# Properties

 Label 4400.2.b.a.4049.1 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.a.4049.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{13} -3.00000i q^{17} -5.00000 q^{19} +3.00000 q^{21} +2.00000i q^{23} +9.00000i q^{27} +5.00000 q^{29} -5.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -18.0000 q^{39} -2.00000 q^{41} -12.0000i q^{43} -2.00000i q^{47} +6.00000 q^{49} -9.00000 q^{51} -13.0000i q^{53} +15.0000i q^{57} +2.00000 q^{59} +1.00000 q^{61} -6.00000i q^{63} +16.0000i q^{67} +6.00000 q^{69} -15.0000 q^{71} +10.0000i q^{73} +1.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} -15.0000i q^{87} -9.00000 q^{89} +6.00000 q^{91} +15.0000i q^{93} +16.0000i q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 2 q^{11} - 10 q^{19} + 6 q^{21} + 10 q^{29} - 10 q^{31} - 36 q^{39} - 4 q^{41} + 12 q^{49} - 18 q^{51} + 4 q^{59} + 2 q^{61} + 12 q^{69} - 30 q^{71} + 4 q^{79} + 18 q^{81} - 18 q^{89} + 12 q^{91} - 12 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 2 * q^11 - 10 * q^19 + 6 * q^21 + 10 * q^29 - 10 * q^31 - 36 * q^39 - 4 * q^41 + 12 * q^49 - 18 * q^51 + 4 * q^59 + 2 * q^61 + 12 * q^69 - 30 * q^71 + 4 * q^79 + 18 * q^81 - 18 * q^89 + 12 * q^91 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 3.00000i − 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i 0.981981 + 0.188982i $$0.0605189\pi$$
−0.981981 + 0.188982i $$0.939481\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ 2.00000i 0.417029i 0.978019 + 0.208514i $$0.0668628\pi$$
−0.978019 + 0.208514i $$0.933137\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 9.00000i 1.73205i
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000i 0.164399i 0.996616 + 0.0821995i $$0.0261945\pi$$
−0.996616 + 0.0821995i $$0.973806\pi$$
$$38$$ 0 0
$$39$$ −18.0000 −2.88231
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ −9.00000 −1.26025
$$52$$ 0 0
$$53$$ − 13.0000i − 1.78569i −0.450367 0.892844i $$-0.648707\pi$$
0.450367 0.892844i $$-0.351293\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 15.0000i 1.98680i
$$58$$ 0 0
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ − 6.00000i − 0.755929i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 16.0000i 1.95471i 0.211604 + 0.977356i $$0.432131\pi$$
−0.211604 + 0.977356i $$0.567869\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −15.0000 −1.78017 −0.890086 0.455792i $$-0.849356\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000i 0.113961i
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 14.0000i 1.53670i 0.640030 + 0.768350i $$0.278922\pi$$
−0.640030 + 0.768350i $$0.721078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 15.0000i − 1.60817i
$$88$$ 0 0
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 15.0000i 1.55543i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ − 16.0000i − 1.50515i −0.658505 0.752577i $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 36.0000i 3.32820i
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ −36.0000 −3.16962
$$130$$ 0 0
$$131$$ −7.00000 −0.611593 −0.305796 0.952097i $$-0.598923\pi$$
−0.305796 + 0.952097i $$0.598923\pi$$
$$132$$ 0 0
$$133$$ − 5.00000i − 0.433555i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 18.0000i − 1.48461i
$$148$$ 0 0
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ 18.0000i 1.45521i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 7.00000i − 0.558661i −0.960195 0.279330i $$-0.909888\pi$$
0.960195 0.279330i $$-0.0901125\pi$$
$$158$$ 0 0
$$159$$ −39.0000 −3.09290
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 1.00000i 0.0783260i 0.999233 + 0.0391630i $$0.0124692\pi$$
−0.999233 + 0.0391630i $$0.987531\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000i 0.0773823i 0.999251 + 0.0386912i $$0.0123189\pi$$
−0.999251 + 0.0386912i $$0.987681\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 30.0000 2.29416
