi$$
0.440302 + 0.897850i $$0.354871\pi$$
$$312$$ 0 0
$$313$$ 810842. 0.467816 0.233908 0.972259i $$-0.424848\pi$$
0.233908 + 0.972259i $$0.424848\pi$$
$$314$$ −544568. −0.311694
$$315$$ 0 0
$$316$$ 1.19053e6 0.670690
$$317$$ −903558. −0.505019 −0.252510 0.967594i $$-0.581256\pi$$
−0.252510 + 0.967594i $$0.581256\pi$$
$$318$$ 0 0
$$319$$ −334440. −0.184010
$$320$$ 270336. 0.147580
$$321$$ 0 0
$$322$$ 422400. 0.227031
$$323$$ 395784. 0.211082
$$324$$ 0 0
$$325$$ −809998. −0.425379
$$326$$ 684496. 0.356720
$$327$$ 0 0
$$328$$ 1.22842e6 0.630465
$$329$$ 3.46368e6 1.76420
$$330$$ 0 0
$$331$$ 1.12197e6 0.562875 0.281438 0.959580i $$-0.409189\pi$$
0.281438 + 0.959580i $$0.409189\pi$$
$$332$$ 103488. 0.0515282
$$333$$ 0 0
$$334$$ −2.70480e6 −1.32669
$$335$$ −1.10906e6 −0.539939
$$336$$ 0 0
$$337$$ −2.75217e6 −1.32008 −0.660041 0.751229i $$-0.729461\pi$$
−0.660041 + 0.751229i $$0.729461\pi$$
$$338$$ −246684. −0.117449
$$339$$ 0 0
$$340$$ 437184. 0.205101
$$341$$ −215520. −0.100369
$$342$$ 0 0
$$343$$ −464288. −0.213085
$$344$$ −852224. −0.388291
$$345$$ 0 0
$$346$$ 532632. 0.239187
$$347$$ −1.91749e6 −0.854889 −0.427445 0.904042i $$-0.640586\pi$$
−0.427445 + 0.904042i $$0.640586\pi$$
$$348$$ 0 0
$$349$$ 1.83659e6 0.807140 0.403570 0.914949i $$-0.367769\pi$$
0.403570 + 0.914949i $$0.367769\pi$$
$$350$$ −866624. −0.378147
$$351$$ 0 0
$$352$$ −61440.0 −0.0264298
$$353$$ 622014. 0.265683 0.132841 0.991137i $$-0.457590\pi$$
0.132841 + 0.991137i $$0.457590\pi$$
$$354$$ 0 0
$$355$$ −403920. −0.170108
$$356$$ 523872. 0.219079
$$357$$ 0 0
$$358$$ −2.77358e6 −1.14376
$$359$$ −3.74062e6 −1.53182 −0.765909 0.642949i $$-0.777711\pi$$
−0.765909 + 0.642949i $$0.777711\pi$$
$$360$$ 0 0
$$361$$ −1.56216e6 −0.630897
$$362$$ −1.50870e6 −0.605104
$$363$$ 0 0
$$364$$ −1.85293e6 −0.733002
$$365$$ −1.68683e6 −0.662733
$$366$$ 0 0
$$367$$ 16232.0 0.00629081 0.00314541 0.999995i $$-0.498999\pi$$
0.00314541 + 0.999995i $$0.498999\pi$$
$$368$$ −153600. −0.0591251
$$369$$ 0 0
$$370$$ 2.23291e6 0.847944
$$371$$ 5.50282e6 2.07563
$$372$$ 0 0
$$373$$ 293606. 0.109268 0.0546340 0.998506i $$-0.482601\pi$$
0.0546340 + 0.998506i $$0.482601\pi$$
$$374$$ −99360.0 −0.0367310
$$375$$ 0 0
$$376$$ −1.25952e6 −0.459447
$$377$$ 3.66769e6 1.32904
$$378$$ 0 0
$$379$$ 3.18012e6 1.13722 0.568611 0.822607i $$-0.307481\pi$$
0.568611 + 0.822607i $$0.307481\pi$$
$$380$$ 1.00954e6 0.358643
$$381$$ 0 0
$$382$$ −1.06138e6 −0.372144
$$383$$ 2.97984e6 1.03800 0.518998 0.854775i $$-0.326305\pi$$
0.518998 + 0.854775i $$0.326305\pi$$
$$384$$ 0 0
$$385$$ 696960. 0.239638
$$386$$ −1.18119e6 −0.403508
$$387$$ 0 0
$$388$$ 2.65731e6 0.896114
$$389$$ −3.45977e6 −1.15924 −0.579620 0.814887i $$-0.696799\pi$$
−0.579620 + 0.814887i $$0.696799\pi$$
$$390$$ 0 0
$$391$$ −248400. −0.0821693
$$392$$ −906816. −0.298060
$$393$$ 0 0
$$394$$ 805176. 0.261307
$$395$$ 4.91093e6 1.58369
$$396$$ 0 0
$$397$$ −3.90416e6 −1.24323 −0.621615 0.783323i $$-0.713523\pi$$
−0.621615 + 0.783323i $$0.713523\pi$$
$$398$$ −2.60979e6 −0.825844
$$399$$ 0 0
$$400$$ 315136. 0.0984800
$$401$$ −5.44115e6 −1.68978 −0.844890 0.534940i $$-0.820334\pi$$
−0.844890 + 0.534940i $$0.820334\pi$$
