# Properties

 Label 450.6.a.u.1.1 Level $450$ Weight $6$ Character 450.1 Self dual yes Analytic conductor $72.173$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ 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 $$\chi$$ $$=$$ 450.1

## $q$-expansion

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

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 118.000 0.910200 0.455100 0.890440i $$-0.349603\pi$$
0.455100 + 0.890440i $$0.349603\pi$$
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −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 $$-0.862044\pi$$
−0.907542 + 0.419961i $$0.862044\pi$$
$$14$$ 472.000 0.643609
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 762.000 0.639488 0.319744 0.947504i $$-0.396403\pi$$
0.319744 + 0.947504i $$0.396403\pi$$
$$18$$ 0 0
$$19$$ −2740.00 −1.74127 −0.870636 0.491928i $$-0.836292\pi$$
−0.870636 + 0.491928i $$0.836292\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −768.000 −0.338302
$$23$$ 1566.00 0.617266 0.308633 0.951181i $$-0.400129\pi$$
0.308633 + 0.951181i $$0.400129\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −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
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5518.00 0.662640 0.331320 0.943519i $$-0.392506\pi$$
0.331320 + 0.943519i $$0.392506\pi$$
$$38$$ −10960.0 −1.23127
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 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 $$-0.467996\pi$$
0.100374 + 0.994950i $$0.467996\pi$$
$$44$$ −3072.00 −0.239216
$$45$$ 0 0
$$46$$ 6264.00 0.436473
$$47$$ 13122.0 0.866474 0.433237 0.901280i $$-0.357371\pi$$
0.433237 + 0.901280i $$0.357371\pi$$
$$48$$ 0 0
$$49$$ −2883.00 −0.171536
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −17696.0 −0.907542
$$53$$ −9174.00 −0.448610 −0.224305 0.974519i $$-0.572011\pi$$
−0.224305 + 0.974519i $$0.572011\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −33722.0 −0.917754 −0.458877 0.888500i $$-0.651748\pi$$
−0.458877 + 0.888500i $$0.651748\pi$$
$$68$$ 12192.0 0.319744
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −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 $$-0.577620\pi$$
−0.241440 + 0.970416i $$0.577620\pi$$
$$74$$ 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 $$0.487028\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1512.00 0.0248323
$$83$$ −109074. −1.73790 −0.868952 0.494896i $$-0.835206\pi$$
−0.868952 + 0.494896i $$0.835206\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 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
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1918.00 0.0206976 0.0103488 0.999946i $$-0.496706\pi$$
0.0103488 + 0.999946i $$0.496706\pi$$
$$98$$ −11532.0 −0.121294
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −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 $$-0.430395\pi$$
0.216932 + 0.976187i $$0.430395\pi$$
$$104$$ −70784.0 −0.641729
$$105$$ 0 0
$$106$$ −36696.0 −0.317215
$$107$$ −1038.00 −0.00876472 −0.00438236 0.999990i $$-0.501395\pi$$
−0.00438236 + 0.999990i $$0.501395\pi$$
$$108$$ 0 0
$$109$$ 206930. 1.66823 0.834117 0.551587i $$-0.185977\pi$$
0.834117 + 0.551587i $$0.185977\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 30208.0 0.227550
$$113$$ 139386. 1.02689 0.513444 0.858123i $$-0.328369\pi$$
0.513444 + 0.858123i $$0.328369\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −299882. −1.64984 −0.824919 0.565252i $$-0.808779\pi$$
−0.824919 + 0.565252i $$0.808779\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −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 $$-0.621946\pi$$
−0.373803 + 0.927508i $$0.621946\pi$$
$$138$$ 0 0
$$139$$ −282100. −1.23841 −0.619207 0.785228i $$-0.712546\pi$$
−0.619207 + 0.785228i $$0.712546\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −280848. −1.16883
$$143$$ 212352. 0.868393
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −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
