# Properties

 Label 1728.3.b.i.1567.6 Level $1728$ Weight $3$ Character 1728.1567 Analytic conductor $47.085$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.116304318664704.2 Defining polynomial: $$x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64$$ x^12 - 2*x^11 + x^10 + 6*x^9 - 9*x^8 - 2*x^7 + 18*x^6 - 4*x^5 - 36*x^4 + 48*x^3 + 16*x^2 - 64*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}\cdot 3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.6 Root $$0.578188 + 1.29062i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1567 Dual form 1728.3.b.i.1567.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.64040i q^{5} +8.30833i q^{7} +O(q^{10})$$ $$q-2.64040i q^{5} +8.30833i q^{7} -18.7629 q^{11} -22.3117i q^{13} +23.4983 q^{17} +9.93333 q^{19} +32.3517i q^{23} +18.0283 q^{25} +8.45525i q^{29} -46.9532i q^{31} +21.9373 q^{35} +28.3219i q^{37} -77.7915 q^{41} -58.4797 q^{43} +54.2933i q^{47} -20.0283 q^{49} +5.81484i q^{53} +49.5416i q^{55} -47.5795 q^{59} +27.7128i q^{61} -58.9118 q^{65} -50.6336 q^{67} -7.34824i q^{71} -29.4383 q^{73} -155.888i q^{77} -43.2050i q^{79} -10.7858 q^{83} -62.0450i q^{85} -136.703 q^{89} +185.372 q^{91} -26.2280i q^{95} -54.6465 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100})$$ 12 * q - 48 * q^17 - 72 * q^25 - 336 * q^41 + 48 * q^49 - 912 * q^65 + 60 * q^73 - 1248 * q^89 - 204 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$5$$ − 2.64040i − 0.528081i −0.964512 0.264040i $$-0.914945\pi$$
0.964512 0.264040i $$-0.0850552\pi$$
$$6$$ 0 0
$$7$$ 8.30833i 1.18690i 0.804870 + 0.593452i $$0.202235\pi$$
−0.804870 + 0.593452i $$0.797765\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −18.7629 −1.70572 −0.852859 0.522141i $$-0.825133\pi$$
−0.852859 + 0.522141i $$0.825133\pi$$
$$12$$ 0 0
$$13$$ − 22.3117i − 1.71628i −0.513415 0.858141i $$-0.671620\pi$$
0.513415 0.858141i $$-0.328380\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 23.4983 1.38225 0.691126 0.722734i $$-0.257115\pi$$
0.691126 + 0.722734i $$0.257115\pi$$
$$18$$ 0 0
$$19$$ 9.93333 0.522807 0.261403 0.965230i $$-0.415815\pi$$
0.261403 + 0.965230i $$0.415815\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 32.3517i 1.40659i 0.710896 + 0.703297i $$0.248290\pi$$
−0.710896 + 0.703297i $$0.751710\pi$$
$$24$$ 0 0
$$25$$ 18.0283 0.721131
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.45525i 0.291560i 0.989317 + 0.145780i $$0.0465692\pi$$
−0.989317 + 0.145780i $$0.953431\pi$$
$$30$$ 0 0
$$31$$ − 46.9532i − 1.51462i −0.653055 0.757310i $$-0.726513\pi$$
0.653055 0.757310i $$-0.273487\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 21.9373 0.626781
$$36$$ 0 0
$$37$$ 28.3219i 0.765457i 0.923861 + 0.382728i $$0.125016\pi$$
−0.923861 + 0.382728i $$0.874984\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −77.7915 −1.89735 −0.948677 0.316246i $$-0.897578\pi$$
−0.948677 + 0.316246i $$0.897578\pi$$
$$42$$ 0 0
$$43$$ −58.4797 −1.35999 −0.679997 0.733215i $$-0.738019\pi$$
−0.679997 + 0.733215i $$0.738019\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 54.2933i 1.15518i 0.816329 + 0.577588i $$0.196006\pi$$
−0.816329 + 0.577588i $$0.803994\pi$$
$$48$$ 0 0
$$49$$ −20.0283 −0.408740
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 5.81484i 0.109714i 0.998494 + 0.0548570i $$0.0174703\pi$$
−0.998494 + 0.0548570i $$0.982530\pi$$
$$54$$ 0 0
$$55$$ 49.5416i 0.900757i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −47.5795 −0.806432 −0.403216 0.915105i $$-0.632108\pi$$
−0.403216 + 0.915105i $$0.632108\pi$$
$$60$$ 0 0
$$61$$ 27.7128i 0.454308i 0.973859 + 0.227154i $$0.0729421\pi$$
−0.973859 + 0.227154i $$0.927058\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −58.9118 −0.906335
$$66$$ 0 0
$$67$$ −50.6336 −0.755725 −0.377862 0.925862i $$-0.623341\pi$$
−0.377862 + 0.925862i $$0.623341\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 7.34824i − 0.103496i −0.998660 0.0517482i $$-0.983521\pi$$
0.998660 0.0517482i $$-0.0164793\pi$$
$$72$$ 0 0
$$73$$ −29.4383 −0.403265 −0.201632 0.979461i $$-0.564625\pi$$
