# Properties

 Label 720.6.f.n.289.7 Level $720$ Weight $6$ Character 720.289 Analytic conductor $115.476$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$115.476350265$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{31}\cdot 3^{2}\cdot 5^{3}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 289.7 Root $$-4.73066i$$ of defining polynomial Character $$\chi$$ $$=$$ 720.289 Dual form 720.6.f.n.289.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(53.0051 - 17.7613i) q^{5} -188.968i q^{7} +O(q^{10})$$ $$q+(53.0051 - 17.7613i) q^{5} -188.968i q^{7} -501.871 q^{11} +1061.48i q^{13} -29.5861i q^{17} +1578.33 q^{19} -1295.86i q^{23} +(2494.07 - 1882.88i) q^{25} -3586.63 q^{29} +3526.32 q^{31} +(-3356.31 - 10016.2i) q^{35} -8413.79i q^{37} -7015.12 q^{41} -22694.2i q^{43} +3501.99i q^{47} -18901.8 q^{49} -27309.1i q^{53} +(-26601.7 + 8913.88i) q^{55} -7925.39 q^{59} -7020.54 q^{61} +(18853.3 + 56264.0i) q^{65} +17631.2i q^{67} +13432.9 q^{71} +39946.8i q^{73} +94837.3i q^{77} -93321.2 q^{79} -58448.5i q^{83} +(-525.488 - 1568.21i) q^{85} -13989.4 q^{89} +200586. q^{91} +(83659.6 - 28033.2i) q^{95} +110640. i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 53.0051 17.7613i 0.948183 0.317724i
$$6$$ 0 0
$$7$$ 188.968i 1.45761i −0.684720 0.728807i $$-0.740075\pi$$
0.684720 0.728807i $$-0.259925\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −501.871 −1.25058 −0.625288 0.780394i $$-0.715018\pi$$
−0.625288 + 0.780394i $$0.715018\pi$$
$$12$$ 0 0
$$13$$ 1061.48i 1.74203i 0.491259 + 0.871013i $$0.336537\pi$$
−0.491259 + 0.871013i $$0.663463\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 29.5861i 0.0248294i −0.999923 0.0124147i $$-0.996048\pi$$
0.999923 0.0124147i $$-0.00395182\pi$$
$$18$$ 0 0
$$19$$ 1578.33 1.00303 0.501515 0.865149i $$-0.332776\pi$$
0.501515 + 0.865149i $$0.332776\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1295.86i 0.510786i −0.966837 0.255393i $$-0.917795\pi$$
0.966837 0.255393i $$-0.0822048\pi$$
$$24$$ 0 0
$$25$$ 2494.07 1882.88i 0.798103 0.602521i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3586.63 −0.791938 −0.395969 0.918264i $$-0.629591\pi$$
−0.395969 + 0.918264i $$0.629591\pi$$
$$30$$ 0 0
$$31$$ 3526.32 0.659048 0.329524 0.944147i $$-0.393112\pi$$
0.329524 + 0.944147i $$0.393112\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3356.31 10016.2i −0.463118 1.38208i
$$36$$ 0 0
$$37$$ 8413.79i 1.01039i −0.863006 0.505193i $$-0.831421\pi$$
0.863006 0.505193i $$-0.168579\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7015.12 −0.651742 −0.325871 0.945414i $$-0.605657\pi$$
−0.325871 + 0.945414i $$0.605657\pi$$
$$42$$ 0 0
$$43$$ 22694.2i 1.87173i −0.352358 0.935865i $$-0.614620\pi$$
0.352358 0.935865i $$-0.385380\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3501.99i 0.231244i 0.993293 + 0.115622i $$0.0368861\pi$$
−0.993293 + 0.115622i $$0.963114\pi$$
$$48$$ 0 0
$$49$$ −18901.8 −1.12464
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 27309.1i 1.33542i −0.744422 0.667710i $$-0.767275\pi$$
0.744422 0.667710i $$-0.232725\pi$$
$$54$$ 0 0
$$55$$ −26601.7 + 8913.88i −1.18578 + 0.397338i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −7925.39 −0.296409 −0.148204 0.988957i $$-0.547349\pi$$
−0.148204 + 0.988957i $$0.547349\pi$$
$$60$$ 0 0
$$61$$ −7020.54 −0.241572 −0.120786 0.992679i $$-0.538541\pi$$