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3718.00 0.0120382 0.00601908 0.999982i $$-0.498084\pi$$
0.00601908 + 0.999982i $$0.498084\pi$$
$$158$$ 18080.0 0.0576177
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 184788. 0.561835
$$162$$ 0 0
$$163$$ 43234.0 0.127455 0.0637274 0.997967i $$-0.479701\pi$$
0.0637274 + 0.997967i $$0.479701\pi$$
$$164$$ 6048.00 0.0175591
$$165$$ 0 0
$$166$$ −436296. −1.22888
$$167$$ 186522. 0.517534 0.258767 0.965940i $$-0.416684\pi$$
0.258767 + 0.965940i $$0.416684\pi$$
$$168$$ 0 0
$$169$$ 851943. 2.29453
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 38944.0 0.100374
$$173$$ −374454. −0.951225 −0.475612 0.879655i $$-0.657774\pi$$
−0.475612 + 0.879655i $$0.657774\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −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
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −146304. −0.305951
$$188$$ 209952. 0.433237
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 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 $$-0.360669\pi$$
0.423876 + 0.905720i $$0.360669\pi$$
$$194$$ 7672.00 0.0146354
$$195$$ 0 0
$$196$$ −46128.0 −0.0857678
$$197$$ −156798. −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$198$$ 0 0
$$199$$ −162520. −0.290920 −0.145460 0.989364i $$-0.546466\pi$$
−0.145460 + 0.989364i $$0.546466\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −310488. −0.535385
$$203$$ −697380. −1.18776
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 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
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −810424. −1.16832
$$218$$ 827720. 1.17962
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −842772. −1.16073
$$222$$ 0 0
$$223$$ 288274. 0.388189 0.194095 0.980983i $$-0.437823\pi$$
0.194095 + 0.980983i $$0.437823\pi$$
$$224$$ 120832. 0.160902
$$225$$ 0 0
$$226$$ 557544. 0.726119
$$227$$ 1.12552e6 1.44974 0.724869 0.688887i $$-0.241900\pi$$
0.724869 + 0.688887i $$0.241900\pi$$
$$228$$ 0 0
$$229$$ −415810. −0.523970 −0.261985 0.965072i $$-0.584377\pi$$
−0.261985 + 0.965072i $$0.584377\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −378240. −0.461368
$$233$$ 770586. 0.929889 0.464945 0.885340i $$-0.346074\pi$$
0.464945 + 0.885340i $$0.346074\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 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
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.03044e6 3.16055
$$248$$ −439552. −0.453817
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −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 $$-0.372556\pi$$
0.389765 + 0.920914i $$0.372556\pi$$
$$258$$ 0 0
$$259$$ 651124. 0.603135
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −31488.0 −0.0283395
$$263$$ 1.36465e6 1.21655 0.608276 0.793726i $$-0.291861\pi$$
0.608276 + 0.793726i $$0.291861\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −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
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −438602. −0.343456 −0.171728 0.985144i $$-0.554935\pi$$
−0.171728 + 0.985144i $$0.554935\pi$$
$$278$$ −1.12840e6 −0.875691
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 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 $$-0.485773\pi$$
0.0446795 + 0.999001i $$0.485773\pi$$
$$284$$ −1.12339e6 −0.826486
$$285$$ 0 0
$$286$$ 849408. 0.614047
$$287$$ 44604.0 0.0319646
$$288$$ 0 0
$$289$$ −839213. −0.591055
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −351776. −0.241440
$$293$$ −2.64209e6 −1.79796 −0.898978 0.437993i $$-0.855689\pi$$
−0.898978 + 0.437993i $$0.855689\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.44756e6 0.876577 0.438288 0.898834i $$-0.355585\pi$$
0.438288 + 0.898834i $$0.355585\pi$$
$$308$$ −362496. −0.217734
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 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 $$-0.730391\pi$$