$$402$$ 0 0
$$403$$ 2.36354e6 0.724936
$$404$$ 352032. 0.107307
$$405$$ 0 0
$$406$$ 3.92410e6 1.18148
$$407$$ −507480. −0.151856
$$408$$ 0 0
$$409$$ 1.96995e6 0.582299 0.291150 0.956678i $$-0.405962\pi$$
0.291150 + 0.956678i $$0.405962\pi$$
$$410$$ 5.06722e6 1.48871
$$411$$ 0 0
$$412$$ −1.26822e6 −0.368089
$$413$$ −4.63584e6 −1.33738
$$414$$ 0 0
$$415$$ 426888. 0.121673
$$416$$ 673792. 0.190894
$$417$$ 0 0
$$418$$ −229440. −0.0642286
$$419$$ −139020. −0.0386850 −0.0193425 0.999813i $$-0.506157\pi$$
−0.0193425 + 0.999813i $$0.506157\pi$$
$$420$$ 0 0
$$421$$ 4.32743e6 1.18994 0.594970 0.803748i $$-0.297164\pi$$
0.594970 + 0.803748i $$0.297164\pi$$
$$422$$ 4.58824e6 1.25419
$$423$$ 0 0
$$424$$ −2.00102e6 −0.540552
$$425$$ 509634. 0.136863
$$426$$ 0 0
$$427$$ −5.47184e6 −1.45232
$$428$$ −3.64781e6 −0.962548
$$429$$ 0 0
$$430$$ −3.51542e6 −0.916867
$$431$$ 2.79936e6 0.725881 0.362941 0.931812i $$-0.381773\pi$$
0.362941 + 0.931812i $$0.381773\pi$$
$$432$$ 0 0
$$433$$ −5.90241e6 −1.51290 −0.756449 0.654052i $$-0.773068\pi$$
−0.756449 + 0.654052i $$0.773068\pi$$
$$434$$ 2.52877e6 0.644443
$$435$$ 0 0
$$436$$ −136480. −0.0343837
$$437$$ −573600. −0.143683
$$438$$ 0 0
$$439$$ −446512. −0.110579 −0.0552894 0.998470i $$-0.517608\pi$$
−0.0552894 + 0.998470i $$0.517608\pi$$
$$440$$ −253440. −0.0624085
$$441$$ 0 0
$$442$$ 1.08965e6 0.265296
$$443$$ −3.49525e6 −0.846193 −0.423096 0.906085i $$-0.639057\pi$$
−0.423096 + 0.906085i $$0.639057\pi$$
$$444$$ 0 0
$$445$$ 2.16097e6 0.517308
$$446$$ −2.80784e6 −0.668398
$$447$$ 0 0
$$448$$ 720896. 0.169698
$$449$$ 1.20613e6 0.282343 0.141171 0.989985i $$-0.454913\pi$$
0.141171 + 0.989985i $$0.454913\pi$$
$$450$$ 0 0
$$451$$ −1.15164e6 −0.266609
$$452$$ 3.12701e6 0.719918
$$453$$ 0 0
$$454$$ 4.94443e6 1.12584
$$455$$ −7.64333e6 −1.73083
$$456$$ 0 0
$$457$$ 233546. 0.0523097 0.0261548 0.999658i $$-0.491674\pi$$
0.0261548 + 0.999658i $$0.491674\pi$$
$$458$$ −423320. −0.0942986
$$459$$ 0 0
$$460$$ −633600. −0.139611
$$461$$ 1.74489e6 0.382398 0.191199 0.981551i $$-0.438762\pi$$
0.191199 + 0.981551i $$0.438762\pi$$
$$462$$ 0 0
$$463$$ −2.91786e6 −0.632576 −0.316288 0.948663i $$-0.602437\pi$$
−0.316288 + 0.948663i $$0.602437\pi$$
$$464$$ −1.42694e6 −0.307689
$$465$$ 0 0
$$466$$ −1.75471e6 −0.374318
$$467$$ 5.31076e6 1.12684 0.563422 0.826169i $$-0.309484\pi$$
0.563422 + 0.826169i $$0.309484\pi$$
$$468$$ 0 0
$$469$$ −2.95750e6 −0.620859
$$470$$ −5.19552e6 −1.08489
$$471$$ 0 0
$$472$$ 1.68576e6 0.348290
$$473$$ 798960. 0.164200
$$474$$ 0 0
$$475$$ 1.17684e6 0.239322
$$476$$ 1.16582e6 0.235839
$$477$$ 0 0
$$478$$ 113856. 0.0227922
$$479$$ −2.34466e6 −0.466918 −0.233459 0.972367i $$-0.575004\pi$$
−0.233459 + 0.972367i $$0.575004\pi$$
$$480$$ 0 0
$$481$$ 5.56536e6 1.09681
$$482$$ −3.57025e6 −0.699972
$$483$$ 0 0
$$484$$ −2.51922e6 −0.488823
$$485$$ 1.09614e7 2.11598
$$486$$ 0 0
$$487$$ 9.81531e6 1.87535 0.937674 0.347517i $$-0.112975\pi$$
0.937674 + 0.347517i $$0.112975\pi$$
$$488$$ 1.98976e6 0.378226
$$489$$ 0 0
$$490$$ −3.74062e6 −0.703806
$$491$$ 5.94520e6 1.11292 0.556458 0.830876i $$-0.312160\pi$$
0.556458 + 0.830876i $$0.312160\pi$$
