Properties

 Label 90.6.a.b.1.1 Level $90$ Weight $6$ Character 90.1 Self dual yes Analytic conductor $14.435$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 90.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.4345437832$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 90.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -118.000 q^{7} -64.0000 q^{8} +O(q^{10})$$ $$q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -118.000 q^{7} -64.0000 q^{8} -100.000 q^{10} -192.000 q^{11} +1106.00 q^{13} +472.000 q^{14} +256.000 q^{16} -762.000 q^{17} -2740.00 q^{19} +400.000 q^{20} +768.000 q^{22} -1566.00 q^{23} +625.000 q^{25} -4424.00 q^{26} -1888.00 q^{28} -5910.00 q^{29} -6868.00 q^{31} -1024.00 q^{32} +3048.00 q^{34} -2950.00 q^{35} -5518.00 q^{37} +10960.0 q^{38} -1600.00 q^{40} +378.000 q^{41} -2434.00 q^{43} -3072.00 q^{44} +6264.00 q^{46} -13122.0 q^{47} -2883.00 q^{49} -2500.00 q^{50} +17696.0 q^{52} +9174.00 q^{53} -4800.00 q^{55} +7552.00 q^{56} +23640.0 q^{58} +34980.0 q^{59} -9838.00 q^{61} +27472.0 q^{62} +4096.00 q^{64} +27650.0 q^{65} +33722.0 q^{67} -12192.0 q^{68} +11800.0 q^{70} -70212.0 q^{71} +21986.0 q^{73} +22072.0 q^{74} -43840.0 q^{76} +22656.0 q^{77} +4520.00 q^{79} +6400.00 q^{80} -1512.00 q^{82} +109074. q^{83} -19050.0 q^{85} +9736.00 q^{86} +12288.0 q^{88} -38490.0 q^{89} -130508. q^{91} -25056.0 q^{92} +52488.0 q^{94} -68500.0 q^{95} -1918.00 q^{97} +11532.0 q^{98} +O(q^{100})$$

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
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ −118.000 −0.910200 −0.455100 0.890440i $$-0.650397\pi$$
−0.455100 + 0.890440i $$0.650397\pi$$
$$8$$ −64.0000 −0.353553
$$9$$ 0 0
$$10$$ −100.000 −0.316228
$$11$$ −192.000 −0.478431 −0.239216 0.970966i $$-0.576890\pi$$
−0.239216 + 0.970966i $$0.576890\pi$$
$$12$$ 0 0
$$13$$ 1106.00 1.81508 0.907542 0.419961i $$-0.137956\pi$$
0.907542 + 0.419961i $$0.137956\pi$$
$$14$$ 472.000 0.643609
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −762.000 −0.639488 −0.319744 0.947504i $$-0.603597\pi$$
−0.319744 + 0.947504i $$0.603597\pi$$
$$18$$ 0 0
$$19$$ −2740.00 −1.74127 −0.870636 0.491928i $$-0.836292\pi$$
−0.870636 + 0.491928i $$0.836292\pi$$
$$20$$ 400.000 0.223607
$$21$$ 0 0
$$22$$ 768.000 0.338302
$$23$$ −1566.00 −0.617266 −0.308633 0.951181i $$-0.599871\pi$$
−0.308633 + 0.951181i $$0.599871\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ −4424.00 −1.28346
$$27$$ 0 0
$$28$$ −1888.00 −0.455100
$$29$$ −5910.00 −1.30495 −0.652473 0.757812i $$-0.726268\pi$$
−0.652473 + 0.757812i $$0.726268\pi$$
$$30$$ 0 0
$$31$$ −6868.00 −1.28359 −0.641795 0.766877i $$-0.721810\pi$$
−0.641795 + 0.766877i $$0.721810\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ 0 0
$$34$$ 3048.00 0.452187
$$35$$ −2950.00 −0.407054
$$36$$ 0 0
$$37$$ −5518.00 −0.662640 −0.331320 0.943519i $$-0.607494\pi$$
−0.331320 + 0.943519i $$0.607494\pi$$
$$38$$ 10960.0 1.23127
$$39$$ 0 0
$$40$$ −1600.00 −0.158114
$$41$$ 378.000 0.0351182 0.0175591 0.999846i $$-0.494410\pi$$
0.0175591 + 0.999846i $$0.494410\pi$$
$$42$$ 0 0
$$43$$ −2434.00 −0.200747 −0.100374 0.994950i $$-0.532004\pi$$
−0.100374 + 0.994950i $$0.532004\pi$$
$$44$$ −3072.00 −0.239216
$$45$$ 0 0
$$46$$ 6264.00 0.436473
$$47$$ −13122.0 −0.866474 −0.433237 0.901280i $$-0.642629\pi$$
−0.433237 + 0.901280i $$0.642629\pi$$
$$48$$ 0 0
$$49$$ −2883.00 −0.171536
$$50$$ −2500.00 −0.141421
$$51$$ 0 0
$$52$$ 17696.0 0.907542
$$53$$ 9174.00 0.448610 0.224305 0.974519i $$-0.427989\pi$$
0.224305 + 0.974519i $$0.427989\pi$$
$$54$$ 0 0
$$55$$ −4800.00 −0.213961
$$56$$ 7552.00 0.321804
$$57$$ 0 0
$$58$$ 23640.0 0.922736
$$59$$ 34980.0 1.30825 0.654124 0.756388i $$-0.273038\pi$$
0.654124 + 0.756388i $$0.273038\pi$$
$$60$$ 0 0
$$61$$ −9838.00 −0.338518 −0.169259 0.985572i $$-0.554137\pi$$
−0.169259 + 0.985572i $$0.554137\pi$$
$$62$$ 27472.0 0.907635
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ 27650.0 0.811730
$$66$$ 0 0
$$67$$ 33722.0 0.917754 0.458877 0.888500i $$-0.348252\pi$$
0.458877 + 0.888500i $$0.348252\pi$$
$$68$$ −12192.0 −0.319744
$$69$$ 0 0
$$70$$ 11800.0 0.287831
$$71$$ −70212.0 −1.65297 −0.826486 0.562957i $$-0.809664\pi$$
−0.826486 + 0.562957i $$0.809664\pi$$
$$72$$ 0 0
$$73$$ 21986.0 0.482880 0.241440 0.970416i $$-0.422380\pi$$
0.241440 + 0.970416i $$0.422380\pi$$
$$74$$ 22072.0 0.468557
$$75$$ 0 0
$$76$$ −43840.0 −0.870636
$$77$$ 22656.0 0.435468
$$78$$ 0 0
$$79$$ 4520.00 0.0814837 0.0407418 0.999170i $$-0.487028\pi$$
0.0407418 + 0.999170i $$0.487028\pi$$
$$80$$ 6400.00 0.111803
$$81$$ 0 0
$$82$$ −1512.00 −0.0248323
$$83$$ 109074. 1.73790 0.868952 0.494896i $$-0.164794\pi$$