−0.662232 + 0.749299i $$0.730391\pi$$
$$314$$ 14872.0 0.00851227
$$315$$ 0 0
$$316$$ 72320.0 0.0407418
$$317$$ 2.73652e6 1.52950 0.764752 0.644324i $$-0.222861\pi$$
0.764752 + 0.644324i $$0.222861\pi$$
$$318$$ 0 0
$$319$$ 1.13472e6 0.624327
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 739152. 0.397278
$$323$$ −2.08788e6 −1.11352
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 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
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.21088e6 1.06045 0.530225 0.847857i $$-0.322108\pi$$
0.530225 + 0.847857i $$0.322108\pi$$
$$338$$ 3.40777e6 1.62248
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 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 $$-0.673615\pi$$
−0.518785 + 0.854905i $$0.673615\pi$$
$$348$$ 0 0
$$349$$ −311290. −0.136805 −0.0684024 0.997658i $$-0.521790\pi$$
−0.0684024 + 0.997658i $$0.521790\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −196608. −0.0845755
$$353$$ −3.08657e6 −1.31838 −0.659189 0.751977i $$-0.729100\pi$$
−0.659189 + 0.751977i $$0.729100\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −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
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −35762.0 −0.0138598 −0.00692989 0.999976i $$-0.502206\pi$$
−0.00692989 + 0.999976i $$0.502206\pi$$
$$368$$ 400896. 0.154316
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.08253e6 −0.408325
$$372$$ 0 0
$$373$$ 1.71525e6 0.638346 0.319173 0.947696i $$-0.396595\pi$$
0.319173 + 0.947696i $$0.396595\pi$$
$$374$$ −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 $$0.687127\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1.42795e6 0.500674
$$383$$ 5.31949e6 1.85299 0.926494 0.376309i $$-0.122807\pi$$
0.926494 + 0.376309i $$0.122807\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −628562. −0.200157 −0.100079 0.994980i $$-0.531909\pi$$
−0.100079 + 0.994980i $$0.531909\pi$$
$$398$$ −650080. −0.205712
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 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 $$0.415254\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 747424. 0.216932
$$413$$ 4.12764e6 1.19077
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −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
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.16088e6 −0.308119
$$428$$ −16608.0 −0.00438236
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 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 $$-0.438740\pi$$
0.191267 + 0.981538i $$0.438740\pi$$
$$434$$ −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 $$0.294350\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.37109e6 −0.820757
$$443$$ −1.86155e6 −0.450678 −0.225339 0.974280i $$-0.572349\pi$$
−0.225339 + 0.974280i $$0.572349\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 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
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.48276e6 1.45201 0.726005 0.687690i $$-0.241375\pi$$
0.726005 + 0.687690i $$0.241375\pi$$
$$458$$ −1.66324e6 −0.370503
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −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 $$-0.890414\pi$$
−0.941321 + 0.337513i $$0.890414\pi$$
$$464$$ −1.51296e6 −0.326236
$$465$$ 0 0
$$466$$ 3.08234e6 0.657531
$$467$$ 6.96412e6 1.47766 0.738829 0.673893i $$-0.235379\pi$$
0.738829 + 0.673893i $$0.235379\pi$$
$$468$$ 0 0
$$469$$ −3.97920e6 −0.835340
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 2.23872e6 0.462535
$$473$$ −467328. −0.0960437
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 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
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5.51808e6 −1.05430 −0.527152 0.849771i $$-0.676740\pi$$
−0.527152 + 0.849771i $$0.676740\pi$$
$$488$$ −629632. −0.119684