$$492$$ 0 0
$$493$$ −2.30764e6 −0.427612
$$494$$ 2.51619e6 0.463902
$$495$$ 0 0
$$496$$ −919552. −0.167831
$$497$$ −1.07712e6 −0.195602
$$498$$ 0 0
$$499$$ 6.47832e6 1.16469 0.582346 0.812941i $$-0.302135\pi$$
0.582346 + 0.812941i $$0.302135\pi$$
$$500$$ −2.00006e6 −0.357782
$$501$$ 0 0
$$502$$ −440496. −0.0780158
$$503$$ −4.71794e6 −0.831444 −0.415722 0.909492i $$-0.636471\pi$$
−0.415722 + 0.909492i $$0.636471\pi$$
$$504$$ 0 0
$$505$$ 1.45213e6 0.253383
$$506$$ 144000. 0.0250027
$$507$$ 0 0
$$508$$ 2.76800e6 0.475890
$$509$$ 1.90771e6 0.326375 0.163188 0.986595i $$-0.447822\pi$$
0.163188 + 0.986595i $$0.447822\pi$$
$$510$$ 0 0
$$511$$ −4.49821e6 −0.762057
$$512$$ −262144. −0.0441942
$$513$$ 0 0
$$514$$ 563208. 0.0940288
$$515$$ −5.23142e6 −0.869164
$$516$$ 0 0
$$517$$ 1.18080e6 0.194290
$$518$$ 5.95443e6 0.975025
$$519$$ 0 0
$$520$$ 2.77939e6 0.450756
$$521$$ −8.01974e6 −1.29439 −0.647196 0.762324i $$-0.724059\pi$$
−0.647196 + 0.762324i $$0.724059\pi$$
$$522$$ 0 0
$$523$$ 1.91162e6 0.305596 0.152798 0.988257i $$-0.451172\pi$$
0.152798 + 0.988257i $$0.451172\pi$$
$$524$$ −2.42016e6 −0.385049
$$525$$ 0 0
$$526$$ −3.75504e6 −0.591766
$$527$$ −1.48709e6 −0.233244
$$528$$ 0 0
$$529$$ −6.07634e6 −0.944068
$$530$$ −8.25422e6 −1.27640
$$531$$ 0 0
$$532$$ 2.69210e6 0.412393
$$533$$ 1.26297e7 1.92563
$$534$$ 0 0
$$535$$ −1.50472e7 −2.27285
$$536$$ 1.07546e6 0.161689
$$537$$ 0 0
$$538$$ −4.45802e6 −0.664028
$$539$$ 850140. 0.126043
$$540$$ 0 0
$$541$$ −1.19900e7 −1.76128 −0.880639 0.473788i $$-0.842886\pi$$
−0.880639 + 0.473788i $$0.842886\pi$$
$$542$$ −2.27082e6 −0.332035
$$543$$ 0 0
$$544$$ −423936. −0.0614190
$$545$$ −562980. −0.0811898
$$546$$ 0 0
$$547$$ 4.45809e6 0.637061 0.318530 0.947913i $$-0.396811\pi$$
0.318530 + 0.947913i $$0.396811\pi$$
$$548$$ 2.05526e6 0.292359
$$549$$ 0 0
$$550$$ −295440. −0.0416450
$$551$$ −5.32874e6 −0.747732
$$552$$ 0 0
$$553$$ 1.30958e7 1.82104
$$554$$ 4.85303e6 0.671798
$$555$$ 0 0
$$556$$ 2.46714e6 0.338459
$$557$$ −9.02612e6 −1.23272 −0.616358 0.787466i $$-0.711393\pi$$
−0.616358 + 0.787466i $$0.711393\pi$$
$$558$$ 0 0
$$559$$ −8.76193e6 −1.18596
$$560$$ 2.97370e6 0.400707
$$561$$ 0 0
$$562$$ 2.75095e6 0.367403
$$563$$ −6.84899e6 −0.910658 −0.455329 0.890323i $$-0.650478\pi$$
−0.455329 + 0.890323i $$0.650478\pi$$
$$564$$ 0 0
$$565$$ 1.28989e7 1.69993
$$566$$ 3.32363e6 0.436086
$$567$$ 0 0
$$568$$ 391680. 0.0509402
$$569$$ 5.46322e6 0.707405 0.353703 0.935358i $$-0.384923\pi$$
0.353703 + 0.935358i $$0.384923\pi$$
$$570$$ 0 0
$$571$$ −1.02324e7 −1.31337 −0.656684 0.754166i $$-0.728041\pi$$
−0.656684 + 0.754166i $$0.728041\pi$$
$$572$$ −631680. −0.0807248
$$573$$ 0 0
$$574$$ 1.35126e7 1.71182
$$575$$ −738600. −0.0931622
$$576$$ 0 0
$$577$$ 1.59437e7 1.99365 0.996825 0.0796186i $$-0.0253702\pi$$
0.996825 + 0.0796186i $$0.0253702\pi$$
$$578$$ 4.99384e6 0.621749
$$579$$ 0 0
$$580$$ −5.88614e6 −0.726542
$$581$$ 1.13837e6 0.139908
$$582$$ 0 0
$$583$$ 1.87596e6 0.228587
$$584$$ 1.63571e6 0.198461
$$585$$ 0 0
$$586$$ −5.25050e6 −0.631622
$$587$$ 9.47713e6 1.13522 0.567612 0.823296i $$-0.307867\pi$$
0.567612 + 0.823296i $$0.307867\pi$$
