# Properties

 Label 2352.2.b.b.1567.2 Level $2352$ Weight $2$ Character 2352.1567 Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1567 Dual form 2352.2.b.b.1567.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +5.19615i q^{11} +6.92820i q^{13} -1.73205i q^{15} +3.46410i q^{17} +2.00000 q^{19} -6.92820i q^{23} +2.00000 q^{25} -1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{31} -5.19615i q^{33} -2.00000 q^{37} -6.92820i q^{39} -3.46410i q^{41} -3.46410i q^{43} +1.73205i q^{45} -3.46410i q^{51} +9.00000 q^{53} -9.00000 q^{55} -2.00000 q^{57} -3.00000 q^{59} +6.92820i q^{61} -12.0000 q^{65} +6.92820i q^{69} -6.92820i q^{71} +6.92820i q^{73} -2.00000 q^{75} +1.73205i q^{79} +1.00000 q^{81} -15.0000 q^{83} -6.00000 q^{85} +9.00000 q^{87} +10.3923i q^{89} -1.00000 q^{93} +3.46410i q^{95} +8.66025i q^{97} +5.19615i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} + 4q^{19} + 4q^{25} - 2q^{27} - 18q^{29} + 2q^{31} - 4q^{37} + 18q^{53} - 18q^{55} - 4q^{57} - 6q^{59} - 24q^{65} - 4q^{75} + 2q^{81} - 30q^{83} - 12q^{85} + 18q^{87} - 2q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$5$$ 1.73205i 0.774597i 0.921954 + 0.387298i $$0.126592\pi$$
−0.921954 + 0.387298i $$0.873408\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.19615i 1.56670i 0.621582 + 0.783349i $$0.286490\pi$$
−0.621582 + 0.783349i $$0.713510\pi$$
$$12$$ 0 0
$$13$$ 6.92820i 1.92154i 0.277350 + 0.960769i $$0.410544\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ − 1.73205i − 0.447214i
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 6.92820i − 1.44463i −0.691564 0.722315i $$-0.743078\pi$$
0.691564 0.722315i $$-0.256922\pi$$
$$24$$ 0 0
$$25$$ 2.00000 0.400000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ − 5.19615i − 0.904534i
$$34$$ 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
$$39$$ − 6.92820i − 1.10940i
$$40$$ 0 0
$$41$$ − 3.46410i − 0.541002i −0.962720 0.270501i $$-0.912811\pi$$
0.962720 0.270501i $$-0.0871893\pi$$
$$42$$ 0 0
$$43$$ − 3.46410i − 0.528271i −0.964486 0.264135i $$-0.914913\pi$$
0.964486 0.264135i $$-0.0850865\pi$$
$$44$$ 0 0
$$45$$ 1.73205i 0.258199i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ − 3.46410i − 0.485071i
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ −9.00000 −1.21356
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 6.92820i 0.887066i 0.896258 + 0.443533i $$0.146275\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 6.92820i 0.834058i
$$70$$ 0 0
$$71$$ − 6.92820i − 0.822226i −0.911584 0.411113i $$-0.865140\pi$$
0.911584 0.411113i $$-0.134860\pi$$
$$72$$ 0 0
$$73$$ 6.92820i 0.810885i 0.914121 + 0.405442i $$0.132883\pi$$
−0.914121 + 0.405442i $$0.867117\pi$$
$$74$$ 0 0
$$75$$ −2.00000 −0.230940
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.73205i 0.194871i 0.995242 + 0.0974355i $$0.0310640\pi$$
−0.995242 + 0.0974355i $$0.968936\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −15.0000 −1.64646 −0.823232 0.567705i $$-0.807831\pi$$
−0.823232 + 0.567705i $$0.807831\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 0 0
$$87$$ 9.00000 0.964901
$$88$$ 0 0
$$89$$ 10.3923i 1.10158i 0.834643 + 0.550791i $$0.185674\pi$$
−0.834643 + 0.550791i $$0.814326\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −1.00000 −0.103695
$$94$$ 0 0