$$172$$ 0 0
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 6.00000i − 0.450988i
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ − 3.00000i − 0.221766i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 3.00000i − 0.219382i
$$188$$ 0 0
$$189$$ −9.00000 −0.654654
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 7.00000i 0.503871i 0.967744 + 0.251936i $$0.0810671\pi$$
−0.967744 + 0.251936i $$0.918933\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.0000i 1.70993i 0.518686 + 0.854965i $$0.326421\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −0.0708881 −0.0354441 0.999372i $$-0.511285\pi$$
−0.0354441 + 0.999372i $$0.511285\pi$$
$$200$$ 0 0
$$201$$ 48.0000 3.38566
$$202$$ 0 0
$$203$$ 5.00000i 0.350931i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 12.0000i − 0.834058i
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ 1.00000 0.0688428 0.0344214 0.999407i $$-0.489041\pi$$
0.0344214 + 0.999407i $$0.489041\pi$$
$$212$$ 0 0
$$213$$ 45.0000i 3.08335i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 5.00000i − 0.339422i
$$218$$ 0 0
$$219$$ 30.0000 2.02721
$$220$$ 0 0
$$221$$ −18.0000 −1.21081
$$222$$ 0 0
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ 11.0000i 0.720634i 0.932830 + 0.360317i $$0.117331\pi$$
−0.932830 + 0.360317i $$0.882669\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 6.00000i − 0.389742i
$$238$$ 0 0
$$239$$ −2.00000 −0.129369 −0.0646846 0.997906i $$-0.520604\pi$$
−0.0646846 + 0.997906i $$0.520604\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 30.0000i 1.90885i
$$248$$ 0 0
$$249$$ 42.0000 2.66164
$$250$$ 0 0
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ 0 0
$$253$$ 2.00000i 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.0000i 1.37232i 0.727450 + 0.686161i $$0.240706\pi$$
−0.727450 + 0.686161i $$0.759294\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −30.0000 −1.85695
$$262$$ 0 0
$$263$$ 13.0000i 0.801614i 0.916162 + 0.400807i $$0.131270\pi$$
−0.916162 + 0.400807i $$0.868730\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 27.0000i 1.65237i
$$268$$ 0 0
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ − 18.0000i − 1.08941i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 16.0000i − 0.961347i −0.876900 0.480673i $$-0.840392\pi$$
0.876900 0.480673i $$-0.159608\pi$$
$$278$$ 0 0
$$279$$ 30.0000 1.79605
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ − 10.0000i − 0.594438i −0.954809 0.297219i $$-0.903941\pi$$
0.954809 0.297219i $$-0.0960592\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 2.00000i − 0.118056i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 48.0000 2.81381
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 9.00000i 0.522233i
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 30.0000i 1.72345i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 22.0000i − 1.25561i −0.778372 0.627803i $$-0.783954\pi$$
0.778372 0.627803i $$-0.216046\pi$$
$$308$$ 0 0
$$309$$ 48.0000 2.73062
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ − 30.0000i − 1.69570i −0.530236 0.847850i $$-0.677897\pi$$
0.530236 0.847850i $$-0.322103\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.00000i 0.168497i 0.996445 + 0.0842484i $$0.0268489\pi$$
−0.996445 + 0.0842484i $$0.973151\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 15.0000i 0.834622i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 30.0000i 1.65900i
$$328$$ 0 0
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 29.0000i − 1.57973i −0.613280 0.789865i $$-0.710150\pi$$
0.613280 0.789865i $$-0.289850\pi$$
$$338$$ 0 0
$$339$$ −48.0000 −2.60700
$$340$$ 0 0
$$341$$ −5.00000 −0.270765
$$342$$ 0 0
$$343$$ 13.0000i 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 2.00000i − 0.107366i −0.998558 0.0536828i $$-0.982904\pi$$
0.998558 0.0536828i $$-0.0170960\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 54.0000 2.88231