$$588$$ 0 0
$$589$$ −3.43395e6 −0.407855
$$590$$ 6.95376e6 0.822412
$$591$$ 0 0
$$592$$ −2.16525e6 −0.253924
$$593$$ −2.45349e6 −0.286515 −0.143258 0.989685i $$-0.545758\pi$$
−0.143258 + 0.989685i $$0.545758\pi$$
$$594$$ 0 0
$$595$$ 4.80902e6 0.556884
$$596$$ −471264. −0.0543436
$$597$$ 0 0
$$598$$ −1.57920e6 −0.180586
$$599$$ 9.29978e6 1.05902 0.529512 0.848302i $$-0.322375\pi$$
0.529512 + 0.848302i $$0.322375\pi$$
$$600$$ 0 0
$$601$$ −1.14617e7 −1.29438 −0.647192 0.762327i $$-0.724057\pi$$
−0.647192 + 0.762327i $$0.724057\pi$$
$$602$$ −9.37446e6 −1.05428
$$603$$ 0 0
$$604$$ −3.26195e6 −0.363819
$$605$$ −1.03918e7 −1.15425
$$606$$ 0 0
$$607$$ 1.12784e7 1.24244 0.621219 0.783637i $$-0.286638\pi$$
0.621219 + 0.783637i $$0.286638\pi$$
$$608$$ −978944. −0.107399
$$609$$ 0 0
$$610$$ 8.20776e6 0.893100
$$611$$ −1.29494e7 −1.40329
$$612$$ 0 0
$$613$$ 93782.0 0.0100802 0.00504009 0.999987i $$-0.498396\pi$$
0.00504009 + 0.999987i $$0.498396\pi$$
$$614$$ −6.76088e6 −0.723740
$$615$$ 0 0
$$616$$ −675840. −0.0717616
$$617$$ 1.49642e7 1.58248 0.791242 0.611504i $$-0.209435\pi$$
0.791242 + 0.611504i $$0.209435\pi$$
$$618$$ 0 0
$$619$$ −5.06888e6 −0.531723 −0.265861 0.964011i $$-0.585656\pi$$
−0.265861 + 0.964011i $$0.585656\pi$$
$$620$$ −3.79315e6 −0.396297
$$621$$ 0 0
$$622$$ −6.00816e6 −0.622681
$$623$$ 5.76259e6 0.594837
$$624$$ 0 0
$$625$$ −1.20971e7 −1.23875
$$626$$ −3.24337e6 −0.330796
$$627$$ 0 0
$$628$$ 2.17827e6 0.220401
$$629$$ −3.50161e6 −0.352892
$$630$$ 0 0
$$631$$ 1.55919e7 1.55892 0.779462 0.626450i $$-0.215493\pi$$
0.779462 + 0.626450i $$0.215493\pi$$
$$632$$ −4.76211e6 −0.474250
$$633$$ 0 0
$$634$$ 3.61423e6 0.357102
$$635$$ 1.14180e7 1.12371
$$636$$ 0 0
$$637$$ −9.32320e6 −0.910367
$$638$$ 1.33776e6 0.130115
$$639$$ 0 0
$$640$$ −1.08134e6 −0.104355
$$641$$ −1.09701e7 −1.05455 −0.527274 0.849695i $$-0.676786\pi$$
−0.527274 + 0.849695i $$0.676786\pi$$
$$642$$ 0 0
$$643$$ −2.83704e6 −0.270607 −0.135303 0.990804i $$-0.543201\pi$$
−0.135303 + 0.990804i $$0.543201\pi$$
$$644$$ −1.68960e6 −0.160535
$$645$$ 0 0
$$646$$ −1.58314e6 −0.149258
$$647$$ 6.05686e6 0.568835 0.284418 0.958700i $$-0.408200\pi$$
0.284418 + 0.958700i $$0.408200\pi$$
$$648$$ 0 0
$$649$$ −1.58040e6 −0.147284
$$650$$ 3.23999e6 0.300788
$$651$$ 0 0
$$652$$ −2.73798e6 −0.252239
$$653$$ 1.08892e6 0.0999341 0.0499671 0.998751i $$-0.484088\pi$$
0.0499671 + 0.998751i $$0.484088\pi$$
$$654$$ 0 0
$$655$$ −9.98316e6 −0.909211
$$656$$ −4.91366e6 −0.445806
$$657$$ 0 0
$$658$$ −1.38547e7 −1.24748
$$659$$ −7.41803e6 −0.665388 −0.332694 0.943035i $$-0.607958\pi$$
−0.332694 + 0.943035i $$0.607958\pi$$
$$660$$ 0 0
$$661$$ 767654. 0.0683379 0.0341690 0.999416i $$-0.489122\pi$$
0.0341690 + 0.999416i $$0.489122\pi$$
$$662$$ −4.48789e6 −0.398013
$$663$$ 0 0
$$664$$ −413952. −0.0364359
$$665$$ 1.11049e7 0.973779
$$666$$ 0 0
$$667$$ 3.34440e6 0.291074
$$668$$ 1.08192e7 0.938110
$$669$$ 0 0
$$670$$ 4.43626e6 0.381794
$$671$$ −1.86540e6 −0.159943
$$672$$ 0 0
$$673$$ 1.42263e6 0.121075 0.0605373 0.998166i $$-0.480719\pi$$
0.0605373 + 0.998166i $$0.480719\pi$$
$$674$$ 1.10087e7 0.933439
$$675$$ 0 0
$$676$$ 986736. 0.0830490