−0.120786 + 0.992679i $$0.538541\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 18853.3 + 56264.0i 0.553483 + 1.65176i
$$66$$ 0 0
$$67$$ 17631.2i 0.479837i 0.970793 + 0.239919i $$0.0771208\pi$$
−0.970793 + 0.239919i $$0.922879\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13432.9 0.316246 0.158123 0.987419i $$-0.449456\pi$$
0.158123 + 0.987419i $$0.449456\pi$$
$$72$$ 0 0
$$73$$ 39946.8i 0.877353i 0.898645 + 0.438677i $$0.144553\pi$$
−0.898645 + 0.438677i $$0.855447\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 94837.3i 1.82286i
$$78$$ 0 0
$$79$$ −93321.2 −1.68234 −0.841168 0.540774i $$-0.818131\pi$$
−0.841168 + 0.540774i $$0.818131\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 58448.5i 0.931275i −0.884975 0.465638i $$-0.845825\pi$$
0.884975 0.465638i $$-0.154175\pi$$
$$84$$ 0 0
$$85$$ −525.488 1568.21i −0.00788888 0.0235428i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −13989.4 −0.187208 −0.0936038 0.995610i $$-0.529839\pi$$
−0.0936038 + 0.995610i $$0.529839\pi$$
$$90$$ 0 0
$$91$$ 200586. 2.53920
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 83659.6 28033.2i 0.951057 0.318687i
$$96$$ 0 0
$$97$$ 110640.i 1.19394i 0.802264 + 0.596969i $$0.203629\pi$$
−0.802264 + 0.596969i $$0.796371\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −198468. −1.93592 −0.967960 0.251103i $$-0.919207\pi$$
−0.967960 + 0.251103i $$0.919207\pi$$
$$102$$ 0 0
$$103$$ 134780.i 1.25179i −0.779906 0.625897i $$-0.784733\pi$$
0.779906 0.625897i $$-0.215267\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 39785.5i 0.335943i −0.985792 0.167971i $$-0.946278\pi$$
0.985792 0.167971i $$-0.0537216\pi$$
$$108$$ 0 0
$$109$$ −92692.6 −0.747272 −0.373636 0.927575i $$-0.621889\pi$$
−0.373636 + 0.927575i $$0.621889\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 66923.1i 0.493037i 0.969138 + 0.246519i $$0.0792866\pi$$
−0.969138 + 0.246519i $$0.920713\pi$$
$$114$$ 0 0
$$115$$ −23016.2 68687.2i −0.162289 0.484318i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −5590.81 −0.0361916
$$120$$ 0 0
$$121$$ 90823.1 0.563940
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 98756.1 144100.i 0.565313 0.824877i
$$126$$ 0 0
$$127$$ 58998.9i 0.324589i −0.986742 0.162295i $$-0.948110\pi$$
0.986742 0.162295i $$-0.0518895\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −209380. −1.06600 −0.533001 0.846115i $$-0.678936\pi$$
−0.533001 + 0.846115i $$0.678936\pi$$
$$132$$ 0 0
$$133$$ 298254.i 1.46203i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 326432.i 1.48591i 0.669344 + 0.742953i $$0.266575\pi$$
−0.669344 + 0.742953i $$0.733425\pi$$
$$138$$ 0 0
$$139$$ −233648. −1.02571 −0.512856 0.858475i $$-0.671412\pi$$
−0.512856 + 0.858475i $$0.671412\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 532727.i 2.17854i
$$144$$ 0 0
$$145$$ −190109. + 63703.2i −0.750902 + 0.251618i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 145336. 0.536299 0.268149 0.963377i $$-0.413588\pi$$
0.268149 + 0.963377i $$0.413588\pi$$
$$150$$ 0 0
$$151$$ −287343. −1.02555 −0.512777 0.858522i $$-0.671383\pi$$
−0.512777 + 0.858522i $$0.671383\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 186913. 62632.0i 0.624899 0.209395i
$$156$$ 0 0
$$157$$ 42626.2i 0.138015i 0.997616 + 0.0690077i $$0.0219833\pi$$
−0.997616 + 0.0690077i $$0.978017\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −244876. −0.744528
$$162$$ 0 0
$$163$$ 221599.i 0.653279i 0.945149 + 0.326639i $$0.105916\pi$$
