# Properties

 Label 18.6.a.c.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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -148.000 q^{7} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -148.000 q^{7} +64.0000 q^{8} +384.000 q^{10} -384.000 q^{11} -334.000 q^{13} -592.000 q^{14} +256.000 q^{16} -576.000 q^{17} -664.000 q^{19} +1536.00 q^{20} -1536.00 q^{22} +3840.00 q^{23} +6091.00 q^{25} -1336.00 q^{26} -2368.00 q^{28} -96.0000 q^{29} -4564.00 q^{31} +1024.00 q^{32} -2304.00 q^{34} -14208.0 q^{35} +5798.00 q^{37} -2656.00 q^{38} +6144.00 q^{40} +6720.00 q^{41} -14872.0 q^{43} -6144.00 q^{44} +15360.0 q^{46} +19200.0 q^{47} +5097.00 q^{49} +24364.0 q^{50} -5344.00 q^{52} -7776.00 q^{53} -36864.0 q^{55} -9472.00 q^{56} -384.000 q^{58} +13056.0 q^{59} +42782.0 q^{61} -18256.0 q^{62} +4096.00 q^{64} -32064.0 q^{65} +36656.0 q^{67} -9216.00 q^{68} -56832.0 q^{70} -64512.0 q^{71} -16810.0 q^{73} +23192.0 q^{74} -10624.0 q^{76} +56832.0 q^{77} +28076.0 q^{79} +24576.0 q^{80} +26880.0 q^{82} +66432.0 q^{83} -55296.0 q^{85} -59488.0 q^{86} -24576.0 q^{88} +81792.0 q^{89} +49432.0 q^{91} +61440.0 q^{92} +76800.0 q^{94} -63744.0 q^{95} -29938.0 q^{97} +20388.0 q^{98} +O(q^{100})$$

## 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
$$5$$ 96.0000 1.71730 0.858650 0.512562i $$-0.171304\pi$$
0.858650 + 0.512562i $$0.171304\pi$$
$$6$$ 0 0
$$7$$ −148.000 −1.14161 −0.570803 0.821087i $$-0.693368\pi$$
−0.570803 + 0.821087i $$0.693368\pi$$
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ 384.000 1.21431
$$11$$ −384.000 −0.956862 −0.478431 0.878125i $$-0.658794\pi$$
−0.478431 + 0.878125i $$0.658794\pi$$
$$12$$ 0 0
$$13$$ −334.000 −0.548136 −0.274068 0.961710i $$-0.588369\pi$$
−0.274068 + 0.961710i $$0.588369\pi$$
$$14$$ −592.000 −0.807238
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −576.000 −0.483393 −0.241696 0.970352i $$-0.577704\pi$$
−0.241696 + 0.970352i $$0.577704\pi$$
$$18$$ 0 0
$$19$$ −664.000 −0.421972 −0.210986 0.977489i $$-0.567668\pi$$
−0.210986 + 0.977489i $$0.567668\pi$$
$$20$$ 1536.00 0.858650
$$21$$ 0 0
$$22$$ −1536.00 −0.676604
$$23$$ 3840.00 1.51360 0.756801 0.653645i $$-0.226761\pi$$
0.756801 + 0.653645i $$0.226761\pi$$
$$24$$ 0 0
$$25$$ 6091.00 1.94912
$$26$$ −1336.00 −0.387590
$$27$$ 0 0
$$28$$ −2368.00 −0.570803
$$29$$ −96.0000 −0.0211971 −0.0105985 0.999944i $$-0.503374\pi$$
−0.0105985 + 0.999944i $$0.503374\pi$$
$$30$$ 0 0
$$31$$ −4564.00 −0.852985 −0.426493 0.904491i $$-0.640251\pi$$
−0.426493 + 0.904491i $$0.640251\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ −2304.00 −0.341810
$$35$$ −14208.0 −1.96048
$$36$$ 0 0
$$37$$ 5798.00 0.696264 0.348132 0.937446i $$-0.386816\pi$$
0.348132 + 0.937446i $$0.386816\pi$$
$$38$$ −2656.00 −0.298380
$$39$$ 0 0
$$40$$ 6144.00 0.607157
$$41$$ 6720.00 0.624323 0.312162 0.950029i $$-0.398947\pi$$
0.312162 + 0.950029i $$0.398947\pi$$
$$42$$ 0 0
$$43$$ −14872.0 −1.22659 −0.613293 0.789855i $$-0.710156\pi$$
−0.613293 + 0.789855i $$0.710156\pi$$
$$44$$ −6144.00 −0.478431
$$45$$ 0 0
$$46$$ 15360.0 1.07028
$$47$$ 19200.0 1.26782 0.633909 0.773408i $$-0.281450\pi$$
0.633909 + 0.773408i $$0.281450\pi$$
$$48$$ 0 0
$$49$$ 5097.00 0.303266
$$50$$ 24364.0 1.37824
$$51$$ 0 0
$$52$$ −5344.00 −0.274068
$$53$$ −7776.00 −0.380248 −0.190124 0.981760i $$-0.560889\pi$$
−0.190124 + 0.981760i $$0.560889\pi$$
$$54$$ 0 0
$$55$$ −36864.0 −1.64322
$$56$$ −9472.00 −0.403619
$$57$$ 0 0
$$58$$ −384.000 −0.0149886
$$59$$ 13056.0 0.488293 0.244146 0.969738i $$-0.421492\pi$$
0.244146 + 0.969738i $$0.421492\pi$$
$$60$$ 0 0
$$61$$ 42782.0 1.47210 0.736049 0.676929i $$-0.236689\pi$$
0.736049 + 0.676929i $$0.236689\pi$$
$$62$$ −18256.0 −0.603151
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −32064.0 −0.941314
$$66$$ 0 0
$$67$$ 36656.0 0.997604 0.498802 0.866716i $$-0.333774\pi$$
0.498802 + 0.866716i $$0.333774\pi$$
$$68$$ −9216.00 −0.241696
$$69$$ 0 0
$$70$$ −56832.0 −1.38627
$$71$$ −64512.0 −1.51878 −0.759390 0.650636i $$-0.774502\pi$$
−0.759390 + 0.650636i $$0.774502\pi$$
$$72$$ 0 0
$$73$$ −16810.0 −0.369199 −0.184600 0.982814i $$-0.559099\pi$$
−0.184600 + 0.982814i $$0.559099\pi$$
$$74$$ 23192.0 0.492333
$$75$$ 0 0
$$76$$ −10624.0 −0.210986
$$77$$ 56832.0 1.09236
$$78$$ 0 0
$$79$$ 28076.0 0.506136 0.253068 0.967448i $$-0.418560\pi$$
0.253068 + 0.967448i $$0.418560\pi$$
$$80$$ 24576.0 0.429325
$$81$$ 0 0
$$82$$ 26880.0 0.441463
$$83$$ 66432.0 1.05848 0.529239 0.848473i $$-0.322477\pi$$
0.529239 + 0.848473i $$0.322477\pi$$
$$84$$ 0 0
$$85$$ −55296.0 −0.830131
$$86$$ −59488.0 −0.867328
$$87$$ 0 0
$$88$$ −24576.0 −0.338302
$$89$$ 81792.0 1.09455 0.547275 0.836953i $$-0.315665\pi$$