$$677$$ 6.16231e6 0.516739 0.258370 0.966046i $$-0.416815\pi$$
0.258370 + 0.966046i $$0.416815\pi$$
$$678$$ 0 0
$$679$$ 2.92304e7 2.43310
$$680$$ −1.74874e6 −0.145028
$$681$$ 0 0
$$682$$ 862080. 0.0709719
$$683$$ −1.50621e7 −1.23548 −0.617739 0.786383i $$-0.711951\pi$$
−0.617739 + 0.786383i $$0.711951\pi$$
$$684$$ 0 0
$$685$$ 8.47796e6 0.690343
$$686$$ 1.85715e6 0.150674
$$687$$ 0 0
$$688$$ 3.40890e6 0.274563
$$689$$ −2.05730e7 −1.65101
$$690$$ 0 0
$$691$$ −5.87636e6 −0.468180 −0.234090 0.972215i $$-0.575211\pi$$
−0.234090 + 0.972215i $$0.575211\pi$$
$$692$$ −2.13053e6 −0.169131
$$693$$ 0 0
$$694$$ 7.66997e6 0.604498
$$695$$ 1.01769e7 0.799199
$$696$$ 0 0
$$697$$ −7.94632e6 −0.619561
$$698$$ −7.34636e6 −0.570734
$$699$$ 0 0
$$700$$ 3.46650e6 0.267390
$$701$$ −3.60077e6 −0.276758 −0.138379 0.990379i $$-0.544189\pi$$
−0.138379 + 0.990379i $$0.544189\pi$$
$$702$$ 0 0
$$703$$ −8.08585e6 −0.617074
$$704$$ 245760. 0.0186887
$$705$$ 0 0
$$706$$ −2.48806e6 −0.187866
$$707$$ 3.87235e6 0.291358
$$708$$ 0 0
$$709$$ 9.22516e6 0.689221 0.344610 0.938746i $$-0.388011\pi$$
0.344610 + 0.938746i $$0.388011\pi$$
$$710$$ 1.61568e6 0.120284
$$711$$ 0 0
$$712$$ −2.09549e6 −0.154912
$$713$$ 2.15520e6 0.158768
$$714$$ 0 0
$$715$$ −2.60568e6 −0.190615
$$716$$ 1.10943e7 0.808758
$$717$$ 0 0
$$718$$ 1.49625e7 1.08316
$$719$$ 2.63923e7 1.90395 0.951975 0.306177i $$-0.0990500\pi$$
0.951975 + 0.306177i $$0.0990500\pi$$
$$720$$ 0 0
$$721$$ −1.39505e7 −0.999426
$$722$$ 6.24865e6 0.446111
$$723$$ 0 0
$$724$$ 6.03478e6 0.427873
$$725$$ −6.86159e6 −0.484819
$$726$$ 0 0
$$727$$ −9.79485e6 −0.687324 −0.343662 0.939093i $$-0.611667\pi$$
−0.343662 + 0.939093i $$0.611667\pi$$
$$728$$ 7.41171e6 0.518311
$$729$$ 0 0
$$730$$ 6.74731e6 0.468623
$$731$$ 5.51282e6 0.381576
$$732$$ 0 0
$$733$$ 4.07584e6 0.280193 0.140096 0.990138i $$-0.455259\pi$$
0.140096 + 0.990138i $$0.455259\pi$$
$$734$$ −64928.0 −0.00444828
$$735$$ 0 0
$$736$$ 614400. 0.0418077
$$737$$ −1.00824e6 −0.0683747
$$738$$ 0 0
$$739$$ −1.65709e7 −1.11618 −0.558089 0.829781i $$-0.688465\pi$$
−0.558089 + 0.829781i $$0.688465\pi$$
$$740$$ −8.93165e6 −0.599587
$$741$$ 0 0
$$742$$ −2.20113e7 −1.46769
$$743$$ −1.44141e7 −0.957892 −0.478946 0.877844i $$-0.658981\pi$$
−0.478946 + 0.877844i $$0.658981\pi$$
$$744$$ 0 0
$$745$$ −1.94396e6 −0.128321
$$746$$ −1.17442e6 −0.0772641
$$747$$ 0 0
$$748$$ 397440. 0.0259727
$$749$$ −4.01259e7 −2.61349
$$750$$ 0 0
$$751$$ 1.67944e7 1.08659 0.543295 0.839542i $$-0.317177\pi$$
0.543295 + 0.839542i $$0.317177\pi$$
$$752$$ 5.03808e6 0.324878
$$753$$ 0 0
$$754$$ −1.46708e7 −0.939776
$$755$$ −1.34556e7 −0.859081
$$756$$ 0 0
$$757$$ 1.32943e7 0.843188 0.421594 0.906785i $$-0.361471\pi$$
0.421594 + 0.906785i $$0.361471\pi$$
$$758$$ −1.27205e7 −0.804137
$$759$$ 0 0
$$760$$ −4.03814e6 −0.253599
$$761$$ 2.14786e6 0.134445 0.0672225 0.997738i $$-0.478586\pi$$
0.0672225 + 0.997738i $$0.478586\pi$$
$$762$$ 0 0
$$763$$ −1.50128e6 −0.0933577
$$764$$ 4.24550e6 0.263145
$$765$$ 0 0
$$766$$ −1.19194e7 −0.733975
$$767$$ 1.73317e7 1.06378
$$768$$ 0 0
$$769$$ −1.31059e7 −0.799193 −0.399596 0.916691i $$-0.630850\pi$$