−0.945149 + 0.326639i $$0.894084\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 334835.i 0.929051i 0.885560 + 0.464525i $$0.153775\pi$$
−0.885560 + 0.464525i $$0.846225\pi$$
$$168$$ 0 0
$$169$$ −755454. −2.03466
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 268475.i 0.682006i 0.940062 + 0.341003i $$0.110767\pi$$
−0.940062 + 0.341003i $$0.889233\pi$$
$$174$$ 0 0
$$175$$ −355803. 471299.i −0.878242 1.16333i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 268052. 0.625297 0.312649 0.949869i $$-0.398784\pi$$
0.312649 + 0.949869i $$0.398784\pi$$
$$180$$ 0 0
$$181$$ −90565.5 −0.205479 −0.102739 0.994708i $$-0.532761\pi$$
−0.102739 + 0.994708i $$0.532761\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −149440. 445973.i −0.321024 0.958031i
$$186$$ 0 0
$$187$$ 14848.4i 0.0310510i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −706725. −1.40174 −0.700870 0.713290i $$-0.747204\pi$$
−0.700870 + 0.713290i $$0.747204\pi$$
$$192$$ 0 0
$$193$$ 236999.i 0.457988i 0.973428 + 0.228994i $$0.0735435\pi$$
−0.973428 + 0.228994i $$0.926456\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 607901.i 1.11601i −0.829838 0.558004i $$-0.811567\pi$$
0.829838 0.558004i $$-0.188433\pi$$
$$198$$ 0 0
$$199$$ 621060. 1.11173 0.555867 0.831272i $$-0.312387\pi$$
0.555867 + 0.831272i $$0.312387\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 677756.i 1.15434i
$$204$$ 0 0
$$205$$ −371837. + 124598.i −0.617971 + 0.207074i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −792118. −1.25437
$$210$$ 0 0
$$211$$ −1.06177e6 −1.64181 −0.820906 0.571063i $$-0.806531\pi$$
−0.820906 + 0.571063i $$0.806531\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −403078. 1.20291e6i −0.594693 1.77474i
$$216$$ 0 0
$$217$$ 666360.i 0.960638i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 31405.2 0.0432534
$$222$$ 0 0
$$223$$ 240721.i 0.324154i −0.986778 0.162077i $$-0.948181\pi$$
0.986778 0.162077i $$-0.0518193\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 202494.i 0.260824i 0.991460 + 0.130412i $$0.0416299\pi$$
−0.991460 + 0.130412i $$0.958370\pi$$
$$228$$ 0 0
$$229$$ 284279. 0.358225 0.179112 0.983829i $$-0.442677\pi$$
0.179112 + 0.983829i $$0.442677\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 345405.i 0.416811i −0.978042 0.208406i $$-0.933173\pi$$
0.978042 0.208406i $$-0.0668274\pi$$
$$234$$ 0 0
$$235$$ 62199.9 + 185623.i 0.0734717 + 0.219262i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.59920e6 −1.81095 −0.905476 0.424398i $$-0.860486\pi$$
−0.905476 + 0.424398i $$0.860486\pi$$
$$240$$ 0 0
$$241$$ −42246.4 −0.0468541 −0.0234270 0.999726i $$-0.507458\pi$$
−0.0234270 + 0.999726i $$0.507458\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1.00189e6 + 335720.i −1.06636 + 0.357324i
$$246$$ 0 0
$$247$$ 1.67537e6i 1.74731i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 802841. 0.804351 0.402175 0.915563i $$-0.368254\pi$$
0.402175 + 0.915563i $$0.368254\pi$$
$$252$$ 0 0
$$253$$ 650354.i 0.638776i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 941153.i 0.888848i −0.895817 0.444424i $$-0.853408\pi$$
0.895817 0.444424i $$-0.146592\pi$$
$$258$$ 0 0
$$259$$ −1.58993e6 −1.47275
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.53130e6i 1.36512i 0.730830 + 0.682560i $$0.239133\pi$$
−0.730830 + 0.682560i $$0.760867\pi$$
$$264$$ 0 0
