# Properties

 Label 4600.2.e.b.4049.2 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.b.4049.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} -1.00000i q^{13} +6.00000 q^{21} -1.00000i q^{23} -9.00000i q^{27} +3.00000 q^{29} +3.00000 q^{31} -8.00000i q^{37} +3.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -11.0000i q^{47} +3.00000 q^{49} +14.0000i q^{53} +8.00000 q^{59} -4.00000 q^{61} +12.0000i q^{63} -4.00000i q^{67} +3.00000 q^{69} +7.00000 q^{71} +9.00000i q^{73} +9.00000 q^{81} -4.00000i q^{83} +9.00000i q^{87} +2.00000 q^{89} -2.00000 q^{91} +9.00000i q^{93} +18.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 12 q^{21} + 6 q^{29} + 6 q^{31} + 6 q^{39} + 6 q^{41} + 6 q^{49} + 16 q^{59} - 8 q^{61} + 6 q^{69} + 14 q^{71} + 18 q^{81} + 4 q^{89} - 4 q^{91}+O(q^{100})$$ 2 * q - 12 * q^9 + 12 * q^21 + 6 * q^29 + 6 * q^31 + 6 * q^39 + 6 * q^41 + 6 * q^49 + 16 * q^59 - 8 * q^61 + 6 * q^69 + 14 * q^71 + 18 * q^81 + 4 * q^89 - 4 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 3.00000i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 9.00000i − 1.73205i
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 11.0000i − 1.60451i −0.596978 0.802257i $$-0.703632\pi$$
0.596978 0.802257i $$-0.296368\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 14.0000i 1.92305i 0.274721 + 0.961524i $$0.411414\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 12.0000i 1.51186i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 0 0
$$73$$ 9.00000i 1.05337i 0.850060 + 0.526685i $$0.176565\pi$$
−0.850060 + 0.526685i $$0.823435\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.00000i 0.964901i
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 9.00000i 0.933257i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 18.0000i 1.82762i 0.406138 + 0.913812i $$0.366875\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 24.0000 2.27798
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 9.00000i 0.811503i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.0000i 0.976092i 0.872818 + 0.488046i $$0.162290\pi$$
−0.872818 + 0.488046i $$0.837710\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.00000i 0.341743i 0.985293 + 0.170872i $$0.0546583\pi$$
−0.985293 + 0.170872i $$0.945342\pi$$
$$138$$ 0 0
$$139$$ 11.0000 0.933008 0.466504 0.884519i $$-0.345513\pi$$
0.466504 + 0.884519i $$0.345513\pi$$
$$140$$ 0 0
$$141$$ 33.0000 2.77910
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 9.00000i 0.742307i
$$148$$ 0 0
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ 0 0
$$159$$ −42.0000 −3.33082
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ − 7.00000i − 0.548282i −0.961689 0.274141i $$-0.911606\pi$$
0.961689 0.274141i $$-0.0883936\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 24.0000i 1.80395i
$$178$$ 0 0
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ − 12.0000i − 0.887066i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −18.0000 −1.30931
$$190$$ 0 0
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 0 0
$$193$$ 1.00000i 0.0719816i 0.999352 + 0.0359908i $$0.0114587\pi$$
−0.999352 + 0.0359908i $$0.988541\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.00000i 0.213741i 0.994273 + 0.106871i $$0.0340831\pi$$
−0.994273 + 0.106871i $$0.965917\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ − 6.00000i − 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000i 0.417029i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0 0
$$213$$ 21.0000i 1.43890i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.00000i − 0.407307i
$$218$$ 0 0
$$219$$ −27.0000 −1.82449
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.00000i 0.132745i 0.997795 + 0.0663723i $$0.0211425\pi$$
−0.997795 + 0.0663723i $$0.978857\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 21.0000i − 1.37576i −0.725826 0.687878i $$-0.758542\pi$$