−0.399596 + 0.916691i $$0.630850\pi$$
$$770$$ −2.78784e6 −0.169450
$$771$$ 0 0
$$772$$ 4.72477e6 0.285323
$$773$$ 2.37154e7 1.42752 0.713759 0.700392i $$-0.246991\pi$$
0.713759 + 0.700392i $$0.246991\pi$$
$$774$$ 0 0
$$775$$ −4.42175e6 −0.264448
$$776$$ −1.06292e7 −0.633648
$$777$$ 0 0
$$778$$ 1.38391e7 0.819707
$$779$$ −1.83495e7 −1.08338
$$780$$ 0 0
$$781$$ −367200. −0.0215415
$$782$$ 993600. 0.0581025
$$783$$ 0 0
$$784$$ 3.62726e6 0.210760
$$785$$ 8.98537e6 0.520430
$$786$$ 0 0
$$787$$ −8.40048e6 −0.483468 −0.241734 0.970343i $$-0.577716\pi$$
−0.241734 + 0.970343i $$0.577716\pi$$
$$788$$ −3.22070e6 −0.184772
$$789$$ 0 0
$$790$$ −1.96437e7 −1.11984
$$791$$ 3.43971e7 1.95470
$$792$$ 0 0
$$793$$ 2.04572e7 1.15522
$$794$$ 1.56166e7 0.879097
$$795$$ 0 0
$$796$$ 1.04392e7 0.583960
$$797$$ −5.41023e6 −0.301696 −0.150848 0.988557i $$-0.548200\pi$$
−0.150848 + 0.988557i $$0.548200\pi$$
$$798$$ 0 0
$$799$$ 8.14752e6 0.451501
$$800$$ −1.26054e6 −0.0696359
$$801$$ 0 0
$$802$$ 2.17646e7 1.19485
$$803$$ −1.53348e6 −0.0839246
$$804$$ 0 0
$$805$$ −6.96960e6 −0.379069
$$806$$ −9.45414e6 −0.512607
$$807$$ 0 0
$$808$$ −1.40813e6 −0.0758776
$$809$$ 2.60777e7 1.40087 0.700436 0.713715i $$-0.252989\pi$$
0.700436 + 0.713715i $$0.252989\pi$$
$$810$$ 0 0
$$811$$ 1.90021e7 1.01449 0.507247 0.861800i $$-0.330663\pi$$
0.507247 + 0.861800i $$0.330663\pi$$
$$812$$ −1.56964e7 −0.835429
$$813$$ 0 0
$$814$$ 2.02992e6 0.107379
$$815$$ −1.12942e7 −0.595608
$$816$$ 0 0
$$817$$ 1.27301e7 0.667231
$$818$$ −7.87978e6 −0.411748
$$819$$ 0 0
$$820$$ −2.02689e7 −1.05268
$$821$$ 3.10173e7 1.60600 0.803001 0.595978i $$-0.203236\pi$$
0.803001 + 0.595978i $$0.203236\pi$$
$$822$$ 0 0
$$823$$ −1.56290e7 −0.804323 −0.402162 0.915569i $$-0.631741\pi$$
−0.402162 + 0.915569i $$0.631741\pi$$
$$824$$ 5.07290e6 0.260278
$$825$$ 0 0
$$826$$ 1.85434e7 0.945667
$$827$$ −1.58421e7 −0.805467 −0.402733 0.915317i $$-0.631940\pi$$
−0.402733 + 0.915317i $$0.631940\pi$$
$$828$$ 0 0
$$829$$ 2.06176e6 0.104196 0.0520980 0.998642i $$-0.483409\pi$$
0.0520980 + 0.998642i $$0.483409\pi$$
$$830$$ −1.70755e6 −0.0860357
$$831$$ 0 0
$$832$$ −2.69517e6 −0.134983
$$833$$ 5.86597e6 0.292905
$$834$$ 0 0
$$835$$ 4.46292e7 2.21515
$$836$$ 917760. 0.0454165
$$837$$ 0 0
$$838$$ 556080. 0.0273544
$$839$$ −3.03900e7 −1.49048 −0.745240 0.666796i $$-0.767665\pi$$
−0.745240 + 0.666796i $$0.767665\pi$$
$$840$$ 0 0
$$841$$ 1.05583e7 0.514760
$$842$$ −1.73097e7 −0.841414
$$843$$ 0 0
$$844$$ −1.83530e7 −0.886850
$$845$$ 4.07029e6 0.196103
$$846$$ 0 0
$$847$$ −2.77114e7 −1.32724
$$848$$ 8.00410e6 0.382228
$$849$$ 0 0
$$850$$ −2.03854e6 −0.0967768
$$851$$ 5.07480e6 0.240212
$$852$$ 0 0
$$853$$ −2.97738e7 −1.40108 −0.700538 0.713615i $$-0.747056\pi$$
−0.700538 + 0.713615i $$0.747056\pi$$
$$854$$ 2.18874e7 1.02695
$$855$$ 0 0
$$856$$ 1.45912e7 0.680624
$$857$$ −8.64100e6 −0.401894 −0.200947 0.979602i $$-0.564402\pi$$
−0.200947 + 0.979602i $$0.564402\pi$$
$$858$$ 0 0
$$859$$ −3.35663e7 −1.55210 −0.776051 0.630670i $$-0.782780\pi$$
−0.776051 + 0.630670i $$0.782780\pi$$
$$860$$ 1.40617e7 0.648323
$$861$$ 0 0
$$862$$ −1.11974e7 −0.513276
