# Properties

 Label 3675.2.a.c.1.1 Level $3675$ Weight $2$ Character 3675.1 Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3675.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -4.00000 q^{16} -2.00000 q^{18} -1.00000 q^{19} +4.00000 q^{22} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{29} -9.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} +2.00000 q^{36} -3.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} -5.00000 q^{43} -4.00000 q^{44} -6.00000 q^{47} -4.00000 q^{48} +2.00000 q^{52} -12.0000 q^{53} -2.00000 q^{54} -1.00000 q^{57} -8.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +18.0000 q^{62} -8.00000 q^{64} +4.00000 q^{66} +5.00000 q^{67} -6.00000 q^{71} -3.00000 q^{73} +6.00000 q^{74} -2.00000 q^{76} -2.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -20.0000 q^{82} +6.00000 q^{83} +10.0000 q^{86} +4.00000 q^{87} -16.0000 q^{89} -9.00000 q^{93} +12.0000 q^{94} +8.00000 q^{96} -6.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 0 0
$$8$$ 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 $$0.597491\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 8.00000 1.41421
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ −8.00000 −1.05045
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 18.0000 2.28600
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −3.00000 −0.351123 −0.175562 0.984468i $$-0.556174\pi$$
−0.175562 + 0.984468i $$0.556174\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −20.0000 −2.20863
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −9.00000 −0.933257
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 8.00000 0.816497
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 24.0000 2.33109
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 8.00000 0.742781
$$117$$ 1.00000 0.0924500
$$118$$ −24.0000 −2.20938
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 20.0000 1.81071
$$123$$ 10.0000 0.901670
$$124$$ −18.0000 −1.61645
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 15.0000 1.33103 0.665517 0.746382i $$-0.268211\pi$$
0.665517 + 0.746382i $$0.268211\pi$$
$$128$$ 0 0
$$129$$ −5.00000 −0.440225
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 12.0000 1.00702
$$143$$ −2.00000 −0.167248
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ −6.00000 −0.493197
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 2.00000 0.159111
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −2.00000 −0.157135
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 20.0000 1.56174
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ −10.0000 −0.762493
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ 8.00000 0.603023
$$177$$ 12.0000 0.901975
$$178$$ 32.0000 2.39850
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 18.0000 1.31982
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −16.0000 −1.13995 −0.569976 0.821661i $$-0.693048\pi$$
−0.569976 + 0.821661i $$0.693048\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 4.00000 0.281439
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −24.0000 −1.64833
$$213$$ −6.00000 −0.411113
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ −3.00000 −0.202721
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 6.00000 0.402694
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 24.0000 1.56227
$$237$$ −1.00000 −0.0649570
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 1.00000 0.0641500
$$244$$ −20.0000 −1.28037
$$245$$ 0 0
$$246$$ −20.0000 −1.27515
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −30.0000 −1.88237
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 10.0000 0.622573
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ −28.0000 −1.72985
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −16.0000 −0.979184
$$268$$ 10.0000 0.610847
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −24.0000 −1.44989
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0000 −0.781094 −0.390547 0.920583i $$-0.627714\pi$$
−0.390547 + 0.920583i $$0.627714\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ −9.00000 −0.538816
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 8.00000 0.471405
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ −6.00000 −0.351123
$$293$$ 8.00000 0.467365 0.233682 0.972313i $$-0.424922\pi$$