$$95$$ 3.46410i 0.355409i
$$96$$ 0 0
$$97$$ 8.66025i 0.879316i 0.898165 + 0.439658i $$0.144900\pi$$
−0.898165 + 0.439658i $$0.855100\pi$$
$$98$$ 0 0
$$99$$ 5.19615i 0.522233i
$$100$$ 0 0
$$101$$ − 13.8564i − 1.37876i −0.724398 0.689382i $$-0.757882\pi$$
0.724398 0.689382i $$-0.242118\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.66025i 0.837218i 0.908166 + 0.418609i $$0.137482\pi$$
−0.908166 + 0.418609i $$0.862518\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 12.0000 1.11901
$$116$$ 0 0
$$117$$ 6.92820i 0.640513i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −16.0000 −1.45455
$$122$$ 0 0
$$123$$ 3.46410i 0.312348i
$$124$$ 0 0
$$125$$ 12.1244i 1.08444i
$$126$$ 0 0
$$127$$ − 15.5885i − 1.38325i −0.722256 0.691626i $$-0.756895\pi$$
0.722256 0.691626i $$-0.243105\pi$$
$$128$$ 0 0
$$129$$ 3.46410i 0.304997i
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ − 1.73205i − 0.149071i
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 22.0000 1.86602 0.933008 0.359856i $$-0.117174\pi$$
0.933008 + 0.359856i $$0.117174\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −36.0000 −3.01047
$$144$$ 0 0
$$145$$ − 15.5885i − 1.29455i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ − 1.73205i − 0.140952i −0.997513 0.0704761i $$-0.977548\pi$$
0.997513 0.0704761i $$-0.0224519\pi$$
$$152$$ 0 0
$$153$$ 3.46410i 0.280056i
$$154$$ 0 0
$$155$$ 1.73205i 0.139122i
$$156$$ 0 0
$$157$$ 6.92820i 0.552931i 0.961024 + 0.276465i $$0.0891631\pi$$
−0.961024 + 0.276465i $$0.910837\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 20.7846i − 1.62798i −0.580881 0.813988i $$-0.697292\pi$$
0.580881 0.813988i $$-0.302708\pi$$
$$164$$ 0 0
$$165$$ 9.00000 0.700649
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −35.0000 −2.69231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ − 13.8564i − 1.05348i −0.850026 0.526742i $$-0.823414\pi$$
0.850026 0.526742i $$-0.176586\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.00000 0.225494
$$178$$ 0 0
$$179$$ 17.3205i 1.29460i 0.762237 + 0.647298i $$0.224101\pi$$
−0.762237 + 0.647298i $$0.775899\pi$$
$$180$$ 0 0
$$181$$ − 3.46410i − 0.257485i −0.991678 0.128742i $$-0.958906\pi$$
0.991678 0.128742i $$-0.0410940\pi$$
$$182$$ 0 0
$$183$$ − 6.92820i − 0.512148i
$$184$$ 0 0
$$185$$ − 3.46410i − 0.254686i
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 19.0000 1.36765 0.683825 0.729646i $$-0.260315\pi$$
0.683825 + 0.729646i $$0.260315\pi$$
$$194$$ 0 0
$$195$$ 12.0000 0.859338
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 0 0
$$207$$ − 6.92820i − 0.481543i
$$208$$ 0 0
$$209$$ 10.3923i 0.718851i
$$210$$ 0 0
$$211$$ 13.8564i 0.953914i 0.878927 + 0.476957i $$0.158260\pi$$
−0.878927 + 0.476957i $$0.841740\pi$$
$$212$$ 0 0
$$213$$ 6.92820i 0.474713i
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 6.92820i − 0.468165i
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ 9.00000 0.597351 0.298675 0.954355i $$-0.403455\pi$$
0.298675 + 0.954355i $$0.403455\pi$$
$$228$$ 0 0
$$229$$ 3.46410i 0.228914i 0.993428 + 0.114457i $$0.0365129\pi$$
−0.993428 + 0.114457i $$0.963487\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 1.73205i − 0.112509i
$$238$$ 0 0
$$239$$ 13.8564i 0.896296i 0.893959 + 0.448148i $$0.147916\pi$$
−0.893959 + 0.448148i $$0.852084\pi$$