$$265$$ −485045. 1.44752e6i −0.424294 1.26622i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 1.25712e6 1.05924 0.529621 0.848234i $$-0.322334\pi$$
0.529621 + 0.848234i $$0.322334\pi$$
$$270$$ 0 0
$$271$$ −843976. −0.698083 −0.349041 0.937107i $$-0.613493\pi$$
−0.349041 + 0.937107i $$0.613493\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.25170e6 + 944961.i −0.998089 + 0.753498i
$$276$$ 0 0
$$277$$ 1.52070e6i 1.19082i −0.803423 0.595408i $$-0.796990\pi$$
0.803423 0.595408i $$-0.203010\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.97975e6 1.49570 0.747848 0.663870i $$-0.231087\pi$$
0.747848 + 0.663870i $$0.231087\pi$$
$$282$$ 0 0
$$283$$ 2.02901e6i 1.50597i −0.658036 0.752986i $$-0.728613\pi$$
0.658036 0.752986i $$-0.271387\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.32563e6i 0.949987i
$$288$$ 0 0
$$289$$ 1.41898e6 0.999384
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.23140e6i 1.51848i −0.650813 0.759238i $$-0.725572\pi$$
0.650813 0.759238i $$-0.274428\pi$$
$$294$$ 0 0
$$295$$ −420086. + 140765.i −0.281050 + 0.0941761i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.37553e6 0.889802
$$300$$ 0 0
$$301$$ −4.28847e6 −2.72826
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −372124. + 124694.i −0.229054 + 0.0767531i
$$306$$ 0 0
$$307$$ 334318.i 0.202448i −0.994864 0.101224i $$-0.967724\pi$$
0.994864 0.101224i $$-0.0322759\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.39090e6 1.40172 0.700858 0.713300i $$-0.252800\pi$$
0.700858 + 0.713300i $$0.252800\pi$$
$$312$$ 0 0
$$313$$ 1.88563e6i 1.08791i −0.839113 0.543957i $$-0.816925\pi$$
0.839113 0.543957i $$-0.183075\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 179636.i 0.100403i 0.998739 + 0.0502014i $$0.0159863\pi$$
−0.998739 + 0.0502014i $$0.984014\pi$$
$$318$$ 0 0
$$319$$ 1.80002e6 0.990378
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 46696.7i 0.0249046i
$$324$$ 0 0
$$325$$ 1.99864e6 + 2.64742e6i 1.04961 + 1.39032i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 661763. 0.337064
$$330$$ 0 0
$$331$$ −168620. −0.0845937 −0.0422968 0.999105i $$-0.513468\pi$$
−0.0422968 + 0.999105i $$0.513468\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 313152. + 934541.i 0.152456 + 0.454974i
$$336$$ 0 0
$$337$$ 1.74240e6i 0.835744i −0.908506 0.417872i $$-0.862776\pi$$
0.908506 0.417872i $$-0.137224\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1.76976e6 −0.824190
$$342$$ 0 0
$$343$$ 395842.i 0.181672i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.89447e6i 1.29046i −0.763987 0.645232i $$-0.776761\pi$$
0.763987 0.645232i $$-0.223239\pi$$
$$348$$ 0 0
$$349$$ −63803.8 −0.0280403 −0.0140202 0.999902i $$-0.504463\pi$$
−0.0140202 + 0.999902i $$0.504463\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2.79206e6i 1.19258i 0.802769 + 0.596290i $$0.203359\pi$$
−0.802769 + 0.596290i $$0.796641\pi$$
$$354$$ 0 0
$$355$$ 712014. 238586.i 0.299859 0.100479i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.48391e6 −0.607674 −0.303837 0.952724i $$-0.598268\pi$$
−0.303837 + 0.952724i $$0.598268\pi$$
$$360$$ 0 0
$$361$$ 15032.0 0.00607083
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 709507. + 2.11738e6i 0.278756 + 0.831892i
$$366$$ 0 0
$$367$$ 1.08188e6i 0.419289i 0.977778 + 0.209644i $$0.0672306\pi$$
−0.977778 + 0.209644i $$0.932769\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5.16053e6 −1.94652
$$372$$ 0 0
$$373$$ 127750.i 0.0475434i −0.999717 0.0237717i $$-0.992433\pi$$