−0.201632 + 0.979461i $$0.564625\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 155.888i − 2.02452i
$$78$$ 0 0
$$79$$ − 43.2050i − 0.546899i −0.961886 0.273450i $$-0.911835\pi$$
0.961886 0.273450i $$-0.0881647\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −10.7858 −0.129950 −0.0649748 0.997887i $$-0.520697\pi$$
−0.0649748 + 0.997887i $$0.520697\pi$$
$$84$$ 0 0
$$85$$ − 62.0450i − 0.729941i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −136.703 −1.53599 −0.767996 0.640454i $$-0.778746\pi$$
−0.767996 + 0.640454i $$0.778746\pi$$
$$90$$ 0 0
$$91$$ 185.372 2.03706
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 26.2280i − 0.276084i
$$96$$ 0 0
$$97$$ −54.6465 −0.563366 −0.281683 0.959508i $$-0.590893\pi$$
−0.281683 + 0.959508i $$0.590893\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 123.165i − 1.21946i −0.792611 0.609728i $$-0.791279\pi$$
0.792611 0.609728i $$-0.208721\pi$$
$$102$$ 0 0
$$103$$ 82.0848i 0.796940i 0.917181 + 0.398470i $$0.130459\pi$$
−0.917181 + 0.398470i $$0.869541\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −180.466 −1.68660 −0.843299 0.537445i $$-0.819390\pi$$
−0.843299 + 0.537445i $$0.819390\pi$$
$$108$$ 0 0
$$109$$ − 22.2137i − 0.203796i −0.994795 0.101898i $$-0.967509\pi$$
0.994795 0.101898i $$-0.0324915\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −139.381 −1.23346 −0.616732 0.787173i $$-0.711544\pi$$
−0.616732 + 0.787173i $$0.711544\pi$$
$$114$$ 0 0
$$115$$ 85.4215 0.742795
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 195.231i 1.64060i
$$120$$ 0 0
$$121$$ 231.046 1.90947
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 113.612i − 0.908896i
$$126$$ 0 0
$$127$$ 80.3083i 0.632349i 0.948701 + 0.316175i $$0.102399\pi$$
−0.948701 + 0.316175i $$0.897601\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 55.7583 0.425636 0.212818 0.977092i $$-0.431736\pi$$
0.212818 + 0.977092i $$0.431736\pi$$
$$132$$ 0 0
$$133$$ 82.5293i 0.620521i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.28641 0.0312877 0.0156438 0.999878i $$-0.495020\pi$$
0.0156438 + 0.999878i $$0.495020\pi$$
$$138$$ 0 0
$$139$$ 7.84615 0.0564471 0.0282236 0.999602i $$-0.491015\pi$$
0.0282236 + 0.999602i $$0.491015\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 418.631i 2.92749i
$$144$$ 0 0
$$145$$ 22.3253 0.153967
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 269.056i 1.80574i 0.429912 + 0.902871i $$0.358545\pi$$
−0.429912 + 0.902871i $$0.641455\pi$$
$$150$$ 0 0
$$151$$ − 179.626i − 1.18958i −0.803881 0.594789i $$-0.797235\pi$$
0.803881 0.594789i $$-0.202765\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −123.976 −0.799842
$$156$$ 0 0
$$157$$ 145.890i 0.929238i 0.885511 + 0.464619i $$0.153809\pi$$
−0.885511 + 0.464619i $$0.846191\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −268.788 −1.66949
$$162$$ 0 0
$$163$$ −100.496 −0.616540 −0.308270 0.951299i $$-0.599750\pi$$
−0.308270 + 0.951299i $$0.599750\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 154.410i 0.924611i 0.886721 + 0.462306i $$0.152978\pi$$
−0.886721 + 0.462306i $$0.847022\pi$$
$$168$$ 0 0
$$169$$ −328.810 −1.94562
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 57.6442i 0.333203i 0.986024 + 0.166602i $$0.0532794\pi$$
−0.986024 + 0.166602i $$0.946721\pi$$
$$174$$ 0 0
$$175$$ 149.785i 0.855913i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −83.2753 −0.465225 −0.232613 0.972569i $$-0.574727\pi$$
−0.232613 + 0.972569i $$0.574727\pi$$
$$180$$ 0 0
$$181$$ − 227.811i − 1.25862i −0.777153 0.629311i $$-0.783337\pi$$
0.777153 0.629311i $$-0.216663\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 74.7813 0.404223
$$186$$ 0 0
$$187$$ −440.896 −2.35773
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 28.0652i 0.146938i 0.997297 + 0.0734692i $$0.0234071\pi$$
−0.997297 + 0.0734692i $$0.976593\pi$$
$$192$$ 0 0
$$193$$ −310.951 −1.61115 −0.805573 0.592496i $$-0.798142\pi$$
−0.805573 + 0.592496i $$0.798142\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 177.071i − 0.898838i −0.893321 0.449419i $$-0.851631\pi$$
0.893321 0.449419i $$-0.148369\pi$$
$$198$$ 0 0