$$240$$ 0 0
$$241$$ − 5.19615i − 0.334714i −0.985896 0.167357i $$-0.946477\pi$$
0.985896 0.167357i $$-0.0535232\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 13.8564i 0.881662i
$$248$$ 0 0
$$249$$ 15.0000 0.950586
$$250$$ 0 0
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 36.0000 2.26330
$$254$$ 0 0
$$255$$ 6.00000 0.375735
$$256$$ 0 0
$$257$$ 3.46410i 0.216085i 0.994146 + 0.108042i $$0.0344582\pi$$
−0.994146 + 0.108042i $$0.965542\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ 17.3205i 1.06803i 0.845476 + 0.534014i $$0.179317\pi$$
−0.845476 + 0.534014i $$0.820683\pi$$
$$264$$ 0 0
$$265$$ 15.5885i 0.957591i
$$266$$ 0 0
$$267$$ − 10.3923i − 0.635999i
$$268$$ 0 0
$$269$$ 1.73205i 0.105605i 0.998605 + 0.0528025i $$0.0168154\pi$$
−0.998605 + 0.0528025i $$0.983185\pi$$
$$270$$ 0 0
$$271$$ −17.0000 −1.03268 −0.516338 0.856385i $$-0.672705\pi$$
−0.516338 + 0.856385i $$0.672705\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 10.3923i 0.626680i
$$276$$ 0 0
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 0 0
$$285$$ − 3.46410i − 0.205196i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ − 8.66025i − 0.507673i
$$292$$ 0 0
$$293$$ − 5.19615i − 0.303562i −0.988414 0.151781i $$-0.951499\pi$$
0.988414 0.151781i $$-0.0485009\pi$$
$$294$$ 0 0
$$295$$ − 5.19615i − 0.302532i
$$296$$ 0 0
$$297$$ − 5.19615i − 0.301511i
$$298$$ 0 0
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 13.8564i 0.796030i
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 19.0526i 1.07691i 0.842653 + 0.538457i $$0.180993\pi$$
−0.842653 + 0.538457i $$0.819007\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −27.0000 −1.51647 −0.758236 0.651981i $$-0.773938\pi$$
−0.758236 + 0.651981i $$0.773938\pi$$
$$318$$ 0 0
$$319$$ − 46.7654i − 2.61836i
$$320$$ 0 0
$$321$$ − 8.66025i − 0.483368i
$$322$$ 0 0
$$323$$ 6.92820i 0.385496i
$$324$$ 0 0
$$325$$ 13.8564i 0.768615i
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 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
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 5.19615i 0.281387i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ 0 0
$$347$$ 31.1769i 1.67366i 0.547459 + 0.836832i $$0.315595\pi$$
−0.547459 + 0.836832i $$0.684405\pi$$
$$348$$ 0 0
$$349$$ − 20.7846i − 1.11257i −0.830990 0.556287i $$-0.812225\pi$$
0.830990 0.556287i $$-0.187775\pi$$
$$350$$ 0 0
$$351$$ − 6.92820i − 0.369800i
$$352$$ 0 0
$$353$$ − 6.92820i − 0.368751i −0.982856 0.184376i $$-0.940974\pi$$
0.982856 0.184376i $$-0.0590263\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 27.7128i 1.46263i 0.682042 + 0.731313i $$0.261092\pi$$
−0.682042 + 0.731313i $$0.738908\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 16.0000 0.839782
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 23.0000 1.20059 0.600295 0.799779i $$-0.295050\pi$$
0.600295 + 0.799779i $$0.295050\pi$$
$$368$$ 0 0
$$369$$ − 3.46410i − 0.180334i
$$370$$ 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
$$375$$ − 12.1244i − 0.626099i
$$376$$ 0 0
$$377$$ − 62.3538i − 3.21139i
$$378$$ 0 0
$$379$$ 17.3205i 0.889695i 0.895606 + 0.444847i $$0.146742\pi$$
−0.895606 + 0.444847i $$0.853258\pi$$
$$380$$ 0 0
$$381$$ 15.5885i 0.798621i
$$382$$ 0 0
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 3.46410i − 0.176090i