0.999717 0.0237717i $$-0.00756748\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.80714e6i 1.37958i
$$378$$ 0 0
$$379$$ −1.61295e6 −0.576798 −0.288399 0.957510i $$-0.593123\pi$$
−0.288399 + 0.957510i $$0.593123\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.96656e6i 1.38171i −0.722993 0.690855i $$-0.757234\pi$$
0.722993 0.690855i $$-0.242766\pi$$
$$384$$ 0 0
$$385$$ 1.68443e6 + 5.02686e6i 0.579165 + 1.72840i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.54852e6 −1.18898 −0.594488 0.804104i $$-0.702645\pi$$
−0.594488 + 0.804104i $$0.702645\pi$$
$$390$$ 0 0
$$391$$ −38339.5 −0.0126825
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.94650e6 + 1.65751e6i −1.59516 + 0.534518i
$$396$$ 0 0
$$397$$ 580854.i 0.184965i 0.995714 + 0.0924827i $$0.0294803\pi$$
−0.995714 + 0.0924827i $$0.970520\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.10413e6 −0.653449 −0.326725 0.945120i $$-0.605945\pi$$
−0.326725 + 0.945120i $$0.605945\pi$$
$$402$$ 0 0
$$403$$ 3.74313e6i 1.14808i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.22263e6i 1.26356i
$$408$$ 0 0
$$409$$ −3.23687e6 −0.956790 −0.478395 0.878145i $$-0.658781\pi$$
−0.478395 + 0.878145i $$0.658781\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.49764e6i 0.432049i
$$414$$ 0 0
$$415$$ −1.03812e6 3.09807e6i −0.295888 0.883020i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2.03059e6 0.565049 0.282525 0.959260i $$-0.408828\pi$$
0.282525 + 0.959260i $$0.408828\pi$$
$$420$$ 0 0
$$421$$ 2.15883e6 0.593625 0.296813 0.954936i $$-0.404076\pi$$
0.296813 + 0.954936i $$0.404076\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −55707.0 73789.9i −0.0149602 0.0198164i
$$426$$ 0 0
$$427$$ 1.32665e6i 0.352118i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.49812e6 1.42568 0.712838 0.701328i $$-0.247409\pi$$
0.712838 + 0.701328i $$0.247409\pi$$
$$432$$ 0 0
$$433$$ 3.87201e6i 0.992470i 0.868188 + 0.496235i $$0.165284\pi$$
−0.868188 + 0.496235i $$0.834716\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.04530e6i 0.512334i
$$438$$ 0 0
$$439$$ 3.15465e6 0.781250 0.390625 0.920550i $$-0.372259\pi$$
0.390625 + 0.920550i $$0.372259\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.07021e6i 0.985389i −0.870202 0.492694i $$-0.836012\pi$$
0.870202 0.492694i $$-0.163988\pi$$
$$444$$ 0 0
$$445$$ −741508. + 248470.i −0.177507 + 0.0594803i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 957718. 0.224193 0.112096 0.993697i $$-0.464243\pi$$
0.112096 + 0.993697i $$0.464243\pi$$
$$450$$ 0 0
$$451$$ 3.52068e6 0.815052
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1.06321e7 3.56267e6i 2.40763 0.806765i
$$456$$ 0 0
$$457$$ 7.38167e6i 1.65335i −0.562682 0.826673i $$-0.690231\pi$$
0.562682 0.826673i $$-0.309769\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.96862e6 0.650582 0.325291 0.945614i $$-0.394538\pi$$
0.325291 + 0.945614i $$0.394538\pi$$
$$462$$ 0 0
$$463$$ 6.84593e6i 1.48416i −0.670313 0.742079i $$-0.733840\pi$$
0.670313 0.742079i $$-0.266160\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4.20816e6i 0.892895i −0.894810 0.446447i $$-0.852689\pi$$
0.894810 0.446447i $$-0.147311\pi$$
$$468$$ 0 0
$$469$$ 3.33172e6 0.699417
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.13895e7i 2.34074i
$$474$$ 0 0
$$475$$ 3.93647e6 2.97181e6i 0.800522 0.604347i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 5.73077e6 1.14123 0.570617 0.821217i $$-0.306704\pi$$