0.233682 + 0.972313i $$0.424922\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 24.0000 1.39028
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 32.0000 1.84139
$$303$$ −2.00000 −0.114897
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.0000 −0.970241 −0.485121 0.874447i $$-0.661224\pi$$
−0.485121 + 0.874447i $$0.661224\pi$$
$$308$$ 0 0
$$309$$ −7.00000 −0.398216
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 28.0000 1.58013
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 24.0000 1.34585
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 9.00000 0.497701
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −3.00000 −0.164399
$$334$$ 28.0000 1.53209
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ 24.0000 1.30543
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 2.00000 0.108148
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −16.0000 −0.860165
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 8.00000 0.428845
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ −16.0000 −0.852803
$$353$$ 34.0000 1.80964 0.904819 0.425797i $$-0.140006\pi$$
0.904819 + 0.425797i $$0.140006\pi$$
$$354$$ −24.0000 −1.27559
$$355$$ 0 0
$$356$$ −32.0000 −1.69600
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 26.0000 1.36653
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 20.0000 1.04542
$$367$$ −9.00000 −0.469796 −0.234898 0.972020i $$-0.575476\pi$$
−0.234898 + 0.972020i $$0.575476\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −18.0000 −0.933257
$$373$$ −23.0000 −1.19089 −0.595447 0.803394i $$-0.703025\pi$$
−0.595447 + 0.803394i $$0.703025\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 3.00000 0.154100 0.0770498 0.997027i $$-0.475450\pi$$
0.0770498 + 0.997027i $$0.475450\pi$$
$$380$$ 0 0
$$381$$ 15.0000 0.768473
$$382$$ −20.0000 −1.02329
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ −5.00000 −0.254164
$$388$$ −12.0000 −0.609208
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 14.0000 0.706207
$$394$$ 32.0000 1.61214
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −9.00000 −0.451697 −0.225849 0.974162i $$-0.572515\pi$$
−0.225849 + 0.974162i $$0.572515\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ −10.0000 −0.498755
$$403$$ −9.00000 −0.448322
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 3.00000 0.146911
$$418$$ −4.00000 −0.195646
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 12.0000 0.581402
$$427$$ 0 0
$$428$$ 16.0000 0.773389
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 31.0000 1.48976 0.744882 0.667196i $$-0.232506\pi$$
0.744882 + 0.667196i $$0.232506\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 6.00000 0.286691
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ −32.0000 −1.51524
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ −20.0000 −0.940721
$$453$$ −16.0000 −0.751746
$$454$$ −36.0000 −1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11.0000 0.514558 0.257279 0.966337i $$-0.417174\pi$$
0.257279 + 0.966337i $$0.417174\pi$$
$$458$$ −38.0000 −1.77562
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −20.0000 −0.931493 −0.465746 0.884918i $$-0.654214\pi$$
−0.465746 + 0.884918i $$0.654214\pi$$
$$462$$ 0 0
$$463$$ 17.0000 0.790057 0.395029 0.918669i $$-0.370735\pi$$
0.395029 + 0.918669i $$0.370735\pi$$
$$464$$ −16.0000 −0.742781
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 2.00000 0.0918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ −12.0000 −0.548867
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 28.0000 1.27537
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ −2.00000 −0.0907218
$$487$$ −31.0000 −1.40474 −0.702372 0.711810i $$-0.747876\pi$$
−0.702372 + 0.711810i $$0.747876\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 20.0000 0.901670
$$493$$ 0 0
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 36.0000 1.61645
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 37.0000 1.65635 0.828174 0.560471i $$-0.189380\pi$$
0.828174 + 0.560471i $$0.189380\pi$$
$$500$$ 0 0
$$501$$ −14.0000 −0.625474
$$502$$ −16.0000 −0.714115
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12.0000 −0.532939
$$508$$ 30.0000 1.33103
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −32.0000 −1.41421