0.868952 + 0.494896i $$0.164794\pi$$
$$84$$ 0 0
$$85$$ −19050.0 −0.285988
$$86$$ 9736.00 0.141950
$$87$$ 0 0
$$88$$ 12288.0 0.169151
$$89$$ −38490.0 −0.515078 −0.257539 0.966268i $$-0.582912\pi$$
−0.257539 + 0.966268i $$0.582912\pi$$
$$90$$ 0 0
$$91$$ −130508. −1.65209
$$92$$ −25056.0 −0.308633
$$93$$ 0 0
$$94$$ 52488.0 0.612689
$$95$$ −68500.0 −0.778720
$$96$$ 0 0
$$97$$ −1918.00 −0.0206976 −0.0103488 0.999946i $$-0.503294\pi$$
−0.0103488 + 0.999946i $$0.503294\pi$$
$$98$$ 11532.0 0.121294
$$99$$ 0 0
$$100$$ 10000.0 0.100000
$$101$$ −77622.0 −0.757149 −0.378575 0.925571i $$-0.623586\pi$$
−0.378575 + 0.925571i $$0.623586\pi$$
$$102$$ 0 0
$$103$$ −46714.0 −0.433864 −0.216932 0.976187i $$-0.569605\pi$$
−0.216932 + 0.976187i $$0.569605\pi$$
$$104$$ −70784.0 −0.641729
$$105$$ 0 0
$$106$$ −36696.0 −0.317215
$$107$$ 1038.00 0.00876472 0.00438236 0.999990i $$-0.498605\pi$$
0.00438236 + 0.999990i $$0.498605\pi$$
$$108$$ 0 0
$$109$$ 206930. 1.66823 0.834117 0.551587i $$-0.185977\pi$$
0.834117 + 0.551587i $$0.185977\pi$$
$$110$$ 19200.0 0.151293
$$111$$ 0 0
$$112$$ −30208.0 −0.227550
$$113$$ −139386. −1.02689 −0.513444 0.858123i $$-0.671631\pi$$
−0.513444 + 0.858123i $$0.671631\pi$$
$$114$$ 0 0
$$115$$ −39150.0 −0.276050
$$116$$ −94560.0 −0.652473
$$117$$ 0 0
$$118$$ −139920. −0.925070
$$119$$ 89916.0 0.582062
$$120$$ 0 0
$$121$$ −124187. −0.771104
$$122$$ 39352.0 0.239369
$$123$$ 0 0
$$124$$ −109888. −0.641795
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ 299882. 1.64984 0.824919 0.565252i $$-0.191221\pi$$
0.824919 + 0.565252i $$0.191221\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ 0 0
$$130$$ −110600. −0.573980
$$131$$ −7872.00 −0.0400781 −0.0200390 0.999799i $$-0.506379\pi$$
−0.0200390 + 0.999799i $$0.506379\pi$$
$$132$$ 0 0
$$133$$ 323320. 1.58491
$$134$$ −134888. −0.648950
$$135$$ 0 0
$$136$$ 48768.0 0.226093
$$137$$ 164238. 0.747605 0.373803 0.927508i $$-0.378054\pi$$
0.373803 + 0.927508i $$0.378054\pi$$
$$138$$ 0 0
$$139$$ −282100. −1.23841 −0.619207 0.785228i $$-0.712546\pi$$
−0.619207 + 0.785228i $$0.712546\pi$$
$$140$$ −47200.0 −0.203527
$$141$$ 0 0
$$142$$ 280848. 1.16883
$$143$$ −212352. −0.868393
$$144$$ 0 0
$$145$$ −147750. −0.583590
$$146$$ −87944.0 −0.341448
$$147$$ 0 0
$$148$$ −88288.0 −0.331320
$$149$$ 388950. 1.43525 0.717626 0.696429i $$-0.245229\pi$$
0.717626 + 0.696429i $$0.245229\pi$$
$$150$$ 0 0
$$151$$ −97948.0 −0.349585 −0.174793 0.984605i $$-0.555926\pi$$
−0.174793 + 0.984605i $$0.555926\pi$$
$$152$$ 175360. 0.615633
$$153$$ 0 0
$$154$$ −90624.0 −0.307923
$$155$$ −171700. −0.574039
$$156$$ 0 0
$$157$$ −3718.00 −0.0120382 −0.00601908 0.999982i $$-0.501916\pi$$
−0.00601908 + 0.999982i $$0.501916\pi$$
$$158$$ −18080.0 −0.0576177
$$159$$ 0 0
$$160$$ −25600.0 −0.0790569
$$161$$ 184788. 0.561835
$$162$$ 0 0
$$163$$ −43234.0 −0.127455 −0.0637274 0.997967i $$-0.520299\pi$$
−0.0637274 + 0.997967i $$0.520299\pi$$
$$164$$ 6048.00 0.0175591
$$165$$ 0 0
$$166$$ −436296. −1.22888
$$167$$ −186522. −0.517534 −0.258767 0.965940i $$-0.583316\pi$$
−0.258767 + 0.965940i $$0.583316\pi$$
$$168$$ 0 0
$$169$$ 851943. 2.29453
$$170$$ 76200.0 0.202224
$$171$$ 0 0
$$172$$ −38944.0 −0.100374
$$173$$ 374454. 0.951225 0.475612 0.879655i $$-0.342226\pi$$
0.475612 + 0.879655i $$0.342226\pi$$
$$174$$ 0 0
$$175$$ −73750.0 −0.182040
$$176$$ −49152.0 −0.119608
$$177$$ 0 0
$$178$$ 153960. 0.364215
$$179$$ −272100. −0.634740 −0.317370 0.948302i $$-0.602800\pi$$
−0.317370 + 0.948302i $$0.602800\pi$$
$$180$$ 0 0
$$181$$ −75418.0 −0.171111 −0.0855556 0.996333i $$-0.527267\pi$$
−0.0855556 + 0.996333i $$0.527267\pi$$
$$182$$ 522032. 1.16820
$$183$$ 0 0
$$184$$ 100224. 0.218236
$$185$$ −137950. −0.296341
$$186$$ 0 0
$$187$$ 146304. 0.305951
$$188$$ −209952. −0.433237
$$189$$ 0 0
$$190$$ 274000. 0.550638
$$191$$ 356988. 0.708060 0.354030 0.935234i $$-0.384811\pi$$
0.354030 + 0.935234i $$0.384811\pi$$
$$192$$ 0 0
$$193$$ −438694. −0.847751 −0.423876 0.905720i $$-0.639331\pi$$
−0.423876 + 0.905720i $$0.639331\pi$$
$$194$$ 7672.00 0.0146354
$$195$$ 0 0
$$196$$ −46128.0 −0.0857678
$$197$$ 156798. 0.287856 0.143928 0.989588i $$-0.454027\pi$$
0.143928 + 0.989588i $$0.454027\pi$$
$$198$$ 0 0
$$199$$ −162520. −0.290920 −0.145460 0.989364i $$-0.546466\pi$$
−0.145460 + 0.989364i $$0.546466\pi$$
$$200$$ −40000.0 −0.0707107
$$201$$ 0 0
$$202$$ 310488. 0.535385
$$203$$ 697380. 1.18776
$$204$$ 0 0
$$205$$ 9450.00 0.0157053
$$206$$ 186856. 0.306788
$$207$$ 0 0
$$208$$ 283136. 0.453771
$$209$$ 526080. 0.833079
$$210$$ 0 0
$$211$$ −181648. −0.280882 −0.140441 0.990089i $$-0.544852\pi$$
−0.140441 + 0.990089i $$0.544852\pi$$