0.570617 + 0.821217i $$0.306704\pi$$
$$480$$ 0 0
$$481$$ 8.93110e6 1.76012
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.96511e6 + 5.86447e6i 0.379343 + 1.13207i
$$486$$ 0 0
$$487$$ 6.63249e6i 1.26723i 0.773650 + 0.633613i $$0.218429\pi$$
−0.773650 + 0.633613i $$0.781571\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.26463e6 −0.611125 −0.305563 0.952172i $$-0.598844\pi$$
−0.305563 + 0.952172i $$0.598844\pi$$
$$492$$ 0 0
$$493$$ 106114.i 0.0196633i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.53839e6i 0.460965i
$$498$$ 0 0
$$499$$ −8.51527e6 −1.53090 −0.765450 0.643495i $$-0.777484\pi$$
−0.765450 + 0.643495i $$0.777484\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.24174e6i 0.571292i 0.958335 + 0.285646i $$0.0922082\pi$$
−0.958335 + 0.285646i $$0.907792\pi$$
$$504$$ 0 0
$$505$$ −1.05198e7 + 3.52505e6i −1.83561 + 0.615088i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1.04569e7 1.78900 0.894499 0.447070i $$-0.147533\pi$$
0.894499 + 0.447070i $$0.147533\pi$$
$$510$$ 0 0
$$511$$ 7.54865e6 1.27884
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2.39387e6 7.14403e6i −0.397725 1.18693i
$$516$$ 0 0
$$517$$ 1.75755e6i 0.289188i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 328142. 0.0529625 0.0264812 0.999649i $$-0.491570\pi$$
0.0264812 + 0.999649i $$0.491570\pi$$
$$522$$ 0 0
$$523$$ 9.42102e6i 1.50606i −0.657984 0.753032i $$-0.728590\pi$$
0.657984 0.753032i $$-0.271410\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 104330.i 0.0163637i
$$528$$ 0 0
$$529$$ 4.75709e6 0.739098
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 7.44643e6i 1.13535i
$$534$$ 0 0
$$535$$ −706642. 2.10883e6i −0.106737 0.318535i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 9.48624e6 1.40644
$$540$$ 0 0
$$541$$ 6.08041e6 0.893182 0.446591 0.894738i $$-0.352638\pi$$
0.446591 + 0.894738i $$0.352638\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −4.91318e6 + 1.64634e6i −0.708551 + 0.237426i
$$546$$ 0 0
$$547$$ 4.32182e6i 0.617587i 0.951129 + 0.308794i $$0.0999253\pi$$
−0.951129 + 0.308794i $$0.900075\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5.66089e6 −0.794338
$$552$$ 0 0
$$553$$ 1.76347e7i 2.45220i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 7.41019e6i 1.01203i 0.862526 + 0.506013i $$0.168881\pi$$
−0.862526 + 0.506013i $$0.831119\pi$$
$$558$$ 0 0
$$559$$ 2.40895e7 3.26061
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.08039e7i 1.43652i 0.695775 + 0.718260i $$0.255061\pi$$
−0.695775 + 0.718260i $$0.744939\pi$$
$$564$$ 0 0
$$565$$ 1.18864e6 + 3.54726e6i 0.156650 + 0.467490i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.77404e6 0.229712 0.114856 0.993382i $$-0.463359\pi$$
0.114856 + 0.993382i $$0.463359\pi$$
$$570$$ 0 0
$$571$$ 70981.0 0.00911070 0.00455535 0.999990i $$-0.498550\pi$$
0.00455535 + 0.999990i $$0.498550\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.43995e6 3.23197e6i −0.307759 0.407660i
$$576$$ 0 0
$$577$$ 4.04410e6i 0.505688i 0.967507 + 0.252844i $$0.0813659\pi$$
−0.967507 + 0.252844i $$0.918634\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.10449e7 −1.35744
$$582$$ 0 0
$$583$$ 1.37056e7i 1.67004i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 2.86862e6i 0.343620i 0.985130 + 0.171810i $$0.0549615\pi$$
−0.985130 + 0.171810i $$0.945038\pi$$
$$588$$ 0 0
