# Properties

 Label 4056.2.c.d.337.1 Level $4056$ Weight $2$ Character 4056.337 Analytic conductor $32.387$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4056.337 Dual form 4056.2.c.d.337.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -4.00000i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -4.00000i q^{5} +1.00000 q^{9} -2.00000i q^{11} +4.00000i q^{15} -2.00000 q^{17} -8.00000i q^{19} -4.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +4.00000i q^{31} +2.00000i q^{33} +6.00000i q^{37} +12.0000i q^{41} -4.00000 q^{43} -4.00000i q^{45} -6.00000i q^{47} +7.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -8.00000 q^{55} +8.00000i q^{57} -14.0000i q^{59} +10.0000 q^{61} +4.00000i q^{67} +4.00000 q^{69} -2.00000i q^{71} -2.00000i q^{73} +11.0000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +8.00000i q^{85} +6.00000 q^{87} -4.00000i q^{93} -32.0000 q^{95} +10.0000i q^{97} -2.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} + 20 q^{61} + 8 q^{69} + 22 q^{75} - 16 q^{79} + 2 q^{81} + 12 q^{87} - 64 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 2 * q^27 - 12 * q^29 - 8 * q^43 + 14 * q^49 + 4 * q^51 - 4 * q^53 - 16 * q^55 + 20 * q^61 + 8 * q^69 + 22 * q^75 - 16 * q^79 + 2 * q^81 + 12 * q^87 - 64 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ − 4.00000i − 1.78885i −0.447214 0.894427i $$-0.647584\pi$$
0.447214 0.894427i $$-0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ − 2.00000i − 0.603023i −0.953463 0.301511i $$-0.902509\pi$$
0.953463 0.301511i $$-0.0974911\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 4.00000i 1.03280i
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ − 8.00000i − 1.83533i −0.397360 0.917663i $$-0.630073\pi$$
0.397360 0.917663i $$-0.369927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −11.0000 −2.20000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000i 0.718421i 0.933257 + 0.359211i $$0.116954\pi$$
−0.933257 + 0.359211i $$0.883046\pi$$
$$32$$ 0 0
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 12.0000i 1.87409i 0.349215 + 0.937043i $$0.386448\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ − 4.00000i − 0.596285i
$$46$$ 0 0
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ 8.00000i 1.05963i
$$58$$ 0 0
$$59$$ − 14.0000i − 1.82264i −0.411693 0.911322i $$-0.635063\pi$$
0.411693 0.911322i $$-0.364937\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ − 2.00000i − 0.237356i −0.992933 0.118678i $$-0.962134\pi$$
0.992933 0.118678i $$-0.0378657\pi$$
$$72$$ 0 0
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 0 0
$$75$$ 11.0000 1.27017
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 14.0000i − 1.53670i −0.640030 0.768350i $$-0.721078\pi$$
0.640030 0.768350i $$-0.278922\pi$$
$$84$$ 0 0
$$85$$ 8.00000i 0.867722i
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ −32.0000 −3.28313
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ − 2.00000i − 0.201008i
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 0 0
$$111$$ − 6.00000i − 0.569495i
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 16.0000i 1.49201i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ 0 0
$$125$$ 24.0000i 2.14663i
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.00000i 0.344265i
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 6.00000i 0.505291i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 24.0000i 1.99309i
$$146$$ 0 0
$$147$$ −7.00000 −0.577350
$$148$$ 0 0
$$149$$ 16.0000i 1.31077i 0.755295 + 0.655386i $$0.227494\pi$$
−0.755295 + 0.655386i $$0.772506\pi$$
$$150$$ 0 0
$$151$$ 20.0000i 1.62758i 0.581161 + 0.813788i $$0.302599\pi$$
−0.581161 + 0.813788i $$0.697401\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 24.0000i 1.87983i 0.341415 + 0.939913i $$0.389094\pi$$
−0.341415 + 0.939913i $$0.610906\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 8.00000i − 0.611775i
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.0000i 1.05230i
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 24.0000 1.76452
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ − 4.00000i − 0.282138i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 48.0000 3.35247