$$199$$ − 183.708i − 0.923156i −0.887100 0.461578i $$-0.847283\pi$$
0.887100 0.461578i $$-0.152717\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −70.2489 −0.346054
$$204$$ 0 0
$$205$$ 205.401i 1.00196i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −186.378 −0.891761
$$210$$ 0 0
$$211$$ 198.513 0.940821 0.470411 0.882448i $$-0.344106\pi$$
0.470411 + 0.882448i $$0.344106\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 154.410i 0.718186i
$$216$$ 0 0
$$217$$ 390.103 1.79771
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 524.286i − 2.37233i
$$222$$ 0 0
$$223$$ − 2.68500i − 0.0120403i −0.999982 0.00602017i $$-0.998084\pi$$
0.999982 0.00602017i $$-0.00191629\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −337.120 −1.48511 −0.742555 0.669786i $$-0.766386\pi$$
−0.742555 + 0.669786i $$0.766386\pi$$
$$228$$ 0 0
$$229$$ 46.9618i 0.205073i 0.994729 + 0.102537i $$0.0326959\pi$$
−0.994729 + 0.102537i $$0.967304\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 86.6965 0.372088 0.186044 0.982541i $$-0.440433\pi$$
0.186044 + 0.982541i $$0.440433\pi$$
$$234$$ 0 0
$$235$$ 143.356 0.610026
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 149.408i 0.625138i 0.949895 + 0.312569i $$0.101189\pi$$
−0.949895 + 0.312569i $$0.898811\pi$$
$$240$$ 0 0
$$241$$ 338.559 1.40481 0.702405 0.711778i $$-0.252110\pi$$
0.702405 + 0.711778i $$0.252110\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 52.8827i 0.215848i
$$246$$ 0 0
$$247$$ − 221.629i − 0.897284i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 117.462 0.467976 0.233988 0.972239i $$-0.424822\pi$$
0.233988 + 0.972239i $$0.424822\pi$$
$$252$$ 0 0
$$253$$ − 607.011i − 2.39925i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −79.7319 −0.310241 −0.155120 0.987896i $$-0.549577\pi$$
−0.155120 + 0.987896i $$0.549577\pi$$
$$258$$ 0 0
$$259$$ −235.308 −0.908524
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 70.1112i − 0.266582i −0.991077 0.133291i $$-0.957445\pi$$
0.991077 0.133291i $$-0.0425546\pi$$
$$264$$ 0 0
$$265$$ 15.3535 0.0579379
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 128.891i − 0.479147i −0.970878 0.239574i $$-0.922992\pi$$
0.970878 0.239574i $$-0.0770077\pi$$
$$270$$ 0 0
$$271$$ 334.953i 1.23599i 0.786182 + 0.617995i $$0.212055\pi$$
−0.786182 + 0.617995i $$0.787945\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −338.262 −1.23005
$$276$$ 0 0
$$277$$ 490.465i 1.77063i 0.464991 + 0.885315i $$0.346058\pi$$
−0.464991 + 0.885315i $$0.653942\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −496.474 −1.76681 −0.883406 0.468608i $$-0.844756\pi$$
−0.883406 + 0.468608i $$0.844756\pi$$
$$282$$ 0 0
$$283$$ 323.319 1.14247 0.571235 0.820787i $$-0.306465\pi$$
0.571235 + 0.820787i $$0.306465\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 646.317i − 2.25198i
$$288$$ 0 0
$$289$$ 263.170 0.910621
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 238.720i 0.814742i 0.913263 + 0.407371i $$0.133554\pi$$
−0.913263 + 0.407371i $$0.866446\pi$$
$$294$$ 0 0
$$295$$ 125.629i 0.425861i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 721.819 2.41411
$$300$$ 0 0
$$301$$ − 485.868i − 1.61418i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 73.1730 0.239912
$$306$$ 0 0
$$307$$ 196.695 0.640699 0.320349 0.947299i $$-0.396200\pi$$
0.320349 + 0.947299i $$0.396200\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 20.0012i 0.0643126i 0.999483 + 0.0321563i $$0.0102374\pi$$
−0.999483 + 0.0321563i $$0.989763\pi$$
$$312$$ 0 0
$$313$$ −341.608 −1.09140 −0.545700 0.837981i $$-0.683736\pi$$
−0.545700 + 0.837981i $$0.683736\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 481.401i − 1.51861i −0.650732 0.759307i $$-0.725538\pi$$
0.650732 0.759307i $$-0.274462\pi$$
$$318$$ 0 0
$$319$$ − 158.645i − 0.497319i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 233.416 0.722651
$$324$$ 0 0
$$325$$ − 402.240i − 1.23766i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −451.086 −1.37108
$$330$$ 0 0
$$331$$ −91.4065 −0.276152 −0.138076 0.990422i $$-0.544092\pi$$