$$589$$ 5.56570e6 0.661046
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.23131e6i 0.377347i 0.982040 + 0.188674i $$0.0604188\pi$$
−0.982040 + 0.188674i $$0.939581\pi$$
$$594$$ 0 0
$$595$$ −296341. + 99300.1i −0.0343163 + 0.0114989i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −3.80078e6 −0.432819 −0.216409 0.976303i $$-0.569435\pi$$
−0.216409 + 0.976303i $$0.569435\pi$$
$$600$$ 0 0
$$601$$ 5.45289e6 0.615801 0.307900 0.951419i $$-0.400374\pi$$
0.307900 + 0.951419i $$0.400374\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 4.81408e6 1.61314e6i 0.534719 0.179177i
$$606$$ 0 0
$$607$$ 1.50483e7i 1.65774i 0.559441 + 0.828870i $$0.311016\pi$$
−0.559441 + 0.828870i $$0.688984\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.71731e6 −0.402833
$$612$$ 0 0
$$613$$ 2.30782e6i 0.248056i −0.992279 0.124028i $$-0.960419\pi$$
0.992279 0.124028i $$-0.0395813\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 381761.i 0.0403718i 0.999796 + 0.0201859i $$0.00642581\pi$$
−0.999796 + 0.0201859i $$0.993574\pi$$
$$618$$ 0 0
$$619$$ −1.07208e7 −1.12460 −0.562302 0.826932i $$-0.690084\pi$$
−0.562302 + 0.826932i $$0.690084\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 2.64354e6i 0.272876i
$$624$$ 0 0
$$625$$ 2.67517e6 9.39207e6i 0.273937 0.961748i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −248931. −0.0250872
$$630$$ 0 0
$$631$$ 4.56189e6 0.456112 0.228056 0.973648i $$-0.426763\pi$$
0.228056 + 0.973648i $$0.426763\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.04790e6 3.12724e6i −0.103130 0.307770i
$$636$$ 0 0
$$637$$ 2.00639e7i 1.95915i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.70983e6 0.260494 0.130247 0.991482i $$-0.458423\pi$$
0.130247 + 0.991482i $$0.458423\pi$$
$$642$$ 0 0
$$643$$ 6.10481e6i 0.582297i −0.956678 0.291149i $$-0.905963\pi$$
0.956678 0.291149i $$-0.0940374\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.22081e7i 1.14654i 0.819367 + 0.573269i $$0.194325\pi$$
−0.819367 + 0.573269i $$0.805675\pi$$
$$648$$ 0 0
$$649$$ 3.97752e6 0.370681
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.26514e6i 0.574973i −0.957785 0.287487i $$-0.907180\pi$$
0.957785 0.287487i $$-0.0928197\pi$$
$$654$$ 0 0
$$655$$ −1.10982e7 + 3.71887e6i −1.01076 + 0.338694i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.85107e7 1.66039 0.830194 0.557474i $$-0.188229\pi$$
0.830194 + 0.557474i $$0.188229\pi$$
$$660$$ 0 0
$$661$$ −8.46041e6 −0.753161 −0.376581 0.926384i $$-0.622900\pi$$
−0.376581 + 0.926384i $$0.622900\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −5.29737e6 1.58089e7i −0.464522 1.38627i
$$666$$ 0 0
$$667$$ 4.64777e6i 0.404511i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.52340e6 0.302104
$$672$$ 0 0
$$673$$ 7.25182e6i 0.617177i −0.951196 0.308588i $$-0.900143\pi$$
0.951196 0.308588i $$-0.0998566\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.74893e6i 0.398221i −0.979977 0.199110i $$-0.936195\pi$$
0.979977 0.199110i $$-0.0638052\pi$$
$$678$$ 0 0
$$679$$ 2.09073e7 1.74030
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1.37712e7i 1.12959i −0.825233 0.564793i $$-0.808956\pi$$
0.825233 0.564793i $$-0.191044\pi$$
$$684$$ 0 0
$$685$$ 5.79786e6 + 1.73025e7i 0.472108 + 1.40891i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2.89882e7 2.32634
$$690$$ 0 0
$$691$$ −1.64305e7 −1.30905 −0.654526 0.756040i $$-0.727132\pi$$