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 2.00000i 0.137038i
$$214$$ 0 0
$$215$$ 16.0000i 1.09119i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 28.0000i − 1.87502i −0.347960 0.937509i $$-0.613126\pi$$
0.347960 0.937509i $$-0.386874\pi$$
$$224$$ 0 0
$$225$$ −11.0000 −0.733333
$$226$$ 0 0
$$227$$ 6.00000i 0.398234i 0.979976 + 0.199117i $$0.0638074\pi$$
−0.979976 + 0.199117i $$0.936193\pi$$
$$228$$ 0 0
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2.00000 0.131024 0.0655122 0.997852i $$-0.479132\pi$$
0.0655122 + 0.997852i $$0.479132\pi$$
$$234$$ 0 0
$$235$$ −24.0000 −1.56559
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ − 2.00000i − 0.129369i −0.997906 0.0646846i $$-0.979396\pi$$
0.997906 0.0646846i $$-0.0206041\pi$$
$$240$$ 0 0
$$241$$ − 18.0000i − 1.15948i −0.814801 0.579741i $$-0.803154\pi$$
0.814801 0.579741i $$-0.196846\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ − 28.0000i − 1.78885i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 14.0000i 0.887214i
$$250$$ 0 0
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ 8.00000i 0.502956i
$$254$$ 0 0
$$255$$ − 8.00000i − 0.500979i
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −20.0000 −1.23325 −0.616626 0.787256i $$-0.711501\pi$$
−0.616626 + 0.787256i $$0.711501\pi$$
$$264$$ 0 0
$$265$$ 8.00000i 0.491436i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ 8.00000i 0.485965i 0.970031 + 0.242983i $$0.0781258\pi$$
−0.970031 + 0.242983i $$0.921874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 22.0000i 1.32665i
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ 4.00000i 0.239474i
$$280$$ 0 0
$$281$$ 20.0000i 1.19310i 0.802576 + 0.596550i $$0.203462\pi$$
−0.802576 + 0.596550i $$0.796538\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 32.0000 1.89552
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 10.0000i − 0.586210i
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ −56.0000 −3.26045
$$296$$ 0 0
$$297$$ 2.00000i 0.116052i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ − 40.0000i − 2.29039i
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ 12.0000i 0.671871i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 16.0000i 0.890264i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.0000i 0.553001i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000i 1.53902i 0.638635 + 0.769510i $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 16.0000 0.874173
$$336$$ 0 0
$$337$$ −30.0000 −1.63420 −0.817102 0.576493i $$-0.804421\pi$$
−0.817102 + 0.576493i $$0.804421\pi$$
$$338$$ 0 0
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ − 16.0000i − 0.861411i
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 6.00000i 0.321173i 0.987022 + 0.160586i $$0.0513385\pi$$
−0.987022 + 0.160586i $$0.948662\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 28.0000i 1.49029i 0.666903 + 0.745145i $$0.267620\pi$$
−0.666903 + 0.745145i $$0.732380\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 30.0000i − 1.58334i −0.610949 0.791670i $$-0.709212\pi$$
0.610949 0.791670i $$-0.290788\pi$$
$$360$$ 0 0
$$361$$ −45.0000 −2.36842
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 12.0000i 0.624695i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ − 24.0000i − 1.23935i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0000i 0.821865i 0.911666 + 0.410932i $$0.134797\pi$$
−0.911666 + 0.410932i $$0.865203\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 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
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ −8.00000 −0.403547
$$394$$ 0 0
$$395$$ 32.0000i 1.61009i
$$396$$ 0 0
$$397$$ − 10.0000i − 0.501886i −0.968002 0.250943i $$-0.919259\pi$$
0.968002 0.250943i $$-0.0807406\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 4.00000i − 0.198762i
$$406$$ 0 0
$$407$$ 12.0000 0.594818
$$408$$ 0 0
$$409$$ − 6.00000i − 0.296681i −0.988936 0.148340i $$-0.952607\pi$$
0.988936 0.148340i $$-0.0473931\pi$$
$$410$$ 0 0
$$411$$ 12.0000i 0.591916i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −56.0000 −2.74893