0.725826 0.687878i $$-0.241458\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.00000 0.0646846 0.0323423 0.999477i $$-0.489703\pi$$
0.0323423 + 0.999477i $$0.489703\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 16.0000 1.00991 0.504956 0.863145i $$-0.331509\pi$$
0.504956 + 0.863145i $$0.331509\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.00000i 0.311891i 0.987766 + 0.155946i $$0.0498425\pi$$
−0.987766 + 0.155946i $$0.950158\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ 12.0000i 0.739952i 0.929041 + 0.369976i $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ −17.0000 −1.03651 −0.518254 0.855227i $$-0.673418\pi$$
−0.518254 + 0.855227i $$0.673418\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ − 6.00000i − 0.363137i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 29.0000i − 1.74244i −0.490892 0.871221i $$-0.663329\pi$$
0.490892 0.871221i $$-0.336671\pi$$
$$278$$ 0 0
$$279$$ −18.0000 −1.07763
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 10.0000i − 0.594438i −0.954809 0.297219i $$-0.903941\pi$$
0.954809 0.297219i $$-0.0960592\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −54.0000 −3.16554
$$292$$ 0 0
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 54.0000i 3.10222i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −29.0000 −1.64444 −0.822220 0.569170i $$-0.807264\pi$$
−0.822220 + 0.569170i $$0.807264\pi$$
$$312$$ 0 0
$$313$$ 20.0000i 1.13047i 0.824931 + 0.565233i $$0.191214\pi$$
−0.824931 + 0.565233i $$0.808786\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 48.0000 2.67910
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 54.0000i 2.98621i
$$328$$ 0 0
$$329$$ −22.0000 −1.21290
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 0 0
$$333$$ 48.0000i 2.63038i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.0000i 1.41631i 0.706057 + 0.708155i $$0.250472\pi$$
−0.706057 + 0.708155i $$0.749528\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 7.00000 0.374701 0.187351 0.982293i $$-0.440010\pi$$
0.187351 + 0.982293i $$0.440010\pi$$
$$350$$ 0 0
$$351$$ −9.00000 −0.480384
$$352$$ 0 0
$$353$$ − 19.0000i − 1.01127i −0.862748 0.505634i $$-0.831259\pi$$
0.862748 0.505634i $$-0.168741\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ − 33.0000i − 1.73205i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 28.0000 1.45369
$$372$$ 0 0
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 3.00000i − 0.154508i
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −33.0000 −1.69064
$$382$$ 0 0
$$383$$ − 30.0000i − 1.53293i −0.642287 0.766464i $$-0.722014\pi$$
0.642287 0.766464i $$-0.277986\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 12.0000i − 0.609994i
$$388$$ 0 0
$$389$$ 16.0000 0.811232 0.405616 0.914044i $$-0.367057\pi$$
0.405616 + 0.914044i $$0.367057\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 27.0000i − 1.36197i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 25.0000i 1.25471i 0.778732 + 0.627357i $$0.215863\pi$$
−0.778732 + 0.627357i $$0.784137\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ − 3.00000i − 0.149441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 11.0000 0.543915 0.271957 0.962309i $$-0.412329\pi$$
0.271957 + 0.962309i $$0.412329\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ − 16.0000i − 0.787309i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 33.0000i 1.61602i
$$418$$ 0 0
$$419$$ −22.0000 −1.07477 −0.537385 0.843337i $$-0.680588\pi$$
−0.537385 + 0.843337i $$0.680588\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ 66.0000i 3.20903i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ − 33.0000i − 1.56788i −0.620838 0.783939i $$-0.713208\pi$$
0.620838 0.783939i $$-0.286792\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 66.0000i 3.12169i
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 21.0000i 0.986666i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.00000i − 0.187112i −0.995614 0.0935561i $$-0.970177\pi$$