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 9.00000i − 0.476331i
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ − 3.00000i − 0.157459i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 12.0000i − 0.626395i −0.949688 0.313197i $$-0.898600\pi$$
0.949688 0.313197i $$-0.101400\pi$$
$$368$$ 0 0
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 13.0000 0.674926
$$372$$ 0 0
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 30.0000i − 1.54508i
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ −24.0000 −1.22956
$$382$$ 0 0
$$383$$ 14.0000i 0.715367i 0.933843 + 0.357683i $$0.116433\pi$$
−0.933843 + 0.357683i $$0.883567\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 72.0000i 3.65997i
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 0 0
$$393$$ 21.0000i 1.05931i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 0 0
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −21.0000 −1.04869 −0.524345 0.851506i $$-0.675690\pi$$
−0.524345 + 0.851506i $$0.675690\pi$$
$$402$$ 0 0
$$403$$ 30.0000i 1.49441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.00000i 0.0495682i
$$408$$ 0 0
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ −36.0000 −1.77575
$$412$$ 0 0
$$413$$ 2.00000i 0.0984136i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −36.0000 −1.75453 −0.877266 0.480004i $$-0.840635\pi$$
−0.877266 + 0.480004i $$0.840635\pi$$
$$422$$ 0 0
$$423$$ 12.0000i 0.583460i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.00000i 0.0483934i
$$428$$ 0 0
$$429$$ −18.0000 −0.869048
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 4.00000i 0.192228i 0.995370 + 0.0961139i $$0.0306413\pi$$
−0.995370 + 0.0961139i $$0.969359\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 10.0000i − 0.478365i
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −36.0000 −1.71429
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 45.0000i − 2.12843i
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 0 0
$$453$$ 54.0000i 2.53714i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 33.0000i − 1.54367i −0.635820 0.771837i $$-0.719338\pi$$
0.635820 0.771837i $$-0.280662\pi$$
$$458$$ 0 0
$$459$$ 27.0000 1.26025
$$460$$ 0 0
$$461$$ 5.00000 0.232873 0.116437 0.993198i $$-0.462853\pi$$
0.116437 + 0.993198i $$0.462853\pi$$
$$462$$ 0 0
$$463$$ − 6.00000i − 0.278844i −0.990233 0.139422i $$-0.955476\pi$$
0.990233 0.139422i $$-0.0445244\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.0000i 1.24941i 0.780860 + 0.624705i $$0.214781\pi$$
−0.780860 + 0.624705i $$0.785219\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −21.0000 −0.967629
$$472$$ 0 0
$$473$$ − 12.0000i − 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 78.0000i 3.57137i
$$478$$ 0 0
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 12.0000i − 0.543772i −0.962329 0.271886i $$-0.912353\pi$$
0.962329 0.271886i $$-0.0876473\pi$$
$$488$$ 0 0
$$489$$ 3.00000 0.135665
$$490$$ 0 0
$$491$$ 13.0000 0.586682 0.293341 0.956008i $$-0.405233\pi$$
0.293341 + 0.956008i $$0.405233\pi$$
$$492$$ 0 0
$$493$$ − 15.0000i − 0.675566i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 15.0000i − 0.672842i
$$498$$ 0 0
$$499$$ −8.00000 −0.358129 −0.179065 0.983837i $$-0.557307\pi$$
−0.179065 + 0.983837i $$0.557307\pi$$
$$500$$ 0 0
$$501$$ 3.00000 0.134030
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 69.0000i 3.06440i
$$508$$ 0 0
$$509$$ −8.00000 −0.354594 −0.177297 0.984157i $$-0.556735\pi$$
−0.177297 + 0.984157i $$0.556735\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 0 0
$$513$$ − 45.0000i − 1.98680i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 2.00000i − 0.0879599i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.0000i 0.653410i
$$528$$ 0 0
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 12.0000i − 0.517838i