$$388$$ 0 0
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 3.00000 0.151330
$$394$$ 0 0
$$395$$ −3.00000 −0.150946
$$396$$ 0 0
$$397$$ − 20.7846i − 1.04315i −0.853206 0.521575i $$-0.825345\pi$$
0.853206 0.521575i $$-0.174655\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 0 0
$$403$$ 6.92820i 0.345118i
$$404$$ 0 0
$$405$$ 1.73205i 0.0860663i
$$406$$ 0 0
$$407$$ − 10.3923i − 0.515127i
$$408$$ 0 0
$$409$$ − 8.66025i − 0.428222i −0.976809 0.214111i $$-0.931315\pi$$
0.976809 0.214111i $$-0.0686854\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 25.9808i − 1.27535i
$$416$$ 0 0
$$417$$ −22.0000 −1.07734
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.92820i 0.336067i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 36.0000 1.73810
$$430$$ 0 0
$$431$$ − 24.2487i − 1.16802i −0.811747 0.584010i $$-0.801483\pi$$
0.811747 0.584010i $$-0.198517\pi$$
$$432$$ 0 0
$$433$$ − 6.92820i − 0.332948i −0.986046 0.166474i $$-0.946762\pi$$
0.986046 0.166474i $$-0.0532382\pi$$
$$434$$ 0 0
$$435$$ 15.5885i 0.747409i
$$436$$ 0 0
$$437$$ − 13.8564i − 0.662842i
$$438$$ 0 0
$$439$$ −25.0000 −1.19318 −0.596592 0.802544i $$-0.703479\pi$$
−0.596592 + 0.802544i $$0.703479\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 22.5167i − 1.06980i −0.844916 0.534899i $$-0.820349\pi$$
0.844916 0.534899i $$-0.179651\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 0 0
$$453$$ 1.73205i 0.0813788i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000 0.795226 0.397613 0.917553i $$-0.369839\pi$$
0.397613 + 0.917553i $$0.369839\pi$$
$$458$$ 0 0
$$459$$ − 3.46410i − 0.161690i
$$460$$ 0 0
$$461$$ − 20.7846i − 0.968036i −0.875058 0.484018i $$-0.839177\pi$$
0.875058 0.484018i $$-0.160823\pi$$
$$462$$ 0 0
$$463$$ − 10.3923i − 0.482971i −0.970404 0.241486i $$-0.922365\pi$$
0.970404 0.241486i $$-0.0776347\pi$$
$$464$$ 0 0
$$465$$ − 1.73205i − 0.0803219i
$$466$$ 0 0
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ − 6.92820i − 0.319235i
$$472$$ 0 0
$$473$$ 18.0000 0.827641
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ 0 0
$$479$$ −42.0000 −1.91903 −0.959514 0.281659i $$-0.909115\pi$$
−0.959514 + 0.281659i $$0.909115\pi$$
$$480$$ 0 0
$$481$$ − 13.8564i − 0.631798i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −15.0000 −0.681115
$$486$$ 0 0
$$487$$ − 15.5885i − 0.706380i −0.935552 0.353190i $$-0.885097\pi$$
0.935552 0.353190i $$-0.114903\pi$$
$$488$$ 0 0
$$489$$ 20.7846i 0.939913i
$$490$$ 0 0
$$491$$ 19.0526i 0.859830i 0.902869 + 0.429915i $$0.141456\pi$$
−0.902869 + 0.429915i $$0.858544\pi$$
$$492$$ 0 0
$$493$$ − 31.1769i − 1.40414i
$$494$$ 0 0
$$495$$ −9.00000 −0.404520
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 17.3205i − 0.775372i −0.921791 0.387686i $$-0.873274\pi$$
0.921791 0.387686i $$-0.126726\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ 0 0
$$507$$ 35.0000 1.55440
$$508$$ 0 0
$$509$$ 19.0526i 0.844490i 0.906482 + 0.422245i $$0.138758\pi$$
−0.906482 + 0.422245i $$0.861242\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 6.92820i 0.305293i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 13.8564i 0.608229i
$$520$$ 0 0
$$521$$ − 31.1769i − 1.36589i −0.730472 0.682943i $$-0.760700\pi$$
0.730472 0.682943i $$-0.239300\pi$$
$$522$$ 0 0