0.995614 0.0935561i $$-0.0298234\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.0000 0.605470 0.302735 0.953075i $$-0.402100\pi$$
0.302735 + 0.953075i $$0.402100\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 42.0000i 1.94353i 0.235954 + 0.971764i $$0.424178\pi$$
−0.235954 + 0.971764i $$0.575822\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 84.0000i − 3.84610i
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ − 6.00000i − 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 25.0000i − 1.13286i −0.824110 0.566429i $$-0.808325\pi$$
0.824110 0.566429i $$-0.191675\pi$$
$$488$$ 0 0
$$489$$ 21.0000 0.949653
$$490$$ 0 0
$$491$$ −31.0000 −1.39901 −0.699505 0.714628i $$-0.746596\pi$$
−0.699505 + 0.714628i $$0.746596\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 14.0000i − 0.627986i
$$498$$ 0 0
$$499$$ 25.0000 1.11915 0.559577 0.828778i $$-0.310964\pi$$
0.559577 + 0.828778i $$0.310964\pi$$
$$500$$ 0 0
$$501$$ −48.0000 −2.14448
$$502$$ 0 0
$$503$$ 14.0000i 0.624229i 0.950044 + 0.312115i $$0.101037\pi$$
−0.950044 + 0.312115i $$0.898963\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 36.0000i 1.59882i
$$508$$ 0 0
$$509$$ −21.0000 −0.930809 −0.465404 0.885098i $$-0.654091\pi$$
−0.465404 + 0.885098i $$0.654091\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 42.0000 1.84360
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 0 0
$$523$$ − 42.0000i − 1.83653i −0.395964 0.918266i $$-0.629590\pi$$
0.395964 0.918266i $$-0.370410\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −48.0000 −2.08302
$$532$$ 0 0
$$533$$ − 3.00000i − 0.129944i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 63.0000i − 2.71865i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.00000 −0.300954 −0.150477 0.988614i $$-0.548081\pi$$
−0.150477 + 0.988614i $$0.548081\pi$$
$$542$$ 0 0
$$543$$ 36.0000i 1.54491i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.0000i 1.49649i 0.663421 + 0.748246i $$0.269104\pi$$
−0.663421 + 0.748246i $$0.730896\pi$$
$$548$$ 0 0
$$549$$ 24.0000 1.02430
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14.0000i 0.593199i 0.955002 + 0.296600i $$0.0958526\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 16.0000i 0.674320i 0.941447 + 0.337160i $$0.109466\pi$$
−0.941447 + 0.337160i $$0.890534\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 18.0000i − 0.755929i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ 6.00000i 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 9.00000i − 0.374675i −0.982296 0.187337i $$-0.940014\pi$$
0.982296 0.187337i $$-0.0599858\pi$$
$$578$$ 0 0
$$579$$ −3.00000 −0.124676
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 33.0000i 1.36206i 0.732257 + 0.681028i $$0.238467\pi$$
−0.732257 + 0.681028i $$0.761533\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −9.00000 −0.370211
$$592$$ 0 0
$$593$$ − 34.0000i − 1.39621i −0.715994 0.698106i $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 12.0000i 0.491127i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ 0 0
$$603$$ 24.0000i 0.977356i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.0000i 0.649420i 0.945814 + 0.324710i $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 0 0
$$609$$ 18.0000 0.729397
$$610$$ 0 0
$$611$$ −11.0000 −0.445012
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 48.0000i − 1.93241i −0.257780 0.966204i $$-0.582991\pi$$
0.257780 0.966204i $$-0.417009\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ −9.00000 −0.361158
$$622$$ 0 0
$$623$$ − 4.00000i − 0.160257i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 0 0
$$633$$ − 48.0000i − 1.90783i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 3.00000i − 0.118864i
$$638$$ 0 0
$$639$$ −42.0000 −1.66149
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ − 34.0000i − 1.34083i −0.741987 0.670415i $$-0.766116\pi$$