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ − 66.0000i − 2.83233i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −25.0000 −1.06504
$$552$$ 0 0
$$553$$ 2.00000i 0.0850487i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ −72.0000 −3.04528
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ 18.0000i 0.758610i 0.925272 + 0.379305i $$0.123837\pi$$
−0.925272 + 0.379305i $$0.876163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 9.00000i 0.377964i
$$568$$ 0 0
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 0 0
$$573$$ 36.0000i 1.50392i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.0000i 0.582828i 0.956597 + 0.291414i $$0.0941257\pi$$
−0.956597 + 0.291414i $$0.905874\pi$$
$$578$$ 0 0
$$579$$ 21.0000 0.872730
$$580$$ 0 0
$$581$$ −14.0000 −0.580818
$$582$$ 0 0
$$583$$ − 13.0000i − 0.538405i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 39.0000i − 1.60970i −0.593477 0.804851i $$-0.702245\pi$$
0.593477 0.804851i $$-0.297755\pi$$
$$588$$ 0 0
$$589$$ 25.0000 1.03011
$$590$$ 0 0
$$591$$ 72.0000 2.96168
$$592$$ 0 0
$$593$$ − 22.0000i − 0.903432i −0.892162 0.451716i $$-0.850812\pi$$
0.892162 0.451716i $$-0.149188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.00000i 0.122782i
$$598$$ 0 0
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ − 96.0000i − 3.90942i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 21.0000i − 0.852364i −0.904638 0.426182i $$-0.859858\pi$$
0.904638 0.426182i $$-0.140142\pi$$
$$608$$ 0 0
$$609$$ 15.0000 0.607831
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 0 0
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 12.0000i − 0.483102i −0.970388 0.241551i $$-0.922344\pi$$
0.970388 0.241551i $$-0.0776561\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ − 9.00000i − 0.360577i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 15.0000i 0.599042i
$$628$$ 0 0
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −15.0000 −0.597141 −0.298570 0.954388i $$-0.596510\pi$$
−0.298570 + 0.954388i $$0.596510\pi$$
$$632$$ 0 0
$$633$$ − 3.00000i − 0.119239i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 36.0000i − 1.42637i
$$638$$ 0 0
$$639$$ 90.0000 3.56034
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ − 7.00000i − 0.276053i −0.990429 0.138027i $$-0.955924\pi$$
0.990429 0.138027i $$-0.0440759\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ 0 0
$$649$$ 2.00000 0.0785069
$$650$$ 0 0
$$651$$ −15.0000 −0.587896
$$652$$ 0 0
$$653$$ − 19.0000i − 0.743527i −0.928327 0.371764i $$-0.878753\pi$$
0.928327 0.371764i $$-0.121247\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 60.0000i − 2.34082i
$$658$$ 0 0
$$659$$ 49.0000 1.90877 0.954384 0.298580i $$-0.0965131\pi$$
0.954384 + 0.298580i $$0.0965131\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 0 0
$$663$$ 54.0000i 2.09719i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.0000i 0.387202i
$$668$$ 0 0
$$669$$ −78.0000 −3.01565
$$670$$ 0 0
$$671$$ 1.00000 0.0386046
$$672$$ 0 0
$$673$$ − 1.00000i − 0.0385472i −0.999814 0.0192736i $$-0.993865\pi$$
0.999814 0.0192736i $$-0.00613535\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 54.0000 2.06928
$$682$$ 0 0
$$683$$ 19.0000i 0.727015i 0.931591 + 0.363507i $$0.118421\pi$$
−0.931591 + 0.363507i $$0.881579\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 18.0000i 0.686743i
$$688$$ 0 0
$$689$$ −78.0000 −2.97156
$$690$$ 0 0
$$691$$ −34.0000 −1.29342 −0.646710 0.762736i $$-0.723856\pi$$
−0.646710 + 0.762736i $$0.723856\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.00000i 0.227266i
$$698$$ 0 0
$$699$$ 33.0000 1.24817
$$700$$ 0 0
$$701$$ −31.0000 −1.17085 −0.585427 0.810725i $$-0.699073\pi$$
−0.585427 + 0.810725i $$0.699073\pi$$
$$702$$ 0 0
$$703$$ − 5.00000i − 0.188579i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 10.0000i − 0.376089i
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 0 0