−0.138076 + 0.990422i $$0.544092\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 133.693i 0.399084i
$$336$$ 0 0
$$337$$ −194.220 −0.576320 −0.288160 0.957582i $$-0.593044\pi$$
−0.288160 + 0.957582i $$0.593044\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 880.979i 2.58352i
$$342$$ 0 0
$$343$$ 240.707i 0.701768i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 68.5829 0.197645 0.0988226 0.995105i $$-0.468492\pi$$
0.0988226 + 0.995105i $$0.468492\pi$$
$$348$$ 0 0
$$349$$ 536.933i 1.53849i 0.638955 + 0.769244i $$0.279367\pi$$
−0.638955 + 0.769244i $$0.720633\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 486.832 1.37913 0.689563 0.724226i $$-0.257803\pi$$
0.689563 + 0.724226i $$0.257803\pi$$
$$354$$ 0 0
$$355$$ −19.4023 −0.0546544
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 31.1271i − 0.0867049i −0.999060 0.0433525i $$-0.986196\pi$$
0.999060 0.0433525i $$-0.0138038\pi$$
$$360$$ 0 0
$$361$$ −262.329 −0.726673
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 77.7291i 0.212956i
$$366$$ 0 0
$$367$$ − 656.463i − 1.78873i −0.447340 0.894364i $$-0.647629\pi$$
0.447340 0.894364i $$-0.352371\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −48.3116 −0.130220
$$372$$ 0 0
$$373$$ 285.162i 0.764508i 0.924057 + 0.382254i $$0.124852\pi$$
−0.924057 + 0.382254i $$0.875148\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 188.651 0.500399
$$378$$ 0 0
$$379$$ 96.1985 0.253822 0.126911 0.991914i $$-0.459494\pi$$
0.126911 + 0.991914i $$0.459494\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 542.017i 1.41519i 0.706619 + 0.707594i $$0.250220\pi$$
−0.706619 + 0.707594i $$0.749780\pi$$
$$384$$ 0 0
$$385$$ −411.608 −1.06911
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 251.966i 0.647729i 0.946104 + 0.323864i $$0.104982\pi$$
−0.946104 + 0.323864i $$0.895018\pi$$
$$390$$ 0 0
$$391$$ 760.209i 1.94427i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −114.079 −0.288807
$$396$$ 0 0
$$397$$ 534.951i 1.34748i 0.738966 + 0.673742i $$0.235314\pi$$
−0.738966 + 0.673742i $$0.764686\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 586.919 1.46364 0.731819 0.681499i $$-0.238672\pi$$
0.731819 + 0.681499i $$0.238672\pi$$
$$402$$ 0 0
$$403$$ −1047.60 −2.59951
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 531.401i − 1.30565i
$$408$$ 0 0
$$409$$ 364.507 0.891214 0.445607 0.895229i $$-0.352988\pi$$
0.445607 + 0.895229i $$0.352988\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 395.306i − 0.957157i
$$414$$ 0 0
$$415$$ 28.4789i 0.0686239i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 181.773 0.433827 0.216913 0.976191i $$-0.430401\pi$$
0.216913 + 0.976191i $$0.430401\pi$$
$$420$$ 0 0
$$421$$ 168.535i 0.400320i 0.979763 + 0.200160i $$0.0641462\pi$$
−0.979763 + 0.200160i $$0.935854\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 423.633 0.996785
$$426$$ 0 0
$$427$$ −230.247 −0.539220
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 373.929i − 0.867585i −0.901013 0.433792i $$-0.857175\pi$$
0.901013 0.433792i $$-0.142825\pi$$
$$432$$ 0 0
$$433$$ 483.636 1.11694 0.558471 0.829524i $$-0.311388\pi$$
0.558471 + 0.829524i $$0.311388\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 321.360i 0.735377i
$$438$$ 0 0
$$439$$ − 449.117i − 1.02305i −0.859270 0.511523i $$-0.829081\pi$$
0.859270 0.511523i $$-0.170919\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −361.663 −0.816396 −0.408198 0.912893i $$-0.633843\pi$$
−0.408198 + 0.912893i $$0.633843\pi$$
$$444$$ 0 0
$$445$$ 360.952i 0.811128i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 589.191 1.31223 0.656115 0.754661i $$-0.272199\pi$$
0.656115 + 0.754661i $$0.272199\pi$$
$$450$$ 0 0
$$451$$ 1459.59 3.23635
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 489.458i − 1.07573i
$$456$$ 0 0
$$457$$ −452.703 −0.990597 −0.495299 0.868723i $$-0.664941\pi$$
−0.495299 + 0.868723i $$0.664941\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 290.060i − 0.629197i −0.949225 0.314598i $$-0.898130\pi$$
0.949225 0.314598i $$-0.101870\pi$$
$$462$$ 0 0
$$463$$ − 389.267i − 0.840749i −0.907351 0.420374i $$-0.861899\pi$$