0.741987 0.670415i $$-0.233884\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 39.0000i − 1.53325i −0.642096 0.766624i $$-0.721935\pi$$
0.642096 0.766624i $$-0.278065\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 18.0000 0.705476
$$652$$ 0 0
$$653$$ 3.00000i 0.117399i 0.998276 + 0.0586995i $$0.0186954\pi$$
−0.998276 + 0.0586995i $$0.981305\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 54.0000i − 2.10674i
$$658$$ 0 0
$$659$$ −8.00000 −0.311636 −0.155818 0.987786i $$-0.549801\pi$$
−0.155818 + 0.987786i $$0.549801\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 3.00000i − 0.116160i
$$668$$ 0 0
$$669$$ −48.0000 −1.85579
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 13.0000i − 0.501113i −0.968102 0.250557i $$-0.919386\pi$$
0.968102 0.250557i $$-0.0806136\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 46.0000i − 1.76792i −0.467559 0.883962i $$-0.654866\pi$$
0.467559 0.883962i $$-0.345134\pi$$
$$678$$ 0 0
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ −6.00000 −0.229920
$$682$$ 0 0
$$683$$ − 35.0000i − 1.33924i −0.742705 0.669619i $$-0.766457\pi$$
0.742705 0.669619i $$-0.233543\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.00000i 0.228914i
$$688$$ 0 0
$$689$$ 14.0000 0.533358
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 63.0000 2.38288
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 36.0000i − 1.35392i
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 3.00000i − 0.112351i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.00000i 0.112037i
$$718$$ 0 0
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 6.00000i 0.223142i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 6.00000i − 0.222528i −0.993791 0.111264i $$-0.964510\pi$$
0.993791 0.111264i $$-0.0354899\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 8.00000i 0.295487i 0.989026 + 0.147743i $$0.0472010\pi$$
−0.989026 + 0.147743i $$0.952799\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −17.0000 −0.625355 −0.312678 0.949859i $$-0.601226\pi$$
−0.312678 + 0.949859i $$0.601226\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.0000i 0.878114i
$$748$$ 0 0
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 0 0
$$753$$ 48.0000i 1.74922i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.0000 1.05125 0.525625 0.850717i $$-0.323832\pi$$
0.525625 + 0.850717i $$0.323832\pi$$
$$762$$ 0 0
$$763$$ − 36.0000i − 1.30329i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 8.00000i − 0.288863i
$$768$$ 0 0
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ −15.0000 −0.540212
$$772$$ 0 0
$$773$$ − 14.0000i − 0.503545i −0.967786 0.251773i $$-0.918987\pi$$
0.967786 0.251773i $$-0.0810135\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 48.0000i − 1.72199i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 27.0000i − 0.964901i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ 0 0
$$789$$ −36.0000 −1.28163
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 12.0000i − 0.425062i −0.977154 0.212531i $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 51.0000i − 1.79529i
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −5.00000 −0.175574 −0.0877869 0.996139i $$-0.527979\pi$$
−0.0877869 + 0.996139i $$0.527979\pi$$
$$812$$ 0 0
$$813$$ − 60.0000i − 2.10429i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ − 33.0000i − 1.15031i −0.818045 0.575154i $$-0.804942\pi$$
0.818045 0.575154i $$-0.195058\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 0 0
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ 87.0000 3.01800
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 27.0000i − 0.933257i
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 18.0000i 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000i 0.755929i
$$848$$ 0 0
$$849$$ 30.0000 1.02960
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 46.0000i − 1.57501i −0.616308 0.787505i $$-0.711372\pi$$