$$713$$ − 10.0000i − 0.374503i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000i 0.224074i
$$718$$ 0 0
$$719$$ 1.00000 0.0372937 0.0186469 0.999826i $$-0.494064\pi$$
0.0186469 + 0.999826i $$0.494064\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ − 54.0000i − 2.00828i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 26.0000i − 0.964287i −0.876092 0.482143i $$-0.839858\pi$$
0.876092 0.482143i $$-0.160142\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −36.0000 −1.33151
$$732$$ 0 0
$$733$$ − 32.0000i − 1.18195i −0.806691 0.590973i $$-0.798744\pi$$
0.806691 0.590973i $$-0.201256\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 16.0000i 0.589368i
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 90.0000 3.30623
$$742$$ 0 0
$$743$$ − 33.0000i − 1.21065i −0.795977 0.605326i $$-0.793043\pi$$
0.795977 0.605326i $$-0.206957\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 84.0000i − 3.07340i
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ 7.00000 0.255434 0.127717 0.991811i $$-0.459235\pi$$
0.127717 + 0.991811i $$0.459235\pi$$
$$752$$ 0 0
$$753$$ − 30.0000i − 1.09326i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ − 10.0000i − 0.362024i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 12.0000i − 0.433295i
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 66.0000 2.37693
$$772$$ 0 0
$$773$$ − 9.00000i − 0.323708i −0.986815 0.161854i $$-0.948253\pi$$
0.986815 0.161854i $$-0.0517473\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 3.00000i 0.107624i
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −15.0000 −0.536742
$$782$$ 0 0
$$783$$ 45.0000i 1.60817i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 48.0000i − 1.71102i −0.517790 0.855508i $$-0.673245\pi$$
0.517790 0.855508i $$-0.326755\pi$$
$$788$$ 0 0
$$789$$ 39.0000 1.38844
$$790$$ 0 0
$$791$$ 16.0000 0.568895
$$792$$ 0 0
$$793$$ − 6.00000i − 0.213066i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ 0 0
$$799$$ −6.00000 −0.212265
$$800$$ 0 0
$$801$$ 54.0000 1.90800
$$802$$ 0 0
$$803$$ 10.0000i 0.352892i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 60.0000i 2.11210i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −1.00000 −0.0351147 −0.0175574 0.999846i $$-0.505589\pi$$
−0.0175574 + 0.999846i $$0.505589\pi$$
$$812$$ 0 0
$$813$$ 72.0000i 2.52515i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 60.0000i 2.09913i
$$818$$ 0 0
$$819$$ −36.0000 −1.25794
$$820$$ 0 0
$$821$$ 50.0000 1.74501 0.872506 0.488603i $$-0.162493\pi$$
0.872506 + 0.488603i $$0.162493\pi$$
$$822$$ 0 0
$$823$$ − 44.0000i − 1.53374i −0.641800 0.766872i $$-0.721812\pi$$
0.641800 0.766872i $$-0.278188\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 52.0000i − 1.80822i −0.427303 0.904109i $$-0.640536\pi$$
0.427303 0.904109i $$-0.359464\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −48.0000 −1.66510
$$832$$ 0 0
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 45.0000i − 1.55543i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ − 54.0000i − 1.85986i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.00000i 0.0343604i
$$848$$ 0 0
$$849$$ −30.0000 −1.02960
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 21.0000i 0.717346i 0.933463 + 0.358673i $$0.116771\pi$$
−0.933463 + 0.358673i $$0.883229\pi$$
$$858$$ 0 0
$$859$$ −22.0000 −0.750630 −0.375315 0.926897i $$-0.622466\pi$$
−0.375315 + 0.926897i $$0.622466\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 46.0000i 1.56586i 0.622111 + 0.782929i $$0.286275\pi$$
−0.622111 + 0.782929i $$0.713725\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 24.0000i − 0.815083i
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ 96.0000 3.25284
$$872$$ 0 0
$$873$$ − 96.0000i − 3.24911i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ − 5.00000i − 0.168263i −0.996455 0.0841317i $$-0.973188\pi$$