0.547275 + 0.836953i $$0.315665\pi$$
$$90$$ 0 0
$$91$$ 49432.0 0.625756
$$92$$ 61440.0 0.756801
$$93$$ 0 0
$$94$$ 76800.0 0.896482
$$95$$ −63744.0 −0.724653
$$96$$ 0 0
$$97$$ −29938.0 −0.323068 −0.161534 0.986867i $$-0.551644\pi$$
−0.161534 + 0.986867i $$0.551644\pi$$
$$98$$ 20388.0 0.214442
$$99$$ 0 0
$$100$$ 97456.0 0.974560
$$101$$ −178656. −1.74267 −0.871333 0.490692i $$-0.836744\pi$$
−0.871333 + 0.490692i $$0.836744\pi$$
$$102$$ 0 0
$$103$$ −115228. −1.07020 −0.535100 0.844789i $$-0.679726\pi$$
−0.535100 + 0.844789i $$0.679726\pi$$
$$104$$ −21376.0 −0.193795
$$105$$ 0 0
$$106$$ −31104.0 −0.268876
$$107$$ 76032.0 0.642003 0.321001 0.947079i $$-0.395981\pi$$
0.321001 + 0.947079i $$0.395981\pi$$
$$108$$ 0 0
$$109$$ −231118. −1.86323 −0.931617 0.363441i $$-0.881602\pi$$
−0.931617 + 0.363441i $$0.881602\pi$$
$$110$$ −147456. −1.16193
$$111$$ 0 0
$$112$$ −37888.0 −0.285402
$$113$$ −142464. −1.04956 −0.524782 0.851237i $$-0.675853\pi$$
−0.524782 + 0.851237i $$0.675853\pi$$
$$114$$ 0 0
$$115$$ 368640. 2.59931
$$116$$ −1536.00 −0.0105985
$$117$$ 0 0
$$118$$ 52224.0 0.345275
$$119$$ 85248.0 0.551845
$$120$$ 0 0
$$121$$ −13595.0 −0.0844143
$$122$$ 171128. 1.04093
$$123$$ 0 0
$$124$$ −73024.0 −0.426493
$$125$$ 284736. 1.62992
$$126$$ 0 0
$$127$$ −988.000 −0.00543560 −0.00271780 0.999996i $$-0.500865\pi$$
−0.00271780 + 0.999996i $$0.500865\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ −128256. −0.665609
$$131$$ −224256. −1.14174 −0.570868 0.821042i $$-0.693393\pi$$
−0.570868 + 0.821042i $$0.693393\pi$$
$$132$$ 0 0
$$133$$ 98272.0 0.481727
$$134$$ 146624. 0.705412
$$135$$ 0 0
$$136$$ −36864.0 −0.170905
$$137$$ 278976. 1.26989 0.634944 0.772558i $$-0.281023\pi$$
0.634944 + 0.772558i $$0.281023\pi$$
$$138$$ 0 0
$$139$$ 177200. 0.777905 0.388953 0.921258i $$-0.372837\pi$$
0.388953 + 0.921258i $$0.372837\pi$$
$$140$$ −227328. −0.980241
$$141$$ 0 0
$$142$$ −258048. −1.07394
$$143$$ 128256. 0.524490
$$144$$ 0 0
$$145$$ −9216.00 −0.0364018
$$146$$ −67240.0 −0.261063
$$147$$ 0 0
$$148$$ 92768.0 0.348132
$$149$$ −236064. −0.871092 −0.435546 0.900166i $$-0.643445\pi$$
−0.435546 + 0.900166i $$0.643445\pi$$
$$150$$ 0 0
$$151$$ −482836. −1.72329 −0.861643 0.507515i $$-0.830564\pi$$
−0.861643 + 0.507515i $$0.830564\pi$$
$$152$$ −42496.0 −0.149190
$$153$$ 0 0
$$154$$ 227328. 0.772416
$$155$$ −438144. −1.46483
$$156$$ 0 0
$$157$$ 381086. 1.23388 0.616941 0.787009i $$-0.288372\pi$$
0.616941 + 0.787009i $$0.288372\pi$$
$$158$$ 112304. 0.357892
$$159$$ 0 0
$$160$$ 98304.0 0.303579
$$161$$ −568320. −1.72794
$$162$$ 0 0
$$163$$ 162920. 0.480292 0.240146 0.970737i $$-0.422805\pi$$
0.240146 + 0.970737i $$0.422805\pi$$
$$164$$ 107520. 0.312162
$$165$$ 0 0
$$166$$ 265728. 0.748457
$$167$$ 566016. 1.57050 0.785249 0.619180i $$-0.212535\pi$$
0.785249 + 0.619180i $$0.212535\pi$$
$$168$$ 0 0
$$169$$ −259737. −0.699547
$$170$$ −221184. −0.586991
$$171$$ 0 0
$$172$$ −237952. −0.613293
$$173$$ 218208. 0.554313 0.277157 0.960825i $$-0.410608\pi$$
0.277157 + 0.960825i $$0.410608\pi$$
$$174$$ 0 0
$$175$$ −901468. −2.22513
$$176$$ −98304.0 −0.239216
$$177$$ 0 0
$$178$$ 327168. 0.773964
$$179$$ 412416. 0.962062 0.481031 0.876704i $$-0.340262\pi$$
0.481031 + 0.876704i $$0.340262\pi$$
$$180$$ 0 0
$$181$$ −25558.0 −0.0579870 −0.0289935 0.999580i $$-0.509230\pi$$
−0.0289935 + 0.999580i $$0.509230\pi$$
$$182$$ 197728. 0.442476
$$183$$ 0 0
$$184$$ 245760. 0.535139
$$185$$ 556608. 1.19569
$$186$$ 0 0
$$187$$ 221184. 0.462540
$$188$$ 307200. 0.633909
$$189$$ 0 0
$$190$$ −254976. −0.512407
$$191$$ −400128. −0.793625 −0.396813 0.917900i $$-0.629884\pi$$
−0.396813 + 0.917900i $$0.629884\pi$$
$$192$$ 0 0
$$193$$ 699650. 1.35203 0.676017 0.736886i $$-0.263705\pi$$
0.676017 + 0.736886i $$0.263705\pi$$
$$194$$ −119752. −0.228443
$$195$$ 0 0
$$196$$ 81552.0 0.151633
$$197$$ −406368. −0.746026 −0.373013 0.927826i $$-0.621675\pi$$
−0.373013 + 0.927826i $$0.621675\pi$$
$$198$$ 0 0
$$199$$ −361996. −0.647994 −0.323997 0.946058i $$-0.605027\pi$$
−0.323997 + 0.946058i $$0.605027\pi$$
$$200$$ 389824. 0.689118
$$201$$ 0 0
$$202$$ −714624. −1.23225
$$203$$ 14208.0 0.0241987
$$204$$ 0 0
$$205$$ 645120. 1.07215
$$206$$ −460912. −0.756746
$$207$$ 0 0
$$208$$ −85504.0 −0.137034
$$209$$ 254976. 0.403770
$$210$$ 0 0
$$211$$ 151856. 0.234815 0.117407 0.993084i $$-0.462542\pi$$
0.117407 + 0.993084i $$0.462542\pi$$
$$212$$ −124416. −0.190124
$$213$$ 0 0
$$214$$ 304128. 0.453965
$$215$$ −1.42771e6 −2.10642
$$216$$ 0 0
$$217$$ 675472. 0.973774
$$218$$ −924472. −1.31751
$$219$$ 0 0
$$220$$ −589824. −0.821610
$$221$$ 192384. 0.264965
$$222$$ 0 0