$$513$$ −1.00000 −0.0441511
$$514$$ −52.0000 −2.29362
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ 8.00000 0.351161
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ −8.00000 −0.350150
$$523$$ 31.0000 1.35554 0.677768 0.735276i $$-0.262948\pi$$
0.677768 + 0.735276i $$0.262948\pi$$
$$524$$ 28.0000 1.22319
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 0 0
$$528$$ 8.00000 0.348155
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 10.0000 0.433148
$$534$$ 32.0000 1.38478
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 2.00000 0.0863064
$$538$$ 12.0000 0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19.0000 −0.816874 −0.408437 0.912787i $$-0.633926\pi$$
−0.408437 + 0.912787i $$0.633926\pi$$
$$542$$ 32.0000 1.37452
$$543$$ −13.0000 −0.557883
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 24.0000 1.02523
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 6.00000 0.254457
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 18.0000 0.762001
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 8.00000 0.337460
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 10.0000 0.417756
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ −17.0000 −0.707719 −0.353860 0.935299i $$-0.615131\pi$$
−0.353860 + 0.935299i $$0.615131\pi$$
$$578$$ 34.0000 1.41421
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 12.0000 0.497416
$$583$$ 24.0000 0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −16.0000 −0.660954
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ 0 0
$$591$$ −16.0000 −0.658152
$$592$$ 12.0000 0.493197
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −24.0000 −0.983078
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 9.00000 0.367118 0.183559 0.983009i $$-0.441238\pi$$
0.183559 + 0.983009i $$0.441238\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ −32.0000 −1.30206
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 34.0000 1.37213
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 14.0000 0.563163
$$619$$ 29.0000 1.16561 0.582804 0.812613i $$-0.301955\pi$$
0.582804 + 0.812613i $$0.301955\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ 2.00000 0.0798723
$$628$$ −28.0000 −1.11732
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 48.0000 1.90632
$$635$$ 0 0
$$636$$ −24.0000 −0.951662
$$637$$ 0 0
$$638$$ 16.0000 0.633446
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ −16.0000 −0.631470
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −40.0000 −1.56174
$$657$$ −3.00000 −0.117041
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 41.0000 1.59472 0.797358 0.603507i $$-0.206231\pi$$
0.797358 + 0.603507i $$0.206231\pi$$
$$662$$ 50.0000 1.94331
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ −28.0000 −1.08335
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ 41.0000 1.58043 0.790217 0.612827i $$-0.209968\pi$$
0.790217 + 0.612827i $$0.209968\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 20.0000 0.768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ −36.0000 −1.37851
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 19.0000 0.724895
$$688$$ 20.0000 0.762493
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 37.0000 1.40755 0.703773 0.710425i $$-0.251497\pi$$
0.703773 + 0.710425i $$0.251497\pi$$
$$692$$ 16.0000 0.608229
$$693$$ 0 0
$$694$$ 64.0000 2.42941
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −28.0000 −1.05982
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 3.00000 0.113147
$$704$$ 16.0000 0.603023
$$705$$ 0 0
$$706$$ −68.0000 −2.55921
$$707$$ 0 0
$$708$$ 24.0000 0.901975
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 6.00000 0.224074
$$718$$ −40.0000 −1.49279
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 36.0000 1.33978
$$723$$ −14.0000 −0.520666
$$724$$ −26.0000 −0.966282
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ −13.0000 −0.482143 −0.241072 0.970507i $$-0.577499\pi$$
−0.241072 + 0.970507i $$0.577499\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −20.0000 −0.739221
$$733$$ −15.0000 −0.554038 −0.277019 0.960864i $$-0.589346\pi$$
−0.277019 + 0.960864i $$0.589346\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ −20.0000 −0.736210
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ 0 0