0.996455 0.0841317i $$-0.0268116\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 40.0000i − 1.34307i −0.740973 0.671534i $$-0.765636\pi$$
0.740973 0.671534i $$-0.234364\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 9.00000 0.301511
$$892$$ 0 0
$$893$$ 10.0000i 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 36.0000i − 1.20201i
$$898$$ 0 0
$$899$$ −25.0000 −0.833797
$$900$$ 0 0
$$901$$ −39.0000 −1.29928
$$902$$ 0 0
$$903$$ − 36.0000i − 1.19800i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 29.0000i − 0.962929i −0.876466 0.481465i $$-0.840105\pi$$
0.876466 0.481465i $$-0.159895\pi$$
$$908$$ 0 0
$$909$$ 60.0000 1.99007
$$910$$ 0 0
$$911$$ −3.00000 −0.0993944 −0.0496972 0.998764i $$-0.515826\pi$$
−0.0496972 + 0.998764i $$0.515826\pi$$
$$912$$ 0 0
$$913$$ 14.0000i 0.463332i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 7.00000i − 0.231160i
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −66.0000 −2.17477
$$922$$ 0 0
$$923$$ 90.0000i 2.96239i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 96.0000i − 3.15305i
$$928$$ 0 0
$$929$$ −59.0000 −1.93573 −0.967864 0.251476i $$-0.919084\pi$$
−0.967864 + 0.251476i $$0.919084\pi$$
$$930$$ 0 0
$$931$$ −30.0000 −0.983210
$$932$$ 0 0
$$933$$ 63.0000i 2.06253i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ 0 0
$$939$$ −90.0000 −2.93704
$$940$$ 0 0
$$941$$ −41.0000 −1.33656 −0.668281 0.743909i $$-0.732970\pi$$
−0.668281 + 0.743909i $$0.732970\pi$$
$$942$$ 0 0
$$943$$ − 4.00000i − 0.130258i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 41.0000i 1.33232i 0.745808 + 0.666160i $$0.232063\pi$$
−0.745808 + 0.666160i $$0.767937\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 0 0
$$951$$ 9.00000 0.291845
$$952$$ 0 0
$$953$$ 29.0000i 0.939402i 0.882826 + 0.469701i $$0.155638\pi$$
−0.882826 + 0.469701i $$0.844362\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 15.0000i − 0.484881i
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 24.0000i 0.773389i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 55.0000i 1.76868i 0.466843 + 0.884340i $$0.345391\pi$$
−0.466843 + 0.884340i $$0.654609\pi$$
$$968$$ 0 0
$$969$$ 45.0000 1.44561
$$970$$ 0 0
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ 0 0
$$973$$ − 4.00000i − 0.128234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 8.00000i 0.255943i 0.991778 + 0.127971i $$0.0408466\pi$$
−0.991778 + 0.127971i $$0.959153\pi$$
$$978$$ 0 0
$$979$$ −9.00000 −0.287641
$$980$$ 0 0
$$981$$ 60.0000 1.91565
$$982$$ 0 0
$$983$$ 14.0000i 0.446531i 0.974758 + 0.223265i $$0.0716716\pi$$
−0.974758 + 0.223265i $$0.928328\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.00000i − 0.190982i
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ − 60.0000i − 1.90404i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 8.00000i − 0.253363i −0.991943 0.126681i $$-0.959567\pi$$
0.991943 0.126681i $$-0.0404325\pi$$
$$998$$ 0 0
$$999$$ −9.00000 −0.284747
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 4400.2.b.a.4049.1 2
4.3 odd 2 2200.2.b.b.1849.2 2
5.2 odd 4 880.2.a.a.1.1 1
5.3 odd 4 4400.2.a.be.1.1 1
5.4 even 2 inner 4400.2.b.a.4049.2 2
15.2 even 4 7920.2.a.e.1.1 1
20.3 even 4 2200.2.a.a.1.1 1
20.7 even 4 440.2.a.d.1.1 1
20.19 odd 2 2200.2.b.b.1849.1 2
40.27 even 4 3520.2.a.a.1.1 1
40.37 odd 4 3520.2.a.bh.1.1 1
55.32 even 4 9680.2.a.a.1.1 1
60.47 odd 4 3960.2.a.f.1.1 1
220.87 odd 4 4840.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.d.1.1 1 20.7 even 4
880.2.a.a.1.1 1 5.2 odd 4
2200.2.a.a.1.1 1 20.3 even 4
2200.2.b.b.1849.1 2 20.19 odd 2
2200.2.b.b.1849.2 2 4.3 odd 2
3520.2.a.a.1.1 1 40.27 even 4
3520.2.a.bh.1.1 1 40.37 odd 4
3960.2.a.f.1.1 1 60.47 odd 4
4400.2.a.be.1.1 1 5.3 odd 4
4400.2.b.a.4049.1 2 1.1 even 1 trivial
4400.2.b.a.4049.2 2 5.4 even 2 inner
4840.2.a.i.1.1 1 220.87 odd 4
7920.2.a.e.1.1 1 15.2 even 4
9680.2.a.a.1.1 1 55.32 even 4