$$223$$ −1.09332e6 −1.47227 −0.736134 0.676836i $$-0.763351\pi$$
−0.736134 + 0.676836i $$0.763351\pi$$
$$224$$ −151552. −0.201810
$$225$$ 0 0
$$226$$ −569856. −0.742154
$$227$$ 566400. 0.729556 0.364778 0.931095i $$-0.381145\pi$$
0.364778 + 0.931095i $$0.381145\pi$$
$$228$$ 0 0
$$229$$ −587206. −0.739949 −0.369974 0.929042i $$-0.620634\pi$$
−0.369974 + 0.929042i $$0.620634\pi$$
$$230$$ 1.47456e6 1.83799
$$231$$ 0 0
$$232$$ −6144.00 −0.00749430
$$233$$ −579456. −0.699247 −0.349624 0.936890i $$-0.613691\pi$$
−0.349624 + 0.936890i $$0.613691\pi$$
$$234$$ 0 0
$$235$$ 1.84320e6 2.17722
$$236$$ 208896. 0.244146
$$237$$ 0 0
$$238$$ 340992. 0.390213
$$239$$ 584448. 0.661837 0.330919 0.943659i $$-0.392641\pi$$
0.330919 + 0.943659i $$0.392641\pi$$
$$240$$ 0 0
$$241$$ −414130. −0.459298 −0.229649 0.973274i $$-0.573758\pi$$
−0.229649 + 0.973274i $$0.573758\pi$$
$$242$$ −54380.0 −0.0596899
$$243$$ 0 0
$$244$$ 684512. 0.736049
$$245$$ 489312. 0.520800
$$246$$ 0 0
$$247$$ 221776. 0.231298
$$248$$ −292096. −0.301576
$$249$$ 0 0
$$250$$ 1.13894e6 1.15253
$$251$$ 1.89965e6 1.90322 0.951610 0.307309i $$-0.0994287\pi$$
0.951610 + 0.307309i $$0.0994287\pi$$
$$252$$ 0 0
$$253$$ −1.47456e6 −1.44831
$$254$$ −3952.00 −0.00384355
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −447744. −0.422860 −0.211430 0.977393i $$-0.567812\pi$$
−0.211430 + 0.977393i $$0.567812\pi$$
$$258$$ 0 0
$$259$$ −858104. −0.794860
$$260$$ −513024. −0.470657
$$261$$ 0 0
$$262$$ −897024. −0.807330
$$263$$ 67584.0 0.0602496 0.0301248 0.999546i $$-0.490410\pi$$
0.0301248 + 0.999546i $$0.490410\pi$$
$$264$$ 0 0
$$265$$ −746496. −0.652999
$$266$$ 393088. 0.340632
$$267$$ 0 0
$$268$$ 586496. 0.498802
$$269$$ −564192. −0.475386 −0.237693 0.971340i $$-0.576391\pi$$
−0.237693 + 0.971340i $$0.576391\pi$$
$$270$$ 0 0
$$271$$ 720308. 0.595792 0.297896 0.954598i $$-0.403715\pi$$
0.297896 + 0.954598i $$0.403715\pi$$
$$272$$ −147456. −0.120848
$$273$$ 0 0
$$274$$ 1.11590e6 0.897946
$$275$$ −2.33894e6 −1.86504
$$276$$ 0 0
$$277$$ −141142. −0.110524 −0.0552620 0.998472i $$-0.517599\pi$$
−0.0552620 + 0.998472i $$0.517599\pi$$
$$278$$ 708800. 0.550062
$$279$$ 0 0
$$280$$ −909312. −0.693135
$$281$$ −584448. −0.441550 −0.220775 0.975325i $$-0.570859\pi$$
−0.220775 + 0.975325i $$0.570859\pi$$
$$282$$ 0 0
$$283$$ 177056. 0.131415 0.0657074 0.997839i $$-0.479070\pi$$
0.0657074 + 0.997839i $$0.479070\pi$$
$$284$$ −1.03219e6 −0.759390
$$285$$ 0 0
$$286$$ 513024. 0.370871
$$287$$ −994560. −0.712732
$$288$$ 0 0
$$289$$ −1.08808e6 −0.766331
$$290$$ −36864.0 −0.0257399
$$291$$ 0 0
$$292$$ −268960. −0.184600
$$293$$ 956832. 0.651128 0.325564 0.945520i $$-0.394446\pi$$
0.325564 + 0.945520i $$0.394446\pi$$
$$294$$ 0 0
$$295$$ 1.25338e6 0.838545
$$296$$ 371072. 0.246166
$$297$$ 0 0
$$298$$ −944256. −0.615955
$$299$$ −1.28256e6 −0.829659
$$300$$ 0 0
$$301$$ 2.20106e6 1.40028
$$302$$ −1.93134e6 −1.21855
$$303$$ 0 0
$$304$$ −169984. −0.105493
$$305$$ 4.10707e6 2.52803
$$306$$ 0 0
$$307$$ 2.88286e6 1.74573 0.872867 0.487958i $$-0.162258\pi$$
0.872867 + 0.487958i $$0.162258\pi$$
$$308$$ 909312. 0.546180
$$309$$ 0 0
$$310$$ −1.75258e6 −1.03579
$$311$$ 2.60045e6 1.52457 0.762285 0.647242i $$-0.224078\pi$$
0.762285 + 0.647242i $$0.224078\pi$$
$$312$$ 0 0
$$313$$ −2.58079e6 −1.48899 −0.744495 0.667628i $$-0.767310\pi$$
−0.744495 + 0.667628i $$0.767310\pi$$
$$314$$ 1.52434e6 0.872487
$$315$$ 0 0
$$316$$ 449216. 0.253068
$$317$$ −2.31101e6 −1.29168 −0.645838 0.763475i $$-0.723492\pi$$
−0.645838 + 0.763475i $$0.723492\pi$$
$$318$$ 0 0
$$319$$ 36864.0 0.0202827
$$320$$ 393216. 0.214663
$$321$$ 0 0
$$322$$ −2.27328e6 −1.22184
$$323$$ 382464. 0.203978
$$324$$ 0 0
$$325$$ −2.03439e6 −1.06838
$$326$$ 651680. 0.339618
$$327$$ 0 0
$$328$$ 430080. 0.220732
$$329$$ −2.84160e6 −1.44735
$$330$$ 0 0
$$331$$ −637024. −0.319585 −0.159792 0.987151i $$-0.551082\pi$$
−0.159792 + 0.987151i $$0.551082\pi$$
$$332$$ 1.06291e6 0.529239
$$333$$ 0 0
$$334$$ 2.26406e6 1.11051
$$335$$ 3.51898e6 1.71319
$$336$$ 0 0
$$337$$ 3.38665e6 1.62441 0.812206 0.583371i $$-0.198267\pi$$
0.812206 + 0.583371i $$0.198267\pi$$
$$338$$ −1.03895e6 −0.494655
$$339$$ 0 0
$$340$$ −884736. −0.415065
$$341$$ 1.75258e6 0.816189
$$342$$ 0 0
$$343$$ 1.73308e6 0.795396
$$344$$ −951808. −0.433664
$$345$$ 0 0
$$346$$ 872832. 0.391959
$$347$$ −2.77824e6 −1.23864 −0.619321 0.785138i $$-0.712592\pi$$
−0.619321 + 0.785138i $$0.712592\pi$$
$$348$$ 0 0
$$349$$ 1.55536e6 0.683545 0.341772 0.939783i $$-0.388973\pi$$
0.341772 + 0.939783i $$0.388973\pi$$
$$350$$ −3.60587e6 −1.57340
$$351$$ 0 0
$$352$$ −393216. −0.169151
$$353$$ −2.11776e6 −0.904565 −0.452283 0.891875i $$-0.649390\pi$$