$$741$$ −1.00000 −0.0367359
$$742$$ 0 0
$$743$$ −42.0000 −1.54083 −0.770415 0.637542i $$-0.779951\pi$$
−0.770415 + 0.637542i $$0.779951\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 46.0000 1.68418
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13.0000 0.474377 0.237188 0.971464i $$-0.423774\pi$$
0.237188 + 0.971464i $$0.423774\pi$$
$$752$$ 24.0000 0.875190
$$753$$ 8.00000 0.291536
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −6.00000 −0.217930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ −30.0000 −1.08679
$$763$$ 0 0
$$764$$ 20.0000 0.723575
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 12.0000 0.433295
$$768$$ 16.0000 0.577350
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ −22.0000 −0.791797
$$773$$ −34.0000 −1.22290 −0.611448 0.791285i $$-0.709412\pi$$
−0.611448 + 0.791285i $$0.709412\pi$$
$$774$$ 10.0000 0.359443
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 12.0000 0.430221
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ 4.00000 0.142948
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −28.0000 −0.998727
$$787$$ 40.0000 1.42585 0.712923 0.701242i $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ −32.0000 −1.13995
$$789$$ −4.00000 −0.142404
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −16.0000 −0.565332
$$802$$ 72.0000 2.54241
$$803$$ 6.00000 0.211735
$$804$$ 10.0000 0.352673
$$805$$ 0 0
$$806$$ 18.0000 0.634023
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −32.0000 −1.12367 −0.561836 0.827249i $$-0.689905\pi$$
−0.561836 + 0.827249i $$0.689905\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000 0.174928
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ −24.0000 −0.837096
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ 0 0
$$829$$ −41.0000 −1.42399 −0.711994 0.702185i $$-0.752208\pi$$
−0.711994 + 0.702185i $$0.752208\pi$$
$$830$$ 0 0
$$831$$ −13.0000 −0.450965
$$832$$ −8.00000 −0.277350
$$833$$ 0 0
$$834$$ −6.00000 −0.207763
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ −9.00000 −0.311086
$$838$$ 60.0000 2.07267
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 14.0000 0.482472
$$843$$ −4.00000 −0.137767
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ 48.0000 1.64833
$$849$$ −11.0000 −0.377519
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −12.0000 −0.411113
$$853$$ 35.0000 1.19838 0.599189 0.800608i $$-0.295490\pi$$
0.599189 + 0.800608i $$0.295490\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −32.0000 −1.09310 −0.546550 0.837427i $$-0.684059\pi$$
−0.546550 + 0.837427i $$0.684059\pi$$
$$858$$ 4.00000 0.136558
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000 1.22616
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 0 0
$$866$$ −62.0000 −2.10685
$$867$$ −17.0000 −0.577350
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ 5.00000 0.169419
$$872$$ 0 0
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ 0 0
$$879$$ 8.00000 0.269833
$$880$$ 0 0
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 0 0
$$883$$ 13.0000 0.437485 0.218742 0.975783i $$-0.429805\pi$$
0.218742 + 0.975783i $$0.429805\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −34.0000 −1.14161 −0.570804 0.821086i $$-0.693368\pi$$
−0.570804 + 0.821086i $$0.693368\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 32.0000 1.07144
$$893$$ 6.00000 0.200782
$$894$$ 24.0000 0.802680
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 36.0000 1.20134
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 32.0000 1.06313
$$907$$ 37.0000 1.22856 0.614282 0.789086i $$-0.289446\pi$$
0.614282 + 0.789086i $$0.289446\pi$$
$$908$$ 36.0000 1.19470
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −12.0000 −0.397142
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 38.0000 1.25556
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 23.0000 0.758700 0.379350 0.925253i $$-0.376148\pi$$
0.379350 + 0.925253i $$0.376148\pi$$
$$920$$ 0 0
$$921$$ −17.0000 −0.560169
$$922$$ 40.0000 1.31733
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −34.0000 −1.11731
$$927$$ −7.00000 −0.229910
$$928$$ 32.0000 1.05045
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −12.0000 −0.393073
$$933$$ 6.00000 0.196431
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 15.0000 0.490029 0.245014 0.969519i $$-0.421207\pi$$