$$523$$ 40.0000 1.74908 0.874539 0.484955i $$-0.161164\pi$$
0.874539 + 0.484955i $$0.161164\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 3.46410i 0.150899i
$$528$$ 0 0
$$529$$ −25.0000 −1.08696
$$530$$ 0 0
$$531$$ −3.00000 −0.130189
$$532$$ 0 0
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ −15.0000 −0.648507
$$536$$ 0 0
$$537$$ − 17.3205i − 0.747435i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 0 0
$$543$$ 3.46410i 0.148659i
$$544$$ 0 0
$$545$$ − 6.92820i − 0.296772i
$$546$$ 0 0
$$547$$ 24.2487i 1.03680i 0.855138 + 0.518400i $$0.173472\pi$$
−0.855138 + 0.518400i $$0.826528\pi$$
$$548$$ 0 0
$$549$$ 6.92820i 0.295689i
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 3.46410i 0.147043i
$$556$$ 0 0
$$557$$ 3.00000 0.127114 0.0635570 0.997978i $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ 0 0
$$565$$ − 10.3923i − 0.437208i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 27.7128i 1.15975i 0.814707 + 0.579873i $$0.196898\pi$$
−0.814707 + 0.579873i $$0.803102\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 13.8564i − 0.577852i
$$576$$ 0 0
$$577$$ 25.9808i 1.08159i 0.841153 + 0.540797i $$0.181877\pi$$
−0.841153 + 0.540797i $$0.818123\pi$$
$$578$$ 0 0
$$579$$ −19.0000 −0.789613
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 46.7654i 1.93682i
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 21.0000 0.866763 0.433381 0.901211i $$-0.357320\pi$$
0.433381 + 0.901211i $$0.357320\pi$$
$$588$$ 0 0
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 20.7846i 0.853522i 0.904365 + 0.426761i $$0.140345\pi$$
−0.904365 + 0.426761i $$0.859655\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 27.7128i 1.13231i 0.824297 + 0.566157i $$0.191571\pi$$
−0.824297 + 0.566157i $$0.808429\pi$$
$$600$$ 0 0
$$601$$ − 15.5885i − 0.635866i −0.948113 0.317933i $$-0.897011\pi$$
0.948113 0.317933i $$-0.102989\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 27.7128i − 1.12669i
$$606$$ 0 0
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 24.0000 0.966204 0.483102 0.875564i $$-0.339510\pi$$
0.483102 + 0.875564i $$0.339510\pi$$
$$618$$ 0 0
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 6.92820i 0.278019i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ − 10.3923i − 0.415029i
$$628$$ 0 0
$$629$$ − 6.92820i − 0.276246i
$$630$$ 0 0
$$631$$ 25.9808i 1.03428i 0.855901 + 0.517139i $$0.173003\pi$$
−0.855901 + 0.517139i $$0.826997\pi$$
$$632$$ 0 0
$$633$$ − 13.8564i − 0.550743i
$$634$$ 0 0
$$635$$ 27.0000 1.07146
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 6.92820i − 0.274075i
$$640$$ 0 0
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ 0 0
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 0 0
$$645$$ −6.00000 −0.236250
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ − 15.5885i − 0.611900i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −21.0000 −0.821794 −0.410897 0.911682i $$-0.634784\pi$$
−0.410897 + 0.911682i $$0.634784\pi$$
$$654$$ 0 0
$$655$$ − 5.19615i − 0.203030i
$$656$$ 0 0
$$657$$ 6.92820i 0.270295i
$$658$$ 0 0
$$659$$ 17.3205i 0.674711i 0.941377 + 0.337356i $$0.109532\pi$$
−0.941377 + 0.337356i $$0.890468\pi$$
$$660$$ 0 0
$$661$$ 38.1051i 1.48212i 0.671440 + 0.741059i $$0.265676\pi$$