−0.452283 + 0.891875i $$0.649390\pi$$
$$354$$ 0 0
$$355$$ −6.19315e6 −2.60820
$$356$$ 1.30867e6 0.547275
$$357$$ 0 0
$$358$$ 1.64966e6 0.680280
$$359$$ −2.17498e6 −0.890673 −0.445337 0.895363i $$-0.646916\pi$$
−0.445337 + 0.895363i $$0.646916\pi$$
$$360$$ 0 0
$$361$$ −2.03520e6 −0.821939
$$362$$ −102232. −0.0410030
$$363$$ 0 0
$$364$$ 790912. 0.312878
$$365$$ −1.61376e6 −0.634026
$$366$$ 0 0
$$367$$ 1.05336e6 0.408235 0.204117 0.978946i $$-0.434568\pi$$
0.204117 + 0.978946i $$0.434568\pi$$
$$368$$ 983040. 0.378400
$$369$$ 0 0
$$370$$ 2.22643e6 0.845483
$$371$$ 1.15085e6 0.434093
$$372$$ 0 0
$$373$$ −677098. −0.251988 −0.125994 0.992031i $$-0.540212\pi$$
−0.125994 + 0.992031i $$0.540212\pi$$
$$374$$ 884736. 0.327065
$$375$$ 0 0
$$376$$ 1.22880e6 0.448241
$$377$$ 32064.0 0.0116189
$$378$$ 0 0
$$379$$ −5.10748e6 −1.82645 −0.913227 0.407452i $$-0.866418\pi$$
−0.913227 + 0.407452i $$0.866418\pi$$
$$380$$ −1.01990e6 −0.362327
$$381$$ 0 0
$$382$$ −1.60051e6 −0.561178
$$383$$ −1.63200e6 −0.568491 −0.284245 0.958752i $$-0.591743\pi$$
−0.284245 + 0.958752i $$0.591743\pi$$
$$384$$ 0 0
$$385$$ 5.45587e6 1.87591
$$386$$ 2.79860e6 0.956032
$$387$$ 0 0
$$388$$ −479008. −0.161534
$$389$$ 4.46563e6 1.49627 0.748133 0.663549i $$-0.230950\pi$$
0.748133 + 0.663549i $$0.230950\pi$$
$$390$$ 0 0
$$391$$ −2.21184e6 −0.731664
$$392$$ 326208. 0.107221
$$393$$ 0 0
$$394$$ −1.62547e6 −0.527520
$$395$$ 2.69530e6 0.869188
$$396$$ 0 0
$$397$$ −611026. −0.194573 −0.0972867 0.995256i $$-0.531016\pi$$
−0.0972867 + 0.995256i $$0.531016\pi$$
$$398$$ −1.44798e6 −0.458201
$$399$$ 0 0
$$400$$ 1.55930e6 0.487280
$$401$$ 6.09158e6 1.89177 0.945887 0.324496i $$-0.105195\pi$$
0.945887 + 0.324496i $$0.105195\pi$$
$$402$$ 0 0
$$403$$ 1.52438e6 0.467552
$$404$$ −2.85850e6 −0.871333
$$405$$ 0 0
$$406$$ 56832.0 0.0171111
$$407$$ −2.22643e6 −0.666229
$$408$$ 0 0
$$409$$ 2.89108e6 0.854578 0.427289 0.904115i $$-0.359469\pi$$
0.427289 + 0.904115i $$0.359469\pi$$
$$410$$ 2.58048e6 0.758125
$$411$$ 0 0
$$412$$ −1.84365e6 −0.535100
$$413$$ −1.93229e6 −0.557438
$$414$$ 0 0
$$415$$ 6.37747e6 1.81773
$$416$$ −342016. −0.0968976
$$417$$ 0 0
$$418$$ 1.01990e6 0.285508
$$419$$ 2.98406e6 0.830373 0.415186 0.909736i $$-0.363716\pi$$
0.415186 + 0.909736i $$0.363716\pi$$
$$420$$ 0 0
$$421$$ 822074. 0.226051 0.113025 0.993592i $$-0.463946\pi$$
0.113025 + 0.993592i $$0.463946\pi$$
$$422$$ 607424. 0.166039
$$423$$ 0 0
$$424$$ −497664. −0.134438
$$425$$ −3.50842e6 −0.942191
$$426$$ 0 0
$$427$$ −6.33174e6 −1.68056
$$428$$ 1.21651e6 0.321001
$$429$$ 0 0
$$430$$ −5.71085e6 −1.48946
$$431$$ −6.32448e6 −1.63995 −0.819977 0.572397i $$-0.806014\pi$$
−0.819977 + 0.572397i $$0.806014\pi$$
$$432$$ 0 0
$$433$$ −851902. −0.218358 −0.109179 0.994022i $$-0.534822\pi$$
−0.109179 + 0.994022i $$0.534822\pi$$
$$434$$ 2.70189e6 0.688562
$$435$$ 0 0
$$436$$ −3.69789e6 −0.931617
$$437$$ −2.54976e6 −0.638698
$$438$$ 0 0
$$439$$ −334732. −0.0828964 −0.0414482 0.999141i $$-0.513197\pi$$
−0.0414482 + 0.999141i $$0.513197\pi$$
$$440$$ −2.35930e6 −0.580966
$$441$$ 0 0
$$442$$ 769536. 0.187358
$$443$$ 1.76218e6 0.426619 0.213309 0.976985i $$-0.431576\pi$$
0.213309 + 0.976985i $$0.431576\pi$$
$$444$$ 0 0
$$445$$ 7.85203e6 1.87967
$$446$$ −4.37330e6 −1.04105
$$447$$ 0 0
$$448$$ −606208. −0.142701
$$449$$ −6.51859e6 −1.52594 −0.762971 0.646433i $$-0.776260\pi$$
−0.762971 + 0.646433i $$0.776260\pi$$
$$450$$ 0 0
$$451$$ −2.58048e6 −0.597392
$$452$$ −2.27942e6 −0.524782
$$453$$ 0 0
$$454$$ 2.26560e6 0.515874
$$455$$ 4.74547e6 1.07461
$$456$$ 0 0
$$457$$ −92074.0 −0.0206227 −0.0103114 0.999947i $$-0.503282\pi$$
−0.0103114 + 0.999947i $$0.503282\pi$$
$$458$$ −2.34882e6 −0.523223
$$459$$ 0 0
$$460$$ 5.89824e6 1.29965
$$461$$ −257568. −0.0564468 −0.0282234 0.999602i $$-0.508985\pi$$
−0.0282234 + 0.999602i $$0.508985\pi$$
$$462$$ 0 0
$$463$$ 3.96228e6 0.858998 0.429499 0.903067i $$-0.358690\pi$$
0.429499 + 0.903067i $$0.358690\pi$$
$$464$$ −24576.0 −0.00529927
$$465$$ 0 0
$$466$$ −2.31782e6 −0.494442
$$467$$ 3.48941e6 0.740388 0.370194 0.928954i $$-0.379291\pi$$
0.370194 + 0.928954i $$0.379291\pi$$
$$468$$ 0 0
$$469$$ −5.42509e6 −1.13887
$$470$$ 7.37280e6 1.53953
$$471$$ 0 0
$$472$$ 835584. 0.172637
$$473$$ 5.71085e6 1.17367
$$474$$ 0 0
$$475$$ −4.04442e6 −0.822475
$$476$$ 1.36397e6 0.275922
$$477$$ 0 0
$$478$$ 2.33779e6 0.467990
$$479$$ −513024. −0.102164 −0.0510821 0.998694i $$-0.516267\pi$$
−0.0510821 + 0.998694i $$0.516267\pi$$
$$480$$ 0 0
$$481$$ −1.93653e6 −0.381647
$$482$$ −1.65652e6 −0.324772
$$483$$ 0 0
$$484$$ −217520. −0.0422071
$$485$$ −2.87405e6 −0.554804
$$486$$ 0 0