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ − 26.0000i − 1.26716i −0.773676 0.633581i $$-0.781584\pi$$
0.773676 0.633581i $$-0.218416\pi$$
$$422$$ 0 0
$$423$$ − 6.00000i − 0.291730i
$$424$$ 0 0
$$425$$ 22.0000 1.06716
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 14.0000i − 0.674356i −0.941441 0.337178i $$-0.890528\pi$$
0.941441 0.337178i $$-0.109472\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ − 24.0000i − 1.15071i
$$436$$ 0 0
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ 8.00000 0.380091 0.190046 0.981775i $$-0.439136\pi$$
0.190046 + 0.981775i $$0.439136\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 16.0000i − 0.756774i
$$448$$ 0 0
$$449$$ − 8.00000i − 0.377543i −0.982021 0.188772i $$-0.939549\pi$$
0.982021 0.188772i $$-0.0604506\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ − 20.0000i − 0.939682i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ − 28.0000i − 1.30409i −0.758180 0.652045i $$-0.773911\pi$$
0.758180 0.652045i $$-0.226089\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 0 0
$$465$$ −16.0000 −0.741982
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 8.00000i 0.367840i
$$474$$ 0 0
$$475$$ 88.0000i 4.03772i
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ 14.0000i 0.639676i 0.947472 + 0.319838i $$0.103629\pi$$
−0.947472 + 0.319838i $$0.896371\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 40.0000 1.81631
$$486$$ 0 0
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ 0 0
$$489$$ − 24.0000i − 1.08532i
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 36.0000i − 1.61158i −0.592200 0.805791i $$-0.701741\pi$$
0.592200 0.805791i $$-0.298259\pi$$
$$500$$ 0 0
$$501$$ − 18.0000i − 0.804181i
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ − 24.0000i − 1.06799i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 12.0000i − 0.531891i −0.963988 0.265945i $$-0.914316\pi$$
0.963988 0.265945i $$-0.0856841\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 8.00000i 0.353209i
$$514$$ 0 0
$$515$$ − 32.0000i − 1.41009i
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 8.00000i − 0.348485i
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ − 14.0000i − 0.607548i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ − 14.0000i − 0.603023i
$$540$$ 0 0
$$541$$ − 6.00000i − 0.257960i −0.991647 0.128980i $$-0.958830\pi$$
0.991647 0.128980i $$-0.0411703\pi$$
$$542$$ 0 0
$$543$$ 6.00000 0.257485
$$544$$ 0 0
$$545$$ −40.0000 −1.71341
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 48.0000i 2.04487i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −24.0000 −1.01874
$$556$$ 0 0
$$557$$ − 8.00000i − 0.338971i −0.985533 0.169485i $$-0.945789\pi$$
0.985533 0.169485i $$-0.0542106\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 4.00000i − 0.168880i
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 72.0000i 3.02906i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 2.00000 0.0838444 0.0419222 0.999121i $$-0.486652\pi$$
0.0419222 + 0.999121i $$0.486652\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 44.0000 1.83493
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ − 26.0000i − 1.08052i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000i 0.165663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 22.0000i − 0.908037i −0.890992 0.454019i $$-0.849990\pi$$
0.890992 0.454019i $$-0.150010\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 12.0000i 0.493614i
$$592$$ 0 0
$$593$$ − 8.00000i − 0.328521i −0.986417 0.164260i $$-0.947476\pi$$
0.986417 0.164260i $$-0.0525237\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ − 28.0000i − 1.13836i
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 0 0
$$615$$ −48.0000 −1.93555
$$616$$ 0 0
$$617$$ − 12.0000i − 0.483102i −0.970388 0.241551i $$-0.922344\pi$$
0.970388 0.241551i $$-0.0776561\pi$$
$$618$$ 0 0
$$619$$ 28.0000i 1.12542i 0.826656 + 0.562708i $$0.190240\pi$$
−0.826656 + 0.562708i $$0.809760\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 16.0000 0.638978
$$628$$ 0 0
$$629$$ − 12.0000i − 0.478471i
$$630$$ 0 0
