# Properties

 Label 1200.2.a.c.1.1 Level $1200$ Weight $2$ Character 1200.1 Self dual yes Analytic conductor $9.582$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} -2.00000 q^{17} +5.00000 q^{19} +3.00000 q^{21} +6.00000 q^{23} -1.00000 q^{27} +10.0000 q^{29} +3.00000 q^{31} +2.00000 q^{33} -2.00000 q^{37} +1.00000 q^{39} -8.00000 q^{41} +1.00000 q^{43} +2.00000 q^{47} +2.00000 q^{49} +2.00000 q^{51} +4.00000 q^{53} -5.00000 q^{57} +10.0000 q^{59} +7.00000 q^{61} -3.00000 q^{63} -3.00000 q^{67} -6.00000 q^{69} +8.00000 q^{71} +14.0000 q^{73} +6.00000 q^{77} +1.00000 q^{81} +6.00000 q^{83} -10.0000 q^{87} +3.00000 q^{91} -3.00000 q^{93} -17.0000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.00000 −0.662266
$$58$$ 0 0
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −10.0000 −1.07211
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 0 0
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.0000 −1.72609 −0.863044 0.505128i $$-0.831445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 4.00000 0.376288 0.188144 0.982141i $$-0.439753\pi$$
0.188144 + 0.982141i $$0.439753\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 8.00000 0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −15.0000 −1.30066
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ −4.00000 −0.317221
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −10.0000 −0.751646
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ 0 0
$$183$$ −7.00000 −0.517455
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ −30.0000 −2.10559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 0 0
$$213$$ −8.00000 −0.548151
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ −3.00000 −0.181568
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000 0.180253 0.0901263 0.995930i $$-0.471273\pi$$
0.0901263 + 0.995930i $$0.471273\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ −9.00000 −0.534994 −0.267497 0.963559i $$-0.586197\pi$$
−0.267497 + 0.963559i $$0.586197\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 17.0000 0.996558
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ −11.0000 −0.621757 −0.310878 0.950450i $$-0.600623\pi$$
−0.310878 + 0.950450i $$0.600623\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000 0.449325 0.224662 0.974437i $$-0.427872\pi$$
0.224662 + 0.974437i $$0.427872\pi$$
$$318$$ 0 0
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −10.0000 −0.556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −5.00000 −0.276501
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ −4.00000 −0.217250
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.00000 −0.317554
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 27.0000 1.40939 0.704694 0.709511i $$-0.251084\pi$$
0.704694 + 0.709511i $$0.251084\pi$$
$$368$$ 0 0
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −10.0000 −0.515026
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −7.00000 −0.351320 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$398$$ 0 0
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ −3.00000 −0.149441
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ −30.0000 −1.47620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −21.0000 −1.01626
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.0000 1.43509
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ 0 0
$$453$$ 7.00000 0.328889
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −38.0000 −1.75843 −0.879215 0.476425i $$-0.841932\pi$$
−0.879215 + 0.476425i $$0.841932\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ −2.00000 −0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4.00000 0.183147
$$478$$ 0 0
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ 18.0000 0.819028
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −13.0000 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 0 0
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −42.0000 −1.85797
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.00000 −0.175920
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 31.0000 1.35554 0.677768 0.735276i $$-0.262948\pi$$
0.677768 + 0.735276i $$0.262948\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −10.0000 −0.431532
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ 0 0
$$543$$ −17.0000 −0.729540
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ 50.0000 2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ 6.00000 0.252870 0.126435 0.991975i $$-0.459647\pi$$
0.126435 + 0.991975i $$0.459647\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 13.0000 0.544033 0.272017 0.962293i $$-0.412309\pi$$
0.272017 + 0.962293i $$0.412309\pi$$
$$572$$ 0 0
$$573$$ 22.0000 0.919063
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.0000 0.541197 0.270599 0.962692i $$-0.412778\pi$$
0.270599 + 0.962692i $$0.412778\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 0 0
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5.00000 −0.204636
$$598$$ 0 0
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ −3.00000 −0.122169
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ −2.00000 −0.0809113
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ 25.0000 1.00483 0.502417 0.864625i $$-0.332444\pi$$
0.502417 + 0.864625i $$0.332444\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 10.0000 0.399362
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 23.0000 0.915616 0.457808 0.889051i $$-0.348635\pi$$
0.457808 + 0.889051i $$0.348635\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 9.00000 0.352738
$$652$$ 0 0
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000 0.546192
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 60.0000 2.32321
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 51.0000 1.95720
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 15.0000 0.572286
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ 0 0
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 18.0000 0.674105