$$487$$ 4.14499e6 0.791956 0.395978 0.918260i $$-0.370406\pi$$
0.395978 + 0.918260i $$0.370406\pi$$
$$488$$ 2.73805e6 0.520465
$$489$$ 0 0
$$490$$ 1.95725e6 0.368261
$$491$$ −2.75866e6 −0.516409 −0.258205 0.966090i $$-0.583131\pi$$
−0.258205 + 0.966090i $$0.583131\pi$$
$$492$$ 0 0
$$493$$ 55296.0 0.0102465
$$494$$ 887104. 0.163552
$$495$$ 0 0
$$496$$ −1.16838e6 −0.213246
$$497$$ 9.54778e6 1.73385
$$498$$ 0 0
$$499$$ 660896. 0.118818 0.0594089 0.998234i $$-0.481078\pi$$
0.0594089 + 0.998234i $$0.481078\pi$$
$$500$$ 4.55578e6 0.814962
$$501$$ 0 0
$$502$$ 7.59859e6 1.34578
$$503$$ 944640. 0.166474 0.0832370 0.996530i $$-0.473474\pi$$
0.0832370 + 0.996530i $$0.473474\pi$$
$$504$$ 0 0
$$505$$ −1.71510e7 −2.99268
$$506$$ −5.89824e6 −1.02411
$$507$$ 0 0
$$508$$ −15808.0 −0.00271780
$$509$$ 7.83773e6 1.34090 0.670449 0.741956i $$-0.266102\pi$$
0.670449 + 0.741956i $$0.266102\pi$$
$$510$$ 0 0
$$511$$ 2.48788e6 0.421480
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ −1.79098e6 −0.299007
$$515$$ −1.10619e7 −1.83785
$$516$$ 0 0
$$517$$ −7.37280e6 −1.21313
$$518$$ −3.43242e6 −0.562051
$$519$$ 0 0
$$520$$ −2.05210e6 −0.332805
$$521$$ −3.29645e6 −0.532049 −0.266025 0.963966i $$-0.585710\pi$$
−0.266025 + 0.963966i $$0.585710\pi$$
$$522$$ 0 0
$$523$$ 6.50238e6 1.03948 0.519742 0.854323i $$-0.326028\pi$$
0.519742 + 0.854323i $$0.326028\pi$$
$$524$$ −3.58810e6 −0.570868
$$525$$ 0 0
$$526$$ 270336. 0.0426029
$$527$$ 2.62886e6 0.412327
$$528$$ 0 0
$$529$$ 8.30926e6 1.29099
$$530$$ −2.98598e6 −0.461740
$$531$$ 0 0
$$532$$ 1.57235e6 0.240863
$$533$$ −2.24448e6 −0.342214
$$534$$ 0 0
$$535$$ 7.29907e6 1.10251
$$536$$ 2.34598e6 0.352706
$$537$$ 0 0
$$538$$ −2.25677e6 −0.336149
$$539$$ −1.95725e6 −0.290184
$$540$$ 0 0
$$541$$ 9.82714e6 1.44356 0.721778 0.692124i $$-0.243325\pi$$
0.721778 + 0.692124i $$0.243325\pi$$
$$542$$ 2.88123e6 0.421289
$$543$$ 0 0
$$544$$ −589824. −0.0854526
$$545$$ −2.21873e7 −3.19973
$$546$$ 0 0
$$547$$ −3.42580e6 −0.489546 −0.244773 0.969580i $$-0.578713\pi$$
−0.244773 + 0.969580i $$0.578713\pi$$
$$548$$ 4.46362e6 0.634944
$$549$$ 0 0
$$550$$ −9.35578e6 −1.31878
$$551$$ 63744.0 0.00894459
$$552$$ 0 0
$$553$$ −4.15525e6 −0.577809
$$554$$ −564568. −0.0781523
$$555$$ 0 0
$$556$$ 2.83520e6 0.388953
$$557$$ 6.43363e6 0.878655 0.439327 0.898327i $$-0.355217\pi$$
0.439327 + 0.898327i $$0.355217\pi$$
$$558$$ 0 0
$$559$$ 4.96725e6 0.672336
$$560$$ −3.63725e6 −0.490120
$$561$$ 0 0
$$562$$ −2.33779e6 −0.312223
$$563$$ −753024. −0.100124 −0.0500620 0.998746i $$-0.515942\pi$$
−0.0500620 + 0.998746i $$0.515942\pi$$
$$564$$ 0 0
$$565$$ −1.36765e7 −1.80242
$$566$$ 708224. 0.0929244
$$567$$ 0 0
$$568$$ −4.12877e6 −0.536970
$$569$$ 1.11481e7 1.44351 0.721755 0.692148i $$-0.243336\pi$$
0.721755 + 0.692148i $$0.243336\pi$$
$$570$$ 0 0
$$571$$ 191024. 0.0245187 0.0122594 0.999925i $$-0.496098\pi$$
0.0122594 + 0.999925i $$0.496098\pi$$
$$572$$ 2.05210e6 0.262245
$$573$$ 0 0
$$574$$ −3.97824e6 −0.503978
$$575$$ 2.33894e7 2.95019
$$576$$ 0 0
$$577$$ 1.03722e7 1.29697 0.648486 0.761227i $$-0.275403\pi$$
0.648486 + 0.761227i $$0.275403\pi$$
$$578$$ −4.35232e6 −0.541878
$$579$$ 0 0
$$580$$ −147456. −0.0182009
$$581$$ −9.83194e6 −1.20837
$$582$$ 0 0
$$583$$ 2.98598e6 0.363845
$$584$$ −1.07584e6 −0.130532
$$585$$ 0 0
$$586$$ 3.82733e6 0.460417
$$587$$ 2.97062e6 0.355838 0.177919 0.984045i $$-0.443063\pi$$
0.177919 + 0.984045i $$0.443063\pi$$
$$588$$ 0 0
$$589$$ 3.03050e6 0.359936
$$590$$ 5.01350e6 0.592941
$$591$$ 0 0
$$592$$ 1.48429e6 0.174066
$$593$$ 7.55827e6 0.882644 0.441322 0.897349i $$-0.354510\pi$$
0.441322 + 0.897349i $$0.354510\pi$$
$$594$$ 0 0
$$595$$ 8.18381e6 0.947683
$$596$$ −3.77702e6 −0.435546
$$597$$ 0 0
$$598$$ −5.13024e6 −0.586658
$$599$$ −6.69158e6 −0.762012 −0.381006 0.924573i $$-0.624422\pi$$
−0.381006 + 0.924573i $$0.624422\pi$$
$$600$$ 0 0
$$601$$ −3.20359e6 −0.361785 −0.180893 0.983503i $$-0.557899\pi$$
−0.180893 + 0.983503i $$0.557899\pi$$
$$602$$ 8.80422e6 0.990147
$$603$$ 0 0
$$604$$ −7.72538e6 −0.861643
$$605$$ −1.30512e6 −0.144965
$$606$$ 0 0
$$607$$ −1.35585e7 −1.49362 −0.746809 0.665038i $$-0.768415\pi$$
−0.746809 + 0.665038i $$0.768415\pi$$
$$608$$ −679936. −0.0745949
$$609$$ 0 0
$$610$$ 1.64283e7 1.78759
$$611$$ −6.41280e6 −0.694936
$$612$$ 0 0
$$613$$ −1.07654e7 −1.15712 −0.578561 0.815639i $$-0.696385\pi$$
−0.578561 + 0.815639i $$0.696385\pi$$
$$614$$ 1.15315e7 1.23442
$$615$$ 0 0
$$616$$ 3.63725e6 0.386208
$$617$$ 9.33504e6 0.987196 0.493598 0.869690i $$-0.335681\pi$$
0.493598 + 0.869690i $$0.335681\pi$$
$$618$$ 0 0
$$619$$ 9.07664e6 0.952135 0.476067 0.879409i $$-0.342062\pi$$
0.476067 + 0.879409i $$0.342062\pi$$