0.907351 0.420374i $$-0.138101\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −175.827 −0.376504 −0.188252 0.982121i $$-0.560282\pi$$
−0.188252 + 0.982121i $$0.560282\pi$$
$$468$$ 0 0
$$469$$ − 420.680i − 0.896972i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1097.25 2.31976
$$474$$ 0 0
$$475$$ 179.081 0.377012
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 321.473i − 0.671134i −0.942016 0.335567i $$-0.891072\pi$$
0.942016 0.335567i $$-0.108928\pi$$
$$480$$ 0 0
$$481$$ 631.909 1.31374
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 144.289i 0.297503i
$$486$$ 0 0
$$487$$ 244.014i 0.501055i 0.968109 + 0.250528i $$0.0806041\pi$$
−0.968109 + 0.250528i $$0.919396\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 239.563 0.487909 0.243955 0.969787i $$-0.421555\pi$$
0.243955 + 0.969787i $$0.421555\pi$$
$$492$$ 0 0
$$493$$ 198.684i 0.403010i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 61.0516 0.122840
$$498$$ 0 0
$$499$$ −134.164 −0.268865 −0.134433 0.990923i $$-0.542921\pi$$
−0.134433 + 0.990923i $$0.542921\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 214.317i 0.426078i 0.977044 + 0.213039i $$0.0683362\pi$$
−0.977044 + 0.213039i $$0.931664\pi$$
$$504$$ 0 0
$$505$$ −325.206 −0.643971
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 453.973i 0.891892i 0.895060 + 0.445946i $$0.147133\pi$$
−0.895060 + 0.445946i $$0.852867\pi$$
$$510$$ 0 0
$$511$$ − 244.583i − 0.478637i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 216.737 0.420849
$$516$$ 0 0
$$517$$ − 1018.70i − 1.97040i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −97.5418 −0.187220 −0.0936102 0.995609i $$-0.529841\pi$$
−0.0936102 + 0.995609i $$0.529841\pi$$
$$522$$ 0 0
$$523$$ 476.762 0.911590 0.455795 0.890085i $$-0.349355\pi$$
0.455795 + 0.890085i $$0.349355\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1103.32i − 2.09359i
$$528$$ 0 0
$$529$$ −517.630 −0.978507
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1735.66i 3.25639i
$$534$$ 0 0
$$535$$ 476.503i 0.890660i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 375.788 0.697195
$$540$$ 0 0
$$541$$ − 987.446i − 1.82522i −0.408826 0.912612i $$-0.634062\pi$$
0.408826 0.912612i $$-0.365938\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −58.6532 −0.107621
$$546$$ 0 0
$$547$$ −396.486 −0.724837 −0.362418 0.932016i $$-0.618049\pi$$
−0.362418 + 0.932016i $$0.618049\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 83.9888i 0.152430i
$$552$$ 0 0
$$553$$ 358.961 0.649117
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 488.922i − 0.877778i −0.898541 0.438889i $$-0.855372\pi$$
0.898541 0.438889i $$-0.144628\pi$$
$$558$$ 0 0
$$559$$ 1304.78i 2.33413i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 417.788 0.742075 0.371037 0.928618i $$-0.379002\pi$$
0.371037 + 0.928618i $$0.379002\pi$$
$$564$$ 0 0
$$565$$ 368.023i 0.651369i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 493.590 0.867469 0.433735 0.901041i $$-0.357196\pi$$
0.433735 + 0.901041i $$0.357196\pi$$
$$570$$ 0 0
$$571$$ −687.730 −1.20443 −0.602215 0.798334i $$-0.705715\pi$$
−0.602215 + 0.798334i $$0.705715\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 583.244i 1.01434i
$$576$$ 0 0
$$577$$ 206.939 0.358647 0.179324 0.983790i $$-0.442609\pi$$
0.179324 + 0.983790i $$0.442609\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 89.6120i − 0.154238i
$$582$$ 0 0
$$583$$ − 109.103i − 0.187141i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 879.623 1.49851 0.749253 0.662284i $$-0.230413\pi$$
0.749253 + 0.662284i $$0.230413\pi$$
$$588$$ 0 0
$$589$$ − 466.402i − 0.791854i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −444.195 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$594$$ 0 0
$$595$$ 515.490 0.866369
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 1025.55i − 1.71211i −0.516888 0.856053i $$-0.672910\pi$$
0.516888 0.856053i $$-0.327090\pi$$
$$600$$ 0 0
$$601$$ −18.5617 −0.0308846 −0.0154423 0.999881i $$-0.504916\pi$$
−0.0154423 + 0.999881i $$0.504916\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 610.055i − 1.00836i