0.616308 0.787505i $$-0.288628\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 41.0000i 1.40053i 0.713881 + 0.700267i $$0.246936\pi$$
−0.713881 + 0.700267i $$0.753064\pi$$
$$858$$ 0 0
$$859$$ 17.0000 0.580033 0.290016 0.957022i $$-0.406339\pi$$
0.290016 + 0.957022i $$0.406339\pi$$
$$860$$ 0 0
$$861$$ 18.0000 0.613438
$$862$$ 0 0
$$863$$ 17.0000i 0.578687i 0.957225 + 0.289343i $$0.0934369\pi$$
−0.957225 + 0.289343i $$0.906563\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 51.0000i 1.73205i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ − 108.000i − 3.65525i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ 0 0
$$879$$ −72.0000 −2.42850
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ 0 0
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 15.0000i − 0.503651i −0.967773 0.251825i $$-0.918969\pi$$
0.967773 0.251825i $$-0.0810309\pi$$
$$888$$ 0 0
$$889$$ 22.0000 0.737856
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 3.00000i − 0.100167i
$$898$$ 0 0
$$899$$ 9.00000 0.300167
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 12.0000i 0.399335i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 0 0
$$909$$ −108.000 −3.58213
$$910$$ 0 0
$$911$$ −44.0000 −1.45779 −0.728893 0.684628i $$-0.759965\pi$$
−0.728893 + 0.684628i $$0.759965\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.0000i 0.594412i
$$918$$ 0 0
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ 60.0000 1.97707
$$922$$ 0 0
$$923$$ − 7.00000i − 0.230408i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 24.0000i − 0.788263i
$$928$$ 0 0
$$929$$ −19.0000 −0.623370 −0.311685 0.950186i $$-0.600893\pi$$
−0.311685 + 0.950186i $$0.600893\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ − 87.0000i − 2.84825i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 44.0000i − 1.43742i −0.695311 0.718709i $$-0.744734\pi$$
0.695311 0.718709i $$-0.255266\pi$$
$$938$$ 0 0
$$939$$ −60.0000 −1.95803
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ 0 0
$$943$$ − 3.00000i − 0.0976934i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 47.0000i − 1.52729i −0.645634 0.763647i $$-0.723407\pi$$
0.645634 0.763647i $$-0.276593\pi$$
$$948$$ 0 0
$$949$$ 9.00000 0.292152
$$950$$ 0 0
$$951$$ 42.0000 1.36194
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 96.0000i 3.09356i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43.0000i 1.38279i 0.722478 + 0.691393i $$0.243003\pi$$
−0.722478 + 0.691393i $$0.756997\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.0000 −0.449281 −0.224641 0.974442i $$-0.572121\pi$$
−0.224641 + 0.974442i $$0.572121\pi$$
$$972$$ 0 0
$$973$$ − 22.0000i − 0.705288i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 48.0000i 1.53566i 0.640656 + 0.767828i $$0.278662\pi$$
−0.640656 + 0.767828i $$0.721338\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −108.000 −3.44817
$$982$$ 0 0
$$983$$ 14.0000i 0.446531i 0.974758 + 0.223265i $$0.0716716\pi$$
−0.974758 + 0.223265i $$0.928328\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 66.0000i − 2.10080i
$$988$$ 0 0
$$989$$ 2.00000 0.0635963
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ − 21.0000i − 0.666415i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 58.0000i 1.83688i 0.395562 + 0.918439i $$0.370550\pi$$
−0.395562 + 0.918439i $$0.629450\pi$$
$$998$$ 0 0
$$999$$ −72.0000 −2.27798
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 4600.2.e.b.4049.2 2
5.2 odd 4 4600.2.a.p.1.1 1
5.3 odd 4 920.2.a.a.1.1 1
5.4 even 2 inner 4600.2.e.b.4049.1 2
15.8 even 4 8280.2.a.d.1.1 1
20.3 even 4 1840.2.a.i.1.1 1
20.7 even 4 9200.2.a.c.1.1 1
40.3 even 4 7360.2.a.a.1.1 1
40.13 odd 4 7360.2.a.ba.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.a.1.1 1 5.3 odd 4
1840.2.a.i.1.1 1 20.3 even 4
4600.2.a.p.1.1 1 5.2 odd 4
4600.2.e.b.4049.1 2 5.4 even 2 inner
4600.2.e.b.4049.2 2 1.1 even 1 trivial
7360.2.a.a.1.1 1 40.3 even 4
7360.2.a.ba.1.1 1 40.13 odd 4
8280.2.a.d.1.1 1 15.8 even 4
9200.2.a.c.1.1 1 20.7 even 4