$$620$$ −7.01030e6 −0.732416
$$621$$ 0 0
$$622$$ 1.04018e7 1.07803
$$623$$ −1.21052e7 −1.24955
$$624$$ 0 0
$$625$$ 8.30028e6 0.849949
$$626$$ −1.03232e7 −1.05288
$$627$$ 0 0
$$628$$ 6.09738e6 0.616941
$$629$$ −3.33965e6 −0.336569
$$630$$ 0 0
$$631$$ −1.13367e7 −1.13348 −0.566741 0.823896i $$-0.691796\pi$$
−0.566741 + 0.823896i $$0.691796\pi$$
$$632$$ 1.79686e6 0.178946
$$633$$ 0 0
$$634$$ −9.24403e6 −0.913352
$$635$$ −94848.0 −0.00933456
$$636$$ 0 0
$$637$$ −1.70240e6 −0.166231
$$638$$ 147456. 0.0143420
$$639$$ 0 0
$$640$$ 1.57286e6 0.151789
$$641$$ −1.55449e7 −1.49432 −0.747159 0.664646i $$-0.768582\pi$$
−0.747159 + 0.664646i $$0.768582\pi$$
$$642$$ 0 0
$$643$$ −8.80026e6 −0.839399 −0.419699 0.907663i $$-0.637864\pi$$
−0.419699 + 0.907663i $$0.637864\pi$$
$$644$$ −9.09312e6 −0.863969
$$645$$ 0 0
$$646$$ 1.52986e6 0.144235
$$647$$ −1.08449e7 −1.01851 −0.509256 0.860615i $$-0.670079\pi$$
−0.509256 + 0.860615i $$0.670079\pi$$
$$648$$ 0 0
$$649$$ −5.01350e6 −0.467229
$$650$$ −8.13758e6 −0.755460
$$651$$ 0 0
$$652$$ 2.60672e6 0.240146
$$653$$ 9.88771e6 0.907429 0.453715 0.891147i $$-0.350099\pi$$
0.453715 + 0.891147i $$0.350099\pi$$
$$654$$ 0 0
$$655$$ −2.15286e7 −1.96070
$$656$$ 1.72032e6 0.156081
$$657$$ 0 0
$$658$$ −1.13664e7 −1.02343
$$659$$ 1.46150e7 1.31095 0.655476 0.755216i $$-0.272468\pi$$
0.655476 + 0.755216i $$0.272468\pi$$
$$660$$ 0 0
$$661$$ 1.57792e7 1.40469 0.702347 0.711834i $$-0.252135\pi$$
0.702347 + 0.711834i $$0.252135\pi$$
$$662$$ −2.54810e6 −0.225980
$$663$$ 0 0
$$664$$ 4.25165e6 0.374229
$$665$$ 9.43411e6 0.827269
$$666$$ 0 0
$$667$$ −368640. −0.0320840
$$668$$ 9.05626e6 0.785249
$$669$$ 0 0
$$670$$ 1.40759e7 1.21140
$$671$$ −1.64283e7 −1.40859
$$672$$ 0 0
$$673$$ 6.64939e6 0.565906 0.282953 0.959134i $$-0.408686\pi$$
0.282953 + 0.959134i $$0.408686\pi$$
$$674$$ 1.35466e7 1.14863
$$675$$ 0 0
$$676$$ −4.15579e6 −0.349774
$$677$$ −4.42550e6 −0.371100 −0.185550 0.982635i $$-0.559407\pi$$
−0.185550 + 0.982635i $$0.559407\pi$$
$$678$$ 0 0
$$679$$ 4.43082e6 0.368816
$$680$$ −3.53894e6 −0.293495
$$681$$ 0 0
$$682$$ 7.01030e6 0.577133
$$683$$ 4.67827e6 0.383737 0.191869 0.981421i $$-0.438545\pi$$
0.191869 + 0.981421i $$0.438545\pi$$
$$684$$ 0 0
$$685$$ 2.67817e7 2.18078
$$686$$ 6.93232e6 0.562430
$$687$$ 0 0
$$688$$ −3.80723e6 −0.306647
$$689$$ 2.59718e6 0.208427
$$690$$ 0 0
$$691$$ −5.54102e6 −0.441463 −0.220731 0.975335i $$-0.570844\pi$$
−0.220731 + 0.975335i $$0.570844\pi$$
$$692$$ 3.49133e6 0.277157
$$693$$ 0 0
$$694$$ −1.11130e7 −0.875853
$$695$$ 1.70112e7 1.33590
$$696$$ 0 0
$$697$$ −3.87072e6 −0.301793
$$698$$ 6.22143e6 0.483339
$$699$$ 0 0
$$700$$ −1.44235e7 −1.11256
$$701$$ 6.53443e6 0.502242 0.251121 0.967956i $$-0.419201\pi$$
0.251121 + 0.967956i $$0.419201\pi$$
$$702$$ 0 0
$$703$$ −3.84987e6 −0.293804
$$704$$ −1.57286e6 −0.119608
$$705$$ 0 0
$$706$$ −8.47104e6 −0.639624
$$707$$ 2.64411e7 1.98944
$$708$$ 0 0
$$709$$ −3.86541e6 −0.288789 −0.144394 0.989520i $$-0.546123\pi$$
−0.144394 + 0.989520i $$0.546123\pi$$
$$710$$ −2.47726e7 −1.84428
$$711$$ 0 0
$$712$$ 5.23469e6 0.386982
$$713$$ −1.75258e7 −1.29108
$$714$$ 0 0
$$715$$ 1.23126e7 0.900708
$$716$$ 6.59866e6 0.481031
$$717$$ 0 0
$$718$$ −8.69990e6 −0.629801
$$719$$ −4.80614e6 −0.346717 −0.173358 0.984859i $$-0.555462\pi$$
−0.173358 + 0.984859i $$0.555462\pi$$
$$720$$ 0 0
$$721$$ 1.70537e7 1.22175
$$722$$ −8.14081e6 −0.581199
$$723$$ 0 0
$$724$$ −408928. −0.0289935
$$725$$ −584736. −0.0413157
$$726$$ 0 0
$$727$$ −1.90590e7 −1.33741 −0.668704 0.743529i $$-0.733150\pi$$
−0.668704 + 0.743529i $$0.733150\pi$$
$$728$$ 3.16365e6 0.221238
$$729$$ 0 0
$$730$$ −6.45504e6 −0.448324
$$731$$ 8.56627e6 0.592923
$$732$$ 0 0
$$733$$ 5.69616e6 0.391582 0.195791 0.980646i $$-0.437273\pi$$
0.195791 + 0.980646i $$0.437273\pi$$
$$734$$ 4.21342e6 0.288666
$$735$$ 0 0
$$736$$ 3.93216e6 0.267570
$$737$$ −1.40759e7 −0.954570
$$738$$ 0 0
$$739$$ 1.84902e7 1.24546 0.622730 0.782437i $$-0.286024\pi$$
0.622730 + 0.782437i $$0.286024\pi$$
$$740$$ 8.90573e6 0.597847
$$741$$ 0 0
$$742$$ 4.60339e6 0.306950
$$743$$ −9.90336e6 −0.658128 −0.329064 0.944308i $$-0.606733\pi$$
−0.329064 + 0.944308i $$0.606733\pi$$
$$744$$ 0 0
$$745$$ −2.26621e7 −1.49593
$$746$$ −2.70839e6 −0.178182
$$747$$ 0 0
$$748$$ 3.53894e6 0.231270
$$749$$ −1.12527e7 −0.732915
$$750$$ 0 0
$$751$$ −9.05914e6 −0.586121 −0.293060 0.956094i $$-0.594674\pi$$
−0.293060 + 0.956094i $$0.594674\pi$$
$$752$$ 4.91520e6 0.316954
$$753$$ 0 0
$$754$$ 128256. 0.00821579
$$755$$ −4.63523e7 −2.95940
$$756$$ 0 0
$$757$$ −1.16677e7 −0.740022 −0.370011 0.929027i $$-0.620646\pi$$
−0.370011 + 0.929027i $$0.620646\pi$$