0.245014 + 0.969519i $$0.421207\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ 28.0000 0.912289
$$943$$ 0 0
$$944$$ −48.0000 −1.56227
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ −2.00000 −0.0649570
$$949$$ −3.00000 −0.0973841
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ 24.0000 0.777029
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ −8.00000 −0.258603
$$958$$ −56.0000 −1.80928
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 6.00000 0.193448
$$963$$ 8.00000 0.257796
$$964$$ −28.0000 −0.901819
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −19.0000 −0.610999 −0.305499 0.952192i $$-0.598823\pi$$
−0.305499 + 0.952192i $$0.598823\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 0 0
$$974$$ 62.0000 1.98661
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 8.00000 0.255812
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 9.00000 0.287348
$$982$$ 56.0000 1.78703
$$983$$ 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
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ −72.0000 −2.28600
$$993$$ −25.0000 −0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 19.0000 0.601736 0.300868 0.953666i $$-0.402724\pi$$
0.300868 + 0.953666i $$0.402724\pi$$
$$998$$ −74.0000 −2.34243
$$999$$ −3.00000 −0.0949158
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 3675.2.a.c.1.1 1
5.4 even 2 147.2.a.b.1.1 1
7.3 odd 6 525.2.i.e.226.1 2
7.5 odd 6 525.2.i.e.151.1 2
7.6 odd 2 3675.2.a.a.1.1 1
15.14 odd 2 441.2.a.a.1.1 1
20.19 odd 2 2352.2.a.w.1.1 1
35.3 even 12 525.2.r.e.499.1 4
35.4 even 6 147.2.e.a.79.1 2
35.9 even 6 147.2.e.a.67.1 2
35.12 even 12 525.2.r.e.424.1 4
35.17 even 12 525.2.r.e.499.2 4
35.19 odd 6 21.2.e.a.4.1 2
35.24 odd 6 21.2.e.a.16.1 yes 2
35.33 even 12 525.2.r.e.424.2 4
35.34 odd 2 147.2.a.c.1.1 1
40.19 odd 2 9408.2.a.k.1.1 1
40.29 even 2 9408.2.a.bz.1.1 1
60.59 even 2 7056.2.a.m.1.1 1
105.44 odd 6 441.2.e.e.361.1 2
105.59 even 6 63.2.e.b.37.1 2
105.74 odd 6 441.2.e.e.226.1 2
105.89 even 6 63.2.e.b.46.1 2
105.104 even 2 441.2.a.b.1.1 1
140.19 even 6 336.2.q.f.193.1 2
140.39 odd 6 2352.2.q.c.961.1 2
140.59 even 6 336.2.q.f.289.1 2
140.79 odd 6 2352.2.q.c.1537.1 2
140.139 even 2 2352.2.a.d.1.1 1
280.19 even 6 1344.2.q.c.193.1 2
280.59 even 6 1344.2.q.c.961.1 2
280.69 odd 2 9408.2.a.bg.1.1 1
280.139 even 2 9408.2.a.cv.1.1 1
280.229 odd 6 1344.2.q.m.193.1 2
280.269 odd 6 1344.2.q.m.961.1 2
315.59 even 6 567.2.g.f.541.1 2
315.94 odd 6 567.2.g.a.541.1 2
315.124 odd 6 567.2.g.a.109.1 2
315.164 even 6 567.2.h.a.352.1 2
315.194 even 6 567.2.h.a.298.1 2
315.229 odd 6 567.2.h.f.298.1 2
315.299 even 6 567.2.g.f.109.1 2
315.304 odd 6 567.2.h.f.352.1 2
420.59 odd 6 1008.2.s.d.289.1 2
420.299 odd 6 1008.2.s.d.865.1 2
420.419 odd 2 7056.2.a.bp.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 35.19 odd 6
21.2.e.a.16.1 yes 2 35.24 odd 6
63.2.e.b.37.1 2 105.59 even 6
63.2.e.b.46.1 2 105.89 even 6
147.2.a.b.1.1 1 5.4 even 2
147.2.a.c.1.1 1 35.34 odd 2
147.2.e.a.67.1 2 35.9 even 6
147.2.e.a.79.1 2 35.4 even 6
336.2.q.f.193.1 2 140.19 even 6
336.2.q.f.289.1 2 140.59 even 6
441.2.a.a.1.1 1 15.14 odd 2
441.2.a.b.1.1 1 105.104 even 2
441.2.e.e.226.1 2 105.74 odd 6
441.2.e.e.361.1 2 105.44 odd 6
525.2.i.e.151.1 2 7.5 odd 6
525.2.i.e.226.1 2 7.3 odd 6
525.2.r.e.424.1 4 35.12 even 12
525.2.r.e.424.2 4 35.33 even 12
525.2.r.e.499.1 4 35.3 even 12
525.2.r.e.499.2 4 35.17 even 12
567.2.g.a.109.1 2 315.124 odd 6
567.2.g.a.541.1 2 315.94 odd 6
567.2.g.f.109.1 2 315.299 even 6
567.2.g.f.541.1 2 315.59 even 6
567.2.h.a.298.1 2 315.194 even 6
567.2.h.a.352.1 2 315.164 even 6
567.2.h.f.298.1 2 315.229 odd 6
567.2.h.f.352.1 2 315.304 odd 6
1008.2.s.d.289.1 2 420.59 odd 6
1008.2.s.d.865.1 2 420.299 odd 6
1344.2.q.c.193.1 2 280.19 even 6
1344.2.q.c.961.1 2 280.59 even 6
1344.2.q.m.193.1 2 280.229 odd 6
1344.2.q.m.961.1 2 280.269 odd 6
2352.2.a.d.1.1 1 140.139 even 2
2352.2.a.w.1.1 1 20.19 odd 2
2352.2.q.c.961.1 2 140.39 odd 6
2352.2.q.c.1537.1 2 140.79 odd 6
3675.2.a.a.1.1 1 7.6 odd 2
3675.2.a.c.1.1 1 1.1 even 1 trivial
7056.2.a.m.1.1 1 60.59 even 2
7056.2.a.bp.1.1 1 420.419 odd 2
9408.2.a.k.1.1 1 40.19 odd 2
9408.2.a.bg.1.1 1 280.69 odd 2
9408.2.a.bz.1.1 1 40.29 even 2
9408.2.a.cv.1.1 1 280.139 even 2