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 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 $$0.555517\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −3.40301e6 −0.602703
$$503$$ 6.73105e6 1.18621 0.593106 0.805124i $$-0.297901\pi$$
0.593106 + 0.805124i $$0.297901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −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
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.51942e6 −0.414548
$$518$$ 2.60450e6 0.426481
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −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 $$-0.309843\pi$$
0.562491 + 0.826804i $$0.309843\pi$$
$$524$$ −125952. −0.0200390
$$525$$ 0 0
$$526$$ 5.45858e6 0.860232
$$527$$ −5.23342e6 −0.820840
$$528$$ 0 0
$$529$$ −3.98399e6 −0.618983
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −5.17312e6 −0.792453
$$533$$ −418068. −0.0637425
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −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
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.44024e6 −0.634510 −0.317255 0.948340i $$-0.602761\pi$$
−0.317255 + 0.948340i $$0.602761\pi$$
$$548$$ −2.62781e6 −0.373803
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 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 $$-0.711071\pi$$
−0.615563 + 0.788088i $$0.711071\pi$$
$$558$$ 0 0
$$559$$ −2.69200e6 −0.364373
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.82679e6 0.778196
$$563$$ −9.81287e6 −1.30474 −0.652372 0.757899i $$-0.726226\pi$$
−0.652372 + 0.757899i $$0.726226\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 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
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −4.50372e6 −0.563160 −0.281580 0.959538i $$-0.590859\pi$$
−0.281580 + 0.959538i $$0.590859\pi$$
$$578$$ −3.35685e6 −0.417939
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −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 $$-0.488066\pi$$
0.0374834 + 0.999297i $$0.488066\pi$$
$$588$$ 0 0
$$589$$ 1.88183e7 2.23508
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.41261e6 0.165660
$$593$$ −2.50385e6 −0.292397 −0.146198 0.989255i $$-0.546704\pi$$
−0.146198 + 0.989255i $$0.546704\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.13772e7 −1.25333 −0.626663 0.779291i $$-0.715580\pi$$
−0.626663 + 0.779291i $$0.715580\pi$$
$$608$$ −2.80576e6 −0.307816
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.45129e7 −1.57272
$$612$$ 0 0
$$613$$ 7.00161e6 0.752570 0.376285 0.926504i $$-0.377201\pi$$
0.376285 + 0.926504i $$0.377201\pi$$
$$614$$ 5.79023e6 0.619833
$$615$$ 0 0
$$616$$ −1.44998e6 −0.153961
$$617$$ 7.90300e6 0.835755 0.417878 0.908503i $$-0.362774\pi$$
0.417878 + 0.908503i $$0.362774\pi$$
$$618$$ 0 0
$$619$$ 4.02362e6 0.422076 0.211038 0.977478i $$-0.432316\pi$$
0.211038 + 0.977478i $$0.432316\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 3.71227e6 0.384737
$$623$$ −4.54182e6 −0.468824
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −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
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.18860e6 0.311352
$$638$$ 4.53888e6 0.441466
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −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 $$-0.576713\pi$$
−0.238676 + 0.971099i $$0.576713\pi$$
$$644$$ 2.95661e6 0.280918
$$645$$ 0 0
$$646$$ −8.35152e6 −0.787380
$$647$$ −8.71928e6 −0.818879 −0.409440 0.912337i $$-0.634276\pi$$
−0.409440 + 0.912337i $$0.634276\pi$$
$$648$$ 0 0
$$649$$ −6.71616e6 −0.625906
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 691744. 0.0637274
$$653$$ −1.58477e6 −0.145440 −0.0727201 0.997352i $$-0.523168\pi$$
−0.0727201 + 0.997352i $$0.523168\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 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
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.25506e6 −0.805498
$$668$$ 2.98435e6 0.258767
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1.88890e6 0.161958