$$631$$ − 20.0000i − 0.796187i −0.917345 0.398094i $$-0.869672\pi$$
0.917345 0.398094i $$-0.130328\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ − 32.0000i − 1.26988i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 2.00000i − 0.0791188i
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ 0 0
$$643$$ 36.0000i 1.41970i 0.704352 + 0.709851i $$0.251238\pi$$
−0.704352 + 0.709851i $$0.748762\pi$$
$$644$$ 0 0
$$645$$ − 16.0000i − 0.629999i
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ −28.0000 −1.09910
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ − 32.0000i − 1.25034i
$$656$$ 0 0
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 38.0000i 1.47803i 0.673690 + 0.739014i $$0.264708\pi$$
−0.673690 + 0.739014i $$0.735292\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 28.0000i 1.08254i
$$670$$ 0 0
$$671$$ − 20.0000i − 0.772091i
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 0 0
$$675$$ 11.0000 0.423390
$$676$$ 0 0
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ − 6.00000i − 0.229920i
$$682$$ 0 0
$$683$$ − 2.00000i − 0.0765279i −0.999268 0.0382639i $$-0.987817\pi$$
0.999268 0.0382639i $$-0.0121828\pi$$
$$684$$ 0 0
$$685$$ −48.0000 −1.83399
$$686$$ 0 0
$$687$$ − 10.0000i − 0.381524i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 20.0000i − 0.760836i −0.924815 0.380418i $$-0.875780\pi$$
0.924815 0.380418i $$-0.124220\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 48.0000i 1.82074i
$$696$$ 0 0
$$697$$ − 24.0000i − 0.909065i
$$698$$ 0 0
$$699$$ −2.00000 −0.0756469
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ 48.0000 1.81035
$$704$$ 0 0
$$705$$ 24.0000 0.903892
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000i 0.0751116i 0.999295 + 0.0375558i $$0.0119572\pi$$
−0.999295 + 0.0375558i $$0.988043\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 2.00000i 0.0746914i
$$718$$ 0 0
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 18.0000i 0.669427i
$$724$$ 0 0
$$725$$ 66.0000 2.45118
$$726$$ 0 0
$$727$$ 24.0000 0.890111 0.445055 0.895503i $$-0.353184\pi$$
0.445055 + 0.895503i $$0.353184\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ 0 0
$$735$$ 28.0000i 1.03280i
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ − 28.0000i − 1.03000i −0.857191 0.514998i $$-0.827793\pi$$
0.857191 0.514998i $$-0.172207\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 34.0000i − 1.24734i −0.781688 0.623670i $$-0.785641\pi$$
0.781688 0.623670i $$-0.214359\pi$$
$$744$$ 0 0
$$745$$ 64.0000 2.34478
$$746$$ 0 0
$$747$$ − 14.0000i − 0.512233i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ 80.0000 2.91150
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 0 0
$$759$$ − 8.00000i − 0.290382i
$$760$$ 0 0
$$761$$ 20.0000i 0.724999i 0.931984 + 0.362500i $$0.118077\pi$$
−0.931984 + 0.362500i $$0.881923\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8.00000i 0.289241i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 26.0000i − 0.937584i −0.883309 0.468792i $$-0.844689\pi$$
0.883309 0.468792i $$-0.155311\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ 0 0
$$773$$ 48.0000i 1.72644i 0.504828 + 0.863220i $$0.331556\pi$$
−0.504828 + 0.863220i $$0.668444\pi$$
$$774$$ 0 0
$$775$$ − 44.0000i − 1.58053i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 96.0000 3.43956
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ 8.00000i 0.285532i
$$786$$ 0 0
$$787$$ − 44.0000i − 1.56843i −0.620489 0.784215i $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ 0 0
$$789$$ 20.0000 0.712019
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 8.00000i − 0.283731i
$$796$$ 0 0
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 12.0000i 0.424529i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −4.00000 −0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −26.0000 −0.915243
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 8.00000i 0.280918i 0.990086 + 0.140459i $$0.0448578\pi$$
−0.990086 + 0.140459i $$0.955142\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ 96.0000 3.36273
$$816$$ 0 0
$$817$$ 32.0000i 1.11954i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 24.0000i − 0.837606i −0.908077 0.418803i $$-0.862450\pi$$