$$212$$ 146784. 0.224305
$$213$$ 0 0
$$214$$ −4152.00 −0.00619759
$$215$$ −60850.0 −0.0897769
$$216$$ 0 0
$$217$$ 810424. 1.16832
$$218$$ −827720. −1.17962
$$219$$ 0 0
$$220$$ −76800.0 −0.106980
$$221$$ −842772. −1.16073
$$222$$ 0 0
$$223$$ −288274. −0.388189 −0.194095 0.980983i $$-0.562177\pi$$
−0.194095 + 0.980983i $$0.562177\pi$$
$$224$$ 120832. 0.160902
$$225$$ 0 0
$$226$$ 557544. 0.726119
$$227$$ −1.12552e6 −1.44974 −0.724869 0.688887i $$-0.758100\pi$$
−0.724869 + 0.688887i $$0.758100\pi$$
$$228$$ 0 0
$$229$$ −415810. −0.523970 −0.261985 0.965072i $$-0.584377\pi$$
−0.261985 + 0.965072i $$0.584377\pi$$
$$230$$ 156600. 0.195197
$$231$$ 0 0
$$232$$ 378240. 0.461368
$$233$$ −770586. −0.929889 −0.464945 0.885340i $$-0.653926\pi$$
−0.464945 + 0.885340i $$0.653926\pi$$
$$234$$ 0 0
$$235$$ −328050. −0.387499
$$236$$ 559680. 0.654124
$$237$$ 0 0
$$238$$ −359664. −0.411580
$$239$$ 595320. 0.674149 0.337074 0.941478i $$-0.390563\pi$$
0.337074 + 0.941478i $$0.390563\pi$$
$$240$$ 0 0
$$241$$ 273902. 0.303775 0.151888 0.988398i $$-0.451465\pi$$
0.151888 + 0.988398i $$0.451465\pi$$
$$242$$ 496748. 0.545253
$$243$$ 0 0
$$244$$ −157408. −0.169259
$$245$$ −72075.0 −0.0767131
$$246$$ 0 0
$$247$$ −3.03044e6 −3.16055
$$248$$ 439552. 0.453817
$$249$$ 0 0
$$250$$ −62500.0 −0.0632456
$$251$$ −850752. −0.852351 −0.426176 0.904640i $$-0.640139\pi$$
−0.426176 + 0.904640i $$0.640139\pi$$
$$252$$ 0 0
$$253$$ 300672. 0.295319
$$254$$ −1.19953e6 −1.16661
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −825402. −0.779530 −0.389765 0.920914i $$-0.627444\pi$$
−0.389765 + 0.920914i $$0.627444\pi$$
$$258$$ 0 0
$$259$$ 651124. 0.603135
$$260$$ 442400. 0.405865
$$261$$ 0 0
$$262$$ 31488.0 0.0283395
$$263$$ −1.36465e6 −1.21655 −0.608276 0.793726i $$-0.708139\pi$$
−0.608276 + 0.793726i $$0.708139\pi$$
$$264$$ 0 0
$$265$$ 229350. 0.200625
$$266$$ −1.29328e6 −1.12070
$$267$$ 0 0
$$268$$ 539552. 0.458877
$$269$$ 113310. 0.0954745 0.0477373 0.998860i $$-0.484799\pi$$
0.0477373 + 0.998860i $$0.484799\pi$$
$$270$$ 0 0
$$271$$ −849628. −0.702758 −0.351379 0.936233i $$-0.614287\pi$$
−0.351379 + 0.936233i $$0.614287\pi$$
$$272$$ −195072. −0.159872
$$273$$ 0 0
$$274$$ −656952. −0.528637
$$275$$ −120000. −0.0956862
$$276$$ 0 0
$$277$$ 438602. 0.343456 0.171728 0.985144i $$-0.445065\pi$$
0.171728 + 0.985144i $$0.445065\pi$$
$$278$$ 1.12840e6 0.875691
$$279$$ 0 0
$$280$$ 188800. 0.143915
$$281$$ 1.45670e6 1.10053 0.550267 0.834989i $$-0.314526\pi$$
0.550267 + 0.834989i $$0.314526\pi$$
$$282$$ 0 0
$$283$$ −120394. −0.0893591 −0.0446795 0.999001i $$-0.514227\pi$$
−0.0446795 + 0.999001i $$0.514227\pi$$
$$284$$ −1.12339e6 −0.826486
$$285$$ 0 0
$$286$$ 849408. 0.614047
$$287$$ −44604.0 −0.0319646
$$288$$ 0 0
$$289$$ −839213. −0.591055
$$290$$ 591000. 0.412660
$$291$$ 0 0
$$292$$ 351776. 0.241440
$$293$$ 2.64209e6 1.79796 0.898978 0.437993i $$-0.144311\pi$$
0.898978 + 0.437993i $$0.144311\pi$$
$$294$$ 0 0
$$295$$ 874500. 0.585066
$$296$$ 353152. 0.234278
$$297$$ 0 0
$$298$$ −1.55580e6 −1.01488
$$299$$ −1.73200e6 −1.12039
$$300$$ 0 0
$$301$$ 287212. 0.182720
$$302$$ 391792. 0.247194
$$303$$ 0 0
$$304$$ −701440. −0.435318
$$305$$ −245950. −0.151390
$$306$$ 0 0
$$307$$ −1.44756e6 −0.876577 −0.438288 0.898834i $$-0.644415\pi$$
−0.438288 + 0.898834i $$0.644415\pi$$
$$308$$ 362496. 0.217734
$$309$$ 0 0
$$310$$ 686800. 0.405907
$$311$$ 928068. 0.544100 0.272050 0.962283i $$-0.412298\pi$$
0.272050 + 0.962283i $$0.412298\pi$$
$$312$$ 0 0
$$313$$ 2.29563e6 1.32446 0.662232 0.749299i $$-0.269609\pi$$
0.662232 + 0.749299i $$0.269609\pi$$
$$314$$ 14872.0 0.00851227
$$315$$ 0 0
$$316$$ 72320.0 0.0407418
$$317$$ −2.73652e6 −1.52950 −0.764752 0.644324i $$-0.777139\pi$$
−0.764752 + 0.644324i $$0.777139\pi$$
$$318$$ 0 0
$$319$$ 1.13472e6 0.624327
$$320$$ 102400. 0.0559017
$$321$$ 0 0
$$322$$ −739152. −0.397278
$$323$$ 2.08788e6 1.11352
$$324$$ 0 0
$$325$$ 691250. 0.363017
$$326$$ 172936. 0.0901242
$$327$$ 0 0
$$328$$ −24192.0 −0.0124162
$$329$$ 1.54840e6 0.788665
$$330$$ 0 0
$$331$$ 3.81879e6 1.91583 0.957913 0.287059i $$-0.0926776\pi$$
0.957913 + 0.287059i $$0.0926776\pi$$
$$332$$ 1.74518e6 0.868952
$$333$$ 0 0
$$334$$ 746088. 0.365952
$$335$$ 843050. 0.410432
$$336$$ 0 0
$$337$$ −2.21088e6 −1.06045 −0.530225 0.847857i $$-0.677892\pi$$
−0.530225 + 0.847857i $$0.677892\pi$$
$$338$$ −3.40777e6 −1.62248
$$339$$ 0 0
$$340$$ −304800. −0.142994
$$341$$ 1.31866e6 0.614109
$$342$$ 0 0
$$343$$ 2.32342e6 1.06633
$$344$$ 155776. 0.0709748
$$345$$ 0 0
$$346$$ −1.49782e6 −0.672618
$$347$$ 2.32724e6 1.03757 0.518785 0.854905i $$-0.326385\pi$$
0.518785 + 0.854905i $$0.326385\pi$$