$$672$$ 0 0
$$673$$ −1.11313e7 −0.947349 −0.473675 0.880700i $$-0.657073\pi$$
−0.473675 + 0.880700i $$0.657073\pi$$
$$674$$ 8.84351e6 0.749851
$$675$$ 0 0
$$676$$ 1.36311e7 1.14727
$$677$$ −235518. −0.0197493 −0.00987467 0.999951i $$-0.503143\pi$$
−0.00987467 + 0.999951i $$0.503143\pi$$
$$678$$ 0 0
$$679$$ 226324. 0.0188389
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 5.27462e6 0.434241
$$683$$ 2.05830e7 1.68833 0.844164 0.536084i $$-0.180097\pi$$
0.844164 + 0.536084i $$0.180097\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −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
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 288036. 0.0224577
$$698$$ −1.24516e6 −0.0967357
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −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 $$0.792334\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2.46336e6 −0.182108
$$713$$ −1.07553e7 −0.792316
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.36740e6 0.306469 0.153235 0.988190i $$-0.451031\pi$$
0.153235 + 0.988190i $$0.451031\pi$$
$$728$$ −8.35251e6 −0.584102
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.85471e6 0.128375
$$732$$ 0 0
$$733$$ 4.05645e6 0.278860 0.139430 0.990232i $$-0.455473\pi$$
0.139430 + 0.990232i $$0.455473\pi$$
$$734$$ −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 $$0.491763\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −4.33013e6 −0.288729
$$743$$ 6.18781e6 0.411211 0.205605 0.978635i $$-0.434084\pi$$
0.205605 + 0.978635i $$0.434084\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −1.93494e7 −1.22724 −0.613618 0.789603i $$-0.710286\pi$$
−0.613618 + 0.789603i $$0.710286\pi$$
$$758$$ −1.24070e7 −0.784318
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 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 $$0.271346\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7.01910e6 0.423876
$$773$$ 3.90895e6 0.235294 0.117647 0.993055i $$-0.462465\pi$$
0.117647 + 0.993055i $$0.462465\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 2.65082e7 1.52561 0.762806 0.646628i $$-0.223821\pi$$
0.762806 + 0.646628i $$0.223821\pi$$
$$788$$ −2.50877e6 −0.143928
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 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 $$-0.402693\pi$$
0.300960 + 0.953637i $$0.402693\pi$$
$$798$$ 0 0
$$799$$ 9.99896e6 0.554100
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 1.08973e7 0.598249
$$803$$ 4.22131e6 0.231025
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 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
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −6.66916e6 −0.349555
$$818$$ 7.12076e6 0.372086
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −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 $$-0.533626\pi$$
−0.105442 + 0.994425i $$0.533626\pi$$
$$824$$ 2.98970e6 0.153394
$$825$$ 0 0
$$826$$ 1.65106e7 0.841999
$$827$$ −1.70652e7 −0.867654 −0.433827 0.900996i $$-0.642837\pi$$
−0.433827 + 0.900996i $$0.642837\pi$$
$$828$$ 0 0
$$829$$ −2.47617e7 −1.25139 −0.625697 0.780066i $$-0.715185\pi$$
−0.625697 + 0.780066i $$0.715185\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.53018e6 −0.226886
$$833$$ −2.19685e6 −0.109695
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.46541e7 −0.701859
$$848$$ −2.34854e6 −0.112153
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.64119e6 0.409025
$$852$$ 0 0
$$853$$ −2.82671e7 −1.33017 −0.665087 0.746765i $$-0.731606\pi$$
−0.665087 + 0.746765i $$0.731606\pi$$
$$854$$ −4.64354e6 −0.217873
$$855$$ 0 0
$$856$$ −66432.0 −0.00309880
$$857$$ 2.60870e7 1.21331 0.606655 0.794966i $$-0.292511\pi$$
0.606655 + 0.794966i $$0.292511\pi$$
$$858$$ 0 0
$$859$$ −3.38111e7 −1.56342 −0.781710 0.623642i $$-0.785652\pi$$
−0.781710 + 0.623642i $$0.785652\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 2.19499e6 0.100615