0.908077 0.418803i $$-0.137550\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ − 22.0000i − 0.765942i
$$826$$ 0 0
$$827$$ − 10.0000i − 0.347734i −0.984769 0.173867i $$-0.944374\pi$$
0.984769 0.173867i $$-0.0556263\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ 72.0000 2.49166
$$836$$ 0 0
$$837$$ − 4.00000i − 0.138260i
$$838$$ 0 0
$$839$$ 30.0000i 1.03572i 0.855467 + 0.517858i $$0.173270\pi$$
−0.855467 + 0.517858i $$0.826730\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ − 20.0000i − 0.688837i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ − 24.0000i − 0.822709i
$$852$$ 0 0
$$853$$ 38.0000i 1.30110i 0.759465 + 0.650548i $$0.225461\pi$$
−0.759465 + 0.650548i $$0.774539\pi$$
$$854$$ 0 0
$$855$$ −32.0000 −1.09438
$$856$$ 0 0
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 2.00000i 0.0680808i 0.999420 + 0.0340404i $$0.0108375\pi$$
−0.999420 + 0.0340404i $$0.989163\pi$$
$$864$$ 0 0
$$865$$ 72.0000i 2.44807i
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 16.0000i 0.542763i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 42.0000i 1.41824i 0.705088 + 0.709120i $$0.250907\pi$$
−0.705088 + 0.709120i $$0.749093\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 56.0000 1.88242
$$886$$ 0 0
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 2.00000i − 0.0670025i
$$892$$ 0 0
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ − 48.0000i − 1.60446i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 24.0000i − 0.800445i
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 24.0000i 0.797787i
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −28.0000 −0.926665
$$914$$ 0 0
$$915$$ 40.0000i 1.32236i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 12.0000i 0.395413i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 66.0000i − 2.17007i
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ − 20.0000i − 0.656179i −0.944647 0.328089i $$-0.893595\pi$$
0.944647 0.328089i $$-0.106405\pi$$
$$930$$ 0 0
$$931$$ − 56.0000i − 1.83533i
$$932$$ 0 0
$$933$$ −12.0000 −0.392862
$$934$$ 0 0
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ − 48.0000i − 1.56476i −0.622804 0.782378i $$-0.714007\pi$$
0.622804 0.782378i $$-0.285993\pi$$
$$942$$ 0 0
$$943$$ − 48.0000i − 1.56310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 42.0000i 1.36482i 0.730971 + 0.682408i $$0.239067\pi$$
−0.730971 + 0.682408i $$0.760933\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 12.0000i − 0.389127i
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 12.0000i − 0.387905i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 104.000 3.34788
$$966$$ 0 0
$$967$$ 28.0000i 0.900419i 0.892923 + 0.450210i $$0.148651\pi$$
−0.892923 + 0.450210i $$0.851349\pi$$
$$968$$ 0 0
$$969$$ − 16.0000i − 0.513994i
$$970$$ 0 0
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ − 10.0000i − 0.319275i
$$982$$ 0 0
$$983$$ 30.0000i 0.956851i 0.878128 + 0.478426i $$0.158792\pi$$
−0.878128 + 0.478426i $$0.841208\pi$$
$$984$$ 0 0
$$985$$ −48.0000 −1.52941
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ − 32.0000i − 1.01447i
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 0 0
$$999$$ − 6.00000i − 0.189832i
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 4056.2.c.d.337.1 2
13.5 odd 4 4056.2.a.a.1.1 1
13.8 odd 4 312.2.a.c.1.1 1
13.12 even 2 inner 4056.2.c.d.337.2 2
39.8 even 4 936.2.a.a.1.1 1
52.31 even 4 8112.2.a.q.1.1 1
52.47 even 4 624.2.a.j.1.1 1
65.34 odd 4 7800.2.a.s.1.1 1
104.21 odd 4 2496.2.a.p.1.1 1
104.99 even 4 2496.2.a.a.1.1 1
156.47 odd 4 1872.2.a.b.1.1 1
312.125 even 4 7488.2.a.cb.1.1 1
312.203 odd 4 7488.2.a.cc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.a.c.1.1 1 13.8 odd 4
624.2.a.j.1.1 1 52.47 even 4
936.2.a.a.1.1 1 39.8 even 4
1872.2.a.b.1.1 1 156.47 odd 4
2496.2.a.a.1.1 1 104.99 even 4
2496.2.a.p.1.1 1 104.21 odd 4
4056.2.a.a.1.1 1 13.5 odd 4
4056.2.c.d.337.1 2 1.1 even 1 trivial
4056.2.c.d.337.2 2 13.12 even 2 inner
7488.2.a.cb.1.1 1 312.125 even 4
7488.2.a.cc.1.1 1 312.203 odd 4
7800.2.a.s.1.1 1 65.34 odd 4
8112.2.a.q.1.1 1 52.31 even 4