$$348$$ 0 0
$$349$$ −311290. −0.136805 −0.0684024 0.997658i $$-0.521790\pi$$
−0.0684024 + 0.997658i $$0.521790\pi$$
$$350$$ 295000. 0.128722
$$351$$ 0 0
$$352$$ 196608. 0.0845755
$$353$$ 3.08657e6 1.31838 0.659189 0.751977i $$-0.270900\pi$$
0.659189 + 0.751977i $$0.270900\pi$$
$$354$$ 0 0
$$355$$ −1.75530e6 −0.739232
$$356$$ −615840. −0.257539
$$357$$ 0 0
$$358$$ 1.08840e6 0.448829
$$359$$ 3.53076e6 1.44588 0.722940 0.690911i $$-0.242790\pi$$
0.722940 + 0.690911i $$0.242790\pi$$
$$360$$ 0 0
$$361$$ 5.03150e6 2.03203
$$362$$ 301672. 0.120994
$$363$$ 0 0
$$364$$ −2.08813e6 −0.826045
$$365$$ 549650. 0.215950
$$366$$ 0 0
$$367$$ 35762.0 0.0138598 0.00692989 0.999976i $$-0.497794\pi$$
0.00692989 + 0.999976i $$0.497794\pi$$
$$368$$ −400896. −0.154316
$$369$$ 0 0
$$370$$ 551800. 0.209545
$$371$$ −1.08253e6 −0.408325
$$372$$ 0 0
$$373$$ −1.71525e6 −0.638346 −0.319173 0.947696i $$-0.603405\pi$$
−0.319173 + 0.947696i $$0.603405\pi$$
$$374$$ −585216. −0.216340
$$375$$ 0 0
$$376$$ 839808. 0.306345
$$377$$ −6.53646e6 −2.36859
$$378$$ 0 0
$$379$$ −3.10174e6 −1.10919 −0.554597 0.832119i $$-0.687127\pi$$
−0.554597 + 0.832119i $$0.687127\pi$$
$$380$$ −1.09600e6 −0.389360
$$381$$ 0 0
$$382$$ −1.42795e6 −0.500674
$$383$$ −5.31949e6 −1.85299 −0.926494 0.376309i $$-0.877193\pi$$
−0.926494 + 0.376309i $$0.877193\pi$$
$$384$$ 0 0
$$385$$ 566400. 0.194747
$$386$$ 1.75478e6 0.599451
$$387$$ 0 0
$$388$$ −30688.0 −0.0103488
$$389$$ −1.16145e6 −0.389158 −0.194579 0.980887i $$-0.562334\pi$$
−0.194579 + 0.980887i $$0.562334\pi$$
$$390$$ 0 0
$$391$$ 1.19329e6 0.394734
$$392$$ 184512. 0.0606470
$$393$$ 0 0
$$394$$ −627192. −0.203545
$$395$$ 113000. 0.0364406
$$396$$ 0 0
$$397$$ 628562. 0.200157 0.100079 0.994980i $$-0.468091\pi$$
0.100079 + 0.994980i $$0.468091\pi$$
$$398$$ 650080. 0.205712
$$399$$ 0 0
$$400$$ 160000. 0.0500000
$$401$$ 2.72432e6 0.846052 0.423026 0.906118i $$-0.360968\pi$$
0.423026 + 0.906118i $$0.360968\pi$$
$$402$$ 0 0
$$403$$ −7.59601e6 −2.32982
$$404$$ −1.24195e6 −0.378575
$$405$$ 0 0
$$406$$ −2.78952e6 −0.839875
$$407$$ 1.05946e6 0.317027
$$408$$ 0 0
$$409$$ 1.78019e6 0.526209 0.263104 0.964767i $$-0.415254\pi$$
0.263104 + 0.964767i $$0.415254\pi$$
$$410$$ −37800.0 −0.0111053
$$411$$ 0 0
$$412$$ −747424. −0.216932
$$413$$ −4.12764e6 −1.19077
$$414$$ 0 0
$$415$$ 2.72685e6 0.777215
$$416$$ −1.13254e6 −0.320865
$$417$$ 0 0
$$418$$ −2.10432e6 −0.589076
$$419$$ −650580. −0.181036 −0.0905181 0.995895i $$-0.528852\pi$$
−0.0905181 + 0.995895i $$0.528852\pi$$
$$420$$ 0 0
$$421$$ −3.54060e6 −0.973579 −0.486790 0.873519i $$-0.661832\pi$$
−0.486790 + 0.873519i $$0.661832\pi$$
$$422$$ 726592. 0.198614
$$423$$ 0 0
$$424$$ −587136. −0.158608
$$425$$ −476250. −0.127898
$$426$$ 0 0
$$427$$ 1.16088e6 0.308119
$$428$$ 16608.0 0.00438236
$$429$$ 0 0
$$430$$ 243400. 0.0634818
$$431$$ 548748. 0.142292 0.0711459 0.997466i $$-0.477334\pi$$
0.0711459 + 0.997466i $$0.477334\pi$$
$$432$$ 0 0
$$433$$ −1.49241e6 −0.382534 −0.191267 0.981538i $$-0.561260\pi$$
−0.191267 + 0.981538i $$0.561260\pi$$
$$434$$ −3.24170e6 −0.826129
$$435$$ 0 0
$$436$$ 3.31088e6 0.834117
$$437$$ 4.29084e6 1.07483
$$438$$ 0 0
$$439$$ 4.86212e6 1.20411 0.602053 0.798456i $$-0.294350\pi$$
0.602053 + 0.798456i $$0.294350\pi$$
$$440$$ 307200. 0.0756466
$$441$$ 0 0
$$442$$ 3.37109e6 0.820757
$$443$$ 1.86155e6 0.450678 0.225339 0.974280i $$-0.427651\pi$$
0.225339 + 0.974280i $$0.427651\pi$$
$$444$$ 0 0
$$445$$ −962250. −0.230350
$$446$$ 1.15310e6 0.274491
$$447$$ 0 0
$$448$$ −483328. −0.113775
$$449$$ −3.73719e6 −0.874841 −0.437421 0.899257i $$-0.644108\pi$$
−0.437421 + 0.899257i $$0.644108\pi$$
$$450$$ 0 0
$$451$$ −72576.0 −0.0168016
$$452$$ −2.23018e6 −0.513444
$$453$$ 0 0
$$454$$ 4.50209e6 1.02512
$$455$$ −3.26270e6 −0.738837
$$456$$ 0 0
$$457$$ −6.48276e6 −1.45201 −0.726005 0.687690i $$-0.758625\pi$$
−0.726005 + 0.687690i $$0.758625\pi$$
$$458$$ 1.66324e6 0.370503
$$459$$ 0 0
$$460$$ −626400. −0.138025
$$461$$ −1.50910e6 −0.330724 −0.165362 0.986233i $$-0.552879\pi$$
−0.165362 + 0.986233i $$0.552879\pi$$
$$462$$ 0 0
$$463$$ 8.68401e6 1.88264 0.941321 0.337513i $$-0.109586\pi$$
0.941321 + 0.337513i $$0.109586\pi$$
$$464$$ −1.51296e6 −0.326236
$$465$$ 0 0
$$466$$ 3.08234e6 0.657531
$$467$$ −6.96412e6 −1.47766 −0.738829 0.673893i $$-0.764621\pi$$
−0.738829 + 0.673893i $$0.764621\pi$$
$$468$$ 0 0
$$469$$ −3.97920e6 −0.835340
$$470$$ 1.31220e6 0.274003
$$471$$ 0 0
$$472$$ −2.23872e6 −0.462535
$$473$$ 467328. 0.0960437
$$474$$ 0 0
$$475$$ −1.71250e6 −0.348254
$$476$$ 1.43866e6 0.291031
$$477$$ 0 0
$$478$$ −2.38128e6 −0.476695
$$479$$ 5.51052e6 1.09737 0.548686 0.836029i $$-0.315128\pi$$