$$758$$ −2.04299e7 −1.29150
$$759$$ 0 0
$$760$$ −4.07962e6 −0.256204
$$761$$ 1.27398e7 0.797444 0.398722 0.917072i $$-0.369454\pi$$
0.398722 + 0.917072i $$0.369454\pi$$
$$762$$ 0 0
$$763$$ 3.42055e7 2.12708
$$764$$ −6.40205e6 −0.396813
$$765$$ 0 0
$$766$$ −6.52800e6 −0.401983
$$767$$ −4.36070e6 −0.267651
$$768$$ 0 0
$$769$$ 1.06783e7 0.651156 0.325578 0.945515i $$-0.394441\pi$$
0.325578 + 0.945515i $$0.394441\pi$$
$$770$$ 2.18235e7 1.32647
$$771$$ 0 0
$$772$$ 1.11944e7 0.676017
$$773$$ 9.18634e6 0.552960 0.276480 0.961020i $$-0.410832\pi$$
0.276480 + 0.961020i $$0.410832\pi$$
$$774$$ 0 0
$$775$$ −2.77993e7 −1.66257
$$776$$ −1.91603e6 −0.114222
$$777$$ 0 0
$$778$$ 1.78625e7 1.05802
$$779$$ −4.46208e6 −0.263447
$$780$$ 0 0
$$781$$ 2.47726e7 1.45326
$$782$$ −8.84736e6 −0.517365
$$783$$ 0 0
$$784$$ 1.30483e6 0.0758166
$$785$$ 3.65843e7 2.11895
$$786$$ 0 0
$$787$$ −9.28209e6 −0.534206 −0.267103 0.963668i $$-0.586066\pi$$
−0.267103 + 0.963668i $$0.586066\pi$$
$$788$$ −6.50189e6 −0.373013
$$789$$ 0 0
$$790$$ 1.07812e7 0.614609
$$791$$ 2.10847e7 1.19819
$$792$$ 0 0
$$793$$ −1.42892e7 −0.806909
$$794$$ −2.44410e6 −0.137584
$$795$$ 0 0
$$796$$ −5.79194e6 −0.323997
$$797$$ −21792.0 −0.00121521 −0.000607605 1.00000i $$-0.500193\pi$$
−0.000607605 1.00000i $$0.500193\pi$$
$$798$$ 0 0
$$799$$ −1.10592e7 −0.612854
$$800$$ 6.23718e6 0.344559
$$801$$ 0 0
$$802$$ 2.43663e7 1.33769
$$803$$ 6.45504e6 0.353273
$$804$$ 0 0
$$805$$ −5.45587e7 −2.96739
$$806$$ 6.09750e6 0.330609
$$807$$ 0 0
$$808$$ −1.14340e7 −0.616125
$$809$$ −1.23085e7 −0.661204 −0.330602 0.943770i $$-0.607252\pi$$
−0.330602 + 0.943770i $$0.607252\pi$$
$$810$$ 0 0
$$811$$ −2.34636e7 −1.25269 −0.626343 0.779547i $$-0.715449\pi$$
−0.626343 + 0.779547i $$0.715449\pi$$
$$812$$ 227328. 0.0120994
$$813$$ 0 0
$$814$$ −8.90573e6 −0.471095
$$815$$ 1.56403e7 0.824806
$$816$$ 0 0
$$817$$ 9.87501e6 0.517586
$$818$$ 1.15643e7 0.604278
$$819$$ 0 0
$$820$$ 1.03219e7 0.536075
$$821$$ 1.44206e7 0.746666 0.373333 0.927697i $$-0.378215\pi$$
0.373333 + 0.927697i $$0.378215\pi$$
$$822$$ 0 0
$$823$$ 3.43419e7 1.76736 0.883679 0.468093i $$-0.155059\pi$$
0.883679 + 0.468093i $$0.155059\pi$$
$$824$$ −7.37459e6 −0.378373
$$825$$ 0 0
$$826$$ −7.72915e6 −0.394168
$$827$$ −2.13327e7 −1.08463 −0.542316 0.840174i $$-0.682453\pi$$
−0.542316 + 0.840174i $$0.682453\pi$$
$$828$$ 0 0
$$829$$ 2.63751e6 0.133293 0.0666465 0.997777i $$-0.478770\pi$$
0.0666465 + 0.997777i $$0.478770\pi$$
$$830$$ 2.55099e7 1.28533
$$831$$ 0 0
$$832$$ −1.36806e6 −0.0685170
$$833$$ −2.93587e6 −0.146597
$$834$$ 0 0
$$835$$ 5.43375e7 2.69702
$$836$$ 4.07962e6 0.201885
$$837$$ 0 0
$$838$$ 1.19363e7 0.587162
$$839$$ 1.00577e7 0.493282 0.246641 0.969107i $$-0.420673\pi$$
0.246641 + 0.969107i $$0.420673\pi$$
$$840$$ 0 0
$$841$$ −2.05019e7 −0.999551
$$842$$ 3.28830e6 0.159842
$$843$$ 0 0
$$844$$ 2.42970e6 0.117407
$$845$$ −2.49348e7 −1.20133
$$846$$ 0 0
$$847$$ 2.01206e6 0.0963679
$$848$$ −1.99066e6 −0.0950619
$$849$$ 0 0
$$850$$ −1.40337e7 −0.666229
$$851$$ 2.22643e7 1.05387
$$852$$ 0 0
$$853$$ −2.30748e7 −1.08584 −0.542919 0.839785i $$-0.682681\pi$$
−0.542919 + 0.839785i $$0.682681\pi$$
$$854$$ −2.53269e7 −1.18833
$$855$$ 0 0
$$856$$ 4.86605e6 0.226982
$$857$$ −3.51646e7 −1.63551 −0.817756 0.575565i $$-0.804782\pi$$
−0.817756 + 0.575565i $$0.804782\pi$$
$$858$$ 0 0
$$859$$ 1.66022e7 0.767684 0.383842 0.923399i $$-0.374601\pi$$
0.383842 + 0.923399i $$0.374601\pi$$
$$860$$ −2.28434e7 −1.05321
$$861$$ 0 0
$$862$$ −2.52979e7 −1.15962
$$863$$ 2.97009e7 1.35751 0.678754 0.734366i $$-0.262520\pi$$
0.678754 + 0.734366i $$0.262520\pi$$
$$864$$ 0 0
$$865$$ 2.09480e7 0.951923
$$866$$ −3.40761e6 −0.154403
$$867$$ 0 0
$$868$$ 1.08076e7 0.486887
$$869$$ −1.07812e7 −0.484303
$$870$$ 0 0
$$871$$ −1.22431e7 −0.546822
$$872$$ −1.47916e7 −0.658753
$$873$$ 0 0
$$874$$ −1.01990e7 −0.451628
$$875$$ −4.21409e7 −1.86073
$$876$$ 0 0
$$877$$ −1.17943e7 −0.517811 −0.258906 0.965903i $$-0.583362\pi$$
−0.258906 + 0.965903i $$0.583362\pi$$
$$878$$ −1.33893e6 −0.0586166
$$879$$ 0 0
$$880$$ −9.43718e6 −0.410805
$$881$$ 2.10378e7 0.913190 0.456595 0.889675i $$-0.349069\pi$$
0.456595 + 0.889675i $$0.349069\pi$$
$$882$$ 0 0
$$883$$ 2.12192e7 0.915855 0.457928 0.888990i $$-0.348592\pi$$
0.457928 + 0.888990i $$0.348592\pi$$
$$884$$ 3.07814e6 0.132482
$$885$$ 0 0
$$886$$ 7.04870e6 0.301665
$$887$$ −2.28818e7 −0.976520 −0.488260 0.872698i $$-0.662368\pi$$
−0.488260 + 0.872698i $$0.662368\pi$$
$$888$$ 0 0
$$889$$ 146224. 0.00620532
$$890$$ 3.14081e7 1.32913
$$891$$ 0 0
$$892$$ −1.74932e7 −0.736134
$$893$$ −1.27488e7 −0.534984
$$894$$ 0 0