−0.671440 + 0.741059i $$0.734324\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 62.3538i 2.41435i
$$668$$ 0 0
$$669$$ 1.00000 0.0386622
$$670$$ 0 0
$$671$$ −36.0000 −1.38976
$$672$$ 0 0
$$673$$ −11.0000 −0.424019 −0.212009 0.977268i $$-0.568001\pi$$
−0.212009 + 0.977268i $$0.568001\pi$$
$$674$$ 0 0
$$675$$ −2.00000 −0.0769800
$$676$$ 0 0
$$677$$ − 25.9808i − 0.998522i −0.866452 0.499261i $$-0.833605\pi$$
0.866452 0.499261i $$-0.166395\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −9.00000 −0.344881
$$682$$ 0 0
$$683$$ − 8.66025i − 0.331375i −0.986178 0.165688i $$-0.947016\pi$$
0.986178 0.165688i $$-0.0529844\pi$$
$$684$$ 0 0
$$685$$ − 20.7846i − 0.794139i
$$686$$ 0 0
$$687$$ − 3.46410i − 0.132164i
$$688$$ 0 0
$$689$$ 62.3538i 2.37549i
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 38.1051i 1.44541i
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −27.0000 −1.01978 −0.509888 0.860241i $$-0.670313\pi$$
−0.509888 + 0.860241i $$0.670313\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 1.73205i 0.0649570i
$$712$$ 0 0
$$713$$ − 6.92820i − 0.259463i
$$714$$ 0 0
$$715$$ − 62.3538i − 2.33190i
$$716$$ 0 0
$$717$$ − 13.8564i − 0.517477i
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 5.19615i 0.193247i
$$724$$ 0 0
$$725$$ −18.0000 −0.668503
$$726$$ 0 0
$$727$$ −17.0000 −0.630495 −0.315248 0.949009i $$-0.602088\pi$$
−0.315248 + 0.949009i $$0.602088\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 31.1769i 1.15155i 0.817610 + 0.575773i $$0.195299\pi$$
−0.817610 + 0.575773i $$0.804701\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 31.1769i − 1.14686i −0.819254 0.573431i $$-0.805612\pi$$
0.819254 0.573431i $$-0.194388\pi$$
$$740$$ 0 0
$$741$$ − 13.8564i − 0.509028i
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ − 10.3923i − 0.380745i
$$746$$ 0 0
$$747$$ −15.0000 −0.548821
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 12.1244i − 0.442424i −0.975226 0.221212i $$-0.928999\pi$$
0.975226 0.221212i $$-0.0710013\pi$$
$$752$$ 0 0
$$753$$ 21.0000 0.765283
$$754$$ 0 0
$$755$$ 3.00000 0.109181
$$756$$ 0 0
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 0 0
$$759$$ −36.0000 −1.30672
$$760$$ 0 0
$$761$$ − 13.8564i − 0.502294i −0.967949 0.251147i $$-0.919192\pi$$
0.967949 0.251147i $$-0.0808078\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ − 20.7846i − 0.750489i
$$768$$ 0 0
$$769$$ 5.19615i 0.187378i 0.995602 + 0.0936890i $$0.0298659\pi$$
−0.995602 + 0.0936890i $$0.970134\pi$$
$$770$$ 0 0
$$771$$ − 3.46410i − 0.124757i
$$772$$ 0 0
$$773$$ − 27.7128i − 0.996761i −0.866959 0.498380i $$-0.833928\pi$$
0.866959 0.498380i $$-0.166072\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 6.92820i − 0.248229i
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 9.00000 0.321634
$$784$$ 0 0
$$785$$ −12.0000 −0.428298
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 0 0
$$789$$ − 17.3205i − 0.616626i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −48.0000 −1.70453
$$794$$ 0 0
$$795$$ − 15.5885i − 0.552866i
$$796$$ 0 0
$$797$$ 8.66025i 0.306762i 0.988167 + 0.153381i $$0.0490162\pi$$
−0.988167 + 0.153381i $$0.950984\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.3923i 0.367194i
$$802$$ 0 0
$$803$$ −36.0000 −1.27041
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 1.73205i − 0.0609711i