0.548686 + 0.836029i $$0.315128\pi$$
$$480$$ 0 0
$$481$$ −6.10291e6 −1.20275
$$482$$ −1.09561e6 −0.214802
$$483$$ 0 0
$$484$$ −1.98699e6 −0.385552
$$485$$ −47950.0 −0.00925623
$$486$$ 0 0
$$487$$ 5.51808e6 1.05430 0.527152 0.849771i $$-0.323260\pi$$
0.527152 + 0.849771i $$0.323260\pi$$
$$488$$ 629632. 0.119684
$$489$$ 0 0
$$490$$ 288300. 0.0542443
$$491$$ 1.51277e6 0.283184 0.141592 0.989925i $$-0.454778\pi$$
0.141592 + 0.989925i $$0.454778\pi$$
$$492$$ 0 0
$$493$$ 4.50342e6 0.834498
$$494$$ 1.21218e7 2.23485
$$495$$ 0 0
$$496$$ −1.75821e6 −0.320897
$$497$$ 8.28502e6 1.50454
$$498$$ 0 0
$$499$$ −1.93042e6 −0.347057 −0.173528 0.984829i $$-0.555517\pi$$
−0.173528 + 0.984829i $$0.555517\pi$$
$$500$$ 250000. 0.0447214
$$501$$ 0 0
$$502$$ 3.40301e6 0.602703
$$503$$ −6.73105e6 −1.18621 −0.593106 0.805124i $$-0.702099\pi$$
−0.593106 + 0.805124i $$0.702099\pi$$
$$504$$ 0 0
$$505$$ −1.94055e6 −0.338607
$$506$$ −1.20269e6 −0.208822
$$507$$ 0 0
$$508$$ 4.79811e6 0.824919
$$509$$ 556650. 0.0952331 0.0476165 0.998866i $$-0.484837\pi$$
0.0476165 + 0.998866i $$0.484837\pi$$
$$510$$ 0 0
$$511$$ −2.59435e6 −0.439517
$$512$$ −262144. −0.0441942
$$513$$ 0 0
$$514$$ 3.30161e6 0.551211
$$515$$ −1.16785e6 −0.194030
$$516$$ 0 0
$$517$$ 2.51942e6 0.414548
$$518$$ −2.60450e6 −0.426481
$$519$$ 0 0
$$520$$ −1.76960e6 −0.286990
$$521$$ −1.01110e7 −1.63192 −0.815962 0.578106i $$-0.803792\pi$$
−0.815962 + 0.578106i $$0.803792\pi$$
$$522$$ 0 0
$$523$$ −7.03719e6 −1.12498 −0.562491 0.826804i $$-0.690157\pi$$
−0.562491 + 0.826804i $$0.690157\pi$$
$$524$$ −125952. −0.0200390
$$525$$ 0 0
$$526$$ 5.45858e6 0.860232
$$527$$ 5.23342e6 0.820840
$$528$$ 0 0
$$529$$ −3.98399e6 −0.618983
$$530$$ −917400. −0.141863
$$531$$ 0 0
$$532$$ 5.17312e6 0.792453
$$533$$ 418068. 0.0637425
$$534$$ 0 0
$$535$$ 25950.0 0.00391970
$$536$$ −2.15821e6 −0.324475
$$537$$ 0 0
$$538$$ −453240. −0.0675107
$$539$$ 553536. 0.0820680
$$540$$ 0 0
$$541$$ −4.23114e6 −0.621533 −0.310766 0.950486i $$-0.600586\pi$$
−0.310766 + 0.950486i $$0.600586\pi$$
$$542$$ 3.39851e6 0.496925
$$543$$ 0 0
$$544$$ 780288. 0.113047
$$545$$ 5.17325e6 0.746057
$$546$$ 0 0
$$547$$ 4.44024e6 0.634510 0.317255 0.948340i $$-0.397239\pi$$
0.317255 + 0.948340i $$0.397239\pi$$
$$548$$ 2.62781e6 0.373803
$$549$$ 0 0
$$550$$ 480000. 0.0676604
$$551$$ 1.61934e7 2.27227
$$552$$ 0 0
$$553$$ −533360. −0.0741665
$$554$$ −1.75441e6 −0.242860
$$555$$ 0 0
$$556$$ −4.51360e6 −0.619207
$$557$$ 9.01448e6 1.23113 0.615563 0.788088i $$-0.288929\pi$$
0.615563 + 0.788088i $$0.288929\pi$$
$$558$$ 0 0
$$559$$ −2.69200e6 −0.364373
$$560$$ −755200. −0.101763
$$561$$ 0 0
$$562$$ −5.82679e6 −0.778196
$$563$$ 9.81287e6 1.30474 0.652372 0.757899i $$-0.273774\pi$$
0.652372 + 0.757899i $$0.273774\pi$$
$$564$$ 0 0
$$565$$ −3.48465e6 −0.459238
$$566$$ 481576. 0.0631864
$$567$$ 0 0
$$568$$ 4.49357e6 0.584414
$$569$$ −1.33152e7 −1.72412 −0.862061 0.506804i $$-0.830827\pi$$
−0.862061 + 0.506804i $$0.830827\pi$$
$$570$$ 0 0
$$571$$ 9.95895e6 1.27827 0.639136 0.769094i $$-0.279292\pi$$
0.639136 + 0.769094i $$0.279292\pi$$
$$572$$ −3.39763e6 −0.434196
$$573$$ 0 0
$$574$$ 178416. 0.0226024
$$575$$ −978750. −0.123453
$$576$$ 0 0
$$577$$ 4.50372e6 0.563160 0.281580 0.959538i $$-0.409141\pi$$
0.281580 + 0.959538i $$0.409141\pi$$
$$578$$ 3.35685e6 0.417939
$$579$$ 0 0
$$580$$ −2.36400e6 −0.291795
$$581$$ −1.28707e7 −1.58184
$$582$$ 0 0
$$583$$ −1.76141e6 −0.214629
$$584$$ −1.40710e6 −0.170724
$$585$$ 0 0
$$586$$ −1.05684e7 −1.27135
$$587$$ −625842. −0.0749669 −0.0374834 0.999297i $$-0.511934\pi$$
−0.0374834 + 0.999297i $$0.511934\pi$$
$$588$$ 0 0
$$589$$ 1.88183e7 2.23508
$$590$$ −3.49800e6 −0.413704
$$591$$ 0 0
$$592$$ −1.41261e6 −0.165660
$$593$$ 2.50385e6 0.292397 0.146198 0.989255i $$-0.453296\pi$$
0.146198 + 0.989255i $$0.453296\pi$$
$$594$$ 0 0
$$595$$ 2.24790e6 0.260306
$$596$$ 6.22320e6 0.717626
$$597$$ 0 0
$$598$$ 6.92798e6 0.792235
$$599$$ 756480. 0.0861451 0.0430725 0.999072i $$-0.486285\pi$$
0.0430725 + 0.999072i $$0.486285\pi$$
$$600$$ 0 0
$$601$$ −1.38565e7 −1.56483 −0.782413 0.622760i $$-0.786011\pi$$
−0.782413 + 0.622760i $$0.786011\pi$$
$$602$$ −1.14885e6 −0.129203
$$603$$ 0 0
$$604$$ −1.56717e6 −0.174793
$$605$$ −3.10468e6 −0.344848
$$606$$ 0 0
$$607$$ 1.13772e7 1.25333 0.626663 0.779291i $$-0.284420\pi$$
0.626663 + 0.779291i $$0.284420\pi$$
$$608$$ 2.80576e6 0.307816
$$609$$ 0 0
$$610$$ 983800. 0.107049
$$611$$ −1.45129e7 −1.57272
$$612$$ 0 0
$$613$$ −7.00161e6 −0.752570 −0.376285 0.926504i $$-0.622799\pi$$
−0.376285 + 0.926504i $$0.622799\pi$$
$$614$$ 5.79023e6 0.619833
$$615$$ 0 0
$$616$$ −1.44998e6 −0.153961