$$606$$ 0 0
$$607$$ − 648.184i − 1.06785i −0.845533 0.533924i $$-0.820717\pi$$
0.845533 0.533924i $$-0.179283\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1211.37 1.98261
$$612$$ 0 0
$$613$$ 616.049i 1.00497i 0.864585 + 0.502487i $$0.167582\pi$$
−0.864585 + 0.502487i $$0.832418\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 321.805 0.521564 0.260782 0.965398i $$-0.416020\pi$$
0.260782 + 0.965398i $$0.416020\pi$$
$$618$$ 0 0
$$619$$ −769.275 −1.24277 −0.621386 0.783505i $$-0.713430\pi$$
−0.621386 + 0.783505i $$0.713430\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 1135.78i − 1.82307i
$$624$$ 0 0
$$625$$ 150.725 0.241160
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 665.516i 1.05805i
$$630$$ 0 0
$$631$$ − 766.730i − 1.21510i −0.794280 0.607551i $$-0.792152\pi$$
0.794280 0.607551i $$-0.207848\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 212.046 0.333931
$$636$$ 0 0
$$637$$ 446.864i 0.701513i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −436.825 −0.681474 −0.340737 0.940159i $$-0.610677\pi$$
−0.340737 + 0.940159i $$0.610677\pi$$
$$642$$ 0 0
$$643$$ 276.284 0.429680 0.214840 0.976649i $$-0.431077\pi$$
0.214840 + 0.976649i $$0.431077\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 795.622i − 1.22971i −0.788640 0.614855i $$-0.789215\pi$$
0.788640 0.614855i $$-0.210785\pi$$
$$648$$ 0 0
$$649$$ 892.729 1.37555
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 305.664i − 0.468091i −0.972226 0.234046i $$-0.924804\pi$$
0.972226 0.234046i $$-0.0751965\pi$$
$$654$$ 0 0
$$655$$ − 147.224i − 0.224770i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 153.889 0.233519 0.116760 0.993160i $$-0.462749\pi$$
0.116760 + 0.993160i $$0.462749\pi$$
$$660$$ 0 0
$$661$$ 38.0847i 0.0576167i 0.999585 + 0.0288084i $$0.00917126\pi$$
−0.999585 + 0.0288084i $$0.990829\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 217.911 0.327685
$$666$$ 0 0
$$667$$ −273.541 −0.410107
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 519.973i − 0.774922i
$$672$$ 0 0
$$673$$ 392.195 0.582757 0.291378 0.956608i $$-0.405886\pi$$
0.291378 + 0.956608i $$0.405886\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 917.262i 1.35489i 0.735573 + 0.677446i $$0.236913\pi$$
−0.735573 + 0.677446i $$0.763087\pi$$
$$678$$ 0 0
$$679$$ − 454.021i − 0.668661i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1280.10 −1.87424 −0.937119 0.349010i $$-0.886518\pi$$
−0.937119 + 0.349010i $$0.886518\pi$$
$$684$$ 0 0
$$685$$ − 11.3179i − 0.0165224i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 129.739 0.188300
$$690$$ 0 0
$$691$$ −1147.79 −1.66106 −0.830531 0.556972i $$-0.811963\pi$$
−0.830531 + 0.556972i $$0.811963\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 20.7170i − 0.0298086i
$$696$$ 0 0
$$697$$ −1827.97 −2.62262
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 450.531i − 0.642697i −0.946961 0.321349i $$-0.895864\pi$$
0.946961 0.321349i $$-0.104136\pi$$
$$702$$ 0 0
$$703$$ 281.331i 0.400186i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1023.30 1.44738
$$708$$ 0 0
$$709$$ 1071.57i 1.51139i 0.654926 + 0.755693i $$0.272700\pi$$
−0.654926 + 0.755693i $$0.727300\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1519.02 2.13046
$$714$$ 0 0
$$715$$ 1105.36 1.54595
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 569.979i − 0.792739i −0.918091 0.396370i $$-0.870270\pi$$
0.918091 0.396370i $$-0.129730\pi$$
$$720$$ 0 0
$$721$$ −681.987 −0.945891
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 152.433i 0.210253i
$$726$$ 0 0
$$727$$ − 699.311i − 0.961914i −0.876744 0.480957i $$-0.840289\pi$$
0.876744 0.480957i $$-0.159711\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1374.17 −1.87985
$$732$$ 0 0
$$733$$ 175.235i 0.239065i 0.992830 + 0.119533i $$0.0381396\pi$$
−0.992830 + 0.119533i $$0.961860\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 950.032 1.28905
$$738$$ 0 0
$$739$$ −975.643 −1.32022 −0.660110 0.751169i $$-0.729490\pi$$
−0.660110 + 0.751169i $$0.729490\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 880.230i 1.18470i 0.805682 + 0.592349i $$0.201799\pi$$