−0.654526 + 0.756040i $$0.727132\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.23845e7 + 4.14990e6i −0.972563 + 0.325893i
$$696$$ 0 0
$$697$$ 207550.i 0.0161823i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.05170e7 0.808342 0.404171 0.914683i $$-0.367560\pi$$
0.404171 + 0.914683i $$0.367560\pi$$
$$702$$ 0 0
$$703$$ 1.32797e7i 1.01345i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 3.75041e7i 2.82182i
$$708$$ 0 0
$$709$$ 1.32691e6 0.0991349 0.0495674 0.998771i $$-0.484216\pi$$
0.0495674 + 0.998771i $$0.484216\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 4.56962e6i 0.336633i
$$714$$ 0 0
$$715$$ −9.46193e6 2.82372e7i −0.692173 2.06565i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −4.65753e6 −0.335995 −0.167998 0.985787i $$-0.553730\pi$$
−0.167998 + 0.985787i $$0.553730\pi$$
$$720$$ 0 0
$$721$$ −2.54691e7 −1.82463
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −8.94531e6 + 6.75318e6i −0.632048 + 0.477159i
$$726$$ 0 0
$$727$$ 1.47884e6i 0.103773i 0.998653 + 0.0518867i $$0.0165235\pi$$
−0.998653 + 0.0518867i $$0.983477\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −671432. −0.0464739
$$732$$ 0 0
$$733$$ 1.41118e7i 0.970115i −0.874482 0.485057i $$-0.838799\pi$$
0.874482 0.485057i $$-0.161201\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.84856e6i 0.600073i
$$738$$ 0 0
$$739$$ −5.94045e6 −0.400137 −0.200068 0.979782i $$-0.564116\pi$$
−0.200068 + 0.979782i $$0.564116\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.46589e7i 0.974156i −0.873359 0.487078i $$-0.838063\pi$$
0.873359 0.487078i $$-0.161937\pi$$
$$744$$ 0 0
$$745$$ 7.70353e6 2.58135e6i 0.508510 0.170395i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −7.51817e6 −0.489674
$$750$$ 0 0
$$751$$ 6.08263e6 0.393543 0.196771 0.980449i $$-0.436954\pi$$
0.196771 + 0.980449i $$0.436954\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.52306e7 + 5.10359e6i −0.972413 + 0.325843i
$$756$$ 0 0
$$757$$ 8.11757e6i 0.514856i 0.966297 + 0.257428i $$0.0828751\pi$$
−0.966297 + 0.257428i $$0.917125\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.63567e7 −1.02384 −0.511921 0.859033i $$-0.671066\pi$$
−0.511921 + 0.859033i $$0.671066\pi$$
$$762$$ 0 0
$$763$$ 1.75159e7i 1.08923i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.41267e6i 0.516352i
$$768$$ 0 0
$$769$$ 2.09586e7 1.27805 0.639024 0.769187i $$-0.279339\pi$$
0.639024 + 0.769187i $$0.279339\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 8.95651e6i 0.539126i −0.962983 0.269563i $$-0.913121\pi$$
0.962983 0.269563i $$-0.0868793\pi$$
$$774$$ 0 0
$$775$$ 8.79489e6 6.63963e6i 0.525989 0.397090i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.10722e7 −0.653717
$$780$$ 0 0
$$781$$ −6.74160e6 −0.395490
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 757097. + 2.25940e6i 0.0438508 + 0.130864i
$$786$$ 0 0
$$787$$ 2.42009e7i 1.39282i 0.717643 + 0.696411i $$0.245221\pi$$
−0.717643 + 0.696411i $$0.754779\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.26463e7 0.718657
$$792$$ 0 0
$$793$$ 7.45219e6i 0.420824i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 9.91659e6i 0.552989i −0.961016 0.276494i $$-0.910827\pi$$
0.961016 0.276494i $$-0.0891728\pi$$
$$798$$ 0 0
$$799$$ 103610. 0.00574164
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2.00481e7i 1.09720i
$$804$$ 0 0
$$805$$ −1.29797e7 + 4.34931e6i −0.705949 + 0.236554i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0