$$617$$ −7.90300e6 −0.835755 −0.417878 0.908503i $$-0.637226\pi$$
−0.417878 + 0.908503i $$0.637226\pi$$
$$618$$ 0 0
$$619$$ 4.02362e6 0.422076 0.211038 0.977478i $$-0.432316\pi$$
0.211038 + 0.977478i $$0.432316\pi$$
$$620$$ −2.74720e6 −0.287019
$$621$$ 0 0
$$622$$ −3.71227e6 −0.384737
$$623$$ 4.54182e6 0.468824
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ −9.18250e6 −0.936538
$$627$$ 0 0
$$628$$ −59488.0 −0.00601908
$$629$$ 4.20472e6 0.423750
$$630$$ 0 0
$$631$$ −1.00227e7 −1.00210 −0.501049 0.865419i $$-0.667052\pi$$
−0.501049 + 0.865419i $$0.667052\pi$$
$$632$$ −289280. −0.0288088
$$633$$ 0 0
$$634$$ 1.09461e7 1.08152
$$635$$ 7.49705e6 0.737830
$$636$$ 0 0
$$637$$ −3.18860e6 −0.311352
$$638$$ −4.53888e6 −0.441466
$$639$$ 0 0
$$640$$ −409600. −0.0395285
$$641$$ −6.37390e6 −0.612718 −0.306359 0.951916i $$-0.599111\pi$$
−0.306359 + 0.951916i $$0.599111\pi$$
$$642$$ 0 0
$$643$$ 5.00457e6 0.477352 0.238676 0.971099i $$-0.423287\pi$$
0.238676 + 0.971099i $$0.423287\pi$$
$$644$$ 2.95661e6 0.280918
$$645$$ 0 0
$$646$$ −8.35152e6 −0.787380
$$647$$ 8.71928e6 0.818879 0.409440 0.912337i $$-0.365724\pi$$
0.409440 + 0.912337i $$0.365724\pi$$
$$648$$ 0 0
$$649$$ −6.71616e6 −0.625906
$$650$$ −2.76500e6 −0.256692
$$651$$ 0 0
$$652$$ −691744. −0.0637274
$$653$$ 1.58477e6 0.145440 0.0727201 0.997352i $$-0.476832\pi$$
0.0727201 + 0.997352i $$0.476832\pi$$
$$654$$ 0 0
$$655$$ −196800. −0.0179235
$$656$$ 96768.0 0.00877955
$$657$$ 0 0
$$658$$ −6.19358e6 −0.557670
$$659$$ −1.26410e7 −1.13388 −0.566940 0.823759i $$-0.691873\pi$$
−0.566940 + 0.823759i $$0.691873\pi$$
$$660$$ 0 0
$$661$$ −3.61572e6 −0.321878 −0.160939 0.986964i $$-0.551452\pi$$
−0.160939 + 0.986964i $$0.551452\pi$$
$$662$$ −1.52752e7 −1.35469
$$663$$ 0 0
$$664$$ −6.98074e6 −0.614442
$$665$$ 8.08300e6 0.708791
$$666$$ 0 0
$$667$$ 9.25506e6 0.805498
$$668$$ −2.98435e6 −0.258767
$$669$$ 0 0
$$670$$ −3.37220e6 −0.290219
$$671$$ 1.88890e6 0.161958
$$672$$ 0 0
$$673$$ 1.11313e7 0.947349 0.473675 0.880700i $$-0.342927\pi$$
0.473675 + 0.880700i $$0.342927\pi$$
$$674$$ 8.84351e6 0.749851
$$675$$ 0 0
$$676$$ 1.36311e7 1.14727
$$677$$ 235518. 0.0197493 0.00987467 0.999951i $$-0.496857\pi$$
0.00987467 + 0.999951i $$0.496857\pi$$
$$678$$ 0 0
$$679$$ 226324. 0.0188389
$$680$$ 1.21920e6 0.101112
$$681$$ 0 0
$$682$$ −5.27462e6 −0.434241
$$683$$ −2.05830e7 −1.68833 −0.844164 0.536084i $$-0.819903\pi$$
−0.844164 + 0.536084i $$0.819903\pi$$
$$684$$ 0 0
$$685$$ 4.10595e6 0.334339
$$686$$ −9.29368e6 −0.754011
$$687$$ 0 0
$$688$$ −623104. −0.0501868
$$689$$ 1.01464e7 0.814265
$$690$$ 0 0
$$691$$ −9.54825e6 −0.760727 −0.380363 0.924837i $$-0.624201\pi$$
−0.380363 + 0.924837i $$0.624201\pi$$
$$692$$ 5.99126e6 0.475612
$$693$$ 0 0
$$694$$ −9.30895e6 −0.733672
$$695$$ −7.05250e6 −0.553836
$$696$$ 0 0
$$697$$ −288036. −0.0224577
$$698$$ 1.24516e6 0.0967357
$$699$$ 0 0
$$700$$ −1.18000e6 −0.0910200
$$701$$ −1.29304e6 −0.0993843 −0.0496921 0.998765i $$-0.515824\pi$$
−0.0496921 + 0.998765i $$0.515824\pi$$
$$702$$ 0 0
$$703$$ 1.51193e7 1.15384
$$704$$ −786432. −0.0598039
$$705$$ 0 0
$$706$$ −1.23463e7 −0.932234
$$707$$ 9.15940e6 0.689157
$$708$$ 0 0
$$709$$ −2.12720e7 −1.58926 −0.794628 0.607097i $$-0.792334\pi$$
−0.794628 + 0.607097i $$0.792334\pi$$
$$710$$ 7.02120e6 0.522716
$$711$$ 0 0
$$712$$ 2.46336e6 0.182108
$$713$$ 1.07553e7 0.792316
$$714$$ 0 0
$$715$$ −5.30880e6 −0.388357
$$716$$ −4.35360e6 −0.317370
$$717$$ 0 0
$$718$$ −1.41230e7 −1.02239
$$719$$ −8.31732e6 −0.600014 −0.300007 0.953937i $$-0.596989\pi$$
−0.300007 + 0.953937i $$0.596989\pi$$
$$720$$ 0 0
$$721$$ 5.51225e6 0.394903
$$722$$ −2.01260e7 −1.43686
$$723$$ 0 0
$$724$$ −1.20669e6 −0.0855556
$$725$$ −3.69375e6 −0.260989
$$726$$ 0 0
$$727$$ −4.36740e6 −0.306469 −0.153235 0.988190i $$-0.548969\pi$$
−0.153235 + 0.988190i $$0.548969\pi$$
$$728$$ 8.35251e6 0.584102
$$729$$ 0 0
$$730$$ −2.19860e6 −0.152700
$$731$$ 1.85471e6 0.128375
$$732$$ 0 0
$$733$$ −4.05645e6 −0.278860 −0.139430 0.990232i $$-0.544527\pi$$
−0.139430 + 0.990232i $$0.544527\pi$$
$$734$$ −143048. −0.00980035
$$735$$ 0 0
$$736$$ 1.60358e6 0.109118
$$737$$ −6.47462e6 −0.439082
$$738$$ 0 0
$$739$$ 768260. 0.0517484 0.0258742 0.999665i $$-0.491763\pi$$
0.0258742 + 0.999665i $$0.491763\pi$$
$$740$$ −2.20720e6 −0.148171
$$741$$ 0 0
$$742$$ 4.33013e6 0.288729
$$743$$ −6.18781e6 −0.411211 −0.205605 0.978635i $$-0.565916\pi$$
−0.205605 + 0.978635i $$0.565916\pi$$
$$744$$ 0 0
$$745$$ 9.72375e6 0.641864
$$746$$ 6.86102e6 0.451379
$$747$$ 0 0
$$748$$ 2.34086e6 0.152976
$$749$$ −122484. −0.00797765
$$750$$ 0 0
$$751$$ 1.81698e7 1.17557 0.587787 0.809016i $$-0.299999\pi$$