$$808$$ 0 0
$$809$$ −24.0000 −0.843795 −0.421898 0.906644i $$-0.638636\pi$$
−0.421898 + 0.906644i $$0.638636\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 0 0
$$813$$ 17.0000 0.596216
$$814$$ 0 0
$$815$$ 36.0000 1.26102
$$816$$ 0 0
$$817$$ − 6.92820i − 0.242387i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.0000 1.57051 0.785255 0.619172i $$-0.212532\pi$$
0.785255 + 0.619172i $$0.212532\pi$$
$$822$$ 0 0
$$823$$ 3.46410i 0.120751i 0.998176 + 0.0603755i $$0.0192298\pi$$
−0.998176 + 0.0603755i $$0.980770\pi$$
$$824$$ 0 0
$$825$$ − 10.3923i − 0.361814i
$$826$$ 0 0
$$827$$ 36.3731i 1.26482i 0.774636 + 0.632408i $$0.217933\pi$$
−0.774636 + 0.632408i $$0.782067\pi$$
$$828$$ 0 0
$$829$$ − 10.3923i − 0.360940i −0.983581 0.180470i $$-0.942238\pi$$
0.983581 0.180470i $$-0.0577618\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ − 20.7846i − 0.719281i
$$836$$ 0 0
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ −18.0000 −0.621429 −0.310715 0.950503i $$-0.600568\pi$$
−0.310715 + 0.950503i $$0.600568\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 60.6218i − 2.08545i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 13.8564i 0.474991i
$$852$$ 0 0
$$853$$ 6.92820i 0.237217i 0.992941 + 0.118609i $$0.0378434\pi$$
−0.992941 + 0.118609i $$0.962157\pi$$
$$854$$ 0 0
$$855$$ 3.46410i 0.118470i
$$856$$ 0 0
$$857$$ 48.4974i 1.65664i 0.560255 + 0.828320i $$0.310703\pi$$
−0.560255 + 0.828320i $$0.689297\pi$$
$$858$$ 0 0
$$859$$ −10.0000 −0.341196 −0.170598 0.985341i $$-0.554570\pi$$
−0.170598 + 0.985341i $$0.554570\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 31.1769i − 1.06127i −0.847599 0.530637i $$-0.821953\pi$$
0.847599 0.530637i $$-0.178047\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ 0 0
$$867$$ −5.00000 −0.169809
$$868$$ 0 0
$$869$$ −9.00000 −0.305304
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 8.66025i 0.293105i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ 0 0
$$879$$ 5.19615i 0.175262i
$$880$$ 0 0
$$881$$ 55.4256i 1.86734i 0.358139 + 0.933668i $$0.383411\pi$$
−0.358139 + 0.933668i $$0.616589\pi$$
$$882$$ 0 0
$$883$$ 31.1769i 1.04919i 0.851353 + 0.524593i $$0.175783\pi$$
−0.851353 + 0.524593i $$0.824217\pi$$
$$884$$ 0 0
$$885$$ 5.19615i 0.174667i
$$886$$ 0 0
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 5.19615i 0.174078i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −30.0000 −1.00279
$$896$$ 0 0
$$897$$ −48.0000 −1.60267
$$898$$ 0 0
$$899$$ −9.00000 −0.300167
$$900$$ 0 0
$$901$$ 31.1769i 1.03865i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ 0 0
$$907$$ 55.4256i 1.84038i 0.391475 + 0.920189i $$0.371965\pi$$
−0.391475 + 0.920189i $$0.628035\pi$$
$$908$$ 0 0
$$909$$ − 13.8564i − 0.459588i
$$910$$ 0 0
$$911$$ 20.7846i 0.688625i 0.938855 + 0.344312i $$0.111888\pi$$
−0.938855 + 0.344312i $$0.888112\pi$$
$$912$$ 0 0
$$913$$ − 77.9423i − 2.57951i
$$914$$ 0 0
$$915$$ 12.0000 0.396708
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 3.46410i − 0.114270i −0.998366 0.0571351i $$-0.981803\pi$$
0.998366 0.0571351i $$-0.0181966\pi$$
$$920$$ 0 0
$$921$$ 16.0000 0.527218
$$922$$ 0 0
$$923$$ 48.0000 1.57994
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 58.8897i 1.93211i 0.258338 + 0.966055i $$0.416825\pi$$