$$863$$ −3.90191e7 −1.78341 −0.891703 0.452621i $$-0.850489\pi$$
−0.891703 + 0.452621i $$0.850489\pi$$
$$864$$ 0 0
$$865$$ −8.78843e6 −0.399366
$$866$$ 2.36097e7 1.06978
$$867$$ 0 0
$$868$$ −1.01151e7 −0.455690
$$869$$ 4.46448e6 0.200549
$$870$$ 0 0
$$871$$ 1.10570e7 0.493848
$$872$$ 545920. 0.0243130
$$873$$ 0 0
$$874$$ 2.29440e6 0.101599
$$875$$ −2.20007e7 −0.971441
$$876$$ 0 0
$$877$$ −1.81382e7 −0.796333 −0.398166 0.917313i $$-0.630353\pi$$
−0.398166 + 0.917313i $$0.630353\pi$$
$$878$$ 1.78605e6 0.0781910
$$879$$ 0 0
$$880$$ 1.01376e6 0.0441294
$$881$$ −3.05312e7 −1.32527 −0.662634 0.748943i $$-0.730562\pi$$
−0.662634 + 0.748943i $$0.730562\pi$$
$$882$$ 0 0
$$883$$ −4.35533e7 −1.87983 −0.939916 0.341405i $$-0.889097\pi$$
−0.939916 + 0.341405i $$0.889097\pi$$
$$884$$ −4.35859e6 −0.187593
$$885$$ 0 0
$$886$$ 1.39810e7 0.598348
$$887$$ 1.34152e7 0.572515 0.286257 0.958153i $$-0.407589\pi$$
0.286257 + 0.958153i $$0.407589\pi$$
$$888$$ 0 0
$$889$$ 3.04480e7 1.29212
$$890$$ −8.64389e6 −0.365792
$$891$$ 0 0
$$892$$ 1.12314e7 0.472629
$$893$$ 1.88141e7 0.789504
$$894$$ 0 0
$$895$$ 4.57641e7 1.90971
$$896$$ −2.88358e6 −0.119995
$$897$$ 0 0
$$898$$ −4.82450e6 −0.199647
$$899$$ 2.00218e7 0.826236
$$900$$ 0 0
$$901$$ 1.29441e7 0.531203
$$902$$ 4.60656e6 0.188521
$$903$$ 0 0
$$904$$ −1.25080e7 −0.509059
$$905$$ 2.48935e7 1.01033
$$906$$ 0 0
$$907$$ 3.10816e6 0.125454 0.0627272 0.998031i $$-0.480020\pi$$
0.0627272 + 0.998031i $$0.480020\pi$$
$$908$$ −1.97777e7 −0.796089
$$909$$ 0 0
$$910$$ 3.05733e7 1.22388
$$911$$ −1.19035e6 −0.0475203 −0.0237602 0.999718i $$-0.507564\pi$$
−0.0237602 + 0.999718i $$0.507564\pi$$
$$912$$ 0 0
$$913$$ 388080. 0.0154079
$$914$$ −934184. −0.0369885
$$915$$ 0 0
$$916$$ 1.69328e6 0.0666792
$$917$$ −2.66218e7 −1.04547
$$918$$ 0 0
$$919$$ −4.71996e7 −1.84353 −0.921764 0.387752i $$-0.873252\pi$$
−0.921764 + 0.387752i $$0.873252\pi$$
$$920$$ 2.53440e6 0.0987201
$$921$$ 0 0
$$922$$ −6.97956e6 −0.270396
$$923$$ 4.02696e6 0.155587
$$924$$ 0 0
$$925$$ −1.04118e7 −0.400103
$$926$$ 1.16715e7 0.447299
$$927$$ 0 0
$$928$$ 5.70778e6 0.217569
$$929$$ −1.33595e6 −0.0507870 −0.0253935 0.999678i $$-0.508084\pi$$
−0.0253935 + 0.999678i $$0.508084\pi$$
$$930$$ 0 0
$$931$$ 1.35456e7 0.512180
$$932$$ 7.01885e6 0.264683
$$933$$ 0 0
$$934$$ −2.12430e7 −0.796800
$$935$$ 1.63944e6 0.0613291
$$936$$ 0 0
$$937$$ 1.47238e7 0.547861 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$938$$ 1.18300e7 0.439014
$$939$$ 0 0
$$940$$ 2.07821e7 0.767131
$$941$$ 2.69196e7 0.991049 0.495525 0.868594i $$-0.334976\pi$$
0.495525 + 0.868594i $$0.334976\pi$$
$$942$$ 0 0
$$943$$ 1.15164e7 0.421733
$$944$$ −6.74304e6 −0.246278
$$945$$ 0 0
$$946$$ −3.19584e6 −0.116107
$$947$$ 3.73160e6 0.135214 0.0676068 0.997712i $$-0.478464\pi$$
0.0676068 + 0.997712i $$0.478464\pi$$
$$948$$ 0 0
$$949$$ 1.68172e7 0.606160
$$950$$ −4.70734e6 −0.169226
$$951$$ 0 0
$$952$$ −4.66330e6 −0.166763
$$953$$ −2.18735e7 −0.780166 −0.390083 0.920780i $$-0.627554\pi$$
−0.390083 + 0.920780i $$0.627554\pi$$
$$954$$ 0 0
$$955$$ 1.75127e7 0.621362
$$956$$ −455424. −0.0161165
$$957$$ 0 0
$$958$$ 9.37862e6 0.330161
$$959$$ 2.26079e7 0.793805