$$863$$ 2.22817e7 1.01841 0.509204 0.860646i $$-0.329940\pi$$
0.509204 + 0.860646i $$0.329940\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 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
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.46748e7 1.52235 0.761177 0.648545i $$-0.224622\pi$$
0.761177 + 0.648545i $$0.224622\pi$$
$$878$$ 1.94485e7 0.851431
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −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 $$-0.199650\pi$$
0.809663 + 0.586895i $$0.199650\pi$$
$$884$$ −1.34844e7 −0.580363
$$885$$ 0 0
$$886$$ −7.44622e6 −0.318677
$$887$$ 4.07657e7 1.73975 0.869873 0.493275i $$-0.164200\pi$$
0.869873 + 0.493275i $$0.164200\pi$$
$$888$$ 0 0
$$889$$ −3.53861e7 −1.50168
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.61238e6 0.194095
$$893$$ −3.59543e7 −1.50877
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 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
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 3.57116e7 1.44142 0.720712 0.693235i $$-0.243815\pi$$
0.720712 + 0.693235i $$0.243815\pi$$
$$908$$ 1.80084e7 0.724869
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 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 $$0.381673\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6.03641e6 −0.233857
$$923$$ 7.76545e7 3.00028
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −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
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.19419e7 0.816441 0.408221 0.912883i $$-0.366149\pi$$
0.408221 + 0.912883i $$0.366149\pi$$
$$938$$ −1.59168e7 −0.590675
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 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 $$-0.305126\pi$$
0.574681 + 0.818378i $$0.305126\pi$$
$$948$$ 0 0
$$949$$ 2.43165e7 0.876468
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 5.75462e6 0.205790
$$953$$ −5.60285e6 −0.199838 −0.0999188 0.994996i $$-0.531858\pi$$
−0.0999188 + 0.994996i $$0.531858\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 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
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.03532e7 0.699949 0.349975 0.936759i $$-0.386190\pi$$
0.349975 + 0.936759i $$0.386190\pi$$
$$968$$ −7.94797e6 −0.272626
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 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 $$-0.756457\pi$$
−0.721303 + 0.692619i $$0.756457\pi$$
$$978$$ 0 0
$$979$$ 7.39008e6 0.246429
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 6.05107e6 0.200241
$$983$$ −4.75003e7 −1.56788 −0.783940 0.620837i $$-0.786793\pi$$
−0.783940 + 0.620837i $$0.786793\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −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
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −2.96332e7 −0.944148 −0.472074 0.881559i $$-0.656495\pi$$
−0.472074 + 0.881559i $$0.656495\pi$$
$$998$$ −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 450.6.a.u.1.1 1
3.2 odd 2 50.6.a.b.1.1 1
5.2 odd 4 450.6.c.f.199.2 2
5.3 odd 4 450.6.c.f.199.1 2
5.4 even 2 90.6.a.b.1.1 1
12.11 even 2 400.6.a.i.1.1 1
15.2 even 4 50.6.b.b.49.1 2
15.8 even 4 50.6.b.b.49.2 2
15.14 odd 2 10.6.a.c.1.1 1
20.19 odd 2 720.6.a.v.1.1 1
60.23 odd 4 400.6.c.i.49.2 2
60.47 odd 4 400.6.c.i.49.1 2
60.59 even 2 80.6.a.c.1.1 1
105.104 even 2 490.6.a.k.1.1 1
120.29 odd 2 320.6.a.f.1.1 1
120.59 even 2 320.6.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 15.14 odd 2
50.6.a.b.1.1 1 3.2 odd 2
50.6.b.b.49.1 2 15.2 even 4
50.6.b.b.49.2 2 15.8 even 4
80.6.a.c.1.1 1 60.59 even 2
90.6.a.b.1.1 1 5.4 even 2
320.6.a.f.1.1 1 120.29 odd 2
320.6.a.k.1.1 1 120.59 even 2
400.6.a.i.1.1 1 12.11 even 2
400.6.c.i.49.1 2 60.47 odd 4
400.6.c.i.49.2 2 60.23 odd 4
450.6.a.u.1.1 1 1.1 even 1 trivial
450.6.c.f.199.1 2 5.3 odd 4
450.6.c.f.199.2 2 5.2 odd 4
490.6.a.k.1.1 1 105.104 even 2
720.6.a.v.1.1 1 20.19 odd 2