0.587787 + 0.809016i $$0.299999\pi$$
$$752$$ −3.35923e6 −0.216618
$$753$$ 0 0
$$754$$ 2.61458e7 1.67484
$$755$$ −2.44870e6 −0.156339
$$756$$ 0 0
$$757$$ 1.93494e7 1.22724 0.613618 0.789603i $$-0.289714\pi$$
0.613618 + 0.789603i $$0.289714\pi$$
$$758$$ 1.24070e7 0.784318
$$759$$ 0 0
$$760$$ 4.38400e6 0.275319
$$761$$ 3.01992e7 1.89031 0.945155 0.326621i $$-0.105910\pi$$
0.945155 + 0.326621i $$0.105910\pi$$
$$762$$ 0 0
$$763$$ −2.44177e7 −1.51843
$$764$$ 5.71181e6 0.354030
$$765$$ 0 0
$$766$$ 2.12779e7 1.31026
$$767$$ 3.86879e7 2.37458
$$768$$ 0 0
$$769$$ 2.15854e7 1.31627 0.658134 0.752901i $$-0.271346\pi$$
0.658134 + 0.752901i $$0.271346\pi$$
$$770$$ −2.26560e6 −0.137707
$$771$$ 0 0
$$772$$ −7.01910e6 −0.423876
$$773$$ −3.90895e6 −0.235294 −0.117647 0.993055i $$-0.537535\pi$$
−0.117647 + 0.993055i $$0.537535\pi$$
$$774$$ 0 0
$$775$$ −4.29250e6 −0.256718
$$776$$ 122752. 0.00731769
$$777$$ 0 0
$$778$$ 4.64580e6 0.275177
$$779$$ −1.03572e6 −0.0611503
$$780$$ 0 0
$$781$$ 1.34807e7 0.790833
$$782$$ −4.77317e6 −0.279119
$$783$$ 0 0
$$784$$ −738048. −0.0428839
$$785$$ −92950.0 −0.00538363
$$786$$ 0 0
$$787$$ −2.65082e7 −1.52561 −0.762806 0.646628i $$-0.776179\pi$$
−0.762806 + 0.646628i $$0.776179\pi$$
$$788$$ 2.50877e6 0.143928
$$789$$ 0 0
$$790$$ −452000. −0.0257674
$$791$$ 1.64475e7 0.934674
$$792$$ 0 0
$$793$$ −1.08808e7 −0.614439
$$794$$ −2.51425e6 −0.141533
$$795$$ 0 0
$$796$$ −2.60032e6 −0.145460
$$797$$ −1.07940e7 −0.601919 −0.300960 0.953637i $$-0.597307\pi$$
−0.300960 + 0.953637i $$0.597307\pi$$
$$798$$ 0 0
$$799$$ 9.99896e6 0.554100
$$800$$ −640000. −0.0353553
$$801$$ 0 0
$$802$$ −1.08973e7 −0.598249
$$803$$ −4.22131e6 −0.231025
$$804$$ 0 0
$$805$$ 4.61970e6 0.251260
$$806$$ 3.03840e7 1.64743
$$807$$ 0 0
$$808$$ 4.96781e6 0.267693
$$809$$ 1.11446e7 0.598675 0.299338 0.954147i $$-0.403234\pi$$
0.299338 + 0.954147i $$0.403234\pi$$
$$810$$ 0 0
$$811$$ −1.14866e7 −0.613253 −0.306626 0.951830i $$-0.599200\pi$$
−0.306626 + 0.951830i $$0.599200\pi$$
$$812$$ 1.11581e7 0.593881
$$813$$ 0 0
$$814$$ −4.23782e6 −0.224172
$$815$$ −1.08085e6 −0.0569995
$$816$$ 0 0
$$817$$ 6.66916e6 0.349555
$$818$$ −7.12076e6 −0.372086
$$819$$ 0 0
$$820$$ 151200. 0.00785267
$$821$$ −3.04347e7 −1.57584 −0.787918 0.615781i $$-0.788841\pi$$
−0.787918 + 0.615781i $$0.788841\pi$$
$$822$$ 0 0
$$823$$ 4.09773e6 0.210884 0.105442 0.994425i $$-0.466374\pi$$
0.105442 + 0.994425i $$0.466374\pi$$
$$824$$ 2.98970e6 0.153394
$$825$$ 0 0
$$826$$ 1.65106e7 0.841999
$$827$$ 1.70652e7 0.867654 0.433827 0.900996i $$-0.357163\pi$$
0.433827 + 0.900996i $$0.357163\pi$$
$$828$$ 0 0
$$829$$ −2.47617e7 −1.25139 −0.625697 0.780066i $$-0.715185\pi$$
−0.625697 + 0.780066i $$0.715185\pi$$
$$830$$ −1.09074e7 −0.549574
$$831$$ 0 0
$$832$$ 4.53018e6 0.226886
$$833$$ 2.19685e6 0.109695
$$834$$ 0 0
$$835$$ −4.66305e6 −0.231448
$$836$$ 8.41728e6 0.416539
$$837$$ 0 0
$$838$$ 2.60232e6 0.128012
$$839$$ −3.16529e7 −1.55242 −0.776208 0.630476i $$-0.782860\pi$$
−0.776208 + 0.630476i $$0.782860\pi$$
$$840$$ 0 0
$$841$$ 1.44170e7 0.702884
$$842$$ 1.41624e7 0.688425
$$843$$ 0 0
$$844$$ −2.90637e6 −0.140441
$$845$$ 2.12986e7 1.02615
$$846$$ 0 0
$$847$$ 1.46541e7 0.701859
$$848$$ 2.34854e6 0.112153
$$849$$ 0 0
$$850$$ 1.90500e6 0.0904373
$$851$$ 8.64119e6 0.409025
$$852$$ 0 0
$$853$$ 2.82671e7 1.33017 0.665087 0.746765i $$-0.268394\pi$$
0.665087 + 0.746765i $$0.268394\pi$$
$$854$$ −4.64354e6 −0.217873
$$855$$ 0 0
$$856$$ −66432.0 −0.00309880
$$857$$ −2.60870e7 −1.21331 −0.606655 0.794966i $$-0.707489\pi$$
−0.606655 + 0.794966i $$0.707489\pi$$
$$858$$ 0 0
$$859$$ −3.38111e7 −1.56342 −0.781710 0.623642i $$-0.785652\pi$$
−0.781710 + 0.623642i $$0.785652\pi$$
$$860$$ −973600. −0.0448884
$$861$$ 0 0
$$862$$ −2.19499e6 −0.100615
$$863$$ −2.22817e7 −1.01841 −0.509204 0.860646i $$-0.670060\pi$$
−0.509204 + 0.860646i $$0.670060\pi$$
$$864$$ 0 0
$$865$$ 9.36135e6 0.425401
$$866$$ 5.96966e6 0.270492
$$867$$ 0 0
$$868$$ 1.29668e7 0.584162
$$869$$ −867840. −0.0389843
$$870$$ 0 0
$$871$$ 3.72965e7 1.66580
$$872$$ −1.32435e7 −0.589810
$$873$$ 0 0
$$874$$ −1.71634e7 −0.760018
$$875$$ −1.84375e6 −0.0814108
$$876$$ 0 0
$$877$$ −3.46748e7 −1.52235 −0.761177 0.648545i $$-0.775378\pi$$
−0.761177 + 0.648545i $$0.775378\pi$$
$$878$$ −1.94485e7 −0.851431
$$879$$ 0 0
$$880$$ −1.22880e6 −0.0534902
$$881$$ −1.42603e7 −0.618998 −0.309499 0.950900i $$-0.600161\pi$$
−0.309499 + 0.950900i $$0.600161\pi$$
$$882$$ 0 0
$$883$$ −3.75177e7 −1.61933 −0.809663 0.586895i $$-0.800350\pi$$
−0.809663 + 0.586895i $$0.800350\pi$$
$$884$$ −1.34844e7 −0.580363
$$885$$ 0 0
$$886$$ −7.44622e6 −0.318677