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.0000 0.746914
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ 23.0000 0.855379
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −43.0000 −1.59478 −0.797391 0.603463i $$-0.793787\pi$$
−0.797391 + 0.603463i $$0.793787\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −2.00000 −0.0739727
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.00000 0.221013
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ 0 0
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23.0000 0.835949 0.417975 0.908459i $$-0.362740\pi$$
0.417975 + 0.908459i $$0.362740\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ −15.0000 −0.543036
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −10.0000 −0.361079
$$768$$ 0 0
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −6.00000 −0.215249
$$778$$ 0 0
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000 0.249523 0.124762 0.992187i $$-0.460183\pi$$
0.124762 + 0.992187i $$0.460183\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ −7.00000 −0.248577
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −52.0000 −1.84193 −0.920967 0.389640i $$-0.872599\pi$$
−0.920967 + 0.389640i $$0.872599\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −28.0000 −0.988099
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10.0000 0.352017
$$808$$ 0 0
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ −27.0000 −0.948098 −0.474049 0.880498i $$-0.657208\pi$$
−0.474049 + 0.880498i $$0.657208\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000 0.174928
$$818$$ 0 0
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ 41.0000 1.42917 0.714585 0.699549i $$-0.246616\pi$$
0.714585 + 0.699549i $$0.246616\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ −3.00000 −0.104069
$$832$$ 0 0
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −3.00000 −0.103695
$$838$$ 0 0
$$839$$ −10.0000 −0.345238 −0.172619 0.984989i $$-0.555223\pi$$
−0.172619 + 0.984989i $$0.555223\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000 0.721569
$$848$$ 0 0
$$849$$ 9.00000 0.308879
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −51.0000 −1.74621 −0.873103 0.487535i $$-0.837896\pi$$
−0.873103 + 0.487535i $$0.837896\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.0000 0.956462 0.478231 0.878234i $$-0.341278\pi$$
0.478231 + 0.878234i $$0.341278\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ −17.0000 −0.575363
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27.0000 −0.911725 −0.455863 0.890050i $$-0.650669\pi$$
−0.455863 + 0.890050i $$0.650669\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ 0 0
$$883$$ 41.0000 1.37976 0.689880 0.723924i $$-0.257663\pi$$
0.689880 + 0.723924i $$0.257663\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 10.0000 0.334637
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ 0 0
$$903$$ 3.00000 0.0998337
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ 0 0
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ 58.0000 1.92163 0.960813 0.277198i $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 36.0000 1.18882
$$918$$ 0 0
$$919$$ −55.0000 −1.81428 −0.907141 0.420826i $$-0.861740\pi$$
−0.907141 + 0.420826i $$0.861740\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 0 0
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ 10.0000 0.327737
$$932$$ 0 0
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.0000 1.07806 0.539032 0.842286i $$-0.318790\pi$$
0.539032 + 0.842286i $$0.318790\pi$$
$$938$$ 0 0
$$939$$ 11.0000 0.358971
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 0 0
$$943$$ −48.0000 −1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ −8.00000 −0.259418
$$952$$ 0 0
$$953$$ −56.0000 −1.81402 −0.907009 0.421111i $$-0.861640\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 20.0000 0.646508
$$958$$ 0 0
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ 10.0000 0.321246
$$970$$ 0 0
$$971$$ −42.0000 −1.34784 −0.673922 0.738802i $$-0.735392\pi$$
−0.673922 + 0.738802i $$0.735392\pi$$
$$972$$ 0 0
$$973$$ 60.0000 1.92351
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ 0 0
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ 12.0000 0.380808
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
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 1200.2.a.c.1.1 1
3.2 odd 2 3600.2.a.j.1.1 1
4.3 odd 2 75.2.a.a.1.1 1
5.2 odd 4 1200.2.f.d.49.2 2
5.3 odd 4 1200.2.f.d.49.1 2
5.4 even 2 1200.2.a.p.1.1 1
8.3 odd 2 4800.2.a.bb.1.1 1
8.5 even 2 4800.2.a.br.1.1 1
12.11 even 2 225.2.a.e.1.1 1
15.2 even 4 3600.2.f.p.2449.1 2
15.8 even 4 3600.2.f.p.2449.2 2
15.14 odd 2 3600.2.a.bk.1.1 1
20.3 even 4 75.2.b.a.49.2 2
20.7 even 4 75.2.b.a.49.1 2
20.19 odd 2 75.2.a.c.1.1 yes 1
28.27 even 2 3675.2.a.b.1.1 1
40.3 even 4 4800.2.f.l.3649.1 2
40.13 odd 4 4800.2.f.y.3649.2 2
40.19 odd 2 4800.2.a.bq.1.1 1
40.27 even 4 4800.2.f.l.3649.2 2
40.29 even 2 4800.2.a.be.1.1 1
40.37 odd 4 4800.2.f.y.3649.1 2
44.43 even 2 9075.2.a.s.1.1 1
60.23 odd 4 225.2.b.a.199.1 2
60.47 odd 4 225.2.b.a.199.2 2
60.59 even 2 225.2.a.a.1.1 1
140.139 even 2 3675.2.a.q.1.1 1
220.219 even 2 9075.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 4.3 odd 2
75.2.a.c.1.1 yes 1 20.19 odd 2
75.2.b.a.49.1 2 20.7 even 4
75.2.b.a.49.2 2 20.3 even 4
225.2.a.a.1.1 1 60.59 even 2
225.2.a.e.1.1 1 12.11 even 2
225.2.b.a.199.1 2 60.23 odd 4
225.2.b.a.199.2 2 60.47 odd 4
1200.2.a.c.1.1 1 1.1 even 1 trivial
1200.2.a.p.1.1 1 5.4 even 2
1200.2.f.d.49.1 2 5.3 odd 4
1200.2.f.d.49.2 2 5.2 odd 4
3600.2.a.j.1.1 1 3.2 odd 2
3600.2.a.bk.1.1 1 15.14 odd 2
3600.2.f.p.2449.1 2 15.2 even 4
3600.2.f.p.2449.2 2 15.8 even 4
3675.2.a.b.1.1 1 28.27 even 2
3675.2.a.q.1.1 1 140.139 even 2
4800.2.a.bb.1.1 1 8.3 odd 2
4800.2.a.be.1.1 1 40.29 even 2
4800.2.a.bq.1.1 1 40.19 odd 2
4800.2.a.br.1.1 1 8.5 even 2
4800.2.f.l.3649.1 2 40.3 even 4
4800.2.f.l.3649.2 2 40.27 even 4
4800.2.f.y.3649.1 2 40.37 odd 4
4800.2.f.y.3649.2 2 40.13 odd 4
9075.2.a.a.1.1 1 220.219 even 2
9075.2.a.s.1.1 1 44.43 even 2