−0.805682 + 0.592349i $$0.798201\pi$$
$$744$$ 0 0
$$745$$ 710.415 0.953578
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 1499.37i − 2.00183i
$$750$$ 0 0
$$751$$ − 48.7538i − 0.0649186i −0.999473 0.0324593i $$-0.989666\pi$$
0.999473 0.0324593i $$-0.0103339\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −474.286 −0.628194
$$756$$ 0 0
$$757$$ − 1194.56i − 1.57802i −0.614382 0.789009i $$-0.710594\pi$$
0.614382 0.789009i $$-0.289406\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1032.85 −1.35722 −0.678612 0.734497i $$-0.737418\pi$$
−0.678612 + 0.734497i $$0.737418\pi$$
$$762$$ 0 0
$$763$$ 184.559 0.241886
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1061.58i 1.38406i
$$768$$ 0 0
$$769$$ −200.830 −0.261157 −0.130579 0.991438i $$-0.541684\pi$$
−0.130579 + 0.991438i $$0.541684\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 949.438i 1.22825i 0.789208 + 0.614126i $$0.210491\pi$$
−0.789208 + 0.614126i $$0.789509\pi$$
$$774$$ 0 0
$$775$$ − 846.486i − 1.09224i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −772.729 −0.991950
$$780$$ 0 0
$$781$$ 137.874i 0.176536i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 385.210 0.490713
$$786$$ 0 0
$$787$$ 646.331 0.821259 0.410630 0.911802i $$-0.365309\pi$$
0.410630 + 0.911802i $$0.365309\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 1158.03i − 1.46400i
$$792$$ 0 0
$$793$$ 618.319 0.779721
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 37.5148i − 0.0470700i −0.999723 0.0235350i $$-0.992508\pi$$
0.999723 0.0235350i $$-0.00749211\pi$$
$$798$$ 0 0
$$799$$ 1275.80i 1.59674i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 552.348 0.687856
$$804$$ 0 0
$$805$$ 709.709i 0.881626i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 572.790 0.708023 0.354011 0.935241i $$-0.384817\pi$$
0.354011 + 0.935241i $$0.384817\pi$$
$$810$$ 0 0
$$811$$ 486.350 0.599692 0.299846 0.953988i $$-0.403065\pi$$
0.299846 + 0.953988i $$0.403065\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 265.350i 0.325583i
$$816$$ 0 0
$$817$$ −580.898 −0.711014
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 311.021i − 0.378832i −0.981897 0.189416i $$-0.939341\pi$$
0.981897 0.189416i $$-0.0606595\pi$$
$$822$$ 0 0
$$823$$ 824.124i 1.00137i 0.865631 + 0.500683i $$0.166918\pi$$
−0.865631 + 0.500683i $$0.833082\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 918.441 1.11057 0.555285 0.831660i $$-0.312609\pi$$
0.555285 + 0.831660i $$0.312609\pi$$
$$828$$ 0 0
$$829$$ − 937.997i − 1.13148i −0.824584 0.565740i $$-0.808591\pi$$
0.824584 0.565740i $$-0.191409\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −470.630 −0.564982
$$834$$ 0 0
$$835$$ 407.705 0.488269
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 612.844i 0.730446i 0.930920 + 0.365223i $$0.119007\pi$$
−0.930920 + 0.365223i $$0.880993\pi$$
$$840$$ 0 0
$$841$$ 769.509 0.914993
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 868.191i 1.02744i
$$846$$ 0 0
$$847$$ 1919.61i 2.26636i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −916.261 −1.07669
$$852$$ 0 0
$$853$$ − 491.372i − 0.576051i −0.957623 0.288026i $$-0.907001\pi$$
0.957623 0.288026i $$-0.0929989\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −642.820 −0.750082 −0.375041 0.927008i $$-0.622371\pi$$
−0.375041 + 0.927008i $$0.622371\pi$$
$$858$$ 0 0
$$859$$ 443.260 0.516018 0.258009 0.966142i $$-0.416934\pi$$
0.258009 + 0.966142i $$0.416934\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 565.494i 0.655265i 0.944805 + 0.327632i $$0.106251\pi$$
−0.944805 + 0.327632i $$0.893749\pi$$
$$864$$ 0 0
$$865$$ 152.204 0.175958
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 810.652i 0.932856i
$$870$$ 0 0
$$871$$ 1129.72i 1.29704i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 943.925 1.07877
$$876$$ 0 0
$$877$$ − 330.215i − 0.376528i −0.982118 0.188264i $$-0.939714\pi$$
0.982118 0.188264i $$-0.0602861\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 358.037 0.406398 0.203199 0.979137i $$-0.434866\pi$$