−0.258338 + 0.966055i $$0.583175\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 6.00000 0.196431
$$934$$ 0 0
$$935$$ − 31.1769i − 1.01959i
$$936$$ 0 0
$$937$$ 50.2295i 1.64093i 0.571700 + 0.820463i $$0.306284\pi$$
−0.571700 + 0.820463i $$0.693716\pi$$
$$938$$ 0 0
$$939$$ − 19.0526i − 0.621757i
$$940$$ 0 0
$$941$$ − 15.5885i − 0.508169i −0.967182 0.254085i $$-0.918226\pi$$
0.967182 0.254085i $$-0.0817742\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 24.2487i − 0.787977i −0.919115 0.393989i $$-0.871095\pi$$
0.919115 0.393989i $$-0.128905\pi$$
$$948$$ 0 0
$$949$$ −48.0000 −1.55815
$$950$$ 0 0
$$951$$ 27.0000 0.875535
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 46.7654i 1.51171i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 8.66025i 0.279073i
$$964$$ 0 0
$$965$$ 32.9090i 1.05938i
$$966$$ 0 0
$$967$$ − 46.7654i − 1.50387i −0.659236 0.751936i $$-0.729120\pi$$
0.659236 0.751936i $$-0.270880\pi$$
$$968$$ 0 0
$$969$$ − 6.92820i − 0.222566i
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ − 13.8564i − 0.443760i
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −54.0000 −1.72585
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 54.0000 1.72233 0.861166 0.508323i $$-0.169735\pi$$
0.861166 + 0.508323i $$0.169735\pi$$
$$984$$ 0 0
$$985$$ 10.3923i 0.331126i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 46.7654i 1.48555i 0.669541 + 0.742775i $$0.266491\pi$$
−0.669541 + 0.742775i $$0.733509\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 27.7128i − 0.878555i
$$996$$ 0 0
$$997$$ 27.7128i 0.877674i 0.898567 + 0.438837i $$0.144609\pi$$
−0.898567 + 0.438837i $$0.855391\pi$$
$$998$$ 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 2352.2.b.b.1567.2 2
3.2 odd 2 7056.2.b.j.1567.1 2
4.3 odd 2 2352.2.b.f.1567.2 2
7.2 even 3 2352.2.bl.k.31.1 2
7.3 odd 6 2352.2.bl.e.607.1 2
7.4 even 3 336.2.bl.f.271.1 yes 2
7.5 odd 6 336.2.bl.b.31.1 2
7.6 odd 2 2352.2.b.f.1567.1 2
12.11 even 2 7056.2.b.f.1567.1 2
21.5 even 6 1008.2.cs.k.703.1 2
21.11 odd 6 1008.2.cs.l.271.1 2
21.20 even 2 7056.2.b.f.1567.2 2
28.3 even 6 2352.2.bl.k.607.1 2
28.11 odd 6 336.2.bl.b.271.1 yes 2
28.19 even 6 336.2.bl.f.31.1 yes 2
28.23 odd 6 2352.2.bl.e.31.1 2
28.27 even 2 inner 2352.2.b.b.1567.1 2
56.5 odd 6 1344.2.bl.g.703.1 2
56.11 odd 6 1344.2.bl.g.1279.1 2
56.19 even 6 1344.2.bl.c.703.1 2
56.53 even 6 1344.2.bl.c.1279.1 2
84.11 even 6 1008.2.cs.k.271.1 2
84.47 odd 6 1008.2.cs.l.703.1 2
84.83 odd 2 7056.2.b.j.1567.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.b.31.1 2 7.5 odd 6
336.2.bl.b.271.1 yes 2 28.11 odd 6
336.2.bl.f.31.1 yes 2 28.19 even 6
336.2.bl.f.271.1 yes 2 7.4 even 3
1008.2.cs.k.271.1 2 84.11 even 6
1008.2.cs.k.703.1 2 21.5 even 6
1008.2.cs.l.271.1 2 21.11 odd 6
1008.2.cs.l.703.1 2 84.47 odd 6
1344.2.bl.c.703.1 2 56.19 even 6
1344.2.bl.c.1279.1 2 56.53 even 6
1344.2.bl.g.703.1 2 56.5 odd 6
1344.2.bl.g.1279.1 2 56.11 odd 6
2352.2.b.b.1567.1 2 28.27 even 2 inner
2352.2.b.b.1567.2 2 1.1 even 1 trivial
2352.2.b.f.1567.1 2 7.6 odd 2
2352.2.b.f.1567.2 2 4.3 odd 2
2352.2.bl.e.31.1 2 28.23 odd 6
2352.2.bl.e.607.1 2 7.3 odd 6
2352.2.bl.k.31.1 2 7.2 even 3
2352.2.bl.k.607.1 2 28.3 even 6
7056.2.b.f.1567.1 2 12.11 even 2
7056.2.b.f.1567.2 2 21.20 even 2
7056.2.b.j.1567.1 2 3.2 odd 2
7056.2.b.j.1567.2 2 84.83 odd 2