$$895$$ 3.95919e7 1.65215
$$896$$ −2.42483e6 −0.100905
$$897$$ 0 0
$$898$$ −2.60744e7 −1.07900
$$899$$ 438144. 0.0180808
$$900$$ 0 0
$$901$$ 4.47898e6 0.183809
$$902$$ −1.03219e7 −0.422420
$$903$$ 0 0
$$904$$ −9.11770e6 −0.371077
$$905$$ −2.45357e6 −0.0995810
$$906$$ 0 0
$$907$$ −9.68692e6 −0.390992 −0.195496 0.980705i $$-0.562632\pi$$
−0.195496 + 0.980705i $$0.562632\pi$$
$$908$$ 9.06240e6 0.364778
$$909$$ 0 0
$$910$$ 1.89819e7 0.759864
$$911$$ −9.38112e6 −0.374506 −0.187253 0.982312i $$-0.559958\pi$$
−0.187253 + 0.982312i $$0.559958\pi$$
$$912$$ 0 0
$$913$$ −2.55099e7 −1.01282
$$914$$ −368296. −0.0145825
$$915$$ 0 0
$$916$$ −9.39530e6 −0.369974
$$917$$ 3.31899e7 1.30341
$$918$$ 0 0
$$919$$ −4.21870e7 −1.64774 −0.823872 0.566775i $$-0.808191\pi$$
−0.823872 + 0.566775i $$0.808191\pi$$
$$920$$ 2.35930e7 0.918994
$$921$$ 0 0
$$922$$ −1.03027e6 −0.0399139
$$923$$ 2.15470e7 0.832497
$$924$$ 0 0
$$925$$ 3.53156e7 1.35710
$$926$$ 1.58491e7 0.607403
$$927$$ 0 0
$$928$$ −98304.0 −0.00374715
$$929$$ 3.04556e7 1.15779 0.578893 0.815404i $$-0.303485\pi$$
0.578893 + 0.815404i $$0.303485\pi$$
$$930$$ 0 0
$$931$$ −3.38441e6 −0.127970
$$932$$ −9.27130e6 −0.349624
$$933$$ 0 0
$$934$$ 1.39576e7 0.523534
$$935$$ 2.12337e7 0.794321
$$936$$ 0 0
$$937$$ 1.47847e7 0.550128 0.275064 0.961426i $$-0.411301\pi$$
0.275064 + 0.961426i $$0.411301\pi$$
$$938$$ −2.17004e7 −0.805304
$$939$$ 0 0
$$940$$ 2.94912e7 1.08861
$$941$$ 9.91997e6 0.365205 0.182602 0.983187i $$-0.441548\pi$$
0.182602 + 0.983187i $$0.441548\pi$$
$$942$$ 0 0
$$943$$ 2.58048e7 0.944977
$$944$$ 3.34234e6 0.122073
$$945$$ 0 0
$$946$$ 2.28434e7 0.829913
$$947$$ −2.91610e6 −0.105664 −0.0528320 0.998603i $$-0.516825\pi$$
−0.0528320 + 0.998603i $$0.516825\pi$$
$$948$$ 0 0
$$949$$ 5.61454e6 0.202371
$$950$$ −1.61777e7 −0.581578
$$951$$ 0 0
$$952$$ 5.45587e6 0.195107
$$953$$ −1.40861e7 −0.502410 −0.251205 0.967934i $$-0.580827\pi$$
−0.251205 + 0.967934i $$0.580827\pi$$
$$954$$ 0 0
$$955$$ −3.84123e7 −1.36289
$$956$$ 9.35117e6 0.330919
$$957$$ 0 0
$$958$$ −2.05210e6 −0.0722410
$$959$$ −4.12884e7 −1.44971
$$960$$ 0 0
$$961$$ −7.79906e6 −0.272417
$$962$$ −7.74613e6 −0.269865
$$963$$ 0 0
$$964$$ −6.62608e6 −0.229649
$$965$$ 6.71664e7 2.32185
$$966$$ 0 0
$$967$$ 1.51949e7 0.522553 0.261276 0.965264i $$-0.415857\pi$$
0.261276 + 0.965264i $$0.415857\pi$$
$$968$$ −870080. −0.0298449
$$969$$ 0 0
$$970$$ −1.14962e7 −0.392306
$$971$$ 5.61220e7 1.91023 0.955113 0.296240i $$-0.0957329\pi$$
0.955113 + 0.296240i $$0.0957329\pi$$
$$972$$ 0 0
$$973$$ −2.62256e7 −0.888062
$$974$$ 1.65800e7 0.559997
$$975$$ 0 0
$$976$$ 1.09522e7 0.368024
$$977$$ −3.45625e7 −1.15843 −0.579214 0.815176i $$-0.696640\pi$$
−0.579214 + 0.815176i $$0.696640\pi$$
$$978$$ 0 0
$$979$$ −3.14081e7 −1.04733
$$980$$ 7.82899e6 0.260400
$$981$$ 0 0
$$982$$ −1.10346e7 −0.365156
$$983$$ −5.56385e7 −1.83650 −0.918252 0.395997i $$-0.870399\pi$$
−0.918252 + 0.395997i $$0.870399\pi$$
$$984$$ 0 0
$$985$$ −3.90113e7 −1.28115
$$986$$ 221184. 0.00724538
$$987$$ 0 0
$$988$$ 3.54842e6 0.115649
$$989$$ −5.71085e7 −1.85656
$$990$$ 0 0
$$991$$ −3.60028e7 −1.16453 −0.582267 0.812998i $$-0.697834\pi$$
−0.582267 + 0.812998i $$0.697834\pi$$
$$992$$ −4.67354e6 −0.150788
$$993$$ 0 0
$$994$$ 3.81911e7 1.22602
$$995$$ −3.47516e7 −1.11280
$$996$$ 0 0
$$997$$ 2.35811e7 0.751322 0.375661 0.926757i $$-0.377416\pi$$
0.375661 + 0.926757i $$0.377416\pi$$
$$998$$ 2.64358e6 0.0840169
$$999$$ 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.c.1.1 yes 1
3.2 odd 2 18.6.a.a.1.1 1
4.3 odd 2 144.6.a.l.1.1 1
5.2 odd 4 450.6.c.c.199.2 2
5.3 odd 4 450.6.c.c.199.1 2
5.4 even 2 450.6.a.k.1.1 1
7.6 odd 2 882.6.a.l.1.1 1
8.3 odd 2 576.6.a.b.1.1 1
8.5 even 2 576.6.a.a.1.1 1
9.2 odd 6 162.6.c.l.109.1 2
9.4 even 3 162.6.c.a.55.1 2
9.5 odd 6 162.6.c.l.55.1 2
9.7 even 3 162.6.c.a.109.1 2
12.11 even 2 144.6.a.a.1.1 1
15.2 even 4 450.6.c.m.199.1 2
15.8 even 4 450.6.c.m.199.2 2
15.14 odd 2 450.6.a.v.1.1 1
21.20 even 2 882.6.a.k.1.1 1
24.5 odd 2 576.6.a.bh.1.1 1
24.11 even 2 576.6.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.a.a.1.1 1 3.2 odd 2
18.6.a.c.1.1 yes 1 1.1 even 1 trivial
144.6.a.a.1.1 1 12.11 even 2
144.6.a.l.1.1 1 4.3 odd 2
162.6.c.a.55.1 2 9.4 even 3
162.6.c.a.109.1 2 9.7 even 3
162.6.c.l.55.1 2 9.5 odd 6
162.6.c.l.109.1 2 9.2 odd 6
450.6.a.k.1.1 1 5.4 even 2
450.6.a.v.1.1 1 15.14 odd 2
450.6.c.c.199.1 2 5.3 odd 4
450.6.c.c.199.2 2 5.2 odd 4
450.6.c.m.199.1 2 15.2 even 4
450.6.c.m.199.2 2 15.8 even 4
576.6.a.a.1.1 1 8.5 even 2
576.6.a.b.1.1 1 8.3 odd 2
576.6.a.bh.1.1 1 24.5 odd 2
576.6.a.bi.1.1 1 24.11 even 2
882.6.a.k.1.1 1 21.20 even 2
882.6.a.l.1.1 1 7.6 odd 2