$$887$$ −4.07657e7 −1.73975 −0.869873 0.493275i $$-0.835800\pi$$
−0.869873 + 0.493275i $$0.835800\pi$$
$$888$$ 0 0
$$889$$ −3.53861e7 −1.50168
$$890$$ 3.84900e6 0.162882
$$891$$ 0 0
$$892$$ −4.61238e6 −0.194095
$$893$$ 3.59543e7 1.50877
$$894$$ 0 0
$$895$$ −6.80250e6 −0.283864
$$896$$ 1.93331e6 0.0804511
$$897$$ 0 0
$$898$$ 1.49488e7 0.618606
$$899$$ 4.05899e7 1.67501
$$900$$ 0 0
$$901$$ −6.99059e6 −0.286881
$$902$$ 290304. 0.0118806
$$903$$ 0 0
$$904$$ 8.92070e6 0.363060
$$905$$ −1.88545e6 −0.0765233
$$906$$ 0 0
$$907$$ −3.57116e7 −1.44142 −0.720712 0.693235i $$-0.756185\pi$$
−0.720712 + 0.693235i $$0.756185\pi$$
$$908$$ −1.80084e7 −0.724869
$$909$$ 0 0
$$910$$ 1.30508e7 0.522437
$$911$$ 2.11389e7 0.843893 0.421947 0.906621i $$-0.361347\pi$$
0.421947 + 0.906621i $$0.361347\pi$$
$$912$$ 0 0
$$913$$ −2.09422e7 −0.831468
$$914$$ 2.59310e7 1.02673
$$915$$ 0 0
$$916$$ −6.65296e6 −0.261985
$$917$$ 928896. 0.0364791
$$918$$ 0 0
$$919$$ 1.85996e7 0.726465 0.363233 0.931698i $$-0.381673\pi$$
0.363233 + 0.931698i $$0.381673\pi$$
$$920$$ 2.50560e6 0.0975983
$$921$$ 0 0
$$922$$ 6.03641e6 0.233857
$$923$$ −7.76545e7 −3.00028
$$924$$ 0 0
$$925$$ −3.44875e6 −0.132528
$$926$$ −3.47360e7 −1.33123
$$927$$ 0 0
$$928$$ 6.05184e6 0.230684
$$929$$ −4.45110e7 −1.69211 −0.846055 0.533096i $$-0.821028\pi$$
−0.846055 + 0.533096i $$0.821028\pi$$
$$930$$ 0 0
$$931$$ 7.89942e6 0.298690
$$932$$ −1.23294e7 −0.464945
$$933$$ 0 0
$$934$$ 2.78565e7 1.04486
$$935$$ 3.65760e6 0.136826
$$936$$ 0 0
$$937$$ −2.19419e7 −0.816441 −0.408221 0.912883i $$-0.633851\pi$$
−0.408221 + 0.912883i $$0.633851\pi$$
$$938$$ 1.59168e7 0.590675
$$939$$ 0 0
$$940$$ −5.24880e6 −0.193749
$$941$$ 7.77722e6 0.286319 0.143160 0.989700i $$-0.454274\pi$$
0.143160 + 0.989700i $$0.454274\pi$$
$$942$$ 0 0
$$943$$ −591948. −0.0216773
$$944$$ 8.95488e6 0.327062
$$945$$ 0 0
$$946$$ −1.86931e6 −0.0679132
$$947$$ −3.17199e7 −1.14936 −0.574681 0.818378i $$-0.694874\pi$$
−0.574681 + 0.818378i $$0.694874\pi$$
$$948$$ 0 0
$$949$$ 2.43165e7 0.876468
$$950$$ 6.85000e6 0.246253
$$951$$ 0 0
$$952$$ −5.75462e6 −0.205790
$$953$$ 5.60285e6 0.199838 0.0999188 0.994996i $$-0.468142\pi$$
0.0999188 + 0.994996i $$0.468142\pi$$
$$954$$ 0 0
$$955$$ 8.92470e6 0.316654
$$956$$ 9.52512e6 0.337074
$$957$$ 0 0
$$958$$ −2.20421e7 −0.775959
$$959$$ −1.93801e7 −0.680470
$$960$$ 0 0
$$961$$ 1.85403e7 0.647601
$$962$$ 2.44116e7 0.850470
$$963$$ 0 0
$$964$$ 4.38243e6 0.151888
$$965$$ −1.09673e7 −0.379126
$$966$$ 0 0
$$967$$ −2.03532e7 −0.699949 −0.349975 0.936759i $$-0.613810\pi$$
−0.349975 + 0.936759i $$0.613810\pi$$
$$968$$ 7.94797e6 0.272626
$$969$$ 0 0
$$970$$ 191800. 0.00654514
$$971$$ 2.34306e7 0.797510 0.398755 0.917057i $$-0.369442\pi$$
0.398755 + 0.917057i $$0.369442\pi$$
$$972$$ 0 0
$$973$$ 3.32878e7 1.12721
$$974$$ −2.20723e7 −0.745505
$$975$$ 0 0
$$976$$ −2.51853e6 −0.0846296
$$977$$ 4.30412e7 1.44261 0.721303 0.692619i $$-0.243543\pi$$
0.721303 + 0.692619i $$0.243543\pi$$
$$978$$ 0 0
$$979$$ 7.39008e6 0.246429
$$980$$ −1.15320e6 −0.0383565
$$981$$ 0 0
$$982$$ −6.05107e6 −0.200241
$$983$$ 4.75003e7 1.56788 0.783940 0.620837i $$-0.213207\pi$$
0.783940 + 0.620837i $$0.213207\pi$$
$$984$$ 0 0
$$985$$ 3.91995e6 0.128733
$$986$$ −1.80137e7 −0.590079
$$987$$ 0 0
$$988$$ −4.84870e7 −1.58028
$$989$$ 3.81164e6 0.123914
$$990$$ 0 0
$$991$$ 2.09231e7 0.676770 0.338385 0.941008i $$-0.390119\pi$$
0.338385 + 0.941008i $$0.390119\pi$$
$$992$$ 7.03283e6 0.226909
$$993$$ 0 0
$$994$$ −3.31401e7 −1.06387
$$995$$ −4.06300e6 −0.130104
$$996$$ 0 0
$$997$$ 2.96332e7 0.944148 0.472074 0.881559i $$-0.343505\pi$$
0.472074 + 0.881559i $$0.343505\pi$$
$$998$$ 7.72168e6 0.245406
$$999$$ 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 90.6.a.b.1.1 1
3.2 odd 2 10.6.a.c.1.1 1
4.3 odd 2 720.6.a.v.1.1 1
5.2 odd 4 450.6.c.f.199.1 2
5.3 odd 4 450.6.c.f.199.2 2
5.4 even 2 450.6.a.u.1.1 1
12.11 even 2 80.6.a.c.1.1 1
15.2 even 4 50.6.b.b.49.2 2
15.8 even 4 50.6.b.b.49.1 2
15.14 odd 2 50.6.a.b.1.1 1
21.20 even 2 490.6.a.k.1.1 1
24.5 odd 2 320.6.a.f.1.1 1
24.11 even 2 320.6.a.k.1.1 1
60.23 odd 4 400.6.c.i.49.1 2
60.47 odd 4 400.6.c.i.49.2 2
60.59 even 2 400.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 3.2 odd 2
50.6.a.b.1.1 1 15.14 odd 2
50.6.b.b.49.1 2 15.8 even 4
50.6.b.b.49.2 2 15.2 even 4
80.6.a.c.1.1 1 12.11 even 2
90.6.a.b.1.1 1 1.1 even 1 trivial
320.6.a.f.1.1 1 24.5 odd 2
320.6.a.k.1.1 1 24.11 even 2
400.6.a.i.1.1 1 60.59 even 2
400.6.c.i.49.1 2 60.23 odd 4
400.6.c.i.49.2 2 60.47 odd 4
450.6.a.u.1.1 1 5.4 even 2
450.6.c.f.199.1 2 5.2 odd 4
450.6.c.f.199.2 2 5.3 odd 4
490.6.a.k.1.1 1 21.20 even 2
720.6.a.v.1.1 1 4.3 odd 2