$$960$$ 0 0
$$961$$ −1.57267e7 −0.549324
$$962$$ −2.22615e7 −0.775561
$$963$$ 0 0
$$964$$ 1.42810e7 0.494955
$$965$$ 1.94897e7 0.673730
$$966$$ 0 0
$$967$$ 1.76025e7 0.605352 0.302676 0.953093i $$-0.402120\pi$$
0.302676 + 0.953093i $$0.402120\pi$$
$$968$$ 1.00769e7 0.345650
$$969$$ 0 0
$$970$$ −4.38456e7 −1.49623
$$971$$ −1.67317e7 −0.569497 −0.284749 0.958602i $$-0.591910\pi$$
−0.284749 + 0.958602i $$0.591910\pi$$
$$972$$ 0 0
$$973$$ 2.71385e7 0.918975
$$974$$ −3.92612e7 −1.32607
$$975$$ 0 0
$$976$$ −7.95904e6 −0.267446
$$977$$ −5.55382e7 −1.86147 −0.930733 0.365699i $$-0.880830\pi$$
−0.930733 + 0.365699i $$0.880830\pi$$
$$978$$ 0 0
$$979$$ 1.96452e6 0.0655088
$$980$$ 1.49625e7 0.497666
$$981$$ 0 0
$$982$$ −2.37808e7 −0.786951
$$983$$ 3.86784e7 1.27669 0.638344 0.769751i $$-0.279620\pi$$
0.638344 + 0.769751i $$0.279620\pi$$
$$984$$ 0 0
$$985$$ −1.32854e7 −0.436299
$$986$$ 9.23054e6 0.302367
$$987$$ 0 0
$$988$$ −1.00648e7 −0.328028
$$989$$ −7.98960e6 −0.259737
$$990$$ 0 0
$$991$$ 9.58498e6 0.310033 0.155016 0.987912i $$-0.450457\pi$$
0.155016 + 0.987912i $$0.450457\pi$$
$$992$$ 3.67821e6 0.118674
$$993$$ 0 0
$$994$$ 4.30848e6 0.138311
$$995$$ 4.30616e7 1.37890
$$996$$ 0 0
$$997$$ −1.03650e7 −0.330242 −0.165121 0.986273i $$-0.552802\pi$$
−0.165121 + 0.986273i $$0.552802\pi$$
$$998$$ −2.59133e7 −0.823561
$$999$$ 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 18.6.a.b.1.1 1
3.2 odd 2 6.6.a.a.1.1 1
4.3 odd 2 144.6.a.j.1.1 1
5.2 odd 4 450.6.c.j.199.1 2
5.3 odd 4 450.6.c.j.199.2 2
5.4 even 2 450.6.a.m.1.1 1
7.6 odd 2 882.6.a.a.1.1 1
8.3 odd 2 576.6.a.i.1.1 1
8.5 even 2 576.6.a.j.1.1 1
9.2 odd 6 162.6.c.e.109.1 2
9.4 even 3 162.6.c.h.55.1 2
9.5 odd 6 162.6.c.e.55.1 2
9.7 even 3 162.6.c.h.109.1 2
12.11 even 2 48.6.a.c.1.1 1
15.2 even 4 150.6.c.b.49.2 2
15.8 even 4 150.6.c.b.49.1 2
15.14 odd 2 150.6.a.d.1.1 1
21.2 odd 6 294.6.e.g.67.1 2
21.5 even 6 294.6.e.a.67.1 2
21.11 odd 6 294.6.e.g.79.1 2
21.17 even 6 294.6.e.a.79.1 2
21.20 even 2 294.6.a.m.1.1 1
24.5 odd 2 192.6.a.o.1.1 1
24.11 even 2 192.6.a.g.1.1 1
33.32 even 2 726.6.a.a.1.1 1
39.38 odd 2 1014.6.a.c.1.1 1
48.5 odd 4 768.6.d.c.385.2 2
48.11 even 4 768.6.d.p.385.1 2
48.29 odd 4 768.6.d.c.385.1 2
48.35 even 4 768.6.d.p.385.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
6.6.a.a.1.1 1 3.2 odd 2
18.6.a.b.1.1 1 1.1 even 1 trivial
48.6.a.c.1.1 1 12.11 even 2
144.6.a.j.1.1 1 4.3 odd 2
150.6.a.d.1.1 1 15.14 odd 2
150.6.c.b.49.1 2 15.8 even 4
150.6.c.b.49.2 2 15.2 even 4
162.6.c.e.55.1 2 9.5 odd 6
162.6.c.e.109.1 2 9.2 odd 6
162.6.c.h.55.1 2 9.4 even 3
162.6.c.h.109.1 2 9.7 even 3
192.6.a.g.1.1 1 24.11 even 2
192.6.a.o.1.1 1 24.5 odd 2
294.6.a.m.1.1 1 21.20 even 2
294.6.e.a.67.1 2 21.5 even 6
294.6.e.a.79.1 2 21.17 even 6
294.6.e.g.67.1 2 21.2 odd 6
294.6.e.g.79.1 2 21.11 odd 6
450.6.a.m.1.1 1 5.4 even 2
450.6.c.j.199.1 2 5.2 odd 4
450.6.c.j.199.2 2 5.3 odd 4
576.6.a.i.1.1 1 8.3 odd 2
576.6.a.j.1.1 1 8.5 even 2
726.6.a.a.1.1 1 33.32 even 2
768.6.d.c.385.1 2 48.29 odd 4
768.6.d.c.385.2 2 48.5 odd 4
768.6.d.p.385.1 2 48.11 even 4
768.6.d.p.385.2 2 48.35 even 4
882.6.a.a.1.1 1 7.6 odd 2
1014.6.a.c.1.1 1 39.38 odd 2