0.203199 + 0.979137i $$0.434866\pi$$
$$882$$ 0 0
$$883$$ 266.351 0.301643 0.150822 0.988561i $$-0.451808\pi$$
0.150822 + 0.988561i $$0.451808\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1375.40i 1.55062i 0.631581 + 0.775310i $$0.282406\pi$$
−0.631581 + 0.775310i $$0.717594\pi$$
$$888$$ 0 0
$$889$$ −667.228 −0.750537
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 539.313i 0.603934i
$$894$$ 0 0
$$895$$ 219.880i 0.245676i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 397.001 0.441603
$$900$$ 0 0
$$901$$ 136.639i 0.151652i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −601.512 −0.664654
$$906$$ 0 0
$$907$$ 106.524 0.117446 0.0587230 0.998274i $$-0.481297\pi$$
0.0587230 + 0.998274i $$0.481297\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 842.877i 0.925222i 0.886561 + 0.462611i $$0.153087\pi$$
−0.886561 + 0.462611i $$0.846913\pi$$
$$912$$ 0 0
$$913$$ 202.373 0.221657
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 463.258i 0.505189i
$$918$$ 0 0
$$919$$ − 1388.07i − 1.51041i −0.655489 0.755205i $$-0.727537\pi$$
0.655489 0.755205i $$-0.272463\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −163.951 −0.177629
$$924$$ 0 0
$$925$$ 510.595i 0.551995i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 557.068 0.599643 0.299821 0.953995i $$-0.403073\pi$$
0.299821 + 0.953995i $$0.403073\pi$$
$$930$$ 0 0
$$931$$ −198.947 −0.213692
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 1164.14i 1.24507i
$$936$$ 0 0
$$937$$ 876.585 0.935523 0.467761 0.883855i $$-0.345061\pi$$
0.467761 + 0.883855i $$0.345061\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 1290.14i − 1.37103i −0.728058 0.685515i $$-0.759577\pi$$
0.728058 0.685515i $$-0.240423\pi$$
$$942$$ 0 0
$$943$$ − 2516.69i − 2.66881i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −850.851 −0.898470 −0.449235 0.893414i $$-0.648303\pi$$
−0.449235 + 0.893414i $$0.648303\pi$$
$$948$$ 0 0
$$949$$ 656.818i 0.692116i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −297.237 −0.311896 −0.155948 0.987765i $$-0.549843\pi$$
−0.155948 + 0.987765i $$0.549843\pi$$
$$954$$ 0 0
$$955$$ 74.1036 0.0775954
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 35.6129i 0.0371355i
$$960$$ 0 0
$$961$$ −1243.61 −1.29408
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 821.037i 0.850815i
$$966$$ 0 0
$$967$$ 943.377i 0.975570i 0.872964 + 0.487785i $$0.162195\pi$$
−0.872964 + 0.487785i $$0.837805\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 709.360 0.730546 0.365273 0.930900i $$-0.380976\pi$$
0.365273 + 0.930900i $$0.380976\pi$$
$$972$$ 0 0
$$973$$ 65.1884i 0.0669973i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 295.503 0.302459 0.151230 0.988499i $$-0.451677\pi$$
0.151230 + 0.988499i $$0.451677\pi$$
$$978$$ 0 0
$$979$$ 2564.95 2.61997
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 559.879i − 0.569561i −0.958593 0.284781i $$-0.908079\pi$$
0.958593 0.284781i $$-0.0919208\pi$$
$$984$$ 0 0
$$985$$ −467.539 −0.474659
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 1891.92i − 1.91296i
$$990$$ 0 0
$$991$$ − 1327.50i − 1.33955i −0.742563 0.669776i $$-0.766390\pi$$
0.742563 0.669776i $$-0.233610\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −485.064 −0.487501
$$996$$ 0 0
$$997$$ − 878.699i − 0.881343i −0.897668 0.440672i $$-0.854740\pi$$
0.897668 0.440672i $$-0.145260\pi$$
$$998$$ 0 0
$$999$$ 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 1728.3.b.i.1567.6 yes 12
3.2 odd 2 1728.3.b.j.1567.8 yes 12
4.3 odd 2 inner 1728.3.b.i.1567.5 12
8.3 odd 2 inner 1728.3.b.i.1567.7 yes 12
8.5 even 2 inner 1728.3.b.i.1567.8 yes 12
12.11 even 2 1728.3.b.j.1567.7 yes 12
24.5 odd 2 1728.3.b.j.1567.6 yes 12
24.11 even 2 1728.3.b.j.1567.5 yes 12

By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.5 12 4.3 odd 2 inner
1728.3.b.i.1567.6 yes 12 1.1 even 1 trivial
1728.3.b.i.1567.7 yes 12 8.3 odd 2 inner
1728.3.b.i.1567.8 yes 12 8.5 even 2 inner
1728.3.b.j.1567.5 yes 12 24.11 even 2
1728.3.b.j.1567.6 yes 12 24.5 odd 2
1728.3.b.j.1567.7 yes 12 12.11 even 2
1728.3.b.j.1567.8 yes 12 3.2 odd 2